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January 18, 2008 11:15 WSPC/148-RMP J070-00324 Reviews in Mathematical Physics Vol. 20, No. 1 (2008) 1–70 c World Sc...

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January

18,

2008 11:15 WSPC/148-RMP

J070-00324

Reviews in Mathematical Physics Vol. 20, No. 1 (2008) 1–70 c World Scientific Publishing Company

SPECTRA OF SELF-ADJOINT EXTENSIONS AND APPLICATIONS TO SOLVABLE ¨ SCHRODINGER OPERATORS

∗ , VLADIMIR GEYLER∗,† ¨ JOCHEN BRUNING and KONSTANTIN PANKRASHKIN∗,‡,§ ∗ Institut

f¨ ur Mathematik, Humboldt-Universit¨ at zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany

† Mathematical

Faculty, Mordovian State University, 430000 Saransk, Russia

‡ D´ epartement

de Math´ ematiques, Universit´ e Paris Nord, 99 av. J.-B. Cl´ ement, 93430 Villetaneuse, France § [email protected] Received 27 March 2007 Revised 5 September 2007

We give a self-contained presentation of the theory of self-adjoint extensions using the technique of boundary triples. A description of the spectra of self-adjoint extensions in terms of the corresponding Krein maps (Weyl functions) is given. Applications include quantum graphs, point interactions, hybrid spaces and singular perturbations. Keywords: Self-adjoint operators; self-adjoint extensions; Weyl function; spectrum; spectral measure; quantum graphs; point perturbations. Mathematics Subject Classification 2000: 47B25, 47A10, 46N50, 81Q10

Contents 0. Introduction

2

1. Abstract Self-Adjoint Boundary Value Problems 1.1. Linear relations 1.2. Boundary triples for linear operators 1.3. Krein’s resolvent formula 1.4. Examples 1.4.1. Sturm–Liouville problems 1.4.2. Singular perurbations 1.4.3. Point interactions on manifolds 1.4.4. Direct sums and hybrid spaces 1

5 5 8 13 22 23 24 26 29

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2. Classiﬁcation of Spectra of Self-Adjoint Operators 2.1. Classiﬁcation of measures 2.2. Spectral types and spectral measures 2.3. Spectral projections

31 31 33 37

3. Spectra and Spectral Measures for Self-Adjoint Extensions 3.1. Problem setting and notation 3.2. Discrete and essential spectra 3.3. Estimates for spectral measures 3.4. Special Q-functions 3.5. Spectral duality for quantum and combinatorial graphs 3.6. Array-type systems

40 40 41 42 47 50 52

4. Isolated Eigenvalues 4.1. Problem setting 4.2. Auxiliary constructions 4.3. Description of eigensubspace

55 55 55 59

0. Introduction In recent two decades, the ﬁeld of applications of explicitly solvable models of quantum mechanics based on the operator extension technique has been expanded considerably. New scopes are presented e.g., in the Appendix by Exner [57] to the second edition of the monograph [7], in the monograph by Albeverio and Kurasov [10], and in the topical issue of the J. Phys. A [45]. A review of papers dealing with the theory of Aharonov–Bohm eﬀects from the point of view of the operator methods is contained in [69, 104]; new methods of analyzing singular perturbations supported by sets with non-trivial geometry are reviewed in [59]. In addition, one should mention the use of such models in the quantum ﬁeld theory [71, 81], including string theory [87], quantum gravity [124], and quantum cosmology (see Novikov’s comment in [75] to results from [74]). Here the two-dimensional δ-like potential, which is a point supported perturbation, is of considerable interest because in this case the Dirac δ-function has the same dimension as the Laplacian, and this property leads to an eﬀective non-perturbative renormalization procedure removing the ultraviolet divergence [41, 83, 84]. Another peculiarity of the two-dimensional case — so-called dimensional transmutation — was observed in [40,42]. The operator extension technique allows to build “toy models” which help better understanding some phenomena in various ﬁelds of mathematics and theoretical physics; as typical examples, we mention here the spectral theory of automorphic functions [33] or renormalization group theory [3]. This technique is applicable not only to self-adjoint operators, it can be used, e.g., in investigating dissipative and accumulative operators as well [86]. Very important applications of the operator extension theory have been found recently in the physics of mesoscopic systems like heterostructures [73], quantum graphs [91,92,94,107] and circuits [1], quantum wells, dots, and wires [82]. It should be stressed that in this case, the corresponding results are not only of qualitative

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character, but allow to give a good quantitative explanation of experimental data (see, e.g., [29, 80]) or explain some discrepancy between experimental data and standard theories [30]. Among the most popular ways of using singular perturbations in the physics literature, one should mention ﬁrst of all various renormalization procedures including the Green function renormalization and cut-oﬀ potentials in the position or momentum representations (see [7] and an informative citation list in [111]). Berezin and Faddeev [21] were ﬁrst who showed that the renormalization approach to singular perturbations is equivalent to searching for self-adjoint extensions of a symmetric operator related to the unperturbed operator in question. At the same time, the mathematical theory of self-adjoint extensions is reduced as a rule to the classical von Neumann description through unitary operators in deﬁciency spaces, which makes its practical use rather diﬃcult. In many cases, self-adjoint operators arise when one introduces some boundary conditions for a diﬀerential expression (like boundary conditions for the Laplacian in a domain), and it would be useful to analyze the operators in terms of boundary conditions directly. Such an approach is common in the physics literature [16, 46]. In the framework of the abstract mathematical theory of self-adjoint (or, more generally, dissipative) extensions, this approach is widely used in the diﬀerential operator theory (see, e.g., [54,72,76] and the historical as well as the bibliographical comments therein). Moreover, there is a series of quantum mechanics problems related to the inﬂuence of topological boundaries, and in this case the above approach is the most adequate [14]. On the other hand, Berezin and Faddeev pointed out that the standard expressions for the Green functions of singularly perturbed Hamiltonians obtained by the renormalization procedure can be easy derived from the so-called Krein resolvent formula [21]. In the framework of the theory of explicitly solvable models with an internal structure, an elegant way to get the Krein resolvent formula with the help of abstract boundary conditions has been proposed by Pavlov [112] (see also [2]), which was applied to the study of numerous applications, see, e.g., [62, 95, 100, 103, 113]. A machinery of self-adjoint extensions using abstract boundary condisitons is presented in a rather detailed form in the monograph [72], but only very particular questions of the spectral theory are adressed. A systematic theory of self-adjoint extensions in terms of boundary conditions, including the spectral analysis, was developed by Derkach and Malamud, who found, in particular, a nice relationship between the parameters of self-adjoint extensions and the Krein resolvent formula, and performed the spectral analysis in terms of the Weyl functions; we refer to the paper [50] summarizing this machinery and containing an extensive bibliography. Nevertheless, one has to admit that the spectral analysis of self-adjoint extensions in such terms is a rarely used tool in the analysis of quantum-mechanical Hamiltonians, especially for operators with inﬁnite deﬁciency indices. On the other hand, the authors’ experience show that the application of the Krein resolvent formula in combination with the boundary values for self-adjoint extensions can

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advance in solutions of some problems related to the applications of singular perturbations [31, 35, 36]. Therefore, it is useful to give a self-contained exposition of the abstract technique of boundary value problems and to analyze some models of mathematical physics using this machinery. This is the ﬁrst aim of the present paper. Using the Krein resolvent formula, it is possible often to reduce the spectral problem for the considered perturbed operator to a problem of ﬁnding the kernel of an analytic family of operators — so-called Krein Q-function — with more simple structure in comparison with the operator in question. Therefore, it would be useful to ﬁnd relations between various parts of the spectrum of the considered operators and the corresponding parts of the spectrum of Q-functions. The second aim of the paper is to describe these relations in a form suitable for applications. Using the corresponding results, we obtain, in particular, new properties of the spectra of equilateral quantum graphs and arrays of quantum dots. Of course, we believe that the technique presented here can be used to analyze much more general systems. It is worth noting that this problem was considered in [25], but the main results were obtained in a form which is diﬃcult to use for our applications. In Sec. 1, we describe the machinery of boundary triples and their applications to self-adjoint extensions. The most results in this section are not new (we give the corresponding references in the text), but we do not know any work where this theory was presented with complete proofs, hence we decided to do it here. We also relate the technique of boundary triples with the so-called Krein Q-functions and Γ-ﬁelds. Some of our deﬁnitions are slightly diﬀerent from the commonly used ones (although we show later that both are equivalent); this is motivated by applied needs. We conclude the section by several examples showing that the machinery of boundary triples include the well-known situations like singular perturbations, point perturbation, hybrid spaces. Section 2 is a summary of a necessary information about the spectra and spectral measures of self-adjoint operators. In Sec. 3, we provide the spectral analysis of self-adjoint extensions with the help of the Krein Q-functions. In particular, we analyze the discrete and essential spectra, and carry out a complete spectral analysis for a special class of Qfunctions, which includes the recently introduced scalar-type functions [6]; these results are new. Using these results, we analyze two classes of quantum-mechanical models: equilateral quantum graphs and arrays of quantum dots, where we perform the complete dimension reduction and describe the spectra of continuous models completely in terms of the associated tight-binding Hamiltonians. Section 4 is devoted to the study of isolated eigenvalues of self-adjoint extensions and generalizes previously known results to the case of operators with inﬁnite deﬁciency indices. The second named author, Vladimir Geyler, passed away on April 2, 2007, several days after the completion and the submission of the manuscript. His untimely death has become a great loss for us.

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1. Abstract Self-Adjoint Boundary Value Problems In this section, we describe the theory of self-adjoint extensions using abstract boundary conditions. Some theorems here are not new, but the existing presentations are spread through the literature, so we decided to provide here the key ideas with complete proofs. 1.1. Linear relations Here we recall some basic facts on linear relations. For a more detailed discussion we refer to [13]. Let G be a Hilbert space. Any linear subspace of G ⊕ G will be called a linear relation in G . For a linear relation Λ in G the sets dom Λ : = {x ∈ G : ∃y ∈ G with (x, y) ∈ Λ)}, ran Λ : = {x ∈ G : ∃y ∈ G with (y, x) ∈ Λ)}, ker Λ : = {x ∈ G : (x, 0) ∈ Λ} will be called the domain, the range, and the kernel of Λ, respectively. The linear relations Λ−1 = {(x, y) ∈ G ⊕ G : (y, x) ∈ Λ}, Λ∗ = {(x1 , x2 ) ∈ G ⊕ G : x1 |y2 = x2 |y1 ∀(y1 , y2 ) ∈ Λ} are called inverse and adjoint to Λ, respectively. For α ∈ C we put αΛ = {(x, αy) : (x, y) ∈ Λ}. For two linear relations Λ , Λ ⊂ G ⊕ G one can deﬁne their sum Λ + Λ = {(x, y + y ) : (x, y ) ∈ Λ , (x, y ) ∈ Λ }; clearly, one has dom(Λ + Λ ) = dom Λ ∩ dom Λ . The graph of any linear operator L with domain in G is a linear relation, which we denote by gr L. Clearly, if L is invertible, then gr L−1 = (gr L)−1 . For arbitrary linear operators L , L one has gr(αL) = α gr L and gr L + gr L = gr(L + L ). Therefore, the set of linear operators has a natural “linear” imbedding into the set of linear relations. Moreover, if L is a densely deﬁned closable operator in G , then gr L∗ = (gr L)∗ , hence, this imbedding commutes with the star-operation. In what follows, we consider mostly only closed linear relations, i.e. which are closed linear subspaces in G ⊕ G . Clearly, this generalizes the notion of a closed operator. Similarly to operators, one introduces the notion of the resolvent set res Λ of a linear relation Λ. By deﬁnition, λ ∈ res Λ if and only if (Λ − λI)−1 is the graph of a certain everywhere deﬁned bounded linear operator (here I ≡ gr idG = {(x, x) : x ∈ G }); this operator will be also denoted as (Λ − λI)−1 . Due to the closed graph theorem, the condition λ ∈ res Λ exactly means that Λ is closed, ker(Λ − λI) = 0, and ran(Λ − λI) = G . The spectrum spec Λ of Λ is deﬁned as spec Λ := C\res Λ.

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A linear relation Λ on G is called symmetric if Λ ⊂ Λ∗ and is called self-adjoint if Λ = Λ∗ . A linear operator L in G is symmetric (respectively, self-adjoint) if and only if its graph is a symmetric (respectively, self-adjoint) linear relation. A selfadjoint linear relation is always maximal symmetric, but the converse in not true; examples are given by the graphs of maximal symmetric operators with deﬁciency indices (n, 0), n > 0. To describe all self-adjoint linear relations we need the following auxiliary result. Lemma 1.1. Let U be a unitary operator in G . Then the operator M : G ⊕ G → G ⊕G, U −1 1 i(1 + U ) (1.1) M= 2 1−U i(1 + U ) is unitary; in particular, 0 ∈ res M . Proof. The adjoint operator M ∗ has the form ∗ ∗ −i(1 + U ) 1 − U 1 M∗ = , 2 U∗ − 1 −i(1 + U ∗ ) and it is easy to show by direct calculation that M ∗ = M −1 . Theorem 1.2. A linear relation Λ in G is self-adjoint iﬀ there is a unitary operator U in G (called the Cayley transform of Λ) such that Λ = {(x1 , x2 ) ∈ G ⊕ G : i(1 + U )x1 = (1 − U )x2 }.

(1.2)

Writing U in the form U = exp(−2iA) with a self-adjoint operator A one can reformulate Theorem 1.2 as follows: Corollary 1.3. A linear relation Λ in G is self-adjoint iﬀ there is a self-adjoint operator A acting in G such that Λ = {(x1 , x2 ) ∈ G ⊕ G : cos A x1 = sin A x2 }. To prove Theorem 1.2, we need the following lemma. Lemma 1.4. Let U be a unitary operator in G and Λ be deﬁned by (1.2), then Λ = {((1 − U )x, i(1 + U )x) : x ∈ G }.

(1.3)

Proof. The linear relation Λ given by (1.2) is closed as it is the null space of the bounded operator G ⊕ G (x1 , x2 ) → i(1 + U )x1 − (1 − U )x2 ∈ G . Denote the set on the right-hand side of (1.3) by Π. Clearly, Π ⊂ Λ. By Lemma 1.1, the operator M ∗ adjoint to M from (1.1) maps closed sets to closed sets.

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In particular, the subspace Π ≡ M ∗ (0 ⊕ G ) is closed. Assume that there exists (y1 , y2 ) ∈ Λ such that (y1 , y2 ) ⊥ Π. The condition (y1 , y2 ) ∈ Λ reads as i(1 + U )y1 − (1 − U )y2 = 0, and (y1 , y2 ) ⊥ Π means that y1 |(1 − U )x + y2 |i(1 + U )x = 0 for all x ∈ G , i.e. that (U − 1)y1 − i(1 + U )y2 = 0. This implies M (y1 , y2 ) = 0. By Lemma 1.1, y1 = y2 = 0. The requested equality Λ = Π is proved. Proof of Theorem 1.2. (1) Let U be a unitary operator in G and Λ be deﬁned by (1.2). By Lemma 1.4, one can represent Λ in the form (1.3). Using this representation, one easily concludes that Λ ⊂ Λ∗ , i.e. that Λ is symmetric. Let (y1 , y2 ) ∈ Λ∗ . The equality x1 |y2 = x1 |y2 for all (x1 , x2 ) ∈ Λ is equivalent to (1 − U )x|y2 = i(1 + U )x|y1 for all x ∈ G , from which −i(1 + U −1 )y1 = (1 − U −1 )y2 and i(1 + U )y1 = (1 − U )y2 , i.e. (y1 , y2 ) ∈ Λ. Therefore, Λ∗ ⊂ Λ, which ﬁnally results in Λ = Λ∗ . (2) Let Λ be a self-adjoint linear relation in G . Set L± := {x1 ± ix2 : (x1 , x2 ) ∈ Λ}. Assume that for some (x1 , x2 ) and (y1 , y2 ) from Λ one has x1 + ix2 = y1 + iy2 , then (x1 − y1 , x2 − y2 ) ∈ Λ and x1 − y1 = −i(x2 − y2 ). At the same time, 0 = Imx1 − y1 |x2 − y2 = Im−i(x2 − y2 )|(x2 − y2 ) = Im ix2 − y2 2 , therefore, x2 = y2 and x1 = y1 . In the same way, one can show that from x1 − ix2 = y1 − iy2 , (x1 , x2 ), (y1 , y2 ) ∈ Λ, it follows that x1 = y1 and x2 = y2 . For x1 + ix2 with (x1 , x2 ) ∈ Λ, set U (x1 + ix2 ) = x1 − ix2 . Clearly, U : L+ → L− is well-deﬁned and bijective. Moreover, U (x1 + ix2 )2 = x1 2 + x2 2 = x1 + ix2 2 , i.e. U is isometric. Show that U is actually a unitary operator, i.e. that L± = G . We consider only L+ ; the set L− can be considered exactly in the same way. Assume that y ⊥ L+ for some y ∈ G , then x1 + ix2 |y = x1 |y − x2 |iy = 0 for all (x1 , x2 ) ∈ Λ. It follows that (iy, y) ∈ Λ∗ = Λ, which implies Imiy|y = − Im iy2 = 0, i.e. y = 0. Therefore, L+ = G . To show that L+ is closed we take an arbitrary sequence (xn1 , xn2 ) ∈ Λ with lim(xn1 + ixn2 ) = y for some y ∈ G , then automatically lim(xn1 − ixn2 ) = y for some y ∈ G , and lim xn1 =

1 (y + y ) =: y1 2

and

lim xn2 =

1 (y − y ) =: y2 . 2i

As we see, the sequence (xn1 , xn2 ) converges, and the limit (y1 , y2 ) lies in Λ as Λ is closed. Therefore, y = y1 + iy2 lies in L+ , L+ is closed, and U is unitary. Clearly, by construction of U , Λ is a subset of the subspace on the right-hand side of (1.2). As shown in item (1), the latter is self-adjoint as well as Λ is, therefore, they coincide. Theorem 1.2 gives only one possible way for parameterizing linear relations with the help of operators. Let us mention some other ways to to this. Proposition 1.5. Let A and B be bounded linear operators in G . Denote Λ := {(x1 , x2 ) ∈ G ⊕ G : Ax1 = Bx2 }. Λ is self-adjoint iﬀ the following two conditions

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are satisﬁed: A ker B

AB ∗ = BA∗ , −B = 0. A

(1.4a) (1.4b)

Proof. Introduce operators L : G ⊕G (x1 , x2 ) → Ax1 −Bx2 ∈ G and J : G ⊕G (x1 , x2 ) → (−x2 , x1 ) ∈ G ⊕ G . There holds Λ∗ = J(Λ⊥ ) and Λ = ker L. Let us show ﬁrst that the condition (1.4a) is equivalent to the inclusion Λ∗ ⊂ Λ. Note that this inclusion is equivalent to J(Λ⊥ ) ⊂ Λ or, due to the bijectivity of J, to Λ⊥ ⊂ JΛ.

(1.5)

Clearly, Λ ≡ ker L is closed, therefore, by the well-known relation, Λ⊥ = ker L⊥ = ran L∗ . As Λ is closed, the condition (1.5) is equivalent to ran L∗ ⊂ J(ker L).

(1.6)

Noting that L∗ acts as G x → (A∗ x, −B ∗ x) ∈ G ⊕ G , we see that (1.6) is equivalent to (1.4a). Now, let Λ be self-adjoint, then J(Λ⊥ ) = Λ or, equivalently, J(Λ) = Λ⊥ ≡ ker L⊥ . Therefore, the restriction of L to J(Λ) is injective. This means that the systems of equations Lz = 0, LJz = 0 has only the trivial solution, which is exactly the condition (1.4b). On the other hand, if (1.4a) and (1.4b) are satisﬁed, then, as shown above, Λ⊥ ⊂ J(Λ). If Λ⊥ = J(Λ), then J(Λ) contains a non-zero element of (Λ⊥ )⊥ ≡ Λ = ker L, i.e. there exists z = 0 with Lz = 0 and LJz = 0, which contradicts (1.4b). For a ﬁnite-dimensional G the condition (1.4b) simpliﬁes, and one arrives at: Corollary 1.6. Let G be ﬁnite dimensional, A, B be linear operators in G . The linear relation Λ := {(x1 , x2 ) ∈ G ⊕ G : Ax1 = Bx2 } is self-adjoint iﬀ the following two conditions are satisﬁed: AB ∗ = BA∗ ,

(1.7a)

det(AA + BB ) = 0 ⇔ the block matrix (A|B) has maximal rank.

(1.7b)

∗

∗

The conditions (1.4a), (1.4b), (1.7a), (1.7b) can be rewritten in many equivalent forms, see, e.g., [4 and 5, Sec. 125] and [32, 48, 108, 119]. 1.2. Boundary triples for linear operators Definition 1.7. Let A be a closed linear operator in a Hilbert space H with the domain dom A. Assume that there exist another Hilbert space G and two linear

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maps Γ1 , Γ2 : dom A → G such that: f |Ag − Af |g = Γ1 f |Γ2 g − Γ2 f |Γ1 g for all f, g ∈ dom A,

(1.8a)

the map (Γ1 , Γ2 ) : dom A → G ⊕ G is surjective,

(1.8b)

the set ker(Γ1 , Γ2 ) is dense in H .

(1.8c)

A triple (G , Γ1 , Γ2 ) with the above properties is called a boundary triple for A. Remark 1.8. This deﬁnition diﬀers slightly from the commonly used one. In [50, 72, 89], one deﬁnes boundary triple only for the case when A∗ is a closed densely deﬁned symmetric operator; the property (1.8c) then holds automatically. In our opinion, in some cases it is more convenient to ﬁnd a boundary triple than to check whether the adjoint operator is symmetric. Below we will see (Theorem 1.12) that these deﬁnitions are actually equivalent if one deals with self-adjoint extensions. In Deﬁnition 1.7, we do not assume any continuity properties of the maps Γ1 and Γ2 , but they appear automatically. Proposition 1.9. Let A be a closed linear operator in a Hilbert space H and (G , Γ1 , Γ2 ) be its boundary triple, then the mapping dom S g → (Γ1 g, Γ2 g) ∈ G ⊕G is continuous with respect to the graph norm of S. Proof. Suppose that a sequence gn ∈ dom A, n ∈ N, converges in the graph norm. As A is closed, there holds g := lim gn ∈ dom A and Ag = lim Agn . Assume that lim(Γ1 gn , Γ2 gn ) = (u, v), where the limit is taken in the norm of G ⊕ G . Let us show that Γ1 g = u and Γ2 g = v; this will mean that the mapping (Γ1 , Γ2 ) is closed and, therefore, continuous by the closed graph theorem. For an arbitrary f ∈ dom A, there holds Γ1 f |Γ2 g − Γ2 f |Γ1 g = f |Ag − Af |g = limf |Agn − Af |gn = limΓ1 f |Γ2 gn − Γ2 f |Γ1 gn = Γ1 f |v − Γ2 f |u. Therefore, Γ1 f |Γ2 g − Γ2 f |Γ1 g = Γ1 f |v − Γ2 f |u and Γ1 f |Γ2 g − v = Γ2 f |Γ1 g − u

(1.9)

for any f ∈ dom A. Using the property (1.8b) from Deﬁnition 1.7, one can take f ∈ dom A with Γ1 f = Γ2 g − v and Γ2 f = 0, then (1.9) reads as Γ2 g − v2 = 0 and Γ2 g = v. Analogously, choosing f ∈ dom A with Γ1 f = 0 and Γ2 f = Γ1 g − u one arrives at Γ1 g = u. Our next aim is to describe situations in which boundary triples exist and are useful. For a symmetric operator A in a Hilbert space H and for z ∈ C, we denote throughout the paper Nz (A) := ker(A∗ − zI) and write sometimes Nz instead of Nz (A), if it does not lead to confusion.

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Is is well-known that A has self-adjoint extensions if and only if dim Ni = dim N−i . The von Neumann theory states a bijection between the self-adjoint extensions and unitary operators from Ni to N−i . More precisely, if U is a unitary operator from Ni to N−i , then the corresponding self-adjoint extension AU has the domain {f = f0 + fi + U fi : f0 ∈ dom A, fi ∈ Ni } and acts as f0 + fi + U fi → Af0 + ifi − iU fi . This construction is diﬃcult to use in practical applications, and our aim is to show that the boundary triples provide a useful machinery for working with self-adjoint extensions. The following proposition is borrowed from [89]. Proposition 1.10. Let A be a densely deﬁned closed symmetric operator in a Hilbert space H with equal deﬁciency indices (n, n), then there is a boundary triple (G , Γ1 , Γ2 ) for the adjoint A∗ with dim G = n. Proof. It is well-known that dom A∗ = dom A + Ni + N−i , and this sum is direct. Let P±i be the projector from dom A∗ to N±i corresponding to this expansion. Let f, g ∈ dom A∗ , then f = f0 +Pi f +P−i f , g = g0 +Pi g+P−i g, f0 , g0 ∈ dom A. Using the equalities A∗ Pi = iPi and A∗ P−i = −iP−i , one obtains f |A∗ g − A∗ f |g = f0 + Pi f + P−i f |Ag0 + iPi g − iP−i g = 2iPi f |Pi g − 2iP−i f |P−i g.

(1.10)

As the deﬁciency indices of A are equal, there is an isomorphism U from N−i onto Ni . Denote G := N−i endowed with the induced scalar product in H , and set Γ1 = iU P−i − iPi , Γ2 = Pi + U P−i , then Γ1 f |Γ2 g − Γ2 f |Γ1 g = 2iPi f |Pi g − 2iU P−i f |U P−i g = 2iPi f |Pi g − 2iP−i f |P−i g.

(1.11)

Comparing (1.10) with (1.11), one shows that (G , Γ1 , Γ2 ) satisfy the property (1.8a) of Deﬁnition 1.7. Due to dom A ⊂ ker(Pi , P−i ) ⊂ ker(Γ1 , Γ2 ) the property (1.8c) is satisﬁed too. To prove (1.8b), take any F1 , F2 ∈ N−i ≡ G and show that the system of equations iU P−i f − iPi f = F1 ,

U P−i f + Pi f = F2 ,

(1.12)

∗

has a solution f ∈ dom A . Multiplying the second equation by i and adding it to the ﬁrst one one arrives at 2iU P−i f = F1 + iF2 . In a similar way, 2iPi f = iF2 − F1 . Therefore, the funtcion 1 1 f = (iF2 − F1 ) + U −1 (F1 + iF2 ) ∈ Ni (A∗ ) + N−i (A∗ ) ⊂ dom A∗ 2i 2i is a possible solution to (1.12), and (1.8b) is satisﬁed. Therefore, (G , Γ1 , Γ2 ) is a boundary triple for A∗ . Let A be a closed densely deﬁned linear operator, A∗ have a boundary triple (G , Γ1 , Γ2 ), Λ be a closed linear relation in G . By AΛ in this subsection we mean the restriction of A∗ to the domain dom AΛ = {f ∈ dom A∗ : (Γ1 f, Γ2 f ) ∈ Λ}.

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The usefulness of boundary triples is described in the following proposition. Proposition 1.11. For any closed linear relation Λ in G one has A∗Λ = AΛ∗ . In particular, AΛ is symmetric/self-adjoint if and only if Λ is symmetric/self-adjoint, respectively. Proof. Clearly, one has A ⊂ AΛ ⊂ A∗ . Therefore, A ⊂ A∗Λ ⊂ A∗ . Moreover, one has gr A∗Λ = {(f, A∗ f ) : f |A∗ g = A∗ f |g ∀g ∈ dom AΛ } = {(f, A∗ f ) : Γ1 f |Γ2 g − Γ2 f |Γ1 g ∀g ∈ dom AΛ } = {(f, A∗ f ) : Γ1 f |x2 − Γ2 f |x1 ∀(x1 , x2 ) ∈ Λ} = {(f, A∗ f ) : (Γ1 f, Γ2 f ) ∈ Λ∗ } = gr AΛ∗ . This proves the ﬁrst part of proposition. The part concerning the self-adjointness of AΛ is now obvious, as AΛ ⊂ AΛ if and only if Λ ⊂ Λ . Theorem 1.12. Let A be a closed densely deﬁned symmetric operator. (1) The operator A∗ has a boundary triple if and only if A admits self-adjoint extensions. (2) If (G , Γ1 , Γ2 ) is a boundary triple for A∗ , then there is a one-to-one correspondence between all self-adjoint linear relations Λ in G and all self-adjoint extensions of A given by Λ ↔ AΛ , where AΛ is the restriction of A∗ to the vectors f ∈ dom A∗ satisfying (Γ1 f, Γ2 f ) ∈ Λ. Proof. (1) Let A∗ have a boundary triple and Λ be any self-adjoint linear relation in G , then according to Proposition 1.11 the operator AΛ is self-adjoint, and AΛ ⊃ A. The converse is exactly Proposition 1.10. (2) If Λ is a self-adjoint linear relation in G , then due to Proposition 1.11 the corresponding operator AΛ is self-adjoint. Now, let B be a self-adjoint extension of A, then A ⊂ B ⊂ A∗ . Denote Λ = {(Γ1 f, Γ2 f ), f ∈ dom B ∗ }, then B = AΛ , and Λ is self-adjoint due to Proposition 1.11. Theorem 1.13. Let a closed linear operator B have a boundary triple (G , Γ1 , Γ2 ), and A := B|ker(Γ1 ,Γ2 ) , then A ⊂ B ∗ . Moreover, the following four conditions are equivalent: (1) (2) (3) (4)

B has at least one restriction which is self-adjoint, B ∗ is symmetric, B ∗ = A, A∗ = B.

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Proof. By construction A is densely deﬁned. By Deﬁnition 1.7 for any f ∈ dom A one has f |Bg − Af |g = 0, which means A ⊂ B ∗ . In particular, B ∗ is densely deﬁned. By Proposition 1.9, A is closed, therefore, (3) and (4) are equivalent. (1) ⇒ (2). Let C be a self-adjoint restriction of B. From C ⊂ B it follows B ∗ ⊂ C ∗ ≡ C ⊂ B ≡ (B ∗ )∗ , i.e. B ∗ is symmetric. (2) ⇒ (3). Let D = B ∗ be symmetric, then D ⊂ B is closed and B = D∗ . Let f ∈ dom D. According to the Deﬁnition 1.7, there exists g ∈ dom D∗ = dom B with Γ1 g = −Γ2 f and Γ2 g = Γ1 f . One has 0 = Df |g − Df |g = f |D∗ g − D∗ f |g ≡ f |Bg − Bf |g = Γ1 f 2 + Γ2 f 2 , from which Γ1 f = Γ2 f = 0. Therefore, dom D ⊂ ker(Γ1 , Γ2 ) ≡ dom A. At the same time, as shown above, A ⊂ B ∗ , which means A = D = B ∗ . (4) ⇒ (1). Let B = A∗ . By Theorem 1.12(1), the operator A has self-adjoint extensions, which are at the same time self-adjoint restrictions of A∗ = B. The proof of Proposition 1.10 gives a possible construction of a boundary triple. Clearly, boundary triple is not ﬁxed uniquely by Deﬁnition 1.7. For a description of all possible boundary triple, we refer to [101, 102]. We restrict ourselves by the following observations. Proposition 1.14. Let A be a closed densely deﬁned symmetric operator with equal deﬁciency indices. For any self-adjoint extension H of A, there exists a boundary triple (G , Γ1 , Γ2 ) for A∗ such that H is the restriction of A∗ to ker Γ1 . Proof. Let (G , Γ1 , Γ2 ) be an arbitrary boundary triple for A∗ . According to Theorem 1.12(2), there exists a self-adjoint linear relation Λ in G such that H is the restriction of A∗ to the vectors f ∈ dom A∗ satisfying (Γ1 f, Γ2 f ) ∈ Λ. Let U be the Cayley transform of Λ (see Theorem 1.2). Set Γ1 :=

1 (i(1 + U )Γ1 + (U − 1)Γ2 ), 2

Γ2 :=

1 ((1 − U )Γ1 + i(1 + U )Γ2 ). 2

By Lemma 1.1, the map (Γ1 , Γ2 ) : dom A∗ → G ⊕ G is surjective and ker(Γ1 , Γ2 ) = ker(Γ1 , Γ2 ). At the same time, one has Γ1 f |Γ2 g − Γ2 f |Γ1 g ≡ Γ1 f |Γ2 g − Γ2 f |Γ1 g, which means that (G , Γ1 , Γ2 ) is a boundary triple for A∗ . It remains to note that the conditions (Γ1 f, Γ2 f ) ∈ Λ and Γ1 f = 0 are equivalent by the choice of U . Proposition 1.15. Let (G , Γ1 , Γ2 ) be an arbitrary boundary triple for A∗ , and L ˜ 2 ) with Γ ˜ 1 = Γ1 and ˜ 1, Γ be a bounded linear self-adjoint operator in G , then (G , Γ ∗ ˜ Γ2 = Γ2 + LΓ1 is also a boundary triple for S . Proof. The conditions of Deﬁnition 1.7 are veriﬁed directly.

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An explicit construction of boundary triples is a rather diﬃcult problem, see, e.g., [125] for the discussion of elliptic boundary conditions. In some cases, there are natural boundary triples reﬂecting some speciﬁc properties of the problem, like in the theory of singular perturbations, see [115] and Sec. 1.4.2 below.

1.3. Krein’s resolvent formula In this subsection, if not speciﬁed explicitly, • S is a densely deﬁned symmetric operator with equal deﬁciency indices (n, n), 0 < n ≤ ∞, in a Hilbert space H , • Nz := ker(S ∗ − z), • G is a Hilbert space of dimension n, • H 0 is a certain self-adjoint extension of S, • for z ∈ res H 0 denote R0 (z) := (H 0 − z)−1 , the resolvent of H 0 . For z1 , z2 ∈ res H 0 , put U (z1 , z2 ) = (H 0 − z2 )(H 0 − z1 )−1 ≡ 1 + (z1 − z2 )R0 (z1 ). It is easy to show that U (z1 , z2 ) is a linear topological isomorphism of H obeying the following properties: U (z, z) = I,

(1.13a)

U (z1 , z2 )U (z2 , z3 ) = U (z1 , z3 ),

(1.13b)

U −1 (z1 , z2 ) = U (z2 , z1 ),

(1.13c)

∗

z1 , z¯2 ), U (z1 , z2 ) = U (¯ U (z1 , z2 )Nz2 (S) = Nz1 (S).

(1.13d) (1.13e)

Definition 1.16. A map γ : res H 0 → L(G , H ) is called a Krein Γ-ﬁeld for (S, H 0 , G ) if the following two conditions are satisﬁed: γ(z) is a linear topological isomorphism of G and Nz for all z ∈ res H 0 , (1.14a) for any z1 , z2 ∈ res H there holds γ(z1 ) = U (z1 , z2 )γ(z2 ) or, equivalently, 0

γ(z1 ) − γ(z2 ) = (z1 − z2 )R0 (z1 )γ(z2 ) = (z1 − z2 )R0 (z2 )γ(z1 ).

(1.14b)

Let us discuss questions concerning the existence and uniqueness of Γ-ﬁelds. Proposition 1.17. For any triple (S, H 0 , G ), there exists a Krein Γ-ﬁeld γ. If γ˜ (z) is another Krein Γ-ﬁeld for (S, H 0 , G˜) with a certain Hilbert space G˜, then there exists a linear topological isomorphism N from G˜ to G such that γ˜ (z) = γ(z)N .

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Proof. Fix any z0 ∈ res H 0 , choose any linear topological isomorphism L : G → Nz0 , and set γ(z0 ) := L. Then property (1.14b) forces to set γ(z) = U (z, z0 )L ≡ L + (z − z0 )R0 (z)L.

(1.15)

On the other hand, the properties (1.13) of U (z1 , z2 ) show that γ(z) deﬁned by (1.15) is a Γ-ﬁeld for (S, H 0 , G ). If γ (z) : G˜ → H , z ∈ res H 0 , is another Γ-ﬁeld for (S, H 0 , G ), then setting N =γ (z0 )γ (−1) (z0 ) where γ (−1) (z0 ) is the inverse to γ(z0 ) : G → Nz0 , and using (1.14b) again, we see that γ (z) = γ(z)N for all z ∈ res H 0 . The following propositions gives a characterization of all Krein Γ-ﬁelds. Proposition 1.18. Let H 0 be a self-adjoint operator in a Hilbert space H , G be another Hilbert space, and γ be a map from res H 0 to L(G , H ), then the following assertions are equivalent: (1) there is a closed densely deﬁned symmetric restriction S of H 0 such that γ is the Γ-ﬁeld for (S, H 0 , G ). (2) γ satisﬁes the condition (1.14b) above and the following additional condition: for some ζ ∈ res H 0 the map γ(ζ) is a linear topological isomorphism of G on a subspace N ⊂ H such that N ∩ dom H 0 = {0}. (1.16) Proof. Clearly, any Γ-ﬁeld satisﬁes (1.16). Conversely, let the conditions (1.16) and (1.14b) be fulﬁlled for a map γ : res H 0 → L(G , H ). Then, in particular, γ(z) is a linear topological isomorphism on a subspace of H for any z ∈ res H 0 . Denote Dz = ker γ ∗ (z)(H 0 − z¯). According to (1.14b) we have for any z1 , z2 ∈ res H 0 γ ∗ (z2 ) = γ ∗ (z1 )U ∗ (z2 , z1 ) = γ ∗ (z1 )(H 0 − z¯1 )(H 0 − z¯2 )−1 . Hence γ ∗ (z2 )(H 0 − z¯2 ) = γ ∗ (z1 )(H 0 − z¯1 ), therefore Dz is independent of z. Denote D := Dz and deﬁne S as the restriction of H 0 to D. Show that D is dense in H . ⊥ ¯ ), this means that R0 (ζ)ϕ|ψ = 0 for each Let ϕ ⊥ D. Since D = Dζ = R0 (ζ)(N ψ ∈ N ⊥ , i.e. we have R0 (ζ)ϕ ∈ N . Hence, R0 (ζ)ϕ = 0, therefore ϕ = 0. Thus, S is densely deﬁned. Let us show that ran(S − z¯) = ker γ ∗ (z)

(1.17)

for any z ∈ res H 0 . Let γ ∗ (z)ϕ = 0; set ψ := (H 0 − z¯)−1 ϕ, then ψ ∈ dom S ≡ D, therefore, ϕ ∈ ran(S − z¯). Conversely, if ϕ ∈ ran(S − z¯), then ϕ = (S − z¯)ψ where γ ∗ (z)(H 0 − z¯)ψ = 0, and (1.17) is proven. In particular, (1.17) implies that S is closed. Moreover, we have from (1.17) Nz = ran(S − z¯)⊥ = ker γ ∗ (z)⊥ = ran γ(z) = ran γ(z). Thus, γ is a Γ-ﬁeld for (S, H 0 , G ).

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Let now the triple (S, H 0 , G ) be endowed with a Γ-ﬁeld γ, γ : res H 0 → L(G , H ). Definition 1.19. A map Q : res H 0 → L(G , G ) is called a Krein Q-function for (S, H 0 , G , γ), if z2 ) = (z1 − z2 )γ ∗ (¯ z2 )γ(z1 ) for any z1 , z2 ∈ res H 0 . Q(z1 ) − Q∗ (¯

(1.18)

Proposition 1.20. For any (S, H 0 , G ) endowed with a Krein Γ-ﬁeld γ, there exists ˜ : G → G , z ∈ res H 0 , is another a Krein Q-function Q : res H 0 → L(G , G ). If Q(z) ˜ Q-function for (S, H 0 , G , γ), then Q(z) = Q(z) + M, where M is a bounded selfadjoint operator in G . Proof. Fix as any z0 ∈ res H 0 and denote x0 := Re z0 , y0 := Im z0 , L := γ(z0 ). If a Q-function exists, then by (1.18) one has Q(z) = Q∗ (z0 ) + (z − z¯0 )L∗ γ(z). On the other hand Q(z0 ) + Q∗ (z0 ) Q(z0 ) − Q∗ (z0 ) − . Q∗ (z0 ) = 2 2 Clearly, Q(z0 ) + Q∗ (z0 ) is a bounded self-adjoint operator in G , denote it by 2C. According to (1.18), Q(z0 ) − Q∗ (z0 ) = 2iy0 L∗ L, and therefore Q(z) = C − iy0 L∗ L + (z − z¯0 )L∗ γ(z).

(1.19)

˜ ˜ We have from (1.19) that if Q(z) is another Q-function for (S, H 0 , G , γ), then Q(z)− Q(z) = M where M is a bounded self-adjoint operator which is independent of z. It remains to show that a function of the form (1.19) obeys (1.18). Take arbitrary z2 − z0 )γ ∗ (z2 )L. Therefore, z1 , z2 ∈ res H 0 . We have Q∗ (z2 ) = C + iy0 L∗ L + (¯ Q(z1 ) − Q∗ (z2 ) = (¯ z0 − z0 )L∗ L + (z − z¯0 )L∗ γ(z1 ) + (z0 − z¯2 )γ ∗ (z2 )L.

(1.20)

By (1.14b), L = γ(z0 ) = γ(z1 ) + (z0 − z1 )R0 (z0 )γ(z1 ) and L∗ = γ ∗ (z0 ) = γ ∗ (z2 ) + (¯ z0 − z¯2 )γ ∗ (z2 )R0 (¯ z0 ). Substituting these expressions in (1.20) we obtain Q(z1 ) − Q∗ (z2 ) = (z1 − z¯2 )γ ∗ (z2 )γ(z1 ) + γ ∗ (z2 ){(¯ z0 − z0 )[(¯ z0 − z¯2 )R0 (¯ z0 ) + (z0 − z1 )R0 (z0 ) z0 )R0 (z0 )] + (z1 − z¯0 )(¯ z0 − z¯2 )R0 (¯ z0 ) + (¯ z0 − z¯2 )(z0 − z1 )R0 (¯ + (z0 − z¯2 )(z0 − z1 )R0 (z0 )}γ(z1 ). The expression in the curly brackets is equal to z0 − z¯2 )R0 (¯ z0 ) + (z1 − z¯0 )(¯ z0 − z¯2 )R0 (¯ z0 ) (¯ z0 − z0 )(¯ z0 − z¯2 )R0 (¯ z0 ) + (¯ z0 − z0 )(z0 − z1 )R0 (z0 ) + (z0 − z1 )(¯ z0 − z¯2 )(z0 − z1 )R0 (z0 ). + (z0 − z¯2 )(z0 − z1 )R0 (z0 ) − (¯ It is easy to see that the latter expression is equal to zero, and we get the result. Below we list some properties of Γ-ﬁelds and Q-functions which follow easily from the deﬁnitions.

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Proposition 1.21. Let γ be a Krein Γ-ﬁeld for (S, H 0 , H ), then γ is holomorphic in res H 0 and satisﬁes d γ(z) = R0 (z)γ(z), dz S ∗ γ(z) = zγ(z),

(1.21b)

γ (z) is a bijection from Nz onto G ,

(1.21c)

(1.21a)

∗

∗

γ (z)f = 0 iﬀ f ⊥ Nz , ∗

(1.21d)

∗

z1 )γ(z2 ) = γ (¯ z2 )γ(z1 ), γ (¯

(1.21e)

ran[γ(z1 ) − γ(z2 )] ⊂ dom H for any z1 , z2 ∈ res H . 0

0

(1.21f)

Let in addition Q be a Krein Q-function for (S, H 0 , G ) and γ, then Q is holomorphic in res H 0 , and the following holds: d Q(z) = γ ∗ (¯ z )γ(z), dz Q∗ (¯ z ) = Q(z), Im Q(z) ≥ cz . for any z ∈ C\R there is cz > 0 with Im z

(1.22a) (1.22b) (1.22c)

Remark 1.22. The property (1.22c) means that Q-function is an operator-valued Nevanlinna function (or Herglotz function). This implies a number of possible relations to the measure theory, spectral theory etc., and such functions appear in many areas outside the extension theory, see, e.g., [48, 50, 63, 65, 105, 106] and references therein. Our next aim is to relate boundary triples in Deﬁnition 1.7 to Krein’s maps from Deﬁnition 1.16. Theorem 1.23. Let S be a closed densely deﬁned symmetric operator in a Hilbert space H with equal deﬁciency indices. (1) For any self-adjoint extension H of S and any z ∈ res H there holds dom S ∗ = dom H + Nz , and this sum is direct. (2) Let (G , Γ1 , Γ2 ) be a boundary triple for S ∗ and H 0 be the restriction of S ∗ to ker Γ1 which is self-adjoint due to Theorem 1.12. Then: (2a) for any z ∈ res H 0 the restriction of Γ1 to Nz has a bounded inverse γ(z) : G → Nz ⊂ H deﬁned everywhere, (2b) this map z → γ(z) is a Krein Γ-ﬁeld for (S, H 0 , G ), (2c) the map res H 0 z → Q(z) = Γ2 γ(z) ∈ L(G , G ) is a Krein Q-function for (S, H 0 , G ) and γ. z )(H 0 − z)f = Γ2 f . (2d) for any f ∈ dom H 0 and z ∈ res H 0 there holds γ ∗ (¯ Proof. (1) Let f ∈ dom S ∗ , Denote f0 := (H − z)−1 (S ∗ − z)f . Clearly, f0 ∈ dom H. For g := f −f0 one has (S ∗ −z)g = (S ∗ −z)f −(S ∗ −z)(H −z)−1 (S ∗ −z)f ≡ (S ∗ − z)f − (H − z)(H − z)−1 (S ∗ − z)f = 0, therefore, g ∈ ker(S ∗ − z) ≡ Nz .

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Now assume that for some z ∈ res H, one has f0 + g0 = f1 + g1 for some f0 , f1 ∈ dom H and g0 , g1 ∈ Nz , then f0 − f1 = g1 − g0 ∈ Nz and (H − z)(f0 − f1 ) = (S ∗ − z)(f0 − f1 ) = 0. As H − z is invertible, one has f0 = f1 and g0 = g1 . (2a) Due to condition (1.8b), Γ1 (dom S ∗ ) = G . Due to Γ1 (dom H 0 ) = 0 and item (1), one has Γ1 (Nz ) = G . Assume that Γ1 f = 0 for some f ∈ Nz , then f ∈ dom H 0 ∩ Nz and f = 0 by item (1). Therefore, Γ1 : Nz → G is a bijection and, moreover, Γ1 is continuous in the graph norm of S ∗ by Proposition 1.9. At the same time, the graph norm of S ∗ on Nz is equivalent to the usual norm in H , which means that the restriction of Γ1 to Nz is a bounded operator. The graph of this map is closed, and the inverse map is bounded by the closed graph theorem. (2b) The property (1.14a) is already proved in item (2a). Take arbitrary z1 , z2 ∈ res H 0 and ξ ∈ G . Denote f = γ(z1 )ξ and g = U (z2 , z1 )f ≡ f + (z2 − z1 )R0 (z2 )f . As R0 (z2 )f ∈ dom H 0 , there holds Γ1 R0 (z2 )f = 0 and Γ1 g = Γ1 f . Clearly, f ∈ Nz1 , and to prove property (1.14b) it is suﬃcient to show that (S ∗ −z2 )g = 0. But this follows from the chain (S ∗ −z2 )g = (S ∗ −z2 )f +(z2 −z1 )(S ∗ − z2 )(H 0 − z2 )−1 f = (S ∗ − z2 )f + (z2 − z1 )(H 0 − z2 )(H 0 − z2 )−1 f = (S ∗ − z1 )f = 0. Therefore, γ satisﬁes both properties (1.14a) and (1.14b) in Deﬁnition 1.16. (2c) As γ(z) is bounded by item (2a) and Γ2 is bounded by Proposition 1.9, the map Q(z) is a bounded linear operator on L(G , G ). To prove property (1.18) z2 )φ, g := γ(z1 )ψ. Clearly, take arbitrary z1 , z2 ∈ res H, φ, ψ ∈ G , and set f := γ(¯ f |S ∗ g − f |S ∗ g − (z1 − z2 )f |g = f |(S ∗ − z1 )g − (S ∗ − z¯2 )f |g = 0. (1.23) At the same time one has z2 )γ(z1 )ψ. f |g = γ(¯ z2 )φ|γ(z1 )ψ = φ|γ ∗ (¯

(1.24)

Moreover, using the equality Γ1 γ(z)ξ = ξ, which holds for all ξ ∈ G and z ∈ res H 0 , one obtains f |S ∗ g − f |S ∗ g = Γ1 f |Γ2 g − Γ2 f |Γ1 g z2 )φ|Γ2 γ(z1 )ψ − Γ2 γ(¯ z2 )φ|Γ1 γ(z1 )ψ = Γ1 γ(¯ = φ|Q(z1 )ψ − Q(¯ z2 )φ|ψ = φ|[Q(z1 ) − Q∗ (¯ z2 )]ψ. Therefore, Eqs. (1.23) and (1.24) read as z2 )]ψ = φ|(z1 − z2 )γ ∗ (¯ z2 )γ(z1 )ψ, φ|[Q(z1 ) − Q∗ (¯ which holds for any φ, ψ ∈ G . This implies (1.18). (2d) For any φ ∈ G , one has z )(H 0 − z)f = γ(¯ z )φ|(H 0 − z)f = γ(¯ z )φ|S ∗ f − zγ(¯ z)φ|f φ|γ ∗ (¯ z )φ|f − zγ(¯ z )φ|f + Γ1 γ(¯ z )φ|Γ2 f − Γ2 γ(¯ z )φ|Γ1 f = S ∗ γ(¯ = (S ∗ − z¯)γ(¯ z )φ|f + φ|Γ2 f = φ|Γ2 f , i.e. Γ2 f = γ ∗ (¯ z )(H 0 − z)f .

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Definition 1.24. The Krein Γ-ﬁeld and Q-function deﬁned in Theorem 1.23 will be called induced by the boundary triple (G , Γ1 , Γ2 ). Remark 1.25. The Q-function induced by a boundary triple is often called the Weyl function [6, 50]. Conversely, starting with given Krein maps one can construct a boundary triple. Proposition 1.26. Let γ be a Krein Γ-ﬁeld for (S, H 0 , G ). For any z ∈ res H 0 , represent f ∈ dom S ∗ as f = fz + γ(z)F,

(1.25)

where fz ∈ dom H 0 , F ∈ G . For a ﬁxed z ∈ res H 0 , deﬁne 1 Γ1 f := F, Γ2 f := (γ ∗ (¯ z )(H 0 − z)fz + γ ∗ (z)(H 0 − z¯)fz¯), 2 then (G , Γ1 , Γ2 ) is a boundary triple for S ∗ , and γ(z) is the induced Γ-ﬁeld. For further references, we formulate a simpliﬁed version of Proposition 1.26 for the case when H 0 has gaps. Corollary 1.27. Let γ be a Krein Γ-ﬁeld for (S, H 0 , G ). Assume that H 0 has a gap, and λ ∈ res H 0 ∩ R. Represent f ∈ dom S ∗ as f = fλ + γ(λ)F, where fλ ∈ dom H 0 , F ∈ G . Deﬁne Γ1 f := F,

Γ2 f := γ ∗ (λ)(H 0 − λ)fλ ,

then (G , Γ1 , Γ2 ) is a boundary triple for S ∗ . Proof of Proposition 1.26 First of all, note that the component F in (1.25) is independent of z. To see that, it is suﬃcient to write f as fz +(γ(z)−γ(λ))F +γ(λ)F and to use the uniqueness of this expansion and the inclusion (γ(z) − γ(λ))F ∈ dom H 0 following from (1.21f). The property (1.8b) of boundary triples is obvious. From the equality (H 0 − z) dom S ⊥ = ker(S ∗ − z¯) and (1.21d), it follows that dom S ⊂ ker(Γ1 , Γ2 ), which proves (1.8c). To show (1.8a) we write 2f |S ∗ g − 2S ∗ f |g = f |(S ∗ − z)g + f |(S ∗ − z¯)g − (S ∗ − z)f |g − (S ∗ − z¯)f |g = fz¯ + γ(¯ z )Γ1 f |(H 0 − z)gz + fz + γ(z)Γ1 f |(H 0 − z¯)gz¯ −(H 0 − z)fz |gz¯ + γ(¯ z )Γ1 g − (H 0 − z¯)fz |gz + γ(z)Γ1 g = fz¯|(H 0 − z)gz + fz |(H 0 − z¯)gz¯ − (H 0 − z¯)fz¯|gz − (H 0 − z)fz |gz¯ + Γ1 f |γ ∗ (¯ z )(H 0 − z)gz + Γ1 f |γ ∗ (z)(H 0 − z¯)gz¯ − γ ∗ (¯ z )(H 0 − z)fz |Γ1 g − γ ∗ (z)(H 0 − z¯)fz , Γ1 g = 2Γ1 f |Γ2 g − 2Γ2 f |Γ1 g.

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To show that this boundary triple induces γ it is suﬃcient to note that Γ1 γ(z) = idG and γ(z)Γ1 = iddom S ∗ . Proposition 1.26 does not use any information on Q-functions, and Q-functions can be taken into account as follows. Proposition 1.28. Let γ be a Γ-ﬁeld for (S, H 0 , G ) and Q be an associated Qfunction, then there exists a boundary triple (G , Γ1 , Γ2 ) for S ∗ which induces γ and Q. Proof. Let (G , Γ1 , Γ2 ) be the boundary triple for S ∗ deﬁned in Proposition 1.26 ˜ be the induced Q-function. By Proposition 1.20, there exists a bounded selfand Q ˜ adjoint operator M on G with Q(z) = Q(z) + M . Clearly, (G , Γ1 , Γ2 ) with Γ1 = Γ1 and Γ2 = Γ2 + M Γ1 is another boundary triple for S ∗ by Proposition 1.15. On the other hand, γ is still the Γ-ﬁeld induced by this new boundary triple, and the ˜ + M , coincides induced Q-function, which is Γ2 γ(z) ≡ Γ2 γ(z) + M Γ1 γ(z) ≡ Q(z) with Q(z). One of the most useful tools for the spectral analysis of self-adjoint extensions is the Krein resolvent formula described in the following theorem. Theorem 1.29. Let S be a closed densely deﬁned symmetric operator with equal deﬁciency indices in a Hilbert space H , (G , Γ1 , Γ2 ) be a boundary triple for S ∗ , H 0 be the self-adjoint restriction of S ∗ to ker Γ1 , γ and Q be the Krein Γ-ﬁeld and Q-function induced by the boundary triple. Let Λ be a self-adjoint linear relation in G and HΛ be the restriction of S ∗ to the functions f ∈ dom S ∗ satisfying (Γ1 f, Γ2 f ) ∈ Λ. (1) For any z ∈ res H 0 , there holds ker(HΛ − z) = γ(z) ker(Q(z) − Λ). (2) For any z ∈ res H 0 ∩ res HΛ , there holds 0 ∈ res(Q(z) − Λ) and z ). (H 0 − z)−1 − (HΛ − z)−1 = γ(z)(Q(z) − Λ)−1 γ ∗ (¯ (3) There holds spec HΛ \spec H 0 = {z ∈ res H 0 : 0 ∈ spec(Q(z) − Λ)}. Proof. (1) Assume that φ ∈ ker(Λ − Q(z)) then there exists ψ ∈ G such that (φ, ψ) ∈ Λ and ψ − Q(z)φ = 0. This means the inclusion (φ, Q(z)φ) ∈ Λ. Consider the vector F = γ(z)φ. Clearly, (S ∗ − z)F = 0. The condition (Γ1 F, Γ2 F ) ≡ (φ, Q(z)φ) ∈ Λ means that F ∈ dom HΛ and (HΛ − z)F = 0. Therefore, γ(z) ker(Q(z) − Λ) ⊂ ker(HΛ − z). Conversely, let F ∈ ker(HΛ − z), z ∈ res H 0 . Then also (S ∗ − z)F = 0 and by Theorem 1.23(1) there exists φ ∈ G with F = γ(z)φ. Clearly, (φ, Q(z)φ) ≡ (Γ1 F, Γ2 F ) ∈ Λ, i.e. there exist ψ ∈ G with (φ, ψ) ∈ Λ and Q(z)φ = ψ. But this means φ ∈ ker(Q(z) − Λ). (2) Let z ∈ res H 0 ∩ res HΛ . Take any h ∈ H and denote f := (HΛ − z)−1 h; clearly, f ∈ dom HΛ , and by Theorem 1.23(1) there exist uniquely determined

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functions fz ∈ dom H 0 and gz ∈ Nz with f = fz + gz . There holds h = (HΛ − z)f = (S ∗ −z)f = (S ∗ −z)fz +(S ∗ −z)gz = (S ∗ −z)fz = (H 0 −z)fz and fz = (H 0 −z)−1 h. Moreover, from Γ1 fz = 0 one has Γ1 f = Γ1 gz , gz = γ(z)Γ1 f , and, therefore, (HΛ − z)−1 h = (H 0 − z)−1 h + γ(z)Γ1 f.

(1.26)

Applying to the both sides of the equality f = fz + γ(z)Γ1 f the operator Γ2 one arrives at Γ2 f = Γ2 fz + Q(z)Γ1 f and Γ2 f − Q(z)Γ1 f = Γ2 fz .

(1.27)

When h runs through the whole space H , then fz runs through dom H 0 and the values Γ2 fz cover the whole space G . At the same time, if f runs through dom HΛ , then the values (Γ1 f, Γ2 f ) cover the whole Λ. It follows then from (1.27) that ran(Λ − Q(z)) = G . On the other hand, by (1) one has ker(Λ − Q(z)) = 0 and 0 ∈ res(Λ − Q(z)). From (1.27), one obtains Γ1 f = (Λ − Q(z))−1 Γ2 fz .

(1.28)

By Theorem 1.23(2d) there holds Γ2 fz = γ ∗ (¯ z )h. Substituting this equality into (1.28) and then into (1.26) one arrives at the conclusion. The item (3) follows trivially from the item (2). Remark 1.30. Note that the operators HΛ and H 0 satisfy dom HΛ ∩ dom H 0 = dom S iﬀ Λ is a self-adjoint operator (i.e. is a single-valued); one says that HΛ and H 0 are disjoint extensions of S. In this case, the resolvent formula conains only operators and has the direct meaning. As we will see below, in this case one can obtain slightly more spectral information in comparison with the case when Λ is a linear relation, so it is useful to understand how to reduce the general case to the disjoint one. Let T be the maximal common part of H 0 and HΛ , i.e. the restriction of S ∗ to dom H 0 ∩ dom HΛ . Clearly, T is a closed symmetric operator, dom T = {f ∈ dom S ∗ : Γ1 f = 0, Γ2 f ∈ L }

(1.29)

where L = ker(Λ−1 ) is a closed linear subspace of G . Lemma 1.31. Let L be a closed linear subspace of G and T be deﬁned by (1.29), then dom T ∗ = {f ∈ dom S ∗ : Γ1 f ∈ L ⊥ }. Proof. It is clear that both T and T ∗ are restrictions of S ∗ . Hence, for any f ∈ dom T and g ∈ dom S ∗ one has W (f, g) := f |S ∗ g − T f |g = Γ1 f |Γ2 g − Γ2 f |Γ1 g = Γ2 f |Γ1 g. As Γ2 (dom T ) = L , one has W (f, g) = 0 for all f iﬀ Γ1 g ⊥ L . Now one can construct a boundary triple for T ∗ starting from the boundary triple for S ∗ .

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Theorem 1.32. Let the assumptions of Theorem 1.29 be satisﬁed. Let L be a ˜ 2 ) is a ˜ 1, Γ closed subset of G and an operator T be deﬁned by (1.29). Then (G˜, Γ ∗ ⊥ ˜ ˜ boundary triple for T , where G := L with the induced scalar product, Γj := P Γj , j = 1, 2, and P is the orthogonal projection onto G˜ in G . The induced Γ-ﬁeld γ˜ and ˜ are γ˜ (z) := γ(z)P, Q(z) ˜ Q-function Q := P Q(z)P considered as maps from G˜ to Nz and in G˜, respectively. Proof. Direct veriﬁcation. Returning to the operators H 0 and HΛ , one sees that, by construction, they are disjoint extensions of T , and in the notation of Theorem 1.32 they are given by the ˜ 2 f = LΓ ˜ 1 f , respectively, where L is a certain ˜ 1 f = 0 and Γ boundary conditions Γ ˜ self-adjoint operator in G . Using Theorem 1.29, one can relate the resolvents of H 0 and HΛ by ˜ − L)−1 γ˜ ∗ (¯ z) (H 0 − z)−1 − (HΛ − z)−1 = γ˜(z)(Q(z) = γ(z)P (P Q(z)P − L)−1 P γ ∗ (¯ z ),

(1.30)

and spec HΛ \spec H 0 = {z ∈ res H 0 : 0 ∈ spec(P Q(z)P − L)}. The operator L can be calculated, for example, starting from the Cayley tranform of Λ (see Proposition 1.2). Namely, let UΛ be the Cayley transform of Λ, then, obviously, G˜ = ker(1 − UΛ )⊥ . The Cayley transform of L is then of the form UL := P UΛ P considered as a unitary operator in G˜, and L = i(1 − UL )−1 (1 + UL ). Remark 1.33. For the case of a simple symmetric operator (that is, having no nontrivial invariant subspaces), one can describe the whole spectrum in terms of the limit values of the Weyl function, and not only the spectrum lying in gaps of a ﬁxed self-adjoint extensions, see [18, 25] for discussion. We note that, neverthless, the simplicity of an operator is a quite rare property in multidimensional problems which is quite diﬃcult to check. Remark 1.34. It seems that the notion of boundary value triple appeared ﬁrst in the papers by Bruk [28] and Kochubei [89], although the idea goes back to the paper by Calkin [39]. The notion of a Γ-ﬁeld and a Q-function appeared ﬁrst in [93, 99], where they were used to describe the generalized resolvents of selfadjoint extensions. The relationship between the boundary triples and the resolvent formula in the form presented in Theorems 1.23 and 1.29 was found by Derkach and Malamud, but it seems that the only existing discussion was in [51], which is hardly available, so we preferred to provide a complete proof here. The same scheme of the proof works in more abstract situations, see, e.g., [47]. The forumula (1.30) is borrowed from [116], but we give a diﬀerent proof. Remark 1.35. Theorem 1.29 shows that one can express the resolvents of all self-adjoint extensions of a certain symmetric operator through the resolvent of a ﬁxed extension, more precisely, of the one corresponding to the boundary condition

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Γ1 f = 0. On the other hand, Proposition 1.14 shows that by a suitable choice of boundary triple one can start with any extension. Formulas expressing Q-functions associated with diﬀerent extensions of the same operator can be found, e.g., in [50, 64, 97]. In view of Proposition 1.5 on the parametrization of linear relations, it would be natural to ask whether one can rewrite the Krein resolvent formula completely in terms of operators without using linear relations. Namely, if a self-adjoint linear relation Λ is given in the form Λ = {(x1 , x2 ) ∈ G ⊕ G ; Ax1 = Bx2 }, where A and B are bounded linear operators satisfying (1.4a) and (1.4b), can one write an analogue of the Krein resolvent formula for HΛ in terms of A and B? We formulate only here the main result referring to the recent work [108] for the proof. Theorem 1.36. Let the assumptions of Theorem 1.29 be satisﬁed and A, B be bounded linear operators in G satisfying (1.4a) and (1.4b). Denote by H A,B the self-adjoint extension of S corresponding to the boundary conditions AΓ1 f = BΓ2 f, then (1) For any z ∈ res H 0 , there holds ker(H A,B − z) = γ(z) ker(BQ(z) − A). (2) For any z ∈ res H 0 ∩ res H A,B the operator BQ(z) − A is injective and z ). (H 0 − z)−1 − (H A,B − z)−1 = γ(z)(BQ(z) − A)−1 Bγ ∗ (¯

(1.31)

(3) If A and B satisfy additionally the stronger condition 0 ∈ res

A B

−B A

,

(1.32)

then 0 ∈ res(BQ(z) − A) for all z ∈ res H 0 ∩ res H A,B , and, respectively, spec H A,B \spec H 0 = {z ∈ res H 0 : 0 ∈ spec(BQ(z) − A)}. Note that the condition (1.32) is satisﬁed if one uses the parametrization by the Cayley transform (Theorem 1.2), i.e. A = i(1 + U ), B = 1 − U with a unitary U , see Proposition 1.1. Therefore, one can perform a “uniform” analysis of all self-adjoint extensions using the single unitary parameter U . Note that the above normalization condition is trivial for ﬁnite deﬁciency indices, hence the Krein formula has a particularly transparent form [11]. We note in conclusion that the resolvent formulas (1.30) and (1.31) provide two diﬀerent ways of working with non-disjoint extensions, and they can be obtained one from another [116].

1.4. Examples Here we consider some situations in which boundary triples arise.

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1.4.1. Sturm–Liouville problems A classical example comes from the theory of ordinary diﬀerential operators. Let V ∈ L2loc (0, ∞) be real-valued and, for simplicity, semibounded below. Denote by d2 ∞ S0 the closure of the operator − dx 2 + V with the domain C0 (0, ∞) in the space 2 H := L (0, ∞). It is well known that the deﬁciency indices of S0 are (1, 1). Using the integration by parts one can easily show that for the adjoint S := S0∗ as a boundary triple one can take (C, Γ1 , Γ2 ), Γ1 f = f (0), Γ2 f = f (0). Denoting for z = C by ψz the unique L2 -solution to −ψz + V ψz = zψz with ψz (0) = 1, we arrive to the induced Krein Γ-ﬁeld, γ(z)ξ = ξψz , and the induced Q-function Q(z) = ψz (0), which is nothing but the Weyl–Titchmarsh function. Determining the spectral properties of the self-adjoint extensions of S0 with the help of this function is a classical problem of the spectral analysis. An analogous procedure can be done for Sturm–Louville operators on a segment. In H := L2 [a, b], −∞ < a < b < ∞ consider an operator S acting by the rule f → −f +V f with the domain dom S = H 2 [a, b]; here we assume that V ∈ L2 [a, b] is real-valued. It is well known that S is closed. By partial integration one easily sees that (G , Γ1 , Γ2 ), (a) f (a) f G = C2 , Γ 1 f = , , Γ2 f = −f (b) f (b) is a boundary triple for S. The distinguished extension H 0 corresponding to the boundary condition Γ1 f = 0 is nothing but the operator −d2 /dx2 + V with the Dirichlet boundary conditions. Let two functions s(·; z), c(·; z) ∈ ker(S − z) solve the equation −f + V f = zf,

z ∈ C,

(1.33)

and satisfy s(a; z) = c (a; z) = 0 and s (a; z) = c(a; z) = 1. Clearly, s, c as well as their derivatives are entire functions of z; these solutions are linearly independent, / and their Wronksian w(z) ≡ s (x; z)c(x; z) − s(x; z)c (x; z) is equal to 1. For z ∈ 0 spec H , one has s(b; z) = 0, and any solution f to (1.33) can be written as f (x; z) =

f (b) − f (a)c(b; z) s(x; z) + f (a)c(x; z), s(b; z)

(1.34)

which means that the Γ-ﬁeld induced by the above boundary triple is ξ2 − ξ1 c(b; z) ξ s(x; z) + ξ1 c(x; z). γ(z) 1 = ξ2 s(b; z) The calculation of f (a) and −f (b) gives 1 f (a; z) f (a; z) −c(b; z) 1 = Q(z) , Q(z) = , f (b; z) −f (b; z) 1 −s (b; z) s(b; z) and Q(z) is the induced Q-function.

(1.35)

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A number of examples of boundary triples in problems concerning ordinary diﬀerential equations as well as their applications to scattering problems can be found, e.g., in [19, 50]. The situation becomes much more complicated when dealing with elliptic differential equations on domains (or manifolds) with boundary. In this case, the construction of a boundary triple involves certain information about the geometry of the domain, namely, the Dirichlet-to-Neumann map, see, e.g., the recent works [17, 116] and the classical paper by Vishik [125], and the question on eﬀective description of all self-adjoint boundary value problems for partial diﬀerential equations is still open, see the discussion in [55, 56] and historical comments in [76]; an explicit construction of boundary triples for the Laplacian in a bounded domain is presented in [116, Example 5.5]. We remark that boundary triples provide only one possible choice of coordinates in the defect subspaces. Another possibility would be to use some generalization of boundary triples, for example, the so-called boundary relations resulting in unbounded Weyl functions [27, 49], but it seems that this technique is rather new and not developed enough for applications.

1.4.2. Singular perurbations Here we discuss the construction of self-adjoint extensions in the context of the so-called singular perturbations; we follow in part the constriction of [115]. Let H 0 be a certain self-adjoint operator in a separable Hilbert space H ; its resolvent will be denoted by R0 (z), z ∈ res H 0 . Denote by H1 the domain dom H0 equiped with the graph norm, f 21 = H 0 f 2 +f 2; clearly, H1 is a Hilbert space. Let G be another Hilbert space. Consider a bounded linear map τ : H1 → G . We assume that τ is surjective and that ker τ is dense in H . By deﬁnition, by a singular perturbation of H 0 supported by τ we mean any self-adjoint extension of the operator S which is the restriction of H 0 to dom S := ker τ . Due to the above restrictions, S is a closed densely deﬁned symmetric operator. It is worthwhile to note that singular perturbations just provide another language for the general theory of self-adjoint extensions. Namely, let S by any closed densely deﬁned symmetric operator with equal deﬁciency indices and H 0 be some its self-adjoint extension. Construct the space H1 as above. Clearly, L := dom S is a closed subspace of H1 , therefore, H1 = L ⊕ L ⊥ . Denoting L ⊥ by G and the orthogonal projection from H1 to L ⊥ by τ , we see the self-adoint extensions of S are exactly the singular perturbations of H 0 supported by τ . At the same time, knowing explicitly the map τ gives a possibility to construct a boundary triple for S. z ))∗ , z ∈ res H 0 , form a Krein Proposition 1.37. The maps γ(z), γ(z) = (τ R0 (¯ 0 Γ-ﬁeld for (S, H , G ).

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Proof. Note that the operator A := τ R0 (z) : H → G is surjective, therefore, ran A∗ = ker A⊥ . In other words, z )⊥ = {f ∈ H : τR0 (¯ z )f = 0}⊥ ran γ(z) = ker τR0 (¯ = {(H 0 − z¯)g : τ g = 0}⊥ = {(S − z¯)g : g ∈ dom S}⊥ = ran(S − z¯)⊥ = ker(S ∗ − z) =: Nz .

(1.36)

Let us show that γ(z) is an isomorphism of G and Nz . First, note that γ(z) is bounded and, as we have shown above, surjective. Moreover, ker γ(z) = ran A⊥ = G ⊥ = {0}. Therefore, γ(z) : G → Nz has a bounded inverse deﬁned everywhere by the closed graph theorem, and the condition (1.14a) is satisﬁed. The condition (1.14b) is a corollary of the Hilbert resolvent identity. Now, one can construct a boundary triple for the operator S ∗ . Proposition 1.38. Take any ζ ∈ res H 0 and represent any f ∈ dom S ∗ in the form f = fζ + γ(ζ)F, fζ ∈ dom H 0 , F ∈ G , where γ is deﬁned in Proposition 1.37. Then (G , Γ1 , Γ2 ), Γ1 f = F, Γ2 f = 12 τ (fζ + fζ¯), is a boundary triple for S ∗ . The induced Γ-ﬁeld is γ(z), and the induced Q-function Q(z) has the form Q(z) =

1 ¯ − 1 τR0 (z)(ζγ(ζ) + ζγ( ¯ ζ)). ¯ zτR0 (z)(γ(ζ) + γ(ζ)) 2 2

Proof. The major part follows from Proposition 1.26. To obtain the formula for Q(z) it is suﬃcient to see that for the function f = γ(z)ϕ, ϕ ∈ G , one has fζ = (γ(z) − γ(ζ))ϕ and to use the property (1.14b). Let us consider in greater detail a special type of the above construction, the so-called ﬁnite rank perturbations [9]. Let H 0 be as above. For α ≥ 0 denote by Hα the domain of the operator ((H 0 )2 + 1)α/2 equiped with the norm f α = ((H 0 )2 + 1)α/2 f . The space Hα becomes a Hilbert space, and this notation is compatible with the above deﬁnition of H1 , i.e. H1 is the domain of H 0 equiped with the graph norm, and H0 = H . Moreover, for α < 0 we denote the completion of H with respect to the norm f α = (H 0 )2 + 1)α/2 f . Clearly, Hα ⊂ Hβ if α > β. Take ψj ∈ H−1 , j = 1, . . . , n. In many problems of mathematical physics one arrives at operators given by formal expressions of the form H = H0 +

n

αjk ψj |·ψk ,

(1.37)

j,k=1

where αjk are certain numbers (“coupling constants”). This sum is not deﬁned / H . At the same time, the operator H given by directly, as generically ψj ∈ this expression is usually supposed to be self-adjoint (and then one has formally αjk = αkj ). Denote by S the restriction of H 0 to the functions f ∈ dom H 0 with ψj |f = 0 for all j; we additionally assume that ψj are linearly indepedent modulo

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H (otherwise S might become non-densely deﬁned). Clearly, for any reasonable deﬁnition, the operators H 0 and H must coincide on the domain of S. Therefore, by deﬁnition, under an operator given by the right-hand side of (1.37) we understand the whole family of self-adjoint extensions of S. The boundary triple for S ∗ can be easily obtained using the above constructions if one set   ψ1 |f τ f :=  · · ·  ∈ Cn . ψn |f The corresponding Γ-ﬁeld from Proposition 1.37 takes the form γ(z)ξ =

n

ξj hj (z),

hj (z) := R0 (z)ψj ∈ H ,

ξ = (ξ1 , . . . , ξn ) ∈ Cn ,

j=1

and the boundary triples and the Q-function are obtained using the formulas of Proposition 1.38. Unfortunately, the above construction has a severe disadvantage, namely, the role of the coeﬃcients αjk in (1.37) remains unclear. The deﬁnition of H using selfadjoint extensions involves self-adjoint linear relations in Cn , and it is diﬃcult to say what is the relationship between these two types of parameters. In some cases, if both H and H 0 have certain symmetries, this relationship can be found using a kind of renormalization technique [96, 98]. The situation becomes more simple if in the above construction one has ψj ∈ H−1/2 and H 0 is semibounded. In this case, one can properly deﬁne H given by (1.37) using the corresponding quadratic form, h(f, g) = h0 (f, g) +

n

αjk f |ψj ψk |g,

j,k=1

where h0 is the quadratic form associated with H 0 , see [90]. Also in this case, one arrives at boundary triples and resolvent formulas. A very detailed analysis of rank-one perturbations of this kind with an extensive bibliography list is given in [122]. We also remark that one can deal with operator of the form (1.37) in the so-called / H−1 ; the corresponding operators H must be constructed supersingular case ψj ∈ then in an extended Hilbert or Pontryagin space, see, e.g., [52,96,120] and references therein. 1.4.3. Point interactions on manifolds Let X be a manifold of bounded geometry of dimension ν, ν ≤ 3. Let A = Aj dxj be a 1-form on X, for simplicity we suppose here Aj ∈ C ∞ (X). The functions Aj can be considered as the components of the vector potential of a magnetic ﬁeld on X. On the other hand, A deﬁnes a connection ∇A in the trivial line bundle X × C → X, ∇A u = du + iuA; by −∆A = ∇∗A ∇A we denote the corresponding

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Bochner Laplacian. In addition, we consider a real-valued scalar potential U of an electric ﬁeld on X. This potential will be assumed to satisfy the following conditions: 0 (X), U+ := max(U, 0) ∈ Lploc

U− := max(−U, 0) ∈ 2 ≤ pi ≤ ∞,

n

Lp i (X),

i=1

0 ≤ i ≤ n;

we stress that pi as well as n are not ﬁxed and depend on U . The class of such potenodinger tials will be denoted by P(X). For the case X = Rn , one can study Schr¨ operators with more general potentials from the Kato class [26, 123]. We denote by HA,U the operator acting on functions φ ∈ C0∞ (X) by the rule HA,U φ = −∆A φ + U φ. This operator is essentially self-adjoint in L2 (X) and semibounded below [37]; its closure will be also denoted by HA,U . It is also known [37] that dom HA,U ⊂ C(X).

(1.38)

In what follows, the Green function GA,U (x, y; ζ) of HA,U , i.e. the integral kernel of the resolvent RA,U (ζ) := (HA,U − ζ)−1 , ζ ∈ res HA,U , will be of importance. The most important its properties for us are the following ones: for any ζ ∈ res HA,U , GA,U is continuous in X × X for ν = 1 and in X × X \{(x, x), x ∈ X} for ν = 2, 3;

(1.39a)

for ζ ∈ res H 0 and y ∈ X, one has GA,U (·, y; ζ) ∈ L2 (X);

(1.39b)

for any f ∈ L (X) and ζ ∈ res HA,U , the function x → GA,U (x, y; ζ) f (y) dy is continuous.

(1.39c)

2

X

We remark that for any f ∈ dom HA,U and ζ ∈ res HA,U one has f = RA,U (ζ)(HA,U − ζ)f . Using the Green function we rewrite this as GA,U (x, y; ζ)(HA,U − ζ)f (y) dy a.e.; f (x) = X

by (1.39c) and (1.38) the both sides are continuous functions of x, therefore, they coincide everywhere, i.e. GA,U (x, y; ζ)(HA,U − ζ)f (y) dy, f ∈ dom HA,U , for all x ∈ X. f (x) = X

(1.40) Fix points a1 , . . . , an ∈ X, aj = ak if j = k, and denote by S the restriction of HA,U on the functions vanishing at all aj , j = 1, . . . , n. Clearly, due to (1.38) this restriction is well-deﬁned, and S is a closed densely deﬁned symmetric operator. By deﬁnition, by a point perturbation of the operator HA,U supported by the points aj , j = 1, . . . , n, we mean any self-adjoint extension of S. Now, we are actually in the situation of Sec. 1.4.2. To simplify notation, we denote H 0 := HA,U and

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change respectively the indices for the resolvent and the Green function. Denote by τ the map   f (a1 ) τ : dom H 0 f →  · · ·  ∈ Cn . f (an ) By (1.40) and (1.39b), τ is bounded in the graph norm of H 0 . Now, let us use z ) is of the form Proposition 1.37. The map τ R0 (¯   0 G (a1 , y; z¯)f (y)dy   X   . · · · f →      0 G (an , y; z¯)f (y)dy X

Calculating the adjoint operator and taking into account the identity G0 (x, y; z) = G0 (y, x; z¯) we arrive at Lemma 1.39. The map γ(ζ) : Cn (ξ1 , . . . , ξn ) →

n

ξj G0 (·, aj ; ζ) ⊂ L2 (X)

(1.41)

j=1

is a Krein Γ-ﬁeld for (S, H 0 , Cn ). Let us construct a boundary triple corresponding to the problem. Use ﬁrst Corollary 1.27. Choose ζ ∈ res H 0 ⊂ R; this is possible because H 0 is semibounded below. For any f ∈ dom S ∗ there are Fj ∈ C such that fζ := f − j Fj G0 (·, aj ; ζ) ∈ dom H 0 . The numbers Fj are ζ-independent, and by Corollary 1.27, the maps ˜ 1 f := (F1 , . . . , Fn ), Γ

˜ 2 f = (fζ (a1 ), . . . , fζ (an )) Γ

(1.42)

form a boundary triple for S ∗ . Nevertheless, such a construction is rarely used in practice due to its dependence on the energy parameter. We modiﬁy the above considerations using some information about the on-diagonal behavior of G0 . Consider the case ν = 2 or 3. As shown in [38], there exists a function F (x, y) deﬁned for x = y such that for any ζ ∈ res H 0 there exists another function G0ren (x, y; ζ) continuous in X × X such that G0 (x, y; ζ) = F (x, y) + G0ren (x, y; ζ),

(1.43)

and we additionally request F (x, y) = F¯ (y, x). It is an important point that under some assumptions the function F can be chosen independent of the magnetic potential Aj and the scalar potential U . For example, if ν = 2, one can always set 1 . In the case ν = 3 the situation becomes more complicated. F (x, y) = log d(x,y) For example, for two scalar potentials U and V satisfying the above conditions one can take the same function F for the operators HA,U and HA,V provided U − V ∈ Lqloc (X) for some q > 3. In paritucular, for any U satisying the above

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conditions and, additionally, U ∈ Lqloc (X), for the operator H0,U one can always 1 . put F (x, y) = 4πd(x,y) For the Schr¨ odinger operator with a uniform magnetic ﬁeld in R3 , H 0 = (i∇ + eiBxy/2 A)2 , where ∇ × A =: B is constant, one can put F (x, y) := 4π|x−y| . For a detailed discussion of on-diagonal singularities, we refer to our paper [38]. Let us combine the representation (1.43) for the Green function and the equality dom S ∗ = dom H 0 + Nζ . Near each point aj , any function f ∈ dom S ∗ has the following asymptotics: f (x) = fj + Fj F (x, aj ) + o(1),

fj , Fj ∈ C.

Proposition 1.40. The triple (Cn , Γ1 , Γ2 ) with Γ1 f = (F1 , . . . , Fn ) ∈ Cn and Γ2 f = (f1 , . . . , fn ) ∈ Cn is a boundary triple for S ∗ . ˜j Proof. Let us ﬁx some ζ res H 0 ∩ R. Comparing the maps Γj with the maps Γ ˜ 1 . Furthermore, Γ2 f = Γ ˜ 2f + BΓ ˜ 1 , where from (1.42) one immediately see Γ1 ≡ Γ B is a n × n matrix, if j = k, G0 (aj , ak ; ζ) Bjk = 0 Gren (aj , aj ; ζ) otherwise. As B = B ∗ , it remains to use Proposition 1.15. Clearly, the map (1.41) is the Krein Γ-ﬁeld induced by the boundary triple (Cn , Γ1 , Γ2 ). The calculation of the corresponding Q-function Q(ζ) gives G0 (aj , ak ; ζ) if j = k, Qjk (ζ) = 0 Gren (aj , aj ; ζ) otherwise. We note that the calculating of the Q-function needs a priori the continuity of the Green function (otherwise the values of the Green function at single points would not be deﬁned). A bibliography concerning the analysis of operators of the above type for particular Hamiltonians H 0 can be found, e.g., in [7]. The above construction can generalized to the case of point perturbations supported by non-ﬁnite (but countable) sets provided some uniform discreteness, we refer to [67] for the general theory, to [8,34,70] for the analysis of periodic conﬁgurations, and to [23,53,78,117] for multidimensional models with random interactions. For analysis of interactions supported by submanifolds of higher dimension, we refer to [20, 43, 44, 60, 61, 114] and references therein. 1.4.4. Direct sums and hybrid spaces Assume that we have a countable family of closed linear operators Sα in some α Hilbert spaces Hα , α ∈ A , having boundary triples (G α , Γα 1 , Γ2 ). Denote by 0 0 Hα the corresponding distinguished extensions, Hα := Sα |ker Γα1 . We impose some

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additional regularity conditions, namely, that: • there exist constants a and b such that for any α ∈ A and fα ∈ dom Sα , there holds Γα 1/2 fα ≤ aSα fα + bfα , 1/2 α • for any (ξα ) ∈ α∈A G , there is (fα ) ∈ α∈A Hα , fα ∈ dom Sα , with 1/2 α Γ1/2 fα = ξα . The above conditions are obviously satisﬁed if, for example, the operators Sα are copies of a ﬁnite set of operators, and the same holds for the boundary triples. Another situation where the conditions are satisﬁed, is provided by the operators d2 2 2 Sα = − dx 2 + Uα acting in L [aα , bα ] with the domains H [aα , bα ] provided that there are constants l1 , l2 , C such that l1 ≤ |aα − bα | < l2 and Uα L2 < C and that the boundary triples are taken as in Sec. 1.4.1, see [109] for details. Under the above conditions, the operator S := α∈A Sα acting in H := α∈A Hα is closed and has a boundary triple (G , Γ1 , Γ2 ), G := Gα , Γj := Γα j = 1, 2. j, α∈A

α∈A

Moreover, the corresponding distinguished extension H 0 and the induced Krein maps γ and Q are also direct sums, i.e. at least Hα0 , γ(z) = γ α (z), Q(z) = Qα (z). H 0 := α∈A

α∈A

α∈A

Note that γ(z) and Q(z) are deﬁned only for z ∈ / spec H 0 ≡ α∈A spec Hα0 . Let us show how this abstract construction can be used to deﬁne Schr¨odinger operators on hybrid spaces, i.e. on conﬁgurations consisting of pieces of diﬀerent dimensions. Let Mα , α ∈ A , be a countable family of manifolds as in Sec. 1.4.3. Fix several points mαj ∈ Mα , j = 1, . . . , nα . We interpret these points as points of glueing. More precisely, we consider a matrix T with the entries T(αj)(βk) such that T(αj)(βk) = 1 if the point mαj is identiﬁed with mβk (i.e. point mαj of Mα is glued to the point mβk of Mβ ), and T(αj)(βk) = 0 otherwise. The obtained topological space is not a manifold as it has singularities at the points of glueing; we will refer it to as a hybrid manifold. Our aim is to show how to deﬁne a Schr¨ odinger operator in such a structure. odinger operators Hα as in Sec. 1.4.3. To On of the manifolds Mα consider Schr¨ satisfy the above regularity conditions we request that these operators are copies of a certain ﬁnite family. For α ∈ A denote by Sα the restriction of Hα to the functions vanishing at all the points mαj and construct a boundary triple (Cmα , Γα1 , Γα2 ) for Sα∗ as in Sec. 1.4.3. Clearly, as a boundary triple for the operator S ∗ , S := α∈A Sα , in the space L2 (M ) := α∈A L2 (Mα ) one take (G , Γ1 , Γ2 ) with G := α∈A Cnα , odinger operator on L2 (M ), one can mean Γj (fα ) = (Γαj fα ), j = 1, 2. Under a Schr¨ any self-adjoint extension of S. To take into account the way how the manifolds are glued with each other, one should restrict the class of possible boundary conditions. A reasonable idea would be to consider boundary conditions of the form

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AΓ1 = BΓ2 such that A(αj)(βk) = B(αj)(βk) = 0 if T(αj)(βk) = 0, i.e. assuming that each boundary condition involves only points glued to each other. The analysis of generic Schr¨odinger operators on hybrid manifolds is hardly possible, as even Schr¨ odinger operators on a single manifold do not admit the complete analysis. One can say some more about particular conﬁguration, for example, if one has only ﬁnitely many pieces Mα and they all are compact [58]. Some additional information can be obtained for periodic conﬁgurations [31, 33]. One can extend the above construction by combining operators from Secs. 1.4.1 and 1.4.3; in this way one arrive at a space with consists of manifolds connected with each other through one-dimensional segments. One can also take a direct sum of operators from Sec. 1.4.1 to deﬁne a Schr¨ odinger operator on a conﬁguration consisting of segments and halﬂines connected with each other; such operators are usually referred to as quantum graphs, and their analysis becomes very popular in the last decades, see, e.g., [22] for the review and recent developments. 2. Classification of Spectra of Self-Adjoint Operators 2.1. Classification of measures Here we recall brieﬂy some concepts of the measure theory. Let B be the set of all the Borel subsets of a locally compact separable metric space X. A mapping µ : B → [0, +∞] is called a positive Borel measure on X if it is σ-additive (i.e. µ( k Bk ) = k µ(Bk ) for every countable family (Bk ) of mutually not-intersecting sets from B) and has the following regularity properties: • µ(K) < ∞ for every compact K ⊂ X; • for every B ∈ B, there holds µ(B) = sup{µ(K) : K ⊂ B, K is compact} = inf{µ(G) : G ⊃ B, G is open}. A complex valued Borel measure on X is a σ-additive mapping µ : B → C such that the variation |µ| of µ deﬁned on B by |µ|(B) = sup |µ(Bk )|, where the supremum is taken over all ﬁnite families (Bk ) of mutually non intersecting sets Bk from B such that Bk ⊂ B, is a Borel measure. For a positive measure µ, one has |µ| = µ. If |µ|(X) < ∞, then µ is called ﬁnite (or bounded) and |µ|(X) is denoted also by µ. We will denote by M (X) (respectively, by M + (X)) the set of all complex Borel measures (respectively, the set of all positive Borel measures) on X; if X = R we write simply M and M + . It is clear that M (X) is a complex vector space (even a complex vector lattice) and the subset M b (X) of all bounded measures from M (X) is a vector subspace of M (X) which is a Banach space with respect to the norm µ. Ona says that a measure µ is concentrated on a Borel set S ∈ B, if µ(B) = µ(B ∩ S) for all B ∈ B. Let µ1 and µ2 be two measures; they are called disjoint or

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mutually singular, if there exists two disjoint Borel set S1 and S2 such that µj is concentrated on Sj (j = 1, 2); we will write µ1 ⊥µ2 if µ1 and µ2 are disjoint. The measure µ1 is said to be subordinated to µ2 (or absolutely continuous with respect to µ2 ) if every |µ2 |-negligible Borel set is simultaneously |µ1 |-negligible. According to the Radon–Nikodym theorem, the following assertions are equivalent: (1) µ1 is subordinated to µ2 ; (2) there exists a Borel function f such that µ1 = f µ2 (in this case f ∈ L1loc (X, µ2 ) and f is called the Radon–Nikodym derivative of µ1 with respect to µ2 ). If µ1 is subordinated to µ2 and simultaneously µ2 is subordinated to µ1 (i.e. if µ1 and µ2 have the same negligible Borel sets), then µ1 and µ2 are called equivalent (in symbols: µ1 ∼ µ2 ). For a subset M ⊂ M (X) we denote M ⊥ = {µ ∈ M (X) : µ ⊥ ν ∀ ν ∈ M }; M ⊥ is a vector subspace of M (X). A subspace M ⊂ M (X) is called a band (or a component) in M (X), if M = M ⊥⊥ . For every subset L ∈ M (X) the set L⊥ is a band; the band L⊥⊥ is called the band generated by L. In particular, if µ ∈ M (X), then the band {µ}⊥⊥ consists of all ν which are subordinated to µ. Moreover, µ1 is subordinated to µ2 if and only if {µ1 }⊥⊥ ⊂ {µ2 }⊥⊥ ; in particular, µ1 ∼ µ2 if and only if {µ1 }⊥⊥ = {µ2 }⊥⊥ . The bands M and N are called disjoint, if µ ⊥ ν for every pair µ ∈ M and ν ∈ N . The family (Lξ )ξ∈Ξ of bands in M (X) such that µ ∈ ( ξ∈Ξ Lξ )⊥ implies µ = 0 is called complete. Let a complete family of mutually disjoint bands Lξ , ξ ∈ Ξ, is given. Then for every µ ∈ M (X), µ ≥ 0, there exists a unique family (µξ )ξ∈Ξ , µξ ∈ Lξ , such that µ = supξ∈Ξ µξ , where the supremum is taken in the vector lattice M (X); µξ is called the component of µ in Lξ . If, in addition, the family (Lξ ) is ﬁnite, then M (X) is the direct sum of (Lξ ) and µ is the sum of its components µξ . In particular, if L is a band, then the pair (L, L⊥ ) is a complete family of mutually disjoint bands; the component of a measure µ in L coincides in this case with the projection of µ onto L parallel to L⊥ and denoted by µL . The measure µL is completely characterized by the following two properties: • µL ∈ L; • (µ − µL )⊥L. A Borel measure µ is called a point or atomic measure, if it is concentrated on a countable subset S ⊂ X. A point s ∈ S such that µ({s}) = 0 is called an atom for µ. For every set B ∈ B there holds µ({s}). µ(B) = s∈B∩S

The set of all complex point Borel measures on X we will denote by Mp (X), this is a band in M (X). A Borel measure µ is called a continuous measure, if µ({s}) = 0 for every s ∈ X. The set of all continuous Borel measures on X we will denote by Mc (X). It is clear that Mc⊥ (X) = Mp (X), Mp⊥ (X) = Mc (X), and M (X) is the direct sum of the bands Mp (X) and Mc (X).

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Let now X be a locally compact separable metric group with the continuous Haar measure. We ﬁx the left Haar measure λ; if X is a compact space, we choose λ to be normalized, in the case X = R we choose λ to be the Lebesgue measure. A measure µ on X is called absolutely continuous, if it is subordinated to λ and singular, if it is disjoint to λ (it is clear that these deﬁnitions are independent on the particular choice of λ). The set of all absolutely continuous Borel measures on X (respectively, the set of all singular Borel measures on X) will be denoted by Mac (X) (respectively, by Ms (X)). In particular, Mp (X) ⊂ Ms (X). It is clear that ⊥ (X) = Ms (X), and M (X) is the direct sum of the bands Ms⊥ (X) = Mac (X), Mac Mac (X) and Ms (X). A Borel measure µ on X is called a singular continuous measure, if it is simultaneously continuous and singular. The set of all singular continuous Borel measures on X we will denote by Msc (X); this is a band in M (X). By deﬁnition, µ ∈ Msc if and only if µ is concentrated on a Borel set of zero Haar measure and µ(S) = 0 for every countable set S. According to the Lebesgue decomposition theorem each Borel measure µ on the group X is decomposable in a unique way into the sum µ = µp + µac + µsc , where µp ∈ Mp (X), µac ∈ Mac (X), µsc ∈ Msc (X). We will denote also µc = µac + µsc and µs = µp + µsc . It is clear that µc ∈ Mc (X), µs ∈ Ms (X). 2.2. Spectral types and spectral measures In this section, A denotes a self-adjoint operator in a Hilbert space H , res A is the resolvent set of A, spec A := C\res A is the spectrum of A. For z ∈ res A, we denote R(z; A) := (A − z)−1 (the resolvent of A). The ﬁrst classiﬁcation of spectra is related to the stability under compact perturbations of A. By deﬁnition, the discrete spectrum of A (it is denoted by specdis A) consists of all isolated eigenvalues of ﬁnite multiplicity, and the essential spectrum of A, specess A, is the complement of the discrete spectrum in the whole spectrum: specess A = spec A\specdis A. By the famous Weyl perturbation theorem, for a point x0 ∈ spec A the following assertions are equivalent • ζ ∈ specess A, • for every compact operator K in H there holds ζ ∈ specess (A + K). The second classiﬁcation is related to the transport and scattering properties of a quantum mechanical system with the Hamiltonian H = A. For Ω ∈ B denote PΩ (A) = χΩ (A), where χΩ is the indicator function of the subset Ω ⊂ R; PΩ (A) is the spectral projector for A on the subset Ω. The mapping B Ω → PΩ (A) is called the projection valued measure associated with A (the resolution of identity). For every pair ϕ, ψ ∈ H , the mapping B Ω → ϕ|PΩ (A)ψ = PΩ (A)ϕ|PΩ (A)ψ

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is a complex Borel measure on the real line R which is called the spectral measure associated with the triple (A, ϕ, ψ) and denoted by µϕ,ψ (or more precisely, by µϕ,ψ (· ; A)). If ϕ = ψ, then µϕ ≡ µϕ,ϕ is a bounded positive Borel measure on R, B Ω → ϕ|PΩ (A)ϕ = PΩ (A)ϕ2 , with the norm µϕ = ϕ2 . Therefore, µϕ,ψ is bounded and |µϕ,ψ |(Ω) ≤ [µϕ (Ω)µψ (Ω)]1/2 . Moreover, supp µϕ,ψ ⊂ spec A. According to the Riesz–Markov theorem, for a bounded complex Borel measure µ on R the following three conditions are equivalent: • µ = µϕ,ψ for some ϕ, ψ ∈ H ; • for every continuous function f on R with compact support f (x) dµ(x); ϕ|f (A)ψ = R

• for every bounded Borel function f on R f (x) dµ(x). ϕ|f (A)ψ = R

The following proposition is obvious. Proposition 2.1. For a Borel subset Ω ⊂ R the following assertions hold: (1) µϕ (Ω) = 0 if and only if PΩ ϕ = 0. (2) µϕ is concentrated on Ω if and only if PΩ ϕ = ϕ. Proposition 2.2. The following assertions take place. (1) (2) (3) (4) (5)

µϕ,ψ and µϕ+ψ are subordinated to µϕ + µψ ; µaϕ = |a|2 µϕ for every a ∈ C; if µϕ ⊥ µψ , then ϕ ⊥ ψ; if µϕ ⊥ µψ , and B = f (A) where f is a bounded Borel function, then µBϕ ⊥ µψ ; if µϕn ⊥ µψ for a sequence ϕn from H , and ϕn → ϕ in H , then µϕ ⊥ µψ .

Proof. (1) For B ∈ B we have: |µϕ,ψ |(B) ≤ PB (ϕ)PB (ψ), 1/2

[µϕ+ψ (B)]

= PB (ϕ + ψ) ≤ PB (ϕ + PB (ψ),

hence |µϕ,ψ |(B) = µϕ+ψ (B) = 0, if µϕ+ψ (B) = 0. (2) Trivial. (3) Let S, T ∈ B, S ∩ T = ∅, µϕ be concentrated on S and µψ be concentrated on T . Then, according to Proposition 2.1, ϕ, ψ = PS ϕ, PT ψ = ϕ, PS PT ψ = 0 . (4) Let S and T be as in item (3). Then PS ϕ = ϕ, PT ψ = ψ. Hence PS f (A)ϕ = f (A)PS ϕ = f (A)ϕ and we can refer to Proposition 2.1

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(5) Let Sn , Tn ∈ B, Sn ∩ Tn = ∅, µϕn be concentrated on Sn and µψ be concentrated on Tn . Set T = Tn , S = R\T . Then µϕn is concentrated on S for every n and µψ is concentrated on T . By Proposition 2.1, PS ϕn = ϕn , PT ψ = ψ. As a result, we have PS ϕ = ϕ, hence µϕ ⊥ µψ by Proposition 2.1. Let L be a band in M . Denote HL ≡ {ψ ∈ H : µψ ∈ L}. Then by Proposition 2.2 HL is a closed A-invariant subspace of H . Moreover, let (Lξ )ξ∈Ξ be a complete family of bands in M . Then H is the closure of the linear span of the family of closed A-invariant subspaces HLξ . If, in addition, Lξ are mutually disjoint then H is the orthogonal sum of HLξ . In particular, HL⊥ = HL⊥ . Moreover, the following proposition is true. Proposition 2.3. Let ϕ ∈ H and ϕL is the orthogonal projection of ϕ onto HL . Then (1) µϕ − µϕL ≥ 0 and is subordinated to µϕ−ϕL ; (2) µϕL = µL ϕ. Proof. (1) First of all we show that µϕ − µϕL ≥ 0. Let B ∈ B, then (µϕ − µϕL )(B) = PB ϕ2 − PB PHL ϕ2 . Since HL is A-invariant, PB PHL = PHL PB , therefore (µϕ − µϕL )(B) = PB ϕ2 − PHL PB ϕ2 ≥ 0. Further, we have for B ∈ B (µϕ − µϕL )(B) = PB ϕ2 − PB ϕL 2 = (PB ϕ + PB ϕL )(PB ϕ − PB ϕL ) ≤ 2ϕPB (ϕ − ϕL ) = 2ϕ[µϕ−ϕL (B)]1/2 , and the proof of the item is complete. (2) µϕL ∈ L, and according to item (1) µϕ − µϕL ∈ L⊥ . Since HL is A invariant, the restriction of A to HL is a self-adjoint operator in HL . The spectrum of this restriction is denoted by specL A and is called L-part of the spectrum of A. It is clear that for a point x0 ∈ R the following assertions are equivalent: • x0 ∈ spec A; • for any ε > 0, there exists ϕ ∈ H such that µϕ (x0 − ε, x0 − ε) > 0. Therefore, we have Proposition 2.4. The following assertions are equivalent: • x0 ∈ specL A; • for any ε > 0, there exists ϕ ∈ H with µL ϕ (x0 − ε, x0 − ε) > 0; • for any ε > 0, there exists ϕ ∈ HL with µϕ (x0 − ε, x0 − ε) > 0.

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Let (Lξ )ξ∈Ξ be a complete family of mutually disjoint bands in M . Then spec A = specLξ A . ξ∈Ξ

Let L be a band in M , N = L⊥ and Ω ∈ B. If B ∩ specM A = ∅, then we say that A has only L-spectrum on Ω (or the spectrum of A on Ω is purely L). Denote now Hj , where j ∈ {p, ac, sc, s, c}, the subspace H ≡ HMj . Then the spectrum of the restriction of A to Hj is denoted specj A. In particular, • H = Hp ⊕ Hac ⊕ Hsc , therefore spec A = specp A ∪ specac A ∪ specsc A. The part specp A is called the point spectrum of A, specac A is called the absolutely continuous spectrum of A and specsc A is called the singular continuous spectrum of A. • H = Hp ⊕ Hc , therefore spec A = specp A ∪ specc A. The part specc A is called the continuous spectrum of A. • H = Hac ⊕ Hs , therefore spec A = specac A ∪ specs A. The part specs A is called the singular spectrum of A. Consider the point part of the spectrum in detail. The set of all eigenvalues of A is denoted by specpp A and is called pure point spectrum of A. In particular, for a point x0 ∈ R the following assertions are equivalent: • x0 ∈ specpp A; • µϕ ({x0 }) > 0 for some ϕ ∈ H . Proposition 2.5. Let δa , where a ∈ R, be the Dirac measure concentrated on a. Then for a ∈ R and ϕ ∈ H the following conditions are equivalent: (1) P{a} ϕ = ϕ; (2) µϕ = ϕ2 δa ; (3) Aϕ = aϕ. Proof. (1) ⇒ (2). For Ω ∈ B we have µϕ (Ω) = PΩ ϕ2 = PΩ P{a} ϕ2 . Therefore, µϕ (Ω) = ϕ2 , if a ∈ Ω and µϕ (Ω) = 0 otherwise. (2) ⇒ (3). We have for a z ∈ res(A) ϕ2 dµϕ (x) = , ϕ|R(z; A)ϕ = a−z R x−z hence, by polarization, R(z; A)ϕ = (a − z)−1 ϕ. (3) ⇒ (1). Indeed, P{a} = χ{a} (A) and χ{a} (a) = 1. As a corollary, we have that if a is an atom for a spectral measure µψ , then a ∈ specpp A. Indeed, if µψ ({a}) > 0, then ϕ = P{a} = 0. On the other hand, P{a} ϕ = ϕ. Proposition 2.6. Hp is the orthogonal direct sum Hpp of the eigensubspaces of A, and specp A = specpp A.

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Proof. It is clear that Hpp ⊂ Hp . To show that Hpp ⊃ Hp it is suﬃcient to prove that if ψ ⊥ Hpp , then µψ has no atoms. Suppose that ψ ⊥ Hpp but µψ ({a}) > 0. Then ϕ = P{a} ψ = 0. Further, ϕ = P{a} ϕ, therefore ϕ ∈ Hpp . On the other hand, ψ, ϕ = ψ, P{a} ψ = µψ ({a}) > 0. It is clear that specpp A ⊂ specp A. Suppose that a ∈ specp A. Take ε > 0, then µψ (a − ε, a + ε) > 0 for some ψ ∈ Hp . Hence, there is an atom s for µψ such that s ∈ (a − ε, a + ε). It remains to remark that s ∈ specpp A. The considered classiﬁcations of spectra are related as follows: • specdis A ⊂ specpp A; • specess A is the union of the following three sets: (1) specc A, (2) {x ∈ R : x is a limiting point of specpp A}, (3) {x ∈ specpp A : x is of inﬁnite multiplicity}. 2.3. Spectral projections Let x, y ∈ R. In what follows we will use often the identities: 1 Imϕ|R(x + iy; A)ϕ = [ϕ|R(x + iy; A)ϕ − R(x + iy; A)ϕ|ϕ] 2i 1 = ϕ|[R(x + iy; A) − R(x − iy; A)]ϕ 2i = yϕ|R(x − iy; A)R(x + iy; A)]ϕ = yR(x + iy; A)ϕ2 .

(2.1)

The following Stone formulas for spectral projections will be very useful, cf. [85, Theorem 42]. Let −∞ < a < b < +∞ and ϕ ∈ H , then b 1 1 [P[a,b] ϕ + P(a,b) ϕ] = lim [R(x + iy; A) − R(x − iy; A)]ϕ dx y→+0 2πi a 2 1 b [Im R(x + iy; A)]ϕ dx = lim y→+0 π a y b = lim R(x − iy; A)R(x + iy; A)ϕ dx. (2.2) y→+0 π a Since µϕ (Ω) = ϕ|PΩ (A)ϕ = PΩ (A)ϕ2 , we have for a, b ∈ R\specpp (A) 1 b µϕ ((a, b)) = µϕ ([a, b]) = lim Imϕ|R(x + iy; A)ϕ dx y→+0 π a b 1 Im = lim ϕ|R(x + iy; A)ϕ dx y→+0 π a y b R(x + iy; A)ϕ2 dx . (2.3) = lim y→+0 π a

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If a ∈ R and ϕ ∈ H , then P{a} (A)ϕ = −i limy→+0 yR(a + iy; A)ϕ, therefore µϕ ({a}) = P{a} (A)ϕ2 = limy→+0 y 2 R(a + iy; A)ϕ2 . The following statement is known [12, 85]: Theorem 2.7. Let ϕ ∈ H . For Lebesgue a.e. x ∈ R, there exists the limit ϕ|R(x + i0; A)ϕ := lim ϕ|R(x + iy; A)ϕ; y→0+

this limit is is ﬁnite and non-zero a.e. and, additionally, using (2.1), −1 (1) µac Fϕ dx, where ϕ =π

Fϕ (x) = Imϕ|R(x + i0; A)ϕ = lim yR(x + iy)ϕ2 . y→0+

(2)

µsϕ

is concentrated on the set {x ∈ R : Imϕ|R(x + i0; A)ϕ = ∞}.

Additionally, for −∞ < a ≤ b < +∞ one has: (3) µac ϕ ([a, b]) = 0 if and only if for some p, 0 < p < 1, b [Imϕ|R(x + iy; A)ϕ]p dx = 0. lim y→0+

a

(4) Assume that for some p, 1 < p ≤ ∞ one has sup{ Imϕ|R(· + iy; A)ϕp : 0 < y < 1} < ∞, where · p stands for the standard norm in the space Lp ([a, b]). Then µsϕ ((a, b)) = 0. (5) Let (a, b) ∩ specs A = ∅. Then there is a dense subset D ⊂ H such that sup{ Imϕ|R(· + iy; A)ϕp : 0 < y < 1} < ∞ for every p, 1 < p ≤ +∞, and every ϕ ∈ D. (6) µpϕ ((a, b)) = 0 if and only if b [Imϕ|R(x + iy; A)ϕ]2 dx = 0. lim y y→0+

a

Lemma 2.8. Let θ be a smooth strictly positive function on [a, b] and a, b ∈ / specpp A. Then b 1 Im ϕ|R(x + iy; A)ϕ dx lim y→+0 π a b 1 Im ϕ|R(x + iyθ(x); A)ϕ dx = lim y→+0 π a y b = lim R(x + iyθ(x); A)ϕ2 θ(x) dx. (2.4) y→+0 π a Proof. The second equality in (2.4) follows from (2.1), so its is suﬃcient to prove the ﬁrst equality only.

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Rewrite the left-hand side of (2.4) as b ϕ|R(x + iy; A)ϕ dx = a

ϕ|R(ζ; A)ϕ dζ,

39

(2.5)

(y)

where the path (y) is given in the coordinates ζ = ξ + iη by the equations: ξ = t, η = y, t ∈ [a, b]. Consider another path λ(y) given by ξ = t, η = yθ(t), t ∈ [a, b] and two vertical intervals: v1 (y): ξ = a, η between y and yθ(a) and v2 (y): ξ = b, η between y and yθ(b). Since the integrand in (2.5) is an analytic function, we can choose the orientation of the intervals v1 (y) and v2 (y) in such a way that ϕ|R(ζ; A)ϕ dζ = ϕ|R(ζ; A)ϕ dζ + ϕ|R(ζ; A)ϕ dζ (y)

λ(y)

v1 (y)

ϕ|R(ζ; A)ϕ dζ.

+

(2.6)

v2 (y)

Suppose θ(a) ≥ 1 (the opposite case is considered similarly). Then yθ(a) ϕ|R(ζ; A)ϕ dζ = ϕ|R(a + iη; A)ϕ dη. y

v1 (y)

Let νϕ be the spectral measure associated with A and ϕ, then by Fubini yθ(a) yθ(a) dνϕ (t) Im dη ϕ|R(a + iη; A)ϕ dη = Im t − a − iη y y R

dηη R

y

=

yθ(a)

= 1 2

ln R

dνϕ (t) (t − a)2 + η 2

(t − a)2 + y 2 θ(a)2 dνϕ (t). (t − a)2 + y 2

Using the estimate ln

(t − a)2 + y 2 θ(a)2 y 2 (θ(a)2 − 1) = ln 1 + ≤ 2 ln θ(a). (t − a)2 + y 2 (t − a)2 + y 2

and the boundedness of νϕ we obtain by the Lebesgue majorization theorem lim Im ϕ|R(ζ; A)ϕ dζ = 0. (2.7a) y→0+

v1 (y)

Exactly in the same way there holds lim y ϕ|R(ζ; A)ϕ dζ = 0. y→0+

(2.7b)

v2 (y)

On the other hand, ϕ|R(ζ; A)ϕ dζ Im λ(y)

= Im a

b

ϕ|R(x + iyθ(x); A)ϕ(1 + iyθ (x)) dx = I1 (y) + iyI2 (y),

(2.8)

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where, by (2.1),

b

I1 (y) := Im

ϕ|R(x + iyθ(x); A)ϕ dx

a

b

≡ y

R(x + iyθ(x); A)ϕ2 dx,

a

b

I2 (y) := Im ≡ y

ϕ|R(x + iyθ(x); A)ϕ θ (x) dx

a b

R(x + iyθ(x); A)ϕ2 θ (x) dx.

a

Denoting c = maxx∈[a,b] |θ (x)|, one immediately obtains |I2 (y)| ≤ c|I1 (y)|. Therefore, passing to the limit y → 0+ in (2.8), we arrive at ϕ|R(ζ; A)ϕ dζ. lim I1 (y) = lim Im y→0+

y→0+

λ(y)

Substituting the latter equality, (2.7a), and (2.7b) in (2.6) results in (2.4). 3. Spectra and Spectral Measures for Self-Adjoint Extensions 3.1. Problem setting and notation In this section we return to self-adjoint extensions. Below • S is a densely deﬁned symmetric operator in H with equal deﬁciency indices in a Hilbert space H , • Nz := ker(S ∗ − z), • (G , Γ1 , Γ2 ) is a boundary triple for S ∗ , • Λ is a self-adjoint operator in G , • H 0 is the self-adjoint restriction of S ∗ to ker Γ1 , • HΛ is the self-adjoint restriction of S ∗ to ker(Γ2 − ΛΓ1 ); due the the condition on Λ, HΛ and H 0 are disjoint, see Remark 1.30. • R0 (z) := (H 0 − z)−1 for z ∈ res H 0 , • RΛ (z) := (HΛ − z)−1 for z ∈ res HΛ , • γ is the Krein Γ-ﬁeld induced by the boundary triple, • Q is the Krein’s Q-function induced by the boundary triple. Recall that the resolvent are connected by the Krein resolvent formula (Theorem 1.29): z ). RΛ (z) = R0 (z) − γ(z)[Q(z) − Λ]−1 γ ∗ (¯

(3.1)

We are interested in the spectrum of HΛ assuming that the spectrum of H 0 is known. Theorem 1.23 and Eq. (3.1) above show the equality spec HΛ \spec H 0 = {E ∈ res H 0 : 0 ∈ spec(Q(E) − Λ)}.

(3.2)

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We are going to reﬁne this correspondence in order to distinguish between diﬀerent spectral types of HΛ in gaps of H 0 . Some of our results are close to that obtained in [25] for simple operators, but are expressed in diﬀerent terms. 3.2. Discrete and essential spectra The aim of the present subsection is to relate the discrete and essential spectra for HΛ with those for Q(z) − Λ. Lemma 3.1. Let A and B be self-adjoint operators in G , and A be bounded and strictly positive, i.e. φ, Aφ ≥ cφ, φ for all φ ∈ dom A with some c > 0. Then 0 is an isolated eigenvalue of B if and only if 0 is an isolated eigenvalue of ABA. Proof. Denote L := ABA. Let 0 is a non-isolated point of the spectrum of B. Then there is φn ∈ dom B such that Bφn → 0 and dist(ker B, φn ) ≥ ε > 0. Set ψn = A−1 φn . Then Lψn → 0. Suppose that lim inf dist(ker L, ψn ) = 0. Then there are ψn ∈ ker L such that lim inf ψn − ψn = 0. It is clear that φn = Aψn ∈ ker B and lim inf φn − φn = lim inf Aψn − Aψn = 0. This contradiction shows that dist(ker L, ψn ) ≥ ε > 0 and 0 is a non-isolated point of the spectrum of L. The converse follows by symmetry, as A−1 is also positive deﬁnite. Theorem 3.2. For E ∈ res H 0 the following assertions are equivalent: (1) E is an isolated point of the spectrum of HΛ ; (2) 0 is an isolated point of the spectrum of Q(E) − Λ. Moreover, if one of these conditions is satisﬁed, then for z in a punctured neighborhood of E there holds c (Q(z) − Λ)−1 ≤ for some c > 0. (3.3) |z − E| Proof. Clearly, one can assume that E is real. Denote Q0 := Q(E), Q1 := Q (E). Both Q0 and Q1 are bounded self-adjoint operators. By (1.22a), there holds Q1 = γ ∗ (E) γ(E), therefore, Q1 is positive deﬁnite. Take any r < dist(E, spec H 0 ∪ spec HΛ \{E}). For |z − E| < r, we have an expansion Q(z) = Q0 + (z − E)Q1 + (z − E)2 S(z),

(3.4)

where S is a holomorphic map from a neighborhood of E to L(G , G ). (1) ⇒ (2). Let E be an isolated point of the spectrum of HΛ . Since E is an isolated point in the spectrum of HΛ , the resolvent RΛ (z) ≡ (HΛ − E)−1 has a ﬁrst order pole at z = E, therefore, as follows from the resolvent formula (3.1), the function z → (Q(z) − Λ)−1 also has a ﬁrst order pole at the same point. Hence, we can suppose that for 0 < |z − E| < r there exists the bounded inverse (Q(z) − Λ)−1 and, moreover, (z − E)(Q(z)− Λ)−1 ≤ c for some constant c > 0. This implies the

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estimate (3.3). By (3.4), we can choose r small enough, such that Q0 −Λ+(z −E)Q1 has a bounded inverse for 0 < |z − E| < r. Representing 1/2

−1/2

Q0 − Λ + (z − E)Q1 = Q1 (Q1

−1/2

(Q(E) − Λ)Q1

1/2

+ (z − E)I)Q1

−1/2

−1/2

we see that 0 is an isolated point in the spectrum of Q1 (Q(E) − Λ)Q1 and hence of Q(E) − Λ in virtue of Lemma 3.1. (2) ⇒ (1). Conversely, let 0 be an isolated point of the spectrum of Q(E) − Λ or, −1/2 −1/2 which is equivalent by Lemma 3.1, in the spectrum of T := Q1 (Q(E)− Λ)Q1 . For suﬃciently small r and 0 < |z − E| < r, the operator M (z) := T + (z − E)I is invertible, and (z − E)M −1 (z) ≤ c for these z for some constant c . For the same 1/2 1/2 z, the operator Q0 − Λ + (z − E)Q1 ≡ Q1 M (z)Q1 is also boundedly invertible, −1 and (z − E)(Q0 − Λ + (z − E)Q1 ) ≤ c . Hence, we can chose r such that Q(z) − Λ = Q0 − Λ + (z − E)Q1 + (z − E)2 S(z) is invertible for 0 < |z − E| < r, which by (3.2) means that z ∈ / res HΛ . Now we are able to reﬁne the relationship (3.2) between the spectra of H 0 and HΛ . This is the main result of the subsection. Theorem 3.3. The spectra of H and HΛ are related by spec• HΛ \spec H 0 = {E ∈ res H 0 : 0 ∈ spec• (Q(E) − Λ)}

(3.5)

with • ∈ {pp, dis, ess}. Proof. By Theorem 1.23(1), Eq. (3.5) holds for • = pp, moreover, the multiplicities of the eigenvalues coincide in this case. Therefore, by Theorem 3.2, the isolated eigenvalues of ﬁnite multiplicities for HΛ correspond to the isolated zero eigenvalues for Q(z) − E, which proves (3.5) for • = dis. By duality this holds for the essential spectra too. It is also useful to write down the spectral projector for HΛ corresponding to isolated eigenvalues lying in res H 0 . Proposition 3.4. Let E ∈ res H 0 be an isolated eigenvalue of HΛ . Then the eigenprojector PΛ for HΛ corresponding to E is given by PΛ = γ(E)(Q (E))−1/2 Π(Q (E))−1/2 γ ∗ (E), where Π is the orthoprojector on ker(Q (E))−1/2 (Q(E) − Λ)(Q (E))−1/2 in G . Proof. Follows from the equality PΛ = −Res[RΛ (z); z = E]. 3.3. Estimates for spectral measures In this subsection we are going to obtain some information on the absolutely continuous, singular continuous, and point spectra of HΛ using the asymptotic behavior

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of (Q(x + iy) − Λ)−1 for x ∈ R and y → 0+. To do this, we need ﬁrst an expression for the resolvent RΛ on the defect subspaces of S. Lemma 3.5. Let ζ, z ∈ C\R, z = ζ, and g ∈ dom Λ. For ϕ = γ(ζ)g there holds 1 [ϕ − γ(z)(Q(z) − Λ)−1 (Q(ζ) − Λ)g]. RΛ (z)ϕ = ζ−z Proof. Substituting identities (1.14b) and (1.21e) into (3.1) we obtain: RΛ (z)ϕ = R0 (z)γ(ζ)g − γ(z)[Q(z) − Λ]−1 γ ∗ (¯ z )γ(ζ)g ¯ = R0 (z)γ(ζ)g − γ(z)[Q(z) − Λ]−1 γ ∗ (ζ)γ(z)g =

Q(z) − Q(ζ) γ(z) − γ(ζ) g − γ(z)[Q(z) − Λ]−1 g z−ζ z−ζ

=

1 [γ(ζ)g − γ(z){I − [Q(z) − Λ]−1 (Q(z) − Λ + Λ − Q(ζ))}g] ζ −z

=

1 [ϕ − γ(z)(Q(z) − Λ)−1 (Q(ζ) − Λ)g]. ζ −z

Theorem 3.6. Fix ζ0 ∈ C\R. Let g ∈ dom Λ; denote h := (Q(ζ0 ) − Λ)g, ϕ := γ(ζ0 )g, and let µϕ be the spectral measure for HΛ associated with ϕ. (1) If [a, b] ⊂ res H 0 ∩ R and a, b ∈ / specpp HΛ , then µϕ ([a, b]) ≡ P[a,b] (HΛ )ϕ2 y b 1 = lim (Q (x))1/2 (Q(x + iy) − Λ)−1 h2 dx. y→+0 π a |ζ0 − x|2 (2) For a.e. x ∈ res H 0 ∩ R, there exists the limit f (x) := lim y(Q (x))1/2 (Q(x + iy) − Λ)−1 h2 , y→+0

and the function F (x) := π|ζ01−x|2 f (x) is the Lebesgue density of the measure ac µac ϕ , i.e. µϕ = F (x) dx. (3) For a ∈ res H 0 ∩ R, the limit lim y 2 (Q (a))1/2 (Q(a + iy) − Λ)−1 h2

y→+0

exists and is equal to µpϕ ({a}). Proof. We start with proving item (2). Using Lemma 3.5, we get for y > 0: RΛ (x + iy)ϕ =

1 1 ϕ− γ(x + iy)[Q(x + iy) − Λ]−1 h, ζ0 − x − iy ζ0 − x − iy

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therefore

√ √ y yRΛ (x + iy)ϕ − ζ0 − x − iy ϕ √ y ≤ γ(x + iy)(Q(x + iy) − Λ)−1 h |ζ0 − x − iy| √ y √ ≤ yRΛ (x + iy)ϕ + ϕ. |ζ0 − x − iy|

√ Hence, if yRΛ (x + iy)ϕ has a limit (ﬁnite or inﬁnite) as y → +0, then also √ yγ(x + iy)(Q(x + iy) − Λ)−1 h does, and in this case lim

y→+0

√ yRΛ (x + iy)ϕ =

√ 1 lim yγ(x + iy)(Q(x + iy) − Λ)−1 h. y→+0 |ζ0 − x|

(3.6)

Let us show that, at ﬁxed x, the ﬁniteness of the limit (3.6) is equivalent to √ (3.7) sup y(Q(x + iy) − Λ)−1 h < ∞. 0
Indeed, since γ(z) is a linear topological isomorphism on its image and is analytic, for a given x ∈ res H 0 there exists c > 0 such that c−1 g ≤ sup0
and lim

y→+0

√ yγ(x + iy)(Q(x + iy) − Λ)−1 h = +∞

√ are equivalent. Assume now limy→+0 yγ(x+iy)(Q(x+iy)−Λ)−1 h < +∞, then, for all 0 < y < 1, one has √ √ | yγ(x + iy)(Q(x + iy) − Λ)−1 h − yγ(x)(Q(x + iy) − Λ)−1 h | √ ≤ yγ(x + iy)(Q(x + iy) − Λ)−1 h − γ(x)(Q(x + iy) − Λ)−1 h ≤ cγ(x + iy) − γ(x), √ where c = sup0
y→+0

√ yRΛ (x + iy)ϕ =

√ 1 lim yγ(x)(Q(x + iy) − Λ)−1 h. |ζ0 − x| y→+0

(3.8)

On the other hand there holds γ(x)(Q(x + iy) − Λ)−1 h2 = γ ∗ (x)γ(x)(Q(x + iy) − Λ)−1 h | (Q(x + iy) − Λ)−1 h, and, due to identities γ ∗ (x)γ(x) ≡ Q (x) and γ(x)(Q(x + iy) − Λ)−1 h2 = (Q (x))1/2 (Q(x + iy) − Λ)−1 h2 , item (2) follows from Proposition 2.7.

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The proof of items (1) and (3) is completely similar to that for item (2); in the case of (1) one should use the norm b 1/2 f (x)2 dx f 2 = a

on the space L ([a, b]; H ) in the above estimates. 2

Below we will use the notation ⊥ ∗ H0 := ker(S − ζ) ,

H1 := H0⊥ .

Im ζ=0

For a subspace L ⊂ G we write H1 (L) := Im ζ=0 γ(ζ)Xζ with Xζ (L) = (Q(ζ) − Λ)−1 L. Note that if span L is dense in G , then also Xζ (L) is, and the linear hull of H0 ∪ H1 (L) is dense in H . If ψ ∈ H0 , then γ ∗ (ζ)ψ = 0 for all ζ ∈ C\R. By (3.1), it follows RΛ (ζ)ψ = 0 R (ζ)ψ, and hence µψ (Ω) = 0 for all Borel sets Ω ⊂ res H 0 ∩ R, where µψ is the spectral measure for HΛ associated with ψ. Proposition 3.7 (cf. [66, Theorem 2]). Let a, b ∈ res H 0 . Suppose that there exists a subset L ⊂ G with dense span L such that sup{(Q(x + iy) − Λ)−1 h : a < x < b, 0 < y < 1} < ∞ for all h ∈ L. Then (a, b) ∩ spec HΛ = ∅. Proof. We can assume that a, b ∈ / specpp HΛ ; otherwise we consider (a, b) as the / specpp HΛ . union of a increasing sequence of intervals (an , bn ), where an , bn ∈ It is suﬃcient to show that P(a,b) (HΛ )H1 (L) = 0. Let ϕ ∈ H1 (L), then there is g ∈ L and ζ ∈ C\R such that ϕ = γ(ζ)(Q(ζ) − Λ)−1 g. Using Lemma 3.5 with z = x + iy we get RΛ (x + iy)ϕ =

1 [ϕ − γ(x + iy)(Q(x + iy) − Λ)−1 g]. ζ − x − iy

(3.9)

Using (2.3) we arrive at P(a,b) (HΛ )ϕ = 0. Proposition 3.8. For any x0 ∈ res H 0 ∩ R the following two assertions are equivalent: / spec HΛ ; (1) x0 ∈ (2) there exist ε > 0 and a subset L ⊂ G with dense span L such that (x0 − ε, x0 + ε) ⊂ res H 0 and x0 +ε (Q(x + iy) − Λ)−1 h2 dx = 0 lim y y→+0

for all h ∈ L.

x0 −ε

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Proof. The implication (1) ⇒ (2) is trivial. Let us prove (2) ⇒ (1). It is suﬃcient to show that P(x0 −ε,x0 +ε) (H Λ )ϕ = 0 for all ϕ ∈ H1 (L). For a given ϕ ∈ γ(ζ)Xζ (L) with Im ζ = 0 we take h ∈ L such that h = (Q(ζ) − Λ)g, ϕ = γ(ζ)g for some g ∈ dom Λ. Then the equality P(x0 −ε,x0 +ε) (H Λ )ϕ = 0 follows from Theorem 3.6(1). Proposition 3.9. Let a, b ∈ res H 0 . Suppose that there exists a subset L ⊂ G with dense span L such that for all h ∈ L and x ∈ (a, b) there holds √ sup{ y(Q(x + iy) − Λ)−1 h : 0 < y < 1} < ∞. Then (a, b) ∩ specs HΛ = ∅. Proof. Let µϕ be the spectral measure associated with ϕ and HΛ . It is suﬃcient to show that µsϕ (a, b) = 0 for all ϕ ∈ H1 (L). Writing any ϕ ∈ H1 (L) in the form ϕ = γ(ζ)(Q(ζ) − Λ)−1 g with g ∈ L and Im ζ = 0 one arrives again at (3.9). Therefore, √ for any x ∈ (a, b) one has supy∈(0,1) yRΛ (x+ iy)ϕ < ∞, and supp µsϕ ∩(a, b) = ∅ by Theorem 2.7(2). Proposition 3.10. Let x0 ∈ resH 0 ∩ R. Then the following assertions are equivalent: / specac H Λ ; (1) x0 ∈ (2) there exist ε > 0 and a subset L ⊂ G with dense span L such that (x0 − ε, x0 + √ ε) ⊂ res H 0 and limy→+0 y(Q(x + iy) − Λ)−1 h = 0 for all h ∈ L and for a.e. x ∈ (x0 − ε, x0 + ε). Proof. The proof of the implication (2) ⇒ (1) is completely similar to that for Proposition 3.8, cf. Theorem 3.6(1) To prove (1) ⇒ (2) we take ε > 0 such that (x0 − ε, x0 + ε) ∩ specac HΛ = ∅. According to Theorem 3.6(2) we have lim y(Q (x))1/2 (Q(x + iy) − Λ)−1 h2 = 0

y→+0

for all h ∈ G , and it is suﬃcient to note that (Q (x))1/2 is a linear topological isomorphism. Proposition 3.11. Let x0 ∈ res H 0 . Then the following assertions are equivalent: (1) x0 ∈ / specp HΛ ; (2) there exist ε > 0 and a subset L ⊂ G with dense span L such that (x0 − ε, x0 + ε) ⊂ res H 0 and limy→+0 y(Q(x + iy) − Λ)−1 h = 0 for all h ∈ L and for every x ∈ (x0 − ε, x0 + ε). Proof. Similar to the proof of Proposition 3.10 using Theorem 3.6(3). Using Propositions 3.10 and 3.11 we get immediately

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Proposition 3.12. Let x0 ∈ res H 0 ∩spec HΛ . If for every ε > 0 there exists h ∈ G such that • limy→+0 y(Q(x + iy) − Λ)−1 h = 0 for all x ∈ (x0 − ε, x0 + ε) and √ • limy→+0 y(Q(x + iy) − Λ)−1 h = 0 for a.e. x ∈ (x0 − ε, x0 + ε), then x0 ∈ specsc HΛ . 3.4. Special Q-functions In this subsection we assume that the expression Q(z) − Λ in the Krein forumula (3.1) has the following special form: Q(z) − Λ =

A − m(z) , n(z)

(3.10)

where • m and n are (scalar) analytic functions at least in C\R, • A is a self-adjoint operator in G . We assume that m and n admit analytic continuation to some interval (a, b) ⊂ res H 0 ∩ R, moreover, they both are real and n = 0 in this interval. Below, in Secs. 3.5 and 3.6 we provide examples where such a situation arises. Our aim is to relate the spectral properties of HΛ in (a, b) to the spectral properties of A. In what follows we denote by J := (inf spec A, sup spec A). Lemma 3.13. If n is constant, then m is monoton in (a, b). If n is non-constant and m (x) = 0 for some x ∈ (a, b), then either m(x) < inf spec A or m(x) > sup spec A. Proof. For any f ∈ dom A consider the function af (x) := Using (1.22a) we write cf 2 ≤ f |Q (x)f ≡ af (x) = −

1 n(x) f |(A

− m(x))f .

m (x) n (x) f 2 − 2 f |(A − m(x))f n(x) n (x)

with some constant c > 0 which is independent of f . (x) ≥ c, i.e. m = 0. For constant n one has n ≡ 0 and − mn(x) If n = 0 and m (x) = 0, then n (x) f |(A − m(x))f ≥ cf 2 n2 (x) for any f , i.e. the operator A − m(x) is either positive deﬁnite or negative deﬁnite. Lemma 3.14. Let K be a compact subset of (a, b) ∩ m−1 (J¯), then there is y0 > 0 such that for x ∈ K and 0 < y < y0 one has (Q(x + iy) − Λ)−1 = n(x + iy) L(x, y)[A − m(x) − iym (x)]−1 ,

(3.11)

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where L(x, y) is a bounded operator and L(x, y) − I → 0 uniformly with respect to x ∈ K as y → 0. Proof. We have (Q(x + iy) − Λ)−1 = n(x + iy)(A − m(x + iy))−1 . Further, A − m(x + iy) = A − m(x) − iym (x) + B(x, y), where B(x, y) = O(y 2 ) uniformly with respect to x ∈ K. Since m (x) = 0 for x ∈ K by Lemma 3.13, the operator A − m(x) − iym (x) has a bounded inverse deﬁned everywhere, and A − m(x + iy) = (A − m(x) − iym (x))[1 + (A − m(x) − iym (x))−1 B(x, y)]. It is easy to see that (A − m(x) − iym (x))−1 = O(|y|−1 ) uniformly with respect to x ∈ K. Therefore, for suﬃciently small y, (A − m(x + iy))−1 = (1 + B1 (x, y))−1 [A − m(x) − iym (x)]−1 with B1 (x, y) = O(|y|) uniformly with respect to x ∈ K. Lemma 3.15. Fix ζ0 with Im ζ0 = 0 and let h ∈ G , ϕ = γ(ζ0 )(Q(ζ0 ) − Λ)−1 h. Denote by µ the spectral measure for the pair (HΛ , ϕ) and by ν the spectral measure for the pair (A, h). There is a constant c > 0 with the following property: for any / specpp HΛ there holds segment K := [α, β] ⊂ (a, b) ∩ m−1 (J¯) such that α, β ∈ µ(K) ≤ cν(m(K)). Proof. Note ﬁrst that m = 0 on [α, β]. To be deﬁnite, we suppose m > 0. According to Theorem 3.6(1) and Lemma 3.14, we have y n2 (x) (Q (x))1/2 (A − m(x) − iym (x))−1 h2 dx. µ(K) = lim y→+0 π K |ζ0 − x|2 Substituting ξ := m(x) and denoting τ (ξ) := m (m−1 (ξ)) we arrive at y n(m−1 (ξ))2 µ(K) = lim y→+0 π m(K) τ (ξ) · |ζ0 − m−1 (ξ)|2 × (Q (ϑ−1 (ξ)))1/2 (A − ξ − iyτ (ξ))−1 h2 dξ. Since

n(m−1 (ξ))2 (Q (ϑ−1 (ξ)))1/2 (A − ξ − iyτ (ξ))−1 h2 dξ, −1 (ξ)|2 m(K) τ (ξ) · |ζ0 − m ≤c (A − ξ − iyτ (ξ))−1 h2 dξ, m(K)

where c is independent of K, we obtain the result with the help of Lemma 2.8. Here is the main result of the subsection. Theorem 3.16. Assume that the term Q(z) − Λ in the Krein resolvent formula (3.1) admits the representation (3.10), then for any x0 ∈ spec HΛ ∩ (a, b)

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and any • ∈ {dis, ess, pp, p, ac, s, sc, c} the conditions: (•) x0 ∈ spec• HΛ , (m − •) m(x0 ) ∈ spec• A are equivalent. Proof. For • = pp, dis, ess see Theorem 3.3. As m is a homeomorphism, the same holds for specp ≡ specpp . For • = ac, use the following sequence of mutually equivalent assertions: / specac A, • m(x0 ) ∈ y→0+

• There is a neighborhood V of m(x0 ) such that y(A − ξ − iy)−1 )h2 −→ 0 for all ξ ∈ V and h ∈ G (use item 1 of Theorem 2.7), y→0+

• There is a neighborhood W of x0 such that y(Q(x + iy) − Λ)−1 )h2 −→ 0 for all ξ ∈ W and h ∈ G (use Lemma 3.14 and replace iym (x) at any ﬁxed x by iy), / specac HΛ (Proposition 3.10). • x0 ∈ Assume now m(x0 ) ∈ specsc A. There exists a neighborhood V of m(x0 ) such that for some h ∈ G we have νhac (V ) = νhp (V ) = 0, where ν stands for the spectral measure for A. Using Lemma 3.14 and Theorem 2.7 one can see that there exists a neighborhood W of x0 such that limy→+0 y 2 (Q(x+iy)−Λ)−1h2 = 0 for all x ∈ W and limy→+0 y(Q(x + iy) − Λ)−1 h2 = 0 for a.e. x ∈ W . By Proposition 3.12, this means that x0 ∈ specsc (HΛ ). Hence, we prove (m − sc) ⇒ (sc). Since specs A = specp A ∪ specsc A, we prove also that (m − s) ⇒ (s). / specs A. To show that x0 ∈ / specs HΛ it is suﬃcient to consider Let now m(x0 ) ∈ the case m(x0 ) ∈ spec A\ specs A. Then by [118, Theorem XIII.20], there exist a dense subset L ⊂ G and a neighborhood V of m(x0 ) such that sup{(A − ξ − iy)−1 h : 0 < y < 1, ξ ∈ V } < ∞ for all h ∈ L. We can assume without loss of generality that m (x0 ) > 0, then by Lemma 3.14 we have for a neighborhood W of x0 and for some y0 , y0 > 0, √ sup{ y(Q(x + iy) − Λ)−1 h : 0 < y < y0 , x ∈ W } < ∞, / specs HΛ by Proposition 3.9. Thus, the equivalence (s) ⇔ (m−s) is proven. and x0 ∈ Now we prove the impication (sc) ⇒ (m − sc). Assume that x0 ∈ specsc (HΛ ) / specsc A. Denote the spectral measure for A by ν and that for HΛ by but m(x0 ) ∈ µ, then there is an interval I containing x0 such that for J = m(I) there holds: νhsc (J) = 0 for all h ∈ G . According to Lemma 3.15, if X is a Borel subset of I such that νh (m(X)) = 0 for all h, then also µϕ (X) = 0 for all ϕ ∈ H1 . In particular, let X be a Borel subset of I of zero Lebesgue measure and containing no eigenvalues of HΛ . Then, m(X) is a Borel subset of J which contains no eigenvalues of A and also has the Lebesgue measure zero. Therefore, νh (m(X)) = 0, and hence, µϕ (X) = 0. We see, that the restriction of µϕ to I is mutually singular with each singular continuous measure on I. Hence, it is true for µϕ with each ϕ ∈ H . This

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contradicts to the assumption x0 ∈ specsc HΛ , and the implication (sc) ⇒ (m − sc) is proven. The equivalence (c) ⇔ (m−c) follows from (sc) ⇔ (m−sc) and (ac) ⇔ (m−ac).

We note that Theorem 3.16 may be considered as an abstract version of the dimension reduction: we reduce the spectrum problem for self-adjoint extensions to a spectral problem “on the boundary”, i.e. in the space G . 3.5. Spectral duality for quantum and combinatorial graphs We have already mentioned that the theory of self-adjoint extensions has obvious applications in the theory of quantum graphs. Here we are going to develop the results of the recent paper [110] concerning the relationship between the spectra of quantum graphs and discrete Laplacians using Theorem 3.16. Actually, this problem was the starting point of the work. Let G be a countable directed graph. The sets of the vertices and of the edges of G will be denoted by V and E, respectively. We do not exclude multiple edges and self-loops. For an edge e ∈ E, we denote by ιe its initial vertex and by τ e its terminal vertex. For a vertex v, the number of outgoing edges (outdegree) will be denoted by outdeg v and the number of ingoing edges (indegree) will be denoted by indeg v. The degree of v is deg v := indeg v + outdeg v. In what follows we assume that the degrees of the vertices of G are uniformly bounded, 1 ≤ deg v ≤ N for all v ∈ V , in particular, there are no isolated vertices. Note that each self-loop at v counts in both indeg v and outdeg v. By identifying each edge e of G with a copy of the segment [0, 1], such that 0 is identiﬁed with the vertex ιe and 1 is identiﬁed with the vertex τ e, one obtain a certain topological space. A magnetic Schr¨odinger operator in such a structure is deﬁned as follows. The state space of the graph is H = e∈E He , He = L2 [0, 1], consisting of functions f = (fe ), fe ∈ He . On each edge consider the same scalar potential U ∈ L2 [0, 1]. Let ae ∈ C 1 [0, 1] be real-valued magnetic potentials on the edges e ∈ E. Associate with each edge a diﬀerential expression Le := (i∂ + ae )2 + U . The maximal operator which can be associated with these diﬀerential expressions 2 H [0, 1]. The integration by parts shows acts as (ge ) → (Le ge ) on functions g ∈ that this operator is not symmetric, and it is necessary to introduce boundary conditions at the vertices to obtain a self-adjoint operator. The standard self-adjoint boundary conditions for magnetic operators are e:ιe=v

ge (1) = gb (0) =: g(v) for all b, e ∈ E with ιb = τ e = v, (ge (0) − iae (0)ge (0)) − (ge (1) − iae (1)ge (1)) = α(v)g(v), e:τ e=v

where α(v) are real numbers, the so-called coupling constants. The gauge trans t formation ge (t) = exp i 0 ae (s)ds fe (t) removes the magnetic potentials from the

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diﬀerential expressions, ((i∂ + ae )2 + U )ge = −fe + U fe , but the 1 magnetic ﬁeld enters the boundary conditions through the parameters β(e) = 0 ae (s) ds in the following way: eiβ(e) fe (1) = fb (0) =: f (v) for all b, e ∈ E with ιb = τ e = v, fe (0) − eiβ(e) fe (1) = α(v)f (v). f (v) := e:ιe=v

(3.12a) (3.12b)

e:τ e=v

The self-adjoint operator in H acting as (fe ) → (−fe + U fe ) on functions (fe ) ∈ 2 H [0, 1] satisfying the boundary conditions (3.12a) and (3.12b) for all v ∈ V will be denoted by H. This is our central object. To describe the spectrum of H let us make some preliminary constructions. We introduce a discrete Hilbert space l2 (G) consisting of functions on V which are summable with respect to the weighted scalar product f, g = v∈V deg vf (v)g(v). Consider an arbitrary function β : E → R and consider the corresponding discrete magnetic Laplacian in l2 (G), 1 −iβ(e) iβ(e) e h(τ e) + e h(ιe) . (3.13) ∆G h(v) = deg v e:ιe=v e:τ e=v This expression deﬁnes a bounded self-adjoint operator in l2 (G). Denote by D the Dirichlet realization of −d2 /dt2 + U on the segment [0, 1], Df = −f + U f , dom D = {f ∈ H 2 [0, 1] : f (0) = f (1) = 0}. The spectrum of D is a discrete set of simple eigenvalues. For any z ∈ C denote by s(·; z) and c(x; z) the solutions to −y + U y = zy satisfying s(0; z) = c (0; z) = 0 and s (0; z) = c(0; z) = 1. Introduce an extension 2 of H, Π, deﬁned by dom Π = {f ∈ H [0, 1] : Eq. (3.12a) holds} and Π(fe ) = (−fe + Uf e ). The following proposition is proved in [110]. Proposition 3.17. The operator Π is closed. For f ∈ dom Π put f (v) Γ1 f = (f (v))v∈V , Γ2 f = deg v v∈V with f (v) and f (v) given by (3.12), then (l2 (G), Γ1 , Γ2 ) is a boundary triple for Π. The induced Γ-ﬁeld γ and Q-function Q are of the form 1 [h(ιe)(s(1; z)c(x; z) − s(x; z)c(1; z)) + e−iβ(e) h(τ e)s(x; z)], (γ(z)h)e (x) = s(1; z) and Q(z)f (v) =

1 (∆G − [outdeg vc(1; z) + indeg vs (1; z)])f (v). deg vs(1; z)

Now, let us make some additional assumptions. We will say that the symmetry condition is satisﬁed if at least one of the following properties holds: indeg v = outdeg v for all v ∈ V or U is even, i.e. U (x) = U (1 − x). The following theorem provides a complete description of the spectrum of the quantum graph H outside spec D in terms of the discrete Laplacian ∆G .

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Theorem 3.18. Let the symmetry condition be satisﬁed and the coupling constants v −1 (spec• ∆G )\spec D α(v) be of the form α(v) = deg 2 α, then spec• Λ\spec D = η 1 for • ∈ {dis, ess, pp, p, ac, s, sc, c}, where η(z) = 2 (s (1; z) + c(1; z) + αs(1; z)). Proof. Let the symmetry conditions be satisﬁed. If U is even, then s (1; z) ≡ c(1; z). If outdeg v = indeg v for all v, then outdeg v = indeg v = 12 deg v. In both

(1;z)−c(1;z) cases one has Q(z) = 2∆G −s2s(1;z) (see [110] for a more detailed discussion). The operator H itself is the restriction of Π to the functions f satisfying Γ2 = α2 Γ1 f with Γ1,2 from Proposition 3.17. The restriction H 0 of S to ker Γ1 is nothing but the direct sum of the operators D over all edges. By Theorem 1.29, the resolvents of H and H0 are related by the Krein resolvent formula and, in particular, the G −η(z) , and we are in the corresponding term Q(z) − Λ has the form Q(z) − Λ = ∆s(1;z) situation of Theorem 3.16.

3.6. Array-type systems Another situation in which Theorem 3.16 becomes useful appears when the Qfunction is of scalar type [6], i.e. when Q(z) is just the multiplication by a certain complex function; such functions are of interest in the invesre spectral problem for self-adjoint extensions [24]. In this case the representation (3.10) holds for any self-adjoint operator Λ, and one has: Proposition 3.19. Let Q be of scalar type, then for any Λ there holds spec• HΛ\spec H 0 = Q−1 (spec• Λ)\spec H 0 with • ∈ {dis, ess, pp, p, ac, s, sc, c}. In other words, the nature of the spectrum of the “perturbed” operator HΛ in the gaps of the “unperturbed” operator H 0 is completely determined in terms of the parameter Λ. Scalar type Q-functions arise, for example, as follows. Let H0 be a separable Hilbert space and S0 be a closed symmetric operator in H0 with the deﬁciency indices (1, 1). Let (C, Γ01 , Γ02 ) be a boundary triple for the adjoint S0∗ , and γ0 (z) and q(z) be the induced Γ-ﬁeld and Q-function. Let D be the restriction of S0∗ to ker Γ01 ; this is a self-adjoint operator. Let A be a certain countable set. Consider the operator S := α∈A Sα in the 2 space H := α∈A He , where Hα H0 and Sα = S0 . Clearly, l (A ), Γ1 , Γ2 0 0 with Γ1 (fα ) = (Γ1 fα ) and Γ2 (fα ) = (Γ2 fα ) becomes a boundary triple for S ∗ . The induced Γ-ﬁeld is γ(z)(ξα ) = (γ0 (z)ξα ) and the Q-function is scalar, Q(z) = q(z) id. It is worthy to note that the corresponding operator H 0 , which is the restriction of S ∗ to ker Γ1 , is just the direct sum of the copies of D over the set A and, in particular, spec H 0 = spec D. Proposition 3.19 becomes especially useful if the spectrum of D is a discrete set, then the spectrum of HΛ is (almost) completely determined in terms of the parametrizing operator Λ. The models of the above type can be used for the construction of solvable models for array of quantum dots and antidots. One of pecularities of such arrays is that

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they involve the microscopic properties of a single point as well as the macroscopic properties of the whole system. We consider for technical simplicity two-dimensional periodic arrays in a uniform magnetic ﬁeld orthogonal to the plane of the system. For a large class of such models, we refer to [68]. Let a1 , a2 be linearly independent vectors of R2 and A be the lattice spanned by them, A := Za1 + Za2 . Assume that each note α of the lattice is occupied by a certain object (quantum dot) whose state space is Hα with a Hamiltonian Hα (their concrete form will be given later). We assume that all quantum dots are identical, i.e. Hα := H0 , Hα = H0 . The system is subjected to a uniform ﬁeld orthogonal to the plane. In our case, the inner state space H0 will be L2 (R2 ). The Hamiltonian H0 will be taken in the form 2 2 ∂ ∂ 1 ω2 2 + πiξy + − πiξx (x + y 2 ). H0 = − + 2 ∂x ∂y 2 Here ξ is the number of magnetic ﬂux quanta through a unit area segment of the plane, and ω is the strength of the quantum dot potential. Note that the spectrum of H 0 is pure point and consists of the inﬁnite degenerate eigenvalues Emn , 1 Emn = (n + m + 1)Ω + (n − m)ξ, Ω := 2 π 2 ξ 2 + ω 2 , m, n ∈ Z, m, n ≥ 0. 2 The Hamiltonian H := ⊕α∈A Hα , describe the array of non-interacting quantum dots. To take into account the interdot interaction we use the restriction-extension procedure. Namely denote by Sα the restriction of Hα to the functions vanishing at the origin. As we have shown in Sec. 1.4.3, these operators are closed and have deﬁciency indices (1, 1). Respectively, one can construct the corrsponding boundary triples for Sα∗ . Namely, for fα ∈ dom Sα∗ we denote π 1 fα (r), b(fα ) := lim f (r) + a(fα ) log |r| . a(fα ) := − lim r→0 log |r| r→0 π According to the constructions of Sec. 1.4.3, (C, a, b) form a boundary triple for Sα∗ , and the corresponding Q-function is 1 z 1 Ω − + 2CE , q(z) = − ψ + log 2π 2 Ω 2π where ψ is the logarithic derivative of the Γ function and CE is the Euler constant. Respectively, the triple (l2 (A ), Γ1 , Γ2 ) with Γ1 (fα ) := (a(fα )),

Γ2 (fα ) := (b(fα )), is a boundary triple for the operator S ∗ , S := Sα , and the induced Q-function is the multiplication by q(z). The above deﬁned operator H corresponds exactly to the boundary condition Γ1 f = 0. For a self-adjoint operator L in l2 (A ) denote by HL the self-adjoint extension of S corresponding to the boundary conditions Γ2 f = LΓ1 f . This operator will be considered as a Hamiltonian of interacting quantum dots, and the way

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how diﬀerent nodes interact with each other is determined by the operator L. To avoid technical diﬃculties, we assume that L is bounded. Furthermore, L must satisfy some additional assumptions in order to take into account the nature of the problem. First, any reasonable deﬁnition of a periodic system with magnetic ﬁeld must include the invariance under the magnetic translation group. In our case this means that the matrix of L in the standard basis of l2 (A ) satisﬁes L(α, α + β) = eπiξα∧β L(0, β) for any α, β ∈ A . Second, we assume that only the nearest neighbors interact with each other, i.e.   λ1 , α = ±a1 , L(α, 0) = λ2 , α = ±a2 , λ1 , λ2 ∈ R\{0},   0, otherwise, Roughly speaking, the above assumptions mean the following: each node interact α with the four nearest nodes α ± aj , j = 1, 2, and the interaction is independent of α. analysis, it is useful to identify l2 (A ) with l2 (Z2 ) by (fn1 a1 +n2 a2 ) ∼ For further f (n1 , n2 ) , n1 , n2 ∈ Z. Then the operator L acts as follows: Lf (n1 , n2 ) ≡ L(η)f (n1 , n2 ) = λ1 [eiπηn2 f (n1 − 1, n2 ) + e−iπηn2 f (n1 + 1, n2 )] + λ2 [e−iπηn1 f (n1 , n2 − 1) + eiπηn1 f (n1 , n2 + 1)],

η = ξa1 ∧ a2 .

This operator L(η) is well known and is called the discrete magnetic Laplacian, and using Proposition 3.19 we can transfer the complete spectral information for L to the Hamiltonian of quantum dots HL . One of interesting moments in the spectral analysis of L is the relationship with the almost Mathieu operator in the space l2 (Z) [121], M (η, θ)f (n) = λ1 [f (n − 1) + f (n + 1)] + 2λ2 cos(2πηn + θ)f (n),

θ ∈ [−π, π).

In particular, spec L(η) =

spec M (η, θ).

θ∈[−π,π)

Elementary constructions of the Bloch analysis show that the spectrum of L(η) is absolutely continuous and has a band structure. At the same time, for irrational η the spectrum of M (η, θ) is independent of θ and hence coincides with the spectrum of L(η). It was shown only recently that the spectrum of M (η, θ) is a Cantor set for all irrational η and non-zero λ1 , λ2 , see [15]. Using our analysis we can claim that, up to the discrete set {Em,n } (a more precise analysis shows that these eigenvalues are all in the spectrum of the array) we can transfer the spectral information for L(η) to the array of quantum dots; in particular, we obtain a Cantor spectrum for irrational η due to the analyticity of the Q-function.

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4. Isolated Eigenvalues 4.1. Problem setting In the previous sections we have analyzed the part of the spectrum of the “perturbed” operator HΛ lying in the resolvent set of the “unperturbed” operator H 0 . If E ∈ spec H 0 , then, in general, it is diﬃcult to determine whether or not E ∈ spec HΛ . Nevertheless, if E is an isolated eigenvalue of H 0 , then the question whether E in the spectrum of HΛ becomes easier in comparison with the general case. (Examples of Secs. 3.5 and 3.6 show that this situation is rather typical for applications.) In this section, we give a necessary and suﬃcient condition for such an E to be an isolated eigenvalue of HΛ and completely describe the corresponding eigensubspace of HΛ (Theorem 4.7). For simplicity, we consider only the case of bounded self-adjoint operator Λ in G . In addition to the notation given in Sec. 3.1, in this section ε0 denotes an eigenvalue of H 0 with the eigensubspace H 0 (which can be inﬁnite-dimensional), P 0 denotes the orthoprojector on H 0 . We denote by V (ε0 ) the set of all open balls O centered at ε0 and such that spec H 0 ∩ O = {ε0 }. By GL(G ) we denote the set of bounded linear operators in G having a bounded inverse. If O ∈ V (ε0 ), then K(O; G ) denotes the space of all analytic mappings V : O → GL(G ) such that z ) = V −1 (z) (the latter condition is equivalent to the following V (ε0 ) = I and V ∗ (¯ one: V (z) is a unitary operator for z ∈ R ∩ O). 4.2. Auxiliary constructions Further we need the following lemma. Lemma 4.1. For any z, ζ ∈ res H 0 there holds: (1) P 0 Nz = P 0 Nζ ; (2) H 0 ∩ dom HΛ = H 0 ∩ Nz⊥ = H 0 ran P 0 γ(ζ); p (3) ker γ ∗ (z)P 0 γ(z) = ker P 0 γ(ζ), i.e., the restriction of γ ∗ (z) to ran P0 γ(ζ) is an injection. In particular, dim ran γ ∗ (z)P 0 γ(z) = dim ran P 0 γ(z). Proof. (1) Recall that P 0 = −i limδ→+0 δR0 (ε0 + iδ) in the weak operator topology. By (1.14b), for any δ > 0 one has γ(z) + (ε0 + iδ − z)R0 (ε0 + iδ)γ(z) = γ(ζ) + (ε0 + iδ − ζ)R0 (ε0 + iδ)γ(ζ).

(4.1)

Multiplying (4.1) with δ and sending δ to 0 we arrive at (ε0 − z)P 0 γ(z) = (ε0 − ζ)P 0 γ(ζ).

(4.2)

Now it is suﬃcient to recall that Nz = ran γ(z) for all z ∈ res H 0 . (2) Let φ ∈ H 0 ∩ dom HΛ and ψ ∈ Nz . As H 0 and HΛ are disjoint, φ ∈ dom S and Sφ = ε0 φ. There holds (ε0 − z)φ|ψ = (S − z¯)φ|ψ = φ|(S ∗ − z)ψ = 0. Hence φ ⊥ Nz .

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Conversely, let φ ∈ H 0 ∩Nz⊥ . By (1.21d), γ ∗ (z)φ = 0. As follows from the Krein z )φ = RΛ (¯ z )φ ∈ dom HΛ . Hence, φ ∈ resolvent formula (3.1), (ε0 − z¯)−1 φ = R0 (¯ dom HΛ , and the ﬁrst equality is proved. The second equality follows immediately from the relations: (a) for any φ ∈ H 0 and ψ ∈ Nz one has φ|ψ = φ, P 0 ψ, (b) Nz = ran γ(z), (c) ran P 0 γ(z) = ran P 0 γ(ζ). (3) Let γ ∗ (z)P 0 γ(ζ)g = 0. By (1.21d), P 0 γ(ζ)g ⊥ Nz . According to (4.2), 0 P γ(ζ)g ⊥ Nζ . It follows from the second equality in item (2) that P 0 γ(ζ)g ⊥ ran P 0 γ(ζ). Hence, P 0 γ(ζ)g = 0. The item (3) of Lemma 4.1 can be generalized as follows. Lemma 4.2. Let εj , j = 1, . . . , m, be distinct eigenvalues of H 0 , P j be orthoprojectors on the corresponding eigensubspaces and P :=

m

Pj .

j=1

Then (I − P )γ(z) is an injection for any z ∈ res H 0 . Proof. Let (I − P )ψ = 0 where ψ = γ(z)φ for some z ∈ res H 0 , φ ∈ G . Then ψ = P ψ ∈ dom H 0 and, therefore, H 0 ψ = zψ. Hence, ψ = 0 and φ = 0. In what follows, z0 denotes a ﬁxed number from res H 0 , x0 := Re z0 , y0 := Im z0 , L := γ(z0 ). Recall that L is a linear topological isomorphism on the deﬁciency subspace N := Nz0 ⊂ H . Since, by deﬁnition, γ(z) = L + (z − z0 )R0 (z)L for any z ∈ res H 0 , the point ε0 is either a regular point for γ or a simple pole with the residue Res[γ(z) : z = ε0 ] = (z0 − ε0 )P 0 L.

(4.3)

Similarly, as Q(z) = C + (z − x0 )L∗ L + (z − z0 )(z − z¯0 )L∗ R0 (z)L, with a bounded self-adjoint operator C (see Proposition 1.20), the point ε0 is either a regular point for Q or a simple pole with the residue: Res[Q(z) : z = ε0 ] = −|ε0 − z0 |2 L∗ P 0 L.

(4.4)

From the equality P 0 Lφ2 = L∗ P 0 Lφ|φ one easily sees that ker P 0 L = ker L∗ P 0 L (see also Lemma 4.1(3)). In particular, P 0 L = 0 if and only if L∗ P 0 L = 0, and there are simple examples where P 0 L = 0. Moreover, the following lemma holds. Lemma 4.3. Let H1 and H2 be two Hilbert spaces and A : H1 → H2 be a bounded linear operator. Then the two conditions below are equivalent: (1) ran A is closed; (2) ran A∗ A is closed. In particular, ran P 0 L is closed if and only if ran L∗ P 0 L is closed.

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Proof. Condition (1) is satisﬁed if and only if there is a constant c > 0 such that Aφ ≥ cφ for all φ ∈ (ker A)⊥ . On the other hand, condition (2) is satisﬁed if and only if there is a constant c > 0 such that A∗ Aφ|φ ≥ c φ2 for all φ ∈ (ker A∗ A)⊥ . Since ker A∗ A = ker A, we get the result. Now, we denote by Gr := ker L∗ P 0 L ⊂ G , G1 := G ⊥ . The orthoprojectors of G on Gr (respectively, on G1 ) are denoted by Πr (respectively, by Π1 ). If A is a bounded operator in G , then we write Ar := Πr AΠr , and this will be considered as an operator in Gr . If z ∈ res H 0 , then γr (z) denotes the operator (I −P 0 )γ(z)Πr acting from Gr to H (to avoid a confusion with the previous notation, we suppose without loss of generality G = H ). Further, we denote by Hr the subspace (I − P 0 )H and by Hr0 the part of H 0 in Hr ; clearly, ε0 ∈ res Hr0 , and both the mappings γr and Qr have analytic continuation to ε0 . Finally, denote G3 = ker(Qr (ε0 ) − Λr ), and G2 = Gr G3 . Lemma 4.4. There exists a closed symmetric densely deﬁned restriction Sr of Hr0 such that γr is a Krein Γ-ﬁeld for the triple (Sr , Hr0 , Gr ), and Qr is a Krein Q-function associated with this triple and γr . Proof. We use Proposition 1.18. Since P 0 and R0 (z) commute for all z ∈ res H 0 , it is clear that γr satisﬁes the condition (1.14b). Further, z0 belongs to res Hr0 and γr (z0 ) = (I − P0 )LΠr . Let us show that the subspace N := ran γr (z0 ) is closed. Let (φn ) ∈ Gr such that ψn := (I − P 0 )Lφn converge to some ψ ∈ Hr . Since φn ∈ Gr , one has L∗ P 0 Lφn = 0, hence P 0 Lφn = 0. On the other hand, Lφn ∈ N by deﬁnition of L. Denote the orthoprojector of H onto N by P , then we have P ψn = Lφn , hence Lφn converge to P ψ. Therefore, the sequence (L∗ Lφn ) converges to L∗ P ψ in G . Since L∗ L is a linear topological automorphism of G , there exists lim φn and this limit belongs to Gr because Gr is closed. Thus, ψ ∈ N and N is closed. By Lemma 4.1(3), γr (z0 ) is injective. By the closed graph theorem, γr (z0 ) is a linear topological isomorphism of Gr onto N . Now, we show that N ∩dom Hr0 = 0. It is suﬃcient to show that ((I −P 0 )N )∩ dom H 0 = 0. Let ψ ∈ ((I − P 0 )N ) ∩ dom H 0 . As ψ ∈ (I − P 0 )N , we have ψ = φ − P 0 φ for some φ ∈ N . Since ψ, P 0 φ ∈ dom H 0 , φ ∈ dom H 0 . Hence φ = 0 and ψ = 0. Thus, by Proposition 1.18, there exists a closed symmetric densely deﬁned restriction of Hr0 such that γr is a Γ-ﬁeld for the triple (Sr , Hr0 , Gr ). Since Q(z) = C − iy0 L∗ L + (z − z¯0 )L∗ γ(z) with a bounded self-adjoint operator C in G (Proposition 1.20), we have Qr (z) = Πr CΠr − iy0 Πr L∗ LΠr + (z − z¯0 )Πr L∗ γ(z)Πr = Πr CΠr − iy0 Πr L∗ (I − P 0 )LΠr + (z − z¯0 )Πr L∗ (I − P 0 )γ(z)Πr − iy0 Πr L∗ P 0 LΠr + (z − z¯0 )Πr L∗ P 0 γ(z)Πr .

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Now, we use the equations Πr L∗ P 0 LΠr = 0 Πr L∗ P 0 γ(z)Πr = 0.

(4.5)

The ﬁrst one follows from deﬁnition of Πr , to prove the second one we note that γ(z) = L + (z − z0 )R0 (z)L, therefore Πr L∗ P 0 γ(z)Πr =

ε0 − z0 Πr L∗ P 0 LΠr = 0. ε0 − z

From (4.5), we obtain Qr (z) = C − iy0 γr∗ (z0 ) γr (z0 ) + (z − z¯0 )γr∗ (z0 ) γr (z), where C = Πr CΠr is a self-adjoint bounded operator in Gr . Hence, Qr is the Krein Q-function associated with the Γ-ﬁeld γr . To prove the main result of the section we need the following lemma. Lemma 4.5. Let S be an analytic function in the disk D = {z ∈ C : |z| < r} with values in the Banach space of all bounded linear operators L(G ) such that there is a bounded inverse S −1 (z) for all z from the punctured disk D\{0} and the function S −1 (z) is meromorphic. If ker S(0) = 0, then S0 := S(0) has the bounded inverse (and, therefore, S −1 has an analytic continuation to the point 0 of the disk). If S0 is self-adjoint and 0 is a pole at most of ﬁrst order for S −1 (z), then ran S0 is closed, i.e. there is a punctured neighborhood of 0 which has no point of spec S0 . Proof. Consider the Laurent expansion S −1 (z) =

∞

Tn z n

n=−m

where m is a natural number. If m ≤ 0, the lemma is trivial. Suppose m > 0. Since S(z)S −1 (z) = I for all z, we have S0 T−m = 0. Let ker S0 = 0, then T−m = 0, and by recursion, Tn = 0 for all n < 0. Then, S0 T0 = T0 S0 = I and the ﬁrst part of the lemma is proved. Now, let now m = 1. Then S0 T−1 = 0 and T−1 S1 + T0 S0 = I, where S1 = S (0). This implies S0 T0 S0 = S0 . Let x ∈ ran S0 , then S0 T0 x = x. Since ran S0 ⊂ (ker S0 )⊥ , there is a linear operator A : ran S0 → ran S0 such that AS0 x = x for all x ∈ ran S0 . From S0 T0 x = x we have A = T0 , i.e. A is bounded. Hence, there is c > 0 such that x ≤ cS0 x for all x ∈ ran S0 and hence for all x ∈ (ker S0 )⊥ . Remark 4.6. If 0 is a second order pole for S −1 (z), then the range of S0 can be non-closed. For example, let A be a self-adjoint operator in a Hilbert space H such that ran A is non-closed. Let G = H ⊕ H , and S(z) is deﬁned as follows: A z S(z) = . z 0

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Then S

−1

1 (z) = 2 z

0 z

59

z . −A

4.3. Description of eigensubspace Theorem 4.7. Let ε0 be an isolated eigenvalue of H 0 and ran P 0 L be closed. Then the following assertions are mutually equivalent. (1) There exists a punctured neighborhood of ε0 that contains no point of spec HΛ (in particular, if ε0 ∈ spec HΛ , then ε0 is an isolated point in the spectrum of HΛ ). (2) The operator Q(z) − Λ has a bounded inverse for all z from a punctured neighborhood of ε0 . (3) ran(Qr (ε0 ) − Λr ) is closed. (4) There is a punctured neighborhood of 0 which contains no point from the spectrum of the operator Qr (ε0 ) − Λr . Let one of the condition (1)–(4) be satisﬁed. Then the eigensubspace HΛ0 := ker(HΛ − ε0 ) is the direct sum, HΛ0 = Hold ⊕ Hnew , where Hold = H 0 ∩ dom HΛ = H 0 ∩ dom S, Hnew = γr (ε0 ) ker[Qr (ε0 ) − Λr ] and dim H 0 Hold = dim G Gr . Therefore, ε0 ∈ spec HΛ if and only if at least one of the following two conditions is satisﬁed: • H 0 ∩ dom HΛ = {0}, • ker[Qr (ε0 ) − Λr ] = {0}. Remark 4.8. Since H 0 ∩ dom HΛ = H 0 ∩ dom S, the component Hold of ker(HΛ − ε0 ) is independent of Λ, i.e. this part is the same for all extensions of S disjoint to H 0 . On the other hand, the component Hnew depends on Λ. Remark 4.9. Clearly, ran P 0 L is closed, if the deﬁciency index of S or dim H 0 are ﬁnite (this simple case is very important in applications of Theorem 4.7). To show that the assumptions are essential for inﬁnite deﬁciency indices, we provide here an example when the range of P 0 L is not closed. (k) Let Hk = l2 (N) for k = 0, 1, . . . and let (en )n≥0 be the standard basis in Hk : (k) en = (δmn )m≥0 . Denote by Hk0 the self-adjoint operator in Hk which is determined (k) (k) (0) by Hk0 en = (n + 1/2)en . Choose a ∈ H0 such that a = 1, a|e0 = 0, 0 (k) = a. Consider in Hk the one-dimensional subspace Nk a ∈ / D(H0 ), and set a (k) generated by e0 + (k + 1)a(k) . Fix z0 ∈ C\ R. By Proposition 1.18, there exists Hk , a symmetric restriction Sk of Hk0 such that Nz0 (Sk ) = Nk . Let now H = Hk0 , S = Sk . Then, the eigensubspace H 0 of H0 corresponding to the H0 = (k) eigenvalue ε0 = 1/2 is the closed linear span of (e0 ), k = 0, 1, . . . , and Nz0 (S) is (k) the closed linear span of (e0 + (k + 1)a(k) ), k = 0, 1, . . . We can choose G := Nz0 , γ(z0 ) = L = I where I is the identical embedding of Nz0 into H . It is clear, that the

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(k) image of P 0 L is the set M of all vectors x from H 0 having the form x = λk e0 2 2 0 0 where (k + 1) |λk | < ∞. Obviously, M is dense in H but M = H , hence M is not closed. Proof of Theorem 4.7. The equivalence (1) ⇔ (2) follows from Theorem 3.2, and the equivalence (3) ⇔ (4) is trivial. Let us prove the implication (1) ⇒ (3). Choose O ∈ V (ε0 ) such that Q(z) − Λ has a bounded inverse for all z ∈ O\{ε0 } and for z ∈ O\{ε0 } consider the mapping T (z) = (z − ε0 )(Q(z) − Λ). Note that • T has an analytic continuation to ε0 by setting T (ε0 ) = −|ε0 − z0 |2 L∗ P 0 L, see Eq. (4.4), and • T has a bounded inverse in O \{ε0 }. Since the operator L∗ P0 L has the closed range, we can apply a result of Kato [88, Secs. VII.1.3 and VII.3.1]. According the mentioned result, there is a mapping V , V ∈ K(O; G ), such that the operator V (z)T (z)V −1 (z) has the diagonal matrix representation with respect to the decomposition G = G1 ⊕ Gr : ! " Tˆ11 (z) 0 −1 . (4.6) V (z)T (z)V (z) = 0 Tˆrr (z) Because the left-hand side of Eq. (4.6) has a bounded inverse for z ∈ O \ {ε0 }, the same is true, in particular for the operator S(z) := (z − ε0 )−1 Tˆrr (z) = Πr V −1 (z)[Q(z) − Λ]V (z)Πr considered in the space Gr . Our next aim to prove that S −1 (z) ≤ c|z − ε0 |−1

(4.7) 0

with a constant c > 0 for all z in a punctured neighborhood of ε . For this purpose we consider together with the decomposition G = G1 ⊕ Gr of the space G , the decomposition H = H1 ⊕ Hr , where H1 = H 0 , Hr = (I − P 0 )H 0 . In virtue of to the Krein resolvent formula (3.1), (z − ε0 )RΛ (z) z) = (z − ε0 )R0 (z) − (z − ε0 )2 γ(z)T −1 (z)γ ∗ (¯ z )V (¯ z )]∗ . = (z − ε0 )R0 (z) − (z − ε0 )2 [γ(z)V (z)]V −1 (z)T −1 (z)V (z)[γ(¯ Represent the operator γ(z)V (z) according to the above mentioned representations of H and G in the matrix form: γˆ11 (z) γˆ1r (z) γ(z)V (z) = . (4.8) γˆr1 (z) γˆrr (z) Since (z−ε0 )RΛ (z) and (z−ε0 )R0 (z) are analytic functions in a neighborhood of ε0, all the matrix term in [γ(z)V (z)]V −1 (z)T −1 (z)V (z)[γ(¯ z )V (¯ z )]∗ are also analytic in the same neighborhood. In particular, we can chose O in such a way that the function ∗ −1 ∗ γr1 (z)Tˆ −1 (z)ˆ γr1 (¯ z ) + γˆrr (z)Tˆrr (z)ˆ γrr (¯ z )) z → (z − ε0 )2 (ˆ 11

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is analytic in O. Since Tˆ11 (ε0 ) ≡ −|z0 − ε0 |2 L∗ P 0 L has a bounded inverse in G1 , −1 the function Tˆ11 (z) is analytic in a neighborhood of ε0 . Therefore, we can chose O 0 2 −1 ∗ (z)ˆ γrr (¯ z ) is analytic in O. Further γˆrr (ε0 ) = γr (ε0 ). such that (z − ε ) γˆrr (z)Tˆrr In virtue of Lemma 4.4 and deﬁnition of the Γ-ﬁeld, we can ﬁnd a constant c > 0 such that γr (ε0 )g ≥ c g for all g ∈ Gr . Therefore we can chose O so small ∗ (¯ z ) is that ˆ γrr (z)g ≥ c g for all z ∈ O, g ∈ Gr with some c > 0. Since γˆrr 0 2 ˆ −1 z ) on Gr , we see that (z − ε ) Trr (z) is bounded in a an isomorphism of ran γr (¯ neighborhood of ε0 . Hence, we obtain (4.7) in a punctured neighborhood of ε0 . By [77, Theorem 3.13.3], S −1 (z) has at point ε0 a pole of the order ≤ 1. Therefore, (1) ⇒ (3) by Lemma 4.5. Now we prove (4) ⇒ (2). Choose O ∈ V (ε0 ) such that Q(z)−Λ has no spectrum in O \ {ε0 }. Moreover, we can use again the representation (4.6). Since V (z) = I +O(z −ε0 ), the function S(z) := Πr V −1 (z)Πr [Q(z)−Λ]Πr V (z)Πr has an analytic continuation at ε0 with the value S(ε0 ) = Qr (ε0 ) − Λr . To proceed further, we need the following auxiliary result. Lemma 4.10. The operator S (ε0 ) is strictly positive on ker[Qr (ε0 ) − Λr ]. Proof of Lemma 4.10. Since V −1 (x) = V ∗ (x) for x ∈ O ∩ R, for the derivative of S one has: S (ε0 ) = Πr (V )∗ (ε0 )Πr [Q(ε0 ) − Λ]Πr + Πr [Q(ε0 ) − Λ]Πr V (ε0 )Πr + Πr Q (ε0 )Πr

(4.9)

(note that Πr Q(ε0 ) and Q(ε0 )Πr are well deﬁned). Let now φ ∈ ker[Qr (ε0 ) − Λr ]. Then we have from (4.9) that φ|S (ε0 )φ = φ|Q (ε0 )φ. Since S (ε0 ) is a selfadjoint operator, we have that S (ε0 )φ = Q (ε0 )φ on ker [Qr (ε0 ) − Λr ]. Therefore, by Lemma 4.4 and (1.22a), S (ε0 )φ = γr∗ (ε0 )γr (ε0 )φ for all φ ∈ ker [Qr (ε0 ) − Λr ],

(4.10)

hence S (ε0 ) is strictly positive on ker[Qr (ε0 ) − Λr ]. To prove the required implication (4) ⇒ (2), it is now suﬃcient to show that S(z) has a bounded inverse in a punctured neighborhood of ε0 . Since S(z) is analytic, it suﬃce to prove that the operator J(z) := S(ε0 ) + S (ε0 )(z − ε0 ) has a bounded inverse in a punctured neighborhood of ε0 with the estimate J(z)−1 ≤ c|z −ε0 |−1 . For this purpose, we represent S (ε0 ) in the matrix form ! " S S 22 23 S (ε0 ) = S32 S33 according to the representation Gr = G2 ⊕G3 . Then, J has the matrix representation " ! S0 + (z − ε0 )S22 (z − ε0 )S23 J(z) = (z − ε0 )S32 (z − ε0 )S33

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where S0 := S(ε0 ). By the assumption of item (4), S0 has a bounded inverse in has a bounded inverse in G3 . Now we use the G2 , and by (4.10) the operator S22 Frobenius formula for the inverse of a block-matrix [79]: ! "−1 ! " −1 −1 −1 A−1 A11 A12 [A11 − A12 A−1 22 A21 ] 11 A12 [A21 A11 A12 − A22 ] = −1 −1 A21 A22 [A21 A−1 A21 A−1 [A22 − A21 A−1 11 A12 − A22 ] 11 11 A12 ] (4.11) which is valid if all the inverse matrices on the right-hand side exist. Using (4.11) it is easy to see that J −1 (z) exists for all z in a punctured neighborhood of ε0 and obeys the estimate J(z)−1 ≤ c|z − ε0 |−1 with some c > 0. Thus, the implication (4) ⇒ (2) and, hence, the equivalence of all the items (1)–(4) are proven. Now, suppose that the conditions of items (1)–(4) are satisﬁed. To determine the eigenspace HΛ0 we ﬁnd the orthoprojector PΛ0 on this space calculating the residue of the resolvent, PΛ0 = −Res[RΛ (z) : z = ε0 ] = P 0 + Res[M (z) : z = ε0 ], where M (z) := γ(z)[Q(z) − Λ]−1 γ ∗ (¯ z ). Using the conditions of item (4), we ﬁnd O ∈ V (ε0 ) and V ∈ K(O, G ) such that for z in O \{ε0 } ! " 0 S1 (z) −1 , V (z)[Q(z) − Λ]V (z) = 0 Sr (z) according to the decomposition G = G1 ⊕ Gr where S1 and Sr have the following properties: Sr is analytic in O with Sr (ε0 ) = Qr (ε0 ) − Λr and S1 (z) = −|ε0 − z0 |2

L∗ P 0 L + F1 (z), z − ε0

where F1 is analytic in O.

(4.12)

Using Lemma 4.10, we ﬁnd a function W ∈ K(O, Gr ) such that for z in O\{ε0 } one has " ! 0 S2 (z) −1 , W (z)Sr (z)W (z) = 0 S3 (z) according to the decomposition Gr = G2 ⊕ G3 where S2 and S3 have the properties: ker S2 (ε0 ) = 0 and S2 (ε0 )φ = [Qr (ε0 ) − Λr ]φ for φ ∈ G2 ,

(4.13)

S3 is analytic in O and has the form S3 (z) = (z − ε )T (z) 0

where T0 := T (ε0 ) is a strictly positive operator in G3 . Denote now

! U (z) := V (z)

I1

0

0

W (z)

" ,

(4.14)

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where the matrices are decomposed according to the representation G = G1 ⊕Gr and ˆ = U −1 (z)[Q(z) − Λ]U (z), I1 is the identity operator on G1 . Further, denote Q(z) −1 ∗ ˆ γ (¯ z ), and for z ∈ O \{ε0 } one has γˆ(z) = γ(z)U (z), then M (z) = γˆ (z)Q (z)ˆ   −1 0 0 S1 (z)  ˆ −1 (z) =  Q S −1 (z) 0 .  0 2

0

S3−1 (z)

0

An important property of γˆ we need is follows γˆ (z) =

z0 − ε0 0 P LU (z) + (I − P 0 )γ(z)U (z), z − ε0

(4.15)

and (I − P 0 )γ is analytic in O. Represent M as the sum M (z) = A1 (z) + A2 (z) + z ); here Πj denote the orthoprojectors A3 (z), where Aj (z) = γˆ (z)Πj Sj−1 (z)Πj γˆ ∗ (¯ of G onto Gj , j = 1, 2, 3. It is clear from (4.12)–(4.15) that at the point z = ε0 , the function Aj (z) has a pole at most of jth order. Let (−j)

Aj (z) = Aj

(z − ε0 )−j + Aj

(−j+1)

(z − ε0 )−j+1 + · · ·

be the Laurent expansion for Aj at the point ε0 . According to the deﬁnition of Aj (z) and formulas (4.12)–(4.15) we have (−j)

Aj

= Cj Bj Cj∗ ,

(−j+1)

Aj

= Cj Bj Dj∗ + Dj Bj Cj∗ + Cj Bj Cj∗ ,

where Cj = (z0 − ε0 )P 0 LΠj ,

B1 = |ε0 − z0 |−2 (Π1 L∗ P 0 LΠ1 )−1 ,

B2 = (Π2 S(ε0 )Π2 )−1 ,

B3 = (Π3 T0 Π3 )−1 ,

and Bj , Cj , Dj are some bounded operators (we need no concrete form of them). By deﬁnition of the spaces Gj , we have Πj L∗ P 0 LΠj = 0 for j = 2, 3, and hence, P 0 LΠj = 0 for the same j’s. As a result we have that A2 (z) has no pole at z = ε0 , i.e. Res[A2 (z) : z = ε0 ] = 0,

(4.16)

and A3 (z) has at this point a pole at least of ﬁrst order. Using (4.15) and taking into consideration P 0 LΠ3 = 0, we obtain Res[A3 (z) : z = ε0 ] =: P3 = (I − P 0 )γ(ε0 )Π3 T0−1 Π3 γ ∗ (ε0 )(I − P 0 ) = γr (ε0 )Π3 T0−1 Π3 γr∗ (ε0 ).

(4.17)

Now, we have according to (4.12) and (4.15) Res[A1 (z) : z = ε0 ] =: −P1 = −P 0 LΠ1 (Π1 L∗ P 0 LΠ1 )−1 Π1 L∗ P 0 .

(4.18)

As a result, we have from (4.16), (4.17), and (4.18) PΛ0 = P 0 − P1 + P3 . Equation (4.18) shows that P1 is an orthoprojector with ran P1 ⊂ ran P 0 . Therefore, P 0 − P1 is an orthoprojector on a subspace of H 0 . Equation (4.17) shows that ran P3 ⊂ ran(I − P 0 ), therefore (P 0 − P1 )P3 = 0. Since P3 is self-adjoint,

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P3 (P 0 − P1 ) = 0. Using (PΛ0 )2 = PΛ0 we see that P32 = P3 , hence P3 is an orthoprojector and P3 ⊥ P 0 . By Lemma 4.1, ran(P 0 − P1 ) = H 0 ∩ dom HΛ ≡ Hold . The relation ran P3 = γr (ε0 ) ker[Qr (ε0 ) − Λr ] ≡ Hnew follows from (4.17) and the deﬁnition of G3 . Theorem 4.7 is proved. Acknowledgments The work was supported in part by the Deutsche Forshungsgemeinschaft (PA 1555/1-1 and 436 RUS 113/785/0-1), the SFB 647 “Space, Time, Matter”, and the German Aerospace Center (Internationales B¨ uro, WTZ Deutschland-Neuseeland NZL 05/001). In course of preparing the manuscript the authors had numerous useful discussions with Sergio Albeverio, Jussi Behrndt, Johannes Brasche, Yves Colin de Verdi`ere, Pavel Exner, Daniel Grieser, Bernard Helﬀer, Peter Kuchment, Hagen Neidhardt, Mark Malamud, Boris Pavlov, Thierry Ramond, Henk de Snoo, ˇˇtov´ıˇcek, which are gratefully acknowledged. and Pavel S References [1] V. Adamyan, Scattering matrices for microschemes, Oper. Theory Adv. Appl. 59 (1992) 1–10. [2] V. M. Adamyan and B. S. Pavlov, Zero-radius potentials and M. G. Krein’s formula for generalized resolvents, J. Soviet Math. 42 (1988) 1537–1550. [3] S. A. Adhikari, T. Frederico and I. D. Goldman, Perturbative renormalization in quantum few-body problems, Phys. Rev. Lett. 74 (1995) 4572–4575. [4] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Vol. I, 3rd edn. (Pitman Adv. Publ., 1981). [5] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Vol. II, 3rd edn. (Pitman Adv. Publ., Boston, 1981). [6] S. Albeverio, J. Brasche, M. M. Malamud and H. Neidhardt, Inverse spectral theory for symmetric operators with several gaps: Scalar-type Weyl functions, J. Funct. Anal. 228(1) (2005) 144–188. [7] S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, 2nd edn. (AMS, Providence, 2005). [8] S. Albeverio and V. Geyler, The band structure of the general periodic Schr¨ odinger operator with point interactions, Commun. Math. Phys. 210 (2000) 29–48. [9] S. Albeverio and P. Kurasov, Finite rank perturbations and distribution theory, Proc. Amer. Math. Soc. 127 (1999) 1151–1161. [10] S. Albeverio and P. Kurasov, Singular Perturbations of Diﬀerential Operators. Solvable Schr¨ odinger Type Operators (Cambridge Univ. Press, Cambridge, 2000). [11] S. Albeverio and K. Pankrashkin, A remark on Krein’s resolvent formula and boundary conditions, J. Phys. A 38 (2005) 4859–4865. [12] R. A. Aleksandryan and R. Z. Mkrtchyan, On qualitative criteria characterizing the specta of arbitrary selfadjoint operators, Sov. J. Contemp. Math. Anal. Arm. Acad. Sci. 19(6) (1984) 22–33. [13] R. Arens, Operational calculus of linear relations, Paciﬁc J. Math. 11 (1961) 9–23. [14] M. Asorey, A. Ibort and S. Marmo, Global theory of quantum boundary conditions and topology change, Int. J. Mod. Phys. A 20 (2005) 1001–1025.

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Reviews in Mathematical Physics Vol. 20, No. 1 (2008) 71–115 c World Scientiﬁc Publishing Company

EDGE CURRENTS FOR QUANTUM HALL SYSTEMS, I. ONE-EDGE, UNBOUNDED GEOMETRIES

PETER D. HISLOP Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027 USA [email protected] ERIC SOCCORSI∗ Universit´ e de la M´ editerran´ ee, Luminy, Case 907, 13288 Marseille, France

Received 27 February 2007 Revised 21 November 2007

Devices exhibiting the integer quantum Hall eﬀect can be modeled by one-electron Schr¨ odinger operators describing the planar motion of an electron in a perpendicular, constant magnetic ﬁeld, and under the inﬂuence of an electrostatic potential. The electron motion is conﬁned to unbounded subsets of the plane by conﬁning potential barriers. The edges of the conﬁning potential barrier create edge currents. In this, the ﬁrst of two papers, we prove explicit lower bounds on the edge currents associated with one-edge, unbounded geometries formed by various conﬁning potentials. This work extends some known results that we review. The edge currents are carried by states with energy localized between any two Landau levels. These one-edge geometries describe the electron conﬁned to certain unbounded regions in the plane obtained by deforming half-plane regions. We prove that the currents are stable under various potential perturbations, provided the perturbations are suitably small relative to the magnetic ﬁeld strength, including perturbations by random potentials. For these cases of one-edge geometries, the existence of, and the estimates on, the edge currents imply that the corresponding Hamiltonian has intervals of absolutely continuous spectrum. In the second paper of this series, we consider the edge currents associated with two-edge geometries describing bounded, cylinder-like regions, and unbounded, strip-like, regions. Keywords: Edge states; quantum Hall eﬀect; Landau Hamiltonians; spectral theory; perturbation theory; asymptotic velocity. Mathematics Subject Classiﬁcation 2000: 47A55, 51Q10, 81Q15

∗ Also

Centre de Physique Th´eorique, Unit´e Mixte de Recherche 6207 du CNRS et des Universit´es Aix-Marseille I, Aix-Marseille II et de l’Universit´ e du Sud Toulon-Var-Laboratoire aﬃli´e ` a la FRUMAM, F-13288 Marseille Cedex 9, France. 71

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Contents 1. Introduction and Main Results 1.1. Related papers 1.2. Contents

72 77 78

2. The 2.1. 2.2. 2.3. 2.4.

78 78 80 86 89

Straight Edge and a Sharp Conﬁning Potential The main results for the unperturbed case Proof of Theorem 2.1 Perturbation theory for the straight edge Localization of the edge current

3. The Straight Edge and Dirichlet Boundary Conditions

92

4. One-Edge Geometries with More General Boundaries

98

5. One-Edge Geometries and the Spectral Properties of H = H0 + V1

102

6. One-Edge Geometries and General Conﬁning Potentials Appendix A. Basic Properties of Eigenfunctions and Eigenvalues of h0 (k) Appendix B. Pointwise Upper and Lower Exponential Bounds on Solutions to Certain ODEs B.1. Basic properties of ψ B.2. Pointwise bounds

103 107 109 110 111

1. Introduction and Main Results The integer quantum Hall eﬀect (IQHE) refers to the quantization of the Hall conductivity in integer multiples of 2πe2 /h. The IQHE is observed in planar quantum devices at zero temperature and can be described by a Fermi gas of noninteracting electrons. This simpliﬁcation reduces the study of the dynamics to the one-electron approximation. Typically, experimental devices consist of ﬁnitely-extended, planar samples subject to a constant perpendicular magnetic ﬁeld B. An applied electric ﬁeld in the x-direction induces a current in the y-direction, the Hall current, and the Hall conductivity σxy is observed to be quantized. Furthermore, the Hall conductivity is a function of the electron Fermi energy, or, equivalently, the electron ﬁlling factor, and plateaus of the Hall conductivity are observed as the ﬁlling factor is increased. It is now accepted that the occurrence of the plateaus is due to the existence of localized states near the Landau levels that are created by the random distribution of impurities in the sample, cf. [1, 2]. Another new phenomenon that arises in the study of these devices exhibiting the IQHE is the occurrence of edge currents associated with the boundaries of quantum devices. These edge currents are the subject of this work. An edge conductance corresponding to edge currents has been deﬁned and extensively studied by several groups, we give references below. The edge conductance has been proven to equal

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the bulk conductance and it is therefore quantized. In order to explain the origin of the edge currents, we recall the theory of an electron in R2 subject to a constant, transverse magnetic ﬁeld. The Landau Hamiltonian HL (B) describes a charged particle constrained to R2 , and moving in a constant, transverse magnetic ﬁeld with strength B ≥ 0. Let px = −i∂x and py = −i∂y be the two free (B = 0) momentum operators. The operator HL (B) is deﬁned on the dense domain C0∞ (R2 ) ⊂ L2 (R2 ) by HL (B) = (−i∇ − A)2 = p2x + (py − Bx)2 ,

(1.1)

in the Landau gauge for which the vector potential is A(x, y) = (0, Bx). The map (1.1) extends to a self-adjoint operator with point spectrum given by {En (B) = (2n+1)B | n = 0, 1, 2, . . .}, called the Landau levels, and each eigenvalue is inﬁnitely degenerate. The perturbation of HL (B) by random Anderson-type potentials Vω in the weak disorder regime for which Vω < C0 B has been extensively studied, cf. [3–6]. It is proved that outside a small interval of size B/ log B about the Landau levels, there are intervals of pure point spectrum with exponentially decaying eigenfunctions. The nature of the spectrum at the Landau levels is unclear. It is now known that there is nontrivial transport near the Landau levels for models on L2 (R2 ) [7]. For a point interaction model on the lattice Z2 , studied in [8], the authors considered the ﬁrst N Landau levels and proved that there exists a BN > 0 so that if B > BN , then the spectrum of Hω below the N th Landau level is pure point almost surely and that each Landau level below the N th is inﬁnitely degenerate. The quantum devices studied with regard to the IQHE may be inﬁnitely extended or ﬁnite, but are distinguished by the fact that there is at least one edge, that can be considered inﬁnitely extended, like in the case of the half-plane, or periodic, as in case of an annulus or cylinder. In all cases, the unperturbed Hamiltonian is a nonnegative, self-adjoint operator on the Hilbert space L2 (R2 ) and having the form H0 = HL (B) + V0 ,

(1.2)

where V0 denotes the conﬁning potential forming the edge (we also consider Dirichlet boundary conditions). The existence of an edge profoundly changes the transport and spectral properties of the quantum system. We consider states ψ ∈ L2 (R2 ) with energy concentration between two successive Landau levels En (B) and En+1 (B). We say that such a state ψ carries an edge current if the expectation of the y-component of the velocity operator Vy ≡ (py − Bx) in the state ψ is nonvanishing. In these two papers, we prove the existence of edge currents carried by these states and provide an explicit lower bound on the strength of the current. This lower bound shows that the edge current persists for all time in that the expectation of the Heisenberg time-dependent current operator Vy (t) ≡ eitH Vy e−itH in the state ψ satisﬁes the same lower bound for all t ∈ R. We will also prove that the states

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that carry edge-currents are well-localized in a neighborhood of the boundary of the region. Our main results, presented in this paper and its sequel [9], concern the following geometries and conﬁning potentials. (1) One-Edge Geometries: We study the half-plane case for which the electron is constrained to the right half-plane x > 0 by a conﬁning potential V0 that has either of the two forms: (a) Hard Conﬁning Potentials, such as the Sharp Conﬁning Potential: V0 (x) = V0 χ{x<0} (x), where V0 > 0 is a constant, or Dirichlet boundary conditions along the edge x = 0. (b) Soft Conﬁning Potentials, such as the Polynomial Conﬁning Potential V0 (x) = V0 |x|p χ{x<0} (x), p ≥ 1, and other rapidly increasing conﬁning potentials. (2) Two-Edge Geometries: We study models for which the electron is conﬁned to the strip SL = [−L/2, L/2] × R by hard or soft conﬁning potentials, such as (a) Sharp Conﬁning Potential V0 (x) = V0 χ{|x|>L/2}(x). (b) Parabolic Conﬁning Potential V0 (x) = V0 (|x| − L/2)2 χ|x|>L/2 (x). (3) Bounded, Two-Edge Geometries: We study models that are topologically a cylinder R × S 1 with conﬁning potential along the x-direction. The present paper deals with the ﬁrst topic of one-edge geometries, and the sequel [9] deals with the second and third topics concerning two-edge geometries. In addition to these results for straight edge geometries, we show that the results are stable under certain perturbations of the straight edge boundaries. Concerning the hard conﬁning potentials, we note that the lower bounds for the sharp conﬁning potential are uniform with respect to the strength of the conﬁning potential V0 . This means that we can take the limit as the size of the conﬁning potential becomes inﬁnite. As a result, our results extend to the case of Dirichlet boundary conditions along the edges. The various soft conﬁning potentials are discussed in Sec. 6. Our strategy in the one-edge case is to analyze the unperturbed operator via the partial Fourier transform in the y-variable. We write fˆ(x, k) for this partial Fourier transform. This decomposition reduces the problem to a study of the ﬁbered operators of the form h0 (k) = p2x + (k − Bx)2 + V0 (x),

(1.3)

2

acting on L (R). Since the eﬀective, nonnegative, potential V (x; k) = (k − Bx)2 + V0 (x) is unbounded as |x| → ∞, the resolvent of h0 (k) is compact and the spectrum is discrete. We denote the eigenvalues of h0 (k) by ωj (k), with corresponding normalized eigenfunctions ϕj (x; k), so that h0 (k)ϕj (x; k) = ωj (k)ϕj (x; k),

ϕj (·; k) = 1.

(1.4)

The properties of the eigenvalue maps k ∈ R → ωj (k) play an important role in the proofs. These maps are called the dispersion curves for the unperturbed

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E4(B)

ω3

E2(B)

ω2

E1(B)

ω1

E0(B)

ω0

75

υo

k 0

Fig. 1.

ωj (k) for j = 1, 2, 3, 4.

Hamiltonian (1.2). The importance of the properties of the dispersion curves comes from an application of the Feynman–Hellmann formula. To illustrate this, let us consider the one-edge geometry of a half-plane with a sharp conﬁning potential that is treated in this paper. It is clear from the form of the eﬀective potential V (x; k) that the dispersion curves are monotone decreasing functions of k, and that limk→+∞ ωn (k) = En (B), and that limk→−∞ ωn (k) = En (B) + V0 , see Fig. 1. For simplicity, we consider in this introduction a closed interval ∆0 ⊂ (B, 3B) and a normalized wave function ψ satisfying ψ = E0 (∆0 )ψ, where E0 is the spectral projection of H0 associated to ∆0 . Such a function admits a decomposition of the form 1 ψ(x, y) = √ 2π

ω0−1 (∆0 )

eiky β0 (k)ϕ0 (x; k)dk,

(1.5)

where the coeﬃcient β0 (k) is deﬁned by ˆ k), ϕ0 (·; k), β0 (k) ≡ ψ(·,

(1.6)

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with ψˆ denoting the partial Fourier transform given by ˆ k) ≡ √1 ψ(x, e−iky ψ(x, y)dy. 2π R

(1.7)

The matrix element of the current operator Vy in such a state is

ψ, Vy ψ = dx dk|β0 (k)|2 (k − Bx)|ϕ0 (x; k)|2 .

(1.8)

From (1.4) and the Feynman–Hellmann Theorem, we ﬁnd that dx(k − Bx)|ϕ0 (x; k)|2 , ω0 (k) = 2

(1.9)

R

ω0−1 (∆0 )

ω0−1 (∆0 )

so that we get

ψ, Vy ψ =

1 2

ω0−1 (∆)

|β0 (k)|2 ω0 (k)dk.

(1.10)

It follows from (1.10) that in order to obtain a lower bound on the expectation of the current operator in the state ψ we need to bound the derivative ω0 (k) from below for k ∈ ω0−1 (∆0 ). The next step of the proof involves relating the derivative ω0 (k) to the trace of the eigenfunction ϕ0 (x; k) on the boundary x = 0. For this, we use the formal commutator expression −i 1 [px , h0 (k)] + V (x). Vˆy (k) ≡ (k − Bx) = 2B 2B 0

(1.11)

Inserting this into the identity (1.9), we ﬁnd ω0 (k) = 2 ϕ0 (·; k), (k − Bx)ϕ0 (·; k) −i −V0

ϕ0 (·; k), [px , h0 (k)]ϕ0 (·; k) + ϕ0 (0; k)2 2B B −V0 ϕ0 (0; k)2 , = B

=

(1.12)

since the commutator term vanishes by the Virial Theorem. Consequently, we are left with the task of estimating the trace of the eigenfunction along the boundary. Much of our technical work is devoted to obtaining lower bounds on quantities of the form V0 ϕn (0; k)2 , for n = 0, 1, 2, . . . . The situation for the two-edge geometries is more complicated since there is an edge current associated with each edge. This analysis of two-edge geometries is the subject of [9]. Let H = HL (B) + V0 + V1 be a perturbation of the one-edge Hamiltonian with spectral family E(·). We consider an energy interval ∆n ⊂ (En (B), En+1 (B)), and |∆n | small. Roughly speaking, the main result of this paper is a uniform lower bound on the expectation of edge currents in all states with energy localized in the interval ∆n . We prove that for each n ∈ N, there exists a ﬁnite constant Cn > 0

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(given precisely below), so that if ψ ∈ E(∆n )L2 (R2 ), and the perturbation V1 is such that V1 ∞ /B is suﬃciently small, then | ψ, Vy ψ| ≥ Cn B 1/2 ψ2 .

(1.13)

We note that the order B 1/2 in (1.13) is optimal as for the unperturbed model, we prove that Cn B 1/2 ψ2 ≤ | ψ, Vy ψ| ≤ (1/Cn )B 1/2 ψ2 .

(1.14)

We make two remarks about this result, one concerning the time-dependent theory, and the second concerning the IQHE. First, we remark that the time-independent estimate (1.13) implies that the current persists with at least the same strength for all times provided that the bulk Hamiltonian Hbulk = HL (B) + V1 has a gap in its spectrum between the Landau levels. That is, the estimate (1.13) remains the same if we replace ψ with ψt = e−iHt ψ, or, equivalently, if we replace the current operator Vy with the Heisenberg current operator Vy (t) = e−iHt Vy eiHt . The edge current also remains localized in a neighborhood of size O(B −1/2 ) near the boundary for all time. Secondly, it has recently been proved that the conductance corresponding to the edge current, called the edge conductance σe , is quantized, and, in fact, equal to the bulk conductance, σb . The edge currents studied in this paper correspond to the edge conductance and we refer to the papers [10–17] for a detailed discussion and proofs. For the importance of edge currents in the IQHE, we refer to the papers [14, 18, 19]. 1.1. Related papers There are several papers on the subject of edge currents for unbounded, one-edge geometries. Macris, Martin and Pul´e [20] studied the half-plane case of one straight edge with soft conﬁning potentials. We extend this work proving the existence of edge currents for a large family of soft conﬁning potentials in Sec. 6. Furthermore, we show that we can interpolate between soft and hard conﬁning potentials. DeBi`evre and Pul´e [21] considered the case of a hard conﬁning potential, that is, Dirichlet boundary conditions (DBC). We treat this case in Secs. 3 and 5 and show that one can interpolate between soft and hard conﬁning potentials. The case of DBC was also treated by Fr¨ ohlich, Graf and Walcher [22] who studied non-straight edges. We consider non-straight edges in Sec. 4. As explained in Sec. 5, these papers [20– 22] linked the spectral properties of the one-edge Hamiltonians to the existence of edge currents through the use of the Mourre commutator method. We discuss this thoroughly in Sec. 5. The main interest in spectral properties is due to the fact that these authors prove that under weak perturbations (relative to B) there is absolutely continuous spectrum in the intervals ∆n . It was pointed out by Exner, Joye and Kovaˇr´ık [23] that absolutely continuous spectrum and edge currents can appear when the edge is simply an inﬁnite array of point interactions. These authors studied the Hamiltonian (1.2) for which V0 (x) = j∈Z αδ(x − j), and proved that

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there are bands of absolutely continuous spectra between the Landau levels and that the Landau levels remain inﬁnitely degenerate. More recently, Buchendorfer and Graf [24] developed a scattering theory for edge states in one-edge geometries. These authors show that edge states acquire a phase due to a bend in the boundary relative to a state propagating along a straight boundary. This work has some similarities with the material in Sec. 4. 1.2. Contents The content of this paper is as follows. Section 2 is devoted the proofs of the edge current estimates for the case of a Sharp Conﬁning Potential and a straight edge. In Sec. 3, we extend these results to the case of Dirichlet boundary conditions along the straight edge. Section 4 is devoted to considering more general boundaries. We introduce the notion of asymptotic edge currents and use scattering theory to prove the stability of these currents. Spectral properties of the Hamiltonians associated with one-edge geometries are studied in Sec. 5 using the Mourre commutator method. In Sec. 6, we extend the results to soft conﬁning potentials. The paper concludes with two appendices. The ﬁrst appendix, Appendix A, presents results on the dispersion curves needed in the proofs. Appendix B, of independent interest, provides explicit pointwise upper and lower bounds on solutions to a certain form of second-order ordinary diﬀerential equations. 2. The Straight Edge and a Sharp Confining Potential In this section, we prove an explicit lower-bound on the edge current formed by a sharp conﬁning potential V0 (x) = V0 χ{x<0} (x) along the straight edge x = 0. The nonperturbed, one-edge geometry Hamiltonian H0 = HL (B) + V0 , is a nonnegative, self-adjoint operator on D(HL (B)). We write E0 (·) for the spectral family of H0 . If a classical electron has energy below V0 , then the corresponding classical Hamiltonian describes the dynamics of the particle in the half-plane x > 0, the classically allowed region. The complementary region is the classically forbidden region for an electron with energy less than V0 . The edge x = 0 reﬂects the cyclotron orbits of these electrons and causes a net drift of the electron along the edge. This is the origin of the edge current. We will later treat a general family of perturbations V1 , and prove the persistence of edge currents, provided V1 ∞ is small enough relative to B (and without assuming that V1 is diﬀerentiable as required by some commutator methods). As discussed in Sec. 5, similar results for more restrictive potentials V1 can be derived from commutator estimates, as obtained by DeBi`evre and Pul´e [21], and by Fr¨ ohlich, Graf and Walcher [22]. 2.1. The main results for the unperturbed case Our main result is an explicit lower-bound on the size of the edge current for halfplane in certain states for the unperturbed Hamiltonian H0 . In order to formulate

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the main theorem, we need to describe these states. Because the edge is straight, we can use the Fourier transform with respect to the y-variable to reduce the problem to a one-dimensional one. The unperturbed operator H0 admits a partial Fourier decomposition with respect to the y-variable, and the Hilbert space L2 (R2 ) can be expressed as a constant ﬁber direct integral over R with ﬁbers L2 (R). For H0 , we write ⊕ h0 (k)dk, (2.1) H0 = R

where h0 (k) = p2x + (k − Bx)2 + V0 (x),

on L2 (R).

(2.2)

As in Sec. 1, we write ϕj (x; k) and ωj (k) for the normalized eigenfunctions and the corresponding eigenvalues. The eigenvalues are nondegenerate (cf. Appendix A) and, consequently, we choose the eigenfunctions ϕj to be real. These eigenfunctions form an orthonormal basis of L2 (R), for any k ∈ R. Because the map k → h0 (k) is operator analytic, the simple eigenvalues ωj (k) are analytic functions of k. We are interested in states that are energy localized in intervals ∆n lying between two consecutive Landau levels, that is ∆n ⊂ (En (B), En+1 (B)). Consider a state ψ having the property that ψ = E0 (∆n )ψ. For such a state ψ, we can take the Fourier transform of ψ with respect to y and, using an eigenfunction expansion, write n 1 ψ(x, y) = √ eiky χω−1 (∆n ) (k)βj (k)ϕj (x; k)dk, j 2π j=0 R

(2.3)

where the coeﬃcients βj (k) are deﬁned by ˆ k), ϕj (·; k), βj (k) ≡ ψ(·,

(2.4)

where the partial Fourier transform is deﬁned in (1.7). The normalization is such ψ2L2 (R2 ) =

n j=0

ωj−1 (∆n )

|βj (k)|2 dk.

(2.5)

Throughout the paper, we will take the interval ∆n ⊂ (En (B), En+1 (B)) to be given by ∆n = [(2n + a)B, (2n + c)B],

for 1 < a < c < 3.

(2.6)

We can now state the main theorem for the unperturbed, single straight edge Hamiltonian H0 with a sharp conﬁning potential. Theorem 2.1. For n ≥ 0, let ∆n be as in (2.6), and suppose that V0 > (2n + 3)B. Let E0 (∆n ) be the spectral projection for H0 and the interval ∆n . Let ψ ∈ L2 (R2 )

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be a state satisfying ψ = E0 (∆n )ψ with an expansion as in (2.3)–(2.5). Then, for c − a > 0 suﬃciently small, if n ≥ 1, so that condition (2.14) is satisﬁed, we have, 1/2 n π 1 |βj (k)|2 − ψ, Vy ψ ≥ 4 −1 2 (n + 1)2 [H(n) ]2 B 7 ω (∆ ) n j j=0 ωj (k) × 1− (2.7) (ωj (k) − En (B))2 (En+1 (B) − ωj (k))2 dk, V0 where the constant H(n) is deﬁned in (2.39). Let us note a simpliﬁcation of the above expression under reasonable conditions. For k ∈ ωj−1 (∆n ), j = 0, . . . , n, we have (ωj (k) − En (B))2 ≥ B 2 (a − 1)2 ,

(En+1 (B) − ωj (k))2 ≥ B 2 (3 − c)2 .

(2.8)

Corollary 2.1. Let us suppose that V0 > (2n + 3)B, for n ≥ 0, is such that for k ∈ ωj−1 (∆n ), we have ωj (k) 1 (2.9) 1− > . V0 2 Then, under this condition, the hypotheses of Theorem 2.1, and recalling (2.8), the edge current satisﬁes the bound − ψ, Vy ψ ≥

π 1/2 (a − 1)2 (3 − c)2 1/2 B ψ2 . 25 (n + 1)2 [H(n) ]2

(2.10)

This result shows that any state with energy between En (B) and En+1 (B) carries an edge current. However, as the energy approaches a Landau level, the state may delocalize away from the edge. 2.2. Proof of Theorem 2.1 In order to prove Theorem 2.1, we note that from the representation (2.3), the matrix element of the edge current can be written as

ψ, Vy ψ = Mn (ψ) + En (ψ), where the main term Mn (ψ) is given by n Mn (ψ) ≡ χω−1 (∆n ) (k)|βj (k)|2 ϕj (·; k), (k − Bx)ϕj (·; k)dk, j=0

R

j

(2.11)

(2.12)

and En (ψ) is the error term involving the cross-terms between diﬀerent Landau levels: En (ψ) ≡

n j=l;j,l=0

R

χω−1 (∆n ) (k)χω−1 (∆n ) (k)β¯l (k)βj (k) ϕl (·; k), (k − Bx)ϕj (·; k)dk. l

j

(2.13) Concerning this term, we have the following result.

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Lemma 2.1. Suppose ∆n ⊂ (En (B), En+1 (B)) has the form given in (2.6). Under the conditions described above, if c − a is suﬃciently small so that condition (2.14) is satisﬁed, then the error term (2.13) for the unperturbed problem is zero: En (ψ) = 0. Proof. The vanishing of En (ψ) follows from the fact that σjl ≡ ωl−1 (∆n ) ∩ ωj−1 (∆n ) = ∅, for j = l and for |∆n | suﬃciently small. Each dispersion curve ωj (k) is strictly monotone decreasing as follows from the representation (1.12), together with the formula (2.16) in Proposition 2.1 and the bound in Lemma 2.3. Furthermore, due to the simplicity of the spectrum of h0 (k) (see Proposition 2.1) the dispersion curves never intersect. Let us suppose that ωj (k) < ωl (k), and let klc be the unique point satisfying ωl (k) = (2n + c)B. Now, it is easy to check that the condition that guarantees that σjl = ∅ is that ((2n + c)B − ωj (klc )) ≥ ((2n + c)B − (2j + 1)B − V0 > (c − a)B.

(2.14)

Since the right side of (2.14) can be made small by taking a close to c, whereas the left-hand side is independent of a, this proves the result. We note that even when the sets σjl are nonempty, the eigenfunctions of the reduced Hamiltonians h0 (k) are spatially localized so that the error term En (ψ) is exponentially small. We therefore have to estimate the main term in (2.11). It is clear that we need to control the matrix element of Vˆy = (k − Bx) in the states ϕj (x; k). The following formal commutator expression plays an important role in the calculation of the current in these eigenstates: −i 1 [px , h0 (k)] + V , Vˆy = (k − Bx) ≡ 2B 2B 0

(2.15)

where V0 is interpreted in the distributional sense. As a ﬁrst step, we note the following basic result that follows from analyticity, the Virial Theorem, the existence of ϕj (0; k) as proved in Proposition A.1, and the expression (2.15). Proposition 2.1. Let ϕj (x; k) be an eigenfunction of h0 (k), with eigenvalue ωj (k). We have V0 ϕj (0; k)2 . (2.16)

ϕj (·; k), Vˆy ϕj (·; k) = − 2B Recall that the matrix element in (2.16) is equal to (1/2)ωj (k). So the problem is to estimate the slope ωj (k) of the dispersion curves from below for k ∈ ωj−1 (∆n ), for j = 1, . . . , n. In light of this estimate, the main term of the edge current in (2.11) can be written as n 1 |βj (k)|2 (V0 ϕj (0; k)2 )dk. (2.17) Mn (ψ) ≡ − 2B j=0 ωj−1 (∆n )

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Our next step is to obtain a lower bound on the trace of the eigenfunction on the edge, so as to be able to estimate V0 ϕj (0; k)2 from below. This will require several steps. STEP 1: Eigenfunction Estimate For the normalized real eigenfunction ϕj (x; k), we deﬁne, for any δ ≥ 0, ηj (δ) ≡ ϕj (−δ; k)2 .

(2.18)

We now obtain exponential decay results on ηj (δ) as δ → ∞. An ordinary diﬀerential equation method allows one to obtain a precise form of the prefactor. Theorem 2.2. Let ϕj (x; k) be the normalized real eigenfunction of h0 (k), deﬁned above, with corresponding eigenvalue ωj (k). Then, for any δ > 0, and for all k ∈ R so that 0 ≤ ωj (k) < V0 , we have √ (2.19) ηj (δ) ≤ ηj (0)e− 2(V0 −ωj (k))δ . Proof. 1. The idea of the proof is to obtain a good lower bound on ηj (δ) and to integrate the result. We refer the reader to Appendix A, Proposition A.1, on the diﬀerentiability of ϕj (x; k). The ﬁrst derivative of ηj (δ) with respect to δ is easily computed ηj (δ) = −2∂x ϕ(−δ; k) ϕ(−δ; k) whence ηj (δ) − 2

−δ

−∞

(∂t2 ϕ)(t; k)ϕ(t; k)dt +

−δ

−∞

(∂t ϕ)(t; k)2 dt .

(2.20)

We use the eigenvalue equation h0 (k)ϕj = ωj (k)ϕj to re-express ∂t2 ϕj for t < 0 as ∂t2 ϕj (t; k) = (k − Bt)2 ϕj (t; k) + (V0 − ωj (k))ϕj (t; k). Substituting this into (2.20), we obtain, −δ −δ 1 2 ϕj (t; k) dt + (∂t ϕj )(t; k)2 dt − ηj (δ) = (V0 − ωj (k)) 2 −∞ −∞ −δ + (k − Bt)2 ϕj (t; k)2 dt.

(2.21)

(2.22)

−∞

2. We now take the derivative with respect to δ of the terms in (2.22). This gives 1 η (δ) = (V0 − ωj (k))η(δ) + (∂x ϕj )(−δ; k)2 2 j + (k + Bδ)2 ϕj (−δ; k)2 .

(2.23)

Since the last two terms on the right-hand side of (2.23) are nonnegative, we have ηj (δ) ≥ 2(V0 − ωj (k))ηj (δ).

(2.24)

As ηj obviously converges to zero at inﬁnity, it follows from (2.24) that ηj (δ) ≤ 0 for any δ ∈ R+ . So multiplying (2.24) by ηj (δ) and integrating along [t, +∞) for

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2

any t ≥ 0 yields ηj (t) ≥ 2(V0 − ωj (k))ηj2 (t). By integrating along [0, δ], for any δ ≥ 0, one ﬁnally obtains (2.19). STEP 2: Harmonic Oscillator Eigenfunction Comparison It is useful to compare the eigenfunctions of h0 (k) to those of the harmonic oscillator Hamiltonian with no conﬁning potential. The harmonic oscillator Hamiltonian hB (k) on L2 (R) is deﬁned as hB (k) ≡ p2x + (k − Bx)2 .

(2.25)

The eigenvalues of this operator are precisely the Landau energies Em (B) and are nondegenerate and independent of k. We will denote the real normalized eigenfunctions by ψm (x; k). These are given by 1/4 √ √ 2 B B 1 e− 2 (x−k/B) Hm (x B − (k/ B)), (2.26) ψm (x; k) = √ m 2 m! π where Hm (u) is the normalized Hermite polynomial with H0 (u) = 1. We expand the eigenfunctions ϕj (x; k) in terms of these eigenfunctions ϕj (x; k) =

∞

α(j) m (k)ψm (x; k),

(2.27)

m=0

where the coeﬃcients are given by α(j) m (k) = ϕj (·; k), ψm (·; k),

(2.28)

and satisfy 2

ϕj (·; k) =

∞

2 |α(j) m (k)| = 1.

(2.29)

m=0 (j)

We occasionally suppress the variable k in the notation and write αm for these coeﬃcients. Lemma 2.2. Let Pn (k) be the projection on the eigenspace spanned by the ﬁrst n (j) eigenfunctions ψm of the harmonic oscillator Hamiltonian hB (k) (2.25). Let αm −1 be the expansion coeﬃcients deﬁned in (2.28). For all k ∈ ωj (∆n ), with ∆n as in (2.6), and for all j = 0, 1, . . . , n, we have n

2 |α(j) m (k)| ≥

m=0

1 (En+1 (B) − ωj (k)) > 0, 2B(n + 1)

(2.30)

and | ϕj (·; k), V0 Pn (k)ϕj (·; k)| ≥

1 (ωj (k) − En (B))(En+1 (B) − ωj (k)) > 0. 2B(n + 1) (2.31)

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Proof. 1. We compute the matrix element ϕj , V0 ϕj using the expansion (2.27),

ϕj , V0 ϕj = ϕj , (h0 (k) − hB (k))ϕj 2 = (ωj (k) − Em (B))|α(j) m (k)| ,

(2.32)

m≥0

using the normalization (2.29). Rearranging the terms in (2.32), we ﬁnd 2 (ωj (k) − Em (B))|α(j) m (k)| = ϕj , V0 ϕj m≤n

+

2 (Em (B) − ωj (k))|α(j) m (k)|

m≥n+1



≥ (En+1 (B) − ωj (k))1 −

 2 . |α(j) m (k)|

m≤n

(2.33) We now assume that k ∈ ωj−1 (∆n ) and j ≤ n. In this case, the coeﬃcient En+1 (B)− ωj (k) > 0. Moving the second term on the right-hand side of (2.33) to the left, we obtain 2 (ωj (k) − Em (B) + En+1 (B) − ωj (k))|α(j) (En+1 (B) − ωj (k)) ≤ m (k)| m≤n

=

2 (En+1 (B) − Em (B))|α(j) m (k)|

m≤n



≤ 2(n + 1)B

 2 . |α(j) m (k)|

(2.34)

m≤n

The result (2.30) follows from (2.34). 2. The calculation of ϕj (·; k), V0 Pn (k)ϕj (·; k), for k ∈ ωj−1 (∆n ), is similar. We write

ϕj (·; k), V0 Pn (k)ϕj (·; k) = ϕj (·; k), (h0 (k) − hB (k))Pn (k)ϕj (·; k) 2 = (ωj (k) − Em (B))|α(j) m (k)| m≤n

≥ (ωj (k) − En (B))

2 |α(j) m (k)|

m≤n

≥

where we used (2.30).

1 (ωj (k) − En (B))(En+1 (B) − ωj (k)), 2B(n + 1) (2.35)

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STEP 3: Lower Bound on the Trace We now use the eigenfunction estimate of Step 1 and the lower bound of Step 2 in order to express the matrix element ϕj (·; k), V0 Pn (k)ϕj (·; k) in terms of the trace of ϕj on the edge. Lemma 2.3. Let ϕj (x; k) be an eigenfunction of h0 (k), as above, for 0 ≤ j ≤ n. Then, for all k ∈ ωj−1 (∆n ), we have V02 ϕj (0; k)2 1/2 π [V0 − ωj (k)] ≥ (ωj (k) − En (B))2 (En+1 (B) − ωj (k))2 , B 8B 2 (n + 1)2 [H(n) ]2 (2.36) where H(n) is deﬁned in (2.39). Proof. We use the expansion of ϕj in the eigenfunctions ψm and obtain 0 V0 α(j) (k) ϕj (x; k)ψm (x; k)dx.

ϕj (·; k), V0 Pn (k)ϕj (·; k) = m m≤n

(2.37)

−∞

To estimate the integral, we use the exponential decay of the eigenfunctions ϕj as given in Theorem 2.2. For x < 0, the main eigenfunction decay estimate (2.19) gives √ (2.38) ϕj (x; k)2 ≤ ϕj (0; k)2 e− 2(V0 −ωj (k))|x| . We recall that ψm (x; k) is given in (2.26), and deﬁne the coeﬃcients  2

Hm ≡ sup Hm (u)e−u

/2

and H(n) ≡ 

u∈R

1/2

1 H2  2m m! m

m≤n

The integral can be bounded above by 0 ≤ C ϕ (·; k)ψ (x; k)dx |ϕ (0; k)|H j m m j m −∞

0

≤ where Cm (B) ≡

B 1/4 π

∞

e−

.

(2.39)

√

2(V0 −ωj (k))x

21/2 Cm (B)|ϕj (0; k)|Hm

, (V0 − ωj (k))

dx (2.40)

(2m m!)−1/2 . From (2.37) and (2.40), we get

| ϕj (·; k), V0 Pn (k)ϕj (·; k)|   1/4 1/2 B 2 V0 |ϕj (0; k)|  1 

√ ≤ Hm |α(j) m (k)| . π (V0 − ωj (k)) m≤n 2m m!

(2.41)

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Applying the Cauchy–Schwarz inequality to the sum in (2.41), and recalling the normalization (2.29), we ﬁnd that 1/4 1/2 B 2 V0 |ϕj (0; k)|H(n) . (2.42) | ϕj (·; k), V0 Pn (k)ϕj (·; k)| ≤ π [V0 − ωj (k)]1/2 We square expression (2.42), and use the bound (2.31) in Lemma 2.4, to obtain the result (2.36). The proof of Theorem 2.1 now follows directly from the expression for the main term Mn (ψ) in (2.17) and the lower bound for the expression V0 ϕj (0; k)2 given in Lemma 2.3. Corollary 2.1 follows directly from the lower bound on the main term. 2.3. Perturbation theory for the straight edge We now consider the perturbation of H0 by a bounded potential V1 (x, y). We prove that the lower bound on the edge current is stable with respect to these perturbations provided V1 ∞ is not too large compared with B. Let ∆n be as in ˜ n , containing ∆n , with the same midpoint (2.6). We consider a larger interval ∆ En = (2n + (a + c)/2)B ∈ ∆n , and of the form ˜ n = [(2n + a ∆ ˜)B, (2n + c˜)B],

for 1 < a ˜ < a < c < c˜ < 3.

(2.43)

In this perturbation argument, we calculate the velocity Vy in states ψ ∈ ˜ n )L2 (R2 ). This closeness is meaE(∆n )L2 (R2 ) that are close to states in E0 (∆ sured by the constant κ > 0 that we now deﬁne. First, we choose the constants a ˜ and c˜ in (2.43) so that c˜ − a ˜ is small enough for Theorem 2.1 to hold for states in ˜ n )L2 (R2 ). Next, we choose a constant Bn > 0 large enough and the constants E0 (∆ a and c, with c − a small enough, so that for all B > Bn , the constant κ deﬁned by 2 2 2 c − a V1 ∞ 2 κ ≡ 1− + , (2.44) c˜ − a ˜ 2 B satisﬁes 0 < κ ≤ 1. Note that if (2.14) holds for B1 , then it holds for all B > B1 since (2j + 1)B < ωj (k) < (2j + 1)B + V0 . Theorem 2.3. Let V1 (x, y) be a bounded potential and let E(∆n ) be the spectral projection for H = H0 + V1 and the interval ∆n as in (2.6). Let ψ ∈ L2 (R2 ) be ˜ n )ψ and ξ ≡ E0 (∆ ˜ cn )ψ, so that a state satisfying ψ = E(∆n )ψ. Let φ ≡ E0 (∆ ψ = φ + ξ. Under the conditions given above on a, c, a ˜, c˜, and for B > Bn , the constant κ, deﬁned in (2.44), satisﬁes 0 < κ ≤ 1 and we have φ ≥ κψ.

(2.45)

Furthermore, we have the lower bound a − 1)2 − F (n, V1 /B))ψ2 , − ψ, Vy ψ ≥ B 1/2 κ2 (Cn (3 − c˜)2 (˜

(2.46)

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∼ ∆n

87

∆n ωn

−1

ωn−1 (∆n)

ωn−1 ω−1 n (∆n) ∼ ωn−1(∆n) −1

Fig. 2.

0

−1 ∼ ωn (∆n)

k

¯ n ), j = n − 1, n. Spectral intervals ωj−1 (∆n ) and ωj−1 (∆

where the constants are deﬁned by Cn =

π 1/2 , 25 (n + 1)2 [H(n) ]2

(2.47)

and 2 1/4

F (n, V1 ∞ /B) = (1 − κ )

1/2

V1 ∞ (2 + 1 − κ2 ) 2n + c + B

a − 1)2 . + Cn (1 − κ2 )(3 − c˜)2 (˜

(2.48)

If we suppose that V1 ∞ < µ0 B, then for a ﬁxed level n, if c − a and µ0 are suﬃciently small (depending on a ˜, c˜, and n), there is a constant Dn > 0 so that for all B, we have − ψ, Vy ψ ≥ Dn κ2 B 1/2 ψ2 .

(2.49)

Proof. With reference to the deﬁnitions (2.6) and (2.43), we write the function ψ as ˜ n )ψ + E0 (∆ ˜ cn )ψ ≡ φ + ξ. ψ = E0 (∆

(2.50)

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We then have

ψ, Vy ψ = φ, Vy φ + 2 Re φ, Vy ξ + ξ, Vy ξ.

(2.51)

The result follows from Theorem 2.1 provided we have a good bound on ξ and on Vy ξ. We ﬁrst note that ˜ cn )(H0 − En )−1 (H − En )ψ + E0 (∆ ˜ cn )(H0 − En )−1 V1 ψ ξ ≤ E0 (∆ 2 (c − a) V1 + ≤ ψ. (2.52) c˜ − a ˜ 2 B The bound (2.45) follows from (2.52) and the orthogonality of φ and ξ. Similarly, we ﬁnd that Vy ξ2 ≤ ξ, H0 ξ ≤ | ψ, Hξ| + V1 ξ ψ ≤ ((2n + c)B + V1 ) ξ ψ.

(2.53)

Combining (2.52) and (2.53), we obtain | ξ, Vy ξ| 3/2 3/2 1/2 2 (c − a) V1 V1 ≤ + B 1/2 ψ2 , 2n + c + c˜ − a ˜ 2 B B (2.54) and | φ, Vy ξ| 1/2 1/2 1/2 2 (c − a) V1 V1 1/2 ≤ 2n + c + B ψ2 . + c˜ − a ˜ 2 B B (2.55) The lower bound on the main term in (2.51) follows from (2.10) and (2.43),   1/2 n π (˜ a − 1)2 (˜ c − 3)2 |βj (k)|2 dk  − φ, Vy φ ≥ B 1/2  −1 ˜ 25 (n + 1)2 [H(n) ]2 ω ( ∆ ) n j j=0 1/2 2 2 π (˜ a − 1) (˜ c − 3) (2.56) B 1/2 (ψ2 − ξ2 ). = 25 (n + 1)2 [H(n) ]2 Combining this lower bound (2.56), with the estimate on ξ in (2.52), and the bounds (2.53)–(2.55), we ﬁnd (2.46) with the constants (2.47) and (2.48). This completes the proof. We remark that if the state ψ ∈ E(∆n )L2 (R2 ) has the property that the corresponding φ = 0, then the right-hand side of (2.56) is zero. It follows from (2.45), however, that if the interval ∆n is small enough, and if the magnetic ﬁeld is large enough, then this cannot happen.

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2.4. Localization of the edge current It follows from the calculations done above that the edge current carried by states ψ of the unperturbed Hamiltonian H0 satisfying ψ = E0 (∆n )ψ are localized within a region of size O(B −1/2 ) near the edge x = 0. This corresponds to the classical cyclotron radius. This is made precise in the following theorem. Theorem 2.4. Let ψ be a normalized edge-current carrying state, i.e. ψ = E0 (∆n )ψ, with ψ = 1. We assume that the interval ∆n as in (2.6) satisﬁes |∆n |/B small, and that V0 > (2n + 3)B, as in Theorem 2.3. Then, for any level n and any real numbers α > −1/2 and β > 0, there are three constants Bn,α,β > 0, Cn,α,β > 0, and Kn,α,β > 0, independent of B, such that 2α+1 dy dx|ψ(x, y)|2 ≤ Cn,α,β e−Kn,α,β B , (2.57) R\[−B −β ,B α ]

R

for all B ≥ Bn,α,β and V0 ≥ (2n + c)B + B 2(2α+β+1) . Proof. Set Iα,β = [−B −β , B α ]. In light of the expansion (2.3)–(2.5), and the normalization of ψ, we have n 2 2 dy dx|ψ(x, y)| = dk|βj (k)| dx|ϕj (x; k)|2 . R

R\Iα,β

j=1

ωj−1 (∆n )

R\Iα,β

(2.58) Hence, it suﬃces to prove that the integrals −B −β ϕj (x; k)2 dx and

+∞

Bα

−∞

ϕj (x; k)2 dx,

(2.59)

are bounded above as in (2.57) for all j = 0, 1, . . . , n and k ∈ ωj−1 (∆n ). The proof consists in four steps. ˜n,δ > 0 such that we have Step 1. For all δ > 1/2 there is a constant B ωj−1 (∆n ) ⊂ (−∞, B δ ),

˜n,δ . for all B ≥ B

(2.60)

To see this, we consider a C 2 (R) function J satisfying J(x) = 0 for x ≤ 0, and J(x) = 1 for x ≥ B −1/2 , with J ∞ ≤ C1 B 1/2 , and J ∞ ≤ C2 B, for two ﬁnite constants C1 , C2 > 0. For all k ∈ R, Jψn (.; k), where ψn (x; k) is given in (2.26), belongs to the domain of h0 (k), and we have (h0 (k) − (2n + 1)B)J(x)ψn (x; k) = [h0 (k), J]ψn (x; k) = −2iJ (x)ψn (x; k) − J (x)ψn (x; k), through a direct computation. Moreover, the function J being supported in [0, B −1/2 ], it follows from this that (h0 (k) − (2n + 1)B)Jψn (.; k) √ ≤ 2C1 BχB ψn (.; k) + C2 BχB ψn (.; k),

(2.61)

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where χB is the characteristic function of [0, B −1/2 ]. Next, for all k ≥ B δ (2.26) > 0, Cn,δ > 0 and Kn,δ > 0 such that assures us there are three constants Bn,δ

χB ψn (.; k) + χB ψn (.; k) ≤ Cn,δ e−Kn,δ B

2δ−1

,

for B ≥ Bn,δ .

This, combined with (2.61) show that |ωn (k) − (2n + 1)B| can be made smaller than (a − 1)B by taking B suﬃciently large. This proves (2.60). Step 2. Let γ be in (−1/2, +∞). The normalization condition ϕj (.; k) = 1 involv Bγ ing B γ /2 ϕ2j (x; k)dx ≤ 1, there is necessarily some x0 = x0 (B, γ) in [B γ /2, B γ ] such that γ −1/2 B ≤ 2B 1/4 . (2.62) ϕj (x0 ; k) ≤ 2 > 0 such that this In light of (2.60), we may also ﬁnd δ in (1/2, γ + 1) and Bn,γ −1 δ . As x0 together with all k ∈ ωj (∆n ) are bounded above by B for all B ≥ Bn,γ 2 2 2 a consequence, we have Wj (x; k) = (Bx − k) − ωj (k) ≥ B (x − x0 ) > 0 and , and hence Wj (x; k) = 2B 2 (x − k/B) > 0 for all x > x0 and B ≥ Bn,γ 2

ϕj (x; k) ≤ ϕj (x0 ; k)e−B/2(x−x0 ) ,

for x ≥ x0

and B ≥ Bn,γ ,

by Lemma B.3 in Appendix B. Bearing in mind (2.62) and recalling that x0 ≤ B γ , this entails ϕj (x; k) ≤ 2B 1/4 e−B/2(x−B

γ 2

)

,

for x ≥ B γ

and B ≥ Bn,γ .

(2.63)

Step 3. Let α be in (−1/2, +∞) and set γ = (α − 1/2)/2. We insert (2.63) in the second integral in (2.59) and get +∞ α γ 2 ϕj (x; k)2 dx ≤ Pn,α (B)e−B/2(B −B ) , for B ≥ Bn,γ , Bα

where Pn,α (B) is a polynomial function of B. This yields that there are three constants Bn,α > 0, Cn,α > 0 and Kn,α ∈ (0, 1) such that +∞ 2α+1 ϕj (x; k)2 dx ≤ Cn,α e−Kn,α B , for B ≥ Bn,α . (2.64) Bα

Step 4. We turn now to estimating the ﬁrst integral in (2.59) for some β > 0. As above, we refer to the normalization condition ϕj (.; k) = 1 to justify the existence of some x1 ∈ (−B −β /2, 0) satisfying √ (2.65) ϕj (x1 ; k) ≤ 2B β/2 . odinger equation ϕj (x; k) = Next, ϕj (.; k) being solution of the Schr¨ Wj (x; k)ϕj (x; k) we choose V0 > (2n + c)B so that Wj (x; k) = (Bx − k)2 + V0 (x) − ωj (k) > 0 for x < 0, and apply Lemma B.3 in Appendix B once more. We get: √ ϕj (x; k) ≤ ϕj (x1 ; k)e V0 −(2n+c)B(x−x1 ) , for x ≤ x1 .

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Since x1 ≥ −B −β /2, this, together with (2.65), entail

−B −β

−∞

ϕj (x; k)2 dx ≤ e−B

2α+1

,

for all α > −1/2, B ≥ 1 and V0 > (2n + c)B + B 2(2α+β+1) . Now, (2.57) follows from this and from (2.64). We now extend this result to the perturbed case. We assume that the conditions guaranteeing the existence of edge current-carrying states for the perturbed Hamiltonian are satisﬁed. In particular, this means that the perturbation V1 satisﬁes a ˜ n lies in the spectral bound V1 ∞ < ν0 B, and that c˜ − ˜a is small enough so that ∆ gap of the bulk Hamiltonian Hbulk = HL (B)+ V1 in the interval (En (B), En+1 (B)). We refer the reader to [3, 5] for a discussion of the properties of Hbulk . Under these conditions, the edge current for the perturbed Hamiltonian remains close to the wall for all time in a strip of width B −α , for any α < 1/2, essentially the cyclotron radius. For any 0 < L0 < ∞, we deﬁne a spatial truncation function 0 ≤ J0 ≤ 1 to be J0 (x) = 0, for x < L0 and J0 (x) = 1 for x > L0 + 1. Theorem 2.5. Consider the perturbed operator H = H0 + V1 with V1 ∞ < ν0 B, ˜ n = [(2n + a for some constant 0 < ν0 < ∞. Let ∆n ⊂ ∆ ˜)B, (2n + c˜)B] lie in the spectral gap of the bulk Hamiltonian Hbulk = HL (B) + V1 in (En (B), En+1 (B)). Let ψ = E(∆n )ψ ∈ L2 (R2 ) be an edge current carrying state so that the results of ˜ are small Theorem 2.3 hold true. In particular, we assume that ν0 and that c˜ − a enough so that the lower bound (2.49) is valid. Then, for any level n, and for any 0 < α < 1/2, there exist constants 0 < Cn , Kn < ∞, independent of B, so that for a strip of width L0 = B −α , we have J0 ψ ≤ Cn e−Kn B

1/2−α

.

(2.66)

Proof. The method of proof is similar to that given in [25]. The resolvent formula for Hbulk and H gives R(z) = Rbulk (z) − Rbulk (z)V0 R(z).

(2.67)

˜ n. Let 0 ≤ f ≤ 1 be a smooth, nonnegative function with f|∆n = 1 and supp f ⊂ ∆ Then, we can write ψ = f (H)ψ. We use the Helﬀer–Sj¨ostrand formula for the operator f (H), cf. [25] or [26]. Let f˜ be an almost analytic extension of f into a ˜ n that vanishes of order two as Im (z) → 0. The small complex neighborhood of ∆ Helﬀer–Sj¨ ostrand formula for f (H) is −1 ∂z¯f˜(z)(H − z)−1 dxdy. (2.68) f (H) = π C Note that since the support of f lies in the spectral gap of Hbulk , formula (2.68) shows that f (Hbulk ) = 0. Then, by the resolvent formula (2.67), and the

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Helﬀer–Sj¨ ostrand formula (2.68), we can write J0 ψ = J0 f (H)ψ −1 = ∂z¯f˜(z)J0 Rbulk (z)V0 R(z)dxdy. π C

(2.69)

The distance between the supports of the conﬁning potential V0 and the localization function J0 is 0 < L0 < ∞. An application of the Combes–Thomas method to Landau Hamiltonians as presented, for example, in [3], results in the following bound for the operator J0 Rbulk (z)V0 for z in the resolvent set of Hbulk . There are constants 0 < C1 , C2 < ∞ so that J0 Rbulk (z)V0 ≤

1/2 C1 e−C2 B L0 . d(σ(Hbulk ), z)

(2.70)

The distance d(σ(Hbulk ), z) is given by the minimum of the distance from the ˜ n to the band edges of the spectrum of Hbulk at En (B) + V1 ∞ larger interval ∆ and En+1 (B) − V1 ∞ . Consequently, if L0 = B −α , for α < 1/2, we obtain the result. 3. The Straight Edge and Dirichlet Boundary Conditions We note that the lower bounds on the edge currents in Theorems 2.1 and 2.3 are independent of the size of the conﬁning potential barrier V0 , provided V0 En+1 (B). This indicates that these lower bounds should remain valid in the limit V0 → ∞. This limit formally corresponds to Dirichlet boundary conditions along the edge at x = 0. In this section, we use the results of Secs. 2.1 and 2.3 to prove lower bounds on the edge current with Dirichlet boundary conditions (DBC) along x = 0. DeBi`evre and Pul´e [21] and Fr¨ ohlich, Graf and Walcher [22] both considered the Landau Hamiltonian with Dirichlet boundary conditions along the edge x = 0 in their articles. Both groups proved the existence of edge currents using the commutator method described in Sec. 5. DeBi`evre and Pul´e [21] avoid the minor technical diﬃculty encountered by Fr¨ ohlich, Graf and Walcher [22] due to the nonselfadjointness of px on a half line by using y as a conjugate operator. We provide an alternate proof of the existence of edge currents in the hard boundary (DBC) case here that does not use commutator estimates. We denote the Landau Hamiltonian HL (B) on the space L2 ([0, ∞) × R) with Dirichlet boundary conditions along x = 0 by H0D . This unperturbed operator admits a direct integral decomposition with respect to the y-variable. We denote D by hD 0 (k) the corresponding ﬁbered operator with eigenvalues ωj (k) and eigenD functions ϕj (x; k). These eigenfunctions provide an eigenfunction expansion of any state, as in (2.3), and we denote the coeﬃcients of this expansion by βjD (k). The eigenfunctions of hD 0 (k) are given explicitly by Whittaker functions. Many properties of the dispersion curves ωjD (k) are derived from the properties and estimates on Whittaker functions, cf. [21]. The perturbed operator is denoted by HD ≡ H0D + V1 , on the same Hilbert space. We let E0D (·) and ED (·) denote the corresponding

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spectral families. As in Sec. 2, the interval ∆n = [(2n + a)B, (2n + c)B], with 1 < a < c < 3. Theorem 3.1. Consider the operators H0D and HD = H0D + V1 , on H ≡ L2 ([0, ∞) × R), with Dirichlet boundary conditions along x = 0. Any state ψ ∈ ED (∆n )H carries an edge current satisfying the lower bounds (2.46), with the same constants (2.47), (2.48), provided (c − a) and V1 ∞ /B are suﬃciently small as stated there. We prove this theorem through a perturbation argument comparing H0D on L ([0, ∞) × R) with H0 = HL (B) + V0 acting on L2 (R2 ) in the large V0 regime. In this regime of very large V0 , the behavior of eigenfunctions with eigenvalues in a ﬁxed energy interval for x < 0 becomes unimportant. We begin with an estimate on the trace of the eigenfunctions ϕj (x; k) of h0 (k) on the line x = 0. 2

Lemma 3.1. Let ϕj (x; k) be a normalized eigenfunction of h0 (k) as in Sec. 2. For any 0 ≤ j ≤ n, and for all k ∈ ωj−1 (∆n ), we have 0 ≤ ϕj (0; k) ≤

2B V0

1/2

[(2n + 3)B]1/4 .

In general, for any eigenfunction ϕl (x; k), and for any k ∈ R, we have 1/2 1/2 2B 2B 1/4 0 ≤ ϕl (0; k) ≤ ωl (k) ≤ [(2l + 1)B + V0 ]1/4 . V0 V0

(3.1)

(3.2)

Proof. One can choose ϕj (x; k) ≥ 0, for x < 0, as discussed in Appendix B, Proposition B.1. From Proposition 2.1, and the consequence of the Feynman–Hellmann Theorem (1.9), (1.10), we have ϕj (0; k)2 = −

2B B

ϕj (·; k), Vˆy (k)ϕj (·; k) = − ωj (k) ≥ 0, V0 V0

(3.3)

as we recall that ωj (k) ≤ 0. A simple calculation now gives |ωj (k)| = | ϕj (·; k), h0 (k)ϕj (·; k)| = 2| ϕj (·; k), (k − Bx)ϕj (·; k)| ≤ 2| ϕj (·; k), (k − Bx)2 ϕj (·; k)|1/2 ≤ 2ωj (k)1/2 ≤ 2[(2n + 3)B]1/2 ,

(3.4)

by positivity of the operator h0 (k), and the fact that k ∈ ωj−1 (∆n ). Combining this with (3.3), we obtain the bound (3.1). The bound (3.2) follows from (3.4) and the structure of the dispersion curves. We next show how Lemma 3.1 implies the convergence of the dispersion curves ωj (k) to ωjD (k) as V0 → ∞. We use an estimate on the eigenvalues ωjD (k) of the Dirichlet problem that follows from an estimate in Lemma 2.1 of De Bi`evre and

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Pul´e [21]. The explicit properties of the eigenfunctions ϕj (x; k) allow one to prove that if j = l, then there is a ﬁnite constant Cjl > 0 so that |ωjD (k) − ωlD (k)| ≥ Cjl B,

∀ k ∈ R.

(3.5)

Lemma 3.2. The dispersion curves ωj (k) are monotonic increasing functions of V0 . For V0 En+1 (B), and for j = 0, . . . , n, and for k ∈ ωj−1 (∆n ), we have 0 ≤ ωjD (k) − ωj (k) ≤

C0 (n, B) 1/2

V0

.

(3.6)

Proof. The Hamiltonians h0 (k) are analytic operators in the parameter V0 . We use the Feynman–Helmann Theorem to compute the variation of the eigenvalues ωj (k) with respect to V0 . This gives ∂ωj (k) = ϕj (x; k)2 dx ≥ 0, (3.7) ∂V0 − R so that the dispersion curves are monotone increasing with respect to V0 . Furthermore, the rate of increase in (3.7) slows as V0 → ∞. This follows from the pointwise upper bound on ϕj (x, k) restricted to x ≤ 0. In particular, from (2.38) and the trace estimate (3.1), we have 0≤

∂ωj (k) ≤ ϕj (0; k)2 ∂V0

0

−∞

√ e−2 (V0 −ωj (k))|x| dx

3/2 B (2n + 3)1/2 ≤ . (V0 − ωj (k)) V0

(3.8)

This shows that the dispersion curve ωjD (k) is an upper bound on the dispersion curves ωj (k). To prove the rate of convergence (3.6), we use the eigenvalue equation −ϕj (x) + (k − Bx)2 ϕj (x) = ωj (k)ϕj (x),

x>0

(3.9)

and take the inner product in R+ with the Dirichlet eigenfunction ϕD l . After integration by parts, and an application of the eigenvalue equation for ϕD l , one obtains, D (ωlD (k) − ωj (k)) ϕD l (·; k), ϕj (·; k) = (ϕl ) (0; k)ϕj (0; k).

(3.10)

The estimate in Lemma 3.1 implies that the right-hand side of (3.10) vanishes as V0 → ∞, that is |ωlD (k)

− ωj (k)|

| ϕD l (·; k), ϕj (·; k)|

≤

|(ϕD l ) (0; k)|

2B V0

1/2

[(2n + 3)B]1/2 . (3.11)

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We next show that | ϕD j (·; k), ϕj (·; k)| is uniformly bounded from below as V0 → ∞, proving the convergence of the eigenvalues. To show this, let χ± denote the characteristic functions onto the left and right half lines (−∞, 0] and [0, ∞), respectively. We ﬁrst note that ϕj (·; k)2 = 1 = χ− ϕj (·; k)2 + χ+ ϕj (·; k)2 ,

(3.12)

and the upper bound on the eigenfunction ϕj on the negative half-axis (2.38), −3/4 , so that together with (3.1), imply that χ− ϕj (·; k) ≤ Cj V0 −3/4

χ+ ϕj (·; k) ≥ 1 − O(V0

),

(3.13)

as V0 → ∞ and k ∈ ωj−1 (∆n ). Now, for l = j, it follows from (3.5) and the monotonicity of the dispersion curves in V0 that |ωlD (k) − ωj (k)| ≥ |ωlD (k) − ωjD (k)| ≥ Clj B.

(3.14)

So it follows from this (3.14) and from (3.11) that for l = j

ϕD l (·; k), ϕj (·; k) → 0,

as V0 → ∞.

(3.15)

ϕD j (·; k), ϕj (·; k) family {ϕD l (·; k)}

If, in addition, the matrix element also vanished as V0 → ∞, is an orthonormal basis. It this would contradict (3.13) as the follows that this matrix element must be bounded from below uniformly in V0 as V0 → ∞. Consequently, the dispersion curves must converge as V0 → ∞ with the speciﬁed rate. The local convergence of the dispersion curves to those for the Dirichlet problem is a key ingredient in proving the convergence of the projection Pj (k), for the eigenvalue ωj (k) of h0 (k), to the projector P0D (k), for the eigenvalue ωjD (k) of hD 0 (k), when V0 tends to inﬁnity (with B ﬁxed). The proof relies on the comparison of the −1 , as V0 → ∞, resolvents R0 (z; k) = (h0 (k) − z)−1 and R0D (z; k) = (hD 0 (k) − z) 3/8 D for z ∈ Γj (V0 ), a contour of radius 1/V0 about ωj (k), for 0 ≤ j ≤ n and k ∈ Σn ≡ ∪nj=0 ωj−1 (∆n ). The comparison of the resolvents relies on a formula derived from Green’s theorem and various trace estimates. This is rather standard; we refer, for example, to the discussion in [27]. This is the content of the next lemma. Lemma 3.3. Let Pj (k), respectively PjD (k), for j = 0, . . . , n, be the projection onto the one-dimensional subspace of h0 (k), respectively hD 0 (k), corresponding to the eigenvalue ωj (k), respectively ωjD (k). Then, there exists a ﬁnite constant C1 (n, B) > 0, such that for all V0 suﬃciently large, and uniformly for k ∈ (ωjD )−1 (∆n ) ∪ ωj−1 (∆n ), we have Pj (k) − PjD (k) ≤

C1 (n, B) 1/4

V0

.

(3.16)

We outline the main ideas of the proof here and refer the reader to the archived version for complete details [28]. We are concerned with the ﬁrst (n + 1)-eigenvalues

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ωj (k) of h0 (k), for j = 0, . . . , n. We ﬁx 0 ≤ j ≤ n, and let Γj (V0 ) be the circle 3/8 of radius 1/V0 about ωjD (k). By Lemma 3.2, there is an amplitude V0∗ 1 −1/8

−1/4

, and dist(z, ωj (k)) ≥ V0 , for V0 > V0∗ . so that |ωjD (k) − ωj (k)| < Cn V0 Moreover, there exists an index N (V0 ) n, such that if l > N (V0 ), we have dist(ωl (k), Γj (V0 )) > V0 . The index N (V0 ) can be chosen to be proportional to V0 since ωl (k) is bounded above by (2l + 1)B + V0 . In order to estimate the diﬀerence of the projectors on the left-hand side in (3.16), we use the contour representation of the projections in terms of the resolvents so that the diﬀerence of the projectors is written as 1 D Pj (k) − Pj (k) = (R0 (z; k) − R0D (z; k))dz. (3.17) 2πi Γj (V0 ) The resolvent formula for the diﬀerence of the two resolvents in (3.17) following from Green’s theorem is R0 (z; k) − R0D (z; k) = R0 (z; k)T0∗ B0 R0D (z; k),

(3.18)

where T0 is the trace map (T0 u)(x) = u(0), and (B0 u)(x) = u (0). The trace map is a bounded map from H 1 (R) → C. Substituting (3.18) into the right-hand side of (3.17), we obtain 1 R0 (z; k)T0∗ B0 R0D (z; k)dz. (3.19) PjD (k) − Pj (k) = 2πi Γj (V0 ) Due to the simplicity of the eigenvalues, the resolvent R0 (z; k) has the expression R0 (z; k) =

∞ j=0

Pj (k) , ωj (k) − z

(3.20)

where Pj (k) projects onto the one-dimensional subspace spanned by ϕj (x; k). We have a similar expression for R0D (z; k). In order to exploit the large V0 regime, we decompose any φ ∈ L2 (R) into a piece φL supported on (−∞, 0], and its complement: φ = φL + φR . With this decomposition applied to any φ, ψ ∈ L2 (R), we write the inner product of the diﬀerence of the resolvents as

φ, (R0 (z; k) − R0D (z; k))ψ = φR , (R0 (z; k) − R0D (z; k))ψ R + ELR (z; k). (3.21) The mixed error term ELR has the form ELR (z; k) = φL , R0 (z; k)ψ R + φ, R0 (z; k)ψ L .

(3.22)

The trace is evaluated using the expansion (3.20) and estimates (3.1), (3.2). As a result of some calculations and these estimates, we ﬁnd that (n, B) C 5

φR , (R0 (z; k) − R0D (z; k))ψ R dz ≤ φ ψ. (3.23) 1/4 Γj (V0 ) V 0

Finally, it remains to estimate the error term ELR in (3.22). This is evaluated by substituting the expansion (3.20) into each inner product of ELR . We then separate

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each sum into three sets of indices. For the ﬁrst two sets of indices, 0 ≤ j ≤ n, and n < j ≤ N (V0 ), the half-line x < 0 is in the classically forbidden region for the eigenfunctions ϕj (x; k), with k ∈ ωj−1 (∆n ). For the third set of indices, we have dist(z, ωl (k)) C0 V0 . For the ﬁrst two sets of indices, that is for 0 ≤ l ≤ N (V0 ), it follows from Appendix B that the eigenfunctions ϕl (x; k) satisfy the bound √ ϕl (x; k) ≤ ϕl (0; k)e− V0 −ωl (k)|x| , for x ≤ 0. (3.24) Combining these exponential decay estimates with the trace estimates (3.1), (3.2), we ﬁnd for the contour integral of the error term ELR , C (n, B) 8 ELR (z; k)dz ≤ φψ. (3.25) 5/8 Γj (V0 ) V 0

This estimate, and the estimate (3.23) of the main term prove the result (3.16). Proof of Theorem 3.1. We begin with the unperturbed case. Let ψ ∈ L2 (R+ ×R) satisfy ψ = E0D (∆n )ψ. We assume that the hypotheses of Lemma 2.1 hold so that there are no cross-terms in the matrix element ψ, Vy ψ. We will use the results of Lemma 3.2 that tell us that ωj (k) → ωjD (k), locally, and that the matrix element

ϕD j (·; k), ϕ(·; k) ≥ D0 , as V0 → ∞. We write − ψ, Vy ψ = − ≥−

n j=0

(ωjD )−1 (∆n )

j=0

(ωjD )−1 (∆n )

n

D D D ˆ dk|βjD (k)|2 ϕD j (·; k), Pj (k)Vy (k)Pj (k)ϕj (·; k)

2 dk|βjD (k)|2 | ϕD j (·; k), ϕj (·; k)|

× ϕj (·; k), Pj (k)Vˆy (k)Pj (k)ϕj (·; k) − R(ψ) n ≥ −D0 dk|βjD (k)|2 ϕj (·; k), Pj (k)Vˆy (k)Pj (k)ϕj (·; k) j=0

(ωjD )−1 (∆n )

− R(ψ).

(3.26)

The remainder R(ψ) is bounded by R(ψ) ≤ 2

n j=0

(ωjD )−1 (∆n )

D dk|βjD (k)|2 {| ϕD j (·; k), (Pj (k)

− Pj (k))Vˆy (k)PjD (k)ϕD j (·; k)|}.

(3.27)

The main term in (3.26) is bounded from below as in Theorem 2.1. Estimates on the diﬀerence of the spectral projectors given in Lemma 3.3 establish the appropriate bounds on the remainder R(ψ). This proves the theorem for the unperturbed case. The perturbation theory of Sec. 2.2 now applies in the same manner as in that section.

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4. One-Edge Geometries with More General Boundaries The previous results were based on the exact calculations for the unperturbed case due to the possibility of taking the partial Fourier transform. Fr¨ ohlich, Graf and Walcher [22] considered more general one-edge geometries for which the boundary satisﬁes some mild regularity conditions. We ﬁrst review these results, and then present some new results based on the notion of the asymptotic velocity of edge currents coming from scattering theory. These results apply to a very general class of perturbations of the half-plane geometry. Fr¨ ohlich, Graf and Walcher [22] studied one-edge, simply connected, unbounded regions Ω ⊂ R2 , with a piecewise C 3 -boundary. The boundary must satisfy some additional geometric conditions so that the edge does not asymptotically become parallel to itself so that the region resembles a two-edge geometry near inﬁnity. If this occurs, the interaction of the classical trajectories in diﬀerent directions may cancel each other. The authors consider the unperturbed Hamiltonian H0D which is the Landau Hamiltonian on Ω with Dirichlet boundary conditions on ∂Ω. The main theorem of [22] is the following. Theorem 4.1. Assume that the region Ω satisﬁes the geometric conditions dis/ 2N + 1 and cussed above and that the perturbation V1 ∈ L∞ (R2 ). Let E/B ∈ suppose that B is taken suﬃciently large so that V1 ∞ /B is suﬃciently small. Then, the spectrum of HΩD = H0D + V1 is absolutely continuous near E. As in the work of DeBi`evre and Pul´e [21], and as we discuss in Sec. 5, Fr¨ ohlich, D Graf and Walcher construct a conjugate operator for the Hamiltonian HΩ on the region Ω. They prove that the commutator, when spectrally localized to a small interval of energies around E, has a strictly positive lower bound. The Mourre theory [30] then implies the existence of absolutely continuous spectrum near E. The Dirichlet boundary conditions on ∂Ω cause some technical complications as px is not self-adjoint on any domain. The conjugate operator is a quantization of a linearization of the classical guiding center trajectory for the classical electron orbit. We introduce another notion to the study of geometrically perturbed regions and use it to prove the persistence of edge currents. The asymptotic velocity is deﬁned for any pair of self-adjoint Schr¨ odinger operators (H0 , H) for which the wave operators exist. The (global) wave operators Ω± for the pair (H0 , H) are deﬁned by Ω± ≡ s − lim eitH e−itH0 Eac (H0 ), t→±∞

(4.1)

where Eac (H0 ) is the projection onto the absolutely continuous spectral subspace for H0 . When the wave operators exist, the range is contained in the absolutely continuous spectral subspace of H, and the wave operators are partial isometries between these spectral subspaces. We will use the local wave operators Ω± (∆) obtained by replacing Eac (H0 ) by the projector E0 (∆) for H0 and an interval ∆ in

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the absolutely continuous subspace of H0 . The asymptotic velocity is deﬁned for any component of the velocity observable. We are interested in velocity asymptotically in the y-direction and for states with energy in an interval ∆. We deﬁne this to be Vy± (∆) ≡ Ω± (∆)Vy Ω∗± (∆).

(4.2)

We note that when H0 commutes with Vy , and the local wave operators exist, the local asymptotic velocity is obtained by the limit Vy± (∆) ≡ s − lim eitH E0 (∆)Vy E0 (∆)e−itH . t→±∞

(4.3)

In the context of potential scattering, we refer to the book of Derezinski and G´erard [29] for a complete discussion of the asymptotic velocity. We consider the geometric perturbation of the straight, one-edge geometry obtained by perturbing the boundary conﬁning potential V0 . We recall that a sharp conﬁning potential V0 is a constant multiple V0 0 of the characteristic function χΩ for a region Ω. In Sec. 2, we treated the case Ω = Ω0 ≡ (−∞, 0] × R, the half-plane. Here, we consider more general Ω obtained by perturbing the half-plane Ω0 . Condition C. The sharp conﬁning potential VΩ is supported in a region Ω so that Ω\Ω0 lies in the strip |y| ≤ R < ∞, for some 0 < R < ∞. We ﬁrst consider the pair of Hamiltonians (H0 , HΩ ), where H0 = HL (B) + V0 is the straight-edge Hamiltonian with sharp conﬁning potential, and HΩ = HL (B) + VΩ , describes the geometric perturbation of the straight-edge boundary satisfying Condition C. We prove that the local wave operators exist for this pair and that the asymptotic velocity observable is bounded from below by B 1/2 . This observable corresponds to the edge current at y = ±∞. Furthermore, the spectrum of the perturbed operator HΩ still has absolutely continuous spectrum between the Landau levels. We then show that this lower bound on the asymptotic velocity observable is stable under a perturbation V1 that is small compared to the ﬁeld strength B. Theorem 4.2. Let H ≡ HL (B) + VΩ + V1 be the perturbed Hamiltonian with sharp conﬁning potential VΩ and a bounded perturbation V1 ∈ L∞ (R2 ). Suppose the region Ω\Ω0 satisﬁes Condition C. Let ∆n be as in (2.6). Let Vy± (∆n ) be the asymptotic velocity for the pair (H0 , HΩ ). Suppose that (c − a) and V1 ∞ /B are suﬃciently small as in Theorem 2.3. For any state ψ = E(∆n )ψ, the asymptotic edge-current velocity Vy± (∆n ) satisﬁes

ψ, Vy± (∆n )ψ ≥ Cn B 1/2 ψ2 .

(4.4)

We remark that is is not required that the new region Ω be connected nor that it be bounded in the x-direction. The basic situation that we have in mind, however, is the one for which the new region Ω represents a distortion of the boundary of the half-plane Ω0 . It is interesting to note that the edge current persists for some states

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even if the boundary extends to +∞ along the x-axis. For example, the right halfplane may actually be disconnected if the perturbation is supported in a cone-type region with vertex at y = 0 and x = +∞. Before we prove Theorem 4.2, we consider the eﬀect of the boundary perturbation with V1 = 0. We deﬁne H0 = HL (B) + V0 and HΩ = HL (B) + VΩ , and we denote the corresponding spectral families by E0 (·) and EΩ (·), respectively. We ﬁrst prove the existence of the local wave operators for the pair (H0 , HΩ ) by the method of stationary phase. This proves the existence of absolutely continuous spectrum in intervals between Landau levels. We then use these local wave operators to prove the persistence of edge currents. We consider the perturbation of the conﬁning potential V0 (x) given by VΩ (x, y) = V0 (χ(−∞,0] (x) + χΩ\Ω0 (x, y)) = V0 (x) + V0 χΩ\Ω0 (x, y),

(4.5)

and we will write δV ≡ VΩ − V0 , so that δV = V0 χΩ\Ω0 (x, y). This perturbation of the conﬁning potential is interpreted as a perturbation of the boundary of the region where the electron can propagate. Proposition 4.1. Let ∆n be as in (2.6) with (c − a) suﬃciently small. Then, the local wave operators Ω± (∆n ) for the pair (H0 , HΩ ) exist. As a consequence, operator HΩ has absolutely continuous spectrum in ∆n . Proof. We use Cook’s method and study the local operators deﬁned by Ω(t; ∆n ) − E0 (∆n ) = eitHΩ e−iH0 t E0 (∆n ) − E0 (∆n ) t eisHΩ δV e−iH0 s E0 (∆n )ds. =i

(4.6)

0

Hence, it suﬃces to prove that for any smooth vector ψ, t2 lim δV e−isH0 E0 (∆n )ψds = 0. t1 ,t2 →∞

(4.7)

t1

In order to prove (4.7), we use the method of stationary phase. Using the partial Fourier transform in (4.7), we have (δV e−isH0 E0 (∆n )ψ)(x, y) n ˆ k)dk. = δV (x, y) e−iωj (k)s+iky χω−1 (∆n ) (k)ψ(x, j

R

j=0

(4.8)

We deﬁne the phase as Φ(k, y, s) ≡ ky − ωj (k)s, and note that the derivative is ∂k Φ(k, y, s) = y − ωj (k)s. Let χR (y) be the characteristic function on the interval [−R, R]. We have the following lower bound |∂k Φ(k, y, s)χω−1 (∆n ) (k)χR (y)| ≥ |ωj (k)s − y|χω−1 (∆n ) (k)χR (y). j

j

(4.9)

In Sec. 2.2, we proved that there is a constant Cn,j > 0 such that −ωj (k)χω−1 (∆n ) (k) ≥ Cn,j Bχω−1 (∆n ) (k). j

j

(4.10)

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Using this lower bound (4.10) in the lower bound (4.9), we obtain |∂k Φ(k, y, s)χω−1 (∆n ) (k)χR (y)| ≥ (Cn,j Bs − R)χω−1 (∆n ) (k)χR (y). j

j

(4.11)

As a consequence, we can diﬀerentiate the phase factor in (4.8) and bound the integral there by n 1 N iΦ(k,y,s) ˆ (∂ e ) ψ(x, k)dk (4.12) , k

sN ωj−1 (∆n ) j=0

where s ≡ (1 + |s|2 )1/2 . The convergence of the integral in (4.7) follows from this decay and integration by parts using the smoothness of ψ. Proposition 4.2. Assume the hypotheses of Proposition 4.1. For any ψ ∈ EΩ (∆n )L2 (R2 ), we have

ψ, Vy± (∆n )ψ ≥ Cn B 1/2 ψ2 ,

(4.13)

where the constant Cn is as in Theorem 2.1. That is, the asymptotic velocity Vy± (∆n ) of the edge current carried by the state ψ = EΩ (∆n )ψ, for the perturbed region, is bounded from below by B 1/2 . Proof. As a consequence of the existence of the wave operators, we have the local intertwining relation Ω± (∆n )∗ EΩ (∆n )ψ = E0 (∆n )Ω± (∆n )∗ ψ.

(4.14)

This intertwining property (4.14) and the deﬁnition (4.2) show that

ψ, Vy± (∆n )ψ = ψ, Ω± (∆n )E0 (∆n )Vy E0 (∆n )Ω∗± (∆n )ψ = E0 (∆n )Ω∗± (∆n )ψ, Vy E0 (∆n )Ω∗± (∆n )ψ.

(4.15)

The lower bound for the right-hand side of (4.15) follows from Theorem 2.1,

E0 (∆n )Ω± (∆n )ψ, Vy E0 (∆n )Ω± (∆n )ψ ≥ Cn B 1/2 E0 (∆n )Ω∗± (∆n )ψ2 ≥ Cn B 1/2 Ω∗± (∆n )ψ2 .

(4.16)

Since the wave operators are partial isometries, we have ψ = Ω∗± (∆n )ψ, which, together with (4.16), proves the lower bound in (4.13). We now prove the stability of the edge current with respect to a small perturbation V1 ∈ L∞ (R2 ). Although we do not necessarily know the spectral type of the perturbed Hamiltonian in intervals between the Landau levels, the edge current is stable. Proof of Theorem 4.2. The proof of Theorem 4.2 follows the same lines of the proof of Theorem 2.3. Given ψ as in the theorem, we decompose it according to

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˜ n containing ∆n . As the spectral projectors for H and a slightly larger interval ∆ in (2.50), we write ˜ n )ψ + EΩ (∆ ˜ cn )ψ ≡ φ + ξ. ψ = EΩ (∆

(4.17)

We then have the decomposition as in (2.51). We bound ξ as in (2.52), and in order to bound Vy± (∆n )ξ, we note that the asymptotic velocity is bounded by deﬁnition, Vy± (∆n ) ≤ [(2n + c)B]1/2 , as follows from (4.3). Finally, we note that the matrix element for φ satisﬁes

φ, Vy± (∆n )φ ≥ C˜n B 1/2 φ2 ,

(4.18)

by Proposition 4.2. A simple calculation as in the proof of Theorem 2.3 allows us to obtain the lower bound c−a 2V1 2 + (4.19) φ ≥ 1 − ψ2 , c˜ − a ˜ B(˜ c−a ˜) so by taking c − a and V1 /B suﬃciently small, we obtain the result (4.4). 5. One-Edge Geometries and the Spectral Properties of H = H0 + V1 The unperturbed operator H0 = HL (B) + V0 has purely absolutely spectrum and σ(H0 ) = [B, ∞). In the paper [21], DeBi`evre and Pul´e proved that perturbations V1 , as in Theorem 2.3, preserve the absolutely continuous spectrum in an interval ∆n , provided |∆n |/B = c − a is suﬃciently small. We mention this result here for completeness, and for comparison with the situation for two-edge geometries where we will use commutator methods. For a review of commutator methods, we refer the reader to [30–32]. The proof in [21] relies on the commutator identity i[H0 , y] = 2Vy .

(5.1)

This commutator shows that an estimate on the edge current is equivalent to an estimate on the positivity of the commutator. This, in turn, provides an estimate on the spectral type of H0 . As we will see, this equivalence, that an estimate on the edge current implies a commutator estimate, no longer holds for two-edge and other, more complicated geometries. This is one of the reasons we presented a diﬀerent approach to the one-edge geometries in the previous sections. Continuing with the perturbation theory of H0 , the commutator on the left in (5.1) is invariant under any perturbation of H0 by a real-valued potential provided V1 and y have a common, dense domain. It follows immediately from the commutator i[H0 + V1 , y] = 2Vy ,

(5.2)

and the techniques of Theorem 2.3, that if c − a is small enough, there exists a ﬁnite constant Kn > 0 such that E(∆n )(i[H, y])E(∆n ) ≥ Kn E(∆n ).

(5.3)

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Since the double commutator is [[H, y], y] = −2i, the following theorem now follows from standard Mourre theory (cf. [30]). Theorem 5.1. Let V1 satisfy the conditions of Theorem 2.3. If c − a and V1 ∞ /B satisfy the smallness conditions of Theorem 2.3 with respect to n and B, then the operator H = H0 + V1 has only absolutely continuous spectrum on ∆n . Thus, in the half-plane case, the existence of edge currents for each ψ ∈ E(∆n )L2 (R2 ) is equivalent to the existence of absolutely continuous spectrum. This need not be the case, however, for more complicated edge geometries. For those situations, there may be edge currents carried by states ψ but the spectrum need not be absolutely continuous (cf. [9, 33–36]). 6. One-Edge Geometries and General Confining Potentials The analysis used in Sec. 2 can be extended to the case of more general conﬁning potentials with a straight edge. These potentials are described as soft potentials, as opposed to the hard potentials such as the Sharp Conﬁning Potential or Dirichlet boundary conditions. In general, the soft conﬁning potential V0 , supported on x ≤ 0, should be rapidly increasing for x < 0. There are two classes of soft conﬁning potentials that we can treat: (1) globally convex potentials, such as monomials V0 (x) = V0 (B 1/2 |x|)p χ(−∞,0) (x),

for p ≥ 1,

(6.1)

and (2) convex-concave potentials that are initially convex and then become asymptotically ﬂat, such as V0 (x) = V0 tanh(B 1/2 |x|)χ(−∞,0) (x).

(6.2)

These two classes of soft conﬁning potentials require that V0 be suﬃciently large, depending on n, where n is the energy level one is studying. For the sake of simplicity we shall restrict ourselves to the potentials given by (6.1) or (6.2), though the results of Theorems 6.1 and 6.2 can be generalized to a wider class of conﬁning potentials. We consider the interval ∆n deﬁned by (2.6). For the unperturbed model H0 = HL + V0 , we have the following result. Theorem 6.1. Let V0 be the globally convex (respectively, convex-concave) conﬁning potential deﬁned by (6.1) (respectively, (6.2)) with 2 −p √ (a − 1)(c − 3) B π , V0 > (2n + c) 4(n + 1)H(n) 2 −1 √ (a − 1)(3 − c) π respectively, V0 > (2n + c) tanh 4(n + 1)H(n) (2n + c) 2p+1

(6.3)

(6.4)

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where the constant H(n) is deﬁned in (2.39). Then, for any ψ = E0 (∆n )ψ having an expansion as in (2.3) with coeﬃcients βj (k), there is a constant Cn > 0 so that for all |∆n |/B small enough, we have   n ˜j (k) V  B 1/2 , dk|βj (k)|2 (6.5) − ψ, Vy ψ ≥ Cn (a−1)2 (3−c)2  2 −1 V (k) j ω (∆ ) n j j=0 where Vj (k) and V˜j (k) are deﬁned by (6.20)–(6.23). Proof. We prove the result for the globally convex potential (6.1), the case of (6.2) being treated in the same way. We assume that the conditions of Lemma 2.1 are satisﬁed so that the cross-terms vanish. We begin with the formula for the matrix element ψ, Vy ψ in (6.5) following from the partial Fourier transform, n −1 0 − ψ, Vy ψ = dx dk|βj (k)|2 ϕj (x; k)2 V0 (x) 2B j=0 −∞ ωj−1 (∆n ) n −1 x∗ dx dk|βj (k)|2 ϕj (x; k)2 V0 (x), (6.6) ≥ −1 2B j=0 −∞ ωj (∆n ) for some x∗ < 0 we will specify below. The strategy is to obtain a lower bound for |ϕj (x∗ ; k)|. We ﬁrst turn to estimating |ϕj (x∗ ; k)|. We use the results of Lemma 2.2. We expand the eigenfunctions ϕj (x; k) in terms of the harmonic oscillator eigenfunctions ψm (x; k) given in (2.26), as in (2.27). We ﬁnd that n

1 (En+1 (B) − ωj (k)), 2B(n + 1)

2 |α(j) m (k)| ≥

m=0

(6.7)

and, with Pn denoting the projector onto the subspace of L2 (R) spanned by the ﬁrst n harmonic oscillator eigenfunctions, | ϕj (·, k), V0 Pn ϕj (·, k)| ≥

1 (ωj (k) − En (B))(En+1 (B) − ωj (k)). (6.8) 2B(n + 1)

We also need an upper bound on this matrix element (6.8). From the deﬁnition of Pn , we obtain n

| ϕj (·, k), V0 Pn ϕj (·, k)| ≤

|α(j) m (k)|{Ij,m (x∗ ; k) + II j,m (x∗ ; k)}

(6.9)

m=0

where the integrals Ij,m and II j,m are given by x∗ V0 (x) |ϕj (x; k)| |ψm (x; k)|dx, Ij,m (x∗ ; k) ≡

(6.10)

−∞

and

IIj,m (x∗ ; k) ≡

0

x∗

V0 (x) |ϕj (x; k)| |ψm (x; k)|dx.

(6.11)

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We turn now to deﬁning x∗ . In light of (6.4) we choose > 0 small enough in such a way that V0 > (2n + c + )2p+1 B

√ π

(a − 1)(c − 3) 4(n + 1)H(n)

2 p .

(6.12)

From this > 0 we deﬁne x∗ = x∗ () as the unique negative real number such that V0 (x∗ ) = (2n + c + )B: x∗ ≡ −B −1/2

(2n + c + )B V0

1/p .

(6.13)

By combining (6.12) with (6.13) we notice that (−x∗ ) <

(a − 1)(3 − c) 4(n + 1)H(n) (2n + c + )

2

π B

1/2 ,

(6.14)

and that the right-hand side of (6.14) is O(B −1/2 ). Having said that, (6.11) can be estimated using the inequalities 0 ≤ V0 (x) ≤ (2n + c + )B for x∗ ≤ x ≤ 0, and the form of the harmonic oscillator wave function (2.26). We get II j,m (x∗ ; k) ≤ (2n + c + )

B 5/4 Hm √ |x∗ |1/2 , π 1/4 2m m!

where the constant Hm is deﬁned by (2.39). This, together with (6.14), entail II j,m (x∗ ; k) ≤

(ωj (k) − En (B))(En+1 (B) − ωj (k)) , 4(n + 1)B

k ∈ ωj−1 (∆n ).

(6.15)

The ﬁrst integral Ij,m is estimated as Ij,m (x∗ ; k) ≤

B π

1/4

Hm √ 2m m!

x∗

−∞

V0 (x)|ϕj (x; k)|dx.

(6.16)

We return to (6.9). In light of the lower bound on the matrix element given in (6.8) and the upper bounds on the integrals given in (6.15) and (6.16), we solve for the integral in (6.16). Bearing in mind (2.39) the sums over m in (6.15), (6.16) are n (j) m ≤ H(n) , so we end up getting bounded by H(n) , m=0 |αm (k)| √H 2m m!

x∗

−∞

V0 (x)|ϕj (x; k)|dx ≥

1 2H(n)

1/4 (ωj (k) − En (B))(En+1 (B) − ωj (k)) π . 2B(n + 1) B

(6.17) (6.18)

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Now |ϕj (x∗ ; k)| can be estimated from this bound and the pointwise upper bound on ϕj in the classically forbidden region and proved in Proposition B.2 of Appendix B, |ϕj (x; k)| ≤ |ϕj (x∗ ; k)|e−

√ B(x∗ −x)

x ≤ x∗ ,

,

(6.19)

since the potential Wj (t; k) ≡ (Bt − k)2 + V0 (t) − ωj (k) ≥ B for any k ∈ ωj−1 (∆n ). In light of this upper bound, we deﬁne a function Vj (k) by Vj (k) ≡

x∗

−∞

V0 (x)e−

√

B(x∗ −x)

dx ≥ 0.

(6.20)

We insert (6.19) into the integral in (6.17), rearrange, and obtain |ϕj (x∗ ; k)| 1 ≥ Vj (k)

1/4 1 (ωj (k) − En (B))(En+1 (B) − ωj (k)) π . 4B(n + 1) B H(n)

(6.21)

We return to the expression for the matrix element of the edge current (6.6). We use the lower bound on the eigenfunction ϕj (x; k) derived in Proposition B.2 of Appendix B: |ϕj (x; k)| ≥ |ϕj (x∗ ; k)|e−

R x∗

√

Sj (t;k)dt

x

,

∀ x ≤ x∗ ,

(6.22)

√ t where Sj (t; k) ≡ Wj (t, k)+ −∞ |Wj (u; k)|e−2 B(t−u) du. We substitute this expression (6.22) into the right-hand side of (6.6). It will be convenient to introduce another constant V˜j (k) deﬁned by

V˜j (k) ≡ −

x∗

−∞

V0 (x)e−2

R x∗ x

√

Sj (t;k)dt

dx ≥ 0.

(6.23)

Notice that both integrals Vj (k) and V˜j (k) converge. Next, using (6.21), we obtain (6.5) with Cn given by (2.47). We now consider the perturbation of H0 by a bounded potential V1 (x, y). As ˜ n given by (2.43) with the same midin Sec. 2.3, we consider a larger interval ∆ point as ∆n , and prove that the edge current survives if V1 ∞ is suﬃciently small relative to B. Theorem 6.2. Let V0 be as in Theorem 6.1. Let V1 (x, y) denote a bounded potential and E(∆n ) be the spectral projection for H = H0 + V1 and the interval ∆n . Let

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ψ ∈ L2 (R2 ) be a state satisfying ψ = E(∆n )ψ, and the following condition. Let ˜ n )ψ have an expansion as in (2.3) with coeﬃcients βj (k) satisfying φ ≡ E0 (∆ n ˜j (k) V dk ≥ (1/2)φ2 , |βj (k)|2 (6.24) 2 (k) −1 V ω (∆ ) j n j j=0 where Vj (k) and V˜j (k) are deﬁned by (6.20)–(6.23). Then, we have, − ψ, Vy ψ ≥ B 1/2 ((Cn /2)(3 − c˜)2 (˜ a − 1)2 − F (n, V1 /B))ψ2 ,

(6.25)

where Cn is deﬁned in (2.47) and 1/2 1/2 1/2 2 (c − a) V1 V1 F (n, V1 /B) = + 2n + c + (˜ c−a ˜) 2 B B 2 (c − a) V1 + × 2+ (˜ c−a ˜) 2 B 2 2 (c − a) V1 2 Cn + (3 − c˜)2 (˜ a − 1)2 . + 2 (˜ c−a ˜) 2 B Proof. As in the proof of Theorem 2.3, we ﬁrst decompose the function ψ as in (2.50) and expand ψ, Vy ψ as in (2.51). Next, we use (2.54) and (2.55) to bound |2 Re φ, Vy ξ| + ξ, Vy ξ|, and deduce from Theorem 6.1 and (6.24) that   n ˜j (k) V  B 1/2 dk|βj (k)|2 − φ, Vy φ ≥ Cn (a − 1)2 (3 − c)2  2 −1 V (k) j ω (∆ ) n j j=0 ≥ (Cn /2)(a − 1)2 (3 − c)2 B 1/2 φ2 . Now, by inserting (2.52) in the identity φ2 = ψ2 − ξ2 we get that 2 2 2 (c − a) V 1 + φ2 ≥ 1 − ψ2 , (˜ c−a ˜) 2 B

(6.26)

so the result follows by elementary computations. Appendix A. Basic Properties of Eigenfunctions and Eigenvalues of h0 (k) After reducing the operator H0 = −∆ + V0 to the operator h0 (k) on L2 (R) due to the y-translational invariance, we are concerned with studying the properties of h0 (k) deﬁned by h0 (k) = p2x + (Bx − k)2 + V0 (x) = p2x + V (x; k),

(A.1)

where p2x = −d2 /dx2 , and the nonnegative potential V0 (x) ∈ L2loc (R). The resolvent of the operator h0 (k) = p2x + V (x; k) is compact since the eﬀective potential V (x; k) = (Bx − k)2 + V0 (x) is unbounded as |x| → ∞, so the spectrum is discrete

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with only ∞ as an accumulation point. We denote the eigenvalues of h0 (k) in increasing order and denote them by ωj (k), j ≥ 0. The normalized eigenfunction associated to ωj (k) is ϕj (x; k). The variational method shows that the domain of h0 (k) is dom(h0 (k)) = {ψ ∈ H 1 (R) ∩ L2 (R; w(x; k)dx), (p2x + V (.; k))ψ ∈ L2 (R)}, (A.2) 2 with w(x; k) = (1 + V (x; k))1/2 . It is a subset of Hloc (R) since the eﬀective potential 2 V (.; k) ∈ Lloc (R). We ﬁrst discuss the regularity properties of the eigenfunctions.

Proposition A.1. The eigenfunctions of h0 (k), given by ϕj (.; k), are continuously diﬀerentiable in R for any j ∈ N and k ∈ R. Furthermore, an eigenfunction ϕj (.; k) ∈ C n+2 (I) for any open subinterval I of R such that V0 ∈ C n (I), n ≥ 0. Proof. The proof of this proposition follows from the Sobolev Embedding Theorem 2 (R) ⊂ C 1 (R), and the fact that the Schr¨ odinger equation which gives Hloc ϕj (x; k) = (V (x; k) − ωj (k))ϕj (x; k), shows that ϕj (x; k) ∈ L2loc (R). In the particular case of the Sharp Conﬁning Potential V0 (x) = V0 χ(−∞,0) (x), Proposition A.1 shows that ϕj (.; k) ∈ C 1 (R) ∩ C ∞ (R\{0}). Notice that ϕj (.; k) is continuously diﬀerentiable at the origin although V0 is discontinuous at this point. For the Parabolic Conﬁning Potential V0 (x) = V0 x2 χ(−∞,0) (x), we have ϕj (.; k) ∈ C 3 (R) ∩ C ∞ (R∗ ) since V0 is only C 1 in any neighborhood of the origin. We next turn to a proof of the simplicity of the eigenvalues of h0 (k). We state Lemma A.1 without proof. It is a simple consequence of the Unique Continuation Theorem for Schr¨ odinger Operators ([37, Theorem XIII.63). We will use this lemma in the proof of Propositions A.2 and B.1. Lemma A.1. Let I be an open (not necessarily bounded) subinterval of R, W ∈ 2 (I) satisfy L2loc (I) and ψ ∈ Hloc ψ (x) = W (x)ψ(x),

a.e. x ∈ I.

Then, if ψ vanishes in the neighborhood of a single point x0 ∈ I, ψ is identically zero in I. Proposition A.2. The eigenvalues ωj (k) of the operator h0 (k) are simple for all k ∈ R. Proof. We consider two L2 -eigenfunctions ϕ and ψ of h0 (k) with same energy E. 2 (R)-solutions of the Schr¨ odinger As follows from Proposition A.1, they are both Hloc

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equation u (x) = (V (x, k) − E)u(x),

a.e. x ∈ R.

(A.3)

By substituting ϕ (respectively, ψ) for u in (A.3), multiplying by ψ (respectively, ϕ), and taking the diﬀerence of the two equalities, we get ϕ (x)ψ(x) − ϕ(x)ψ (x) = (ϕ ψ − ϕψ ) (x) = 0,

a.e. x ∈ R.

Consequently, the function (ϕ ψ − ϕψ ) is a constant for a.e. x in R, and this constant is zero since the function is in L2 (R) as follows from Proposition A.1, (ϕ ψ − ϕψ )(x) = 0,

∀ x ∈ R.

(A.4)

We notice there is always a real number a such that the potential V (x; k) − E > 0 for a.e. x > a (since V (x; k) → ∞ as x → ∞) and ψ(a) = 0 (ψ would be identically zero in R by Lemma A.1 otherwise) so ψ(x) = 0 for any x > a by part 1 of Proposition B.1. Hence (A.4) implies (ϕ/ψ) (x) = 0,

∀ x > a,

so we have ϕ = λψ on (a, +∞) for some constant λ ∈ R. The function ϕ − λψ is 2 (R)-solution to (A.3) which vanishes in (a, +∞). It is also identically also an Hloc zero in R by Lemma A.1 hence {ϕ, ψ} is a one-dimensional manifold of L2 (R). Appendix B. Pointwise Upper and Lower Exponential Bounds on Solutions to Certain ODEs We obtain pointwise, exponential, upper and lower bounds on solutions to the ordinary diﬀerential equation (ODE) ψ = W ψ, with W > 0. We apply these results in the next section to the eigenfunctions ϕj (.; k) of h0 (k) in the classically forbidden region where Wj (x; k) ≡ V (x; k) − ωj (k) > 0. We consider the following general situation. We let ψ denote a real H 1 ((−∞, a))-solution to the system

ψ (x) = W (x)ψ(x),

a.e. x < a

lim ψ(x) = ψ(a) > 0,

(B.1)

x→a−

for some a ∈ R, where W ∈ L2loc ((−∞, a)) is such that: W (x) > 0,

a.e. x < a.

(B.2)

Standard arguments already used in the proof of Proposition A.1, assure us that 2 ((−∞, a)) so ψ ∈ C 1 ((−∞, a)). Moreover, ψ is left continuous the solution ψ ∈ Hloc at a, according to (B.1).

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B.1. Basic properties of ψ We prove the following basic result that characterizes the behavior of the solution ψ in the classically forbidden region where W (x) > 0. Proposition B.1. Any real H 1 ((−∞, a))-solution ψ to (B.1) satisﬁes: (1) ψ(x) > 0 and ψ (x) > 0, for any x < a; (2) lim W (x)ψ 2 (x) = 0. x→−∞

We prove the ﬁrst part of Proposition B.1 in two elementary lemmas. Lemma B.1. Under the hypotheses of Proposition B.1, suppose that ψ(x0 )ψ (x0 ) < 0, for some x0 < a. If ψ(x0 ) > 0, we have ψ(x) > ψ(x0 ), for any x < x0 , and if ψ(x0 ) < 0, we have ψ(x0 ) > ψ(x), for any x < x0 . Consequently, we have ψ(x)ψ (x) ≥ 0, for any x < a. Proof. We assume that ψ(x0 ) > 0 so that the hypothesis implies that ψ (x0 ) < 0. The case ψ(x0 ) < 0, implying ψ (x0 ) > 0, is treated in the same manner. Notice that E = {δ > 0 | ψ(x) > ψ(x0 ), for x ∈ (x0 − δ, x0 )} = ∅, since ψ (x0 ) < 0, so δ0 = sup E > 0. If δ0 < ∞, then x1 = x0 − δ0 satisﬁes ψ(x) > ψ(x0 ) ∀ x ∈ (x1 , x0 ) ψ(x1 ) = ψ(x0 ). Thus for a.e. x ∈ [x1 , x0 ), we have ψ (x) = W (x)ψ(x) ≥ W (x)ψ(x0 ) > 0 hence ψ (x) < ψ (x0 ) < 0 for all x ∈ [x1 , x0 ), so we ﬁnally get ψ(x1 ) > ψ(x0 ). Actually ψ(x1 ) = ψ(x0 ), hence δ0 = +∞ and the ﬁrst result follows. Finally, if there is some x0 < a such that ψ(x0 )ψ (x0 ) < 0, then the ﬁrst result implies that |ψ(x)| ≥ |ψ(x0 )| > 0, for any x ≤ x0 . This is impossible since ψ ∈ L2 ((−∞, a)). We next consider the possibility that the wave function has zeros in the classically forbidden region. Lemma B.2. Under the hypotheses of Proposition B.1, we have ψ(x) > 0 for any x < a. Proof. 1. We ﬁrst show that ψ(x)ψ (x) > 0, for any x < a such that ψ(x) = 0. We assume that ψ(x) > 0 (the case ψ(x) < 0 being treated in the same way) so ψ(t) > 0 for any t ∈ (x − δ, x) for some δ > 0 and ψ (t) = W (t)ψ(t) > 0 for a.e. t in (x − δ, x). If ψ (x) = 0 we have ψ (t) < 0 and also ψ(t)ψ (t) < 0 for each t ∈ (x − δ, x). This is impossible according to Lemma B.1. Hence ψ (x) > 0 since ψ (x) ≥ 0 by Lemma B.1.

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2. Next we show that if ψ(x0 ) = 0, for some x0 < a, then ψ (x0 ) = 0. We assume that ψ(x0 ) = 0 and ψ (x0 ) > 0 (the case ψ (x0 ) < 0 being treated in the same manner). In this case we can ﬁnd some δ > 0 such that ψ(x) < 0 and ψ (x) > 0, for any x ∈ (x0 − δ, x0 ), which is impossible according to Lemma B.1. 3. To complete the proof, we assume that there is a real number x0 < a such that ψ(x0 ) = 0. We also have ψ (x0 ) = 0 by Part 2 and sup{x < x0 | ψ(x) = 0} = x0 , since ψ would be zero on (−∞, a) otherwise by Lemma A.1. Thus, we can ﬁnd some δ > 0 such that ±ψ(x) > 0, for all x ∈ (x0 − δ, x0 ), so ±ψ (x) = W (x)(±ψ(x)) > 0 a.e. in (x0 − δ, x0 ). This implies that ±ψ (x) < 0, and, consequently, that ψ(x)ψ (x) < 0, for any x ∈ (x0 − δ, x0 ). This is impossible according to Lemma B.1.

To justify the second part of Proposition B.1, we multiply (B.1) by ψ, and integrate over [x, x0 ], for some x0 < a and x < x0 . We obtain x0 x0 ψ (u)ψ(u)du = W (u)ψ 2 (u)du. (B.3) x

x

Integrating by parts in the left-hand side of (B.3), we get x0 x0 ψ(x0 )ψ (x0 ) − ψ(x)ψ (x) − ψ 2 (u)du = W (u)ψ 2 (u)du, x

(B.4)

x

so by taking the limit x → −∞ in (B.4), we obtain the inequality: x0 x0 0≤ W (u)ψ 2 (u)du ≤ ψ(x0 )ψ (x0 ) − ψ 2 (u)du < ∞. −∞

−∞

2

1

Hence, the function W ψ ∈ L ((−∞, x0 )), and the result follows. B.2. Pointwise bounds We ﬁrst compute an upper bound to an H 1 ((−∞, a))-solution to (B.1) for a potential W bounded from below. Lemma B.3. If W ∈ L2loc ((−∞, a)) is bounded from below, W (x) ≥ Wm > 0,

a.e. x < a,

(B.5)

then any real H 1 ((−∞, a))-solution of (B.1) satisﬁes: 1/2

ψ(x) ≤ ψ(x0 )e−Wm

(x0 −x)

,

∀ x ≤ x0 ≤ a.

(B.6)

Proof. We multiply (B.1) by ψ (u) so we get ψ (u)ψ (u) = W (u)ψ(u)ψ (u) ≥ Wm ψ(u)ψ (u),

a.e. u < a,

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according to (B.5) and Part 1 of Proposition B.1. Next, we integrate this inequality over [x, t] for x < t < a, get ψ 2 (t) − ψ 2 (x) ≥ Wm (ψ 2 (t) − ψ 2 (x)), and take the limit as x → −∞: ψ 2 (t) ≥ Wm ψ 2 (t),

∀ t < a.

1/2

This leads to ψ (t) ≥ Wm ψ(t) for any t < a, by Part 1 of Proposition B.1. By integrating over [x, x0 ], x ≤ x0 < a, we ﬁnally obtain 1/2

ψ(x) ≤ ψ(x0 )e−Wm

(x0 −x)

.

This result continues to hold for x0 = a since ψ is left continuous at a. We then examine the behavior of an H 1 ((−∞, a))-solution to (B.1) for a potential 1 ((−∞, a)). W ∈ Hloc

(B.7)

The main result on L2 -solutions of the Eq. (B.1) is the following: Proposition B.2. Let W satisfy (B.5)–(B.7) together with the condition:

a

−∞

1/2

|W (u)|e2Wm

u

du < ∞.

(B.8)

Then any real H 1 ((−∞, a))-solution ψ to (B.1) satisﬁes ψ(x0 )e−

Rx x

0

√

where S(t) = W (t) +

S(t)dt

t −∞

1/2

≤ ψ(x) ≤ ψ(x0 )e−Wm 1/2

|W (u)|e−2Wm

(t−u)

(x0 −x)

,

for x ≤ x0 ≤ a,

du, for all t ≤ a.

Proof. The left inequality being already given by Lemma B.3 we only need to prove right one. To do that we multiply (B.1) by ψ (x) and integrate over [u, t], for u < t < a:

t

u

ψ 2 (t) − ψ 2 (u) = ψ (x)ψ (x)dx = 2

t

W (x)ψ(x)ψ (x)dx.

u

Next, integrating by parts, the right-hand side of this equality gives ψ 2 (t) − ψ 2 (u) = W (t)ψ 2 (t) − W (u)ψ 2 (u) −

t

u

W (x)ψ 2 (x)dx,

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the above integral being well deﬁned since W ∈ L2loc ((−∞, a)) and ψ is bounded in [u, t]. Taking the limit as u → −∞ in the previous equality leads to t ψ 2 (t) = W (t)ψ 2 (t) − W (u)ψ 2 (u)du, ∀ t < a, (B.9) −∞

according to part 2 of Proposition B.1. Now we insert the inequality (B.6) written for u < t < a 1/2

ψ(u) ≤ ψ(t)e−Wm

(t−u)dv

,

into the following obvious consequence of (B.9), (B.8): t ψ 2 (t) ≤ W (t)ψ 2 (t) + |W (u)|ψ 2 (u)du, −∞

∀ t < a,

getting ψ 2 (t) ≤ S(t)ψ 2 (t),

t < a.

Thus ψ (t) ≤ S(t)ψ(t) for all t < a, by part 1 of Proposition B.1, so we get Rx √ 0 S(t)dt , ∀ x ≤ x0 < a, (B.10) ψ(x) ≥ ψ(x0 )e− x by integrating over [x, x0 ]. Taking account of the left continuity of ψ at a we extend this result at x0 = a by taking the limit in (B.10) as x0 → a. Acknowledgment We thank J.-M. Combes for many discussions on edge currents and their role in the IQHE. We also thank F. Germinet, G.-M. Graf, E. Mourre and H. Schulz-Baldes for fruitful discussions. Some of this work was done when ES was visiting the Mathematics Department at the University of Kentucky and he thanks the Department for its hospitality and support. We thank the referees for a careful reading of the manuscript and helpful comments. The ﬁrst-named author was supported in part by NSF grant DMS-0503784. References [1] J. Bellissard, A. van Elst and H. Schulz-Baldes, The noncommutative geometry of the quantum Hall eﬀect. Topology and physics, J. Math. Phys. 35(10) (1994) 5373–5451. [2] H. Kunz, The quantum Hall eﬀect for electrons in random potentials, Comm. Math. Phys. 112 (1987) 121–145. [3] J. M. Combes and P. D. Hislop, Landau Hamiltonians with random potentials: Localization and the density of states, Comm. Math. Phys. 177 (1996) 603–629. [4] T. C. Dorlas, N. Macris and J. V. Pul´e, Localization in single Landau bands, J. Math. Phys. 177(4) (1996) 1574–1595. [5] F. Germinet and A. Klein, Explicit ﬁnite volume criteria for localization in continuous random media and applications, GAFA 13 (2003) 1201–1238.

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[6] W.-M. Wang, Microlocalization, percolation, and Anderson localization for the magnetic Schr¨ odinger operator with a random potential, J. Funct. Anal. 146 (1997) 1–26. [7] F. Germinet, A. Klein and J. Schenker, Dynamical delocalization in random Landau Hamiltonians, to appear in Ann. Math. [8] T. C. Dorlas, N. Macris and J. Pul´e, Characterization of the spectrum of the Landau Hamiltonian with delta impurities, Comm. Math. Phys. 204 (1999) 367–396. [9] P. D. Hislop and E. Soccorsi, Edge currents for quantum Hall systems, II. Two-edge bounded and unbounded geometries, arXiv:math-phy/0702093. [10] J. M. Combes and F. Germinet, Stability of the edge conductivity in quantum Hall systems, Comm. Math. Phys. 256 (2005) 159–180. [11] J. M. Combes, F. Germinet and P. D. Hislop, On the quantization of Hall currents in presence of disorder, in Mathematical Physics of Quantum Mechanics, Lecture Notes in Physics, Vol. 690 (Springer, Berlin, 2006), pp. 307–323. [12] P. Elbau and G. M. Graf, Equality of bulk and edge Hall conductance revisited, Comm. Math. Phys. 229(3) (2002) 415–432. [13] A. Elgart, G. M. Graf and J. H. Schenker, Equality of the bulk and edge Hall conductances in a mobility gap, Comm. Math. Phys. 259 (2005) 185–221. [14] J. Kellendonk, T. Richter and H. Schulz-Baldes, Edge channels and Chern numbers in the integer quantum Hall eﬀect, Rev. Math. Phys. 14 (2002) 87–119. [15] J. Kellendonk and H. Schulz-Baldes, Boundary maps for C ∗ -crossed products with R with an application to the quantum Hall eﬀect, Comm. Math. Phys. 249 (2004) 611–637. [16] J. Kellendonk and H. Schulz-Baldes, Quantization of edge currents for continuous magnetic operators, J. Funct. Anal. 209 (2004) 388–413. [17] H. Schulz-Baldes, J. Kellendonk and T. Richter, Simultaneous quantization of the edge and bulk Hall conductivity, J. Phys. A 33 (2000) L27–L32. [18] B. I. Halperin, Quantized Hall conductance, current carrying edge states, and the existence of extended states in a two-dimensional disordered potential, Phys. Rev. B 25 (1982) 2185–2190. [19] O. Heinonen and P. L. Taylor, Current distributions in the quantum Hall eﬀect, Phys. Rev. B 32 (1985) 633–639. [20] N. Macris, P. A. Martin and J. Pul´e, On edge states in a semi-inﬁnite quantum Hall system, J. Phy. A 32 (1999) 1985–1996. [21] S. De Bi`evre and J. V. Pul´e, Propagating edge states for a magnetic Hamiltonian, Math. Phys. Elec. J. 5 (1999), Paper 3, 17 pp. [22] J. Fr¨ ohlich, G. M. Graf and J. Walcher, On the extended nature of edge states of quantum Hall Hamiltonians, Ann. H. Poincar´e 1 (2000) 405–444. [23] P. Exner, A. Joye and H. Kovarik, Edge currents in the absence of edges, Phys. Lett. A 264 (1999) 124–130. [24] C. Buchendorfer and G. M. Graf, Scattering of magnetic edge states, Ann. Henri Poincar´e 7 (2006) 303–333. [25] J.-M. Combes, P. D. Hislop and E. Soccorsi, Edge states for quantum Hall Hamiltonians, Contemp. Math. 307 (2002) 69–81. [26] E. B. Davies, Spectral Theory and Diﬀerential Operators, Cambridge Studies in Advanced Mathematics, Vol. 42 (Cambridge University Press, Cambridge, 1995). [27] P. D. Hislop and A. Martinez, Scattering resonances of a Helmholtz resonator, Indiana Univ. Math. J. 40 (1991) 767–788. [28] P. D. Hislop and E. Soccorsi, Edge currents for quantum Hall systems, I. One-edge unbounded geometries, arXiv:math-phy/0702092.

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[29] J. Derezinski and C. G´erard, Scattering Theory of Classical and Quantum N -Particle Systems, Texts and Monographs in Physics (Springer-Verlag, Berlin, 1997). [30] H. Cycon, R. Froese, W. Kirsch and B. Simon, Schr¨ odinger Operators with Application to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics, Springer Study Edition (Springer-Verlag, Berlin, 1987). [31] W. Amrein, A. Boutet de Monvel and V. Georgescu, C0 -Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians (Birkh¨ auser, 1996). [32] E. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators, Comm. Math. Phys. 78 (1981) 519–567. [33] C. Ferrari and N. Macris, Spectral properties of ﬁnite quantum Hall systems, in Operator Algebras and Mathematical Physics (Constancta, 2001), Theta, Bucharest (2003), pp. 115–122. [34] C. Ferrari and N. Macris, Intermixture of extended edge and localized bulk energy levels in macroscopic Hall systems, J. Phys. A 35(30) (2002) 6339–6358. [35] P. Exner, A. Joye and H. Kovarik, Magnetic transport in a straight parabolic channel, J. Phys. A 34(45) (2001) 9733–9752. [36] C. Ferrari and N. Macris, Extended edge states in ﬁnite Hall systems, J. Math. Phys. 44(9) (2003) 3734–3751. [37] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators (Academic Press, 1978).

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Reviews in Mathematical Physics Vol. 20, No. 1 (2008) 117–118 c World Scientific Publishing Company

ERRATA NONCOMMUTATIVE KdV HIERARCHY [Reviews in Mathematical Physics, Vol. 19, No. 7 (2007) 677–724] FRANC ¸ OIS TREVES Mathematics Department, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA [email protected]

The proof of Lemma 2.20 in the article “Noncommutative KdV Hierarchy” is corrected.

Errors in the proofs of Lemma 2.20 and Proposition 2.25 in the article [1] stem from the misconception that if F ∈ P, i.e. F is a polynomial in the noncommuting ∂F vanishes identically for all j ∈ Z+ then F is indeterminates ξ0 , ξ1 , . . . , and if ∂ξ j a constant. Counter-example: [ξ0 , ξ1 ]. Actually, the set of polynomials F such that dF vanishes identically form the subring (for ordinary addition and multiplication) of P generated by the constants and by the Lie algebra generated by the monomials ξi , i ∈ Z+ . We restate and prove Lemma 2.20. Lemma 2.20 [1]. If the polynomial F ∈ P is such that P F ∈ [P, P] + dP for every P ∈ P then F vanishes identically. Proof. Suppose F ∈ P satisﬁes the hypothesis in the lemma. Since the subspace [P, P] + dP is stable under the standard partial derivative ∂ξ∂ 0 we derive that P

∂F ∂ ∂P = (P F ) − F ∈ [P, P] + dP ∂ξ0 ∂ξ0 ∂ξ0

∂F for every P ∈ P. Induction on the degree of F allows us to conclude that ∂ξ 0 vanishes identically. But obviously ξ0 F has the same properties as F and therefore F = ∂ξ∂0 (ξ0 F ) vanishes identically.

Proof of Proposition 2.25 [1] requires a diﬀerent approach; we hope to present it in a future article. Actually, whereas Lemma 2.20 is pivotal in the construction of the KdV hierarchy, Proposition 2.25 is not used in the remainder of the article. 117

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The conjectured “Theorem” 4.1 is false. A proof of this fact will appear in an article under preparation. Reference [1] F. Treves, Noncommutative KdV hierarchy, Rev. Math. Phys. 19(7) (2007) 677–724.

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Reviews in Mathematical Physics Vol. 20, No. 2 (2008) 119–172 c World Scientiﬁc Publishing Company

REMOVAL OF VIOLATIONS OF THE MASTER WARD IDENTITY IN PERTURBATIVE QFT

FERDINAND BRENNECKE Institut f¨ ur Quantenelektronik, ETH Z¨ urich, CH-8093 Z¨ urich, Switzerland [email protected] ¨ MICHAEL DUTSCH Institut f¨ ur Theoretische Physik, Universit¨ at Z¨ urich, CH-8057 Z¨ urich, Switzerland and Max Planck Institute for Mathematics in the Sciences, D-04103 Leipzig, Germany [email protected] Received 22 May 2007 Revised 22 October 2007 We study the appearance of anomalies of the Master Ward Identity, which is a universal renormalization condition in perturbative QFT. The main insight of the present paper is that any violation of the Master Ward Identity can be expressed as a local interacting ﬁeld; this is a version of the well-known Quantum Action Principle of Lowenstein and Lam. Proceeding in a proper ﬁeld formalism by induction on the order in , this knowledge about the structure of possible anomalies as well as techniques of algebraic renormalization are used to remove possible anomalies by ﬁnite renormalizations. As an example, the method is applied to prove the Ward identities of the O(N ) scalar ﬁeld model. Keywords: Perturbative renormalization; symmetries. Mathematics Subject Classiﬁcation 2000: 81T15, 70S10

Contents 1. Introduction

120

2. Classical Field Theory for Localized Interactions

122

3. Perturbative Quantum Field Theory

127

4. Proper Vertices for T -Products 4.1. Diagrammatics and deﬁnition of the 1-particle-irreducible part T 1PI of the time ordered product

133

119

133

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4.2. Deﬁnition and basic properties of the vertex function ΓT 4.3. Comparison with the literature 5. The 5.1. 5.2. 5.3.

Master Ward Identity The classical MWI in the oﬀ-shell formalism Structure of possible anomalies of the MWI in QFT The MWI in the proper ﬁeld formalism, and the Quantum Action Principle 5.4. Removal of violations of the MWI 5.4.1. Fulﬁllment of the MWI to ﬁrst order in S 5.4.2. Removal of possible anomalies by induction on the order in 5.4.3. Assumption: Localized oﬀ-shell version of Noether’s Theorem 5.4.4. Proof of the Ward identities in the O(N ) scalar ﬁeld model

136 140 141 141 142 152 154 154 155 157 161

6. Conclusions and Outlook

164

Appendix A. Proper Vertices for R-Products A.1. Deﬁnition and basic properties A.2. Comparison of the vertex functions in terms of T - and R-products

165 165 168

1. Introduction In the quantization of a classical ﬁeld theory symmetries and corresponding conservation laws are in general not maintained: due to the distributional character of quantum ﬁelds the arguments valid for classical ﬁeld theory are not applicable. Therefore, in perturbative quantum ﬁeld theory (pQFT) symmetries and conservation laws play the role of renormalization conditions (the “Ward identities”). In [1] a universal formulation of Ward identities was studied and termed Master Ward Identity (MWI). This identity — which originally was proposed in [2] — can be derived in the framework of classical ﬁeld theory simply from the fact that classical ﬁelds can be multiplied pointwise. However, in pQFT the MWI serves as a highly non-trivial renormalization condition, which cannot be fulﬁlled in general due to the well-known anomalies appearing in QFT. In traditional renormalization theory (e.g. BPHZ renormalization or dimensional renormalization) the question whether certain Ward identities can be fulﬁlled, is usually treated by means of algebraic renormalization.a This method relies on the Quantum Action Principle (QAP), which was derived by Lowenstein and Lam in the early seventies [3, 4] and proved in several renormalization schemes [5]. The QAP characterizes the possible violations of Ward identities and, hence, allows one to derive algebraic conditions whose solvability guarantees the existence of a renormalization maintaining the Ward identities. These conditions often lead to cohomological problems involving the inﬁnitesimal symmetry operators which a There is a huge literature about algebraic renormalization. For brevity, we cite only some of the founding articles, a few textbooks and reviews, which should suﬃce to understand this paper.

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appear in the considered Ward identities. The application of this procedure to Yang–Mills theories lead to a detailed study of BRST cohomology [6–11]. Traditionally, perturbation theory is done in the functional formulation of QFT which starts from the path integral. However, we take the point of view of algebraic pQFT [12–16], which is based on causal perturbation theory (Bogoliubov [17], Epstein and Glaser [18]) and concentrates on the algebraic structure of interacting ﬁelds. Starting with some well deﬁned free QFT one separates UV-problems from IR-problems by considering solely interactions with compact support. Whereas the UV-problem concerns the construction of time ordered or retarded products (in terms of which interacting ﬁelds are formulated), the IR-problem appears only in the construction of states on the algebra of local observables. The restriction on compactly supported interactions leaves it possible to construct the whole net of local observables [12]. Therefore, this approach seems to be well suited for a rigorous perturbative construction of quantum Yang–Mills theories, for example, where an adiabatic limit seems to be out of reach. However, in the non-Abelian case the construction of the net of local observables is still an open problem within the framework of algebraic pQFT. As it was worked out in [2], the decisive input to reach this goal is the MWI, respectively, certain cases of it. Motivated by this, the aim of our present work is to transfer techniques from algebraic renormalization theory into the framework of algebraic pQFT in order to gain more insight into the violations of the MWI and to ﬁnd concrete conditions for the solvability of the MWI in relevant cases. The paper is organized as follows: Sec. 2 deals with classical ﬁeld theory for localized interactions. We generalize the treatment given in [1, Sec. 2] to the oﬀshell formalism, i.e. the values of the retarded products are oﬀ-shell ﬁelds. In Sec. 3 we summarize the quantization of perturbative classical ﬁeld theory worked out in [16] and give some completions. Algebraic renormalization proceeds in terms of the “vertex functional” (or “proper function”) Γ (which is usually derived in the functional formulation of QFT along a Legendre transformation). Hence, to make accessible techniques of algebraic renormalization, we develop a proper ﬁeld formalism, which describes the combinatorics of 1-particle-irreducible (1PI) diagrams in a purely algebraic setting (Sec. 4). This is done by reformulating perturbative QFT as a classical ﬁeld theory with a non-local interaction Γ which is a formal power series in . After these preparatory sections we turn in Sec. 5 to the maintenance of the MWI in the process of renormalization. Starting from a derivation of the MWI in the oﬀ-shell formalism [16], we prove an identity (“anomalous MWI”) which gives a characterization of the possible violations of the MWI. More precisely, we ﬁnd that the most general violation can be expressed in terms of a local interacting ﬁeld. Translation of the anomalous MWI into the proper ﬁeld formalism (introduced in Sec. 4) yields an identity which contains solely the “quantum part” (loop part) of the original version and which is shown to be formally equivalent to the QAP. Crucial properties of the violating local terms appearing in the anomalous MWI are

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proved, in particular an upper bound for the mass dimension. Equipped with this new insight, we transfer basic ideas of algebraic renormalization into the setting of causal perturbation theory. Usually, in the latter the maintenance of Ward identities in renormalization is proved by induction on the power of the coupling constant. In contrast, we proceed by induction on the power of similarly to algebraic renormalization. We ﬁnd explicit conditions, whose solvability guarantees the existence of a renormalization prescription satisfying corresponding cases of the MWI. In addition, we prove that the MWI can always be fulﬁlled to ﬁrst order in the coupling constant (that is to second order of the corresponding time ordered products). We apply these results to models fulﬁlling (classically) a localized oﬀ-shell version of Noether’s Theorem and ﬁnd simpliﬁcations of the mentioned conditions. Finally, as a simple application of the method, we prove the Ward identities of the O(N ) scalar ﬁeld model by using cohomological arguments.

2. Classical Field Theory for Localized Interactions In this section we generalize the formalism developed in [1, Sec. 2] to oﬀ-shell ﬁelds. This will provide us with the necessary framework to derive the oﬀ-shell version of the MWI in Sec. 5.1. In order to keep the formulas as simple as possible, we study the theory of a real scalar ﬁeld ϕ on d dimensional Minkowski space M, d > 2. We interpret ϕ and partial derivatives thereof as evaluation functionals on the conﬁguration space C ≡ C ∞ (M, R) : (∂ a ϕ)(x)(h) = ∂ a h(x), a ∈ Nd0 . Let F (C) be the space of all functionals F (ϕ) : C → C,

F (ϕ)(h) = F (h),

(2.1)

which are localized polynomialsb in ϕ: F (ϕ) =

N

dx1 · · · dxn ϕ(x1 ) · · · ϕ(xn )fn (x1 , . . . , xn ) =:

fn , ϕ⊗n ,

(2.2)

n

n=0

where N < ∞ and the fn ’s are C-valued distributions with compact support, which are symmetric under permutations of the arguments and whose wave front sets satisfy the conditionc WF(fn ) ∩ Mn × (V¯+n ∪ V¯−n ) = ∅

(2.3)

and f0 ∈ C. F is a commutative algebra with the classical product (F1 · F2 )(h) := F1 (h) · F2 (h). By the support of a functional F ∈ F we mean the support of δF δϕ . bA

generalization to non-polynomial (localized) interactions is possible, see e.g. [19]. denotes the closure of the forward and backward light-cones, respectively, and V¯±n their n-fold direct products.

cV ¯±

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The space of local functionals Floc (C) ⊂ F(C) is deﬁned as N N def Floc (C) = dx Ai (x)hi (x) ≡ Ai (hi ) | Ai ∈ P , hi ∈ D(M) , i=1

123

(2.4)

i=1

where P is the space of all polynomials of the ﬁeld ϕ and its partial derivatives. This representation of local functionals as smeared ﬁelds can be made unique by introducing the subspace of balanced ﬁelds Pbal ⊂ P [16, 20]: def Pbal = P (∂1 , . . . , ∂n )ϕ(x1 ) · · · ϕ(xn )|x1 =···=xn =x | P (∂1 , . . . , ∂n ) ∈ Pnrel , n ∈ N0 (2.5) where is the space of all polynomials in the “relative derivatives” (∂k − ∂l ), 1 ≤ k < l ≤ n. With that it holds: given F ∈ Floc there exists a unique h ∈ D(M, Pbal ) with h|ϕ=0 = 0 and F − F (0) = dx h(x). Proofs are given in [16, Proposition 3.1] and in [20, Lemma 1]. Since we are mainly interested in perturbation theory we consider action funcdef tionals of the form Stot = S0 + λ S where S0 = dx 12 (∂µ ϕ∂ µ ϕ − m2 ϕ2 ) denotes the free action, λ a real parameter and S ∈ F(C) is some compactly supported interaction, which may be non-local.d We denote by ∆ret Stot the retarded Green function corresponding to the action Stot , which is deﬁned by δ 2 Stot δ 2 Stot ret = δ(x − z) = dy ∆ret (y, z) (2.6) dy ∆Stot (x, y) δϕ(y)δϕ(z) δϕ(x)δϕ(y) Stot Pnrel

and ∆ret Stot (x, y) = 0 for x suﬃciently early. In the following we will assume that for all actions Stot under consideration the retarded Green function exists and is unique in the sense of formal power series in λ. Analytic expressions for ∆ret Stot are in general unknown. However, perturbatively the retarded solution of (2.6) can be ret given in terms of the (unique) retarded Green function ∆ret S0 (x, y) = ∆m (x − y) of the Klein Gordon operator: Lemma 1. In the sense of formal power series in λ, the retarded Green function ∆ret S0 +λS is given by the following formula [1]: ∞ ret ret n (−λ) ∆S0 +λS (x, y) = ∆S0 (x, y) + d(u1 , . . . , un )d(v1 , . . . , vn )∆ret S0 (x, u1 ) n=1

δ S δ2S · · · ∆ret ∆ret (vn , y). (2.7) S0 (vn−1 , un ) δϕ(u1 )δϕ(v1 ) δϕ(un )δϕ(vn ) S0 Its support is contained in the set

δS δS (x, y)|x ∈ supp + V¯+ ∧ y ∈ supp + V¯− ∪ {(x, y)|x ∈ y + V¯+ }. δϕ δϕ (2.8) ·

2

that the free action is not an element of F (C). Therefore we interpret S0 as functional on the subspace of compactly supported functions in conﬁguration space C. However, this restriction 0 is an element of P. is not necessary (and, hence, will not be done) for the free ﬁeld equation: δS δϕ

d Note

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In case of a local interaction S ∈ Floc (C) the support can be limited even stronger: ret δ∆S0 +S ⊂ {(x, y)|x ∈ y + V¯+ }. supp δϕ

(2.9)

Our assumption ensures that the pointwise product of distributions in (2.7) exists. Proof. We multiply the left-hand side equation in (2.6) by ∆ret S0 (z, u) and integrate over z afterwards to obtain the relation ret ∆ret S0 +λS (x, u) = ∆S0 (x, u) − λ

dy dz ∆ret S0 +λS (x, y)

δ2S ∆ret (z, u), δϕ(y)δϕ(z) S0

(2.10)

which can be solved by recursion on the powers of λ (2.7). For local interactions δ∆ret S ∈ Floc (C), the support property (2.9) follows immediately from supp δϕS0 ⊂ δ2 S {(x, y)|x ∈ y + V¯+ } and δϕ(u)δϕ(v) = 0 for u = v. In the general case S ∈ F(C) δ2 S δS we use supp δϕ2 ⊂ supp δϕ × supp δS that x has to lie in the δϕ to conclude δS future of u1 ∈ supp δϕ and y in the past of vn ∈ supp δS δϕ to get a non-vanishing contribution of the integral in (2.7). The space of all smooth solutions of the Euler–Lagrange equation with respect to the action Stot will be denoted by CStot ⊂ C. Interacting ﬁelds FS , corresponding to some functional F ∈ F(C), could be deﬁned by restricting F to the space def of solutions CS0 +S : FS = F |CS0 +S . However, the idea of perturbative algebraic classical ﬁeld theory is to introduce interacting ﬁelds as functionals on the space CS0 of free solutions. This corresponds to the usual interaction picture known from QFT where interacting ﬁelds are constructed as operators on the Fock space of the underlying free theory. Therefore one introduces retarded wave operators which map solutions of the free theory to solutions of the interacting theory [1]. As it turns out, it is convenient to construct such a map on the space C of all ﬁeld conﬁgurations (oﬀ-shell formalism) and not only on the space of solutions, as it was done in [1]: Deﬁnition 1. A retarded wave operator is a family of maps (rS0 +S,S0 )S∈F (C) from C into itself with the properties: (i) rS0 +S,S0 (f )(x) = f (x) for x suﬃciently early 0 +S) 0 ◦ rS0 +S,S0 = δS (ii) δ(Sδϕ δϕ . Lemma 2. The retarded wave operator (rS0 +S,S0 )S∈F (C) exists and is unique and invertible in the sense of formal power series in the interaction S.

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Proof. To determine rS0 +S,S0 we multiply S in Deﬁnition 1(ii) by a real parameter λ and diﬀerentiate with respect to this parameter, ending up with d δS δ(S0 + λS) ◦ rS0 +λS,S0 · ϕ(y) ◦ rS +λS,S0 + ◦ rS0 +λS,S0 = 0. dy δϕ(x)δϕ(y) dλ 0 δϕ(x) After multiplication of this equation with ∆ret S0 +λS (z, x) and integration over x, we obtain the following diﬀerential equation for rS0 +λS,S0 d δS ret rS +λS,S0 = − dx ∆S0 +λS (z, x) · ϕ(z) ◦ (2.11) ◦ rS0 +λS,S0 . dλ 0 δϕ(x) Finally, integration over λ leads to the equation 1 δS ϕ(z) ◦ rS0 +S,S0 = ϕ(z) − dλ dx ∆ret (z, x) · ◦ rS0 +λS,S0 , S0 +λS δϕ(x) 0

(2.12)

which can be solved iteratively in the sense of formal power series in the interaction S. We deﬁne the retarded wave operator rS0 +S1 ,S0 +S2 connecting two interacting theories by rS0 +S1 ,S0 +S2 := rS0 +S1 ,S0 ◦ (rS0 +S2 ,S0 )−1 . Obviously it fulﬁlls Deﬁnition 1(i) and

δ(S0 +S1 ) δϕ

◦ rS0 +S1 ,S0 +S2 =

rS0 +S1 ,S0 +S2 ◦ rS0 +S2 ,S0 +S3 = rS0 +S1 ,S0 +S3 .

(2.13) δ(S0 +S2 ) δϕ

and (2.14)

Retarded ﬁelds (to the interaction S and the free theory S0 and corresponding to the functional F ∈ F(C)) are deﬁned by = F ◦ rS0 +S,S0 : C → C. FSret 0 ,S def

(2.15)

A crucial property of classical interacting ﬁelds — which does not hold anymore for interacting quantum ﬁelds — is their factorization with respect to the classical product ret ret (F · G)ret S0 ,S = FS0 ,S · GS0 ,S .

(2.16)

This is why certain symmetry properties of classical ﬁeld theory in general cannot be transferred directly into quantum ﬁeld theory (see Sec. 5.1). In classical ﬁeld theory retarded products Rcl are deﬁned as coeﬃcients in the expansion (with respect to the interaction) of interacting retarded ﬁelds [1]: dn def ⊗n ⊗n F ◦ rS0 +λS,S0 . (2.17) Rcl : F (C) ⊗ F(C) → F (C), Rcl (S , F ) = dλn λ=0

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Interacting ﬁelds can then be written as FSret 0 ,S

∞ 1 Rcl (S ⊗n , F ) ≡ Rcl (eS⊗ , F ) n! n=0

(2.18)

where the right-hand side of is interpreted as a formal power series in S (we do not care about convergence of the series). In the last expression Rcl is viewed as a linear map ∞

Rcl : TF (C) ⊗ F(C) → F (C),

(2.19)

where TV = C ⊕ n=1 V ⊗n denotes the tensor algebra corresponding to some vector space V. By introducing the diﬀerential operator δ δS def , (2.20) DS0 ,S (λ) = − dx dy ∆ret S0 +λS (x, y) δϕ(y) δϕ(x) def

we obtain from (2.12) the following explicit expression for the interacting ﬁeld: λn−1 λ1 ∞ 1 FSret F + dλ dλ · · · dλn DS0 ,S (λn ) · · · DS0 ,S (λ1 )F. (2.21) 1 2 0 ,S n=1

0

0

0

To ﬁrst order in S this formula reads (see also [1]) δS δF Rcl (S, F ) = − dx dy ∆ret . S0 (x, y) δϕ(y) δϕ(x)

(2.22)

We can now endow classical ﬁelds with a Poisson structure: we introduce the (oﬀ-shell) Poisson bracket using Peierls deﬁnition [21] (see also [1, 22]) Deﬁnition 2. The Poisson bracket associated to the action S ∈ F(C) is the map {·, ·}S : F (C) ⊗ F(C) → F (C)

(2.23)

{F, G}S = RS (F, G) − RS (G, F )

(2.24)

def

where

d G ◦ rS+λF,S . RS (F, G) = dλ λ=0 def

(2.25)

The properties δG δF • RS (F, G) = − dx dy δϕ(x) ∆ret S (x, y) δϕ(y) (which is a generalization of (2.22)) and • {·, ·}S is indeed a Poisson bracket, i.e. it satisﬁes the Leibniz rule and the Jacobi identity, are proved in [1] for the on-shell restrictions RSon-shell (F, G) = RS (F, G)|CS and -shell = {F, G} | . These proofs can easily be generalized to R and {·, ·} . {F, G}on S CS S S S

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The Jacobi identity is derived from

d {RS (H, F ), G}S + {F, RS (H, G)}S = RS (H, {F, G}S ) + {F, G}S+λ H , dλ λ=0 (2.26)

which is the inﬁnitesimal version of the statement that the map F (C) F → F ◦ rS1 ,S2 ∈ F(C) is a canonical transformation: {F ◦ rS1 ,S2 , G ◦ rS1 ,S2 }S2 = {F, G}S1 ◦ rS1 ,S2 .

(2.27)

In [1] only the proof of the inﬁnitesimal version (2.26) is given. We are going to show that integration of (2.26), written in a suitable form, yields indeed (2.27).e First note d d F ◦ rS,S+λH = − F ◦ rS+λH,S = −RS (H, F ). (2.28) dλ dλ λ=0

λ=0

Let H := S2 − S1 , S(λ) := S1 + λH and Fλ := F ◦ rS1 ,S(λ) . With that we obtain d {F ◦ rS1 ,S(λ) , G ◦ rS1 ,S(λ) }S(λ) ◦ rS(λ),S1 dλ λ=λ0 d = ({Fλ0 ◦ rS(λ0 ),S(λ) , Gλ0 ◦ rS(λ0 ),S(λ) }S(λ) ◦ rS(λ),S(λ0 ) ) ◦ rS(λ0 ),S1 dλ λ=λ0 = 0

(2.29)

by using (2.25), (2.28) and the inﬁnitesimal version (2.26). Integrating this equation over λ0 from λ0 = 0 to λ0 = 1 it results the assertion. Due to the perturbative expansion around the free theory only the Poisson bracket associated to the free action, {·, ·}cl ≡ {·, ·}S0 , will be used in the following sections. As one can easily check, the retarded products (2.17) have the same properties as the on-shell retarded products in [1] (which are related to (2.17) by on−shell (S ⊗n , F ) = Rcl (S ⊗n , F )|CS0 ). These are the properties which are used Rcl to deﬁne retarded products in perturbative QFT in an axiomatic way. 3. Perturbative Quantum Field Theory We summarize the quantization of perturbative classical ﬁelds as it is worked out in [16] on the basis of causal perturbation theory [17, 18] and work of Steinmann [23]. Since the direct quantization of an interacting theory is in general not solved, we quantize the free theory, around which the perturbative expansion is done (see Sec. 2), by using deformation quantization: we replace F (C) and def

def

Floc (C) by F = F (C)[[]] and Floc = Floc (C)[[]], respectively, (i.e. all functionals are formal power series in ) and deform the classical product into the -product, e This

proof is due to Klaus Fredenhagen.

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: F × F → F, which is still associative but non-commutative and is deﬁned by ∞ n δnF (F G)(ϕ) := dx1 · · · dxn dy1 · · · dyn n! δϕ(x1 ) · · · δϕ(xn ) n=0 ·

n

Hm (xi − yi )

i=1

δnG . δϕ(y1 ) · · · δϕ(yn )

(3.1)

There is a freedom in the choice of the 2-point function Hm (x): it is required to diﬀer from the Wightman 2-point function ∆+ m (x) by a smooth and even function of x, to be Lorentz invariant and to satisfy the Klein Gordon equation. The “vacuum state” is the map (see (2.2)) ω0 : F → C[[]],

F → F (0) = f0 .

(3.2)

For the interacting quantum ﬁeld FG , (F, G ∈ Floc ) one makes the ansatzf of a formal power series in the interaction G: FG =

∞ 1 Rn,1 G⊗n , F ≡ R(eG ⊗ , F ). n! n=0

(3.3)

⊗n The “retarded product” Rn,1 is a linear map, from Floc ⊗ Floc into F which is symmetric in the ﬁrst n variables. The last expression in (3.3) is understood analogously to (2.19). We interpret R(A1 (x1 ), . . . ; An (x n )), A1 , . . . , An ∈ P, as F n valued distributions on D(M ), which are deﬁned by: dx h(x) R(. . . , A(x), . . .) := R(· · · ⊗ A(h) ⊗ · · ·) ∀h ∈ D(M). Interacting ﬁelds are deﬁned by the following axioms [16], which are motivated by the principle that we want to maintain as much as possible of the classical structure in the process of quantization:

Basic axioms: Initial Condition. R(F ) ≡ R0,1 (1, F ) = F . δH ¯ Causality. FG+H = FG if supp δF δϕ ∩ supp δϕ + V+ = ∅. GLZ Relation. In the classical GLZ Relation we replace the Poisson bracket {·, ·}cl by 1 [·, ·] = {·, ·}cl + O() i (where [H, F ] ≡ H F − F H). This gives d (FG+λH − HG+λF ). {FG , HG } = dλ def

{·, ·} =

(3.4)

(3.5)

λ=0

Based on these requirements, the retarded products Rn,1 can be constructed by induction on n. However, in each inductive step one is free to add a local f With

respect to factors of , our conventions (R) diﬀer from [16] (RDF ), namely R(eG ⊗, F ) =

G/ RDF (e⊗ , F ).

However, for the T -products we use the same conventions.

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functional, which corresponds to the usual renormalization ambiguity. This ambiguity is reduced by imposing the following further axioms: Renormalization conditions: Unitarity. (FG )∗ = FG∗ ∗ ↑ Poincare Covariance. For L ∈ P+ we set ϕL (x) := ϕ(L−1 x) and hL (x) := −1 h(L x), h ∈ C, and deﬁne an automorphism

βL (fn , ϕ⊗n ) = fn , (ϕL−1 )⊗n ,

βL : F → F ;

(3.6)

↑ that is (βL F )(h) = F (hL−1 ). P+ -covariance of the interacting ﬁelds means: ↑ . βL (FG ) = (βL F )βL G ∀L ∈ P+

Field Independence. The interacting ﬁeld FG depends on ϕ only through F and δR = 0. This condition is equivalent to the requirement that R fulﬁlls the G: δϕ(x) causal Wick expansion [18], that is Rn−1,1 (A1 (x1 ) ⊗ · · · ⊗ An−1 (xn−1 ), An (xn ))   1 = ω0 Rn−1,1 · · · l1 ! · · · ln ! a ···a l1 ,...,ln

·

li n

i1

 li

ili

∂ Ai (xi ) · · · ∂(∂ ai1 ϕ) · · · ∂(∂ aili ϕ)

∂ aiji ϕ(xi )

(3.7)

i=1 ji =1

with multi-indices aiji ∈ Nd0 . Field Equation.

ϕG (x) = ϕ(x) −

∆ret m (x

− y)

δG δϕ(y)

dy,

∀G ∈ Floc .

(3.8)

G

Smoothness in m. Through the GLZ condition the interacting ﬁelds depend on the 2-point function Hm and with that they depend on the mass m of the free ﬁeld: FG ≡ (FG )Hm . We require that the maps 0 ≤ m → (FG )Hm ,

F, G ∈ Floc ,

(3.9)

are smooth. In even dimensional spacetime this excludes the 2-point function ∆+ m due to logarithmic singularities at m = 0; more generally, homogeneous scaling of Hm is not compatible with smoothness in m ≥ 0. As in [16] we work with the µ (and the corresponding Feynman propagator) which is distin2-point function Hm µ depends on an additional mass guished by almost homogeneous scaling [19]. Hm parameter µ > 0 and is explicitly given in [16, Appendix A]. For the corresponding star product, retarded product and interacting ﬁelds we write m,µ , R(m,µ) and µ (FG )(m,µ) ≡ (FG )Hm , respectively.

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µ-Covariance. The -products (m,µ )µ>0 (and the -product with respect to ∆+ m) are equivalent, that is there exists an invertible operator (µ2 /µ1 )Γ which intertwines these products (see e.g. [14]): F m,µ2 G = (µ2 /µ1 )Γ (((µ2 /µ1 )−Γ F ) m,µ1 ((µ2 /µ1 )−Γ G)), ∞ 1 def where rΓ = 1 + k=1 k! (log(r) · Γ)k (for r > 0) and δ2 def Γ ≡ Γ(m) = . dx dy md−2 f (m2 (x − y)2 ) δϕ(x)δϕ(y) (m)

(m)

(m)

(3.10)

(3.11)

(The smooth function f is explicitly given by [16, formula (A.9)].) The axiom µCovariance requires that (µ2 /µ1 )Γ intertwines also the retarded productsg : (m)

R(m,µ2 ) = (µ2 /µ1 )Γ

◦ R(m,µ1 ) ◦ T(µ2 /µ1 )−Γ

(m)

.

(3.13)

Scaling. The mass dimension of a monomial in P is deﬁned by the conditions d−2 + |a| and dim(A1 A2 ) = dim(A1 ) + dim(A2 ) (3.14) 2 for all monomials A1 , A2 ∈ P. The mass dimension of a polynomial in P is the maximum of the mass dimensions of the contributing monomials. We denote by Phom the set of all ﬁeld polynomials which are homogeneous in the mass dimension. A scaling transformation σρ is introduced as an automorphism of F (considered as an algebra with the classical product) by n(2−d) ⊗n def 2 dx1 · · · dxn fn (x1 , . . . , xn )ϕ(x1 /ρ) · · · ϕ(xn /ρ). (3.15) σρ (fn , ϕ ) = ρ dim(∂ a ϕ) =

For A ∈ Phom we obtain ρdim(A) σρ (A(ρx)) = A(x) . Our condition of almost homogeneous scaling states that (ρ−1 m,µ)

σρ ◦ Rn,1

◦ (σρ−1 )⊗(n+1) ,

n ∈ N0 ,

m ≥ 0,

µ > 0,

(3.16)

is a polynomial in (log ρ). The construction of the retarded products proceeds in terms of the distributions R(A1 (x1 ), . . . ; An (xn )) , A1 , . . . , An ∈ P. Since the retarded products depend only on the functionals (and not on how the latter are written as smeared ﬁelds (2.4)), they must satisfy the Action Ward Identity (AWI) [16]: ∂µx Rn−1,1 (. . . , Ak (x), . . .) = Rn−1,1 (. . . , ∂µ Ak (x), . . .).

(3.17)

The AWI can simply be fulﬁlled by constructing R(A1 (x1 ), . . . ; An (xn )) ﬁrst only for balanced ﬁelds Ak ∈ Pbal ∀k, and by using the AWI and linearity for the extension to general ﬁelds Ak ∈ P. a linear map f : V → V (where V is a vector space) we deﬁne 0 1 ∞ ∞ M M Tf : TV → TV ; (Tf ) @c ⊕ (vj1 ⊗ · · · ⊗ vjj )A = c ⊕ (f (vj1 ) ⊗ · · · ⊗ f (vjj )).

g Given

j=1

j=1

(3.12)

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The axioms smoothness in m, µ-covariance and scaling can be replaced by the weaker axiom Scaling Degree, which requires that “renormalization may not make the interacting ﬁelds more singular” (in the UV-region). Usually this is formulated in terms of Steinmann’s scaling degree [24]:

def δ sd(f ) = inf δ ∈ R lim ρ f (ρx) = 0 , f ∈ D (Rk ) or f ∈ D (Rk \{0}). ρ↓0

(3.18) Namely, one requires sd(ω0 (R(A1 , . . . ; An ))(x1 − xn , . . .)) ≤

n

dim(Aj ),

∀Aj ∈ Phom ,

(3.19)

j=1

where Translation Invariance is assumed. In the inductive construction of the sequence (Rn,1 )n∈N (given in [16]), the problem of renormalization appears as the extension of C[[]]-valued distributions from D(Rdn \{0}) to D(Rdn ). This extension has to be done in the sense of formal power series in , that is individually in each order in . With that it holds lim R = Rcl .

→0

(3.20)

Namely, the GLZ Relation is the only axiom which depends explicitly on and in the classical limit it goes over into the classical GLZ Relation, due to (3.4). The retarded product, having two diﬀerent kinds of arguments, can be derived from the time ordered product (“T -product”) T : TFloc → F , which is totally symmetric i.e. it has only one kind of arguments. The corresponding relation is Bogoliubov’s formula: d iS/ S −1 S(S + τ F ), S(S) ≡ T (e⊗ ). (3.21) R(e⊗ , F ) = −i S(S) dτ τ =0 The axioms for retarded products translate directly into corresponding axioms for T -products, see [16, Appendix E]. There is no axiom corresponding to the GLZ Relation. The latter can be interpreted as “integrability condition” for the “vector potential” R(eS⊗ , F ), that is it ensures the existence of the “potential” S(S) fulﬁlling (3.21); for details see [19]. (A derivation of the GLZ Relation from (3.21) is given in [13].) In [16] it is shown that there exist retarded products which fulﬁll all axioms. The non-uniqueness of solutions is characterized by the “Main Theorem”; we use the version given in [16]: ˆ be retarded products which fulﬁll the basic axioms Theorem 3. (a) Let R and R ↑ -Covariance, Field Independence and the renormalization conditions Unitarity, P+ and Field Equation. Then there exists a unique, symmetric and linear map D : TFloc → Floc

(3.22)

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with D(1) = 0, D(F ) = F (∀F ∈ Floc ), such that for all F, S ∈ Floc the following intertwining relation holds (in the sense of formal power series in λ) λS

ˆ λS , F ) = R(eD(e⊗ ) , D(eλS ⊗ F )). R(e ⊗ ⊗ ⊗

(3.23)

In addition, D satisﬁes the conditions: δD(F1 ⊗ · · · ⊗ Fn ) δFi (i) supp ⊂ i∈n supp , Fi ∈ Floc δϕ δϕ δD(eF δF ⊗) = D eF (ii) ⊗ ⊗ δϕ δϕ (iii) D(eF ⊗ ⊗ ϕ(h)) = ϕ(h), ∀F ∈ Floc (iv) D(F ⊗n )∗ = D((F ∗ )⊗n )

↑ (v) βL ◦ D = D ◦ TβL , ∀L ∈ P+ ˆ (m,µ) are smooth in m ≥ 0 and satisfy the axioms µ-Covariance (vi) (A) If R(m,µ) , R and Scaling, then the corresponding D(m,µ) is also smooth in m, invariant under scaling −1

σρ ◦ D(ρ

m,µ)

◦ Tσρ−1 = D(m,µ)

(3.24)

and µ-covariant h (m)

D(m,µ2 ) = (µ2 /µ1 )Γ

◦ D(m,µ1 ) ◦ T(µ2 /µ1 )−Γ

(m)

.

(3.25)

ˆ satisfy the axiom Scaling degree, then (B) Alternatively, if R and R sd(ω0 (D(A1 , . . . , An ))(x1 − xn , . . .)) ≤

n

dim(Aj ),

∀Aj ∈ Phom .

j=1

(3.26) (b) Conversely, given R and D as above, Eq. (3.23) gives a new retarded product ˆ which satisﬁes the axioms. R Since the classical limit of the axioms has a unique solution (which is Rcl ), the map D is trivial to lowest order in , i.e. D(eS⊗ ) = S + O() and D(eS⊗ ⊗ F ) = F + O().

(3.27)

The relation (3.23) can equivalently be expressed in terms of time ordered products, i D(eS )/ iS/ Tˆ(e⊗ ) = T e⊗ ⊗ , (3.28) ˆ respectively, where T and Tˆ are the time ordered products belonging to R and R, according to (3.21). h The

claim in [16] that D (m,µ) is independent of µ, is wrong.

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4. Proper Vertices for T -Products Proper verticesi are an old and standard tool in perturbative QFT. In terms of R-products the basic idea is the following: a perturbative QFT can be rewritten as classical ﬁeld theory with a non-local interaction (“proper interaction”) which agrees to lowest order in with the original local interaction. Since Rcl is the sum of all (connected) tree diagrams (as explained in Appendix A), this rewriting means that we interpret each diagram as tree diagram with non-local vertices (“proper vertices”) given by the 1-particle-irreducible (1PI) subdiagrams. This structural decomposition of Feynman diagrams can just as well be done for T -products and it is this latter form of proper vertices which is well known in the literature. Since T -products are totally symmetric, it is simpler to introduce proper vertices in terms of T -products than in terms of R-products and, hence, we work with the former (for the introduction of proper vertices for R-products see Appendix A). A main motivation to introduce proper vertices is that the renormalization of an arbitrary diagram reduces to the renormalization of its 1PI-subdiagrams. Indeed, due to the validity of the MWI for tree diagrams (i.e. in classical FT), the MWI can equivalently be formulated in terms of proper vertices (i.e. in terms of 1PI-diagrams), see Sec. 5.3. This “proper MWI” formally coincides with the usual formulation of Ward identities in the functional approach to QFT (for an overview see e.g. [10]). 4.1. Diagrammatics and deﬁnition of the 1-particle-irreducible part T 1PI of the time ordered product To introduce proper vertices we need the tree part Ttree for non-local entries and, for later purpose, the 1PI part T 1PI of the time ordered product T . The deﬁnition of Ttree can obviously be given in terms of Feynman diagrams; but in case of T 1PI we are faced with the problem that for loop diagrams the decomposition of T (A1 (x1 ), . . .) into contributions of Feynman diagrams is non-unique, due to the local terms coming from renormalization. To motivate the deﬁnition of T 1PI we ﬁrst study a “smooth and symmetric product”: let f ∈ C ∞ (R4 , C) with f (x) = f (−x), ∀x. We deﬁne f : F ⊗ F → F by replacing in the deﬁnition (3.1) of the -product the 2-point function Hm by f . This product, f , is associative and commutative. By deﬁnition f satisﬁes “Wick’s theorem”. Due to that, nf,j=1 Fj ≡ F1 f · · · f Fn can uniquely be viewed as a sum of diagrams. In spite of the possible non-locality of the Fj ’s, we symbolize each Fj by one vertex. The contractions are symbolized by inner lines connecting the i In the literature proper vertices (or the “proper interaction”) are sometimes called “eﬀective vertices” (or “eﬀective interaction”, respectively). However, diﬀerently to what we are doing here, the notion “eﬀective ﬁeld theory” usually means an approximation to the perturbation series. For this reason we omit the word “eﬀective” and use the terminology of [25, Sec. 6-2-2].

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vertices, which are not oriented due to f (x) = f (−x). (We do not draw any external lines.) The contribution of the connected diagrams (denoted by (nf,j=1 Fj )c ) is obtained from nf,j=1 Fj by subtraction of the contributions of all disconnected diagrams; this gives the recursion relation (see e.g. [15]) (f,j∈J Fj )c , (4.1) (nf,j=1 Fj )c = nf,j=1 Fj − |P |≥2 J∈P

where the sum runs over all partitions P of {1, . . . , n} in at least two subsets and means the classical product. One easily sees that the linked cluster theorem applies to f : F eF f = exp• (ecf ),

where

def

eF f = 1 +

∞ F f n , n! n=1

def

eF cf =

∞ (F f n )c n! n=1

(4.2)

(with F f n being the n-fold product F f · · ·f F ) and exp• denotes the exponential function with respect to the classical product. Analogously to (4.1) the contribution of all 1PI-diagrams to nf,j=1 Fj (denoted by (nf,j=1 Fj )1PI ), is obtained from the connected diagrams (nf,j=1 Fj )c by subtracting the contributions of all connected one-particle-reducible diagrams. To formulate this we need the contribution of all connected tree diagrams to (nf,j=1 Fj )c , which we denote by (nf,j=1 Fj )ctree . This diagrammatic deﬁnition of (nf,j=1 Fj )ctree fulﬁlls the following unique and independent characterizations: • By recursion: One easily ﬁnds that the connected tree diagrams satisfy the recursion relation n δ k Fn+1 c F ) = · · · dx dy · · · dy dx (n+1 j 1 k 1 k tree f,j=1 δϕ(x1 ) · · · δϕ(xk ) k=1 k j=1

·

f (xj − yj )

1 k!

I1 ··· Ik ={1,...,n}

δ (f,j∈I1 Fj )ctree · · · · δϕ(y1 )

δ (f,j∈Ik Fj )ctree , δϕ(yk )

(4.3)

where Ij = ∅ ∀j, means the disjoint union. (Note that in the sum over I1 , . . . , Ik the succession of I1 , . . . , Ik is distinguished and, hence, there is a factor 1 .) of k! • By the power in : As explained in [15, Sec. 5.2] it holds (nf,j=1 Fj )c = O(n−1 )

for F1 , . . . , Fn ∼ 0 ,

(4.4)

and the contribution of all tree diagrams is given by the terms of lowest order in (nf,j=1 Fj )ctree = n−1 lim −(n−1) (nf,j=1 Fj )c . →0

(4.5)

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The contribution of all tree diagrams to nf,j=1 Fj , which we denote by (nf,j=1 Fj )tree , is related to the connected tree diagrams by the linked cluster theorem: F eF f,tree = exp• (ecf,tree ). (This follows immediately from (4.2) by selecting all tree diagrams.) We now give the (above announced) unique recursive characterization of the 1PI-diagrams: (f,J∈P (f,j∈J Fj )1PI )ctree . (4.6) (nf,j=1 Fj )1PI = (nf,j=1 Fj )c − |P |≥2

The formulas (4.1), (4.4) and (4.5) hold also for the usual -product (i.e. with Hm instead of f ) [15]; but (4.6) needs to be reﬁned, because Hm is not symmetrical and, hence, Hm is not commutative (in particular (F1 Hm · · · Hm Fn )ctree is not symmetrical). Turning to the time ordered product T , we will use (4.1) and (4.6) as motivation for the (recursive) deﬁnition of the connected part T c and the 1PI-part T 1PI of T , respectively. So we deﬁne [15] def T c (⊗j∈J Fj ). (4.7) T c (⊗nj=1 Fj ) = T (⊗nj=1 Fj ) − |P |≥2 J∈P

It follows that T and T c are related by the linked cluster theorem (4.2): T (eiF ⊗ ) = )). exp• (T c (eiF ⊗ c of T applies also to The following deﬁnition of the connected tree part Ttree j non-local entries : c c : F ⊗n → F ; Ttree,n (⊗nj=1 Fj ) = (F1 ∆F · · · ∆F Fn )ctree , Ttree,n

(4.8)

i.e. we replace in the deﬁnition of (f . . .)ctree the smooth function f by the Feynman propagator µ 0 µ 0 µ ∆F (z) ≡ ∆F m (z) = Θ(z )Hm (z) + Θ(−z )Hm (−z) µ F = −i ∆ret m (z) + Hm (−z) = ∆ (−z).

(4.9)

For tree diagrams the resulting expressions are well deﬁned, since pointwise prodc fulﬁlls the recursion ucts of Feynman propagators do not appear. Obviously Ttree j In QCD the interaction S = κS + κ2 S is a sum of a term of ﬁrst order in the coupling constant 1 2 R R κ, S1 ∼ gAA∂A (g ∈ D(M, R)), and a term of second order in κ, S2 ∼ g 2 AAAA. One can achieve that the order in κ agrees with the order of the T - (or R-) product [26]. Namely, one starts with T1 (S1 ), the R term S2 is generated by a non-trivial renormalization of a certain tree diagram: in T2 (S1⊗2 ) ∼ dx dy gAA(x) gAA(y) ∂∂∆F (x−y)+· · · the propagator ∂ µ ∂ ν ∆F (x−y) is replaced by ∂ µ ∂ ν ∆F (x−y)−1/2 g µν δ(x−y). Due to the inductive procedure of causal perturbation theory this additional term propagates to higher orders such that this modiﬁed T -product, T N , yields i(κS +κ2 S )

2 1 the same S-matrix: T N (eiκS ) = T (e⊗ 1 ) in the sense of formal power series in κ. (The ⊗ c (4.8) and corresponding renormalization map D (3.22) is given in [27].) Our deﬁnitions of Ttree Ttree (4.10) do not contain this 1/2 g µν δ-term, in agreementPwith the deﬁnition of Rcl (2.17). Generally, in this paper all terms Sn of the interaction S = n≥1 κn Sn enter the perturbative construction of the S-matrix (or interacting ﬁeld) already to ﬁrst order of the T - (or R-) product.

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c relation (4.3). In case of local entries another unique characterization of Ttree is c possible: doing renormalization individually in each order in , T satisﬁes (4.4) c is the classical limit of −(n−1) Tnc similarly to (4.5). (This is the and −(n−1) Ttree,n translation of (3.20) into T -products, see [15, Sec. 5.2].) Analogously one deﬁnes the tree part Ttree of T by def

Ttree,n (⊗nj=1 Fj ) = (F1 ∆F · · · ∆F Fn )tree ,

Fj ∈ F ∀j.

(4.10)

Obviously the linked cluster theorem for (f,tree , cf,tree ) is valid also for f = ∆F : c iF Ttree (eiF ⊗ ) = exp• (Ttree (e⊗ )). c Since Ttree is totally symmetric we may deﬁne T 1P I in analogy to (4.6) by the recursive formula T 1PI (F ⊗n ) = T c (F ⊗n ) − def

n

k=2 l1 +···+lk =n, lj ≥1 ∀j

n! k! l1 ! · · · lk !

c 1PI · Ttree, (F ⊗l1 ) ⊗ · · · ⊗ T 1PI (F ⊗lk )). k (T

(4.11) c (m,µ)

c The renormalization conditions listed in Sec. 3 are satisﬁed by Ttree ≡ Ttree , (m,µ) Ttree ≡ Ttree and, provided that T fulﬁlls these conditions, also by T 1PI ≡ T 1PI (m,µ) (apart from the Field Equation). This can be veriﬁed by using the deﬁnitions (4.8), (4.10), (4.11) and corresponding properties of the Feynman propagator. c by the Or, in case of local interactions, these properties can be derived for Ttree −(n−1) c Tn ; the linked cluster theorem implies then their validity classical limit of for Ttree .

4.2. Deﬁnition and basic properties of the vertex function ΓT Note that T and Ttree satisfy the relations T(tree)(1) = 1,

T(tree) (F ) = F,

T(tree) n+1 (1 ⊗ F1 ⊗ · · ·) = T(tree) n (F1 ⊗ · · ·), (4.12)

which imply the following conclusions for Ttree and T T(tree)

P∞

e⊗

n n=1 Fn λ

⊗

∞

Gn λn

= 0 ⇒ Gn = 0

∀n

(4.13)

n=0

P∞ P∞ F λn G λn = T(tree) e⊗ n=1 n ⇒ Fn = Gn T(tree) e⊗ n=1 n

∀n

(4.14)

(where Fn , Gn ∈ F are independent of λ), as one obtains by proceeding by induction on the order in λ.

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R- and T -products can be obtained from each other by Bogoliubov’s formula (3.21), which we will use in the more explicit form −iS/ iS/ R(eS⊗ , F ) = T¯ (e⊗ ) T (e⊗ ⊗ F ),

(4.15)

where the anti-chronological product T¯ is deﬁned by iG −1 T¯ (e−iG = ⊗ ) ≡ T (e⊗ )

∞

n (1 − T (eiG ⊗ )) .

(4.16)

n=0 −1 (T (eiG is the inverse with respect to the -product. Although R contains solely ⊗ ) connected diagrams (see Appendix A), disconnected diagrams of T and T¯ contribute to (4.15). Unitarity reads iG∗ ∗ T¯(tree) (e−iG ⊗ ) = T(tree) (e⊗ ) ;

(4.17)

in this form it holds for the tree diagrams separately, where T¯tree is deﬁned by (4.10) with ∆F replaced by the Anti-Feynman propagator ∆AF (x) = ∆F (x)∗ . However, iG∗ ∗ note that Ttree (eiG ⊗ ) Ttree (e⊗ ) is not equal to 1. We deﬁne the “vertex function” ΓT implicitly by the following proposition: Proposition 4. There exists a totally symmetric and linear map ΓT : TFloc → F

(4.18)

which is uniquely determined by iS/

T (e⊗

iΓT (eS ⊗ )/

) = Ttree (e⊗

).

(4.19)

To zeroth and ﬁrst order in S we obtain ΓT (1) = 0,

ΓT (S) = S.

(4.20)

The deﬁning relation (4.19) also implies iS/

T (e⊗

iΓT (eS ⊗ )/

⊗ F ) = Ttree (e⊗

⊗ ΓT (eS⊗ ⊗ F )).

(4.21)

For S = 0 this gives F = ΓT (1 ⊗ F ). The proposition remains true if, in (4.19), we replace the time ordered product (T, Ttree ) by the anti-chronological product ¯T . (T¯, T¯tree ) and i by (−i); we denote the corresponding vertex function by Γ Proof. We construct ΓT (⊗nj=1 Fj ) by induction on n, starting with (4.20). Let ΓT of less than n factors be constructed. Then, (4.19) and the requirements total symmetry and linearity determine ΓT (⊗nj=1 Fj ) uniquely: ΓT (⊗nj=1 Fj ) = (i/)n−1 T (⊗nj=1 Fj ) −

|P |−1

(i/)

|P |≥2

Ttree

ΓT (⊗j∈J Fj ) ,

(4.22)

J∈P

where P is a partition of {1, . . . , n} in |P | subsets J. Obviously the so constructed ΓT is totally symmetric and linear.

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c The proof applies also to the connected parts (T c , Ttree ). Hence, a totally symmetric and linear map Γc can be deﬁned analogously to (4.18), (4.19), that is by iS/

T c (e⊗

iΓc (eS ⊗ )/

c ) = Ttree (e⊗

).

(4.23)

The linked cluster theorem for T and Ttree and the deﬁnitions of ΓT and Γc give iΓc (eS ⊗ )/

Ttree (e⊗

iΓT (eS ⊗ )/

) = Ttree (e⊗

)

(4.24)

and with (4.14) we conclude ΓT = Γc .

(4.25)

Therefore, on the right-hand side of (4.22) we may replace the time ordered products T and Ttree, k by their connected parts: ΓT (S ⊗n ) = (i/)n−1 T c (S ⊗n ) −

n

k=2 l1 + · · · + lk =n

(i/)k−1 n! k! l1 ! · · · lk !

lj ≥1 ∀j c ⊗l1 · Ttree ) ⊗ · · · ⊗ ΓT (S ⊗lk )). , k (ΓT (S

(4.26)

c (S ⊗n ) = O(n ) we Now let S ∼ 0 and F ∼ 0 . From (4.26) and T c(S ⊗n ) − Ttree inductively conclude

ΓT (eS⊗ ) = S + O(),

ΓT (eS⊗ ⊗ F ) = F + O().

(4.27)

Motivated by this relation and (4.19) we call ΓT (eS⊗ ) the “proper interaction”. By comparing the recursion relation (4.26) for ΓT with the recursive deﬁnition of T 1PI (4.11) we conclude: Corollary 5. iS/

ΓT (eS⊗ ) = (/i) T 1PI(e⊗

).

(4.28)

Analogously to the Main Theorem it holds: Lemma 6. The validity of the renormalization conditions for T (≡ T (m,µ) ) implies (m,µ) ): corresponding properties of ΓT (≡ ΓT ↑ ↑ -Covariance: βL ◦ ΓT = ΓT ◦ TβL for all L ∈ P+ ; • P+ S S ¯ • Unitarity: ΓT (e⊗ ) = ΓT (e⊗ ); δ ΓT (eS ⊗) S • Field Independence: = ΓT δS δϕ δϕ ⊗ e⊗ ;

• Field Equation: ΓT (eS⊗ ⊗ ϕ(h)) = ϕ(h); (m,µ)

• Smoothness in m ≥ 0: ΓT • µ-Covariance:

(m,µ ) ΓT 2

is smooth in m ≥ 0; (m,µ1 )

= (µ2 /µ1 )Γ ◦ ΓT

◦ T(µ2 /µ1 )−Γ ;

• Almost Homogeneous Scaling: In contrast to the map D of the Main Theorem (ρ−1 m,µ) (m,µ) ΓT scales only almost homogeneously; σρ ◦ ΓT ◦ T σρ−1 = ΓT + O(log ρ) is a polynomial in log ρ.

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• If, instead of Smoothness in m, µ-Covariance and Almost homogeneous Scaling, T satisﬁes the axiom Scaling Degree, then sd(ω0 (ΓT (A1 , . . . , An ))(x1 − xn , . . .)) ≤

n

dim(Aj ),

∀Aj ∈ Phom .

(4.29)

j=1

Proof. Each property can be proved for ΓT (S ⊗n ) (or ΓT (S ⊗n ⊗ϕ(h)), respectively) by induction on n: we work with the recursion relation (4.22) and use that T and Ttree satisfy the corresponding axiom. In case of the property Scaling Degree we take into account that ω0 (Ttree (· · ·)) is a tensor product of distributions tj and apply sd(⊗j tj ) = j sd(tj ). Only the Field Equation is somewhat more involved. We use that T and Ttree fulﬁl the Field Equation and the Field Independence. This implies T (ϕ(h) ⊗ F1 ⊗ · · · ⊗ Fn ) = c

dx dy h(x)∆F (x − y)

δ T c (F1 ⊗ · · · ⊗ Fn ) δϕ(y) (4.30)

c and the same equation for Ttree . In the latter case F1 , . . . , Fn may be non-local. (It is not necessary to work with the connected parts, but this simpliﬁes the formulas.) With the recursion relation (4.26) and the inductive assumption we obtain

ΓT (ϕ(h) ⊗ F1 ⊗ · · · ⊗ Fn ) = T (ϕ(h) ⊗ F1 ⊗ · · · ⊗ Fn ) − c

P

c Ttree

ϕ(h) ⊗

ΓT (FJ )

J∈P

δ = dx dy h(x) ∆ (x − y) T c (F1 ⊗ · · · ⊗ Fn ) δϕ(y) c − Ttree ΓT (FJ ) = 0,

F

P

(4.31)

J∈P

where FJ ≡ ⊗j∈J Fj and P runs through all partitions of {1, . . . , n}. Analogously to the conventions for R and T -products we sometimes write dx g(x)ΓT (A(x) ⊗ F2 · · ·) for ΓT ( dx g(x)A(x) ⊗ F2 · · ·). Since ΓT depends only on the functionals, it fulﬁlls the AWI: ∂xµ ΓT (A(x) ⊗ F2 · · ·) = ΓT (∂ µ A(x) ⊗ F2 · · ·). In the proper vertex formalism a ﬁnite renormalization T → Tˆ of the T ˆ T of the corresponding products is reﬂected in a ﬁnite renormalization ΓT → Γ ˆ T (eS )/ iΓ iS/ vertex functions. To derive this we insert Tˆ(e⊗ ) = Ttree (e⊗ ⊗ ) and (4.19)

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into (3.28) and obtain ˆ T (eS )/ iΓ ⊗

Ttree (e⊗

D(eS ⊗)

iΓT (e⊗

) = Ttree (e⊗

)/

).

(4.32)

(Note that the tree part Ttree is independent of the normalization of the T -product.) By using (4.14) we conclude S

ˆ T (eS ) = ΓT (eD(e⊗ ) ). Γ ⊗ ⊗

(4.33)

4.3. Comparison with the literature Deﬁnition of the “vertex functional” (or “proper function”) Γ in the literature, see e.g. [10]. Usually Γ(h) , h ∈ S(M, R), is deﬁned as the Legendre transformed j → h of the generating functional Z(j) of the connected Green’s functions (where j is the “classical source” of ϕ). With that Γ(h) is the generating functional of the 1PI-diagrams of T (eiS ⊗ ) (see [28]). We are going to express the latter fact in our formalism. To simplify the notations we study a scalar ﬁeld ϕ with free action S0 (ϕ) = 1/2 dx ((∂ϕ(x))2 − m2 (ϕ(x))2 ). Green’s functions are obtained by the Gell-Mann Low formula [29], which contains the adiabatic limit g → 1: ω0 (T (ϕ(x1 ) · · · ϕ(xn ) eiS(g)/ )) , g→1 ω0 (T (eiS(g)/ ))

G(x1 , . . . , xn ) = lim

(4.34)

∞ where S(g) = n=1 κn dx(g(x))n Ln (x) and κ is the coupling constant. All diagrams with vacuum-subdiagrams are divided out. These diagrams are disconnected and, hence, not of interest for our purposes. Namely, to obtain the vertex functional Γ one selects all diagrams of G(x1 , . . . , xn ) which are 1PI after amputation of the external legs. The contribution of these diagrams is given by G1PI (x1 , . . . , xn ) = lim dy1 · · · dyn ∆F (x1 − y1 ) · · · ∆F (xn − yn ) g→1

· ω0

δn T 1PI (eiS(g)/ ) + δn,2 ∆F (x1 − x2 ). δϕ(y1 ) · · · δϕ(yn ) (4.35)

From this expression Γ(h) is obtained by replacing each external leg ∆F (xl − yl ) by the classical ﬁeld h(yl ). In addition one multiplies with (−i)/n! and sums over n ≥ 1k : 1 dy1 · · · dyn h(y1 ) · · · h(yn ) Γ(h) = S0 (h) + lim i g→1 n! n≥1 δn 1PI iS(g)/ T (e ) . (4.36) · ω0 δϕ(y1 ) · · · δϕ(yn ) k Usually

it is assumed that limg→1 ω0 (ϕgL (x)) = 0, which implies runs only over n ≥ 2.

δΓ(h) | δh(x) h=0

= 0,i.e. the sum

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Note that the term S0 (h) comes from the δn,2 ∆F -term in (4.35). S

Relation to our proper interaction ΓT (e ⊗ (g )): We compare (4.36) with S(g)

the Taylor expansion in ϕ of ΓT (e⊗ ): 1 δn S(g) S(g) dy1 · · · dyn ϕ(y1 ) · · · ϕ(yn )ω0 ΓT (e⊗ ) ΓT (e⊗ ) = n! δϕ(y1 ) · · · δϕ(yn ) n≥0

(4.37) and use Corollary 5. This yields S(g)

S(g)

Γ(h) = S0 (h) + lim (ΓT (e⊗ )(h) − ω0 (ΓT (e⊗ ))). g→1

(4.38)

S(g)

(On the right-hand side the functionals S0 , ΓT (e⊗ ) ∈ F are evaluated on the classical ﬁeld conﬁguration h ∈ S(M, R).) 5. The Master Ward Identity 5.1. The classical MWI in the oﬀ-shell formalism In [1] the MWI for on-shell ﬁelds (i.e. the retarded products are restricted to the solutions of the free ﬁeld equation(s)) was derived in the framework of classical ﬁeld theory. Since here, we work throughout in a general oﬀ-shell formalism [16], we shall derive an oﬀ-shell version of the classical MWI. In addition, we give an equivalent formulation of the classical MWI in terms of Ttree -products that will be useful for the proper ﬁeld formulation of the MWI in Sec. 5.3. The classical oﬀ-shell MWI follows from the factorization (2.16) and the deﬁnition of the retarded wave operators. Let J be the ideal generated by the free ﬁeld equation(s), N δS0 def fn (x1 , . . . , xn ) ⊂ F, dx1 · · · dxn ϕ(x1 ) · · · ϕ(xn−1 ) J = δϕ(xn ) n=1 with N < ∞ and the fn ’s being deﬁned as in (2.2). Every A ∈ J can be written as δS0 def A = dx Q(x) , (5.1) δϕ(x) where Q is of the form Q(x) =

N

dx1 · · · dxn ϕ(x1 ) · · · ϕ(xn )fn+1 (x1 , . . . , xn , x).

(5.2)

n=0

Note that in the present framework of classical ﬁeld theory Q does not need to be a local functional. Given A ∈ J we introduce a corresponding derivation [1] δ def . (5.3) δA = dx Q(x) δϕ(x)

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From the deﬁning property of the retarded wave operators Deﬁnition 1(ii) we obtain δ(S0 + S) ◦ rS0 +S,S0 (A + δA S) ◦ rS0 +S,S0 = dx Q(x) ◦ rS0 +S,S0 δϕ(x) δS0 , (5.4) = dx Q(x) ◦ rS0 +S,S0 δϕ(x) which reads perturbatively Rcl (eS⊗ , A

+ δA S) =

δS0 ∈ J. dx Rcl eS⊗ , Q(x) δϕ(x)

(5.5)

This is the MWI written in the general oﬀ-shell formalism. Indeed, by restricting (5.5) on solutions of the free ﬁeld equation, the right-hand side vanishes and we obtain the on-shell version of the MWI, as it was derived in [1]. Note that for the simplest case Q = 1 the MWI reduces to the oﬀ-shell version of the (interacting) ﬁeld equation δS0 S δ(S0 + S) , (5.6) = Rcl e⊗ , δϕ(x) δϕ(x) which is an alternative formulation of the axiom Field Equation in Sec. 3. The classical ﬁeld equation (5.6) can be expressed in the time ordered formalism: δ(S0 + S) δS0 iS · Ttree (eiS (5.7) Ttree e⊗ ⊗ = ⊗ ). δϕ(x) δϕ(x) This identity holds even for non-local entries and can be obtained easily by using the deﬁnition of Ttree given in (4.10) and the fact that ∆F is a Green’s function of the Klein Gordon operator. Similarly to Rcl , the tree diagrams of the time ordered product factorize (cf. [30]), that is iS iS iS Ttree (eiS ⊗ ⊗ F ) · Ttree (e⊗ ⊗ G) = Ttree (e⊗ ⊗ F G) · Ttree (e⊗ ).

(5.8)

We now multiply the ﬁeld equation for Ttree with Ttree (eiS ⊗ ⊗ Q(x)). This yields the MWI in the time ordered formalism: δS0 Ttree (eiS ⊗ (A + δ S)) = dx Ttree (eiS . (5.9) A ⊗ ⊗ ⊗ Q(x)) · δϕ(x) We point out that the MWI for Ttree (5.9) holds also for non-local entries S, Q(x) and A. 5.2. Structure of possible anomalies of the MWI in QFT The classical MWI was derived for arbitrary interaction S ∈ F and arbitrary A ∈ J . For local functionals A ∈ Jloc ≡ J ∩ Floc and S ∈ Floc it can be transferred formally into pQFT (by the replacement Rcl → R), where it serves as an additional, highly non-trivial renormalization condition. It is impossible to fulﬁll this condition for all A ∈ Jloc . We aim to ﬁnd a general expression for the possible

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violations (“anomalies”) of the MWI. Later we will use this result as starting point for a proof of relevant cases of the MWI. This procedure is motivated by algebraic renormalization, where the QAP serves as the crucial input to study the possibility to fulﬁll some Ward identities (see [10] and references cited therein). The main insight into the structure of possible anomalies of the MWI is the fact that they can be expressed in terms of a local interacting ﬁeld: Theorem 7. Given a retarded product R fulﬁlling the basic axioms Initial Condition, Causality and GLZ Relation and given a local functional A=

dx h(x)Q(x)

δS0 ∈ Jloc , δϕ(x)

h ∈ D(M),

Q ∈ P,

(5.10)

there exists a unique, linear and symmetric map ∆A : TFloc → Floc

(5.11)

F1 ⊗ · · · ⊗ Fn → ∆A (F1 ⊗ · · · ⊗ Fn ) which is implicitly deﬁned by the “anomalous MWI” R eS⊗ , A + δA S + ∆A (eS⊗ ) =

dy h(y)R(eS⊗ , Q(y))

δS0 . δϕ(y)

(5.12)

As a consequence of (5.12) the map ∆A has the following properties: (i) ∆A depends linearly on A; (ii) locality expressed by the two relations: n δFi δA supp (a) ω0 (∆A (⊗nj=1 Fj )) = 0 if ∩ supp = ∅, δϕ δϕ i=1 n δ∆A (⊗nj=1 Fj ) δFi δA supp ⊂ ∩ supp ; (b) supp δϕ δϕ δϕ i=1 (iii) ∆A (1) = 0; (iv) ∆A ≡ 0 ⇔

R(eS⊗ , A

dx h(x)R(eS⊗ , Q(x))

+ δA S) =

(v) ∆A (F1 ⊗ · · · ⊗ Fn ) = O()

∀n > 0,

δS0 , δϕ(x)

∀S ∈ Floc ;

Fi ∼ 0 ,

(5.13)

and (vi) We set ∆nA ≡ ∆A |F ⊗n . For gj ∈ D(M) , Lj ∈ P it holds loc

∆nA (L1 (g1 )

⊗ · · · ⊗ Ln (gn )) =

dx1 · · · dxn dy g1 (x1 ) · · · gn (xn )h(y) · ∆n (L1 (x1 ) ⊗ · · · ⊗ Ln (xn ); Q(y)),

(5.14)

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where the distributional kernel ∆n (L1 (x1 ) ⊗ · · · ; Q(y)) is inductively given by ∆n (L1 (x1 ) ⊗ · · · ⊗ Ln (xn ); Q(y)) δS0 = −R ⊗nj=1 Lj (xj ); Q(y) · δϕ(y) n R ⊗j( =l) Lj (xj ); Q(y) (∂ a δ)(xl − y) − l=1

−

a

∂Ll (xl ) ∂(∂ a ϕ)

R(⊗i∈I Li (xi ); ∆|I | (⊗j∈I c Lj (xj ); Q(y))) c

I⊂{1,...,n} , I =∅

δS0 . + R ⊗nj=1 Lj (xj ); Q(y) · δϕ(y)

(5.15)

Note that (5.12) diﬀers from the MWI (5.5) only by the local term ∆A (eS⊗ ), which clearly depends on the chosen normalization of the retarded products. Therefore, property (iv) means that the MWI for A is fulﬁlled if and only if the corresponding map ∆A vanishes identically. Proof. To show the existence and uniqueness of ∆A we construct its components ∆nA by induction on n using (5.12). In this inductive procedure we also prove the def properties (i)–(iii) and (vi). To lowest order in S the condition (5.12) gives ∆A (1) = 0. Given n > 0, we assume the existence and uniqueness of linear and symmetrical ⊗k → Floc , 0 < k < n, which depend linearly on A, are local and maps ∆kA : Floc satisfy (vi), such that (5.12) is fulﬁlled to all lower orders in S: R(S ⊗k , A) + kR(S ⊗k−1 , δA S) + =

k k R(S ⊗k−l , ∆lA (S ⊗l )) l l=0

dx h(x)R(S ⊗k , Q(x))

δS0 δϕ(x)

(5.16)

for all k < n. We deﬁne ∆nA in terms of the inductively known ∆kA , k < n: δS0 def ∆nA (F1 ⊗ · · · ⊗ Fn ) = dx h(x)R(F1 ⊗ · · · ⊗ Fn , Q(x)) δϕ(x) n R(⊗i∈n\{k} Fi , δA Fk ) − R(F1 ⊗ · · · ⊗ Fn , A) + k=1

+



|J| R(⊗i∈I Fi , ∆A (⊗j∈J Fj )) ,

(5.17)

I J=n, |J|
where we used the notation n = {1, . . . , n} and I J for the disjoint union of I and J. For F1 = · · · = Fn = S the formula (5.17) agrees with (5.12) to order n in S. def

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Obviously, ∆nA is linear and symmetrical and it is uniquely determined by these properties and (5.12). We also see that ∆nA is linear in Q, respectively, A. The main task is to show the locality (ii) of the right-hand side of (5.17) (which also implies ∆nA (F1 ⊗ · · · ⊗ Fn ) ∈ Floc ). To this end we denote by Mn,1 (L1 (x1 ), . . . , Ln (xn ); Q(y)) δS0 def = R ⊗i∈n Li (xi ), Q(y) δϕ(y) n δLk (xk ) R ⊗i∈n\{k} Li (xi ), Q(y) + δϕ(y) k=1 R(⊗i∈I Li (xi ), ∆|J| (⊗j∈J Lj (xj ); Q(y))) + I J=n , |J|
the distributional kernel of the three terms in brackets in equation (5.17), where |J| Fi = dx gi (x)Li (x) (gi ∈ D(M), Li ∈ P) and we use (5.14) for ∆A to lower orders |J| < n. We will show that Mn,1 (L1 (x1 ), . . . ; Q(y)) coincides outside the total diagonal Dn+1 = {(x1 , . . . , xn+1 ) ∈ Mn+1 , x1 = · · · = xn+1 } with the distriδS0 of the ﬁrst term in (5.17). This butional kernel R(L1 (x1 ) ⊗ · · · ⊗ Ln (xn ), Q(y)) δϕ(y) shows n

supp ∆

n

Lj ; Q

⊂ Dn+1 ,

(5.18)

j=1

where ∆n (⊗nj=1 Lj ; Q) is deﬁned by (5.15). (By construction ∆n (⊗nj=1 Lj ; Q) is the distributional kernel of ∆nA (L1 (g1 ) ⊗ · · · ) (5.17), which proves (vi).) The support property (5.18) implies locality (ii(a)). To derive the second locality statement (ii(b)) we additionally use that δR

k−1 j=1

Aj (xj ); Ak (xk ) δϕ(z)

=0

if z = xj

∀j = 1, . . . , k,

as explained in [30]. By means of (5.15) we conclude

supp

δ∆n

n j=1

δϕ

Lj ; Q

⊂ Dn+2 ,

which is equivalent to locality (ii(b)). It follows also ∆nA (⊗nj=1 Lj (gj )) ∈ Floc . We turn to the proof of (5.18). Since Mn,1 has retarded support supp(δMn,1 /δϕ) ⊂ {(x1 , . . . , xn , y) ∈ Mn+1 ; x1 , . . . , xn ∈ y + V¯− }

(5.19)

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and is symmetrical under permutation of the ﬁrst n entries, it is thereby suﬃcient to consider the particular case where xn ∈ y + V¯− \{y}. In this case it holds Mn,1 (L1 (x1 ), . . . ; Q(y)) δS0 = R ⊗i∈n−1 Li (xi ) ⊗ Ln (xn ), Q(y) δϕ(y) n−1 δLk (xk ) R ⊗i∈n\{k} Li (xi ) ⊗ Ln (xn ), Q(y) + δϕ(y) k=1 R ⊗i∈I Li (xi ) ⊗ Ln (xn ), ∆|J| (⊗j∈J Lj (xj ); Q(y)) + I J=n−1

where the locality of the distributions ∆|J| (⊗j∈J Lj (xj ); Q(y)) for |J| < n is used. Now we apply the GLZ Relation to each term and take the support property of the retarded products into account. This yields Mn,1 (L(x1 ), . . . ; Q(y))

δS0 = R(⊗i∈I Li (xi ), Ln (xn )), R ⊗j∈J Lj (xj ), Q(y) δϕ(y) I J=n−1

+

n−1

δLk (xk ) R(⊗i∈I Li (xi ), Ln (xn )), R ⊗j∈J Lj (xj ), Q(y) δϕ(y)

k=1 I J {k}=n−1

+

{R(⊗i∈K Li (xi ), Ln (xn )) ,

I J=n−1 K L=I

R(⊗l∈L Ll (xl ), ∆|J| (⊗j∈J Lj (xj ); Q(y)))}.

(5.20)

To transform the last term we use the induction hypothesis (5.16) written in distributional form, R(⊗l∈L Ll (xl ), ∆|J| (⊗j∈J Lj (xj ); Q(y))) L J=H

δS0 δS0 − R ⊗i∈H Li (xi ), Q(y) δϕ(y) δϕ(y) δLj (xj ) R ⊗i∈H\{j} Li (xi ), Q(y) − δϕ(y)

= R(⊗i∈H Li (xi ), Q(y))

(5.21)

j∈H

for all H ⊂ n − 1. After rearranging the sums the last term in (5.20) is equal to

δS0 − R(⊗i∈I Li (xi ), Ln (xn )), R ⊗j∈J Lj (xj ), Q(y) δϕ(y) I J=n−1

−

n−1

k=1 I J {k}=n−1

δLk (xk ) R(⊗i∈I Li (xi ), Ln (xn )), R ⊗j∈J Lj (xj ), Q(y) δϕ(y)

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I J=n−1

δS0 R(⊗i∈I Li (xi ), Ln (xn )), R(⊗j∈J Lj (xj ), Q(y)) . δϕ(y)

147

(5.22)

The last term in (5.22) can be transformed by using the identity

δS0 δS0 , F, P (x) = {F, P (x)} δϕ(x) δϕ(x)

F, P (x) ∈ F

(5.23)

(which follows from the fact that the 2-point function Hm is a solution of the free ﬁeld equation) and by applying the GLZ Relation: I J=n−1

=

δS0 R(⊗i∈I Li (xi ), Ln (xn )), R(⊗j∈J Lj (xj ), Q(y)) δϕ(y)

I J=n−1

δS0 R(⊗i∈I Li (xi ), Ln (xn )), R(⊗j∈J Lj (xj ), Q(y)) δϕ(y)

= R(L1 (x1 ) ⊗ · · · ⊗ Ln (xn ), Q(y))

δS0 . δϕ(y)

Summing up the ﬁrst two terms in (5.20) and the ﬁrst two terms in (5.22) cancel and we get the desired result Mn,1 (L1 (x1 ), . . . , Q(y)) = R(L1 (x1 ) ⊗ · · · ⊗ Ln (xn ), Q(y))

δS0 δϕ(y)

(5.24)

∀(x1 , . . . , xn , y) ∈ / Dn+1 which proves (5.18). The conclusion “⇒” of property (iv) is obvious from (5.12) and “⇐” follows inductively for ∆nA (S ⊗n ), because the right-hand side of (5.17) (for F1 = · · · = Fn = S) vanishes in that case if ∆kA ≡ 0 ∀k < n. And ∆A (eS⊗ ) = 0 ∀S implies ∆A ≡ 0 by the polarization identity. The crucial property that ∆A (F1 ⊗ · · · ⊗ Fn ) = O() for all n > 0 and Fi ∼ 0 , follows immediately from the validity of the classical MWI and lim→0 R = Rcl by using property (iv). Mostly we will omit the index n of ∆nA and its kernel ∆n . Up to here we only assumed that the R-product satisﬁes the basic axioms. If it fulﬁlls renormalization conditions, then corresponding properties of ∆A are implied. ↑ Lemma 8. (i) The axioms P+ -Covariance, Unitarity, Field Independence and Field Equation, respectively, imply corresponding properties of ∆A : βL S ↑ ), ∀L ∈ P+ ; — P ↑+ -Covariance βL ∆A (eS⊗ ) = ∆βL A (e⊗

— Unitarity ∆A (eS⊗ ) = ∆A (eS⊗ );

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— Field Independence δS δ ⊗n ⊗n ⊗(n−1) ∆A (S ) = ∆Ax (S ) + n ∆A S ⊗ , δϕ(x) δϕ(x) where Ax is obtained from A (5.10) by replacing Q(y) by def

Ax =

dy h(y)

(5.25)

δQ(y) δϕ(x) :

δQ(y) δS0 ∈ JS0 ; δϕ(x) δϕ(y)

(5.26)

— Field Equation ∆A ≡ 0

if

A=

dx h(x)

δS0 , δϕ(x)

∀ h ∈ D(M);

(5.27)

(ii) Let gj ∈ D(M), Lj ∈ P. Assuming that the R-products satisfy the axioms Translation Invariance and Field Independence, there exist linear maps Pan : P ⊗(n+1) → P (where a runs through a ﬁnite subset of (Nd0 )n ), which are symmetric in the ﬁrst n factors, such that ∆nA can be written as ∆nA (L1 (g1 ) ⊗ · · · ⊗ Ln (gn )) = dx h(x) ∂ a1 g1 (x) · · · ∂ an gn (x) Pan (L1 ⊗ · · · ⊗ Ln ; Q)(x).

(5.28)

n a∈(Nd 0)

Part (ii) can be proved without assuming Field Independence of the R-products, see [30]. Proof. (i) We prove all properties for the components ∆nA of ∆A by induction on n. We verify that the right-hand side of (5.17) fulﬁlls the assertion by using the pertinent property of the R-product and the inductive assumption. In case of the Field Independence we additionally take δA = Ax + δϕ(x)

dy h(y)Q(y)

δ(δA S) δS = δ Ax S + δ A δϕ(x) δϕ(x)

δ 2 S0 , δϕ(y) δϕ(x)

(5.29) (5.30)

into account. δS0 the MWI reduces to the oﬀ-shell Field Equation: For A = dx h(x) δϕ(x) ﬁeld equation in diﬀerential form, which is equivalent to the integrated version (3.8). The assertion follows by means of property (iv) of ∆A in Theorem 7. (ii) The statement is a simpliﬁed version of [16, Proposition 4.3]. The proof, which is given there in words, is carried out here explicitly, since these formulas will be used below in the proof of part (ii) of Proposition 10.

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The Field Independence (5.25) translates into a corresponding relation for the distributional kernel (5.14): δ ∆(⊗nj=1 Lj (xj ); Q(y)) δϕ(x) n δLl (xl ) δQ(y) n = ; Q(y) + ∆ ⊗j=1 Lj (xj ); ∆ ⊗j( =l) Lj (xj ) ⊗ . δϕ(x) δϕ(x) l=1

(5.31) We insert this identity into the Taylor expansion of ∆(⊗nj=1 Lj (xj ); Q(y)) with respect to ϕ. This yields the causal Wick expansion ∆(⊗nj=1 Lj (xj ); Q(y)) Cal111...;l = ···a1l l1 ,...;l aiji ,aj

1

,...;a1 ···al

∂ li L i ∂lQ (xi ); (y) ∂(∂ ai1 ϕ) · · · ∂(∂ aili ϕ) ∂(∂ a1 ϕ) · · · ∂(∂ al ϕ)   li n l  · ∂ aiji ϕ(xi ) · ∂ aj ϕ(y), (5.32) · ω0 ∆ ⊗ni=1

i=1

ji =1

j=1

where Cal111...;l ···a1l1 ,...;a1 ···al is a combinatorial factor. Now we crucially use the l1 L l Q 1 (x1 ) ⊗ · · · ; ∂∂··· (y) . Hence this distribution is of locality (ii(a)) of ω0 ∆ ∂ ∂··· the form l1 ∂ L1 ∂lQ (x1 ) ⊗ · · · ; (y) = Cb (∂ b δ)(x1 − y, . . . , xn − y), (5.33) ω0 ∆ ∂ ··· ∂··· b

with constant numbers Cb (due to Translation Invariance) which depend on the ﬁeld polynomials in the argument of ∆. The term with index b gives the following contribution to ∆A (L1 (g1 ) ⊗ · · · ⊗ Ln (gn )) (5.14): |b| dx1 · · · dxn dy δ(x1 − y, . . . , xn − y) (−1)  · h(y) · ∂xb 1 ···xng1 (x1 ) · · · gn (xn ) ·

n i=1

 

li ji =1

 ∂ aiji ϕ(xi ) ·

l

 ∂ aj ϕ(y),

j=1

(5.34) where we have omitted constant factors. By reordering the sums we write ∆A (L1 (g1 ) ⊗ · · · ⊗ Ln (gn )) in the form (5.28). Since ∆A (L1 (g1 ) ⊗ · · · ⊗ Ln (gn )) is multilinear in the ﬁelds L1 , . . . , Ln , Q and symmetric in L1 , . . . , Ln , the maps Pan must satisfy corresponding properties.

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We turn to the scaling behaviour of ∆A . Assuming that the R-products satisfy the axioms Smoothness in m ≥ 0, µ-Covariance and almost homogeneous Scal(m,µ) (S ⊗n ) does not scale almost homogeing (3.16), the map (S ⊗n ⊗ A) → ∆A neously in general. Namely, for non-vanishing mass we are faced with m-dependent (m)

δS

(m)

0 ≡ S0 ∈ Phom , δϕ

∈ Phom and, hence, inhomogeneous polynomials: S0 (m) ≡ A is in general not in Phom . In typical applications of the MWI (see A e.g. the O(N )-model in Sec. 5.4.4 or current conservation in QED) the simpliﬁδS0 cation appears that Qj is independent of m and the m-dependent terms of δϕ j (ρ−1 m,µ) −1 ⊗n 0 cancel in A = dx j Qj (x) δϕδS . But even with that, σ ∆ ((σ S) ) ρ −1 ρ j (x) σ A ρ

is in general not a polynomial in (log ρ): proceeding inductively the ﬁrst term on the right-hand side of (5.17) does not scale almost homogeneously due to the mass 0 l term in δS δϕ . For this reason, we assume here the axiom Scaling Degree (3.19) instead of Smoothness in m, µ-Covariance and almost homogeneous Scaling. With this weaker assumption we are going to derive a corresponding scaling degree property of ∆A and an upper bound for the mass dimension of ∆A (S ⊗n ) which does not depend on n if S is a renormalizable interaction. For the latter purpose we have to deﬁne the mass dimension of a local functional. We use that every F ∈ Floc can uniquely be written in the form F = (5.35) dx gi (x)Pi (x), gi ∈ D(M), Pi ∈ Pbal . i

With that we deﬁne def

dim F = maxi dim Pi

(5.36)

This deﬁnition is minimal in the following sense: Lemma 9. Let F =

dx gi (x)Bi (x),

gi ∈ D(M),

Bi ∈ P.

(5.37)

i

Then it holds dim F ≤ maxi dim Bi .

(5.38)

Proof. Every Bi can uniquely be written as Bi = j pij (∂)Pij , Pij ∈ Pbal , where pij (∂) is a polynomial in the partial derivatives [16]. Inserting this into (5.37) and l In

detail: from the derivation of the classical MWI (5.4), (5.5) it follows that the scaling of the right-hand side of the anomalous MWI (5.12) must be such that the mass m in (m) R δS0 has the same value that is the scale transformation reads dy h(y)σρ ◦ R(m,µ) and in δϕ(y)

R(ρ

−1

−1

m,µ) (eσρ ⊗

S

, σρ−1 Q(y))

(ρ−1 m)

δS0 δϕ(y)

.

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shifting the derivatives to the test function we obtain the unique representation (5.35) of F . The assertion follows from dim Pij ≤ dim Bi ∀i, j. δS0 From A = dy h(y)Q(y) δϕ(y) we conclude dim(A) ≤ dim(Q) + dim

δS0 δϕ

where

δS0 d+2 . = dim(( + m2 )ϕ) = dim δϕ 2 δS Analogously the relation δA S = dy h(y)Q(y) δϕ(y) implies dim(δA S) ≤ dim(Q) + dim(S) − dim(ϕ) = dim(Q) + dim(S) −

(5.39)

d−2 . 2

(5.40)

With these tools we are ready to formulate and prove the following proposition. Proposition 10. (i) Scaling Degree. If the R-products fulﬁll the axiom Scaling Degree (3.19), then the scaling degree of the “vacuum expectation value” of ∆(L1 (x1 ) ⊗ · · · ; Q(y)) is bounded by n d+2 dim Li + dim Q + . sd ω0 ∆(⊗nj=1 Lj (xj ); Q(y)) ≤ 2 i=1

(5.41)

(ii) Mass Dimension. If the R-products fulﬁll the axioms Field Independence and Scaling Degree (3.19), then the mass dimension of ∆A (F1 ⊗ · · · ⊗ Fn ) is bounded by dim ∆A (F1 ⊗ · · · ⊗ Fn ) ≤

n

dim(Fi ) + dim Q +

i=1

d+2 − dn. 2

(5.42)

For a renormalizable interaction, that is dim(S) ≤ d, this implies dim ∆A (eS⊗ ) ≤ dim Q +

d+2 . 2

(5.43)

Note that for a renormalizable interaction the upper bounds on the mass dimension of A (5.39), δA S (5.40) and ∆A (eS⊗ ) agree, as one expects because the sum (A + δA S + ∆A (eS⊗ )) appears in Theorem 7. Proof. (i) Proceeding by induction on n we apply ω0 to (5.15) and estimate the scaling degree of the resulting terms on the right-hand side by using n sd ω0 R(⊗n−1 L (x ); L (x )) ≤ dim Lj , j j n n j=1

(5.44)

j=1

which follows from (3.19). (In contrast to (3.19) we do not assume Lj ∈ Phom .)

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For example we obtain sd ω0 R ⊗j( =l) Lj (xj ); Q(y)(∂ a δ)(xl − y)

∂Ll (xl ) ∂(∂ a ϕ) ∂Ll ≤ dim Lj + dim Q + dim + sd(∂ a δ) ∂(∂ a ϕ) j( =l)

=

n

dim Lj + dim Q +

j=1

d+2 . 2

(5.45)

The same bound results for all other terms. (The vacuum expectation value of the last term vanishes.) (ii) We may assume Fj = Lj (gj ) with Lj a balanced ﬁeld, Lj ∈ Pbal . With that it holds dim(Lj ) = dim(Fj ). We use the formulas derived in the proof of part (ii) of Lemma 8, in particular the causal Wick expansion of ∆(⊗j Lj (xj ); Q(y)) (5.32). (In the formulas given below the indices have precisely the same meaning as in that proof.) Looking at (5.33), we get an upper bound for the scaling degree of the left-hand side by means of (5.41). This implies that the range of b on the right-hand side of (5.33) is restricted by   li n d−2  dim Li − |aiji | + |b| + dn ≤ 2 i=1 j =1 i

+ dim Q −

l j=1

d−2 |aj | + 2

+

d+2 . 2

(5.46)

All terms of ∆A (L1 (g1 ) ⊗ · · · ⊗ Ln (gn )) are of the form (5.34) where |b| is bounded by (5.46). Due to Lemma 9 the mass dimension of the functional (5.34) is bounded by li n l d−2 d−2 + + |b| |aiji | + |aj | + 2 2 i=1 j =1 j=1 i

≤

n

dim Li + dim Q +

i=1

d+2 − dn. 2

(5.47)

This implies the assertion (5.42). 5.3. The MWI in the proper ﬁeld formalism, and the Quantum Action Principle In the literature the Quantum Action Principle (QAP) is formulated in terms of time ordered products. Hence, to be able to compare our results with the QAP, we

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ﬁrst rewrite Theorem 7 using T -products: Lemma 11. Using the assumptions and notations of Theorem 7 the anomalous MWI (5.12) can equivalently be written in terms of the time-ordered product: iS/ δS0 iS/ . (5.48) T e⊗ ⊗ (A + δA S + ∆A (eS⊗ )) = dy h(y)T (e⊗ ⊗ Q(y)) δϕ(y) Proof. Using the formula of Bogoliubov (4.15) and the identity δS0 δS0 (F G) · = F G· , ∀F, G ∈ F δϕ δϕ

(5.49)

(cf. (5.23)), where · denotes the classical product, we obtain iS/ −iS/ T¯ (e⊗ ) T e⊗ ⊗ (A + δA S + ∆A (eS⊗ )) δS0 −iS/ iS/ = T¯(e⊗ ) dy h(y)T (e⊗ ⊗ Q(y)) · . δϕ(y) iS/

Multiplication by T (e⊗

(5.50)

) from the left yields (5.48).

As shown in (5.9) the MWI is satisﬁed by Ttree , i.e. it can be violated only by the contribution of the loop diagrams. To concentrate the study of the solvability of the MWI to that terms for which the MWI is a non-trivial condition, we reformulate the anomalous MWI (5.12) within the proper ﬁeld formalism introduced in Sec. 4. Corollary 12. The anomalous MWI in Theorem 7 can equivalently be expresses in terms of the vertex function ΓT (deﬁned in Proposition 4) : δ(S0 + ΓT (eS⊗ )) = ΓT eS⊗ ⊗ (A + δA S + ∆A (eS⊗ )) . dx h(x)ΓT (eS⊗ ⊗ Q(x)) δϕ(x) (5.51) Proof. Applying (4.19) on both sides of (5.48) we obtain iΓT (eS )/ Ttree e⊗ ⊗ ⊗ ΓT eS⊗ ⊗ A + δA S + ∆A eS⊗ iΓT (eS )/ δS0 . = dy h(y)Ttree e⊗ ⊗ ⊗ ΓT eS⊗ ⊗ Q(y) δϕ(y) On the right-hand side we use the classical MWI in terms of T -products (5.9) and obtain δ(S0 + ΓT (eS⊗ )) iΓT (eS ⊗ )/ S ⊗ dy h(y)ΓT (e⊗ ⊗ Q(y)) · · · = Ttree e⊗ , δϕ(y) which leads, by means of (4.13), to the assertion (5.51). def

To compare this result with the literature, we introduce Γ(S0 , S) = S0 + ΓT (eS⊗ ) and, motivated by (4.27) and (4.38), interpret Γ(S0 , S) as the proper total action associated with the total classical action Stot = S0 + S. In addition, we introduce “insertions” (see [10]) in our algebraic formalism: let

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P ∈ C ∞ (M, P) and ρ ∈ D(M) some test function (in the literature often called as “external ﬁeld”). We deﬁne the “insertion of P (x)”, usually denotedm by P (x) · Γ(S0 , S), as followsn : δ def P (x) · Γ(S0 , S) = Γ S0 , S + dxρ(x)P (x) = ΓT eS⊗ ⊗ P (x) . (5.52) δρ(x) ρ≡0 def We denote by S = S + dxρ(x)Q(x), ρ ∈ D(M) the modiﬁed classical interaction containing a coupling to the external ﬁeld ρ, and write ∆A (eS⊗ ) = ˜ S⊗ ; Q(x)) ∈ D(M, P) (Lemma 8(ii)). Using this nota˜ S⊗ ; Q(x)) with ∆(e dx h(x)∆(e tion and the fact that the anomalous MWI (5.51) is valid for all h ∈ D(M), we rewrite (5.51) in terms of the proper total action: δΓ(S0 , S ) δΓ(S0 , S ) = ∆(x) · Γ(S0 , S), (5.53) δρ(x) δϕ(x) ρ≡0 def 0 +S) ˜ S where the local ﬁeld ∆(x) is given by ∆(x) = Q(x) δ(S δϕ(x) + ∆(e⊗ ; Q(x)). This formulation of the anomalous MWI is formally equivalent to the general formulation of the QAP in [10]. Note that due to property (v) in Theorem 7 and (4.27), the -expansion of the right-hand side of (5.53) starts with

δ(S0 + S) + O() δϕ(x) δ(S0 + S ) δ(S0 + S ) ≡ + O(), δρ(x) δϕ(x) ρ=0

∆(x) · Γ(S0 , S) = Q(x)

(5.54)

which is completely analogous to the expansion of the right-hand side of the QAP in [10]. Moreover, for a renormalizable interaction S, the mass dimension of the local insertion ∆(x) is bounded by dimQ + d+2 2 = dimQ − dimϕ + d (Proposition 10), in agreement with [10]. 5.4. Removal of violations of the MWI 5.4.1. Fulﬁllment of the MWI to ﬁrst order in S Before we are going to investigate the question whether the MWI can be fulﬁlled in a concrete model (i.e. for a given A ∈ Jloc and a given interaction S ∈ Floc ), we will show that the MWI can always be fulﬁlled to ﬁrst order in the interaction S.o Apart from the interest in its own, this result will be needed in our proof of the Ward identities of the scalar O(N )-model (Sec. 5.4.4). m The dot does not mean n Note, that in the setting

the classical product here! of causal perturbation theory the introduction of external ﬁelds in order to express “insertions” or nonlinear symmetry transformations is — in contrast to conventional perturbation theory — not necessary. o E.g. the axial anomaly of QED is of second order in the interaction.

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δS0 Proposition 13. Given A = dy h(y)Q(y) δϕ(y) (with Q ∈ P) and S ∈ Floc there exists a retarded product R1,1 which fulﬁlls all renormalization conditions (speciﬁed in Sec. 3) and δS0 . (5.55) R1,1 (S, A) + δA S = dy h(y)R1,1 (S, Q(y)) δϕ(y) Proof. We start with R-products satisfying all renormalization conditions. For simplicity we work in the proper ﬁeld formalism. From the anomalous MWI (Corollary 12) we obtain to ﬁrst order in S δS0 δS0 ∆A (S) = − dy h(y) ΓT S ⊗ Q(y) − ΓT (S ⊗ Q(y)) . (5.56) δϕ(y) δϕ(y) By inserting the Wick expansion (3.7) of ΓT (which holds since ΓT fulﬁlls Field 0 Independence), we ﬁnd that the terms containing no contraction of δS δϕ cancel. It remains: δS0 ¯ ¯ ⊗ Q(y) ω0 ΓT L(x) ∆A (S) = − dx dy g(x)h(y) L(x)Q(y) δϕ(y) ¯ ¯ L⊂L,Q⊂Q

(5.57)

where we write S = dx g(x)L(x), g ∈ D(M) and sum over all subpolynomials l ∂lP 1 aj P¯ = ∂(∂ a1 ϕ)···∂(∂ al j=1 ∂ ϕ ϕ) of P ∈ P (for P = L, Q), denoting by P = l! the corresponding “counterpart” (see (3.7)). Due to locality of ∆A (S) the distriδS0 ¯ ¯ ⊗ Q(y) butions ω0 (ΓT (L(x) δϕ(y) )) are supported on the diagonal {(x, y)|x = y}, cancellations of non-local terms on the right-hand side of (5.57) are impossible. We ˆ T which removes the violating term are searching a ﬁnite renormalization ΓT → Γ ∆A (S). Due to (4.33) a renormalization of ΓT (S ⊗ F ) must be of the form ˆ T (S ⊗ F ) = ΓT (S ⊗ F ) + D(S ⊗ F ), Γ

∀F ∈ Floc ,

(5.58)

with D satisfying the properties listed in the Main Theorem. To obtain a ﬁeld independent D we construct it from its vacuum expectation values by the causal Wick expansion (3.7). Obviously, setting δS0 δS0 def ¯ ¯ ¯ ¯ = −ω0 ΓT L(x) ⊗ Q(y) ω0 D L(x) ⊗ Q(y) , (5.59) δϕ(y) δϕ(y) ˆ T satisﬁes the MWI to ﬁrst order in S. Here we essentially use the corresponding Γ that the distribution on the right-hand side has local support. One veriﬁes easily ˆ T ) satisﬁes all renormalization conditions. that D (and hence the resulting Γ 5.4.2. Removal of possible anomalies by induction on the order in The formal equivalence of the anomalous MWI (written in the form of Corollary 12) and the QAP makes it possible to apply basic techniques of algebraic renormalization within the framework of causal perturbation theory. The main idea underlying

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algebraic renormalization is to start with an arbitrary renormalization prescription; with that the considered Ward identities are in general broken. Proceeding by induction on the power of , the question whether one can ﬁnd a ﬁnite renormalization which removes the possible anomaly (i.e. the local terms violating the Ward identity) then amounts — by means of the QAP — to purely algebraic (often cohomological) problems. We will apply this strategy to the MWI in the proper ﬁeld formalism (which is given by (5.51) with ∆A (eS⊗ ) = 0): given A ∈ Jloc and S ∈ Floc , we are going to investigate whether possible anomalies of the corresponding MWI can be removed by ﬁnite renormalizations of ΓT (4.33). Proceeding by induction on the power of (k) , we assume that for a given map ΓT (which satisﬁes the renormalization con↑ ditions Unitarity, P+ -Covariance, Field Independence, Field Equation and Scaling Degree, see Lemma 6) the MWI is violated only by terms of order k ; that is in the (k) anomalous MWI (Corollary 12) for ΓT , (k) (k) (k) ΓT eS⊗ ⊗ (A + δA S) + ΓT (eS⊗ ⊗ ∆A (eS⊗ )) S δ(S0 + Γ(k) (k) S T (e⊗ )) = dy h(y)ΓT e⊗ ⊗ Q(y) , (5.60) δϕ(y) the violating term can be written as (k)

(k)

(k)

ΓT (eS⊗ ⊗ ∆A (eS⊗ )) = ∆A (eS⊗ ) + O(k+1 ),

(k)

∆A (eS⊗ ) = O(k ),

(5.61)

where (4.27) is used. Note that, due to Theorem 7(v) and (4.27), any vertex function (1) ΓT fulﬁlls this assumption (5.60), (5.61) for k = 1 and, hence, can be used as ΓT . To fulﬁll the MWI to kth order in (and to maintain the other renormalization conditions), we have to ﬁnd a renormalization map D(k) (with the properties (i)–(v) and (vi)(B) of the Main Theorem) such that (k+1) S ΓT e⊗ ⊗ (A + δA S) + O(k+1 ) δ(S0 + Γ(k+1) (eS⊗ )) (k+1) S T = dy h(y) ΓT , (5.62) e⊗ ⊗ Q(y) δϕ(y) (k+1)

with ΓT

given by (k+1)

ΓT

(k) F (k) D (e⊗ ) e⊗

(eF ⊗ ) = ΓT

∀F ∈ Floc .

(5.63)

To maintain the validity of the MWI to lower orders in we choose D(k) to be of the form (k)

F D(k) (eF ⊗ ) = F + D>1 (e⊗ ),

(k)

k D>1 (eF ⊗ ) = O( ),

∀F ∈ Floc ,

F ∼ 0 .

(5.64)

This implies D(k) (eF ⊗)

e⊗

(k)

F F k+1 = eF ) ⊗ + e⊗ ⊗sym D>1 (e⊗ ) + O(

(5.65)

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(where ⊗sym denotes the symmetrized tensor product) and (k+1)

ΓT (k+1)

ΓT

(k)

(k)

(eS⊗ ) = ΓT (eS⊗ ) + D>1 (eS⊗ ) + O(k+1 ), (k)

(5.66)

(k)

(eS⊗ ⊗ F ) = ΓT (eS⊗ ⊗ F ) + D>1 (eS⊗ ⊗ F ) + O(k+1 ).

(5.67)

We insert the latter two equations into our requirement (5.62) and use the inductive assumption (5.60), (5.61). It results (k) (k) (k) ∆A (eS⊗ ) = D>1 eS⊗ ⊗ (A + δA S) − δA D>1 (eS⊗ ) δ(S0 + S) (k) − dy h(y)D>1 (eS⊗ ⊗ Q(y)) + O(k+1 ) . (5.68) δϕ(y) (k)

(k)

The violating term ∆A (eS⊗ ) is inductively given by ΓT . If we succeed to ﬁnd a (k) corresponding map D>1 fulﬁlling (5.68) and the properties of a renormalization (k) map, then the pertinent ﬁnite renormalization removes the “anomaly” ∆A (eS⊗ ). (k) However, since D>1 appears in (5.68) several times with diﬀerent arguments it seems almost impossible to discuss the existence of solutions in general. 5.4.3. Assumption: Localized oﬀ-shell version of Noether’s Theorem (k)

In various important applications of the MWI the search for solutions D>1 of (5.68) is simpliﬁed due to the validity of the following assumption. In the given model the total actionp S0 + S(g)  with S(g) =

κn Sn (g), Sn (g) =

 dx(g(x))n Ln (x), g ∈ D(M), Ln ∈ P

n≥1

is invariant with respect to the symmetry transformation δ δA = dy h(y)Q(y) = κ n δ An δϕ(y) n≥0 κn An , An ∈ Jloc corresponding to A = n≥0

in the following way: there exist • a current j µ (g) = n≥0 κn jnµ (g), jnµ (g)(x) = (g(x))n jnµ (x) (with jnµ ∈ P) and (1)µ (1)µ n (1)µ • a “Q-vertex”q L(1)µ (g) = (g), Ln (g)(x) = (g(x))n−1 Ln (x) n≥1 κ Ln (1)µ

(with Ln pκ

∈ P)

denotes the coupling constant. name “Q-vertex” is due to “perturbative gauge invariance” [31], which is related to BRSTsymmetry.

q The

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such that

δA (S0 + S(g)) ≡ A + δA S(g) =

dx h(x)(∂µ j µ (g)(x) − L(1)µ (g)(x)∂µ g(x)) (5.69)

in the sense of formal power series in κ. To zeroth order in κ the assumption (5.69) reads (5.70) δA0 S0 ≡ A0 = dx h(x)∂µ j0µ (x), where j0 is the symmetry current of the underlying free theory. This simplifying assumption can be interpreted as the validity of an oﬀ-shell version of Noether’s Theorem for the case that the interaction and the symmetry transformation are localized. It is satisﬁed e.g. for the scalar O(N )-model treated in Sec. 5.4.4 and for the BRST-symmetry of (massless) Yang–Mills theories, massive spin-1 ﬁelds and massless spin-2 ﬁelds (gravity). For the interacting scalar O(N )-model (see Sec. 5.4.4) the simpliﬁcations A = A0 , δA S = 0, L(1) = 0 and j = j0 appear. However, for the BRST-symmetry δA and j are generically non-trivial deformations of δA0 and j0 , and δA S and L(1) do not vanish. Example. BRST-symmetry. We are going to verify that the above mentioned models satisfy the assumption (5.69). Our argumentation is based on conservation of the classical BRST-current. For constant couplings (i.e. g(x) = 1 ∀x) there is a con served Noether current j = n≥0 κn jn , due to the BRST-invariance of the total action. We use these jn ’s to construct the BRST-current of the corresponding model with localized coupling κg(x) (g ∈ D(M)): we set j(g)(x) := n≥0 (κg(x))n jn (x). The violation of the conservation of j(g) is expressed in terms of the Q-vertex [12, 1, 30]: in [27, Secs. 3 and 4] it is shown that for the considered models there exists a Q-vertex L(1) (g) such that S(g) (1)µ (g)(x)∂µ g(x)) ∈ J . (5.71) Rcl e⊗ , dx h(x)(∂j(g)(x) − L Proceeding by induction on the order in κ and using the MWI (as it is worked out in formulas in [1, (190), (191) and (152)–(157)]) one ﬁnds that (5.71) is equivalent to the sequence of relations A0 := dx h(x)∂j0 (x) ∈ Jloc (5.72) and An : = −

n−1

δAl Sn−l (g) +

dx h(x)(∂jn (g)(x) − L(1) n (g)(x)∂g(x)) ∈ Jloc

l=0

(5.73) for n ≥ 1. This yields our assumption: with (5.72) the condition (5.70) holds true and (5.73) implies An = δAn S0 and with that (5.73) gives (5.69) to nth order in κ. δA = n≥0 κn δAn is a localized version of the usual BRST-transformation.

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We point out that the nilpotency of the BRST-transformation is not directly used. Every model fulﬁlling the local current conservation (5.71) (with S(g), j(g) and L(1) (g) of the above form) satisﬁes our assumption (5.69) if the symmetry transformation δA is deﬁned by (5.72), (5.73). Assuming now the validity of (5.69), we are going to derive a simpliﬁed version of (5.68). For shortness and coincidence with the notations of the preceding sections we write S, j and L(1) for S(g), j(g) and L(1) (g), respectively. For the time being, we additionally assume that the test function h satisﬁes h(x) = 1 for all x ∈ (k) supp(δ S/δϕ). With that ∆A (eS⊗ ) is independent of the choice of h within this class and h can be replaced by the number 1 (see Lemma 8(ii)). h does not appear also on the right-hand side of (5.68): namely, due to our assumption and the locality (k) of the map D>1 (see Theorem 3(i)) we obtain (k)

D>1 (eS⊗ ⊗ (A + δA S))     (k) (k) = dy ∂µy D>1 eS⊗ ⊗ j µ (y) −D>1 eS⊗ ⊗ L(1) µ (y) ∂µ g(y) (5.74) !" # =0 for y ∈supp(δS/δϕ) / (k)

and, using again the locality of D>1 , (5.68) simpliﬁes to the condition (k)

∆

(eS⊗ )

=−

δ(S0 + S) (k) S (k) S (1) + D>1 e⊗ ⊗ L (y) ∂g(y) dy D>1 e⊗ ⊗ Q(y) δϕ(y) (k)

− δD>1 (eS⊗ ) + O(k+1 ), def

(5.75)

(k)

where ∆(k) (eS⊗ ) = ∆A (eS⊗ )|h≡1 and δ is the non-localized version of δA : δ def . dx Q(x) δ= δϕ(x)

(5.76)

(k)

It is much easier to ﬁnd a solution D>1 for (5.75) than for (5.68), since in (5.75) the localization of the derivation δA is removed (i.e. h is replaced by 1) and since (k) the D>1 (eS⊗ ⊗ ∂j)-term vanishes. However, we want to solve the MWI for general h ∈ D(M).r To investigate whether (5.75) is also suﬃcient for the more involved condition (5.68) for (k) arbitrary h, let a solution D>1 of (5.75) be given. We point out that the map (k) (k) D>1 is not completely determined by that, only the combination of D>1 (eS⊗ ), (k) S (k) S D>1 (e⊗ ⊗Q) and D>1 (e⊗ ⊗L(1) ) appearing on the right-hand side of (5.75) is ﬁxed. (k) We claim that, given such a D>1 , there exists a linear map K µ : TFloc → D(M, P) r For example, this is used in the derivation of our version of the QAP (5.53) from the anomalous MWI.

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with the property (k) ∆A (eS⊗ )

δ(S0 + S) (k) = dy h(y) ∂µ K µ (eS⊗ )(y) − D>1 (eS⊗ ⊗ Q(y)) δϕ(y) (k) (k) − D>1 eS⊗ ⊗ L(1) (y) ∂g(y) − δA D>1 (eS⊗ ) + O(k+1 ) (5.77)

for all h ∈ D(M) and with K µ (1) = 0. The latter condition is compatible with (5.77), because to zeroth order in κ the condition (5.77) reduces to h ∂K(1) = (k) (k) O(k+1 ), due to D>1 (1 ⊗ F ) = D>1 (F ) = 0 (∀F , see (5.64)). To show the existence of K µ , we ﬁrst prove the following lemma, which describes the diﬀerence between δA and δ with respect to their action on local functionals: Lemma 14. Let be given F ∈ Floc , l ∈ D(M, P) and a localized derivation δh = δ dx h(x)Q(x) δϕ(x) , h ∈ D(M), Q ∈ P, such that the corresponding non-localized δ derivation δ = dx Q(x) δϕ(x) satisﬁes δF = dx l(x). (5.78) Then there exists a k µ ∈ D(M, P) such that the following localized version of (5.78) holds true: (5.79) δh F = dx h(x)(l(x) + ∂µ k µ (x)). Proof. Let f ∈ D(M, P) with F = dy f (y). Carrying out the functional derivative in (5.78) we conclude ∂f l(x) = (x) + ∂µ k1µ (x) ∂ a Q(x) (5.80) a ϕ) ∂(∂ d a∈N0

for some k1µ ∈ D(M, P). On the other hand we obtain ∂f (x) ∂ a h(x)Q(x) δh F = dx ∂(∂ a ϕ) d =

a∈N0

dx h(x)

∂ a Q(x)

a∈Nd 0

∂f (x) + ∂(∂ a ϕ)

dx h(x)∂µ k2µ (x)

(5.81)

for some other k2µ ∈ D(M, P). Hence, setting k µ = −k1µ +k2µ we obtain the assertion (5.79). (k)

To prove (5.77) we use that ∆A (eS⊗ ) can be written as (k) S ˜ (k) (eS ; Q(x)) with ∆ ˜ (k) (eS ; Q(x)) ∈ D(M, P), (5.82) ∆A (e⊗ ) = dx h(x)∆ ⊗ ⊗ due to Lemma 8(ii). Hence, (5.75) can be written in the form δ F =

l + O(k+1 )

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(k)

(5.78) with F = D>1 (eS⊗ ) and δ(S0 + S) δϕ(x) (k) + D>1 eS⊗ ⊗ L(1) (x) ∂g(x) (k)

−l(x) = D>1 (eS⊗ ⊗ Q(x))

˜ (k) (eS ; Q(x)) ∈ D(M, P). +∆ ⊗

(5.83)

With that Lemma 14 yields our assertion (5.77). We conclude that a solution of (5.68) can be obtained from a solution of (5.75) by setting def (k) (5.84) D>1 eS⊗ ⊗ j µ (y) = K µ (eS⊗ )(y), provided this does not lead to any contradictions with the partial ﬁxing of D in (k) (k) (k) terms of D>1 (eS⊗ ), D>1 (eS⊗ ⊗ Q) and D>1 (eS⊗ ⊗ L(1) ). Due to the causal Wick (k) expansion (3.7) and the Field Equation D>1 (eS⊗ ⊗ ∂ a ϕ) = 0 (see Theorem 3(iii)), this is the case whenever the intersection of the subpolynomials of j with the subpolynomials of Q, L(1) or Ln (∀n ∈ N) contains only numbers and terms which are linear in the ﬁeld ϕ itself or partial derivatives thereof.s Note that (5.84) satisﬁes (k) the condition D>1 (j) = 0. Remark. If Q is linear in ϕ, the maintenance of the Field Equation requires (k) D>1 (eS⊗ ⊗ Q) = 0 (Theorem 3). With that the right-hand side of (5.75) vanishes to ﬁrst order in κ up to terms of order k+1 . That is the condition (5.75) can only be satisﬁed if ∆(k) (S1 ) = O(k+1 ). Hence, following the proof of Proposition 13, we ﬁrst perform a ﬁnite renormalization which maintains the considered renormal(k) ization conditions and removes the term ∼ k of ∆A (S1 ). This can be done such (k) (k) that ∆A (eS⊗ ) = O(k ) is preserved. Namely, since ∆A (S1 ) = O(k ) the vacuum expectation values on the right-hand side of (5.57) are of order k and, hence, the pertinent renormalization map D (5.59) can be chosen of the form (5.64). 5.4.4. Proof of the Ward identities in the O(N ) scalar ﬁeld model In the case of compact internal symmetry groups, covariance can be obtained by integration over the group. To illustrate the developed formalism we proceed alternatively. We will prove that an oﬀ-shell generalization of the Ward identities expressing current conservation in a scalar O(N )-model can be fulﬁlled to all orders of perturbation theory. Our strategy is based partially on techniques of algebraic renormalization described in detail in [10]. s In QED, this condition is not satisﬁed; one has to discuss the individual cases to see that the deﬁnition (5.84) does not lead to contradictions (see e.g. [13]).

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We consider a multiplet of N scalar ﬁelds ϕi (x), i = 1, . . . , N , transforming under the deﬁning representation of O(N ) — the group of orthogonal N × N matricest — ϕi → Aij ϕj ,

A ∈ O(N ).u

(5.85)

Let {X a | a = 1, . . . , 12 N (N − 1)} be a basis of the Lie algebra o(N ) of O(N ), and f abc the corresponding structure constants, [X a , X b ] = f abc X c . The dynamics of our model is given by the free action 1 S0 = dx(∂ µ ϕi (x)∂µ ϕi (x) − m2 ϕi (x)ϕi (x)) 2 and the localized, O(N )-invariant interaction 2 S = dx g(x) ϕi (x)ϕi (x) with g ∈ D(M).

(5.86)

(5.87)

(5.88)

Since the free action S0 is invariant under the transformation (5.85), there exist 1 a a 2 N (N − 1) conserved Noether currents jµ = Xij ϕj ∂µ ϕi , i.e. the local functionals def Aa = dx h(x)∂ µ jµa (x) (with arbitrary h ∈ D(M)) are elements of the ideal J generated by the free ﬁeld equations: δ a . (5.89) ϕj (x) Aa = δAa S0 ∈ J , with δAa = dx h(x)Xij δϕi (x) Essential simpliﬁcations of this model are the validity of (5.69) in simpliﬁed form and additionally that Q (5.10) is linear in ϕ. The conservation of the interacting currents (jµa )S = R(eS⊗ , jµa ) follows from the MWI for the given Aa and interaction S: δS0 a . (5.90) dx h(x)∂ µ R(eS⊗ , jµa (x)) ≡ R(eS⊗ , Aa ) = dx h(x)R(eS⊗ , Xij ϕj (x)) δϕi (x) Regarding the question whether (5.90) can be fulﬁlled to all orders, we start with ↑ an R-product satisfying the renormalization conditions Unitarity, P+ -Covariance, Field Independence, Field Equation and Scaling Degree. Then, the Ward identities (5.90) may be violated; however, Theorem 7 guarantees the existence of local maps ∆Aa : TFloc → Floc such that δS0 a . (5.91) R eS⊗ , Aa + ∆Aa (eS⊗ ) = dx h(x)R(eS⊗ , Xij ϕj (x)) δϕi (x) To ﬁnd a ﬁnite renormalization of the R-product which removes R(eS⊗ , ∆Aa (eS⊗ )), we follow the technique described in the preceding sections: we assume setting ϕ = √1 (ϕ1 + iϕ2 ) and ϕ ¯ = √1 (ϕ1 − iϕ2 ) the O(2) model can be seen to be equivalent 2 2 to the well known U (1) model of a complex scalar ﬁeld ϕ. u Repeated indices are summed over. t By

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(k)

the existence of a ΓT such that (5.90) is fulﬁlled up to terms of order k , i.e. (k) S e⊗

ΓT

(k) ⊗ Aa + ∆Aa (eS⊗ ) + O(k+1 ) (k) δ(S0 + ΓT (eS⊗ )) a = dy h(y)Xij ϕj (y) δϕi (y)

(k)

(5.92)

(k)

where ∆Aa (eS⊗ ) = O(k ). (The Field Equation for ΓT (Lemma 6) is taken into (k) account.) First we perform a ﬁnite renormalization which maintains ∆Aa (eS⊗ ) = O(k ) and the mentioned renormalization conditions and which removes the terms (k) (k) ∼ k of ∆Aa (S). Due to the requirement D>1 (eS⊗ ⊗ ϕ) = 0 (Theorem 3(iii)), the condition (5.75) simpliﬁes to (k)

S S k+1 ), ∆(k) a (e⊗ ) = −δa D>1 (e⊗ ) + O(

(5.93)

def def (k) (k) a ϕj (x) δϕiδ(x) . To fulﬁll where we set ∆a (eS⊗ ) = ∆Aa (eS⊗ )h≡1 and δa = dx Xij k+1 , we have to solve (5.93) and to extend the deﬁnition (5.90) up to terms of order (k) of this D>1 in such a way that condition (5.68) (for general h) holds true. The latter can be done by means of (5.84), because the intersection of the non-trivial subpolynomials of jµa with the subpolynomials of Q or (ϕi ϕi )2 is a subset of C ϕ. It remains to show the solvability of (5.93). For this purpose we temporarily restrict the functionals (2.1) in (5.92) to the space D(M) of compactly supported test functions on Minkowski space. This permits us to perform the limitv h → 1 in (5.92), ending up with the equation (k)

S k+1 ). δa (S0 + ΓT (eS⊗ )) = ∆(k) a (e⊗ ) + O(

(5.94)

Furthermore, using (5.86) we obtain the identity [δa , δb ] = fabc δc ,

(5.95)

(k)

which we insert into [δa , δb ](S0 + ΓT (eS⊗ )). This yields the consistency condition (k)

S (k) S k+1 δa ∆b (eS⊗ ) − δb ∆(k) ). a (e⊗ ) = fabc ∆c (e⊗ ) + O(

(5.96)

(k)

Due to the compact support of S, and the locality of ∆a , each term in (5.96) has compact support as well. Therefore, this equation holds true on the entire conﬁguration space C(M, R), i.e. the restriction of the functionals to D(M) can be omitted. v This limit is done as follows: let h ∈ D(M) such that there is a neighborhood U of 0(∈ M) with h|U = 1. Then we replace h(x) by h (x) ≡ h(x) ( > 0) and perform the limit → 0.

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The consistency condition (5.96) is the cocycle condition in the Lie algebra cohomology corresponding to the Lie algebra generated by the derivations {δa } acting on Floc . Trivial solutions are the coboundaries, S ˆ (k) (eS⊗ ) + O(k+1 ) ∆(k) a (e⊗ ) = −δa ∆

(5.97)

S ˆ (k) : TFloc → Floc . If ∆(k) for some linear, symmetric and local map ∆ a (e⊗ ) is a coboundary, the condition (5.93) can be solved by setting (k)

def

D>1 (1) = 0,

(k)

def

D>1 (S) = 0

def (k) ˆ (k) (S ⊗j ) ∀j ≥ 2, and D>1 (S ⊗j ) = ∆

(5.98)

(k) ∆a (S)

due to = O(k+1 ). Hence, we only have to show that the present cohomology is trivial. In the literature (see e.g. [10] and references therein) it is shown that every Lie algebra cohomology corresponding to some semi-simple Lie group and some ﬁnite dimensional representation is trivial. This result applies to our problem. Namely, O(N ) (k) is semi-simple for N > 2 and, since the mass dimension of ∆a (eS⊗ ) is bounded (5.43), the anomaly terms indeed span a ﬁnite dimensional representation of o(N ). (k) It does not matter that our functionals are local. Note that D>1 (eS⊗ ) is not uniquely deﬁned by this procedure. 6. Conclusions and Outlook In algebraic renormalization the QAP is used to remove possible anomalies of Ward identities by induction on . We have worked out an analogous procedure for the MWI in the diﬀerent framework of causal perturbation theory. The main diﬀerence is that we work solely with compactly supported interactions S and localized symmetry transformations δA .w Our main result gives a crucial insight into the structure of possible anomalies of the MWI, in particular with respect to the deformation parameter , and allows the transfer of techniques from algebraic renormalization into causal perturbation theory. This yields a general method to fulﬁll the MWI for a given model. A ﬁrst non-trivial application is worked out (Sec. 5.4.4). The developed method seems to be applicable to many models (as suggested by [2, 10, 11]). Together with the powerful tool of BRST cohomology it should make possible a proof of that cases of the MWI which are needed for the construction of the net of local observables of Yang–Mills type QFTs. (This would complete the construction given in [2].) A main advantage of this approach to quantum Yang–Mills theories is that there seems to be no serious obstacle for the generalization to curved spacetimes where the techniques developed for scalar ﬁelds in [12, 32] can be used. For recent and farreaching progress in the construction of renormalized quantum Yang–Mills ﬁelds in curved spacetime see [33]; this paper uses a generalization of the oﬀ-shell Master w Algebraic renormalization applies to global and local symmetries; examples for local symmetries which have been dealt are current algebras of σ-models and current algebras of gauge theories in which one keeps external ﬁelds (e.g. antiﬁelds).

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BRST Identity (i.e. the MWI for the symmetry transformation δA0 with A0 given by (5.72), see [1, 2] for the on-shell version) to models with antiﬁelds. Appendix A. Proper Vertices for R-Products A.1. Deﬁnition and basic properties Before we introduce proper vertices in terms of R-products, we shortly consider the diagrammatics of retarded products R(A1 (x1 ), . . . ; An (xn )). For the unrenormalized expressions (i.e. for xi = xj , ∀i = j) the diagrammatic interpretation is unique. One can show that there are two kinds of inner lines µ , respectively, and are oriented. For tree diagrams only which symbolize ∆ret and Hm ret ∆ appears and all inner lines are pointing to the distinguished vertex An (xn ). Solely connected diagrams contribute to R; and Rcl is precisely the contribution of all tree diagrams. Both statements follow from the inductive construction of the (Rn,1 )n∈N [16]. The decisive step is the GLZ Relation: {FG , HG } = · · · . In the quantum case there is at least one contraction between FG and HG , and in classical FT there is precisely one contraction in {FG , HG }cl . For the renormalized retarded product we use these results as deﬁnition of the connected and tree part: Rc ≡ R and Rtree ≡ Rcl . Analogously to (4.12)–(4.14) the property R(cl) (1, F ) = F implies the following conclusions ∞ P∞ n n n=1 Fn λ R(cl) e⊗ , Gn λ (A.1) = 0 ⇒ Gn = 0 ∀n, n=0

P∞

R(cl) e⊗

n=1

Fn λn

∞ δFn n λ , δϕ n=0

= R(cl)

P∞

e⊗

n=1

Gn λn

∞ δGn n λ , δϕ n=0

∧ ω0 (Fn ) = ω0 (Gn )

∀n ⇒ Fn = Gn ∀n, (A.2)

which hold for the R-products of classical FT (Rcl ) and of QFT (R). In classical FT F (λ) the statement (A.1) can be proved also non-perturbatively: 0 = Rcl (e⊗ , G(λ)) = G(λ) ◦ rS0 +F (λ),S0 implies G(λ) = G(λ) ◦ rS0 +F (λ),S0 ◦ rS0 ,S0 +F (λ) = 0. The concept of proper vertices has a clear physical interpretation when applied to R-products, since Rtree = Rcl . As explained in Sec. 2 the entries of Rcl may be non-local. We want to rewrite an interacting QFT-ﬁeld R(eS⊗ , F ) as a classical ﬁeld ΓR (eS )

Rcl (e⊗ ⊗ , Γret (eS⊗ , F )) where the “proper interaction” ΓR (eS⊗ ) and the “proper retarded ﬁeld” Γret (eS⊗ , F ) are non-local and agree to lowest order in with the original local functionals S and F respectively. This is indeed possible: Proposition 15. (a) There exist — a totally symmetric and linear map ΓR : TFloc → F

(A.3)

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— and a linear map Γret : TFloc ⊗ Floc → F

(A.4)

which is totally symmetric in the former entries (i.e. Γret (⊗nj=1 Fπj , F ) = Γret (⊗nj=1 Fj , F )), which are uniquely determined by the conditions ΓR (eS ) R(eS⊗ , F ) = Rcl e⊗ ⊗ , Γret (eS⊗ , F ) , Γret (eS⊗ , ϕ(h)) = ϕ(h), ΓR (1) = 0,

(A.5)

h ∈ D(M),

(A.6)

ω0 (ΓR (eS⊗ )) = 0.

(A.7)

(b) ΓR and Γret are related by δΓR (eS⊗ ) δS = Γret eS⊗ , , δϕ(x) δϕ(x)

(A.8)

that is, with (A.7), ΓR is uniquely determined by Γret . Compared with the deﬁning condition (4.19) for ΓT , there is more ﬂexibility in (A.5) since it contains two kinds of “vertex functions”, ΓR and Γret . To deﬁne the latter uniquely, we additionally require (A.6) and (A.7). Proof. (b) First we show that the deﬁning conditions for ΓR , Γret given in part (a) imply the statement in part (b). The oﬀ-shell ﬁeld equation δS0 δS δS0 − R(cl) eS⊗ , R(cl) eS⊗ , = , (A.9) δϕ(x) δϕ(x) δϕ(x) holds for R (QFT) and Rcl (classical FT), in the latter case even for non-local entries. With that and using the conditions (A.5) and (A.6) and ﬁnally (A.1) we obtain the assertion (A.8). (a) By expanding (A.5) in powers of S and using (A.8) and (A.7) we ﬁnd an inductive construction of ΓR and Γret in terms of R and Rcl : Γret (1, F ) = R(F ) ≡ F, δΓR (S) δS δS = Γret 1, ⇒ ΓR (S) = S, = δϕ(x) δϕ(x) δϕ(x)

(A.10a) (A.10b)

Γret (S, F ) = R(S, F ) − Rcl (S, F ), (A.10c) δΓR (S ⊗2 ) δS δS δS = 2Γret S, = 2R S, − 2Rcl S, , δϕ(x) δϕ(x) δϕ(x) δϕ(x) (A.10d) Γret (S ⊗2 , F ) = R(S ⊗2 , F ) − Rcl (S ⊗2 , F ) − Rcl (ΓR (S ⊗2 ), F ) − 2Rcl(S, Γret (S, F )).

(A.10e)

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We explicitly see that Γret is not totally symmetric, it is retarded with respect to the last entry. Now let ΓR (eS⊗ ) and Γret (eS⊗ , F ) be constructed up to order n and (n − 1), respectively, in S. Then, the condition (A.5) determines Γret (S ⊗n , F ) uniquely: k n n d ΓR (eλS ⊗ ) ⊗n ⊗n ⊗n−k e , Γ (S , F ) Γret (S , F ) = R(S , F ) − Rcl ret dλk λ=0 ⊗ k k=1

= R(S

⊗n

,F) −

n k n k=1

· Rcl

j

k

j=1

k

1 j!l1 ! · · · lj !

l1 ,...,lj =1 l1 +···+lj =k

ΓR (S ⊗li ), Γret (S ⊗n−k , F ) .

(A.11)

i=1

From that and with (A.8) and (A.7) we uniquely get ΓR (S ⊗n+1 ). Remark. The roles of the conditions (A.6) and (A.8) can be exchanged. In the list (A.5)–(A.7) of deﬁning conditions, (A.6) can be replaced by (A.8). Then, (A.6) can be derived from (A.5), (A.7) and (A.8) analogously to (4.30) and (4.31): proceeding inductively we use (A.11), the integrated ﬁeld equation for R and Rcl and (A.8). Following the construction (A.10), (A.11) we inductively prove the following properties of ΓR , Γret : • -Dependence: Γret (eS⊗ , F ) = F + O() ΓR (eS⊗ ) = S + O()

if F, S ∼ 0 .

(A.12)

∗ S • P ↑+ -Covariance, Unitarity (ΓR (eS ⊗ ) = ΓR (e⊗ ) and similarly for Γret ), Field Independence, Smoothness in m ≥ 0, µ-Covariance and Almost (m,µ) (m,µ) and Γret ≡ Γret (or, alternaHomogeneous Scaling of ΓR ≡ ΓR tively, Scaling Degree). ∗

In the proof of (A.12) we use R = Rcl + O() , Rcl ∼ 0 . The other properties rely on the validity of the corresponding axioms for R and Rcl , analogously to Lemma 6. We point out that ΓR (eS⊗ ⊗ ϕ(h)) diﬀers in general from ϕ(h), in contrast to the Field equation for ΓT and (A.6). Namely, inserting (A.5), (A.6) into the GLZrelation for [R(eS⊗ , ϕ(h)), R(eS⊗ , ϕ(g))] we obtain ΓR (eS ) & i % ΓR (eS⊗ ) Rcl e⊗ , ϕ(h) , Rcl e⊗ ⊗ , ϕ(g) ΓR (eS ) = Rcl e⊗ ⊗ ⊗ ΓR (eS⊗ ⊗ ϕ(g)), ϕ(h) ΓR (eS ) − Rcl e⊗ ⊗ ⊗ ΓR (eS⊗ ⊗ ϕ(h)), ϕ(g) .

(A.13)

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Due to the GLZ Relation for Rcl , the left-hand side is equal to ΓR (eS ) ΓR (eS ) Rcl e⊗ ⊗ ⊗ ϕ(g), ϕ(h) − Rcl e⊗ ⊗ ⊗ ϕ(h), ϕ(g) +

ΓR (eS ) & i % ΓR (eS⊗ ) Rcl e⊗ , ϕ(h) , Rcl e⊗ ⊗ , ϕ(g) (≥)2 ,

(A.14)

where i [·, ·](≥2) ≡ i [·, ·] − {·, ·}cl. The assertion follows from the non-vanishing of the [·, ·](≥2) -term. ˆ To express the corresponding Finally we study a ﬁnite renormalization R → R. renormalizations of ΓR and Γret in terms of the corresponding map D of the Main Theorem we insert the deﬁning relation (A.5) into both sides of (3.23). In the resulting equation Γˆ R (eS ) ˆ ret (eS⊗ , F ) Rcl e⊗ ⊗ , Γ Γ eD(eS⊗ ) D(eS ) R = Rcl e⊗ ⊗ , Γret e⊗ ⊗ , D(eS⊗ ⊗ F ) (A.15) we choose F = D(eS ⊗)

Γret (e⊗

δS δϕ .

ˆ ret (eS⊗ , F ) by With that we may replace Γ

, D(eS⊗ ⊗ F )) by

D(eS ) δΓR (e⊗ ⊗ )

δϕ

ˆ R (eS ) δΓ ⊗ δϕ

and

. By means of (A.2) we conclude S

ˆ R (eS ) = ΓR (eD(e⊗ ) ). Γ ⊗ ⊗ We insert this into (A.15) and apply (A.1). This yields S ˆ ret (eS⊗ , F ) = Γret eD(e⊗ ) , D(eS⊗ ⊗ F ) . Γ ⊗

(A.16)

(A.17)

A.2. Comparison of the vertex functions in terms of T - and R-products The vertex functions ΓT and ΓR deﬁned in terms of T - and R-products, respectively, are both totally symmetric, nevertheless they do not agree. This follows from the diﬀerent forms of the unitarity property or, alternatively, from the non-validity of ΓT (eS⊗ ⊗ ϕ(h)) = ϕ(h) for ΓR . We are going to compare ΓT with ΓR to lowest orders in S. By using the deﬁnitions of ΓR , Γret (A.5) and ΓT (4.19), as well as (4.15), (4.16) we obtain ΓR (eS ) Rcl e⊗ ⊗ , Γret (eS⊗ , F ) =

∞ iΓT (eS )/ n iΓT (eS )/ 1 − Ttree e⊗ ⊗ Ttree e⊗ ⊗ ⊗ ΓT (eS⊗ ⊗ F ) . n=0

(A.18) If we interpret ΓR , Γret and ΓT as one vertex, then the left-hand side contains solely tree diagrams, but on the right-hand side there appear also loop diagrams! This

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indicates that the relation of ΓR to ΓT is rather involved. To zeroth and ﬁrst order in S we obtain Γret (1, F ) = ΓT (1 ⊗ F ) = F, Γret (S, F ) + Rcl (S, F ) = i/ Ttree (S ⊗ F ) + ΓT (S ⊗ F ) − i/ S F.

(A.19) (A.20)

The terms ∼ −1 and ∼ 0 of (4.15) read Rcl (S, F ) = i/ Ttree (S ⊗ F ) − i/ S (≤1) F, where S

(≤1)

(A.21)

n δnS F = dx1 · · · dxn dy1 · · · dyn n! δϕ(x1 ) · · · δϕ(xn ) n≤1

·

n i=1

µ Hm (xi − yi )

δn F . δϕ(y1 ) · · · δϕ(yn )

(A.22)

In the same way we deﬁne S (≥2) F (i.e. S F = S (≤1) F + S (≥2) F ) and e.g. S (2) F . With that (A.20) reads ΓT (S ⊗ F ) = Γret (S, F ) + i/ S (≥2) F =

1 (Γret (S, F ) + Γret (F, S) + i/ (S (≥2) F + F (≥2) S)). 2

Using additionally the Field Independence of ΓT and (A.10d) we ﬁnd δ i (≥2) δF (≥2) δS (ΓT (S ⊗ F ) − ΓR (S ⊗ F )) = +F S . δϕ δϕ δϕ

(A.23)

(A.24)

Selecting the terms of second order in S from (A.18) we ﬁnd 1/2 Γret(S ⊗2 , F ) + Rcl (S, Γret (S, F )) + 1/2 Rcl(S ⊗2 , F ) + 1/2 Rcl(ΓR (S ⊗2 ), F ) = 1/2 ΓT (S ⊗2 ⊗ F ) + i/ Ttree (S ⊗ ΓT (S ⊗ F )) + i/2 Ttree (ΓT (S ⊗2 ) ⊗ F ) − 1/22 Ttree (S ⊗2 ⊗ F ) − i/ S ΓT (S ⊗ F ) + 1/2 S Ttree (S ⊗ F ) − i/2 ΓT (S ⊗2 ) F + 1/22 Ttree (S ⊗2 ) F − 1/2 S S F.

(A.25)

To simplify this formula and to eliminate all vertex functions Γret , ΓR , ΓT with two arguments, we use (A.23), (A.10d), the Field Independence, as well as δH δG ∆ret (x − y) (A.26) Rcl (G, H) = dx dy δϕ(x) δϕ(y) c and the corresponding expression for Ttree (G, H), in addition (A.21) and the corresponding identity

1/2 Rcl (S ⊗2 , F ) =

1 (−1/2 Ttree(S ⊗2 ⊗ F ) + (S T (S ⊗ F ))tree 2 + 1/2(T (S ⊗2) F )tree + (S S F )tree ),

(A.27)

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and also Ttree (S ⊗ F ) − i ΓT (S ⊗ F ) = T (S ⊗ F )

(A.28)

(which follows from Corollary 5 and (4.11)). It results 1/2 Γret (S ⊗2 , F ) − ΓT (S ⊗2 ⊗ F ) δF i ret (≥2) δS dx dy ∆ (x − y) S = δϕ(x) δϕ(y) i 1 + Rcl (S , S (≥2) F ) − 2 (S (≥a) S (≥b) F )c a+b=3

1 ((S (≥2) T (S ⊗ F ))c − (S (2) (S · F ))tree ) 2 1 + 2 ((T (S ⊗2 ) (≥2) F )c − ((S · S) (2) F )tree ). 2

+

(A.29)

In comparison with (A.25) a main simpliﬁcation is that on the right-hand side solely connected diagrams contribute and the cancellation of all tree diagrams is obvious (i.e. the right-hand side is manifestly of order ). We have not succeeded to generalize the results (A.23), (A.24) and (A.29) to a general formula relating ΓT to Γret or ΓR . Acknowledgment This paper is to a large extent based on the diploma thesis of one of us (F.B.) [34], which was supervised by Klaus Fredenhagen. We proﬁtted from discussions with him in many respects: he gave us important ideas, technical help and also suggestions for the presentation of the material. We are grateful also to Raymond Stora for valuable and detailed comments on the manuscript, which we used to improve some formulations. References [1] M. D¨ utsch and K. Fredenhagen, The Master Ward identity and generalized Schwinger–Dyson equation in classical ﬁeld theory, Commun. Math. Phys. 243 (2003) 275–314. [2] M. D¨ utsch and F. M. Boas, The Master Ward identity, Rev. Math. Phys. 14 (2002) 977–1049. [3] J. H. Lowenstein, Diﬀerential vertex operations in Lagrangian ﬁeld theory, Commun. Math. Phys. 24 (1971) 1–21. [4] Y.-M. P. Lam, Perturbation Lagrangian theory for scalar ﬁelds: Ward–Takahasi identity and current algebra, Phys. Rev. D 6 (1972) 2145–2161. [5] P. Breitenlohner and D. Maison, Dimensional renormalization and the action principle, Commun. Math. Phys. 52 (1977) 11–38. [6] C. Becchi, A. Rouet and R. Stora, Renormalization of the abelian Higgs–Kibble model, Commun. Math. Phys. 42 (1975) 127–162.

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[7] C. Becchi, A. Rouet and R. Stora, Renormalization of gauge theories, Ann. Phys. 98 (1976) 287–321. [8] J. Zinn-Justin, Renormalization of gauge theories and master equation, Modern Phys. Lett. A 14 (1999) 1227–1236. [9] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems (Princeton University Press, Princeton, New Jersey, 1992). [10] O. Piguet and S. P. Sorella, Algebraic Renormalization: Perturbative Renormalization, Symmetries and Anomalies, Lect. Notes Phys., Vol. M28 (Springer, 1995). [11] G. Barnich, F. Brandt and M. Henneaux, Local BRST cohomology in gauge theories, Phys. Rep. 338 (2000) 439–569. [12] R. Brunetti and K. Fredenhagen, Microlocal analysis and interacting quantum ﬁeld theories: Renormalization on physical backgrounds, Commun. Math. Phys. 208 (2000) 623–661. [13] M. D¨ utsch and K. Fredenhagen, A local (perturbative) construction of observables in gauge theories: The example of QED, Commun. Math. Phys. 203 (1999) 71–105. [14] M. D¨ utsch and K. Fredenhagen, Perturbative algebraic ﬁeld theory, and deformation quantization, Fields Inst. Commun. 30 (2001) 151–160. [15] M. D¨ utsch and K. Fredenhagen, Algebraic quantum ﬁeld theory, perturbation theory, and the loop expansion, Commun. Math. Phys. 219 (2001) 5–30. [16] M. D¨ utsch and K. Fredenhagen, Causal perturbation theory in terms of retarded products, and a proof of the Action Ward identity, Rev. Math. Phys. 16 (2004) 1291–1348. [17] N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (Wiley Interscience, New York, 1959). [18] H. Epstein and V. Glaser, The role of locality in perturbation theory, Ann. Poincare Phys. Theor. A 19 (1973) 211–295. [19] R. Brunetti, M. D¨ utsch and K. Fredenhagen, Retarded products versus time-ordered products: A geometrical interpretation, work in preparation. [20] M. D¨ utsch and K. Fredenhagen, Action Ward identity and the St¨ uckelberg– Petermann renormalization group, in Rigorous Quantum Field Theory, eds. A. Boutet de Monvel, D. Iagolnitzer and U. Moschella (Birkh¨ auser, 2006), pp. 113–123. [21] R. Peierls, The commutation laws of relativistic ﬁeld theory, Proc. Roy. Soc (London) A 214 (1952) 143–157. [22] D. M. Marolf, The generalized Peierls bracket, Ann. Phys. 236 (1994) 392–412. [23] O. Steinmann, Perturbative Expansion in Axiomatic Field Theory, Vol. 11, Lect. Notes Phys. (Springer, 1971). [24] O. Steinmann, Perturbative Quantum Electrodynamics and Axiomatic Field Theory (Springer, 2000). [25] C. Itzykson and J.-B. Zuber, Quantum Field Theory (Mc Graw-Hill, 1980). [26] M. D¨ utsch, T. Hurth, F. Krahe and G. Scharf, Causal construction of Yang–Mills theories. I, N. Cimento A 106 (1993) 1029–1041. [27] M. D¨ utsch, Proof of perturbative gauge invariance for tree diagrams to all orders, Ann. Phys. (Leipzig) 14 (2005) 438–461. [28] G. Jona-Lasinio, Relativistic ﬁeld theories with symmetry-breaking solutions, N. Cimento 34 (1964) 1790–1795. [29] M. Gell-Mann and F. Low, Bound states in quantum ﬁeld theory, Phy. Rev. 84 (1951) 350–354. [30] F. Brennecke and M. D¨ utsch, The quantum action principle in the framework of causal perturbation theory, hep-th/0801.1408. [31] M. D¨ utsch, T. Hurth, F. Krahe and G. Scharf, Causal construction of Yang–Mills theories. II, N. Cimento A 107 (1994) 375–406.

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[32] S. Hollands and R. M. Wald, Conservation of the stress tensor in interacting quantum ﬁeld theory in curved spacetimes, Rev. Math. Phys. 17 (2005) 227–312. [33] S. Hollands, Renormalized quantum Yang–Mills ﬁelds in curved spacetime (2007), gr-qc/0705.3340. [34] F. Brennecke, Investigations to the anomaly problem of the Master Ward identity, Diploma Thesis (in German) (2005); http://www.desy.de/uni-th/lqp/psﬁles/diplbrennecke.ps.gz.

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Reviews in Mathematical Physics Vol. 20, No. 2 (2008) 173–198 c World Scientific Publishing Company

SYMMETRIES AND INVARIANTS OF TWISTED QUANTUM ALGEBRAS AND ASSOCIATED POISSON ALGEBRAS

A. I. MOLEV∗ and E. RAGOUCY† ∗School

of Mathematics and Statistics, University of Sydney, NSW 2006, Australia [email protected] †LAPTH, Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux Cedex, France [email protected]

Received 2 April 2007 We construct an action of the braid group BN on the twisted quantized enveloping algebra Uq (oN ) where the elements of BN act as automorphisms. In the classical limit q → 1, we recover the action of BN on the polynomial functions on the space of upper triangular matrices with ones on the diagonal. The action preserves the Poisson bracket on the space of polynomials which was introduced by Nelson and Regge in their study of quantum gravity and rediscovered in the mathematical literature. Furthermore, we construct a Poisson bracket on the space of polynomials associated with another twisted quantized enveloping algebra Uq (sp2n ). We use the Casimir elements of both twisted quantized enveloping algebras to reproduce and construct some well-known and new polynomial invariants of the corresponding Poisson algebras. Keywords: Poisson algebra; braid group action; quantized enveloping algebra. Mathematics Subject Classification 2000: 17B37, 17B63, 81R50

1. Introduction The deformations of the commutation relations of the orthogonal Lie algebra o3 were considered by many authors. The earliest reference we are aware of is Santilli [28]. Such deformed relations can be written as qXY − YX = Z,

qYZ − ZY = X,

qZX − XZ = Y.

(1.1)

More precisely, regarding q as a formal variable, we consider the associative algebra Uq (o3 ) over the ﬁeld of rational functions C(q) in q with the generators X, Y, Z and deﬁning relations (1.1). From an alternative viewpoint, relations (1.1) deﬁne a family of algebras depending on the complex parameter q. The same algebras were also deﬁned by Odesskii [26], Fairlie [9] and Nelson, Regge and Zertuche [23]. Putting

173

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q = 1 in (1.1), we get the deﬁning relations of the universal enveloping algebra U(o3 ). The algebra Uq (o3 ) should be distinguished from the quantized enveloping algebra Uq (o3 ) ∼ = Uq (sl2 ). The latter is a deformation of U(o3 ) in the class of Hopf algebras; see, e.g., Chari and Pressley [4, Sec. 6]. Introducing the generators x = (q − q −1 )X,

y = (q − q −1 )Y,

z = (q − q −1 )Z,

we can write the deﬁning relations of Uq (o3 ) in the equivalent form qxy − yx = (q − q −1 )z, qyz − zy = (q − q −1 )x, qzx − xz = (q − q −1 )y. Note that the element x2 + q −2 y 2 + z 2 − xyz belongs to the center of Uq (o3 ). This time, putting q = 1 into the deﬁning relations we get the algebra of polynomials C[x, y, z]. Moreover, this algebra can be equipped with a Poisson bracket in a usual way f g − gf . {f, g} = 1 − q q=1 Thus, C[x, y, z] becomes a Poisson algebra with the bracket given by {x, y} = xy − 2z,

{y, z} = yz − 2x,

{z, x} = zx − 2y.

(1.2)

These formulas are contained in the paper by Nelson, Regge and Zertuche [23]. In the classical limit q → 1, the central element x2 + q −2 y 2 + z 2 − xyz becomes the Markov polynomial x2 + y 2 + z 2 − xyz which is an invariant of the bracket. The Poisson bracket (1.2) was rediscovered by Dubrovin [8], where x, y, z are interpreted as the entries of 3 × 3 upper triangular matrices with ones on the diagonal (the Stokes matrices)   1 x y 0 1 z  . 0 0 1 For an arbitrary N the twisted quantized enveloping algebra Uq (oN ) was introduced by Gavrilik and Klimyk [11] which essentially coincides with the algebra of Nelson and Regge [20]. Both in the orthogonal and symplectic case the twisted analogues of the quantized enveloping algebras were introduced by Noumi [24] using an R-matrix approach. In the orthogonal case, this provides an alternative presentation of Uq (oN ). The ﬁnite-dimensional irreducible representations of the algebra Uq (oN ) were classiﬁed by Iorgov and Klimyk [14]. In the limit q → 1, the twisted quantized enveloping algebra Uq (oN ) gives rise to a Poisson algebra of polynomial functions PN on the space of Stokes matrices. The corresponding Poisson bracket was given in [20]. The same bracket was also

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found by Ugaglia [29], Boalch [1] and Bondal [2, 3]. This Poisson structure was studied by Xu [30] in the context of Dirac submanifolds, while Chekhov and Fock [6] considered it in relation with the Teichm¨ uller spaces. A quantization of the Poisson algebra of Stokes matrices leading to the algebra Uq (oN ) was constructed by Ciccoli and Gavarini [7] in the context of the general “quantum quality principle”; see also Gavarini [10]. It was shown by Odesskii and Rubtsov [27] that the Poisson bracket on the space of Stokes matrices is essentially determined by its Casimir elements. Automorphisms of both the algebra Uq (oN ) and the Poisson bracket on PN were given in [21, 22], although the explicit group relations between them were only discussed in the classical limit for N = 6. An action of the braid group BN on the Poisson algebra PN was given by Dubrovin [8] and Bondal [2]. In this paper, we produce a “quantized” action of BN on the twisted quantized enveloping algebra Uq (oN ), where the elements of BN act as automorphisms. Since Uq (oN ) is a subalgebra of the quantized enveloping algebra Uq (glN ), one could expect that Lusztig’s action of BN on Uq (glN ) (see [16]) leaves the subalgebra Uq (oN ) invariant. However, this turns out not to be true, and the action of BN on Uq (oN ) can rather be regarded as a q-version of the natural action of the symmetric group SN on the universal enveloping algebra U(oN ). The relationship between Uq (oN ) and the Poisson algebra PN can also be exploited in a diﬀerent way. Some families of Casimir elements of Uq (oN ) were produced by Noumi, Umeda and Wakayama [25], Gavrilik and Iorgov [12] and Molev, Ragoucy and Sorba [19]. This gives the respective families of Casimir elements of the Poisson algebra. We show that the Casimir elements of [19] specialize precisely to the coeﬃcients of the characteristic polynomial of Nelson and Regge [22]. This polynomial was rediscovered by Bondal [2] who also produced an algebraically independent set of generators of the subalgebra of invariants of the Poisson algebra PN . Furthermore, using [12, 25] we obtain new Pfaﬃan type invariants and analogues of the Gelfand invariants. In a similar manner, we use the twisted quantized enveloping algebra Uq (sp2n ) associated with the symplectic Lie algebra sp2n to produce a symplectic version of the above results. First, we construct a Poisson algebra associated with Uq (sp2n ) by taking the limit q → 1 and thus produce explicit formulas for the Poisson bracket on the corresponding space of matrices. Then, using the Casimir elements of Uq (sp2n ) constructed in [19], we produce a family of invariants of the Poisson algebra analogous to [2, 22]. We also show that some elements of the braid group B2n preserve the subalgebra Uq (sp2n ) of Uq (gl2n ). We conjecture that there exists an action of the semi-direct product Bn Z n on Uq (sp2n ) analogous to the BN action on Uq (oN ). We show that the conjecture is true for n = 2. After we prepared the ﬁrst version of our paper, we learned of a recent preprint by Chekhov [5] where he produces (without detailed proofs) an action of the braid group BN on Uq (oN ) equivalent to ours.

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2. Braid Group Action We start with some deﬁnitions and recall some well-known results. Let q be a formal variable. The quantized enveloping algebra Uq (glN ) is an algebra over C(q) generated by elements tij and t¯ij with 1 ≤ i, j ≤ N subject to the relations 1 ≤ i < j ≤ N, tij = t¯ji = 0, ¯ ¯ tii tii = tii tii = 1, 1 ≤ i ≤ N, RT¯1 T¯2 = T¯2 T¯1 R, RT1 T2 = T2 T1 R, Here T and T¯ are the matrices tij ⊗ Eij , T =

T¯ =

i,j

(2.1) RT¯1 T2 = T2 T¯1 R.

t¯ij ⊗ Eij ,

(2.2)

i,j

which are regarded as elements of the algebra Uq (glN ) ⊗ End C N , the Eij denote the standard matrix units and the indices run over the set {1, . . . , N }. Both sides of each of the R-matrix relations in (2.1) are elements of Uq (glN ) ⊗ End C N ⊗ End C N and the subscripts of T and T¯ indicate the copies of End C N , e.g., tij ⊗ Eij ⊗ 1, T2 = tij ⊗ 1 ⊗ Eij , T1 = i,j

i,j

while R is the R-matrix Eii ⊗ Eii + Eii ⊗ Ejj + (q − q −1 ) Eij ⊗ Eji . R=q i

(2.3)

i<j

i=j

In terms of the generators, the deﬁning relations between the tij can be written as q δij tia tjb − q δab tjb tia = (q − q −1 )(δb
(2.4)

where δi<j equals 1 if i < j, and 0 otherwise. The relations between the t¯ij are obtained by replacing tij by t¯ij everywhere in (2.4), while the relations involving both tij and t¯ij have the form q δij t¯ia tjb − q δab tjb t¯ia = (q − q −1 )(δb
(2.5)

The braid group BN is generated by elements β1 , . . . , βN −1 subject to the deﬁning relations βi βi+1 βi = βi+1 βi βi+1 ,

i = 1, . . . , N − 2

and β i β j = βj β i ,

|i − j| > 1.

The group BN acts on the algebra Uq (glN ) by automorphisms; see Lusztig [16]. Explicit formulas for the images of the generators are found from [16] by re-writing

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the action in terms of the presentation (2.1). For any i = 1, . . . , N − 1, we have βi : tii → ti+1,i+1 , βi : ti+1,i → q −1 t¯i,i+1 t2ii tik → qtik ti+1,i t¯ii − ti+1,k , tli →

q −1 t¯i,i+1 tli tii

− tl,i+1 ,

tkl → tkl

ti+1,i+1 → tii ,

tkk → tkk

ti+1,k → q −1 tik ,

if k ≤ i − 1

tl,i+1 → qtli ,

if l ≥ i + 2

if k = i, i + 1,

in all remaining cases,

and βi : t¯i,i+1 → q t¯2ii ti+1,i t¯ki → q −1 tii t¯i,i+1 t¯ki − t¯k,i+1 , t¯k,i+1 → q t¯ki , if k ≤ i − 1 −1 t¯i+1,l → q t¯il , if l ≥ i + 2 t¯il → q t¯ii t¯il ti+1,i − t¯i+1,l , in all remaining cases. t¯kl → t¯kl Following Noumi [24], we deﬁne the twisted quantized enveloping algebra Uq (oN ) as the subalgebra of Uq (glN ) generated by the matrix elements sij of the matrix S = T T¯ t so that N sij = tik t¯jk . k=1

Equivalently,

Uq (oN )

is generated by the elements sij subject only to the relations sij = 0,

1 ≤ i < j ≤ N,

(2.6)

sii = 1,

1 ≤ i ≤ N,

(2.7)

RS1 Rt S2 = S2 Rt S1 R, t

(2.8)

t1

where R := R denotes the element obtained from R by the transposition in the ﬁrst tensor factor: Eii ⊗ Eii + Eii ⊗ Ejj + (q − q −1 ) Eji ⊗ Eji . (2.9) Rt = q i

i=j

i<j

In terms of the generators, the relations (2.8) take the form q δjk +δik sij skl − q δjl +δil skl sij = (q − q −1 )q δji (δl<j − δi
= = = = = =

0 0 (q − q −1 )(skj sil − sik sjl ) (q − q −1 )sil (q − q −1 )slj (q − q −1 )ski

if if if if if if

i>j>k>l i>k>l>j i>k>j>l i>j>l i>l>j k > i > j.

(2.11)

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In this form the relations were given by Nelson and Regge [20]. An analogue of the Poincar´e–Birkhoﬀ–Witt theorem for the algebra Uq (oN ) was proved in [13]; see also [17, 19] for other proofs. This theorem implies that at q = 1 the algebra Uq (oN ) specializes to the algebra of polynomials in N (N − 1)/2 variables. More precisely, set A = C[q, q −1 ] and consider the A-subalgebra UA of Uq (oN ) generated by the elements sij . Then we have an isomorphism UA ⊗A C ∼ = PN ,

(2.12)

where the action of A on C is deﬁned via the evaluation q = 1 and PN denotes the algebra of polynomials in the independent variables aij with 1 ≤ j < i ≤ N . The elements aij are respective images of the sij under the isomorphism (2.12). Furthermore, the algebra PN is equipped with the Poisson bracket {·, ·} deﬁned by ˜−h ˜ f˜ f˜h , (2.13) {f, h} = 1 − q q=1 ˜ are elements of U whose images in PN under the where f, h ∈ PN and f˜ and h A specialization q = 1 coincide with f and h, respectively. Indeed, write the element ˜−h ˜ f˜ ∈ U as a linear combination of the ordered monomials in the generators f˜h A ˜ −h ˜ f˜ in PN is zero, all the coeﬃcients with coeﬃcients in A. Since the image of f˜h are divisible by 1 − q. Clearly, the element {f, h} ∈ PN is independent of the choice ˜ and of the ordering of the generators of U . Obviously, (2.13) does deﬁne of f˜ and h A a Poisson bracket on PN . By deﬁnition, sij skl − skl sij {aij , akl } = . 1−q q=1 Hence, using the deﬁning relations (2.11), we get {aij , akl } {aij , akl } {aij , akl } {aij , ajl } {aij , ail } {aij , akj }

=0 =0 = 2(aik ajl − akj ail ) = aij ajl − 2ail = aij ail − 2alj = aij akj − 2aki

if if if if if if

i>j>k>l i>k>l>j i>k>j>l i>j>l i>l>j k > i > j.

(2.14)

This coincides with the Poisson brackets of [2, 21, 29], up to a constant factor if we interpret aij as the jith entry of the upper triangular matrix. We shall also use the presentation of the algebra Uq (oN ) due to Gavrilik and Klimyk [11]. An isomorphism between the presentations was given by Noumi [24], a proof can be found in Iorgov and Klimyk [13]. Set si = si+1,i for i = 1, . . . , N − 1. Then the algebra Uq (oN ) is generated by the elements s1 , . . . , sN −1 subject only to the relations sk s2k+1 − (q + q −1 )sk+1 sk sk+1 + s2k+1 sk = −q −1 (q − q −1 )2 sk , s2k sk+1 − (q + q −1 )sk sk+1 sk + sk+1 s2k = −q −1 (q − q −1 )2 sk+1 ,

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for k = 1, . . . , N − 2 (the Serre type relations), and s k sl = sl sk ,

|k − l| > 1.

It is easy to see that the subalgebra Uq (oN ) ⊂ Uq (glN ) is not preserved by the action of the braid group BN on Uq (glN ) described above. Nevertheless, we have the following theorem. Theorem 2.1. For i = 1, . . . , N − 1 the assignment 1 (qsi+1 si − si si+1 ) βi : si+1 → q − q −1 si−1 →

1 (si si−1 − qsi−1 si ) q − q −1

si → −si sk → sk

if k = i − 1, i, i + 1,

deﬁnes an action of the braid group BN on Uq (oN ) by automorphisms. Proof. We verify ﬁrst that the images of the generators s1 , . . . , sN −1 under βi satisfy the deﬁning relations of Uq (oN ). A nontrivial calculation is only required to verify that the images of the pairs of generators βi (sk ) and βi (sk+1 ) with k = i − 2, i − 1, i, i + 1 satisfy both Serre type relations, and that the images βi (si−1 ) and βi (si+1 ) commute. Observe that by (2.11), the image of si+1 can also be written as βi : si+1 → si+2,i . Hence, for k = i + 1 we need to verify that si+2,i s2i+3,i+2 − (q + q −1 )si+3,i+2 si+2,i si+3,i+2 + s2i+3,i+2 si+2,i = −q −1 (q − q −1 )2 si+2,i . We shall verify the following more general relation in Uq (oN ), sij s2ki − (q + q −1 )ski sij ski + s2ki sij = −q −1 (q − q −1 )2 sij ,

(2.15)

where k > i > j. Indeed, the left hand side equals −(qski sij − sij ski )ski + q −1 ski (qski sij − sij ski ).

(2.16)

However, by (2.11) we have qski sij − sij ski = (q − q −1 )skj so that (2.16) becomes −q −1 (q − q −1 )(qskj ski − ski skj ) which equals −q −1 (q−q −1 )2 sij by (2.11) thus proving (2.15). The second Serre type relation for the images βi (si+1 ) and βi (si+2 ) follows from a more general relation in Uq (oN ), s2ij ski − (q + q −1 )sij ski sij + ski s2ij = −q −1 (q − q −1 )2 ski ,

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where k > i > j, and which is veriﬁed in the same way as (2.15). Next, the Serre type relations for the images βi (si ) and βi (si+1 ) follow respectively from the relations s2ij skj − (q + q −1 )sij skj sij + skj s2ij = −q −1 (q − q −1 )2 sij and s2ij skj − (q + q −1 )sij skj sij + skj s2ij = −q −1 (q − q −1 )2 skj , where k > i > j, which both are implied by (2.11). The Serre type relations for the pairs βi (si−1 ), βi (si ) and βi (si−2 ), βi (si−1 ) can now be veriﬁed by using the involutive automorphism ω of Uq (oN ) which is deﬁned on the generators by sk → sN −k ,

k = 1, . . . , N − 1.

(2.17)

We have ω : βi (si−2 ) → βN −i (sN −i+2 ), βi (si−1 ) → −βN −i (sN −i+1 ), βi (si ) → βN −i (sN −i ), and so the desired relations are implied by the Serre type relations for the pairs of the images βj (sj ), βj (sj+1 ) and βj (sj+1 ), βj (sj+2 ) with j = N − i. Now, we verify that the images βi (si−1 ) and βi (si+1 ) commute, that is, (si si−1 − qsi−1 si )(qsi+1 si − si si+1 ) = (qsi+1 si − si si+1 )(si si−1 − qsi−1 si ). (2.18) By the Serre type relations, we have s2i si+1 − (q + q −1 )si si+1 si + si+1 s2i = −q −1 (q − q −1 )2 si+1 and s2i si−1 − (q + q −1 )si si−1 si + si−1 s2i = −q −1 (q − q −1 )2 si−1 . Multiply the ﬁrst of these relations by si−1 and the second by si+1 from the left. Taking the diﬀerence we come to si−1 s2i si+1 − (q + q −1 )si−1 si si+1 si = si+1 s2i si−1 − (q + q −1 )si+1 si si−1 si . Now, repeat the same calculation but multiply the Serre type relations by si−1 and si+1 , respectively, from the right. This gives si−1 s2i si+1 − (q + q −1 )si si−1 si si+1 = si+1 s2i si−1 − (q + q −1 )si si+1 si si−1 . Hence, si−1 si si+1 si − si+1 si si−1 si = si si−1 si si+1 − si si+1 si si−1 and (2.18) follows.

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Thus, each βi with i = 1, . . . , N −1 deﬁnes a homomorphism Uq (oN ) → Uq (oN ). Now observe that βi is invertible with the inverse given by 1 βi−1 : si+1 → (si+1 si − qsi si+1 ) q − q −1 si−1 →

1 (qsi si−1 − si−1 si ) q − q −1

si → −si sk → sk if k = i − 1, i, i + 1, and so βi and βi−1 are mutually inverse automorphisms of Uq (oN ). Finally, we verify that the automorphisms βi satisfy the braid group relations. It suﬃces to check that for each generator sk we have βi βi+1 βi (sk ) = βi+1 βi βi+1 (sk )

(2.19)

for i = 1, . . . , N − 2, and βi βj (sk ) = βj βi (sk )

(2.20)

for |i − j| > 1. Clearly, the only nontrivial cases of (2.19) are k = i − 1, i, i + 1, i + 2 while (2.20) is obvious for all cases except for j = i + 2 and k = i + 1. Take k = i − 1 in (2.19). We have βi+1 (si−1 ) = si−1 while 1 si si−1 − qsi−1 si = qsi+1,i−1 − qsi+1,i si,i−1 , βi : si−1 → q − q −1 where we have used (2.11). Furthermore, using again (2.11), we ﬁnd βi+1 βi : si−1 → q 2 si+2,i−1 − q 2 si+2,i+1 si+1,i−1 − q 2 si+2,i si,i−1 + q 2 si+2,i+1 si+1,i si,i−1 . It remains to verify with the use of (2.11) that this element is stable under the action of βi . The remaining cases of (2.19) and (2.20) are veriﬁed with similar and even simpler calculations. Corollary 2.2. In terms of the generators skl of the algebra Uq (oN ), for each index i = 1, . . . , N − 1 the action of βi is given by −si+1,i βi : si+1,i → sik → qsi+1,k − qsi+1,i sik , −1

sli → q sl,i+1 − sli si+1,i , skl skl →

si+1,k → sik ,

if k ≤ i − 1

sl,i+1 → sli , in all remaining cases.

if l ≥ i + 2

Proof. This follows from the deﬁning relations (2.11). Indeed, the elements skl can be expressed in terms of the generators s1 , . . . , sN −1 by induction, using the relations 1 (qskj sjl − sjl skj ), k > j > l. (2.21) skl = q − q −1

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This determines the action of βi on the elements skl and the formulas are veriﬁed by induction. Remark 2.3. It is possible to prove that the formulas of Corollary 2.2 deﬁne an action of the braid group BN on Uq (oN ) by automorphisms only using the presentation (2.11). However, this leads to a slightly longer calculations as compared with the proof of Theorem 2.1. Note also that the universal enveloping algebra U(oN ) can be obtained as a specialization of Uq (oN ) in the limit q → 1; see [19] for a precise formulation. In this limit, the elements sij /(q − q −1 ) with i > j specialize to the generators Fij of oN , where Fij = Eij − Eji . Hence, the action of BN on Uq (oN ) specializes to the action of the symmetric group SN on U(oN ) by permutations of the indices of the Fij . The mapping (2.17) can also be extended to the entire algebra Uq (oN ) as an anti-automorphism. This is readily veriﬁed with the use of the Serre type relations. We denote this involutive anti-automorphism of Uq (oN ) by ω . Proposition 2.4. The action of ω on the generators skl is given by ω : skl → sN −l+1,N −k+1 ,

1 ≤ l < k ≤ N.

(2.22)

Moreover, we have the relations −1 ω βi ω = β N −i ,

i = 1, . . . , N − 1,

(2.23)

where the automorphisms βi of Uq (oN ) are deﬁned in Theorem 2.1. Proof. The deﬁning relations (2.11) imply that the mapping (2.22) deﬁnes an antiautomorphism of Uq (oN ). Obviously, the images of the generators sk are found by (2.17). The second part of the proposition is veriﬁed by comparing the images of the generators sk under the automorphisms on both sides of (2.23). Observe that the image of the matrix S under ω is given by ω : S → S , where the prime denotes the transposition with respect to the second diagonal. Now consider the involutive automorphism ω of Uq (oN ) deﬁned by the mapping (2.17). Proposition 2.5. The image of the matrix S under ω is given by ω : S → (1 − q −1 )I + q −1 D(S −1 ) D−1 , 2

(2.24) N

where I is the identity matrix and D = diag(−q, (−q) , . . . , (−q) ). In terms of the generators, this can be written as (−1)p sN −l+1,r1 sr1 r2 · · · srp ,N −k+1 , ω : skl → (−q)k−l−1 N −l+1>r1 >···>rp >N −k+1

k > l, summed over p ≥ 0 and the indices r1 , . . . , rp .

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Proof. The elements skl can be expressed in terms of the generators s1 , . . . , sN −1 by (2.21). The formula for ω(skl ) is then veriﬁed by induction on k − l. The matrix form (2.24) is implied by the relation (−1)p+1 sk,r1 sr1 r2 · · · srp ,l , k > l, (2.25) (S −1 )kl = k>r1 >···>rp >l

summed over p ≥ 0 and the indices r1 , . . . , rp . For any diagonal matrix C = diag(c1 , . . . , cN ), the relation (2.8) is preserved by the transformation S → CSC. Indeed, the entries of S are then transformed as sij → sij ci cj and the claim is immediate from (2.10). This implies that if c2i = 1 for all i then the mapping ς : S → CSC deﬁnes an automorphism of Uq (oN ). Therefore, Propositions 2.4 and 2.5 imply the following corollary. Corollary 2.6. The mapping ρ : S → (1 − q −1 )I + q −1 HS −1 H −1 ,

(2.26)

where H = diag(q, q 2 , . . . , q N ), deﬁnes an involutive anti-automorphism of Uq (oN ). Proof. We obviously have ρ = ς ◦ ω ◦ ω for an appropriate automorphism ς. Hence ρ is an anti-automorphism. We have ρ : sk → −sk ,

k = 1, . . . , N − 1,

and so ρ is involutive. We can now recover the braid group action on the algebra PN ; see Dubrovin [8], Bondal [2]. Corollary 2.7. The braid group BN acts on the algebra PN by βi : ai+1,i aik ali akl

→ → → →

−ai+1,i ai+1,k − ai+1,i aik , al,i+1 − ali ai+1,i , akl

ai+1,k → aik , al,i+1 → ali , in all remaining cases,

if k ≤ i − 1 if l ≥ i + 2

where i = 1, . . . , N − 1. Moreover, the Poisson bracket on PN in invariant under this action. Proof. This is immediate from Corollary 2.2. We combine the variables aij into the lower triangular matrix A = [aij ] where we set aii = 1 for all i and aij = 0 for i < j.

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Corollary 2.8. The mappings : A → A−1

and

: A → A

(2.27)

deﬁne anti-automorphisms of the Poisson bracket on PN . Explicitly, the image of akl under is given by : akl → (−1)p+1 akr1 ar1 r2 · · · arp ,l , k > l, k>r1 >···>rp >l

summed over p ≥ 0 and the indices r1 , . . . , rp . Proof. This follows from Proposition 2.4 and Corollary 2.6 by taking q = 1. 3. Casimir Elements of the Poisson Algebra PN Using the relationship between the twisted quantized enveloping algebra Uq (oN ) and the Poisson algebra PN , we can get families of invariants of PN by taking the classical limit q → 1 in the constructions of [19, 12, 25]. First, we recall the construction of Casimir elements for the algebra Uq (oN ) given in [19]. Consider the q-permutation operator P q ∈ End (C N ⊗ C N ) deﬁned by Pq = Eii ⊗ Eii + q Eij ⊗ Eji + q −1 Eij ⊗ Eji . (3.1) i

i>j

i<j

Introduce the multiple tensor product Uq (oN ) ⊗ (End C N )⊗r . The action of the symmetric group Sr on the space (C N )⊗r can be deﬁned by setting σi → Pσqi := q for i = 1, . . . , r − 1, where σi denotes the transposition (i, i + 1). If σ = Pi,i+1 σi1 · · · σil is a reduced decomposition of an element σ ∈ Sr we set Pσq = Pσqi1 · · · Pσqi . l We denote by Aqr the q-antisymmetrizer Aqr = sgn σ · Pσq . (3.2) σ∈Sr

Now take r = N . We have the relation t t t t AqN S1 (u)R12 · · · R1N S2 (uq −2 )R23 · · · R2N S3 (uq −4 ) t −2N +2 ) × · · · RN −1,N SN (uq t −4 t t )R2N · · · R23 S2 (uq −2 ) = SN (uq −2N +2 )RN −1,N · · · S3 (uq

(3.3)

t t × R1N · · · R12 S1 (u)AqN ,

where the following notation was used. The matrix S(u) is deﬁned by ¯ S(u) = S + q −1 u−1 S, where u is a formal variable and S¯ is the upper triangular matrix with ones on the diagonal whose ijth entry is s¯ij = qsji for i < j. Furthermore, t t Rij = Rij (u−1 q 2i−2 , uq −2j+2 )

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with Rt (u, v) = (u − v)

Eii ⊗ Ejj + (q −1 u − qv)

i=j

+ (q −1 − q)u

Eii ⊗ Eii

i

Eji ⊗ Eji + (q −1 − q)v

i>j

185

Eji ⊗ Eji .

(3.4)

i<j

The subscripts in (3.3) indicate the copies of End C N in Uq (oN ) ⊗ (End C N )⊗N which are labeled by 1, . . . , N ; cf. (2.1). The element (3.3) equals AqN sdet S(u), where sdet S(u) is a rational function in u (the Sklyanin determiant ) valued in the center of Uq (oN ); see [19, Theorem 3.8 and Corollary 4.3]. Recall that the Poisson algebra PN is the algebra of polynomials in the variables aij with i > j which are combined into the matrix A = [aij ] with aii = 1 for all i and aij = 0 for i < j. The following theorem was proved in diﬀerent ways by Nelson and Regge [22] and Bondal [2]. Theorem 3.1. The coeﬃcients of the polynomial det(A + λAt ) = f0 + f1 λ + · · · + fN λN are Casimir elements of the Poisson algebra PN . Proof. We use the centrality of the Sklyanin determinant sdet S(u) in Uq (oN ). Note that at q = 1 the q-antisymmetrizer AqN becomes the antisymmetrizer in (C N )⊗N , the element Rt (u−v) becomes u−v times the identity. Since the images of the elements sij in PN coincide with aij , the image of the matrix S(u) is A+u−1 At . Hence, at q = 1 the Sklyanin determinant sdet S(u) becomes γ(u) det(A + u−1 At ), where γ(u) = (u−1 − u)N (N −1)/2 .

(3.5)

Therefore, replacing u with λ−1 we thus prove that all coeﬃcients of det(A + λAt ) are Casimir elements for the Poisson bracket on PN . Note that, as was proved in [2, 22], the polynomial det(A + λAt ) is invariant under the action of the braid group BN . Now we recall the construction of Casimir elements given in [12]. For all i > j, deﬁne the elements s+ ij of Uq (oN ) by induction from the formulas s+ ij =

1 (s+ s − qsj+1,j s+ i,j+1 ), q − q −1 i,j+1 j+1,j

i > j + 1,

and s+ j+1,j = sj+1,j for j = 1, . . . , N − 1. A straightforward calculation shows that these elements can be equivalently deﬁned by i−j−1 s+ (S −1 )ij , ij = −q

i > j,

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where the entries of the inverse matrix are found from (2.25). Let k be a positive integer such that 2k ≤ N . For any subset I = {i1 < i2 < · · · < i2k } of {1, . . . , N } introduce the elements ΦI and Φ+ I of Uq (oN ) by ΦI = (−q)−(σ) siσ(2) iσ(1) · · · siσ(2k) iσ(2k−1) σ∈S2k

and Φ+ I =

σ∈S2k

+ (−q)(σ) s+ iσ(2) iσ(1) · · · siσ(2k) iσ(2k−1) ,

where (σ) is the length of the permutation σ, and the sums are taken over those permutations σ ∈ S2k which satisfy the conditions iσ(2) > iσ(1) , . . . , iσ(2k) > iσ(2k−1)

and iσ(2) < iσ(4) < · · · < iσ(2k) .

Then according to [12], for each k the element φk = q i1 +i2 +···+i2k Φ+ I ΦI I,|I|=2k

belongs to the center of Uq (oN ). Moreover, in the case N = 2n both elements ΦI0 and Φ+ I0 with I0 = {1, . . . , 2n} are also central. Remark 3.2. Our notation is related to [12] by − sij = −q −1/2 (q − q −1 )Iij ,

+ −1/2 s+ (q − q −1 )Iij , ij = −q

i > j.

Note also that the elements φk are q-analogues of the Casimir elements for the orthogonal Lie algebra oN constructed in [18]; see also [15]. Now, we return to the Poisson algebra PN . Recall that the Pfaﬃan of a 2k × 2k skew symmetric matrix H is given by Pf H =

1 2k k!

sgn σ · Hσ(1),σ(2) · · · Hσ(2k−1),σ(2k) .

σ∈S2k

Given a lower triangular N × N matrix B and a 2k-element subset I of {1, . . . , N } as above, we denote by Pf I (B) the Pfaﬃan of the 2k × 2k submatrix (B t − B)I of B t − B whose rows and columns are determined by the elements of I. Theorem 3.3. For each positive integer k such that 2k ≤ N the element ck = (−1)k Pf I (A) Pf I (A−1 )

(3.6)

I,|I|=2k

is a Casimir element of PN . Moreover, in the case N = 2n both Pf I0 (A) and Pf I0 (A−1 ) with I0 = {1, . . . , 2n} are also Casimir elements.

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Proof. Observe that in the limit q → 1 the elements ΦI and Φ+ I specialize respectively to the Pfaﬃans ΦI → Pf I (A),

k −1 Φ+ ). I → (−1) Pf I (A

Hence, the central element φk specializes to ck . Example 3.4. As the matrix elements of the inverse matrix A−1 are found by the formula of Corollary 2.8, we have the following explicit formula for c1 , (−1)p aij air1 ar1 r2 · · · arp j . c1 = i>r1 >···>rp >j

For N = 3, it gives the Markov polynomial. Corollary 3.5. The algebra of Casimir elements of PN is generated by c1 , . . . , cn for N = 2n + 1, and by c1 , . . . , cn−1 , Pf I0 (A) if N = 2n. In both cases, the families of generators are algebraically independent. Moreover, Pf I0 (A−1 ) = (−1)n Pf I0 (A). Proof. Since det(A + λAt ) = λN det(A + λ−1 At ), we have the relations fN −i = fi . Moreover, f0 = fN = 1 since det A = 1. It was proved in [2] that if N = 2n + 1 is odd then the coeﬃcients f1 , . . . , fn are algebraically independent generators of the algebra of Casimir elements of PN . If N = 2n is even then det(A − At ) = Pf I0 (A)2 .

(3.7)

In this case, a family of algebraically independent generators of the algebra of Casimir elements of PN is obtained by replacing any one of the elements f1 , . . . , fn with Pf I0 (A). The claim will be implied by the following identity det(A + λAt ) =

n

(−λ)k (1 + λ)N −2k ck .

(3.8)

k=0

Indeed, by the identity, the elements f1 , . . . , fn can be expressed as linear combinations of c1 , . . . , cn . In order to verity (3.8), we use the observation of [2] that the Casimir elements of PN are determined by their restrictions on a certain subspace H of matrices. If N = 2n, then H consists of the matrices of the form

I O , (3.9) D I where I and O are the identity and zero n × n matrices, respectively, while D = diag(d1 , . . . , dn ) is an arbitrary diagonal matrix. If N = 2n+1, then H consists of the matrices obtained from (3.9) by inserting an extra row and column in the middle of the matrix whose only nonzero entry is 1 at their intersection. So, by Theorems 3.1

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and 3.3, we only need to verify (3.8) for the matrices A ∈ H. However, in this case the element ck coincides with the elementary symmetric polynomial d2r1 · · · d2rk , ck = r1 <···
while det(A + λAt ) =

n (1 + λ)2 − λd2i i=1

if N = 2n, and n (1 + λ)2 − λd2i det(A + λA ) = (1 + λ) t

i=1

if N = 2n + 1. This gives (3.8). To verify the last statement of the corollary, put λ = −1 into (3.8) with N = 2n. Together with (3.7) this gives cn = Pf I0 (A)2 , so that the statement follows from (3.6) with k = n. Finally, we consider the invariants of the Poisson bracket on PN which can obtained from the construction of the Casimir elements of Uq (oN ) given in [25]. Theorem 3.6. The elements tr(A−1 At )k ,

k = 1, 2, . . . ,

are Casimir elements of PN . Proof. This follows by taking the classical limit of the Casimir elements of [25]. Alternatively, this is also implied by Theorem 3.1 and the Liouville formula ∞

(−1)k−1 λk−1 tr H k =

k=1

d ln det(1 + λH) dλ

which holds for any square matrix H. We apply it to the matrix H = A−1 At and observe that det(A + λAt ) = det(1 + λH) since det A = 1. 4. A New Poisson Algebra Here we use the symplectic version of the twisted quantized enveloping algebra introduced by Noumi [24] to deﬁne a new Poisson algebra and calculate its Casimir elements. The twisted quantized enveloping algebra Uq (sp2n ) is an associative algebra generated by elements sij , i, j ∈ {1, . . . , 2n} and s−1 i,i+1 , i = 1, 3, . . . , 2n − 1. The generators sij are zero for j = i + 1 with even i, and for j ≥ i + 2 and all i. We combine

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the sij into a matrix S as in (2.2), S=

sij ⊗ Eij ,

189

(4.1)

i,j

so that S has a block-triangular form with n diagonal 2 × 2-blocks,   s12 0 0 ··· 0 0 s11   s s22 0 0 ··· 0 0   21   s s32 s33 s34 ··· 0 0   31   s42 s43 s44 ··· 0 0 s41 . S=   . .. .. .. .. .. ..   . .   . . . . . .   s2n−1,1 s2n−1,2 s2n−1,3 s2n−1,4 · · · s2n−1,2n−1 s2n−1,2n  s2n,1 s2n,2 s2n,3 s2n,4 ··· s2n,2n−1 s2n,2n The deﬁning relations of Uq (sp2n ) have the form of a reﬂection equation (2.8) together with −1 si,i+1 s−1 i,i+1 = si,i+1 si,i+1 = 1

(4.2)

si+1,i+1 sii − q 2 si+1,i si,i+1 = q 3

(4.3)

and

for i = 1, 3, . . . , 2n − 1. More explicitly, the relations (2.8) have exactly the same form (2.10) as in the orthogonal case. Recall the quantized enveloping algebra Uq (gl2n ) deﬁned in Sec. 2. Introduce the block-diagonal 2n × 2n matrix G by   0 q ··· 0 0 −1 0 · · · 0 0    .. .. . . .. ..  . G= . . . . .    0 0 ··· 0 q 0

0 · · · −1 0

We can regard Uq (sp2n ) as a subalgebra of Uq (gl2n ) by setting S = T GT¯ t, or in terms of generators, sij = q

n k=1

ti,2k−1 t¯j,2k −

n

ti,2k t¯j,2k−1 ;

(4.4)

k=1

see [24, 19] for the proofs. ˆ (sp2n ) as follows. Deﬁne the extended twisted quantized enveloping algebra U q This is an associative algebra generated by elements sij , i, j ∈ {1, . . . , 2n} where sij = 0 for j = i + 1 with even i, and for j ≥ i + 2 and all i. The deﬁning relations are given by (2.8) or, equivalently, by (2.10). We use the same symbols as for the generators of Uq (sp2n ); a confusion should be avoided as we indicate which algebra is considered at any moment. This deﬁnition essentially coincides with the original

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one due to Noumi [24]. Note that, in comparison with Uq (sp2n ), we neither require the elements si,i+1 with odd i to be invertible, nor we impose the relations (4.3). ˆ (sp2n ) An analogue of the Poincar´e–Birkhoﬀ–Witt theorem for the algebra U q follows from [17, Corollary 3.4]. As with the algebra Uq (oN ), this theorem implies ˆ q (sp2n ) specialthat at q = 1 the extended twisted quantized enveloping algebra U 2 ˆ izes to the algebra P2n of polynomials in 2n +2n variables. We denote the variables by aij with the same restrictions on the indices i, j as for the elements sij , so that sij specializes to aij . We shall combine the variables aij into a matrix A which has a block-triangular form with n diagonal 2 × 2-blocks,   a11 a12 0 0 ··· 0 0   a a22 0 0 ··· 0 0   21   a a32 a33 a34 ··· 0 0   31   a a a · · · 0 0 a .  42 43 44 A =  41  .. .. .. .. .. ..   .. .   . . . . . .   a2n−1,1 a2n−1,2 a2n−1,3 a2n−1,4 · · · a2n−1,2n−1 a2n−1,2n  a2n,1 a2n,2 a2n,3 a2n,4 ··· a2n,2n−1 a2n,2n Theorem 4.1. The algebra Pˆ2n possesses the Poisson bracket deﬁned by {aij , akl } = δik + δjk − δil − δjl aij akl − 2 δl<j − δi
Remark 4.2. Both in the orthogonal and symplectic case, the Poisson brackets of P = PN or P = Pˆ2n can be written in a uniform way in a matrix form. Introducing the elements of P ⊗ End C N ⊗ End C N by A1 = aij ⊗ Eij ⊗ 1, A2 = aij ⊗ 1 ⊗ Eij , i,j

i,j

we have {A1 , A2 } = [r, A1 A2 ] + A1 rt A2 − A2 rt A1 , where r=

i

Eii ⊗ Eii + 2

i<j

Eij ⊗ Eji ,

rt =

Eii ⊗ Eii + 2

i

This follows from (2.8) and the observation that R − I ⊗ I . r= q − 1 q=1

i<j

Eji ⊗ Eji .

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Theorem 4.3. The elements ai+1,i+1 aii − ai+1,i ai,i+1 ,

i = 1, 3, . . . , 2n − 1,

(4.5)

and the coeﬃcients of the polynomial det(A + λAt ) = f0 + f1 λ + · · · + f2n λ2n are Casimir elements of the Poisson algebra Pˆ2n . Proof. For any i = 1, 3, . . . , 2n − 1 the element si+1,i+1 sii − q 2 si+1,i si,i+1 ˆ (sp2n ); see [19, Sec. 2.2]. This implies the belongs to the center of the algebra U q claim for the elements (4.5). We proceed as in the proof of Theorem 3.1. The relation (3.3) holds in the same form with the matrix S(u) now given by ¯ S(u) = S + qu−1 S, where the matrix elements s¯ij of S¯ are deﬁned as follows. For any i = 1, 3, . . . , 2n−1 we have s¯ii = −q −2 sii , s¯i+1,i = −q −1 si,i+1 ,

s¯i+1,i+1 = −q −2 si+1,i+1 , s¯i,i+1 = −q −1 si+1,i + (1 − q −2 )si,i+1 ,

while s¯kl = −q −1 slk for k < l except for the pairs k = i, l = i + 1, with odd i, and the remaining entries of S¯ are equal to zero. The element (3.3) equals AqN sdet S(u), where sdet S(u) is the Sklyanin determiant of the matrix S(u). This is a rational function in u valued in the (extended) twisted quantized enveloping algebra. When the values are considered in the algebra Uq (sp2n ), they are contained in the center of Uq (sp2n ), as proved in [19, Theorem 3.15 and Corollary 4.3]. The same property holds for the algebra ˆ (sp2n ), that is, when the values of the function sdet S(u) are regarded as elements U q ˆ (sp2n ) (see the ˆ (sp2n ), they belong to the center of U of the extended algebra U q q proof in the Appendix A). At q = 1, the matrix S(u) becomes A− u−1 At . Hence, the Sklyanin determinant sdet S(u) becomes γ(u) det(A−u−1 At ), where γ(u) is deﬁned in (3.5) with N = 2n. Therefore, replacing u with −λ−1 we thus prove that all coeﬃcients of det(A+ λAt ) are Casimir elements for the Poisson bracket on Pˆ2n . As in the orthogonal case, we have f2n−i = fi for all i = 0, 1, . . . , 2n. Note also that f0 = f2n = det A and so we have the following relation between the Casimir elements n (a2k,2k a2k−1,2k−1 − a2k,2k−1 a2k−1,2k ). f0 = k=1

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Conjecture 4.4. The algebra of Casimir elements of Pˆ2n is generated by the family of elements provided by Theorem 4.3 and the Pfaﬃan Pf(A − At ). In the rest of this section we work with the twisted quantized enveloping algebra Recall the action of the braid group B2n on the quantized enveloping algebra Uq (gl2n ); see Sec. 2.

Uq (sp2n ).

Proposition 4.5. The subalgebra Uq (sp2n ) ⊂ Uq (gl2n ) is stable under the action of the elements β1 , β3 , . . . , β2n−1 of B2n . Proof. Observe that the algebra Uq (sp2n ) is generated by the elements sii ,

si+1,i+1 ,

si,i+1 ,

s−1 i,i+1

for i = 1, 3, . . . , 2n − 1

(4.6)

and si+3,i+1

for i = 1, 3, . . . , 2n − 3.

(4.7)

Indeed, si+1,i for odd i can be expressed in terms of the elements (4.6) from (4.3). Furthermore, the remaining generators can be expressed in terms of the elements (4.6) and si+2,i ,

si+2,i+1 ,

si+3,i

si+3,i+1

for i = 1, 3, . . . , 2n − 3

(4.8)

by induction from the relations (q − q −1 )skl = s−1 i,i+1 (sk,i+1 sil − sil sk,i+1 ),

k > i + 1,

i > l,

i odd,

which are implied by the deﬁning relations (2.10). However, for each i as in (4.8) we have (q − q −1 )si+3,i = s−1 i,i+1 (si+3,i+1 sii − sii si+3,i+1 ), (q − q −1 )si+2,i+1 = s−1 i+3,i+2 (si+3,i+1 si+2,i+2 − si+2,i+2 si+3,i+1 ), (q − q −1 )si+2,i = s−1 i+3,i+2 (si+3,i si+2,i+2 − si+2,i+2 si+3,i ). Hence, it suﬃces to verify that the images of the elements (4.6) and (4.7) under the action of β1 , β3 , . . . , β2n−1 are contained in Uq (sp2n ). These images can be explicitly calculated from (4.4). For any odd j the elements (4.6) with i = j are ﬁxed by the action of βj , while βj : sjj → s−2 j,j+1 sj+1,j+1 ,

sj+1,j+1 → q −2 sjj ,

sj,j+1 → q 2 s−1 j,j+1 .

Moreover, the elements (4.7) with i = j − 2, j are ﬁxed by the action of βj , while βj : sj+1,j−1 → q −1 sj,j−1 ,

sj+3,j+1 → q −1 sj+3,j .

All these relations are veriﬁed by direct calculation with the use of the deﬁning relations of Uq (gl2n ). In particular, the restrictions of the action of β1 , β3 , . . . , β2n−1 to the subalgebra Uq (sp2n ) yield automorphisms of the latter.

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Now, observe that the elements γ1 , γ3 , . . . , γ2n−3 of B2n given by γ2k−1 = β2k β2k−1 β2k+1 β2k ,

k = 1, . . . , n − 1

generate a subgroup of B2n isomorphic to Bn . The braid relations for the γ2k−1 are easily veriﬁed with the use of their geometric interpretation. Indeed, if we regard βj as the braid r

r

r

r

1

2

r @

···

r j

r @r

···

j+1

r

r

r

r

2n − 1

2n

then each γ2i−1 is just an elementary braid on the doubled strands: r

r

r

r

1

2

r r r r H H H H H H r r H r Hr

···

2i − 1

2i

2i + 1

···

2i + 2

r

r

r

r

2n − 1

.

2n

For each odd i, the elements (4.6) generate a subalgebra of Uq (sp2n ) isomorphic to Uq (sp2 ). The next proposition shows that the elements γi permute these subalgebras. Proposition 4.6. The images of the elements (4.6) under the action of the automorphisms γ1 , γ3 , . . . , γ2n−3 belong to Uq (sp2n ). Proof. This is veriﬁed with the use of (4.4). For any odd j the elements (4.6) with i = j, j + 2 are ﬁxed by the action of γj , while γj : sjj → sj+2,j+2 ,

sj+1,j+1 → sj+3,j+3 ,

sj,j+1 → sj+2,j+3

γj : sj+2,j+2 → sjj ,

sj+3,j+3 → sj+1,j+1 ,

sj+2,j+3 → sj,j+1 .

and

This follows from the formulas for the action of the βi on Uq (gl2n ) which imply, for instance, relations of the type βj βj+1 : tj+1,j → tj+2,j+1 . tj+3,j+2 . The images of the Since γj = βj+1 βj+2 βj βj+1 , this gives γj : tj+1,j → remaining elements of the form tjj , t¯j,j+1 , tj+1,j+1 are calculated in a similar way which gives the desired formulas. It can be shown that Proposition 4.6 is not extended to the remaining generators (4.8) of the algebra Uq (sp2n ). Observe that the elements βi and γi of B2n with odd i satisfy the relations γi−1 βj γi = βj

if j = i, i + 2

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while γi−1 βi γi = βi+2

and γi−1 βi+2 γi = βi .

The elements βi generate a subgroup of B2n isomorphic to Z n . We shall identify Z n with this subgroup. These observations suggest the following deﬁnition. Consider and the usual deﬁning relations the braid group Bn with generators γ1 , γ3 , . . . , γ2n−3 γi γi+2 γi = γi+2 γi γi+2 ,

i = 1, 3, . . . , 2n − 5

and γi γj = γj γi ,

|i − j| > 2.

Deﬁne the group Γn as the semidirect product Γn = Bn Z n where the action of Bn on Z n is deﬁned by γ

βj i = βj

if j = i, i + 2

while γ

βi i = βi+2

γ

i and βi+2 = βi .

Note that the Weyl group W (Cn ) = Sn Z n2 of type Cn may be regarded as a classical counterpart of Γn . Conjecture 4.7. There exists an action of the group Γn on the algebra Uq (sp2n ) by automorphisms which corresponds to the action of W (Cn ) on U(sp2n ). Our ﬁnal theorem shows that the conjecture holds for n = 2. Theorem 4.8. Let the generators β1 and β3 of the group Γ2 act on Uq (sp4 ) as in Proposition 4.5 and let the generator γ1 act on the elements (4.6) with i = 1, 3 as γ1 . Then, together with the assignment γ1 : s32 → s41 , this deﬁnes an action of Γ2 on

s41 → s32 , Uq (sp4 )

s31 → s31 ,

s42 → s42

by automorphisms.

Proof. It is easy to verify that γ1 respects the deﬁning relations of Uq (sp4 ). For instance, the following relations are clearly respected by γ1 s33 s32 = s32 s33 ,

s11 s32 = s32 s11 + (q −1 − q)s12 s31 ,

s31 s32 = q −1 s32 s31 + (q − q −1 )(q −1 s21 s33 − s12 s33 ), and s11 s41 = s41 s11 ,

s33 s41 = s41 s33 + (q −1 − q)s34 s31 ,

s31 s41 = q −1 s41 s31 + (q − q −1 )(q −1 s43 s11 − s34 s11 ), together with s32 s41 = s41 s32 + (q − q −1 )(s12 s43 − s34 s21 ), and this holds for the remaining relations as well. The deﬁning relations of the group Γ2 are also easily veriﬁed.

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Appendix A. Sklyanin Determinant for Extended Quantized Enveloping Algebras Here, we prove that the Sklyanin determinant sdet S(u) is central in the extended ˆ (spN ) with N = 2n; see the proof of Theorem 4.3. We need to introduce algebra U q some more notation. Following [19], introduce the trigonometric R-matrix Eii ⊗ Ejj + (q −1 u − qv) Eii ⊗ Eii R(u, v) = (u − v) i=j

+ (q

−1

− q)u

i

Eij ⊗ Eji + (q

−1

i>j

− q)v

Eij ⊗ Eji

(A.1)

i<j

and a rational function in independent variables u1 , . . . , ur , q valued in (End C N )⊗r by Rij (ui , uj ), (A.2) R(u1 , . . . , ur ) = i<j

where the product is taken in the lexicographical order on the pairs (i, j). We have ˆ (spN ) ⊗ (End C N )⊗r , the following relation in the algebra U q t t t t t R(u1 , . . . , ur )S1 (u1 )R12 · · · R1r S2 (u2 )R23 · · · R2r S3 (u3 ) · · · Rr−1,r Sr (ur ) t t t t t · · · S3 (u3 )R2r · · · R23 S2 (u2 )R1r · · · R12 S1 (u1 )R(u1 , . . . , ur ); = Sr (ur )Rr−1,r

(A.3) t t t see [19], where Rij = Rij (u−1 i , uj ) with R (u, v) deﬁned in (3.4). Now take r = N +1 ˆ q (spN ) ⊗ (End C N )⊗(N +1) and label the copies of End C N in the tensor product U with the indices 0, 1, . . . , N . Furthermore, specialize the parameters ui in (A.3) as follows:

u0 = v,

ui = q −2i+2 u

for i = 1, . . . , N.

Then by [19, Proposition 4.1], the element (A.2) will take the form →

R(v, u, . . . , q −2N +2 u) = α(u)

i=1,...,N

where α(u) = uN (N −1)/2

R0i (v, q −2i+2 u)AqN ,

(q −2i+2 − q −2j+2 ).

1≤i<j≤N

We shall now be verifying that → i=1,...,N

R0i (v, q −2i+2 u)AqN = δ(u, v)AqN

where δ(u, v) = (q

−1

v − qu)

N −1

(v − q −2i u).

i=1

(A.4)

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The R-matrix R(u, v) satisﬁes the Yang–Baxter equation R12 (u, v)R13 (u, w)R23 (v, w) = R23 (v, w)R13 (u, w)R12 (u, v). Using this relation repeatedly, we derive the identity R(u1 , . . . , ur ) = Rij (ui , uj ), i<j

where the product is taken in the order opposite to the lexicographical order on the pairs (i, j). Taking here r = N + 1 and specializing the variables ui as above, we arrive at → ← R0i (v, q −2i+2 u)AqN = AqN R0i (v, q −2i+2 u). (A.5) i=1,...,N

i=1,...,N

Hence, for the proof of (A.4), it now suﬃces to compare the images of the operators on both sides at the basis vectors of the form vk = ek ⊗ ei1 ⊗ · · · ⊗ eiN with k = 1, . . . , N , where the ei denote the canonical basis vectors of C N and {i1 , . . . , iN } is a ﬁxed permutation of {1, . . . , N }. Our next observation is the fact that for any i, j ∈ {1, . . . , N } the expression R(u, v)(ei ⊗ ej ) is a linear combination of ei ⊗ ej and ej ⊗ ei . This implies that for each k, AqN

← i=1,...,N

R0i (v, q −2i+2 u)vk = δk (u, v)AqN vk ,

(A.6)

for some scalar function δk (u, v) which is independent of the permutation {i1 , . . . , iN }. It remains to show that δk (u, v) = δ(u, v) for all k. However, this is immediate from (A.6) if for a given k we choose a permutation {i1 , . . . , iN } with i1 = k, thus completing the proof of (A.4). Now apply the transposition t on the 0th copy of End C N and combine (A.4) and (A.5) to derive another identity AqN

→

t R0i (v, q −2i+2 u) =

i=1,...,N

← i=1,...,N

t R0i (v, q −2i+2 u)AqN = δ(u, v)AqN .

Thus, (A.3) becomes δ(u, v)δ(u, v −1 )AqN S0 (v) sdet S(u) = δ(u, v)δ(u, v −1 )AqN sdet S(u)S0 (v), ˆ (spN ). proving that sdet S(u) lies in the center of U q As a ﬁnal remark, note that the above argument applies to more general matrices S(u). The only property of S(u) used above is the fact that S(u) satisﬁes the reﬂection equation R(u, v)S1 (u)Rt (u−1 , v)S2 (v) = S2 (v)Rt (u−1 , v)S1 (u)R(u, v). AqN

(A.7)

This implies that (3.3) equals sdet S(u) for some formal series sdet S(u) called the Sklyanin determinant. Then sdet S(u) is central in the algebra with the deﬁning relations (A.7). In particular, this applies to the (extended) twisted q-Yangians associated with the orthogonal and symplectic Lie algebras; see [19].

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Acknowledgment This work was inspired by Alexei Bondal’s talk at the Prague’s conference ISQS 2006. We would like to thank Alexei for many stimulating discussions. The ﬁnancial support of the Australian Research Council is acknowledged. The second author is grateful to the University of Sydney for the warm hospitality during his visit.

References [1] P. P. Boalch, Stokes matrices, Poisson-Lie groups and Frobenius manifolds, Invent. Math. 146 (2001) 479–506. [2] A. I. Bondal, A symplectic groupoid of triangular bilinear forms and the braid group, Izv. Math. 68 (2004) 659–708. [3] A. I. Bondal, Symplectic groupoids related to Poisson–Lie groups, Proc. Steklov Inst. Math. 246 (2004) 34–53. [4] V. Chari and A. Pressley, A Guide to Quantum Groups (Cambridge University Press, Cambridge, 1994). [5] L. O. Chekhov, Teichm¨ uller theory of bordered surfaces, preprint ITEP/TH-53/06, math.AG/0610872. [6] L. O. Chekhov and V. V. Fock, Observables in 3D gravity and geodesic algebras, Czechoslovak J. Phys. 50 (2000) 1201–1208. [7] N. Ciccoli and F. Gavarini, A quantum duality principle for coisotropic subgroups and Poisson quotients, Adv. Math. 199 (2006) 104–135. [8] B. Dubrovin, Geometry of 2D topological ﬁeld theory, in Integrable Systems and Quantum Groups, eds. M. Francaviglia and S. Greco, Lect. Notes. Math., Vol. 1620 (Springer, 1996), pp. 120–348. [9] D. B. Fairlie, Quantum deformations of SU (2), J. Phys. A 23 (1990) L183–L187. [10] F. Gavarini, Presentation by Borel subalgebras and Chevalley generators for quantum enveloping algebras, Proc. Edinburgh Math. Soc. 49 (2006) 291–308. [11] A. M. Gavrilik and A. U. Klimyk, q-Deformed orthogonal and pseudo-orthogonal algebras and their representations, Lett. Math. Phys. 21 (1991) 215–220. [12] A. M. Gavrilik and N. Z. Iorgov, On Casimir elements of q-algebras Uq (son ) and their eigenvalues in representations, in Symmetry in Nonlinear Mathematical Physics, Proc. Inst. Mat. Ukr. Nat. Acad. Sci., Vol. 30 (BITP, Kiev, 1999), pp. 310–314. [13] N. Z. Iorgov and A. U. Klimyk, The nonstandard deformation Uq (son ) for q a root of unity, Methods Funct. Anal. Topology 6 (2000) 15–29. [14] N. Z. Iorgov and A. U. Klimyk, Classiﬁcation theorem on irreducible representations of the q-deformed algebra Uq (son ), Int. J. Math. Sci. 2 (2005) 225–262. [15] M. Itoh and T. Umeda, On central elements in the universal enveloping algebras of the orthogonal Lie algebras, Compos. Math. 127 (2001) 333–359. [16] G. Lusztig, Finite-dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990) 257–296. [17] A. I. Molev, Representations of the twisted quantized enveloping algebra of type Cn , Moscow Math. J. 6 (2006) 531–551. [18] A. Molev and M. Nazarov, Capelli identities for classical Lie algebras, Math. Ann. 313 (1999) 315–357. [19] A. Molev, E. Ragoucy and P. Sorba, Coideal subalgebras in quantum aﬃne algebras, Rev. Math. Phys. 15 (2003) 789–822. [20] J. E. Nelson and T. Regge, 2 + 1 quantum gravity, Phys. Lett. B 272 (1991) 213–216.

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[21] J. E. Nelson and T. Regge, 2 + 1 gravity for genus > 1, Comm. Math. Phys. 141 (1991) 211–223. [22] J. E. Nelson and T. Regge, Invariants of 2 + 1 gravity, Comm. Math. Phys. 155 (1993) 561–568. [23] J. E. Nelson, T. Regge and F. Zertuche, Homotopy groups and (2 + 1)-dimensional quantum de Sitter gravity, Nucl. Phys. B 339 (1990) 516–532. [24] M. Noumi, Macdonald’s symmetric polynomials as zonal spherical functions on quantum homogeneous spaces, Adv. Math. 123 (1996) 16–77. [25] M. Noumi, T. Umeda and M. Wakayama, Dual pairs, spherical harmonics and a Capelli identity in quantum group theory, Compos. Math. 104 (1996) 227–277. [26] A. Odesskii, An analogue of the Sklyanin algebra, Funct. Anal. Appl. 20 (1986) 152–154. [27] A. Odesskii and V. Rubtsov, Polynomial Poisson algebras with regular structure of symplectic leaves, Theor. Mat. Phys. 133 (2002) 3–24. [28] M. R. Santilli, A realization of the Uq,1 algebra in terms of quantum-mechanical operators, Nuovo Cimento A (10) 51 (1967) 74–88. [29] M. Ugaglia, On a Poisson structure on the space of Stokes matrices, Int. Math. Res. Not. 6 (1999) 473–493. ´ [30] P. Xu, Dirac submanifolds and Poisson involutions, Ann. Sci. Ecole Norm. Sup. (4) 36 (2003) 403–430.

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Reviews in Mathematical Physics Vol. 20, No. 2 (2008) 199–231 c World Scientiﬁc Publishing Company

EXISTENCE OF SPECTRAL GAPS, COVERING MANIFOLDS AND RESIDUALLY FINITE GROUPS

´∗ FERNANDO LLEDO Department of Mathematics, University Carlos III Madrid, Avda. de la Universidad 30, E-28911 Leganes (Madrid), Spain [email protected] OLAF POST Institut f¨ ur Mathematik, Humboldt-Universit¨ at zu Berlin, Rudower Chaussee 25, D-12489 Berlin, Germany [email protected] Dedicated to Volker Enß on his 65th Birthday Received 19 September 2007 Revised 9 December 2007 In the present paper we consider Riemannian coverings (X, g) → (M, g) with residually ﬁnite covering group Γ and compact base space (M, g). In particular, we give two general procedures resulting in a family of deformed coverings (X, gε ) → (M, gε ) such that the spectrum of the Laplacian ∆(Xε ,gε ) has at least a prescribed ﬁnite number of spectral gaps provided ε is small enough. If Γ has a positive Kadison constant, then we can apply results by Br¨ uning and Sunada to deduce that spec ∆(X,gε ) has, in addition, band-structure and there is an asymptotic estimate for the number N (λ) of components of spec ∆(X,gε ) that intersect the interval [0, λ]. We also present several classes of examples of residually ﬁnite groups that ﬁt with our construction and study their interrelations. Finally, we mention several possible applications for our results. Keywords: Covering manifolds; spectral gaps; residually ﬁnite groups; min-max principle. Mathematics Subject Classiﬁcation 2000: 58J50, 57M10, 35J20, 20E26

1. Introduction Spectral properties of the Laplacian on a compact manifold is a well-established and still active ﬁeld of research. Much less is known on the spectrum of non-compact ∗ Institut f¨ ur Reine und Angewandte Mathematik, Rheinisch-Westf¨ alische Technische Hochschule Aachen, Templergraben 55, D-52062 Aachen, Germany (on leave). E-mail: [email protected].

199

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manifolds. We restrict ourselves here to the class of non-compact covering manifolds X → M with compact quotient M , in which the covering group Γ plays an important role. In the open problem section of [49, Chap. IX, Problem 37], Yau posed the question about the nature and the stability of the (purely essential) spectrum of such a covering X → M . The aim of this paper is to provide a large class of examples of Riemannian coverings X → M having spectral gaps in the essential spectrum of its Laplacian ∆X . Here, a spectral gap is a non-void open interval (α, β) with (α, β) ∩ spec ∆X = ∅ and α, β ∈ spec ∆X . The manifolds X and M are d-dimensional, d ≥ 2, and we denote by D a fundamental domain associated to this covering. The main idea for producing spectral gaps is to construct a family of Riemannian metrics (gε )ε>0 on X such that the length scale with respect to the metric gε is of order ε at the boundary of a fundamental domain D and unchanged elsewhere (cf. Fig. 1). If such a fundamental domain exists, we say that the family of metrics (gε ) decouples the manifold X. The covering X → M with a decoupling family of metrics (gε ) “converges” in a sense to be speciﬁed below to a limit covering consisting of the inﬁnite disjoint (“decoupled”) union of the limit quotient manifold N which are again d-dimensional (see Secs. 1.3 and 3 for details). We stress that the curvature does not remain bounded as ε → 0; in contrast to degeneration of Riemannian metrics under curvature bounds developed e.g. in [14]. All groups Γ are assumed to be discrete and ﬁnitely generated throughout the present article. 1.1. Statement of the main results Main Theorem 1 (cf. Theorem 6.8). Suppose that X → M is a Riemannian covering with residually ﬁnite covering group Γ and metric g. Then by a local deformation of g we construct a family of metrics (gε ) decoupling X, such that for each n ∈ N there exists εn > 0 where spec ∆(X,gεn ) has at least n gaps, i.e. n + 1 components as subset of [0, ∞). Basically, we will give two diﬀerent constructions for the family of manifolds (X, gε ): ﬁrst, “adding small handles” to a given manifold (N, g) and second, a conformal perturbation of g. As a set, (X, gε ) converges to a limit manifold consisting of inﬁnitely many disjoint copies of the limit quotient manifold N as ε → 0.

Fig. 1. A covering manifold X with fundamental domain D. The junctions between diﬀerent translates of D are of order ε.

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A residually ﬁnite group is a countable discrete group such that the intersection of all its normal subgroups of ﬁnite index is trivial. Roughly speaking, a residually ﬁnite group has many normal subgroups of ﬁnite index. Geometrically, a covering with a residually ﬁnite covering group can be approximated by a sequence of ﬁnite coverings Mi → M (a tower of coverings). The class of residually ﬁnite groups is very large, containing e.g. ﬁnitely generated abelian groups, type I groups (i.e. ﬁnite extensions of Zr ), free groups or ﬁnitely generated subgroups of the isometries of the d-dimensional hyperbolic space Hd (cf. Sec. 6). Denote by N (g, λ) the number of components of spec ∆(X,g) which intersect the interval [0, λ]. Our result gives a lower bound on N (g, λ), in particular, we can reformulate the Main Theorem 1 as follows: For each n ∈ N there exists g = gεn such that N (g, λ) ≥ n + 1, provided λ is large enough. Using the Weyl eigenvalue asymptotic on the limit d-dimensional manifold (N, g) associated to the decoupling family (gε ) on X → M , we obtain the following asymptotic lower bound on the number of gaps (where ωd denotes the volume of the d-dimensional Euclidean unit ball): Main Theorem 2 (cf. Theorem 7.5). Assume that the covering group is residually ﬁnite and that the spectrum of the Laplacian on the limit manifold (N, g) is simple, i.e. all eigenvalues have multiplicity one. Then for each λ ≥ 0 there exists ε(λ) > 0 such that N (gε(λ) , λ) −d λ→∞ (2π) ωd vol(N, g)λd/2

lim inf

≥ 1.

The assumption on the spectrum of (N, g) is natural since N (g, λ) counts components in the spectrum without multiplicity. A priori, the number of gaps N (g, λ) could be inﬁnite, e.g. if spec ∆(X,g) contains a Cantor set. But Br¨ uning and Sunada showed in [11] that for covering groups Γ with positive Kadison constant C(Γ) > 0 (cf. Sec. 7) the asymptotic upper bound lim sup λ→∞

N (g, λ) 1 ≤ C(Γ) (2π)−d ωd vol(M, g)λd/2

holds. In particular, N (g, λ) is ﬁnite, and the spectrum of ∆(X,g) does not contain Cantor-like subsets. Applying these results to our situation we give a partial answer on the question of Yau of the nature of the spectrum: Main Theorem 3 (cf. Theorem 7.5). Suppose that X → M is a Riemannian Γ-covering with decoupling family of metrics (gε ), where Γ is a residually ﬁnite group that has positive Kadison constant C(Γ) > 0. Then spec ∆(X,gε ) has bandstructure, i.e. N (gε , λ) < ∞ for any λ ≥ 0 and N (gε , λ) can be made arbitrary large provided ε is small and λ is large enough. Some examples of groups with positive Kadison constant and which are residually ﬁnite are ﬁnitely generated, abelian groups, the free (non-abelian) group in r ≥ 2 generators or fundamental groups of compact, orientable surfaces (see also Sec. 8).

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1.2. Motivation and related work A main motivation for our work comes from the spectral theory of Schr¨ odinger operators H = −∆ + V on Rd , d ≥ 2, with V periodic with respect to the action of a discrete abelian group Γab = Zd on Rd . For such operators, it is a well known fact that if V has high barriers near the boundary of a fundamental domain D, then gaps appear in the spectrum of H. In this way, the potential V essentially decouples the fundamental domain D from its neighbouring domains (see [24] for an overview on this subject). A natural generalization into a geometric context is to replace the periodic structure (Rd , Zd ) by a Riemannian covering X → M with a discrete (in general non-abelian) group Γ. Our work shows that the decoupling eﬀect of the potential V can be replaced purely by geometry, in particular by the decoupling family of metrics (gε ) on X → M . From a quantum mechanical or probabilistic point of view, the correspondence seems to be natural: One has a small probability to ﬁnd a particle (with low energy) in a region with a high potential barrier or where the manifold (X, gε ) is very thin and the absolute value of the curvature is very large. It was already observed by, e.g. Br¨ uning, Gruber, Kobayashi, Ono and Sunada [11, 23, 51, 32] that many properties of the spectrum of a periodic Schr¨ odinger operator (e.g. band-structure, Bloch’s property etc.) generalize to the context of Riemannian coverings. An important diﬀerence is the existence of L2 -eigenvalues in the context of manifolds (cf. [32]). Such eigenvalues cannot occur in the spectrum of a periodic Schr¨ odinger operator on Rd (cf. [51]). The existence of (covering) manifolds with spectral gaps has also been established by Br¨ uning, Exner, Geyler and Lobanov in [6, 7]. They couple compact manifolds by points or line-segments with certain boundary condition at the coupling points; the point coupling corresponds to the case ε = 0 in our situation (with decoupled boundary condition). The case of abelian smooth coverings has been established in [42] (cf. also the references therein). Spectral gaps of Schr¨ odinger operators on the hyperbolic space have been analyzed in [33]. For other manifolds with spectral gaps (not necessarily periodic), we refer to [19, 43]. Under certain topological restrictions on the middle degree homology group one can show the existence of spectral gaps also for the diﬀerential form Laplacian on a Z-covering (see [1]). Some further interesting results on the group Γ and spectral properties of a Riemannian Γ-covering were shown by Brooks [9], e.g. that Γ is amenable iﬀ 0 ∈ spec ∆X . Moreover, Brooks [10] provided a combinatorial criterion whether the second eigenvalue of ∆Mi is bounded from below as i → ∞, where Mi → M is a tower of coverings. For physical applications of our results we refer to Sec. 9. Let us ﬁnish with two consequences of our result giving partial answers to the question of Yau on the nature and stability of the spectrum of ∆X : Consequence 1 (Manifold with Given Spectrum). First, we can solve the following inverse spectral problem: Given a compact (connected) manifold N of

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dimension d ≥ 3 and a sequence of numbers 0 = λ1 (0) < · · · < λn (0) it is possible to construct a metric g on N having exactly the numbers λk (0) as ﬁrst n eigenvalues with multiplicity 1 (cf. [12]). Then, applying our Main Theorem 3 and using the relation between spec ∆(X,gε ) and spec ∆(N,g) we can construct a covering X → M with decoupling family (gε ) having band spectrum close to the given points {λk (0)}, k = 1, . . . , n. The covering (X, gε ) → (M, gε ) is obtained roughly by joining copies of N through small, thin cylinders (see ﬁrst construction mentioned below). In particular, we have constructed a covering manifold with approximatively given spectrum in a ﬁnite spectral interval [0, λ], independently of the covering group! Consequence 2 (Instability of Gaps). Suppose X = Hd is the d-dimensional (d ≥ 3) hyperbolic space (or more generally, a simply connected, complete, symmetric space of non-compact type) with its natural metric g. It is known, that ∆(X,g) has no spectral gaps, in particular spec ∆(X,g) = [λ0 , ∞) for some constant λ0 ≥ 0 (see e.g. [18]). Let Γ be a ﬁnitely generated subgroup of the isometries of X such that M = X/Γ is compact. Note that such groups are residually ﬁnite. The second construction described below allows us to ﬁnd a decoupling family (gε ) on X where gε = ρ2ε g is conformally equivalent to g. We then apply Main Theorem 1 and obtain for each n ∈ N a metric gεn such that the corresponding Laplacian has at least n gaps. In particular, the number of gaps is not stable, even under uniform conformal changes of the metric. Note that the conformal factor ρε can be chosen in such a way that ρε → ρε0 uniformly as ε → ε0 provided ε0 > 0. Nevertheless, the band-gap structure remains invariant due to Main Theorem 3, once Γ has a positive Kadison constant.

1.3. An outline of the argument In the rest of the introduction we will present the main ideas of the construction of the decoupling metrics and mention the strategy for showing the existence of spectral gaps. The ﬁrst construction starts from a compact Riemannian manifold N of dimension d ≥ 2 (for simplicity without boundary) and a group Γ with generators γ1 , . . . , γr . We choose 2r diﬀerent points x1 , y1 , . . . , xr , yr . For each generator, we endow xi and yi with a cylindrical end of radius and length of order ε > 0 (by changing the metric appropriately on D := N \{x1 , y1 , . . . , xr , yr }). If we join Γ copies of these decorated manifolds (D, gε ) according to the Cayley graph of Γ associated to γ1 , . . . , γr , we obtain a Γ-covering X → M with a decoupling family of metrics (gε ) (cf. Fig. 1). The second construction starts with an arbitrary covering (X, g) → (M, g) (with compact quotient) of dimension d ≥ 3 and changes the metric conformally, i.e. gε := ρ2ε g, in such a way, that ρε is still periodic and of order ε close to the boundary of a fundamental domain D; more details can be found in Sec. 3. In the case of abelian coverings these constructions have already been used in [42].

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Once the construction of the family of decoupling metrics (gε ) has been done, the strategy to show the existence of spectral gaps goes as follows. We consider ﬁrst the Dirichlet (+) and Neumann (−) eigenvalues λ± k (ε) of the Laplacian on the ± fundamental domain (D, gε ). One can show that λk (ε) converges to the eigenvalues λk (0) of the Laplacian on the limit manifold (N, g) (see [42] and references therein). In other words, the Dirichlet–Neumann intervals + Ik (ε) := [λ− k (ε), λk (ε)]

converge to a point as ε → 0. Therefore, if ε is small enough, the union I(ε) := Ik (ε) k∈N

is a closed set having at least n gaps, i.e. n + 1 components as a subset of [0, ∞). The rest of the argument depends on the properties of the covering group Γ: (i) For abelian groups Γab , the inclusion spec ∆(X,gε ) ⊂ I(ε) is given by the Floquet theory (cf. Sec. 4 or [34, 50]). Basically, one shows that ∆(X,gε ) is unitary equivalent to a direct integral of operators on (D, gε ) acting on ρ-equivariant functions, where ρ runs through the set of irreducible unitary representations ab (characters). Note that in the abelian case all ρ are one-dimensional and Γ Γab is homeomorphic to (disjoint copies of) the torus Tr . The Min-max principle ensures that the kth eigenvalue of the equivariant operator lies in Ik (ε). (ii) If the group is non-abelian but still has only ﬁnite-dimensional irreducible representations, then one can show that the spectrum of the ρ-equivariant Laplacian is still included in I(ε). In this case the (non-abelian) Floquet theory guarantees again that spec ∆(X,gε ) ⊂ I(ε). The class of groups which satisfy the previous condition are type I groups, i.e. ﬁnite extensions of abelian groups. which is a nice measure space (smooth in These groups have a dual object Γ the terminology of [37, Chap. 2]). (iii) If the group is residually ﬁnite (a much wider class of groups including type I groups), then one can construct a so-called tower of coverings consisting of ﬁnite coverings Mi → M “converging” to the original covering X → M . The inclusion of the spectrum of ∆(X,gε ) in the closure of the union over all spectra of ∆(Mi ,gε ) was shown in [4, 2]. For the ﬁnite coverings Mi → M we again have the inclusion spec ∆(Mi ,gε ) ⊂ I(ε). (iv) For non-amenable groups (i.e. groups, for which spec ∆(M,gε ) is not included in spec ∆(X,gε ) ), cf. Remark 5.3, we have to assure that any of the intervals Ik (ε) intersects spec ∆X non-trivially. This will be done in Theorem 3.3. 1.4. Organization of the paper In the following section we set up the problem, present the geometrical context and state some results and conventions that will be needed later. In Sec. 3 we present in detail the two procedures for constructing covering manifolds with a

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decoupling family of metrics. In this case the set I(ε) deﬁned above will have at least a prescribed ﬁnite number of spectral gaps. Each procedure is well adapted to a given initial geometrical context (cf. Remark 3.8 as well as Examples 8.3 and 8.4). In Sec. 4 we show the inclusion of the spectrum of equivariant Laplacians into the union of the Dirichlet–Neumann intervals Ik (ε) and review brieﬂy the Floquet theory for non-abelian groups. The Floquet theory is applied in Sec. 5 for coverings with type I groups. In Sec. 6 we study a class of covering manifolds with residually ﬁnite groups. In Sec. 7 we consider residually ﬁnite groups Γ that in addition have a positive Kadison constant. In Sec. 8 we illustrate the results obtained with some classes of examples and point out their mutual relations. Section 8.3 contains an interesting example of a covering with an amenable, not residually ﬁnite group which cannot be treated with our methods. We expect though that in this case one can still generate spectral gaps by the construction presented in Sec. 3. Finally, we conclude mentioning several possible applications for our results. 2. Geometrical Preliminaries: Covering Manifolds and Laplacians We begin ﬁxing our geometrical context and recalling some results that will be useful later on. We denote by X a non-compact Riemannian manifold of dimension d ≥ 2 with a metric g. We also assume the existence of a ﬁnitely generated (inﬁnite) discrete group Γ of isometries acting properly discontinuously and cocompactly on X, i.e. for each x ∈ X there is a neighborhood U of x such that the sets γU and γ U are disjoint if γ = γ and M := X/Γ is compact. Moreover, the quotient M is a Riemannian manifold which also has dimension d and is locally isometric to X. In other words, π : X → M is a Riemannian covering space with covering group Γ. We call such a manifold Γ-periodic or simply periodic. All groups Γ appearing in this paper will satisfy the preceding properties. We also ﬁx a fundamental domain D, i.e. an open set D ⊂ X such that γD ¯ = X. We always assume that and γ D are disjoint for all γ = γ and γ∈Γ γ D ¯ D is compact and that ∂D is piecewise smooth. If not otherwise stated we also assume that D is connected. Note that we can embed D ⊂ X isometrically into the quotient M . In the sequel, we will not always distinguish between D as a subset of X or M since they are isometric. For details we refer to [45, § 6.5]. As a prototype for an elliptic operator we consider the Laplacian ∆X on a Riemannian manifold (X, g) acting on a dense subspace of the Hilbert space L2 (X) with norm · X . For the formulation of the Theorems 5.4 and 6.8 and at other places, it is useful to denote explicitly the dependence on the metric, since we deform the manifold by changing the metric. In this case we will write ∆(X,g) for ∆X or L2 (X, g) for L2 (X). The positive self-adjoint operator ∆X can be deﬁned in terms of a suitable quadratic form qX (see e.g. [29, Chap. VI], [48] or [15]). Concretely we have qX (u) := du 2X = |du|2 , u ∈ C∞ (2.1) c (X) X

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where the integral is taken with respect to the volume density measure of (X, g). In coordinates we write the pointwise norm of the 1-form du as g ij (x)∂i u(x)∂j u(x), |du|2 (x) = i,j

where (g ij ) is the inverse of the metric tensor (gij ) in a chart. Taking the closure of the quadratic form we can extend qX onto the Sobolev space H1 (X) = H1 (X, g) = {u ∈ L2 (X) | qX (u) < ∞}. As usual the operator ∆X is related with the quadratic form by the formula ∆X u, u = qX (u), u ∈ C∞ c (X). Since the metric on X is Γ-invariant, the Laplacian ∆X (i.e. its resolvent) commutes with the translation on X given by (Tγ u)(x) := u(γ −1 x),

u ∈ L2 (X),

γ ∈ Γ.

(2.2)

Operators with this property are called periodic. For an open, relatively compact subset D ⊂ X with suﬃciently smooth boundary ∂D (e.g. Lipschitz) we deﬁne the Dirichlet (respectively, Neumann) Laplacian − + − ∆+ D (respectively, ∆D ) via its quadratic form qD (respectively, qD ) associated to ∞ the closure of qD on Cc (D), the space of smooth functions with compact support, ¯ the space of smooth functions with continuous derivatives (respectively, C∞ (D), + (respectively, up to the boundary). We also use the notation H1◦ (D) = dom qD − 1 H (D) = dom qD ). Note that the usual boundary condition of the Neumann Lapla¯ is cian occurs only in the operator domain via the Gauß–Green formula. Since D + + compact, ∆D has purely discrete spectrum λk , k ∈ N. It is written in ascending order and repeated according to multiplicity. The same is true for the Neumann Laplacian and we denote the corresponding purely discrete spectrum by λ− k , k ∈ N. One of the advantages of the quadratic form approach is that one can easily read oﬀ from the inclusion of domains an order relation for the eigenvalues. In fact, by the the min-max principle we have λ± k = inf Lk

± qD (u) , 2

u

u∈Lk \{0}

sup

(2.3)

where the inﬁmum is taken over all k-dimensional subspaces Lk of the corresponding ± , cf. e.g. [15]. Then the inclusion quadratic form domain dom qD + − = H1◦ (D) ⊂ H1 (D) = dom qD dom qD

(2.4)

implies the following important relation between the corresponding eigenvalues − λ+ k ≥ λk .

(2.5)

This means, that the Dirichlet kth eigenvalue is in general larger than the kth Neumann eigenvalue and this justiﬁes the choice of the labels +, respectively, −.

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3. Construction of Periodic Manifolds In the present section we will give two diﬀerent construction procedures (labeled by the letters “A” and “B”) for covering manifolds, such that the corresponding Laplacian will have a prescribed ﬁnite number of spectral gaps. In contrast with [42] (where only abelian groups were considered) we will base the construction on the speciﬁcation of the quotient space M = X/Γ. By doing this, the spectral convergence result in Theorem 3.1 becomes manifestly independent of the fact whether Γ is abelian or not. Both constructions are done in two steps: ﬁrst, we specify in two ways the quotient M together with a family of metrics gε . Second, we construct in either case the covering manifold with covering group Γ which has r generators. In the last section we will localize the spectrum of the covering Laplacian in certain intervals given by an associated Dirichlet, respectively, Neumann eigenvalue problem. Some reasons for presenting two diﬀerent methods (A) and (B) are formulated in a ﬁnal remark of this section. 3.1. Construction of the quotient In the following two methods we deﬁne a family of Riemannian manifolds (M, gε ) that converge to a Riemannian manifold (N, g) of the same dimension (cf. Fig. 2). In each case we will also specify a domain D ⊂ M (in the following section D will become a fundamental domain of the corresponding covering): (1A) Attaching r handles: We construct the manifold M by attaching r handles diﬀeomorphic with C := (0, 1) × Sd−1 to a given d-dimensional compact orientable manifold N with metric g. For simplicity we assume that N has

(A)

(B)

Fig. 2. Two constructions of a family of manifold (M, gε ), ε > 0: In both cases, the grey area has a length scale of order ε in all directions. (A) We attach r handles (here r = 1) of diameter and length of order ε to the manifold (N, g). We also denoted the two cycles α1 and β1 . (B) We change the metric conformally to gε = ρ2ε g. The grey area D\N (with Fermi coordinates in the upper left corner) shrinks conformally to a point as ε → 0 whereas N remains ﬁxed. Note that the opposite sides of the square are identiﬁed (to obtain a torus as manifold M ).

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no boundary. Concretely, for each handle we remove two small discs of radius ε > 0 from N , denote the remaining set by Rε and identify {0} × Sd−1 with the boundary of the ﬁrst hole and {1} × Sd−1 with the boundary of the second hole. We denote by D the open subset of M where the mid section {1/2} × Sd−1 of each handle is removed. One can ﬁnally deﬁne a family of metrics (gε )ε , ε > 0, on M such that the diameter and length of the handle is of order ε (see e.g. [42, 13]). In this situation the handles shrink to a point as ε → 0. Note that (Rε , g) can be embedded isometrically into (N, g), respectively, (M, gε ). This fact will we useful for proving Theorem 3.3. (1B) Conformal change of metric: In the second construction, we start with an arbitrary compact d-dimensional Riemannian manifold M with metric g. We consider only the case d ≥ 3 (for a discussion of some two-dimensional examples see [42]). Moreover, we assume that N and D are two open subsets ¯ ⊂ D, (iii) D ¯ = M and (iv) D\N can of M such that (i) ∂N is smooth, (ii) N completely be described by Fermi coordinates (i.e. coordinates (r, y), r being the distance from N and y ∈ ∂N ) up to a set of measure 0 (cf. Fig. 2(B)). The last assumption assures that N is in some sense large in D. Suppose in addition, that ρε : M → (0, 1], ε > 0, is a family of smooth functions such that ρε (x) = 1 if x ∈ N and ρε (x) = ε if x ∈ M \N and dist(x, ∂N ) ≥ εd . Then ρε converges pointwise to the characteristic function of N . Furthermore, the Riemannian manifold (M, gε ) with gε := ρ2ε g converges to (N, g) in the sense that M \N shrinks to a point in the metric gε . Now we can formulate the following spectral convergence result which was proven in [42]: Theorem 3.1. Suppose (M, gε ) and D ⊂ M are constructed as in parts (1A) or (1B) above. In Case (1B) we assume in addition that d ≥ 3. Then λ± k (ε) → λk (0) as ε → 0 for each k. Here, λ± k (ε) denotes the kth Dirichlet, respectively, Neumann eigenvalue of the Laplacian on (D, gε ) whereas λk (0) is the kth eigenvalue of (N, g) (with Neumann boundary conditions at ∂N in Case (1B)). 3.2. Construction of the covering spaces Given (M, gε ) and D as in the previous subsection, we will associate a Riemannian covering π : (X, gε ) → (M, gε ) with covering group Γ such that D is a fundamental ˜ := π −1 (D). domain. Note that we identify D ⊂ M with a component of the lift D Moreover, Γ is isomorphic to a normal subgroup of the fundamental group π1 (M ). (2A) Suppose that Γ is a discrete group with r generators γ1 , . . . , γr . We will construct a Γ-covering (X, gε ) → (M, gε ) with fundamental domain D where D and (M, gε ) are given as in Part (1A) of the previous subsection. Roughly

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speaking, we glue together Γ copies of D along the handles according to the Cayley graph of Γ with respect to the generators γ1 , . . . , γr . For convenience of the reader, we specify the construction: The fundamental group of M is given by π1 (M ) = π1 (N ) ∗ Z∗r in the case d ≥ 3. Here, G1 ∗ G2 denotes the free product of G1 and G2 , and Z∗r is the free group in r generators α1 , . . . , αr . If d = 2 we know from the classiﬁcation result for two-dimensional orientable manifolds that N is diﬀeomorphic to an s-holed torus. In this case the fundamental group is given by π1 (M ) = α1 , β1 , . . . , αr+s , βr+s | [α1 , β1 ], . . . , [αr+s , βr+s ] = e ,

(3.1)

where [α, β] := αβα−1 β −1 is the usual commutator. We may assume that αi represents the homotopy class of the cycle transversal to the section of the ith handle and that βi represents the section itself (i = 1, . . . , r) (cf. Fig. 2(A)). One easily sees that there exists an epimorphism ϕ : π1 (M ) → Γ which maps αi ∈ π1 (M ) to γi ∈ Γ (i = 1, . . . , r) and all other generators to the unit element e ∈ Γ. Note that this map is also well-deﬁned in the case d = 2, since the relation in (3.1) is trivially satisﬁed in the case when the βi ’s are mapped to e. Finally, Γ ∼ = π1 (M )/ ker ϕ, and X → M is the associated covering with ˜ → M (considered as a principal bundle respect to the universal covering M with discrete ﬁber Γ) and the natural action of Γ on π1 (M ). Then X → M is a normal Γ-covering with fundamental domain D constructed as in (1A) of the preceding subsection. Here we use the fact that αi is transversal to the section of the handle in dimension 2. (2B) Suppose (X, g) → (M, g) is a Riemannian covering with fundamental domain ¯ = M , where we have embedded D such that ∂D is piecewise smooth. Then D D into the quotient, cf. [45, Theorem 6.5.8]. According to (1B) we can conformally change the metric on M , to produce a new covering (X, gε ) → (M, gε ) that satisﬁes the required properties. In both cases, we lift for each ε > 0 the metric gε from M to X and obtain a Riemannian covering (X, gε ) → (M, gε ). Note that the set D speciﬁed in the ﬁrst step of the previous construction becomes a fundamental domain after the speciﬁcation of the covering in the second step. The following statement is a direct consequence of the spectral convergence result in Theorem 3.1: Theorem 3.2. Suppose (X, gε ) → (M, gε ) (ε > 0) is a family of Riemannian coverings with fundamental domain D constructed as in the previous parts (2A) or (2B). Then for each n ∈ N there exists ε = εn > 0 such that I(ε) :=

k∈N

Ik (ε),

with

+ Ik (ε) := [λ− k (ε), λk (ε)],

(3.2)

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is a closed set having at least n gaps, i.e. n+1 components as subset of [0, ∞). Here, λ± k (ε) denotes the kth Dirichlet, respectively, Neumann eigenvalue of the Laplacian on (D, gε ). Proof. First, note that {λ± k (ε) | k ∈ N}, ε ≥ 0, has no ﬁnite accumulation point, since the spectrum is discrete. Second, Theorem 3.1 shows that the intervals Ik (ε) reduce to the point {λk (0)} as ε → 0. Therefore, I(ε) is a locally ﬁnite union of compact intervals, hence closed. 3.3. Existence of spectrum outside the gaps In the following subsection we will assure that each Neumann–Dirichlet interval Ik (ε) contains at least one point of spec ∆(X,gε ) provided ε is small enough. In our general setting described below (cf. Theorems 5.4 and 6.8) we will show the inclusion Ik (ε). (3.3) spec ∆(X,gε ) ⊂ k∈N

It is a priori not clear that each Ik (ε) intersects the spectrum of the Laplacian on (X, gε ), i.e. that gaps in k∈N Ik (ε) are also gaps in spec ∆(X,gε ) . If the covering group is amenable, the kth eigenvalue of the Laplacian on the quotient (M, gε ) is always an element of Ik (ε) ∩ spec(∆X , gε ) (see Remark 5.3). In general, this need not to be true. Therefore, we need the following theorem which will be used in Theorems 6.8 and 7.5: Theorem 3.3. With the notation of the previous theorem, we have Ik (ε) ∩ spec ∆(X,gε ) = ∅

(3.4)

for all k ∈ N. We begin with a general criterion which will be useful to detect points in the spectra of a parameter-dependent family of operators using only its sesquilinear form. A similar result is also stated in [31, Lemma 5.1]. Suppose that Hε is a self-adjoint, non-negative, unbounded operator in a Hilbert space Hε for each ε > 0. Denote by Hε1 := dom hε the Hilbert space of the corresponding quadratic form hε associated to Hε with norm u 1 := (hε (u)+ u Hε )1/2 and by Hε−1 the dual of Hε1 with norm · −1 . Note that Hε : Hε1 → Hε−1 is continuous. In the next lemma we characterize for each ε certain spectral points of Hε . Lemma 3.4. Suppose there exist a family (uε ) ⊂ Hε1 and constants λ ≥ 0, c > 0 such that

(Hε − λ)uε −1 → 0

as

ε→0

(3.5)

and uε ≥ c > 0 for all ε > 0, then there exists δ = δ(ε) → 0 as ε → 0 such that λ + δ(ε) ∈ spec Hε .

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Proof. Suppose that the conclusion is false. Then there exist a sequence εn → 0 and a constant δ0 > 0 such that Iλ ∩ spec Hεn = ∅ with

Iλ := (λ − δ0 , λ + δ0 )

for all n ∈ N. Denote by Et the spectral resolution of Hε . Then (t − λ)2 2 dEt uε , uε

(Hε − λ)uε −1 = R+ \Iλ (t + 1) cδ02 δ02 dEt uε , uε ≥ ≥ λ + δ0 + 1 R+ \Iλ λ + δ0 + 1 since Iλ does not lie in the support of the spectral measure. But this inequality contradicts (3.5). Remark 3.5. Equation (3.5) is equivalent to the inequality |hε (uε , vε ) − λuε , vε | ≤ o(1) vε 1

for all vε ∈ Hε1

(3.6)

as ε → 0. Note that o(1) could depend on uε . The advantage of the criterion in the previous lemma is that one only needs to ﬁnd a family (uε ) in the domain of the quadratic form hε . We will need the following lemma in order to deﬁne a cut-oﬀ function with convergent L2 -integral of its derivative. Its proof is straightforward. Lemma 3.6. Denote by h(r) := r−d+2 if d ε ∈ (0, 1) deﬁne    0,   h(r) − h(ε) √ , χε (r) := h( ε) − h(ε)     1, then χε ∈ H1 ((0, 1)) and

χε 2

:=

1

≥ 3 and h(r) = ln r if d = 2. For 0
(3.7)

√ ε≤r

|χε (r)|2 rd−1 dr = o(1)

0

as ε → 0. Remember that (N, g) is the unperturbed manifold as in Fig. 2. In Case A of Sec. 3.1, we denoted by Rε the manifold N with a closed ball of radius ε removed around each point where the handles have been attached (note that Rε is also contained in D) and denote by (r, y) the polar coordinates around such a point (r = ε corresponds to a component of ∂Rε ). Proof of Theorem 3.3. Let ϕ be the kth eigenfunction of the limit operator ∆N with eigenvalue λ = λk (0). We will treat Cases A and B of Sec. 3.1 separately.

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(3A) Set uε (r, y) := χε (r)ϕ(r, y) in the polar coordinates described above and uε := ϕ on R√ε . Now, ϕ 2R√ε ≥ c since ϕ 2R√ε → ϕ 2N > 0 as ε → 0. In addition, uε ∈ H1◦ (Rε ) ⊂ H1 (X, gε ) and |duε , dvε − λuε , vε |

dϕ, d(χε vε ) − λϕχε v + =

Rε

ϕdχε , dvε −

Rε

Rε

v¯dϕ, dχε

for all vε ∈ H (Dε ). Now the ﬁrst integral vanishes since ϕ is the eigenfunction with eigenvalue λ on N . Note that χε v ∈ H1◦ (Rε ) can be interpreted as function in H1 (N ). The second and third integral can be estimated from above by sup |ϕ(x)| + |dϕ(x)| χε

vε 1 = o(1) vε 1 1

x∈N

since ϕ is a smooth function on an ε-independent space and due to Lemma 3.6.

ε (r)ϕ(0, y), r > 0, i.e. on D\N with (3B) Set uε := ϕ on N and uε (r, y) := χ √ χ

ε (r) := χε ( ε + εd − r), where χε is deﬁned in (3.7) with d = 2. Note that χ

ε (r) = 0 only for those r = dist(x, ∂N ) where the conformal factor ρε (x) = ε. Now, uε ∈ H1◦ (D, gε ) ⊂ H1 (X, gε ). Furthermore, for vε ∈ H1 (D, gε ) we have |duε , dvε − λuε , vε | ≤ [| χε (r)ϕ(0, y)∂r vε |ρd−2 ε D\N

+ | χε (r)dy ϕ(0, y), dy vε |ρd−2 ε + λ χε (r)|ϕ(0, y)vε |ρdε ] dr dy  √ d ε+ε −ε

≤C

εd

√

| χε (r)|2 εd−2 dr 12

ε

| χε (r)|2 ρd−2 ε

+ 0

+

√

ε

12

dr

12  | χε (r)|2 ρdε dr  vε 1

0

where we have used that ϕ is the Neumann eigenfunction on N . Furthermore, C depends on the supremum of ϕ and dϕ and on λ. Note that the conformal factor

ε , therefore, the ﬁrst integral converges to 0 since ρε equals ε on the support of χ d ≥ 3. Finally, estimating χ

ε and ρε by 1, the second and third integrals are bounded by ε1/4 . We ﬁnally can deﬁne formally the meaning of “decoupling”: Definition 3.7. We call a family of metrics (gε )ε on X → M decoupling, if the conclusions of Theorems 3.2 and 3.3 hold, i.e. if there exists a fundamental domain D such that for each n there exists εn > 0 such that I(εn ) in (3.2) has at least n + 1 components and if (3.4) holds for all k ∈ N.

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Remark 3.8. In the present section we have speciﬁed two constructions of decoupling families of metrics on covering manifolds, such that the corresponding Laplacians will have at least a prescribed number of spectral gaps (cf. Secs. 5 and 6). The construction speciﬁed in method (A) is feasible for every given covering group Γ with r generators. Note that this method produces fundamental domains that have smooth boundaries (see e.g. Example 8.3 below). The construction in (B) applies for every given Riemannian covering (X, g) → (M, g), since, by the procedure described, one can modify conformally this covering in order to satisfy the spectral convergence result of Theorem 3.1 (cf. Example 8.4).

4. Floquet Theory for Non-Abelian Groups The aim of the present section is to state a spectral inclusion result (cf. Theorem 4.3) and the direct integral decomposition of ∆X (cf. Theorem 4.5) for certain nonabelian discrete groups Γ. These results will be used to prove the existence of spectral gaps in the situations analyzed in the next two sections. A more detailed presentation of the results in this section may be found in [36]. 4.1. Equivariant Laplacians We will introduce next a new operator that lies “between” the Dirichlet and Neumann Laplacians and that will play an important role in the following. Suppose ρ is a unitary representation of the discrete group Γ on the Hilbert space H, i.e. ρ : Γ → U(H) is a homomorphism. We ﬁx a fundamental domain D for the Γ-covering X → M . We now introduce the space of smooth ρ-equivariant functions ∞ C∞ ρ (D, H) := {hD | h ∈ C (X, H), h(γx) = ργ h(x), γ ∈ Γ, x ∈ X}.

(4.1)

This deﬁnition coincides with the usual one for abelian groups, cf. [36]. Note that we need vector-valued functions h : X → H since the representation ρ acts on the Hilbert space H, which, in general, has dimension greater than 1. We deﬁne next the so-called equivariant Laplacian (with respect to the representation ρ) on L2 (D, H) ∼ = L2 (D) ⊗ H: Let a quadratic form be deﬁned by

dh(x) 2H dX(x) (4.2)

dh 2D := D

for h ∈

C∞ ρ (D, H),

where the integrand is locally given by

dh(x) 2H = g ij (x) ∂i h(x), ∂j h(x) H , x ∈ D. i,j

This generalizes Eq. (2.1) to the case of vector-valued functions. We denote the domain of the closure of the quadratic form by H1ρ (D, H). The corresponding non-negative, self-adjoint operator on L2 (D, H), the ρ-equivariant Laplacian, will be denoted by ∆ρD,H (cf. [29, Chap. VI]).

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4.2. Dirichlet–Neumann bracketing We study in this section the spectrum of a ρ-equivariant Laplacian ∆ρ associated with a ﬁnite-dimensional representation ρ. In particular, we show that spec ∆ρ is contained in a suitable set determined by the spectrum of the Dirichlet and Neumann Laplacians on D. The key ingredient in dealing with non-abelian groups is the observation that this set is independent of ρ. We begin with the deﬁnition of certain operators acting in L2 (D, H) and its ρ + eigenvalues. We denote by λ− m (H), λm (H), respectively, λm (H) the mth eigenvalue ρ − + of the operator ∆D,H , ∆D,H , respectively, ∆D,H corresponding to the quadratic form (4.2) on H1◦ (D, H), H1ρ (D, H), respectively, H1 (D, H). Recall that H1◦ (D, H) is the H1 -closure of the space of smooth functions h : D → H with support away from ∂D and H1 (D, H) is the closure of the space of smooth functions with derivatives continuous up to the boundary. The proof of the next lemma follows, as in the abelian case (cf. Eqs. (2.4) and (2.5)), from the reverse inclusions of the quadratic form domains H1 (D, H) ⊃ H1ρ (D, H) ⊃ H1◦ (D, H)

(4.3)

and the min-max principle (2.3). Lemma 4.1. We have ρ + λ− m (H) ≤ λm (H) ≤ λm (H)

for all m ∈ N. From the deﬁnition of the quadratic form in the Dirichlet, respectively, Neumann case we have that the corresponding vector-valued Laplacians are a direct sum of the scalar operators. Therefore the eigenvalues of the corresponding vector-valued Laplace operators consist of repeated eigenvalues of the scalar Laplacian. We can arrange the former in the following way: Lemma 4.2. If n := dim H < ∞ then ± λ± m (H) = λk ,

where

λ± k

m = (k − 1)n + 1, . . . , kn,

denotes the (scalar) kth Dirichet/Neumann eigenvalue on D.

Proof. Note that ∆± D,H is unitarily equivalent to an n-fold direct sum of the scalar on L (D) since there is no coupling between the components on the operator ∆± 2 D boundary. + Recall the deﬁnition of the intervals Ik := [λ− k , λk ] in Eq. (3.2) (for simplicity, we omit in the following the index ε). From the preceding two lemmas we may collect the n eigenvalues of ∆ρD,H which lie in Ik :

Bk (ρ) := {λρm (H) | m = (k − 1)n + 1, . . . , kn} ⊂ Ik ,

n := dim H.

(4.4)

Therefore, we obtain the following spectral inclusion for equivariant Laplacians. This result will be applied in Theorems 5.4 and 6.8 below.

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Theorem 4.3. If ρ is a unitary representation on a ﬁnite-dimensional Hilbert space H then spec ∆ρD,H = Bk (ρ) ⊆ Ik k∈N

where

∆ρD,H

k∈N

denotes the ρ-equivariant Laplacian.

4.3. Non-abelian Floquet transformation Consider ﬁrst the right, respectively, left regular representation R, respectively, L on the Hilbert space 2 (Γ): (Rγ a)eγ = aeγ γ ,

a = (aγ )γ ∈ 2 (Γ),

(Lγ a)eγ = aγ −1 eγ ,

γ, γ ∈ Γ.

(4.5)

Using standard results we introduce the following unitary map (see e.g. [36, Sec. 3 and the Appendix] and references cited therein) ⊕ F : 2 (Γ) → H(z) dz (4.6) Z

for a suitable measure space (Z, dz). The map F is a generalization of the Fourier transformation in the abelian case. Moreover, it transforms the right regular representation R into the following direct integral representation ⊕ γ = F Rγ F ∗ = R Rγ (z) dz, γ ∈ Γ. (4.7) Z

Remark 4.4. Let R be the von Neumann algebra generated by all unitaries Rγ , γ ∈ Γ, i.e. R = {Rγ | γ ∈ Γ} ,

(4.8)

where R denotes the commutant of R in L( 2 (Γ)). Then we decompose R with respect to a maximal abelian von Neumann subalgebra A ⊂ R (for a concrete example, see Example 4.6). The space Z is the compact Hausdorﬀ space associated, by Gelfand’s isomorphism, to a separable C ∗ -algebra C, which is strongly dense in A. Furthermore, dz is a regular Borel measure on Z. We may identify the algebra A with L∞ (Z, dz) and since it is maximal abelian, the ﬁbre representations R(z) are irreducible a.e. (see [55, Sec. 14.8 ﬀ.]). The generalized Fourier transformation introduced in Eq. (4.6) can be used to decompose L2 (X) into a direct integral. In particular, we deﬁne for a.e. z ∈ Z: (U u)(z)(x) := u(γx)Rγ −1 (z)v(z), (4.9) γ∈Γ

C∞ c (X)

and x ∈ D. The map U extends to a uniwhere v := F δe ∈ 2 (Γ), u ∈ tary map ⊕ ⊕ ∼ L2 (D, H(z)) dz = H(z) dz ⊗ L2 (D), U : L2 (X) → Z

Z

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the so-called Floquet or partial Fourier transformation. Moreover, operators commuting with the translation T on L2 (X) are decomposable, in particular, we can decompose ∆X since its resolvent commutes with all translations (2.2). We denote by C∞ eq (D, H(z)) the set of smooth R(z)-equivariant functions deﬁned eq in (4.1) and ∆D (z) is the R(z)-equivariant Laplacian in L2 (D, H(z)). One can show in this context (cf. [50, 36]): ⊕ ∞ Theorem 4.5. The operator U maps C∞ c (X) into Z Ceq (D, H(z)) dz. Moreover, ⊕ ∆X is unitary equivalent to Z ∆eq D (z) dz and spec ∆X ⊆ spec ∆eq (4.10) D (z). z∈Z

If Γ is amenable (cf. Remark 5.3), then we have equality in (4.10). Example 4.6. Let us illustrate the above direct integral decomposition in the case of the free group Γ = Z ∗ Z generated by α and β. Let A ∼ = Z be the cyclic subgroup generated by α. We can decompose the algebra R given in (4.8) with respect to the abelian algebra A := {La ∈ L( 2 (Γ)) | a ∈ A} ⊂ R , and, in this case, we have / A, the algebra is Z = S1 . Since the set {aγa−1 | a ∈ A} is inﬁnite provided γ ∈ maximal abelian in R (i.e. A = A ∩ R ), and therefore, each ﬁber representation R(z) is irreducible in H(z). Moreover, since La ∈ A (a ∈ A) we can also decompose these operators with respect to the previous direct integral. We can give a more concrete realization of the abstract Fourier transformation F = FΓ (see e.g. [47, Sec. 19]): We interprete Γ → A\Γ as covering space with abelian covering group A acting on Γ from the left; the corresponding translation action Ta on 2 (Γ) coincides with the left regular representation La (a ∈ A). The (abelian) Floquet transformation U = UA gives a direct integral decomposition ⊕ H(χ) dχ, FΓ = UA : 2 (Γ) → b A

where H(χ) ∼ = 2 (A\Γ) is the space of χ-equivariant sequences in 2 (Γ). Note that H(χ) is inﬁnite dimensional. A straightforward calculation shows that ⊕ ⊕ ∼ R (χ) dχ and L La (χ) dχ, Rγ ∼ = γ a = b A

b A

γ ) = u( γ γ) and La (χ)u( γ ) = χ(a)u( γ ) for u ∈ H(χ). Note that Lγ , where Rγ (χ)u( γ∈ / A, does not decompose into a direct integral over Z since it mixes the ﬁbres. Furthermore, one sees that v = (U δe )(χ) is the unique normalized eigenvector of Ra (χ) with eigenvalue χ(a). This follows from the fact that the set of cosets {Aγa | a ∈ A} ⊂ A\Γ is inﬁnite provided γ ∈ / A. From the previous facts one can directly check that each R(χ) is an irreducible representation of Γ in H(χ) and that these representations are mutually inequivalent. Finally, R(χ) is also inequivalent to any irreducible component of the direct integral decomposition obtained from a diﬀerent maximal abelian subgroup B = A.

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5. Spectral Gaps for Type I Groups We will present in this section the ﬁrst method to show that the Laplacian of the manifolds constructed in Sec. 3 with (in general non-abelian) type I covering groups have an arbitrary ﬁnite number of spectral gaps. We begin recalling the deﬁnition of type I groups in the context of discrete groups. Definition 5.1. A discrete group Γ is of type I if Γ is a ﬁnite extension of an abelian group, i.e. if there is an exact sequence 0 → A → Γ → Γ0 → 0, where A Γ is abelian and Γ0 ∼ = Γ/A is a ﬁnite group. Remark 5.2 (i) In the previous deﬁnition we have used a simple characterization of countable, discrete groups of type I due to Thoma, cf. [53]. Moreover, all irreducible representations of a type I group Γ are ﬁnite-dimensional and have a uniform bound on the dimension (see [53, 41]). Therefore, the following properties are all equivalent: (a) there is a uniform bound on the dimensions of irreducible representations of Γ, (b) all irreducible representations of Γ are ﬁnite-dimensional, (c) Γ is a ﬁnite extension of an abelian group, (d) Γ is CCR (completely continuous representation, cf. [55, Chap. 14]), (e) Γ is of type I. Recall also that Γ is of type I iﬀ the von Neumann algebra R generated by Γ (cf. Eq. (4.8)) is of type I (cf. [28]). Note that for our application it would be enough if Γ has a decomposition over a measure space (Z, dz) as in Remark 4.4 such that almost every representation ρ(z) is ﬁnite-dimensional. But such a group is already of type I: indeed, if the set {z ∈ Z | dim H(z) = ∞} has measure 0, then it follows from [17, Sec. II.3.5] that the von Neumann Algebra R (cf. Eq. (4.8)) is of type I. By the above equivalent characterisation this implies that Γ is of type I. (ii) The following criterion (cf. [28, 26]) will be used in Examples 8.4 and 8.5 to decide that a group is not of type I: The von Neumann algebra R is of type II1 iﬀ Γfcc has inﬁnite index in Γ. Here, Γfcc := {γ ∈ Γ | Cγ is ﬁnite}

(5.1)

is the set of elements γ ∈ Γ having ﬁnite conjugacy class Cγ . In particular such a group is not of type I. Even worse: Almost all representations in the direct integral decomposition (4.7) are of type II1 [17, Sec. II.3.5] and therefore inﬁnite-dimensional (see e.g. Example 4.6). Remark 5.3. The notion of amenable discrete groups will be useful at diﬀerent stages of our approach. For a deﬁnition of amenability of a discrete group Γ see e.g. [16] or [9]. We will only need the following equivalent characterisations: (a) Γ is amenable. (b) 0 ∈ spec ∆X [9]. (c) spec ∆M ⊂ spec ∆X [50, Propositions 7 and 8]. Here, X → M is a covering with covering group Γ. Note that discrete type I

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groups are amenable since they are ﬁnite extensions of abelian groups (extensions of amenable groups are again amenable, cf. [16, Sec. 4]). We want to stress that Theorem 3.3 is no contradiction to the fact that Γ is amenable iﬀ 0 ∈ spec ∆(X,gε ) although the ﬁrst interval I1 (gε ) = [0, λ+ k (gε )] tends to 0 as ε → 0. Note that we have only shown that I1 (gε ) ∩ spec ∆(X,gε ) = ∅ and not 0 = λ1 (M, gε ) ∈ spec ∆(X,gε ) which is only true in the amenable case. is the set of equivalence classes of unitary The dual of Γ, which we denote by Γ, irreducible representations of Γ. We denote by [ρ] the (unitary) equivalence class of a unitary representation ρ on H. Note that the spectrum of a ρ-equivariant Laplacian and dim H only depend on the equivalence class of ρ. becomes a nice measure space (“smooth” in If Γ is of type I, then the dual Γ as measure space in the terminology of [37, Chap. 2]). Furthermore, we can use Γ the direct integral decomposition deﬁned in Sec. 4.3. In particular, combining the results of Secs. 2 and 4 we obtain the main result for type I groups: Theorem 5.4. Suppose X → M is a Riemannian Γ-covering with fundamental domain D, where Γ is a type I group and denote by g the Riemannian metric on X. Then spec ∆(X,g) ⊂

Ik (g)

and

Ik (g) ∩ spec ∆(X,g) = ∅,

k ∈ N,

k∈N + where Ik (g) := [λ− k (D, g), λk (D, g)] is the Neumann–Dirichlet interval deﬁned as in (3.2). In particular, for each n ∈ N there exists a metric g = gεn constructed as in Sec. 3.2 such that spec ∆(X,g) has at least n gaps, i.e. n + 1 components as subset of [0, ∞).

Proof. We have spec ∆X =

b [ρ]∈Γ

spec ∆ρD,H ⊆

k∈N

Ik (g) =

Ik (g),

k∈N

for the ﬁrst equality and Theorem 4.3 where we used the Theorem 4.5 with Z = Γ for the inclusion. Note that Γ is amenable and that the latter theorem applies since all (equivalence classes of) irreducible representations of a type I group are ﬁnite dimensional (cf. Remark 5.2(i)). The existence of gaps in k Ik (g) follows from Theorem 3.2. Since Γ is amenable, spec ∆M ⊂ spec ∆X (cf. (c) in Remark 5.3). Moreover, from Eq. (4.4) with ρ the trivial representation on H = C, we have that λk (M ) ∈ Ik . Note that functions on M correspond to functions on D with periodic boundary conditions. Therefore, we have shown that every gap of the union k Ik (g) is also a gap of spec ∆X .

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6. Spectral Gaps for Residually Finite Groups In this section, we present a new method to prove the existence of a ﬁnite number of spectral gaps of ∆X . The present approach is applicable to so-called residually ﬁnite groups Γ, which is a much larger class of groups containing type I groups (cf. Sec. 8). Roughly speaking, residually ﬁnite means that Γ has a lot of normal subgroups with ﬁnite index. Geometrically, this implies that one can approximate the covering π : X → M with covering group Γ by ﬁnite coverings pi : Mi → M , where the Mi ’s are compact. Since the present section is central to the paper we will give for completeness proofs of known results, namely for Theorem 6.6 (see [4, 2]).

6.1. Subcoverings and residually finite groups Suppose that π : X → M is a covering with covering group Γ (as in Sec. 2). Corresponding to a normal subgroup Γi Γ we associate a covering πi : X → Mi such that X πi Mi

Γi

@ @ π @ Γ @ @ R pi - M Γ/Γi

(6.1)

is a commutative diagram. The groups under the arrows denote the corresponding covering groups. Definition 6.1. A (countable, inﬁnite) discrete group Γ is residually ﬁnite if there exists a monotonous decreasing sequence of normal subgroups Γi Γ such that Γ = Γ0 Γ 1 · · · Γ i · · · , Γi = {e} and Γ/Γi is ﬁnite. (6.2) i∈N

Denote by RF the class of residually ﬁnite groups. Suppose now that Γ is residually ﬁnite. Then there exists a corresponding sequence of coverings πi : X → Mi such that pi : Mi → M is a ﬁnite covering (cf. Diagram (6.1)). Such a sequence of covering maps is also called tower of coverings. Remark 6.2. We recall also the following equivalent deﬁnitions of residually ﬁnite groups (see e.g. [38] or [46, Sec. 2.3]). (i) A group Γ is called residually ﬁnite if for all γ ∈ Γ\{e} there is a group homomorphism Ψ : Γ → G such that Ψ(γ) = e and Ψ(Γ) is a ﬁnite group.

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(ii) Let F denote the class of ﬁnite groups. Then Γ is residually ﬁnite, iﬀ the so-called F -residual N (6.3) RF (Γ) := N Γ Γ/N ∈F

is trivial, i.e. RF (Γ) = {e}. Next we give some examples for residually ﬁnite groups (cf. the survey article [38]): Example 6.3. (i) Abelian and ﬁnite groups are residually ﬁnite. (ii) Free products of residually ﬁnite groups are residually ﬁnite, in particular, the free group in r generators Z∗r is residually ﬁnite. (iii) Finitely generated linear groups are residually ﬁnite (for a simple proof of this fact cf. [3]; a group is called linear iﬀ it is isomorphic to a subgroup of GLn (C) for some n ∈ N). In particular, SLn (Z), fundamental groups of closed, orientable surfaces of genus g or, more generally, ﬁnitely generated subgroups of the isometry group on the hyperbolic space Hd are residually ﬁnite. Next we need to introduce a metric on the discrete space Γ: Definition 6.4. Let G be a set which generates Γ. The word metric d = dG on Γ is deﬁned as follows: d(γ, e) is the minimal number of elements in G needed to express γ as a word in the alphabet G; d(e, e) := 0 and d(γ, γ ) := d(γ γ −1 , e). Geometrically, residually ﬁniteness means that, given any compact set K ⊂ X, there exists a ﬁnite covering pi : Mi → M and a covering πi : X → Mi which is injective on K (cf. [10]). This idea is used in the following lemma: Lemma 6.5. Fix a fundamental domain D for the covering π : X → M and suppose that πi : X → Mi (i ∈ N) is a tower of coverings as above. Then for each covering πi : X → Mi there is a fundamental domain Di (not necessarily connected) such that Di = X. D0 := D ⊂ D1 ⊂ · · · ⊂ Di ⊂ · · · and i∈N

Proof. It is enough to show the existence of a family of representants Ri ⊂ Γ of Γ/Γi , i ∈ N, satisfying R0 := {e} ⊂ R1 ⊂ · · · ⊂ Ri ⊂ · · · and Ri = Γ. i∈N

In this case the fundamental domains are given explicitly by ¯ Di := int r−1 D, r∈Ri

where int denotes the topological interior.

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Let d be the word metric on Γ with respect to the set of generators G := ¯ ∩D ¯ = ∅}, which is naturally adapted to the fundamental domain D. {γ ∈ Γ | γ D ¯ is compact (cf. [45, Theorems 6.5.10 Note that G is ﬁnite and generates Γ since D and 6.5.11]). We choose a set of representants Ri of Γ/Γi that have minimal distance in the word metric to the neutral element, i.e. if r ∈ Ri , then d(r, e) ≤ d(rΓi , e). Note that since Γi+1 ⊂ Γi we have Ri+1 ⊃ Ri . To conclude the proof we have to show that every γ ∈ Γ is contained in some Ri , i ∈ N. Since Γ is ﬁnitely generated, there exists n ∈ N such that γ ∈ Bn := {γ ∈ Γ | d(γ, e) ≤ n}. Moreover, since B2n is ﬁnite and Γ residually ﬁnite we also have B2n ∩ Γi = {e} for i large enough. Therefore, any other element γ

= γγi−1 in the class γΓi with γi ∈ Γi \{e} has a distance greater than n, since d( γ , e) = d(γγi−1 , e) = d(γ, γi ) ≥ d(e, γi ) − d(γ, e) > 2n − n = n. This implies that γ ∈ Ri by the minimality condition in the choice of the representants. Theorem 6.6. Suppose Γ is residually ﬁnite with the associated sequence of coverings πi : X → Mi and pi : Mi → M as in (6.1). Then spec ∆X ⊆ spec ∆Mi , i∈N

and the Laplacian ∆Mi with respect to the ﬁnite covering pi : Mi → M has discrete spectrum. Equality holds iﬀ Γ is amenable. Proof (cf. [2]). If λ ∈ spec ∆X , then for each ε > 0 there exists u ∈ C∞ c (X) such that

(∆X − λ)u 2X < ε.

u 2X Applying Lemma 6.5 there is an i = i(ε) such that supp u ⊂ Di . Furthermore, since Di → Mi = X/Γi is an isometry, u can be written as the lift of a smooth f on Mi , i.e. f ◦ πi = u. Therefore,

(∆Mi − λ)f 2Mi

(∆X − λ)u 2X = < ε, 2

f Mi

u 2X which implies λ ∈ i∈N spec ∆Mi . Finally, since Mi → M is a ﬁnite covering and M is compact, spec ∆Mi is discrete. For the second assertion cf. [2] or [4]. One basically uses the characterisation due to [9] that Γ is amenable iﬀ 0 ∈ spec ∆X (cf. Remark 5.3). Next we analyze the spectrum of the ﬁnite covering Mi → M . Note that D is also isometric to a fundamental domain for each ﬁnite covering Mi → M , i ∈ N.

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Lemma 6.7. We have

spec ∆Mi =

spec ∆ρD,H(ρ) ,

ci [ρ]∈G

where ∆ρ is the equivariant Laplacian introduced in Sec. 4.1 and Gi := Γ/Γi is a i its dual. ﬁnite group and G Proof. Applying the results of Sec. 4.3 to the ﬁnite group Gi and the ﬁnite measure i with the counting measure all direct integrals become direct sums. space Z := G By Peter–Weyl’s theorem (see e.g. [25, § 27.49]) we also have n(ρ)H(ρ), F : 2 (Gi ) → ci [ρ]∈G

where each multiplicity satisﬁes n(ρ) = dim H(ρ) < ∞. Finally, ρ ∆D,H(ρ) ∆Mi ∼ = ci [ρ]∈G

and the result follows. We now can formulate the main result of this section: Theorem 6.8. Suppose X → M is a Riemannian Γ-covering with fundamental domain D, where Γ is a residually ﬁnite group and denote by g the Riemannian metric on X. Then Ik (g), Ik (g) ∩ spec ∆(X,g) = ∅, k ∈ N, spec ∆(X,g) ⊂ k∈N + where Ik (g) := [λ− k (D, g), λk (D, g)] is deﬁned as in (3.2). In particular, for each n ∈ N there exists a metric g = gεn , constructed as in Sec. 3.2, such that spec ∆(X,g) has at least n gaps, i.e. n + 1 components as subset of [0, ∞).

Proof. We have spec ∆X ⊆

i∈N

spec ∆Mi =

i∈N ci [ρ]∈G

spec ∆ρD,H(ρ) ⊆

k∈N

Ik (g) =

Ik (g),

k∈N

where we used Theorem 6.6, Lemma 6.7 and Theorem 4.3. Note that the latter theorem applies since all (equivalence classes of) irreducible representations of the ﬁnite groups Gi , i ∈ N, are ﬁnite-dimensional. The existence of gaps in k Ik (g) follows from Theorem 3.2. Finally, by Theorem 3.3, a gap of k Ik (g) is in fact a gap of spec ∆X .

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7. Kadison Constant and Asymptotic Behavior In the present section we will combine our main result stated in Theorem 6.8 with some results by Sunada and Br¨ uning (cf. [52, Theorem 1] or [11]), to give a more complete description of the spectrum of the Laplacian ∆X , where X → M is the Γ-covering constructed in Sec. 3. For this, we need a further deﬁnition: Definition 7.1. Let Γ be a ﬁnitely generated discrete group. The Kadison constant of Γ is deﬁned as ∗ (Γ, K)}, C(Γ) := inf{trΓ (P ) | P non-trivial projection in Cred ∗ where trΓ (·) is the canonical trace on Cred (Γ, K) , the tensor product of the reduced group C ∗ -algebra of Γ and the algebra K of compact operators on a separable Hilbert space of inﬁnite dimension (see [52, Sec. 1] for more details).

In this section, we assume that Γ is is residually ﬁnite and has a strictly positive Kadison constant, i.e. C(Γ) > 0. For example, the free product Z∗r ∗Γ1 ∗· · ·∗Γa with ﬁnite groups Γi satisﬁes both properties (cf. e.g. [38], [52, Appendix]). Another such group is the fundamental group (cf. Eq. (3.1)) of a (compact, orientable) surface of genus g (see [40]). Remark 7.2. Suppose that K is an integral operator on L2 (X) commuting with the group action, having smooth kernel k(x, y) and satisfying k(x, y) = 0

for all x, y ∈ X with d(x, y) ≥ c

∗ (Γ, K) for some constant c > 0. Then K can be interpreted as an element of Cred and one can write the Γ-trace as k(x, x) dx trΓ K = D

(see [52, Sec. 1] as well as [5] for further details), where D is a fundamental domain of X → M . ⊕ λ dE(λ), then If we consider the spectral resolution of the Laplacian ∆X ∼ = it follows that ∗ (Γ, K) E(λ2 ) − E(λ1 ) ∈ Cred

if λ1 < λ2 and λ1 , λ2 ∈ spec ∆X (cf. [52, Sec. 2]). Denote by N (g, λ) the number of components of spec ∆(X,g) ∩ [0, λ]. From [11, 52] we obtain the following asymptotic estimate on N (g, λ): Theorem 7.3. Suppose (X, g) → (M, g) is a Riemannian Γ-covering where Γ has a positive Kadison constant, i.e. C(Γ) > 0 then lim sup λ→∞

N (g, λ) 1 . ≤ d/2 C(Γ) d vol(M, g)λ

(2π)−d ω

(7.1)

In particular, the spectrum of ∆X has band-structure, i.e. N (g, λ) < ∞ for all λ ≥ 0.

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Remark 7.4. Note that Theorem 7.3 only gives an asymptotic upper bound on the number of components of spec ∆X ∩ [0, λ], not on the whole spectrum itself. Therefore, we have no assertion about the so-called Bethe–Sommerfeld conjecture stating that the number of spectral gaps for a periodic operator in dimensions d ≥ 2 remains ﬁnite. Combining Theorem 7.3 with our result on spectral gaps we obtain more information on the spectrum and a lower asymptotic bound on the number of components: Theorem 7.5. Suppose (X, g) → (M, g) is a Riemannian Γ-covering where Γ is a residually ﬁnite group and where g = gε is the family of decoupling metrics constructed in Sec. 3. Then we have: (i) For each n ∈ N there exists g = gεn such that spec ∆(X,g) has at least n gaps. If in addition C(Γ) > 0 then there exists λ0 > 0 such that n + 1 ≤ N (g, λ) < ∞ for all λ ≥ λ0 , i.e. spec ∆(X,g) has band-structure. (ii) Suppose in addition that the limit manifold (N, g) has simple spectrum, i.e. all eigenvalues λk (0) have multiplicity 1 (cf. Theorem 3.1). Then for each λ ≥ 0 there exists ε(λ) > 0 such that N (gε(λ) , λ) −d λ→∞ (2π) ωd vol(N, g)λd/2

lim inf

≥ 1.

Here, gε denotes the metric constructed in Sec. 3. Proof. (i) follows immediately from Theorems 6.8 and 7.3. (ii) Suppose λ ∈ / spec ∆N , then λk (0) < λ < λk+1 (0) for some k ∈ N. Let ε = ε(λ) ∈ (0, 1] be the largest number such that N (λ, gε ) is (at least) k, in other words, N (λ, gε ) ≥ k = N (λ, ∆N ) where the latter number denotes the number of eigenvalues of ∆N below λ. We conclude with the Weyl theorem, lim

λ→∞

N (λ, ∆N ) = 1, (2π)−d ωd vol(N, g)λd/2

where ωd denotes the volume of the d-dimensional Euclidean unit ball. To conclude the section we remark that generically, ∆(N,g) has simple spectrum (cf. [54]). The assumption on the spectrum of (N, g) is natural since N (g, λ) counts the components without multiplicity. 8. Examples 8.1. Relation between the approaches presented in Secs. 5 and 6 We begin comparing the two main approaches presented in this paper which assure the existence of spectral gaps (cf. Secs. 5 and 6).

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One easily sees from Deﬁnition 6.1 that a ﬁnite extension of a residually ﬁnite group is again residually ﬁnite. In particular, type I groups are residually ﬁnite as ﬁnite extensions of abelian groups (cf. Deﬁnition 5.1). Therefore, for type I groups one can also produce spectral gaps by the approximation method with ﬁnite coverings introduced in Sec. 6. Nevertheless we believe that the direct integral method will be useful when analyzing further spectral properties: Example 8.1. One of the advantages of the method described in Sec. 5 is that one has more information about the bands. Suppose Γ is ﬁnitely generated and abelian, is the disjoint i.e. Γ ∼ = Zr ⊕ Γ0 , where Γ0 is the torsion subgroup of Γ. Then Γ r union of ﬁnitely many copies of T . From the continuity of the map ρ → λρk (cf. [8] or [51]), we can simplify the characterisation of the spectrum in Theorem 4.5 and obtain ⊆ Ik , spec ∆X = Bk , where Bk := {λρk | ρ ∈ Γ} (8.1) k∈N

is compact, Bk is also compact, but in general, Bk need not to the kth band. Since Γ is connected iﬀ Γ is torsion free, i.e. Γ = Zr ). Note also be connected (recall that Γ that Bk has only ﬁnitely many components. For non-abelian groups this approach may be generalised in the direction of Hilbert C*-modules (cf. [23]). In principle one could also consider a combination of the methods of Secs. 5 and 6: denote by T1 the class of type I groups and by RT1 the class of residually type I groups, i.e. Γ ∈ RT1 iﬀ the T1 -residual RT1 (Γ) is trivial (cf. Eq. (6.3)). Similarly we denote by RF the class of residually ﬁnite groups (cf. Deﬁnition 6.1). If we consider a covering with a group Γ ∈ RT1 , then instead of the ﬁnite covering pi : Mi → M considered in Eq. (6.1) we would have a covering with a type I group. For these groups, we can replace Lemma 6.7 by the direct integral decomposition of Theorem 4.5. Nevertheless the following lemma shows that the class of residually ﬁnite and residually type I groups coincide. Lemma 8.2. From the inclusion F ⊂ T1 ⊂ RF it follows that the corresponding residuals for the group Γ coincide, i.e. RF (Γ) = RT1 (Γ). Moreover, RF = RT1 . Proof. From the inclusion F ⊂ T1 it follows immediately that RF (Γ) ⊃ RT1 (Γ). To show the reverse inclusion one uses the following characterization: a group is residually F iﬀ it is a subcartesian product of ﬁnite groups (cf. [46, Sec. 2.3.3]). Finally, from the equality of the residuals it follows that RF = RT1 . 8.2. Examples with residually finite groups In the rest of this subsection we present several examples of residually ﬁnite groups which are not type I. They show diﬀerent aspects of our analysis. For the next example recall the construction (A) described in Sec. 3.

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Example 8.3 (Fundamental groups of oriented, closed surfaces). Suppose that N := S2 is the two-dimensional sphere with a metric such that ∆N has simple spectrum (cf. [54] for the existence of such metrics). Suppose, in addition, that M is obtained by adding r handles to N as described in Sec. 3, Case A. The fundamental group Γ of M (cf. Eq. (3.1) with s = 0) is residually ﬁnite (recall Example 6.3(iii)). Moreover, from the proof of [40, Proposition 2.16], it follows that Γ has a positive Kadison constant. Therefore, Theorem 7.5 applies to the universal → M with the metric gε speciﬁed in Sec. 3. cover X := M The following example uses the construction (B) in Sec. 3. Example 8.4 (Heisenberg group). Let Γ := H3 (Z) be the discrete Heisenberg group, where H3 (R) denotes the set of matrices   1 x y Ax,y,z := 0 1 z  (8.2) 0 0 1 with coeﬃcients x, y, z in the ring R. A covering with group Γ is given e.g. by X := H3 (R) with compact quotient M := H3 (R)/H3 (Z). Note that X is diﬀeomorphic to R3 . Clearly, Γ is a ﬁnitely generated linear group and therefore residually ﬁnite (cf. Example 6.3(iii)). Now, by Theorem 6.8 one can deform conformally a Γ-invariant metric g as in Case (B) of Sec. 3, such that spec ∆X has at least n spectral gaps, n ∈ N. In this case, Γ is also amenable as an extension of amenable groups (cf. Remark 5.3). In fact, Γ is isomorphic to the semi-direct product Z Z2 , where 1 ∈ Z acts on Z2 by the matrix 1 1 . 0 1 Therefore, we have equality in the characterization of spec ∆X in Theorems 4.5 and 6.6. Note ﬁnally that the group Γ is not of type I since Γfcc = {A0,y,0 | y ∈ Z} has inﬁnite index in Γ (cf. Remark 5.2). Thus, our method in Sec. 5 does not apply since the measure dz in (4.6) is supported only on inﬁnite-dimensional Hilbert spaces. Curiously, one can construct a ﬁnitely additive measure on the group dual supported by the set of ﬁnite-dimensional representations of Γ (cf. [44]). The Γ group dual Γ is calculated e.g. in [27, Beispiel 1]. Example 8.5 (Free groups). Let Γ = Z∗r be the free group with r > 1 generators. Then Γ is residually ﬁnite (recall Example 6.3(ii)) and has positive Kadison constant (cf. [52, Appendix]). Therefore, Theorem 7.5 applies to the Γ-coverings X → M speciﬁed in Sec. 3. Note that Γ is not of type I since Γfcc = {e} (cf. Remark 5.2). Such groups are called ICC (inﬁnite conjugacy class) groups. Again, for any direct integral decomposition (4.6), almost all Hilbert spaces H(z) are inﬁnite-dimensional. Finally, Γ is not amenable.

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8.3. An example with an amenable, non-residually finite group Kirchberg mentioned in [30, Sec. 5] an interesting example of a ﬁnitely generated amenable group which is not residually ﬁnite: Denote by S0 the group of permutations of Z which leave unpermuted all but a ﬁnite number of integers. We call A0 the normal subgroup of even permutations in S0 . Let Z act on S0 as shift operator. Then the semi-direct product Γ := Z S0 is (ﬁnitely) generated by the shift n → n + 1 and the transposition interchanging 0 and 1. Note that Γ and S0 are ICC groups. Lemma 8.6. The group Γ is amenable. Moreover, RF (Γ) = A0 , hence Γ is not residually ﬁnite. Proof. The group S0 is amenable as inductive limit of amenable groups; therefore, Γ is amenable as semi-direct product of amenable groups (cf. [16, Sec. 4]). The equality RF (Γ) = A0 follows from the fact that A0 is simple. Proposition 8.7. Every ﬁnite-dimensional unitary representation ρ of Γ leaves A0 elementwise invariant, i.e. ρ(γ) = ½ for all γ ∈ A0 . Proof. Let E be the class of countable subgroups of U(n), n ∈ N, and F G the class of ﬁnitely generated groups. Note that F ⊂ E ∩ F G and that ﬁnitely generated linear groups are residually ﬁnite (cf. Example 6.3(iii)), i.e. E ∩ F G ⊂ RF . Arguing as in the proof of Lemma 8.2 we obtain from the inclusions F ⊂ E ∩ F G ⊂ RF that RE∩F G (Γ) = RF (Γ). Now by Lemma 8.6 the F -residual of Γ is A0 . Finally, since Γ itself is ﬁnitely generated (i.e. Γ ∈ F G), we have RE (Γ) = RE∩F G (Γ) = A0 . This concludes the proof since ρ is a ﬁnite-dimensional unitary representation iﬀ im(ρ) ∼ = Γ/ ker ρ ∈ E, i.e. RE (Γ) is the intersection of all ker ρ, where ρ are the ﬁnite-dimensional, unitary representations of Γ. In conclusion, we cannot analyze the spectrum of ∆X by none of the above methods since Γ is not residually ﬁnite (and therefore neither of type I). Nevertheless, equality holds in (4.10), but we would need inﬁnite-dimensional Hilbert spaces H(z) in the direct integral decomposition in order to describe the spectrum of the whole covering X → M and not only of the subcovering X/A0 → M (with covering group Z × Z2 , cf. Diagram (6.1)). Remark 8.8. Coverings with transformation groups as in the present subsection cannot be treated with the methods developed in this paper. It seems though reasonable that even for non-residually ﬁnite groups the construction speciﬁed in Sec. 3 still produces at least n spectral gaps, n ∈ N. To show this one needs to replace the techniques of Sec. 4 that use the min-max principle in order to prove the existence of spectral gaps for these types of covering manifolds.

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9. Conclusions and Applications Given a Riemannian covering (X, g) → (M, g) with a residually ﬁnite transformation group Γ we constructed a deformed Γ-covering (X, gε ) → (M, gε ) such that spec ∆(X,gε ) has n spectral gaps, n ∈ N. Intuitively one decouples neighboring fundamental domains by deforming the metric g → gε in such a way that the junctions of the fundamental domains are scaled down (cf. Fig. 1). Therefore, our construction may serve as a model of how to use geometry to remove unwanted frequencies or energies in certain situations which may be relevant for technological applications. For instance, the Laplacian on (X, gε ) may serve to give an approximate description of the energy operator of a quantum mechanical particle moving along the periodic space X. Usually, the energy operator contains additional potential terms coming form the curvature of the embedding in some ambient space, cf. [21], but, nevertheless, ∆(X,gε ) is still a good approximation for describing properties of the particle. A spectral gap in this context is related to the transport properties of the particle in the periodic medium, e.g. an insulator has a large ﬁrst spectral gap. Another application are photonic crystals, i.e. optical materials that allow only certain frequencies to propagate. Usually, one has to consider diﬀerential forms in order to describe the propagation of classical electromagnetic waves in a medium. Nevertheless, if we assume that the Riemannian density is related to the dielectric constant of the material, one can use the scalar Laplacian on a manifold as a simpliﬁed model. For more details, we refer to [35, 20] and the references therein. A further interesting line of research would be to consider the opposite situation as in the present paper; that means the use of geometry to prevent the appearance + of spectral gaps (cf. [22, 39]). In fact, these authors proved that λ− k+1 (D) ≤ λk (D) for all k ∈ N, i.e. that Ik ∩ Ik+1 = ∅ for all k ∈ N provided D is an open subset of Rn or a Riemannian symmetric space of non-compact type. On such a space, we have a priori no information on the existence of gaps. It would also be interesting to connect the number of gaps with geometric quantities, e.g. isoperimetric constants or the curvature. We want to stress that the curvature of (X, gε ) is not bounded as ε → 0 (cf. [42]) in contrast to the degeneration of Riemannian metrics under curvature bounds (cf. e.g. [14]). In the present paper we have considered ∆X as a prototype of an elliptic operator and have avoided the use of a potential V . In this way we isolate the eﬀect of geometry on spec ∆X . Of course, our methods and results may also be extended to more general periodic structures that have a “reasonable” Neumann Laplacian as a lower bound and satisfy the spectral “localization” result in Theorem 4.3. For example, one can also study periodic operators like ∆X + V , operators on quantum wave guides, more general periodic elliptic operators or operators on metric graphs (cf. e.g. [19] for examples of periodic metric graphs with spectral gaps). Finally, we conclude mentioning that we can not apply directly our result to disprove the Bethe–Sommerfeld conjecture on manifolds, which says that the number of spectral gaps for a periodic operator in dimensions d ≥ 2 remains ﬁnite. Even if

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we know that the spectrum of the Laplacian on (X, gε ) converges to the discrete set {λk | k ∈ N} as ε → 0, we cannot expect a uniform control of the spectral convergence on the whole interval [0, ∞) since there are topological obstructions (cf. [13]). Note that a uniform convergence would immediately imply that spec ∆(X,gε ) would have an inﬁnite number of spectral gaps. Nevertheless, we hope that our construction will contribute to the clariﬁcation of the status of this conjecture. Acknowledgments It is a pleasure to thank Mohamed Barakat for helpful discussions on residually ﬁnite groups. We are also grateful to David Krejˇciˇr´ık and Norbert Peyerimhoﬀ for useful comments. Finally, we would like to thank Volker Enß, Christopher Fewster, Luka Grubiˇsi´c and Vadim Kostrykin for valuable remarks and suggestions on the manuscript. References [1] C. Ann´e, G. Carron and O. Post, Gaps in the diﬀerential forms spectrum on cyclic coverings, preprint (2007), arXiv:0708.3981. [2] T. Adachi, On the spectrum of periodic Schr¨ odinger operators and a tower of coverings, Bull. London Math. Soc. 27 (1995) 173–176. [3] R. C. Alperin, An elementary account of Selberg’s lemma, Enseign. Math. (2) 33 (1987) 269–273. [4] T. Adachi, T. Sunada and P. W. Sy, On the regular representation of a group applied to the spectrum of a tower, in Analyse Alg´ebrique des Perturbations Singuli` eres, II (Marseille-Luminy, 1991), Travaux en Cours, Vol. 48 (Hermann, Paris, 1994), pp. 125–133. [5] M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Asterisque 32–33 (1976) 43–72. [6] J. Br¨ uning, P. Exner and V. A. Geyler, Large gaps in point-coupled periodic systems of manifolds, J. Phys. A 36 (2003) 4875–4890. [7] J. Br¨ uning, V. Geyler and I. Lobanov, Spectral properties of Schr¨ odinger operators on decorated graphs, Mat. Zametki 77(1) (2005) 152–156. [8] O. Bratteli, P. E. T. Jørgensen and D. W. Robinson, Spectral asymptotics of periodic elliptic operators, Math. Z. 232 (1999) 621–650. [9] R. Brooks, The fundamental group and the spectrum of the Laplacian, Comment. Math. Helv. 56 (1981) 581–598. [10] R. Brooks, The spectral geometry of a tower of coverings, J. Diﬀerential Geom. 23 (1986) 97–107. [11] J. Br¨ uning and T. Sunada, On the spectrum of periodic elliptic operators, Nagoya Math. J. 126 (1992) 159–171. [12] Y. Colin de Verdi`ere, Construction de laplaciens dont une partie ﬁnie du spectre est ´ donn´ee, Ann. Sci. Ecole Norm. Sup. (4) 20(4) (1987) 599–615. [13] I. Chavel and E. A. Feldman, Spectra of manifolds with small handles, Comment. Math. Helv. 56 (1981) 83–102. [14] J. Cheeger, Degeneration of Riemannian Metrics Under Ricci Curvature Bounds, Fermi Lectures (Scuola Normale Superiore, Pisa, 2001).

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SINGULAR FERMI SURFACES I. GENERAL POWER COUNTING AND HIGHER DIMENSIONAL CASES

JOEL FELDMAN∗,‡ and MANFRED SALMHOFER∗,†,§ ∗Mathematics

Department, The University of British Columbia, 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2 †Theoretische Physik, Universit¨ at Leipzig, Postfach 100920, 04009 Leipzig, Germany ‡[email protected] §[email protected]

Received 12 June 2007 Revised 3 December 2007 We prove regularity properties of the self-energy, to all orders in perturbation theory, for systems with singular Fermi surfaces which contain Van Hove points where the gradient of the dispersion relation vanishes. In this paper, we show for spatial dimensions d ≥ 3 that despite the Van Hove singularity, the overlapping loop bounds we proved together with E. Trubowitz for regular non-nested Fermi surfaces [J. Stat. Phys. 84 (1996) 1209–1336] still hold, provided that the Fermi surface satisfies a no-nesting condition. This implies that for a fixed interacting Fermi surface, the self-energy is a continuously differentiable function of frequency and momentum, so that the quasiparticle weight and the Fermi velocity remain close to their values in the noninteracting system to all orders in perturbation theory. In a companion paper, we treat the more singular two-dimensional case. Keywords: Fermion systems; Fermi surfaces; Van Hove singularities; renormalization. Mathematics Subject Classification 2000: 81T15, 81T17, 81T08, 82D35, 82D40

1. Introduction In 1953, Van Hove published a general argument implying the occurrence of singularities in the photon and electron spectrum of crystals [1]. The core of his argument is an application of Morse theory [2] — a sufficiently smooth function defined on the torus and having only nondegenerate critical points must have saddle points. In the independent-electron approximation, the dispersion relation k 7→ ²(k) of the electrons plays the role of the Morse function, and the Van Hove singularities (VHS) manifest themselves in the electronic density of states Z dd k ρ(E) = δ(E − ²(k)) (1) (2π)d 233

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at those values of the energy where the level set {k : ²(k) = E} contains one (or more) of the saddle points, the Van Hove points. The nature of these singularities in ρ depends on the dimension. In two dimensions, ρ has a logarithmic singularity. In three dimensions, ρ is continuous but its derivative has singularities. In all dimensions, these singularities have observable consequences, although they occur only at discrete values of the energy. In mean-field theories for symmetry breaking, the density of states plays an important role because it enters the self-consistency equations for the order parameter. For instance, in BCS theory, the superconducting gap ∆ is determined as a function of the temperature T = β −1 as the solution to the equation Z ∆=g∆

ρ(E)

dE p tanh 2 (E − EF )2 + ∆2

β

p

(E − EF )2 + ∆2 2

(2)

where g > 0 is the coupling constant that determines the strength of the meanfield interaction between Cooper pairs and EF is the Fermi energy determined by the electron density. (We have written the equation for an s-wave superconductor.) The properties of ρ(E) for E near to EF obviously influence the temperaturedependence of ∆, as well as the value of the critical temperature Tc , defined as the largest value of T below which (2) has a nonzero solution. If ρ is smooth, the small-g asymptotics of Tc is Tc ∼ e−ρ(EF )/g . A logarithmic divergence in ρ W of the form ρ(E) = K ln |E−E (with fixed constants K and W ) enhances the V H| √

critical temperature to Tc ∼ e−K/ g if EF = EVH . Similarly, Van Hove singularities cause ferromagnetism in mean-field theory at arbitrarily small couplings g ¿ 1. In a true many-body theory, all this becomes much less clear-cut. Besides the obvious remark that in two dimensions, there is no long-range order at positive temperatures [4], hence the above discussion is restricted to mean-field theory, the question whether Van Hove singularities indeed occur in interacting systems and if so, what their influence on observable quantities is, remains open and important. The theoretical quantity related to the electron spectrum and the density of states of the interacting system is the interacting dispersion relation or the spectral function, obtained from the full propagator, hence ultimately from the electron self-energy. The VHS might cease to exist in the interacting system for various reasons. The interacting Fermi surface may turn out to avoid the saddle points, or the singularity caused by the saddle points of the dispersion relation may be smoothed out by more drastic effects, such as the opening of gaps in the vicinity of the saddle points. On the other hand, the VHS might also become more generic because the Fermi surface may get pinned at the Van Hove points, and the singularity might also get stronger due to interaction effects. A lot of research has gone into these questions because Van Hove singularities were invoked as a possible explanation of high-temperature superconductivity (see, e.g. [5] and references therein). In

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particular, there are competition effects between superconductivity, ferromagnetism and antiferromagnetism [6–9], as well as interesting phenomena connected to Fermi surface fluctuations [10, 11], to mention but a few results. The above speculations as to the fate of the Fermi surface and the VHS have been discussed widely in the literature [5]. In this paper, we begin a mathematical study of Fermi surfaces that contain Van Hove points, but that satisfy a no-nesting condition away from these points, with the aim of understanding some of the above questions. We prove regularity properties of the electron self-energy to all orders of perturbation theory using the multiscale techniques of [13–16], which are closely related to the renormalization group techniques used in [6–8]. In the present paper, we give bounds that apply in all dimensions d ≥ 2 and then consider the case d ≥ 3 in more detail. In a companion paper [12], we focus on the two-dimensional case, and in particular on the question of the renormalization of the quasiparticle weight and the Fermi velocity. Our motivation for imposing the no-nesting condition is twofold. First, an example of a dispersion relation in d = 2 with a Fermi surface that contains Van Hove points and satisfies our no-nesting condition is the (t, t0 ) Hubbard model with t0 6= 0 and t, t0 < 0 at the Van Hove density. For t0 = 0, the Van Hove density is at half-filling, and the Fermi surface becomes flat, hence nested under our definition. However, there is ample evidence that in the Hubbard model it is the parameter range t0 6= 0 and electron density near to the van Hove density that is relevant for high-Tc superconductivity (see, e.g. [5–8]). Second, nesting causes additional singularities, and to get a clear picture of which property of the Fermi surface causes what kind of phenomena, it is useful to disentangle the effects of the VHS from those of nesting. We now give an overview of the technical parts of the present paper and state our main result about the self-energy and the correlation functions. In Sec. 2, we prove bounds for volumes of thin shells in momentum space close to the Fermi surface. These volume bounds are the essential ingredient for power counting bounds. In Lemma 2.3, we show that these volume bounds are not changed by the introduction of the most common singularities in d ≥ 3 and increase by a logarithm of the scale in d = 2. This implies by the general bounds of [13] that the superficial power counting of the model is unchanged for d ≥ 3 and changes “only” by logarithms in d = 2. Lemma 2.4 contains a refinement of these bounds in which one restricts to small balls near the singular points. In Sec. 3, we turn to the finer aspects of power counting that are necessary to understand the regularity of the self-energy, for spatial dimensions d ≥ 3. We define a weak no-nesting condition which is essentially identical to that of [13] and prove that the volume improvement estimate (1.34) of [13] carries over unchanged (Proposition 3.6). By [13, Theorem 2.40], this implies that the bulk of the conclusions of [13, Theorems 1.2–1.8] carry over to the situation with VHS in d ≥ 3.

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Namely, Theorem 1.1. Let d ≥ 3, and let the dispersion relation k 7→ e(k) satisfy • F = {k ∈ Rd | e(k) = 0} is compact, • e(k) is C 3 , • ∇e(k) vanishes only at isolated points of F. We shall call them singular points, 2 • if e(k) = 0 and ∇e(k) = 0, then [ ∂∂ki ∂kj e(k)]1≤i,j≤d is nonsingular and has at least one positive eigenvalue and at least one negative eigenvalue. • There is no nesting, in the precise sense of Hypothesis NN in Sec. 3.1 Let the interaction be short-range in the sense that the Fourier transform k 7→ vˆ(k) of the two-body interaction is twice continuously differentiable in k. Introduce the counterterm function k 7→ K(k) as in [13, Sec. 2], but using the localization operator (`T )(q0 , q) = T (0, q) in place of the localization operator of [13, Definition 2.6], to renormalize the perturbation expansion. Then 1. To any fixed order in renormalized perturbation theory, the electronic self-energy (i.e. the sum of the values of all two-legged one-particle-irreducible Feynman graphs) is continuously differentiable in the frequency and momentum variables. There is an ε > 0 so that all first derivatives of the self-energy are H¨ older continuous of degree ε. The counterterm function K has the same regularity properties. 2. To any fixed order in renormalized perturbation theory, all correlation functions are well-defined, locally integrable functions of the external momenta. 3. To any fixed order in renormalized perturbation theory, the only contributions to the four-point function that fail to be bounded and continuous are the (generalized) ladder diagrams

Here each vertex

is an arbitrary connected four-legged subdiagram and each

line is a string whose vertices (if any) are arbitrary one particle irreducible two-legged subdiagrams. 4. For each natural number r, denote by λr Kr (k) the order r contribution to the renormalized perturbation expansion of the counterterm function K(k). For each PR natural number R, the map e 7→ e + r=1 λr Kr is locally injective. The precise meaning of and hypotheses for this statement are given in the paragraph containing (21). The above statements are proven in Sec. 3.3 at temperature T = 0. However, the same methods show that they extend to small T ≥ 0, with the change that for T > 0, singularities are replaced by finite values that, however, diverge as T → 0.

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As explained in detail in [13], the counterterm K fixes the Fermi surface, so that all our results are about the model with a fixed interacting Fermi surface. Whether the situation that the Fermi surface contains zeroes of the gradient of e can indeed be achieved is related to the question whether there is an inversion theorem generalizing that of [16] to the situation with VHS, i.e. which provides existence of a solution of the equation e + K(e) = E for the present situation (item 4 of the above theorem only gives local uniqueness). This is a difficult question which is still under investigation (see also [12]). A natural question is the relation between these statements to all orders in perturbation theory and results obtained from truncated renormalization group flows in applied studies, which are often claimed to be “nonperturbative”. The allorder results are statements about an iterative solution to a full renormalization group flow. The solution of renormalization group flows obtained by truncating the infinite hierarchy to a finite hierarchy creates scale-dependent approximations to the Green functions. These approximations give the leading order behavior if the truncation has been done appropriately. Often, the results indicate instabilities of the flow, which signal that the true state of the system is not well-described by an action of the form assumed in the flow. A true divergence in the solution occurs only when the regime of validity of the truncation is left. (In the simplest situations, such singularities coincide with the divergence of a geometric series.) In careful studies, the equations are never integrated to the point where anything diverges. In that case, the regularity bounds obtained by all-order estimates are more accurate than those obtained from the solution of the flow equations truncated at finite order. There is one case where the integration of the renormalization group equations gives an effect within the validity of the truncation, but qualitatively different from all-order theory: this is when the flow satisfies infrared asymptotic freedom, i.e. the coupling function becomes screened at low scales. For instance, in the repulsive Hubbard model, the ladders with the bare interaction lead to a screening of the superconducting interaction, corresponding to g < 0 and hence to no solution in the BCS gap equation. (However, in this case, an attractive Cooper interaction is generated in second order, and it then grows in the flow to lower scales.) Such screening effects can only make terms smaller. Hence the upper bounds provided by the all-order analysis are still as good as the integration of truncations to the same order, as far as regularity properties are concerned. In practice, the truncations done in the RG equations are of very low order, so that the all-order analysis includes many contributions that are not taken into account in these truncations. A nonperturbative mathematical proof involves bounding the remainders created in the expansion (or truncation). This is possible in d = 2 using the sector method of [18] (see [19–30]), but a full construction has not yet been achieved in d ≥ 3. Because the graphical structures used in our arguments only require one overlap of loops, we expect that a suitable adaptation will be possible in constructive studies. In addition to the above-mentioned problem with constructive arguments in d ≥ 3, the important question of the inversion theorem should also be addressed.

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2. General Power Counting Bounds in d ≥ 2 2.1. Analytic structure of the one-body problem Here we discuss briefly the properties of the one-body problem, to show that the Fermi surface of the noninteracting system is given as the zero set of an analytic function, hence no-nesting in a polynomial sense is a generic condition. For lattice models, analyticity of the dispersion relation e is obvious for hopping amplitudes that decay exponentially with distance (or are even of finite range). For continuum Schr¨odinger operators, it follows from the statements below, which even hold for the case with a magnetic field. Let d ≥ 2 and Γ be a lattice in Rd of maximal rank. Let r > d. Define ( ) ¯Z ¯ r d d ¯ A = A = (A1 , . . . , Ad ) ∈ (LR (R /Γ)) ¯ A(x)dx = 0 , Rd /Γ ( ) ¯Z ¯ r/2 d V (x)dx = 0 . V = V ∈ LR (R /Γ) ¯¯ d R /Γ

For (A, V ) ∈ A × V set Hk (A, V ) = (i∇ + A(x) − k)2 + V (x). When d = 2, 3, the operator Hk (A, V ) describes an electron in Rd with quasimomentum k moving under the influence of the magnetic field with periodic vector potential A(x) = R (A1 (x), . . . , Ad (x)) and R electric field with periodic potential V (x). The conditions Rd /Γ A(x)dx = 0 and Rd /Γ V (x)dx = 0 are included purely for convenience and can always be achieved by translating k and shifting the zero point of the energy scale. The following theorem is proven in [17]. Theorem 2.1. Let (

) ¯Z ¯ AC = A = (A1 , . . . , Ad ) ∈ (L (R /Γ)) ¯¯ A(x)dx = 0 , Rd /Γ ( ) ¯Z ¯ VC = V ∈ Lr/2 (Rd /Γ) ¯¯ V (x)dx = 0 d r

d

d

R /Γ

be the complexifications of A and V, respectively. There exists an analytic function F on Cd × C × AC × VC such that, for k, A, V real, λ ∈ Spec(Hk (A, V ))

⇐⇒

F (k, λ, A, V ) = 0.

The theorem is proven by providing a formula for F . Write (i∇ + A(x) − k)2 + V (x) − λ = 1l − ∆ + u(k, λ) + w(k, A, V ) with u(k, λ) = −2ik · ∇ + k2 − λ − 1l, w(k, A, V ) = i∇ · A + iA · ∇ − 2k · A + A2 + V.

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Then the function F (k, λ, A, V ) of the above theorem is a suitably regularized 1 1 1 1 u(k, λ) √1l−∆ + √1l−∆ w(k, A, V ) √1l−∆ . determinant of 1l + √1l−∆ 2.2. Volumes of shells around singular Fermi surfaces Suppose that the energy eigenvalues for the one-body problem with quasimomentum k are the solutions of an equation F (k, λ) = 0. That is, the bands e1 (k) ≤ e2 (k) ≤ e3 (k) ≤ · · · all obey F (k, en (k)) = 0. Our analysis of the regularity properties of the self-energy and correlation functions depends on having good bounds on the volume of the set of all quasimomenta k for which there are very low energy bands. More precisely, fix any M > 1 and let j ≤ 0. We need to know the volume of the set of all quasimomenta k for which there is at least one band with |en (k)| ≤ M j . The following lemma provides a useful simplification. Lemma 2.2. Let K be a compact subset of Rd and F : K × [−1, 1] → R be C 1 . Then there is a constant C such that Vol{k ∈ K | F (k, λ) = 0 for some |λ| ≤ M j } ≤ Vol{k ∈ K | |F (k, 0)| ≤ CM j } for all j ≤ 0. In particular, if all bands en (k) obey F (k, en (k)) = 0 then, Vol{k ∈ K | |en (k)| ≤ M j for some n} ≤ Vol{k ∈ K | |F (k, 0)| ≤ CM j }. Proof. Since F is C 1 on the compact set K × [−1, 1], ¯ ¯ ¯ ∂F ¯ ¯ ¯ C≡ sup ¯ ∂λ (k, λ)¯ < ∞. (k,λ)∈K×[−1,1] Hence, if for some k ∈ K and some |λ| ≤ M j , we have F (k, λ) = 0, then, for that same k, |F (k, 0)| = |F (k, λ) − F (k, 0)| ≤ C|λ| ≤ CM j . Hence {k ∈ K | F (k, λ) = 0 for some |λ| ≤ M j } ⊂ {k ∈ K | |F (k, 0)| ≤ CM j }. We now, and for the rest of this paper, focus on a single band k 7→ ²(k), and assume that the chemical potential µ, used to fix the density, is such that e(k) = ²(k) − µ has a nonempty zero set, the Fermi surface, which has also not degenerated to a point. In the scale analysis, momentum space is cut up in shells around the Fermi surface. Here we take the convention of labeling these shells by negative integers j ≤ 0. The shell number j contains momenta k and Matsubara frequencies k0 with 1 j j 2 M ≤ |ik0 − e(k)| ≤ M . Here M > 1 is fixed once and for all. For the (standard)

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details about the scale decomposition and the corresponding renormalization group flow, obtained by integrating over degrees of freedom in the shell number j successively, downwards from j = 0, see, e.g. [13, Sec. 2]. The next lemma contains the basic volume bound for the scale analysis. In the case without VHS, the bound is of order M j . The lemma implies that this bound remains unchanged for d ≥ 3, and that there is an extra logarithm in d = 2. Lemma 2.3. Let K be a compact subset of Rd and e : K → R be C 2 . Assume that for every point p ∈ K at least one of • e(p) 6= 0, • ∇e(p) 6= 0, 2 • det[ ∂∂ki ∂kj e(p)]1≤i,j≤d 6= 0, is true. Then there is a constant C such that j

Vol{k ∈ K | |e(k)| ≤ M } ≤ CM

j

( |j|

if d = 2

1

if d > 2

for all j ≤ −1. Proof. Since K is compact, it suffices to prove that, for each p ∈ K there are constants R > 0 and C (depending on p) such that for all j ≤ −1, VR,j (p) = Vol{k ∈ K | |e(k)| ≤ M j , |k − p| ≤ R} ( j |j| if d = 2 ≤ CM 1 if d > 2. Case 1: e(p) 6= 0. We are free to choose R sufficiently small that |e(k)| ≥ 12 |e(p)| for all k ∈ K with |k − p| ≤ R. Then {k ∈ K | |e(k)| ≤ M j , |k − p| ≤ R} is empty unless M j ≥ 12 |e(p)| and it suffices to take C=

2 Vol{k ∈ K | |k − p| ≤ R} |e(p)|

Case 2: e(p) = 0, ∇e(p) 6= 0. By translating and permuting indices, we may ∂e assume that p = 0 and that ∂k (p) 6= 0. Then, if R is small enough, 1 x = X(k) = (e(k), k2 , . . . , kd ) is a C 2 diffeomorphism from KR = {k ∈ K | |k − p| ≤ R} to some bounded subset ∂e X of Rd . The Jacobian of this diffeomorphism is ∂k (k) and is bounded away from 1

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zero, say by c1 . Then VR,j (p) = Vol{k ∈ K | |e(k)| ≤ M j , |k − p| ≤ R} Z 1(|e(k)| ≤ M j )dd k = KR

Z = X

¯ ¯−1 ¯∂e ¯ (X −1 (x))¯¯ dd x 1(|x1 | ≤ M j ) ¯¯ ∂k1 Z

≤

c−1 1

1(|x1 | ≤ M j )dd x X

j ≤ c−1 1 c2 M .

Here 1(E) denotes the indicator function of the event E, i.e. 1(E) = 1 if E is true and 1(E) = 0 otherwise. 2

∂ e(p)]1≤i,j≤d 6= 0. By translating, Case 3: e(p) = 0, ∇e(p) = 0, det[ ∂k i ∂kj we may assume that p = 0. Then, if R is small enough, the Morse lemma [3, Theorem 8.3bis] implies that there exists a C 1 diffeomorphism, X(k), from KR to some bounded subset X of Rd such that

e(X −1 (x)) = Qm (x) = x21 + · · · + x2m − x2m+1 − · · · − x2d for some 0 ≤ m ≤ d. Then Z VR,j (p) = 1(|e(k)| ≤ M j )dd k KR

Z

¯−1 ¯ ¸ · ¯ ¯ ∂Xi −1 ¯ ¯ 1(|Qm (x)| ≤ M ) ¯det (X (x)) ¯ dd x ¯ ¯ ∂kj 1≤i,j≤d j

= X

Z ≤

c−1 1

1(|Qm (x)| ≤ M j )dd x. X

If m = 0 or m = d, Z Z 1(|Qm (x)| ≤ M j )dd x = 1(|x21 + · · · + x2d | ≤ M j )dd x X

X

Z ≤ Rd

1(|x21 + · · · + x2d | ≤ M j )dd x

= cd M dj/2 so it suffices to consider 1 ≤ m ≤ d − 1. Go to spherical coordinates separately in x1 , . . . , xm and xm+1 , . . . , xd , using q q u = x21 + · · · + x2m , v = x2m+1 + · · · + x2d .

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If R is small enough Z VR,j (p) ≤ c−1 1 cm,d

1(|u2 − v 2 | ≤ M j )um−1 v d−m−1 dudv. 0≤u,v≤1

Now make the change of variables x = u + v, y = u − v. Then Z 1(|u2 − v 2 | ≤ M j )um−1 v d−m−1 dudv 0≤v≤u≤1

Z

Z

2

≤

1

dx 0

Z

dy 1(xy ≤ M j )(x + y)m−1 |x − y|d−m−1

0

Z

2

≤

1

dx 0

dy 1(xy ≤ M j )(x + y)d−2

0

and the lemma follows from Z 2 Z dx 0

min{1,M j /x}

dy = M j + M j ln

0

2 Mj

and, for n ≥ 1, Z

Z

2

min{1,M j /x}

dx 0

0

dy xn ≤

1 M (n+1)j + 2n M j n+1

and Z

Z

2

dx 0

min{1,M j /x}

dy y n ≤ M j .

0

That Vol{k ∈ K | |e(k)| ≤ M j } ≤ CM j |j| and that this bound suffices to yield a well-defined counterterm and well-defined correlation functions, to all orders of perturbation theory, was also proven in [31]. We now refine Lemma 2.3 a little. Lemma 2.4. Let e : Rd → R be C 2 . Assume that • e(0) = 0, • ∇e(0) = 0, 2 • det[ ∂∂ki ∂kj e(0)]1≤i,j≤d 6= 0, 2

• [ ∂∂ki ∂kj e(0)]1≤i,j≤d has at least one positive eigenvalue and at least one negative eigenvalue. Then there are C, C 0 > 0 such that for all q ∈ Rd , j ≤ 0 and 0 < ε < 12 , Vol{k ∈ Rd | |e(k)| ≤ M j , |k − q| ≤ M εj , |k| ≤ C 0 } ( 1 + (1 − 2ε)|j| if d = 2 j ≤ CM M (d−2)εj if d > 2.

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Proof. By the Morse lemma, we can assume without loss of generality that 2 2 e(k) = k12 + · · · + km − km+1 − · · · − kd2

for some 1 ≤ m ≤ d − 1. Go to spherical coordinates separately in k1 , . . . , km and km+1 , . . . , kd , using q q 2 2 , u = k12 + · · · + km v = km+1 + · · · + kd2 . For any fixed u > 0, the condition |k − q| ≤ M εj restricts (k1 , . . . , km ) to lie on a spherical cap of diameter at most 2M εj on the sphere of radius u. This cap has an area of at most an m-dependent constant times min{u, M εj }m−1 . Similarly, for any fixed v > 0, the condition |k − q| ≤ M εj restricts (km+1 , . . . , kd ) to run over an area of at most a constant times min{v, M εj }d−m−1 . The condition |k − q| ≤ M εj also restricts u and v to run over intervals I1 , I2 of length at most 2M εj . Thus Vol{k | |e(k)| ≤ M j , |k − q| ≤ M εj } Z ≤ cm,d 1(|u2 − v 2 | ≤ M j ) min{u, M εj }m−1 min{v, M εj }d−m−1 dudv I1 ×I2

Z 1(|u2 − v 2 | ≤ M j ) min{max{u, v}, M εj }d−2 dudv.

≤ cm,d I1 ×I2

It suffices to consider the case 0 ≤ v ≤ u. Make the change of variables x = u + v, y = u − v. Then x and y are restricted to run over intervals J1 , J2 of length at most 4M εj and Vol{k | |e(k)| ≤ M j , |k − q| ≤ M εj } Z ≤ 2cm,d J ×J 1(xy ≤ M j ) min{x, M εj }d−2 dxdy. 1

2

0≤y≤x

In the event that J1 ⊂ [M εj , ∞], then on the domain of integration, x ≥ M εj and the condition xy ≤ M j forces y ≤ M j /M εj , so that Z M (1−ε)j Z Vol{k | |e(k)| ≤ M j , |k − q| ≤ M εj } ≤ 2cm,d dy dx M (d−2)εj 0

J1

≤ 2cm,d M (1−ε)j 4M εj M (d−2)εj = 8cm,d M j M (d−2)εj . If J1 ∩ [0, M εj ] 6= ∅, the domain of integration is contained in 0 ≤ y ≤ x ≤ 5M εj and Z 5M εj Z 5M εj j εj Vol{k | |e(k)| ≤ M , |k − q| ≤ M } ≤ 2cm,d dx dy 1(xy ≤ M j )xd−2 . 0

0

For d = 2, the lemma follows from Z 5M εj Z min{5M εj ,M j /x} dx dy = M j (1 + ln 25 + (1 − 2ε)|j| ln M ). 0

0

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For d > 2, Z

Z

5M εj

min{5M εj ,M j /x}

dx 0

dy xd−2 ≤ 5d−1 M j M (d−2)εj .

0

3. Improved Power Counting From now on we assume that d ≥ 3, and that • • • •

F = {k ∈ Rd | e(k) = 0} is compact e(k) is C 3 ∇e(k) vanishes only at isolated points of F. We shall call them singular points. 2 if e(k) = 0 and ∇e(k) = 0, then [ ∂∂ki ∂kj e(k)]1≤i,j≤d is nonsingular and has at least one positive eigenvalue and at least one negative eigenvalue.

In addition, we make an assumption that there is no nesting. In general, this means that any nontrivial translate of F or −F only has intersections with F of at most some fixed finite degree. Here we only require a weak form of no-nesting — namely that there is only polynomial flatness. This assumption, which is essentially the same as [13, Hypothesis A3], is introduced and discussed in detail in the following. 3.1. A no-nesting hypothesis and its consequences To make precise the “only polynomial flatness” hypotheses, let n : F → Rd ,

ω 7→ n(ω) =

∇e (ω) |∇e|

be the unit normal to the Fermi surface. It is defined except at singular points, which are isolated. For ω, ω 0 ∈ F, define the angle between n(ω) and n(ω 0 ) by θ(ω, ω 0 ) = arccos(n(ω) · n(ω 0 )). Let D(ω) = {ω 0 ∈ F | |n(ω) · n(ω 0 )| = 1} = {ω 0 ∈ F | n(ω) = ±n(ω 0 )}

(3)

and denote the (d − 1)-dimensional measure of A ⊂ F by Vold−1 A. Also, for any A ⊂ Rd and β > 0 denote by Uβ (A) = {p ∈ Rd | distance(p, A) < β} the open β-neighborhood of A. We assume: Hypothesis NN. There are strictly positive numbers Z0 , β0 and κ such that for all β ≤ β0 and all ω ∈ F, p Vold−1 {ω 0 ∈ F | |sin θ(ω, ω 0 )| = 1 − (n(ω 0 ) · n(ω))2 ≤ β} ≤ Z0 β κ .

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To verify this hypothesis, it suffices to find strictly positive numbers z0 , z1 , ρ0 , β0 and κ0 such that for all for all β ≤ β00 and all ω ∈ F, 0

(i) Vold−1 (Uβ (D(ω)) ∩ F) ≤ z0 β κ , p 0 (ii) if ω 0 ∈ 6 Uβ (D(ω)) ∩ F, then |sin θ(ω, ω 0 )| = 1 − (n(ω) · n(ω 0 ))2 ≥ z1 β ρ . Then κ =

κ0 ρ0 ,

−κ0 /ρ0

Z0 = z0 z1

ρ0

and β0 = z1 β00 .

2 Example. As an example, take √ d ≥ 3, 1 ≤ m < d and e(k) = k12 + · · · + km − 2 2 km+1 − · · · − kd , say with |k| ≤ 2. The corresponding Fermi surface, F, is the 2 2 + · · · + kd2 , which we may parametrize by = km+1 (truncated) cone k12 + · · · + km m−1 k = (rθ, rφ) with 0 ≤ r ≤ √ 1, θ ∈ S and φ ∈ S d−m−1 . The volume element on F d−2 m−1 in this parametrization is 2r dr d θ dd−m−1 φ, where dm−1 θ and dd−m−1 φ are the volume elements on S m−1 and S d−m−1 respectively. The unit normals to F at k = (rθ, rφ) are ± √12 (θ, −φ). Now fix any ω = (rθ, rφ) with 0 < r ≤ 1. Then

D(ω) = {(tθ, tφ) | 0 < |t| ≤ 1}. If (t0 θ 0 , t0 φ0 ) ∈ Uβ (D(ω)) ∩ F the there is a t such that q 2 2 |(t0 θ 0 , t0 φ0 ) − (tθ, tφ)| < β =⇒ |t0 θ 0 − tθ| + |t0 φ0 − tφ| < β =⇒ |t0 θ 0 − tθ| < β, |t0 φ0 − tφ| < β, |t − t0 | < β =⇒ |t0 θ 0 − t0 θ| < 2β, |t0 φ0 − t0 φ| < 2β. For each fixed t0 the volume of the t0 θ 0 s in t0 S m−1 for which |θ 0 − θ| < 2β/|t0 | is at most a constant, depending only on m, times |t0 |m−1 min{1, ( |tβ0 | )m−1 } ≤ β m−1 and the volume of the t0 φ0 s in t0 S d−m−1 for which |φ0 −φ| < 2β/|t0 | is at most a constant, depending only on d − m − 1, times |t0 |d−m−1 min{1, ( |tβ0 | )d−m−1 } ≤ β d−m−1 . Hence Z

1

Vold−1 (Uβ (D(ω)) ∩ F) ≤ cd,m

dt0 β d−2 = cd,m β d−2 .

0

Thus condition (i) of Hypothesis NN is satisfied with κ0 = d − 2. If ω 0 = (t0 θ 0 , t0 φ0 ) ∈ / Uβ (D(ω)) ∩ F then, for every |t| ≤ 1, |(t0 θ 0 , t0 φ0 ) − (tθ, tφ)| ≥ β. In particular, |(t0 θ 0 , t0 φ0 ) ± (t0 θ, t0 φ)| ≥ β =⇒ |(θ 0 , φ0 ) ± (θ, φ)| ≥ β. The angle between n(t0 θ 0 , t0 φ0 ) = ± √12 (θ 0 , −φ0 ) and n(rθ, rφ) = ± √12 (θ, −φ) is the same (±π) as the angle between (θ 0 , φ0 ) and (θ, φ) (measured at the origin).

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By picking signs appropriately, we may assume that 0 ≤ θ(ω, ω 0 ) ≤ π2 . Thus ¯ ¯ ¯ ¯ 1 1 1 |sin θ(ω, ω 0 )| ≥ ¯¯sin θ(ω, ω 0 )¯¯ = √ |(θ 0 , φ0 ) ± (θ, φ)| ≥ √ β 2 2 2 2 2 and condition (ii) of Hypothesis NN is satisfied with ρ0 = 1. So κ = d − 2. Proposition 3.1. Let d ≥ 3 and let e : Rd → R be C 3 . Assume that • e(0) = 0, • ∇e(0) = 0, 2 • det[ ∂∂ki ∂kj e(0)]1≤i,j≤d 6= 0, 2

• [ ∂∂ki ∂kj e(0)]1≤i,j≤d has m ≥ 1 positive eigenvalues and d − m ≥ 1 negative eigenvalues. Then there is a c > 0 and constants β0 > 0 and Z0 such that for every unit vector a ∈ Rd , p Vold−1 {k ∈ F | 1 − (n(k) · a)2 ≤ β} ≤ Z0 β max{m−1,d−m−1} where F = {k ∈ Rd | |k| < c, e(k) = 0} for all 0 < β < β0 . Proof. By a rotation, followed by a permutation of indices, we may assume that ∂2 [ ∂k e(0)]1≤i,j≤d is a diagonal matrix, with diagonal entries 2λ1 , 2λ2 , . . . , 2λd i ∂kj that are in decreasing order. By hypothesis, λj > 0 for 1 ≤ j ≤ m and λj < 0 for m + 1 ≤ j ≤ d. Replace λj by −λj for j > m. Then, 2 2 e(k) = λ1 k12 + · · · + λm km − λm+1 km+1 − · · · − λd kd2 + G(k)

with G(k) a C 3 function having a third order zero at 0. Define q 2 , R1 (k) = λ1 k12 + · · · + λm km q 2 R2 (k) = λm+1 km+1 + · · · + λd kd2 , q R(k) = λ1 k12 + · · · + λd kd2 . Also use 2 S˜1m−1 = {(k1 , . . . , km ) | λ1 k12 + · · · + λm km = 1}, 2 S˜2d−m−1 = {(km+1 , . . . , kd ) | λm+1 km+1 + · · · + λd kd2 = 1}

to denote “unit” (m−1)-dimensional and (d−m−1)-dimensional ellipsoids, respecd tively. For each r > 0, the surface R(k) = r is a d−1 dimensional p p ellipsoid in R with smallest semi-axis r/ maxj λj and largest semi-axis r/ minj λj . We now concentrate on the intersection of F and that ellipsoid. The proof of Proposition 3.1 will continue following the proof of Lemma 3.4.

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Lemma 3.2. Suppose that |G(k)| ≤ g0 R(k)3 ,

|∇G(k)| ≤ g1 R(k)2

and that c is small enough (depending only on g0 , g1 and the λi ’s). (a) For each θ 1 ∈ S˜m−1 , θ 2 ∈ S˜d−m−1 and r ≥ 0 such that the ellipsoid {k ∈ Rd | R(k) = r} is contained in the sphere {k | |k| < c}, there is a unique (r1 , r2 ) such that r1 , r2 ≥ 0,

r12 + r22 = r2

and

(r1 θ 1 , r2 θ 2 ) ∈ F .

2

Furthermore |r1 − r2 | ≤ g0 r . (b) F is a C 3 manifold, except for a singularity at k = 0. Proof. (a) The point (r1 θ 1 , r2 θ 2 ) is on F if and only if 0 = r12 − r22 + G(r1 θ 1 , r2 θ 2 ) = [r1 − r2 ][r1 + r2 ] + G(r1 θ 1 , r2 θ 2 ) or G(r1 θ 1 , r2 θ 2 ) . r1 + r2 For each −r ≤ s ≤ r there are unique r1 (s) ≥ 0 and r2 (s) ≥ 0 such that r1 − r2 = −

r1 (s) − r2 (s) = s, r2

s = −r

(4)

r1 (s)2 + r2 (s)2 = r2 . s=0 s=r r1

Furthermore r10 (s) − r20 (s) = 1 and r1 (s)r10 (s) + r2 (s)r20 (s) = 0 gives that r10 (s) = r2 (s) r1 (s) 0 r1 (s)+r2 (s) and r2 (s) = − r1 (s)+r2 (s) have magnitude at most 1. Since r1 (s)+r2 (s) ≥ 1 ,r2 (s)θ 2 ) r, H(s) = − G(rr11(s)θ obeys (s)+r2 (s)

|H(s)| ≤ g0 r2 and

¯ ¯ ¯ [r1 (s) + r2 (s)]∇G(r1 (s)θ 1 , r2 (s)θ 2 ) · (r0 (s)θ 1 , r0 (s)θ 2 ) ¯ 1 2 ¯ ¯ ¯ ¯ − [r10 (s) + r20 (s)]G(r1 (s)θ 1 , r2 (s)θ 2 ) ¯ ¯ 0 |H (s)| = ¯ ¯ ¯ ¯ [r1 (s) + r2 (s)]2 ¯ ¯ ¯ ¯ 1 1 g1 r2 |(r10 (s)θ 1 , r20 (s)θ 2 )| + 2 2g0 r3 r r ≤ r[g1 |(θ 1 , θ 2 )| + 2g0 ] r · ¸ 2 ≤ r g1 max + 2g0 1≤i≤d λi <1 ≤

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provided c is small enough. Consequently the function s − H(s) increases strictly monotonically from −r − H(−r) ≤ −r + g0 r2 to r − H(r) ≥ r − g0 r2 as s increases from −r to r. So this function has a unique zero and (4) has a unique solution and the solution obeys |r1 − r2 | ≤ g0 r2 . (b) Since ∇e(k) = 2(λ1 k1 , . . . , λm km , −λm+1 km+1 , . . . , −λd kd ) + ∇G(k) and |∇G(k)| ≤ g1 R(k)2 ≤ g1 (maxi λi ) |k|2 , the only zero of ∇e(k) is at k = 0, assuming that c has been chosen small enough. Lemma 3.3. Define, for each a, b ∈ Rd \{0}, θ(a, b) ∈ [0, π] to be the angle between a and b. Let P1 : Rd → Rm and P2 : Rd → Rd−m be the orthogonal projections onto the first m and last d − m components of Rd , respectively. Assume that the hypotheses of Lemma 3.2 are satisfied. There is a constant g2 (depending only on the λi ’s) such that if 0 6= ω ∈ F, 0 6= a ∈ Rd with |sin θ(n(ω), a)| ≤ β, then |sin θ(P1 n(ω), P1 a)| ≤ g2 β,

|sin θ(P2 n(ω), P2 a)| ≤ g2 β.

(5)

Proof. We will prove the first bound of (5). We may assume that a is a unit vector. Possibly replacing a by −a, we may also assume that the angle between a and n(ω) is at most π2 . By part (a) of Lemma 3.4, below, 1 |a − n(ω)| = 2 sin θ(a, n(ω)) ≤ 2 sin θ(a, n(ω)) ≤ 2β. 2 So, by part (a) of Lemma 3.4, sin θ(P1 a, P1 n(ω)) ≤

|P1 a − P1 n(ω)| |a − n(ω)| 2β ≤ ≤ . |P1 n(ω)| |P1 n(ω)| |P1 n(ω)|

Thus it suffices to prove that |P1 n(ω)| is bounded away from zero. Recall that ∇e(ω) = n1 (ω) + ∇G(ω) where n1 (ω) = 2(λ1 k1 , . . . , λm km , −λm+1 km+1 , . . . , −λd kd ). Use α ∼ γ to designate that there are constants c, C > 0, depending only on the λi ’s, such that c|γ| ≤ |α| ≤ C|γ|. In this notation |n1 (ω)| ∼ |ω|,

|P1 n1 (ω)| ∼ |P1 ω| ∼ |R1 (ω)|,

|P2 n1 (ω)| ∼ |P2 ω| ∼ |R2 (ω)|.

By part (a) of Lemma 3.2, since ω ∈ F, |R1 (ω) − R2 (ω)| ≤ g0 R(ω)2 ,

R1 (ω)2 + R2 (ω)2 = R(ω)2 .

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As the maximum of R1 (ω) and R2 (ω) must be at least R(ω) ≥ R1 (ω),

√1 R(ω), 2

249

we have

1 1 R2 (ω) ≥ √ R(ω) − g0 R(ω)2 ≥ R(ω) 2 2

if c is small enough. So |P1 n1 (ω)|, |P2 n1 (ω)| ∼ |R(ω)| ∼ |ω|. As

³ ´ |∇G(ω)| ≤ g1 R(ω)2 ≤ g1 max λi |ω|2 i

we have that |P1 ∇e(ω)|, |P2 ∇e(ω)| ∼ |ω| and hence that |P1 ∇e(ω)|

|P1 n(ω)| = p

|P1 ∇e(ω)|2 + |P2 ∇e(ω)|2

is bounded away from zero. Lemma 3.4. Let a, b ∈ Rd \{0}. 1 |a−b| 2 |a| . sin θ(a, b) ≤ |a−b| |a| .

(a) If |a| = |b|, then sin 21 θ(a, b) = (b) For all a, b ∈ Rd \{0},

Proof. Part (a) is obvious from the figure on the left below. For part (b), in the notation of the figure on the right below, we have, by the sin law sin θ sin φ |c| |b − a| = =⇒ sin θ = sin φ ≤ . |c| |a| |a| |a| a−b

c φ

b

a

b

a θ

Proof of Proposition 3.1 (continued). Fix k2 ∈ Rd−m−1 . If k = (k1 , k2 ) ∈ F, then P1 n(k) is normal to Fk2 = {k1 ∈ Rm | (k1 , k2 ) ∈ F} because both n(k) and P2 n(k) are perpendicular to any vector (t, 0) that is tangent to F at k. The matrix · 2 ¸ ¸ · 2 ∂ e ∂ G (k1 , k2 ) (k1 , k2 ) = [2λi δi,j ]1≤i,j≤m + ∂ki ∂kj ∂ki ∂kj 1≤i,j≤m 1≤i,j≤m 2

∂ G is strictly positive definite (assuming that c is small enough) because ∂k (k) = i ∂kj O(|k|). So the slice Fk2 is strictly convex. The solution (r1 , r2 ) of Lemma 3.2

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depends continuously on θ 1 , θ 2 and r, so, assuming that m > 1, Fk2 is connected. Hence, for any fixed nonzero vector P1 a, there are precisely two points of Fk2 at which |sin θ(P1 n(k1 , k2 ), P1 a)| = 0. And at other points k1 ∈ Fk2 , |sin θ(P1 n(k1 , k2 ), P1 a)| is larger than a constant times the distance from k1 to the nearest of those two points. So Vold−1 {k ∈ F | |sin θ(n(k), a)| ≤ β} ≤ const sup Volm−1 {k1 ∈ Fk2 | |sin θ(P1 n(k), P1 a)| ≤ g2 β} k2

≤ const β m−1 . The bound Vold−1 {k ∈ F | |sin θ(n(k), a)| ≤ β} ≤ const β d−m−1 is proven similarly.

¤

Remark 3.5. The exponent κ = max{m − 1, d − m − 1} of Proposition 3.1 is not optimal, unless m = 1 or m = d − 1. Suppose that 2 ≤ m ≤ d − 2. As we observed in the proof of Proposition 3.1, for each fixed k2 there are precisely two distinct points of Fk2 at which sin θ(P1 n(k), P1 a) = 0. That is, at which P1 n(k) is parallel or antiparallel to P1 a. Hence {k ∈ F \{0} | sin θ(P1 n(k), P1 a) = 0} [ = {(k1 , k2 ) | k1 ∈ Fk2 , sin θ(P1 n(k), P1 a) = 0} k2 6=0

consists of two disjoint d − m dimensional submanifolds of F and {k ∈ F\{0} | |sin θ(P1 n(k), P1 a)| < g2 β} consists of two tubes of thickness of order β, and volume of order β m−1 , about those submanifolds. Similarly, {k ∈ F\{0} | |sin θ(P2 n(k), P2 a)| < g2 β} consists of two tubes of thickness of order β, and volume of order β d−m−1 , about two disjoint m dimensional submanifolds. In the “free” case, when G = 0, F = {(rθ 1 , rθ 2 ) | |(rθ 1 , rθ 2 )| ≤ c, θ 1 ∈ S˜m−1 , θ 2 ∈ S˜d−m−1 } and n(rθ 1 , rθ 2 ) k (Λ1 θ 1 , −Λ2 θ 2 )

where Λ1 = [λi δi,j ]1≤i,j≤m ,

Λ2 = [λi δi,j ]m
So M1 = {k ∈ F \{0} | sin θ(P1 n(k), P1 a) = 0} = {(rθ 1 , rθ 2 ) | 0 < |(rθ 1 , rθ 2 )| ≤ c, θ 2 ∈ S˜d−m−1 , θ 1 k Λ−1 1 P1 a}, M2 = {k ∈ F \{0} | sin θ(P2 n(k), P2 a) = 0} = {(rθ 1 , rθ 2 ) | 0 < |(rθ 1 , rθ 2 )| ≤ c, θ 1 ∈ S˜m−1 , θ 2 k Λ−1 2 P2 a}

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intersect in the lines −1 M1 ∩ M2 = {(rθ 1 , rθ 2 ) | 0 < |(rθ 1 , rθ 2 )| ≤ c, θ 1 k Λ−1 1 P1 a, θ 2 k Λ2 P2 a}

and otherwise cross transversely. (If the λi ’s are all the same, they cross perpendicularly.) So even when G is nonzero, the tubes will cross transversely (for sufficiently small c) and the volume of intersection will be of the order of the product β m−1 β d−m−1 = β κ with κ = d − 2. 3.2. The overlapping loop bound for d ≥ 3 In this section we prove the overlapping loop bound. It generalizes the analogous bound of [13, Proposition 1.1] to singular Fermi surfaces in d ≥ 3. The overlapping loop bound implies [13] that the first order derivatives of Σ are bounded continuous functions of momentum and frequency, to all orders in the renormalized expansion in the interaction, and that the same holds for the counterterm function K. Proposition 3.6. Let d ≥ 3, and let the dispersion relation k 7→ e(k) satisfy the generic assumptions stated at the beginning of Sec. 3 as well as the no-nesting hypothesis NN. Let K, Kq be any compact subsets of R2d and R, respectively. There are constants ε > 0 and const such that for all j1 , j2 , j3 < 0 and all q ∈ Kq , Vol{(k, p) ∈ R2d ∩ K | |e(k)| ≤ M j1 , |e(p)| ≤ M j2 , |e(q ± k ± p)| ≤ M j3 } ≤ const M jπ(1) M jπ(2) M εjπ(3) where π is a permutation of {1, 2, 3} with jπ(3) = max{j1 , j2 , j3 }. Proof. We may assume without loss of generality that j3 = max{j1 , j2 , j3 }. Otherwise make a change of variables with k0 = q ± k ± p, p0 = k or p. By compactness, ˜ p ˜ p ˜ and q ˜ with (k, ˜ ) ∈ K and q ˜ ∈ Kq , there are it suffices to show that for any k, ˜ ˜ and q ˜ , but independent of the constants c and ε > 0 (possibly depending on k, p ji ’s) such that ˜ ≤ c, |e(p)| ≤ M j2 , |p − p ˜ | ≤ c, Vol{(k, p)||e(k)| ≤ M j1 , |k − k| |e(q ± k ± p)| ≤ M j3 } ≤ const M j1 M j2 M εj3

(6)

˜ | ≤ c and all j1 , j2 , j3 < 0 with j3 = max{j1 , j2 , j3 }. for all q with |q − q ˜ e(˜ ˜±p ˜ ) is nonzero, the left-hand side of (6) is If any one of e(k), p), e(˜ q±k exactly zero for all sufficiently small c and sufficiently large |j3 | (which also forces |j1 | and |j2 | to be sufficiently large). On the other hand, for any bounded set of j3 ’s, (6) follows from ˜ ≤ c} ≤ const M j1 Vol{k ∈ Rd | |e(k)| ≤ M j1 , |k − k| ˜ | ≤ c} ≤ const M j2 Vol{p ∈ Rd | |e(p)| ≤ M j2 , |p − p ˜ = e(˜ ˜ p ˜ ) = 0. which holds by Lemma 2.3. So it suffices to consider e(k) p) = e(˜ q ± k±

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˜ is a singular point, then, for any 0 ≤ η < 1 , By Lemma 2.4, if k 2 ˜ ≤ M ηj , |k − k| ˜ ≤ c} ≤ const M j M (d−2)ηj . Vol{k ∈ Rd | |e(k)| ≤ M j , |k − k| ˜ is a regular point (that is, if ∇e(k) ˜ 6= 0). Clearly, the same bound applies when k j3 j3 By replacing (j, η) with (j1 , j1 η) (observe that j1 η is still between 0 and 12 ), we have ˜ ≤ M ηj3 , |k − k| ˜ ≤ c} Vol{k ∈ Rd | |e(k)| ≤ M j1 , |k − k| ≤ const M j1 M (d−2)ηj3 and hence ˜ ≤ M ηj3, |e(p)| ≤ M j2, |p − p ˜ | ≤ c, Vol{(k, p) | |e(k)| ≤ M j1 , |k − k| |e(q ± k ± p)| ≤ M j3} ˜ | ≤ c} ≤ const M j1 M (d−2)ηj3 Vol{p ∈ Rd | |e(p)| ≤ M j2 , |p − p ≤ const M j1 M j2 M (d−2)ηj3 . Similarly, ˜ ≤ c, |e(p)| ≤ M j2, |p − p ˜ | ≤ M ηj3 , Vol{(k, p) | |e(k)| ≤ M j1, |k − k| |e(q ± k ± p)| ≤ M j3} ≤ const M j1 M j2 M (d−2)ηj3 and ˜ ≤ c, |e(p)| ≤ M j2 , |p − p ˜ | ≤ c, Vol{(k, p)| |e(k)| ≤ M j1 , |k − k| j3 ˜∓p ˜∓k ˜ | ≤ M ηj3 } |e(q ± k ± p)| ≤ M , |q ± k ± p − q ˜ ≤ c, |k − k ˜ 0 | ≤ M ηj3 } ≤ const M j2 sup Vol{k | |e(k)| ≤ M j1 , |k − k| j1

˜0 k j2

≤ const M M M (d−2)ηj3 . Hence it suffices to prove that there is are ε˜ > 0 and 0 < η <

1 2

such that

˜ ≤ c, Vol{(k, p)||e(k)| ≤ M j1 , M ηj3 ≤ |k − k| ˜| ≤ c |e(p)| ≤ M j2 , M ηj3 ≤ |p − p ˜∓p ˜∓k ˜ | ≤ 3c} |e(q ± k ± p)| ≤ M j3 , M ηj3 ≤ |q ± k ± p − q ≤ const M j1 M j2 M ε˜j3 .

(7)

2

˜ 1≤i,j≤d is nonsingular for every singular point k. ˜ But, by hypothesis, [ ∂∂ki ∂kj e(k)] ηj ηj ˜ ≥ M 3 for all singular points k, ˜ then |∇e(k)| ≥ CM 3 and if Hence, if |k − k| ˜ | ≥ M ηj3 for all singular points p ˜ , then |∇e(p)| ≥ CM ηj3 and if |q ± k ± |p − p 0 ηj 3 ˜|≥M ˜ 0 , then |∇e(q ± k ± p)| ≥ CM ηj3 . So, by p−q for all singular points q

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Proposition 3.7 below, with δ = CM ηj3 , ε1 = M j1 , ε2 = M j2 and ε3 = M j3 , ˜ ≤ c, Vol{(k, p)||e(k)| ≤ M j1 , M ηj3 ≤ |k − k| ˜| ≤ c |e(p)| ≤ M j2 , M ηj3 ≤ |p − p ˜∓p ˜∓k ˜ | ≤ 3c} |e(q ± k ± p)| ≤ M j3 , M ηj3 ≤ |q ± k ± p − q 1 ≤ const 4 M j1 M j2 M ²j3 δ = const M j1 M j2 M (²−4η)j3 . If we choose η = with

² d+2 ,

then (d − 2)η = ² − 4η = d−2 d+2

ε=

²=

d−2 d+2 ²

and the proposition follows

d−2 κ . d+21+κ

We can now prove the volume improvement estimate that generalizes the one from [13, Proposition 1.1] to our situation. Proposition 3.7. Let Kk , Kp and Kq be compact subsets of Rd and v1 , v2 ∈ {+1, −1}. There are constants Cvol and Cδ such that the following holds. Assume that there are δ, κ, ρ > 0 such that (A1) for all k ∈ Kk , p ∈ Kp and q ∈ Kq : |∇e(k)| ≥ δ, |∇e(p)| ≥ δ, and |∇e(v1 k+ v2 p + q)| ≥ δ, (A2) the “only polynomial flatness” condition of Hypothesis NN is satisfied. Set ²= Let

κ . 1+κ

(8)

Z dd kdd p 1(|e(k)| ≤ ε1 )1(|e(p)| ≤ ε2 )

I2 (ε1 , ε2 , ε3 ) = sup

q∈Kq

Kk ×Kp

× 1(|e(v1 k + v2 p + q)| ≤ ε3 ).

(9)

Then, for all 0 < ε1 ≤ 1, 0 < ε2 ≤ 1, max{ε1 , ε2 } ≤ ε3 ≤ 1 with δ ≥ √ √ Cδ max{ ε1 , ε2 } I2 (ε1 , ε2 , ε3 ) ≤ Cvol

1 ε1 ε2 ε²3 . δ4

(10)

Proof. By compactness it suffices to assume that Kk is contained either in the ball ˜ ≤ c} for some k ˜ ∈ F with ∇e(k) ˜ 6= 0 (i.e. k ˜ is a regular point) or {k ∈ Rd | |k − k| d 0 ˜ ≤ c} for some k ˜ ∈ F with ∇e(k) ˜ = 0 (i.e. k ˜ in the annulus {k ∈ R | c δ ≤ |k − k| 0 ˜ is a singular point). We are free to choose c, c > 0, depending on k. We may make similar assumptions about Kp and the allowed values of v1 k + v2 p + q.

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Make a change of variables from k to (ρ1 , ω1 ), with ρ1 = e(k). We may assume that Kk is covered by a single such coordinate patch, with Jacobian const . |J1 (ρ1 , ω1 )| ≤ δ ˜ is a singular point, we would use the Morse lemma, to provide a In the case that k diffeomorphism k(x) such that e(k(x)) = x21 + · · · + x2m − x2m+1 − · · · − x2d . On the inverse image of Kk ,

¯ ¯ ¯ ¯ ∂k 2|x| = |∇x e(k(x))| = ¯¯(∇k e)(k(x))t (x)¯¯ ≥ const δ. ∂x

So we may first change variables from k to x, with Jacobian bounded and bounded away from zero (uniformly in δ) and then, in the region where, for example |x1 | ≥ const max{|x2 |, . . . , |xd |}, change variables from x to (ρ, ω) = (x21 + · · · + x2m − x2m+1 − · · · − x2d , x2 , . . . , xd ). The second change of variables has Jacobian 2|x1 | ≥ const δ. Observe that, under this change of variables, the matrix  1 x2 xd   −1 − ... 2x1 2x2 . . . −2xd x1 x1   2x1   ∂k  ∂k  ∂k  0   0  =  .  =   .   ∂(ρ, ω) ∂x  . ∂x  . 1l  ..  1l 0 0 |ρ| has operator norm bounded by const δ . So |k(ρ, ω) − k(0, ω)| ≤ const δ . Make a similar change of variables from p to (ρ2 , ω2 ), with ρ2 = e(p). Again, we may assume that Kp is covered by a single such coordinate patch, with Jacobian |J2 (ρ2 , ω2 )| ≤ const δ . Then Z Z Z ε2 Z ε1 dρ2 dω2 J2 (ρ2 , ω2 ) dρ1 dω1 J1 (ρ1 , ω1 ) I2 (ε1 , ε2 , ε3 ) ≤ sup q∈Kq

−ε1

S1

−ε2

S2

× 1(|e(v1 k(ρ1 , ω1 ) + v2 p(ρ2 , ω2 ) + q)| ≤ ε3 ) Z Z 1 ≤ const ε1 ε2 2 sup sup dω1 dω2 δ q∈Kq |ρ1 |,|ρ2 |≤ε3 S1 S2 × 1(|e(v1 k(ρ1 , ω1 ) + v2 p(ρ2 , ω2 ) + q)| ≤ ε3 ). By the mean value theorem |e(v1 k(ρ1 , ω1 ) + v2 p(ρ2 , ω2 ) + q) − e(v1 k(0, ω1 ) + v2 p(0, ω2 ) + q)| ≤ const for all ρ1 , ρ2 with |ρi | ≤ ε3 . Thus |e(v1 k(ρ1 , ω1 ) + v2 p(ρ2 , ω2 ) + q)| ≤ ε3

ε3 δ

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implies |e(v1 k(0, ω1 ) + v2 p(0, ω2 ) + q)| ≤ const and I2 (ε1 , ε2 , ε3 ) ≤ const ε1 ε2 with

Z

µ ¶ ε3 1 W const δ2 δ

Z

W (ζ) = sup q∈Kq

ε3 δ

dω1 S1

dω2 1(|e(v1 k(0, ω1 ) + v2 p(0, ω2 ) + q)| ≤ ζ). S2

We claim that |∇e(v1 k(0, ω1 ) + v2 p(0, ω2 ) + q)| ≥ const δ for all ω1 ∈ S1 and ω2 ∈ S2 . This will be used in the proof of the following Lemma, which generalizes [13, Lemma A.1] and which implies the bound (10). We have assumed that Kk , Kp ˜ p ˜ and q ˜ , respectively. and Kq are contained in small balls or annuli centered on k, ˜ + v2 p ˜+q ˜ is a regular point, simple continuity yields that |∇e(v1 k(0, ω1 ) + If v1 k v2 p(0, ω2 )+q)| ≥ const provided we choose c small enough. So it suffices to consider ˜ + v2 p ˜ +q ˜ is a singular point. the case that r = v1 k The constraint |ρ1 | < ε1 ensures that |k(ρ1 , ω1 ) − k(0, ω1 )| ≤ const εδ1 and the constraint |ρ2 | < ε2 ensures that |k(ρ2 , ω2 ) − k(0, ω2 )| ≤ const εδ2 . So the original condition that |∇e(v1 k(ρ1 , ω1 ) + v2 p(ρ2 , ω2 ) + q)| ≥ δ implies that |v1 k(ρ1 , ω1 ) + v2 p(ρ2 , ω2 ) + q − r| ≥ const δ and hence |v1 k(0, ω1 ) + v2 p(0, ω2 ) + q − r| ≥ const δ − √ √ provided δ ≥ const max{ ε1 , ε2 }. So

ε1 ε2 − ≥ const δ δ δ

|∇e(v1 k(0, ω1 ) + v2 p(0, ω2 ) + q)| ≥ const δ as desired. Lemma 3.8. W (ζ) ≤ Z3 1δ ζ ² where ² =

κ 1+κ .

Proof. Let γ ∈ (0, 1), T = {(ω1 , ω2 ) ∈ F × F |

p

1 − (n(ω1 ) · n(ω2 ))2 ≥ ζ 1−γ }

be the set where the intersection is transversal and E = F × F\T its complement. We shall choose γ at the end. Split W (ζ) = T (ζ) + E(ζ) into the contributions from these two sets. The contribution from the set of exceptional momenta E is bounded using Hypothesis NN. For each ω1 ∈ S1 , let p Eω1 = {ω2 ∈ S2 | 1 − (n(ω1 ) · n(ω2 ))2 < ζ 1−γ }.

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Then by Hypothesis NN Z Z E(ζ) ≤ dω1 S1

Eω1 ∩S2

Z dω1 Z0 ζ κ(1−γ) = const ζ κ(1−γ) .

dω2 ≤ S1

Now we bound T . We start by introducing a cover of F by coordinate patches. ˜ of F, O ˜ be the open neighborhood of k ˜ that is the Let, for each singular point k k image of {|x| < 1} under the Morse diffeomorphism k(x). If e(k(x)) = x21 + · · · + x2m − x2m+1 − · · · − x2d √ write x = (rθ 1 , rθ 2 ) with 0 ≤ r ≤ 1/ 2, θ 1 ∈ S m−1 and θ 2 ∈ S d−m−1 . Introduce “roughly orthonormal” coordinate patches on S m−1 . Here is what we mean by the statement that θ 1 (α1 , . . . , αm−1 ) is “roughly orthonormal”. Let · ¸ ∂θ 1 ∂θ 1 A(α1 , . . . , αm−1 ) = (α1 , . . . , αm−1 ), . . . , (α1 , . . . , αm−1 ) ∂α1 ∂αm−1 be the m × m − 1 matrix whose columns are the tangent vectors to the coordinate axes at θ 1 (α1 , . . . , αm−1 ). The columns of this matrix span the tangent space to S m−1 at θ 1 (α1 , . . . , αm−1 ). Let V (α1 , . . . , αm−1 ) be an (m − 1) × (m − 1) matrix such that the columns of A(α1 , . . . , αm−1 )V (α1 , . . . , αm−1 ) are mutually orthogonal unit vectors. Those columns form an orthonormal basis for the tangent space to S m−1 at θ 1 (α1 , . . . , αm−1 ). “Roughly orthogonal” signifies that V and its inverse are uniformly bounded on the domain of the coordinate patch. The only consequence of rough orthonormality that we will use is that, if v is any vector in the tangent space to S m−1 at θ 1 (α1 , . . . , αm−1 ), then, because kvk = kv t [A(α1 , . . . , αm−1 )V (α1 , . . . , αm−1 )]k ≤ kv t A(α1 , . . . , αm−1 )kkV (α1 , . . . , αm−1 )k implies kv tA(α1 , . . . , αm−1 )k ≥ kV (α1 , . . . , αm−1 )k−1 kvk we have

¯ ¯ ¯ ∂θ 1 ¯ 1 ¯ max v · (α1 , . . . , αm−1 )¯¯ ≥ √ kV (α1 , . . . , αm−1 )k−1 kvk. 1≤j≤m−1 ¯ ∂αj m−1

(11)

Also introduce a “roughly orthonormal” coordinate patch θ 2 (αm , . . . , αd−2 ) on S and parametrize (a patch on) the cone x21 +· · ·+x2m −x2m+1 −· · ·−x2d = 0 by d−m−1

x(α1 , . . . , αd−1 ) = (αd−1 θ 1 (α1 , . . . , αm−1 ), αd−1 θ 2 (αm , . . . , αd−2 )) and the corresponding patch on Ok˜ by k(x(α1 , . . . , αd−1 )). Denote ω1 (α1 , . . . , αd−1 ) = k(x(α1 , . . . , αd−1 )).

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For patches away from the singular points, any roughly orthonormal coordinate systems will do. Observe that, if v is any vector in the tangent space to F at ω1 (α1 , . . . , αd−1 ), then ( ¯ ¯ ¯ ¯ ∂ω const regular patch max ¯¯v · (α1 , . . . , αd−1 )¯¯ ≥ kvk (12) 1≤j≤d−1 ∂αj const αd−1 singular patch. Now fix any q ∈ Kq and consider the contribution to ZZ dω1 dω2 1(|e(v1 k(0, ω1 ) + v2 p(0, ω2 ) + q)| ≤ ζ) S1 ×S2 ∩T

from one pair, ω1 (α1 , . . . , αd−1 ) and ω2 (β1 , . . . , βd−1 ), of coordinate patches as ω1 described above. The Jacobian ∂∂α1 ···∂α is bounded by a constant, in the regular d−1 d−2 case, and a constant times αd−1 , in the singular case. Denote by θ(ω1 , ω2 ) the angle between n(ω1 ) and n(ω2 ). By the transversality condition, sin θ(ω1 , ω2 ) ≥ ζ 1−γ . Consequently, for at least one i ∈ {1, 2} the sine of the angle between n(ωi ) and ∇e(v1 k(0, ω1 ) + v2 p(0, ω2 ) + q) is at least

1 1 1 sin θ(ω1 , ω2 ) ≥ sin θ(ω1 , ω2 ) ≥ ζ 1−γ 2 2 2 and the length of the projection of ∇e(v1 k(0, ω1 ) + v2 p(0, ω2 ) + q) on Tωi F must be at least 21 ζ 1−γ |∇e(v1 k(0, ω1 ) + v2 p(0, ω2 ) + q)| ≥ const δ ζ 1−γ . Suppose that i = 1. Define ρ = e(v1 k(0, ω1 ) + v2 p(0, ω2 ) + q) viewed as a function of α1 , . . . , αd−1 and β1 , . . . , βd−1 . By (12), there must be a 1 ≤ j ≤ d − 1 such that ¯ ¯ ¯ ¯ ¯∂ ρ ¯ ¯ ¯ ¯ ¯ = ¯∇e(v1 k(0, ω1 ) + v2 p(0, ω2 ) + q) · ∂ ω (α1 , . . . , αd−1 )¯ ¯ ∂αj ¯ ¯ ¯ ∂αj ( const regular patch ≥ const δ ζ 1−γ const αd−1 singular patch. Make a final change of variables replacing αj by ρ. The Jacobian for the composite change of variables from (ω1 , ω2 ) to (α1 , . . . , αd−1 , β1 , . . . , βd−1 ) and then to ((αi ) 1≤i≤d − 1 , (βi )1≤i≤d−1 , ρ) is bounded by i6=j

1 const ζ γ−1 δ

( const const

regular patch d−3 αd−1

singular patch

We thus have 1 T (ζ) ≤ const ζ γ−1 δ

Z

)

1 ≤ const ζ γ−1 . δ

ζ

1 dρ ≤ const ζ γ . δ −ζ

The optimal bound is when κ(1 − γ) = γ, that is, γ = κ/(1 + κ).

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3.3. The proof of Theorem 1.1 Proof of Theorem 1.1. Now that we have Proposition 3.6, the proof of Theorem 1.1 is almost identical to the corresponding proofs of [13]. The main change is that our current choice of localization operator simplifies the argument. Several proofs in this paper and its companion paper [12] are variants of the arguments of [13]. So we have provided, in Appendix A, a complete, self-contained proof that the value, G(q), of each renormalized 1PI, two-legged graph is C 1−ε , using the simplest form of the argument in question. In particular, it does not use “volume improvement” bounds like Proposition 3.6. We here show how to use Proposition 3.6 to upgrade C 1−ε to C 1+ε . This is a good time to read that Appendix, since we shall just explain the modifications to be made to it. As in Appendix A, use (22) to introduce a scale expansion for each propagator and express G(q) in terms of a renormalized tree expansion (24). We shall prove, by induction on the depth, D, of GJ , the bound X sup |∂qs00 ∂qs1 GJ (q)| J∈J (j,t,R,G)

q

≤ constn |j|

3n−2

M

(1−s0 −s1 )j

( M εj

if s0 + s1 ≥ 1

1

if s0 = s1 = 0

(13)

for s0 , s1 ∈ {0, 1, 2}. Here ε was specified in Proposition 3.6 and the other notation is as in Appendix A: n is the number of vertices in G and J (j, t, R, G) is the set of all assignments J of scales to the lines of G that have root scale j, that give forest t and that are compatible with the assignment R of renormalization labels to the two-legged forks of t. (This is explained in more detail just before (24).) If s0 +s1 = 1, the right-hand side becomes constn |j|3n−2 M εj , which is summable over P P P j < 0, implying that G(q) = R j<0 J∈J (j,t,R,G) GJ (q) is C 1 . To show that the first order derivatives of G(q) are H¨older continuous of any degree strictly less than ε, just observe that if kfj k∞ ≤ constn |j|3n−2 M εj

and

kfj0 k∞ ≤ constn |j|3n−2 M εj M −j

then |fj (x) − fj (y)| ≤ min{2kfj k∞ , kfj0 k∞ |x − y|} ≤ constn |j|3n−2 M εj min{2, M −j |x − y|} ≤ constn |j|3n−2 M εj M −ηj |x − y|η

for any 0 ≤ η ≤ 1

is summable over j < 0 for any 0 < η < ε. If s0 = s1 = 0, (13) is contained in Proposition A.1, so it suffices to consider s0 + s1 ≥ 1. As in Appendix A, if D > 0, decompose the tree t into a pruned tree t˜ and insertion subtrees τ 1 , . . . , τ m by cutting the branches beneath all minimal Ef = 2 forks f1 , . . . , fm . In other words each of the forks f1 , . . . , fm is an Ef = 2 fork having no Ef = 2 forks, except φ, below it in t. Each τi consists of the fork fi and all of t that is above fi . It has depth at most D − 1 so the corresponding

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subgraph Gfi obeys (13). Think of each subgraph Gfi as a generalized vertex in the ˜ = G/{Gf , . . . , Gf }. Thus G ˜ now has two as well as four-legged vertices. graph G 1 m P These two-legged vertices have kernels of the form Ti (k) = jf ≤jπ(f ) `Gfi (k) when i i P fi is a c-fork and of the form Ti (k) = jf >jπ(f ) (1l − `)Gfi (k) when fi is an r-fork. i i At least one of the external linesa of Gfi must be of scale precisely jπ(fi ) so the momentum k passing through Gfi lies in the support of Cjπ(fi ) . In the case of a c-fork f = fi we have, as in (27) and using the same notation, by the inductive hypothesis, ¯ 0 ¯ X X ¯ s ¯ J sup ¯∂k1 `Gf f (k)¯ jf ≤jπ(f ) Jf ∈J (jf ,tf ,Rf ,Gf ) k

≤

X

0

constnf |jf |3nf −2 M jf M −s1 (1−ε)jf

jf ≤jπ(f ) 0

≤ constnf |jπ(f ) |3nf −2 M jπ(f ) M −s1 (1−ε)jπ(f ) (14) for s01 = 0, 1. Note that the sum in the analog of (14) diverges when s01 = 2, so J it is essential that no more than one derivative act on any c-fork. As `Gf f (k) is independent of k0 , derivatives with respect to k0 may not act on it. In the case of an r-fork f = fi , we have, as in (29), using the mean value theorem in the case s00 = 0, X X J s0 s0 sup 1(Cjπ(f ) (k) 6= 0)|∂k00 ∂k1 (1l − `)Gf f (k)| jf >jπ(f ) Jf ∈J (jf ,tf ,Rf ,Gf ) k

≤

X

X

0

max{1,s00 } s01 Jf ∂k Gf (k)|

M (1−min{1,s0 })jπ(f ) sup |∂k0 k

jf >jπ(f ) Jf ∈J (jf ,tf ,Rf ,Gf ) 0

≤ constnf M (1−min{1,s0 })jπ(f )

X

0

0

|jf |3nf −2 M −(max{1,s0 }+s1 −1−ε)jf

jf >jπ(f ) 0

0

≤ constnf |jπ(f ) |3nf −2 M jπ(f ) M −s0 jπ(f ) M −s1 jπ(f ) .

(15)

˜ of the scale assignment J. We bound G ˜ J˜, Denote by J˜ the restriction to G which again is of the form (31), by a variant of the six step procedure followed in Appendix A. In fact the first five steps are almost identical. ˜ with the property that T˜ ∩ G ˜ J˜ is a connected 1. Choose a spanning tree T˜ for G f ˜ J˜). tree for every f ∈ t(G 2. Apply any q-derivatives. By the product rule each derivative may act on any line or vertex on the “external momentum path”. It suffices to consider any one a Note that the root fork, ∅, of (24) does not carry an r, c label so that G ˜ may not be simply a single two-legged c- or r-vertex. At least one external line of each Gfi must be an internal line ˜ of G.

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such action. Ensure, through a judicious use of integration by parts, that at most one derivative acts on any single c-fork. To do so, observe that a derivative with respect to the external momentum acting on a c-fork is, up to a sign, equal to the derivative with respect to any loop momentum that flows through the fork. So replace one external momentum derivative by a loop momentum derivative and integrate by parts to move the latter off of the c-fork. 3. Bound each two-legged renormalized subgraph (i.e. r-fork) by (15) and each two-legged counterterm (i.e. c-fork) by (14). Observe that when s00 k0 derivatives and s01 k-derivatives act on the vertex, the bound is no worse than 0 0 M −s0 jπ(f ) M −s1 jπ(f ) times the bound with no derivatives. (We shall not need the 0 factor M s1 εjπ(f ) in (14). So we simply discard it.) As we have already observed, one of the external lines of the two-legged vertex must be of scale precisely jπ(f ) . 0 0 0 0 We write M −s0 jπ(f ) M −s1 jπ(f ) = M −s0 j` M −s1 j` , where ` is that line. 4. Bound all of the remaining vertex functions, (suitably differentiated) by their suprema in momentum space. We have already observed that if s0 = s1 = 0, the target bound (13) is contained in Proposition A.1, with s0 = s = 0. In the event that s0 + s1 ≥ 1, but all derivatives act on four-legged vertex functions, Proposition A.1, again with s0 = s = 0 but with one or two four-legged vertex functions replaced by differentiated functions, again gives (13). So it suffices to consider the case that at least one derivative acts on a propagator or on a c- or r-fork. 5. Bound each propagator s0

s0

0

0

|∂k00 ∂k1 Cj` (k)| ≤ const M −(1+s0 +s1 )j` 1(|ik0 − e(k)| ≤ M j` ). s00

(16)

s01

Once again, when k0 -derivatives and k-derivatives act on the propagator, 0 0 the bound is no worse than M −(s0 +s )j` times the bound with no derivatives. ˜ J˜(q)| bounded, uniformly in q by We now have |∂qs00 ∂qs1 G constn

Y

M −j`d

×

|jπ(fi ) |3nfi −2 M jπ(fi )

i=1

d

Z

m Y

Y ˜ T˜ `∈G\

dk ¯ `

Y

Y

M −j`

˜ `∈G

1(|ik`0 − e(k` )| ≤ M j` ).

(17)

˜ `∈G

Here d runs over the s0 + s1 ≥ 1 derivatives in ∂qs00 ∂qs1 and `d refers to the specific line on which the derivative acted (or, in the case that the derivative acted on a c- or r-fork, the external line specified in step 3). For ` ∈ T˜, the momentum k` is a signed sum of the loop momenta and external momentum flowing through `. In Appendix A, we discarded the factors of the Q integrand `∈G˜ 1(|ik`0 − e(k` )| ≤ M j` ) with ` ∈ T˜ at this point. Then the integrals over the loop momenta factorized and we bounded them by the volumes of their domains of integration, using Lemma 2.3. We now deviate from the argument of Appendix A by exploiting the constraint that one factor 1(|ik`0 − e(k` )| ≤ M j` ) with ` ∈ T˜ imposes on the domain of integration.

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We have reduced consideration to cases in which at least one derivative with respect to the external momentum acts either on a propagator in T˜ or on a twolegged c- or r-vertex in T˜, so that the associated line `d ∈ T˜. Select any such `d0 . ˜ T˜ is associated with a loop Λ` that consists of ` and Recall that any line ` ∈ G\ ˜ the linear subtree of T joining the vertices at the ends of `. By [12, Lemma 4.3], ˜ T˜ such that `d ∈ Λ` ∩ Λ` . By [12, Lemma 4.4], there exist two lines `1 , `2 ∈ G\ 0 1 2 Q j`1 , j`2 ≤ j`d0 . Now discard all of the factors `∈G˜ 1(|ik`0 − e(k` )| ≤ M j` ) in the integrand of (17) with ` ∈ T˜\{`d0 }. Choose the order of integration in (17) so that k`1 and k`2 are integrated first. By Proposition 3.6, Z Y Y dk ¯ ` 1(|ik`0 − e(k` )| ≤ M j` ) ≤ const M 2j`1 M 2j`2 M εjd0 . `∈{`1 ,`2 }

`∈{`1 ,`2 ,`d0 }

(18) ˜ T˜ ∪{`1 , `2 }) as in step Finally, integrate over the remaining loop momenta k` , ` ∈ G\( 6 of Proposition A.1. The integral over each such k` is bounded by vol{k0` | |k0` | ≤ M j` } ≤ 2M j` times the volume of {k` | |e(k` )| ≤ M j` }, which, by Lemma 2.3, ˜ J˜(q)| bounded, is bounded by a constant times |j` |M j` . We now have |∂qs00 ∂qs1 G uniformly in q, by constn M

εj`d

Y 0

M −j`d

m Y

|jπ(fi ) |3nfi −2 M jπ(fi )

i=1

d

Y

M −j`

˜ `∈G

Y

|j` |M 2j` .

˜ T˜ `∈G\

For every derivative d, j`d ≥ j = jφ , so that Y Y εj −(1−ε)j`d 0 M `d0 M −j`d = M M −j`d ≤ M −s0 j−s1 j M εj . d

d6=d0

Bounding each |jπ(fi ) |3nfi −2 ≤ |jπ(fi ) |3nfi −1 , we come to the conclusion that ˜ J˜(q)| is bounded, uniformly in q, by |∂qs00 ∂qs1 G constn M −s0 j−s1 j M εj

m Y

|jπ(fi ) |3nfi −1 M jπ(fi )

i=1

Y ˜ `∈G

M −j`

Y

|j` |M 2j` .

(19)

˜ T˜ `∈G\

This is exactly M −s0 j−s1 j M εj times the bound (33)|s0 =s=0 of Appendix A. So (36)|s0 =s=0 of Appendix A now gives (13). This completes the proof that the value of each graph contributing to the self-energy is C 1+η in the external momentum, for every η strictly less than the ε of Proposition 3.6. We may also apply this technique to connected four-legged graphs. There is no need for an induction argument because we already have all of the needed bounds ˜ argument once. When we do on c- and r-forks. We just need to go through the G so, there are three changes: 1

• The overall power counting factor M 2 j(4−Eφ ) in (34), which took the value M j for two-legged graphs, now takes the value 1 for four-legged graphs.

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• We may only apply the overlapping loop bound (18) when we can find a line ˜ T˜ with `3 ∈ Λ` ∩ Λ` . By [13, Lemma 2.34], `3 ∈ T˜ and two lines `1 , `2 ∈ G\ 1 2 ˜ is overlapping, as defined in [13, Definition 2.19]. this is the case if and only if G By [13, Lemma 2.26], four-legged connected graphs fail to be overlapping if and only if they are dressed bubble chains, as defined in [13, Definition 2.24]. • To convert the M εj`3 from the overlapping loop integral (18) into the M εj that ˜ f and write we want in the final bound, we set f3 to the highest fork with `3 ∈ G 3 Y εj`3 εjf3 εj ε(jf −jπ(f ) ) M =M =M M . f ∈t˜ φ
The extra factors M ε(jf −jπ(f ) ) , which are all at least one, are easily absorbed by (35) provided Ef > 4 for all forks f between the root φ and f3 . We may choose `3 so that this is the case precisely when G is not a generalized ladder. To see ˆ J = GJ /{GJ | Ef = 2, 4} be the diagram G, but with both two- and this, let G f four-legged subdiagrams Gf viewed as generalized vertices. Then we can find a ˆ J is overlapping which in turn is the case if and only suitable `3 if and only if G J ˆ if G is not a dressed bubble chain, which in turn is the case, for all labelings J, if and only if G is not a generalized ladder. Thus when G is a connected four-legged graph, the right-hand side of (13) is replaced by a constant times a power of j times ( M εj if G is not a generalized ladder (−s0 −s1 )j M 1 if G is a generalized ladder for s0 , s1 ∈ {0, 1}. This implies that four-legged graphs, other than generalized ladders, are C η functions of their external momenta for all η strictly smaller than ε. For graphs G contributing to the higher correlation functions, we may once again repeat the same argument, but with s0 = s1 = 0 and without having to exploit overlapping loops, provided we use the L1 norm, rather than the L∞ norm, on the momentum space kernel of G. In [13], this norm was denoted | · |0 and was defined in (1.46). See [13, (2.27) and Theorem 2.47] for the proof. Denote by K(e, q) the counterterm function for the dispersion relation e(k) and by Cj (e, k) =

f (M −2j |ik0 − e(k)|2 ) ik0 − e(k)

the scale j propagator for the dispersion relation e(k). Observe that, for all j` < 0 and s00 ∈ {0, 1}, ¯ ¯ ¯ ¯ ∂ s0 ¯ ¯ ¯ ∂ 0 Cj (e + th, k)¯ ¯ ≤ constkhk∞ M −(2+s00 )j` 1(|ik0 − e(k)| ≤ M j` ). ¯ ∂t k0 ` ¯ ¯ t=0 Thus the effect of a directional derivative with respect to the dispersion relation in direction h is to multiply (16) by khk∞ M −j` , which is khk∞ times the effect of a

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∂k0 derivative. So the same argument that led to (13) gives ¯ ¯ ¯ X ¯ ¯ ¯∂ supq ¯¯ ∂qs00 GJ (e + th, q)¯¯ ¯¯ ≤ constn |j|3n−2 M −s0 j M εj khk∞ ∂t t=0 J∈J (j,t,R,G)

for s0 ∈ {0, 1}. When s0 = 0, this is summable over j < 0 so that ¯ " R ¯ #¯¯ ¯∂ X ¯ ¯ ¯ r sup ¯ λ Kr (e + th, q)¯¯ ¯ ≤ constdKde |λ|khk∞ . ¯ q ¯ ∂t t=0 r=1

(20)

The constant constdKde = constdKde (e, v) depends on R and the various parameters in the hypotheses imposed by Theorem 1.1 on the dispersion relation e and twobody interaction v, like the C 3 norm of e, the eigenvalues of the Hessian of e at singular points, the C 2 norm of v and the constants Z0 , β0 and κ of Hypothesis NN. Fix a two-body interaction v and a constant A > 0. Denote by EA the set of dispersion relations such that constdKde (e, v) ≤ A. If the dispersion relations e, e0 and all interpolants (1 − t)e + te0 , 0 ≤ t ≤ 1 are in EA , and if |λ| < A1 , then e+

R X r=1

r

0

λ Kr (e) = e +

R X

λr Kr (e0 ) =⇒ e = e0 .

r=1

(21) ¤

Acknowledgment The first-named author was supported by NSERC of Canada. The second-named author was supported by DFG-grant Sa-1362/1-1, an ESI senior research fellowship, and NSERC of Canada. Appendix A. Bounding General Diagrams — A Review For the convenience of the reader, we here provide a review of the general diagram bounding technique of [13]. As a concrete example of the technique, we consider models in d ≥ 2 for which the interaction v has C 1 Fourier transform and the dispersion relation e and its Fermi surface F = {k | e(k) = 0} obey: H10 {k | |e(k)| ≤ 1} is compact; H20 e(k) is C 1 ; ˜ = 0 and ∇e(k) ˜ = 0 simultaneously only for finitely many k’s, ˜ called H30 e(k) singular points; ˜ is a singular point then [ ∂ 2 ˜ H40 If k ∂ki ∂kj e(k)]1≤i,j≤d is nonsingular; and we prove that, any graph contributing to the proper self-energy is C s for any s < 1. Note that, in this appendix, we do not require the no-nesting condition of Hypothesis NN. The same methods apply to graphs with more than two legs as well.

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Let G be any two-legged 1PI graph. We also use the symbol G to stand for the value of the graph G. Singularities of the Fermi surface have no influence on the ultraviolet regime, so we introduce a fixed ultraviolet cutoff by choosing a compactly supported C ∞ function U (k) that is identically one on a neighborhood of {0} × F (k) and use the propagator C(k) = ik0U−e(k) . If M > 1 and f is a suitable C0∞ function −4 that is supported on [M , 1], we have the partition of unity [13, §2.1] X U (k) = f (M −2j |ik0 − e(k)|2 ) j<0

and hence C(k) =

X

Cj (k)

where

Cj (k) =

j<0

f (M −2j |ik0 − e(k)|2 ) . ik0 − e(k)

(22)

Note that f (M −2j |ik0 − e(k)|2 ) and Cj (k) vanish unless M j−2 ≤ |ik0 − e(k)| ≤ M j . First, suppose that G is not renormalized. Expand each propagator of G using P C = j<0 Cj to give G=

X

GJ .

J

The sum runs over all possible labelings of the graph G, with each labeling consisting of an assignment J = {j` < 0 | ` ∈ G} of scales to the lines of G. We now construct a natural hierarchy of subgraphs of GJ . This family of subgraphs will be a forest, meaning that if Gf , Gf 0 are in the forest and intersect, either Gf ⊂ Gf 0 or Gf 0 ⊂ Gf . First let, for each j < 0, G(≥j) = {` ∈ GJ | j` ≥ j} be the subgraph of GJ consisting of all lines of scale at least j. (Think of an interaction line as a generalized four-legged vertex

rather than a line.) There is no need for G(≥j) to be connected. The forest t(GJ ) is the set of all connected subgraphs of GJ that are components of some G(≥j) . This forest is naturally partially ordered by containment. In order to make t(GJ ) look like a tree with its root at the bottom, we define, for f, f 0 ∈ t(GJ ), f > f 0 if Gf ⊂ Gf 0 . We denote by π(f ) the highest fork of t(GJ ) below f and by φ the root element, i.e. the element with Gφ = G. To each Gf ∈ t(G) there is naturally associated the scale jf = min{j` | ` ∈ Gf }. In the example below j4 > j3 > j1 and j2 > j1 . External lines are in gray while internal lines are in black.

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j3

j3 j4

Gφ = j1

265

j4

Gf4 =

j4

Gf3 =

j1 j 1

f4

j2

Gf2 = j2

t(GJ ) =

j2

f3 φ

j2

Reorganize the sum over J using X X Y 1 G= bf ! j<0 t∈F (G) f ∈t

f2 .

X

GJ

(23)

J∈J (j,t,G)

where F(G) = the set of forests of subgraphs of G, bf = the number of upward branches at the fork f , J (j, t, G) = {labelings J of G | t(GJ ) = t, jφ = j}. A given labeling J of G is in J (j, t, G) if and only if • for each f ∈ t, all lines of Gf \ ∪ f 0 ∈t Gf 0 have the same scale. Call the common f 0 >f scale jf ; • if f > f 0 then jf > jf 0 ; • jφ = j. It is a standard result [13, (2.72)] that renormalization of the dispersion relation may be implemented by modifying (23) as follows: • Each ∅ 6= f ∈ t for which Gf has two external lines is assigned a “renormalization label”. This label can take the values r and c. The set of possible assignments of renormalization labels, i.e. the set of all maps from {f ∈ t | Gf has two external legs} to {r, c}, is denoted R(t). • In the definition of the renormalized value of the graph G, the value of each subgraph Gf with renormalization label r is replaced by (1l − `)Gf (k). Here ` is the localization operator, which we takeb to be simply evaluation at k0 = 0. For these r-forks, the constraint jf > jπ(f ) still applies. b The

(1l−`)G (k)

f main property that the localization operator should have is that ik −e(k) should be 0 bounded for any (sufficiently smooth) Gf . Here is another possible localization operator for d = 2. In a neighborhood of a regular point of the Fermi surface, define `Gf (k) = Gf (k0 = 0, P k) where P k is any reasonable projection of k onto the Fermi surface, as in [13, Sec. 2.2]. In a neighborhood of a singular point, use a coordinate system in which e(x, y) = xy and, in this coordinate system, define `Gf (k0 , x, y) = Gf (0, x, 0) + Gf (0, 0, y) − Gf (0, 0, 0). Use a partition of unity to patch the different neighborhoods together.

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• In the definition of the renormalized value of the graph G, the value of each subgraph Gf with renormalization label c is replaced by `Gf (k). For these c forks the constraint jf > jπ(f ) is replaced by jf ≤ jπ(f ) . Given a graph G, a forest t of subgraphs of G and an assignment R of renormalization labels to the two-legged forks of t, we define J (j, t, R, G) to be the set of all assignments of scales to the lines of G obeying: • for each f ∈ t, all lines of Gf \ ∪ f 0 ∈t Gf 0 have the same scale. Call the common f 0 >f scale jf ; • if Gf is not two-legged then jf > jπ(f ) ; • if Gf is two-legged and Rf = r, then jf > jπ(f ) ; • if Gf is two-legged and Rf = c then jf ≤ jπ(f ) ; • jφ = j. Then, the value of the graph G with all two-legged subdiagrams correctly renormalized is G=

X

X Y 1 X bf !

j<0 t∈F (G) f ∈t

X

GJ .

(24)

R∈R(t) J∈J (j,t,R,G)

To derive bounds on G, when we are not interested in the dependence of those bounds on G and in particular on the order of perturbation theory, it suffices to P P derive bounds on j≤0 J∈J (j,t,R,G) GJ for each fixed t and R. Proposition A.1. Assume that the interaction has C 1 Fourier transform and the dispersion relation obeys H10 –H40 above. Let G be any two-legged 1PI graph of order n. Let t be a tree corresponding to a forest of subgraphs of G. Let R be an assignment of r, c labels to all forks f > φ of t for which Gf is two-legged. Let J (j, t, R, G) be the set of all assignments of scales to the lines of G that have root scale j and are consistent with t and R. Let s ∈ (0, 1). Then there is a constant const, depending on s but independent of j, such that X sup |GJ (q)| ≤ constn |j|3n−2 M j , J∈J (j,t,R,G)

X J∈J (j,t,R,G)

X J∈J (j,t,R,G)

sup q,p

q

sup |∂q0 GJ (q)| ≤ constn |j|3n−2 , q

1 |GJ (q + p) − GJ (q)| ≤ constn |j|3n−2 M (1−s)j . |p|s

Remark A.2. Note that here the root scale is not summed over and Gφ is not renormalized. But all internal scales are summed over and internal two-legged subgraphs that correspond to r and c forks are renormalized and localized, respectively.

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Proof. The proof is by induction on the depth of the graph, which is defined by D = max{n | ∃ forks f1 > · · · > fn > φ with Gf1 , . . . , Gfn all two-legged}. The inductive hypothesis is that X

sup |∂qs00 GJ (q)| ≤ constn |j|3n−2 M (1−s0 )j ,

J∈J (j,t,R,G)

X

sup

J∈J (j,t,R,G)

q,p

q

1 |∂ s0 GJ (q + p) − ∂qs00 GJ (q)| ≤ constn |j|3n−2 M (1−s−s0 )j , |p|s q0

for s0 = 0, 1 and all s ∈ (0, 1) (with the constant depending on s). If D > 0, decompose the tree t into a pruned tree t˜ and insertion subtrees 1 τ , . . . , τ m by cutting the branches beneath all minimal Ef = 2 forks f1 , . . . , fm . In other words each of the forks f1 , . . . , fm is an Ef = 2 fork having no Ef = 2 forks, except φ, below it in t. Each τi consists of the fork fi and all of t that is above fi . It has depth at most D − 1 so the corresponding subgraph Gfi obeys the conclusion of this proposition. Think of each subgraph Gfi as a generalized vertex ˜ = G/{Gf , . . . , Gf }. Thus G ˜ now has two- as well as four-legged in the graph G 1 m vertices. These two-legged vertices have kernels of the form X Ti (k) = `Gfi (k) (25) jfi ≤jπ(fi )

when fi is a c-fork and of the form X Ti (k) =

(1l − `)Gfi (k)

(26)

jfi >jπ(fi )

when fi is an r-fork. At least one of the external lines of Gfi must be of scale precisely jπ(fi ) so the momentum k passing through Gfi lies in the support of Cjπ(fi ) . In the case of a c-fork f = fi we have, by the inductive hypothesis, X X J sup |`Gf f (k)| jf ≤jπ(f ) Jf ∈J (jf ,tf ,Rf ,Gf ) k

≤

X

X

J

sup |Gf f (k)|

jf ≤jπ(f ) Jf ∈J (jf ,tf ,Rf ,Gf ) k

≤

X

constnf |jf |3nf −2 M jf

jf ≤jπ(f )

≤ constnf M jπ(f )

X

(|jπ(f ) | + i)3nf −2 M −i

i≥0

≤ constnf |jπ(f ) |3nf −2 M jπ(f )

X

(i + 1)3nf −2 M −i

i≥0

≤ constnf |jπ(f ) |3nf −2 M jπ(f ) .

(27)

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Here nf is the number of vertices of Gf and tf and Rf are the restrictions of t and R, respectively, to forks f 0 ≥ f . Hence Jf runs over all assignments of scales to the lines of Gf consistent with the original t and R and with the specified value of jf . Similarly, X X 1 J J sup s |`Gf f (k + p) − `Gf f (k)| k,p |p| jf ≤jπ(f ) Jf ∈J (jf ,tf ,Rf ,Gf )

≤ constnf |jπ(f ) |3nf −2 M (1−s)jπ(f ) .

(28)

J

Note that `Gf f (k) is independent of k0 so that ∂k0 may never act on it. In the case of an r-fork f = fi , we have |(1l − `)G(k)| = |G(k0 , k) − G(0, k)| ≤ |k0 | sup |∂k0 G(k)|. k

Hence, by the inductive hypothesis and, when s0 = 0, the mean value theorem, X X J sup 1(Cjπ(f ) (k) 6= 0)|∂ks00 (1l − `)Gf f (k)| jf >jπ(f ) Jf ∈J (jf ,tf ,Rf ,Gf ) k

≤

X

X

J

M (1−s0 )jπ(f ) sup |∂k0 Gf f (k)| k

jf >jπ(f ) Jf ∈J (jf ,tf ,Rf ,Gf )

≤ constnf M (1−s0 )jπ(f )

X

|jf |3nf −2

jf >jπ(f )

≤ constnf |jπ(f ) |3nf −1 M (1−s0 )jπ(f ) .

(29)

Similarly, for |k0 | ≤ M jπ(f ) , X X 1 J J sup s |∂ks00 (1l − `)Gf f (k + p) − ∂ks00 (1l − `)Gf f (k)| |p| k,p j >j f

π(f )

≤

Jf ∈J (jf ,tf ,Rf ,Gf )

X

X

M (1−s0 )jπ(f ) sup k,p

jf >jπ(f ) Jf ∈J (jf ,tf ,Rf ,Gf )

≤ constnf M (1−s0 )jπ(f )

X

1 J J |∂k G f (k + p) − ∂k0 Gf f (k)| |p|s 0 f

|jf |3nf −2 M −jf s

jf >jπ(f )

≤ constnf |jπ(f ) |3nf −2 M (1−s0 −s)jπ(f ) .

(30)

˜ J˜, where J˜ is the restriction of J to G. ˜ It is both We are now ready to bound G convenient and standard to get rid of the conservation of momentum delta functions ˜ J˜ by integrating out some momenta. Then, instead of having arising in the value of G one (d + 1)-dimensional integration variable k for each line of the diagram, there is one for each momentum loop. Here is a convenient way to select these loops. ˜ A spanning tree is a subgraph of G ˜ that is a Pick any spanning tree T˜ for G. ˜ ˜ T˜ the tree and contains all the vertices of G. We associate to each line ` of G\

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“internal momentum loop” Λ` that consists of ` and the unique path in T˜ joining the ends of `. The “external momentum path” is the unique path in T˜ joining the external legs. It carries the external momentum q. The loop Λ` carries momentum k` . The momentum k`0 of each line `0 ∈ T˜ is the signed sum of all loop and external momenta passing through `0 . ˜ J˜(q) is then The form of the integral giving the value of G Z Y Y Y dd+1 k J˜ ˜ uv where dk ¯ = G (q) = dk ¯ ` Cj` (k` ) . (31) (2π)d+1 v ˜ T˜ `∈G\

˜ `∈G

˜ The loops are labeled by the lines of G\ ˜ T˜. Here T˜ is any spanning tree for G. ˜ For each ` ∈ T , the momentum k` is a signed sum of loop momenta and external Q ˜ and uv is the vertex momentum q. The product v runs over the vertices of G function for v. If v is one of the original interaction vertices then uv is just v evaluated at the signed sum of loop and external momenta passing through v. If v is a two-legged vertex, then uv is given either by (25) or by (26). ˜ in six steps. We are now ready to bound G ˜ with the property that T˜ ∩ G ˜ J˜ is a connected 1. Choose a spanning tree T˜ for G f ˜ ˜ J ). T˜ can be built up inductively, starting with the smallest tree for every f ∈ t(G ˜ f , because, by construction, every G ˜ f is connected and t(G ˜ J ) is a subgraphs G forest. Such a spanning tree is illustrated below for the example given just before (23) with j4 > j3 > j1 , j2 > j1 . j3 j4 j1

j1 j 1

j2

j2 2. Apply any q-derivatives. By the product rule, or, in the case of a “discrete derivative”, the “discrete product rule” f (k + q)g(k 0 + q) − f (k)g(k 0 ) = [f (k + q) − f (k)]g(k 0 ) + f (k + q)[g(k 0 + q) − g(k 0 )], each derivative may act on any line or vertex on the “external momentum path”. It suffices to consider any one such action. 3. Bound each two-legged renormalized subgraph (i.e. r-fork) by (29), (30) and each two-legged counterterm (i.e. c-fork) by (27), (28). Observe that when s0 k0 -derivatives and s k-derivatives act on the vertex, the bound is no worse than M −(s0 +s)j times the bound with no derivatives, because we necessarily have j ≤ jπ(f ) < 0. 4. Bound all remaining vertex functions, uv , (suitably differentiated) by their suprema in momentum space.

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5. Bound each propagator |∂ks00 Cj` (k)| ≤ const M −(1+s0 )j` 1(|ik0 − e(k)| ≤ M j` ), 1 |∂ s0 Cj (k + p) − ∂ks00 Cj` (k)| ≤ const M −(1+s0 +s)j` . |p|s k0 `

(32)

Once again, when s0 k0 -derivatives and s k-derivatives act on the propagator, the bound is no worse than M −(s0 +s)j times the bound with no derivatives, because ˜ J˜(q)| and 1 s |∂ s0 G ˜ J˜(q + p) − we necessarily have j ≤ j` < 0. We now have |∂ks00 G k0 |p| ˜ J˜(q)| bounded, uniformly in q and p by ∂ s0 G k0

constn M −(s0 +s)j

m Y

|jπ(fi ) |3nfi −1 M jπ(fi )

i=1

Z

Y

×

˜ T˜ `∈G\

dk ¯ `

Y

M −j`

˜ `∈G

Y

1(|ik`0 − e(k` )| ≤ M j` )

˜ T˜ `∈G\

˜ J˜(q)| we may with s = 0 in the first case. We remark that for the bound on |∂ks00 G Q Q Q j` replace the `∈G\ ˜ T˜ in ˜ T˜ 1(|ik`0 − e(k` )| ≤ M ) by ˜ . These extra `∈G\ `∈G integration constraints are not used in the current naive bound, but are used in other bounds that exploit “overlapping loops”. ˜ T˜ is 6. Integrate over the remaining loop momenta. Integration over k` with ` ∈ G\ j` j` bounded by vol{k0` | |k0` | ≤ M } ≤ 2M times the volume of {k` | |e(k` )| ≤ M j` }, which, by Lemma 2.3, is bounded by a constant times |j` |M j` . ˜ J˜(q)| and The above six steps give that |∂ks00 G bounded, uniformly in q and p by ˜

B J = constn M −(s0 +s)j

m Y

s0 ˜ J˜ 1 |p|s |∂k0 G (q

|jπ(fi ) |3nfi −1 M jπ(fi )

i=1

Y

M −j`

˜ `∈G

˜ J˜(q)| are + p) − ∂ks00 G Y ˜ T˜ `∈G\

again with s = 0 in the first case. Define the notation ˜f , T˜f = number of lines of T˜ ∩ G ˜ f = number of internal lines of G ˜f , L nf = number of vertices of Gf , Ef = number of external lines of Gf , Ev = number of lines hooked to vertex v. Applying M αj` = M αjφ

Y f ∈t f >φ `∈Gf

M α(jf −jπ(f ) )

|j` |M 2j`

(33)

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to each M −j` and M 2j` and

Y

M jπ(fi ) = M jφ

271

M jf −jπ(f )

f ∈t˜ φ
Y

1

= M − 2 (Efi −4)jφ

1

M − 2 (Efi −4)(jf −jπ(f ) )

f ∈t˜ f >φ ˜f f i ∈G

˜ gives for each 1 ≤ i ≤ m (thinking of fi as a vertex of G) ˜

˜

˜

P

B J ≤ constn M −(s0 +s)j |j|Lφ −Tφ + ×

Y

M

(3nfi −1)

˜ f −2T˜f −P ˜ (jf −jπ(f ) )(L v∈G

˜

˜

M j(Lφ −2Tφ −

1 (Ev −4)) f 2

P

1 ˜ 2 (Ev −4)) v∈G

.

f ∈t˜ f >φ

The sums tices. As

P ˜ v∈G

and

P ˜f v∈G

run over two- as well as four-legged generalized ver-

  X 1 ˜f =  L Ev − Ef  2

and

T˜f =

˜f v∈G



1 4 − Ef + 2

X

(Ev − 4)

˜f v∈G

and we have ˜

˜

1−1

˜f v∈G

 ˜ f − 2T˜f = =⇒ L

X

˜

B J ≤ constn M −(s0 +s)j |j|Lφ −Tφ +

P

(3nfi −1)

Y

1

M 2 j(4−Eφ )

1

M 2 (jf −jπ(f ) )(4−Ef )

f ∈t˜ f >φ ˜

˜

= constn M −(s0 +s)j |j|Lφ −Tφ +

P

(3nfi −1)

Mj

Y

1

M 2 (jf −jπ(f ) )(4−Ef )

(34)

f ∈t˜ f >φ

since Eφ = 2. The scale sums are performed by repeatedly applying   if Ef = 4 |j| X 1 M 2 (jf −jπ(f ) )(4−Ef ) ≤ 1   if Ef > 4 jf M −1 jf >jπ(f )

(35)

˜ φ − 1 additional factors of |j| starting with the highest forks, and give at most L since ˜ J˜), f 6= φ} ≤ L ˜ φ − 1. #{f ∈ t(G

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Thus X

˜

˜

˜

P

B J ≤ constn |j|2Lφ −Tφ −1+

(3nfi −1)

M (1−s0 −s)j

˜ ˜ J∈J (j,t˜,G)

≤ constn |j|3n−2 M (1−s0 −s)j

(36)

˜ since, using n ˜ 4 to denote the number of four-legged vertices in G, ˜ φ − T˜φ − 1 + 2L

m X

(3nfi − 1)

i=1

m X 1 = 2 (4˜ n4 + 2m − 2) − (˜ n4 + m − 1) − 1 + 3 nfi − m 2 i=1

= 3˜ n4 + 3

m X

nfi − 2

i=1

= 3n − 2. This is the desired bound. Corollary A.3. Assume that the interaction has C 1 Fourier transform and the dispersion relation obeys H10 –H40 above. Let G(q) be any graph contributing to the proper self-energy. Then, for every 0 < s < 1, sup |G(q)| < ∞, q

1 sup s |G(q + p) − G(q)| < ∞. q,p |p| Proof. Both bounds are immediate from Proposition A.1. One merely has to sum over j, t and R. The bound on supq |G(q)| was also proven by these same methods in [31]. References [1] L. Van Hove, The occurrence of singularities in the elastic frequency distribution of a crystal, Phys. Rev. 89 (1953) 1189–1193. [2] M. Morse, Relations between the critical points of a real function of n independent variables, Trans. Amer. Math. Soc. 27 (1925) 345–396. [3] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, Vol. 74 (Springer-Verlag, 1989). [4] T. Koma and H. Tasaki, Decay of superconducting and magnetic correlations in oneand two-dimensional Hubbard models, Phys. Rev. Lett. 68 (1992) 3248–3251. [5] R. S. Markiewicz, A Survey of the Van Hove scenario for high Tc superconductivity with special emphasis on pseudogaps and stripes phases, J. Phys. Chem. Solids 58 (1997) 1179–1310. [6] C. J. Halboth and W. Metzner, Renormalization-group analysis of the twodimensional Hubbard model, Phys. Rev. B 61 (2000) 7364–7377.

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[7] C. J. Halboth and W. Metzner, d-wave superconductivity and Pomeranchuk instability in the two-dimensional Hubbard model, Phys. Rev. Lett. 85 (2000) 5162–5165. [8] C. Honerkamp, M. Salmhofer, N. Furukawa and T. M. Rice, Breakdown of the Landau–Fermi liquid in two dimensions due to Umklapp scattering, Phys. Rev. B 63 (2001) 035109. [9] C. Honerkamp and M. Salmhofer, Magnetic and superconducting instabilities of the Hubbard model at the Van Hove filling, Phys. Rev. Lett. 87 (2001) 187004. [10] A. Neumayr and W. Metzner, Renormalized perturbation theory for Fermi systems: Fermi surface deformation and superconductivity in the two-dimensional Hubbard model, Phys. Rev. B 67 (2003) 035112. [11] W. Metzner, D. Rohe and S. Andergassen, Soft Fermi surfaces and breakdown of Fermi-liquid behavior, Phys. Rev. Lett. 91 (2003) 066402. [12] J. Feldman and M. Salmhofer, Singular Fermi surfaces II. The two-dimensional case, Rev. Math. Phys. 20 (2008) 275–334. [13] J. Feldman, M. Salmhofer and E. Trubowitz, Perturbation theory around nonnested Fermi surfaces. I. Keeping the Fermi surface fixed, J. Stat. Phys. 84 (1996) 1209– 1336. [14] J. Feldman, M. Salmhofer and E. Trubowitz, Regularity of the moving Fermi surface: RPA contributions, Comm. Pure Appl. Math. 51 (1998) 1133–1246. [15] J. Feldman, M. Salmhofer and E. Trubowitz, Regularity of interacting nonspherical Fermi surfaces: The full self-energy, Comm. Pure Appl. Math. 52 (1999) 273–324. [16] J. Feldman, M. Salmhofer and E. Trubowitz, An inversion theorem in Fermi surface theory, Comm. Pure Appl. Math. 53 (2000) 135–1384. [17] J. Feldman, H. Kn¨ orrer and E. Trubowitz, Asymmetric Fermi surfaces for magnetic Schr¨ odinger operators, Comm. Partial Differential Equations 25 (2000) 319–336. [18] J. Feldman, J. Magnen, V. Rivasseau and E. Trubowitz, An infinite volume expansion for many fermion Green’s functions, Helv. Phys. Acta 65 (1992) 679–721. [19] M. Disertori and V. Rivasseau, Interacting Fermi liquid at finite temperature: Part I: Convergent attributions, Comm. Math. Phys. 215 (2000) 251–290. [20] J. Feldman, H. Kn¨ orrer and E. Trubowitz, Single scale analysis of many fermion systems, Part 1: Insulators, Rev. Math. Phys. 15 (2003) 949–993. [21] J. Feldman, H. Kn¨ orrer and E. Trubowitz, Single scale analysis of many fermion systems, Part 2: The first scale, Rev. Math. Phys. 15 (2003) 9995–1037. [22] J. Feldman, H. Kn¨ orrer and E. Trubowitz, Single scale analysis of many fermion systems, Part 3: Sectorized Norms, Rev. Math. Phys. 15 (2003) 1039–1120. [23] J. Feldman, H. Kn¨ orrer and E. Trubowitz, Single scale analysis of many fermion systems, Part 4: Sector counting, Rev. Math. Phys. 15 (2003) 1121–1169. [24] J. Feldman, H. Kn¨ orrer and E. Trubowitz, A two dimensional Fermi liquid, Part 1: Overview, Comm. Math. Phys. 247 (2004) 1–47. [25] J. Feldman, H. Kn¨ orrer and E. Trubowitz, A two dimensional Fermi liquid, Part 2: Convergence, Comm. Math. Phys. 247 (2004) 49–111. [26] J. Feldman, H. Kn¨ orrer and E. Trubowitz, A two dimensional Fermi liquid, Part 3: The Fermi surface, Comm. Math. Phys. 247 (2004) 113–177. [27] J. Feldman, H. Kn¨ orrer and E. Trubowitz, Particle-hole ladders (summary version), Comm. Math. Phys. 247 (2004) 179–194. [28] J. Feldman, H. Kn¨ orrer and E. Trubowitz, Convergence of perturbation expansions in fermionic models, Part 1: Nonperturbative bounds, Comm. Math. Phys. 247 (2004) 195–242.

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[29] J. Feldman, H. Kn¨ orrer and E. Trubowitz, Convergence of perturbation expansions in fermionic models, Part 2: Overlapping loops, Comm. Math. Phys. 247 (2004) 243–319. [30] S. Afchain, J. Magnen and V. Rivasseau, Renormalization of the 2-point function of the Hubbard Model at half-filling, Ann. Henri Poincar´e 6 (2005) 399–449. [31] D. Brox, Renormalization of many body fermion models with singular fermi surfaces, Thesis (M.Sc.), University of British Columbia (2005).

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Reviews in Mathematical Physics Vol. 20, No. 3 (2008) 275–334 c World Scientific Publishing Company °

SINGULAR FERMI SURFACES II. THE TWO-DIMENSIONAL CASE

JOEL FELDMAN∗,‡ and MANFRED SALMHOFER∗,†,§ ∗Mathematics

Department, The University of British Columbia, 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2 †Theoretische Physik, Universit¨ at Leipzig, Postfach 100920, 04009 Leipzig, Germany ‡[email protected] §[email protected]

Received 12 June 2007 Revised 3 December 2007 We consider many-fermion systems with singular Fermi surfaces, which contain Van Hove points where the gradient of the band function k 7→ e(k) vanishes. In a previous paper, we have treated the case of spatial dimension d ≥ 3. In this paper, we focus on the more singular case d = 2 and establish properties of the fermionic self-energy to all orders in perturbation theory. We show that there is an asymmetry between the spatial and frequency derivatives of the self-energy. The derivative with respect to the Matsubara frequency diverges at the Van Hove points, but, surprisingly, the self-energy is C 1 in the spatial momentum to all orders in perturbation theory, provided the Fermi surface is curved away from the Van Hove points. In a prototypical example, the second spatial derivative behaves similarly to the first frequency derivative. We discuss the physical significance of these findings. Keywords: Fermion systems; Fermi surface; Van Hove singularities; renormalization. Mathematics Subject Classification 2000: 81T15, 81T17, 81T08, 82D35, 82D40

Contents 1. Introduction

276

2. Main Results 279 2.1. Hypotheses on the dispersion relation . . . . . . . . . . . . . . . . . 279 2.2. Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 2.3. Heuristic explanation of the asymmetry . . . . . . . . . . . . . . . . 281 3. Fermi Surface 3.1. Normal form for e(k) near a singular point 3.2. Length of overlap estimates . . . . . . . . . 3.2.1. Length of overlap — Special case . . 3.2.2. Length of overlap — General case . 275

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4. Regularity 4.1. The gradient of the self-energy . . . . . . 4.1.1. The second order contribution . . 4.1.2. The general diagram . . . . . . . . 4.2. The frequency derivative of the self-energy

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6. Interpretation 6.1. Asymmetry and Fermi velocity suppression 6.2. Inversion problem . . . . . . . . . . . . . . . Appendix A. Interval Lemma . . . . . . . . . . . . Appendix B. Signs etc . . . . . . . . . . . . . . . .

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1. Introduction In this paper, we continue our analysis of (all-order) perturbative properties of the self-energy and the correlation functions in fermionic systems with a fixed nonnested singular Fermi surface. That is, the Fermi surface contains Van Hove points, where the gradient of the dispersion function vanishes, but satisfies a no-nesting condition away from these points. In a previous paper [1], we treated the case of spatial dimensions d ≥ 3. Here we focus on the two-dimensional case, where the effects of the Van Hove points are strongest. We have given a general introduction to the problem and some of the main questions in [1]. As discussed in [1], the no-nesting hypothesis is natural from a theoretical point of view, because it separates effects coming from the saddle points and nesting effects. Moreover, generically, nesting and Van Hove effects do not occur at the same Fermi level. In the following, we discuss those aspects of the problem that are specific to two dimensions. As already discussed in [1], the effects caused by saddle points of the dispersion function lying on the Fermi surface are believed to be strongest in two dimensions (we follow the usual jargon of calling the level set the Fermi “surface” even though it is a curve in d = 2). Certainly, the Van Hove singularities in the density of states of the noninteracting system are strongest in d = 2. As concerns many-body properties, we have shown in [1] that for d ≥ 3, the overlapping loop estimates of [11] carry over essentially unchanged, which implies

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differentiability of the self-energy and hence a quasiparticle weight (Z-factor) close to 1 to all orders in renormalized perturbation theory. In this paper, we show that for d = 2, there are more drastic changes. Namely, there is an asymmetry between the derivatives of the self-energy Σ(q0 , q) with respect to the frequency variable q0 and the spatial momentum q. We prove that the spatial gradient ∇Σ is a bounded function to all orders in perturbation theory if the Fermi surface satisfies a no-nesting condition. By explicit calculation, we show that for a standard saddle point singularity, even the second-order contribution ∂0 Σ2 (q0 , qs ) diverges as (log|q0 |)2 at any Van Hove point qs (if that point is on the Fermi surface). This asymmetric behavior is unlike the behavior in all other cases that are under mathematical control: in one dimension, both ∂0 Σ2 and ∂1 Σ2 diverge like log|q0 | at the Fermi point. This is the first indication for vanishing of the Z-factor and the occurrence of anomalous decay exponents in this model. The point is, however, that once a suitable Z-factor is extracted, Z∂1 Σ2 remains of order 1 in one dimension, while for two-dimensional singular Fermi surfaces, the p-dependent function Z(p)∇Σ(p) vanishes at the Van Hove points. In higher dimensions d ≥ 2, and with a regular Fermi surface fulfilling a no-nesting condition very similar to that required here, Σ is continuously differentiable both in q0 and in q. Thus it is really the Van Hove points on the Fermi surface that are responsible for the asymmetry. In the last section of this paper, we point out some possible (but as yet unproven) consequences of this behavior. Our analysis is partly motivated by the two-dimensional Hubbard model, a lattice fermion model with a local interaction and a dispersion relation k 7→ e(k) which, in suitable energy units, reads e(k) = −cos k1 − cos k2 + θ(1 + cos k1 cos k2 ) − µ.

(1)

The parameter µ is the chemical potential, used to adjust the particle density, and θ is a ratio of hopping parameters. As we shall explain now, the most interesting parameter range is µ ≈ 0 and 0 < θ < 1. The zeroes of the gradient of e are at (0, 0), (π, π) and at (π, 0), (0, π). The first two are extrema, and the last two are the saddle points relevant for Van Hove singularities (VHS). For µ = 0, both saddle points are on the Fermi surface. For θ = 1 the Fermi surface degenerates to the pair of lines {k1 = 0} ∪ {k2 = 0}, so we assume that θ < 1. For θ = 0 and µ = 0, the Fermi surface becomes the so-called Umklapp surface U = {k : k1 ± k2 = ±π}, which is nested since it has flat sides. This case has been studied in [2–4]. There, it was shown that for a local Hubbard interaction of strength λ, perturbation theory converges in the region of (β, λ) where |λ| is small and |λ|(log β)2 ¿ 1. We shall discuss this result further in Sec. 6. For 0 < θ < 1 the Fermi surface at µ = 0 has nonzero curvature away from the Van Hove points (π, 0) and (0, π). Viewed from the point (π, π), it encloses a strictly

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convex region (as a subset of R2 ). There is ample evidence that in the Hubbard model, it is the parameter range θ > 0 and electron density near to the Van Hove density (µ ≈ 0) that is relevant for high-Tc superconductivity (see, e.g. [5–10]). In this parameter region, an important kinematic property is that the two saddle points at (π, 0) and (0, π) are connected by the vector Q = (π, π), which has the property that 2Q = 0 mod 2πZ2 . This modifies the leading order flow of the fourpoint function strongly (Umklapp scattering, [7–10]). The bounds we discuss here hold both in presence and absence of Umklapp scattering. The interaction of the fermions is given by λˆ v , where λ is the coupling constant and vˆ is the Fourier transform of the two-body potential defining a density-density interaction. For the special case of the Hubbard model, two fermions interact only if they are at the same lattice point, so that vˆ(k) = 1. Despite the simplicity of the Hamiltonian, little is known rigorously about the low-temperature phase diagram of the Hubbard model, even for small |λ|. In this paper, we do perturbation theory to all orders, i.e. we treat λ as a formal expansion parameter. For a discussion of the relation of perturbation theory to all orders to renormalization group flows obtained from truncations of the RG hierarchies, see the Introduction of [1]. Although our analysis is motivated by the Hubbard model, it applies to a much more general class of models. In this paper, we shall need only that the band function e has enough derivatives, as stated below, and a similar condition on the interaction. In fact, the interaction is allowed to be more general than just a density-density interaction: it may depend on frequencies, as well as the spin of the particles. See [11–14] for details. As far as the singular points of e are concerned, we require that they are nondegenerate. The precise assumptions on e will be stated in detail below. We add a few remarks to put these assumptions into perspective. No matter if we start with a lattice model or a periodic Schr¨odinger operator describing Bloch electrons in a crystal potential, the band function given by the Hamiltonian for the one-body problem is, under very mild conditions, a smooth, even analytic function. In such a class of functions, the occurrence of degenerate critical points is nongeneric, i.e. measure zero. In other words, if e(ks + Rk) = −ε1 k12 + ε2 k22 + · · · around a Van Hove point ks (here R is a rotation that diagonalizes the Hessian at ks ), getting even one of the two prefactors εi to vanish in a Taylor expansion requires a fine-tuning of the hopping parameters, in addition to the condition that the VH points are on S. Thus, in a one-body theory, an extended VHS, where the critical point becomes degenerate because, say, ε1 vanishes, is nongeneric. On the other hand, experiments suggest [6] that ε1 is very small in some materials, which seem to be modeled well by Hubbard-type band functions. On the theoretical side, in a renormalized expansion with counterterms, it is not the dispersion relation of the noninteracting system, but that of the interacting system, which appears in all fermionic covariances. It is thus an important theoretical question to decide what

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effects the interaction has on the dispersion relation and in particular whether an extended VHS can be caused by the interaction. We shall discuss this question further in Sec. 6. 2. Main Results In this section we state our hypotheses on the dispersion function and the Fermi surface, and then state our main result. 2.1. Hypotheses on the dispersion relation We make the following hypotheses on the dispersion relation e and its Fermi surface F = {k | e(k) = 0} in d = 2. H1 {k | |e(k)| ≤ 1} is compact. H2 e(k) is C r with r ≥ 7. ˜ = 0 and ∇e(k) ˜ = 0 simultaneously only for finitely many k’s, ˜ called Van H3 e(k) Hove points or singular points.£ ¤ ∂2 ˜ is a singular point then ˜ H4 If k ∂ki ∂kj e(k) 1≤i,j≤d is nonsingular and has one positive eigenvalue and one negative eigenvalue. ˜ ∈ F. H5 There is at worst polynomial flatness. This means the following. Let k Suppose that k2 − k˜2 = f (k1 − k˜1 ) is a C r−2 curve contained in F in a neigh˜ (If k ˜ is a singular point, there can be two such curves.) Then borhood of k. some derivative of f (x) at x = 0 of order at least two and at most r − 2 does not vanish. Similarly if the roles of the first and second coordinates are exchanged. ˜ ∈ F H6 There is at worst polynomial nesting. This means the following. Let k ˜ ˜ ˜ ˜ ∈ F with k 6= p ˜ . Suppose that k2 − k2 = f (k1 − k1 ) is a C r−2 and p ˜ and k2 − p˜2 = g(k1 − p˜1 ) is curve contained in F in a neighborhood of k r−2 ˜ . Then some derivative a C curve contained in F in a neighborhood of p of f (x) − g(x) at x = 0 of order at most r − 2 does not vanish. Similarly if the roles of the first and second coordinates are exchanged. If e(k) is not even, we further assume a similar nonvanishing when f gives a curve in F in ˜ ∈ F and g gives a curve in −F in a neighborhood of a neighborhood of any k ˜ ∈ −F. any p We denote by n0 the largest nonflatness or nonnesting order plus one, and assume that r ≥ 2n0 + 1. The Fermi surface for the Hubbard model with 0 < θ < 1 and µ = 0, when viewed from (π, π), encloses a convex region. See the figure below. It has nonzero curvature except at the singular points. If one writes the equation of (one branch of) the Fermi surface near the singular point (0, π) in the form k2 − π = f (k1 ),

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then f (3) (0) 6= 0. So this Fermi surface satisfies the nonflatness and no-nesting conditions with n0 = 4.

2.2. Main theorem In the following, we state our main results about the fermionic self-energy. A discussion will be given at the end of the paper, in Sec. 6. Theorem 2.1. Let B = R2 /2πZ2 and e ∈ C 7 (B, R). Assume that the Fermi surface S = {k ∈ B : e(k) = 0} contains points where ∇e(k) = 0, and that the Hessian of e at these points is nonsingular. Moreover, assume that away from these points, the Fermi surface can have at most finite-order tangencies with its (possibly reflected) translates and is at most polynomially flat. (These hypotheses have been spelled out in detail in H1–H6 above.) As well, the interaction v is assumed to be short-range, so that its Fourier transform vˆ is C 2 . Then there is a counterterm function K ∈ C 1 (B, R), given as a formal power P r series K = r≥1 Kr λ in the coupling constant λ, such that the renormalized expansion for all Green functions, at temperature zero, is finite to all orders in λ. P (1) The self-energy is given as a formal power series Σ = r≥1 Σr λr , where for all r ∈ N and all ω ∈ R, the function k 7→ Σr (ω, k) ∈ C 1 (B, C). Specifically, we have kΣr k∞ ≤ const k∇Σr k∞ ≤ const with the constants depending on r. Moreover, the function ω 7→ Σr (ω, k) is C 1 in ω for all k ∈ B\V¯ , where V denotes the integer lattice generated by all Van Hove points, and the bar means the closure in B. (2) For e given by the normal form e(k) = k1 k2 , which has a Van Hove point at k = 0, the second order contribution Σ2 to the self-energy obeys Im ∂ω Σ2 (ω, 0) = −a1 (log|ω|)2 + O(|log|ω||) Re ∂k1 ∂k2 Σ2 (ω, 0) = a2 (log|ω|)2 + O(|log|ω||).

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∂k21 Σ2 (ω, 0) and ∂k22 Σ2 (ω, 0) grow at most linearly in log|ω|. The explicit values of a1 > 0 and a2 > 0 are given in Lemmas 5.1 and 5.2, below. Theorem 2.1 is a statement about the zero temperature limit of Σ. That is, Σr (ω, k) and its derivatives are computed at a positive temperature T = β −1 , where they are C 2 in ω and q, and then the limit β → ∞ is taken. (Because only one-particleirreducible graphs contribute to Σr , it is indeed a regular function of ω for all ω ∈ R at any inverse temperature β < ∞.) The bounds to all orders stated in item 1 of Theorem 2.1 generalize to low positive temperatures in an obvious way: the length-of-overlap estimates and the singularity analysis done below only use the spatial geometry of the Fermi surface for e, which is unaffected by the temperature. The other changes are merely to replace some derivatives with respect to frequency by finite differences, which only leads to trivial changes. Our explicit computation of the asymptotics in the model case of item 2 of Theorem 2.1 uses that several contributions to these derivatives vanish in the limit β → ∞, and that certain cancellations occur in the remaining terms. For this reason, the result stated in item 2 is a result at zero temperature. (In particular, the coefficients in the O(log|ω|) terms are just numbers.) However, we do not expect any significant change in the asymptotics at low-temperature and small ω to occur. That is, we expect the low-temperature asymptotics to contain only terms whose supremum over |ω| ≥ π/β is at most of order (log β)2 , and the square of the logarithm to be present. 2.3. Heuristic explanation of the asymmetry We refer to the different behavior of ∇Σ (which is bounded to all orders) and ∂ω Σ (which is log2 -divergent in second order) as the asymmetry in the derivatives of Σ. In the case of a regular Fermi surface, no-nesting implies that Σr is in C 1+δ with a H¨older exponent δ that depends on the no-nesting assumption. A similar bound was shown in [1] for Fermi surfaces with singularities in d ≥ 3 dimensions. In [1], we formulated a slight generalization of the no-nesting hypothesis of [11], and again proved a volume improvement estimate, which implies the above-mentioned H¨older continuity of the first derivatives. In the more special case of a regular Fermi surface with strictly positive curvature, we have given, in [12, 13], bounds on certain second derivatives of the self-energy with respect to momentum. We briefly review that discussion for the second-order contribution, to motivate why there is a difference between the spatial and the frequency derivatives. For simplicity, we assume a local interaction, and consider the infrared part of the two-loop contribution ® I(q0 , q) = C(ω1 , e(p1 ))C(ω2 , e(p2 ))C(ω, e˜)

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where ω = q0 + v1 ω1 + v2 ω2 , vi = ±1 and e˜ = e(q + v1 p1 + v2 p2 ). The angular brackets denote integration of p1 and p2 over the two-dimensional Brillouin zone and Matsubara summation of ω1 and ω2 , over the set πβ (2Z + 1). By infrared part we mean that the fermion propagators are of the form C(ω, E) =

U (ω 2 + E 2 ) iω − E

where U is a suitable cutoff function that is supported in a small, fixed neighborhood of zero. The third denominator depends on the external momentum (q0 , q) and derivatives with respect to the external momentum increase the power of that denominator, which may lead to bad behavior as β → ∞. The main idea why some derivatives behave better than expected by simple counting of powers (see [12]) is that in dimension two and higher, there are, in principle, enough integrations to make a change of variables so that e˜, e(p1 ) and e(p2 ) all become integration variables. This puts all dependence on the external variable q into the Jacobian J of this change of variables. If J were C k with uniform bounds, I(q0 , q) would be C k in the spatial momentum q, and (by integration by parts) also in q0 . However, J always has singularities, and the leading contributions to the derivatives of I(q0 , q) come from the vicinity of these singularities. It was proven in [12, 13] that if the Fermi surface is regular and has strictly positive curvature, these singularities of the Jacobian are harmless, provided derivatives are taken tangential to the Fermi surface. To explain the change of variables, we first show it for the case without Van Hove singularities and then discuss the changes required when Van Hove singularities are present. In a neighborhood of the Fermi surface, we introduce coordinates ρ and θ, so that p = P (ρ, θ). The coordinates are chosen such that ρ = e(p) and that P (ρ, θ + π) is the antipode of P (ρ, θ). (We are assuming that the Fermi surface is strictly convex — see [12].) Doing this for p1 and p2 , with corresponding Jacobian J = det P 0 , we have ·Z ¸ I(q0 , q) = dθ1 dθ2 J(ρ1 , θ1 )J(ρ2 , θ2 )C(ω, e˜) 1,2

where [F ]1,2 now denotes multiplying F by C(ω1 , ρ1 )C(ω2 , ρ2 ) and integrating over ρ1 and ρ2 and summing over the frequencies. To remove the q-dependence from C(ω, e˜), one would now want to change variables from θ1 or θ2 to e˜. This works except near points where ∂˜ e ∂˜ e = = 0. ∂θ1 ∂θ2 These equations determine the singularities of the Jacobian. The detailed analysis of their solutions is in [12]. Essentially, if one requires that the momenta p1 and p2

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are on the FS, i.e. ρ1 = ρ2 = 0, that q = P (0, θq ) is on the FS and that the sum q+v1 P (0, θ1 )+v2 P (0, θ2 ) is on the FS, then the only solutions are θ1 ∈ {θq , θq +π} and θ2 ∈ {θq , θq + π}. (The general case of momenta near to the Fermi surface is then treated by a deformation argument which requires that ∇e 6= 0 and that the curvature be nonzero.) A detailed analysis of the singularity in J, in which strict convexity enters again, then implies that the self-energy is regular. The conditions needed for the above argument fail at the Van Hove points. But, introducing a partition of unity on the Fermi surface, they still hold away from the singular points. So the only contributions that may fail to have derivatives come from q + v1 p1 + v2 p2 in a small neighborhood of the singular points. When a derivative with respect to q0 is taken, the integrand contains a factor of −i(iω − e˜)−2 . When a derivative with respect to q is taken, the integrand contains a factor ∇e(q + v1 p1 + v2 p2 )(iω − e˜)−2 . Because we are in a small neighborhood of the singular point, the numerator, ∇e(q + v1 p1 + v2 p2 ), in the latter expression is small, and vanishes at the singular point. This suggests that the first derivative with respect to q may be better behaved than the first derivative with respect to q0 , as is indeed the case — see item 1 of Theorem 2.1. A second derivative with respect to q may act on the numerator, ∇e(q + v1 p1 + v2 p2 ), and eliminate its zero. This suggests that the second derivative with respect to q behaves like the first derivative with respect to q0 , as is indeed the case — see item 2 of Theorem 2.1. The above heuristic discussion is only to provide a motivation as to why the asymmetry in the derivatives of Σ2 occurs. The proof does not make use of the idea of the change of variables to e˜, but rather of length-of-overlap estimates, which partially replace the overlapping loop estimates, away from the singular points. This allows us to show the convergence of the first q-derivative under conditions H1–H6, which are significantly weaker than strict convexity, and it also allows us to treat the situation with Umklapp scattering, which had to be excluded in second order in [12], and which is the reason for the restriction on the density in [12]. 3. Fermi Surface In this section we prove bounds on the size of the overlap of the Fermi surface with translates of a tubular neighborhood of the Fermi surface. These bounds make precise the geometrical idea that for non-nested surfaces (here: curves), the nonflatness condition H5 strongly restricts such lengths of overlap. 3.1. Normal form for e(k) near a singular point ˜=0 Lemma 3.1. Let d = 2 and assume H2–H5 with r ≥ n0 + 1. Assume that k is a singular point of e. Then there are • integers 2 ≤ ν1 , ν2 < n0 , • a constant, nonsingular matrix A and • C r−2−max{ν1 ,ν2 } functions a(k), b(k) and c(k) that are bounded and bounded away from zero

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such that in a neighborhood of the origin ¡ ¢¡ ¢ e(Ak) = a(k) k1 − k2ν1 b(k) k2 − k1ν2 c(k) .

(2)

£ 2 ¤ ˜ = Proof. Let λ1 and λ2 be the eigenvalues of ∂k∂i ∂kj e(0) 1≤i,j≤2 and set λ p r−2 |λ2 /λ1 |. By the Morse lemma [15, Chap. 6, Lemma 1.1], there is a C diffeomorphism x(k) with x(0) = 0 such that e(k) = λ1 x1 (k)2 + λ2 x2 (k)2 ˜ 2 x2 (k)2 ) = λ1 (x1 (k)2 − λ ˜ 2 (k))(x1 (k) + λx ˜ 2 (k)) = λ1 (x1 (k) − λx = λ1 (a1 k1 + a2 k2 − x ˜1 (k))(b1 k1 + b2 k2 − x ˜2 (k)) with x ˜1 (k) and x ˜2 (k) vanishing to at least order two at k = 0. Here a1 k1 +a2 k2 and ˜ 2 (k) and b1 k1 + b2 k2 are the degree one parts of the Taylor expansions of x1 (k) − λx ˜ x1 (k) + λx2 (k), respectively. Since h the i Jacobian det Dx(0) of the diffeomorphism ˜ ˜ 6= 0, we have at the origin is nonzero and det 11 −λλ˜ = 2λ · a det 1 b1

¸ ½· a2 1 = det b2 1

¸ ¾ ˜ −λ ˜ Dx(0) 6= 0. λ

Setting · a A= 1 b1

a2 b2

¸−1

we have ¡ ¢¡ ¢ e(Ak) = λ1 k1 − x ˜1 (Ak) k2 − x ˜2 (Ak) . Write ˜1 (Ak) = k1 − k1 f1 (k) − k2ν1 g1 (k2 ) k1 − x with ¯ k1 f1 (k) = x ˜1 (Ak) − x ˜1 (Ak)¯k1 =0 ¯ k2ν1 g1 (k2 ) = x ˜1 (Ak)¯k1 =0 . Since x ˜1 (Ak) vanishes to order at least two at k = 0, ¸ Z 1· ∂ x ˜1 (Ak) dt f1 (k) = ∂k1 0 k=(tk1 ,k2 ) is C r−3 and vanishes to order at least one at k = 0 and, in particular, |f1 (k)| ≤ 12 for all k in a neighborhood of the origin. We choose ν1¯ to be the power of the first nonvanishing term in the Taylor expansion of x ˜1 (Ak)¯k =0 . Since this function 1 must vanish to order at least two in k2 , we have that ν1 ≥ 2. By the nonflatness

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condition, applied to the curve implicitly determined by k1 = x ˜1 (Ak), ν1 ≤ r−2. So g1 (k2 ) is C r−2−ν1 and is bounded and bounded away from zero in a neighborhood of k2 = 0. In a similar fashion, write k2 − x ˜2 (Ak) = k2 − k2 f2 (k) − k1ν2 g2 (k1 ) with ¯ k2 f2 (k) = x ˜2 (Ak) − x ˜2 (Ak)¯k2 =0 ¯ k1ν2 g2 (k1 ) = x ˜2 (Ak)¯k2 =0 . Then we have the desired decomposition (2) with ¡ ¢¡ ¢ a(k) = λ1 1 − f1 (k) 1 − f2 (k) b(k) =

g1 (k2 ) , 1 − f1 (k)

c(k) =

g2 (k1 ) . 1 − f2 (k)

We remark that it is possible to impose weaker regularity hypotheses by exploiting that k2ν1 b(k), respectively k1ν2 c(k), is a C r−3 function whose k2 , respectively k1 , derivatives of order strictly less than ν1 , respectively ν2 , vanish at k2 = 0, respectively k1 = 0. 3.2. Length of overlap estimates It follows from the normal form derived in Lemma 3.1 that under the hypotheses H2–H5 the curvature of the Fermi surface may vanish as one approaches the singular points. Thus, even if the Fermi surface is curved away from these points, there is no uniform lower bound on the curvature. Curvature effects are very important in the analysis of regularity estimates, and in a situation without uniform bounds these curvature effects improve power counting only at scales lower than a scale set by the rate at which the curvature vanishes. Thus it becomes natural to define, at a given scale, scale-dependent neighborhoods of the singular points, outside of which curvature improvements hold. The estimates for the length of overlaps that we prove in this section allow us to make this idea precise. They hold under much more general conditions than a nonvanishing curvature, namely the nonnesting and nonflatness assumptions H5 and H6 suffice. We first discuss the special case corresponding to the normal form in the vicinity of a singular point, and then deal with the general case. 3.2.1. Length of overlap — Special case Lemma 3.2. Let ν1 ≥ 2 and ν2 ≥ 2 be integers and ¡ ¢¡ ¢ e(x, y) = x − y ν1 b(x, y) y − xν2 c(x, y)

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with b and c bounded and bounded away from zero and with b, c ∈ C ν2 +1 . Let u(x) obey ¡ ¢ u(x) = xν2 c x, u(x) for all x in a neighborhood of 0. That is, y = u(x) lies on the Fermi curve e(x, y) = 0. There are constants C and D > 0 such that for all ε > 0 and 0 < δ ≤ |(X, Y )| ≤ D µ ¶1/ν2 ¯ ¡ ¢¯ ε Vol{x ∈ R | |x| ≤ D, ¯e X + x, Y + u(x) ¯ ≤ ε} ≤ C . δ Proof. Write ¡ ¢ e X + x, Y + u(x) = F (x, X, Y )G(x, X, Y )

(3)

with ¡ ¢ F (x, X, Y ) = X + x − (Y + u(x))ν1 b X + x, Y + u(x) ¡ ¢ G(x, X, Y ) = Y + u(x) − (X + x)ν2 c X + x, Y + u(x) © ª ¡ ¢ = Y − (X + x)ν2 − xν2 c X + x, Y + u(x) © ¡ ¢ ¡ ¢ª − xν2 c X + x, Y + u(x) − c x, u(x) . Observe that, for all allowed x, X and Y , ¯ ¯ ¯ ∂ ¯ 1 99 |F (x, X, Y )| ≤ , ¯¯ F (x, X, Y )¯¯ ≥ 100 ∂x 100 since x, X, Y and u(x) all have to be O(D) small. For our analysis of G(x, X, Y ) we consider two separate cases. ¡ Case 1. |Y¢| ≥ ¡κ|X| with κ a constant to be chosen shortly. Since c X + ¢ x, Y + u(x) − c x, u(x) vanishes to first order in (X, Y ), for all x ¯ ν © ¡ ¢ ¡ ¢ª¯ £ ¤ ¯x 2 c X + x, Y + u(x) − c x, u(x) ¯ ≤ 1 |X| + |Y | 100 ¯ ¯ ¯ ¯ ∂ £ ν © ¡ ¢ ¡ ¢ª¤ ¤ 1 £ 2 ¯ ¯ ¯ ∂x x c X + x, Y + u(x) − c x, u(x) ¯ ≤ 100 |X| + |Y | . Since (X + x)ν2 − xν2 vanishes to first order in X, for all x ¯© ª ¡ ¢¯ ¯ (X + x)ν2 − xν2 c X + x, Y + u(x) ¯ ≤ κ ˜ |X| ¯ ¯ ¯ ∂ £© ª ¡ ¢¤¯ ν2 ν2 ¯ ˜ |X|. ¯ ¯ ∂x (X + x) − x c X + x, Y + u(x) ¯ ≤ κ We choose κ = max{2, 200˜ κ}. Then |G(x, X, Y )| ≥

98 |Y |, 100

¯ ¯ ¯ ∂ ¯ ¯ G(x, X, Y )¯ ≤ 2 |Y |. ¯ ∂x ¯ 100

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Thus, by (3) and the product rule, ¯ µ ¯ ¶ ¯ ∂ ¡ ¢¯ 1 ¯ e X + x, Y + u(x) ¯ ≥ 99 98 − 1 2 |Y | ≥ |Y | ¯ ¯ ∂x 100 100 100 100 2 and, by Lemma A.1, ¯ ¡ ¢¯ Vol{x ∈ R | |x| ≤ D, ¯e X + x, Y + u(x) ¯ ≤ ε} ≤ 4

ε ε ≤ 16 . |Y |/2 δ

Case 2. |Y | ≤ κ|X|. In this case we bound the ν2 th x-derivative away from zero. We claim that the dominant term comes from one derivative acting on F and ν2 − 1 derivatives acting on G. Observe that for |X|, |Y |, |x| ≤ D with D sufficiently small ¯ m ¯ ¯ d ¯ ¯ ¯ ¯ dxm u(x)¯ ≤ O(D), for 0 ≤ m < ν2 ¡ ¢ since u(x) = xν2 c x, u(x) and consequently |F (x, X, Y )| ≤ O(D) ¯ ¯ ¯ ∂ ¯ ¯ F (x, X, Y )¯ ≥ 1 − O(D) ¯ ∂x ¯ ¯ m ¯ ¯ ∂ ¯ ¯ ¯ ≤ O(D), for 1 < m ≤ ν2 . F (x, X, Y ) ¯ ∂xm ¯ ¡ ¢ ¡ ¢ Furthermore, since c X + x, Y + u(x) − c x, u(x) vanishes to first order in (X, Y ), for all x, ¯ m ¯ ¯ ∂ £ ν © ¡ ¢ ¡ ¢ª¤¯ £ ¤ 2 ¯ ¯ ¯ ∂xm x c X + x, Y + u(x) − c x, u(x) ¯ ≤ O(D) |X| + |Y | for 0 ≤ m < ν2 and ¯ m ¯ ¯ ∂ £ ν © ¡ ¢ ¡ ¢ª¤¯ £ ¤ 2 ¯ ¯ ≤ O(1) |X| + |Y | x c X + x, Y + u(x) − c x, u(x) ¯ ∂xm ¯ for m = ν2 . Since (X + x)ν2 − ν2 Xxν2 −1 − xν2 vanishes to second order in X, for all x, ¯ m ¯ ¯ ∂ £© ª ¡ ¢¤¯ ν2 ν2 −1 ν2 ¯ − x c X + x, Y + u(x) ¯¯ ≤ O(|X|2 ) ¯ ∂xm (X + x) − ν2 Xx ≤ O(D)|X| for all 0 ≤ m ≤ ν2 . Finally ¯ m ¯ ¯ ∂ £ ¡ ¢¤¯ ν2 −1 ¯ ¯ ≤ O(D)|X| ν Xx c X + x, Y + u(x) for 0 ≤ m < ν2 − 1 ¯ ∂xm 2 ¯ ¯ ¯ m ¯ ∂ £ ¡ ¢¤¯ ν2 −1 ¯ ≥ κ0 |X| − O(D)|X| for m = ν2 − 1 ¯ ν Xx c X + x, Y + u(x) ¯ ¯ ∂xm 2 ¯ m ¯ ¯ ∂ £ ¡ ¢¤¯ ν2 −1 ¯ ¯ ≤ O(1)|X| ν Xx c X + x, Y + u(x) for m = ν2 ¯ ∂xm 2 ¯

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with κ0 = ν2 ! inf |c(x, y)| > 0. Consequently, ¯ ν ¯ ¯ ∂ 2 ¡ ¢¯ ¯ ¯ e X + x, Y + u(x) ¯ ∂xν2 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ν2 µ ¶ m ¯ ∂F ∂ ν2 −1 G ν2 −m ¯ X G¯ ν2 ∂ F ∂ ¯ = ¯ν2 + ¯ ¯ ∂x ∂xν2 −1 m ∂xm ∂xν2 −m ¯ ¯ ¯ m=0 ¯ ¯ m6=1 ¡ ¢¡ ¢ £ ¤ ≥ 1 − O(D) κ0 − O(D) |X| − O(D) |X| + |Y | ¡ ¢¡ ¢ ≥ 1 − O(D) κ0 − O(D) |X| − O(D)(1 + κ)|X| ≥

κ0 |X| 2

if D is small enough. Hence, by Lemma A.1, ¯ ¡ ¢¯ Vol{x ∈ R | |x| ≤ D, ¯e X + x, Y + u(x) ¯ ≤ ε} Ã √ !1/ν2 ¶1/ν2 µ 2 1 + κ2 ε ε ν2 +1 ν2 +1 ≤2 . ≤2 κ0 |X|/2 κ0 δ 3.2.2. Length of overlap — General case Proposition 3.3. Assume H1–H6 with r ≥ 2n0 + 1. There is a constant D > 0 such that for all 0 < δ < 1 and each sign ± the measure of the set of p ∈ R2 such that µ j ¶1/n0 M `({k ∈ F | |e(p ± k)| ≤ M j }) ≥ for some j < 0 δ is at most Dδ 2 . Here ` is the Euclidean measure (length) on F. Recall that n0 is the largest nonflatness or nonnesting order plus one. ˜ ∈ F, there are constants ˜ ∈ R2 and k Lemma 3.4. Let r ≥ 2n0 + 1. For each p 0 ˜ ˜ and k) such that for each sign ±, all j < 0, and d, D > 0 (possibly depending on p ˜| ≤ d all p ∈ R2 obeying |p − p ¡ ¢ ˜ ≤ d, |e(p ± k)| ≤ M j } ≤ D0 ` {k ∈ F | |k − k|

µ

Mj ˜| |p − p

¶1/n0 .

˜ ∈ F, ˜ ∈ R2 and k Proof of Proposition 3.3, assuming Lemma 3.4. For each p 0 let dp˜ ,k˜ , Dp˜ ,k˜ be the constants of the Lemma and set ˜ < d ˜ }. ˜ | < dp˜ ,k˜ , |k − k| Op˜ ,k˜ = {(p, k) ∈ R2 × F | |p − p ˜ ,k p

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Since F = {k | e(k) = 0} and {k ¯| |e(k)| ≤ ¡ ¢¯ 1} are compact, there is an R > 0 such that if |p| > R, then {k ∈ F | ¯e p ± k ¯ ≤ M j } is empty for all j < 0. Since ˜ 1 ), . . . , (˜ ˜ N ) such that {p ∈ R2 | |p| ≤ R} × F is compact, there are (˜ p1 , k pN , k 2

{p ∈ R | |p| ≤ R} × F ⊂

N [

Op˜ i ,k˜ i .

i=1

˜ i | > δi Fix any 0 < δ < 1 and set, for each 1 ≤ i ≤ N , δi = (N Dp0˜ ,k˜ )n0 δ. If |p − p i i for all 1 ≤ i ≤ N , then for all j < 0 ¡ ¢ ` {k ∈ F | |e(p ± k)| ≤ M j }     ≤ ` 

1≤i≤n |p−˜ pi |≤dp ˜

≤

N X i=1

 ¯ ¡ ¢¯ ˜ ˜ | ≤ d ˜ , ¯e p ± k ¯ ≤ M j } {k ∈ F | |k − k  ˜ ˜ p i , ki pi ,ki 

[

µ Dp0˜

˜

i , ki

˜ i ,ki

Mj ˜ i| |p − p

¶1/n0 <

N X

µ Dp0˜

i=1

˜

i ,ki

Mj δi

¶1/n0

µ ¶1/n0 µ j ¶1/n0 N X 1 Mj M ≤ = . N δ δ i=1 Consequently the measure of the set of p ∈ R2 for which µ j ¶1/n0 M j for some j < 0 `({k ∈ F | |e(p ± k)| ≤ M }) ≥ δ is at most N X i=1

πδi2

≤ Dδ

2

where D =

N X

π(N Dp0˜

i=1

˜

i , ki

)2n0 .

Proof of Lemma 3.4. We give the proof for p + k. The other case is similar. ˜ ∈ ˜ +¯ k In the event that p ¡ / F,¢¯there is a d > 0 and an integer j0 < 0 such that ˜ ¯ ˜ | ≤ d and j < j0 . So {k ∈ F | |k − k| ≤ d, e p + k ¯ ≤ M j } is empty for all |p − p ˜ ∈ F. ˜ +k we may assume that p ˜ is not a singular point. By a rotation and translation of the k Case 1. p ˜+k ˜ = 0 and that the tangent line to F at p ˜ is ˜+k ˜+k plane, we may assume that p k2 = 0. Then, as in Sec. 3.1, there are • ν ∈ N with 2 ≤ ν < n0 and • C r−ν functions a(q) and b(q) that are bounded and bounded away from zero such that ¡ ¢ e(q) = a(q) q2 − q1ν b(q)

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¯ in a neighborhood of 0. (Choose q2 a(q) = e(q) − e(q)¯q2 =0 and q1ν a(q)b(q) = ˜ (when k ˜ is a singular point the tangent e(q)|k =0 .) If the tangent line to F at k q

line of one branch) is not parallel to k1 = 0 and if d is small enough, we can write ˜ (when k ˜ is a singular point, the the equation of F for k within a distance d of k 0 ˜ equation of the branch under consideration) as k2 − k2 = (k1 − k˜1 )ν c(k1 − k˜1 ) for some 1 ≤ ν 0 < n0 and some C r−n0 −1 function c that is bounded and bounded away ˜ is a singular point, by Lemma 3.1). Then, for k ∈ F, we have, from zero (when k ˜ = (X, Y ) and k1 − k˜1 = x, writing p − p ¡ ¢ ˜ = e (X, Y ) + (x, xν 0 c(x)) ˜ + k − k) e(p + k) = e(p − p ¡ ¢ 0 = A(x, X, Y ) Y + xν c(x) − (X + x)ν B(x, X, Y ) where ¡ ¢ 0 A(x, X, Y ) = a X + x, Y + xν c(x) ¡ ¢ 0 B(x, X, Y ) = b X + x, Y + xν c(x) and c(x) are bounded and bounded away from zero. 0 Observe that y = xν c(x) is the equation of a fragment of F translated so as ˜ to 0 and y = xν b(x, y) is the equation of a fragment of F translated to move k ˜ to 0. If p ˜+k ˜ = 0 these two fragments may be identical. That is so as to move¡p ¢ ν0 ν ν0 ˜ 6= 0, the nonnesting x c(x) ≡ x b x, x c(x) . (Of course, in this case ν = ν 0 .) If p condition says that there is an n ∈ N such that if y = xν b(x, y) is rewritten in the 0 form y = xν C(x), then the nth derivative of xν C(x) − xν c(x) must not vanish at x = 0. Let n < n0 be the smallest such natural number. Since derivatives of xν C(x) at x = 0 of order strictly lower than n agree with the corresponding derivatives of 0 0 xν c(x), the nth derivatives at x = 0 of xν b(x, xν C(x)) and xν b(x, xν c(x)) coincide. ¡ ¢ 0 0 Since xν C(x) ≡ xν b(x, xν C(x)), the nth derivative of xν c(x) − xν b x, xν c(x) = 0 xν c(x) − xν B(x, 0, 0) must not vanish at x = 0. • If ν 0 < ν, 0

¢ dν ¡ ν0 c(x) − (X + x)ν B(x, X, Y ) = ν 0 ! c(x) + O(d) = ν 0 ! c(0) + O(d) 0 Y + x ν dx is uniformly bounded away from zero, if d is small enough. • If ν 0 > ν, ¢ 0 dν ¡ Y + xν c(x) − (X + x)ν B(x, X, Y ) = −ν! B(0, 0, 0) + O(d) ν dx is uniformly bounded away from zero, if d is small enough.

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0

• If ν 0 = ν and xν c(x) 6≡ xν B(x, 0, 0), then, as the function xν B(x, 0, 0) − (X + x)ν B(x, X, Y ) vanishes for all x if X = Y = 0, ¤ 0 dn £ Y + xν c(x) − (X + x)ν B(x, X, Y ) n dx 0 dn £ = n Y + xν c(x) − xν B(x, 0, 0) dx £ ¤¤ + xν B(x, 0, 0) − (X + x)ν B(x, X, Y ) ¢ dn ¡ ν 0 x c(x) − xν B(x, 0, 0) + O(|X| + |Y |) n dx ¢¯ dn ¡ 0 = n xν c(x) − xν B(x, 0, 0) ¯x=0 + O(d) + O(|X| + |Y |) dx =

is uniformly bounded away from zero, if d is small enough. 0 • If ν 0 = ν and xν c(x) ≡ xν B(x, 0, 0) and |Y | ≤ |X| 0

Y + xν c(x) − (X + x)ν B(x, X, Y ) = Y − νXxν−1 B(x, X, Y ) − {(X + x)ν − νXxν−1 − xν }B(x, X, Y ) − xν {B(x, X, Y ) − B(x, 0, 0)} so that ¢ 0 dν−1 ¡ Y + xν c(x) − (X + x)ν B(x, X, Y ) dxν−1 £ ¤ ¡ ¢ = −ν! X B(0, 0, 0) + O(d) + O(d)O |X| + |Y | ¯ ¯ is bounded away from zero by 12 ν! ¯XB(0, 0, 0)¯, if d is small enough. In all of the above cases, by Lemma A.1, ¯ ¡ ¡ ¢¯ ¢ q ˜ ≤ d, ¯e p + k ¯ ≤ M j } ≤ 1 + c2 2ν0 +1 ` {k ∈ F | |k − k| 1

µ

c0 M j ρ

¶1/ν0

where ν0 is one of ν 0 , ν, n or ν − 1, the constant c0 is the inverse of the infimum of ˜ and ρ is a(k), the constant c1 is the maximum slope of F within a distance d of k ˜ |. either a constant or a constant times X with X at least a constant times |p − p ˜ (when There are two remaining possibilities. One is that the tangent line to F at k ˜ is a singular point the tangent line of one branch) is parallel to k1 = 0. This k case is easy to handle because the two fragments of F are almost perpendicular, so that ¯ ¡ ¡ ¢¯ ¢ ˜ ≤ d, ¯e p + k ¯ ≤ M j } ≤ const M j . ` {k ∈ F | |k − k|

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The final possibility is 0

• If ν 0 = ν and xν c(x) ≡ xν B(x, 0, 0) and |Y | ≥ |X| 0

Y + xν c(x) − (X + x)ν B(x, X, Y ) = Y − {(X + x)ν − xν }B(x, X, Y ) − xν {B(x, X, Y ) − B(x, 0, 0)} so that ¯ ¯ ¯Y + xν 0 c(x) − (X + x)ν B(x, X, Y )¯ ≥ |Y | − O(d)O(|X| + |Y |) ≥ 1 |Y | 2 ¯ ¡ ¢¯ ˜ ¯ ¯ if d is small enough. As a result {k ∈ F | |k − k| ≤ d, e p + k ≤ M j } is empty if |Y | is larger than some constant times M j . On the other hand, if |Y | Mj is smaller than a constant times M j , then |p−˜ p| is larger than some constant. ˜ is a singular point. By Lemma 3.1, Case 2. p ˜+k ¡ ¢¡ ¢ ˜ + Mq) = a(q) q1 − q ν1 b(q) q2 − q ν2 c(q) e(˜ p+k 2 1 where 2 ≤ ν1 , ν2 < n0 are integers, M is a constant, nonsingular matrix and a(k), b(k) and c(k) are C r−2−max{ν1 ,ν2 } functions that are bounded and bounded away from zero. ˜ (when k ˜ is a singular point, the Suppose that the tangent line to M−1 F at k ◦ tangent line of one branch) makes an angle of at most 45 with the x-axis. Otherwise exchange the roles of the q1 and q2 coordinates. If d is small enough, we can write ˜ (when k ˜ is a singular point, the the equation of F for k within a distance d of k equation of the branch under consideration) as ¡ −1 ¢ ¡ ¢ 0 ¡¡ ¢ ¢ ˜ ˜ ν v M−1 (k − k) ˜ M (k − k) = M−1 (k − k) 2 1 1 for some 1 ≤ ν 0 < n0 and some C r−n0 −1 function v that is¡ bounded and ¢ bounded ˜ ˜ ) = (X, Y ) and M−1 (k − k) away from zero. Then, writing M−1 (p − p = x and 1 assuming that k ∈ F, ¡ ¢ ˜ + MM−1 (p − p ˜ ˜ +k ˜ + k − k) e(p + k) = e p ¢ ¡ ˜ + M(X + x, Y + xν 0 v(x)) ˜ +k =e p = A(x, X, Y )F (x, X, Y )G(x, X, Y ) where ¡ ¢ 0 A(x, X, Y ) = a X + x, Y + xν v(x) 0

F (x, X, Y ) = X + x − (Y + xν v(x))ν1 B(x, X, Y ) 0

G(x, X, Y ) = Y + xν v(x) − (X + x)ν2 C(x, X, Y ) ¡ ¢ 0 B(x, X, Y ) = b X + x, Y + xν v(x) ¡ ¢ 0 C(x, X, Y ) = c X + x, Y + xν v(x) .

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The functions A(x, X, Y ), B(x, X, Y ), C(x, X, Y ) and v(x) are all C r−n0 −1 and bounded and bounded away from zero. 0 As in case 1, y = xν v(x) is the equation of a fragment of M−1 F translated so as ˜ to 0 and y = xν2 c(x, y) is the equation of a fragment of M−1 F translated to move k ˜ to 0. If p ˜ +k ˜ = 0, these two fragments may be identical in which so as to move p ¡ ¢ 0 ν0 ν2 case x v(x) ≡ x c x, xν v(x) and ν2 = ν 0 . This case has already been dealt with in Lemma 3.2. Otherwise, the nonnesting condition says that there is an n ∈ N such that if y = xν2 c(x, y) is rewritten in the form y = xν2 V (x), then the nth derivative 0 of xν2 V (x) − xν v(x) must not vanish at ¡ x = 0. Let ¢ n < n0 be the smallest such natural number. Since xν2 V (x) ≡ xν2 c x, xν2 V (x) and since derivatives at 0 of 0 xν2 V (x) of order lower than n agree¡ with the corresponding derivatives of xν v(x), ¢ 0 0 0 the nth derivative of xν v(x) − xν2 c x, xν v(x) = xν v(x) − xν2 C(x, 0, 0) must not vanish at x = 0. So the remaining cases are: • If ν 0 < ν2 , then 0

dν G(x, X, Y ) = ν 0 ! v(0) + O(d). dxν 0 Since F (x, X, Y ) = O(d),

d F (x, X, Y ) = 1 + O(d) dx

and applying zero to ν 0 − 1 x-derivatives to G(x, X, Y ) gives O(d), we have 0

dν +1 F (x, X, Y )G(x, X, Y ) = (ν 0 + 1)! v(0) + O(d) dxν 0 +1 uniformly bounded away from zero, if d is small enough. • If ν 0 > ν2 , dν2 G(x, X, Y ) = −ν2 ! C(0, 0, 0) + O(d). dxν2 Again d F (x, X, Y ) = 1 + O(d) dx and applying zero to ν2 − 1 x-derivatives to G(x, X, Y ) gives O(d), so that F (x, X, Y ) = O(d),

dν2 +1 F (x, X, Y )G(x, X, Y ) = −(ν2 + 1)! C(0, 0, 0) + O(d) dxν2 +1 is uniformly bounded away from zero, if d is small enough. 0 • If ν 0 = ν2 and xν v(x) 6≡ xν2 C(x, 0, 0) 0 dn £ dn G(x, X, Y ) = Y + xν v(x) − xν2 C(x, 0, 0) n dx dxn £ ¤¤ + xν2 C(x, 0, 0) − (X + x)ν2 C(x, X, Y ) =

¢¯ dn ¡ ν 0 x v(x) − xν2 C(x, 0, 0) ¯x=0 + O(d) + O(|X| + |Y |) n dx

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and applying strictly fewer than n derivatives gives O(d). As in the last two cases ¢¯ dn ¡ ν 0 dn+1 F (x, X, Y )G(x, X, Y ) = n x v(x) − xν2 C(x, 0, 0) ¯x=0 + O(d) n+1 n dx dx is uniformly bounded away from zero, if d is small enough. The lemma now follows by Lemma A.1, as in Case 1. 4. Regularity 4.1. The gradient of the self-energy 4.1.1. The second order contribution Let C(k) =

U (k) ik0 − e(k)

where the ultraviolet cutoff U (k) is a smooth compactly supported function that is identically one for all k with |ik0 − e(k)| sufficiently small. We consider the value Z F (q) = d3 k (1) d3 k (2) d3 k (3) δ(k (1) + k (2) − k (3) − q)C(k (1) )C(k (2) )C(k (3) ) × V (k (1) , k (2) , k (3) , q) of the diagram

.

The function V is a second order polynomial in the interaction function vˆ. For details, as well as the generalization to frequency-dependent interactions, see [12]. For the purposes of the present discussion, all we need is a simple regularity assumption on V . Lemma 4.1. Assume H1–H6. If V (k (1) , k (2) , k (3) , q) is C 1 , then F (q) is C 1 in the spatial coordinates q. Proof. Introduce our standard partition of unity of a neighborhood of the Fermi surface [11, § 2.1] X U (k) = f (M −2j |ik0 − e(k)|2 ) j<0

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where f (M −2j |ik0 − e(k)|2 ) vanishes unless M j−2 ≤ |ik0 − e(k)| ≤ M j . We have X f (M −2j |ik0 − e(k)|2 ) C(k) = Cj (k) where Cj (k) = ik0 − e(k) j<0 and

Z

X

F (q) =

d3 k (1) d3 k (2) d3 k (3) δ(k (1) + k (2) − k (3) − q)

j1 ,j2 ,j3 <0

× Cj1 (k (1) )Cj2 (k (2) )Cj3 (k (3) ) V. Route the external momentum q through the line with smallest |ji | (i.e. use the delta function to evaluate the integral over the k (i) corresponding to the smallest |ji |) and apply ∇q . Rename the remaining integration variables k (i) to k and p. Permute the indices so that j1 ≤ j2 ≤ j3 . If the ∇q acts on V , the estimate is easy. For each fixed j1 , j2 and j3 , • the volume of the domain of integration is bounded by a constant times |j1 |M 2j1 |j2 |M 2j2 , by [1, Lemma 2.3]. (The k0 and p0 components contribute M j1 M j2 to this bound.) • and the integrand is bounded by const M −j1 M −j2 M −j3 . so that

Z

X

¯ ¯ d3 kd3 p¯Cj1 (k)Cj2 (p)Cj3 (±k ± p ± q)∇q V ¯

j1 ≤j2 ≤j3 <0

X

≤ const

|j1 |M j1 |j2 |M j2 M −j3

j1 ≤j2 ≤j3 <0

X

≤ const

|j1 |M j1 |j2 |

j1 ≤j2 <0

≤ const

X

|j1 |3 M j1

j1 <0

is uniformly bounded. So assume that the ∇q acts on Cj3 (±k ± p ± q). The terms of interest are now of the form Z d3 kd3 pVC j1 (k)Cj2 (p)∇q Cj3 (±k ± p ± q) with j1 ≤ j2 ≤ j3 < 0 and ∇q Cj3 (±k ± p ± q) = ±

"

˜ ∇e(k) ˜ 2) f (M −2j3 |ik˜0 −e(k)| ˜ 2 [ik˜0 − e(k)]

˜ ˜ 2M −2j3 e(k)∇e( k) ˜ 2) + f 0 (M −2j3 |ik˜0 −e(k)| ˜ ik˜0 − e(k)

# . ˜ k=±k±p±q

(4)

¯ ¯ ˜ ≥ const M j3 and Observe that ¯∇q Cj3 (±k ± p ± q)¯ ≤ const M −2j3 since |ik˜0 − e(k)| 0 j3 −2j3 ˜ 2 ˜ ˜ |e(k)| ≤ const M on the support of f (M |ik0 − e(k)| ). Choose three small

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0 −1 constants η, η˜, ε > 0 such that 0 < ε ≤ 2n1 0 , 0 < η < 2n ˜ ≤ 2n02η+ε 2n0 +2 ε and η +2η+ε . Here n0 is the integer in Proposition 3.3. For example, if n0 = 3, we can choose 1 5 1 ε = 61 , η = 10 < 48 and η˜ = 20 .

Reduction 1. For any η˜ > 0, it suffices to consider j1 ≤ j2 ≤ j3 ≤ (1 − η˜)j1 . For the remaining terms, we simply bound ¯Z ¯ ¯ ¯ ¯ d3 kd3 pVC j (k)Cj2 (p)∇q Cj3 (±k ± p ± q)¯ 1 ¯ ¯ µ ¶ µ ¶ Z |ik0 − e(k)|2 |ip0 − e(p)|2 −j1 ≤ const d3 kd3 pf M f M −j2M −2j3 M 2j1 M 2j2 ≤ const|j1 |M j1 |j2 |M j2M −2j3 and

X

|j1 |M j1 |j2 |M j2M −2j3

j1 ≤j2 ≤j3 <0 j3 ≥(1−˜ η )j1

X

≤ const

|j1 |M j1 |j2 |M j2M −j3 −(1−˜η)j1

j1 ≤j2 ≤j3 <0

X

≤ const

|j1 | |j2 |M j1M −(1−˜η)j1

j1 ≤j2 <0

X

= const

|j1 |3 M η˜j1 < ∞.

j1 <0

˜| ≥ Reduction 2. For any η > 0, it suffices to consider (k, p) with | ± k ± p ± q − q ηj3 ˜ . Let Ξj3 (k) be the characteristic function of the set M for all singular points q ˜ | ≥ M ηj3 for all singular points q ˜ }. {k ∈ R2 | |k − q ¯ ¯ ˜ | ≤ M ηj3 for some singular point q ˜ , then ¯∇e(±k ± p ± q)¯ ≤ If | ± k ± p ± q − q const M ηj3 and we may bound ¯Z ¯ ¯ ¡ ¢¯ ¯ d3 kd3 pVC j (k)Cj2 (p)∇q Cj3 (±k ± p ± q) 1 − Ξj3 (±k ± p ± q) ¯ 1 ¯ ¯ ¶ µ ¶ µ Z |ip0 − e(p)|2 |ik0 − e(k)|2 −j1 M f M −j2M −(2−η)j3 ≤ const d3 kd3 pf M 2j1 M 2j2 ≤ const |j1 |M j1 |j2 |M j2M −(2−η)j3 and

X

|j1 |M j1 |j2 |M j2M −(2−η)j3

j1 ≤j2 ≤j3 <0

≤ const

X j1 ≤j2 <0

|j1 |M j1 |j2 |M j2M −(2−η)j2

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X

≤ const

297

|j1 |2 M j1M −j2 +ηj2

j1 ≤j2 <0

X

≤ const

|j1 |2 M j1M −j1 +ηj1

j1 <0

X

= const

|j1 |2 M ηj1 < ∞.

j1 <0

Current status. It remains to bound ¯Z ¯ X ¯ ¯ ¯ d3 kd3 pVC j (k)Cj (p)∇q Cj (±k ± p ± q)Ξj (±k ± p ± q)¯ 2 3 3 1 ¯ ¯ j1 ≤j2 ≤j3 < 0 j3 ≤(1−˜ η )j1

≤ const

X

µ

Z d3 kd3 pf

j1 ≤j2 ≤j3 <0 j3 ≤(1−˜ η )j1

|ik0 − e(k)|2 M 2j1

¶

µ M −j1 f

|ip0 − e(p)|2 M 2j2

¶ M −j2

M −2j3 Ξj3 (±k ± p ± q)χj3 (±k ± p ± q) Z X −2j3 ≤ const M d2 kd2 pχj1 (k)χj2 (p)(χj3 Ξj3 )(±k ± p ± q) j1 ≤j2 ≤j3 <0 j3 ≤(1−˜ η )j1

≤ const

X

Z M

−2j3

d2 kd2 pχj1 (k)χj2 (p)(χj3 Ξj3 )(±k ± p ± q)

j1 , j2 , j3 <0 j3 1−η ˜ ≤j1 , j2 ≤j3

where χj (k) is the characteristic function of the set of k’s with |e(k)| ≤ M j . Make a change of variables with ±k ± p ± q becoming the new k integration variable. This gives Z X const M −2j3 d2 kd2 pχj1 (±k ± p ± q)χj2 (p)χj3 (k) Ξj3 (k). j1 , j2 , j3 <0 j3 1−η ˜ ≤j1 , j2 ≤j3

˜ p ˜ and q ˜ in R2 , there are Reduction 3. It suffices to show that, for each fixed k, ˜ ˜, q ˜ dependent, but ji independent) constants c and C such that (possibly k, p Z Z 2 d k d2 pχj1 (±k ± p ± q)χj2 (p)χj3 (k) Ξj3 (k) ˜ |k−k|≤c

|p−˜ p|≤c

0

≤ C|j2 |M j2 M (1−η)j3 M εj3 −η j3

(5)

˜ | ≤ c. The constant η 0 will be chosen later and will obey for all q obeying |q − q 0 ε − η − η > 0. Since {(k, p, q) ∈ R6 | |e(k)| ≤ 1, |e(p)| ≤ 1, |e(±k ± p ± q)| ≤ 1}

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is compact, once this is proven we will have the bound Z X −2j3 M d2 kd2 pχj1 (±k ± p ± q)χj2 (p)χj3 (k)Ξj3 (k) const j1 , j2 , j3 <0 j3 1−η ˜ ≤j1 , j2 ≤j3

X

≤ const

0

M −2j3 |j2 |M j2 M (1−η)j3 M εj3 −η j3

j1 , j2 , j3 <0 j3 1−η ˜ ≤j1 , j2 ≤j3

≤ const

X µ |j3 | ¶3 0 M (ε−η−η )j3 1 − η˜ j <0 3

≤ const ˜ or p ˜±p ˜ or ±k ˜±q ˜ does not lie on F, we can since ε > η + η 0 . Furthermore, if k ˜ | ≤ c and choose c sufficiently small that the integral of (5) vanishes whenever |q − q ˜ p ˜±p ˜ and ±k ˜ ±q ˜ all |j1 |, |j2 |, |j3 | are large enough. So it suffices to require that k, lie on F. ˜ is not a singular point, make a change of variables to ρ = e(k) Reduction 4. If k and an “angular” variable θ. So the condition χj3 (k) 6= 0 forces |ρ| ≤ const M j3 . ˜ is a singular point, the condition Ξj (k) 6= 0 forces |k − k| ˜ ≥ const M ηj3 and If k 3 this, in conjunction with the condition that χj3 (k) 6= 0, forces k to lie fairly near ˜ at least a distance const M ηj3 from k. ˜ Using one of the two branches of F at k ¡ ¢ Lemma 3.1, we can make a change of variables such that e k(ρ, θ) = ρθ and either |θ| ≥ const M ηj3 or |ρ| ≥ const M ηj3 . Possibly exchanging the roles of ρ and θ, we may, without loss of generality assume the former. Then the condition χj3 (k) 6= 0 ˜ is singular or not, forces |ρ| ≤ const M (1−η)j3 . Thus, regardless of whether k Z Z 2 d k d2 pχj1 (±k ± p ± q)χj2 (p)χj3 (k)Ξj3 (k) ˜ |k−k|≤c

≤ const

|p−˜ p|≤c

Z

Z

|θ|≤1 |ρ|≤const M (1−η)j3

dρdθ |p−˜ p|≤c

Z ≤ const

|θ|≤1 |ρ|≤const M (1−η)j3

Z

≤ const M

(1−η)j3

Z dρdθ |p−˜ p|≤c

0

d2 pχj 0 (±k(0, θ) ± p ± q)χj2 (p)

Z

dθ |θ|≤1

d2 pχj1 (±k(ρ, θ) ± p ± q)χj2 (p)

|p−˜ p|≤c

d2 pχj 0 (±k(0, θ) ± p ± q)χj2 (p)

where M j = M j1 + const M (1−η)j3 ≤ const M (1−η)j3 . Thus it suffices to prove that Z Z 0 d2 p dθχj 0 (±k(0, θ) ± p ± q)χj2 (p) ≤ C|j2 |M j2 M εj3 −η j3 |p−˜ p|≤c

|θ|≤1

˜ | ≤ c. for all q obeying |q − q

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The Final Step. We apply Proposition 3.3 with j = j 0 and δ = M (1+ε)j3 /2 . Denote by χ(p) ˜ the characteristic function of the set of p’s with !1/n0 Ã j0 ¯ ¡ ¡ ¢¯ 0 ¢ M j µ {−1 ≤ θ ≤ 1 | ¯e ± p ± q ± k(0, θ) ¯ ≤ M } ≥ c1 M (1+ε)j3 /2 where c1 is the supremum of dθ ds (s is arc length). If ±p ± q is not in the set of measure Dδ 2 specified in Proposition 3.3, then Ã !1/n0 0 ¯ ¡ ¡ ¢¢¯ ¢ Mj j0 ¯ ¯ ` {k ∈ F | e ± p ± q ± k ≤M } < δ and hence

¯ ¡ ¡ ¢¯ 0 ¢ µ {−1 ≤ θ ≤ 1 | ¯e ± p ± q ± k(0, θ) ¯ ≤ M j } Z 1 ¡¯ ¡ ¢¯ 0¢ = dθχ ¯e ± p ± q ± k(0, θ) ¯ ≤ M j Z

−1

¢¯ 0¢ dθ ¡¯¯ ¡ χ e ± p ± q ± k ¯ ≤ Mj ds ¯ ¡ ¡ ¢¢¯ 0 ¢ ≤ c1 ` {k ∈ F | ¯e ± p ± q ± k ¯ ≤ M j } !1/n0 Ã 0 Mj < c1 M (1+ε)j3 /2 =

ds

so that χ(p) ˜ = 0. Thus χ(p) ˜ vanishes except on a set of measure DM (1+ε)j3 and Z Z d2 p dθχj 0 (±k(0, θ) ± p ± q)χj2 (p) |θ|≤1

Z

d2 pχ(p)χ ˜ j2 (p)

≤

Z +

Z dθχj 0 (±k(0, θ) ± p ± q) |θ|≤1

¡ ¢ d2 p 1 − χ(p) ˜ χj2 (p)

Z ≤2

Z dθχj 0 (±k(0, θ) ± p ± q) |θ|≤1

Ã

Z 2

d pχ(p) ˜ + const

Mj

2

d pχj2 (p) Ã

≤ const M (1+ε)j3 + const|j2 |M j2

0

!1/n0

M (1+ε)j3 /2 Mj

0

!1/n0

M (1+ε)j3 /2

¡ ¢1/n0 1+ε ≤ const M j3 M εj3 + const|j2 |M j2 M (1−η)j3 − 2 j3 ¡ ¢1/n0 η ˜ 1 ε ≤ const M j2 M (ε− 1−η˜ )j3 + const|j2 |M j2 M ( 2 −η− 2 )j3 since j3 ≤ (1 − η˜)j2 ≤ j2 − 0

≤ const|j2 |M j2 M εj3 −η j3

η˜ j3 1 − η˜

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provided ε ≤

1 2n0 , 0

η˜ η ε 0 1−˜ η and η ≥ n0 + 2n0 . η ε n0 + 2n0 . Then the remaining

η0 ≥

We choose η =

ε > η + η0 ⇔ ε > η +

conditions

ε 2n0 − 1 n0 + 1 2n0 − 1 η ε + ⇔ ε> η⇔η< n0 2n0 2n0 n0 2n0 + 2

(6)

and η0 η˜ 2η + ε ⇔ η˜ ≤ = 1 − η˜ 1 + η0 2n0 + 2η + ε are satisfied because of the conditions we imposed when we chose η, η˜ and ε just before Reduction 1. We have now verified that ∇q F (q) is Ra uniformly convergent sum (over j1 , j2 , and j3 ) of the continuous functions ∇q d3 kd3 pVC j1 (k)Cj2 (p)Cj3 (±k ± p ± q). Hence F (q) is C 1 in q. η0 ≥

4.1.2. The general diagram The argument of the last section applies equally well to general diagrams. Lemma 4.2. Let G(q) be the value of any two-legged 1PI graph with external momentum q. Then G(q) is C 1 with respect to the spatial components q. Proof. We shall simply merge the argument of the last section with the general bounding argument of [1, Appendix A]. This is a good time to read that appendix, since we shall just explain the modifications to be made to it. In addition to the small constants η, η 0 , η˜, ε > 0 of Lemma 4.1, we choose a small constant ε¯ > 0 and requirea that 1 2n0 − 1 η ε 0<ε≤ , 0<η< ε, η 0 = + 2n0 2n0 + 2 n0 2n0 η˜ and < min{η 0 , ε − η − η 0 } 1 − η˜ and ε¯ ≤ min{η, η˜, (1 + ε − η − η 0 )(1 − η˜) − 1} with n0 being the integer in Proposition 3.3. All of these conditions may be satisfied by 1 2n0 and then, 0 −1 < η < 2n 2n0 +2 ε (by (6), η˜ < η˜ < 1 so that 1−˜ η 0

• choosing 0 < ε ≤

this ensures that ε − η − η 0 > 0) and then, © ª < min η 0 , ε − η − η 0 (this ensures that the • choosing 0 expression (1 + ε − η − η )(1 − η˜) − 1 > 0) and then, • choosing ε¯ > 0 so that ε¯ ≤ min{η, η˜, (1 + ε − η − η 0 )(1 − η˜) − 1}. • choosing 0

a The

first three conditions as well as the condition that Lemma 4.1. The other conditions are new.

η ˜ 1−˜ η

≤ η 0 were already present in

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As in [1, Appendix A], use [1, (22)] to introduce a scale expansion for each propagator and express G(q) in terms of a renormalized tree expansion [1, (24)]. We shall prove, by induction on the depth, D, of GJ , the bound X ¯ ¯ sup¯∂qs00 ∂qs1 GJ (q)¯ ≤ constn |j|3n−2 M j M −s0 j M −s1 (1−¯ε)j (7) J∈J (j,t,R,G)

q

for s0 , s1 ∈ {0, 1}. The notation is as in [1, Appendix A]: n is the number of vertices in G and J (j, t, R, G) is the set of all assignments J of scales to the lines of G that have root scale j, that give forest t and that are compatible with the assignment R of renormalization labels to the two-legged forks of t. (This is explained in more detail just before [1, (24)].) If s0 = 0 and s1 = 1, the right-hand side becomes constn |j|3n−2 M ε¯j , which is summable over j < 0, implying that G(q) is C 1 with respect to the spatial components q. If s1 = 0, (7) is contained in [1, Proposition A.1], so it suffices to consider s1 = 1. As in [1, Appendix A], if D > 0, decompose the tree t into a pruned tree t˜ and insertion subtrees τ 1 , . . . , τ m by cutting the branches beneath all minimal Ef = 2 forks f1 , . . . , fm . In other words each of the forks f1 , . . . , fm is an Ef = 2 fork having no Ef = 2 forks, except φ, below it in t. Each τi consists of the fork fi and all of t that is above fi . It has depth at most D − 1 so the corresponding subgraph Gfi obeys (7). Think of each subgraph Gfi as a generalized vertex in the graph ˜ = G/{Gf , . . . , Gf }. Thus G ˜ now has two as well as four-legged vertices. These G 1 m P two-legged vertices have kernels of the form Ti (k) = jf ≤jπ(f ) `Gfi (k) when fi i i P is a c-fork and of the form Ti (k) = jf >jπ(f ) (1 − `)Gfi (k) when fi is an r-fork. i i At least one of the external lines of Gfi must be of scale precisely jπ(fi ) so the momentum k passing through Gfi lies in the support of Cjπ(fi ) . In the case of a c-fork f = fi we have, as in [1, (27)] and using the same notation, by the inductive hypothesis, X X ¯ ¯ J sup¯∂ s1 `G f (k)¯ jf ≤jπ(f ) Jf ∈J (jf ,tf ,Rf ,Gf ) k

≤

X

k

f

constnf |jf |3nf −2 M jf M −s1 (1−¯ε)jf

jf ≤jπ(f )

≤ constnf |jπ(f ) |3nf −2 M jπ(f ) M −s1 (1−¯ε)jπ(f ) J

(8)

for s1 = 0, 1. As `Gf f (k) is independent of k0 derivatives with respect to k0 may not act on it. In the case of an r-fork f = fi , we have, as in [1, (29)], X X ¯ ¡ ¢¯ J sup 1 Cjπ(f ) (k) 6= 0 ¯∂ks00 ∂ks1 (1 − `)Gf f (k)¯ jf >jπ(f ) Jf ∈J (jf ,tf ,Rf ,Gf ) k

≤

X

X

jf >jπ(f ) Jf ∈J (jf ,tf ,Rf ,Gf )

¯ ¯ J M (1−s0 )jπ(f ) sup¯∂k0 ∂ks1 Gf f (k)¯ k

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≤ constnf M (1−s0 )jπ(f )

X

|jf |3nf −2 M −s1 (1−¯ε)jf

jf >jπ(f )

≤ constnf |jπ(f ) |3nf −1 M jπ(f ) M −s0 jπ(f ) M −s1 (1−¯ε)jπ(f ) . Denote by J˜ the restriction to which again is of the form [1, (31)], in [1, Appendix A]. In fact the first

(9)

˜ of the scale assignment J. We bound G ˜ J˜, G by a variant of the six-step procedure followed four steps are identical.

˜ with the property that T˜ ∩ G ˜ J˜ is a connected 1. Choose a spanning tree T˜ for G f ˜ J˜). tree for every f ∈ t(G 2. Apply any q-derivatives. By the product rule each derivative may act on any line or vertex on the “external momentum path”. It suffices to consider any one such action. 3. Bound each two-legged renormalized subgraph (i.e. r-fork) by (9) and each twolegged counterterm (i.e. c-fork) by (8). Observe that when s00 k0 -derivatives and 0 0 s01 k-derivatives act on the vertex, the bound is no worse than M −s0 j M −s1 (1−¯ε)j times the bound with no derivatives, because we necessarily have j ≤ jπ(f ) < 0. 4. Bound all remaining vertex functions, uv , (suitably differentiated) by their suprema in momentum space. We have already observed that if s1 = 0, the bound (7) is contained in [1, Proposition A.1], with s = 0. In the event that s1 = 1, but the spatial gradient acts on a vertex, [1, Proposition A.1], again with s = 0 but with either one v replaced by its gradient or with an extra factor of M −(1−¯ε)j coming from Step 3, again gives (7). So it suffices to consider the case that s1 = 1 and the spatial gradient acts on a propagator of the “external momentum path”. It is in this case that we apply the arguments of Lemma 4.2. The heart of those arguments was the observation that, when the gradient acted on a line `3 of scale j3 , the line `3 also lay on distinct momentum loops, Λ`1 and Λ`2 , generated by lines, `1 and `2 of scales j`1 and j`2 with j`1 , j`2 ≤ j`3 . This is still the case and is proven in Lemmas 4.3 and 4.4 below. So we may now apply the procedure of Lemma 4.1. Reduction 1. It suffices to consider j ≤ j`3 ≤ (1 − η˜)j. For the remaining terms, we simply bound the differentiated propagator, as in the argument following (4), by ¯ s0 ¡ ¢¯ ¯∂q00 ∂q Cj k` (q) ¯ ≤ const M −2j`3 M −s00 j`3 3 `3 0

≤ const M −j`3 M −s0 j`3 M −(1−˜η)j 0

≤ const M −j`3 M −s0 j`3 M −(1−¯ε)j . So for the terms with j`3 > (1 − η˜)j, the effect of the spatial gradient is to degrade the s1 = 0 bound by at most a factor of const M −(1−¯ε)j and we may apply the rest of [1, Proposition A.1], starting with Step 5, without further modification. ¡ ¢ Reduction 2. It suffices to consider loop momenta k` `∈G\ ˜ T˜ , in the domain of integration of [1, (31)], for which the momentum k`3 flowing through `3 (which is

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a linear combination of q and various loop momenta) remains a distance at least ηj`3 M ηj`3 from ¯all singular from some singular `3 is at most a distance M ¯ points. If k ηj point then ¯∂k e(k`3 )¯ ≤ const M `3 and we may use the ∇e in the numerator of (4) to improve the bound on the `3 propagator to ¯ s0 ¡ ¢¯ ¯∂q00 ∂q Cj k` (q) ¯ ≤ const M −2j`3 M −s00 j`3 M ηj`3 3 `3 0

≤ const M −j`3 M −s0 j`3 M −(1−η)j 0

≤ const M −j`3 M −s0 j`3 M −(1−¯ε)j . Once again, in this case, the effect of the spatial gradient is to degrade the s1 = 0 bound by at most a factor of const M −(1−¯ε)j and we may apply the rest of the proof [1, Proposition A.1], starting with Step 5, without further modification. Reduction 3. Now apply Step 5, that is, bound every propagator. The extra ∂q acting on Cj`3 gives a factor of M −s1 j`3 ≤ M −s1 j worse than the bound achieved in Step 5 of [1, Proposition A.1]. Prepare for the application of Step 6, the integration over loop momenta, by ordering the integrals in such a way that the two integrals executed first (that is the two innermost integrals) are those over k`1 and k`2 . The momentum flowing through `3 is of the form k`3 = ±k`1 ± k`2 + q 0 , where q 0 is some linear combination of the external momentum q and possibly various other loop momenta. Make a change of variables from k`1 and k`2 to k = k`3 = ±k`1 ±k`2 +q0 ˜ p ˜ and q ˜ 0 in R2 , there and p = k`2 . It now suffices to show that, for each fixed k, ˜ p ˜, q ˜ 0 dependent, but j`i independent) constants c and C such that are (possibly k, Z Z d2 k d2 pχj`1(±k ± p ± q0 )χj`2 (p)χj`3 (k)Ξj`3(k) ˜ |k−k|≤c

|p−˜ p|≤c

0

≤ C|j`2 |M j`2 M (1−η)j`3 M εj`3 −η j`3

(10)

˜ 0 | ≤ c. Recall that χj (k) is the characteristic function of for all q0 obeying |q0 − q the set of k’s with |e(k)| ≤ M j and Ξj (k) is the characteristic function of the set ˜ | ≥ M ηj for all singular points q ˜ }. {k ∈ R2 | |k − q Once proven, the bound (10) (together with the usual compactness argument) replaces the bound Z Z 2 d k`1 d2 k`2 χj`1 (k`1 )χj`2 (k`2 ) ≤ const|j`1 |M j`1 |j`2 |M j`2 used in Step 6 of [1, Proposition A.1]. Since j ≤ j`1 , j`2 ≤ j`3 ≤ (1 − η˜)j, the bound (10) constitutes an improvement by a factor of 0

const M j`2 M (1−η)j`3 M εj`3 −η j`3 M −j`1 M −j`2 0

≤ const M (1−η−η +ε)j`3 M −j`1 0

≤ const M [(1−η−η +ε)(1−˜η)−1]j ≤ const M ε¯j .

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So once we proven the bound (10), we may continue with the rest of the proof of [1, Proposition A.1], without further modification, and show that the inductive hypothesis (7) is indeed preserved. In fact the bound (10) has already been proven in Reduction 4 and The Final Step of the proof of Lemma 4.1. Lemma 4.3. Let G be any two-legged 1PI graph with each vertex having an even number of legs. Let T be a spanning tree for G. Assume that the two external legs of G are hooked to two distinct vertices and that `3 is a line of G that is in the linear subtree of T joining the external legs. Recall that any line ` not in T is associated with a loop Λ` that consists of ` and the linear subtree of T joining the vertices at the ends of `. There exist two lines `1 and `2 , not in T such that `3 ∈ Λ`1 ∩ Λ`2 . Proof. Since T is a tree, T \{`3 } necessarily contains exactly two connected components T1 and T2 (though one could consist of just a single vertex). On the other hand, since G is 1PI, G\{`3 } must be connected. So there must be a path in G\T that connects the two components of T \{`3 }. Since T is a spanning tree, every line of G\T joins two vertices of T , so we may alway choose the connecting path to consist of a single line. Let `1 be any such line. Then `3 ∈ Λ`1 . If G\{`1 , `3 } is still connected, then there must be a second line `2 6= `1 of G\T that connects the two components of T \{`3 }. Again `2 ∈ Λ`1 . If G\{`1 , `3 } is not connected, it consists of two connected components G1 and G2 with G1 containing T1 and G2 containing T2 . Each of T1 and T2 must contain exactly one external vertex of G. So each of G1 and G2 must have exactly one external leg that is also an external leg of G. As `1 and `3 are the remaining external legs of both G1 and G2 , each has three external legs, which is impossible. Lemma 4.4. Let G and T be as in Lemma 4.3. Let J be an assignment of scales to the lines of G such that T ∩ GJf is a connected tree for every fork f ∈ t(GJ ). (See Step 1 of [1, Proposition A.1].) Let `3 ∈ T . Let ` ∈ G\T connect the two components of T \{`3 }, then j` ≤ j`3 . Proof. Let G0 be the connected component containing ` of the subgraph of G consisting of lines having scales j ≥ j` . By hypothesis, G0 ∩ T is a spanning tree for G0 . So the linear subtree of T joining the vertices of ` is completely contained in G0 . But `3 is a member of that line’s subtree and so has scale j`3 ≥ j` . 4.2. The frequency derivative of the self-energy We show to all orders that singularities in this derivative can only occur at the closure of the lattice generated by the Van Hove points. That the singularities really occur is shown by explicit calculations in model cases in the following sections. Lemma 4.5. Let G(q) be the value of any two-legged 1PI graph with external momentum q. Let B be the closure of the set of momenta of the form (0, q) with

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Pn ˜ i where n ∈ N and, for each 1 ≤ i ≤ n, si ∈ {0, 1} and q ˜ i is a q = i=1 (−1)si q singular point. Then G(q) is C 1 with respect to q0 on R3 \B. Proof. The proof is similar to that of Lemma 4.2. Introduce scales in the standard way and denote the root scale j. Choose a spanning tree T in the standard way. View c- and r-forks as vertices. The external momentum is always routed through the spanning tree so the derivative may only act on vertices and on lines of the spanning tree. The cases in which the derivative acts on an interaction vertex or cfork are trivial. If the derivative acts on an r-fork, the effect on the bound [1, (29)], namely a factor of M −jπ(f ) , is the same as the effect on the bound [1, (32)] when the derivative acts on a propagator attached to the r-fork. So suppose that the derivative acts on a line `3 of the spanning tree that has scale j3 . We know from Lemma 4.3 in the last section that there exist two different lines `1 and `2 , not in T such that `3 lies on the loops associated to `1 and `2 . We also know, from Lemma 4.4, that the scales j 0 of all loop momenta running through `3 , including the scales j1 and j2 of the two lines chosen, obey j 0 ≤ j3 . Reduction 1. It suffices to consider j ≤ j1 , j2 ≤ j3 ≤ (1 − η˜)j. For the remaining terms, we simply bound ¯ ¯ ¯ d ¯ ¯ ¯ ≤ const M −2j3 ≤ const M −j3 M −(1−˜η)j . C (±q ± internal momenta) ¯ dq0 j3 ¯ After one sums over all scales except the root scale j, one ends up with const|j|n M j M −(1−˜η)j = const|j|n M η˜j which is still summable. Reduction 2. Denote by k, p and ±k ± p + q 0 the momenta flowing in the lines `1 , `2 and `3 , respectively. Here q 0 is plus or minus the external momentum q possibly plus or minus some some other loop momenta. In this reduction we prove ˜ ≤ M ηj and |p − p ˜ | ≤ M ηj and that it suffices to consider (k, p) with |k − k| ˜ p ˜ | ≤ M ηj for some singular points k, ˜ and q ˜. |±k ± p ± q0 − q 0 Suppose that at least one of k, p, ±k±p±q is farther than M ηj from all singular points. We can make a change of variables (just for the purposes of computing the volume of the domain of integration) such that k is at least a distance M ηj from all singular points. After the change of variables, the indices j1 , j2 and j3 are no longer ordered, but all are still between j and (1 − η˜)j. Let Ξj (k) be the characteristic function of the set ˜ ≥ M ηj for all singular points k}. ˜ {k ∈ R2 | |k − k| We claim that Vol{(k, p) ∈ R4 | χj1 (k)Ξj (k)χj2 (p)χj3 (±k ± p ± q0 ) 6= 0} ≤ C|j2 |M j1 −ηj M j2 M εj−ηj . This would constitute a volume improvement of M (ε−2η)j and would provide summability if 2η < ε. By the usual compactness arguments, it suffices to show

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˜ p ˜ p ˜ and q ˜ in R2 , there are (possibly k, ˜, q ˜ dependent, but that, for each fixed k, j, ji independent) constants c and C such that Z

Z 2

˜ |k−k|≤c

d k |p−˜ p|≤c

d2 pχj1 (k)Ξj (k)χj2 (p)χj3 (±k ± p ± q0 )

≤ C|j2 |M j1 −ηj M j2 M εj−ηj ˜ or p ˜ ±p ˜ | ≤ c. Furthermore, if k ˜ or ±k ˜ ±q ˜ 0 does not lie on for all q0 obeying |q0 − q ˜| ≤ c F, we can choose c sufficiently small that the integral vanishes whenever |q0 − q ˜ ˜ ˜ and ±k ± p ˜ ±q ˜0 and |j1 |, |j2 |, |j3 | are large enough. So it suffices to require that k, p all lie on F. ˜ is not a singular point, make a change of variable to ρ = e(k) and an If k ˜ is a singular point, the condition Ξj (k) 6= 0 forces |k− k| ˜ ≥ “angular” variable θ. If k ¡ ¢ ηj M . We can make a change of variables such that e k(ρ, θ) = ρθ and either |θ| ≥ const M ηj or |ρ| ≥ const M ηj . Possibly exchanging the roles of ρ and θ, we may, without loss of generality assume the former. In all of these cases, the condition χj1 (k) 6= 0 forces |ρ| ≤ const M j1 −ηj . Thus Z

Z

˜ |k−k|≤c

d2 k |p−˜ p|≤c

d2 pχj1 (k)Ξj (k)χj2 (p)χj3 (±k ± p ± q0 )

Z

≤ const Z ≤ const

Z

|θ|≤1 |ρ|≤constM j1 −ηj

|θ|≤1 |ρ|≤constM j1 −ηj

dρdθ |p−˜ p|≤c

Z dρdθ

Z

≤ const M j1 −ηj

|p−˜ p|≤c

d2 pχj 0 (±k(0, θ) ± p ± q0 )χj2 (p)

Z

dθ |θ|≤1

d2 pχj2 (p)χj3 (±k(ρ, θ) ± p ± q0 )

|p−˜ p|≤c

d2 pχj 0 (±k(0, θ) ± p ± q0 )χj2 (p)

0

where M j = M j3 + const M j1 −ηj ≤ const M (1−η−˜η)j . Thus it suffices to prove that Z

Z d2 p |p−˜ p|≤c

|θ|≤1

dθχj 0 (±k(0, θ) ± p ± q0 )χj2 (p) ≤ C|j2 |M j2 M εj−ηj

˜ | ≤ c. for all q0 obeying |q0 − q We again apply Proposition 3.3 with j = j 0 and δ = M (1+ε)j2 /2 . If we denote by χ(p) ˜ the characteristic function of the set of p’s with ¯ ¡ ¡ ¢¯ 0 ¢ µ {−1 ≤ θ ≤ 1 | ¯e ± p ± q0 ± k(0, θ) ¯ ≤ M j } ≥ c1

Ã

Mj

0

M (1+ε)j2 /2

!1/n0

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where c1 is the supremum of dθ ˜ vanishes except on a ds (s is arc length), then χ(p) set of measure DM (1+ε)j2 and Z Z d2 p dθχj 0 (±p ± q0 ± k(0, θ))χj2 (p) |θ|≤1

Z

Z

d2 pχ(p)χ ˜ j2 (p)

≤

Z +

dθχj 0 (±p ± q0 ± k(0, θ)) |θ|≤1

¡ ¢ d2 p 1 − χ(p) ˜ χj2 (p)

Z ≤2

Z dθχj 0 (±p ± q0 ± k(0, θ)) |θ|≤1

Ã

Z d2 pχ(p) ˜ + const

d2 pχj2 (p) Ã

≤ const M

(1+ε)j2

+ const|j2 |M

Mj

!1/n0

M (1+ε)j2 /2 Mj

j2

0

!1/n0

0

M (1+ε)j2 /2

¡ 1+ε ¢1/n0 ≤ const M j2 M ε(1−˜η)j + const|j2 |M j2 M (1−η−˜η)j− 2 j 1

1

≤ const M j2 M ε(1−˜η)j + const|j2 |M j2 M n0 ( 3 −η)j ≤ const|j2 |M j2 M εj−ηj © ª 1 provided η˜ ≤ η ≤ 12 and ε ≤ min 16 , 3n1 0 . End stage of proof. The momentum flowing through the differentiated line is of the form ±q ± k1 ± · · · ± kn where q is the external momentum and each of the ki ’s is a loop momentum of a line not in the spanning tree whose scale is no closer to zero than the scale of the differentiated line. / B, then either q0 is nonzero or ¯ ¯ If q ∈ ˜1 ± · · · ± k ˜n − k ˜ n+1 ¯, with the k ˜ i ’s running over all singular the infimum of ¯ ± q ± k points, is nonzero. So there exists a j0 such that when the root scale obeys j ≤ j0 , either the zero component of one of k1 , . . . , kn , ±q ± k1 ± · · · ± kn has magnitude larger than const M j , in which case the corresponding covariance vanishes, or the distance of one of k1 , . . . , kn , ±q ± k1 ± · · · ± kn to the nearest singular point is at least M ηj , in which case we can apply the argument of Reduction 2. (If it is one of k1 , . . . , kn whose distance to the nearest singular point is at least M ηj , we may choose as the `1 of Lemma 4.3 the line initiating that loop momentum.) 5. Singularities In this section, we do a two-loop calculation for a typical case to show that the frequency derivative of the self-energy is indeed divergent in typical situations, and to calculate the second spatial derivative at the singular points. By the Morse lemma, there are coordinates (x, y) such that in a neighborhood of the Van Hove singularity, the dispersion relation becomes e(k) = e˜(x, y) = xy.

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k1 k2 Here we consider the case of a Van Hove singularity at k = 0 with e(k) = (2π) 2. (In particular the Van Hove singularity is on the Fermi surface for µ = 0.) The nonlinearities induced by the changes of variables are absent in this example, and k1 k2 moreover, the curvature is zero on the Fermi surface. We rescale to x = 2π , y = 2π and, for definiteness, take the integration region for each variable to be [−1, 1]. For this case, we determine the asymptotics of derivatives of the two-loop contribution to the self-energy as a function of q0 for small q0 . We find that again, the gradient of the self-energy is bounded (in fact, the correction is zero in that case) but that the q0 -derivative is indeed divergent. Power counting by standard scales 3 suggests that this derivative diverges at zero temperature like |log q0 | . However, there is a cancellation of the leading singularity which is not seen when taking absolute values, so that the true behavior is only (log q0 )2 . We then also calculate the asymptotics of the second spatial derivative and find that it is of the same order as the first frequency derivative. Finally, we do the calculation for the one-loop contributions to the four-point function, to compare the coefficients of different divergences in perturbation theory. The physical significance of these results will be discussed in Sec. 6.

5.1. Preparations We restrict to a local potential. Since we have only considered short-range interactions, the potential is smooth in momentum space. For differentiability questions, a momentum dependence could only make a difference if the potential vanished at the singular points or other special points, so the restriction to a local potential, which is constant in momentum space, is not a loss of generality. There are two graphs contributing, one of vertex correction type and the other of vacuum polarization type (the graphs with insertion of first-order self-energy graphs have been eliminated by renormalization through a shift in µ).

The latter gets a (−1) from the fermion loop and a 2 from the spin sum. Thus the total contribution is Z 1 X 1 d2 k1 d2 k2 d2 k3 δ(q − k1 + k2 − k3 ) Σ2 (q0 , q) = − 2 β ω ,ω (2π)4 1

2

× C(ω1 , e(k1 ))C(ω2 , e(k2 ))C(q0 − ω1 + ω2 , e(k3 )) with C(ω, E) =

1 . iω − E

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Call Ei = e(ki ) and Z hF iq =

dρq (k)F (k)

R 2 1 where k = (k1 , k2 , k3 ) and dρq (k) = (2π) d k1 d2 k2 d2 k3 δ(q − k1 + k2 − k3 ). The 4 frequency summation gives ¿ À (fβ (E1 ) + bβ (E2 − E3 )) (fβ (E2 ) − fβ (E3 )) Σ2 (q0 , q) = − (11) iq0 + (E2 − E3 − E1 ) q where fβ (E) = (1 + eβE )−1 is the Fermi function and bβ (E) = (eβE − 1)−1 is the Bose function. Since £ ¤ £ ¤ bβ (E2 − E3 ) fβ (E2 ) − fβ (E3 ) = fβ (E2 ) fβ (E3 ) − 1 the numerator is bounded in magnitude by 2. The denominator is bounded below in magnitude by |q0 |, so the integrand is C ∞ in q0 , for all q0 6= 0 and all β ≥ 0. Because the integral is over a compact region of momenta, the same holds for the integral. The limit β → ∞ exists and has the same property. The structure of the denominator may suggest that for it to almost vanish requires only the combination E2 − E3 − E1 to get small, but a closer look reveals that each Ei has to be small: at T = 0, the Fermi functions become step functions, fβ (E) → Θ(−E) and bβ (E) → −Θ(−E), and then the factors in the numerator imply that all summands in E2 − E3 − E1 really have the same sign, i.e. |E2 − E3 − E1 | = |E2 | + |E3 | + |E1 |, so all |Ei | must be small for the energy difference to be small. At finite β, when the Ei ’s are “of the wrong sign” the exponential suppression provided by the numerator compensates for the |q0 | ≥ πβ in the denominator. Because iq0 − e(q) − λ2 Σ2 (q0 , q) = iq0 (1 + iλ2 ∂0 Σ2 (0, q)) − (e(q) + λ2 Σ2 (0, q)) + · · · (12) and because ∂0 Σ2 (0, q) is purely imaginary, we are interested in ¡ ¢−1 Z2 (q) = 1 − λ2 Im ∂0 Σ2 (0, q) .

(13)

The value q0 = 0 is not an allowed fermionic Matsubara frequency at T > 0. We shall keep q0 6= 0 in the calculations. In discussions about temperature dependence, we shall replace |q0 | by π/β. The second order contribution to Im ∂0 Σ is Im ∂0 Σ2 (q0 , q) ® = Φq0 (E2 − E3 − E1 )[fβ (E1 ) + bβ (E2 − E3 )] [fβ (E2 ) − fβ (E3 )] q

(14)

with Φq0 (ε) = Re

1 ε2 − q02 = 2 . 2 (iq0 + ε) (ε + q02 )2

(15)

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5.2. q0 -derivative The above expression (14) for Im ∂0 Σ2 (q0 , q) shows that it is an even function of q0 and that q0 serves as a regulator so that even at zero temperature, a singularity can develop only in the limit q0 → 0. We therefore calculate it as a function of q0 at zero temperature. In the following, we take q0 > 0. As β → ∞, the Fermi function fβ (E) → Θ(−E) and for E 6= 0, the Bose function bβ (E) → −Θ(−E). In this limit, the integrand vanishes except when E2 E3 < 0 and E1 (E2 − E3 ) < 0. This reduces to the two cases E1 > 0,

E2 < 0,

E3 > 0 and

E1 < 0,

E2 > 0,

E3 < 0.

(16)

In both cases, the combination of fβ ,bβ ’s in the numerator is −1. Thus Im ∂0 Σ2 (q0 , q) = − Φq0 (E2 − E3 − E1 )[1(E1 > 0 ∧ E2 < 0 ∧ E3 > 0) ® + 1(E1 < 0 ∧ E2 > 0 ∧ E3 < 0)] q . Recall that we are considering the case of a Van Hove point at k = (x, y) = 0 for the dispersion relation e(k) = xy with x = k1 /2π and y = k2 /2π, so that d2 k (2π)2 = dxdy. Moreover we set q = 0, and use the delta function to fix E1 in terms of E2 and E3 , so that E3 = x0 y 0 ,

E2 = xy,

E1 = (x − x0 )(y − y 0 )

and ε = E2 − E3 − E1 = xy 0 − 2x0 y 0 + x0 y. Recall also that, for definiteness, we are taking the integration region for each variable to be [−1, 1]. The sign conditions on the Ei impose conditions on the variables x, . . . , which are listed in Appendix B, and which we use to transform the integration region to [0, 1]4 . At T = 0, only n ∈ M = {1, 2, 3, 4, 9, 10, 11, 12} from the table in Appendix B contribute. Thus Z X Im ∂0 Σ2 (q0 , 0) = − dxdydx0 dy0 1(ρn )Φq0 (εn (x, y, x0 , y0 )) [0,1]4

n∈M

with εn and ρn given in the table in Appendix B. By (31)–(33), and because Φq0 is even in ε, all eight terms give the same contribution. Hence Im ∂0 Σ2 (q0 , 0) = −2I(q0 )

(17)

where Z I(q0 ) = 4

Z

1

dy 0

Z

1

dy 0

0

1

Z

x0

0

dx 0

0

dxΦq0 ((2x0 − x)y 0 + yx0 ).

Lemma 5.1. Let I be defined as in (18) and 0 < q0 < 21 . Then 2

Im ∂0 Σ2 (q0 , 0) = −4 log2|log q0 | − 2C1 |log q0 | + B(q0 )

(18)

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where B is a bounded function of q0 , and µ ¶ Z 1 1 + 2x dx 2 C1 = 2(log 2) − 4 log . x 1+x 0 Proof. By (17), it suffices to show that ˜ 0) I(q0 ) = 2 log2|log q0 | + C1 |log q0 | + B(q 2

˜0 . We rewrite the argument ε of Φq as ε = x0 (2y 0 + y) − xy 0 . We with bounded B 0 first bound the contribution of y 0 ≤ q0 . To do this, bound |Φq0 (ε)| ≤

1 . + ε2

q02

Use that this bound is decreasing in ε and that ε ≥ x0 y. Therefore Z 1 Z q0 Z 1 Z x0 0 0 4 dy dy dx dxΦq0 ((2x0 − x)y 0 + yx0 ) 0

0

0

Z

Z

1

≤4

0

Z

q0

dy 0

dy 0

Z

1

= 4q0

dx0

0

Z

dx0

0

0

x0

dx

dx

0 1

0

q02

1 + (x0 y)2

0

dy 0

1

= 4q0

Z

Z

1

0

q02

x + (x0 y)2

1 x0 arctan q0 q0

≤ 2π. Thus it suffices to calculate the asymptotic behavior of Z 1 Z 1 Z 1 Z x0 0 0 ˜ I(q0 ) = 4 dy dy dx dxΦq0 (x0 (2y 0 + y) − xy 0 ) 0

q0

0

0

for small q0 > 0. By (15) Φq0 (ε) = − Because

∂ε ∂x

∂ ε . 2 ∂ε q0 + ε2

= −y 0 , Z 0

x0

· ¸(y0 +y)x0 1 ε dx Φq0 (ε(x)) = 0 2 . y ε + q02 (2y0 +y)x0

0

The integral over x can now be done, using µ ¶ Z 1 αx0 α2 1 dx0 log 1 + = . (αx0 )2 + q02 2α q02 0 Thus ˜ 0 ) = I˜1 − I˜2 I(q

(19)

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with Z

1

I˜j = 4

q0

Z

1 q0

=2

Z

dy 0 y0

dy 0

dη η

1

1

Z

1 q0

jη

µ ¶ (jy 0 + y)2 1 log 1 + 2(jy 0 + y) q02

+jη

dξ log(1 + ξ 2 ) ξ jy 0 +y q0 ,

where we have made the change of variables ξ = change of variables η =

0

y q0 ,

dη = Z

˜ 0) = I(q

1 0 q0 dy .

= 1

with

Z JA = 2 A

1 q0 dy

followed by the

Thus

q0−1

dη − J[2η,q−1 +2η] ) (J −1 0 η [η,q0 +η]

q0−1

dη (J[η,2η] − J[q−1 +η,q−1 +2η] ) 0 0 η

1

Z

dξ =

dξ log(1 + ξ 2 ) ≥ 0 for A ⊂ [0, ∞). ξ

We have Z J[a,b] = 2

a

b

¡ ¢ dξ log 1 + ξ 2 = ξ

Z

b2

a2

dt log(1 + t). t

(20)

In our case both integration intervals for J are subsets of [1, ∞), so we can expand the logarithm, to get · µ ¶¸ Z b2 dt 1 J[a,b] = log(t) + log 1 + t a2 t ¤ X (−1)n −2n 1£ (log b2 )2 − (log a2 )2 − (a − b−2n ) = 2 n2 n≥1

£ ¤ X (−1)n −2n = 2 (log b)2 − (log a)2 − (a − b−2n ) n2 n≥1

b X (−1)n −2n = 2 log(ab) log − (a − b−2n ). a n2 n≥1

The final integral over η gives, for the first term, Z 1

q0−1

dη J[η,2η] = 2 log 2|log q0 |2 + 2(log 2)2 |log q0 | η 1 X (−1)n − (1 − 4−n )(1 − q02n ) 2 n3 n≥1

(21)

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with the last term analytic, and hence bounded, for |q0 | < 1. The second term gives two contributions: Z

q0−1

1

dη −1 J −1 = W −R η [q0 +η,q0 +2η]

with (here M = q0−1 ) Z R= 1

M

dη X(η), η

X(η) =

X (−1)n [(M + η)−2n − (M + 2η)−2n ] n2

n≥1

and Z

M

dη [(log(M + αη))2 ]α=2 α=1 . η

W =2 1

For |q0 | ≤ 1 the series for X converges absolutely and gives X (−1)n £ ¤ q 2n (1 + ηq0 )−2n − (1 + 2ηq0 )−2n n2 0

X=

n≥1

so |X| ≤

P

2 2n n≥1 q0 /n

≤ 2q02 , hence for |q0 | ≤ 1, R ≤ 2q02 | ln q0 | ≤ q0 .

In W , scaling back to x = q0 η gives Z

1

W =2 q0

Z

1

=2 q0

dx [(log q0−1 + log(1 + αx))2 ]α=2 α=1 x dx [2 log q0−1 log(1 + αx) + (log(1 + αx))2 ]α=2 α=1 . x

Because αx ≥ 0, 0 ≤ log(1+αx) ≤ αx, so W is bounded by a constant times log q0−1 . The integral of the same function from 0 to q0 is of order q0 | log q0 |, therefore Z W =

4 log q0−1

0

1

µ ¶ dx 1 + 2x ˜ 0) log + B(q x 1+x

˜ is a bounded function. where B 5.3. First spatial derivatives q In our model case, we have E2 = xy and E3 = x0 y 0 . Moreover we take 2π = (ξ, η) so 0 0 that, fixing k1 by momentum conservation, E1 = (ξ + x − x )(η + y − y ). It is clear from (11) that the spatial derivatives also act on the Fermi function fβ (E1 ), so we

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get two terms. The derivative of the Fermi function fβ (x) is minus the approximate delta δβ (x) = β/(4 cosh2 (βx/2)) so that ¸À · (f1 + b23 )(f2 − f3 ) f2 − f3 + (∂i e) −δβ (E1 ) iq0 + ε (iq0 + ε)2 q

¿ −∂i Σ2 (q0 , q) =

(22)

where ∂i e has argument q + k2 − k3 , fi = fβ (Ei ), b23 = bβ (E2 − E3 ) and, as before, ε = E2 − E3 − E1 . At q = 0 −

∂ Σ2 (q0 , 0) = S1 (β, q0 ) + S2 (β, q0 ) ∂ξ

with Z S1 (β, q0 ) =

dxdydx0 dy 0 (y − y 0 )

fβ (xy) − fβ (x0 y 0 ) (−δβ )((x − x0 )(y − y 0 )) iq0 + ε

dxdydx0 dy 0 (y − y 0 )

fβ (xy) − fβ (x0 y 0 ) (iq0 + ε)2

[−1,1]4

Z S2 (β, q0 ) =

[−1,1]4

× [fβ ((x − x0 )(y − y 0 )) + bβ (xy − x0 y 0 )]. Here ² = xy − x0 y 0 − (x − x0 )(y − y 0 ) = xy 0 + x0 y − 2x0 y 0 . Now consider S1 and apply the reflection (x, y, x0 , y 0 ) → (−x, −y, −x0 , −y 0 ) to the integration variables. The domain of integration is invariant. The only noninvariant factor is y − y 0 and it changes its sign. Thus S1 vanishes. By the same argument, S2 vanishes as well. By symmetry, the same holds for the η-derivative. Thus ∇Σ2 (q0 , 0) = 0. 5.4. The second spatial derivatives q Let 2π = (ξ, η). The real part of Σ2 (ξ, η) is a correction to e(ξ, η) = ξη. Since ∂ξ ∂η e(ξ, η) = 1, we calculate the correction that Σ2 gives to that quantity.

Lemma 5.2. For small q0 6= 0 lim Re

β→∞

∂2 Σ2 (q0 , 0) = (2 + 4 log 2)(log|q0 |)2 + O(| log|q0 ||). ∂ξ∂η

(23)

Proof. By symmetry it suffices to consider q0 > 0. In general, the second order spatial derivatives are (i,j)

−∂i ∂j Σ2 (q0 , q) = Z1

(i,j)

+ Z2

(i,j)

+ Z3

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(i,j)

Z2

(i,j)

Z3

315

¶ À ¿µ (∂i e)(∂j e) (f1 + b23 )(f2 − f3 ) , = (∂i ∂j e) + 2 iq0 + ε (iq0 + ε)2 q À ¿ ¡ ¢ f2 − f3 0 , = (∂i e)(∂j e) (−δβ )(E1 ) + (∂i ∂j e)(−δβ )(E1 ) iq0 + ε q À ¿ f2 − f3 = 2(∂i e)(∂j e)(−δβ )(E1 ) . (iq0 + ε)2 q

Here e = e(k2 − k3 + q) = E1 and ε = E2 − E3 − E1 . Denote ∂2 Σ2 (q0 , 0, 0). ∂ξ∂η In the xy case, the two integration momenta are denoted by (x, y) and (x0 , y 0 ), and E2 = xy, E3 = x0 y 0 , and E1 = e(k2 − k3 + q) = (ξ + x − x0 )(η + y − y 0 ). Since 2 ∂ξ e |ξ=η=0 = (y − y 0 ), ∂η e |ξ=η=0 = (x − x0 ), and ∂ξη e = 1, ζ(q0 ) = −

ζ = ζ1 + ζ2 + ζ3 with

µ ¶ (x − x0 )(y − y 0 ) (f1 + b23 )(f2 − f3 ) ζ1 = d X 1+2 0 0 iq0 + ε(x, y, x , y ) (iq0 + ε(x, y, x0 , y 0 ))2 [−1,1]4 Z ¢ ¡ f2 − f3 ζ2 = d4 X −E1 δβ0 (E1 ) − δβ (E1 ) iq + ε(x, y, x0 , y 0 ) 4 0 [−1,1] Z f2 − f3 ζ3 = d4 X2(−E1 )δβ (E1 ) . (iq + ε(x, y, x0 , y 0 ))2 4 0 [−1,1] Z

4

Here X = (x, y, x0 , y 0 ) and d4 X = dxdydx0 dy 0 , and ε(x, y, x0 , y 0 ) = xy − x0 y 0 − E1 = xy − x0 y 0 − (x − x0 )(y − y 0 ). Using the decomposition given in Appendix B, Z 4 µ X 0 0 ζ1 −→ − 2 dxdydx dy 1+2 β→∞

[0,1]4

j=1

Fj iq0 + εj

¶

1 1(ρj ). (iq0 + εj )2

(The limit can be taken under the integral. Decomposing according to the signs of x, x0 . . . , only j ∈ {1, 2, 3, 4, 9, 10, 11, 12} can contribute. Using the symmetry (31), one obtains the above. The minus sign arises from the combination of Fermi functions fβ , as discussed around (16).) We write ζ1 = −(ζ11 + ζ12 ). The term involving the 1 is Z 4 X 1 ζ11 = 2 dxdydx0 dy0 1(ρi ). (iq + εi )2 4 0 [0,1] i=1 By (32) and (33),

Z ζ11 = 8

dx dy dx0 dy0 Re [0,1]4

1 1(ρ1 ). (iq0 + ε1 )2

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This is the same (up to a sign) as (17), (18), hence ζ11 = 4 log 2(log |q0 |)2 + O(|log |q0 ||). The term ζ12 involving the Fj gives another contribution, Z 4 X Fj ζ12 = 2 dxdydx0 dy0 2 1(ρj ). 3 (iq 4 0 + εj ) [0,1] j=1 The summand for j = 2 gives the same integral as that for j = 1 because the integrand is related by the exchange x ↔ y and x0 ↔ y0 . Ditto for j = 4 and j = 3. Thus Z X Fj ζ12 = 8 dxdydx0 dy0 1(ρj ). (iq0 + εj )3 [0,1]4 j=1,3 Because F1 = −F3 and ε1 = −ε3 , and because ρ1 ⇔ ρ3 , X Fj 1 1(ρj ) = 1(ρ1 )F1 2 Re . 3 (iq + ε ) (iq + ε1 )3 0 j 0 j=1,3 Thus

Z

Z

1

ζ12 = 16 Re

dy

Z

1

dy

0

Z

1

0

x0

0

dx

0

0

dx 0

(x − x0 )(y + y0 ) . (iq0 + x0 (y + y0 ) + y0 (x0 − x))3

Let b = x0 (y + y0 ), change integration variables from x to u = x0 − x ∈ [0, x0 ], and use Z x0 u (x0 )2 du = . (iq0 + b + y0 u)3 2(iq0 + b + x0 y0 )2 (iq0 + b) 0 Renaming to z = x0 , we have Z 1 Z 1 Z 0 ζ12 = −8 dy dy 0

The bound

0

1

dz Re

0

(y + y0 ) z 2 . (iq0 + z(y + y0 ))(iq0 + z(y + 2y0 ))2

¯ ¯ ¯ ¯ (y + y0 ) z 2 z ¯ ¯ ¯ (iq0 + z(y + y0 ))(iq0 + z(y + 2y0 ))2 ¯ ≤ q 2 + z 2 (y + 2y0 )2 0

for the integrand implies that Z 1 Z |ζ12 | ≤ 4 dy 0

0

1

µ ¶ (y + 2y0 )2 1 ln 1 + dy (y + 2y0 )2 q02 0

which shows that |ζ12 | ≤ const[log |q0 |]2 for small |q0 |. We now calculate the coefficient of the log2 . First rewrite 1 z2 (y + y0 ) z 2 = 0 0 2 0 2 (iq0 + z(y + y )) (iq0 + z(y + 2y )) (y + 2y ) (z + iA)(z + iB)2 with A=

q0 y + y0

B=

q0 . y + 2y0

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Partial fractions give β b2 z2 α + +i = (z + iA)(z + iB)2 z + iA z + iB (z + iB)2 with

¶2 y + 2y0 y0 µ ¶2 B(B − 2A) y + 2y0 β= =1− (B − A)2 y0 A2 = (B − A)2

b2 =

B2 y + y0 =B . A−B y0

α, β and b2 are real, so we need Z 1 Re 0

and

Z Re ib2 0

With this, we have Z 1 Z 1 ζ12 = −4 dy dy0 0

0

µ

α=

dz 1 = ln(1 + A−2 ) z + iA 2 1

b2 dz = . 2 (z + iB) B(1 + B 2 )

· ¸ 1 2b2 −2 −2 α ln(1 + A ) + β ln(1 + B ) + . (y + 2y0 )2 B(1 + B 2 )

Collecting terms and renaming y0 = η gives Z 1 Z 1 · 1 q 2 + (y + η)2 y+η 1 ζ12 = −4 dy dη 2 ln 20 +2 2 2 η q0 + (y + 2η) η q0 + (y + 2η)2 0 0 µ ¶¸ 1 (y + 2η)2 + ln 1+ . (y + 2η)2 q02 Although the first summands individually contain nonintegrable singularities at η = 0, these singularities cancel in the sum. A convenient way to implement this is to use that q02 + (y + η)2 1 y 1 ln +2 2 η 2 q02 + (y + 2η)2 η q0 + (y + 2η)2 · · ¸ ¸ 1 q 2 + (y + η)2 2(y + η) 1 2(y + 4η) ∂ − ln 20 + − . = ∂η η q0 + (y + 2η)2 η q02 + (y + η)2 q02 + (y + 2η)2 Moreover

µ ¶ (y + 2η)2 1 ln 1 + (y + 2η)2 q02 · µ ¶¸ 1 (y + 2η)2 1 ∂ 2 − ln 1 + = + 2 . 2 ∂η y + 2η q02 q0 + (y + 2η)2

(24)

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Thus

µ ¶¸η=1 1 q02 + (y + η)2 1 1 (y + 2η)2 dy − ln 2 − ln 1 + = −4 η q0 + (y + 2η)2 2 y + 2η q02 0 η=0 · ¸ Z 1 Z 1 2 y + 2η y+η −4 dy dη 2 + (y + η)2 − q 2 + (y + 2η)2 . η q 0 0 0 0 Z

ζ12

1

·

Evaluation at η = 1 gives one bounded term and one term of order log|q0 |. The terms at η = 0 give ¶¸ µ Z 1 · 1 y2 2y − −4 dy 2 ln 1 + 2 . q0 + y 2 2y q0 0 The first summand integrates to O(ln|q0 |). By (21), the second term gives 2(ln|q0 |)2 + bounded. The remaining integrals are Z 1 Z −8 dy 0

Z

=8

·

1

dη

0

Z

1

dy 0

2 1 − 2 q02 + (y + η)2 q0 + (y + 2η)2

2

dη 1

¸

1 ≤8 q02 + (y + η)2

since the integrand is bounded by 1, and · ¸ Z 1 Z 1 2 y y −4 dy dη − . η q02 + (y + η)2 q02 + (y + 2η)2 0 0 The integrand is bounded by a constant times 2

1 q02 +y2 +η 2

so the integral is of order

log|q0 |. Thus ζ12 = 2(log|q0 |) + bounded terms, so that ζ1 = −(ζ11 + ζ12 ) = −(4 log2 + 2)(log|q0 |)2 + less singular terms. We now show that ζ2 and ζ3 vanish in the limit β → ∞. Consider ζ2 first. Let Gβ (E1 ) = −E1 δβ0 (E1 ) − δβ (E1 ). (Note that Gβ (E) = βG1 (βE).) At ξ = η = 0, E1 = (x − x0 )(y − y 0 ) is invariant under the exchange (x, y) ↔ (x0 , y 0 ), so µ ¶ Z 1 1 ζ2 = d4 XGβ (E1 )fβ (xy) − . iq0 + ε(x, y, x0 , y 0 ) iq0 + ε(x0 , y 0 , x, y) [−1,1]4 Because ε(x, y, x0 , y 0 ) = xy 0 + x0 y − 2x0 y 0 , ε(x0 , y 0 , x, y) − ε(x, y, x0 , y 0 ) = 2(x0 y 0 − xy) = 2[x(y 0 − y) + (x0 − x)y 0 ]. Hence ζ2 = 2

Z [−1,1]4

Consider first Z T1 = 2

[−1,1]4

d4 XGβ (E1 )fβ (xy)

d4 XGβ (E1 )fβ (xy)

x(y 0 − y) + (x0 − x)y 0 . (iq0 + ε(x, y, x0 , y 0 ))(iq0 + ε(x0 , y 0 , x, y))

(iq0 +

x(y 0 − y) . + ε(x0 , y 0 , x, y))

ε(x, y, x0 , y 0 ))(iq0

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Because [−1, 1]4 and the integrand are invariant under the reflection R which maps (x, y, x0 , y 0 ) to (−x, −y, −x0 , −y 0 ) and R{(x, y, x0 , y 0 ) | y ≥ y 0 } = {(x, y, x0 , y 0 ) | y ≤ y 0 }, we can put in a factor 2 1(y > y 0 ). Changing variables from x0 to E1 , so that x0 = x − gives

E1 , y − y0

Z T1 = 4

[−1,1]3

Z

dxdydy 0 1(y − y 0 > 0)(−x)fβ (xy)

(x+1)(y−y 0 )

(x−1)(y−y 0 )

dE1

Gβ (E1 ) . (iq0 + ε(x, y, x0 , y 0 ))(iq0 + ε(x0 , y 0 , x, y))

Change variables one last time, from E1 to u = βE1 , to get Z Z 0 T1 = 4 dxdydy du [−1,1]3

R

0

1(y − y > 0)1(β(x − 1)(y − y 0 ) ≤ u ≤ β(x + 1)(y − y 0 )) (−x)fβ (xy)

(iq0 +

G1 (u) 0 ε(x, y, x , y 0 )) (iq0

+ ε(x0 , y 0 , x, y))

u where x0 = x − β(y−y 0 ) → x as β → ∞. The integrand is bounded in magnitude by 1 the L function 1 (x, y, y 0 , u) 7→ G1 (u) 2 q0

and it converges almost everywhere (namely, for x 6= ±1, xy 6= 0) to 1(y − y 0 > 0)xΘ(−xy)

£ ¤ 1 (−u)δ10 (u) − δ1 (u) . q02 + x2 (y − y 0 )2

By dominated convergence, the limit β → ∞ can be taken under the integral. The limiting range of integration for u is R. By the fundamental theorem of calculus Z Z ¤ d £ duG1 (u) = du −uδ1 (u) = 0, du R R so T1 vanishes as β → ∞. The calculation of the limit β → ∞ of Z (x0 − x)y 0 T2 = 2 d4 XGβ (E1 )fβ (xy) 0 (iq0 + ε(x, y, x , y 0 ))(iq0 + ε(x0 , y 0 , x, y)) [−1,1]4 is similar and gives 0 as well. Thus ζ2 → 0 as β → ∞. By the same arguments Z 2iq0 (˜ ε − ε) + (˜ ε − ε)(˜ ε + ε) ζ3 = d4 X2(−E1 )δβ (E1 )fβ (xy) 2 (iq0 + ε) (iq0 + ε˜)2 [−1,1]4

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where ε = ε(x, y, x0 , y 0 ) and ε˜ = ε(x0 , y 0 , x, y). After the same limiting argument as before, the u-integral is now Z uδ1 (u)du = 0 R

because the integrand is odd. Thus ζ3 → 0 as β → ∞, too. Lemma 5.3. The real part of the second derivative of the self-energy with respect to ξ grows at most logarithmically as q0 → 0 : there are constants A and B such that for all |q0 | < 1 ¯ ¯ 2 ¯ ¯ ¯Re ∂ Σ2 (q0 , 0)¯ ≤ A + B log 1 . ¯ ∂ξ 2 ¯ |q0 | The same holds by symmetry for the second derivative with respect to η. Proof. We need to bound ¿ À (fβ (E1 ) + bβ (E2 − E3 )) (fβ (E2 ) − fβ (E3 )) 2 ∂ξ iq0 + ε at ξ = η = 0 where all ξ-dependence is in E1 = (ξ + x − x0 )(η + y − y 0 ) and in ε = E2 − E3 − E1 = xy 0 + x0 y − 2x0 y 0 . We proceed as for the mixed derivative ∂ 2 /∂ξ∂η, but now some terms are different because ∂ξ2 E1 = 0. We obtain ∂ 2 Σ2 (q0 , 0, 0) = X1 + X2 + X3 ∂ξ 2 with ¿

(fβ (E1 ) + bβ (E2 − E3 )) (fβ (E2 ) − fβ (E3 )) X1 = 2 (y − y ) (iq0 + ε)3 + * δβ0 (E1 ) (y − y 0 )2 (fβ (E2 ) − fβ (E3 )) , X2 = iq0 + ε ¿ À −δβ (E1 ) (y − y 0 )2 X3 = 2 (fβ (E2 ) − fβ (E3 )) . (iq0 + ε)2

À

0 2

,

We first calculate the zero-temperature limit of X1 . Using the notations of Appendix B, Z (0) X1

= lim X1 = −4 β→∞

dxdx0 dydy0 [0,1]4

4 X

Dn2 1(ρn ). (iq0 + εn )3 n=1

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We use that D12 = D32 and ε3 = −ε1 , to combine n = 1 and n = 3 in one term, and (0) (0) (0) n = 2 and n = 4 in another, so that X1 = X1,1 + X1,2 with Z (0) X1,1

Z

1

= −8i

dy 0

Z

0

0

dy (y + y) Z

y

dy0

0

Z

1

2

x0

0

dx

0 1

(0)

X1,2 = −8i

Z

1

0

dx Im

0

0

Z

Z

dy(y0 − y)2

0

0

1

dx0

1

dx Im 0

(iq0 +

x0 y

1 , + y0 (2x0 − x))3

1 . (iq0 + xy0 + x0 (2y0 − y))3

Thus X1 does not contribute to Re ∂ξ2 Σ2 . We now show that X3 is imaginary in the limit β → ∞, because after a rewriting of terms, the difference fβ (E2 ) − fβ (E3 ) effectively implies taking the imaginary part. Because (y − y 0 )2 δβ (E1 ) is invariant under (x, y) ↔ (x0 , y 0 ) (and denoting ε˜(x, y, x0 , y 0 ) = ε(x0 , y 0 , x, y)) ·

Z

1 1 X3 = −2 d X(y − y ) δβ (E1 )fβ (xy) − (iq0 + ε)2 (iq0 + ε˜)2 Z Z Z = −4 dy dy 0 1(y − y 0 > 0) dxfβ (xy)(y − y 0 ) 4

¸

0 2

·

Z dx0 (y − y 0 )δβ (E1 )

¸ 1 1 − . (iq0 + ε)2 (iq0 + ε˜)2

For the second equality, we used invariance under the reflection (x, y, x0 , y 0 ) → (−x, −y, −x0 , −y 0 ). The convergence argument used in the analysis of the T1 con2 tribution to ∂ξη Σ2 can be summarized in the following lemma. Lemma 5.4. Let β0 ≥ 0 and F : [β0 , ∞) × [−1, 1]4 → C be bounded, and µ lim F β, x, y, x −

β→∞

u , y0 β(y − y 0 )

¶ = f (x, y, y 0 )

a.e. in (x, y, u, y 0 ). Let E1 = (x − x0 )(y − y 0 ). Then Z F (β, X)δβ (E1 )(y − y 0 )1(y > y 0 )d4 X

lim

β→∞

Z =

Z

Z dy 0 1(y > y 0 )

dy

dxf (x, y, y 0 ).

Applying Lemma 5.4, and using that ε(x, y, x, y 0 ) = x(y − y 0 ) = −˜ ε(x, y, x0 y 0 ), we get Z β→∞

Z

1

lim X3 = −4

y

dy −1

−1

Z dy 0 (y − y 0 )

1 −1

dxΘ 21 (−xy)I(x, y, y 0 )

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where Θ 12 (x) = limβ→∞ fβ (−x) is the Heaviside function, except that Θ 21 (0) = 1/2, · ¸ 1 1 I(x, y, y 0 ) = − (iq0 + x(y − y 0 ))2 (iq0 − x(y − y 0 ))2 1 . = 2i Im (iq0 + x(y − y 0 ))2 Thus, in the limit β → ∞, X3 does not contribute to the real part of ∂ξ2 Σ2 either. It remains to bound X2 . Here we integrate by parts, to remove the derivative from the approximate delta function, and then take β → ∞. We first do the standard rewriting Z X2 = 2 d4 X1(y − y 0 > 0)(y − y 0 )2 ·

× fβ (xy)δβ0 ((x

¸ 1 1 − x )(y − y )) − . iq0 + ε iq0 + ε˜ 0

0

We integrate by parts in x, using that (y − y 0 )δβ0 ((x − x0 )(y − y 0 )) = x0 )(y − y 0 )). This gives the boundary term Z Z Z X 0 0 B1/β = 2 x dy dy 1(y − y > 0) dx0 fβ (xy) x=±1

·

× (y − y 0 )δβ ((x − x0 )(y − y 0 ))

1 1 − iq0 + ε iq0 + ε˜

∂ ∂x δβ ((x

−

¸

and two integral terms, namely Z Z Z Z I1 = 2 dy dy 0 1(y − y 0 > 0) dx dx0 yδβ (xy) · × (y − y 0 )δβ ((x − x0 )(y − y 0 )) and

Z I2 = 2

Z dy

Z 0

0

dy 1(y − y > 0)

1 1 − iq0 + ε iq0 + ε˜

¸

Z dx ·

× (y − y 0 )δβ ((x − x0 )(y − y 0 ))

dx0 fβ (xy)

¸ y0 y 0 − 2y − . (iq0 + ε)2 (iq0 + ε˜)2

An obvious variant of Lemma 5.4, where x is summed over ±1 instead of integrated, applies to the boundary term and gives Z 1 Z y X B0 = lim B1/β = 2 x dy dy 0 Θ 21 (−xy) β→∞

x=±1

·

×

−1

−1

¸ 1 1 − . iq0 + x(y − y 0 ) iq0 − x(y − y 0 )

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Thus B0 is real. Because Θ 12 (y) + Θ 12 (−y) = 1, we get Z 1 Z y y − y0 B0 = 4 dy dy 0 2 . q0 + (y − y 0 )2 −1 −1 With z = y − y 0 , we have Z 1 Z B0 = 4 dy

Z 1 z 1 dz 2 =4 dy ln(q02 + z 2 )|1+y 0 2 q + z 2 −1 0 −1 0 ¶ µ ¶ µ Z 2 Z 1 η2 (1 + y)2 dη ln 1 + 2 =2 =2 dy ln 1 + q02 q0 0 −1 1+y

= O(log|q0 |). In I1 , we change variables from (x0 , x) to (u, v) where u = β(x − x0 )(y − y 0 ) and v = βx|y| and get v Z 1 Z y Z β|y| Z ( |y| +β )(y−y 0 ) 0 I1 = 2 dy dy sgn(y) dvδ1 (v) duδ1 (u) v −1 −1 −β|y| −β )(y−y 0 ) ( |y|   × 

µ iq0 + ε

1 ¶ v u 0 v β|y| , y, β|y| − β(y − y 0 ) , y 

iq0 + ε

 ¶ . u v 0 − , y , , y β(y − y 0 ) β|y| 1

µ

−

v β|y|

The integral converges in the limit β → ∞ by dominated convergence. The last factor vanishes in that limit, so I1 → 0 as β → ∞. (0) Finally, Lemma 5.4 implies that I2 = limβ→∞ I2 exists and equals Z 1 Z y Z 1 (0) I2 = 2 dy dy 0 dxΘ 21 (−xy) −1

−1

−1

¸ y 0 − 2y y − . × (iq0 + x(y − y 0 ))2 (iq0 − x(y − y 0 ))2 ·

0

In the real part, the terms with y 0 in the numerator cancel, so that Z 1 Z 1 Z y 2y (0) Re I2 = 2 dxΘ 12 (−xy) Re dy dy 0 (iq − x(y − y 0 ))2 0 −1 −1 −1 Z 1 Z 1 Z 1+y 2y =2 dy dxΘ 12 (−xy) Re dz (iq − xz)2 0 −1 −1 0 Z 1 Z 1 1+y = −4 ydy dxΘ 12 (−xy) 2 2 (1 + y)2 q + x −1 −1 0

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µ ¶ vy 1 ydy dvΘ 12 − = −4 2 1 + y q0 + v 2 −1 −(1+y) Z 1+y Z 1 dv ydy = −4 2 + v2 . q 0 1−y 0 Z

Z

1

1+y

The contribution from the region v ≥ 1 is obviously bounded. The remaining integral is Z 1 Z 1 Z 1 Z 1 dv dv ydy = ydy 2 2 2 2 0 1−y q0 + v 0 q0 + v 1−v µ ¶ Z 1 v2 dv v − = = O(| log|q0 | |). 2 2 2 0 q0 + v 5.5. One-loop integrals for the xy case For the discussion in Sec. 6, it is useful to calculate the lowest order contributions to the four-point function, the so-called bubble integrals. Again we restrict to the xy-type singularity. Because the fermionic bubble integrals are not continuous at zero temperature, it is best to calculate them by setting q0 = 0 first, then letting the spatial part q tend to 0, all at a fixed inverse temperature β, and then calculate the asymptotics as β → ∞.

5.5.1. The particle-hole bubble Write x0 = x + ξ, y 0 = y + η. The bubble is Z 1X 1 1 Bph (q0 , ξ, η) = dxdy β iω − xy i(q + ω) − x0 y 0 2 0 [−1,1] ω Z fβ (xy) − fβ (x0 y 0 ) = dxdy iq0 + xy − x0 y 0 [−1,1]2 Z 1 Z xy − x0 y 0 = dt dxdy (−δβ )(txy + (1 − t)x0 y 0 ). 0 y0 iq + xy − x 2 0 0 [−1,1]

(25)

Obviously, at q0 6= 0, Bph (q0 , ξ, η) → 0 as (ξ, η) → 0. So set q0 = 0, i.e. consider (0)

Bph =

lim Bph (0, ξ, η).

(ξ,η)→0

(0)

Lemma 5.5. The large β asymptotics of Bph is (0)

where K =

R∞ 0

Bph = −2 ln β + 2K + O(e−β ) du 2 cosh2

u 2

ln u.

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Proof. By (25),

Z

Z

(0)

Bph =

[−1,1]2

Z

1

= −4 0

dxdy(−δβ )(xy) = 4 dx x Z

Z

∞

= −2 ln β 0

[0,1]2

dxdy(−δβ )(xy)

Z

β du β = −4 ln 2 u 2 u u 0 0 4 cosh 4 cosh 2 2 Z ∞ du du u + 2K − ln . 2 u 2 u β β 2 cosh cosh 2 2 βx

du

The last integral is exponentially small in β because of the decay of 1/ cosh2 . 5.5.2. The particle-particle bubble This time write x0 = x − ξ, y 0 = y − η. The bubble is Z 1X 1 1 Bpp (q0 , ξ, η) = dxdy β iω − xy i(q − ω) − x0 y 0 2 0 [−1,1] ω Z fβ (−xy) − fβ (x0 y 0 ) = dxdy −iq0 + xy + x0 y 0 [−1,1]2 µ ¶ Z 1 β −→ dxdy tanh xy . −iq0 + 2xy 2 (ξ,η)→0 [−1,1]2 Again, we set q0 = 0 and keep β < ∞. Then (0) Bpp = Bpp (0, 0, 0) µ ¶ µ ¶ Z Z 1 β 1 β = dxdy tanh xy = 2 dxdy tanh xy 2xy 2 xy 2 [−1,1]2 [0,1]2 Z 1 Z Z 1 dx x dE β ln E β =2 tanh E = −2 dE tanh E x 0 E 2 E 2 0 0 · ¸1 Z 1 β β = −(ln E)2 tanh E + dE(ln E)2 β 2 0 0 2 cosh2 E 2 ¶2 Z β/2 µ 2v 1 = dv ln β cosh2 v 0 ¶2 ¶2 Z ∞ µ Z ∞ µ 2v 1 2v 1 = dv ln − dv ln . 2 β β cosh v cosh2 v 0 β/2

Thus we have Lemma 5.6.

where K =

R∞ 0

¢ ¡ (0) Bpp = (ln β)2 − 2K ln β + K 0 + O e−β R∞ 2 ln(2v) dv and K 0 = 0 (ln(2v)) dv. cosh2 v cosh2 v

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6. Interpretation Let us discuss these results a bit more informally. The above calculations for the xy case can be summarized as follows. Evidently, the derivatives we were looking at diverge in the limit q0 → 0 at zero temperature. To proceed, we discuss positive temperatures and replace q0 by π/β, and thus translate everything into β-dependent quantities. We have proven that to all orders r in λ, |∇Σr | ≤ const

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(where the constant depends on the order r in λ). In the model computations of the last section, we have seen that to second order in the coupling constant λ Im ∂0 Σ ∼ −4 ln 2(λ ln β)2 , Re

∂2 Σ ∼ (2 + 4 ln 2)(λ ln β)2 , ∂ξ∂η

Re

∂2 Σ ∼ O(λ2 ln β), ∂ξ 2

Bph (0) ∼ −2 λ ln β, α (λ ln β)2 . 2 We have redefined Bph and Bpp to include the appropriate coupling constant dependence. The line for Bpp includes second order corrections to the superconducting vertex, where the coefficient given by the loop integral is α. If λ is negative (attractive interaction), the superconducting instability driven by the λ(ln β)2 term is always strongest, but if λ > 0 (repulsive bare interaction), the λ(ln β)2 term suppresses the leading order Cooper pair interaction, and the higher order term proportional to α is only of order (λ ln β)2 . In this case, all terms that can drive instabilities are linear or quadratic in λ ln β. A first attempt to weigh the relative strength of these divergences is to look at the prefactors. Here it seems that the second derivative gets the largest contribution. The asymmetry between this logarithmic divergence and the boundedness of the gradient is striking. We now put the results of [2–4] for the half-filled, t0 = 0, Hubbard model into context. It was proven there that perturbation theory in the coupling constant λ converges in the regime Wβ,λ = {(λ, β) : |λ| ¿ 1, |λ|(log β)2 ¿ 1}. Our analysis, while presently restricted to all-order perturbation theory, is not restricted to Wβ,λ . Indeed our results are most interesting at temperatures lower than those given (at small fixed λ > 0) by Wβ,λ : in the regime Wβ,λ all the interesting effects that we summarized at the beginning of this section are still O(λ) and hence small. As mentioned in [1], we expect that some of our results can be proven nonperturbatively, using sector techniques, provided the flow of the four-point function can be tracked in enough detail to see the differences between the various attractive and repulsive initial interactions in the bounds. Bpp (0) ∼ λ (ln β)2 −

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A nonperturbative treatment of the half-filled, t0 = 0, Hubbard model at temperatures below those permitted by Wβ,λ will require great care. The restriction |λ|(log β)2 ¿ 1 of Wβ,λ eliminates the singularity that occurs for λ < 0 in the sum over particle-particle ladders. The square of the logarithm arises from the Van Hove singularity, as discussed in the Introduction of [1]. For λ > 0, however, there is no singularity in the flow of the four-point function when λ(log β)2 becomes of order one. (The coupling constant for s-wave superconductivity is suppressed rather than enhanced by the flow.) The O(λ2 ) terms in Bpp discussed above can generate singularities in the sum of particle-particle ladders, but this happens only when λ log β becomes of order 1. In this regime, all the other effects we have studied here, as well as nesting effects, come into play and compete with each other. In the exactly half-filled, t0 = 0 case, the singularities in the particlehole channel drive N´eel antiferromagnetism. For t0 6= 0, antiferromagnetism is weakened because nesting is destroyed. In the exactly half-filled, t0 = 0, λ > 0 case, N´eel antiferromagnetism is believed to be the true ordering. However this is unproven. Indeed, at the present time, even Fermi liquid behavior has not been definitively ruled out. The divergence of the second derivative of the selfenergy with respect to the frequency ω was proven by a second-order calculation in [4]. While the property that Σ is C 2 appears in the sufficient condition for Fermi liquid behavior of [21], and while the second spatial derivative enters in the curvature of the Fermi surface and hence needs to be controlled carefully (unless the Fermi surface is fixed by a symmetry), only a divergence in the first ω-derivative will change the asymptotic behavior at small frequency and result in a breakdown of Fermi liquid behavior. But the first ω-derivative still remains small in the regime Wβ,λ , so that the Z factor stays close to 1 there. In the following we further discuss the physical significance of our findings. This discussion is not rigorous, but it reveals some interesting possibilities that should be studied further. We first note that the above results were achieved in renormalized perturbation theory, that is, the above all-order results are in the context of an expansion where a counterterm is used to fix the Fermi surface. (The second order explicit calculations assume only that the first order corrections can be taken into account by a shift in µ, which is true because the interaction is local.) Using a counterterm is the only way to get an all-order expansion that is well-defined in the limit of zero temperature. A scheme where the scale decomposition adapts to the Fermi surface movement was outlined in [16] and developed mathematically in 17–19; but in any such scheme the expansions have to be done iteratively and cannot be cast in the form of a single renormalized expansion, because the singularity moves in every adjustment of the Fermi surface [16]. The meaning of the counterterm was explained in detail in [11–14]. In brief, the dispersion relation we are using in our propagators is not the bare one, but the renormalized one, whose zero level set is the Fermi surface of

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the interacting system. Thus, we have in fact made the assumption that the Fermi surface of the interacting system has the properties H1–H6. In particular it contains singular points and these singular points are nondegenerate. We shall first discuss our results under this assumption and then speculate about how commonly it will be valid. In the case of a nonsingular, strictly convex, curved Fermi surface, there was a similar assumption, which was, however, mathematically justified by our proof of an inversion theorem [14] that gives a bijective relation between the free and interacting dispersion relation. 6.1. Asymmetry and Fermi velocity suppression First note that the regularized (discrete-time) functional integral for many-body systems has a symmetry that allows one to make an arbitrary nonzero rescaling of the field variables. This is based on the behavior of the measure under ψi → gi ψi and ψ¯i → g˜i ψ¯i . The result is Ã !−1 Z Y Y Y −1 ¯ −1 ¯ −A( ψ,ψ) gi g˜i dψ¯i dψi e−A(˜g ψ,g ψ) . dψ¯i dψi e = (27) i

i

i

Source terms get a similar rescaling. This can, of course, be used to remove a factor Z −1 from the quadratic term in the fields, but the factor Z (see (13)) will then reappear in the interaction and source terms. Note that Z depends on momentum. In the limit of very small Z, some terms may get greatly enhanced or suppressed. The Fermi velocity is defined as vF (p) = Z∇Σ on the Fermi surface. If one extrapolates the above formula for ∂0 Σ2 to β → ∞, the Z factor, defined in (13), becomes zero at the Van Hove points. There is a crucial difference between the one- and two-dimensional cases. In dimension one, both ∂0 Σ2 and ∇Σ2 behave like λ2 log β, so that, after extracting the field strength renormalization, the Fermi velocity retains its original value. In our two-dimensional situation, however, ∇Σ remains bounded, and thus the Fermi velocity gets suppressed in a neighborhood of the Van Hove singularity because it ¯ 0ψ contains a factor of Z. Because the time derivative term in the action is Z −1 ψik −1 and k0 is an odd multiple of the temperature T = β , one can also interpret Z(k)−1 T = T (k) heuristically as a “momentum-dependent temperature” that varies over the Fermi surface and that increases as one approaches the Van Hove points (“hot spots”). This behavior is illustrated in Fig. 1. 6.2. Inversion problem There is, however, also a more basic problem that is exhibited by these results. It arises when one starts questioning the assumption that the interacting Fermi surface contains singularities. The second derivative of the self-energy in spatial directions

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Fig. 1. Comparison of the derivative of the Fermi function, where on the right, the inverse temperature β is scaled by an angle-dependent factor that vanishes near the points (π, 0) and (0, π). The dark regions correspond to small derivatives, the bright ones to large derivatives.

is divergent at zero temperature. Thus even the lowest nontrivial correction may change the structure of e significantly. This is related to the inversion problem, which was solved for strictly convex Fermi surfaces in [14]. In all of the following discussion, we concentrate on two dimensions and assume that by adjusting µ, we can arrange things such that the interacting Fermi surface still contains a point where the gradient of the dispersion relation vanishes. In the general theory of expansions for many-fermion systems, the most singular terms are those created by the movement of the Fermi surface. Counterterms need to be used to make perturbation theory well-defined, or an adaptive scale decomposition has to be chosen. To show that the model with counterterms corresponds to a bare model in some desired class, one needs to prove an inversion theorem, which requires regularity estimates. To get an idea, it is instructive to consider a neighborhood of a regular point on the Fermi surface. By an affine transformation to coordinates (u, v) in momentum space, i.e. shifting the origin to that point and rotating so that the tangent plane to the Fermi surface is given by u = 0, the function e can be transformed to a function e˜(u, v) = u + κ2 v 2 + · · · , where κ denotes the curvature and . . . the higher order terms. (By an additional change of coordinates, the error terms can be made to vanish, but this is not important here.) Let Σ0 (p) be the self-energy at frequency zero (or ±π/β). Then (e + Σ0 )∼ (u, v) = u +

κ 2 v + λσ(u, v). 2

In order for the correction λσ not to overwhelm the zeroth order contribution e, we need ∂u σ to be bounded and ∂v2 σ to be bounded. Then the correction is small when λ is small. Note that different regularity is needed in normal and tangential

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directions. Indeed, even in the absence of Van Hove singularities, the function σ is not twice differentiable in u at T = 0, but it is C 2 in v if e has the same property [12]. Thus, the mere divergence of a derivative does not tell us about a potential problem with renormalization; the relevant question is whether the correction is larger than the zeroth order term. In the strictly convex, curved case, there is no problem. However, in two dimensions, the normal form for the Van Hove singularity, e˜(uv) = uv gets changed significantly because ¯ ∂2 ζ˜2 = [e(u, v) + σ ˜ (u, v)] ¯u=v=0 ∂u∂v

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diverges at zero temperature. To leading order in q0 , u and v, our result for the inverse full propagator to second order takes the form ζ2 iq0 − ζ˜2 uv with ζ2 = 1 + ϑ2 (λ ln β)2 , ζ˜2 = 1 + ϑ˜2 (λ ln β)2 . Because ζ˜2 diverges in the zero-temperature limit, it is not even in second order consistent to assume that the type of (u, v) = (0, 0) as a critical point of e and E = e+σ ˜ is the same. One can attempt to fix this problem by rescaling the 1/2 ˜ fields by ζ . Then the iq0 term gets rescaled similarly because ζ depends on the ˜ It should, however, be noted that, as in same combination of λ and ln β as ζ. the above-discussed suppression of the Fermi velocity, a corresponding rescaling of all interaction and source terms also occurs. In particular, σ, which gives the Fermi surface shift, itself gets replaced by σ ζ˜−1 . Because σ remains bounded (only its derivatives diverge), this would imply that the Fermi surface shift gets scaled down in the rescaling transformation, indicating a “pinning” of the Fermi surface at the Van Hove points. In our case, ϑ2 = 4 ln 2 and ϑ˜2 = 2 + 4 ln 2. With these numerical values, ˜ ζ2 is larger than ζ2 , so ζ˜2 appears as the natural factor for the rescaling. In the Hubbard model case, the transformations leading to the normal form depend on the parameter θ in (1), so that one can expect ζ and ζ˜ to depend on θ. The study of these dependencies, as well as the interplay between the two critical points that contribute, is left to future work. One point of criticism of the Van Hove scenario has always been that it appears nongeneric because the logarithmic singularity gets weakened very fast as one moves away from the Van Hove densities. If the above speculations can be substantiated by careful studies, the singular Fermi surface scenario may turn out to be much more natural than one would naively assume. Moreover, there is a natural way how extended Van Hove singularities may arise by interaction effects. Acknowledgments J. F. was supported by NSERC of Canada. M. S. was supported by DFG grant Sa-1362, an ESI senior research fellowship, and NSERC of Canada.

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Appendix A. Interval Lemma The following standard result is included for the convenience of the reader. Lemma A.1. Let ², η be strictly positive real numbers and k be a strictly positive integer. Let I ⊂ R be an interval (not necessarily compact) and f a C k function on I obeying |f (k) (x)| ≥ η

for all x ∈ I.

Then k+1

Vol{x ∈ I | |f (x)| ≤ ²} ≤ 2 Proof. Denote α =

¡ ² ¢1/k η

µ ¶1/k ² . η

. In terms α, we must show

² for all x ∈ I =⇒ Vol{x ∈ I | |f (x)| ≤ ²} ≤ 2k+1 α. αk Define ck inductively by c1 = 2 and ck = 2 + 2ck−1 . Because bk = 2−k ck obeys b1 = 1 and bk = 2−k+1 + bk−1 we have bk ≤ 2 and hence ck ≤ 2k+1 . We shall prove ² |f (k) (x)| ≥ k for all x ∈ I =⇒ Vol{x ∈ I | |f (x)| ≤ ²} ≤ ck α α by induction on k. Suppose that k = 1 and let x and y be any two points in {x ∈ I | |f (x)| ≤ ²}. Then |x − y| |f (x) − f (y)| 2² |x − y| = |f (x) − f (y)| = ≤ 0 |f (x) − f (y)| |f 0 (ζ)| |f (ζ)| |f (k) (x)| ≥

for some ζ ∈ I. As |f 0 (ζ)| ≥ α² we have |x − y| ≤ 2α. Thus {x ∈ I | |f (x)| ≤ ²} is contained in an interval of length at most 2α as desired. Now suppose that the induction hypothesis is satisfied for k − 1 and that ² |f (k) (x)| ≥ α²k on I. As in the last paragraph the set {x ∈ I | |f (k−1) (x)| ≤ αk−1 } is contained in a subinterval I0 of I of length at most 2α. Then I is the union ² of I0 and at most two other intervals I+ , I− on which |f (k−1) (x)| ≥ αk−1 . By the inductive hypothesis X Vol{x ∈ I | |f (x)| ≤ ²} ≤ Vol(I0 ) + Vol{x ∈ Ii | |f (x)| ≤ ²} i=±

≤ 2α + 2ck−1 α = ck α. Appendix B. Signs etc The restrictions (16) are summarized in the following table E2 = xy E3 = x0 y 0 E1 = (x − x0 )(y − y 0 ) sf + − − (−1) . − + + (−1)

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In both cases, the product of indicator functions resulting from the limit of Fermi functions is −1. Let R be the reflection at zero, R(x, y, x0 , y 0 ) = (−x, −y, −x0 , −y 0 ).

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The function ε = ε(x, y, x0 , y 0 ) = xy 0 + x0 y − 2x0 y 0 satisfies ε(x, y, x0 , y 0 ) = ε(−x, −y, −x0 , −y 0 ). The function D(x, y, x0 , y 0 ) = y − y 0 satisfies D(R(x, y, x0 , y 0 )) = −D(x, y, x0 , y 0 ). The function F (x, y, x0 , y 0 ) = (x − x0 )(y − y 0 ) is invariant under R. In the following we list all cases of signs for x, x0 , y and y 0 , together with ε, D, F written as functions of x = |x|,

y = |y|,

x0 = |x0 |,

y0 = |y 0 |

to be able to restrict the integrals to [0, 1] whenever this is convenient (by transforming to x, . . . , y0 as integration variables), and to exhibit some important sign changes. In the last column, we list the condition ρn obtained from the restriction on the sign of (x − x0 )(y − y 0 ) in (29). n x y x0 y 0 1 + + +− 2 + + −+ 3 + − ++ 4 + − −− 5 + + ++ 6 + + −− 7 + − +− 8 + − −+ 9 − − −+ 10 − − +− 11 − + −− 12 − + ++ 13 − − −− 14 − − ++ 15 − + −+ 16 − + +−

εn = xy 0 + x0 y − 2x0 y 0 ε1 = x0 y + (2x0 − x)y0 ε2 = xy0 + (2y0 − y)x0 ε3 = xy0 − (2y0 + y)x0 ε4 = x0 y − (2x0 + x)y0 ε5 = xy0 − x0 (2y0 − y) ε6 = −(xy0 + x0 (2y0 + y)) ε7 = −(xy0 − x0 (2y0 − y)) ε8 = xy0 + x0 (2y0 + y) ε1 ε2 ε3 ε4 ε5 ε6 ε7 ε8

Dn = y − y 0 Fn D1 = y + y0 (x − x0 )(y + y0 ) D2 = y − y0 (x + x0 )(y − y0 ) D3 = −(y + y0 ) −(x − x0 )(y + y0 ) D4 = −(y − y0 ) −(x + x0 )(y − y0 )

−D1 −D2 −D3 −D4

ρn x < x0 y < y0 x < x0 y < y0

ρ1 ρ2 ρ3 ρ4

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Cases 1–4 and 9–12 obey the restrictions (29). Cases 5–8 and 13–16 do not because there, the signs of xy and x0 y 0 are the same. They are used to discuss some terms at finite β. Since the first two restrictions are not satisfied, the column for the last restriction, ρ, is left empty in these cases. Case n + 8 is obtained from n by the reflection R. Thus εn+8 = εn ,

Dn+8 = −Dn ,

Fn+8 = Fn ,

ρn+8 = ρn .

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Moreover, ε1 = −ε3

and

ε2 = −ε4

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and ε2 (y, x, y0 , x0 ) = ε1 (x, y, x0 , y0 ) and

ρ2 (y, x, y0 , x0 ) ⇔ ρ1 (x, y, x0 , y0 ).

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References [1] J. Feldman and M. Salmhofer, Singular Fermi surfaces I. General power counting and higher dimensions, Rev. Math. Phys. 20 (2008) 233–274. [2] V. Rivasseau, The two dimensional Hubbard model at half-filling. I. Convergent contributions, J. Stat. Phys. 106 (2002) 693–722. [3] S. Afchain, J. Magnen and V. Rivasseau, Renormalization of the 2-point function of the Hubbard model at half-filling, Ann. Henri Poincar´e 6 (2005) 399–448. [4] S. Afchain, J. Magnen and V. Rivasseau, The Hubbard model at half-filling. III. The lower bound on the self-energy, Ann. Henri Poincar´e 6 (2005) 449–483. [5] R. S. Markiewicz, A survey of the Van Hove scenario for high-Tc superconductivity with special emphasis on pseudogaps and striped phases, J. Phys. Chem. Solids 58 (1997) 1179–1310. [6] A. A. Kordyuk and S. V. Borisenko, ARPES on high-temperature superconductors: Simplicity vs. complexity, Fiz. Nizk. Temp. 32 (2006) 401–410. [7] N. Furukawa, T. M. Rice and M. Salmhofer, Truncation of a 2-dimensional Fermi surface due to quasiparticle gap formation at the saddle points, Phys. Rev. Lett. 81 (1998) 3195–3198. [8] C. J. Halboth and W. Metzner, Renormalization-group analysis of the twodimensional Hubbard model, Phys. Rev. B 61 (2000) 7364–7377. [9] C. J. Halboth and W. Metzner, d-wave superconductivity and Pomeranchuk instability in the two-dimensional Hubbard model, Phys. Rev. Lett. 85 (2000) 5162–5165. [10] C. Honerkamp, M. Salmhofer, N. Furukawa and T. M. Rice, Breakdown of the Landau–Fermi liquid in two dimensions due to Umklapp scattering, Phys. Rev. B 63 (2001) 035109. [11] J. Feldman, M. Salmhofer and E. Trubowitz, Perturbation theory around non-nested Fermi surfaces I. Keeping the Fermi surface fixed, J. Stat. Phys. 84 (1996) 1209–1336. [12] J. Feldman, M. Salmhofer and E. Trubowitz, Regularity of the moving Fermi surface: RPA contributions, Comm. Pure Appl. Math. 51 (1998) 1133–1246. [13] J. Feldman, M. Salmhofer and E. Trubowitz, Regularity of interacting nonspherical Fermi surfaces: The full self-energy, Comm. Pure Appl. Math. 52 (1999) 273–324. [14] J. Feldman, M. Salmhofer and E. Trubowitz, An inversion theorem in Fermi surface theory, Comm. Pure Appl. Math. 53 (2000) 1350–1384. [15] M. W. Hirsch, Differential Topology (Springer Verlag, New York, 1976).

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[16] W. Pedra and M. Salmhofer, Fermi systems in two dimensions and Fermi surface flows, in Proceedings of the 14th International Congress of Mathematical Physics, Lisbon, Portugal (2003), pp. 663–664. [17] W. Pedra, Zur mathematischen Theorie der Fermifl¨ ussigkeiten bei positiven Temperaturen, PhD Thesis, University of Leipzig and MPI-MIS Leipzig (2005). [18] W. Pedra and M. Salmhofer, Mathematical Theory of Fermi Liquids at Positive Temperatures I, to appear. [19] W. Pedra and M. Salmhofer, Mathematical Theory of Fermi Liquids at Positive Temperatures II, to appear. [20] D. Brox, Renormalization of many body fermion models with singular Fermi surfaces, Thesis (M.Sc.), University of British Columbia (2005). [21] M. Salmhofer, Continuous renormalization for fermions and Fermi liquid theory, Commun. Math. Phys. 194 (1998) 249–295.

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Reviews in Mathematical Physics Vol. 20, No. 3 (2008) 335–365 c World Scientific Publishing Company °

FREE ENERGY DENSITY FOR MEAN FIELD PERTURBATION OF STATES OF A ONE-DIMENSIONAL SPIN CHAIN

´ MOSONYI† , HIROMICHI OHNO‡ FUMIO HIAI∗ , MILAN ´ and DENES PETZ§ ∗,†Graduate

School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan ∗[email protected] †[email protected] ‡Graduate

School of Mathematics, Kyushu University, 1-10-6 Hakozaki, Fukuoka 812-8581, Japan [email protected] §Alfr´ ed

R´ enyi Institute of Mathematics, H-1364 Budapest, POB 127, Hungary [email protected] Received 30 July 2007 Revised 12 January 2008

Dedicated to Professor Walter Thirring on his 80th birthday Motivated by recent developments on large deviations in states of the spin chain, we reconsider the work of Petz, Raggio and Verbeure in 1989 on the variational expression of free energy density in the presence of a mean field type perturbation. We extend their results from the product state case to the Gibbs state case in the setting of translationinvariant interactions of finite range. In the special case of a locally faithful quantum Markov state, we clarify the relation between two different kinds of free energy densities (or pressure functions). Keywords: Free energy density; mean relative entropy; interactions; Gibbs states; KMS states; finitely correlated states; quantum Markov states; Legendre transform. Mathematics Subject Classification 2000: 82B10, 82B20

1. Introduction The theoretical description of the statistical mechanics of quantum spin chains was the first success of the operator algebraic approach to quantum physics. A oneN dimensional spin chain is described by a quasi-local C*-algebra A := k∈Z Ak which is the infinite tensor product of full matrix algebras Ak = Md (C) and the N limit of the local algebras AΛ := k∈Λ Ak , where Λ ⊂ Z is finite. A state ϕ of the spin chain is uniquely specified by its local restrictions ϕΛ := ϕ|AΛ . A local 335

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state ω of AΛ can equivalently be given by its density matrix D(ω) satisfying ω(A) = Tr D(ω)A, A ∈ AΛ . A translation-invariant interaction Φ of the spins determines a local Hamiltonian X HΛ (Φ) := Φ(X) (1.1) X⊂Λ

with corresponding local Gibbs state e−HΛ (Φ) (1.2) Tr e−HΛ (Φ) for all finite Λ ⊂ Z. The local Gibbs state is the unique maximizer of the functional ω 7→ −ω(HΛ (Φ)) + S(ω), where ω is an arbitrary state of AΛ and S(ω) is the von Neumann entropy S(ω) := −Tr D(ω) log D(ω). Furthermore, D(ϕG Λ ) :=

log Tr e−HΛ (Φ) = max{−ω(HΛ (Φ)) + S(ω) : ω state of AΛ }.

(1.3)

One of the main problems in the statistical mechanics of the spin chain is the determination of the global equilibrium states of A for a given interaction. When Φ is of relatively short range, it is well known [11, 22] that the variational formula (1.3) holds in the asymptotic limit: P (Φ) = max{−ω(AΦ ) + s(ω) : ω translation-invariant state of A},

(1.4)

where 1 log Tr e−HΛ (Φ) , Λ→Z |Λ|

P (Φ) := lim s(ω) := lim

Λ→Z

AΦ :=

1 S(ω|AΛ ), |Λ|

X Φ(X) |X|

(1.5) (1.6) (1.7)

X30

are the pressure (or free energy density) of Φ, the mean entropy of ω and the mean energy of Φ, respectively. (Here note that the term “free energy” should be used with minus sign in the exact sense of physics.) Maximizers of the right-hand side of (1.4) are the equilibrium states for the interaction Φ. If Φ is of finite range, then the equilibrium state is unique. One of the main subjects of the present paper is an extension of the free energy density (1.5) when the interaction is perturbed by a mean field term. Let γ be the right-translation automorphism of A and set X 1 γ k (A) ∈ A[1,n] sn (A) := n Λ+k⊂[1,n]

for a fixed A ∈ Asa Λ with a finite Λ ⊂ Z. We will study the limit 1 lim log Tr exp(−H[1,n] (Φ) − nf (sn (A))), (1.8) n→∞ n where f is a real continuous function. This kind of problem was initiated by Petz, Raggio and Verbeure [33] in the particular case when there is no interaction between

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the spins. The motivation came from mean field models and the extension of large deviation theory for quantum chains [32]. An important tool was Størmer’s quantum version of the de Finetti theorem for symmetric states. The subject was treated in details in the monograph [31] under the name “perturbational limits” by using the concept of approximately symmetric sequences [36]. Since the interaction Φ in the general situation is not invariant under the permutation of the spins, our method in the general case is the extremal decomposition theory for translationinvariant states that is standard in quantum statistical mechanics, see [10]. In the present paper we will show that the limit is expressed by a variational formula generalizing (1.4). The limit (1.8) has a direct physical meaning in the case when f (x) = x2 and A = A0 ∈ A0 . Then −H[1,n] (Φ) −

n 1 X Ai Aj n i,j=1

is a mean field perturbation of the interaction Φ, where Aj := γ j (A0 ). The limit is the free energy density for the mean field model and the variational formula has an important physical interpretation. The limit density (1.8) can be considered in a different way as well. Given a translation-invariant state ϕ, we can study the limit ¡ ¢ 1 pϕ (A, f ) := lim log Tr exp log D(ϕ|A[1,n] ) − nf (sn (A)) (1.9) n→∞ n and its variational expression under the duality between the observable space Asa and the translation-invariant state space Sγ (A). In particular, when f (x) = x, the limit (1.9) becomes a simply perturbed free energy density function (or pressure function) ¡ ¢ 1 pϕ (A) := lim log Tr exp log D(ϕ|A[1,n] ) − nsn (A) n→∞ n for local observables A in Asa (if the limit exists). The dual function of the function pϕ (A) is the mean relative entropy 1 (1.10) S(ω|A[1,n] , ϕ|A[1,n] ) n with respect to ϕ defined for ω ∈ Sγ (A). The existence of the mean relative entropy and its properties were worked out in [18, 20, 21]. When Φ is a translation-invariant interaction of finite range and ϕ is the equilibrium state for Φ, the limits (1.8) and (1.9) are the same (up to an additive term P (Φ)), but (1.9) can also be studied for a wider class of translation-invariant states, for example, for finitely correlated states which were introduced by Fannes, Nachtergaele and Werner [14]. A slightly different concept of quantum Markov states was formerly introduced by Accardi and Frigerio [3]. A translation-invariant and locally faithful quantum Markov state in the sense of Accardi and Frigerio is known to be a finitely correlated state as well as the equilibrium state for a nearest-neighbor SM (ω, ϕ) := lim

n→∞

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interaction [4, 30]. Remarkably, a Markovian structure similar to the special quantum Markov state just mentioned appears in the recent characterization [15, 28] of the quantum states which saturate the strong subadditivity of the von Neumann entropy. A similar but different version of the free energy density function pϕ (A) is Ã Ã n !! X ¡ nsn (A) ¢ 1 1 k p˜ϕ (A) := lim log ϕ e = lim log ϕ exp γ (A) , n→∞ n n→∞ n k=1

which gives the logarithmic moment generating function for a sequence of compactly supported probability measures on the real line. Large deviations governed by this generating function have recently been studied in [17, 26, 29] for example. In fact, our first motivation of the present paper came from large deviation results in [26, 29] with respect to Gibbs-KMS states. It is not known in general for pϕ to have the interpretation as the logarithmic moment generating function as p˜ϕ does. Indeed, this question is nothing more than the so-called BMV-conjecture [9]. On the other hand, since p˜ϕ is not a convex function in general, it is impossible for p˜ϕ to enjoy such a variational expression as pϕ does. The paper is organized as follows. Section 2 is a preliminary on translationinvariant interactions and Gibbs-KMS equilibrium states of the one-dimensional spin chain. In Sec. 3 the existence of the functional free energy density (1.9) and its variational expression are obtained when ϕ is the Gibbs state for a translationinvariant interaction of finite range. In Sec. 4 the existence of the density pϕ (A) is proven for a general finitely correlated state ϕ, and the exact relation between the functionals pϕ and p˜ϕ introduced above is clarified in the special case when ϕ is a locally faithful quantum Markov state. Section 5 is a brief guide to how our results for a Gibbs state ϕ can be extended to the case of arbitrary dimension.

2. Preliminaries A one-dimensional spin chain is described by the infinite tensor product C ∗ -algebra N A := k∈Z Ak of full matrix algebras Ak := Md (C) over Z. The right-translation automorphism of A is denoted by γ. We denote by Sγ (A) the set of all γ-invariant states of A. The C ∗ -subalgebra of A corresponding to a subset X of Z is AX := N k∈X Ak with convention A∅ := C1, where 1 is the identity of A. If X ⊂ Y ⊂ Z, then AX ⊂ AY by a natural inclusion. The local algebra is the dense ∗-subalgebra S∞ Aloc := n=1 A[−n,n] of A. The self-adjoint parts of Aloc and A are denoted by sa Asa loc and A , respectively. The usual trace on AX for each finite X ⊂ Z is denoted by Tr without referring to X since it causes no confusion. An interaction Φ in A is a mapping from the nonempty finite subsets of Z into A such that Φ(X) = Φ(X)∗ ∈ AX for each finite X ⊂ Z. Given an interaction Φ and a finite subset Λ ⊂ Z, we have the local Hamiltonian HΛ (Φ) given in (1.1) and

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the surface energy WΛ (Φ) WΛ (Φ) :=

X

{Φ(X) : X ∩ Λ 6= ∅, X ∩ Λc 6= ∅}

whenever the sum converges in norm. We always assume that Φ is γ-invariant, i.e. γ(Φ(X)) = Φ(X + 1) for every finite X ⊂ Z, where X + 1 := {k + 1 : k ∈ X}. We denote by B0 (A) the set of all γ-invariant interactions Φ in A such that X kΦk0 := kΦ(X)k + sup kW[1,n] (Φ)k < +∞. n≥1

X30

It is easy to see that B0 (A) is a real Banach space with the usual linear operations and the norm kΦk0 . Associated with Φ ∈ B0 (A) we have a strongly continuous one-parameter automorphism group αΦ of A given by αtΦ (A) =

lim

m→−∞,n→∞

eitH[m,n] (Φ) Ae−itH[m,n] (Φ)

(A ∈ A).

Then it is known [6, 24] that there exists a unique αΦ -KMS state (at β = −1) ϕ of A, which is automatically faithful and ergodic (i.e. an extremal point of Sγ (A)). The KMS state ϕ is characterized by the Gibbs condition and so it is also called the (global) Gibbs state for Φ. The state ϕ is also characterized by the variational principle s(ϕ) = ϕ(AΦ ) + P (Φ), the equality case of the expression (1.4), where P (Φ), s(ϕ) and AΦ are given in (1.5)–(1.7). See [11, 22] for details on these equivalent characterizations of equilibrium states. In the rest of this section, assume that Φ is a γ-invariant interaction of finite range, i.e. there is an N0 ∈ N such that Φ(X) = 0 whenever the diameter of X is greater than N0 . Of course, Φ ∈ B0 (A). Let ϕ be the αΦ -KMS state (at β = −1) of A. The next lemma will play an essential role in our discussions below; the proof can be found in [5, 7, 8]. Lemma 2.1. There is a constant λ ≥ 1 (independent of n) such that λ−1 ϕn ≤ ϕG n ≤ λϕn for all n ∈ N, where ϕG n is the local Gibbs state (1.2) with Λ = [1, n]. For ω ∈ Sγ (A) and Ψ ∈ B0 (A) we write for short ωn and Hn (Ψ) for ω|A[1,n] and H[1,n] (Ψ), respectively. Lemma 2.1 gives ¯ ¯1 ¯ log Tr exp(log D(ϕn ) − Hn (Ψ)) ¯n ¯ ¯ log λ 1 G − log Tr exp(log D(ϕn ) − Hn (Ψ))¯¯ ≤ . n n Since ¡ ¢ Tr e−Hn (Φ+Ψ) Tr exp log D(ϕG ) − H (Ψ) = , n n Tr e−Hn (Φ)

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we have Lemma 2.2. For every Ψ ∈ B0 (A) the limit Pϕ (Ψ) := lim

n→∞

¡ ¢ 1 log Tr exp log D(ϕn ) − Hn (Ψ) n

exists and Pϕ (Ψ) = P (Φ + Ψ) − P (Φ). For every ω ∈ Sγ (A) the mean relative entropy (1.10) exists and SM (ω, ϕ) = lim

n→∞

1 1 S(ωn , ϕn ) = lim S(ωn , ϕG n ), n→∞ n n

(2.1)

see [20, p. 710]. In fact, since −Hn (Φ) S(ωn , ϕG n ) = −S(ωn ) + ω(Hn (Φ)) + log Tr e

and lim

n→∞

ω(Hn (Φ)) = ω(AΦ ), n

we have Lemma 2.3. For every ω ∈ Sγ (A), SM (ω, ϕ) = −s(ω) + ω(AΦ ) + P (Φ). Hence, the function ω 7→ SM (ω, ϕ) is affine and lower semicontinuous in the weak* topology on Sγ (A). Theorem 2.4. (a) For every Ψ ∈ B0 (A), Pϕ (Ψ) = max{−ω(AΨ ) − SM (ω, ϕ) : ω ∈ Sγ (A)}. (b) For every ω ∈ Sγ (A), SM (ω, ϕ) = sup{−ω(AΨ ) − Pϕ (Ψ) : Ψ ∈ B0 (A)}. (c) The function Pϕ on B0 (A) is Gˆ ateaux-differentiable at any Ψ ∈ B0 (A), i.e. the limit Pϕ (Ψ + tΨ0 ) − Pϕ (Ψ) t→0 t

∂(Pϕ )Ψ (Ψ0 ) := lim

exists for every Ψ0 ∈ B0 (A). Moreover, when ϕΨ is the unique αΦ+Ψ -KMS state, ∂(Pϕ )Ψ (Ψ0 ) = −ϕΨ (AΨ0 ). Proof. The variational expressions in (a) and (b) are just rewriting of (1.4) and s(ω) = inf{ω(AΨ ) + P (Ψ) : Ψ ∈ B0 (A)}

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thanks to Lemmas 2.2 and 2.3 (see [22, § II.3] for the above expression of s(ω) complementary to (1.4)). Note also that the maximum in (a) is attained by the unique Gibbs state for Φ + Ψ. The differentiability of Pϕ in (c) was essentially shown in [26, Corollary 3.5]; we give the proof for completeness. Let B0 (A)∗ be the dual Banach space of B0 (A). For each ω ∈ Sγ (A) define fω ∈ B0 (A)∗ by fω (Ψ) := −ω(AΨ ). Then ω 7→ fω is an injective and continuous (in the weak* topologies) affine map [22, Lemma II.1.1]; hence Γ := {fω : ω ∈ Sγ (A)} is a weak* compact convex subset of B0 (A)∗ and ( SM (ω, ϕ) if f = fω with ω ∈ Sγ (A), F (f ) := +∞ otherwise is a well-defined function on B0 (A)∗ which is convex and weakly* lower semicontinuous. The assertion (a) means that Pϕ is the conjugate function of F , which in turn implies that the conjugate function of Pϕ on B0 (A) is F . By the general theory of conjugate functions (see [13, Proposition I.5.3] for example), Pϕ is Gˆateaux-differentiable at Ψ ∈ B0 (A) if and only if there is a unique f ∈ B0 (A)∗ such that (Pϕ )∗ (f ) = f (Ψ) − Pϕ (Ψ), that is, there is a unique ϕΨ ∈ Sγ (A) such that SM (ϕΨ , ϕ) = −ϕΨ (AΨ ) − Pϕ (Ψ).

(2.2)

By Lemmas 2.2 and 2.3 the above equality is equivalent to the variational principle s(ϕΨ ) = ϕΨ (AΦ+Ψ ) + P (Φ + Ψ), which is equivalent to ϕΨ being the αΦ+Ψ -KMS state. Hence the differentiability assertion of Pϕ follows. Moreover, by (a) we get Pϕ (Ψ + tΨ0 ) ≥ −ϕΨ (AΨ+tΨ0 ) − SM (ω, ϕ) for any Ψ0 ∈ B0 (A) and t ∈ R. This together with equality (2.2) for t = 0 gives the formula ∂(Pϕ )Ψ (Ψ0 ) = −ϕΨ (AΨ0 ). sa Corollary 2.5. (1) For every A ∈ Asa loc so that A ∈ AΛ with a finite Λ ⊂ Z, the free energy density   X 1 pϕ (A) := lim log Tr exp log D(ϕn ) − γ k (A) (2.3) n→∞ n Λ+k⊂[1,n]

exists (independently of the choice of Λ). ateaux-differentiable at any A ∈ Asa (2) The function pϕ on Asa loc in the sense loc is Gˆ that the limit pϕ (A + tB) − pϕ (A) lim t→0 t exists for every B ∈ Asa loc . In particular, the function t ∈ R 7→ pϕ (tA) is differentiable for every A ∈ Asa loc .

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(3) The above function pϕ on Asa loc uniquely extends to a function (denoted by the same pϕ ) on Asa which is convex and Lipschitz continuous with |pϕ (A) − pϕ (B)| ≤ kA − Bk,

A, B ∈ Asa .

(4) For every A ∈ Asa , pϕ (A) = max{−ω(A) − SM (ω, ϕ) : ω ∈ Sγ (A)}. (5) For every ω ∈ Sγ (A), SM (ω, ϕ) = sup{−ω(A) − pϕ (A) : A ∈ Asa loc } = sup{−ω(A) − pϕ (A) : A ∈ Asa }. Proof. To show (1), we may assume A ∈ Asa [1,`(A)] with some `(A) ∈ N, and set a γ-invariant interaction ΨA of finite range (hence ΨA ∈ B0 (A)) by ½ k γ (A) if X = [k + 1, k + `(A)], k ∈ Z, ΨA (X) := 0 otherwise. Pn−`(A) k Since k=0 γ (A) = Hn (ΨA ), the limit (2.3) exists by Lemma 2.2 and its independence of the choice of Λ is obvious. The differentiability in (2) immediately follows from Theorem 2.4(c). (In fact, the derivative of pϕ at A is ∂(pϕ )A (B) = A Φ+ΨA −ϕA (B) for every B ∈ Asa -KMS state.) Moreover, loc , where ϕ is the unique α since AΨA

`(A) 1 X −k = γ (A) `(A) k=1

so that ω(AΨA ) = ω(A) for all ω ∈ Sγ (A), Theorem 2.4(a) implies the variational expression in (4) for any A ∈ Asa loc . The Lipschitz inequality in (3) for every A, B ∈ Asa loc is immediately seen from the formula (2.3). Hence pϕ uniquely extends to a Lipschitz continuous function on Asa , and the convexity of pϕ on Asa is obvious. To prove (4) for general A ∈ Asa let {An } be a sequence in Asa loc such that kAn − Ak → 0. It is clear by convergence that pϕ (A) ≥ −ω(A) − SM (ω, ϕ) for all ω ∈ Sγ (A). Let ωn be the maximizer of the right-hand side of (4) for An ; here it may be assumed that {ωn } converges to ω ∈ Sγ (A) in the weak* topology. Then we get pϕ (A) = lim {−ωn (An ) − SM (ωn , ϕ)} ≤ −ω(A) − SM (ω, ϕ) n→∞

by Lemma 2.3 (the weak* lower semicontinuity), which proves (4). Finally, (5) follows from Lemma 2.3 and the duality theorem for conjugate functions [13, Proposition I.4.1]. For each A ∈ Asa we have the convex and continuous function t 7→ pϕ (tA) on R by Corollary 2.5(3). We now introduce the function IA (x) := inf{SM (ω, ϕ) : ω ∈ Sγ (A), ω(A) = x}

(x ∈ R).

(2.4)

Obviously, IA (x) = +∞ for x 6∈ [λmin (A), λmax (A)], where λmin (A) and λmax (A) are the minimum and the maximum of the spectrum of A. The next proposition

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says that pϕ (tA) and IA (x) are the Legendre transforms of each other, which are the contractions of the expressions in the above (5) and (4) into the real line via ω 7→ ω(A). Proposition 2.6. For every A ∈ Asa , IA (x) = sup{−tx − pϕ (tA) : t ∈ R}, pϕ (tA) = max{−tx − IA (x) : x ∈ [λmin (A), λmax (A)]},

x ∈ R, t ∈ R.

Proof. We have IA (x) =

min sup{t(−x + ω(A)) + SM (ω, ϕ)}

ω∈Sγ (A) t∈R

= sup min {t(−x + ω(A)) + SM (ω, ϕ)} t∈R ω∈Sγ (A)

= sup{−tx − pϕ (tA)} t∈R

by Corollary 2.5(4). In the above, the second equality follows from Sion’s minimax theorem [35] thanks to Lemma 2.3. (The elementary proof in [25] for real-valued functions can also work for functions with values in (−∞, +∞].) The second formula follows from the first by duality. Remark 2.7. An alternative notion of free energy density Ã Ãn−1 !! X 1 k p˜ϕ (A) := lim γ (A) log ϕ exp n→∞ n

(2.5)

k=0

was recently studied in [17, 26, 29] in relation with large deviation problems on the spin chain. The function t ∈ R 7→ p˜ϕ (tA) is the so-called logarithmic moment generating function [12] of a sequence of probability measures and existence of the limit guarantees large deviation upper bound to hold, while if the limit is even differentiable that provides full large deviation principle. The existence of the limit was proven for any A ∈ Asa loc when ϕ is the unique Gibbs state of a translationinvariant interaction of finite range [26] and when ϕ is a finitely correlated state [17]. Differentiability was shown in [29] and [17] for certain special cases. The Golden– Thompson inequality shows that pϕ (A) ≤ p˜ϕ (A)

(2.6) N

holds for any A ∈ Asa loc . For instance, for a product state ϕ = Z ρ with D(ρ) = e−H and a one-site observable A, since p˜ϕ (A) = log Tr(e−H e−A ) while pϕ (A) = log Tr(e−H−A ), the equality pϕ (A) = p˜ϕ (A) occurs only when A commutes with H (see [16]). Although the Lipschitz continuity of p˜ϕ on Asa loc and its variational expression as in the above (4) are impossible, it might be possible to get the variational expression as in (5) with p˜ϕ in place of pϕ . This is equivalent to saying that pϕ on Asa is the lower semicontinuous convex envelope of p˜ϕ on Asa loc , as will be shown in a special case in Sec. 4 (see Corollary 4.10).

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Remark 2.8. An equivalent formulation of the celebrated conjecture due to Bessis, Moussa and Villani [9] (the so-called BMV-conjecture) is stated as follows [27]: If H0 and H1 are N × N Hermitian matrices with H1 ≥ 0, then there exists a positive measure µ on [0, ∞) such that Z ∞ e−ts dµ(s), t > 0; Tr eH0 −tH1 = 0 H0 −tH1

or equivalently, the function Tr e the BMV-conjecture held true with

H0 := log D(ϕn ),

on t > 0 is completely monotone. Now if

H1 :=

1 n

X

γ k (A),

Λ+k⊂[1,n]

Asa Λ

where A ∈ with a finite Λ ⊂ Z, we would have a probability measure µn supported in [λmin (A), λmax (A)] such that   Z ∞ X k   Tr exp log D(ϕn ) − γ (tA) = e−nts dµn (s), t ∈ R. −∞

Λ+k⊂[1,n]

(The restriction on the support of µn easily follows from the Paley–Wiener theorem.) In this situation, the free energy density pϕ (tA) is the logarithmic moment generating function of the sequence of measures (µn ), and Corollary 2.5 and Proposition 2.6 combined with the G¨artner–Ellis theorem [12, Theorem 2.3.6] yield that (µn ) satisfies the large deviation principle with the good rate function IA (x) given in (2.4). 3. Perturbation of Gibbs States When the reference state ϕ is a product state and A is a one-site observable, the variational expression of functional free energy density ¡ ¢ 1 log Tr exp log D(ϕn ) − nf (sn (A)) lim n→∞ n ½ ¾ = sup − lim ω(f (sn (A))) − SM (ω, ϕ) ω

n→∞

was obtained in [33], where ω runs over the symmetric (or permutation-invariant) states. A comprehensive exposition on the subject is also found in [31, §13], which contains a generalization of the above expression though ϕ is still a product state. In this section we consider the case when the reference state ϕ is the Gibbs state for a translation-invariant interaction Φ of finite range. sa Let A ∈ Asa loc . We may assume without loss of generality that A ∈ A[1,`(A)] with some `(A) ∈ N, and set n−`(A) 1 X k sn (A) := γ (A) ∈ A[1,n] . n k=0

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Given A and a continuous function f : [λmin (A), λmax (A)] → R the functional free energy density is defined as the limit 1 log Zϕ (n, A, f ) lim n→∞ n for ¡ ¢ Zϕ (n, A, f ) := Tr exp log D(ϕn ) − nf (sn (A)) as n → ∞. We will show the existence of the limit in Theorem 3.4. The extreme boundary ex Sγ (A) of the set Sγ (A) consists of the ergodic states. It is known that ex Sγ (A) is a Gδ -subset of Sγ (A) (see [34, Proposition 1.3]). Since (A, γ) is asymptotically Abelian in the norm sense, Sγ (A) is a so-called Choquet simplex (see [10, Corollary 4.3.11]) so that each ω ∈ Sγ (A) has a unique extremal decomposition Z ω= ψdνω (ψ) ex Sγ (A)

with a probability Borel measure νω on ex Sγ (A) (see [34, p. 66] and [10, Theorem 4.1.15]). Lemma 3.1. For every continuous f : [λmin (A), λmax (A)] → R and for every ω ∈ Sγ (A) the limit EA,f (ω) := lim ω(f (sn (A))) n→∞

exists and

Z EA,f (ω) =

f (ψ(A))dνω (ψ) ex Sγ (A)

for the extremal decomposition ω =

R

ex Sγ (A)

ψdνω (ψ).

Proof. The first assertion is contained in [31, Proposition 13.2]. However, we use a different method to prove the two statements together. First let ψ ∈ ex Sγ (A) and (πψ , Hψ , Uψ , Ωψ ) be the GNS construction associated with ψ, i.e., πψ is a representation of A on Hψ with a cyclic vector Ωψ and Uψ is a unitary on Hψ such that ψ(A) = hπψ (A)Ωψ , Ωψ i and πψ (γ(A)) = Uψ πψ (A)Uψ∗ for all A ∈ A. Thanks to the asymptotic Abelianness, the extremality of ψ means (see [10, Theorem 4.3.17]) that the set of Uψ -invariant vectors in Hψ is the onedimensional subspace CΩψ . Hence the mean ergodic theorem implies that n−`(A) 1 X πψ (sn (A))Ωψ = Uψk πψ (A)Ωψ n k=0

converges in norm to ψ(A)Ωψ as n → ∞. The case f (x) = xm easily follows from this, and by approximating f by polynomials, we get lim kπψ (f (sn (A)))Ωψ − f (ψ(A))Ωψ k = 0

n→∞

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so that lim ψ(f (sn (A))) = f (ψ(A)).

n→∞

Finally, for a general ω ∈ Sγ (A) with the extremal decomposition ω = R ψdν ω (ψ), the Lebesgue convergence theorem gives ex Sγ (A) Z lim ω(f (sn (A))) = lim ψ(f (sn (A)))dνω (ψ) n→∞

n→∞

Z

ex Sγ (A)

=

f (ψ(A))dνω (ψ), ex Sγ (A)

as required. In the following proofs we will often use a state perturbation technique. For the convenience of the reader, we here summarize some basic properties of state perturbation restricted to the simple case of matrix algebras. See [11, 31] for the general theory of the subject matter. Let ρ be a faithful state of B := MN (C) with density matrix e−H . For each h ∈ B sa define the perturbed functional ρh by ρh (A) := Tr e−H−h A

(A ∈ B)

and the normalized version [ρh ](A) :=

ρh (A) Tr e−H−h A = ρh (1) Tr e−H−h

(A ∈ B).

The state [ρh ] is characterized as the unique minimizer of the functional ω 7→ S(ω, ρ) + ω(h) on the states of B. It is plain to see the chain rule: [[ρh ]k ] = [ρh+k ] for all h, k ∈ B sa . For each state ω of B, from the equality S(ω, [ρh ]) = S(ω, ρ) + ω(h) + log ρh (1) and the Golden–Thompson inequality ρh (1) ≤ ρ(e−h ), the following are readily seen: log ρh (1) ≥ −ω(h) − S(ω, ρ), h

|S(ω, ρ) − S(ω, [ρ ])| ≤ 2khk.

(3.1) (3.2)

Lemma 3.2. For every continuous f : [λmin (A), λmax (A)] → R and for every ω ∈ Sγ (A), lim inf n→∞

1 log Zϕ (n, A, f ) ≥ sup{−EA,f (ω) − SM (ω, ϕ) : ω ∈ Sγ (A)} n

holds. Proof. For n ∈ N write hn := nf (sn (A)) for simplicity. The perturbed functional ϕhnn of ϕn on A[1,n] has the density exp(log D(ϕn )−hn ) and so Zϕ (n, A, f ) = ϕhnn (1).

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Hence it follows from (3.1) that log Zϕ (n, A, f ) ≥ −ωn (hn ) − S(ωn , ϕn ),

ω ∈ Sγ (A).

By Lemma 3.1 and (2.1) we have 1 lim inf log Zϕ (n, A, f ) ≥ −EA,f (ω) − SM (ω, ϕ) n→∞ n for all ω ∈ Sγ (A). Lemma 3.3. For every continuous f : [λmin (A), λmax (A)] → R, sup{−EA,f (ω) − SM (ω, ϕ) : ω ∈ Sγ (A)} = sup{−f (ψ(A)) − SM (ψ, ϕ) : ψ ∈ ex Sγ (A)} = max{−f (ω(A)) − SM (ω, ϕ) : ω ∈ Sγ (A)} = max{−f (x) − IA (x) : x ∈ [λmin (A), λmax (A)]}. R Proof. For every ω ∈ Sγ (A) let ω = ex Sγ (A) ψdνω (ψ) be the extremal decomposition of ω. By Lemma 2.3 it follows from [34, Lemma 9.7] that Z SM (ω, ϕ) = SM (ψ, ϕ)dνω (ψ). ex Sγ (A)

This together with Lemma 3.1 shows that Z −EA,f (ω) − SM (ω, ϕ) = (−f (ψ(A)) − SM (ψ, ϕ))dνω (ψ) ex Sγ (A)

≤ sup{−f (ψ(A)) − SM (ψ, ϕ) : ψ ∈ ex Sγ (A)}. Therefore, sup{−EA,f (ω) − SM (ω, ϕ) : ω ∈ Sγ (A)} ≤ sup{−f (ψ(A)) − SM (ψ, ϕ) : ψ ∈ ex Sγ (A)}, and the converse inequality is obvious. Hence the first equality follows. The last equality immediately follows from the definition (2.4). To prove the second equality, let ω ˜ be a maximizer of ω 7→ −f (ω(A))−SM (ω, ϕ) on Sγ (A). For each m ∈ N with m > `(A) we introduce a product state O ω ˜m ψ := of the re-localized spin chain average

N i∈Z

Z

A[im+1,(i+1)m] and define ψ¯ ∈ Sγ (A) to be the

m−1 1 X ψ ◦ γk. ψ¯ := m k=0

First we prove that ψ¯ is γ-ergodic. For every B1 , B2 ∈ Aloc choose an i0 ∈ N such that B1 , B2 ∈ A[−i0 m,(i0 −1)m] . Let n ∈ N be given so that n = jm + r with j ∈ N, j > 2i0 and 0 ≤ r < m. When i ≥ 2i0 , 1 ≤ t ≤ m and 0 ≤ k ≤ m − 1, we have ψ(γ k (B1 )γ k+im+t (B2 )) = ψ(γ k (B1 ))ψ(γ k+im+t (B2 )) = ψ(γ k (B1 ))ψ(γ k+t (B2 )),

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because γ k (B1 ) ∈ A(−∞,i0 m] and γ k+im+t (B2 ) ∈ A[(i−i0 )m+1,∞) with i0 ≤ i − i0 . Hence for every i ≥ 2i0 we get m X

m m−1 X X ¯ 1 γ im+t (B2 )) = 1 ψ(B ψ(γ k (B1 ))ψ(γ k+im+t (B2 )) m t=1 t=1 k=0 ! Ã m m−1 X X 1 ψ(γ k+t (B2 )) = ψ(γ k (B1 )) m t=1 k=0

=

m−1 X

¯ 2 ) = mψ(B ¯ 1 )ψ(B ¯ 2 ). ψ(γ k (B1 ))ψ(B

k=0

Therefore, n

1X¯ ψ(B1 γ t (B2 )) n t=1 Ã2i m ! j−1 X jm+r m 0 X X 1 X ¯ 1 γ t (B2 )) + 1 ¯ 1 γ im+t (B2 )) = ψ(B + ψ(B n t=1 t=jm+1 n i=2i t=1 0

=

Ã2i m 0 1 X n

t=1

jm+r X

+

!

t=jm+1

¯ 1 γ t (B2 )) + (j − 2i0 )m ψ(B ¯ 1 )ψ(B ¯ 2 ), ψ(B n

which obviously implies that n

1X¯ ¯ 1 )ψ(B ¯ 2 ). ψ(B1 γ t (B2 )) = ψ(B n→∞ n t=1 lim

By [10, Theorems 4.3.17 and 4.3.22] this is equivalent to ψ¯ ∈ ex Sγ (A). Furthermore, since m − `(A) + 1 1 ¯ ψ(A) = ω ˜ (A) + m m

m−1 X

ψ(γ k (A)),

k=m−`(A)+1

we get ¯ |ψ(A) −ω ˜ (A)| ≤

2`(A)kAk . m

(3.3)

Now for m greater than both the range of Φ and `(A), we set a product state O ϕG φ(m) := (3.4) m N

Z

of the re-localized i∈Z A[im+1,(i+1)m] , where ϕG m is the local Gibbs state of A[1,m] for Φ. We also set X W := {Φ(X) : X ∩ (−∞, 0] 6= ∅, X ∩ [1, ∞) 6= ∅}, (3.5) X K := {kΦ(X)k : X ∩ (−∞, 0] 6= ∅, X ∩ [1, ∞) 6= ∅} (≥ kW k). (3.6)

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For each j ∈ N, since Hjm (Φ) =

j−1 X

γ im (Hm (Φ)) +

i=0 (m)

it is clear that φ

|A[1,jm] =

Nj

j O

G 1 ϕm

j−1 X

γ im (W ),

i=1

is the perturbed state of ϕG jm as follows:

£ G −W (m) ϕG ], m = (ϕjm )

(3.7)

1

where W (m) :=

Pj−1 i=1

γ im (W ). Hence by Lemma 2.1 and (3.2) we get

S(ψjm , ϕjm ) ≤ S(ψjm , ϕG jm ) + log λ ! Ã j j O O + 2(j − 1)K + log λ ϕG ω ˜m, ≤S m 1

=

jS(˜ ωm , ϕG m)

1

+ 2(j − 1)K + log λ

≤ jS(˜ ωm , ϕm ) + 2(j − 1)K + (j + 1) log λ.

(3.8)

Since ϕ can be considered as the Gibbs state for an interaction of finite range in N the re-localized i∈Z A[im+1,(i+1)m] , Lemma 2.3 (the affine property) implies that ¯ ϕ) = 1 lim 1 S(ψ| ¯A SM (ψ, , ϕ|A[1,jm] ) [1,jm] m j→∞ j m−1 1 X 1 = 2 lim S(ψ ◦ γ k |A[1,jm] , ϕ|A[1,jm] ) j→∞ j m k=0

1 1 = lim S(ψ|A[1,jm] , ϕ|A[1,jm] ) m j→∞ j

(3.9)

similarly to [31, (13.29)]. Therefore, 2K + log λ ¯ ϕ) ≤ 1 S(˜ SM (ψ, ωm , ϕm ) + . m m From (3.3) and (3.10) together with (2.1), for any ε > 0 we have

(3.10)

¯ ¯ ϕ) ≥ −f (˜ −f (ψ(A)) − SM (ψ, ω (A)) − SM (˜ ω , ϕ) − ε, whenever m is sufficiently large. With ψ¯ ∈ ex Sγ (A) this proves the second equality.

The next theorem showing the variational expression of the functional free energy density with respect to the state ϕ is a generalization of [33, Theorem 12] N as well as [31, Theorem 13.11]. In fact, when ϕ is a product state Z ρ and A is a one-site observable in A0 , one can easily see that max{−f (ω(A)) − SM (ω, ϕ) : ω ∈ Sγ (A)} = max{−f (σ(A)) − S(σ, ρ) : σ state of A0 }

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and IA (x) = min{S(σ, ρ) : σ state of A0 , σ(A) = x} = sup{−tx − log ρtA (I) : t ∈ R} so that Theorem 3.4, together with Lemma 3.3, exactly becomes [33, Theorem 12]. A typical case is the quadratic function f (x) = x2 , which is familiar in quantum models of mean field type as remarked in [33] (also in Introduction). The proof below is a modification of that of [31, Theorem 13.11]. Here it should be noted that the quantities c(ϕ, nf (sn (A)) in [31, §13] and Zϕ (n, A, f ) here are in the relation c(ϕ, nf (sn (A))) = − log Zϕ (n, A, f ) as long as ϕ is a product state. Theorem 3.4. For every continuous f : [λmin (A), λmax (A)] → R the limit 1 pϕ (A, f ) := lim log Zϕ (n, A, f ) n→∞ n exists and pϕ (A, f ) = sup{−EA,f (ω) − SM (ω, ϕ) : ω ∈ Sγ (A)} = max{−f (x) − IA (x) : x ∈ [λmin (A), λmax (A)]}. Proof. By Lemmas 3.2 and 3.3 we only have to show that 1 lim sup log Zϕ (n, A, f ) ≤ sup{−EA,f (ω) − SM (ω, ϕ) : ω ∈ Sγ (A)}. n→∞ n To prove this, we may assume by approximation that f is a polynomial. For each N m ∈ N greater than both `(A) and the range of Φ, let φ(m) := Z ϕG m , a product N state of the re-localized i∈Z A[im+1,(i+1)m] as in (3.4). Furthermore, we set A(m) :=

1 m

m−`(A)

X

γ k (A) ∈ A[1,m] .

k=0

According to [33, Theorem 1] (or [31, Proposition 13.8]), for any ε > 0 there exists N a symmetric (hence γ m -invariant) state ψ of i∈Z A[im+1,(i+1)m] such that lim

j→∞

1 (m) (m) (m) log Zφ(m) (j, A(m) , mf ) < −EA(m) ,mf (ψ) − SM (ψ, φ(m) ) + ε, j

where

Ã

(m) Zφ(m) (j, A(m) , mf )

:= Tr exp log Ã

(m) EA(m) ,mf (ψ)

:= lim ψ mf j→∞

(m)

SM (ψ, φ(m) ) := lim

j→∞

Ã j O Ã

! D(ϕG m)

Ã − jmf

1

!! j−1 1 X im (m) γ (A ) , j i=0

¢ 1 ¡ S ψ|A[1,jm] , φ(m) |A[1,jm] . j

(3.11)

!! j−1 1 X im (m) γ (A ) , j i=0

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Pm−1 1 k Then one can define an ω ∈ Sγ (A) by ω := m k=0 ψ ◦ γ . Since we assumed that f is a polynomial, there is a constant M > 0 (depending on kAk) such that kf (B1 ) − f (B2 )k ≤ M kB1 − B2 k for all B1 , B2 ∈ Asa with kB1 k, kB2 k ≤ kAk. For each n ∈ N with n ≥ m, write n = jm + r where j ∈ N and 0 ≤ r < m. Since m is greater than the range of Φ, one can write Hn (Φ) =

j−1 X

γ

im

(Hm (Φ)) +

i=0

j−1 X

γ im (W ) + Wj ,

i=1

where W is given in (3.5) and X Wj := {Φ(X) : X ⊂ [1, n], X ∩ [jm + 1, jm + r] 6= ∅}. We have by Lemma 2.1 log D(ϕn ) ≤ log D(ϕG n ) + log λ =−

j−1 X

γ

im

i=0

(Hm (Φ)) −

− log Tr exp − ≤ log

γ im (W ) − Wj

i=1

Ã

Ã j O

j−1 X

j−1 X i=0

! D(ϕG m)

γ

im

(Hm (Φ)) −

j−1 X

! γ

im

(W ) − Wj

+ log λ

i=1

+ 2jK + 2kWj k + log λ

1

with K given in (3.6). Here it is clear that kWj k ≤ mL with L := Furthermore, it is readily seen that ° ° µ ¶ j−1 ° 1 X im (m) ° 2 `(A) ° ° γ (A )° ≤ + kAk °sn (A) − ° ° j i=0 j m

P

and hence ° Ã j−1 !° µ ¶ ° 1 X im (m) ° 2 `(A) ° ° + M kAk. γ (A ) ° ≤ °f (sn (A)) − f ° ° j i=0 j m

X30

kΦ(X)k.

(3.12)

Therefore, ° Ã j−1 !° ° 1 X im (m) ° ° ° γ (A ) ° ≤ mkf k∞ + (2m + j`(A))M kAk, °nf (sn (A)) − jmf ° ° j i=0

where kf k∞ is the sup-norm of f on [λmin (A), λmax (A)]. From the above estimates we get 1 1 (m) log Zϕ (n, A, f ) ≤ log Zφ(m) (j, A(m) , mf ) n n ª 1© + 2jK + 2mL + log λ + mkf k∞ + (2m + j`(A))M kAk n

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so that 1 log Zϕ (n, A, f ) n 1 2K `(A)M kAk 1 (m) ≤ lim log Zφ(m) (j, A(m) , mf ) + + . m j→∞ j m m

lim sup n→∞

(3.13)

Next, thanks to (3.12) we get ¯ Ã Ã j−1 !!¯ ¯ ¯ 1 X im (m) ¯ ¯ γ (A ) ¯ω(f (sn (A))) − ψ f ¯ ¯ ¯ j i=0

° !° Ã j−1 ° 1 X im (m) ° ° ° γ (A ) ° kf (γ (sn (A))) − f (sn (A))k + °f (sn (A)) − f ° ° j i=0 k=0 µ ¶ m−1 2 `(A) M X k kγ (sn (A)) − sn (A)k + + M kAk ≤ m j m k=0 µ ¶ 4 `(A) ≤ + M kAk. j m 1 ≤ m

m−1 X

k

Therefore, ¯ ¯ ¯ ¯ ¯EA,f (ω) − 1 EA(m) ,mf (ψ)¯ ≤ `(A)M kAk . ¯ ¯ m m

(3.14)

Furthermore, we get S(ψjm , ϕjm ) ≤ S(ψjm , ϕG jm ) + log λ ≤ S(ψ|A[1,jm] , φ(m) |A[1,jm] ) + 2(j − 1)K + log λ similarly to (3.8) using the state perturbation (3.7). Since SM (ω, ϕ) =

1 1 lim S(ψ|A[1,jm] , ϕ|A[1,jm] ) m j→∞ j

in the same way as (3.9), it follows that SM (ω, ϕ) ≤

1 (m) 2K S (ψ, φ(m) ) + . m M m

(3.15)

Inserting (3.13)–(3.15) into (3.11) gives lim sup n→∞

1 1 log Zϕ (n, A, f ) ≤ −EA,f (ω) − SM (ω, ϕ) + (ε + 4K + 2`(A)M kAk), n m

implying the required inequality because m and ε are arbitrary. The following is a straightforward consequence of Theorem 3.4. Corollary 3.5. For every continuous f : R → R, the function pϕ (·, f ) on Asa loc uniquely extends to a continuous function (denoted by the same pϕ (·, f )) on Asa

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satisfying pϕ (A, f ) = max{−f (x) − IA (x) : x ∈ [λmin (A), λmax (A)]} for all A ∈ Asa . Moreover, for every continuous f, g : R → R and every A ∈ Asa , |pϕ (A, f ) − pϕ (A, g)| ≤ max{|f (x) − g(x)| : x ∈ [λmin (A), λmax (A)]}. Remark 3.6. Suppose the “semi-classical” case where the observable A ∈ Asa loc commutes with all Φ(X). Since αtΦ (A) = A for all t ∈ R, A belongs to the centralizer of ϕ, i.e. ϕ(AB) = ϕ(BA) for all B ∈ A. (To see this, apply [23, p. 617] in the GNS von Neumann algebra πϕ (A)00 having the modular automorphism group which extends αtΦ .) This implies that sn (A) commutes with D(ϕn ) for every n ∈ N. As stated in Remark 2.8, pϕ (tA) becomes the logarithmic moment generating function of (µn ) satisfying the large deviation principle with the good rate function IA (x) in (2.4). For any continuous f : R → R we have Z λmax (A) Zϕ (n, A, f ) = ϕn (exp(−nf (sn (A)))) = e−nf (s) dµn (s). λmin (A)

Now Varadhan’s integral lemma [12, Theorem 4.3.1] can be applied to obtain 1 log Zϕ (n, A, f ) = max{−f (x) − IA (x) : x ∈ [λmin (A), λmax (A)]}. n→∞ n The exact large deviation principle is not formulated in our noncommutative setting as long as the BMV-conjecture remains unsolved (see Remark 2.8); nevertheless Varadhan’s formula is valid as stated in Theorem 3.4. lim

4. Perturbation of Finitely Correlated States The notion of (C ∗ -)finitely correlated states was introduced by Fannes, Nachtergaele and Werner in [14]. Let B be a finite-dimensional C ∗ -algebra, E : A0 ⊗ B → B (A0 = Md (C)) a completely positive unital map and ρ a state of B such that ρ(E(I ⊗ b)) = ρ(b) for all b ∈ B. For each A ∈ A0 define a map EA : B → B by EA (b) := E(A ⊗ b), b ∈ B. Then the finitely correlated state ϕ determined by the triple (B, E, ρ) is the γ-invariant state of A given by ϕ(A0 ⊗ A1 ⊗ · · · ⊗ An ) := ρ(EA0 ◦ EA1 ◦ · · · ◦ EAn (1B )) (Ai ∈ Ai , 0 ≤ i ≤ n). As was shown in the proof of [17, Proposition 4.4], a finitely correlated state has the following upper factorization property, which will be useful in our discussions below. Lemma 4.1. If ϕ is a finitely correlated state of A, then there exists a constant α ≥ 1 such that ¡ ¢ ¡ ¢ ϕ ≤ α ϕ|A(−∞,0] ⊗ ϕ|A[1,∞) . The next proposition is a generalization of [31, Proposition 11.2].

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Proposition 4.2. Let ϕ be a finitely correlated state of A. For every ω ∈ Sγ (A) the mean relative entropy 1 S(ωn , ϕn ) n exists. Moreover, the function ω ∈ Sγ (A) 7→ SM (ω, ϕ) is affine and weakly* lower semicontinuous on Sγ (A). SM (ω, ϕ) = lim

n→∞

Proof. The proof of the first assertion is a slight modification of that of [20, Theorem 2.1] while it will be repeated below for the convenience of the remaining proof. For each n, m ∈ N with n ≥ m, write n = jm + r with j ∈ N and 0 ≤ r < m. Lemma 4.1 implies that ! Ã j−1 O¡ ¢ ¡ ¢ j ϕ|A[im+1,(i+1)m] ⊗ ϕ|A[jm+1,jm+r] . ϕn ≤ α (4.1) i=0

N Consider the product state φ(m) := Z ϕm of the re-localized spin chain N A . For every ω ∈ S (A) we have γ i∈Z [im+1,(i+1)m] ! Ã j O ϕm − j log α S(ωn , ϕn ) ≥ S(ωjm , ϕjm ) ≥ S ωjm , (4.2) 1

due to the monotonicity of relative entropy and (4.1). Dividing (4.2) by n and letting n → ∞ with m fixed we get lim inf n→∞

1 (m) log α 1 S(ωn , ϕn ) ≥ SM (ω, φ(m) ) − , n m m

(m)

where SM (ω, φ(m) ) denotes the mean relative entropy in the re-localized N (m) (m) ) ≥ S(ωm , ϕm ) by [18, (2.1)], i∈Z A[im+1,(i+1)m] as in (3.11). Since SM (ω, φ we further get 1 1 log α S(ωn , ϕn ) ≥ S(ωm , ϕm ) − . n m m Since m ∈ N is arbitrary, this shows the existence of SM (ω, ϕ) and the above inequalities become lim inf n→∞

1 log α S(ωm , ϕm ) − . (4.3) m m The affinity of ω 7→ SM (ω, ϕ) is a consequence of the general property [31, Proposition 5.24]. Assume that ω, ω (k) ∈ Sγ (A) and ω (k) → ω weakly*. Then from (4.3) we have SM (ω, ϕ) ≥

lim inf SM (ω (k) , ϕ) ≥ k→∞

log α 1 log α 1 (k) lim inf S(ωm , ϕm ) − ≥ S(ωm , ϕm ) − k→∞ m m m m

thanks to the lower semicontinuity of relative entropy (in fact, ω 7→ S(ωm , ϕm ) is continuous due to finite dimensionality). Letting m → ∞ shows the lower semicontinuity of ω 7→ SM (ω, ϕ).

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Next we show the existence of the free energy density with respect to a finitely correlated state ϕ. Since ϕ is not assumed to be locally faithful in the sense that D(ϕn ) ¡is strictly positive for every n ∈ N, we need to be careful in defining ¢ Tr exp log D(ϕn ) − B for B ∈ Asa [1,n] . Let D be a nonzero positive semidefinite matrix and B a Hermitian matrix in MN (C). It is known [19, Lemma 4.1] that lim elog(D+εI)−B = P (eP (log D)P −P BP )P,

ε&0

where P is the support projection of D. Hence one can define Tr elog D−B by Tr elog D−B := lim Tr elog(D+εI)−B = Tr P eP (log D)P −P BP . ε&0

(4.4)

Proposition 4.3. Let ϕ be a finitely correlated state of A. For every A ∈ Asa loc so sa that A ∈ AΛ with a finite Λ ⊂ Z, the free energy density Ã ! X 1 k γ (A) pϕ (A) := lim log Tr exp log D(ϕn ) − n→∞ n Λ+k⊂[1,n]

exists (independently of the choice of Λ). Moreover, pϕ is convex and Lipschitz continuous with |pϕ (A) − pϕ (B)| ≤ kA − Bk, and therefore it uniquely extends to a function on Asa with the same properties. Proof. To prove the first assertion we may assume that A ∈ Asa [1,`(A)] with some `(A) ∈ N. For each n, m ∈ N with n ≥ m > `(A), write n = jm + r with 0 ≤ r < m. From (4.1) we get D(ϕn ) ≤ αj

j−1 Y

γ im (D(ϕm ))

i=0

with a constant α ≥ 1 independent of n, m. For any ε > 0 this implies that D(ϕn ) + εj I ≤ αj

j−1 Y

γ im (D(ϕm ) + εI).

i=0

Furthermore, it is immediately seen that Ãm−`(A) ! n−`(A) j−1 X X X k im k γ (A) ≥ γ γ (A) − (j(`(A) − 1) + r)kAk. k=0

i=0

k=0

Pn−`(A)

Set hn := k=0 γ k (A). From the above two inequalities we get ¡ ¢ Tr exp log(D(ϕn ) + εj I) − hn © ¡ ¢ªj ¡ ¢ ≤ Tr exp log(D(ϕm ) + εI) − hm exp j log α + (j(`(A) − 1) + r)kAk . In view of the definition (4.4), letting ε & 0 gives ¡ ¢ Tr exp log D(ϕn ) − hn © ¡ ¡ ¢ ¢ªj ≤ Tr exp log D(ϕm ) − hm exp j log α + (j(`(A) − 1) + r)kAk

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so that ¡ ¢ 1 log Tr exp log D(ϕn ) − hn n n→∞ ¡ ¢ log α `(A) − 1 1 log Tr exp log D(ϕm ) − hm + + kAk. ≤ m m m Since m (> `(A)) is arbitrary, this shows the existence of the limit pϕ (A). It is obvious that pϕ (A) is independent of the choice of Λ. It is also clear that pϕ (A) on sa Asa loc is convex and satisfies |pϕ (A) − pϕ (B)| ≤ kA − Bk for all A, B ∈ Aloc , from which the second part of the proposition follows. lim sup

Remark 4.4. The limit p˜ϕ (A) similar to pϕ (A) was referred to in Remark 2.7 from the viewpoint of large deviations. In [17] the limit p˜ϕ (A) was shown to exist for any A ∈ Asa loc when ϕ is a finitely correlated state (as well as when ϕ is a Gibbs state). In fact, the proof for p˜ϕ in [17] was given in a more general setting and is more involved than the above for pϕ , which relies on the estimate in [26, Theorem 3.7] related to Gibbs state perturbation. Once we had Propositions 4.2 and 4.3, it is natural to expect that SM (ω, ϕ) and pϕ (A) enjoy the same Legendre transform formulas as (4) and (5) of Corollary 2.5 in the Gibbs state case. But this is still unsolved while the following inequality is easy as Lemma 3.2. For the proof use [20, (4.2)] or [31, Proposition 1.11], the extended version of (3.1). Proposition 4.5. Let ϕ be a finitely correlated state of A. For every A ∈ Asa , pϕ (A) ≥ max{−ω(A) − SM (ω, ϕ) : ω ∈ Sγ (A)}. Remark 4.6. Suppose that ϕ satisfies the lower factorization property ¡ ¢ ¡ ¢ ϕ ≥ β ϕ|A(−∞,0] ⊗ ϕ|A[1,∞) for some β > 0 (the opposite version of Lemma 4.1). (In fact, it is enough to suppose a slightly weaker version of lower factorization as in [17, Definition 4.1].) Then all the results in Sec. 3 are true for ϕ. The proofs can be carried out similarly to those in Sec. 3; in fact, they are even easier without the state perturbation technique. However, the lower factorization property for finitely correlated states is quite strong; for example, one can easily see that a classical irreducible Markov chain has this property if and only if its transition stochastic matrix is strictly positive (i.e., all entries are strictly positive), which is stronger than the strong mixing property. More details are in [17]. In the rest of this section, we assume that ϕ is a γ-invariant quantum Markov state of Accardi and Frigerio type [3], and further assume that ϕ is locally faithful. According to [4, 30], there exists a conditional expectation E from Md (C) ⊗ Md (C) into Md (C) such that ϕ0 ◦ E(I ⊗ A) = ϕ0 (A) for all A ∈ Md (C) and ϕ(A0 ⊗ A1 ⊗ · · · ⊗ An ) = ϕ0 (E(A0 ⊗ E(A1 ⊗ · · · ⊗ E(An−1 ⊗ An ) · · ·)))

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for all A0 , A1 , . . . , An ∈ Md (C), where ϕ0 := ϕ|A0 . Set B := E(Md (C) ⊗ Md (C)), a subalgebra of Md (C), E := E|Md (C)⊗B and ρ := ϕ0 |B . Then ϕ is a finitely correlated state with the triple (B, E, ρ). Let q1 , . . . , qk be the minimal central projections of B; then Bqi ∼ = Mdi (C) and B is decomposed as B=

k M

Bqi =

i=1

k M ¡

¢ Mdi (C) ⊗ Imi ,

i=1

where mi is the multiplicity of Mdi (C) in Md (C). Let B0 be the relative commutant Lk of B in Md (C) so that B0 = i=1 Idi ⊗ Mmi (C). For each m, n ∈ Z, m ≤ n, set Ae[m,n] := B 0 ⊗ A[m+1,n−1] ⊗ B (⊂ A[m,n] ) Lk with convention Ae[n,n] := CI (⊂ An ). Let C := i=1 Mdi (C) ⊗ Mmi (C) (⊂ Md (C)) Pk and EC be the pinching A ∈ Md (C) 7→ i=1 qi Aqi ∈ C (or the conditional expectation onto C with respect to the trace). The following properties were shown in [4, 30]: (i) There exist positive linear functionals ρij on Mmi (C) ⊗ Mdj (C), 1 ≤ i, j ≤ k, such that Ã k ! M idMdi (C) ⊗ ρij ◦ (EC ⊗ idB ). E= i,j=1

(ii) Let Tij be the density matrices of ρij for 1 ≤ i, j ≤ k. Then the density matrix of ϕ|Ae[m,n] is M e [m,n] := D ρ(qim )Tim im+1 ⊗ Tim+1 im+2 ⊗ · · · ⊗ Tin−1 in . (4.5) im ,im+1 ,...,in

e [m,n] have a simple form of product type. Since Tij is The density matrices D strictly positive in Mmi (C) ⊗ Mdj (C) for each i, j due to the local faithfulness of ϕ, a γ-invariant nearest-neighbor interaction Φ can be defined by Φ([0, 1]) := −

k X

log Tij ∈ B 0 ⊗ B ⊂ A[0,1] ,

Φ([n, n + 1]) := γ n (Φ([0, 1])).

i,j=1

Then the density of the local Gibbs state of A[m,n] for Φ is M Tim im+1 ⊗ · · · ⊗ Tin−1 in , im ,...,in

and the automorphism group αtΦ is given by αtΦ (A) =

lim

m→−∞,n→∞

e −it AD e it D [m,n] [m,n]

(A ∈ A).

(4.6)

Hence ϕ is the αΦ -KMS state (or the Gibbs state for Φ) and so all the results in Secs. 2 and 3 hold for ϕ. Below let us further investigate the relation between pϕ (A) in (2.3) and p˜ϕ (A) in (2.5).

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The centralizer of ϕ is given by Aϕ := {A ∈ A : ϕ(AB) = ϕ(BA) for all B ∈ A}, which is a γ-invariant C ∗ -subalgebra of A. For each m, n ∈ Z with m ≤ n, we also define (Ae[m,n] )ϕ := {A ∈ Ae[m,n] : ϕ(AB) = ϕ(BA) for all B ∈ Ae[m,n] }. Lemma 4.7. If m0 ≤ m ≤ n ≤ n0 in Z, then (Ae[m,n] )ϕ ⊂ (Ae[m0 ,n0 ] )ϕ ⊂ Aϕ . S∞ Moreover, Aeϕ,loc := n=1 (Ae[−n,n] )ϕ is a dense ∗-subalgebra of Aϕ . e [m,n] } in Ae[m,n] , the first Proof. Since (Ae[m,n] )ϕ is the relative commutant of {D e assertion is immediately seen from the form (4.5) of D[m,n] . Furthermore, it is also obvious from (4.6) that αtΦ (Ae[m,n] ) = Ae[m,n] , t ∈ R, for any m ≤ n. By [37] applied in the GNS von Neumann algebra πϕ (A)00 with the modular automorphism group extending αtΦ , there exists the conditional expectation E[m,n] : A → Ae[m,n] with ϕ ◦ E[m,n] = ϕ. Then it is clear that kE[m,n] (A) − Ak → 0 as m → −∞ and n → ∞ for any A ∈ A. Now let A ∈ Aϕ . Since ϕ(E[m,n] (A)B) = ϕ(AB) = ϕ(BA) = ϕ(BE[m,n] (A)),

B ∈ Ae[m,n] ,

we have E[m,n] (A) ∈ (Ae[m,n] )ϕ for any m ≤ n, implying the latter assertion. Lemma 4.8. For every ω ∈ Sγ (A), SM (ω, ϕ) = lim

n→∞

¢ 1 ¡ S ω|(Ae[1,n] )ϕ , ϕ|(Ae[1,n] )ϕ n

and hence SM (ω, ϕ) is determined by ω|Aϕ . Moreover, if ω, ω (i) ∈ Sγ (A), i ∈ N, and ω (i) |Aϕ → ω|Aϕ in the weak* topology, then SM (ω, ϕ) ≤ lim inf SM (ω (i) , ϕ). i→∞

Proof. The proof of the first assertion is essentially the same as that of [18, TheoPLij rem 2.1] as will be sketched below. Let Tij = `=1 λij` eij` be the spectral decomposition of Tij for 1 ≤ i, j ≤ k, and Θ be the set of all (i, j, `) with 1 ≤ i, j ≤ k and 1 ≤ ` ≤ Lij . For each n ∈ N let Kn be the set of all tuples (nθ )θ∈Θ of nonnegP ative integers such that θ∈Θ nθ = n − 1. For each 1 ≤ i ≤ k and (nθ ) ∈ Kn we denote by Ii,(nθ ) the set of all (i1 , i2 , . . . , in ; `1 , `2 , . . . , `n−1 ) such that i1 = i and #{r ∈ [1, n − 1] : (ir , ir+1 , `r ) = θ} = nθ for all θ ∈ Θ, and define the projection Pi,(nθ ) in Ae[1,n] and λi,(nθ ) ∈ R by X Pi,(nθ ) := ei1 i2 `1 ⊗ ei2 i3 `2 ⊗ . . . ⊗ ein−1 in `n−1 , (i1 ,...,in ;`1 ,...,`n−1 )∈Ii,(nθ )

λi,(nθ ) := ρ(qi )

Y

θ∈Θ

λnθ θ

where λθ := λij` for θ = (i, j, `).

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Then

Pk

P

i=1

(nθ )∈Kn

359

e [1,n] is written as Pi,(nθ ) = I and D e [1,n] = D

k X

X

λi,(nθ ) Pi,(nθ ) .

i=1 (nθ )∈Kn

Now, for each ω ∈ Sγ (A), the proof of [18, Theorem 2.1] implies that ¢ ¡ ¢ ¡ S(ωn−2 , ϕn−2 ) ≤ S ω|Ae[1,n] , ϕ|Ae[1,n] ≤ S ω|(Ae[1,n] )ϕ , ϕ|(Ae[1,n] )ϕ + log k + log #Kn for every n ≥ 3. Since #Kn ≤ n#Θ , we get SM (ω, ϕ) ≤ lim inf n→∞

¢ 1 ¡ S ω|(Ae[1,n] )ϕ , ϕ|(Ae[1,n] )ϕ , n

which proves the first assertion. Set γ := 1/ min1≤i≤k ρ(qi ). For each m, m0 ∈ N, since it follows from (4.5) that ¡ ¢ ¡ ¢ ϕ|(Ae[1,m] )ϕ ⊗(Ae ≤ γ ϕ|(Ae[1,m] )ϕ ⊗ ϕ|(Ae , 0 )ϕ 0 )ϕ [m+1,m+m ]

[m+1,m+m ]

we get ¡ S ω|(Ae

[1,m+m0 ] )ϕ

¢

, ϕ|(Ae

[1,m+m0 ] )ϕ

¡ ¢ ≥ S ω|(Ae[1,m] )ϕ ⊗(Ae , ϕ|(Ae[1,m] )ϕ ⊗(Ae − log γ [m+1,m+m0 ] )ϕ [m+1,m+m0 ] )ϕ ¡ ¢ ¡ ¢ ≥ S ω|(Ae[1,m] )ϕ , ϕ|(Ae[1,m] )ϕ + S ω|(Ae 0 )ϕ , ϕ|(Ae 0 )ϕ − log γ [1,m ]

[1,m ]

due to the monotonicity and the superadditivity of relative entropy [31, Corollary 5.21]. Let ω and ω (i) be given as stated in the lemma. For any m ∈ N and n = jm + r with j ∈ N and 0 ≤ r < m, the above inequality gives ¡ ¢ ¡ ¢ S ω (i) |(Ae[1,n] )ϕ , ϕ|(Ae[1,n] )ϕ ≥ jS ω (i) |(Ae[1,m] )ϕ , ϕ|(Ae[1,m] )ϕ − j log γ. Dividing this by n and letting n → ∞ with m fixed we get SM (ω (i) , ϕ) ≥

¢ log γ 1 ¡ (i) S ω |(Ae[1,m] )ϕ , ϕ|(Ae[1,m] )ϕ − m m

and hence lim inf SM (ω (i) , ϕ) ≥ i→∞

¢ log γ 1 ¡ S ω|(Ae[1,m] )ϕ , ϕ|(Ae[1,m] )ϕ − . m m

Letting m → ∞ gives the latter assertion. In addition to the variational expression in Corollary 2.5(5) we have Theorem 4.9. For every ω ∈ Sγ (A), SM (ω, ϕ) = sup{−ω(A) − pϕ (A) : A ∈ Asa ϕ} = sup{−ω(A) − p˜ϕ (A) : A ∈ Asa loc }, where p˜ϕ (A) is given in (2.5).

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Proof. The proof of the first equality is a simple duality argument. Set Γ := ∗ {ω|Asa : ω ∈ Sγ (A)}, which is a weakly* compact and convex subset of (Asa ϕ ) , the ϕ sa dual Banach space of the real Banach space Aϕ . From Lemma 4.8 one can define ∗ F : (Asa ϕ ) → [0, +∞] by ½ with some ω ∈ Sγ (A), SM (ω, ϕ) if f = ω|Asa ϕ F (f ) := +∞ otherwise, ∗ which is affine and weakly* lower semicontinuous on (Asa ϕ ) by Proposition 4.2 and Lemma 4.8. Corollary 2.5(4) says that ∗ pϕ (A) = max{−f (A) − F (f ) : f ∈ (Asa ϕ ) },

A ∈ Asa ϕ.

Hence it follows by duality [13, Proposition I.4.1] that F (f ) = sup{−f (A) − pϕ (A) : A ∈ Asa ϕ },

∗ f ∈ (Asa ϕ) .

For every ω ∈ Sγ (A) this means the first equality, which also gives SM (ω, ϕ) = sup{−ω(A) − pϕ (A) : A ∈ Aesa ϕ,loc }

(4.7)

thanks to Lemma 4.7. To prove the second equality, we show that pϕ (A) = p˜ϕ (A) for all A ∈ Aesa ϕ,loc . Thanks to Lemma 4.7 and the γ-invariance of pϕ and p˜ϕ , we may assume that A ∈ (Ae[1,m] )sa ϕ for some m ∈ N. For each n ∈ N and 0 ≤ k ≤ n − m, we have ¢ ¡ Pn−m k e γ (A) ∈ (A[1+k,m+k] )ϕ ⊂ (Ae[1,n] )ϕ so that exp − k=0 γ k (A) ∈ (Ae[1,n] )ϕ . Furthermore, since Ae[1,n] ⊂ A[1,n] ⊂ Ae[0,n+1] , it is easy to see by Lemma 4.7 that ¡ Pn−m k ¢ (Ae[1,n] )ϕ¡⊂ (A[1,n] )ϕ . Hence ¢ we get exp − k=0 γ (A) ∈ (A[1,n] )ϕ , which implies Pn−m k that exp − k=0 γ (A) commutes with the density D(ϕn ) so that Ã Ã n−m !! Ã ! n−m X X k k ϕ exp − γ (A) = Tr exp log D(ϕn ) − γ (A) , k=0

k=0

showing pϕ (A) = p˜ϕ (A) by definitions (2.3) and (2.5). From this and (4.7) we get SM (ω, ϕ) ≤ sup{−ω(A) − p˜ϕ (A) : A ∈ Asa loc } ≤ sup{−ω(A) − pϕ (A) : A ∈ Asa loc } = SM (ω, ϕ) thanks to (2.6) and Corollary 2.5(5), implying the second equality. Corollary 4.10. The function pϕ on Asa is the lower semicontinuous convex envelope of p˜ϕ on Asa loc in the sense that pϕ is the largest among lower semicontinuous and convex functions q on Asa satisfying q ≤ p˜ϕ on Asa loc . Proof. Let q be as stated in the corollary. Define Q : (Asa )∗ → (−∞, +∞] by Q(f ) := sup{−f (A) − q(A) : A ∈ Asa }

(f ∈ (Asa )∗ ).

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Let us prove that ½

Q(ω) ≥ SM (ω, ϕ) Q(f ) = +∞

if ω ∈ Sγ (A), if f ∈ (Asa )∗ \Sγ (A).

(4.8)

For ω ∈ Sγ (A) Theorem 4.9 gives Q(ω) ≥ sup{−ω(A) − p˜ϕ (A) : A ∈ Asa loc } = SM (ω, ϕ). For f ∈ (Asa )∗ \Sγ (A) we may consider the following three cases: (a) f (A) < 0 for some positive A ∈ Aloc , (b) f (1) 6= 1, (c) f (A) = 6 f (γ(A)) for some A ∈ Asa . In case (a), since q(αA) ≤ p˜ϕ (αA) ≤ 0 for α > 0, we get −f (αA) − q(αA) ≥ −αf (A) → +∞ as α → +∞. In case (b), since q(α1) ≤ p˜ϕ (α1) = −α, we get −f (α1) − q(α1) ≥ −α(f (1) − 1) → +∞ as α → +∞ or −∞ according as f (1) < 1 or f (1) > 1. Finally in case (c), since n 1 log ϕ(e−α(A−γ (A)) ) = 0, n→∞ n

q(α(A − γ(A))) ≤ p˜ϕ (α(A − γ(A))) = lim

we get −f (α(A − γ(A))) − q(α(A − γ(A))) ≥ −αf (A − γ(A)) → +∞ as α → +∞ or −∞ according as f (A) < f (γ(A)) or f (A) > f (γ(A)). Hence (4.8) follows. By duality this implies that q ≤ pϕ on Asa . N In particular, when ϕ is the product state Z ρ of a not necessarily faithful ρ, all the variational expressions in Corollary 2.5 and Theorem 4.9 are valid for ϕ, and so Corollary 4.10 holds for ϕ. Although we have no strong evidence, it might be conjectured that Corollary 4.10 is true generally for the Gibbs-KMS state ϕ treated in Secs. 2 and 3. 5. Concluding Remarks: Guide to the Case of Arbitrary Dimension In this paper we confined ourselves to the one-dimensional spin chain case for the following reasons. First, our main motivation came from recent developments on large deviations in spin chains, where the differentiability of logarithmic moment generating functions is crucial. The corresponding functions in our setting are free energy density functions so that we wanted to provide their differentiability (see Theorem 2.4(c) and Corollary 2.5(2)), and the one dimensionality is essential for this. Secondly, finitely correlated states treated in the latter half are defined only in a one-dimensional spin chain though some attempts to multi-dimensional extension were made for similar states of quantum Markov type (see [1, 2] for example). However, all the discussions (except the differentiability assertions) presented for a

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Gibbs state of one dimension in Secs. 2 and 3 can also work well in the setting of arbitrary dimension but in high temperature regime, which we outline below. N Consider a ν-dimensional spin chain A := k∈Zν Ak , Ak = Md (C), with the N translation automorphism group γk , k ∈ Zν , and local algebras AΛ := k∈Λ Ak for finite Λ ⊂ Zν . We denote by B(A) the set of all translation-invariant interacP tions Φ in A of relatively short range, i.e. |||Ψ||| := X30 kΨ(X)k/|X| < +∞, which is a real Banach space with the norm |||Ψ|||. Let Φ ∈ B(A) and assume further that Φ is of finite body, i.e. N (Φ) := sup{|X| : Φ(X) 6= 0} < +∞ (weaker than the assumption of finite range). Then Φ is automatically of short range, i.e. P kΦk := X30 kΦ(X)k < +∞. It is well known [11, 22] that the one-parameter automorphism group αtΦ of A is defined and all of the αΦ -KMS condition, the Gibbs condition and the variational principle for states ϕ ∈ Sγ (A) are equivalent. The pressure (1.5) and the mean entropy (1.6), the main ingredients in the variational principle, can be defined in the van Hove limit of Λ → ∞ (see [22, p. 12] or [11, p. 287]), but in our further discussions we may simply restrict to the parallelepipeds Λ = {(k1 , . . . , kν ) : 1 ≤ ki ≤ ni , 1 ≤ i ≤ ν} with Λ → ∞ meaning ni → ∞ for 1 ≤ i ≤ ν. A crucial point in the arbitrary dimensional setting is the following generalization of Lemma 2.1 given in [8] in high temperature regime with an inverse temperature β. Lemma 5.1. Let Φ be given as above and r(Φ) := {2kΦk(N (Φ) − 1)}−1 (meant +∞ if N (Φ) ≤ 1). Assume that 0 < β < 2r(Φ) and ϕ ∈ Sγ (A) satisfies the Gibbs condition for βΦ (equivalently, the αΦ -KMS condition at −β). Then there are constants λΛ such that β,G λ−1 ≤ λΛ ϕΛ Λ ϕΛ ≤ ϕΛ

and lim

Λ→∞

log λΛ = 0, |Λ|

(5.1)

where ϕβ,G is the local Gibbs state of AΛ for βΦ. Λ Even though a Gibbs state ϕ ∈ Sγ (A) for βΦ is not necessarily unique and constants λΛ are depending on Λ, property (5.1) is enough for us to show all the results in Sec. 2 (except the differentiability assertions mentioned above) in the same way under the situation where Φ is replaced by βΦ with β as in Lemma 5.1 and B0 (A) is replaced by B(A). In particular, it was formerly observed in [20, pp. 710–711] that for every ω ∈ Sγ (A) the mean relative entropy (2.1) exists and furthermore SM (ω, ϕ) = 0 if and only if ω is a Gibbs state for βΦ too. In fact, the latter assertion is immediate from the formula in Lemma 2.3 due to the equivalence of the Gibbs condition and the variational principle. sa Next let A ∈ Asa loc so that we may assume that A ∈ AΛ0 with some parallelepiped ν Λ0 ⊂ Z of the form mentioned above. Let f be a real continuous function on

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[λmin (A), λmax (A)]. For each parallelepiped Λ of the same form, we set X 1 sΛ (A) := γk (A) |Λ| Λ0 +k⊂Λ

and ¡ ¢ Zϕ (Λ, A, f ) := Tr exp log D(ϕΛ ) − |Λ|f (sΛ (A)) . Then Lemmas 3.1 and 3.2 hold true in the same way as before. Moreover, the proof of Lemma 3.3 can easily be carried out in the present framework with slight modifications, for example, with replacing the uniform boundedness of surface energies by the asymptotic property 1 X {kΦ(X)k : X ∩ Λ 6= ∅, X ∩ Λc 6= ∅} → 0 |Λ| as Λ → ∞ of parallelepipeds Λ. In fact, this property holds in general for translationinvariant interactions of short range. Finally, we can prove the existence of the functional free energy density pϕ (A, f ) := lim

Λ→∞

1 log Zϕ (Λ, A, f ) |Λ|

and its variational expressions in the same way as in Theorem 3.4. A key point in proving this is that the result for the product state case in [33] (or [31]) used in the proof of Theorem 3.4 can be applied as well since the dimension of the integer lattice is irrelevant in the situation of product/symmetric states. In this way, all the proofs in Sec. 3 of one dimension can easily be adapted to the present framework by using Lemma 5.1 and the property of short range for Φ, and the condition of finite range is not necessary. Acknowledgments The authors are grateful to a referee for valuable comments that helped to improve the final version of the paper. This work was partially supported by Grant-inAid for Scientific Research (B) 17340043 (F. H.), Grant-in-Aid for JSPS Fellows 18·06916 (M. M.), Grant-in-Aid for JSPS Fellows 19·2166 (H. O.) and the Hungarian Research Grant OTKA T068258 (D. P., M. M.). References [1] L. Accardi and F. Fidaleo, Quantum Markov fields, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003) 123–138. [2] L. Accardi and F. Fidaleo, Non-homogeneous quantum Markov states and quantum Markov fields, J. Funct. Anal. 200 (2003) 324–347. [3] L. Accardi and A. Frigerio, Markovian cocycles, Proc. Roy. Irish Acad. 83A(2) (1983) 251–263. [4] L. Accardi and V. Liebscher, Markovian KMS-states for one-dimensional spin chains, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2 (1999) 645–661.

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[5] H. Araki, Gibbs states of a one-dimensional quantum lattice, Comm. Math. Phys. 14 (1969) 120–157. [6] H. Araki, On uniqueness of KMS states of one-dimensional quantum lattice systems, Comm. Math. Phys. 44 (1975) 1–7. [7] H. Araki, Radon–Nikodym theorems, relative Hamiltonian and the Gibbs condition in statistical mechanics. An application of the Tomita–Takesaki theory, in C*-Algebras and their Applications to Statistical Mechanics and Quantum Field Theory (Proc. Internat. School of Physics “Enrico Fermi ”, Course LX, Varenna, 1973), ed. D. Kastler (North Holland, Amsterdam, 1976), pp. 64–100. [8] H. Araki and P. D. F. Ion, On the equivalence of KMS and Gibbs conditions for states of quantum lattice systems, Comm. Math. Phys. 35 (1974) 1–12. [9] D. Bessis, P. Moussa and M. Villani, Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics, J. Math. Phys. 16 (1975) 2318–2325. [10] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, 2nd edn. (Springer-Verlag, 2002). [11] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2, 2nd edn. (Springer-Verlag, 1997). [12] A. Dembo and O. Zeitouni, Large Deviation Techniques and Applications, 2nd edn. (Springer, New York, 1998). [13] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Studies in Mathematics and its Applications, Vol. 1 (North-Holland, Amsterdam-Oxford, 1976). [14] M. Fannes, B. Nachtergaele and R. F. Werner, Finitely correlated states on quantum spin chains, Comm. Math. Phys. 144 (1992) 443–490. [15] P. Hayden, R. Jozsa, D. Petz and A. Winter, Structure of states which satisfy strong subadditivity of quantum entropy with equality, Comm. Math. Phys. 246 (2004) 359–374. [16] F. Hiai, Equality cases in matrix norm inequalities of Golden–Thompson type, Linear Multilinear Algebra 36 (1994) 239–249. [17] F. Hiai, M. Mosonyi and T. Ogawa, Large deviations and Chernoff bound for certain correlated states on the spin chain, J. Math. Phys. 48 (2007) 123301, 1–19. [18] F. Hiai and D. Petz, The proper formula for relative entropy and its asymptotics in quantum probability, Comm. Math. Phys. 143 (1991) 99–114. [19] F. Hiai and D. Petz, The Golden–Thompson trace inequality is complemented, Linear Algebra Appl. 181 (1993) 153–185. [20] F. Hiai and D. Petz, Entropy densities for Gibbs states of quantum spin systems, Rev. Math. Phys. 5 (1993) 693–712. [21] F. Hiai and D. Petz, Entropy densities for algebraic states, J. Funct. Anal. 125 (1994) 287–308. [22] R. B. Israel, Convexity in the Theory of Lattice Gases (Princeton Univ. Press, Princeton, 1979). [23] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. II. Advanced Theory (Amer. Math. Soc., Providence, RI, 1997). [24] A. Kishimoto, On uniqueness of KMS states of one-dimensional quantum lattice systems, Comm. Math. Phys. 47 (1976) 167–170. [25] H. Komiya, Elementary proof for Sion’s minimax theorem, Kodai Math. J. 11 (1988) 5–7. [26] M. Lenci and L. Rey-Bellet, Large deviations in quantum lattice systems: one-phase region, J. Stat. Phys. 119 (2005) 715–746.

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[27] E. H. Lieb and R. Seiringer, Equivalent forms of the Bessis–Moussa–Villani conjecture, J. Stat. Phys. 115 (2004) 185–190. [28] M. Mosonyi and D. Petz, Structure of sufficient quantum coarse-grainings, Lett. Math. Phys. 68 (2004) 19–30. [29] K. Neto˘cn´ y and F. Redig, Large deviations for quantum spin systems, J. Stat. Phys. 117 (2004) 521–547. [30] H. Ohno, Translation-invariant quantum Markov states, Interdiscip. Inform. Sci. 10 (2004) 53–58. [31] M. Ohya and D. Petz, Quantum Entropy and Its Use, 2nd edn. (Springer-Verlag, 2004). [32] D. Petz, First steps towards a Donsker and Varadhan theory in operator algebras, in Quantum Probability and Applications IV, Lecture Notes in Math., Vol. 1442 (Springer, 1990), pp. 311–319. [33] D. Petz, G. A. Raggio and A. Verbeure, Asymptotic of Varadhan-type and the Gibbs variational principle, Comm. Math. Phys. 121 (1989) 271–282. [34] R. R. Phelps, Lectures on Choquet’s Theorem (Van Nostrand, New York-TorontoLondon-Melbourne, 1966). [35] M. Sion, On general minimax theorems, Pacific J. Math. 8 (1958) 171–176. [36] G. A. Raggio and R. F. Werner, Quantum statistical mechanics of general mean field systems, Helvetica Phys. Acta 62 (1989) 980–1003. [37] M. Takesaki, Conditional expectations in von Neumann algebras, J. Funct. Anal. 9 (1972) 306–321.

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Reviews in Mathematical Physics Vol. 20, No. 4 (2008) 367–406 c World Scientific Publishing Company °

IONIZATION BY QUANTIZED ELECTROMAGNETIC FIELDS: THE PHOTOELECTRIC EFFECT

HERIBERT ZENK Mathematisches Institut, Ludwig-Maximilians-Universit¨ at, Theresienstraße 39, 80333 M¨ unchen, Germany [email protected] Received 22 March 2007 Revised 23 November 2007 In this paper, we explain the photoelectric effect in a variant of the standard model of non relativistic quantum electrodynamics, which is in some aspects more closely related to the physical picture, than the one studied in [5]. Now, we can apply our results to an electron with more than one bound state and to a larger class of electron-photon interactions. We will specify a situation, where the second order of ionization probability is a weighted sum of single photon terms. Furthermore, we will see that Einstein’s equality Ekin = hν − 4E > 0 for the maximal kinetic energy Ekin of the electron, energy hν of the photon and ionization gap 4E is the crucial condition, for these single photon terms to be nonzero. Keywords: Photoelectric effect; scattering theory; QED. Mathematics Subject Classification 2000: 81Q10, 81V10, 47N50

1. A Mathematical Model for the Photoelectric Effect 1.1. Introduction In the first years after the discovery of the photoelectric effect, it has been a big challenge to obtain more and finer experimental results. Parallel to experiment, there were changes in the theoretical interpretation, which had to be verified in the experiment: • In 1887, Hertz [13] observed, that the length of a flame in a “Funkenstrecke” depends on the light falling on the apparatus. Most remarkable is his intuition, that this effect depends on the ultraviolet part of the incident light. • A year later, Hallwachs [12] saw that an isolated, negatively charged metal plate loses its charge, when it is enlighted with ultraviolet light. This is the simplest setup for the photoeffect we know it already from our physics lessons. • Although there were a couple of experimental results in the next years, every attempt for a theoretical description failed. There was no theory based on classical 367

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physics, which could explain the existence of a minimal frequency ν0 of the incoming light needed for the photoelectric effect to take place. • A turning point in our way of describing nature, is Einstein’s paper [8] from 1905, where he takes a look at Wien’s radiation formula from the viewpoint of statistical mechanics and thermodynamic. He concludes: Monochromatische Strahlung von geringer Dichte (innerhalb des G¨ ultigkeitsbereiches der Wienschen Strahlungsformel) verh¨alt sich in w¨armetheoretischer Beziehung so, wie wenn sie aus voneinander unabh¨angigen Energiequanten von der Gr¨oße hν best¨ unde and applies this conclusion for the photoelectric effect. The “Energiequanten” are nowadays called photons and the photoelectric effect is in this picture a consequence of the absorption of photons by electrons in the metal. An electron inside the metal needs a minimum amount of energy 4E to leave the metal. If one electron is allowed to absorb only one photon, then due to conservation of energy it may escape from the metal, provided hν > 4E. This explains the minimal frequency ν0 = proposed the bound

(1.1) 4E h ,

Ekin = hν − 4E

but at the same time this model (1.2)

for the maximal kinetic energy Ekin of an escaping electron. • The experimental verification of (1.2) was done by Millikan [19, 20] in 1916. It confirms that Einstein’s model is appropriate to describe the photoelectric effect. The goal of this article is to explain the photoelectric effect in some variants of the standard model of non relativistic quantum electrodynamics, which are more closely related to the physical picture than the model studied in [5]. Now we can apply our results to an electron with more than one bound state and to a larger class of electron-photon interactions. The paper is organized as follows: We start with a short overview of the photoelectric effect, motivate the definition of the two quantities, which we call zeroth and second order of ionization probability and describe our main results. In Sec. 2, we introduce the model(s) under consideration stating all definitions and model assumptions. In particular, this includes a description of the electron and the photon subsystems and the total interacting systems in terms of Hamiltonians generating the dynamics. A description of the photoelectric effect needs some special initial states, which model a bound state plus some incoming photons. For this initial states we derive an asymptotic expansion of the full interacting time evolution in terms of free Heisenberg time evolutions in Sec. 3. This asymptotic expansion is the key ingredient in the definition of the zeroth and second order of ionization probability. These two quantities are modifications towards a limit of weak coupling of the transported charge in [5]. The results for zeroth and

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second order of ionization probability are stated and proven in Sec. 4. Finally the Appendix contains some of the often used technical tools. 1.2. Zeroth and second order of ionization probability, photon clouds and photoelectric effect For a Pauli–Fierz operator of non relativistic quantum electrodynamics Hg = H0 + gW (1) + g 2 W (2) = Hel ⊗ 1 + 1 ⊗ Hf + gW (1) + g 2 W (2) with ground state Φg and ground state energy Eg (see Sec. 2 for precise definitions and model assumptions), we want to see the relationship of this model to the photoelectric effect we know from standard physics textbooks. The experimental setup consists in the simplest form of a source, emitting a beam of photons, which are absorbed in a “target”. A detector measures the current of the electrons emitted from the target. If there is any effect at all, it is seen “immediately”, which is within about 10−9 s, see [21, p. 48]. How can we relate this experiment with the above Hamiltonian? The quantum mechanical model under consideration should cover all effects of non relativistic quantum electrodynamics, in particular Compton scattering. The borderline between Compton scattering and photoeffect in this model is hard to define; it depends on the initial state: In Compton, scattering an electron, which is not bound to an atom is scattered in the presence of the photon field. On the other side, a bound state, which is ionized by photons is the setup of the photoelectric effect. Hence for a description of the photoeffect, we have to choose some initial states, which model a bound state plus some photons. 1.2.1. Choice of initial state Choose f1 , . . . , fN ∈ C0∞ (R3 \{0}) (to avoid infrared and ultraviolet divergencies or complicated integrability conditions) and λ1 , . . . , λN ∈ Z2 , define the creation operator A :=

N Y

a∗λj (fj ),

(1.3)

j=1

and its free time evolution Aτ := e−iτ H0 Aeiτ H0

(1.4)

and take as the initial state A(t)Φg := e−itHg eitH0 Ae−itH0 eitHg Φg = e−itHg

N Y

a∗λj (eitω fj )eitHg Φg

(1.5)

j=1

in the limit t → ∞. The following points motivate our choice of the initial state: • Φg as bound state: In a similar model, where the interacting Hamiltonian is also called Hg and under some conditions specified in [3, Theorem I.2 and Corollary III.5] the spectrum of Hg is purely absolute continuous outside a

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O(g)-neighborhood of all eigenvalues and thresholds of Hel . Moreover the spectrum is absolutely continuous in those neighborhoods of the energies e1 , e2 , . . . corresponding to the exited states of Hel below the ionization threshold. So the ground state Φg is the only eigenstate of Hg below a O(g) neighborhood of the ionization threshold in this slightly different model. • We want to decide, if the photoeffect is either: — a collective effect of many photons and depends, e.g., on the sum of all photon energies, — or if it can be explained as a result of some single photon processes and depends, e.g., on the maximum of all photon energies. For this purpose, we have a look at N > 1 incoming photons, otherwise we would not see any difference in the two cases. • In [5] we have seen, that a single photon result like (1.2) is a result of the preparation of the initial state: In Einstein’s model the interaction is essentially turned on and off by hand, hence the energy balance is the noninteracting one. If we add a photon cloud at time zero to the ground state by just applying creation operators AΦg =

N Y

a∗λj (fj )Φg ,

j=1

then due to the interaction, we would expect a modified energy balance compared to the noninteracting case. On the other hand, if we observe exponential decay of Φg , then we could hope to mimic such an almost free energy balance by adding the photons wide inside the exponential tail of Φg , where the wavefunction is tiny and the interaction may be negligible. A way to write this vague idea in precise formulas is to use an incoming scattering state A(t)Φg in the limit t → ∞. We will see in Sec. 3.3, as a little corollary of Secs. 3.1 and 3.2, that this limit actually exists. 1.2.2. Definition of zeroth order of ionization probability Note, that a definition of any kind of “ionization probability” for the photoelectric effect must take into account some dynamical aspects. So we introduce the orthogonal projection FR := 1{|x|≥R} ⊗ 1F

(1.6)

onto the functions in the electron space Hel with support outside the ball of radius R > 0. As a first guess and with the huge distance between target and detector and the 10−9 seconds in mind one is probably tempted to define some kind of “ionization probability” as lim lim kFR e−iτ Hg A(t)Φg k2

R→∞ t→∞

for some fixed τ (approximately 10−9 s). But as limt→∞ A(t)Φg exists and FR converges strongly to 0, this expression is for some fixed τ just 0 and in contrast to

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our definition of Q(0) (A), there is no chance to see this as a zeroth order quantity in g. Another idea is to choose a g-dependent τ , such that τ (g) % ∞ as g & 0 and to have a look at Q(0) (A) := lim lim lim kFR e−iτ (g)Hg A(t)Φg k2 , R%∞ g&0 t→∞

(1.7)

the zeroth order of the ionization probability. The choice τ (g) % ∞ as g & 0 should be seen as a weak coupling limit: The weaker the interaction (smaller g) the longer you will have to wait until you see an effect (larger τ ). In fact, we will see in Theorem 4.1, that Q(0) (A) = 0 provided τ (g) % ∞ as g & 0. 1.2.3. Definition of the second order of ionization probability Additionally in the proof of Theorem 4.1 we get a decomposition of the vector (expressed in terms of the free time evolution Aτ := e−iτ H0 Aeiτ H0 , see (3.1)) FR e−iτ (g)(Hg −Eg ) A(t)Φg Z = FR Aτ (g) Φg − igFR

t+τ (g)

0

dse−is(Hg −Eg ) [W (1) + gW (2) , Aτ (g)−s ]Φg , (1.8)

which has the same norm square as FR e−iτ (g)Hg A(t)Φg . In (1.8) the first term does not depend on t and vanishes in the limit lim supR→∞ limg&0 , see Lemma 3.8 for details. The second term carries an explicit prefactor g, so in order to see the contributions in second order of g, we subtract FR Aτ (g) Φg and eliminate the prefactor dividing by g, i.e. we define Q(2) (A) := lim lim g −2 lim kFR e−iτ (g)(Hg −Eg ) A(t)Φg − FR Aτ (g) Φg k2 R%∞ g&0

t→∞

° Z °2 ° ° t+τ (g) ° ° −is(Hg −Eg ) (1) (2) = lim lim lim °FR dse [W + gW , Aτ (g)−s ]Φg ° ° R%∞ g&0 t→∞ ° 0 (1.9) as the second order of ionization probability. We will see, that the second order of ionization probability is suitable for describing the photoelectric effect; we will not comment on forth or other order of ionization probability. 1.3. Main results The asymptotic expansion of e−iτ (Hg −Eg ) A(t)Φg in terms of the free time evolution in Theorem 3.9 is used for calculations on Q(0) (A) and Q(2) (A) in Secs. 4.1–4.3. In Theorem 4.1, we will prove Q(0) (A) = 0. Assuming g −α < τ (g) < g −1 for some α ∈ ]0, 1[, we will then prove: • a decoupling property for orthonormal photon wave functions (Theorems 4.2 and 4.3): If m1 , . . . , mη , n1 , . . . , nη ∈ N0 and ϕ1 , . . . , ϕη ∈ C0∞ (R3 \{0}) are

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orthonormal, then ∗ Q(2) (a∗+ (ϕ1 )m1 a∗− (ϕ1 )n1 · · · a∗+ (ϕη )mη a− (ϕη )nη ) m1 ! · · · mη !n1 ! · · · nη !

=

η X ¡

(2)

(2)

nj Q− (ϕj ) + mj Q+ (ϕj )

¢

j=1 (2)

for some photon quantities Qλ (ϕj ) depending on the photon polarization λ and the momentum wave functions ϕj . • Theorem 4.5: If g −α < τ (g), then ¯Z ¯2 Z ¯ ¯ ¯ ¯ (2) Qλ (ϕj ) = dp ¯ (1.10) dµp2 −e0 (k)ˆ ρλ (p, k)ϕj (k)¯ ¯ ¯ 3 2 2 R S (p −e0 ) is the contribution of a single photon with wave function ϕj and polarization λ. Here e0 < 0 denotes the ground state energy of Hel , p is the electron momentum and ρˆλ (p, k) can be calculated from electron Hamiltonian Hel , electron ground state and electron-photon interaction. Formula (1.10) reflects just Einstein’s condition (1.2): Instead of having an electron with fixed momentum and a photon with one frequency, we have an electron and a photon Rwavefunction, viewed in Fourier space with momentum as variable. The R integral R3 dp takes into account all possible electron momenta, the integrals S 2 (p2 −e0 ) dµp2 −e0 (k) over the sphere S 2 (p2 − e0 ) of radius p2 − e0 pose the condition p2 − e0 − ω(k) = 0. So (1.10) is the integrated version of (1.2) for the free kinetic energy p2 of the electron and for ionization gap 4E = |e0 |. These theorems specify a typical situation, where the reduction of the photon field as a multi-particle system to an effective one photon system is justified: A cloud of photons in Einstein’s model with momenta k1 , . . . , km should be “approximated” in our model by photons with a smooth momentum distribution ϕj,ε of compact ε→0

support in {|k−kj | < ε} and ϕj,ε → δ(k−kj ) as distributions. If ε is small enough, then ϕj,ε ⊥ ϕl,ε for j 6= l and we may apply Theorem 4.3 for this approximation. As a summary note: The second order of ionization probability is additive for orthonormal photons. If the second order of ionization probability is not zero, this is not a collective effect of the whole system; it depends on the single photons. Each individual photon term of the second order of ionization probability satisfies the analogon (1.10) of Einstein’s condition (1.2). Therefore we can see these results on the second order of ionization probability as a way to derive Einstein’s model of photoeffect as an effective one-electron-one-photon approximation coming from quantum electrodynamics. 2. Definitions, Model Assumptions and First Conclusions Now we give a precise definition of the model(s) under consideration including all the model assumptions and give references to literature.

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2.1. The subsystem of the electron We start with a non relativistic, spinless electron whose dynamics is given by a Schr¨odinger operator Hel = −4 + V

(2.1)

in Hel = L2 (R3 ). Hypothesis 1. V is relatively −4-bounded with bound < 1, thus Hel = −4 + V defines a self-adjoint operator on the domain D(−4) of −4. Hel has a non degenerate ground state ϕ0 ∈ Hel with energy e0 < 0: Hel ϕ0 = e0 ϕ0 .

(2.2)

The singular continuous spectrum σsc (Hel ) = ∅ is empty. 2.2. Photons We couple the electron described above to a quantized photon field. The Hilbert space F carrying the photon degrees of freedom is the bosonic Fock space F = Fb (L2 (R3 × Z2 )) over the one-photon Hilbert space L2 (R3 × Z2 ). R3 × Z2 is viewed as photon momentum space, the two components describe the two independent transversal polarizations of the photon (in radiation gauge) M F= F (n) , (2.3) n∈N0

where the vacuum sector F (0) is a one-dimensional subspace spanned by the normalized Fock vacuum Ω and the n-photon sectors F (n) are the subspaces of L2 ((R3 × Z2 )n ) containing totally symmetric vectors. The Hamiltonian in F representing the energy of the free photon field is given by XZ Hf = dkω(k)a∗λ (k)aλ (k), (2.4) λ∈Z2

R3

where ω(k) := |k|

(2.5)

a∗λ

is the photon dispersion and and aλ are the standard creation- and annihilation operators in F, which fulfil the canonical commutation relations [aλ (k), aµ (k 0 )] = [a∗λ (k), a∗µ (k 0 )] = 0 [aλ (k), a∗µ (k 0 )] = δλ,µ δ(k − k 0 ) aλ (k)Ω = 0

(2.6) (2.7) (2.8)

in the sense of operator valued distributions. In other words, Hf is the second quantization of the multiplication operator with the photon dispersion ω(k) = |k| restricted to F. For some of the estimates, we introduce cutoff parameters 0 ≤ r˜ < r ≤ ∞ and define the regularized dispersion ω(˜r,r) (k) := ω(k)1{˜r≤ω(k)≤r} (k),

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the regularized free field Hf,(˜r,r) =

XZ λ∈Z2

R3

dkω(˜r,r) (k)a∗λ (k)aλ (k)

and regularized number operator XZ dk1{˜r≤ω(k)≤r} (k)a∗λ (k)aλ (k). N(˜r,r) = λ∈Z2

(2.9)

(2.10)

R3

2.3. The interaction between electron and photons The Hilbert space of states for the electron-photon system is the Hilbert space ˆ F. In H the dynamics is given by tensor product H = Hel ⊗ Hg = H0 + W,

(2.11)

introducing the non-interacting dynamics H0 = Hel ⊗ 1F + 1Hel ⊗ Hf .

(2.12)

The spectral measure of H0 = Hel ⊗ 1 + 1 ⊗ Hf can be described very explicit in terms of the spectral measures of Hel and Hf , see e.g. [27, Chap. 8.5], in particular Φ0 = ϕ0 ⊗ Ω is the ground state of H0 with ground state energy E0 = inf σ(H0 ) = e0 = inf σ(Hel ). In the interaction W = gW (1) + g 2 W (2) = gW (1,0) + gW (0,1) + g 2 W (2,0) + g 2 W (0,2) + g 2 W (1,1) , with W (1,0) = W

(0,1)

=

W (2,0) =

XZ λ∈Z2

R3

λ∈Z2

R3

XZ X

λ1 ,λ2 ∈Z2

W (0,2) =

X λ1 ,λ2 ∈Z2

W (1,1) =

X λ1 ,λ2 ∈Z2

(2.13)

dkw(1,0) (k, λ)a∗λ (k),

(2.14)

dkw(0,1) (k, λ)aλ (k),

(2.15)

Z Z R3

R3

dk1 dk2 w(2,0) (k1 , λ1 ; k2 , λ2 )a∗λ1 (k1 )a∗λ2 (k2 ),

(2.16)

dk1 dk2 w(0,2) (k1 , λ1 ; k2 , λ2 )aλ1 (k1 )aλ2 (k2 ),

(2.17)

dk1 dk2 w(1,1) (k1 , λ1 ; k2 , λ2 )a∗λ1 (k1 )aλ2 (k2 ),

(2.18)

Z Z R3 R3

Z Z R3 R3

the upper index indicates the total number of created and annihilated photons, respectively, a pair of upper indices indicates the number of created and annihilated

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photons. In order to get at least a symmetric interaction we have to require w(1,0) (k, λ) = (w(0,1) (k, λ))∗

(2.19)

w(2,0) (k1 , λ1 , k2 , λ2 ) = (w(0,2) (k1 , λ1 , k2 , λ2 ))∗ .

(2.20)

Note at this point, that no W (2) term shows up in the asymptotic expansion (3.14) in Theorem 3.9 and in all following results on the photoeffect formulated in terms of zeroth and second order of ionization probability — and this is in some sense part of the result: If you think like in Einstein’s model “photoeffect = absorption of a single photon” this is quite intuitive, because W (2) includes two photons. If you think in terms of naive power counting, then W (2) carries a prefactor g 2 , so after taking the norm square in the second order of ionization probability, we will end up with a higher order g 4 -term. Hence the assumptions on W (2) define a situation, where a rigorous proof of the above intuition is possible. As usual, we assume, that the interactions w(m,n) , m + n = 2 can be factorized: Let µ be the measure on the Borel sets of R3 × Z2 , which is the sum of the measures with Lebesgue density 1 1 + ω(·) on R3 . Let L(Hel ) denote the bounded operators on Hel . Hypothesis 2. There is a G ∈ L2 ((R3 × Z2 , µ), L(Hel )3 ), i.e.   G1 (k, λ) G(k, λ) =  G2 (k, λ)  G3 (k, λ) consisting of bounded operators G1 (k, λ), G2 (k, λ), G3 (k, λ) on Hel for µ-almost every (k, λ) ∈ R3 × Z2 , and XZ dkkG(k, λ)k2 (1 + ω(k)) < ∞, (2.21) λ∈Z2

R3

such that w(2,0) (k, λ, k 0 , λ0 ) =

3 X

Gι (k, λ)Gι (k 0 , λ0 )

(2.22)

ι=1

w(1,1) (k, λ, k 0 , λ0 ) =

3 X ¡ ¢ Gι (k, λ)∗ Gι (k 0 , λ0 ) + Gι (k, λ)Gι (k 0 , λ0 )∗ .

(2.23)

ι=1

Hypothesis 2 is quite natural: In [3, 4] it can be seen how the Hamiltonian Hg of the form specified in (2.11)–(2.20) is related to the standard model of quantum electrodynamics and some of its approximations. (2.22) and (2.23) are part of this type of models. (2.21) is still true in the usual minimal coupling model, where κ(k) −ikx e ελ (k) G(k, λ) = p ω(k)

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with some ultraviolet cutoff function κ (choosing a Schwarz function or the characteristic function of some box for κ) and vectors ε− (k), ε+ (k) ∈ R3 , such that k form an oriented orthonormal basis of R3 . ε− (k), ε+ (k), |k| Hypothesis 3. Fix some b < e0 = inf σ(Hel ), then there is a ζ ≥ 2, such that: (1) For ι = 1, 2, 3 and (m, n) = (1, 0) or (0, 1): 1

Gι (·, λ), w(m,n) (·, λ)(Hel − b)− 2 , ∈ C ζ (R3 \{0}, L(Hel )). 1

(2) ∂kα Gι (·, λ), ∂kα w(m,n) (·, λ)(Hel − b)− 2 ∈ L2 (K, L(Hel )) for ι = 1, 2, 3, (m, n) = (0, 1) or (1, 0) and any index α ∈ N30 with |α| ≤ ζ and compact sets K ⊆ R3 \{0}. Ignoring electron spin, the coupling functions in minimal coupled Pauli–Fierz models take the form w(1,0) (k, λ) = −2G(k, λ) · (−i∇x ), so Hypothesis 3 does not apply for the standard minimal coupling model: First of all, Hypothesis 3 requires some smoothness of G, but it is not possible to define differentiable vectorfields ε± on the sphere S 2 without a zero, see [26, p. 209]. More obvious, the partial derivatives ∂kα produce unbounded xα operators in ∂kα Gι (·, λ). On the other hand if we drop the Coulomb gauge condition and choose some smooth ˜± on S 2 (with zeros) and introduce spacial cutoffs µ ∈ S(R3 , R) or vectorfields ε χ ∈ S(R3 , R3 ), then µ ¶ κ(k) k ˜λ G(k, λ) = p ε e−ikx µ(x) |k| ω(k) or κ(k) ˜λ G(k, λ) = p ε ω(k)

µ

k |k|

¶ e−ikχ(x)

satisfy Hypothesis 3. Hypothesis (Hel , γ): For |α| ≤ ζ and for compact sets K ⊆ R3 \{0} Z γ γ+1 k(Hel − b) 2 ∂kα w(0,1) (k, λ)(Hel − b)− 2 k2 dk < ∞.

(2.24)

K

Before stating the next hypothesis, we fix some notation: For β, γ ≥ 0 we define: γ γ+1 XZ k(Hel − b) 2 w(m,n) (k, λ)(Hel − b)− 2 k2 (1) Λβ,γ := max (1 + ω(k))β , dk ω(k) m, n∈N0 R3 m+n=1 λ∈Z2

˜ (1) := max Λ β,γ

XZ

m, n∈N0 m+n=1 λ∈Z2

γ

R3

dkk(Hel − b) 2 w(m,n) (k, λ)(Hel − b)−

γ+1 2

k2 (1 + ω(k))β ,

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(2)

Λβ,γ := ˜ (2) := Λ β,γ

XZ

dk

λ∈Z2

R3

λ∈Z2

R3

XZ

γ

γ

γ

γ

k(Hel − b) 2 G(k, λ)(Hel − b)− 2 k2 (1 + ω(k))β , ω(k)

dkk(Hel − b) 2 G(k, λ)(Hel − b)− 2 k2 (1 + ω(k))β .

Hypothesis (Hel , β, γ): (1) (2) ˜ (1) ˜ (2) ˜ (2) Given β, γ ∈ N0 , then Λβ 0 ,γ 0 , Λ β 0 ,γ 0 , Λβ 0 ,γ 0 , Λβ 0 ,γ 0 and Λ β 0 0

0

377

0

0

β , γ ∈ N0 with β ≤ β and γ ≤ γ. If η ≥

α+β+γ 2

β

α

2

,γ 0

(2.25)

are finite for any

+ 2, then γ

(Hf,(˜r,r) + 1) 2 (Hf + 1) 2 +1 (Hel − b) 2 +1 (Hg − Eg + 1)−η is bounded. In [9] boundedness of (Hf + 1)n (H + i)−n has been proven for any n ∈ N and the Hamiltonian H of the standard model. 2.3.1. Self-adjointness and semiboundedness of Hg W is a relatively H0 bounded operator, more precisely: (2) ˜ (2) (1) ˜ (1) (1) is infinitesimally H0 Lemma 2.1. Let Λ0,0 , Λ 0,0 , Λ0,0 , Λ0,0 < ∞, then W (2) bounded and W is relatively H0 bounded, satisfying

kW (2) (H0 − b + 1)−1 k h i1 i1 h (2) ˜ (2) + Λ(2) )Λ(2) 2 + 2 (Λ(2) )2 + 4Λ(2) Λ ˜ (2) ˜ (2) 2 2 . (2.26) ≤ Λ0,0 + 2 (Λ 0,0 0,0 0,0 0,0 0,0 0,0 + (Λ0,0 ) In particular if g is small enough, such that i1 i1 i h h h (2) ˜ (2) 2 2 < 1, ˜ (2) + Λ(2) )Λ(2) 2 + 2 (Λ(2) )2 + 4Λ(2) Λ ˜ (2) g 2 Λ0,0 + 2 (Λ 0,0 0,0 0,0 0,0 0,0 0,0 + (Λ0,0 )

(2.27)

then Hg is self-adjoint on D(H0 ) and bounded from below. Proof. The estimates

q 1

1

1

1

(1)

kW (0,1) (Hf + 1)− 2 (Hel − b)− 2 k ≤ q kW (1,0) (Hf + 1)− 2 (Hel − b)− 2 k ≤

Λ0

˜ (1) + Λ(1) Λ 0 0

are special cases of Lemma A.5. Hence ˜ (1) + 2Λ(1) )k(Hel − b) 12 (Hf + 1) 12 Ψk2 . kW (1) Ψk2 ≤ 2(Λ 0 0 For any ε > 0 ° °2 ° 1 ° ° 0 ≤ °ε(Hel − b)(Hf + 1)Ψ − Ψ° ε ° = ε2 k(Hel − b)(Hf + 1)Ψk2 +

1 kΨk2 − 2
(2.28)

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Hel − b and Hf + 1 are self-adjoint positive commuting operators, so by spectral calculus the last inequality implies: 1

1

k(Hel − b) 2 (Hf + 1) 2 Ψk2 =
k(Hel − b)(Hf + 1)Ψk ≤ a1 k(H0 − b + 1)Ψk + a2 kΨk for any Ψ ∈ D(H0 ), see e.g. [17, Proposition 13.2]. In particular W (1) is infinitesimally H0 -bounded. Hf + 1 ≥ 0 and H0 − b + 1 ≥ 0 commute, so 1

1

0 ≤ h(H0 − b + 1)− 2 Ψ, (Hf + 1)(H0 − b + 1)− 2 Ψi = hΨ, (Hf + 1)(H0 − b + 1)−1 Ψi 1

1

≤ h(H0 − b + 1)− 2 Ψ, (Hf + 1 + Hel − b)(H0 − b + 1)− 2 Ψi = hΨ, (H0 − b + 1)(H0 − b + 1)−1 Ψi = kΨk2 , and estimate (2.26) follows from (2)

kW (2,0) (Hf + 1)−1 k ≤ Λ0,0 ˜ (2) + Λ(2) )Λ(2) ] 21 kW (1,1) (Hf + 1)−1 k ≤ 2[(Λ 0,0 0,0 0,0 (2) (2) ˜ (2) ˜ (2) 2 21 kW (0,2) (Hf + 1)−1 k ≤ 2[(Λ0,0 )2 + 4Λ0,0 Λ 0,0 + (Λ0,0 ) ]

which result from Lemma A.6. When (2.27) is fulfilled, then W = gW (1) + g 2 W (2) is H0 -bounded with bound < 1 and Kato–Rellich Theorem implies self-adjointness of Hg on D(H0 ) and in particular, Hg is bounded from below. 2.3.2. Properties of the ground state Hypothesis 4. The interacting Hamiltonian possesses a normalized ground state Φg ∈ H, kΦg k = 1. The infimum of the spectrum Eg := inf σ(Hg ) is an eigenvalue of Hg with corresponding eigenvector Φg : Hg Φg = Eg Φg

(2.30)

and Eg < e1 := inf σ(Hel )\{e0 }. There is some compact neighborhood U of 0, such that g 7→ Eg is continuous on U The projection PΦ⊥0 onto the orthogonal complement of the one-dimensional space spanned by Φ0 satisfies: kPΦ⊥0 Φg k ≤ c1 g

(2.31)

for some c1 < ∞. Hypothesis 5. Let f1 , . . . , fN ∈ C0∞ (R3 \{0}) be the functions in (1.3), let r˜ < SN SN inf{ω(k) : k ∈ j=1 supp fj } and r > sup{ω(k) : k ∈ j=1 supp fj } and U as in

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Hypothesis 4, then: N

lim sup sup k1{|x|≥R} (Hf,(˜r,r) + 1) 2 Φg k = 0

(2.32)

R→∞ g∈U

or N

lim sup sup k1{|x|≥R} (N(˜r,r) + 1) 2 Φg k = 0.

(2.33)

R→∞ g∈U

Existence of a ground state has been proven for many variants of the Pauli–Fierz model; an incomplete list is [1, 2, 10, 11, 14, 15, 18]. An existence proof for Φg , that gives the overlap (2.31) with the vacuum Φ0 and an exponential decay keα|x| Φg k < ∞ (for some α > 0) is found in [3, 4]. In [16] decay of powers of the k photon-number k(N 2 Φg )(x)kF with respect to the electron coordinate x is proven. A stronger version of (2.31) is needed for our problem; the proof of Lemma 2.2 combines the relative bounds of Lemmas A.5 and A.6 with some ideas from the proof of exponential decay in [3]: Lemma 2.2. Let 0 ≤ r˜ < r ≤ ∞ and α, β, γ ∈ N0 , such that Hypothesis 1, 2, 4 and (Hel , β, γ) hold true. Then there is a c2 = c2 (α, β, γ) < ∞, such that γ

β

α

k(Hel − b) 2 (Hf + 1) 2 (Hf,(˜r,r) + 1) 2 (Φg − Φ0 )k ≤ c2 g. Proof. By Hypothesis 4 kPΦ⊥0 Φg k = kΦg − hΦg , Φ0 iΦ0 k ≤ c1 g, hence |hΦg , Φ0 i| = 1 − O(g) and kΦg − Φ0 k ≤ kΦg − hΦg , Φ0 iΦ0 k + kΦ0 k|1 − hΦg , Φ0 i| = O(g).

(2.34)

Let e0 be the ground state energy of Hel and e1 = inf σ(Hel )\{e0 } the energy of the first excited state, if there are more bound electron states, respectively the ionization threshold of Hel , if there is just one bound electron state and choose e0 ∈ ]e0 , e1 [, such that e0 > Eg . From Hf ≥ 0 and e0 < e1 we conclude 1]−∞,e0 [ (H0 ) = 1{e0 } (Hel )1]−∞,e0 [ (H0 ) = 1[0,|e0 −e0 |[ (Hf )1{e0 } (Hel )1]−∞,e0 [ (H0 ), hence γ

(Hel − b) 2 (Hf + 1)

α+β 2

1]−∞,e0 [ (H0 )

γ 2

= (Hel − b) 1{e0 } (Hel )(Hf + 1)

α+β 2

1[0,|e0 −e0 |[ (Hf )1]−∞,e0 [ (H0 )

is bounded. Hf , Hf,(˜r,r) , Hel and H0 commute, so from Hf,(˜r,r) ≤ Hf we conclude γ

β

α

0 ≤ k(Hel − b) 2 (Hf + 1) 2 (Hf,(˜r,r) + 1) 2 1]−∞,e0 [ (H0 )Ψk2 = hΨ, (Hel − b)γ (Hf + 1)β (Hf,(˜r,r) + 1)α 1]−∞,e0 [ (H0 )Ψi ≤ hΨ, (Hel − b)γ (Hf + 1)α+β 1]−∞,e0 [ (H0 )Ψi γ

= k(Hel − b) 2 (Hf + 1)

α+β 2

1]−∞,e0 [ (H0 )Ψk2 ,

so by (2.34) it is enough to prove γ

β

α

k(Hel − b) 2 (Hf + 1) 2 (Hf,(˜r,r) + 1) 2 1[e0 ,∞[ (H0 )(Φg − Φ0 )k ≤ O(g).

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But as Φ0 = ϕ0 ⊗ Ω = 1{e0 } (H0 )Φ0 and e0 < e0 , we get 1[e0 ,∞[ (H0 )(Φg − Φ0 ) = 1[e0 ,∞[ (H0 )Φg . 0

By e > Eg we can choose χ ∈ C0∞ (]−∞, e0 [), such that χ(Eg ) = 1, hence χ(Hg )Φg = Φg and 1[e0 ,∞[ (H0 )χ(H0 ) = 0. Choose an almost analytic extension χ ˜ of χ in some compact set M ⊆ ]−∞, e0 [ +iR, such that for z = x + iy µ ¶ ∂χ ˜ 1 ∂χ ˜ ∂χ ˜ := +i ∂z 2 ∂x ∂y satisfies ¯ ∂χ ¯ ¯ ˜¯ (2.35) ¯ ¯ ≤ O(|=z|2 ), ∂z see e.g. [6 Chap. 2.2] for the explicit construction. Introducing the complex measure dµ(z) := π1 ∂∂zχ˜ dxdy spectral calculus implies γ

β

α

k(Hel − b) 2 (Hf + 1) 2 (Hf,(˜r,r) + 1) 2 1[e0 ,∞[ (H0 )Φg k γ

β

α

= k(Hel − b) 2 (Hf + 1) 2 (Hf,(˜r,r) + 1) 2 1[e0 ,∞[ (H0 )(χ(Hg ) − χ(H0 ))Φg k ° ° β γ α 2 2 2 =° °(Hel − b) (Hf,(˜r,r) + 1) 1[e0 ,∞[ (H0 )(Hf + 1) Z −1

×

dµ(z)[(Hg − z)

−1

− (H0 − z)

M

° ° ]Φg ° °

° ° β γ α 2 2 2 =° °1[e0 ,∞[ (H0 )(Hel − b) (Hf + 1) (Hf,(˜r,r) + 1) Z −1

×

dµ(z)(H0 − z)

−1

W (Hg − z)

M

° ° Φg ° °.

(2.36)

The eigenvalue equation Hg Φg = Eg Φg implies Φg = (Hg − Eg + 1)l Φg for any l ∈ Z. Choose η ∈ N, such that η ≥ α+β+γ + 2, then 2 β

α

γ

(Hf,(˜r,r) + 1) 2 (Hf + 1) 2 +1 (Hel − b) 2 +1 (Hg − Eg + 1)−η β

is bounded. So commuting (Hf + 1) 2 and (H0 − z)−1 , °Z ° ° ° γ β α −1 −1 ° dµ(z)(Hel − b) 2 (H0 − z) (Hf + 1) 2 (Hf,(˜r,r) + 1) 2 W (Hg − z) Φg ° ° ° M °Z ° β γ α =° dµ(z)(Hel − b) 2 (H0 − z)−1 (Hf + 1) 2 (Hf,(˜r,r) + 1) 2 W ° M

−η

× (Hg − Eg + 1)

−1

(Hg − z) β

γ

° ° Φg ° ° α

α

β

≤ k(Hel − b) 2 (Hf + 1) 2 (Hf,(˜r,r) + 1) 2 W (Hf,(˜r,r) + 1)− 2 (Hf + 1)− 2 −1 Z γ × (Hel − b)− 2 −1 k dµ(z)k(H0 − z)−1 k|(Eg − z)−1 | M α 2

β

γ

k(Hf,(˜r,r) + 1) (Hf + 1) 2 +1 (Hel − b) 2 +1 (Hg − Eg + 1)−η k.

(2.37)

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The integrand is bounded by |=z|−2 , so by compactness of M and the bound (2.35), this integral is finite, and due to Lemmas A.5 and A.6 γ

β

α

γ

α

k(Hel −b) 2 (Hf +1) 2 (Hf,(˜r,r) +1) 2 W (j) (Hf,(˜r,r) +1)− 2 (Hf +1)−β−1 (Hel −b)− 2 −1 k remains bounded for j = 1, 2. If we now remember W = gW (1) + g 2 W (2) , we see, that the right-hand side of (2.37) is O(g). Corollary 2.3. Let the assumptions Lemma 2.2 be satisfied and suppose, that (2.27) is true for all g ∈ U . Then there is a c3 = c3 (α, β, γ, U ) < ∞, such that γ

β

α

sup k(Hel − b) 2 (Hf + 1) 2 (Hf,(˜r,r) + 1) 2 Φg k ≤ c3 .

(2.38)

g∈U

Proof. Choosing e0 as in the proof of Lemma 2.2, γ

β

α

(Hel − b) 2 (Hf + 1) 2 (Hf,(˜r,r) + 1) 2 1]−∞,e0 [ (H0 ) is a bounded operator. Due to the normalization condition kΦg k = 1 γ

β

α

sup k(Hel − b) 2 (Hf + 1) 2 (Hf,(˜r,r) + 1) 2 1]−∞,e0 [ (H0 )Φg k

g∈U

γ

β

α

≤ k(Hel − b) 2 (Hf + 1) 2 (Hf,(˜r,r) + 1) 2 1]−∞,e0 [ (H0 )k gives a uniform bound. Resolvent equation implies (Hg − Eg + 1)−1 − (Hh − Eh + 1)−1 = (Hh − Eh + 1)−1 [(h − g)W (1) + (h2 − g 2 )W (2) + Eg − Eh ](Hg − Eg + 1)−1, so due to the relative bounds on the interaction and continuity of the ground state energies, g 7→ (Hg − Eg + 1)−1 is continuous. Therefore the g-dependent terms in (2.37) can be estimated uniform on U .

3. Asymptotic Expansions In this section we develop asymptotic expressions for the full interacting dynamics applied to photon clouds plus ground state. For this purpose, we define the free Heisenberg time evolution Zt := e−itH0 ZeitH0

(3.1)

on the domain D(Zt ) := {Ψ ∈ H : eitH0 Ψ ∈ D(Z)} of an operator Z in H. Using this free time evolution, we find an asymptotic expansion of e−iτ (Hg −Eg ) A(t)Φg for intermediate times g −α < τ < g −β as g & 0 with 0 < α < β < 1. 3.1. Rewriting the time evolution As a part of this program, we have to supply two kind of technical lemma. The first kind, allows us to rewrite e−iτ (Hg −Eg ) A(t)Φg in terms of Aτ Φg plus an integral, where several commutators of the free Heisenberg time evolution of the interaction W and the photon cloud A come into play.

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(1) ˜ (1) (2) ˜ (2) Theorem 3.1. Let Λ0,0 , Λ 0,0 , Λ0,0 , Λ0,0 < ∞, n ∈ N and let (2.27) be satisfied. Let Z ∈ L(D(H0n ), D(H0 )) be a bounded operator from D(H0n ) into D(H0 ). Given τ ∈ R and Ψ ∈ Ran(Hg − i)−n the map

hτ,Z : R −→ H s 7−→ e−isHg eisH0 Zτ e−isH0 eisHg Ψ

(3.2)

is differentiable with derivative h0τ,Z (s) = −ie−isHg eisH0 [Ws , Zτ ]e−isH0 eisHg Ψ.

(3.3)

Proof. From the general definitions of sums and products of operators, we conclude D(Hgl ) ⊆ D(H0l ) for all l ∈ N. The unitary groups (e−isHg )s∈R and (e−isH0 )s∈R leave D(Hgl ), respectively, D(H0l ) invariant, hence we get e−isH0 eisHg Ψ ∈ D(H0n ) and Zτ e−isH0 eisHg Ψ ∈ D(H0 ). From Lemma 2.1, we know D(Hg ) = D(H0 ), so this subspace is invariant under e±isHg and e±isH0 and the function hτ,Z is well defined. We look at the restrictions of the operators e±isH0 and e±isHg to D(H0 ) as bounded operators on D(H0 ) and apply results on one parameter unitary groups, see [25, (13.35)] to get on D(H0 ) = D(Hg ): d ±isH0 (e ) = ±iH0 e±isH0 , ds Now the chain rule implies

d ±isHg (e ) = ±iHg e±isHg . ds

d ∓isHg ±isH0 (e e ) = ∓ie∓isHg (Hg − H0 )e±isH0 , ds d ±isH0 ∓isHg (e e ) = ±ie±isH0 (H0 − Hg )e∓isHg . ds The assumption Z ∈ L(D(H0n ), D(H0 )) implies Zτ ∈ L(D(H0n ), D(H0 )), hence for each Ψ ∈ Ran(Hg − i)−n , the map hτ,Z : R → H s 7→ e−isHg eisH0 Zτ e−isH0 eisHg Ψ is differentiable, see [7, (8.1.4)], with derivative h0τ,Z (s) = −ie−isHg (Hg − H0 )eisH0 Zτ e−isH0 eisHg Ψ + ie−isHg eisH0 Zτ e−isH0 (Hg − H0 )eisHg Ψ = −ie−isHg eisH0 [(Hg − H0 )s , Zτ ]e−isH0 eisHg Ψ = −ie−isHg eisH0 [Ws , Zτ ]e−isH0 eisHg Ψ. Corollary 3.2. Under the assumptions of Theorem 3.1 the time evolution is Z t+τ −iτ Hg iτ Hg e Z(t)e Ψ = Zτ Ψ − i dse−isHg eisH0 [Ws , Zτ ]e−isH0 eisHg Ψ, Z e−iτ Hg eiτ H0 Ze−iτ H0 eiτ Hg Ψ = ZΨ − i 0

0 τ

dse−isHg eisH0 [Ws , Z]e−isH0 eisHg Ψ.

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Proof. Using Theorem 3.1, the differentiability of hτ,Z implies e−iτ Hg Z(t)eiτ Hg Ψ = e−i(t+τ )Hg ei(t+τ )H0 Zτ e−i(t+τ )H0 ei(t+τ )Hg Ψ Z t+τ dsh0τ,Z (s) = Zτ Ψ + hτ,Z (s)|s=t+τ = Z Ψ + τ s=0 Z

0

t+τ

= Zτ Ψ − i

dse−isHg eisH0 [Ws , Zτ ]e−isH0 eisHg Ψ

(3.4)

0

and in the same way e−iτ Hg eiτ H0 Ze−iτ H0 eiτ Hg Ψ = ZΨ + h0,Z (s)|s=τ s=0 Z τ = ZΨ − i dse−isHg eisH0 [Ws , Z]e−isH0 eisHg Ψ. 0

3.2. Commutator estimates The second kind of lemma, which we are going to prove now, establishes some control on the time decay of the commutators in Corollary 3.2. These results are needed for an error bound of the asymptotic expansions. (1) (1) ˜ (1) Lemma 3.3. Suppose Λ0,γ , Λβ,γ , Λ β,γ < ∞ and that Hypothesis 1, 3 and (Hel , γ) hold true. Let

r˜ < inf{ω(k) : k ∈ supp fj , j = 1, . . . , N }, r > sup{ω(k) : k ∈ supp fj , j = 1, . . . , N },

(3.5)

and λ1 , . . . , λN ∈ Z2 , then for A = a∗λ1 (f1 ) · · · a∗λN (fN ) and all s ∈ R γ

β

(Hf + 1) 2 (Hel − b) 2 [W (1) , As ](Hel − b)−

γ+1 2

(Hf + 1)−

β+1 2

N

(Hf,(˜r,r) + 1)− 2

defines a bounded operator on H and there is c4 = c4 (β, γ) < ∞, which can be chosen independent of s, such that β

γ

k(Hf + 1) 2 (Hel − b) 2 [W (1) , As ](Hel − b)−

γ+1 2

(Hf + 1)−

β+1 2

≤ c4 (1 + |s|)−ζ .

N

(Hf,(˜r,r) + 1)− 2 k (3.6)

Proof. We apply the pull-through formula to obtain the free time evolution of A: As = a∗λ1 (e−isω f1 ) · · · a∗λN (e−isω fN ), − γ+1 2

− β+1 2

then for Φ ∈ Ran(Hel − b) (Hf + 1) mutator applied to Φ all terms like β

(Hf,(˜r,r) + 1)

γ

(3.7) −N 2

writing out the com-

(Hf + 1) 2 (Hel − b) 2 W (1) As Φ ¡ β γ γ+1 β+1 ¢ = (Hf + 1) 2 (Hel − b) 2 W (1) (Hel − b)− 2 (Hf + 1)− 2   N β+1 Y β+1 N × (Hf + 1) 2 a∗λj (e−isω fj )(Hf + 1)− 2 (Hf,(˜r,r) + 1)− 2  j=1

× (Hel − b)

γ+1 2

(Hf + 1)

β+1 2

N

(Hf,(˜r,r) + 1) 2 Φ

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or the other way around are well defined by Corollary A.3 and Lemma A.5. Inserting the two commutators   N Y a∗λ (k), a∗λj (kj ) = 0, j=1

 aλ (k),

N Y

 a∗λj (kj ) =

N X

j=1

N Y

δ(k − kj )δλ,λj

j=1

a∗λl (kl ),

l=1 l6=j

we get γ

β

γ+1

β+1

N

(Hf + 1) 2 (Hel − b) 2 [W (1) , As ](Hel − b)− 2 (Hf + 1)− 2 (Hf,(˜r,r) + 1)− 2 Ψ Z Z XZ = dk dk1 · · · dkN e−isω(k1 ) f1 (k1 ) · · · e−isω(kN ) fN (kN ) R3

λ∈Z2

R3

R3

γ+1

γ

β

× (Hel − b) 2 w(0,1) (k, λ)(Hel − b)− 2 (Hf + 1) 2   N Y β+1 N × aλ (k), a∗λj (kj ) (Hf + 1)− 2 (Hf,(˜r,r) + 1)− 2 Ψ j=1

=

N Z X j=1

R3

γ

dk(Hel − b) 2 w(0,1) (k, λj )(Hel − b)−

× (Hf + 1)

β 2

N Y

a∗λl (e−isω fl )(Hf + 1)−

β+1 2

γ+1 2

e−isω(k) fj (k)

N

(Hf,(˜r,r) + 1)− 2 Ψ.

(3.8)

l=1 l6=j

From Corollary A.3 we conclude, that β

Ψj,s := (Hf + 1) 2

N Y

a∗λl (e−isω fl )(Hf + 1)−

β+1 2

N

(Hf,(˜r,r) + 1)− 2 Ψ

l=1 l6=j

is a well-defined element of F with sup

s∈R j=1, . . . , N

kΨj,s k < c(kf1 kω , . . . , kfN kω )kΨk,

with a finite constant c(kf1 kω , . . . , kfN kω ) depending only on the weighted L2 norms given by µ ¶ Z 1 2 2 kfj kω := |fj (k)| 1 + dk, ω(k) R3 which are finite for fj ∈ C0∞ (R3 \{0}). Due to Hypothesis (Hel , γ) γ

T (k, λ) := (Hel − b) 2 w(0,1) (k, λ)(Hel − b)−

γ+1 2

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and all partial k-derivatives of order ≤ ζ are square integrable on each compactum K ⊆ R3 \{0}. Then the commutator takes the form γ

β

(Hf + 1) 2 (Hel − b) 2 [W (1) , As ](Hel − b)− =

N Z X j=1

R3

γ+1 2

(Hf + 1)−

β+1 2

N

(Hf,(˜r,r) + 1)− 2 Ψ

dke−isω fj (k)T (k, λj )Ψj,s .

(3.9)

The support of fj is located away from the origin. So ∇k ω(k) = i k s |k|

· ∇k e

−isω(k)

=e

−isω(k)

k |k|

and

on supp fj . For Φ ∈ H by ζ times partial integration,

¯* +¯¯ ¯ N Z X ¯ ¯ ¯ Φ, dke−isω(k) T (k, λj )fj (k)Ψj,s ¯¯ ¯ 3 ¯ ¯ j=1 R ¯ ! +¯¯ * Ã· ¯N Z ¸ζ ¯ ¯X i k · ∇k e−isω(k) T (k, λj )fj (k)Ψj,s ¯¯ = ¯¯ dk Φ, s |k| ¯ ¯ j=1 R3 ¯ ¯ * +¯ ¯N Z · ¸ζ ¯ ¡ ¢ k 1 ¯¯X −isω(k) ∇k T (k, λj )fj (k) Ψj,s ¯¯ dk Φ, e = ζ¯ s ¯ j=1 R3 |k| ¯ °· ° ¸ζ Z N ° ° X k 1 ° ° kΨj,s k dk ° ∇k (T (k, λj )fj (k))° . ≤ ζ kΦk ° ° s |k| supp f j j=1 γ

(3.10)

γ+1

k ζ (0,1) 1 [∇k |k| ] (T (k, λ)fj (k)) is a sum of (Hel −b) 2 ∇α (k, λ)(Hel −b)− 2 multiplied k w α3 2 k with some derivatives (∇α k |k| )(∇k fj ) for |α1 |, |α2 |, |α3 | ≤ ζ. So all these terms are integrable on the support of fj and

¯* +¯¯ ¯ N Z X ¯ ¯ ¯ Φ, dke−isω(k) T (k, λj )fj (k)Ψj,s ¯¯ ≤ c4 (1 + |s|)−ζ kΨkkΦk ¯ 3 ¯ ¯ j=1 R α3 2 k for some c4 depending on β, γ, kf1 kω , . . . , kfN kω , k(∇α k |k| )(∇k fj )kL2 (supp fj ) γ

γ+1

(0,1) 1 and k(Hel − b) 2 ∇α (·, λ)(Hel − b)− 2 kL2 (supp fj ) for |α1 |, |α2 |, |α3 | ≤ ζ, k w j = 1, . . . , N but chosen independent of s ∈ R. (2) ˜ (2) Lemma 3.4. Suppose Hypothesis 1– 3 hold true and Λ0,0 , Λ 0,0 < ∞. Let f1 , . . . , ∞ 3 fN ∈ C0 (R \{0}), λ1 , . . . , λN ∈ Z2 and choose r˜, r as in (3.5), then for A = a∗λ1 (f1 ) · · · a∗λN (fN ) N

[W (2) , As ](Hf + 1)−1 (Hf,(˜r,r) + 1)− 2

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defines a bounded operator on H and there is a constant c5 < ∞, which can be chosen independent of s ∈ R, such that N

k[W (2) , As ](Hf + 1)−1 (Hf,(˜r,r) + 1)− 2 k ≤ c5 (1 + |s|)−ζ .

(3.11)

Proof. The proof needs some additional commutator gymnastics, but it follows the basic lines of the previous one. Corollary 3.5. Under the hypothesis of Lemmas 3.3 and 3.4 and with the constants c4 , c5 from there, for any (s, t) ∈ R2 β

γ

(1)

k(Hf + 1) 2 (Hel − b) 2 [Wt , As ](Hel − b)−

γ+1 2

(Hf + 1)−

β+1 2

N

(Hf,(˜r,r) + 1)− 2 k

≤ c4 (1 + |t − s|)−ζ ,

(3.12)

(2)

k[Wt , As ](Hf + 1)−1 (Hf,(˜r,r) + 1) (j)

−N 2

k ≤ c5 (1 + |t − s|)−ζ .

(3.13)

(j)

Proof. eitH0 [Wt , As ]e−itH0 = [Wt , As ]−t = [W (j) , As−t ] and H0 commutes with Hel , Hf and Hf,(˜r,r) , so due to unitarity of e±itH0 (2)

N

k[Wt , As ](Hf + 1)−1 (Hf,(˜r,r) + 1)− 2 k (2)

N

= keitH0 [Wt , As ](Hf + 1)−1 (Hf,(˜r,r) + 1)− 2 e−itH0 eitH0 k N

≤ k[W (2) , As−t ](Hf + 1)−1 (Hf,(˜r,r) + 1)− 2 k ≤ c5 (1 + |t − s|)−ζ , the estimate for the W (1) commutator is proven the same way. Corollary 3.6. Under the Hypothesis 1– 3, (Hel , 1, 1) and (Hel , 1) there is c6 < ∞, which can be chosen independent of (s, t) ∈ R2 , such that (1)

N

3

k[W, [Wt , As ]](Hel − b)−1 (Hf,(˜r,r) + 1)− 2 (Hf + 1)− 2 k ≤ c6 (1 + |t − s|)−ζ , (1)

N

k[W (1) , [Wt , As ]](Hel − b)−1 (Hf,(˜r,r) + 1)− 2 (Hf + 1)−1 k ≤ c6 (1 + |t − s|)−ζ , (1)

1

N

3

k[W (2) , [Wt , As ]](Hel − b)− 2 (Hf,(˜r,r) + 1)− 2 (Hf + 1)− 2 k ≤ c6 (1 + |t − s|)−ζ . Proof. Inserting identity in form of positive and negative powers of Hf + 1, Hf,(˜r,r) + 1 and Hel − b for example (1)

N

[W (1) , [Wt , As ]](Hel − b)−1 (Hf,(˜r,r) + 1)− 2 (Hf + 1)−1 1

1

1

(1)

1

= (W (1) (Hel − b)− 2 (Hf + 1)− 2 )((Hel − b) 2 (Hf + 1) 2 [Wt , As ] N

× (Hel − b)−1 (Hf,(˜r,r) + 1)− 2 (Hf + 1)−1 ) (1)

1

N

1

1

− ([Wt , As ](Hel − b)− 2 (Hf,(˜r,r) + 1)− 2 (Hf + 1)− 2 )((Hel − b) 2 N

1

N

× (Hf,(˜r,r) + 1) 2 (Hf + 1) 2 W (1) (Hel − b)−1 (Hf,(˜r,r) + 1)− 2 (Hf + 1)−1 )

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so the commutator terms are estimated by Corollary 3.5 and the W (1) terms by Lemma A.5. 3.3. Existence of incoming scattering states QN Lemma 3.7. Suppose Hypothesis 1– 4 and let A = j=1 a∗λj (fj ) be as in (1.3), then for any g ∈ U A(∞)Φg = lim A(t)Φg t→∞

exists. Proof. By Corollary 3.2 for t, s ∈ R, s ≤ t: °Z t ° Z t ° ° −iq(Hg −Eg ) ° kA(t)Φg − A(s)Φg k = ° dqe [W, A−q ]Φg ° k[W, A−q ]Φg kdq. °≤ s

s

Choose r˜, r as in (3.5). An application of the commutator estimates from Corollaries 3.5 and 2.3 imply: Z t Z t s, t→∞ 2 k[W, A−q ]Φg kdq ≤ c3 (N, 2, 1, U )(gc4 + g c5 ) (1 + |q|)−ζ dq −−−−→ 0. s

s

So A(t)Φg is Cauchy and A(∞)Φg = limt→∞ A(t)Φg exists. 3.4. An asymptotic expansion, that is correct in second order Lemma 3.8. Suppose Hypothesis 5 and τ (g) % ∞ as g & 0, then lim sup sup kFR Aτ (g) Φg k = 0. R→∞ g∈U

Proof. Hel = Hel ⊗ 1F commutes with A = 1Hel ⊗ A, so Aτ = 1 ⊗ e−iτ Hf Aeiτ Hf and pull through formula implies e−iτ Hf Aeiτ Hf = a∗λ1 (e−iτ ω f1 ) · · · a∗λN (e−iτ ω fN ). In particular Aτ and FR commute. Let U be the compact neighborhood of 0 from Hypothesis 5. Choose the regularization parameters 0 < r˜ < r < ∞ such that r˜ < inf{ω(k) : k ∈ supp (fj ), j = 1, . . . , N }, r > sup{ω(k) : k ∈ supp (fj ), j = 1, . . . , N }, then N

N

kFR Aτ (g) Φg k ≤ kAτ (g) (Hf,(˜r,r) + 1)− 2 k k(Hf,(˜r,r) + 1) 2 FR Φg k. N

Corollary A.3 implies supτ ∈R kAτ (Hf,(˜r,r) + 1)− 2 k ≤ Ckf1 kω · · · kfN kω < ∞, so in case of (2.32): lim sup sup kFR Aτ (g) Φg k ≤ Ckf1 kω R→∞ g∈U

N

· · · kfN kω lim sup sup kFR (Hf,(˜r,r) + 1) 2 Φg k = 0. R→∞ g∈U

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In case of (2.33), we can use Corollary A.4 instead and get: lim sup sup kFR Aτ (g) Φg k ≤ Ckf1 k R→∞ g∈U

N

· · · kfN k lim sup sup kFR (N(˜r,r) + 1) 2 Φg k = 0. R→∞ g∈U

Theorem 3.9. Suppose Hypothesis 1– 4, (Hel , 1) and (Hel , 1, 1), let Hg be selfadjoint and let 0 < α < β < 1 and g −α < τ = τ (g) < g −β as g & 0, then there is some R(τ (g), t) ∈ H, such that e−iτ (Hg −Eg ) A(t)Φg Z = Aτ Φg − ig

∞

e−i(τ −r)(H0 −E0 ) [W (1) , Ar ]Φ0 dr + R(τ, t)

(3.14)

−∞

and sup kR(τ (g), t)k ≤ o(g). t≥g −1

Proof. Choose β, γ, such that α < β < γ < 1. The time evolution in Corollary 3.2 gives e−iτ (Hg −Eg ) A(t)Φg Z τ +t = Aτ Φg − i dse−isHg eisH0 [Ws , Aτ ]e−isH0 eisHg Φg 0

Z

τ +t

= Aτ Φg − i

dse−is(Hg −Eg ) [W, Aτ −s ]Φg

0

Z

g −γ

= Aτ Φg − ig 0

dse−isHg eisH0 [Ws(1) , Aτ ]e−isH0 eisHg Φg + o(g),

−1

because t ≥ g ensures t + τ ≥ g −γ , so due to Lemma A.1, the commutator estimates in Lemma 3.3 (with c4 = c4 (0, 0)) and Corollary 2.3, the following estimate ° Z t+τ ° ° ° −is(Hg −Eg ) (1) °g e [W , Aτ −s ]Φg ds° ° ° g −γ

Z

≤ gc3 c4

∞

(1 + s − τ )−ζ ds = o(g)

(3.15)

g −γ

is uniform in t ≥ g −1 . The estimate for the W (2) commutator in Lemma 3.4 implies ° ° Z t+τ ° ° 2 −is(Hg −Eg ) (2) °g e [W , Aτ −s ]Φg ds° ° ° 0

≤ g 2 c3 c5

Z

∞

−∞

min{1, |τ − s|−ζ }ds = O(g 2 )

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uniform in t. Corollary 3.2 applied again yields Z g−γ g e−isHg eisH0 [Ws(1) , Aτ ]e−isH0 eisHg Φg ds 0

Z

g −γ

=g 0

Z

s

× 0

Z 0

g −γ

ds 0

dqe−iqHg eiqH0 [Wq , [Ws(1) , Aτ ]]e−iqH0 eiqHg Φg

g −γ

=g

Z [Ws(1) , Aτ ]Φg ds − ig

Z [Ws(1) , Aτ ]Φg ds − ig

Z

g −γ

s

ds 0

0

(1)

dqe−iq(Hg −Eg ) [W, [Ws−q , Aτ −q ]]Φg

and due to Corollary 3.6 and the choice τ < g −γ with γ < 1 °Z −γ Z ° ° g ° s ° (1) 2° −iq(Hg −Eg ) (1) g ° ds dqe [W , [Ws−q , Aτ −q ]]Φg ° ≤ O(g 2 τ ) = o(g), ° 0 ° 0 (1)

and the [W (2) , [Ws−q , Aτ −q ]] term is estimated similar and gives an o(g 2 )-term, hence ° ° Z g−γ ° ° ° −iτ (Hg −Eg ) ° (3.16) A(t)Φg − Aτ Φg + ig ds[Ws(1) , Aτ ]Φg ° ≤ o(g). °e ° ° 0 According to Corollary 3.5 and Lemma 2.2 1

1

N

k[Ws(1) , Aτ ](Φg − Φ0 )k ≤ k[Ws(1) , Aτ ](Hel − b)− 2 (Hf + 1)− 2 (Hf,(˜r,r) + 1)− 2 k 1

1

N

× k(Hel − b) 2 (Hf + 1) 2 (Hf,(˜r,r) + 1) 2 (Φg − Φ0 )k ≤ gc2 c4 (1 + |τ − s|)−ζ , and even in the worst case ζ = 2 °Z −γ ° Z g−γ ° g ° ° ° (1) g° ds[Ws , Aτ ](Φg − Φ0 )° ≤ g 2 c2 c4 (1 + |τ − s|)−2 ds = O(g 2 ) ° 0 ° 0 is only a term of lower order, so in (3.16) we may replace Φg by Φ0 : ° ° Z g−γ ° ° ° −iτ (Hg −Eg ) ° (1) A(t)Φg − Aτ Φg + ig ds[Ws , Aτ ]Φ0 ° ≤ o(g). °e ° ° 0

(3.17)

(1)

In the domain of the commutator [Ws , Aτ ] = e−isH0 [W (1) , Aτ −s ]eisH0 , so Z g−γ Z τ (1) g ds[Ws , Aτ ]Φ0 = g dre−i(τ −r)(H0 −E0 ) [W (1) , Ar ]Φ0 . 0

τ −g −γ

Due to the choice of γ and the assumption on τ , from which we concluded τ < g −β with β < γ, we get g −γ − τ ≥ g −γ (1 − g γ−β ) = O(g −γ ) so in analogy to (3.15) °Z ° ° τ −g−γ ° ° ° g° dre−i(τ −r)(H0 −E0 ) [W (1) , Ar ]Φ0 ° ≤ o(g) ° −∞ °

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and the bound g −α < τ and the analogon of (3.15) implies °Z ∞ ° ° ° −i(τ −r)(H0 −E0 ) (1) ° dre g° [W , A ]Φ r 0 ° ≤ o(g). ° τ

Plugging this estimates into (3.17), we get: ° ° Z ∞ ° −iτ (H −E ) ° −i(τ −r)(H0 −E0 ) (1) g g °e ° ≤ o(g), A(t)Φ − A Φ + ig e [W , A ]Φ dr g τ g r 0 ° ° −∞

uniform in t ≥ g −1 . 4. Formulas for Zeroth and Second Order of Ionization Probability The goal of this section is a derivation of Einstein’s description of the photoelectric effect out of our quantum electrodynamical model. In a few steps, we will see, in which aspects this simple model is adequate. 4.1. Zeroth order of ionization probability vanishes Theorem 4.1. Suppose Hypothesis 1– 5 and (Hel , 1, 1) and let Hg be self-adjoint. Let τ (g) % ∞ when g & 0, then the zeroth order of ionization probability vanishes: Q(0) (A) = lim lim lim kFR e−iτ (g)Hg A(t)Φg k2 = 0. R%∞ g&0 t%∞

Proof. Due to Corollary 3.2 −iτ (Hg −Eg )

e

Z

A(t)Φg = Aτ Φg − ig

t+τ

(4.1)

dse−is(Hg −Eg ) [W (1) , Aτ −s ]Φg

0

Z

t+τ

− ig 2

dse−is(Hg −Eg ) [W (2) , Aτ −s ]Φg

0

and the commutator estimates imply °Z t+τ ° ° ° −is(Hg −Eg ) (1) ° ° dse [W , A ]Φ τ −s g° ° 0 Z ∞ ≤ c3 (N, 1, 1, U )c4 (0, 0) (1 + r)−ζ dr = O(1)

(4.2)

−∞

and a similar bound for the W (2) commutator. In kFR e−iτ (g)(Hg −Eg ) A(t)Φg k2 °Z °2 ° t+τ (g) ° ° ° −is(Hg −Eg ) (1) ≤ 3kFR Aτ (g) Φg k + 3g ° dse [W , Aτ −s ]Φg ° ° 0 ° 2

2

°Z °2 ° t+τ (g) ° ° ° −is(Hg −Eg ) (2) + 3g ° dse [W , Aτ −s ]Φg ° ° 0 ° 4

the first term does not depend on t, so it vanishes in the limit limR→∞ limg&0 according to Lemma 3.8, the last two integrals are O(1) uniform in t like in (4.2), hence they vanish in limg&0 limt→∞ and (4.1) is proven.

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4.2. A formula for second order of ionization probability Theorem 4.2. Suppose Hypothesis 1– 4, (Hel , 1) and (Hel , 1, 1), let Hg be selfadjoint. Let 0 < α < β < 1 and g −α < τ (g) < g −β for g & 0 and set Z t (1) Ψ := Ψ(A) = lim ds[W−s , A]Φ0 , (4.3) t→∞

−t

then the second order of ionization probability is ° Z °2 ° ° t+τ (g) ° ° Q(2) (A) = lim lim lim °FR dse−is(Hg −Eg ) [W (1) + gW (2) , Aτ (g)−s ]Φg ° ° R%∞ g&0 t→∞ ° 0 = k1ac (Hel ) ⊗ 1F Ψ(A)k2 . Proof. Application of Corollary 3.5 and Hf Ω = 0 implies (1)

1

k[W−s , A]Φ0 k ≤ c4 (0, 0)(1 + |s|)−ζ k(Hel − b) 2 (Hf + 1)

N +1 2

Φ0 k

1

= c4 (1 + |s|)−ζ |e0 − b| 2 , which shows convergence of Z

t

(1)

Ψ = Ψ(A) = lim

t→∞

−t

ds[W−s , A]Φ0 .

Due to Corollary 3.2 and Theorem 3.9 Z t+τ (g) dse−is(Hg −Eg ) [W (1) + gW (2) , Aτ (g)−s ]Φg 0

Z

∞

=

˜ (g), t), dre−i(τ (g)−r)(H0 −E0 ) [W (1) , Ar ]Φ0 + R(τ

−∞

˜ (g), t)k ≤ o(1) as g & 0. So for any g > 0 where supt≥g−1 kR(τ ˜ (g), t)k ≤ sup kR(τ ˜ (g), t)k ≤ o(1) lim kFR R(τ

t→∞

t≥g −1

and therefore ˜ (g), t)k = 0 = lim lim lim kFR R(τ ˜ (g), t)k2 . lim lim lim kFR R(τ

R→∞ g&0 t→∞

R→∞ g&0 t→∞

Furthermore ° Z ∞ ° °Z ° ° ° −i(τ (g)−r)(H0 −E0 ) (1) °FR ° dre [W , Ar ]Φ0 ° ° °≤° −∞

so

∞

−∞

¯ ¿ Z ¯ ¯ lim lim lim < FR ¯R→∞ g&0 t→∞

∞

−∞

= kΨk,

À¯ ¯ ˜ (g), t) ¯ dre −i(τ (g)−r)(H0 −E0 ) [W (1) , Ar ]Φ0 , FR R(τ ¯

˜ (g), t)k = 0 ≤ kΨk lim lim lim kFR R(τ R→∞ g&0 t→∞

°

° (1) [W−r , A]Φ0 dr° °

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and we conclude "° Z ° Q(2) (A) = lim lim lim ° FR R→∞ g&0 t→∞ °

∞

dre

−i(τ (g)−r)(H0 −E0 )

[W

(1)

−∞

°2 ° , Ar ]Φ0 ° °

˜ (g), t)k2 + kFR R(τ ¿ Z ∞ À¸ −i(τ (g)−r)(H0 −E0 ) (1) ˜ + 2< FR dre [W , Ar ]Φ0 , FR R(τ (g), t) −∞

° Z ° = lim lim ° °FR R→∞ g&0

∞

−∞

°2 ° dre −i(τ (g)−r)(H0 −E0 ) [W (1) , Ar ]Φ0 ° °

= lim lim kFR e−iτ (g)H0 Ψ(A)k2 .

(4.4)

R→∞ g&0

Apart from τ (g) any other g dependence has disappeared from (4.4), so Q(2) (A) = lim lim kFR e−iτ H0 Ψ(A)k2 .

(4.5)

R→∞ τ →∞

The algebraic tensor-product Hel ⊗ F is dense in H and 1pp (Hel )Hel is the closure of finite linear combinations of eigenfunctions of Hel . So for any ε > 0, there are M ∈ N, φ1 , . . . , φm ∈ F, h1 , . . . , hM ∈ Hel , such that ° ° ° ° M X ° ° ε °Ψ − hj ⊗ φj ° ° °< 2 ° ° j=1 and furthermore mj ∈ N and eigenfunctions ηj,l of Hel corresponding to the eigenvalues ej,l , j = 1, . . . , M , l = 1, . . . , mj , such that ° ° mj ° ° X ε ° ° . ηj,l ° < °1pp (Hel )hj − ° ° 2M kφj k l=1

kFR e−iτ H0 k ≤ 1 so ° ° ° ° M X ° ° ε −iτ H0 ° kFR e−iτ H0 1pp (Hel )Ψk ≤ ° F e 1 (H )h ⊗ φ pp el j j° + ° R ° ° 2 j=1 ≤

M X

k1{|x|≥R} e−iτ Hel 1pp (Hel )hj k kφj k +

j=1

≤

M X

k1{|x|≥R} e−iτ Hel

j=1

≤

mj M X X j=1 l=1

mj X

ε 2

ηj,l k kφj k + ε

l=1

k1{|x|≥R} ηj,l k kφj k + ε.

(4.6)

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The right-hand side of (4.6) does not depend on τ , so sup kFR e−iτ H0 1pp (Hel )Ψk ≤ τ ∈R

mj M X X

k1{|x|≥R} ηj,l k kφj k + ε

j=1 l=1

and 1{|x|≥R} converges strongly to 0 for R → ∞, so lim sup kFR e−iτ H0 1pp (Hel )Ψk ≤

R→∞ τ ∈R

mj M X X j=1 l=1

lim k1{|x|≥R} ηj,l k kφj k + ε = ε,

R→∞

hence lim sup sup kFR e−iτ H0 1pp (Hel ) ⊗ 1F Ψk = 0.

(4.7)

R→∞ τ ∈R+

Due to Hypothesis 1 the singular continuous spectrum σsc (Hel ) = ∅ is empty, hence 1Hel = 1pp (Hel ) + 1ac (Hel ) and in combination with (4.7) we get: lim lim kFR e−iτ H0 Ψk2 = lim lim kFR e−iτ H0 1ac (Hel ) ⊗ 1F Ψk2

R→∞ τ →∞

R→∞ τ →∞

2

= k1ac (Hel ) ⊗ 1F Ψk − lim lim k(1 − FR )e−iτ H0 1ac (Hel ) ⊗ 1F Ψk2 . R→∞ τ →∞

(4.8)

D(Hel ) is dense in Hel , so for each ε > 0 there are ϕ1 , . . . , ϕn ∈ D(Hel ) and Pn φ1 , . . . , φn ∈ F , such that k j=1 ϕj ⊗ φj − Ψk < ε, hence lim lim k(1 − FR )e−iτ H0 1ac (Hel ) ⊗ 1F Ψk

R→∞ τ →∞

≤

n X j=1

lim lim k1{|x|
R→∞ τ →∞

(4.9)

For the last estimate we used • limτ →∞ k1{|x| 0. Putting together these results and with Ψ(A) as in (4.3), finally we get: Q(2) (A) = k1ac (Hel ) ⊗ 1F Ψ(A)k2 . Note, that in this RAGE-type theorem, we have to do the finite rank approximations of Ψ “by hand”, because FR = 1{|x|≥R} ⊗ 1F destroys relative H0 compactness.

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Theorem 4.3. Under the assumptions of Theorem 4.2 let (ϕn )n∈N be an orthonormal family in C0∞ (R3 \{0}), such that hϕj , ϕl iL2 = δjl . Suppose (m1 , . . . , mη ), (n1 , . . . , nη ) ∈ Nη with m1 + · · · + mη + n1 + · · · + nη = N, then the second order of ionization probability Q(2) (a∗+ (ϕ1 )m1 a∗− (ϕ1 )n1 · · · a∗+ (ϕη )mη a∗− (ϕη )nη ) by a photon cloud A = a∗+ (ϕ1 )m1 a∗− (ϕ1 )n1 · · · a∗+ (ϕη )mη a∗− (ϕη )nη ,

(4.10)

is given by Q(2) (a∗+ (ϕ1 )m1 a∗− (ϕ1 )n1 · · · a∗+ (ϕη )mη a∗− (ϕη )nη ) m1 ! · · · mη !n1 ! · · · nη ! =

η X (2) (2) (nj Q− (ϕj ) + mj Q+ (ϕj ))

(4.11)

j=1

with one photon terms ° Z ° (2) Qλ (ϕ) := ° 1 (H ) ac el °

Z

∞

ds

−∞

dke

is(Hel −e0 )

w

(0,1)

(k, λ)e

−isω(k)

R3

°2 ° ϕ(k)ϕ0 ° ° . (4.12)

Proof. (3.8) implies k(1ac (Hel ) ⊗ 1F )Ψ(A)k2 ° °2 Z ∞ ° ° is(H0 −e0 ) (1) ° =° (1 (H ) ⊗ 1 ) dse [W , A ]Φ F s 0° ° ac el −∞ °  ° Z ∞ Z η X ° =° (1 (H ) ⊗ 1 ) ds n dke is(Hel −e0 ) w(0,1) (k, −) F j ° ac el  3 −∞ R ° j=1 × e−isω(k) ϕj (k)ϕ0

η Y

a∗+ (ϕl )ml a∗− (ϕl )nl −δjl Ω

l=1

+

η X

Z mj

j=1

×

η Y l=1

R3

dke is(Hel −e0 ) w(0,1) (k, +)e−isω(k) ϕj (k)ϕ0

°2 ° ° a∗+ (ϕl )ml −δjl a∗− (ϕl )nl Ω ° . ° °

(4.13)

Commuting creation and annihilation operators, the canonical commutation relations together with the orthonormality of ϕ1 , . . . , ϕm imply aλ (ϕj )a∗λ0 (ϕl ) = a∗λ0 (ϕl )aλ (ϕj ) + δλ,λ0 δjl .

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By induction aλ (ϕj )q a∗λ0 (ϕl )q Ω = δλ,λ0 δjl q!Ω for q ∈ N, where a(ϕj )Ω = 0 was used. As a generalization of the last result * η + η η Y Y Y 0 0 ∗ ql ∗ rl ∗ ql ∗ rl (4.14) a+ (ϕl ) a− (ϕl ) Ω, a+ (ϕl ) a− (ϕl ) Ω = δql ql0 δrl rl0 ql ! rl ! l=1

l=1

l=1

(q1 , . . . , qη ), (q10 , . . . , qη0 ), (r1 , . . . , rη ), (r10 , . . . , rη0 )

for ∈ Nη . When we use these orthogonality relations in expanding the sum under the norm square in (4.13) k(1ac (Hel ) ⊗ 1F )Ψ(A)k2 m1 ! · · · mη !n1 ! · · · nη ! ° Z η X ° nj ° = 1 (H ) ° ac el j=1

+

η X

°2 ° dke is(Hel −e0 ) w(0,1) (k, −)e−isω(k) ϕj (k)ϕ0 ° °

Z

∞

ds R3

−∞

° Z ° ° mj °1ac (Hel )

Z

∞

ds

dke

w

(0,1)

(k, +)e

R3

−∞

j=1

is(Hel −e0 )

−isω(k)

°2 ° ϕj (k)ϕ0 ° °

which is the desired result. 4.3. Expansion in generalized eigenfunctions and the “explicit” calculation of Q(2) (ϕj ) To see an analogon of (1.2) in our model, we need a more explicit calculation of Q(2) (ϕj ). Such an “explicit” calculation of Q(2) (ϕj ) for some given momentum distribution ϕj ∈ C0∞ (R3 \{0}), needs some results from scattering theory of the electron Hamiltonian Hel , in particular an expansion in (generalized) eigenfunctions. For the application of eigenfunction expansion to the calculation of Q(2) (ϕj ) we assume: Hypothesis 6. The wave operators Ω± (−4, Hel ) := s- lim eit(−4) e−itHel 1ac (Hel ) t→∓∞

3

exist. For compact K ⊆ R \{0}, α ∈ N30 , |α| ≤ ζ, there is some θ ∈ L2 (R3 ), such that ¯ ¯ sup ¯hxi2 (∂kα w(0,1) (k, λ)ϕ0 )(x)¯ ≤ |θ(x)| (4.15) k∈K λ∈Z2

and for s ∈ R, k ∈ R3 \{0} and λ ∈ Z2 ws(0,1) (k, λ)ϕ0 = e−is(Hel −e0 ) w(0,1) (k, λ)ϕ0 ∈ D(h·i2 ).

(4.16)

Theorem 4.4. Suppose Hypothesis 1– 4, 6, (Hel , 1) and (Hel , 1, 1) are satisfied, then there is a function ρˆλ : R3 × R3 → C, (p, k) 7→ ρˆλ (p, k)

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such that for |α| ≤ 2 all partial derivatives ∂kα ρˆλ exist on R3 \{0}, Z Z dp dk|∂kα ρˆλ (p, k)|2 < ∞ R3

K

for each compact set K ⊆ R3 \{0} and for ϕj ∈ C0∞ (R3 \{0}) the second order of ionization probability is Z (2)

Qλ (ϕj ) = lim

t→∞

R3

¯Z ¯ dp ¯¯

Z

t −t

2

dke is(p

ds

−e0 −ω(k))

R3

¯2 ¯ ρˆλ (p, k)ϕj (k)¯¯ .

(4.17)

Proof. The absolute continuous subspace 1ac (Hel )Hel of Hel is a reducing subspace for Hel . Due to the assumptions, the wave operators Ω± (−4, Hel ) := s- lim eit(−4) e−itHel 1ac (Hel ) t→∓∞

exist. Using the intertwining properties of Ω± and conjugation with Fourier transform F, we obtain Hel |1ac (Hel )Hel as a multiplication operator with p2 : FΩ± (−4, Hel )Hel |1ac (Hel )Hel (FΩ± (−4, Hel ))∗ = p2 1Ran(FΩ± ) . So for an application of [22, Theorems 2.2 and Theorem 2.3] we may take H = Hel , M = 1ac (Hel )Hel , X = R3 , dρ(p) = dp, h(p) = p2 and U := FΩ± (−4, Hel ) : 1ac (Hel )Hel → L2 (R3 ). We further fix z ∈ C\R and define γ(λ) := (λ − z)−2 , then γ(h(p)) = (p2 − z)−2 6= 0 1 for each p ∈ R3 and γ(h(p)) = (p2 − z)2 remains bounded on each compact subset of R3 . Hence the limiting arguments in the proof of [22, Theorem 2.2] can be done for the σ-compact space R3 with Borel measure as in the case of a σ-finite measure space with γ finite on sets of finite measure. [22, Theorem 3.6] is applicable due to the relative −4-bound of V in Hypothesis 1, hence it implies, that we can choose T = h·i2 and S = 1 − 4, so that ¡ ¢∗ γ(H)T −1 S = (Hel − z)−2 h·i−2 (1 − 4) ⊆ (1 − 4)h·i−2 (Hel − z)−2 ¡ ¢∗ is the restriction of the Hilbert–Schmidt operator (1 − 4)h·i−2 (Hel − z)−2 . According to Hypothesis 6 for any k ∈ R3 \{0}, λ ∈ Z2 and s ∈ R (0,1)

eis(Hel −e0 ) w(0,1) (k, λ)ϕ0 = w−s (k, λ)ϕ0 ∈ D(h·i2 ), hence Z

Z

t

Ψ(t) :=

ds −t

R3

dke is(Hel −e0 −ω(k)) ϕj (k)w(0,1) (k, λ)ϕ0 ∈ D(h·i2 )

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and by [22, Theorems 2.2 and 2.3] we get for (U 1ac (Hel )Ψ(t))(p) : k1ac (Hel )Ψ(t)k2 Z Z dp|(U 1ac (Hel )Ψ(t))(p)|2 = =

R3

R3

dp|hϕ(p), Ψ(t)i∓ |2

¯¿ À ¯¯2 Z t Z ¯ ¯ ¯ is(Hel −e0 −ω(k)) (0,1) dke ϕj (k)w (k, λ)ϕ0 ¯ dp ¯ ϕ(p), ds = ¯ ¯ −t R3 R3 ∓ Z

Z = R3

¯Z ¯ dp ¯¯

ds R3

−t

¯Z Z ¯ = dp ¯¯ 3 R

Z

t

¯2 ¯ dke−isω(k) ϕj (k)hϕ(p), eis(Hel −e0 ) w(0,1) (k, λ)ϕ0 i∓ ¯¯

Z

t

ds

dke

is(p2 −e0 −ω(k))

R3

−t

ϕj (k)hϕ(p), w

(0,1)

¯2 ¯ (k, λ)ϕ0 i∓ ¯¯ .

(4.18)

Now we define ρˆλ (p, k) := hϕ(p), w(0,1) (k, λ)ϕ0 i∓ ,

(4.19)

and note, that the construction of the generalized eigenfunctions ϕ(p) in [22] and our choice of S and T implies 2 ϕ(p) ∈ Ran(h·i2 (1 − 4)−1 ) = H−2 (R3 )

≡ {f : R3 → C measurable , h·i−2 f ∈ H 2 (R3 )}, so due to w(0,1) (k, λ)ϕ0 ∈ D(h·i2 ) and the definition of the dual pairing h·, ·i∓ in [23], in (4.19) it boils down to the following integral Z hϕ(p), w(0,1) (k, λ)ϕ0 i∓ = ϕ(p, x)(w(0,1) (k, λ)ϕ0 )(x)dx. (4.20) R3

(2)

By (4.18) and (4.19), we get the same type of formula for Qλ (ϕj ) as in [5]: (2)

Qλ (ϕj ) = lim k1ac (Hel )Ψ(t)k2 t→∞

Z = lim

t→∞

¯Z ¯ dp ¯¯ 3

R

Z

t

ds

dke R3

−t

is(p2 −e0 −ω(k))

¯2 ¯ ρˆλ (p, k)ϕj (k)¯¯ .

(4.21)

Due to Hypothesis 6 |ϕ(p, x)(∂kα w(0,1) (k, λ)ϕ0 )(x)| ≤ |ϕ(p, x)hxi−2 ||θ(x)| is dominated by the L1 function on the right hand side, so dominated convergence theorem implies Z α α ∂k ρˆλ (p, k) = ∂k dxϕ(p, x)(w(0,1) (k, λ)ϕ0 )(x) Z = R3

R3

dxϕ(p, x)(∂kα w(0,1) (k, λ)ϕ0 )(x)

= hϕ(p), ∂kα w(0,1) (k, λ)ϕ0 i∓ .

(4.22)

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For further applications, now we check the regularity properties of ρˆλ and its derivatives. ∂kα w(0,1) (k, λ)ϕ0 ∈ D(h·i2 ) according to Hypothesis 6, so Z Z Z Z α 2 dp|hϕ(p), ∂kα w(0,1) (k, λ)ϕ0 i∓ |2 dp|∂k ρˆλ (p, k)| = dk dk K

R3

K

Z =

Z

R3

dk R3

K

Z =

K

dp|U 1ac (Hel )∂kα w(0,1) (k, λ)ϕ0 |2 (p)

dkk1ac (Hel )∂kα w(0,1) (k, λ)ϕ0 k2 Z 1

≤ (e0 − b) K

dkk∂kα w(0,1) (k, λ)(Hel − b)− 2 k2 ,

3

which is finite for any compact K ⊆ R \{0} and α ∈ N30 with |α| ≤ 2 according to Hypothesis 3. Theorem 4.5. Let the assumptions of Theorem 4.4 be satisfied and let µr be the Lebesgue measure on the sphere S 2 (r) := {k ∈ R3 : |k| = r}. Then the second order of ionization probability for a single photon with wavefunction ϕj is given by ¯Z ¯2 Z ¯ ¯ ¯ ¯ (2) Qλ (ϕj ) = dp ¯ (4.23) dµp2 −e0 (k)ˆ ρλ (p, k)ϕj (k)¯ . ¯ S 2 (p2 −e0 ) ¯ R3 Proof. As ϕj ∈ C0∞ (R3 \{0}), the dispersion ω is differentiable on the support of ϕj , hence " #2 3 X 2 i ∂ ∂ω is(p2 −e0 −ω(k)) ϕj (k)e = ϕj (k) (k) eis(p −e0 −ω(k)) . (4.24) 2 s|∇ω| (k) ∂kl ∂kl l=1

Two times integration by parts of (4.24) shows us, that there are C0∞ (R3 \{0}) functions fα , α ∈ N30 , |α| ≤ 2, such that for |s| > 1 we obtain: Z 2 dkeis(p −e0 −ω(k)) ρˆλ (p, k)ϕj (k) R3

=

1 s2

Z dk R3

X

2

(∂kα ρˆλ )(p, k)fα (k)eis(p

−e0 −ω(k))

.

|α|≤2

By (4.25) and Schwarz inequality, we obtain: ¯Z t ¯2 Z ¯ ¯ is(p2 −e0 −ω(k)) ¯ ¯ ds dke ρ ˆ (p, k)ϕ (k) λ j ¯ ¯ 3 −t

R

¯2 ¯ ¯ ¯Z −1 Z X ¯ ¯ 2 1 α is(p −e0 −ω(k)) ¯ ¯ dk (∂k ρˆλ )(p, k)fα (k)e ≤¯ ds 2 ¯ s R3 ¯ ¯ −t |α|≤2 ¯Z ¯ + ¯¯

Z

1

ds −1

dk e R3

is(p2 −e0 −ω(k))

¯2 ¯ ρˆλ (p, k)ϕj (k)¯¯

(4.25)

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¯2 ¯ ¯ ¯Z t Z X ¯ ¯ 2 1 α is(p −e0 −ω(k)) ¯ ¯ dk (∂k ρˆλ )(p, k)fα (k)e +¯ ds 2 ¯ s 3 R ¯ ¯ 1 |α|≤2 ¯Z ¯ ≤ ¯¯

Z

1

ds −1

¯2 ¯ dk|ˆ ρλ (p, k)| |ϕj (k)|¯¯ 3

R

¯ ¯2 ¯Z ∞ ¯ Z X ¯ ¯ ds α ¯ + 2 ¯¯ ρ ˆ )(p, k)| |f (k)| dk |(∂ α k λ ¯ 2 ¯ 1 s R3 ¯ |α|≤2 Z Z ≤4 dk|ϕj (k)|2 dk|ˆ ρλ (p, k)|2 R3

supp ϕj

Z

+ 200 max

|α|≤2

R3

Z 2

dk|fα (k)|2 max

|α|≤2

K

dk |∂kα ρˆλ (p, k)| ,

(4.26)

S where K := |α|≤2 supp fα is a compact subset of R3 \{0}. In the last estimate, we R∞ integrated 1 s−2 ds = 1 and used, that there are 10 multi-indices α ∈ N30 with |α| ≤ 2. Thus by Theorem 4.4, the bound on the right-hand side of (4.26), which is uniform in t, is integrable in (R3 , dp), so by dominated convergence ¯Z t Z ¯2 Z ¯ ¯ (2) is(p2 −e0 −ω(k)) ¯ Qλ (ϕj ) = lim dp ¯ ds dke ρˆλ (p, k)ϕj (k)¯¯ t→∞

Z = R3

R3

¯ Z ¯ dp ¯¯ lim t→∞

R3

−t

R3

Z

t

dk

dse is(p

−t

2

−e0 −ω(k))

¯2 ¯ ρˆλ (p, k)ϕj (k)¯¯

¯2 ¯ Z Z 2 2 ¯ ¯ eit(p −e0 −ω(k)) − e−it(p −e0 −ω(k)) ¯ ¯ = dp ¯ lim ρ ˆ (p, k)ϕ (k) dk ¯ λ j ¯ ¯t→∞ R3 i(p2 − e0 − ω(k)) R3 ¯2 ¯ Z ∞ it(p2 −e0 −r) −it(p2 −e0 −r)Z ¯ ¯ e −e ¯ ¯ = dp ¯ lim dµ (k)ˆ ρ (p, k)ϕ (k) dr ¯. r λ j ¯ ¯t→∞ 0 i(p2 − e0 − r) S 2 (r) R3 Z

(4.27) In the last step we changed to polar coordinates for the k-integration and used the Lebesgue measure µr on the sphere S 2 (r) := {k ∈ R3 : |k| = r}, which is normalized as µr (S 2 (r)) = 4πr2 . Passing to the new integration variable y := p2 − e0 − r and Z up (y) := dµp2 −e0 −y (k)ˆ ρλ (p, k)ϕj (k), (4.28) S 2 (p2 −e0 −y)

we see, that the differentiability of ρˆλ in k and ϕj ∈ C0∞ (R3 \{0}) imply up ∈ C01 (R), hence we can perform the y-integration in the limit t → ∞ explicit: ¯2 Z ¯ Z Z ¯ ¯ 2 sin(ty) (2) 2 up (y)¯¯ = dp |2πup (0)| Qλ (ϕj ) = dp ¯¯ lim dy t→∞ y 3 3 R R R ¯Z ¯2 Z ¯ ¯ ¯ ¯ = dp ¯ dµp2 −e0 (k)ˆ ρλ (p, k)ϕj (k)¯ . ¯ ¯ 3 2 2 R S (p −e0 )

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Appendix A. (Regularized) Field Energy, Relative Bounds for Creation- and Annihilation Operators Let 0 ≤ r˜ be an infrared and r > r˜ be an ultraviolet regularization parameter and define the regularized dispersion relation ω(˜r,r) (k) := ω(k)1{˜r≤ω(k)≤r} (k) and the regularized free field Hf,(˜r,r) :=

XZ

λ∈Z2

R3

dkω(˜r,r) (k)a∗λ (k)aλ (k).

(A.1)

(A.2)

Hf,(˜r,r) is the second quantization dΓ(ω(˜r,r) ) of the multiplication with ω(˜r,r) , so as in the non-regularized case r˜ = 0 and r = ∞ the pull-through formula aλ (k)F (Hf,(˜r,r) ) = F (Hf,(˜r,r) + ω(˜r,r) (k))aλ (k)

(A.3)

F (Hf,(˜r,r) )a∗λ (k)

(A.4)

=

a∗λ (k)F (Hf,(˜r,r)

+ ω(˜r,r) (k))

hold true for measurable F : R → C. The restriction of F (Hf,(˜r,r) ) to the n-photon sector Sn (L2 (R3 × Z2 )n ) is the multiplication operator Ψ(k1 , λ1 , . . . , kn , λn ) 7→ F (ω(˜r,r) (k1 ) + · · · + ω(˜r,r) (kn ))Ψ(k1 , λ1 , . . . , kn , λn ). Lemma A.1. For 0 ≤ s˜ ≤ r˜ < r ≤ s ≤ ∞ and 0 ≤ β ≤ α k(Hf,(˜r,r) + 1 + ω(˜r,r) (k))β (Hf,(˜s,s) + 1 + ω(˜s,s) (k))−α k ≤ 1,

(A.5)

k(Hf,(˜r,r) + 1 + ω(˜r,r) (k) + ω(˜r,r) (k 0 ))β (Hf,(˜s,s) + 1 + ω(˜s,s) (k) + ω(˜s,s) (k 00 ))−α k ≤ (1 + ω(˜r,r) (k 0 ))β .

(A.6)

Proof. These two operators leave the n-photon sectors F (n) invariant: Applied to Ψn ∈ F (n) = Sn (L2 (R3 × Z2 )n ) in the n photon sector the operator Hf,(˜r,r) + 1 + ω(˜r,r) (k) is just the multiplication operator with the function ω(˜r,r) (k1 ) + · · · + ω(˜r,r) (kn ) + 1 + ω(˜r,r) (k), furthermore (ω(˜r,r) (k1 ) + · · · + ω(˜r,r) (kn ) + 1 + ω(˜r,r) (k) + ω(˜r,r) (k 0 ))β (ω(˜s,s) (k1 ) + · · · + ω(˜s,s) (kn ) + 1 + ω(˜s,s) (k) + ω(˜s,s) (k 00 ))α µ ¶β ω(˜r,r) (k 0 ) − ω(˜r,r) (k 00 ) ≤ 1+ ω(˜s,s) (k1 ) + · · · + ω(˜s,s) (kn ) + 1 + ω(˜s,s) (k) + ω(˜s,s) (k 00 ) ≤ (1 + ω(˜r,r) (k 0 ))β , which proves (A.6). Lemma A.2. Let 0 ≤ s˜ ≤ r˜ < r ≤ s ≤ ∞ and f : {k ∈ R3 : s˜ ≤ ω(k) ≤ s} × Z2 → C

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or f : {k ∈ R3 : s˜ ≤ ω(k) ≤ s} × Z2 → L(Hel ) such that XZ

ϑ0 :=

λ∈Z2

dk

{˜ s≤ω(k)≤s}

kf (k, λ)k2 < ∞, ω(˜s,s) (k)

then for any l, m ∈ N0 l

m

k(Hf + 1) 2 (Hf,(˜r,r) + 1) 2 aλ (f )(Hf,(˜s,s) + 1)− If for some n ∈ N0 XZ θn : = λ∈Z2

µ dk 1 +

{˜ s≤ω(k)≤s}

m+1 2

l

(Hf + 1)− 2 k ≤

p

ϑ0 .

(A.7)

¶

1

(1 + ω(˜r,r) (k))n kf (k, λ)k2 < ∞

ω(˜s,s) (k)

then n

k(Hf,(˜r,r) + 1) 2 a∗λ (f )(Hf,(˜s,s) + 1)−

n+1 2

k≤

p

θn .

(A.8)

If moreover s < ∞ and ϑ :=

µ 1+

XZ λ∈Z2

{˜ s≤ω(k)≤s}

¶

1 ω(˜s,s) (k)

kf (k, λ)k2 < ∞,

then for any m, n ∈ N0 n

m

k(Hf + 1) 2 (Hf,(˜r,r) + 1) 2 a∗λ (f )(Hf,(˜s,s) + 1)−

n+1 2

m

(Hf + 1)− 2 k ≤

√

ϑ(1 + s)

m+n 2

.

(A.9) Proof. Definition of Hf,(˜s,s) as a quadratic form and H¨older inequality imply °Z °2 ° ° ° ° ∗ kaλ (f )Ψk = ° dkf (k, λ) aλ (k)Ψ° ° {˜s≤ω(k)≤s} ° # " XZ kf (k, λ)k2 ≤ dk ω(˜s,s) (k) s≤ω(k)≤s} λ∈Z2 {˜ " # XZ × dkω(˜s,s) (k)kaλ (k)Ψk2 2

λ∈Z2 1

{˜ s≤ω(k)≤s}

2 2 ≤ ϑ0 kHf,(˜ s,s) Ψk , 1

1

(A.10)

so kaλ (f )(Hf,(˜s,s) + 1)− 2 k ≤ ϑ02 . If there are powers of Hf + 1, Hf,(˜r,r) + 1 and Hf,(˜s,s) + 1 on both sides of aλ (f ), the pull through formula allows us to

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shift them to one side, rearrange them because [Hf , Hf,(˜s,s) ] = [Hf , Hf,(˜r,r) ] = [Hf,(˜r,r) , Hf,(˜s,s) ] = 0 and finally use Lemma A.1: m+1

m

l

l

k(Hf + 1) 2 (Hf,(˜r,r) + 1) 2 aλ (f )(Hf,(˜s,s) + 1)− 2 (Hf + 1)− 2 Ψk °Z ° m+1 l m ° dk(Hf + 1) 2 (Hf,(˜r,r) + 1) 2 aλ (k)f (k, λ)∗ (Hf,(˜s,s) + 1)− 2 =° ° {˜s≤ω(k)≤s} ° ° − 2l ° × (Hf + 1) Ψ° ° °Z ° m l m ° =° dk(Hf,(˜r,r) + 1) 2 (Hf,(˜s,s) + 1 + ω(˜s,s) (k))− 2 (Hf + 1) 2 ° {˜s≤ω(k)≤s} ° ° 1 l ° × (Hf + 1 + ω(k))− 2 aλ (k)f (k, λ)∗ (Hf,(˜s,s) + 1)− 2 Ψ° ° m

m

≤ sup k(Hf,(˜r,r) + 1) 2 (Hf,(˜s,s) + 1 + ω(˜s,s) (k))− 2 k {|k|≤s}

Ã l 2

× sup k(Hf + 1) (Hf + 1 + ω(k))

− 2l

{|k|≤s}

Ã × =

p

λ∈Z2

kf (k, λ)k2 dk ω(˜s,s) (k) {˜ s≤ω(k)≤s}

!21

! 21

XZ

− 21

dkω(˜s,s) (k)kaλ (k)(Hf,(˜s,s) + 1)

{˜ s≤ω(k)≤s}

λ∈Z2

k

XZ

1

1

−2 2 ϑ0 kHf,(˜ Ψk ≤ s,s) + 1) s,s) (Hf,(˜

p

2

Ψk

ϑ0 kΨk,

(A.11)

proving (A.7). The canonical commutation relations allow us to convert creation into annihilation operators plus some extra terms, so ka∗λ (f )Ψk2 °Z °2 ° ° ° ° ∗ =° dkf (k, λ)aλ (k)Ψ° ° {˜s≤ω(k)≤s} ° Z Z = dk1 {˜ s≤ω(k1 )≤s}

{˜ s≤ω(k2 )≤s}

× dk2 hf (k1 , λ)Ψ, (a∗λ (k2 )aλ (k1 ) + δ(k1 − k2 ))f (k2 , λ)Ψi Z = dkkf (k, λ)Ψk2 {˜ s≤ω(k)≤s}

Z

+

Z dk1

{˜ s≤ω(k1 )≤s}

dk2 hf (k1 , λ)aλ (k2 )Ψ, f (k2 , λ)aλ (k1 )Ψi {˜ s≤ω(k2 )≤s}

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Ã

Z 2

2

≤ kΨk

dkkf (k, λ)k + {˜ s≤ω(k)≤s}

Z 2 {˜ s≤ω(k)≤s}

XZ λ∈Z2

dkkf (k, λ)kkaλ (k)Ψk

{˜ s≤ω(k)≤s}

dkkf (k, λ)k2

≤ kΨk +

!2

XZ

λ∈Z2

403

dk {˜ s≤ω(k)≤s}

Z kf (k, λ)k2 X dkω(˜s,s) (k)kaλ (k)Ψk2 ω(˜s,s) (k) {˜ s≤ω(k)≤s} λ∈Z2

1 2

1

≤ θ0 kΨk2 + θ0 kHf,(˜s,s) Ψk2 = θ0 k(Hf,(˜s,s) + 1) 2 Ψk2 .

(A.12)

If there are powers of Hf,(˜r,r) + 1 and Hf,(˜s,s) + 1 on both sides of a∗λ (f ), the pull through formula allows us to shift them to one side and by the canonical commutation relations we convert the creation into annihilation operators, so a straightforward calculation using Lemma A.1 gives (A.8). If s < ∞, then (A.9) follows with the additional estimate ω(k) ≤ s on {|k| ≤ s} along the same lines. Corollary A.3. Let f1 , . . . , fN ∈ C0∞ (R3 \{0}), λ1 , . . . , λN ∈ Z2 and 0 ≤ r˜ < inf{ω(k) : k ∈ supp fj : j = 1, . . . , N } ∞ > r > sup{ω(k) : k ∈ supp fj : j = 1, . . . , N } then for any m, γ ∈ N0 and t ∈ R γ

N

m

(Hel − b) 2 (Hf + 1) 2 e−itH0 a∗λ1 (f1 ) · · · a∗λN (fN )eitH0 (Hf,(˜r,r) + 1)− 2 γ

m

× (Hf + 1)− 2 (Hel − b)− 2 defines a bounded operator on H and moreover γ

m

N

sup k(Hel − b) 2 (Hf + 1) 2 e−itH0 a∗λ1 (f1 ) · · · a∗λN (fN )eitH0 (Hf,(˜r,r) + 1)− 2 t∈R

m

γ

N

× (Hf + 1)− 2 (Hel − b)− 2 k ≤ kf1 kω · · · kfN kω (1 + r) 4 (2m+N −1) < ∞. Proof. The creation operators a∗λ1 (f1 ), . . . , a∗λN (fN ) act on the photon Fock space F and e±itH0 = e±itHel ⊗ e±itHf , so e−itH0 a∗λ1 (f1 ) · · · a∗λN (fN )eitH0 = e−itHf a∗λ1 (f1 ) · · · a∗λN (fN )eitHf = a∗λ1 (e−itω f1 ) · · · a∗λN (e−itω fN ). Now Hel commutes with all other terms, so γ

m

N

(Hel − b) 2 (Hf + 1) 2 a∗λ1 (e−itω f1 ) · · · a∗λN (e−itω fN )(Hf,(˜r,r) + 1)− 2 m

γ

× (Hf + 1)− 2 (Hel − b)− 2 m

N

m

= (Hf + 1) 2 a∗λ1 (e−itω f1 ) · · · a∗λN (e−itω fN )(Hf,(˜r,r) + 1)− 2 (Hf + 1)− 2 .

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Due to the choice of f1 , . . . , fN ∈ C0∞ (R3 \{0}) ¶ Z µ 1 2 1+ kfj kω = kfj (k, λ)k2 < ∞. ω(˜r,r) (k) R3 j

m

j

m

Inserting identities as 1F = (Hf,(˜r,r) + 1)− 2 (Hf + 1)− 2 (Hf + 1) 2 (Hf,(˜r,r) + 1) 2 and applying Lemma A.2, Eq. (A.9) one gets: γ

m

N

k(Hel − b) 2 (Hf + 1) 2 a∗λ1 (e−itω f1 ) · · · a∗λN (e−itω fN )(Hf,(˜r,r) + 1)− 2 γ

m

× (Hf + 1) 2 (Hel − b)− 2 k ≤

N Y

m

k(Hf + 1) 2 (Hf,(˜r,r) + 1)

j−1 2

j

m

a∗λj (e−itω fj )(Hf,(˜r,r) + 1)− 2 (Hf + 1)− 2 k

j=1 N

≤ kf1 kω · · · kfN kω (1 + r) 4 (2m+N −1) < ∞ independent of t. With the same techniques one can prove: Corollary A.4. In the situation of Corollary A.3 N

sup ke−itH0 a∗λ1 (f1 ) · · · a∗λN (fN )eitH0 (N(˜r,r) + 1)− 2 k ≤ 2

N (N −1) 2

t∈R

kf1 k · · · kfN k.

(1) ˜ (1) (1) Lemma A.5. If Hypothesis 1 is satisfied and Λ0,γ , Λβ,γ , Λ β,γ < ∞, then for any α ∈ N0 and 0 ≤ s˜ ≤ r˜ < r ≤ s < ∞ β

α

γ

γ+1

β+1

k(Hf,(˜r,r) + 1) 2 (Hf + 1) 2 (Hel − b) 2 W (1) (Hel − b)− 2 (Hf + 1)− 2 q q nq o α α (1) (1) ˜ (1) . × (Hf,(˜s,s) + 1)− 2 k ≤ Λ0,γ + (1 + r) 2 max Λβ,γ , Λ β,γ

Proof. The operators Hf,(˜s,s) , Hf and Hel commute, so for the W (0,1) term pull(1)

through formula, (A.6), H¨older inequality and the definition of Λ0,γ in (2.25) gives: β

α

γ

k(Hf,(˜r,r) + 1) 2 (Hf + 1) 2 (Hel − b) 2 W (0,1) (Hel − b)−

γ+1 2

(Hf + 1)−

α

β+1 2

× (Hf,(˜s,s) + 1)− 2 Ψk ° β °X Z α α (Hf + 1) 2 ° =° dk(Hf,(˜r,r) + 1) 2 (Hf,(˜s,s) + 1 + ω(˜s,s) (k))− 2 β ° (Hf + 1 + ω(k)) 2 R3 λ∈Z2

γ 2

× (Hel − b) w(0,1) (k, λ)(Hel − b)

− γ+1 2

aλ (k)(Hf + 1)

− 12

° ° ° Ψ° °

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≤

XZ λ∈Z2

Ã ≤

(1)

Λ0,γ

γ

R3

dkk(Hel − b) 2 w(0,1) (k, λ)(Hel − b)−

XZ λ∈Z2

R3

γ+1 2

405

1

aλ (k)(Hf + 1)− 2 Ψk

! 12 dkω(k)kaλ (k)(Hf + 1)

− 12

Ψk2

q q 1 1 (1) (1) = kHf2 (Hf + 1)− 2 Ψk Λ0,γ ≤ kΨk Λ0,γ < ∞.

(A.13)

For the next term we use pull-through plus commutation relations to bring it in an appropriate form substituting creation operators by annihilation operators, then along the same line as in (A.13) we are done. (2) ˜ (2) ˜ (2) ˜ (2) (2) Lemma A.6. If Λ0,γ , Λβ,γ , Λ 0,γ , Λ β ,γ , Λβ,γ < ∞ and Hypothesis 1 and 2 are sat2

isfied, then for any α ∈ N0 and 0 ≤ s˜ ≤ r˜ < r ≤ s < ∞ β

α

γ

γ

β

k(Hf,(˜r,r) + 1) 2 (Hf + 1) 2 (Hel − b) 2 W (2) (Hel − b)− 2 (Hf + 1)− 2 −1 α

× (Hf,(˜s,s) + 1)− 2 k q (2) ˜ (2) + Λ(2) )Λ(2) (1 + r) α2 + 2(1 + 2r) α2 max{1, 2 β2 −1 } ≤ Λ0,γ + 2 (Λ 0,γ β,γ β,γ q h i 12 (2) ˜ (2) (2) (2) (2) (2) ˜ (2) (2) ˜ (2) ˜ (2) ˜ (2) Λ Λ + Λ + Λ . × Λβ,γ Λ0,γ + 2 Λβ,γ Λ0,γ Λ + Λ Λ β β 0,γ 0,γ 0,γ β,γ β,γ ,γ ,γ 2

2

Acknowledgments The author thanks Laszlo Erd¨os and Marcel Griesemer for various helpful discussions and comments. The introduction of g −γ in the proof of Theorem 3.9, which simplified my former proof, comes from a discussion with Marcel Griesemer. References [1] A. Arai and M. Hirokawa, On the existence and uniqueness of ground states of a generalized spin-boson model, J. Funct. Anal. 151 (1997) 455–503. [2] A. Arai and M. Hirokawa, Ground states of a general class of quantum field Hamiltonians, Rev. Math. Phys. 12 (2000) 1085–1135. [3] V. Bach, J. Fr¨ ohlich and I. M. Sigal, Quantum electrodynamics of confined nonrelativistic particles, Adv. Math. 137 (1998) 299–395. [4] V. Bach, J. Fr¨ ohlich and I. M. Sigal, Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field, Comm. Math. Phys. 207 (1999) 249–290. [5] V. Bach, F. Klopp and H. Zenk, Mathematical analysis of the photoelectric effect, Adv. Theoret. Math. Phys. 5 (2001) 969–999. [6] E. B. Davis, Spectral Theory and Differential Operators (Cambridge University Press, 1995). [7] J. Dieudonn´e, Foundations of Modern Analysis, Vol. 1 (Academic Press, 1960).

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¨ [8] A. Einstein, Uber einen die Erzeugung und Verwandlung des Lichts betreffenden heuristischen Gesichtspunkt, Ann. Phys. 17 (1905) 132–148. [9] J. Fr¨ ohlich, M. Griesemer and B. Schlein, Asymptotic electromagnetic fields in models of quantum-mechanical matter interacting with the quantized radiation field, Adv. Math. 164 (2001) 349–398. [10] C. Gerard, On the existence of ground states for massless Pauli–Fierz Hamiltonians, Ann. Henri Poincare 1 (2000) 443–459. [11] M. Griesemer, E. H. Lieb and M. Loss, Ground states in non-relativistic quantum electrodynamics, Invent. Math. 145 (2001) 557–595. [12] W. Hallwachs, Ueber den Einfluss des Lichtes auf electrostatisch geladene K¨ orper, Ann. Phys. Chem. 33 (1888) 301–312. [13] H. Hertz, Ueber einen Einfluss des ultravioletten Lichtes auf die electrische Entladung, Ann. Phys. Chem. 31 (1887) 983–1000. [14] F. Hiroshima, Ground states of a model in nonrelativistic quantum electrodynamics I, J. Math. Phys. 40 (1999) 6209–6222. [15] F. Hiroshima, Ground states of a model in nonrelativistic quantum electrodynamics II, J. Math. Phys. 41 (2000) 661–674. [16] F. Hiroshima, Localization of the number of photons of ground states in nonrelativistic QED, Rev. Math. Phys. 15 (2003) 271–312. [17] P. Hislop and M. I. Sigal, Introduction to Spectral Theory, Applied Mathematical Sciences, Vol. 113 (Springer, 1996). [18] E. H. Lieb and M. Loss, Existence of atoms and molecules in non-relativistic quantum electrodynamics, Adv. Theoret. Math. Phys. 7 (2003) 667–710. [19] R. Millikan, A direct photoelectric determination of Planck’s h, Phys. Rev. 7 (1916) 355–388. [20] R. Millikan, Einstein’s photoelectric equation and contact electromotive force, Phys. Rev. 7 (1916) 18–32. [21] W. Nolting, Grundkurs Theoretische Physik, Band 5, Quantenmechanik, Teil 1: Grundlagen, 2. Auflage (Verlag Zimmermann-Neufang, 1994). [22] T. Poerschke and G. Stolz, On eigenfunction expansions and scattering theory, Math. Z. 212 (1993) 337–357. [23] T. Poerschke, G. Stolz and J. Weidmann, Expansions in generalized eigenfunctions of selfadjoint operators, Math. Z. 202 (1989) 397–408. [24] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Scattering Theory, Vol. 3 (Academic Press, San Diego, 1979). [25] W. Rudin, Functional Analysis, 2nd edn. (McGraw-Hill, 1991). [26] R. Walter, Differentialgeometrie, 2. Auflage (BI Wissenschaftsverlag, 1989). [27] J. Weidmann, Linear Operator in Hilbert Spaces (Springer-Verlag, 1980).

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Reviews in Mathematical Physics Vol. 20, No. 4 (2008) 407–449 c World Scientific Publishing Company °

INVARIANT DIFFERENTIAL OPERATORS FOR NON-COMPACT LIE GROUPS: PARABOLIC SUBALGEBRAS

V. K. DOBREV ∗ Institute

for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria

and The Abdus Salam International Center for Theoretical Physics, P. O. Box 586, Strada Costiera 11, 34014 Trieste, Italy [email protected]

Received 1 July 2007 Revised 15 December 2007 In the present paper, we start the systematic explicit construction of invariant differential operators by giving explicit description of one of the main ingredients — the cuspidal parabolic subalgebras. We explicate also the maximal parabolic subalgebras, since these are also important even when they are not cuspidal. Our approach is easily generalized to the supersymmetric and quantum group settings and is necessary for applications to string theory and integrable models. Keywords: Invariant differential operators; non-compact Lie groups; parabolic subalgebras. Mathematics Subject Classification 2000: 17B10, 22E47, 81R05

1. Introduction Invariant differential operators play very important role in the description of physical symmetries — starting from the early occurrences in the Maxwell, d’Allembert, Dirac equations, (for more examples cf., e.g., [1]), to the latest applications of (super-)differential operators in conformal field theory, supergravity and string theory, (for a recent review, cf., e.g., [2]). Thus, it is important for the applications in physics to study systematically such operators. In the present paper, we start with the classical situation, with the representation theory of semisimple Lie groups, where there are lots of results by both mathematicians and physicists, cf., e.g., [3–44]. We shall follow a procedure in representation theory in which such operators appear canonically [27] and which has ∗ Permanent

address. 407

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been generalized to the supersymmetry setting [45] and to quantum groups [46–50]. We should also mention that this setting is most appropriate for the classification of unitary representations of superconformal symmetry in various dimensions, [51– 58], for generalization to the infinite-dimensional setting [59, 60], and is also an ingredient in the AdS/CFT correspondence, cf. [61]. (For a recent paper with more references, cf. [62].) Although the scheme was developed some time ago there is still missing explicit description of the building blocks, namely, the parabolic subgroups and subalgebras from which the representations are induced. Just in passing, we shall mention that parabolic subalgebras found applications in quantum groups, (in particular, for the quantum deformations of noncompact Lie algebras), cf., e.g., [46–50, 63–67], and in integrable systems, cf., e.g., [68–71]. In the present paper, the focus will be on the role of parabolic subgroups and subalgebras in representation theory. In the next section we recall the procedure of [27] and the preliminaries on parabolic subalgebras. Then, in Secs. 3–11, we give the explicit classification of the cuspidal parabolic subalgebras which are the relevant ones for our purposes. The cuspidal parabolic subalgebras are also summarized in table form in the Appendix. 2. Preliminaries 2.1. General setting Let G be a noncompact semisimple Lie group. Let K denote a maximal compact subgroup of G. Then we have an Iwasawa decomposition G = KAN , where A is abelian simply connected, a vector subgroup of G, N is a nilpotent simply connected subgroup of G preserved by the action of A. Further, let M be the centralizer of A in K. Then the subgroup P0 = MAN is a minimal parabolic subgroup of G. A parabolic subgroup P = M 0 A0 N 0 is any subgroup of G (including G itself) which contains a minimal parabolic subgroup. The number of non-conjugate parabolic subgroups is 2r , where r = rank A, cf., e.g., [7]. Note that in general M 0 is a reductive Lie group with structure: M 0 = Md Ms Ma , where Md is a finite group, Ms is a semisimple Lie group, Ma is an abelian Lie group central in M 0 . The importance of the parabolic subgroups stems from the fact that the representations induced from them generate all (admissible) irreducible representations of G, see [8]. (For the role of parabolic subgroups in the construction of unitary representations, we refer to [11, 14].) In fact, it is enough to use only the so-called cuspidal parabolic subgroups P = M 0 A0 N 0 , singled out by the condition that rank M 0 = rank M 0 ∩ K [9, 15], so that M 0 has discrete series representations [3].a However, often induction from a non-cuspidal parabolic is also convenient. Let P = M 0 A0 N 0 be a parabolic subgroup. Let ν be a (non-unitary) character of A0 , ν ∈ A0∗ , where A0 is the Lie algebra of A0 . If P is cuspidal, let µ fix a discrete a The simplest example of cuspidal parabolic subgroup is P when M 0 = M is compact. In all 0 other cases, M 0 is non-compact.

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series representation Dµ of M 0 on the Hilbert space Vµ , or the so-called limit of a discrete series representation (cf. [26]).b Although not strictly necessary, sometimes it is convenient to induce from noncuspidal P (especially if P is a maximal parabolic). In that case, we use any non-unitary finite-dimensional irreducible representation Dµ of M 0 on the linear space Vµ . More than this, except in the case of induction from limits of discrete series, we can always work with finite-dimensional representations Vµ by the so-called translation. Namely, when P is non-minimal and cuspidal, then instead of the inducing discrete series representation of M 0 we can consider the finite-dimensional irrep of M 0 which lies on the same orbit of the Weyl group (in other words, has the same Casimirs). We call the induced representation χ = IndG P (µ ⊗ ν ⊗ 1) an elementary representation of G [17]. (These are called generalized principal series representations (or limits, thereof) in [26].) Their spaces of functions are: Cχ = {F ∈ C ∞ (G, Vµ ) | F(gman) = e−ν(H) · Dµ (m−1 )F(g)}

(2.1)

where a = exp(H) ∈ A0 , H ∈ A0 , m ∈ M 0 , n ∈ N 0 . The special property of the functions of Cχ is called right covariance [17, 27] (or equivariance).c Because of this covariance the functions F actually do not depend on the elements of the parabolic subgroup P = M 0 A0 N 0 . The elementary representation (ER) T χ acts in Cχ as the left regular representation (LRR) by: (T χ (g)F)(g 0 ) = F(g −1 g 0 ),

g, g 0 ∈ G.

(2.2)

One can introduce in Cχ a Fr´echet space topology or complete it to a Hilbert space (cf. [7]). We shall need also the infinitesimal version of LRR: . d (XL F)(g) = F(exp(−tX)g)|t=0 , dt

(2.3)

where, F ∈ Cχ , g ∈ G, X ∈ G; then we use complex linear extension to extend the definition to a representation of G C . The ERs differ from the LRR (which is highly reducible) by the specific representation spaces Cχ . In contrast, the ERs are generically irreducible. The reducible ERs form a measure zero set in the space of the representation parameters µ, ν. (Reducibility here is topological in the sense that there exist nontrivial (closed) b In

general, µ is a actually a triple (², σ, δ), where ² is the signature of the character of Md , σ gives the unitary character of Ma , δ fixes a discrete or finite-dimensional irrep of Ms on Vµ (the latter depends only on δ). c It is well known that when V is finite-dimensional C can be thought of as the space of smooth µ χ sections of the homogeneous vector bundle (called also vector G-bundle) with base space G/P and fiber Vµ , (which is an associated bundle to the principal P -bundle with total space G). We shall not need this description for our purposes.

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invariant subspace.) The irreducible components of the ERs (including the irreducible ERs) are called subrepresentations. The other feature of the ERs which makes them important for our considerations is a highest (or lowest) weight module structure associated with them [27]. For this we shall use the right action of G C (the complexification of G) by the standard formula: . d (XR F)(g) = F(g exp(tX))|t=0 , (2.4) dt where F ∈ Cχ , g ∈ G, X ∈ G; then we use complex linear extension to extend the definition to a representation of G C . Note that this action takes F out of Cχ for some X but that is exactly why it is used for the construction of the intertwining differential operators. We can show this property in all cases when Vµ is a highest weight module, e.g., the case of the minimal parabolic subalgebra and when (M 0 , M 0 ∩K) is a Hermitian symmetric pair. In fact, we agreed that, except when inducing from limits of discrete series, the space Vµ will be finite-dimensional. Then Vµ has a highest weight vector v0 . Using this we introduce C-valued realization T˜ χ of the space Cχ by the formula: ϕ(g) ≡ hv0 , F(g)i,

(2.5) 0

where h , i is the M -invariant scalar product in Vµ . (If M = M0 is abelian or discrete then Vµ is one-dimensional and C˜χ coincides with Cχ ; so we set ϕ = F.) On these functions the right action of G C is defined byd : (XR ϕ)(g) ≡ hv0 , (XR F)(g)i.

(2.6)

Part of the main result of our paper [27] is: Proposition. The functions of the C-valued realization T˜ χ of the ER Cχ satisfy: XR ϕ = Λ(X) · ϕ,

X ∈ HC ,

XR ϕ = 0,

C X ∈ G+ ,

Λ ∈ (HC )∗ ,

(2.7a) (2.7b)

C where Λ = Λ(χ) is built canonically from χ,e G± are from the standard triangular C C C C f decomposition G = G+ ⊕ H ⊕ G− .

Note that conditions (2.7) are the defining conditions for the highest weight vector of a highest weight module (HWM) over G C with highest weight Λ. Of course, it is enough to impose (2.7b) for the simple root vectors Xj+ . d In

the geometric language we have replaced the homogeneous vector bundle with base space G/P and fiber Vµ with a line bundle again with base space G/P (also associated to the principal P -bundle with total space G). The functions ϕ can be thought of as smooth sections of this line bundle. e It contains all the information from χ, except about the character ² of the finite group M . In d the case of G being a complex Lie group, we need two weights to encode χ, cf. Sec. 3. f Note that we are working here with highest weight modules instead of the lowest weight modules used in [27]; also the weights are shifted by ρ with respect to the notation of [27].

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Furthermore, special properties of a class of highest weight modules, namely, Verma modules, are immediately related with the construction of invariant differential operators. To be more specific let us recall that a Verma module is a highest weight module Λ V with highest weight Λ, induced from one-dimensional representations of the C C Borel subalgebra B = HC ⊕ G+ . Thus, V Λ ∼ )v0 , where v0 is the highest = U (G− C C g weight vector, U (G− ) is the universal enveloping algebra of G− . Verma modules are universal in the following sense: every irreducible HWM is isomorphic to a factor-module of the Verma module with the same highest weight. Generically, Verma modules are irreducible, however, we shall be mostly interested in the reducible ones since these are relevant for the construction of differential equations. We recall the Bernstein–Gel’fand–Gel’fand [6] criterion (for semisimple Lie algebras) according to which the Verma module V Λ is reducible iff 2hΛ + ρ, βi − mhβ, βi = 0,

(2.8)

holds for some β ∈ ∆+ , m ∈ N, where ∆+ denotes the positive roots of the root system (G C , HC ), ρ is half the sum of the positive roots ∆+ . Whenever (2.8) is fulfilled there exists [13] in V Λ a unique vector vs , called singular vector, which has the properties (2.7) of a highest weight vector with shifted weight Λ − mβ: Xv s = (Λ − mβ)(X) · vs , Xv s = 0,

X ∈ HC , X∈

C G+ .

(2.9a) (2.9b)

The general structure of a singular vector is [27]: vs = Pmβ (X1− , . . . , X`− )v0 ,

(2.10)

where Pmβ is a homogeneous polynomial in its variables of degrees mki , where P ki ∈ Z+ come from the decomposition of β into simple roots: β = ki αi , αi ∈ ∆S , the system of simple roots, Xj− are the root vectors corresponding to the negative roots (−αj ), αj being the simple roots, ` = rankC G C = dimC HC is the (complex) rank of G C .h It is obvious that (2.10) satisfies (2.9a), while conditions (2.9b) fix the coefficients of Pmβ up to an overall multiplicative nonzero constant. Now we are in a position to define the differential intertwining operators for semisimple Lie groups, corresponding to the singular vectors. Let the signature χ of an ER be such that the corresponding Λ = Λ(χ) satisfies (2.8) for some β ∈ ∆+ and some m ∈ N.i Then there exists an intertwining g For

more mathematically precise definition, cf. [13]. C. singular vector may also be written in terms of the full Cartan–Weyl basis of G− i If β is a real root, (i.e. β| = 0, where H is the Cartan subalgebra of M), then some conditions C m Hm are imposed on the character ² representing the finite group Md [19].

hA

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differential operator [27] 0 Dmβ : T˜ χ −→ T˜ χ ,

0

0

0

(2.11)

0

where χ is such that Λ = Λ (χ ) = Λ − mβ. The most important fact is that (2.11) is explicitly given by [27]: Dmβ ϕ(g) = Pmβ ((X1− )R , . . . , (X`− )R )ϕ(g),

(2.12)

where Pmβ is the same polynomial as in (2.10) and (Xj− )R denotes the action (2.4). One important simplification is that in order to check the intertwining properties of the operator in (2.12) it is enough to work with the infinitesimal versions of (2.1) and (2.2), i.e. work with representations of the Lie algebra. This is important for using the same approach to superalgebras and quantum groups, and to any other (infinite-dimensional) (super-)algebra with triangular decomposition. 2.2. Generalities on parabolic subalgebras Let G be a real linear connected semisimple Lie group.j Let G be the Lie algebra of G, θ be a Cartan involution in G, and G = K ⊕ P be a Cartan decomposition of G, so that θX = X, X ∈ K, θX = −X, X ∈ P; K is a maximal compact subalgebra of G. Let A be a maximal subspace of P which is an abelian subalgebra of G; r = dim A is the split (or real) rank of G, 1 ≤ r ≤ ` = rank G. The subalgebra A is called a Cartan subspace of P. Let ∆A be the root system of the pair (G, A): . λ λ . = {X ∈ G | [Y, X] = λ(Y )X, ∀ Y ∈ A}. ∆A = {λ ∈ A∗ | λ 6= 0, GA 6= 0}, GA (2.13) λ GA

The elements of ∆A are called A-restricted roots. For λ ∈ ∆A , are called λ A-restricted root spaces, dimR GA ≥ 1. Next we introduce some ordering (e.g., the lexicographic one) in ∆A . Accordingly the latter is split into positive and neg− ative restricted roots: ∆A = ∆+ A ∪ ∆A . Now we can introduce the corresponding nilpotent subalgebras: . λ N ± = ⊕ GA . (2.14) λ∈∆± A

With this data we can introduce the Iwasawa decomposition of G: G = K ⊕ A ⊕ N,

N = N ±.

(2.15)

. Next let M be the centralizer of A in K, i.e. M = {X ∈ K | [X, Y ] = 0, ∀ Y ∈ A}. In general M is a compact reductive Lie algebra, and we can write M = . Ms ⊕ Ma , where Ms = [M, M] is the semisimple part of M, and Ma is the abelian subalgebra central in M. j The

results are easily extended to real linear reductive Lie groups with a finite number of components.

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We mention also that a Cartan subalgebra Hm of M is given by: Hm = Hs ⊕Ma , where Hs is a Cartan subalgebra of Ms . Then a Cartan subalgebra H of G is given by: H = Hm ⊕ A.k Next we recall the Bruhat decomposition [72]: G = N + ⊕ M ⊕ A ⊕ N −,

(2.16) . − and the subalgebra P0 = M ⊕ A ⊕ N called a minimal parabolic subalgebra of G. (Note that we may take equivalently N + instead of N − .) Naturally, the G-subalgebras K, A, N ± , M, Ms , Ma , P0 are the Lie algebras of the G-subgroups introduced in the previous subsection K, A, N ± , M, Ms , Ma , P0 , respectively. We mention an important class of real Lie algebras, the split real forms. For these we can use the same basis as for the corresponding complex simple Lie algebra G C , but over R. Restricting C → R one obtains the Bruhat decomposition of G (with M = 0) from the triangular decomposition of G C = G + ⊕ HC ⊕ G − , and obtains the minimal parabolic subalgebras P0 from the Borel subalgebra B = HC ⊕ G + , (or G − instead of G + ). Furthermore, in this case dimR K = dimR N ± . A standard parabolic subalgebra is any subalgebra P 0 of G containing P0 . The number of standard parabolic subalgebras, including P0 and G, is 2r . Remark. In the complex case a standard parabolic subalgebra is any subalgebra P 0 of G C containing B. The number of standard parabolic subalgebras, including B and G C , is 2` , ` = rankC G. Thus, if r = 1 the only nontrivial parabolic subalgebra is P0 . Thus, further in this section r > 1. Any standard parabolic subalgebra is of the form: P 0 = M0 ⊕ A0 ⊕ N 0− ,

(2.17)

so that M0 ⊇ M, A0 ⊆ A, N 0− ⊆ N − ; M0 is the centralizer of A0 in G (mod A0 ); N 0− is comprised from the negative root spaces of the restricted root system ∆A0 of (G, A0 ). The decomposition (2.17) is called the Langlands decomposition of P 0 . One also has the analogue of the Bruhat decomposition (2.16): G = N 0+ ⊕ A0 ⊕ M0 ⊕ N 0− , 0+

(2.18)

0−

where N = θN . The standard parabolic subalgebras may be described explicitly using the − + restricted simple root system ∆SA = ∆+ A ∪ ∆A , such that if λ ∈ ∆A , (respectively λ ∈ ∆− A ), one has: λ=

r X

n i λi ,

λi ∈ ∆SA ,

all ni ≥ 0,

(respectively all ni ≤ 0).

(2.19)

i=1

k Note that H is a θ-stable Cartan subalgebra of G such that H∩P = A. It is the most noncompact among the non-conjugate Cartan subalgebras of G.

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We shall follow Warner [7], where one may find all references to the original mathematical work on parabolic subalgebras. For a short formulation one may say that the parabolic subalgebras correspond to the various subsets of ∆SA — hence their number 2r . To formalize this, let us denote: Sr = {1, 2, . . . , r}, and let Θ denote any subset of Sr . Let ∆± Θ ∈ ∆A denote all positive/negative restricted roots which are linear combinations of the simple restricted roots λi , ∀ i ∈ Θ. Then a standard parabolic subalgebra corresponding to Θ will be denoted by PΘ and is given explicitly as: . λ PΘ = P0 ⊕ N + (Θ), N + (Θ) = ⊕ GA (2.20) . λ∈∆+ Θ

Clearly, P∅ = P0 , PSr = G, since N + (∅) = 0, N + (Sr ) = N + . Further, we need to bring (2.20) in the form (2.17). First, define G(Θ) as the algebra generated by . . N + (Θ) and N − (Θ) = θN + (Θ). Next, define A(Θ) = G(Θ) ∩ A, and AΘ as the orthogonal complement (relative to the Euclidean structure of A) of A(Θ) in A. Then A = A(Θ) ⊕ AΘ . Note that dim A(Θ) = |Θ|, dim AΘ = r − |Θ|. Next, define: . . λ , NΘ− = θNΘ+ . GA NΘ+ = ⊕ (2.21) + λ∈∆+ A −∆Θ

. Then N ± = N ± (Θ) ⊕ NΘ± . Next, define MΘ = M ⊕ A(Θ) ⊕ N + (Θ) ⊕ N − (Θ). Then MΘ is the centralizer of AΘ in G (mod AΘ ). Finally, we can derive: PΘ = P0 ⊕ N + (Θ) = M ⊕ A ⊕ N − ⊕ N + (Θ) = M ⊕ A(Θ) ⊕ AΘ ⊕ N − (Θ) ⊕ NΘ− ⊕ N + (Θ) = (M ⊕ A(Θ) ⊕ N − (Θ) ⊕ N + (Θ)) ⊕ AΘ ⊕ NΘ− = MΘ ⊕ AΘ ⊕ NΘ− .

(2.22)

Thus, we have rewritten explicitly the standard parabolic PΘ in the desired form (2.17). The associated (generalized) Bruhat decomposition (2.18) is given now explicitly as: G = N + ⊕ P0 = NΘ+ ⊕ N + (Θ) ⊕ P0 = NΘ+ ⊕ PΘ = NΘ+ ⊕ MΘ ⊕ AΘ ⊕ NΘ− .

(2.23)

Another important class are the maximal parabolic subalgebras which correspond to Θ of the form: Θmax = Sr \{j}, j

1 ≤ j ≤ r,

(2.24)

dim A(Θmax ) = r − 1, dim AΘmax = 1. j j Reminder 1. We recall for further use the fundamental result of Harish-Chandra [3] that G has discrete series representations iff rank G = rank K. Reminder 2. We recall for further use the well known fact that (G, K) is a Hermitian symmetric pair when the maximal compact subalgebra K contains a

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u(1) factor. Then G has highest and lowest weight representations. All these algebras have discrete series representations. 3. The Complex Simple Lie Algebras Considered as Real Lie Algebras Let Gc be a complex simple Lie algebra of dimension d and (complex) rank `. We need the triangular decomposition: Gc = N + ⊕ H ⊕ N − .

(3.1)

We have dimC Gc = d, rankC Gc = dimC H = `, dimC N ± = (d − `)/2. Considered as real Lie algebras we have: dimR Gc = 2d, rankR Gc = dimR H = 2`, dimR K = d, rankR K = `, dimR N ± = d − `. Note that the maximal compact subalgebra K of Gc is isomorphic to the compact real form Gk of Gc . Thus, the complex simple Lie algebras do not have discrete series representations (and highest/lowest weight representations over R). Let Hj , j = 1, . . . , `, be a basis of H, i.e. H = c.l.s. {Hj , j = 1, . . . , `}, (where c.l.s. stands for complex linear span), such that each ad(Hj ) has only real eigenval. ues. Let A = HR = r.l.s. {Hj , j = 1, . . . , `}, where r.l.s. stands for real linear span. Then the Iwasawa decomposition of Gc is: Gc = K ⊕ A ⊕ N ,

N = N ±.

(3.2)

The commutant M of A in K is given by: M = u(1) ⊕ · · · ⊕ u(1),

` factors.

(3.3)

In fact, the basis of M consists of the vectors {i Hj , j = 1, . . . , `}. The Bruhat decomposition of Gc is: Gc = N + ⊕ M ⊕ A ⊕ N − .

(3.4)

Comparing (3.1) and (3.4) we see that H = M ⊕ A.

(3.5)

The restricted root system (Gc , A) looks the same as the complex root system (Gc , H), but the restricted roots have multiplicity 2, since dimR N ± = 2 dimC N ± . Let Θ be a string subset of S` of length s. The MΘ -factor of the corresponding parabolic subalgebra is: MΘ = Gs ⊕ u(1) ⊕ · · · ⊕ u(1),

` − s factors,

(3.6)

where Gs is a complex simple Lie algebra of rank s isomorphic to a subalgebra of Gc . Thus, the complex simple Lie algebras, considered as real noncompact Lie algebras, do not have non-minimal cuspidal parabolic subalgebras. Thus, it is enough to consider elementary representations induced from the minimal parabolic subgroup P0 = MAN , where M ∼ = U (1) × · · · × U (1), (` factors),

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A ∼ = SO(1, 1) × · · · × SO(1, 1), (` factors), N ∼ = exp N ± .l Thus, the signature χ = [µ, ν], consists of ` integer numbers µi ∈ Z giving the unitary character µ = (µ1 , . . . , µ` ) of M , and of ` complex numbers νi ∈ C giving the charP acter ν = (ν1 , . . . , ν` ) of A, νj = ν(Hj ). Thus, if H = j σj Hj , σj ∈ R, is a generic element of A, then for the corresponding factor in (2.1) we have P P eν(H) = exp j σj νj . Analogously, if m = exp i j φj Hj ∈ M , φj ∈ R, then we P have Dµ (m−1 ) = exp i j φj µj . Thus, the right covariance property (2.1) becomes: X F(gman) = exp (σj νj + iφj µj ) · F(g). (3.7) j

To relate with the general setting of the previous subsection we must introduce ˜ j . Let us use (3.5) ˜ such that Λ(Hj ) = λj , Λ(H ˜ j) = λ two weight functionals: Λ, Λ, P and H = j (σj + iφj )Hj ∈ H. Thus the elementary representations (in particular, the right covariance conditions) for a complex semisimple Lie group Gc are given by: ½ ˜ H)) ¯ · F(g) CΛ,Λ˜ = F ∈ C ∞ (Gc ) | F(gman) = exp(Λ(H) + Λ( ¾ X ˜ j ) · F(g) , (3.8) = exp ((σj + iφj )λj + (σj − iφj )λ j

˜j , νj = λj + λ

˜j ∈ Z µ j = λj − λ

and the last condition in (3.8) stresses that we have uniqueness on the compact subgroup M of the Cartan subgroup Hc = M A of Gc . ˜ = 0 are called holomorphic, and those for which Λ = 0 are The ERs for which Λ called antiholomorphic. Thus, we see that the complex case is richer than the real one. Indeed, there are two Verma modules associated with an ER, one “holomorphic” V Λ and one ˜ ˜ “antiholomorphic” V Λ . The ER is reducible when either V Λ or V Λ are reducible, ˜ i.e. when (2.8) holds for either Λ or Λ. More information can be found in [9] from where we mention some important statements: All irreducible representations of a complex semisimple Lie group are obtained as subrepresentations of the elementary representations induced from the minimal parabolic subgroup. All finite-dimensional irreps are obtained as subrep˜ j ∈ Z+ . resentations when all λj , λ The maximal parabolic subalgebras have MΘ -factors as follows MΘ = Gi ⊕ u(1),

i = 1, . . . , `,

(3.9)

where Gi is a complex simple Lie algebra of rank ` − 1 which may be obtained from Gc by deleting the ith node of the Dynkin diagram of Gc . l We

should note that the minimal parabolic subgroup P0 is isomorphic to a Borel subgroup of Gc , due to the obvious isomorphism between the abelian subgroup MA and the Cartan subgroup H of Gc .

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4. AI: SL(n, R) In this section G = SL(n, R), the group of invertible n × n matrices with real elements and determinant 1. Then G = sl(n, R) and the Cartan involution is given explicitly by: θX = − t X, where t X is the transpose of X ∈ G. Thus, K ∼ = so(n), and is spanned by matrices (r.l.s. stands for real linear span): K = r.l.s.{Xij ≡ eij − eji , 1 ≤ i < j ≤ n},

(4.1)

where eij are the standard matrices with only nonzero entry (= 1) on the ith row and jth column, (eij )k` = δik δj` . Note that G does not have discrete series representations if n > 2. Indeed, the rank of sl(n, R) is n − 1, and the rank of its maximal compact subalgebra so(n) is [n/2] and the latter is smaller than n − 1 unless n = 2. Further, the complementary space P is given by: P = r.l.s.{Yij ≡ eij + eji , 1 ≤ i < j ≤ n, Hj ≡ ejj − ej+1,j+1 , 1 ≤ j ≤ n − 1}. (4.2) The split rank is r = n − 1, and from (4.2) it is obvious that in this setting one has: A = r.l.s.{Hj , 1 ≤ j ≤ n − 1 = r}.

(4.3)

Since G is a maximally split real form of G C = sl(n, C), then M = 0, and the minimal parabolic subalgebra and the Bruhat decomposition, respectively, are given as a Borel subalgebra and triangular decomposition of G C , but over R: G = N + ⊕ A ⊕ N −,

P0 = A ⊕ N − ,

(4.4)

where N + , N − , respectively, are upper, lower, triangular, respectively: N + = r.l.s.{eij , 1 ≤ i < j ≤ n},

N − = r.l.s.{eij , 1 ≤ j < i ≤ n}.

(4.5)

The simple root vectors are given explicitly by: . Xj+ = ej,j+1 ,

. Xj− = ej+1,j ,

1 ≤ j ≤ n − 1 = r.

(4.6)

Note that matters are arranged so that [Xj+ , Xj− ] = Hj ,

[Hj , Xj± ] = ±2Xj± ,

(4.7)

and further we shall denote by sl(2, R)j the sl(2, R) subalgebra of G spanned by Xj± , Hj . The parabolic subalgebras may be described by the unordered partitions of n.m Ps . Explicitly, let ν¯ = {ν1 , . . . , νs }, s ≤ n, be a partition of n: j=1 νj = n. Then the m The

parabolic subalgebras may also be described by the various flags of Rn , F., e.g., [7], but we shall not use this description.

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set Θ corresponding to the partition ν¯ and denoted by Θ(¯ ν ) consists of the numbers of the entries νj that are bigger than 1: Θ(¯ ν ) = {j | νj > 1}.

(4.8)

Note that in the case s = n all νj are equal to 1 — this is the partition ν¯0 = {1, . . . , 1} corresponding to the empty set: Θ(¯ ν0 ) = ∅ (corresponding to the minimal parabolic). Then the factor MΘ(¯ν ) in (2.22) and (2.23) is: MΘ(¯ν ) =

⊕ 1≤j≤s νj >1

sl(νj , R) =

⊕

1≤j≤s

sl(νj , R),

sl(1, R) ≡ 0.

(4.9)

Certainly, some partitions give isomorphic (though non-conjugate!) MΘ(¯ν ) subalgebras. The parabolic subalgebras in these cases are called associated, and this is an equivalence relation. The parabolic subalgebras up to this equivalence relation correspond to the ordered partitions of n. The most important for us cuspidal parabolic subalgebras correspond to those partitions ν¯ = {ν1 , . . . , νs } for which νj ≤ 2, ∀ j. Indeed, if some νj > 2 then MΘ(¯ν ) will not have discrete series representations since it contains the factor sl(νj , R). A more explicit description of the cuspidal cases is given as follows. It is clear that the cuspidal parabolic subalgebras are in one-to-one correspondence with the sequences of r numbers: . n ¯ = {n1 , . . . , nr }, (4.10) such that nj = 0, 1, and if for fixed j we have nj = 1, then nj+1 = 0, (clearly from the latter follows also nj−1 = 0, but we shall use this notation also in other contexts). In the language above to each nj = 1 there is an entry νj = 2 in ν¯ bringing an sl(2, R) factor to MΘ , i.e. Θ(¯ n) = {j | nj = 1, nj+1 = 0}.

(4.11)

More explicitly, the cuspidal parabolic subalgebras are given as follows: MΘ(¯n) =

⊕

1≤t≤k

sl(2, R)jt ,

njt = 1,

1 ≤ j1 < j2 < · · · < jk ≤ r,

jt < jt+1 − 1. (4.12)

± The corresponding AΘ(¯n) and NΘ(¯ n) have dimensions:

dim AΘ(¯n) = n − 1 − k,

± dim NΘ(¯ n) =

1 n(n − 1) − k, 2

(4.13)

where k = |Θ(¯ n)| was introduced in (4.12). Note that the minimal parabolic subalgebra is obtained when all nj = 0, n ¯0 = {0, . . . , 0}, then Θ(¯ n0 ) = ∅, MΘ(¯n0 ) = 0, k = 0. Interlude. The number of cuspidal parabolic subalgebras of sl(n, R), n ≥ 2, including also the case P = M0 = sl(n, R) when n = 2, is equal to F (n + 1), where F (n), n ∈ Z+ , are the Fibonacci numbers.

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Proof. First we recall that the Fibonacci numbers are determined through the relations F (m) = F (m − 1) + F (m − 2), m ∈ 2 + Z+ , together with the boundary values: F (0) = 0, F (1) = 1. We shall count the number of sequences of r numbers ni , introduced above (r = n − 1). Let us denote by N (r) the number of the abovedescribed sequences. Let us divide these sequences in two groups: the first with n1 = 1 and the others with n1 = 0. Obviously the number of sequences with n1 = 1 is equal to the N (r − 2) since n2 = 0, and then we are left with the above-described sequences but of r − 2 numbers. Analogously, the number of sequences with n1 = 0 is equal to the N (r − 1) since we are left with all above-described sequences of r − 1 numbers. Thus, we have proved that N (r) = N (r − 1) + N (r − 2). This is the Fibonacci recursion relation and we have only to adjust the boundary conditions. We have N (1) = 2, N (2) = 3, i.e. N (r) = F (r + 2), or in terms of n = r + 1: N (n − 1) = F (n + 1). For further use we recall that there is explicit formula for the Fibonacci numbers: ¶ [(n−1)/2] µ X n xn − (1 − x)n 1−n √ 5s , F (n) = =2 (4.14) 2s + 1 5 s=0 √ where x is the golden ratio: x2 = x + 1, i.e. x = (1 ± 5)/2. Finally, we mention that the maximal parabolic subalgebras corresponding to Θ from (2.24) have the following factors: MΘj = sl(j, R) ⊕ sl(n − j, R), dim AΘj = 1,

dim

NΘ±j

1 ≤ j ≤ n − 1, = j(n − j).

(4.15)

(Note that the cases j and n − j are isomorphic, or coinciding when n is even and j = 12 n.) Only one of the maximal ones is cuspidal, namely, for G = sl(4, R), n = 4 and j = 2 we have MΘ2 = sl(2, R) ⊕ sl(2, R).

(4.16)

5. AII: SU ∗ (2n) The group G = SU ∗ (2n), n ≥ 2, consists of all matrices in SL(2n, C) which commute with a real skew-symmetric matrix times the complex conjugation operator C: ½ µ ¶¾ 0 1n . SU ∗ (2n) = g ∈ SL(2n, C) | Jn Cg = gJn C, Jn ≡ . (5.1) −1n 0 The Lie algebra G = su∗ (2n) is given by: . su∗ (2n) = {X ∈ sl(2n, C) | Jn CX = XJn C} !¯ ( Ã ) a b ¯¯ = X= ¯) = 0 , ¯ a, b ∈ gl(n, C), tr(a + a −¯b a ¯ ¯ dimR G = 4n2 − 1.

(5.2)

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We consider n ≥ 2 since su∗ (2) ∼ = su(2), and we note that the case n = 2 (of split rank 1) will appear also below: su∗ (4) ∼ = so(5, 1), cf. the corresponding section. The Cartan involution is given by: θX = −X † . Thus, K ∼ = sp(n): ( Ã !¯ ) ¯ a b ¯ † t K= X= (5.3) ¯ a, b ∈ gl(n, C), a = −a, b = b . −b† − t a ¯ Note that su∗ (2n) does not have discrete series representations (rank K = n < rank su∗ (2n) = 2n − 1). The complimentary space P is given by: ( Ã !¯ ) a b ¯¯ † t P = X = † t ¯ a, b ∈ gl(n, C), a = a, b = −b, tr a = 0 . (5.4) b a ¯ The split rank is n − 1 and the abelian subalgebra A is given explicitly by: ( Ã !¯ ) a 0 ¯¯ A= X= ¯ a = diag(a1 , . . . , an ), aj ∈ R, tr a = 0 . 0 a ¯

(5.5)

The subalgebras N ± which form the root spaces of the root system (G, A) are of real dimension 2n(n − 1). The subalgebra M is given by: ( Ã !¯ a b ¯¯ M= X= ¯ a = i diag(φ1 , . . . , φn ), φj ∈ R, −¯b −a ¯ ) (5.6) b = diag(b1 , . . . , bn ), bj ∈ C ∼ = su(2) ⊕ · · · ⊕ su(2),

n factors.

Claim. All non-minimal parabolic subalgebras of su∗ (2n) are not cuspidal. Proof. Necessarily n > 2. Let Θ enumerate a connected string of restricted simple roots: Θ = Sij = {i, . . . , j}, where 1 ≤ i ≤ j < n. Then the corresponding subalgebra MΘ is given by: Mij = su∗ (2(s + 1)) ⊕ su(2) ⊕ · · · ⊕ su(2),

n − s − 1 factors,

s ≡ j − i + 1. (5.7)

In general Θ consists of such strings, each string of length s produces a factor su∗ (2(s + 1)), the rest of MΘ consists of su(2) factors. The maximal parabolic subalgebras, cf. (2.24), 1 ≤ j ≤ n − 1, contain MΘ subalgebras of the form: Mmax = su∗ (2j) ⊕ su∗ (2(n − j)). j

(5.8)

(For j = 1 or j = n − 1 (8.8) coincides with (5.7) for s = n − 2 (and using su∗ (2) ∼ = su(2)).

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6. AIII, AIV: SU (p, r) In this section G = SU (p, r), p ≥ r, which standardly is defined as follows: ( Ã ! ) 1p 0 . † SU (p, r) = g ∈ GL(p + r, C) | g β0 g = β0 , β0 ≡ , det g = 1 , 0 −1r

(6.1)

where g † is the Hermitian conjugate of g. We shall use also another realization of G differing from (6.1) by unitary transformation:     1p−r 0 0 1p−r 0 0 1     β0 7→ β2 ≡  0 (6.2) 0 1r  = U β0 U −1 , U ≡ √  0 1 r 1r  . 2 0 1r 0 0 −1r 1r The Lie algebra G = su(p, r) is given by (β = β0 , β2 ): . su(p, r) = {X ∈ gl(p + r, C) | X † β + β X = 0, tr X = 0}.

(6.3)

The Cartan involution is given explicitly by: θX = βXβ. Thus, K ∼ = u(1) ⊕ su(p) ⊕ su(r), and more explicitly is given as (β = β0 ): ( Ã !¯ ) u1 0 ¯¯ † K= X= (6.4) ¯ uj = −u, j = 1, 2; tr u1 + tr u2 = 0 . 0 u2 ¯ Note that su(p, r) has discrete series representations since rank K = 1 + rank su(p) + rank su(r) = p + r − 1 = rank su(p, r), and highest/lowest weight representations. The split rank is equal to r and the abelian subalgebra A may be given explicitly by (β = β2 ): A = r.l.s.{Hju ≡ ep−r+j,p−r+j − ep+j,p+j , 1 ≤ j ≤ r}.

(6.5)

At this moment we need to consider the cases p = r and p > r separately, since the minimal parabolic subalgebras are different. 6.1. The case SU (n, n), n > 1 In this subsection G = SU (n, n). We consider n > 1 since SU (1, 1) ∼ = SL(2, R), which was already treated. The subalgebra M ∼ = u(1) ⊕ · · · ⊕ u(1), (n − 1 factors), and is explicitly given as (β = β2 ): !¯ ( Ã ) u 0 ¯¯ M= X= (6.6) ¯ u = i diag(φ1 , . . . , φn ), φj ∈ R; tr u = 0 . 0 u ¯ The subalgebras N ± which form the root spaces of the root system (G, A) are of real dimension n(2n − 1). The simple root system (G, A) looks as that of the symplectic algebra Cn , however, the root spaces of the short roots have multiplicity 2.

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Further, we choose the long root of the Cn simple root system to be αn . Claim. The nontrivial cuspidal parabolic subalgebras are given by Θ of the form: Θj = {j + 1, . . . , n},

1 ≤ j < n.

(6.7)

Proof. First note that we exclude j = 0 since Θ0 = Sn . Consider now any Θ which contains a subset Sij = {i, . . . , j}, where 1 ≤ i ≤ j < n. Then the simple roots corresponding to Sij form a string subset πij of the simple root system of An−1 , but each root has multiplicity 2. Because of this multiplicity this string of simple roots will produce a subalgebra sl(j − i + 2, C) of MΘ . Since the simple Lie algebras sl(n, C) do not have discrete series representations, then PΘ is not cuspidal. Now it remains to note that PΘj is cuspidal for all j since MΘj ∼ = su(n − j, n − j) ⊕ u(1) ⊕ · · · ⊕ u(1),

j factors

(6.8)

cf. the remark above. , j = 1, . . . , n, The maximal parabolic subalgebras correspond to the sets Θmax j cf. (2.24). The corresponding MΘ subalgebras are of the form: Mmax = sl(j, C) ⊕ su(n − j, n − j) ⊕ u(1) ⊕ · · · ⊕ u(1), j

j factors,

(6.9)

= Θ1 and that the only where we use the convention: sl(1, C) = 0. Note that Θmax 1 cuspidal maximal parabolic subalgebra is PΘ1 . 6.2. The case SU (p, r), p > r ≥ 1 In this subsection G = SU (p, r). We include also the case r = 1 although we noted that the case of split rank 1 is clear in general. The subalgebra M ∼ = su(p − r) ⊕ u(1) ⊕ · · · ⊕ u(1), (r factors), and is explicitly given as (β = β2 ):  ¯  up−r 0 0 ¯¯    ¯ M= X= 0 u 0 ¯ u†p−r = −up−r , u = i diag(φ1 , . . . , φn ), φj ∈ R,  ¯  0 0 u ¯    tr up−r + 2tr u = 0

 

.

(6.10)

The subalgebras N ± which form the root spaces of the root system (G, A) are of real dimension r(2p − 1). The restricted simple root system (G, A) looks as that of the orthogonal algebra Br , however, the root spaces of the long roots have multiplicity 2, the short simple root, say αr , has multiplicity 2(p − r), and there is also a root 2αr with multiplicity 1.

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Similarly to the su(n, n) case one can prove that the nontrivial cuspidal parabolic subalgebras are given by Θ of the form: Θj = {j + 1, . . . , r},

1 ≤ j < r,

r > 1.

(6.11)

The corresponding cuspidal parabolic subalgebras contain the subalgebras MΘj ∼ = su(p − j, r − j) ⊕ u(1) ⊕ · · · ⊕ u(1),

j factors.

(6.12)

The maximal parabolic subalgebras, (cf. (2.24)), contain the MΘ subalgebras are of the form: Mmax = sl(j, C) ⊕ su(p − j, r − j) ⊕ u(1) ⊕ · · · ⊕ u(1), j

j factors.

(6.13)

Thus, the only cuspidal maximal parabolic subalgebra is PΘ1 . 7. BDI, BDII: SO(p, r) In this section G = SO(p, r), p ≥ r, which standardly is defined as follows: ( Ã !) 1p 0 . † SO(p, r) = g ∈ SO(p + r, C) | g β0 g = β0 , β0 ≡ , 0 −1r

(7.1)

where g † is the Hermitian conjugate of g. We shall use also another realization of G differing from (7.1) by unitary transformation:     1p−r 0 0 1p−r 0 0 1     β0 7→ β2 ≡  0 (7.2) 0 1r  = U β0 U −1 , U ≡ √  0 1r 1r  . 2 0 1r 0 0 −1r 1r The Lie algebra G = so(p, r) is given by (β = β0 , β2 ): . so(p, r) = {X ∈ so(p + r, C) | X † β + β X = 0}.

(7.3)

The Cartan involution is given explicitly by: θX = βXβ. Thus, K ∼ = so(p) ⊕ so(r), and more explicitly is given as (β = β0 ): ( Ã !¯ ) u1 0 ¯¯ K= X= (7.4) ¯ u1 ∈ so(p), u2 ∈ so(r) . 0 u2 ¯ Note that so(5, 1) ∼ = su∗ (4), so(4, 2) ∼ = su(2, 2), so(3, 3) ∼ = sl(4, R), so(4, 1) ∼ = ∼ ∼ sp(1, 1), so(3, 2) ∼ sp(2, R), so(3, 1) sl(2, C), so(2, 2) sl(2, R) ⊕ sl(2, R), = = = so(2, 1) ∼ = sl(2, R), (so(1, 1) is abelian). Thus, below we can restrict to p + r > 4, since the cases p + r = 5 are not treated yet. Note that so(p, r) has discrete series representations except when both p, r are odd numbers, since then rank K = rank so(p) + rank so(r) = 21 (p + r − 2) < rank so(p, r) = 12 (p + r). It has highest/lowest weight representations when p ≥ r = 2 and p = 2, r = 1.

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The split rank is equal to r and the abelian subalgebra A may be given explicitly by (β = β2 ): A = r.l.s.{Hju ≡ ep−r+j,p−r+j − ep+j,p+j , 1 ≤ j ≤ r}. The subalgebra M ∼ = so(p − r) and is explicitly given as (β = β2 ): ( Ã !¯ ) u 0 ¯¯ M= X= ¯ u ∈ so(p − r) . 0 0 ¯

(7.5)

(7.6)

The subalgebras N ± which form the root spaces of the root system (G, A) are of real dimension r(p − 1). Except in the case p = r the restricted simple root system (G, A) looks as that of the orthogonal algebra Br , however, the short simple root, say αr , has multiplicity p − r. Thus, we consider first the case p > r > 1. First we note that the parabolic subalgebras given by Θj = {j}, j < r contain a factor: MΘj = sl(2, R) ⊕ so(p − r). More generally, if r ∈ / Θ then all possible cuspidal parabolic subalgebras are like those of sl(r, R), adding the compact subalgebra so(p−r). Suppose now, that r ∈ Θ. In that case, Θ will include a set Θj of the form: Θj = {j + 1, . . . , r},

1 ≤ j < r.

(7.7)

That would bring a MΘ factor of the form so(p−j, r−j). Thus, all possible cuspidal parabolic subalgebras are obtained for those j, for which the number (p − j)(r − j) is even and for fixed such j they would be like those of sl(j, R), adding the noncompact subalgebra so(p − j, r − j). Clearly, if both p, r are even (odd), then also j must be even (odd), while if one of p, r is even and the other odd, i.e. p + r is odd, then j takes all values from (7.7). To be more explicit we first introduce the notation: . n ¯ s = {n1 , . . . , ns }, 1 ≤ s ≤ r, (7.8) (note that n ¯r = n ¯ from (4.10)). Then we shall use the notation introduced for the sl(n, R) case, namely, Θ(¯ ns ) from (4.11). Then the cuspidal parabolic subalgebras are given by the noncompact factors MΘ from (4.12):   s = 1, 2, . . . , r − 1 p + r odd Ms = MΘ(¯ns ) ⊕ so(p − s, r − s), (7.9) s = 2, 4, . . . , r − 2 p, r even .   s = 1, 3, . . . , r − 2 p, r odd Next we note that we can include the case when the second factor in MΘ is compact by just extending the range of s to r. Thus, all cuspidal parabolic subalgebras of so(p, r) in the case p > r will be determined by the following MΘ subalgebras:   s = 1, 2, . . . , r p + r odd Ms = MΘ(¯ns ) ⊕ so(p − s, r − s), (7.10) s = 2, 4, . . . , r p, r even .   s = 1, 3, . . . , r p, r odd

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The algebras MΘ have highest/lowest weight representations only when s = r − 2 or s = r, since then the second factor is so(p − r + 2, 2), so(p − r), respectively. Finally, we note that the maximal parabolic subalgebras corresponding to (2.24) have MΘ -factors given by: Mmax = sl(j, R) ⊕ so(p − j, r − j), j

j = 1, 2, . . . , r.

(7.11)

Thus, the maximal parabolic subalgebras are cuspidal (and can be found in (7.10)) when j = 1, 2 and the number (p − j)(r − j) is even. In addition, Mmax have j highest/lowest weight representations only when r − j = 0, 2, (or p − j = 2). Now we consider the split cases p = r ≥ 4. (Note that the other split-real cases, i.e. when p = r + 1, were considered above without any peculiarities. The split cases p = r < 4 are not representative of the situation and were treated already: so(3, 3) ∼ = sl(4, R), so(2, 2) ∼ = so(2, 1) ⊕ so(2, 1), so(1, 1) is not semisimple.) We accept the convention that the simple roots αr−1 and αr form the fork of the so(2r, C) simple root system, while αr−2 is the simple root connected to the simple roots αr−3 , αr−1 and αr . Special care is needed only when Θ includes these four special roots, i.e., . ˆs = Θ⊃Θ {s, . . . , r}, 1 ≤ s ≤ r − 4. (7.12) In these cases, we have M factor of the form so(r − s, r − s), i.e. there will be no cuspidal parabolic if r − s is odd. For all other Θ the parabolic subalgebras would be like those of sl(r, R), when r ∈ / Θ or r − 1 ∈ / Θ, or like those of sl(r − 2, R) with possible addition of one or two sl(2, R) factors, (when r − 2 ∈ / Θ). To describe the latter cases we need a modification of the notation (7.8): Θo (¯ n) = {j | nj = 1, nj+1 = 0 if j 6= r − 1}.

(7.13)

Thus, the cuspidal parabolic subalgebras are determined by the following MΘ factors:  ½ s = 2, 4, . . . , r − 4 r even  MΘ(¯n ) ⊕ so(r − s, r − s), Θ ⊃ Θ ˆ s, s s = 1, 3, . . . , r − 4 r odd . MΘ =   ˆs MΘo (¯n) , Θ⊃ /Θ (7.14) Only the second subcase, namely, MΘo (¯n) , has highest/lowest weight representations. The maximal parabolic subalgebras corresponding to (2.24) have MΘ -factors given by:  sl(r, R) j = r − 1, r     sl(r − 2, R) ⊕ sl(2, R) ⊕ sl(2, R) j = r − 2 Mmax = (7.15) . j  sl(r − 3, R) ⊕ sl(4, R) j =r−3     sl(j, R) ⊕ so(r − j, r − j) j ≤r−4

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Thus the maximal parabolic subalgebras which are cuspidal occur for j = 1 and odd r ≥ 5, (4th case), or j = 2 and either r = 4, (2nd case), or even r ≥ 6, (4th case): Mmax = so(r − 1, r − 1), r = 5, 7, . . . , 1 ( sl(2, R) ⊕ sl(2, R) ⊕ sl(2, R) r = 4, Mmax = 2 sl(2, R) ⊕ so(r − 2, r − 2) r = 6, 8, . . . .

(7.16)

Of these, only Mmax for r = 4 has highest/lowest weight representations (it belongs 2 to the second subcase of (7.14)). 8. CI: Sp(n, R), n > 1 In this section G = Sp(n, R) — the split real form of Sp(n, C). Both are standardly defined by: . Sp(n, F ) = {g ∈ GL(2n, F ) | t gJn g = Jn , det g = 1},

F = R, C.

(8.1)

Correspondingly, the Lie algebras are given by: sp(n, F ) = {X ∈ gl(2n, F ) | t XJn + Jn X = 0}.

(8.2)

Note that dimF sp(n, F ) = n(2n + 1). The general expression for X ∈ sp(n, F ) is Ã ! A B X= , A, B, C ∈ gl(n, F ), t B = B, t C = C. (8.3) C − tA A basis of the Cartan subalgebra H of sp(n, C) is: Ã ! Ai 0 Hi = , i = 1, . . . , n − 1, Ai = diag(0, . . . 0, 1, −1, 0, . . . , 0), 0 −Ai (8.4) Ã ! A0n 0 Hn = , A0n = (0, . . . , 0, 2). 0 −A0n The same basis over R spans the subalgebra A of G = sp(n, R), since rankF sp(n, F ) = n. Note that sp(2, R) ∼ = so(3, 2), sp(1, R) ∼ = sl(2, R). The maximal compact subalgebra of G = sp(n, R) is K ∼ = u(n), thus sp(n, R) has discrete series representations (and highest/lowest weight representations). Explicitly, ( Ã !¯ ) ¯ A 0 ¯ K= X= (8.5) ¯ A ∈ u(n) . 0 − tA ¯ The subalgebras N ± which form the root spaces of the root system (G, A) are of real dimension n2 . Further, we choose the long root of the Cn simple root system to be αn .

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The parabolic subalgebras corresponding to Θ such that n ∈ / Θ are the same as the parabolic subalgebras of sl(n, R). The parabolic subalgebras corresponding to Θ such that n ∈ Θ contain a string Θ0s = {s + 1, . . . , n}. This string brings in MΘ a factor sp(n − s, R), which has discrete series representations. Thus cuspidality depends on the rest of the possible choices and are the same as the parabolic subalgebras of sl(j, R). Thus, we have: Θs = Θ(¯ ns−1 ) ∪ Θ0s ,

s = 1, . . . , n,

(8.6)

Θ0n

with the convention that Θ(¯ n0 ) = ∅, = ∅. Then the MΘ -factors of the cuspidal parabolic subalgebras of sp(n, R) are given as follows: MΘs = MΘ(¯ns−1 ) ⊕ sp(n − s, R),

s = 1, . . . , n.

(8.7)

The minimal parabolic subalgebra for which MΘ = 0 is obtained for s = n since then MΘ(¯nn−1 ) enumerates all cuspidal parabolic subalgebras of sl(n, R), including the minimal case MΘ = 0. The maximal parabolic subalgebras, cf. (2.24), 1 ≤ j ≤ n, contain MΘ subalgebras of the form: = sl(j, R) ⊕ sp(n − j, R), Mmax j

(8.8)

i.e. the only maximal cuspidal are those for j = 1, 2. 9. CII: Sp(p, r) In this section G = Sp(p, r), p ≥ r, which standardly is defined as follows: Ã ! β0 0 . † Sp(p, r) = {g ∈ Sp(p + r, C) | g γ0 g = γ0 }, γ0 = , 0 β0 and correspondingly the Lie algebra G = sp(p, r) is given by . sp(p, r) = {X ∈ sp(p + r, C) | X † γ0 + γ0 X = 0}.

(9.1)

(9.2)

The Cartan involution is given explicitly by: θX = γ0 Xγ0 . Thus, K ∼ = sp(p) ⊕sp(r), and G has discrete series representations (but not highest/lowest weight representations). More explicitly: 8 > > > > < K=

0

u1

B 0 B X=B B−v † > > @ 1 > > : 0

0

0

u2

0

0

t

− u1

−v2†

0

9 1˛ ˛ > ˛ > > ˛ > = v2 C C˛ t t C˛ u1 ∈ u(p), u2 ∈ u(r), v1 = v1 , v2 = v2 . C ˛ > 0 A˛ > > ˛ > ; t ˛ − u2 0

(9.3) The split rank is equal to r and the abelian subalgebra A may be given explicitly by: A = r.l.s.{Hjs ≡ ep−r+j,p−r+j − ep+j,p+j − e2p+j,2p+j + e2p+r+j,2p+r+j , 1 ≤ j ≤ r}.

(9.4)

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The subalgebras N ± which form the root spaces of the root system (G, A) are of real dimension r(4p − 1). The subalgebra M ∼ = sp(p − r) ⊕ sp(1) ⊕ · · · ⊕ sp(1), r factors. Here, just for a moment we distinguish the cases p > r and p = r, since the restricted root systems are different. When p > r the restricted simple root system (G, A) looks as that of Br , however, the short simple root say, αr , has multiplicity 4(p − r), the long roots have multiplicity 4. When p = r > 1 the restricted simple root system (G, A) looks as that of Cr , however, the long root say, αr , has multiplicity 3, the short roots have multiplicity 4. (We consider r > 1 since sp(1, 1) ∼ = so(4, 1).) In spite of these differences from now on we can consider the two subcases together, i.e. we take p ≥ r. There are two types of parabolic subalgebras depending on whether r ∈ / Θ or r ∈ Θ. Let r ∈ / Θ. Then the parabolic subalgebras are like those of su∗ (2n). Let Θ enumerate a connected string of restricted simple roots: Θ = Sij = {i, . . . , j}, where 1 ≤ i ≤ j < r. Then the corresponding subalgebra MΘ is given by: Mij = su∗ (2(s + 1)) ⊕ sp(1) ⊕ · · · ⊕ sp(1),

r − s − 1 factors,

s ≡ j − i + 1. (9.5)

In general Θ consists of such strings, each string of length s produces a factor su∗ (2(s + 1)), the rest of MΘ consists of sp(1) ∼ = su(2) factors. All these parabolic subalgebras are not cuspidal. Let r ∈ Θ and consider the various strings containing r: Θj = {j + 1, . . . , r},

1 ≤ j < r.

(9.6)

The corresponding factor in MΘ is given by algebra sp(p − j, r − j) which has discrete series representations. If Θ contains in addition some other string then it would bring some su∗ (2(s + 1)) factor and the corresponding MΘ will not have discrete series representations. Thus the nontrivial cuspidal parabolic subalgebras are given by Θj from (9.6) and the corresponding MΘ is: Mj ∼ = sp(p − j, r − j) ⊕ sp(1) ⊕ · · · ⊕ sp(1),

j factors.

(9.7)

All these Mj do not have highest/lowest weight representations. The other factors in the cuspidal parabolic subalgebras have dimensions: dim Aj = j, dim Nj± = j(4p + 4r − 4j − 1). Extending the range of j we include the minimal parabolic subalgebra for j = r and the case M0 = P = G for j = 0. The maximal parabolic subalgebras corresponding to (2.24) have MΘ -factors given by: Mmax = su∗ (2j) ⊕ sp(p − j, r − j), j

1 ≤ j ≤ r.

(9.8)

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The N ± factors in the maximal parabolic subalgebras have dimensions: dim (Nj± )max = j(4p+4r −6j +1). The only cuspidal maximal parabolic subalgebra is PΘ1 , using su∗ (2) ∼ = su(2) ∼ = sp(1) and noting that (9.7) and (9.8) coincide for j = 1. 10. DIII: SO ∗ (2n) The group G = SO∗ (2n) consists of all matrices in SO(2n, C) which commute with a real skew-symmetric matrix times the complex conjugation operator C: . SO∗ (2n) = {g ∈ SO(2n, C) | Jn Cg = gJn C}. (10.1) The Lie algebra G = so∗ (2n) is given by: . so∗ (2n) = {X ∈ so(2n, C) | Jn CX = XJn C} ( Ã !¯ ) a b ¯¯ = X= ¯ a, b ∈ gl(n, C), t a = −a, b† = b , −¯b a ¯ ¯

(10.2)

dimR G = n(2n − 1), rank G = n. Note that so∗ (8) ∼ = = so(3)⊕so(2, 1), so∗ (2) ∼ = su(3, 1), so∗ (4) ∼ = so(6, 2), so∗ (6) ∼ so(2). Further, we can restrict to n ≥ 4 since the other cases are not representative. The Cartan involution is given by: θX = −X † . Thus, K ∼ = u(n): ( Ã !¯ ) a b ¯¯ t † ¯ K= X= (10.3) a, b = b = b , ¯ a, b ∈ gl(n, C), a = −a = −¯ −b a ¯ and G = so∗ (2n) has discrete series representations (and highest/lowest weight representations). The complimentary space P is given by: ( Ã !¯ ) a b ¯¯ t † ¯ P= X= (10.4) ¯ , b = b = −b , ¯ a, b ∈ gl(n, C), a = −a = a b −a ¯ dimR P = n(n − 1). The split rank is r ≡ [n/2]. The subalgebras N ± which form the root spaces of the root system (G, A) are of real dimension n(n − 1) − [n/2]. Here, just for a moment we distinguish the cases n even and n odd since the subalgebras M and the restricted root systems are different. For n = 2r the split rank is equal to r ≥ 2, and the restricted root system is as that of Cr , but the short roots have multiplicity 4, while the long simple root αr has multiplicity 1. The subalgebra M ∼ = so(3) ⊕ · · · ⊕ so(3), r factors. For n = 2r + 1 the split rank is equal to r ≥ 2, and the restricted root system is as that of Br , but all simple roots have multiplicity 4, and there is a restricted root 2αr of multiplicity 1, where αr is the short simple root. The subalgebra M ∼ = so(2) ⊕ so(3) ⊕ · · · ⊕ so(3), r factors. In spite of these differences from now on, we can consider the two subcases together.

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There are two types of parabolic subalgebras depending on whether r ∈ / Θ or r ∈ Θ. If r ∈ / Θ then the parabolic subalgebras are like those of su∗ (2r), and they are not cuspidal. Let r ∈ Θ and consider the various strings containing r: Θj = {j + 1, . . . , r},

1 ≤ j < r = [n/2].

(10.5)

The corresponding factor in MΘ is given by the algebra so∗ (2n − 4j) which has discrete series representations. Thus, all cuspidal parabolic subalgebras are enumerated by (10.5) and are: Mj = so∗ (2n − 4j) ⊕ so(3) ⊕ · · · ⊕ so(3),

j factors,

j = 1, . . . , r − 1.

(10.6)

All these Mj have highest/lowest weight representations. The other factors in the cuspidal parabolic subalgebras have dimensions: dim Aj = j, dim Nj± = j(4n − 4j − 3). Extending the range of j we include the minimal parabolic case for j = r = [n/2], and the case M0 = P = G for j = 0 which is also cuspidal. from (2.24) have The maximal parabolic subalgebras enumerated by Θmax j MΘ -factors as follows: Mmax = so∗ (2n − 4j) ⊕ su∗ (2j), j

j = 1, . . . , r.

(10.7)

The N ± factors in the maximal parabolic subalgebras have dimensions: dim (Nj± )max = j(4n − 6j − 1). Only the case j = 1 is cuspidal, noting that Mmax 1 coincides with M1 from (10.6), (su∗ (2) ∼ = so(3)). = su(2) ∼ 11. Real Forms of the Exceptional Simple Lie Algebras We start with the real forms of the exceptional simple Lie algebras. Here we cannot be so explicit with the matrix realizations. To compensate this we use the Satake diagrams [7, 73], which we omitted until now.n A Satake diagram has a starting point the Dynkin diagram of the corresponding complex form. For a split real form it remains the same. In the other cases some dots are painted in black — these considered by themselves are Dynkin diagrams of the compact semisimple factors M of the minimal parabolic subalgebras. Further, there are arrows connecting some nodes which use the Z2 symmetry of some Dynkin diagrams. Then the reduced root systems are described by Dynkin–Satake diagrams which are obtained from the Satake diagrams by dropping the black nodes, identifying the arrow-related nodes, and adjoining all nodes in a connected Dynkin-like diagram, but in addition noting the multiplicity of the reduced roots (which is in general different from 1). More details can be seen in [7], and we have tried to make the exposition transparent (by repeating things). n We

could do this, since by being explicit we could consider simultaneously cases with different Satake diagrams.

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11.1. EI: E60 The split real form of E6 is denoted as E60 , sometimes as E6(+6) . The maximal compact subgroup is K ∼ = sp(4), dimR P = 42, dimR N ± = 36. This real form does not have discrete series representations. For a split real form the Satake diagram coincides with the Dynkin diagram of the corresponding complex Lie algebra [7]. In the present case this Dynkin–Satake diagram is taken as follows: ◦α6 | ◦ −−− ◦ −−− ◦ −−− ◦ −−− ◦

α1

α2

α3

α4

α5

(11.1)

Taking into account the above enumeration of simple roots the cuspidal parabolic subalgebras have MΘ -factors as follows:  0 Θ = ∅, minimal       sl(2, R)j , Θ = {j}, j = 1, . . . , 6     sl(2, R)j ⊕ sl(2, R)k , Θ = {j, k} : j + 1 < k, {j, k} 6= {3, 6};   MΘ = (11.2) (j, k) = (5, 6),     sl(2, R)j ⊕ sl(2, R)k Θ = {j, k, `}, (j, k, `) = (1, 3, 5), (1, 4, 6),      ⊕ sl(2, R)` , (1, 5, 6), (2, 4, 6), (2, 5, 6)     so(4, 4), Θ = {2, 3, 4, 6}, where sl(2, R)j denotes the sl(2, R) subalgebra of G spanned by Xj± , Hj , (using the same notation as in the Section on sl(n, R)). All these MΘ , except the last case (so(4, 4)), have highest/lowest weight representations. The dimensions of the other factors are, respectively:   6 36             5 35 ± dim AΘ = 4 , dim NΘ = 34 . (11.3)     3 33         2 24 Taking into account (2.24) the maximal parabolic subalgebras are determined by: ∼ ∼ Mmax = Mmax = so(5, 5), 1 5

dim (NΘ± )max = 16,

∼ ∼ Mmax = Mmax = sl(5, R) ⊕ sl(2, R), 2 5

dim (NΘ± )max = 25,

∼ Mmax = sl(3, R) ⊕ sl(3, R) ⊕ sl(2, R), dim (NΘ± )max = 29, 3 ∼ Mmax = sl(6, R), 6

dim (NΘ± )max = 21.

Clearly, no maximal parabolic subalgebra is cuspidal.

(11.4)

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11.2. EII: E600 Another real form of E6 is denoted as E600 , sometimes as E6(+2) . The maximal compact subgroup is K ∼ = su(6) ⊕ su(2), dimR P = 40, dimR N ± = 36. This real form has discrete series representations. The split rank is equal to 4, while M ∼ = u(1) ⊕ u(1). The Satake diagram is: ◦α6 | ◦ −−− ◦ −−− ◦ −−− ◦ −−− ◦ α1 α2 α3 α4 α5 | {z } {z } |

(11.5)

Thus, the reduced root system is presented by a Dynkin–Satake diagram looking like the F4 Dynkin diagram: ◦ −−− ◦ =⇒ ◦ −−− ◦

λ1

λ2

λ3

(11.6)

λ4

but the short roots have multiplicity 2 (the long — multiplicity 1). It is obtained from (11.5) by identifying α1 and α5 and mapping them to λ4 , identifying α2 and α4 and mapping them to λ3 , while the roots α3 , α6 are mapped to the F4 -like long simple roots λ2 , λ1 , respectively. Using the above enumeration of F4 simple roots we give the MΘ -factors of all parabolic subalgebras:

MΘ =

  u(1) ⊕ u(1),      sl(2, R)j ⊕ u(1) ⊕ u(1),       sl(2, C)j ⊕ u(1),        sl(3, R) ⊕ u(1) ⊕ u(1),      sl(2, R)1 ⊕ sl(2, C)j ⊕ u(1),      sl(4, R) ⊕ u(1),   sl(2, R)2 ⊕ sl(2, C)4 ⊕ u(1),       sl(3, C),       so(4, 4) ⊕ u(1),        sl(3, R) ⊕ u(1) ⊕ sl(2, C)4 ,       sl(2, R)1 ⊕ sl(3, C),      sl(6, R),

Θ = ∅,

minimal

Θ = {j},

j = 1, 2

Θ = {j},

j = 3, 4

Θ = {1, 2} Θ = {1, j}, Θ = {2, 3} Θ = {2, 4} Θ = {3, 4} Θ = {1, 2, 3} Θ = {1, 2, 4} Θ = {1, 3, 4} Θ = {2, 3, 4}

j = 3, 4 .

(11.7)

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The dimensions of the other factors are, respectively:

dim AΘ =

 4      3       3      2      2     2  2      2       1      1     1     1,

,

dim NΘ±

 36      35       34      33      33     30 = .  33      30       24      31      29     21

(11.8)

The maximal parabolic subalgebras are given the last four lines in the above lists corresponding to Θmax , j = 4, 3, 2, 1, cf. (2.24). j The cuspidal parabolic subalgebras are those containing:  u(1) ⊕ u(1), Θ = ∅, minimal    MΘ = sl(2, R)j ⊕ u(1) ⊕ u(1), Θ = {j}, j = 1, 2 . (11.9)    so(4, 4) ⊕ u(1), Θ = {1, 2, 3} All these MΘ , except the last case (containing so(4, 4)), have highest/lowest weight representations. 11.3. EIII: E6000 Another real form of E6 is denoted as E6000 , sometimes as E6(−14) . The maximal compact subgroup is K ∼ = so(10) ⊕ so(2), dimR P = 32, dimR N ± = 30. This real form has discrete series representations (and highest/lowest weight representations). The split rank is equal to 2, while M ∼ = so(6) ⊕ so(2). The Satake diagram is: ◦α6 | ◦ −−− • −−− • −−− • −−− ◦ α1 α2 α3 α4 α5 {z } |

(11.10)

Thus, the reduced root system is presented by a Dynkin–Satake diagram looking like the B2 Dynkin diagram but the long roots (including λ1 ) have multiplicity 6,

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while the short roots (including λ2 ) have multiplicity 8, and there are also the roots 2λ of multiplicity 1, where λ is any short root. It is obtained from (11.10) by dropping the black nodes, (they give rise to M), identifying α1 and α5 and mapping them to λ2 , while the root α6 is mapped to the long simple root λ1 . The non-minimal parabolic subalgebras are given by: ( MΘ =

so(7, 1) ⊕ so(2),

Θ = {1}

sl(6, R),

Θ = {2}

,

dim

NΘ±

=

( 24 21

.

(11.11)

Both are maximal (dim AΘ = 1) and not cuspidal. 11.4. EIV: E6iv Another real form of E6 is denoted as E6iv , sometimes as E6(−26) . The maximal compact subgroup is K ∼ = f4 , dimR P = 26, dimR N ± = 24. This real form does not have discrete series representations. The split rank is equal to 2, while M ∼ = so(8). The Satake diagram is: •α6 | ◦ −−− • −−− • −−− • −−− ◦

α1

α2

α3

α4

(11.12)

α5

Thus, the reduced root system is presented by a Dynkin–Satake diagram looking like the A2 Dynkin diagram but all roots have multiplicity 8. It is obtained from (11.12) by dropping the black nodes, while α1 , α5 , respectively, are mapped to the A2 -like simple roots λ1 , λ2 . The two non-minimal parabolic subalgebras are isomorphic and given by: MΘ = so(9, 1),

Θ = {j},

j = 1, 2,

dim NΘ± = 16.

(11.13)

Both are maximal (dim AΘ = 1) and not cuspidal. 11.5. EV: E70 The split real form of E7 is denoted as E70 , sometimes as E7(+7) . The maximal compact subgroup K ∼ = su(8), dimR P = 70, dimR N ± = 63. This real form has discrete series representations. We take the Dynkin–Satake diagram as follows: ◦α7 | ◦ −−− ◦ −−− ◦ −−− ◦ −−− ◦ −−− ◦

α1

α2

α3

α4

α5

α6

(11.14)

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Taking into account the above enumeration of simple roots the cuspidal parabolic subalgebras have MΘ -factors as follows:  0 Θ = ∅, minimal      sl(2, R)j , Θ = {j}, j = 1, . . . , 7       sl(2, R)j ⊕ sl(2, R)k , Θ = {j, k} : j + 1 < k,      {j, k} 6= {3, 7}; {j, k} = {6, 7},      sl(2, R)j ⊕ sl(2, R)k Θ = {j, k, `} = {1, 3, 5}, {1, 3, 6}, {1, 4, 6},      ⊕ sl(2, R) , {1, 4, 7}, {1, 5, 7}, {1, 6, 7}, {2, 4, 6}, `   {2, 4, 7}, {2, 5, 7}, {2, 6, 7} MΘ =   sl(2, R)1 ⊕ sl(2, R)4       ⊕ sl(2, R)6 ⊕ sl(2, R)7      sl(2, R)2 ⊕ sl(2, R)4      ⊕ sl(2, R)6 ⊕ sl(2, R)7       so(4, 4), Θ = {2, 3, 4, 7},    so(6, 6), Θ = {2, 3, 4, 5, 6, 7}. (11.15) All these MΘ , except the last two cases (so(4, 4), so(6, 6)), have highest/lowest weight representations. The dimensions of the other factors are, respectively:   7 63           6 62           5 61         4 60 dim AΘ = (11.16) , dim NΘ± = . 3 59           3 59             3 51       1 33 Taking into account (2.24) the maximal parabolic subalgebras are determined by: ∼ Mmax = so(6, 6), 1 max ∼ M = sl(6, R) ⊕ sl(2, R),

2 max M3 Mmax 4 max M5 Mmax 6 max M7

∼ = sl(4, R) ⊕ sl(3, R) ⊕ sl(2, R), ∼ = sl(5, R) ⊕ sl(3, R), ∼ = so(5, 5) ⊕ sl(2, R), ∼ = E60 , ∼ = sl(7, R),

dim (NΘ± )max = 33 dim (NΘ± )max = 47 dim (NΘ± )max = 53 dim (NΘ± )max = 60 dim (NΘ± )max = 42 dim (NΘ± )max = 27 dim (NΘ± )max = 42.

(11.17)

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Clearly, the only maximal cuspidal parabolic subalgebra is the one containing Mmax . 1 11.6. EVI: E700 Another real form of E7 is denoted as E700 , sometimes as E7(−5) . The maximal compact subgroup is K ∼ = so(12) ⊕ su(2), dimR P = 64, dimR N ± = 60. This real form has discrete series representations. The split rank is equal to 4, while M ∼ = su(2) ⊕ su(2) ⊕ su(2). The Satake diagram is: •α7 | ◦ −−− ◦ −−− ◦ −−− • −−− ◦ −−− •

α1

α2

α3

α4

α5

α6

(11.18)

Thus, the reduced root system is presented by a Dynkin–Satake diagram looking like the F4 Dynkin diagram, cf. (11.6), but the short roots have multiplicity 4 (the long — multiplicity 1). Going to this Dynkin–Satake diagram we drop the black nodes, (they give rise to M), while α1 , α2 , α3 , α5 , are mapped to λ1 , λ2 , λ3 , λ4 , respectively, of (11.6). Using the above enumeration of F4 simple roots we shall give the MΘ -factors of all parabolic subalgebras:  M = su(2) ⊕ su(2), Θ = ∅, minimal     ⊕ su(2)        sl(2, R)j ⊕ M, Θ = {j}, j = 1, 2       su∗ (4) ⊕ su(2)j+3 , Θ = {j}, j = 3, 4       sl(3, R) ⊕ M, Θ = {1, 2}       sl(2, R)1 ⊕ su∗ (4) Θ = {1, j}, j = 3, 4     ⊕ su(2) ,  j+3    Θ = {2, 3} MΘ = so(6, 2) ⊕ su(2), (11.19) .   ∗   sl(2, R)2 ⊕ su (4) Θ = {2, 4}     ⊕ su(2)7 ,      ∗  su (6), Θ = {3, 4}        so(7, 3) ⊕ su(2)6 , Θ = {1, 2, 3}       sl(3, R) ⊕ su∗ (4) ⊕ su(2)7 , Θ = {1, 2, 4}       sl(2, R)1 ⊕ su∗ (6), Θ = {1, 3, 4}      ∗ so (12), Θ = {2, 3, 4}

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The dimensions of the other factors are, respectively:   4 60           3 59             3 56         2  57           2 55         2 50 dim AΘ = , dim NΘ± = . 2  55           2 48             1 42         1  53           1 47         1 33

437

(11.20)

The maximal parabolic subalgebras are the last four in the above list corre, j = 4, 3, 2, 1, cf. (2.24). sponding to Θmax j The cuspidal parabolic subalgebras are those containing  M = su(2) ⊕ su(2) ⊕ su(2), Θ = ∅, minimal     sl(2, R) ⊕ M, Θ = {j}, j = 1, 2 j MΘ = (11.21)  so(6, 2) ⊕ su(2), Θ = {2, 3}     ∗ so (12), Θ = {2, 3, 4} the last one being also maximal. All these MΘ have highest/lowest weight representations. 11.7. EVII: E7000 Another real form of E7 is denoted as E7000 , sometimes as E7(−25) . The maximal compact subgroup is K ∼ = e6 ⊕ so(2), dimR P = 54, dimR N ± = 51. This real form has discrete series representations (and highest/lowest weight representations). The split rank is equal to 3, while M ∼ = so(8). The Satake diagram is: •α7 | ◦ −−− • −−− • −−− • −−− ◦ −−− ◦ (11.22) α1

α2

α3

α4

α5

α6

Thus, the reduced root system is presented by a Dynkin–Satake diagram looking like the C3 Dynkin diagram: ◦ =⇒ ◦ −−− ◦

λ1

λ2

λ3

(11.23)

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but the short roots have multiplicity 8 (the long — multiplicity 1). Going to the C3 diagram we drop the black nodes, (they give rise to M), while α1 , α5 , α6 , are mapped to λ3 , λ2 , λ1 , respectively, of (11.23). Using the above enumeration of C3 simple roots we shall give the MΘ -factors of all parabolic subalgebras:  so(8), Θ = ∅, minimal      so(9, 1), Θ = {j}, j = 1, 2     sl(2, R) ⊕ so(8), Θ = {3} 3 MΘ = (11.24) . iv  Θ = {1, 2} e6 ,      sl(2, R)3 ⊕ so(9, 1), Θ = {1, 3}     so(10, 2), Θ = {2, 3} The dimensions of the other factors are, respectively:   3 51           2 43         2 50 dim AΘ = , dim NΘ± = .   1 27           1 42         1 33

(11.25)

The last three give rise to the maximal parabolic subalgebras. The cuspidal parabolic subalgebras are those containing  Θ = ∅, minimal  so(8), MΘ = sl(2, R)3 ⊕ so(8), Θ = {3}   so(10, 2), Θ = {2, 3}

(11.26)

the last one being also maximal. All these MΘ have highest/lowest weight representations. 11.8. EVIII: E80 The split real form of E8 is denoted as E80 , sometimes as E8(+8) . The maximal compact subgroup K ∼ = so(16), dimR P = 128, dimR N ± = 120. This real form has discrete series representations. We take the Dynkin–Satake diagram as follows: ◦α8 | ◦ −−− ◦ −−− ◦ −−− ◦ −−− ◦ −−− ◦ −−− ◦

α1

α2

α3

α4

α5

α6

α7

(11.27)

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Taking into account the above enumeration of simple roots the cuspidal parabolic subalgebras have MΘ -factors as follows:  0      sl(2, R)j ,       sl(2, R)j ⊕ sl(2, R)k ,            sl(2, R)j ⊕ sl(2, R)k       ⊕ sl(2, R)` ,                           sl(2, R)1 ⊕ sl(2, R)3       ⊕ sl(2, R)5 ⊕ sl(2, R)7       sl(2, R)1 ⊕ sl(2, R)4    ⊕ sl(2, R)6 ⊕ sl(2, R)8 MΘ =    sl(2, R)1 ⊕ sl(2, R)4      ⊕ sl(2, R)7 ⊕ sl(2, R)8       sl(2, R)1 ⊕ sl(2, R)5       ⊕ sl(2, R)7 ⊕ sl(2, R)8       sl(2, R)2 ⊕ sl(2, R)4      ⊕ sl(2, R)6 ⊕ sl(2, R)8       sl(2, R)2 ⊕ sl(2, R)4      ⊕ sl(2, R)7 ⊕ sl(2, R)8       sl(2, R)2 ⊕ sl(2, R)5       ⊕ sl(2, R)7 ⊕ sl(2, R)8       so(4, 4),       so(6, 6),     0 E7 ,

Θ = ∅,

minimal

Θ = {j},

j = 1, . . . , 8

Θ = {j, k} : j + 1 < k, {j, k} 6= {3, 8}; {j, k} = {7, 8}, Θ = {j, k, `} = {1, 3, 5}, {1, 3, 6}, {1, 3, 7}, {1, 4, 6}, {1, 4, 7}, {1, 4, 8}, {1, 5, 7}, {1, 5, 8}, {1, 6, 8}, {2, 4, 6}, {2, 4, 7}, {2, 4, 8}, {2, 5, 7}, {2, 5, 8}, {2, 6, 8}, {2, 7, 8}, {3, 5, 7}, {4, 6, 8}, {4, 7, 8}, {5, 7, 8}

.

Θ = {2, 3, 4, 8} Θ = {2, 3, 4, 5, 6, 8} Θ = {1, 2, 3, 4, 5, 6, 8} (11.28)

All these MΘ , except the last three cases (so(4, 4), so(6, 6), E70 ), have highest/lowest weight representations.

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The dimensions of the other factors are, respectively:   8 120           7 119           6 118           5  117           4 116           4 116         4 116 dim AΘ = , dim NΘ± = .   4 116             4 116       4 116           4 116           4 108             2 90       1 57

(11.29)

Taking into account (2.24) the maximal parabolic subalgebras are determined by: ∼ Mmax dim (NΘ± )max = 78 = so(7, 7), 1 Mmax ∼ dim (N ± )max = 98 = sl(7, R) ⊕ sl(2, R), 2

Θ

∼ Mmax = sl(5, R) ⊕ sl(3, R) ⊕ sl(2, R), 3 Mmax ∼ = sl(5, R) ⊕ sl(4, R), 4

∼ Mmax = so(5, 5) ⊕ sl(3, R), 5 max ∼ M = E 0 ⊕ sl(2, R), 6

dim (NΘ± )max = 106 dim (NΘ± )max = 104 dim (NΘ± )max = 97

(11.30)

dim (NΘ± )max = 83

6

∼ Mmax = E70 , 7 Mmax ∼ = sl(8, R),

dim (NΘ± )max = 57 dim (NΘ± )max = 92.

8

Clearly, the only maximal cuspidal parabolic subalgebra is the one containing Mmax . 7 11.9. EIX: E800 Another real form of E8 is denoted as E800 , sometimes as E8(−25) . The maximal compact subgroup is K ∼ = e7 ⊕ su(2), dimR P = 54, dimR N ± = 51. This real form has discrete series representations. The split rank is equal to 4, while M ∼ = so(8). The Satake diagram •α8 | ◦ −−− • −−− • −−− • −−− ◦ −−− ◦ −−− ◦ (11.31) α1

α2

α3

α4

α5

α6

α7

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Thus, the reduced root system is presented by a Dynkin–Satake diagram looking like the F4 Dynkin diagram, cf. (11.6), but the short roots have multiplicity 8 (the long — multiplicity 1). Going to the F4 diagram we drop the black nodes, (they give rise to M), while α1 , α5 , α6 , α7 are mapped to λ4 , λ3 , λ2 , λ1 , respectively, of (11.6). Using the above enumeration of F4 simple roots we shall give the MΘ -factors of all parabolic subalgebras:

MΘ =

 M = so(8),       sl(2, R)j ⊕ M,       so(9, 1),      sl(3, R) ⊕ M,      sl(2, R)1 ⊕ so(9, 1),     so(10, 2),  sl(2, R)2 ⊕ so(9, 1),     eiv ,   6      so(11, 3),      sl(3, R) ⊕ so(9, 1),      iv   sl(2, R)1 ⊕ e6 ,   000 e7 ,

Θ = ∅,

minimal

Θ = {j},

j = 1, 2

Θ = {j},

j = 3, 4

Θ = {1, 2} Θ = {1, j},

j = 3, 4

Θ = {2, 3} Θ = {2, 4}

.

(11.32)

Θ = {3, 4} Θ = {1, 2, 3} Θ = {1, 2, 4} Θ = {1, 3, 4} Θ = {2, 3, 4}

The dimensions of the other factors are, respectively:  4      3       3      2      2     2 dim AΘ = , 2      2       1      1      1     1

dim NΘ±

 108      107       100      105      99     90 = . 99      84       78      97      83     57

(11.33)

The maximal parabolic subalgebras are the last four in the above lists corresponding to Θmax , j = 4, 3, 2, 1, cf. (2.24). j

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The cuspidal ones arise from:  so(8),     sl(2, R) ⊕ so(8), j MΘ =  so(10, 2),     000 e7 ,

Θ = ∅,

minimal

Θ = {j},

j = 1, 2

Θ = {2, 3}

(11.34)

Θ = {2, 3, 4}

the last one being also maximal. All these MΘ have highest/lowest weight representations. 11.10. FI: F40 The split real form of F4 is denoted as F40 , sometimes as F4(+4) . The maximal compact subgroup K ∼ = sp(3) ⊕ su(2), dimR P = 28, dimR N ± = 24. This real form has discrete series representations. Taking into account the enumeration of simple roots as in (11.6) the parabolic subalgebras have MΘ -factors as follows:   0 Θ = ∅, minimal      Θ = {j}, j = 1, 2, 3, 4 sl(2, R)j ,      sl(3, R)jk , Θ = {j, k} = {1, 2}, {3, 4}     sl(2, R)j ⊕ sl(2, R)k , Θ = {j, k} = {1, 3}, {1, 4}, {2, 4} MΘ = (11.35) . sp(2, R), Θ = {2, 3}       so(4, 3), Θ = {1, 2, 3}       sl(3, R) ⊕ sl(2, R), Θ = {1, 2, 4}, {1, 3, 4}     sp(3, R), Θ = {2, 3, 4} The dimensions of the other factors are, respectively:

dim AΘ =

 4      3      2     2  2      1       1    1

,

dim NΘ± =

 24      23      21     22  20      15       20    15

.

(11.36)

The maximal parabolic subalgebras are in the last three lines in the above lists corresponding to Θmax , j = 4, 3, 2, 1, cf. (2.24). j

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The cuspidal parabolic subalgebras arise from:  0 Θ = ∅, minimal       Θ = {j}, j = 1, 2, 3, 4 sl(2, R)j ,  MΘ = sl(2, R)j ⊕ sl(2, R)k , Θ = {j, k} = {1, 3}, {1, 4}, {2, 4}    sp(2, R), Θ = {2, 3}     sp(3, R), Θ = {2, 3, 4}

443

(11.37)

the last one being also maximal. All these MΘ have highest/lowest weight representations. 11.11. FII: F400 Another real form of F4 is denoted as F400 , sometimes as F4(−20) . The maximal compact subgroup K ∼ = so(9), dimR P = 16, dimR N ± = 15. This real form has discrete series representations. The split rank is equal to 1, while M ∼ = so(7). The Satake diagram is: • −−− • =⇒ • −−− ◦

α1

α2

α3

(11.38)

α4

Thus, the reduced root system is presented by a Dynkin–Satake diagram looking like the A1 Dynkin diagram but the roots have multiplicity 8. Going to the A1 Dynkin diagram we drop the black nodes, (they give rise to M), while α4 becomes the A1 diagram. 11.12. G: G02 The split real form of G2 is denoted as G02 , sometimes as G2(+2) . The maximal compact subgroup K ∼ = su(2) ⊕ su(2), dimR P = 8, dimR N ± = 6. This real form has discrete series representations. The non-minimal parabolic subalgebras have MΘ -factors as follows: MΘ = sl(2, R)j ,

Θ = {j},

j = 1, 2.

They are cuspidal and maximal. All MΘ representations.

(11.39)

have highest/lowest weight

12. Summary and Outlook In the present paper we have started the systematic explicit construction of the invariant differential operators by giving explicit description of one of the main ingredients in our setting — the cuspidal parabolic subalgebras. We explicated also the maximal parabolic subalgebras, since these are important even when they are not cuspidal. In sequels of this paper [74] we shall present the construction of the invariant differential operators and expand the scheme to the supersymmetric case, in view of applications to conformal field theory and string theory.

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Acknowledgments The author would like to thank for hospitality the Abdus Salam International Center for Theoretical Physics, where part of the work was done. This work was supported in part by the Bulgarian National Council for Scientific Research, grant F-1205/02, the Alexander von Humboldt Foundation in the framework of the Clausthal-Leipzig-Sofia Cooperation, and the European RTN “Forces-Universe”, contract MRTN-CT-2004-005104. The author would like to thank the anonymous referee for numerous remarks which contributed to improving the exposition. Appendix. Table of Cuspidal Parabolic Subalgebras G

MΘ

GC dimC GC = d rankC GC = `

u(1) ⊕ · · · ⊕ u(1) ` factors

sl(n, R)

MΘ(¯ n) =

`

d−`

n−1−k

1 n(n − 1) − k 2

minimal: k = 0, MΘ = 0

n−1

1 n(n − 1) 2

su(2) ⊕ · · · ⊕ su(2) n factors

n−1

2n(n − 1)

su(p − j, r − j) ⊕ u(1) ⊕ · · · ⊕ u(1) j factors, 1 ≤ j < r minimal: j = r from above minimal: u(1) ⊕ · · · ⊕ u(1) r − 1 factors

j

j(2(p + r − j) − 1)

r r

r(2p − 1) r(2r − 1)

MΘ(¯ ns ) ⊕ so(p − s, r − s) s = 1, 2, . . . , r, p + r odd s = 1, 3, . . . , r, p, r odd s = 2, 4, . . . , r, p, r even MΘo (¯ n) minimal: so(p − r)

≤s

⊕

1≤t≤k

sl(2, R)jt

0 ≤ k ≤ [n/2] jt < jt+1 − 1 1 ≤ j1 , jk ≤ n − 1

su∗ (2n) su(p, r) p≥r p>r p=r so(p, r) p≥r

p=r p≥r sp(n, R)

± dimR NΘ

dimR AΘ

r

r(p − 1)

MΘ(¯ ns−1 ) ⊕ sp(n − s, R) s = 1, . . . , n minimal: MΘ = 0, (s = n)

≤s n

n2

sp(p, r) p≥r

sp(p − j, r − j) ⊕ sp(1) ⊕ · · · ⊕ sp(1) j factors, j = 1, . . . , r minimal: j = r

j

j(4p + 4r − 4j − 1)

so∗ (2n)

so∗ (2n − 4j) ⊕ so(3) ⊕ · · · ⊕ so(3) j factors, j = 1, . . . , r ≡ [n/2] minimal: j = r ≡ [n/2]

j

j(4n − 4j − 3)

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Appendix (Continued) G

MΘ

EI ∼ = E60

dimR AΘ

± dimR NΘ

0, minimal sl(2, R)j j = 1, . . . , 6 sl(2, R)j ⊕ sl(2, R)k j + 1 < k, {j, k} 6= {3, 6}; (j, k) = (5, 6) sl(2, R)j ⊕ sl(2, R)k ⊕ sl(2, R)` (j, k, `) = (1, 3, 5), (1, 4, 6), (1, 5, 6), (2, 4, 6), (2, 5, 6) so(4, 4)

6 5

36 35

4

34

3

33

2

24

u(1) ⊕ u(1), minimal sl(2, R)j ⊕ u(1) ⊕ u(1), j = 1, 2 so(4, 4) ⊕ u(1)

4 3 1

36 35 24

EIII ∼ = E6000

so(6) ⊕ so(2)

2

30

EIV ∼ = E6iv ∼ E0 EV =

so(8) 0, minimal sl(2, R)j , j = 1, . . . , 7 sl(2, R)j ⊕ sl(2, R)k j + 1 < k, {j, k} 6= {3, 7}; (j, k) = (6, 7) sl(2, R)j ⊕ sl(2, R)k ⊕ sl(2, R)` (j, k, `) = (1, 3, 5), (1, 3, 6), (1, 4, 6), (1, 4, 7), (1, 5, 7), (1, 6, 7), (2, 4, 6), (2, 4, 7), (2, 5, 7), (2, 6, 7) sl(2, R)1 ⊕ sl(2, R)4 ⊕ sl(2, R)6 ⊕ sl(2, R)7 sl(2, R)2 ⊕ sl(2, R)4 ⊕ sl(2, R)6 ⊕ sl(2, R)7 so(4, 4) so(6, 6)

2 7 6 5

24 63 62 61

4

60

3 3 3 1

59 59 51 33

EVI ∼ = E700

su(2) ⊕ su(2) ⊕ su(2), minimal sl(2, R)j ⊕ su(2) ⊕ su(2) ⊕ su(2), j = 1, 2 so(6, 2) ⊕ su(2) so∗ (12)

4 3 2 1

60 59 50 33

EVII ∼ = E7000

so(8), minimal sl(2, R)3 ⊕ so(8) so(10, 2)

3 2 1

51 50 33

FI ∼ = F40

0, minimal sl(2, R)j , j = 1, 2, 3, 4 sl(2, R)j ⊕ sl(2, R)k , (j, k) = (13), (14), (24) sp(2, R) sp(3, R)

4 3 2 2 1

24 23 22 20 15

FII ∼ = F400

so(7)

1

15

G∼ =

0, minimal sl(2, R)j , j = 1, 2 0, minimal sl(2, R)j , j = 1, . . . , 8 sl(2, R)j ⊕ sl(2, R)k j + 1 < k, {j, k} 6= {3, 8}; (j, k) = (7, 8) sl(2, R)j ⊕ sl(2, R)k ⊕ sl(2, R)` (j, k, `) = (1, 3, 5), (1, 3, 6), (1, 3, 7), (1, 4, 6), (1, 4, 7), (1, 4, 8), (1, 5, 7), (1, 5, 8), (1, 6, 8), (2, 4, 6), (2, 4, 7), (2, 4, 8), (2, 5, 7), (2, 5, 8), (2, 6, 8), (2, 7, 8), (3, 5, 7), (4, 6, 8), (4, 7, 8), (5, 7, 8),

2 1 8 7 6

6 5 120 119 118

5

117

EII ∼ =

E600

7

G02

EVIII ∼ = E80

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EIX ∼ = E800

± dimR NΘ

MΘ

dimR AΘ

sl(2, R)j ⊕ sl(2, R)k ⊕ sl(2, R)` ⊕ sl(2, R)m (j, k, `, m) = (1, 3, 5, 7), (1, 4, 6, 8), (1, 4, 7, 8), (1, 5, 7, 8), (2, 4, 6, 8), (2, 4, 7, 8), (2, 5, 7, 8) so(4, 4) so(6, 6) E70

4

116

4 2 1

108 90 57

so(8), minimal sl(2, R)j ⊕ so(8), j = 1, 2 so(10, 2) E70

4 3 2 1

108 107 90 57

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Reviews in Mathematical Physics Vol. 20, No. 4 (2008) 451–492 c World Scientific Publishing Company °

SELF-ORGANIZING BIRTH-AND-DEATH STOCHASTIC SYSTEMS IN CONTINUUM

YURI KONDRATIEV Department of Mathematics, Bielefeld University, 33615 Bielefeld, Germany [email protected] ROBERT MINLOS∗ and ELENA ZHIZHINA† Institute for Information Transmission Problems, Bolshoy Karetny per. 19, 127994 Moscow, Russia ∗[email protected] †[email protected] Received 24 June 2007 Revised 18 December 2007 We consider birth-and-death stochastic particle systems in continuum which are under a self-regulation mechanism controlling configurations of particles via a pairwise interaction between them. The latter is reflected in a potential perturbation of the free generator. We show that the ground state renormalization scheme in the considered model leads to an invariant measure, a renormalized generator and resulting equilibrium birthand-death stochastic dynamics for the system. The proof is based on the Gibbs-type representation for related path space measure. This measure has OS-positivity property and is constructed via the cluster expansion method. Keywords: Spatial birth-and-death processes; Feynman–Kac formula; OS-positivity; generators; spectral gap. Mathematics Subject Classification 2000: 82C21, 60J25

1. Introduction We consider an infinite system of particles in continuum under a stochastic evolution corresponding to a heuristic generator H = L0 − αU.

(1)

Here L0 is the generator of a non-interacting birth-and-death process (a Glaubertype dynamics), and U is an operator of multiplication by a function (equals to a sum of pair interactions over the configuration of points of the system), 0 < α ¿ 1 is a coupling constant. The goal of the paper is to construct and to study a process properly associated with operator (1). We will use here an approach similar to used 451

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for the investigation of an infinite system of quantum anharmonic oscillators [8] or the quantum Heisenberg model [2]. Our approach is based on the Feynman–Kac formula: Z xt =y2 Rt etH (y1 , y2 ) = e−α 0 U (xs )ds dPz0 (x), (2) x0 =y1

where the integration in (2) is over the distribution Pz0 on a space of trajectories x = {xs , s ∈ R1 } of a Glauber-type free stochastic dynamics with the generator L0 (so-called Surgailis process, see [19, 20]). A rigorous meaning this formula has only under some regularity assumptions on the potential U . The expression in the right-hand side of (2) up to a multiplicative constant coincides with a Gibbs reconstruction of the measure Pz0 . The latter gives us a hope to apply well-known methods from statistical physics to the construction and investigation of semigroup (2). Let us note that the Feynman–Kac formula is in common use for the study of models in quantum statistical physics and quantum field theory, when Pz0 is a measure on trajectories of a free process, usually defined by a Schr¨odinger operator, see [5, 17]. We briefly describe now our constructions and state main results of the paper. Initially, we consider truncated (over the space) potential UΛ , where Λ ⊂ Rd is a bounded domain of the space Rd . It means that we consider a system where particles interact only if they are inside of domain Λ. Then the operator HΛ = L0 − αUΛ is defined correctly and it is unitary equivalent up to an additive constant to the generator of a stationary Markov process GΛ = {γt , t ∈ R1 },

γt ∈ Γ

with values in a space Γ of locally finite configurations in Rd . The path space measure of the process GΛ may be obtained R Tas the limit when T → ∞ of the Gibbs reconstructions by the energy function −T UΛ (γs )ds of the reference measure corresponding to the Surgailis process [19, 20] with the generator L0 . Then taking the thermodynamic limit as Λ%Rd we get the limit path space measure and the stochastic process G∞ . We prove that the limit process meets the condition of OS-positivity (Osterwalder–Schrader positivity, see, e.g., [16]). Using that fact we can construct in canonical way corresponding Hilbert space H and a semigroup of self-adjoint operators in H associated with the process G∞ and generated by ˆ Thus through the use of the operator H ˆ heuristic expression (1) an operator H. ˆ should be considered as a correct regularization of gains rigorous meaning, and H the operator (1). In addition, we prove the existence of the spectral gap for the ˆ using estimates on decay of correlations for the limit process G∞ . The operator H main technique we use here is based on cluster expansion methods for point fields developed in [10, 12, 7, 4]. Let us discuss possible interpretations of our results in individual based models of spatial economics. In this case, a configuration should be considered as a set of economic units (points of the configuration) located in the space. A pure birth

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Markov process corresponds to an economic growth model in which the density of units is linearly growing in time. Assuming additionally random life time of any unit (independent and exponentially distributed for each existing one), we will arrive in the Surgailis process mentioned above. The equilibrium measure of this process is Poisson one and its Markov generator is a self-adjoint operator in corresponding Poisson L2 -space, see [19, 20]. This generator admits a nice and easy spectral decomposition that relates one to the Fock space number operator in quantum field theory, see, e.g. [1]. Actually, the Surgailis process can also be considered as a free Glauber-type stochastic dynamics in continuum, see, e.g. [6]. In a more realistic model, we should take into account a competition between units. One way to include this notion is related with a modification of the death rate in the generator s.t. the growing density of the configuration will increase the intensity of death. Another possibility is based on the consideration of a rate functional which should play the role of a regulation mechanism in the economic society. Namely, configurations of units with high rate must have less chances to survive in the stochastic evolution of the system. This rate functional is included as the potential U in the model considered in this paper. The main question which appears here is the existence of an equilibrium state in such economic models as well as the construction of related equilibrium stochastic process of the economic development with described regulation based on a local interaction between units. The results of the present paper give a positive answer to this problem. 2. The Model and Main Results 2.1. Free Glauber dynamics (Surgailis process) The configuration space Γ = Γ(Rd ) of the model is the set of all locally finite subsets of Rd : γ ⊂ Rd . The space Γ is naturally endowed with a topology, namely the P weakest topology on Γ with respect to which all maps Γ 3 γ 7→ hf, γi := x∈γ f (x), f ∈ C0 (Rd ), are continuous (here, C0 (Rd ) is the space of all continuous real-valued functions on Rd with compact support). We denote by B(Γ) the Borel σ-algebra on Γ generated by this topology, and let πz be the Poisson measure on (Γ, B(Γ)) with activity z, z > 0, see e.g. [10]. We define a stationary Markov process on Γ with the invariant measure πz . A generator of the corresponding stochastic semigroup St0 acting in the functional space L2 (Γ, dπz ) has a form Z X (F (γ\x) − F (γ)) + z (F (γ ∪ x) − F (γ))dx. (L0 F )(γ) = (3) x∈γ

Rd

The operator L0 is defined on local bounded functions F (γ), and the expression (3) can be extended to a self-adjoint operator in L2 (Γ, dπz ). The corresponding process is a birth-and-death process on Γ, we also call it the free Glauber dynamics or the Surgailis process, see [19,20]. The process can be described as follows: each particle in the configuration can disappear after an exponentially distributed life time and

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new particles can appear in the configuration with intensity z uniformly over the space. We denote by Pz0 the distribution of the Surgailis process (on the trajectory space). 2.2.

Glauber dynamics with interaction

The generator of the dynamics with interaction is given heuristically as follows: (HF )(γ) = (L0 F )(γ) − αU (γ)F (γ) where U (γ) =

X

(4)

ϕ(x − y).

(5)

{x,y}⊂γ

Here ϕ(u), u ∈ Rd is a stable potential (even real-valued function), see [15], with a fast decreasing on the infinity (we give the precise conditions on ϕ later), α > 0 is a small enough real constant. We will introduce below a dynamics with the generator (4) as a limit of dynamics given in bounded regions Λ ⊂ Rd . Namely, let us consider the operator (HΛ F )(γ) = (L0 F )(γ) − αUΛ (γ)F (γ), with UΛ (γ) =

X

F ∈ L2 (Γ, dπz )

ϕ(x − y).

(6)

(7)

{x,y}⊂γ∩Λ

Theorem 1. Let ϕ be a bounded stable potential with a fast decreasing (34) on the infinity, and α is small enough. Then we have for all bounded Λ ⊂ Rd : (1) the operator HΛ is self-adjoint and bounded from above; (2) a non-degenerate ground state of HΛ exists, i.e. a unique normalized eigenvector ΨΛ of the operator HΛ such that ΨΛ (γ) > 0 exists, and corresponding eigenvalue λ0Λ is the same as the upper boundary of the spectrum of HΛ . Proof. See Sec. 6. We apply below the general scheme of the ground state transformation for potential perturbations of Markov generators, see e.g. [8]. Assign a new measure on Γ in the following way: dνzΛ 2 (γ) = (ΨΛ (γ)) . dπz

(8)

Define an unitary transformation WΛ : L2 (Γ, νzΛ ) 7→ L2 (Γ, πz ) : (WΛ F )(γ) = ΨΛ (γ)F (γ), ˜ Λ in and the operator H

F ∈ L2 (Γ, νzΛ )

(9)

L2 (Γ, νzΛ ) ˜ Λ = W −1 (HΛ − λ0 I)WΛ H Λ Λ

(10)

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which is unitary equivalent to HΛ − λ0Λ I, where I is the identity operator in ˜ Λ has the following form: L2 (Γ, πz ). It follows from (3) and (6) that operator H Z X ΨΛ (γ\x) ΨΛ (γ ∪ y) ˜ Λ F )(γ) = (H (F (γ\x) − F (γ)) + z (F (γ ∪ y) − F (γ))dy. ΨΛ (γ) Rd ΨΛ (γ) x∈γ Clearly, ˜ Λ 1 = 0, H

(11)

where 1 ∈ L2 (Γ, νzΛ ) is the constant function equals to 1. We denote by StΛ = exp{tHΛ }

and

˜ Λ} S˜tΛ = exp{tH

(12)

semigroups acting in the spaces L2 (Γ, πz ) and L2 (Γ, νzΛ ), correspondingly. We have the following representation for the kernel of the semigroup etHΛ using the Feynman–Kac formula ½ ¾ Z Z t StΛ (γ1 , γ2 ) = R(γ1 , γ2 ) = γ(0)=γ exp −α UΛ (γ(τ ))dτ dPz0 (γ), (13) 1

0

γ(t)=γ2

where Pz0 is the distribution of the Surgailis process. It follows from representation (13) and the strict positivity of ΨΛ that semigroups (12) improve positivity, see [14, Vol. 4, Theorem XIII.44]. Moreover, relation (11) implies that ˜ Λ }1 = 1. exp{tH

(14)

Consequently, S˜tΛ is the Markov semigroup and the associated process is a Markov stationary process with the invariant measure νzΛ . Thus, the following theorem holds. ˜ Λ } is the Markov semigroup. The process Theorem 2. Semigroup S˜tΛ = exp{tH GΛ = {γt , t ∈ R1 },

γt ∈ Γ

(15)

associated with the semigroup S˜tΛ is the stationary reversible Markov process on Γ with the invariant measure νzΛ . We denote by PΛ,z the distribution of the process GΛ . As any stationary reversible Markov process, process (15) has property of OS-positivity. We remind this notion and also some related facts, see [16, Chap. II]. Let x = {xt , t ∈ R1 } be a stationary reversible process on the space X and P be a distribution on trajectories of the process. Introduce the time reflection transform ϑ in the space of trajectories Ω = X R: (ϑx)t = x−t , x ∈ Ω.

(16)

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Since the process is reversible, ϑ preserves the distribution P. The unitary representation for ϑ in the space L2 (Ω, P) we denote θ: (θf )(x) = f (ϑx),

f ∈ L2 (Ω, P), x ∈ Ω.

Let H+ ⊂ L2 (Ω, P) be the subspace of functions on Ω depending on the process “at present and in future”: f ∈ H+ :

f (x) = f ({xt }, t ∈ [0, ∞)).

Then the reversible process x called OS-positive if for any f ∈ H+ the quadratic form (θf, f )L2 (Ω,P) ≡ (f, f )H+ = (f, f )+ ≥ 0

(17)

is non-negative. We notice that any Markov stationary reversible process is always OS-positive since (f, f )+ = kP (H0 )f k2 ,

f ∈ H+ ,

where P (H0 ) is the projection to the space H0 of functions depending only on values of the process x0 at zero time. For the stationary reversible OS-positive process x, we can construct a semigroup which is similar to the stochastic semigroup for a Markov process. If I0 ⊂ H+ is the kernel of the quadratic form (17): I0 = {f ∈ H+ : (f, f )+ = 0}, then I0 is the closed subspace of H+ and we can consider the factor-space G = H+ /I0 . The scalar product in G is defined by the following way ([f1 ], [f2 ])G = (f1 , f2 )+ ,

(18)

where [f ] ∈ G is the class of the element f ∈ H+ . The space G is usually called the physical space of the process x. We denote by Ut a unitary operator of a time shift acting in L2 (Ω, P) in the following way (Ut f )(x) = f (s−t x), where sτ are shifts in the space of trajectories (sτ x)t = xt−τ . Clear, Ut H+ ⊂ H+ for any t > 0. Then the permutation relation θUt = U−t θ together with the unitarity of Ut in L2 (Ω, P) imply that the operators Ut , t > 0 are symmetrical with respect to the quadratic form (17): (θUt f1 , f2 )L2 (Ω,P) = (θf1 , Ut f2 )L2 (Ω,P) ,

f1 , f2 ∈ H+ .

(19)

Proposition 1. For any f ∈ H+ and t ≥ 0 the following inequality holds (Ut f, Ut f )+ ≤ (f, f )+ . Proof. See Attachment in Sec. 9.

(20)

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Inequality (20) implies that Ut I0 ⊂ I0 for t > 0, consequently the semigroup ˆt , t ≥ 0 of operators in G: {Ut , t ≥ 0} in H+ generates the semigroup U ˆt [f ] = [Ut f ], U

t > 0,

f ∈ H+ .

ˆt is the self-adjoint contraction semigroup. In It follows from (19) and (20) that U ˆt is called a addition, it is strongly-continuous by construction. This semigroup U transfer-matrix of the process x. The Stone theorem, see e.g. [14, Vol. 1, Chap. VIII], ˆt have the form: implies that the operators U ˆt = eth , U where h is a non-positive self-adjoint operator in G. Note, that the element e = [1] is the normalized ground state of the operator h with the eigenvalue 0. Let F be a local function on the space of trajectories {γt , t ∈ R} of the process, that means there exist a finite interval I ⊂ R1 and a bounded domain Λ0 ⊂ Rd such that the function F depends only on {γt ∩ Λ0 , t ∈ I}, i.e. on the part of trajectories {γt , t ∈ R} lying inside of a bounded domain Λ0 × I = M in Rd+1 . The domain M = Λ0 × I is called the localization domain for F . Remind that the weak convergence of the processes PΛ,z ⇒ P∞,z means that for any local bounded function F the following holds as Λ%Rd hF iPΛ,z 7→ hF iP∞,z ,

(21)

where h·iPΛ,z means the average over the distribution PΛ,z and the same for h·iP∞,z . Theorem 3. Under conditions of Theorem 1 distributions PΛ,z of the processes (15) converge weakly as Λ%Rd to the distribution P∞,z of a stationary reversible OS-positive process G∞ = {γt , t ∈ R1 }

(22)

with values γt ∈ Γ. Moreover, the stationary distributions νzΛ converge weakly to the marginal distribution νz∞ of the process (22). The weak convergence of measures on the space Γ of locally-finite configurations is defined by the same way as in (21) with local functions F depending on the part of the configuration γ ∈ Γ in a bounded domain Λ0 ⊂ Rd : F (γ) = F (γ|Λ0 ). Theorem 3 is a corollary of Theorem 4 below, a construction and investigations of process (22) will be done in the proof of Theorem 4. Remark. Using as above OS-positivity of the process (22) with the distribution ˆ acting in the P∞,z we can introduce the generator of its transfer-matrix h = H corresponding physical space G. The generator can be treated as a correctly defined limit Hamiltonian (up to an additive constant) associated to the formal Hamiltonian ˆ can be considered as a regularized limit for the operators H ˆ Λ. (4). The operator H Conjecture. Although we proved here only that the limit process has the property of OS-positivity, we believe that the limit process should be Markov.

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3. Euclidean Representation 3.1. Path space measure for free Glauber dynamics We denote by Υ a space of configurations η = (γ, lγ ) = {(x, lx )|x ∈ γ} of a marked point field in the space Rd × R1 = Rd+1 . Here γ ⊂ Rd+1 is a locallyfinite configuration of points in Rd+1 : x ∈ γ : x = (s, t) ∈ Rd+1 ,

s ∈ Rd ,

t ∈ R1 ,

and lx ∈ (0, ∞) is a value of the mark at the point x. The distribution Pz0 of the marked point field can be described by the following way: point configurations γ form the Poisson field in Rd+1 with an intensity z > 0 (the corresponding distribution is denoted by Πz , see [19, 20]), and under a fixed point configuration γ the conditional distribution of marks is conditionally independent and exponential: Pr(l > u) = e−u ,

u ≥ 0.

(23)

A configuration η ∈ Υ can be visually depicted as a configuration η of rods ξ ∈ η lying in Rd+1 and directed along the positive direction of the time axis t. Here x = x(ξ) = (s, t) ∈ Rd+1 is the origin of the rod, and l = l(ξ) is the length of the rod. Let K be a space of all rods in Rd+1 lying along a given (time) direction, then 1 K is the same as Rd+1 × R+ , and the described above configuration space Υ of rods 1 ) of could be identified with a subset of all locally finite configurations Γ(Rd+1 × R+ d+1 1 points in R × R+ . This permits to introduce a topology and the Borel σ-algebra on Υ generated by this topology. We shall say that a configuration of rods η is locally finite if any bounded subset of Rd+1 has an intersection only with a finite number of rods from this configuration. Let us denote by Υ0 ⊂ Υ a set of locally finite configurations composed of pairwise disjoint rods. Lemma 1. The set Υ0 of locally finite configurations of pairwise disjoint rods form a set of the full measure Pz0 . For the proof, see Attachment in Sec. 9. For any configuration η ∈ Υ0 and any τ ∈ R1 we consider a section η ∩ Yτ ⊂ d+1 R of rods from the configuration η by the hyperplane Yτ = {x = (s, t) : t = τ }. Then by Lemma 1 we have that the projection of the section to the space Rd is a locally finite set γτ ∈ Γ(Rd ). Thus, any configuration of rods η ∈ Υ0 generates a curve γ = {γτ , τ ∈ R} in the space Γ(Rd ), and different curves correspond to different configurations of rods. Let a set of these curves will be Σ. Then the distribution Pz0 can be regarded as a distribution on Σ, in this case call it Pˆz0 . Thus, using Lemma 1 we get the following Lemma 2. The above curves γ = {γτ , τ ∈ R} ∈ Σ form the full measure set of trajectories of the free Glauber dynamics from Sec. 2.1 with the generator (3), and the distribution Pˆz0 on Σ is the same as the distribution of the Glauber dynamics.

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In the representation for the trajectory γ = {γτ , τ ∈ R} in the form of a rod configuration η the origin x = (s, t) of the rod ξ ∈ η, ξ = (x, lx ) marks the position s and the time t of the birth of a new particle in the point configuration, and the length lx is the life time of the particle. Remark. Reversibility of the free Glauber dynamics implies that the above field of rods is also reversible in time. Indeed, ends of rods under reflection in time come to origins of the reflected rods, but the point field corresponding to the ends of all rods is also the Poisson field in Rd+1 with the intensity z. This fact has been discussed earlier, see, for instance, [3]. 3.2. Euclidean representation for dynamics of interacting particles (ensemble of rods) For any bounded Λ ⊂ Rd and any 0 < T < ∞ we consider a new probability measure PˆΛ,T,z on Σ using Feynmann–Kac representation: ( ) Z T dPˆΛ,T,z 1 (γ) = exp −α UΛ (γτ )dτ , γ = {γτ , τ ∈ R} (24) ZΛ,T dPˆz0 −T with the normalization factor ( Z Z ZΛ,T = exp −α Σ

)

T

UΛ (γτ )dτ

−T

dPˆz0 .

(25)

RT Since −∞ < −T UΛ (γτ )dτ < ∞ on a set of the full measure, then 0 < ZΛ,T < ∞ and relation (24) is correctly defined. We denote by PΛ,T,z a measure on the configuration space of rods which is corresponding to PˆΛ,T,z . Then the probability density (24) is rewritten as     X dPΛ,T,z 1 T exp −α Φ (ξ , ξ ) , (η) = (26) j j 1 2   dPz0 ZΛ,T {ξj1 ,ξj2 }⊂ηΛ,T

where ηΛ,T ⊆ η is a subset of rods from configuration η which have intersection with Λ × [−T, T ] ⊂ Rd+1 , and ΦT (ξ1 , ξ2 ) = ϕ(s1 − s2 ) ∆T (ξ1 , ξ2 ),

(27)

with ξi = ( (si , ti ), li ), i = 1, 2, and ∆T (ξ1 , ξ2 ) = |(t1 , t1 + l1 ) ∩ (t2 , t2 + l2 ) ∩ (−T, T )|

(28)

is a length of the common part of the projections to the axis t of the rods ξ1 and ξ2 which are inside of [−T, T ]. We introduce the following notation ∆(ξ1 , ξ2 ) = lim ∆T (ξ1 , ξ2 ) = |(t1 , t1 + l1 ) ∩ (t2 , t2 + l2 )|. T →∞

(29)

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Thus the measure PΛ,T,z on the space Υ0 of rods is the Gibbs reconstruction of the measure Pz0 by means of the following pair interaction X UΛ,T (η) = ΦT (ξ1 , ξ2 ). {ξ1 ,ξ2 }⊂ηΛ,T

For any M ⊂ Rd+1 we consider a set Gint M of all rods intersecting M and a set int ⊂ G of all rods with origins in M . We say that a set of rods G ⊂ K is Gloc M M bounded if there exists a bounded set M ⊂ Rd+1 such that G ⊆ Gint M , and is 0 0 strictly bounded if G ⊆ Gloc . For any G ⊂ K let Υ (G) ⊆ Υ be a set of all locally M finite configurations of pairwise disjoint rods from G. In the case of bounded G the space Υ0 (G) contains finite configurations η. For any G ⊂ K we can represent configurations η ∈ Υ0 as η = (ηG , ηG0 ),

ηG ∈ Υ0 (G),

ηG0 ∈ Υ0 (G0 )

with G0 = K\G.

This representation implies the following decomposition of Υ0 to the Cartesian product Υ0 = Υ0 (G) × Υ0 (G0 ). We say that a function F = F (η) defined on Υ0 is local if there exists a bounded set M ⊂ Rd+1 such that F depends only on ηGint ⊆ η : F (η) = F (ηGint ). If a M M function F depends only on ηGloc ⊆ η : F (η) = F (η ) we denote the function F loc GM M d+1 strongly local. In any case, the set M ⊂ R is called the localization domain of the function F . Let us note, that each local function on the space of trajectories Σ with the localization domain Λ0 × I = M regarded as a function on the configuration space of rods Υ0 with the localization domain M . Theorem 4. We assume that ϕ is a stable integrable potential, and α is a small enough. Then (1) the distributions PΛ,T,z converge weakly as T %∞ to the distribution PΛ,∞,z = PΛ,z PΛ,∞,z = w − lim PΛ,T,z , T →∞

(30)

and the corresponding distribution PˆΛ,∞,z = PˆΛ,z on the space Σ is a distribution of a stationary reversible Markov process on Γ GΛ = {γt , t ∈ R1 }

(31)

with the invariant measure νzΛ and associated stochastic semigroup S˜tΛ generated ˜ Λ . Thus, process (31) is exactly the same as process (15) from Theorem 2, by H that has been constructed by the different way.

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(2) There exists a weak limit P∞,z = w − lim PΛ,∞,z = w − Λ%Rd

lim

T →∞,Λ%Rd

PΛ,T,z .

(32)

The corresponding distribution Pˆ∞,z on the space of trajectories Σ is a distribution of a stationary reversible OS-positive process G∞ = {γt , t ∈ R1 }

(33) νz∞

− limΛ%Rd νzΛ .

with values γt ∈ Γ and the marginal distribution =w the process (33) is the same as limit process (22) from Theorem 3.

Thus,

Here, as above the weak convergence of distributions, means the convergence of averages over corresponding distributions for any bounded local function F defined on Υ0 . Remark. In the next section, we will prove the crucial point of our reasoning — Lemma 5 — for a general stable integrable potential ϕ. However, in what follows we will consider the case of a nonnegative potential to make our reasoning more simple. The general case can be considered in a similar way with evident modifications in the proof, using the estimates from Lemma 5. We formulate next results on decay of correlations for the distributions PΛ,z and the limit distribution P∞,z . We assume that for all large enough |u| the potential ϕ meets one of the following estimates: c (1) |ϕ(u)| < (34) , m > d, (2) |ϕ(u)| < ce−k|u| , (1 + |u|)2m with constants c > 0, m > d, k > 0, and introduce the following metrics in the space Rd × R1 :  1    m ln(1 + |s1 − s2 |) + |t1 − t2 |, in the case (1), 2 %((s1 , t1 ), (s2 , t2 )) = (35)    k |s1 − s2 | + 1 |t1 − t2 |, in the case (2) 2 2 with s1 , s2 ∈ Rd , t1 , t2 ∈ R1 . Theorem 5. For any strongly local bounded functions F1 , F2 depending on the process G∞ (or GΛ ) with localization domains Mi = Λi ×Ii , i = 1, 2 correspondingly the following estimate holds: ¯ ¯ ¯hF1 · F2 iP∞,z − hF1 iP∞,z hF2 iP∞,z ¯ < C |M1 |+|M2 | (|M1 | + |M2 |) e−d(M1 ,M2 ) , (36) and the analogous one is true for the processes GΛ . Here C > 0 is a constant, |Mi | are (d + 1)-dimensional volumes of the domains Mi , and d(M1 , M2 ) is the distance between the domains in the metric (35): d(M1 , M2 ) =

inf

(si ,ti )∈Mi ,i=1,2

%((s1 , t1 ), (s2 , t2 )).

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ˆ has a spectral gap, i.e. there exists a gap between 0 Corollary. The operator H ˆ and the spectrum of H in the orthogonal complement to the ground state e ∈ G. In particular, that implies the uniqueness of the ground state. For computational convenience we modify the definition of the measure PΛ,T,z . We call G0Λ,T = Gint Λ×(−T,T ) ⊂ K a set of rods which have an intersection with Λ × (−T, T ) ⊂ Rd+1 , G± Λ,T ⊂ K a set of rods entirely belonging to the region Λ × (T, +∞) (in the case +) and correspondingly, entirely belonging to the region Λ×(−∞, −T ) (in the case −), GΛ0 ,∞ ⊂ K a set of rods lying inside Λ0 ×(−∞, ∞) ⊂ Rd+1 with Λ0 = Rd \Λ. Obviously, these sets are mutually disjoint and their union is the same as K. Then any configuration η ∈ Υ0 is the sum of 4 mutually disjoint configurations η = ηG0Λ,T ∪ ηG+ ∪ ηG− ∪ ηGΛ0 ,∞ ,

(37)

Λ,T

Λ,T

and the configuration space Υ0 is the Cartesian product of the spaces − 0 0 Υ0 = Υ0 (G0Λ,T ) × Υ0 (G+ Λ,T ) × Υ (GΛ,T ) × Υ (GΛ0 ,∞ ).

(38)

We denote by 0 PG 0

Λ,T ,z

0 , PG +

Λ,T ,z

0 , PG −

Λ,T ,z

0 , PG Λ0 ,∞ ,z

(39)

distributions on the spaces of corresponding configurations, i.e. restrictions of the distribution Pz0 to the sets − 0 0 Υ0 (G0Λ,T ), Υ0 (G+ Λ,T ), Υ (GΛ,T ), Υ (GΛ0 ,∞ ),

respectively. We consider next a general random field Π(M, ζ, p). Here M ⊂ Rd+1 is a domain in Rd+1 , ζ = ζ(x), x ∈ M is a positive function defined on M , the function ζ(x) specifies an activity (non-homogeneous, in general) of the Poisson field of the rods origins x ∈ M , p = {px (l) = Pr(lx > l), l > 0, x ∈ M } is a family of distribution functions marked by points x ∈ M for the length of a rod with the origin at x ∈ M . Under fixed origins of the rods their lengths have conditionally-independent distributions with densities px (l). Lemma 3. (1) All components ηG0Λ,T , ηG+ , ηG− , ηGΛ0 ,∞ in decomposition (37) Λ,T Λ,T are independent, i.e. 0 Pz0 = PG 0

Λ,T ,z

0 × PG +

Λ,T ,z

0 × PG −

Λ,T ,z

0 × PG . Λ0 ,∞ ,z

(40)

(2) Each distribution from (39) is a distribution of the form Π(M, ζ, p), namely, 0 (a) in the case PG Λ0 ,∞ ,z

M = Λ0 ×(−∞, +∞),

ζ(x) ≡ z,

px (l) = e−l ,

l > 0,

x ∈ M;

(41)

0 (b) in the case PG +

Λ,T ,z

M = Λ × (T, +∞),

ζ(x) ≡ z,

px (l) = e−l ,

l > 0,

x ∈ M;

(42)

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0 (c) in the case PG 0

Λ,T ,z

½

z, ze−τ ,

x ∈ Λ × (−T, T ) , x ∈ Λ × (−∞, −T )

e−l , l > 0, eτ e−l χ(τ, ∞) + χ(0, τ ),

x ∈ Λ × (−T, T ) , x ∈ Λ × (−∞, −T )

M = Λ × (−∞, T ), ½ px (l) =

ζ(x) =

(43)

where τ = −T − t > 0 is a distance from x = (s, t) ∈ Λ × (−∞, −T ) to the hyperplane Y−T = {x : t = −T }, χ(a, b) is the characteristic function of the interval (a, b); 0 , (d) in the case PG − ,z Λ,T

M = Λ × (−∞, −T ), px (l) = e−l χ(0, τ ),

ζ(x) = z(1 − e−τ ), l > 0,

(44)

x ∈ M.

Proof of Lemma 3. We consider a decomposition of a configuration η ∈ Υ0 to four configurations η = ηˆΛ,0 ∪ ηˆΛ,+ ∪ ηˆΛ,− ∪ ηˆΛ0 .

(45)

Here ηˆΛ0 is a configuration of rods which are entirely outside of the cylinder Λ × (−∞, +∞), ηˆΛ,− is a configuration from rods with origins in Λ × (−∞, −T ), ηˆΛ,0 is a configuration from rods with origins in Λ × (−T, T ) and ηˆΛ,+ is a configuration of rods which are entirely inside of the region Λ × (T, +∞). Since any rod ξ ∈ η belongs to one of the sub-configurations ηˆΛ,0 , ηˆΛ, + , ηˆΛ,− , ηˆΛ0 , and it is determined only by the origin of the rod ξ, all these configurations are configurations of a marked Poisson field in the corresponding volumes Λ × (−T, T ), Λ × (T, ∞), Λ × (−∞, −T ), Λ0 × (−∞, ∞). That implies independence of all configurations, and comparing (37) with (45) we have: ηˆΛ0 = ηGΛ0 ,∞ ,

ηˆΛ,+ = ηG+ , Λ,T

ηˆΛ,0 ∪ ηˆΛ,− = ηG0Λ,T ∪ ηG− . Λ,T

The configuration ηˆΛ,− can be decomposed into two configurations ηˆΛ,− = η˘Λ,− ∪ η˘Λ,0 , where η˘Λ,− is a configuration of rods which are entirely inside of the region Λ × (−∞, −T ), η˘Λ,0 is a configuration of rods with origins in Λ × (−∞, −T ) which have intersection with Λ × (−T, T ). Since the question about belonging a rod ξ ∈ ηˆΛ,− to one of the configurations η˘Λ,− , η˘Λ,0 depends on the length of the rod, the configurations η˘Λ,− and η˘Λ,0 are independent, and the activity of the rods origins equals to the product of z on the probability to reach (in the case η˘Λ,0 ) or not to

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reach (in the case η˘Λ,− ) the level −T . Distributions for the length of the rods are also properly changed. Moreover, η˘Λ,− = ηG− ,

ηG0Λ,T = η˘Λ,0 ∪ ηˆΛ,0 .

Λ,T

Thus, configurations η˘Λ,− form a field (M, ζ, p) defined by (44) and the union of configurations η˘Λ,0 ∪ ηˆΛ,0 form again the field (M, ζ, p) defined by (43). ¤ 0 Formula (26) implies that only the distribution PG of the component ηG0Λ,T 0 Λ,T ,z is subjected to a reconstruction: 0 PΛ,T,z = PG0Λ,T ,z × PG +

Λ,T ,z

0 × PG −

Λ,T ,z

0 , × PG Λ0 ,∞ ,z

(46)

where PG0Λ,T ,z is assigned by a probability density analogous to (26): dPG0Λ,T ,z

(ηGΛ,T ) 0 dPG 0 ,z Λ,T 0

=

1 ZΛ,T

   exp −α   {ξj

X

ΦT (ξj1 , ξj2 )

,ξ }⊂ηG0 1 j2

    

.

(47)

Λ,T

We denote by Υ0 =

∞ [

Υ0n

(48)

n=0

a space of finite configurations of rods in Rd+1 . Here Υ0n is a set of all n rods configurations, and Υ00 = {∅}. Then the Lebesgue–Poisson measure λζ,p with intensity ζ = ζ(x) and distribution function p = {px (l)} for the length of rods can be considered on the space Υ0 . This measure on each Υ0n , n = 0, 1, 2, . . . is defined as follows Z Υ0n

f (η)dλζ,p

1 = n!

Z Kn

f˜(ξ1 , . . . , ξn )

n Y

(−dpxi (li ))

i=1

n Y

ζ(xi )dxi ,

(49)

i=1

where ξi = (xi , li ) are rods, f (η) is a bounded function on Υ0 with a finite support and f˜(ξ1 , . . . , ξn ) is a symmetrical extension of f (η) to the space K n of ordered sequences of rods (ξ1 , . . . , ξn ), ξi ∩ ξj = ∅, i 6= j. We will use the notation λζ,p = λz in the case when ζ ≡ z, px (l) = e−l , l > 0, ∀x. For any G ⊂ K let Υ0 (G) ⊂ Υ0 be a set of all finite configurations of rods from G, and λG ζ,p be a restriction of the measure λζ,p to the set Υ0 (G). Then for any two non-intersecting domains G1 , G2 ⊂ K we have Υ0 (G1 ∪ G2 ) = Υ0 (G1 ) × Υ0 (G2 ),

G2 1 ∪G2 1 λG = λG ζ,p ζ,p × λζ,p .

(50)

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Moreover, for any integrable functions F (η), φi (η), i = 1, . . . , m defined on the space Υ0 the following equality holds, see for example [10],   Z (η) X F (η) φ1 (η1 ) . . . φm (ηm ) dλζ,p (η) Υ0

(η1 ,...,ηm )

Z

Z =

... Υ0 | {z

Υ0

F (η1 ∪ · · · ∪ ηm )

}

m Y

φi (ηi )

i=1

m Y

(51)

dλζ,p (ηi ).

i=1

m times

Here the sum in the left-hand side of (51) is taken over all ordered sets (η1 , . . . , ηm ) of m nonempty finite configurations ηi ∈ Υ0 such that ηi ∩ ηj = ∅, i 6= j, and ∪m 1 ηi = η. In what follows we will take G ⊂ G0Λ,T and will write for simplicity λG = λG ζ,p , G 0 where ζ(x), p = {px (l)} are defined by formulas (43). The measure λ (Υ (G)) for any bounded domain G ⊂ G0Λ,T is equal to ½Z ¾ λG (Υ0 (G)) = exp ζ(x)(−dpx (l))dx . (52) G

In the case G =

G0Λ,T Z

¡ ¢ = Gint M , M = Λ × (−T, T )

G0Λ,T

ζ(x)(−dpx (l))dx = (2T + 1)|Λ|z,

(53)

where |Λ| is a d-dimensional volume of the domain Λ ⊂ Rd . In the case G = Gloc M =Λ×(−T,T ) Z ζ(x)(−dpx (l))dx = 2T |Λ|z. Gloc M

0 Lemma 4. The probability measure PG 0

on Υ0G0

Λ,T ,z

0 PG 0

Λ,T

defined by (43) is equal to

0

Λ,T ,z

= e−(2T +1)|Λ|z λGΛ,T .

(54)

For the proof, see Attachment in Sec. 9. Next we consider the probability density pˆΛ,T,z (η) =

dPG0Λ,T ,z 0

dλGΛ,T

(η)

(55)

0

0 for PG0Λ,T ,z with respect to the measure λGΛ,T (instead of the measure PG 0

Λ,T ,z

).

Density (55) can be defined again by formula (47) where the new normalizing factor ZˆΛ,T is related with the normalizing factor from (47) by the following way ZˆΛ,T = ZΛ,T e(2T +1)|Λ|z . Let us note that for any bounded G ⊂ K the sets Υ0G and Υ0 (G) are the same up 0 to a set with zero λG measure, consequently the distributions PG and PG0Λ,T ,z 0 ,z could be considered as distributions on Υ0 (G0Λ,T ).

Λ,T

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4. Related Cluster Expansion We will follow here constructions from the book [10]. The density (55) pˆΛ,T,z (η), η ∈ Υ0 (G0Λ,T ) ≡ ΥΛ,T has the following representation  −1  ZˆΛ,T , η = ∅,    (η) m X Y pˆΛ,T,z (η) = (56)  ˆ −1 Z K T (ηi ), η 6= ∅,  Λ,T   {η1 ,...,ηm } i=1

P(η)

where the sum {η1 ,...,ηm } is taken over all partitions of the finite configuration of rods η, i.e. over all unordered sets of mutually-disjoint configurations η1 , . . . , ηm , ηi ⊆ η, i = 1, . . . , m, m = 1, 2, . . . , such that ∪ηi = η. We will use further the following designation: {η1 , . . . , ηm } for unordered sets and (η1 , . . . , ηm ) for ordered sets of configurations. The cluster weight K T (η) is equal to  1, |η| = 1,    (η) K T (η) = X (57)   κTσ , |η| ≥ 2  σ

P(η) where |η| is the number of rods in the configuration η, the sum σ is taken over all connected graphs σ with the set of nodes V (σ) = {ξ1 , . . . , ξs } = η, and Y T κTσ = (e−α Φ (ξi ,ξj ) − 1), (58) hξi ,ξj i∈σ

where the product is over all edges hξi , ξj i of the graph σ. Lemma 5. For any stable potential ϕ the cluster weight meets the following bound as |η| ≥ 2 T

αB

|K (η)| ≤ e

P ξ∈η

l(ξ)

(η) X

Y

T

hξi ,ξj i∈T

(1 − e−α |Φ

T

(ξi ,ξj )|

),

(59)

where the sum is taken over all trees with the vertex set V (T ) = η, l(ξ) is the length of the rod ξ, B is a constant. Under ϕ ≥ 0 (B = 0) bound (59) can be rewritten as |K T (η)| ≤

(η) X

Y

T

hξi ,ξj i∈T

|(e−α Φ

T

(ξi ,ξj )

− 1)|

(60)

where the sum is taken over all trees with the same vertex set V (T ) = η as above. For the proof, see Attachment. We formulated Lemma 5 for the general case of a stable potential because bound (59) is the crucial point in the proof of Theorem 4. Further line of the proof is well

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adapted to the general case, and we will consider in what follows the case of a non-negative potential ϕ ≥ 0 to simplify our reasoning. Then Z ZˆΛ,T = 1 +

(η) X

m Y

Υ0 (G0Λ,T )\∅ {η ,...,η } i=1 1 m

0

K T (ηi )dλGΛ,T (η).

(61)

Using equality (51), we have ZˆΛ,T = 1 +

ÃZ

∞ X 1 m! m=1

!m T

K (η)dλ

G0Λ,T

(η)

Υ0 (G0Λ,T )\∅

(Z

) T

= exp

K (η)dλ

G0Λ,T

(η) .

(62)

Υ0 (G0Λ,T )\∅

For any bounded set G ⊂ G0Λ,T the space Υ0 (G0Λ,T ) can be written as Υ0 (G0Λ,T ) = Υ0 (G) × Υ0 (G0Λ,T \G).

(63)

We denote a distribution on Υ0 (G) generated by the distribution PG0Λ,T ,z (47) as G PΛ,T,z . Lemma 6. The probability density pG Λ,T,z (η) =

pG Λ,T,z (η)

=

G fΛ,T

 1,      

G dPΛ,T ,z , dλG

η ∈ Υ0 (G) is equal to η = ∅,

(η)

m Y

X

G rΛ,T (ηi ),

(64)

η 6= ∅,

{η1 ,...,ηm } i=1

where the sum is taken over partitions of the configuration η, and Z G rΛ,T (η) =

G fΛ,T

0

K T (η ∪ η¯) dλGΛ,T \G (¯ η ),

(65)

Υ0 (G0Λ,T \G)

( Z = exp −

) Υ0 (G)\∅

( Z = exp −

G rΛ,T (η)dλG (η)

)

Z G

T

dλ (η) Υ0 (G)\∅

G0Λ,T \G

K (η ∪ η¯)dλ Υ0 (G0Λ,T \G)

Proof of Lemma 6 follows the same line as in [10].

(¯ η) .

(66)

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It follows from (64) that the average of any bounded function F on Υ0 (G) with a bounded set G ⊂ G0Λ,T equals to G hF iPΛ,T ,z = hF iPΛ,T ,z

 G  = fΛ,T F (∅) +

" G = fΛ,T

×

m Y

G = fΛ,T

×

m Y

(η) X

F (η) Υ0 (G)\{∅}

m Y

G (ηi ) rΛ,T

m Y





G rΛ,T (ηi ) dλG (η)

{η1 ,...,ηm } i=1

Z Z ∞ X 1 F (∅) + ... F m! Υ0 (G)\{∅} Υ0 (G)\{∅} m=1

i=1

"



Z

Ãm [

! ηi

i=1

# G

dλ (ηi )

i=1

Z Z ∞ X 1 F (∅) + F m! (Υ0 (G)\{∅})m (Υ0 (G0Λ,T \G))m m=1 m Y

T

K (ηi ∪ η¯i )

i=1

G

dλ (ηi )

i=1

m Y

Ãm [

! ηi

i=1

# G0Λ,T \G

dλ

(67)

(¯ ηi ) .

i=1

For any bounded set Λ ⊂ Rd we introduce a “tube” 1 ⊂K GΛ,∞ = Λ × {−∞, ∞} × R+

in the space of rods. Lemma 7. There exist the following limits as T → ∞: (1) for any bounded domain G ⊂ GΛ,∞ : G λG ζ,p → λz ,

(68) λG z

λG ζ,p

is the Lebesque–Poisson measure defined by (43), and is the where Lebesque–Poisson measure with parameters ζ(x) ≡ z and px (l) = e−l , see (42); (2) for any finite configuration η ∈ Υ0 K T (η) → K(η),

(69)

where K(η) is defined similarly to K T (η), see (57), (58) with Φ(ξ1 , ξ2 ) instead of the function ΦT (ξ1 , ξ2 ), see (27); (3) for any bounded domain G ⊂ GΛ,∞ : G G fΛ,∞ = lim fΛ,T T →∞ ( Z

)

Z

= exp −

K(η ∪ Υ0 (G)\{∅}

G where fΛ,T is defined in (66);

Υ0 (GΛ,∞ \G)

GΛ,∞ \G η¯)dλG (¯ η) z (η)dλ

(70)

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G lim hF iPΛ,T = hF iPΛ,∞,z ,z

469

(71)

T →∞

for any bounded domain G ⊂ GΛ,∞ and a bounded local function F . Here Ã Z m ∞ Y X 1 G hF iPΛ,∞,z = fΛ,∞ F (∅) + dλG (ηi ) m! 0 m (Υ (G)\{∅}) i=1 m=1 Ãm ! m ! Z m Y [ Y GΛ,∞ \G × dλ (¯ ηi )F ηi K(ηi ∪ η¯i ) . (Υ0 (GΛ,∞ \G))m i=1

i=1

i=1

Proof. The statements of items (1) and (2) are clear. (3) The integral in (66) can be bounded from above by ¯Z ¯ Z ¯ ¯ G0Λ,T \G ¯ ¯ G T dλz (η) K (η ∪ η¯)dλz (¯ η )¯ ¯ ¯ Υ0 (G)\{∅} ¯ 0 0 Υ (GΛ,T \G) 

Z

Z dλG z (η)

< Υ0 (G)\{∅}

Z

Υ0 (GΛ,∞ \G)

Z

=z

Υ0 (G)

Z

χG (ξ) |K T (η ∪ η¯)|

ξ∈η

Z

dξ G

Λ,∞ \G (¯ η)  dλG z

 X

dλG z (η)

Υ0 (GΛ,∞ \G)

dλzGΛ,∞ \G (¯ η )|K T (ξ ∪ η ∪ η¯)|

Z

=z

dξ Υ0 (GΛ,∞ )

G

dλzGΛ,∞ (η)|K T (ξ ∪ η)|.

(72)

Here χG (ξ), ξ = (x, l) is the characteristic function of G, dξ = dx(−dp(l)) = dxe−l dl and we used in (72) general formula (50) and equality (51). Next using estimate T

|e−αΦ

(ξi ,ξj )

− 1| ≤ |e−αΦ(ξi ,ξj ) − 1|

and estimate (60) we can continue Z Z z dξ dλzGΛ,∞ (η)|K T (ξ ∪ η)| Υ0 (GΛ,∞ )

G

Z = z|G| + z

dξ0 G

< z|G| +

×

Z ∞ X z n−1 |K T (ξ0 , ξ1 , . . . , ξn−1 )|dξ1 · · · dξn−1 (n − 1)! n−1 K n=2

∞ X

zn (n − 1)! n=2

Y hξi ,ξj i∈T

|e−αΦ

T

X T :V (T ) = {ξ0 , ξ1 , . . . , ξn−1 }

(ξi ,ξj )

− 1|

Z

Z dξ0

G

K n−1

dξ1 · · · dξn−1

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≤ z|G| +

∞ X

zn (n − 1)! n=2

Y

×

Z

X

Z dξ0

T : V (T ) = {ξ0 , ξ1 , . . . , ξn−1 }

G

K n−1

dξ1 · · · dξn−1

|e−αΦ(ξi ,ξj ) − 1|

hξi ,ξj i∈T

≤ z|G| +

∞ X

zn (n − 1)! n=2

Z ×

Z dξ0

G

n−1 Y

K n−1 i=1

Here the summation in

P

X

X

κ0 , . . . , κn−1 : T :V (T )={ξ0 , . . . , ξn−1 } κ0 + · · · +κn−1 =2(n−1) r(i)=κi , i=0, . . . , n−1

dξi

Y

|e−αΦ(ξi ,ξj ) − 1|.

(73)

hξi ,ξj i∈T

T :V (T )={ξ0 , . . . , ξn−1 } r(i)=κi , i=0, . . . , n−1

is over trees T with a set of points

of the tree {ξ0 , . . . , ξn−1 } and vertex degrees r(i) = κi > 0, i = 0, . . . , n − 1. We notice that any tree with n points has exactly n − 1 edges and the sum of the vertex degrees equals to 2(n − 1). The integration over variables ξ0 , ξ1 , . . . , ξn−1 will operate recurrently. On the first step we will integrate over all “end variables”, i.e. the variables ξi associated with the end points of the tree T 0 = T except the variable ξ0 . Then we pass on to a new tree T (1) , which is constructed as a result of eliminating of all integrated on the first step points together with corresponding edges. If we continue this procedure we will integrate over all variables ξ1 , . . . , ξn−1 step by step, and ξ0 will be the last variable for integration. Let us estimate a result of the first integration over the “end variables” for the following integral (under given tree T 0 ): Z Z n−1 Y Y I(T 0 ) = dξ0 (74) dξi |e−αΦ(ξi ,ξj ) − 1|. G

K n−1 i=1

hξi ,ξj i∈T 0

We will show next that for a fixed rod ξ ∈ K the integral Z ¯ ¯ J(ξ) = |e−αΦ(ξ,ξ) − 1|dξ.

(75)

K

meets the following estimate J(ξ) < 4αRl(ξ) (76) R ¯ ≥ with a length l(ξ) of the rod ξ, and R = Rd ϕ(u)du. We remind that Φ(ξ, ξ) ¯ ¯ ¯ 0, α > 0, then putting ξ = {(¯ s, t), l} we have Z Z ¯ sdt¯e−¯l d¯l < αR ¯ t¯e−¯l d¯l, J(ξ) < α ϕ(s − s¯) ∆(ξ, ξ)d¯ ∆(ξ, ξ)d (77) K

1 R1 ×R+

¯ does not depend on s and s¯. We consider now 4 cases to get an estimate since ∆(ξ, ξ) on the integral in (77).

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(1) A projection of the rod ξ¯ to the time axis t is entirely covered by a projection of the rod ξ, i.e. in the notation ξ = {(s, t), l} we have t < t¯ < t¯ + ¯l < t + l. In this case Z l Z Z l−¯l ¯ t¯e−¯l d¯l = ¯le−¯l d¯l ∆(ξ, ξ)d dt¯ 1 R1 ×R+

0

Z

=o

0 l

¯l(l − ¯l)e−¯l d¯l < l

Z

0

∞

¯le−¯l d¯l

0

= l.

(78)

(2) In the projection to the axis t the rods ξ and ξ¯ are overlapping in such way that t < t¯ < t + l < t¯ + ¯l. Then Z Z l Z ∞ ¯ t¯e−¯l d¯l = ∆(ξ, ξ)d ue−u du e−m dm < l. (79) 0

0

We used here new variables ¯l = u + m,

¯ u = ∆(ξ, ξ),

t¯ = l − u.

(3) The similar case: the projections of rods ξ and ξ¯ are overlapping in such way that t¯ < t < t¯ + ¯l < t + l. Then we have the same estimate as (79). (4) A projection of the rod ξ to the time axis t is entirely covered by a projection ¯ i.e. t¯ < t < t + l < t¯ + ¯l. In this case of the rod ξ, Z Z ∞ Z ∞ −¯ l ¯ −l −m1 ¯ ¯ ∆(ξ, ξ)dte dl < le e dm1 e−m2 dm2 < l. (80) 0

0

Thus, (77)–(80) immediately imply (76). For any node i of the reminder tree, i.e. for nodes of a new tree T 1 , we denote by K (1) (i) the number of bonds of the tree T 0 incident to the node i and eliminated on the first step. The above estimates imply the following bound on the integral I(T 0 ) Z Z Y 0 I(T ) < dξ0 |e−αΦ(ξi ,ξj ) − 1| G

×

K n1 −1

Y

hξi ,ξj i∈T (1)

(4Rαli )K

(1)

(i)

i∈V (T (1) )

Y

dξi ,

(81)

i ∈ V (T (1) ) i6=0

where n1 is a number of nodes of the tree T (1) . We notice that for end nodes i ∈ V (T (1) ) of the tree T (1) , i.e. nodes eliminated on the second step of our procedure, we have K (1) (i) = κi − 1, where κi is the degree of the node i in the original tree T (0) . Thus, to continue the estimation of integral I(T 0 ) after the second step we have to find a bound for the

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following integral Z

¯ |e−αΦ(ξ,ξ) − 1|(4Rα¯l)κ−1 dξ¯ Z ¯ ¯lκ−1 e−¯l dt¯d¯l. < (4Rα)κ−1 Rα ∆(ξ, ξ)

Jκ (ξ) =

(82)

Let us consider again 4 cases as above. (1) In the first case Z

Z

¯ ¯lκ−1 e−¯l dt¯d¯l = ∆(ξ, ξ)

l

¯lκ (l − ¯l)e−¯l d¯l < l

Z

0

∞

¯lκ e−¯l d¯l = lκ!

(83)

0

(2–3) Using the same change of variables as above ¯l = u + m,

¯ u = ∆(ξ, ξ),

t¯ = l − u,

we have Z

¯ ¯lκ−1 e−¯l dt¯d¯l = ∆(ξ, ξ)

Z

Z

l

−u

∞

ue 0

(m + u)κ−1 e−m dmdu

0

Z

∞

Z

∞

Z

(m + u)κ−1 e−m−u dmdu

0 ∞

=l

v κ e−v dv

0

= lκ!

(84)

(4) Here Z

¯ ¯lκ−1 e−¯l dt¯d¯l = l ∆(ξ, ξ)

Z

∞

0

Z

Z

∞

(m1 + m2 + l)κ−1 e−m1 −m2 −l dm1 dm2

0 ∞

Z sκ−1 e−s ds

0

Z
s−l

dm1 0

∞

sκ e−s ds

0

= lκ!

(85)

where s = m1 + m2 + l. Thus Jκ (ξ) < (4Rα)κ lκ!,

l = l(ξ),

(86)

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and consequently, Z Z 0 dξ0 I(T ) ≤ G

×

Y K n2 −1

|e−αΦ(ξi ,ξj ) − 1|

hξi ,ξj i∈T (2)

Y

Y

(4Rαli )K

(2)

473

(i)

i∈V (T (2) )

Y

κj !(4Rα)κj −1

j∈V (T (1) \T (2) )

dξi .

(87)

i ∈ V (T (2) ) i6=0

Here K (2) (i) is a number of bonds incident to the node i ∈ V (T (2) ) and eliminated on the second step, V (T (1) )\V (T (2) ) is a set of nodes of the tree T (1) eliminated on the second step. If we continue this procedure in the same way we get eventually the following estimate with the last node ξ0 Z Y ¡ ¢ I(T 0 ) ≤ l0κ dξ0 (4Rα)κ0 κi !(4Rα)κi −1 G

i∈V (T (0) ) i6=0

Z n−1

< (4Rα)

G

Y

l0κ dξ0

κi !.

(88)

i∈V (T (0) ) i6=0

P Here we used that i∈V (T (0) ) κi = 2n − 2. Let us estimate now Z l0κ dξ0 . G

We denote by M ⊂ Rd+1 a bounded closed subset of Rd+1 such that all rods from G have some intersection with M . And let GM , G ⊆ GM ⊂ K be a set of all rods from K intersecting M . We suppose that set M has no “time holes”, i.e. each straight line parallel to the time axis intersects M at a point or in a segment. Let Λ ⊂ Rd be a projection of M to the space Rd . Then the following bound holds Z Z l0κ dξ0 ≤ l0κ dξ0 = (κ + 1)!|Λ| + κ!|M |, (89) G

GM

where |Λ| is a d-dimensional volume of Λ, and |M | is a (d + 1)-dimensional volume of M . Indeed, we can represent GM as a union of two nonoverlapping sets GM = G0 ∪ G1 ,

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where G0 is a set of all rods with origins in M , and G1 is a set of rods with origins outside of M (in the “past” of M ). Then Z Z ∞ Z l0κ dξ0 ≤ dx lκ e−l dl = κ!|M | (90) G0

M

0

and Z

Z

∞Z ∞

(r + s)κ e−(r+s) drds = (κ + 1)!|Λ|.

(91)

Thus, estimate (88) together with (89) and bound κ + 1 ≤ n implies Y I(T 0 ) ≤ κi !(4Rα)n−1 (|M | + n|Λ|).

(92)

G1

l0κ dξ0 = |Λ|

0

0

i∈V (T (0) )

The number of all trees T with n vertices and fixed vertex degrees {κi , i ∈ V (T )} is equal, see [10, 13, 18], 2n (n − 2)! (n − 2)! Y < Y . (κi − 1)! κi ! i∈V (T )

i∈V (T )

The number of ordered set {κi } from n integer positive numbers such that κ1 + · · · + κn = 2n − 2, n < 22n−3 . Substituting these bounds to (73) can be estimated from above by C2n−3 we get for small enough α Z Z Λ,∞ z dξ dλG (η)|K(ξ ∪ η)| z G

Υ0 (GΛ,∞ )

≤ z|G| +

∞ X z n (n − 2)! (4Rα)n−1 23n−3 (|M | + n|Λ|) (n − 1)! n=2

< (|M | + |Λ|)C(α, z),

(93)

where C(α, z) is a constant does not depending on G. Finally, relations (69), (72), (93) and dominated convergence theorem imply (70). Let us consider now the last statement of the lemma. The expressions under the integral in the square brackets at two last lines in formula (67) converge as T → ∞ to the expressions Ãm ! m [ Y F ηi K(ηi ∪ η¯i ). i=1

i=1

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475

Consequently, the integral in the square brackets in expression (67) is majorized by the following sum Z Z ∞ X 1 max |F (η)| η m! (Υ0 (G)\{∅})m (Υ0 (G0Λ,∞ \G))m m=1 ×

m Y

K(ηi ∪ η¯i )

i=1

m Y i=1

∞ X 1 < max |F (η)| η m! m=1

dλG (ηi ) ÃZ Z

m Y

0

dλGΛ,∞ \G (¯ ηi )

i=1

!m K(ξ ∪ η)dλGΛ,∞ (η)dξ

Υ0 (GΛ,∞ )

G

(94)

< max |F (η)| exp{(|M | + |Λ|) C(α, z)}. η

Thus Lemma 7 is proved completely. Corollary. Formulae (64)–(66) imply that hF i is the average of the function F over G a probability distribution PΛ,∞,z on the set Υ0 (G), where the probability density with G respect to the measure λz is given by   fG , η = ∅,   Λ,∞  (η) X Y pG (95) Λ,∞,z (η) = G G  (ηi ), η 6= ∅, fΛ,∞ rΛ,∞    {η1 ,...,ηs } i

with

Z G (η) = rΛ,∞

G fΛ,∞

Υ0 (G

Λ,∞ \G)

η ), K(η ∪ η¯)dλzGΛ,∞ \G (¯

( Z = exp −

Υ0 (G)

) G rΛ,∞ (η)dλG z (η) .

Let G0 ⊂ K is a set of rods intersecting hyperplane Y0 = {(s, t) ∈ Rd+1 : t = 0}. We can introduce in G0 new coordinates ξ = (s, l− , l+ ), where s = ξ ∩ Y0 ∈ Λ ⊂ Rd is the point of intersection of the rod ξ with the hyperplane Y0 , l− = |t|, l+ = l − |t| are lengths of two parts of the rod ξ lying to the left and to the right of the point s respectively. Then dξ = dse−l− dl− e−l+ dl+ , and the space Υ0 (G0 ) can be considered as a space of finite configurations of pairs η = {(ξ− , ξ+ )i }i of rods with corresponding lengths l− , l+ . These pairs are situated on the different sides of the hyperplane Y0 and have the common end s ∈ Λ lying 0 0 on Y0 . The measure λG z on Υ (G0 ) can be written as Y Y Λ 0 e−l+ (s) dl+ (s), η ∈ Υ0 (G0 ), e−l− (s) dl− (s) dλG (96) z (η) = dµz (γ) s∈γ

s∈γ

where γ = γ(η) = {si } is a configuration of points of intersection of the rods ξ ∈ η with the hyperplane Y0 , dµΛ z (γ) is d-dimensional Lebesgue–Poisson measure with

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activity z on the set of finite configurations in Λ; l− (s), l+ (s) are lengths of the corresponding parts of the rod ξ “attached” at point s ∈ γ. 0 0 We denote by ΠG Λ,∞ a probability distribution on the set Γ (Λ) of finite configG0 urations inside Λ induced by the distribution PΛ,∞ on Υ0 (G0 ). From (96) it is seen that the density ω ˜ Λ,∞,z (γ) =

0 dΠG Λ,∞

dµΛ z

of this distribution with respect to d-dimensional Lebesgue–Poisson measure µΛ z on Γ0 (Λ) is equal to Z 0 ω ˜ Λ,∞,z (γ) = pG Λ,∞ (γ, {l− (s), s ∈ γ}, {l+ (s), s ∈ γ}) ×

Y

e−l− (s) dl− (s)

s∈γ

Y

e−l+ (s) dl+ (s),

(97)

s∈γ

G 0 where pG Λ,∞ (γ, {l− (s), s ∈ γ}, {l+ (s), s ∈ γ}) is the same density pΛ,∞,z (η) as in formula (95) rewritten in the new variables. Formulas (95) and (97) imply that   1, γ = ∅,   (γ) m X Y ω ˜ Λ,∞,z (γ) = ϕΛ × (98) %Λ (γi ), γ 6= ∅,    {γ1 ,...,γm } i=1

where

P(γ) {γ1 ,...,γm }

is the same as above sum over all partitions of the configuration γ, ( Z ϕΛ = exp −

)

Γ0 (Λ)

%Λ (γ)dµΛ z (γ)

,

and %Λ (γ) is defined as Z Y Y G0 e−l+ (s) dl+ (s). %Λ (γ) = rΛ,∞ e−l− (s) dl− (s) (γ, {l− (s)}, {l+ (s)}) s∈γ

(99)

s∈γ

5. Representation for Ground State ΨΛ 0 We find now another representation for the probability distribution ΠG Λ,∞ on the G0 ˆ Λ,T ), then the density ω set Γ0 (Λ). Let us consider the trajectory space Υ(G ˆ Λ,T of G0 the distribution ΠΛ,T on the space of configurations γˆ = γt=0 ⊂ Λ with respect to the Poisson measure πz on Γ0 (Λ) can be written as

G0 ω ˆ Λ,T (ˆ γ) =

ZΛ,(−T,0) (ˆ γ )ZΛ,(0,T ) (ˆ γ) , ZΛ,(−T,T )

(100)

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where partition function ZΛ,(−T,T ) = ZΛ,T is defined by (25), and Z RT 0,+ e−α 0 UΛ (γ(t))dt dPΛ,z ({γ(t), t > 0}|ˆ γ ), ZΛ,(0,T ) (ˆ γ) = γ) Υ+ Λ (ˆ

Z ZΛ,(−T,0) (ˆ γ) =

Υ− γ) Λ (ˆ

e−α

R0 −T

UΛ (γ(t))dt

0,− dPΛ,z ({γ(t), t < 0}|ˆ γ ),

with ZΛ,(−T,0) (ˆ γ ) = ZΛ,(0,T ) (ˆ γ ) and Z ZΛ,(−T,T ) = ZΛ,(−T,0) (ˆ γ )ZΛ,(0,T ) (ˆ γ )dπz (ˆ γ ).

477

(101) (102)

(103)

Γ0 (Λ)

Here Υ+ γ ) is the space of trajectories {γ(t), t > 0} of the free Glauber dynamics Λ (ˆ considered in a “semi-tube” of future Λ × (0, ∞) with condition γ(t = 0) = γˆ .

(104)

The space Υ− γ ) is defined by the analogous way. Conditional distributions Λ (ˆ 0,± 0 γ ) are generated by the distribution PΛ,z on the space of trajectories of PΛ,z (·|ˆ the free Glauber dynamics under condition (104), and the conditional distributions are given on the spaces Υ± γ ), respectively; πz is the stationary distribution on Λ (ˆ Γ0 (Λ) of the free Glauber dynamics considered in the domain Λ × (−∞, ∞), i.e. the Poisson measure on Γ(Λ) with the intensity z. As before we rewrite expressions (101), (102) for partition functions using the ensemble of rods. Any configuration from Υ+ Λ,T can be decomposed into a pair of configurations of rods (˜ ηγˆ , η), where η˜γˆ = {ξs , s ∈ γˆ } is a configuration of rods ξs attached to a corresponding point s ∈ γˆ of the point configuration γˆ ⊂ Λ, and η is a configuration of “free” rods with origins from Λ × (0, T ) ⊂ Rd+1 . Similarly, + the space of configurations Υ+ Λ,∞ = ΥΛ can be represented as a space of pairs of rods (˜ ηγˆ , η), where η˜γˆ = {ξs , s ∈ γˆ } is defined as above, and a “free” configuration η of rods with origins from Λ × (0, ∞) can be again decomposed into a pair of configurations with origins in Λ × (0, T ) and in Λ × [T, ∞) respectively. Thus, the 0,+ distribution PΛ,z (·|ˆ γ ) on Υ+ Λ,∞ is represented as a product 0,+ 0,+ 0,+ PΛ,z (·|ˆ γ ) = PΛ,T (·|ˆ γ ) × PΛ,[T,∞) (·|ˆ γ ),

and G+

0,+ dPΛ,T ((˜ ηγˆ , η)|ˆ γ ) = dλz Λ,T (η)

Y

e−l(s) dl(s)e−|Λ|T z ,

s∈ˆ γ

where G+ Λ,T is the set of rods with origins in Λ × (0, T ). Then we get Z Y Z P G+ −α {ξ ,ξ }⊂η˜ ∪η ΦT (ξi ,ξj ) −|Λ|T z −l(s) i j γ ˆ ZΛ,(0,T ) (ˆ γ) = e dl(s) dλz Λ,T (η)e e . s∈ˆ γ

(105)

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Using the same reasoning as above the exponent in (105) can be rewritten as

e

−α

P {ξi ,ξj }⊂η ˜γ ˆ ∪η

ΦT (ξi ,ξj )

=

η˜γˆ ∪η

X

m Y

X

K T (˜ ηi ∪ ηi )

m {(˜ η1 ,η1 ),...,(˜ ηm ,ηm )} i=1

=

η˜γˆ X

XX

η X

η1 X

r Y

(1)

K T (˜ ηi ∪ ηi )

m r≤m {˜ η1 ,...,˜ ηr } (η1 ,η2 ) (η (1) ,...,η (1) ) i=1 r

1

η2 X

m−r Y

(2) (2) {η1 ,...,ηm−r }

j=1

×

(2)

K T (ηj ).

(106)

Pη˜γˆ Here the sum ˜γˆ , the sum {˜ η1 ,...,˜ ηr } is taken over all unordered partition of η Pη is the sum over ordered decomposition of η into two configurations (η1 ,η2 ) Pη1 is taken over all ordered partitions of η1 into r η1 ∪ η2 , the sum (1) (1) (η1 ,...,ηr ) Pη2 (1) sub-configurations ηi (which can be empty), and the last sum is (2) (2) {η1 ,...,ηm−r }

defined by the similar way. Using decomposition (106) we have

ZΛ×[0,T ] (ˆ γ) =

X

(ˆ γ) X

m Y

 



Z K T (˜ ηγˆi ∪

G+ η)dλz Λ,T (η)

m {ˆ γ1 ,...,ˆ γm } i=1

Y

e−ls dls 

s∈ˆ γi

(Z

) G+

K T (η)dλz Λ,T (η) − z|Λ|T

× exp Υ0 (Λ,T )

=

X

(ˆ γ) X

m Y

%ˆΛ,T (ˆ γi )

m {ˆ γ1 ,...,ˆ γm } i=1

(Z

) G+ K T (η)dλz Λ,T (η)

× exp

,

(107)

e−ls dls

(108)

− z|Λ|T

Υ0 (Λ,T )

where Z %ˆΛ,T (ˆ γ) =

G+

|ˆ γ|

Υ0 (Λ,T )×R+

K T (˜ ηγˆ ∪ η)dλz Λ,T (η)

Y s∈ˆ γ

with η˜γˆ = {(s, ls ), s ∈ γˆ }. Then (103), (107) together with dπz (ˆ γ ) = dµΛ γ )e−z|Λ| z (ˆ

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imply that ZΛ×(−T,T )

( Z = exp 2 Z ×



) K

T

G+ (η)dλz Λ,T (η)

Υ0 (Λ,T )



479

(ˆ γ) X

X

m Y

− 2|Λ|T z − |Λ|z

2 %ˆΛ,T (ˆ γi ) dµΛ γ ). z (ˆ

(109)

m {ˆ γ1 ,...,ˆ γm } i=1 G0 (ˆ γ) Thus using (100) we get the following expression for the density ω ˆ Λ,∞

ZΛ×(0,T ) (ˆ γ )ZΛ×(−T,0) (ˆ γ) ZΛ×(−T,T )  2 (ˆ γ) m X X Y  %ˆΛ,∞ (ˆ γi ) ez|Λ|

G0 ω ˆ Λ,∞ (ˆ γ ) = lim

T →∞

=

m {ˆ γ1 ,...,ˆ γm } i=1

 Z X 

(ˆ γ) X

m Y

(110)

2 γ) %ˆΛ,∞ (ˆ γi ) dµΛ z (ˆ

m {ˆ γ1 ,...,ˆ γm } i=1

with

Z

%ˆΛ,∞ (ˆ γ ) = lim %ˆΛ,T (ˆ γ) = T →∞

G+

|ˆ γ|

Υ0 (Λ,∞)×R+

K(˜ ηγˆ ∪ η)dλz Λ,∞ (η)

Y

e−ls dls . (111)

s∈ˆ γ

Thus, we get G0 0 dΠG γ) = ω ˆ Λ,∞ (ˆ γ )dµΛ γ )e−z|Λ| . z (ˆ Λ,∞ (ˆ

On the other hand, as follows from the Feynman–Kac formula, we have for semigroup exp{tHΛ } ( Z ) Z T ZΛ×(−T,T ) = exp −α UΛ (γ(t))dt dPz0 (σ) = (e2T HΛ 1, 1)µΛz e−z|Λ| , (112) ΥΛ,∞

−T

where (·, ·)µΛz is the scalar product in L2 (Γ0 (Λ), µΛ z ). Similar to (112) we get ZΛ×(0,T ) (ˆ γ ) = ZΛ×(−T,0) (ˆ γ ) = (eT HΛ 1)(ˆ γ ).

(113)

Thus (110) implies that (eT HΛ 1)(ˆ γ)

ΨΛ (ˆ γ ) ≡ lim

= lim

T →∞ keT HΛ 1kL2 (Γ0 (Λ),µΛ ) z

T →∞

ZΛ×(0,T ) (ˆ γ) (ZΛ×(−T,T ) )1/2

(114)

exists and equals to X ΨΛ (ˆ γ) =  Z  

(ˆ γ) X

m Y

1

%ˆΛ,∞ (ˆ γi )e 2 z|Λ|

m {ˆ γ1 ,...,ˆ γm } i=1

  Γ0 (Λ)

X

(ˆ γ)

X

m Y

m {ˆ γ1 ,...,ˆ γm } i=1

2

1/2 .

 %ˆΛ,∞ (ˆ γi ) dµΛ γ ) z (ˆ

(115)

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G0 (ˆ γ ) = Ψ2Λ (ˆ γ ), and finally we have Consequently, (110) and (115) imply that ω ˆ Λ,∞ G0 Λ Λ ΠΛ,∞ = νz , where the measure νz was defined by formula (8).

6. Proof of Theorem 1 0

Using decomposition L2 (Γ, dπz ) = L2 (Γ0 (Λ), dπzΛ ) ⊗ L2 (Γ0 (Λ0 ), dπzΛ ) and equality −z|Λ| we can reduce the problem of self-adjointness for the operator HΛ dπzΛ = dµΛ ze on L2 (Γ, dπz ) to the same problem for the operator HΛ0 on HΛ = L2 (Γ0 (Λ), µΛ z) acting as follows (HΛ0 Φ)n (s1 , . . . , sn )

=

n X

Φn−1 (s1 , . . . , s˘i . . . sn ) − (αUΛ (s1 , . . . , sn )

i=1

+ n + z|Λ|)Φn (s1 , . . . , sn ) Z +z Φn+1 (s1 , . . . , sn , s)ds, n ≥ 1, Λ Z (HΛ0 Φ)0 = −zΦ0 + z Φ1 (s)ds, n = 0.

(116)

Λ

Here s˘i means that the variable si is omitted, Φ = (Φ0 , Φ1 (s1 ), . . . Φn (s1 , . . . , sn ), . . .) ∈ L2 (Γ0 (Λ)),

(117)

Φn (s1 , . . . , sn ) is the value of Φ(γ) on the stratum Γ0n (Λ) ⊂ Γ0 (Λ) of n-points configurations γ = (s1 , . . . , sn ), si 6= sj , and UΛ (s1 , . . . , sn ) ≥ 0 (or in the general case, UΛ (s1 , . . . , sn ) ≥ −Bn). The operator HΛ0 is a symmetrical operator on the set Df in ⊂ HΛ of finite vectors, i.e. vectors with Φn = 0 for all n > N = N (Φ). Thus, HΛ0 is a closable operator. For the closure, we will use the same notation HΛ0 . To complete the proof of the self-adjointness we should check that for large enough ξ > 0 the range Ran(HΛ0 − ξE) = HΛ , or what is equivalent that the equation (HΛ0 − ξE)F = G

(118)

is solvable for large enough ξ > 0 and all G ∈ HΛ . We can rewrite HΛ0 as HΛ0 = T +R separating the diagonal (over number of variables si ) part T of HΛ0 . Then positivity of T implies that (118) is equivalent to the equation (T − ξE)(E + (T − ξE)−1 R)F = G.

(119)

k(T − ξE)−1 R)k < 1,

(120)

If

then the solution of Eq. (119) can be found as F = (E + (T − ξE)−1 R)−1 (T − ξE)−1 G. Thus, we should prove inequality (120).

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481

Denote by HΛ,n ⊂ HΛ a subspace of sequences (117) with Φk = 0, k 6= n, then the norm in HΛ,n is defined as Z zn Φ2 (s1 , . . . , sn )ds1 · · · dsn . kΦn k2HΛ,n = n! Λn n For any n ≥ 1 we have ° ° n ° ° X ° ° Φn−1 (s1 , . . . , s˘i · · · sn ) ° ° ° ° i=1 ° ° ° (αUΛ (s1 , . . . , sn ) + n + z|Λ| + ξ) ° ° ° ° ° ° °

p <

nz|Λ|kΦn−1 kHΛ,n−1 . n + z|Λ| + ξ

HΛ,n

Consequently, °2 ° n ° ° X ° ° Φn−1 (s1 , . . . , s˘i · · · sn ) ° ° ° ° i=1 ° ° ° (αUΛ (s1 , . . . , sn ) + n + z|Λ| + ξ) ° ° ° ° ° ° °

<

nz|Λ| kΦn−1 k2HΛ,n−1 (n + z|Λ| + ξ)2

HΛ,n

< max n

nz|Λ| kΦn−1 k2HΛ,n−1 . (n + z|Λ| + ξ)2 (121)

Similarly, we get Z ° °2 ° ° ° ° z Φn+1 (s1 , . . . , sn , s)ds ° ° Λ ° ° ° (αUΛ (s1 , . . . , sn ) + n + z|Λ| + ξ) ° ° ° ° °

<

(n + 1)z|Λ| kΦn+1 k2HΛ,n+1 (n + z|Λ| + ξ)2

HΛ,n

< max n

(n + 1)z|Λ| kΦn+1 k2HΛ,n+1 . (n + z|Λ| + ξ)2 (122)

Finally (116), (121), (122) imply that s k(T − ξE)

−1

Rk < 2

max n

(n + 1)z|Λ| <1 (n + z|Λ| + ξ)2

for all large enough ξ. Let us notice that the general case can be considered in the same way using stability of the potential and the assumption that α is small enough. The first statement of Theorem 1 is proved. We will use the following proposition in the proof of the second statement of the theorem.

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Proposition [9]. Let H be a bounded from above self-adjoint operator in a Hilbert space L2 (Ω, µ), where (Ω, µ) is a space with a finite measure µ, the semigroup exp{tH} meets the condition of improving positivity (see [14]), and the limit eT H 1 =Ψ T →∞ keT H 1k lim

(123)

exists, such that (Ψ, 1) > 0. Then Ψ is a unique ground state of the operator H and Ψ > 0. In our case we see from (114), (115) that (123) holds and (Z (ΨΛ (ˆ γ ), 1)L2 (Γ0 (Λ),µΛz ) =   

Z

) 1 exp %ˆΛ,∞ (ˆ γ )dµΛ γ ) + z|Λ| z (ˆ 2 Γ0 (Λ) 1/2  2 (ˆ γ) m X X Y   %ˆΛ,∞ (ˆ γi ) dµΛ γ ) z (ˆ

Γ0 (Λ)

m {ˆ γ1 ,...,ˆ γm } i=1

>0 consequently, the operator HΛ has a unique ground state ΨΛ (γ) > 0. Thus, the second state of Theorem 1 is proved. 7. Proof of Theorem 4 7.1. First statement G As follows from representation (95) probability measures {PΛ,∞,z , G ⊂ GΛ,∞ } constructed in Lemma 7 form an consistent family of measures, i.e. for any two bounded sets G1 ⊂ G2 ⊂ GΛ,∞ we have: G2 G1 PΛ,∞,z |Υ0 (G1 ) = PΛ,∞,z .

That implies, see [11], that a probability measure P˜Λ,∞,z exists on Υ(GΛ,∞ ), such that for any local function FG with G ⊆ Gint M ⊂ GΛ,∞ (with some bounded set M ⊂ Λ × (−∞, ∞)): G lim hFG iPΛ,T ,z = hFG iPΛ,∞,z = hFG iP˜Λ,∞,z .

T →∞

If we consider a measure on the space Υ of the following form PΛ,∞,z = P˜Λ,∞,z × PΛ0 0 ,∞,z

(124)

where PΛ0 0 ,∞,z is the distribution of the free dynamics on Λ0 , we get (30). We will show next that a distribution PˆΛ,∞,z of the process associated with the ˜ Λ } (i.e. the Markov process GΛ from Theorem 2) is the semigroup S˜tΛ = exp{tH same as the distribution PΛ,∞,z (124).

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Lemma 8. The following asymptotics hold as T → ∞ 0

ZΛ,T (γ) = eλΛ T ΨΛ (γ)(ΨΛ (γ), 1)πz + δ)

(125)

with kδkL2 (Γ0 (Λ),πz ) = o(1), 0

ZΛ×(−T,T ) = e2λΛ T ((ΨΛ (γ), 1)2πz + o(1)),

(126)

where λ0Λ is the eigenvalue corresponding to the ground state ΨΛ , see Theorem 1. Proof. Let H⊥ ⊂ L2 (Γ0 (Λ), πz ) be the orthogonal complement to the vector ΨΛ , and HΛ⊥ be a restriction of the operator HΛ to the space H⊥ . Then formula (112) implies 0

⊥

(eT HΛ 1)(γ) = eλΛ T ΨΛ (γ)(ΨΛ (γ), 1)πz + eHΛ T 1⊥ (γ), where 1⊥ is the projection of the vector 1 to the space H⊥ . We estimate now the norm of the second term: Z λ0Λ ⊥ keHΛ T 1⊥ k2 = e2λT dσ1⊥ (λ) −∞ ÃZ 0 ! Z λ0Λ λΛ −a 2λ0Λ T 2(λ−λ0Λ )T 2(λ−λ0Λ )T =e e dσ1⊥ (λ) + e dσ1⊥ (λ) . λ0Λ −a

−∞

(127) Here σ1⊥ (∆) is the spectral measure of the operator HΛ⊥ on the vector 1⊥ . Then the first term in the bracket in (127) is less then e−2aT and the second term is estimated from above by σ1⊥ (λ0Λ − a, λ0Λ ). Since the ground state is unique, then the measure σ1⊥ (λ) is continuous at the point λ = λ0Λ . Consequently, taking a = √1T we get that both terms in (127) tend to 0 as T → ∞. Using the same reasoning and formula (112) we obtain asymptotics (126). Lemma is proved completely. We remaind that PˆΛ,∞,z is a distribution on the space of trajectories of the process GΛ = {γ(t), t ∈ R1 } ˜ Λ }. For any finite time intervals t0 < associated with Markov semigroup exp{tH t1 < · · · < tn and any bounded functions f0 , f1 , . . . , fn on Γ0 (Λ) the average over PˆΛ,∞,z can be written as * n + Z Y fi (γ(ti )) = fn (γn )Qtn −tn−1 (γn , γn−1 ) · · · f1 (γ1 ) i=0

ˆΛ,∞,z P

(Γ0 (Λ))n

× Qt1 −t0 (γ1 , γ0 )f0 (γ0 )

n Y i=0

dνzΛ (γi ),

(128)

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˜ Λ } in the space L2 (Γ0 (Λ), νzΛ ): where Qt (γ, γ 0 ) is the kernel of the operator exp{tH Z ˜ Λ }f )(γ) = Qt (γ, γ 0 )f (γ 0 )dνzΛ (γ 0 ). (exp{tH (129) Γ0 (Λ)

On the other hand, the average over PΛ,∞,z , see (124), is calculated as the limit ( ) Z Y Z T n * n + UΛ (γ(τ ))dτ dPz0 fi (γ(ti )) exp −α Y −T i=0 . fi (γ(ti )) = lim T →∞ ZΛ×[−T,T ] i=0 PΛ,∞,z

(130) Using the Feynman–Kac representation (13) for the kernel of the semigroup etHΛ and asymptotics (125), (126) we can rewrite (130) as follows Z 0 fn (γn )ΨΛ (γn )Rtn −tn−1 (γn , γn−1 ) e−λΛ (tn −tn−1 ) (Γ0 (Λ))n

0

0

· · · e−λΛ (t2 −t1 ) f1 (γ1 )Rt1 −t0 (γ1 , γ0 )e−λΛ (t1 −t0 ) ΨΛ (γ0 )f0 (γ0 )

n Y

dπz (γi ).

(131)

i=0

It follows from (10) and (8) that 0 1 1 Qt (γ1 , γ2 ) = e−λΛ t Rt (γ1 , γ2 ) , ΨΛ (γ1 ) ΨΛ (γ2 )

dνzΛ (γ) = Ψ2Λ (γ)dπz (γ),

and consequently, averages (128) and (131) are the same. Thus, all finite dimensional distributions for the measures PΛ,∞,z and PˆΛ,∞,z coincide, and consequently, the measures are the same. The first assertion of Theorem 4 is completely proved. 7.2. Second statement Repeating arguments from the proof of Lemma 5 we get that for any bounded subset G ⊂ K and bounded local function FG the following limits exist and can be written as lim hFG iPΛ,∞,z = hFG i Ã

Λ%Rd

= fG FG (∅) + Z

Z ∞ m X Y 1 dλG (ηi ) m! 0 (G)\∅)m (Υ m=1 i=1 m Y

×

dλ

(Υ0 (K\G))m i=1

with

( Z fG = exp −

Υ0 (G)\∅

( Z = exp −

K\G

(¯ ηi )FG (∪¯ ηi )

m Y

K(ηi ∪ η¯i )

(132)

i=1

)

Z G

K(η ∪ η¯)dλ (η)dλ Υ0 (K\G)

η∈Υ0 (K):η∩G6=∅

!

K\G

(¯ η)

)

K(η)dλz (η) .

(133)

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By analogy with our reasoning in the proof of the first part of the theorem using corollary of Lemma 7 we get that limits (132) define a system of compatible probaG bility distributions {P∞,z } on the space Υ0 , and thereby a limit distribution {P∞,z } on Υ0 is defined. The limit distribution is invariant with respect to the space translations in Rd and with respect to reflections in time. Moreover, this distribution meets the property of OS positivity. Really, for any local bounded function F dependent on the process as t ≥ 0 we have (θF · F )P∞,z = lim (θF, F )PΛ,∞z ≥ 0, Λ%Rd

(134)

since distributions PΛ,∞,z are the distributions of the Markov processes for any bounded Λ ⊂ Rd+1 . Consequently, relation (134) is also valid for any function F ∈ L2 (Υ0 , P∞,z ) which is also dependent on the process values as t ≥ 0. Thus we construct the limit measure P∞,z and establish properties of this measure. Theorem 4 is completely proved. 8. Proof of Theorem 5 Let us consider strongly local functions FMi , i = 1, 2 depending on the process G (or GΛ ) with bounded localization domain Mi ⊂ Rd+1 , i = 1, 2. Denote by Gi = Gloc Mi ⊂ K a set of rods starting at Mi . Then the functions FMi can be considered as strongly local functions FGi on the space Υ0 with the same localization domains Mi , i = 1, 2 correspondingly. We will use here the following formula for correlations (36), see [7]: hFG1 · FG2 iP∞,z − hFG1 iP∞,z hFG2 iP∞,z ˆ

= fG1 · fG2 ((e∆(G1 ,G2 ) − 1)(FG1 (∅)FG2 (∅) + FG1 (∅)I2 + FG2 (∅)I1 + I1 I2 ) ˆ + e∆(G1 ,G2 ) I1,2 + FG1 (∅)Iˆ2 + FG2 (∅)Iˆ1 + I1 Iˆ2 + I2 Iˆ1 + Iˆ1 Iˆ2 ). (135) Here fG1 , fG2 are defined by (133), Z ˆ 1 , G2 ) = ∆(G

K(η)dλz (η),

(136)

η:η∩G1 6=∅,η∩G2 6=∅

Z ∞ n n X Y Y 1 i i i i i Ij = dλGj (ηG )dλG3 (¯ ηG )FGj (∪ηG ) K(ηG ∪¯ ηG ) j 3 j j 3 n! 0 (G )\∅×Υ0 (G ))n (Υ j 3 n=0 i=1 i=1 with G3 = K\(G1 ∪ G2 ), j = 1, 2; Z n n ∞ Y Y X 1 i i i i i i i ˆ FG1 (∪ηG1 ) K(ηG1 ∪ηG2 ∪ηG3 ) dλG1 (ηG )dλG2 (ηG )dλG3 (ηG ), I1 = 1 2 3 n! n=0 i=1 i=1 where the integration is taken over all sets from n triplets ¡ i ¢ i i η G1 , η G , ηG , i = 1, . . . , n, 2 3 i i such that ηG 6= ∅ for each i = 1, . . . , n, and at least in one of the triplets: ηG 6= ∅. 1 2 ˆ We can define I2 in the analogous way.

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Further, I1,2

Z ∞ X 1 i i = FG1 (∪ηG )FG2 (∪ηG ) 1 2 n! n=0 ×

n Y

i i i K(ηG ∪ ηG ∪ ηG ) 1 2 3

i=1

n Y

i i i dλG1 (ηG )dλG2 (ηG )dλG3 (ηG ). 1 2 3

i=1

Here the integration is over all sets from n triplets ¡ i ¢ i i ηG1 , ηG , ηG , i = 1, . . . , n, 2 3 i i such that at least in one of the triplets: ηG 6= ∅ and ηG 6= ∅. 1 2 ˆ Let us estimate ∆(G1 , G2 ) in the case when ϕ meets condition (1) in (34), the case 34(2) can be studied in the same way. We can rewrite (60) as follows

|K(η)| ≤

µ

X

Y

T

hξi ,ξj i∈T

Y

×

¡

1 1 + |si − sj |2m

1 + |si − sj |

¶ 12

Y

1

e − 2 li

i∈V (T )

Y

¢1 2m 2

1

e 2 li

i∈V (T )

hξi ,ξj i∈T

Y

|e−α Φ(ξi ,ξj ) − 1|,

(137)

hξi ,ξj i∈T

with ξi = ((si , ti ), li ), i = 1, . . . , n. Then we have for any tree T in (137) µ

Y hξi ,ξj i∈T

1 1 + |si − sj |2m

¶ 12

Y

1

e − 2 li < e

−

P hξi ,ξj i∈T

m ln(1+|si −sj |)− 21

P

i∈V (T ) li

i∈V (T )

< e−diam η˜, where η˜ = ∪ξ˜i ⊂ Rd+1 is a subset of Rd+1 which is a union of all rods from configuration η, and the diameter of η˜ is calculated in the metrics 35(1). Thus, X Y Y 1 1 |K(η)| < e−diam η˜ ((1 + |si − sj |2m ) 2 |e−α Φ(ξi ,ξj ) − 1|) e 2 li . T

hξi ,ξj i∈T

i∈V (T )

Using bound (34) on the potential ϕ we can apply here the above reasoning and finally get the following estimate Z ˆ K(η)dλz (η) < e−dist(M1 ,M2 ) max{υz (G1 ), υz (G2 )}C(α, z), (138) η : η ∩ G 6= ∅, 1

η ∩ G2 6= ∅

where

Z υz (G) =

z(−dpx (l))dx, G

ˆ and a constant C(α, z) doesn’t depend on G1 and G2 . Thus, we have ˆ ˆ −dist(M1 ,M2 ) max{υz (G1 ), υz (G2 )}eCˆ max{υz (G1 ),υz (G2 )} . |e∆(G1 ,G2 ) − 1| < Ce

(139)

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We estimate next I1 : ∞ X 1 |I1 | = max |FG1 | n! n=0

< max |FG1 |

ÃZ

!n G1

dλ Υ0 (G1 )\∅×Υ0 (G3 )

G3

(ηG1 )dλ

(ηG3 )|K(ηG1 ∪ ηG3 )|

µZ ¶n ∞ X 1 ˆ z (G1 )} |K(η)|dλ(η) < max |FG1 | exp{Cυ n! η:η∩G1 6=∅ n=0

ˆ In the same way we can obtain the upper bound on I2 . with a constant C. Let us estimate now Iˆ1 : Z ∞ X 1 |Iˆ1 | < max |FG1 | K(ηG1 ∪ ηG2 ∪ ηG3 ) n! ηG1 6=∅,ηG2 6=∅ n=0 µZ ¶n−1 G1 G2 G3 × dλ (ηG1 )dλ (ηG2 )dλ (ηG3 ) K(η)dλ(η) . (140) η:η∩G1 6=∅

The first integral in (140) can be estimated in the same manner as above, and we get ˆ

|Iˆ1 | < e−dist(M1 ,M2 ) max{υz (G1 ), υz (G2 )} max |FG1 |eCυz (G1 ) . The analogous estimate is valid also for Iˆ2 and I1,2 . Finally representation (133) implies ½Z ¾ ˆ z (G)}. |fG | < exp |K(η)|dλz (η) < exp{Cυ η:η∩G6=∅

Since υz (G) = z|M | in the case G = Gloc M , then after substitution all above estimates to (135) we get the main estimate (36) of Theorem 4. 8.1. Proof of Corollary from Theorem 5 We study here the generator h of the time translations in the physical space H and prove that the operator h has a spectral gap. Let µ = sup σ(h|H⊥ ) be the supremum of a spectrum of the restriction of the operator h to the invariant subspace H⊥ ˆ ⊂ H a dense set containing vectors from H orthogonal to vector e. We denote by H ˆ contain strongly local bounded functions in H such that classes of elements from H ˆ For any ε > 0 there exists an element ϕ ∈ H ˆ such that on Υ. (1) (ϕ, e) = 0,

(2) (ϕ, ϕ) = 1,

(3) σϕ (µ − ε, µ) >

1 , 2

(141)

where σϕ (∆) = (Eh (∆)ϕ, ϕ)H is the spectral measure of the element ϕ, and Eh (∆) is the resolution of identity of the operator h. Condition (3) implies that Z µ 1 (eth ϕ, ϕ)H = eλt dσϕ (λ) > e(µ−ε)t . (142) 2 −∞

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Let Φ ∈ H+ is a strongly local bounded function (not a constant) such that the class [Φ] of Φ is the same as ϕ. Then conditions (141) is rewritten as 1 (1) hΦiP∞,z = 0, (2) hθΦ · ΦiP∞,z = 1, (3) hθUt Φ · ΦiP∞,z > e(µ−ε)t . (143) 2 d+1 Localization domain M1 6= ∅ of the function Φ is in the right half-space R+ = d 1 R × R+ and localization domain M2 of the function θUt Φ is in the left half-space d+1 1 R− = R d × R− . Moreover the distance in the metrics (35) between M1 and M2 is not less then 2t . Then using the result of Theorem 4 we have t

(θUt Φ, Φ)H = hθUt Φ · ΦiP∞,z < c1 e− 2 ,

(144)

with a constant c1 does not depending on t. Comparing (142) with (144) it is easy to see that 1 (µ − ε)t < − t + c2 (145) 2 with an absolute constant c2 . Since inequality (145) holds for any ε > 0 and any t > 0, we get µ ≤ − 12 . That means that the operator h has a spectral gap and the unique ground state. 9. Attachment 9.1. Proof of Lemma 1 Let M ⊂ Rd+1 be a bounded set. Without loss of generality we can take M = Λ×I, where Λ ⊂ Rd is a bounded domain in Rd and I = [T1 , T2 ] ⊂ R1 is a segment. Then for a.e. configurations η of rods a number of rods with origins inside of M is finite. For the proof, it remained to show that the mean value of the number of rods from η with origins x = (s, t) at the “past” to M (i.e. with t < T1 ) intersecting M is also finite. (n) We denote GMn ⊂ K the set of rods beginning at Mn = Λ × In where In = (T1 − (n + 1), T1 − n), of length not less than n, and let GM Mn ⊂ K be the set of rods (n)

M with origins in Mn intersecting M . It is clear that GM Mn ⊂ GMn and Υ(GMn ) ⊂ (n)

Υ(GMn ). On the other hand, the probability that a rod ξ ∈ Gloc Mn belongs to a (n)

configuration ηn ∈ Υ(GMn ) equals to e−n , consequently the set of origins of rods (n)

from Υ(GMn ) forms a Poisson field in Mn with intensity ze−n . We denote ηnM ⊆ ηn 0 a sub-configuration of ηn such that ηnM ∈ Υ(GM Mn ). Averaging over Pz we get h|ηnM |iPz0 < h|ηn |iPz0 = z|Λ|e−n . Thus the mean value of a number of rods intersecting M with beginnings at the “past” of M can be bounded from above by ∞ X z|Λ| e−n < ∞. n=0

That means that configurations with infinite number of rods intersecting M form a set of zero measure. Since the space Rd+1 can be covered by a countable family

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of bounded sets of the form M = Λ × I, then almost all configurations of rods are locally finite. The lemma is proved. 9.2. Proof of Proposition 1 Let us consider a quadratic form for any F ∈ H+ and t > 0: qF (t) = (F, Ut F )+ . Using Cauchy–Buniakovskyi–Schwarz inequality n times and equality (Ut F, F )+ = (F, Ut F )+ we get 1

1

1

1

qF (t) ≤ (F, F )+2 (Ut F, Ut F )+2 = (F, F )+2 (F, U2t F )+2 1

+ 41

≤ (F, F )+2

1

1− 21n

(F, U4t F )+4 ≤ · · · ≤ (F, F )+

1

(F, U2n t F )+2n .

(146)

For any t > 0 and any n (F, U2n t F )+ = (θF, U2n t F )L2 (Ω,P) ≤ kF k2L2 . Since qF (t) does not depend on n, we take a limit in (146) as n → ∞ and obtain for any t qF (t) ≤ (F, F )+ . Finally, (Ut F, Ut F )+ = (F, U2t F )+ = qF (2t) ≤ (F, F )+ . 9.3. Proof of Lemma 4 Note, that by virtue of (52), (53) the measure in the right-hand side of (54) is the probabilistic one. Under decomposition G0Λ,T = G1 ∪G2 on two nonintersecting sets this measure by (50) could be written as a product of two probabilistic measures: λG1 e−S(G1 ) × λG2 e−S(G2 ) R with S(G) = G ξ(x)px (l)dldx. For the probability that the number of rods |ηGi | in the configuration ηGi ⊂ Υ0 (Gi ) equals k, we get: Pr(|ηGi | = k) =

1 (S(Gi ))k e−S(Gi ) , k!

i.e. the probability has a Poisson form. That means that the measure in (54) is the distribution of the Poisson field of the form Π(M, ζ, {px }), where M, ζ, {px } are the same as in formula (43).

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9.4. Proof of Lemma 5 Let us start with the case when ϕ ≥ 0. Inequality (60) is evident for a set of nodes η = {ξ1 , ξ2 , ξ3 }. We assume that (60) holds for all sets η (m) = {ξ1 , . . . , ξm } of nodes with 2 ≤ m ≤ n, and prove the statement of the Lemma for a vertex set V (σ) = η (n+1) = {ξ0 , . . . , ξn } using the induction assumption. We note first that κTσ defined by (58) can be rewritten for any given connected graph σ with a vertex set V (σ) = η (n+1) as κTσ

k Y

=

Y

κTσi

i=1

T

(e−α Φ

(ξ0 ,ξj )

− 1).

(147)

ξj ∈mi

Here σi , i = 1, . . . , k are connected subgraphs of the graph σ with a vertex set V (σi ) ⊆ η (n) = {ξ1 , . . . , ξn }, such that σ1 , . . . , σk form a decomposition of the rest of σ after removing all edges incident to the node ξ0 , and mi ⊆ V (σi ) is a subset of V (σi ) governing a subset of incident to ξ0 edges with ends from V (σi ). Then using (147) we have   (η (n+1) ) (η (n) ) n k X X Y X X Y X T  (e−α Φ (ξ0 ,ξj ) − 1) κTσ = κTσi σ

k=1 {η1 ,...,ηk } i=1

≤

n X

(η (n) )

k Y

X

σi :V (σi )=ηi

mi ⊆ηi ,mi 6=∅ ξj ∈mi

 X



k=1 {η1 ,...,ηk } i=1

Y

|e−αΦ

T

(ξα ,ξβ )

− 1|

Ti :V (Ti )=ηi hξα ,ξβ i∈Ti

 ×

X

|e−α Φ

T

(ξ0 ,ξj )

− 1| ,

(148)

ξj ∈ηi

where in (148) the internal sum is taken over all trees Ti with vertex set ηi . In the last bound we applied the induction assumption, the identity P X Y T T (e−α Φ (ξ0 ,ξ) − 1) = e−α ξ∈η Φ (ξ0 ,ξ) − 1 m⊆η,m6=∅ ξ∈m

and the following inequality |e−α

P ξ∈η

ΦT (ξ0 ,ξ)

− 1| ≤

X

T

|e−αΦ

(ξ0 ,ξ)

− 1|

(149)

ξ∈η

which is valid for ΦT ≥ 0. Connecting all possible trees from Ti , i = 1, . . . , k with all possible edges (ξ0 , ξj ), ξj ∈ ηi from the last sum in (148) we obtain all possible trees T with V (T ) = η (n+1) , consequently the sum in (148) is the same as (η (n+1) )

X

Y

T

hξi ,ξj i∈T

T

|(e−αΦ

(ξi ,ξj )

− 1)|.

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If ϕ is a general stable potential, then the Ruelle condition for n-points configuration γn X

U (γn ) ≡

{x,y}⊂γn

1 ϕ(x − y) ≥ − Bn 2

can be rewritten as a corresponding condition on rod configurations η ∈ Υ0 (G0Λ,T ): X {ξ,ξ 0 }⊂η

1 X Φ(ξ, ξ 0 ) ≥ − B l(ξ), 2 ξ∈η

where l(ξ) is the length of the rod ξ. This estimate implies existence of such ξ0 ∈ η that X Φ(ξ0 , ξ) ≥ −Bl(ξ0 ). (150) ξ∈η\ξ0

We take in the configuration η (n+1) such rod ξ0 ∈ η (n+1) which was indicated in inequality (150), and prove the modification of inequality (149). Let us consider a decomposition of η into two sub-configurations: η = η+ ∪ η− with η+ = {ξ ∈ η : ΦT (ξ0 , ξ) ≥ 0},

η− = {ξ ∈ η : ΦT (ξ0 , ξ) < 0}.

Then using inequalities (150) and (149) (the last one holds for ΦT ≥ 0) we have |e−α

P ξ∈η

≤ |e

ΦT (ξ0 ,ξ)

−α

≤ e−α

P

− 1|

ξ∈η+ ∪η−

P ξ∈η

≤ eαBl(ξ0 )

ΦT (ξ0 ,ξ)

ΦT (ξ0 ,ξ)

X

−e α

|1 − e

|1 − eαΦ

T

−α

P ξ∈η+

P ξ∈η−

(ξ0 ,ξ)

|+

ξ∈η−

≤ eαBl(ξ0 )

X

(1 − e−α|Φ

ΦT (ξ0 ,ξ)

ΦT (ξ0 ,ξ)

X

| + |e

| + |e T

|e−αΦ

−α

−α

P

P

(ξ0 ,ξ)

ξ∈η+

ξ∈η+

ΦT (ξ0 ,ξ)

ΦT (ξ0 ,ξ)

− 1|

− 1|

− 1|

ξ∈η+ T

(ξ0 ,ξ)|

).

(151)

ξ∈η

Repeating above reasoning under the induction assumption and revised estimate (151) we obtain estimate (59) in the general case of a stable potential ϕ. The lemma is proved. Acknowledgments The authors gratefully acknowledge the financial support of the SFB 701, Bielefeld University. R. M. and E. Zh. acknowledge the partial financial support of the RFBR grant 05-01-00449 and the CRDF grant RUM1-2693-MO-05.

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References [1] S. Albeverio, Yu. G. Kondratiev and M. Rockner, Analysis and geometry on configuration spaces, J. Funct. Anal. 154 (1998) 444–500. [2] N. Angelescu, R. A. Minlos and V. A. Zagrebnov, The one-particle branch of the energy spectrum of weakly interacting quantum lattice fields with values on the twodimensional sphere, J. Math. Phys. 41 (2000) 2–23. [3] J. L. Doob, Stochastic Processes (John Wiley and Sons, 1967). [4] M. Duneau, D. Jagolnitzer and B. Souillard, Decrease properties of truncated correlation functions and analyticity properties for classical lattices and continuous systems, Comm. Math. Phys. 31 (1973) 191–208. [5] D. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View (Springer-Verlag, 1981). [6] Yu. G. Kondratiev and E. Lytvynov, Glauber dynamics of continuous particle systems, Ann. Inst. H. Poincar´e Probab. Statist. 41 (2005) 685–702. [7] Yu. G. Kondratiev, R. A. Minlos, M. Rockner and G. V. Shchepanuk, Exponential mixing for classical continuous systems, in Stochastic Processes, Physics and Geometry: New Interplays, I (Leipzig, 1999), Canadian Math. Society Conference Proceedings, Vol. 28 (Amer. Math. Soc., Providence, R.I., 2000), pp. 243–253. [8] Yu. G. Kondratiev, R. A. Minlos and E. A. Zhizhina, Lower branches of the spectrum of Hamiltonians for infinite quantum system with compact spin, Trans. Moscow Math. Soc. 60 (1998) 259–302. [9] J. Lorinczi, R. A. Minlos and H. Spohn, The infrared behavior in Nelson’s model of a quantum particle coupled to a massless scalar field, Ann. Henri Poincar´e 3 (2002) 269–295. [10] V. A. Malyshev and R. A. Minlos, Gibbs Random Fields (Kluwer Academic Publishers, 1991). [11] R. A. Minlos, Limiting Gibbs distribution, Funktsional Anal. i Prilozhen. 1(2) (1967) 60–73. [12] R. A. Minlos and S. K. Pogosyan, Estimates on the Ursell functions, group functions and their derivatives, Teore. Mat. Fiz. 31(2) (1977) 199–213. [13] O. Ore, Theory of Graphs, AMS Colloquium Publ., Vol. 38 (Amer. Math. Soc., 1962). [14] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vols. 1–4 (Academic Press, 1978). [15] D. Ruelle, Statistical Mechanics. Rigorous Results (Benjamins, Amsterdam, 1969). [16] B. Simon, The P (ϕ)2 Euclidean Quantum Field Theory (Princeton University Press, 1974). [17] B. Simon, Functional Integration and Quantum Physics (Academic Press, 1979). [18] B. Simon, The Statistical Mechanics of Lattice Gas, Vol. 1 (Princeton University Press, 1993). [19] D. Surgailis, On multiple Poisson stochastic integrals and associated Markov semigroups, Probab. Math. Statist. 3 (1984) 217–239. [20] D. Surgailis, On Poisson multiple stochastic integrals and associated equilibrium Markov processes, in Theory and Application of Random Fields (Bangalore, 1982), Lecture Notes in Control and Inform. Sci., Vol. 49 (Springer, 1983), pp. 233–248.

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Reviews in Mathematical Physics Vol. 20, No. 5 (2008) 493–527 c World Scientiﬁc Publishing Company

COMBINATORIAL BETHE ANSATZ AND GENERALIZED PERIODIC BOX-BALL SYSTEM

ATSUO KUNIBA Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo, Komaba, Tokyo 153-8902, Japan [email protected] REIHO SAKAMOTO Department of Physics, Graduate School of Science, University of Tokyo, Hongo, Tokyo 113-0033, Japan [email protected] Received 28 August 2007

We reformulate the Kerov–Kirillov–Reshetikhin (KKR) map in the combinatorial Bethe ansatz from paths to rigged conﬁgurations by introducing local energy distribution in crystal base theory. Combined with an earlier result on the inverse map, it completes b 2 ). As an application, we solve the crystal interpretation of the KKR bijection for Uq (sl an integrable cellular automaton, a higher spin generalization of the periodic box-ball system, by an inverse scattering method and obtain the solution of the initial value problem in terms of the ultradiscrete Riemann theta function. Keywords: Bethe ansatz; box-ball system; tropical theta functions; crystal basis. Mathematics Subject Classiﬁcation 2000: 35Q51, 82B23, 17B37, 05E10, 14K25

1. Introduction The Kerov–Kirillov–Reshetikhin (KKR) bijection [12,13] is a combinatorial version of the Bethe ansatz. It gives a one-to-one correspondence between rigged conﬁgurations and highest paths, which are combinatorial analogues of the Bethe roots and the associated Bethe vectors in integrable spin chains.a The relevant problem of state counting stemmed from Bethe’s original work [3], was developed further in the KKR theory, and has been formulated as the X = M conjecture for arbitrary aﬃne Lie algebra [16]. See [18, 30] for a recent status. a The

original KKR bijection concerns semistandard tableaux rather than highest paths. The KKR bijection in this paper is to be understood as the composition of the original one with the Robinson–Schensted–Knuth correspondence between semistandard tableaux and highest paths. 493

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In [20, 28], the KKR map φ−1 from rigged conﬁgurations to highest paths was identiﬁed with a certain composition of combinatorial R in crystal base theory [10, 11, 14]. It provided a long sought representation theoretical meaning with φ−1 and opened a connection with the integrable cellular automata called the box-ball system [33, 34] and its generalizations [17, 7, 15]. They are identiﬁed with solvable vertex models [2] associated with the quantum group Uq at q = 0. In this context, the KKR theory is regarded as the inverse scattering formalism of the generalized box-ball systems, where the rigged conﬁgurations and φ−1 play the roles of scattering data and inverse scattering transform, respectively. Precise descriptions are available either in [20, Proposition 2.6], [25, Sec. 3.2 and Appendix E], and Lemma B.9 in this paper. 2 ) case. In In this paper we study two closely related problems concerning Uq (sl the ﬁrst part (Sec. 2), we give a crystal theoretical interpretation of the opposite KKR map φ from paths to rigged conﬁgurations. It is done by introducing the local energy distribution of paths, which provides a bird’s-eye view of the whole combinatorial procedures involved in the KKR algorithm. In terms of generalized box-ball systems, φ is a direct scattering map and separates the dynamical degrees of freedom into action-angle variables, which are amplitudes and phase of solitons. The local energy distribution makes it possible to grasp these data from a global viewpoint. See Example 2.4. Together with the earlier result on φ−1 , we complete 2 ). the crystal interpretation of the KKR bijection φ±1 for Uq (sl The results mentioned so far are concerned with generalized box-ball systems on (semi) inﬁnite lattice. In the second part of the paper (Sec. 3), we launch the inverse scattering formalism in the periodic case. This was achieved in [21] for the simplest spin 1/2 system called the periodic box-ball system [40, 39], and subsequently in [26]. Here we treat the general spin s/2 case based on the crystal base theory. Here is an example of time evolution (T4 ) of an s = 3 case. t = 0 : 122 · 122 · 112 · 112 · 111 · 122 · 111 · 111 · 112 t = 1 : 112 · 112 · 122 · 122 · 112 · 111 · 122 · 111 · 111 t = 2 : 111 · 112 · 112 · 112 · 122 · 122 · 111 · 122 · 111 t = 3 : 111 · 111 · 112 · 112 · 112 · 112 · 222 · 111 · 122 t = 4 : 122 · 111 · 111 · 112 · 112 · 112 · 111 · 222 · 112 t = 5 : 112 · 222 · 111 · 111 · 112 · 112 · 112 · 111 · 122 t = 6 : 122 · 111 · 222 · 112 · 111 · 112 · 112 · 112 · 111 t = 7 : 111 · 122 · 111 · 122 · 122 · 111 · 112 · 112 · 112 t = 8 : 112 · 111 · 122 · 111 · 112 · 222 · 111 · 112 · 112 t = 9 : 112 · 112 · 111 · 122 · 111 · 111 · 222 · 112 · 112 t = 10 : 112 · 112 · 112 · 111 · 122 · 111 · 111 · 122 · 122 t = 11 : 122 · 122 · 112 · 112 · 111 · 122 · 111 · 111 · 112.

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A local spin s/2 state is an s array of 1 and 2 which are arranged not to decrease to the right. Each local state is regarded as a capacity s box. Local states, say 111, 112, 122 and 222 for s = 3, represent an empty box and those containing 1, 2 and 3 balls, respectively. An array of such local states are called paths. The above paths are of length 9. A path of length L can naturally be viewed as an element of Bs⊗L , the tensor 2 ). product of the crystal Bs of the s-fold symmetric tensor representation of Uq (sl A wealth of notions and combinatorial operations on Bs are provided by the crystal base theory. We make use of them to characterize a certain class of paths that are ˆ (A(1) ) and the commuting family of invariant under extended aﬃne Weyl group W 1 invertible time evolutions {Tl }. This is an important non-trivial step characteristic to the s > 1 situation. We introduce action-angle variables which correspond to those paths bijectively and linearize the dynamics. These features are integrated in Theorem 3.11. As corollaries of it, generic period and a counting formula of the paths are obtained in terms of conserved quantities in (3.28) and (3.26), respectively. For example (3.28) tells that the period of the above paths under T4 is indeed 11. (Notice that the t = 0 and t = 11 paths are the same.) These results agree with the conjecture in the most general setting [22]. The initial value problem is solved either by a combinatorial algorithm or by an explicit formula (3.36), (3.34) involving the ultradiscrete Riemann theta function (3.30), generalizing the s = 1 results in [23, 24]. These expressions follow rather straightforwardly from the ultradiscrete tau function studied in [25]. For the background idea of ultra-discretization and relevant issues in tropical geometry, see [35] and [27]. Several characteristic features in quasi-periodic solutions to soliton equations [4, 6] will be demonstrated in the ultradiscrete setting. In particular our actionangle variables live in the set (3.15) which is an ultradiscrete analogue of the Jacobi variety. For a reduced case (3.29) with s = 1, the underlying tropical hyperelliptic curve has been identiﬁed recently [9]. The action-angle variables are essentially solutions of the string center equation, which is a version of the Bethe equation at q = 0 [19]. In this sense, the inverse scattering formalism in this paper connects the Bethe ans¨atze at q = 1 [13, 12] and q = 0 [19] to the algebraic geometry techniques of soliton theory at a combinatorial level. Our crystal interpretation of the KKR map φ has stemmed from an attempt to formulate the direct scattering map in the generalized periodic box-ball system. In fact, we will show in Sec. 3.3 that the idea of local energy distribution is eﬃcient also in the periodic setting. The paper is organized as follows. In Sec. 2, the KKR map φ is identiﬁed with a procedure based on the local energy distribution in Theorem 2.2. We illustrate it along a few instructive examples. The proof will be given in [29]. Section 3 is devoted to the generalized periodic box-ball system. Section 4 is a summary. Appendix A recalls the basic facts on crystal base theory [10,11,14]. Appendix B is an exposition of the KKR bijection including the non-highest case [5, 30].

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2. Local Energy Distribution and the KKR Bijection In this section, we reformulate the combinatorial procedure of the KKR map φ in terms of the energy functions of crystal base theory. See Appendix A for the basic facts on crystal base theory. Consider the relation a ⊗ b1 b1 ⊗ a and the energy function e1 = H(a ⊗ b1 ) under the combinatorial R. We depict them by the vertex diagram: b1 e1

a

a .

b1 Successive applications of the combinatorial R a ⊗ b1 ⊗ b2 b1 ⊗ a ⊗ b2 b1 ⊗ b2 ⊗ a , with e2 = H(a ⊗ b2 ) is expressed by joining two vertices: b1 a

b2

e1

a

e2

b1

a. b2

Given a path b = b1 ⊗ b2 ⊗ · · · ⊗ bL , its local energy El,j is deﬁned by (j−1) (j−1) ⊗ bj ), where ul is speciﬁed by the following diagram with El,j := H(ul (0) the convention ul = ul (A.6). b1 ul

El,1

b2 (1)

ul

bL

El,2

b1

(2)

ul

(L−1)

· · · · · · · · · · ul

b2

El,L

(L)

ul . bL

We set E0,j = 0 for all 1 ≤ j ≤ L. We deﬁne Tl and El by Tl (b) = b1 ⊗ b2 ⊗ · · · ⊗ bL and El :=

L

El,j .

(2.1)

j=1 R

(L)

In other words, ul [0] ⊗ b Tl (b) ⊗ ul [El ], where we have omitted modes for b and Tl (b). (L+Λ) = ul for Given a path b = b1 ⊗ b2 ⊗ · · · ⊗ bL (bi ∈ Bλi ), we always have ul any l for a modiﬁed path b = b ⊗ 1

⊗Λ

if Λ > λ1 + · · ·+ λL . In such a circumstance,

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El (Tk (b )) = El (b ) is known to hold ([7, Theorem 3.2] and [15, Sec. 3.4]). Namely the sum El is a conserved quantity of the box-ball system on semi-inﬁnite lattice. Here we need more detailed information such as El,j . Lemma 2.1. For a path b = b1 ⊗ b2 ⊗ · · · ⊗ bL , we have El,j − El−1,j = 0 or 1, for all l > 0 and for all 1 ≤ j ≤ L. Proof. When l = 1, this is clear from the deﬁnition E0,j = 0 and the fact H(x⊗y) = 0 or 1 for any x ∈ B1 . Now we investigate possible values for El,j − El−1,j . We show that the diﬀerence (j) (j) (j) between tableaux for ul and ul−1 is only one letter, namely, if ul−1 = (x1 , x2 ), (j)

(j)

then ul = (x1 + 1, x2 ) or ul = (x1 , x2 + 1). We show the claim by induction on j. (0) (0) For j = 0, it is true because ul−1 = ul−1 = (l − 1, 0) and ul = ul = (l, 0). Suppose (k)

that the above claim holds for all j < k for some k. In order to compare ul−1 and (k)

(k−1)

(k)

(k−1)

ul , consider the relations ul−1 ⊗ bk bl−1,k ⊗ ul−1 and ul (k−1)

(k−1)

(k)

⊗ bk bl,k ⊗ ul .

is one letter. Recall By the assumption, the diﬀerence between ul−1 and ul that in calculating the combinatorial R by the graphical rule (Sec. A.2), order of (k−1) ⊗ bk , ﬁrst we can make all pairs making pairs is arbitrary. Therefore, in ul (k−1) that appear in ul−1 ⊗ bk , and then make the remaining one pair. This means the diﬀerence of number of unwinding pairs, i.e. El,k − El−1,k is 0 or 1. To make the (k) (k) induction proceed, note that this fact means the diﬀerence between ul−1 and ul is also one letter. Let b = b1 ⊗ · · · ⊗ bL ∈ Bλ1 ⊗ · · · ⊗ BλL be an arbitrary (either highest or not) path. Set N = E1 (b). We determine the pair of numbers (µ1 , r1 ), (µ2 , r2 ), . . . , (µN , rN ) by Steps (i)–(iv). (i) Draw a table containing (El,j − El−1,j = 0, 1) at the position (l, j), i.e. at the lth row and the jth column. We call this table local energy distribution. (ii) Starting from the rightmost 1 in the l = 1st row, pick one 1 from each successive row. The one in the (l + 1)th row must be weakly right of the one selected in the lth row. If there is no such 1 in the (l + 1)th row, the position of the lastly picked 1 is called (µ1 , j1 ). Change all the selected 1 into 0. (iii) Repeat Step (ii) (N − 1) times to further determine (µ2 , j2 ), . . . , (µN , jN ) thereby making all 1 into 0. (iv) Determine r1 , . . . , rN by rk =

j k −1

min(µk , λi ) + Eµk ,jk − 2

i=1

jk

Eµk ,i .

(2.2)

i=1

One may replace the procedure (ii) by (ii)’ Starting from any one of the lowest 1, pick one 1 from each preceding row. The one in the (l − 1)th row must be weakly left and nearest of the one selected in

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the lth row. The position of the ﬁrstly picked 1 is called (µ1 , j1 ). Change all the selected 1 into 0. Our main result in this section is the following theorem, which gives a crystal theoretic reformulation of the KKR map φ. Theorem 2.2. The above procedure (i)–(iv) is well defined and (λ, (µ, r)) coincides with the (unrestricted) rigged configuration φ(b). The procedure (i), (ii)’, (ii), (iv) is also well defined and leads to the same rigged configuration up to a permutation of (µk , jk )’s. The proof will be given in [29]. Example 2.3. Consider the path which will also be treated in Example B.5: b = 1111 ⊗ 11 ⊗ 22 ⊗ 12 ⊗ 2 ⊗ 122 ⊗ 122 ⊗ 1112 . According to Step (i), the local energy distribution is given in the following table (j stands for column coordinate of the table).

E1,j E2,j E3,j E4,j E5,j E6,j E7,j

− E0,j − E1,j − E2,j − E3,j − E4,j − E5,j − E6,j

1111 0 0 0 0 0 0 0

11 0 0 0 0 0 0 0

22 1 1 0 0 0 0 0

12 0 0 1 0 0 0 0

2 1 0 0 0 0 0 0

122 0 0 0 1 1 0 0

122 1 0 0 0 0 1 0

1112 0 1 0 0 0 0 0

Following Steps (ii) and (iii), letters 1 contained in the above table are classiﬁed into 3 groups, as indicated in the following table. 1111 E1,j E2,j E3,j E4,j E5,j E6,j E7,j

− E0,j − E1,j − E2,j − E3,j − E4,j − E5,j − E6,j

11

22 3 3

12

2 2∗

122

122 1

1112 1∗

3 3 3 3∗

The cardinalities of the 3 groups are 2, 1 and 6, respectively. From the positions marked with ∗, we ﬁnd (µ1 , j1 ) = (2, 8), (µ2 , j2 ) = (1, 5) and (µ3 , j3 ) = (6, 7). Now

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we evaluate riggings ri according to the rule (2.2). r1 =

8−1

min(2, λi ) + E2,8 − 2

i=1

8

E2,i

i=1

= (2 + 2 + 2 + 2 + 1 + 2 + 2) + 1 − 2(0 + 0 + 2 + 0 + 1 + 0 + 1 + 1) = 4, r2 =

5−1

min(1, λi ) + E1,5 − 2

i=1

5

E1,i

i=1

= (1 + 1 + 1 + 1) + 1 − 2(0 + 0 + 1 + 0 + 1) = 1, r3 =

7−1

min(6, λi ) + E6,7 − 2

i=1

7

E6,i

i=1

= (4 + 2 + 2 + 2 + 1 + 3) + 2 − 2(0 + 0 + 2 + 1 + 1 + 2 + 2) = 0. Therefore we obtain (µ1 , r1 ) = (2, 4), (µ2 , r2 ) = (1, 1) and (µ3 , r3 ) = (6, 0), which coincide with the calculation in Example B.5. The reader should compare the above local energy distribution and box adding procedure fully exhibited in Example B.5. Then it will be observed that the complicated combinatorial procedure in Deﬁnition B.4 is reduced to rather automatic applications of the combinatorial R and energy functions. Example 2.4. Theorem 2.2 provides a panoramic view on the combinatorial procedure of the KKR bijection from energy distribution. To show a typical example, we pick the following long path (length 30). 22 ⊗ 2 ⊗ 2 ⊗ 1 ⊗ 1122 ⊗ 112 ⊗ 1 ⊗ 11 ⊗ 222 ⊗ 12 ⊗ 11 ⊗ 2 ⊗ 2 ⊗ 2 ⊗ 22 ⊗ 2 ⊗ 1122 ⊗ 22 ⊗ 2 ⊗ 222 ⊗ 1 ⊗ 112 ⊗ 1 ⊗ 12 ⊗ 1222 ⊗ 11122 ⊗ 2 ⊗ 22 ⊗ 2 ⊗ 2 . Then, the local energy distribution takes the following form. E1,j − E0,j

s s

E6,j − E5,j E11,j − E10,j E16,j − E15,j

?

5s s

s

s

s

s10 s 15 s s s s s s s

30 j s 20 s s25 s s s s s s s s s

s s

s

s s s

s

s

s

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In the above table, letters 1 in the local energy distribution are represented by “•”, and letters 0 are suppressed. According to Steps (ii) and (iii), • belonging to the same group are joined by thick lines. We see there are 8 groups whose cardinalities are 5, 2, 16, 3, 2, 1, 4, 4 from left to right, respectively. By using the formula (2.2), we get the unrestricted rigged conﬁguration as follows: (µ1 , r1 ) = (4, 8), (µ2 , r2 ) = (4, 8), (µ3 , r3 ) = (1, 10), (µ4 , r4 ) = (2, 8), (µ5 , r5 ) = (3, 2), (µ6 , r6 ) = (16, −15), (µ7 , r7 ) = (2, 0), (µ8 , r8 ) = (5, −5). The vacancy numbers for each row is p16 = −15, p5 = 7, p4 = 10, p3 = 14, p2 = 16 and p1 = 14. Note that since the path in this example is not highest, the resulting unrestricted rigged conﬁguration has negative riggings and vacancy numbers. 3. Generalized Periodic Box-Ball System Here we extend the inverse scattering formalism [21] of the simplest periodic boxball system [40, 39] to general higher spins. The relevant time evolutions and associated energy will be denoted by Tl and El for distinction from Tl and El for the non-periodic case. Most of the proofs will be omitted as they are similar (but somewhat more involved) to [21]. Our new algorithm for the KKR map (Theorem 2.2), adapted to the periodic boundary condition, serves as a simple algorithm for the direct scattering transform. 3.1. Time evolution Fix the integer L, s ∈ Z≥1 throughout. Set P = Bs⊗L .

(3.1)

⊗L

We will also write Aﬀ(P) = Aﬀ(Bs ) . An element of P is called a path. A path b is highest if e˜1 b = 0. The weight of a path b = b1 ⊗ · · · ⊗ bL is given by wt(b) = wt(b1 ) + · · · + wt(bL ). We write wt(b) > 0 (wt(b) < 0) when it belongs to Z>0 Λ1 (Z<0 Λ1 ). Our generalized periodic box-ball system is a dynamical system on a subset of P equipped with the commuting family of time evolutions T1 , T2 , . . . . Let b = b1 ⊗ · · · ⊗ bL ∈ P be a path and l ∈ Z≥1 . For vl ∈ Bl , suppose ζ 0 vl ⊗ (ζ 0 b1 ⊗ · · · ⊗ ζ 0 bL ) (ζ −d1 ˜b1 ⊗ · · · ⊗ ζ −dL ˜bL ) ⊗ ζ e vl

(3.2)

holds under the isomorphism Aﬀ(Bl ) ⊗ Aﬀ(P) Aﬀ(P) ⊗ Aﬀ(Bl ), where the righthand side is unambiguously determined from the left-hand side. (e = d1 + · · · + dL .) We say that b is Tl -evolvable if the following (i) existence and (ii) uniqueness are satisﬁed: (i) there exists vl ∈ Bl such that vl = vl . (ii) if there are more than one such vl , ˜b1 ⊗ · · · ⊗ ˜bL is independent of their choice. If b is Tl -evolvable, we deﬁne Tl (b) = ˜b1 ⊗ · · · ⊗ ˜bL (∈ P). Otherwise we set Tl (b) = 0. In this sense, we will also write Tl (b) = 0 to mean that b is Tl -evolvable.

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Lemma 3.1. If b = b1 ⊗ · · · ⊗ bL is Tl -evolvable, not only ˜b1 ⊗ · · · ⊗ ˜bL but also d1 , . . . , dL and e in (3.2) are independent of the possibly nonunique choices of vl = vl . Thanks to this lemma we are entitled to deﬁne El (b) = e(∈ Z≥0 ) by (3.2) for a Tl -evolvable path b. Actually vl can be nonunique only if l > s and wt(p) = 0. The operations T1 , T2 , . . . form a family of time evolution operators associated with the energy E1 , E2 , . . . . These deﬁnitions can be summarized in ζ 0 vl ⊗ b Tl (b) ⊗ ζ El (b) vl

(3.3)

up to the spectral parameter for Tl -evolvable b. Pictorially, (3.2) looks as vl = v(0)

b1

v(1)

b2

v(2) · · ·

bL−1 v(L − 1) ˜bL−1

bL

v(L) = vl .

(3.4) ˜b1 ˜b2 ˜bL Clearly the time evolutions are weight preserving, i.e. wt(Tl (b)) = wt(b) when Tl (b) = 0. Since the combinatorial R is trivial on Bs ⊗ Bs (see (A.7)), we have the unique choice vs = vs = bL in (3.2), saying that a path is always Ts -evolvable and Ts acts as a cyclic shift: Ts (b1 ⊗ b2 ⊗ · · · ⊗ bL ) = bL ⊗ b1 ⊗ · · · ⊗ bL−1 .

(3.5)

If s = 1, all the paths are Tl -evolvable for any l ≥ 1 [21]. However this is no longer the case for s > 1. A similar situation is known also in the higher rank extensions [22]. Here we treat such a subtlety characteristic in the periodic setting. We simply say that b ∈ P is evolvable if it is Tl -evolvable for all l ∈ Z≥1 . We warn that “b is Tl -evolvable” is diﬀerent from “Tl (b) is evolvable”. The former means Tl (b) = 0 whereas the latter does Tk Tl (b) = 0 for all k ≥ 1. Here is a characterization of evolvable paths. Proposition 3.2. A path b = b1 ⊗ · · · ⊗ bL is evolvable if and only if bi = 1 . . . 1 or bi = 2 . . . 2 for some i. The proof of the proposition also tells the way to construct vl that makes (3.3) hold for a given path b. For l ≥ s, determine vl ∈ Bl by (see (A.6) for ul ) ul ⊗ b b ⊗vl ω(ul ) ⊗ b b ⊗vl

if wt(b) ≥ 0, if wt(b) < 0,

(3.6)

where b ∈ P is another path. So obtained vl is shown to satisfy vl ⊗ b Tl (b) ⊗ vl under Bl ⊗ P P ⊗ Bl . One may either use the latter relation in (3.6) to deﬁne vl when wt(b) = 0. For l < s, one has the unique vl from v ⊗ b b ⊗ vl for arbitrary v ∈ Bl if b is evolvable. Then vl ⊗ b Tl (b) ⊗ vl is again valid. Theorem 3.3. Suppose b, Tl (b) and Tk (b) are evolvable. Then the commutativity Tl Tk (b) = Tk Tl (b) and the energy conservation El (Tk (b)) = El (b), Ek (Tl (b)) = Ek (b) hold.

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Proof. Take vk for b and vl for Tk (b) as in (3.6). Set R(ζ 0 vl ⊗ ζ 0 vk ) = ζ −δ v¯k ⊗ ζ δ v¯l and regard b as an element of Aﬀ(P). By using the combinatorial R, one can reorder ζ 0 vl ⊗ ζ 0 vk ⊗ b in two ways along the isomorphism Aﬀ(Bl ) ⊗ Aﬀ(Bk ) ⊗ Aﬀ(P) Aﬀ(P) ⊗ Aﬀ(Bk ) ⊗ Aﬀ(Bl ) as follows:

where the equality is due to the Yang–Baxter equation. The outputs have been identiﬁed with Tk Tl (b), ζ Ek (Tl (b))−δ v¯k , etc. In particular the uniqueness (ii) stated under (3.2) guarantees that v¯k ⊗ Tl (b) Tk Tl (b) ⊗ v¯k and v¯l ⊗ b Tl (b) ⊗ v¯l up to the spectral parameter. The sought relations Tl Tk (b) = Tk Tl (b) and El (Tk (b)) = El (b), Ek (Tl (b)) = Ek (b) are obtained by comparing the two sides. Let s0 , s1 be the Weyl group operators (A.3) and ω be the involution (A.4) ˆ (A(1) ) = ω, s0 , s1 forms the extended aﬃne Weyl acting on the crystal P. Then W 1 (1) group of type A1 . The time evolutions Tl and the energy El enjoy the symmetry ˆ (A(1) ). under W 1 ˆ (A(1) ), w(b) Proposition 3.4. Let b be an evolvable path. Then for any w ∈ W 1 is also evolvable and the commutativity wTl (b) = Tl (w(b)) and the invariance El (w(b)) = El (b) are valid. In particular, the relation Tl = ω ◦ Tl ◦ ω

(3.7)

exchanging the letters 1 ↔ 2 is useful. Any path is Tl -evolvable for l ≥ s. In fact, for l suﬃciently large the time evolution Tl and the energy El admit a simple description as follows. Proposition 3.5. For any path b ∈ P, there exists k ≥ s such that Tl (b) and El (b) are independent of l for l ≥ k. Denoting them by T∞ (b) and E∞ (b), one has T∞ (b) = ω(s0 (b)), wt(b) = p∞ Λ1 T∞ (b) = ω(s1 (b)), wt(b) = −p∞ Λ1

if wt(b) ≥ 0, if wt(b) ≤ 0,

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where p∞ = Ls−2E∞ (b) according to (3.10). In particular, T∞ (b) = ω(b) if wt(b) = 0. Example 3.6. For b = 112 ⊗ 111 ⊗ 222 ⊗ 122 ⊗ 112 having a positive weight, we have T1 (b) = 122 ⊗ 111 ⊗ 122 ⊗ 222 ⊗ 111 , T2 (b) = 112 ⊗ 112 ⊗ 112 ⊗ 222 ⊗ 112 , T3 (b) = 112 ⊗ 112 ⊗ 111 ⊗ 222 ⊗ 122 , T4 (b) = 122 ⊗ 112 ⊗ 111 ⊗ 122 ⊗ 122 , Tl (b) = 122 ⊗ 122 ⊗ 111 ⊗ 112 ⊗ 122

(l ≥ 5) .

So T∞ (b) = T5 (b). On the other hand, 0-signature and reduced 0-signature of b read 112 ⊗ 111 ⊗ 222 ⊗ 122 ⊗ 112 .

−−+ −−

−−− −−

+++ ++

−++

−−+ +

Thus s0 (b) = 112 ⊗ e˜0 111 ⊗ 222 ⊗ 122 ⊗ 112 , which coincides with ω(T∞ (b)). For an evolvable path b ∈ P, we have the time evolution Tl (b) ∈ P and the associated energy El (b) ∈ Z≥0 for all l ≥ 1. This leads us to introduce the “isolevel” set     ˆ P(m) = b ∈ P | b : evolvable, El (b) = min(l, k)mk (3.8)   k≥1

labeled with the sequence m = {mk | k ≥ 1}. We shall always take it for granted that {mk } and {El } are in one-to-one correspondence via El = min(l, k)mk , mk = −Ek−1 + 2Ek − Ek+1 (E0 = 0). (3.9) k≥1

We also use the vacancy number pj = L min(s, j) − 2Ej .

(3.10)

The following result is due to T. Takagi. ˆ Proposition 3.7 ([32]). For any path b ∈ P(m) with wt(b) ≥ 0, its time evolution dl b l Tl )(b) becomes highest under appropriate choices of {dl }. Such {dl } is not unique. Cyclic shift Tsds is not enough to achieve this in general. From Proposition 3.7 one can show ˆ Proposition 3.8. P(m) = ∅ if and only if ∀pj ≥ 0. ˆ Henceforth we assume ∀pj ≥ 0. If b belongs to P(m) and Tl (b) is evolvable, then ˆ must hold because of Ek (Tl (b)) = Ek (b) by Theorem 3.3. However, Tl (b) ∈ P(m) b Actually

Ts and Ts−1 have been shown to suﬃce.

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the point here is that even if a path b is evolvable, it is not guaranteed in general that its time evolution Tl (b) is again evolvable. Example 3.9. b1 = 11 ⊗ 22 and b2 = 22 ⊗ 11 are evolvable, but T1 (b1 ) = T1 (b2 ) = 12 ⊗ 12 is not. See Proposition 3.2. The situation is depicted as 11 ⊗ 22 HT1 j 12 ⊗ 12 H * 22 ⊗ 11 T1

T1 0.

ˆ can contain non-evolvable paths in general. On the other Thus the set Tl (P(m)) ˆ must share the same energy spectrum hand, all the evolvable paths in Tl (P(m)) ˆ {El } as P(m) by virtue of Theorem 3.3. Therefore, what holds in general is ˆ ˆ ˆ Tl (P(m)) = (subset of P(m)) {Tl (b) : non-evolvable | b ∈ P(m)}. ˆ A natural question is to ﬁnd a pleasant situation where Tl acts on P(m) as a bijection. This is answered in: ˆ ˆ = P(m) holds for all l if and only if (E1 , E2 ) = Proposition 3.10. Tl (P(m)) (L/2, L). So this is always satisﬁed if L is odd. For an evolvable path b with even length L, the condition (E1 , E2 ) = (L/2, L) is equivalent to b = 11(c1 ) ⊗ (c2 )22 ⊗ · · · ⊗ (cL )22 or b = (c1 )22 ⊗ 11(c2 ) ⊗ · · · ⊗ 11(cL ) for some ci ∈ Bs−2 , where 11 and 22 alternate. Here for example, 11(c) = 11122 and (c)22 = 12222 for c = 122 ∈ B3 . (Thus such b can exist only for s ≥ 2.) The two paths in Example 3.9 correspond to the case (E1 , E2 ) = (1, 2) with L = 2. For ˆ P(m) such that (E1 , E2 ) = (L/2, L), the inverse time evolution is given by Tl−1 = ◦ Tl ◦ ,

(3.11)

where is deﬁned by (b1 ⊗ b2 ⊗ · · · ⊗ bL ) = bL ⊗ · · · ⊗ b2 ⊗ b1 . ˆ To summarize Theorem 3.3, Propositions 3.4 and 3.10, each set P(m) of evolv able paths is characterized by the conserved quantity El = k≥1 mink (l, k)mk called energy, and enjoys the invariance under ˆ (A(1) ), (i) the extended aﬃne Weyl group W 1 (ii) the commuting family of invertible time evolutions {Tl | l ≥ 1}. ˆ P(m) is non-empty if ∀pj ≥ 0. The invariance (ii) is valid if (E1 , E2 ) = (L/2, L) is further satisﬁed.

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3.2. Action-angle variable From now on, we assume that m = {mj } satisﬁes ∀pj ≥ 1.

(3.12)

See (3.9) and (3.10). This fulﬁlls the conditions in Propositions 3.8 and 3.10 since ˆ is decomposed into a disjoint union of ﬁxed weight p1 = L − 2E1 . The set P(m) subsets: ˆ P(m) = P(m) ω(P(m)), ˆ P(m) = {b ∈ P(m) | wt(b) = p∞ Λ1 }.

(3.13)

ˆ In view of (3.7), dynamics on P(m) is reduced to the commuting family of invertible time evolutions {Tl } on the ﬁxed (positive) weight subset P(m): Tl : P(m) → P(m).

(3.14)

We present the inverse scattering transform that linearizes the dynamics (3.14) and an explicit solution of the initial value problem. For a general background on the inverse scattering method, see [1,8]. In our approach the direct scattering transform is formulated either by a modiﬁed KKR bijection as in the s = 1 case [21] or by an appropriate extension of the procedure in Theorem 2.2 to a periodic setting. First we introduce the action-angle variables. For the paths belonging to P(m), the action variable is just the conserved quantity m = {mj } or equivalently {El } (3.9). It may also be presented as the Young diagram µ having mj rows with length j. Let H = {j1 < · · · < jg } be the set of distinct row lengths of µ, namely, j ∈ H ↔ mj > 0. The set J (m) of angle variables is deﬁned by J (m) = ((Zmj1 × · · · × Zmjg )/Γ − ∆)sym , Γ = A(Zmj1 × · · · × Zmjg ).

(3.15)

Here A = (Ajα,kβ ) is the matrix of size mj1 + · · · + mjg having a block structure: Ajα,kβ = δj,k δα,β (pj + mj ) + 2 min(j, k) − δj,k ,

(3.16)

where j, k ∈ H and 1 ≤ α ≤ mj , 1 ≤ β ≤ mk . A is symmetric and positive deﬁnite under the assumption (3.12) [19]. ∆ is the subset of (Zmj1 × · · · × Zmjg )/Γ having coincident components within a block: ∆ = {(Ij,α )j∈H,1≤α≤mj | Ij,α = Ij,β for some j ∈ H, 1 ≤ α = β ≤ mj }.

(3.17)

In (3.15), −∆ signiﬁes the complement of ∆. The subscript sym means the identiﬁcation under the exchange of components within blocks via the symmetric group Smj1 × · · · × Smjg . We introduce the time evolution of the angle variables by Tl : J (m) → J (m), (Ij,α ) → (Ij,α + min(l, j)),

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which makes sense because it obviously preserves the set (Zmj1 ×· · ·×Zmjg )/Γ −∆. We shall simply write this as Tl (I) = I + hl .

(3.18)

Namely, hl = (min(j, l))j∈H,1≤α≤mj ∈ Zmj1 +···+mjg is the velocity of the angle variable I = (Ij,α ) under the time evolution Tl . 3.3. Direct scattering We introduce the direct scattering map Φ : P(m) → J (m). A quick formulation is due to a modiﬁed KKR bijection as done in [21] for s = 1. Note that Proposition 3.7 tells that P(m) is the {Tl }-orbit of highest paths having thedlKKR conﬁguration m = {mj }. Thus we express a given b ∈ P(m) as b = l Tl )(b+ ) in terms of a highest path b+ ∈ P(m). Let (µ, r) be the rigged conﬁguration for b+ , where the appearance of the µ corresponding to m = {mj } is due to Theorem 3.3. Let the rigging attached to the length j(∈ H) rows of µ be 0 ≤ rj,1 ≤ · · · ≤ rj,mj ≤ pj . Consider the element I+ dl hl mod Γ ∈ J (m), where I = (rj,α + α − 1)j∈H,1≤α≤mj . (3.19) l

rj,α + α − 1 is strictly increasing with α, therefore I + l dl hl mod Γ belongs to (Zmj1 × · · · × Zmjg )/Γ − ∆. Given b ∈ P(m), the choice of {dl } and the highest

dl path b+ such that b = l Tl )(b+ ) is not unique in general. This non-uniqueness is to be cancelled by mod Γ. In fact we have Theorem 3.11. The rule Φ(b) = I+ l dl hl mod Γ specified by (3.19) is a bijection Φ : P(m) → J (m), and the following commutative diagram is valid: Φ

P(m) −−−−→ J (m)   T .  Tl

l

(3.20)

Φ

P(m) −−−−→ J (m) An alternative way to deﬁne the direct scattering map Φ is obtained by a periodic extension of the procedure (i), (ii)’, (iii), (iv) in Theorem 2.2. This option is valid under a certain condition which we shall explain after (3.24). It is more direct

dl than the above one in that the relation b = l Tl )(b+ ) need not be found. Here we illustrate it along the example: b = 122 ⊗ 122 ⊗ 112 ⊗ 112 ⊗ 111 ⊗ 122 ⊗ 111 ⊗ 111 ⊗ 112 ∈ B3⊗9 .

(3.21)

This path has appeared as the t = 0 case in the introduction. Let vl ∈ Bl be the element satisfying vl ⊗b Tl (b)⊗vl . It is unique under the condition (3.12) and can

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be found by (3.6). The energy is given by E1 = 4, E2 = 7, E3 = 8, El = 9 (l ≥ 4). So the action variable is µ=

.

Local energy El,k = H(v(k − 1) ⊗ bk ) is determined by using v(k) in (3.4). The distribution of δEl,k = El,k − El−1,k looks as δE1,k δE2,k δE3,k δE4,k

122 0 1 1 0

122 1 0 0 1

112 0 1 0 0

112 1 0 0 0

111 0 0 0 0

122 1 1 0 0

111 0 0 0 0

111 0 0 0 0

112 1 0 0 0

We group 1’s by a periodic analogue of the procedure (i), (ii)’, (iii), (iv) in Theorem 2.2. Pick a lowest 1, say δEl,k = 1 at the lth row. If there are more than one such k, any choice is possible. Let the rightmost 1 in δEl−1,k+1 , . . . , δEl−1,L−1 , δEl−1,L , δEl−1,1 , . . . , δEl−1,k−1 , δEl−1,k

(3.22)

be δEl−1,k = 1. Namely, k is the position of the rightmost 1 satisfying k ≤ k cyclically. Then connect δEl,k to δEl−1,k . Repeat this until the successive connection reaches some δE1,k on the ﬁrst row. This completes one group. Erase all the 1’s in it and repeat the same procedure starting from a lowest 1 in the rest to form other groups until all the initial 1’s are exhausted. 122 122 112 112 111 122 111 111 112 s s s s .. .. . s . s s s s A group consisting of l dots will be called a soliton of length l. Make a cyclic shift Ts−d so that all the solitons stay within the left and the right boundary. Namely, no soliton sits across the boundary. In the above, we take, for example, d = 3. 112 111 122 111 111 112 122 122 112 s s s s s

s s

s s

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Computing the rigging of each soliton according to (2.2), we ﬁnd 112 111 122 111 111 112 122 122 112 s s s s −1 s0 s s3 s s7 These values are for Ts−d (b). The rigging for b in question is deﬁned to be their shift +d min(s, j) for length j solitons, leading to (s = d = 3 in this example) 122 122 112 112 111 122 111 111 112 s s2 s s .. .. . s . s s 9 6 s s 16 Order the so obtained rigging for length j solitons as rj,1 ≤ · · · ≤ rj,mj and set J = (Jj,α )j∈H,1≤α≤mj ,

where Jj,α = rj,α + α − 1.

(3.23)

In the present example, H = {1, 2, 4}, (m1 , m2 , m4 ) = (1, 2, 1), (p1 , p2 , p4 ) = (1, 4, 9) and 

p1 +m1 +1 2 min(1, 2) A= 2 min(1, 2) 2 min(1, 4)

2 min(1, 2) p2 +m2 +3 3 2 min(2, 4)

2 min(1, 2) 3 p2 +m2 +3 2 min(2, 4)

  3 2 min(1, 4) 2 2 min(2, 4) = 2 min(2, 4) 2 p4 +m4 +7 2

2 9 3 4

2 3 9 4

 2 4 , 4 17

so the angle variable is            J1,1 3 2 2 2 2 J2,1   6          2 9 3 4            4 J2,2  = 9 + 1 mod AZ = Z 2 ⊕ Z 3 ⊕ Z 9 ⊕ Z  4 , J4,1 2 16 4 4 17 

(3.24)

where +1 is the contribution of α − 1 in Jj,α = rj,α + α − 1. This procedure for the direct scattering map Φ works provided that there is a cyclic shift Ts−d (b) such that no soliton stays across the boundary. The case s = 1 [21] is such an example. We conjecture it for general s under the assumption (3.12). One can show that J mod Γ is independent of the possible non-uniqueness of such cyclic shifts. The diﬀerence caused by such choices belong to Γ. This can be observed, for example, by comparing (3.24) and (3.25).

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Let us re-derive the result (3.24) from (3.19). The latter starts, for example, from the relation b = T3−5 (b+ ), where b+ = 111 ⊗ 122 ⊗ 111 ⊗ 111 ⊗ 112 ⊗ 122 ⊗ 122 ⊗ 112 ⊗ 112 is a highest path corresponding to the rigged conﬁguration 6 3 0 1 Thus (3.19) is evaluated as        I1,1 1 1 −4 I2,1   0  2 −10       I − 5h3 =  I2,2  = 3 + 1 − 5 2 =  −6 I4,1

6

   mod AZ4 . 

−9

3

This certainly coincides with the result (3.24) since the diﬀerence    2 −4  6 −10  − 10  −6 16



        6 2 2 2  16 9 3  4 = = + +   16 3 9  4

−9

25

4

4

(3.25)

17

belongs to AZ4 . 3.4. Inverse scattering According to Theorem 3.11, the dynamics of the generalized periodic box-ball system is transformed to a straight motion (3.18) in the set J (m) of angle variables. To complete the inverse scattering method, one needs the inverse scattering map Φ−1 from J (m) back to paths P(m). Under the condition (3.12), it is easy to show that any element I ∈ J (m) has a (not necessarily unique) representative form I = l dl hl + (rj,α + α − 1)j∈H,1≤α≤mj such that 0 ≤ rj,1 ≤ · · · ≤ rj,mj ≤ pj by using the equivalence under Γ. If µ denotes the Young diagram for m = {mk } Letting b+ and r = (rj,α )j∈H,1≤α≤mj , then (µ, r) becomes a rigged conﬁguration. dl (b be the highest path corresponding to it, Φ−1 (I) := T ) is independent of + l l the choice of the representative form and yields the inverse of Φ. Actually, one can always take dl = 0 for l = 1. Our solution of the initial value problem is achieved by the commutative diagram (3.20), namely the composition: Φ

{Tl }

Φ−1

P(m) −→ J (m) −→ J (m) −→ P(m),

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where the number of computational steps is independent of the time evolution. As an illustration we derive T21000 (b) = 111 ⊗ 111 ⊗ 112 ⊗ 112 ⊗ 122 ⊗ 122 ⊗ 112 ⊗ 111 ⊗ 122 , T41000 (b) = 112 ⊗ 112 ⊗ 112 ⊗ 111 ⊗ 122 ⊗ 111 ⊗ 111 ⊗ 122 ⊗ 122 for b given in (3.21). From (3.24) and (3.18), the angle variables for T21000 (b) and T41000 (b) are given by             1002 0 + 82 3 2 2 2 2006  0 + 82  2 9 3  4             2010 = 3 + 1 + 82 + 108 2 + 129 3 + 129 9 + 40  4 , 2016 6 + 82 2 4 4           1002 0 + 222 3 2 2 2006  0 + 222  2 9 3             2010 = 3 + 1 + 222 + 28 2 + 84 3 + 84 9 + 180  

4016

6 + 222

2

4

4

17  2 4 . 4

17

4

In the right-hand sides, the last four terms belong to AZ , hence can be neglected. The ﬁrst terms correspond to T182 and T1222 of the rigged conﬁguration (+1 is removed as the “α − 1 part”) 6 3 0 0 which is mapped, under the KKR bijection, to the highest path b+ := 111 ⊗ 122 ⊗ 111 ⊗ 112 ⊗ 111 ⊗ 122 ⊗ 122 ⊗ 112 ⊗ 112 . One can check T182 (b+ ) = T21000 (b) and T1222 (b+ ) = T41000 (b) completing the derivation. 3.5. State counting and periodicity The matrix A (3.16) was originally introduced in the study of the Bethe equation at q = 0 [19]. From this connection we have Theorem 3.12 ( [19, Theorems 3.5 and 4.9]). 1 pj + mj − 1 |J (m)| = (det F ) . mj mj − 1

(3.26)

j∈H

F = (Fj,k )j,k∈H ,

Fj,k = δj,k pj + 2 min(j, k)mk .

(3.27)

Combined with Theorem 3.11, this yields a formula for |P(m)|, namely, the number of states characterized by the conserved quantity. For B3⊗9 and m corresponding to the Young diagram (4, 2, 2, 1) considered above, we have |J (m)| = 990.

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From (3.15), all the paths b ∈ P(m) obey the relation TlNl (b) = b if

Nl hl ∈ Γ.

Writing Nl hl = An, components of the vector n = (nj,α ) are given by nj,α = Nl

det A[jα] , det A

where A[jα] is obtained by replacing (jα)th column of A by hl . It is elementary to check det F [j] det A[jα] = . det A det F F [j] is obtained from F by replacing the jth column by the l-dependent vector (min(l, k))k∈H . The independence on α reﬂects the symmetry of A (3.16) within blocks. Thus the generic period of P(m), namely the minimum Nl such that Nl hl ∈ Γ is det F det F ,..., Nl = LCM , (3.28) det F [j1 ] det F [jg ] where by LCM(r1 , . . . , rg ) for rational numbers r1 , . . . , rg , we mean the smallest positive integer in Zr1 ∩ · · · ∩ Zrg . When det F [jk ] = 0, the entry det F /det F [jk ] is to be excluded. For B3⊗9 and the Young diagram (4, 2, 2, 1) in the above example, we have         3 4 2 1 4 2 3 1 2 3 4 1 F = 2 12 4, F [1] = 1 12 4, F [2] = 2 1 4, F [4] = 2 12 1 2 8 17 1 8 17 2 1 17 2 8 1 396 for l = 1, leading to N1 = LCM 99 28 , 13 , 99 = 396. Similar calculations yield N1 = 396,

N2 = 99,

N3 = 9,

Nl = 11 (l ≥ 4).

TlNl (b) = b can be directly checked for b in (3.21). In fact T411 (b) = b has been demonstrated in the introduction. For the fundamental period, formally the same closed formula as Eq. (4.26) in [21] is valid. The formula (3.28) agrees with the most (1) general conjecture on An [22]. For s = 1 and l = ∞, it was originally obtained in [39] by a combinatorial argument. 3.6. Ultradiscrete Riemann theta lattice Let us present an explicit formula for the inverse scattering map Φ−1 : J (m) → P(m) in terms of the ultradiscrete Riemann theta function. We keep assuming the condition (3.12). For general {mj }, [24, Theorem 5.1] remains valid if the vacancy number pj is replaced by (3.10) in this paper. Here we restrict ourselves to the case mj = 1

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for all j ∈ H = {j1 < · · · < jg } for simplicity. Thus J (m) (3.15) and A (3.16) reduce to J (m) = Zg /AZg ,

(3.29)

A = (Aj,k )j,k∈H ,

Aj,k = δj,k pj + 2 min(j, k).

Following [23], we introduce the ultradiscrete Riemann theta function by t nAn/2 + t nz Θ(z) = lim log exp − →+0 g n∈Z

(3.30)

= − ming {t nAn/2 + t nz}, n∈Z

which enjoys the quasi-periodicity: Θ(z + v) = t vA−1 (z + v/2) + Θ(z)

for any v ∈ Γ = AZg .

(3.31)

Consider the planar square lattice Bs1 z0,0

Bs2 z0,1

Bs3

BsL

z0,2

z0,L ...

Bl1 z1,0

z1,1

z1,2

z1,L ...

Bl2 z2,0

z2,1 .. .

z2,2 .. .

z2,L .. .

.. .

where we assume s1 = · · · = sL = s for the time being. zt,k is deﬁned by p zt,k = I − + hl1 + · · · + hlt − hs1 − · · · − hsk , (3.32) 2 where I ∈ J (m) is the angle variable of a path b = b1 ⊗ · · · ⊗ bL ∈ Bs1 ⊗ · · · ⊗ BsL , and p = t (pj1 , . . . , pjg ). To each edge, either vertical or horizontal, we attach a number Θ(z, z ) via the rule: z

z z ,

z

→ Θ(z, z ) := Θ(z) − Θ(z ) − Θ(z + h∞ ) + Θ(z + h∞ ). (3.33)

The rule (3.33) can also be described neatly in terms of a two-layer cubic lattice whose sites are assigned with zt,k or zt,k + h∞ . Assign the 2-component integer vectors xt,k = (lt − Θ(zt−1,k−1 , zt,k−1 ), Θ(zt−1,k−1 , zt,k−1 )), yt,k = (sk − Θ(zt−1,k , zt−1,k−1 ), Θ(zt−1,k , zt−1,k−1 ))

(3.34)

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to the edges as follows: yt,k zt−1,k−1

zt−1,k

xt,k

xt,k+1 zt,k−1

zt,k

yt+1,k By the same argument as [23], one can show that the ultradiscrete tau function in [25] for the periodic system coincides essentially with Θ (3.30) here. From this fact and Theorem 3.11 we obtain: Theorem 3.13. xt,k ∈ Blt , yt,k ∈ Bsk ,

R(xt,k ⊗ yt,k ) = yt+1,k ⊗ xt,k+1 ,

(3.35)

where R denotes the combinatorial R. The path b is reproduced from I ∈ J (m) by b = y1,1 ⊗ y1,2 ⊗ · · · ⊗ y1,L . The assertion xt,k ∈ Blt means that Θ(zt−1,k−1 , zt,k−1 ) is an integer in the range [0, lt ], and similarly for yt,k ∈ Bsk . The periodic boundary condition in the horizontal direction xt,0 = xt,L is valid for any t. This can easily be checked by using zt,0 − zt,L = hs1 + · · · + hsL = At (1, 1 . . . , 1) and the quasi-periodicity (3.31). Theorem 3.13 tells that xt,k and yt,k obey the local dynamics (combinatorial R) of the generalized periodic box-ball system. As a result, we get Tlt · · · Tl1 (b) = yt+1,1 ⊗ yt+1,2 ⊗ · · · ⊗ yt+1,L .

(3.36)

In this way the solution of the initial value problem under arbitrary time evolutions {Tlt } is written down explicitly. Note that we have given an explicit formula not only for yt,k but also xt,k which is often called “carrier” [34]. These aspects of the periodic box-ball system on P = Bs⊗L admit a generalization to the inhomogeneous case P = Bs1 ⊗ · · · ⊗ BsL . A typical example is a combinatorial version of Yang’s system [38, 36], which is endowed with the family of time evolutions {Ts1 , . . . , TsL }. The well-known relation Ts1 Ts2 · · · TsL = Id is understood from hs1 + · · · + hsL = At (1, 1 . . . , 1) ∈ Γ in our linearization scheme. To adapt the formalism to the inhomogeneous situation P = Bs1 ⊗ · · · ⊗ BsL , we employ (3.23) to specify Φ and replace the vacancy number (3.10) with pj =

L i=1

min(si , j) − 2Ej .

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Under the condition (3.12), we conjecture that Theorem 3.13 remains valid. For an illustration, we consider the path b = 11 ⊗ 2 ⊗ 1 ⊗ 2 ⊗ 122 ⊗ 1 ⊗ 12 ⊗ 2 ⊗ 1.

(3.37)

The action-angle variable is depicted as −1 1

.

0 So the vacancy numbers p, the period matrix A and the angle variable I are given by         p1 3 5 2 2 0 p = p2  = 2 , A = 2 6 4 , I =  1  . p3 1 2 4 7 −1 We set (s1 , . . . , s9 ) = (2, 1, 1, 1, 3, 1, 2, 1, 1) according to (3.37). Let us take (l1 , l2 , l3 ) = (2, 1, 3) and consider the time evolution b → Tl1 (b) → Tl2 Tl1 (b) → Tl3 Tl2 Tl1 (b). Then the edge variables exhibit the following pattern: 11 12

2 11

12 2

12 1

1 22

111

1

12 1

1

122

112

12

1 2

11

12

2

112

1

12 122

1

1 22

1

2 112

2

11

2

112 1

12

2

112

111 2

1

122

2 1

2

122

11 2

1

11

2

2 1

1 122

12

12

2 1

112 2

111 . 2

In terms of the edge variable Θ(z, z ) (3.33) representing the number of the letter 2 in tableaux on edges, this looks as 0 1

1 0

1 1

1 0

0 2

0

0

0

1

1

1

0

0

1

1

1

2

0

1 1

0 0

2 1

0

0

1 2

0

1

0

1 1

1

1

1

1 1

0

0

2

1 0

1

2

0

0 1

1

1

1

0 2

0

0

1 0

1 1

0 1 (3.38)

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The table of the values of the ultradiscrete Riemann theta Θ(zt,k ) is as follows: 0

0

1

2

4

8

10

14

17

20

0

0

0

1

2

5

7

10

12

15

0

0

0

0

1

3

5

8

10

12

2

0

0

0

0

1

2

4

6

8

Similarly the values of Θ(zt,k + h∞ ) are given as follows: 0

0

0

1

2

4

6

9

11

14

1

0

0

0

1

2

3

6

8

10

2

0

0

0

0

1

2

4

6

8

4

2

1

0

0

0

1

2

3

4

From these values of Θ(zt,k ) and Θ(zt,k + h∞ ), the table in (3.38) is reproduced by the rule (3.33). 4. Summary We have introduced the local energy distribution of paths in Sec. 2 and reformulated the KKR map φ in Theorem 2.2. Combined with the result for φ−1 [20, 28], it 2 ). completes the crystal interpretation of the KKR bijection for Uq (sl In Sec. 3, the generalized periodic box-ball system on Bs⊗L is studied. Under ˆ the condition (3.12), the set of paths P(m) (3.8) characterized by conserved quantities enjoy all the properties (i) and (ii) stated under (3.11). The action-angle variables are introduced in Sec. 3.2. The inverse scattering formalism, i.e. simultaneous linearization of the commuting family of time evolutions, is established in Theorem 3.11. It leads to the formulas for state counting (Theorem 3.12), generic period (3.28), and an algorithm for solving the initial value problem. According to Theorem 3.13, the solution of the initial value problem (3.36) is expressed in terms of the ultradiscrete Riemann theta function (3.30). A similar formula has been obtained also for the carrier variable xt,k simultaneously. These results extend the s = 1 case [21, 23] and agree with the conjecture on the most general case [22].

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Appendix A. Crystals and Combinatorial R A.1. Crystals We recapitulate the basic facts in the crystal base theory [10, 11, 14]. Let Bl the (1) crystal of the l-fold symmetric tensor representation of Uq (A1 ). As a set it is 2 given by Bl = {x = (x1 , x2 ) ∈ (Z≥0 ) | x1 + x2 = l}. The element (x1 , x2 ) will also be expressed as the length l rowshape semistandard tableau containing the letter i xi times. For example, B1 = 1 , 2 , B2 = 11 , 12 , 22 . (We shall often omit the frames of the tableaux.) The action of Kashiwara operators f˜i , e˜i : B → ei x)j = xj + δj,i − δj,i+1 , B {0} (i = 0, 1) reads (f˜i x)j = xj − δj,i + δj,i+1 and (˜ where all the indices are in Z2 , and if the result does not belong to (Z≥0 )2 , it should be understood as 0. The classical part of the weight of x = (x1 , x2 ) ∈ Bl is wt(x) = lΛ1 − x2 α1 = (x1 − x2 )Λ1 , where Λ1 and α1 = 2Λ1 are the fundamental weight and the simple root of A1 . For any b ∈ B, set εi (b) = max{m ≥ 0 | e˜m i b = 0},

ϕi (b) = max{m ≥ 0 | f˜im b = 0}.

By the deﬁnition one has εi (x) = xi+1 and ϕi (x) = xi for x = (x1 , x2 ) ∈ Bl . Thus εi (x) + ϕi (x) = l is valid for any x ∈ Bl . For two crystals B and B , one can deﬁne the tensor product B ⊗ B = {b ⊗ b | b ∈ B, b ∈ B }. The operators e˜i , f˜i act on B ⊗ B by e˜i b ⊗ b if ϕi (b) ≥ εi (b ) e˜i (b ⊗ b ) = (A.1) b ⊗ e˜i b if ϕi (b) < εi (b ), f˜i b ⊗ b if ϕi (b) > εi (b ) (A.2) f˜i (b ⊗ b ) = b ⊗ f˜i b if ϕi (b) ≤ εi (b ). Here 0 ⊗ b and b ⊗ 0 should be understood as 0. The tensor product Bl1 ⊗ · · · ⊗ Blk is obtained by repeating the above rule. The classical part of the weight of b ∈ B for any B = Bl1 ⊗ · · · ⊗ Blk is given by wt(b) = (ϕ1 (b) − ε1 (b))Λ1 = (ε0 (b) − ϕ0 (b))Λ1 . Let si (i = 0, 1) be the Weyl group operator [10] acting on any crystal B as ϕ (b)−ε (b) i (b) ϕi (b) ≥ εi (b), f˜i i (A.3) si (b) = εi (b)−ϕi (b) e˜i (b) ϕi (b) ≤ εi (b), for b ∈ B. Let ω : (x1 , x2 ) → (x2 , x1 )

(A.4)

be the involutive Dynkin diagram automorphism of Bl . We extend it to any B = ˆ (A(1) ) = ω, s0 , s1 acts Bl1 ⊗ · · · ⊗ Blk by ω(B) = ω(Bl1 ) ⊗ · · · ⊗ ω(Blk ). Then W 1 (1) on B as the extended aﬃne Weyl group of type A1 . The action of f˜i , e˜i and si is determined in principle by (A.1) and (A.2). Here we explain the signature rule to ﬁnd the action on any Bl1 ⊗ · · · ⊗ Blk , which is of great εi (b)

ϕi (b)

! ! practical use. The i-signature of an element b ∈ Bl is the symbol − · · · − + · · · +.

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The i-signature of the tensor product b1 ⊗ · · · ⊗ bk ∈ Bl1 ⊗ · · · ⊗ Blk is the array of the i-signature of each bj . Here is an example from B5 ⊗ B2 ⊗ B1 ⊗ B4 : 0-signature 1-signature

11112 ⊗ 12 ⊗ 2 ⊗ 1122

−−−−+ −++++

−+ −+

+ −

−−++ −−++

where 1122 for example represents 1122 ∈ B4 and not 1 ⊗ 1 ⊗ 2 ⊗ 2 ∈ B1⊗4 , etc. In the i-signature, one eliminates the neighboring pair +− (not −+) α

β

! ! successively to ﬁnally reach the pattern − · · · − + · · · + called reduced i-signature. The result is independent of the order of the eliminations when it can be done simultaneously in more than one places. The reduced i-signature tells that εi (bi ⊗ · · · ⊗ bk ) = α and ϕi (bi ⊗ · · · ⊗ bk ) = β. In the above example, we get 11112 ⊗ 12 ⊗ 2 ⊗ 1122

0-signature −−−− 1-signature −+

++ ++

Thus ε0 = 4, ϕ0 = 2, ε1 = 1 and ϕ1 = 3. Finally f˜i hits the component that is responsible for the leftmost + in the reduced i-signature making it −. Similarly, e˜i hits the component corresponding to the rightmost − in the reduced i-signature making it +. If there is no such + or − to hit, the result of the action is 0. The α

β

! ! Weyl group operator si acts so as to change the reduced i-signature − · · · − + · · · + β

α

! ! into − · · · − + · · · +. In the above example, we have p = 11112 ⊗ 12 ⊗ 2 ⊗ 1122 ˜ f0 (p) = 11112 ⊗ 12 ⊗ 2 ⊗ 1112 f˜1 (p) = 11122 ⊗ 12 ⊗ 2 ⊗ 1122 e˜0 (p) = 11122 ⊗ 12 ⊗ 2 ⊗ 1122 e˜1 (p) = 11111 ⊗ 12 ⊗ 2 ⊗ 1122 s0 (p) = 11222 ⊗ 12 ⊗ 2 ⊗ 1122 s1 (p) = 11122 ⊗ 12 ⊗ 2 ⊗ 1222. For both i = 0 and 1, note that wt(si (p)) = −wt(p) for any p, and si (p) = p if wt(p) = 0. In order that e˜1 p = 0 to hold for any p ∈ Bl1 ⊗ · · · ⊗ Blk , it is necessary and suﬃcient that the reduced 1-signature of p to become + · · · +. A.2. Combinatorial R The crystal Bl admits the aﬃnization Aﬀ(Bl ). It is the inﬁnite set Aﬀ(Bl ) = ei b)[d + δi,0 ], {b[d] | b ∈ Bl , d ∈ Z} endowed with the crystal structure e˜i (b[d]) = (˜ d ˜ ˜ fi (b[d]) = (fi b)[d − δi,0 ]. The element b[d] will also be denoted by ζ b to save the space, where ζ is an indeterminate.

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The isomorphism of the aﬃne crystal Aﬀ(Bl ) ⊗ Aﬀ(Bk ) → Aﬀ(Bk ) ⊗ Aﬀ(Bl ) is the unique bijection that commutes with Kashiwara operators (up to a constant shift of H below). It is the q = 0 analogue of the quantum R and called the combinatorial R. Explicitly it is given by [15, 37] R(x[d] ⊗ y[e]) = y˜[e − H(x ⊗ y)] ⊗ x ˜[d + H(x ⊗ y)], where x˜ = (˜ xi ), y˜ = (˜ yi ) are given by x ˜i = xi + Qi (x, y) − Qi−1 (x, y), Qi (x, y) = min(xi+1 , yi ),

y˜i = yi + Qi−1 (x, y) − Qi (x, y), H(x ⊗ y) = Q0 (x, y).

Here x ⊗ y y˜ ⊗ x ˜ under the isomorphism Bl ⊗ Bk Bk ⊗ Bl , which is also called (classical) combinatorial R. H is called the local energy function. It is characterized by the recursion relation: H(˜ ei (x ⊗ y))  H(x ⊗ y) − 1 = H(x ⊗ y) + 1  H(x ⊗ y)

y⊗x ˜) = e˜0 y˜ ⊗ x˜, if i = 0, e˜0 (x ⊗ y) = e˜0 x ⊗ y, e˜0 (˜ y⊗x ˜) = y˜ ⊗ e˜0 x˜, if i = 0, e˜0 (x ⊗ y) = x ⊗ e˜0 y, e˜0 (˜ otherwise

(A.5)

together with the connectedness of the crystal Bl ⊗ Bk . (The original H [11] is −H here.) Q0 (x, y) is a solution of (A.5) normalized so as to attain the minimum at Q0 (ul ⊗ uk ) = 0 and ranges over 0 ≤ Q0 ≤ min(l, k) on Bl ⊗ Bk . Here, ul = (l, 0) = 1 · · · 1 ∈ Bl

(A.6) denotes the highest element with respect to the sl2 subcrystal concerning e˜1 , f˜1 . y⊗x ˜) holds. When l = k, the classical part of the The invariance Qi (x ⊗ y) = Qi (˜ combinatorial R is trivial: R(ζ d x ⊗ ζ e y) = ζ e−H(x⊗y) x ⊗ ζ d+H(x⊗y) y

on Bl ⊗ Bl .

(A.7)

The combinatorial R has the following properties: (ω ⊗ ω)R = R(ω ⊗ ω)

on Bl ⊗ Bk ,

(A.8)

R = R

on Bl ⊗ Bk ,

(A.9)

(1 ⊗ R)(R ⊗ 1)(1 ⊗ R) = (R ⊗ 1)(1 ⊗ R)(R ⊗ 1)

on Aﬀ(Bj ) ⊗ Aﬀ(Bl ) ⊗ Aﬀ(Bk ).

(A.10)

Here ω is the involutive automorphism (A.4). (b1 ⊗ · · · ⊗ bk ) = bk ⊗ · · · ⊗ b1 is the reverse ordering of the tensor product for any k. (A.10) is the Yang–Baxter relation. To calculate the combinatorial R, it is convenient to use the graphical rule ([14, Rule 3.11]). Consider the two elements x = (x1 , x2 ) ∈ Bk and y = (y1 , y2 ) ∈ Bl . Draw the following diagram to express the tensor product x ⊗ y. y1

! • •···•

x1

! • • ···•

x2

! • • ···•

! • •···•

y2

.

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The combinatorial R and energy function H for x ⊗ y ∈ Bk ⊗ Bl (with k ≥ l) are found by the following rule. (i) Pick any dot, say •a , in the right column and connect it with a dot •a in the left column by a line. The partner •a is chosen from the dots whose positions are higher than that of •a . If there is no such a dot, go to the bottom, and the partner •a is chosen from the dots in the lower row. In the former case, we call such a pair “unwinding,” and, in the latter case, we call it “winding.” (ii) Repeat procedure (i) for the remaining unconnected dots (l − 1) times. (iii) The isomorphism is obtained by moving all unpaired dots in the left column to the right horizontally. We do not touch the paired dots during this move. (iv) The energy function H is given by the number of unwinding pairs. The number of winding (unwinding) pairs is called the winding (unwinding) number. It is known that the result is independent of the order of making pairs ([14, Propositions 3.15 and 3.17]). In the above description, we only consider the case k ≥ l. The other case k ≤ l can be done by reversing the above procedure, noticing the fact R2 = id. For more properties, including that the above deﬁnition indeed satisﬁes the axiom, see [14]. Example A.1. Corresponding to the tensor product 12222 ⊗ 1122 , we draw diagram of the left-hand side of the following.

t

t

t

t t t t

t

t

t

t t t

t

t

t t t

.

By moving one unpaired dot to the right, we obtain 12222 ⊗ 1122 1222 ⊗ 11222 . Since we have one unwinding pair, the energy function is H( 12222 ⊗ 1122 ) = 1.

Appendix B. Kerov–Kirillov–Reshetikhin Bijection B.1. Rigged configurations The Kerov–Kirillov–Reshetikhin (KKR) bijection gives a one to one correspondence between rigged conﬁgurations and highest paths. The latter are elements of Bλ1 ⊗ · · · ⊗ BλL annihilated by e˜1 . See around (3.1).

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Let us deﬁne the rigged conﬁgurations. Consider a pair (λ, µ), where both λ and µ are positive integer sequences: λ = (λ1 , λ2 , . . . , λL ),

(L ∈ Z≥0 , λi ∈ Z>0 ),

µ = (µ1 , µ2 , . . . , µN ),

(N ∈ Z≥0 , µi ∈ Z>0 ),

we use usual Young diagrammatic expression for these integer sequences, although λ, µ are not necessarily assumed to be weakly decreasing. Definition B.1. (1) For a given diagram ν, we introduce coordinates (row, column) of each boxes just like matrix entries. For a box α of ν, col(α) is the column coordinate of α. Then we deﬁne the following subsets: ν|≤j := {α | α ∈ ν , col(α) ≤ j},

ν|≥j := {α | α ∈ ν , col(α) ≥ j} . (a)

(2) For the diagrams (λ, µ), we deﬁne Qj (0) Qj

:=

L

min(j, λk ),

(a = 0, 1) by

(1) Qj

:=

k=1

N

min(j, µk ),

(B.1)

k=1

i.e. the number of boxes in λ|≤j and µ|≤j . Then the vacancy number pj for rows of µ is deﬁned by (0)

(1)

pj := Qj − 2Qj ,

(B.2)

where j is the width of the corresponding row. Definition B.2. Consider the following data: N RC := λ, (µ, r) = (λi )L i=1 , (µi , ri )i=1 . (1) Calculate the vacancy numbers with respect to the pair (λ, µ). If all vacancy numbers for rows of µ are nonnegative, i.e. 0 ≤ pµi , (1 ≤ i ≤ N ), then RC is called a configuration. (2) If the integer ri satisﬁes the condition 0 ≤ ri ≤ pµi ,

(B.3)

then ri is called a rigging associated with the row µi . For the rows of equal widths, i.e. µi = µi+1 , we assume that ri ≤ ri+1 . (3) If RC is a conﬁguration and if all integers ri are riggings associated with row µi , then RC is called sl2 rigged conﬁguration. In the rigged conﬁguration, λ is sometimes called a quantum space which determines the shape of the corresponding path, as we will see in the next subsection. In the diagrammatic expression of rigged conﬁgurations, the riggings are attached to the right of the corresponding row. Definition B.3. For a given rigged conﬁguration, consider a row µi and the corresponding rigging ri . If they satisfy the condition ri = pµi , then the row µi is called singular.

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B.2. Definition of the KKR bijection Here we explain the original combinatorial procedure to obtain a rigged conﬁguration RC φ : b → RC = λ, (µ, r) from a given path b = b1 ⊗ b2 ⊗ · · · ⊗ bL ∈ Bλ1 ⊗ Bλ2 ⊗ · · · ⊗ BλL . The appearance of λi in the right-hand side is clear from the following deﬁnition. Definition B.4. Under the above setting, the image of the KKR map φ is deﬁned by the following procedure. (i) We start from the empty rigged conﬁguration RC0 := ∅, (∅, ∅) and construct RC1 , . . . , RC|λ| successively as follows (note that |λ| = λi ). (ii) Set b1,0 := b1 ∈ Bλ1 for the path b = b1 ⊗ · · · . From b1,0 , we recursively construct b1,1 , b1,2 , . . . , b1,λ1 and RC1 , RC2 , . . . , RCλ1 . b1,i+1 and RCi+1 are constructed from b1,i := (x1 , x2 ) ∈ Bλ1 −i and RCi as follows: (a) First, assume that x2 = 0. Then we set b1,i+1 = (x1 − 1, 0). If i = 0, we create a new row to the quantum space as follows: RC1 = 1, (∅, ∅) . If i > 0, then we add one box to the row of the quantum space which is lengthened when we construct RCi . (b) On the contrary, assume that x2 > 0. Then we set b1,i+1 = (x1 , x2 − 1). We add a box to the quantum space by the same procedure as in the case x2 = 0. Operation on (µ, r) part of RCi is as follows. Calculate the vacancy numbers of RCi and determine all the singular rows. If there is no singular row in µ|≥i , then create a new row in µ. On the other hand, assume that there is at least one singular row in µ|≥i . Then, among these singular rows, we choose one of the longest singular rows arbitrary, and let us tentatively call it µs . We add one box to the row µs and do not change the other parts, which gives µ of new RCi+1 . As for the riggings, calculate the vacancy numbers of RCi+1 . Then we choose rs , i.e. the rigging associated to the lengthened row µs , so as to make the row µs singular in RCi+1 . Other riggings are chosen to be the same as the corresponding ones in RCi . (c) Repeat the above Step (b) for all letters 2 contained in b1 , then we do Step (a) for the rest of letters 1 in b1 . (iii) Do the same procedure of Step (ii) for b2 , . . . , bL . Each time when we start with a new element bi , we create a new row in the quantum space, and apply Step (ii). The result gives the image RC = RC|λ| of the map φ. It is known that all RCi in the above procedure are rigged conﬁgurations.

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Example B.5. Consider the following path: b = 1111 ⊗ 11 ⊗ 22 ⊗ 12 ⊗ 2 ⊗ 122 ⊗ 122 ⊗ 1112 . From the above path, we obtain the sequence of letters 1111 · 11 · 22 · 21 · 2 · 221 · 221 · 2111. Then the calculation of φ(b) proceeds as follows: ∅

∅

d

1

−→ d

∅

1

∅

d

1

−→

1

−→

2 d 1 −→

−→

d

1

∅

−→

∅

−→

2

2 d 2 −→

2

d

d 1 d 2 −→

2

2

2 −→

3

d

d 2 1

2

d 1

2 −→

2 2

d

2 d 2 −→

1

d 1 2

1 d 1 −→

2 2

1

d

2

1 −→ 1

d 1 1 1

d

−→ 1

∅

d

d

1

∅

2

1 −→

0 1 1

1 d 1

d

1 d 0 −→

1 1

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0 −→

1 1 1

0 2 4

1 1

c

523

1

0 −→ 1 c 4

c

1

0 −→

1 2 5

2 2 5

1 4

c

1

0 −→ 1 4

c 3 2 5

0 1 4

d In the above diagrams, newly added boxes are indicated by circles “◦”, and vacancy numbers are listed on the left of the corresponding rows in order to facilitate the calculations. This example, fully worked out here, will be revisited in Example 2.3 by using Theorem 2.2 for comparison. B.3. Basic properties of the KKR bijection It is known that the inverse map φ−1 can be described by a similar combinatorial procedure. We will use both φ and φ−1 in later arguments. Theorem B.6. The inverse map φ−1 : RC = λ, (µ, r) → b, is obtained by the following procedure (λ = (λ1 , λ2 , . . . , λL )). (i) We construct RC|λ| , RC|λ|−1 , . . ., RC1 , RC0 = ∅, (∅, ∅) and bL , bL−1 , . . . , b1 (bi ∈ Bλi ) as follows.

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(ii) We start from λL of λ, and set RC|λ| := RC and bL,λL = (0, 0). In order to obtain the quantum space of RC|λ|−1 , RC|λ|−2 , . . . , we remove boxes of row λL of the quantum space one by one. RCi−1 and bL,i−1 are constructed from RCi and bL,i . We call the rightmost box of the row of length λL − i in the quantum space as α. (col(α) = λL − i.) Let us tentatively denote the “µ part” of RCi by ν. (a) Assume that there is no singular row in ν|≥col(α) . Then RCi−1 is obtained by removing the box α from the quantum space. In this case, bL,i−1 is obtained by adding letter 1 to bL,i . (b) Assume on the contrary that there are singular rows in ν|≥col(α) . Among these singular rows, we choose one of the shortest rows arbitrary, and denote the rightmost box of the chosen row by β. Then RCi−1 is obtained by removing the two boxes α and β from RCi . New riggings are specified as follows. For the row from which the box β is removed, take new rigging so that it becomes singular in RCi−1 . On the contrary, for the other rows riggings are kept unchanged from the corresponding ones in RCi . Finally, bL,i−1 is obtained by adding letter 2 to bL,i . (c) By doing Steps (a) and (b) for the rest of boxes of λL in the quantum space, we obtain bL ∈ BλL . Here, the orderings of letters 1 and 2 within bL is chosen so that bL becomes semi-standard Young tableau. (iii) By doing Step (ii) for the rest of rows λL−1 , . . . , λ1 , we obtain bL−1 , . . . , b1 respectively. Finally, we obtain the path b = b1 ⊗ b2 ⊗ · · · ⊗ bL as the image of the map φ−1 . Example B.7. For an example of the calculation of φ−1 , one can use Example B.5, that is, reverse all arrows “→” to “←”. The above map φ−1 depends on ordering of the quantum space λ = (λ1 , λ2 , . . . , λL ). The dependence is described by Theorem B.8 ( [18, Lemma 8.5]). Let α, β be any successive two rows in the quantum space of a rigged configuration. Suppose the removal of α first and β next by φ−1 lead to the tableaux a1 and b1 , respectively. Suppose similarly that the removal of β first and α next lead to b2 and a2 , respectively. If the order of the other removal is the same, b 1 ⊗ a1 a2 ⊗ b 2 is valid under the isomorphism of the combinatorial R. The KKR bijection φ±1 , originally designed only for highest paths, is known to admit an extension which covers all the paths. In fact one can apply the same combinatorial procedure as φ to obtain φ(b) for any b ∈ Bλ1 ⊗ · · · ⊗ BλL . The resulting object is a unrestricted rigged configuration, where the condition (B.3) is relaxed to −µi ≤ ri ≤ pµi [31,5]. Obviously rigged conﬁgurations are special case of

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unrestricted ones. For unrestricted rigged conﬁgurations, combinatorial procedure in Theorem B.6 also works to deﬁne the inverse map φ−1 . Let us write φ(b) = (λ, (µ, r)). Then, |λ| represents the total number of letters 1 and 2 contained in the path b, whereas |µ| is the number of letter 2 in b. Note in particular that |λ| ≥ |µ| holds for unrestricted rigged conﬁgurations. Given a non-highest path b, one can always make b := 1

⊗Λ

⊗b

highest by taking Λ ≥ λ1 + · · · + λL . Under these notations, we have Lemma B.9. Let the unrestricted rigged configuration corresponding to b be N (λi )L (B.4) i=1 , (µj , rj )j=1 . Then the rigged configuration corresponding to the highest path b is given by Λ N (λi )L (B.5) i=1 ∪ (1 ), (µj , rj + Λ)j=1 . Proof. Let the vacancy number of a row µj of the pair (λ, µ) of (B.4) be pµj . Then the vacancy number of the row µj of (B.5) is equal to pµj + Λ. Now we apply φ−1 on (B.5). From the quantum space λ ∪ (1Λ ), we remove λ ﬁrst, and next remove (1Λ ). Recall that the combinatorial procedure in Theorem B.6 only refers to corigging (:= vacancy number − rigging), rather than the riggings. Therefore, when we remove λ from the quantum space of (B.5), we get b as the corresponding part of the image. Then, the remaining rigged conﬁguration has the quantum space (1Λ ) without µ part. Since the map φ−1 becomes trivial on it, we obtain b as the image of (B.5). Acknowledgments The authors thank Rei Inoue, Tomoki Nakanishi, Akira Takenouchi for discussion and Taichiro Takagi for collaboration in an early stage of the work and for communicating the result in [32]. A. K. is supported by Grants-in-Aid for Scientiﬁc No. 19540393. R. S. is a research fellow of the Japan Society for the Promotion of Science. References [1] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM Studies in Appl. Math., Vol. 4 (SIAM, 1981). [2] R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982). [3] H. A. Bethe, Zur Theorie der Metalle, I. Eigenwerte und Eigenfunktionen der linearen Atomkette, Z. Physik 71 (1931) 205–231. [4] E. Date and S. Tanaka, Periodic multi-soliton solutions of Korteweg–de Vries equation and Toda lattice, Progr. Theoret. Phys. Suppl. 59 (1976) 107–125.

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[5] L. Deka and A. Schilling, New fermionic formula for unrestricted Kostka polynomials, J. Combin. Theory Ser. A 113 (2006) 1435–1461; math.CO/0509194. [6] B. A. Dubrovin, V. B. Matveev and S. P. Novikov, Nonlinear equations of Korteweg– de Vries type, ﬁnite-band linear operators and Abelian varieties, Russian Math. Surveys 31 (1976) 59–146. [7] K. Fukuda, M. Okado and Y. Yamada, Energy functions in box-ball systems, Int. J. Mod. Phys. A 15 (2000) 1379–1392; math.QA/9908116. [8] C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg–de Vries equation, Phys. Rev. Lett. 19 (1967) 1095–1097. [9] R. Inoue and T. Takenawa, Tropical spectral curves and integrable cellular automata; arXiv:0704.2471. [10] M. Kashiwara, Crystal bases of modiﬁed quantized universal enveloping algebra, Duke Math. 73 (1994) 383–413. [11] S.-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima and A. Nakayashiki, Aﬃne crystals and vertex models, Int. J. Mod. Phys. A 7(suppl. 1A) (1992) 449–484. [12] S. V. Kerov, A. N. Kirillov and N. Yu. Reshetikhin, Combinatorics, Bethe ansatz, and representations of the symmetric group, Zap. Nauch. Semin. LOMI. 155 (1986) 50–64. [13] A. N. Kirillov and N. Yu. Reshetikhin, The Bethe ansatz and the combinatorics of Young tableaux, J. Soviet Math. 41 (1988) 925–955. [14] A. Nakayashiki and Y. Yamada, Kostka polynomials and energy functions in solvable lattice models, Selecta Math. (N.S.) 3 (1997) 547–599. (1) [15] G. Hatayama, K. Hikami, R. Inoue, A. Kuniba, T. Takagi and T. Tokihiro, The AM automata related to crystals of symmetric tensors, J. Math. Phys. 42 (2001) 274–308. [16] G. Hatayama, A. Kuniba, M. Okado, T. Takagi and Z. Tsuboi, Paths, crystals and fermionic formulae, Progr. Math. Phys. 23 (2002) 205–272. [17] G. Hatayama, A. Kuniba and T. Takagi, Soliton cellular automata associated with crystal bases, Nucl. Phys. B 577 (2000) 619–645. [18] A. N. Kirillov, A. Schilling and M. Shimozono, A bijection between Littlewood– Richardson tableaux and rigged conﬁgurations, Selecta Math. (N.S.) 8 (2002) 67–135. [19] A. Kuniba and T. Nakanishi, The Bethe equation at q = 0, the M¨ obius inversion formula, and weight multiplicities: I. The sl(2) case, Progr. Math. 191 (2000) 185– 216. [20] A. Kuniba, M. Okado, R. Sakamoto, T. Takagi and Y. Yamada, Crystal interpretation of Kerov–Kirillov–Reshetikhin bijection, Nucl. Phys. B 740 (2006) 299–327. [21] A. Kuniba, T. Takagi and A. Takenouchi, Bethe ansatz and inverse scattering transform in a periodic box-ball system, Nucl. Phys. B 747 (2006) 354–397. [22] A. Kuniba and A. Takenouchi, Periodic cellular automata and Bethe ansatz, in Differential Geometry and Physics, eds. M.-L. Ge and W. Zhang, Nankai Tracts in Math. (World Scientiﬁc, 2006), pp. 293–302. [23] A. Kuniba and R. Sakamoto, The Bethe ansatz in a periodic box-ball system and the ultradiscrete Riemann theta function, J. Stat. Mech. Theory Exp. 9 (2006) P09005. [24] A. Kuniba and R. Sakamoto, Combinatorial Bethe ansatz and ultradiscrete Riemann theta function with rational characteristics, Lett. Math. Phys. 80 (2007) 199–209. [25] A. Kuniba, R. Sakamoto and Y. Yamada, Tau functions in combinatorial Bethe ansatz, Nucl. Phys. B 786 (2007) 207–266. [26] J. Mada, M. Idzumi and T. Tokihiro, On the initial value problem of a periodic box-ball system, J. Phys. A 39 (2006) L617–L623. [27] G. Mikhalkin and I. Zharkov, Tropical curves, their Jacobians and Theta functions, arXiv:math/0612267.

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[28] R. Sakamoto, Crystal interpretation of Kerov–Kirillov–Reshetikhin bijection II. Proof for sln Case, to appear in J. Algebraic Combin. math.QA/0601697. [29] R. Sakamoto, A crystal theoretic method for ﬁnding rigged conﬁgurations from paths, preprint (2007). [30] A. Schilling, X = M Theorem: Fermionic formulas and rigged conﬁgurations under review, MSJ Memoirs 17 (2007) 75–104. [31] A. Schilling, Crystal structure on rigged conﬁgurations, Int. Math. Res. Not. (2006) Article ID 97376, 1–27. [32] T. Takagi, Creation of ballot sequences in a periodic cellular automaton, preprint (2007); arXiv:0708.0705. [33] D. Takahashi and J. Satsuma, A soliton cellular automaton, J. Phys. Soc. Japan 59 (1990) 3514–3519. [34] D. Takahashi and J. Matsukidaira, Box and ball system with a carrier and ultradiscrete modiﬁed KdV equation, J. Phys. A 30 (1997) L733–L739. [35] T. Tokihiro, D. Takahashi, J. Matsukidaira and J. Satsuma, From soliton equations to integrable cellular automata through a limiting procedure, Phys. Rev. Lett. 76 (1996) 3247–3250. [36] A. P. Veselov, Yang–Baxter maps: Dynamical point of view, MSJ Memoirs 17 (2007) 145–167. [37] Y. Yamada, A birational representation of Weyl group, combinatorial R-matrix and discrete Toda equation, in Physics and Combinatorics 2000, eds. A. N. Kirillov and N. Liskova (World Scientiﬁc, 2001), pp. 305–319. [38] C. N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967) 1312–1314. [39] D. Yoshihara, F. Yura and T. Tokihiro, Fundamental cycle of a periodic box-ball system, J. Phys. A 36 (2003) 99–121. [40] F. Yura and T. Tokihiro, On a periodic soliton cellular automaton, J. Phys. A 35 (2002) 3787–3801.

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Reviews in Mathematical Physics Vol. 20, No. 5 (2008) 529–595 c World Scientiﬁc Publishing Company

PHASE TRANSITIONS AND QUANTUM STABILIZATION IN QUANTUM ANHARMONIC CRYSTALS

ALINA KARGOL Instytut Matematyki, Uniwersytet Marii Curie-Sklodowskiej, 20-031 Lublin, Poland [email protected] YURI KONDRATIEV Fakult¨ at f¨ ur Mathematik, Universit¨ at Bielefeld, D-33615 Bielefeld, Germany [email protected] YURI KOZITSKY Instytut Matematyki, Uniwersytet Marii Curie-Sklodowskiej, 20-031 Lublin, Poland [email protected] Received 10 October 2007 Revised 5 January 2008 A uniﬁed theory of phase transitions and quantum eﬀects in quantum anharmonic crystals is presented. In its framework, the relationship between these two phenomena is analyzed. The theory is based on the representation of the model Gibbs states in terms of path measures (Euclidean Gibbs measures). It covers the case of crystals without translation invariance, as well as the case of asymmetric anharmonic potentials. The results obtained are compared with those known in the literature. Keywords: Quantum crystal; Euclidean Gibbs state; phase transition; quantum eﬀect. Mathematics Subject Classiﬁcation 2000: 82B10, 82B26, 81S40

Contents 1. Introduction and Setup

530

2. Euclidean Gibbs States 2.1. Local Gibbs states 2.2. Path spaces 2.3. Local Euclidean Gibbs measures 2.4. Tempered conﬁgurations 2.5. Local Gibbs speciﬁcation

534 534 536 538 540 542

529

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2.6. Tempered Euclidean Gibbs measures 2.7. Periodic Euclidean Gibbs measures 2.8. The pressure

544 547 549

3. Phase Transitions 3.1. Phase transitions and order parameters 3.2. Infrared bound 3.3. Phase transition in the translation and rotation invariant model 3.4. Phase transition in the symmetric scalar models 3.5. Phase transition in the scalar model with asymmetric potential 3.6. Comments

553 553 560 564 569 570 574

4. Quantum Stabilization 4.1. The stability of quantum crystals 4.2. Quantum rigidity 4.3. Properties of the quantum rigidity 4.4. Decay of correlations in the scalar case 4.5. Decay of correlations in the vector case 4.6. Suppression of phase transitions 4.7. Comments

575 576 577 579 582 587 588 590

1. Introduction and Setup Recently there appeared a number of publications describing inﬂuence of quantum eﬀects on phase transitions in quantum anharmonic crystals, where the results were obtained by means of path integrals, see [2, 3, 6, 8–10, 46, 48, 49, 51, 58, 59, 67, 86]. Their common point is a statement that the phase transition (understood in one or another way) is suppressed if the model parameters obey a certain condition (more or less explicitly formulated). The existence of phase transitions in quantum crystals of certain types was proven earlier, see [16, 17, 24, 45, 63], also mostly by means of path integral methods. At the same time, by now only two works, [7] and [55], have appeared where both these phenomena are studied in one and the same context. In the latter paper, a more complete and extended version of the theory of interacting systems of quantum anharmonic oscillators based on path integral methods has been elaborated, see also [11–13] for more recent development, and [54] where the results of [55] were announced. The aim of the present article is to reﬁne and extend the previous results and to develop a uniﬁed and more or less complete theory of phase transitions and quantum eﬀects in quantum anharmonic crystals, also in the light of the results of [54, 55]. Note that, in particular, with the help of these results we prove here phase transitions in quantum crystals with asymmetric anharmonic potentials,a what could hardly be done by other methods. The quantum crystal studied in this article is a system of interacting quantum anharmonic oscillators indexed by the elements of a crystal lattice L, which for a This

result was announced in [40].

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simplicity we assume to be a d-dimensional simple cubic lattice Zd . The quantum anharmonic oscillator is a mathematical model of a quantum particle moving in a potential ﬁeld with possibly multiple minima, which has a suﬃcient growth at inﬁnity and hence localizes the particle. Most of the models of interacting quantum oscillators are related with solids such as ionic crystals containing localized light particles oscillating in the ﬁeld created by heavy ionic complexes, or quantum crystals consisting entirely of such particles. For instance, a potential ﬁeld with multiple minima is seen by a helium atom located at the center of the crystal cell in bcc helium, see [44, p. 11]. The same situation exists in other quantum crystals, He, H2 and to some extent Ne. An example of the ionic crystal with localized quantum particles moving in a double-well potential ﬁeld is a KDP-type ferroelectric with hydrogen bounds, in which such particles are protons or deuterons performing onedimensional oscillations along the bounds, see [19, 76, 82, 83]. It is believed that in such substances phase transitions are triggered by the ordering of protons. Another relevant physical object of this kind is a system of apex oxygen ions in YBaCuOtype high-temperature superconductors, see [28, 60, 78, 79]. Quantum anharmonic oscillators are also used in models describing interaction of vibrating quantum particles with a radiation (photon) ﬁeld, see [33, 34, 61], or strong electron-electron correlations caused by the interaction of electrons with vibrating ions, see [26, 27], responsible for such phenomena as superconductivity, charge density waves, etc. Finally, we mention systems of light atoms, like Li, doped into ionic crystals, like KCl. The quantum particles in this system are not necessarily regularly distributed. For more information on this subject, we refer to the survey [35]. Note that a mathematical theory of Gibbs states of such systems was developed in [54, 55]. To be more concrete, we assume that our model describes an ionic crystal and thus adopt the ferroelectric terminology. In the corresponding physical substances, the quantum particles carry electric charge; hence, the displacement of the particle from its equilibrium point produces dipole moment. Therefore, the main contribution into the two-particle interaction is proportional to the product of the displacements of particles and is of long range. According to these arguments our model is described by the following formal Hamiltonian 1 J · (q , q ) + H . (1.1) H=− 2 ,

Here the sums run through the lattice L = Zd , d ∈ N, the displacement, q , of the oscillator attached to a given ∈ L is a ν-dimensional vector. In general, we do not assume that the interaction intensities J have ﬁnite range. By (·, ·) and | · | we denote the scalar product and norm in Rν , Rd . The one-site Hamiltonian def

H = Hhar + V (q ) =

1 a |p |2 + |q |2 + V (q ), 2m 2

a > 0,

(1.2)

describes an isolated quantum anharmonic oscillator. Its part Hhar corresponds to a ν-dimensional harmonic oscillator of rigidity a. The mass parameter m includes

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Planck’s constant, that is, m = mph /2 ,

(1.3)

where mph is the physical mass of the particle. Therefore, the commutation relation for the components of the momentum and displacement operators takes the form (j) (j )

p q

(j ) (j)

− q p = −ıδ δjj ,

j, j = 1, . . . , ν.

(1.4)

For a detailed discussion on how to derive a model like (1.1), (1.2) from physical models of concrete substances, we refer to the survey [76]. The theory of phase transitions is one of the most important and spectacular parts of equilibrium statistical mechanics. For classical lattice models, a complete description of the equilibrium thermodynamic properties is given by constructing their Gibbs states as probability measures on appropriate conﬁguration spaces. Usually, it is made in the Dobrushin–Lanford–Ruelle (DLR) approach which is now well-elaborated, see Georgii’s monograph [31] and the references therein. In general, the quantum case does not permit such a universal description. For some systems with bounded one-site Hamiltonians, e.g., quantum spin models, the Gibbs states are deﬁned as positive normalized functionals on algebras of quasi-local observables obeyng the condition of equilibrium between the dynamic and thermodynamic behavior of the model (KMS condition), see [20]. However, this algebraic way cannot be applied to the model (1.1), (1.2) since the construction of its dynamics in the whole crystal L is beyond the technical possibilities available by this time. In 1975, an approach employing path integral methods to the description of thermodynamic properties of models like (1.1), (1.2) has been initiated in [1]. Its main idea was to pass from real to imaginary values of time, similarly as it was done in Euclidean quantum ﬁeld theory, see [32, 71], and thereby to describe the dynamics of the model in terms of stochastic processes. Afterwards, this approach, also called Euclidean, has been developed in a number of works. Its latest and most general version is presented in [54, 55], where the reader can also ﬁnd an extensive bibliography on this subject. The methods developed in these works will mostly be used in the present study. According to a commonly adopted physical interpretation, in the substances modeled by (1.1), (1.2) phase transitions occur at low temperatures when the oscillations of the particles become strongly correlated that produces macroscopic ordering. The mathematical theory of phase transitions in such models is based on quantum versions of the method of infrared estimates developed in [29]. The ﬁrst publication where the infrared estimates were applied to quantum spin models seems to be [25]. After certain modiﬁcations this method, combined with path integral techniques, was applied in [16,17,24,45,63] to particular versions of our model. Their main characteristic feature was a symmetry, broken by the phase transition. In classical systems, ordering is achieved in competition with thermal ﬂuctuations only. However, in quantum systems quantum eﬀects play a signiﬁcant disordering role, especially at low temperatures. This role was ﬁrst discussed in [69].

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Later on a number of publications dedicated to the study of quantum eﬀects in such systems had appeared, see e.g., [58, 59, 86] and the references therein. For better understanding, illuminating exactly solvable models of systems of interacting quantum anharmonic oscillators were introduced and studied, see [66, 77, 84, 85]. In these works, the quantity m−1 = 2 /mph was used as a parameter describing the rate of quantum eﬀects. Such eﬀects became strong in the small mass limit, which was in agreement with the experimental data, e.g., on the isotopic eﬀect in the ferroelectrics with hydrogen bounds, see [19, 83], see also [60] for the data on the isotopic eﬀect in the YBaCuO-type high-temperature superconductors. However, in these works no other quantum eﬀects, e.g., those connected with special properties of the anharmonic potentials, were discussed. At the same time, experimental data, see e.g., the table on p. 11 in the monograph [19] or the article [80], show that high hydrostatic pressure applied to KDP-type ferroelectrics prevents them from ordering. It is believed that the pressure shortens the hydrogen bounds and thereby changes the anharmonic potential. This makes the tunneling motion of the quantum particles more intensive, which is equivalent to diminishing the particle mass. In [6, 9, 10], a theory of such quantum eﬀects in the model (1.1), (1.2), which explains both mentioned mechanisms, was built up. Its main conclusion is that the quantum dynamical properties, which depend on the mass m, the interaction intensities J , and the anharmonic potentials V , can be such that the model is stable with respect to phase transitions at all temperatures. As was mentioned above, the aim of this article is to present a uniﬁed description of phase transitions and quantum stabilization in the model (1.1), (1.2), mostly by means of methods of [54, 55]. We also give here complete proofs of a number of statements announced in our previous publications. The article is organized as follows. In Sec. 2, we brieﬂy describe those elements of the theory developed in [54, 55] which we then apply in the subsequent sections. In Sec. 3, we present the theory of phase transitions in the model (1.1), (1.2). We begin by introducing three deﬁnitions of a phase transition in this model and study the relationships between them. Then we develop a version of the method of infrared estimates adapted to our model, which is more transparent and appropriate than the one employed in [7]. Afterwards, we obtain a suﬃcient conditions for the phase transitions to occur in a number of versions of (1.1), (1.2). This includes also the case of asymmetric anharmonic potentials V which was never studied before. At the end of the section, we make some comments on the results obtained and compare them with similar results known in the literature. Section 4 is dedicated to the study of quantum stabilization, which we understand as the suppression of phase transitions by quantum eﬀects. Here we discuss the problem of stability of quantum crystals and the ways of its description. In particular, we introduce a parameter (quantum rigidity), responsible for the stability and prove a number of statements about its properties. Then we show that under the stability condition which we introduce here the correlations decay “in a proper way”, that means the absence of phase transitions. The relationship between the quantum stabilization and phase transitions are also

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analyzed. In the simplest case, where the model is translation invariant, scalar (ν = 1), and with the interaction of nearest neighbor type, this relation looks as follows. The key parameter is 8dmJϑ2∗ , where d is the lattice dimension, J > 0 is the interaction intensity, and ϑ∗ > 0 is determined by the anharmonic potential V (the steeper is V the smaller is ϑ∗ ). Then the quantum stabilization condition (respectively, the phase transition condition) is 8dmJϑ2∗ < 1, see (4.32), (respectively, 8dmJϑ2∗ > φ(d), d ≥ 3, see (3.70) and (4.33)). Here φ is a function, such that φ(d) > 1 and φ(d) → 1 as d → +∞. We conclude the section by commenting the results obtained therein. 2. Euclidean Gibbs States The main element of the Euclidean approach is the description of the equilibrium thermodynamic properties of the model (1.1), (1.2) by means of Euclidean Gibbs states, which are probability measures on certain conﬁguration spaces. In this section, we brieﬂy describe the main elements of this approach which are then used in the subsequent parts of the article. The details can be found in [55]. 2.1. Local Gibbs states Let us begin by specifying the properties of the model described by the Hamiltonian (1.1). The general assumptions regarding the interaction intensities J are def J < ∞. (2.1) J = J ≥ 0, J = 0, Jˆ0 = sup

In view of the ﬁrst of these properties the model is ferroelectric. Regarding the anharmonic potentials we assume that each V : Rν → R is a continuous function, which obeys AV |x|2r + BV ≤ V (x) ≤ V (x),

(2.2)

with a continuous function V and constants r > 1, AV > 0, BV ∈ R. In certain cases, we shall include an external ﬁeld term in the form V (x) = V0 (x) − (h, x),

h ∈ Rν .

(2.3)

Deﬁnition 2.1. The model is translation invariant if V = V for all , and the interaction intensities J are invariant under the translations of L. The model is rotation invariant if for every orthogonal transformation U ∈ O(ν) and every , V (U x) = V (x). The interaction has ﬁnite range if there exists R > 0 such that J = 0 whenever | − | > R. If V ≡ 0 for all , one gets a quantum harmonic crystal. It is stable if Jˆ0 < a, see Remark 2.1 below. By Λ we denote subsets of the lattice L; we write Λ L if Λ is non-void and ﬁnite. For such a Λ, by |Λ| we denote its cardinality. A sequence of subsets Λ L

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is called coﬁnal if it is ordered by inclusion and exhausts the lattice L. If we say that something holds for all , we mean it holds for all ∈ L; sums like mean + ∈L . We also use the notations R = [0, +∞) and N0 = N ∪ {0}, N being the set of positive integers. Given Λ L, the local Hamiltonian of the model is HΛ = −

1 J · (q , q ) + H , 2 , ∈Λ

(2.4)

∈Λ

which by the assumptions made above is a self-adjoint and lower bounded operator in the physical Hilbert space L2 (Rν|Λ| ). For every β = 1/kB T , T being absolute temperature, the local Gibbs state in Λ L is Λ (A) = trace[A exp(−βHΛ )]/ZΛ ,

A ∈ CΛ ,

(2.5)

where ZΛ = trace[exp(−βHΛ )] < ∞

(2.6)

is the partition function, and CΛ is the algebra of all bounded linear operators on L2 (Rν|Λ| ). Note that adjective local will always stand for a property related with a certain Λ L, whereas global will characterize the whole inﬁnite system. The dynamics of the subsystem located in Λ is described by the time automorphisms CΛ A → aΛ t (A) = exp(ıtHΛ )A exp(−ıtHΛ ),

(2.7)

where t ∈ R is time. Given n ∈ N and A1 , . . . , An ∈ CΛ , the corresponding Green function is Λ Λ GΛ A1 ,...,An (t1 , . . . , tn ) = Λ at1 (A1 ) · · · atn (An ) ,

(2.8)

which is a complex valued function on Rn . Each such a function can be looked upon, see [1, 7], as the restriction of a function GΛ A1 ,...,An analytic in the domain Dβn = {(z1 , . . . , zn ) ∈ Cn | 0 < (z1 ) < · · · < (zn ) < β},

(2.9)

and continuous on its closure. The corresponding statement is known as the multiple-time analyticity theorem, see [1, 7], as well as [42] for a more general consideration. For every n ∈ N, the subset {(z1 , . . . , zn ) ∈ Dβn | (z1 ) = · · · = (zn ) = 0}

(2.10)

is an inner uniqueness set for functions analytic in Dβn , see [70, p. 101 and p. 352]. This means that two such functions which coincide on this set should coincide everywhere on Dβn .

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For a bounded continuous function F : Rν|Λ| → C, the corresponding multiplication operator F ∈ CΛ acts as follows (F ψ)(x) = F (x)ψ(x),

ψ ∈ L2 (Rν|Λ| ).

Let FΛ ⊂ CΛ be the set of all such operators. One can prove (the density theorem, see [52, 53]) that the linear span of the products Λ aΛ t1 (F1 ) · · · atn (Fn ),

with all possible choices of n ∈ N, t1 , . . . , tn ∈ R, and F1 , . . . , Fn ∈ FΛ , is dense in CΛ in the σ-weak topology in which the state (2.5) is continuous as a linear functional. Thus, the latter is determined by the set of Green functions GΛ F1 ,...,Fn with n ∈ N and F1 , . . . , Fn ∈ FΛ . The restriction of the Green functions GΛ F1 ,...,Fn to the imaginary-time sets (2.10) are called Matsubara functions. For τ1 ≤ τ2 ≤ · · · ≤ τn ≤ β,

(2.11)

Λ ΓΛ F1 ,...,Fn (τ1 , . . . , τn ) = GF1 ,...,Fn (ıτ1 , . . . , ıτn ).

(2.12)

they are

Since (2.10) is an inner uniqueness set, the collection of the Matsubara functions (2.12) with all possible choices of n ∈ N and F1 , . . . , Fn ∈ FΛ determines the state (2.5). The extensions of the functions (2.12) to [0, β]n are deﬁned as Λ ΓΛ F1 ,...,Fn (τ1 , . . . , τn ) = ΓFσ(1) ,...,Fσ(n) (τσ(1) , . . . , τσ(n) ),

where σ is the permutation such that τσ(1) ≤ τσ(2) ≤ · · · ≤ τσ(n) . One can show that for every θ ∈ [0, β], Λ ΓΛ F1 ,...,Fn (τ1 + θ, . . . , τn + θ) = ΓF1 ,...,Fn (τ1 , . . . , τn ),

(2.13)

where addition is modulo β. 2.2. Path spaces By (2.8), the Matsubara function (2.12) can be written as ΓΛ F1 ,...,Fn (τ1 , . . . , τn ) = trace[F1 e−(τ2 −τ1 )HΛ F2 e−(τ3 −τ2 )HΛ · Fn e−(τn+1 −τn )HΛ ]/ZΛ ,

(2.14)

where τn+1 = β + τ1 and the arguments obey (2.11). This expression can be rewritten in an integral form ΓΛ (τ , . . . , τ ) = F1 (ωΛ (τ1 )) · · · Fn (ωΛ (τn ))νΛ (dωΛ ), (2.15) n F1 ,...,Fn 1 ΩΛ

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that is the main point of the Euclidean approach. Here νΛ is a probability measure on the path space ΩΛ which we introduce now. The main single-site path space is the space of continuous periodic paths (temperature loops) Cβ = {φ ∈ C([0, β] → Rν ) | φ(0) = φ(β)}.

(2.16)

It is a Banach space with the usual sup-norm · Cβ . For an appropriate φ ∈ Cβ , we set Kσ (φ) = β σ ·

sup τ,τ ∈[0,β] τ =τ

|φ(τ ) − φ(τ )| , |τ − τ |σβ

σ > 0,

(2.17)

where |τ − τ |β = min{|τ − τ |; β − |τ − τ |}

(2.18)

is the periodic distance on the circle Sβ ∼ [0, β]. Then the set of H¨ older-continuous periodic functions, Cβσ = {φ ∈ Cβ | Kσ (φ) < ∞},

(2.19)

can be equipped with the norm

φ Cβσ = |φ(0)| + Kσ (φ),

(2.20)

which turns it into a Banach space. Along with the spaces Cβ , Cβσ , we shall also use the Hilbert space L2β = L2 (Sβ → Rν , dτ ), equipped with the inner product (·, ·)L2β and norm · L2β . By B(Cβ ), B(L2β ) we denote the corresponding Borel σ-algebras. In a standard way, see [62, p. 21] and the corresponding discussion in [55], it follows that Cβ ∈ B(L2β ) and B(Cβ ) = B(L2β ) ∩ Cβ .

(2.21)

Given Λ ⊆ L, we set ΩΛ = {ωΛ = (ω )∈Λ | ω ∈ Cβ }, Ω = ΩL = {ω = (ω )∈L | ω ∈ Cβ }.

(2.22)

These path spaces are equipped with the product topology and with the Borel σ-algebras B(ΩΛ ). Thereby, each ΩΛ is a complete separable metric space (Polish space), its elements are called conﬁgurations in Λ. For Λ ⊂ Λ , the juxtaposition ωΛ = ωΛ × ωΛ \Λ deﬁnes an embedding ΩΛ → ΩΛ by identifying ωΛ ∈ ΩΛ with ωΛ × 0Λ \Λ ∈ ΩΛ . By P(ΩΛ ), P(Ω) we denote the sets of all probability measures on (ΩΛ , B(ΩΛ )), (Ω, B(Ω)), respectively.

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2.3. Local Euclidean Gibbs measures Now we construct the measure νΛ which appears in (2.15). A single harmonic oscillator is described by the Hamiltonian, cf., (1.2), 1 =− 2m j=1 ν

Hhar

∂ (j) ∂x

2

a + |x |2 . 2

(2.23)

It is a self-adjoint operator in the space L2 (Rν ), the properties of which are well known. The operator semigroup exp(−τ Hhar ), τ ∈ Sβ , deﬁnes a Gaussian β-periodic Markov process, see [43]. In quantum statistical mechanics, it ﬁrst appeared in Høegh-Krohn’s paper [36]. The canonical realization of this process on (Cβ , B(Cβ )) is described by the path measure which can be introduced as follows. In the space L2β , we deﬁne the following self-adjoint Laplace–Beltrami type operator d2 A = −m 2 + a ⊗ I, dτ

(2.24)

where I is the identity operator in Rν . Its spectrum consists of the eigenvalues λl = m(2πl/β)2 + a,

l ∈ Z.

(2.25)

Therefore, the inverse A−1 is a trace-class operator on L2β and the Fourier transform

1 −1 exp[ı(ψ, φ)L2β ]χ(dφ) = exp − (A ψ, ψ)L2β 2 L2β

(2.26)

deﬁnes a zero mean Gaussian measure χ on (L2β , B(L2β )), see [75]. Employing the eigenvalues (2.25) one can show that, for any p ∈ N,

2p

|ω(τ ) − ω(τ )| Cβ

Γ(ν/2 + p) χ(dω) ≤ Γ(ν/2)

2 m

p

· |τ − τ |pβ .

(2.27)

Therefrom, by Kolmogorov’s lemma ([72, p. 43]) it follows that χ(Cβσ ) = 1,

for all σ ∈ (0, 1/2).

(2.28)

Thereby, χ(Cβ ) = 1; hence, with the help of (2.21) we redeﬁne χ as a measure on (Cβ , B(Cβ )), possessing the property (2.28). We shall call it Høegh-Krohn’s measure. An account of the properties of χ can be found in [7]. Here we present the following two of them. The ﬁrst property is obtained directly from Fernique’s theorem (see [23, Theorem 1.3.24]).

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Proposition 2.1 (Fernique). For every σ ∈ (0, 1/2), there exists λσ > 0, which can be evaluated explicitly, such that exp(λσ φ 2Cβσ )χ(dφ) < ∞. (2.29) L2β

The second property follows from the estimate (2.27) by the Garsia–Rodemich– Rumsey lemma, see [30]. For ﬁxed σ ∈ (0, 1/2), we set

|ω(τ ) − ω(τ )|2 Ξϑ (ω) = , ϑ ∈ (0, β/2), ω ∈ Cβσ . sup (2.30) |τ − τ |2σ τ,τ : 0<|τ −τ |β <ϑ β One can show that, for each σ and ϑ, it can be extended to a measurable map Ξϑ : Cβ → [0, +∞]. Proposition 2.2 (Garsia–Rodemich–Rumsey Estimate). Given σ ∈ (0, 1/2), let p ∈ N be such that (p − 1)/2p > σ. Then Ξpϑ (ω)χ(dω) ≤ D(σ, p, ν)m−p ϑp(1−2σ) , (2.31) Cβ

where m is the mass (1.3) and D(σ, p, ν) =

23(2p+1) (1 + 1/σp)2p 2p Γ(ν/2 + 1) · . (p − 1 − 2σp)(p − 2σp) Γ(ν/2)

(2.32)

The Høegh-Krohn measure is the local Euclidean Gibbs measure for a single harmonic oscillator. The measure νΛ ∈ P(ΩΛ ), which is the Euclidean Gibbs measure corresponding to the system of interacting anharmonic oscillators located in Λ L, is deﬁned by means of the Feynman–Kac formula as a Gibbs modiﬁcation νΛ (dωΛ ) = exp[−IΛ (ωΛ )]χΛ (dωΛ )/NΛ

(2.33)

of the “free measure” χΛ (dωΛ ) =

χ(dω ).

(2.34)

∈Λ

Here β 1 J (ω , ω )L2β + V (ω (τ ))dτ IΛ (ωΛ ) = − 2 0 , ∈Λ

(2.35)

∈Λ

is the energy functional which describes the interaction of the paths ω , ∈ Λ. The normalizing factor exp[−IΛ (ωΛ )]χΛ (dωΛ ) (2.36) NΛ = ΩΛ

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is the relative partition function, whereas the Feynman–Kac representation of the partition function (2.6) is ZΛ = NΛ ZΛhar , where

def ZΛhar =

trace exp −β

=

(2.37) Hhar

∈Λ

ν|Λ| exp[−(β/2) a/m] . 1 − exp(−β a/m)

Now let us summarize the connections between the description of the subsystem located in Λ L in terms of the states (2.5) and of the Euclidean Gibbs measures (2.33). By the density theorem, the state Λ is fully determined by the Green functions (2.8) corresponding to all choices of n ∈ N and F1 , . . . , Fn ∈ FΛ . Then the multiple-time analyticity theorem leads us from the Green functions to the Matsubara functions (2.12), which then are represented as integrals over path spaces with respect to the local Euclidean Gibbs measures, see (2.15). On the other hand, these integrals taken for all possible choices of bounded continuous functions F1 , . . . , Fn fully determine the measure νΛ . Thereby, we have a one-to-one correspondence between the local Gibbs states (2.5) and the states on the algebras of bounded continuous functions determined by the local Euclidean Gibbs measures (2.33). Our next aim is to extend this approach to the global states. To this end we make more precise the deﬁnition of the path spaces in inﬁnite Λ, e.g., in Λ = L. 2.4. Tempered configurations To describe the global thermodynamic properties we need the conditional distributions πΛ (dω|ξ), Λ L. For models with inﬁnite-range interactions, the construction of such distributions is a nontrivial problem, which can be solved by imposing a priori restrictions on the conﬁgurations deﬁning the corresponding conditions. In this and in the subsequent subsections, we present the construction of such distributions performed in [55]. The distributions πΛ (dω|ξ) are deﬁned by means of the energy functionals IΛ (ω|ξ) describing the interaction of the conﬁguration ω with the conﬁguration ξ, ﬁxed outside Λ. Given Λ L, such a functional is J (ω , ξ )L2β , ω ∈ Ω, (2.38) IΛ (ω | ξ) = IΛ (ωΛ ) − ∈Λ, ∈Λc

where IΛ is given by (2.35). Recall that ω = ωΛ × ωΛc ; hence, IΛ (ω | ξ) = IΛ (ωΛ × 0Λc | 0Λ × ξΛc ).

(2.39)

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The second term in (2.38) makes sense for all ξ ∈ Ω only if the interaction has ﬁnite range, see Deﬁnition 2.1. Otherwise, one has to impose appropriate restrictions on the conﬁgurations ξ, such that, for all and ω ∈ Ω, J · |(ω , ξ )L2β | < ∞. (2.40)

These restrictions are formulated by means of special mappings (weights), which deﬁne the scale of growth of { ξ L2β }∈L . Their choice depends on the asymptotic properties of J , | − | → +∞, see (2.1). If for a certain α > 0, J exp(α| − |) < ∞, (2.41) sup

then the weights {wα (, )}α∈I are chosen as wα (, ) = exp(−α| − |),

I = (0, α ¯ ),

(2.42)

where α ¯ is the supremum of α > 0, for which (2.41) holds. If the latter condition does not hold for any α > 0, we assume that J · (1 + | − |)αd < ∞, (2.43) sup

for a certain α > 1. Then we set α ¯ to be the supremum of α > 1 obeying (2.43) and wα (, ) = (1 + ε| − |)−αd ,

(2.44)

where ε > 0 is a technical parameter. In the sequel, we restrict ourselves to these two kinds of J . More details on this item can be found in [54, 55]. Given α ∈ I and ω ∈ Ω, we set

ω α =

1/2

ω 2L2 wα (0, )

,

(2.45)

β

and Ωα = {ω ∈ Ω | ω α < ∞}.

(2.46)

Thereby, we endow Ωα with the metric ρα (ω, ω ) = ω − ω α +

2−||

ω − ω Cβ , 1 + ω − ω Cβ

(2.47)

which turns it into a Polish space. The set of tempered conﬁgurations is deﬁned to be Ωt = Ωα . (2.48) α∈I

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We endow it with the projective limit topology, which turns it into a Polish space as well. For every α ∈ I, the embeddings Ωt → Ωα → Ω are continuous; hence, Ωα , Ωt ∈ B(Ω) and the Borel σ-algebras B(Ωα ), B(Ωt ) coincide with the ones induced on them by B(Ω). 2.5. Local Gibbs specification Let us turn to the functional (2.38). By standard methods, one proves that, for every α ∈ I, the map Ωα × Ωα → IΛ (ω | ξ) is continuous. Furthermore, for any ball Bα (R) = {ω ∈ Ωα | ρα (0, ω) < R}, R > 0, one has inf ω∈Ω, ξ∈Bα (R)

IΛ (ω | ξ) > −∞,

sup

|IΛ (ω | ξ)| < +∞.

ω,ξ∈Bα (R)

Therefore, for Λ L and ξ ∈ Ωt , the conditional relative partition function NΛ (ξ) = exp[−IΛ (ωΛ × 0Λc | ξ)]χΛ (dωΛ ) (2.49) ΩΛ

is continuous in ξ. Furthermore, for any R > 0 and α ∈ I, inf ξ∈Bα (R)

NΛ (ξ) > 0.

For such ξ and Λ, and for B ∈ B(Ω), we set 1 πΛ (B | ξ) = NΛ (ξ)

exp[−IΛ (ωΛ × 0Λc | ξ)]IB (ωΛ × ξΛc )χΛ (dωΛ ),

(2.50)

ΩΛ

where IB stands for the indicator of B. We also set πΛ (· | ξ) ≡ 0,

for ξ ∈ Ω\Ωt .

(2.51)

From these deﬁnitions one readily derives the following consistency property πΛ (B | ω)πΛ (dω | ξ) = πΛ (B | ξ), Λ ⊂ Λ , (2.52) Ω

which holds for all B ∈ B(Ω) and ξ ∈ Ω. The local Gibbs speciﬁcation is the family {πΛ }ΛL . Each πΛ is a measure kernel, which means that, for a ﬁxed ξ ∈ Ω, πΛ (·|ξ) is a measure on (Ω, B(Ω)), which is a probability measure whenever ξ ∈ Ωt . For any B ∈ B(Ω), πΛ (B | ·) is B(Ω)-measurable. By Cb (Ωα ) (respectively, Cb (Ωt )) we denote the Banach spaces of all bounded continuous functions f : Ωα → R (respectively, f : Ωt → R) equipped with the supremum norm. For every α ∈ I, one has a natural embedding Cb (Ωα ) → Cb (Ωt ).

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Given α ∈ I, by Wα we denote the usual weak topology on the set of all probability measures P(Ωα ) deﬁned by means of Cb (Ωα ). By W t we denote the weak topology on P(Ωt ). With these topologies the sets P(Ωα ) and P(Ωt ) become Polish spaces ([62, Theorem 6.5, p. 46]). By standard methods one proves the following, see [55, Lemma 2.10], Proposition 2.3 (Feller Property). For every α ∈ I, Λ L, and any f ∈ Cb (Ωα ), the function Ωα ξ → πΛ (f |ξ) 1 def = f (ωΛ × ξΛc ) exp[−IΛ (ωΛ × 0Λc | ξ)]χΛ (dωΛ ), NΛ (ξ) ΩΛ

(2.53)

belongs to Cb (Ωα ). The linear operator f → πΛ (f | ·) is a contraction on Cb (Ωα ). Note that by (2.50), for ξ ∈ Ωt , α ∈ I, and f ∈ Cb (Ωα ), πΛ (f | ξ) =

f (ω)πΛ (dω | ξ).

(2.54)

Ω

We remind that the particular cases of our model were speciﬁed by Deﬁnition 2.1. For B ∈ B(Ω) and U ∈ O(ν), we set U ω = (U ω )∈L ,

U B = {U ω | ω ∈ B}.

Furthermore, for 0 ∈ L, Λ L, and ω ∈ Ω, we set Λ + 0 = { + 0 | ∈ Λ};

t0 (ω) = (ξ0 )∈L ,

ξ0 = ω−0 .

(2.55)

Thereby, for B ∈ B(Ω), t (B) = {t (ω) | ω ∈ B}.

(2.56)

Clearly, t (B) ∈ B(Ω) and t (Ωt ) = Ωt for all . Deﬁnition 2.2. A probability measure µ ∈ P(Ω) is said to be translation invariant if for every and B ∈ B(Ω), one has µ(t (B)) = µ(B). From the deﬁnition (2.50) one readily gets that if the model possesses the corresponding symmetry, then πΛ (U B | U ξ) = πΛ (B | ξ),

πΛ+ (t (B) | t (ξ)) = πΛ (B | ξ),

(2.57)

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which ought to hold for all U , , B, and ξ. One observes that the translation invariance of the Gibbs speciﬁcation does not mean that each probability kernel πΛ as a measure is translation invariant.

2.6. Tempered Euclidean Gibbs measures Deﬁnition 2.3. A measure µ ∈ P(Ω) is called a tempered Euclidean Gibbs measure if it satisﬁes the Dobrushin–Lanford–Ruelle (equilibrium) equation πΛ (B | ω)µ(dω) = µ(B), for all Λ L and B ∈ B(Ω). (2.58) Ω

By G t we denote the set of all tempered Euclidean Gibbs measures of our model existing at a given β. The elements of G t are supported by Ωt . Indeed, by (2.50) and (2.51) πΛ (Ω\Ωt | ξ) = 0 for every Λ L and ξ ∈ Ω. Then by (2.58), µ(Ω\Ωt ) = 0.

(2.59)

µ({ω ∈ Ωt | ∀ ∈ L : ω ∈ Cβσ }) = 1,

(2.60)

Furthermore,

which follows from (2.28), (2.29). From Proposition 2.3, one readily gets the following important fact. Proposition 2.4. For each α ∈ I, every Wα -accumulation point µ ∈ P(Ωt ) of the family {πΛ (· | ξ) | Λ L, ξ ∈ Ωt } is an element of G t . If the model is translation and/or rotation invariant, then, for every U ∈ O(ν) and ∈ L, the corresponding transformations preserve G t . That is, for any µ ∈ G t , ΘU (µ) = µ ◦ U −1 ∈ G t , def

t θ (µ) = µ ◦ t−1 ∈G . def

(2.61)

In particular, if G t is a singleton, its unique element should be invariant in the same sense as the model. Consider B inv = {B ∈ B(Ω) | ∀ : t (B) = B},

(2.62)

which is the set of all translation invariant events. By construction, Ωt ∈ B inv . We say that µ ∈ P(Ω) is trivial on B inv if for every B ∈ B inv , one has µ(B) = 0 or µ(B) = 1. By P inv (Ω) we denote the set of translation invariant probability measures on (Ω, B). Deﬁnition 2.4. A probability measure µ ∈ P inv (Ω) is said to be ergodic (with respect to the group L) if it is trivial on B inv .

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Ergodic measures are characterized by a mixing property, which we formulate here according to [73, Theorem III.1.8, p. 244]. For L ∈ N, we set ΛL = (−L, L]d ∩ Zd ,

(2.63)

which is called a box. For a measure µ and an appropriate function f , we write (2.64) f µ = f dµ Proposition 2.5 (Von Neumann Ergodic Theorem). Given µ ∈ P inv (Ω), the following statements are equivalent : (i) µ is ergodic; (ii) for all f, g ∈ L2 (Ω, µ), 1 lim f (ω)g(t (ω))µ(dω) − f µ · gµ = 0. L→+∞ |ΛL | Ω

(2.65)

∈ΛL

Proposition 2.6. If the model is translation invariant and G t is a singleton, its unique element is ergodic. Now we give a number of statements describing the properties of G t . More details can be found in [55]. Proposition 2.7. For every β > 0, the set of tempered Euclidean Gibbs measures G t is non-void, convex, and W t -compact. Recall that the H¨ older norm · Cβσ was deﬁned by (2.20). Proposition 2.8. For every σ ∈ (0, 1/2) and κ > 0, there exists a positive constant C such that, for any and for all µ ∈ G t , exp(λσ ω 2Cβσ + κ ω 2L2 )µ(dω) ≤ C, (2.66) Ω

β

where λσ is the same as in (2.29). In view of (2.66), the one-site projections of each µ ∈ G t are sub-Gaussian. The constant C does not depend on and is the same for all µ ∈ G t , though it may depend on σ and κ. The estimate (2.66) plays a crucial role in the theory of G t . According to [31, Comment (7.8), p. 119], certain Gibbs states correspond to the thermodynamic phases of the underlying physical system. Thus, in our context multiple phases exist only if G t has more than one element for appropriate values of β and the model parameters. On the other hand, a priori one cannot exclude that this set always has multiple elements, which would make it useless for describing

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phase transitions. The next statement which we present here,b clariﬁes the situation. Let us decompose V = V1, + V2, , where V1, ∈ C 2 (Rν ) is such that

(2.67)

V1, (x)y, y /|y|2 < ∞.

(2.68)

As for the second term, we set

def 0 ≤ δ = sup sup V2, (x) − infν V2, (x) ≤ ∞.

(2.69)

def

−a ≤ b = inf

inf ν

x,y∈R , y=0

x∈R

x∈Rν

Its role is to produce multiple minima of the potential energy responsible for eventual phase transitions. Clearly, the decomposition (2.67) is not unique; its optimal realizations for certain types of V are discussed in [14, Sec. 6]. Recall that the interaction parameter Jˆ0 was deﬁned in (2.1). Proposition 2.9. The set G t is a singleton if eβδ < (a + b)/Jˆ0 .

(2.70)

Remark 2.1. The latter condition surely holds at all β if δ=0

and Jˆ0 < a + b.

(2.71)

If the oscillators are harmonic, δ = b = 0, which yields the stability condition Jˆ0 < a.

(2.72)

The condition (2.70) does not contain the particle mass m; hence, the property stated holds also in the quasi-classical limitc m → +∞. By the end of this subsection we consider the scalar case ν = 1 only. Let us introduce the following order on G t . As the components of the conﬁgurations ω ∈ Ω are continuous functions ω : Sβ → R, one can set ω ≤ ω ˜ if ω (τ ) ≤ ω ˜ (τ ) for all and τ . Thereby, def

ω), if ω ≤ ω ˜ }, K+ (Ωt ) = {f ∈ Cb (Ωt ) | f (ω) ≤ f (˜

(2.73)

which is a cone of bounded continuous functions. Proposition 2.10. If for given µ, µ ˜ ∈ G t , one has f µ = f µ˜ ,

for all f ∈ K+ (Ωt ),

then µ = µ ˜. This fact allows for introducing the FKG-order. b Cf.

[55, Theorem 3.4], [15, Theorem 2.1], and [14, Theorem 4.1]. details on this limit can be found in [4, 6].

c More

(2.74)

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Deﬁnition 2.5. For µ, µ ˜ ∈ G t , we say that µ ≤ µ ˜ , if f µ ≤ f µ˜ ,

for all f ∈ K+ (Ωt ).

(2.75)

Proposition 2.11. The set G t possesses maximal µ+ and minimal µ− elements in the sense of Deﬁnition 2.5. These elements are extreme; they also are translation invariant if the model is translation invariant. If V (−x) = V (x) for all , then µ+ (B) = µ− (−B) for all B ∈ B(Ω). The proof of this statement follows from the fact that, for f ∈ K+ (Ωt ) and any Λ L, f πΛ (·|ξ) ≤ f πΛ (·|ξ ) ,

whenever ξ ≤ ξ ,

(2.76)

which one obtains by the FKG inequality, see [7, 55]. By means of this inequality, one also proves the following Proposition 2.12. The family {πΛ (· | 0)}ΛL has only one W t -accumulation point, µ0 , which is an element of G t . 2.7. Periodic Euclidean Gibbs measures If the model is translation invariant, there should exist ϕ : Nd0 → R+ such that J = ϕ(|1 − 1 |, . . . , |d − d |).

(2.77)

For the box (2.63), we set Λ J = ϕ(|1 − 1 |L , . . . , |d − d |L ), def

(2.78)

where |j − j |L = min{|j − j |; 2L − |j − j |}, def

j = 1, . . . , d.

For , ∈ Λ, we introduce the periodic distance | − |Λ = |1 − 1 |2L + · · · + |d − d |2L .

(2.79)

(2.80)

With this distance the box Λ turns into a torus, which one can obtain by imposing periodic conditions on its boundaries. Now we set, cf., (2.35), β 1 Λ per IΛ (ωΛ ) = − J (ω , ω )L2β + V (ω (τ ))dτ, (2.81) 2 0 , ∈Λ

∈Λ

and thereby, cf., (2.33), νΛper (dωΛ ) = exp[−IΛper(ωΛ )]χΛ (dωΛ )/NΛper , NΛper = exp[−IΛper (ωΛ )]χΛ (dωΛ ). ΩΛ

(2.82)

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By means of (2.78) we introduce the periodic Hamiltonian HΛper = −

1 Λ J · (q , q ) + H , 2 , ∈Λ

(2.83)

∈Λ

and the corresponding periodic local Gibbs state per per per Λ (A) = trace[A exp(−βHΛ )]/trace[exp(−βHΛ )],

A ∈ CΛ .

(2.84)

The relationship between the measure νΛper and this state is the same as in the case of νΛ and Λ . Set, cf., (2.50), 1 per (B) = per exp[−IΛper(ωΛ )] IB (ωΛ × 0Λc )χΛ (dxΛ ), (2.85) πΛ NΛ ΩΛ which is a probability measure on Ωt . Then per (d(ωΛ × ωΛc )) = νΛper (dωΛ ) πΛ

δ0 (dx ),

(2.86)

∈Λc

where 0 is the zero element of the Banach space Cβ . Note that the projection of per onto ΩΛ is νΛper . πΛ Let Lbox be the sequence of all boxes (2.63). Arguments similar to those used in the proof of [55, Lemma 4.4] yield the following Lemma 2.1. For every α ∈ I and σ ∈ (0, 1/2), there exists a constant C > 0 such that, for all boxes Λ, Ωt

2

ω 2Cβσ wα (0, )

per πΛ (dω) ≤ C.

(2.87)

per Thereby, the family {πΛ }Λ∈Lbox is W t -relatively compact. per }Λ∈Lbox . Let Mper be the family of W t -accumulation points of {πΛ

Proposition 2.13. It follows that Mper ⊂ G t . The elements of Mper , called periodic Euclidean Gibbs measures, are translation invariant. The proof of this statement is similar to the proof of Proposition 2.4. It can be done by demonstrating that each µ ∈ Mper solves the DLR equation (2.58). To this end, for chosen Λ L, one picks the box ∆ containing this Λ, and shows that per per πΛ (· | ξ)π∆ (dξ) ⇒ µ(·), if π∆ ⇒ µ in W t . Ω

Here both convergence are taken along a subsequence of Lbox .

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2.8. The pressure In the translation invariant case, one can introduce a thermodynamic function, which contains important information about the thermodynamic properties of the model. This is the pressure, which in our case up to a factor coincides with the free energy density. As our special attention will be given to the dependence of the pressure on the external ﬁeld h, cf. (2.3), we indicate this dependence explicitly. For Λ L, we set, see (2.49), pΛ (h, ξ) =

1 log NΛ (h, ξ), |Λ|

ξ ∈ Ωt .

(2.88)

To simplify notations we write pΛ (h) = pΛ (h, 0). Thereby, for µ ∈ G t , we set pµΛ (h) =

pΛ (h, ξ)µ(dξ).

(2.89)

1 log NΛper (h). |Λ|

(2.90)

Ωt

Furthermore, we set pper Λ (h) =

If, for a coﬁnal sequence L, the limit def

pµ (h) = lim pµΛ (h),

(2.91)

L

exists, we call it the pressure in the state µ. We shall also consider def

p(h) = lim pΛ (h), L

def

pper (h) = lim pper Λ (h). Lbox

(2.92)

Given l = (l1 , . . . , ld ), l = (l1 , . . . , ld ) ∈ L = Zd , such that lj < lj for all j = 1, . . . , d, we set Γ = { ∈ L | lj ≤ j ≤ lj , for all j = 1, . . . , d}.

(2.93)

For this parallelepiped, let G(Γ) be the family of all pair-wise disjoint translates of Γ which cover L. Then for Λ L, we let N− (Λ | Γ) (respectively, N+ (Λ | Γ)) be the number of the elements of G(Γ) which are contained in Λ (respectively, which have non-void intersections with Λ). Deﬁnition 2.6. A coﬁnal sequence L is a van Hove sequence if for every Γ, (a) lim N− (Λ | Γ) = +∞; L

(b) lim (N− (Λ | Γ)/N+ (Λ | Γ)) = 1. L

(2.94)

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One observes that Lbox is a van Hove sequence. It is known, see [55, Theorem 3.10], that Proposition 2.14. For every h ∈ R and any van Hove sequence L, it follows that the limits (2.91) and (2.92) exist, do not depend on the particular choice of L, and are equal, that is p(h) = pper (h) = pµ (h) for each µ ∈ G t . Let the model be rotation invariant, see Deﬁnition 2.1. Then the pressure depends on the norm of the vector h ∈ Rν . Therefore, without loss of generality one can choose the external ﬁeld to be (h, 0, . . . , 0), h ∈ R. For the measure (0) (2.33), by νΛ we denote its version with h = 0. Then β (1) (0) NΛ (h) = NΛ (0) exp h ω (τ )dτ νΛ (dωΛ ). (2.95) ΩΛ

∈Λ

0

The same representation can also be written for NΛper (h). One can show that the pressures pΛ (h) and pper Λ (h), as functions of h, are analytic in a subset of C, which contains R. Thus, one can compute the derivatives and obtain ∂ pΛ (h) = βMΛ (h), ∂h

∂ per p (h) = βMΛper (h), ∂h Λ

(2.96)

where def

MΛ (h) =

1 (1) Λ [q ], |Λ|

def

(1)

MΛper (h) = per Λ [q ]

(2.97)

∈Λ

are local polarizations, corresponding to the zero and periodic boundary conditions respectively. Furthermore, ∂2 pΛ (h) ∂h2 1 = 2|Λ|

ΩΛ

ΩΛ

∈Λ

0

β

2 (1) (ω (τ )

−

(1) ω ˜ (τ ))dτ

νΛ (dωΛ )νΛ (d˜ ωΛ ) ≥ 0. (2.98)

The same can be said about the second derivative of pper Λ (h). Therefore, both pΛ (h) and pper Λ (h) are convex functions. For the reader convenience, we present here the corresponding properties of convex functions following [73, pp. 34–37]. For a function ϕ : R → R, by ϕ± (t) we denote its one-side derivatives at a given t ∈ R. By at most countable set we mean the set which is void, ﬁnite, or countable. Proposition 2.15. For a convex function ϕ : R → R, it follows that : (a) the derivatives ϕ± (t) exist for every t ∈ R; the set {t ∈ R | ϕ+ (t) = ϕ− (t)} is at most countable;

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(b) for every t ∈ R and θ > 0, ϕ− (t) ≤ ϕ+ (t) ≤ ϕ− (t + θ);

(2.99)

(c) the point-wise limit ϕ of a sequence of convex functions {ϕn }n∈N is a convex function; if ϕ and all ϕn ’s are diﬀerentiable at a given t, ϕn (t) → ϕ (t) as n → +∞. Proposition 2.16. The pressure p(h), see Proposition 2.14, is a convex function of h ∈ R. Therefore, the set R = {h ∈ R | p− (h) < p+ (h)} def

(2.100)

is at most countable. For any h ∈ Rc and any van Hove sequence L, it follows that lim MΛ (h) = lim MΛper (h) = β −1 p (h) = M (h). def

L

Lbox

(2.101)

By this statement, for any h ∈ Rc , the limiting periodic state is unique. In the scalar case, one can tell more on this item. The following result is a consequence of Propositions 2.14 and 2.11. Proposition 2.17. If ν = 1 and p(h) is diﬀerentiable at a given h ∈ R, then G t is a singleton at this h. Returning to the general case ν ∈ N we note that by Proposition 2.16 the global polarization M (h) is a nondecreasing function of h ∈ Rc ; it is continuous on each open connected component of Rc . That is, M (h) is continuous on the intervals (a− , a+ ) ⊂ Rc , where a± are two consecutive elements of R. At each such a± , the global magnetization is discontinuous. One observes however that the set Rc may have empty interior; hence, M (h) may be nowhere continuous. In the sequel, to study phase transitions in the model with the anharmonic potentials V of general type, we use the regularity of the temperature loops and Proposition 2.2. Let the model be just translation invariant. i.e. the anharmonic potential has the form (2.3), where V 0 is independent of . Let us consider the following measure on Cβ : β 1 0 exp − V (ω(τ ))dτ χ(dω), λ(dω) = Nβ 0 Nβ = Cβ

exp − 0

β

V 0 (ω(τ ))dτ χ(dω),

(2.102)

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where χ is Høegh-Krohn’s measure. For a box Λ, we introduce the following functions on ΩΛ YΛ (ωΛ ) =

ν 1 Λ β (j) (j) J ω (τ )ω (τ )dτ, 2 0 j=1 , ∈Λ

(j) XΛ (ωΛ )

=

∈Λ

β

0

(j) ω (τ )dτ,

(2.103)

j = 1, . . . , ν.

Then from (2.90) one gets pper Λ (h) = log Nβ

    ν   1 (j) + expYΛ (ωΛ ) + h(j) XΛ (ωΛ ) λ(dω ) . log   ΩΛ |Λ| j=1

(2.104)

∈Λ

As the measure (2.102) is a perturbation of the Høegh-Krohn measure, we can study the regularity of the associated stochastic process by means of Proposition 2.2. Fix some p ∈ N\{1} and σ ∈ (0, 1/2 − 1/2p). Thereby, for ϑ ∈ (0, β), one obtains Cβ

Ξpϑ (ω)λ(dω) ≤ e−βBV · Ξpϑ χ /Nβ ,

BV being as in (2.2). By Proposition 2.2 this yields Ξpϑ λ ≤ DV (σ, ν, p)m−p ϑp(1−2σ) ,

(2.105)

where, see (2.32), def

DV (σ, ν, p) =

2p exp(−βBV )Γ(ν/2 + p) 23(2p+1) (1 + 1/σp)2p · . (p − 1 − 2pσ)(p − 2pσ) Nβ Γ(ν/2)

For c > 0 and n ∈ N, n ≥ 2, we set C ± (n; c) = {ω ∈ Cβ | ± ω (j) (kβ/n) ≥ c,

j = 1, . . . , ν;

k = 0, 1, . . . n}. (2.106)

For every n ∈ N, j1 , . . . , jn ∈ {1, . . . , ν}, and τ1 , . . . , τn ∈ [0, β], the joint distribution of ω (j1 ) (τ1 ), . . . , ω (jn ) (τn ) induced by Høegh–Krohn’s measure χ is Gaussian. Therefore, χ(C ± (n; c)) > 0. Clearly, the same property has the measure (2.102). Thus, we have ! def Σ(n; c) = min λ C + (n; c) ; λ C − (n; c) > 0.

(2.107)

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For ε ∈ (0, c), we set A(c; ε) = {ω ∈ Cβ | Ξβ/n (ω) ≤ (c − ε)2 (β/n)−2σ }, B ± (ε, c) = A(c; ε) ∩ C ± (n; c).

(2.108)

Then for any τ ∈ [0, β], one ﬁnds k ∈ N such that |τ − kβ/n| ≤ β/n, and hence, for any j = 1, . . . , ν, 1/2 (β/n)σ , |ω (j) (τ ) − ω (j) (kβ/n)| ≤ Ξβ/n (ω) which yields ±ω (j) (τ ) ≥ ε if ω ∈ B ± (ε, c). Let us estimate λ[B ± (ε, c)]. By (2.105) and Chebyshev’s inequality, one gets λ(Cβ \A(c; ε)) ≤ ≤

β 2σp Ξp λ − ε)2p β/n

n2σp (c

β p DV (σ, ν, p) . [mn(c − ε)2 ]p

Thereby, λ B ± (ε, c) = λ[C ± (n; c)\(Cβ \A(c; ε))] ≥ Σ(n; c) − λ(Cβ \A(c; ε)) ≥ Σ(n; c) −

β p DV (σ, ν, p) [mn(c − ε)2 ]p

def

= γ(m),

(2.109)

which is positive, see (2.107), for all def

m ≥ m∗ =

β · n(c − ε)2

DV (σ, ν, p) Σ(n; c)

1/p .

(2.110)

This result will be used for estimating the integrals in (2.104). 3. Phase Transitions There exist several approaches to describe phase transitions. Their common point is that the macroscopic equilibrium properties of a statistical mechanical model can be diﬀerent at the same values of the model parameters. That is, one speaks about the possibility for the multiple states to exist rather than the transition (as a process) between these states or between their uniqueness and multiplicity. 3.1. Phase transitions and order parameters We begin by introducing the main notion of this section. Deﬁnition 3.1. The model described by the Hamiltonians (1.1), (1.2) has a phase transition if |G t | > 1 at certain values of β and the model parameters.

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Note that here we demand the existence of multiple tempered Euclidean Gibbs measures. For models with ﬁnite range interactions, there may exist Euclidean Gibbs measures, which are not tempered. Such measures should not be taken into account. Another observation is that in Deﬁnition 3.1 we do not assume any symmetry of the model, the translation invariance including. If the model is rotation invariant (symmetric for ν = 1, see Deﬁnition 2.1), the unique element of G t should have the same symmetry. If |G t | > 1, the symmetry can be “distributed” among the elements of G t . In this case, the phase transition is connected with a symmetry breaking. In the sequel, we consider mostly phase transitions of this type. However, in Sec. 3.5 we study the case where the anharmonic potentials V have no symmetry and hence there is no symmetry breaking connected with the phase transition. If the model is translation invariant, the multiplicity of its Euclidean Gibbs states is equivalent to the existence of non-ergodic elements of G t , see Corollary 2.6. Thus, to prove that the model has a phase transition it is enough to show that there exists an element of G t , which fails to obey (2.65). In the case where the model is not translation invariant, we employ a comparison method, based on correlation inequalities. Its main idea is that the model has a phase transition if the translation invariant model with which we compare it has a phase transition. Let us consider ﬁrst the translation and rotation invariant case. Given and j = 1, . . . , ν, we set Λ D

"

β

=β

(ω (τ ), ω (τ ))

#

0

per νΛ

dτ .

(3.1)

The right-hand side in (3.1) does not depend on τ due to the property (2.13). To introduce the Fourier transformation in the box Λ we employ the conjugate set Λ∗ (Brillouin zone), consisting of the vectors p = (p1 , . . . , pd ), such that pj = −π +

π sj , L

sj = 1, . . . , 2L,

j = 1, . . . , d.

(3.2)

Then the Fourier transformation is (j)

ω (τ ) =

1 ω ˆ (j) (τ )eı(p,) , |Λ|1/2 p∈Λ p ∗

(j)

ω ˆ p (τ ) =

1 |Λ|1/2

ω (τ )e−ı(p,) . (j)

∈Λ

(j)

In order that ω (τ ) be real, the Fourier coeﬃcients should satisfy (j)

(j)

ω ˆ p (τ ) = ω ˆ −p (τ ).

(3.3)

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By the rotation invariance of the state · νΛper , as well as by its invariance with respect to the translations of the torus Λ, it follows that (j) (j ) (j) ωp (τ )νΛper = δjj δ(p + p ) ω (τ )ω (τ )νΛper eı(p, −) . (3.4) ˆ ωp(j) (τ )ˆ ∈Λ

Thus, we set ˆΛ = D p

Λ ı(p, −) D , e

∈Λ Λ D =

(3.5)

1 ˆ Λ ı(p,− ) Dp e . |Λ| p∈Λ∗

ˆ Λ can be extended to all p ∈ (−π, π]d . Furthermore, One observes that D p ˆΛ = D ˆΛ = D p −p

Λ D cos(p, − ),

(3.6)

∈Λ

and Λ D =

1 ˆ Λ ı(p,− ) 1 ˆΛ Dp e Dp cos(p, − ). = |Λ| |Λ| p∈Λ∗

(3.7)

p∈Λ∗

For uΛ = (u )∈Λ , u ∈ R, def

(uΛ , DΛ uΛ )l2 (Λ) =

Λ D u u

, ∈Λ

=

ν

$

j=1

∈Λ

u 0

β

2 % (j) ω (τ )dτ

≥ 0.

(3.8)

per νΛ

Thereby, the operator DΛ : l2 (Λ) → l2 (Λ) is strictly positive; hence, all its eigenˆ pΛ are also strictly positive. values D ˆ : (−π, π]d → (0, +∞] Suppose now that we are given a continuous function B with the following properties: ˆ B(p)dp < ∞, (3.9) (i) (−π,π]d

ˆ pΛ ≤ B(p), ˆ (ii) D

for all p ∈ Λ∗ \{0},

holding for all boxes Λ. Then we set 1 ˆ B(p) cos(p, − )dp, B = (2π)d (−π,π]d

, ∈ L,

(3.10)

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and Λ B =

1 |Λ|

ˆ B(p) cos(p, − ),

, ∈ Λ.

(3.11)

p∈Λ∗ \{0}

Λ c We also set B = 0 if either of , belongs to Λ . ΛL Proposition 3.1. For every , , it follows that B → B as L → +∞.

ˆ Proof. By (3.9), B(p) cos(p, − ) is an absolutely integrable function in the sense of improper Riemann integral. The right-hand side of (3.11) is its integral sum; thereby, the convergence stated is obtained in a standard way. From claim (i) of (3.9) by the Riemann–Lebesgue lemma, see [57, p. 116], one obtains lim

|− |→+∞

B = 0.

Lemma 3.1. For every box Λ and any , ∈ Λ, it follows that Λ Λ Λ Λ D ≥ D − B + B .

(3.12)

(3.13)

Proof. By (3.7), (3.11), and claim (ii) of (3.9), one has 2 Λ Λ ˆ Λ sin2 (p, − ) D − D D = p |Λ| p∈Λ∗ \{0}

≤

2 |Λ|

=

Λ B

ˆ B(p) sin2 (p, − )

p∈Λ∗ \{0} Λ − B ,

which yields (3.13). For µ ∈ G t , we set, cf., (3.1), β µ = β (ω (τ ), ω (τ ))µ dτ . D

(3.14)

0

Corollary 3.1. For every periodic µ ∈ G t , it follows that µ µ D ≥ (D − B ) + B ,

(3.15)

holding for any , . Proof. For periodic µ ∈ G t , one ﬁnds the sequence {Ln }n∈N ⊂ N, such that per πΛ ⇒ µ as n → +∞, see Proposition 2.13. This fact alone does not yet mean Ln Λ

µ that DL n → D , what we would like to get. To prove the latter convergence one employs Lemma 2.1 and proceeds as in the proof of [55, Lemma 5.2, Claim (b)]. Then (3.15) follows from (3.13) and Proposition 3.1.

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One observes that the ﬁrst summand in (3.15) is independent of . Suppose now that there exists a positive ϑ such that, for any box Λ, Λ ≥ ϑ. D

(3.16)

Then, in view of (3.12), the phase transition occurs if ϑ > B .

(3.17)

ˆ obeying the conditions For certain versions of our model, we ﬁnd the function B (3.9) and the bound (3.17). Note that under (3.16) and (3.17) by (3.15) it follows that 1 µ 1 µ D = lim D (3.18) lim > 0. 2 L→+∞ |ΛL | L→+∞ |ΛL | ∈ΛL

, ∈ΛL

Let us consider now another possibilities to deﬁne phase transitions in translation invariant versions of our model. For a box Λ, see (2.63), we introduce 1 Λ D 2 (β|Λ|) , ∈Λ &2 && β & & 1 & = ω (τ )dτ & νΛper (dωΛ ), & & & β|Λ| 0 ΩΛ

PΛ =

(3.19)

∈Λ

and set def

P = lim sup PΛL .

(3.20)

L→+∞

Deﬁnition 3.2. The above P is called the order parameter. If P > 0 for given values of β and the model parameters, then there exists a long range order. By standard arguments one proves the following Proposition 3.2. If (3.16) and (3.17) hold, then P > 0. The appearance of the long range order, which in a more “physical” context is identiﬁed with a phase transition, does not imply the phase transition in the sense of Deﬁnition 3.1. At the same time, Deﬁnition 3.1 describes models without translation invariance. On the other hand, Deﬁnition 3.2 is based upon the local states only and hence can be formulated without employing G t . Yet another “physical” approach to phase transitions in translation invariant models like (1.1), (1.2) is based on the properties of the pressure p(h), which by Proposition 2.14 exists and is the same in every state. It does not employ the set G t and is based on the continuity of the global polarization (2.101), that is, on the diﬀerentiability of p(h). Deﬁnition 3.3 (Landau Classiﬁcation). The model has a ﬁrst order phase transition if p (h) is discontinuous at a certain h∗ . The model has a second order

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phase transition if there exists h∗ ∈ Rν such that p (h) is continuous but p (h) is discontinuous at h = h∗ . Remark 3.1. Like in Deﬁnition 3.1, here we do not assume any symmetry of the model (except for the translation invariance). As p(h) is convex, p (h) is increasing; hence, p (h) ≥ 0. The discontinuity of the latter mentioned in Deﬁnition 3.3 includes the case p (h∗ ) = +∞, where the polarization M (h) at h = h∗ grows inﬁnitely fast, but still is continuous. The relationship between the ﬁrst order phase transition and the long range order is established with the help of the following result, the proof of which can be done by a slight modiﬁcation of the arguments used in [25], see Theorem 1.1 and its corollaries. Let {µn }N ∈N (respectively, {Mn }n∈N ) be a sequence of probability measures on R (respectively, positive real numbers, such that lim Mn = +∞). We also suppose that, for any y ∈ R, 1 log n→+∞ Mn

eyu µn (du)

f (y) = lim

(3.21)

(0), exists and is ﬁnite. As the function f is convex, it has one-sided derivatives f± see Proposition 2.15.

Proposition 3.3 (Griﬃths). Let the sequence of measures {µn }N ∈N be as above. (0) = f− (0) = φ (i.e. f is diﬀerentiable at y = 0), then If f+ lim

g(u/Mn )µn (du) = g(φ),

n→+∞

(3.22)

for any continuous g : R → R, such that |g(u)| ≤ λeκ|u| with certain λ, κ > 0. Furthermore, for each such a function g, lim sup n→+∞

g(u/Mn )µn (du) ≤

max

(0),f (0)] z∈[f− +

g(z).

(3.23)

(0) = −f+ (0), then for any k ∈ N, In particular, if f−

f+ (0)

≥ lim sup

1/2k . (u/Mn ) µn (du) 2k

n→+∞

(3.24)

Write, cf., (2.95), NΛper (h)

=

NΛper (0)

exp h

ΩΛ

∈Λ

0

β

(1) ω (τ )dτ

νΛ0,per (dωΛ ),

(3.25)

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where νΛ0,per is the local periodic Euclidean Gibbs measure with h = 0. Now let } {Ln }n∈N ⊂ N be the sequence such that the sequences of local measures {νΛ0,per Ln 0 } converge to the corresponding periodic Euclidean Gibbs measures µ and {νΛper Ln and µ respectively. Set &

& β (1) & Xn = ωΛLn ∈ ΩΛLn & ∃u ∈ R : ω (τ )dτ = u . (3.26) & 0 ∈ΛLn

Clearly, each such Xn is measurable and isomorphic to R. Let µn , n ∈ N, be the } onto this Xn . Then projection of {νΛ0,per Ln p(h) = p(0) + f (h),

(3.27)

where f is given by (3.21) with such µn and Mn = |ΛLn | = (2Ln )d . Thereby, we apply (3.24) with k = 1 and obtain p+ (0) ≥ β lim sup PΛLn . n→+∞

Thus, in the case where the model is just rotation and translation invariant, the existence of the long range order implies the ﬁrst order phase transition. Consider now the second order phase transitions in the rotation invariant case. For α ∈ [0, 1], we set, cf., (3.19), &2 && β & β −2 & & (α) ω (τ )dτ & νΛper (dωΛ ), (3.28) PΛ = & 1+α & & |Λ| 0 ΩΛ ∈Λ

(1)

where Λ is a box. Then PΛ = PΛ and, as we just have shown, the existence of a positive limit (3.20) yields a ﬁrst order phase transition. Proposition 3.4. If there exists α ∈ (0, 1), such that for a sequence {Ln }, there exists a ﬁnite limit (α) def

lim PΛLn = P (α) > 0.

n→+∞

(3.29)

Then the model has at h = 0 a second order phase transition. Proof. We observe that (α)

PΛ

= νpΛ (0)/β 2 |Λ|α .

Then by (3.29) there exists c > 0, such that pΛLn (0) ≥ c|ΛLn |α ,

for all n ∈ N.

As each pΛ is continuous, one ﬁnds the sequence {δn }n∈N such that δn ↓ 0 and pΛLn (h) ≥

1 c|ΛLn |α , 2

for all h ∈ [0, δn ] and n ∈ N.

(3.30)

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If p (0) were ﬁnite, see Remark 3.1, one would get p (0) = lim [pΛLn (δn ) − pΛLn (0)]/δn , n→+∞

which contradicts (3.30). Proposition 3.4 remains true if one replaces in (3.28) the periodic local measure νΛper by the one corresponding to the zero boundary condition, i.e. by νΛ . Then the limit in (3.29) can be taken along any van Hove sequence L. We remind that Proposition 3.4 describes the rotation invariant case. The existence of a positive P (α) with α > 0 may be interpreted as follows. According to the central limit theorem for independent identically distributed random variables, for our model with J = 0 and V = V , the only possibility to have a ﬁnite positive limit in (3.29) is to set α = 0. If P (0) < ∞ for nonzero interaction, one can say that the dependence between the temperature loops is weak; it holds for small Jˆ0 . Of course, in this case P (α) = 0 for any α > 0. If P (α) gets positive for a certain α ∈ (0, 1), one says that a strong dependence between the loops appears. In this case, the central limit theorem holds with an abnormal normalization. However, this dependence is not so strong to make p discontinuous, which occurs for α = 1, where a new law of large numbers comes to power. In statistical physics, the point at which P (α) > 0 for α ∈ (0, 1) is called a critical point. The quantity P (0) is called susceptibility, it gets discontinuous at the critical point. Its singularity at this point is connected with the value of α for which P (α) > 0. The above analysis opens the possibility to extend the notion of the critical point to the models which are not translation invariant. Deﬁnition 3.4. The rotation invariant model has a critical point if there exist a van Hove sequence L and α ∈ (0, 1) such that &2 && β & 1 & & ω (τ )dτ (3.31) lim & & νΛ (dωΛ ) > 0 L |Λ|1+α Ω & & 0 Λ ∈Λ

at certain values of the model parameters, including h and β. Note that by Proposition 3.4, it follows that in the translation invariant case the notions of the critical point and of the second order phase transition coincide. 3.2. Infrared bound Here, for the version of our model which is translation and rotation invariant, we ˆ obeying (3.9). ﬁnd the function B For a box Λ, let E be the set of all unordered pairs , , , ∈ Λ, such that | − |Λ = 1, see (2.80). Suppose also that the interaction intensities (2.78) are such Λ that J = J > 0 if and only if , ∈ E and hence the measure (2.82) can be

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written  J 1 exp− νΛper (dωΛ ) = YΛ (0) 2



ω − ω 2L2 σΛ (dωΛ ),

(3.32)

β

, ∈E

where σΛ (dωΛ ) = exp Jd

ω 2L2 − β

∈Λ

∈Λ

β

V (ω (τ ))dτ χΛ (dωΛ ),

(3.33)

0

and 

J YΛ (0) = exp− 2 ΩΛ



ω − ω 2L2 σΛ (dωΛ ). β

, ∈E

(3.34)

With every pair , ∈ E we associate b ∈ L2β and consider



J YΛ (b) = exp− 2 ΩΛ



ω − ω − b 2L2 σΛ (dωΛ ).

(3.35)

β

, ∈E

By standard arguments, see [47] and the references therein, one proves the following Lemma 3.2 (Gaussian Domination). For every b = (b ), ∈E , b ∈ L2β , it follows that YΛ (b) ≤ YΛ (0).

(3.36)

Let XE be the real Hilbert space XE = {b = (b ), ∈E | b ∈ L2β },

(3.37)

with scalar product (b, c)XE =

, ∈E

(b , c )L2β .

(3.38)

To simplify notations we write e = , . A bounded linear operator Q : XE → XE may be deﬁned by means of its kernel Qjj ee (τ, τ ), j, j = 1, . . . , ν, e, e ∈ E, and τ, τ ∈ [0, β]. That is (Qb)(j) e (τ ) =

d j =1 e ∈E

0

β

(j )

Qjj ee (τ, τ )be (τ )dτ .

(3.39)

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Let us study the operator with the following kernel

(j )

(j )

per Qjj 1 , 2 , (τ, τ ) = [ω1 (τ ) − ω (τ )] · [ω2 (τ ) − ω (τ )]νΛ , (j)

1

(j) 1

2

2

(3.40)

where the expectation is taken with respect to the measure (3.32). This operator in positive. Indeed,  (b, Qb)XE = 

2

, ∈E

(ω − ω , b )L2β 

≥ 0. per νΛ

The kernel (3.40) can be expressed in terms of the Matsubara functions; thus, as a function of τ, τ , it has the property (2.13). We employ it by introducing yet another Fourier transformation. Set K = {k = (2π/β)κ | κ ∈ Z},  −1/2 cos kτ, if k > 0;  β −1/2 sin kτ, if k < 0; ek (τ ) = −β   2/β, if k = 0.

(3.41)

(3.42)

The transformation we need is (j)

ω ˆ (k) = (j)

ω (τ ) =

β

(j)

ω (τ )ek (τ )dτ,

0

(j)

ω ˆ (k)ek (τ ).

(3.43)

k∈K

Then the property (2.13) yields, cf., (3.4) (j )

ω (k )νΛper = 0 ˆ ω (k)ˆ (j)

if

k = k ,

or j = j .

Taking this into account we employ in (3.40) the transformation (3.43) and obtain

Qjj 1 , 2 , (τ, τ ) = δjj 1

2

ˆ , , (k)ek (τ )ek (τ ), Q 1 1 2 2

(3.44)

k∈K

with (j) (j) (j) (j) ˆ , , (k) = [ˆ Q ω1 (k) − ω ˆ (k)] · [ˆ ω2 (k) − ω ˆ (k)]νΛper . 1 1 2 2 1

2

(3.45)

In view of the periodic conditions imposed on the boundaries of the box Λ the latter kernel, as well as the one given by (3.40), are invariant with respect to the translations of the corresponding torus. This allows us to “diagonalize” the kernel (3.45) by means of a spatial Fourier transformation (3.2), (3.3). Then the spacial

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periodicity of the state · νΛper yields ˆ ω (j) (p, k)ˆ ω (j) (p , k)νΛper = 0 if p + p = 0.

(3.46)

Taking this into account we obtain ˆ , , (k) = Q ˆ ω (j) (p, k)ˆ ω (j) (−p, k)νΛper × (eı(p,1 ) − eı(p,1 ) )/|Λ|1/2 1 1 2 2 p∈Λ∗

× (e−ı(p,2 ) − eı(−p,2 ) )/|Λ|1/2 .

(3.47)

Since the summand corresponding to p = 0 equals zero, the sum can be restricted to Λ∗ \{0}. This representation however cannot serve as a spectral decomposition similar to (3.44) because the eigenfunctions here are not normalized. Indeed, (eı(p,) − eı(p, ) )/|Λ|1/2 × (e−ı(p,) − e−ı(p, ) )/|Λ|1/2 = 2E(p) , ∈E

where def

E(p) =

d

[1 − cos pj ].

(3.48)

j=1

Then we set

σ (p) = (eı(p,) − eı(p, ) )/

2|Λ|E(p),

p ∈ Λ∗ \{0},

(3.49)

and ˆ k) = 2E(p)ˆ Q(p, ω (j) (p, k)ˆ ω (j) (−p, k)νΛper ,

p ∈ Λ∗ \{0}.

(3.50)

Thereby, Q1 ,1 2 ,2 (τ, τ ) =

ˆ k)σ (p)σ (−p)ek (τ )ek (τ ), Q(p, 1 1 2 2

(3.51)

p∈Λ∗ \{0} k∈K

which is the spectral decomposition of the operator (3.39). Now we show that the eigenvalues (3.50) have a speciﬁc upper bound.d Lemma 3.3. For every p ∈ Λ∗ \{0} and k ∈ K, the eigenvalues (3.50) obey the estimate ˆ k) ≤ 1/J, Q(p,

(3.52)

where J is the same as in (3.32). From this estimate one gets ˆ ω (j) (p, k)ˆ ω (j) (−p, k)νΛper ≤ d Their

1 , 2JE(p)

p ∈ Λ∗ \{0}.

natural lower bound is zero as the operator (3.39) is positive.

(3.53)

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Proof. The estimate in question will be obtained from the Gaussian domination (3.36). For t ∈ R and a given b ∈ XE , we consider the function φ(t) = YΛ (tb). By Lemma 3.2, φ (0) ≤ 0. Computing the derivative from (3.35) we get φ (0) = J(b, Qb)XE − b 2XE , where the operator Q is deﬁned by its kernel (3.40). Therefrom one gets the estimate (3.52). By (3.3), (3.44), and (3.50), we readily obtain ν ˆ k) cos[k(τ − τ )], Q(p, ˆ −p (τ ))νΛper = (ˆ ωp (τ ), ω 2βE(p)

p = 0,

k∈K

which yields, see (3.5) and (3.52), ˆΛ = D p

βν βν ˆ , Q(p, 0) ≤ 2E(p) 2JE(p)

p = 0.

(3.54)

Comparing this estimate with (3.9) we have the following Corollary 3.2. If the model is translation and rotation invariant with the nearest neighbor interaction, then the infrared estimate (3.9) holds with ˆ B(p) =

βν , 2JE(p)

p ∈ (−π, π]d \{0},

ˆ B(0) = +∞.

(3.55)

3.3. Phase transition in the translation and rotation invariant model In this subsection, we consider the model described by Corollary 3.2. First we obtain the lower bounds for (ω (τ ), ω (τ ))νΛper , from which we then obtain the bounds (3.16). In the case where the anharmonic potential has the form V (u) = −b|u|2 + b2 |u|4 ,

b > a/2,

b2 > 0,

(3.56)

a being the same as in (1.1), the bound (3.16) can be calculated explicitly. We begin by considering this special case. Lemma 3.4. Let V be as in (3.56). Then, for every Λ L, (ω (τ ), ω (τ ))νΛper ≥

(2b − a)ν def = ϑ∗ . 4b2 (ν + 2)

(3.57)

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Proof. Let A be a self-adjoint operator, such that the expressions below make sense. Then per per Λ ([A, [HΛ , A]]) per per per per = per Λ (AHΛ A + AHΛ A − AAHΛ − HΛ AA) 1 = per |Ass |2 (Esper − Esper ){exp[−βEsper ] − exp[−βEsper ]} Zβ,Λ s,s ∈N

≥ 0.

(3.58)

Here Esper , s ∈ N are the eigenvalues of the periodic Hamiltonian (2.83), Ass are the corresponding matrix elements of A, and per Λ is the periodic local Gibbs state (2.84). By the Euclidean representation, (ω (τ ), ω (τ ))νΛper =

ν

(j)

(ω (0))2 νΛper =

j=1

ν

(j)

2 per Λ [(q ) ].

j=1

(j)

Then we take in (3.58) A = p , j = 1, . . . , ν, make use of the commutation relation (1.4), take into account the rotation invariance, and arrive at (j)

per per 2 2 per Λ ([A, [HΛ , A]]) = Λ (−2b + a + 2b2 |q | + 4b2 (q ) ) (j)

= −2b + a + 4b2 (ν + 2)[ω (0)]2 νΛper ≥ 0,

(3.59)

which yields (3.57). Now we consider the case where V is more general as to compare with (3.56). Lemma 3.5. Let the model be translation and rotation invariant, with nearest neighbor interaction. Then, for every θ > 0, there exist positive m∗ and J∗ , which may depend on β, θ, and on the potential V, such that, for m > m∗ and J > J∗ , (ω (τ ), ω (τ ))νΛper ≥ θ.

(3.60)

Proof. Let us rewrite (2.104) pper Λ (J)

1 log = log Nβ + |Λ|

exp[YΛ (ωΛ )] ΩΛ

λ(dω ) ,

(3.61)

∈Λ

where we indicate the dependence of the pressure on the interaction intensity and have set h = 0 since the potential V should be rotation invariant. Clearly, pper Λ (J)

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is convex; its derivative can be computed from (3.61). Then we get J |Λ|

" # ∂ per pΛ (J) (ω , ω )L2β ν per = J Λ ∂J

, ∈E

per ≥ pper Λ (J) − pΛ (0)

1 log = exp[YΛ (ωΛ )] λ(dω ) , |Λ| ΩΛ

(3.62)

∈Λ

where E is the same as in (3.37). By the translation invariance and (2.13), one gets (ω , ω )L2β νΛper ≤ ((ω , ω )L2β νΛper + (ω , ω )L2β νΛper )/2 = (ω , ω )L2β νΛper = β(ω (τ ), ω (τ ))νΛper . Then we choose ε, c, and n as in (2.109), apply this estimate in (3.62), and obtain

βJd(ω (τ ), ω (τ ))νΛper

1 log ≥ |Λ|

[B + (ε;c)]ν|Λ|

exp[YΛ (ωΛ )]

λ(dω )

∈Λ

≥ βJνdε2 + ν log γ(m).

(3.63)

(2.110), γ(m) > 0 and the latter estimate makes sense. Given For m > m∗ given by θ > 0, one picks ε > θ/ν and then ﬁnds J∗ such that the right-hand side of the latter estimate equals θ for J = J∗ . To convert (3.57) and (3.60) into the bound (3.16) we need the function f : [0, +∞) → [0, 1) deﬁned implicitly by f (u tanh u) = u−1 tanh u,

for u > 0;

and f (0) = 1.

(3.64)

It is diﬀerentiable, convex, monotone decreasing on (0, +∞), such that tf (t) → 1. For t ≥ 6, f (t) ≈ 1/t to ﬁve-place accuracy, see [25, Theorem A.2]. By direct calculation, f (uτ ) 1 τ − u(1 − τ 2 ) =− · , f (uτ ) uτ τ + u(1 − τ 2 )

τ = tanh u.

(3.65)

Proposition 3.5. For every ﬁxed α > 0, the function φ(t) = tαf (t/α),

t>0

is diﬀerentiable and monotone increasing to α2 as t → +∞.

(3.66)

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Proof. By (3.65), φ (t) =

2ατ (1 − τ 2 ) > 0, τ + u(1 − τ 2 )

uτ = u tanh u = t/α.

The limit α2 is obtained from the corresponding asymptotic property of f . Next, we need the following fact, known as Inequality of Bruch and Falk, see [73, Theorem IV.7.5, p. 392] or [25, Theorem 3.1]. Proposition 3.6. Let A be as in (3.58). Let also b(A) = β −1

0

β

per per per Λ {A exp[−τ HΛ ]A exp[τ HΛ ]}dτ,

2 g(A) = per Λ (A ); per c(A) = per Λ {[A, [βHΛ , A]]},

Then

b(A) ≥ g(A)f

c(A) 4g(A)

,

(3.67)

dp , E(p)

(3.68)

with f deﬁned by (3.64). Set 1 J (d) = (2π)d

(−π,π]d

where E(p) is given by (3.48). The exact value of J (3) can be expressed in terms of complete elliptic integrals, see [87] and also [38] for more recent developments. For our aims, it is enough to have the following property, see [24, Theorem 5.1]. Proposition 3.7. For d ≥ 4, one has 1 1 1 < J (d) < < , d − 1/2 d − α(d) d−1

(3.69)

where α(d) → 1/2 as d → +∞. Recall that m is the reduced particle mass (1.3). Theorem 3.1. Let d ≥ 3, the interaction be of nearest neighbor type, and the anharmonic potential be of the form (3.56), which deﬁnes the parameter ϑ∗ . Let also the following condition be satisﬁed 8mϑ2∗ J > J (d).

(3.70)

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Then for every β > β∗ , where the latter is the unique solution of the equation 2βJϑ∗ f (β/4mϑ∗ ) = J (d),

(3.71)

the model has a phase transition in the sense of Deﬁnition 3.1. Proof. One observes that (j)

(j)

[q , [HΛper , q ]] = 1/m, (j)

Then we take in (3.67) A = q b(A) ≥

∈ Λ.

and obtain

(j) (ω (0))2 νΛper f

β (j)

4m(ω (0))2 νΛper

(3.72)

.

By Proposition 3.5, ϑf (β/4mϑ) is an increasing function of ϑ. Thus, by (3.57) and (3.1), Λ ≥ β 2 νϑ∗ f (β/4mϑ∗ ), D

(3.73)

which yields the bound (3.16). Thereby, the condition (i) in (3.17) takes the form ϑ∗ f (β/4mϑ∗ ) > J (d)/2βJ.

(3.74)

By Proposition 3.5, the function φ(β) = 2βJϑ∗ f (β/4mϑ∗ ) is monotone increasing and hits the level J (d) at certain β∗ . For β > β∗ , the estimate (3.74) holds, which yields |G t | > 1. One observes that f (β/4mϑ∗ ) → 1 as m → +∞. In this limit, the condition (3.70) turns into the corresponding condition for a classical model of φ4 anharmonic oscillators, Now let us turn to the general case. Theorem 3.2. Let d ≥ 3, the interaction be of nearest neighbor type, and the anharmonic potential be rotation invariant. Then, for every β > 0, there exist m∗ and J∗ > 0, which may depend on β and on the anharmonic potential, such that |G t | > 1 for m > m∗ and J > J∗ . Proof. Given positive β and θ, the estimate (3.60) holds for big enough m and J. Then one applies Proposition 3.6, which yields that the condition (i) in (3.17) is satisﬁed if θf (β/4mθ) > J (d)/2βJ.

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Then one sets m∗ to be as in (2.110) and J∗ to be the smallest value of J for which both (3.60) and the latter inequality hold. 3.4. Phase transition in the symmetric scalar models In the case ν = 1, we can extend the above results to the models without translation invariance and with much more general J and V . However, certain assumptions beyond (2.1) and (2.2) should be made. Suppose also that the interaction between the nearest neighbors is uniformly nonzero, i.e. inf

|− |=1

def

J = J > 0.

(3.75)

Next we suppose that all V ’s are even continuous functions and the upper bound in (2.79) can be chosen to obey the following conditions: (a) for every , V (u ) − V (u ) ≤ V (˜ u ) − V (˜ u ),

whenever u2 ≤ u ˜2 ;

(3.76)

(b) the function V has the form V (u ) =

r

b(s) u2s ;

2b(1) < −a;

b(s) ≥ 0,

s ≥ 2,

(3.77)

s=1

where a is as in (1.1) and r ≥ 2 is either positive integer or inﬁnite; (c) if r = +∞, the series Φ(ϑ) =

+∞ s=2

(2s)! b(s) ϑs−1 , − 1)!

2s−1 (s

(3.78)

converges at some ϑ > 0. Since 2b(1) + a < 0, the equation a + 2b(1) + Φ(ϑ) = 0,

(3.79)

has a unique solution ϑ∗ > 0. By the above assumptions, all V are “uniformly double-welled”. If V (u ) = v (u2 ) and v are diﬀerentiable, the condition (3.76) can be formulated as an upper bound for v . Note that the pressure as a uniﬁed characteristics of all Euclidean Gibbs states makes senses for translation invariant models only. Thus, the notions mentioned in Deﬁnition 3.3 are not applicable to the versions of the model which do not possess this property. The main result of this subsection is contained in the following statement. Theorem 3.3. Let the model be as just described. Let also the condition (3.70) with ϑ∗ deﬁned by Eq. (3.76) and J deﬁned by (3.75) be satisﬁed. Then for every β > β∗ , where β∗ is deﬁned by Eq. (3.70), the model has a phase transition in the sense of Deﬁnition 3.1. If the model is translation invariant, the long range order and the ﬁrst order phase transition take place at such β.

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Proof. The proof is made by comparing the model under consideration with a reference model, which is the scalar model with the nearest neighbor interaction of intensity (3.75) and with the anharmonic potential (3.77). Thanks to the condition (3.76), the reference model is more stable; hence, the phase transition in this model implies the same for the model considered. The comparison is conducted by means of correlation inequalities. The reference model is translation invariant and hence can be deﬁned by its local periodic Hamiltonians Hhar + V (q ) − J q q , (3.80) HΛlow = , ∈E

∈Λ

where Λ is a box and E is the same as in (3.32); Hhar is as in (1.1). For this model, we have the infrared estimate (3.54) with ν = 1. Let us obtain the lower bound, see (3.57). To this end we use the inequalities (3.58), (3.59) and obtain 0 ≤ a + 2b(1) +

r

2s(2s − 1)b(s) [ω (0)]2(s−1) νΛlow

s=2

≤ a + 2b(1) +

r s=2

2s(2s − 1)

" # s−1 (2s − 2)! · b(s) (ω (0))2 ν low . Λ − 1)!

2s−1 (s

(3.81)

Here νΛlow is the periodic Gibbs measure for the model (3.80). To get the second line we used the Gaussian upper bound inequality, see [55, p. 1031] and [7, p. 1372], which is possible since all b(s) , s ≥ 2 are nonnegative. The solution of the latter inequality is (ω (0))2 νΛlow ≥ ϑ∗ .

(3.82)

Then the proof of the phase transitions in the model (3.80) goes along the line of > 0, where arguments used in proving Theorem 3.1. Thus, for β > β∗ , ω (0)µlow + µlow + is the corresponding maximal Euclidean Gibbs measure, see Proposition 2.11. But, , ω (0)µ+ > ω (0)µlow +

(3.83)

see [55, Lemma 7.7]. At the same time ω (0)µ = 0 for any periodic µ ∈ G t , which yields the result to be proven. 3.5. Phase transition in the scalar model with asymmetric potential The phase transitions proven so far have a common feature — the spontaneous symmetry breaking. This means that the symmetry, e.g., rotation invariance, possessed by the model and hence by the unique element of G t is no longer possessed by the multiple Gibbs measures appearing as its result. In this subsection, we show

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that the translation invariant scalar version o the model (1.1), (1.2) has a phase transition without symmetry breaking. However, we restrict ourselves to the case of ﬁrst order phase transitions, see Deﬁnition 3.3. The reason for this can be explained µ as follows. The fact that D does not decay to zero as | − | → +∞, see (3.18), implies that µ is non-ergodic only if µ is symmetric. Otherwise, to show that µ is non-ergodic one should prove that the diﬀerence µ D

−β

β

(ω (τ )µ , ω (τ )µ )dτ

0

does not decay to zero, which cannot be done by means of our methods based on the infrared estimate. In what follows, we consider the translation invariant scalar version of the model (1.1), (1.2) with the nearest neighbor interaction. The only condition imposed on the anharmonic potential is (2.2). Obviously, we have to include the external ﬁeld, that is the anharmonic potential is now V (u)−hu. Since we are not going to impose any conditions on the odd part of V , we cannot apply the GKS inequalities, see [7, 55], the comparison methods are based on, see (3.83). In view of this fact we suppose that the interaction is of nearest neighbor type. Thus, for a box Λ, the periodic local Hamiltonian of the model has the form (3.80). In accordance with Deﬁnition 3.3, our goal is to show that the model parameters (except for h) and the inverse temperature β can be chosen in such a way that the set R, deﬁned by (2.100), is non-void. The main idea on how to do this can be µ explained as follows. First we ﬁnd a condition, independent of h, under which D does not decay to zero for a certain periodic µ. Next we prove the following Lemma 3.6. There exist h± , h− < h+ , which may depend on the model parameters and β, such that the magnetization (2.101) has the property: M (h) < 0,

for

h ∈ Rc ∩ (−∞, h− );

M (h) > 0,

for

h ∈ Rc ∩ (h+ + ∞).

Thereby, if R were void, one would ﬁnd h∗ ∈ (h− , h+ ) such that M (h∗ ) = 0. At such h∗ , the aforementioned property of Dµ would yield the non-ergodicity of µ and hence the ﬁrst order phase transition, see Theorem 3.3. µ In view of Corollary 3.1, D does not decay to zero if (3.16) holds with big enough ϑ. By Proposition 3.6, the lower bound (3.16) can be obtained from the estimate (3.60). The only problem with the latter estimate is that it holds for h = 0. Lemma 3.7. For every β > 0 and θ, there exist positive m∗ and J∗ , which may depend on β > 0 and θ but are independent of h, such that, for any box Λ and any h ∈ R, [ω (0)]2 νΛper ≥ θ,

if

J > J∗

and

m > m∗ .

(3.84)

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Proof. For h ∈ R, we set β 1 λ (dω) = h exp h ω(τ )dτ λ(dω), Nβ 0 h

exp h

Nβh =

β

(3.85)

ω(τ )dτ λ(dω),

0

Cβ

where λ is as in (2.102). Then for ±h > 0, we get the estimate (3.63) in the following form βJd[ω (0)]2 νΛper ≥ βJdε2 + log λh [B ± (ε, c)],

(3.86)

where B ± (ε, c) is as in (2.108), (2.109). Let us show now that, for ±h ≥ 0, λh [B ± (ε, c)] ≥ λ[B ± (ε, c)].

(3.87)

For h ≥ 0, let I(ω) be the indicator function of the set C + (n; c), see (2.106). For δ > 0 and t ∈ R, we set  t ≤ c, 0 ιδ (t) = (t − c)/δ t ∈ (c, c + δ],  1 c ≥ c + δ. Thereby, def

Iδ (ω) =

n

ιδ [ω(kβ/n)].

k=0

By Lebesgue’s dominated convergence theorem Nβh λh [C + (n; c)]

=

I(ω) exp h

ω(τ )dτ λ(dω) 0

Cβ

= lim δ↓0

β

Iδ (ω) exp h

Cβ

β

ω(τ )dτ λ(dω).

(3.88)

0

As the function Iδ is continuous and increasing, by the FKG inequality, see [7, Theorem 6.1], it follows that β h Iδ (ω) exp h ω(τ )dτ λ(dω) ≥ Nβ Iδ (ω)λ(dω). Cβ

0

Passing here to the limit we obtain from (3.88) λh [C + (n; c)] ≥ λ[C + (n; c)],

Cβ

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which obviously yields (3.87). For h ≤ 0, one just changes the signs of h and ω. Thereby, we can rewrite (3.86) as follows, cf., (3.63), [ω (0)]2 νΛper ≥ ε2 + [log γ(m)]/βJd. Then one applies the arguments from the very end of the proof of Lemma 3.5. Proof of Lemma 3.6. Suppose that h > 0. Then restricting the integration in (2.104) to [B + (ε, c)]Λ , we get 1 2 Λ J + log λ[B + (ε, c)] pper Λ (h) ≥ hβε + log Nβ + βε 2 ∈Λ

≥ hβε + log Nβ + log γ(m).

(3.89)

As the right-hand side of the latter estimate is independent of Λ, it can be extended to the limiting pressure p(h). For any positive h ∈ Rc , by the convexity of p(h) one has M (h) ≥ [p(h) − p(0)]/βh ≥ ε+

1 {−p(0) + log Nβ + log γ(m)}. βh

Picking big enough h we get the positivity stated. The negativity can be proven in the same way. Now we are at a position to prove the main statement of this subsection. Theorem 3.4. Let the model be scalar, translation invariant, and with the nearestneighbor interaction. Let also d ≥ 3. Then for every β, there exist m∗ > 0 and J∗ > 0 such that, for all m > m∗ and J > J∗ , there exists h∗ ∈ R, possibly dependent on m, β, and J, such that p (h) gets discontinuous at h∗ , i.e. the model has a ﬁrst order phase transition. Proof. Let m∗ be as in (2.110) and J∗ , θ be as in Lemma 3.7. Fix any β > 0 and m > m∗ . Then, for J > J∗ , the estimate (3.84) holds, which yields the validity of (3.73) for all boxes Λ with such β, m, and ν = 1. Thereby, we increase J, if necessary, up to the value at which (3.74) holds. Afterwards, all the parameters, except for h, are set ﬁxed. In this case, there exists a periodic state µ ∈ G t such µ that the ﬁrst summand in (3.15) is positive; hence, D does not decay to zero as | − | → +∞, see (3.12) and (3.15). If p(h) is everywhere diﬀerentiable, i.e. if R = ∅, then by Lemma 3.6 there exists h∗ such that M (h∗ ) = 0; hence, the state µ with such h∗ is non-ergodic, which yields |G t | > 1 and hence a ﬁrst order phase transition. Otherwise, R = ∅.

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3.6. Comments • Section 3.1: According to Deﬁnition 3.1, the phase transition corresponds to the existence of multiple equilibrium phases at the same values of the model parameters and temperature. This is a standard deﬁnition for theories, which employ Gibbs states, see [31, p. 28 and p. 119]. In the translation invariant case, one can prove phase transitions by showing the existence of non-ergodic elements of G t . For classical lattice systems, it was realized in [29] by means of infrared estimates. More or less at the same time, an alternative rigorous theory of phase transitions in classical lattice spin models based on contour estimates has been proposed. This is the Pirogov–Sinai theory elaborated in [64, 65], see also [74]. Later on, this theory was essentially extended and generalized into an abstract sophisticated method, applicable also to classical (but not quantum) models with unbounded spins, see [88] and the references therein. For quantum lattice models, the theory of phase transitions has essential peculiarities, which distinguish it from the corresponding theory of classical systems. Most of the results in this domain were obtained by means of quantum versions of the method of infrared estimates. The ﬁrst publication in which such estimates were applied to quantum spin models seems to be the article [25]. After certain modiﬁcations this method was applied to a number of models with unbounded Hamiltonians [7,16,17,24,45,63]. In our approach, the quantum crystal is described as a system of “classical” inﬁnite dimensional spins. This allows for applying here the original version of the method of infrared estimates elaborated in [29] adapted to the inﬁnite dimensional case, which has been realized in the present work. Among others, the adaptation consists in employing such tools as the Garsia–Rodemich–Rumsey lemma, see [30]. This our approach is more eﬀective and transparent than the one used in [7, 16, 17, 45]. It also allows for comparing the conditions (3.16), (3.17) with the stability conditions obtained in the next section. In the physical literature, there exist deﬁnitions of phase transitions alternative to Deﬁnition 3.1, based directly on the thermodynamic properties of the system. These are the deﬁnition employing the diﬀerentiability of the pressure (Deﬁnition 3.3, which is applicable to translation invariant models only), and the deﬁnition based on the long range order. The relationship between the latter two notions is established by means of the Griﬃths theorem, Proposition 3.3, the proof of which can be found in [25]. For translation invariant models with bounded interaction, non-diﬀerentiability of the pressure corresponds to the nonuniqueness of the Gibbs states, see [37,73]. We failed to prove this for our model. In the language of limit theorems of probability theory, the appearance of the long range order corresponds to the fact that a new law of large numbers comes to power, see Theorem 3.3 and the discussion preceding Deﬁnition 3.4. The critical point of the model corresponds to the case where the law of large numbers still holds in its original form (in the translation invariant case this means absence of

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the ﬁrst order phase transitions), but the central limit theorem holds true with an abnormal normalization. For a hierarchical version of the model (1.1), (1.2), the critical point was described in [5]. Algebras of abnormal ﬂuctuation operators were studied in [21]. In application to quantum crystals, such operators were discussed in [84, 85], where the reader can ﬁnd a more detailed discussion of this subject as well as the corresponding bibliography. • Section 3.2: As was mentioned above, the method of infrared estimates was originated in [29]. The version employed here is close to the one presented in [47]. We note that, in accordance with the conditions (3.9), (3.16), and (3.17), the infrared bound was obtained for the Duhamel function, see (3.54), rather than for (ω (τ ), ω (τ ))νΛper · cos(p, − ), ∈Λ

which was used in [7, 16, 17, 45]. • Section 3.3: The lower bound (3.57) was obtained in the spirit of [24, 63]. The estimate stated in Lemma 3.5 is completely new; the key element of its proving is the estimate (2.105), obtained by means of Proposition 2.2. The suﬃcient condition for the phase transition obtained in Theorem 3.1 is also new. Its significant feature is the appearance of a universal parameter responsible for the phase transition, which includes the particle mass m, the anharmonicity parameter ϑ∗ , and the interaction strength J. This is the parameter on the left-hand side of (3.70). The same very parameter will describe the stability of the model studied in the next section. Theorem 3.2 is also new. • Section 3.4: Here we mostly repeat the corresponding results of [55], announced in [54]. • Section 3.5: The main characteristic feature of the scalar model studied in [7, 16, 17,24,45,63], as well the the one described by Theorem 3.3, was the Z2 -symmetry broken by the phase transition. This symmetry allowed for obtaining estimates like (3.82), crucial for the method. However, in classical models, for proving phase transitions by means of the infrared estimates, symmetry was not especially important, see [29, Theorem 3.5] and the discussion preceding this theorem. There might be two explanations of such a discrepancy: (a) the symmetry was the key element but only of the methods employed therein, and, like in the classical case, its lack does not imply the lack of phase transitions; (b) the symmetry is crucial in view of e.g. quantum eﬀects, which stabilize the system, see the next section. So far, there has been no possibility to check which of these explanations is true. Theorem 3.4 solves this dilemma in favor of explanation (a). Its main element is again an estimate, obtained by means of the Garsia–Rodemich–Rumsey lemma. The corresponding result was announced in [40]. 4. Quantum Stabilization In physical substances containing light quantum particles moving in multi-welled potential ﬁelds phase transitions are experimentally suppressed by application of

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strong hydrostatic pressure, which makes the wells closer to each other and increases the tunneling of the particles. The same eﬀect is achieved by replacing the particles with the ones having smaller mass. The aim of this section is to obtain a description of such eﬀects in the framework of the theory developed here and to compare it with the theory of phase transitions presented in the previous section. 4.1. The stability of quantum crystals Let us look at the scalar harmonic version of the model (1.1) — a quantum harmonic crystal. For this model, the one-particle Hamiltonian includes the ﬁrst two terms of (1.2) only. Its spectrum consists of the eigenvalues Enhar = (n + 1/2) a/m, n ∈ N0 . The parameter a > 0 is the oscillator rigidity. For reasons which become clear in a while, we consider the following gap parameter har ). ∆har = min(Enhar − En−1 n∈N

Then ∆har =

a/m;

a = m∆2har .

(4.1)

(4.2)

The set of tempered Euclidean Gibbs measures of the harmonic crystal can be constructed similarly as it was done in section 2, but with one exception. Such measures exist only under the stability condition (2.71), which might now be rewritten Jˆ0 < m∆2har .

(4.3)

In this case, G t is a singleton at all β, that readily follows from Theorem 2.9. As the right-hand side of (4.3) is independent of m, this stability condition is applicable also to the classical harmonic crystal which is obtained in the classical limit m → +∞, see [4, 7]. According to (2.2) the anharmonic potentials V have a super-quadratic growth due to which the tempered Euclidean Gibbs measures of anharmonic crystals exist for all Jˆ0 . In this case, the instability of the crystal is connected with phase transitions. A suﬃcient condition for some of the models described in the previous section to have a phase transition may be derived from the Eq. (3.74). It is 2βJϑ∗ f (β/4mϑ∗ ) > J (d),

(4.4)

which in the classical limit m → +∞ takes the form 2βJϑ∗ > J (d). The latter condition can be satisﬁed by picking big enough β. Therefore, the corresponding classical anharmonic crystals always have phase transitions — no matter how small is the interaction intensity. For ﬁnite m, the left-hand side of (4.4) is bounded by 8mϑ2∗ J, and the bound is achieved in the limit β → +∞. If for given values of the interaction parameter J, the mass m, and the parameter ϑ∗ which characterizes the anharmonic potential, this bound does not exceed J (d),

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the condition (4.4) will never be satisﬁed. Although this condition is only suﬃcient, one might expect that the phase transition can be eliminated at all β if the compound parameter 8mϑ2∗ J is small enough. Such an eﬀect, if really exists, could be called quantum stabilization as it is impossible in the classical analog of the model.

4.2. Quantum rigidity In the harmonic case, big values of the rigidity a ensure the stability. In this subsection, we introduce and stugy quantum rigidity, which plays a similar role in the anharmonic case. Above the suﬃcient condition (4.4) for a phase transition to occur was obtained for a simpliﬁed version of the model (1.1), (1.2) — nearest neighbor interactions, polynomial anharmonic potentials of special kind (3.77), ect. Then the results were extended to more general models via correlation inequalities. Likewise here, we start with a simple scalar version of the one-particle Hamiltonian (1.1), which we take in the form Hm =

1 2 a 2 p + q + V (q), 2m 2

(4.5)

where the anharmonic potential is, cf., (3.77), V (q) = b(1) q 2 + b(2) q 4 + · · · + b(r) q 2r ,

b(r) > 0,

r ∈ N\{1}.

(4.6)

The subscript m in (4.5) indicates the dependence of the Hamiltonian on the mass. Recall that Hm acts in the physical Hilbert space L2 (R). Its relevant properties are summarized in the following Proposition 4.1. The Hamiltonian Hm is essentially self-adjoint on the set C0∞ (R) of inﬁnitely diﬀerentiable functions with compact support. The spectrum of Hm has the following properties: (a) it consists of eigenvalues En , n ∈ N0 only; (b) to each En there corresponds exactly one eigenfunction ψn ∈ L2 (R); (c) there exists γ > 1 such that n−γ En → +∞,

as n → +∞.

(4.7)

Proof. The essential self-adjointness of Hm follows from the Sears theorem, see [18, Theorem 1.1, p. 50] or [68, Theorem X.29]. The spectral properties follow from [18, Theorem 3.1, p. 57, (Claim (a))] and [18, Proposition 3.3, p. 65, (Claim (b))]. To prove Claim (c) we employ a classical formula, see [81, Eq. (7.7.4), p. 151], which in our context reads un 1 1 2√ 2m En − V (u)du = n + + O , (4.8) π 2 n 0

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where n, and hence En , are big enough so that the equation (4.9)

V (u) = En have the unique positive solution un . Then 1 1 π 1 r+1 2r un φn (t) − t dt = √ n+ +O , (r) 2 n 2 2mb 0

(4.10)

where φn (t) =

En u2−2r u−2 n n (1) 2 − (b + a/2)t − · · · − b(r−1) t2(r−1) . (r) (r) b(r) u2r b b n

Note that φn (1) = 1 for all n, which follows from (4.9). Thus, En (r) b u2r n

→ 1,

as n → +∞.

(4.11)

Thereby, we have def

1

cn =

φn (t) − t2r dt →

0

3 1 Γ 2 2r . = 3 1 + 2rΓ 2 2r

1

1 − t2r dt

0

Γ

(4.12)

Then combining (4.12) with (4.9) and (4.11) we get

2r  r+1 3 1 , (r) -1/(r+1)  πrΓ 2 + 2r  b  · n+ 1  · + o(1), En =  r 3 1 (2m) 2  Γ Γ 2 2r



(4.13)

which readily yields (4.7) with any γ ∈ (1, 2r/(r + 1)). Thus, in view of the property (4.13) we introduce the gap parameter ∆m = min(En − En−1 ),

(4.14)

Rm = m∆2m ,

(4.15)

n∈N

and thereby, cf., (4.2),

which can be called quantum rigidity of the oscillator. One might expect that the stability condition for quantum anharmonic crystals, at least for their scalar versions with the anharmonic potentials independent of , is similar to (4.3). That is, it has

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the form Jˆ0 < Rm .

(4.16)

4.3. Properties of the quantum rigidity Below f ∼ g means that lim(f /g) = 1. Theorem 4.1. For every r ∈ N, the gap parameter ∆m , and hence the quantum rigidity Rm corresponding to the Hamiltonian (4.5), (4.6), are continuous functions of m. Furthermore, ∆m ∼ ∆0 m−r/(r+1) ,

Rm ∼ ∆20 m−(r−1)/(r+1) ,

m → 0,

(4.17)

with a certain ∆0 > 0. Proof. Given α > 0, let Uα : L2 (R) → L2 (R) be the following unitary operator √ (4.18) (Uα ψ) (x) = αψ(αx). Then by (1.4) Uα−1 pUα = αp,

Uα−1 qUα = α−1 q.

Fix any m0 > 0 and set ρ = (m/m0 )1/(r+1) , α = ρ1/2 . Then ˜ m def H = Uα−1 Hm Uα = ρ−r T (ρ),

(4.19)

where T (ρ) = Hm0 + Q(ρ) =

1 2 p + ρr−1 (b(1) + a/2)q 2 + ρr−2 b(2) q 4 + · · · + b(r) q 2r , 2m0

(4.20)

Q(ρ) = (ρ − 1)[pr−1 (ρ)(b(1) + a/2)q 2 + pr−2 (ρ)b(2) q 4 + · · · + pr−s (ρ)b(s) q 2s + · · · + b(r−1) q 2(r−1) ],

(4.21)

and pk (ρ) = 1 + ρ + ρ2 + · · · + ρk−1 .

(4.22)

˜ m , are unitary equivalent, their gap parameters (4.14) As the operators Hm , H ˜ m and T (ρ), ρ > 0 possess the properties established by coincide. The operators H Proposition 4.1. In particular, they have the property (4.7) with one and the same γ. Therefore, there exist ε > 0 and k ∈ N such that for |ρ − 1| < ε, the gap parameters ˜ m and T (ρ) are deﬁned by the ﬁrst k eigenvalues of these operators. As (4.14) for H an essentially self-adjoint operator, T (ρ) possesses a unique self-adjoint extension Tˆ(ρ), the eigenvalues of which coincide with those of T (ρ). Furthermore, for complex

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ρ, Tˆ(ρ) is a closed operator, its domain Dom[Tˆ (ρ)] does not depend on ρ. For every ψ ∈ Dom[Tˆ(ρ)], the map C ζ → Tˆ (ζ)ψ ∈ L2 (R) is holomorphic. Therefore, {Tˆ (ρ) | |ρ − 1| < ε} is a self-adjoint holomorphic family. Hence, the eigenvalues Θn (ρ), n ∈ N0 of Tˆ(ρ) are continuous functions of ρ ∈ (1 − ε, 1 + ε), see [41, ˆ m0 . Since we have given Chap. VII, Sec. 3]. At ρ = 1 they coincide with those of H k ∈ N such that, for all ρ ∈ (1 − ε, 1 + ε), min[Θn (ρ) − Θn−1 (ρ)] = n∈N

[Θn (ρ) − Θn−1 (ρ)],

min

n∈{1,2,...,k}

the function def ˜ ∆(ρ) = min ρ−r [Θn (ρ) − Θn−1 (ρ)] n∈N

(4.23)

is continuous. But by (4.19) 1/(r+1) ˜ ), ∆m = ∆((m/m 0)

(4.24)

which proves the continuity stated since m0 > 0 has been chosen arbitrarily. To prove the second part of the theorem we rewrite (4.20) as follows (0) + R(ρ), T (ρ) = Hm 0

(4.25)

where (0) Hm = 0

1 2 p + b(r) q 2r , 2m0

and R(ρ) = ρ(ρr−2 (b(1) + a/2)q 2 + ρr−3 b(2) q 4 + · · · + b(r−1) q 2(r−1) ). Repeating the above perturbation arguments one concludes that the self-adjoint family {Tˆ(ρ) | |ρ| < ε} is holomorphic at zero; hence, the gap parameter of (4.25) (0) tends, as ρ → 0, to that of Hm0 , i.e. to ∆0 . Thereby, the asymptotics (4.17) for ∆m ˜ m. follows from (4.19) and the unitary equivalence of Hm and H Our second result in this domain is the quasi-classical analysis of the parameters (4.14), (4.15). Here we shall suppose that the anharmonic potential V has the form (4.6) with b(s) ≥ 0 for all s = 2, . . . , r − 1, cf., (3.77). We remind that in this case the parameter ϑ∗ > 0 is the unique solution of Eq. (3.78). Theorem 4.2. Let V be as in (3.77). Then the gap parameter ∆m and the quantum rigidity Rm of the Hamiltonian (4.5) with such V obey the estimates ∆m ≤

1 , 2mϑ∗

Rm ≤

1 . 4mϑ2∗

(4.26)

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Proof. Let m be the local Gibbs state (2.5) corresponding to the Hamiltonian (4.5). Then by means of the inequality (3.58) and the Gaussian upper bound we get, see (3.81), a + 2b(1) + Φ(m (q 2 )) ≥ 0, by which m (q 2 ) ≥ ϑ∗ .

(4.27)

Let ψn , n ∈ N0 be the eigenfunctions of the Hamiltonian Hm corresponding to the eigenvalues En . By Proposition 4.1, to each En there corresponds exactly one ψn . Set Qnn = (ψn , qψn )L2 (R) ,

n, n ∈ N0 .

Obviously, Qnn = 0 for any n ∈ N0 . Consider Γ(τ, τ ) = m [q exp(−(τ − τ )Hm )q exp(−(τ − τ )Hm )],

τ, τ ∈ [0, β],

which is the Matsubara function corresponding to the state m and the operators F1 = F2 = q. Set

β

Γ(0, τ ) cos kτ dτ,

u ˆ(k) =

k ∈ K = {(2π/β)κ | κ ∈ Z}.

(4.28)

0

Then uˆ(k) =

+∞ En − En 1 |Qnn |2 2 {exp(−βEn ) − exp(−βEn )}, Zm k + (En − En )2

(4.29)

n,n =0

where Zm = trace exp(−βHm ). The term (En − En )2 in the denominator can be estimated by means of (4.14), which yields u ˆ(k) ≤

+∞ 1 1 · |Qnn |2 (En − En ){exp(−βEn ) − exp(−βEn )} k 2 + ∆2m Zm n,n =0

≤ =

k2

1 · m ([q, [Hm , q]]) + ∆2m

1 . m(k 2 + ∆2m )

(4.30)

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By this estimate we get m (q 2 ) = Γ(0, 0) =

1 u(k) β k∈K

1 1 1 = ≤ coth (β∆m /2) . 2 2 β m(k + ∆m ) 2m∆m

(4.31)

k∈K

Combining the latter estimate with (4.27) we arrive at ∆m tanh(β∆m /2) < 1/(2mϑ∗ ), which yields (4.26) in the limit β → +∞. Now let us analyze the quantum stability condition (4.16) in the light of the latter results. The ﬁrst conclusion is that, unlike to the case of harmonic oscillators, this condition can be satisﬁed for all Jˆ0 by letting the mass be small enough. For the nearest-neighbor interaction, one has Jˆ0 = 2dJ; hence, if (4.16) holds, then 8dmϑ2∗ J < 1.

(4.32)

This can be compared with the estimate 8dmϑ2∗ J > dJ (d),

(4.33)

guaranteeing a phase transition, which one derives from (4.4). For ﬁnite d, dJ (d) > 1, see Proposition 3.7; hence, there is a gap between the latter estimate and (4.32), which however diminishes as d → +∞ since lim dJ (d) = 1.

d→+∞

In the remaining part of this section, we show that for the quantum crystals, both scalar and vector, a stability condition like (4.16) yields a suﬃcient decay of the pair correlation function. In the scalar case, this decay guaranties the uniqueness of tempered Euclidean Gibbs measures. However, in the vector case it yields a weaker result — suppression of the long range order and of the phase transitions of any order in the sense of Deﬁnition 3.3. The discrepancy arises from the fact that the uniqueness criteria based on the FKG inequalities are applicable to scalar models only. 4.4. Decay of correlations in the scalar case In this subsection, we consider the model (1.1), (1.2) which is (a) translation invariant; (b) scalar; (c) the anharmonic potential is V (q) = v(q 2 ) with v being convex on R+ . Let Λ be the box (2.63) and Λ∗ be its conjugate (3.2). For this Λ, we let " # Λ def ω (τ )ω (τ ) ν per (4.34) K (τ, τ ) = Λ

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be the periodic correlation function. Recall that the periodic interaction potential Λ ˆ(k) be as J was deﬁned by (2.78). For the one-particle Hamiltonian (1.2), let u in (4.28). Theorem 4.3. Let the model be as just describes. If u ˆ(0)Jˆ0 < 1,

(4.35)

then Λ K (τ, τ ) ≤

1 exp[ı(p, − ) + ık(τ − τ )] , β|Λ| [ˆ u(k)]−1 − Jˆ0Λ + ΥΛ (p) p∈Λ∗ k∈K

(4.36)

where

Jˆ0Λ =

Λ J ,

ΥΛ (p) = Jˆ0Λ −

∈Λ

Λ J exp[ı(p, − )].

(4.37)

∈Λ

Proof. Along with the periodic local Gibbs measure (2.82) we introduce νΛper (dωΛ | t)

  β   t 1 Λ J V (ω (τ ))dτ = per exp χΛ (dωΛ ), (ω , ω )L2 − β 2  NΛ (t) 0 , ∈Λ

∈Λ

(4.38) where t ∈ [0, 1] and NΛper (t) is the corresponding normalization factor. Thereby, we set X (τ, τ | t) = ω (τ )ω (τ )νΛper (·|t) ,

, ∈ Λ.

(4.39)

By direct calculation β ∂ 1 Λ X (τ, τ | t) = J1 2 R 1 2 (τ, τ , τ , τ | t)dτ ∂t 2 0 1 ,2 ∈Λ β + JΛ1 2 X1 (τ, τ | t)X2 (τ , τ | t)dτ , 1 ,2 ∈Λ

(4.40)

0

where R1 2 3 4 (τ1 , τ2 , τ3 , τ4 | t) = ω1 (τ1 )ω2 (τ2 )ω3 (τ3 )ω4 (τ4 )νΛper (·|t) − ω1 (τ1 )ω2 (τ2 )νΛper (·|t) · ω3 (τ3 )ω4 (τ4 )νΛper (·|t) − ω1 (τ1 )ω3 (τ3 )νΛper (·|t) · ω2 (τ2 )ω4 (τ4 )νΛper (·|t) − ω1 (τ1 )ω4 (τ4 )νΛper (·|t) · ω2 (τ2 )ω3 (τ3 )νΛper (·|t) .

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By the Lebowitz inequality, see [7], we have R1 2 3 4 (τ1 , τ2 , τ3 , τ4 |t) ≤ 0,

(4.41)

holding for all values of the arguments. Let us consider (4.40) as an integrodiﬀerential equation subject to the initial condition X (τ, τ | 0) = δ Γ(τ, τ ) = (δ /β)

u ˆ(k) cos k(τ − τ ).

(4.42)

k∈K

Besides, we also have Λ X (τ, τ | 1) = K (τ, τ | p).

(4.43)

Along with the Cauchy problem (4.40), (4.42) let us consider the following equation - β , ε ∂ Y (τ, τ | t) = Y1 (τ, τ | t)Y2 (τ , τ | t)dτ , (4.44) JΛ1 2 + ∂t |Λ| 0 1 ,2 ∈Λ

where ε > 0 is a parameter, subject to the initial condition Y (τ, τ | 0) = X (τ, τ | 0) u ˆ(k) cos k(τ − τ ). = (δ /β)

(4.45)

k∈K

Let us show that under the condition (4.35) there exists ε0 > 0 such that, for all ε ∈ [0, ε0 ), the problem (4.44), (4.45), t ∈ [0, 1], has the unique solution Y (τ, τ | t) =

exp[ı(p, − ) + ık(τ − τ )] 1 , β|Λ| [ˆ u(k)]−1 − t[Jˆ0Λ + εδp,0 ] + tΥΛ (p) p∈Λ∗ k∈K

(4.46)

where Jˆ0 , ΥΛ (p) are the same as in (4.37) and δp,0 is the Kronecker symbol with respect to each of the components of p. By means of the Fourier transformation Y (τ, τ | t) =

1 ˆ Y (p, k | t) exp[ı(p, − ) + ık(τ − τ )], β|Λ| p∈Λ∗ k∈K

Yˆ (p, k | t) =

∈Λ

β

(4.47)

Y (τ, τ | t) exp[−ı(p, − ) − ık(τ − τ )]dτ ,

0

we bring (4.44), (4.45) into the following form ∂ ˆ Y (p, k | t) = [JˆΛ (p) + εδp,0 ] · [Yˆ (p, k | t)]2 , ∂t

Yˆ (p, k | 0) = u ˆ(k),

(4.48)

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where, see (4.37), JˆΛ (p) =

Λ Λ ˆΛ J exp[ı(p, − )] = J0 − Υ (p).

(4.49)

∈Λ

ˆ(0). Then in view of (4.35), one ﬁnds Clearly, Jˆ0Λ ≤ Jˆ0 , |JˆΛ (p)| ≤ Jˆ0Λ , and uˆ(k) ≤ u ε0 > 0 such that, for all ε ∈ (0, ε0 ), the following holds [JˆΛ (p) + εδp,0 ]ˆ u(k) < 1, for all p ∈ Λ∗ and k ∈ K. Thus, the problem (4.48) can be solved explicitly, which via the transformation (4.47) yields (4.46). Given θ ∈ (0, 1), we set Y (τ, τ | t) = Y (τ, τ | t + θ), (θ)

t ∈ [0, 1 − θ].

(4.50)

Obviously, the latter function obeys Eq. (4.44) on t ∈ [0, 1 − θ] with the initial condition Y (τ, τ | 0) = Y (τ, τ | θ) > Y (τ, τ | 0) = X (τ, τ | 0). (θ)

(4.51)

The latter inequality is due to the positivity of both sides of (4.44). Therefore, Y (τ, τ | t) > 0, (θ)

(4.52)

for all , ∈ Λ, τ, τ ∈ [0, β], and t ∈ [0, 1 − θ]. Let us show now that under the condition (4.35), for all θ ∈ (0, 1) and ε ∈ (0, ε0 ), X (τ, τ | t) < Y (τ, τ | t), (θ)

(4.53)

also for all , ∈ Λ, τ, τ ∈ [0, β], and t ∈ [0, 1 − θ]. To this end we introduce ± Z (τ, τ | t) = Y (τ, τ | t) ± X (τ, τ | t), def

(θ)

t ∈ [0, 1 − θ].

(4.54)

Then one has from (4.40), (4.44) β ∂ − 1 Λ + Z (τ, τ | t) = J1 2 {Z (τ, τ | t)Z− 2 (τ , τ | t) 1 ∂t 2 0 1 ,2 ∈Λ

− + Z (τ, τ | t)Z+ 2 (τ , τ | t)}dτ 1 β ε (θ) (θ) + Y1 (τ, τ | t)Y 2 (τ , τ | t)dτ |Λ| 0 1 ,2 ∈Λ

− S (τ, τ | t),

(4.55)

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where S (τ, τ | t) stands for the ﬁrst term on the right-hand side of (4.40). By (4.54) and (4.51) − Z (τ, τ | 0) = Y (τ, τ | θ) − X (τ, τ | 0) > 0,

(4.56)

which holds for all , ∈ Λ, τ, τ ∈ [0, β]. For every , ∈ Λ, both Y (τ, τ | t), ± | t) are continuous functions of their arguX (τ, τ | t) and, hence, Z (τ, τ ments. Set ! − ζ(t) = inf Z (τ, τ | t) | , ∈ Λ, τ, τ ∈ [0, β] .

(4.57)

By (4.56), it follows that ζ(0) > 0. Suppose now that ζ(t0 ) = 0 at some t0 ∈ [0, 1−θ] − and ζ(t) > 0 for all t ∈ [0, t0 ). Then by the continuity of Z , there exist , ∈ Λ and τ, τ ∈ [0, β] such that − − and Z for all t < t0 . Z (τ, τ | t0 ) = 0 (τ, τ | t) > 0 − For these , ∈ Λ and τ, τ ∈ [0, β], the derivative (∂/∂t)Z | t) at t = t0 (τ, τ is positive since on the right-hand side of (4.55) the third term is positive and the remaining terms are non-negative. But a diﬀerentiable function, which is positive at t ∈ [0, t0 ) and zero at t = t0 , cannot increase at t = t0 . Thus, ζ(t) > 0 for all t ∈ [0, 1 − θ], which yields (4.53), and thereby

X (τ, τ | 1 − θ) < Y (τ, τ | 1) 1 exp[ı(p, − ) + ık(τ − τ )] = . β|Λ| [ˆ u(k)]−1 − t[Jˆ0Λ + εδp,0 ] + tΥΛ (p) p∈Λ∗ k∈K All the function above depend on θ and ε continuously. Hence, passing here to the limit θ = ε ↓ 0 and taking into account (4.43) we obtain (4.36). By means of Proposition 2.13, the result just proven can be extended to all periodic elements of G t . For µ ∈ G t , we set µ K (τ, τ ) = ω (τ )ω (τ )µ .

(4.58)

Theorem 4.4. Let the stability condition (4.16) be satisﬁed. Then for every periodic µ ∈ G t , the correlation function (4.58) has the bound µ K (τ, τ ) ≤ Y (τ, τ ) exp[ı(p, − ) + ık(τ − τ )] 1 def = dp, d β(2π) d [ˆ u(k)]−1 − Jˆ0 + Υ(p) k∈K (−π,π]

(4.59)

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where Υ(p) = Jˆ0 −

J exp[ı(p, − )],

p ∈ (−π, π]d .

(4.60)

µ0 t The same bound has also the correlation function K is as (τ, τ ), where µ0 ∈ G in Proposition 2.12.

Remark 4.1. By (4.30), [ˆ u(k)]−1 ≥ m(∆2m + k 2 ). The upper bound in (4.59) with −1 [ˆ u(k)] ] replaced by m([∆har ]2 +k 2 ) turns into the inﬁnite volume correlation function for the quantum harmonic crystal discussed at the beginning of Sec. 4.1. Thus, under the condition (4.35) the decay of the correlation functions in the periodic states is not less than it is in the stable quantum harmonic crystal. As we shall see in the next subsection, such a decay stabilizes also anharmonic ones. For Υ(p) ∼ Υ0 |p|2 , Υ0 > 0, as p → 0, the asymptotics of the bound in (4.59) as | − |2 + |τ − τ |2 → +∞ will be the same as for the (d + 1)-dimensional free ﬁeld, which is well known, see claim (c) of [32, Proposition 7.2.1, p. 162]. Thus, we have the following Proposition 4.2. If the function (4.60) is such that Υ(p) ∼ Υ0 |p|2 , Υ0 > 0, as p → 0, the upper bound in (4.59) has an exponential spacial decay.

4.5. Decay of correlations in the vector case In the vector case, the eigenvalues of the Hamiltonian (4.5) are no longer simple; hence, the parameter (4.14) deﬁnitely equals zero. Therefore, one has to pick another parameter, which can describe the quantum rigidity in this case. If the model is rotation invariant, its dimensionality ν is just a parameter. Thus, one can compare the stability of such a model with the stability of the model with ν = 1. This approach was developed in [50], see also [7,39]. Here we present the most general result in this domain, which is then used to study the quantum stabilization in the vector case. We begin by introducing the corresponding class of functions. A function f : R → R is called polynomially bounded if f (x)/(1 + |x|k ) is bounded for some k ∈ N. Let F be the set of continuous polynomially bounded f : R → R which are either odd and increasing or even and positive. Proposition 4.3. Suppose that the model is rotation invariant and for all ∈ Λ, Λ L, V (x) = v (|x|2 ) with v being convex on R+ . Then for any τ1 , . . . , τn ∈ [0, β], 1 , . . . , n ∈ Λ, j = 1, . . . , ν, f1 , . . . fn ∈ F, (j)

(j)

f1 (ω1 (τ1 )) · · · fn (ωn (τn ))νΛ ≤ f1 (ω1 (τ1 )) · · · fn (ωn (τn ))ν˜Λ ,

(4.61)

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where ν˜Λ is the Euclidean Gibbs measure (2.33) of the scalar model with the same J as the model considered and with the anharmonic potentials V (q) = v (q 2 ). By this statement one immediately gets the following fact. Theorem 4.5. Let the model be translation invariant and such as in Proposition 4.3. Let also ∆m be the gap parameter (4.14) of the scalar model with the same interaction intensities J and with the anharmonic potentials V (q) = v(q 2 ). Then if the stability condition (4.16) is satisﬁed, the longitudinal correlation function µ (τ )µ , K (τ, τ ) = ω (τ )ω (j)

(j)

j = 1, 2, . . . , ν,

(4.62)

corresponding to any of the periodic states µ ∈ G t , as well as to any of the accumulation points of the family {πΛ (· | 0)}ΛL , obeys the estimate (4.59) in which uˆ(k) is calculated according to (4.29) for the one-dimensional anharmonic oscillator of mass m and the anharmonic potential v(q 2 ). 4.6. Suppression of phase transitions From the “physical” point of view, the decay of correlations (4.59) already corresponds to the lack of any phase transition. However, in the mathematical theory, one should show this as a mathematical fact basing on the deﬁnition of a phase transition. The most general one is Deﬁnition 3.1 according to which the suppression of phase transitions corresponds to the uniqueness of tempered Euclidean Gibbs states. Properties like the diﬀerentiability of the pressure, cf., Deﬁnition 3.3, or the lack of the order parameter, see Deﬁnition 3.2, may also indicate the suppression of phase transitions, but in a weaker sense. The aim of this section is to demonstrate that the decay of correlations caused by the quantum stabilization yields the two-times diﬀerentiability of the pressure, which in the scalar case yields the uniqueness. This result is then extended to the models which are not necessarily translation invariant. In the scalar case, the most general result is the following statement, see [55, Theorem 3.13]. Theorem 4.6. Let the anharmonic potentials V be even and such that there exists a convex function v : R+ → R, such that, for any V , V (x ) − v(x2 ) ≤ V (˜ x ) − v(˜ x2 )

whenever

x2 < x ˜2 .

(4.63)

For such v, let ∆m be the gap parameter of the one-particle Hamiltonian (1.1) with the anharmonic potential v(q 2 ). Then the set of tempered Euclidean Gibbs measures of this model is a singleton if the stability condition (4.16) involving ∆m and the interaction parameter Jˆ0 of this model is satisﬁed. The proof of this theorem is conducted by comparing the model with the translation invariant reference model with the anharmonicity potential V (q) = v(q 2 ). By

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Proposition 2.11, for the model considered and the reference model, there exist maximal elements, µ+ and µref + , respectively. By means of the symmetry V (q) = V (−q) and the FKG inequality, one proves that, for both models, the uniqueness occurs if = 0, ω (0)µref +

ω (0)µ+ = 0,

for all .

(4.64)

By the GKS inequalities, the condition (4.63) implies , 0 ≤ ω (0)µ+ ≤ ω (0)µref +

(4.65)

which means that the reference model is less stable with respect to the phase transitions than the initial model. The reference model is translation invariant. By means of a technique employing this fact, one proves that the decay of correlations in the reference model which occurs under the stability condition (4.16) yields, see Theorem 4.3, = 0, ω (0)µref + and therefrom (4.64) by (4.65). The details can be found in [55]. As was mentioned above, in the vector case we failed to prove that the decay of correlations implies the uniqueness. The main reason for this is that the proof of Theorem 4.6 was based on the FKG inequality, which can be proven for scalar models only. In the vector case, we get a weaker result, by which the decay of correlations yields the normality of thermal ﬂuctuations. To this end we introduce the ﬂuctuation operators 1 (j) (j) q , QΛ = |Λ| ∈Λ

Λ L,

j = 1, . . . , ν.

(4.66)

Such operators correspond to normal ﬂuctuations. Deﬁnition 4.1. The ﬂuctuations of the displacements of oscillators are called normal if the Matsubara functions (2.12) for the operators F1 = Q(j1 ) , . . . , Fn = Q(jn ) , remain bounded as Λ L. If Λ is a box, the parameter (3.28) can be written (α) PΛ

ν 1 β β β,Λ = 2 α Γ (j) (j) (τ, τ )dτ dτ . β |Λ| j=1 0 0 QΛ ,QΛ

(4.67)

Thus, if the ﬂuctuations are normal, phase transitions of the second order (and all the more of the ﬁrst order) do not occur. Like in the proof of Theorem 4.5, the model is compared with the scalar ferromagnetic model with the same mass and the anharmonic potential v(q 2 ). Then the gap parameter ∆m is the one calculated for the latter model.

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Theorem 4.7. Let the model be the same as in Theorem 4.5 and let the stability condition involving the interaction parameter Jˆ0 of the model and the gap parameter ∆m corresponding to its scalar analog be satisﬁed. Then the ﬂuctuations of the displacements of the oscillators remain normal at all temperatures.

4.7. Comments • Section 4.1: In an ionic crystal, the ions usually form massive complexes the dynamics of which determine the physical properties of the crystal, including its instability with respect to structural phase transitions, see [22]. Such massive complexes can be considered as classical particles; hence, the phase transitions are described in the framework of classical statistical mechanics. At the same time, in a number of ionic crystals containing localized light ions certain aspects of the phase transitions are apparently unusual from the point of view of classical physics. Their presence can only be explained in a quantum-mechanical context, which points out on the essential role of the light ions. This inﬂuence of the quantum eﬀects on the phase transition was detected experimentally already in the early 1970’s. Here we mention the data presented in [19,80] on the KDP-type ferroelectrics and in [60] on the YBaCuO-type superconductors. These data were then used for justifying the corresponding theoretical models and tools of their study. On a theoretical level, the inﬂuence of quantum eﬀects on the structural phase transitions in ionic crystals was ﬁrst discussed in the paper [69], where the particle mass was chosen as the only parameter responsible for these eﬀects. The conclusion, obtained there was that the long range order, see Deﬁnition 3.2, gets impossible at all temperatures if the mass is suﬃciently small. Later on, a number of rigorous studies of quantum eﬀects inspired by this result as well as by the corresponding experimental data have appeared, see [58,59,86] and the references therein. Like in [69], in these works the reduced mass (1.3) was the only parameter responsible for the eﬀects. The result obtained was that the long range order is suppressed at all temperatures in the light mass limit m → 0. Based on the study of the quantum crystals performed in [2,3,6,8,10], a mechanism of quantum eﬀects leading to the stabilization against phase transitions was proposed, see [9]. • Section 4.2: According to [9] the key parameter responsible for the quantum stabilization is Rm = m∆2m , see (4.15). In the harmonic case, m∆2m is merely the oscillator rigidity and the stability of the crystal corresponds to large values of this quantity. That is why the parameter m∆2m was called quantum rigidity and the eﬀect was called quantum stabilization. If the tunneling between the wells gets more intensive (closer minima), or if the mass diminishes, m∆2m gets bigger and the particle “forgets” about the details of the potential energy in the vicinity of the origin (including instability) and oscillates as if its equilibrium at zero is stable, like in the harmonic case. • Section 4.3: Theorems 4.1 and 4.2 are new. Preliminary results of this kind were obtained in [3, 51]. With regard to Theorem 4.2 we note the article [56] where an

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interesting technique of deriving estimates of the eigenvalue diﬀerences by means of commutators was developed. • Section 4.4: Theorems 4.3, 4.4 and 4.2 were proven in [46]. • Section 4.5: Various scalar domination estimates were obtained in [48–50]. • Section 4.6: Theorem 4.6 was proven in [55]. The proof of Theorem 4.7 was done in [50]. The suppression of abnormal ﬂuctuations in the hierarchical version of the model (1.1), (1.2) was proven in [2]. Acknowledgments The authors are grateful to M. R¨ockner and T. Pasurek for valuable discussions. The ﬁnancial support by the DFG through the project 436 POL 113/115/0-1 and through SFB 701 “Spektrale Strukturen und topologische Methoden in der Mathematik” is cordially acknowledged. A. Kargol is grateful for the support by the KBN under the Grant N N201 0761 33. References [1] S. Albeverio and R. Høegh-Krohn, Homogeneous random ﬁelds and quantum statistical mechanics, J. Funct. Anal. 19 (1975) 242–279. [2] S. Albeverio, Y. Kondratiev and Y. Kozitsky, Absence of critical points for a class of quantum hierarchical models, Comm. Math. Phys. 187 (1997) 1–18. [3] S. Albeverio, Y. Kondratiev and Y. Kozitsky, Suppression of critical ﬂuctuations by strong quantum eﬀects in quantum lattice systems, Comm. Math. Phys. 194 (1998) 493–512. [4] S. Albeverio, Y. Kondratiev and Y. Kozitsky, Classical limits of Euclidean Gibbs states for quantum lattice models, Lett. Math. Phys. 48 (1999) 221–233. [5] S. Albeverio, Y. Kondratiev, A. Kozak and Y. Kozitsky, A hierarchical model of quantum anharmonic oscillators: Critical point convergence, Comm. Math. Phys. 251 (2004) 1–25. [6] S. Albeverio, Y. Kondratiev, Y. Kozitsky and M. R¨ ockner, Uniqueness for Gibbs measures of quantum lattices in small mass regime, Ann. Inst. H. Poincar´ e 37 (2001) 43–69. [7] S. Albeverio, Y. Kondratiev, Y. Kozitsky and M. R¨ ockner, Euclidean Gibbs states of quantum lattice systems, Rev. Math. Phys. 14 (2002) 1335–1401. [8] S. Albeverio, Y. Kondratiev, Y. Kozitsky and M. R¨ ockner, Gibbs states of a quantum crystal: Uniqueness by small particle mass, C. R. Math. Acad. Sci. Paris 335 (2002) 693–698. [9] S. Albeverio, Y. Kondratiev, Y. Kozitsky and M. R¨ ockner, Quantum stabilization in anharmonic crystals, Phys. Rev. Lett. 90(17) (2003) 170603-1–4. [10] S. Albeverio, Y. Kondratiev, Y. Kozitsky and M. R¨ ockner, Small mass implies uniqueness of Gibbs states of a quantum crystall, Comm. Math. Phys. 241 (2003) 69–90. [11] S. Albeverio, Y. Kondratiev, T. Pasurek and M. R¨ ockner, Euclidean Gibbs measures on loop lattices: Existence and a priori estimates, Ann. Probab. 32 (2004) 153–190. [12] S. Albeverio, Y. Kondratiev, T. Pasurek and M. R¨ ockner, Euclidean Gibbs measures of quantum crystals: Existence, uniqueness and a priori estimates, in Interacting Stochastic Systems, eds. J. D. Deuschel and A. Greven (Springer, Berlin, 2005), pp. 29–54.

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Reviews in Mathematical Physics Vol. 20, No. 5 (2008) 597–623 c World Scientific Publishing Company

UNIFORM ERGODIC THEOREMS ON APERIODIC LINEARLY REPETITIVE TILINGS AND APPLICATIONS

ADNENE BESBES Math´ ematiques, Universit´ e Paris Diderot, 175 rue du Chevaleret, 75013 Paris, France ∗ [email protected]

Received 10 July 2007 Revised 15 February 2008 The paper is concerned with aperiodic linearly repetitive tilings. For such tilings, we establish a weak form of self-similarity that allows us to prove general (sub)additive ergodic theorems. Finally, we provide applications to the study of lattice gas models. Keywords: Tiling; ergodic; linear repetitivity. Mathematics Subject Classification 2000: 37B50, 5C23, 82B20, 82B05

1. Introduction Aperiodic tilings with long range order have attracted much attention in recent years (see e.g. the monographs and survey volumes [1, 8, 14, 17]). This is partly due to the actual discovery of quasicrystals i.e. physical substances exhibiting such a form of (dis)order, by Shechtman, Blech, Gratias and Cahn [18] in 1984. Moreover, this is also due to the intrinsic interest in structures lying exactly at the border between order and disorder. Tilings arising from primitive substitutions, constitute a special class of linearly repetitive tilings, that have been studied in several contexts [4, 6, 13, 19, 20], including random Schr¨ odinger operators and lattice gas models. It is this application to lattice gas theory that we focus on in this paper. More exactly we will be concerned with existence of thermodynamic quantities and their relationship. This has been investigated by Geerse/Hof for tilings associated to primitive substitutions in [4]. Their work relies heavily on the self-similarity induced in the system by the substitution. This self-similarity has two consequences: It gives a canonical way of decomposing the tiling into bigger and bigger pieces and it gives lower and upper ∗ UFR

de Math´ ematiques, Universit´e Paris Diderot, 175 rue du Chevaleret, F-75013, Paris, France. 597

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bounds on frequencies of these pieces. For other types of systems such a study is still missing, due to the lack of any (possibly weak) form of self-similarity. In order to establish a weak form of self-similarity, we use results of Priebe [15] on return tiles and derived Vorono˘ı tilings, see also [16]. These results give analogues for tilings of results of Durand [3]. They allow us to decompose the system into bigger and bigger pieces. This decomposition does not rely on linear repetitivity. However, linear repetitivity implies suitable bounds on frequencies. Given decomposition and bounds on frequencies we can follow the strategy of [4] and prove certain additive and subadditive ergodic theorems. These results can then in turn be used to study lattice gas models as in [4]. Our ergodic theorems and their proofs are also related to the corresponding investigations of [2] and [12], respectively. In fact, [12] already uses the derived Vorono˘ı construction to prove an additive ergodic theorem for aperiodically ordered sets outside the context of substitutions, our additive ergodic theorem can be derived from it. The paper [2] contains a subadditive ergodic theorem for linearly repetitive systems. This theorem is somewhat weaker than our result, as they require “asymptotic translation invariance” for all large box-patterns whereas we only need it for certain patterns. In order to carry over the methods of [4] we need our results. The outline of the paper is as follows. In Sec. 2, we review basic facts on tilings and ﬁx the notations. In Sec. 3, we present some speciﬁc features of linearly repetitive tilings. In Sec. 4, we discuss a suitable form of weak self-similarity induced by return tiles introduced by Priebe in [15], see also [16]. In Sec. 5, we derive a suitable form of (sub)additive theorems to deal with lattice gas models. In Sec. 6, we apply our theorems to the study of lattice gas models. Finally, in Sec. 7, we show that the supremum in the variational principle is attained by the unique Gibbs measure for interactions that are suﬃciently weak and decay suﬃciently fast.

2. Preliminaries The reader is referred to [4] and [19] for terminology and facts concerning tilings and tiling dynamical systems. A tile is a compact set which is the closure of its interior. A collection of tiles with disjoint interiors whose union is the whole space is called a tiling. A vector x ∈ Rd is a period of the tiling T , if T + x = T . A tiling is called crystallographic if it has d linearly independent periods, and aperiodic if it has no nonzero periods. Two tiles of T are called equivalent if one can be mapped onto the other by translation. The equivalence class of a tile m ∈ T is called a prototile of T , it is denoted by m. ˜ For the remainder of this paper we only consider tilings constructed by a ﬁnite set of prototiles. A ﬁnite set of tiles in a tiling T is called a patch. Let P, P two patches, if every tile in P belongs to P we write P ⊂ P . Two patches P and P are said to be equivalent if there is a translation that maps every tile in P to a tile in P and vice

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versa, we then write P P . The equivalence class of P is called a tilepattern and is denoted by P˜ . For a patch P we deﬁne: • S(P ) = m∈P m ⊂ Rd , • rin (P ) = max{r ∈ R+ | rmthere exists x ∈ Rd , B(x, r) ⊂ S(P )}, ) = Rout (P ) = min{r ∈ R+ | there exists x ∈ Rd , S(P ) ⊂ B(x, r)}, • diam(P 2 where B(x, r) denotes the closed ball centered at x with radius r. The functions rin and Rout can be extended to tilepatterns in the obvious way. Given a tiling T of Rd , we deﬁne • [F ]T = {m ∈ T | m ∩ F = ∅} for F ⊂ Rd . For m ∈ T the patch [m]T is called the T -corona of the tile m. In some cases, it is useful to consider marked tilings. A marked tiling is a tiling in which each tile has been assigned an element of some set G, called a set of markings. So every tile is a pair m = (S(m), l(m)) with S(m) ⊂ Rd and l(m) ∈ G. Two tiles are equivalent if they have the same marking and one can be mapped onto the other by translation. Two patches P and P are equivalent if a translation maps every tile in P to a tile in P that has the same marking and vice versa. For a n n patch P = i=1 (S(mi ), l(mi )) the support of P is deﬁned by S(P ) = i=1 S(mi ). The tilings considered here are built with a ﬁnite set of prototiles which we ˜ i ﬁx a point ym denote by {m ˜ i }, i = 1, . . . , K. For each prototile m ˜ i in the interior. We then obtain a set of marked prototiles deﬁned by {(m ˜ i , ym ˜ i )}, i = 1, . . . , K. Now a point y is a vertex in the tiling T if the equivalence class of ([y]T , y) is equal to some marked prototile (m ˜ i , ym ˜ i ), i = 1, . . . , K. The set of ﬁnite sets of vertices in T is denoted by ϕ. The set of vertices contained in a subset Q ⊂ Rd is denoted by Q∗ and the cardinality of Q∗ is denoted by #Q∗ . We deﬁne an equivalence relation on ϕ in the following way: two ﬁnite sets of vertices X and X are equivalent if one can be mapped onto the other by translation, we write then X ≈ X . The ˆ The set of all equivalence class of X ∈ ϕ is called vertexpattern, it is denoted by X. vertexpatterns is denoted by ϕ. ˆ Remark 2.1. By the deﬁnition of vertices it is clear that equivalent patches give rise to equivalent set of vertices. Let T0 be a tiling in Rd . We deﬁne the tiling space XT0 as the set of all tilings T of Rd with the property that every patch occurring in T is equivalent to some patch in T0 . The tiling space XT0 has ﬁnite local complexity (FLC) if for any R > 0, the tilepatterns of diameter less than R occurring in one of the tilings in XT0 are ﬁnite. The tiling space will be equipped with the following metric d(T, T ) = min{1, I(T, T )},

where I(T, T ) = inf{ε : ∃ x, y ∈ B(0, ε) s.t [B(0, 1/ε)]T −x = [B(0, 1/ε)]T −y }. It is well known that (XT0 , d) is compact whenever XT0 has FLC, see [19, Lemma 1.1], see also [10, Lemma 2.2] for Delone sets. The additive group Rd acts on XT0 by

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translations Γt : T → T − t, t ∈ Rd . The function Γt is a homeomorphism of XT0 . (XT0 , Rd ) will be called a tiling dynamical system. The set {T − t | t ∈ Rd } is the orbit of T . The dynamical system is called minimal if every orbit is dense. The tiling dynamical system (XT0 , Rd ) is called aperiodic if every tiling in XT0 is aperiodic. Definition 2.1 (Linear Repetitivity). A tiling dynamical system (XT0 , Rd ) is called linearly repetitive if there is a constant CLR > 0 such that for every patch P occurring in T ∈ XT0 there is a copy of P in every patch P ⊂ T satisfying rin (P ) ≥ CLR Rout (P ). For the remainder of the paper we will write LRTDS for Linearly Repetitive Tiling Dynamical System. Remark 2.2. A LRTDS (XT0 , Rd ) has ﬁnite local complexity, as all the possible R in T0 . tilepatterns of diameter less than R are contained in any ball of radius CLR 2 Linear repetitivity of the tiling dynamical system implies strict ergodicity, i.e. minimality and unique ergodicity. The minimality is a direct consequence of [19, Lemma 1.2], see [9] as well, and the unique ergodicity was shown in [9], see [2, Corollary 4.6] as well. The next lemma gives us an upper bound of the number of vertices in some measurable set. Lemma 2.1. Let (XT0 , Rd ) be a tiling dynamical system which has FLC, then there exists r > 0 such that for every tiling T ∈ XT0 the distance between two diﬀerent vertices in T is greater than 2r. In particular, for all Q ∈ B(Rd ) the following holds #(Q∗ ) ≤

|Qr | . |B(0, r)|

Proof. By the deﬁnition of vertices it is clear that the set of vertices in T is a Delone set, i.e. there exists exists r > 0 such that d(x, x ) ≥ 2r, for any vertices x and x in T . Now for Q ∈ B(Rd ), the balls with radius r around diﬀerent vertices in Q are disjoint and contained in Qr which ends the proof. Definition 2.2. Let (XT0 , Rd ) be a minimal tiling dynamical system with ﬁnite local complexity, and P a patch in T0 . We deﬁne the packing radius with respect to P by r(P ) =

1 inf{ q − q | P + q ⊂ T and P + q ⊂ T, T ∈ XT0 } 2

and the occurrence radius with respect to P by R(P ) = inf{R > 0 | NPT (B(y, R)) ≥ 1 for every T ∈ XT0 and y ∈ Rd }. Here NPT (B(y, R)) denotes the number of patches in T which are equivalent to P and contained in B(y, R). Note that r(P ) > 0 by FLC and R(P ) < ∞ by minimality.

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Definition 2.3 (van Hove Sequence). Let B(Rd ) denote the set of bounded measurable sets in Rd . A sequence (Qn )n∈N in B(Rd ) is called a van Hove sequence if for all h > 0 |Qhn \Qn,h | = 0, n→∞ |Qn | lim

with Qh = {x ∈ Rd | dist(x, Q) ≤ h}, Qh = {x ∈ Q | dist(x, ∂Q) ≥ h} where ∂Q denotes the boundary of Q. Definition 2.4 (Cube-Like Sequence). A van Hove sequence (Qn )n∈N is called cube-like if there exist a sequence of cubes (Cn )n∈N and δ > 0 such that for all n∈N |Qn | ≥ δ. Qn ⊂ Cn and |Cn | 3. Specific Features of Linearly Repetitive Tilings This section contains some combinatorial results concerning linearly repetitive tilings. For aperiodic LRTDS we need the next lemma, this result is proven for self-similar tilings in [20, Lemma 2.4]. The proof can be easily carried on to our setting. For the convenience of the readers we include it here. Lemma 3.1. Let (XT0 , Rd ) be an aperiodic LRTDS. There exists N ≥ 1 such that for any x = 0 satisfying (a) P ⊂ T, P + x ⊂ T for T ∈ XT0 , (b) S(P ) ⊃ B(y, r) for some y ∈ Rd , then x ≥

r . N

Proof. Let dM denote the maximal diameter of tiles in XT0 , and let η > 0 such that every tile in XT0 contains a ball of diameter η in its interior. Let CLR (3η −1 dM + 1), (3.1) 2 where CLR is the linear repetitivity constant of XT0 from Deﬁnition 2.1. Let P be a patch in T ∈ XT0 such that P + x ⊂ T and S(P ) contains a ball of radius r, suppose that x < Nr . We need to show that x is a period of T , i.e. T + x = T . Observe that T + x = T is equivalent to the implication m ∈ T ⇒ m + x ∈ T . Let m be an arbitrary tile in T and consider the patch N≥

π(m) = {m} ∪ [m + x]T .

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We have diam(m ∪ (m + x)) ≤ x + dM , hence diam(π(m)) ≤ x + 3dM . By Deﬁnition 2.1 every ball of radius C2LR ( x + 3dM ) contains a translate of π(m). As x = 0 then x ≥ η, as having both P ⊂ T and P + x ⊂ T is impossible (because a tile in P and its translate by a vector less than η in norm will have intersecting interiors). By assumption and (3.1), r ≥ N x ≥

CLR CLR (3η −1 dM + 1) x ≥ ( x + 3dM ). 2 2

Since S(P ) contains a ball of radius r, it contains a ball of radius C2LR ( x + 3dM ), hence a translated copy of π(m), call it π(m) + g ⊂ P . This patch contains the tile m = m + g. We have m ∈ P , so m + x ∈ T . Moreover, m + x ∈ π(m) + g since m + x = m + x + g ⊂ S(π(m)) + g. It follows that m + x = (m + x) − g ∈ π(m) ⊂ T . Since m was arbitrary x is a period of T . But XT0 is aperiodic, then x = 0. This contradicts the fact that x = 0.

Lemma 3.2. Let (XT0 , Rd ) be an aperiodic LRTDS and P a patch in T0 then the following holds r(P ) ≥

rin (P ) 2N

R(P ) ≤ CLR Rout (P ),

and

where N is the constant from Lemma 3.1. Proof. Let T ∈ XT0 be arbitrary and q1 = q2 such that there exist P1 , P2 ⊂ T with P1 = P + q1 and P2 = P + q2 so P2 = P1 − q1 + q2 and since P contains a ball of radius rin (P ) by Lemma 3.1 we obtain that q1 − q2 ≥

rin (P ) . N

As T was arbitrary then r(P ) ≥

rin (P ) . 2N

As XT0 is linearly repetitive every ball of radius R = CLR Rout (P ) contains a translate of P , hence NPT (B(y, R)) ≥ 1, for all y ∈ Rd . As T was arbitrary, R(P ) ≤ CLR Rout (P ). ˆ be a vertexpattern Theorem 3.1. Let (XT0 , Rd ) be an aperiodic LRTDS, and X that occurs in XT0 . Then there exists a positive number nXˆ such that for every van Hove sequence (Qn )n∈N and every tiling T ∈ XT0 , nXˆ = lim

n→∞

T NX ˆ (Qn )

|Qn |

.

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T ˆ Here NX ˆ (Qn ) denotes the number of representatives of X in T that are contained in Qn .

Proof. As discussed above (XT0 , Rd ) is strictly ergodic. Now the result is standard (see e.g. [10]). Remark 3.1. Naturally, Theorem 3.1 remains true if the word “vertexpattern” is replaced by “tilepattern” or by “patch”. Definition 3.1 (Frequency). Let (XT0 , Rd ) be an aperiodic LRTDS. Then nXˆ ˆ (respectively, P ). (respectively, nP ) is called the frequency of X Lemma 3.3. For α > 0 let P(XT0 , α) denote the set of patches P occurring in XT0 such that rin (P )/Rout (P ) ≥ α. For each α > 0 there exists C(α) > 0 such that for every P ∈ P(XT0 , α) the following inequalities hold : nP |P | ≥ nP |B(0, rin (P ))| ≥ C(α). Proof. From deﬁnition of nP we can take a sequence of cubes (Qn ) such that the sidelength of Qn is 2n CLR Rout (P ). Partitioning each side of Qn into n parts of equal length, the cube Qn can be decomposed into nd cubes of sidelength 2 CLR Rout (P ). By the linear repetitivity each of these cubes contains a patch equivalent to P . Combining this we have nd |B(0, 1)|rin (P )d |B(0, 1)|αd NPT (Qn ) |P | ≥ ≥ = C(α). d |Qn | (2n CLR Rout (P )) (2CLR )d Taking the limit n → ∞ gives us the desired result. ˆ be a vertexpatProposition 3.1. Let (XT0 , Rd ) be an aperiodic LRTDS, and let X tern that occurs in XT0 . Then for any δ > 0, there exists c > 0 such that for every cube-like sequence (Qn )n∈N with parameter δ and every tiling T ∈ XT0 T NX ˆ (Qn )

|Qn |

≤ c nXˆ .

Proof. We ﬁrst prove that there exists c > 0 such that our statement holds for ˆ any cube. Consider a cube Q of sidelength l(Q) containing S representatives of X. Then, by linear repetitvity any cube of sidelength CLR l(Q) contains a translate of Q ˆ Considering a sequence of cubes and hence contains at least S representatives of X. d n) (Cn )n∈N with l(Cn ) → ∞ when n → ∞, each Cn contains at least E Cl(C LR l(Q) disjoint cubes of sidelength l(Q), where E(x) denotes the greatest integer smaller than x. Hence we have d l(Cn ) d E S T NXˆ (Cn ) 1 1 CLR l(Q) T ≥ − ≥ NX ˆ (Q). |Cn | l(Cn )d CLR l(Q) l(Cn )

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Theorem 3.1 applied to the sequence (Cn ) gives us nXˆ ≥

T NX ˆ (Q)

d |Q| CLR

T NX ˆ (Q)

=

c |Q|

.

Now for a cube-like sequence (Qn ) there is δ > 0 and a sequence of cubes (Cn ) such n| that Qn ⊂ Cn and |Q |Cn | ≥ δ for all n ∈ N. Hence we obtain T NX ˆ (Qn )

|Qn |

≤

T NX ˆ (Cn ) |Cn |

|Cn |

|Qn |

Consequently, the statement follows with c =

≤

c n ˆ. δ X

c δ.

4. Derived Vorono˘ı Tiling In this section we discuss a way of decomposing tilings into big pieces. Basic ideas are taken from [15] (which in turn is a higher dimensional analogue of the symbolic dynamic case considered in [11]). Let (XT0 , Rd ) be a minimal tiling dynamical system with FLC. A patch P is called central if it contains the origin in its interior. Fixing a central nonempty patch P in T0 , we deﬁne the locator set LP (T ) by LP (T ) = {q ∈ Rd | there exists P ⊂ T with P = P − q}. The elements of this set are the locator points of equivalent copies of P in the tiling T ∈ XT0 . As noted in Deﬁnition 2.2 LP (T ) forms a Delone set. This is exactly the type of set for which it is possible to form a normal Vorono˘ı tesselation, see e.g. [17]. The Vorono˘ı cell for q ∈ LP (T ) is given by Vq = {x ∈ Rd | x − q ≤ x − q for all q ∈ LP (T )}. Vq is the intersection of a ﬁnite number of closed half-spaces, so Vq is a convex polytope. The following lemma gives us an approximation of the volume of the Vorono˘ı cells (see [17, Corollary 5.2] for related result). Lemma 4.1. Let (XT0 , Rd ) be a minimal tiling dynamical system with ﬁnite local complexity, and let P be a central patch in T0 . Let R(P ) and r(P ) deﬁned as above, then one has for T ∈ XT0 and any q ∈ LP (T ), B(q, r(P )) ⊂ Vq ⊂ B(q, R(P )). Thus all points in LP (T ) which are neighbors of q (i.e. their Vorono˘ı cells share edges with Vq ) are contained in B(q, 2R(P )). Proof. Let q ∈ LP (T ) and x ∈ Vq so that x − q ≤ x − q for all q ∈ LP (T ). Suppose that x − q > R(P ), then x − q > R(P ), so there are no translates of P in B(x, R(P )). This contradicts the deﬁnition of R(P ) and shows that for all x ∈ Vq , x − q ≤ R(P ) which implies that S(Vq ) ⊂ B(q, R(P )).

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For the second inclusion, let x ∈ B(q, r(P )) and q ∈ LP (T ), q = q. So we have x − q = x − q − (q − q) ≥ q − q − x − q ≥ 2r(P ) − r(P ) = r(P ) ≥ x − q . As q was arbitrary x must belong to Vq . Lemma 4.1 ensures that all locator points neighboring q in LP (T ) appear in a ball of radius 2R(P ), so the equivalence class of B(q, 2R(P )), and thus on the equivalence class of the patch [B(q, R)]T for R = 2R(P ) + Rout (P ). As the tiling dynamical system considered here has FLC, the set of equivalence classes in T of the form P = [B(q, R)]T , with R = 2R(P ) + Rout (P ) and q ∈ LP (T ) is ﬁnite. 1 (T ), . . . , P Denote its cardinality by N (P ) and its elements by P N (P ) (T ) (notice that these elements are the same for all T ∈ XT0 ). Definition 4.1 (Derived Vorono˘ı Tiling). Let (XT0 , Rd ) be a minimal tiling dynamical system with FLC, P a central patch in T0 and R = 2R(P ) + Rout (P ). For any q ∈ LP (T ) with T ∈ XT0 , the return tile mq is deﬁned to be the pair mq = (Vq , l(Vq )) where l(Vq ) = i if [B(q, R)]T ∈ P˜i (T ). S(mq ) = Vq denotes the support of mq and l(Vq ) its marking. The derived Vorono˘ı tiling for the patch P ∈ T is given by TP (R) = {mq | q ∈ LP (T )}. It is clear from the deﬁnition of R that if l(Vq ) = l(Vq ) for certain q, q ∈ LP (T ), then there is a translation that maps Vq into Vq . In this case mq and mq are equivalent. So the set of prototiles in TP (R) is ﬁnite. Denote its ele˜ N (P ) (T ). Similarly the equivalence class of the TP (R)-corona ments by m ˜ 1 (T ), . . . , m [mq ]TP (R) , q ∈ LP (T ) only depends on the tilepattern of the patch [B(q, 4R(P ) + Rout (P ))]T . As XT0 has ﬁnite local complexity, the set of equivalence classes of the TP (R)-corona {[mq ]TP (R) , q ∈ LP (T )} is also ﬁnite. Denote its cardinality by J(P ) and its elements by C˜1 (T ), . . . , C˜J(P ) (T ) (notice that these elements are the same for all T ∈ XT0 ). A collection of sets is a partition of Rd if these sets are pairwise disjoint and their union is the whole space Rd . The collection TP = {Vq | q ∈ LP (T )} is a tiling of Rd however it is not a partition of Rd . Hence for the remainder of this paper it is desirable to associate to every return tile mq = (Vq , l(Vq )) a tile Mq = (M (Vq ), l(Vq )) such that the interiors of M (Vq ) and Vq coincide, and the collection {M (Vq ) | q ∈ LP (T )} is a partition of Rd . We follow [4] to construct such a partition, the details are explained in the next proposition. Proposition 4.1. Under hypotheses of Deﬁnition 4.1, for every return tile mq there is a subset M (Vq ) ⊂ Vq such that (i) the interiors of Vq and M (Vq ) coincide; (ii) the collection {M (Vq )|q ∈ LP (T )} is a partition of Rd ;

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(iii) if mq and mq are equivalent then M (Vq ) and M (Vq ) are translates of each other. Proof. We will just explain how to construct the set M (Vq ). For the proof of statements (i)–(iii) we refer the reader to [4, Proposition 14]. Since the number of prototiles in TP (R) which are polytopes is ﬁnite, all the (d − 1)-dimensional faces of tiles in TP (R) are parallel to a ﬁnite number of hyperplanes h1 , . . . , hn . Assume that all hyperplanes pass through 0. Each hyperplane separates the Euclidean space in two open half-spaces. For every hyperplane hi we choose one of the two open half-spaces, denoted by Hi , in such a way that Q0 = n i=1 Hi = ∅. We deﬁne M (Vq ) by M (Vq ) = {x ∈ Vq | Vq ∩ Qx = ∅}. We deﬁne the tiling corresponding to the partition constructed in Proposition 4.1 by M(TP (R)) = {Mq = (M (Vq ), l(Vq )) | q ∈ LP (T )}. Remark 4.1. By the statement (iii) of Proposition 4.1 equivalent tiles in TP (R) give rise to equivalent tiles in M(TP (R)). The same holds for equivalent patches. ˜ N (P ) (T ) will denote the proHence for the remainder of this paper m ˜ 1 (T ), . . . , m totiles in M(TP (R)), and C˜1 (T ), . . . , C˜J(P ) (T ) will denote the equivalence classes of the M(TP (R))-corona. Proposition 4.2. Let (XT0 , Rd ) be an aperiodic LRTDS. Let (Pk )k∈N the sequence of patches in T0 deﬁned by Pk = [B(0, k)]T0 for all k ∈ N\{0}. Then for every k ∈ N\{0} we have k , (i) r(Pk ) ≥ 2N (ii) R(Pk ) ≤ C1 k with C1 = CLR (1 + dM ), (iii) C2 k ≥ Rk = 2R(Pk ) + Rout (Pk ) with C2 = (2CLR + 1)(1 + dM ).

Here N is the constant from Lemma 3.1, and dM denotes the maximal diameter of tiles in XT0 . Proof. Remark that rin (Pk ) ≥ k and Rout (Pk ) ≤ k + dM . So Lemma 3.2 gives us (i) and (ii). Moreover (iii) follows immediately from (ii). Definition 4.2. Under assumptions of Proposition 4.2 and for T ∈ XT0 we denote k by {m ˜ k1 (T ), . . . , m ˜ kN (k) (T )} and by {C˜1k (T ), . . . , C˜J(k) (T )} the set of possibly prototiles in M(TPk (Rk )) and the set of possibly equivalence classes of M(TPk (Rk ))corona, respectively. 5. Ergodic Theorems In this section, we prove ergodic (sub)additive theorems using ideas from the last section and methods developed in [4, 12]. For related results, we refer the reader

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to [2, 12]. However, our setting is somewhat diﬀerent, our functions are deﬁned on the whole space instead of on patterns and satisfying a rather weak form of asymptotic translation invariance (conditions (), (∗) and (∗∗) below). We start with an additive theorem. First, we need the following deﬁnition. Definition 5.1. Let (B, · ) be a Banach space. A function F : B(Rd ) → B is called additive if there exist a function b: B(Rd ) → R+ and a constant dF > 0 such that n n n (A1 ) F i=1 Qi − i=1 F (Qi ) ≤ i=1 b(Qi ), where the Qi are disjoint. (A2 ) F (Q) ≤ dF |Q| + b(Q) for all Q ∈ B(Rd ). (A3 ) There exists hb > 0, Cb > 0 such that b(Q) ≤ Cb |Qhb \Qhb | for all Q ∈ B(Rd ). Now we state our additive theorem. Theorem 5.1. Let (XT0 , Rd ) be an aperiodic LRTDS. Let T ∈ XT0 and an additive function F : B(Rd ) → B satisfying max

lim

k→∞ 1≤i≤N (k)

where

V (i, k) =

sup M,M ∈m ˜k i (T )

V (i, k) = 0

()

F (S(M )) − F (S(M )) . |S(M )|

Then the limit lim

n→∞

F (Qn ) |Qn |

exists for any van Hove sequence (Qn )n∈N . Proof. The proof is a variant of the proof of [12, Theorem 1]. Thus we only sketch ˜ ki (T ) deﬁned it. For k ∈ N\{0} and i = 1, . . . , N (k) denote by nki the frequency of m k (Qn )/|Qn | for every van Hove sequence (Qn )n∈N . Here by ni = limn→∞ Nm ˜k i (T ) Nm k (Q ) is the number of occurrences of m ˜ ki (T ) in Qn and the existence of the n ˜ i (T ) frequency is guaranteed by the strict ergodicity. Deﬁne ˜ ki (T )}, Fik = sup{F (S(M )) | M ∈ m

N (k)

Fk =

nki Fik ,

i=1

c(k) =

max

1≤i≤N (k)

b(S(M )) k M ∈m ˜ i (T ) . |S(M )|

Let (Qn )n∈N be a van Hove sequence. Following the proof of [12, Theorem 1], for n ∈ N and k ∈ N\{0} we have

F (Qn ) k

− F

≤ D1 (n, k) + D2 (n, k),

|Qn |

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where D1 and D2 have the property that for arbitrary ε > 0, there exist k(ε) > 0 and n0 ∈ N such that ε for k ≥ k(ε) and n ≥ n0 , D1 (n, k) ≤ 2 and by the deﬁnition of F k , the addivity of F and the assumption () there exist k (ε) > 0 and n1 ≥ n0 such that D2 (n, k) ≤

ε 2

for k ≥ k (ε)

and n ≥ n1 .

Combining all these together we infer that for k ≥ max(k(ε), k (ε)) and n ≥ n1

F (Qn )

k

|Qn | − F ≤ D1 (n, k) + D2 (n, k) ≤ ε. (Qn ) As ε was arbitrary this implies that F|Q is a Cauchy sequence. As B is a n∈N n| Banach space, we have the desired result. We can now heading towards an ergodic theorem for subadditive functions. To do so we need some preparations. Definition 5.2. A function F : B(Rd ) → R is called subadditive if there exists a constant dF > 0 such that for any Q, Q ∈ B(Rd ): • Q ∩ Q = ∅ ⇒ F (Q ∪ Q ) ≤ F (Q) + F (Q ). • F (Q) ≤ dF |Q|. Lemma 5.1. Let (XT0 , Rd ) be an aperiodic LRTDS. For each α > 0 there exists a constant C (α) > 0 such that for every return tile MV = (M (V ), l(V )) of some P ∈ P(XT0 , α) with frequency nMV the next inequality holds nMV |V | ≥ C (α). Proof. For P ∈ P(XT0 , α) and MV a return tile of P . Note that the occurring of MV only depends on patch P of diameter D = 4R(P ) + 2Rout (P ) and by Lemma 3.2, D ≤ (4CLR + 2)Rout (P ). On the other hand, by Lemmas 3.2 and 4.1, (P ) (P ) . Hence nMV |V | ≥ nP B(0, rin2N ) . As we have rin (V ) ≥ rin2N rin (P ) α ≥ , 2N Rout (P ) 2N (2 CLR + 1) Lemma 3.3 gives us the desired result. Lemma 5.2. Let (XT0 , Rd ) be an aperiodic LRTDS and T ∈ XT0 . For all k ∈ N, k ≥ 1 and 1 ≤ i ≤ N (k), deﬁne ˜ ki (T ))|, pki = nki |S(m

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where nki = limn→∞ Nm (Qn )/|Qn | for an arbitrary van Hove sequence (Qn ). ˜k i (T ) Then

N (k)

pki = 1

(5.1)

i=1

and there exists a constant C > 0 such that lim inf pki(k) ≥ C

(5.2)

k→∞

for any sequence (i(k))k with 1 ≤ i(k) ≤ N (k). Proof. Notice that |S(m ˜ ki (T ))| = |S(M )| for each M ∈ m ˜ ki (T ). Let (Qn )n∈N be a van Hove sequence and k ∈ N\{0} be arbitrary. Qn can be decomposed into disjoint sets in the following way   (k) N  Qn =   i=1

M∈m ˜k i (T ) S(M)⊂Qn

  k) S(M ) ∪ S with S ⊂ (Q2R(P \Qn,2R(Pk ) ). n 

So,

N (k)

i=1 M∈m ˜k i (T ) S(M)⊂Qn

As limn→∞

|S| |Qn |

N (k) Nm (Qn ) |S| |S(M )| |S| ˜k i (T ) + = |S(m ˜ ki (T ))| + = 1. |Qn | |Qn | |Q | |Q n n| i=1

= 0 then Nm (Qn ) ˜k i (T )

n→∞

i=1

|Qn |

N (k)

N (k)

lim

|S(m ˜ ki (T ))| =

pki = 1.

i=1

m ˜ ki (T )

is the equivalence class of some return tile of Let us now turn to (5.2). As the patch Pk = [B(0, k)]T0 . As rin (Pk )/Rout (Pk ) ≥ k/(k + dM ), remark that there is some α0 > 0 satisfying k/(k + dM ) ≥ α0 for all k ∈ N\{0}. Lemma 5.1 gives us the desired result. Definition 5.3. Let (XT0 , Rd ) be an aperiodic LRTDS. For T ∈ XT0 deﬁne

F (S(M )) (k) k Fi = sup (T ) , M ∈ m ˜ i |S(M )|

N (k)

F (k) =

(k)

Fi pki ,

i=1

F¯ =

inf

k∈N\{0}

F (k) .

Remark 5.1. Here F¯ = −∞ is possible.

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Lemma 5.3. Let F : B(Rd ) → R be a subadditive function, then for every van Hove sequence (Qn )n∈N we have lim sup n→∞

F (Qn ) ≤ F¯ . |Qn |

Proof. The proof is an adaptation of the proof of [4, Lemma 2] to our context. Choose k ∈ N\{0}, Qn can be decomposed into disjoint sets in the following way   (k) N  Qn =   i=1

  S(M ) ∪ S 

M∈m ˜k i (T ) S(M)⊂Qn

with

k) \Qn,2R(Pk ) ). S ⊂ (Q2R(P n

So by subadditivity of F , we infer N (k) F (Qn ) ≤ |Qn | i=1

M∈m ˜k i (T ) S(M)⊂Qn

N (k)

≤

F (S(M )) |S(M )| F (S) + |S(M )| |Qn | |Qn |

(Qn ) ˜k (k) Nm i (T )

Fi

|Qn |

i=1 |S| limn→∞ |Q n|

As = 0, we have lim supn→∞ have the desired result.

|S(m ˜ ki (T ))| + dF

F (Qn ) |Qn |

|S| . |Qn |

≤ F (k) . As k was arbitrary we

Lemma 5.4. Let (XT0 , Rd ) be an aperiodic LRTDS, T ∈ XT0 and F a subadditive function. Let (i(n)) and (k(n)) be sequences with k(n) → ∞ when n → ∞ and i(n) ∈ {1, . . . , N (k(n))}, n ∈ N. Then (k(n))

lim Fi(n)

n→∞

= F¯ .

Proof. As the previous Lemma a simple adaptation of the proof of [4, Lemma 1] to our context gives us the result. See also [11, Lemma 4.4]. Lemma 5.5. Let (Qn ) and (Qn ) be van Hove sequences satisfying (i) Qn ⊂ Qn , (ii) there exists δ > 0 such that (iii) limn→∞ F (Qn ) = F¯ .

|Qn | |Qn |

≥ δ,

|Qn |

Then we have limn→∞

F (Qn ) |Qn |

= F¯ .

Proof. The proof is a simple adaptation of the proof of the Lemma 3 in [4] to our context. See also [11, Lemma 4.5].

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Theorem 5.2. Let (XT0 , Rd ) be an aperiodic LRTDS, T ∈ XT0 . Assume that F is subadditive and satisﬁes

F (S(M )) − F (S(M )) k lim max sup ˜ i (T ) = 0, (∗) M, M ∈ m k→∞ 1≤i≤N (k) |S(M )|

F (S(C)) − F (S(C )) k ˜ lim max sup (∗∗) C, C ∈ Ci (T ) = 0. k→∞ 1≤i≤J(k) |S(C)| Then limn→∞

F (Qn ) |Qn |

= F¯ holds for every cube-like sequence (Qn )n∈N .

See Deﬁnition 2.4 for the deﬁnition of cube-like sequence. Proof. As in [4], Theorem 5.2 will be proven in three steps: (a) The statement holds for sequence (Qn ) in which Qn is a support of a tile in M(TPk(n) (Rk(n) )), with k(n) → ∞ when n → ∞. (b) The statement holds for sequences (Qn ) in which Qn is the support of a corona in M(TPk(n) (Rk(n) )), with k(n) → ∞ when n → ∞. (c) The statement holds for every cube-like sequence (Qn ). Step (a). By Lemma 5.4 and assumption (∗) the statement holds for sequences (Qn ) in which Qn is the support of a tile in M(TPk(n) (Rk(n) )). Step (b). Let (Qn )n∈N be a sequence such that for all n ∈ N, Qn = S(Cn ) with Cn a corona in M(TPk(n) (Rk(n) )), k(n) → ∞ when n → ∞. Let k = 2N CLR (2C2 + dM )k(n), choose an arbitrary Mn ∈ M(TPk (Rk )) then k ≥ CLR (4R(Pk(n) ) + Rout (Pk(n) ) + dM ) 2N ≥ CLR Rout ([B(q, 4R(Pk(n) ) + Rout (Pk(n) ))]T ).

rin (S(M )) ≥

Then by linear repetitivity, there is a corona C˜n equivalent to Cn in M(TPk(n) (Rk(n) )) with S(C˜n ) ⊂ S(Mn ). By Lemma 4.1 and Proposition 4.2, we can estimate |S(C˜n )| |B(0, k(n)/2N )| 1 ≥ ≥ = δ > 0. |S(Mn )| |B(0, C1 k )| (4N 2 C1 (2C2 + dM ))d By step (a) the statement holds for (S(Mn ))n∈N and by Lemma 5.5 |F (S(C˜n ))| = F¯ . n→∞ |S(C˜n )| lim

Now assumption (∗∗) gives us |F (Qn )| |F (S(C˜n ))| = lim = F¯ . n→∞ n→∞ |Qn | |S(C˜n )| lim

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Step (c). Let (Qn )n∈N be a cube-like sequence. There exists δ > 0 and a sequence of cubes (Qn )n∈N such that for all n ∈ N Qn ⊂ Qn

and

|Qn | ≥ δ. |Qn |

It is easy to see that for all k ∈ N and for all x ∈ Rd the ball B(x, k/2N ) must be contained in the corona of some tile in M(TPk (Rk )). Now let n ∈ N arbitrary. For the cube Qn there exists k ∈ N such that Qn is not contained in the support of any corona of tile in M(TPk (Rk )) and it is contained in the support of a corona Cn of some tile in M(TPk+1 (Rk+1 )). So Rout (Qn ) must be greater than k/2N . Remarking that a cube C in Rd satisﬁes Rout (C)/rin (C) = c(d) > 0 we ﬁnd rin (Qn ) ≥ k/2N c(d). By Lemma 4.1 and Proposition 4.2, we can estimate |Qn | |B(0, k/2Nc(d))| 1 ≥ ≥ . |S(Cn )| |B(0, 2C1 (k + 1))| (2Nc(d)(2C1 + 1))d Finally we have Qn ⊂ S(Cn ) and δ |Qn | ≥ = δ. |S(Cn )| (2Nc(d)(2C1 + 1))d As limn→∞

F (S(Cn )) |S(Cn )|

= F¯ by step (b), we have the result by Lemma 5.5.

6. Lattice Models In this section we consider some applications of the ergodic theorems from the preceeding section. Following [4], we study lattice gas models on aperiodic linearly repetitive tilings, we will be concerned with existence of thermodynamic quantities (pressure, mean energy, entropy) and their relationship. We start by introducing the constituents of lattice models and investigate existence of thermodynamic limits. First, we ﬁx some notations. Let E be a compact metric space, ξ its Borel σ-algebra, and λ a probability measure on E. The space (E, ξ) is called the single spin space and λ is called the a priori measure. With every vertex x in T we associate a copy of (E, ξ, λ) denoted by (Ex , ξx , λx ). For Q ∈ B(Rd ) or Q = Rd , we deﬁne E Q as the product space x∈Q∗ Ex . This space is equipped with the product topology, the Borel σ algebra ξ Q and the product measure λQ = x∈Q∗ λx . Integration with respect to λQ is denoted by · Q . There is a canonical projection σQ : E R → E Q given by σQ ((ux )) = (ux )x∈Q∗ . The set of continuous functions on E Q is denoted by C(E Q ) and it is equipped d ∗ with the supremum norm · ∞ . σQ induces a canonical embedding σQ : C(E R ) → ∗ (f ) = f ◦ σQ . In the sequel we tacitly identify C(E Q ) with a C(E Q ) given by σQ d

subspace of C(E R ). Now for (XT0 , Rd ) an aperiodic LRTDS, ﬁx T a tiling belonging to XT0 . The spaces C(E Q ) and C(E Q ) can be identiﬁed in a natural way if Q∗ ≈ Q∗ . If after this identiﬁcation, A ∈ C(E Q ) and B ∈ C(E Q ) are equal, we write A ∼ B. d

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An interaction is a function that assigns to every ﬁnite set of vertices X a function in C(E X ). An interaction Φ is called vertexpattern invariant if Φ(X) ∼ Φ(X ) whenever X ≈ X . An interaction is said to be of ﬁnite range if there exists D > 0 such that Φ(X) = 0 if the diameter of X exceeds D. The inﬁnimum over these D is called the range of Φ. The space of ﬁnite range, vertexpattern invariant interactions is denoted by B0 . Denote by B, B∼ and Bs the spaces of vertexpattern invariant interactions with respectively ˆ ∞ < ∞, nXˆ Φ(X) Φ = ˆ ϕ X∈ ˆ

Φ ∼ =

ˆ ˆ ∞ < ∞, #(X) Φ( X)

ˆ ϕ X∈ ˆ

Φ(X) ∞ < ∞. #(X) x∈L

Φ s = sup

X x∈X

Let L be the set of all vertices in T . We know that B0 is dense in B, also B0 ⊂ Bs , but we do not know if B0 is · s -dense in Bs . It is well known that B∼ ⊂ Bs ⊂ B, see [4, Proposition 19] for further details. The Hamiltonian for an interaction Φ in a bounded Q ⊂ Rd is the function on Q E deﬁned by Φ = Φ(X). HQ X⊂Q∗ Φ For u ∈ E Q , the quantity HQ (u) is called the energy of u. If Φ is vertexpattern Φ Φ ∗ ∗ invariant interaction, HQ ∼ HQ if Q ≈ Q . Φ ∞ viz For Φ ∈ Bs there is a useful estimate on HQ Φ HQ ∞ ≤

X⊂Q∗

Φ(X) ∞ =

Φ(X) ∞ ≤ #(Q∗ ) Φ s . #(X) ∗ ∗

(6.1)

x∈Q X⊂Q x∈X

The pressure PQ (Φ) of an interaction Φ in a bounded Q ⊂ Rd is deﬁned by Φ Q ) . PQ (Φ) = logexp(−HQ

The set of all probability measure on E R is denoted by P(E R ). For ρ ∈ P(E R ) −1 A) for and Q ∈ B(Rd ), the restriction ρQ of ρ to Q is deﬁned by ρQ = ρ(σQ all A ∈ ξ Q . If this measure is absolutely continuous with respect to λQ , we write ρQ λQ , and its Radon–Nikodym density will be denoted by ρ(Q) . d The entropy SQ (ρ) of ρ ∈ P(E R ) in a bounded Q ⊂ Rd is deﬁned by  if Q∗ = ∅, 0 SQ (ρ) = −ρ(Q) log ρ(Q) Q if ρQ λQ ,  −∞ otherwise. d

d

d

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Here we set x log x = 0 for x = 0. Finally, we will deﬁne the set of probability measures for which we can prove the existence of the mean entropy and the mean energy. To proceed, we will need to introduce the distance in variation between measures ρ and µ on E Q ρ − µ = sup ρ(A) − µ(A) . A⊂ξ Q

Definition 6.1. The set PB (E R ) of balanced probability measures on E R consists d of all elements ρ ∈ P(E R ) satisfying conditions (B1) and (B2): d

d

(B1) For every sequence (Qn ), in which Qn is a tile or a corona in M(TPk(n) (Rk(n) )) with k(n) → ∞ when n → ∞: lim

sup ρS(Q) − ρS(Qn ) = 0.

n→∞ Q Qn

(B2) The measure ρ is absolutely continuous with respect to λR and there exist constants K > 0 and K > 1 such that for every sequence (Qn ) as in (B1), there exists hρ ≥ 0 such that d

sup |χA(Q) ρ(S(Q)) log ρ(S(Q)) S(Q) | < K

Q Qn

holds, where A(Q) = {u ∈ E S(Q) |ρ(S(Qn )) (u) > K #(S(Q) ρ

(S(Q))

(u) > K

#(S(Q)hρ )∗

hρ ∗ )

or

}.

In this deﬁnition we have tacitly identiﬁed the two spaces E S(Q) and E S(Qn ) . Proposition 6.1. For every Φ ∈ B and every cube-like sequence (Qn )n∈N , 1 T ˆ ∞. Φ = lim NXˆ (Qn ) Φ(X) n→∞ |Qn | ˆ ϕ X∈ ˆ

Proof. The result is obvious for ﬁnite range interactions. For a given cube-like sequence (Qn )n∈N , by Proposition 3.1, there exists a constant c > 0 such that for all Φ, Ψ ∈ B and for all n ∈ N, 1 T T ˆ ˆ NXˆ (Qn ) Φ(X) ∞ − NXˆ (Qn ) Ψ(X) ∞ ≤ c Φ − Ψ . |Qn | ˆ ϕ ˆ ϕ X∈ ˆ X∈ ˆ As B0 is dense in B we are done. Proposition 6.2. For arbitrary interactions Φ and Ψ and all bounded Q subsets of Rd we have the estimate Φ Ψ |PQ (Φ) − PQ (Ψ)| ≤ HQ − HQ ∞ .

Proof. Follows from [7, Lemma I.2.2].

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We can now prove existence of the mean pressure. Theorem 6.1 (Mean Pressure). For every Φ ∈ B there exists P(Φ) such that PQn (Φ) → P(Φ) |Qn | for every cube-like sequence (Qn ). Moreover, P: B → R is convex and satisﬁes P(Φ) − P(Ψ) ≤ Φ − Ψ . Proof. This can be proven as [4, Theorem 3]. We only give a brief sketch: For Φ ∈ B0 the function deﬁned by F (Q) = PQ (Φ) is additive. Moreover, it obviously satisﬁes () of Theorem 5.1, as PQ (Φ) only depends on the equivalence class of [Q]T . This implies existence of the limit for these Φ by Theorem 5.1. For arbitrary Φ ∈ B, as (Qn )n∈N is a cube-like-sequence, Propositions 3.1 and 6.2 give Φ Φ |Qn |−1 |PQn (Φ) − PQn (Ψ)| ≤ |Qn |−1 HQ − HQ n n ∞ T ˆ ˆ ≤ |Qn |−1 NX ˆ (Qn ) Φ(X) − Ψ(X) ∞

≤c

ˆ ϕ X∈ ˆ

ˆ − Ψ(X) ˆ ∞ nXˆ Φ(X)

ˆ ϕ X∈ ˆ

≤ c Φ − Ψ . As B0 is dense in B, this estimate and the fact that the theorem holds for arbitrary Φs in B0 ensure us the existence of the limit for all Φ ∈ B. older inequality. As P(Φ) is the limit The convexity of PQ : B0 → R follows by H¨ of |Qn |−1 PQn (Φ), P is also convex. Finally, taking the limit n → ∞ in ˆ − Ψ(X) ˆ ∞ |Qn |−1 |PQn (Φ) − PQn (Ψ)| ≤ |Qn |−1 N Tˆ (Qn ) Φ(X) X

ˆ ϕ X∈ ˆ

gives the continuity of P. Now, we will prove existence of the mean energy. Theorem 6.2 (Mean Energy). For Φ ∈ B, ρ ∈ PB (E R ) there exists eΦ (ρ) such that d

Φ ) ρ(HQ n → eΦ (ρ) |Qn |

for every cube-like sequence (Qn )n∈N . Proof. This can be proven as the corresponding part of [4, Theorem 4]: For Φ ∈ B0 d Φ and ρ ∈ P(E R ), the function deﬁned by F (Q) = ρ(HQ ) is additive. Moreover, using (B1) of Deﬁnition 6.1 it satisﬁes () of Theorem 5.1. Thus, existence of the limit follows for these Φ from Theorem 5.1.

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Existence of the limit for arbitrary Φ ∈ B follows again by a density argument. As (Qn )n∈N is a cube-like sequence, we have for arbitrary Φ ∈ B by Proposition 6.1 lim sup n→∞

Φ Ψ Φ Ψ |ρ(HQ ) − ρ(HQ )| HQ − HQ n n n n ∞ ≤ lim sup |Qn | |Qn | n→∞ ≤ Φ − Ψ .

As B0 is dense in B, the limit exists for all Φ ∈ B. Now, we will show existence of the mean entropy. Theorem 6.3 (Mean Entropy). For ρ ∈ PB (E R ), there exists s(ρ) such that d

SQn (ρ) → s(ρ) |Qn | for every cube-like sequence (Qn ). Moreover, s: PB (E R ) → R is aﬃne. d

Proof. As in the corresponding part of [4, Theorem 4], it can be shown that the function deﬁned by F (Q) = SQ (ρ) is subadditive. Moreover, using (B1) and (B2) of Deﬁnition 6.1 it can be shown to satisfy (∗) and (∗∗) of Theorem 5.2. This Theorem gives us existence of the limit. Aﬃnity follows by standard arguments (see [7, Theorem II.2.3]). We close this section with a variational principle. Before stating the theorem, we introduce the notion of Gibbs measure. Definition 6.2 (Gibbs Measure). For every bounded Q ⊂ Rd and every interQ Q action Φ, let ρΦ Q be the probability measure on (E , ξ ) that has Radon–Nikodym Φ Φ Q )/exp(−HQ ) relative to λQ . This measure is called the Gibbs density exp(−HQ measure for Φ in Q. The importance of the Gibbs measure derives from the following proposition. Proposition 6.3. For every bounded Q ⊂ Rd and every interaction Φ the following inequality Φ ) PQ (Φ) ≥ SQ (ρ) − ρ(HQ

holds for every probability measure ρ ∈ P(E R ). The equality holds if and only if ρ = ρΦ Q. d

Proof. See [7, Sec. II.3]. Theorem 6.4 (Variational Principle). For every Φ ∈ B the following holds P(Φ) =

sup ρ∈PB

s(ρ) − eΦ (ρ).

(E Rd )

Proof. Proposition 6.3 implies that for all Φ ∈ B and all ρ ∈ PB (E R ) d

P(Φ) ≥ s(ρ) − eΦ (ρ).

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So it remains to show that P(Φ) is the smallest upper bound. Because P(Φ) and eΦ (ρ) are continuous in Φ, and the continuity of eΦ (ρ) is uniform in ρ, it suﬃces to prove the statement for Φ ∈ B0 . Let ε > 0 and Φ ∈ B0 . For k ∈ N\{0}, deﬁne ρk by ρk = ρΦ S(M) . M∈M(TPk (Rk ))

We ﬁrst show that ρk ∈ PB (E R ) for all k ∈ N\{0}. Let k ∈ N\{0} and let (Qn ) be a sequence as in (B1) of Deﬁnition 6.1. Then there exists n0 ∈ N such that k(n) ≥ 4C2 N k for all n ≥ n0 . Now by Proposition 4.2 d

R(Pk(n) ) ≥ r(Pk(n) ) ≥ 2C2 k ≥ 2Rk . M(TPk (Rk ))

[S(Qn )]M(TPk (Rk )) if Q Qn and n ≥ n0 . By the This gives us [S(Q)] deﬁnition of ρk we have ρk,S(Q) − ρk,S(Qn ) = 0 whenever n ≥ n0 . Thus, condition (B1) is satisﬁed. We next show (B2). For M ∈ M(TPk (Rk )) it is clear that ∗

Φ ∗ #S(M) ρΦ , S(M) (u) ≤ exp(2 HQ ∞ ) ≤ exp(2#S(M ) Φ s ≤ K

where K = exp(2 Φ s ). For a patch P ⊂ M(TPk (Rk )) we have ∗ ∗ ρΦ K #S(M) = K #S(P ) . ρk,S(P ) (u) = S(M) (σS(M) (u)) ≤ M∈P

M∈P

Now by Proposition 4.3 for all Q ∈ B(R ), S(P ) ⊂ Q2C1 k where P is deﬁned by P = [Q]M(TPk (Rk )) . Then we have ∗ 2C1 k)∗ ρk,Q (u) = ρk,S(P ) (u, u )dλS(P )\Q (u ) ≤ K #S(P ) ≤ K #(Q . d

E S(P )\Q

Hence for hρk = 2C1 k the set A(Q) in (B2) of Deﬁnition 6.1 is empty and the condition is satisﬁed. Let (Qn )n∈N be a cube-like sequence. Then by Theorems 6.1, 6.2 and 6.3 there exists n0 such that for all n ≥ n0 the three next inequalities hold: P(Φ) − PQn (Φ) ≤ ε, (6.2) |Qn | Φ ) ρk (HQ Φ n (6.3) ≤ ε, e (ρk ) − |Qn | s(ρk ) − SQn (ρk ) ≤ ε. (6.4) |Qn | Without loss of generality, we can choose Qn as a disjoint union of supports of elements on M(TPk (Rk )). So SS(M ) (ρΦ (6.5) SQn (ρk ) = S(M) ). M∈M(TPk (Rk )) S(M)⊂Qn

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On the other hand, from the proofs of Theorems 6.1 and 6.2 we have |Qn |−1 PQn (Φ) − PS(M) (Φ) ≤ c(k), M∈M(TPk (Rk )) S(M)⊂Qn −1 Φ Φ Φ |Qn | ρk (HQn ) − ρS(M) (HS(M) ) ≤ c(k), M∈M(TPk (Rk )) S(M)⊂Qn

(6.6)

(6.7)

where c(k) is the function introduced in the proof of Theorem 5.1. Since limk→∞ c(k) = 0 there exists k0 such that c(k) ≤ ε, for all k ≥ k0 . Combining (6.2) to (6.7) with the fact that Φ Φ M ∈ M(TPk (Rk )) ⇒ PS(M) (Φ) = SS(M ) (ρΦ S(M) ) − ρS(M) (HS(M) ),

we can see that |P(Φ) − s(ρk ) + eΦ (ρk )| ≤ 5ε, whenever k ≥ k0 . As ε was arbitrary, this proves the theorem. 7. Existence of Balanced Gibbs Measures In this section we show that Gibbs measures are balanced for interactions that are suﬃciently weak, i.e. which satisfy Dobrushin’s condition for uniqueness, and decay suﬃciently fast. This is relevant because for balanced Gibbs measures the supremum on the variational principle is attained. First we deﬁne the Gibbs measure, next we introduce some results from the litterature on Dobrushin’s uniqueness condition. Finally, we derive some auxiliary propositions and prove the theorem. Assume that Φ ∈ B∼ and for Q ∈ B(Rd ) denote it’s complementary in Rd by d c c Q . The Hamiltonian with boundary condition u ∈ E R is the function on E Q × E Q that for ζ ∈ E Q is deﬁned by Φ HQ (ζ | uQc ) = Φ(X)(ζ × uQc ). X:X∩Q∗ =∅

By (6.1) and the fact that Φ s ≤ Φ ∼ we have Φ HQ ( · | · ) ∞ = sup

Φ sup |HQ (ζ | uQc )| ≤ #Q∗ Φ ∼ .

c u∈E Rd ζ∈E Q

The partition function is the function on E R deﬁned by Φ Φ ZQ (u) = exp[−HQ (ζ | uQc )]dλQ (ζ). d

EQ

(7.1)

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It is clear by inequality (7.1) that the partition function is ﬁnite for all u. For d d d Φ ( · |u) on (E R , ξ R ) by u ∈ E R deﬁne a probablity measure γQ Φ Φ −1 Φ γQ (A | u) = ZQ (u) exp[−HQ (ζ | uQc )]χA (ζ × uQc )dλQ (ζ). EQ

It is called the Gibbs distribution in Q with boundary condition uQc . The collection Φ ’s with Q bounded and measurable is called the Gibbs speciﬁcation for Φ. For of γQ

Φ f by f ∈ C(E R ) we deﬁne a continuous function γQ Φ Φ γQ f (u) = f (ζ)dγQ (ζ | u). d

E Rd

By the proof of [5, Proposition 4.19] for Φ ∈ B0 and Ψ ∈ B∼ we have Φ−Ψ Φ Ψ γQ f − γQ f ∞ ≤ 2 f ∞ (exp HQ ( · | · ) ∞ − 1).

(7.2)

A measure µ on (E R , ξ R ) is called a Gibbs measure for Φ ∈ B∼ if Φ γQ (A | u)dµ(u), µ(A) = d

d

E Rd

for all A ∈ ξ R and all bounded Q. As the set E is complete and separable, a Gibbs measure exists for every Φ ∈ B∼ see [5, Theorem 4.23]. d A Gibbs measure µ for Φ ∈ B∼ is absolutely continuous with respect to λR . For every bounded Q and every A ∈ ξ Q the restriction of µ to Q is given by Φ −1 µQ (A) = γQ (σQ A | u)dµ(u) d

E Rd

(see Sec. 6 as well). Using (7.1) we show that the Radon–Nikodym density of µQ relative to λQ satisﬁes |µ(Q) | ≤ exp(2#Q∗ Φ ∼ ).

(7.3)

As above the set of all vertices in T will be denoted by L. |x − y| denotes the Euclidean distance between two points x and y, d(Q) the Euclidean diameter of a set Q and d(Q, Q ) the Euclidean distance between Q and Q . For α ≥ 0 deﬁne Bα as the Banach space of vertexpattern invariant interactions for which ˆ X) ˆ + 1)α Φ(X) ˆ ∞ < 0. #X(d( Φ α = ˆ X

It is clear that Φ ∼ ≤ Φ α , which implies that Bα ⊂ B∼ . We remark that for larger α the decay of Φ is quicker. Let γ be the Gibbs speciﬁcation for an interaction Φ ∈ B∼ . For i ∈ L and d u ∈ E R we deﬁne the probability measure on (E, ξ) by −1 A | u), γi0 (A | u) = γ{i} (σ{i}

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for all A ∈ ξ. Dobrushin’s interdependence matrix C(γ) = {Cij (γ)}i,j∈L is deﬁned by Cij (γ) =

sup

u,u ∈E L : uL\{j} =uL\{j}

γi0 ( · | u) − γi0 ( · | u ) ,

where · denotes the distance in variation. Deﬁne Cij (γ). c(γ) = sup i∈L j∈L

If c(γ) < 1 the speciﬁcation is said to satisfy Dobrushin’s condition. For such speciﬁcation deﬁne the matrix D(γ) = {Dij (γ)}i,j∈L by C n (γ), D(γ) = n≥0

this sum is ﬁnite. For Q, Q ⊂ Rd deﬁne D(Q, Q , γ) =

Dij (γ),

i∈Q∗ , j∈Qc∗

which is ﬁnite for bounded Q. The following proposition gives us the uniqueness of Gibbs measure for interactions satisfying Dobrushin’s condition. We refer the reader to [5, Theorem 8.23] for a proof and for references to original literature. Proposition 7.1. If the Gibbs speciﬁcation γ for Φ ∈ B∼ satisﬁes Dobrushin’s condition, then it admits a unique Gibbs measure µ and for bounded Q, Q ⊂ Rd we have sup A∈ξ Q ,

u∈E Rd

|γQ (A | u) − µ(A)| ≤ D(Q, Q , γ).

The next proposition gives us a suitable bound on D(Q, Q , γ) in terms of Q and Q . Proposition 7.2. Let Φ ∈ B∼ and suppose that γ, the Gibbs speciﬁcation of Φ, satisﬁes Dobrushin’s condition. Then if for some p > 0 |i − j|p Cij (γ) = 0, (7.4) lim sup R→∞ i∈L

j∈L: |i−j|>R

there exists a constant C > 0 such that for all bounded Q, Q ⊂ Rd D(Q, Q , γ) ≤ C #Q∗ d(Q, Qc )−p . Still assuming that Φ satisﬁes Dobrushin’s condition, Φ ∈ Bp+d implies (7.4). Proof. See the proof of [4, Proposition 9]. We also need to compare a speciﬁcation for a given interaction with that for truncated interaction. A suitable result is given by the following proposition.

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Proposition 7.3. Let α > 0, R > 0 and Φ ∈ Bα . Deﬁne Ψ by Φ(X) if d(X) < R Ψ(X) = 0 otherwise. Then, for all Q sup A∈ξ Q ,

u∈E Rd

Φ Ψ |γQ (A | u) − γQ (A | u)| ≤ 2(exp(#Q∗ R−α Φ α ) − 1).

Proof. See the proof of [4, Proposition 10]. Theorem 7.1. Let Φ ∈ Bp+d for some p > 0 and suppose that γ the Gibbs speciﬁcation for Φ, satisﬁes Dobrushin’s condition. Assume (a) either lim sup

R→∞ i∈L

|i − j|p+d Cij (γ) = 0,

j∈L: |i−j|>R

(b) or Φ ∈ Bp+2d . Then the unique Gibbs measure admitted by γ is balanced. Proof. By Proposition 7.1, γ admits admits a unique Gibbs measure µ. Taking K = exp(2 Φ ∼ ) and hµ = 0 by (7.3) the set A(Q) in condition (B2) of Deﬁnition 6.1 is empty, and then µ satisﬁes (B2). It remains to show that µ satisﬁes (B1). Let (Qn ) a sequence in which Qn is a tile or a corona in M(TPk(n) (Rk(n) )) with k(n) → ∞ when n → ∞. If A ∈ ξ S(Qn ) and Qn Qn we denote by A the corresponding element of ξ S(Qn ) . We have to show that lim

sup

sup

n→∞ Q Qn A∈ξ S(Qn ) n k(n)

|µ(A) − µ(A )| = 0. k(n)

As Qn Qn then (S(Qn ) 2N )∗ ≈ (S(Qn ) 2N )∗ . Let rn = k(n) 4N . Deﬁne ∆n = rn rn rn ∗ rn ∗ S(Qn ) and ∆n = S(Qn ) . Thus (∆n ) ≈ (∆n ) . 1 )k(n)) for some x ∈ Rd . Hence by By Proposition 4.2, ∆n ⊂ B(x, (C2 + 4N Lemma 2.1 d 1 B x, C2 + 1 k(n) + r (C2 + 4N )k(n) + r ∗ 4N ≤ . #(∆n ) ≤ |B(0, r)| rd Fix ε > 0. As k(n) → ∞ when n → ∞, there exists n0 ∈ N such that for all n ≥ n0 ε (7.5) 2(exp(#(∆n )∗ rn−(p+d) Φ p+d ) − 1) < , 8 and as we have d(S(Qn ), ∆cn ) = rn , Proposition 7.2 gives ε (7.6) D(Qn , ∆n , γ) < . 8 Estimates (7.5) and (7.6) are valid for all copies of Qn .

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Choose a Qn Qn such that d(S(Qn ), S(Qn )) > 4rn . So the sets S(Qn )2rn and d S(Qn )2rn are disjoint, and then we can choose a conﬁguration u ∈ E R such that uS(Qn )2rn is equal to uS(Qn )2rn . By Proposition 7.1 we have for n ≥ n0 : |µ(A) − µ(A )| ≤ |µ(A) − γ∆n (A | u)| + |γ∆n (A | u) − γ∆n (A | u)| + |γ∆n (A | u) − µ(A )|

≤ D(Qn , ∆n , γ) + |γ∆n (A | u) − γ∆n (A | u)| + D(Qn , ∆n , γ) ε ≤ + |γ∆n (A | u) − γ∆n (A | u)|. 4 Deﬁne a vertexpattern invariant ﬁnite range interaction Φn by Φ(X) if d(X) < rn , Φn (X) = 0 if d(X) > rn . Then

Φn Φn Φn Φ Φ Φ |γ∆ (A | u) − γ∆ (A | u)| ≤ |γ∆ (A | u) − γ ∆n (A | u)| + |γ∆n (A | u) − γ∆n (A | u)| n n n Φn Φ + |γ∆ (A | u) − γ∆ (A | u)|. n n

The second term on the right-hand side is zero since ∆n and ∆n have the same environments up to a radius rn which is the range of Φn . The other terms are smaller than 8ε by (7.5) and Proposition 7.3. Combining all these we have shown that ε |µ(A) − µ(A )| < 2 for all A ∈ ξ S(Qn ) and all Qn Qn such that d(S(Qn ), S(Qn )) > 4rn . If d(S(Qn ), S(Qn )) ≤ 4rn , there is always a Qn Qn such that d(S(Qn ), S(Qn )) > 4rn and d(S(Qn ), S(Qn )) > 4rn . This implies that sup

sup

Qn Qn A∈ξ S(Qn )

|µ(A) − µ(A )| < ε

for all n ≥ n0 . Acknowledgments The author would like to thank Daniel Lenz, who ﬁrst introduced him to quasicrystallographic systems and for his helpful comments and guiding suggestions. The author would also like to thank Peter Stollmann for many helpful discussions and Anne Boutet de Monvel for many hints. References [1] M. Baake and U. Grimm, A guide to quasicrystal literature, in Directions in Mathematical Quasicrystals, eds. M. Baake and R. V. Moody, CRM Monogr. Ser. (Amer. Math. Soc., Providence, RI, 2000), pp. 371–373.

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[2] D. Damanik and D. Lenz, Linear Repetitivity, I. Uniform subadditive ergodic theorems and apllications, Discrete Comput. Geom. 26 (2001) 411–428. [3] F. Durand, Linearly recurrent subshifts have a ﬁnite number of non-periodic subshift factors, Ergodic Theory Dynam. Systems 20 (2000) 1061–1078; Corrigendum and addendum, ibid. 23 (2003) 663–669. [4] C. P. M. Geerse and A. Hof, Lattice gas models on self-similar aperiodic tilings, Rev. Math. Phys. 3 (1991) 163–221. [5] H. O. Georgii, Gibbs Measures and Phase Transitions (Walter de Gruyter & Co., Berlin, 1988). [6] A. Hof, Some remarks on discrete aperiodic Schr¨ odinger operators, J. Statist. Phys. 72 (1993) 1353–1374. [7] R. B. Israel, Convexity in the Theory of Lattice Gases (Princeton University Press, Princeton, N.J., 1979). [8] C. Janot, Quasicrystals: A Primer (Oxford University Press, Oxford, 1997). [9] J. C. Lagarias and P. A. B. Pleasants, Repetitive Delone sets and quasicrystals, Ergodic Theory Dynam. Systems 23 (2003) 831–867. [10] J.-Y. Lee, R. V. Moody and B. Solomyak, Pure point dynamical and diﬀraction spectra, Ann. Henri Poincar´e 3 (2002) 1003–1018. [11] D. Lenz, Hierarchical structures in Sturmian dynamical systems, Theoret. Comput. Sci. 303 (2003) 463–490. [12] D. Lenz and P. Stollmann, An ergodic theorem for Delone dynamical systems and existence of the integrated density of states, J. Anal. Math. 97 (2005) 1–24. [13] W. Lunnon and P. A. B. Pleasants, Quasicrystallographic tilings, J. Math. Pures Appl. (9) 66 (1987) 217–263. [14] J. Patera (ed.), Quasicrystals and Discrete Geometry, Fields Institute Monographs (American Mathematical Society, Providence, RI, 1998). [15] N. Priebe, Towards a characterization of self-similar tilings in terms of derived Vorono˘ı tesselations, Geom. Dedicata 79 (2000) 239–265. [16] N. Priebe and B. Solomyak, Characterization of planar pseudo-self-similar tilings, Discrete Comput. Geom. 26 (2001) 289–306. [17] M. Senechal, Quasicrystals and Geometry (Cambridge University Press, Cambridge, 1995). [18] D. Shechtman, I. Blech, D. Gratias and J. W. Cahn, Metallic phase with long-range orientational order and no translation symmetry, Phys. Rev. Lett. 53 (1984) 1951– 1953. [19] B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems 17 (1997) 695–738; Corrections, ibid. 19 (1999) 1685. [20] B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally ﬁnite tilings, Discrete Comput. Geom. 20 (1998) 265–279.

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Reviews in Mathematical Physics Vol. 20, No. 6 (2008) 625–706 c World Scientiﬁc Publishing Company

ON APPROXIMATE SOLUTIONS OF SEMILINEAR EVOLUTION EQUATIONS II. GENERALIZATIONS, AND APPLICATIONS TO NAVIER–STOKES EQUATIONS

CARLO MOROSI Dipartimento di Matematica, Politecnico di Milano, P.za L. da Vinci 32, I-20133 Milano, Italy [email protected] LIVIO PIZZOCCHERO Dipartimento di Matematica, Universit` a di Milano, Via C. Saldini 50, I-20133 Milano, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Italy [email protected] Received 11 September 2007 In our previous paper [12], a general framework was outlined to treat the approximate solutions of semilinear evolution equations; more precisely, a scheme was presented to infer from an approximate solution the existence (local or global in time) of an exact solution, and to estimate their distance. In the ﬁrst half of the present work, the abstract framework of [12] is extended, so as to be applicable to evolutionary PDEs whose nonlinearities contain derivatives in the space variables. In the second half of the paper, this extended framework is applied to the incompressible Navier–Stokes equations, on a torus Td of any dimension. In this way, a number of results are obtained in the setting of the Sobolev spaces Hn (Td ), choosing the approximate solutions in a number of different ways. With the simplest choices we recover local existence of the exact solution for arbitrary data and external forces, as well as global existence for small data and forces. With the supplementary assumption of exponential decay in time for the forces, the same decay law is derived for the exact solution with small (zero mean) data and forces. The interval of existence for arbitrary data, the upper bounds on data and forces for global existence, and all estimates on the exponential decay of the exact solution are derived in a fully quantitative way (i.e. giving the values of all the necessary constants; this makes a diﬀerence with most of the previous literature). Next, the Galerkin approximate solutions are considered and precise, still quantitative estimates are derived for their Hn distance from the exact solution; these are global in time for small data and forces (with exponential time decay of the above distance, if the forces decay similarly). Keywords: Diﬀerential equations; theoretical approximation; Navier–Stokes equations; Galerkin method. Mathematics Subject Classiﬁcation 2000: 35A35, 35Q30, 65M60

625

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Contents 1. Introduction

626

2. Introducing the Abstract Setting

631

3. Approximate Solutions of the Volterra and Cauchy Problems: The Main Result

635

4. Proof of Proposition 3.7

639

5. Applications of Proposition 3.7 to Systems with Quadratic Nonlinearity: Local and Global Results

644

6. The Navier–Stokes (NS) Equations on a Torus

654

7. The NS Equations in the General Framework for Evolution Equations with Quadratic Nonlinearity

668

8. Results for the NS Equations Arising from the Previous Framework

672

9. Galerkin Approximate Solutions of the NS Equations

676

10. Numerical Examples

683

Appendix A. Proof of Lemma 2.6

685

Appendix B. A Scheme to Solve Numerically the Control Inequality (5.14)

685

Appendix C. Proof of Lemmas 6.1 and 6.2

691

Appendix D. Proof of Propositions 6.3 and 6.5

699

Appendix E. Derivation of the NS Equations (6.87)

700

Appendix F. Proof of Eq. (6.100)

702

Appendix G. Proof of Proposition 7.2, Item (iv)

703

Appendix H. The Constants K2 and K4 in Dimension d = 3

704

1. Introduction This is a continuation of our previous paper [12] on the approximate solutions of semilinear Cauchy problems in a Banach space F, and on their use to get fully quantitative estimates on: (i) the interval of existence of the exact solution; (ii) the distance at any time between the exact and the approximate solution. In [12], we mentioned the potential interest of (i) and (ii) in relation to the equations of ﬂuid dynamics. Here we treat speciﬁcally the incompressible Navier– Stokes (NS) equations on a torus Td of any dimension d ≥ 2, taking for F a Sobolev space of vector ﬁelds over Td . To be precise, we consider the Sobolev space Hn (Td ) ≡ Hn of the “velocity ﬁelds” v : Td → Rd whose derivatives of order ≤ n are square integrable; then we choose F := HnΣ0 , where the subscripts Σ0 indicate the subspace of Hn formed by the divergence free, zero mean velocity ﬁelds f : Td → Rd (of course, the condition of zero divergence represents incompressibility; the mean

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velocity can always be supposed to vanish, passing to a convenient moving frame). We always take n > d/2. The choice of Td as a space domain allows a rather simple treatment, based on Fourier analysis; we presume that the results of this paper could be extended to bounded domains of Rd , with suitable boundary conditions. In our notations, the NS Cauchy problem is written ϕ(t) ˙ = ∆ϕ(t) − L(ϕ(t) • ∂ϕ(t)) + ξ(t),

ϕ(0) = f0 ,

(1.1)

where f = ϕ(t) is the velocity ﬁeld at time t, L the Leray projection on the divergence free vector ﬁelds and ξ(t) is the external forcing at time t (more precisely, what remains of the external force ﬁeld after applying L and subtracting the mean value). We can regard (1.1) as a realization of the abstract semilinear Cauchy problem ϕ(t) ˙ = Aϕ(t) + P(ϕ(t), t),

ϕ(t0 ) = f0

(1.2)

where A : f → Af is a linear operator and P : (f, t) → P(f, t) is a nonlinear map. Of course, in the NS case A is the Laplacian ∆ and P(f, t) := −L(f • ∂f ) + ξ(t). By a standard method, both (1.1) and its abstract version (1.2) can be reformulated as a Volterra integral equation, involving the semigroup (etA )t≥0 . In [12], a general setting was proposed for semilinear Volterra problems, when the nonlinearity P(·, t) is a suﬃciently smooth map of a Banach space F into itself. This setting cannot be applied to Cauchy problems like (1.1). In fact, due to the presence of the derivatives ∂f , the map f → L(f • ∂f ) cannot be seen as a smooth map of a Sobolev space, say HnΣ0 , into itself; on the contrary, the above map is smooth from HnΣ0 to Hn−1 Σ0 . The external forcing ξ : t → ξ(t) ﬁts well to this situation if we require it to be a suﬃciently smooth map from [0, +∞) to Hn−1 Σ0 . In view of the applications to (1.1), in the ﬁrst half of the present paper (Secs. 2– 5) we extend the abstract framework of [12] to the case where, at each time t, P(·, t) is a smooth map between F and a larger Banach space F− . A general scheme to treat approximate solutions is developed along these lines; this could be applied not only to (1.1), but also to other evolutionary PDEs (essentially, of parabolic type) with space derivatives in the nonlinear part. In the second half of the paper (Secs. 6–10) we ﬁx the attention on the NS equations, in the framework of the above mentioned HnΣ0 spaces (incidentally, we wish to point out that other function spaces could be used to analyze the same equations within our general scheme). Some technicalities related to either the ﬁrst or the second half are presented in Appendices A–H. Of course, there is an enormous literature on NS equations, their approximation methods and the intervals of existence of the exact solutions: references [2, 3, 5–11, 14, 15] are examples including seminal works, classical treatises and recent contributions. Some diﬀerences between the present analysis and most of the published

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literature are the following: (i) Our discussion of the NS approximate solutions is part of a more general framework, in the spirit of the ﬁrst half of the paper. (ii) Our analysis is fully quantitative: any function, numerical constant, etc., appearing in our estimates on the solutions is given explicitly. In the end, our approach gives bounds on the interval of existence of the exact NS solution and on its distance from the approximate solution in terms of fully computable numbers; such computations are exempliﬁed in a number of cases. (iii) If compared with other contributions, our approach seems to be more suitable to derive the existence of global exact solutions from suitable approximate solutions, under speciﬁc conditions (typically, of small initial data); a comment on this point appears in Remark 8.7(iii). Hereafter we give more details about the contents of the paper. First half: A general setting for the approximate solutions of (1.2). We have just mentioned the assumption P(·, t) : F → F− . We furtherly suppose A : F+ → F− where F+ is a dense subspace of F, to be equipped with the graph norm of A; in the end, this gives a triple of spaces F+ ⊂ F ⊂ F− . To go on, we require A to generate a semigroup on F− , with the fundamental regularizing property etA (F− ) ⊂ F for all t > 0. A more precise description of all these assumptions is given in Sec. 2: here we suppose, amongst else, the availability of an upper bound u− (t) ∈ (0, +∞) for the operator norm of etA , regarding the latter as a map from F− to F. The bound u− is allowed to diverge (mildly) for t → 0+ , an indication that etA F− ⊂ F for t = 0 : the precise assumption is u− (t) = O(1/t1−σ ) with 0 < σ ≤ 1. In applications to the NS system, F = HnΣ0 and t∆ F∓ = Hn∓1 Σ0 ; the semigroup (e ) of the Laplacian has the prescribed regularizing features, with σ = 1/2. In Sec. 3, we present a general theory of the approximate solutions, for an abstract Cauchy (or Volterra) problem of the type sketched above. The basic idea is to associate to any approximate solution ϕap : [t0 , T ) → F of the problem an integral control inequality for an unknown function R : [t0 , T ) → [0, +∞); this has the form t E(t) + ds u− (t − s)(R(s), s) ≤ R(t) (1.3) t0

where E : [t0 , T ) → [0, +∞) is an estimator for the (integral) error of ϕap , and is a function describing the growth of P from ϕap . The main result in this framework is the following: if the control inequality is fulﬁlled by some function R on [t0 , T ), then the semilinear Volterra problem has an exact solution ϕ : [t0 , T ) → F, and

ϕ(t) − ϕap (t) ≤ R(t) for all t in this interval ( is the norm of F). When F− = F, we recover from here the framework of [12]. Similarly to the result of [12], the present theorem about R, ϕap and ϕ can be considered as the

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abstract and unifying form of many statements, appearing in the literature about speciﬁc systems. The available literature would suggest to prove the above theorem along this path: (i) derive an existence theorem for ϕ on small intervals; (ii) use some nonlinear Gronwall lemma to prove that ϕ(t) − ϕap (t) ≤ R(t) on any interval [t0 , T ) ⊂ [t0 , T ) where ϕ is deﬁned; (iii) show the existence of ϕ on the full domain [t0 , T ) of R by the following reductio ad absurdum: if not so, ϕ(t) − ϕap (t) would diverge before T and its upper bound via R(t) would be violated. Our proof of the theorem on R, ϕap and ϕ, presented in Sec. 4, replaces the above strategy with a more constructive approach. The main idea is to interpret the control inequality (1.3) as individuating a tube of radius R = R(t) around ϕap , invariant under the action of the semilinear Volterra operator J for our problem. This makes possible to construct the solution by an iteration of Peano–Picard type, starting from ϕap ; the result is a Cauchy sequence of functions ϕk = J k (ϕap ) on [t0 , T ), (k = 0, 1, 2, . . .), whose k → +∞ limit is an exact solution of the given Volterra problem. From this viewpoint, existence of the solution on a short time interval, with any datum ϕ(t0 ) = f0 , is a very simple corollary of the previous theorem based the choice ϕap (t) := constant = f0 . Even though there is a basic analogy with [12], proving the main theorem on approximate solutions is technically more diﬃcult in the present case, mainly due to the divergence of u− (t) for t → 0+ . Such a divergence is also relevant in applications: in fact, diﬀerently from [12], Eq. (1.3) with ≤ replaced by = cannot be reduced to an ordinary diﬀerential equation. Our assumption u− (t) = O(1/t1−σ ) relates (1.3) to the framework of singular integral equations of fractional type (which could be interpreted in terms of the so-called “fractional diﬀerential calculus”). In spite of these pathologies, solving (1.3) is rather simple when the semigroup (etA ) and ϕap have suitable features, and the nonlinear function P has the (aﬃne) quadratic structure P(f, t) = P(f, f ) + ξ(t),

(1.4)

with P : F− × F− → F a continuous bilinear form and ξ : [0, +∞) → F− a (locally Lipschitz) map; this is the subject of Sec. 5 (where the datum f0 of (1.2) is always speciﬁed at t0 = 0). The section starts from a fairly general statement on the control inequality (1.3), which is subsequently applied with speciﬁc choices of the approximate solution. First of all, we consider the choice ϕap (t) := 0. In this case, for any datum f0 and external forcing ξ, we construct for the control inequality a solution R with domain a suitable interval [0, T ); this implies the existence on [0, T ) of the solution ϕ of (1.2), and gives an estimate ϕ(t) ≤ R(t) on the same interval. If f0 and ξ are suﬃciently small, T = +∞ and so ϕ is global. With the stronger assumption that ξ(t) decays exponentially for t → +∞, we derive for the control inequality a solution t → R(t) which is also exponentially decaying; so, the same can be said for ϕ(t) . Next we

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consider, for a small f0 and a small, exponentially decaying ξ, the approximate solution ϕap obtained solving the linear Cauchy problem ϕ˙ ap (t) = Aϕap (t) + ξ(t), ϕap (0) = f0 . In this case the control equation still possesses a global, exponentially decaying solution R, giving a precise estimate on the distance ϕ(t) − ϕap (t) . Second half: Applications to the NS equations. In Sec. 6, we review the Sobolev spaces of vector ﬁelds on Td , and the Leray formulation of the incompressible NS equations within this framework; furthermore, we show that the Cauchy problem with mean initial velocity m0 can be reduced to an equivalent Cauchy problem where the initial velocity has zero mean, by a change of spacetime coordinates (x, t) → (x − h(t), t), where the function t → h(t) is suitably determined. In the same section we give explicitly the constants Knd ≡ Kn such that

f • ∂g n−1 ≤ Kn f n g n for all velocity ﬁelds f, g on Td , n and n−1 denoting the Sobolev norms of orders n and n − 1. The study of these constants, inspired by our previous work [13], prepares the fully quantitative application of the methods presented in the ﬁrst half of the paper. Section 7 starts from the formulation (1.1) of the Cauchy problem, in the already mentioned Sobolev spaces F = HnΣ0 , F∓ = Hn∓1 Σ0 . We check that (1.1) fulﬁlls all requirements of the general theory for quadratic nonlinearities, and construct the estimator u− for the semigroup (et∆ ). In Sec. 8, we rephrase for the NS equations all the results of Sec. 5 on the abstract quadratic case (1.4). The estimates on the time of existence T , for arbitrary data and forcing, have a fully explicit form; the same happens for the bounds on the norms f0 n , ξ(t) n−1 which ensure global existence and, possibly, exponential decay of ϕ(t) for t → +∞. In Sec. 9, we discuss the approximate NS solutions provided by the Galerkin method. More precisely, for each ﬁnite set G( 0) of wave vectors we consider the ikx (k ∈ G), and the projection on HG subspace HG Σ0 spanned by the exponentials e Σ0 of the NS Cauchy problem; this has a solution t → ϕG (t) (in general, on a suﬃciently small interval; with special assumptions, also involving the forcing, ϕG is global and decays exponentially for t → +∞). Applying the framework of Sec.7 with ϕap = ϕG we derive the following results (with p > n and |G| := inf k∈Zd0 \G 1 + |k|2 ). (i) For any initial datum f0 ∈ HpΣ0 of the NS Cauchy problem (1.1), and each G and ϕ exist on a suitable external forcing ξ with values in Hp−1 Σ0 , both ϕ interval [0, T ), and there is an estimate

ϕ(t) − ϕG (t) n ≤

Wnp |G| (t) |G|p−n

for t ∈ [0, T );

T can be +∞, if the datum and the forcing are suﬃciently small. Both T and the function t → Wnp |G| (t) are given explicitly.

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(ii) If f0 ∈ HpΣ0 is suﬃciently small and there is a small, exponentially decaying forcing t → ξ(t) ∈ Hp−1 Σ0 , then

ϕ(t) − ϕG (t) n ≤

Wnp |G| −t e |G|p−n

for t ∈ [0, +∞);

the upper bounds for f0 , ξ and the coeﬃcient Wnp |G| are also given explicitly. The results (i) and (ii) imply convergence of ϕG to ϕ as |G| → +∞, on the time interval where the previous estimates hold (which can be [0, +∞), as pointed out). In Sec. 10, we exemplify our estimates giving the numerical values of T and of the error estimators in (i) and (ii) for certain data and forcing, with d = 3 and n = 2, p = 4. 2. Introducing the Abstract Setting Notations. (i) All Banach spaces considered in this paper are over the same ﬁeld, which can be R or C. (ii) If X and Y are Banach spaces, we write X → Y

(2.1)

to indicate that X is a dense vector subspace of Y and that its natural inclusion into Y is continuous (i.e. x Y ≤ constant x X for all x ∈ X). (iii) Consider two sets Θ, X and a function χ : Θ → X, t → χ(t). The graph of χ is gr χ := {(χ(t), t) | t ∈ Θ} ⊂ X × Θ.

(2.2)

If X = [0, +∞], we deﬁne the subgraph of χ as sgr χ := {(r, t) | t ∈ Θ, r ∈ [0, χ(t))} ⊂ [0, +∞) × Θ.

(2.3)

(iv) Consider a function χ : Θ → X, where Θ is a real interval and X a Banach space. This function is locally Lipschitz if, for each compact subset I of Θ, there is a constant M = M (I) ∈ [0, +∞) such that

χ(t) − χ(t ) X ≤ M |t − t | for all t, t ∈ I.

(2.4)

As usually, we denote with C 0,1 (Θ, X) the set of these functions. General assumptions. Throughout the section, we will consider a set (F+ , F, F− , A, u, u− , P)

(2.5)

with the following properties. (P1) F+ , F and F− are Banach spaces with norms + , and − , such that F+ → F → F− .

(2.6)

Here and in the sequel, B(f0 , r) will denote the open ball {f ∈ F | f − f0 < r} (the radius r can be +∞, and in this case B(f0 , r) = F).

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(P2) A is a linear operator such that A : F+ → F− ,

f → Af.

(2.7)

Viewing F+ as a subspace of F− , the norm + is equivalent to the graph norm f ∈ F+ → f − + Af − . (P3) Viewing A as a densely deﬁned linear operator in F− , it is assumed that A generates a strongly continuous semigroup (etA )t∈[0,+∞) on F− (of course, from the standard theory of linear semigroups, we have etA (F+ ) ⊂ F+ for all t ≥ 0). (P4) One has etA (F) ⊂ F for t ∈ [0, +∞);

(2.8)

the function (f, t) → e f gives a strongly continuous semigroup on F (i.e. it is continuous from F × [0, +∞) to F). Furthermore, u ∈ C([0, +∞), (0, +∞)) is a function such that tA

etA f ≤ u(t) f for t ≥ 0, f ∈ F;

(2.9)

this function will be referred to as an estimator for the semigroup (etA ) with respect to the norm of F. (P5) One has etA (F− ) ⊂ F for t ∈ (0, +∞);

(2.10)

the function (f, t) → e f is continuous from F− × (0, +∞) to F (in a few words: for all t > 0, etA regularizes the vectors of F− , sending them into F continuously). Furthermore, u− ∈ C((0, +∞), (0, +∞)) is a function such that tA

etA f ≤ u− (t) f − 1 u− (t) = O 1−σ t

for t > 0, f ∈ F− ; for t → 0+ ,

σ ∈ (0, 1].

(2.11) (2.12)

The function u− will be referred to as an estimator for the semigroup etA with respect to the norms of F and F− ; Eq. (2.12) ensures its integrability in any right neighborhood of t = 0. (P6) One has P : Dom P ⊂ F × R → F− ,

(f, t) → P(f, t),

(2.13)

and the domain of P is semi-open in F × R : by this we mean that, for any (f0 , t0 ) ∈ Dom P, there are δ, r ∈ (0, +∞] such that B(f0 , r) × [t0 , t0 + δ) ⊂ Dom P. Furthermore, P is Lipschitz on each closed, bounded subset C of F × R such that C ⊂ Dom P; by this, we mean that there are constants L = L(C) and M = M (C) ∈ [0, +∞) such that

P(f, t) − P(f , t ) − ≤ L f − f + M |t − t | for all (f, t), (f , t ) ∈ C.

(2.14)

Remark 2.1. As anticipated, our aim is to discuss the Cauchy problem ϕ(t) ˙ = Aϕ(t)+P(ϕ(t), t), ϕ(t0 ) = f0 (and its equivalent formulation as a Volterra problem)

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for a system (F+ , F, F− , A, u, u− , P) with the previously mentioned properties (P1)–(P6). In comparison with the present work, the analysis of [12] corresponds to the special case F− = F,

u− = u

(2.15)

in which, by the continuity of u at t = 0, Eq. (2.12) is fulﬁlled with σ = 1.a Preliminaries to the analysis of the Cauchy and Volterra problems. (i) In the sequel, whenever we consider an interval [t0 , T ), we intend −∞ < t0 < T ≤ +∞.b (ii) Let us consider a function ω ∈ C([t0 , T ), F− ) and the function t ds e(t−s)A ω(s). (2.16) Ω : t ∈ [t0 , T ) → Ω(t) := t0

Using the regularizing properties (P5) of the semigroup with respect to the spaces F and F− , one easily proves that for each ﬁxed t the function s → e(t−s)A ω(s) belongs to L1 ((t0 , t), dt, F) and Ω ∈ C([t0 , T ), F). (iii) All the results in (ii) apply in particular to the case ω(s) := P(ψ(s), s) where ψ ∈ C([t0 , T ), F) and gr ψ ⊂ Dom P. Functions of this form will often appear in the forthcoming analysis of Volterra problems. (iv) Many facts stated in the sequel depend on the basic identity, here recalled for future citation, t ˙ ds e(t−s)A [ψ(s) − Aψ(s)], (2.17) ψ(t) = e(t−t0 )A ψ(t0 ) + t0

holding for any function ψ ∈ C([t0 , T ), F+ ) ∩ C 1 ([t0 , T ), F− ) and each t in this interval. Formal deﬁnitions of the Cauchy and Volterra problems. These deﬁnitions are similar to the ones adopted in [12], with slight changes due to the present use of two diﬀerent spaces F, F− . Deﬁnition 2.2. Consider a pair (f0 , t0 ) ∈ Dom P, with f0 ∈ F+ . The Cauchy problem CP(f0 , t0 ) with datum f0 at time t0 is the following one: Find ϕ ∈ C([t0 , T ), F+ ) ∩ C 1 ([t0 , T ), F− ) such that gr ϕ ⊂ Dom P and ϕ(t) ˙ = Aϕ(t) + P(ϕ(t), t)

for all t ∈ [t0 , T ),

ϕ(t0 ) = f0 .

(2.18)

[12], etA was written U (t), and F+ was simply indicated with Dom A; furthermore, we assumed Dom P to be open in R × F. b In [12], we also considered solutions of the Cauchy or Volterra problems with domain a closed, bounded interval [t0 , T ]; the symbol [t0 , T | was employed to denote an interval of either type. Here we only consider the ﬁrst case (semiopen, possibly unbounded), simply to avoid tedious distinctions. a In

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We note that C([t0 , T ), F+ ) ⊂ C([t0 , T ), F). This fact, with the properties of A and P, implies the following: if ϕ ∈ C([t0 , T ), F+ ) and gr ϕ ⊂ Dom P, the right-hand side of the diﬀerential equation in (2.18) deﬁnes a function in C([t0 , T ), F− ). Deﬁnition 2.3. Consider a pair (f0 , t0 ) ∈ Dom P. The Volterra problem VP(f0 , t0 ) with datum f0 at time t0 is the following one: Find ϕ ∈ C([t0 , T ), F) such that gr ϕ ⊂ Dom P and t ds e(t−s)A P(ϕ(s), s) for all t ∈ [t0 , T ). ϕ(t) = e(t−t0 )A f0 +

(2.19)

t0

Proposition 2.4. For (f0 , t0 ) ∈ Dom P and f0 ∈ F+ , we have the following: (i) a solution ϕ of CP(f0 , t0 ) is also solution of VP(f0 , t0 ); (ii) a solution ϕ of VP(f0 , t0 ) is also a solution of CP(f0 , t0 ), if F− is reﬂexive. Proof. It is based on (2.17): see [1]. The derivation of (ii), which is the most technical part, uses the Lipschitz property (P6) of P and the reﬂexivity of F− to show that a solution of VP(f0 , t0 ) has the necessary regularity to fulﬁll CP(f0 , t0 ). Proposition 2.5 (Uniqueness theorem for the Volterra problem). Consider a pair (f0 , t0 ) ∈ Dom P, and assume that VP(f0 , t0 ) has two solutions ϕ ∈ C([t0 , T ), F), ϕ ∈ C([t0 , T ), F). Then ϕ(t) = ϕ (t)

for t ∈ [t0 , min(T, T )).

(2.20)

Proof. We consider any τ ∈ [t0 , min(T, T )), and show that ϕ = ϕ in [t0 , τ ]. To this purpose, we subtract Eq. (2.19) for ϕ from the analogous equation for ϕ ; taking the norm and using Eqs. (2.11), (2.12) and (2.14), for each t ∈ [t0 , τ ] we obtain: t

ϕ(t) − ϕ (t) ≤ ds u− (t − s) P(ϕ(s), s) − P(ϕ (s), s) − t0

≤ UL

t

ds t0

ϕ(s) − ϕ (s)

. (t − s)1−σ

(2.21)

In the above: L ≥ 0 is a constant fulﬁlling the Lipschitz condition (2.14) for P on the set C := gr (ϕ [t0 , τ ]) ∪ gr (ϕ [t0 , τ ]); U ≥ 0 is a constant such that 1−σ for all t ∈ (0, τ ] (which exists due to (2.12)). u− (t ) ≤ U/t Equation (2.21) implies ϕ(t) − ϕ (t) = 0 for all t ∈ [t0 , τ ]; in fact, this result follows applying to the function z(t) := ϕ(t) − ϕ (t) the forthcoming lemma. Lemma 2.6. Consider a function z ∈ C([t0 , τ ], [0, +∞)) (with −∞ < t0 < τ < +∞), and assume there are Λ ∈ [0, +∞), σ ∈ (0, 1] such that t z(s) ds for t ∈ [t0 , τ ]. (2.22) z(t) ≤ Λ (t − s)1−σ t0 Then, z(t) = 0 for all t ∈ [t0 , τ ].

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Proof. See Appendix A. Remark 2.7. For VP(f0 , t0 ) we will grant existence as well, on suﬃciently small time intervals (see the forthcoming Proposition 3.10, where local existence is obtained as a simple application of the general theory of approximate solutions). The Volterra integral operator. This is the (nonlinear) integral operator appearing in problem VP(f0 , t0 ). More precisely, let us state the following. Deﬁnition 2.8. Let (f0 , t0 ) ∈ Dom P. The Volterra integral operator J(f0 ,t0 ) ≡ J associated to this pair is the following map: (i) Dom J is made of the functions ψ ∈ C([t0 , T ), F) (with arbitrary T ∈ (t0 , +∞]) such that gr ψ ⊂ Dom P; (ii) for each ψ in this domain, J (ψ) ∈ C([t0 , T ), F) is the function t t ∈ [t0 , T ) → J (ψ)(t) := e(t−t0 )A f0 + ds e(t−s)A P(ψ(s), s). (2.23) t0

3. Approximate Solutions of the Volterra and Cauchy Problems: The Main Result Throughout the section, we consider again a set (F+ , F, F− , A, u, u− , P), with the properties (P1)–(P6) of the previous section. The deﬁnitions that follow generalize similar notions, introduced in [12]. Approximate solutions, and their errors. We introduce them in the following way. Deﬁnition 3.1. Let (f0 , t0 ) ∈ Dom P. (i) An approximate solution of VP(f0 , t0 ) is any function ϕap ∈ C([t0 , T ), F) such that gr ϕap ⊂ Dom P. (ii) The integral error of ϕap is the function (3.1) E(ϕap ) := ϕap − J (ϕap ) ∈ C([t0 , T ), F), t i.e. E(ϕap )(t) = ϕap (t) − e(t−t0 )A f0 − t0 ds e(t−s)A P(ϕap (s), s). An integral error estimator for ϕap is a function E ∈ C([t0 , T ), [0, +∞)) such that, for all t in this interval,

E(ϕap )(t) ≤ E(t).

(3.2)

Deﬁnition 3.2. Let (f0 , t0 ) ∈ Dom P, and f0 ∈ F+ . (i) An approximate solution of CP(f0 , t0 ) is any function ϕap ∈ C([t0 , T ), F+ ) ∩ C 1 ([t0 , T ), F− ) such that gr ϕap ⊂ Dom P.

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(ii) The datum error for ϕap is the diﬀerence d(ϕap ) := ϕap (t0 ) − f0 ∈ F+ ⊂ F;

(3.3)

a datum error estimator for ϕap is a nonnegative real number δ such that

d(ϕap ) ≤ δ.

(3.4)

(iii) The diﬀerential error of ϕap is the function e(ϕap ) ∈ C([t0 , T ), F− ), t → e(ϕap )(t) := ϕ˙ ap (t) − Aϕap (t) − P(ϕap (t), t); (3.5) a diﬀerential error estimator for ϕap is a function ∈ C([t0 , T ), [0, +∞)) such that, for t in this interval,

e(ϕap )(t) − ≤ (t).

(3.6)

Remark 3.3. (i) A function ϕap as in Deﬁnition 3.1 (respectively, Deﬁnition 3.2) is a solution of VP(f0 , t0 ) (respectively, of CP(f0 , t0 )) if and only if E(ϕap ) = 0 (respectively, d(ϕap ) = 0 and e(ϕap ) = 0). (ii) Of course, the previous deﬁnitions of the error estimators can be fulﬁlled setting E(t) := E(ϕap )(t) , δ := d(ϕap ) , (t) := e(ϕap )(t) − . Lemma 3.4. Let (f0 , t0 ) ∈ Dom P, f0 ∈ F+ , and ϕap be an approximate solution of CP(f0 , t0 ) with datum and diﬀerential errors d(ϕap ), e(ϕap ). Then: (i) ϕap is also an approximate solution of VP(f0 , t0 ), with integral error t E(ϕap )(t) = e(t−t0 )A d(ϕap ) + ds e(t−s)A e(ϕap )(s).

(3.7)

t0

(ii) If δ, are datum and diﬀerential error estimators for ϕap , an integral error estimator for ϕap is t E(t) := u(t − t0 ) δ + ds u− (t − s) (s) for all t ∈ [t0 , T ). (3.8) t0

Proof. (i) To derive Eq. (3.7), use the deﬁnitions of E(ϕap ), d(ϕap ), e(ϕap ) and the identity (2.17) with ψ := ϕap . (ii) To derive the estimator (3.8), apply to both sides of (3.7), using Eqs. (2.9) and (2.11) for u, u− , and Eqs. (3.4) and (3.6) for δ, . Growth of P from a curve. To introduce this notion, we need some notations. Let us consider a function ρ ∈ C([t0 , T ), (0, +∞]); we recall that, according to (2.3), the subgraph of ρ is the set sgr ρ := {(r, t) | t ∈ [t0 , T ), r ∈ [0, ρ(t))}. Furthermore,

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let φ ∈ C([t0 , T ), F). We deﬁne the ρ-tube around φ as the set T(φ, ρ) := {(f, t) | f ∈ F, t ∈ [t0 , T ), f − φ(t) < ρ(t)};

(3.9)

of course, the above tube is the whole space F if ρ(t) = +∞ for all t.c Deﬁnition 3.5. Let φ ∈ C([t0 , T ), F), with gr φ ⊂ Dom P. A growth estimator for P from φ is a function with these features. (i) The domain of is the subgraph of some function ρ ∈ C([t0 , T ), (0, +∞]), and ∈ C(sgr ρ, [0, +∞)),

(r, t) → (r, t);

(3.10)

is nondecreasing in the ﬁrst variable, i.e. (r, t) ≤ (r , t) for r ≤ r and any t. (ii) The function ρ in (i) is such that T(φ, ρ) ⊂ Dom P. For all (f, t) ∈ T(φ, ρ), it is

P(f, t) − P(φ(t), t) − ≤ ( f − φ(t) , t).

(3.11)

Remark 3.6. Consider any tube T(φ, ρ ) ⊂ Dom P. Using the Lipschitz property (2.14) of P, one can easily construct a growth estimator of domain sgr (ρ /2), depending linearly on r: (r, t) = λ(t)r. The main result on approximate solutions. This is contained in the following: Proposition 3.7. Let (f0 , t0 ) ∈ Dom P, and consider the problem VP(f0 , t0 ). Suppose that: (i) ϕap ∈ C([t0 , T ), F) is an approximate solution of VP(f0 , t0 ), E ∈ C([t0 , T ), [0, +∞)) is an estimator for the integral error E(ϕap ); (ii) ∈ C(sgr ρ, [0, +∞)) is a growth estimator for P from ϕap (for a suitable ρ ∈ C([t0 , T ), (0, +∞]). Consider the following problem: Find R ∈ C([t0 , T ), [0, +∞)) such that gr R ⊂ sgr ρ, and t ds u− (t − s) (R(s), s) ≤ R(t) for t ∈ [t0 , T ). E(t) +

(3.12)

t0

If (3.12) has a solution R on [t0 , T ), then VP(f0 , t0 ) has a solution ϕ with the same domain, and

ϕ(t) − ϕap (t) ≤ R(t)

for t ∈ [t0 , T ).

(3.13)

The solution ϕ is constructed by a Peano–Picard iteration of J , starting from ϕap . Proof. See the next section. Deﬁnition 3.8. Equation (3.12) will be referred to as the control inequality. c In

[12], this notion was presented in the case ρ = constant; the present generalization is harmless, and could have been employed in our previous work as well.

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Remark 3.9. (i) It is worthwhile stressing the following: the estimators u− , , E in the control inequality (3.12) depend on A, P, ϕap , and should be regarded as known when the Volterra problem and the approximate solution are speciﬁed. So, (3.12) is a problem in one unknown R, that one sets up using only informations about ϕap . After R has been found, it is possible to draw conclusions about the (exact) solution ϕ of VP(f0 , t0 ). In the usual language: the control inequality allows predictions on ϕ through an a posteriori analysis of ϕap . (ii) (Extending to the present framework a comment in [12].) Typically, one meets this situation: ϕap , E, are deﬁned for t in some interval [t0 , T ), and the control inequality (3.12) has a solution R on an interval [t0 , T ) ⊂ [t0 , T ); in this case one renames ϕap , E, etc. the restrictions of the previous functions to [t0 , T ), and applies Proposition 3.7 to them (as an example, this occurs essentially in the proof of the forthcoming result). A ﬁrst implication of Proposition 3.7: Local existence. The most general and simple consequence of Proposition 3.7 is the fact anticipated in Remark 2.7, i.e. the local existence for the Volterra problem. Here we formulate this statement precisely. Proposition 3.10. Let (f0 , t0 ) ∈ Dom P. Then, there are R , T , E, such that (i)–(iii) hold: (i) R ∈ (0, +∞], T ∈ (t0 , +∞] and B(f0 , R ) × [t0 , T ) ⊂ Dom P; (ii) E ∈ C([t0 , T ), [0, +∞)) and, for all t ∈ [t0 , T ), t (t−s)A ≤ E(t), E(t0 ) = 0; f0 − e(t−t0 )A f0 − ds e P(f , s) 0

(3.14)

t0

(iii) ∈ C([0, R ) × [t0 , T ), [0, +∞)), (r, t) → (r, t); this function is non decreasing in the ﬁrst variable and, for (f, t) ∈ B(f0 , R ) × [t0 , T ),

P(f, t) − P(f0 , t) − ≤ ( f − f0 , t).

(3.15)

Given R , T , E, with properties (i)–(iii), we have (a) and (b): (a) there are R ∈ (0, R ) and T ∈ (t0 , T ] such that, for all t ∈ [t0 , T ), t ds u− (t − s)(R, s) ≤ R; (3.16) E(t) + t0

(b) if T and R are as in item (a), VP(f0 , t0 ) has a solution ϕ of domain [t0 , T ) and, for all t in this interval,

ϕ(t) − f0 ≤ R.

(3.17)

Proof. Step 1. Existence of R , T , E, . A pair (R , T ) as in (i) exists because Dom P is semi-open (see the explanations in (P6)). A function E as in (ii) can be constructed setting E(t) := the left-hand side of the inequality in (3.14). A function

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as in (iii) is constructed putting (r, t) := supf ∈B(f ¯ 0 ,r) P(f, t) − P(f0 , t) − ; this sup is proved to be ﬁnite using the Lipschitz type inequality (2.14) for P on each ¯ 0 , r) × {t}. One checks that (r, t) is continuous in (r, t) and non set C := B(f decreasing in r. Step 2. Proof of (a). Let us pick up any R ∈ (0, R ), and deﬁne t G : [t0 , T ) → [0, +∞), t → G(t) := E(t) + ds u− (t − s)(R, s).

(3.18)

t0

Then G is continuous and G(t0 ) = 0; this fact, with the positivity of R, implies the existence of T ∈ [t0 , T ) such that G(t) ≤ R for all t ∈ [t0 , T ); the last inequality is just the thesis (3.16). Step 3. Proof of (b). We apply Proposition 3.7 to the approximate solution ϕap (t) := f0

for all t ∈ [t0 , T ).

(3.19)

Due to (i)–(iii), the function E [t0 , T ) is an integral error estimator for ϕap , and the function [0, R ) × [t0 , T ) is a growth estimator for P from ϕap (the function ρ in the general Deﬁnition 3.5 of growth estimator is given in this case by ρ(t) = constant = R ). Equation (3.16) tells us that the general control inequality (3.12) is fulﬁlled by the function R(t) := constant = R for all t ∈ [t0 , T ). So, Proposition 3.7 implies the existence of a solution ϕ of VP(f0 , t0 ) on [t0 , T ), and also gives the inequality (3.17). 4. Proof of Proposition 3.7 Let us make all the assumptions in the statement of the proposition. We begin the proof introducing an appropriate topology for the space of continuous functions [t0 , T ) → F. Deﬁnition 4.1. From now on, C([t0 , T ), F) will be viewed as a (Hausdorﬀ, complete) locally convex space with the topology of uniform convergence on all compact subintervals [t0 , τ ] ⊂ [t0 , T ). By this, we mean the topology induced by the seminorms ( τ )τ ∈[t0 ,T ) , where

τ : C([t0 , T ), F) → [0, +∞),

ψ → ψ τ := sup ψ(t)

(4.1)

t∈[t0 ,τ ]

( is the usual norm of F). Remark 4.2. The uncountable family of seminorms ( τ ) is topologically equivalent to the countable subfamily ( τn ), where (τn ) is any sequence of points of [t0 , T ) such that limn→+∞ τn = T . Therefore, C([t0 , T ), F) is a Fr´echet space. To go on we introduce a basic set, whose deﬁnition depends on ϕap and on the function R in the control inequality (3.12).

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Deﬁnition 4.3. We put D := {ψ ∈ C([t0 , T ), F) | ψ(t) − ϕap (t) ≤ R(t) for t ∈ [t0 , T )}.

(4.2)

Lemma 4.4. (i) D is a closed subset of C([t0 , T ), F), in the topology of Deﬁnition 4.1. (ii) For all ψ ∈ D, one has gr ψ ⊂ Dom P (and so, J (ψ) is well deﬁned). Proof. (i) Suppose ψ ∈ C([t0 , T ), F) and ψ = limn→∞ ψn , where (ψn ) is a sequence of elements of D. Then, for all t ∈ [t0 , T ) we have ψ(t) − ϕap (t) = limn→∞ ψn (t) − ϕap (t) ≤ R(t). (ii) Let us consider the function ρ ∈ C([t0 , T ), (0, +∞]) mentioned in the statement of Proposition 3.7. Then, ψ ∈ D ⇒ ψ(t) − ϕap (t) ≤ R(t) < ρ(t) for all t ∈ [t0 , T ) ⇒ gr ψ ⊂ T(ϕap , ρ) ⊂ Dom P. From now on, our attention will be focused on the map D → C([t0 , T ], F),

ψ → J (ψ).

(4.3)

Of course, for ϕ ∈ D, we have the equivalence ϕ solves VP(f0 , t0 ) ⇔ J (ϕ) = ϕ.

(4.4)

To clarify the sequel, let us recall that σ ∈ (0, 1] is the constant appearing in Eq. (2.12). Lemma 4.5. (i) For each τ ∈ [t0 , T ) there is a constant Λτ ∈ [0, +∞) such that, for all ψ, ψ ∈ D, t

ψ(s) − ψ (s)

J (ψ)(t) − J (ψ )(t) ≤ Λτ ds for t ∈ [t0 , τ ]. (4.5) (t − s)1−σ t0 (ii) For all τ ∈ [t0 , T ) and ψ, ψ ∈ D, the above equation implies a Lipschitz type inequality

J (ψ) − J (ψ ) τ ≤

Λτ (τ − t0 )σ

ψ − ψ τ σ

(4.6)

(which ensures, amongst else, the continuity of J on D). Proof. (i) Let ψ, ψ ∈ D. We consider Eq. (2.23) for J (ψ)(t), and subtract from it the analogous one for J (ψ )(t). After applying the norm of F and using (2.11), we infer t

J (ψ)(t) − J (ψ )(t) ≤ ds u− (t − s) P(ψ(s), s) − P(ψ (s), s) − (4.7) t0

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for all t ∈ [t0 , T ). To go on, we ﬁx τ ∈ [t0 , T ) and deﬁne Cτ := {(f, s) ∈ F × [t0 , τ ] | f − ϕap (s) ≤ R(s)}.

(4.8)

This is a closed, bounded subset of F × R, and Cτ ⊂ T(ϕap , ρ) ⊂ Dom P. Therefore, by the Lipschitz property (P6) of P, there is a nonnegative constant L(Cτ ) ≡ Lτ such that

P(f, s) − P(f , s) − ≤ Lτ f − f

for (f, s), (f , s) ∈ Cτ .

(4.9)

Furthermore, recalling Eq. (2.12) for u− , we see that there is another constant Uτ such that u− (t ) ≤

Uτ t 1−σ

for t ∈ (0, τ − t0 ].

(4.10)

Inserting Eqs. (4.9) and (4.10) into (4.7), we obtain the thesis (4.5) with Λτ := Uτ L τ . (ii) For each t ∈ [t0 , τ ], Eq. (4.5) implies t ds (t − t0 )σ . = Λ

ψ − ψ

J (ψ)(t) − J (ψ )(t) ≤ Λτ ψ − ψ τ τ τ 1−σ σ t0 (t − s) Taking the sup over t, we obtain the thesis (4.6). Lemma 4.6 (Main Consequence of the Control Inequality for R). One has J (D) ⊂ D.

(4.11)

Proof. Let ψ ∈ D; then J (ψ) − ϕap = [J (ϕap ) − ϕap ] + [J (ψ) − J (ϕap )] = −E(ϕap ) + [J (ψ) − J (ϕap )].

(4.12)

We write this equality at any t ∈ [t0 , T ), explicitating J (ψ)(t) − J (ϕap )(t); this gives t ds e(t−s)A J (ψ)(t) − ϕap (t) = −E(ϕap )(t) + t0

× [P(ψ(s), s) − P(ϕap (s), s)].

(4.13)

Now, we apply the norm of F to both sides and use Eq. (3.2) for E(ϕap )(t), (2.11) for e(t−s)A , (3.11) for the growth of P from ϕap : in this way we obtain t

J (ψ)(t) − ϕap (t) ≤ E(t) + ds u− (t − s) ( ψ(s) − ϕap (s) , s). (4.14) t0

On the other hand, ψ(s) − ϕap (s) ≤ R(s) implies ( ψ(s) − ϕap (s) , s) ≤ (R(s), s); inserting this into (4.14), and using the control inequality (3.12) for

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R, we conclude

J (ψ)(t)−ϕap (t) ≤ E(t)+

t

ds u− (t−s)(R(s), s) ≤ R(t),

(4.15)

t0

i.e. J (ψ) ∈ D. The invariance of D under J is a central result; with the previously shown properties of J , it allows to set up the Peano–Picard iteration and get ultimately a ﬁxed point of this map. Deﬁnition 4.7. (ϕk ) (k ∈ N) is the sequence in D deﬁned recursively by ϕ0 := ϕap ,

ϕk := J (ϕk−1 ) (k ≥ 1).

(4.16)

Lemma 4.8. Let τ ∈ [t0 , T ). For all k ∈ N, one has

ϕk+1 (t) − ϕk (t) ≤ Στ

Λkτ Γ(σ)k (t − t0 )kσ Γ(kσ + 1)

for t ∈ [t0 , τ ],

(4.17)

where Λτ is the constant of Eq. (4.5) and Στ := maxt∈[t0 ,τ ] E(t). So,

ϕk+1 − ϕk τ ≤ Στ

Θkτ σ , Γ(kσ + 1)

Θτ σ := Λτ Γ(σ) (τ − t0 )σ .

(4.18)

Proof. Equation (4.18) is an obvious consequence of (4.17). We prove (4.17) by recursion, indicating with a subscript k the thesis at a speciﬁed order. We have ϕ1 − ϕ0 = J (ϕap ) − ϕap = −E(ϕap ), whence ϕ1 (t) − ϕ0 (t) ≤ E(t) ≤ Στ ; this gives (4.17)0 . Now, we suppose (4.17)k to hold and infer its analogue of order k + 1. To this purpose, we keep in mind Eq. (4.5) and write

ϕk+2 (t) − ϕk+1 (t) = J (ϕk+1 )(t) − J (ϕk )(t)

t

ϕk+1 (s) − ϕk (s)

≤ Λτ ds (t − s)1−σ t0 ≤ Λ τ Στ

Λkτ Γ(σ)k Γ(kσ + 1)

t

ds t0

(s − t0 )kσ . (t − s)1−σ

On the other hand, we haved t (s − t0 )kσ Γ(kσ + 1)Γ(σ) (t − t0 )(k+1)σ ; ds = 1−σ (t − s) Γ((k + 1)σ + 1) t0

(4.19)

(4.20)

inserting (4.20) into (4.19) we obtain the thesis (4.17)k+1 . check this, make in the integral the change of variable s = t0 + x(t − t0 ), with x ∈ [0, 1], and then use the general identity Z 1 Γ(α)Γ(β) dx xα−1 (1 − x)β−1 = for α, β > 0. Γ(α + β) 0 d To

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The next (and ﬁnal) Lemma is a generalization of the inequality (4.18), based on the Mittag–Leﬄer function Eσ (see, e.g., [17]). For any σ > 0, this is the entire function deﬁned by Eσ : C → C,

z → Eσ (z) :=

+∞ =0

z . Γ(σ + 1)

(4.21)

In particular, Eσ (z) ∈ [1, +∞) for all z ∈ [0, +∞) and E1 (z) = ez for all z ∈ C. Lemma 4.9. For all τ ∈ [t0 , T ) and k, k ∈ N,

ϕk − ϕk τ ≤ Στ

Θhτσ Eσ (Θτ σ ), Γ(hσ + 1)

h := min(k, k )

(4.22)

(Θτ σ being deﬁned by (4.18)); this implies that (ϕk ) is a Cauchy sequence. Proof. It suﬃces to consider the case k > k (so that h = k). Writing ϕk − ϕk = k −1 j=k (ϕj+1 − ϕj ) and using Eq. (4.18) we get

ϕk − ϕk τ ≤ Στ

−1 k

j=k

Θjτ σ . Γ(jσ + 1)

(4.23)

On the other hand, for each z ≥ 0, k −1

j=k

zj zj z ≤ = zk Γ(jσ + 1) Γ(jσ + 1) Γ((k + )σ + 1) +∞

+∞

j=k

≤

=0

k

z Γ(kσ + 1)

+∞ =0

zk z = Eσ (z) Γ(σ + 1) Γ(kσ + 1)

(4.24)

(the last inequality depends on the relation Γ(α + β + 1) ≥ Γ(α + 1)Γ(β + 1) for α, β ≥ 0). With z = Θτ σ , from (4.23) and (4.24) we obtain (4.22). This equation, with the obvious fact that z h /Γ(hσ + 1) → 0 for h → ∞ and ﬁxed z ∈ C, implies

ϕk − ϕk τ → 0

for (k, k ) → ∞,

(4.25)

for each ﬁxed τ ∈ [t0 , T ). In conclusion, (ϕk ) is a Cauchy sequence. Proof of Proposition 3.7. (ϕk ) being a Cauchy sequence, limk→∞ ϕk := ϕ exists in C([t0 , T ), F); ϕ belongs to D, because this set is closed. By the continuity of J , we have J (ϕ) = lim J (ϕk ) = lim ϕk+1 = ϕ. k→∞

Now, recalling (4.4) we get the thesis.

k→∞

(4.26)

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Remark 4.10. The Mittag–Leﬄer function Eσ on [0, +∞) is strictly related to a linear integral equation. More precisely, given σ > 0 let us consider the following problem: ﬁnd G ∈ C([0, +∞), R) such that t G(s) 1 ds for all t ∈ [0, +∞). (4.27) G(t) = 1 + Γ(σ) 0 (t − s)1−σ This has a unique solution G(t) := Eσ (tσ ) for t ∈ [0, +∞).

(4.28)

One checks directly that the above G solves (4.27)e ; uniqueness of the solution follows from the linearity of the problem and from Lemma 2.6. Integral equations like (4.27) are related to the so-called “fractional diﬀerential equations” (see e.g. [4], also mentioning Eσ ). 5. Applications of Proposition 3.7 to Systems with Quadratic Nonlinearity: Local and Global Results The setting. Throughout this section we consider a set (F+ , F, F− , A, u, u− , P, ξ) with the following features. F+ , F, F− are Banach spaces, A is an operator and u, u− are semigroup estimators fulﬁlling conditions (P1)–(P5). Furthermore: (Q1) P is a bilinear map such that P : F × F → F− ,

(f, g) → P(f, g);

(5.1)

we assume continuity of P, which is equivalent to the existence of a constant K ∈ (0, +∞) such that

P(f, g) − ≤ K f

g

(5.2)

ξ ∈ C 0,1 ([0, +∞), F− ),

(5.3)

for all f, g ∈ F.f (Q2) We have

(recall that C 0,1 stands for the locally Lipschitz maps). Having made the above assumptions, let us ﬁx some notations. e To

this purpose one inserts into (4.27) the series expansion coming from (4.21), and then uses the identity in the previous footnote. f Of course, in the trivial case P = 0 we could fulﬁll (5.2) with K = 0 as well. In the sequel, we always assume K > 0, to avoid tedious speciﬁcations in many subsequent statements and formulas. In any case, such statements and formulas could be extended to K = 0 by elementary limiting procedures.

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Deﬁnition 5.1. From now on: (i) U ∈ C([0, +∞), [0, +∞)) is a nondecreasing function such that t ds u− (s) ≤ U(t) for t ∈ [0, +∞), U(0) = 0

(5.4)

0

t (e.g., U(t) := 0 ds u− (s); in any case, (5.4) and the positivity of u− imply U(t) > 0 for all t > 0). We put U(+∞) := limt→+∞ U(t) ∈ (0, +∞]. (ii) Ξ− ∈ C([0, +∞), [0, +∞)) is any nondecreasing function such that

ξ(t) − ≤ Ξ− (t) for t ∈ [0, +∞)

(5.5)

(e.g., Ξ− (t) := sups∈[0,t] ξ(s) − ). We put Ξ− (+∞) := limt→+∞ Ξ− (t) ∈ [0, +∞]. (iii) P is the (aﬃne) quadratic map induced by P and ξ in the following way: P : F × [0, +∞) → F− ,

(f, t) → P(f, t) := P(f, f ) + ξ(t).

(5.6)

Properties of P. Let us analyze this map, so as to match the schemes of the previous sections. First of all we note that Dom P is semiopen in F × R, in the sense deﬁned in (P6). Furthermore, we have the following. Proposition 5.2. For all (f, t) and (f , t ) ∈ F × [0, +∞),

P(f, t) − P(f , t ) − ≤ 2K f f − f + K f − f 2 + ξ(t) − ξ(t ) − . (5.7) Proof. For the sake of brevity, we deﬁne h := f − f . Then P(f, t) = P(f + h, f + h) + ξ(t) = P(f , f ) + P(f , h) + P(h, f ) + P(h, h) + ξ(t); subtracting P(f , t ), we get P(f, t) − P(f , t ) = P(f , h) + P(h, f ) + P(h, h) + ξ(t) − ξ(t ). We apply − to both sides, taking into account Eq. (5.2); this gives P(f, t) − P(f , t ) − ≤ 2K f h + K h 2 + ξ(t) − ξ(t ) − , yielding the thesis (5.7). The previous proposition has two straightforward consequences. Corollary 5.3. For each bounded set C of F × [0, +∞), there are two constants L = L(C), M = M (C) such that

P(f, t) − P(f , t ) − ≤ L f − f + M |t − t |

for (f, t), (f , t ) ∈ C;

(5.8)

so, P fulﬁlls condition (P6). Let us denote with B and I the projections of C on F and [0, +∞), respectively. Then we can take L := 4K B , where B := supf ∈B f ;

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furthermore, we can take for M any constant such that ξ(t) − ξ(t ) − ≤ M |t − t | for t, t ∈ I. Proof. In Eq. (5.7), we substitute the relations f ≤ B , f − f 2 ≤ ( f +

f ) f − f ≤ 2 B

f − f , and the inequality deﬁning M . Corollary 5.4. Let us consider any function φ ∈ C([0, T ), F). Then

P(f, t) − P(φ(t), t) − ≤ ( f − φ(t) , t) : [0, +∞) × [0, T ) → [0, +∞),

for all f ∈ F, t ∈ [0, T ),

(5.9)

(r, t) → (r, t) := 2K φ(t) r + Kr2 .

(5.10)

The function is a growth estimator for P from φ, in the sense of Deﬁnition 3.5 (with a radius of the tube ρ(t) := +∞ for all t). Proof. Use Eq. (5.7) with (f , t ) := (φ(t), t). Cauchy and Volterra problems; approximate solutions. These problems will always be considered taking t0 := 0 as the initial time; we will write VP(f0 ) := VP(f0 , 0),

CP(f0 ) := CP(f0 , 0) for each f0 .

(5.11)

For the above problems, we have the following results. (a) If f0 ∈ F+ and F− is reﬂexive, VP(f0 ) is equivalent to CP(f0 ) (see Proposition 2.4). (b) For any f0 ∈ F, uniqueness and local existence are granted for VP(f0 ): see Propositions 2.5 and 3.10. (c) We can apply to VP(f0 ) Proposition 3.7 on approximate solutions, choosing arbitrarily the approximate solution ϕap ; as an error estimator, we can use at will the function in Corollary 5.4 (or any upper bound for it). This yields the following statement. Proposition 5.5. Let f0 ∈ F and T ∈ (0, +∞]. Let us consider for VP(f0 ) an approximate solution ϕap ∈ C([0, T ), F), and suppose there are functions E, D, R ∈ C([0, T ), [0, +∞)) such that (i)–(iii) hold: (i) ϕap has the integral error estimate

E(ϕap (t)) ≤ E(t)

for t ∈ [0, T );

(5.12)

(ii) one has

ϕap (t) ≤ D(t)

for t ∈ [0, T );

(5.13)

(iii) with K as in (5.2) and U as in (5.4), R solves the control inequality t E(t) + K ds u− (t − s)(2D(s)R(s) + R2 (s)) ≤ R(t) for t ∈ [0, T ). (5.14) 0

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Then, (a) and (b) hold: (a) VP(f0 ) has a solution ϕ : [0, T ) → F; (b) one has

ϕ(t) − ϕap (t) ≤ R(t)

for t ∈ [0, T ).

(5.15)

Proof. We refer to the control inequality (3.12) in Proposition 3.7 (with t0 = 0). Due to Corollary 5.4, the growth of P from ϕap has the quadratic estimator (r, t) := 2K ϕap (t) r + Kr2 ; binding ϕap (t) via (5.13) we get another estimator, that we call again , of the form (r, t) = 2KD(t)r + Kr2 .

(5.16)

With this choice of , the control inequality (3.12) takes the form (5.14) and (a) and (b) follow from Proposition 3.7. How to handle the control inequality (5.14). Rephrasing Remark 3.9 in the present case, we repeat that R is the only unknown in (5.14). In fact, the functions E, D appearing therein can be determined when ϕap is given, and u− , K can be obtained from A, P (as an example, the computation of u− and K for the NS equations will be presented in Secs. 6 and 7). Two basic strategies to ﬁnd a function R solving (5.14) on an interval [0, T ), if it exists, can be introduced: (a) the analytical strategy: one makes some ansatz for R, substitutes it into (5.14) and checks whether the inequality is fulﬁlled; (b) the numerical strategy. Let us mention that a numerical approach was presented in [12], for the simpler control inequality considered therein; in that case it was possible to transform the control equality (with ≤ replaced by =) into an equivalent Cauchy problem for R, and then solve it by a standard package for ODEs. The approach of [12] cannot be used for (5.14) due to the singularity of u− (t) for t → 0+ ; a diﬀerent numerical attack could be used, but this is not so simple and its features suggest to treat it extensively elsewhere. For the above reasons, in the present work we only give an introductory sketch of the numerical strategy: see Appendix B. In the rest of the paper, starting from the next paragraph, we will use the analytical strategy (a). Some special results on VP(f0 ). All these results will be derived solving the control inequality (5.14) by analytic means, in special cases. Proposition 5.6. Let f0 ∈ F and T ∈ (0, +∞]. Let us consider for VP(f0 ) an approximate solution ϕap ∈ C([0, T ), F), and suppose there are functions E, D ∈ C([0, T ], [0, +∞)) such that (i)–(iii) hold: (i) E is nondecreasing and binds the integral error as in (5.12); (ii) D is nondecreasing and binds ϕap as in (5.13);

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(iii) with K as in (5.2) and U as in (5.4),g 2

KU(T )E(T ) + 2KU(T )D(T ) ≤ 1.

(5.17)

Then VP(f0 ) has a solution ϕ : [0, T ) → F and, for all t ∈ [0, T ),

ϕ(t) − ϕap (t) ≤ R(t),  1−2KU(t)D(t) − (1−2KU(t)D(t))2 − 4KU(t)E(t) R(t) := 2KU(t)  E(0)

if t ∈ (0, T ), if t = 0; (5.18)

the above prescription gives a well deﬁned, nondecreasing function R ∈ C([0, T ), [0, +∞)). Proof. We refer to Proposition 5.5, and try to fulﬁll the control inequality (5.14) with a nondecreasing R ∈ C([0, T ), [0, +∞)). Noting that R(s) ≤ R(t), D(s) ≤ D(t) for s ∈ [0, t], we have E(t) + K

t

ds u− (t − s)(2D(s)R(s) + R2 (s)) t ds u− (t − s) ≤ E(t) + K(2D(t)R(t) + R2 (t)) 0

0

≤ E(t) + K(2D(t)R(t) + R2 (t))U(t);

(5.19)

t t the last inequality follows from 0 ds u− (t − s) = 0 ds u− (s) ≤ U(t). Due to (5.19), (5.14) holds if E(t) + K(2 U(t)D(t)R(t) + U(t)R2 (t)) ≤ R(t), i.e. KU(t)R(t)2 − (1 − 2KU(t)D(t))R(t) + E(t) ≤ 0.

(5.20)

This inequality is fulﬁlled as an equality if we deﬁne R as in (5.18), provided that 2 KU(t)E(t) + 2KU(t)D(t) ≤ 1; this happens for each t ∈ [0, T ) due to the assumption (5.17).h The function R on [0, T ) deﬁned by (5.18) is continuous and nonnegative; to conclude the proof, we must check it to be nondecreasing. To this purpose, we note

course, in the case T = +∞ (5.17) implies U (+∞) < +∞. enough, we take for R(t) the deﬁnition (5.18) since this gives the smallest nonnegative solution of Eq. (5.20). g Of

h Obviously

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that R(t) = Υ(KU(t), D(t), E(t)),   1 − 2µδ − (1 − 2µδ)2 − 4µ Υ(µ, δ, ) := 2µ 

if µ > 0,

(5.21)

if µ = 0.

The above function Υ has domain √ Dom Υ := {(µ, δ, ) | µ, δ, ≥ 0, 2µδ + 2 µ ≤ 1},

(5.22)

containing all triples (KU(t), D(t), E(t)) due to (5.17); one checks by elementary means (e.g., computing derivatives) that Υ is a nondecreasing function of each one of the variables µ, δ, , when the other two are ﬁxed. Remark 5.7. The inequality (5.17) is certainly fulﬁlled if (E(T ), D(T )) or T are suﬃciently small (recall that U(T ) vanishes for T → 0+ ). An example: The zero approximate solution. For simplicity, we suppose u(t) ≤ 1

for all t ∈ [0, +∞).

(5.23)

Let f0 ∈ F, T ∈ (0, +∞]; we apply Proposition 5.6, choosing for VP(f0 ) the trivial approximate solution ϕap (t) := 0 for t ∈ [0, T ). Lemma 5.8. ϕap := 0 has the integral error t tA E(ϕap )(t) = −e f0 − ds e−(t−s)A ξ(s);

(5.24)

(5.25)

0

with Ξ− as in Deﬁnition 5.5, E(ϕap )(t) has the estimate

E(ϕap )(t) ≤ F(t),

F(t) := f0 + Ξ− (t)U(t).

(5.26)

Proof. Equation (5.25) follows from the general deﬁnition (3.1) of integral error, and from the fact that P(ϕap (s), s) = ξ(s). Having Eq. (5.25), we derive (5.26) in the following way. First of all, t

E(ϕap )(t) ≤ u(t) f0 + ds u− (t − s) ξ(s) − ; (5.27) 0

but u(t) ≤ 1, ξ(s) − ≤ Ξ− (t) for s ∈ [0, t], so t ds u− (t − s) ≤ f0 + Ξ− (t)U(t).

E(ϕap )(t) ≤ f0 + Ξ− (t) 0

From the previous lemma and Proposition 5.6, we infer the following result.

(5.28)

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Proposition 5.9. With u, F as in (5.23) and (5.26), assume 4KU(T )F(T ) ≤ 1.

(5.29)

Then VP(f0 ) has a solution ϕ : [0, T ) → F and, for all t in this interval,

ϕ(t) ≤ F(t)X (4KU(t)F(t)). Here X ∈ C([0, 1], [1, 2]) is the increasing function deﬁned by  √ 1− 1−z for z ∈ (0, 1], X (z) := (z/2)  1 for z = 0.

(5.30)

(5.31)

Proof. According to the previous lemma, we can apply Proposition 5.6 with ϕap = 0, E = F; obviously, we have for ϕap (t) the estimator D(t) := 0. Eqs. (5.17) and (5.18) yield the present relations (5.29), (5.30) and (5.31); in particular, the function R in (5.18) is given by R(t) = (1 − 1 − 4KU(t)F(t))/2KU(t) = F(t) X (4KU(t)F(t)). Of course, the basic inequality (5.29) is fulﬁlled if (f0 , Ξ− (T )) or T are suﬃciently small. Further results (global in time) for VP(f0 ), under special assumptions. We keep the assumptions (P1)–(P5) of Sec. 2 and (Q1) and (Q2) at the beginning of this section, and put more speciﬁc requirements on the semigroup estimators u, u− . More precisely, we add to (P4) and (P5) the following conditions, involving two constants B ∈ [0, +∞),

N ∈ (0, +∞).

(5.32)

(P4 ) The semigroup estimator u has the form u(t) = e−Bt

for t ∈ [0, +∞).

(5.33)

(P5 ) The semigroup estimator u− has the form u− (t) = µ− (t)e−Bt

for t ∈ (0, +∞), (5.34) 1 µ− ∈ C((0, +∞), (0, +∞)), µ− (t) = O 1−σ for t → 0+ (σ ∈ [0, 1)), (5.35) t t dsµ− (t − s) e−Bs ≤ N for t ∈ [0, +∞). (5.36) 0

Proposition 5.10. Given f0 ∈ F, let us consider for VP(f0 ) an approximate solution ϕap ∈ C([0, +∞), F). Suppose there are constants E, D ∈ [0, +∞)

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such that : (i) ϕap admits the integral error estimate

E(ϕap )(t) ≤ Ee−Bt

for t ∈ [0, +∞);

(5.37)

(ii) for all t as above,

ϕap (t) ≤ De−Bt ; (iii) with N as in (5.36) and K as in (5.2), one has √ 2 KNE + 2KND ≤ 1.

(5.38)

(5.39)

Then VP(f0 ) has a global solution ϕ : [0, +∞) → F and, for all t ∈ [0, +∞),

ϕ(t) − ϕap (t) ≤ Re−Bt , 1 − 2KND − (1 − 2KND)2 − 4KNE R := . 2KN

(5.40)

Proof. Step 1. The control inequality. We use again Proposition 5.5 and the control inequality (5.14). With the notations of the cited proposition, we have E(t) := Ee−Bt , D(t) := De−Bt and u− has the expression (5.34). So, (5.14) takes the form t −Bt Ee +K ds µ− (t − s)e−B(t−s) (2De−Bs R(s) + R2 (s)) ≤ R(t), (5.41) 0

that we regard as an inequality for an unknown nonnegative function R. Step 2. Searching for a global solution R of (5.41). We try to fulﬁll (5.41) with R(t) := Re−Bt

for all t ∈ [0, +∞),

(5.42)

with R ≥ 0 an unknown constant. Then, the left-hand side of (5.41) is

t −Bt 2 −Bs e ds µ− (t − s)e E + K(2DR + R ) 0

≤ e−Bt (E + K(2DR + R2 )N ),

(5.43)

where the inequality depends on (5.36). The last expression is bounded by R(t) if R fulﬁlls the inequality E + K(2DR + R2 )N ≤ R, i.e. KNR 2 − (1 − 2KND)R + E ≤ 0.

(5.44)

This condition is fulﬁlled as an equality if we deﬁne R as in (5.40); this R is well deﬁned and nonnegative due to the assumption (5.39).i Due to the above considerations, the thesis is proved. i And,

in fact, is the smallest nonnegative solution of (5.44).

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Applications to cases with exponentially decaying forcing. From here to the end of the section, we add to (P1)–(P5 ), (Q1) and (Q2) the following condition: (Q3) There is a constant J ∈ [0, +∞) such that (with B as in (P4 ) and (P5 ))

ξ(t) − ≤ Je −2Bt

for all t ∈ [0, +∞).

(5.45)

Two cases where (Q3) holds are: (i) the trivial case ξ(t) = 0 for all t; (ii) situations where the external forcing is switched oﬀ in the future. Hereafter we present two applications of Proposition 5.10, corresponding to diﬀerent choices for ϕap . Both of them give global existence for the exact solution ϕ of VP(f0 ) when the datum f0 is suﬃciently small, with suitable estimates of the form (5.40). Example: The zero approximate solution. Let f0 ∈ F; we reconsider, from the viewpoint of Proposition 5.10, the VP(f0 ) approximate solution ϕap (t) := 0

for t ∈ [0, +∞).

(5.46)

Lemma 5.11. ϕap := 0 has the integral error estimator

E(ϕap )(t) ≤ Fe −Bt ,

F := f0 + N J.

(5.47)

Proof. The integral error E(ϕap ) was already computed, see Eq. (5.25). From this equation and the assumptions (5.33) and (5.34) on u, u− we infer t −Bt

E(ϕap )(t) ≤ e

f0 + ds e−B(t−s) µ− (t − s) ξ(s) − ; (5.48) 0

inserting here the bound (5.45) for ξ(s) − , and using Eq. (5.36) for µ− we get t

E(ϕap )(t) ≤ e−Bt f0 + Je−Bt ds µ− (t − s)e−Bs 0

≤ e−Bt f0 + Je−Bt N.

(5.49)

From the previous lemma and Proposition 5.10, we infer the following. Proposition 5.12. With F as in (5.47), let 4KNF ≤ 1.

(5.50)

Then VP(f0 ) has a global solution ϕ : [0, +∞) → F and, for all t ∈ [0, +∞),

ϕ(t) ≤ F X (4KNF )e−Bt ,

(5.51)

with X as in (5.31). Proof. We apply Proposition 5.10 with ϕap := 0. The constants of the cited proposition are E = F and D = 0 (the ﬁrst equality follows from the previous lemma,

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the second one is obvious). Equations (5.39) and (5.40) yield the present relations (5.50), √ (5.51) and (5.31); in particular, the constant in (5.40) is given by R = (1 − 1 − 4KNF )/(2KN ) = F X (4KN F ). Example. The “A -ﬂow” approximate solution. Given f0 ∈ F, we consider for VP(f0 ) the approximate solution t ϕap (t) := etA f0 + ds e(t−s)A ξ(s) for t ∈ [0, +∞) (5.52) 0

(i.e. we use the ﬂow of the linear equation f˙ = Af + ξ(t)). Lemma 5.13. Let F be as in (5.47). The above ϕap has the integral error estimator

E(ϕap )(t) ≤ KNF 2 e−Bt

(5.53)

and fulﬁlls for all t ∈ [0, +∞) the norm estimate

ϕap (t) ≤ F e−Bt .

(5.54)

Proof. We ﬁrst derive Eq. (5.54). To this purpose, we note that the deﬁnition (5.52) of ϕap , the assumptions (5.33) and (5.34) on u, u− and Eq. (5.45) for

ξ(·) − , (5.36) for µ− imply t ds e−B(t−s) µ− (t − s) ξ(s) −

ϕap (t) ≤ e−Bt f0 + 0

≤ e−Bt f0 + Je−Bt

t

ds µ− (t − s)e−Bs

0

≤e

−Bt

f0 + Je

−Bt

N;

(5.55)

by comparison with the deﬁnition (5.47) of F , we get the thesis (5.54). Let us pass to the proof of (5.53). To this purpose we note that the deﬁnition (3.1) of integral error gives, in the present case, t ds e(t−s)A P(ϕap (s), ϕap (s)). (5.56) E(ϕap )(t) = − 0

From here and from Eq. (5.34) for u− , (5.2) for P we infer t

E(ϕap )(t) ≤ ds µ− (t − s)e−B(t−s) P(ϕap (s), ϕap (s)) − 0

≤K

t

ds µ− (t − s)e−B(t−s) ϕap (s) 2 .

(5.57)

0

In the last inequality, we insert the bound (5.54) and then recall (5.36). This gives t ds e−Bs µ− (t − s) ≤ KNF 2 e−Bt . (5.58)

E(ϕap )(t) ≤ KF 2 e−Bt 0

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Let us return to Proposition 5.10; with the previous lemma, this implies the following result. Proposition 5.14. Let us keep the deﬁnition (5.47) for F , and the assumption (5.50) 4KNF ≤ 1. The global solution ϕ : [0, +∞) → F of VP(f0 ) is such that, for all t ∈ [0, +∞),

ϕ(t) − etA f0 ≤ KNF 2 X(4KNF )e−Bt ;

(5.59)

here X ∈ C([0, 1], [1, 4]) is the increasing function deﬁned by √   1 − (z/2) − 1 − z X(z) := (z 2 /8)  1

for z ∈ (0, 1],

(5.60)

for z = 0.

Proof. According to the previous Lemma, we can apply Proposition 5.10 with E = KNF 2 and D = F . Equations (5.39) and (5.40) yield the present relations (5.50), (5.59) and (5.60); in particular, the constant in Eq. (5.40) is R = (1 − 2KNF −

√ 1 − 4KNF )/(2KN ) = KNF 2 X(4KNF ).

Remark 5.15. Most of the results presented in this section could be extended to the case P(f, t) = P(f, . . . , f ) + ξ(t), where P : Fm → F− is continuous and mlinear for some integer m ≥ 3. In this case, the growth of P from any approximate solution admits an estimator (r, t) more general than (5.10), which is polynomial of degree m in r. One could extend the analysis as well to the case P(f, t) = P(f, . . . , f, t)+ξ(t), involving a time dependent multilinear map P : Fm ×[0, +∞) → F− , (f1 , . . . , fm , t) → P(f1 , . . . , fm , t). These generalizations are not written only to save space.

6. The Navier–Stokes (NS) Equations on a Torus From here to the end of the paper, we work in any space dimension d ≥ 2.

(6.1)

Preliminaries: Distributions on Td , Fourier series and Sobolev spaces. Throughout this section, we use r, s as indices running from 1 to d and employ for them the Einstein summation convention on repeated, upper and lower indices; δrs or δ rs is the Kronecker symbol. Elements a, b, . . . of Rd or Cd will be written with upper or lower indices, according to convenience: (as ) or (as ), (bs ) or (bs ). In any case, a • b is the sum of product of the components of a and b, that will be written in diﬀerent ways to accomplish

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with the Einstein convention. Two examples, corresponding to diﬀerent positions for the indices of a, are a • b = as b s ,

a • b = δrs ar bs .

(6.2)

Let us consider the d-dimensional torus Td := T × · · · × T

(d times),

T := R/(2πZ).

(6.3)

A point of Td will be generally written x = (xr )r=1,...,d . We also consider the “dual” lattice Zd of elements k = (kr )r=1,...,d and the Fourier basis (ek )k∈Zd , made of the functions 1 eik • x (6.4) ek : Td → C, ek (x) := (2π)d/2 (k • x := kr xr makes sense as an element of Td ). We introduce the space of periodic ∞ , which is the dual of C ∞ (Td , C) ≡ CC (equipping distributions D (Td , C) ≡ DC the latter with the topology of uniform convergence of all derivatives); we write ∞ on a test function f ∈ CC . The v, f for the action of a distribution v ∈ DC ∞ weak topology on DC is the one induced by the seminorms pf (f ∈ CC ), where pf (v) := |v, f |. has a unique (weakly convergent) series expansion Each v ∈ DC v= vk ek , (6.5) k∈Zd

with coeﬃcients vk ∈ C for all k, given by vk = v, e−k .

(6.6)

and the The “Fourier series transformation” v → (vk ) is one-to-one between DC d space of sequences s (Z , C) ≡ sC , where

sC := {c = (ck )k∈Zd | ck ∈ C, |ck | = O(|k|p ) as k → ∞, for some p ∈ R}. (6.7) , which is In the sequel, we often use the mean of a distribution v ∈ DC

v :=

1 1 v, 1 = v0 d (2π) (2π)d/2

(6.8)

(in the ﬁrst passage, v, 1 means the action of v on the test function 1; the second relation follows from (6.6) with k = 0, noting that e0 = 1/(2π)d/2 . Of course, v, 1 = Td v(x)dx if v is an ordinary, integrable function). is the unique distribution v¯ The complex conjugate of a distribution v ∈ DC ∞ ¯ v , f for each f ∈ CC ; one has v¯ = k∈Zd vk e−k . such that v, f = ¯ From now on we will be mainly interested in the space of real distributions D (Td , R) ≡ D , deﬁned as follows: | v¯ = v} = {v ∈ DC | vk = v−k for all k ∈ Zd }; D := {v ∈ DC

(6.9)

we note that v ∈ D implies v ∈ R. The weak topology on D is the one inherited . from DC

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Let us write ∂s (s = 1, . . . , d) for the distributional derivative with respect to the coordinate xs ; from these derivatives, we deﬁne the distributional Laplacian → DC . Of course ∂s ek = iks ek , ∆ek = −|k|2 ek for each k. For ∆ := δ rs ∂r ∂s : DC , this implies any v ∈ DC ∂s v = i ks vk ek , ∆v = − |k|2 vk ek ; (6.10) k∈Td

k∈Td

(1 − ∆)m v =

(1 + |k|2 )m vk ek

(6.11)

k∈Td

for m ∈ 0, 1, 2, 3, . . . . For any m ∈ R, we will regard (6.11) as the deﬁnition of into itself. Comparing the previous Fourier (1 − ∆)m as a linear operator from DC expansions with (6.8), we ﬁnd ∂s v = 0,

∆v = 0.

(6.12)

All the above diﬀerential operators leave invariant the space of real distributions, more interesting for us; in the sequel we will ﬁx the attention on the maps ∂s , ∆, (1 − ∆)m : D → D . Let us consider the real Hilbert space L2 (Td , R, dx) ≡ L2 , i.e. v 2 (x)dx < +∞ L2 := v : Td → R Td  

  = v ∈ D |vk |2 < +∞ ;   d

(6.13)

k∈Z

this has the inner product and the associated norm v | wL2 := v(x)w(x)dx = vk wk , Td

v L2 :=

v 2 (x)dx = Td

(6.14)

k∈Zd

|vk |2 .

(6.15)

k∈Zd

To go on, we introduce the Sobolev spaces H n (Td , R) ≡ H n . For each n ∈ R, H n := {v ∈ D | (1 − ∆)n/2 v ∈ L2 }     (1 + |k|2 )n |vk |2 < +∞ ; = v ∈ D   d

(6.16)

k∈T

this is also a real Hilbert space with the inner product v | wn := (1 − ∆)n/2 v | (1 − ∆)n/2 wL2 = (1 + |k|2 )n vk wk k∈Td

(6.17)

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and the corresponding norm

v n := (1 − ∆)

n/2

v

L2

=

(1 + |k|2 )n |vk |2 .

(6.18)

k∈Td

One proves that

n ≥ n ⇒ H n → H n ,

n ≤ n .

(6.19)

In particular, H 0 is the space L2 and contains H n for each n ≥ 0; for any real n, ∆ is a continuous map of H n into H n−2 . Finally, let us recall that H n → D n

H → C

q

for each n ∈ R;

(6.20)

if q ∈ N, n ∈ (q + d/2, +∞).

(6.21)

In the above H n carries its Hilbertian topology, and D the weak topology; C q stands for the space C q (Td , R), with the topology of uniform convergence of all derivatives up to order q. Obviously enough, we could deﬁne as well the complex Hilbert spaces L2C and n HC ; however, these are never needed in the sequel. Spaces of vector valued functions on Td . To deal with the NS equations, we need vector extensions of all the above spaces and mappings. Let us stipulate the following: if V (Td , R) ≡ V is any vector space of real functions or distributions on Td , then V := V d = {v = (v 1 , . . . , v d ) | v r ∈ V for all r}.

(6.22)

This notation allows to deﬁne the spaces D , L2 , Hn . Any v = (v r ) ∈ D will be referred to as a vector ﬁeld on Td . We note that v has a unique Fourier series expansion (6.5) with coeﬃcients vk = (vkr )r=1,...,d ∈ Cd ,

vkr := v r , e−k ;

(6.23)

again, the reality of v ensures vk = v−k . We deﬁne componentwisely the mean v ∈ Rd of any v ∈ D (see Eq. (6.8)), the derivative operators ∂s : D → D , their iterates and, consequently, the Laplacian ∆. The prescription (6.11) gives a map (1−∆)m : D → D for all real m. Whenever V is made of ordinary functions, a d-uple v ∈ V can be identiﬁed with a function v : Td → Rd , x → v(x) = (v r (x))r=1,...,d . L2 is a real Hilbert space. Its inner product is as in (6.14), with v(x)w(x) and vk wk replaced by v(x) • w(x) = δrs v r (x) ws (x),

vk • wk = δrs vkr wks ;

the corresponding norm is as in (6.15), replacing v 2 (x) with |v(x)|2 = d and intending |vk |2 = r=1 |vkr |2 .

(6.24) d

r=1 v

r

(x)2

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For any real n, the Sobolev space Hn is made of all d-uples v with components v ∈ H n ; an equivalent deﬁnition can be given via Eq. (6.16), replacing therein L2 with L2 . Hn is a real Hilbert space with the inner product r

v | wn := (1 − ∆)n/2 v | (1 − ∆)n/2 wL2 =

(1 + |k|2 )n vk • wk .

(6.25)

k∈Td

The corresponding norm n is given, verbatim, by Eq. (6.18); Eq. (6.19) holds as well for Hn , Hn and their norms. Let us consider the Laplacian operator ∆ : D → D; for any real n ∆Hn ⊂ Hn−2 ,

(6.26)

and ∆ is continuous with respect to the norms n , n−2 . The embeddings (6.19)–(6.21) have obvious vector analogues. Zero mean vector ﬁelds. The space of these vector ﬁelds is D0 := {v ∈ D | v = 0}

(6.27)

(of course, v = 0 is equivalent to the vanishing of the Fourier coeﬃcient v0 ). Divergence free vector ﬁelds. Let us consider the divergence operator (linear, weakly continuous) div : D → D ,

v → div v := ∂r v r ;

(6.28)

we put DΣ := {v ∈ D | div v = 0}

(6.29)

and refer to this as to the space of divergence free (or solenoidal) vector ﬁelds. The description of these objects in terms of Fourier transform is obvious, namely: div v = i

(k • vk )ek

for all v =

k∈Zd

vk ek ∈ D ;

(6.30)

k∈Zd d

DΣ = {v ∈ D | k • vk = 0 for all k ∈ Z } = {v ∈ D | vk ∈≺ k ⊥ for all k ∈ Zd },

(6.31)

where ≺ k is the subspace of Cd spanned by k, and ⊥ is the orthogonal complement with respect to the inner product (b, c) ∈ Cd × Cd → ¯b • c. In the sequel, we will consider as well the subspace DΣ0 := DΣ ∩ D0 .

(6.32)

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Gradient vector ﬁelds. Let us consider the gradient operator (again linear, weakly continuous)

if p =

∂ : D → D ;

k∈Zd

p → ∂p := (∂s p)s=1,...,d ;

pk ek , then ∂p = i

(6.33)

kpk ek .

(6.34)

DΓ = {∂p | p ∈ D }

(6.35)

k∈Zd

The image

is a linear subspace of D , hereafter referred to as the space of gradient vector ﬁelds; for any vector ﬁeld w, comparison with (6.34) gives DΓ = {w ∈ D | wk ∈≺ k for all k ∈ Zd }.

(6.36)

Of course, if w is in this subspace, the distribution p such that w = ∂p is deﬁned up to an additive constant. We put ∂ −1 w := unique p ∈ D such that w = ∂p and p0 = 0.

(6.37)

This gives a linear map ∂ −1 : DΓ → D .

(6.38)

The Leray projection. Using the Fourier representations (6.31) and (6.36), one easily proves the following facts. (i) One has D = DΣ ⊕ DΓ

(6.39)

in algebraic sense, i.e. any v ∈ D has a unique decomposition as the sum of a divergence free and a gradient vector ﬁeld. (ii) The projection L : D → DΣ ,

v → Lv

(6.40)

corresponding to the decomposition (6.39) is given by Lv = (Lk vk )ek for all v = vk ek ∈ D, k∈Zd

k∈Zd d

Lk := orthogonal projection of C onto ≺ k ⊥ ;

(6.41)

more explicitly, for all c ∈ Cd , L0 c = c,

Lk c = c −

(k • c)k |k|2

for k ∈ Zd ,

k = 0.

(6.42)

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As usually, we refer to L as to the Leray projection; this operator is weakly continuous. From the Fourier representations of L, of the mean and of the derivatives one easily infers, for all v ∈ D , Lv = v,

L(∂s v) = ∂s (Lv),

L(∆v) = ∆(Lv).

(6.43)

A Sobolev framework for the previous decomposition. For n ∈ R, let us deﬁne HnΣ := Hn ∩ DΣ = {v ∈ Hn | div v = 0};

(6.44)

HnΓ := Hn ∩ DΓ = {w ∈ Hn | w = ∂p, p ∈ D };

(6.45)

∂H n := {∂p | p ∈ H n }.

(6.46)

Then the following holds for each n: (i) HnΓ is a closed subspace of the Hilbert space (Hn , | n ) (because div is continuous between this Hilbert space and D with the weak topology). (ii) One has HnΓ = ∂H n+1

(6.47)

and HnΓ is also a closed subspace of Hn . The map ∂ −1 of Eq. (6.37) is continuous between HnΓ and H n+1 . (iii) Denoting with ⊥n the orthogonal complement in (Hn , | n ), we have HnΣ ⊥n = HnΓ

(6.48)

and L Hn is the orthogonal projection of Hn onto HnΣ ; so, as usual for Hilbertian orthogonal projections,

Lv n ≤ v n

for all v ∈ Hn .

(6.49)

Other Sobolev spaces of vector ﬁelds. For n ∈ R, we put Hn0 := Hn ∩ D0 := {f ∈ Hn | f = 0};

(6.50)

HnΣ0 := HnΣ ∩ Hn0 := {f ∈ Hn | divf = 0, f = 0}.

(6.51)

Then, Hn0 is a closed subspace of Hn (by the continuity of : Hn → C); the same holds for HnΣ0 , since this is the intersection of two closed subspaces. The space (6.51) plays an important role in the sequel; we will often use the Fourier representation     (1 + |k|2 )n |fk |2 < +∞, k • fk = 0 for all k ∈ Zd , f0 = 0 . HnΣ0 = f ∈ D   d k∈Z

(6.52)

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Some inclusions. We note that the relations ∆Hn ⊂ Hn−2 and div(∆v) = ∆(div v), ∆v = 0 for all v ∈ D imply the following, for each real n: , ∆Hn ⊂ Hn−2 0

∆HnΣ ⊂ Hn−2 Σ0 .

(6.53)

A digression: Estimates on certain series. Let us deﬁne Zd0 := Zd \{0};

(6.54)

throughout the section, n is a real number such that n>

d . 2

(6.55)

The series considered hereafter are used shortly afterwards to derive quantitative estimates on the fundamental bilinear map appearing in the NS equations: by this we mean the map sending two vector ﬁelds v, w on Td into the vector ﬁeld v • ∂w (see the next paragraph). The estimates we give are also useful for the numerical computation of those series. In both lemmas hereafter, Zd := Zd

or Zd0 .

(6.56)

Lemma 6.1. One has 1 1 < +∞. (2π)d (1 + |h|2 )n d

Σn :=

(6.57)

h∈Z

√ For any real “cutoﬀ ” λ ≥ 2 d, one has Sn (λ) < Σn ≤ Sn (λ) + δSn (λ)

(6.58)

where Sn (λ) :=

δSn (λ) :=

1 (2π)d

h∈Zd ,|h|<λ

1 , (1 + |h|2 )n

(1 + d)n √ . 2d−1 π d/2 Γ(d/2)(2n − d) (λ − d)2n−d

(6.59)

(6.60)

Proof. See Appendix C. Lemma 6.2. For k ∈ Zd , deﬁne Kn (k) ≡ Knd (k) :=

|k − h|2 (1 + |k|2 )n−1 ; (2π)d (1 + |h|2 )n (1 + |k − h|2 )n d h∈Z

then, (i) and (ii) hold.

(6.61)

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(i) One has Kn (k) < +∞ for all k ∈ Zd ; furthermore, with Σn as in (6.57), Kn (k) → Σn

for k → ∞.

(6.62)

Thus, sup Kn (k) < +∞.

(6.63)

k∈Zd

√ (ii) Let us choose any “cutoﬀ function” Λ : Zd → [2 d, +∞) and deﬁne  if Λ(k) < |k|,  1 + |k|2 d λ : Z → (0, +∞), k → λ(k) := 1 + |k|2  if Λ(k) ≥ |k|. 1 + (Λ(k) − |k|)2

(6.64)

Then, for all k ∈ Zd , Kn (k) < Kn (k) ≤ Kn (k) + δKn (k),

(6.65)

where Kn (k) :=

δKn (k) :=

(1 + |k|2 )n−1 (2π)d

h∈Zd ,|h|<Λ(k)

|k − h|2 , (1 + |h|2 )n (1 + |k − h|2 )n

(1 + d)n λ(k)n−1 2d−1 π d/2 Γ(d/2)(2n − d) (Λ(k) −

√ . d)2n−d

(6.66)

(6.67)

Finally, suppose the cutoﬀ Λ has the property α|k| ≤ Λ(k) ≤ β|k| for all k ∈ Zd such that |k| ≥ χ (χ ≥ 0, then

1 < α ≤ β);

Kn (k) → Σn ,

δKn (k) = O

1 |k|2n−d

(6.68)

→0

for k → ∞.

(6.69)

Proof. See Appendix C. The fundamental bilinear map. Let v, w ∈ Hn . For each r, s ∈ {1, 2, . . . , d}, v r ∈ H n and ∂r ws ∈ H n−1 are ordinary real functions: note that v r ∈ C 0 by the embedding (6.21), and ∂r ws ∈ L2 since n − 1 > d/2 − 1 ≥ 0. These functions can be multiplied pointwisely, and this allows to deﬁne v • ∂w := (v • ∂ws )s=1,...,d ,

v • ∂ws := v r ∂r ws : Td → C.

(6.70)

Proposition 6.3. (i) Consider v, w ∈ Hn . The vector ﬁeld v • ∂w has Fourier coeﬃcients i [(vh • (k − h)]wk−h . (6.71) (v • ∂w)k = d/2 (2π) d h∈Z

Furthermore, v • ∂w ∈ Hn−1 .

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(ii) The bilinear map Hn × Hn → Hn−1 ,

(v, w) → v • ∂w

(6.72)

admits an estimate (indicating continuity)

v • ∂w n−1 ≤ Kn v n w n

(6.73)

for all v, w as above, with a suitable constant Kn ≡ Knd ∈ (0, +∞). For the latter one can take any constant such that sup Kn (k) ≤ Kn , (6.74) k∈Zd

Kn (k) :=

|k − h|2 (1 + |k|2 )n−1 d 2 (2π) (1 + |h| )n (1 + |k − h|2 )n d

(6.75)

h∈Z

(as in (6.61), with Zd = Zd ). Proof. See Appendix D. Lemma 6.4. Let v ∈ HnΣ , w ∈ Hn . Then v • ∂w = 0;

(6.76)

combined with Proposition 6.3, this gives v • ∂w ∈ Hn−1 . 0 Proof. For s = 1, . . . , d, integration by parts and the assumption 0 = div v = ∂r v r give 1 1 s r s v • ∂w = dx v ∂r w = − dx (∂r v r )ws = 0. d d (2π) Td (2π) Td Proposition 6.5. The bilinear map HnΣ0 × HnΣ0 → Hn−1 , 0

(f, g) → f • ∂g

(6.77)

admits an estimate

f • ∂g n−1 ≤ Kn f n g n

(6.78)

for all f, g ∈ HnΣ0 , with a suitable constant Kn ≡ Knd ∈ (0, +∞). One can take for the latter any constant such that sup Kn (k) ≤ Kn , (6.79) k∈Zd 0

Kn (k) :=

|k − h|2 (1 + |k|2 )n−1 (2π)d (1 + |h|2 )n (1 + |k − h|2 )n d h∈Z0

(as in (6.61), with Zd = Zd0 ).

(6.80)

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Proof. See Appendix D. To conclude, we report another integral identity frequently used in the sequel. Lemma 6.6. For any v ∈ HnΣ , one has v | v • ∂vL2 = 0

(6.81)

Proof. We have (with q, s, r ∈ {1, . . . , d}), v | v • ∂vL2 = dx δqs v q (v r ∂r v s ).

(6.82)

Td

From here we infer, integrating by parts, dx δqs ∂r (v q v r )v s v | v • ∂vL2 = − Td

=−

dx δqs (∂r v )v v − q

Td

r s

=−

Td

Td

dx δqs (∂r v q )v r v s

dx δqs v q (∂r v r )v s (6.83)

since div v = 0; by comparison with (6.82) we obtain v | v • ∂vL2 = −v | v • ∂vL2 ,

(6.84)

whence the thesis (6.81). Remark 6.7. The inequality (6.73) (or (6.78)) is known from the literature, in this form or in some variant: see, for example, one of the standard references on NS equations cited in the Introduction. To our knowledge, the novelty of Proposition 6.3 (or Proposition 6.5) with respect to the already published material is the rule (6.74) (or (6.79)) to determine Kn , that can be used with Lemmas 6.1 and 6.2 to provide a numerical value for this constant. Examples of this computation appear in Sec. 10 and Appendix H. The method employed in Appendix D to prove Propositions 6.3 and 6.5 is very similar to one employed in [13] to estimate the product of two scalar functions in H n (Rd ). In that paper, already mentioned in the Introduction, we have given a rule similar to (6.74) (or (6.79)) to ﬁnd a constant Cnd ≡ Cn such that pq H n (Rd ) ≤ Cn p H n (Rd ) q H n (Rd ) ; furthermore, using convenient trial functions p, q we have shown this constant to be very close to the smallest one fulﬁlling the inequality. Due to the similarities with [13], the constant Kn provided by (6.74) (or (6.79)) is hopingly close to the smallest one for the inequality (6.73) (or (6.78)).

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Functions on Td , depending on time. Suppose we have a function ρ : [t0 , T ) ⊂ R → V

or V,

t → ρ(t)

(6.85)

where V , V stand for some spaces of R or Rd valued functions on Td . At each “time” t ∈ [t0 , T ), this gives a function x ∈ Td → ρ(t)(x); of course, we will use the more common notation ρ(x, t) := ρ(t)(x).

(6.86)

The incompressible NS equations, in the Leray formulation. Let us recall that C 0,1 indicates the locally Lipschitz maps. Deﬁnition 6.8. The incompressible NS Cauchy problem with initial datum v0 ∈ Hn+1 and forcing term η ∈ C 0,1 ([0, +∞), Hn−1 Σ Σ ), in the Leray formulation, is the following. n−1 1 Find ν ∈ C([0, T ), Hn+1 Σ ) ∩ C ([0, T ), HΣ ), such that ν(t) ˙ = ∆ν(t) − L ν(t) • ∂ν(t) + η(t) for t ∈ [0, T ),

ν(0) = v0

(6.87)

(for some T ∈ (0, +∞]). Remark 6.9. (i) The requirement ν ∈ C([0, T ), Hn+1 Σ ) ensures by itself that the right-hand side of the above diﬀerential equation is in C([0, T ), Hn−1 Σ ). (ii) The diﬀerential equation in (6.87) can be interpreted as the usual NS equation for an incompressible ﬂuid, in a convenient adimensional formulation. For each t ∈ [0, T ), ν(t) : x ∈ Td → ν(x, t) is the velocity ﬁeld of the ﬂuid at time t; η(t) : x → η(x, t) is the Leray projection of the density of external forces. Of to fulﬁll the condition of course, ν(t) is taken in the divergence free space Hn+1 Σ incompressibility; the pressure gradient does not appear in (6.87), having been eliminated by application of L. For completeness, all these facts are surveyed in Appendix E. The statement that follows refers to the time evolution of the functions t → ν(t) = −d 2 (2π) Td ν(x, t)dx and t → (1/2) ν(t) L2 . Up to dimensional factors, the ﬁrst one gives the mean value of the velocity ﬁeld or, equivalently, the total momentum; the second one gives the total kinetic energy. Proposition 6.10. Suppose ν fulﬁlls (6.87) on an interval [0, T ). Then, for all t in this interval, we have the following relations: (i) Balance of momentum: d ν(t) = η(t). dt

(6.88)

1 d

ν(t) 2L2 = ν(t) | ∆ν(t)L2 + ν(t) | η(t)L2 ≤ ν(t) | η(t)L2 . 2 dt

(6.89)

(ii) Balance of energy:

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Proof. (i) Let us observe that d ν = ν ˙ = ∆ν − L(ν • ∂ν) + η; dt

(6.90)

the means of ∆ν and L(ν • ∂ν) vanish due to Eqs. (6.12), (6.43) and (6.76), so we get the thesis (6.88). (ii) Let us write 1 d

ν 2L2 = ν | ν ˙ L2 = ν | ∆νL2 − ν | L(ν • ∂ν)L2 + ν|ηL2 ; (6.91) 2 dt on the other hand by the symmetry of L, the equality Lν = ν and Lemma 6.6, ν | L(ν • ∂ν)L2 = Lν | ν • ∂νL2 = ν | ν • ∂νL2 = 0;

(6.92)

these facts yield the equality in (6.89). The subsequent inequality in (6.89) follows via elementary integration by parts: r s v ∆v = −δrs ∂ν r • ∂ν s ≤ 0. (6.93) ν | ∆νL2 = δrs Td

Td

Reducing the NS equations to the case of a zero mean velocity ﬁeld. Let us recall the notation HnΣ0 for the space of divergence free, zero mean vector ﬁelds (see Eq. (6.51) and subsequent comments); we regard this as a Hilbert space, with the inner product | n and the norm n inherited from Hn . The purpose of this paragraph is to show that the general Cauchy problem (6.87) can be reduced to a Cauchy problem for zero mean vector ﬁelds; let us deﬁne the latter precisely. and ξ ∈ C 0,1 ([t0 , +∞), Hn−1 Deﬁnition 6.11. Let f0 ∈ Hn+1 Σ0 Σ ). The incompressible, zero mean NS Cauchy problem with initial datum v0 and forcing term ξ is the following. n−1 1 Find ϕ ∈ C([0, T ), Hn+1 Σ0 ) ∩ C ([0, T ), HΣ0 ) such that ϕ(t) ˙ = ∆ϕ(t) − L ϕ(t) • ∂ϕ(t) + ξ(t) for t ∈ [0, T ), ϕ(0) = f0 (6.94) (for some T ∈ (0, +∞]). Let us connect this problem with the previous one (6.87), for given v0 and η. To this purpose, we need a bit more regularity on the forcing η; to be precise, we assume v0 ∈ Hn+1 Σ ,

n η ∈ C 0,1 ([0, +∞), Hn−1 Σ ) ∩ C([0, +∞), HΣ ).

(6.95)

Let us deﬁne from v0 and η the following objects: m0 := v0 ∈ Rd ; m ∈ C([0, +∞), Rd ),

(6.96)

t

t → m(t) := m0 +

ds η(s);

0

h ∈ C ([0, +∞), R ), 1

d

t → h(t) :=

t

ds m(s); 0

(6.97)

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f0 := v0 − m0 ∈ Hn+1 Σ0 ; ξ : [0, T ) → HnΣ0 ,

t → ξ(t) such that ξ(x, t) := η(x + h(t), t) − η(t).

667

(6.98) (6.99)

The statement ξ(t) ∈ HnΣ0 for each t is evident from the deﬁnition. In Appendix F we will prove that ξ ∈ C 0,1 ([0, +∞), Hn−1 Σ0 ).

(6.100)

Proposition 6.12. Let us consider : (i) the Cauchy problem (6.87), with any datum v0 and forcing η as in (6.95); (ii) the above deﬁnitions of m0 , m, h, f0 , ξ and the Cauchy problem (6.94). A function ν of domain [0, T ) fulﬁlls (6.87) if and only if there is a function ϕ on [0, T ) fulﬁlling (6.94) such that, for all x ∈ Td and t ∈ [0, T ), ν(x, t) = m(t) + ϕ(x − h(t), t).

(6.101)

Proof. Step 1. We suppose (6.94) with the above datum f0 to have a solution ϕ on [0, T ); we deﬁne ν as in (6.101) and prove that it solves problem (6.87). It is clear that ν is in the functional space prescribed by (6.87), and that the following holds: ν(x, 0) = m0 + ϕ(x, 0) = m0 + f0 (x) = v0 (x); ˙ • ∂ϕ(x − h(t), t) ν(x, ˙ t) = η(t) + ϕ(x ˙ − h(t), t) − h(t) = η(t) + ϕ(x ˙ − h(t), t) − m(t) • ∂ϕ(x − h(t)t, t); ∆ν(x, t) = ∆ϕ(x − h(t), t); (ν • ∂ν)(x, t) = (m(t) • ∂ϕ)(x − h(t), t) + (ϕ• ∂ϕ)(x − h(t), t).

(6.102) (6.103) (6.104) (6.105)

The Leray projection L commutes with space translations, so the last equation implies L(ν • ∂ν)(x, t) = L(m(t) • ∂ϕ)(x − h(t), t) + L(ϕ• ∂ϕ)(x − h(t), t).

(6.106)

To conclude, we note that L(m(t) • ∂ϕ) = m(t) • ∂ϕ;

(6.107)

to prove this, it suﬃces to check that m(t) • ∂ϕ is divergence free. In fact, div(m(t) • ∂ϕ) = ∂s (mr (t)∂r ϕs ) = mr (t)∂r (∂s ϕs ) = mr (t)∂r (div ϕ) = 0, (6.108) since ϕ is divergence free at all times. From Eq. (6.102) we see that ν fulﬁlls the initial condition in (6.87); from Eqs. (6.103), (6.104), (6.106), (6.107) and (6.94), we see that ν fulﬁlls the evolution equation in (6.87).

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Step 2. Let us consider function ν on [0, T ), fulﬁlling (6.87); we will prove the existence of a function ϕ on [0, T ) fulﬁlling (6.94), such that ν and ϕ are related by (6.101). To this purpose, let us deﬁne a function ϕ by ϕ(x, t) := ν(x + h(t), t) − m(t);

(6.109)

then, at each time t ∈ [0, T ), ϕ(t) = ν(t) − m(t) = 0

(6.110)

on account of Eq. (6.88) for ν and of the deﬁnition (6.97) of m. Besides having zero mean, ϕ belongs to the function spaces prescribed by (6.87) due to the properties of ν. Now, computations very similar to the ones of Step 1 prove that ϕ fulﬁlls the initial condition and the evolution equation in (6.94). Remark 6.13. Let us regard Eqs. (6.98) and (6.99) as deﬁning a transformation n+1 0,1 T : Hn+1 × C 0,1 ([0, +∞), Hn−1 ([0, +∞), Hn−1 Σ Σ ) → HΣ0 × C Σ0 ),

(v0 , η) → (f0 , ξ) = T (v0 , η).

(6.111)

The map T is onto, due to the trivial equality (f0 , ξ) = T (f0 , ξ). 7. The NS Equations in the General Framework for Evolution Equations with Quadratic Nonlinearity Basic notations. The zero mean version (6.94) is the ﬁnal form for the NS Cauchy problem, to which we stick from now on. Let us recall that d ∈ {2, 3, . . .}; throughout the section, we ﬁx n>

d , 2

ξ ∈ C 0,1 ([0, +∞), Hn−1 Σ0 )

(7.1)

(the function ξ is regarded to be given by itself, independently of any function η as in the previous section). Our aim is to apply the formalism of Sec. 5 to the Cauchy problem (6.94) (and to the equivalent Volterra problem); in this case F± ≡ (F± , ± ) := (Hn±1 Σ0 , n±1 ),

F ≡ (F, ) := (HnΣ0 , n ),

n−1 A := ∆ : Hn+1 Σ0 → HΣ0 ,

P : HnΣ0 × HnΣ0 → Hn−1 Σ0 ,

f → ∆f ;

P(f, g) := −L(f • ∂g);

(7.2) (7.3)

ξ as in (7.1).

(7.4)

From P and ξ, we construct the function P : HnΣ0 × [0, +∞) → Hn−1 Σ0 , which appears in (6.94).

P(f, t) := P(f, f ) + ξ(t) = −L(f • ∂f ) + ξ(t),

(7.5)

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For subsequent reference, we record the Fourier representations

f n±1 = (1 + |k|2 )n±1 |fk |2 , f n = (1 + |k|2 )n |fk |2 , k∈Zd 0

k∈Zd 0

(∆f )k = −|k| fk . 2

(7.6)

Veriﬁcation of properties (P1)–(P5 ). The above set (F− , F, F+ , A), completed with suitable semigroup estimators u, u− , has the properties (P1)–(P5) prescribed in Sec. 2, and (P4 )–(P5 ) of Sec. 5. We will indicate which parts of the proof are obvious, and give details on the nontrivial parts. and A = ∆ fulﬁll conditions (P1) Proposition 7.1. F = HnΣ0 , F∓ = Hn∓1 Σ0 and (P2). Proof. Everything follows easily from the Fourier representations. Proposition 7.2. (i) ∆ generates a strongly continuous semigroup on Hn−1 Σ0 , given by 2 et∆ f = e−t|k| fk ek , for f ∈ Hn−1 (7.7) Σ0 , t ∈ [0, +∞). k∈Zd 0

So, (P3) holds. (ii) For f ∈ HnΣ0 and t ∈ [0, +∞) one has et∆ f ∈ HnΣ0 ,

et∆ f n ≤ u(t) f n , u(t) := e−t ;

(7.8)

the function (f, t) → et∆ f gives a strongly continuous semigroup on HnΣ0 . (iii) For f ∈ Hn−1 Σ0 and t ∈ (0, +∞) one has et∆ f ∈ HnΣ0 ,

et∆ f n ≤ u− (t) f n−1 , (7.9)  2t 1   √e for 0 < t ≤ , −t 4 (7.10) u− (t) := e µ− (t), µ− (t) := √ 2et 1   2 for t > ; 4 √ + note that u− (t), µ− (t) = O(1/ t) for t → 0 . The function (f, t) → et∆ f is n continuous from Hn−1 Σ0 × (0, +∞) to HΣ0 . t With u− as in (7.10), the function t → U(t) := 0 ds u− (s) is given by  γ(t) 1   √ for 0 < t ≤ ,   2 4 U(t) := (7.11)   γ(1/4) √ −1/4 1 −t   √ + 2(e − e ) for < t ≤ +∞, 4 2 t 1 es γ(t) := for 0 ≤ t ≤ . (7.12) ds √ s 4 0 √ √ (In particular, U(+∞) = γ(1/4)/ 2 + 2 e−1/4 ∈ (1.872, 1.873)).

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(iv) With µ− as in (7.10), one has t √ ds µ− (t − s)e−s = 2. sup t∈[0,+∞)

(7.13)

0

(v) In conclusion, (P4), (P4 ) and (P5), (P5 ) hold with B = 1,

σ=

1 , 2

N=

√

2.

(7.14)

Proof. (i) This follows basically from Eq. (7.7) for ∆. (ii) and (iii) We only give details on the derivation of Eqs. (7.8)–(7.11). Let f ∈ HnΣ0 , t ∈ [0, +∞). Then, 2 (1 + |k|2 )n |(et∆ f )k |2 = (1 + |k|2 )n e−2t|k| |fk |2 k∈Zd 0

k∈Zd 0

≤ e−2t

(1 + |k|2 )n |fk |2 ,

(7.15)

k∈Zd 0

since |k| ≥ 1 for k ∈ Zd0 ; this yields Eq. (7.8). Now, let f ∈ Hn−1 Σ0 and t ∈ (0, +∞); then, 2 (1 + |k|2 )n |(et∆ f )k |2 = (1 + |k|2 )e−2t|k| (1 + |k|2 )n−1 |fk |2 k∈Zd 0

k∈Zd 0

  Ut (ϑ)  (1 + |k|2 )n−1 |fk |2  ,

≤

sup ϑ∈[1,+∞)

Ut (ϑ) := (1 + ϑ)e

−2tϑ

k∈Zd 0

, (7.16)

and an elementary computation gives  1 e2t   Ut −1 = 2t 2et sup Ut (ϑ) =  ϑ∈[1,+∞)  U (1) = 2e−2t

for 0 < t ≤

1 , 4

1 for t > . t 4 ! From (7.17) we infer Eq. (7.9) with u− (t) := supϑ∈[1,+∞) Ut (ϑ), i.e.  t e   √ 2et u− (t) = √ −t    2e

for 0 < t ≤ for t >

1 ; 4

1 , 4

(7.17)

(7.18)

this deﬁnition of u− (t) agrees with Eq. (7.10), and (7.11) follows trivially.

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(iv) See Appendix G. (v) Obvious consequence of items (i)–(iv). Analysis of P and P. We turn the attention to the functions in Eqs. (7.4) and (7.5). Proposition 7.3. (i) P is a bilinear map and admits the estimate (implying continuity)

P(f, g) n−1 ≤ Kn f n g n

(7.19)

for all f, g ∈ HnΣ0 , with Kn ≡ Knd any constant fulﬁlling (6.78) (so, condition (Q1) holds for this map). (ii) As a consequence of (i), P fulﬁlls the Lipschitz condition (P6). Furthermore, for each function φ ∈ C([0, T ), HnΣ0 ), the growth of P from φ admits the estimate

P(f, t) − P(φ(t), t) n−1 ≤ n (t, f − φ(t) n )

(7.20)

for t ∈ [0, T ) and f ∈ HnΣ0 , where n : [0, +∞) × [0, T ) → [0, +∞), (r, t) → n (r, t) := 2Kn φ(t) r + Kn r2 . (7.21) Proof. (i) The bilinearity is obvious, the estimate follows from Eqs. (6.49) for L and (6.78) for the map (f, g) → f • ∂g. (ii) Use Corollaries 5.3 and 5.4 on quadratic maps. The function Ξn−1. From here to the end of the paper, we denote in this way any function in C([0, +∞), [0, +∞)) such that Ξn−1 ∈ C([0, +∞), [0, +∞)) nondecreasing,

(7.22)

ξ(t) n−1 ≤ Ξn−1 (t) for t ∈ [0, +∞) (e.g., Ξn−1 (t) := sups∈[0,t] ξ(s) n−1 ). Cauchy and Volterra problems. Deﬁnition 7.4. For any f0 ∈ Hn+1 Σ0 , CPn (f0 ) is the Cauchy problem (6.94), i.e. 1 ) ∩ C ([0, T ), Hn−1 Find ϕ ∈ C([0, T ), Hn+1 Σ0 Σ0 ) such that ϕ(t) ˙ = ∆ϕ(t) + P(ϕ(t), t)

for all t ∈ [0, T ),

ϕ(0) = f0 .

For any f0 ∈ HnΣ0 , VPn (f0 ) is the Volterra problem: Find ϕ ∈ C([0, T ), HnΣ0 ) such that t t∆ ϕ(t) = e f0 + ds e(t−s)∆ P(ϕ(s), s) for all t ∈ [0, T ). 0

(7.23)

(7.24)

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Remark 7.5. (i) If f0 ∈ Hn+1 Σ0 , VPn (f0 ) is equivalent to CPn (f0 ) by the reﬂexivity of the Hilbert space Hn−1 Σ0 (see once more Proposition 2.4). n (ii) For any f0 ∈ HΣ0 , uniqueness and local existence are granted for VPn (f0 ) (Propositions 2.5 and 3.10). 8. Results for the NS Equations Arising from the Previous Framework We keep the assumption (7.1) and all notations of the previous section; furthermore, we ﬁx an initial datum f0 ∈ HnΣ0 .

(8.1)

The analysis of the previous section allows us to identify VPn (f0 ) with a Volterra problem of the general type discussed in Sec. 5, with semigroup estimators u, u− of the form considered therein and a quadratic nonlinearity P. Due to Propositions 7.2 and 7.3, the constant K and the functions u, u− , µ− , U, Ξ− of Sec. 5 can be taken as follows: K = a constant Kn fulﬁlling (6.78), µ− as in (7.10),

u(t) := e−t , u− (t) = µ− (t)e−t ,

U as in (7.11),

(8.2)

Ξ− = Ξn−1 as in (7.22).

Hereafter, we rephrase Propositions 5.5 and 5.6 with the above speciﬁcations. Proposition 8.1. Let us consider for VPn (f0 ) an approximate solution ϕap ∈ C([0, T ), HnΣ0 ), where T ∈ (0, +∞]. Suppose there are functions En , Dn , Rn ∈ C([0, T ), [0, +∞)) such that (i)–(iii) hold: (i) ϕap has the integral error estimate

E(ϕap (t)) n ≤ En (t)

for t ∈ [0, T );

(8.3)

(ii) one has

ϕap (t) n ≤ Dn (t)

for t ∈ [0, T );

(8.4)

(iii) Rn solves the control inequality t En (t) + Kn ds u− (t − s)(2Dn (s)Rn (s) + R2n (s)) ≤ Rn (t) 0

for t ∈ [0, T ).

(8.5)

Then, (a) and (b) hold: (a) VP(f0 ) has a solution ϕ : [0, T ) → HnΣ0 ; (b) one has

ϕ(t) − ϕap (t) n ≤ Rn (t)

for t ∈ [0, T ).

(8.6)

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Proposition 8.2. Let us consider for VPn (f0 ) an approximate solution ϕap ∈ C([0, T ), HnΣ0 ), where T ∈ (0, +∞]. Suppose there are functions En , Dn ∈ C([0, T ], [0, +∞)) such that (i)–(iii) hold: (i) En is nondecreasing, and binds the integral error as in (8.3); (ii) Dn is nondecreasing and binds ϕap as in (8.4); (iii) one has 2 Kn U(T )En (T ) + 2Kn U(T )Dn (T ) ≤ 1.

(8.7)

Then VPn (f0 ) has a solution ϕ : [0, T ) → HnΣ0 and, for all t ∈ [0, T ),

ϕ(t) − ϕap (t) n ≤ Rn (t),  1 − 2Kn U(t)Dn (t)     − (1 − 2Kn U(t)Dn (t))2 − 4Kn U(t)En (t) Rn (t) :=   2Kn U(t)   En (0)

if t ∈ (0, T ), if t = 0; (8.8)

the above prescription gives a well deﬁned, nondecreasing function Rn ∈ C([0, T ), [0, +∞)). In Sec. 5, from Proposition 5.6 we have inferred Proposition 5.9, corresponding to the approximate solution ϕap := 0; in the present situation, this reads as follows. Proposition 8.3. Let Fn (t) := f0 n + Ξn−1 (t) U(t).

(8.9)

Suppose T ∈ [0, +∞], and 4Kn U(T )Fn (T ) ≤ 1.

(8.10)

Then VPn (f0 ) has a solution ϕ : [0, T ) → HnΣ0 and, for all t ∈ [0, T ),

ϕ(t) n ≤ Fn (t) X (4Kn U(t)Fn (t)) (where, as in (5.31): X (z) :=

√

1− 1−z (z/2)

(8.11)

for z ∈ (0, 1], X (0) := 1).

The other results of Sec. 5 were about global existence and exponential decay, under speciﬁc assumption. In the present framework the constants B, σ, N of the cited section are given by Eq. (7.14); this allows to rephrase Proposition 5.10 in this way. Proposition 8.4. Let us consider for VPn (f0 ) an approximate solution ϕap ∈ C([0, +∞), HnΣ0 ). Suppose there are constants En , Dn ∈ [0, +∞) such that: (i) ϕap admits the integral error estimate

E(ϕap )(t) n ≤ En e−t

for t ∈ [0, +∞);

(8.12)

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(ii) for all t as above,

ϕap (t) n ≤ Dn e−t ; (iii) one has 2

4

(8.13)

√ √ 2 Kn En + 2 2Kn Dn ≤ 1.

(8.14)

Then VPn (f0 ) has a global solution ϕ : [0, +∞) → HnΣ0 and, for all t ∈ [0, +∞),

ϕ(t) − ϕap (t) n ≤ Rn e−t , ! √ √ √ 1 − 2 2Kn Dn − (1 − 2 2Kn Dn )2 − 4 2Kn En √ . Rn := 2 2Kn

(8.15)

The applications of Proposition 5.10 considered in Sec. 5 were based on the assumption of exponential decay for the external forcing, that in the present framework must be formulated in this way: (Q3)n There is a constant Jn−1 ∈ [0, +∞) such that

ξ(t) n−1 ≤ Jn−1 e−2t

for all t ∈ [0, +∞).

(8.16)

The above mentioned applications in Sec. 5 were Proposition 5.12 (corresponding to the choice ϕap := 0) and Proposition 5.14 (with ϕap the A-ﬂow approximate solution). These can be restated, respectively, in the following way: Proposition 8.5. Assume (Q3)n , and deﬁne √ Fn := f0 n + 2Jn−1 ; furthermore, assume

(8.17)

√ 4 2Kn Fn ≤ 1.

(8.18)

Then VPn (f0 ) has a global solution ϕ : [0, +∞) → HnΣ0 and, for all t ∈ [0, +∞), √ (8.19)

ϕ(t) n ≤ Fn X (4 2Kn Fn ) e−t (with X as in (5.31)). Proposition 8.6. Deﬁne ϕap ∈

C([0, +∞), HnΣ0 ),

ϕap (t) := e

t∆

f0 +

t

ds e(t−s)∆ ξ(s).

(8.20)

0

Furthermore, let us keep the assumptions and deﬁnitions (Q3)n , (8.17) and (8.18). Then the global solution ϕ : [0, +∞) → HnΣ0 of VPn (f0 ) is such that, for all t ∈ [0, +∞), √ √

ϕ(t) − ϕap (t) n ≤ 2Kn Fn2 X(4 2Kn Fn ) e−t (8.21) (where, as in (5.60): X(z) :=

√ 1−(z/2)− 1−z (z 2 /8)

for z ∈ (0, 1], X(0) := 1).

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Remark 8.7. (i) Condition (8.10) can be fulﬁlled with either f0 , ξ small, T large or f0 , ξ large, T small. Condition (8.18) is fulﬁlled if f0 and ξ are suﬃciently small. (ii) As a special case, suppose the external forcing ξ to be identically zero; then we can take Ξn−1 = 0 and Jn−1 = 0. Equations (8.10) and (8.18) ensure global existence if the datum fulﬁlls the conditions 4Kn U(+∞) f0 n ≤ 1 and √ 4 2Kn f0 n ≤√1, respectively. The less restrictive condition on f0 is the second one, since 2 < U(+∞). (iii) Let us return to Proposition 8.1; this states, amongst else, that the solution ϕ of VPn (f0 ) exists on [0, T ) if the inequality (8.5) has a solution R : [0, T ) → [0, +∞). Let us compare this statement with a result presented in the recent work [2], that we rephrase here in our notations. Let us consider the (incompressible, zero mean) NS equations with external forcing ξ and initial datum f0 ; when we refer to [2] a solution of this Cauchy problem means a strong solution, as deﬁned therein. Now suppose ϕap to be an approximate solution on an interval [0, T ), and

ϕap (0) − f0 n−1 ≤ δn−1 ,

ϕ˙ ap (t) − ∆ϕap (t) − P(ϕap (s), s) n−1 ≤ n−1 (t),

ϕap (t) n−1 ≤ Dn−1 (t),

(8.22)

ϕap (t) n ≤ Dn (t)

for all t ∈ [0, T ), for suitable estimators δn−1 ≥ 0, n−1 , Dn−1 , Dn : [0, T ) → [0, +∞). According to [2, pp. 065204–10], the NS Cauchy problem has a solution ϕ : [0, T ) → HnΣ0 if δn−1 +

T

ds n−1 (s) < 0

1 Cn T

e−Cn

RT 0

ds (Dn−1 (s)+Dn (s))

(8.23)

(with Cn > 0 a constant not computed explicitly, whose role is analogous to the one of Kn ); (8.23) is an inequality involving only the approximate solution, and plays a role similar to our (8.5) to grant the existence of an exact solution on [0, T ). Seemingly, Eq. (8.23) is not suited to obtain results of global existence for the exact solution. To explain this statement, suppose ϕap and its estimators to be deﬁned on [0, +∞), with δn−1 = 0 or n−1 non identically zero; then (8.23) surely fails for large T , even in the most favorable situation where all integrals therein converge for T → +∞. In fact, +∞ T →+∞ ds n−1 (s) ∈ (0, +∞], l.h.s. of (8.23) −−−−−→ δn−1 + 0 (8.24) 1 T →+∞ 0 < r.h.s. of (8.23) ≤ −→ 0. Cn T On the contrary, our control inequality (8.5) can be used in certain cases to derive the existence of ϕ (and bind its distance from ϕap ) up to T = +∞; some applications

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of this type have appeared in the present section, further examples will be given in the next one on the Galerkin approximations.j 9. Galerkin Approximate Solutions of the NS Equations Throughout this section, we consider a set G with the following features: G ⊂ Zd0 ,

k ∈ G ⇔ −k ∈ G.

G ﬁnite,

(9.1)

Hereafter we write ≺ ek k∈G for the linear subspace of D spanned by the functions ek for k ∈ G. Galerkin subspaces and projections. We deﬁne them as follows. Deﬁnition 9.1. The Galerkin subspace and projection corresponding to G are HG Σ0 := DΣ0 ∩ ≺ ek k∈G = vk ek | vk ∈ Cd , vk = v−k , k • vk = 0 for all k .

(9.2)

k∈G

PG : DΣ0 → HG Σ0 ,

v=

vk ek → PG v :=

k∈Zd 0

vk ek .

(9.3)

k∈G

It is clear that ∞ ∩ DΣ0 , HG Σ0 ⊂ C

HG Σ0

⊂

Hm Σ0 ,

P

G

(Hm Σ0 )

=

G ∆(HG Σ0 ) ⊂ HΣ0 ;

(9.4)

HG Σ0

(9.5)

for all m ∈ R.

The following result will be useful in the sequel. Lemma 9.2. Let n, p ∈ R, n ≤ p and v ∈ HpΣ0 . Then,

v p

(1 − PG )v n ≤ , |G| := inf 1 + |k|2 . |G|p−n k∈Zd 0 \G Proof. We have (1 − PG )v =

vk ek , k∈Zd 0 \G

(9.6)

implying

(1 + |k|2 )p |v |2 2 )p−n k (1 + |k| k∈Zd k∈Zd 0 \G 0 \G   1  ≤ sup (1 + |k|2 )p |vk |2  2 )p−n (1 + |k| k∈Zd \G d 0

(1 − PG )v 2n =

(1 + |k|2 )n |vk |2 =

k∈Z0 \G

≤

1 |G|2(p−n)

v 2p ,

(9.7)

whence the thesis. j To conclude this remark we wish to point out that, under special assumptions, some global existence results could perhaps be derived from the approach of [2], with a diﬀerent analysis of the diﬀerential inequalities proposed by the authors to infer Eq. (8.23). A discussion of this point, and of other interesting features of [2], would occupy too much space here.

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Galerkin approximate solutions. Let ξ ∈ C([0, +∞), DΣ0 ),

f0 ∈ DΣ0

(9.8)

(of course, in the sequel P(f, t) := −L(f • ∂f ) + ξ(t) whenever this makes sense). Deﬁnition 9.3. The Galerkin approximate solution of NS corresponding to G, G with external forcing ξ and datum f0 , is the maximal solution ϕG of the f0 ≡ ϕ G following Cauchy problem, in the ﬁnite dimensional space HΣ0 : Find ϕG ∈ C 1 ([0, TG ), HG Σ0 ) such that ϕ˙ G (t) = ∆ϕG (t) + PG P(ϕG (t))

for all t,

ϕG (0) = PG f0 .

(9.9)

Of course, “maximal” means that [0, TG ) is the largest interval of existence. In certain cases, one can prove that TG = +∞ and derive estimates of ϕG (we return on this in the sequel). In this section we use the functions U as in Eq. (7.11),

X as in Eq. (5.31).

(9.10)

Let us consider any real number m, and assume ξ ∈ C 0,1 ([0, +∞), Hm−1 Σ0 ). We will use the notation Ξm−1 to indicate any function in C([0, +∞), [0, +∞)) fulﬁlling Eq. (7.22) with n replaced by m; in the sequel that equation will be referred to as (7.22)m . When necessary we will make the assumption (Q3)m of exponential decay for the external forcing; this is like (Q3)n with n → m, thus involving a constant Jm−1 ∈ [0, +∞). The forthcoming proposition gives estimates on the interval of existence of the Galerkin solution and on its norm m , which are in fact independent of G. m Proposition 9.4. Let m > d/2, ξ ∈ C 0,1 ([0, +∞), Hm−1 Σ0 ) and f0 ∈ HΣ0 ; then, (i) and (ii) hold.

(i) Deﬁne (similarly to (8.9)) Fm (t) := f0 m + Ξm−1 (t) U(t);

(9.11)

furthermore, let T ∈ (0, +∞], and assume the inequality 4Km U(T )Fm (T ) ≤ 1.

(9.12)

Then the Galerkin solution ϕG with this datum exists on [0, T ) and fulﬁlls

ϕG (t) m ≤ Dm (t)

for t ∈ [0, T ),

(9.13)

Dm (t) := Fm (t)X (4Km U(t)Fm (t)).

(9.14)

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(ii) Alternatively, assume (Q3)m ; deﬁne (similarly to (8.17)) Fm := f0 m +

√ 2Jm−1 ,

(9.15)

and suppose √ 4 2Km Fm ≤ 1.

(9.16)

Then the Galerkin solution ϕG with this datum is global, and fulﬁlls

ϕG (t) m ≤ Dm e−t Dm

for t ∈ [0, +∞), √ := Fm X (4 2Km Fm )

(9.17) (9.18)

(the above equations will be referred to in the sequel as (9.11)m , (9.12)m , etc.). Proof. We refer to the framework of Sec. 5 on systems with quadratic nonlinG earities. In the present case (F, ) := (HG Σ0 , m ), (F∓ , ∓ := (HΣ0 , m∓1 ) (we have three copies of the same ﬁnite dimensional space, but equipped with different, though equivalent, norms); the operator A is ∆ HG Σ0 , and the bilinear G G G G G map is P P : HΣ0 × HΣ0 → HΣ0 , (f, g) → P P(f, g); the function ξ of Sec. 5 is G G PG ξ ∈ C 0,1 ([0, +∞), HG Σ0 ); the initial datum is P f0 ∈ HΣ0 . G For the operator ∆ HΣ0 we use the same estimates given for ∆ in Proposition 7.2, with n → m; √ this justiﬁes using the scheme of Sec. 5 with U as in (7.11) and B = 1, N = 2. To estimate PG P, we use the inequalities on P in Proposition 5.2 with n → m, and the obvious relation PG · m−1 ≤ · m−1 ; this gives PG P(f, g) m−1 ≤ Km f m g m, and so the constant K of Sec. 5 is, in this case, Km . For the initial datum PG f0 and for PG ξ we use the estimates

PG f0 m ≤ f0 m ,

P ξ(t) m−1 ≤ ξ(t) m−1 ≤ Ξm−1 (t) G

(9.19) or Jm−1 e

−2t

;

(9.20)

of course, the bound via Ξm−1 refers to case (i) and the bound via Jm is for case (ii). Applying to this framework Propositions 5.9 and 5.12 we get the statements in (i) and (ii), respectively. Remark 9.5. Global existence of ϕG could be proved under much weaker conditions than the ones in item (ii) of the above proposition. In fact, using for ϕG an energy balance relation similar to (6.89), one can derive global existence and boundedness of ϕG (t) L2 when f0 is arbitrary and the external forcing makes ﬁnite both +∞ +∞ integrals 0 dt ξ(t) L2 , 0 dt ξ(t) 2L2 : see, e.g., [15, 16]. However, the energetic approach does not allow to derive estimates of the speciﬁc type appearing in Proposition 9.4.

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The distance between the exact NS solution and the Galerkin approximations. From here to the end of the paragraph, we ﬁx two real numbers p≥n>

d ; 2

(9.21)

we also ﬁx ξ ∈ C 0,1 ([0, +∞), Hp−1 Σ0 ),

f0 ∈ HpΣ0

(9.22)

and denote with ϕG the Galerkin approximate solution with such forcing and datum, for any G as before. This will be compared with the solution ϕ of the NS equations with the same forcing and datum. Lemma 9.6. Let us regard ϕG as an approximate solution of VPn (f0 ); then the following holds. (0) The integral error of ϕG is

t ds e(t−s)∆ P(ϕG (s)) . E(ϕG )(t) = −(1 − PG ) et∆ f0 +

(9.23)

0

(i) Let us introduce the deﬁnitions or assumptions (9.11)p and (9.12)p , for some T ∈ (0, +∞] (implying existence of ϕG on [0, T )). Then, for all t ∈ [0, T ) we have Yp (t) , (9.24)

E(ϕG )(t) n ≤ |G|p−n Yp (t) := Fp (t)[1 + Kp U(t)Fp (t)X 2 (4Kp U(t)Fp (t))].

(9.25)

The function Yp is nondecreasing. (ii) Alternatively, introduce the deﬁnitions or assumptions (Q3)p , (9.15)p and (9.16)p (implying that ϕG is global). Then, for all t ∈ [0, +∞) we have Yp e−t , |G|p−n √ √ Yp := Fp [1 + 2Kp Fp X 2 (4 2Kp Fp )].

E(ϕG )(t) n ≤

Proof. Derivation of (9.23). By deﬁnition t E(ϕG )(t) = ϕG (t) − et∆ f0 − ds e(t−s)∆ P(ϕG (s));

(9.26) (9.27)

(9.28)

0

on the other hand, the Cauchy problem (9.9) deﬁning ϕG has the integral reformulation t G t∆ G ϕ (t) = e P f0 + ds e(t−s)∆ PG P(ϕG (s)); (9.29) 0

inserting this into (9.28) we get

E(ϕG )(t) = −et∆ (1 − PG )f0 − 0

t

ds e(t−s)∆ (1 − PG )P(ϕG (s)).

(9.30)

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Finally, the operator 1 − PG commutes with ∆ and its semigroup (as made evident by the Fourier representations); so, 1 − PG can be factored out and we obtain the thesis (9.23). Some preliminaries to the proof of (i) and (ii). From (9.23), the estimates (9.6) on 1 − PG and (7.8)–(7.10) on the semigroup of ∆ we get

t 1 G t∆ (t−s)∆ G

E(ϕ )(t) n ≤ ds e P(ϕ (s)) p

e f0 p + |G|p−n 0

t 1 −t −(t−s) G

f

+ ds e µ (t − s) P(ϕ (s))

≤ e . 0 p − p−1 |G|p−n 0 (9.31) Proof of (i). From P(ϕG (s)) = P(ϕG (s), ϕG (s)) + ξ(s) we infer the following, for s ∈ (0, t):

P(ϕG (s)) p−1 ≤ Kp ϕG (s) 2p + ξ(s) p−1 ≤ Kp Dp2 (s) + Ξp−1 (s) ≤ Kp Dp2 (t) + Ξp−1 (t)

(9.32)

(in the above, we have used (9.13)p and (7.22)p and the relation Dp (s) ≤ Dp (t)). insert the result (9.32) Eq. (9.31); in this way we are left with an integral t into t We−(t−s) −s −t ds e µ (t − s) = ds e µ ≤ 1 we − − (s) ≤ U(t). From this bound and e 0 0 obtain

E(ϕG (t)) n ≤ =

1 [ f0 p + U(t)(Kp Dp2 (t) + Ξp−1 (t))] |G|p−n 1 [Fp (t) + U(t)Kp Dp2 (t)], |G|p−n

(9.33)

where the last passage follows from deﬁnition (9.11)p ; now, explicitating Dp (t) we get the thesis (9.24) and (9.25). Finally, Yp is nondecreasing because Fp , U and X are so. Proof of (ii). In this case, from the inequality P(ϕG (s)) p−1 ≤ Kp ϕG (s) 2p +

ξ(s) p−1 we infer, by means of Eqs. (9.17)p and (Q3)p ,

P(ϕG (s)) p−1 ≤ Kp Dp2 e−2s + Jp−1 e−2s . Inserting this result √ −tinto (9.31) we are left with a term e is bounded by 2e due to (7.13); the conclusion is

E(ϕG (t)) n ≤ =

−t t 0

(9.34) ds e−s µ− (t− s), which

√ e−t [ f0 p + 2(Kp Dp2 + Jp−1 )] p−n |G| √ e−t [Fp + 2Kp Dp2 ] p−n |G|

(9.35)

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(the last equality following from (9.15)p ). Now, explicitating Dp we get the thesis (9.26) and (9.27). The following proposition contains the main result of the section. Proposition 9.7. (i) Let T ∈ (0, +∞]; make the assumptions and deﬁnitions (9.11)n , (9.12)n and (9.11)p , (9.12)p (implying the existence of ϕG on [0, T )). Finally, with Dn and Yp deﬁned by (9.14)n and (9.25), assume Kn U(T )Yp (T ) + 2Kn U(T )Dn (T ) ≤ 1. (9.36) 2 |G|p−n Then VPn (f0 ) has a solution ϕ of domain [0, T ) and, for all t in this interval,

ϕ(t) − ϕG (t) n ≤ Wnp|G| (t) :=

Yp (t) X 1 − 2Kn U(t)Dn (t)

Wnp |G| (t) , |G|p−n

4Kn U(t)Yp (t) . (1 − 2Kn U(t)Dn (t))2 |G|p−n

(9.37) (9.38)

The function t → Wnp|G| (t) is nondecreasing; a rough, |G|-independent bound for it is Wnp|G| (t) ≤

2Yp (T ) 1 − 2Kn U(T )Dn (T )

(9.39)

for all t ∈ [0, T ). (ii) Alternatively, make the assumptions and deﬁnitions (Q3)n , (9.15)n , (9.16)n and (Q3)p , (9.15)p , (9.16)p (implying global existence of ϕG ). Finally, with Dn and Yp deﬁned by (9.18)n and (9.27), assume √ √ Kn Yp 4 2 2 + 2 2Kn Dn ≤ 1. (9.40) p−n |G| Then VPn (f0 ) has a solution ϕ of domain [0, +∞) and, for all t in this interval,

ϕ(t) − ϕG (t) n ≤

Wnp|G|

Yp √ := X 1 − 2 2Kn Dn

Wnp |G| −t e , |G|p−n

√ 4 2Kn Yp √ . (1 − 2 2Kn Dn )2 |G|p−n

(9.41)

(9.42)

The above constant has the rough, |G|-independent bound Wnp|G| ≤

2Yp √ . 1 − 2 2Kn Dn

(9.43)

Proof. (i) A simple application of Proposition 8.2, with ϕap = ϕG ; Dn as in (9.14); En (t) =

Yp (t) . |G|p−n

(9.44)

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The condition (8.7) in the cited proposition takes the form (9.36). The proposition ensures that ϕ is deﬁned on [0, T ), and gives the estimate

ϕ(t) − ϕG (t) n ≤ Rn (t),

(9.45)

involving the nondecreasing, continuous function 1 − 2Kn U(t)Dn (t) − (1 − 2Kn U(t)Dn (t))2 − 4Kn U(t)Yp (t)/|G|p−n Rn (t) := , 2Kn U(t) Rn (0) :=

f0 p . |G|p−n

(9.46)

We note that we can write Rn (t) =

Yp (t) (1 − 2Kn U(t)Dn (t))|G|p−n 4Kn U(t)Yp (t) ×X ; (1 − 2Kn U(t)Dn (t))2 |G|p−n

(9.47)

this yields the thesis (9.37) and (9.38). The fact that Wnp|G| is a nondecreasing function of time is apparent from its deﬁnition. The bound (9.39) for it follows from the nondecreasing nature of the function t → Yp (t)/(1 − 2Kn U(t)Dn (t)) and from the inequality X (z) ≤ 2 for all z ∈ [0, 1]). (ii) A simple application of Proposition 8.4, with ϕap = ϕG ,

Dn as in (9.18),

En =

Yp . |G|p−n

(9.48)

The condition (8.14) in the cited proposition takes the form (9.40). The proposition ensures that ϕ is deﬁned on [0, +∞), and gives the estimate

ϕ(t) − ϕG (t) n ≤ Rn e−t , ! √ √ √ 1 − 2 2Kn Dn − (1 − 2 2Kn Dn )2 − 4 2Kn Yp /|G|p−n √ Rn := . 2 2Kn

(9.49)

(9.50)

We note that we can write Yp √ Rn = X (1 − 2 2Kn Dn )|G|p−n

√ 4 2Kn Yp √ , (1 − 2 2Kn Dn )2 |G|p−n

(9.51)

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yielding the thesis (9.41) and (9.42); the rough bound (9.43) follows again from the inequality X (z) ≤ 2. Remark 9.8. Of course, if p > n the previous proposition implies convergence of the Galerkin solution ϕG to the exact solution ϕ of VPn (f0 ). More precisely, with the assumptions in (i) we infer from (9.36) and (9.39) that 1 G → 0 for |G| → +∞; (9.52) sup ϕ(t) − ϕ (t) n = O |G|p−n t∈[0,T ) in case (ii), we infer from (9.41) and (9.43) that 1 sup et ϕ(t) − ϕG (t) n = O →0 |G|p−n t∈[0,+∞)

for |G| → +∞.

(9.53)

10. Numerical Examples Given the necessary constants Kn , the datum norms and some bounds on the external forcing, the framework of Secs. 8 and 9 yields informations on the time of existence of the solution ϕ of VPn (f0 ), and on its HnΣ0 distance from an approximate solution. In the sequel we exemplify such estimates referring to Sec. 9, i.e. to the Galerkin approximations. Throughout the section, we take d = 3;

n = 2,

p = 4.

(10.1)

The constants K2 and K4 involved in calculations can be obtained from Lemmas 6.1 and 6.2 and Proposition 6.3; a MATLAB computation illustrated in Appendix H yields the values K2 = 0.20,

K4 = 0.067.

(10.2)

The other calculations mentioned hereafter have been performed using MATHEMATICA. An application of Proposition 9.7, item (i). We suppose the external forcing has bounds (7.22)2 and (7.22)4 with Ξ1 (t) = const. ≡ Ξ1 and Ξ3 (t) = const. ≡ Ξ3 for all t ∈ [0, +∞). Conditions (9.12)2 and (9.12)4 are satisﬁed with T = +∞ if

f0 2 + 1.88 Ξ1 < 0.667,

f0 4 + 1.88 Ξ3 < 1.99;

(10.3)

under the above inequalities for f0 and the forcing, the Galerkin solution ϕG exists on [0, +∞) for each G. As an example, conditions (10.3) are satisﬁed in the case

f0 2 = 0.15, f0 4 = 1.50,

Ξ1 = 0.025, Ξ3 = 0.25,

(10.4)

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to which we stick hereafter. In the above case, condition (9.36) with n = 2, p = 4 and T = +∞ becomes 0.161 + 2.31/|G| ≤ 1, which is fulﬁlled if |G| ≥ 2.76.

(10.5)

Assuming (10.4) and (10.5), the solution ϕ of VP2 (f0 ) is also global, and

ϕ(t) − ϕG (t) 2 ≤

W24 |G| (t) 8.71 ≤ |G|2 |G|2

for all G as above, t ∈ [0, +∞).

(10.6)

The numerical value of W24 |G| (t) can be computed at will from deﬁnition (9.38); here we have used the rough bound W24 |G| (t) ≤ 8.71, coming from (9.39). Another application of Proposition 9.7, item (i). We maintain the assumptions Ξ1 (t) = const. ≡ Ξ1 and Ξ3 (t) = const. ≡ Ξ3 for all t ∈ [0, +∞). We take

f0 2 = 0.20, f0 4 = 2.00,

Ξ1 , Ξ3 as in (10.4).

(10.7)

Now conditions (10.3) are not fulﬁlled, indicating that (9.12)2 and (9.12)4 are not satisﬁed with T = +∞. On the contrary, (9.12)2 and (9.12)4 are found to hold with T = 1.51,

(10.8)

i.e. the Galerkin solution ϕG exists for any G on the time interval [0, 1.51). To go on, we note that condition (9.36) with n = 2, p = 4 and T as above becomes 0.163 + 2.41/|G| ≤ 1, which is fulﬁlled if |G| ≥ 2.88.

(10.9)

Under the assumption (10.9) the solution ϕ of VP2 (f0 ) exists on the same interval, and

ϕ(t)−ϕG (t) 2 ≤

W24 |G| (t) 11.1 ≤ |G|2 |G|2

for all G as above, t ∈ [0, 1.51).

(10.10)

Again, we can compute the numerical value of W24 |G| (t) from the deﬁnition (9.38); here we have used the bound W24 |G| (t) ≤ 11.1, coming from (9.39). An application of Proposition 9.7, item (ii). Let us recall that this case refers to exponentially decaying forcing. From the datum norms f0 m and the constants√Jm−1 in the forcing bounds, as in (9.15) we deﬁne the coeﬃcients Fm :=

f0 m + 2Jm−1 for m = 2, 4. Conditions (9.16) for m = 2, 4 become, respectively, 1.14F2 ≤ 1 and 0.380F4 ≤ 1; these are fulﬁlled if F2 ≤ 0.877,

F4 ≤ 2.63,

(10.11)

and in this case the Galerkin solution ϕG is global for each G. As an example, let us suppose F2 = 0.20,

F4 = 2.00;

(10.12)

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then, condition (9.40) becomes 0.121 + 1.75/|G| ≤ 1, which is fulﬁlled if |G| ≥ 2.00.

(10.13)

With these assumptions the exact solution ϕ of VP2 (f0 ) is global, and

ϕ(t) − ϕG (t) 2 ≤

W24 |G| e−t 6.10 e−t ≤ for all G as above, t ∈ [0, +∞). |G|2 |G|2 (10.14)

The expression of W24 |G| is provided by (9.42); here we have used the rough bound W24 |G| ≤ 6.10, coming from (9.43). Appendix A. Proof of Lemma 2.6 First of all, we put Z := sup z(t);

(A.1)

t∈[t0 ,τ ]

we continue in two steps. Step 1. For all k ∈ N, one has z(t) ≤ Z

Λk Γ(σ)k (t − t0 )kσ Γ(kσ + 1)

for t ∈ [t0 , τ ].

(A.2)

To prove this, we write (A.2)k for the above equation at order k, and proceed by recursion. Equation (A.2)0 is just the inequality z(t) ≤ Z. Now, we suppose that (A.2)k holds and infer from it Eq. (A.2)k+1 . To this purpose, we substitute (A.2)k into the basic inequality (2.22), which gives t Γ(σ)k (s − t0 )kσ z(t) ≤ ZΛk+1 ds ; Γ(kσ + 1) t0 (t − s)1−σ expressing the integral via the known identity (4.20), we get the thesis (A.2)k+1 . Step 2. z(t) = 0 for all t ∈ [t0 , τ ]. In Eq. (A.2), let us ﬁx t and send k to ∞; the right-hand side of this inequality vanishes in this limit, yielding the thesis. Appendix B. A Scheme to Solve Numerically the Control Inequality (5.14) Notations. In this appendix we often write {0, . . . , M} where M is an integer or +∞. If M is a nonnegative integer this will mean, as usually, the set of integers 0, 1, 2, . . . , M. If M is a negative integer, we will intend {0, . . . , M} := ∅. If M = +∞, {0, . . . , M} will mean the set N of all natural numbers. We often consider ﬁnite or inﬁnite sequences of real numbers of the form (tm )m∈{0,...,M} ; if M = +∞, we intend tM := limm→+∞ tm whenever the limit exists.

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The numerical scheme. Let us be given an approximate solution ϕap ∈ C([0, T ), F) of VP(f0 ), where T ∈ (0, +∞]; in the sequel we always intend E(t) := E(ϕap )(t) ,

for t ∈ [0, T ).

D(t) := ϕap (t)

(B.1)

Hereafter we outline a numerically implementable algorithm to construct a solution R of the integral inequality (5.14) on some interval [0, T ) ⊂ [0, T ); this solution R will be piecewise linear. In order to construct the algorithm, we choose a sequence of instants (tm )m=0,...,M , where M is a positive integer or +∞. We assume 0 = t 0 < t1 < t 2 < · · · < tM = T .

(B.2)

Furthermore, we denote with Em , Dm , Hmk , Imk , Nmk some constants such that E(t) ≤ Em ,

sup t∈[tm ,tm+1 )

tk+1

sup t∈[tm ,tm+1 )

ds u− (t − s)

tk

tk+1

sup t∈[tm ,tm+1 )

tk+1

sup t∈[tm ,tm+1 )

sup

D(t) ≤ Dm ;

s − tk tk+1 − tk

ds u− (t − s)

tk

ds u− (t − s) ≤ Nmk

2 ≤ Hmk ,

s − tk ≤ Imk , tk+1 − tk for m ∈ {1, . . . , M − 1},

tk

k ∈ {0, . . . , m − 1};

t

ds u− (t − s)

sup t∈[tm ,tm+1 )

tm

t

sup t∈[tm ,tm+1 )

t

sup t∈[tm ,tm+1 )

(B.3)

t∈[tm ,tm+1 )

s − tm tm+1 − tm

ds u− (t − s)

tm

ds u− (t − s) ≤ Nmm

(B.4)

2 ≤ Hmm ,

s − tm ≤ Imm , tm+1 − tm for m ∈ {0, . . . , M − 1}.

tm

Finally, for m, k as above and all a, x ∈ R, we deﬁne Φmk (a, x) := (Hmk + Nmk − 2Imk )a2 + 2(Imk − Hmk )ax + Hmk x2 + 2(Nmk − Imk )Dk a + 2Imk Dk x.

(B.5)

Proposition B.1. Suppose there is a ﬁnite or inﬁnite sequence of nonnegative reals (Rm )m∈{0,...,M} (with 1 ≤ M ≤ M ) such that Em + K

m

Φmk (Rk , Rk+1 ) ≤ min(Rm , Rm+1 )

k=0

for m ∈ {0, . . . , M − 1}.

(B.6)

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Let R ∈ C([0, tM ), [0, +∞)) be the unique piecewise linear map with values Rm at the times tm , i.e. R(t) = Rm + (Rm+1 − Rm ) for t ∈ [tm , tm+1 ),

t − tm tm+1 − tm

m ∈ {0, . . . , M − 1}.

(B.7)

Then, R solves the integral inequality (5.14) on [0, tM ). Proof. Let R be deﬁned as above, and t in some subinterval [tm , tm+1 ) (m ∈ {0, . . . , M − 1}). Then

t

l.h.s. of (5.14) ≤ Em + K

ds u− (t − s)(2D(s) + R(s))R(s)

0

≤ Em + K

m−1 k=0

= Em + K

t

ds u− (t − s)(2Dk + R(s))R(s)

+ tm

tk

m−1 k=0

tk+1

tk+1

tk

t

+

ds u− (t − s)

tm

s − tk × 2Dk + Rk + (Rk+1 − Rk ) tk+1 − tk s − tk × Rk + (Rk+1 − Rk ) tk+1 − tk ≤ Em + K

m

Φmk (Rk , Rk+1 ),

(B.8)

k=0

the last passage following from the inequalities (B.4) and the deﬁnition (B.5) of Φmk . From here and from (B.6) we infer, for t in the same interval, l.h.s. of (5.14) ≤ min(Rm , Rm+1 ) ≤ R(t) = r.h.s. of (5.14).

(B.9)

In conclusion, (B.6) ensures R to fulﬁll (5.14) on [0, tM ). Remark B.2. (i) A sequence of constants (Ek ) fulﬁlling the ﬁrst inequality (B.3) is easily obtained if ϕap ∈ C([0, T ), F+ ) ∩ C 1 ([0, T ), F− ), and there are suitable estimators for (the semigroup and) for the datum and diﬀerential errors d(ϕap ), e(ϕap ). More precisely suppose that

d(ϕap ) ≤ δ,

(B.10)

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and that, for m ∈ {0, . . . , M − 1}, sup

u(t) ≤ um ,

t∈[tm ,tm+1 )

e(ϕap (t)) − ≤ m ,

sup

(B.11)

t∈[tm ,tm+1 )

(um )m=0,...,M −1 and ( m )m=0,...,M −1 being sequences of nonnegative reals. From Lemma 3.4 on the integral error we obtain, for t ∈ [tm , tm+1 ), m−1 tk+1 t E(t) ≤ u(t)δ + ds u− (t − s) e(ϕap (s)) − ; + (B.12) k=0

tm

tk

now, from (B.11) and (B.4) we infer, for m ∈ {0, 1, . . . , M − 1}, E(t) ≤ Em if t ∈ [tm , tm+1 ),

Em := um δ +

m

Nmk k .

(B.13)

k=0

(In fact, one could extend this result to the case where ϕap is continuous from [t0 , T ) to F+ and piecewise C 1 from [t0 , T ) to F− : this typically occurs for the approximate solutions deﬁned by ﬁnite diﬀerence schemes in time.) (ii) For any m, Eq. (B.6) holds if and only if either

Rm+1 ∈ [0, Rm ),

Em + K

m

Φmk (Rk , Rk+1 ) ≤ Rm+1 ,

(B.14)

k=0

or Rm+1 ∈ [Rm , +∞),

Em + K

m

Φmk (Rk , Rk+1 ) ≤ Rm ;

(B.15)

k=0

note that each Φmk in these inequalities is a quadratic polynomial. For m = 0, Eqs. (B.14) and (B.15) deﬁne a problem for two unknowns (R0 , R1 ); for m > 0, we can see (B.14) and (B.15) as a problem to determine recursively Rm+1 from R0 , . . . , Rm . For each m, if the problem has solutions it seems convenient to choose for Rm+1 the smallest admissible value. This criterion could be applied for m = 0 as well, choosing among all solutions (R0 , R1 ) the one with the smallest R1 . (iii) In practical computations, the determination of Rm+1 from R0 , . . . , Rm goes on until problem (B.14) or (B.15) have solutions. The iteration ends if, for some ﬁnite M , both the above inequalities for RM+1 have no solutions. In this case, we have a function R solving (5.14) on the interval [0, tM ). Alternatively, the iteration might go on indeﬁnitely. (iv) The recursive scheme (B.6) has a typical feature of the iterative methods to solve integral equations or inequalities of the Volterra type: to ﬁnd Rm+1 one must compute a “memory term” involving R0 , . . . , Rm . The memory term depends nontrivially on m (through the coeﬃcients Imk , etc.), so it must be fully redetermined at each step; this makes the computation more and more expensive while m grows. An exception to this framework occurs if the semigroup estimator t → u− (t) is (a constant ×) an exponential, at least for t greater than some ﬁxed time ϑ;

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this is just the case of the NS equations, see the forthcoming Remark B.4(ii). In this special situation, Eq. (B.6) can be rephrased as a pair of recursion relations for two real sequences (Rm ), (Sm ); at each step, computation of Rm+1 and Sm+1 does not involve the whole previous history, but only the values of Sm and of Rk for tm − ϑ < tk+1 ≤ tm+1 . The forthcoming Proposition explains all this. Proposition B.3. Let us suppose there are ϑ ≥ 0 and A, B > 0 such that u− (t) = Ae−Bt

for t ∈ (ϑ, +∞);

(B.16)

furthermore, let us intend that k always means an integer in {0, . . . , M − 1}. Then, (i) and (ii) hold. (i) Let m ∈ {0, . . . , M },

tk+1 ≤ tm − ϑ;

(B.17)

then, conditions (B.4) are fulﬁlled with Hmk := Ae−Btm Hk , Hk :=

2(eBtk+1 − eBtk ) − 2BeBtk+1 (tk+1 − tk ) + B 2 eBtk+1 (tk+1 − tk )2 ; B 3 (tk+1 − tk )2

Imk := Ae−Btm Ik ,

Ik :=

eBtk − eBtk+1 + BeBtk+1 (tk+1 − tk ) ; B 2 (tk+1 − tk )

Nmk := Ae−Btm Nk ,

Nk :=

eBtk+1 − eBtk . B

(B.18)

Consequently, for all real a, x one has Φmk (a, x) = Ae−Btm Φk (a, x), Φk (a, x) := (Hk + Nk − 2Ik )a2 + 2(Ik − Hk )ax + Hk x2

(B.19)

+ 2(Nk − Ik )Dk a + 2Ik Dk x. (ii) Consider a sequence (Rm )m∈{0,...,M} of nonnegative reals. Then, (Rm ) fulﬁlls Eq. (B.6) if and only if there is a sequence of reals (Sm )m∈{0,...,M−1} such that Sm + Φk (Rk , Rk+1 ) ≤ Sm+1 for m ∈ {0, . . . , M − 2}; {k|tm −ϑ
(B.20) Em + K

Φmk (Rk , Rk+1 ) + KAe−Btm Sm

{k|tm −ϑ
≤ min(Rm , Rm+1 )

for m ∈ {0, . . . , M − 1}.

(B.21)

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(iii) In particular, suppose the instants tm and ϑ to be integer multiples of a basic spacing τ > 0: tm = mτ

for m ∈ {0, . . . , M };

for some L ∈ N.

ϑ = Lτ

(B.22)

Then, for m ∈ {0, . . . , M − 1}, {k | tm − ϑ < tk+1 ≤ tm+1 } =

{m − L, . . . , m}

if m ≥ L,

{0, . . . , m}

if m < L,

{k | tm − ϑ < tk+1 ≤ tm+1 − ϑ} =

{m − L}

if m ≥ L,

∅

if m < L.

(B.23)

(B.24)

Proof. (i) Let t ∈ [tm , tm+1 ). For s ∈ [tk , tk+1 ) one has t − s > tm − tk+1 ≥ ϑ, implying u− (t − s) = Ae−B(t−s) ; so,

tk+1

ds u− (t − s)

tk

s − tk tk+1 − tk

2

= Ae−Bt

tk+1

ds eBs

tk

s − tk tk+1 − tk

2

= Ae−Bt Hk ≤ Ae−Btm Hk = Hmk

as in (B.18).

(B.25)

In conclusion, deﬁning Hmk as in (B.18) we fulﬁll the ﬁrst inequality in (B.4) (note that k ≤ m − 1 due to tk+1 ≤ tm ). Similarly, the other inequalities (B.4) are fulﬁlled with Imk , Nmk as in (B.18). Finally, inserting Eq. (B.18) into Eq. (B.5) for Φmk we obtain the thesis (B.19). (ii) Let us rephrase Eq. (B.6) in the case under examination. To this purpose, we reexpress the sum therein writing m

=

{k|tm −ϑ
k=0

+

,

(B.26)

{k|tk+1 ≤tm −ϑ}

and then use Eq. (B.19) for the summands with tk+1 ≤ tm − ϑ; in this way, Eq. (B.6) becomes Em + K

Φmk (Rk , Rk+1 )

{k|tm −ϑ
+ KAe−Btm

Φk (Rk , Rk+1 )

{k|tk+1 ≤tm −ϑ}

≤ min(Rm , Rm+1 ) for m ∈ {0, . . . , M − 1}.

(B.27)

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Let us consider any sequence (Rm )m∈{0,...,M} of nonnegative reals. If (Rm ) fulﬁlls (B.27), deﬁne

Sm :=

Φk (Rk , Rk+1 ) for m ∈ {0, . . . , M − 1};

(B.28)

{k|tk+1 ≤tm −ϑ}

then (B.20) and (B.21) follow immediately (in fact, with ≤ replaced by = in (B.20)). Conversely, suppose there is a sequence of reals (Sm )m∈{0,...,M−1} fulﬁlling Eqs. (B.20) and (B.21) with (Rm ); then {k|tk+1 ≤tm −ϑ} Φk (Rk , Rk+1 ) ≤ Sm , and it is easy to infer Eq. (B.27) for (Rm ). (iii) Obvious. Remark B.4. (i) Equation (B.20) does not prescribe S0 . It is convenient to choose S0 := 0; with this position Eq. (B.21) with m = 0 is a problem for two unknowns (R0 , R1 ), for which we could repeat the comments of Remark B.2(ii). After these initial steps, we can use Eqs. (B.20) and (B.21) as recursion relations to obtain S1 , R2 , S2 , R3 , and so on. (ii) As anticipated, Proposition B.3 can be applied to the NS equations, in the framework of Sec. 7. Equation (7.10) of the cited section √ √ gives the semigroup So, estimator u− (t) := et / 2et for t ≤ 1/4, and u− (t) := 2e−t for all t > 1/4.√ the conditions of the previous proposition are fulﬁlled with ϑ = 1/4, A = 2, B = 1. These values must be substituted into Eq. (B.18) for Hmk , Imk and Nmk giving, for example, Nmk :=

√ −t 2e m Nk ,

Nk := etk+1 − etk

for tk+1 ≤ tm − 1/4.

(B.29)

Explicit expressions could be derived as well for Hmk , Imk and Nmk when tm − 1/4 < tk+1 ≤ tm , using elementary bounds on u− derived from the expression (7.10). However, this requires a tedious analysis of a number of cases, since the parameter t − s in Eq. (B.4) can be smaller or greater that 1/4; details will be given elsewhere, when we will treat systematically the approach outlined in this appendix.

Appendix C. Proof of Lemmas 6.1 and 6.2 We begin with two auxiliary lemmas. √ Lemma C.1. Let us consider two radii √ ρ, ρ1 such that 2 d ≤ ρ < ρ1 ≤ +∞, and a nonincreasing function χ ∈ C([ρ − 2 d, ρ1 ), [0, +∞)). Then, h∈Zd ,ρ≤|h|<ρ1

χ(|h|) ≤

2π d/2 Γ(d/2)

ρ1 √ dt (t ρ−2 d

+

√ d−1 d) χ(t).

(C.1)

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Proof. Let us introduce the cubes h + [0, 1]d

(h ∈ Zd )

(C.2)

and the annulus √ √ √ √ A(ρ − d, ρ1 + d) := {q ∈ Rd | ρ − d ≤ |q| < ρ1 + d};

(C.3)

we claim that "

(h + [0, 1]d ) ⊂ A(ρ −

√ √ d, ρ1 + d).

(C.4)

h∈Zd, ρ≤|h|<ρ1

√ √ In fact, q ∈ h√+ [0, 1]d implies√|h| − d ≤ |q| ≤ |h| + d; now, if ρ ≤ |h| < ρ1 we conclude ρ − d ≤ |q| < ρ1 + d. The inclusion (C.4) implies √ √ dq χ(|q| − d) ≤ d). (C.5) √ √ dq χ(|q| − h∈Zd ,ρ≤|h|<ρ1

h+[0,1]d

A(ρ− d,ρ1 + d)

√ On the other hand, √ for h as in the above sum and q ∈ h + [0, 1]d , we have |q| − d ≤ |h|, whence χ(|q| − d) ≥ χ(|h|); this implies √ dq χ(|q| − d) ≥ χ(|h|) dq = χ(|h|). (C.6) h+[0,1]d

h+[0,1]d

From here and from (C.5) we obtain χ(|h|) ≤

√

dq χ(|q| −

√

√ d).

(C.7)

A(ρ− d,ρ1 + d)

h∈Zd ,ρ≤|h|<ρ1

The right-hand side of (C.7) can be expressed in terms of the one-dimensional variable r = |q|; as well known, dq = (2π d/2 /Γ(d/2))rd−1 dr, so h∈Zd ,ρ≤|h|<ρ1

2π d/2 χ(|h|) ≤ Γ(d/2)

now, a change of variables t = r −

√ ρ1 + d √

√ d);

(C.8)

for ν, λ ∈ (0, +∞).

(C.9)

dr rd−1 χ(r −

ρ− d

√ d gives the thesis (C.1).

To go on, let us recall the convention (6.56) Zd := Zd or Zd0 . Lemma C.2. (i) Generalizing (6.59), let Sν (λ) :=

1 (2π)d

h∈Zd ,|h|<λ

1 (1 + |h|2 )ν

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Then, √ (1 + d)ν Fν (λ) for ν > 0, λ > 2 d, d−1 d/2 2 π Γ(d/2) 1 1 Sν := , (2π)d (1 + |h|2 )ν √ h∈Zd ,|h|<2 d  1  1 1   √ − if ν = d/2,   2ν − d √ 2ν−d (λ + d)2ν−d d Fν (λ) := √   λ+ d   √ if ν = d/2.  log d Sν (λ) ≤ Sν +

(C.10)

For ﬁxed ν and λ → +∞, this implies   O(1) Sν (λ) = O(log λ)  O(λd−2ν )

if ν > d/2, if ν = d/2, if 0 < ν < d/2.

(C.11)

(ii) Let ∆Sν (λ) :=

1 (2π)d

h∈Zd ,|h|≥λ

1 (1 + |h|2 )ν

for ν > d/2,

λ > 0.

(C.12)

Then ∆Sν (λ) ≤ δSν (λ)

for ν > d/2,

√ λ > 2 d,

(C.13)

where (generalizing (6.60)) δSν (λ) :=

(1 + d)ν √ . 2d−1 π d/2 Γ(d/2)(2ν − d) (λ − d)2ν−d

(C.14)

√ Proof. (i) Let ν > 0, λ > 2 d. Dividing in two parts the sum deﬁning Sν (λ), we get Sν (λ) = Sν +

1 (2π)d

√ h∈Zd ,2 d≤|h|<λ

1 . (1 + |h|2 )ν

(C.15)

√ Now, we bind the sum using (C.1) with χ(t) := 1/(1 + t2 )ν , ρ := 2 d and ρ1 := λ, yielding Sν (λ) ≤ Sν +

1 2d−1 π d/2 Γ(d/2)

λ

dt 0

√ (t + d)d−1 . (1 + t2 )ν

(C.16)

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On the other hand, one establishes by elementary means the inequality 1 1+d √ ≤ for t ∈ [0, +∞) (C.17) 1 + t2 (t + d)2 √ (holding as an equality when t = 1/ d). Inserting this into (C.16), we get λ (1 + d)ν 1 √ dt . (C.18) Sν (λ) ≤ Sν + d−1 d/2 2 π Γ(d/2) 0 (t + d)2ν−d+1 The last integral equals Fν (λ), so (C.10) is proved. Having this result, the statement (C.11) on √ the limit λ → +∞ is obvious. (ii) Let ν > d/2, λ > 2 d. To bind ∆Sn (λ) we use (C.1) with χ(t) := 1/(1 + t2 )ν , ρ := λ and ρ1 := +∞, and subsequently employ the inequality (C.17). This gives √ +∞ 1 (t + d)d−1 dt ∆Sν (λ) ≤ d−1 d/2 (1 + t2 )ν 2 π Γ(d/2) λ−2√d +∞ (1 + d)ν 1 √ ≤ d−1 d/2 dt , (C.19) 2 π Γ(d/2) λ−2√d (t + d)2ν−d+1 and computing the last integral we justify Eqs. (C.13) and (C.14). From here to the end of the appendix we ﬁx a real number n, fulﬁlling the relation (6.55) n>

d ; 2

here are the proofs of Lemmas 6.1 and 6.2.

√ Proof of Lemma 6.1. We take any λ ≥ 2 d; with the notations of the previous Lemma, we have Σn = Sn (λ) + ∆Sn (λ).

(C.20)

Both terms in the right-hand side are ﬁnite; the term ∆Sn (λ) has the upper bounds (C.13) and the obvious lower bound ∆Sn (λ) > 0. From these facts we infer the ﬁniteness of Σn , and the bounds (6.58) for it. √ Proof of Lemma 6.2. We consider any cutoﬀ function Λ : Zd → [2 d, +∞). For k ∈ Zd , we introduce the decomposition Kn (k) = Kn (k) + ∆Kn (k);

(C.21)

here Kn (k) is deﬁned by (6.66), and ∆Kn (k) =

(1 + |k|2 )n−1 (2π)d

h∈Zd ,|h|>Λ(k)

|k − h|2 . (1 + |h|2 )n (1 + |k − h|2 )n

(C.22)

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For the term Kn (k), we furtherly introduce a decomposition Kn (k) = Kn (k) + Kn (k), Kn (k) := Kn (k) :=

(1 + |k|2 )n−1 (2π)d

(1 + |k|2 )n−1 (2π)d

h∈Zd ,|h|<|k|/2

(1 +

|k − h|2 , + |k − h|2 )n

|h|2 )n (1

h∈Zd ,|k|/2≤|h|<Λ(k)

(C.23)

(1 +

|k − h|2 + |k − h|2 )n

|h|2 )n (1

(C.24)

(C.25)

(the sum deﬁning K (k) is meant to be zero if Λ(k) ≤ |k|/2). In the sequel we analyze separately ∆Kn , Kn and Kn ; we will frequently use the inequalities |k − h|2 1 ≤ 2 n (1 + |k − h| ) (1 + |k − h|2 )n−1 1+z 1 ≤ max 1, 1 + ηz η

for all k, h ∈ Zd ;

for all η > 0, z ≥ 0.

(C.26)

(C.27)

Step 1. For all k ∈ Zd , one has 0 < ∆Kn (k) ≤ δKn (k);

(C.28)

here, as in (6.67), δKn (k) :=

(1 + d)n λ(|k|)n−1 2d−1 π d/2 Γ(d/2)(2n − d) (Λ(k) −

√ , d)2n−d

λ being deﬁned by (6.64). If Λ fulﬁlls (6.68), the above upper bound is such that δKn (k) = O

1 |k|2n−d

→0

for k → ∞.

(C.29)

The inequality ∆Kn (k) > 0 is obvious. To prove the rest we start from Eq. (C.26), implying ∆Kn (k) ≤

(1 + |k|2 )n−1 (2π)d

h∈Zd ,|h|≥Λ(k)

1 ; (1 + |h|2 )n (1 + |k − h|2 )n−1

(C.30)

setting µ(k) :=

inf

h∈Zd ,|h|≥Λ(k)

|k − h|,

(C.31)

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we infer from (C.30) that 1 (2π)d

∆Kn (k) ≤

1 + |k|2 1 + µ2 (k)

1 + |k|2 1 + µ(k)2

=

n−1

n−1

h∈Zd ,|h|≥Λ(k)

1 (1 + |h|2 )n

∆Sn (Λ(k)),

(C.32)

where ∆Sn is deﬁned following Eq. (C.12). To go on we claim that, for all k ∈ Zd , # 0 if Λ(k) < |k|, µ(k) ≥ (C.33) Λ(k) − |k| if Λ(k) ≥ |k|. The above inequality is trivial if Λ(k) < |k|; if Λ(k) ≥ |k|, it follows noting that |h| ≥ Λ(k) implies |k − h| ≥ |h| − |k| ≥ Λ(k) − |k|. Having proved (C.33), we insert it into (C.32); this gives the inequality ∆Kn (k) ≤ λ(k)n−1 ∆Sn (Λ(k))

(C.34)

and substituting therein the upper bound (C.13) for ∆Sn we get the upper bound in (C.28). To go on, suppose Λ fulﬁlls (6.68); then Λ(k) ≥ α|k| ≥ |k|, 1 1 + |k|2 ≤ max 1, λ(k) ≤ 1 + (α − 1)2 |k|2 (α − 1)2

(C.35) for |k| ≥ χ

(the last inequality follows from (C.27), with z = |k|2 and η = (α − 1)2 ). So, 1 1 √ =O , λ(k) = O(1) for k → ∞; (C.36) |k| Λ(k) − d inserting this in the deﬁnition of δKn , we infer Eq. (C.29). Step 2. With Σn as in (6.57), one has Kn (k) → Σn

for k → ∞.

(C.37)

To prove this, we write Kn (k) =

cnk (h),

h∈Zd

θ(|k|/2 − |h|)|k − h|2 (1 + |k|2 )n−1 , (2π)d (1 + |h|2 )n (1 + |k − h|2 )n # 1 if z ∈ (0, +∞), θ(z) := 0 if z ∈ (−∞, 0].

cnk (h) :=

(C.38)

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For any ﬁxed h ∈ Zd , we have cnk (h) →

1 1 d (2π) (1 + |h|2 )n

for k → +∞;

(C.39)

this implies Kn (k) →

1 1 = Σn , d (2π) (1 + |h|2 )n d

(C.40)

h∈Z

if the limit k → ∞ can be exchanged with the sum over h. By the Lebesgue theorem of dominated convergence, the exchange is possible if cnk (h) is bounded from above by a summable function of h, uniformly in k; indeed this occurs, since cnk (h) ≤

1 4n−1 d (2π) (1 + |h|2 )n

for all h,

k ∈ Zd .

(C.41)

Let us prove (C.41). The thesis is obvious if |h| ≥ |k|/2, since in this case cnd (k) = 0; hereafter we assume |h| < |k|/2. First of all, we note that cnk (h) = ≤

|k − h|2 (1 + |k|2 )n−1 (2π)d (1 + |h|2 )n (1 + |k − h|2 )n (2π)d (1

(1 + |k|2 )n−1 ; + |h|2 )n (1 + |k − h|2 )n−1

secondly, from |h| < |k|/2 we infer |k − h| ≥ |k| − |h| ≥ |k|/2, whence n−1 1 + |k|2 1 1 1 4n−1 ≤ cnk (h) ≤ (2π)d 1 + |k|2 /4 (1 + |h|2 )n (2π)d (1 + |h|2 )n

(C.42)

(C.43)

(the last inequality follows from (C.27), with η = 1/4 and z = |k|2 ). Step 3. Suppose Λ fulﬁlls (6.68); then, for k → ∞,  1   O if n > d/2 + 1,   |k|2     log |k| Kn (k) = O if n = d/2 + 1,  |k|2      1  O if d/2 < n < d/2 + 1 |k|2n−d

→ 0.

(C.44)

To prove this, let us take any k ∈ Zd such that |k| ≥ χ. First of all, from (C.26) and Λ(k) ≤ β|k| we infer Kn (k) ≤

≤

(1 + |k|2 )n−1 (2π)d (1 + |k|2 )n−1 (2π)d

h∈Zd ,|k|/2≤|h|<Λ(k)

1 (1 + |h|2 )n (1 + |k − h|2 )n−1

h∈Zd ,|k|/2≤|h|<β|k|

(1 +

|h|2 )n (1

1 . + |k − h|2 )n−1

(C.45)

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For |h| ≥ |k|/2 we have 1/(1 + |h|2 ) ≤ 1/(1 + |k|2 /4), whence Kn (k) ≤

=

≤

1 (1 + |k|2 )n−1 (2π)d (1 + |k|2 /4)n 1 (2π)d

1 + |k|2 1 + |k|2 /4

1 4n−1 d (2π) 1 + |k|2 /4

h∈Zd ,|k|/2≤|h|<β|k|

n−1

1 1 + |k|2 /4

h∈Zd ,|k|/2≤|h|<β|k|

1 (1 + |k − h|2 )n−1

h∈Zd ,|k|/2≤|h|<β|k|

1 (1 + |k − h|2 )n−1

1 (1 + |k − h|2 )n−1

(C.46)

(the last passage uses again (C.27), with η = 1/4 and z = |k|2 ). Now, a change of variable h = q − k in the last sum gives Kn (k) ≤

1 4n−1 (2π)d 1 + |k|2 /4

q∈Zd ,|k|/2≤|q−k|<β|k|

1 . (1 + |q|2 )n−1

(C.47)

To go on we note that, for all q ∈ Zd , |q − k| < β|k| ⇒ |q| ≤ |q − k| + |k| < (β + 1)|k|.

(C.48)

Thus, Kn (k) ≤

1 4n−1 (2π)d 1 + |k|2 /4

q∈Zd ,|q|<(β+1)|k|

1 (1 + |q|2 )n−1

n−1

=

4 Sn−1 ((β + 1)|k|), 1 + |k|2 /4

(C.49)

where Sn−1 is deﬁned following Eq. (C.9). The k → ∞ behavior of Sn−1 ((β + 1)|k|) is inferred from Eq. (C.11); this function is multiplied by 4n−1 /(1 + |k|2 /4) = O(1/|k|2 ), and the conclusion is Eq. (C.44). Step 4. Conclusion of the proof. For any cutoﬀ Λ, the decomposition (C.21) and the bounds (C.28) give the inequalities (6.65) for Kn (k), also implying the ﬁniteness of this quantity for arbitrary k. From now on Λ is supposed to fulﬁll (6.68), and we consider the limit k → ∞. Then, the decomposition Kn = Kn + Kn and Steps 2 and 3 give Kn (k) → Σn , as claimed in (6.69); the other statement in (6.69), namely, δKn (k) = O(1/|k|2n−d ) → 0, is known from (C.29). The decomposition Kn = Kn + ∆Kn , and Eqs. (C.28) and (6.69) also give Kn (k) → Σn . Summing up, all statements in items (i) and (ii) of this lemma are proved.

(C.50)

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Appendix D. Proof of Propositions 6.3 and 6.5 Proof of Proposition 6.3. We consider two vector ﬁelds v, w ∈ Hn with n > d/2, and proceed in two steps. Step 1. Proof of Eq. (6.71). We have

vr =

vhr eh ,

∂r ws = i

h∈Zd

r ws e ;

(D.1)

∈Zd

this implies v • ∂ws = v r ∂r ws = i v • ∂w = i

r s h,∈Zd (vh r w )(eh e )

or, in vector form,

(vh • )w (eh e ).

(D.2)

h,∈Zd

On the other hand eh e = eh+ /(2π)d/2 , so  i  v • ∂w = (2π)d/2 d d k∈Z

 (vh • )w  ek

h,∈Z , h+=k



 i  = [vh • (k − h)]wk−h  ek . (2π)d/2 d d k∈Z

(D.3)

h∈Z

The term multiplying ek in the last expression is the Fourier coeﬃcient (v • ∂w)k ; so, Eq. (6.71) is proved. Step 2. Proof that v • ∂w is in Hn−1 and fulﬁlls (6.73). For any k ∈ Zd , Eq. (6.71) implies |(v • ∂w)k | ≤

1 |vh ||k − h||wk−h | d/2 (2π) d h∈Z

=

Now, H¨older’s inequality |

1 |k − h| n n d/2 2 (2π) 1 + |h| 1 + |k − h|2 h∈Zd n n × 1 + |h|2 |vh | 1 + |k − h|2 |wk−h |.

2 h ah b h |

≤

h

|ah |2

h

(D.4)

|bh |2 gives

|(v • ∂w)k |2 ≤ ck pk , ck := pk :=

|k − h|2 1 , d 2 (2π) (1 + |h| )n (1 + |k − h|2 )n d

h∈Zd

h∈Z

(1 + |h|2 )n |vh |2 (1 + |k − h|2 )n |wk−h |2 .

(D.5)

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The last inequality implies (1 + |k|2 )n−1 |(v • ∂w)k |2 ≤ (1 + |k|2 )n−1 ck pk k∈Zd

k∈Zd

≤

sup (1 + |k|2 )n−1 ck

k∈Zd

pk .

(D.6)

k∈Zd

On the other hand, (1 + |k|2 )n−1 ck = Kn (k),

sup (1 + |k|2 )n−1 ck ≤ Kn2

(D.7)

k∈Zd

with Kn , Kn as in Eqs. (6.75) and (6.74); the ﬁniteness of Kn (k) for any k, and of its sup over k are known from Lemma 6.2. To go on, we note that    pk =  (1 + |h|2 )n |vh |2   (1 + |h|2 )n |wh |2  = v 2n w 2n . (D.8) k∈Zd

h∈Zd

h∈Zd

Inserting Eqs. (D.7) and (D.8) into (D.6), we see that |k|2 )n−1 |(v • ∂w)k |2 < +∞, implying v • ∂w ∈ Hn−1 . Furthermore,

v • ∂w 2n−1 = (1 + |k|2 )n−1 |(v • ∂w)k |2 ≤ Kn2 v 2n w 2n ,

k∈Zd (1

+

(D.9)

k∈Zd

yielding Eq. (6.73). Proof of Proposition 6.5. This is a simple variant of the proof given for Proposition 6.3. One takes into account the following facts: if f, g ∈ HnΣ0 , then their zero order Fourier coeﬃcients are f0 = 0, g0 = 0; furthermore, (f • ∂g)0 = 0, since this function has mean zero by Lemma 6.4. Appendix E. Derivation of the NS Equations (6.87) The NS equations in physical and adimensional units. In the space L of (oriented) lengths we ﬁx some positive length λ, determining the size of the system in consideration. The “space domain” of the system is modeled as a torus Ld /(2πλ)d ; we write x for any point in this domain, and t for the “physical” time. The NS equations are (E.1) ρ v(t) ˙ + v(t) • ∂v(t) = −∂p(t) + η∆v(t) + k(x, t) where v(t) : x → v(x, t), p(t) : x → p(x, t) are the velocity and the pressure ﬁelds, while k(t) : x → k(x, t) is the density of external forces; in the above, · indicates the partial derivative with respect to time t. The coeﬃcients ρ > 0, η > 0 are the density and viscosity of the ﬂuid, respectively. The velocity ﬁeld v(t) is required to be divergence free, to fulﬁll the condition of incompressibility.

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One passes to the adimensional form introducing three functions ν(x, t), π(x, t), κ(x, t) (x ∈ Td , t ∈ [0, T ) ⊂ R) via the equations v(x, t) =

η ν(x, t), ρλ

η2 η2 k(x, t) = κ(x, t) 2 π(x, t), ρλ ρλ3 ηt x for x = , t = . λ ρλ2 p(x, t) =

(E.2)

With these positions, Eq. (E.1) is equivalent to ν(t) ˙ + ν(t) • ∂ν(t) = −∂π(t) + ∆ν(t) + κ(t). Hereafter we formalize this adimensional version, specifying the necessary functional spaces. First of all, we suppose κ ∈ C 0,1 ([0, +∞), Hn−1 );

(E.3)

secondly, we stipulate the following. Deﬁnition E.1. The incompressible NS Cauchy problem with initial datum v0 ∈ Hn+1 Σ , in the pressure formulation, is the following. n−1 1 n Find ν ∈ C([0, T ), Hn+1 Σ ) ∩ C ([0, T ), HΣ ), π ∈ C([0, T ), H ) such that

ν(t) ˙ + ν(t) • ∂ν(t) = −∂π(t) + ∆ν(t) + κ(t)

for t ∈ [0, T ),

ν(0) = v0 (E.4)

(for some T ∈ (0, +∞]). n We note the following: the requirements ν ∈ C([0, T ), Hn+1 Σ ) and π ∈ C([0, T ), H ) are suﬃcient for the right-hand side of the above diﬀerential equation to be in C([0, T ), Hn−1 Σ ).

The equivalence between the pressure formulation (E.4) and the Leray formulation (6.87) of the Cauchy problem. Let us introduce the function η ∈ C 0,1 ([0, +∞), Hn−1 Σ ),

t → η(t) := Lκ(t).

(E.5)

For convenience, we report here the Cauchy problem in the form (6.87): n−1 1 Find ν ∈ C([0, T ), Hn+1 Σ ) ∩ C ([0, T ), HΣ ), such that

ν(t) ˙ = ∆ν(t) − L ν(t) • ∂ν(t) + η(t)

for t ∈ [0, T ),

ν(0) = v0

(for some T ∈ (0, +∞]). The above mentioned equivalence can be stated as follows. Proposition E.2. A function ν of domain [0, T ) fulﬁlls (6.87) if and only if there is a function π with the same domain, such that (ν, π) fulﬁlls (E.4).

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Proof. Suppose a pair (ν, π) fulﬁlls (E.4), and apply the projector L to both sides of the diﬀerential equation. We have Lν(t) = ν(t), and L commutes with both the time derivative ˙ and the Laplacian ∆; ﬁnally, L∂π(t) = 0. These facts yield Eq. (6.87). Conversely, suppose a function ν fulﬁlls (6.87) and deﬁne t → γ(t) := ∆ν(t) − ν(t) • ∂ν(t) + κ(t) − ν(t); ˙

γ ∈ C([0, T ), Hn−1 ),

(E.6)

for all t. Let then from (6.87) one infers Lγ(t) = 0, i.e. γ(t) ∈ Hn−1 Γ π : t ∈ [0, T ) → π(t) := ∂ −1 γ(t)

(E.7)

with ∂ −1 as in (6.37); then π ∈ C([0, T )H n ) (since ∂ −1 maps continuously Hn−1 Γ into H n ). One easily checks that (ν, π) fulﬁlls (E.4). Appendix F. Proof of Eq. (6.100) Let us rephrase the deﬁnition (6.99) of ξ as ξ(t) = ζ(t) − η(t),

(F.1)

having put ζ : [0, T ) → HnΣ ,

t → ζ(t)

such that

ζ(x, t) := η(x + h(t), t).

(F.2)

Clearly, the thesis (6.100) follows if we prove that ζ ∈ C 0,1 ([0, +∞), Hn−1 Σ0 ).

(F.3)

To this purpose, we consider the Fourier coeﬃcients ηk (t), ζk (t) of η(t), ζ(t) and note that (F.2) implies ζk (t) = ηk (t) eik•h(t)

(k ∈ Zd , t ∈ [0, T )).

(F.4)

Let t, t ∈ [0, T ). We have ζk (t) − ζk (t ) = αk (t, t ) + βk (t, t )

αk (t, t ) := ηk (t)eik•h(t ) (eik•(h(t)−h(t )) − 1),

ik•h(t )

(F.5)

(ηk (t) − ηk (t )). βk (t, t ) := e 2 n−1 |ζ (t) − ζ (t )|2 has the bound Therefore ζ(t) − ζ(t ) n−1 = k k k∈Zd (1 + |k| )

ζ(t) − ζ(t ) n−1 ≤ A(t, t ) + B(t, t ), (1 + |k|2 )n−1 |αk (t, t )|2 , A(t, t ) :=

(F.6)

k∈Zd

B(t, t ) :=

(1 + |k|2 )n−1 |βk (t, t )|2 .

k∈Zd

On the other hand, Eq. (F.5) and the elementary inequality |eik•y − 1| ≤ |k||y| (for all y ∈ Rd ) give |αk (t, t )| ≤ |k| |ηk (t)| |h(t) − h(t )|,

|βk (t, t )| = |ηk (t) − ηk (t )|.

(F.7)

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Inserting these bounds into the expressions (F.6) of A(t, t ), B(t, t ) (and using (1 + |k|2 )n−1 |k|2 ≤ (1 + |k|2 )n ) we get

ζ(t) − ζ(t ) n−1 ≤ η(t) n |h(t) − h(t )| + η(t) − η(t ) n−1 .

(F.8)

Now, let us consider any compact subset I of [0, +∞). Then, the assumptions (6.95) on η and the C 1 nature of h ensure the existence of constants Q, M1 , M2 such that

η(t) n ≤ Q, |h(t)−h(t )| ≤ M1 |t−t | and η(t)−η(t ) n−1 ≤ M2 |t−t | for t, t ∈ I. This implies

ζ(t) − ζ(t ) n−1 ≤ (QM1 + M2 )|t − t |,

(F.9)

and (F.3) is proved. Appendix G. Proof of Proposition 7.2, Item (iv) Our aim is to derive Eq. (7.13); we write this as √ sup N (t) = 2, t∈[0,+∞)

N : [0, +∞) → [0, +∞),

(G.1)

t → N (t) :=

t

ds µ− (t − s)e−s .

(G.2)

0

Equation (G.1) will follow from Steps 1 and 2, giving separate estimates on N (t) for 0 ≤ t ≤ 1/4 and t > 1/4. √ Step 1. For 0 ≤ t ≤ 1/4 one has N (t) < 2. In fact, with this range for t Eq. (7.10) for µ− implies t t √ ds e2t e2t √ e−3s 1 e2t √ ≤√ = √ 2 t ≤ √ < 2. ds √ N (t) = √ t−s t−s 2e 0 2e 0 2e 2 Step 2. One has supt>1/4 N (t) = for µ− we get

√ 2. In fact, for all t > 1/4, using again Eq. (7.10)

t−1/4

N (t) =

t

ds + 0

ds µ− (t − s)e−s

t−1/4

√ = 2 (1 − e1/4−t ) + Ce−t √ √ = 2 − ( 2 e1/4 − C)e−t , 1/4 e3s ds √ . C := 2 es 0 √ √ One has the estimate C ≤ 0.6 < 2 e1/4 , implying N (t) < 2. From the above √ expression for N (t), we also get limt→+∞ N (t) = 2; these facts yield the thesis.

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Appendix H. The Constants K2 and K4 in Dimension d = 3 The above constants are needed for the numerical examples in Sec. 10; the route to compute them is outlined in Lemmas 6.1 and 6.2, and Proposition 6.3. Computing K2 . We can take for it any constant such that sup K2 (k) ≤ K2 .

(H.1)

k∈Zd 0

We have the bounds (6.65): K2 (k) < K2 (k) ≤ K2 (k) + δK2 (k), with the explicit expressions (6.66) for K2 (k) and (6.67) for δK2 (k) (and Zd = Zd0 therein). Both K2 and δK2 are deﬁned in terms of some cutoﬀ function Λ2 , that we choose in this way: Λ2 (k) := 24 if |k| < 4, and Λ2 (k) := 6|k| if |k| ≥ 4. The above setting can be employed to evaluate K2 (k) for some set of values of k; computations performed for all k’s with |ki | ≤ 10 (i = 1, 2, 3) seem to indicate that sup K2 (k) = lim K2 (k). k→∞

k∈Zd 0

(H.2)

On the other hand, according to (6.62), lim K2 (k) = Σ2 ,

k→∞

(H.3)

where Σ2 is the series (6.57) with n = 2 and Zd = Zd0 . To estimate this series, we use the bounds (6.58): S2 (λ) < Σ2 ≤ S2 (λ) + δS2 (λ) with S2 (λ) and δS2 (λ) as in Eqs. (6.59) and (6.60); these depend on a cutoﬀ λ, to be chosen as large as possible to reach a good precision. Taking λ = 250, we get 0.03607 ≤ Σ2 ≤ 0.03934, which implies, taking square roots, 0.1899 ≤ sup K2 (k) ≤ 0.1984. k∈Zd 0

Retaining only two meaningful digits, we take as a ﬁnal upper bound for the quantity

(H.4)

(H.5) √ sup K2

K2 := 0.20.

(H.6) ! Computing K4 . We can take for it any constant such that supk∈Zd0 K4 (k) ≤ K4 .

We use again the bounds (6.65) K4 (k) ≤ K4 (k) + δK4 (k); the cutoﬀ Λ4 deﬁning K4 and δK4 is chosen setting Λ4 (k) := 10 if |k| < 10/3 and Λ4 (k) := 3|k| if |k| ≥ 10/3. Computing (K4 + δK4 )(k) for |ki | ≤ 6 (i = 1, 2, 3) we obtain numerical evidence that supk∈Zd0 (K4 + δK4 )(k) is attained at k = (3, 0, 0). So, sup K4 (k) ≤ (K4 + δK4 )(3, 0, 0) ≤ 0.004383. k∈Zd 0

(H.7)

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We also have sup K4 (k) ≥ K4 (3, 0, 0) ≥ 0.004382,

(H.8)

k∈Zd 0

and in conclusion, taking square roots, 0.06619 ≤ sup K4 (k) ≤ 0.06621. k∈Zd 0

Retaining only two meaningful digits, we take as a ﬁnal upper bound for the quantity K4 := 0.067.

(H.9) √ sup K4 (H.10)

Acknowledgments We are grateful to G. Furioli and E. Terraneo for useful bibliographical indications. This work was partly supported by INdAM and by MIUR, PRIN 2006 Research Project “Geometrical methods in the theory of nonlinear waves and applications”. References [1] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations (Oxford Univ. Press, New York, 1998). [2] S. I. Chernyshenko, P. Constantin, J. C. Robinson and E. S. Titi, A posteriori regularity of the three-dimensional Navier–Stokes equations from numerical computations, J. Math. Phys. 48(6) (2007) 065204/10. [3] P. Costantin and C. Foias, Navier–Stokes Equations (University of Chicago Press, Chicago, 1988). [4] D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional diﬀerential equation, J. Math. Anal. Appl. 204 (1996) 609–625. [5] C. Foias and G. Prodi, Sur le comportement global des solutions non stationnaires des equations de Navier–Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova 39 (1967) 1–34. [6] C. Foias and R. Temam, Asymptotic numerical analysis for the Navier–Stokes equations I, in Nonlinear Dynamics and Turbulence, eds. G. I. Barenblatt, G. Iooss and D. D. Joseph (Pitman, London, 1983), pp. 139–155. [7] G. Gallavotti, Foundations of Fluid Dynamics (Springer, Berlin, 2002). [8] T. Kato and H. Fujita, On the nonstationary Navier–Stokes system, Rend. Sem. Mat. Univ. Padova 32 (1962) 243–260. [9] H. Fujita and T. Kato, On the Navier–Stokes initial value problem I, Arch. Ration. Mech. Anal. 16 (1964) 269–315. [10] T. Kato, Strong Lp -solutions of the Navier–Stokes equation in Rm , with applications to weak solutions, Math. Z. 187(4) (1984) 471–480. [11] P. G. Lemari´e-Rieusset, Recent Developments in the Navier–Stokes Problem (Chapman & Hall, Boca Raton, 2002). [12] C. Morosi and L. Pizzocchero, On approximate solutions of semilinear evolution equations, Rev. Math. Phys. 16(3) (2004) 383–420. [13] C. Morosi and L. Pizzocchero, On the constants for multiplication in Sobolev spaces, Adv. Appl. Math. 36(4) (2006) 319–363.

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[14] Ya. Sinai, Power series for solutions of the 3D Navier–Stokes system on R3 , J. Stat. Phys. 121(5/6) (2005) 779–803. [15] R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis (NorthHolland, Amsterdam, 1979). [16] R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis (SIAM, Philadelphia, 1983). [17] E. Weisstein, World of Science, http://scienceworld.wolfram.com; see the entry “Mittag–Leﬄer function” and the related bibliography.

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Reviews in Mathematical Physics Vol. 20, No. 6 (2008) 707–724 c World Scientific Publishing Company

ON HAAG DUALITY FOR PURE STATES OF QUANTUM SPIN CHAINS

M. KEYL Institute for Scientific Interchange Foundation, Viale S. Severo 65, 10133 Torino, Italy [email protected] TAKU MATSUI Graduate School of Mathematics, Kyushu University, 1-10-6 Hakozaki, Fukuoka 812-8581, Japan [email protected] D. SCHLINGEMANN Institute for Scientific Interchange Foundation, Viale S. Severo 65, 10133 Torino, Italy [email protected] R. F. WERNER Institut f¨ ur Mathematische Physik, TU Braunschweig, Mendelssohnstr.3, 38106 Braunschweig, Germany [email protected] Received 14 May 2007 Revised 28 January 2008 In this note, we consider quantum spin chains and their translationally invariant pure states. We prove Haag duality for quasilocal observables localized in semi-infinite intervals (−∞, 0] and [1, ∞) when the von Neumann algebra generated by observables localized in [0, ∞) is non-type I. Keywords: UHF algebra; pure state; translational invariance; Haag duality; Cuntz algebra. Mathematics Subject Classification 2000: 82B10

1. Introduction In local Quantum Field Theory, Haag duality is a crucial notion in structure analysis. (See [9].) In this article, we consider the Haag duality for quantum spin systems on a one-dimensional lattice in an irreducible representation. By Haag duality we mean that the von Neumann algebra MΛ generated by observables localized in an inﬁnite subset Λ of Z is the commutant of the von Neumann algebra MΛc generated 707

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by observables localized in the complement Λc of Λ. This duality plays a crucial role in analysis of entanglement property of states of inﬁnite spin chain. See [10, 11]. In [10], an inﬁnite dimensional analogue of maximally entangled states is constructed, and, in [11], we investigated the condition of possibility of producing inﬁnitely many copies of maximally entangled q-bits in an inﬁnite bipartite quantum system (= inﬁnite one copy entanglement). It turns out that impossibility of splitting two mutually commuting von Neumann algebras into a tensor product is a necessary condition for the inﬁnite one copy entanglement. To apply our results to quantum spin chains, we required Haag duality for the algebra of local observables. If the von Neumann algebra MΛ is of type I, the duality is easy to show. However, even if the representation of a whole quasi-local algebra is irreducible, the restriction to an inﬁnite region may give rise to a non-type I von Neumann sub-algebra. For example, the restriction of the ground state of massless XY model to the semiinﬁnite interval [1, ∞) gives rise to a type III von Neumann algebra and we believe that the same is true for the spin 1/2 massless antiferromagnetic XXZ chain. Though Haag duality is a basic concept, it seems that the proof of Haag duality is not obtained so far for the general case when both Λ and its complement Λc are inﬁnite sets. We will see that the duality holds when the representation contains a vector state which is translationally invariant and Λ = [1, ∞). To explain our results more precisely, we introduce our notation now. By A, we denote the UHF C ∗ -algebra d∞ (the inﬁnite tensor product of d by d matrix algebras): A=

C∗

Md (C)

.

Z

Each component of the tensor product above is speciﬁed with a lattice site j ∈ Z. By Q(j) we denote the element of A with Q in the jth component of the tensor product and the identity in any other component. For a subset Λ of Z, AΛ is deﬁned as the C ∗ -subalgebra of A generated by elements Q(j) with all j in Λ. We set AΛ Aloc = Λ⊂Z:|Λ|<∞

where the cardinality of Λ is denoted by |Λ|. We call an element of Aloc a local observable or a strictly local observable. When ϕ is a state of A, the restriction of ϕ to AΛ will be denoted by ϕΛ : ϕΛ = ϕ|AΛ . We set AR = A[1,∞) ,

AL = A(−∞,0] ,

ϕR = ϕ[1,∞) ,

ϕL = ϕ(−∞,0] .

By τj we denote the automorphism of A determined by τj (Q(k) ) = Q(j+k) for any j and k in Z. τj is referred to as the lattice translation of A.

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Given a representation π of A on a Hilbert space, the von Neumann algebra generated by π(AΛ ) is denoted by MΛ . We set MR = M[1,∞) = π(AR ) ,

ML = M(−∞,0] = π(AL ) .

For a state ψ of a C ∗ -algebra A we denote the GNS triple by {πψ (A), Ωψ , Hψ } where πψ is the GNS representation and Ωψ is the GNS cyclic vector in the GNS Hilbert space πψ . The main result of this paper is described as follows. Theorem 1.1. Let ϕ be a translationally invariant pure state of the UHF algebra A. and let {πϕ (A), Ωϕ , Hϕ } be the GNS triple for ϕ. Then, the Haag duality holds: MR = ML .

(1.1)

Remark 1.2. We consider the situation that the state ϕR may not be faithful. In [11, Proposition 4.2], we have shown that MR appearing in our context cannot be a type II1 factor. Precise statement and its proof is included here in Lemma 2.2. We are not aware of any example of MR which is of type II∞ . MR is of type III in generic cases. For example, when the state ϕR is faithful, MR is of type III1 due to [13, Theorem 4]. More precisely, an endomorphism τˆ of a factor is strongly asymptotically abelian if τ n (Q), R] = 0 lim[ˆ n

in strong operator topology for any Q and R in M. If ϕR is faithful, the restriction of the (normal extension) shift τ1 of M to MR is an strongly asymptotic abelian endomorphism of MR and MR is a type III1 factor. See [13]. Remark 1.3. In our proof of Haag duality, we consider a gauge invariant extension of the state ϕ to a state of the tensor product Od ⊗ Od of Cuntz algebras and show Haag duality at this level. We use ideas of [5] in our proof, though, our way of proof is diﬀerent from [5]. In [5], Bratteli, Jorgensen, Kishimoto and Werner focus on dilation of Popescu systems to representations of Cuntz algebras and their pure states while our starting point is a pure state of (two-sided inﬁnite) UHF algebras and go down to Popescu systems. At ﬁrst look, [5, Sec. 7] may give an impression that Proof of [5, Theorem 7.1] implies Haag duality. (cf. Lemmas 7.7 and 7.8.) However, for the KMS state of the standard U (1) gauge action of Od the assumption of [5, Theorem 7.1] are satisﬁed both Lemmas 7.7 and 7.8 do not hold. Let Sj be the Cuntz generator and consider the gauge action γz deﬁned in Sec. 2. Then the β = ln d KMS state ψ is unique, in particular it is faithful and the gauge invariant extension of the trace of MR . Then the assumption of [5, Theorem 7.1] is satisﬁed for the GNS representation {πψ , Hψ } of Od associated with ψ if we set

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K = Hψ . Vj = πψ (Sj ). Then, V˜j = d1 Jπψ (Sj∗ )J and Hψ = H0 ,

E|K = P = IK .

Nevertheless, the state ω is not pure and the equivalence of conditions (i) and (iii) of [5, Theorem 7.1] is valid. We prove that [5, Lemma 7.6] is valid when the state of A is pure, and for that purpose we introduce new ideas in Sec. 3. Our ideas are based on the observation that the translation τ1 is an inner automorphism of Od ⊗ Od . We do not use [5, Commuting Lifting Theorem] for our proof of Haag duality. 2. Split Property One key word in our analysis is split property or split inclusion. Let M1 and M2 be factors acting on a Hilbert space H satisfying M1 ⊂ M2 . We say the inclusion is split if and only if there exists an intermediate type I factor N such that M1 ⊂ N ⊂ M2 . The split inclusion is introduced for analysis of local QFT and of von Neumann algebras in 1980’s (cf. [8]). In [12], Longo used this notion of splitting for his solution to the factorial Stone–Weierstrass conjecture. If mutually commuting factors M1 and M2 are acting on a Hilbert space H and, the inclusion is split, the von Neumann algebra M3 generated by M1 and M2 is isomorphic to the tensor product of M1 and M2 . Thus the split inclusion is a weak notion of independence of two quantum systems. In the case of quantum spin chains, we set M1 = π(AΛ ) , M2 = π(AΛc ) . Motivated by this fact, we introduce the split property for states. Definition 2.1. A state ϕ of the UHF algebra A for a one-dimensional quantum spin system has split property with respect to Λ and Λc if and only if ϕ is quasiequivalent to the product state ϕΛ ⊗ ϕΛc . When ϕ is pure and split, ϕ is unitarily equivalent to a pure product state and the von Neumann algebra πϕ (AΛ ) is of type I. When ϕ is pure, πϕ (AΛ ) is of type I, if and only if ϕ has the split property. Thus if the von Neumann algebra MΛ generated πϕ (AΛ ) is of type I, the Haag duality is very easy to see. Lemma 2.2. Let ϕ be a pure state of A. If the von Neumann algebra MΛ generated by πϕ (AΛ ) is of type I, then MΛ = MΛc . Proof. As the pure state ϕ of A is split with respect to Λ and Λc , ϕ is unitarily equivalent to ψ1 ⊗ ψ2 where ψ1 (respectively, ψ2 ) is a state of AΛ (respectively, AΛc ). The GNS Hilbert space Hϕ associated with ϕ is unitarily equivalent to the tensor product Hψ1 ⊗ Hψ2 and MΛ = B(Hψ1 ) ⊗ 1Hψ2 , MΛc = 1Hψ1 ⊗ B(Hψ2 ) = MΛ . As a consequence, in our proof of Haag duality, we concentrate on pure states ϕ which are not quasi-equivalent to ϕΛ ⊗ ϕΛc . Existence of a translationally invariant

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pure without the split property for Λ = [1, ∞), is highly non-trivial. In [14], we have shown that ground states of some spin 1/2 systems satisfy these requirements. When ϕ is a translationally invariant factor state of A, ϕR gives rise to a shift of the von Neumann algebra MR in the following way. As there exists a unitary U implementing the shift τ1 speciﬁed with U π(Q)Ωϕ = π(τ1 (Q))Ωϕ for Q in A. Ad(U ) gives rise to an endomorphism on the factor MR generated by πϕ (AR ). We denote this endomorphism of MR by τˆ1 : U QU ∗ = τˆ1 (Q) (Q ∈ MR ). By deﬁnition, ∞

τˆ1n (MR ) = C1.

n=0

Lemma 2.3. Let ϕ be a translationally invariant pure state and let MR be the von Neumann algebra generated by πϕ (AR ). MR cannot be of type II1 . Proof. Suppose that MR is of type II1 and let tr be its unique normal tracial state. The shift endomorphism of AR is a limit of cyclic permutations of (1, 2, . . . , n) of lattice site which is implemented by unitary Un , τ1 (Q) = lim Un QUn∗ . It turns out that the trace is invariant under τˆ1 because tr(ˆ τ πϕ (Q)) = tr(πϕ (τ1 (Q)) = lim tr(πϕ (Un QUn∗ )) = tr(πϕ (Q)). n→∞

Thus, as ϕ is the unique normal shift invariant state, ϕR = tr. Then, the two sided translationally invariant extension of tr to A is a trace and this contradicts with our assumption that ϕ is pure. If a translationally invariant pure state ϕ has the split property, the endomorphism ΘR of AR deﬁned as the restriction of τ1 to AR is weakly inner on the GNS subspace associated with ϕ. More precisely, let ΘR be an endomorphism of A determined by ΘR (Q) = τ1 (Q) for Q ∈ AR and ΘR (Q) = Q for Q ∈ AL . If ϕ is a translationally invariant pure state of A with the split property, there exist isometries Sj (j = 1, 2, . . . , d) acting on the GNS space associated with ϕ satisfying generating relations of the Cuntz algebra (cf. the next section) Sj∗ Si = d δij 1, k=1 Sk Sk∗ = 1 and d

Sj πϕ (Q)Sj∗ = πϕ (ΘR (Q)),

Sj ∈ πϕ (A)

( Q ∈ A).

(2.1)

j=1

As a consequence of weakly inner property of ΘR , ϕ and ϕ ◦ ΘR are mutually quasi-equivalent. When ϕ is a state without the split property ϕ and ϕ ◦ ΘR may not be mutually quasi-equivalent. For example, the (unique) inﬁnite volume ground state of the massless XY model with spin 1/2 (d = 2) gives rise to such non-equivalence. For simplicity of exposition we consider the isotropic case which is called the XX model in some literatures.

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Proposition 2.4. Let ϕ be the unique infinite volume ground state of the massless XY model with the following Hamiltonian H: {σx(j) σx(j+1) + σy(j) σy(j+1) } (2.2) H=− j∈Z (j)

(j)

where σx and σy are Pauli spin matrices at the site j in one-dimensional integer lattice Z. Then, ΘR cannot be weakly inner in the sense specified in (2.1). In other words, the representations of A associated with ϕ and ϕ ◦ ΘR are disjoint. Does non-split property of a translationally invariant pure state imply impossibility of obtaining a representation of the Cuntz algebra implementing ΘR on A? At the moment we are not able to prove it. For the proof of Haag duality we do not need an answer to this question, though, we have to keep Proposition 2.4 in mind. Sketch of Proof 2.4. The XY model is formally equivalent to the free Fermion on the one-dimensional lattice Z. Our proof of Proposition 2.4 relies deeply on C ∗ algebraic methods of [4] and results on quasifree states of CAR algebras. As these topics are not related to the proof of Haag duality we present here only a sketch of proof of Proposition 2.4. Let cj and c∗j be the creation annihilation operators of Fermions on Z satisfying Canonical Anti-Commutation Relations (CAR), {cj , c∗k } = δjk etc. For f = f (j) ∗ ∗ ∗ in l2 (Z) we set c∗ (f ) = j∈Z cj fj and c(f ) = (c (f )) . By ACAR we denote ∗ ∗ the C -algebra generated by cj and ck . We introduce the parity automorphism Θparity of ACAR and the spin algebra A determined by Θparity (cj ) = −cj and (j) (j) Θparity (σx,y ) = −σx,y . We set A± CAR = {Q ∈ ACAR | Θparity (Q) = ±Q},

A± = {Q ∈ A | Θparity (Q) = ±Q}.

A gauge invariant quasifree state ψ of ACAR is determined by the covariance operator A deﬁned by ψ(c∗ (f )c(g)) = (g, Af )l2 (Z) where the right-hand side is the inner product of l2 (Z). Any bounded selfadjoint operator A on l2 (Z) satisfying 0 ≤ A ≤ 1 gives rise to a quasifree state in this way, so by ψA we denote the gauge invariant quasifree state of ACAR determined by ψA (c∗ (f )c(g)) = (g, Af )l2 (Z) . Via Jordan–Wigner transformation and Z2 cross product, Pauli spin matrices (on + Z) are written in terms of cj and c∗j and A+ CAR = A . The inﬁnite volume ground state ϕ of the XY model (2.2) is Θparity invariant and is determined by a quasifree state ψp of ACAR : ϕ|A+ = ψp . In this formula, with help of Fourier series, l2 (Z) is identiﬁed with L2 ([−π, π]) and p is the multiplication operator of the characteristic function χ[0,π] .

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To show that ΘR is not weakly inner on the GNS space of the ground state ϕ of the XY model, it suﬃces to show that ϕ and ϕ ◦ ΘR are not quasi-equivalent. To prove this claim, we focus our attention to the representation of A+ CAR . The on the GNS space associated with ϕ has decomposition representation of A+ CAR into two components, both of which are irreducible. Now look at ϕ ◦ ΘR |A+ = ψu∗ pu |A+ where u is an isometry on l2 (Z). On L2 ([−π, π]), u∗ pu is an operator with a kernel function. If ϕ ◦ ΘR and ϕ both restricted to A+ are quasi-equivalent, the quasifree states ψp and ψu∗ pu of ACAR must be quasi-equivalent. (See the argument on the top of page 99 in [16].) So p − (u∗ pu)1/2 and (1 − p) − (1 − u∗ pu)1/2 are of the Hilbert Schmidt class. These conditions imply that X = p − u∗ pu is a Hilbert Schmidt operator. However, the kernel k(θ1 , θ2 ) for the operator X has a singularity of order |θ1 − θ2 |−2 at the diagonal part. Thus Tr(X ∗ X) = Tr((p − u∗ pu)2 ) = ∞. Thus (2.3) leads a contradiction if ϕ and ϕ ◦ ΘR are quasi-equivalent.

(2.3)

3. Od ⊗ Od Our basic strategy to prove Theorem 1.1 is the following. We consider the gauge invariant extension ψ¯ of the state ϕ to the Cuntz algebra Od ⊗ Od and examine conditions of factoriality of ϕ. Then, we consider a pure state ψ of Od ⊗ Od which is an extension of ϕ and prove Haag duality at the level of the Cuntz algebra. Next we introduce our notation for the Cuntz algebra Od . The Cuntz algebra Od is a simple C ∗ -algebra generated by isometries S1 , S2 , . . . , Sd satisfying Sk∗ Sl = δkl 1, d ∗ k=1 Sk Sk = 1. The gauge action γU of the group U (d) of d by d unitary matrices is deﬁned via the following formula: γU (Sk ) =

d

Ulk Sl

l=1

where Ukl is the k × l matrix element for U in U (d). Consider the diagonal circle group U (1) = {z ∈ C | |z| = 1} and γz on Od , γz (Sj ) = zSj , (j = 1, 2, . . . , d). The ﬁxed point algebra Od U(1) for this action of U (1) is the UHF algebra d∞ which we will identify with AR = A[1,∞) as follows: Let I and J be m-tuples of ordered indices, I = (i1 , i2 , i3 , . . . , im ), J = (j1 , j2 , j3 , . . . , jm ) (ik , jl ∈ {1, 2, . . . , d}) and set SI = Si1 Si2 · · · Sim , SJ = Sj1 Sj2 · · · Sjm . Then, we identify the matrix unit of AR and the U (1) gauge invariant part of Od via the following equation: (1)

(2)

(m)

SI SJ∗ = ei1 j1 ei2 j2 · · · eim jm

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where eij is the matrix unit of the one-site matrix algebra. The canonical endomorphism Θ of Od is determined by Θ(Q) =

d

Sk QSk∗ ,

Q ∈ Od .

k=1

It is easy to see that the restriction of Θ to AR is the lattice translation τ1 . Lemma 3.1. Let ϕ be a translationally invariant factor state of A. Consider the restriction ϕR of ϕ to AR . Let ψ˜ be the U (1) gauge invariant extension of ϕR to Od . Suppose further that ψ˜ is not factor. Then, there exists a positive k such that τk acting on AR is weakly inner on the ˜ More precisely, there exists a representation GNS spaces associated with ϕ and ψ. π ˜ (·) of the Cuntz algebra Od×k on the GNS space Hψ˜ such that π ˜ (Sl ) ∈ πψ˜ (AR ) ,

dk

π ˜ (Sl )πψ˜ (Q)˜ π (Sl∗ ) = πψ˜ (τk (Q)).

(3.1)

l=1

π ˜ (Sl ) implements the canonical endomorphism of Od×k as well. Conversely, if there exist operators Tj in πψ˜ (AR ) satisfying d

Tj πψ˜ (Q)Tj∗ = πψ˜ (Θ(Q))

j=1

for any Q in Od , ψ˜ is not a factor. Proof. Let ψ˜ be the U (1) gauge invariant extension of ϕ to Od and {π(Od ), Ω, H} ˜ (Ω is the GNS cyclic vector.) There be the GNS representation associated with ψ. exists a unitary representation Uz of U (1) satisfying Uz Ω = Ω,

Uz π(Q)Uz∗ = π(γz (Q)) for Q ∈ Od .

(3.2)

We set N = π(Od ) and C = N ∩ N . Using Uz we have introduced the normal extension γz of U (1) action to the von Neumann algebra N . (By abuse of notation we use the same symbol γz for this action.) Let Q be an element of N = π(Od ) . and consider the Fourier expansion of Q: ∞ Qk , Qk = dzz −k Uz QUz∗ . (3.3) Q= k=−∞

Let Nk be the subspace generated by operators Qk : Nk = {Q ∈ N | γz (Q) = z k Q},

N0 = π(AR ) .

(3.4)

Let Ck = Nk ∩ C = {Q ∈ C | γz (Q) = z k Q}. As we assumed that N is not a factor, we can ﬁnd a non-trivial self-adjoint element c of the center C. As N0 is a factor on N0 Ω and Ω is cyclic for N , c0 is a scalar

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multiple of the identity, i.e. c0 = c1. As ck c−k and ck c∗k belong to C, and since we assume that C is self-adjoint ck c−k = ck c∗k is scalar. By the same reason, c−k ck and c∗k ck are scalars as well. Thus by rescaling we can assume that any non-vanishing ck is a unitary. Moreover if Ck is not 0, it is one-dimensional. To see this take another central element c1 and consider its Fourier component c1k . As c1k c−k belongs to C0 , it is a scalar. Take the smallest positive k such that C0 is one-dimensional and for a multiindex I with |I| = k, we set π ˜ (SI ) = π(SI )c∗k ,

π ˜ (SI∗ ) = π(SI∗ )ck .

Both π ˜ (SI ) and π ˜ (SI )∗ are γz invariant and their restriction to HϕR satisﬁes (3.1). Next let {Tj } be a set of operators in πψ˜ (AR ) implementing the canonical endomorphism Θ of Od . Then the operator πψ˜ (Sj∗ )Ti commutes with any element of πψ˜ (Od ) because of πψ˜ (Sj∗ )πψ˜ (Θ(Q)) = πψ˜ (Q)πψ˜ (Sj∗ ). Thus πψ˜ (Od ) is not a factor. The following lemma is known. (See, for example, [5, Lemmas 6.10 and 6.11].) Lemma 3.2. Let ϕ be a translationally invariant factor state of A. Suppose that for a positive k, the restriction τk to AR is implemented by a representation π ˜ (Od×k ) of the Cuntz algebra Od×k on the GNS space HϕR and the gauge invariant part of π ˜ (Od×k ) coincides with AR . More precisely, (1)

π ˜ (Sl Sk∗ ) = πϕR (ekl ). Suppose that the gauge action γz does not admit a normal extension to the von Neumann algebra π ˜ (Od×k ) for any z. Then, τk is weakly inner in the sense of (3.1), namely π ˜ (Od×k ) = AR . Proof. By abuse of notation ϕR is regarded as a state of the ﬁxed point subalgebra (Od×k )U(1) . Consider a vector state ψ0 of (Od×k )U(1) associated with the GNS vector for ϕR and let ψ be the U (1) invariant extension of ϕR to (Od×k )U(1) . Then, ψ0 ◦ γz dz = ψ and at the level of the GNS representation, Hψ =

⊕

Hψ0 dz = Hψ0 ⊗ L2 (S 1 ),

πψ =

⊕

πψ0 ◦ γz dz.

Due to our assumption that γz does not admit any normal extension to π ˜ (Od×k ) for any z, the von Neumann algebra N = πψ (Od×k ) is isomorphic to M ⊗ L∞ (S 1 )

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where the gauge action acts as the rotation on S 1 . πψ (AR ) is the commutant of the unitaries implementing the rotation. πψ (AR ) = M ⊗ 1.

(3.5)

By deﬁnition, πψ (Q) = πψ0 (Q) ⊗ 1 for Q in AR and we have πψ (AR ) = πϕR (AR ) ⊗ 1.

(3.6)

Looking at each ﬁber of Eqs. (3.5) and (3.6) , we conclude that π ˜ (Od×k ) = AR . (L)

(R)

Next we consider a pair of Cuntz algebras denoted by Od and Od and we (L) (R) (L) (R) set B = Od ⊗ Od . The Cuntz generators are denoted by SI and SI etc. The algebra B is naturally equipped with the U (1) ⊗ U (1) gauge action γzL ,zR = γzL ⊗ γzR : (L)

|I|

(L)

γzL ,zR (SI ) = zL SI ,

(R)

|I|

(R)

γzL ,zR (SI ) = zR SI

(zL , zR ∈ U (1)).

As A = AL ⊗ AR we identify A with the U (1) ⊗ U (1) ﬁxed point sub-algebra B = OdL ⊗ OdR . The canonical endomorphisms of B is deﬁned via the following equation: Θk,l = ΘkL ⊗ ΘlR (L)

where ΘL (respectively, OdR ) is the canonical endomorphism of Od (respectively, (R) Od ). The lattice translation automorphism τ1 has an extension to B as an inner automorphism. To see this, set V =

d

(L)

(R)

(L)

(L) ∗

(Sj )∗ Sj

.

(3.7)

j=1

Then, V satisﬁes V V ∗ = V ∗ V = 1,

(0)

V ekl V ∗ = V Sk (Sl

(R)

(R) ∗

) V ∗ = Sk (Sl

(1)

) = ekl

(3.8)

which shows that Ad(V )(Q) = τ1 (Q) Q ∈ A.

(3.9)

We extend τ1 to B via the above Eq. (3.9). Let k be a positive integer and we regard Od×k as a subalgebra of Od which is generated by SI and SJ∗ with |I| = kn, |J| = km (n, m = 1, 2, . . .). Set L R ⊗ Od×k ⊂ B. B k = Od×k

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Lemma 3.3. Let ϕ be a pure state of A. Suppose that there exists a representation π ˜ of B k on the GNS space Hϕ associated with ϕ such that (L)

(L)

(0)

(−1)

(R)

(R)

(1) (2)

(−k+1)

π ˜ (SI (SJ )∗ ) = πϕ (ei1 j1 ei2 j2 · · · eik jk

), (3.10)

(k)

π ˜ (SI (SJ )∗ ) = πϕ (eij ei1 j1 · · · eik jk ). (L)

(R)

Then, π ˜ (Od×k ) ⊂ πϕ (AL ) and π ˜ (Od×k ) ⊂ πϕ (AR ) . Proof. Due to Lemma 3.2, we have only to show the gauge action γ does not have a normal extension to the von Neumann algebra π ˜ (B) . Any normal homorophism of a type I factor is implemented by a unitary. As π˜ (B) is irreducible, we suppose there exists a unitary W such that (L)

Wπ ˜ (Si

(L)

)W ∗ = zL Si ,

(R)

Wπ ˜ (Si

(R)

)W ∗ = zR Si

.

(3.11)

Then, due to (3.10) W commutes with the gauge invariant part AL and AR . As ϕ is pure, W is a scalar multiple of the identity and zL = zR = 1. Lemma 3.4. Let ϕ be a translationally invariant pure state of A and let ψ¯ be the U (1) × U (1) gauge invariant extension of ϕ to B. ¯ γzL zR (Q)dzL dzR Q ∈ B. (3.12) ψ(Q) =ϕ U(1)×U(1)

ψ¯ is not a pure state. ¯ be the GNS triple. As ψ¯ is γzL zR invariant, there exists a Proof. Let {¯ π (B), Ω, H} unitary UzL zR satisfying UzL zR π ¯ (Q)Uz∗L zR = π ¯ (γzL zR (Q)),

UzL zR Ω = Ω.

¯ with We consider the restriction of π ¯ to A and the Fourier decomposition of H respect to UzL zR . ¯= ¯ kl . H ⊕ H k,l∈Z

If ψ¯ is pure, π ¯ (A) = π ¯ (B) ∩ C = C where C is the abelian von Neumann algebra generated by UzL zR . As π ¯ (A) is the commutant of C, the center of π ¯ (A) is C. Each irreducible representation π(A) appearing in π ¯ (A) as a subrepresentation is of the form π ¯ (Q)P where P is a central ¯ (A) is decomposed into irreducible representations πkl projection of π ¯ (A) . Thus π ¯ kl . πkl and πnm are equivalent if and only if k = n, and l = m. π00 is equivalent on H to the GNS representation associated with ϕ. However, the operator π ¯ (V ) gives rise to unitary equivalence between π00 and π1−1 , which implies contradiction. Thus ψ¯ cannot be pure.

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By the same line of argument in Lemma 3.1, we can show that the Fourier component Cij of C in Lemma 3.4, is either one or zero dimensional. Furthermore, C is generated by Ck,−k for some k when the canonical endomorphism is not weakly inner in πϕ (A) . We show this claim rather implicitly in the next step. We introduce the diagonal action γzd of U (1) on B via the equation: γzd = γz,z and similarly the diagonal action γzd,k of U (1) on B k . Set D = {Q ∈ B | γzd (Q) = Q for any z}. Lemma 3.5. (i) D is generated by A and V , hence D is isomorphic to the crossed product of A by the action τj of Z. (ii) Let ϕ be a translationally invariant state of A. There exists a state ϕ˜ of D satisfying ϕ(V ˜ ) = 1,

ϕ(Q) ˜ = ϕ(Q),

Q ∈ A.

(3.13)

The state ϕ˜ of D satisfying (3.13) is unique. (iii) ϕ˜ is pure if ϕ is factor. (L)

(R)

Proof. (i) D is generated by SI (SJ )∗ Q where multi-indices I and J satisfy |I| − |J| = 0 and Q is an element of A. By direct calculation, we have (L) (R) (R) (R) V Si (Sj )∗ = Si (Sj )∗ . Thus (L)

(R) ∗

Si (Sj

(L)

(R) ∗

) Q = V ∗ V Si (Sj

(L)

(R)

) Q = V ∗ Si

(R) ∗

(Sj

) Q

(R)

which shows that SI (SJ )∗ Q is written by a product of V and elements in A. (ii) Consider the GNS triple {πϕ (A), Ω, Hϕ } associated with ϕ. As the state ϕ is translationally invariant we have a unitary W implementing τ1 and W Ω = Ω. Then we set πϕ (V ) = W the vector state ϕ˜ of D associated with Ω satisﬁes (3.13). Conversely, if a state ϕ˜ satisﬁes (3.13), the GNS cyclic vector Ωϕ˜ is invariant under πϕ˜ (V ) due to the identity: ˜ ) − ϕ(V ˜ ∗ ) = 2 − 1 − 1 = 0. (πϕ˜ (V ) − 1)Ωϕ˜ 2 = 2 − ϕ(V Thus W = πϕ˜ (V ). (iii) As ϕ is factor, for Q ∈ A w − lim πϕ (τk (Q)) = ϕ(Q)1. k→∞

Suppose P commutes with πϕ˜ (V ) and πϕ (A). Then, (Ω, πϕ (Q)P Ω) = (Ω, πϕ (Q)P πϕ˜ (V −k )Ω) = (Ω, πϕ (Q)πϕ˜ (V −k )P Ω) = (Ω, πϕ (Q)πϕ˜ (V −k )P Ω) = (Ω, πϕ (τk (Q))P Ω) = lim (Ω, πϕ (τk (Q))P Ω) = ϕ(Q)(Ω, P Ω) k→∞

which implies P Ω = (Ω, P Ω)Ω, P = (Ω, P Ω)1.

(3.14)

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Lemma 3.6. Let ϕ be a translationally invariant pure state of A. Then, for a positive k there exists a pure state extension ψ of ϕ to B k such that ψ is invariant under τk and (L) (R) ψ((SI )∗ SI ) = 1. (3.15) |I|=k

Furthermore, one of the following mutually exclusive conditions is valid. (i) ψ is invariant under γzd,k . (ii) ψ ◦ γzd,k is not equivalent to ψ for any z. When (ii) is valid, the assumptions of Lemma 3.3 are satisfied. Proof. Consider the state ϕ˜ of D satisfying (3.13). Let ψ˜ be the γ d invariant extension of ϕ˜ to B. If ψ˜ is pure, we set ψ˜ = ψ and as ϕ is translationally invariant, there exists a unitary W on Hϕ = H0 satisfying W πϕ (Q)W ∗ = π(τ1 (Q)),

W Ω ϕ = Ωϕ .

Then the operator πϕ˜ (V )W ∗ acting on Hϕ commutes with πϕ (A). This shows that (L) πϕ˜ (V )W ∗ is a scalar. After a gauge transformation of Od , we have πϕ˜ (V )Ωϕ˜ = Ωϕ˜ which is equivalent to Eq. (3.15). By deﬁnition the state ψ is γzd,k invariant. Next we consider the case that ψ˜ is not pure. Let U (z) be the unitary on the GNS space Hψ˜ associated with ψ˜ such that πψ˜ (γzd (Q)) = U (z)πψ˜ (Q)U (z)∗ ,

U (z)Ωψ˜ = Ωψ˜ .

The GNS representation πψ˜ restricted to D is a direct sum of πj (D) on Hj : Hψ˜ = Hj , U (z)|Hj = z j 1, j∈Z

πψ˜ =

⊕πj ,

πj = πψ˜ |Hj .

j∈Z

Note that the representations πj and πi are disjoint when i = j. The Fourier component of the commutant C of πψ˜ (B) is denoted by Cj . For Q in C = πψ˜ (B) ∞ Q= Qk , Qk = dzz −k Uz QUz∗ . k=−∞

For each integer k, let Ck be the subspace generated by operators Qk : Ck = {Q ∈ C | γz (Q) = z k Q}. As the state ϕ is pure C0 is one dimensional, C0 = C1 because C0 commutes with πϕ (A). By the similar argument in proof of Lemma 3.1, it is possible to show the

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dimension of Ck is zero or one and C is generated by a single unitary U in Ck for some k. Now we introduce a representation π(Bk ) of Bk on H0 = Hϕ determined by (L)

(L)

π(SI ) = eiθ πψ˜ (SI )U ∗ |H0 ,

(R)

(R))

π(SJ ) = πψ˜ (SJ

)U ∗ |H0

for |I| = |J| = k

where the phase factor eiθ is determined later. By deﬁnition π(Q) = πϕ (Q) for Q in A while on H0 , πϕ (A) acts irreducibly. Let ψ be the vector state of B k associated with Ωϕ . As ϕ is translationally invariant, there exists a unitary W on Hϕ = H0 satisfying W πϕ (Q)W ∗ = π(τk (Q)), Set V (k) =

(L)

W Ω ϕ = Ωϕ . (R)

(SI )∗ SI .

|I|=k

Then the operator V k W ∗ commutes with πϕ (A) . This shows that π(V (k) )W ∗ is a scalar. By suitably choosing the phase factor eiθ we have π(V (k) ) = W,

π(V (k) )Ωϕ = Ωϕ .

ψ is the state satisfying our requirement. 4. Proof of Theorem 1.1 We consider Haag duality for the Cuntz algebras Od ⊗ Od ﬁrst. The same duality (Proposition 4.2) is stated in [5]. However, due to the reason stated in the introduction of this paper, we present our proof here. To show the Haag duality for the Cuntz algebra we apply the Tomita–Takesaki Theory. The state ϕR or its extension to Od may not be faithful so we consider reduction of the von Neumann algebra generated by Od by the support projection and employ the Tomita modular conjugation to obtain the (reduced) commutant. Then, we use the following lemma. Lemma 4.1. Let M1 ⊂ M2 be a pair of factor-subfactor on a separable Hilbert space H. Suppose that there exists a non-trivial projection P ( = 0) in M1 such that P M1 P = P M2 P . Then, M1 and M2 coincide: M1 = M2 . Proof. Suppose that we have a matrix unit eij (i, j = 1, 2, . . .) in M1 such that e11 = P, ejj = 1. (4.1) j=1

Let Q be an element of M2 . Then eii Qejj is an element of M1 because eii Qejj = ei1 e1i Qej1 e1j ,

e1i Qej1 ∈ M1 .

Thus if we have a matrix unit satisfying (4.1) any Q in M2 is an element of M1 . When M1 has a tracial state tr and 1/tr P is not an integer, the matrix

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unit satisfying j=1 ejj = 1 does not exist. In such a case, we consider another projection q in M1 such that q ≤ P and 1/trq is a positive integer. Then we apply the above argument to qM1 q = qM2 q. Without loss of generality, we assume that k = 1 in Lemma 3.6 for the proof of Haag duality. Let ϕ be a translationally invariant state. From now on, ψ is the pure state extension of ϕ to B such that ψ is invariant under τ1 . Recall that due to the Eq. (3.15), πψ (V )Ωψ = Ωψ . Hence, (R)

(L)

(SI )∗ V = (SI )∗ ,

(R)

(R)

(L)

(R)

(L)

πψ (SI )∗ Ωψ = πψ (SI )∗ πψ (V )Ωψ = πψ (SI )∗ Ωψ . πψ (SI )∗ Ωψ = πψ (SI )∗ Ωψ .

(4.2)

As a consequence the Hilbert space Hψ is generated by the following vectors: (L)

(R)

(R)

πψ (SI )πψ (SJ )πψ (Q)πψ (SJ )∗ Ωψ ,

Q ∈ AR .

(4.3)

Proposition 4.2. Suppose ψ is the pure state extension of ϕ to B such that ψ is invariant under τ1 . Then, (L)

(R)

πψ (Od ) = πψ (Od ) .

(4.4)

Equation (4.2) is crucial in our proof of (4.4). We need some preparation for our proof of (4.4). (R)

Proof. Let ER be the support projection of ψ for πψ (Od ) and EL be the sup(L) port projection of ψ for πψ (Od ) . By ER we denote the projection with range (R) (R) [πψ (Od ) Ωψ ] where [πψ (Od ) Ωψ ] is the closed subspace of Hψ generated by (R) (L) πψ (Od )Ωψ . Similarly, by EL we denote the projection to [πψ (Od ) Ωψ ]. (R) Set P = ER ER and K = P H. The range of P is [ER πψ (Od ) Ωψ ] . (R) Now we denote the von Neumann algebra ER πψ (Od ) ER by N. Ωψ is a cyclic and separating vector for N acting on K. Let ∆ and J be the Tomita modular (R) operator and the modular conjugation associated with Ωψ for ER πψ (Od ) ER . (R) Set vj = P πψ (Sj )P . As (R)

(Ωψ , ((P πψ (Sj

(R)

)P − P πψ (Sj

(R) ∗

))(P πψ (Sj

(R) ∗

) P − πψ (Sj

we have (R)

P πψ (Sj

(R)

)P = P πψ (Sj

and d j=1

vj vj∗ = 1.

)

) P )Ωψ ) = 0,

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Set v˜j = J∆−1/2 vj∗ ∆1/2 J and v˜j∗ = J∆1/2 vj ∆−1/2 J. The closure of v˜j and v˜j∗ are bounded operators satisfying d

v˜j v˜j∗ = 1

j=1

because d

˜ vj QΩψ 2 =

j=1

d

Q˜ vj Ωψ 2

j=1

=

d

(R) ∗

Qπψ (Sj

) Ωψ 2 = ψ(τ1 (Q∗ Q)) = QΩψ 2

j=1 (R)

for Q ∈ P πψ (Od )P . Moreover, (L)

P πψ (Sj )∗ P = v˜j∗ .

(4.5)

(4.5) follows from the fact that Ωψ is separating for the commutant of ER NER = (R) P πψ (Od ) P and the following equations: (L)

(L)

(R) ∗

P πψ (Sj )∗ P Ωψ = P πψ (Sj )∗ Ωψ = P πψ (Sj

(R) ∗

) Ωψ = πψ (Sj

(R) ∗

v˜j∗ Ωψ = πψ (Sj

) Ωψ .

) Ωψ ,

(4.6) (4.7)

Lemma 4.3. Let N be the von Neumann algebra on K generated by vj as in Proposition 4.2 and let N1 be the von Neumann algebra on K generated by v˜j . Then, N = N1 .

(4.8)

Proof. N is generated by Jvj J and N1 ⊂ N . The modular operator ∆1 and the conjugation J1 of N1 acting on [N1 Ωψ ] are the restriction of those for N on K. Then Jvj J = ∆−1/2 v˜j ∆−1/2 is in N1 . Lemma 4.4. (R)

(L)

[ER πψ (Od )Ωψ ] = [EL πψ (Od )Ωψ ].

(4.9) (R)

Proof. By Lemma 4.3, the commutant of N = ER πψ (Od ) ER acting on K is (L) P πψ (Od ) P . Obviously P = ER ER ≤ ER EL . Then, (R)

(L)

[ER πψ (Od ) Ωψ ] = [P πψ (Od ) Ωψ ] (L)

(L)

⊂ [ER EL πψ (Od ) Ωψ ] = [EL πψ (Od ) Ωψ ]. The above inclusion tells us (R)

(L)

[ER πψ (Od )Ωψ ] ⊂ [EL πψ (Od )Ωψ ]. By the symmetry of L and R we have reverse inclusion.

(4.10)

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Lemma 4.5. P = ER EL .

(4.11)

Proof. We show P = ER ER ≥ ER EL . Due to (4.2), the Hilbert space Hψ is (L) (R) (R) generated by the vectors πψ (SI )πψ (SJ )πψ (SJ )∗ Ωψ . It suﬃces to show that (L) (R) (R) the vector ξ = ER EL πψ (SI )πψ (SJ )πψ (SJ )∗ Ωψ is in K (= the range of P ). (L) Due to the previous Lemma, η = EL πψ (SI )Ωψ is in K. Thus, (L)

(R)

(R)

(R)

(R)

(R)

(R)

(R)

(R)

(L)

ER EL πψ (SI )πψ (SJ )πψ (SJ )∗ Ωψ = ER πψ (SJ )πψ (SJ )∗ EL πψ (SI )Ωψ = ER πψ (SJ )πψ (SJ )∗ η = ER πψ (SJ )πψ (SJ )∗ ER η ∈ K.

(4.12)

Now we return to proof of Proposition 4.2. First we look at the commutant of (R) (L) (R) ER πψ (Od ) ER on ER Hψ . Obviously, ER πψ (Od ) ER ⊂ (ER πψ (Od )ER ) on ER Hψ . By Lemmas 4.3 and 4.5, (L)

(R)

ER EL πψ (Od ) EL ER = (ER EL πψ (Od )EL ER ) . (L)

(R)

Then, due to Lemma 4.1 πψ (Od ) = πψ (Od ) on ER Hψ . (R) (L) Next we consider the inclusion πψ (Od ) ⊂ πψ (Od ) on Hψ . As we already (R) (L) know that ER πψ (Od ) ER = ER πψ (Od ) ER we apply Lemma 4.1 again and (R) (L) conclude that πψ (Od ) = πψ (Od ) . Proof of Theorem 1.1. Now recall Lemma 3.6. We have two cases (i) and (ii). In the case (ii), our previous analysis shows that the pair of Cuntz algebras Bk is in the von Neumann algebra πϕ (A) and the duality follows from Proposition 4.2. Hence we consider the case where the pure state ψ of B is invariant under γzd . Let Uz be the unitary implementing γzd and satisfying Uz Ωψ = Ωψ . We use the previous notation in our proof for Proposition 4.2. By the duality for Cuntz algebras (Proposition 4.2), ER = EL , EL = ER . ER commutes with Uz due to γzd invariance of ψ. As a result, the support projection of ϕ for πϕ (AR ) is the ER restricted to Hϕ . So we use the same notation ER (respectively, EL ) for the support projection of ϕ for πϕ (AR ) (respectively, πϕ (AL ) ). To show Haag duality we proceed as before. Taking into account of P = ER EL and πϕ (AL ) ⊂ πϕ (AR ) , it suﬃces to show P πϕ (AL ) P = P πϕ (AR ) P.

(4.13)

On K0 = P Hϕ we apply the Tomita–Takesaki theorem. P πϕ (AR ) P is generated ∗ J0 . where I and K are multi-indices satisfying |I| = |K| and J0 is the by J0 vI vK ∗ J are restriction of J to K0 . By Haag duality for Cuntz algebras, JvI J and JvK

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approximated in strong operator topology by elements wα and xα of P πψ (Od )P . Using the Fourier decomposition (with help of Uz ) we may assume that Uz wα Uz∗ = z |I|wα ,

Uz xα Uz∗ = z |K| xα .

∗ As a consequence, JvI vK J is approximated ∗ U (1)} . Thus JvI vK J is contained in

by elements of

(4.14) (L) P πψ (Od )P

∩ {Uz | z ∈

(L)

P πψ (Od ) P ∩ {Uz | z ∈ U (1)} = P πψ (AL ) P on K. By taking restriction to K0 , we see (4.13).

References [1] H. Araki, On quasifree states of CAR and Bogoliubov automorphisms, Publ. Res. Inst. Math. Sci. 6 (1970/71) 385–442. [2] H. Araki, Bogoliubov automorphisms and Fock representations of canonical anticommutation relations, in Operator Algebras and Mathematical Physics (Iowa City, Iowa, 1985), eds. Palle E. T. Jorgensen and Paul S. Muhly, Contemp. Math., Vol. 62 (Amer. Math. Soc., Providence, R.I., 1987), pp. 23–141. [3] H. Araki, On the XY -model on two-sided infinite chain, Publ. Res. Inst. Math. Sci. 20(2) (1984) 277–296. [4] H. Araki and T. Matsui, Ground states of the XY -model, Comm. Math. Phys. 101(2) (1985) 213–245. [5] O. Bratteli, P. Jorgensen, A. Kishimoto and R. F. Werner, Pure states on Od , J. Operator Theory 43(1) (2000) 97–143. [6] O. Bratteli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics I, 2nd edn. (Springer, 1987). [7] O. Bratteli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics II, 2nd edn. (Springer, 1997). [8] S. Doplicher and R. Longo, Standard and split inclusions of von Neumann algebras, Invent. Math. 75 (1984) 493–536. [9] R. Haag, Local Quantum Physics (Springer-Verlag, 2nd edn. 1996) [10] M. Keyl, D. Schlingemann and R. F. Werner, Infinitely entangled states, Quantum Inf. Comput. 3(4) (2003) 281–306. [11] M. Keyl, T. Matsui, D. Schlingemann and R. F. Werner, Entanglement, Haag-duality and type properties of infinite quantum spin chains, Rev. Math. Phys. 18 (2006) 935– 970. [12] R. Longo, Solution to the factorial Stone-Weierstrass conjecture. An application of standard split W ∗ -inclusion, Invent. Math. 76 (1984) 145–155. [13] R. Longo, Algebraic and modular structure of von Neumann algebras of physics, in Operator Algebras and Applications, ed. R. V. Kadison, Proc. Sympos. Pure Math., Vol. 38, Part 2 (Amer. Math. Soc. Providence, R. I., 1982), pp. 551–566. [14] T. Matsui, The split property and the symmetry breaking of the quantum spin chain, Commun. Math. Phys. 218 (2001) 393–416. [15] T. Matsui, Factoriality and quasi-equivalence of quasifree states for Z2 and U (1) invariant CAR algebras, Rev. Roumaine Math. Pure Appl. 32 (1987) 693–700. [16] S. Stratila and D. Voiculescu, On a class of KMS states for the unitary group U (∞), Math. Ann. 235 (1978) 87–110.

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Reviews in Mathematical Physics Vol. 20, No. 6 (2008) 725–764 c World Scientiﬁc Publishing Company

ON THE STABILITY OF PERIODICALLY TIME-DEPENDENT QUANTUM SYSTEMS

‡ and M. VITTOT§ ˇTOV ˇ ´ICEK ˇ P. DUCLOS∗ , E. SOCCORSI† , P. S ∗,†,§Centre de Physique th´ eorique de Marseille UMR 6207, Unit´ e Mixte de Recherche du CNRS et des Universit´ es Aix-Marseille I, Aix-Marseille II et de l’ Universit´ e du Sud Toulon-Var, Laboratoire aﬃli´ e a ` la FRUMAM, France ‡Department

of Mathematics, Faculty of Nuclear Science, Czech Technical University, Trojanova 13, 120 00 Prague, Czech Republic ∗[email protected] †[email protected] ‡[email protected]ﬁ.cvut.cz §[email protected] Received 26 September 2007 Revised 20 February 2008

The main motivation of this article is to derive suﬃcient conditions for dynamical stability of periodically driven quantum systems described by a Hamiltonian H(t), i.e. conditions under which it holds true supt∈R |ψt , H(t)ψt | < ∞ where ψt denotes a trajectory at time t of the quantum system under consideration. We start from an analysis of the domain of the quasi-energy operator. Next, we show, under certain assumptions, that if the spectrum of the monodromy (Floquet) operator U (T, 0) is pure point then there exists a dense subspace of initial conditions for which the mean value of the energy is uniformly bounded in the course of time. Further, we show that if the propagator admits a diﬀerentiable Floquet decomposition then H(t)ψt is bounded in time for any initial condition ψ0 , and one employs the quantum KAM algorithm to prove the existence of this type of decomposition for a fairly large class of H(t). In addition, we derive bounds uniform in time on transition probabilities between diﬀerent energy levels, and we also propose an extension of this approach to the case of a higher order of diﬀerentiability of the Floquet decomposition. The procedure is demonstrated on a solvable example of the periodically time-dependent harmonic oscillator. Keywords: Time-dependent quantum system; Floquet operator; dynamical stability. Mathematics Subject Classiﬁcation 2000: 81Q10, 81Q50, 35B30

1. Introduction We discuss several topics related to the dynamical properties of periodically timedependent quantum systems. Such a system is described by a Hamiltonian H(t) in a Hilbert space H depending on t periodically with a period T , and we suppose that the propagator U (t, s) associated with the Hamiltonian H(t) exists. 725

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We start our exposition from an analysis of the domain of the Floquet Hamiltonian (the quasi-energy operator). The quasi-energy operator is a basic tool in the theory of time-dependent quantum systems and is closely related to the monodromy operator U (T, 0) (also called the Floquet operator or the propagator over the interval ]0, T [ in the sequel), see [1, 2]. This is a common belief that the dynamical properties are essentially determined by the spectral properties of U (T, 0). It is shown in [3] that ψ belongs to H pp (U (T, 0)) (the subspace in H corresponding to the pure point spectrum of U (T, 0)) if and only if the trajectory {ψt ; t ≥ 0} is precompact (where ψt = U (t, 0)ψ). Under the assumptions that H(0) is positive, discrete and unbounded, and that the perturbation H(t) − H(0) is uniformly bounded, it is observed in [4] that if the mean value of the energy, ψt , H(t)ψt , is bounded then the corresponding trajectory {ψt ; t ≥ 0} is precompact. Jointly this implies that if the mean value of the energy is bounded for any initial condition then U (T, 0) has a pure point spectrum. To our knowledge, the inverse implication is not clariﬁed yet. In the present paper we show, under certain assumptions, that if the spectrum of U (T, 0) is pure point then there exists, in H , a dense subspace of initial conditions for which the mean value of the energy is bounded. However it has been shown very recently in [5] that there exist situations when some trajectories may lead to unbounded energy in spite of pure pointness of U (T, 0). There is no doubt that the knowledge of evolution of the mean value of the energy in the case of time-dependent systems is important from the physical point of view. This is also our basic topic in this paper. More precisely, instead of treating directly the mean value of the energy we consider the quantity H(t)ψt . Naturally, this type of problems attracted attention in the past though the results are less numerous than one might expect. Let us mention some of them that motivated us though in no way we attempt to provide an exhaustive list. Assuming a growing gap structure of the spectrum of H(0) it is shown in [6] with the aid of adiabatic methods that ψt , H(t)ψt = O(tδ ) where δ > 0 is inversely proportional to the order of diﬀerentiability of H(t). An upper bound of this type is also derived in [7] under rather mild assumptions on the gap structure of the spectrum and without diﬀerentiability of H(t). On the other hand, the latter result is directly applicable only provided the perturbation is in certain sense small when compared to H(0). For example, in the case of simple spectrum the operator H(0)q (H(t) − H(0)) is required to be Hilbert–Schmidt for some q ≥ 1/2. Some extensions and applications can be also found in [8]. These estimates on the growth of the energy were derived without assuming the periodicity. Let us also mention [9] where bounds on the energy growth are derived in the case of shrinking gaps in the spectrum. A stronger result is known for periodically time-dependent systems [10]. It suggests that for a large class of periodic systems one can expect uniform boundedness of the mean value of the energy for any initial condition ψ ∈ Dom H(0). Further, in [11] the energy is shown to be uniformly bounded in time in the particular case when the harmonic oscillator is driven by quasi-periodically time-dependent

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Gaussian potentials for suitable non-resonant frequencies and a small enough coupling constant. It is proposed in [12] to call this property dynamical stability. We adopt this terminology in the current paper. Though the ideas concerning the dynamical stability are developed in [10] on a particular example of the driven ring it is indicated there that they are valid also under more general settings. The proof is based on two observations. First, if the propagator admits a diﬀerentiable Floquet decomposition in the sense that it can be written in the form U (t) = UF (t) exp(−itHF ) where HF is self-adjoint and UF (t) is a periodic and strongly diﬀerentiable family of unitary operators then the system is dynamically stable. According to the second observation one can use the quantum KAM (Kolmogorov–Arnold–Moser) algorithm to show that the propagator actually admits this type of decomposition in the case when H(0) is a semi-bounded discrete operator obeying a gap condition, and provided the frequency is non-resonant and the time-dependent perturbation is suﬃciently small. In particular, the result in [10] is based on a formulation of the quantum KAM theorem presented in [13]. In the current paper we wish to further develop the basic ideas from [10] and particularly to work out the proofs in full detail when considering applications of these ideas to more general systems. In addition, we derive uniform bounds on transition probabilities between diﬀerent energy levels. Moreover, we propose an extension to the case when the Floquet decomposition is p times continuously differentiable in the strong sense. Restricting the perturbation V (t) = H(t) − H(0) to a certain class of operator-valued functions by requiring the multiple commutators with H(0) to be bounded up to some order one can show that H(t)p ψt is bounded in time. Furthermore, the basic procedure is demonstrated on the solvable example of the periodically time-dependent harmonic oscillator. For the purposes of this example, we collect in the Appendix some useful formulas for the propagator. Finally we combine the procedure based on the diﬀerentiability of the Floquet decomposition with an improved version of the quantum KAM theorem that was presented in [14].

2. The Floquet Hamiltonian Let us make more precise the assumptions on the Hamiltonian. Let {H(t); t ∈ R} be a family of self-adjoint operators such that the domain Dom(H(t)) does not depend on time. Further we assume that the propagator U (t, s) associated with H(t) exists. This means that U (t, s) is a function with values in B(H ) which is strongly continuous jointly in t and s, U (t, t) = I, the domain Dom(H(0)) is invariant under the action of U (t, s) for all t, s, and ∀ψ ∈ Dom(H(0)),

i∂t U (t, s)ψ = H(t)U (t, s)ψ.

Then the propagator is unique, unitary and satisﬁes the Chapman–Kolmogorov equation: U (t, r)U (r, s) = U (t, s) for all t, r, s ∈ R.

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Let us recall that usually one imposes a standard suﬃcient condition that guarantees the existence of the evolution operator. Namely, if the mapping (t, s) →

1 ((H(t) + i)(H(s) + i)−1 − I) t−s

can be extended for t = s to a strongly continuous mapping R2 → B(H ) then the propagator exists [15]. For more general suﬃcient conditions one can consult the monographs [16] and [17]. But as already stated, we assume directly the existence of the propagator without bothering about particular hypotheses that guarantee it. Since the Hamiltonian H(t) is assumed to be T -periodic the same is true for the propagator. This means that ∀t, s,

U (t + T, s + T ) = U (t, s).

(1) −1

is Notice also that by the closed graph theorem the operator H(t) (H(0) + i) bounded. In addition, in this section we impose the following two assumptions: R t → H(t)(H(0) + i)−1 ∀ψ ∈ Dom(H(0)),

is locally bounded,

R t → H(0)U (t, 0)ψ is locally square integrable.

(2) (3)

In fact, hypothesis (2) means that H(t)(H(0) + i)−1 is bounded uniformly in t since we are considering the periodic case. An important tool when investigating time-dependent quantum systems is the Floquet Hamiltonian (also called the quasi-energy operator) [1, 2]. It acts in the Hilbert space K = L2 ([ 0, T ], H , dt) ≡ L2 ([ 0, T ], dt) ⊗ H . If convenient we shall regard the elements of K as T -periodic vector-valued functions on R with values in H . A unique Floquet Hamiltonian is associated with any strongly continuous propagator via the Stone theorem according to the prescription ∀f ∈ K ,

∀σ ∈ R,

for a.a. t ∈ R, (e−iσK f )(t) = U (t, t − σ)f (t − σ).

(4)

Hence f belongs to Dom(K) if and only if the derivative i∂σ U (t, t − σ)f (t − σ)|σ=0 exists in K . Morally the Floquet Hamiltonian can be regarded as −i∂t +H(t) but in general this formal expression should be interpreted in a weak sense. The following remarks aim to provide some details about the deﬁnition of K. In the particular case when the Hamiltonian does not depend on time and equals H0 for all t it holds true U (t, t − σ) = exp(−iσH0 ) and one easily ﬁnds from (4) that the associated Floquet Hamiltonian K0 is nothing but the closure of the operator −i∂t ⊗ 1 + 1 ⊗ H0 deﬁned on the algebraic tensor product Dom(i∂t ) ⊗ Dom(H0 ). Here and everywhere in what follows the time derivative is automatically considered with the periodic boundary conditions. This is to say that the orthonormal basis {T −1/2 exp(2πikt/T ); k ∈ Z} in L2 ([ 0, T ], dt) is formed by eigenfunctions of i∂t .

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Let us denote by CT∞ (R) the space of T -periodic smooth functions on R and let CT∞ (R) ⊗ Dom(H(0)) = span{η(t)ψ; η ∈ CT∞ (R),

ψ ∈ Dom(H(0))} ⊂ K

be the algebraic tensor product. It is straightforward to see that CT∞ (R) ⊗ Dom(H(0)) ⊂ Dom(K) and K(η ⊗ ψ)(t) = −iη (t)ψ + η(t)H(t)ψ, for every η ∈ CT∞ (R) and ψ ∈ Dom(H(0)). Set K 0 = K|CT∞ (R)⊗Dom(H(0)) . It follows that K 0 is a symmetric operator, K 0 ⊂ K ⊂ (K 0 )∗ . Let K 1 be another operator acting in K and deﬁned by the prescription: f ∈ Dom(K 1 ) if and only if, for every ψ ∈ Dom(H(0)), the function t → ψ, f (t)H is absolutely continuous and there exists g ∈ K such that ∀ψ ∈ Dom(H(0)),

−i∂t ψ, f (t)H + H(t)ψ, f (t)H = ψ, g(t)H ,

(5)

(the last equality is valid, of course, almost everywhere on R). In that case g is unique and we set K 1 f = g. From the deﬁnition it is obvious that K 0 ⊂ K 1 . Hence K and K 1 coincide on ∞ CT (R) ⊗ Dom(H(0)). We shall show that K and K 1 are actually equal. Let us make a remark on the notation used below and everywhere in the remainder of the paper: the natural numbers N start from 1 while Z+ stands for non-negative integers. Lemma 1. For all ψ ∈ H and f ∈ Dom(K 1 ), the function U (t, 0)ψ, f (t)H is absolutely continuous and it holds true that −i∂t U (t, 0)ψ, f (t)H = U (t, 0)ψ, g(t)H

for a.e. t ∈ R,

(6)

where g = K 1 f . Proof. Let us ﬁrst suppose that ψ ∈ Dom(H(0)). Let P be the projector-valued measure for H(0) and set Pn = P ([−n, n]), n ∈ N. Then Pn → I strongly as n → ∞ and therefore the following limit is true in the space of distributions D (R) (actually in L1loc (R)): lim U (t, 0)ψ, Pn f (t)H = U (t, 0)ψ, f (t)H .

n→∞

We shall compute the time derivative of U (t, 0)ψ, f (t)H in the sense of distributions when making use of the fact that −i∂t is continuous on D (R). Choose an orthonormal basis in H called {ϕk }. The series U (t, 0)ψ, ϕk H ϕk , Pn f (t)H U (t, 0)ψ, Pn f (t)H = k

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converges in D (R) (actually in L1loc (R)) since it converges absolutely and is majorized by ψH f (t)H , a locally integrable function. Then, in the sense of distributions, H(t)U (t, 0)ψ, ϕk H ϕk , Pn f (t)H −i∂t U (t, 0)ψ, Pn f (t)H = k

− U (t, 0)ψ, ϕk H H(t)Pn ϕk , f (t)H + U (t, 0)ψ, ϕk H ϕk , Pn g(t)H .

(7)

Here we have used the deﬁnition of K 1 (note that Pn ϕk ∈ Dom(H(0))). The right-hand side in (7) splits into three sums each of them can be summed in D (R). To see it let us note that with the aid of the Schwarz inequality and the Parseval equality one can estimate |H(t)U (t, 0)ψ, ϕk H ϕk , Pn f (t)H | ≤ H(t)U (t, 0)ψH f (t)H . (8) k

Furthermore, f (t)H is square integrable and H(t)U (t, 0)ψH ≤ H(t)(H(0) + i)−1 (H(0) + i)U (t, 0)ψH is locally square integrable due to (2) and (3). Hence the right-hand side of (8) is locally integrable. As far as the second sum is concerned let us note that Gn (t) := H(t)Pn = H(t)(H(0) + i)−1 (H(0) + i)Pn is a bounded operator and Gn (t) is locally bounded according to hypothesis (2). Finally, the third sum does not cause any problem. Consequently, the right-hand side of (7) equals H(t)U (t, 0)ψ, Pn f (t)H − U (t, 0)ψ, Gn (t)∗ f (t)H + U (t, 0)ψ, Pn g(t)H . (9) Thus −i∂t U (t, 0)ψ, f (t)H is equal to the limit of (9) as n → ∞. Since for every ϕ ∈ Dom(H(0)) one has H(0)Pn ϕ → H(0)ϕ and U (t, 0)ψ ∈ Dom(H(0)), in the second term in (9) we get lim Gn (t)U (t, 0)ψ

n→∞

= lim H(t)(H(0) + i)−1 (H(0) + i)Pn U (t, 0)ψ = H(t)U (t, 0)ψ. n→∞

The point-wise limits of the ﬁrst and the third term in (9) are obvious. To justify the convergence in D (R) one can apply once more assumptions (2) and (3) to show that each term has a locally integrable majorant which is independent of n. Thus sending n → ∞ one ﬁnds that equality (6) holds true in the sense of distributions. The right-hand side is a locally integrable function. By a standard

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result of the theory of distributions this implies that the function U (t, 0)ψ, f (t)H is absolutely continuous and that equality (6) holds true in the usual sense. Finally let us show that the condition ψ ∈ Dom(H(0)) from the beginning of the proof can be relaxed. Actually, if h ∈ K and ψk → ψ in H then U (t, 0)ψk , h(t)H is locally integrable and this sequence of functions converges to U (t, 0)ψ, h(t)H in the L1 norm on every bounded interval and hence in the sense of distributions. For any ψ ∈ H one can choose a sequence ψk ∈ Dom(H(0)) such that ψk → ψ and then send k → ∞ in the equality −i∂t U (t, 0)ψk , f (t)H = U (t, 0)ψk , g(t)H

in D (R).

Since the function U (t, 0)ψ, g(t)H is locally integrable the function U (t, 0)ψ, f (t)H can be redeﬁned on a measure zero set so that it is absolutely continuous and equality (6) holds true in the usual sense. Lemma 2. K 1 is symmetric. Proof. Suppose that f ∈ Dom(K 1 ), K 1 f = g and ψ ∈ H . According to Lemma 1 we have 2

−i∂t |U (t, 0)ψ, f (t)H | = 2i Im(f (t), U (t, 0)ψH U (t, 0)ψ, g(t)H ) . Let {ψk } be an orthonormal basis in H . Then, for almost all s ∈ R and all k, ψk , U (s + T, 0)−1 f (s + T )H 2 − ψk , U (s, 0)−1 f (s)H 2

= −2 Im

s

s+T

U (t, 0)−1 f (t), ψk H ψk , U (t, 0)−1 g(t)H dt .

Summing in k one can commute the sum and the integral. Consequently, for almost all s, 2 2 − f (s)H = −2 Im f (s + T )H

s

s+T

f (t), g(t)H dt = −2 Im(f, gK ).

Since f (t)H is periodic the LHS vanishes almost everywhere. We ﬁnd that ∀f ∈ Dom(K 1 ), Im(f, K 1 f K ) = 0. This shows that K 1 is symmetric. Lemma 3. (K 0 )∗ = K 1 . Consequently, K 1 is closed and K 0 is essentially selfadjoint. Proof. By deﬁnition, f ∈ Dom((K 0 )∗ ) if and only if there exists g ∈ K such that ∀h ∈ Dom(K 0 ),

K 0 h, f K = h, gK .

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Moreover, in that case g is unique and (K 0 )∗ f = g. Setting h = η ⊗ ψ we ﬁnd that for all ψ ∈ Dom(H(0)) it is true that T ∞ (iη (t)ψ, f (t)H + η(t)H(t)ψ, f (t)H )dt ∀η ∈ CT (R), 0

T

η(t)ψ, g(t)H dt .

= 0

The last statement can be rewritten as equality (5) valid in the sense of distributions. Since both functions H(t)ψ, f (t)H and ψ, g(t)H belong to L1loc (R) (using again (2) in the former case) the standard results of the theory of distributions tell us that ψ, f (t)H is actually absolutely continuous and equality (5) holds true in the usual sense. Thus we conclude that f ∈ Dom(K 1 ) and K 1 f = g. Hence (K 0 )∗ ⊂ K 1 . Now it suﬃces to apply Lemma 2. Actually, the relations (K 0 )∗ ⊂ K 1 ⊂ (K 1 )∗ ⊂ (K 0 )∗∗ = K 0 ⊂ (K 0 )∗ imply that K 1 = (K 0 )∗ is closed and (K 0 )∗ = (K 0 )∗ = K 0 . Proposition 4. Assuming (2) and (3), it holds true that K = K 1 = K 0. In particular, CT∞ (R) ⊗ Dom(H(0)) is a core of K. Proof. According to Lemmas 2 and 3, it holds true that K 0 ⊂ K 0 ⊂ K = K ∗ ⊂ (K 0 )∗ = K 0 and K 1 = (K 0 )∗ = K 0 . The proposition follows immediately. Let us note that if a vector-valued function f (t) from the domain of K is in addition known to be continuously diﬀerentiable (in the strong sense) then necessarily f (t) ∈ Dom(H(0)) for all t. Under this additional assumption we actually have (Kf )(t) = −i∂t f (t) + H(t)f (t) = g(t). In the general case, however, one should use the weaker form (5). The relation between K ⊕and the formal expression −i∂t + H(t) can be also expressed as follows. H(t) dt be the self-adjoint operator in K with the domain formed by Let H = T those f ∈ K satisfying f (t) ∈ Dom(H(0)) for a.a. t and 0 H(t)f (t)2 dt < ∞, with (Hf )(t) = H(t)f (t). Clearly, Dom(K) ⊃ Dom(−i∂t ⊗ 1) ∩ Dom(H) ⊃ CT∞ (R) ⊗ Dom(H(0)) and therefore, according to Proposition 4, Dom(−i∂t ⊗ 1) ∩ Dom(H) is a core of K. Hence K = −i∂t ⊗ 1 + H .

(10)

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3. Boundedness of the Energy for a Dense Set of Initial Conditions In this section we consider slightly more general periodically time-dependent Hamiltonians H(t), t ∈ R, than those presented in the beginning of Sec. 2, at least among those which are bounded below. We suppose that the Hamiltonian H(t) is associated with a closed, densely deﬁned and positive sesquilinear form q(t), with a domain independent of t: Dom q(t) = Dom q(0),

∀t ∈ R.

(11)

Assuming that the spectrum of U (T, 0) is pure point we wish to construct a rich set of initial conditions for which the mean value of the energy is uniformly bounded in time. It turns out that this is possible if the eigenvectors of U (T, 0) belong to the form domain Dom q(0). The space Dom q(0) endowed with the scalar product u, v1 = u, vH + q(0)(u, v) is a Hilbert space denoted by H1 , and we recall that H(t)u, vH = q(t)(u, v),

∀u ∈ Dom H(t),

∀v ∈ H1 ,

where Dom H(t) = {u ∈ H1 ; ∃Cu ≥ 0 s.t. ∀v ∈ H1 , |q(t)(u, v)| ≤ Cu vH }. We call H−1 the dual space of H1 , that is to say the vector space of continuous conjugate linear forms on H1 . For any u ∈ H , the functional v → v, uH belongs to H−1 since |v, uH | ≤ uH vH ≤ uH v1 , and we can also regard H as a subspace of H−1 with u−1 =

|u, vH | ≤ uH . v1 v∈H1 , v=0 sup

Thus H1 ⊂ H ⊂ H−1 , where the symbol ⊂ means a topological embedding. Actually, H(t) can be extended into an operator mapping H1 into H−1 provided there exists a constant Ct ≥ 0 such that ∀u ∈ H1 ,

q(t)(u, u) ≤ Ct u12 .

Let us denote by ·, ·−1,1 the dual pairing between H−1 and H1 . This pairing is conjugate linear in the ﬁrst and linear in the second argument. In other words, the embedding H ⊂ H−1 means that ψ, g−1,1 = ψ, gH for all ψ ∈ H and g ∈ H1 , and the mapping H(t) : H1 → H−1 is deﬁned so that H(t)u, v−1,1 = q(t)(u, v) for all u, v ∈ H1 . In the remainder of this section, we will refer to the propagator U (t, 0) associated with the family of Hamiltonians H(t), t ∈ R. Its existence is implied by the following result which can be found in [16] and that we reproduce below for the reader’s convenience.

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Theorem 5. We assume that q(t) satisﬁes (11) and that there is a constant C ≥ 1 such that the operator H(t) satisﬁes, for all t ∈ R: 1. C −1 (H(0) + 1) ≤ H(t) ≤ C(H(0) + 1). d H(t)−1 exists in the norm sense and 2. The derivative dt

H(t) d H(t)−1 ≤ C. H(t) dt Then, for any ψ0 ∈ H1 there is a unique function R t → ψ(t) ∈ H1 such that: 1. ψ is H1 -weakly continuous, i.e., for all g ∈ H−1 , t → g, ψ(t)−1,1 is a continuous function. 2. ψ is a weak solution of the Schr¨ odinger equation in the following sense: d g, ψ(t)H + q(t)(g, ψ(t)) = 0 and dt 3. For all s ∈ R, we have ψ(t) − ψ(s) + iH(t)ψ(t) lim = 0. t→s t−s ∀g ∈ H1 ,

−i

ψ(0) = ψ0 .

−1

4. ψ(t)H = ψ0 H for all t ∈ R and t → ψ(t) is continuous in the norm topology in H . The propagator U : (s, t) ∈ R2 → U (t, s) associated with the Hamiltonian H(t) is deﬁned by U (t, s)ψ(s) = ψ(t). It is unitary and strongly continuous according to point 4. For the proof of the main result of this section, Proposition 7, we need the following lemma. Lemma 6. Let ψ ∈ H1 be an eigenfunction of the Floquet operator U (T, 0). Then the function Fψ (t) := H(t)U (t, 0)ψ−1 is bounded in R: Fψ ∞ := sup Fψ (t) < +∞. t∈R

Proof. First, we notice that the function t → |H(t)U (t, 0)ψ, g−1,1 |, with g ∈ H1 , is periodic with period T . This can be seen from the equality U (t + T, 0)ψ = U (t + T, T )U (T, 0)ψ = λU (t, 0)ψ, and |H(t + T )U (t + T, 0)ψ, g−1,1 | = |λ||H(t + T )U (t, 0)ψ, g−1,1| = |H(t)U (t, 0)ψ, g−1,1 | since |λ| = 1 and H(t + T ) = H(t).

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Moreover, the H−1 -valued function t → H(t)U (t, 0)ψ is weakly continuous on R. Indeed, for any given real numbers s and t, we derive from the following obvious decomposition, with g ∈ H1 , H(t)ψ(t), g−1,1 − H(s)ψ(s), g−1,1 ψ(t) − ψ(s) ψ(t) − ψ(s) ,g ,g = H(t)ψ(t) − i − H(s)ψ(s) − i , t−s t−s −1,1 −1,1 that |H(t)ψ(t), g−1,1 − H(s)ψ(s), g−1,1 | ψ(t) − ψ(s) g1 ≤ H(t)ψ(t) − i t−s −1 ψ(t), gH − ψ(s), gH + q(s)(ψ(s), g) − i t−s

.

Applying respectively points 3 and 2 of Theorem 5 one ﬁnds that both terms on the right-hand side of the preceding inequality tend to zero as t tends to s. This implies that for every g ∈ H1 the function t → |H(t)U (t, 0)ψ, g−1,1 | is bounded on R (since we just checked that it is periodic). From the uniform boundedness principle it follows that Fψ (t) =

sup g∈H1 , g 1 =1

|H(t)U (t, 0)ψ, g−1,1 |

is bounded on R as well. Proposition 7. Let us suppose that the Floquet operator U (T, 0) has a pure point spectrum and admits a basis B formed by eigenfunctions belonging to H1 . Then the energy of the quantum system, when starting from any initial state ψ ∈ span B, the set of ﬁnite linear combinations of vectors from B, is bounded in the course of time: sup |H(t)U (t, 0)ψ, U (t, 0)ψ−1,1 | = sup |q(t) U (t, 0)ψ, U (t, 0)ψ | < ∞. t∈R

t∈R

Proof. Recall that by our assumptions H(t)−1 is a bounded operator on H (see Theorem 5). If u ∈ H1 , v ∈ H , then H(t)−1 v ∈ Dom H(t) and q(t)(u, H(t)−1 v) = u, vH . Consequently, H(t)u, H(t)−1 v−1,1 = q(t)(u, H(t)−1 v) = u, vH . We can assume that the basis B is orthonormal. For any given ψ in B ⊂ H1 , we ﬁrst notice that U (t, 0)ψ1 is bounded by Fψ (t) deﬁned in Lemma 6 up to a multiplicative constant C. Indeed, for any g ∈ H we have |U (t, 0)ψ, gH | = |H(t)U (t, 0)ψ, H(t)−1 g−1,1 | ≤ H(t)U (t, 0)ψ−1 H(t)−1 g1 ,

(12)

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with H(t)−1 g21 = H(t)−1 g, (H(0) + 1)H(t)−1 gH ≤ CH(t)−1 g, gH ≤ CH(t)−1 g1 g−1, according to assumption 1 in Theorem 5. Thus H(t)−1 g1 ≤ Cg−1 and (12) becomes |g, U (t, 0)ψ−1,1 | = |U (t, 0)ψ, gH | ≤ CH(t)U (t, 0)ψ−1 g−1 = CFψ (t)g−1 .

(13)

Furthermore, H being dense in H−1 in the norm topology, inequality (13) remains valid for any g ∈ H−1 implying U (t, 0)ψ1 =

|g, U (t, 0)ψ−1,1 | ≤ CFψ (t). g−1 g∈H−1 ,g=0 sup

(14)

N To complete the proof we pick a function ϕ in span B, ϕ = i=1 ci ψi , with ψi ∈ B and ci ∈ C for i = 1, 2, . . . , N . The energy function of the quantum system with the initial condition ϕ decomposes as Eϕ (t) ≡ H(t)U (t, 0)ϕ, U (t, 0)ϕ−1,1 =

N

ci cj H(t)U (t, 0)ψi , U (t, 0)ψj −1,1 .

i,j=1

Therefore, |Eϕ (t)| ≤

N

|ci ||cj |H(t)U (t, 0)ψi −1 U (t, 0)ψj 1

i,j=1

≤C

N

|ci ||cj |Fψi (t)Fψj (t) ≤ C max Fψi (t)

2

1≤i≤N

i,j=1

N

2 |ci |

,

i=1

according to (14), so we ﬁnally obtain |Eϕ (t)| ≤ CN ϕ2 max Fψi 2∞ , 1≤i≤N

by combining the Cauchy–Schwarz inequality with Lemma 6. 4. Bounds on the Energy and Transition Probabilities The only assumptions needed in this section are that the domain Dom H(t) of a T -periodic family of self-adjoint operators is time-independent and that the propagator U (t, s) associated with H(t) exists in the usual sense, as recalled at the beginning of Sec. 2. By the spectral theorem, the Floquet (monodromy) operator U (T, 0) can be written in the form U (T, 0) = exp(−iTH F ) where HF is a self-adjoint operator. Of

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course, the choice of HF is highly ambiguous. Let UF (t) be the family of unitary operators deﬁned by the equality U (t, 0) = UF (t)e−itHF .

(15)

Then UF (0) = I and from the periodicity of U (t, s) (see (1)) it follows that UF (t) also depends on t periodically. Relation (15) is known as the Floquet decomposition. Deﬁnition 8. We shall say that a Floquet decomposition is r times continuously diﬀerentiable in the strong sense for some r ∈ N if this is case for the family UF (t). Furthermore, we shall say that a Floquet decomposition is relatively continuously diﬀerentiable in the strong sense if the family UF (t)(HF + i)−1 is continuously diﬀerentiable in the strong sense. Equivalently this means that for all ψ ∈ Dom HF the vector-valued function UF (t)ψ is continuously diﬀerentiable. Assume that the propagator U (t, s) admits a Floquet decomposition which is relatively continuously diﬀerentiable in the strong sense. Set SF (t) = iUF (t)−1 ∂t UF (t),

Dom SF (t) = Dom HF .

(16)

By the uniform boundedness principle, SF (t) is HF -bounded for all t ∈ R. Using the periodicity of UF (t) and applying again the uniform boundedness principle one ﬁnds that SF (t)(HF + i)−1 is bounded uniformly in t. Moreover, SF (t) is a symmetric operator. If the Floquet decomposition is continuously diﬀerentiable in the strong sense then SF (t) will be naturally supposed to be deﬁned on the entire space H . Referring again to the uniform boundedness principle, in this case we have SF (t) ∈ B(H ). Using the periodicity of UF (t) and applying the uniform boundedness principle once ⊕ SF (t)dt is a more one ﬁnds that SF (t) is bounded uniformly in t. Hence SF := bounded operator in K whose norm equals SF = sup SF (t). t∈R

Moreover, SF (t) is a Hermitian operator. Lemma 9. Assume that a Floquet decomposition (15) is relatively continuously diﬀerentiable in the strong sense and that the relative bound of SF (t) with respect to HF is less than one for all t. Then (17) ∀t ∈ R, H(t) = UF (t) HF + SF (t) UF (t)−1 . In particular, H(0) = HF + SF (0). Furthermore, Dom(HF ) = Dom(H(0)) and this domain is UF (t) invariant.

(18)

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Proof. By the assumptions and the Kato–Rellich theorem (see, for example, [15]), ˜ H(t) := UF (t)(HF + SF (t))UF (t)−1 ,

˜ Dom H(t) = UF (t)(Dom HF ),

is a T -periodic family of self-adjoint operators. From (15) it follows that U (t, s) = UF (t)e−i(t−s)HF UF (s)−1 . From this relation it is obvious that ˜ ˜ U (t, s)(Dom H(s)) = Dom H(t). Suppose that ϕ ∈ UF (s)(Dom HF ) and thus ϕ = UF (s)ψ for some ψ ∈ Dom HF . A straightforward computation yields i∂t U (t, s)ϕ = i∂t UF (t)e−i(t−s)HF ψ ˜ = UF (t)(HF + SF (t))e−i(t−s)HF ψ = H(t)U (t, s)ϕ. ˜ Hence U (t, s) is a propagator associated with the family H(t). Using the property of self-adjointness one can easily see that the uniqueness of the relation between a Hamiltonian and a propagator applies also in the following direction: if two (in general time-dependent) Hamiltonians generate the same prop˜ agator then they are equal. In our case this means that H(t) = H(t) for all t, i.e. equality (17) holds. Consequently, UF (t)(Dom HF ) = Dom H(t) = Dom H(0) and setting t = 0 we have Dom HF = Dom H(0). Next we shall show that the relative continuous diﬀerentiability of UF (t) implies the dynamical stability. Proposition 10. Under the same assumptions as in Lemma 9, the energy of the system described by the Hamiltonian H(t) is uniformly bounded for any initial condition. More precisely, ∀ψ ∈ Dom(H(0)),

sup H(t)U (t, 0)ψ ≤ Cψ t∈R

where Cψ = HF ψ + supSF (t)(HF + i)−1 (HF + i)ψ. t

Proof. From equalities (15) and (17) it follows that H(t)U (t, 0)ψ = (HF + SF (t))e−itHF ψ ≤ Cψ . Remark. Proposition 10 even implies that the mean value of the square of the energy, H(t)2 , is uniformly bounded. Another application is an estimate of transition probabilities under the assumption of the strong diﬀerentiability of UF (t). To this end we shall need the following lemma.

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Lemma 11. Assume that X, Y ∈ B(H ), A and B are bounded Hermitian operators on H such that AX − XB = Y.

(19)

If there exist two disjoint closed intervals containing respectively Spec(A) and Spec(B) then X ≤

Y . dist(Spec(A), Spec(B))

Proof. For the sake of deﬁniteness let us suppose that inf Spec(B) > sup Spec(A). The solution X of Eq. (19) is unique and given by the formula 1 X= (A − z)−1 Y (B − z)−1 dz . (20) 2πi γ After a usual limit procedure we can choose for the integration path γ in (20) the line which is parallel to the imaginary axis and intersects the real axis in the point (sup Spec(A) + inf Spec(B))/2. Integral (20) admits a simple estimate leading to the desired inequality. Remark. Let us note that an estimate of this sort still exists when the spectra of A and B are interlaced provided dist(Spec(A), Spec(B)) > 0. In the general case, as discussed in article [18], it holds true that X ≤

Y π . 2 dist(Spec(A), Spec(B))

Proposition 12. Assume that the propagator U (t, s) admits a Floquet decomposition (15) which is continuously diﬀerentiable in the strong sense. Let P (t, ·) be the projector-valued measure from the spectral decomposition of H(t). Let ∆1 , ∆2 ⊂ R be two intervals such that dist(∆1 , ∆2 ) > 0. Then it holds true that ∀s, t ∈ R,

P (t, ∆1 )U (t, s)P (s, ∆2 ) ≤

2SF . dist(∆1 , ∆2 )

In particular, if En (t) and Em (s) are eigenvalues of H(t) and H(s), respectively, En (t) = Em (s), and if Pn (t) and Pm (s) denote the projectors onto the corresponding eigenspaces then Pn (t)U (t, s)Pm (s) ≤

2SF . |En (t) − Em (s)|

Proof. Using relation (17) one veriﬁes the equality H(t)U (t, s) − U (t, s)H(s) = U (t, 0) eitHF SF (t)e−itHF − eisHF SF (s)e−isHF U (0, s)

(21)

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which is valid on Dom(H(0)). In particular, the left-hand side of (21) extends to an operator bounded on H whose norm may be estimated from above by 2SF . Setting A = H(t)P (t, ∆1 ), B = H(s)P (s, ∆2 ) and X = P (t, ∆1 )U (t, s)P (s, ∆2 ), one easily ﬁnds that AX − XB = P (t, ∆1 )(H(t)U (t, s) − U (t, s)H(s))P (s, ∆2 ). If the intervals ∆1 , ∆2 are bounded then Lemma 11 implies that X ≤ 2SF / dist(∆1 , ∆2 ). If the intervals are not bounded one can use a limit procedure.

5. Extension: A Higher Order of Diﬀerentiability of the Floquet Decomposition Under assumptions on higher order diﬀerentiability in the strong sense of the operator-valued function UF (t) in (15) one can extend the conclusions of Propositions 10 and 12. To this end, as an auxiliary tool we ﬁrst need to state some basic facts concerning the multiple commutators. 5.1. Multiple commutators Deﬁnition 13. Let A be a selfadjoint operator in H , X ∈ B(H ) and n ∈ Z+ . The sesquilinear form n

n (−1)k XAk ξ, An−k ηH αn (ξ, η) = k k=0

n

is well deﬁned on Dom(A ). If it is bounded then there exists a unique bounded operator, denoted by adAn X, such that ∀ξ, η ∈ Dom(An ),

αn (ξ, η) = (adAn X)ξ, ηH .

If this is the case we shall say that (the n-multiple commutator) adAn X exists in B(H ). Remark. Some elementary facts follow immediately from the deﬁnition. Suppose that B = B ∗ is bounded. Then adBn X ∈ B(H ) exists for all n ∈ Z+ and one has n

n adBn X = (−1)k B n−k XB k . k k=0

Moreover, in this case adA+B X ∈ B(H ) exists if and only if adA X ∈ B(H ) exists and then adA+B X = adA X + adB X. Deﬁnition 14. Suppose that A = A∗ in H . For every n ∈ Z+ we introduce the linear subspace Cn (A) ⊂ B(H ) formed by those bounded operators X for which the commutators adAk X ∈ B(H ) exist for all k = 0, 1, . . . , n.

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Remark. Clearly, adA0 X = X and C0 (A) = B(H ). From the deﬁnition it is also obvious that the vector spaces are nested, i.e. C0 (A) ⊃ C1 (A) ⊃ C2 (A) ⊃ · · · .

(22)

Lemma 15. Suppose that A = A∗ and X, Y ∈ B(H ). If the commutators adA X, adA Y ∈ B(H ) exist then there also exist adA X ∗ , adA (XY ) ∈ B(H ) and the following relations hold: (i) X(Dom A) ⊂ Dom A, (ii) adA X ∗ = −(adA X)∗ , (iii) adA (XY ) = (adA X)Y + X adA Y . Proof. To show (i) choose ξ ∈ Dom A. By deﬁnition, for all η ∈ Dom A we have Xξ, Aη = XAξ, η + (adA X)ξ, η. Hence Xξ belongs to Dom A∗ = Dom A. Point (ii) follows from the equality X ∗ ξ, Aη − X ∗Aξ, η = −Xη, Aξ − XAη, ξ = −(adA X)∗ ξ, η which is valid for all ξ, η ∈ Dom A. For ξ, η from the same domain we know, by points (i) and (ii), that Y ξ, X ∗ η ∈ Dom A. Thus we have the equality XY ξ, Aη − XYAξ, η = XY ξ, Aη − XAY ξ, η + Y ξ, AX ∗ η − YAξ, X ∗ η with the right-hand side being equal to (adA X)Y ξ, η + (adA Y )ξ, X ∗ η. Point (iii) follows. Remark. Lemma 15 implies that adA X ∈ B(H ) exists if and only if Dom(A) is invariant with respect to X and the operator AX − XA is bounded on this domain. If this is the case then adA X = AX − XA on Dom(A). Lemma 16. Let {Xn }n be a sequence of bounded operators in H such that the commutators adA Xn ∈ B(H ) exist for all n. If the sequence {Xn }n converges weakly to a bounded operator X and the sequence {adA Xn }n converges weakly to a bounded operator Y then adA X ∈ B(H ) exists and equals Y . Remark. From Lemma 16 it follows that the linear operator adA on B(H ), with Dom(adA ) = C1 (A), is closed. Proof. Let ξ, η ∈ Dom(A) be arbitrary vectors. By deﬁnition, for all n, Xn ξ, Aη − Xn Aξ, η = (adA Xn )ξ, η. It suﬃces to send n to inﬁnity.

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Proposition 17. The following statements are true for all X ∈ B(H ) and n ∈ Z+ : (i) If X ∈ Cn (A) then X(Dom Ak ) ⊂ Dom Ak for all k = 0, 1, . . . , n. (ii) X ∈ Cn+1 (A) if and only if adA X ∈ B(H ) exists and belongs to Cn (A). Moreover, if this is the case then adAk (adA X) = adAk+1 X

for k = 0, 1, . . . , n.

(iii) Cn (A) is a ∗-subalgebra of B(H ). Proof. (i) We shall show that, for a given k ∈ Z+ , the domain Dom Ak is invariant with respect to all X ∈ Cn (A) as long as n ≥ k. Recalling that the spaces Cn (A) are nested it suﬃces to consider the case of n = k. To this end, we shall proceed by induction in k. For k = 0 the statement is trivial. Suppose that the statement holds true for all , 0 ≤ ≤ k. Choose X ∈ Ck+1 (A). The induction hypothesis implies that for any ξ ∈ Dom(Ak+1 ) and 1 ≤ ≤ k + 1, XA ξ ∈ Dom(Ak+1− ). By the deﬁnition of adAk+1 X we have the equality k+1

Xξ, A

η =

k+1 =1

k+1 (−1)+1 Ak+1− XA ξ, η + (adAk+1 )Xξ, η,

valid for all η ∈ Dom(Ak+1 ). Hence Xξ ∈ Dom(Ak+1 ). (ii) By the very deﬁnition, if X ∈ Cn+1 (A) then adA X ∈ B(H ) exists. If 0 ≤ m ≤ n and ξ, η ∈ Dom(Am+1 ) then simple algebraic manipulations lead to the equality m

m (−1)k (adA X)Ak ξ, Am−k η = (adAm+1 X)ξ, η. k

(23)

k=0

The both sides in (23) extend in a unique way to the domain ξ, η ∈ Dom(Am ). It follows that adAm (adA X) ∈ B(H ) exists and equals adAm+1 X. Hence adA X ∈ Cn (A). Conversely, suppose that adA X ∈ Cn (A). For any m, 0 ≤ m ≤ n, and ξ, η ∈ Dom(Am+1 ), one ﬁnds, again with the aid of simple algebraic manipulations, that adAm (adA

X)ξ, η =

m+1 k=0

m+1 (−1)k XAk ξ, Am+1−k η. k

Hence adAm+1 X ∈ B(H ) exists and thus X ∈ Cn+1 (A). (iii) First, let us show that X ∗ ∈ Cn (A) provided the same is true for X. We shall proceed by induction in n. The case n = 0 is obvious. Suppose that the claim is true for n. If X ∈ Cn+1 (A) then, by the already proved point (ii) of the current proposition, adA X ∈ Cn (A). By the induction hypothesis and Lemma 15(ii)

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we have adA X ∗ = −(adA X)∗ ∈ Cn (A). Referring once more to point (ii) of the current proposition we conclude that indeed X ∗ ∈ Cn+1 (A). Finally, let us show that XY ∈ Cn (A) provided X, Y ∈ Cn (A). We shall proceed by induction in n. The case n = 0 is again obvious. Suppose that the claim is true for n. If X, Y ∈ Cn+1 (A) then, by point (ii) of the current proposition, adA X, adA Y ∈ Cn (A). By the induction hypothesis and Lemma 15(iii) we have adA (XY ) = (adA X)Y + X adA Y ∈ Cn (A). Referring again to point (ii) of the current proposition we conclude that XY ∈ Cn+1 (A). Remark. As an immediate consequence of Proposition 17(i) one has Dom(Ak ) = Dom((A + B)k ) for k = 0, 1, . . . , p, provided B ∈ Cp−1 (A) for some p ∈ N. Deﬁnition 18. Let A be a self-adjoint operator on H and X(t) ∈ B(H ) be an operator-valued function, with the variable t running over R, and let n ∈ Z+ . We shall say that X(t) is in the algebra Cn (A) uniformly if X(t) ∈ Cn (A) for all t ∈ R and n sup adAk X(t) < ∞. t∈R

k=0

Remarks. Of course, the operator-valued function X(t) may be constant. From Proposition 17(ii) one immediately deduces that an operator-valued function X(t) ∈ B(H ) is in Cn+1 (A) uniformly if and only if X(t) is uniformly bounded and adA X(t) is in Cn (A) uniformly. Moreover, a straightforward induction procedure based on this observation jointly with Lemma 15’s (ii) and (iii) implies that if X(t) and Y (t) are in Cn (A) uniformly then also X(t)∗ and X(t)Y (t) are in Cn (A) uniformly. Lemma 19. Let A be a self-adjoint operator and B ∈ Cp−1 (A) be a Hermitian operator for some p ∈ N. Then an operator-valued function X(t) ∈ B(H ), with t ∈ R, is in Cp (A) uniformly if and only if X(t) is in Cp (A + B) uniformly. Proof. Clearly it suﬃces to prove only one implication since the other one follows after replacing A by A + B and B by −B (while making use of the simple fact that m B = adAm B). We shall proceed by induction in p. adA+B As far as the case p = 1 is concerned we assume that B ∈ C0 (A) and X(t) is in C1 (A) uniformly. This in particular means that X(t) is a uniformly bounded operator-valued function and hence the same is true for adB X(t) = BX(t)−X(t)B. Now it suﬃces to take into account the equality adA+B X(t) = adA X(t)+adB X(t). Let us now assume that the lemma has been proved for some p ∈ N, and that B ∈ Cp (A) and X(t) is in Cp+1 (A) uniformly. Now we can repeatedly apply the remarks following Deﬁnition 18. Firstly, adB X(t) = BX(t) − X(t)B is in Cp (A)

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uniformly. Secondly, X(t) is uniformly bounded and adA X(t) is in Cp (A) uniformly. Consequently, adA+B X(t) is in Cp (A) uniformly as well. By the induction hypothesis, adA+B X(t) is in Cp (A + B) uniformly. This in turn implies that X(t) is in Cp+1 (A + B) uniformly. In the particular case when the operator-valued function X(t) is constant Lemma 19 reduces to the following statement. Lemma 20. Let A be a self-adjoint operator on H and B ∈ Cp−1 (A) for some p ∈ N, and suppose that B = B ∗ . Then Ck (A) = Ck (A + B) for k = 0, 1, . . . , p. We shall also need the following algebraic lemma. Lemma 21. Suppose that A = A∗ and B ∈ Cp (A) for some p ∈ Z+ . Then the following claims are true: (i) On Dom(Ap ), Ap B =

p

p (adAp−k B)Ak . k

(24)

k=0

(ii) There exist polynomials Fp,k (x0 , x1 , . . . , xp−k−1 ), k = 0, 1, . . . , p − 1, in noncommutative variables xj , with non-negative integer coeﬃcients and such that p

p

(A + B) = A +

p−1

Fp,k B, adA B, . . . , adAp−k−1 B Ak

(25)

k=0

on Dom(Ap ). Proof. (i) By Proposition 17(i) the both sides of (24) are well deﬁned on Dom(Ap ). To verify (24) one can proceed by induction in p which amounts to simple algebraic manipulations. We omit the details. (ii) Again, the both sides of (25) are well deﬁned on Dom(Ap ). One can proceed by induction in p. Set, by convention, Fp,p = 1. To carry out the induction step let us write (A + B)p+1 =

p

Fp,k B, adA B, . . . , adAp−k−1 B Ak (A + B)

k=0

and apply claim (i) of the current lemma to manage the term Ak B on the right-hand side. By comparison one arrives at the recursion rule Fp+1,k (x0 , x1 , . . . , xp−k ) = Fp,k−1 (x0 , x1 , . . . , xp−k ) p

+ Fp, (x0 , x1 , . . . , xp−−1 )x−k k =k

from which claim (ii) easily follows.

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5.2. Diﬀerentiable Floquet decompositions In this section we shall assume that V (t) := H(t) − H(0) is a uniformly bounded operator-valued function. Of course, it is Hermitian and T -periodic. We also assume that we are given a Floquet decomposition (15) of the corresponding propagator U (t, s). For p ∈ Z+ let us set A0p = Cp (H0 ),

Ap = Cp (HF ).

Here and everywhere in this section we write shortly H0 = H(0) and S0 = SF (0). Thus we have H0 = HF + S0 (see (16) and (17)). If the Floquet decomposition is continuously diﬀerentiable in the strong sense and S0 ∈ Ap−1 for some p ∈ N then Lemma 20 tells us that Ak = A0k for k = 0, 1, . . . , p. Lemma 22. Let us assume that p ∈ N and V (t) ∈ C p−1 (R) in the strong sense, and that the propagator U (t, s) admits a Floquet decomposition (15) which is p times continuously diﬀerentiable in the strong sense. If V (k) (t) is in Ap−1−k uniformly

for k = 0, 1, . . . , p − 1,

(26)

then (k)

UF (t) is in Ap−k uniformly

for k = 0, 1, . . . , p.

(27)

Moreover, Ak = A0k for k = 0, 1, . . . , p, and SF (t) = i UF (t)−1 ∂t UF (t) is in Ap−1 uniformly. Proof. For the proof we shall need the relation adHF UF (t) = UF (t)SF (t) − S0 + V (t) UF (t).

(28)

Here UF (t) preserves the domain Dom(HF ) = Dom(H0 ). Equality (28) follows from (17) and the substitution H(t) = H0 + V (t) = S0 + HF + V (t). From the diﬀerentiability of UF (t) it follows that SF (t) belongs to C p−1 (R) in the strong sense. Thus all derivatives of SF (t) up to the order p − 1 are uniformly bounded (due to the periodicity). With the aid of Lemma 16 we derive from (28) that (k)

adHF UF (t) =

dk (UF (t)SF (t) − (S0 + V (t))UF (t)) ∈ B(H ) dtk for k = 0, 1, . . . , p − 1,

(29)

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(with all derivatives taken in the strong sense). Moreover, adHF UF (t) is uniformly bounded for 0 ≤ k ≤ p − 1. Note also that (26) can be rewritten in the form V (k) (t) is in Ap−1− uniformly

if 0 ≤ k ≤ ≤ p − 1

(30)

since the algebras Ar are nested, A0 ⊃ A1 ⊃ A2 ⊃ · · ·, (see (22)). We shall verify that, for = 0, 1, . . . , p, (k)

UF (t) is in Ap− uniformly

if 0 ≤ k ≤ ≤ p.

(31)

Since (k)

SF (t) =

dk (iUF (t)∗ UF (t)) dtk

and An is a ∗-algebra relation (31) implies that, for = 1, . . . , p, (k)

SF (t) is in Ap− uniformly

if 0 ≤ k ≤ − 1 ≤ p − 1.

(32)

To show (31) we shall proceed by a ﬁnite descending induction in . According to (k) the assumptions of the lemma, UF (t) is uniformly bounded for 0 ≤ k ≤ p and the case = p follows. Suppose now that (31) is valid for some , 1 ≤ ≤ p. Then for the same , (32) is valid as well. Moreover, replacing by − 1 in (30) one knows that V (k) (t) is in Ap− uniformly for 0 ≤ k ≤ − 1. Thus if 0 ≤ k ≤ − 1 then from the fact that Ap− is an algebra and from the induction hypothesis one deduces that the right-hand side of (29) is in Ap− uniformly. This implies, in (k) virtue of Proposition 17(ii), that UF (t) is in Ap−+1 uniformly. This completes the induction step and relation (31) is veriﬁed. Setting k = in (31) one obtains (27). Setting k = 0 and = 1 in (32) one ﬁnds that SF (t) is Ap−1 uniformly. In particular, S0 ≡ SF (0) belongs to Ap−1 = Cp−1 (HF ). Since H0 = HF + S0 from Lemma 20 we know that Ak = Ck (HF ) = Ck (H0 ) = A0k for k = 0, 1, . . . , p. Corollary 23. Lemma 22 remains still true if Ap−1−k is replaced by A0p−1−k in the condition (26). Proof. We shall proceed by induction in p. For p = 1 we have A00 = A0 = B(H ) and thus replacing Ap−1−k by A0p−1−k in (26) does not mean any change. Let us now suppose that the claim is true for some p ∈ N. And we assume that V (k) (t) is in A0p−k uniformly for k = 0, 1, . . . , p. Of course, the other assumptions of Lemma 22, except of the condition (26), are satisﬁed as well, namely V (t) ∈ C p (R) in the strong sense, and the propagator U (t, s) admits a Floquet decomposition (15) which is p + 1 times continuously diﬀerentiable in the strong sense. Since A0p−k ⊂ A0p−1−k , V (k) (t) is in A0p−1−k uniformly for k = 0, 1, . . . , p − 1. By the induction hypothesis, Lemma 22 is applicable for the value p and therefore, in particular, Ak = A0k for

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k = 0, 1, . . . , p. Hence V (k) (t) is in Ap−k uniformly for k = 0, 1, . . . , p, which is nothing but condition (26) with p being replaced by p + 1. It follows that the conclusions of Lemma 22 hold true for the value p + 1 as well. Proposition 24. Let us assume that p ∈ N and V (t) ∈ C p−1 (R) in the strong sense, and that the propagator U (t, s) admits a Floquet decomposition (15) which is p times continuously diﬀerentiable in the strong sense. If V (k) (t) is in A0p−1−k uniformly

for k = 0, 1, . . . , p − 1,

(33)

then U (t, 0), t ∈ R, preserves the domain Dom(H0p ) and ∀ψ ∈ Dom(H0p ),

sup H(t)p U (t, 0)ψ < ∞. t∈R

Proof. From Corollary 23 we know that UF (t) is in Ap uniformly and SF (t) is in Ap−1 uniformly. Since S0 ∈ Ap−1 and H0 = HF + S0 , Lemma 20 tells us that Ak = A0k for 0 ≤ k ≤ p. From the relations V (t) ∈ A0p−1 and S0 ∈ Ap−1 it also follows that Dom(H0k ) = Dom(H(t)k ) = Dom(HFk )

for k = 0, 1, . . . , p,

(34)

see Proposition 17(i). Furthermore, from the Floquet decomposition (15) and the above observation on UF (t) one deduces that U (t, 0) is in Ap = A0p uniformly and therefore U (t, 0)(Dom(H0p )) ⊂ Dom(H0p ). Suppose that ψ ∈ Dom(H0p ). From (15) and (17) one ﬁnds that p H(t)p U (t, 0)ψ = UF (t) HF + SF (t) e−itHF ψ. With the aid of equality (25) of Lemma 21(ii) one derives the estimate p

H(t) U (t, 0)ψ ≤

p

Fp,k (S0 , S1 , . . . , Sp−k−1 ) HFk ψ

k=0

where k SF (t), Sk := sup adH F t∈R

k = 0, 1, . . . , p − 1.

The proposition follows. Lemma 25. Under the same assumptions as in Proposition 24, the operators n

n (35) Xn (t, s) = (−1)k H(t)n−k U (t, s)H(s)k , n = 0, 1, . . . , p, k k=0

are well deﬁned on Dom(H0n ). Moreover, Xn (t, s) extends in a unique way to a bounded operator on H which is in Ap−n uniformly with respect to the variables (t, s) ∈ R2 .

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Proof. In the same way as in the proof of Proposition 24, we deduce from the assumptions that equalities (34) hold true as well as that UF (t) is in Ap = A0p uniformly and SF (t) is in Ap−1 uniformly. Moreover, Proposition 17(i) tells us that UF (t) preserves Dom(HFk ) for k = 0, 1, . . . , p. From (15) and (17) it follows that Xn (t, s) = UF (t)Zn (t, s)UF (s) −1 where Zn (t, s) =

n

n−k −i(t−s)HF k n HF + SF (s) . e (−1)k HF + SF (t) k k=0

It suﬃces to show that Zn (t, s) is well deﬁned on Dom(HFn ) and extends to a bounded operator on H which is in Ap−n uniformly. To verify it we proceed by induction in n. k Z0 (t, s) = 0 for all k ≥ 1 and so it For n = 0, Z0 (t, s) = e−i(t−s)HF fulﬁlls adH F is in Ap uniformly. To carry out the induction step observe that Zn+1 (t, s) = (HF + SF (t))Zn (t, s) − Zn (t, s)(HF + SF (s)) = adHF Zn (t, s) + SF (t)Zn (t, s) − Zn (t, s)SF (s). The induction hypothesis and Proposition 17(ii) (see also Remarks following Deﬁnition 18) imply that adHF Zn (t, s) is in Ap−n−1 uniformly. Recalling Proposition 17(iii) it also holds true that SF (t)Zn (t, s) and Zn (t, s)SF (s)are in Ap−n−1 uniformly. This veriﬁes the induction step and concludes the proof of the lemma. Proposition 26. Under the same assumptions as in Proposition 24 (including condition (33)), let P (t, ·) be the projection-valued measure from the spectral decomposition of H(t). Then there exists a constant Cp ≥ 0 such that for any couple of intervals ∆1 , ∆2 ⊂ R whose distance dist(∆1 , ∆2 ) is positive it holds true ∀s, t ∈ R,

P (t, ∆1 )U (t, s)P (s, ∆2 ) ≤

Cp p . dist(∆1 , ∆2 )

(36)

Proof. It suﬃces to verify the assertion for bounded intervals. The general case then follows by a limit procedure. Set Yn (t, s) = P (t, ∆1 )Xn (t, s)P (s, ∆2 ) where Xn (t, s) is deﬁned in (35), and Q1 (t) = H(t)P (t, ∆1 ), Q2 (s) = H(s)P (s, ∆2 ). In particular, Y0 (t, s) = P (t, ∆1 )U (t, s)P (s, ∆2 ). From Lemma 25 we know that the operator-valued functions Xn (t, s) are uniformly bounded. If 0 ≤ n < p then Q1 (t)Yn (t, s) − Yn (t, s)Q2 (s) = Yn+1 (t, s). By Lemma 11 we have the estimate Yn (t, s) ≤

Yn+1 (t, s) Yn+1 (t, s) ≤ . dist(Spec(Q1 (t)), Spec(Q2 (s))) dist(∆1 , ∆2 )

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Applying this estimate consecutively for n = 0, 1, . . . , p − 1, we ﬁnd that (36) holds true with Cp = sup(t,s)∈R2 Xp (t, s). 6. A Solvable Example: The Time-Dependent Harmonic Oscillator Let us consider the time-dependent harmonic oscillator H(t) = Hω + f (t)x,

1 ω 2 x2 , Hω = − ∂x2 + 2 2

in H = L2 (R, dx) where the function f (t) is supposed to be continuous and T periodic. The Hamiltonians quadratic in x and p turn out to be quite attractive in various situations since they allow for explicit computations. For example, a classical result is a formula for the Green function computed in the framework of the Feynman path integral [19], see also [20] and comments on the literature therein. For purposes of the present paper we need some of the results derived in [3] and concerned with the dynamical properties of H(t), see also an additional analysis in [21, Chap. 5]. Let us also mention that in [22] it has been shown that the Floquet operator associated with a time-dependent quadratic Hamiltonian can only have either a pure point spectrum or a purely absolutely transient continuous spectrum. As pointed out in [3], it holds true U (t, 0)−1 xU (t, 0) = x cos(ωt) +

1 p sin(ωt) − ϕ2 (t, 0), ω ω

U (t, 0)−1 pU (t, 0) = − ωx sin(ωt) + p cos(ωt) + ϕ1 (t, 0), where the functions ϕ1 (t, s) and ϕ2 (t, s) are given in (A.9). Assume for a moment that ϕ1 (t, 0) and ϕ2 (t, 0) are uniformly bounded. Under this assumption it is obvious that if an initial condition ψ belongs to the Schwartz space S then the quantity U (t, 0)ψ, P (p, x)U (t, 0)ψ is uniformly bounded in time for any polynomial P (p, x) in the non-commuting variables p = −i∂x and x. In particular, for such an initial condition, the mean value of the energy is bounded uniformly. As stated in [3, Proposition 4.1], it follows that all trajectories {U (t, 0)ψ; t ∈ R}, for any initial condition ψ ∈ H , are precompact subsets in H . This in turn implies that the spectrum of the monodromy (Floquet) operator U (T, 0) is pure point (see [3, Theorem 2.3]). The fact that the mean value of the energy is bounded for all initial conditions from a total set has also the following consequence (see [3, Lemma 3.3]): ∀ψ ∈ H ,

lim sup F (Hω > R)U (t, 0)ψ = 0

R→∞ t∈R

where the symbol F stands for the projection-valued measure from the spectral decomposition of the operator indicated in the argument and taken for a subset of the real line which is indicated in the argument as well. In the second part of [3] the particular case f (t) = sin(2πt/T ) is analyzed in detail. In that case a simple computation shows that the functions ϕ1 (t, 0) and ϕ2 (t, 0) are bounded if and only if 2π/T = ω.

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Let us now examine how Proposition 10 can be applied to this example. We consider the non-resonant case T ∈ /

2π N. ω

Let us write T =

2π N + ∆, ω

with

N ∈ Z+ ,

2π ∆ ∈ 0, . ω

As a ﬁrst step one has to make a choice of a self-adjoint operator HF so that U (T, 0) = exp(−iTH F ). According to Proposition A.3, the monodromy operator corresponding to H(t) can be expressed in the form

µ(T, 0) p + iν(T, 0)x + iσ(T, 0) (37) U (T, 0) = (−1)N exp −i∆Hω + i ω where the functions µ(t, s) and ν(t, s) are given in (A.12) and σ(t, s) is given in (A.13). We shall seek HF in the form H F = Hω −

β γ α p− x+ ωT T T

for some α, β, γ ∈ R. Then one has

α exp(−iTH F ) = e−iγ exp − iTH ω + i p + iβx ω

2 2 α +β β α = exp −iγ + i x exp −i 2 p exp(−iTH ω ) exp i 2ω 2 T ωT ω T

α β x . × exp i 2 p exp − i ω T ωT Here we have used that eisx Hω e−isx = Hω − sp +

s2 , 2

eisp Hω e−isp = Hω + sω 2 x +

s2 ω 2 . 2

(38)

By the well-known spectral properties of Hω , exp(−iTH ω ) equals (−1)N exp(−i∆Hω ), and so one ﬁnally arrives at the expression

β∆ α2 + β 2 α∆ ∆ p + i x . + i (−1)N exp −iγ + i 1 − exp −i∆H ω 2ω 2 T T ωT T Equating this expression to the right-hand side of (37) one has to set

T T α2 + β 2 ∆ α= µ(T, 0), β = ν(T, 0), −γ + 1 − = σ(T, 0). ∆ ∆ 2ω 2 T T

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Thus our choice of HF reads µ(T, 0) ν(T, 0) σ(T, 0) µ(T, 0)2 + ν(T, 0)2 p− x− + πN . (39) ω∆ ∆ T ω 3 ∆2 T As a next step one has to compute the T -periodic family of unitary operators UF (t) = U (t, 0) exp(itH F ). With the aid of Lemma A.1 one can express µ(T, 0)2 + ν(T, 0)2 t σ(T, 0)t − iπN exp(−itH F ) = exp −iφ(t) + i T ω 3 ∆2 T

ξ(t) × exp i p exp(iη(t)x) exp(−itH ω ) ω HF = H ω −

where

2 ωt ωt ωt sin cos µ(T, 0) − sin ν(T, 0) , ω∆ 2 2 2

ωt ωt ωt 2 sin sin µ(T, 0) + cos ν(T, 0) , η(t) = ω∆ 2 2 2 ξ(t) =

and φ(t) = −

1 4ω 3 ∆2

(2ωt − 4 sin(ωt) + sin(2ωt))µ(T, 0)2

+ (2 − 4 cos(ωt) + 2 cos(2ωt))µ(T, 0)ν(T, 0) + (2ωt − sin(2ωt))ν(T, 0)2 . Using relations (A.12) for µ(T, 0) and ν(T, 0) this can be rewritten as

ωt

T sin t+T 2

−u f (u)du, sin ω ξ(t) = ωT 2 0 sin 2

ωt

T sin t+T 2

η(t) = − −u f (u)du, cos ω ωT 2 0 sin 2

(40)

and it also holds true that (compare to (A.7)) φ(t) =

1 ωt − sin(ωt) 2 2 ξ(t)η(t) − 2 ξ(t) + η(t) . 2ω ωt 8ω sin 2

Expressing the propagator U (t, 0) according to formula (A.8) due to Enss and Veseli`c one ﬁnally arrives at the sought equality UF (t) = eiΦ(t) eiF2 (t)x ei(F1 (t)/ω)p where F1 (t) = ϕ2 (t, 0) − ξ(t),

F2 (t) = −ϕ1 (t, 0) − η(t),

(41)

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and

µ(T, 0)2 + ν(T, 0)2 t ϕ2 (t, 0)η(t) σ(T, 0)t + πN − Φ(t) = −ψ(t, 0) + φ(t) − T ω 3 ∆2 T ω

(ψ(t, s) is given in (A.10)). After some elementary manipulations this can be rewritten as

T 1 T

cos ω u − F1 (t) = f (t − u) − f (u) du, ωT 2 0 2 sin 2

T 1 T

sin ω u − F2 (t) = f (t − u) + f (u) du, ωT 2 0 2 sin 2 and Φ(t) = −

1 2

t

(ϕ1 (v, 0)2 − ϕ2 (v, 0)2 )dv + 0

t 2T

T

(ϕ1 (v, 0)2 − ϕ2 (v, 0)2 )dv 0

1 t ϕ2 (t, 0)η(t) + ξ(t)η(t) − ϕ1 (T, 0)ϕ2 (T, 0) − 2ω 2ωT ω (ωT − sin(ωT ))t ωt − sin(ωt) 2 2 ϕ1 (T, 0)2 + ϕ2 (T, 0)2 . − 2 ξ(t) + η(t) +

2 ωt ωT 8ω sin 8ω sin T 2 2 In the last equality one has to substitute for ϕ1 (t, 0) and ϕ2 (t, 0) from (A.9), and for ξ(t) and η(t) from (40). It is of importance to observe that the functions F1 (t), F2 (t) and Φ(t) entering formula (41) are continuously diﬀerentiable. In addition, they are necessarily T -periodic. Furthermore, the operators x and p are inﬁnitesimally small with respect to Hω . This is a well-known fact which is also brieﬂy recalled at the beginning of the Appendix. From equality (39), one can see that Dom HF = Dom Hω . Moreover, from the commutation relations (38) it follows that the unitary groups {exp(isx); s ∈ R} and {exp(isp); s ∈ R} preserve the domain Dom Hω . Hence one can diﬀerentiate UF (t) given in (41) on any vector ψ ∈ Dom HF . Computing SF (t) according to (16) one ﬁnds that SF (t) = −

F1 (t) F1 (t)F2 (t) p − F2 (t)x + − Φ (t). ω ω

Consequently, SF (t) is inﬁnitesimally small with respect to HF for any t. Thus all assumptions of Proposition 10 are fulﬁlled and one concludes that H(t)U (t, 0)ψ is bounded in time for any ψ ∈ Dom(H(0)) = Dom(Hω ). From the explicit form of H(t) and from the inﬁnitesimal smallness of x with respect to Hω it follows that the quantity Hω U (t, 0)ψ is bounded in time as well. Let us recall once more the consequences of this observation. Firstly, as stressed in

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[4, Proposition 4], since F (Hω < R) is a ﬁnite rank projector for any R > 0 it is true that all trajectories {U (t, 0)ψ; t ∈ R} are precompact. Secondly, in virtue of Theorem 2.3 in [3], the monodromy operator U (T, 0) has a pure point spectrum. Finally, let us shortly discuss the resonant case T = (2π/ω)N , N ∈ N. Using again formula (A.8) we have

ϕ2 (T, 0) p . U (T, 0) = (−1)N e−iψ(T,0) exp(−iϕ1 (T, 0)x) exp i ω

(42)

Notice that the unitary operator eiαx eiβp , with α, β ∈ R, is either the identity if α = β = 0 or it has a purely absolutely continuous spectrum. For example, if β = 0 then we have the commutation relation

α 2 α 2 i x exp iβp − αβ exp i x . eiαx eiβp = exp −i 2β 2 2β Hence the spectrum of eiαx eiβp coincides with that of e−iαβ/2 eiβp . In the case α = 0 one can argue in a similar way. Thus when applying this observation to (42) we have to distinguish the case ϕ1 (T, 0) = ϕ2 (T, 0) = 0. Recalling deﬁning relations (A.9) we denote by 1 fk = T

T

exp(−i 0

2π kt)f (t)dt, T

k ∈ Z,

the Fourier coeﬃcients of f (t). We conclude that if f−N = fN = 0 then the monodromy operator U (T, 0), with T = 2πN/ω, is a multiple of the identity. If |f−N | + |fN | > 0 then U (T, 0) has a purely absolutely continuous spectrum. This in turn implies that, in the latter case, the quantity H(t)U (t, 0)ψ cannot happen to be bounded in time for all ψ ∈ Dom Hω . 7. An Application of the Quantum KAM Method The quantum KAM method was originally proposed by Bellissard [23] and it has been later reconsidered and in some respects improved several times, see, for example, [13, 14, 24–26]. When discussing an application of the quantum KAM method to our problem we shall stick to the presentation given in [14] but the notation will be partially modiﬁed. A particularity of the method is that the frequency ω = 2π/T should be considered as a parameter. Usually the method is used to show that for a large subset of so called non-resonant frequencies the spectrum of the Floquet Hamiltonian is pure point. Here we would like to point out, following some ideas from [10], that the method provides a more detailed information which can be used to reveal the structure of the propagator. Let us ﬁrst recall the main theorem from [14]. Let H0 be a self-adjoint operator in H with a discrete spectrum, Spec(H0 ) = {hm }∞ m=1 , and such that the multiplicities

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Mm = dim Ker(H0 − hm ) are ﬁnite. Suppose also that ∆0 = inf |hm − hn | > 0. m=n

Furthermore, let V (t) be a 2π-periodic uniformly bounded operator-valued function deﬁned on R and with values in B(H ). Set 2π 1 e−ikt Qn V (t)Qm dt Vknm = 2π 0 where Qn is the orthogonal projector onto Ker(H0 −hn ). As already mentioned, the frequency ω = 2π/T , T > 0, is regarded as a parameter. Set K = L2 ([ 0, T ], H , dt) and let V ∈ B(K ) be the operator acting via multiplication by V (ωt), (Vf )(t) = V (ωt)f (t). Let K0 be the closure of −i∂t ⊗ 1 + 1 ⊗ H0 . Theorem 27. Fix J > 0 and set Ω0 = [ 89 J, 98 J ]. Assume that there exists σ > 0 such that Mm Mn < ∞. (43) (hm − hn )σ m,n∈N hm −hn >J/2

Then for every r > σ + 12 there exist positive constants (depending on σ, r, ∆0 and J but independent of V ), and δ , with the property: if V := sup (1 + |k|)r Vknm < (44) n∈N

m∈N k∈Z

then there exists a measurable subset Ω∞ ⊂ Ω0 such that |Ω∞ | ≥ |Ω0 | − δ V (here |Ω∗ | stands for the Lebesgue measure of Ω∗ ) and the operator K0 + V has a pure point spectrum for all ω ∈ Ω∞ . The proof of Theorem 27 is somewhat lengthy and tedious because one has to eliminate the resonant frequencies. The basic idea is, however, rather simple and is based on an iterative procedure as described in the following proposition. It is formulated at the level of Banach spaces but afterwards we shall again work with Hilbert spaces. Let Φ(x) be the analytic function deﬁned by

ex − 1 1 ex − . Φ(x) = x x Proposition 28. Assume that K is a Banach space, K0 is a closed operator in K , V ∈ B(K ) and D ∈ B(B(K )). Assume further that V = lim Vs in B(K ) where {Vs }∞ s=0 is a sequence of bounded operators in K . If there exist sequences

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∞ {As }∞ s=0 and {Gs }s=0 , As , Gs ∈ B(K ), fulﬁlling the following recurrence relations for all s ∈ Z+ :

G0 = V0 , Gs+1 = Gs + exp(adAs ) · · · exp(adA0 )(Vs+1 − Vs ) + adAs Φ(adAs )(1 − D)(Gs − Gs−1 ),

(45)

As Dom(K0 ) ⊂ Dom(K0 ), [A0 , K0 + D(G0 )] = −(1 − D)(G0 ), [As+1 , K0 + D(Gs+1 )] = −(1 − D)(Gs+1 − Gs ), (46)

∞ and such that s=0 As < ∞ and the limit lim Gs = G∞ exists in B(K ) then there exists W ∈ B(K ) such that W −1 ∈ B(K ) and W (K0 + V )W −1 = K0 + D(G∞ ).

(47)

Here, as usual, ad means the adjoint action, adA X = [A, X] and exp(adA )X = eA Xe−A. For s = 0 in (45) we set G−1 = 0. The proof of Proposition 28 is immediate. If we set Ws = eAs−1 · · · eA0 ,

Ws −1 = e−A0 · · · e−As−1 ,

then the recurrence relations (45), (46) exactly mean that ∀s ∈ Z+ ,

Ws (K0 + Vs )Ws −1 = K0 + D(Gs ) + (1 − D)(Gs − Gs−1 ).

Now it suﬃces to send s to inﬁnity. In the applications of Proposition 28, and this is also the case for Theorem 27, K is a separable Hilbert space, K0 = K0∗ , V = V ∗ , the spectrum of K0 is pure point and D(X) is the diagonal part of a bounded operator X with respect to the spectral ∗ = G∞ , D(G∞ )∗ = D(G∞ ) and W ∗ = W −1 . The decomposition of K0 . Then G∞ operator K0 +D(G∞ ) has obviously a pure point spectrum and relation (47) implies that the same is true for K0 + V . Let us note that technically the basic problem of the entire method is the commutator equation (46) whose solution is complicated by the fact that, generically, the eigenvalues of K0 are dense in R. This leads to the famous problem of small denominators in this context. There is another feature concerning the application of the recursive procedure (45) and (46) in the proof of Theorem 27. Let M ∈ B(K ) be the multiplication operator deﬁned by the relation ∀f ∈ K ,

(Mf )(t) = eiωt f (t).

(48)

Since V ∈ B(K ) is a multiplication operator it commutes with M . Also the sequence {Vs } is chosen in such a way that M commutes with all Vs . Furthermore, the eigenvalues of K0 are kω + hm , k ∈ Z and m ∈ N, and so they are

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linear in k. Using these facts it is readily seen from the recursive relations that M commutes with both As and Gs for all s. Then necessarily M commutes with G∞ and W as well. This implies that there exists a bounded Hermitian operator G on H such that G Dom(H0 ) ⊂ Dom(H0 ), [H0 , G] = 0 and (D(G∞ )f )(t) = Gf (t),

∀f ∈ K ,

a.a. t ∈ R,

and there exists a T -periodic operator-valued function t → W (t) with values in unitary operators on H such that equality (47) is satisﬁed with (Wf )(t) = W (t)f (t),

∀f ∈ K ,

a.a. t ∈ R.

Moreover, an information about the regularity of W is also available. More precisely, one knows that sup Wknm < ∞ (49) n∈N m∈N

k∈Z

where again Wknm

1 = T

T

e−ikωt Qn W (t)Qm dt .

0

Particularly, the operator-valued function W (t) is continuous in the operator norm. Equality (47) can be rewritten in terms of propagators. It exactly means that ∀t, s ∈ R,

U (t, s) = W (t)∗ e−i(t−s)(H0 +G) W (s).

(50)

By a closer look at the proof of Theorem 27, one ﬁnds that the result can be partially improved. In the course of the proof one constructs a directed sequence of Banach spaces {Xs }, ⊕ ∞ B(Hm , Hn ) , Ωs × Z × N × N, Xs ⊂ L n∈N m∈N

with the norms Xs =

sup

sup

ω,ω ∈Ωs n∈N m∈N k∈Z ω=ω

˜ knm (ω, ω ))e|k|/Es (Xknm (ω) + ϕs ∂X

(51)

where X = {Xknm (ω)} ∈ Xs , i.e. Xknm (ω) ∈ B(Hm , Hn ) for all ω ∈ Ωs and (k, n, m) ∈ Z × N × N. Here Hm := Ker(H0 − hm ) = Ran Qm , {Ωs } is a decreasing sequence of subsets of the interval Ω0 , {ϕs } and {Es } are respectively decreasing and strictly increasing sequences of positive numbers such that lim ϕs = 0, 1 ≤ Es and lim Es = +∞. The symbol ∂˜ designates the discrete derivative in ω, X(ω) − X(ω ) ˜ . ∂X(ω, ω) = ω − ω

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For ω ∈ Ω∞ = Ωs ﬁxed one applies the limit procedure s → ∞ and arrives at equality (47) with the objects G∞ and W belonging to the Banach space ⊕ ∞ Z × N × N, X∞ ⊂ L B(Hm , Hn ) n∈N m∈N

where the norm is deﬁned by X∞ = sup

n∈N m∈N k∈Z

Xknm (ω) .

This is also how one obtains the information about the regularity of W expressed in (49). The announced improvement consists in modifying the norms (51) by an additional weight (1 + |k|)ν where ν should be chosen in the range 0≤ν
1 . 2

(52)

Recall that r determines the regularity of V in (44), σ comes from the “gap condition” (43) and one requires that r > σ + 12 . The modiﬁed norm reads Xs = sup sup (1 + |k|)ν (Xknm (ω) ω,ω ∈Ωs n∈N m∈N k∈Z ω=ω

˜ knm (ω, ω ))e|k|/Es , + ϕs ∂X and the limit procedure results in a norm in X∞ , X∞ = sup (1 + |k|)ν Xknm (ω) . n∈N m∈N k∈Z

(53)

(54)

Let us note that restriction (52) comes from the lower estimate of Lebesgue measure of the set Ω∞ (see [14, relation (77)] and the derivation preceding it where one has to replace r by r − ν if using the modiﬁed norm (53)). After this modiﬁcation, Theorem 27 is valid exactly in the same formulation as before, its proof requires no additional changes, only the constants and δ should be modiﬁed correspondingly. The interest of the modiﬁcation is that we get a better information about the regularity of W . Namely, for ω ∈ Ω∞ (the set of non-resonant frequencies) W is regular in the sense that W ∞ < ∞ with the norm given by (54). In particular, if r > σ + 32 then one can choose ν ≥ 1. In that case the property W ∞ < ∞ implies that W (t) belongs to the class C 1 in the operator norm and supt ∂t W (t) < ∞. This discussion shows that Theorem 27 can be reformulated in the following way. Theorem 29. Under the same assumptions as in Theorem 27 suppose that r > σ + 32 . Then there exist positive constants (independent of V ), and δ , with the property: if V < then there exists a measurable subset Ω∞ ⊂ Ω0 such that |Ω∞ | ≥ |Ω0 | − δ V , and for every ω ∈ Ω∞ there exist a bounded Hermitian operator G

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commuting with H0 and a T -periodic function W (t) with values in unitary operators and belonging to the class C 1 in the operator norm such that the propagator obeys equality (50). From relation (50) it follows that the propagator admits a Floquet decomposition (15) with HF = W (0)∗ (H0 + G)W (0),

UF (t) = W (t)∗ W (0).

Moreover, formula (16) implies that SF (t) = −iW (0)∗ (∂t W (t))W (t)∗ W (0). In particular, if W (t) is known to be C 1 in the operator norm then the Floquet decomposition is continuously diﬀerentiable in the strong sense and, consequently, the assumptions both of Propositions 10 and 12 are satisﬁed. These arguments prove the following theorem. Theorem 30. Under the same assumptions as in Theorem 27 suppose that r > σ + 32 . Then there exist positive constants (independent of V ), and δ , with the property: if V < (with V deﬁned in (44)) then there exists a measurable subset Ω∞ ⊂ Ω0 such that |Ω∞ | ≥ |Ω0 | − δ V and for every ω ∈ Ω∞ the energy of the quantum system described by the time-dependent Hamiltonian H0 + V (ωt) is bounded uniformly in time. More precisely, ∀ψ ∈ Dom(H(0)),

sup (H0 + V (ωt))U (t, 0)ψ < ∞. t∈R

Moreover, there exists a constant c ≥ 0 such that for any couple of intervals ∆1 , ∆2 ⊂ R fulﬁlling dist(∆1 , ∆2 ) > 0 it holds true that sup P (t, ∆1 )U (t, s)P (s, ∆2 ) ≤

s,t∈R

c dist(∆1 , ∆2 )

where P (t, ·) is the spectral measure of H(t). In particular, if En (t) and Em (s) are two distinct eigenvalues of H(t) and H(s), respectively, and Pn (t) and Pm (s) are the corresponding orthogonal projectors then Pn (t)U (t, s)Pm (s) ≤

c . |En (t) − Em (s)|

Acknowledgments ˇ wishes to acknowledge gratefully the support from the grant No. 201/05/0857 P. S. of Grant Agency of the Czech Republic.

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Appendix. The Propagator for the Time-Dependent Harmonic Oscillator Let Hω = (1/2)(p 2 + ω 2 x2 ), with p = −i∂x , be the Hamiltonian of the harmonic oscillator in H = L2 (R, dx). It is a known fact that the operators x and p are relatively bounded with respect to Hω with the relative bound zero. One can see it also directly from the following inequality which is easy to verify on the Schwartz space S : 2ω 2 x2 ≤ ε−2 + ε2 ω 4 x4 + ε2 (p4 + 2ω 2 px2 p) = ε−2 + 2ε2 ω 2 + 4ε2 Hω2 . Hence it is true that

xψ ≤ ε2 + 2

1 2ω 2 ε2

ψ2 +

2ε2 2 Hω ψ ω2

for all ψ ∈ S and consequently for all ψ ∈ Dom Hω . A bound for the operator p can be derived analogously. It follows that for any α, β ∈ R, the domain Dom(Hω + αx + βp) coincides with Dom Hω . For ϕ, ψ ∈ Dom(Hω ) set x(t) = e−itHω ϕ, xe−itHω ψ,

p(t) = e−itHω ϕ, pe−itHω ψ.

As a standard exercise one derives, by diﬀerentiating and using the canonical commutation relation, that the quantities x(t) and p(t) obey the classical evolution ˙ ˙ equations, i.e. x(t) = p(t), p(t) = −ω 2 x(t). It follows that for all ψ ∈ Dom(Hω ), eitHω xe−itHω ψ = cos(ωt)xψ +

1 sin(ωt)pψ, ω

(A.1)

eitHω pe−itHω ψ = −ω sin(ωt)xψ + cos(ωt)pψ. Lemma A.1. For µ, ν, t ∈ R it holds true

µ ξ exp −itHω + i p + iνx = e−iφ exp i p exp(iηx) exp(−itH ω ) ω ω where

ωt

2 sin ωt ωt 2 ξ= cos µ − sin ν , ωt 2 2

ωt

2 sin ωt ωt 2 η= sin µ + cos ν , ωt 2 2

(A.2)

(A.3)

and φ=−

1 ((2ωt − 4 sin(ωt) + sin(2ωt))µ2 4ω 3 t2

+ (2 − 4 cos(ωt) + 2 cos(2ωt))µν + (2ωt − sin(2ωt))ν 2 ).

(A.4)

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Proof. Set

ξ W1 = exp i p exp(iηx) exp(−itH ω ) , ω

µ W2 = exp − itHω + i p + iνx . ω

Clearly, both W1 and W2 leave the domain of Hω invariant. Using (A.1) one ﬁnds that for all ψ ∈ Dom(Hω ), ξ 1 sin(ωt)pψ − ψ, ω ω W1−1 pW1 ψ = −ω sin(ωt)xψ + cos(ωt)pψ + ηψ.

W1−1 xW1 ψ = cos(ωt)xψ +

(A.5)

On the other hand, −itH ω + i

t 2 µ2 + ν 2 µ p + iνx = −i p˜ + ω 2 x ˜2 + i ω 2 2 ω2t

where x ˜=x−

ν , ω2t

p˜ = p −

µ . ωt

Since p˜ and x˜ also obey the canonical commutation relation we have, analogously to (A.1), ˜ W2 ψ = cos(ωt)˜ xψ + W2−1 x

1 sin(ωt)˜ pψ, ω

W2−1 p˜ W2 ψ = −ω sin(ωt)˜ xψ + cos(ωt)˜ pψ. for all ψ ∈ Dom(Hω ). This can be rewritten as 1 1 sin(ωt)pψ + 2 (ν − ν cos(ωt) − µ sin(ωt))ψ, ω ω t 1 W2−1 pW2 ψ = −ω sin(ωt)xψ + cos(ωt)pψ + (µ + ν sin(ωt) − µ cos(ωt))ψ. ωt (A.6)

W2−1 xW2 ψ = cos(ωt)xψ +

Comparing (A.5) to (A.6) one ﬁnds that for all ψ ∈ Dom(Hω ) it holds true W1−1 xW1 ψ = W2−1 xW2 ψ and W1−1 pW1 ψ = W2−1 pW2 ψ provided ξ=−

ν µ 1 − cos(ωt) + sin(ωt), ωt ωt

η=

ν µ sin(ωt) + 1 − cos(ωt) ωt ωt

(which is nothing but (A.3)). Hence W = W2 W1−1 fulﬁlls W −1 xW ψ = xψ and W −1 pW ψ = pψ. Since Dom(Hω ) is a core both for x and p this implies that W −1 xW = x and W −1 pW = p. If follows that W is a multiple of the unity, i.e. W2 = e−iφ W1 for some φ ∈ R. It remains to determine φ. To this end it suﬃces to take the mean value of the corresponding operators at the ground state of Hω which is ψ0 (x) = exp(−ωx2 /2)

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(unnormalized). Writing

2

µ µ + ν2 ν W2 = exp i x e−itHω exp −i 2 p exp i 2 ω2t ω t ωt

ν µ x exp i 2 p × exp −i ωt ω t the equality W2 = e−iφ W1 becomes 2

ν µ + ν2 µ −itHω exp i x exp i 2 p exp −i e 2 ω2t ωt ω t

ν ξ + = exp −iφ + i η 2 ω t ω

ξ ν µ + × exp i η − x exp i p e−itHω . ωt ω ω2t The mean value at ψ0 of both sides of the last equality then yields

ν µ2 + ν 2 ξ µ ν exp i φ + − + x ψ (x), exp −i x + η ψ 0 0 2 ω2t ω2t ω ωt ω2t

ν µ ξ = ψ0 (x), exp i η − x ψ0 x + + 2 . ωt ω ω t A straightforward computation then leads to the value (A.4). Conversely, one can express µ and ν in terms of ξ, η and t provided t is suﬃciently small. In other words, one can read equality (A.2) from the right to the left. The restriction on smallness of t should not be considered as surprising since the reversed equality is in fact an application of the Baker–Campbell–Haussdorf formula which is known to be guaranteed only locally. Lemma A.2. For ξ, η, t ∈ R, |t| < 2π/ω, it holds true

µ ξ iφ exp i p exp(iηx) exp(−itH ω ) = e exp −itH ω + i p + iνx ω ω where ωt µ= 2

ωt cot ξ+η , 2

ωt ν= 2

ωt −ξ + cot η , 2

and φ=

ωt − sin(ωt) 2 1 2 ξη − 2 ξ + η . 2ω ωt 8ω sin 2

(A.7)

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Set H(t) = Hω + f (t)x where the function f (t) is supposed to be continuous and T periodic. In [3] a formula has been derived for the propagator corresponding to the Hamiltonian H(t):

ϕ2 (t, s) p exp −i(t − s)Hω − iψ(t, s) (A.8) U (t, s) = exp −i ϕ1 (t, s)x exp i ω where

ϕ1 (t, s) =

and ψ(t, s) =

1 2

s

t

cos(ω(t − u))f (u)du,

s

ϕ2 (t, s) =

t

(A.9)

t

s

sin(ω(t − u))f (u)du,

ϕ1 (v, s)2 − ϕ2 (v, s)2 dv

(A.10)

(be aware of a sign error in the deﬁnition of ψ(t, s) in [3]). Our goal is to transform formula (A.8) due to Enss and Veseli`c into another one expressing the propagator as a single exponential function of an operator. Proposition A.3. For t, s ∈ R, (t − s) ∈ / (2π/ω)Z, set 2π ω(t − s) 2π ω(t − s) ∈ Z, ∆ = ∈ 0, N= 2π ω 2π ω (where [x] and {x} are respectively the integer part and the fractional part of x). Then it holds true

µ(t, s) N p + iν(t, s)x + iσ(t, s) (A.11) U (t, s) = (−1) exp −i∆Hω + i ω where ω∆

µ(t, s) = ω(t − s) 2 sin 2 ν(t, s) = −

ω∆

ω(t − s) 2 sin 2

and 1 σ(t, s) = − 2

t+s −u f (u)du, sin ω 2

t

s

s

t

t+s −u f (u)du, cos ω 2

t s

(ϕ1 (v, s)2 − ϕ2 (v, s)2 )dv +

(A.12)

1 ϕ1 (t, s)ϕ2 (t, s) 2ω

ω∆ − sin(ω(t − s)) 2 2 −

2 ϕ1 (t, s) + ϕ2 (t, s) . ω(t − s) 8ω sin 2

(A.13)

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Proof. We have t − s = (2π/ω)N + ∆. Since the spectrum of Hω equals (ω/2) + ωZ+ one has

2π exp −i N Hω = (−1)N . ω Combining (A.8) with Lemma A.2 we get

ϕ2 (t, s) U (t, s) = (−1)N exp −i ϕ1 (t, s)x exp i p exp −i∆Hω − iψ(t, s) ω

µ(t, s) N iφ(t,s) p + iν(t, s)x = (−1) e exp −i∆Hω + i ω where

ω∆ cot ϕ2 (t, s) − ϕ1 (t, s) , 2

ω∆ ω∆ ν(t, s) = ϕ2 (t, s) + cot ϕ1 (t, s) , 2 2

µ(t, s) =

ω∆ 2

and φ(t, s) =

ω∆ − sin(ω∆) 1 2 2 ϕ1 (t, s)ϕ2 (t, s) −

2 ϕ1 (t, s) + ϕ2 (t, s) − ψ(t, s). 2ω ω∆ 8ω sin 2

After some simple manipulations one arrives at the desired formula (A.11). References [1] J. S. Howland, Scattering theory for Hamiltonians periodic in time, Indiana J. Math. 28 (1979) 471–494. [2] K. Yajima, Scattering theory for Schr¨ odinger equations with potential periodic in time, J. Math. Soc. Japan 29 (1977) 729–743. [3] V. Enss and K. Veseli´c, Bound states and propagating states for time-dependent Hamiltonians, Ann. Inst. Henri Poincar´e A Phys. Theor. 39 (1983) 159–191. [4] C. R. de Oliveira, Some remarks concerning stability for nonstationary quantum systems, J. Stat. Phys. 78 (1995) 1055–1066. [5] C. R. de Oliveira and M. S. Simsen, A Floquet operator with pure point spectrum and energy instability, Ann. Henri Poincar´e 8 (2007) 1255–1277. [6] G. Nenciu, Adiabatic theory: Stability of systems with increasing gaps, Ann. Inst. Henri Poincar´e 67 (1997) 411–424. [7] A. Joye, Upper bounds for the energy expectation in time-dependent quantum mechanics, J. Stat. Phys. 85 (1996) 575–606. [8] J. M. Barbaroux and A. Joye, Expectation values of observables in time-dependent quantum systems, J. Stat. Phys. 90 (1998) 1225–1249. ˇˇtov´ıˇcek, On the energy growth of some periodically driven [9] P. Duclos, O. Lev and P. S quantum systems with shrinking gaps in the spectrum, J. Stat. Phys. 130 (2008) 169–193. [10] J. Asch, P. Duclos and P. Exner, Stability of driven systems with growing gaps, quantum rings, and Wannier ladders, J. Stat. Phys. 92 (1998) 1053–1070.

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[11] W.-M. Wang, Pure point spectrum of the Floquet Hamiltonian for the quantum harmonic oscillator under time quasi-periodic perturbations, Commun. Math. Phys. 227 (2008) 459–496. [12] S. De Bi`evre and G. Forni, Transport properties of kicked and quasiperiodic Hamiltonians, J. Stat. Phys. 90 (1998) 1201–1223. ˇˇtov´ıˇcek, Floquet Hamiltonian with pure point spectrum, Commun. [13] P. Duclos and P. S Math. Phys. 177 (1996) 327–247. ˇˇtov´ıˇcek and M. Vittot, Weakly regular Floquet Hamiltonians [14] P. Duclos, O. Lev, P. S with pure point spectrum, Rev. Math. Phys. 14 (2002) 531–568. [15] M. Reed and B. Simon, Methods of Modern Mathematical Physics II (Academic Press, New York, 1975). [16] B. Simon, Quantum Mechanics for Hamiltonians Deﬁned as Quadratic Forms (Princeton University Press, Princeton, 1971). [17] S. G. Krein, Linear Diﬀerential Equations in Banach Space (Amer. Math. Soc., Providence, R. I., 1971). [18] R. Bhatia and P. Rosenthal, How and why to solve the operator equation AX −XB = Y , Bull. London Math. Soc. 29 (1997) 1–21. [19] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGrawHill, New York, 1965). [20] L. S. Schulman, Techniques and Applications of Path Integration (John Wiley, New York, 1981). [21] L. Bunimovich, H. R. Jauslin, J. L. Lebowitz, A. Pellegrinotti and P. Nielaba, Diffusive energy growth in classical and quantum driven oscillators, J. Stat. Phys. 62 (1991) 793–817. [22] G. A. Hagedorn, M. Loss and J. Slawny, Non-stochasticity of time-dependent quadratic Hamiltonians and the spectra of canonical transformations, J. Phys. A 19 (1986) 521–531. [23] J. Bellissard, Stability and instability in quantum mechanics, in Trends and Developments in the Eighties, ed. Albeverio Blanchard (World Scientiﬁc, Singapore, 1985), pp. 1–106. [24] M. Combescure, The quantum stability problem for time-periodic perturbations of the harmonic oscillator, Ann. Inst. Henri Poincar´e 47 (1987) 62–82; Erratum, ibid. 47 (1987) 451–454. [25] P. M. Bleher, H. R. Jauslin and J. L. Lebowitz, Floquet spectrum for two-level systems in quasi-periodic time-dependent ﬁelds, J. Stat. Phys. 68 (1992) 271–310. [26] D. Bambusi and S. Graﬃ, Time quasi-periodic unbounded perturbations of Schr¨ odinger operators and KAM methods, Commun. Math. Phys. 219 (2001) 465–480.

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Reviews in Mathematical Physics Vol. 20, No. 7 (2008) 765–786 c World Scientific Publishing Company

ON A MINIMIZING PROPERTY OF THE HOPF SOLITON IN THE FADDEEV–SKYRME MODEL

TAKESHI ISOBE Department of Mathematics, Graduate School of Science and Engineering, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan [email protected] Received 5 April 2007 We study local minimizing solutions for √ the Faddeev–Skyrme problem on the 3-sphere with radius R. We show that for R < 2 the Hopf soliton is a local minimizing solution and all the other local minimizing solutions near the Hopf soliton are obtained as the composition of isometries of S 3 with the Hopf soliton. Keywords: Faddeev–Skyrme model; Hopf soliton; stability; local minimizing soliton; uniqueness. Mathematics Subject Classification 2000: 53Z05, 58E15, 81V35

1. Introduction In [5], Faddeev proposed a (3 + 1)-dimensional ﬁeld theory model which possibly describes stable closed strings as topological solitons. This model, so-called the Faddeev–Skyrme model, is a O(3)-sigma model modiﬁed by adding a fourth order derivative terms in the standard sigma model Lagrangian. It involves a map u : M 3 → S 2 from a 3-dimensional manifold M to the 2-sphere S 2 and can be obtained from the Skyrme model [15] by restricting the ﬁeld values to an equatorial S 2 ⊂ S 3 . More precisely, the Faddeev–Skyrme functional FS is deﬁned for a ﬁeld u : M → S 2 as 1 |du|2 dvolM + |du ∧ du|2 dvolM . (1.1) FS(u) = 4 M M In (1.1), du(x) : Tx M → Tu(x) S 2 is the diﬀerential of u and it is regarded as a section of the vector bundle T ∗ M ⊗ u∗ T S 2 → M . Its norm |du(x)| is deﬁned from the metrics on M and S 2 . More precisely, if e1 , e2 , e3 is an orthonormal frame of T M at x and f1 , f2 an orthonormal frame of T S 2 at u(x), then with respect to these 3 2 α i 1 2 3 frames du(x) is written as du(x) = i=1 α=1 ∂u ∂xi θ ⊗ fα , where (x , x , x ) and 1 2 2 (u , u ) are local coordinates of M at x and S at u(x) with respect to the frames 765

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ei and fα , respectively and θi is the dual co-frame of ei . Then |du(x)| is deﬁned by 3 2 α 2 . Since Tu(x) S 2 ⊂ R3 , we have du(x) ∈ Tx∗ M ⊗R3 and |du(x)|2 = i=1 α=1 ∂u i 2 ∗ ∂x 3 α ∂uβ i j Tx M ⊗R is deﬁned as du(x)∧du(x) = ∂u du(x)∧du(x) ∈ ∂xi ∂xj θ ∧ θ ⊗(fα ×fβ ), where fα × fβ is the vector product of fα and fβ in R3 . Its norm |du(x) ∧ du(x)| is deﬁned form the metric on 2 Tx∗ M ⊗ R3 . In this model, one is interested in the value Iα = inf{FS(u) : u ∈ α} and maps u which attain Iα for each homotopy class α ∈ [M, S 2 ] ([M, S 2 ] denotes the set of homotopy classes of maps from M to S 2 ). Recently, this problem attracted a lot of attention, see [1, 11, 12, 17, 18]. For compact M , this problem is studied in [1] and satisfactory existence result is obtained. However, one of the important cases is a non-compact case M = R3 . In this case, ﬁnite action FS(u) < +∞ imposes a rapid decay at inﬁnity on u, i.e. u(x) → c for some c ∈ S 2 rapidly as |x| → ∞. So a ﬁnite action conﬁguration u is considered as a map R3 ∪ {∞} ∼ = S3 → S2. 3 2 Homotopy classes of maps from S to S are classiﬁed by the Hopf invariant Q which associates to each homotopy class α ∈ [S 3 , S 2 ] ∼ = π3 (S 2 ) an integer Q(α) ∈ Z 2 ∼ and gives an isomorphism π3 (S ) = Z. In this case, the results of [11,17] assert that there is a bound of the form C1 |Q(α)|3/4 ≤ Iα ≤ C2 |Q(α)|3/4 for some constants C1 , C2 > 0. Since FS is invariant under the R3 -action (a, u) → a · u := u(· + a) and R3 is non-compact, the variational problem Iα is non-compact and it is a diﬃcult problem to ﬁnd a minimizing solution of Iα . The existence result for this case is established in [11,12]. An interesting feature of this problem is that for large |Q(α)|, energy minimizing conﬁgurations have knot like structures, see [2, 3, 6]. However, a mathematically rigorous proof of such knot-like structures for solutions obtained in [11] is still lacking. To obtain further insight about Iα for the case M = R3 , Ward [18] considered 3 , the 3-sphere of radius R, which approximates R3 as R → ∞. the model on M = SR 3 → S2 One of the most important solution in this model is the Hopf soliton H : SR which has Hopf invariant Q(H) = 1. The Hopf soliton has constant energy density and does not indicate a particle-like behavior. Thus, it is natural to conjecture that for large value of R, the Hopf soliton is an unstable critical √ point of FS. 2. In the same In fact, Ward [18] showed that it is indeed unstable for R > √ paper [18], Ward also conjectured that for R < 2 it is stable and, in fact, it is conjectured that the Hopf soliton is an absolute minimum among maps of unit charge. The analogous problem for the Skyrme problem was studied in [9, 10, 13]. In this direction, it would be very interesting to see the structure of the minimizing solutions for general compact M . It would also be interesting to explicitly describe minimizing solutions on special Riemannian manifold such as S 3 , tori T 3 , etc. In this paper, we partially answer these √ problems, in particular, for the case of the 3-sphere. We prove that for 0 < R ≤ 2, the Hopf soliton is weakly stable, i.e. the Hessian d2 FS(H) of FS at the Hopf√soliton H is non-negative; d2 FS(H) ≥ 0. Moreover, we show that for 0 < R < 2, it is strongly stable in the sense that it locally minimizes the Faddeev–Skyrme functional in the conﬁguration space and

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identify all local minimizing solutions near the Hopf soliton, i.e. we show that near the Hopf soliton, all the other local minimizing solutions are given as the composition of isometries of S 3 with the Hopf soliton. More precisely, we shall prove the following: √ Theorem 1.1. For 0 < R ≤ 2, the Hopf soliton is a weak√stable critical point of the Faddeev–Skyrme functional FS. Moreover, for 0 < R < 2 the Hopf soliton is a local minimizing solution among fields of Hopf charge 1 in the sense that there √ 3 , S2) exists = (R) depending only on 0 < R < 2 such that for any u ∈ C 1 (SR with Q(u) = 1 and u − HC 1 := maxx∈SR3 {|u(x) − H(x)| + |du(x) − dH(x)|} < , there holds FS(H) ≤ FS(u). Moreover, the equality FS(u) = FS(H) holds for such 3 3 → SR such that u = H ◦ ϕ. a map u if and only if there exists an isometry ϕ : SR After the completion of the present work, the author of the present paper is aware of the paper by Speight and Svensson [16] where the ﬁrst part of the above theorem, i.e. the weak stability of the Hopf soliton has already been established by a diﬀerent method. Since our approach gives some more insight about a minimizing property of the Hopf soliton as stated in the theorem, it would be of independent interest. Also, we hope that the method in this paper makes a contribution towards the proof of the global minimizing property of the Hopf soliton as conjectured by Ward [18]. As a corollary of our result, we see that the moduli space of (locally) minimizing solution of the Faddeev–Skyrme problem is locally isomorphic to SO(4)/S 1 = 3 3 )/S 1 near the Hopf soliton, where S 1 ⊂ SO(4) acts on SR as Isometry(SR iθ iθ iθ 3 e · (z0 , z1 ) = (e z0 , e z1 ) (SR is regarded as the sphere of radius R in C2 : 3 SR = {(z0 , z1 ) : |z0 |2 + |z1 |2 = R2 }). In particular, the dimension of the component of the moduli space of minimizing solutions containing H is 5. In general, we expect that the component of the global moduli space of minimizing solutions containing the Hopf soliton is isomorphic to SO(4)/S 1 . We hope that our approach provides the ﬁrst step to a rigorous proof of this. Our approach is based on a simple observation that the fourth-order derivative ∗ |u ωS 2 |2 dvolM and term of the Lagrangian (1.1) is written as M the ﬁrst term of the Lagrangian M |du|2 dvolM is equal to or larger than 2 M |u∗ ωS 2 | dvolM , where ωS 2 is the canonical volume form of S 2 , see Sec. 2. For the Hopf soliton H, we ﬁrst study a functional we have |dH|2 = 2|H ∗ ωS 2 |. From these observations, 3 , E(α) = S 3 |α| dvolSR3 + 12 S 3 |α|2 dvolSR3 , and E deﬁned on closed 2-forms on SR R R study the minimizing property of H ∗ ωS 2 among closed 2-forms satisfying a certain constraint, see Sec. 2. A similar observation is used before by Rivi`ere [14] for the proof of the weak stability of the Hopf map for a related but diﬀerent conformally invariant variational problem associated to the functional S 3 |du|3 dvolS 3 deﬁned on C 1 (S 3 , S 2 ). We will see that this somewhat indirect approach gives some useful information about minimizing solutions for the Faddeev–Skyrme problem.

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2. Some Fundamental Inequalities In this section, we give some simple but fundamental inequalities which play a crucial role in this paper. We ﬁrst rewrite the fourth-order derivative term of FS. For this, recall that the canonical volume 2-form on S 2 is a 2-form ωS 2 on S 2 deﬁned by ωS 2 = ι ∂ (dx1 ∧ dx2 ∧ dx3 )|S 2 ∂r

= x1 dx2 ∧ dx3 + x2 dx3 ∧ dx1 + x3 dx1 ∧ dx2 , ∂ ∂ ∂ 3 ∂ where ∂r = x1 ∂r +x2 ∂x 2 +x ∂x3 |S 2 and ι ∂ ωS 2 denotes the contraction of ω by ∂r For u = (u1 , u2 , u3 ) : M → S 2 a smooth map, we have

and

∂ ∂r .

u∗ ωS 2 = u1 du2 ∧ du3 + u2 du3 ∧ du1 + u3 du1 ∧ du2

(2.1)

 2  du ∧ du3   du ∧ du = 2 du3 ∧ du1  .

(2.2)

du1 ∧ du2 From (2.1) and (2.2), we have 1 u · (du ∧ du) 2

∂u ∂u = u· × dxj ∧ dxk . ∂xj ∂xk

u ∗ ωS 2 =

(2.3)

1≤j
If the local coordinate x = (x1 , x2 , x3 ) is chosen such that it is normal at x, then we have 2

∂u ∂u u · × (2.4) |u∗ ωS 2 |2 = ∂xj ∂xk 1≤j
2 ∗ T M is taken with respect to the Riemannian at x, where the norm of u∗ ωS 2 ∈ metric on M . ∂u ∂u ∂u ∂u ∂u ∂u u and u · u · j = u · ∂xk = 0, we have ∂xj × ∂xk = j × ∂xk ∂x ∂x Since u · ∂uj × ∂uk = ∂uj × ∂uk . Thus from (2.4), we obtain ∂x ∂x ∂x ∂x 2

∂u ∂u |u∗ ωS 2 |2 = × . (2.5) ∂xj ∂xk 1≤j
On the other hand, we have du ∧ du = 2

1≤j
∂u ∂u × k ∂xj ∂x

and |du ∧ du|2 = 4

1≤j
⊗ (dxj ∧ dxk )

2 ∂u ∂u × . ∂xj ∂xk

(2.6)

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Combining (2.5) and (2.6), we obtain |du ∧ du|2 = 4|u∗ ωS 2 |2 .

(2.7)

Therefore, the Faddeev–Skyrme functional FS is written as FS(u) = |du|2 dvolM + |u∗ ωS 2 |2 dvolM . M

M 3 SR

We, in particular, consider the case M = := {x ∈ R3 : |x|2 = (x1 )2 + (x2 )2 + 3 2 2 3 3 ) argument shows that this (x ) = R }. Simple re-scaling (S x → Rx ∈ SR is equivalent to considering the following functional FSR deﬁned on maps u : S3 → S2: 1 2 |du| dvolS 3 + 2 |u∗ ωS 2 |2 dvolS 3 . FSR (u) = R S3 S3 As for the ﬁrst term of the Lagrangian, we have Lemma 2.1. Let u : S 3 → S 2 be a C 1 -map. Then we have the following inequality |u∗ ωS 2 (x)| ≤

1 |du(x)|2 2

at any x ∈ S 3 . The equality holds at x ∈ S 3 if and only if there exists a 2dimensional plane Vx ∈ Gr2 (Tx S 3 ) such that du|Vx : Vx → Tu(x) S 2 is conformal. Proof. Let x ∈ S 3 be arbitrary. Choose an orthonormal basis e1 , e2 , e3 of Tx S 3 and an orthonormal basis f1 , f2 of Tu(x) S 2 . As before we denote (x1 , x2 , x3 ) and (u1 , u2 ) the corresponding normal coordinates at x ∈ S 3 and u(x) ∈ S 2 , respectively. Also let θ1 , θ2 , θ3 be the dual co-frame of e1 , e2 , e3 . Since du(x) : Tx S 3 → Tu(x) S 2 , we always have dim ker du(x) ≥ 1 and we may assume without loss of generality that e3 ∈ ker du(x). Under this assumption, we have u∗ ωS 2 = u∗ ωS 2 (e1 , e2 )θ1 ∧ θ2 + u∗ ωS 2 (e2 , e3 )θ2 ∧ θ3 + u∗ ωS 2 (e1 , e3 )θ1 ∧ θ3 = u∗ ωS 2 (e1 , e2 )θ1 ∧ θ2

(2.8)

at x. On the other hand, we have du(x)(e1 ) =

∂u1 ∂u2 f1 + 1 f2 , 1 ∂x ∂x

and obtain u∗ ωS 2 (e1 , e2 ) = ωS 2 =

du(x)(e2 ) =

∂u1 ∂u2 f1 + 2 f2 2 ∂x ∂x

∂u1 ∂u2 ∂u1 ∂u2 f1 + 1 f2 , 2 f1 + 2 f2 1 ∂x ∂x ∂x ∂x

∂u1 ∂u2 ∂u2 ∂u1 − ∂x1 ∂x2 ∂x1 ∂x2

(2.9)

at x. From (2.8), (2.9) and the Cauchy–Schwartz inequality, we obtain an inequality |u∗ ωS 2 (x)| ≤ 12 |du(x)|2 . The equality |u∗ ωS 2 (x)| = 12 |du(x)|2 holds if and only if

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∂u1 ∂x1

2

1

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∂u ∂u = ± ∂u ∂x2 , ∂x2 = ∓ ∂x1 . That is, du(x) restricted to the 2-plane spanned by e1 , e2 is a conformal map. This completes the proof.

We are primarily interested in the Hopf map. We show that the equality in Lemma 2.1 holds for the Hopf map. To see this, we ﬁrst recall the deﬁnition of the Hopf map. We write the standard 3-sphere S 3 as S 3 = {(u, v) ∈ C2 : |u|2 +|v|2 = 1}. Then the Hopf map H is deﬁned as a map S 3 (u, v) → uv −1 ∈ C ∪ {∞} followed by the inverse of the stereographic projection from the north pole of S 2 . Thus H(u, v) = (2uv, |u|2 − |v|2 ). We use the following parametrization of S 3 and S 2 : S 3 is parametrized by [0, π/2]×S 1 ×S 1 (t, eiθ1 , eiθ2 ) → (eiθ1 sin t, eiθ2 cos t) ∈ S 3 . With this parametrization, the round metric gS 3 on S 3 is written as gS 3 = dt2 + (sin t)2 dθ12 + (cos t)2 dθ22 . S 2 is parametrized by [0, π/2] × S 1 (t, eiθ ) → (eiθ sin 2t, − cos 2t) ∈ S 2 . With this parametrization, the round metric gS 2 on S 2 is written as gS 2 = 4dt2 + (sin 2t)2 dθ2 . With these coordinates on S 3 and S 2 , the Hopf map is written as H : [0, π/2] × S 1 × S 1 (t, eiθ1 , eiθ2 ) → (t, ei(θ1 −θ2 ) ) ∈ [0, π/2] × S 1 .

(2.10)

Choose an orthonormal frame e1 , e2 , e3 at any point (t, eiθ1 , eiθ2 ) of S 3 as e1 =

∂ ∂ + , ∂θ1 ∂θ2

e2 =

∂ , ∂t

e3 = cot t

∂ ∂ − tan t . ∂θ1 ∂θ2

Notice that e1 is the fundamental vector ﬁeld of the S 1 -action of S 3 and we have dH(e1 ) = 0,

dH(e2 ) =

∂ , ∂t

dH(e3 ) =

2 ∂ , sin 2t ∂θ

(2.11)

where dH : T S 3 → T S 2 is the diﬀerential of H and θ = θ1 − θ2 . ∂ ∂ and sin12t ∂θ form an orthonormal frame of S 2 , we see that dH Since 12 ∂t restricted to the orthogonal complement of the vertical subbundle of the Hopf ﬁbration is conformal (with conformal factor 4). From these calculations we also obtain |H ∗ ωS 2 | = 4

(2.12)

√ |dH| = 2 2.

(2.13)

and

3. A Variational Problem on the Closed 2-Forms on S 3 Motivated by the observation made in the previous section, we consider a variational problem on the closed 2-forms on S 3 related to the functional FSR . To introduce such a variational problem, we ﬁrst recall the analytical expression of the Hopf 2 (S 3 ) = 0, there exists a 1-form invariant for a smooth map u : S 3 → S 2 . Since HdR

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β = β(u) ∈ C ∞ (S 3 , T ∗ S 3 ) such that u∗ ωS 2 = dβ(u). Then the Hopf invariant Q(u) of u is given by the formula (see [4] for details) 1 H(u) = β(u) ∧ dβ(u). (3.1) 16π 2 S 3 We consider the functional ER deﬁned on the closed 2-forms on S 3 deﬁned by 1 ER (α) = |α| dvolS 3 + |α|2 dvolS 3 . (3.2) 2R2 S 3 S3 1 ∞ 3 ∗ 3 Since the integral Q(β) := 16π 2 S 3 β ∧ dβ (β ∈ C (S , T S )) depends only on dβ, the functional Q is also considered as deﬁned on the closed 2-forms on S 3 . When considered as a functional on closed 2-forms we denote it as Q(α) for α a closed 2-form, i.e. Q(α) = Q(β) for α = dβ. We are interested in the following variational problem 2 2 3 ∗ 3 T S , dα = 0 and Q(α) = 1 . (3.3) ER = inf ER (α) : α ∈ L S , Here and in the following, for a vector bundle E → M , Lp (M, E) denotes the space of Lp -sections of E → M and W k,p (M, E) denotes the space of W k,p -sections of E → M , where W k,p is the Sobolev space of functions in Lp whose distributional derivatives up to order k are in Lp . We ﬁrst prove: Proposition 3.1. ER is attained for some α ∈ L2 (S 3 , 2 T ∗ S 3 ) with dα = 0 and Q(α) = 1. Proof. For β ∈ W 1,2 (S 3 , T ∗ S 3 ), we obviously have α := dβ ∈ L2 S 3 , 2 T ∗ S 3 . 2 ∗ 3 T S with dα = 0 On the other hand, by the Hodge theory any α ∈ L2 S 3 , 2 1,2 3 ∗ 3 T S . Thus the variational can be written as α = dβ for some β ∈ W S , problem (3.3) is equivalent to the following variational problem on 1-forms: ER = inf{FR (β) : β ∈ W 1,2 (S 3 , T ∗ S 3 ) and Q(β) = 1},

(3.4)

where we set FR (β) = ER (dβ). Let {βn } ⊂ W 1,2 (S 3 , T ∗ S 3 ) be a minimizing sequence for (3.4), i.e. FR (βn ) → ER as n → ∞. Since the functional FR has an invariance property FR (β + dϕ) = FR (β) for any ϕ ∈ W 2,2 (S 3 ) and any β ∈ W 1,2 (S 3 , T ∗ S 3 ) and there exists ϕ ∈ W 2,2 (S 3 ) such that d∗ (β + dϕ) = 0, we may assume at the beginning that each βn satisﬁes the Coulomb condition d∗ βn = 0 in S 3 . Since {dβn } is bounded in 2 ∗ 3 1 T S , βn is in Coulomb gauge and HdR (S 3 ) = 0, the elliptic estimate L2 S 3 , ∗ for an elliptic operator d + d yields the following: βn W 1,2 (S 3 ) ≤ C(d + d∗ )βn L2 (S 3 ) ≤ Cdβn L2 (S 3 ) ≤ C for some constant C > 0 independent of n.

(3.5)

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From (3.5), the weak compactness of a bounded sequence in a Hilbert space W 1,2 (S 3 , T ∗ S 3 ) and the Rellich’s lemma we see that there exists a subsequence of {βn } (still denoted by {βn }) and β∞ ∈ W 1,2 (S 3 , T ∗ S 3 ) such that βn β∞ weakly in W 1,2 (S 3 , T ∗ S 3 ), βn → β∞ strongly in L2 (S 3 , T ∗ S 3 ) and βn → β∞ a.e. in S 3 . Under these conditions, we have FR (β∞ ) ≤ lim inf FR (βn ) = ER n→∞

(3.6)

and

1 1 1 = Q(βn ) = βn ∧ dβn → β∞ ∧ dβ∞ = Q(β∞ ) (3.7) 16π 2 S 3 16π 2 S 3 as n → ∞. (3.6) and (3.7) imply that β∞ is a solution to the problem (3.4). This completes the proof. ˜ R deﬁned on {α ∈ L2 (S 3 , 2 T ∗ S 3 ) : dα = We introduce another functional E 0, Q(α) > 0} as α ˜ ER (α) := ER . (3.8) Q(α)1/2 ˜R as We deﬁne E 2 2 3 ∗ 3 ˜ ˜ ER := inf ER (α) : α ∈ L S , T S , dα = 0, Q(α) > 0 .

(3.9)

˜R = ER : A solution of (3.3) is automatically a solution It is easily veriﬁed that E of (3.9) and conversely a solution α of (3.9) gives a solution α/Q(α)1/2 of (3.3). In particular, by Proposition 3.1, (3.9) has a solution. To relate variational problems (3.3) and (3.9) with the minimizing property of the Hopf map for the Faddeev–Skyrme problem, we ﬁrst calculate H ∗ ωS 2 . To calculate this, recall that the Hopf map is obtained as a composition of two maps: S 3 (z0 , z1 ) → [z0 , z1 ] ∈ CP 1 and CP 1 [z0 , z1 ] → π −1 (z0 /z1 ) ∈ S 2 , where S 3 is identiﬁed (z0 , z1 ) ∈ C2 with |z0 |2 + |z1 |2 = 1 and π : S 2 → C ∪ {∞} is the stereographic projection from the north pole. Denoting CP 1 [z0 , z1 ] → π −1 (z0 /z1 ) ∈ S 2 2y x2 +y 2 −1 also as π −1 , we have π −1 ([z0 , z1 ]) = 1+x2x 2 +y 2 , 1+x2 +y 2 , 1+x2 +y 2 , where z0 /z1 = z = x + iy. We deﬁne α = −(π −1 )∗ ωS 2 . This is a positive generator of H 2 (CP 1 ; Z) 2idz∧dz and an easy calculation shows that it is given as α = (1+|z| 2 )2 . Since z = z0 /z1 , we also have, in terms of the homogeneous coordinate [z0 , z1 ], α=

2i(z1 dz0 − z0 dz1 ) ∧ (z 1 dz 0 − z 0 dz 1 ) . (|z0 |2 + |z1 |2 )2

(3.10)

H ∗ ωS 2 is calculated by substituting z0 and z1 with z0 = x1 + ix2 , z1 = x3 + ix4 which satisfy |z0 |2 + |z1 |2 = (x1 )2 + (x2 )2 + (x3 )2 + (x4 )2 = 1 in (3.10). In this way, we arrive at H ∗ ωS 2 = 4(dx1 ∧ dx2 + dx3 ∧ dx4 ).

(3.11)

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Here and in the following, we set 1 2 3 4 α+ 0 = dx ∧ dx + dx ∧ dx ,

β0 = x1 dx2 + x3 dx4 .

(3.12)

Thus H ∗ ωS 2 = 4α+ 0 = 4dβ0 . In the remaining of this section, we show that 4α+ 0 is a critical point of ER in 2 α ∈ L2 S 3 , = 1 . Furthermore, we shall show that it is T ∗ S 3 : dα = 0, Q(α) √ a stable critical point provided R ≤ 2. More precisely, we shall prove: Lemma 3.1. For all R > 0, 4α+ of ER in S := α ∈ 0 is a critical point √ 2 ∗ 3 T S : dα = 0, Q(α) = 1 . Moreover, if R ≤ 2, it is a weakly stable L2 S 3 , critical point. Notice that the above proposition is equivalent that for all R > to the statement ˜ R in S˜ = α ∈ L2 S 3 , 2 T ∗ S 3 : dα = 0, Q(α) > 0 is a critical point of E 0, 4α+ 0 √ and it is stable if R ≤ 2. Thus to prove Lemma 3.1, we need to calculate the ﬁrst ˜ R in S. ˜ and the second variations of E + + ˜ ˜ ˜ The ﬁrst variation of ER in S at 4α0 is deﬁned as dER (4α0 )(dϕ) := d ˜ Thus it is an element dt t=0 ER (4α0 + tdϕ) for any closed (hence exact) 2-form dϕ. 2 ∗ 3 + ∗ ∼ ˜ ˜ of the dual of the tangent space to S at 4α0 ; T4α+ S = α ∈ L2 S 3 , T S : 0 + ˜ dα = 0 . The second variation of ER at 4α0 is a symmetric bilinear form on ˜ R (4α+ )(dϕ, dϕ) = d22 E ˜ R (4α+ + tdϕ). We will T ∗ + S˜ × T ∗ + S˜ deﬁned by d2 E 4α0

0

4α0

dt

ER (4α+ 0 )(dϕ, dϕ)

2˜

use a notation Q4α+ (dϕ) := d 0

t=0

0

for simplicity. By deﬁnition, 4α+ 0

˜ R if dE ˜ R (4α0 )(dϕ) = 0 and Q + (dϕ) ≥ 0 for is a weakly stable critical point of E 4α0 2 2 3 ∗ 3 any closed 2-form dϕ in L S , T S . ˜ R (4α+ + tdϕ) To calculate Q4α+ (dϕ), we give the second-order expansion of E 0 0 with respect to t. For this calculation, we easily check the facts: |α+ 0|= 1

(3.13)

1 and β0 ∧ α+ 0 = 2 ωS 3 . Therefore we have

∗α+ 0 = 2β0 .

(3.14)

In (3.13) and (3.14), the norm and the Hodge star ∗ are taken with respect to the round metric on S 3 . By (3.13), some easy calculation shows that + 2 |4α0 + tdϕ| dvolS 3 = 8π + t (α+ 0 , dϕ), dvolS 3 S3

+

t2 8

S3

S3

2 (|dϕ|2 − (α+ 0 , dϕ) ) dvolS 3

where (·, ·) denotes the inner product on

2

+ R1 (t), (3.15)

T ∗ S 3 and R1 (t) = O(t3 ) as t → 0.

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We also have + + 2 2 2 |4α0 + tdϕ| dvolS 3 = 32π + 8t (α0 , dϕ) dvolS 3 + t S3

S3

S3

|dϕ|2 dvolS 3 (3.16)

and S3

(4β0 + tϕ) ∧ d(4β0 + tϕ) dvolS 3 = 16π 2 + 8t

S3

β0 ∧ dϕ + t2

S3

ϕ ∧ dϕ. (3.17)

(3.15)–(3.17) and some computation using (3.14) show that 2 1 1 ˜ R (4α+ + tdϕ) = 8π 2 + 16π + t2 + E |dϕ|2 dvolS 3 0 R2 8 2R2 S3 2 1 1 1 + 3 + 2 − ϕ ∧ dϕ + (α , dϕ) dvol S 0 4 R 16π 2 S3 S3 1 2 3 − (α+ , dϕ) dvol (3.18) + R2 (t), S 8 S3 0 where R2 (t) = O(t3 ) as t → 0. ˜ R (hence ˜ R (4α+ )(dϕ) = 0, i.e. 4α+ is a critical point of E From (3.18), we have dE 0 0 a critical point of ER on S). This proves the ﬁrst part of Lemma 3.1. It remains to prove the second assertion of Lemma 3.1. From (3.18) we have 1 1 2 + |dϕ| dvolS 3 − 2 ϕ ∧ dϕ Q4α+ (dϕ) = 0 4 R2 S3 S3 2 1 1 3 + 2 (α+ , dϕ) dvol − (α+ , dϕ)2 dvolS 3 . (3.19) S 0 8π 4 S3 0 S3 We shall study spectral properties of Q4α+ on closed 2-forms on S 3 . This requires 0 a study of spectral properties of the Laplacian on closed 2-forms on S 3 . 2 Since HdR (S 3 ) = 0, we have ker d ∩ C ∞ (S 3 , 2 S 3 ) = dC ∞ (S 3 , T ∗ S 3 ). On ker d ∩ C ∞ (S 3 , 2 T ∗ S 3 ), we have ∆ = (d + d∗ )2 = dd∗ . Spectral properties of ∆ on ker d ∩ C ∞ (S 3 , 2 T ∗ S 3 ) is well known, see [7,8]. The kth eigenvalue is (k + 2)2 and the corresponding eigenfunctions are given by the restriction to S 3 of closed and co-closed homogeneous degree k harmonic 2-forms on R4 . Thus in particular, the possible smallest (in absolute value) eigenvalue of the squire root of ∆, ∆1/2 = d∗ , 2 ∗ 3 T S ) is ±2. We show that ±2 are indeed eigenvalues of d∗ on ker d ∩ C ∞ (S 3 , 2 on ker d ∩ C ∞ (S 3 , T ∗ S 3 ) and identify corresponding eigenfunctions. 2 ∗ 3 T S ) are the restrictions The ﬁrst eigenfunctions of ∆ in ker d ∩ C ∞ (S 3 , to S 3 of closed and co-closed homogeneous degree 0 harmonic 2-forms on R4 , i.e. they are restrictions to S 3 of constant 2-forms on R4 . Thus they are identiﬁed 2 4 R . Recall that there is an orthogonal decomposition (with respect to the with 2 4 + 4 − 4 + 4 R = R ⊕ R , where R is the set of self-dual ﬂat metric on R4 )

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− 4 2-(co)vectors on R4 and R the set of anti-self-dual 2-(co)vectors on R4 , i.e. ± 4 R if and only if ∗0 ϕ = ±ϕ (∗0 is the Hodge star with respect to the ﬂat ϕ∈ metric on R4 ). Using the standard orthonormal basis of R4 and the corresponding coordinate, bases of ± R4 are given, respectively, as 1 2 3 4 α± 0 : = dx ∧ dx ± dx ∧ dx ,

1 3 2 4 α± 1 : = dx ∧ dx ∓ dx ∧ dx , 1 4 2 3 α± 2 : = dx ∧ dx ± dx ∧ dx .

We now claim ± Claim 3.1. For k = 0, 1, 2, we have d(∗α± k ) = ±2αk .

Proof. We only prove “+” case. The other case is similar. We use the variational characterization of the smallest positive eigenvalue λ0 of d∗ on ker d ∩ C ∞ (S 3 , 2 T ∗ S 3 ),     2   |dβ| dvolS 3   3 ∞ 3 ∗ 3 S : β ∈ C (S , T S ), λ0 = inf β ∧ dβ > 0 . (3.20)   S3     β ∧ dβ S3

We have already seen that λ0 ≥ 2. Deﬁne one forms βi (i = 0, 1, 2) as β0 = x1 dx2 + x3 dx4 , β1 = x1 dx3 − x2 dx4 , β2 = x1 dx4 + x2 dx3 . One can easily calculate that |dβi |2 dvolS 3 = S3

S3

(3.21)

S3

dvolS 3 = 2π 2 ,

βi ∧ dβi =

S3

=

S3

= π2

β0 ∧ dβ0 (x1 dx2 ∧ dx3 ∧ dx4 + x3 dx1 ∧ dx2 ∧ dx4 ) (3.22)

for i = 0, 1, 2. R |dβ|2 From (3.21) and (3.22), we have R S33 β∧dβ = 2. Thus βi (i = 0, 1, 2) attains λ0 S in (3.20) and therefore satisﬁes d(∗dβi ) = 2dβi . This completes the proof. 2 ∗ 3 2 4 Since dim ker(∆ − 4) ∩ ker d ∩ C ∞ (S 3 , T S ) = dim R = 6, (ker(d ∗ −2) ⊕ 2 ∗ 3 2 ∗ 3 ∞ 3 T S ) ⊂ ker(∆ − 4) ∩ ker d ∩ C ∞ (S 3 , T S ) ker(d ∗ +2)) ∩ ker d ∩ C (S , 2 ∗ 3 T S ) ≥ 6 by the above and dim(ker(d ∗ −2) ⊕ ker(d ∗ +2)) ∩ ker d ∩ C ∞ (S 3 ,

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2 ∗ 3 claim, we have (ker(d ∗ −2) ⊕ ker(d ∗ +2)) ∩ ker d ∩ C ∞ (S 3 , T S ) = ker(∆ − 2 ∞ 3 ∗ 3 T S ) and 4) ∩ ker d ∩ C (S , 2 ± ± ∞ 3 ∗ 3 ker(d ∗ ±2) ∩ ker d ∩ C T S (3.23) S , = span{α± 0 , α1 , α2 }. ± ± In the following we set E1± = span{α± 0 , α1 , α2 }. We are now ready to complete the proof of Lemma 3.1. Complete proof of Lemma 3.1. Let dϕ ∈ ker d∩L2 S 3 , 2 T ∗ S 3 be an arbitrary closed 2-form on S 3 . We expand dϕ as a Fourier series of the eigenfunctions of ∆1/2 ; + + dϕ = a0 α+ 0 + a1 α1 + a2 α2 + dψ,

(3.24)

where ai ∈ R (i = 0, 1, 2) and dψ ∈ (E1+ )⊥ ((E1+ )⊥ is the L2 -orthogonal complement 2 ∗ 3 T S )). of E1+ in L2 (S 3 , Four integrals in the expression of Q4α+ in (3.19) are written as 0

1 2π 2

S3

1 2π 2

|dϕ|2 dvolS 3 = a20 + a21 + a22 +

1 2π 2

S3

|dψ|2 dvolS 3 ,

a21 a22 1 a20 + + + 2 ϕ ∧ dϕ = ψ ∧ dψ, 2 2 2 2π S 3 S3 1 (α+ , dϕ) dvolS 3 = a0 2π 2 S 3 0

(3.25) (3.26) (3.27)

and 1 2π 2

S3

2 2 (α+ 0 , dϕ) dvolS 3 = a0 +

1 2π 2

S3

2 (α+ 0 , dψ) dvolS 3 ,

+ where we have used (α+ 0 , dϕ) = a0 + (α0 , dψ) in (3.28). From (3.25)–(3.28), we obtain 1 1 2 + |dψ| dvolS 3 − 2 ψ ∧ dψ Q4α+ (dϕ) = 0 4 R2 S3 S3 1 − (α+ , dψ)2 dvolS 3 . 4 S3 0

(3.28)

(3.29)

∞ Here dψ ∈ (E1+ )⊥ = E1− i=2 (Ei+ ⊕ Ei− ) and Ei± = {dψ ∈ ker d ∩ 2 ∗ 3 T S ) : d ∗ dψ = ±(i + 2)dψ} is the ith eigenspace of d∗ in ker d ∩ C ∞ (S 3 , 2 ∗ 3 ∞ 3 T S ). C (S , For dψ ∈ E1− , we have 2 3 |dψ| dvolS − 2 ψ ∧ dψ = 2 |dψ|2 dvolS 3 (3.30) S3

S3

S3

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and for dψ ∈

∞

+ i=2 (Ei

⊕ Ei− ), we have

S3

2

|dψ| dvolS 3

≥ 3

S

ψ ∧ dψ . 3

From (3.30) and (3.31), for dψ ∈ (E1+ )⊥ , we obtain 1 2 |dψ| dvolS 3 − 2 ψ ∧ dψ ≥ |dψ|2 dvolS 3 . 3 S3 S3 S3 On the other hand, we have from (3.13) 2 3 (α+ , dψ) dvol ≤ S 0 S3

S3

|dψ|2 dvolS 3

and therefore we obtain from (3.32) and (3.33) that 1 1 1 1 Q4α+ (dψ) ≥ + 2 |dψ|2 dvolS 3 − |dψ|2 dvolS 3 0 3 4 R 4 S3 S3 1 1 ≥ − |dψ|2 dvolS 3 . 3R2 6 S3 From (3.34), we see that Q4α+ (dψ) ≥ 0 if 1/3R2 − 1/6 ≥ 0, i.e. R ≤ 0 completes the proof of Lemma 3.1.

777

(3.31)

(3.32)

(3.33)

(3.34) √ 2. This

√ The next proposition improves the assertion of Lemma 3.1 in the case R < 2. √ Lemma 3.2. Assume 0 < R < 2. There exist > 0 and C > 0 such that if dϕ ∈ ker d ∩ C ∞ (S 3 , 2 T ∗ S 3 ) satisfies dϕ − 4α+ 0 C 0 < , then we have ˜ R (dϕ) ≥ E ˜ R (4α+ ) + C E |dψ|2 dvolS 3 , 0 S3

where dψ is the L2 -orthogonal projection of dϕ in (E1+ )⊥ . In particular, 4α+ 0 is a ˜ R in S. ˜ local minimizing critical point of E 2 ∗ 3 Proof. Assume dϕ ∈ ker d ∩ C ∞ (S 3 , T S ) and dϕ − 4α+ 0 C 0 is small (to be speciﬁed in the course of the proof). As in the proof of Lemma 3.1, we write + + + dϕ = 4α+ 0 + a0 α0 + a1 α1 + a2 α2 + dψ,

(3.35)

where ai ∈ R and dψ ∈ (E1+ )⊥ . We may assume that ψ in (3.35) satisﬁes d∗ ψ = 0

(3.36)

in S 3 after considering ψ + dγ for a suitable γ instead of ψ if necessary. For simplicity, we set + + + α+ = 4α+ 0 + a0 α0 + a1 α1 + a2 α2 .

(3.37)

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A simple computation shows that 2 ˜ R (α+ ) = 8π 2 + 16π E R2

(3.38)

˜ R (α+ ) = 0. dE

(3.39)

and

Therefore, we have ˜ R (dϕ) = E ˜ R (α+ + dψ) E ˜ R (α+ ) + 1 Qα+ (dψ) + R(dψ), =E 2

d2 + ˜ dt2 t=0 ER (α 1 d3 + ˜ 3! dt3 t=θ ER (α

where Qα+ (dψ) =

(3.40)

+ tdψ) and the remainder R(dψ) is given by the

+ tdψ) Taylor’s formula for some 0 < θ < 1. To estimate R(dψ), we set g(t) = S 3 |α+ + tdψ| dvolS 3 and f (t) = ( S 3 (β + tψ) ∧ (α+ + tdψ))1/2 , where β = 4β0 + a0 β0 + a1 β1 + a2 β2 . Then g(t)/f (t) is the ˜ R (α+ + tdψ). To estimate the third derivative of g/f at t = θ, we ﬁrst term of E notice that g(t) and f (t) are bounded from below and above if dϕ − 4α+ 0 C 0 is small and |t| ≤ 1. Then a simple computation shows that there exists a constant C > 0 independent of ψ and 0 < θ < 1 such that g (3) (θ) ≤ C(|g (3) (θ)| + |f (3) (θ)| + |g (2) (θ)||f (θ)| f + |g (θ)||f (2) (θ)| + |f (2) (θ)||f (θ)| + |g (θ)||f (θ)|2 ) and |g (θ)| ≤ C |g

(2)

|g (3) (θ)| ≤ C |f (θ)| ≤ C |f

(2)

S3

(θ)| ≤ C

S3

S3

|f (3) (θ)| ≤ C

|dψ| dvolS 3 ,

(3.42)

|dψ|2 dvolS 3 ,

(3.43)

|dψ|3 dvolS 3 ,

(3.44)

|dψ| dvolS 3 ,

(3.45)

|dψ|2 dvolS 3 ,

(3.46)

|dψ|3 dvolS 3 .

(3.47)

S3

(θ)| ≤ C

(3.41)

S3

S3

In the estimates of (3.46) and (3.47), we have used (3.36) so that by the elliptic estimate we have ψW 1,p (S 3 ) ≤ CdψLp (S 3 ) for 1 < p < ∞ and ψL∞ (S 3 ) is small if dψL∞ (S 3 ) is small (this is the case if dϕ − 4α+ 0 C 0 is small).

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From (3.41)–(3.47), we obtain g (3) (θ) ≤ C|dψ|L∞ (S 3 ) |dψ|2 dvolS 3 . f S3

779

(3.48)

˜ R (α+ + tdψ) at t = θ is estimated The third derivative of the second term of E similarly and we obtain |R(dψ)| ≤ C|dψ|L∞ (S 3 ) |dψ|2 dvolS 3 . (3.49) S3

On the other hand, from (3.38) and (3.40) we have ˜ R (dϕ) = E ˜ R (4α+ ) + 1 Q + (dψ) + 1 (Qα+ (dψ) − Q + (dψ)) + R(dψ). E 0 4α0 2 4α0 2

(3.50)

Here as in the derivation of (3.29), we have after some calculation 4 16 2 3 + + 2 +2 |dψ| dvolS − 2 ψ ∧ dψ Qα (dψ) = |α+ |2 R |α | S3 S3 2 2 4 + + +4 2 (α , dψ) dvolS 3 − + 4 (α+ , dψ)2 dvolS 3 . |α | π |α | S 3 S3 (3.51) From (3.29) and (3.51), if > 0 is small we obtain |dψ|2 dvolS 3 |Qα+ (dψ) − Q4α+ (dψ)| ≤ C 0

(3.52)

S3

for some C > 0 independent of ψ. Combining (3.49), (3.50) and (3.52), there exist > 0 and C > 0 such that if dϕ − 4α+ 0 C 0 < we have ˜ R (dϕ) ≥ E ˜ R (4α+ ) + C E |dψ|2 dvolS 3 . (3.53) 0 S3

This completes the proof. 4. Proof of the Main Theorem In this section we prove our main theorem, Theorem 1.1. We decompose its proof into two parts. We ﬁrst prove that the Hopf map is a local minimizing solution in its homotopy class and then prove that it is the unique local minimizing solution in a neighborhood of the Hopf map up to the isometry group action of S 3 . 4.1. Local minimizing property of the Hopf soliton In this subsection, we prove that the Hopf soliton is a local √ minimizing solution of the Faddeev–Skyrme functional FSR provided 0 < R < 2. Its weak stabil√ ity property for 0 < R ≤ 2 follows from this. More precisely, we shall prove

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the following: √ Proposition 4.1. Assume 0 < R < 2. There exists > 0 such that if u ∈ C 1 (S 3 , S 2 ) satisfies Q(u) = 1 and u − HC 1 < , then we have FSR (u) ≥ FSR (H). In particular, the Hopf map H is a local minimizing solution of the Faddeev–Skyrme problem in its homotopy class. Proof. For u ∈ C 1 (S 3 , S 2 ), we can write u∗ ωS 2 = α+ + dψ,

(4.1)

where α+ ∈ E1+ and dψ ∈ (E1+ )⊥ . + If u satisﬁes u − HC 1 < and > 0 is small, we can write α+ = 4α+ 0 + a0 α0 + + + a1 α1 + a2 α2 with |ai | ≤ C (i = 0, 1, 2) for some C > 0. Therefore by Lemmas 2.1 and 3.2, we have FSR (u) ≥ 2ER (u∗ ωS 2 ) ˜ R (u∗ ωS 2 ) = 2E ˜ R (4α+ ) + 2C ≥ 2E 0 = FSR (H) + 2C

S3

S3

|dψ|2 dvolS 3

|dψ|2 dvolS 3

≥ FSR (H)

(4.2)

if > 0 is small. This completes the proof. We recall that the Hessian of FSR at H, denoted as d2 FSR (H), is deﬁned 3 (s, t, x) → for any two parameter family of maps F : (−δ, δ) × (−δ, δ) × SR 2 ∂ Hs,t (x) ∈ S 2 satisfying H0,0 = H by d2 FS(H)(V, W ) = ∂s∂t (s,t)=(0,0) FS(Hs,t ), where V = ∂ Hs,0 and W = ∂ H0,t . By deﬁnition, H is called weakly ∂s s=0

∂t t=0

stable if d2 FSR (H) is non-negative for any such F ; d2 FSR (H)(V, W ) ≥ 0. √ By Proposition 4.1, H is a weakly stable critical point √ of FSR for 0 < R < 2. By continuity, it is also weakly stable for the case R = 2. Thus we have √ Corollary 4.1. H is a weakly stable critical point of FSR for 0 < R ≤ 2. 4.2. Uniqueness of the local minimizing solution

We prove in this subsection the uniqueness assertion of Theorem 1.1. We have seen in the previous subsection that the Hopf map H is a local minimizing solution in a C 1 -neighborhood of H. Thus the result follows from the following: √ Proposition 4.2. Suppose 0 < R < 2. There exists > 0 such that if u ∈ {u ∈ C 1 (S 3 , S 2 ) : u − HC 1 < } satisfies FSR (u) = FSR (H), then there exists an isometry of S 3 , ψ ∈ Isometry(S 3 ), such that u = H ◦ ψ.

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The proof of the above proposition requires some preparations. The ﬁrst result we need is: Lemma 4.1. There exists > 0 such that if u ∈ C 1 (S 3 , S 2 ) satisfies u−HC 1 < , then there exists φ1 ∈ Isometry(S 3 ) such that (u◦φ1 )∗ (ωS 2 ) is of the following form: + (u ◦ φ1 )∗ (ωS 2 ) = 4α+ 0 + a0 α0 + dψ,

where a0 ∈ R and dψ ∈ (E1+ )⊥ . Proof. Consider the vector ﬁelds Xij (1 ≤ i < j ≤ 4) on S 3 deﬁned by ∂ ∂ − xj i , (4.3) ∂xj ∂x where x = (x1 , x2 , x3 , x4 ) is the standard coordinate of R4 . Notice that Xij (1 ≤ i < j ≤ 4) generate the Killing ﬁelds on S 3 . We calculate the Lie derivative LXij α+ 0 . By the Cartan formula LX = dιX + ιX d (ιX is the contraction with X), we have Xij = xi

+ + LX12 α+ 0 = d(ιX12 α0 ) + ιX12 (dα0 )

= d(ιX12 α+ 0)

= d(−x1 dx1 − x2 dx2 ) = 0.

(4.4)

Similarly, we have + + LX13 α+ 0 = −LX24 α0 = α2 ,

(4.5)

+ + LX14 α+ 0 = LX23 α0 = −α1 ,

(4.6)

and LX34 α+ 0 = 0.

(4.7)

Set X1 = −X14 and X2 = X13 . Then X1 and X2 are Killing ﬁelds and we have from (4.4)–(4.7) that + LXi α+ 0 = αi

(i = 1, 2).

(4.8)

Deﬁne V to be a two-dimensional vector space over R spanned by X1 and X2 . We equip V a metric such that X1 and X2 form an orthonormal basis of V . The unit sphere with respect to this metric is denoted by S 1 (V ) := {y1 X1 + y2 X2 ∈ V : y12 + y22 = 1}. For y = t (y1 , y2 ) ∈ R2 , deﬁne a vector ﬁeld X(y) on S 3 by X(y) = y1 X1 + y2 X2 . By (4.8) we have + + LX(y) α+ 0 = y1 α1 + y2 α2 .

(4.9)

For u ∈ C 1 (S 3 , S 2 ) with u − HC 1 < ( > 0 is to be speciﬁed below), we write u∗ ωS 2 = 4α+ 0 + dφ

+ + + = 4α+ 0 + a0 α0 + a1 α1 + a2 α2 + dψ,

where ai ∈ R (i = 0, 1, 2) and dψ ∈ (E1+ )⊥ .

(4.10)

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In (4.10), we may assume (a1 , a2 ) = (0, 0) (otherwise there is nothing to prove). We denote by φtX(y) the isometry of S 3 generated by the Killing ﬁled X(y), i.e. d t t 0 3 dt φX(y) = X(y) ◦ φX(y) and φX(y) = idS . We then have from (4.9) + (φtX(y) )∗ (4α+ 0 + dφ) − 4α0 + + = a0 α+ 0 + (a1 + 4ty1 )α1 + (a2 + 4ty2 )α2 + dψ + R(y, t),

(4.11)

where R(y, t) is a continuous function of (y, t) such that R(y, t) = (φtX(y) )∗ dφ − dφ + R (y, t)

(4.12)

|R (y, t)| ≤ C|t|2

(4.13)

and

for some C > 0 independent of t with |t| < 1 and y = (y1 , y2 ) ∈ S 1 . Therefore we have + + + + + t ∗ (((φtX(y) )∗ (4α+ 0 + dφ) − 4α0 , α1 )L2 (S 3 ) , (φX(y) ) (4α0 + dφ) − 4α0 , α2 )L2 (S 3 ) )

˜ y), = 2π 2 (a1 , a2 ) + 8π 2 t(y1 , y2 ) + R(t,

(4.14)

where from (4.12) and (4.13) we have ˜ y) = ((R(t, y), α+ )L2 (S 3 ) , (R(t, y), α+ )L2 (S 3 ) ) R(t, 1 2

+ t ∗ 2 = (((φtX(y) )∗ dφ − dφ, α+ 1 )L2 (S 3 ) , ((φX(y) ) dφ − dφ, α2 )L2 (S 3 ) ) + O(|t| )

(4.15) uniformly for t with |t| < 1 and y = (y1 , y2 ) ∈ S 1 . As for the ﬁrst term of (4.15), we claim that there exists C(t) ≥ 0 with C(t) → 0 as t → 0 uniformly for y = (y1 , y2 ) ∈ S 1 such that + t ∗ |(((φtX(y) )∗ dφ − dφ, α+ 1 )L2 (S 3 ) , ((φX(y) ) dφ − dφ, α2 )L2 (S 3 ) )| ≤ C(t)dφL2 (S 3 ) .

(4.16) To see this we shall prove a more general assertion that there exists C(t) ≥ 0 as above such that the estimate + t ∗ |(((φtX(y) )∗ ω − ω, α+ 1 )L2 (S 3 ) , ((φX(y) ) ω − ω, α2 )L2 (S 3 ) )| ≤ C(t)ωL2 (S 3 ) (4.17) holds for all ω ∈ L2 S 3 , 2 T ∗ S 3 and y ∈ S 1 . prove (4.17), assume contrary that there exist C > 0, tn → 0, ωn ∈ To3 2 ∗ 3 2 T S with ωn L2 (S 3 ) = 1 and yn ∈ S 1 such that L S , tn + ∗ n )∗ ωn − ωn , α+ |(((φtX(y 1 )L2 (S 3 ) , ((φX(yn ) ) ωn − ωn , α2 )L2 (S 3 ) )| ≥ C. n)

(4.18)

Passing a subsequence if necessary, we may assume that there exists ω ∈ 3 to 2 ∗ 3 2 T S and y ∈ S 1 such that ωn ω weakly in L2 (S 3 ) and yn → y L S , n → idS 3 in C ∞ (S 3 ) as n → ∞, it is easy to see that as n → ∞. Since φtX(y n) tn (φX(yn ) )∗ ωn ω weakly in L2 (S 3 ) as n → ∞. However, this contradicts (4.18). Thus (4.17) and hence (4.16) is proved.

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Deﬁne F (t, y) =

1 + + (((φtX(y) )∗ (4α+ 0 + dφ) − 4α0 , α1 )L2 (S 3 ) , 2π 2 + + ((φtX(y) )∗ (4α+ 0 + dφ) − 4α0 , α2 )L2 (S 3 ) ).

(4.19)

Writing a = (a1 , a2 ) and y = (y1 , y2 ), we have F (0, y) = a and 1 ˜ y) F (t, y) = a + 4ty + 2 R(t, 2π a 1 ˜ = 4t y + + 2 R(t, y) 4t 8π t

(4.20)

for 0 < t < 1. By our assumption u − HC 1 < , we have |ai | ≤ C and dψL2 (S 3 ) ≤ C for some C > 0 and we obtain from (4.15) and (4.16) a 1 ˜ 1/2 , y) ≤ C1/2 R( + (4.21) 41/2 2 1/2 8π for some C > 0 independent of y ∈ S 1 and > 0. Thus if > 0 is small enough, ˜ 1/2 , y) = 0 for all y ∈ S 1 . For such small > 0, if we have y + 4a1/2 + 8π211/2 R( F (t, y) = 0 for all t ∈ [0, 1/2 ] and y ∈ S 1 , we have a homotopy F (·, ·) : [0, 1/2 ] × S 1 → S 1 |F (·, ·)|

(4.22)

between F (0, ·)/|F (0, ·)| = a/|a| and F (1/2 , ·)/|F (1/2 , ·)|. On the other hand, by the homotopy (0 ≤ s ≤ 1) a 1 ˜ 1/2 , y) R( + y+s 41/2 8π 2 1/2 , (4.23) (s, y) → a 1 ˜ 1/2 , y) y + s R( + 41/2 8π 2 1/2 we see that F (1/2 , ·)/|F (1/2 , ·)| is homotopic to the identity of S 1 . This is a contradiction. Thus there exist y = (y1 , y2 ) ∈ S 1 and t ∈ [0, 1/2 ] such that F (t, y) = 0. Deﬁne φ1 = φtX(y) . This φ1 satisﬁes the desired property. Suppose u ∈ C 1 (S 3 , S 2 ) satisﬁes u − HC 1 < ( > 0 is as in Lemma 4.1). By Lemma 4.1, after considering u ◦ φ for some φ ∈ Isometry(S 3 ) if necessary, we may assume that u satisﬁes + u∗ ωS 2 = 4α+ 0 + a0 α0 + dψ

(E1+ )⊥ .

for some a0 ∈ R and dψ ∈ is small if u − HC 1 is small.

(4.24)

Notice that we may choose φ so that φ − idC 1

Lemma 4.2. Let u ∈ C 1 (S 3 , S 2 ) be as above and assume also that u satisfies FSR (u) = FSR (H). Then we have (1) |u∗ ωS 2 | = 12 |du|2 , (2) u∗ ωS 2 = 4α+ 0 and

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(3) du : (T V )⊥ → T S 2 is conformal, where V → S 2 is the vertical subbundle of the Hopf fibration S 3 → S 2 . Proof. We have from Lemmas 2.1 and 3.2 2 −2 FSR (u) = |du| dvolS 3 + R S3

≥2

S3

≥2

S3

|u∗ ωS 2 | + R−2 ∗

|H ω | + R S2

S3

−2

S3

|u∗ ωS 2 |2 dvolS 3

|u∗ ωS 2 |2 dvolS 3

S3

|H ∗ ωS 2 |2 dvolS 3

= FSR (H).

(4.25)

Therefore, by our assumption, all inequalities in (4.25) are equalities and we obtain (1) of the lemma. + + ⊥ Since we already have u∗ ωS 2 = 4α+ 0 + a0 α0 + dψ, dψ ∈ (E1 ) , we have from Lemma 3.2 2 |u∗ ωS 2 | dvolS 2 + R−2 |u∗ ωS 2 |2 dvolS 3 S3

≥ 2

S3

S3

|H ∗ ωS 2 | dvolS 3 + R−2

S3

|H ∗ ωS 2 |2 dvolS 3 + C

S3

|dψ|2 dvolS 3 . (4.26)

Combining (4.26) with (4.25) and the assumption FSR (u) = FSR (H), we have dψ = 0.

(4.27)

From (4.27), u∗ ωS 2 is of the following form u∗ ωS 2 = (4 + a0 )α+ 0

(4.28)

for some a0 ∈ R. Inserting (4.28) in (4.25) and using FSR (u) = FSR (H) again, we obtain (4 + a0 )2 8 2 2 4π 4 + a0 + = 4π 4 + 2 . (4.29) 2R2 R From this, we have a0 = 0 or a0 = −8 − 2R2 < −8. Since |a0 | ≤ C, the second alternative is excluded if > 0 is chosen small enough. Thus for small > 0, we have (2) of the lemma. To prove (3), let e0 ∈ T S 3 be the fundamental vertical vector ﬁeld, i.e. it is the generating vector ﬁeld of the S 1 -action S 1 × S 3 (eit , (u, v)) → (eit u, eit v) ∈ S 3 and e2 , e3 be orthonormal vector ﬁelds on S 3 which are also orthogonal to e0 . By (2) and dH(e0 ) = 0, we have u∗ ωS 2 (e0 , ej ) = H ∗ ωS 2 (e0 , ej ) = 0 for j = 0, 1, 2.

(4.30)

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Since dH(e1 ) and dH(e2 ) span TS 2 (see the proof of Lemma 2.1), if > 0 is small enough du(e1 ) and du(e2 ) also span TS 2 and since ωS 2 is non-degenerate, we have from (4.30) du(e0 ) = 0.

(4.31)

From (4.31) and (1), (3) follows from Lemma 2.1. We are now ready to complete the proof of Proposition 4.2. Proof of Proposition 4.2. By Lemma 4.1, after considering u ◦ φ1 for some φ1 ∈ Isometry(S 3 ) if necessary, we may assume that u satisﬁes u − HC 1 < and (4.24). Consider the pullback of the Hopf ﬁbration H : S 3 → S 2 by u : S 3 → S 2 , u∗ S 3 = {(x, y) ∈ S 3 × S 3 : u(x) = H(y)} → S 3 : (x, y) → x. This is a S 1 bundle over S 3 and it is trivial since S 3 is 2-connected. Therefore, there exists a section of the bundle u∗ S 3 → S 3 and thus there exists ϕ ∈ C 1 (S 3 , S 3 ) such that u(x) = H(ϕ(x)) for all x ∈ S 3 . As we have seen in the proof of Lemma 4.2, we have 0 = du(e0 ) = dH ◦ dϕ(e0 ) = 0 and therefore dϕ(e0 ) is a vertical vector ﬁled of the Hopf ﬁbration. Thus ϕ : S 3 → S 3 maps ﬁbers to ﬁbers and there exists a smooth map ψ : S 2 → S 2 such that u = H ◦ ϕ = ψ ◦ H.

(4.32) ⊥

(4.32) implies that du = dψ ◦ dH and since dH : (T V ) → T S and du : (T V )⊥ → TS 2 are conformal isomorphisms if > 0 is small (by Lemma 4.2(3)), dψ : TS 2 → T S 2 is also conformal. Moreover by (2) of Lemma 4.2, we have ∗ ∗ ∗ ∗ 4α+ 0 = H ωS 2 = u ωS 2 = H ψ ωS 2 .

2

(4.33)

Denoting by λ2 the conformal factor of ψ, i.e. ψ ∗ gS 2 = λ2 gS 2 , and f1 , f2 an orthonormal frame of T S 2 , we see that λ−1 dψ(f1 ), λ−1 dψ(f2 ) is an orthonormal frame of S 2 and therefore ψ ∗ ωS 2 (f1 , f2 ) = ωS 2 (dψ(f1 ), dψ(f2 )) = λ2 = λ2 ωS 2 (f1 , f2 ), i.e. ψ ∗ ωS 2 = λ2 ωS 2 . Combining this with (4.33), we obtain λ2 = 1, i.e. ψ : S 2 → S 2 is an isometry. This shows that under the assumption of Proposition 4.2, there exist φ1 ∈ Isometry(S 3 ) and ψ ∈ Isometry(s2 ) such that u = ψ ◦ H ◦ φ1 . The ﬁnal step to complete the proof is to show that any isometry of S 2 (such as ψ) is induced from the isometry of S 3 . To see this, recall that in quartanionic notation, the Hopf map is written as H(p) = pip∗ , where p ∈ S 3 = {x0 + ix1 + jx2 + kx3 : x20 +x21 +x22 +x23 = 1} is a unit quartanion and for q = x0 +ix1 +jx2 +kx3 ∈ H, we set q ∗ = x0 − ix1 − jx2 − kx3 . S 2 is identiﬁed with the unit sphere in Im H, S 2 = {ix1 + jx2 + kx3 : x21 + x22 + x23 = 1}. Any isometry ψ ∈ Isometry(S 2 ) is expressed as ψ(x) = ψg (x) := gxg ∗ for some g ∈ S 3 . For g ∈ S 3 , deﬁne an isometry

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φg ∈ Isometry(S 3 ) as φg (p) = gp. Then we have H ◦ φg (p) = H(gp) = (gp)i(gp)∗ = ψg ◦ H(p). Therefore H ◦ φg = ψg ◦ H. This completes the proof of Proposition 4.2. Combining Proposition 4.1, Corollary 4.1 and Proposition 4.2, we complete the proof of Theorem 1.1. Remark 4.1. The isotropy subgroup Isometry(S 3 )H := {φ ∈ Isometry(S 3 ) : H ◦ φ = H} at H is S 1 . Thus the moduli space of local minimizing solitons of the Faddeev–Skyrme problem is locally isomorphic near the Hopf map to SO(4)/S 1 . References [1] D. Auckly and L. Kapitanski, Analysis of S 2 -valued maps and Faddeev’s model, Comm. Math. Phys. 256 (2005) 611–620. [2] R. A. Battye and P. M. Sutcliﬀe, Knots as stable soliton solutions in a threedimensional classical ﬁeld theory, Phys. Rev. Lett. 81 (1998) 4798–4801. [3] R. A. Battye and P. M. Sutcliﬀe, Solitons, links and knots, Proc. Roy. Soc. London A 455 (1999) 4305–4331. [4] R. Bott and L. W. Tu, Diﬀerential Forms in Algebraic Topology, Graduate Texts in Math., Vol. 82 (Springer, 1982). [5] L. D. Faddeev, Some comments on the many dimensional solitons, Lett. Math. Phys. 1 (1976) 289–293. [6] L. D. Faddeev and A. J. Niemi, Stable knot-like structures in classical ﬁeld theory, Nature 387 (1997) 58–61. [7] S. Gallot and D. Meyer, Operateur de courbure et Laplacien des formes diﬀ´erentielles d’une vari´et´e riemannienne. J. Math. Pures Appl. 54 (1975) 259–299. [8] A. Ikeda and Y. Taniguchi, Spectra and eigenforms of the Laplacian on S n and P n (C), Osaka J. Math. 15 (1978) 515–546. [9] E. H. Lieb, Remarks on the Skyrme model, Proc. Sympos. Pure Math. 54 (1993) 379–384. [10] M. Loss, The Skyrme model on Riemannian manifolds, Lett. Math. Phys. 14 (1987) 149–156. [11] F. H. Lin and Y. Yang, Existence of energy minimizers as stable knotted solitons in the Faddeev model, Comm. Math. Phys. 249 (2004) 273–303. [12] F. H. Lin and Y. Yang, Energy splittings, substantial inequality, and minimization for the Faddeev and Skyrme models, Comm. Math. Phys. 269 (2007) 137–152. [13] N. Manton, Geometry of Skyrmions, Comm. Math. Phys. 111 (1987) 469–478. [14] T. Riviere, Minimizing ﬁbrations and p-harmonic maps in homotopy classes from S 3 into S 2 , Comm. Anal. Geom. 6 (1998) 427–483. [15] T. H. Skyrme, A nonlinear ﬁeld theory, Proc. Roy. Soc. London A 260 (1961) 127–138. [16] J. M. Speight and M. Svensson, On the strong coupling limit of the Faddeev–Hopf model, Comm. Math. Phys. 272 (2007) 751–773. [17] A. F. Vakulenko and L. V. Kapitanski, Stability of solitons in S 2 nonlinear σ-model, Sov. Phys. Dokl. 24 (1979) 433–434 . [18] R. S. Ward, Hopf solitons on S 3 and R3 , Nonlinearity 12 (1999) 241–246.

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Reviews in Mathematical Physics Vol. 20, No. 7 (2008) 787–800 c World Scientific Publishing Company

ON THE SELF-ADJOINTNESS AND DOMAIN OF PAULI–FIERZ TYPE HAMILTONIANS

D. HASLER∗ and I. HERBST† Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA ∗[email protected] †[email protected] Received 9 September 2007 We prove a general theorem about the self-adjointness and domain of Pauli–Fierz type Hamiltonians. Our proof is based on commutator arguments which allow us to treat fields with non-commuting components. As a corollary, it follows that the domain of the Hamiltonian of non-relativistic QED with Coulomb interactions is independent of the coupling constant. Keywords: Pauli–Fierz; non-relativistic QED; self-adjointness. Mathematics Subject Classification 2000: 81Q10, 81V10, 81T08

1. Introduction Pauli–Fierz Hamiltonians are at the foundation of a mathematically consistent description of non-relativistic quantum mechanical matter interacting with the quantized electromagnetic ﬁeld. For a Hamilton operator to describe a unitary dynamics, it must be self-adjoint. Thus the question of self-adjointness is intimately related to physics. Knowing the domain of self-adjointness turns out to be of technical relevance for proving various properties about the Hamiltonian. In this paper, we prove a general theorem stating that the domains of Pauli–Fierz type Hamiltonians are independent of the coupling strength. Our proof is based on elementary commutator arguments, which allow us to treat ﬁelds of general form. Thus our theorem does not require the components of the ﬁelds to commute (see Theorem 6). In such a case functional integral methods are typically not applicable. As a corollary, we show that the domain of the Hamiltonian of non-relativistic QED with Coulomb interactions is independent of the coupling constant. Such a result has been obtained previously using functional integral methods, see [1–5]. However, an operator theoretic proof has, so far, been lacking in the literature. The paper is organized as follows. First, we introduce deﬁnitions and collect some elementary properties in lemmas. Although these properties are well known, 787

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a proof is given in the Appendix for the convenience of the reader. The Hamiltonian of the interacting system is realized as the self-adjoint operator associated to a semibounded quadratic form. In a ﬁrst step, we show using a commutator argument that the domain of the free Hamiltonian is an operator core for the interacting Hamiltonian (see Lemma 11). In a second step, we show using operator inequalities that the free Hamiltonian is operator bounded by the interacting Hamiltonian on a suitable core for the free Hamiltonian (see Lemma 12). Our result then follows as an application of the closed graph theorem. 2. Model and Statement of Result Consider the Hilbert space L2 (Rn ). For a measurable function f : Rn → C, we deﬁne the multiplication operator Mf ϕ := f ϕ for all ϕ in the domain D(Mf ) = {ϕ ∈ L2 (Rn ) | f ϕ ∈ L2 (Rn )}. If f is real-valued, then Mf is self-adjoint. Let pj be the operator deﬁned by, pj ϕ := −i∂j ϕ := −i(∂j ϕ)dist , for ϕ in the domain D(pj ) := {ψ ∈ L2 (Rn ) | (∂j ψ)dist ∈ L2 (Rn )}, where (·)dist stands for the distributional derivative and ∂j stands for the partial derivative with respect to the jth coordinate in Rn . The Laplacian is deﬁned by n −∆ := p2 := j=1 p2j with domain D(p2 ) := H 2 (Rn ). The operators pj and p2 are self-adjoint on their domains. In this paragraph, we review some standard conventions about tensor products, which can be found, for example, in [6]. The algebraic tensor product V ⊗ W of the vector spaces V and W consists of all ﬁnite linear combinations of vectors of the form ϕ ⊗ η with ϕ ∈ V and η ∈ W . For H and K, the two Hilbert spaces, the tensor product of Hilbert spaces is the closure of the algebraic tensor product of H and K in the topology induced by the inner product. We adopt the standard convention that V ⊗ W denotes the tensor product of Hilbert spaces if V and W are Hilbert spaces; if V or W is a non-complete inner product space then V ⊗ W denotes the algebraic tensor product. For A and B, closed operators in the Hilbert spaces H and K, respectively, we denote by A ⊗ 1 the closure of A ⊗ 1 D(A) ⊗ K and by 1 ⊗ B the closure of 1 ⊗ B H ⊗ D(B). If A is essentially self-adjoint on Da , then A ⊗ 1 is essentially self-adjoint on Da ⊗ K. An analogous statement holds for 1 ⊗ B. For notational convenience, the operators A ⊗ 1 and B ⊗ 1 are written as A and B, respectively. No confusion should arise, since it should be clear from the context in which space the operator acts. By associativity and bilinearity of the tensor product, the above deﬁnitions, conventions, and properties generalize in a straightforward way to multiple tensor products [6]. Let h be a separable complex Hilbert space and let ⊗n h = h ⊗ h ⊗ · · · ⊗ h denote the n-fold tensor product of h with itself. We deﬁne the Hilbert spaces F :=

∞ n=0

Fn ,

F0 := C,

Fn := Sn (⊗n h),

n ≥ 1,

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where Sn denotes the orthogonal projection onto totally symmetric tensors, i.e. the 1 projection satisfying Sn (ϕ1 ⊗ ϕ2 ⊗ · · · ⊗ ϕn ) = n! σ∈Sn ϕσ(1) ⊗ ϕσ(2) ⊗ · · · ⊗ ϕσ(n) , with Sn being the set of permutations of the numbers 1 through n. By deﬁnition, a vector ψ ∈ F is a sequence (ψ(n) )n≥0 of vectors ψ(n) ∈ Fn such that its norm ∞ ( n=0 ψ(n) 2 )1/2 is ﬁnite. Let Ω = (1, 0, 0, . . .), and let Fﬁn = {ψ ∈ F | ψ(n) = 0 except for ﬁnitely many n} denote the subspace consisting of states containing only ﬁnitely many “particles.” Let A be a self-adjoint operator on h with domain D(A). The second quantization dΓ(A) is an operator in F deﬁned as follows. Let A(n) be the closure of (A ⊗ 1 ⊗ · · · ⊗ 1 + 1 ⊗ A ⊗ 1 ⊗ · · · ⊗ 1 + · · · + 1 ⊗ · · · ⊗ 1 ⊗ A) Sn (⊗n D(A)). Deﬁne (dΓ(A)ψ)(n) = A(n) ψ(n) for all ψ in the domain D(dΓ(A)) := {ψ ∈ F | ψ(n) ∈ ∞ 2 D(A(n) ), n=0 A(n) ψ(n) < ∞}. It follows from the deﬁnition that dΓ(A) is selfadjoint. For each h ∈ h we deﬁne the creation operator a∗ (h) by a∗ (h)ϕ = (n + 1)1/2 Sn+1 h ⊗ ϕ,

∀ϕ ∈ Fn ,

and extend a∗ (h) to be an operator in F by taking the closure. Let a(h) be the adjoint of a∗ (h). The annihilation operator a(h) acts on F0 as the zero operator and on vectors Sn (ϕ1 ⊗ · · · ⊗ ϕn ) ∈ Fn , with n ≥ 1, as a(h)Sn (ϕ1 ⊗ ϕ2 ⊗ · · · ⊗ ϕn ) = n−1/2 Sn−1

n

(h, ϕi )ϕ1 ⊗ · · · ϕi−1 ⊗ ϕi+1 ⊗ · · · ⊗ ϕn .

i=1

For h ∈ h, we introduce the ﬁeld operator on Fﬁn ˆ φ(h) = 2−1/2 (a(h) + a∗ (h)). ˆ This operator is symmetric, and hence closable. Let φ(h) denote the closure of φ(h). 2 d p d We shall henceforth assume that h = L (R ; C ) and that ω : R → [0, ∞) is a measurable function which is a.e. non-zero. The ﬁeld energy, deﬁned by, Hf = dΓ(Mω ) is self-adjoint in F . It is notationally convenient to deﬁne the Hilbert space hω := √ {h ∈ h | hω < ∞} with norm hω := (h2 + h/ ω2 )1/2 . In the next lemma we collect some basic and well known properties. A proof of the lemma can be found in the Appendix. Lemma 1. The following statements hold: (a) For g, h ∈ h, [φ(g), φ(h)] = i Im(g, h)

on Fﬁn .

(1)

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(b) If h ∈ hω , then D(Hf ) ⊂ D(φ(h)) and φ(h)(Hf + 1)−1/2 ≤ 21/2 hω .

(2)

If g, h ∈ hω , then D(Hf ) ⊂ D(φ(g)φ(h)) and φ(g)φ(h)(Hf + 1)−1 ≤ 4gω hω .

(3)

(c) If h, ωh ∈ h, then [Hf , φ(h)] = −iφ(iωh)

on Fﬁn ∩ D(Hf ).

(4)

Now we will extend the above deﬁnition to the tensor product H = L2 (Rn ) ⊗ F of Hilbert spaces. We will use the natural isomorphism of Hilbert spaces, H := L2 (Rn ) ⊗ F ∼ = L2 (Rn ; F ), and we introduce the space Hﬁn = {ψ ∈ H | ψ(n) = 0, except for ﬁnitely many n}. Let L∞ (Rn ; h) and L∞ (Rn ; hω ) denote the Banach spaces of measurable functions from Rn to h and hω with norms G∞ := ess supx∈Rn G(x) and ˆ for Gω,∞ := ess supx∈Rn G(x)ω , respectively. For G ∈ L∞ (Rn ; h) deﬁne Φ(G) ψ ∈ Hﬁn by ˆ (Φ(G)ψ)(x) = φ(G(x))ψ(x). ˆ Note that Φ(G) is a symmetric operator and hence closable. Let Φ(G) denote the ˆ closure of Φ(G). Remark. Although not needed for the proof of the theorem, we note that φ(f ) and Φ(G) are essentially self-adjoint on Fﬁn and Hﬁn , respectively. This can be shown using, for example, Nelson’s analytic vector theorem, see [7]. Lemma 2. Let G ∈ L∞ (Rn ; hω ). Then D(Hf ) ⊂ D(Φ(G)) and 1/2

Φ(G)(Hf + 1)−1/2 ≤ 21/2 Gω,∞

(5)

Proof. Follows from inequality (2). Lemma 3. Let (Gi )ni=1 ⊂ L∞ (Rn ; hω ), and Aj := Φ(Gj ). Then the quadratic form q(ϕ, ψ) :=

n

1/2

1/2

((pj + Aj )ϕ, (pj + Aj )ψ) + (Hf ϕ, Hf ψ),

(6)

j=1

deﬁned on the form domain Q(q) :=

i

1/2

D(pi )∩D(Hf ) is non-negative and closed.

The proof of this Lemma is given in the Appendix. Definition 4. For (Gj )nj=1 ⊂ L2 (Rn ; hω ), let TA be the unique self-adjoint operator associated to the quadratic form (6). For Gj ∈ L2 (Rn ; h), we will set Aj := Φ(Gj ).

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Remark. By the ﬁrst representation theorem for quadratic forms, TA is characterized as follows: (TA ϕ, ψ) = q(ϕ, ψ),

∀ψ ∈ Q(q),

(7)

for all ϕ in the domain D(TA ) = {ϕ ∈ Q(q) | ∃η ∈ H, ∀ψ ∈ C, q(ϕ, ψ) = (η, ψ)}, where C is any form core for q. Definition 5. G ∈ L∞ (Rn ; h) is said to be weakly ∂j -diﬀerentiable if there is a K ∈ L∞ (Rn ; h) such that for all v ∈ h and all f ∈ C0∞ (Rn ) ∂j f (x)(v, G(x))dx = − f (x)(v, K(x))dx; in that case we write ∂j G = K. Hypothesis (G). (Gj )nj=1 ⊂ L∞ (Rn ; hω ) is a collection of functions such that Gj n is weakly ∂j -diﬀerentiable and ωGj , l=1 ∂l Gl ∈ L∞ (Rn ; hω ). We will adopt standard conventions for the sum and the composition of two operators: D(S + R) = D(S) ∩ D(R) and D(SR) = {ψ ∈ D(R) | Rψ ∈ D(S)}. Since p2 and Hf are commuting positive operators, p2 + Hf is self-adjoint on the domain D(p2 + Hf ) = D(p2 ) ∩ D(Hf ). Theorem 6. Let Hypothesis (G) hold. Then TA is essentially self-adjoint on any operator core for p2 + Hf and D(TA ) = D(p2 + Hf ). The next theorem relates TA with a natural deﬁnition. By (p + A)2 we denote the operator sum j (pj + Aj )2 . By deﬁnition ϕ ∈ D((p + A)2 ) if and only if ϕ ∈ D(pj ) ∩ D(Aj ) and (pj ϕ + Aj ϕ) ∈ D(pj ) ∩ D(Aj ),

∀ j = 1, . . . , n.

Moreover, D((p + A)2 + Hf ) = D((p + A)2 ) ∩ D(Hf ). Theorem 7. Suppose Gj ∈ L∞ (Rn ; hω ) is weakly ∂j -diﬀerentiable and ∂j Gj ∈ L∞ (Rn ; hω ) for j = 1, . . . , n. Then D(p2 + Hf ) ⊂ D(TA ) ∩ D((p + A)2 + Hf ). Furthermore, (p + A)2 ϕ + Hf ϕ = TA ϕ,

if ϕ ∈ D(p2 + Hf ).

The proofs of Theorems 6 and 7 are given in Sec. 4. 3. Applications Let H = ⊗N (L2 (R3 ) ⊗ C2 )) ∼ = L2 (R3N ; ⊗N C2 ) be the Hilbert space, describing N spin-1/2 particles. Let xj denote the coordinate of the jth particle having mass mj > 0, and let x = (x1 , . . . , xn ) ∈ R3N . Let σ j,a = 1 ⊗ · · · 1 ⊗ σa ⊗ 1 · · · ⊗ 1, where σa , the ath Pauli matrix, acts on the jth factor of ⊗N C2 . Let h = L2 (R3 ; C2 ), and let ε(1, k) and ε(2, k) be normalized vectors in C3 depending measurably on

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k/|k| such that (ε(i, k), k) = 0, for i = 1, 2 and (ε(1, k), ε(2, k)) = 0. Let ω(k) = √ m2ph + k 2 for some mph ≥ 0. Let ρ(k) be a function such that ρ/ω, ωρ ∈ L2 (R3 ). For a = 1, 2, 3 and j = 1, . . . , N , let ρ(k) −ik·xj e [Gj,a (x)](k, λ) = εa (λ, k), ω(k) −iρ(k) −ik·xj [Ej,a (x)](k, λ) = (k ∧ ε(λ, k))a , e ω(k)

(8)

and Aj,a = Φ(Gj,a ) and Bj,a = Φ(Ej,a ). Let Vc : R3N → R be a function which is inﬁnitesimally bounded with respect to −∆ := p2 . For example this is the case, if 3 for cj,l , zj,J ∈ R and (RJ )M J=1 ⊂ R , Vc =

j=l

cj,l zj,J + . |xj − xl | j=1 |xj − RJ | N

M

J=1

We want to point out that one usually imposes the constraint that ρ(k) = ρ(−k), which is not needed for the corollary below to hold. Moreover, note that [Aa,l , Ab,j ] = 0 is satisﬁed only if |ρ(k)| = |ρ(−k)| (see Lemma 9). Corollary 8. The operator ej 1 (−i∇j − ej Aj )2 + Hf + σ j · Bj + Vc , 2mj 2mj j j with ej ∈ R, is well deﬁned on D( j

(9)

−∆j 2mj

+ Hf ). It is self-adjoint with this domain, −∆j essentially self-adjoint on any core for j 2mj + Hf , and bounded from below. Clearly the same result holds if we restrict the operators to subspaces taking into account certain particle statistics. The statement of this corollary has been previously obtained using functional integral methods [5]. Proof. After rescaling the particle coordinates and the functions (8), we can assume that mj = 1 and ej = −1. The Gj,a (possibly a rescaled version thereof) satisfy the assumptions of Theorems 6 and 7. Thus by Theorem 7, (p+A)2 +Hf is well deﬁned on D(p2 + Hf ). Moreover, for ϕ ∈ D(p2 + Hf ), we have (p + A)2 ϕ + Hf ϕ = TA ϕ. By Theorem 6, D(p2 + Hf ) = D(TA ) and therefore TA is p2 + Hf bounded. Since σ j · Bj and Vc are inﬁnitesimally small with respect to p2 + Hf , the claim follows now from Kato’s Theorem. 4. Proofs We use the convention that [S, R] stands for the operator SR − RS deﬁned on the domain D([S, R]) = {ψ ∈ D(R) ∩ D(S) | Sψ ∈ D(R), Rψ ∈ D(S)}.

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Lemma 9. The following statements hold: (a) For F, G ∈ L∞ (Rn ; h), [Φ(F ), Φ(G)] = i Im(F, G)h

on Hﬁn ,

where the right-hand side is a multiplication operator and the inner product is taken in h. 1/2 (b) If F, G ∈ L∞ (Rn ; hω ), then D(Hf ) ⊂ D(Φ(G)) and Φ(G)(Hf + 1)−1/2 ≤ 21/2 Gω,∞ moreover, D(Hf ) ⊂ D(Φ(F )Φ(G)) and Φ(F )Φ(G)(Hf + 1)−1 ≤ 4F ω,∞ Gω,∞ . (c) For G, ωG ∈ L∞ (Rn ; h), [Hf , Φ(G)] = −iΦ(iωG) ∞

on Hﬁn ∩ D(Hf ).

(d) Let G ∈ L (R ; h) be weakly ∂j -diﬀerentiable. Then Φ(G) leaves Hﬁn ∩ D(pj ) invariant and n

[pj , Φ(G)] = −iΦ(∂j G),

on Hﬁn ∩ D(pj ).

Proof. All statements up to and including (c) follow directly from the deﬁnition and corresponding statements in Lemma 1. (d) Follows from Lemma 13 in the Appendix. In the proof, we will use certain commutator identities which can be easily veriﬁed on a suitable core, which we shall now introduce. Let ∞

∞ n n ω Sn (⊗ C ) , C := C0 (R ) ⊗ n=0

where C denotes the set of functions f in L2 (Rd ; Cp ) with supp f ⊂ ∞ ∞ n ω m=0 {k | ω(k) ≤ m}, and n=0 Sn (⊗ C ) denotes the set of all sequences ∞ n ω (ψ(n) )n=0 such that ψ(n) ∈ Sn (⊗ C ) and ψ(n) = 0 for all but ﬁnitely many ∞ n. Note that C ⊂ m=1 D(Hfm ). If Gj ∈ L∞ (Rn ; h) is weakly ∂j -diﬀerentiable, we have by Lemma 9(d), ω

C ⊂ D(p2j ) ∩ D(Aj pj ) ∩ D(pj Aj ) ∩ D(A2j ).

(10)

Lemma 10. Let (Gj )nj=1 ⊂ L∞ (Rn ; hω ). The set C is a form core for q. Proof. By deﬁnition we have to show that C is dense in (Q(q), · + ), where 1/2 (pj + Aj )ϕ2 + Hf ϕ2 . (11) ϕ2+1 := ϕ2 + j

For ψ ∈ Q(q), there exists a sequence (ψn )∞ n=0 ⊂ C such that ψn → ψ, pj ψn → pj ψ, 1/2 1/2 and Hf ψn → Hf ψ. This and Lemma 9(b) imply that (pj +Aj )ψn → (pj +Aj )ψ. Thus C is dense in (Q(q), · + ).

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Part (c) of the next lemma immediately implies Theorem 7. Parts (a), (b) and (d) will be used to prove Theorem 6. Lemma 11. Suppose for j = 1, . . . , n, Gj ∈ L∞ (Rn ; h) is weakly ∂j -diﬀerentiable. Then following statements are true: (a) For all ϕ ∈ C, TA ϕ = (p + A)2 ϕ + Hf ϕ. (b) Let Gj , l ∂l Gl ∈ L∞ (Rn ; hω ) for all j = 1, . . . , n. Then D(p2 + Hf ) ⊂ D(TA ). (c) Let Gj , ∂j Gj ∈ L∞ (Rn ; hω ) for all j = 1, . . . , n. Then D(p2 + Hf ) ⊂ D((p + A)2 ) ∩ D(TA ) and for all ϕ ∈ D(p2 + Hf ), TA ϕ = (p + A)2 ϕ + Hf ϕ. (d) If Hypothesis (G) holds, the set D(p2 + Hf ) is an operator core for TA . Proof. (a) Using (10), we see that for ϕ, ψ ∈ C, q(ϕ, ψ) = j ((p + A)2j ϕ, ψ) + (Hf ϕ, ψ). This shows (a). (b) Let ϕ ∈ D(p2 +Hf ). By deﬁnition ϕ ∈ D(p2 )∩D(Hf ). By Lemma 9, ϕ ∈ D(A2l ). 1/2 Since Hf pl ϕ2 ≤ Hf ϕ2 + p2l ϕ2 , we have pl ϕ ∈ D(Al ). Thus it follows that for ψ ∈ C, {(p2j ϕ, ψ) + (Aj pj ϕ, ψ) + (A2j ϕ, ψ) + (ϕ, Aj pj ψ)} + (Hf ϕ, ψ). q(ϕ, ψ) = j

(12) Now using Lemma 9(d), we see that the summation over the last term in the sum yields, (ϕ, Aj pj ψ) = (ϕ, iΦ(Σj ∂j Gj )ψ) + (ϕ, pj Aj ψ) j

j

= (−iΦ(Σj ∂j Gj )ϕ, ψ) +

(Aj pj ϕ, ψ).

j

Thus, there exists an η ∈ H, such that for all ψ ∈ C, q(ϕ, ψ) = (η, ψ). This shows (b). (c) In view of (b) and (12), we only need to show Aj ϕ ∈ D(pj ). By Lemma 9(d), for ϕn = χ[0,n] (dΓ(1))ϕ with χ[0,n] denoting the characteristic function of the set [0, n], pj Aj ϕn = −iφ(∂j Gj )ϕn + Aj pj ϕn . Since the limit of the right-hand side exists and Aj ϕn converges, it follows that Aj ϕ ∈ D(pj ). (d) Let α > 0. For notational compactness, we set Rα := (αHf + 1)−1 and Πj := Φ(iωGj ). Moreover, observe that D(|p|) = ∩j D(pj ). Step 1: For ϕ ∈ D(TA ), and for all ψ ∈ C, q(Rα ϕ, ψ) = q(ϕ, Rα ψ) + (Fα ϕ, ψ) + 2

(Eα,j (pj + Aj )ϕ, ψ), j

(13)

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where Eα,j and Fα are bounded operators deﬁned by Eα,j = −iRα αΠj Rα , 2 2α2 Rα (Πj Rα )2 − αRα (ωGj , Gj )h + i[Rα , Φ(Σj ∂j Gj )]. Fα = − j

j

To show this, let ϕ ∈ D(TA ) and ψ ∈ C. By deﬁnition ϕ ∈ D(|p|). Since Rα is bounded and acts on a diﬀerent factor of the tensor product it leaves D(|p|) invariant. It follows that Rα ϕ ∈ Q(q). By the deﬁnition of the quadratic form, 1/2 1/2 q(Rα ϕ, ψ) = (ϕ, Rα (pj + Aj )2 ψ) + (Hf ϕ, Hf Rα ψ). (14) j

We write the summand in the ﬁrst expression on the right as (ϕ, Rα (pj + Aj )2 ψ) = (ϕ, [Rα , (pj + Aj )2 ]ψ) + (ϕ, (pj + Aj )2 Rα ψ) = (ϕ, [Rα , (pj + Aj )2 ]ψ) + ((pj + Aj )ϕ, (pj + Aj )Rα ψ). Inserting this into (14) we ﬁnd q(Rα ϕ, ψ) = q(ϕ, Rα ψ) +

(ϕ, [Rα , (pj + Aj )2 ]ψ).

j

We calculate the commutator (ϕ, [Rα , (pj + Aj )2 ]ψ) j

=

{(ϕ, [[Rα , Aj ], pj + Aj ]ψ) + (ϕ, 2(pj + Aj )[Rα , Aj ]ψ)}

j

=

{(ϕ, [[Rα , Aj ], Aj ]ψ) + (ϕ, [Rα , [Aj , pj ]]ψ) − (2(pj + Aj )ϕ, Eα,j ψ)}

j

= (Fα ϕ, ψ) + 2

(Eα,j (pj + Aj )ϕ, ψ), j

where we used that on Hﬁn , Eα,j = [Aj , Rα ],

2 [[Rα , Aj ], Aj ] = −2α2 Rα (Πj Rα )2 − αRα (ωGj , Gj )h .

Step 2: For all ϕ ∈ D(TA ), Rα ϕ ∈ D(TA ) and limα↓0 TA Rα ϕ = TA ϕ. From Eq. (13), it follows that Rα ϕ ∈ D(TA ) and that Eα,j (pj + Aj )ϕ. TA Rα ϕ = Rα TA ϕ + Fα ϕ + 2

(15)

j

By the spectral theorem s- limα↓0 Rα = 1. Using the estimate α1/2 Πj (αHf + 1)−1/2 ≤ max(1, α1/2 )Πj (Hf + 1)−1/2 , we see that Eα,j and the ﬁrst term of Fα converge to 0 for α ↓ 0. Moreover, Φ(Σj ∂j Gj )Rα ϕ = Φ(Σj ∂j Gj )(Hf + 1)−1/2 Rα (Hf + 1)1/2 ϕ → Φ(Σj ∂j Gj )ϕ, as α ↓ 0, which implies limα↓0 [Rα , Φ(Σj ∂j Gj )]ϕ = 0. Thus the right-hand side of Eq. (15) converges for α ↓ 0 to TA ϕ.

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Step 3: For ϕ ∈ D(TA ), and α > 0, Rα ϕ ∈ D(p2 ) ∩ D(Hf ). It is clear that Rα ϕ ∈ D(Hf ). Let ϕ ∈ D(TA ). Then by (13) there exists an η ∈ H, such that for all ψ ∈ C, ((pj + Aj )Rα ϕ, (pj + Aj )ψ) (η, ψ) = j

=

{(pj Rα ϕ, pj ψ) + (A2j Rα ϕ, ψ) + (pj Rα ϕ, Aj ψ) + (Aj Rα ϕ, pj ψ)}.

j

Furthermore, using j (Aj Rα ϕ, pj ψ) = (Rα ϕ, iΦ(Σj ∂j Gj )ψ) + j (Rα ϕ, pj Aj ψ), ϕ ∈ D(|p|), and Lemma 9, we see that there exists an η1 ∈ H, such that (pj Rα ϕ, pj ψ) = (η1 , ψ), ∀ψ ∈ C. j

This implies Rα ϕ ∈ D(p2 ), since C is a form core for p2 . Lemma 12. Let Hypothesis (G) hold. Then there exists constants C1 , C2 such that for all ϕ ∈ C, (p2 + Hf )ϕ2 ≤ C1 ((p + A)2 + Hf )ϕ2 + C2 ϕ2 .

(16)

Proof. The proof will be based on the relations given in Lemma 9. First observe that (p2 + Hf )ϕ2 ≤ 2p2 ϕ2 + 2Hf ϕ2 ,

∀ϕ ∈ C.

(17)

The lemma will follow as a direct consequence of Inequality (17) and Steps 1 and 2, below. Step 1: There exist constants c1 , c2 , c3 such that p2 ϕ2 ≤ c1 (p + A)2 ϕ2 + c2 Hf ϕ2 + c3 ϕ2 ,

∀ϕ ∈ C.

We have p2 ϕ2 = ((p + A)2 − A · (p + A) − (p + A) · A + A2 )ϕ2 ≤ 3(p + A)2 ϕ2 + 3(A · (p + A) + (p + A) · A)ϕ2 + 3A2 ϕ2 , writing A2 = A · A. We estimate the middle term using the notation [p, A] := j [pj , Aj ], (A · (p + A) + (p + A) · A)ϕ2 = (2A · (p + A) + [p, A])ϕ2 ≤ 8A · (p + A)ϕ2 + 2[p, A]ϕ2 .

(18)

The second term on the last line is estimated using [p, A]ϕ2 ≤ C(Hf + 1)1/2 ϕ2 . Here and below C denotes a constant which may change from one inequality to the next. The ﬁrst term in (18) is estimated as follows: Aj (pj + Aj )ϕ2 ≤ C (Hf + 1)1/2 (pj + Aj )ϕ2 . A · (p + A)ϕ2 ≤ C j

j

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Further, using a commutator (Hf + 1)1/2 (pj + Aj )ϕ2 j

=

((pj + Aj )ϕ, (Hf + 1)(pj + Aj )ϕ)

j

=

(((pj + Aj )2 ϕ, (Hf + 1)ϕ) + ((p + A)j ϕ, [Hf , Aj ]ϕ))

j

≤ C((p + A)2 ϕ2 +

(pj + Aj )ϕ2 + (Hf + 1)ϕ2 )

j

≤ C((p + A) ϕ + (Hf + 1)ϕ2 ). 2

2

Collecting the above estimates yields Step 1. Step 2: There exists a constant C such that 1 (p + A)2 ϕ2 + Hf ϕ2 ≤ ((p + A)2 + Hf )ϕ2 + Cϕ2 , 2

∀ϕ ∈ C.

Calculating a double commutator, we see that 1 Hf ϕ2 + (Hf ϕ, (p + A)2 ϕ) + ((p + A)2 ϕ, Hf ϕ) 2 1 = Hf ϕ2 + 2((pj + Aj )ϕ, Hf (pj + Aj )ϕ) 2 j +

((ϕ, [Aj , [Aj , Hf ]]ϕ) + ([Aj , pj ]ϕ, Hf ϕ) + (Hf ϕ, [Aj , pj ]ϕ))

j

≥

1 Hf ϕ2 − C(Hf + 1)1/2 ϕ2 4

≥ −bϕ2 , for some b. Step 2 follows from this. Proof of Theorem 6. By Lemma 11(b), we know the inclusion D(p2 + Hf ) ⊂ D(TA ). From the closed graph theorem it follows that TA is p2 + Hf bounded. This, Lemma 11(d), and the fact that C is an operator core for p2 + Hf , imply that C is an operator core for TA . From this, Lemma 11(a), and Inequality (16) we conclude that D(TA ) ⊂ D(p2 + Hf ). The statement about the core holds for any closed operators having equal domain. Appendix Proof of Lemma 1. (a) Relation (1) follows from the following relations on Fﬁn , [a(f ), a(g)] = 0, [a∗ (f ), a∗ (g)] = 0, and [a(f ), a∗ (g)] = (f, g), for all f, g ∈ h.

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(b) We will use the natural isomorphism ⊗n h ∼ = L2 ((Rd × Cp )n ). We set Kn = d p n (k1 , λ1 , . . . , kn , λn ) ∈ (R × C ) and write dKn for pλ1 ,···λn =1 dk1 . . . dkn . For ψ ∈ Fﬁn and f ∈ hω ,

∞

2 1/2

a(f )ψ = f¯(k˜1 )ω(k˜1 )−1/2 ω(k˜1 )1/2

(n + 1)

n=0 ˜ λ 1

2

˜ 1 , Kn )dk˜1

dKn × ψ(n+1) (k˜1 , λ

∞ √ ≤ f / ω2

n=0

˜ 1 , Kn )|2 dk˜1 dKn (n + 1)ω(k˜1 )|ψn+1 (k˜1 , λ

˜1 λ

√ = f / ω2 (ψ, Hf ψ). Thus

1/2 D(Hf )

(19)

⊂ D(a(f )). For ϕ ∈ Fﬁn ,

∗

a (f )ϕ2 = (a∗ (f )ϕ, a∗ (f )ϕ) = f 2ϕ2 + a(f )ϕ2 . By this and (19) we ﬁnd D(Hf ) ⊂ D(a∗ (f )) and √ 1/2 a∗ (f )ψ2 ≤ f 2 ψ2 + f / ω2 Hf ψ2 . 1/2

If ψ ∈ Fﬁn and f, g ∈ hω , then with cn := (n + 1)(n + 2), a(f )a(g)ψ2

∞

1/2 ˜ 1 )¯ ˜2 )

c = g (k˜2 , λ f¯(k˜1 , λ

n

n=0 ˜ ,λ ˜ λ 1

2

2

˜ 1 , k˜2 , λ ˜ 2 , Kn )dk˜1 dk˜2

dKn × ψ(n+2) (k˜1 , λ

∞ f 2 g 2 ≤ √ √ cn ω(k˜1 )ω(k˜2 ) ω ω n=0

˜ 1 ,λ ˜2 λ

˜ 1 , k˜2 , λ ˜ 2 , Kn )|2 dk˜1 dk˜2 dKn × |ψ(n+2) (k˜1 , λ √ 2 √ 2 ≤ f / ω g/ ω Hf ψ2 .

(20)

Now using the commutation relations, linearity, and the triangle inequality we can reduce Inequalities (2) and (3) to the estimates (19) and (20). (c) This follows from the identities [Hf , a(f )] = −a(ωf ) and [Hf , a∗ (f )] = a∗ (ωf ) on Fﬁn ∩ D(Hf ), which in turn follow from the deﬁnition. Proof of Lemma 3. By Lemma 2, the right-hand side of (6) is well deﬁned. By deﬁnition q is closed if and only if Q(q) is complete under the norm (11). We write (an ) as a shorthand notation for the sequence (an )∞ n=0 . Let (ϕn ) ⊂ Q(q) be a Cauchy

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sequence with respect to the norm · +1 , see (11). We see that the sequences 1/2

(ϕn ), (Hf ϕn ), ((pj + Aj )ϕn ),

j = 1, 2, 3,

(21)

1/2

are Cauchy sequences in H. Since (Hf ϕn ) is Cauchy in H, it follows from Lemma 2 that (Aj ϕn ) is also Cauchy in H. Hence also (pj ϕn ) is Cauchy in H. Since pj and Hf are closed, it follows that the limit ϕ = limn→∞ ϕn is in the domain of Q(q). We conclude ϕ − ϕn +1 → 0 as n tends to inﬁnity. Lemma 13. Assume G ∈ L∞ (Rn ; h) is weakly ∂j -diﬀerentiable. Then for ψ ∈ D(pj ) ∩ Hﬁn , Φ(G)ψ is in the domain of pj and pj Φ(G)ψ = −iΦ(∂j G)ψ + Φ(G)pj ψ. Proof. Suppose ψ1 = f1 ⊗ ξ1 , ξ1 = a∗ (h1 ) · · · a∗ (hM )Ω, ψ2 = f2 ⊗ ξ2 , ξ2 = a∗ (g1 ) · · · a∗ (gN )Ω, with fj ∈ C0∞ (Rn ) and gi , hi ∈ h. Then (ipj ψ1 , Φ(G)ψ2 ) = (ipj f1 ⊗ ξ1 , Φ(G)f2 ⊗ ξ2 ) = (∂j f1 (x))[(ξ1 , 2−1/2 (a∗ (G(x)) + a(G(x)))ξ2 )f2 (x)]dx −1/2 =2 (ξ1 , (G(x), gl )a∗ (g1 ) · · · a∗ (ˆ gl ) · · · Ω)f2 (x) ∂j f1 (x) l

+

ˆ l ) · · · Ω, ξ2 )f2 (x) dx, ∂j f1 (x)((G(x), hl )a∗ (h1 ) · · · a∗ (h

l

ˆ l ) stands for the omission of the term a∗ (hl ). We ﬁnd, where a∗ (h (ξ1 , (gl , ∂j G(x))a∗ (g1 ) · · · a∗ (ˆ gl ) · · · Ω)f2 (x) (ipj ψ1 , Φ(G)ψ2 ) = −2−1/2 f1 (x) +

l

(ξ1 , (gl , G(x))a∗ (g1 ) · · · a∗ (ˆ gl ) · · · Ω)∂j f2 (x)

l

+

ˆ l ) · · · Ω, ξ2 )f2 (x) ((hl , ∂j G(x))a∗ (h1 ) · · · a∗ (h

l

+

ˆ l ) · · · Ω, ξ2 )∂j f2 (x) dx ((hl , G(x))a∗ (h1 ) · · · a∗ (h

l

= (ψ1 , −Φ(∂j G)ψ2 ) + (ψ1 , Φ(G)(−ipj )ψ2 ).

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Since linear combinations of vectors of the form ψ1 constitute a core for pj and pj is self-adjoint we ﬁnd Φ(G)ψ2 ∈ D(pj ) and pj (Φ(G)ψ2 ) = −iΦ(∂j G)ψ2 + Φ(G)pj ψ2 . This equation now follows for any ψ2 ∈ D(pj ) ∩ Hﬁn by taking linear combinations and then limits. References [1] B. Simon, The P (φ)2 Euclidean (Quantum) Field Theory, Princeton Series in Physics (Princeton University Press, Princeton, N.J., 1974). [2] J. Glimm and A. Jaﬀe, Quantum Physics. A Functional Integral Point of View (Springer-Verlag, New York, 1987). [3] C. Feﬀerman, J. Fr¨ ohlich and G. M. Graf, Stability of ultraviolet-cutoﬀ quantum electrodynamics with non-relativistic matter, Comm. Math. Phys. 190(2) (1997) 309–330. [4] F. Hiroshima, Essential self-adjointness of translation-invariant quantum ﬁeld models for arbitrary coupling constants, Comm. Math. Phys. 211(3) (2000) 585–613. [5] F. Hiroshima, Self-adjointness of the Pauli–Fierz Hamiltonian for arbitrary values of coupling constants, Ann. Henri Poincar´e 3(1) (2002) 171–201. [6] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis (Academic Press, Inc., New York, 1980). [7] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics. 2. Equilibrium States. Models in Quantum Statistical Mechanics, Texts and Monographs in Physics (Springer-Verlag, Berlin, 1997).

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Reviews in Mathematical Physics Vol. 20, No. 7 (2008) 801–833 c World Scientiﬁc Publishing Company

ON THE LAGRANGIAN AND HAMILTONIAN FORMULATION OF A SCALAR FREE FIELD THEORY AT NULL INFINITY

CLAUDIO DAPPIAGGI II. Institut f¨ ur Theoretische Physik, Luruper Chaussee, 149 D-22763 Hamburg, Germany [email protected] Received 3 April 2007 Revised 6 March 2008 In the framework of asymptotically ﬂat spacetimes, we discuss the notion of a covariant scalar free ﬁeld at null inﬁnity. We construct the associated equations of motion exploiting either the theory of induced representations or the white noise analysis. Eventually, we prove that these equations can also be derived as the minimum of a suitable Lagrangian and that a real scalar ﬁeld theory at null inﬁnity admits a Hamiltonian formulation. Keywords: Asymptotically ﬂat spacetimes; BMS group; ﬁeld theory at null inﬁnity; white noise calculus. Mathematics Subject Classiﬁcation 2000: 81R10, 83C30, 60H40

1. Introduction The quest to understand, to clarify and, to a certain extent, also to develop more in detail the classical and quantum ﬁeld theory over curved backgrounds has heavily relied on the holographic principle in the last decade. Originally introduced by ’t Hooft in [1] to study black hole backgrounds and the related information paradox, it has been only heuristically formulated as follows: any ﬁeld theory living on a D-dimensional manifold M — possibly including gravity — can be described by means of a suitable second ﬁeld theory living on a codimension one submanifold of M . On a practical ground, even if one is inclined to believe in such a conjecture, it is straightforward to realize that the above formulation does not provide any concrete mean or hint on how to rigorously implement the holographic paradigm. Amending such lack has been one of the main guidelines in theoretical highenergy and mathematical physics research for the past few years; at present a cornerstone is represented by the widely accepted realization of holography for ﬁeld

801

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theories living in asymptotically anti-de-Sitter spacetimes (see [2] or [3, 4] for an analysis within the framework of algebraic quantum ﬁeld theory). In this paper, we will try instead to focus on a diﬀerent kind of backgrounds, namely we consider spacetimes which are asymptotically ﬂat at null inﬁnity. Beside being a solution of vacuum Einstein’s equations with vanishing cosmological constant, they also admit a conformal completion and hence they can be endowed with a natural notion of boundary where to encode the data from a bulk ﬁeld theory. Since Minkowski spacetime is the prototype of such a class of manifolds, from a physical perspective, we hope that a deeper understanding of holography in this scenario could either enhance our comprehension or to eventually shed some light on some unsolved puzzles of quantum ﬁeld theory. From a mere mathematical perspective, our analysis on classical ﬁeld theory at null inﬁnity shows a deep-rooted, and a priori unpredictable, connection with recently developed techniques in functional analysis such as, in particular, white noise calculus. Often binded to play a somehow ancillary role in classical or quantum ﬁeld theory (see [5, 6] and references therein), this seems to be the natural framework to develop a mathematically rigorous holographic theory at null inﬁnity thus providing a further interesting motivation for this kind of research, as already envisaged in [7]. Referring to “holography at null inﬁnity in asymptotically ﬂat spacetimes”, such theory is not in its infancy (see, for example, [8]) and this paper follows the path drawn in [10, 11] where it was ﬁrst advocated to holographically encode the data of a ﬁeld theory living on an asymptotically ﬂat spacetime in a free ﬁeld theory at null inﬁnity. In the related analysis a distinguished role is played by the Bondi– Metzner–Sachs (BMS) group which is an inﬁnite-dimensional preferred subgroup of the diﬀeomorphism group of inﬁnity, i.e. the semidirect product between the homogeneous orthocronous component of the Lorentz group and the set of smooth functions over the 2-sphere thought as an Abelian group under addition. Exploiting group theoretical techniques in [9, 10], unitary and irreducible representations of the BMS group were classiﬁed and the related induced or canonical wave functions constructed. These results have been recast in terms of an holographic correspondence in [12] exploiting the algebraic formulation of quantum ﬁeld theory. Particularly it has been shown the existence of a 1:1 correspondence between a massless scalar ﬁeld conformally coupled to gravity in any asymptotically ﬂat, globally hyperbolic spacetime and a BMS invariant induced wave function at null inﬁnity [13, 14]. Unfortunately the whole approach advocated in [12] suﬀers of two main drawbacks; the ﬁrst, which will not be tackled in this paper, concerns bulk massive ﬁelds, namely there is no known meaningful projection of a solution of the Klein–Gordon equation with m = 0 to a suitable function on null inﬁnity unless the background is Minkowski spacetime [15]. The second deﬁciency consists in the complete lack of interactions in the boundary theories; one of the underlying reasons can be traced back to the employed approach in [10, 12], i.e. Wigner programme. Such analysis provides a construction of the relevant free ﬁelds and of their equations of

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motion without deriving the latter from a variational principle. Thus, at present, it is unknown whether the BMS ﬁeld theory admits a Lagrangian or Hamiltonian formulation. From the point of view of interactions and in particular of gauge interactions it is known that it possible to rigorously construct the coupling between a free ﬁeld theory and gauge ﬁelds by means of a symplectic deformation of the free Hamiltonian system (see [16, 17] and in particular [18]). Wishing ultimately to follow a similar route in the framework of BMS ﬁeld theory, our aim in this paper is to cover the ﬁrst part of the above sketched programme, namely to identify if a Hamiltonian system can be associated with a scalar BMS free ﬁeld, leaving to a future paper the symplectic deformation leading to gauge coupling. As a side remark, we also wish to point out that our attempt to construct a dynamical system at null inﬁnity has been preceded by a now more than twentyﬁve-year-old attempt by Ashtekar and Streubel in [19]. Unfortunately, since the whole approach shows no apparent direct connection with our intrinsic formulation of a BMS invariant free ﬁeld theory on null inﬁnity, a direct comparison between the two analysis is not an easy task. To conclude the introduction, it is worth to stress that, although, our main driving principle is holography, the reconstruction of bulk information by means of scattering data at null inﬁnity is not a novel idea and it has been already studied in detail also within the framework of conformal compactiﬁcation (see e.g. [20]). We wish to emphasize to a potential reader that, since in this line of research, often spatial inﬁnity plays a distinguished role, a direct application of our results to such a framework might not be immediate. As a matter of fact in the deﬁnition of asymptotic ﬂatness we employ, it is future time inﬁnity which has a prominent role since it is known that part of the data of bulk ﬁelds ﬂow through this preferred point in the Penrose conformal diagram. Therefore, a clear control of the geometric properties of spacetime in its neighborhood is mandatory in order to avoid a possible loss of information. Outline of the paper. The paper is divided, besides the introduction and the conclusions, into four main sections. In the next section, we will brieﬂy sketch the deﬁnition of an asymptotically ﬂat spacetime at future (or past) null inﬁnity and we will introduce the Bondi– Metzner–Sachs (BMS) group. The main issue of the section will be a novel proof that the BMS group is a nuclear Lie group. In Sec. 3, we will brieﬂy sketch the notion of BMS irreducible representations and of induced wave functions. Particularly, we will discuss in detail the examples of the massless and the massive scalar ﬁeld and we will introduce the Casimir invariant which plays the role of the mass for a ﬁeld at null inﬁnity. In Sec. 4, the main part of the paper, we will discuss covariant ﬁelds and we will show how this notion is intertwined with white noise calculus, due to the inﬁnite-dimensional nature of the BMS group. Within this framework, we will develop the key functional spaces associated to a BMS ﬁeld and, in particular, the

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“BMS counterpart” of the Schwartz space of rapidly decreasing test functions and of distributions. At the end of the section, we prove that Wigner programme is well deﬁned also in this setting, namely the equations of motion for BMS ﬁelds can be interpreted as suitable operators acting on the above mentioned functional spaces. Eventually in the ﬁfth and last section, we will start from these results in order to solve the inverse Lagrangian problem. Exploiting an analysis due to Gotay and Nester on presymplectic Lagrangian systems, we prove that the constructed Lagrangian is almost regular and thus it admits an associated Hamiltonian system. 2. Asymptotically Flat Spacetimes and the BMS Group Throughout this paper we will refer to a spacetime as a four-dimensional smooth (Hausdorﬀ second countable) manifold M equipped with a Lorentzian metric gµν assumed to be everywhere smooth; ﬁnally M is supposed to be time orientable and time oriented. A vacuum spacetime is a spacetime satisfying vacuum Einstein equations. In detail (M, gµν ) is called an asymptotically ﬂat and vacuum spacetime with ˆ , gˆµν ) future time inﬁnity i+ if it exists a second four-dimensional spacetime (M + ˆ with a preferred point i , a diﬀeomorphism λ : M → λ(M ) ⊂ M and a scalar function Ω ≥ 0 on λ(M ) such that gˆ = Ω2 λ∗ g and the following holds [21]: ˆ ) is closed and λ(M ) = J − (i+ )\∂J − (i+ ; M ˆ ). Moreover ∂λ(M ) = (1) J − (i+ ; M + + + . − + ˆ + ∪ i where = ∂J (i ; M )\{i } is future null inﬁnity. (2) λ(M ) is strongly causal. ˆ. (3) Ω can be extended to a smooth function on M + ˆ ν Ω(i+ ) = ˆ µ∇ and dΩ(i+ ) = 0, but ∇ (4) Ω|∂J − (i+ ;M) ˆ = 0, dΩ(x) = 0 for x ∈ + − 2ˆ gµν (i ). . ˆ ν Ω, it exists a strictly positive smooth function ω, deﬁned (5) Calling nµ = gˆµν ∇ ˆ µ (ω 4 nµ ) = 0 on + , such that the in a neighborhood of + and satisfying ∇ −1 µ integral curves of ω n are complete on + . From now we shall refer to λ(M ) simply as M since no confusion will arise in the manuscript due to this identiﬁcation. Furthermore we point out that, with minor adaption, the above deﬁnition can be recast for spacetimes which are asymptotically ﬂat with past time inﬁnity i− and all our results hold identically for − . For this reason, we shall drop from now on the pedex ± when referring to null inﬁnity. Thus let us consider any asymptotically ﬂat spacetime as per the previous deﬁnition; the metric structure of future null inﬁnity is not uniquely determined but it is aﬀected by a gauge freedom in the choice of the compactiﬁcation factor namely, if we rescale Ω as ωΩ with ω ∈ C ∞ (, R+ ), the topology and the diﬀerentiable structure of future null inﬁnity is left unchanged. Hence the diﬀerence between the possible geometries for the conformal boundary is caught by equivalence classes of the following triplet of data (, na , hab ) where stands for the S2 × R topology of . ˆa ˆ the covariant derivative with respect to gˆab ) and Ω (being ∇ null inﬁnity, na = ∇

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. hab = gˆab | . Two triplets (, na , hab ) and (, na , hab ) are called equivalent if and only if it exists a gauge factor ω such that hab = ω 2 hab whereas na = ω −1 na . The set of all these equivalence classes is universal in the sense that, given any two asymptotically ﬂat spacetimes M1 and M2 with associated triplets (1 , n1a , hab 1 ) ), it always exists a diﬀeomorphism γ ∈ Diﬀ( , ) such that and (2 , n2a , hab 1 2 2 ab γ ∗ hab 2 = h1 and γ∗ n1a = n2a . Definition 2.1. The set of all group elements γ ∈ Diﬀ(, ) mapping a triplet into a gauge equivalent onea is called the Bondi–Metzner–Sachs group (BMS), GBMS [19, 22–24]. It is always possible to choose ω in such a way that on null inﬁnity we can introduce the so-called Bondi frame (u, z, z¯) where u is the aﬃne parameter along the null complete geodesics generating and (z, z¯) are the complex coordinates constructed out of a stereographic projection on C from (θ, ϕ) ∈ S2 . Accordingly the BMS group GBMS is topologically SO(3, 1) × C ∞ (S2 ): u → u = KΛ (z, z¯)(u + α(z, z¯)), ¯z¯ + ¯b . a . az + b , z¯ → Λ¯ z= , z → Λz = cz + d c¯z¯ + d¯ where Λ is identiﬁed with the matrix ac db satisfying ad − bc = 1, whereas KΛ (z, z¯) =

(2.1) (2.2)

1 + |z|2 . |az + b|2 + |cz + d|2

2.1. Group theoretical data Starting from (2.1) and (2.2), in a ﬁxed Bondi frame, GBMS can be viewed as a regular semidirect product of SO(3, 1)↑ , the proper orthocronous subgroup of the Lorentz group and the Abelian additive group C ∞ (S2 ) i.e. GBMS = SO(3, 1)↑ C ∞ (S2 ). Particularly, if denotes the product in GBMS , ◦ the composition of functions, · the ¯ as said in the right-hand pointwise product of scalar functions and Λ acts on (ζ, ζ) sides of (2.2): ¯ ¯ ¯ KΛ (Λ(ζ, ζ))K Λ (ζ, ζ) = KΛ Λ (ζ, ζ), (Λ , α ) (Λ, α) = (Λ Λ, α + (KΛ−1 ◦ Λ) · (α ◦ Λ)) .

(2.3) (2.4)

In the forthcoming discussion concerning the construction of ﬁeld theories on , the GBMS group is going to play a key role and thus it is necessary to better understand a The

constraint we impose is equivalent to require that the bulk geometry is left unchanged i.e. we are working on a ﬁxed background.

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and characterize its structure. To this avail, the ﬁrst step consists of a throughout analysis of the GBMS subgroups. Particularly it is worth to develop a more detailed analysis for C ∞ (S 2 ) whose elements are usually referred to in the physical literature as supertranslations. As a subgroup it is straightforward to realize that it is an inﬁnite-dimensional Abelian ideal of GBMS ; hence C ∞ (S2 ) as well as the full GBMS group are not ordinary Lie groups. Nonetheless, in the class of inﬁnite-dimensional groups, they lie in a rather privileged class, the nuclear groups ﬁrst introduced by Gelfand and Vilenkin [25]: Definition 2.2. A group G is a nuclear Lie group if it exists a neighborhood of the unit element in G which is homeomorphic to a neighborhood of a countably Hilbertb nuclear space. To prove that GBMS satisﬁes the hypothesis of this deﬁnition, we shall make use, also for later convenience, of a speciﬁc construction for nuclear spaces [6,26,27] diﬀerent from the one proposed in [9]: Proposition 2.1. Let H be any real separable Hilbert space with norm , and let A be any self-adjoint densely deﬁned operator on H such that it exists an orthonormal base {ei } (i ∈ N) of H satisfying the conditions: (1) Aei = λi ei , ∀i ∈ N, (2) 1 < λ1 ≤ · · · ≤ λn ≤ · · · , ∞ ≤ ∞. (3) ∃α ∈ R+ such that j=1 λ−α j Then if we introduce for any natural number p the subspace of H Ep = {ψ ∈ Dom(A) ⊆ H ψp = Ap ψ < ∞},

(2.5)

we can close each Ep to an Hilbert space with respect to the norm , p and we can introduce the projective limit space E = p Ep . Let us equip E with the projective limit topology τp i.e. an open neighborhood of the origin in E is given by the choice > 0, n ∈ N and by the set U,n = {ψ ∈ H, ψn < }. Then a sequence {ψm }m∈N is said to converge to ψ ∈ E iﬀ it converges to ψ in every Hilbert space Ep . The pair (E, τp ) is metrizable and complete thus it is a Fr´echet space; furthermore, the inclusion map Ep+ α2 → Ep is Hilbert–Schmidt, i.e. E is also a nuclear space. Theorem 2.1. C ∞ (S 2 )c is an inﬁnite-dimensional nuclear Lie group. The proof of the theorem is in the Appendix. topological vector space over C endowed with a family of inner product norms {| · |p , p ∈ N, p ≥ 1} is called a countably Hilbert space if it is complete with respect to the topology induced by the norms. c Within this paper, α(ζ, ζ) ¯ ∈ C ∞ (S 2 ) will stand for a representative of the equivalence class ¯ where α(ζ, ζ) ¯ ∼ α(ζ, ¯ if they diﬀer for a function of zero measure with respect to the [α(ζ, ζ)] ˜ ζ) bA

canonical measure over S 2 .

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Remark 2.1. To each Ep it is possible to associate the topological dual space Ep i.e. the set of continuous linear functionals from Ep to R. It can be closed to Hilbert space with respect to the norm , −p such that ψ−p = A−p ψ being , the L2 (S2 )-norm. We may now deﬁne E = p Ep which is the topological dual space of E and thus we will also refer to it as the space of real distributions on S2 . Consequently we end up with the following Gelfand triplet E ⊂ L2 (S2 ) ⊂ E

(2.6)

together with the set of continuous inclusions E → Ep → L2 (S2 ) → Ep → E . We shall denote with ( , ) the natural pairing between E and C ∞ (S2 ) and it will be subject to the compatibility condition: ¯ = α(ζ, ζ), ¯ α (ζ, ζ) ¯ L2 , ¯ α (ζ, ζ)) (α(ζ, ζ),

∞

(2.7)

¯ ∈ C (S ). In (2.7) , L2 stands for the ¯ ∈ L (S ) and any α (ζ, ζ) for any α(ζ, ζ) internal product in L2 (S2 ). 2

2

2

Remark 2.2. If we adopt the standard topology for SO(3, 1)↑ and the product topology for GBMS , then one can straightforwardly conclude that the conditions of Deﬁnition 2.2 are fulﬁlled; hence the whole BMS group becomes a nuclear Lie group. To conclude the section we wish to state a last theorem concerning the Abelian ideal of the GBMS group which will be exploited in the discussion of the covariant wave functions as in Sec. 4. Theorem 2.2. Referring to T 4 as the closed subspace of C ∞ (S2 ) out of the real ¯ (with l = 0, 1 linear combinations of the ﬁrst four spherical harmonics Ylm (ζ, ζ) ∞ 2 and m = −l, . . . , l) and to ST as the closed subspace of C (S ) out of the linear ¯ l>1 , the following holds: combinations of the spherical harmonics {Ylm (ζ, ζ)} C ∞ (S2 ) = T 4 ⊕ ST, where ⊕ stands for the direct sum. Closure is here deﬁned with respect to to τp . The proof of the theorem is in the Appendix. 3. BMS Free Field Theory Although BMS invariant free ﬁeld theories have been already discussed in previous papers, we shall recast some of the already known results as they appeared in [10,12] since they are a necessary tool in the next sections. 3.1. BMS unitary and irreducible representations Since the GBMS group is an inﬁnite-dimensional nuclear Lie group with a semidirect product structure, its irreps. can be constructed by means of Mackey’s induction techniques (see in particular [30, 31] and the review in [32]) which have been extended to a semidirect product with an inﬁnite-dimensional Abelian ideal in [33].

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To this avail we replace SO(3, 1)↑ , with its universal cover SL(2, C). At a level of theory of representations this amounts only to introduce a further irrep. induced from the Z2 subgroup. Hence, from now on we will shall refer to G BMS = SL(2, C) ∞ 2 C (S ) which is still a nuclear Lie group. The second step in Mackey machinery consists of constructing a “character” by means of the following proposition, proved in [12, Sec. 3.2]: Proposition 3.1. Given an Abelian topological group A, a character is a continuous group homomorphism χ : A → U (1), the latter being equipped with the natural topology induced by C. If A = E ≡ C ∞ (S2 ) then it exists a unique real distribution ¯ = exp[i(β, α(ζ, ζ))] ¯ for any α(ζ, ζ) ¯ ∈ C ∞ (S2 ). Here β ∈ E such that χ(α(ζ, ζ)) ∞ 2 ¯ (β, α(ζ, ζ)) stands for the natural dual pairing between C (S ) and its topological dual E constructed in Remark 2.1. ¯ equipped with the product operation Remark 3.1. The set of characters A, ¯ = χ1 (α(ζ, ζ))χ ¯ 2 (α(ζ, ζ)) ¯ (χ1 χ2 )(α(ζ, ζ))

∀α ∈ E

is an Abelian group called the dual character group. The third step in Mackey’s machinery, applied to a regular semidirect product, consists of the identiﬁcation of three key structures: Definition 3.1. Consider G = B A as the regular semidirect product between ¯ we may a topological Abelian group A and any group B. Then for any χ ∈ A, associate: • The orbit Oχ ⊂ A¯ as the set Oχ = {χ ∈ A¯ | ∃g ∈ G with χ = gχ}, where gχ(a) = χ(g −1 a) for any a ∈ A and for any g ∈ G. . • The isotropy group Hχ = {g ∈ G | gχ = χ}. • The little group Lχ ⊂ Hχ as the subset {g ∈ Hχ | g = (Λ, 0) ∈ G}. Remark 3.2. Referring to G BMS , the construction of the structures outlined in Deﬁnition 3.1 only requires the identiﬁcation of the little groups. As shown in [9,12], given a ﬁxed character χ, its isotropy group is Hχ = Lχ C ∞ (S2 ) whereas the

BMS associated orbit Oχ is the coset GH ∼ SL(2,C) . Furthermore it turns out that Lχ χ all the possible little groups Lχ are closed subgroups of SL(2, C) namely SU (2), SO(2), ∆ the double cover of E(2), SL(2, R) and the set of all cyclic, alternating and dihedral ﬁnite-dimensional groups of order n ≥ 2.

Remark 3.3. According to the above deﬁnition the orbit Oχ should be thought as embedded in the space of characters and generated by the action of SL(2,C) Lχ on χ ¯ where Lχ satisﬁes Lχ χ ¯ = χ. ¯ Equivalently, exploiting either Proposition 3.1 ¯ according to which χ(α) = ei(β,α(ζ,ζ)) for a unique choice of β ∈ E either

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¯ ¯ = χ(Λ−1 α(ζ, ζ)) ¯ = ei(Λβ,α(ζ,ζ)) Λχ(α(ζ, ζ)) for any Λ ∈ SL(2, C), the identity ¯ ¯ Lχ χ = χ can be traded Lχ β = β. Therefore the orbit Oχ can be seen as canoni¯ From on β. cally isomorphic to the locus in E generated by the action of SL(2,C) Lχ now on we shall stick to this more convenient perspective Oχ → E though we will retain the pedex χ for later convenience.

Let us now still focus our attention speciﬁcally to G BMS . Our aim is to introduce the notion of a function transforming under a unitary and irreducible representation of the said group. This topic has been addressed in detail in [12] with the powerful means of ﬁber bundle techniques. Being a rather lengthy and technical topic, we will not discuss it here in detail leaving an interested reader to the mentioned reference. Conversely, we will recall the main results, namely, bearing in mind Deﬁnition 3.1, Proposition 3.1 and Remark 3.2, we can introduce the space ∞ ˜ dµ(p)Φ(p), Φ(p) < ∞ , (3.1) Hµ = Φ ∈ C (Oχ , H) Oχ

where p ∈ Oχ , µ is any representative of the unique quasi-invariant measure class [µ] associated to Oχ (see [12, Theorem 3.1]) whereas H is a suitable target Hilbert ˜ µ inherits the natural space endowed with the scalar product , . Each element in H G BMS -action

dµ(g −1 p) (gΦ)(g −1 p), ∀g = (Λ, α) ∈ SL(2, C) C ∞ (S2 ) (3.2) (gΦ)(p) = dµ(p) which can be rewritten in the more common and convenient form [30]:

dµ(Λ−1 p) σ(s(p)−1 Λs(Λ−1 p))Φ(Λ−1 p), (ΛΦ)(p) = dµ(p) (αΦ)(p) = χ(α)Φ(p), where

−1

dµ(Λ p) dµ(p)

(3.3) (3.4)

is the Radon–Nikodym derivative and where, bearing in mind

Remark 3.3, χ(α) = ei(p,α) . Here s is a global Borel section for the bundle (G BMS , Oχ , π) with π : GBMS → Oχ . We summarize the above discussion with the following statement: Definition 3.2. We call G BMS induced wave function (or GBMS free ﬁeld) any map in (3.1) which satisﬁes (3.2) i.e. it is a square integrable function over Oχ with values in a suitably chosen target Hilbert space H and it transforms under a unitary and irreducible induced representation of the G BMS -group. In order to complete the analysis of free ﬁelds exploiting inducing techniques it is also necessary to construct the full set of Casimir invariants for the unitary G BMS representations. Bearing in mind the example of the Poincar´e group, one hopes to give a group-theoretical deﬁnition to the notion of mass for an induced G BMS -ﬁeld and to univocally characterize the orbit by the lone value of the invariants.

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We exploit the following proposition (see [29, Chap. 4] for the proof): Proposition 3.2. Given any subspace V of a locally convex linear topological space Ψ with Ψ as dual space, any linear continuous functional β : V → C can be extended to a functional on all Ψ . Furthermore if we introduce the annihilator of V as: V 0 = {β ∈ Ψ | (β, v) = 0, ∀v ∈ V },

(3.5)

the following holds

(1) the factor space VΨ0 is the dual space of V, (2) if V is a d-dimensional subspace of Ψ (with d < ∞), then also dimensional.

Ψ V0

is d-

In the G BMS setting, this proposition can be exploited considering the subspace consisting of the real linear combinations of the ﬁrst four real spherical harmonics ¯ ⊂ C ∞ (S2 ) with l = 0, 1 and m = −l, . . . , l. This is a four-dimensional Ylm (ζ, ζ) subspace which we will refer to as T 4 and which, furthermore, is invariant under the SL(2, C) action induced by (2.4). Thus, since, according to Theorem 2.2, C ∞ (S2 ) is a nuclear space and thus a locally convex linear topological space, we can introduce the projection π : E →

E ∼ (T 4 ) , (T 4 )0

(3.6)

where the isomorphism between (T 4 ) and (TE4 )0 is SL(2, C) invariant. The map (3.6) enjoys the following remarkable properties whose demonstration is given in [9, 12] (though with slightly diﬀerent techniques and nomenclatures): ∗ } (with l = 0, 1 and m = −l, . . . , l) be Proposition 3.3. Let β ∈ E and let {Ylm 4 ∗ , Yl m ) = δll δmm where (, ) the base of (T ) constructed in such a way that (Ylm refers to the natural pairing between C ∞ (S2 ) and E . Then

π(β) =

l 1

∗ alm Ylm ,

l=0 m=−l

from which we can extract the four vector 3 ˆ (a00 , a1−1 , a10 , a11 ). π(β)µ = − 4π Moreover if one deﬁnes the real bilinear form B on E such that ˆ 1 ) π(β ˆ 2) , B(β1 , β2 ) = η µν π(β µ ν

∀β1 , β2 ∈ E

(3.7)

then B turns out to be SL(2, C) invariant and a Casimir invariant for the G BMS unitary and irreducible representations. Remark 3.4. In analogy with the Poincar´e counterpart, we will refer to (3.7) as 2 the deﬁning relation for the G BMS squared mass m . Furthermore such proposition

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justiﬁes a posteriori the reason for the name of space of supermomenta for E which is common in the physical literature. Remark 3.5. In [9], McCarthy showed that the values of m2 together with the ˆ sign of π(β) 0 univocally characterize only the orbits for the little group SU (2). Furthermore, as in the Poincar´e theory, there is only one connected subgroup of SL(2, C) which admits m2 = 0 namely ∆ the double cover of E(2). Though we have fully characterized the full set of G BMS induced free ﬁelds, we need to remember that ultimately our goal is to describe them as a Lagrangian/Hamiltonian dynamical system. We shall not deal contemporary with all the possible cases, but we restrict ourselves to the example of the G BMS scalar ﬁeld. Let us thus distinguish between two cases [12, 9]: 2 SL(2,C) (1) The G BMS real massive scalar field which is a map Φ ∈ L ( SU(2)χ , µ) ¯ whose orbit is generated by the action of SL(2,C) SU(2) on the real distribution β =

4π ∗ 4 ¯ 3 mY00 . Furthermore, since β ∈ (T ) , we can exploit the SL(2, C) invariant isomorphism on the right-hand side of (3.6) to conclude that the whole orbit is contained in (T 4 ) . If we now choose µ as the SL(2, C)-invariant measure on the hyperboloid SL(2,C) SU(2)χ , then Φ transforms under a GBMS action as

(gΦ)(β) = eiβ(α) Φ(Λ−1 β),

SL(2, C) ¯ ∀g = (Λ, α) ∈ G β. BMS ∧ β ∈ SU (2)

(3.8)

2 SL(2,C) (2) The G BMS real massless scalar field which is a map Φ ∈ L ( ∆χ , µ) whose orbit is generated by the SL(2,C) action on the real distribution β¯ = ∆

(Cδ + Kδ (2,2) + S|z|−6 )(1 + |z|2 )3 where δ (2,2) represents the derivative of the δ function twice respect to the variable ζ and ζ¯ whereas K, S ∈ R and C ∈ R−{0}. The ﬁxed point β¯ lies in (T 4 ) and thus we may exploit (3.6) to conclude that the whole orbit lies in (T 4 ) . If we choose µ as an SL(2, C)-invariant measure on the light-cone SL(2,C) ∆χ , then Φ transforms under a GBMS action as SL(2, C) ¯ (gΦ)(β) = ei(β,α) Φ(Λ−1 β) ∀g = (Λ, α) ∈ G β. (3.9) BMS ∧ β ∈ ∆ We recall that, according to [12, Theorem 3.2], only the ﬁeld living on the orbit with K = S = 0 can coincide with the projection on null inﬁnity of a solution for the bulk massless Klein–Gordon equation conformally coupled to gravity. For this reason when we shall refer from now on only to this case. 4. The Covariant Wave Function and the Associated Functional Spaces The aim of this section is to ﬁll a gap in the discussion of ﬁeld theory at future null inﬁnity as it is appeared up to now in the literature. In [10, 12] the building block

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of a BMS invariant theory is an induced wave function, i.e. a map transforming under a unitary and irreducible representation of the BMS group. On the opposite the covariant perspective, though fully equivalent to the canonical one and more common in physics, was introduced but not discussed in detail. Since this latter point of view is ultimately the most natural one to deal with a Lagrangian or an Hamiltonian formulation of the a free ﬁeld theory, we need to amend such a lack. The starting point is to deﬁne the notion of a covariant ﬁeld at null inﬁnity. It ﬁnds a natural formulation within the framework of ﬁber bundles as already discussed in [12]. Nonetheless, as a starting point, we need to ﬁll in a gap of previous discussions i.e. we need to prove the following lemma: Lemma 4.1. Fixing the nuclear space E = C ∞ (S2 ) and its topological dual space E along the lines of Remark 2.1, the complex valued function i

¯

¯ = e− 2 α(ζ,ζ)L2 ϕ(α(ζ, ζ))

(4.1)

is the characteristic function of a unique probability measure ν on E . Proof. The reasoning is similar to the standard one for the Schwartz space of real-valued rapidly decreasing functions on R and it relies on the Minlos’ theorem (see [6] and in particular [29]). In detail we need to show that ϕ is continuous, equal to 1 if evaluated at 0 and positive deﬁnite. The ﬁrst two steps are a straightforward consequence of (4.1). Therefore we need only to verify the positivity of (4.1). Let us thus consider any ﬁnite n-tuple of complex numbers {zi }ni=1 and let us call with C ⊂ C ∞ (S 2 ) the subspace (with norm , L2 ) spanned by any but ﬁxed n-tuple of smooth functions over S2 , say ¯ n . Referring to the standard Gaussian measure on C with µC , then any {αi (ζ, ζ)} i=1 ¯ ∈ C satisﬁes α(ζ, ζ) ¯ ¯ ¯ dµC (α )ei(α (ζ,ζ),α(ζ,ζ)) = eiα(ζ,ζ)L2 , C

where ( , ) is the internal product in L2 (S2 ). Consequently n n ¯ ¯ ¯ ¯ ¯ zj z¯k ϕ(αi (ζ, ζ) − αk (ζ, ζ)) = dµC (α )ei(α (ζ,ζ),αj (ζ,ζ)−αk (ζ,ζ)) j,k=1

j,k=1

= C

C

dµC (α )

n

¯

¯

|ei(α (ζ,ζ),αj (ζ,ζ)) |2 ≥ 0,

j=1

which grants us that ϕ satisﬁes the conditions of Minlos’ theorem. The pair (E , ν) plays in the BMS ﬁeld theory the same role that the space of momenta (R4 , d4 k) plays for a Poincar´e invariant ﬁeld theory over Minkowski spacetime. It is thus natural to ask ourselves if we can deﬁne a natural counterpart in the BMS setting also for L2 (R4 , d4 k) as well for S(R4 ), the set of rapidly decreasing

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test function over R4 and the space of tempered distributions S (R4 ). In order to deal with this question we employ white noise distribution theory [6, 26]. Definition 4.1. We call the space of square-integrable functions over the supermomenta the set of equivalence class of maps ˜ ν) = ψ : E → H ˜ ψ(β), ψ(β)dν(β) < ∞ , (4.2) L2 (E , H, E

˜ Two functions are equivalent if they agree where , is the internal product on H. everywhere except in a set of zero measure. The above space is also referred to as (L2 )H˜ or, in [6, 27] in a slightly diﬀerent context, “white noise space”. Eventually we can deﬁne 2 Definition 4.2. A G ˜ , which transBMS covariant field is a function ψ ∈ (L )H forms under a unitary representation of the G group as: BMS

[U (g)ψ](β) = ei(β,α) D(Λ)ψ(Λ−1 β),

¯ ∈G ∀g = (Λ, α(ζ, ζ)) BMS

(4.3)

where D(Λ) is a unitary SL(2, C) representation. As in the induced scenario we shall work with a speciﬁc example namely: Definition 4.3. A G BMS real scalar covariant field is a map ψ which lies in (L2 )R ≡ (L2 ) which transforms as: [U (g)ψ](β) = ei(β,α) ψ(Λ−1 β),

¯ ∈G ∀g = (Λ, α(ζ, ζ)) BMS .

(4.4)

The Deﬁnitions 4.2 and 4.3 are at this stage useless until two important aspects are clariﬁed. The ﬁrst concerns the relation of (4.3) with the induced wave function (3.2). Following the seminal work of Wigner for the Poincar´e group, such a problem has been tackled in [10, 12] where the two approaches are shown to be equivalent provided that suitable constraints are imposed to (4.3). Nonetheless these should be interpreted as suitable operators acting on (L2 )H˜ and their deﬁnition requires the introduction of a suitable space of test functions and of generalized functions associated with L2 H˜ . The theory we refer to is developed in [6, 26] together with detailed discussions and proofs of the main statements. Conversely we shall provide the relevant notions adapted to the speciﬁc scenario we are interested in. ˜ = C (or R), we recall that, As a starting point and choosing for simplicity H according to the Itˆo–Wiener theorem, each function ψ ∈ (L2 ) can be decomposed as ψ(β) =

∞

In (fn ),

fn ∈ C ∞ (S2 )⊗n c ˆ

(4.5)

n=0 ˆ where C ∞ (S2 )⊗n represents the complexiﬁcation of the n-times symmetric tensor c product of C ∞ (S2 ) whereas In is the multiple Wiener integral deﬁned as the linear

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ˆ functional In : C ∞ (S2 )⊗n → C such that for any n1 + n2 + · · · = n c

¯ ⊗n1 ⊗ ¯ ⊗n2 ⊗ ¯ ¯ ˆ α2 (ζ, ζ) ˆ · · ·)(·) = Fn1 [(·, α1 (ζ, ζ))]F In (α1 (ζ, ζ) n2 [(·, α2 (ζ, ζ))] . . . , (4.6) x2

x2

where Fn [x] = (−)n e 2 ∂xn e− 2 . A further interesting presentation of an element in (L2 ) consists of showing that the Itˆo–Wiener decomposition (4.5) is ultimately equivalent to ([6, §5]): ψ(β) =

∞

(:β ⊗n :, fn )

(4.7)

n=0

where ( , ) refers to the canonical pairing between C ∞ (S2 ) and E whereas :β ⊗n : stands for the Wick tensor [n/2] n ˆ τ ⊗k , :β ⊗n : = (2k − 1)!!β ⊗(n−2k) ⊗ 2k k=0

ˆ is the symmetrized tensor product and τ : Ec⊗2 → C, is the trace operator where ⊗ mapping two elements η, ξ in the complexiﬁcation of C ∞ (S2 ) into (τ, η ⊗ ξ) = η, ξ, being , the internal product in L2 (S 2 ). Proposition 4.1. Given Γ(A) the densely deﬁned operator on (L2 ) such that Γ(A)ψ =

∞

In (A⊗n fn ),

n=0

we introduce for any p ∈ N the set (E)p = {ψ ∈ (L2 ) | Γ(A)p ψ ∈ (L2 )}.

(4.8)

Closing (E)p to Hilbert space with respect to the norm ψp = Γ(A)p ψ(L2 ) then we may ﬁx (E) = p (E)p as the projective limit of the sequence (E)p and (E )p , (E ) as the topological dual space respectively of (E)p and of (E). Then (E) is a nuclear space with an associated Gelfand triplet (E) ⊂ (L2 ) ⊂ (E ), and with the following series of continuous inclusions (E) → (E)p → (L2 ) → (E )p → (E ), where (E )p is now the completion of (L2 ) with respect to the norm ψ−p = Γ(A)−p ψ(L2 ) . The spaces (E) — endowed with the projective limit topology — and (E ) are, respectively, called the space of Hida testing functionals and of Hida distributions.

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We can endow (E) and (E ) with the natural pairing , subject to the compatibility condition ψ(β), ψ (β) = dν(β)ψ(β)ψ (β), (4.9) for any ψ(β) ∈ (E) and ψ (β) ∈ (L2 ). Nonetheless, in order to correctly identify the constraints which reduce the covariant to the induced wave function, we need now to introduce the concept of multiplication operator. ¯ ∈ C ∞ (S2 ), we call multiplication operator Definition 4.4. Given any α(ζ, ζ) (along the α-direction) the continuous operator Qα : (E) → (E) such that ¯ Qα ϕ(β) = (β, α(ζ, ζ))ϕ(β),

¯ ∈ C ∞ (S2 ). ∀ϕ ∈ (E) ∧ ∀α(ζ, ζ)

(4.10)

˜ α as the continuous extension of Qα to (E ) which is Furthermore we refer to Q deﬁned in analogy with (4.10). Bearing in mind the above deﬁnitions, we are now facing the following situation: 2 aG BMS covariant ﬁeld is, per Proposition 4.1, an element of (L ) and hence a Hida distribution. At the same time a G BMS free ﬁeld is per (3.1) a function whose support is Oχ embedded in E . In the case of a Poincar´e invariant free ﬁeld, according to Wigner seminal work, one would need to impose suitable constraints on the covariant ﬁeld in order to reduce it to the induced counterpart. We can extend such prescription also to the BMS scenario though one needs to resort to Hida distributions. To be precise, as a starting point, let us introduce a ﬁnite-dimensional counterpart of the elements in (E) as stated in [34]. Definition 4.5. Let C ∞ (S2 ) ⊂ H ⊂ E be the Gelfand triplet constructed in Remark 2.1 out of which the space of Hida distributions (E ) has been constructed as in Proposition 4.1. If we choose any k-tuple {e1 , . . . , ek } ⊂ L2 (S2 ) with k < ∞ and if we refer to V as the real linear space spanned by e1 , . . . , ek , we may introduce . the space (E )V as the (E )-closure of all polynomials in ·, e = (·, e1 , . . . , ·, ek ). Then we call ψ ∈ (E ) a finite-dimensional Hida distribution if ψ ∈ (E )V for . some ﬁnite-dimensional subspace V constructed as above. We call (E)V = (E) ∩ (E )V the space of finite-dimensional Hida test functions. The above deﬁnition clearly underlines that certain speciﬁc Hida distributions could be interpreted as ﬁnite-dimensional distributions. Unfortunately the Gelfand triplet S(Rk ) ⊂ L2 (Rk ) ⊂ S (Rk ) is not the best candidate to work with. Thus we need to introduce a new auxiliary Gelfand triplet. Let us consider P(Rk ), the space of polynomials in xµ = (x1 , . . . , xk ) ∈ Rk with k < ∞ and, calling µk the standard ¯ k ) with respect to the Gaussian measure on Rk , its closure to Hilbert space P(R

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inner product

(F, G) = Rk

F (xµ )G(xµ )dµk (xµ ),

∀F, G ∈ P(Rk ).

Let us introduce now the Ornstein–Uhlenbeck operator on Rk i.e. L = ∇2 − k ∂ k 2 i=1 xi ∂xi , where xi are the Cartesian coordinates on R whereas ∇ is the Laplak cian operator on R . As shown in [34] we can now exploit Proposition (2.1) with ¯ k ) given by the vector Hn (xµ ) = k Hn (xµ ) respect to the basis for L in P(R i i=1 being Hni the ni th Hermite polynomial. Thus we introduce the sequence of spaces — with respect to the parameter t ∈ R — ¯ k ) | exp(−tL)ψ ∈ P(R ¯ k )}. It (Rk ) = {ψ ∈ P(R

(4.11)

Closing this space to Hilbert space with respect to the internal product (F, G)t = (exp(−tL)F, exp(−tL)G),

∀F, G ∈ It (Rk ),

one ends up for any real positive value of t with the sequence of continuous inclusions ¯ k ) → I−t (Rk ). It (Rk ) → P(R

Considering now the projective limit space I(Rk ) = t It (Rk ), endowed as in Proposition 2.1 with the projective limit topology, we end up with the Gelfand triplet ¯ k ) ⊂ I (Rk ), I(Rk ) ⊂ P(R

(4.12)

where I (Rk ) is the topological dual space of I(Rk ). We can now formulate a characterization theorem for ﬁnite-dimensional Hida distributions whose demonstration has been given in [34]: Theorem 4.1. Referring to the Gelfand triplet C ∞ (S2 ) ⊂ L2 (S2 ) ⊂ E let us choose, as in Deﬁnition 4.5, a k-tuple {ei }ki=1 ∈ C ∞ (S2 ) whose elements are mutually orthogonal with respect to the inner product in L2 (S2 ) and let us call V = span{ei }ki=1 . Then for any ﬁnite-dimensional Hida distribution ψ ∈ (E)V , it exists a function F ∈ I(Rk ) such that ϕ = F (xµ ) where xµ = ( · , e1 , . . . , · , ek ). k Furthermore let πV : E → V be the map πV (β) = i=1 (β, ei )ei where ( , ) stands for the pairing between C ∞ (S2 ) and E . Then πV automatically induces a . projection ΠV : (E ) → (E )V such that ΠV = Γ(πV ) maps any ψ ∈ (E ) in ∞ ∞ ΠV ψ = ΠV In (fn ) = In (πv⊗n fn ), (4.13) n=0

n=0

being In the multiple Wiener integral as in (4.5). Thus it holds that ψ ∈ (E )V iﬀ ΠV ψ = ψ.

(4.14)

The above theorem grants us that any Hida testing functionals which satisﬁes (4.14) naturally identiﬁes a function lying in I(Rk ). Let us also remember

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that [34]: 1

Proposition 4.2. If a function F (xµ ) lies in I(Rk ) then F (xµ )e− 4 δ S(Rk ) being δ µν the Kronecker delta.

µν

xµ xν

lies in

We have now all the ingredient to exploit the theory of ﬁnite-dimensional Hida distributions to construct the equations of motion for the BMS free ﬁeld. Bearing in mind that both the orbit of the massive and massless real scalar G BMS ﬁeld lies 4 4 ¯ in (T ) , it is natural to choose V = T = spanR {Ylm (ζ, ζ)}l=0,1 . Lemma 4.2. The orbit/support of a covariant real (massive or massless) scalar ﬁeld ψ lies in (T 4 ) iﬀ ΠT 4 ψ(β) = ψ(β).

(4.15)

Proof. We exploit the Itˆo–Wiener decomposition of a generic functional in (L2 ) ⊗n :, fn ) and (4.13). According to this latter equation and introas ψ(β) = ∞ n=0 (:β ¯ . . . , Y11 (ζ, ζ)), ¯ (4.15) reads: ducing eµ = (Y00 (ζ, ζ), ∞ 4 ∞ ⊗n (eµ , fn )eµ = (:β ⊗n :, fn ), :β :, n=0

µ=0

n=0

which is satisﬁed iﬀ β ∈ (T 4 ) or fn lies in (T 4 )⊗n for any n. In this latter case we shall make use of Proposition 3.2 (see also its proof in [29]) to conclude that, ¯ ∈ T 4, whenever a generic distribution β ∈ E is evaluated with a test function α(ζ, ζ) it is equivalent to extract from β a representative in an equivalence class of (TE4 )0 ¯ Bearing in mind the SL(2, C)-invariant isomorphism and evaluate it with α(ζ, ζ). between (TE4 )0 with (T 4 ) , the statement of the theorem holds. ˆ

For later convenience it is interesting to notice that the above equation of motion can be written as Lemma 4.3. Bearing in mind the decomposition in Theorem 2.2, a ﬁeld ψ ∈ (L2 ) satisﬁes (4.15) iﬀ Qα(ζ,ζ) ¯ ψ(β) = 0,

¯ ∈ ST. ∀α(ζ, ζ)

(4.16)

The proof is given in the Appendix. We have now identiﬁed the class of covariant G BMS scalar ﬁelds ψ which are 4 supported on (T ) . The last step consists of choosing suitable constraints which grant us that ψ is supported either on the hyperboloid SL(2,C) SU(2) either on the light cone

SL(2,C) . ∆

As one could forsee, it holds:

scalar ﬁeld ψ has support on the orbit generated Lemma 4.4. A G BMS covariant

SL(2,C) 3 by action on β¯1 = mY ∗ or on that generated by SL(2,C) action on SU(2)

4π

00

∆

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β¯2 = Cδ iﬀ, besides (4.15), ψ satisﬁes 0 for the little group ∆ µν η Qeµ Qeν ψ(β) = , m2 ψ(β) for the little group SU (2) ¯ . . . , Y11 (ζ, ζ)) ¯ and η µν = diag(−1, 1, 1, 1). where eµ = (Y00 (ζ, ζ),

(4.17)

The proof is given in the Appendix. We have almost completed our task. According to Lemmas 4.2 and 4.4 we have 2 satisfying (4.4) can be reduced to shown that a G BMS covariant scalar ﬁeld ψ ∈ L a function on the mass hyperboloid or on the light cone transforming under a scalar G BMS unitary and irreducible representation iﬀ it satisﬁes Eqs. (4.15) and (4.17). The tricky point is the following: can we conclude that this function is square integrable with respect to the measure on each orbit? At this stage this is deﬁnitely not possible since Theorem 4.1 grants us that a map ψ ∈ (L2 ) such that ΠT 4 ψ(β) = ψ(β) is in 1:1 correspondence with the functions — say ψ(pµ ) with pµ = β, eµ — ¯ 4 ) as per (4.12). Hence we need to exploit Theorem 4.2 to claim in I(R4 ) ⊂ P(R δµν p p ˜ µ ) = e− 4µ ν ψ(pµ ) lies in S(R4 ) ⊂ L2 (R4 , d4 p). The remaining constraint that ψ(p (4.17) does not harm the previous reasoning since it corresponds in the space I(R4 ) to impose the usual equation

[η µν pµ pν − m2 ]ψ(pµ ) = 0, ˜ µ ). Thus we can summarize the full conwhich is also identically satisﬁed by ψ(p struction in the following theorem: Theorem 4.2. A covariant G BMS (massive or massless) scalar ﬁeld ψ : E → R which transforms as (4.4) and which satisﬁes Eqs. (4.15) and (4.17) corresponds to a G BMS induced scalar ﬁeld ((3.8) or (3.9)) up to the rescaling of the latter by

e−

δµν pµ pν 4

.

4.1. G BMS equations of motion as evolution equations To conclude this section it is natural to deal with the following remark: the equations of motion for a G BMS scalar ﬁeld are constraint equations as their counterpart in a Poincar´e invariant free ﬁeld in the momenta space. We now wonder whether it is possible to recast them as an evolution equation. To answer this query, we ﬁrst introduce the notion of derivative operator as discussed in [6, 26]: Definition 4.6. Let us consider any Hida testing functional ψ(β) on (E); we deﬁne the Gateaux derivative of ψ(β) along the direction β˜ ∈ E as the continuous operator Dβ˜ : (E) → (E) such that ∞ ˜ − ψ(β) ψ(β + β) ˜ fn )). = (:β ⊗(n−1) :, (β, →0 n=1

Dβ˜ ψ(β) = lim

˜ ˜ : (E ) → (E ). The operator Dβ˜ admits a unique continuous extension to D β

(4.18)

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We are going to state now an important lemma, proved in [6], which shows that, opposite to the usual behavior, the multiplication operator is a sort of “derivative operator” i.e. it obeys a Leibnitz rule. ¯ ∈ C ∞ (S2 ), the following equality holds: Lemma 4.5. For any α(ζ, ζ) ∗ Qα(ζ,ζ) ¯ = Dα(ζ,ζ) ¯ + Dα(ζ,ζ) ¯ ,

which is meant as a continuous operator form (E) into itself. Furthermore, for any β ∈ E , it also holds: Qβ = Dβ + Dβ∗ , which is meant as a continuous operator from (E) into (E ). In order to switch from a constraint equation such as (4.16) and (4.17) to an evolution equation, the natural step in the canonical formulation of quantum ﬁeld theory over Minkowski background consists of performing a Fourier transform F˜ . In the framework of white noise calculus, we shall use a slightly less intuitive concept ﬁrst discussed in [6, 35]: Definition 4.7. We call Fourier–Gauss transform the continuous linear operator Ga,b : (E) → (E) such that, being a, b ∈ C − {0}, Ga,b ψ(β) = ψ(aβ + bβ)dµ(β ), ∀ψ ∈ (E). (4.19) E

¯ ∈ C ∞ (S2 ), it holds that Furthermore, for all a, b ∈ C − {0} and for all α(ζ, ζ) −1 Dα(ζ,ζ) Ga,b Dα(ζ,ζ) ¯ = b ¯ Ga,b ,

(4.20)

2 −1 Ga,b Qα(ζ,ζ) Dα(ζ,ζ) ¯ = a b ¯ Ga,b + bQα(ζ,ζ) ¯ Ga,b ,

(4.21)

where Dη and Qη are the multiplication and derivative operators respectively introduced in Deﬁnitions 4.4 and 4.6. Moreover, bearing in mind the spaces ((E)p , , p ) as introduced in (4.8), if a2 + b2 = 1 and |b| = 1, then Ga,b ψ(β)p = ψ(β)p for all ψ ∈ (E)p and for all p ≥ 0. We seek now to single out a preferred Ga,b within the set of Fourier–Gauss transforms parametrized by the complex numbers a, b. The criterion, we shall refer to, consists of requiring that the kernel of the operator η µν Qeµ Qeν is mapped into the kernel of a new but symmetric operator. Bearing in mind that, for a linear operator, such a condition coincides with the request of self-adjointness, it holds: Proposition 4.3. There are only two Fourier–Gauss transforms, namely G√2,i and G√2,−i , such that Ga,b η µν Qeµ Qeν = A(a, b)Ga,b where A(a, b) is a linear continuous selfadjoint operator on (E) which admits an extension to a unitary operator on (L2 ). Furthermore √ A( 2, ±i) = ∓η µν (Qeµ − 2Deµ )(Qeν − 2Deν ), where the plus stands for b = −i whereas the minus for b = i.

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Proof. According to Deﬁnition 4.7, for any non vanishing a, b ∈ C the Fourier– Gauss transform is a continuous linear operator from (E) into itself such that, exploiting (4.20), Ga,b η µν Qeµ Qeν = η µν [a4 b−2 Deµ Deν + b−2 Qeµ Qeν + a2 (Deµ Qeν + Qeµ Deν )]Ga,b . In order to realize when A(a, b) = η µν [a4 b−2 Deµ Deν + b−2 Qeµ Qeν + a2 (Deµ Qeν + Qeµ Deν )] is self-adjoint on (E) we refer to Lemma 4.5 and to the relation Q∗α(ζ,ζ) ¯ ¯ = Qα(ζ,ζ) ¯ ∈ C ∞ (S2 ) according to which: on (E) for any α(ζ, ζ) A∗ (a, b) = η µν [a4 b−2 Deµ Deν + (a4 b−2 + b2 + 2a2 )Qeµ Qeν − (a4 b−2 + a2 )(Qeµ Deν + Deµ Qeν )]. Thus, on (E), A∗ (a, b) = A(a, b) iﬀ a2 = −2b2 . If we require that Ga,b could also be extended to a unitary operator on (L2 ), then, according to Deﬁnition √ 4.7, we also impose |b| = 1 and a2 + b2 = 1 which imply that b = ±i and a = ± 2. To conclude we refer to [6, Theorem 11.28] and remarks below according to which Ga,b = Gc,d iﬀ a = ±c and b = d. Thus G√2,±i = G−√2,±i on (E). Remark 4.1. The arbitrariness in the choice of the Fourier–Gauss transform which arises from the previous theorem is only apparent. If we exploit [6, Theorem 11.30], according to which, ∀a, b, c, d ∈ C − {0}, Gc,d Ga,b = G±√a2 +b2 c2 ,bd , we end up with −1 , G√2,i = G√ 2,−i

and viceversa. √ Furthermore, since, according to the previous proposition, √ √ A( 2, i) = −A( 2, −i) it √ is immediate to conclude that ψ(β) ∈ Ker(A( 2, i)) ⊂ (E) iﬀ ψ(β) ∈ Ker(A( 2, −i)) and viceversa. For this reason we deal only with one of the two choices and, from now, G will stand for G√2,i whereas G −1 = G√2,−i . To conclude the section we summarize the latter results i.e. if we start from (4.17) and (4.16) and if we perform the Fourier–Gauss transform G, we end up with:

η µν (−2Deµ

G ¯ ∈ ST, ∀α(ζ, ζ) (−2Dα(ζ,ζ) ¯ + Qα(ζ,ζ) ¯ )ψ (β) = 0, 0 for ∆ G , + Qeµ )(−2Deν + Qeν )ψ (β) = m2 ψ G (β) for SU (2)

where ψ G (β) =

E

√ dµ(β )ψ( 2β + iβ).

(4.22) (4.23)

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5. The Lagrangian and Hamiltonian Formulation of G BMS Scalar Field Theory In the previous discussions we have developed the covariant approach to G BMS scalar ﬁeld theory and the related equations of motion. Unfortunately, these have not been derived from a variational principle and now we shall amend such deﬁciency. The steps we will perform are the following: ﬁrst we construct a suitable “Lagrangian” functional whose extremum provides (4.22) and (4.23) and then we derive the Hamiltonian function by means of standard techniques. Remark 5.1. In order to construct the above mentioned Lagrangian, the starting point consists of introducing a suitable space of kinematically allowed conﬁgurations. In an inﬁnite-dimensional setting, there are two commonly accepted and widely exploited choices: the tangent bundle and the ﬁrst jet bundle. In the latter case we should deal with equivalence classes of sections of an associated bundle over E , but alas the characterization of a jet over the space of distribution over S 2 is rather tricky. On the other hand it is more convenient to identify a tangent bundle over a suitable space of functions and an associated Lagrangian. echet In the setting proper of G BMS covariant ﬁeld theory we deal with a Fr´ manifold i.e. (E) the space of Hida testing functionals. It would be possible to associate to it a notion of tangent space identifying (E) with an Abelian ILH (inverse limit of Hilbert) group [46] but we shall follow a path closer to the one employed in a Poincar´e invariant setting. To wit we shall consider the set of Hida testing functionals as continuously embedded in (L2 ) and, hence, exploiting Riesz theorem we end up with T (L2 ) ≡ (L2 ) × (L2 ). A potential reader should notice that all the relevant operators which appeared in the previous sections, namely Qα(ζ,ζ) ¯ , Dα(ζ,ζ) ¯ and the Fourier–Gauss G transform admit a unique continuous extension from their natural space of deﬁnition, (E), to (L2 ). In the case of G such extension identiﬁes a unitary operator. Starting from these premises, we shall now solve the inverse “Lagrangian” problem i.e. we shall seek a functional L : (L2 ) → R whose extremum is (4.22) and (4.23). The strategy we follow consists of ignoring at the beginning (4.22) requiring only that our G BMS real massive or massless scalar ﬁeld satisﬁes (4.23). Hence, as a ﬁrst result, we need to prove a useful lemma. Before stating it, let us remind the reader that, given a Banach space X and its dual space X , an operator F : X → X is called potential on some subset H ⊂ X iﬀ it exists a functional f on X such that, for all x ∈ H, F (x) = ∇f (x) where ∇ is the gradient of the functional f . Lemma 5.1. Referring to Q and to D as the unique continuous extension of the multiplication and derivative operator from (E) to (L2 ) and referring to eµ , eν as ¯ . . . , Y11 (ζ, ζ)}, ¯ then the operator A = η µν (Qeµ − 2Deµ )(Qeν − 2Deν ) : {Y00 (ζ, ζ), 2 2 (L ) → (L ) is potential and the unique functional Sdyn : (L2 ) → R, whose value

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in ψ0 ∈ (L2 ) is S0 , is

Sdyn (ψ) = S0 +

1

dt η µν (Qeµ − 2Deµ )(Qeν − 2Deν )

0

× (ψ0 + t(ψ − ψ0 )), ψ − ψ0 (L2 ) ,

(5.1)

where , (L2 ) is the internal product on (L2 ) as introduced in (4.9). Proof. Our strategy to prove that the operator A is potential, consists in employing Vainberg’s theorem ([36, §5]). Hence, identifying the Hilbert space (L2 ) with its dual by means of the Riesz theorem, A is a map from (L2 ) to (L2 ) . It admits a continuous Gateaux derivatived for all ψ ∈ (L2 ) and along all directions ψ ∈ (L2 ) since, per deﬁnition (4.18) and, being A linear, Dψ [η µν (Qeµ − 2Deµ )(Qeν − 2Deν )]ψ = η µν (Qeµ − 2Deµ )(Qeν − 2Deν )ψ . Furthermore, on (L2 ) the operator under analysis is, according to Proposition 4.3, selfadjoint thus symmetric. The hypotheses of Vainberg’s theorem are fulﬁlled and η µν (Qeµ − 2Deµ )(Qeν − 2Deν ) is potential. Even uniqueness of the functional, i.e. (5.1), whose gradient satisﬁes Eq. (4.22) is a direct consequence of Vainberg’s theorem which grants us that Dψ S(ψ) = η µν (Qeµ − 2Deµ )(Qeν − 2Deν )ψ, ψ (L2 ) . Thus for any ψ in a ball D = {ψ ∈ (L2 ) | ψ − ψ0 (L2 ) < r} centered in ψ0 and of ﬁxed radius r, and for any t ∈ [0, 1] the last equality translates as d S(ψ0 + t(ψ − ψ0 )) dt = η µν (Qeµ − 2Deν )(Qeν − 2Deν )(ψ0 + t(ψ − ψ0 )), ψ − ψ0 . An integration in the t variable shows that (5.1) is the unique functional whose gradient is our equation η µν (Qeµ − 2Deµ )(Qeν − 2Deν )ψ(β) = 0. Remark 5.2. Setting the initial condition as ψ0 = 0, S0 = 0 and adding the mass term in the case of a massive G BMS real scalar ﬁeld, then (5.1) is: 1 [η µν (Qeµ − 2Deµ )(Qeν − 2Deν ) + m2 ]ψ, ψ(L2 ) 2 1 = dµ(β)η µν [−4Deµ ψ(β)Deν ψ(β) + (β, eµ )(β, eν )ψ 2 (β) 2 E + 4(β, eµ )ψ(β)Deν ψ(β) + m2 ψ 2 (β)],

LKG (ψ) =

(5.2)

where we exploited the deﬁnition of multiplication operator and Lemma 4.5 whereas (β, eµ ) stands for the canonical pairing between E and C ∞ (S2 ). d The deﬁnition Gateaux derivative on a functional from (L2 ) to R is a straightforward adaptation of Deﬁnition 4.6. For this reason, we shall still employ the symbol D.

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We will refer to this term as the Klein–Gordon part of the Lagrangian for the G BMS massive or massless scalar ﬁeld. To conclude we need also to implement (4.23). A direct inspection shows that (4.23) is a family of constraint on the covariant ﬁelds and thus we seek to implement them in terms of Lagrange multipliers: Proposition 5.1. The functional S : (L2 ) → R whose associated Euler equations are (4.22) and (4.23) is dµ(β)λi (β)[(−2Dei + Qei )ψ(β)]2 , (5.3) S(ψ, λi ) = SKG (ψ) + i

E

¯ l>1 whereas λi (β) ∈ (L2 ) are Lagrange mulwhere SKG is (5.2), ei = {Ylm (ζ, ζ)} tipliers. The operator Q refers to the unique continuous extension to (L2 ) of the corresponding Gateaux derivative and multiplication operator on (E). The theorem is proved in the Appendix. Having solved the inverse Lagrangian problem, we can now formulate the free G BMS ﬁeld theory in an Hamiltonian framework. While the Lagrangian analysis is best performed in the tangent space of a suitably chosen conﬁguration space, the Hamiltonian counterpart is naturally developed in the cotangent space which, exploiting Riesz theorem, can be identiﬁed as T ∗ (L2 ) = (L2 ) × (L2 ). Following the nomenclatures of [38, 39] we deﬁne: . Definition 5.1. We call Γ = (L2 ) × (L2 ) the phase space of a G BMS real (massive or massless) scalar ﬁeld associated to the conﬁguration space (L2 ). The vector space Γ endowed with the continuous bilinear map (with respect to the product topology) Ω : Γ × Γ → R Ω((ψ1 , Ψ1 ), (ψ2 , Ψ2 )) = Ψ2 , ψ1 − Ψ1 , ψ2

(5.4)

is a symplectic vector space. Here , is the internal product on (L2 ) as per (4.9). Unfortunately the construction of the Hamiltonian function is not a straightforward calculation since (5.3) is a singular Lagrangian. Thus we need to resort to the theory of constraints and in particular to the algorithm developed by Gotay, Nester and Hinds in [40, 41] which is a geometrization and a generalization of the canonical Dirac–Bergman theory. Particularly here we will adapt to our Hilbert conﬁguration manifold the analysis of a Lagrangian system with Lagrange multipliers performed in [42, 43] for ﬁnite-dimensional conﬁguration spaces. Since the constraints (4.23) are globally deﬁned on T (L2 ), the ﬁrst natural step consists of promoting the multipliers in (5.3) to dynamical variables thus switching from T (L2 ) to TP ≡ T [(L2 ) × (L2 )N ] where (L2 )N means that we consider as many copies of (L2 ) as the number of needed Lagrange multipliers. In the case under consideration this is equal to the number of spherical harmonics with l > 1. Local coordinates on P are given by (ψ, De0 ψ, λi , De0 λi ) where now D stands for

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the Fr´echet derivative on (L2 ). Within this framework we are entitled to consider (4.23) as a map S : TP → R and thus we can introduce the ﬁbre derivative F L : TP → T ∗ P such that d ψ , F L(ψ)(L2 ) = S(ψ + tψ )|t=0 = Dψ S(ψ) = DS, ψ (L2 ) . dt In local coordinates such a transformation becomes F L(ψ, De0 ψ, λi , De0 λi ) = (ψ, DDψ S, λi , 0) = (ψ, 4De0 ψ − 2Qe0 ψ, λi , 0). (5.5) The above equality simply restates that the Lagrangian function is not hyperregular and thus the ﬁber derivative is not a diﬀeomorphism. Consequently we are obstructed to introduce the Hamiltonian as H = E ◦ F L−1 where E represents the energy function E = ψ, F L(ψ)(L2 ) − S(ψ). On the opposite we might still construct it implicitly on the image of F L(T P ) as H ◦ FL = E which is a reasonable deﬁnition iﬀ for any two points p, p ∈ T P such that F L(p) = F L(p ) then E(p) = E(p ). As discussed mainly in [41], such last condition is satisﬁed if the Lagrangian under analysis is almost regular i.e. F L is a submersion onto T ∗ P and, for any p ∈ TP , the ﬁbers (F L)−1 {F L(p)} are connected submanifolds of TP . Proposition 5.2. The functional (5.3) is an almost regular Lagrangian. The proof is in the Appendix. As a consequence of this last theorem, we know that the energy function in constant along the ﬁbres of F L and thus it induces on the manifold M1 a well deﬁned Hamiltonian function as 3 1 2 dµ(β)Π2 (β) − [Qek ψ(β) − 2Dek ψ(β)] H(ψ, λi , Π) = 2 E k=1 1 − dµ(β)λi (β)[(−2Dei + Qei )ψ(β)]2 , (5.6) 2 i E where Π = −2De0 ψ(β) + Qe0 ψ(β) is the conjugate momentum of De0 ψ(β) whereas ek refers to the three spherical harmonics with l = 1. 6. Comments and Conclusions The origin of this paper could be traced back to the observation in [10] that the inﬁnite-dimensional nature of the supertranslations leads the G BMS ﬁelds to be functionals instead of functions as usual in quantum ﬁeld theory over a curved background. Consequently it appeared that only the purely group theoretical Wigner programme could shed some light on the kinematically and dynamically allowed conﬁgurations for a BMS invariant ﬁeld theory living at null inﬁnity; the paradigm of

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equations of motion as an extremum out of a variational principle was thus a priori discarded. In this paper we wished to overcome the above deﬁciency ad we managed to associate to a scalar G BMS ﬁeld theory a genuine Hamiltonian system. To achieve such a goal we followed the path to rigorously deﬁne and analyse the covariant formulation of a G BMS invariant theory. Within this framework each ﬁeld arises as an element in a suitably constructed space of Hida testing functionals. This novel point of view leads us to write the equations of motion for a G BMS ﬁeld as suitable operators acting on the above mentioned space. Afterwards each equation of motion for a real massive or massless scalar ﬁeld has been interpreted as the Euler–Lagrange equation of a suitable functional. Alas, such a Lagrangian turned out not be hyperregular and thus the ﬁber derivative from the tangent to the relevant cotangent space conﬁguration is not a diﬀeomorphism. Exploiting the geometric description of the constraint algorithm originally due to Nester and Gotay for presymplectic manifolds, we have nonetheless managed to show that on a suitable connected submanifold of the symplectic cotangent bundle, we could identify an Hamiltonian function. From a future perspective, one could claim that, on a physical ground, the results achieved put us into the position to discuss without further ado if an holographic correspondence between bulk and boundary (Yang–Mills) gauge theories really exists in an asymptotically ﬂat spacetime. We envisage that, as a useful tool we shall employ symplectic techniques in order to construct a G BMS interacting ﬁeld theory. From a pure mathematical point of view it appears that the realization of BMS ﬁeld theory as a dynamical system can be coherently and fully described in terms of white noise analysis. The only nuisance to the date consists in the “tangent bundle” approach. In a ﬁnite-dimensional counterpart, it is more common to formulate classical ﬁeld theory in terms of jet bundle which allow to treat on the same ground time and spatial derivatives. Such a problem clearly arises also in a G BMS framework where one wishes to encompass in a unique setting all the Gateaux derivatives along E -directions. Unfortunately, as outlined in Secs. 4 and 5, it appears to be rather diﬃcult to coherently introduce, within the G BMS framework, the notion of (ﬁrst) jet bundle; the most promising road within this direction lies in a sheaf theoretical formulation of the Hamiltonian theory though it would possibly forbid us to deal with global issues. We will analyze in detail such a problem in a future paper. Appendix A. Proofs of the Main Theorems Proof of Theorems 2.1. Let us consider H = L2 (S 2 ) i.e. the space of square integrable functions over the two sphere with respect to the canonical volume element on S 2 and let us consider the operator A = L2 + kI on H where k is any but ﬁxed real number greater than 1 and where L2 is the angular momentum 2 operator. Since S 2 can be identiﬁed with SO(3) SO(2) and since L is the second order

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Casimir operator of SO(3), it coincides with the Laplace Beltrami operator on S 2 up to a factor −2. Thus, with respect to the canonical local chart (θ, ϕ) on S2 , 2 ∂2 ∂2 L2 = −2 ∂θ 2 + sin θ ∂ϕ2 ; hence A is a second-order diﬀerential elliptic operator. If we choose the basis of spherical harmonics {Ylm (θ, ϕ), l ≥ 0, m = −l, . . . , l}, A admits a dense linear manifold of analytic vectors out of ﬁnite linear combinations of spherical harmonics. Therefore A is essentially self-adjoint and, also being A symmetric in L2 (S2 ), it admits a unique self-adjoint extension which will be denoted by the same symbol A. Thus AYlm (θ, ϕ) = λlm Ylm (θ, ϕ) = [l(l + 1) + k]Ylm (θ, ϕ). Furthermore 1 < λ00 < λ1−1 ≤ λ10 < · · · and it holds that l ∞ l=0 m=−l

λ−α lm =

∞

(2l + 1)[l(l + 1) + k]−α < k −α + 4

l=0

∞

l−2α+1 ,

l=1

which implies that the ﬁrst sum is certainly convergent for α > 12 . Thus the hypotheses of the Proposition 2.1 are satisﬁed and we may construct for each p ∈ N the space Ep as in (2.5) with A = L2 + kI. Furthermore Ep ⊂ Eq for α any p > q ≥ 0 and the inclusion map of Ep+ α2 in Ep is given by the operator A− 2 α ∞ which is an Hilbert–Schmidt operator with A− 2 2HS = l=0 (2l+1)[l(l+1)+k]−α. Accordingly, given E = p Ep and τp , the induced limit topology deﬁned in Proposition 2.1, the space (E, τp ) is a nuclear (Fr´echet) space. We now need to show that E coincides with C ∞ (S2 ). Supposee ψ ∈ E1 ; as a consequence, if: ψ = Aψ,

(A.1)

both ψ , ψ ∈ L2 (S2 ). Notice that every ψ ∈ L2 (S2 ) is locally L1 (S2 ) and thus it individuates a distribution of D (S2 ). In this way (A.1) can be interpreted as a diﬀerential equation in weak sense determining the distribution ψ. This is because, for every real-valued h ∈ C0∞ (S2 ) = C ∞ (S2 ), ψ ∈ L2 (S2 ), (A.1) and self-adjointness of A together entail, where A is a proper diﬀerential operator: hψ = (Ah)ψ. S2

S2

A is strongly elliptic ([44, p. 112]) in an open neighborhood Ω (viewed as a neighborhood in R2 ) of every ﬁxed point in S2 barring the points with (θ, φ) ∈ [0, π] × {π}. However those points can be included as well by rotating the coordinate system (using the fact that A is rotation invariant). Since ψ ∈ L2 (Ω) = W0 (Ω) in (A.1) and A is second order, Friedrichs elliptic regularity theorem ([44, p. 112]) entails that ψ ∈ W2 (Ω). The procedure can be iterated twice if ψ ∈ E2 : In this case g1 = A(Aψ) ∈ W0 (Ω) so that g = Aψ ∈ W0+2 (Ω) and thus ψ ∈ W0+2+2 (Ω). Hence ψ ∈ W4 (Ω). By induction ψ ∈ E implies ψ ∈ W∞ (Ω). Eventually, exploiting Sobolev’s lemma ([45, Theorem 7.25]) one has that ψ ∈ E equals a function of e We

are grateful to V. Moretti for pointing out this argument.

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C ∞ (Ω) almost everywhere. Arbitrariness of Ω implies that ψ ∈ E coincides with a function of C ∞ (S2 ) almost everywhere. Proof of Theorem 2.2. The statement of the theorem can be straightforwardly proved in several diﬀerent ways if we refer to L2 (S2 ). One of the simplest consists of recognizing that the spherical harmonics are an orthonormal complete system of L2 (S2 ) constructed according to standard harmonic functions techniques once S 2 is identiﬁed with the symmetric space SO(3) SO(2) (see [28, Chap. 10, §3]). 2 2 4 Thus we may claim that L (S ) = T ⊕ ST and that, since C ∞ (S2 ) ⊂ L2 (S2 ), ¯ ¯ ∈ C ∞ (S2 ) can be univocally decomposed as ∞ l any α(ζ, ζ) l=0 m=−l αlm Ylm (ζ, ζ) ¯ with respect to the topology of L2 (S2 ). Furthermore take which converges to α(ζ, ζ) ¯ ∈ C ∞ (S2 ). into account that, per construction, each Ylm (ζ, ζ) We now show that the same sum converges in the topology of E ≡ C ∞ (S2 ) as constructed in Theorem 2.1. Let us choose > 0 such that, for any n ∈ N 2greater ¯ − n l ¯ < , being , the L -norm. α Y (ζ, ζ) than a ﬁxed n ¯ , α(ζ, ζ) l=0 m=−l lm lm 2 Let us now consider the operator A = L + kI (k > 1) and let us evaluate     2 ¯ ¯ ¯ ¯ − A α(ζ, ζ) + k,    α Y (ζ, ζ) α Y (ζ, ζ) α(ζ, ζ) − ≤ L lm lm lm lm l,m l,m

l,m

¯ = α (ζ, ζ)

in this proof stands for

n l

us now remember that, according to harmonic function theory, for any α (ζ, ζ) ∈ L2 SO(3) SO(2) , the sum

where

∞

¯ where αlm Ylm (ζ, ζ)

m=−l . Let ¯

l=0

αlm =

SO(3) SO(2)

l=0

¯ (ζ, ζ)Y ¯ lm (ζ, ζ) ¯ dµ(ζ, ζ)α

converges in the topology of L2 SO(3) SO(2) and the decomposition is unique. Per linearity of L2 we know that n n l l 2 2 ¯ ¯ ¯ ¯ L α(ζ, ζ) − αlm Ylm (ζ, ζ) = L α(ζ, ζ) + l(l + 1)αlm Ylm (ζ, ζ). l=0 m=−l

¯ ∈ Furthermore, since [L α](ζ, ζ) 2

L2 SO(3) SO(2)

l=0 m=−l

≡ L (S ), then ﬁxing 2

2

∞

. 2 ¯ = ¯ = ¯ α (ζ, ζ) [L α](ζ, ζ) αlm Ylm (ζ, ζ); l=0

¯ in spherical harmonics, that we conclude, by means of the decomposition of α(ζ, ζ) αlm = l(l + 1)αlm . Consequently   2 2 ¯ ¯ ¯ ¯ − L α(ζ, ζ)  αlm Ylm (ζ, ζ) = [L α(ζ, ζ)] − l(l + 1)αlm Ylm (ζ, ζ) ≤ , l,m l,m

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for suﬃciently large n; it descends n l ¯ − ¯ αlm Ylm (ζ, ζ) ≤ (k + 1), A α(ζ, ζ) n l

l=0 m=−l

¯ ¯ i.e. the sum l=0 m=−l [l(l + 1) + k]αlm Ylm (ζ, ζ) converges to Aα(ζ, ζ) in the 2 2 topology of L (S ). The same reasoning leads to the same conclusion with respect to l n p ¯ ¯ − A α(ζ, ζ) [l(l + 1) + k]αlm Ylm (ζ, ζ) l=0 m=−l

n l ¯ converges to for any integer p. Consequently the series l=0 m=−l αlm Ylm (ζ, ζ) ¯ α(ζ, ζ) with respect to , p in each Ep as introduced in Theorem 2.1. Thus, per ¯ in E ≡ C ∞ (S2 ) with respect to the deﬁnition, the series converges as well to α(ζ, ζ) induced topology τp . The uniqueness of the decomposition is accordingly traded from L2 (S2 ) to ∞ 2 C (S ) and the hypotheses of closure in C ∞ (S2 ) for ST is justiﬁed. ¯ Proof of Lemma 4.3. According to the Deﬁnition 4.4, Qα (ζ, ζ)ψ(β) = ¯ (β, α(ζ, ζ))ψ(β); if (4.15) holds, then Lemma 4.2 grants us that β can be chosen in (T 4 ) and, unless ψ(β) is identically vanishing, (4.16) is zero iﬀ (T 4 ) ⊂ (ST )0 which is the annihilator of ST . At the same time, if we suppose that (4.16) holds, then β ∈ (ST )0 and (4.15) holds iﬀ (ST )0 ⊂ (T 4 ) . We need only to prove that it exists an isomorphism between (T 4 ) and (ST )0 . The starting point consists of exploiting Theorem 2.2 according to which the ∞ 2 factor space C ST(S ) is isomorphic to the subspace T 4 ⊂ C ∞ (S2 ). Accordingly, per ∞ 2 duality, also (T 4 ) is isomorphic to C ST(S ) . Furthermore any β ∈ (T 4 ) can be extended according to Theorem 3.2 to a functional β˜ on E in such a way that, given ¯ α (ζ, ζ) ¯ ∈ C ∞ (S2 ), β(α(ζ, ˜ ¯ = β(α ˜ (ζ, ζ)) ¯ if α(ζ, ζ) ¯ − α (ζ, ζ) ¯ ∈ any two α(ζ, ζ), ζ)) ˜ ¯ ¯ ST . Per linearity of the elements in E , it implies β(α(ζ, ζ) − α (ζ, ζ)) vanishes i.e. β˜ ∈ (ST )0 and (T 4 ) ⊆ (ST )0 . To show the opposite inclusion let us start from any β ∈ (ST )0 . We can now exploit Theorem 2.2 according to which ST is a subspace of C ∞ (S2 ) and thus, according to Theorem 3.2, β can be extended to a functional in E . Choose any ¯ ∈ C ∞ (S2 ). Still according such extension, say β˜ and evaluate it on any α(ζ, ζ) ¯ ∈ T 4 and ¯ can be univocally split in the sum of α (ζ, ζ) to Theorem 2.2, α(ζ, ζ) ¯ ∈ ST . Thus, per linearity, α(ζ, ˜ ζ) ˜ ¯ = β(α ˜ (ζ, ζ)) ¯ + β(˜ ˜ α(ζ, ζ)) ¯ = β(α ˜ (ζ, ζ)), ¯ β(α(ζ, ζ)) where the last equality holds since β˜ must agree with β on ST . Thus the above equation grant us theta β˜ ∈ (T 4 ) i.e. (ST )0 ⊆ (T 4 ) , which concludes the demonstration. Proof of Lemma 4.4. Suppose that ψ(β) is supported on the orbit generated by SL(2,C) on β¯i where i = 1, 2 and L1 = SU (2) and L2 = ∆. Then, for any point β on Li

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one of the two orbits, it exists Λ ∈ SL(2, C) such that β = Λβi and the following chain of identities holds: η µν Qeµ Qeν ψ(β) = η µν Qeµ Qeν ψ(Λβ¯i ) = η µν (eµ , Λβ¯i )(eν , Λβ¯i )ψ(Λβ¯i ) = B(Λβ¯i , Λβ¯i )ψ(Λβ¯i ) = m2 ψ(β), where m2 is either 0 or diﬀerent from 0 depending on the chosen little group. In the above chain of identities we used the Deﬁnition 4.4 of multiplication operator whereas, in the last two identities, we refer to Proposition 3.3 and, in particular, to the deﬁnition of the real bilinear form (3.7) and its SL(2, C) invariance. To prove the converse, suppose a covariant scalar G BMS ﬁeld satisﬁes (4.15). 4 Then β ∈ (T ) and (4.17) becomes [η µν (β, eµ )(β, eν ) − m2 ]ψ(β) = 0,

β ∈ (T 4 ) .

Thus, unless ψ(β) is identically vanishing, η µν (β, eµ )(β, eν ) − m2 = 0 which is, depending on the chosen value for m2 , the deﬁning equation for the mass hyperboloid or for the light cone realized in R4 . We need at last to show that the orbit is necessarily generated by the ﬁxed point β¯i . Suppose that m2 = 0, then we need to exploit that η µν (β, eµ )(β, eν ) is the SL(2, C) invariant bilinear form B(β, β) as in (3.7). Thus we may ﬁnd Λ ∈ SL(2, C) such that B(β, β) = B(Λβ, Λβ) = m2 ¯ β should and B(Λβ, Λβ) = (β , e0 )(β , e0 ) where β = Λβ. Since e0 = Y00 (ζ, ζ), ∗ be equal to a constant times Y00 plus a term lying in the annihilator of e0 , say, (Y00 )0 . To be rigorous one now should exploit Proposition 3.2 to show that it exists (T 4 ) an isomorphism between (Y 0 and the space dual to one-dimensional subspace of 00 ) T 4 generated by Y00 . Thus one can always choose the representative in such factor group in such a way that it coincides with β¯1 i.e. the distribution generating the orbit for the massive canonical G BMS scalar ﬁeld. An identical procedure leads to the same conclusion for the massless case and thus the statement is proved. Proof of Proposition 5.1. The ﬁrst step in the demonstration consists of showing ¯ ∈ that an element ψ(β) ∈ (L2 ) satisﬁes (−2Dα(ζ,ζ) ¯ +Qα(ζ,ζ) ¯ )ψ(β) = 0 for all α(ζ, ζ) ST iﬀ (−2Dei + Qei )ψ(β) = 0 for each ei . This statement straightforwardly holds since, according to Theorem 2.2, ST is the closed set of real linear combinations of the real spherical harmonics with l > 1 and since the operator −2D + Q, seen as a ¯ ψ(β)) into (−2D map from C ∞ (S2 )×(L2 ) → (L2 ) mapping the pair (α(ζ, ζ), ¯ + α(ζ,ζ) Qα(ζ,ζ) ¯ )ψ(β), is linear in the ﬁrst argument. The remaining part of the proof will be structured as follows: we will calculate the variation with respect to ψ of a generic functional dµ(β)L(β, ψ(β), Dβ ψ(β)), S(ψ) = E

where L : (L2 ) → (L2 ) is a Lagrangian “density” depending both on the ﬁelds and on its derivative along any direction. The ﬁnal result will be the “Euler–Lagrange”

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equation associated to a functional deﬁned on a space endowed with a Gaussian measure. Eventually we will apply the result to (5.3). Let us thus perform the following variation: pick any φ(β) ∈ (L2 ), then δS 1 , φ(β) = lim [S(β, ψ + φ, Dβ (ψ + φ)) − S(β, ψ, Dβ ψ)] →0 δψ = dµ(β)Dψ L(β, ψ, Dβ ψ)φ(β) E

+ E

dµ(β)DDβ ψ L(β, ψ, Dβ ψ)Dβ φ(β).

The second element in the right-hand side of the last equality can be written in the more convenient form DDβ ψ L(β, ψ, Dβ ψ), Dβ φ(β) = Dβ∗ [DDβ ψ L(β, ψ, Dβ ψ)], φ(β) = (−Dβ + Qβ )DDβ ψ L(β, ψ, Dβ ψ), φ(β), where Dβ∗ is the adjoint derivative operator and where we have exploited the relation Dβ∗ + Dβ = Qβ as in Lemma 4.5. Thus in order for the variation of (5.3) to vanish for any choice of φ(β), we end up with the following Euler–Lagrange equation: Dψ L(β, ψ, Dβ ψ) − (Dβ − Qβ )DDβ ψ L(β, ψ, Dβ ψ) = 0,

(A.4)

where the term with the multiplication operator is a feature typical due to the presence of a Gaussian measure µ on E . This formula can be straightforwardly extended when, as in the scenario under consideration, the functional depends upon more than one ﬁeld. Thus an application of (A.4) to (5.3) shows that the variation of functional for the scalar G BMS ﬁeld with respect to the Lagrange multipliers provides that (−2Dei + Qei ) ψ(β) = 0 for all ei whereas a variation with respect to ψ provides, once the constraints are imposed, Eq. (4.17). Proof of Proposition 5.2. The demonstration is divided in two parts: ﬁrst we show that F L(T P ) is a submersion and than we prove that the ﬁbers are connected submanifolds. To prove the ﬁrst assertion we exploit [47, Proposition 2.2, Chap. II, §2] according to which a class C p (p ≥ 0) morphism f between two manifolds of class C p X, Y modelled over Banach spaces is a submersion at x ∈ X iﬀ it exists a chart (U, ϕ) at x and a second chart (V, φ) at f (x) ∈ Y such that DfV,U (ϕ(x)) is surjective and the kernel splits. In this proposition both TP and T ∗ P are Hilbert spaces; thus can choose any chart out of a norm induced open ball centred on a point p ∈ TP . Furthermore, since T ∗ P ≡ T P per Riesz theorem, the same choice can be performed for any point in the cotangent bundle. Thus a direct inspection of (5.5) shows either that

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the ﬁber derivative is a surjection on its image either that the kernel of F L is the set of real linear combinations of vectors (0, 0, 0, Dλi ). Thus Ker(F L(TP )) is isomorphic to (L2 )N and T P = Ker(F L) + M1 where M1 = (L2 ) × (L2 ) × (L2 )N with M1 ∩ Ker(F L) = {0}. This latter decomposition induces a natural map from T P into the Cartesian product Ker(F L) × M1 which is a (toplinear) isomorphism and thus the kernel splits. Concerning the second part of the demonstration, consider any point q ∈ T ∗ P such that q = F L(¯ p) with p¯ ∈ TP . Pick any two points, say, p1 , p2 lying in F L−1 (q). Referring to τ : TP → P as the tangent bundle projection map and to π : T ∗ P → P as the cotangent bundle counterpart, we may conclude from the compatibility condition π ◦ F L = τ that τ (p1 ) = τ (p2 ) i.e. (ψ1 , λi1 ) = (ψ2 , λi2 ). To conclude the demonstration it is suﬃcient now to exploit (5.5) and the hypothesis F L(p1 ) = F L(p2 ) according to which (ψ1 , 4(De0 ψ)1 − 2Qe0 ψ1 , λi1 , 0) = (ψ2 , 4(De0 ψ)2 − 2Qe0 ψ2 , λi2 , 0). It implies that the two points p1 , p2 diﬀer at most for an element in Ker(F L) thus the ﬁbers are connected submanifolds. Acknowledgments The author is in great debt to V. Moretti, M. Carfora and O. Maj for several useful discussions during the realization of this manuscript. The work has been supported partly by a grant from the Department of Theoretical and Nuclear Physics of Pavia University, partly by a grant from INDAM (Istituto Nazionale Di Alta Matematica) under the project “Olograﬁa in spazi asintoticamente piatti: Un approccio rigoroso” and partly by a grant of the Von Humboldt Foundation. References [1] G. ’t Hooft, Dimensional reduction in quantum gravity, arXiv:gr-qc/9310026. [2] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, Large N ﬁeld theories, string theory and gravity, Phys. Rept. 323 (2000) 183–386; arXiv:hepth/9905111. [3] K. H. Rehren, Algebraic holography, Ann. Henri Poincar´e 1 (2000) 607–623; arXiv:hep-th/9905179. [4] M. Duetsch and K. H. Rehren, Generalized free ﬁelds and the ads-cft correspondence, Ann. Henri Poincar´e 4 (2003) 613–635; arXiv:math-ph/0209035. [5] S. Albeverio, A. Hahn and A. N. Sengupta, Rigorous Feynman path integrals with applications to quantum theory, gauge ﬁelds and topological invariants, in Stochastic Analysis and Mathematical Physics, eds. R. Rebolledo, J. Rezende and J.-C. Zambrini (World Scientiﬁc, 2004), pp. 1–60. [6] H.-H. Kuo, White Noise Distribution Theory (CRC Press, 1996). [7] C. Dappiaggi, BMS ﬁeld theory and holography in asymptotically ﬂat space-times, J. High Energy Phys. 0411 (2004) 011, 36 pp.; arXiv:hep-th/0410026. [8] J. de Boer and S. N. Solodukhin, A holographic reduction of Minkowski spacetime, Nucl. Phys. B 665 (2003) 545–593; arXiv:hep-th/0303006.

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[9] P. J. McCarthy, The Bondi–Metzner–Sachs in the nuclear topology, Proc. R. Soc. London A 343 (1975) 489–523. [10] G. Arcioni and C. Dappiaggi, Exploring the holographic principle in asymptotically ﬂat spacetimes via the BMS group, Nucl. Phys. B 674 (2003) 553–592; arXiv:hepth/0306142. [11] G. Arcioni and C. Dappiaggi, Holography in asymptotically ﬂat spacetimes and the BMS group, Class. Quant. Grav. 21 (2004) 5655–5674. [12] C. Dappiaggi, V. Moretti and N. Pinamonti, Rigorous steps towards holography in asymptotically ﬂat spacetimes, Rev. Math. Phys. 18 (2006) 349–416; arXiv:grqc/0506069. [13] V. Moretti, Uniqueness theorem for BMS-invariant states of scalar QFT on the null boundary of asymptotically ﬂat spacetimes and bulk-boundary observable algebra correspondence, Comm. Math. Phys. 268 (2006) 727–756; arXiv:gr-qc/0512049. [14] V. Moretti, Quantum grand states holographically induces by asymptotic ﬂatness: Invariance under spacetime symmetries, energy positivity and Hadamard property, Comm. Math. Phys. 279 (2008) 31–75; arXiv:gr-qc/0610143. [15] C. Dappiaggi, Projecting massive scalar ﬁelds to null inﬁnity, to appear in Ann. Henri Poincar´e; arXiv:0705.0284 [gr-qc]. [16] A. Weinstein, A universal phase space for particles in Yang–Mills Field, Lett. Math. Phys. 2 (1978) 417–420. [17] V. P. Guillemin and S. Sternberg, Symplectic Techniques in Physics (Cambridge University Press, 1984). [18] N. P. Landsman, Mathematical Topics between Classical and Quantum Mechanics (Springer, 1998). [19] A. Ashtekar and M. Streubel, Symplectic geometry of radiative modes and conserved quantities at null inﬁnity, Proc. Roy. Soc. Lond. A 376 (1981) 585–607. [20] J. Dimock and B. S. Kay, Classical wave operators and asymptotic quantum ﬁeld operators on curved space-times, Ann. Poincar´e Phys. Theor. 37 (1982) 93–114. [21] H. Friedrich, On static and radiative space-times, Comm. Math. Phys. 119 (1988) 51–73. [22] R. Penrose, Asymptotic properties of ﬁelds and spacetimes, Phys. Rev. Lett. 10 (1963) 66–68. [23] R. Penrose, Relativistic symmetry group, in Group Theory in Non-Linear Problems, ed. A. O. Barut (Reidel, Dordrecht, 1974), Chap. 1, pp. 1–58. [24] R. Geroch, Asymptotic structure of space-time, in Asymptotic Structure of Spacetime, eds. P. Esposito and L. Witten (Plenum, New York, 1977), pp. 1–105. [25] I. M. Gel’fand et al., Generalized Functions: Integral Geometry and Representation Theory, Vol. 5 (Academic Press, 1966). [26] T. Hida, H.-H. Kuo and N. Obata, Transformations for white noise functionals, J. Funct. Anal. 111 (1993) 259–277. [27] T. Hida, H.-H. Kuo, J. Potthoﬀ and L. Streit, White Noise: An Inﬁnite-Dimensional Calculus (Kluwer Academics Publishers, 1993). [28] A. O. Barut and R. Raczka, Theory of Group Representation and Applications, 2nd edn. (World Scientiﬁc, 1986). [29] I. M. Gel’fand et al., Generalized Functions: Application of Harmonic Analysis, Vol. 4 (Acadmic Press, 1966). [30] D. J. Simms, Lie Groups and Quantum Mechanics (Springer-Verlag, 1968). [31] G. W. Mackey, Unitary Group Representations in Physics, Probability and Number Theory (Addison-Wesley Publishing, 1989).

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[32] F. LLedo, Massless relativistic wave equations and quantum ﬁeld theory, Ann. Henri Poincar´e 5 (2004) 607–670. [33] A. Piard, Unitary representations of semidirect product groups with inﬁnite dimensional Abelian normal Subgroup, Rept. Math. Phys. 11 (1977) 259–278. [34] I. Kubo and H.-H. Kuo, Finite dimensional Hida distributions, J. Funct. Anal. 128 (1995) 1–47. [35] J. Y. Lee, Integral transforms of analytic functions on abstract Wiener spaces, J. Funct. Anal. 47 (1982) 153–164. [36] M. M. Vainberg, Variational Methods for the Study of Nonlinear Operators (HoldenDay, Inc., 1964). [37] F. Bampi and A. Morro, The inverse problem of the calculus of variations applied to continuum physics, J. Math. Phys. 23 (1982) 2312–2321. [38] P. R. Chernoﬀ and J. E. Marsden, Properties of Inﬁnite Dimensional Hamiltonian Systems (Springer-Verlag, 1974). [39] R. Schmid, Inﬁnite-Dimensional Hamiltonian Systems (Bibliopolis, 1987). [40] M. Gotay, J. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac– Bergmann theory of constraints, J. Math. Phys. 19 (1978) 2388–2399. [41] M. Gotay and J. M. Nester, Presymplectic Lagrangian systems I: The constraint algorithm and the equivalence theorem, Ann. Inst. Henri Poincar´e A 30 (1979) 129–142. [42] J. F. Carinena and M. F. Ranada, Comments on the presymplectic formalism and the theory of regular Lagrangians with constraints, J. Phys. A 28 (1995) L91–L97. [43] S. Martinez, J. Cortes and M. de Leon, The geometrical theory of constraints applied to the dynamics of vakonomic mechanical systems: The vakonomic bracket, J. Math. Phys. 41 (2000) 2090–2120. [44] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 2 (Academic Press, San Diego, 1975). [45] W. Rudin, Functional Analysis, 2nd edn. (McGraw Hill, Boston, 1991). [46] H. Omori, Inﬁnite-Dimensional Lie Groups (Springer-Verlag, 1974). [47] S. Lang, Diﬀerential and Riemannian Manifolds (Springer, 1996).

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Reviews in Mathematical Physics Vol. 20, No. 7 (2008) 835–872 c World Scientiﬁc Publishing Company

IONIZATION IN A 1-DIMENSIONAL DIPOLE MODEL

O. COSTIN∗ , J. L. LEBOWITZ† and C. STUCCHIO‡ ∗Department of Mathematics, The Ohio State University, 100 Math Tower, Columbus, OH 43210-1174, USA [email protected] †Department of Mathematics and Physics, Rutgers, The State University of New Jersey, Hill Center, Busch Campus, New Brunswick, New Jersey 08901, USA [email protected] ‡Courant

Institute, New York University, New York, NY 10012-1185, USA [email protected] [email protected] Received 19 September 2007 Revised 23 March 2008

We study the evolution of a one-dimensional model atom with δ-function binding potential, subjected to a dipole radiation ﬁeld E(t)x with E(t) a 2π/ω-periodic real-valued function. We prove that when E(t) is a trigonometric polynomial, complete ionization occurs, i.e. the probability of ﬁnding the electron in any ﬁxed region goes to zero as t → ∞. For ψ(x, t = 0) compactly supported and general periodic ﬁelds, we decompose ψ(x, t) into uniquely deﬁned resonance terms and a remainder. Each resonance is 2π/ω periodic in time and behaves like the exponentially growing Green’s function near x = ±∞. The remainder is given by an asymptotic power series in t−1/2 with coeﬃcients varying with x. Keywords: Floquet theory; ionization; Gamow vectors; resonances. Mathematics Subject Classiﬁcation 2000: 47A53, 35P25, 81V45, 35B34, 35B10, 35Q40.

1. Introduction The ionization of an atom by an electromagnetic ﬁeld is one of the central problems of atomic physics. There are a variety of approximate methods for treating this problem, including perturbation theory (Fermi’s golden rule), numerical integration of the time-dependent Schr¨ odinger equation and semi-classical phase space analysis leading to stochastic ionization [3, 5, 18, 21, 22, 25]. Rigorous approaches include Floquet theory and complex dilations [18, 19, 37, 2]. Despite this, there are few 835

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exact results available for the ionization of a bound particle by a realistic time periodic electric ﬁeld of dipole form E(t)· x (an AC-Stark ﬁeld) for ﬁelds of arbitrary strength. The most realistic results we are aware of are based on complex scaling [18,19,37] and show ionization (for small electric ﬁeld) of certain bound states of the Coulomb atom as well as deﬁning resonances in some regions of the complex energy plane. The lack of rigorous results for large electric ﬁelds is true not only for realistic systems with Coulombic binding potential, but even for model systems with short range binding potentials [3, 5, 17]. The most idealized version of the latter has an attractive δ-function potential in 1 dimension. The unperturbed Hamiltonian H0 = −∂x2 − 2δ(x) has a bound state φ0 (x) = e−|x| with energy E0 = −1, and explicitly known continuum states [9]. This model has been studied extensively in the literature, but the only rigorous results (known to us) concerning ionization involve short range external forcing potentials rather than dipole interaction; see however [4, 15, 24] for some rigorous bounds on the ionization probability by a dipole potential for ﬁnite time pulses. Detailed results for compactly supported forcings were obtained in [8, 9, 28, 6]. In this work we develop techniques to deal with physically realistic dipole interactions. We consider the time evolution of a particle in one dimension governed by the Schr¨ odinger equation (in appropriate units) with a time-periodic dipole ﬁeld: ∂2 (1.1a) i∂t ψ(x, t) = − 2 − 2δ(x) ψ(x, t) + E(t)xψ(x, t) ∂x ψ(x, 0) = ψ0 (x) ∈ L2 (R).

(1.1b)

We prove the following result. Theorem 1 (Ionization). Suppose E(t) is a trigonometric polynomial, i.e. E(t) =

N

(En einωt + En e−inωt ).

(1.2)

n=1

Then for any ψ0 (x) ∈ L2 (R) ionization occurs, i.e. for ψ(x, t) solving (1.1), L |ψ(x, t)|2 dx = 0, ∀ L ∈ R+ . (1.3) lim t→∞

−L

If ψ0 (x) ∈ L (R) ∩ L (R), then the approach to zero is at least as fast as t−1 . 1

2

When E(t) is not a trigonometric polynomial (i.e. N = ∞ in (1.2)), the Floquet Hamiltonian may have time-dependent bound states and ionization may fail. This is uncommon, but there are examples of time periodic operators where such bound states exist [9, 27]. A key part of the proof of Theorem 1 is a theorem characterizing the structure of ψ(x, t). This result holds even if N = ∞ in (1.2). We ﬁrst need a deﬁnition.

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Definition 1.1. For the system described by (1.1), a Gamow vector is a classical solution of Floquet eigenvalue equation with ˇ σk ∈ [0, ω)a :

∂2 ˇk Φk,0 (x, t) = 0, −i∂t − 2 − 2δ(x) + E(t)x − σ ∂x Φk,0 (x, t) = Φk,0 (x, t + 2π/ω),

(1.4a) (1.4b)

subject to a radiation boundary condition at x = ±∞:

Φk,0 (x, t) =

 √   ψnL ei σˇk +nωx e−inωt e−ib(t)x−ia(t)  

x≤0

√   ψnR e−i σˇk +nωx e−inωt e−ib(t)x−ia(t)  

x ≥ 0.

n

(1.4c)

n

The functions a(t) and b(t) are deﬁned in (1.8), and are necessary to account for the presence of the E(t)x potential. The square root is chosen to have a branch cut on −iR+ and maps R+ into R+ . A generalized Gamow vector Φk,j (x, t) solves the equation:

∂2 ˇk Φk,j (x, t) = Φk,j−1 (x, t). −i∂t − 2 − 2δ(x) + E(t)x − σ ∂x

(1.5)

By deﬁnition, Φk,−1 (x, t) = 0. Remark 1.2. For ˇ σk ≥ 0, (1.4c) implies that solutions to (1.4) revert to ordinary L2 ([0, 2π/ω] × R)-eigenvalues of the Floquet Hamiltonian. This implies that solutions of (1.4) with ˇ σk > 0 are impossible; if such solutions existed, that would 2 contradict the L -self-adjointness of (1.4a). When ˇ σk = 0, Gamow vectors revert to being L2 ([0, 2π/ω] × R) solutions of (1.4a), and ψnR,L = 0 for n ≥ 0 (otherwise σk < 0 the terms in (1.4c) become exponenΦk,0 (x, t) would not be in L2 ). When ˇ tially growing near x = ±∞, and no exponentially decaying terms are present. This is what distinguishes Gamow vectors from other solutions to (1.4a) and (1.4b). In fact, one can construct exponentially growing solutions of (1.4a) for all σ. Such solutions would have both growing and decaying terms on at least one side of zero, and therefore fail to satisfy (1.4c). This makes the radiation boundary condition (1.4c) crucial in deﬁning Gamow vectors and resonances [25, 31, 20, 32, 36]. These boundary conditions deﬁne σ ˇk and Φk,j (x, t) uniquely. Theorem 2 (below) shows that these Gamow vectors are directly connected to the time behavior of ψ(x, t). a We

make this restriction due to the time-modulation invariance of (1.4). If Φk,0 (x, t) solves (1.4), ˇk by σ ˇk + nω. then einωt Φk,0 (x, t) also solves (1.4) if we replace σ

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Theorem 2. Suppose ψ0 (x) is compactly supported and in H 1 and E(t) is smooth and time periodic. Then, the solution ψ(x, t) of (1.1), can be uniquely (for ﬁxed M ) decomposed as: ψ(x, t) =

nk M

αk,j tj e−iˇσk t Φk,j (x, t) + ΨM (x, t).

(1.6)

k=0 j=0

The resonance energies σ ˇk satisfy ˇ σk ≤ 0, and are ordered according to ˇ σk+1 ≤ ˇ σk . The resonant states Φk,j (x, t) are the Gamow vectors (generalized, if j > 0), as per Deﬁnition 1.1. In (1.6), we collect the M resonances with ˇ σk closest to zero, and M must σk = 0. The number of be large enough so that we collect all resonances σ ˇk with ˇ Gamow vectors (and resonances) may be inﬁnite, but we can only include ﬁnitely many of them in (1.6). The remainder ΨM (x, t) has the following asymptotic expansion in time: ΨM (x, t) ∼

j∈Z

eijωt

∞

Dj,n (x)t−n/2 .

(1.7)

n=1

This expansion is uniform on compact sets in x, but not in L2 . In general, Dj,n (x) is not in L2 . Uniqueness of the decomposition is deﬁned relative to the analytic structure of ψ(x, t): the Zak transform of ψ(x, t) has poles at σ = σ ˇk (with residues proportional to Φk,j (x, t)), while ΨM (x, t) has Zak transform that is analytic on the region {σ : σ ∈ (0, ω), σ > ˇ σM }. The Zak transform is deﬁned in Sec. 3. Gamow vectors have a long history in quantum mechanics, dating back to [16, 13, 25]. They were ﬁrst introduced by Gamow, who used them to study tunneling rates. Deﬁnition 1.1 is an extension of the usual deﬁnition of Gamow vectors; a Gamow vector for a compactly supported potential (on [−L, L]) is typically deﬁned as a classical solution of the equation [−∂x2 + V (x) −√µ]Φ(x) = 0 having √ L i µx the behavior Φ(x) = ψ e for x < −L and Φ(x) = ψ R e−i µx for x > L (with µ the corresponding complex eigenvalue). The usual interpretation (see [32, 25] for an operator theoretic perspective) states that the real part of µ is the energy of a Gamow vector, while the imaginary part is the decay rate. In spite of their age and usefulness in making experimental predictions, rigorous justiﬁcation of the use of Gamow vectors is still lacking (see, however [32]). Theorem 2 provides a rigorous deﬁnition of resonances and ionization rates for the case we consider, and is thus a step towards making Gamow vectors completely rigorous. The remainder term ΨM (x, t), which we shall term the dispersive part, incorporates any resonances σ ˇk with k > M (if such resonances exist), as well as the integral around a branch point which gives rise to the polynomially decaying component. The resonant states Φk,j (x, t) are the residues of the poles of that same function (M cannot be greater than the total number of poles). This is equivalent

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(via (3.7)) to the requirement that the Laplace transform of ΨM (x, t) is analytic for p > −γM except for an array of branch points on the periodic array p = nω, n ∈ Z (with p dual to t). We believe the dispersive part ΨM (x, t) is Borel summable, although this does not follow from our results. To show this, one needs to ﬁnd exponential bounds on Z[ψ(0, t)](σ, t) as σ → −∞, which would also show that there is only one resonance σ ˇ0 , the analytic continuation of the bound state. Remark 1.3. Deﬁne γk = −ˇ σk ; for small values of γk , 2γk gives the dominant part of the ionization rate for the kth resonance. The smallest rate, γ0 , gives the overall ionization rate for most experimentally relevant times [3]. We are only aware of one experiment where the dispersive part of the wavefunction has actually been observed experimentally [29], although under signiﬁcantly diﬀerent physical conditions.b In most experiments the dispersive part is small enough to be safely neglected, and is in fact very diﬃcult to measure. Remark 1.4. If ψ0 (x) is not compactly supported a similar decomposition to (1.6) can be computed, but with extra terms coming from singularities of the Fourier 2 transform of ei∂x t ψ0 (x) (with respect to t). These terms are present even in the absence of a potential, and are therefore not resonances. 1.1. Small field limit Replacing E(t) by E(t), σ ˇ0 and Φ0,0 (x, t) have convergent power series expansions in when ω −1 ∈ N. When → 0, we have e−iˇσ0 t Φ0,0 (x, t) → eit e−|x| (pointwise), the bound state of H0 and Ψ1 (x, t) goes to the projection of ψ(x, t) on the continuum states of H0 . This shows that the ﬁrst resonance is the analytic continuation in of the bound state. This rigorously justiﬁes some standard physics calculations in [13, 16, 25] (see also the forthcoming work [36], from which we drew inspiration). The Fermi golden rule and multiphoton generalizations can be recovered in our formalism through perturbation theory. All other resonances come from σ = −i∞, i.e. as → 0, γk → ∞ for k ≥ 1. We conjecture that states with k > 0 (which do not exist when = 0) do not exist regardless of . Indeed, in all other cases considered [8, 9], such states do not exist, but our technique does not rule them out. See Remark 3.15 for more details on this point. 1.2. Equivalent formulations Here we describe some equivalent formulations of (1.1). This material is essentially taken from [11, Chap. 7]. We will use (1.10) in the proof of Theorem 1 and (1.9) in b In

[29], the authors studied luminescent decay of dissolved organic materials after a pulsed laser excitation, and observed polynomial decay after all exponential terms had vanished.

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the proof of Theorem 2. We ﬁrst deﬁne some auxiliary functions:

t

a(t) =

b(s)2 ds ≡ a0 t + av (t)

0

b(t) =

∞ En n=1

c(t) = 2

inω

einωt +

¯n E e−inωt −inω

(1.8a) (1.8b)

∞ ∞ ¯n

E En inωt −inωt Cn einωt + C¯n e−inωt e + e ≡ 2 2 (inω) (−inω) n=1 n=1 (1.8c)

2π/ω where av (t) is 2π/ω periodic and has mean 0, and a0 = (ω/2π) 0 b(s)2 ds. Note that (1/2)c (t) = b (t) = E(t). Deﬁne ψv (x, t) ≡ e+ia(t) e+ib(t)(x−c(t)) ψ(x − c(t), t); then the following equation for ψv is equivalent to (1.1): ∂2 (1.9) i∂t ψv (x, t) = − 2 − 2δ(x − c(t)) ψv (x, t). ∂x This is the velocity gauge, and the equivalence can be veriﬁed by a computation.c Similarly, there is an equivalent equation in the magnetic gauge. We obtain it by setting ψB (x, t) = e+ia(t) e+ib(t)x ψ(x, t): ∂2 i∂t ψB (x, t) = − 2 − 2δ(x) + 2ib(t)∂x ψB (x, t). (1.10) ∂x Remark 1.5. Suppose that either ψB (x, t) or ψv (x, t) are time-periodic solutions of (1.10) or (1.9). Then ψ(x, t) is a time quasi-periodic solution of (1.1), and eia0 t ψ(x, t) is time-periodic. These computations are formal, and we must show that at least one of (1.1), (1.9) or (1.10) are well posed. This is shown in Appendix C. Once one is well posed, all are, simply by applying the unitary gauge transformations.

1.3. Organization of the paper The paper is organized in the following way. In Sec. 2, we assume Theorem 2 to be true and use it to prove Theorem 1. In Sec. 3, we prove Theorem 2. In Sec. 4, we make some concluding remarks, and discuss possible directions of future research. Some technical material is presented in the appendices. R (1.9) diﬀers from what one ﬁnds in [11]. In [11], the authors take ˜b(t) = 0t E(s)ds and c˜(t) = 0 b(t)dt, which imply that c˜(t) = c(t) + c0 + cv t. This does not change the essential feature that (1/2)c (t) = b (t) = E(t).

c Equation

Rt

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2. Ionization Based on Theorem 2, we will show that the Floquet equation (1.4) in the magnetic gauge has no nonzero solutions with σ = 0 which decay at x = ±∞. This implies ionization for compactly supported initial data. Ionization follows for all initial data in L2 (R) by a simple application of the following well-known result to the operator family T (t) = 1[−L,L](x)U (t) (with U (t) the propagator for (1.1)): Proposition 2.1. If T (t) is a uniformly bounded family of operators on L2 (R), and if T (t)u → 0 for u in a dense subset of L2 (R), then T (t)u → 0 for all u ∈ L2 (R). In Sec. 2.1, we solve (1.10) without a binding potential (the −2δ(x) term) and characterize the solutions. We then assume that a bound state Φk,0 (x, t) exists, expand it in an appropriate basis, and derive necessary conditions on the coeﬃcients to meet the boundary conditions (decay at x = ±∞ and continuity at x = 0). In Sec. 2.2, we use the characterization of solutions we constructed in Sec. 2.1 and show for E(t) a trigonometric polynomial that there are no continuous, nonzero solutions to (1.10) which vanish at x = ±∞. The basic technique is to analytically continue, in the t variable, both ψB (0− , t) and ψB (0+ , t) (which must coincide) and use the Phragmen–Lindel¨ of theorem to show that an associated function must be entire and bounded (and therefore constant). This implies that any localized solution to (1.4) is zero, and ionization occurs. 2.1. Solutions to the free problem By Theorem 2, we need to show that (1.4) has no nontrivial solutions. In the magnetic gauge, this is the same as showing that if Φk,0 (x, t) solves σ ˇk Φk,0 (x, t) = (−i∂t − ∂x2 − 2δ(x) + 2ib(t)∂x )Φk,0 (x, t),

(2.1)

is time periodic and decays at x = ±∞, then Φk,0 (x, t) = 0. We begin by solving (2.1) without the δ-function binding potential (and letting σ=σ ˇk , which causes no confusion in this section), σψ(x, t) = (−i∂t − ∂x2 + 2ib(t)∂x )ψ(x, t). Taking ψ(x, t) = eλx ϕλ (t) as an ansatz, we obtain an ODE for ϕλ (t):

∂t ϕλ (t) = −i −σ − λ2 + 2iλb(t) ϕλ (t).

(2.2)

(2.3)

This has the following family of solutions (recalling that c (t) = 2b(t)): ϕλ (t) = e−iEλ t eλc(t) , Eλ = −σ − λ2 .

(2.4)

To ensure 2π/ω periodicity in time, we must have (−σ − λ2 ) = mω, m ∈ Z. This √ √ implies that λ = ±i mω + σ (with the branch cut of z taken to be −iR+ ).

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Therefore, (2.2) has the family of solutions: ϕm,± (x, t) = e±λm x e−imωt e±λm c(t) , √ λm = −i σ + mω.

(2.5a) (2.5b)

2.2. Matching solutions Given the family of solutions to (2.2), we can attempt to solve (1.10). Applying Theorem 2, we have three boundary conditions to satisfy: Φk,0 (0, t) = Φk,0 (0− , t) = Φk,0 (0+ , t),

(2.6a)

∂x Φk,0 (0+ , t) − ∂x Φk,0 (0− , t) = −2Φk,0 (0, t),

(2.6b)

lim Φk,0 (−x, t) = lim Φk,0 (+x, t) = 0.

(2.6c)

x→∞

x→∞

Consider now a solution Φk,0 (x, t). We can expand (formally) Φk,0 (x, t) in terms of the functions ϕm,± in the regions x < 0 and x > 0 separatelyd :    L L  (ψm,+ ϕm,+ (x, t) + ψm,− ϕm,− (x, t)), x≤0   m∈Z Φk,0 (x, t) = (2.7)  R R   (ψ ϕ (x, t) + ψ ϕ (x, t)), x ≥ 0. m,+ m,−  m,+ m,−  m∈Z−

For m ≥ 0 (recalling σ ˇk ∈ [0, ω) and examining (2.5b)), the functions ϕm,± (x, t) L,R (m ≥ 0) were not are oscillatory in x as x → ±∞. Thus, if the coeﬃcients ψm,± zero, then Φk,0 (x, t) would not decay as x → ±∞, violating (2.6c). Similarly, we observe that ϕm,+ (x, t) are exponentially growing when m < 0 as R must similarly be zero. The same argument applied to the region x → +∞, so ψm,+ L must be zero when m < 0. Therefore after dropping the ± x < 0 shows that ψm,+ L,R in the coeﬃcients ψm,± , we obtain the result we seek. Thus, we ﬁnd that we can actually write Φk,0 (x, t) as:    L  ψm ϕm,+ (x, t), x≤0   Φk,0 (x, t) = m<0 (2.8)  R   ψm ϕm,− (x, t), x ≥ 0   m<0−

L,R ψm

2

with both sequences in l . Although this derivation is purely formal, it is proved in Appendix B. It also motivates (1.4c). Substituting (2.8) into the continuity condition (2.6a) yields: L −imωt λm c(t) R −imωt −λm c(t) ψm e e = ψm e e . (2.9) m<0

d The

m<0

validity of the expansion is proved in Lemma B.2 in Appendix B.

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Proposition 2.2. Suppose E(t) is a trigonometric polynomial with highest mode N ¯n e−inωt ). Set z = e−iωt . Then Φk,0 (0, t) has N, that is E(t) = n=1 (En einωt + E the decomposition: Φk,0 (0, t) = f (z) + g(z −1 ).

(2.10)

The functions f (·) and g(·) are entire functions of exponential order 2N, and g(0) = 0. This shows in particular that Φk,0 (0, t) is continuous. The correspondence between Φk,0 (0, t), f (z) and g(z) is as follows. Let ψj denote the j th Fourier coeﬃcient of Φk,0 (0, t), that is Φk,0 (0, t) = j ψj eijωt . Then letting fj , gj be the Taylor coeﬃcients of f (z), g(z), we ﬁnd fj = ψ−j for j ≥ 0 and gj = ψj for j < 0. The proof of this fact uses results from Sec. 3, and is deferred to Appendix A. Finally, we state a result we use, proved in most complex analysis textbooks, e.g. [35]. Theorem 3 (Phragmen–Lindel¨ of ). Let f (z) be an analytic function of expo 2N nential order 2N, that is |f (z)| ≤ CeC |z| . Let S be a sector of opening smaller than π/2N . Then: sup |f (z)| ≥ sup |f (z)|. z∈∂S

z∈S

We are now prepared to prove the main result. Proof of Theorem 1. We describe ﬁrst the case N = 1 now (i.e. E(t) = E cos(ωt); the case of arbitrary N is treated below). The key idea is that we can use (2.8) and (2.9) to obtain an asymptotic expansion of Φk,0 (0+ , t) and Φk,0 (0− , t) in the open right and left half planes in the variable z = e−iωt . To leading order as |z| → ∞ L m −C|z| z e and in the left and right half planes (respectively), Φk,0 (0− , t) ∼ ψm R m −C|z| z e (note that m and C may be diﬀerent). This asymptotic Φk,0 (0+ , t) ∼ ψm expansion shows that f (z) decays exponentially along any ray z = reiφ in the open left or right half planes. In fact, the asymptotic expansion allows us to observe that f (z) (the part of Φk,0 (0, t) which is analytic in z) must be bounded except possibly on the line iR. Theorem 3 combined with Proposition 2.2 allow us to conclude that f (z) is bounded on the line iR. This shows f (z) is bounded on C and hence zero. Since f (z) is zero, Φk,0 (0, t) = g(z) ∼ gM z −M for some M ∈ N (since g(z) L m −C|z| z e . Two is analytic). But we previously showed also that Φk,0 (0, t) ∼ ψm asymptotic expansions must agree to leading order; the only way this can happen is if g(z) = Φk,0 (0, t) = 0. The main diﬀerence between the case N = 1 (monochromatic ﬁeld) and N > 1 (polychromatic ﬁeld) is that instead of the exponential asymptotic expansions being valid in the left and right half planes, they are valid in sectors of opening π/N ; to show this we need to apply Theorem 3 to the boundaries of these sectors.

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We now go through the details. Step 1: Setup Let Φk,0 (x, t) be a solution to (1.4). By the hypothesis of Theorem 1, we let E(t) be a nonzero trigonometric polynomial of order N . Let z = e−iωt . Let N −j ¯ j ) where the Cj are the coeﬃcients from (1.8c). We C(z) = j=1 (Cj z + Cj z apply Proposition 2.2 to Φk,0 (0, t) and (2.9) to obtain: Φk,0 (0, t) = f (z) + g(z −1 ) L m +λm C(z) R m −λm C(z) = ψm z e = ψm z e . m<0

(2.11)

m<0

The ﬁrst equality holds by (2.10), the second by (2.8) with x = 0. A priori, equality holds only when |z| = 1. However, both of the latter two sums are analytic in any neighborhood of the unit circle in which they are uniformly convergent. Thus, f (z) + g(z −1 ) is the analytic continuation of the sum if the sum is convergent in some neighborhood containing part of the unit disk. For the rest of this proof, we make the following convention. The functions ψ L,R (z) are deﬁned by L m +λm C(z) ψ L (z) = ψm z e (2.12a) m<0 R

ψ (z) =

R m −λm C(z) ψm z e

(2.12b)

m<0

for those z for which the sum is convergent. Step 2: Convergence of the sum We show now that the sum in (2.11) is convergent in a suﬃciently large region. R m −λm C(z) z e . In this For |z| ≥ 1 and C(z) > 0, consider the sum m<0 ψm λm C(z) L,R ≤ 1. The coeﬃcients ψm are bounded region, since C(z) > 0, we ﬁnd that e uniformly in m (since they form an l2 sequence). For |z| > 1, z m is geometrically decaying as m → −∞. Therefore the series is absolutely convergent when |z| > 1 and C(z) > 0. L m +λm C(z) in the region where The same statement holds with m<0 ψm z e C(z) < 0. Let us deﬁne the following sets: S + = Connected component of S 1 in {z ∈ C : |z| ≥ 1, C(z) > 0}, S − = Connected component of S 1 in {z ∈ C : |z| ≥ 1, C(z) < 0}. A plot indicating the structure of these sectors (for a particular choice of C(z)) is shown in Fig. 1 for the case where N = 2. By Proposition 2.2, we see that ψ R (z) is analytic in S + and ψ L (z) is analytic in S − , since the sum in (2.12) is convergent there. We now show that S + and S − must be unbounded since C(z) is not constant. z −1 ). As in the Schwarz reﬂection principle, deﬁne First, note that C(z) = C(¯

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Fig. 1. A plot of C(z), for the speciﬁc choice of C(z) = z 2 − 5z − 5z −1 + z −2 . As is apparent from the ﬁgure, the regions C(z) > 0 and C(z) < 0 are approximately sectors for suﬃciently large |z|.

B = S + ∪ (S¯+ )−1 . Clearly, C(z) = 0 for z ∈ ∂B. If S + is bounded, then B is bounded as well. By the real max modulus principle, C(z) must be zero inside B, and hence C(z) is bounded everywhere, which is impossible. Finally we show that the regions S + and S − “ﬁll out” to open sectors as |z| → ∞. That is to say, if S is some sector in which z N > 0, then for any ray {reiθ : r > 1} contained in S, there exists R = R(θ) so that the truncated ray {reiθ : r > R(θ)} ⊂ S + . Without loss of generality,e let us suppose that CN ∈ R+ . For very large |z|, −j ¯ j = C¯N z N + O(z N −1 ). Then setting z = reiθ , we write C(z) = N j=1 Cj z + Cj z −N iθ iN θ + O(r−1 ). Thus, for r suﬃciently large and we ﬁnd that r C(re ) = C¯N e −N N θ = (2m + 1)π/2, we ﬁnd that r C(reiθ ) has either strictly positive real part or strictly negative real part. In particular, if |N θ ∓ π/2| > , then there exists an R = R(, θ) so that r−N C(reiθ ) is bounded strictly away from zero. Motivated by the above, we deﬁne the following subsets of C (with j = 0, . . . , N − 1): iθ A+ j, = {re : r ≥ R(, θ),

θ ∈ [−π/2N + 2πj/N + , π/2N + 2πj/N − ]}.

e Suppose

(2.13a)

CN = ρeiθ . Then rather than choosing z = eiωt , we would substitute z = ei(ωt−θ/N) .

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θ ∈ [−π/2N + 2π(j + 1/2)/N + , π/2N + 2π(j + 1/2)/N − ]}. (2.13b) − + − Clearly, for suﬃciently large R, A+ j, \BR ⊂ S and Aj, \BR ⊂ S . Here, BR is the ball of radius R about z = 0.

Step 3: Asymptotics of f (z) We now show that f (z) = 0. We begin by writing f (z) as follows: f (z) =

∞

fn z n = −

n=0

f (z) =

∞

∞

gn z −n +

n=1

fn z n = −

n=0

∞

R m −λm C(z) ψm z e ,

z ∈ S+,

(2.14a)

L m +λm C(z) ψm z e ,

z ∈ S−.

(2.14b)

m<0

gn z −n +

n=1

m<0

We let Sk , k = 0, . . . , 2N + 1 be a set of sectors of opening π/(2N + 1) arranged in such a way that the boundaries of Sk avoid the rays reiπ(2j+1)/2N . Therefore, − for suﬃciently large |z|, the boundaries of Sk are contained in either A+ j, or Aj, except for a compact region. On ∂Sk , f (z) is decaying as |z| → ∞, by a simple examination of (2.14). Since f (z) is entire (unlike ψ(z)), f (z) is also bounded on ∂Sk even for small z. of We have shown that f (z) is bounded on ∂Sk . Applying the Phragmen–Lindel¨ theorem, f (z) is therefore bounded on Sk . Since ∪2n+1 S = C, we ﬁnd f (z) is k k=0 , f (z) is decreasing, constant. Since we know that along any ray contained in A± j,ε we know f (z) = 0. Step 4: Asymptotics of g(z) We now show that g(z) = 0. We rewrite (2.14) with g(z) on the left-hand side. ∞

gn z −n =

n=1 ∞

R m −λm C(z) ψm z e ,

z ∈ S+,

(2.15a)

L m +λm C(z) ψm z e ,

z ∈ S−.

(2.15b)

m<0

gn z −n =

n=1

m<0

Since the left-hand sides of (2.15a) and (2.15b) are (convergent) asymptotic power series (for suﬃciently large |z|), while the right-hand sides of (2.15a) and (2.15b) are (convergent) asymptotic series of exponentials, we ﬁnd that the righthand side decays much faster than the left-hand side. This is impossible unless both sides are zero. 3. The Floquet Formulation In this section we prove Theorem 2. To do so we consider the function Y (t) = ψ(0, t) and derive a closed integral equation for it via Duhamel’s formula. Computing

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the Zak transform in time of this equation yields an integral equation of compact Fredholm type for Z[Y ](σ, t), the Zak transform of Y (t). The integral operator is shown to be analytic in σ; the analytic Fredholm alternative to this equation shows that Z[Y ](σ, t) is meromorphicf in σ 1/2 for σ ∈ [0, ω). The poles corresponds to resonances or bound states, while the branch point at σ = 0 corresponds to the dispersive part of the solution. In Sec. 3.3, we extend these results from x = 0 to the entire real line. We show that the wavefunction can be decomposed in the form (1.6). If ˇ σk = 0, then ˇ σk ∈ (0, ω) and Φk,0 (x, t) corresponds to a Floquet bound state. The remainder ΨM (x, t) decays with time, in particular |ΨM (x, t)| = O(t−1/2 ) (or faster) as t → ∞, though not uniformly in x. 3.1. Setting up the problem We work in the velocity gauge. We rewrite (1.9) in Duhamel form, using the Green’s 2 function for the free Schr¨ odinger equation, (4πit)−1/2 eix /4t : ψv (x, t) = ψv,0 (x, t) t i(x − x )2 dt + 2i exp δ(x − c(t ))ψv (x , t )dx 4(t − t ) 4πi(t − t ) 0 R where we have deﬁned: ψv,0 (x, t) = e

i∂x2 t

ψv (x, 0) =

2

(4πit)−1/2 ei|x−x |

/4t

R

(3.1)

ψv (x , 0)dx .

Computing the x integral explicitly and changing variables to s = t − t yields: ψv (x, t) = ψv,0 (x, t) t i(x − c(t − s))2 ds + 2i exp . ψv (c(t − s), t − s) √ 4s 4πis 0

(3.2)

We now substitute x = c(t), to obtain a closed equation for ψv (c(t), t): ψv (c(t), t) = ψv,0 (c(t), t) t i(c(t) − c(t − s))2 ds i + exp ψv (c(t − s), t − s) √ . π 0 4s s

(3.3)

Letting Y0 (t) = ψv,0 (c(t), t) and Y (t) = ψ(c(t), t) for t ≥ 0 (both are set equal to 0 for t < 0) we obtain: t i(c(t) − c(t − s))2 i ds Y (t) = Y0 (t) + exp (3.4) Y (t − s) √ . π 0 4s s The main tool of our analysis will be the Zak transform. f The branch point at σ = 0 is repeated at σ = nω, n ∈ Z, due to the pseudo-periodicity of the Zak transform, cf. (3.6c). This also makes it suﬃcient to consider only σ ∈ [0, ω).

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Definition 3.1. Let f (t) = 0 for t < 0 and |f (t)| ≤ Ceαt for some α > 0. Then f (t) is said to be Zak transformable. The Zak transform of f (t) is deﬁned (for σ > α) by: eiσ(t+2πj/ω) f (t + 2πj/ω) (3.5) Z[f ](σ, t) = j∈Z

and by the analytic continuation of (3.5) when σ < α, provided that the analytic continuation exists (treating Z[f ](σ, t) as a function of σ taking values in L2 ([0, 2π/ω], dt)). Proposition 3.2. Z[f ](σ, t) has the following properties: iβ+ω f (t) = ω −1 e−iσt Z[f ](σ, t)dσ.

(3.6a)

iβ

If Z[f ](σ, t) is singular for σ = β, this integral is interpreted as the limit of integrals over the contours [i(β + ), i(β + ) + ω] as → 0 from above. Z[f ](σ, t + 2π/ω) = Z[f ](σ, t),

(3.6b)

Z[f ](σ + ω, t) = eiωt Z[f ](σ, t).

(3.6c)

If p(t) is 2π/ω-periodic, then: Z[pf ](σ, t) = p(t)Z[f ](σ, t).

(3.6d)

With the exception of (3.6a), these results all follow immediately from (3.5). See Remark 3.5 for an explanation of (3.6a). Remark 3.3. Suppose f (t) is Zak transformable, and uniformly bounded in time (α = 0). Suppose further that the analytic continuation of Z[f ](σ, t) has a singularity (say at σ = 0). Then (3.6c) still holds, in the sense that for any direction θ, Z[f ](σ + ω + 0eiθ , t) = eiωt Z[f ](σ + 0eiθ , t). Remark 3.4. More information on the Zak transform can be found in, e.g., [12, pp. 109–110]. Our deﬁnition diﬀers slightly from that in [12] by allowing σ to take complex values. Remark 3.5. One can relate the Zak and Fourier transforms as follows. Let fˆ(k) = ikt e f (t)dt denote the Fourier transform of f (t). Then: ω ˆ (3.7) f (σ + nω)e−inωt . Z[f ](σ, t) = 2π n∈Z

The Poisson summation formula, applied to (3.5), yields (3.7). Equation (3.6a) follows immediately from (3.7). This relation implies that our approach is equivalent to the Fourier/Laplace transform analysis done in [7, 8, 1]. The Zak transform is used simply for algebraic convenience.

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We proceed as follows. Applying the Zak transform to (3.4) yields an integral equation of the form y(σ, t) = y0 (σ, t) + K(σ)y(σ, t)

(3.8)

with y(σ, t) = Z[Y ](σ, t), y0 (σ, t) = Z[Y0 ](σ, t) and K(σ) the Zak transform of the integral operator in (3.4). K(σ) will be shown to be meromorphic in σ as a compact operator family from L2 (S 1 , dt) → L2 (S 1 , dt), except for a branch point at σ = 0. We then use the Fredholm alternative theorem to invert (1 − K(σ)). Once this is done, we ﬁnd: y(σ, t) = (1 − K(σ))−1 y0 (σ, t).

(3.9)

The poles of (1 − K(σ))−1 correspond to resonances, and a branch point at σ = 0 corresponds to the dispersive part of the solution, i.e. the part with polynomial decay in t−1/2 . To begin, we determine the analyticity properties of Z[Y0 ](σ, t). Proposition 3.6. Suppose ψ0 (x) is smooth and compactly supported. Then near σ = 0, y0 (σ, t) has the expansion: 1 Z[Y0 ](σ, t) = y0 (σ, t) = σ −1/2 ψ0 (x)dx + f (σ, t). (3.10) 2 R The function f (σ, t) is analytic in σ 1/2 for σ ∈ [0, ω) (there are similar branch points at σ = nω), and is in L2 (S 1 , dt) for each σ. Also, for some constants C1 and C2 , we have Z[Y0 ](σ, t)L2 (Sω1 ,dt) ≤ C1 eC2 |σ| . Here, Sω1 = R/(2π/ω)Z is the set [0, 2π/ω] with periodic boundaries. The same conclusion follows for Z[ψ(x + c(t), t)](σ, t) for any ﬁxed x. Proof. We can write out Y0 (t) using the Fourier transform as: 2 Y0 (t) = χR+ (t) eikc(t) eik t ψˆ0 (k)dk. R

Computing the Zak transform yields: 2 Z[Y0 ](σ, t) = eiσ(t−2πj/ω) χR+ (t − 2πj/ω) eikc(t) eik (t−2πj/ω) ψˆ0 (k)dk R

j∈Z

  2 = eikc(t) ψˆ0 (k)  eiσ(t−2πj/ω) eik (t−2πj/ω) χR+ (t − 2πj/ω) dk R

= R

eikc(t) ψˆ0 (k)

j∈Z

n∈Z

e−inωt dk i(k 2 + σ + nω)

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= σ −1/2 (1/2)

R

ψ0 (y)dy + σ −1/2 (1/2)

e−inωt √ + 2 σ + nω n=0

e−

√ σ+nω|c(t)−y|

R

(e−

√ σ|c(t)−y|

R

− 1)ψ0 (y)dy

ψ0 (y)dy.

(3.11)

The interchange of the sum and integral between lines 1 and 2 is justiﬁed (for σ > 0 and t ﬁxed) since the sum over j is absolutely convergent, as the integral over k is bounded and uniformly convergent with respect to k. The change inside the square brackets between lines 2 and 3 comes from the Poisson summation formula in the t variable, and the fact that the Fourier transform 2 of χR+ (t)ei(k +σ)t is −i(k 2 + σ + ζ)−1 (with ζ dual to t). The ﬁrst term on the right-hand side of (3.11) agrees with that in (3.10). Since √ (e− σ|c(t)−y| − 1) is analytic in σ 1/2 and has no constant term, the second term is analytic in σ 1/2 . The third term is analytic in σ. When added together, these terms become f (σ, t) which is analytic in σ 1/2 . The result is valid for arbitrary σ by analytic continuation. Since ψ0 (x) is supported on a compact region, |c(t) − y| is bounded (say by C2 ) and exponential growth follows. The result follows for all x by translation invariance 2 of ei∂x t . We now determine the Zak transform of the integral operator in (3.4) and compute the resolvent of it. 3.2. Construction of the resolvent We now apply the Zak transform to (3.4) to construct an equivalent integral equation. Proposition 3.7. Let f (t) be Zak transformable. Consider the integral operator: t i (c(t) − c(t − s))2 ds KV f (t) = (3.12) exp i f (t − s) √ . π 0 4s s For σ > 0, deﬁne Z[KV f ](σ, t) = (K(σ)Z[f ])(σ, t). Then: ∞ i (c(t) − c(t − s))2 ds K(σ)f (σ, t) = exp i eiσs Z[f ](σ, t − s) √ . π 0 4s s

(3.13)

For σ ≤ 0, we deﬁne K(σ) to be the analytic continuation of K(σ) (where it exists). Proof. We rewrite (3.12) as: t i (c(t) − c(t − s))2 ds exp i f (t − s) √ π 0 4s s i (c(t) − c(t − s))2 ds = exp i f (t − s)χR+ (s) √ . π R 4s s

(3.14)

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Applying Z to both sides of (3.14) yields Z[KV f ](σ, t) =

eiσ(t+2πj/ω) [KV f ](t + 2πj/ω)

j∈Z

=

i iσ(t+2πj/ω) (c(t) − c(t − s))2 e exp i π 4s R j∈Z

ds × f (t + 2πj/ω − s)χR+ (s) √ s i (c(t) − c(t − s))2 = exp i eiσs π R 4s   ds × eiσ(t−s+2πj/ω) f (t − s + 2πj/ω) χR+ (s) √ s =

j∈Z

i π

0

∞

(c(t) − c(t − s))2 ds exp i eiσs Z[f ](σ, t − s) √ . 4s s

(3.15)

This is what we wanted to show. We now show that the operator K(σ), constructed above, is compact. We decompose K(σ) as KF (σ) + KL (σ) (deﬁned shortly), and treat each piece separately. Proposition 3.8. Deﬁne KF (σ) : L2 (S 1 , dt) → L2 (S 1 , dt) by: KF (σ)f (t) =

i π

0

∞

ds eiσs f (t − s) √ . s

Then, KF (σ) is compact and analytic for σ > 0. It can be analytically continued to σ ≤ 0, σ = 0, and the continuation has a σ −1/2 branch point at σ = 0. Proof. We compute this exactly by expanding f (t) in Fourier series and interchanging the order of summation and integration:

∞ i ds f √ n fn e−inωt ei(σ+nω)s √ = e−inωt . π s σ + nω 0 n∈Z n∈Z

(3.16)

This is valid for σ > 0, as well as σ = 0 but in this case we must treat the integral as improper. Thus, in the basis e−inωt , this operator is diagonal multiplication by (σ + nω)−1/2 . Compactness follows since the diagonal elements decay in both directions. Analyticity for σ = 0 follows by inspection of the right-hand side of (3.16), √ and choosing the branch cut of σ + nω to lie on the negative real line.

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Proposition 3.9. Deﬁne KL (σ) : L2 (S 1 , dt) → L2 (S 1 , dt) as: KL (σ)f (t) =

i π

∞

0

ds (c(t) − c(t − s))2 exp i − 1 eiσs f (t − s) √ . 4s s

(3.17)

Then KL (σ) is compact for σ ≥ 0 and analytic for σ > 0. It has continuous limiting values at σ = 0. Proof. We rewrite (3.17) as: ∞ 2π/ω

0

k=0

iσ(s+2πk/ω) e (c(t) − c(t − s))2 f (t − s)ds. exp i −1 4(s + 2πk/ω) s + 2πk/ω

(3.18)

If σ ≥ 0, the summands decay at least as fast as k −3/2 as k → ∞. Each term in the sum is continuous. Thus the sum is absolutely convergent to a smooth function in t and s, which is analytic in σ (thus the limit is analytic except when σ = 0). The region of integration is compact, and so is KL (σ). We now analytically continue KL (σ) to the strip 0 < σ < ω. Proposition 3.10. Let K (σ) be the integral operator deﬁned by: K (σ)f (t) =

0

kσ (t, s)

ω = 2πi

C

2π/ω

kσ (t, s)f (t − s)ds

(c(t) − c(t − s))2 dp eσp exp −1 √ 1 − eωp−iωs 4p p

(3.19a) (3.19b)

where C is some contour along the real line in the upper half plane which avoids the singularities of the integrand at p = 0 and p = i(s + 2πn/ω) (see the proof for a speciﬁc example). Then K (σ) is an analytic (in σ) family of compact operators for 0 < σ < ω, and vanishes as σ → +∞. Furthermore, K (σ) is the analytic continuation of KL (σ). Finally, for σ = −iλ (with λ < 0) or σ = −iλ + ω, K(σ) is analytic in the parameter λ1/2 or (σ − ω)1/2 . Proof. Step 1: Analyticity To perform the integral in (3.19b), we let γR (t) = tωR/2π for t ∈ R\[−2π/ω, 2π/ω], and γR (t) = Rei[π−(ωt+2π)/4] for t ∈ [−2π/ω, 2π/ω]. That is, γR (t) travels along the real line, and circles upward around the disk of radius R. The integral is then deﬁned as limR→0 γR (·) dp.

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To compute the behavior of the integral, simply take R = 2π/ω:

ω −1 kσ (t, s)

(c(t) − c(t − s))2 dp eσp = exp −1 √ ωp−iωs 1 − e 4p p R+0i (c(t) − c(t − s))2 dp eσp = exp − 1 √ ωp−iωs 4p p γ 1−e (c(t) − c(t − s))2 1 2πi iσs e (3.20) + exp −1 √ . ω 4is is

The integrand in the ﬁrst term is analytic in t and s since p stays away from 0 (thus avoiding the essential singularity at p = 0). It is exponentially decaying both for large positive p (at the rate e(σ−ω)p ) and for large negative p (at the rate e−σp ). If σ = 0 or σ = ω, the integrand still decays at the rate p−3/2 , which is integrable. The last term is integrable at s = 0 and analytic (in s) elsewhere. Thus, kσ (t, s) has only a singularity of order s−1/2 , and is analytic elsewhere. This shows that K (σ) is a compact family of operators, analytic on σ. Step 2: Vanishing of the operator as σ → +∞ We examine (3.20). The ﬁrst term vanishes as σ → ∞ by the Riemann–Lebesgue lemma. The second term vanishes since eiσs does. Thus, kσ (t, s) → 0, and so does K (σ). Step 3: Continuation of KL (σ) To show that K (σ) = KL (σ) if σ > 0, we simply move the contour of integration in (3.19b) upward and collect residues: (c(t) − c(t − s))2 dp eσp exp −1 √ ωp−iωs 4p p R+0i 1 − e  (c(t) − c(t − s))2 dp eσp  = lim exp −1 √ ωp−iωs N →∞ 4p p R+i2πN/ω 1 − e

+

N 2πi j=0

=

ω

eiσ(s+2πj/ω)

 (c(t) − c(t − s))2 1  exp −1 4i(s + 2πj/ω) i(s + 2πj/ω)

(c(t) − c(t − s))2 1 eiσ(s+2πj/ω) exp . −1 ω 4i(s + 2πj/ω) i(s + 2πj/ω

∞ 2πi j=0

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We then integrate this kernel against an L2 (S 1 , dt) function f (t) and obtain: 2π/ω ∞ i ω 2πi iσ(s+2πj/ω) e π 2πi j=0 ω 0 (c(t) − c(t − s))2 1 f (t − s)ds × exp −1 4i(s + 2πj/ω) i(s + 2πj/ω) ∞ i ds (c(t) − c(t − s))2 = exp i − 1 eiσs f (t − s) √ . π 0 4s s

This is in agreement with (3.17). Hence, K (σ) = KL (σ) for σ > 0, σ ∈ (0, ω) and therefore K (σ) is the analytic continuation of KL (σ). Step 4: Singularity at σ = 0, ω √ We now wish to show that K (−iλ) is analytic in λ for σ = −iλ, and similarly that K(−iλ+ω) is analytic in λ. To do this, we proceed as in Step 3, but push the contour down instead of up. We rotate the contour γ1 ∪ γ2 ∪ γ3 , with γ1 = [−i∞ − R, −R], γ2 which goes around the unit circle of radius R in the upper half plane (as in step 1), and γ3 which is [R, R−i∞]. This lets us avoid concerning ourselves with the singularities of the integrand; the important behavior is the decay near p = −i∞. Note that the integral kernel of K (−iλ) is given by (c(t) − c(t − s))2 dp e−iλp k−iλ (t, s) = exp −1 √ ωp−iωs 4p p γ1 ∪γ2 ∪γ3 1 − e while that of K (−iλ + ω) is given by: (c(t) − c(t − s))2 e−iλp eωp dp (t, s) = k−iλ+ω exp − 1 √ . ωp−iωs 1 − e 4p p γ1 ∪γ2 ∪γ3 First, observe that the integral over γ2 is analytic in λ, provided R = ωs. Thus, choosing a diﬀerent R for ωs < (3/4)π and ωs > (1/4)π shows analyticity in λ. We consider the case σ = −iλ, the case σ = −iλ + ω being treated similarly. We now observe that, for p = R (the same argument applies to p = −R), the integrand (over γ3 or γ1 ) becomes a Laplace transform: (c(t) − c(t − s))2 dp e−iλp exp −1 √ ωp−iωs 1 − e 4p p γ3 −i∞ (c(t) − c(t − s))2 dp e−iλp −iλR . exp = e −1 √ ω(p+R)−iωs 4(p + R) 1−e p+R 0 (3.21) We then observe that we can rewrite (c(t) − c(t − s))2 1 exp = (p + R)−3/2 H(c(t), c(t − s), p + R) −1 √ 4(p + R) p+R

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with H(c(t), c(t − s), p + R) analytic in p for p = R, and therefore analytic on the contour in (3.21). This follows since ez − 1 = O(z) near z = 0. We now substitute this back into (3.21) and change variables to iλp = z, to obtain: ∞ 1 H(c(t), c(t − s), −iz/λ + R) dz e−z (3.21) = −ie−iλR ω(−iz/λ+R)−iωs λ 1−e (−iz/λ + R)3/2 0 ∞ 1 H(c(t), c(t − s), −iz/λ + R) e−z dz. = −iλ1/2 e−iλR ω(−iz/λ+R)−iωs 1 − e (−iz + Rλ)3/2 0 (3.22) The integrand is analytic in λ, and absolutely convergent. The power of λ1/2 makes the net result a ramiﬁed analytic function. The same argument can be applied to (t, s) is analytic in λ1/2 . This γ1 , replacing R by −R. Thus, we have shown that k−iλ 1/2 implies that K(−iλ) is analytic in λ . As remarked before, the case K(−iλ + ω) is identical, so the proof is complete. Now we have shown that K = KL . In addition, now that KL (σ) and KF (σ) are deﬁned, it is clear that KF (σ) + KL(σ) = K(σ). Thus, K(σ) = KF (σ) + KL (σ) can be analytically continued to the region σ ≤ 0. Next we show that K(σ) grows at most exponentially as σ → ±∞. Proposition 3.11. K(σ) vanishes as σ → ∞. Proof. We break K(σ) up as K(σ) = KF (σ) + KL (σ). The ﬁrst term, KF (σ) is bounded (away from σ = 0) simply by inspecting (3.16). The second vanishes near σ = ∞ by Proposition 3.10. We have now shown that K(σ) : L2 (Sω1 , dt) → L2 (Sω1 , dt) is an analytic (in σ) family of compact operators. This allows us to construct the resolvent. Proposition 3.12. The operator (1 − K(σ))−1 is a meromorphic (in σ) family of bounded operators. This implies that if (1 − K(σ))−1 has a pole of order n at a point ˇk : σ=σ ˇk , we then have the following asymptotic expansion as σ → σ n k Yk,j (t)Yk,j (t)|· + D(σ) (3.23) (1 − K(σ))−1 = (σ − σ ˇk )j+1 j=0 σk ))Yk,j (t) = Yk,j−1 (t) (with where D(σ) is analytic near σ ˇk . Yk,j (t) solves (1 − K(ˇ Yk,−1 (t) = 0). The functions Yk,j (t) are all L2 (Sω1 ) functions. If σ ˇk = 0, then the same result holds, except that the poles are in the variable √ σ instead of (σ − σ ˇk ). An additional result (which we use later) is that if (1 − K(σ))−1 has no poles in the variable σ 1/2 near σ = 0, then P0 y(0, t) = (1/2) R ψ0 (x)dx, where P0 is projection onto the zero’th Fourier coeﬃcient. Remark 3.13. Note that we do not assume that dim Ker K(ˇ σk ) = 1. We choose the convention that if the dimension of the kernel is greater than 1, we consider this

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to mean that σ ˇk = σ ˇk for some k = k . Thus, one can have degenerate resonances as well as degenerate eigenvalues. Proof. This is merely the analytic Fredholm alternative theorem. There is only one technical point regarding the behavior near σ = 0 due to the fact that K(σ) is singular there. This can be remedied as follows. Note that K(σ) = KF (σ) + KL (σ). For small σ, we use the following resolvent identity: (1 − K(σ)−1 = (1 − KF (σ) − KL (σ))−1 = [1 − KF (σ)]−1 (1 − KL (σ)[1 − KF (σ)]−1 )−1 .

(3.24)

The operator [1 − KF (σ)]−1 is analytic in σ 1/2 (for small σ), being deﬁned by −1 fj e−ijωt = (1 − σ −1/2 )−1 fj e−ijωt . [1 − Kf (σ)] j

j

Since KL (σ) is compact, the Fredholm alternative applies to the resolvent of (1 − KL (σ)[1 − KF (σ)]−1 ), implying that (1 − KL (σ)[1 − KF (σ)]−1 )−1 and therefore (1 − K(σ))−1 is meromorphic in σ 1/2 . If this operator has no poles in the variable σ 1/2 near σ = 0, then y(σ) is analytic in σ 1/2 for small σ. We then rewrite (3.8) as: (1 − KF (σ)(1 − P0 ) − KL (σ))y(σ, t) + σ −1/2 P0 y(σ, t) −1/2 = σ (1/2) ψ0 (x)dx + f (σ 1/2 , t). R

Matching coeﬃcients to order σ −1/2 as σ (1/2) R ψ0 (x)dx.

→ 0 shows that P0 y(0, t) =

Proposition 3.14. Deﬁne K (σ) as K(σ) with c(t) replaced by c(t) (so in particular, K1 (σ) = K(σ)). Then the position of the poles of K (σ) are ramiﬁed analytic functions of , the ﬁeld strength, except possibly near σ ˇk = −i∞. For small , there is only one pole σ ˇ0 near the real axis (corresponding to the dressed bound state), and all other poles are located near σ = −i∞. Proof. This is basically the analytic implicit function theorem, using the fact that K (σ) is analytic in , and K0 (σ) = KF (σ) (cf. Proposition 3.8). We ﬁrst show that no poles form spontaneously. Consider a compact set, bounded by the curve γ. Then deﬁne Rγ, = [1 − K (σ)]−1 dσ. γ −1

Provided [1 − K (σ)] is analytic on γ (in σ and jointly), then Rγ, is analytic in . For = 0, we ﬁnd that: −1 fn e−inωt = [1 − (σ + nω)−1/2 ]−1 fn e−inωt (3.25) [1 − K0 (σ)] n

n

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which has one pole on [0, ω), and no others. Since K (σ) − K0 (σ) ≤ C (with C depending in γ and σ), [1 − K (σ)]−1 is analytic on and inside γ for small . Thus Rγ, = 0 for small , and by analyticity is zero on any interval ∈ [0, δ) on which it is analytic. Rγ, is analytic as varies from = 0, until [1 − K (σ)]−1 becomes singular on γ. We therefore ﬁnd that Rγ, becomes nonzero only after poles of K (σ) have crossed γ, i.e. no poles formed spontaneously inside γ. The same argument shows that spontaneous poles of higher order do not form k if we use Rγ, = γ fk (σ)[1 − K (σ)]−1 dσ (for fk (σ) a function with nonvanishing kth derivative) instead of Rγ, . This argument implies that any poles which are not present for = 0 must come from σ = −i∞ as is “switched on”. Analyticity of σ ˇk follows immediately from Theorems 1.7 and 1.8 in [23, pp. 368– 370] (see also the discussion following Theorem 1.7). These results show that any eigenvalue λ(, σ) of K (σ) is an analytic function. Poles occur where λ(, σ) = 1. ˇk () is ramiﬁed analytic. By the implicit function theorem, σ ˇk = σ Remark 3.15. The only obstacle to proving the absence of poles moving in from σ = −i∞ is a lack of bounds on the norm of [1 − K(σ)]−1 . If we had such bounds, it would be possible to show that the only pole of [1 − K(σ)]−1 is the analytic continuation of the bound state for E(t) = 0. In other cases we have considered [8, 9] such bounds were proved, and there is no fundamental reason it should not be true in this case as well. 3.3. Time behavior of ψ(x, t) We have now shown that K(σ) is a compact analytic operator. By the Fredholm alternative, (1 − K(σ))−1 is a meromorphic operator family. By deforming the contour in (3.6a), we can determine the behavior of Y (t). Once this is complete, we can calculate Φk,j (x, t) and ΨM (x, t) and ﬁnish the proof of Theorem 2. Proposition 3.16. The function Y (t) has the expansion: Y (t) =

nk M−1

αk,j tj e−iˇσk t Yk,j (t) + DM (t)

(3.26)

k=0 j=0

with Yk,j (t) the residue of [1−K(σ)]−1 at σ ˇk and αj,k = (2π/ω)y0 (ˇ σk , t)|Yk,j (t)/j!. ˇk = 0, then M must not be greater than the number of poles of [1 − K(σ)]−1 . If no σ DM (t) has the asymptotic expansion: DM (t) ∼

n∈Z

e−inωt

∞

Dj,n t−j/2 .

(3.27)

j=3

ˇk = 0 for some k, Y (t) = In particular this shows that |DM (t)| = O(t−3/2 ). If σ DM (t) except that in (3.27) the sum starts at j = 1 rather than j = 3.

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Proof. Because (1 − K(σ))−1 is meromorphic in σ, y(σ, t) can be written as y(σ, t) = (1 − K(σ))−1 y0 (σ, t) =

nk Yk,j (t)Yk,j (t)|y0 (σ, t)

(σ − σb )j

j=0

+ D(σ)y0 (σ, t).

(3.28)

We compute Y (t) using (3.6a), and shifting the contour: ω− e−iσt y(σ, t)dσ Y (t) = ω −1 0+

= ω −1

−iK(M)+0+

e−iσt y(σ, t)dσ + ω −1

e−iσt y(σ, t)dσ

−iK(M)

0+

+ ω −1

−iK(M)+ω

ω−

e−iσt y(σ, t)dσ + Residues

−iK(M)+ω−

= ω −1

−iK(M)+ω

e−iσt y(σ, t)dσ

−iK(M)

+ ω −1

−iK(M)+0

e−iσt y(σ + 0+ , t)

0

−e

−(iσ+ω− )t

y(σ + ω− , t)dσ + Residues .

(3.29)

If [1 − K(σ)]−1 has more than M poles, then we make K(M ) suﬃciently large to collect M of them; otherwise, we simply collect all the poles. The residue term is given by: nk M−1

αk, j tj e−iˇσk t Yk,j (t)

k=0 j=0

stemming from the M poles with ˇ σk > −K(M ). By (3.6c), we can change the integral in the second to last line of (3.29) to: −iK(M) −1 e−iσt (y(σ + 0+ , t) − y(σ + 0− , t))dσ. (3.30) ω 0

Note that y(σ, t) is analytic in σ 1/2 , and thus y(σ + 0+ , t) − y(σ + 0− , t) can be expanded in a Puiseux series in σ 1/2 (and a Fourier series in t). Watson’s Lemma yields: −iK(M) ∞ e−iσt e−inωt Dj,n σ j/2 dσ (3.30) = ω −1 0

∼ ω −1

n∈Z

n∈Z

This is what we wanted to show.

einωt

∞ j=3

j=0

Dj,n Γ(j/2)t−j/2 .

(3.31)

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When σ ˇk = 0, the result follows simply by noting that the sum over j in (3.30) starts from j = −1 rather than j = 0, thereby letting the sum on the right-hand side of (3.30) start at j = 1 instead of j = 3. The integral from −iK(M ) to −iK(M )+ω decays at least as fast as O(e−K(M)t ), and is included in Dk (t). We now reconstruct the wavefunction in the magnetic gauge. The basic idea M nk αk,j tj e−iˇσk t Yk,j (t). is as follows. We know that ψB (0, t) = DM (t) + k=0 j=0 Using the fact that δ(x)ψB (x, t) = δ(x)ψB (0, t), we ﬁnd that ψB (x, t) satisﬁes the following equation: i∂t ψB (x, t) = (−∂x2 + 2ib(t)∂x )ψB (x, t) − 2δ(x − c(t))ψB (x, t) = (−∂x2 + 2ib(t)∂x )ψB (x, t) − 2δ(x − c(t))ψB (0, t) = (−∂x2 + 2ib(t)∂x )ψB (x, t) − 2δ(x − c(t))Y (t).

(3.32)

We will use, for σ < 0 a solution operator for the Floquet problem G(σ) (described shortly) to extend (3.26) to all x, thereby recovering (1.6). The Gamow vectors will come from applying G(σ) to Yk,j (t), while the dispersive part will come from applying this operator to DM (t). Proposition 3.17. Let G(σ) be the solution operator for the equation: (σ + i∂t + ∂x2 − b(t)∂x )u(x, t) = −2δ(x)f (t)

(3.33)

so that for σ > 0, u(x, t) decays as x → ±∞. By “solution operator”, we mean that G(σ) maps f (t) → u(x, t). Then G(σ) can be analytically continued to the region σ ≤ 0. The function u(x, t) = G(σ)[−2δ(x)f (t)] has the expansion:  −imωt ∓λm,− c(t)  um,R eλm,− x 2−1/2 λ−1 e , x ≥ 0,  m,− e  m (3.34a) u(x, t) = −imωt ∓λm,+ c(t)   um,R eλm,+ x 2−1/2 λ−1 e , x ≤ 0, m,+ e  m

√ λm,± = ∓i σ + mω,

(3.34b)

where f (t) → {um,R , um,L } is a mapping from L2 (Sω1 ) → l2 (Z × {L, R}). G(σ) is also a continuous map, analytic in σ 1/2 from L2 (Sω1 ) → L2 (BR ×Sω1 ) with BR = {x : |x| < R} for any (ﬁxed) R. Near σ = 0, we have G(σ)δ(x)f (t) = σ −1/2 (1/2)P0 f (t)+ O(1), with P0 f (t) the projection onto the zero’th Fourier coeﬃcient of f (t) and the O(1) term being analytic in σ 1/2 . This result is proved in Appendix B. We are now ready to compute the resonance decomposition, (1.6). Proposition 3.18. The expansion (1.6) holds. Proof. We work in the magnetic gauge, to simplify this part of the problem. Note that ψB (x, t) = ψv (x + c(t), t), so in particular, ψB (0, t) = ψv (c(t), t) = Y (t).

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Moreover, recall that the Zak transform commutes with periodic operators, such as the coordinate transform (x, t) → (x + c(t), t). Additionally, in what follows, the notation A(σ 1/2 ) denotes a function analytic in σ 1/2 taking values in L2 (Sω1 , dt). Note that A(σ 1/2 ) may vary from line to line and from equation to equation. 2 odinger equaDeﬁne ψI (x, t) = ei∂x t eic(t)∂x ψ0 (x) to be the solution of the Schr¨ tion in the magnetic gauge with no binding potential, and initial condition ψ0 (x). Deﬁne ψS (x, t) to be the scattered part of ψ(x, t), i.e. the solution of (3.32) with ψS (x, 0) = 0. This implies that ψB (x, t) = ψI (x, t) + ψS (x, t). By Zak transforming (3.32), we obtain the following equation for ΨS (σ, x, t) = Z[ψS ](σ, x, t): (σ + i∂t )ΨS (σ, x, t) = [−∆ + 2ib(t)∂x ] y(σ, t)ΨS (σ, x, t) − 2δ(x)y(σ, t). Equivalently: −1

ΨS (σ, x, t) = − [+σ + i∂t + ∆ − b(t)∂x ]

2δ(x)y(σ, t).

(3.35)

This has the solution ΨS (σ, x, t) = −G(σ)2δ(x)y(σ, t). Note that by Proposition 3.6, for each x, ΨI (σ, x, t) = Z[ψI ](σ, x, t) takes the form (1/2)σ −1/2 ψ0 (x)dx + f (σ 1/2 , x, t) with f (σ 1/2 , x, t). We can now reconstruct ψB (x, t) by inverting the Zak transform of Ψ(σ, x, t) = ΨI (σ, x, t) + ΨS (σ, x, t): ω ψB (x, t) = ω −1 e−iσt Ψ(σ, x, t)dσ

0 −iK(M)

=

e 0

−iσt

−iK(M)+ω

Ψ(σ, x, t)dσ +

e−iσt Ψ(σ, x, t)dσ

−iK(M) ω

+

e−iσt Ψ(σ, x, t)dσ + Residues.

(3.36)

−iK(M)+ω

Note that both ΨI (σ, x, t) and ΨS (σ, x, t) are bounded on for ﬁxed x, and for σ = −iK(M ). Thus, the integral over the contour [−iK(M ), −iK(m) + ω] decays like O(e−K(M)t ). Thus, (3.36) becomes: −iK(M)+ω −iK(M) e−iσt Ψ(σ, x, t)dσ − e−iσt Ψ(σ, x, t)dσ ψ(x, t) = 0

+ Residues + O(e

ω −K(M)t

).

(3.37)

We show that the contour integral in (3.37) gives rise to the dispersive part, while residues give rise to the resonance. The Residue Term, σ ˇ k = 0 By substituting (3.23) into (3.36), we ﬁnd that when σ ˇk = 0, the residue term (for each pole) takes the form: 2 σ, t) = αj,k e−iˇσk t tj Φk,j (x, t) (3.38) G(ˇ σk )δ(x)Yk,j (t)Yk,j (t)|y0 (ˇ −e−iˇσk t tj ωj!

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with αk,j = Yk,j (t)|y0 (ˇ σ, t) and Φk,j (x, t) = G(ˇ σk )δ(x)Yk,j (t). This follows because y(σ, t) has a pole at σ = σ ˇk with residue Yk,j (t), and analyticity of G(σ) (recall Proposition 3.17, in particular (3.34a)) shows that Ψ(σ, x, t) has a pole at σ = σ ˇk with residue Φk,j (x, t). Thus we have proved (1.4c). The Dispersive Part To compute the integral term of (3.37), note that we must compute: −iK(M) −iK(M)+ω e−iσt Ψ(σ, x, t)dσ − e−iσt Ψ(σ, x, t)dσ ΨM (x, t) = 0

ω −iK(M)

= 0

e−iσt ΨI (σ, x, t)dσ − e−i(σ+ω)t ΨI (σ + ω, x, t)dσ

−iK(M)

−2

e−iσt G(σ)δ(x)y(σ, t)

0

− e−i(σ+ω)t G(σ + ω)δ(x)y(σ + ω, t)dσ + O(e−K(M)t ). Since Z[f ](σ + ω, t) = e e

−iσt

(ΨI (σ, x, t) − e

iωt

−iωt

Z[f ](σ, t), we ﬁnd that:

ψI (σ + ω, x, t)) = e−iσt (ΨI (σ, x, t) − ΨI (σ − 0, x, t)).

Using the fact that ΨI (σ, x, t) = σ tion 3.6) we ﬁnd that: (3.40) = e

(3.39)

−iσt

−1/2

−1/2

(σ + 0)

(1/2) ψ0 (x)dx + A(σ

1/2

) (by Proposi-

(1/2)

− e−iσt (σ − 0)−1/2 (1/2) = e−iσt σ −1/2

(3.40)

ψ0 (x)dx + A(σ 1/2 ) ψ0 (x)dx + A(σ 1/2 )

ψ0 (x)dx + e−iσ A(σ 1/2 ).

Plugging this into (3.39) yields: −iK(M) −1/2 e−iσt A(σ 1/2 )dσ (3.39) = t ψ0 (x)dx + 0

−iK(M)

−2

e−iσt G(σ)δ(x)y(σ, t) − e−i(σ+ω)t G(σ + ω)δ(x)y(σ + ω, t)dσ

0

+ O(e−K(M)t ).

(3.41)

Again using the identity Z[y](σ + ω, t) = e Z[y](σ, t), we ﬁnd: −iK(M) e−iσt A(σ 1/2 )dσ (3.41) = t−1/2 ψ0 (x)dx + iωt

−2

0 −iK(M)

e−iσt [G(σ + 0)δ(x)y(σ + 0, t) − G(σ − 0)δ(x)y(σ − 0, t)]dσ

0

+ O(e−K(M)t ).

(3.42)

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Since G(σ)δ(x) = σ −1/2 (1/2)P0 + (C + A(σ 1/2 )) near σ = 0, the second to last line of (3.42) becomes: −iK(M) e−iσt [(σ + 0)−1/2 (1/2)P0 y(σ + 0, t) + (C + A(σ 1/2 ))y(σ + 0, t) 0

− (σ − 0)−1/2 (1/2)P0 y(σ, t) + (C + A(σ 1/2 ))δ(x)y(σ − 0, t)]dσ −iK(M) e−iσt σ −1/2 P0 y(0, t) + e−iσt A(σ 1/2 )dσ. (3.43) = 0

If (1 − K(σ)) has no poles near σ = 0, then P0 y(0, t) = (1/2) ψ0 (x)dx (see Proposition 3.12), and plugging (3.43) into (3.41) yields: −iK(M) e−iσt A(σ 1/2 )dσ ΨM (x, t) = t−1/2 ψ0 (x)dx + −1

− =

0 −iK(M)

e−iσt σ −1/2

ψ0 (x)dx dσ +

0 −iK(M)

−iK(M)

e−iσt A(σ 1/2 )dσ

0

e−iσt A(σ 1/2 )dσ + O(e−K(M)t ).

(3.44)

0

The integral in the last line of (3.44) is a Laplace-type integral, and A(σ 1/2 ) is analytic in σ 1/2 . Watson’s lemma therefore yields (1.7), and analyticity in σ 1/2 shows that the sum in (1.7) starts at n = 3. If (1 − K(σ))−1 has poles near σ = 0 (i.e. σ ˇk = 0 for some k), then the sum will begin from n = 1 (see below). The Residue Term, σ ˇk = 0 In the event that (1 − K(σ))−1 has poles in σ 1/2 near σ = 0, the only diﬀerence in the above analysis is that the σ −1/2 terms coming from y0 (σ, t) will not cancel the σ −1/2 terms coming from G(σ)δ(x)y(σ, t). Thus, (3.44) instead becomes:   −iK(M) M d (x, t) j ΨM (x, t) = e−iσt  + A(σ 1/2 ) dσ + O(e−iK(M)t ). −n/2 σ 0 n=1 The order of the pole, M , cannot be larger than 2 since this would imply that ΨM (x, t) grows at the rate tM /2−1 , which would contradict conservation of probability. If the order of the pole is 2, this corresponds to a Floquet bound state at zero energy, and if the order is 1, this corresponds to a zero energy resonance, and (1.6) holds with the sum starting from n = 1 in (1.7). We have thus far proved all of Theorem 2 except for the fact that Φk,0 (x, t) decays at x = ±∞ if ˇ σk = 0. Proposition 3.19. Suppose that ˇ σk = 0. Then Φk,0 (x, t) decays at x = ±∞ and ψnL,R = 0 for all n < 0. Furthermore, the pole is of order 1.

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Proof. It is clear that unitary evolution implies: 2π/ω R ω |ψ(x, t)|2 dxdt ≤ 1. 2π 0 −R

863

(3.45)

If the pole is of order greater than 1, then: ψ(x, t) =

nk

tj αk,j e−iˇσk t Φk,j (x, t) +

tj αk ,j e−iˇσk t Φk ,j (x, t) + ΨM (x, t).

k =k

j=0

But the second two terms decay, while the ﬁrst grows with time. This contradicts unitary evolution, unless nk = 0. Thus the pole must be of ﬁrst order. Now suppose that in the expansion of Φk,0 (x, t), at least one ψnL,R = 0 with n < 0. Then Φk,0 (x, t) will oscillate with x rather than decay. This implies that: ω 2π

0

2π/ω R −R

|Φk,j (x, t)|2 dxdt ≥ CR

for suﬃciently large R and some C > 0. On the other hand, the rest of ψ(x, t) (the dispersive part, and the exponentially decaying poles) which we denote R(x, t) decays with time. √This implies that for t ≥ tR (with TR chosen large enough so that |R(x, t)| ≤ / 2R that: √ ψ(x, t) = Φk,0 (x, t) + R(x, t) ≥ Φk,0 (x, t) − R(x, t) ≥ CR − . Selecting R > (2 + )/C causes ψ(x, t) ≥ 1, contradicting unitary evolution. Intuitively, what this means is the following. The modes ψnL,R with n < 0 correspond to radiation modes. If such a mode is nonzero, then Φk,0 (x, t) will be emitting “radiation” without decaying, which is clearly impossible. 4. Concluding Remarks In this paper we studied the interaction of a simple model atom with a dipole radiation ﬁeld of arbitrary strength. We obtained a resonance expansion, in which resonances can be resolved regardless of their complex quasi-energy. In particular, we obtained a rigorous deﬁnition of the ionization rate γ = −2ˇ σk and Stark-shifted energy, ˇ σk for the kth resonance. We further showed that complete ionization occurs (γ > 0) when E(t) is a trigonometric polynomial. Some possible future directions of research include: 4.1. Perturbative and numerical calculations The main feature of our method is that it turns a time dependent problem on R into a compact analytic Fredholm integral equation. This implies that a family of ﬁnite dimensional approximations can be used (in the Zak domain) to approximate solutions to the time dependent Schr¨ odinger equation.

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We believe that the quasi-energy methodology used here and in related papers [6, 9, 8] can be used for quantitative calculations of realistic physical systems. Perturbative calculations along these lines have recovered Fermi’s Golden Rule and the multiphoton eﬀect. 4.2. Extension to 3 dimensions In the case of H0 = −∆ − 2δ(x) with x ∈ R3 , a similar equation to (3.4) can be derived. Due to the fact that δ(x) is not in H −1 (R3 ), ψ(x, t) becomes singular at t = 0+ . This can be remedied by considering weak solutionsg [30], and an equation similar in most respects to (3.4) can be derived which governs the evolution [14]. For this reason, we believe most results can be adapted to this case, as has been done for H0 = −∆ − 2δ(x) + E(t)δ(x) [9, 6, 26]. Appendix A. Proof of Proposition 2.2 We observe that by the results of Sec. 3, if a bound state exists, then: ψB (0, t) = ea(t)/4 e−ia(t) Yk (t). Setting z = e−iωt , and y(z) = Yk (t), we wish to show that y(z) = f (z) + g(z) with f, g both entire of exponential order 2n. This is equivalent to showing that: |Yk (t + iα)| ≤ C exp[C exp(|2N ωα|)]. The function Yk (t) satisﬁes the equation: 2π/ω Yk (t) = k (t, s)Yk (t − s)ds = − 0

2π/ω

k (t, t − s)Yk (s)ds

0

with k (t, s) as deﬁned in (3.19b). Thus we obtain the bound: 2π/ω |Yk (t + iα)| ≤ |k (t + iα, t + iα − s)||Yk (s)|ds

(A.1)

0

and it suﬃces to bound |k (t + iα, t + iα − s)|. From the deﬁnition of k (t, s), we ﬁnd: k (t + iα, t + iα − s) (c(t + iα) − c(s))2 ω dp eσp = exp − 1 √ . 2πi R+0i 1 − eωp+α−iω(t−s) 4p p Supposing α/ω > 1 (we are interested in the behavior as α → ∞), the integrand is analytic for z = reiθ , 0 < r < 1 and 0 ≤ θ ≤ π. Thus, we can deform the contour from R + 0i to γ = ∂{z : z < 0 or |z| < 1}. considers the operator H0 restricted to the domain D = {f ( x) : f (x) ∈ H 1 (R3 ) and f (0) = 0}. From this domain, one can construct a self-adjoint extension of H0 , thus allowing the evolution to be deﬁned.

g One

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Note that for some constant C, |c(t + iα)| ≤ CeN ω|α| , since c(t) is a trigonometric polynomial of order N . We ﬁnd that there are three regions of integration which contribute to k (t + iα, t + iα − s). The regions of integration contributing come from the region near 1 − eωp+α−iω(t−s) = 0 (the pole of the integrand), large p and small p. If the pole is closer to R than π/ω, we deform γ up to encircle it at a distance piω. Otherwise, we ignore it. Therefore, in any case, for z ∈ γ, 1 − eωp+α−iω(t−s) is uniformly bounded away from zero. We then split γ = γ< ∪ γ> ∪ γα where γ< = {p ∈ γ : |p| < (CeN ω|α| + c(s)L∞ )2 } and γ> = γ \ γ< . We therefore ﬁnd that: |k (t + iα, t + iα − s)| ≤ |residue|C

dp (c(t + iα) − c(s))2 exp − 1 ωp+α−iω(t−s) 4p 1 − e |p| γ< σp 2 dp e (c(t + iα) − c(s)) +C − 1 1 − eωp+α−iω(t−s) exp 4p |p| γ> eσp

≤ C. The residue can be bounded by: |residue| (c(t + iα) − c(s))2 1 σ(−α+iω(t−s))/ω ≤ C e exp −1 4(−α + iω(t − s))/ω (−α + iω(t − s))/ω ≤ C exp(C|c(t + iα)|2 ) ≤ C exp(C exp(2N ω|α|)). We bound the integral over the compact region γ< simply by taking absolute values: dp eσp (c(t + iα) − c(s))2 − 1 1 − eωp+α−iω(t−s) exp 4p |p| γ< ≤ |γ< |C exp(C exp(2N ω|α|)). For the integral over γ> , we use the fact that if |z| < 1, |ez − 1| ≤ e|z|: dp eσp (c(t + iα) − c(s))2 exp − 1 1 − eωp+α−iω(t−s) |p| 4p γ> (CeN ω|α| + c(s)L∞ )2 dp eσp × |p| ωp+α−iω(t−s) |p| γ> 1 − e eσp 2N ω|α| −3/2 ≤ Ce 1 − eωp+α−iω(t−s) p dp ≤ C exp(C exp(2N ω|α|)). γ> Combining these estimates, we ﬁnd that k (t+iα, t+iα−s) has the required growth as α → ∞, hence Yk (t) does. The same argument applies as α → −∞.

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Appendix B. Proof of Proposition 3.17 We state a few results we need. Theorem 4 (Kato, [23, p. 368]). If a family T (σ) of closed operators on X depending on σ holomorphically has a spectrum consisting of two separated parts, the subspaces of X corresponding to the separated parts also depend on σ holomorphically. Remark B.1. A few words of explanation are in order. In [23], they are given in the commentary following the theorem. The analytic dependence of the separated parts of the spectrum means the following. Let Mσ , Mσ be the spectral subspaces of T (σ), related to the two separated parts. Then there exists an analytic function U (σ) (called the transformation function), with analytic inverse, so that Mσ = U (σ)M0 and Mσ = U (σ)M0 . For ﬁxed σ, both U (σ) and U −1 (σ) are bounded operators on the Hilbert space. In addition, the spectral projections PM (σ) and PM (σ) can be written as: PM (σ) = U (σ)PM (0)U −1 (σ),

(B.1a)

PM (σ) = U (σ)PM (0)U −1 (σ).

(B.1b)

We now prove a Lemma which allows us to reconstruct Ψ(σ, x, t) given solely information about Ψ(σ, 0, t). The basic idea is to treat the Schr¨odinger equation as an evolution equation in x, with a “Hamiltonian” that is periodic in t. Lemma B.2. Deﬁne the Hilbert space H = H 1/2 (Sω1 , dt) ⊕ L2 (Sω1 , dt). Then there exists a sequence Nm with 0 < inf m |Nm | ≤ supm |Nm | so that −imωt ∓λm,± c(t) e 2−1/2 λ−1 def m,± e φm,± = Nm (B.2a) 2−1/2 e−imωt e∓λm,± c(t) is a Riesz basis for H. Here, λm,± is deﬁned as: √ def λm,± = ∓i σ + mω. Furthermore, the operator

def

H =

0 σ + i∂t

1 0 + 0 0

(B.2b) 0 b(t)

is diagonal in this basis, with Hφm,± = λm,± φm,± . Moreover, if we deﬁne H+ as the span of {φm,+ }m∈Z and H− as the span of {φm,− }m∈Z then exH is deﬁned, bounded and analytic in σ for σ ∈ [0, ω) (except at σ = 0) on H+ for x ≤ 0, and on H− for x ≥ 0. We are nearly ready to prove Lemma B.2. First a minor technical point. √ Remark B.3. Consider the sequence σ + nω, with σ ∈ (0, ω). For n negative,

√ √ σ + nω grows like |n|. For n positive, σ + nω = O(n−1/2 ), and is uniformly bounded below.

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Proof of Lemma B.2. It is a simple calculation to show φm,± are eigenvectors of H with eigenvalues λm,± . To show that {φ± m } is a Riesz basis for H, we show that H is a bounded perturbation of a normal operator. Consider the family of operators (analytic in ζ) on H: u 0 1 0 0 u u = + . Hζ ux σ + i∂t 0 0 ζb(t) ux ux Consider also the family of vectors (parametrized by ζ): −1/2 −1 −imωt ζλm,± c(t) 2 λ e e m,± Nm,ζ . {φ± m,ζ } = 2−1/2 e−imωt e∓ζλm,± c(t) m∈Z Nm,ζ is a normalizing constant which is deﬁned implicitly; we discuss it below. For ζ = 0, Nm,ζ = 1. A simple calculation shows that (φ± m,ζ , λm,± ) are eigenvector/eigenvalue pairs of Hζ . In particular, each λm,± is separate from all the others. For ζ = 0, they are ± (ζ) be the associated spectral projection operators, also orthonormal in H. Let Pm given by ± (Hζ − z)−1 dz Pm (ζ) = γm

where γm is a closed curve containing only λ± m,ζ , and no other eigenvalue of Hζ . Let U (ζ) be the transformation function of Theorem 4 (see also Remark B.1 and Eq. (B.1)). Since each λm,± is separated from all the others and varies analytically (except near σ = 0), Theorem 4 (using λm,± as one of the separated parts of the spectrum and {λm ,± }m =m as the other) implies that: ± ± ± (ζ) = U (ζ)Pm (0)U −1 (ζ) = U (ζ)−1 · |φ± Pm m,0 U (ζ)φm,0 ± = · |[U (ζ)−1 ]∗ φ± m,0 U (ζ)φm,0 .

We know that U (ζ)φ± m,0 is a vector in the direction −imωt ∓ζλm,± c(t) 2−1/2 λ−1 e m,± e 2−1/2 e−imωt e∓ζλm,± c(t) but this determines U (ζ)φ± m,0 only up to a constant (not necessarily real), denoted −1 by Nm,ζ . Since U (ζ) is bounded above and below, 0 < U (ζ)−1 ≤ |Nm,ζ | ≤

U (ζ). To compute the expansion of a function ψ(t) in this basis, we use the ± ± = U (1)−1 ψ|φ± formula ψm m,0 . Since φm,0 is an orthonormal basis, this set of −1 coeﬃcients is clearly in l2 , with l2 norm bounded below by U (1) ψ(0, t)H and above by U (1) ψ(0, t)H . + − (ζ) + Pm (ζ) = 1, interpreting the sum in Finally, we need to show that m Pm ± (0) the strong topology. The sum is strongly convergent when ζ = 0, since the Pm

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are orthogonal projections. Multiplying on the left and right by the continuous operators U (ζ) and U (ζ)−1 yields: −1 + − 1 = U (ζ)U (ζ) = U (ζ) Pm (0) + Pm (0) U (ζ)−1 =

m + U (ζ)[Pm (0)

+

− Pm (0)]U (ζ)−1

=

m

+ − Pm (ζ) + Pm (ζ).

m

This proves the Riesz basis property. To show boundedness of exH , simply note that the real part of the eigenvalues of H is bounded above on H+ and bounded below on H− for σ in compact regions not containing σ = 0. Thus, exH is bounded on H+ . Analyticity follows by observing that the eigenvalues and eigenfunctions are analytic in σ 1/2 , except near σ = 0. We are now prepared to prove Proposition 3.17. Proof of Proposition 3.17. Note that (3.33) can be rewritten as: u u ∂x =H . ux ux Away from x = 0, the solution u(x, t) can be written (formally) as: u(0± , t) u(x, t) xH =e , ±x < 0. ∂x u(0± , t) ∂x u(x, t)

(B.3)

At x = 0, the two matching conditions need be satisﬁed: u(0+ , t) − u(0− , t) = 0

(Continuity),

−

∂x u(0 , t) − ∂x u(0 , t) = −2f (t) (Diﬀerentiability). +

For σ > 0, λm,+ always has positive real part and λm,− always has negative real part (recall (B.2b)). Thus, if u(x, t) is to vanish as x → ±∞, we ﬁnd that: u(0− , t) um,R φm,− (t), = ∂x u(0− , t) m u(0+ , t) um,L φm,+ (t). = ∂x u(0+ , t) m Since φm,± is a Riesz basis and [0, f (t)] ∈ H, we can write: 0 fm,+ φm,+ + fm,− φm,− . = f (t) m

(B.4)

Choosing um,R = fm,− and um,L = −fm,+ solves (3.33), at least on a formal level. Since u(0+ , t) ∈ H+ and u(0− , t) ∈ H− , (B.3) makes sense. Since exH is bounded and analytic provided |x| < R, this is thus an analytic mapping from L2 (Sω1 ) → L2 (BR × Sω1 ).

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Now observe that both (B.3) and (B.4) can be analytically continued in σ, and the continuation also solves (3.33), therefore G(σ) can be analytically continued in σ as well. We now need only determine the behavior near σ = 0. By Taylor-expanding (B.2a) in σ 1/2 , we ﬁnd that: −1/2 O(1) −1/2 ∓σ φ0,± = N0 2 + . (B.5) 1 O(σ 1/2 ) while

φm,± = Nm 2

−1/2

−imωt λ−1 m,± e

e−imωt

O(1) + . O(σ 1/2 )

Thus, near σ = 0, we ﬁnd to leading order (plugging (B.5) into (B.4)) that: fm,+ − fm,− = 0, N0 2

−1/2

(fm,+ + fm,− ) = P0 f (t),

with P0 f (t) projection onto the zeroth Fourier coeﬃcient. This implies that f0,± = u0,R = −u0,L = 21/2 N0−1 (1/2)P0 f (t)+O(σ 1/2 ). On all other coeﬃcients, the behavior is analytic in σ since λm,± is analytic in σ for m = 0. Thus for small σ: σ −1/2 u(x, t) = (1/2)(P0 f (t)) + O(1) 1 and therefore G(σ)δ(x)f (t) = (1/2)σ −1/2 [P0 f (t)] + O(1) near σ = 0. Appendix C. Wellposedness Wellposedness of (1.10) (and by extension (1.9) and (1.1)) is sketched in an example at the end of [33], though we sketch a proof for completeness. Let H0 be a self adjoint operator and let V (t) be a time-dependent quadratic form. Suppose the following conditions hold: D(H0 ) ⊆ D(V (t)).

(C.1a)

There exist constants a ∈ (0, 1), b ∈ (0, ∞) such that: 1/2

1/2

|V (t)(f, f )| ≤ aH0 f |H0 f + bf |f .

(C.1b)

The function (H0 + 1)−1/2 V (t)(H0 + 1)−1/2 is norm diﬀerentiable with derivative (H0 + 1)−1/2 V (t)(H0 + 1)−1/2 and |V (t)(f, f )| ≤ aH0 f |H0 f + bf |f . 1/2

1/2

(C.1c)

If H0 , V (t) satisfy these conditions, then by [33, Theorem 11], there is a constant C so that H0 + V (t) + C is a K-generator (deﬁned in [33]). It is then shown in [33, Theorem 8] that a K-generator generates a unitary propagator U (t, s), i.e. ∂t f |U (t, s)ψ0 = (H0 + V (t) + C)1/2 f |(H0 + V (t) + C)1/2 U (t, s)ψ0 .

(C.2)

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We let H0 = −∂x2 and V (t) = ib(t)∂x − 2δ(x). (C.1a) clearly holds; D(ib(t)∂x ) = ⊇ H 1 , while Sobolev embedding shows that D(−2δ(x)) ⊇ H 1 . (C.1c) follows H simply by noting that V (t) = E(t)∂x , which is again −∂x2 -bounded for suﬃciently large b. (C.1b) can be veriﬁed with a = 1/2. Clearly, ib(t)∂x is −∂x2 -bounded with a = 1/4 (or any other a). Note that f (0) = fˆ(k)dk. Thus: 1/2

2 2 (b + k 2 /4)1/2 ˆ ˆ f (k)dk f |2δ(x)f = 2|f (0)| = 2 f (k)dk = 2 (b + k 2 /4)1/2 2 2 ≤ 2 (b + k 2 /4)1/2 fˆ(k) (b + k 2 /4)−1/2 L2 L2 2 1/2 1/2 = 2 (b + k 2 /4)−1/2 2 [(1/4)H0 f |H0 f + bf |f ]. 2

L

2 We can make (b + k 2 /4)−1/2 L2 ≤ 1/2 by choosing b large; thus −2δ(x) is −∂x2 bounded with a = 1/4. Adding the results together veriﬁes (C.1b). Thus, (1.10) is well posed. Applying the unitary transformations described in Sec. 1.2 shows that (1.1) and (1.9) are well posed as well. Acknowledgments We thank A. Soﬀer and M. Kiessling for useful discussions. J.L.L. and O.C. would like to thank the IHES in Bures-sur-Yvette and the IAS in Princeton where part of the work was done. We also thank an anonymous referee for a very careful reading. Work supported by NSF Grants DMS-0100495, DMS-0406193, DMS0600369, DMS01-00490, DMR 01-279-26 and AFOSR grant AF-FA9550-04. Any opinions, ﬁndings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reﬂect the views of the National Science Foundation. References [1] A. Jensen A. Galtbayar and K. Yajima, Local time-decay of solutions to Schr¨ odinger equations with time-periodic potentials, J. Statist. Phys. 116 (2004) 231–282. [2] S. Agmon and M. Klein, Analyticity properties in scattering and spectral theory for Schr¨ odinger operators with long-range radial potentials, Duke Math. J. 68(2) (1992) 337–399. [3] J. Dupont-Roc, C. Cohen-Tannoudji and G. Gryndberg, Atom-Photon Interactions (Wiley, 1992). [4] A. Fring, C. Figueira de Morisson Faria and R. Schrader, Analytical treatment of stabilization, arxiv:physics/9808047 (1998). [5] S. L. Chin and P. Lambropoulos, Multiphoton Ionization of Atoms (Academic Press, 1984). [6] M. Correggi, G. Dell’Antonio, R. Figari and A. Mantile, Ionization for three dimensional time-dependent point interactions, Commun. Math. Phys. 257(1) (2005) 169–192.

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[7] O. Costin, R. D. Costin and J. L. Lebowitz, Transition to the continuum of a particle in time-periodic potentials, in Advances in Diﬀerential Equations and Mathematical Physics (Birmingham, AL, 2002), ed. Y. Karpeshina, Contemp. Math., Vol. 327 (Amer. Math. Soc., Providence, RI, 2003), pp. 75–86. [8] O. Costin, R. D. Costin and J. L. Lebowitz, Time asymptotics of the Schr¨ odinger wave function in time-periodic potentials, J. Statist. Phys. 116(1–4) (2004) 283–310. [9] O. Costin, R. D. Costin, J. L. Lebowitz and A. Rokhlenko, Evolution of a model quantum system under time periodic forcing: Conditions for complete ionization, Commun. Math. Phys. 221(1) (2001) 1–26. [10] O. Costin and A. Soﬀer, Resonance theory for Schr¨ odinger operators, Commun. Math. Phys. 224(1) (2001) 133–152; Dedicated to Joel L. Lebowitz. [11] H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon, Schr¨ odinger Operators with Application to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics (Springer-Verlag, 1987). [12] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61 (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992). [13] R. de la Madrid and M. Gadella, A pedestrian introduction to Gamow vectors, Amer. J. Phys. 70 (2002) 626–638. [14] G. F. Dell’Antonio, R. Figari and A. Teta, The Schr¨ odinger equation with moving point interactions in three dimensions, in Stochastic Processes, Physics and Geometry: New Interplays, I (Leipzig, 1999), eds. S. Albeverio, F. Gesztesy and H. Holden, CMS Conf. Proc., Vol. 28 (Amer. Math. Soc., Providence, RI, 2000), pp. 99–113. [15] A. Fring, V. Kostrykin and R. Schrader, Ionization probabilities through ultra-intense ﬁelds in the extreme limit, J. Phys. A 30(24) (1997) 8599–8610. [16] G. Gamow, Zur quantentheori de atomkernus, Z. Phys. 51 (1928) 204–212. [17] S Geltman, Multiphoton ionization of atoms, J. Phys. 10 (1974) 831–835. [18] S. Graﬃ, V. Grecchi and H. J. Silverstone, Resonances and convergence of perturbation theory in n-body atomic systems in external ac-ﬁelds, Ann. Inst. H. Poincare 42 (1985) 215–234. [19] S. Graﬃ and K. Yajima, Exterior complex scaling and the ac-stark eﬀect in a coulomb ﬁeld, Commun. Math. Phys. 89 (1983) 277–301. [20] J. Howland, Perturbation of embedded eigenvalues, Bull. Amer. Math. Soc. 7 (1972) 280–283. [21] R. V. Jensen, Stochastic ionization of surface state electrons: Classical theory, Phys. Rev. A. 30 (1984) 386–397. [22] R. V. Jensen and I. B. Bernstein, Semiclassical theory of relativistic electrons in space and time varying electromagnetic ﬁeld, Phys. Rev. A 29 (1984) 282–289. [23] T. Kato, Perturbation Theory for Linear Operators, A Series of Comprehensive Studies in Mathematics, Vol. 132 (Springer Verlag, 1976); Corrected printing of the 2nd edn. [24] V. Kostrykin and R. Schrader, Ionization of atoms and molecules by short, strong laser pulses, J. Phys. A 30(1) (1997) 265–275. [25] L. D. Landau and E. M Lifshitz, Quantum Mechanics Non-Relativistic Theory (Pergamon Press, 1977). [26] G. F. DellAntonio and M. Correggi, Decay of a bound state under a time-periodic perturbation: A toy case, J. Phys. A 38(22) (2005) 4769–4781. [27] P. D. Miller, A. Soﬀer and M. I. Weinstein, Metastability of breather modes of timedependent potentials, Nonlinearity 13(3) (2000) 507–568. [28] A. Rokhlenko, O. Costin and J. L. Lebowitz, Decay versus survival of a localized state subjected to harmonic forcing: Exact results, J. Phys. A 35(42) (2002) 8943–8951.

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[29] C. Rothe, S. I. Hintschich and A. P. Monkman, Violation of the exponential-decay law at long times, Phys. Rev. Lett. 96(16) (2006) 163601. [30] R. Hegh-Krohn, H. Holden, S. A. Al-beverio and F. Gesztesy, Solvable Models in Quantum Mechanics (Springer-Verlag, 1988). [31] B. Simon, Resonances and complex scaling: A rigorous overview, Int. J. Quant. Chem. 14 (1978) 529–542. [32] E. Skibsted, Truncated Gamow functions, α-decay and the exponential law, Commun. Math. Phys. 104(4) (1986) 591–604. [33] A. D. Sloan, The strong convergence of Schr¨ odinger propagators, Trans. Amer. Math. Soc. 264(2) (1984) 557–570. [34] A. Soﬀer and M. I. Weinstein, Time dependent resonance theory, Geom. Funct. Anal. 8(6) (1998) 1086–1128. [35] M. Stein and R. Shakarchi, Complex Analysis (Princeton University Press, 2003). [36] Y. Strauss, I. M. Sigal and A. Soﬀer, From Gamow states to resonances for Schr¨ odinger operators, in preparation (2007). [37] K. Yajima, Resonances for the ac-stark eﬀect, Commun. Math. Phys. 87 (1982) 331–352.

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Reviews in Mathematical Physics Vol. 20, No. 7 (2008) 873–900 c World Scientific Publishing Company

¨ HOLDER CONTINUITY OF THE INTEGRATED DENSITY OF STATES FOR MATRIX-VALUED ANDERSON MODELS

HAKIM BOUMAZA Keio University, Department of Mathematics, Hiyoshi 3-14-1, Kohoku-ku 223-8522, Yokohama, Japan [email protected] Received 14 January 2008 Revised 12 June 2008 We study a class of continuous matrix-valued Anderson models acting on L2 (Rd ) ⊗ CN . We prove the existence of their Integrated Density of States for any d ≥ 1 and N ≥ 1. Then, for d = 1 and for arbitrary N , we prove the H¨ older continuity of the Integrated Density of States under some assumption on the group GµE generated by the transfer matrices associated to our models. This regularity result is based upon the analoguous regularity of the Lyapounov exponents associated to our model, and a new Thouless formula which relates the sum of the positive Lyapounov exponents to the Integrated Density of States. In the final section, we present an example of matrix-valued Anderson model for which we have already proved, in a previous article, that the assumption on the group GµE is verified. Therefore, the general results developed here can be applied to this model. Keywords: Integrated Density of States; Lyapounov exponents; Anderson model; Thouless formula. Mathematics Subject Classification 2000: 47B80, 37H15

1. Introduction We will study the question of the existence of the Integrated Density of States and its regularity for continuous matrix-valued Anderson models of the form: HA (ω) = −∆d ⊗ IN + Vω(n) (x − n) (1.1) n∈Zd

acting on L2 (Rd ) ⊗ CN , where d and N are non-negative integers, IN is the identity matrix of order N and ∆d denotes the d-dimensional continuous Laplacian. Let (Ω, A, P) be a complete probability space and ω ∈ Ω. For every n ∈ Z, the functions (n) x → Vω (x) will be symmetric matrix-valued functions, supported in [0, 1]d, and

873

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bounded uniformly on x, n and ω. We also set: ∀x ∈ Rd , Vω (x) = Vω(n) (x − n) n∈Zd

and denote by Vω the maximal multiplication operator by x → Vω (x). The function x → Vω (x) is uniformly bounded on R in x and in ω. The potential Vω will also be such that the operator HA (ω) is Zd -ergodic. As a bounded perturbation of −∆d ⊗ IN , the operator HA (ω) is self-adjoint on the Sobolev space H 2 (Rd ) ⊗ CN . We want to deﬁne a function of the real variable which will count the number of proper energy states of HA (ω) below a ﬁxed energy E. For systems like (1.1), such a deﬁnition will usually lead to an inﬁnite function as the operators we study act on an inﬁnite-dimensional Hilbert space and thus have inﬁnitely many spectral values. To avoid this problem, we will deﬁne our function, the Integrated Density of States or IDS, as a thermodynamical limit as explained in Sec. 2. It will lead to a problem of existence of such a thermodynamical limit. We will prove the existence of the IDS in Sec. 2 for any d and any N . This existence proof will be based upon a matrix-valued Feynman–Kac formula proven in [2] and the adaptation of the argument of Carmona in [7] to matrix-valued operators. Once we have proven the existence of the IDS, we will study its regularity as a function of the energy parameter E. For this second step, we will restrict ourselves to the case where d = 1 and N is arbitrary, and to be able to use the tools coming from the theory of ODEs such as the notion of transfer matrix. We will prove in Sec. 4 that under some assumption on Vω , or, more precisely, on the group generated by the transfer older continuous. This result matrices associated to HA (ω), the IDS is locally H¨ will come from the analoguous regularity result on Lyapounov exponents proved in Sec. 3, and from a Thouless formula proven in Sec. 4 which relates the IDS to the Lyapounov exponents. To prove this Thouless formula, we use results of Kotani and Simon in [20] and Kotani in [19]. The regularity result on Lyapounov exponents is based upon the results of Carmona and Lacroix in [9] and Lacroix, Klein and Speis in [17]. We also need to prove estimates on the transfer matrices for our model (1.1) (for d = 1) similar to those proven in [11] in the scalar-valued case. In a ﬁnal section, we present an example of continuous matrix-valued Anderson model for which the needed assumption on the group generated by the transfer matrices is veriﬁed. This example is the following matrix-valued Anderson–Bernoulli model: HAB (ω) d2 = − 2 ⊗ I2 + dx

0 1

1 0

+

n∈Z

(n)

ω1 χ[0,1] (x − n) 0

0 (n)

ω2 χ[0,1] (x − n) (1.2)

(n)

(n)

acting on L2 (R) ⊗ C2 , with (ω1 )n∈Z and (ω2 )n∈Z two independent sequences of independent and identically distributed (i.i.d.) random variables with common law ν such that {0, 1} ⊂ supp ν. This model has already been studied by the author in [3]

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as an improvement of a result by Stolz and the author in [5]. We proved in [5] the absence of absolutely continuous spectrum and pointed out that the improvement made in [3] was necessary to be able to prove local H¨older continuity of the IDS. The study of the regularity of the IDS is an important step to prove Anderson localization by using a multiscale analysis scheme. It is the key ingredient to prove a Wegner estimate as was done in [8] and to adapt it to the case of scalar-valued continuous Anderson model in [11]. We believe that once we have proved a Wegner estimate and an Initial Length Scale Estimate for model (1.1) for d = 1 and arbitrary N , it will be possible to adapt existing multiscale analysis schemes to the case of matrix-valued operators. We will then be able to prove Anderson and dynamical localization for this model as explained in [24]. The question of localization for one-dimensional continuous matrix-valued Anderson model is based on a more general problem on Anderson models. Localization for continuous Anderson models in dimension d ≥ 2 at all energies is still an open problem if one looks for arbitrary disorder, including Bernoulli randomness. A possible approach to the localization for d = 2 is to discretize one direction, which leads to considering a one-dimensional Anderson model, that is no longer scalarvalued, but N × N matrix-valued as we have here for d = 1. What is already well understood is the case of dimension one scalar-valued continuous Schr¨ odinger operators with arbitrary randomness (see [11]) and discrete matrix-valued Schr¨ odinger operators, also for arbitrary randomness (see [14, 17]). We want to combine here techniques of [11, 17] to get the local H¨ older continuity of the IDS for continuous matrix-valued models. We ﬁnish by mentioning that diﬀerent methods have been used in [16] to prove localization properties for random operators on discrete strips. They are based upon the use of spectral averaging techniques which did not allow us to handle with singular distributions of the random parameters like in our model (1.2). 2. Existence of the IDS In this section, we will deﬁne the IDS associated to the operator HA (ω) and prove its existence. The proof of the existence for the IDS will strongly rely on a matrixvalued Feynman–Kac formula which we will present after the deﬁnition of the IDS. As we have already noticed in the introduction, the operator HA (ω) is selfadjoint and Zd -ergodic. But, in some parts of the following proofs, and also in Sec. 4, we will need a stronger assumption of Rd -ergodicity for HA (ω) instead of only Zd -ergodicity. To avoid this lack of Rd -ergodicity in general, we can refer to the suspension procedure developed by Kirsch in [15]. This procedure allows us to con˜ A (˜ ω ), deﬁned on a bigger probability space, which struct from HA (ω) an operator H d ˜ ω ) is also constructed in a way such that its IDS and Lyapounov is R -ergodic. HA (˜ exponents exist if and only if those of HA (ω) exist, and in this case they are equal for both operators. Considering the use of this suspension procedure we will work in the following with HA (ω) as if it is Rd -ergodic instead of being only Zd -ergodic.

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2.1. Definition of the IDS We aim at deﬁning a function that will give us the mean number per unit volume of spectral values of HA (ω) situated below a ﬁxed real number E. In order to deﬁne this function, we will ﬁrst restrict HA (ω) to cubes of ﬁnite volume of Rd . Let L be a strictly positive integer and D = [−L, L]d ⊂ Rd be the cube centered at 0 and of length 2L. We set: (D) (D) Vω(n) (x − n) (2.1) HA (ω) = −∆d ⊗ IN + n∈Zd

the restriction of HA (ω) acting on L2 (D) ⊗ CN with Dirichlet boundary conditions on D. Definition 1. The Integrated Density of States, or IDS, associated to HA (ω) is the function from R to R+ , E → N (E) where N (E) for E ∈ R is deﬁned as the following thermodynamical limit: 1 (D) #{λ ≤ E | λ ∈ σ(HA (ω))} L→+∞ |D|

N (E) = lim

(2.2)

where |D| is the volume of D. Here we have a double problem of existence in the expression (2.2). First, we (D) have to prove that the cardinal #{λ ≤ E | λ ∈ σ(HA (ω))} is ﬁnite for each ﬁxed E and then we have to show the existence of the limit. The answer to each one of these problems relies on the existence of an L2 -kernel for the one-parameter semigroup (D) (e−tHA (ω) )t>0 . 2.2. A matrix-valued Feynman–Kac formula We will ﬁrst present a matrix-valued Feynman–Kac formula for the one-parameter semigroup (e−tHA (ω) )t>0 due to Boulton and Restuccia ([2]). We will then deduce (D) a Feynman–Kac formula for (e−tHA (ω) )t>0 . Let W = C(R+ , R) be the space of continuous functions from R+ to R. For every t ≥ 0 we consider the coordinate function: Xt :

W→R w → Xt (w) = w(t).

Let W be the smallest σ-algebra on W for which all the applications Xt are measurable. For s, t ≥ 0 and x, y ∈ Rd we denote by Ws,x,t,y the conditional Wiener measure, deﬁned on (W, W), associated to the Brownian motion starting from x at the time s and arriving on y at the time t. We also denote by Es,x,t,y the expectation value associated to the measure Ws,x,t,y . For a construction of such conditional Wiener measure and for a construction of the path integral associated to, we refer to [22, Chap. 2].

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We now study the one-parameter semigroup (e−tHA (ω) )t>0 . We ﬁx t > 0 and ω ∈ Ω. By the Lie–Trotter formula we have: e−tHA (ω) f = lim (e−(−∆d ⊗IN ) n e−Vω n )n f. t

∀f ∈ L2 (Rd ) ⊗ CN ,

t

n→+∞

(2.3)

For a ﬁxed n ∈ N, we can use [13, Corollary 3.1.2, p. 47] to get that the operator: (e−(−∆d ⊗IN ) n e−Vω n )n t

t

has an integral kernel given by the following path integral: n

jt

jt

e−( n )·Vω (w( n )) dW0,x,t,y (w).

(2.4)

j=1

But when n tends to inﬁnity we ﬁnd, by deﬁnition of the time-ordered exponential (see [12]): lim

n

n→+∞

t jt jt e−( n )·Vω (w( n )) = expord − Vω (w(s)) ds .

(2.5)

0

j=1

Then by Lebesgue’s dominated convergence theorem, we have that: Kt (x, y)f (y) dx ∀f ∈ L2 (Rd ) ⊗ CN , ∀x ∈ Rd , e−tHA (ω) f (x) =

(2.6)

Rd

where: ∀x, y ∈ R , d

∀t > 0,

Kt (x, y) =

t expord − Vω (w(s)) ds dW0,x,t,y (w). 0

(2.7) So we have just proven that e−tHA (ω) has an integral kernel, Kt (x, y). Let us see how (D) to deduce from this integral kernel, the existence of an integral kernel for e−tHA (ω) . We denote by TD (w) the time of the ﬁrst exit from D of the path w ∈ W: / D}. TD (w) = inf{t > 0, Xt (w) ∈

(2.8) (D)

Then the fact that we used Dirichlet boundary conditions to deﬁne HA (ω) allows us to use results on killed Brownian motions (see [18]) which lead us to the following formula: (D)

∀f ∈ L2 (Rd ) ⊗ CN , ∀x ∈ Rd , e−tHA (ω) f (x) t 1 = √ Vω (Xs (w)) ds χ{t
∀t > 0,

× dW0,x,t,y (w)e−

|x−y|2 2t

f (y) dy.

(2.9)

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So we have the following proposition: (D)

Proposition 1. For every t > 0, e−tHA formula:

(ω)

has an integral kernel given by the

(D)

∀x, y ∈ Rd , ∀t > 0, Kt (x, y) t 2 1 − |x−y| 2t = √ χ{t
and Kt

is in L2 (D2 ) ⊗ MN (C) for every t > 0.

Proof. The ﬁrst assertion and the formula (2.10) come from (2.9). Then D is a (D) compact domain in Rd and for a ﬁxed t > 0, (x, y) → Kt (x, y) is continuous. As (D) in (2.10), t is bounded by TD (w), we have that Kt is in L2 (D2 ) ⊗ MN (C) as it 2 is a bounded continuous function on D . This proposition will be the main ingredient to prove the existence of the IDS associated to HA (ω). 2.3. Existence of the IDS (D)

From Proposition 2.10, we deduce that for every t > 0, the operator e−tHA Hilbert–Schmidt on L2 (D) ⊗ CN . Thus, its spectrum is of the form: (D)

{e−tλj (D)

where (λj

(ω)

(ω)

is

, j ≥ 0}

(ω))j≥0 is an increasing sequence of real numbers, bounded from below (D)

and tending to +∞. This sequence is the spectrum of HA (ω). In particular, for a ﬁxed E ∈ R: (D)

(D)

#{λ ≤ E | λ ∈ σ(HA (ω))} = #{λj

(ω) ≤ E} < +∞.

This answers the ﬁrst part of of N (E). It remains for us the problem of existence (D) 1 #{λj (ω) ≤ E} converges to a real number to prove that the sequence |D| L≥1

independent of ω : N (E). To that end, we introduce the counting measure of the (D) eigenvalues of HA (ω): nD,ω =

1 δλ(D) (ω) j |D|

(2.11)

j≥0

(D)

where δλ(D) (ω) is the Dirac measure at λj

(ω). Then we have:

j

Proposition 2. The sequence of measures (nD,ω )L≥1 converges vaguely to a measure n independent of ω as L tends to inﬁnity for P-almost every ω in Ω. Moreover,

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the Laplace transform of this measure n is given by: ∀t > 0, t 1 L(n)(t) = √ TrCN expord − Vω (Xs (w)) ds dωdW0,0,t,0 (w). 2πt 0 Ω (2.12) Corollary 1. For every E ∈ R, the limit: N (E) = lim

L→+∞

1 (D) #{λ ≤ E | λ ∈ σ(HA (ω))} |D|

exists and is P-almost surely independent of ω. The function E → N (E) is the distribution function of n: ∀E ∈ R,

N (E) = n((−∞, E]).

Before proving this proposition, we need to prove a lemma which gives the expression of the trace of an operator with matrix-valued integral kernel. We adapt here a result of Simon proven in [23, Theorem 3.9, p. 35]. Lemma 1. Let H be a self-adjoint operator acting on L2 (D) ⊗ CN where D ⊂ Rd is a compact set. We assume that for all t > 0 the operator e−tH is class-trace and has a matrix-valued integral kernel Kt . Then: −tH Tr(e )= TrCN Kt (x, x) dx D

where TrCN denotes the usual trace on N × N matrices. Proof. Let n ∈ N, m ∈ {0, . . . , 2n } and k ∈ {1, . . . , N }. We set:  t (0, . . . , 0, 2 n2 , 0, . . . , 0) if ∀i ∈ {1, . . . N }, −L · m − 1 ≤ x < L · m i 2n 2n φn,m,k (x) = t (0, . . . , 0) otherwise n

where 2 2 is at the kth position. Then the family {φn,m,k }1≤k≤N n∈N,0≤m≤2n is a Hilbert 2 N basis of the Hilbert space L (D) ⊗ C . Let Pn be the projection on the subspace spanned by the 2n N functions φn,m,k for n ﬁxed and m ∈ {0, . . . , 2n }, k ∈ {1, . . . , N }. Then one can construct a Hilbert basis (ψ1 , ψ2 , . . .) of L2 (D) ⊗ CN such that: ∀n ∈ N,

ψ1 , . . . , ψ2n N ∈ Im Pn .

Then we have: Tr(e−tH ) = lim Tr(Pn e−tH Pn ) n→+∞

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by [23, Theorem 3.1, p. 31]. But: ∀n ∈ N, 2 N n

Tr(Pn e

−tH

Pn ) =

φn,m,k , e−tH φn,m,k

k=1 m=1

=

N 2n k=1 m=1

=

D

t

φn,m,k (x)Kt (x, y)φn,m,k (y) dxdy

D

2n m −L· m−1 2n ≤xi ,yi
m=1

×

n

N

n

22 ·22

(0, . . . , 1, . . . , 0)Kt (x, y)t (0, . . . , 1, . . . , 0) dxdy

k=1

TrCN (Kt (x,y))

=2

n

2n m −L· m−1 2n ≤xi ,yi
m=1

TrCN (Kt (x, y)) dxdy.

Then by uniform continuity of Kt on the compact set D2 : lim 2

n→+∞

= D

n

2n m=1

−L· m−1 ≤xi ,yi
TrCN (Kt (x, y)) dxdy

TrCN (Kt (x, x)) dx.

Proof of Proposition 2. We ﬁx t > 0. We have: e−Et nD,ω (E) L(nD,ω )(t) = R

=

1 −λ(D) e j (ω)t |D| j≥0

(D) 1 Tr(e−tHA (ω) ) |D| 1 = Tr N (Kt (x, x)) dx |D| D C 1 1 √ = χ{t
=

0

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by (2.10). We set: t 1 1 √ AD = Vω (Xs (w)) ds dW0,x,t,x (w) dx TrCN expord − |D| 2πt D 0 and: BD

1 1 √ = χ{t≥TD (w)} (w) TrCN expord |D| 2πt D t × − Vω (Xs (w)) ds dW0,x,t,x (w) dx.

881

(2.13)

(2.14)

0

Using Birkhoﬀ’s theorem when L → +∞ in AD , we get: t 1 lim AD = √ TrCN expord − Vω (Xs (w)) ds dωdW0,0,t,0 (w). L→+∞ 2πt 0 Ω (2.15) Let n be the measure on R (with the Borel σ-algebra) such that: t 1 L(n)(t) = √ TrCN expord − Vω (Xs (w)) ds dωdW0,0,t,0 (w). 2πt 0 Ω

(2.16)

To prove that nD,ω converges vaguely to n as L tends to inﬁnity, it remains to prove that BD → 0 and that the convergence of AD and BD happens on a set Ω1 independent of t and of measure 1. Actually, for the rest of the proof, we can refer to the proof of Carmona in [7, Theorem V1, pp. 66 and 67]. Indeed, as Vω is uniformly bounded on R in x and in ω, the function: Ω×W → C

t (ω, w) → TrCN expord − Vω (Xs (w)) ds

(2.17)

0

for t > 0 ﬁxed is in every L (Ω × W, P ⊗ W0,0 ) for all r > 1. Here W0,0 is the Wiener measure deﬁned on (W, W) associated to the Brownian motion starting from 0 at time 0. Thus, function (2.17) has the same properties as the function: r

Ω×W → C

t (ω, w) → exp − q − (Xs (w), ω) ds

(2.18)

0

in [7, Theorem V1]. Then, changing (2.18) by (2.17), one can rewrite the proof of [7]. Remark. In the proof of Proposition 2, we did not verify that the limit measure (D) n does not depend on the choice of boundary conditions for HA (ω). This choice appears in formula (2.9) by introducing the characteristic function χ{t
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motion (see [18, Chap. 4]). The rest of the proofs is unchanged and the expression of n does not depend on χ{t
(2.19)

where E is the expectation value associated to the probability measure P. Proof. If B ⊂ R is a bounded Borel set of R, then there exist strictly positives constants C and t such that: ∀x ∈ R,

χB (x) ≤ Ce−tx .

(2.20)

Let {fk }k≥1 be a Hilbert basis of L2 (Rd ) ⊗ CN . Let f be a positive, continuous, compactly supported function on Rd , such that f L2(Rd ) = 1. Then:     E  (Mf EHA (ω) (B)Mf )fk , fk  ≤ CE  e−tHA (ω) (ffk ), (ffk )  k≥1

k≥1

by the spectral theorem applicated to χB , the inequality (2.20) and the fact that Mf is self-adjoint as f is real-valued. But:   E  e−tHA (ω) (ffk ), (ffk )  = E(Tr(Mf e−tHA (ω) Mf )). k≥1

Let L be large enough for D = [−L, L]d to contain the support of f . Then using Lemma 1: −tHA (ω) 2 Mf )) = E f (x) TrCN Kt (x, x) dx E(Tr(Mf e supp f

=E

f (x) TrCN Kt (x, x) dx 2

R

(2.21)

with Kt given by (2.7). Then, using the Rd -ergodicity of HA (ω) at the second equality: 2 E f (x) TrCN Kt (x, x) dx Rd

1 =√ E 2πt

2

f (x) Rd

TrCN

t expord − Vω (w(s)) ds dW0,x,t,x (w) dx 0

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1 E = √ 2πt 1 = √ E 2πt 1 = √ E 2πt

TrCN

t expord − Vω (x+w(s)) ds dW0,0,t,0 (w) dx

TrCN

t expord − Vω (w(s)) ds dW0,0,t,0 (w) dx

2

f (x) Rd

f (x)2

Rd

TrCN

883

0

0

t expord − Vω (w(s)) ds dW0,0,t,0 (w) .

(2.22)

0

And this last expectation value is ﬁnite by Proposition 2. So we have proved that:   E  (Mf EHA (ω) (B)Mf )fk , fk  k≥1

1 ≤ C√ E TrCN expord 2πt t × − Vω (w(s)) ds dW0,0,t,0 (w) < +∞

(2.23)

0

which means that the operator Mf EHA (ω) (B)Mf is trace class P-almost surely on ω ∈ Ω. It also proves that B → E(Tr(Mf EHA (ω) (B)Mf )) deﬁnes a Radon measure on R whose Laplace transform is: L(E(Tr(Mf EHA (ω) (·)Mf )))(t) = E(Tr(Mf e−tHA (ω) Mf )) = L(n)(t)

(2.24)

by (2.22), (2.21) and (2.12). By injectivity of the Laplace transform, we have that for every bounded Borel set B ⊂ R: n(B) = E(Tr(Mf EHA (ω) (B)Mf )). All the results of this section were valid for HA (ω) acting on L2 (Rd ) ⊗ CN for every d and every N . In the next few sections, we will restrict our presentation to the case of d = 1 and N arbitrary, N ≥ 1. It will allow us to introduce the Lyapounov exponents associated to HA (ω). We want to study the regularity of the function E → N (E). As an increasing function we already know that it has left and right limits at each point of the real line. We will actually prove that the IDS is locally H¨ older continuous. To prove this, we will prove the same regularity property for the Lyapounov exponents associated to HA (ω) and show that the IDS and the Lyapounov exponents are related to each other trough an harmonic analysis formula, a Thouless formula. 3. Lyapounov Exponents 3.1. Definition and integral representation We start with a review of some results about Lyapounov exponents. These results holds for general sequences of independent and identically distributed (i.i.d.) random symplectic matrices. Let N be a positive integer. Let SpN (R) denote the group

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of 2N × 2N real symplectic matrices. It is the subgroup of GL2N (R) of matrices M satisfying t

M JM = J,

where J is the matrix of order 2N deﬁned by J =

0 IN

−IN 0

.

Definition 2. Let (Aω n )n∈N be a sequence of i.i.d. random matrices in SpN (R) with E(log+ Aω 0 ) < ∞. The Lyapounov exponents γ1 , . . . , γ2N associated with (Aω n )n∈N are deﬁned inductively by p

γi = lim

n→∞

i=1

1 ω E(log ∧p (Aω n−1 · · · A0 ) ) n

(3.1)

for every p ∈ {1, . . . , 2N }. Here, ∧p M denotes the pth exterior power of the matrix M , acting on the pth exterior power of R2N . One has γ1 ≥ · · · ≥ γ2N . Moreover, the random matrices (Aω n )n∈N being symplectic, we have the symmetry property γ2N −i+1 = −γi , for every i ∈ {1, . . . , N } (see [1, Proposition 8.2, p. 89]. Let µ be a probability measure on SpN (R). We denote by Gµ the smallest closed subgroup of SpN (R) which contains the topological support of µ, supp µ. We also deﬁne for every p ∈ {1, . . . , 2N }, the p-Lagrangian submanifold Lp of R2N , as the subspace of ∧p R2N spanned by {M e1 ∧ · · · ∧ M ep | M ∈ SpN (R)}, where (e1 , . . . , e2N ) is the canonical basis of R2N . We can now give a generalization of F¨ urstenberg’s theorem for N > 1. For the deﬁnitions of Lp -strong irreducibility and p-contractivity we refer to [1, Deﬁnitions A.IV.3.3 and A.IV.1.1], respectively. Proposition 4. Let (Aω n )n∈N be a sequence of i.i.d. random symplectic matrices of order 2N and p be an integer, p ∈ {1, . . . , 2N }. Let µ be the common distribution of the Aω n . If (a) Gµ is p-contracting and Lp -strongly irreducible, (b) E(log Aω 0 ) < ∞, then the following holds: (i) γp > γp+1 . (ii) For any non zero x in Lp : p 1 p ω ω lim E log (∧ An−1 . . . A0 )x = γi . n→∞ n i=1

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(iii) There exists a unique µ-invariant probability measure νp on P(Lp ) = {¯ x ∈ P(∧p R2N ) | x ∈ Lp } such that: p i=1

γi =

log SpN (R)×P(Lp )

(∧p M )x dµ(M ) dνp (¯ x).

x

Proof. This is [1, Proposition 3.4]. It remains to deﬁne the Lyapounov exponents associated to the operator HA (ω) for d = 1 and N ≥ 1. For E ∈ R we can consider the second order diﬀerential system: HA (ω)u = Eu ⇔ −u + Vω u = Eu

(3.2)

with u = (u1 , . . . , uN ) a function taking values in CN . We introduce the transfer matrix Aω n (E) from n to n + 1, deﬁned by the relation: u(n + 1, E) u(n, E) ω (E) = A . (3.3) n u (n + 1, E) u (n, E) Then one can verify that (Aω n (E))n∈N is a sequence of i.i.d. random symplectic matrices because the system (3.2) is Hamiltonian. So we can deﬁne the Lyapounov exponents associated to the operator HA (ω) as the Lyapounov exponents of the sequence of transfer matrices (Aω n (E))n∈N . Since the transfer matrices depend on a real parameter E, so will the Lyapounov exponents of HA (ω) and so do the measure µE (the common law of the Aω n (E)), the group GµE and the µE -invariant probability measure νp,E of Proposition 4. 3.2. Regularity of the Lyapounov exponents We want to study the regularity of the function E → γp (E) for p ∈ {1, . . . , N }. According to the integral representation obtained at Proposition 4, we have to understand the regularity of E → νp,E for any p ∈ {1, . . . , N } and to control the term ∧p M in the integral, which depends on E as µE depends on E. We will now give a general theorem for the regularity of the Lyapounov exponents of sequences of i.i.d. random symplectic matrices depending on a real parameter. Theorem 1. Let (Aω n (E))n∈N be a sequence of i.i.d. random symplectic matrices depending on a real parameter E. Let µE be the common distribution of the Aω n (E). We ﬁx a compact interval I in R and we assume that for E ∈ I we have: (i) GµE is p-contracting and Lp -strongly irreducible for every p ∈ {1, . . . , N }. (ii) There exist C1 > 0, C2 > 0 independent of n, ω, E such that for every p ∈ {1, . . . , N }: 2

∧p Aω n (E) ≤ exp(pC1 + p|E| + p) ≤ C2 .

(3.4)

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(iii) There exists C3 > 0 independent of n, ω, E such that for every E, E ∈ I and every p ∈ {1, . . . , N }: p ω

∧p Aω n (E) − ∧ An (E ) ≤ C3 |E − E |.

(3.5)

Then there exist two real numbers α > 0 and 0 < C < +∞ such that: ∀p ∈ {1, . . . , N },

∀E, E ∈ I,

|γp (E) − γp (E )| ≤ C|E − E |α .

Proof. The methods to prove this theorem can be found in [9, Chap. V]. In this reference this regularity result is written for transfer matrices associated to matrixvalued discrete Schr¨odinger operators. But this restriction to discrete operators only concerns the estimates (3.4) and (3.5). They are obviously veriﬁed in the case of transfer matrices of discrete Schr¨odinger operators as it is explained in [9, p. 279]. For a presentation using estimates (3.4) and (3.5), one can read [11] where it is done in the case of transfer matrices associated to scalar-valued continuous Schr¨ odinger operators. The main steps of the proof are the following. First we prove continuity of the Lyapounov exponents on I by proving continuity of the function: I × P(Lp ) → R Φp,E :

(∧p Aω n (E))x x) = E log (E, x ¯) → Φp,E (¯

x

for every p ∈ {1, . . . , N }. We only use estimates (3.4) and (3.5) to prove this continuity. Then we prove weak continuity of the function E → νp,E using Banach– Alaoglu theorem and the unicity of the µE -invariant measure νp,E as stated in point (iii) of Proposition 4. Combining these two continuity properties and noting that: γ1 (E) + · · · + γp (E) = νp,E (Φp,E ) we get the continuity of the Lyapounov exponents. To prove the H¨ older continuity of the Lyapounov exponents we need a result on negative cocyles as stated in [9, Proposition IV 3.5, p. 187]. We also need estimates on Laplace operators on H¨ older spaces like [9, Proposition V 4.13, p. 277] which relies on estimates (3.4) and (3.5). Finally using the decomposition given in [9, Proposition IV 3.12, p. 192] one can prove the H¨ older continuity of E → νp,E on I. For a complete presentation of this proof in the case of transfer matrices for continuous matrix-valued Schr¨ odinger operators, with proofs showing the role of the pth exterior powers, we refer to [4, Chap. 6]. We will now use this general result to prove the following theorem: Theorem 2. Let I be a compact interval in R. We assume that the potential Vω in HA (ω) for d = 1 and N ≥ 1 is such that the group GµE associated to the transfer matrices of HA (ω) is p-contracting and Lp -strongly irreducible for every p ∈ {1, . . . , N } and all E ∈ I. Then the Lyapounov exponents associated to HA (ω)

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are H¨ older continuous on I, i.e. there exist two real numbers α > 0 and 0 < C < +∞ such that: ∀p ∈ {1, . . . N },

∀E, E ∈ I,

|γp (E) − γp (E )| ≤ C|E − E |α .

According to Theorem 1 we only have to show that the transfer matrices Aω n (E) associated to HA (ω) verify estimates (3.4) and (3.5). They already verify point (i) of Theorem 1 by assumption. Before proving (3.4) and (3.5) we will give two lemmas which are the analog for matrix-valued operators of [11, Lemmas A.1 and A.2]. Lemma 2. Let V be a matrix-valued function in L1loc (R, MN (R)) and u a solution of −u + V u = 0. Then for every x, y ∈ R: max(x,y) 2 2 2 2

V (t) + 1 dt .

u(x) + u (x) ≤ ( u(y) + u (y) ) exp min(x,y)

Proof. Let R(t) = u(t) 2 + u (t) 2 . We have: R (t) = u(t), u (t) + u (t), u(t) + u (t), u (t) + u (t), u (t) = 2 Re(u(t), u (t) ) + 2 Re(u (t), V (t)u(t) ) = 2 Re(u (t), (V (t) + 1)u(t) ) ≤ 2 Re( u (t) V (t) + 1 u(t) )

u(t) 2 + u (t) 2 ≤ 2 V (t) + 1 2 = V (t) + 1 R(t). We have used the Cauchy–Schwarz inequality and the arithmetico-geometric inequality. Finally, we have the inequality: R (t) ≤ V (t) + 1 R(t) which by integration gives us the expected inequality. Lemma 3. For i = 1, 2, let Vi ∈ L1loc (R, MN (R)) and ui a solution of −u +Vi u = 0 such that: ∃y ∈ R,

u1 (y) = u2 (y)

and

u1 (y) = u2 (y).

Then, for every x ∈ R: 1

u1 (x) − u2 (x) 2 + u1 (x) − u2 (x) 2 2 1 ≤ u1 (y) 2 + u1 (y) 2 2 max(x,y) × exp

V1 (t) + V2 (t) + 2 dt × min(x,y)

max(x,y) min(x,y)

V1 (t) − V2 (t) dt.

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Proof. Without loss of generality, we can assume that y ≤ x. We have, because of the assumptions made on the solutions u1 and u2 :

u1 (x) − u2 (x) u1 (x) − u2 (x)

0 dt (V1 (t) − V2 (t))u1 (t) x y 0 I u1 (t) − u2 (t) + dt. u1 (t) − u2 (t) V2 (t) 0 x

=

y

We take the norm of the two sides of the equality: y u1 (x) − u2 (x) ≤

V1 (t) − V2 (t) u1 (t) dt u (x) − u (x) x 1 2 y u1 (t) − u2 (t) dt. + ( V2 (t) + 1) u1 (t) − u2 (t) x Then by Gronwall lemma: y u1 (x) − u2 (x) ≤

V1 (t) − V2 (t) u1 (t) dt u (x) − u (x) x 1 2 y × exp ( V2 (t) + 1) dt .

(3.6)

x

But by Lemma 2, for every t ∈ [y, x]:

u1 (t) 2 ≤ u1 (t) 2 + u1 (t) 2 ≤ u1 (y) 2 + u1 (y) 2 y ( V1 (s) + 1) ds . × exp x

So: 1

u1 (t) ≤ u1 (y) 2 + u1 (y) 2 2 exp

y 1 ( V1 (s) + 1) ds . 2 x

We put this in (3.6): 1

u1 (x) − u2 (x) 2 + u1 (x) − u2 (x) 2 2 max(x,y) 1 1 1 2 2 2

V1 (t) + + V2 (t) + 1 dt ≤ u1 (y) + u1 (y) exp 2 min(x,y) 2 ×

max(x,y)

V1 (t) − V2 (t) dt.

min(x,y)

And we have ﬁnished the proof because: 12 V1 (t) +

1 2

≤ V1 (t) + 1.

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Notation. Let u1 , . . . , u2N be solutions of (3.2) with initial conditions:     1 0 1 0 . 2N u (n, E) u (n, E)   . =  . ,..., =  . . 1 2N (u ) (n, E) (u ) (n, E)  ..  0 0

(3.7)

1

Then the transfer matrix Aω n (E) has the expression: u1 (n + 1, E) · · · u2N (n + 1, E) Aω (E) = . n (u1 ) (n + 1, E) · · · (u2N ) (n + 1, E)

(3.8)

Proof of Theorem 2. We start by proving (3.4). Let t (ui (n+ 1, E) (ui ) (n+ 1, E)) be the column of Aω n (E) of maximal norm. Then: 2 i 2 i 2

Aω n (E) = u (n + 1, E) + (u ) (n + 1, E) .

Applying Lemma 2 with x = n + 1 and y = n one gets:

ui (n + 1, E) 2 + (ui ) (n + 1, E) 2 ≤ ui (n, E) 2 + (ui ) (n, E) 2 exp

n+1

Vω (t) − E + 1 dt .

n

But due to (3.7) we have: ui (n, E) 2 + (ui ) (n, E) 2 = 1. We also have that x → Vω (x) is 1-periodic. Thus:

n+1

1

Vω (t) − E + 1 dt =

n

Vω (t) − E + 1 dt

0

≤

sup Vω (t)

+ |E| + 1.

t∈[0,1]

But Vω being uniformly bounded on x and ω, there exists C1 > 0 independent of ω, n and E such that: sup Vω (t)

≤ C1 .

t∈[0,1]

Then: 2

Aω n (E) ≤ exp(C1 + |E| + 1).

As I is compact, |E| is also bounded and so there exists C˜2 > 0 independent of ω, n and E such that: exp(C1 + |E| + 1) ≤ C˜2 . Finally, we use that for every p ∈ {1, . . . , 2N } and for every M ∈ GL2N (R), ∧p M ≤ M p . Applying it to M = Aω n (E), we obtain (3.4).

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To prove (3.5) we ﬁrst prove it for p = 1. Let E, E ∈ I. First there exists i ∈ {1, . . . , 2N } such that: i i u (n + 1, E) u (n + 1, E ) ω (E) − A (E ) =

Aω − n n (ui ) (n + 1, E) (ui ) (n + 1, E ) i n+1 u (n, E) ≤

V (t) − E − (V (t) − E ) dt ω ω (ui ) (n, E) n n+1

Vω (t) − E + (Vω (t) − E ) + 2 dt . × exp n

by Lemma 3. Thus:

Aω n (E)

−

Aω n (E )

≤ |E − E | exp

1

2 Vω (t) + |E| + |E | + 2 dt

0

≤ |E − E | exp(2C1 + 2 + 2 (I)) ≤ C˜3 |E − E | with C˜3 independent of n, ω and E. Now for p ≥ 1 we use the following estimate valid for M, N ∈ GL2N (R) and p ∈ {1, . . . , 2N }:

∧p M − ∧p N ≤ N − M ( N p−1 + M · N p−2 + · · · + M p−1 ). It is a direct computation (see [4, p. 118] for details). Applying it to M = Aω n (E) ω and N = An (E ) one gets: p−1 ˜ p ω C3 |E − E |

∧p Aω n (E) − ∧ An (E ) ≤ pC2

and C3 = pC2p−1 C˜3 is independent of n, ω, E and E . We have checked (ii) and (iii) in Theorem 1 and (i) is an assumption in Theorem 2. Therefore we can apply Theorem 1 to have the H¨older continuity on I of the Lyapounov exponents associated to HA (ω). 4. H¨ older Continuity of the IDS 4.1. Kotani’s w function We start by introducing the w function of Kotani as deﬁned in [20] for matrix-valued Schr¨ odinger operators. For this, we ﬁrst have to deﬁne the m-functions associated to such operators. We follow [20] and we will refer to this article for all proofs of this paragraph. Let C+ denote the half upper plane {z ∈ C | Im(z) > 0} and C− the lower half plane {z ∈ C | Im(z) < 0}. Proposition 5. Let E ∈ C+ ∪ C− . We ﬁx ω ∈ Ω. Then there exists a unique function x → F+ (x, E) with values in MN (C) (respectively x → F− (x, E)) satisfying: ∞

F+ (x, E) 2 dx < +∞ −F+ + Vω F+ = EF+ , F+ (0, E) = I and 0

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respectively: −F− + Vω F− = EF− ,

F− (0, E) = I

0

and −∞

F− (x, E) 2 dx < +∞.

Proof. See [20, Corollary 2.2]. Definition 3. For E ∈ C+ ∪ C− we deﬁne the m-functions M+ and M− associated to HA (ω) by: M+ (E) =

d F+ (x, E)|x=0 dx

and M− (E) = −

d F− (x, E)|x=0 . dx

With these functions we can compute the Green kernel of the resolvant of HA (ω). Proposition 6. Let E ∈ C+ ∪ C− . Then (HA (ω) − E)−1 has a continuous integral kernel GE (x, y, ω) given by: −F− (x)(M+ + M− )−1 t F+ (y) if x ≤ y GE (x, y, ω) = −F+ (x)(M+ + M− )−1 t F− (y) if y ≤ x. Proof. See [20, Theorem 3.2]. We can now deﬁne the w function of Kotani. This function will be the link between the Lyapounov exponents and the IDS. Indeed, its real part will be the sum of the N positive Lyapounov exponents while its imaginary part will tend to πN (E) when E tends to the real line. Definition 4. Let E ∈ C+ ∪ C− . We deﬁne the w function of Kotani by: w(E) =

1 E(Tr(M+ (E) + M− (E))). 2

Then the w function has the following properties: Proposition 7. For E ∈ C+ ∪ C− : (i) w(E) = E(Tr(M+ (E))) = E(Tr(M− (E))). d w(E) = E(Tr(GE (0, 0, ω))). (ii) dE (iii) −Re γN )(E). w(E) = (γ1 + · · · + N )(E) = 2(γ1 +···+γ . (iv) E Tr(Im M± (E, ω)−1 ) = − 2 ReImw(E) E Im E Proof. See [20, Theorem 6.2C]. In point (iii) we have to precise that the formula: γ1 (E) + · · · + γN (E) = lim

n→∞

makes sense for every E ∈ C.

1 ω E(log ∧N (Aω n−1 (E) . . . A0 (E)) ) n

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We can now generalize results of harmonic analysis of the w function presented in the case of scalar-valued Schr¨ odinger operators by Kotani in [19] to the case of matrix-valued Schr¨ odinger operators. First we introduce the space of Herglotz functions: H = {h | h is holomorphic on C+ and h : C+ → C+ }. Then we deﬁne a subspace of H: W = {w ∈ H | w, w , −iw ∈ H}. Proposition 8. The Kotani’s function w is in W. Proof. First, as HA (ω) is self-adjoint, its spectrum is included in R and E → M+ (E) is holomorphic on C\R and so is E → Tr(M+ (E)). If Im E > 0, by [20, Proposition 2.3(a)], one has: +∞ Im M+ (E) = (Im E) F+ (x, E)∗ F+ (x, E) > 0. 0

Thus, E → Tr(M+ (E)) is in H and w ∈ H. Then by Proposition 7(ii), w (E) = E(Tr(GE (0, 0, ω))). But GE (0, 0, ω) is holomorphic away from the spectrum of HA (ω) and so is Tr(GE (0, 0, ω)). If Im E > 0, then the operator Im(HA (ω) − E)−1 is a positive deﬁnite operator and Im Tr(GE (0, 0, ω)) > 0. Then Im w (E) = Im Tr(GE (0, 0, ω)) > 0 and w ∈ H. Finally, −iw is holomorphic on C+ as w is. If E ∈ C+ : Im(−iw(E)) = −Re w(E) = (Im E)E(Tr(Im M+ (E, ω)−1 )) by Proposition 7(iv). But if E ∈ C+ , Tr(Im M+ (E, ω)−1 ) > 0 and then Im(−iw(E)) > 0. Therefore, −iw ∈ H. 4.2. A Thouless formula Let n be the measure deﬁned in Proposition 2. Proposition 9. ∀E ∈ C\R,

E(Tr GE (0, 0, ω)) =

R

dn(E ) . E − E

(4.1)

Proof. As R is a limit of bounded Borel sets and the Dirac distribution at 0, δ0 , can be approached by compactly supported continuous functions, positives and of L2 -norm equal to 1, using Proposition 3 we have: dn(E ) 1 = dE Tr(δ0 , EHA (ω) ((−∞, E ])δ0 ) . R E −E R E −E

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Then applying the spectral theorem to the self-adjoint operator HA (ω): dn(E ) 1 = E Tr dδ0 , EHA (ω) ((−∞, E ])δ0 R E −E R E −E 1 dE = E Tr δ0 , ((−∞, E ]) δ0 HA (ω) R E −E = E Tr(δ0 , (HA (ω) − E)−1 δ0 ) = E(Tr(GE (0, 0, ω))) . With this proposition, we can express the imaginary part of w in terms of the IDS, E → N (E). Proposition 10. ∀E ∈ R,

lim Im w(E + ia) = πN (E).

a→0+

(4.2)

Proof. First, by Proposition 7(ii): ∀z ∈ C\R,

w (z) = E(Tr(Gz (0, 0, ω))).

Then, we can apply Proposition 9: ∀z ∈ C\R,

w (z) =

R

dn(E ) E − z

R

N (E ) dE (E − z)2

=

by integrating by parts. Then by integrating this expression, there exists a constant c ∈ C such that: 1 + Ez N (E ) dE . w(z) = c + (4.3) 2 R (E − z)(1 + E ) But if z ∈ R is not in the spectrum of HA (ω) then w(z) ∈ R (see [7, Lemma 5.10, p. 84]). Thus we must have c ∈ R. Then, taking imaginary part in (4.3) and writing for z ∈ C+ , z = E + ia, E ∈ R, a > 0: N (E ) Im w(E + ia) = a dE 2 2 R (E − E) + a N (E + au) = du 1 + u2 R where u = and so:

E −E a .

∀E ∈ R,

But N (E) being a distribution function, it is right continuous lim Im w(E + ia) = N (E)

a→0+

R

1 du = πN (E). 1 + u2

We have an analoguous proposition for the real part of w(E).

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Proposition 11. For Lebesgue-almost every E in R, we have: lim Re w(E + ia) = −(γ1 + · · · + γN )(E).

a→0+

(4.4)

Moreover, if I ⊂ R is an interval on which E → −(γ1 + · · · + γN )(E) is continuous then (4.4) holds for every E ∈ I. Proof. First by Proposition 7(iii), we have: ∀z ∈ C\R,

Re w(z) = −(γ1 + · · · + γN )(z).

(4.5)

The function z → −(γ1 + · · · + γN )(z) is subharmonic (see [10]) and so for almost every E in R the following limit exists: lim (γ1 + · · · + γN )(E + ia) = (γ1 + · · · + γN )(E).

a→0

(4.6)

Let E be a real number such that (4.6) holds. Then setting z = E + ia with a > 0 in (4.5) one gets the existence of the following limit: lim Re w(E + ia) = −(γ1 + · · · + γN )(E).

a→0+

(4.7)

Moreover, if I is an interval on which E → (γ1 + · · · + γN )(E) is continuous, the relation (4.7) holds for every E in I as it holds for almost every E ∈ I. Now we can prove a Thouless formula adapted to matrix-valued continuous Schr¨ odinger operators. As (γ1 +· · ·+γN )(E) and N (E) are respectively the real and imaginary part of the function w which lies in W, the harmonic analysis developed in [19] says that these two functions are linked by an integral relation. Theorem 3 (Thouless Formula). For Lebesgue-almost every E ∈ R we have: E − E dn(E ) (4.8) (γ1 + · · · + γN )(E) = −α + log E −i R

where α is a real number independent of E and n is the measure of which the IDS E → N (E) is the distribution function. Moreover, if I ⊂ R is an interval on which E → −(γ1 + · · · + γN )(E) is continuous then (4.8) holds for every E ∈ I. Proof. As w ∈ W, we can apply to w the [19, Lemma 7.7]. In particular, using also Proposition 10, we have: E −i ∀z ∈ C\R, w(z) = w(i) + log (4.9) dn(E ). −z E R Then:

Re w(z) = Re w(i) + R

E −i dn(E ). log E − z

(4.10)

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Let z = E + ia with E ∈ R such that (4.4) holds and a > 0. Then when a goes to 0, by Proposition 11 we have: E −i dn(E ). −(γ1 + · · · + γN )(E) = Re w(i) + log (4.11) E −E R

If we set α = Re w(i) we ﬁnally get (4.8) for every E in R such that (4.4) holds, i.e. for almost every E in R. Then if I is an interval on which E → (γ1 + · · · + γN )(E) is continuous, by Proposition 11, (4.8) will hold for every E in I. We can now use this Thouless formula to prove that the IDS, E → N (E), has the same regularity as the Lyapounov exponents. 4.3. Local H¨ older continuity of the IDS We start by a quick review of the Hilbert transform and its main properties. For the proofs we refer to [21, Chap. 3]. Definition 5. If ψ ∈ L2 (R), its Hilbert transform is the function deﬁned on R by: 1 ψ(t) dt. (T ψ)(x) = lim+ ε→0 π |x−t|>ε x − t Proposition 12. Let ψ ∈ L2 (R). (i) Then T 2 ψ(x) = −ψ(x) for Lebesgue-almost every x in R. (ii) If ψ is H¨ older continuous on the interval [x0 − a, x0 + a], a > 0, then T ψ is H¨ older continuous on the interval [x0 − a2 , x0 + a2 ]. Now we can prove the following result of regularity of the IDS. ˜ We Theorem 4. Let I be a compact interval in R and I˜ be an open interval, I ⊂ I. assume that the potential Vω in HA (ω) for d = 1 and N ≥ 1 is such that the group GµE associated to the transfer matrices of HA (ω) is p-contracting and Lp -strongly ˜ Then the IDS associated to irreducible for every p ∈ {1, . . . , N } and every E ∈ I. older continuous on I. HA (ω) is H¨ Proof. First, the application E → log EE −E −i is n-integrable on R. Indeed the renormalization term E − i at the denominator balances the fact that the support of n is non-compact. Thus, we have: E+ε E − E dn(E ) = 0 (4.12) ∀E ∈ R, lim log E −i ε→0+ E−ε from which we deduce that: ∀E ∈ R,

lim |log(ε)|(N (E + ε) − N (E − ε)) = 0.

ε→0+

(4.13)

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It implies that E → N (E) is continuous on R. Let E0 ∈ I be ﬁxed and a > 0 such ˜ Then, by Theorem 3, for E ∈ ] E0 − 4a, E0 + 4a[: that [E0 − 4a, E0 + 4a] ⊂ I. E − E dn(E ) (γ1 + · · · + γN )(E) + α − log E − i |E −E0 |>4a E0 +4a E − E dn(E ). log = E −i E0 −4a Then:

E − E dn(E ) log E −i

E0 +4a

E0 −4a

E−ε

= lim+ ε→0

E0 −4a

E0 +4a

+

log |E − E|dn(E )

log |E − E|dn(E )

E+ε

We set: I(E0 ) =

1 2

E0 +4a

−

1 2

E0 +4a

log(1 + (E )2 ) dn(E ).

E0 −4a

log(1 + (E )2 ) dn(E ).

E0 −4a

Then, integrating by parts the ﬁrst two integrals leads to: E0 +4a E − E dn(E ) log E − i E0 −4a = lim+ [N (E ) log |E − E|]E−ε E0 −4a ε→0

N (E ) E0 +4a dE + [N (E ) log |E − E|]E+ε E0 −4a E − E E0 +4a N (E ) dE − I(E0 ). − E − E E+ε −

E−ε

We set ψ(E) = N (E)χ{|E−E0 |≤4a} ∈ L2 (R). By deﬁnition of the Hilbert transform: E0 +4a log |E − E| dn(E ) E0 −4a

= π(T ψ)(E)+ lim [(N (E − ε)−N (E + ε)) log ε+N (E0 + 4a) log |E0 −E + 4a| ε→0+

− N (E0 − 4a) log |E0 − E − 4a|] − I(E0 ) = π(T ψ)(E) + N (E0 + 4a) log |E0 − E + 4a| − N (E0 − 4a) log |E0 − E − 4a| − I(E0 )

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by (4.13). We ﬁnally get:

897

E − E dn(E ) log π(T ψ)(E) = (γ1 + · · · + γN )(E) + α − E −i |E −E0 |>4a

− N (E0 + 4a) log |E0 − E +4a|+N (E0 − 4a) log |E0 − E − 4a|+I(E0 ) E − E dn(E )+I(E0 ). log = (γ1 + · · ·+ γN )(E)+ α − E −i |E −E0 |≥4a ˜ E → (γ1 + · · · + γN )(E) is H¨older continuous on But as [E0 − 4a, E0 + 4a] ⊂ I ⊂ I, [E0 − 4a, E0 + 4a] by Theorem 2. Moreover, E → |E −E0 |≥4a log EE −E −i dn(E ) is H¨older continuous of order 1 on the interval ]E0 − 4a, E0 + 4a[. Then T ψ is H¨older continuous on every compact interval included in ]E0 − older continuous on [E0 − 2a, E0 + 2a]. Thus 4a, E0 + 4a[, in particular it is H¨ 2 by Proposition 12(ii), T ψ is H¨older continuous on [E0 − a, E0 + a]. But by Proposition 12(i) and by continuity of E → N (E) (by (4.13)), we have: ∀E ∈ [E0 − a, E0 + a],

(T 2 ψ)(E) = −N (E).

Then E → N (E) is H¨older continuous on [E0 − a, E0 + a]. But I being compact, it can be covered by a ﬁnite number of intervals ]E0 − a, E0 + a[ ⊂ I˜ with E0 ∈ I. Thus, E → N (E) is H¨older continuous on I. The H¨older continuity of the Lyapounov exponents and of the IDS relies on the assumptions of p-contractivity and Lp -strong irreducibility for every p ∈ {1, . . . , N } made on GµE . But, for arbitrary potential Vω , we do not know if these assumptions are veriﬁed or not. In the next section we will present a ﬁrst example of continuous matrix-valued Anderson model for which these assumptions are veriﬁed. 5. Anderson Model on Two Coupled Strings We will now see how to apply Theorem 4 to a particular case of HA (ω), which is the following operator: (n) d2 0 1 ω1 χ[0,1] (x − n) 0 HAB (ω) = − 2 ⊗ I2 + + (n) 1 0 dx 0 ω2 χ[0,1] (x − n) n∈Z

acting on L (R) ⊗ C . Here, χ[0,1] denotes the characteristic function of the interval (n) (n) [0, 1] and (ω1 )n∈Z and (ω2 )n∈Z are two independent sequences of i.i.d. random variables with common law ν such that {0, 1} ⊂ supp ν. This operator is a bounded d2 2 2 perturbation of (− dx 2 ) ⊗ I2 and thus self-adjoint on the Sobolev space H (R) ⊗ C . For the operator HAB (ω), we have the following result: 2

2

Theorem 5. The Integrated Density of States N (E) associated to HAB (ω) exists for every E ∈ R. Moreover, there exists a discrete subset SB ⊂ R such that for every older continuous compact interval I ⊂ (2, +∞)\SB , the function E → N (E) is H¨ on I.

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According to Theorem 4, we only have to prove that there exists a discrete subset SB ⊂ R such that for every E ∈ (2, +∞)\SB , the group GµE associated to the transfer matrices of HAB (ω) is p-contracting and Lp -strongly irreducible for p ∈ {1, 2}. It has already been proved in a previous article of the author, [3], and we will only give here the outlines of the proof and some comments. To prove that an explicit group is p-contracting and Lp -strongly irreducible can be very complicated. It has been done in [11] for the case of a scalar-valued continuous Anderson model, but their proof relies on properties of reﬂection and transmission coeﬃcients which no longer holds in the matrix-valued case. In the case of a discrete matrix-valued Anderson model, a more algebraic approach has been successfully used by Gol’dsheid and Margulis in [14]. We follow here this approach and adapt it to the case of continuous matrix-valued Anderson models. It is based on the following criterion: Theorem 6 ([14]). If a subgroup G of SpN (R) is dense for the Zariski topology in SpN (R) then it is p-contracting and Lp -strongly irreducible for every p ∈ {1, . . . , N }. In the case of a discrete matrix-valued Anderson model, the transfer matrices have a simple enough expression to make possible a direct construction of the Zariski closure of the group GµE generated by these transfer matrices. And so it can be proved that for every E ∈ R, GµE is Zariski dense in SpN (R). In our case, the transfer matrices associated to HAB (ω), even if they are still explicit, are complicated enough to not allow a direct reconstruction of the Zariski closure of GµE for every E except those in a discrete set. It is due to the fact that E and the ωi ’s are not separated in the expressions of these transfer matrices. A direct reconstruction of the Zariski closure of GµE is in fact possible, but only for values of E away from a dense countable subset of R, as shown in [5]. It leads to the impossibility to ﬁnd an interval of values of E such that GµE is p-contracting and Lp -strongly irreducible and makes it impossible to apply Theorem 4. The idea in [3], to improve the result of [5], is to combine the criterion of Gol’dsheid and Margulis to a recent result of Breuillard and Gelander on Lie groups: Theorem 7 ([6]). Let G be a real, connected, semisimple Lie group, whose Lie algebra is g. Then there exists a neighborhood O of 1 in G, on which log = exp−1 is a well deﬁned diﬀeomorphism, such that g1 , . . . , gm ∈ O generate a dense subgroup whenever log g1 , . . . , log gm generate g. Using this theorem leads us to: (i) Prove that we can ﬁnd suitable powers of the transfer matrices which lies in an arbitrary neighborhood of the identity in Sp2 (R). These powers will be our “g1 , . . . , gm ”. To construct these powers we use simultaneous diophantine approximation which can be used only for E > 2 in our model, as explained in [3, Sec. 4.1].

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(ii) Compute the logarithms of these powers of transfer matrices. It leads to a ﬁrst discrete set of E’s in R on which these logarithms are not deﬁned. (iii) Out of this discrete set of E’s, prove that these logarithms generates the Lie algebra sp2 (R) of Sp2 (R), except for E’s in an other discrete subset of R which corresponds to zeros of some determinants (see [3, Sec. 4.3]). This part of the proof is constructive and for the moment it was not possible to do it for N strictly larger than 2. So ﬁnally, in [3], we were able to prove that there exists a discrete set SB ⊂ R such that for every E in SB , E > 2, the closed group GµE is dense and, therefore, equal to Sp2 (R). So we can apply Theorem 4, because any compact interval I ⊂ (2, +∞)\SB is also included in an interval I˜ ⊂ (2, +∞)\SB on which GµE is pcontracting and Lp -strongly irreducible for p ∈ {1, 2}. This ﬁnishes the proof of Theorem 5. To conclude the study of the operator HAB (ω), we would like to precise that the approach used here to prove the density of GµE in Sp2 (R), and based on Breuillard and Gelander’s result, relies on the fact that the transfer matrices can be expressed as exponentials of matrices. This fact is no longer true if instead of considering characteristic functions in the deﬁnition of HAB (ω), we consider more general functions. If we do, the transfer matrices becomes time-ordered exponentials instead of exponentials and the Breuillard and Gelander’s result no longer applies directly. But we hope that, by some perturbation argument, the case of more general functions could still be handled.

Acknowledgments The author is supported by JSPS Grant P07728. This paper was written during his post-doctoral stay at Keio University in Yokohama, Japan. He would like to thank Yoshiaki Maeda, his host in this university, to allow him the opportunity of working in excellent conditions at the Keio Mathematics Department. The author would also like to thank Anne Boutet de Monvel and G¨ unter Stolz for numerous references and suggestions, and also for their constant encouragements. Finally, he would like to thank Florent Schaﬀhauser for reading over this article and helping him to improve its writing.

References [1] P. Bougerol and J. Lacroix, Products of Random Matrices with Applications to Schr¨ odinger Operators, Progr. Probab. Statist., Vol. 8 (Birkh¨ auser, Boston, 1985). [2] L. Boulton and A. Restuccia, The heat kernel of the compactiﬁed D = 11 supermembrane with non-trivial winding, Nuclear Phys. B 724(1–2) (2005) 380–396. [3] H. Boumaza, Positivity of Lyapunov exponents for a continuous matrix-valued Anderson model, Math. Phys. Anal. Geom. 10(2) (2007) 97–122; DOI:10.1007/s11040-0079023-6.

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[4] H. Boumaza, Lyapunov exponents and Integrated Density of States for matrix-valued continuous Schr¨ odinger operators, Th`ese de l’Universit´e Denis Diderot-Paris 7 (2007); http://www.institut.math.jussieu.fr/∼boumaza/these-boumaza.pdf. [5] H. Boumaza and G. Stolz, Positivity of Lyapunov exponents for Anderson-type models on two coupled strings, Electron. J. Diﬀerential Equations 47 (2007) 1–18. [6] E. Breuillard and T. Gelander, On dense free subgroups of Lie groups, J. Algebra 261(2) (2003) 448–467. ´ [7] R. Carmona, Random Schr¨ odinger Operators, in Ecole d’´et´e de Probabilit´es de SaintFlour (XIV), 1984, Lecture Notes in Math., Vol. 1180 (Springer, Berlin, 1986), pp. 1–124. [8] R. Carmona, A. Klein and F. Martinelli, Anderson localization for Bernoulli and other singular potentials, Comm. Math. Phys. 108 (1987) 41–66. [9] R. Carmona and J. Lacroix, Spectral Theory of Random Schr¨ odinger Operators, Probability and Its Applications (Birkh¨ auser, Boston, 1990). [10] W. Craig and B. Simon, Subharmonicity of the Lyaponov index, Duke Math. J. 50(2) (1983) 551–560. [11] D. Damanik, R. Sims and G. Stolz, Localization for one-dimensional, continuum, Bernoulli–Anderson models, Duke Math. J. 114 (2002) 59–99. [12] J. Dollard and C. Friedman, Product Integration with Applications to Diﬀerential Equations, Encyclopedia of Mathematics and Its Application, Vol. 10 (AddisonWesley, 1979). [13] J. Glimm and A. Jaﬀe, Quantum Physics, A Functional Integral Point of View, 2nd edn. (Springer-Verlag, 1987). [14] I. Ya. Gol’dsheid and G. A. Margulis, Lyapunov indices of a product of random matrices, Russian Math. Survey 44(5) (1989) 11–71. [15] W. Kirsch, On a class of random Schr¨ odinger operators, Adv. Appl. Math. 6 (1985) 177–187. [16] W. Kirsch, S. Molchanov, L. Pastur and B. Vainberg, Quasi 1D localization: Deterministic and random potentials, Markov Process. Related Fields 9 (2003) 687–708. [17] A. Klein, J. Lacroix, A. Speis, Localization for the Anderson model on a strip with singular potentials, J. Funct. Anal. 94 (1990) 135–155. [18] F. Knight, Brownian Motion and Diﬀusion, Mathematical Survey, Vol. 18 (American Mathematical Society, 1981). [19] S. Kotani, Lyapounov indices determine absolutely continuous spectra of stationary random one-dimensional Schr¨ odinger operators, in Proc. Taniguchi Symp. on PMMP, Katata, Japan (1983), pp. 225–247. [20] S. Kotani, B. Simon, Stochastic Schr¨ odinger operators and Jacobi matrices on the strip, Comm. Math. Phys. 119(3) (1988) 403–429. [21] U. Neri, Singular Integrals, Lecture Notes in Mathematics, Vol. 200 (Springer-Verlag, 1971). [22] G. Roepstorﬀ, Path Integral Approach to Quantum Physics, An Introduction, Texts and Monographs in Physics (Springer-Verlag, Berlin, 1994). [23] B. Simon, Trace Ideals and Their Applications, Mathematical Surveys and Monographs, Vol. 120, 2nd edn. (American Mathematical Society, Providence, RI, 2005). [24] P. Stollmann, Caught by Disorder — Bound States in Random Media, Progress in Mathematical Physics, Vol. 20 (Birkh¨ auser, 2001)

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Reviews in Mathematical Physics Vol. 20, No. 8 (2008) 901–932 c World Scientific Publishing Company

¨ WEYL ASYMPTOTICS FOR MAGNETIC SCHRODINGER OPERATORS AND DE GENNES’ BOUNDARY CONDITION

AYMAN KACHMAR Universit´ e Paris-Sud, D´ epartement de Math´ ematique, Bˆ at. 425, F-91405 Orsay, France [email protected] Received 6 September 2007 Revised 24 June 2008 This paper is concerned with the discrete spectrum of the self-adjoint realization of the semi-classical Schr¨ odinger operator with constant magnetic field and associated with the de Gennes (Fourier/Robin) boundary condition. We derive an asymptotic expansion of the number of eigenvalues below the essential spectrum (Weyl-type asymptotics). The methods of proof rely on results concerning the asymptotic behavior of the first eigenvalue obtained in a previous work [10]. Keywords: Magnetic Schr¨ odinger operator; spectral function; discrete spectrum; semiclassical analysis. Mathematics Subject Classification 2000: 81Q10, 35J10, 35P15, 82D55

1. Introduction and Main Results Let Ω ⊂ R2 be an open domain with regular and compact boundary. Given a smooth function γ ∈ C ∞ (∂Ω; R) and a number α ≥ 12 , we consider the Schr¨odinger operator with magnetic ﬁeld: α,γ = −(h∇ − iA)2 , Ph,Ω

(1.1)

whose domain is, α,γ D(Ph,Ω ) = {u ∈ L2 (Ω) : (h∇ − iA)j ∈ L2 (Ω), j = 1, 2,

ν · (h∇ − iA)u + hα γu = 0 on ∂Ω}.

(1.2) 1

2

Here ν is the unit outward normal vector of the boundary ∂Ω, A ∈ H (Ω; R ) is a α,γ satisfy vector ﬁeld and curl A is the magnetic ﬁeld. Functions in the domain of Ph,Ω the de Gennes boundary condition. α,γ arises from the analysis of the onset of superconductivity for The operator Ph,Ω a superconductor placed adjacent to another material. For the physical motivation and the mathematical justiﬁcation of considering this type of boundary condition and not the usual Neumann condition (γ ≡ 0), we invite the interested reader to 901

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see the book of de Gennes [6] and the papers [10–13]. We would like to mention α,γ has been the subject of many papers, see [4] that when γ ≡ 0, the operator Ph,Ω and the references therein. We shall restrict ourselves with the case of constant magnetic ﬁeld, namely when curl A = 1 in Ω. It follows from the well-known inequality 2 |(h∇ − iA)u| dx ≥ h |u|2 dx, Ω

Ω

(1.3)

∀ u ∈ C0∞ (Ω),

(1.4)

and from a “magnetic” Persson’s Lemma (cf. [16,1]), that the bottom of the essential α,γ is above h. Assuming that the boundary of Ω is smooth and spectrum of Ph,Ω compact, then it follows from [10] that (in the parameter regime α ≥ 12 ), the α,γ has discrete spectrum below h. Thus, given b0 < 1, one is led to operator Ph,Ω estimate the size of the discrete spectrum below b0 h, i.e. we look for the asymptotic behavior of the number N (α, γ; b0 h)

(1.5)

α,γ (taking multiplicities into account) included in the interval of eigenvalues of Ph,Ω ]0, b0 h]. For the case with non-constant magnetic ﬁeld and Neumann boundary condition, this problem has been analyzed by Frank [5] (related questions are also treated in [2, 8, 18, 19]). As we shall see, depending on the type of the boundary condition, one can produce many additional eigenvalues below the essential spectrum. To state the results concerning N (α, γ ; b0 h), we need to introduce some notation. Let us introduce the smooth functions, which arise from the analysis of the model-operator in the half-plane (see [10, Sec. II]),

R × R+ (γ, ξ) → µ1 (γ, ξ), where

R γ → Θ(γ),

(1.6)

µ1 (γ, ξ) =

inf

u∈B 1 (R+ ), u≡0

(|u (t)|2 + |(t − ξ)u(t)|2 ) dt + γ |u(0)|2 R+ , |u(t)|2 dt R+

Θ(γ) = inf µ1 (γ, ξ), ξ∈R

and the space B 1 (R+ ) consists of functions in the space H 1 (R+ ) ∩ L2 (R+ ; t2 dt). Actually, µ1 (γ, ξ) is the ﬁrst eigenvalue of the self-adjoint operator −∂t2 + (t − ξ)2

in L2 (R+ )

associated with the boundary condition u (0) = γ u(0). The eigenvalues of this operator form an increasing sequence which we denote (µj (γ, ξ))j∈N , see Sec. 2.1 for more details.

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When γ = 0, we write as in the usual case (see [4]) µ1 (ξ) := µ1 (0, ξ),

Θ0 := Θ(0).

Furthermore, we denote by, 1 (1 + γ Θ(γ) + γ 2 )2 Θ (γ), 3 We are ready now to state our main results. C1 (γ) =

C1 = C1 (0).

Theorem 1.1. Assume that α > 12 and Θ0 < b0 < 1. Then as h → 0, |∂Ω| √ |{ξ ∈ R : µ1 (ξ) < b0 }| (1 + o(1)). N (α, γ; b0 h) = 2π h

(1.7)

(1.8)

On the other hand, if α = 12 and Θ(γ0 ) < b0 < 1, then as h → 0,   ∞ 1 |{ξ ∈ R : µj (γ(s), ξ) < b0 }| ds (1 + o(1)). N (α, γ; b0 h) =  √ 2π h ∂Ω j=1 (1.9) Here γ0 = min γ(s). s∈∂Ω

If we suppose furthermore that γ0 ≥ 0, then (1.9) simplifies to 1 √ |{ξ ∈ R : µ1 (γ(s), ξ) < b0 }| ds (1 + o(1)). N (α, γ; b0 h) = 2π h ∂Ω

(1.10)

(1.11)

By taking γ ≡ 0, we recover in Theorem 1.1 the result of Frank [5]. We notice that when α = 12 and γ is constant, we have additional eigenvalues than the usual case of Neumann boundary condition if γ < 0 and less eigenvalues if γ > 0. This is natural as we apply the variational min-max principle. However, when γ0 = 0 or α > 12 , Theorem 1.1 fails to give a comparison with the Neumann case, i.e. we have no more information about the size of the diﬀerence: N (α, γ; b0 h) − N (0; b0 h). This is at least a motivation for some of the next results, where we take b0 = b0 (h) asymptotically close to Θ0 , each time with an appropriate scale (this will cover also the case b0 = Θ0 ). 1 2

< α < 1 then for all a ∈ R, 1 1 N (α, γ; hΘ0 + 3aC1 hα+ 2 ) =

(a − γ(s)) ds (1 + o(1)). + √ 3 ∂Ω π h 2 −α Θ0

Theorem 1.2. If

(1.12)

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In the particular case when the function γ is constant, the leading order term in (1.12) will vanish when a is taken equal to γ. In this speciﬁc regime, Theorem 1.5 (more precisely the formula in (1.16)) will substitute Theorem 1.2. Theorem 1.3. Assume that α = 1/2. Let 0 < < 12 , ζ0 > 0, h0 > 0 and ]0, h0 ] h → c0 (h) ∈ R+ a function such that limh→0 c0 (h) = ∞. If c0 (h)h1/2 ≤ |λ − Θ(γ0 )| ≤ ζ0 h ,

∀ h ∈ ]0, h0 ],

then we have the asymptotic formula,

[λ − Θ(γ(s))]+ 1 N (α, γ; hλ) = ds (1 + o(1)), π ∂Ω hΘ (γ(s)) Θ(γ(s)) + γ(s)2

(1.13)

where the function Θ(·) being introduced in (1.6). Remark 1.4. Theorem 1.3 becomes of particular interest when the function γ has a unique non-degenerate minimum and λ = Θ(γ0 ) + ahβ , for some a ∈ R+ and β ∈ ]0, 12 [. In this case, we have in the support of [λ − Θ(γ(s))]+ , Θ(γ(s)) = Θ(γ0 ) + c1 s2 + O(h3β/2 ), for an explicit constant c1 > 0 determined by the functions γ and Θ. Therefore, the asymptotic expansion (1.13) reads in this case, for some explicit constant c2 > 0, √ 1 (1.14) N (α, γ; hλ) = c2 a a hβ− 2 (1 + o(1)). The next theorem deals with the regime where the scalar curvature becomes eﬀective in the asymptotic expansions. Theorem 1.5. (1) Assume that α = 1. Then, for all a ∈ R, the following asymptotic expansion holds as h → 0, N (1, γ; hΘ0 + aC1 h3/2 ) 1 = (κr (s) − 3γ(s) + a)+ ds (1 + o(1)), √ π 3h1/2 Θ0 ∂Ω

(1.15)

where κr is the scalar curvature of ∂Ω. (2) If the function γ is constant, then for all α > 1/2 and a ∈ R, we have the asymptotic expansion, N (α, γ; hΘ(hα−1/2 γ) + aC1 h3/2 ) 1 (κr (s) + a)+ ds (1 + o(1)). = √ π 3h1/2 Θ0 ∂Ω

(1.16)

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(3) If the function γ is constant and α = 1/2, then for all a ∈ R, we have N (α, γ; hΘ(γ) + aC1 (γ)h3/2 ) 1 + γ Θ(γ) + γ 2 (κr (s) + a)+ ds (1 + o(1)). =

∂Ω π 3h1/2 Θ(γ) + γ 2

(1.17)

Here C1 (γ) has been defined in (1.7). The proof of Theorems 1.1–1.5 is through careful estimates in the semi-classical regime of the quadratic form α,γ (u) = |(h∇ − iA)u|2 dx + h1+α γ(s)|u(s)|2 ds. u → qh,Ω Ω

∂Ω

These estimates are essentially obtained in [7] when γ ≡ 0, then adapted to situations involving the de Gennes boundary condition in [10,14]. We shall follow closely the arguments of [5] but we also require to use various properties of the function γ → Θ(γ) established in [10]. The paper is organized in the following way. Section 2 is devoted to the analysis of the model operator in a half-cylinder when the function γ is constant. Section 3 extends the result obtained for the model case in a half-cylinder for a general domain by which we prove Theorem 1.1. Section 4 deals with model operators on weighted L2 spaces which serve in proving Theorems 1.2–1.5.

2. Analysis of the Model Operator 2.1. A family of one-dimensional diﬀerential operators Let us recall the main results obtained in [9, 10] concerning a family of diﬀerential operators with Robin boundary condition. Given (γ, ξ) ∈ R × R, we deﬁne the quadratic form, B 1 (R+ ) u → q[γ, ξ](u) = (|u (t)|2 + |(t − ξ)u(t)|2 ) dt + γ|u(0)|2 , (2.1) R+

where, for a positive integer k ∈ N and a given interval I ⊆ R, the space B k (I) is deﬁned by: B k (I) = {u ∈ H k (I); tj u(t) ∈ L2 (I), ∀ j = 1, . . . , k}.

(2.2)

By Friedrichs Theorem, we can associate to the quadratic form (2.1) a self adjoint operator L[γ, ξ] with domain, D(L[γ, ξ]) = {u ∈ B 2 (R+ ); u (0) = γu(0)},

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and associated to the diﬀerential operator, L[γ, ξ] = −∂t2 + (t − ξ)2 .

(2.3)

We denote by {µj (γ, ξ)}+∞ j=1 the increasing sequence of eigenvalues of L[γ, ξ]. When γ = 0 we write, µj (ξ) := µj (0, ξ),

∀ j ∈ N,

LN [ξ] := L[0, ξ].

(2.4)

+∞ We also denote by {µD j (ξ)}j=1 the increasing sequence of eigenvalues of the Dirichlet 2 realization of −∂t + (t − ξ)2 . By the min-max principle, we have,

µ1 (γ, ξ) =

inf

u∈B 1 (R+ ),u=0

q[γ, ξ](u) . u2L2 (R+ )

(2.5)

Let us denote by ϕγ,ξ the positive (and L2 -normalized) ﬁrst eigenfunction of L[γ, ξ]. It is proved in [10] that the functions (γ, ξ) → µ1 (γ, ξ),

(γ, ξ) → ϕγ,ξ ∈ L2 (R+ )

are regular (i.e. of class C ∞ ), and we have the following formulae, ∂ξ µ1 (γ, ξ) = −(µ1 (γ, ξ) − ξ 2 + γ 2 )|ϕγ,ξ (0)|2 ,

(2.6)

∂γ µ1 (γ, ξ) = |ϕγ,ξ (0)|2 .

(2.7)

Notice that (2.7) will yield that the function (γ, ξ) → ϕγ,ξ (0) is also regular of class C ∞ . We deﬁne the function: Θ(γ) = inf µ1 (γ, ξ). ξ∈R

(2.8)

It is a result of [3] that there exists a unique ξ(γ) > 0 such that, Θ(γ) = µ1 (γ, ξ(γ)),

Θ(γ) < 1,

(2.9)

and ξ(γ) satisﬁes (cf. [10]), ξ(γ)2 = Θ(γ) + γ 2 .

(2.10)

Moreover, the function Θ(γ) is of class C ∞ and satisﬁes, Θ (γ) = |ϕγ (0)|2 ,

(2.11)

where ϕγ is the positive (and L2 -normalized) eigenfunction associated to Θ(γ): ϕγ = ϕγ,ξ(γ) .

(2.12)

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When γ = 0, we write, Θ0 := Θ(0),

ξ0 := ξ(0).

(2.13)

It is a consequence of (2.11) that the constant C1 introduced in (1.7) can be deﬁned by the alternative manner, |ϕ0 (0)|2 . (2.14) 3 Let us recall an important consequence of standard Sturm–Liouville theory (cf. [5, Lemma 2.1]). C1 :=

Lemma 2.1. For all ξ ∈ R, we have µ2 (ξ) > µD 1 (ξ) > 1. Let us also introduce, Θk (γ) = inf µk (γ, ξ), ξ∈R

∀ k ∈ N.

(2.15)

Another consequence of Sturm–Liouville theory that we shall need is the following result on Θ2 (γ). Lemma 2.2. For any γ ∈ R, we have, Θ2 (γ) > Θ(γ). Proof. Let us introduce the continuous function f (γ) = Θ2 (γ) − Θ(γ). Using the min-max principle, it follows from (2.9) and Lemma 2.1 that f (0) > 0. It is then suﬃcient to prove that the function f never vanish. Suppose by contradiction that there is some γ0 = 0 such that Θ2 (γ0 ) = Θ(γ0 ). By the same method used in [10] one is able to prove that there exists ξ2 (γ0 ) > 0 such that Θ2 (γ0 ) = µ2 (γ0 , ξ2 (γ0 )), and that ξ2 (γ0 )2 = Θ2 (γ0 ) + γ02 . Therefore we get ξ2 (γ0 ) = ξ(γ0 ). Now, by Sturm–Liouville theory, the eigenvalues of the operator L[γ0 , ξ(γ0 )] are all simple, whereas, by the above, we get a degenerate eigenvalue µ1 (γ0 , ξ(γ0 )) = Θ(γ0 ) = µ2 (γ0 , ξ(γ0 )), which is the desired contradiction. One more useful result in Sturm–Liouville theory is the following. Lemma 2.3. Let γ ∈ R− and k ∈ N. Then Θk (γ) < 2k + 1 and for all b0 ∈ ]Θk (γ), 2k + 1[, the equation µk (γ, ξ) = b0 has exactly two solutions ξk,− (γ, b0 ) and ξk,+ (γ, b0 ). Moreover, {ξ ∈ R : µk (γ, ξ) < b0 } = ]−ξk,− (γ, b0 ), ξk,+ (γ, b0 )[ .

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Proof. We can study the variations of the function ξ → µk (γ, ξ) using exactly the same method of [10, 14, 3]. We obtain that the function ξ → µk (γ, ξ) attains a unique non-degenerate minimum at the point ξk (γ) = Θk (γ) + γ 2 , and analogous formulae to (2.6) and (2.7) continue to hold for (γ, ξ) → µk (γ, ξ). Moreover, limξ→−∞ µk (γ, ξ) = ∞ and limξ→∞ µk (γ, ξ) = 2k + 1. For instance, the restrictions of the function ξ → µk (γ, ξ) to the intervals ]−∞, ξk (γ)[ and ]ξk (γ), ∞[ are invertible. It is a result of the variational min-max principle that the function γ → Θk (γ) is continuous, see [10, Proposition 2.5] for the case k = 1. Thus the set Uk = {(γ, b) ∈ R × R : Θk (γ) < b < 2k + 1} is open in R2 .

(2.16)

Lemma 2.4. The functions Uk (γ, b) → ξk,± (γ, b) admit continuous extensions R × ]−∞, 2k + 1[ → ξ k,± (γ, b). Proof. Using the regularity of µk (γ, ξ), the implicit function theorem applied to Uk × R (γ, b, ξ) → µk (γ, ξ) − b near (γ0 , b0 , ξk,± (γ0 , b0 )) (for an arbitrary point (γ0 , b0 ) ∈ Uk ) permits to deduce that the functions Uk (γ, b) → ξk,± (γ, b) are C 1 . We then deﬁne the following continuous extensions of ξk,± , ξk,± (γ, b), if Θk (γ) < b < 2k + 1, ξ k,± (γ, b) = if Θk (γ) ≥ b, ξk (γ), where ξk (γ) is the unique non-degenerate minimum of ξ → µk (γ, ξ). The next lemma justiﬁes that the sum on the right-hand side of (1.9) is indeed ﬁnite. Lemma 2.5. For each M > 0 and b0 ∈ ]0, 1[, there exists a constant C > 0 such that, for all γ ∈ ]−M, M [ and b ∈ ]Θ(γ), b0 [, we have ∞ {ξ ∈ R : µj (γ, ξ) < b} ≤ C. j=1

Proof. Let us notice that for all j ≥ 1, {ξ ∈ R : µj (γ, ξ) < b} ⊂ {ξ ∈ R : µ1 (γ, ξ) < b0 },

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and for all γ ∈ ]−M, M [ (using the monotonicity of η → µ1 (η, ξ)), {ξ ∈ R : µ1 (γ, ξ) < b0 } ⊂ {ξ ∈ R : µ1 (−M, ξ) < b0 }. > 0 such that Consequently, there exists a constant M , M ], {ξ ∈ R : µj (γ, ξ) < b} ⊂ [−M

∀ b ≤ b0 , ∀ γ ∈ ]−M, M [.

Since the functions ξ → µj (γ, ξ) (j ∈ N) , M ] by are regular, we introduce constants (ξj (M ))j∈N ⊂ [−M µj (−M, ξj (M )) =

min

fM f] ξ∈[−M,

µj (−M, ξ).

We claim that lim µj (−M, ξj (M )) = ∞.

(2.17)

j→∞

Once this claim is proved, we get the result of the lemma, since by monotonicity µj (γ, ξ) ≥ µ1 (−M, ξ),

∀ γ ≥ −M,

∀ ξ ∈ R.

Let us assume by contradiction that the claim (2.17) were false. Then we may ﬁnd a constant M > 0 and a subsequence (jn ) such that µjn (−M, ξjn (M )) ≤ M,

∀ n ∈ N.

(2.18)

≤ ξjn (M ) ≤ M for all n, we get a subsequence, denoted again by ξjn (M ), Since −M such that , M ]. lim ξjn (M ) = ζ(M ) ∈ [−M

n→∞

It is quite easy, by comparing the corresponding quadratic forms, to prove the existence of a constant C > 0 such that, for all ε ∈ ]0, 12 [ and n ∈ N, we have the estimate µjn (−M, ξjn (M )) ≥ (1 − ε)µjn (−2M, ζ(M )) − C(ε + ε−1 |ξjn (M ) − ζ(M )|2 ). (2.19) We shall provide some details concerning the above estimate, but we would like ﬁrst to achieve the proof of the lemma. Notice that, since the operator L[−M, ζ(M )] has compact resolvent, then lim µjn (−2M, ζ(M )) = ∞.

n→∞

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Upon choosing ε = |ξjn (M ) − ζ(M )|, we get from (2.19) that lim µjn (−M, ξjn (M )) = ∞,

n→∞

contradicting thus (2.18). We conclude by some wards concerning the proof of (2.19). Notice that, for a normalized L2 -function u, we have by Cauchy–Schwarz inequality: ∞ (ζ − ξjn (M ))(t − ζ)|u|2 dt ≤ 2|ζ − ξjn (M )| × (t − ζ)uL2 (R+ ) 2 0

≤ ε(t − ζ)u2L2 (R+ ) + ε−1 |ζ − ξjn (M )|2 , for any ε > 0. On the other hand, writing (t − ξjn (M ))2 = (t − ζ)2 + (ξ − ξjn (M ))2 + 2(ζ − ξjn (M ))(t − ζ), we get the following comparison of the quadratic forms q[−M, ξjn (M )](u) ≥ q[−M, ζ](u) − ε(t − ζ)u2L2 (R+ ) − ε−1 |ζ − ξjn (M )|2 , where q[−M, ·] has been introduced in (2.1). Noticing that for ε ∈ ]0, 12 [, −M 1−ε ≥ −2M , the application of the min-max principle permits then to conclude the desired bound (2.19). Remark 2.6. Once the asymptotic expansion (1.9) is proved, the formula (1.11) becomes a consequence of Lemma 2.1. The next lemma will play a crucial role in establishing the main results of this paper. Lemma 2.7. The function S : R× ]−∞, 1[ (γ, b) →

∞

|{ξ ∈ R : µj (γ, ξ) < b}|

j=1

is locally uniformly continuous. Proof. Let b0 ∈ ]0, 1[ and m > 0. It is suﬃcient to establish, |S(γ + τ, b + δ) − S(γ, b)| → 0 as (τ, δ) → 0. sup

(2.20)

|γ|≤m, b≤b0

Let τ1 = 1 − b0 > 0. By monotonicity, for all τ, δ ∈ [−τ1 , τ1 ], the following holds {ξ ∈ R : µj (γ + τ, ξ) < b + δ} ⊂ {ξ ∈ R : µ1 (−m − τ1 , ξ) < b + τ1 },

∀ j ∈ N.

Therefore, we may ﬁnd a constant M > 0 depending only on m and b0 such that {ξ ∈ R : µj (γ + τ, ξ) < b + δ} ⊂ [−M, M ],

∀ τ, δ ∈ [−τ1 , τ1 ],

∀ j ∈ N.

(2.21)

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So deﬁning ξj (M ) as in the proof of Lemma 2.5, i.e. ∀ ξ ∈ [−M, M ],

∀ τ ∈ [−τ1 , τ1 ],

µj (γ + τ, ξ) ≥ µj (−m − τ1 , ξj (M )),

we get as in (2.17): lim µj (−m − τ1 , ξj (M )) = ∞.

j→∞

Hence, we may ﬁnd j0 ≥ 1 depending only on m and b0 such that µj (−m − 1, ξj (M )) ≥ b0 + 2τ1 ,

∀ j ≥ j0 ,

and consequently, for |τ | ≤ τ1 , |δ| ≤ τ1 , we get ∞

|{ξ ∈ R : µj (γ + τ, ξ) < b + δ}| =

j=1

j0

|{ξ ∈ R : µj (γ + τ, ξ) < b + δ}|.

j=1

Therefore, we deal only with a ﬁnite sum of j0 terms, j0 being independent from τ , δ, γ and b. So given k ∈ {1, . . . , j0 } and setting Sk (γ, b) = |{ξ ∈ R : µk (γ, ξ) < b}|, it is suﬃcient to show that sup |Sk (γ + τ, b + δ) − Sk (γ, b)| = 0. (2.22) lim (τ,δ)→0 |τ |+|δ|≤τ1

|γ|≤m, b≤b0

The above formula is only a direct consequence of Lemmas 2.3 and 2.4. 2.2. The model operator on a half-cylinder α,γ We treat now the operator Ph,Ω = −(h∇ − iA0 )2 , where ΩS is the half-cylinder S

ΩS = ]0, S[ × ]0, ∞[, S > 0 and γ ∈ R are constants. The magnetic potential A0 is taken in the canonical way A0 (s, t) = (−t, 0),

∀ (s, t) ∈ [0, S] × [0, ∞[.

(2.23)

α,γ satisfy the periodic conditions Functions in the domain of Ph,Ω S

u(0, ·) = u(S, ·) on R+ , and the de Gennes boundary condition at t = 0, h ∂t u|t=0 = hα γu|t=0 . We shall from now on use the following notation. For a self-adjoint operator T and a real number λ < inf σess (T ), we denote by N (λ, T ) the number of eigenvalues of T (counted with multiplicity) included in ]−∞, λ].

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Lemma 2.8. For each M > 0, there exists a constant C > 0 such that, for all b0 ∈ ]−∞, 1[, α ≥ 12 , S > 0, γ ∈ [−M, M ] and h ∈ ]0, 1[, we have,   ∞ −1/2 Sh α−1/2 N (b0 h, P α,γ ) −  ≤ C.  |{ξ ∈ R : µ (h γ, ξ) ≤ b }| j 0 h,ΩS 2π j=1 α,γ Proof. By separation of variables (cf. [17]) and a scaling we may decompose Ph,Ω S as a direct sum: d2 1/2 −1 2 h − 2 + (2πnh S + τ ) L2 (R+ ), in dτ n∈Z

n∈Z

α−1/2

γ u(0) at τ = 0. with the boundary condition u (0) = h Consequently we obtain: α,γ σ(Ph,Ω ) = {hµj (hα−1/2 γ, 2πh1/2 S −1 n) : n ∈ N}, S

(2.24)

j∈N

and each eigenvalue is of multiplicity 1. Thus, putting fj (ξ) = 1{ξ∈R:µj (hα−1/2 γ,ξ)
ξn = 2πh1/2 S −1 n,

we obtain α,γ α,γ N (b0 h, Ph,Ω ) = Card(σ(Ph,Ω )∩ ]0, b0 h]) S S

=

∞

Card{n ∈ N : µj (hα−1/2 γ, 2πh1/2 S −1 n) ≤ b0 }

j=1

=

∞

fj (ξn )

j=1 n∈Z

=

j0

fj (ξn ).

j=1 n∈Z

Notice that the last step is due to Lemma 2.5 (and its proof) which yields the existence of j0 ∈ N, depending only on γ, h and b0 , such that fj ≡ 0 for j ≥ j0 . Now, by deﬁnition of ξn , n∈Z

fj (ξn ) =

Sh−1/2 (ξn+1 − ξn )fj (ξn ), 2π n∈Z

and we can verify easily the following estimate (thanks in particular to Lemma 2.3), −2πh1/2 S −1 + fj (ξ) dξ ≤ (ξn+1 − ξn )fj (ξn ) ≤ fj (ξ) dξ + 2πh1/2 S −1 . R

n∈Z

R

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Therefore, we conclude upon noticing that, by the deﬁnition of the function fj , fj (ξ) dξ = |{ξ ∈ R : µj (hα−1/2 γ, ξ) < b0 }|. R

2.3. The model operator on a Dirichlet strip α,γ We consider now the operator Ph,Ω = −(h∇ − iA0 )2 , where ΩS,T is the strip S,T

ΩS,T = ]0, S[ × ]0, T [, S, T > 0 and γ ∈ R are constants. The magnetic potential A0 was deﬁned in (2.23). α,γ satisfy the de Gennes condition h∂t u = hα γ u Functions in the domain of Ph,Ω S,T at t = 0 and Dirichlet condition on the other sides of the boundary. α,γ The next lemma gives a comparison between the counting function of Ph,Ω S,T α,γ and that of Ph,ΩS . Lemma 2.9. There exists a constant c > 0 such that, ∀ S > 0,

∀ T > 0,

∀ γ ∈ R,

∀ δ ∈ ]0, S/2],

∀ b0 ∈ ]−∞, 1[,

we have, 1 α,γ α,γ N (b0 h − ch2 (δ −2 + T −2 ), Ph,Ω ) ≤ N (b0 h, Ph,Ω ) S,T 2(S−δ) 2 α,γ ). ≤ N (b0 h, Ph,Ω S α,γ is Proof. Since the extension by zero of a function in the form domain of Ph,Ω S,T α,γ included in that of Ph,ΩS , and the values of the quadratic forms coincide for such a function, we get the upper bound of the lemma by a simple application of the variational principle. We turn now to the lower bound. The argument is like the one used in [2,5] but we explain it because it illustrates in a simple case the arguments of this paper. Let us introduce two partitions of unity (ϕδi ) and (ψjT ) such that:

(ϕδ1 )2 + (ϕδ2 )2 = 1

in [0, 2(S − δ)],

  supp ϕδ1 ⊂ [0, S]         supp ϕδ ⊂ [S − δ, 2(S − δ)]   2

    

2

|(ϕδi ) |2

≤ cδ

−2

    

and

(ψ0T )2 + (ψ1T )2 = 1 in R+ ,   supp ψ0T ⊂ [T /2, ∞[         supp ψ T ⊂ [0, T ]   1 1        |(ψiT ) |2 ≤ cT −2   

i=1

i=0

where c > 0 is a constant independent from S, T and δ. Upon putting δ T χδ,T i (s, t) = ϕi (s)ψ1 (t) (i = 1, 2),

T χδ,T 0 (s, t) = ψ0 (t),

we get a partition of unity of Ω2(S−δ) = ]0, 2(S − δ)[ ×R+ .

,

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α,γ Let us take a function u in the form domain of Ph,Ω . Then, by the IMS 2(S−δ) decomposition formula: |(h∇ − iA0 )u|2 dx Ω2(S−δ)

=

2 i=0

≥

Ω2(S−δ)

2 i=0

Ω2(S−δ)

|(h∇ −

2 iA0 )χδ,T i u| dx

−h

2

2 i=0

2 |∇χδ,T i |uL2 (ΩS,T )

2 −2 |(h∇ − iA0 ) χδ,T + T −2 )h2 u2L2 (ΩS,T ) . i u| dx − c(δ

α,γ δ,T Notice that χδ,T 1 u is in the form domain of Ph,ΩS,T and χ2 u is in that of α,γ Ph,]S−δ,2(S−δ)[ × ]0,T [ (this last operator, thanks to translational invariance with

α,γ respect to s, is unitary equivalent to Ph,Ω ). Also, χδ,T 0 u is in the form domain of S,T D the Ph,Ω2(S−δ) , the Dirichlet realization of −(h∇ − iA0 )2 in Ω2(S−δ) . α,γ Since the form domain of Ph,Ω can be viewed ina 2(S−δ) α,γ α,γ FD(Ph,Ω ) ⊕ FD(Ph,]S−δ,2(S−δ)[ S,T

× ]0,T [ )

D ⊕ FD(Ph,Ω ) 2(S−δ)

δ,T δ,T via the isometry u → (χδ,T 1 u, χ2 u, χ0 u), we get upon applying the variational principle (see [17, Sec. XII.15]), α,γ α,γ D N (b0 h − ch2 (δ −2 + T −2 ), Ph,Ω ) ≤ 2N (b0 h, Ph,Ω ) + N (b0 h, Ph,Ω ). 2(S−δ) 2(S−δ) 2(S−δ)

(2.25) D has no spectrum below b0 h when Notice that, by (1.4), the operator Ph,Ω 2(S−δ) b0 < 1. Hence, D ) = 0. N (b0 h, Ph,Ω 2(S−δ)

Coming back to (2.25), we get the lower bound stated in the lemma. 3. Proof of Theorem 1.1 We come back to the case of a general smooth domain Ω whose boundary is compact. We introduce the following quadratic forms: α,γ (u) = |(h∇ − iA)u|2 dx + h1+α γ(s)|u(s)|2 ds, (3.1) qh,Ω Ω

qh,Ω (u) =

a For

Ω

∂Ω

|(h∇ − iA)u|2 dx,

an operator A, FD(A) denotes its form domain.

(3.2)

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deﬁned for functions in the magnetic Sobolev space: 1 Hh,A (Ω) = {u ∈ L2 (Ω) : (h∇ − iA)u ∈ L2 (Ω)}.

(3.3)

Here, we recall that A ∈ C 2 (Ω) is such that curl A = 1 in Ω. We shall recall in the Appendix a standard coordinate transformation valid in a suﬃciently thin neighborhood of the boundary: Φt0 : Ω(t0 ) x → (s(x), t(x)) ∈ [0, |∂Ω|[ × [0, t0], where for t0 > 0, Ω(t0 ) is the tubular neighborhood of ∂Ω: Ω(t0 ) = {x ∈ Ω : dist(x, ∂Ω) < t0 }. Let us mention that t(x) = dist(x, ∂Ω) measures the distance to the boundary and s(x) measures the curvilinear distance in ∂Ω. Using the coordinate transformation Φt0 , we associate to any function u ∈ deﬁned in [0, |∂Ω|[ × [0, t0] by, L2 (Ω), a function u u (s, t) = u(Φ−1 t0 (s, t)).

(3.4)

α,γ (u) The next lemma states a standard approximation of the quadratic form qh,Ω by the canonical one in the half-plane, provided that the function u is supported near the boundary.

Lemma 3.1. There exists a constant C > 0, and for all S1 ∈ [0, |∂Ω|[,

S2 ∈ ]S1 , |∂Ω|[,

there exists a function φ ∈ C 2 ([S1 , S2 ] × [0, t0 ]; R) such that, for all S1 ∈ [S1 , S2 ],

T ∈ ]0, t0 [,

ε ∈ [CT, Ct0 ],

1 (Ω) satisfying and for all u ∈ Hh,A

supp u ⊂ [S1 , S2 ] × [0, T ], one has the following estimate, α,b γ1 iφ/h (1 − ε)qh,Ω (e u ) − Cε−1 (((S2 − S1 )2 + T 2 )2 + h2 ) u2L2 (Ω1 ) 1 α,γ α,e γ1 iφ/h ≤ qh,Ω (u) ≤ (1 + ε)qh,Ω (e u ) + Cε−1 (((S2 − S1 )2 + T 2 )2 + h2 ) u2L2 (Ω1 ) . 1

Here Ω1 = [S1 , S2 ] × [0, T ],

γ 1 =

γ(S1 ) + C(S2 − S1 ) , 1+ε

and the function u is associated to u by (3.4).

γ 1 =

γ(S1 ) − C(S2 − S1 ) . 1−ε

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Proof. Notice that when γ ≡ 0, the result follows from [5, Lemma 3.5], which reads explicitly in the form: α,0 |(h∇ − iA0 )eiφ/h u |2 ds dt qh,Ω (u) − [S1 ,S2 ]×[0,T ] ≤ε

[S1 ,S2 ]×[0,T ]

|(h∇ − iA0 )eiφ/h u |2 ds dt

+ Cε−1 ((T 2 + (S2 − S1 )2 )2 + h2 ) u2L2 . Since u restricted to the boundary is supported in [S1 , S2 ], we get as an immediate consequence the following two-sided estimate for non-zero γ: α,γ (1 + ε)−1 qh,Ω (u) ≤

[S1 ,S2 ]×[0,T ]

|(h∇ − iA0 )e

iφ/h

h1+α u | ds dt + 1+ε 2

γ(s)| u(s, 0)|2 ds

[S1 ,S2 ]

u2L2 , + C(1 + ε)−1 ε−1 ((T 2 + (S2 − S1 )2 )2 + h2 )

(3.5)

and α,γ (1 − ε)−1 qh,Ω (u) ≥

[S1 ,S2 ]×[0,T ]

|(h∇ − iA0 )eiφ/h u |2 ds dt +

−1 −1

− C(1 − ε)

ε

2

2 2

((T + (S2 − S1 ) ) + h

h1+α 1−ε 2

γ(s)| u(s, 0)|2 ds

[S1 ,S2 ]

) u2L2 .

(3.6)

The idea now is to approximate γ by a constant in a simple manner without needing an estimate of the boundary integral. Actually, Taylor’s formula applied to the function γ near S1 leads to the estimate |γ(s) − γ(S1 )| ≤ C(S2 − S1 ),

∀ s ∈ [S1 , S2 ],

where the constant C > 0 is possibly replaced by a larger one. Having this estimate in hand, we get: 2 2 γ(s)| u(s, 0)| ds − γ(S1 ) | u(s, 0)| ds [S1 ,S2 ] [S1 ,S2 ] ≤ C(S2 − S1 ) | u(s, 0)|2 ds. [S1 ,S2 ]

Recalling the deﬁnition of γ and γ in Lemma 3.1 (they actually depend on ε, S1 , S2 and hence account to all possible errors), we infer directly from the previous

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estimate,

(1 − ε) γ1

2

[S1 ,S2 ]

| u(s, 0)| ds ≤

917

γ(s)| u(s, 0)|2 ds

[S1 ,S2 ]

≤ (1 + ε) γ1

[S1 ,S2 ]

| u(s, 0)|2 ds.

(3.7)

Substituting the lower and upper bound of (3.7) in (3.5) and (3.6) respectively, and recalling the hypothesis that u (s, 0) is supported in [S1 , S2 ], we obtain the desired estimates of the lemma. We shall divide a thin neighborhood of ∂Ω into many small sub-domains, and in each sub-domain, we shall apply Lemma 3.1 to approximate the quadratic form. α,γ ) in terms of the spectral counting This will yield a two-sided estimate of N (λ, Ph,Ω functions of model operators on half-cylinders. Let us put N = [h−3/8 ], the greatest positive integer below h−3/8 . Let |∂Ω| , sn = nS, n ∈ {0, 1, . . . , N }, N and we emphasize that these quantities depend on h. We put further S=

ΩS = ]0, S[ × R. With these notations, the proof of Theorem 1.1 is given by the following proposition. Proposition 3.2. Let b0 ∈ ]−∞, 1[. There exist constants C, h0 > 0 such that for all h ∈ ]0, h0 ],

δ ∈ ]0, S/2],

Sn ∈ [sn−1 , sn ],

n ∈ {1, 2, . . . , N },

one has the following estimate on the spectral counting function, N 1 α,e γn α,γ N (hb0 − Ch2 δ −2 , Ph,Ω ) ≤ N (hb0 , Ph,Ω ) 2(S−δ) 2 n=1

≤

N n=1

α,b γn N (hb0 + Ch2 δ −2 , Ph,Ω ). S+2δ

Here γ n =

γ(Sn ) + CS 1 + h1/4

and

γ n =

γ(Sn ) − CS . 1 − h1/4

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Before proving Proposition 3.2, let us see how it serves for obtaining the conclusion of Theorem 1.1. Proof of Theorem 1.1. We keep the notation introduced for the statement of Proposition 3.2. The proof is in two steps. Step 1. Let us establish the asymptotic formula (as h → 0): α,γ )= h1/2 N (b0 h, Ph,Ω

1 2π

∞ {ξ ∈ R : µj (hα−1/2 γ(s), ξ) < b0 } ds + o(1),

∂Ω j=1

(3.8) where b0 ∈ ]−∞, 1[ . From the lower bound in Proposition 3.2, we get upon applying Lemma 2.8, a > 0 such that constant C α,γ ) h1/2 N (b0 h, Ph,Ω

≥

N ∞ 1 1/2 . (S − δ) |{ξ ∈ R : µj (hα−1/2 γ n , ξ) < b0 − Chδ −2 }| − Ch 2π n=1 j=1

Since α ≥ 1/2 and ∂Ω is bounded, it is a result of Lemmas 2.5 and 2.7 that there exist constants C > 0 and h0 > 0 together with a function ]0, h0 ] h → (h) tending to 0 as h → 0 such that, for all h ∈ ]0, h0 ] and n ∈ {1, 2, . . . , N }, we have (provided that hδ −2 is suﬃciently small), γn , b0 ) ≤ C, S(hα−1/2 n , b0 − Chδ −2 ) − S(hα−1/2 γn , b0 )| ≤ (h), |S(hα−1/2 γ γn − CS = γ(Sn ). where the function S is introduced in 2.7, and γn = (1 + h1/4 ) −3/8 ], we get upon choosing δ = 1/N 2 , Recalling that S = |∂Ω|/N and N = [h α,γ h1/2 N (b0 h, Ph,Ω )   N ∞ 1  S|{ξ ∈ R : µj (hα−1/2 γn , ξ) < b0 }| − (h) − Ch3/8 , ≥ 2π n=1 j=1

where the leading order term on the right-hand side is a Riemann sum. We get then the following lower bound, h

1/2

α,γ N (b0 h, Ph,Ω )

1 ≥ 2π

∞

∂Ω j=1

|{ξ ∈ R : µj (hα−1/2 γ(s), ξ) < b0 }| ds + o(1).

This is the lower bound in (3.8). In a similar manner we obtain an upper bound.

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919

Step 2. If α = 1/2, the asymptotic formula (3.8) is just the conclusion of Theorem 1.1. We turn to the case when α > 1/2. Again, it results from Lemma 2.7 the existence of a constant h0 and a function ]0, h0 ] h → 1 (h) tending to 0 as h → 0 such that for all h ∈ ]0, h0 ] and s ∈ ∂Ω,   ∞ ≤ 1 (h), S(hα−1/2 γ(s), b0 ) −   |{ξ ∈ R : µ (ξ) < b | j 0 j=1 where µj (ξ) = µj (0, ξ). Moreover, by Lemma 2.1, it holds that ∞

|{ξ ∈ R : µj (ξ) < b0 }| = 0.

j=2

Now we can infer from (3.8) the asymptotic formula announced in (1.8). α,γ Proof of Proposition 3.2. Let us establish the lower bound. Let PN,h,Ω be the α,γ α,γ restriction of the operator Ph,Ω for functions u in D(Ph,Ω ) that vanishes on the set

{x ∈ Ω : t(x) ≥ T } ∪

N

{x ∈ Ω : 0 ≤ t(x) ≤ T, s(x) = sn },

n=1

where T > is to be speciﬁed later. The important remark is that the spectrum of α,γ α,γ is below that of PN,h,Ω . Ph,Ω α,γ . Applying Lemma 3.1 Let us take a function u in the form of domain of PN,h,Ω with T = h3/8 and ε = h1/4 , we get the estimate, α,γ qh,Ω (u) ≤ (1 + h1/4 )

N

α,e γn −iφn /h (qh,Ω (e u ) + Ch5/4 e−iφn /h u 2L2 (Ωn ) ), n

n=1

e

Sn )+CS n = γ(1+h . Then, by the variational princiwhere Ωn = ]sn−1 , sn [ × ]0, T [ and γ 1/4 α,γ α,γ is below that of PN,h,Ω ), ple, we obtain (recall that the spectrum of Ph,Ω α,γ N (λ, Ph,Ω )

≥

N n=1

N

λ − Ch5/4 α,eγn , Ph,ΩS,T , 1 + h1/4

where ΩS,T = ]0, S[ × ]0, T [. Applying Lemma 2.9, this is suﬃcient to conclude the lower bound announced in Proposition 3.2. The upper bound is derived by introducing a partition of unity attached to the sub-domains Ωn and by using the IMS decomposition formula. The analysis is similar to that presented for the lower bound above and also to that in the proof of Lemma 2.9, so we omit the proof. For the details, we refer to [5, Proposition 3.6].

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4. Curvature Eﬀects 4.1. A family of ordinary diﬀerential operators on a weighted L2 space A ﬁner approximation of the quadratic form (3.1) leads to the analysis of a family of ordinary diﬀerential operators on a weighted L2 space that takes into account the curvature eﬀects of the boundary. We shall recall in this section the main results for the lowest eigenvalue problem concerning this family of operators. These results were obtained in [7] for the Neumann problem and then generalized in [10] for situations involving de Gennes’ boundary condition. Let us introduce, for technical reasons that will be clariﬁed later, a positive parameter δ ∈ ] 14 , 12 [. Let us also consider parameters h > 0 and β ∈ R such that |β|hδ <

1 . 3

We deﬁne the family of quadratic forms (indexed by ξ ∈ R) 2 hδ−1/2 2 α,η 2 1/2 1/2 t qh,β,ξ (u) = |u (t)| + (1 + 2βh t) t − ξ − βh u(t) 2 0 × (1 − βh1/2 t) dt + hα−1/2 η|u(0)|2 ,

(4.1)

deﬁned for functions u in the space: α,η D(qh,β,ξ ) = {u ∈ H 1 (]0, hδ−1/2 [) : u(hδ−1/2 ) = 0}.

(4.2)

α,η Let us denote by Hh,β,ξ the self-adjoint realization associated to the quadratic form α,η ))j the increasing (4.1) by Friedrich’s’ theorem. Let us denote also by (µj (Hh,β,ξ α,η sequence of eigenvalues of Hh,β,ξ . For each α ≥ 12 and η ∈ R we introduce the positive numbers: 1 1 d2 , η = ξ(η)Θ (η), d2 (α, η) = ξ0 Θ (0) α> , 2 2 (4.3) 1 1 1 1 2 , η = (ηξ(η) + 1) Θ (η), d3 (α, η) = Θ (0) α> d3 . 2 3 3 2 α,η in the next theorem has The result concerning the lowest eigenvalue of Hh,β,ξ been proved in [10].

Theorem 4.1. Suppose that δ ∈ ] 14 , 12 [ and α ≥ 12 . Let  1 1   if α =  δ − 4, 2 , η = hα−1/2 η, ρ0 =  1 1 1   , α − min δ − , if α >  4 2 2

and

0 < ρ < ρ0 .

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For each M > 0 and ζ0 > 0, there exists a constant ζ1 > 0, and for each ζ ≥ ζ1 , there exist positive constants C, h0 and a function ]0, h0 ] h → (h) ∈ R+ with limh→0 (h) = 0 such that, ∀ η, β ∈ ]−M, M [,

∀ h ∈ ]0, h0 ],

the following assertions hold: • If |ξ − ξ( η )| ≤ ζhρ , then α,η,D |µ1 (Hh,β,ξ ) − {Θ( η) + d2 (α, η)(ξ − ξ( η ))2 − d3 (α, η)βh1/2 }|

≤ C[h1/2 |ξ − ξ( η )| + hδ+1/2 + h1/2 (h)],

(4.4)

and α,η,D µ2 (Hh,β,ξ ) ≥ Θ( η ) + ζ0 h2ρ .

(4.5)

• If |ξ − ξ( η )| ≥ ζhρ , then α,η,D µ1 (Hh,β,ξ ) ≥ Θ( η ) + ζ0 h2ρ .

(4.6)

Here, the parameters d2 (α, η) and d3 (α, η) has been introduced in (4.3). η )| ≥ ζ1 hρ Proof. The existence of ζ1 so that the lower bound (4.6) holds for |ξ − ξ( has been established in [10, Lemma V.8]. Now, for ζ ≥ ζ1 , (4.6) obviously holds. Under the hypothesis |ξ − ξ( η )| ≤ ζhρ , the existence of the constants C, h0 and the estimate (4.4) have been established in [10, Lemmas V.8 and V.9]. So we only need to establish (4.5). We start with the case α = 12 . It results from the min-max principle (see [10] or [14, Lemma 4.2.1] for details), α,η,D α,η,D 2δ− 12 (1 + µ2 (H α,η,D )), ) − µ2 (Hh,0,ξ )| ≤ Ch |µ2 (Hh,β,ξ h,0,ξ

depends only on M . where the constant C It results again from the min-max principle, α,η,D µ2 (Hh,0,ξ ) ≥ µ2 (L[η, ξ]),

where L[η, ξ] is the operator introduced in (2.3). We get then the following lower bound, α,η,D 2δ− 12 )µ2 (L[η, ξ]) − Ch 2δ− 12 µ2 (Hh,β,ξ ) ≥ (1 − 2Ch

2δ− 12 )Θ2 (η) − Ch 2δ− 12 ≥ (1 − 2Ch ≥ Θ(η) + ζ0 h2ρ . Here, we recall the deﬁnition of Θ2 (η) in (2.15). Let us also point out that the ﬁnal conclusion above follows by Lemma 2.2 upon taking h ∈ ]0, h0 ] with h0 chosen so 1 2δ− 2 (Θ2 (η) + 1) < 1 (Θ2 (η) − Θ(η)). small that ζ0 h2ρ + 2Ch 0

0

2

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When α > 12 , the result follows from the above argument upon using the continuity of our spectral functions with respect to small perturbations. 4.2. Spectral function for the model operator on a half-cylinder Let us consider parameters h > 0, δ > 0, S > 0 and β ∈ R s.t. |β|hδ <

1 . 2

α,η the self adjoint operator in We denote by L h,β,S L2 (]0, S[ × ]0, hδ [ ; (1 − βt) ds dt) associated with the quadratic form 2 S hδ t2 α,η 2 Qh,β,S (u) = |h∂t u| + (1 + 2βt) h∂s + t − β u 2 0

0

× (1 − βt) ds dt + h1+α η

0

S

|u(s, 0)|2 ds,

deﬁned for functions u in the form domain α,η ) = {u ∈ H 1 (]0, S[ × ]0, hδ [) : u(·, hδ ) = 0, u(0, ·) = u(S, ·)}. D(Q h,β,S We recall again the notation that for a self adjoint operator A and a number λ < inf σess (A), we denote by N (λ, A) the number of eigenvalues of A (counted with multiplicity) below λ. Proposition 4.2. With the notation and hypotheses of Theorem 4.1, let ζ0 > 0, h0 > 0 and λ = λ(h) such that |λ − Θ( η )| < ζ0 h2ρ ,

∀ h ∈ ]0, h0 ].

(4.7)

Then, for each M > 0, there exist constants C > 0 and h0 > 0 and a function ]0, h0 ] h → 0 (h) ∈ R+ with limh→0 0 (h) = 0 such that, for all h ∈ ]0, h0 ] and S, η, β ∈ ]−M, M [,

−1/4 S h α,η −1/2 ) − (d (α, η)β + h [λ − Θ( η )]) N (hλ, L 3 + h,β,S π d2 (α, η) ≤

h−1/4 S π d2 (α, η)

(d3 (α, η)β + h−1/2 [λ − Θ( η )])+ 0 (h).

(4.8)

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923

Proof. By separation of variables and by performing the scaling τ = h−1/2 t, we α,η as a direct sum, decompose L h,β,S n∈Z

α,η h Hh,β,2πnh 1/2 S −1

in

L2 (]0, hδ−1/2 [; (1 − βh1/2 t) dt).

n∈N

Consequently, by Theorem 4.1 and the hypothesis λ − Θ( η ) < ζ0 h2ρ , we obtain, α,η ) = Card({n ∈ Z; µ1 (H α,η N (hλ, L ) ≤ λ}). h,β,S h,β,2πnh1/2 S −1 Again, Theorem 4.1 yields the existence of a positive constant ζ (that we may choose suﬃciently large as we wish) and a function h → (h) such that, upon deﬁning the subsets η )| ≤ ζhρ , Θ( η) + d2 (α, η)(2πnh1/2 S −1 S± = {n ∈ Z : |2πnh1/2 S −1 − ξ( − ξ( η ))2 − d3 (α, η)βh1/2 ± h1/2 (h) < λ}, one gets the inclusion, α,η S+ ⊂ {n ∈ Z; µ1 (Hh,β,2πnh 1/2 S −1 ) ≤ λ} ⊂ S− .

(4.9)

Therefore, we deduce that α,η ) ≤ Card S− . Card S+ ≤ N (hλ, L h,β,S On the other hand, thanks to (4.7), we may choose ζ > 0 suﬃciently large so that Θ( η ) + d2 (α, η)(2πnh1/2 S −1 − ξ( η ))2 − d3 (α, η)βh1/2 ± h1/2 (h) < λ ⇒ |2πnh1/2 S −1 − ξ( η )| ≤ ζhρ . With this choice of ζ, one can rewrite S± in the following equivalent form S± = {n ∈ Z : d2 (α, η)(2πnS −1 − h−1/2 ξ( η ))2 ≤ h−1/2 (d3 (α, η)β + h−1/2 [λ − Θ( η )] ± (h))+ }, from which one obtains a positive function 0 (h) 1 such that

h−1/4 S (d3 (α, η)β + h−1/2 [λ − Θ( η )])+ Card S± − π d2 (α, η) ≤

h−1/4 S π d2 (α, η)

(d3 (α, η)β + h−1/2 [λ − Θ( η )])+ 0 (h),

when S varies in a bounded interval ]−M, M [. This ﬁnishes the proof of the proposition.

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4.3. The model operator on a Dirichlet strip We continue to work in the framework of the previous subsection by keeping our choice of parameters β, η, α ≥ 12 , h > 0, δ ∈ ] 14 , 12 [ and S. Let us consider the α,η operator Lα,η h,β,S obtained from Lh,β,S by imposing additional Dirichlet boundary conditions at s ∈ {0, S}, i.e. α,η α,η Lα,η h,β,η : D(Lh,β,S ) u → Lh,β,S u

with α,η D(Lα,η h,β,S ) = {u ∈ D(Lh,β,η ) : u(0, ·) = u(S, ·) = 0}. 2 δ δ Actually, Lα,η h,β,S is the self-adjoint operator in L (]0, S[ × ]0, h [; (1 − βh ) ds dt) associated with the quadratic form, 2 S hδ t2 α,η 2 Qh,β,S (u) = |h∂t u| + (1 + 2βt) h∂s + t − β u 2 0 0 S × (1 − βt) ds dt + h1+α η |u(s, 0)|2 ds, 0

deﬁned for functions u in the form domain 1 δ δ D(Qα,η h,β,S ) = {u ∈ H (]0, S[ × ]0, h [) : u(·, h ) = u(0, ·) = u(S, ·) = 0}.

Using the same reasoning as that for the proof of Lemma 2.9, we get in the next lemma an estimate of the spectral counting function of the operator Lα,η h,β,S . Lemma 4.3. For each M > 0, there exist constants C > 0 and h0 > 0 such that, for all β, η, S ∈ ]−M, M [,

ε0 ∈ ]0, S/2[,

λ ∈ R,

h ∈ ]0, h0 [,

one has the estimate: 1 α,η 2 α,η α,η N (λ − Cε−2 0 h , Lh,β,2(S−ε0 ) ) ≤ N (λ, Lh,β,S ) ≤ N (λ, Lh,β,S ). 2 4.4. Spectral counting function in general domains We return in this subsection to the case of a general smooth domain Ω whose boundary is compact. 4.4.1. An energy estimate Let us recall the notation that κr denotes the scalar curvature of the boundary ∂Ω. As was ﬁrst noticed in [7], since the magnetic ﬁeld is constant, the quadratic form (3.2) can be estimated with a high precision by showing the inﬂuence of the scalar curvature. This is actually the content of the next lemma, which we quote from [5, Lemma 4.7]. Before stating the estimate, let us recall that to a given function 1 (Ω), we associate by means of boundary coordinates a function u , see (3.4). u ∈ Hloc

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Lemma 4.4. Let δ ∈ ] 14 , 12 [. There exists a constant C > 0, and for all S ∈ [0, |∂Ω|[,

S ∈ ]0, S[,

there exists a function φ ∈ C 2 ([0, S] × [0, Chδ ]; R) such that, for all ε ∈ [Ch δ , 1], 1 (Ω) satisfying and for all u ∈ Hh,A supp u ⊂ [0, S] × [0, Chδ ], one has the following estimate, |qh,Ω (u) − Qh,eκ,S (eiφ/h u )| ≤ C(hδ S Qh,eκ,S (eiφ/h u ) + (h2+δ + Sh3δ )eiφ/h u 2L2 ). and the function u is associated to u by (3.4). Here κ = κr (S), Let us mention that we omit α and η from the notation in Sec. 4.3 when η ≡ 0. 4.4.2. Estimates of the counting function As in Sec. 3, we introduce a partition of a thin neighborhood of ∂Ω: Given N ∈ N (that will be chosen later as a function of h) such that N = N (h) 1 (h → 0), we put |∂Ω| , sn = nS, κn = κr (sn ), n ∈ {0, 1, . . . , N }. N By this way, we are able to estimate the spectral counting function of the operator α,γ by those of the operators Lα,γ Ph,Ω h,κn ,S . S=

Proposition 4.5. Let δ ∈ ] 14 , 12 [, γ0 = minx∈∂Ω γ(x) and λ = λ(h) such that |λ − Θ(hα−1/2 γ0 )| 1

(h → 0).

(4.10)

There exist constants C > 0 and h0 > 0 such that, for all ε0 ∈ ]0, S/2[,

h ∈ ]0, h0 [,

Sn ∈ [sn−1 , sn ],

n ∈ {1, . . . , N },

one has the estimate N 1 2 α,eγn N (hλ − C(Sh3δ + ε−2 0 h ), Lh,e κn ,2(S+ε0 ) ) 2 n=1 α,γ ≤ N (hλ, Ph,Ω )

≤

N n=1

2 α,bγn N (hλ + C(Sh3δ + ε−2 0 h ), Lh,e κn ,S+2ε0 ).

Here κ n = κr (Sn ),

γ n =

γ(Sn ) + CS , 1 + Chδ S

γn =

γ(Sn ) − CS . 1 − Chδ S

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The proof is exactly as that of Proposition 3.2 and we omit it. Let us only mention the main points. Thanks to a partition of unity associated with the intervals [sn−1 , sn ] and the variational principle, the result follows from local estimates of the quadratic form. For this sake, the procedure consists of the implementation of γn and Lemma 4.4, bounding γ(s) from above and below respectively by (1 + Chδ S) γn in each [sn−1 , sn ] (thus getting errors of order S) and ﬁnally of the (1 − Chδ S) application of Lemma 4.3. 4.4.3. An asymptotic formula of the counting function Let us take constants ζ0 > 0, c0 > 0, δ ∈ ] 14 , 12 [, and let us recall that we introduce a parameter ρ such that: 1 1 if α = , 0<ρ<δ− 4 2 1 1 1 0 < ρ < min δ − , α − if α > . 4 2 2 We take h0 > 0 and λ = λ(h) such that, c0 h1/2 ≤ |λ − Θ(hα−1/2 γ0 )| < ζ0 h2ρ ,

∀ h ∈ ]0, h0 ].

(4.11)

Here γ0 = min γ(x) x∈∂Ω

∞

and γ ∈ C (∂Ω; R) is the function in de Gennes’ boundary condition that we α,γ , see (1.2). impose on functions in the domain of the operator Ph,Ω 5 1 , 2 [. With the above notations, we have the following Theorem 4.6. Let δ ∈ ] 12 asymptotic formula as h → 0, h−1/4 α,γ N (hλ, Ph,Ω ) = ∂Ω π d2 (α, γ(s))

−1/2 α−1/2 [λ − Θ(h γ(s))])+ ds (1 + o(1)). × (d3 (α, γ(s))κr (s) + h

Here, for a given (α, η) ∈ R × R, the parameters d2 (α, η) and d3 (α, η) have been introduced in (4.3). Proof. The proof is similar to that of the asymptotic formula (3.8). There we have shown how to establish a lower bound, so we show here how to establish an upper bound. For each n ∈ {1, . . . , N }, let us introduce the function (see Lemma 4.3): α−1/2 n ) κn + h−1/2 [λ + C(Sh3δ−1 + ε−2 γ n )]. fn (δ, S, ε0 ) = d3 (α, γ 0 h) − Θ(h

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Here γ n and κ n are given by Proposition 4.5. Then, combining Propositions 4.2 and 4.5, we get, N

h−1/4 (S + 2ε0 ) α,γ [fn (δ, S, ε0 )]+ (1 + 0 (h)), (4.12) N (hλ, Ph,Ω ) ≤ n ) n=1 π d2 (α, γ where the function 0 is independent of N and satisﬁes lim 0 (h) = 0.

h→0

We recall also that S =

|∂Ω| N .

We make the following choice of ε0 ∈ ]0, S/2[:

ε0 = S 1+ς

with ς > 0.

Then we pose the following condition on S as h → 0, γn )|, Sh3δ−1 + S −2−2ς h |λ − Θ(hα−1/2

∀ n ∈ {1, . . . , N }.

By the hypothesis in (4.11), it suﬃces to choose S in the following way: Sh3δ−1 + S −2−2ς h h1/2

(h → 0).

This yields, 1

1

h 4(1+ς) S h3( 2 −δ) and we notice that a choice of ς > 0 such that 1 1 >3 −δ 4(1 + ς) 2 5 1 is possible only if δ ∈ ] 12 , 2 [ (this will yield when α = 12 that ρ0 ∈ ] 16 , 14 [, ρ0 being introduced in Theorem 4.1). With this choice, the upper bound (4.12) becomes (for a possibly diﬀerent 0 (h) 1), N ! h−1/4 S α,γ N (hλ, Ph,Ω ) ≤ 1+ 0 (h) [gn (λ, α)]+ , n ) n=1 π d2 (α, γ

(4.13)

with gn (λ, α) = d3 (α, γn ) κn + h−1/2 [λ − Θ(hα−1/2 γ n )]. Replacing γ n by γn = γ(Sn ) in (4.13) will yield an error of the order O(S), and the sum on the right-hand side of (4.13) becomes a Riemann sum. We therefore conclude the following upper bound ! h−1/4 α,γ 0 (h) [g(λ, α; s)]+ ds, N (hλ, Ph,Ω ) ≤ 1 + ∂Ω π d2 (α, γ(s)) with κ(s) + h−1/2 [λ − Θ(hα−1/2 γ(s))]. g(λ, α; s) = d3 (α, γ(s)) By a similar argument, we get a lower bound.

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Remark 4.7. When relaxing the hypotheses of Theorem 4.6 by allowing λ to satisfy (compare with (4.11)): |λ − Θ(hα−1/2 γ0 )| = o(h1/2 )

as h → 0,

the result for the counting function becomes (as can be checked by adjusting the proof of Theorem 4.6)

h−1/4 α,γ (d3 (α, γ(s))κr (s))+ ds (1 + o(1)). N (hλ, Ph,Ω ) = ∂Ω π d2 (α, γ(s)) Proof of Theorem 1.2. We recall in this case that 1

λ(h) = Θ0 + 3aC1 hα− 2

1 2

< α < 1 and that

with a ∈ R\{γ0 }.

Here C1 > 0 is the universal constant introduced in (1.7). In this speciﬁc regime, (4.11) is veriﬁed when making a choice of ρ ∈ ]0, 12 min(δ− 1 1 4 , α − 2 )[. The leading order term of the integrand in the asymptotic formula of Theorem 4.6 is, up to a multiplication by a positive constant,

h−1/2 [λ − Θ(hα−1/2 γ(s))]+ . We write by using the asymptotic expansion of Θ(·) given by Taylor’s formula (see (2.11)–(2.14)): Θ(hα−1/2 γ(s)) = Θ0 + 3C1 γ(s)hα−1/2 + O(h2α−1 ),

(h → 0).

Therefore, it results from Theorem 4.6, 1

3 h 2 (α− 2 ) α,γ √ N (hλ, Ph,Ω ) = (a − γ(s))+ ds (1 + o(1)). π ξ0 ∂Ω When a = γ0 , we may encounter the regime of Remark 4.7, hence by using the result of that remark and noticing that when 12 < α < 1 1

3

h−1/4 h 2 (α− 2 ) h−1/2

as h → 0,

we recover the asymptotic expansion announced in Theorem 1.3 in the present case. Proof of Theorem 1.3. In this case α =

1 2

h1/2 |λ − Θ(γ0 )| ≤ ζ0 h

and with 0 < <

1 . 2

5 1 Taking ρ = /2, then we may choose δ ∈ ] 12 , 2 [ such that (4.11) is satisﬁed. Thus, the asymptotic formula of Theorem 4.6 is still valid in this regime, and the leading

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order term of the integrand is, up to a multiplicative constant, " h−1/2 # # [λ −Θ(γ(s))]+ . # 1 π $ , γ(s) d2 2 This proves the theorem. Proof of Theorem 1.5. Again, the proof follows from Theorem 4.6 and the properties of the function Θ(·). Acknowledgments The author would like to express his thanks to R. Frank for the fruitful discussions around the subject, and also to B. Helﬀer for his helpful remarks. He wishes also to thank the anonymous referees for their careful reading of the paper, for pointing out many corrections and for their many helpful suggestions. This work has been partially supported by the European Research Network “Postdoctoral Training Program in Mathematical Analysis of Large Quantum Systems” with contract number HPRN-CT-2002-00277 and by the ESF Scientiﬁc Programme in Spectral Theory and Partial Diﬀerential Equations (SPECT). Appendix A. Boundary coordinates We recall now the deﬁnition of the standard coordinates that straightens a portion of the boundary ∂Ω. Given t0 > 0, let us introduce the following neighborhood of the boundary, Nt0 = {x ∈ R2 ; dist(x, ∂Ω) < t0 }.

(A.1)

|∂Ω| ]− |∂Ω| 2 , 2 ]

As the boundary is smooth, let s ∈ → M (s) ∈ ∂Ω be a regular parametrization of ∂Ω that satisﬁes:   s is the oriented “arc length” between M (0) and M (s). T (s) := M (s) is a unit tangent vector to ∂Ω at the point M (s).  The orientation is positive, i.e. det(T (s), ν(s)) = 1. We recall that ν(s) is the unit outward normal of ∂Ω at the point M (s). The scalar curvature κr is now deﬁned by: T (s) = κr (s)ν(s).

(A.2)

When t0 is suﬃciently small, the map: Φ : ]−|∂Ω|/2, |∂Ω|/2] × ]−t0 , t0 [ (s, t) → M (s) − tν(s) ∈ Nt0 ,

(A.3)

is a diﬀeomorphism. For x ∈ Nt0 , we write, Φ−1 (x) := (s(x), t(x)),

(A.4)

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where t(x) = dist(x, ∂Ω) if x ∈ Ω

and t(x) = −dist(x, ∂Ω) if x ∈ Ω.

The Jacobian of the transformation Φ−1 is equal to, a(s, t) = det(DΦ−1 ) = 1 − tκr (s).

(A.5)

To a vector ﬁeld A = (A1 , A2 ) ∈ H 1 (R2 ; R2 ), we associate the vector ﬁeld = (A 1 , A 2 ) ∈ H 1 (]−|∂Ω|/2, |∂Ω|/2] × ]−t0 , t0 [; R2 ) A by the following relations: 1 (s, t) = (1 − tκr (s))A(Φ(s, A t)) · M (s),

2 (s, t) = A(Φ(s, A t)) · ν(s).

(A.6)

We get then the following change of variable formulae. 1 Proposition A.1. Let u ∈ HA (R2 ) be supported in Nt0 . Writing u (s, t) = u(Φ(s, t)), then we have:

Ω

2

|(∇ − iA)u| d x =

|∂Ω| 2

− |∂Ω| 2

t0 0

1 ) [|(∂s − iA u|2 + a−2 |(∂t − iA˜2 ) u|2 ]a dsdt, (A.7)

Ωc

|(∇ − iA)u|2 dx =

|∂Ω| 2

− |∂Ω| 2

0

−t0

1 ) 2 ) [|(∂s − iA u|2 + a−2 |(∂t − iA u|2 ]a dsdt, (A.8)

and

2

R2

|u(x)| dx =

|∂Ω| 2

− |∂Ω| 2

t0

−t0

| u(s, t)|2 a dsdt.

(A.9)

We have also the relation: 2 − ∂t A 1 )a−1 ds ∧ dt, (∂x1 A2 − ∂x2 A1 )dx1 ∧ dx2 = (∂s A which gives, curl(x1 ,x2 ) A = (1 − tκr (s))−1 curl(s,t) A. We give in the next proposition a standard choice of gauge. 1 Proposition A.2. Consider a vector field A = (A1 , A2 ) ∈ Cloc (R2 ; R2 ) such that

curl A = 1

in R2 .

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For each point x0 ∈ ∂Ω, there exist a neighborhood Vx0 ⊂ Nt0 of x0 and a smooth real-valued function φx0 such that the vector field Anew := A − ∇φx0 satisfies in Vx0 :

and,

2 = 0, A new

(A.10)

1 = −t 1 − t κr (s) , A new 2

(A.11)

1 , A 2 ). new = (A with A new new

References [1] V. Bonnaillie, On the fundamental state energy for a Schr¨ odinger operator with magnetic ﬁeld in domains with corners, Asymptot. Anal. 41(3–4) (2005) 215–258. [2] Y. Colin de Verdi`ere, L’asymptotique de Weyl pour les bouteilles magn´etiques, Comm. Math. Phys. 105 (1986) 327–335. [3] M. Dauge and B. Helﬀer, Eigenvalues variation I, J. Diﬀerential Equations 104 (1993) 243–262. [4] S. Fournais and B. Helﬀer, Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian, Ann. Inst. Fourier. 56(1) (2006) 1–67. [5] R. Frank, On the asymptotic number of edge states for magnetic Schr¨ odinger operators, Proc. London Math. Soc. (3) 95(1) (2007) 1–19. [6] P. G. de Gennes, Superconductivity of Metals and Alloys (Benjamin, New York, 1966). [7] B. Helﬀer and A. Morame, Magnetic bottles in connection with superconductivity, J. Funct. Anal. 181(2) (2001) 604–680. [8] V. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics, Springer Monographs in Mathematics (Springer-Verlag, Berlin, 1998). [9] A. Kachmar, On the ground state energy for a magnetic Schr¨ odinger operator and the eﬀect of the de Gennes boundary condition, C. R. Math. Acad. Sci. Paris 332 (2006) 701–706. [10] A. Kachmar, On the ground state energy for a magnetic Schr¨ odinger operator and the eﬀect of the De Gennes boundary condition, J. Math. Phys. 47(7) (2006) 072106, 32 pp. [11] A. Kachmar, On the stability of normal states for a generalized Ginzburg–Landau model, Asymptot. Anal. 55(3–4) (2007) 145–201. [12] A. Kachmar, On the perfect superconducting solution for a generalized Ginzburg– Landau equation, Asymptot. Anal. 54(3–4) (2007) 125–164. [13] A. Kachmar, Magnetic Ginzburg–Landau functional with discontinuous constraint, C. R. Math. Acad. Sci. Paris 346(5–6) (2008) 297–300. [14] A. Kachmar, Probl`emes aux limites issus de la supraconductivit´e, Ph.D. thesis, University Paris-Sud/Orsay (2007); www.math.u-psud.fr/∼kachmar. [15] T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1995). [16] A. Persson, Bounds for the discrete part of the spectrum of a semi-bounded Schr¨ odinger operator, Math. Scand. 8 (1960) 143–153.

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[17] M. Reed and B. Simon, Methods of Modern Mathematical Physics VI: Analysis of Operators (Academic Press, New York, 1979). [18] H. Tamura, Asymptotic distribution of eigenvalues for Schr¨ odinger operators with magnetic ﬁelds, Nagoya Math. J. 105 (1987) 49–69. [19] F. Truc, Semiclassical asymptotics for magnetic bottles, Asymptot. Anal. 15(3–4) (1997) 385–395.

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Reviews in Mathematical Physics Vol. 20, No. 8 (2008) 933–949 c World Scientific Publishing Company

EXISTENCE OF ASYMPTOTIC EXPANSIONS IN NONCOMMUTATIVE QUANTUM FIELD THEORIES

C. A. LINHARES∗,‡ , A. P. C. MALBOUISSON† and I. RODITI†,§ ∗Instituto de F´ ısica, Universidade do Estado do Rio de Janeiro, Rua S˜ ao Francisco Xavier, 524, 20559-900 Rio de Janeiro, R.J., Brazil †Centro

Brasileiro de Pesquisas F´ısicas, Rua Dr. Xavier Sigaud, 150, 22290-180 Rio de Janeiro, R.J., Brazil ‡[email protected] §[email protected] Received 7 March 2007 Revised 27 June 2008

Starting from the complete Mellin representation of Feynman amplitudes for noncommutative vulcanized scalar quantum field theory, introduced in a previous publication, we generalize to this theory the study of asymptotic behaviors under scaling of arbitrary subsets of external invariants of any Feynman amplitude. This is accomplished in both convergent and renormalized amplitudes. Keywords: Feynman amplitudes; Mellin representation; noncommutative field theory. Mathematics Subject Classification 2000: 81T18, 81T75

1. Introduction The possibility of studying both the ultraviolet and infrared behaviors of Feynman amplitudes in quantum ﬁeld theories, obtained directly without the need of ﬁrst calculating explicitly the complete expressions for them, is a subject that is still ﬁnding new applications. In particular, several groups, working in a wide range of theories that goes from QCD phenomenology to supersymmetric Yang–Mills theory, have been very recently employing various techniques involving asymptotic expansions based on the Mellin–Barnes transform [1–5]. The establishment on a rigorous basis of the determination of asymptotic behaviors of Feynman amplitudes in more general frameworks is bound to be useful for further developments. This is the case, for instance, of noncommutative ﬁeld theories, which is the subject of our present interest. In this paper, we investigate the existence of asymptotic expansions for noncommutative theories known in the literature as of the “vulcanized” type, that is, 933

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those which incorporate suitable modiﬁcations in order to avoid the occurrence of the ultraviolet-infrared divergence mixing, and thus become renormalizable [6–11]. The direct approach to asymptotic behaviors was formulated for commuting theories in the 1970’s in the papers [12–14] within the Bogoliubov–Parasiuk–Hepp– Zimmermann renormalization scheme. It is based on the Feynman–Schwinger parametric representation of amplitudes, expressed in terms of Symanzik polynomials in the Schwinger parameters [15, 16]. However, in vulcanized noncommutative theories, propagators are based on the Mehler kernel, instead of the heat kernel of commutative theories. This leads to propagators that are quadratic in the position space, so that the noncommutative parametric representation involves integration over position and momentum variables, which can be performed. It results that one obtains hyperbolic polynomials in the Schwinger parameters, not just the Symanzik polynomials of the commutative case [17, 18]. See also the reviews [19–21]. In [12, 13, 22], the Mellin transform technique was applied in order to prove theorems implying the existence of asymptotic expansions of the amplitudes and in [23] the concept of “FINE” polynomials was introduced, that is, those having the property of being factorizable in each Hepp sector [24] of the variables (a complete ordering of the Schwinger parameters). Under scaling by a parameter λ of (at least a few of) external invariants associated to a diagram, the Mellin transform with respect to this scaling parameter leads, as λ is taken to inﬁnity, to asymptotic series in powers of λ and powers of logarithms of λ. This was possible because for amplitudes having the FINE property the Mellin transform may be “desingularized”, which means that the integrand of the inverse Mellin transform, which gives back the Feynman amplitude as a function of λ, has a meromorphic structure, so that the residues of its various poles generate the asymptotic expansion. However, this is not the case under arbitrary scaling, as the FINE property simply does not occur in many diagrams. For those non-FINE diagrams, it was introduced in [23] the so-called “multiple Mellin” representation, which consists in splitting the Symanzik polynomials in a certain number of pieces, each one of which having the FINE property. Then, after scaling by the parameter λ, an asymptotic expansion can be obtained as a sum over all Hepp sectors. This is always possible to do if one adopts, as was done in [25–27], the extreme point of view to split the Symanzik polynomials in all its monomials, which leads to the so-called “complete Mellin” (CM) representation. The CM representation provides a general framework to the study of asymptotic expansions of Feynman amplitudes. Moreover, the integrations over the Schwinger parameters can be explicitly performed without any division of the integral into Hepp sectors, and we are left with the pure geometrical study of convex polyhedra in the Mellin variables [25]. Also, the CM representation allows a uniﬁed treatment of the asymptotic behavior of both ultraviolet convergent and divergent amplitudes. This happens because, as shown in [25, 26], the renormalization procedure does not alter the algebraic structure of integrands in the CM representation. It only changes

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the set of relevant integration domains in the Mellin variables. The method allows the study of dimensional regularization [26, 27] and of the infrared behavior of amplitudes relevant to critical phenomena [28, 29]. With the CM representation, one is also able to prove the existence of asymptotic expansions for most useful commutative ﬁeld theories, including gauge theories in an arbitrary gauge [30]. In what regards noncommutative ﬁeld theories, one expects that an adaptation of the general results of all these references could be developed. In fact, recently [31], the CM representation has been extended to the “vulcanized” noncommutative φ4 massless theory and a proof of dimensional meromorphy of its Feynman amplitudes has been presented. Our choice of a massless theory is due to the fact that the CM representation becomes less explicit and less appealing in the massive model. In any case, masses are not essential for vulcanized noncommutative ﬁeld theories which have no “infrared divergences” and only “half-a-direction” for their renormalization group. Based on [31], we intend to show in the present paper that asymptotic expansions exist for this noncommutative theory, in a similar way as the analogous result for the respective commutative theory. We also study explicitly the case of divergent noncommutative amplitudes in the CM representation, by adapting to this context the renormalization procedure of subtraction of suitably truncated Taylor expansions of amplitude integrand functions along the lines of [12, 25, 26, 32]. We ﬁnd that the renormalization procedure in the CM representation, as already mentioned for commutative theories, also does not alter the algebraic structure of integrands for the noncommutative Feynman amplitudes, only the set of relevant integration domains in the Mellin variables changes. This allows to transpose to divergent Feynman integrals the machinery used in the convergent case and prove the existence of asymptotic expansions for renormalized amplitudes. The paper is organized as follows. In Sec. 2, we very brieﬂy recall the main features of the complete Mellin representation for commutative scalar theories. Next, in Sec. 3, we review the CM representation for the vulcanized φ4 theory. In Secs. 4 and 5, we present the generalizations to the noncommutative theory of the respective theorems on the existence of the asymptotic expansions for the convergent and renormalized amplitudes. In the last section, we summarize our conclusions. 2. Complete Mellin Representation in the Commutative Scalar Case Let us ﬁrst consider the simpler case of a Feynman amplitude in a commutative massive scalar theory. The amplitude related to an arbitrary diagram G, with I internal lines, V vertices, and L loops, reads in d spacetime dimensions, AG = CG 0

I ∞

dα

=1 dL/2

(4π)

U d/2 (α)

e−

P

α m2 −N (sk ;α)/U(α)

e

,

(1)

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where CG is a constant, U and N are homogeneous polynomials in the α variables, known in the literature as the Symanzik polynomials, which are written as I I uj α ≡ Uj , N (α) = sk αn k ≡ Nk , (2) U (α) = j

j

=1

k

=1

k

where j runs over the set of 1-trees and k over the set of 2-trees of the diagram G; sk are O(d)-invariants given by the square of the sum of all external momenta at one of the components of the 2-tree k; also, 0 if the line belongs to the 1-tree j, (3) uj = 1 otherwise, and nk =

0 if the line belongs to the 2-tree k, 1 otherwise.

(4)

The complete Mellin representation for AG , following the steps shown in [25, 26, 30], is given by Γ(−xj ) y j   skk Γ(−yk ) (m2 )−φ Γ(φ ), (5) AG (sk , m2 ) = δ k Γ − xj  j

where φ =

uj xj +

j

nk yk + 1.

(6)

k

Im x yk The symbol δ means integration over the independent variables 2πi j , Im 2πi in the convex domain δ deﬁned by (σ and τ standing respectively for Re xj and Re yk )   d     σ < 0; τk < 0; xj + yk = − ;   j   2 j k . (7) δ = σ, τ ∀i, Re φ =     u σ + n σ + 1 > 0   i ij j ik k   j

k

This domain δ is nonempty as long as d is positive and small enough so that every subdiagram of G has a convergent power counting [25]; hence, in particular for the φ4 theory it is always nonempty for any diagram for 0 < d < 2. Let us denote collectively by ζµ the arguments of the Γ-functions: −xj , −yk , φ . Also we call collectively tµ the set of invariants sk and the squared masses m2 . A general asymptotic regime is then deﬁned as the scaling tµ → λbµ tµ , in such a way

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that the amplitude (5) becomes a function of λ written in the convenient form [25] 1 µ , t−ζ (8) AG (λ) = λζ µ Γ(ζµ )  δ µ Γ − xj  j

with ζ = µ bµ ζµ . This representation can be extended to complex values of d. For instance, for a massive φ4 diagram, it is analytic in d for Re d < 2 and meromorphic in d in the whole complex plane with singularities at rational values; furthermore, its dimensional analytic continuation has the same unchanged CM integrand but translated integration contours. Also, it is valid without change in the form of the integrand for renormalized amplitudes [25, 26]. Using the meromorphic properties of the integrand of Eq. (8), an asymptotic expansion in powers of λ and powers of logarithms of λ is obtained for AG (λ) in [25]. 3. Complete Mellin Representation for Noncommutative Scalar Theories In order to establish notation, we review in this section the results of [31], which we take as the starting point of the study of asymptotic behaviors and renormalization, to be developed in the following sections, and which constitutes the main subject of the present paper. According to the analysis exposed in [17], the amplitude related to a ribbon diagram G with L internal lines, by choosing a particular root vertex V¯ , has a parametric representation in terms of the variable t = tanh α /2, where α is the former Schwinger parameter, as 1 Ω dt (1 − t2 )d/2−1 dx dp exp − XGX t , AG ({xe }, pV¯ ) = KG (9) 2 0

where KG is a constant, d is the spacetime dimension, Ω is the Grosse–Wulkenhaar vulcanization coeﬃcient, X summarizes all positions and hypermomenta and G is a certain quadratic form. Calling xe and pV¯ the external variables and xi , pi the internal ones, we decompose G into an internal quadratic form Q, an external one M and a coupling part P , so that M P G= . (10) X = (xe pV¯ xi pi ), Pt Q Performing the Gaussian integration over all internal variables, one gets the noncommutative parametric representation given by 1 e−HVG,V¯ (t,xe ,pV¯ )/HUG,V¯ (t) dt (1 − t2 )d/2−1 , (11) AG ({xe }, pV¯ ) = KG [HUG,V¯ (t)]d/2 0 where new polynomials, in the t variables ( = 1, . . . , L), HUG,V¯ and HVG,V¯ , have been introduced, which are the analogs of the Symanzik polynomials U and N of

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the commutative case. It has been shown in [17] that for the Grosse–Wulkenhaar φ4 model we have HUG,V¯ = s2g−kKU n2KU t t KU =I∪J; n+|KU | odd

=

aK U

KU

uKU

≡

t

∈I /

∈J

HUKU ,

(12)

KU

where I is a subset of the ﬁrst L indices, with |I| elements, and J a subset of the next L indices, with |J| elements; s = 1/4ΘΩ is a constant containing the noncommutative parameter Θ and the vulcanization coeﬃcient Ω; g is the genus of the diagram, aKU = s2g−kKU n2KU , kKU = |KU | − L − F − 1, F being the number of faces of the diagram; nKU = Pf(BKˆ U ), where B is the antisymmetric part of the quadratic form Q restricted by omitting hypermomenta, so nKU is the Pfaﬃan of the antisymmetric matrix obtained from B by deleting the lines and columns in the set KU = I ∪ J; ﬁnally,  / J,   0 if ∈ I and ∈ uKU = 1 if ∈ (13) / I and ∈ / J,   2 if ∈ / I and ∈ J. The second polynomial HV has both a real part HV R and an imaginary part / I which HV I . We need to introduce besides I and J as above a particular line τ ∈ is the analog of a 2-tree cut. Then it is shown in [17] that  2 R t t  xe1 Pe1 τ KV τ Pf(BKˆ V τˆ ) HVG, ¯ = V ∈J

KV =I∪J ∈I /

=

sR KV

KV

L

vKV

t

≡

sR KV

HVKRV ,

(14)

KV

=1

where

τ ∈K / V

e1

 2 = xe Peτ KV τ Pf(BKˆ V τˆ )

(15)

τ ∈K / V

e

and vKV is given by the same formula as uKU . The imaginary part involves pairs of lines τ , τ and corresponding signatures [17, 18]: I HVG, = t t KV Pf(BKˆ V ) ¯ V KV =I∪J ∈I /



∈J

    Pe1 τ KV τ τ Pf(BKˆ V τˆτ )Pe2 τ  xe1 σxe2  × e1 ,e2

=

KV

sIKV

τ,τ

L =1

vK t V

≡

KV

HVKI V ,

(16)

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where



sIKV = KV Pf(BKˆ V ) 

  Peτ KV τ τ Pf(B ˆ

e,e



KV τˆτ )Pe τ

τ,τ

939



 xe σxe  ,

(17)

where σ = σ02 σ02 and σ2 is the second Pauli matrix. The main diﬀerences of the noncommutative parametric representation with respect to the commutative case are the presence of the constants aKU in HU (which contains the noncommutative quantity s = 1/4ΘΩ), the presence of the imaginary part iHV I in HV , and the fact that the parameters uj and vk in the formulas above can have also the value 2 (and not only 0 and 1). In order to proceed, we now introduce the Mellin parameters. For the real part HV R of HV , we use the identity [31] e

R −HVK V

/HUKU

=

R Γ −yK V

R τK

V

where

R τK V

is a short notation for

+∞

HVKRV HUKU

R d(Im yK ) V , 2π −∞

R y K

V

,

(18)

R R with Re yK ﬁxed at τK < 0. V V

However, for the imaginary part one cannot apply anymore the same identity. It nevertheless remains true in the sense of distributions. More precisely, we have for I < 0 (see [31]) HVKRV /HUKU > 0 and −1 < τK V e

I −HVK /HUKU V

=

I τK

I Γ −yK V

V

i HVKI V HUKU

I y K

V

,

(19)

which introduces another set of Mellin parameters. The distributional sense of the formula above is a major diﬀerence with respect to the commutative case. For the polynomial HU one can use the formula [31] P R I d xK − K ( yK +yK −d/2 ) V V V (HUKU ) y KV + = Γ(−xKU )HUKUU . (20) Γ 2 σ KV

KU

As in the commutative case, we now insert the distribution formulas (18)–(20) into the general form of the Feynman amplitude. This gives xK aKUU Γ(−xKU ) R R yK KU R V Γ −yKV sK V AG = KG ∆ KV Γ − xKU KU

I I yK I V Γ −yKV × sK V KV

L 1

0 =1

dt (1 − t2 )d/2−1 tφ −1 ,

(21)

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where φ ≡

uKU xKU +

KU

R I y R + vK yI vK + 1. V KV V KV

(22)

KV

Im y R Im y I Im x Here ∆ means integration over the variables 2πiKU , 2πiKV and 2πiKV , where ∆ is the convex domain   R I σKU < 0; τK < 0; −1 < τK < 0;   V V         R   d   I   ; y = − x + + y   K U K K V V   2     KU KV R I (23) ∆ = σ, τ , τ  ∀, Re φ =  uKU xKU         KU         R R I I   v + 1 > 0 + y + v y   K K K K V V V V   K V

R I and σ, τ R and τ I stand for Re xKU , Re yK and Re yK . The t integrations in V V (21) may be performed using the representation for the beta function

d φ Γ 2 2 . = φ + d 2Γ 2

1

0

dt (1 − t2 )d/2−1 tφ −1 =

φ d 1 B , 2 2 2

Γ

(24)

The representation is convergent for 0 < Re d < 2. Therefore, we can claim that any Feynman amplitude of a φ4 diagram is analytic at least in the strip 0 < Re d < 2, where it admits the following CM representation [31] AG = KG ∆

KU

xK

aKUU Γ(−xKU )

Γ −

xKU

R yK R R V sK V Γ −yKV KV

KU

  L Γ φ Γ d yI I  2 2   , × sIKV KV Γ −yK V  φ + d  KV =1 2Γ 2

(25)

which holds as a tempered distribution of the external invariants. We have thus obtained the complete Mellin representation of Feynman amplitudes for a noncommutative quantum ﬁeld theory. The beta functions, which result from the t -integrations, lead to the appearance of gamma functions that were not present in the commutative case. We will comment about this in the next section.

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4. Asymptotic Expansions for Convergent Amplitudes I A general asymptotic regime is deﬁned by scaling the invariants sR KV , sKV and aKU , bKV R sR sK V KV → λ

sIKV → λcKV sIKV aK U → λ

dKU

(26)

aK U ,

where bKV , cKV and dKU may have positive, negative or null values, and letting λ go to inﬁnity. We then obtain under these scalings xK aKUU Γ(−xKU ) R yK R KU R V sK V Γ −yKV AG (λ) = KG ∆ KV Γ − xKU KU

  L Γ φ Γ d yI I  2 2    λψ , sIKV KV Γ −yK × V  φ + d  KV =1 2Γ 2 where the exponent of λ is a linear function of the Mellin variables: R I ψ= b KV y K + cK V y K + dKU xKU . V V

(27)

(28)

KV ,KU

d φ +d #−1 ! "L in the integrand of Eq. (27) does Notice that the factor =1 Γ 2 2Γ 2 not aﬀect the meromorphic structure of the amplitude (27). Moreover, for strictly positive dimensions d > 0 and φ ∈ ∆, this factor also does not introduce zeroes in the integrand. From the above expressions, we can show that the proof of the theorem given in [25] can be extended for the noncommutative case. To do this, let us rewrite the above expression for AG (λ) in a convenient way. Let us denote collectively the variables {xKU , yKV } as {zK }, whereas the arguments of the gamma functions R I , −yK , and φ2 will be renamed ψν (zK ). The leading to singularities, −xKU , −yK V V convex domain ∆ can then be rewritten simply as ∆ = {zK such that Re ψν (zK ) > 0, for all ν} .

(29)

Let us deﬁne the set of quantities {sν } such that it includes the quantity aKU , which are functions of the objects Θ and Ω having no correspondents in ordinary commutative ﬁeld theory,   aKU if zK = xKU     R  if zK = yK  sR KV V (30) sν = sI I if zK = yKV  KV     φ  1 if zK = . 2

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Also, we introduce the factors fν (in general functions of the variables xKU and yKV ) such that  $ %−1 L  d φ + d  Γ 2Γ fν = 2 2  =1  1

if ψν = φ /2

(31)

otherwise.

Therefore the expression for AG (λ) in (27) can be simpliﬁed to 1 ν . λψ fν s−ψ Γ(ψν ) AG (λ) = KG ν ∆ ν Γ − xKU

(32)

KU

Equation (32) has exactly the same singularity structure as (8), the factors fν only modify the residues at the poles. Thus we can translate to the present situation all the steps of the proof of the asymptotics theorem of [25], since it relies entirely on displacements of the integration contours crossing the singularities of the gamma functions Γ(ψν ) (Γ(ζµ ) in the commutative counterpart of Eq. (8)). For completeness, this demonstration is given in the Appendix. Thus the result of [25] remains valid mutatis mutandis for the the vulcanized φ4 theory and we are allowed to state the following theorem: Theorem 4.1. Let us consider a ribbon diagram G of the vulcanized φ4 theory, and its related amplitude AG (λ) under the general scaling of its invariants, bKV R sK V sR KV → λ

sIKV → λcKV sIKV aKU → λdKU aKU ,

(33)

where bKV , cKV and dKU may have positive, negative or null values, and as λ → ∞. Then there exists an asymptotic expansion of AG (λ) of the form AG (λ) =

−∞

qmax (p)

p=pmax

q=0

q I p Apq (sR KV , sKV , aKU ) λ ln λ,

(34)

where p runs over the rational values of a decreasing arithmetic progression, with pmax as a “leading power”, and q, for a given p, runs over a finite number of nonnegative integer values. I The coeﬃcients Apq (sR KV , sKV , aKU ) of the expansion in (34) are functions only of the invariants associated to the hyperbolic polynomials. Notice, in particular, that the invariants aKU contain the noncommutative entities Θ and Ω encoded.

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943

5. Renormalized Amplitudes Let us now consider the complete Mellin representation of divergent amplitudes. The analysis follows the steps taken in [25, 26] for the commutative case, and the recent results of [32] for noncommutative theories. We have to go back to the amplitude given in Eq. (11), for the occurrence of ultraviolet divergences in an expression such as this one prevents the interchange of the integral over the t variables with the one over the domain ∆. It means that the t -integral cannot be performed and a renormalization prescription is therefore required. For this, we use the method of subtracting the ﬁrst terms of a generalized Taylor expansion corresponding to the inﬁnities of the divergent subdiagrams [12, 13], as adapted to the t integrations [32]. Each t variable belonging to a divergent subdiagram S is scaled by a parameter ρ2 , t∈S → ρ2 t∈S , and the integral in Eq. (11) becomes a function of ρ, which we call g(ρ). Next, following the steps of [12], we deﬁne the generalized Taylor operator of order n, τ n [ρν g(ρ)] = ρν Tρn−E[ν] [g(ρ)],

(35)

with E[ν] being the smallest integer greater than ν, and Tρq [g(ρ)] being the (truncated) usual Taylor operator over a function g(ρ), Tρq [g(ρ)] =

q ρk k=0

k!

g (k) (0),

(36)

which makes sense only for q ≥ 0. The generalized Taylor operator acts on the t integrand, so that for each primitively divergent subdiagram S of G one associates a subtraction operator τS−2lS , where lS is the number of lines in the subdiagram S. The τS operator is equivalent to the introduction of counterterms in the theory, in that it is deﬁned in order to suppress the ultraviolet divergent terms from the integrand; the t variables associated to the subdiagram S are ﬁrst scaled by the parameter ρ, and the ﬁrst few terms of the generalized Taylor expansion in ρ are kept in τS . In fact, this corresponds to the Taylor operator in Eq. (36), truncated at the order q, which is the superﬁcial degree of convergence (the negative of the superﬁcial degree of divergence) of the subdiagram S, q = dLS − 2lS , LS being the number of loops of S. At the end of the computation, one takes ρ = 1. Now, a crucial point, argued in [32] is that since one is interested in the region of " ultraviolet divergences, the factor dt (1 − t2 )d/2−1 in Eq. (11) can be bounded in such a way that it cannot contribute to divergences and so it is included in the integration mesure. This factor plays exactly the same rˆ ole of the integration " measure dα exp (− m α ) in the massive commutative case. Thus the action of the generalized Taylor operator on the integrand is    −HVG /HUG −HVG /HUG e e  . (37) = τ −2lS  τS−2lS d/2 d/2 HUG HUG 2 tS →ρ tS ρ=1

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The renormalized amplitude is deﬁned by introducing the operator R [12], R=

(1 − τS−2lS ),

(38)

S

which satisﬁes the identity [32]

R = 1+

(−τS−2lS ) =

F S∈F

1 − τS−2lS ,

(39)

S

where F is the set of all nonempty forests of primitively divergent subdiagrams. Then the renormalized amplitude is Aren ¯) G ({xe }, pV

=K

0

1

& dt (1 −

t2 )d/2−1 R

e−HVG,V¯ (t,xe ,pV¯ )/HUG,V¯ (t) ! #d/2 HUG,V¯ (t)

' . (40)

Now, within the context of the complete Mellin representation, we have, from (21), Aren G = KG ∆

xK

aKUU Γ(−xKU ) R yK R KU R V sK V Γ −yKV KV Γ − xKU KU

I I yK I V × Γ −yKV sK V

L 1

0 =1

KV

! # dt (1 − t2 )d/2−1 R tφ −1 .

(41)

This is the analogous of the starting point of the analysis of [26] on renormalized amplitudes in the complete Mellin representation. We see that the renormalization operator R acts on the t -variables, exactly in the same way as it acts on the αl variables in the commutative situation of [25, 26], the only diﬀerence (which does not aﬀect the validity of the theorems in [26]) being in the integration measure over the t -variables. Therefore the theorems stated in [26] remain valid in the vulcanized noncommutative case. Then we can follow the same steps as in [26], that is, we deﬁne cells C such that inf inf Re(φ ) ≤ 0. (42) sup Re(φ ) > 0 , ∀; C

S

C

∈S

The eﬀect of the R operator in Eq. (41) is to split the factor R(tφ −1 ) into a set of terms {µC tφ −1 , φ ∈ C}, where µC are numerical coeﬃcients. This allows the t integral to be evaluated just as in the convergent case. The renormalized amplitude

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in the CM representation is then given by R I Aren = K µ IC (xKU , yK , yK ), G C G V V where we have deﬁned the integrands xK aKUU Γ(−xKU ) IC =

Γ −

(43)

∆C

C

KU

945

R R yK R V Γ −yKV sK V KV

xKU

KU

L I φ y I Γ × sIKV KV Γ −yK . V 2 KV

(44)

=1

We now have a set of integration domains given by   σKU < 0; τ R < 0; −1 < τ I < 0;   KV KV           d   R I   ; y = − x + + y   KU KV KV   2     KU KV R I , ∆C = σ, τ , τ ∈ C   ∀, Re φ = uKU xKU         KU         R R I I   v + 1 > 0 + y + v y   KV KV KV KV   K

(45)

V

instead of the single one (∆) of the convergent amplitude. As in the commutative case, we see that the renormalization procedure only changes the relevant integration domains in the Mellin variables. The structure of the integrands IC in the cells C remains exactly of the same form as for convergent amplitudes. This then implies that we can apply in each cell the machinery used in the previous section and we can state the following theorem: R I , yK ) Theorem 5.1. Under the scaling of Eq. (33), each integral ∆C IC (xKU , yK V V has an asymptotic expansion of the same form of the the one of Eq. (34); therefore the amplitude Aren G (λ) has an asymptotic expansion of the form µC IC (λ), (46) Aren G (λ) = C

with IC (λ) =

−∞

C qmax (p)

p=pC max

q=0

q R I p AC pq (sKV , sKV , aKU )λ ln λ

(47)

and where in each cell C, p runs over the rational values of a decreasing arithmetic progression, with pC max as a “leading power”, and q, for a given p, runs over a finite number of nonnegative integer values.

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R I As in the convergent case, the coeﬃcients AC pq (sKV , sKV , aKU ) of the expansion in IC (λ) are functions only of the invariants associated to the hyperbolic polynomials.

6. Conclusions We have shown in this paper that all the steps in the proofs of the theorems in [25, 26] may be reproduced in the context of the vulcanized noncommutative scalar φ∗4 model. In particular, the proof of the existence of asymptotic expansions for Feynman amplitudes in commutative ﬁeld theories done in [25] may be transposed to the present situation, for both convergent and renormalized amplitudes. The resulting theorems take into account the inﬂuence of the speciﬁcities of the noncommutative generalization of the theory in the details of the proof. In particular, it was crucial to observe that the parameters aKU , within which the noncommutative entities Θ and Ω are encoded, and are of course inexistent in the commutative case, may be deﬁned as part of the “invariants” sν and therefore are related to the meromorphic structure of the amplitude and its asymptotic behavior can be studied. Another diﬀerence with respect to the commutative case is the fact that I the next set of invariants sν , sKV , have real and imaginary parts (sR KV and sKV ), and they contribute separately. Also, as the ﬁeld we are considering is massless, the sν related to the functions φ are trivial. In principle, the explicit calculation of the coeﬃcients of the expansions, in both Theorems 4.1 and 5.1, is possible but, for a general amplitude, is an extremely hard task. Nevertheless, those corresponding to the leading terms can be evaluated (see Appendix) along the same lines as in the commutative case in [25].

Appendix. Proof of Theorem 4.1 In this Appendix we perform a “translation” for the noncommutative theory of the proof of the asymptotics theorem of [25]. In Eq. (32), when ψν ∈ ∆, the integral is absolutely convergent, so we have a ﬁrst bound: AG (λ) < const. λpmax + ;

pmax = inf (Re ψ(zK )), ∆

(48)

where is an arbitrary small number. Therefore, the function ψ(zK ) − pmax is positive in ∆, and reaches zero on its boundary. It then ensues that there exist nonnegative coeﬃcients dν such that ψ(zK ) − pmax = ν dν ψν , which implies dν 1 1 ≡ . ψ(zK ) − pmax ν ψν ψν ν

ν =ν

(49)

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For a given ν, if the subset {ψν , ν = ν} still generates ψ(zK ) − pmax , we can repeat the procedure, which is iterated until we obtain dE 1 1 ≡ . (50) qE +1 (ψ(zK ) − pmax ) ψν ψν E ν

ν∈E

For each E ⊂ {ν}, ψ(zK ) − pmax does not belong to the convex domain deﬁned by the subset {ψν ≥ 0, ν ∈ E} and it becomes negative somewhere in 0 if ν ∈ E (51) ∆E = {zK such that ψν + θνE > 0, for all ν} ; θνE = 1 otherwise. Therefore, the amplitude AG (λ) in Eq. (32) becomes, λψ ME (zK ) AG (λ) = dE , qE +1 ∆E ; Re(ψ(zK )−pmax )>0 (ψ(zK ) − pmax )

(52)

E

where we have deﬁned the function ME (z) =

ν

ν fν s−ψ Γ(ψν + θνE ) ν

Γ −

,

(53)

xKU

KU

which is analytical in ∆E . The integration path can be moved up to a point where ψ(zK ) − pmax < 0, and applying Cauchy’s integral formula we obtain & ' qE λψ ME (zK ) q pmax E dE λ Apmax q ln λ + , (54) AG (λ) = qE +1 ∆E (ψ(zK ) − pmax ) q=0 E where in ∆E , Re (ψ(zK ) − pmax ) < 0 and 1 E Apmax q = ∇qE −q ME (zK ), q!(qE − q)! ∆E ; ψ(zK )−pmax =0

(55)

with ∇ being the diﬀerential operator along any direction crossing the plane ψ = pmax . The integral in the second term of (54) is bounded, being less than a constant times λpmax −bE + , where pmax − bE = Inf ∆E (Re ψ(zK )), in which bE is a strictly positive rational. The remaining gamma-function singularities are treated in a similar fashion, by applying the identity Γ(ψν + θνE ) =

1 Γ(ψν + θνE + 1) ψν + θνE

(56)

in the second term of (54), leading to an analogous term with λpmax −aE , and another integral in the next analiticity strip, and so forth. In this way, a complete asymptotic expansion is produced.

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References [1] S. Friot, D. Greynat and E. de Rafael, Asymptotics of Feynman diagrams and the Mellin–Barnes representation, Phys. Lett. B 628 (2005) 73. [2] J.-Ph. Aguilar, E. de Rafael and D. Greynat, Muon anomaly from lepton vacuum polarization and the Mellin–Barnes representation, Phys. Rev. D 77 (2008) 093010. [3] R. Kaiser and J. Schweizer, The expansion by regions in πk scattering, J. High Energy Phys. 06 (2006) 009, 20 pp. [4] Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower and V. A. Smirnov, Four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric Yang–Mills theory, Phys. Rev. D 75 (2007) 085010. [5] F. Cachazo, M. Spradlin and A. Volovich, Hidden beauty in multiloop amplitudes, J. High Energy Phys. 07 (2006) 007. [6] H. Grosse and R. Wulkenhaar, Power-counting theorem for non-local matrix models and renormalisation, Comm. Math. Phys. 254 (2005) 91–27. [7] H. Grosse and R. Wulkenhaar, Renormalization of φ4 -theory on noncommutative R2 in the matrix base, J. High Energy Phys. 12 (2003) 019. [8] H. Grosse and R. Wulkenhaar, Renormalization of φ4 -theory on noncommutative R4 in the matrix base, Comm. Math. Phys. 256 (2005) 305–374. [9] V. Rivasseau, F. Vignes-Tourneret and R. Wulkenhaar, Renormalization of noncommutative φ4 -theory by multi-scale analysis, Comm. Math. Phys. 262 (2006) 565–594. [10] R. Gurau, J. Magnen, V. Rivasseau and F. Vignes-Tourneret, Renormalization of non-commutative φ44 field theory in x space, Comm. Math. Phys. 267 (2006) 515– 542. [11] F. Vignes-Tourneret, Renormalization of the orientable non-commutative Gross– Neveu model, Ann. H. Poincar´e 8 (2007) 427–474. [12] M. C. Berg`ere and J.-B. Zuber, Renormalization of Feynman amplitudes and parametric integral representation, Comm. Math. Phys. 35 (1974) 113–140. [13] M. C. Berg`ere and Y.-M. P. Lam, Asymptotic expansion of Feynman amplitudes. Part I. The convergent case, Comm. Math. Phys. 39 (1974) 1–32. [14] M. C. Berg`ere and Y.-M. P. Lam, Bogolubov–Parasiuk theorem in the alphaparametric representation, J. Math. Phys. 17 (1976) 1546–1557. [15] N. Nakanishi, Graph Theory and Feynman Integrals (Gordon and Breach, New York, 1971). [16] C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980). [17] R. Gurau and V. Rivasseau, Parametric representation of noncommutative field theory, Comm. Math. Phys. 272 (2007) 811–835. [18] V. Rivasseau and A. Tanas˘ a, Parametric representation of “critical” noncommutative QFT models, Comm. Math. Phys. 279 (2008) 355–379. [19] V. Rivasseau and F. Vignes-Tourneret, Renormalisation of non-commutative field theories, arXiv: hep-th/0702068. [20] V. Rivasseau and F. Vignes-Tourneret, Non-commutative renormalization, in Rigorous Quantum Field Theory: A Festschrift for Jacques Bros, eds. A. B. de Monvel, D. Buchholz, D. Iagolnitzer and U. Moschella (Birkh¨ auser, Basel, 2006), pp. 271–281; arXiv: hep-th/0409312. [21] V. Rivasseau, Non-commutative renormalization, S´emi. Poincar´e 10 (2007) 1–81; arXiv: 0705.0705 [hep-th]. [22] M. C. Berg`ere and Y.-M. P. Lam, Asymtpotic expansion of Feynman amplitudes: Part II — The divergent case, preprint, Freie Universit¨ at, Berlin, HEP (May 1979/9) (unpublished).

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[23] M. C. Berg`ere, C. de Calan and A. P. C. Malbouisson, A theorem on asymptotic expansion of Feynman amplitudes, Comm. Math. Phys. 62 (1978) 137–158. [24] K. Hepp, Proof of the Bogoliubov–Parasiuk theorem on renormalization, Comm. Math. Phys. 2 (1966) 301–326. [25] C. de Calan and A. P. C. Malbouisson, Complete Mellin representation and asymptotic behaviors of Feynman amplitudes, Ann. Inst. Henri Poincar´e 32 (1980) 91–107. [26] C. de Calan, F. David and V. Rivasseau, Renormalization in the complete Mellin representation of Feynman amplitudes, Comm. Math. Phys. 78 (1981) 531–544. [27] C. de Calan and A. P. C. Malbouisson, Infrared and ultraviolet dimensional meromorphy of Feynman amplitudes, Comm. Math. Phys. 90 (1983) 413–416. [28] A. P. C. Malbouisson, A convergence theorem for asymptotic expansions of Feynman amplitudes, J. Phys. A 33 (2000) 3587–3595. [29] A. P. C. Malbouisson, Critical behavior of correlation functions and asymptotic expansions of Feynman amplitudes, in Fluctuating Paths and Fields: Festschrift Dedicated to Hagen Kleinert, eds. W. Janke, A. Pelster, H.-J. Schmidt and M. Bachmann (World Scientific, Singapore, 2001), pp. 259–270. [30] C. A. Linhares, A. P. C. Malbouisson and I. Roditi, Asymptotic expansions of Feynman amplitudes in a generic covariant gauge, to appear in Int. J. Mod. Phys.; arXiv: hep-th/0612010. [31] R. Gurau, A. P. C. Malbouisson, V. Rivasseau and A. Tanas˘ a, Non-commutative complete Mellin representation for Feynman amplitudes, Lett. Math. Phys. 81 (2007) 161–175. [32] R. Gurau and A. Tanas˘ a, Dimensional regularization and renormalization of noncommutative QFT, arXiv:0706.1147 [math-ph].

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Reviews in Mathematical Physics Vol. 20, No. 8 (2008) 951–978 c World Scientific Publishing Company

TIME OPERATORS OF A HAMILTONIAN WITH PURELY DISCRETE SPECTRUM

ASAO ARAI∗ and YASUMICHI MATSUZAWA† Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan ∗[email protected] †[email protected]

Received 7 January 2008 Revised 14 July 2008 We develop a mathematical theory of time operators of a Hamiltonian with purely discrete spectrum. The main results include boundedness, unboundedness and spectral properties of them. In addition, possible connections of a time operator of H with regular perturbation theory are discussed. Keywords: Canonical commutation relation; Hamiltonian; time operator; time-energy uncertainty relation; phase operator; spectrum; regular perturbation theory. Mathematics Subject Classification 2000: 81Q10, 47N50

1. Introduction This paper is concerned with mathematical theory of time operators in quantum mechanics [2–4, 6, 12]. There are some types of time operators which are not necessarily equivalent to each other. For the reader’s convenience, we ﬁrst recall the deﬁnitions of them with comments. Let H be a complex Hilbert space. We denote the inner product and the norm of H by ·, · (antilinear in the ﬁrst variable) and · , respectively. For a linear operator A on a Hilbert space, D(A) denotes the domain of A. Let H be a self-adjoint operator on H and T be a symmetric operator on H. The operator T is called a time operator of H if there is a (not necessarily dense) subspace D = {0} of H such that D ⊂ D(T H) ∩ D(HT ) and the canonical commutation relation (CCR) [T, H] := (T H − HT ) = i

(1.1)

holds on D (i.e. [T, H]ψ = iψ, ∀ψ ∈ D), where i is the imaginary unit. In this case, T is called a canonical conjugate to H too. ∗ Corresponding

author. 951

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The name “time operator” for the operator T comes from the quantum mechanical context where H is taken to be the Hamiltonian of a quantum system and the heuristic classical-quantum correspondence based on the structure that, in the classical relativistic mechanics, time is a canonical conjugate variable to energy in each Lorentz frame of coordinates. Note also that the dimension of T is that of time if the dimension of H is that of energy in the original unit system where the righthand side of (1.1) takes the form i with being the Planck constant h divided by 2π. We remark, however, that this name is somewhat misleading, because, in the framework of the standard quantum mechanics, time is not an observable, but just a parameter assigning the time when a quantum event is observed. But we follow the convention in this respect. By the same reason as just remarked, T is not necessarily (essentially) self-adjoint. But this does not mean that it is “unphysical” [2, 12]. Note also that we do not require the denseness of the subspace D in the deﬁnition stated above. This is more general. In fact, there is an example of the pair (T, H) satisfying (1.1) on a non-dense subsapce D [8, 10]. From a representation theoretic point of view, the pair (T, H) is a symmetric representation of the CCR with one degree of freedom. But one should remember that, as for this original form of representation of the CCR, the von Neumann uniqueness theorem ([13] and [14, Theorem VIII.14]) does not necessarily hold. In other words, (T, H) is not necessarily unitarily equivalent to a direct sum of the Schr¨ odinger representation of the CCR with one degree of freedom. Indeed, for example, it is obvious that, if T or H is bounded below or bounded above, then (T, H) cannot be unitarily equivalent to a direct sum of the Schr¨ odinger representation of the CCR with one degree of freedom. A classiﬁcation of pairs (T, H) with T being a bounded self-adjoint operator has been done by Dorfmeister and Dorfmeister [7]. We remark, however, that the class discussed in [7] does not cover the pairs (T, H) considered in this paper, because the paper [7] treats only the case where T is bounded and absolutely continuous. A weak form of time operator is deﬁned as follows. We say that a symmetric operator T is a weak time operator of H if there is a subspace Dw = {0} of H such that Dw ⊂ D(T ) ∩ D(H) and T ψ, Hφ − Hψ, T φ = ψ, iφ,

ψ, φ ∈ Dw ,

i.e. (T, H) satisﬁes the CCR in the sense of sesquilinear form on Dw . Obviously, a time operator T of H is a weak time operator of H. But the converse is not true (it is easy to see, however, that, if T is a weak time operator of H and Dw ⊂ D(T H) ∩ D(HT ), then T is a time operator). An important aspect of a weak time operator T of H is that a time-energy uncertainty relation is naturally derived [2, Proposition 4.1]: for all unit vectors ψ in Dw ⊂ D(T ) ∩ D(H), (∆T )ψ (∆H)ψ ≥

1 , 2

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where, for a linear operator A on H and φ ∈ D(A) with φ = 1, (∆A)φ := (A − φ, Aφ)φ, called the uncertainty of A in the vector φ. In contrast to the weak form of time operator, there is a strong form. We say that T is a strong time operator of H if, for all t ∈ R, e−itH D(T ) ⊂ D(T ) and T e−itH ψ = e−itH (T + t)ψ,

ψ ∈ D(T ).

(1.2)

We call (1.2) the weak Weyl relation [2]. From a representation theoretic point of view, we call a pair (T, H) obeying the weak Weyl relation a weak Weyl representation of the CCR. This type of representation of the CCR was extensively studied by Schm¨ udgen [17,18]. It is shown that a strong time operator of H is a time operator of H [12]. But the converse is not true. In fact, the time operators considered in the present paper are not strong ones. There is a generalized version of strong time operator [2]. We say that T is a generalized time operator of H if, for each t ∈ R, there is a bounded self-adjoint operator K(t) on H with D(K(t)) = H, e−itH D(T ) ⊂ D(T ) and a generalized weak Weyl relation (GWWR) T e−itH ψ = e−itH (T + K(t))ψ

(∀ψ ∈ D(T ))

(1.3)

holds. In this case, the bounded operator-valued function K(t) of t ∈ R is called the commutation factor of the GWWR under consideration. We now come to the subject of the present paper. In his interesting paper [9], Galapon showed by an explicit construction that, for every self-adjoint operator H (a Hamiltonian) on an abstract Hilbert space H which is bounded below and has purely discrete spectrum with some growth condition, there is a time operator T1 on H, which is a bounded self-adjoint operator under an additional condition (for the deﬁnition of T1 , see (2.12) below). To be deﬁnite, we call the operator T1 introduced in [9] the Galapon time operator. An important point of Galapon’s work [9] is in that it disproved the longstanding belief or folklore among physicists that there is no self-adjoint operator canonically conjugate to a Hamiltonian which is bounded below (for a historical survey, see [9, Introduction]). Motivated by work of Galapon [9], we investigate, in this paper, properties of time operators of a self-adjoint operator H with purely discrete spectrum. In Sec. 2, we introduce a densely deﬁned linear operator T whose restriction to a subspace yields the Galapon time operator T1 and prove basic properties of T and T1 , in particular the closedness of T . It follows that, if T is bounded, then T is self-adjoint with D(T ) = H and a time operator of H. We denote by T # one of T1 , T and T ∗ (the adjoint of T ). In Sec. 3, we discuss some general properties of T # . Moreover, the reﬂection symmetry of the spectrum of T # with respect to the imaginary axis is proved. Sections 4–6 are the main parts of this paper. In Sec. 4, we

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present a general criterion for T to be bounded with D(T ) = H, while, in Sec. 5, we give a suﬃcient condition for T to be unbounded. In Sec. 6, we present a necessary and suﬃcient condition for T to be Hilbert–Schmidt. In Sec. 7, we show that, under some condition, the Galapon time operator is a generalized time operator of H, too. We also discuss non-diﬀerentiability of the commutation factor K in the GWWR for (T1 , H). In the last section, we consider a perturbation of H by a symmetric operator and try to draw out physical meanings of T1 and K in the context of regular perturbation theory. 2. Time Operators In this section, we recapitulate some basic aspects of the Galapon time operator in more apparent manner than in [9]. Let H be a complex Hilbert space and H be a self-adjoint operator on H which has the following properties (H.1) and (H.2): (H.1) The spectrum of H, denoted σ(H), is purely discrete with σ(H) = {En }∞ n=1 , where each eigenvalue En of H is simple and 0 < En < En+1 for all n ∈ N (the set of positive integers). ∞ 1 < ∞. (H.2) E 2 n=1 n Throughout the present paper we assume (H.1) and (H.2). Remark 2.1. The positivity condition En > 0 for the eigenvalues of H does not lose generality, because, if H is bounded below, but not strictly positive, then one ˜ := H + c with a constant c > − inf σ(H), needs only to consider, instead of H, H which is a strictly positive self-adjoint operator. Property (H.2) implies that En → ∞ (n → ∞).

(2.1)

Let en be a normalized eigenvector of H belonging to eigenvalue En : Hen = En en ,

n ∈ N.

(2.2)

Then, by property (H.1), the set {en }∞ n=1 is a complete orthonormal system (C.O.N.S.) of H. Lemma 2.1. (i) For all m ∈ N, ∞ n=m

1 < ∞. (En − Em )2

(2.3)

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In particular, for each m ∈ N, ∞ n=m

1 en En − Em

converges in H. (ii) For all n ∈ N and vectors ψ in H, the inﬁnite series ∞ em , ψ En − Em

(2.4)

m=n

absolutely converges. Proof. (i) By (2.1), we have 1 Cm := sup 2 < ∞. n=m Em 1− En

(2.5)

Hence we have ∞ n=m

∞ 1 1 ≤ C < ∞. m |En − Em |2 En2 n=m

(ii) By the Cauchy–Schwarz inequality, the Parseval equality and part (i), we have  12  1  2 2 ∞ ∞ ∞ em , ψ 1 2   |em , ψ|   En − Em ≤ En − Em m=n

m=n

m=n

1 2 2 ∞ 1  < ∞. ≤ ψ  En − Em 

(2.6)

m=n

By Lemma 2.1(ii), one can deﬁne a linear operator T on H as follows:   2 ∞   e , ψ   m <∞ , D(T ) := ψ ∈ H   E − Em   n=1 m=n n  ∞ e , ψ m   en , T ψ := i E − E n m n=1 ∞

(2.7)



ψ ∈ D(T ).

(2.8)

m=n

Note that

2 ∞ ∞ em , ψ . T ψ2 = E − Em n=1 m=n n

(2.9)

For a subset D ⊂ H, we denote by l.i.h.(D) the subspace algebraically spanned by the vectors of D.

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The subspace D0 := l.i.h.({en }∞ n=1 )

(2.10)

is dense in H. Lemma 2.2. The operator T is densely deﬁned with D0 ⊂ D(T ) and T ek = i

∞ n=k

1 en , En − Ek

k ∈ N.

(2.11)

Proof. To prove D0 ⊂ D(T ), it is suﬃcient to show that ek ∈ D(T ), k ∈ N. Putting ∞ em , ek , En − Em

cn (k) :=

m=n

we have ck (k) = 0 and cn (k) = 1/(En − Ek ) for n = k. Hence, by Lemma 2.1(i), we have ∞ ∞ 1 |cn (k)|2 = < ∞. (En − Ek )2 n=1 n=k

Hence ek ∈ D(T ) and (2.11) holds. In general, it is not clear whether or not T is a symmetric operator. But a restriction of T to a smaller subspace gives a symmetric operator. Indeed, we have the following fact: Lemma 2.3 ([9]). The operator T1 := T |D0

(2.12)

(the restriction of T to D0 ) is symmetric. Proof. It is enough to show that, for all ψ ∈ D0 , ψ, T ψ is real. For a complex number z ∈ C (the set of complex numbers), we denote its complex conjugate by z ∗ . We have ∞ ∞ em , ψ ψ, T ψ = i ψ, en . E n − Em n=1 m=n

Hence ψ, T ψ∗ = i

∞

en , ψ

n=1

∞ ψ, em . Em − En

m=n

Since ψ is in D0 , the double sum on m, n with m = n is a sum consisting of a ﬁnite term. Hence we can exchange the sum on n and that on m to obtain ∞ ∞ en , ψ ψ, em = ψ, T ψ. ψ, T ψ∗ = i Em − En m=1 n=m

Hence ψ, T ψ is real.

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The operator T1 deﬁned by (2.12) is the time operator introduced by Galapon in [9]. Obviously we have T1 ⊂ T.

(2.13)

T ∗ ⊂ T1∗ .

(2.14)

Hence

Remark 2.2. It is asserted in [9] that T1 is essentially self-adjoint without additional conditions. But, unfortunately, we ﬁnd that this is not conclusive, because the proof of it given in [9, pp. 2678–2679] has some gap: the interchange of the double sum in [9, Eq. (2.30), p. 2678] may not be justiﬁed, at least, by the reasoning given there. The assertion is true in the case where T1 becomes a bounded operator under an additional condition for {En }∞ n=1 , as we show below in the present paper. But, in the case where T1 is unbounded, it seems to be very diﬃcult to prove or disprove the essential self-adjointness of T1 . We leave this problem for future study. Lemma 2.4. The subspace Dc := l.i.h.({en − em ∈ H|n, m ≥ 1})

(2.15)

is dense in the Hilbert space H. Moreover, Dc ⊂ D0 ⊂ D(T ).

(2.16)

Proof. Let ψ ∈ D⊥ c (the orthogonal complement of Dc ). Then, for all m, n ≥ 1, |en , ψ|2 = |em , ψ|2 . By the Parseval equality, ψ2 = limN →∞ N |em , ψ|2 . This implies that |em , ψ|2 = 0 for all m ≥ 1 and ψ = 0. Hence ψ = 0. Thus Dc is dense in H. Inclusion relation (2.16) is obvious. Theorem 2.5 ([9]). It holds that Dc ⊂ D(T1 H) ∩ D(HT1 )

(2.17)

ψ ∈ Dc .

(2.18)

and [T1 , H]ψ = iψ,

Theorem 2.5 shows that T1 is a time operator of H. Remark 2.3. It is easy to see that, for all k ∈ N, T1 ek ∈ D(H). Hence D0 ⊂ D(HT1 ). Therefore one cannot consider the commutation relation [T1 , H] on D0 . Moreover, by direct computation, we have T1 ek , He − Hek , T1 e = −i(1 − δk ),

k, ∈ N.

(2.19)

This means that (T1 , H) does not satisfy the CCR in the sense of sesquilinear form on D0 (a weak form of the CCR), either. These facts suggest that the pair (T1 , H) is very sensitive to the domain on which their commutation relation is applied.

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In concluding this section we discuss shortly non-uniqueness of time operators of H. We introduce a set of symmetric operators associated with H: {H}Dc := {S|S is a symmetric operator on H such that Dc ⊂ D(SH) ∩ D(HS) and SHψ = HSψ, ∀ψ ∈ Dc },

(2.20)

which may be viewed as a commutant of {H} in a restricted sense. It is easy to see that, for all real-valued continuous function f on R, the operator f (H) deﬁned via the functional calculus is in {H}Dc . Proposition 2.6. For all S ∈ {H}Dc , Dc ⊂ D((T1 + S)H) ∩ D(H(T1 + S)) and [T1 + S, H]ψ = iψ,

ψ ∈ Dc .

(2.21)

Proof. A direct computation using Theorem 2.5 and (2.20). Proposition 2.7. Let T2 be a time operator of H such that Dc ⊂ D(T2 H)∩D(HT2 ) and [T2 , H]ψ = iψ,

∀ψ ∈ Dc .

Then T2 = T1 + S with some S ∈ {H}Dc . Proof. We need only to show that S := T2 − T1 is in {H}Dc . But this is obvious.

3. General Properties 3.1. Closedness of T and symmetry of T ∗ Lemma 3.1. D0 ⊂ D(T ∗ ) and T ∗ |D0 = T1 , i.e., T1 ⊂ T ∗ . Proof. It is enough to show that, for all k ∈ N, ek ∈ D(T ∗ ) and T ∗ ek = T ek (= T1 ek ). It is easy to see that, for all ψ ∈ D(T ), ek , T ψ = i

∞ em , ψ . Ek − Em

(3.1)

m=k

By Lemma 2.2, the right-hand side is equal to T ek , ψ. Hence ek ∈ D(T ∗ ) and T ∗ ek = T ek . Proposition 3.2. The operator T is closed and T ∗ ⊂ T.

(3.2)

In particular, if T is bounded, then T is self-adjoint with D(T ) = H. Proof. Let ψk ∈ D(T ), k ∈ N and ψk → ψ ∈ H, T ψk → φ ∈ H as k → ∞. Then supk≥1 T ψk < ∞. Hence, by (2.9), there exists a constant C > 0 independent

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of k ∈ N such that

959

2 ∞ e , ψ m k ≤ C. E − Em n=1 n=m n

By (2.6), we have lim

k→∞

Hence it follows that

∞ ∞ em , ψk em , ψ = . En − Em En − Em

n=m

(3.3)

n=m

2 ∞ ∞ em , ψ ≤ C. E − E n m n=1 n=m

Therefore ψ ∈ D(T ). By (3.1) and (3.3), we have for all ∈ N lim e , T ψk = e , T ψ.

k→∞

Hence e , φ = e , T ψ, ∈ N, implying φ = T ψ. Thus T is closed. To prove (3.2), let ψ ∈ D(T ∗ ). Putting η = T ∗ ψ, we have η, χ = ψ, T χ for all χ ∈ D(T ). Taking χ = ek (k ∈ N), we have η, ek = i

∞ ψ, en , En − Ek

(3.4)

n=k

which implies that

2 ∞ ∞ ψ, en = η2 < ∞. En − Ek k=1 n=k

Hence ψ ∈ D(T ). Then, by (3.1), the right-hand side of (3.4) is equal to T ψ, ek . Hence η = T ψ. Thus (3.2) holds. Let T be bounded. Then, by the denseness of D(T ) and the closedness of T , D(T ) = H. Hence D(T ∗ ) = H. Thus, by (3.2), T ∗ = T , i.e. T is self-adjoint. Corollary 3.3. The operator T ∗ is symmetric. Proof. By Lemma 3.1, T ∗ is densely deﬁned. Hence, by Proposition 3.2, T ∗ ⊂ T = (T ∗ )∗ . Thus T ∗ is symmetric. Thus we have T1 ⊂ T ∗ ⊂ T. Corollary 3.3 shows that T ∗ also is a time operator of H. ¯ For a closable operator A on a Hilbert space, we denote its closure by A. Proposition 3.4. T¯1 = T ∗ .

(3.5)

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Proof. Note that T¯1 = T ∗ if and only if T1∗ = T . By (3.5), we have T¯1 ⊂ T ∗ . Hence T ⊂ T1∗ . Thus it is enough to show that D(T1∗ ) ⊂ D(T ). For all ψ ∈ D(T1∗ ), we have ψ, en . T1∗ ψ, el = ψ, T1 el = i En − El n=l

Hence we obtain

2 ∞ ψ, e n ∗ 2 ∗ 2 ∞ > T1 ψ = |T1 ψ, el | = , En − El l=1 l=1 n=l ∞

implying that ψ ∈ D(T ). Thus D(T1∗ ) ⊂ D(T ). 3.2. Absence of invariant dense domains for T under some condition We ﬁrst note the following general fact: Proposition 3.5. Let Q be a bounded self-adjoint operator on H and P be a selfadjoint operator on H. Suppose that there is a dense subspace D in H such that the following (i)–(iii) hold : (i) QD ⊂ D ⊂ D(P ). (ii) D is a core of P . (iii) The pair (Q, P ) obeys the CCR on D : [Q, P ]ψ = iψ, ∀ψ ∈ D. Then σ(P ) = R. Proof. Since Q is a bounded self-adjoint operator, we have for all t ∈ R ∞ (itQ)k eitQ = k! k=0

in operator norm. Conditions (i) and (iii) imply that, for all k ∈ N and ψ ∈ D Qk P ψ − P Qk ψ = ikQk−1 ψ. Hence, for all t ∈ R and vectors ψ in D, we have ∞ (it)k k Q Pψ eitQ P ψ = P ψ + k! k=1

= Pψ +

∞ (it)k k=1

= Pψ +

k!

(P Qk + ikQk−1 )ψ

∞ (itQ)k−1 (itQ)k ψ−t ψ . P k! (k − 1)!

k=1

It follows from the closedness of P that eitQ ψ is in D(P ) and P eitQ ψ = eitQ P ψ + teitQ ψ.

(3.6)

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By condition (ii), this equality extends to all ψ ∈ D(P ) with eitQ ψ ∈ D(P ), ∀t ∈ R, ∀ψ ∈ D(P ). Hence the operator equality e−itQ P eitQ = P + t follows. Thus σ(P ) = σ(P + t) for all t ∈ R. This implies that σ(P ) = R. Theorem 3.6. If T is bounded (hence self-adjoint by Proposition 3.2), then there is no dense subspace D in H such that the following (i)–(iii) hold : (i) T D ⊂ D ⊂ D(H). (ii) D is a core of H. (iii) The pair (T, H) obeys the CCR on D. Proof. Suppose that there were such a dense subspace D as stated above. Then we can apply Proposition 3.5 with (Q, P ) = (T, H) to conclude that σ(H) = R. But this is a contradiction. Remark 3.1. A special case of this theorem was established in [7, Theorem 9.5]. 3.3. Reﬂection symmetry of the spectrum of T1 , T ∗ and T We ﬁrst recall the deﬁnition of the spectrum of a general linear operator (not necessarily closed). For a linear operator A on a Hilbert space K, the resolvent set of A, denoted ρ(A), is deﬁned by ρ(A) := {z ∈ C|Ran(A − z) (the range of A − z) is dense in K and A − z is injective with (A − z)−1 bounded}. Then the set σ(A) := C\ρ(A) is called the spectrum of A. The set of eigenvalues of A, called the point spectrum of A, is denoted σp (A). We denote by T # any of T1 , T ∗ and T . We deﬁne a conjugation J on H by ∞ ψ, en en , ψ ∈ H. (3.7) Jψ := n=1

Proposition 3.7. The spectrum σ(T # ) of T # is reﬂection symmetric with respect to the imaginary axis, i.e., if z ∈ σ(T # ), then −z ∗ ∈ σ(T # ). In particular, if T is self-adjoint, then σ(T ) is reﬂection symmetric with respect to the origin of the real axis. Moreover, for all z ∈ σp (T # ), −z ∗ is in σp (T # ) and J ker(T # − z) = ker(T # + z ∗ ),

∀z ∈ σp (T # ).

(3.8)

Proof. It is easy to see that operator equality JT #J = −T # holds (JD(T # ) = D(T # )). Hence, for all z ∈ C, we have J(T # − z)J = −(T # + z ∗ ) · · · (∗). This implies that, if z ∈ ρ(T # ), then −z ∗ ∈ ρ(T # ). Thus the same holds for the spectrum σ(T # ) = C\ρ(T # ). Equation (3.8) follows from (∗).

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4. Boundedness of T In this section, we present a general criterion for the operator T to be bounded. For mathematical generality and for later use, we consider a more general class of operators than that of T . Let b := {bnm }∞ n,m=1 be a double sequence of complex numbers such that b∞ := sup |bnm | < ∞.

(4.1)

n,m≥1

Then, in the same way as in Lemma 2.1(ii), for all ψ ∈ H, the inﬁnite series ∞ m=n

bnm em , ψ En − Em

absolutely converges. Hence one can deﬁne a linear operator Tb on H as follows:   2 ∞ ∞     bnm , (4.2) D(Tb ) := ψ ∈ H e , ψ < ∞ m   E − Em   n=1 m=n n

Tb ψ := i

∞ n=1

 

∞ m=n

 bnm em , ψ en , En − Em

ψ ∈ D(Tb ).

(4.3)

Obviously T = Tb with b satisfying bnm = 1 for all n, m ∈ N. In the same way, as in the case of T , one can prove the following fact: Lemma 4.1. The operator Tb is closed. The following lemma is probably well known (but, for the completeness, we give a proof): Lemma 4.2. Let A be a densely deﬁned linear operator on a Hilbert space K. Suppose that there exist a dense subspace D in K and a constant C > 0 such that D ⊂ D(A) and |ψ, Aψ| ≤ Cψ2 ,

ψ ∈ D.

¯ ≤ 2C, where A¯ is the closure of A. Then A is bounded with A ¯ ≤ C. If A is symmetric in addition, then A Proof. Let ψ, φ ∈ D. Then, by the polarization identity 1 ψ, Aφ = (ψ + φ, A(ψ + φ) − ψ − φ, A(ψ − φ) 4 + iψ − iφ, A(ψ − iφ) − iψ + iφ, A(ψ + iφ)), we have |ψ, Aφ| ≤

C (ψ + φ2 + ψ − φ2 + ψ − iφ2 + ψ + iφ2 ) 4

= C(ψ2 + φ2 ).

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Replacing ψ = 0 by φψ/ψ we have |ψ, Aφ| ≤ 2Cψφ. For ψ = 0, this inequality trivially holds. Since D is dense, it follows from the Riesz representation theorem that Aφ ≤ 2Cφ, φ ∈ D. Thus the ﬁrst half of the lemma follows. Let A be symmetric. Then, ψ, Aψ ∈ R for all ψ ∈ D(A). Hence 1 |ψ, Aφ| = |ψ + φ, A(ψ + φ) − ψ − φ, A(ψ − φ)| 4 C ≤ (ψ2 + φ2 ), ψ ∈ D. 2 We write ψ, Aφ = |ψ, Aφ|eiθ with θ ∈ R. Then |ψ, Aφ| = eiθ ψ, Aφ. Hence C |ψ, Aφ| = eiθ ψ, Aφ ≤ (eiθ ψ2 + φ2 ) 2 C 2 2 = (ψ + φ ). 2 Thus, in the same manner as above, we can obtain |ψ, Aφ| ≤ Cψφ, ψ, φ ∈ D.

The next lemma is easily proven by elementary calculus. Therefore, we omit proof of it. Lemma 4.3. For all s > 1 and n ≥ 2, n−1

1 log n 1 ≤ s−1 + . s − ms n n s(n − 1)s−1 m=1 Lemma 4.4. Let s > 1. Then n−1 sup

1 s n − ms m=1

n≥2

and

sup

m≥1

∞

(4.4)

1 s n − ms n=m+1

<∞

(4.5)

< ∞.

(4.6)

Proof. Property (4.5) follows from Lemma 4.3. To prove (4.6), we write ∞ ∞ 1 1 1 ≤ dx + . s s s s n −m (m + 1)s − ms m+1 x − m n=m+1 We ﬁx a constant R > 2 (≥ (m + 1)/m). By the change of variable x = my, we have ∞ R 1 1 1 ds + CR , dx = s−1 s s s m m+1 x − m (m+1)/m y − 1

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where

CR :=

∞

1 ds < ∞. −1

ys

R

Using the well-known inequality xs − 1 ≥ s(x − 1), we have

R

(m+1)/m

1 dy ≤ s y −1 =

Hence

∞ m+1

s ≥ 1,

x > 0,

R

(m+1)/m

(4.7)

1 dy s(y − 1)

1 (log(R − 1) + log m) . s

1 log m log(R − 1) 1 dx ≤ + + s−1 CR . xs − ms sms−1 sms−1 m

Thus (4.6) follows. Let cH (n) :=

dH (m) :=

n−1

En , (E − Em )Em n m=1 ∞

Em , (En − Em )En n=m+1

n ≥ 2,

(4.8)

m ≥ 1.

(4.9)

Since cH (n) and dH (m) are positive (recall that En > 0, ∀n ∈ N), one can deﬁne constants cH := sup cH (n),

(4.10)

dH := sup dH (m),

(4.11)

n≥2

m≥1

which are ﬁnite or inﬁnite. Theorem 4.5. Suppose that there exist constants α > 1, C > 0 and a > 0 such that En − Em ≥ C(nα − mα ),

n > m > a.

Then Tb is a bounded operator with D(Tb ) = H and Tb ≤ 4b∞ cH dH .

(4.12)

(4.13)

Moreover, if b∗nm = bmn for all m, n ∈ N, then Tb is a bounded self-adjoint operator with D(Tb ) = H and Tb ≤ 2b∞ cH dH . (4.14) In particular, T is a bounded self-adjoint operator with D(T ) = H.

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Proof. By Lemma 4.2, it is enough to show that cH and dH are ﬁnite and |ψ, Tb ψ| ≤ 2b∞ cH dH ψ2 , ψ ∈ D0 . (4.15) Then Tb is bounded with (4.13). Since Tb is densely deﬁned and closed, it follows that D(Tb ) = H. As in the case of T , one can show that, if b∗nm = bmn for all m, n ∈ N, then Tb |D0 is symmetric and hence Tb is a bouned self-adjoint operator with D(Tb ) = H and (4.14) holds. Therefore the desired result follows. To prove (4.15), we ﬁrst note that, for ψ ∈ D0 , ∞

ψ, Tb ψ = i

bnm ψ, en em , ψ. En − Em

m,n=1,m=n

Hence |ψ, Tb ψ| ≤ 2b∞A(ψ), where A(ψ) :=

|em , ψ||ψ, en | . En − Em

n>m≥1

Inserting 1 = Em /En · En /Em into the summand on the right-hand side and using the Cauchy–Schwarz inequality, we have A(ψ)2 ≤ B(ψ)C(ψ) with

B(ψ) =

n>m≥1

|en , ψ|2 En · , En − Em Em

|em , ψ|2 Em · . En − Em En

C(ψ) =

n>m≥1

One can rewrite and estimate B(ψ) as follows: B(ψ) =

∞

|en , ψ|2 cH (n)

n=2

≤ ψ2 cH . Similarly we have C(ψ) ≤ ψ2 dH . Hence |ψ, Tb ψ| ≤ 2b∞

cH dH ψ2 .

Therefore, we need only to prove that cH and dH are ﬁnite.

(4.16)

(4.17)

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We can write cH (n) =

n−1

n−1 1 1 + . E − Em m=1 Em m=1 n

By assumption (4.12), we have 1 1 , ≤ En − Em C(nα − mα )

n > m > a.

(4.18)

Since we have ∞ 1 < ∞, α n n=1

it follows that ε1 :=

∞ 1 < ∞. E m=1 m

Thus cH (n) ≤

n−1

1 + ε1 . E − Em m=1 n

Let n0 ≥ 2 be a natural number such that n0 > a. Then, for all n > n0 n n−1 0 −1 1 1 1 1 ≤ + . α − mα E − E E − E C n n m n m m=n m=1 m=1 n−1

0

By (4.4), the right-hand side is uniformly bounded in n. Thus, we have cH < ∞. To prove dH < ∞, we write for m > a ∞ ∞ 1 1 dH (m) = − (E − E ) E n m n=m+1 n=m+1 n ∞

≤

1 (E − Em ) n n=m+1

≤

1 C

∞ n=m+1

nα

1 . − mα

Hence, by (4.6) in Lemma 4.4, we have sup dH (m) < ∞.

m>a

Thus it follows that dH < ∞. Example 4.1. Let λ > 0, α > 1 and P (x) be a real polynomial of x ∈ R with degree p < α. Then it is easy to see that the sequense {En }∞ n=1 deﬁned by En := λnα + P (n) satisﬁes the assumptions (H.1), (H.2) and (4.12). Thus, by Theorem 4.5, in the present example, Tb is bounded.

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We remark that Theorem 4.5 does not cover the case En = λn+µ with constants λ > 0 and µ ∈ R. For this case, we have the following theorem: Theorem 4.6. Suppose that there exist constants λ > 0, µ ∈ R and a > 0 such that En = λn + µ,

n > a.

(4.19)

Then T is a bounded self-adjoint operator with D(T ) = H. Proof. Let k0 be the greatest integer such that k0 ≤ a. Let an := en , ψ (ψ ∈ H). 2 2 Then, by the Parseval equality, we have ∞ n=1 |an | = ψ . Let ψ ∈ D0 . Then we can write: ψ, T ψ = SI + S2 + S3 + S4 , where S1 := i

k0 k0 n=1 m=n

S2 := i

k0

a∗n am , En − Em

∞

n=1 m≥k0 +1 ∞

S3 := i S4 := i

n≥k0 +1 ∞

1 λ

a∗n am , En − Em

k0

a∗n am , E − Em m=1 n ∞

n=k0 +1 m=n,m≥k0 +1

a∗n am . n−m

By the Schwarz inequality, we have |Sj | ≤ Cj ψ2 ,

j = 1, 2, 3,

where Cj > 0 is a constant. To estimate |S4 |, we use the following well known inequality [11, Theorem 294]: ∞ ∞ ∞ x y n m 2 2 ≤π xn ym n,m=1,n=m n − m n=1 m=1 ∞ for all real sequences {xn }∞ n=1 and {yn }n=1 . Hence

|S4 | ≤ πψ2 . Therefore it follows that |ψ, T ψ| ≤ const.ψ2 . Thus T is bounded. Example 4.2. A physically interesting case is the case where En = ω(n + 12 ), n ∈ {0} ∪ N with a constant ω > 0. In this case, by Theorem 4.6, T is a bounded

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self-adjoint operator with D(T ) = H and takes the form   ∞ ∞ i em , ψ   en , ψ ∈ H. Tψ = ω n=1 n−m m=n

Moreover, one can prove that σ(T ) = [−π/ω, π/ω] ([4, Theorem 2.1]). ˆ := ω −1 H − 1/2 and θˆ := ωT . Then it follows that Let N ˆ ) = {0} ∪ N, σ(N ˆN ˆ ]ψ = iψ, [θ,

ˆ = [−π, π], σ(θ)

(4.20)

ψ ∈ Dc .

(4.21)

As is well known, in the context of quantum mechanics, the sequence {ω(n + appears as the spectrum of the one-dimensional quantum harmonic oscillator Hamiltonian with mass m > 0 1 ∞ 2 )}n=1

1 p2 + mω 2 q 2 2m 2 in the Schr¨ odinger representation (q, p) of the CCR, where p := −iD with D being the generalized partial diﬀerential operator on L2 (R) and q is the multiplication ˆ is called the number operator by the variable x ∈ R. In this context, the operator N operator and, in view of (4.20) and (4.21), the operator θˆ is interpreted as a phase operator [7]. Hos :=

5. Unboundedness of T As for the unboundedness of T , we have the following fact: Theorem 5.1. If {En }∞ n=1 satisﬁes inf (En+1 − En ) = 0,

n∈N

(5.1)

then T is unbounded. ∞ Proof. By (5.1), there exists a subsequence {Epk }∞ k=1 of {Ep }p=1 such that

lim (Epk +1 − Epk ) = 0.

k→∞

Hence we have

2 2 ∞ ∞ ∞ e , e 1 m pk 2 T epk = = E − E E − E n m n p k n=1 m=n n=pk 2 1 → ∞ (k → ∞). ≥ Epk +1 − Epk

Thus T is unbounded.

(5.2)

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Example 5.1. Let En = λnα + µ with constants λ > 0, α > 1/2 and µ ∈ R. Then {En }∞ n=1 satisﬁes the assumptions (H.1) and (H.2). As we have already seen, T is bounded if α ≥ 1. Let 1/2 < α < 1. Then, one easily sees that lim (En+1 − En ) = 0.

n→∞

Hence inf n∈N (En+1 − En ) = 0. Therefore, in this case, T is unbounded. Thus T is bounded if and only if α ≥ 1. 6. Hilbert–Schmidtness of T In this section we investigate Hilbert–Schmidtness of the operator T . Proposition 6.1. The operator T is Hilbert–Schmidt if and only if ∞ ∞ n=1 m=n

1 < ∞. (En − Em )2

(6.1)

In that case, T is self-adjoint with T 22 =

∞ ∞ n=1 m=n

1 , (En − Em )2

(6.2)

where · 2 denotes Hilbert–Schmidt norm. In particular, there exist a C.O.N.S. {fn }∞ n=1 of H and real numbers tn , n ∈ N such that T fn = tn fn and tn → 0 (n → ∞). Proof. Suppose that T is Hilbert–Schmidt. Then hand, we have ∞ n=1

T en2 =

∞ ∞ n=1 m=n

∞

n=1

1 (En − Em )2

T en 2 < ∞. On the other

(6.3)

Hence (6.1) follows with (6.2). ∞ 2 Conversely, (6.1) holds. Hence, by (6.3), n=1 T en < ∞. Therefore T is Hilbert–Schmidt. The last statement follows from the Hilbert–Schmidt theorem (e.g., [14, Theorem VI.16]). Remark 6.1. In Proposition 6.1, the number tn = 0 is an eigenvalue of T with a ﬁnite multiplicity. Since T is self-adjoint in the present case, it may be an observable in the context of quantum mechanics. If this is the case, then Proposition 6.1 shows that the observable described by T (“time” in any sense?) is quantized (discretized) in the quantum system whose Hamiltonian is H with eigenvalues {En }∞ n=1 satisfying (6.1).

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The next theorem gives a class of H such that T is Hilbert–Schmidt: Theorem 6.2. Suppose that (4.12) in Theorem 4.5 holds with α > 3/2. Then T is Hilbert–Schmidt and self-adjoint. Proof. Since 1/(En − Em )2 is symmetric in n and m, it is suﬃcient to show that 2 n>m≥1 1/(En − Em ) < ∞. By the present assumption, we need only to show that 1 <∞ Σ := α (n − mα )2 n>m≥1

for all α > 3/2. We have Σ= ≤

∞ n−1 n=2 m=1 ∞

(nα

1 − mα )2

n−1 1 1 . · α − (n − 1)α α − mα ) n (n n=2 m=1

Using (4.4) and the elementary inequality nα

1 1 ≤ , α − (n − 1) s α(n − 1)α−1

we obtain Σ≤

∞

∞

1 log n 1 + . α−1 nα−1 2 α(n − 1) α (n − 1)2(α−1) n=2 n=2

Each inﬁnite series on the right-hand side converges for all α > 3/2. Thus, the desired result follows. 7. The Galapon Time Operator as a Generalized Time Operator It is shown that every self-adjoint operator which has a strong time operator is absolutely continuous [12, 17]. Hence the Galapon time operator T1 is not a strong time operator of H. But it may be a generalized time operator of H. In this section, we investigate this aspect. 7.1. An operator-valued function on R In the same way as in Lemma 2.1(ii), one can show that, for all ψ ∈ H, n ∈ N and all t ∈ R, the inﬁnite series ∞ eit(En −Em ) − 1 em , ψ En − Em

m=n

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absolutely converges. Hence, for each t ∈ R, one can deﬁne a linear operator K(t) as follows:   2   ∞ ∞   eit(En −Em ) − 1 , (7.1) e , ψ < ∞ D(K(t)) := ψ ∈ H m   En − Em   n=1 m=n   ∞ ∞ it(En −Em ) e − 1  K(t)ψ := i em , ψ en , ψ ∈ D(K(t)). (7.2) E − E n m n=1 m=n

It is easy to see that, for all t ∈ R, D0 ⊂ D(K(t)) and K(t)ek = i

eit(En −Ek ) − 1 en , En − Ek

(7.3)

k ∈ N.

(7.4)

n=k

The correspondence K : R t → K(t) gives an operator-valued function on R. In the notation in Sec. 4, K(t) is the operator Tb with bnm = eit(En −Em ) −1, n, m ∈ N. Remark 7.1. Equation (7.4) shows that K(t) = tI|D0 . Hence T cannot be a strong time operator of H, as already remarked based on the general theory of strong time operators. Proposition 7.1. For all t ∈ R, K(t) is a densely deﬁned closed operator. Proof. Similar to the proof of Proposition 3.2. Proposition 7.2. For all t ∈ R, K(t)|D0 is symmetric. Proof. Similar to the proof of Lemma 2.3. Theorem 7.3. For all ψ ∈ D(T1 )(= D0 ) and t ∈ R, e−itH ψ ∈ D(T1 ) and T1 e−itH ψ = e−itH (T1 + K(t))ψ.

(7.5)

Proof. We need only to prove the statement in the case ψ = ek (∀k ∈ N). Since e−itH ek = e−itEk ek , it follows that e−itH ek ∈ D(T1 ) with ∞ i T1 e−itH ek = e−itEk en . En − Ek n=k

We have e−itH T1 ek = i

∞ e−itEn en . En − Ek

n=k

It follows from these equations that T1 e−itH ek − e−itH T1 ek = e−itH K(t)ek . Thus, the desired result follows.

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Corollary 7.4. Suppose that, for all t ∈ R, K(t) is bounded. Then T1 is a generalized time operator of H with commutation factor K. Proof. This follows from Theorem 7.3, Propositions 7.1 and 7.2. In view of Corollary 7.4, we need to investigate conditions for K(t) to be bounded. Proposition 7.5. Suppose that (4.12) holds with α > 1. Then, for all t ∈ R, K(t) is a bounded self-adjoint operator with D(K(t)) = H. Proof. This follows from an application of Theorem 4.5 to the case where bnm = eit(En −Em ) − 1, n, m ∈ N. Proposition 7.6. Suppose that (6.1) holds. Then, for all t ∈ R, K(t) is HilbertSchmidt and self-adjoint with 2 ∞ it(En −Ek ) ∞ e − 1 2 (7.6) K(t)2 = En − Ek . k=1 n=k

Proof. Similar to the proof of Proposition 6.1. 7.2. Non-diﬀerentiability of K From the viewpoint of the theory of generalized time operators [2], it is interesting to examine diﬀerentiability of the operator-valued function K. Proposition 7.7. For all k ∈ N, the H-valued function: R t → K(t)ek is not strongly diﬀerentiable on R. Proof. We ﬁrst show that K(t)ek is not strongly diﬀerentiable at t = 0. Since K(0)ek = 0, we have for all t ∈ R\{0} and N > k ∞ K(t)ek − K(0)ek 2 |eit(En −Ek ) − 1|2 = t t2 |En − Ek |2 n=k

≥

N +1 n=k

Hence

|eit(En −Ek ) − 1|2 . t2 |En − Ek |2

+1 K(t)ek − K(0)ek 2 N ≥ lim inf 1 = N. t→0 t n=k

Since N > k is arbitrary, it follows that K(t)ek − K(0)ek 2 = +∞. lim t→0 t This implies that K(t)ek is not strongly diﬀerentiable at t = 0.

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We next show that K(t)ek is not strongly diﬀerentiable at each t = 0. By (7.5), we have for all s ∈ R\{0}, K(t + s)ek − K(t)ek K(s)ek = eit(H−Ek ) . s s Hence

K(t + s)ek − K(t)ek K(s)ek = s . s

By the preceding result, the right-hand side diverges to +∞ as s → 0. Therefore, K(t)ek is not strongly diﬀerentiable at t. Remark 7.2. We have

 it(E −E )  i e k − 1 ; = k E − Ek e , K(t)ek =  0; = k.

(7.7)

Hence, for all k, ∈ N, e , K(t)ek is diﬀerentiable in t ∈ R and d e , K(t)ek = (δk − 1)eit(E −Ek ) . dt

(7.8)

Proposition 7.7 tells us some singularity of K(t) acting on D0 . But, as shown in the next proposition, K(t) restricted to Dc is strongly diﬀerentiable at t = 0. Proposition 7.8. For all ψ ∈ Dc , the H-valued function K(t)ψ is strongly diﬀerentiable at t = 0 with d K(t)ψ|t=0 = ψ. dt

(7.9)

Proof. We need only to prove the statement for ψ of the form ψ = ek − e (k, ∈ N, k = ). For all t ∈ R\{0}, we have K(t)(ek − e ) = A(t) + B(t), t where eit(Ek −E ) − 1 eit(E −Ek ) − 1 e − i ek , t(E − Ek ) t(Ek − E ) ∞ it(En −Ek ) e − 1 eit(En −E ) − 1 − B(t) := i en . t(En − Ek ) t(En − E ) A(t) := i

n=k,

It is easy to see that lim A(t) = ek − e .

t→0

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As for B(t), we have B(t)2 =

∞

|Fn (t)|2 ,

n=k,

where Fn (t) :=

eit(En −Ek ) − 1 eit(En −E ) − 1 − . t(En − Ek ) t(En − E )

It is easy to see that lim Fn (t) = 0.

t→0

Moreover, one can show that |Fn (t)| ≤

C , |En − Ek |

n = k, ,

∞ where C > 0 is a constant independent of n and t. Since n=k 1/|En − Ek |2 < ∞, one can apply the dominated convergence theorem to conclude that limt→0 B(t)2 = 0. Thus K(t)(ek − e ) is strongly diﬀerentiable at t = 0 and (7.9) with ψ = ek − e holds. Proposition 7.9. For all k, ∈ N with k = , the H-valued function K(t)(ek − e ) is not strongly diﬀerentiable at t ∈ {2πn/(Ek − E )|n ∈ Z}. Proof. Let t = 2πn/(Ek − E ) (n ∈ Z) and s ∈ R\{0}. Then, by (7.5), we have (K(t + s) − K(t))(ek − e ) K(s) −itH = eitH e (ek − e ). s s Hence

(K(t + s) − K(t))(ek − e ) = u(s) s

with u(s) :=

K(s) −itEk (e ek − e−itE e ). s

We write u(s) = u1 (s) + u2 (s) with K(s) K(s) (ek − e ), u2 (s) := (e−itEk − e−itE ) e . s s By Proposition 7.8, we have lims→0 u1 (s) = e−itEk (ek − e ). On the other hand, we have from the proof of Proposition 7.7 and the assumed condition for t u1 (s) := e−itEk

lim u2 (s) = +∞.

s→0

Hence lims→0 u(s) = +∞. Thus the desired result follows.

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8. Possible Connections with Regular Perturbation Theory We consider a perturbation of H by a symmetric operator HI on H: H(λ) := H + λHI ,

(8.1)

where λ ∈ R is a perturbation parameter. For simplicity, we assume that HI is H-bounded: D(H) ⊂ D(HI ) and there exist constants a, b ≥ 0 such that HI ψ ≤ aHψ + bψ,

ψ ∈ D(H).

Then, by the Kato–Rellich theorem (e.g., [15, Theorem X.12]), for all λ ∈ R satisfying a|λ| < 1,

(8.2)

H(λ) is self-adjoint and bounded below. In what follows, we assume (8.2). 8.1. Eigenvalues of H(λ) We ﬁx n ∈ N arbitrarily. By a general theorem in regular perturbation theory (e.g., [16, Theorem XII.9]), there exists a constant cn > 0 such that, for all |λ| < cn , H has a unique, isolated non-degenerate eigenvalue En (λ) near En . Moreover, En (λ) is analytic in λ with Taylor expansion En (λ) = En + En(1) λ + En(2) λ2 + · · · ,

(8.3)

where En(1)

:= en , HI en ,

En(2)

∞ |en , HI em |2 := . En − Em

(8.4)

m=n

As an eigenvector of H(λ) with eigenvalue En (λ), one can take a vector ψn (λ) analytic in λ with Taylor expansion ψn (λ) = en + e(1) n λ + ···,

(8.5)

where e(1) n :=

∞ em , HI en em . En − Em

(8.6)

m=n

By Lemma 2.2, we have

 

i ; n = m, E − Em en , T em = n  0; n = m. (2)

Hence En can be written En(2) = (−i)

∞ m=1

|en , HI em |2 en , T em .

(8.7)

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To rewrite the right-hand side only in terms of en and linear operators on H, we note that ∞

|en , HI em |2 = HI en 2 < ∞

m=1

by the Parseval equality. Hence ∞

|en , HI em |4 < ∞.

m=1

Therefore the inﬁnite series fn :=

∞

|en , HI em |2 em

(8.8)

m=1

strongly converges and deﬁnes a vector in H. Thus we can deﬁne a linear operator V on H as follows: D(V ) := D0 , ∞ V ψ := −i en , ψfn ,

(8.9) ψ ∈ D0

(8.10)

n=1

where the right-hand side of (8.10) is a sum over a ﬁnite term. It is easy to see that V is a symmetric operator. Proposition 8.1. For all n ∈ N, En(2) = T en , V en .

(8.11)

Proof. We have V en = −ifn . Hence T en , V en = −iT en , fn , which is equal to the right-hand side of (8.7). This proposition suggests some role of the time operator T1 = T |D0 in the perturbation expansions of the eigenvalues of H. (1) As for the ﬁrst order term en λ of the eigenvector ψn (λ), we have e(1) n = (−i)

∞

em , HI en en , T em em .

(8.12)

m=1

8.2. Transition probability amplitudes In the context of quantum mechanics where H(λ) is the Hamiltonian of a quantum system, the complex number φ, e−itH(λ) ψ with unit vectors φ, ψ ∈ H is called the transition probability amplitude for the probability such that the state of the quantum system at time t is found in the state φ under the condition that the state of the quantum system at time zero is ψ.

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Lemma 8.2. Let φ, ψ ∈ D(H). Then, for all t ∈ R, t −itH(λ) −itH φ, e ψ = φ, e ψ − iλ ei(t−s)H φ, HI e−isH ψds + O(λ2 ).

977

(8.13)

0

Proof. By a simple application of a general formula for the unitary group generated by a self-adjoint operator ( [5, Lemma 5.9]), we have t e−i(t−s)H(λ) HI e−isH ψ ds, (8.14) e−itH(λ) ψ = e−itH ψ − iλ 0

where the integral is taken in the strong sense. Hence t −itH(λ) −itH φ, e ψ = φ, e ψ − iλ ei(t−s)H(λ) φ, HI e−isH ψ ds 0

= φ, e−itH ψ − iλ where

R(λ) := −iλ

t

0

0

t

ei(t−s)H φ, HI e−isH ψ ds + R(λ),

(ei(t−s)H(λ) − ei(t−s)H )φ, HI e−isH ψ ds.

Using (8.14) again, we have t −(t−s) R(λ) = −λ2 ds ds ei(t−s+s )H(λ) HI e−is H φ, HI e−isH ψ. 0

0

Hence 2

|R(λ)| ≤ λ

0

|t|

ds

|t−s| 0

ds HI e−is H φHI e−isH ψ

2

Therefore R(λ) = O(λ ). Thus (8.13) holds. Applying (8.13) with φ = em and ψ = en (n = m), we have em , e−itH(λ) en = −λ

e−itEn − e−itEm em , HI en + O(λ2 ), Em − En

(8.15)

which, combined with (7.7), gives em , e−itH(λ) en = iλen , e−itH K(t)em em , HI en + O(λ2 ),

m = n.

(8.16)

This suggests a physical meaning of the commutation factor K. By Theorem 7.3, one can rewrite the ﬁrst term on the right-hand side in terms of T1 and e−itH , obtaining em , e−itH(λ) en = iλen , [T1 , e−itH ]em em , HI en + O(λ2 ),

m = n.

This also is suggestive on physical meaning of the time operator T1 .

(8.17)

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Acknowledgment The work is supported by the Grant-in-Aid No. 17340032 for Scientiﬁc Research from Japan Society for the Promotion of Science (JSPS). References [1] Y. Aharonov and D. Bohm, Time in the quantum theory and the uncertainty relation for time and energy, Phys. Rev. 122 (1961) 1649–1658. [2] A. Arai, Generalized weak Weyl relation and decay of quantum dynamics, Rev. Math. Phys. 17 (2005) 1071–1109. [3] A. Arai, Spectrum of time operators, Lett. Math. Phys. 80 (2007) 211–221. [4] A. Arai, Some aspects of time operators, in Quantum Bio-Informatics, eds. L. Accardi, W. Freudenberg and M. Ohya (World Scientific, 2008), pp. 26–35. [5] A. Arai, Heisenberg operators, invariant domains and Heisenberg equations of motion, Rev. Math. Phys. 19 (2007) 1045–1069. [6] A. Arai and Y. Matsuzawa, Construction of a Weyl representation from a weak Weyl representation of the canonical commutation relation, Lett. Math. Phys. 83 (2008) 201–211. [7] G. Dorfmeister and J. Dorfmeister, Classification of certain pairs of operators (P, Q) satisfying [P, Q] = −iId, J. Funct. Anal. 57 (1984) 301–328. [8] E. A. Galapon, Pauli’s theorem and quantum canonical pairs: The consistency of a bounded, self-adjoint time operator canonically conjugate to a Hamiltonian with non-empty point spectrum, Proc. R. Soc. Lond. A 458 (2002) 451–472. [9] E. A. Galapon, Self-adjoint time operator is the rule for discrete semi-bounded Hamiltonians, Proc. R. Soc. Lond. A 458 (2002) 2671–2689. [10] E. A. Galapon, R. F. Caballar and R. T. Bahague Jr., Confined quantum time of arrivals, Phys. Rev. Lett. 93 (2004) 180406. [11] G. H. Hardy, J. E. Littlewood and G. P´ olya, Inequalities (Cambridge University Press, London, 1934). [12] M. Miyamoto, A generalized Weyl relation approach to the time operator and its connection to the survival probability, J. Math. Phys. 42 (2001) 1038–1052. [13] J. von Neumann, Die Eindeutigkeit der Schr¨ odingerschen Operatoren, Math. Ann. 104 (1931) 570–578. [14] M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis (Academic Press, New York, 1972). [15] M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness (Academic Press, New York, 1975). [16] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators (Academic Press, New York, 1978). [17] K. Schm¨ udgen, On the Heisenberg commutation relation. I, J. Funct. Anal. 50 (1983) 8–49. [18] K. Schm¨ udgen, On the Heisenberg commutation relation. II, Publ. Res. Inst. Math. Sci. 19 (1983) 601–671.

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Reviews in Mathematical Physics Vol. 20, No. 8 (2008) 979–1006 c World Scientific Publishing Company

THE NONCOMMUTATIVE GEOMETRY OF THE QUANTUM PROJECTIVE PLANE

FRANCESCO D’ANDREA∗ , LUDWIK DA ¸ BROWSKI† ‡ and GIOVANNI LANDI ∗D´ epartement

de Math´ ematique, Universit´ e Catholique de Louvain, Chemin du Cyclotron 2, B-1348, Louvain-La-Neuve, Belgium [email protected] †Scuola

Internazionale Superiore di Studi Avanzati, Via Beirut 2-4, I-34014, Trieste, Italy [email protected]

‡Dipartimento

di Matematica e Informatica, Universit` a di Trieste, Via A. Valerio 12/1, I-34127, Trieste, Italy and INFN, Sezione di Trieste, Trieste, Italy [email protected] Received 29 February 2008

We study the spectral geometry of the quantum projective plane CP2q , a deformation of the complex projective plane CP2 , the simplest example of spinc manifold which is not spin. In particular, we construct a Dirac operator D which gives a 0+ -summable triple, equivariant under Uq (su(3)). The square of D is a central element for which left and right actions on spinors coincide, a fact that is exploited to compute explicitly its spectrum. Keywords: Noncommutative geometry; quantum groups; quantum homogeneous spaces; spectral triples. Mathematics Subject Classification 2000: 58B34, 17B37

1. Introduction The geometry of quantum spaces — whose coordinate algebras are noncommutative — can be studied, following Connes [3], by means of a spectral triple. The latter is the datum (A, H, D), where A is a unital, involutive, associative (but not necessarily commutative) C-algebra with a faithful representation, π : A → B(H), on a separable Hilbert space H, and D is a selfadjoint operator on H with compact resolvent and such that [D, a] is bounded for all a ∈ A. The operator D is called (a generalized) Dirac operator. In addition, the spectral triple is called even if H = H+ ⊕ H− is Z2 -graded, the representation of A is diagonal and the operator D is oﬀ-diagonal for this decomposition. The requirement of compact resolvent for 979

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the Dirac operator guarantees, for example, that in the even case the twisting of the operator D± = D|H± with projections (describing classes in the K-theory of A) are unbounded Fredholm operators: the starting point for the construction of “topological” invariants via index computations. Roughly, the bounded commutators condition says that the specrum of D does not grow too rapidly, while the compact resolvent one says that the specrum of D does not grow too slowly. It is the interplay of the two that (together with further requirements) imposes stringent restrictions on the geometry and produces spectacular consequences. For quantum homogeneous spaces (that is spaces which are “homogeneous” for quantum groups, see, e.g., [11]), a possible strategy consists to deﬁne a Dirac operator by its spectrum, in a suitable basis of “harmonic” spinors, and to prove that the commutators [D, a] are bounded by the use of quantum groups representation theory. In this manner one usually ﬁnds Dirac operators with spectrum growing at most polynomially (cf. [2, 4, 5, 8]). A diﬀerent occurrence is for the standard Podle´s quantum sphere where also a Dirac operator exists [6] with a spectrum growing exponentially, deﬁning then a 0+ -summable spectral triple (a behavior on the opposite hand to that of thetasummability). This operator has a particular geometrical meaning as it can be constructed [18] by using the action of certain generators of Uq (su(2)) which act as derivations on the standard Podle´s sphere. Along this line, a general construction of Dirac operators D on quantum irreducible ﬂag manifolds, including projective spaces, was given in [13]. These operators were used to realize the diﬀerential calculi of [10] by expressing the exterior derivative as a commutator with D. However, in [13] there is no computation of any spectrum of D and thus no addressing, among other things, of the compact resolvent requirement for the Dirac operator, an essential feature of spectral triples as mentioned above. Furthermore, the construction there depends on the choice of a morphism γ ([13, Proposition 2]) that appears to be neither unique nor canonical. In the present paper, as a ﬁrst step for a general strategy, we work out from scratch the spectral geometry of a basic example (besides the standard Podle´s sphere), that is the quantum complex projective plane CP2q . This is deﬁned as a q-deformation with real parameter (that we restrict to q ∈ (0, 1)) of the complex projective plane CP2 seen as the four-dimensional real manifold S 5 /S 1 = SU (3)/SU (2) × U (1). Our example is particularly important in that it is a deformation of a manifold which is not a spin manifold but only spinc . In analogy with the standard Podle´s sphere, we ﬁnd a Dirac operator D on CP2q with exponentially growing spectrum — a q-deformation of the spectrum of the Dolbeault–Dirac operator on undeformed CP2 (for the latter, cf. [9]) — thus giving a 0+ -dimensional spectral triple. The spectrum is explicitly computed by relating the square of D to a quantum Casimir element for which a left and a right action on spinors coincide. As motivated in Sec. 2, to get this quantum Casimir element we need to enlarge the symmetry algebra. The use of this technique to compute the spectrum via left/right actions seem to be, to the best of our knowledge, a novel one. There remain open

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problems, notably the issue of regularity for the present spectral geometry, which might hold at most in the “twisted sense” of [15]; their analysis is postponed to future work. The plan of the paper is the following. In Sec. 2, we introduce the Hopf algebra Uq (su(3)), which describes the “inﬁnitesimal” symmetries of CP2q , and in Sec. 3, the dual Hopf algebra A(SUq (3)), whose elements are representative functions on the quantum SU (3) group. The coordinate algebra of CP2q is deﬁned in Sec. 4 as the ﬁxed point subalgebra of A(SUq (3)) for the action of a suitable Hopf subalgebra Uq (u(2)) ⊂ Uq (su(3)). In Sec. 5, we describe the q-analogue of antiholomorphic forms and use them to construct ﬁrst a diﬀerential calculus and then a spectral triple on CP2q in Sec. 6. The appendix contains the description of antiholomorphic forms as equivariant maps on bundles over the undeformed CP2 , a description which was the motivation for an analogous identiﬁcation on the quantum CP2q . 2. The Symmetry Hopf Algebra Uq (su(3)) Let Uq (su(3)) be the Hopf ∗-algebra generated (as a ∗-algebra) by Ki , Ki−1 , Ei , Fi , i = 1, 2, with Ki = Ki∗ , Fi = Ei∗ , and relations [Ki , Kj ] = 0,

Ki Ei Ki−1 = qEi ,

Ki Ej Ki−1 = q −1/2 Ej ,

[Ei , Fi ] = (q − q −1 )−1 (Ki2 − Ki−2 ), [Ei , Fj ] = 0,

if i = j,

and (Serre relations) Ei2 Ej − (q + q −1 )Ei Ej Ei + Ej Ei2 = 0,

∀ i = j.

(2.1)

We can restrict the real deformation parameter to the interval 0 < q < 1; for q > 1 we get isomorphic algebras. In the appendix, we shall also brieﬂy decribe the “classical limit” U (su(3)). In the notation of [11, Sec. 6.1.2] the above Hopf ˘q (su(3)), the “compact” real form of the Hopf algebra denoted algebra is denoted U ˘ Uq (sl(3, C)) there. With the q-commutator deﬁned as [a, b]q := ab − q −1 ba, relations (2.1) can be rewritten as [Ei , [Ej , Ei ]q ]q = 0 or [[Ei , Ej ]q , Ei ]q = 0. Coproduct, counit and antipode are given by (with i = 1, 2) ∆(Ki ) = Ki ⊗ Ki , (Ki ) = 1,

(Ei ) = 0,

∆(Ei ) = Ei ⊗ Ki + Ki−1 ⊗ Ei , S(Ki ) = Ki−1 ,

S(Ei ) = −qEi .

The opposite Hopf ∗-algebra Uq (su(3))op is deﬁned to be isomorphic to Uq (su(3)) as ∗-coalgebra, but equipped with opposite multiplication and with antipode S −1 . There is a Hopf ∗-algebra isomorphism ϑ : Uq (su(3)) → Uq (su(3))op given on generators by ϑ(Ki ) := Ki , and satisfying ϑ2 = id.

ϑ(Ei ) := Fi ,

ϑ(Fi ) := Ei ,

i = 1, 2,

(2.2)

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We denote (for obvious reasons) by Uq (su(2)) the sub Hopf ∗-algebra of Uq (su(3)) generated by the elements {K1 , K1−1 , E1 , F1 } and by Uq (u(2)) the Hopf ∗-algebra generated by Uq (su(2)), K1 K22 and (K1 K22 )−1 . Notice that K1 K22 commutes with all elements of Uq (su(2)). Irreducible ﬁnite dimensional ∗-representations of Uq (su(3)) are classiﬁed by two nonnegative integers n1 , n2 (see, e.g., [11]). The representation space V(n1 ,n2 ) has dimension dim V(n1 ,n2 ) =

1 (n1 + 1)(n2 + 1)(n1 + n2 + 2), 2

and orthonormal basis |n1 , n2 , j1 , j2 , m , with labels satisfying the constraints ji = 0, 1, 2, . . . , ni ,

1 (j1 + j2 ) − |m| ∈ N. 2

(2.3)

The generators of Uq (su(3)) act on V(n1 ,n2 ) as follows: K1 |n1 , n2 , j1 , j2 , m := q m |n1 , n2 , j1 , j2 , m , 3

1

K2 |n1 , n2 , j1 , j2 , m := q 4 (j1 −j2 )+ 2 (n2 −n1 −m) |n1 , n2 , j1 , j2 , m , 1 1 (j1 + j2 ) − m (j1 + j2 ) + m + 1 E1 |n1 , n2 , j1 , j2 , m := 2 2 × |n1 , n2 , j1 , j2 , m + 1 , 1 1 (j1 + j2 ) − m + 1 Aj1 ,j2 n1 , n2 , j1 + 1, j2 , m − E2 |n1 , n2 , j1 , j2 , m := 2 2 1 1 (j1 + j2 ) + m Bj1 ,j2 n1 , n2 , j1 , j2 − 1, m − + , 2 2 with coeﬃcients given by [n1 − j1 ][n2 + j1 + 2][j1 + 1] , Aj1 ,j2 := [j1 + j2 + 1][j1 + j2 + 2]    [n1 + j2 + 1][n2 − j2 + 1][j2 ] [j1 + j2 ][j1 + j2 + 1] Bj1 ,j2 :=   1

if j1 + j2 = 0, if j1 + j2 = 0.

As usual, [z] := (q z − q −z )/(q − q −1 ) denotes the q-analogue of z ∈ C. The highest weight vector of V(n1 ,n2 ) is |n1 , n2 , n1 , 0, 12 n1 , corresponding to the weight (q n1 /2 , q n2 /2 ). There are additional ∗-representations of Uq (su(3)) that we do not need in the present paper. Up to a relabeling, the basis we use is the Gelfand–Tsetlin basis ([11, Sec. 7.3.3]). One can pass to the notations of [1] with the replacement

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Ei = ei , Fi = fi , Ki = q hi /2 and n1 = p13 − p23 − 1, n2 = p23 − p33 − 1, j1 = p12 − p23 − 1,

j2 = p23 − p22 ,

2m = 2p11 − p12 − p22 − 1.

The fundamental representation V(0,1) will be needed later on in Sec. 3 to construct a pairing of Uq (su(3)) with a dual Hopf algebra. Its matrix form, σ : Uq (su(3)) → Mat3 (C), is  −1/2 0 q σ(K1 ) =  0 q 1/2 0 0   0 0 0 σ(E1 ) = 1 0 0 , 0 0 0

0 0 1

 ,

  1 0 0 σ(K2 ) = 0 q −1/2 0 , 1/2 0 0 q   0 0 0 σ(E2 ) = 0 0 0 , 0 1 0

(2.4a)

(2.4b)

having identiﬁed |0, 1, − 21 with (1, 0, 0)t , |0, 1, 12 with (0, 1, 0)t and |0, 0, 0 with (0, 0, 1)t . In order to have a Casimir operator for the algebra Uq (su(3)) one needs to enlarge it. The minimal extension is obtained by adding the element H := (K1 K2−1 )2/3 and its inverse; by a slight abuse of notation we continue to use the symbol Uq (su(3)) for this extension. Such a Casimir element appeared already in [17, Eq. 48] but in the framework of formal power series. In our notations, it reads Cq = (q − q −1 )−2 ((H + H −1 ){(qK1 K2 )2 + (qK1 K2 )−2 } + H 2 + H −2 − 6) + (qHK22 + q −1 H −1 K2−2 )F1 E1 + (qH −1 K12 + q −1 HK1−2 )F2 E2 + qH[F2 , F1 ]q [E1 , E2 ]q + qH −1 [F1 , F2 ]q [E2 , E1 ]q ,

(2.5)

satisﬁes Cq∗ = ϑ(Cq ) = Cq and commutes with all elements of Uq (su(3)) as can also be checked by a straightforward computation. Moreover the restriction of Cq to the irreducible representation V(n1 ,n2 ) is proportional to the identity (by Schur’s lemma) with the constant readily found (by acting on the highest weight vector v := |n1 , n2 , n1 , 0, 12 n1 ) to be given by

Cq |V(n1 ,n2 )

2 2 2 1 1 1 (n1 − n2 ) + (2n1 + n2 ) + 1 + (n1 + 2n2 ) + 1 . (2.6) = 3 3 3

3. The Quantum SU (3) Group The deformation A(SUq (3)) of the Hopf ∗-algebra of representative functions of SU (3) is given in [16] (cf. [11, Sec. 9.2]). As a ∗-algebra it is generated by 9 elements

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uij (i, j = 1, . . . , 3) with commutation relations uik ujk = qujk uik , uki ukj = qukj uki , [uil , ujk ]

[uik , ujl ]

= 0,

and a cubic relation

= (q − q

∀ i < j, −1

)uil ujk ,

∀ i < j, k < l,

(−q)(π) u1π(1) u2π(2) u3π(3) = 1,

p∈S3

where the sum is over all permutations π of three elements and (π) is the length of π. The ∗-structure is given by (uij )∗ = (−q)j−i (ukl11 ukl22 − qukl21 ukl12 ) with {k1 , k2 } = {1, 2, 3}\{i} and {l1 , l2 } = {1, 2, 3}\{j} (as ordered sets). Thus for example (u11 )∗ = u22 u33 − qu23 u32 . Coproduct, counit and antipode are the usual ones: uik ⊗ ukj , (uij ) = δji , S(uij ) = (uji )∗ . ∆(uij ) = k

Using the fundamental representation σ : Uq (su(3)) → Mat3 (C), given by (2.4), one deﬁnes a nondegenerate dual pairing (cf. [11, Sec. 9.4]) h, uij := σji (h).

, : Uq (su(3)) × A(SUq (3)) → C, With this pairing — using Sweedler notation ∆(a) = a(1) ⊗a (2) for the coproduct — one gets left and right canonical actions ha = a(1) h, a(2) and a h = h, a(1) a(2) , explicitly given by uik σjk (h), uij h = σki (h)ukj , h uij = k

k

and which make A(SUq (3)) an Uq (su(3))-bimodule ∗-algebra. It is convenient to convert the right action into a second left action commuting with the action . This is done by using the map ϑ given by (2.2): h a := a ϑ(h), for all h ∈ Uq (su(3)) and a ∈ A(SUq (3)). Since ϑ is a Hopf ∗-algebra isomorphism from Uq (su(3)) to Uq (su(3))op the action is compatible with the coproduct and the antipode of Uq (su(3)). Thus, these two left actions make A(SUq (3)) a left Uq (su(3)) ⊗ Uq (su(3))-module ∗-algebra. Explicitly, the actions of generators of Uq (su(3)) on generators of A(SUq (3)) are: 1

1

K1 ui1 = q − 2 ui1 , K1 ui2 = q 2 ui2 , − 12

ui2 ,

K1 ui3 = ui3 , 1

K2 ui1 = ui1 ,

K2 ui2 = q

K2 ui3 = q 2 ui3 ,

E1 ui1 = ui2 ,

E1 ui2 = 0,

E1 ui3 = 0,

F1 ui1 = 0,

F1 ui2 = ui1 ,

F1 ui3 = 0,

E2 ui1 = 0,

E2 ui2 = ui3 ,

E2 ui3 = 0,

F2 ui1 = 0,

F2 ui2 = 0,

F2 ui3 = ui2 ,

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and 1

1

K1 u1j = q − 2 u1j , K1 u2j = q 2 u2j , − 12

K1 u3j = u3j , 1

K2 u1j = u1j ,

K2 u2j = q

u2j , K2 u3j = q 2 u3j ,

E1 u1j = u2j ,

E1 u2j = 0,

E1 u3j = 0,

F1 u1j = 0,

F1 u2j = u1j ,

F1 u3j = 0,

E2 u1j = 0,

E2 u2j = u3j ,

E2 u3j = 0,

F2 u1j = 0,

F2 u2j = 0,

F2 u3j = u2j .

When computing the spectrum of the “exponential Dirac operator” on CP2q in Sec. 6 below, we shall use the fact that the “white” and “black” actions of the Casimir element concide. For the sake of clarity, we state this fact as a lemma. Lemma 3.1. Let Cq be the Casimir element defined in (2.5), then Cq a = Cq a,

f or all a ∈ A(SUq (3)).

(3.1)

Proof. Since ϑ(Cq ) = Cq , this statement is equivalent to Cq a = a Cq , for all a ∈ A(SUq (3)), an equality that follows from a simple characterization of the center of U. In fact, if U and A are any two Hopf ∗-algebras with a nondegenerate dual pairing , and left and right canonical actions h a = a(1) h, a(2) corresponding and a h = h, a(1) a(2) , for h ∈ U and a ∈ A, the center of U coincides with Z(U) := {h ∈ U | h a = a h, ∀ a ∈ A}. Indeed, from the deﬁnition of the actions, and nondegeneracy of the pairing, the proposition {h a = a h} is equivalent to the proposition { h ⊗ h, ∆(a) = h ⊗ h , ∆(a) , ∀ h ∈ U}; this follows from the equalities

h ⊗ h, ∆(a) = h , h a ,

and

h ⊗ h , ∆(a) = h , a h .

Then h ∈ Z if and only if h ⊗ h , ∆(a) = h ⊗ h, ∆(a) , for all h ∈ U and a ∈ A. In turn, this is equivalent to [h, h ], a = 0, for all h ∈ U, a ∈ A, and non-degeneracy of the pairing makes this equivalent to [h, h ] = 0, for all h ∈ U, that is h is in the center of U. Below we shall need an explicit basis of “harmonic functions” for the coordinate algebra on the quantum 5-sphere, and for some “equivariant line bundles” on the quantum projective plane. It follows from general facts (cf. [11, Sec. 11], see also [12]) that the algebra A(SUq (3)) is an Uq (su(3)) ⊗ Uq (su(3))-module ∗-algebra and the Peter– Weyl theorem states that it is the multiplicity-free direct sum of all irreducible representations of Uq (su(3)) ⊗ Uq (su(3)) with highest weight (λ, λ), where λ runs over all highest weights of Uq (su(3)). These representations are ∗-representations

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with respect to the inner product (a, b) = ϕ(a∗ b) induced by the Haar state ϕ. “Dually”, A(SUq (3)) is the direct sum of all its irreducible corepresentations, with multiplicity being the corresponding dimension. Indeed, we can construct (almost) explicitly the corresponding “harmonic” orthonormal basis. The element {(u11 )∗ }n1 (u33 )n2 is annihilated by both Ei and Ei and satisﬁes Ki {(u11 )∗ }n1 (u33 )n2 = Ki {(u11 )∗ }n1 (u33 )n2 = q ni /2 {(u11 )∗ }n1 (u33 )n2 . Then the highest weight vector in A(SUq (3)) corresponding to the weight λ = (n1 , n2 ) is cn1 ,n2 {(u11 )∗ }n1 (u33 )n2 , with cn1 ,n2 a normalization constant. The remaining vectors of the basis are computed using the following Lemma. Recall that the q-factorial is deﬁned by [n]! := [n][n − 1] · · · [2][1] for n a positive integer, while [0]! := 1. The q-binomial is given by [n]! n . := m [m]![n − m]! Lemma 3.2. With |n1 , n2 , j1 , j2 , m the basis of the irreducible representation V(n1 ,n2 ) of Uq (su(3)) described in Sec. 2, we have that 1 n1 ,n2 |n1 , n2 , j1 , j2 , m = Xj1 ,j2 ,m n1 , n2 , n1 , 0, n1 , 2 where 2 Xjn11,j,n2 ,m

:=

2 Njn11,j,n2 ,m

n 1 −j1 k=0

q −k(j1 +j2 +k+1) [j1 + j2 + k + 1]!

– 1 » (j +j )−m+k n −j [F2 , F1 ]nq 1 −j1 −k F2j2 +k × 1 k 1 F12 1 2

and 2 Njn11,j,n2 ,m :=

[j1 + j2 + 1] j +j 2 1 +m ! [n2 − j2 ]![j1 ]! [n1 + j2 + 1]![n2 + j1 + 1]! 2 . × j1 + j2 [n1 − j1 ]![j2 ]! [n1 ]![n2 ]![n1 + n2 + 1]! −m ! 2

Proof. Consider the map T ∈ Aut(V(n1 ,n2 ) ) deﬁned by 2 n1 , n2 , n1 , 0, 1 n1 . T |n1 , n2 , j1 , j2 , m = Xjn11,j,n2 ,m 2 One checks that [T, h]v = 0 for any v ∈ V(n1 ,n2 ) and any h ∈ Uq (su(3)). It is enough to do the check for h ∈ {Hi , Ei , Fi }i=1,2 . Thus for example, if h = F1 we have j1 + j2 j1 + j2 n1 ,n2 2 F1 Xj1 ,j2 ,m = +m − m + 1 Xjn11,j,n2 ,m−1 2 2

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j1 + j2 2 − m + 1 Xjn11,j,n2 ,m−1 2 1 × n1 , n2 , n1 , 0, n1 2 j1 + j2 j1 + j2 = +m − m + 1 T |n1 , n2 , j1 , j2 , m − 1 2 2

F1 T |n1 , n2 , j1 , j2 , m =

j1 + j2 +m 2

= T F1 |n1 , n2 , j1 , j2 , m . The remaining cases are either straightforward (if h = K1 , K2 ) or can be derived in a similar manner using the following commutation rules (proved by induction on n): [E1 , F1n ] = [n]F1n−1 (q − q −1 )(q −n+1 K12 − q n−1 K1−2 ), [E1 , [F2 , F1 ]nq ] = −[n]q n−2 [F2 , F1 ]n−1 F2 K1−2 , q [E2 , F2n ] = [n]F2n−1 (q − q −1 )(q −n+1 K22 − q n−1 K2−2 ), [E2 , [F2 , F1 ]nq ] = [n]F1 [F2 , F1 ]n−1 K22 , q F2 F1n − q −n F1n F2 = [n]F1n−1 [F2 , F1 ]q . By Schur’s Lemma, T is then proportional to the identity in every irreducible repre,n2 = 1, sentation V(n1 ,n2 ) , with some proportionality constant An1 ,n2 . Since Xnn1,0, 1 1 2 n1 1 1 T |n1 , n2 , n1 , 0, 2 n1 = |n1 , n2 , n1 , 0, 2 n1 and we deduce that An1 ,n2 = 1. This means 1 n1 ,n2 Xj1 ,j2 ,m n1 , n2 , n1 , 0, n1 = T |n1 , n2 , j1 , j2 , m = |n1 , n2 , j1 , j2 , m , 2 which concludes the proof. From this lemma and the Peter–Weyl decomposition, we deduce that an orthonormal basis of A(SUq (3)) is given by the elements n1 ,n2 n1 ,n2 1 ∗ n1 3 n2 t(n1 , n2 )lj11,l,j22,k (3.2) ,m := cn1 ,n2 Xj1 ,j2 ,m Xl1 ,l2 ,k {(u1 ) } (u3 ) and that the linear isometry Q : A(SUq (3)) →

V(n1 ,n2 ) ⊗ V(n1 ,n2 ) ,

(n1 ,n2 )∈N2

Q(t(n1 , n2 )lj11,l,j22,k ,m ) := |n1 , n2 , j1 , j2 , m ⊗ |n1 , n2 , l1 , l2 , k is an Uq (su(3)) ⊗ Uq (su(3))-module map, that is for all h ∈ Uq (su(3)) Q(h t(n1 , n2 )lj11,l,j22,k ,m ) = h |n1 , n2 , j1 , j2 , m ⊗ |n1 , n2 , l1 , l2 , k , Q(h t(n1 , n2 )lj11,l,j22,k ,m ) = |n1 , n2 , j1 , j2 , m ⊗ h |n1 , n2 , l1 , l2 , k .

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From now on, we will identify t(n1 , n2 )lj11,l,j22,k ,m with its image |n1 , n2 , j1 , j2 , m ⊗ |n1 , n2 , l1 , l2 , k . 4. The Quantum Projective Plane CP2q The quantum complex projective plane, which we denote by CP2q , was studied already in [14] (see also [20]). The most natural way to come to CP2q is via the 5-dimensional sphere Sq5 . We shall therefore start by studying the algebra A(Sq5 ) of coordinate functions on the latter. The algebra A(Sq5 ) is made of elements of A(SUq (3)) which are Uq (su(2))invariant, A(Sq5 ) := {a ∈ A(SUq (3)) | h a = (h)a ∀ h ∈ Uq (su(2))} and, as such, it is the ∗-subalgebra generated by elements {u3i , i = 1, . . . , 3} of the last “row”. In [19], it is proved to be isomorphic, through the identiﬁcation zi = u3i , to the abstract ∗-algebra with generators zi , zi∗ and relations: zi zj = qzj zi [z1∗ , z1 ] = 0,

zi∗ zj = qzj zi∗

∀ i < j,

[z2∗ , z2 ] = (1 − q 2 )z1 z1∗ ,

∀ i = j,

[z3∗ , z3 ] = (1 − q 2 )(z1 z1∗ + z2 z2∗ ),

z1 z1∗ + z2 z2∗ + z3 z3∗ = 1. Since K1 K22 is in the commutant of Uq (su(2)), in addition to the “white” action of Uq (su(3)), the algebra A(Sq5 ) carries a “black” action of the Hopf ∗-algebra generated by K1 K22 and its inverse, which we denote by Uq (u(1)). Thus, A(Sq5 ) is an Uq (su(3)) ⊗ Uq (u(1))-module ∗-algebra. A vector |n1 , n2 , l1 , l2 , k of the Gelfand–Tsetlin basis of V(n1 ,n2 ) is invariant for the action of Uq (su(2)) if and only if k = 0 = (l1 + l2 )/2. Last equality implies l1 = l2 = 0. Then an orthonormal basis of A(Sq5 ) is given by t(n1 , n2 )0,0,0 j1 ,j2 ,m

(4.1)

where the elements t’s are given by (3.2), with n1 , n2 nonnegative integers and labels j1 , j2 , m restricted as in (2.3). Thus, we have the decomposition: A(Sq5 )

V(n1 ,n2 ) ,

(n1 ,n2 )∈N2

into irreducible representations of Uq (su(3)) ⊗ Uq (u(1)), with the generator K1 K22 acting on V(n1 ,n2 ) as q n2 −n1 times the identity map. The algebra A(CP2q ) of the quantum projective plane CP2q can be deﬁned either as a subalgebra of A(Sq5 ) or (equivalently) as a subalgebra of A(SUq (3)). Both versions will be used when constructing (anti)-holomorphic forms on CP2q later on.

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We remind that K1 K22 is the generator of the Hopf ∗-algebra denoted Uq (u(1)) above. Then, we deﬁne A(CP2q ) := {a ∈ A(Sq5 ) | K1 K22 a = a} = {a ∈ A(SUq (3)) | h a = (h)a, ∀ h ∈ Uq (u(2))}. The ∗-algebra A(CP2q ) is generated by elements pij := (u3i )∗ u3j = zi∗ zj , j = 1, 2, 3, with ∗-structure (pij )∗ = pji . The relations split in commutation rules pii pjk = q sign(i−j)+sign(k−i) pjk pii pii pij = q

sign(j−i)+1

2

pij pii − (1 − q )

if i, j, k are distinct, q

6−2k

pkk pij

k
pij pik = q sign(k−j) pik pij sign(i−j)+sign(k−j)+1

2

pjk pij − (1 − q ) pil plk l<j   = (1 − q 2 )  pjl plj − pil pli 

pij pjk = q pij pji

l
if i = j, if i ∈ / {j, k}, if i, j, k are distinct, if i = j,

l<j

(here sign(0) := 0) and “projective plane” conditions pjk pkl = pjl , q 4 p11 + q 2 p22 + p33 = 1.

(4.2)

k

The relations above are obtained straightforwardly from those of A(Sq5 ). There cannot be additional generators: since K1 K22 zi = qzi and K1 K22 zj∗ = q −1 zj∗ , a monomial in the algebra A(Sq5 ) is invariant if and only if it contains the same number of zi and zi∗ ’s, which can be reordered using the commutation relations of A(Sq5 ). The elements pij are assembled in a 3 × 3 matrix P which by the ﬁrst relation in (4.2) is an idempotent, P 2 = P ; it is indeed a projection since P = P ∗ with the given ∗-structure. By the second relation in (4.2) it has q-trace Trq (P ) := q 4 p11 + q 2 p22 + p33 = 1. At the classical value, q = 1, of the parameter, the algebra A(CP2 ) is the algebra of (polynomial) functions on the space of size 3 rank 1 complex projections. This space is diﬀeomorphic to the projective plane CP2 by identifying each line in C3 with the range of a projection. Commutativity of the actions and entails that A(CP2q ) is an Uq (su(3))module ∗-algebra for the action with a decomposition of A(CP2q ) into irreducible representations of Uq (su(3)): A(CP2q ) V(n,n) . n∈N

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5 2 Indeed, a vector t(n1 , n2 )0,0,0 j1 ,j2 ,m ∈ A(Sq ) is annihilated by K1 K2 if and only if n1 = n2 . Thus an orthonormal basis of A(CP2q ) is given by the vectors:

t(n, n)0,0,0 j1 ,j2 ,m , with n a nonnegative integer and labels j1 , j2 , m again restricted as in (2.3). 5. The Dolbeault Complex The algebra inclusion A(CP2q ) → A(Sq5 ) is a noncommutative analogue of the U (1)principal bundle S 5 → CP2 , and as in the classical case, “modules of sections of line bundles over CP2q ” can be constructed, as equivariant maps, via the characters of U (1). For N ∈ Z, we deﬁne LN := {a ∈ A(Sq5 ) | K1 K22 a = q N a};

(5.1)

in particular L0 = A(CP2q ). Being A(Sq5 ) the subalgebra of A(SUq (3)) made of Uq (su(2))-invariant elements, LN can be equivalently described as LN = {a ∈ A(SUq (3)) | K1 K22 a = q N a, h a = (h)a, ∀ h ∈ Uq (su(2))}. (5.2) Each LN is a A(CP2q )-bimodule. Moreover, since the actions and commute, each LN is a Uq (su(3))-equivariant (left) A(CP2q )-module, that is it carries a left action of the crossed product A(CP2q ) Uq (su(3)). Using the orthonormal basis 5 2 {t(n1 , n2 )0,0,0 j1 ,j2 ,m } of A(Sq ) given by (4.1) on whose elements the generator K1 K2 acts as q n2 −n1 times the identity map, we argue that the vectors t(n, n + N )0,0,0 j1 ,j2 ,m

if N ≥ 0,

or t(n − N, n)0,0,0 j1 ,j2 ,m

if N < 0,

form a linear basis of LN , with n ∈ N and (j1 , j2 , m) satisfying the usual constraints (2.3). Thus, we have the decomposition into irreducible representations of Uq (su(3)): LN V(n,n+N ) , if N ≥ 0, n∈N

LN

V(n−N,n) ,

if N < 0.

n∈N

In the commutative (q → 1) limit, using the K¨ ahler structure of CP2 the Hilbert (0,k) , spaces of chiral spinors can be written as the completion of Ω(0,•) := kΩ (0,k) are antiholomorphic k-forms. As sections of equivariant vector bunwhere Ω dles (see e.g. [9, Sec. 2.4]), Ω(0,0) is isomorphic to A(CP2 ) and Ω(0,2) to the commutative limit of L3 . Contrary to 0 and 2 antiholomorphic forms, 1-forms are not associated with the principal bundle S 5 → CP2 but rather with the U (2)-bundle SU (3) → CP2 , via a suitable 2-dimensional representation of U (2). For the sake of completeness and clarity, we re-derive these classical results in the appendix.

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Motivated by the discussion above, in the deformed, q = 1, case we define antiholomorphic 0- and 2-forms as elements of the bimodules Ω(0,0) := L0 = A(CP2q ),

Ω(0,2) := L3 .

Instead for 1-forms we use the ∗-representation τ : Uq (su(2)) → Mat2 (C) given by 1/2 q 0 1 0 τ (K1 ) = , τ (E1 ) = , 0 0 0 q −1/2 and deﬁne Ω(0,1) as the equivariant A(CP2q )-bimodule 3

Ω(0,1) := {v ∈ A(SUq (3))2 | K1 K22 v = q 2 v, (h(1) v)τ (S(h(2) )) = (h)v, ∀ h ∈ Uq (su(2))}.

(5.3)

That is, v = (v+ , v− ) ∈ A(SUq (3))2 belongs to the subspace Ω(0,1) if and only if 3

K1 K22 (v+ , v− ) = q 2 (v+ , v− ),

1

1

K1 (v+ , v− ) = (q 2 v+ , q − 2 v− ),

E1 (v+ , v− ) = (0, v+ ),

F1 (v+ , v− ) = (v− , 0).

(5.4a) (5.4b)

Also the bimodule Ω(0,1) carries a representation of Uq (su(3)) given by the “white” action, and its decomposition into irreducible representations of Uq (su(3)) is readily found. With the basis (3.2), we ﬁnd that highest weight vectors of the spin 1/2 1,0,+ 1

0,1,+ 1

2 2 and t(n1 , n2 )j1 ,j2 ,m (in representation of Uq (su(2)) have the form t(n1 , n2 )j1 ,j2 ,m the former case (2.3) forces n1 ≥ 1, in the latter n2 ≥ 1). These are eigenvectors of 3 3 3 K1 K22 with eigenvalue q n2 −n1 + 2 and q n2 −n1 − 2 , respectively. To get a factor q 2 we need n2 = n1 , respectively, n2 = n1 + 3. Thus Ω(0,1) is the linear span of the vectors 1,0,+ 12 1,0,− 12 0,1,+ 12 0,1,− 12 t(n, n)j1 ,j2 ,m , t(n, n + 3)j1 ,j2 ,m , , t(n, n)j1 ,j2 ,m , t(n, n + 3)j1 ,j2 ,m

and we have the decomposition into irreducible representations of Uq (su(3)): V(n,n) ⊕ V(n,n+3) . Ω(0,1) n≥1

n≥0

We are ready to construct a cochain complex ∂¯

∂¯

Ω(0,0) → Ω(0,1) → Ω(0,2) → 0, with dual chain complex ∂¯†

∂¯†

0 ← Ω(0,0) ← Ω(0,1) ← Ω(0,2) . Proposition 5.1. Let X and Y be the operators X := F2 F1 − 2[2]−1 F1 F2 ,

Y := E2 E1 − 2[2]−1 E1 E2 .

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The maps ∂¯ : Ω(0,0) → Ω(0,1) ,

¯ := (X ∗ a, E2 a), ∂a

(5.5a)

∂¯ : Ω(0,1) → Ω(0,2) ,

¯ := −E2 v+ − Y v− , ∂v

(5.5b)

¯ with v = (v+ , v− ), are well defined and their composition is ∂¯2 = 0, that is (Ω(0,•) , ∂) is a cochain complex. Similarly, the maps ∂¯† : Ω(0,2) → Ω(0,1) ,

∂¯† b := (−F2 b, −Y ∗ b),

(5.5c)

∂¯† : Ω(0,1) → Ω(0,0) ,

∂¯† v := X v+ + F2 v− ,

(5.5d)

are well defined and their composition is (∂¯† )2 = 0, that is (Ω(0,•) , ∂¯† ) is a chain complex. Before we prove this proposition we remark that Serre relations for Uq (su(3)) read E1 Y + X ∗ E1 = 0,

E2 X ∗ + Y E2 = 0.

(5.6)

Moreover, from the commutation relations of Uq (su(3)) we get 3

1

K1 K22 X ∗ = q 2 X ∗ K1 K22 , K1 X ∗ = q 2 X ∗ K1 , 3

K1 K22 E2 = q 2 E2 K1 K22 ,

1

K1 E2 = q − 2 E2 K1 .

(5.7)

Later on, we shall also need their coproducts: ∆X = X ⊗ K1 K2 + (K1 K2 )−1 ⊗ X +

1 − q2 (F2 K1−1 ⊗ K2 F1 − K2−1 F1 ⊗ F2 K1 ), 1 + q2 −1

∆Y = Y ⊗ K1 K2 + (K1 K2 )

(5.8)

⊗Y

2

+

1−q (E2 K1−1 ⊗ K2 E1 − K2−1 E1 ⊗ E2 K1 ). 1 + q2

Proof of Proposition 5.1. We start with ∂¯ and we ﬁrst prove that it is well ¯ satisﬁes deﬁned. For any a ∈ Ω(0,0) = A(CP2q ) we want to show that (v+ , v− ) := ∂a (0,1) . Deﬁnition (5.5a) gives v+ = X ∗ a and the deﬁning properties (5.4) of Ω ¯ v− = E2 a. These, together with the invariance of a, proves that (v+ , v− ) = ∂a satisﬁes (5.4a). Next, we consider the action of E1 and F1 . As E1 a = 0, v+ = X ∗ a = E1 E2 a = E1 v− , and since [F1 , E2 ] = 0 and F1 a = 0, we have also F1 v− = E2 F1 a = 0. Thus, two of conditions (5.4b) are satisﬁed. From relations (2.1) we get 0 = (E12 E2 − [2]E1 E2 E1 + E2 E12 ) a = E12 E2 a = E1 v+ ,

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having used E1 a = 0. The commutation rule [F1 , X ∗ ] = [2]−1 E2 (q −1 K12 + qK1−2 ) yields: F1 v+ = [F1 , X ∗ ] a = [2]−1 E2 (q −1 K12 + qK1−2 ) a = E2 a = v− . Hence, all conditions (5.4) are proved and the map ∂¯ sends 0-forms to 1-forms. Next, we prove that (5.5b) is well deﬁned, i.e. for all v = (v+ , v− ) satisfying ¯ = E2 v+ + Y v− is in Ω(0,2) = L3 . It is Uq (su(2))(5.4), the element b := −∂v invariant: the ﬁrst identity in (5.6) gives E1 b = E1 E2 v+ + E1 Y v− = E1 E2 v+ − X ∗ E1 v− , and from E1 v− = v+ (and using also E1 v+ = 0), we get E1 b = (E1 E2 − X ∗ ) v+ = 2[2]−1 E2 E1 v+ = 0. Thus b is the highest weight vector of a representation of Uq (su(2)). Using (5.4a), we get 1

1

K1 b = K1 E2 v+ + K1 Y v− = q − 2 E2 K1 v+ + q 2 Y K1 v− = E2 v+ + Y v− = b that is the highest weight is zero and b carries the trivial representation h b = (h)b. In a similar fashion, one proves that K1 K22 b = q 3 b. We conclude that b ∈ L3 and (5.5b) maps 1-forms to 2-forms. To prove that ∂¯2 = 0 it is enough to compute the action of ∂¯2 on a 0-form a. Composition of (5.5a) and (5.5b) yields ∂¯2 a = −(E2 X ∗ + Y E2 ) a, which is zero by (5.6). We omit the proof for ∂¯† which goes along similar lines. In the commutative case, Ω(0,•) is a graded associative graded-commutative algebra. For q = 1, we know how to multiply 0-forms by 1-forms and by 2-forms (Ω(0,1) and Ω(0,2) are bimodules for A(CP2q ) = Ω(0,0) ), but we still do not know how to multiply two 1-forms. The next lemma shows how it is done. Lemma 5.2. The product of two 1-forms v = (v+ , v− ) and w = (w+ , w− ), defined by 1 2 1 (q 2 v+ w− − q − 2 v− w+ ), v ∧q w := [2] is a 2-form, that is an element of Ω(0,2) = L3 . 3

Proof. Clearly K1 (v ∧q w) = v ∧q w and K2 (v ∧q w) = q 2 v ∧q w. Further by (5.4), which are satisﬁed by both v and w, 1

1

2−1 [2]E1 (v ∧q w) = q 2 (K1−1 v+ )(E1 w− ) + q − 2 (E1 v− )(K1 w+ ) = v+ w+ − v+ w+ = 0.

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Hence v ∧q w is the highest weight vector of the trivial representation of Uq (su(2)), which in particular means that F1 (v ∧q w) = 0. If ω = (a, v, b) is a general element of Ω(0,•) , with a of degree zero, v of degree 1 and b of degree 2, the algebra structure of Ω(0,•) is ω1 · ω2 = (a1 a2 , a1 v2 + v1 a2 , a1 b2 + b1 a2 + v1 ∧q v2 ). It is easy to see that this product is associative, thus making Ω(0,•) a graded associative algebra (clearly it is not graded-commutative). This algebra carries a left action of Uq (su(3)): the white action acting on components; it is a module ∗algebra for this action. Using the faithful Haar functional ϕ of SUq (3) we deﬁne a nondegenerate inner product on forms, ∗ ∗ v2+ + v1− v2− ) + ϕ(b∗1 b2 ),

ω1 , ω2 := ϕ(a∗1 a2 ) + ϕ(v1+

(5.9)

with respect to which the action of Uq (su(3)) is unitary, that is it corresponds to a ∗-representation (see, [7, Lemma 2.5]), and the decomposition Ω(0,•) := n Ω(0,n) is orthogonal. The operators ∂¯ and ∂¯† , being deﬁned via the black action, clearly commute with the above action of Uq (su(3)) on forms. It also follows from [7, Lemma 2.5] that h∗ v = (h )† v for all vectors v with entries in A(SUq (3)) and with respect to the inner product coming from the Haar state, and this easily implies that ∂¯† is ¯ the Hermitian conjugate of ∂. Proposition 5.3. The map ∂¯ is a graded-derivation: ¯ ¯ + (∂a)b, ¯ ∂(ab) = a(∂b)

¯ ¯ ¯ ∧q v + a(∂v), ∂(av) = (∂a)

¯ ¯ ¯ ∂(va) = (∂v)a − v ∧q (∂a),

while ∂¯† satisfy: [∂¯† , a]v = 2[2]−1 (F2 a)v− + q(X a)v+ ,

3

[∂¯† , a]c = −q 2 (F2 a , F1 F2 a)c,

for all a, b ∈ Ω(0,0) , v ∈ Ω(0,1) , c ∈ Ω(0,2) . Proof. From the formula (5.8) for the coproducts of X and Y , and by covariance ¯ of the action, w = ∂(ab) has components ∗ ∗ ∗ ∗ w+ = (X(1) a)(X(2) b) = (X a)b + a(X b),

w− = (E2 a)(K2 b) + (K2 a)(E2 b) = (E2 a)b + a(E2 b), ¯ ¯ Next, and so w = (∂a)b + a(∂b). ¯ −∂(av) = E2 (av+ ) + Y (av− ) 1

1

= q 2 (E2 a)v+ + a(E2 v+ ) + q 2 (Y a)v− + a(Y v− ) +

1 − q2 (E2 a)(K2 E1 v− ) 1 + q2

¯ + 2[2]−1 q − 12 (E2 a)v+ + q 12 (Y a)v− ; = −a(∂v)

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but Y a = E2 E1 a = −2[2]−1 X ∗ a and so ¯ ¯ − 2[2]−1 {q − 12 (E2 a)v+ − q 12 (X ∗ a)v− } ∂(av) = a(∂v) ¯ + (∂a) ¯ ∧q v. = a(∂v) Similarly, ¯ −∂(va) = E2 (v+ a) + Y (v− a) 1 1 ¯ = −(∂v)a + q − 2 v+ (E2 a) + q − 2 v− (Y a)

−

q −1 − q −1 (K E1 v− )(E2 K1 a) q −1 + q 2

1 1 ¯ = −(∂v)a + 2[2]−1 q 2 v+ (E2 a) + q − 2 v− (Y a) 1 1 ¯ = −(∂v)a + 2[2]−1 {q 2 v+ (E2 a) − q − 2 v− (X ∗ a)}

¯ ¯ = −(∂v)a + v ∧q (∂a). In the same manner one proves the identities involving [∂¯† , a]. ¯ give a left-covariant diﬀerential calculus; it is of Hence, the data (Ω(0,•) , ∂) “dimension” 2 since we are considering only the “antiholomorphic” forms. 6. The Spectral Triple One could try to deﬁne a “Dolbeault–Dirac” operator D on CP2q as ∂¯ + ∂¯† ; on a compact K¨ahler spin manifold this is proportional to the Dirac operator of the Levi–Civita connection. We start with a more general one, ¯ + s∂¯† b, s∂v), ¯ Dω := (∂¯† v, ∂a

(6.1)

where ω = (a, v, b) is a diﬀerential form, and s ∈ R+ is arbitrary for the time being. We shall be able to check the compact resolvent condition only for s = [2]/2. As shown below, for this value the square of the operator D is related to the Casimir Cq of Uq (su(3)) given in (2.5), and whose spectrum is in (2.6). Denote with H+ the Hilbert space completion of Ω(0,0) ⊕ Ω(0,2) and with H− the completion of Ω(0,1) , with respect to the inner product (5.9). Let H := H+ ⊕ H− with grading γ := 1 ⊕ −1. Proposition 6.1. For s = [2]/2 in (6.1), the datum (A(CP2q ), H, D, γ) is a 0+ dimensional Uq (su(3))-equivariant even spectral triple. The aim of this section is to prove this proposition. We have A(CP2q ) ⊂ A(SUq (3)) and H ⊂ L2 (SUq (3), ϕ)4 . The diagonal lift of the left regular representation of A(SUq (3)) to L2 (SUq (3), ϕ)4 is bounded, thus the representation of

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A(CP2q ) is bounded too. By Proposition 5.3, the commutator [D, a] acts on forms via left multiplication by elements of A(SUq (3)), ¯ a]b + s[∂¯† , a]c, s[∂, ¯ a]v , ∀ ω = (b, v, c), [D, a]ω = [∂¯† , a]v, [∂, and is bounded for any a ∈ A(CP2q ). Equivariance holds because forms are deﬁned as equivariant A(CP2q )-modules, and the operators ∂¯ and ∂¯† are Uq (su(3))-invariant. Last step is to check that D has a compact resolvent: we do this by diagonalizing it, which also guarantees the existence a self-adjoint extension. Classically, the K¨ahler Laplacian D2 is half the Laplace–Beltrami operator ∆ = † dd + d† d, which in turn, on a symmetric space is related to the quadratic Casimir of the symmetry algebra. A similar property holds in the present case. Proposition 6.2. For s = [2]/2, the operator ∆∂¯ := D2 is given by ∆∂¯ ω = [2]−1 (Cq − 2) ω, for all ω ∈ Ω(0,•) . Proof. Let a be a 0-form, v a 1-form and b a 2-form. We need to show that ¯ = [2]−1 (Cq − 2) a, ∆∂¯a = ∂¯† ∂a

(6.2a)

∆∂¯b = s2 ∂¯∂¯† b = [2]−1 (Cq − 2) b,

(6.2b)

¯ = [2]−1 (Cq − 2) v. ∆∂¯ v = (∂¯∂¯† + s2 ∂¯† ∂)v

(6.2c)

In the following, when acting with elements of Uq (su(3)) on forms, the black action is understood and the symbol is often omitted. Step 1: Proof of (6.2a). From the deﬁnition (5.5a) and (5.5d) we have ∂¯† ∂¯Ω(0,0) = F2 E2 + XX ∗ , while, using the invariance of 0-forms: Ki a = a, E1 a = F1 a = 0, and neglecting terms that vanish on 0-forms, we rewrite the restriction of Cq to Ω(0,0) as Cq |Ω(0,0) 2 + [2]F2 E2 + (q[F2 , F1 ]q − [F1 , F2 ]q )E1 E2 2 + [2](F2 E2 + XX ∗ ). This proves (6.2a). Step 2: Proof of (6.2b). From the deﬁnition (5.5c) and (5.5b) ∂¯∂¯† |Ω(0,2) = E2 F2 + Y Y ∗ , 3

while, using the symmetry properties of 2-forms: K1 b = b, K2 b = q 2 b and E1 b = F1 b = 0, and neglecting terms that vanish on 2-forms, we have Cq |Ω(0,2) 1 + [2]2 + [3]2 + (q 2 + q −2 )F2 E2 + ([F2 , F1 ]q − q[F1 , F2 ]q )E1 E2 1 + [2]2 + [3]2 + (q 2 + q −2 )F2 E2 + 2−1 [2]2 Y ∗ Y.

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To compare the last two equations, we need the commutator [2][Y, Y ∗ ] = [2]E2 [E1 , Y ∗ ] − 2[E1 , Y ∗ ]E2 + [2][E2 , Y ∗ ]E1 − 2E1 [E2 , Y ∗ ] = E2 F2 (qK12 + q −1 K1−2 ) − 2[2]−1 F2 (qK12 + q −1 K1−2 )E2 − F1 (K22 + K2−2 )E1 + 2[2]−1 E1 F1 (K22 + K2−2 ), which, modulo operators vanishing on Ω(0,2) , becomes [2][Y, Y ∗ ] [2]E2 F2 − 4[2]−1 F2 E2 . This yields (using −1 + [2]2 + [3]2 = [2]2 [3]) [2]2 ¯ ¯† −1 + [2]2 + [3]2 + (q 2 + q −2 )F2 E2 Cq − ∂∂ − 2 2 Ω(0,2) − 2−1 [2]2 ([Y, Y ∗ ] + E2 F2 ) K22 − K2−2 2 2 0. [2] ([3] − [E2 , F2 ]) [2] [3] − q − q −1 Then on 2-forms, Cq − 2 = 2−1 [2]2 ∂¯∂¯† , which gives (6.2b) iff s2 = [2]/2. Step 3: Proof of (6.2c). From now on, s = [2]/2 is ﬁxed. Let w := (∂¯∂¯† + ¯ Then by deﬁnition (5.5), and using v− = F1 v+ and v+ = E1 v− , we s2 ∂¯† ∂)v. get w+ = (X ∗ X + s2 F2 E2 ) v+ + (X ∗ F2 + s2 F2 Y ) v− = {X ∗ (X + F2 F1 ) + s2 F2 (E2 + Y F1 )} v+ , w− = (E2 X + s2 Y ∗ E2 ) v+ + (E2 F2 + s2 Y ∗ Y ) v− = {E2 (XE1 + F2 ) + s2 Y ∗ (E2 E1 + Y )} v− . Using Ki v+ = q 1/2 v+ and E1 v+ = 0, we get [2][X ∗ , F2 F1 ] v+ = (q 2 + q −2 − 2F2 E2 ) v+ ,

[Y, F1 ] v+ = E2 v+ ,

as well as (the action v+ is omitted) [2][X, X ∗ ] = [2][X, E1 ]E2 − 2E2 [X, E1 ] + [2]E1 [X, E2 ] − 2[X, E2 ]E1 = F2 (K12 + K1−2 )E2 − 2[2]−1 E2 F2 (K12 + K1−2 ) − E1 F1 (qK22 + q −1 K2−2 ) + 2[2]−1 F1 (qK22 + q −1 K2−2 )E1 = 2[F2 , E2 ] − (q 2 + q −2 )[E1 , F1 ] = −2 − q 2 − q −2 = −[2]2 , that is X ∗ X v+ = (XX ∗ + [2]) v+ . Therefore, [2]w+ = {(q 2 + q −2 )F2 E2 + [2]XE1 E2 + 2[3]} v+ .

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In turn, using K1 v− = q −1/2 v− , K2 v− = qv− and F1 v− = 0, we get [Y, Y ∗ ] v− = v− ,

[X, E1 ] v− = F2 v− ,

∗

[Y , E2 E1 ] v− = (1 − 2[2]−1 E2 F2 ) v− . Thus [2]w− = {[2]E2 F2 + {(q 2 + q −2 )E2 E1 − [2]E1 E2 }F1 F2 } v− . On the other hand, for the action of Cq on v+ we get (Cq − 2) v+ = {2[2]2 − 2 + (q 2 + q −2 )F2 E2 + (q[F2 , F1 ]q − [F1 , F2 ]q )E1 E2 } v+ = {2[3] + (q 2 + q −2 )F2 E2 + [2]XE1 E2 } v+ = [2]w+ , while for the action on v− we get (Cq − 2) v− = {2(q 2 + q −2 − 1) + [2]F2 E2 + [F2 , F1 ]q [E1 , E2 ]q + q 2 [F1 , F2 ]q [E2 , E1 ]q } v− . To simplify last equation we need some extra work. Firstly, [[F2 , F1 ]q , [E1 , E2 ]q ] = (F2 E2 − q −2 E2 F2 )K1−2 − (E1 F1 − q −2 F1 E1 )K22 qF2 E2 − q −1 E2 F2 + 1, [[F1 , F2 ]q , [E2 , E1 ]q ] = (F1 E1 − q −2 E1 F1 )K2−2 − (E2 F2 − q −2 F2 E2 )K12 q −2 − q −1 E2 F2 + q −3 F2 E2 , where now the symbol “” means that we are neglecting operators vanishing on v− . Using these commutation relations, we arrive at (Cq − 2) v− = {[2]E2 F2 + [E1 , E2 ]q [F2 , F1 ]q + q 2 [E2 , E1 ]q [F1 , F2 ]q } v− = {[2]E2 F2 + {(q 2 + q −2 )E2 E1 − [2]E1 E2 }F1 F2 } v− = [2]w− . This concludes the proof. From now on, s =

[2]/2 is ﬁxed.

Lemma 6.3. The kernel of D are the constant 0-forms, while its non-zero eigenvalues are 2 [n][n + 2] with multiplicity (n + 1)3 , ± [2] 1 ± [n + 1][n + 2] with multiplicity n(n + 3)(2n + 3), 2 for all n ≥ 1.

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Proof. The crucial property is Cq ψ = Cq ψ, cf. Lemma 3.1. For the action “ ” we have a decomposition into irreducible representations of Uq (su(3)) as Ω(0,0)

V(n,n) ,

n≥0

Ω(0,1)

n≥1

Ω(0,2)

V(n,n) ⊕

V(n,n+3) ,

n≥0

V(n,n+3) .

n≥0

These two observations allow us to compute the spectrum of the operator [2]D2 = (Cq − 2) . Its eigenvalues are {0, αn , βm }n≥1,m≥0 , given with their multiplicities by (cf. eq. (2.6)) 0,

mult. = 1, 2

αn := 2[n + 1] − 2 = 2[n][n + 2], mult. = 2(n + 1)3 , βm := [m + 2]2 + [m + 3]2 − 1

mult. = (m + 1)(m + 4)(2m + 5).

= [2][m + 2][m + 3], Since D is odd, its spectrum is symmetric (Dv = λv implies Dγv = −λγv). Thus, ker D = ker D2 is the subspace V(0,0) made of constant 0-forms, and pos1/2 1/2 itive roots and negative roots ±αn and ±βm appear in the spectrum with the same multiplicity. Since the eigenvalues of D grows exponentially (counting multiplicities), we conclude that (D+i)− is of trace class for any > 0 and the metric dimension is 0+ . In particular, D has compact resolvent. This concludes the proof of Proposition 6.1. We stress that the spectrum of D is a q-deformation of the classical one [9]. The Connes’ diﬀerential calculus associated with D is left-covariant, and it would be interesting to compare it with the ﬁrst order covariant diﬀerential calculi studied in [20]. As a byproduct of Lemma 6.3 we compute the cohomology H∂•¯ (CP2q ) of the ¯ in Proposition 5.1. The property that allows us to compute it complex (Ω(0,•) , ∂) is an analogue of Hodge decomposition theorem. We call harmonic n-forms the collection Hn := {ω ∈ Ω(0,n) | Dω = 0}; . ¯ and ∂¯† ω with Since for a homogeneous form ω, Dω is the sum of two pieces ∂ω diﬀerent degree, both must vanish in order for Dω to be zero. Thus, ω is harmonic ¯ = ∂¯† ω = 0. iff ∂ω

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Proposition 6.4. For all n, there is an orthogonal decomposition ¯ (0,n−1) ⊕ ∂¯† Ω(0,n+1) . Ω(0,n) = Hn ⊕ ∂Ω

(6.3)

In particular, this means that there is exactly one harmonic form for each cohomology class: H∂n¯ (CP2q ) Hn = ker D|Ω(0,n) . Proof. Given two forms ω1 , ω2 of degree n − 1 and n + 1, respectively, we have that ¯ 1 , ∂¯† ω2 = ∂¯2 ω1 , ω2 = 0, ∂ω ¯ (0,n−1) and ∂¯† Ω(0,n+1) are with the inner product deﬁned in (5.9). Thus ∂Ω orthogonal. ¯ (0,n−1) and It remains to show that an n-form η is orthogonal to both ∂Ω † (0,n+1) ¯ iff it is harmonic. This follows from nondegeneracy of the inner product: ∂ Ω we have ¯ 1 = ∂¯† η, ω1 = 0, η, ∂ω

¯ ω2 = 0, η, ∂¯† ω2 = ∂η,

¯ = ∂¯† η = 0, that is iff η is harmonic. for all ω1 ∈ Ω(0,n−1) and ω2 ∈ Ω(0,n+1) , iff ∂η This establishes the orthogonal decomposition in (6.3). ¯ (0,n−1) are ∂-closed ¯ by construction. On the Forms in the subspace Hn ⊕ ∂Ω † (0,n+1) ¯ ¯ must be harmonic. Orthogonality of other hand, a ∂-closed form ω ∈ ∂ Ω the decomposition forces it to vanish. It follows that ¯ (0,n−1) /∂Ω ¯ (0,n−1) = Hn , H∂n¯ (CP2q ) = Hn ⊕ ∂Ω and this concludes the proof. An immediate consequence of this proposition and of Lemma 6.3, is that H∂0¯ (CP2q ) = C,

H∂1¯ (CP2q ) = H∂2¯ (CP2q ) = 0.

Acknowledgments The work of FD was partially supported by the ‘Belgian project IAP — NOSY’. The work of LD and GL was partially supported by the ‘Italian project PRIN06 — Noncommutative geometry, quantum groups and applications’. LD acknowledges partial support from the grant PL N201177033.

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Appendix. Antiholomorphic Forms as Equivariant Maps In this appendix, we describe the identiﬁcation of antiholomorphic forms on the undeformed CP2 with suitable equivariant maps on bundles over this manifold. In Sec. 5 this was the motivation to deﬁne antiholomorphic forms on CP2q as equivariant maps. We denote by A(SU (3)) ⊂ C ∞ (SU (3)) the algebra of polynomials in the coordinate functions u = (ukj )k,j=1,2,3 which associate to g ∈ SU (3) its matrix entries: ukj (g) := gjk . Abstractly, A(SU (3)) is the ∗-algebra generated by elements ukj for k, j = 1, 2, 3, with relations ukj uhl = uhl ukj ,

(−1)|π| u1π(1) u2π(2) u3π(3) = 1,

π∈S3

where S3 are all permutations π of three elements and |π| is the sign of π. The real structure is (ukj )∗ = (−1)j−k (ukl11 ukl22 − ukl21 ukl12 ), where {k1 , k2 } = {1, 2, 3}\{k} and {l1 , l2 } = {1, 2, 3}\{j} (as ordered sets). The above relations are just the statements that uu† = u† u = 1. As a Hopf algebra, A(SU (3)) has usual coproduct, counit and antipode, which are obtained by dualizing the group operations. The Hopf-algebra U (sl(3)) is generated by six elements H1 , H2 , E1 , E2 , F1 , F2 subject to the relations coming from Serre’s presentation: [Hk , Ek ] = 2Ek ,

[Hk , Fk ] = −2Fk ,

[Ek , Fk ] = Hk ,

[Hk , Hj ] = 0,

[Ek , Fj ] = 0,

[Hk , Ej ] = −Ej ,

[Hk , Fj ] = Fj ,

(ad Ek )2 (Fj ) = 0,

(ad Fk )2 (Ej ) = 0,

for all k, j = 1, 2, with k = j. Coproduct, counit and antipode are the standard ones for a universal enveloping algebra. The ∗-structure corresponding to the real form U (su(3)) of U (sl(3)) is given by Hk∗ := Hk and Ek∗ := Fk . The Lie algebra su(3) is recovered as the set of primitive elements satisfying h∗ = −h. Thus, Hk , Ek and Fk generate the Lie algebra sl(3) while su(3) is the linear span of iHk , i(Ek + Fk ) and (Ek − Fk ). Of course this is an example of a general statement. The Hopf algebra U (su(3)) is (cum grano salis) the “classical limit q → 1” of the Hopf algebra Uq (su(3)) described in Sec. 2 and can be obtained from it at the level of formal power series in := log q, by setting Kk = q Hk /2 and truncating at the 0th order in .

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The fundamental ∗-representation σ : U (su(3)) → Mat3 (C) is given by     −1 0 0 0 0 0 σ(H1 ) =  0 1 0, σ(H2 ) = 0 −1 0, 0 0 0 0 0 1     0 0 0 0 0 0 σ(E1 ) = 1 0 0, σ(E2 ) = 0 0 0. 0 0 0 0 1 0 With these, the pairing , : U (su(3)) × C ∞ (SU (3)) → C deﬁned by d

X, f = f (etσ(X) ), for all X ∈ su(3) dt t=0 becomes X, ukj = σjk (X) on generators, and is nondegenerate when restricted to A(SU (3)). The actions of U (su(3)) on C ∞ (SU (3)) via left (respectively, right) invariant vector ﬁelds are given by d d tσ(X) f (g e ), (X f )(g) = f (e−tσ(X) g), (X f )(g) = dt dt t=0

t=0

and are the q → 1 limit of the corresponding actions of Uq (su(3)) described in Sec. 3, as one can see by computing them for a pair of generators. Note that a left (respectively, right) invariant vector ﬁeld generates a right (respectively, left) multiplication on the group but a left (respectively, right) action on functions. In the limit the map ϑ in (2.2) is simply the ∗-structure on the real vector space su(3), extended as a linear antimultiplicative map to the whole of U (su(3)); thus σ(ϑ(X)) = σ(X)∗ = −σ(X) for all X ∈ su(3). Functions on the sphere S 5 are identiﬁed with functions on SU (3) which are annihilated by the action of H1 , E1 , F1 . They are generated by zk = u3k , k = 1, 2, 3, for which one has that k zk zk∗ = det(u) = 1. Functions on CP2 = S 5 /S 1 are identiﬁed with functions on S 5 which are annihilated by the action of H2 . They are generated by pkj = zk∗ zj and correspond to the identiﬁcation of CP2 as a real manifold with the space of 3 × 3 projections of rank 1; we denote A(CP2 ) the coordinate ∗-algebra generated by p = (pkj ). Homogeneous coordinates on CP2 are classes [x1 , x2 , x3 ], where (x1 , x2 , x3 ) ∈ 3 C \{0} and [x1 , x2 , x3 ] = [λx1 , λx2 , λx3 ] for λ ∈ C∗ . We can always choose a representative (x1 , x2 , x3 ) ∈ S 5 . Local coordinates are given by xj /xk , in the chart (k) (k) Uk deﬁned by xk = 0. Local coordinate functions Uk are the functions {Z1 , Z2 } associating to each point its local coordinate, thus (1)

Z1

(2)

Z1

(3)

Z1

(1)

= z2 /z1 , Z2

(2)

= z1 /z2 , Z2

(3)

= z1 /z3 , Z2

= z3 /z1 , on U1 , = z3 /z2 , on U2 , = z2 /z3 , on U3 ,

with zk the generators of A(S 5 ). Transition functions are clearly holomorphic.

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An antiholomorphic 1-form is written as a collection ω = (ω (j) ) and on each chart Uj , (j)

(j)

(j)

(j)

ω (j) = f1 dZ¯1 + f2 dZ¯2 , (j)

(j)

(j)

(j)

where the coeﬃcients (f1 , f2 ) are smooth complex functions (of Z1 , Z2 , (j) (j) Z¯1 , Z¯2 ) that must satisfy — in order for ω to be uniquely deﬁned — on each (j) (j) (k) (k) overlap Uj ∩ Uk , the conditions (f1 , f2 )gjk = (f1 , f2 ), with gjk : Uj ∩ Uk → GL(2, C) given by ! (j) " (k) (j) (k) dZ¯1 /dZ¯1 dZ¯1 /dZ¯2 . gjk = (j) (k) (j) (k) dZ¯2 /dZ¯1 dZ¯2 /dZ¯2 Explicitly: g12 =

−1 g21

=

z¯2 /¯ z12

−¯ z2 −¯ z3

0 , z¯1

g23 =

−1 g31 = g13 = z¯1 /¯ z32

0 z¯3

−1 g32

=

z¯3 /¯ z22

z¯2 0

−¯ z1 . −¯ z2

−¯ z1 , −¯ z3

(j)

The functions fk can be extended to global C ∞ -functions on CP2 vanishing when (1) zj = 0. For example in the limit z2 → 0, corresponding to Z1 → 0, the functions (1) (1) (f1 , f2 ) are well deﬁned and ﬁnite while g12 vanishes; thus from the equality (2) (2) (1) (1) (2) (2) (f1 , f2 ) = (f1 , f2 )g12 we deduce that (f1 , f2 ) vanish too for z2 → 0. We conclude that, as a C ∞ (CP2 )-bimodule, # (j) (j) (j) Ω(0,1) (fk )i=1,2, j=1,2,3 | fk ∈ C ∞ (CP2 ), fk |zj =0 = 0, $ (j) (j) (k) (k) (f1 , f2 )gjk = (f1 , f2 ), ∀ i, j, k . With τ the spin 1/2 representation of the algebra U (su(2)) generated by H1 , E1 , F1 , consider now the C ∞ (CP2 )-bimodule: Γ := {v = (v+ , v− ) ∈ C ∞ (SU (3))2 |(H1 + 2H2 ) v = 3v, (h(1) v)τ (S(h(2) )) = (h)v, ∀ h ∈ U (su(2))}; namely, elements of Γ are vectors v = (v+ , v− ) ∈ C ∞ (SU (3))2 satisfying the conditions: H1 (v+ , v− ) = (v+ , −v− ), F1 (v+ , v− ) = (v− , 0),

E1 (v+ , v− ) = (0, v+ ), (H1 + 2H2 ) (v+ , v− ) = 3(v+ , v− ).

(A.1a) (A.1b)

The bimodule Γ is the q → 1 limit of the bimodule in the right-hand side of (5.3). The following result is just the motivation for the identiﬁcation of that bimodule as the bimodule of antiholomorphic 1-forms.

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Proposition A.1. There is an isomorphism of C ∞ (CP2 )-bimodules ψ : Γ → Ω(0,1) given by (j)

where P (j)

(j)

(v+ , v− ) → (f1 , f2 ) = ψ(v+ , v− )(j) := (v+ , v− )P (j) , ∈ Mat2 C ∞ (SU (3)) are the following matrices: 1 u1l uk P (j) := z¯j , j = 1, 2, 3, −u2k −u2l

with {j, k, l} the permutation of {1, 2, 3} with k < l. Under this isomorphism the operator ∂¯ becomes the q → 1 limit of the operator (5.5a), that is: (j)

(j)

ψ([E1 , E2 ] a, E2 a)(j) = (∂a/∂ Z¯k , ∂a/∂ Z¯2 ),

(A.2)

for all a ∈ C ∞ (CP2 ). Proof. Since the algebra of functions is commutative, ψ is a bimodule map (rather (j) than just a left module map). A priori, ψ maps (v+ , v− ) ∈ Γ into functions fk ∈ (j) (j) C ∞ (SU (3)), with (f1 , f2 ) := (v+ , v− )P (j) . As we shall prove presently, the image (0,1) . of ψ is indeed in Ω (j) Firstly, the function fk vanishes for zj = 0, since the matrix P (j) vanishes (j) (j) (k) (k) there. The relation (f1 , f2 )gjk = (f1 , f2 ) follows from the property P (k) = P (j) gjk , which is straightforward to check; for instance: 1 u13 −¯ z2 0 u2 P (1) g12 = z¯2 /¯ z1 −u22 −u23 −¯ z3 z¯1 −¯ z2 u12 − z¯3 u13 z¯1 u13 = z¯2 /¯ z1 , z¯2 u22 + z¯3 u23 −¯ z1 u23 and the last matrix is just P (2) From the properties: −1 (j) H1 P = 0

since

3 ¯k ujk k=1 z

=

3 k=1

u ¯3k ujk = 0 for j = 3.

0 0 −1 (j) (j) P , E1 P = P (j) , 1 0 0 0 0 F1 P (j) = P (j) , −1 0

and (H1 + 2H2 ) P (j) = −3P (j) , together with the relations (A.1) we deduce that (j) the functions fk are annihilated by K1 , K2 , E1 , F1 : they are functions on CP2 , which proves Im(ψ) ⊂ Ω(0,1) . zj )3 , the matrix P (j) is invertible on Uj and ψ is an Since det P (j) = (−1)j (¯ isomorphism. By Leibniz rule, it is enough to prove (A.2) in the case a = pkl . Being U1 dense in CP2 , two forms are equal iff they are equal on U1 , and we can also assume

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j = 1. On the other hand, ([E1 , E2 ], E2 ) pkl = (¯ u1k , −¯ u2k )zl and the left-hand side of (A.2) is u1k , −¯ u2k )zj P (1) = p1j (δi2 − pi2 , δi3 − pi3 ). ψ([E1 , E2 ] pij , E2 pij )(1) = (¯ Now, in local coordinates the projection p = (pij ) is given by,   1 1  ¯ (1)  (1) (1) p= 1 Z1 Z2 . Z (1) 2 (1) 2  1  (1) 1 + |Z1 | + |Z2 | Z¯2 Thus

 0 ∂p  = p11 /p21 (1) ∂ Z¯ 1

To show that

 1

 p − p12 p,

 0 ∂p  = p11 /p31 (1) ∂ Z¯

 0

 p − p13 p.

2 0 1 ∂ , ¯∂(1) p = ψ([E1 , E2 ] p, E2 p)(1) is now a simple algebraic ¯ (1)

∂ Z1

∂ Z2

manipulation. This concludes the proof. Since antiholomorphic 2-forms are the wedge product of two antiholomorphic 1-forms, its easy to identify them with equivariant maps: they are invariant under U (su(2)) since ∧2 τ = is the trivial representation, while H1 + 2H2 acts as multiplication by 6, that is: Ω(0,2) {a ∈ A(S 5 ) | (H1 + 2H2 ) a = 6a}. This is just the identiﬁcation of antiholomorphic 2-forms with the q → 1 limit of the bimodule L3 deﬁned in (5.1) as mentioned in Sec. 5. References [1] D. Arnaudon, Non-integrable representations of the restricted quantum analogue of sl(3), J. Phys. A 30(10) (1997) 3527–3541; doi:10.1088/0305-4470/30/10/027. [2] P. S. Chakraborty and A. Pal, Equivariant spectral triples on the quantum SU (2) group, K-Theory 28(2) (2003) 107–126; doi:10.1023/A:1024571719032; arxiv:math/0201004. [3] A. Connes, Noncommutative Geometry (Academic Press, 1994). [4] L. D¸abrowski, G. Landi, M. Paschke and A. Sitarz, The spectral geometry of the equatorial Podle´s sphere, C. R. Acad. Sci. Paris 340(11) (2005) 819–822; doi:10.1016/j.crma.2005.04.003; arxiv:math/0408034v2. [5] L. D¸abrowski, G. Landi, A. Sitarz, W. van Suijlekom and J. C. V´ arilly, The Dirac operator on SUq (2), Comm. Math. Phys. 259(3) (2005) 729–759; doi:10.1007/s00220005-1383-9; arxiv:math/0411609. [6] L. D¸abrowski and A. Sitarz, Dirac operator on the standard Podle´s quantum sphere, in Noncommutative Geometry and Quantum Groups, Vol. 61 (Banach Center Publ., 2003), pp. 49–58; arxiv:math/0209048. [7] F. D’Andrea, L. D¸abrowski and G. Landi, The isospectral dirac operator on the 4-dimensional orthogonal quantum sphere, Commun. Math. Phys. 279(1) (2008) 77– 116; doi:10.1007/s00220-008-0420-x; arxiv:math.QA/0611100.

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[8] F. D’Andrea, L. D¸abrowski, G. Landi and E. Wagner, Dirac operators on all Podle´s spheres, J. Noncommut. Geom. 1(2) (2007) 213–239; arxiv:math/0606480. [9] H. Grosse and A. Strohmaier, Noncommutative geometry and the regularization problem of 4D quantum ﬁeld theory, Lett. Math. Phys. 48(2) (1999) 163–179; doi:10.1023/A:1007518622795; arxiv:hep-th/9902138. [10] I. Heckenberger and S. Kolb, The locally ﬁnite part of the dual coalgebra of quantized irreducible ﬂag manifolds, Proc. London Math. Soc. 89(2) (2005) 457–484; doi:10.1112/S0024611504014777; arxiv:math/0301244. [11] A. Klimyk and K. Schm¨ udgen, Quantum Groups and Their Representations, Texts and Monographs in Physics (Springer, 1997). [12] T. H. Koornwinder, General compact quantum groups and q-special functions, in Representations of Lie Groups and Quantum Groups, Pitman Research Notes in Mathematics Series, Vol. 311 (Longman Scientiﬁc & Technical, 1994), pp. 46–128; arxiv:hep-th/9401114 and arxiv:math.CA/9403216. [13] U. Kr¨ ahmer, Dirac operators on quantum ﬂag manifolds, Lett. Math. Phys. 67(1) (2004) 49–59; doi:10.1023/B:MATH.0000027748.64886.23; arxiv:math/0305071. [14] U. Meyer, Projective quantum spaces, Lett. Math. Phys. 35(2) (1995) 91–97; doi:10.1007/BF00750759; arxiv:hep-th/9410039. [15] S. Neshveyev and L. Tuset, A local index formula for the quantum sphere, Comm. Math. Phys. 254(2) (2005) 323–341; doi:10.1007/s00220-004-1154-z; arxiv:math.QA/ 0309275. [16] N. Yu. Reshetikhin, L. Takhtadzhyan and L. D. Fadeev, Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990) 193–225. [17] M. J. Rodr´ıguez-Plaza, Casimir operators of Uq (sl(3)), J. Math. Phys. 32(8) (1991) 2020–2027; doi:10.1063/1.529497. [18] K. Schm¨ udgen and E. Wagner, Dirac operator and a twisted cyclic cocycle on the standard Podle´s quantum sphere, J. Reine Angew. Math. 574 (2004) 219–235; doi:10.1515/crll.2004.072; arxiv:math/0305051. [19] L. Vaksman and Ya. Soibelman, The algebra of functions on the quantum group SU (n+1) and odd-dimensional quantum spheres, Leningrad Math. J. 2 (1991) 1023– 1042. [20] M. Welk, Diﬀerential calculus on quantum projective spaces, Czech. J. Phys. 50(1) (2000) 219–224; doi:10.1023/A:1022870308859; arxiv:math/9908069.

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Reviews in Mathematical Physics Vol. 20, No. 8 (2008) 1007–1032 c World Scientific Publishing Company

STATIONARY STATES OF NONLINEAR DIRAC EQUATIONS WITH GENERAL POTENTIALS

YANHENG DING∗ and JUNCHENG WEI† ∗Institute

of Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100080, P. R. China and

International Center for Theoretical Physics, Trieste, Italy †Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong ∗[email protected] †[email protected]

Received 27 June 2008 Revised 31 July 2008 We establish the existence of stationary states for the following nonlinear Dirac equation 8 3 > X > <−i αk ∂k u + aβu + M (x)u = g(x, |u|)u for x ∈ R3 , k=1 > > :u(x) → 0 as |x| → ∞, with real matrix potential M (x) and superlinearity g(x, |u|)u both without periodicity assumptions, via variational methods. Keywords: Nonlinear Dirac equation; non-periodic potential; superlinearity. Mathematics Subject Classifications 2000: 35Q40, 49J35

1. Introduction and Main Results Of concern is the existence of solutions to the following nonlinear Dirac equations  3  −i α ∂ u + aβu + M (x)u = g(x, |u|)u for x ∈ R3 , k k (1.1) k=1   u(x) → 0 as |x| → ∞, where x = (x1 , x2 , x3 ) ∈ R3 , u(x) ∈ C4 , ∂k = ∂x∂ k , a is a positive constant, α1 , α2 , α3 and β are 4 × 4 complex matrices (in 2 × 2 blocks): I 0 0 σk β= , αk = , k = 1, 2, 3 0 −I σk 0 1007

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with

0 σ1 = 1

1 , 0

0 −i σ2 = , i 0

1 0 σ3 = , 0 −1

M (x) denotes a 4 × 4 real symmetric matrix valued function, and g ∈ C(R3 × R+ , R+ ), R+ := [0, ∞). In physics, M (x) represents the external potential. See [38]. Problem (1.1) arises in the study of stationary solutions to the nonlinear Dirac equation which models extended relativistic particles in external ﬁelds and has been used as eﬀective theories in atomic, nuclear and gravitational physics (see [9, 29, 18, 16]). Its most general form is −i∂t ψ = ic

3

αk ∂k ψ − mc2 βψ − P (x)ψ + Gψ (x, ψ).

(1.2)

k=1

Here c denotes the speed of light, m > 0 is the mass of the electron, denotes Planck’s constant, the 4 × 4 real symmetric matrix P (x) stands for the external ﬁeld, and the nonlinearity G : R3 × C4 → R represents a nonlinear self-coupling. A solution ψ : R × R3 → C4 of (1.2), with ψ(t, ·) ∈ L2 (R3 , C4 ), is a wave function which represents the state of a relativistic electron. Assuming that G satisﬁes G(x, eiθ ψ) = G(x, ψ) for all θ ∈ [0, 2π], one is ﬁnding solutions of (1.2) with the iθt form ψ(t, x) = e u(x) which may be regarded as “particle-like solutions” (see [29]): they propagate without changing their shape and thus have a soliton-like behavior. Then u : R3 → C4 satisﬁes the equation −i

3

˜ u (x, u) αk ∂k u + aβu + M (x)u = G

for x ∈ R3

(1.3)

k=1

˜ u (x, u) = Gu (x, u)/c. with a = mc/, M (x) = P (x)/c + θI4 and G Mathematically, there are new diﬃculties in using the Calculus of Variations to ﬁnd solutions to Problem (1.3). First, the energy functional (see (1.6) below) is strongly indeﬁnite: it is unbounded from below and all its critical points have indeﬁnite Morse index. The second diﬃculty is the lack of compactness: the Palais– Smale condition is not satisﬁed due to the unboundedness of the domain R3 . The combination of the above types of diﬃculties poses a challenge in the Calculus of Variations. As a result, many authors have developed new methods and techniques to study (1.3). We summarize the research on the existence (and multiplicity) of solutions to problems of form (1.3) for particularly the following three cases. Case 1: The autonomous system, that is, M = ωI4 (ω a constant and I4 the 4 × 4 ˜ does not depend on x. In [4, 5, 10, 27] the authors studied identity matrix), and G ˜ having the form the problem with ω ∈ (−a, 0) and G 1 ˜ uu), G(u) = H(˜ 2

H ∈ C 2 (R, R),

H(0) = 0;

here u ˜u := (βu, u)C4 ,

(1.4)

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by using shooting methods. (This is the so-called Soler model [35].) Finkelstein ˜ of the form et al. [19] considered the nonlinearity G 1 ˜ G(u) = |˜ uαu|2 , uu|2 + b|˜ 2

u ˜αu := (βu, αu)C4 ,

α := α1 α2 α3

(1.5)

˜ with b > 0. In [17] Esteban and S´er´e treated the equation with G(u) of form (1.4) under the main additional assumption that H (s) s ≥ θ H(s) for some θ > 1, and all s ∈ R. [17] also considered nonlinearities of type (1.5), however with a weaker growth ˜ G(u) = µ|˜ uu|τ + b|˜ uαu|σ ,

1 < τ, σ <

3 , 2

µ, b > 0;

˜ they also investigated a more general G(u) growing likely |u|p , p ∈ (2, 3), as |u| → ∞. ˜ u) depend periodically on x. Case 2: The periodic system, that is, M (x) and G(x, In [7] Bartsch and Ding investigated this case with additionally M (x) = βV (x), V ∈ ˜ u) which may be superquadratic or asympC(R3 , R). They treated functions G(x, totically quadratic in u as |u| → ∞ and they obtained inﬁnitely many solutions if ˜ is additionally even in u. G ˜ u (x, u) is asymptotically linear as |u| → ∞. Case 3: Non periodic system but G Recently, in their paper [16], Ding and Ruf considered this case with either the vector potentials M (x) of Coulomb-type (see (M1 ) below) or the scalar potentials of the form M (x) = βV (x) satisfying roughly lim inf |x|→∞ βV (x) > 0. Under suitable assumptions they obtained the existence and multiplicity of solutions of (1.3). Now one of the remaining cases is that the potential M (x) depends explicitly ˜ u) grows on x without periodicity assumption and the nonlinear interaction G(x, super-quadratically as |u| → ∞. Indeed, as far as we know there is no existence results on (1.3) with the Coulomb-type potential (see (M1 ) below) and “powerlike” interaction |u|p , p > 2 (the so-called Soler model [35]). In the present paper we consider this case. In the following, for convenience, any real function U (x) will be regarded as the symmetric matrix U (x)I4 . For a symmetric real matrix function L(x), let λL (x) ¯ L (x)) be the minimal (respectively, the maximal) eigenvalue of L(x), (respectively, λ ¯ L (x)|}, |L|∞ := ¯L (x), |L(x)| := max{|λ (x)|, |λ inf L := inf x λL (x), sup L := supx λ L ess supx |L(x)|, and L(∞) := lim|x|→∞ L(x) if and only if |L(x) − L(∞)| → 0 as |x| → ∞. For two given symmetric real matrix functions L1 (x) and L2 (x), we write L1 (x) ≤ L2 (x) if and only if max

ξ∈C4 ,|ξ|=1

(L1 (x) − L2 (x)) ξ · ξ¯ ≤ 0.

Associated with (1.1) is the following energy functional deﬁned by

3 1 −i αk ∂k + aβ + M (x) u · u ¯ − F (x, u) dx ΦM (u) := 2 R3 k=1

(1.6)

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where

|u|

g(x, s)s ds.

F (x, u) := 0

Let cM := inf{ΦM (u) : u = 0 is a solution of (1.1)}. A solution u0 = 0 with ΦM (u0 ) = cM is called a least energy solution. Let SM denote the set of all least energy solutions of (1.1). We start with the following typical problem:  3  −i α ∂ u + aβu + M (x)u = q(x)|u|p−2 u for x ∈ R3 k k (1.7)   k=1 u(x) → 0 as |x| → ∞ with p ∈ (2, 3). We assume (q0 ) q ∈ C(R3 , R) with q(x) ≥ q0 > 0

for all x

where

q0 := lim|y|→∞ q(y).

For the vector external potentials we consider ﬁrstly the Coulomb-type: (M1 ) M is a symmetric continuous real 4 × 4-matrix function on R3 \{0} with κ where κ < 12 . 0 > M (x) ≥ − |x| Our ﬁrst theorem concerns the regularity of solutions to (1.7). Theorem 1.1. Assume that p ∈ (2, 3) and (M1 ) and (q0 ) are satisfied. Then Eq. (1.7) has at least one least energy solution u ∈ W 1,q (R3 , C4 ) for all q ≥ 2. Moreover, SM is compact in H 1 (R3 , C4 ). Theorem 1.1 applies to the physically relevant case when q(x) ≡ 1 and M (x) κ . We refer to [38, Chap. 4] for discussions on is Coulomb potential: M (x) = − |x| external ﬁelds. The restriction on κ is technical. See [7]. Next we consider the following potential: (M2 ) M is a symmetric continuous real 4×4-matrix function on R3 with |M |∞ < a, M (x) < M (∞) for all x, and either (i) M (∞) ≤ 0 or (ii) M (∞) = m∞ I4 a constant. In what follows, for describing the exponential decay of solutions, we restrict ourselves to consider the scalar potential M (x) = V (x)β or M (x) = V (x), where V ∈ W 1,∞ (R3 , R). Denote 2

E(a, M (x)) = (a + V (x)) + iβ

3

αk ∂k V (x)

if M (x) = V (x)β

k=1

and E(a, M (x)) = a2 − V (x)2

if M (x) = V (x)I4 .

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We say that E(a, M (x)) is real positive definite at infinity if there exist τ > 0 and R > 0 such that ¯ ≥ τ |ξ|2 [E(a, M (x))ξ · ξ]

for all |x| ≥ R and ξ ∈ C4 .

Assume (M3 ) Either M (x) = V (x)β or M (x) = V (x)I4 with E(a, M (x)) being real positive deﬁnite at inﬁnity. Conditions (M2 ) and (M3 ) are technical conditions which are needed for concentration-compactness. Note that the Coulomb potential satisﬁes (M2 ), (M3 ). Our second theorem concerns the exponential decay of solutions to (1.7). Theorem 1.2. Assume that p ∈ (2, 3) and (M2 ) and (q0 ) are satisfied. Then (i) Equation (1.7) has at least one least energy solution u ∈ W 1,q (R3 , C4 ) for all q ≥ 2; (ii) SM is compact in H 1 (R3 , C4 ); (iii) If additionally (M3 ) holds and q is of W 1,∞ then there exist C, c > 0 such that |u(x)| ≤ C exp(−c|x|)

for all x ∈ R3 ,

u ∈ SM .

Finally, we consider the existence of ground states under more general nonlinearities of (1.1). Assume (g1 ) g(x, s) ≥ 0, g(x, s) = o(s) as s → 0 uniformly in x, and there exist p ∈ (2, 3), c1 > 0 such that g(x, s) ≤ c1 (1 + sp−2 ); (g2 ) there is µ > 2 such that 0 < µF (x, u) ≤ g(x, |u|)|u|2 if u = 0; (s) > 0 for s > 0 such that g(x, s) → g∞ (s) (g3 ) there is g∞ ∈ C 1 (R+ , R+ ) with g∞ as |x| → ∞ uniformly on bounded sets of s, and g∞ (s) ≤ g(x, s) for all (x, s). |u| (s) = dg∞ (s)/ds and F∞ (u) := 0 g∞ (s)s ds. Here g∞ Conditions (g1 ), (g3 ) are called Ambrosetti–Rabinowitz conditions, which are assumed in saddle-type critical point theory. In particular, the nonlinearity in the so-called Soler model [35] (see (1.4)) satisﬁes (g1 )−(g3 ). Our last theorem gives the existence of ground states. Theorem 1.3. Let (g1 )−(g3 ) and either (M1 ) or (M2 ) be satisfied. Then (i) Equation (1.1) has at least one least energy solution u ∈ W 1,q (R3 , C4 ) for all q ≥ 2; (ii) SM is compact in H 1 (R3 , C4 ); (iii) If (M2 ) and (M3 ) are satisfied and g(x, s) is additionally of class C 1 on R3 × (0, ∞), then there exist C, c > 0 such that |u(x)| ≤ C exp(−c|x|)

for all x ∈ R3 ,

u ∈ SM .

It is clear that Theorems 1.1 and 1.2 are consequences of Theorem 1.3.

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Our argument is variational, which can be outlined as follows. The solutions of (1.1) are obtained as critical points of the energy functional ΦM on the space H 1/2 (R3 , C4 ). ΦM possesses the linking structure, however it does not satisfy the Palais–Smale condition in general. Thus we consider certain auxiliary problem related to the “limit equation” of (1.1) which is autonomous and whose least energy solutions with least energy Cˆ are known. It will be proved that ΦM satisﬁes the ˆ We then show that the minimax value Cerami condition (C)c at all levels c < C.

M based on the linking structure of ΦM satisﬁes 0 < M < Cˆ via a recent critical point theorem and obtain ﬁnally the solutions. For the corresponding nonlinear Schr¨ odinger equation h2 ∆u − V (x)u + f (u) = 0,

u ∈ H 1 (RN ),

(1.8)

results similar to Theorems 1.2 and 1.3 have been established previously by Rabinowitz [28] and Sirakov [37]. The existence of spike layer solutions in the semiclassical limit (i.e. h → 0) has been established under various conditions of V (x). See [2,8,13,14,20,22–24,26] and the references therein. Due to the strong indeﬁnite structure of Dirac operator, our results in this paper seem to be the ﬁrst of such for nonlinear Dirac operators. The paper is organized as follows. In Sec. 2, we formulate the variational setting and recall some critical point theorems required. We then, in Sec. 3, discuss the least energy solutions of the associated limit equation, in particular, characterize the least energy in three versions (the results of this section seems useful also for dealing with semiclassical solutions of some singularly perturbed Dirac equations). And ﬁnally, in Sec. 4, we complete the proof of the main results. 2. The Variational Setting In what follows by | · |q we denote the usual Lq -norm, and (·, ·)2 the usual L2 -inner 3 product. Let H0 = −i k=1 αk ∂k +aβ denote the selfadjoint operator on L2 (R3 , C4 ) with domain D(H0 ) = H 1 (R3 , C4 ). For any symmetric real matrix function M set HM := H0 + M with its spectrum and continuous spectrum denoted by σ(HM ) and σc (HM ), respectively. Lemma 2.1. Let M be a symmetric real matrix function. (1) σ(H0 ) = σc (H0 ) = R\(−a, a); (2) If M satisfies (M1 ) then HM is selfadjoint with D(HM ) = H 1 (R3 , C4 ) and σ(HM ) ⊂ R\(−(1 − 2κ)a, (1 − 2κ)a); (3) If M satisfies (M2 ) then HM is selfadjoint with D(HM ) = H 1 (R3 , C4 ) and σ(HM ) ⊂ R\(−a + |M |∞ ), a − |M |∞ ). Proof. (1) follows from a standard argument of Fourier analysis. We now check (2). Setting Vκ (x) := κ/|x|, it follows from (M1 ) that |M u|22 ≤ |Vκ u|22 . Since a > 0, the Kato’s inequality implies that |Vκ u|22 ≤ 4κ2 |∇u|22 ≤

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4κ2 |H0 u|22 (see [12]). Since 2κ < 1, it follows from the Kato–Rellich theorem that HM is selfadjoint. Furthermore, |HM u|2 ≥ |H0 u|2 − |M u|2 ≥ (1 − 2κ)|H0 u|2 ≥ (1 − 2κ)a|u|2 , thus, σ(HM ) ⊂ R\(−(1 − 2κ)a, (1 − 2κ)a). Similarly, one checks (3) easily. It follows from (1) of Lemma 2.1 that the space L2 possesses the orthogonal decomposition: L2 = L− ⊕ L+ ,

u = u− + u+

so that H0 is negative deﬁnite (respectively, positive deﬁnite) in L− (respectively, L+ ). Let |H0 | denote the absolute value, |H0 |1/2 the squared root, and take E = D(|H0 |1/2 ). E is a Hilbert space equipped with the inner product (u, v) = (|H0 |1/2 u, |H0 |1/2 v)2 and the induced norm u = (u, u)1/2 . E possesses the following decomposition E = E− ⊕ E+

with E ± = E ∩ L± ,

orthogonal with respect to both (·, ·)2 and (·, ·) inner products. The following lemma can be found in [7] or [15]. Lemma 2.2. E embeds continuously into H 1/2 (R3 , C4 ), hence, E embeds continuously into Lq for all q ∈ [2, 3] and compactly into Lqloc for all q ∈ [1, 3). Assuming (g1 )−(g3 ) are satisﬁed and either (M1 ) or (M2 ) holds, we deﬁne on E the following functional 1 1 + 2 u − u− 2 + M (x)u¯ u − Ψ(u) (2.1) ΦM (u) = 2 2 R3 where

Ψ(u) :=

F (x, u). R3

Then ΦM ∈ C 1 (E, R) and a standard argument shows that critical points of ΦM are solutions of (1.1). Using the operator HM one may give ΦM another representation as follows. Note that, by the (2) and (3) of Lemma 2.1, E = D(|HM |1/2 ) with the equivalent inner product (u, v)M := (|HM |1/2 u, |HM |1/2 v)2 1/2

and norm u M := (u, u)M . Then as above there is a decomposition − + ⊕ EM , E = EM

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and ΦM can be represented as 1 + 2 u M − u− 2M − Ψ(u). (2.2) 2 In order to ﬁnd critical points of ΦM we will use the following abstract theorem which is taken from [6, 15]. Let E be a Banach space with direct sum decomposition E = X ⊕ Y, u = x + y and corresponding projections PX , PY onto X, Y , respectively. For a functional Φ ∈ C 1 (E, R) we write Φa = {u ∈ E : Φ(u) ≥ a}. Recall that a sequence (un ) ⊂ E is said to be a (C)c -sequence (respectively, (P S)c -sequence) if Φ(un ) → c and (1 + un )Φ (un ) → 0 (respectively, Φ (un ) → 0). Φ is said to satisfy the (C)c condition (respectively, (P S)c -condition) if any (C)c -sequence (respectively, (P S)c sequence) has a convergent subsequence. Now we assume that X is separable and reﬂexive, and we ﬁx a countable dense subset S ⊂ X ∗ . For each s ∈ S there is a semi-norm on E deﬁned by ΦM (u) =

ps : E → R,

ps (u) = |s(x)| + y for u = x + y ∈ X ⊕ Y.

We denote by TS the induced topology. Let w∗ denote the weak*-topology on E ∗ . Suppose: (Φ0 ) There exists ζ > 0 such that u < ζ PY u for all u ∈ Φ0 . (Φ1 ) For any c ∈ R, Φc is TS -closed, and Φ : (Φc , TS ) → (E ∗ , w∗ ) is continuous. (Φ2 ) There exists ρ > 0 with κ := inf Φ(Sρ Y ) > 0 where Sρ Y := {u ∈ Y : u = ρ}. The following theorem is a special case of [6, Theorem 3.4]; see also [15, Theorem 4.3]. Theorem 2.3. Let (Φ0 )–(Φ2 ) be satisfied and suppose there are R > ρ > 0 and e ∈ Y with e = 1 such that sup Φ(∂Q) ≤ κ where Q = {u = x + te : x ∈ X, t ≥ 0, u < R}. Then Φ has a (C)c -sequence with κ ≤ c ≤ sup Φ(Q). The following lemma is useful to verify (Φ1 ) (see [6] or [15]). Lemma 2.4. Suppose Φ ∈ C 1 (E, R) is of the form Φ(u) =

1 ( y 2 − x 2 ) − Ψ(u) 2

for u = x + y ∈ E = X ⊕ Y

such that (i) Ψ ∈ C 1 (E, R) is bounded from below ; (ii) Ψ : (E, Tw ) → R is sequentially lower semicontinuous, that is, un u in E implies Ψ(u) ≤ lim inf Ψ(un ); (iii) Ψ : (E, Tw ) → (E ∗ , Tw∗ ) is sequentially continuous. (iv) ν : E → R, ν(u) = u 2 , is C 1 and ν : (E, Tw ) → (E ∗ , Tw∗ ) is sequentially continuous. Then Φ satisfies (Φ1 ).

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3. Autonomous Equation-Limit Problem In this section we study the following autonomous equation  3  −i α ∂ u + aβu + (b + L)u = g (|u|)u for x ∈ R3 ,  

k k

∞

k=1

u(x) → 0

(3.1)

as |x| → ∞,

where g∞ is the function from assumption (g3 ), b is a real number and L is a symmetric real constant matrix with b ∈ (−a, a) and b − a < L ≤ 0.

(3.2)

Without loss of generality we may assume b ≥ 0 because otherwise we consider ˜b = 0 and L ˜ = b + L (which ≤ 0 if b < 0) replacing b and L. Remark that by (3.2) the minimal eigenvalue 0 ≥ λL > b − a, hence |L| < a − b.

(3.3)

In our later application, we are interested in the situation b = 0 and L = 0 in the case (M1 ); the situation b = 0 and L = M (∞) in the case (i) of (M2 ); and the situation b = m∞ and L = 0 in the case (ii) of (M2 ). Equation (3.1) may be regarded as a “limit equation” related to (1.1). The main consideration services to constructing linking levels of the functional ΦM in the proof of main results. Although the existence of stationary solutions of (3.1) is known, we would also like to provide some minimax characterization of its least energy which seems useful in studying “semiclassical solutions” of singularly perturbed Dirac equation. Let Hb := H0 + b, a selfadjoint operator in L2 with D(Hb ) = H 1 and σ(Hb ) ⊂ R\(−a + b, a + b). We introduce on E = H 1/2 the equivalent inner product (u, v)b := (|Hb |1/2 u, |Hb |1/2 v)2

(3.4)

with the deduced norm u b := ||Hb |1/2 u|2 . Note that the decomposition E = E − ⊕ E + is also orthogonal with respect to the inner product (·, ·)b , and u± 2b = u± 2 ± b|u± |22

for u± ∈ E ±

and (3.5) u 2b ≥ (a − b)|u|22 . |u| Setting F∞ (u) = F∞ (|u|) := 0 g∞ (s)s ds, it follows from the assumption on ˆ ∈ (2, µ) such that g∞ that there is µ 0<µ ˆF∞ (u) ≤ g∞ (|u|)|u|2

whenever u = 0.

Indeed, one has g∞ (|u|)|u|2 − µ ˆ F∞ (u) = lim (g(x, |u|)|u|2 − µ ˆF (x, u)) |x|→∞

= lim (g(x, |u|)|u|2 − µF (x, u)) |x|→∞

ˆ )F (x, u) + lim (µ − µ |x|→∞

>0

(3.6)

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if u = 0. By (3.6), for any δ > 0 there is cδ > 0 such that F∞ (u) ≥ cδ |u|µˆ

for all |u| ≥ δ;

(3.7)

and, for any ε > 0, there exist Cε , cε > 0 with g∞ (|u|)|u| ≤ ε|u| + Cε |u|p−1

(3.8)

F∞ (u) ≤ ε|u|2 + Cε |u|p

(3.9)

and

for all u ∈ C4 . Moreover, setting F˜∞ (u) := 12 g∞ (|u|)|u|2 − F∞ (u), there holds µ ˆ−2 µ ˆ−2 F˜∞ (u) ≥ g∞ (|u|)|u|2 ≥ F∞ (u). 2ˆ µ 2 Note also that, σ := p/(p − 2) > 3, and for any δ > 0, if |u| ≥ δ then g∞ (|u|) ≤ cδ |u|p−2 so g∞ (|u|)σ−1 ≤ cδ |u|2 and σ g∞ (|u|)|u|2 (g∞ (|u|)|u|2 )σ−1 g∞ (|u|)σ = = g∞ (|u|)|u|2 2 |u| |u|2σ =

g∞ (|u|)σ−1 g∞ (|u|)|u|2 ≤ cδ g∞ (|u|)|u|2 |u|2

≤ cδ F˜∞ (u). Therefore, for any ε > 0 there exist ρε > 0 and cε > 0 such that g∞ (|u|) ≤ ε if |u| ≤ ρε

and g∞ (|u|) ≤ cε F˜∞ (u)1/σ

Set

if |u| ≥ ρε .

(3.10)

Ψ∞ (u) :=

R3

F∞ (u)

and deﬁne the functional 1 1 1 Φb (u) := u+ 2 − u− 2 + (b + L)u¯ u − Ψ∞ (u) 2 2 2 R3 1 + 2 1 − 2 1 Lu¯ u − Ψ∞ (u) = u b − u b + 2 2 2 R3 for u = u− + u+ ∈ E − ⊕ E + . It follows from the assumption on g∞ that Φb ∈ C 1 (E, R) and its critical points are solutions of (3.1). By (3.2), (3.7) and (3.9) it is not diﬃcult to verify that Φb possesses the linking structure, that is, for any ﬁnite dimensional subspace Z ⊂ E + , Φb (u) → −∞ as u ∈ E − ⊕ Z,

u → ∞,

and there are r > 0 and ρ > 0 such that Φb |Br ∩E + ≥ 0 and Φb |∂Br ∩E + ≥ ρ.

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Let Kb := {u ∈ E : Φb (u) = 0} be the critical set of Φb . The following lemma is an easy consequence of [7, 17]. Lemma 3.1. Kb \{0} = ∅ and Kb ⊂ q≥2 W 1,q . Denote cb := inf{Φb (u): u ∈ Kb \{0}}. Lemma 3.2. cb > 0. In particular, 0 is an isolated critical point of Φb . Proof. If u ∈ Kb , one has

1 Φb (u) = Φb (u) − Φb (u)u = 2

R3

F˜∞ (u) ≥ 0.

For proving cb > 0, assume by contradiction that cb = 0. Let uj ∈ Kb \{0} be such that Φb (uj ) → 0. Then it is not diﬃcult to check that (uj ) is bounded. We can suppose uj u ∈ Kb . Then F˜∞ (uj ) → 0. Φb (uj ) = R3

Since 0 =

Φb (uj )(u+ j

−

u− j ),

uj 2b = − ≤

(3.3) and (3.5) imply that ¯ u− + ¯ − Luj u+ − g∞ (|uj |)uj u+ j j j − uj

R3

|L| uj 2b + a−b

R3

R3

¯ − g∞ (|uj |)uj u+ j − uj .

By (3.10) and using H¨ older inequality (1/σ + 1/σ = 1, σ = p/(p − 2)), one sees

|L| ¯ − 2 1− + g∞ (|uj |)uj u+ uj b ≤ j − uj a−b |uj |≤ρε |uj |>ρε − F˜∞ (uj )1/σ |uj ||u+ ≤ ε|uj |22 + cε j − uj | R3

≤

ε|uj |22

+ c1 cε

R3

1/σ ˜ F∞ (uj ) |uj |2p

≤ c2 ε uj 2b + c3 cε Φb (uj )1/σ uj 2b hence 1 ≤ c4 ε + o(1), a contradiction. Remark 3.3. Let Sˆb denote the set of all least energy solutions u with |u(0)| = |u|∞ . Remark that as before (4.11) keeps true for u ∈ Sˆb with C0 independent of x and u ∈ Sˆb . Although we do not know if the least energy solution of (3.1) is unique up to translation, we can show the following a substitute (which will not be used for proving our main result): Lemma 3.4. Sˆb is compact in H 1 (R3 , C4 ).

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Proof. Let uj ∈ Sˆb with uj u0 in H 1 . It is clear that u0 ∈ Kb and uj → u0 in Lqloc for any q < 5. We claim ﬁrst that u0 = 0. In fact, suppose u0 = 0 and uj → 0 in Lqloc . Then 0 . But this contradicts with |uj (0)| = |uj |∞ ≥ (4.11) implies that uj → 0 in Cloc δ > 0. Thus u0 = 0 and hence Φb (u0 ) ≥ cb . Since there is no nonzero critical value of Φb less than cb and u0 = 0, it is standard to show that Φb (uj − u0 ) → cb − Φb (u0 ), Φb (uj − u0 ) → 0, and uj − u0 b → 0 (see, e.g., [15]). Denoting A = H0 + b + L and using the equation for uj and u0 , |A(uj − u0 )|2 = |g∞ (|uj |)uj − g∞ (|u0 |)u0 |2 ≤ |g∞ (|uj |)(uj − u0 )|2 + |(g∞ (|uj |) − g∞ (|u0 |))u0 |2 . Since |uj |∞ ≤ C and uj → u0 in E, |g∞ (|uj |)2 |uj − u0 |2 ≤ C|uj − u0 |2 → 0, R3

and since |u0 (x)| → 0 as |x| → ∞, |(g∞ (|uj |) − g∞ (|u0 |))u0 |2 R3

=

+ |x|
|x|≥R

|(g∞ (|uj |) − g∞ (|u0 |))u0 |2 → 0.

Therefore, one sees that |A(uj − u0 )|2 → 0 which, together with (3.3), implies |H0 (uj − u0 )|2 → 0 i.e. uj → u0 in H 1 . This proves the ﬁrst conclusion. Following Ackermann [1], for ﬁxed u ∈ E + , we introduce the functional φu: E → R by −

φu (v) := Φb (u + v) 1 ¯ v − Ψ∞ (u + v) (b + L)(u + v)u + = u 2 − v 2 + 2 R3 1 1 2 2 ¯ v − Ψ∞ (u + v). L(u + v)u + = ( u b − v b ) + 2 2 R3 One has φu (v)[w, w] = − w 2b + = − w 2b + −

R3

R3

Lww ¯ − Ψ∞ (u + v)[w, w]

R3

Lww ¯−

g∞ (|u + v|)|w|2

R3

g∞ (|u + v|) ([(u + v)w]) ¯ 2 |u + v|

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for all v, w ∈ E − , which implies that φu (·) is strictly concave (recalling that L ≤ 0). Moreover φu (v) ≤

1 ( u 2b − v 2b ) → −∞ as v → ∞. 2

Plainly, φu is weakly sequentially upper semicontinuous. Thus there is a unique strict maximum point hb (u) for φu (·), which is also the only critical point of φu on E − and satisﬁes: v = hb (u) ⇔ Φb (u + v) < Φb (u + hb (u)) − (hb (u), w)b + L(u + hb (u))w¯ =

R3

(3.11)

R3

g∞ (|u + hb (u)|)(u + hb (u))w¯

(3.12)

for all u ∈ E + and v, w ∈ E − . As [1, Lemma 5.6], we have the following: Lemma 3.5. There hold the following: (i) hb is R3 -invariant, i.e. hb (a ∗ u) = hb (u) where (a ∗ u)(x) := u(x + a) for all a ∈ R3 ; (ii) hb ∈ C 1 (E + , E − ) and hb (0) = 0; (iii) hb is a bounded map; (iv) If un u in E + , then hb (un ) − hb (un − u) → hb (u) and hb (un ) hb (u). The same is true for |hb (u)|22 . Now we deﬁne the reduce functional Ib : E + → R by Ib (u) := Φb (u + hb (u)) =

1 1 1 u 2b − hb (u) 2b + 2 2 2

R3

¯ b (u) − Ψ ˜ ∞ (u). L(u + hb (u))u + h

Observe that Ib (u)v = (u, v)b − (hb (u), hb (u)v)b + −

R3

R3

L(u + hb (u))v + h¯b (u)v

g∞ (|u + hb (u)|)(u + hb (u))v + h¯b (u)v

= Φb (u + hb (u))(v + hb (v))

(by (3.11)) ¯ b (v) = (u, v)b − (hb (u), hb (v))b + L(u + hb (u))v + h −

R3

R3

¯ b (v) g∞ (|u + hb (u)|)(u + hb (u))v + h

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for all u, v ∈ E + , and critical points of Ib and Φb are in one to one correspondence via the injective map u → u + hb (u) from E + into E, that is, letting Kb+ := {u ∈ E + : Ib (u) = 0}, one has Kb = {u + hb (u) : u ∈ Kb+ }. Next we discuss the mountain pass geometry of functional Ib . Lemma 3.6. Ib possesses the mountain pass geometry : (1) There is ρ > 0 such that inf Ib (E + ∩ ∂Bρ ) > 0; (2) For any finite dimensional subspace X ⊂ E + , Ib (u) → −∞ as u ∈ X, u → ∞. Proof. (1) We have 1 1 1 Ib (w) = w 2b − hb (w) 2b + 2 2 2

R3

¯ b (w) L(w + hb (w))w + h

− Ψ∞ (w + hb (w))

1 1 = w 2b + Lww ¯ + Φb (w + hb (w)) − Φb (w) − Ψ∞ (w) 2 2 R3 1 1 2 Lww ¯ − Ψ∞ (w) ≥ w b + 2 2 R3 ≥

|L| 2 1 w 2b − |w|2 − Ψ∞ (w) 2 2

for all w ∈ E + . The desired conclusion now follows from (3.3), (3.5) and (3.9). (2) Let P : Lµˆ → X denote the natural projection. Then there is c1 > 0 such that c1 |P v|µµˆˆ ≤ |v|µµˆˆ for all v ∈ Lµˆ . Let u ∈ X. One has by (3.3), (3.5) and (3.7), for any ε > 0, 1 1 1 2 2 ¯ b (u) L(u + hb (u))u + h Ib (u) = u b − h(u) b + 2 2 2 R3 − F∞ (u + h(u)) R3

1 1 u 2b − h(u) 2b + ε|u + hb (u)|22 − cε |u + hb (u)|µµˆˆ 2 2 1 1 ε u + hb (u) 2b − c1 cε |u|µµˆˆ ≤ u 2b − h(u) 2b + 2 2 a−b 1 1 ε ε 2 + − = u b − hb (u) 2b − cε u µbˆ , 2 a−b 2 a−b

≤

hence the conclusion is true.

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Lemma 3.2 implies 0 is an isolated critical point of Ib . Therefore there is a ν > 0 such that w ≥ ν for all nontrivial critical points w of Ib . Let Nb+ := {u ∈ E + \{0} : Ib (u)u = 0}. Lemma 3.7. For each u ∈ E + \{0}, there is a unique t = t(u) > 0 such that tu ∈ Nb+ . Proof. See [1]. We outline its proof as follows. Observe that, if z ∈ E + \{0} with Ib (z)z = 0, it is not diﬃcult to check Ib (z)[z, z] = (∇2 Ib (z)z, z)b < 0.

(3.13)

+

Let now u ∈ E \{0}. Setting f (t) = Ib (tu), one has f (0) = 0, f (t) > 0 for t > 0 suﬃciently small, and f (t) → −∞ as t → ∞ by Lemma 3.6. Thus, there is t(u) > 0 such that Ib (t(u)u) = sup Ib (tu). t≥0

It is clear that dIb (tu) dt

t=t(u)

= Ib (t(u)u)u =

1 I (t(u)u)(t(u)u) = 0 t(u) b

and consequently by (3.13) Ib (t(u)u)(t(u)u) < 0. One sees that such t(u) > 0 is unique. Set b1 := inf{Ib (u) : u ∈ Nb+ }, b2 := inf{Ib (u) : u ∈ Kb+ \{0}}, b3 := inf max Ib (γ(t)), γ∈Γb t∈[0,1]

where Γb := {γ ∈ C([0, 1], E) : γ(0) = 0, Ib (γ(1)) < 0}. Lemma 3.8. cb := b1 = b2 = b3 . Proof. We check b1 ≤ b2 ≤ b3 ≤ b1 . • b1 ≤ b2 . This holds because Kb+ \{0} ⊂ Nb+ . • b2 ≤ b3 . Let (uj ) be a Mountain-Pass sequence: Ib (uj ) → b3 and Ib (uj ) → 0. It is not diﬃcult to check that (uj ) is bounded in E. By the concentration compactness principle, a standard argument shows that (uj ) is non-vanishing, that is, there exist r, η > 0 and (aj ) ⊂ R3 with lim sup |uj |2 ≥ η. j→∞

Br (aj )

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Set vj := aj ∗ uj . It follows from the invariance of the norm and of the functional under the ∗-action that Ib (vj ) → b3 , Ib (vj ) → 0. Therefore, vj v in E with v = 0 and I (v) = 0. Additionally, a standard argument yields that Ib (vj − v) → b3 − Ib (v), Ib (vj − v) → 0 and b3 − Ib (v) ≥ 0 ([15]). Therefore b2 ≤ Ib (v) = b3 . • b3 ≤ b1 . Take U ∈ Sˆb and deﬁne γ(t) := tU (x) for t ≥ 0. Then since Ib (U ) = 0 one has t(U ) = 1. Then γ ∈ Γb and b3 ≤ max Ib (γ(t)) = Ib (U ) = cb . t∈[0,1]

The proof is completed. Let u0 ∈ E + be such that Ib (u0 ) < 0, and set Γ0 := {γ ∈ C([0, 1], E + ) : γ(0) = 0, γ(1) = u0 } b0 := inf max Ib (γ(t)). γ∈Γ0 t∈[0,1]

Lemma 3.9. There holds b0 = b3 . Proof. Since Γ0 ⊂ Γb it is clear that b3 ≤ b0 . Let γ ∈ Γb . Then as before Ib (tγ(1)) and Ib (tu0 ) are strictly decreasing for t ≥ 1, and Ib (tγ(1)) → −∞, Ib (tu0 ) → −∞ as t → ∞. Let (s) be a cure in the two-dimensional subspace span{γ(1), u0 } jointing γ(1) and u0 such that Ib ( (s)) < 0 for 1 ≤ s ≤ 2 (such a cure exists because of Lemma 3.6(2)). Deﬁne γˆ(t) by γˆ (t) = γ(2t) for t ∈ [0, 1/2] and γˆ(t) = (2t) for γ (t)) = maxt∈[0,1] Ib (γ(t)). Thus b0 ≤ b3 . t ∈ [1/2, 1]. Then γˆ ∈ Γ0 and maxt∈[0,1] Ib (ˆ Lemma 3.10. Let u ∈ Kb+ be such that Ib (u) = cb , and set Eu = E − ⊕ Ru. Then sup Φb (w) = Ib (u).

w∈Eu

Proof. For any w = v + su ∈ Eu , by (3.11), 1 1 1 2 2 ¯ Φb (w) = su b − v b + L(v + su)v + su − F∞ (v + su) 2 2 2 R3 R3 ≤ Φb (su + hb (su)) = Ib (su). Thus since u ∈ Nb+ , sup Φb (w) ≤ sup Ib (su) = Ib (u).

w∈Eu

s≥0

4. Proof of the Main Result We are now going to prove the main result. Observe that (g1 ) and (g2 ) imply that F (x, u) ≥ c1 |u|µ

for all |u| ≥ 1;

1 µ−2 F˜ (x, u) := g(x, |u|)|u|2 − F (x, u) ≥ g(x, |u|)|u|2 2 2µ

(4.1) for all u

(4.2)

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hence F˜ (x, u) > 0 if u = 0 and F˜ (x, u) → ∞ as |u| → ∞; and σ |g(x, |u|)u| p > 3, ≤ c2 F˜ (x, u) for all |u| ≥ 1. σ := p−2 |u|

1023

(4.3)

By (g2 ), for |u| ≥ 1, g(x, |u|) ≤ a1 |u|p−2 , so g(x, |u|)σ−1 ≤ a2 |u|2 and consequently σ |g(x, |u|)u| = g(x, |u|)σ ≤ a2 g(x, |u|)|u|2 ≤ a2 F˜ (x, u). |u| Furthermore, for each ε > 0 there is Cε > 0 such that |g(x, |u|)u| ≤ ε|u| + Cε |u|p−1

(4.4)

F (x, u) ≤ ε|u|2 + Cε |u|p

(4.5)

and

for all (x, u). Now consider the functional ΦM deﬁned by (2.1), or equivalently (2.2). Let KM := {u ∈ E : ΦM (u) = 0} be the critical set of ΦM and recall that cM := inf{ΦM (u) : u ∈ KM \{0}}. Using the same iterative argument of [17, Proposition 3.2] one obtains easily the following Lemma 4.1. If u ∈ KM with |ΦM (u)| ≤ C1 and |u|2 ≤ C2 , then, for any q ∈ [2, ∞), u ∈ W 1,q (R3 ) with u W 1,q ≤ Λq where Λq depends only on C1 , C2 and q. Let SM be the set of all least energy solutions u. If u ∈ SM then ΦM (u) = cM and, by (g1 )−(g2 ), a standard argument shows that SM is bounded in E, hence, |u|2 ≤ C2 for all u ∈ SM , some C2 > 0. Therefore, as a consequence of Lemma 4.1 we see that, for each q ∈ [2, ∞), there is Λq > 0 such that u W 1,q ≤ Λq

for all u ∈ SM .

(4.6)

This, together with the Sobolev embedding theorem, implies that there is C∞ > 0 with |u|∞ ≤ C∞

for all u ∈ SM .

(4.7)

Lemma 4.2. ΨM is weakly sequentially lower semicontinuous and ΦM is weakly sequentially continuous. Proof. In virtue of (4.4) and (4.5) the lemma follows easily because E embeds continuously into Lq (R3 , C4 ) for q ∈ [2, 3] and compactly into Lqloc (R3 , C4 ) for q ∈ [1, 3) by Lemma 2.2. Lemma 4.3. There exist r > 0 and ρ > 0 such that ΦM |Br+ (u) ≥ 0 and ΦM |Sr+ ≥ ρ where Br+ = {u ∈ E + : u ≤ r} and Sr+ = {u ∈ E + : u = r}.

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Proof. We only check the Coulomb potential case because the other case can be treated similarly. Assume (M1 ) is satisﬁed. Recall that |Vκ u|22 ≤ 4κ2 |H0 u|22 = |(2κH0 )u|22 , thus |Vκ1/2 u|22 ≤ (2κH0 )|1/2 u|22 = 2κ H0 |1/2 u|22 , that is

κ u · u¯ ≤ 2κ||H0 |1/2 u|22 = 2κ u 2 . |x|

R3

By (M1 ),

−

For u ∈ E + one has

R3

M (x)u · u ¯ ≤ 2κ||H0 |1/2 u|22 = 2κ u 2.

1 1 M (x)u · u¯ − F (x, u) u 2 + 2 2 R3 R3 1 2 ≥ − κ u − F (x, u) 2 R3 1 − κ u 2 − ε|u|22 − Cε |u|33 ≥ 2 1 − κ u 2 − c1 ε u 2 − c2 Cε u 3 ≥ 2

ΦM (u) =

so the conclusion follows. For continuing our arguments, some further notations are in order. In the sequel, for the case of (M1 ), the Coulomb-type potential, we consider b = 0 and L = 0 in (3.1), and denote the corresponding functional by 1 F∞ (u), Φ0 (u) := ( u+ 2 − u− 2 ) − 2 R3 the critical set by K0 = {u ∈ E : Φ0 (u) = 0}, the least energy by Cˆ0 = min{Φ0 (u): u ∈ K0 \{0}}, the least energy solution set by Sˆ0 = {u ∈ K0 : Φ0 (u) = Cˆ0 }, and the induced map from E + → E − by h0 . In the case (M2 )(i) we consider b = 0 and L = M (∞) in (3.1), and denote the functional by 1 1 + 2 − 2 M (∞)u¯ u− F∞ (u) ΦI (u) := u − u + 2 2 R3 R3 with the critical set KI , the least energy CˆI , the least energy solution set SˆI , and the induced map hI : E + → E − . Similarly in the case of (M2 )(ii) we take b = m∞

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and L = 0 in (3.1), and denote correspondingly

1 m∞ 2 + 2 − 2 |u|2 − ΦII (u) := ( u − u ) + F∞ (u) 2 2 R3 1 = ( u+ 2m∞ − u− 2m∞ ) − F∞ (u) 2 R3

(where · m∞ denotes the norm given by (3.4) with b = m∞ ) with notations KII , CˆII , SˆII and hII . Sometimes, if no confusion arises, we shall write simply ˆ Sˆ and h standing for one of the cases. Φ, K, C, Lemma 4.4. There is R > 0 such that, for any e ∈ E + and Ee = E − ⊕ R e, ΦM (u) < 0

for all u ∈ Ee \BR .

(4.8)

Proof. This follows from the following facts: if M satisﬁes (M1 ) then 1 1 ΦM (u) = ( u+ 2 − u− 2 ) + M (x)u¯ u− F (x, u) 2 2 R3 R3 1 + 2 − 2 ≤ ( u − u ) − F∞ (u) = Φ0 (u), 2 R3 similarly if (M2 )(i) appears then ΦM (u) ≤ ΦI (u) +

1 2

R3

(M (x) − M (∞))u¯ u ≤ ΦI (u)

and if (M2 )(ii) is satisﬁed then

1 m∞ 2 1 + 2 − 2 |u|2 + ΦM (u) = ( u − u ) + (M − m∞ )u¯ u− F (x, u) 2 2 2 R3 R3 1 ≤ ΦII (u) + (M − m∞ )u¯ u ≤ ΦII (u), 2 R3

and Φn veriﬁes (4.8) by Lemma 3.6 for n = 0, I and II . Let Un ∈ Sˆn for n = 0, I and II . Set e ≡ Un+ and Ee ≡ E − ⊕ Re. Lemma 4.5. We have ˆ d := sup{ΦM (u) : u ∈ Ee } < C. Proof. Observe that by Lemma 4.3 and the linking property we have d ≥ ρ. Assume (M1 ) is satisﬁed. Since by (M1 ), M (x) < 0 and ΦM (u) ≤ Φ0 (u) for all u = v + sU0+ , and Φ0 (u) = Φ0 (v + sU0+ ) ≤ Φ0 (sU0+ + h0 (sU0+ )) = Cˆ0 , hence d ≤ Cˆ0 . Assume by contradiction that d = Cˆ0 . Let wj = v + sj U0+ ∈ Ee be such that d − 1j ≤ ΦM (wj ) → d. It follows from Lemma 4.4 that wj is bounded and we can assume wj w in E with vj v ∈ E − and sj → s. It is clear that s > 0

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(otherwise there should appear the contradiction that d = 0). Then 1 1 d − ≤ ΦM (wj ) ≤ Φ0 (wj ) + M (x)wj w ¯j j 2 R3 1 ˆ ≤ C0 + M (x)wj w ¯j . 2 R3 Taking the limit yields Cˆ0 ≤ Cˆ0 + 12 R3 M (x)ww¯ which implies that w = 0, a contradiction. Similarly, if (M2 )(i) holds, for u = v + sUI+ ∈ Ee , ΦM (u) ≤ ΦI (u) ≤ ΦI (sUI+ + hI (sUI+ )) hence d ≤ CˆI , and as above 1 (M (x) − M (∞))u¯ u ΦM (u) ≤ ΦI (u) + 2 R3 hence d < CˆI ; if (M2 )(ii) appears, for u = v + sU + ∈ Ee , ΦM (u) ≤ ΦII (u) ≤ II

+ + ΦII (sUII + hII (sUII )) and

ΦM (u) ≤ ΦII (u) +

1 2

R3

(M (x) − m∞ )u¯ u

hence d < CˆII . Set Q0 := {u = u− + sU0+ : u− ∈ E − , s ≥ 0, u < R}, QI := {u = u− + sUI+ : u− ∈ E − , s ≥ 0, u < R} and + QII := {u = u− + sUII : u− ∈ E − , s ≥ 0, u < R}.

Letting Q stand for one of Qn , n = 0, I, II , as a consequence of Lemma 4.5 one has the following ˆ Lemma 4.6. sup ΦM (Q) < C. We now turn to the analysis on (C)c sequences. Firstly we have Lemma 4.7. Any (C)c sequence for ΦM is bounded. Proof. It can be shown along the way of proof of [7, Lemma 7.3] by using (4.1)– (4.3) together with the representation (2.2) of ΦM . In what follows let (zj ) denote a (C)c -sequence for ΦM . By Lemma 4.7, it is bounded, hence along a subsequence denoted again by (zj ), zj zM . It is obvious that zM is a critical point of ΦM . Moreover there holds the following Lemma 4.8. Either (i) zj → zM , or (ii) c ≥ Cˆ and there exist a positive integer , points z¯1 , . . . , z¯ ∈ K\{0}, a subsequence denoted again by (zj ), and sequences (aij ) ⊂ Z3 , such that,

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as j → ∞,

1027

(aij ∗ z¯i ) → 0, zj − zM − i=1

|aij |

→ ∞,

|aij

− akj | → ∞

if i = k

and ΦM (zM ) +

Φ(¯ zi ) = c.

i=1

Proof. Remark that, since (zj ) is bounded, it is a (P S)c sequence. The proof is well known (see, for example, Alama–Li [3]), which can be outlined as follows. Firstly observe that c ≥ 0 which follows by taking the limit in 1 1 1 − g(x, |zj |)|zj |2 ≥ 0. ΦM (zj ) − ΦM (zj )zj ≥ 2 2 µ R3 Assume (i) is false. It is easy to see that zj1 := zj − zM is a (PS )c1 sequence for Φ with c1 = c − ΦM (zM ) and zj1 0. Note that Φ is invariant under the ∗-action of R3 . A standard argument of concentration compactness principle implies that there exist a sequence a1j ∈ R3 with |a1j | → ∞ and a critical point z¯1 = 0 of Φ satisfying a1j ∗ zj1 z¯1 and Φ(a1j ∗ zj1 ) → c − ΦM (zM ) − Φ(¯ z1 ) ≥ 0. ˆ one sees that c ≥ C. ˆ Since ΦM (zM ) ≥ 0 and Φ(¯ z1 ) ≥ C, 1 1 If aj ∗ zj → z¯1 then we are done. Otherwise, repeating the above argument, after at most ﬁnitely many steps we ﬁnish the proof. As a straight consequence of Lemma 4.8 we have the following ˜ Lemma 4.9. ΦM satisfies the (C)c condition for all c < C. We now in a position to complete the proof of Theorem 1.3. Proof of Theorem 1.3. Firstly we prove Existence. It is clear that ΦM checks (Φ0 ) because of the form (2.2) and because of F (x, u) ≥ 0. The combination of Lemmas 4.2 and 2.4 implies that ΦM veriﬁes (Φ1 ). Lemma 4.3 is nothing but (Φ2 ). Lemma 4.4 shows that the linking condition of Theorem 2.3 is satisﬁed. These, together with Lemma 4.6, yield a (C)c sequence (uj ) with c < Cˆ for ΦM . Now by virtue of Lemma 4.9, uj → u so that ΦM (u) = 0 and ΦM (u) ≥ ρ. Along the same lines of proof of Lemma 3.2, it is easy to check that cM > 0. ˆ we have uj → u in Let uj be such that ΦM (uj ) → cM , ΦM (uj ) = 0. Since cM < C, E with ΦM (u) = cM and ΦM (u) = 0, hence SM = ∅.

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Compactness of SM . Now we prove that SM is compact in H 1 . Let uj ∈ SM : ˆ ΦM (uj ) = cM and ΦM (uj ) = 0. Hence (uj ) is a (C)cM sequence. Since cM < C, it follows from Lemma 4.9 that uj → u along a subsequence in E with clearly u ∈ SM . By H0 u = −M (x)u + g(x, |u|)u one has |H0 (uj − u)|2 ≤ |M (uj − u)|2 + |g(·, |uj |)uj − g(·, |u|)u|2 ≤ o(1) + |g(·, |uj |)(uj − u)|2 + |g(·, |uj |) − g(·, |u|)u|2 . Since |uj |∞ ≤ C and uj → u in E, |g(x, |uj |)2 |uj − u|2 ≤ C|uj − u|22 → 0, R3

and since |u(x)| → 0 as |x| → ∞, |(g(x, |uj |) − g(x, |u|))u|2 = R3

+ |x|
|x|≥R

|(g(x, |uj |) − g(x, |u|))u|2 .

→ 0. Therefore, one sees that |H0 (uj − u)|2 → 0, i.e. uj → u in H 1 . Exponential Decay. Let (g1 )−(g3 ) and (M2 ), (M3 ) be satisﬁed. Assume g : R3 × (0, ∞) → R is of class C 1 . Write D = −i

3

αk ∂k

and H0 = D + aβ.

k=1

Using the relationship H02 = −∆ + a2 and H0 u = −M u + g(x, |u|)u, one gets −∆u + a2 u = H0 (−M u + g(x, |u|)u). Hence there holds ∆u = a2 u + H0 (M u) − H0 (g(x, |u|)u) = a2 u + D(M u) + aβM u − H0 (g(x, |u|)u) = E(a, M )u + rM (x, u)u where if M = V β rV β (x, u) = −g(x, |u|)2 + i

3 k=1

αk

u ¯ ∂g(x, |u|) ∂k u , + gs (x, |u|) ∂xk |u|

and if M = V rV (x, u) = −i

3 k=1

αk ∂k V + 2V g(x, |u|) + rV β (x, u).

(4.9)

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Letting

u  ¯ sgn u = |u|  0

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if u = 0 if u = 0,

by the Kato’s inequality ([12]), (4.9) and the real positivity of E(a, M ), there exist R > 0 and τ > 0 such that ∆|u| ≥ [∆u(sgn u)] u ¯ u ¯ = E(a, M )u + rM (x, u)u |u| |u| u ¯ ≥ τ |u| + rM (x, u)u |u| for all |x| ≥ R. Observe that, for fk : R3 → R, k = 1, 2, 3,  0 0 3  0 0 Bf := αk fk =   f3 f1 − if2 k=1 −f3 f1 + if2

(4.10)

 f1 − if2 −f3   0  0

f3 f1 + if2 0 0

is a Hermitian matrix. Applying to fk = ∂g/∂xk , gs [u¯ u/|u| ], ∂k V , respectively, one sees plainly that, if M = V β u ¯ rM (x, u)u = −g(x, |u|)2 |u|, |u| and if M = V

u ¯ rM (x, u)u = (2V − g(x, |u|))g(x, |u|)|u|. |u|

Remark that, by assumption on V we see E(a, M ) ∈ L∞ , and by (4.7), g(x, |u|) is bounded uniformly in u ∈ SM . Thus the sub-solution estimate [36] implies that |u(x)| ≤ C0 |u(y)| dy (4.11) B1 (x)

with C0 independent of x and u ∈ SM . Since SM is compact in H 1 , |u(x)| → 0 as |x| → ∞ uniformly in u ∈ SM . In fact, if not, then by (4.11) there exist κ > 0, uj ∈ SM and xj ∈ R3 with |xj | → ∞ such that κ ≤ |uj (xj )| ≤ C0 B1 (xj ) |uj |. One may assume uj → u ∈ SM in H 1 and to get |uj | ≤ C0 |uj − u| + C0 |u| κ ≤ C0 B1 (xj )

≤ C

R3

|uj − u|2

B1 (xj )

1/2

B1 (xj )

+ C0

B1 (xj )

|u| → 0,

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a contradiction. Now since g(x, s) → 0 as s → 0 uniformly in x, one may take 0 < δ < τ /2 and R > 0 such that |u(x)| ≤ δ and ¯ τ rM (x, u)u u ≤ |u| |u| 2 for all |x| ≥ R, u ∈ SM . This, together with (4.10), implies ∆|u| ≥ δ|u| for all |x| ≥ R,

u ∈ SM .

Let Γ(y) = Γ(y, 0) be a fundamental solution to −∆ + δ (see, e.g., [30–34]). Using the uniform boundedness, one may choose Γ so that |u(y)| ≤ δΓ(y) holds on |y| = R, all u ∈ SM . Let w = |u| − δΓ. Then ∆w = ∆|u| − δ∆Γ ≥ δ|u| − δ 2 Γ = δ(|u| − δΓ) = δw. By the maximum principle we can conclude that w(y) √ ≤ 0 on |y| ≥ R. It is well known that there is C > 0 such that Γ(y) ≤ C exp(− δ|y|) on |y| ≥ 1. We see that √ |u(y)| ≤ C exp(− δ|y|) for all y ∈ R3 and all u ∈ SM . The proof is completed. Acknowledgment The research of the ﬁrst-named author is partially supported by The Institute of Mathematical Sciences of The Chinese University of Hong Kong, the National Natural Science Foundation, 973 Program (2007CB814800), and the key fund from Beijing in China. This work was done while he was visiting IMS of CUHK. He wishes to thank the IMS and Professor Z. P. Xin for the hospitality during his stay in Hong Kong. The research of the second author is partially supported by an Earmarked Grant from RGC of HK. References [1] N. Ackermann, A nonlinear superposition principle and multibump solutions of peridic Schr¨ odinger equations, J. Funct. Anal. 234 (2006) 423–443. [2] A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Sch¨ odinger equations, Arch. Ration. Mech. Anal. 140 (1997) 285–300. [3] A. Alama and Y. Y. Li, On “multibump” bound states for certain semilinear elliptic equations, Indiana Univ. Math. J. 41 (1992) 983–1026. [4] M. Balabane, T. Cazenave, A. Douady and F. Merle, Existence of excited states for a nonlinear Dirac field, Comm. Math. Phys. 119 (1988) 153–176. [5] M. Balabane, T. Cazenave and L. Vazquez, Existence of standing waves for Dirac fields with singular nonlinearities, Comm. Math. Phys. 133 (1990) 53–74.

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[6] T. Bartsch and Y. H. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nachr. 279 (2006) 1267–1288. [7] T. Bartsch and Y. H. Ding, Solutions of nonlinear Dirac equations, J. Diﬀerential Equations 226 (2006) 210–249. [8] J. Byeon and Z. Q. Wang, Standing waves with a critical frequency for nonlinear Schr¨ odinger equations, II, Calc. Var. Partial Diﬀerential Equations 18 (2003) 207–219. [9] J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields (McGraw-Hill, 1965). [10] T. Cazenave and L. Vazquez, Existence of local solutions for a classical nonlinear Dirac field, Comm. Math. Phys. 105 (1986) 35–47. [11] V. Coti-Zelati and P. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on Rn , Comm. Pure Appl. Math. 46 (1992) 1217–1269. [12] R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 3 (Springer-Verlag, Berlin, 1990). [13] M. del Pino and P. Felmer, Multipeak bound states of nonlinear Schr¨ odinger equations, Ann. Inst. H. Poincar´ e Anal. Lin´eaire 15 (1998) 127–149. [14] M. del Pino and P. Felmer, Semi-classical states of nonlinear Schr¨ odinger equations: A variational reduction method, Math. Ann. 324 (2002) 1–32. [15] Y. H. Ding, Variational Methods for Strongly Indeﬁnite Problems, Interdisciplinary Math. Sci., Vol. 7 (World Scientific Publ., 2007). [16] Y. H. Ding and B. Ruf, Solutions of a nonlinear Dirac equation with external fields, Arch. Ration. Mech. Anal., in press; doi: 10.1007/s00205-008-0163-z. [17] M. J. Esteban and E. S´er´e, Stationary states of the nonlinear Dirac equation: A variational approach, Comm. Math. Phys. 171 (1995) 323–350. [18] M. J. Esteban and E. S´er´e, An overview on linear and nonlinear Dirac equations, Discrete Contin. Dyn. Syst. 8 (2002) 281–397. [19] R. Finkelstein, C. F. Fronsdal and P. Kaus, Nonlinear spinor field theory, Phys. Rev. 103 (1956) 1571–1579. [20] A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schr¨ odinger equation with a bounded potential, J. Funct. Anal. 69 (1986) 397–408. [21] W. T. Grandy, Relativistic Quantum Mechanics of Leptons and Fields, Fund. Theories of Physics, Vol. 41 (Kluwer Acad. Publisher, 1991). [22] C. Gui, Existence of multi-bump solutions for nonlinear Schr¨ odinger equations via variational method, Comm. Partial Diﬀerential Equations 21 (1996) 787–820. [23] L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. Partial Diﬀerential Equations 21 (2004) 287–318. [24] X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schr¨ odinger equations, Adv. Diﬀerential Equations 5 (2000) 899–928. [25] W. Kryszewki and A. Szulkin, Generalized linking theorem with an application to semilinear Schr¨ odinger equation, Adv. Diﬀerential Equations 3 (1998) 441–472. [26] Y. Y. Li, On singularly perturbed elliptic equation, Adv. Diﬀerential Equations 2 (1997) 955–980. [27] F. Merle, Existence of stationary states for nonlinear Dirac equations, J. Diﬀerential Equations 74 (1988) 50–68. [28] P. H. Rabinowitz, On a class of nonlinear Schr¨ odinger equations, Z. Angew. Math. Phys. 43 (1992) 270–291. [29] A. F. Ranada, Classical nonlinear Dirac field models of extended particles, in Quantum Theory, Groups, Fields and Particles, ed. A. O. Barut (Reidel, Amsterdam, 1982), pp. 271–291.

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[30] M. Reed and B. Simon, Methods of Mathematical Physics, Vol. I (Academic Press, 1978). [31] M. Reed and B. Simon, Methods of Mathematical Physics, Vol. II (Academic Press, 1978). [32] M. Reed and B. Simon, Methods of Mathematical Physics, Vol. III (Academic Press, 1978). [33] M. Reed and B. Simon, Methods of Mathematical Physics, Vol. IV (Academic Press, 1978). [34] M. Reed and B. Simon, Methods of Mathematical Physics, Vol. V (Academic Press, 1978). [35] M. Soler, Classical stable nonlinear spinor field with positive rest energy, Phys. Rev. D 1 (1970) 2766–2769. [36] B. Simon, Schr¨ odinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982) 447–526. [37] B. Sirakov, Standing wave solutions of the nonlinear Schr¨ odinger equation in RN , Ann. Mat. Pura Appl. (4) 181(1) (2002) 73–83. [38] B. Thaller, The Dirac Equation, Texts and Monographs in Physics (Springer, Berlin, 1992).

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Reviews in Mathematical Physics Vol. 20, No. 9 (2008) 1033–1172 c World Scientiﬁc Publishing Company

RENORMALIZED QUANTUM YANG–MILLS FIELDS IN CURVED SPACETIME

STEFAN HOLLANDS School of Mathematics, Cardiﬀ University, Cardiﬀ, Wales, CF24 4AG, UK and Institut f¨ ur Theoretische Physik, Universit¨ at G¨ ottingen, D-37077 G¨ ottingen, Germany HollandsS@Cardiﬀ.ac.uk Received 15 June 2007 Revised 28 February 2008 Dedicated to K. Fredenhagen on the Occasion of his 60th Birthday We present a proof that the quantum Yang–Mills theory can be consistently deﬁned as a renormalized, perturbative quantum ﬁeld theory on an arbitrary globally hyperbolic curved, Lorentzian spacetime. To this end, we construct the non-commutative algebra of observables, in the sense of formal power series, as well as a space of corresponding quantum states. The algebra contains all gauge invariant, renormalized, interacting quantum ﬁeld operators (polynomials in the ﬁeld strength and its derivatives), and all their relations such as commutation relations or operator product expansion. It can be viewed as a deformation quantization of the Poisson algebra of classical Yang–Mills theory equipped with the Peierls bracket. The algebra is constructed as the cohomology of an auxiliary algebra describing a gauge ﬁxed theory with ghosts and antiﬁelds. A key technical diﬃculty is to establish a suitable hierarchy of Ward identities at the renormalized level that ensures conservation of the interacting BRST-current, and that the interacting BRST-charge is nilpotent. The algebra of physical interacting ﬁeld observables is obtained as the cohomology of this charge. As a consequence of our constructions, we can prove that the operator product expansion closes on the space of gauge invariant operators. Similarly, the renormalization group ﬂow is proved not to leave the space of gauge invariant operators. The key technical tool behind these arguments is a new universal Ward identity that is formulated at the algebraic level, and that is proven to be consistent with a local and covariant renormalization prescription. We also develop a new technique to accomplish this renormalization process, and in particular give a new expression for some of the renormalization constants in terms of cycles. Keywords: Quantum ﬁeld theory on curved spacetime; Yang–Mills theory; perturbative renormalization; Ward identities. Mathematics Subject Classiﬁcation 2000: 81T20, 81T13, 81T15

1033

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Contents 1. Introduction 1.1. Generalities 1.2. Renormalization of theories without local gauge invariance 1.3. The problem of local gauge invariance 1.4. Summary of the report 1.5. Guide to the literature

1035 1035 1037 1039 1041 1041

2. Generalities Concerning Classical Field Theory 2.1. Lagrange formalism 2.2. Yang–Mills theories, consistency conditions, cohomology 2.3. Proof of the Algebraic Poincare Lemma, and the Thomas Replacement Theorem

1042 1042 1048 1058

3. Quantized Field Theories on Curved Spacetime: Renormalization 3.1. Deﬁnition of the free ﬁeld algebra W0 for scalar ﬁeld theory 3.2. Renormalized Wick products and their time-ordered products 3.3. Inductive construction of time-ordered products 3.4. Examples 3.5. Ghost ﬁelds and vector ﬁelds 3.6. Renormalization ambiguities of the time-ordered products 3.7. Perturbative construction of interacting quantum ﬁelds

1063 1063 1067 1073 1087 1093 1095 1098

4. Quantum Yang–Mills Theory 4.1. General outline of construction 4.1.1. Free ﬁelds 4.1.2. Interacting ﬁelds 4.1.3. Operator product expansions and RG-ﬂow 4.2. Free gauge theory 4.3. Interacting gauge theory 4.4. Inductive proof of Ward identities T12a, T12b, and T12c 4.4.1. Proof of T12a 4.4.2. Proof of T12b 4.4.3. Proof of T12c 4.5. Formal BRST-invariance of the S-matrix 4.6. Proof that dJI = 0 4.7. Proof that Q2I = 0 4.8. Proof that [QI , ΨI ] = 0 when Ψ is gauge invariant 4.9. Relation to other perturbative formulations of gauge invariance

1102 1102 1102 1104 1106 1108 1113 1117 1130 1134 1138 1138 1139 1140 1145 1145

5. Summary and Outlook 5.1. Matter ﬁelds, anomalies 5.2. Other gauge ﬁxing conditions 5.3. Background independence

1149 1151 1151 1152

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Appendix A. U (1)-Gauge Theory Without Vector Potential

1153

Appendix B. Eﬀective Actions in Curved Spacetime

1158

Appendix C. Wave Front Set and Scaling Degree

1159

Appendix D. Hadamard Parametrices D.1. Scalar Hadamard parametrix D.2. Vector Hadamard parametrix

1161 1162 1165

Appendix E. Hadamard States

1166

1. Introduction The known interactions of elementary particles seem to be well-described by quantized ﬁeld theories with local gauge invariance such as QCD. Such theories have been extensively investigated in the context of ﬂat Minkowski spacetime from a variety of diﬀerent angles. It has in particular been demonstrated that these quantum ﬁeld theories are internally consistent, at least to all orders in the renormalized perturbation expansion. The early Universe on the other hand is described by a strongly curved spacetime, and important new quantum ﬁeld theory eﬀects arise in this situation — an important example being the generation of primordial ﬂuctuations that have left an imprint in the CMB as well as the large scale structure of the universe. For this reason, it is obviously important to study quantum gauge theories in curved Lorentzian spacetimes such as the expanding Universe. The question how to consistently construct such theories in arbitrary curved, globally hyperbolic spacetimes is an open problem. As a ﬁrst step in this direction, we will prove in this paper that perturbative non-abelian pure Yang–Mills theory can be consistently quantized on any globally hyperbolic spacetime, to all orders in perturbation theory, and any gauge group G that is a direct product of U (1)l and a semi-simple Lie group. The essence of our proof is the inductive construction of an explicit renormalization prescription for the perturbatively deﬁned interacting ﬁeld quantities that preserves gauge invariance, and that depends locally and covariantly upon the spacetime metric. The proof of this statement is rather complicated, and it relies partly on auxiliary constructions that have been previously given in the literature. Some of these constructions are not so widely known as the renormalization techniques in ﬂat spacetime, and there is at present no comprehensive review. We therefore found it appropriate to present these constructions in the form of a report. 1.1. Generalities Quantum ﬁeld theory in curved spacetime is a natural generalization of ﬂat space quantum ﬁeld theory in which one considers quantized ﬁelds propagating on a rigidly ﬁxed, non-dynamical, Lorentzian spacetime rather than ﬂat Minkowski spacetime. In order to have a well-deﬁned propagation of such ﬁelds (even at the classical level), one usually assumes that the spacetime does not have any gross

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causal pathologies such as closed time-like curves, (a typical assumption is that the spacetime is “globally hyperbolic”) but otherwise no restrictions on the metric are placed. In particular, one does not have to (and does not want to) assume that the metric has any isometries, or that it is a solution to a particular ﬁeld equation. As quantum ﬁeld theory on ﬂat spacetime, quantum ﬁeld theory on curved spacetime is in general only believed to be an eﬀective theory with a limited range of validity. It is expected to lose predictive power when the spacetime curvatures become as large as the inverse Planck length, or in quantum states where typical quantum ﬁeld observables such as the quantum stress energy operator have expectation values or variances (ﬂuctuations) of the order of the Planck length. On the other hand, the theory is expected to be a very good approximation when the spacetime curvatures are of the order (or below) the scale of elementary particle physics such as ΛQCD , or even the grand uniﬁcation (GUT) scale, which is expected to be the relevent scale during inﬂation. Naturally, it is also in this regime (as well as in the case of black holes) that the most interesting physical eﬀects predicted by the theory occur. Independent of those questions regarding the limits of physical applicability of quantum ﬁeld theory in curved spacetime, one may ask whether this theory, in itself, has a consistent mathematical formulation or not — just as it is a relevant question whether classical mechanics has a well-deﬁned mathematical formulation even though it clearly has a limited range of validity as a physical theory. Unfortunately, this question is a very diﬃcult one, which has not been answered in a satisfactory manner for interacting quantum ﬁeld theory models even in ﬂat spacetime (in 4 dimensions). Nevertheless, there exist perturbative approaches to interacting quantum ﬁeld theory in Minkowski spacetime, and it is by now well understood how to calculate, in principle, terms of arbitrary high order in the perturbation expansion. In particular, one has a good understanding how to systematically deal with the problem of renormalization that needs to be addressed at each order to get meaningful expressions, and it is known how to calculate quantities of physical interest for, say, the purposes of collider physics. In fact, this approach is at present by far the most powerful method to obtain theoretical predictions for particle physics experiments, and to test quantum ﬁeld theory. In quantum ﬁeld theories in curved spacetime, new conceptual problems arise because one no longer has a preferred vacuum state in time-dependent spacetimes, as may be understood from the familiar fact that time-dependent background ﬁelds tend to give rise to particle creation. Thus, a state that may be thought of as a vacuum at one time may fail to be the vacuum at later time. This suggests to use an S-matrix formulation of the theory, but such a formulation also does not make sense in general if the spacetime does not have any asymptotically timeindependent regions in the far past or future, or if the metric approaches a timeindependent metric too slowly. At the technical level, one no longer has a clear cut relation between quantum ﬁeld theory on Lorentzian spacetimes and Riemannian spacetimes, because a general (even analytic) Lorentzian spacetime will not be a

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real section in a complexiﬁed manifold that also has a real, Riemannian section. Furthermore, familiar ﬂat space techniques such as momentum space, dimensional regularization, the Euclidean path integral, are not available on a curved manifold. As had been realized for some time, these conceptual problems can in principle be overcome by shifting the emphasis to the local quantum ﬁeld operators, which can be unambiguously deﬁned on any (globally hyperbolic) Lorentzian spacetime. The key insight was that the algebraic relations between the quantum ﬁelds (such as commutators, or the “operator product expansion”) have an invariant meaning for any such spacetime, even if there are no states with a deﬁnite particle interpretation. Nevertheless, it remained an unsolved problem how to construct in practice interesting (non-free) quantum ﬁeld theories perturbatively on a general globally hyperbolic spacetime, mainly because of the very complicated issues related to renormalization on a curved manifold. A fully satisfactory construction of perturbative, renormalized quantum ﬁeld theory on curved space was ﬁnally given in a series of papers [20, 21, 66–68] where it was shown that the algebras of local observables (interacting local ﬁelds) can always be constructed at the level of formal power series in the coupling, independent of the asymptotic behavior of the metric at inﬁnity. It was shown in detail how to perform the renormalization process in a local and covariant way, and it was thereby seen that the remaining ﬁnite renormalization ambiguities correspond to the possibility of adding ﬁnite local terms (possibly with curvature couplings) to the Lagrangian, and to the possibility of making ﬁnite ﬁeld-redeﬁnitions (“operator mixing with curvature”). These constructions also provided a completely new, geometrical understanding of the nature of the singularities of multi-point operator products and their expectation values in terms of “microlocal analysis” [76, 18, 19, 100–102], and thereby provided a geometric generalization of the usual spectrum condition in Minkowski spacetime quantum ﬁeld theory to curved manifolds. By considering the behavior of the theory under a rescaling of the metric g → µ2 g, a deﬁnition of the renormalization group could be given [68], and detailed results about the (poly-logarithmic) scaling behavior of products of interacting ﬁeld operators were thereby obtained. It is also understood how to construct the operator product expansion from the algebra of interacting ﬁelds in curved space, and this gives direct information about the interplay between quantum ﬁeld interactions and spacetime curvature at small scales [72].

1.2. Renormalization of theories without local gauge invariance The building blocks in the renormalized perturbation series for the interacting ﬁelds are the time-ordered products Tn (O1 ⊗· · ·⊗On ) of composite ﬁelds in the underlying free ﬁeld theory. In standard approaches in ﬂat spacetime, these objects are typically viewed as operators on a Hilbert space (“Fock-space”), but in curved spacetime there is no preferred Hilbert-space representation. In this context, it is more useful to view them instead as members of an abstract algebra, which may in the end be

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represented on a Hilbert space (typically in inﬁnitely many inequivalent ways). The ﬁrst step in the renormalization program therefore is to deﬁne a suitable abstract algebra, and this can indeed be done using the techniques of the “wave front set”. The next step is to actually construct the time-ordered products as speciﬁc elements in this algebra. A naive deﬁnition leads to inﬁnite meaningless expressions, but one can show that it is possible to obtain meaningful objects by a process called “renormalization”. Conceptually, the best approach here is to ﬁrst formulate a set of conditions (“renormalization conditions”) on the time-ordered products to be constructed, and then show via an explicit construction that these properties can be satisﬁed. It turns out that the conditions do not uniquely ﬁx the time ordered products, but there remain certain ﬁnite renormalization ambiguities. In curved spacetime, it is a major challenge to formulate suﬃciently strong renormalization conditions in order to guarantee that these ambiguities only consist in adding ﬁnite “contact terms” at each order n, which are covariant expressions of the Riemann curvature and the ﬁelds of a suitable dimension. A key condition to guarantees this is that the Tn should themselves be local and covariant [66], and a precise formulation of that condition naturally leads to a formulation of quantum ﬁeld theory in the language of category theory [22]. The condition of locality and covariance is a rather strong one, and it is correspondingly non-trivial to ﬁnd a renormalization method that will ensure that this condition is indeed satisﬁed. Such a scheme was found in [66, 67] for interacting scalar ﬁeld theory, based on key earlier work of [21, 20], and also on the work [40, 41], where an algebraic variant of perturbation theory in ﬂat space was developed. We will present these constructions in Sec. 3 of the paper. Here we follow the general steps proposed in these references, but we develop a new technique to perform the actual renormalization (extension) step. Our new method (described in the proof of Lemma 6) is more explicit than previous constructions, and also gives an interesting new formula for some of the renormalization constants describing the departure from homogeneous scaling in terms of an integral of a closed form of a cycle in R4n , see Proposition 1. In quantum ﬁeld theory, one typically wants certain ﬁelds to have special properties. For example, an important observable in any theory with a metric is the stress energy tensor, which is conserved at the classical level if the metric is the only background ﬁeld (as we assume). One would like the corresponding quantum ﬁeld to be conserved as well. In perturbative quantum ﬁeld theory, it is far from obvious that the corresponding interacting quantum ﬁeld quantity is also conserved, and indeed there exist theories where this fails to be the case [2]. In general, one can formulate a set of renormalization conditions on the time-ordered products (the “principle of perturbative agreement” [70]) that will guarantee conservation to all orders in the perturbation expansion. In [70], it was shown that the question whether or not these identities can be satisﬁed is equivalent to the question whether a certain cohomological class on the space of all metric deﬁned by the ﬁeld theory is trivial or not. The obstruction sometimes cannot be lifted, and then the renormalization condition is impossible to satisfy: There are anomalies. Similarly, in gauge

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theories, one wants certain currents to be conserved at the quantum level and it is important to ensure that there are no anomalies. 1.3. The problem of local gauge invariance In fact, the perturbative construction of renormalized ﬁeld theories on curved space without local gauge invariance does not carry over straightforwardly to theories with local gauge invariance, and the construction of such models was therefore up to now an important open problem. The key obstacle is that the ﬁeld equations of local gauge theories, such as e.g. the pure Yang–Mills theory studied in this paper, are not globally hyperbolic in nature even if the underlying spacetime is globally hyperbolic. This, however, is a basic assumption in the constructions [20,21,66,67]. In theories with local gauge invariance, the ﬁeld equations fail to be hyperbolic in nature precisely due to local gauge invariance, because it implies that solutions to the ﬁeld equations are not entirely determined by their initial data on some Cauchy surface as required by hyperbolicity, but also on an arbitrary choice of local gauge. At the classical level, this problem can be dealt with by simply ﬁxing a suitable gauge. However, at the quantum level, it is problematical to base the theory on a gauge-ﬁxed formulation, because gauge ﬁxing typically has non-local features. This causes severe problems, e.g., for the renormalization process. An elegant and very successful approach circumventing these problems is the BRST-method [11, 12]. This method consists in replacing the original action by a new action containing additional dynamical ﬁelds. That new action yields hyperbolic ﬁeld equations, and has an invariance under a nilpotent so-called “BRST transformation”, s, on ﬁeld space. Gauge invariant ﬁeld observables are precisely those that are annihilated by s, or more precisely, the cohomology classes of s. Furthermore, the classical Poisson (or Peierls) brackets [96, 92, 30, 40] of the gauge ﬁxed theory are invariant under s. Thus, as ﬁrst suggested by [42] (based on [88, 89]), one can try to proceed by ﬁrst quantizing the brackets of the gauge ﬁxed action (in the sense of deformation quantization [40, 41, 9, 10]), promote the diﬀerential s to a graded derivation at the quantum level leaving the quantized brackets invariant, and then at the end deﬁne the algebra of physical observables to be the kernel (or rather cohomolgy) of the quantum BRST-diﬀerential. As we will prove in this paper, this program can be carried out successfully for renormalized Yang–Mills theory in curved spacetime, at the level of formal power series in the coupling constant. Thus, the ﬁrst step consists in ﬁnding an appropriate gauge ﬁxed and BRST invariant modiﬁed action, S, for pure Yang–Mills theory in curved space involving the gauge ﬁeld, and new auxiliary ﬁelds (“anti-ﬁelds”). This step is completely analogous to Yang–Mills theory in ﬂat space. Next, one needs to “quantize” the brackets associated with the new action S. It is not known presently how to do this non-perturbatively even in ﬂat space, but one can proceed in a perturbative fashion as in theories without local gauge invariance. The ﬁnal step special to gauge theories is now to deﬁne a quantum BRST derivation acting on the quantum interacting ﬁelds. This derivation should (a) leave the

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product invariant, (b) square to 0, and which (c) go over to the classical BRST transformation s in the classical limit. The natural strategy for constructing the quantum BRST transformation is to consider the quantum Noether current corresponding to the classical BRST-transformation. One then deﬁnes a corresponding charge, and deﬁnes BRST-derivation via the graded commutator in the star-product with this charge. While this deﬁnition automatically satisﬁes (a), it is highly non-obvious that it would also satisfy properties (b) and (c). In fact, it is even unclear whether that the quantum Noether current operator associated with the BRST-transformations would be conserved, as would be required in order to yield a conserved charge. The basic reason why it is a non-trivial challenge to establish conservation of the quantum BRST current, as well as (b) and (c), is that the construction of the time ordered products Tn used to deﬁne the interacting quantum ﬁelds via the Bogoliubov formula involve renormalization. It is far from obvious that a renormalization prescription exists such that interacting BRST current would be conserved, and such that (b) and (c) would hold. In fact, as we will show, these properties follow from a new inﬁnite hierarchy of Ward identities for the time-ordered products [see Eq. (336) for a generating functional of these identities], which are violated for a generic renormalization prescription. We will show that there nevertheless exists a renormalization prescription compatible with locality and covariance such that these Ward identities are satisﬁed in curved space, to all orders in the renormalized perturbation expansion, when the gauge group is a product of U (1)l and a semi-simple group. Thus, we can deﬁne an algebra of interacting quantum ﬁelds as the cohomology of the quantum BRST-diﬀerential, and this deﬁnes perturbative quantum Yang–Mills theory. In a second step, we then deﬁne quantum states (i.e. representations) of this algebra by a deformation argument. Here we rely on a construction invented in [42]. As a by-product of our constructions, we can also show that the operator product expansion in curved space [72] closes among gaugeinvariant operators, and that the renormalization group ﬂow likewise closes among gauge-invariant operators. Our approach has several virtues also in the context in ﬂat spacetime. The key virtue is that, since our constructions are entirely local, there is a clear separation between issues related to the ultra-violet (UV) and infra-red (IR) behavior of the theory. In particular, in our approach, the identities reﬂecting gauge invariance may be formulated and proved entirely independently from the infrared behavior of the theory, while the infra-red cutoﬀ is only removed in the very end in an entirely welldeﬁned manner at the algebraic level (“algebraic adiabatic limit” [21]). In this way, infra-red divergences are neither encountered at the level of the interacting ﬁeld algebras, nor in fact at the level of quantum states, i.e. representations.a In this respect, our approach is diﬀerent from traditional treatments based on Feynman a However, we would encounter the familiar infra-red divergences if we were to try to construct scattering states. Actually, it is clear that those types of states cannot be deﬁned in a generic curved spacetime anyway even for massive ﬁelds, so we do not see this as a problem.

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diagrams or eﬀective actions, which are only formal in as far as the treatment of the IR-problems are concerned. We explain in some more detail the relation of our approach to those treatments in Sec. 4.9. A local approach that is similar to ours in spirit has previously been taken in the context of QED on ﬂat spacetime in [42], and in [38, 39] for non-abelian gauge theories on ﬂat spacetime. Note, however, that the “Master Ward identity” expressing the conditions for local gauge invariance in [38] was taken as an axiom and has not been shown to be consistent yet,b as opposed to the Ward identities of our paper, which are shown to hold. Also, our Ward identities (336) appear to be diﬀerent from those expressed in the Master Ward Identity of [38, 39]. 1.4. Summary of the report This paper is organized as follows. In Sec. 2, we ﬁrst review basic notions from classical ﬁeld theory, including classical BRST-invariance and associated cohomological constructions. The material in this secion is well known and serves mainly to set up the notations and provide basic results that are needed in later sections. In Sec. 3, we review the perturbative construction of interacting quantum ﬁeld theory on curved spacetime. We focus on theories without local gauge invariance. We explicitly describe scalar ﬁeld theory, and we brieﬂy mention the changes that have to be made for ghost and vector ﬁelds (in the Lorentz gauge). We give a detailed renormalization prescription for the time-ordered products, their renormalization ambiguities, and describe how interacting ﬁelds may be constructed from them. We also show how the method works in some concrete examples. The material presented in this section is to some extent taken from [21, 66, 67, 40, 39, 37], but there are also some important new developments. In Sec. 4, we perturbatively construct renormalized quantum Yang–Mills theory. We ﬁrst give an outline of the basic strategy, and then ﬁll in the technical details in the later sections. We present our new Ward identities in Sec. 4.3, and then prove them in Sec. 4.4. We prove in Sec. 4.5 that our identities formally imply the BRST-invariance of the S-matrix, in Sec. 4.5 that they imply the conservation of the interacting BRST-current, and in Sec. 4.6 that they imply the nilpotency of the interacting BRST-charge operator. We conclude and name open problems in Sec. 6. Appendix A contains a treatment of free U (1)-theory avoiding the introduction of the vector potential and an explanation of the new superselection sectors arising in this context. The Appendices B to E contain deﬁnitions and various constructions that are omitted from the main part of the paper. 1.5. Guide to the literature A standard introduction to the theory of quantum ﬁelds on a curved space is [115], which gives an in-depth discussion of the conceptual problems of the theory, as b For

recent progress in analyzing the validity of the Master Ward identity, see [17].

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well as the Hawking and Unruh-eﬀect, at the level of free quantum ﬁelds. The generalization of the latter eﬀect to certain black-hole spacetimes — emphasizing especially the role of the so-called “Hadamard condition” — is discussed in the review-style article [84]. Other monographs are [52, 13]. The perturbative construction of interacting scalar quantum ﬁeld theories on curved spaces was given in the series of papers [21, 20, 66, 67, 70]. Important contributions to the understanding of Hadamard states in terms of microlocal analysis, which were a key input in these papers, were made by Radzikowski [98,99]. These results are reviewed and extended in the very readable paper [81]. A complete characterization of the state space of perturbative quantum ﬁeld theory using microlocal analysis is given in [69]. A deﬁnition and analysis of the renormalization group in curved space was given in [68]. The generalization of the Wilson operator product expansion in curved spacetime was constructed to all orders in perturbation theory in [72]. Perturbative scalar quantum ﬁeld theory on Riemannian spaces was treated in [23] using the BPHZ method, and by [86] using the method of ﬂow equations. General theorems about quantum ﬁeld theory in curved spacetime within a model-independent setting were obtained in [71] (PCT-theorem), and by [114] (spin and statistics theorem). The literature on the quantization of gauge theory, and especially Yang–Mills theory in ﬂat spacetime is huge. The use of ghost ﬁelds was proposed ﬁrst by [48], and the early approaches to prove gauge invariance at the renormalized level used the method of Feynman graphs, together with special regularization techniques [73–75]. More recent discussions based on the Hopf-algebra structure behind renormalization [24, 25, 87] may be found in [112, 113]. With the discovery of the BRST-method [11, 12], cohomological methods were developed and used to argue that gauge invariance can be maintained at the perturbative level in ﬂat spacetime. Comprehensive reviews containing many references are [28, 95, 64, 5], see also, e.g., [110, 111, 44–46]. There are also other approaches to quantum gauge invariance in ﬂat space, based on the Epstein–Glaser method [47] for renormalization. These are described in the monographs [103, 104] and also in [109], which also contain many references. For a related approach, see [108]. The idea to formulate quantum gauge theory at the level of observables, and to implement the gauge invariance in the operator setting was developed in ﬂat space in [42, 39, 38], building on earlier work of [88, 89]. A somewhat more detailed comparison between the various approaches to the gauge invariance problem and our solution is given in Sec. 4.9, where additional references are given. 2. Generalities Concerning Classical Field Theory 2.1. Lagrange formalism Most, though not all, known quantum ﬁeld theories have a classical counterpart that is described in terms of a classical Lagrangian ﬁeld theory. This is especially true for the gauge theories studied in this paper, so we collect some basic notions and results from Lagrangian ﬁeld theory in this subsection that we will need later.

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Not surprisingly, for perturbative quantum ﬁeld theories derived from a classical Lagrangian, many formal aspects can be formulated using the language of classical ﬁeld theory, but we emphasize that, from the physical viewpoint, quantum ﬁelds are really fundamentally diﬀerent from classical ﬁelds. To specify a classical ﬁeld theory on an n-dimensional manifold M , we ﬁrst need to specify its ﬁeld content. We will generally divide the ﬁelds into background ﬁelds, collectively denoted Ψ, and dynamical ﬁelds, collectively denoted Φ. Both background and dynamical ﬁelds are viewed as sections in a certain ﬁber bundle, B → M , over the spacetime manifold. We will assume that the background ﬁelds always comprise a Lorentzian metric g = gµν dxµ dxν over M (which is a section in the bundle of non-degenerate symmetric tensors in T ∗ M ⊗ T ∗ M of signature (− + + · · · +)). More generally, the background ﬁelds may comprise a non-abelian background gauge connection, or varions external sources. We will also admit Grassmann-valued ﬁelds, which are described in more detail below. The dynamical ﬁelds will typically satisfy equations of motion, which are derived from an action principle. By contrast, the background ﬁelds will never be subject to any equations of motion. To set up an action principle, we need to specify a Lagrangian. The Lagrangians that we will consider have the property that they are locally and covariantly constructed out of the dynamical ﬁelds Φ, and the background ﬁelds Ψ. In particular, they do not depend implicitly on additional background structure such as the speciﬁcation of a coordinate system. Since such functionals will play an important role in perturbation theory, it is worth deﬁning the notion that a quantity is locally and covariantly out of a set of dynamical and non-dynamical ﬁelds Φ, Ψ with some care. Let us denote by B → M the “total bundle” in which the dynamical and non-dynamical ﬁelds live. For example, in case all the ﬁelds are tensor ﬁelds, the total bundle is simply the direct sum of all the tensor bundles corresponding to the various types of ﬁelds. If x ∈ M , we let Jxk (B) denote the space of “k-jets” over M . This is deﬁned as the equivalence class of all sections σ = (Φ, Ψ) : M → B, with the equivalence relation σ1 ∼ σ2 if ∇q σ1 |x = ∇q σ2 |x for all q ≤ k, where ∇ is any aﬃne connection in the bundle B, and where we have put ∇k σ = dxµ1 ⊗ · · · ⊗ dxµk ∇(µ1 · · · ∇µk ) σ.

(1)

We say that a p-form O = Oµ1 ...µp dxµ1 ∧ · · · ∧ dxµk is constructed out of σ = (Φ, Ψ) and its ﬁrst k derivatives if O is a map O:

Jxk (B)

→

p

Tx∗ M

(2)

for each x ∈ M , which we will also write as O(x) = O[σ(x), ∇σ(x), . . . , ∇k σ(x)]. Now let ψ : M → M be an immersion that lifts to a bundle map B → B denoted by the same symbol, and let σ and σ be sections in B → M , respectively B → M , such that σ = ψ ∗ σ . We will say that O is a p-form that is locally constructed out

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of the ﬁelds σ if we have O[σ(x), ∇σ(x), . . . , ∇k σ(x)] = ψ ∗ O[σ (x ), ∇σ (x ), . . . , ∇k σ (x )],

ψ(x) = x , (3)

for any x and any such embedding ψ. This condition makes precise the idea that O is only constructed out of σ = (Φ, Ψ) and ﬁnitely many of its derivatives, but depends on “nothing else”. For example, if the ﬁelds are a background metric, g, and a set of dynamical tensor or spinor ﬁelds Φ, then one can show that O can depend upon the metric only via the curvature, i.e. it may be written in the form O(x) = O[Φ(x), ∇Φ(x), . . . , ∇k Φ(x), g(x), R(x), ∇R(x), . . . , ∇k−2 R(x)]

(4)

where ∇ is now the Levi–Civita (or spin-) connection associated with g, and R = Rµνσρ (dxµ ∧dxν )⊗(dxσ ∧dxρ ) is the curvature tensor. This result is sometimes called the “Thomas replacement theorem”, and a proof may be found in [80]. The second example relevant to this work is when the background ﬁelds contain in addition a background gauge connection ∇ in a principal ﬁber bundle, such as B = M × G. Then the lift of ψ to a bundle map B → B , with B = M × G incorporates the speciﬁcation of a map γ : M → G that provides the identiﬁcation of the ﬁbers, i.e. a local gauge transformation. The condition that ∇ = ψ ∗ ∇ then means that ∇ = ∇+γ −1 dγ, and the condition of local covariance of a functional O now implies that O can depend on the connection only via its curvature F and its covariant derivatives ∇F, . . . , ∇k−2 F . More generally, if in addition there are dynamical ﬁelds Φ valued in an associated bundle B×G V (with V a representation of G), then O can only depend on gauge invariant combinations of Φ, ∇Φ, . . . , ∇k Φ. These statements can be proved by the same type arguments as in [80]. For completeness, we give a proof of this generalization of the Thomas replacement theorem incorporating gauge ﬁelds in Sec. 2.3 below. In our later application to Yang–Mills theory, Φ will consist of Lie-algebra valued vector and ghost ﬁelds, in which case V is the Lie-algebra of G, on which G acts via the adjoint representation. We denote the space of all locally covariant p-form functionals (2) by Pp (M ), or simply by Pp , and we deﬁne P(M ) =

n

Pp (M ).

(5)

p=0

We also assume for technical reasons that the expressions in P have at most polynomial dependence upon the dynamical ﬁelds Φ, and an analytic dependence upon the background ﬁelds Ψ. These deﬁnitions can easily be generalized to the case when (Φ, Ψ) are not ordinary ﬁelds valued in some bundle, but instead Grassmann valued ﬁelds. A Grassmann valued ﬁeld is by deﬁnition simply a ﬁeld that is valued in the inﬁnite dimensional exterior algebra E, which is the graded vector space E = Ext(V ) =

n

En ,

En =

n

V

(6)

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with V some inﬁnite dimensional complex vector space. The space E is equipped with the wedge product ∧ : En × Em → Em+n , which has the property that en em = (−1)nm em en for en ∈ En , em ∈ Em , and en em = 0 for all en if and only if em = λen . The elements en in En are assigned Grassmann parity ε(en ) = n modulo 2. Thus, when Grassmann valued ﬁeld are present, expressions O ∈ P p are no longer valued in the p-forms over M , but instead in the set of p-forms over M , tensored with E. A Grassmann valued ﬁeld consequently has a formal expansion of the form en Φn (x), en ∈ En , (7) Φ(x) = n≥0

where each Φn is an ordinary p-form ﬁeld. A Lagrangian is a (possibly E-valued) n-form L = L[Φ, Ψ] that is locally and covariantly constructed out of the dynamical ﬁelds Φ, the background ﬁelds Ψ, and ﬁnitely many of its derivatives. For manifolds M carrying an orientation, which we shall assume to be given from now on, one can deﬁne a canonical volume n-form ε = εµ1 ···µn dxµ1 ∧ · · · ∧ dxµn by the standard formula √ (8) dx = ε = −gdx0 ∧ · · · ∧ dxn−1 √ where x0 , . . . , xn−1 is right-handed, and where −g is the square root of minus the determinant of gµν . Using the volume n-form, one deﬁnes the Hodge dual of a form by ∗αµ1 ···µn−p =

(−1)p ν1 ···νp ε µ1 ···µn−p αν1 ···νp (n − p)!

(9)

and it is thereby possible to convert the Lagrangian into a scalar. This is more standard in the physics literature, but for our purposes it will be slightly more advantageous to view L as an n-form. For compactly supported ﬁeld conﬁgurations, we may form an associated action by integrating the Lagrangian n-form over M , L. (10) S= M

We deﬁne the left and right variations, δL S/δΦ(x), respectively, δR S/δΦ(x), with respect to the dynamical ﬁelds by the relation d δR S δL S d S[Φt ; Ψ]|t=0 = δΦ(x) = , δΦ(x) = Φt (x)|t=0 . δΦ(x) dt δΦ(x) δΦ(x) dt M M (11) The left and right derivatives may diﬀer from each other only for Grassmann-valued ﬁelds Φ, and we adopt the convention that the left derivative is meant by default if the subscript is suppressed. In terms of the Lagrangian n-form, the variational

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derivative is given by δS = (−1)q ∇(µ1 ···µq ) δΦ(x) q=0 k

∂L ∂(∇(µ1 ···µq ) Φ(x))

,

(12)

where we use the abbreviation ∇(µ1 ···µk ) for the k-fold symmetrized derivative in Eq. (1). The quantity δS/δΦ(x) is an n-form that is locally and covariantly constructed out of the dynamical ﬁelds and the background ﬁelds and their derivatives, and may hence be viewed as a diﬀerential operator acting on Φ. Field conﬁgurations Φ satisfying the diﬀerential equation δS =0 δΦ(x)

(13)

are said to satisfy the equations of motion associated with S, or to be “on shell”. A symmetry is an inﬁnitesimal ﬁeld variation sΦ = δΦ of the dynamical ﬁelds such that sL = dB for some locally constructed (n − 1)-form B. The existence of symmetries implies the existence of a conserved Noether current, J, deﬁned by J(Φ) = θ(Φ, sΦ) − B(Φ),

(14)

where θ is the (n − 1) form deﬁned by θν1 ··· νn−1 (Φ, δΦ) =

k−1

∇(µ1 ··· µq ) δΦ

q=0

∂Lν1 ··· νn−1 σ ∂(∇(µ1 ··· µq σ) Φ)

,

(15)

where we are suppressing the dependence upon the background ﬁelds. θ is the boundary term that would arise if L is varied under an integral sign. As a consequence of the deﬁnition, we have δS , (16) dJ = sΦi δΦi so J is indeed conserved on shell. In the context of perturbation theory studied in this paper, the Lagrangian is a power series L = L0 + λL1 + λ2 L2 + · · · ,

(17)

where L0 is called the “free Lagrangian” and contains only terms at most quadratic in the dynamical ﬁelds Φ, hence giving rise to linear equations of motion. If the symmetry is also a formal power series s = s0 + λs1 + λ2 s2 + · · · ,

(18)

then there is obviously an expansion J = J0 + λJ1 + λ2 J2 + · · · ,

(19)

s0 is a symmetry of the free Lagrangian L0 with corresponding conserved Noether current J0 when the equations of motion hold for L0 . The theories that we will deal with in this paper all have the property that L0 contains the highest derivative terms in the dynamical ﬁelds Φ. In this case,

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it is natural to assign a “canonical dimension” to each of the dynamical ﬁelds as follows. Let us assume that the background ﬁelds consist of a metric, g, and a covariant derivative operator, ∇, which acts like the Levi–Civita connection on tensors. Consider a rescaling of the metric by a constant conformal factor, µ2 g, where µ ∈ R. Then there exists typically a unique rescaling Φi → µd(Φi ) Φi , Ψi → µd(Ψi ) Ψi and ci → µd(ci ) ci of the dynamical ﬁelds, the background ﬁelds, and the coupling constants in L0 such that L0 → L0 . The numbers d(Φi ), d(Ψi ) and d(ci ) are called the “engineering dimensions” of the ﬁelds and the couplings, respectively. The corresponding dimension of composite objects in P is given by the counting operators Nf , Nc , Nr : P(M ) → P(M ) Nf = Nc = Nr =

(d(Φi ) + k)∇k Φi d(ci )ci

∂ , ∂(∇k Φi )

∂ , ∂ci

(d(Ψi ) + k)∇k Ψi

(20) (21)

∂ . ∂(∇k Ψi )

(22)

Not for all S, and not for all choices of the background ﬁelds Ψ do the equations of motion (13) possess a well-posed initial value formulation, which is a key requirement for a physically reasonable theory. For ﬁrst order diﬀerential equations, one can formulate general conditions under which the equations will possess a wellposed initial value formulation. For example, for ﬁrst order systems of so-called “symmetric hyperbolic type”, the initial value problem is well-posed in the sense that, given initial data for Φ on a suitably chosen (n − 1)-dimensional hypersurface, there exists a unique solution for suﬃciently short “times”, i.e. in some open neighborhood of Σ. Furthermore, the propagation of disturbances is “causal” in a well-deﬁned sense, see, e.g., [59]. Equations of motion of higher diﬀerential order can always be reduced to ones of ﬁrst order by picking suitable auxiliary ﬁeld variables, but it is not obvious in a given example which choice will lead to a symmetric hyperbolic system. Fortunately, the equations of motion that we will study in this paper will all be of the form of a simple wave-equation. Actually, since we only consider perturbation theory, we will only be concerned with the existence of solutions for the “free theory”, deﬁned by S0 . For the actions considered in this paper, the corresponding equations are linear, and of the form 0=

δS0 = Φ + (lower order terms) δΦ

(23)

where = g µν ∇µ ∇ν is the wave operator in curved space. Such equations do possess a well-posed initial value formulation if the metric does not have any gross causal pathologies, such as closed timelike curves. A typical such equation (for a real scalar ﬁeld Φ = φ) is the Klein–Gordon equation ( − m2 )φ = j,

(24)

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where m2 is a constant. For that equation, the initial value problem is well-posed globally for example if the spacetime manifold (M, g) is “globally hyperbolic”, meaning by deﬁnition that there exists a (necessarily spacelike) “Cauchy-surface”, Σ, i.e. a surface which has the property that any inextendible timelike curve hits Σ precisely once. We will always assume in this work that (M, g) is globally hyperbolic. Then, given any f0 , f1 ∈ C0∞ (Σ), there exists a unique solution to Eq. (24) such that φ|Σ = f0 , and nµ ∇µ φ|Σ = f1 , where n is the timelike normal to Σ. The well-posedness of the initial value problem for the Klein–Gordon equation directly leads to the existence of advanced and retarded propagators, which are the uniquely determined distributions ∆A , ∆R on M × M with the properties ( − m2 )∆A (x, y) = δ(x, y) = ( − m2 )∆R (x, y)

(25)

and the support properties supp ∆A,R ⊂ {(x, y) ∈ M × M | y ∈ J ∓ (x)},

(26)

where J ± (S) denotes the causal future/past of a set S ⊂ M and is deﬁned as the set of points x ∈ M with the property that there is a future/past directed timelike or null curve γ connecting x with a point in S. 2.2. Yang–Mills theories, consistency conditions, cohomology The theory that we are considering in this paper is pure Yang–Mills theory, classically described by the action 1 F I ∧ ∗FI . (27) Sym = − 2 M Here, Fµν = (i/λ)[Dµ , Dν ] is the 2-form ﬁeld strength tensor of a gauge connection D in some principal G-bundle over M , where G is a direct product of U (1)l and a semi-simple Lie group. λ is a coupling constant that could be omitted at the classical level. For the sake of simplicity, we will assume that the principal bundle is toplogically trivial, i.e. of the form M × G. We denote the generators of the I dxµ ∧ dxν for gauge Lie algebra by TI , I = 1, . . . , dim(G), and we write F = TI Fµν the components of the ﬁeld strength and similarly for any other Lie algebra-valued ﬁeld. Lie algebra indices I are raised an lowered with the Cartan–Killing metric kIJ deﬁned by Tr ad(TI ) ad(TJ ) for the generators of the semi-simple part, and by 1 for the abelian factors. The classical ﬁeld equations for the connection D derived from is action are g µν [Dµ , [Dν , Dσ ]] = 0,

(28)

or, written in more conventional form, D[µ ∗Fνσ] = 0.

(29)

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As is particularly clear from the ﬁrst formulation, the connection D is the dynamical ﬁeld variable in this equation. It is convenient to decompose it into a ﬁxed background connection ∇, plus λ times a Lie algebra-valued 1-form ﬁeld A = TI AIµ dxµ , D = ∇ + iλA,

λ ∈ R.

(30)

The 1-form ﬁeld A is now the dynamical variable. The coupling constant λ is redundant at the classical level and may be absorbed in A, but it is useful as an explicit perturbation parameter when one wants to study the theory perturbatively. The coupling constant λ acquires a new role at the quantum level due to renormalization eﬀects as we will see below. It is convenient to deﬁne ∇ on tensor ﬁelds to be the standard Levi–Civita connection of the metric. The background derivative operator then has the curvature tensor I R(TI )kσ [∇µ , ∇ν ]kσ = Rµνσ ρ kρ + fµν

(31)

where R is the representation of the Lie algebra associated with kµ , and f = I dxµ ∧ dxν is the curvature of the background gauge connection. In Minkowski TI fµν space, it is typically assumed that ∇ = ∂, implying that f = 0. For simplicity, we will assume that the background gauge connection has been chosen as the standard ﬂat connection in the bundle M × G, so that f = 0 on our manifold M . The advantage of this choice is that all quantities that are locally and covariantly constructed out of the ﬁeld A and the background structure g = gµν dxµ dxν and ∇ can be written in the form Eq. (4) with Φ = A, without any explicit appearance of the background curvature f . If f = 0, we would have to include everywhere explicitly the background curvature. With the decomposition D = ∇ + iλA, the curvature F is given by I = ∇µ AIν − ∇ν AIµ + iλf IJK AJµ AK Fµν ν

(32)

f IJK

where are the structure constants of the Lie algebra deﬁned by [TI , TJ ] = f KIJ TK . The equations of motion, when written in terms of A, are not hyperbolic, in the sense that the highest derivative term is not of the form of a wave equation. Thus, the equations of motion for Yang–Mills theory do not straightforwardly admit an initial value formulation. This feature is a consequence of the fact that the Yang– Mills Lagrangian and equations of motion is invariant under the group of local gauge transformations acting on the dynamical ﬁelds by D → γ(x)−1 Dγ(x), where g : M → G is any smooth function valued in the group, or equivalently by i −1 γ dγ. (33) λ Since such local gauge transformations allow one to make local changes to the dynamical ﬁeld variables, it is clear that those are not entirely speciﬁed by initial conditions. However, the freedom of making local gauge transformation can be used to set some components of A to zero, so that the remaining components satisfy a hyperbolic equation and consequently admit a well-posed initial value formulation, as described, e.g., in [1]. Later, we want to perturbatively construct a quantum ∇ → ∇,

A → γ −1 Aγ −

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version of Yang–Mills theory, and for this purpose, another approach seems to be much more convenient. This approach consists in adding further ﬁelds to the theory which render the equations of motion hyperbolic, and which can, at a ﬁnal stage, be removed by a symmetry called “BRST-symmetry”. In the BRST approach, one introduces additional dynamical Grassmann Liealgebra valued ﬁelds C = TI C I , C¯ = TI C¯ I , and a Lie-algebra valued ﬁeld B = TI B I , and one deﬁnes a new theory with action Stot by Stot = Sym + Sgf + Sgh ,

(34)

where Sgf is a “gauge ﬁxing” term deﬁned by 1 Sgf = B I iGI + BI 2 M

(35)

with a local covariant “gauge ﬁxing” functional G of the ﬁeld A, and where Sgh is the “ghost” term, deﬁned by δ(G I C¯I ) Dµ C J ε. (36) Sgh = i δAJµ M The total set of dynamical ﬁelds is denoted Φ = (AI , C I , C¯ I , B I ), and their assignment of ghost number, Grassmann parity, dimension, and form degree is summarized in the following table Φ Dimension Ghost Number Form Degree Grassman Parity

AI 1 0 1 0

CI 0 1 0 1

C¯ I 2 −1 0 1

BI 2 0 0 0

The assignments of the dimensions are given for the case when the spacetime M is 4-dimensional, to which we now restrict attention for deﬁniteness. To state the relation between the auxiliary theory and the original Yang–Mills theory, one ﬁrst observes that the action Stot of the auxiliary theory is invariant under the following so-called BRST-transformations [11, 12]: sAI = dC I + iλf IJK AJ C K , sC I = −

iλ I f CI CK , 2 JK

(37) (38)

sC¯ I = B I ,

(39)

sB I = 0.

(40)

The assignment of the various numbers to the ﬁelds are done in such a way that s has dimension 0, ghost number +1, Grassmann parity +1, and form degree 0. It is declared on arbitrary local covariant functionals O ∈ P(M ) of the dynamical ﬁelds ¯ B and the background ﬁelds by the rules ∇◦s−s◦∇ = 0 = dxµ ◦s+s◦dxµ , A, C, C,

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and on (wedge) products via the graded Leibniz rule, s(Op ∧ Oq ) = sOp ∧ Oq + (−1)p+ε(Op ) Op ∧ sOq . With these deﬁnitions, it follows that s2 = 0,

sd + ds = 0.

(41)

The key equation is sStot = 0,

(42)

which one may verify by writing Stot in the form Stot = Sym + sΨ where

C¯ I

Ψ= M

1 BI + iGI 2

(43) ε.

(44)

Indeed, Sym is invariant because s just acts like an ordinary inﬁnitesimal gauge transformation on A, while s annihilates the second term because s2 = 0. In this paper, we choose the gauge ﬁxing functional as G I = ∇µ AIµ .

(45)

Then the equation of motion for B I is algebraic, B I = −i∇µ AIµ . Inserting this into the equation of motion for AIµ , one sees that this equation is of the form (23). Indeed, this special choice of the gauge ﬁxing function eﬀectively eliminates a term of the form ∇µ ∇ν AIµ (which would spoil hyperbolicity) from the equations of motion for the gauge ﬁeld, thus leaving only the wave operator. The remaining equations for C I , C¯ I are also of the form (23). Thus, the equations of motion for the total action Stot are of wave equation type. They consequently possess a well-posed initial value formulation at the linear level, which is suﬃcient for perturbation theory, and in fact also at the non-linear level [1]. Given that Stot deﬁnes a classical theory with a well-posed initial value formulation, we may deﬁne an associated graded Peierls bracket [38, 37, 96, 30, 92] {O1 , O2 }P.B. , for any pair of localc functionals O1 , O2 ∈ P. Since the action Stot is invariant under s, it follows that the (graded) Peierls bracket is also invariant under s, in the sense that s{O1 , O2 }P.B. = {sO1 , O2 }P.B. + (−1)ε(O1 )+deg(O1 ) {O1 , sO2 }P.B. ,

(46)

(−1)ε(O1 ) denoting the Grassmann parity of a functional of the ﬁelds, and deg(O1 ) the form degree. The connection between the classical auxiliary theory associated with Stot , and Yang–Mills theory with action Sym is based on the following c The

Peierls bracket may also be deﬁned for certain non-local functionals. The consideration of such functionals is necessary in order to obtain a set of functionals that is stable under the bracket.

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key lemma: Lemma 1. Let O ∈ P be a local covariant functional of the background connection, ¯ B). Let sO = 0. Then, up to a the background metric, and the ﬁelds Φ = (A, C, C, term of the form sO , O is a linear combination of elements of the form O= rtk (g, R, ∇R, . . . , ∇k R) pri (C) Θrj (F, DF, . . . , Dl F ), (47) i

k

j

where pr , Θs are invariant polynomials of the Lie-algebra of G, where F = Fµν dxµ ∧ dxν , and where rt is a local functional of the metric g, and the Riemann tensor R and its derivatives. The lemma is essentially a standard result in BRST-cohomolgy, see, e.g., [5] and the references cited there. The only diﬀerence to the formulation given in [5] is that, in the present setting, the coeﬃcients rt can only depend locally and covariantly upon the metric (as opposed to being an arbitrary form on spacetime). The fact that rt has to be a functional of the Riemann tensor and its derivatives follows again from the “Thomas replacement argument”, see, e.g. [80], and the next subsection. Thus, at zero ghost number, the local and covariant functionals in the kernel of s are precisely the local gauge invariant observables of Yang–Mills theory modulo an element in the image of s, so the equivalence classes of the kernel of s modulo the image of s at zero ghost number, {class. gauge. inv. ﬁelds} =

Kernel s Image s

(at zero ghost number),

(48)

are in one-to-one correspondence with the gauge invariant observables. Furthermore, by (46), the brackets are well-deﬁned on the cohomology classes, and the Yang–Mills equations of motion hold modulo s. Thus, the theory whose observables are deﬁned by the equivalence classes of s (at zero ghost number), and whose bracket is deﬁned by the Peierls bracket may be viewed as a deﬁnition of classical Yang–Mills theory. The BRST-transformation s plays a crucial role also in the perturbative quantum ﬁeld theory associated with Yang–Mills theory, where its role is among other things to derive certain consistency conditions on the terms in the renormalized perturbation series. We therefore discuss some of the relevant facts about the BRSTtransformation in some more detail. Since s2 = 0, the BRST transformation deﬁnes a “diﬀerential”, or, more precisely, a diﬀerential complex s : P0 → P1 → · · · → PN → · · ·

(49)

where a subscript denotes the grading of the functionals in P by the ghost number, deﬁned by the ghost number operator Ng counting the ghost number of an element in P by the formula ∂ ∂ . (50) − ∇k C¯ I ∇k C I Ng = ∂(∇k C I ) ∂(∇k C¯ I )

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Thus, P is doubly graded space, by the form degree and ghost number, and we write Pqp for the subspace of elements with form degree q and ghost number p. We deﬁne the cohomology ring H p (s, Pq ) to be the set of all local covariant q-form functionals O of ghost number p, and sO = 0, modulo the set of a q-form functionals O = sO with ghost number p, i.e. H p (s, Pq (M )) =

{Kernel s : Pqp → Pqp+1 } . {Image s : Pqp−1 → Pqp }

(51)

The above lemma may be viewed as the determination of the space H q (s, Pp ) for all q, p. We will also encounter another cohomology ring, consisting of all s-closed local covariant functionals modulo exact local covariant functionals. To describe this ring more precisely, it is useful to know the following result, sometimes called “algebraic Poincare lemma”, or “fundamental lemma of the calculus of variations”: Lemma 2 (Algebraic Poincare Lemma). Let α = α[Φ, Ψ] be a p-form on an n-dimensional manifold M, which is locally and covariantly constructed out of a number of dynamical ﬁelds Φ, and background ﬁelds Ψ. Assume that dα[Φ, Ψ] = 0 for all Ψ, and that each Ψ is pathwise connected to a reference Ψ0 for which α[Φ, Ψ0 ] = 0. Then α = dβ for some β = β[Φ, Ψ] which is locally constructed out of the ﬁelds. The proof is given for convenience in the next subsection. Consider now a Oq ∈ Pq such that sOq = dOq−1 , i.e. Oq is s-closed modulo d. Then, by s2 = 0 and ds + sd = 0, the form sOq−1 is d-closed, and hence d-exact by the fundamental lemma, so sOq−1 = dOq−2 . We can now repeat this procedure until we have reached the forms of degree 0, thereby arriving at what is called a “decent-equation”, or a “ladder”: sOq = dOq−1 ,

(52)

sOq−1 = dOq−2 ,

(53)

···

(54)

sO1 = dO0 ,

(55)

sO0 = 0.

(56)

Note that, within each ladder, the form degree plus the ghost number is constant. We denote the space of Oq that are s-closed modulo d at ghost number p, factored by elements that are s-exact modulo d by H p (s|d, Pq ) ≡ H p (s, H q (d, P)). In practice, ladders can be used to determine the cohomology of s modulo d. For the purpose of perturbative quantum ﬁeld theory, it will be convenient to consider yet another cohomology ring related to s that incorporates also the equations of motion. Let us add to the theory a further set of background ﬁelds (“BRST sources”, or “anti-ﬁelds” [6]) Φ‡ = (A‡I , CI‡ , C¯I‡ , BI‡ ) corresponding to the

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dynamical ﬁelds Φ = (AI , C I , C¯ I , B I ): Φ‡ Dimension Ghost Number Form Degree Grassmann Parity

A‡I 3 −1 3 1

CI‡ 4 −2 4 0

C¯I‡ 2 0 4 0

Consider now the action S[Φ, Φ‡ ] = Sym + Sgf + Sgh + Ssc ,

Ssc =

BI‡ 2 −1 4 1

sΦi ∧ Φ‡i .

(57)

M

The new action is still BRST-closed, sS = 0, because it is given by the sum of Stot and a BRST-exact term, and it satisﬁes in addition (S, S) = 0, where the “anti-bracket” (., .) is deﬁned by the equation [6]

δR F1 δL F2 δL F2 deg(Φ‡i ) δR F1 ∧ − (−1) ∧ (F1 , F2 ) = . (58) δΦ‡i (x) δΦ‡i (x) δΦi (x) M δΦi (x) The local anti-bracket satisﬁes the graded Jacobi-identity (−1)ε3 ε1 ((F1 , F2 ), F3 ) + (−1)ε2 ε1 ((F2 , F3 ), F1 ) + (−1)ε3 ε2 ((F3 , F1 ), F2 ) = 0 (59) and as a consequence (F, (F, F )) = 0 for any F . The diﬀerential incorporating the equations of motion is deﬁned by sˆF = (S, F ).

(60)

It satisﬁes sˆ2 = 0 as a consequence of (S, S) = 0 and the Jacobi identity, as well as, sˆd + dˆ s = 0. It diﬀers from the BRST-diﬀerential s by the “Koszul–Tatediﬀerential” σ sˆ = s + σ,

(61)

where σ 2 = 0, and σ is anticommuting with s. It acts on the ﬁelds by σΦi = 0,

σΦ‡i =

δS . δΦi

(62)

Thus, acting with σ on a monomial in P containing an anti-ﬁeld automatically gives an expression containing a factor of the equations of motion, i.e. an on-shell quantity. This will be useful in the context of perturbative quantum ﬁeld theory in order to keep track of such terms. Starting from the diﬀerential sˆ, one can s, Pq ) and H p (ˆ s|d, Pq ). The ring H 0 (ˆ s, Pq ) is again deﬁne cohomology rings H p (ˆ p still described by Lemma 2, because one can prove in general that H (s, Pq ) and s, Pq ) are isomorphic, see, e.g., [5]. The relative cohomology rings H p (ˆ s|d, Pq ) H p (ˆ appear in the analysis of gauge invariance in quantum Yang–Mills theory. They are also known, but they depend somewhat upon the choice of the gauge group G.

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They are described by the following theorem, see, e.g., [5]: Theorem 1. Let the Lie-group G be semi-simple with no abelian factors, and let n = dim(M ). Then each class in H(ˆ s|d, Pn ) is a linear combination of expressions O of the form (47), and representatives O of the form O = n-form part of rtk (R, ∇R, . . . , ∇nk R) qri (C + A, F ) fsj (F ), i

k

j

(63) where qri (A + C, F ) are the Chern–Simons forms, 1 qr (A + C, F ) = Tr((C + A)[tF + λt(t − 1)(C + A)2 ]m(r)−1 )dt

(64)

0

where fs are strictly gauge-invariant monomials of F containing only the curvature, F, but not its derivatives. The numbers m(r) are the degrees of the independent Casimir elements of G, and the trace is in some representation. The rt are taken to be a basis of closed forms drt = 0 that are analytic functions of the metric and the covariant derivatives of the Riemann tensor. For p < n, a basis of H(ˆ s|d, Pp ) is given by the O at form degree p, together with all elements O of the form (47), for any Lie-group H = U (1)l × G, with G semi-simple. Remarks. (1) The statement of the theorem given in [5] only asserts that the rt are closed forms on M . To obtain that the rt in fact have to be analytic functions of R, ∇R, ∇2 R, . . . , one has to use that, as we are assuming, the elements in Pq are locally and covariantly constructed out of the metric in the sense described above, with an analytic dependence upon the spacetime metric. It then follows from the “Thomas replacement argument” [80] that the rt have to be analytic functions of the curvature tensor and its derivatives. It furthermore follows that the rt may be chosen to be characteristic classes rt = Tr(R ∧ · · · ∧ R),

(65)

where Tr is the trace in a representation of the Lie-algebra of SO(n − 1, 1), and ab where R = Tab Rµν dxµ ∧ dxν is the curvature 2-form of the metric, identiﬁed with a 2-form valued in the Lie-algebra of SO(n − 1, 1) via a tetrad ﬁeld eaµ dxµ . (2) There are more elements in H(ˆ s|d, Pn ) when the group G has abelian factors, see, e.g., for a discussion [64]. In pure Yang–Mills theory, abelian factors decouple and hence can be treated separately. In perturbation theory, we expand S as S = S0 + λS1 + λ2 S2 , and we correspondingly expand the Lagrangian as 1 L0 = dAI ∧ ∗dAI − idC¯ I ∧ ∗dCI 2 1 I + B id∗AI + ∗BI + s0 AI ∧ A‡I , 2

(66)

(67)

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L1 =

1 fIJK ∗dAI ∧ AJ ∧ AK + fIJK C¯ I ∧ AJ ∧ ∗dC K 2 + s1 AI ∧ A‡I + s1 CI ∧ C ‡I + s1 C¯I ∧ C¯ ‡I ,

(68)

1 I f JK fILM AJ ∧ AK ∗ (AL ∧ AM ), (69) 4 in our choice of gauge (45). We correspondingly have an expansion of the Slavnov– s1 + λ2 sˆ2 , and similarly of the Koszul–Tate diﬀerenTaylor diﬀerential as sˆ = sˆ0 + λˆ 2 tial as σ = σ0 + λσ1 + λ σ2 . The zeroth order parts of these expansions still deﬁne diﬀerentials. The free Slavnov–Taylor diﬀerential sˆ0 O = (S0 , O), decomposed as L2 =

sˆ0 = s0 + σ0 ,

(70)

will play an important role in perturbative quantum ﬁeld theory. Its action is given explicitly by sˆ0 AI = dC I ,

sˆ0 C I = 0,

sˆ0 C¯ I = B I ,

sˆ0 B I = 0

(71)

on the ﬁelds, where it coincides with that of s0 . Its action on the anti-ﬁelds is given by δS0 δS0 δS0 δS0 , sˆ0 CI‡ = , sˆ0 C¯I‡ = ¯ I , sˆ0 BI‡ = (72) I I δA δC δB I δC where it coincides with that of σ0 . The actions of s0 and σ0 are summarized in the following table: sˆ0 A‡I =

Field I

s0

σ0 I

A BI CI C¯ I

dC 0 0 BI

0 0 0 0

A‡I

0

−d ∗ dAI − i ∗ dBI

BI‡ CI‡ C¯I‡

0 0

−id ∗ AI id ∗ dC¯I − dA‡

0

id ∗ dCI

I

In perturbation theory, if F = F0 + λF1 + λ2 F2 + · · · is a local functional of the ﬁelds, equations like sF = 0 are understood in terms perturbative sense, as the hierarchy of identities obtained by expanding the terms out in λ. This makes no diﬀerence with regard to the above 2 cohomological lemmas, which now also have to be interpreted in the sense of formal power series (in fact, the proof of those lemmas is perturbative). We ﬁnally metion a few identities satisﬁed by the BRST-current J deﬁned above that we will need later. First, from the expression for the diﬀerential of the BRST current, we have (S, Φi (x))(Φ‡i (x), S). (73) dJ(x) = i

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Applying the diﬀerential sˆ = (S, .) and using the Jacobi identity for the anti-bracket as well as (S, S) = 0, we get dˆ sJ = 0,

(74)

so by Lemma 3, we have the identity sˆJ = dK,

(75)

for some (n − 2)-form K, which is the beginning of a cohomologically trivial lads|d, P3 ). If we expand this identity in λ, der, i.e. J is the zero element in H 1 (ˆ we get sˆ0 J0 = dK0 ,

sˆ1 J0 + sˆ0 J1 = dK1 ,

etc.

(76)

The free BRST-current J0 and its perturbation J1 are given by J0 = ∗dAI ∧ dCI − iB I ∗dCI = sˆ0 (dAI ∧ AI − iC¯ I ∧ ∗dCI ),

i J1 = ifIJK C JAK ∧ ∗AI + ∗dC¯ IC JC K + iB IC J ∗AK 2 1 J K ‡I 1 I J K − C C A + dC ∧ ∗(A ∧ A ) , 2 2

(77)

and the Ki are given by K0 = 0, and −i fIJK dAIC JC K , 2 1 K2 = fIJK f I MN AI ∧ AKC MC N . 2 K1 =

(78) (79)

Another fact that we will need later is the equation sˆL = dJ,

(80)

where ψ = (Φ, Φ‡ ) collectively denotes the ﬁelds and anti-ﬁelds, and where we recall that J[ψ] = θ[ψ, sΦ], where θ was deﬁned in Eq. (15). To prove this relation, we recall that sˆ = s + σ, so sˆL[ψ] = σL[ψ] +

δS[ψ] ∧ sΦ + dθ[ψ, sΦ] δΦ

= σL[ψ] + σΦ‡ ∧ sΦ + dθ[ψ, sΦ] = dθ[ψ, sΦ],

(81)

using in the last line that σL = −δS/δΦ ∧ sΦ. The equation we have just derived may be expanded in powers of λ, leading to the relations sˆ0 L0 = dJ0 ,

(82)

sˆ0 L1 + sˆ1 L0 = dJ1 .

(83)

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2.3. Proof of the Algebraic Poincare Lemma, and the Thomas Replacement Theorem Lemma 3 (Algebraic Poincare Lemma). Let α = α[Φ, Ψ] be a p-form on an n-dimensional manifold M, which is locally and covariantly constructed out of a number of dynamical ﬁelds Φ, and background ﬁelds Ψ. Assume that dα[Φ, Ψ] = 0 for all Ψ, and that each Ψ is pathwise connected to a reference Ψ0 for which α[Φ, Ψ0 ] = 0. Then α = dβ for some β = β[Φ, Ψ] which is locally constructed out of the ﬁelds. The algebraic Poincare lemma has been rediscovered many times, and diﬀerent proofs exist in the literature. Here we follow the proof given in [116], for other accounts see, e.g., [5]. Proof. One ﬁrst considers the case when α[Φ, Ψ] is linear in Ψ, i.e. of the form αµ1 ...µp =

k

Ai µ1 ···µνp1 ···νi (Φ)∇(ν1 · · · ∇νi ) Ψ,

(84)

i=0

where we may assume that Ai is totally symmetric in the upper indices, and totally anti-symmetric in the lower indices. The condition that dα = 0 implies the condition Ak [µ1 ···µp ν1 ···νk δγ] δ ∇(δ ∇ν1 · · · ∇νk ) Ψ = 0.

(85)

At each x ∈ M , ∇(ν1 · · · ∇νk ) Ψ|x can be chosen to be an arbitrary totally symmetric tensor, so we must have Ak [µ1 ···µp (ν1 ···νk δγ] δ) = 0.

(86)

Contracting over δ, γ and using the symmetries of Ak , one ﬁnds

n k p + − Ak µ1 ···µp ν1 ···νk (k + 1)(p + 1) (k + 1)(p + 1) (k + 1)(p + 1) −

kp Ak γ[µ2 ···µp γ(ν2 ···νk δµ1 ] ν1 ) = 0 (k + 1)(p + 1)

(87)

and therefore that Ak µ1 ···µp ν1 ...νk =

kp γ(ν2 ···νk δµ1 ] ν1 ) . Ak (k + 1)(p + 1) γ[µ2 ···µp

(88)

For k = 0, this condition simply reduces to A0 = 0 and hence α = 0, thus proving that the lemma is trivially fulﬁlled when k = 0 and when α depends linearly on Ψ. For k > 0, one may proceed inductively. Thus, assume that the statement has been shown for all k ≤ m − 1. Deﬁne mp Am γ[µ2 ···µp ] γν2 ···νm ∇(ν2 · · · ∇νm ) Ψ, (89) τµ2 ···µm = (m + 1)(p + 1) and let α = α − dτ.

(90)

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Then α is still closed and locally constructed from Φ, Ψ, linear in Ψ, but by (194), it only contains terms with a maximum number m − 1 of derivatives on Φ. For such α , we inductively know that α = dγ for a locally constructed γ. Thus, α = d(γ + τ ), thereby closing the induction loop. Thus, we have proved the lemma when α depends linearly upon Ψ. Consider now the case when α[Φ, Ψ] is non-linear in Ψ. Let τ → Ψτ be a smooth d Ψ|τ =0 = δΨ, we have path in ﬁeld space with Ψ0 = Ψ. Putting dτ

k d ∂α[Φ, Ψ] d α[Φ, Ψτ ]|τ =0 = d ∇(µ1 · · · ∇µi ) δΨ = 0. (91) dτ ∂(∇(µ1 · · · ∇µi ) Ψ) i=1 Since this must hold for all paths, the identity holds for all Φ, Ψ, δΨ. Thus, since this expression is linear in δΨ and must hold for all δΨ, we can ﬁnd a γ such that d α[Φ, Ψτ ]|τ =0 = dγ[Φ, Ψ, δΨ]. dτ

(92)

where γ is constructed locally out of the ﬁelds. Thus, for any path in ﬁeld space, we have τ

d α[Φ, Ψτ ] = α[Φ, Ψ0 ] + d γ Φ, Ψt , Ψt dt . (93) dt 0 Consequently, for any ﬁeld conﬁguration Ψ that can be reached by a diﬀerentiable path from a reference conﬁguration Ψ0 for which α(Φ, Ψ0 ), we can write α(Φ, Ψ) = dβ(Φ, Ψ). We next give the precise statement and proof of the Thomas replacement theorem in the case that we have gauge ﬁelds, a metric and background ﬁelds. We consider spacetime manifolds (M, g), and G principal ﬁber bundles B → M over M with an arbitrary but ﬁxed structure group G. On B, we consider gauge connections D. As above, if we have any section k in (T ∗ M )⊗m ⊗ (T M )⊗n times B ×G V , where V is a vector space with an action of G, then we let D act on the “tensor part” of k by the Levi–Civita connection ∇ of g, and on the “ﬁber bundle part” by D. We denote by jxp (g, D, Ψ) the p-jet of the metric and the gauge connection and background ﬁeld, and we consider functionals O(x) = O[jxp (g, D, Ψ)].

(94)

Let ψ : B → B be a bundle morphism, i.e. a diﬀeomorphism of B which is compatible with the G-action on B in the sense that gψ(y) = ψ(gy). Let g, D be a metric and connection on B, and let ψ ∗ g, ψ ∗ D be the pull-backs. If O depends locally and covariantly upon the metric and connection, then we have ψ ∗ O[j p (ψ ∗ g, ψ ∗ D, Ψ)] = O[j p (ψ ∗ g, ψ ∗ D, Ψ)],

(95)

where we note that ψ ∗ does not act on the background ﬁelds on the right-hand side. This equation is to hold for all g, D, and some choice of the background ﬁeld(s) Ψ.

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If B = M × G, then the above condition can be stated somewhat more explicitly as follows. We may identify the p-jet of the background ﬁelds Ψ with a collection of tensor ﬁelds on M (including the derivatives of any background ﬁeld), which we again denote by Ψ for simplicity. Let us introduce an arbitrary background derivative operator D (no relation to D is assumed), and consider ﬁrst the case when ψ is a “pure diﬀeomorphism”, i.e. ψ = f × idG , with f a diﬀeomorphism of M . Let us decompose D as D = D + iλA, with A a Lie-algebra valued 1-form on M . Then the above condition can be written as f ∗ O[g, . . . , Dp g, A, . . . , Dp A, Ψ] = O[f ∗ g, . . . , Dp f ∗ g, f ∗ A, . . . , Dp f ∗ A, Ψ]

(96)

where as usual we denote by Dk = dxµ1 ⊗ · · · dxµk D(µ1 · · · Dµk ) the symmetrized k-fold derivative. Note that, in the above expression, f ∗ does not act on any of the background ﬁelds Ψ, nor on D. (The background ﬁelds Ψ may in particular include the curvature of the background connection D, and their derivatives are regarded as “independent ﬁelds”.) Secondly, let ψ be a “pure gauge transformation”, i.e. a transformation of the form ψ = idM × γ, where γ : M → G is a local gauge transformation. Let Aγ = γ −1 Aγ − (i/λ)γ −1 dγ. Then the above condition (95) becomes O[g, . . . , Dp g, A, . . . , Dp A, Ψ] = O[g, . . . , Dp g, Aγ , . . . , Dp Aγ , Ψ].

(97)

Lemma 4 (Thomas Replacement Theorem). If O is a functional satisfying Eq. (95) (or equivalently Eqs. (96) and (97) when B = M × G), then it can be written as O(x) = O[g(x), R(x), . . . , ∇p−2 R(x), F (x), . . . , Dp−2 F (x)].

(98)

In particular, there cannot be any dependence upon the background ﬁelds Ψ. Proof. The proof follows [80], with a slight generalization due to the presence of gauge ﬁelds that were not considered in that reference. We ﬁrst consider the case B = M × G. Then our covariance condition implies the conditions £ξ O =

p k=0

∂O ∂O k D Dk£ξ A £ g + ξ ∂(Dk g) ∂(Dk A) p

(99)

k=0

for any vector ﬁeld ξ on M , and 0=

p k=0

∂O Dk Dh ∂(Dk A)

(100)

for any Lie-algebra valued function h on M . We ﬁrst analyze the ﬁrst of these conditions, following [80]. First, we rewrite all D-derivatives of A in terms of ∇derivatives (where ∇ is the Levi–Civita connection of g), plus additional terms

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involving D-derivatives of g. Thus, we write O(x) = O[g(x), . . . , Dp g(x), A(x), . . . , ∇p A(x), Ψ(x)].

(101)

Next, we eliminate Dk g in favor of C and its D-derivatives, where C is the tensor ﬁeld deﬁned by 1 C µ νσ = − g µα (Dα gνσ − 2D(ν gσ)α ). 2

(102)

O(x) = O[g(x), C(x), . . . , Dp−1 C(x), A(x), . . . , ∇p A(x), Ψ(x)].

(103)

We thereby obtain

Next, we observe that the symmetrized derivatives of C can be rewritten as D(α1 · · · Dαl ) C µ γδ = D(α1 · · · Dαl C µ γδ) l+3 αi · · · ∇α (Rµ γαi δ + Rµ δαi γ ) + ∇(α1 · · · ∇ l 4(l + 1)(l + 2) i +

3l + 4 αj · · · ∇α ) Rµ αi δαj αi ∇ (∇(γ ∇α1 · · · ∇ l 8(l + 1)(l + 2) i=j

αj · · · ∇α ) Rµ αi γαj ) αi ∇ + ∇(δ ∇α1 · · · ∇ l + terms with less than l derivatives on C.

(104)

By iterating this substitutions, we can achieve that all derivatives of C µ νσ in O only appear in totally symmetrized form D(α1 · · · Dαl C µ νσ) , at the expense of possibly having an additional dependence upon the curvature tensor Rµ αβγ of the metric and its covariant derivatives. In other words, we may assume that O is given as O = O[gµν , C µ νσ , . . . , D(α1 · · · Dαp−1 C µ νσ) , Rµ νσρ , . . . , ∇(α1 · · · ∇αp−2 ) Rµ νσρ , Aµ , . . . , ∇(α1 · · · ∇αp ) Aµ ; Φ].

(105)

We now apply the condition (99) to this expression. We ﬁnd p−1 k=0

∂O ∂O £ξ D(α1 · · · Dαk C µ νσ) + £ξ Ψ µ ∂(D(α1 · · · Dαk C νσ) ) ∂Ψ =

p−1 k=0

∂O D(α1 · · · Dαk δC µ νσ) ∂(D(α1 · · · Dαk C µ νσ) )

(106)

where δC µ νσ is the variation arising from the variation δgµ = £ξ gµν = 2∇(µ ξν) under an inﬁnitesimal diﬀeomorphism, δC α βγ = g αδ (D(β Dγ) ξδ − Rδ(βγ)ρ ξ ρ ) − 2D(α ξ δ) gδρ C ρ βγ ,

(107)

with Rµνσρ the curvature of D. The terms in the above equation arising from an inﬁnitesimal variation of gµν , Aµ , Rµνσρ and their ∇-derivatives cancel out. The key point about the above equation is now that, on the left-hand side, there appears

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no more than one derivative of ξ µ , while on the right-hand side there can appear as many as p + 1 symmetrized derivatives of ξ µ . Since the symmetrized derivatives of ξ µ can be chosen independently at each given point x in M , it follows that a necessary condition for Eq. (112) to hold is that ∂O =0 ∂(D(α1 · · · Dαk C µ νσ) )

(108)

for k = 0, . . . , p − 1. Thus, our expression for O must have the form O = O[gµν , Rµ νσρ , . . . , ∇(α1 · · · ∇αp−2 ) Rµ νσρ , Aµ , . . . , ∇(α1 · · · ∇αp ) Aµ ; Ψ]. (109) We also get the condition that ∂O/∂Ψ · £ξ Ψ = 0. If Ψ only consists of scalar ﬁelds, then it follows immediately that O cannot have any dependence on Ψ. If Ψ contains tensor ﬁelds, then we may reduce this to the situation of only scalar ﬁelds by picking a coordinate system, and by treating the coordinate components of Ψ as scalars. We ﬁnally use the condition to show that the A-dependence of O can only be through the ﬁeld strength tensor and its covariant derivatives. To show this, we rewrite ∇(α1 · · · ∇αl ) Aµ = ∇(α1 · · · ∇αl Aµ) +

l D · · · Dαl−1 Fαl )µ l + 1 (α1

+ terms with no more than l − 1 derivatives of A. (110) By repeatedly substituting this relation into O, we can rewrite it as O = O[gµν , Fµν , . . . , D(α1 · · · Dαp−1 ) Fµν , Rµ νσρ , . . . , ∇(α1 · · · ∇αp−2 ) Rµ νσρ , Aµ , . . . , ∇(α1 · · · ∇αp Aµ) ].

(111)

We now substitute this into the (inﬁnitesimal version) of our condition (100), to get 0=

p k=0

+

∂O ∇(α1 · · · ∇αk Dµ) h ∂(∇(α1 · · · ∇αk Aµ) )

p−2 k=0

∂O [h, D(α1 · · · Dαk ) Fµν ], ∂(D(α1 · · · Dαk ) Fµν )

(112)

for all Lie-algebra valued functions h. Note that, in the second sum, we have no derivatives of h, while in the ﬁrst sum we have at least one symmetrized derivative of h. Since the symmetrized derivatives of h are independent at each point, the above equation can only hold if ∂O =0 ∂(∇(α1 · · · ∇αk Aµ) )

(113)

for all k. This proves the Thomas replacement theorem in the case when B = M ×G. But, since it is a local statement and any principal ﬁber bundle is locally trivial, it must in fact hold for any principal ﬁber bundle.

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3. Quantized Field Theories on Curved Spacetime: Renormalization 3.1. Definition of the free field algebra W0 for scalar field theory Consider a classical scalar ﬁeld φ described by the quadratic Lagrangian L0 =

1 (dφ ∧ ∗dφ − m2 ∗φ2 ). 2

(114)

The quantity m2 is a real parameter (we do not assume m2 ≥ 0). In this section, we explain how to quantize such a theory in curved spacetime, and how to deﬁne Wick powers and time-ordered products of φ at the quantum level. We assume only that (M, g) is globally hyperbolic and we assume for the rest of the paper that the spacetime dimension is 4. We do not assume that (M, g) has any symmetries. As discussed above, if (M, g) is globally hyperbolic, then the Klein–Gordon equation has a well-posed initial value formulation and unique retarded and advanced propagators ∆R and ∆A . A fundamental object in the quantization of φ is the commutator function, ∆ = ∆A − ∆R

(115)

which is anti-symmetric, ∆(x, y) = −∆(y, x). We want to deﬁne a non-commutative product between classical ﬁeld observables such that φ(x) φ(y) − φ(y) φ(x) = i∆(x, y)1.

(116)

This formula is motivated by the fact that, as → 0, we would like the above commutator divided by i to go to the classical Peierls bracket. The classical Peierls bracket for a linear scalar ﬁeld with Lagrangian L0 , however, is given by {φ(x), φ(y)}P.B. = ∆(x, y), see, e.g., [42]. To deﬁne the desired “deformation quantization”, we proceed as follows. We ﬁrst consider the free *-algebra generated by the expressions φ(f ), where f is any smooth compactly supported testfunction, to be thought of informally as the integral expressions φ(x)f (x)dx. We now simply factor this free algebra by the relation (116). This deﬁnes the desired deformation quantization algebra W00 . Evidently, the construction of W00 only depends upon the spacetime (M, g) and its orientations, because these data uniquely determine the retarded and advanced propagators. The algebra W00 by itself is too small to serve as an arena for renormalized perturbation theory. It does not, for example, even contain the Wick-powers of the free ﬁeld, or other quantized composite ﬁelds, which are a minimal input to even deﬁne interactions at the quantum level. More generally, to do perturbation theory we need an algebra that also contains the time-ordered products of composite ﬁelds, and these are, of course, not contained in W00 either. Thus, our ﬁrst task is to deﬁne an algebra that is suﬃciently large to contain such quantities. The key input in the construction of such an algebra is an arbitrary, but ﬁxed 2-point function ω(x, y) on M × M of “Hadamard type” which serves to deﬁne a suitable completion of

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W00 . This is by deﬁnition a distribution on M × M which is (a) a bisolution to the equations of motion, that is, ( − m2 )x ω(x, y) = ( − m2 )y ω(x, y) = 0,

(117)

ω(x, y) − ω(y, x) = i∆(x, y)

(118)

which (b) satisﬁes

and which (c) has a wave front set [76] of “Hadamard type” [98] WF(ω) = {(x1 , k1 , x2 , k2 ) ∈ T ∗ M × T ∗ M ; x1 and x2 can be joined by null-geodesic γ ˙ and k2 = −γ(1), ˙ and k1 ∈ V¯ + }. k1 = γ(0)

(119)

The wave front set completely characterizes the singularity structure of ω, and its deﬁnition and properties are recalled in Appendix C. It can be shown that, on any globally hyperbolic spacetime (M, g), there exist inﬁnitely many distributions ω of Hadamard type [82,53,54,85]. Using ω, we now deﬁne the following set of generators of W00 , where u = f1 ⊗ · · · ⊗ fn : ··· f1 (x1 ) · · · fn (xn ) : φ(x1 ) · · · φ(xn ) :ω dx1 · · · dxn F (u) = M

M

  dn   = exp i τj φ(fj ) + τi τj ω(fi , fj ) dτ1 · · · dτn 2 i,j j

. (120)

si =0

The commutator property of ω implies that the quantities : φ(x1 ) · · · φ(xn ) :ω are symmetric in its arguments. In fact, these quantities are nothing but the “normal ordered ﬁeld products” (with respect to ω), but we note that we do not think of these objects as operators deﬁned on a Hilbert space as is usually done when introducing normal ordered expressions. So far, we have done nothing but to introduce a new set of expressions in W00 that generate this algebra. We can express the product between to elements F (u), F (v) of the form (120) as k F (u ⊗k v) (121) F (u) F (v) = k

where u ⊗k v is the k-times contracted tensor product of distributions u, v in n, respectively, m spacetime variables. It is deﬁned by (u ⊗k v)(x1 , . . . , xn+m−2k ) n!m! = u(xπ(1) , . . . , y1 , . . .)v(xπ(n−k+1) , . . . , yk+1 , . . .) k! π ×

k i=1

ω(yi , yk+i )dy1 · · · dy2k ,

(122)

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where the sum is over all permutations of n + m − 2k elements. A somewhat more symbolic, but more compact and suggestive way to write the product is 1 (123) F (u) F (v) = : F (u) exp < D> F (v) :ω 2 where

< D>

is the bi-diﬀerential operator deﬁned by δR δL ω(x, y) dxdy. < D> = δφ(x) δφ(y)

(124)

The superscripts on the functional derivatives indicate that the ﬁrst derivative acts to the left, and the second one to the right factor in a tensor product. These functional derivatives are to be understood to act on an expression like : φ(x1 ) · · · φ(xn ) :ω a classical product of classical ﬁelds in P(M ). The point is now that the product can still be deﬁned on a much larger class of expressions. These expressions are of the form (125) F (u) = u(x1 , . . . , xn ) : φ(x1 ) · · · φ(xn ) :ω dx1 · · · dxn (n ≥ 1), where u is now a distribution on M n , rather than the product of n smooth functions on M as above in Eq. (120). To make the product well deﬁned, we only need to impose a mild wave front set condition on u [42]: [(V¯x+ )×n ∪ (V¯x− )×n ] = ∅, (126) WF(u) ∩ x∈M

V¯x±

with denoting the closure of the future/past lightcone at x. The reason for imposing this condition is that it ensures, together with (119), that the distributional products in the contracted tensor products that arise when carrying out the product F G of two expressions of the type (125) make sense. The point is that in such a product, there appear distributional products of u, v, ω in the contracted tensor product of u, v, see Eq. (122). Normally, the product of distributions does not make sense, but because of our wave front set conditions on u, v, ω, the relevant products exist due to the fact that vectors in the wave front set of ω, u, v can never add up to 0, see Appendix C for details. We deﬁne the desired enlarged algebra, W0 , to be the algebra generated by (125), with the product . It can be viewed in a certain sense as the closure of W00 , because the distributions u in Eq. (125) can be approximated, to arbitrarily good precision, by sums of smooth functions of the form f1 ⊗ · · · ⊗ fn as in (120) (in the Hormander topology [76]). The algebra W0 will turn out to be large enough to serve as an arena for perturbation theory. For example, it can be seen immediately that W0 contains normal ordered Wick-powers of φ(x): Namely, since the wave front set of the delta-distribution on M n is WF(δ) = (x, k1 , . . . , x, kn ); x ∈ M, ki ∈ Tx∗ M, ki = 0 (127)

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it follows that u(y, x1 , . . . , xn ) = f (y)δ(y, x1 , . . . , xn ) satisﬁes the wave front condition (126). The corresponding generator F as in (125) may be viewed as the normal ordered Wick power : φn (x) :ω , smeared with f (x). As it stands, the Klein–Gordon equation is not implemented in the algebra (W0 , ). This could easily be incorporated by factoring W0 by an appropriate ideal (i.e. a linear subspace that is stable under -multiplication by any F ∈ W0 ). The ideal for the ﬁeld equation is simply the linear space δS0 · · · φ(xn ) :ω dx1 · · · dxn , J0 = F = u(x1 , . . . , xn ) : φ(x1 ) · · · δφ(xi )

+ ×n − ×n ¯ ¯ for some u of compact support, WF(u) ∩ [(V ) ∪ (V ) ] = ∅ x

x

x∈M

(128) of generators containing a factor of the wave equation. This space is stable under the adjoint operation and -products with any F ∈ W0 by Eq. (117) and so indeed an ideal. If we consider the factor algebra pr : W0 → F0 = W0 /J0 ,

(129)

then within F0 , the ﬁeld equation ( − m )φ(x) = 0 holds. The factor algebra F0 is the algebra of physical interest for free ﬁeld theory. For physical applications, one is interested in representations of F0 as operators on a Hilbert space, H0 , and in n-point functions of observables in F0 in physical states. However, in the context of perturbation theory, it will be much more useful to work with the algebra W0 at intermediate stages. To make physical predictions, one ﬁnally needs to represent the algebra of observables F0 as linear operators with a dense, invariant domain on a Hilbert space H0 . A vector state |Ψ in H0 is said to be of Hadamard form if its n-point functions 2

GΨ n (x1 , . . . , xn ) = Ψ|π0 (φ(x) ) · · · π0 (φ(xn ))|Ψ

(130)

are of “Hadamard form”. By this one means that the 2-point function has a wave front set of Hadamard form (119), and that its truncated n-point functionsd are smooth for n = 2. A Hadamard representation is a representation containing a dense, invariant domain of Hadamard states. Hadamard representations may be constructed on any globally hyperbolic spacetime as one may show using the deformation argument of [52, 85] (or the construction of [82], and combining these with those of [69]). We describe the deformation construction below in Sec. 4.2 in the context of gauge theories. It is clear that, since W0 (M, g) was obtained as the completion of the algebra W00 (M, g), also W0 (M, g) depends locally and covariantly upon the metric. truncated n-point functions of a hierarchy of n-point distributions {hn } are deﬁned by the P generating functional hc (ef⊗ ) = log h(ef⊗ ), where h(ef⊗ ) = n hn (f, f, . . . , f )/n!.

d The

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Because this fact will be of key importance when we formulate the local and covariance condition of renormalized time-ordered products, we now explain more formally what exactly we mean by this statement. Consider two oriented and timeoriented spacetimes (M, g) and (M , g ) and a map ψ : M → M which is an orientation and causality preservinge isometric embedding. Then there is a corresponding isomorphism αψ : W0 (M, g) → W0 (M , g ),

(131)

which behaves naturally under composition of embeddings. This map is simply deﬁned on W00 (M, g) by setting αψ (φM,g (f )) = φM ,g (ψ∗ f ), where ψ∗ f (x ) = f (x) for x = ψ(x ). Since, as explained above, W0 (M, g) is essentially the closure of W00 (M, g), we can deﬁne αψ on W0 (M, g) by continuity. The action of αψ on F of the form (120) may be calculated straightforwardly from the deﬁnition. However, we note that its form will depend on the choices ω and ω for the Hadamard bidistributions on M respectively M , and will look somewhat involved if ω and ω are such that ψ ∗ ω = ω. These expressions are given in [66], but will not be needed here. 3.2. Renormalized Wick products and their time-ordered products In the previous section we have laid the groundwork for the construction of linear quantum ﬁeld theory in curved spacetime by giving the deﬁnition of an algebra W0 (M, g) associated with a free Lagrangian L0 that can be viewed as a deformation quantization of the algebra of classical observables with the Peierls bracket. In this section we shall identify, within W0 (M, g), the various objects that have the interpretation of the various Wick powers in the theory, and their time-ordered products. Those objects will be the quantities of prime interest in the perturbative constructions in the subsequent sections. For simplicity, we again address the case when L0 describes a linear, hermitian scalar ﬁeld φ, see Eq. (114). Actually, for reasons that we will explain below, it is convenient to adopt a uniﬁed viewpoint on the Wick products and their time-ordered products. We deﬁne a time-ordered product with n factors (where n ≥ 1) to be a linear map Tn : Pk1 (M ) ⊗ · · · ⊗ Pkn (M ) → D (M n ; ∧k1 T ∗ M × · · · × ∧kn T ∗ M ) ⊗ W0 , (132) n

taking values in the distributions over M with target space W0 . Thus, the linear map Tn takes as arguments the tensor product of n local covariant classical forms O1 , . . . , On , and it gives an expression Tn (O1 (x1 ) ⊗ · · · ⊗ On (xn )), which is itself a distribution in n spacetime variables x1 , . . . , xn , with values in e An

isometric embedding may be such that the intrinsic notion of causality is not the same as the notion of causality inherited from the ambient space. Examples of this sort may be constructed by embedding suitable regions of Minkowski spacetime into Minkowski space with periodic identiﬁcations in one or more spatial directions.

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W0 , i.e. Tn (O1 (x1 ) ⊗ · · · ⊗ On (xn )) is itself a map that needs to be smeared with n-test forms f1 (x1 ), . . . , fn (xn ), where the ith test form is an element in i (M ) over M . The set the set of compactly supported smooth forms fi ∈ Ω4−k 0 n k1 ∗ kn ∗ D (M ; ∧ T M × · · · × ∧ T M ) denotes the dual space (in the standard distri1 n (M ) × · · · × Ω4−k (M ). bution topology [76]) of the space of forms Ω4−k 0 0 The time-ordered products Tn are characterized abstractly by certain properties which we will list. We deﬁne the Wick powers of a ﬁeld to be the time-ordered products with 1 factor, i.e. n = 1. We will formulate the properties of the timeordered products in the form of axioms in this section, but we will see in the following section that one can turn these properties into a concrete constructive algorithm for these quantities. In fact, as we will see, the properties that we wish the time-ordered products to have do not uniquely characterize them, but leave a certain ambiguity. This ambiguity corresponds precisely to the renormalization ambiguity in other approaches in ﬂat spacetime, with the addition of couplings to curvature. However, we note that our time-ordered products are rigorously deﬁned, by contrast to the corresponding quantities in other approaches to renormalization in ﬂat spacetime, where they are a priori only formal (i.e. inﬁnite) objects. T1 Locality and Covariance. The time ordered products are locally and covariantly constructed in terms of the metric. This means that, if ψ : M → M is a causality preserving isometric embedding between two spacetimes preserving the causal structure, and αψ denotes the corresponding homomorphism W0 (M, g) → W0 (M , g ), see Eq. (131), then we have αψ ◦ Tn = Tn ◦

n

ψ∗

(133)

where Tn denotes the time-ordered product on (M, g), while Tn denotes the timeordered product on (M , g ). The mapping ψ∗ : P(M ) → P(M ) is the natural pushforward map. Thus, the local and covariance condition imposes a relation between the construction of time-ordered products on locally isometric spacetimes. Written more explicitly (in the case of scalar operators), the local covariance condition is αψ [Tn (φk1 (x1 ) ⊗ · · · ⊗ φkn (xn ))] = Tn (φk1 (x1 ) ⊗ · · · ⊗ φkn (xn )),

ψ(xi ) = xi . (134)

In particular, if n = 1, then the Wick products T1 (O(x)) are local covariant ﬁelds in one variable. As we will see more clearly in the next subsection, the requirement of locality and covariance is a non-trivial renormalization condition already in the case of 1 factor. It is instructive to consider the local covariance requirement for the special case where M = M is Minkowksi spacetime, with g = g the Minkowski metric −dt2 + dx2 + dy 2 + dz 2 . In that case, the causality and orientation preserving isometric embeddings are just the proper, orthochronous Poincare transformations ψ = (Λ, a) ∈ P+↑ , while the map αψ may be implemented by Ad(U0 (Λ, a))

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in the vacuum Hilbert space representation π0 of the algebra W0 (we need to assume m2 ≥ 0 to have that representation), with U0 (Λ, a) the unitary representative of the proper orthochronous Poincare transformation (Λ, a) on the Hilbert space of the representation π0 . The local covariance condition (135) reduces in that case to Ad[U0 (Λ, a)]π0 (Tn (φk1 (x1 ) ⊗ · · · ⊗ φkn (xn ))) = π0 (Tn (φk1 (Λx1 − a) ⊗ · · · ⊗ φkn (Λxn − a)))

(135)

which is the standard transformation law for the time ordered product (and in fact any relativistic ﬁeld) in Minkowski spacetime. T2 Scaling. We would like the time-ordered products to satisfy a certain scaling relation. For distributions u(x), x ∈ Rn on ﬂat space, it is natural to consider the scaled distribution u(µx), µ ∈ R+ . Such a distribution is then said to scale homogeneously with degree D if u(µx) = µD u(x), in the sense of distributions, which is equivalent to the diﬀerential relation ∂ − D u(µx) = 0. (136) µ ∂µ More generally, it is said to scale “polyhomogeneously” or “homogeneously up to logarithms” if instead only N ∂ ∂N µ −D u(µx) = [µDu(µx)] = 0 (137) ∂µ ∂(log µ)N holds for some N ≥ 2, which gives the highest power +1 of the logarithmic corrections. For the quantities in the quantum ﬁeld theory associated with the Lagrangian L0 on a generic curved spacetime without dilation symmetry, we do not expect a simple scaling behavior under rescalings in an arbitrarily chosen coordinate system. However, we know that the Lagrangian L0 has an invariance under a rescaling g → µ2 g,

m2 → µ−2 m2 ,

φ → µ−1 φ.

(138)

It is therefore natural to expect that the time-ordered products can be constructed so as to have a simple scaling behavior under such a rescaling. However, due to quantum eﬀects, one cannot expect an exactly homogeneous scaling, but only a homogeneous scaling behavior that is modiﬁed by logarithms. To describe this behavior, we must ﬁrst take into account that the time-ordered products associated with the spacetime metric g live in a diﬀerent algebra than the time-ordered products associated with µ2 g, so we must ﬁrst identify these algebras. This is achieved by the linear map σµ : φ → µφ, which may be checked to deﬁne an isomorphism between W0 (M, g, m2 ) and W0 (M, µ2 g, µ−2 m2 ). The desired polyhomogeneous

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scaling behavior is then formulated as follows. Let Tn [µ] = σµ−1 ◦ Tn ◦

n

exp(ln µ · Nd )

(139)

where Nd is the dimension counter, deﬁned as Nd := Nc + Nf + Nr , where Nc , Nf , Nr : P(M ) → P(M ) are the number counting operators for the coupling constants, ﬁelds, and curvature terms, deﬁned for Klein–Gordon theory in 4 spacetime dimensions by Nf :=

(1 + k)(∇k φ)

k

Nc := 2m2 Nr :=

k

∂ , ∂(∇k φ)

∂ , ∂m2

(k + 2)(∇k R)

(140) (141)

∂ . ∂(∇k R)

(142)

For example Tn [µ](φk1 (x1 ) ⊗ · · · ⊗ φkn (xn )) = µk1 +···+kn σµ−1 Tn (φk1 (x1 ) ⊗ · · · ⊗ φkn (xn )). (143) Because we have put the identiﬁcation map σµ on the right-hand side, Tn [µ] deﬁnes a new time ordered product in the algebra associated with the unscaled metric, g, and coupling constants. In the absence of scaling anomalies, this would be equal to the original Tn for all µ ∈ R+ . As we have said, it is not possible to achieve this exactly homogeneous scaling behavior, so we only postulate the polyhomogeneous scaling behavior ∂N Tn [µ] = 0. ∂(log µ)N

(144)

T3 Microlocal Spectrum Condition. Consider a time ordered product Tn (O1 (x1 ) ⊗ · · · ⊗ On (xn )) as an W0 valued distribution on M n . Then we require that WF(Tn ) ⊂ CT (M, g),

(145)

where the set CT (M, g) ⊂ T ∗M n \0 is described as follows (we use the graph theoretical notation introduced in [20, 21]): Let G(p) be a “decorated embedded graph” in (M, g). By this we mean an embedded graph ⊂ M whose vertices are points x1 , . . . , xn ∈ M and whose edges, e, are oriented null-geodesic curves. Each such null geodesic is equipped with a coparallel, cotangent covectorﬁeld pe . If e is an edge in G(p) connecting the points xi and xj with i < j, then s(e) = i is its source and / J ± (xt(e) ). t(e) = j its target. It is required that pe is future/past directed if xs(e) ∈

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With this notation, we deﬁne   CT (M, g) = (x1 , k1 ; . . . ; xn , kn ) ∈ T ∗ M n \0 | ∃ decorated graph G(p) with    pe − pe ∀i . (146) vertices x1 , . . . , xn such that ki =  e:s(e)=i

e:t(e)=i

T4 Smoothness. The functional dependence of the time ordered products on the spacetime metric, g, is such that if the metric is varied smoothly, then the time ordered products vary smoothly, in the sense described in [66]. T5 Analyticity. Similarly, we require that, for an analytic family of analytic metrics (depending analytically upon a set of parameters), the expectation value of the time-ordered products in an analytic family of statesf varies analytically in the same sense as in T4. T6 Symmetry. The time ordered products are symmetric under a permutation of the factors, Tn (O1 (x1 ) ⊗ · · · ⊗ On (xn )) = Tn (Oπ1 (xπ1 ) ⊗ · · · ⊗ Oπn (xπn ))

(147)

for any permutation π. T7 Unitarity. Let T¯n (⊗i Oi (xi )) = [Tn (⊗i Oi (xi )∗ )]∗ be the “anti-time-ordered” product. Then we require n Oi (xi ) T¯n i=1

=

(−1)n+j T|I1 |

I1 ··· Ij =n

i∈I1

 Oi (xi )

· · · T|Ij | 

 Oj (xj ) , (148)

j∈Ij

where the sum runs over all partitions of the set {1, . . . , n} into pairwise disjoint subsets I1 , . . . , Ij . T8 Causal Factorization. The “product” Tn is time ordered in the sense that the following causal factorization property is to be satisﬁed. Let {x1 , . . . , xi } ∩ J − ({xi+1 , . . . , xn }) = ∅. Then we have Tn (O1 (x1 ) ⊗ · · · ⊗ On (xn )) = Ti (O1 (x1 ) ⊗ · · · ⊗ Oi (xi )) Tn−i (Oi+1 (xi+1 ) ⊗ · · · ⊗ On (xn )). f As

(149)

explained in [66, Remark (2), p. 311], it suﬃces to consider a suitable analytic family of linear functionals on W0 that do not necessarily satisfy the positivity condition required for states.

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For the case of 2 factors, this means T1 (O1 (x1 )) T1 (O2 (x2 )) T2 (O1 (x1 ) ⊗ O2 (x2 )) = T1 (O2 (x2 )) T1 (O1 (x1 ))

when x1 ∈ / J − (x2 ); when x2 ∈ / J − (x1 ).

(150)

T9 Commutator. The commutator of a time-ordered product with a free ﬁeld is given by lower order time-ordered products times suitable commutator functions, namely ! n " Tn Oi (xi ) , φ(x) i

= i

n k=1

δOk (xk ) ⊗ · · · ⊗ On (xn ) , Tn O1 (x1 ) ⊗ · · · ⊗ ∆(x, y) δφ(y)

(151)

where ∆ is the causal propagator. T10 Field equation. The free ﬁeld equation δS0 /δφ holds in the sense that n δS0 ⊗ Tn+1 Oi (xi ) δφ(x) i δOi (xi ) ⊗ · · · ⊗ On (xn ) Tn O1 (x1 ) ⊗ · · · ⊗ mod J0 . (152) = δφ(x) i T11 Action Ward identity. If dk = dxµk ∧ on the kth spacetime variable, then we have

∂ ∂xµ k

is the exterior diﬀerential acting

Tn (O1 (x1 ) ⊗ · · · ⊗ dk O(xk ) ⊗ · · · ⊗ On (xn )) = dk Tn (O1 (x1 ) ⊗ · · · ⊗ O(xn )). (153) Thus, derivatives can be freely pulled inside the time-ordered products. Condition T11 can be stated as saying that Tn may alternatively be viewed as a linear map Tn : A⊗n → W0 for each n, where A is the space of all local action functionals, i.e. all expressions of the form F = O ∧ f , where f ∈ Ωp0 (M ) is any pform of compact support, and where O ∈ P4−p .To explain how this comes about, consider the integrated ﬁeld polynomial F = f ∧ dO. It may equivalently be written as − (df ) ∧ O, so the time ordered product should give the same result for either choice. T11 means that the time ordered products f (xi )Tn (· · · ⊗ di O(xi ) ⊗ · · ·) and − di f (xi )Tn (· · · ⊗ O(xi ) ⊗ · · ·) are equal, where the exterior derivative di = dxµi ∧ ∂/∂xµi acts on the ith spacetime argument. This means that Tn may be viewed as a functional taking as arguments the integrated functionals (or “actions”) in A, because it does not matter how F is represented. This is the origin of the name “action Ward identity” for T11. The action Ward identity also means that we may apply the Leibniz rule for derivative of quantum Wick powers, i.e. time ordered

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products with one factor, which is why the same condition was called “Leibniz rule” in [70]. 3.3. Inductive construction of time-ordered products In the previous subsection, we have given a list of properties of the local Wick powers and their time-ordered products. We now present an algorithm showing how these can be constructed, and thus in particular demonstrating that axioms T1 through T11 are not empty. We shall reduce the problem to successively simpler problems by a series of reduction steps. These steps are as follows: 1. First, construct the time-ordered products with one factor. 2. Assuming inductively that time-ordered products with n factors have been constructed, we show, following the ideas of “causal perturbation theory” [47, 14, 109, 108] that the time-ordered products with n + 1 factors are already uniquely ﬁxed, apart from points on the total diagonal, by the lower order time-ordered products. 3. The problem of extending the time-ordered products at order n + 1 to the total diagonal is reduced to that of extending certain scalar distributions to the total diagonal. 4. The problem of reducing the scalar functions on M n+1 to the diagonal is reduced to that of extending a set of distributions on the (n + 1)-fold Cartesian power of Minkowski space via a curvature expansion. 5. The extension of the Minkowski distributions is performed. This step corresponds to renormalization. Thus, we shall proceed inductively in the number of factors, n, appearing in the time-ordered product Tn (O1 (x1 ) ⊗ · · · ⊗ On (xn )). To keep our discussion as simple as possible, we now restrict attention to the case when the ﬁelds Oi ∈ P in the time ordered product contain no spacetime derivatives, i.e. Oi = φki for some natural numbers ki . We brieﬂy explain how to deal with the general case in the end. Time-Ordered Products with 1 Factor. For n = 1, the time-ordered products are just the local covariant Wick powers, i.e. T1 (φk (x)) is a local covariant ﬁeld in one spacetime variable, interpreted as the kth local covariant Wick power of φ. These Wick powers may be constructed as follows. Let H(x, y) be the “local Hadamard parametrix”, for the Klein–Gordon operator, given by u(x, y) 1 + v(x, y) log(σ + it0) . (154) H(x, y) = 2 2π σ + it0 Here, σ(x, y) is the signed squared geodesic distance between two points x, y in a convex normal neighborhood of M , and u, v are smooth kernels that are locally constructed in terms of the metric, which are determined by the Hadamard recursion relations [27], which are obtained by demanding that H be a bi-solution (modulo a smooth remainder) of the Klein–Gordon equation. Their construction is

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recalled in Appendix D. The quantity t(x, y) = T (x) − T (y) is deﬁned in terms of an arbitrary global time coordinate T . Consider now, for any k ≥ 1, the “locally normal ordered expressions” : φ(x1 ) · · · φ(xk ) :H δk exp i = k f (x)φ(x) + H(x, y)f (x)f (y) . i δf (x1 ) · · · δf (xk ) 2 M×M M f =0 (155) Because H is deﬁned locally and covariantly in terms of the metric, it follows that : φ(x1 ) · · · φ(xk ) :H are local and covariant ﬁelds that are deﬁned in a convex normal neighborhood of the diagonal ∆k , where ∆k = {(x, x, . . . , x) | x ∈ M } ⊂ M k .

(156)

The following lemma shows that the normal ordered quantities (155) diﬀer from the quantities : φ(x1 ) · · · φ(xn ) :ω only by a smooth function (valued in W0 ). Lemma 5. Let ω(x, y) be a 2-point function of Hadamard form, i.e. the wave front set WF(ω) is given by (119). Then locally (i.e. where H is deﬁned ), ω − H is smooth, i.e. u(x, y) 1 + v(x, y) log(σ + it0) + (smooth function in x, y). ω(x, y) = 2π 2 σ + it0 (157) Furthermore, any two Hadamard states can at most diﬀer by a globally smooth function in x, y. The proof is given in Appendix E. Because the normal ordered products may be smeared with a δ-function (or derivatives thereof), we may deﬁne T1 (φk (x)) = : φk (x) :H

(158)

which is a well-deﬁned element in W0 after smearing with any testfunction f ∈ C0∞ (M ). This deﬁnes our time-ordered products with one factor. It follows from the deﬁnition of H that T1 (φk (x)) is a local covariant ﬁeld, i.e. it satisﬁes T1 for n = 1. The other properties T2–T11 are also seen to be satisﬁed using the properties of H described in Appendix D. Time-Ordered Products with n > 1 Factors. We have deﬁned the timeordered products with n = 1 factor, and we may inductively assume that time ordered products with properties T1–T11 have been deﬁned for any number of factors ≤ n. The key idea of causal perturbation theory [47, 14, 108, 109] is that the

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time ordered products with n + 1 factors are then already uniquely determined as algebra-valued distributions on the manifold M n+1 minus its total diagonal ∆n+1 = {(x, x, . . . , x) ∈ M n+1 } by the causal factorization requirement T8. The construction of the time ordered products at order n + 1 is then equivalent to the task of extending this distribution in a suitable way compatible with the other requirements T1–T10. In order to perform this task in an eﬃcient way, it is useful to derive a number of properties that hold at all orders m ≤ n as a consequence of T1–T10. The ﬁrst property is a local Wick expansion for time ordered products [67]. This is a key simpliﬁcation, because it will enable one to reduce the problem of extending algebra-valued quantities to one of ﬁnding an extension of c-number distributions. In the simplest case, when none of the Oi contain derivatives of φ, we have in an open neighborhood of ∆m Tm (φk1 (x1 ) ⊗ · · · ⊗ φkm (xm )) ki tj1 ,...,jm (x1 , . . . , xm ) : φk1 −j1 (x1 ) · · · φkm −jm (xm ) :H = ji 0≤ji ≤ki

i

(159) for all 1 < m ≤ n, where tj1 ,...,jm are c-number distributions. The Wick expansion when derivatives are present is analogous. The Wick expansion formula can be proved from axiom T9. Because the time-ordered products are local and covariant, the c-number distributions in the Wick expansion have the same property, in the sense that if ψ : (M , g ) → (M, g) is an isometric, causality and orientation preserving embedding, so that if ψ ∗ g = g , then tj1 ,...,jm [ψ ∗ g; x1 , . . . , xm ] = tj1 ,...,jm [g; ψ(x1 ), . . . , ψ(xm )].

(160)

Because H and the local normal ordered products are in general only deﬁned in a neighborhood of the diagonal, it it follows that also the c-number distributions are only deﬁned on a neighborhood of the diagonal, but this will turn out to be suﬃcient for our purposes. It follows from the scaling property T2 and the corresponding scaling properties of H that ∂N {µj1 +···+jm tj1 ,...,jm [µ−2 m2 , µ2 g; x1 , . . . , xm ]} = 0 ∂(log µ)N

(161)

for some N . This relation, together with the condition of locality and covariance and the analytic dependence of the time ordered products on the metric, can be used to derive a subsequent “scaling-” or “curvature expansion” [67] of each of the distributions tj1 ,...,jm in powers of the Riemann tensor and the coupling constants

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(in our case only m2 ) at a reference point: Proposition 0. The distributions t := tj1 ,...,jm have the asymptotic expansion t(expy ξ1 , . . . , expy ξm−1 , y) =

S

Cµk1 ···µt (y)uµk 1 ···µt (ξ1 , . . . , ξm ) + rS (y, ξ1 , . . . , ξm−1 )

(162)

k=0

in an open neighborhood of the diagonal ∆m . The terms have the following properties: (i) The remainder rS is a distribution of scaling degree (see Appendix C for the mathematical deﬁnition of this concept) strictly lower than the scaling degree of any term in the sum. (ii) Each uk is a Lorentz invariant distribution on (R4 )m−1 , i.e. uµk 1 ···µt (Λξ1 , . . . , Λξm ) = Λµν11 · · · Λµνtt uνk1 ···νt (ξ1 , . . . , ξm )

∀Λ ∈ SO0 (3, 1). (163)

(iii) Each distribution uk scales almost homogeneously under a coordinate rescaling, i.e. ∂N [µρ uµk 1 ···µt (µξ1 , . . . , µξm−1 )] = 0 ∂(log µ)N with ρ ∈ N. The scaling condition can be rewritten equivalently as m−1 N ν ∂ ξi ν − ρ uµk 1 ···µt (ξ1 , . . . , ξm−1 ) = 0. ∂ξ i i=1

(164)

(165)

(iv) Each term C k is a polynomial in m2 and the covariant derivatives of the Riemann tensor, Cµk1 ···µt (y) = Cµk1 ···µt [m2 , R(y), ∇R(y), . . . , ∇l R(y)]. (v) The scaling degree ρ = sd(uk ) is given by ji − Nr (C k ), sd(uk ) =

(166)

(167)

i

where Nr is the dimension counting operator for curvature terms and dimensionful coupling constants (in our case only m2 ), see Eq. (140). By the above proposition, we see that, by including suﬃciently (but ﬁnitely many) terms in the scaling expansion (162) (i.e. choosing S suﬃciently large), one can achieve that the remainder rS has arbitrarily low scaling degree. It does not mean that the sum is convergent in any sense (it is not).

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Having stated the detailed properties of the time ordered products with ≤ n factors, we are now resume the main line of the argument and perform the construction of the time-ordered products with n + 1 factors. Let I be a proper subset of {1, 2, . . . , n + 1}, and let UI be the subset of M n+1 deﬁned by UI = {(x1 , x2 , . . . , xn+1 ) | xi ∈ / J − (xj ) for all i ∈ I, j ∈ / I}.

(168)

It can be seen [21] that the sets UI are open and that the collection {UI } of these sets covers the manifold M n+1 \ ∆n+1 . We can therefore deﬁne an algebra-valued distribution Tn+1 on this manifold by declaring it for each (x1 , . . . , xn+1 ) ∈ UI by Tn+1 (φk1 (x1 ) ⊗ · · · ⊗ φkn+1 (xn+1 )) = T|I| (⊗i∈I φki (xi )) Tn+1−|I| (⊗j∈n+1\I φkj (xj ))

∀(x1 , . . . , xn+1 ) ∈ UI . (169)

To avoid a potential inconsistency in this deﬁnition for points in UI ∩ UJ = ∅ for diﬀerent I, J, we must show that the deﬁnition agrees for diﬀerent I, J. This can be achieved using the causal factorization property T8 of the time-ordered products with less or equal than n factors [47, 21]. Property T8 applied to the time-ordered products with n + 1 factors also implies that the restriction of Tn+1 to M n+1 \∆n+1 must agree with (169). Thus, property T8 alone determines the time ordered products up to the total diagonal, as we desired to show, see [21] for details. In fact — assuming that time-ordered products with less or equal than n factors have been deﬁned so as to satisfy properties T1–T11 on M n — one can argue in a relatively straightforwardly way that the ﬁelds deﬁned by Eq. (169) with n + 1 factors automatically satisfyg the restrictions of properties T1–T9 to M n+1 \∆n+1 , while T10 and T11 are empty in the present case for time ordered products without derivatives. Our remaining task is to ﬁnd an extension of each of the algebra-valued distributions Tn+1 in n + 1 factors from M n+1 \∆n+1 to all of M n+1 in such a way that properties T1–T9 continue to hold for the extension. This step, of course, corresponds to renormalization. Condition T8 does not impose any additional conditions on the extension, so we need only satisfy T1–T7 and T9. However, it is not diﬃcult to see that if an extension Tn+1 is deﬁned that satisﬁes T1–T5 and T9, then that extension can be modiﬁed, if necessary, so as to also satisfy the symmetry and unitarity conditions, T6 and T7, see [66]. Thus, we have reduced the problem of deﬁning time-ordered products to the problem of extending the distributions Tn+1 deﬁned by (169) from M n+1 \∆n+1 to all of M n+1 so that properties T1–T5 and T9 continue to hold for the extension. To ﬁnd that extension, we now make a Wick expansion of Tn+1 , which follows from course, if any Tn+1 failed to satisfy any of these properties on M n+1 \∆n+1 , we would have a proof that no deﬁnition of time ordered products could exist that satisﬁes T1–T9.

g Of

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the Wick expansion at lower orders. That Wick expansion will contain c-number distribution coeﬃcients, t, that are deﬁned as distributions on a neighborhood of ∆n+1 in M n+1 \∆n+1 . They possess a scaling expansion analogous to (162), with distributions uk that are deﬁned on (R4 )n \0. As we have just argued, time-ordered products satisfying all of our conditions will exist if and only if the c-number distributions t deﬁned away from ∆n+1 appearing in the Wick expansion for Tn+1 analogous (159) can be extended to distributions deﬁned on an open neighborhood of ∆n+1 in such a way that the distribution Tn+1 deﬁned by (159) continues to satisfy properties T1–T5, and T9. It is straightforward to check that this will be the case if and only if the extensions t satisfy the following ﬁve corresponding conditions: t1 Locality/Covariance. The distributions t = tj1 ,...,jn+1 are locally constructed from the metric in a covariant manner in the following sense. Let ψ : M → M be a causality-preserving isometric embedding, so that ψ ∗ g = g. Then Eq. (160) holds for m = n + 1. t2 Scaling. The extended distributions t scale homogeneously up to logarithmic terms, in the sense that there is an N ∈ N such that (161) holds for m = n + 1. t3 Microlocal Spectrum Condition. The extension satisﬁes the wave front set condition that the restriction of WF(t) to the diagonal ∆n+1 is contained in # {(x, k1 , . . . , x, kn+1 ) | ki = 0}. t4 Smoothness. t depends smoothly on the metric. t5 Analyticity. For analytic spacetimes t depends analytically on the metric. In summary, we have reduced the problem of deﬁning time-ordered products to the following question: Assume that time-ordered products involving ≤ n factors have been constructed so as to satisfy our requirements T1–T9. Deﬁne Tn+1 by (169) and deﬁne the distributions t on M n+1 \∆n+1 by the analogy of (159) for Tn+1 , in a neighborhood of the diagonal. Can each t be extended to a distribution deﬁned on a neighborhood of ∆n+1 so as to satisfy requirements t1–t5? The answer to this question is “yes”, and we shall now show how the desired extension of t(x1 , . . . , xn+1 ) may be found. The idea is that, since the remainder in the scaling expansion (162) for t has an arbitrary low scaling degree for suﬃciently large m by item (v), it can be extended to the diagonal ∆n+1 by continuity [21], i.e. there is no need to “renormalize” the remainder for suﬃciently large but ﬁnite S. In fact, by [21, Theorem 5.3], it is suﬃcient to choose any S ≥ d − 4n for this purpose. Furthermore, each term in the sum in the scaling expansion (162) can be written as C k (y) · uk (ξ1 , . . . , ξn ) by (i). Each uk is an almost homogeneous, Lorentz invariant n-point distribution on (R4 )n \0. As we will see presently in [67, Lemma 6], this Minkowski distribution can be extended to a distribution on (R4 )n with the

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same properties [possibly with a higher N than that appearing (165)], by techniques in Minkowski space. It is this step that corresponds to the renormalization. As a consequence of the properties satisﬁed by the extension u, the corresponding extension t can be seen to satisfy t1–t5, thus solving the renormalization problem ki for the time-ordered products Tn+1 (⊗n+1 i=1 φ (xi )) with n + 1 factors. Lemma 6. Let u ≡ uµ1 ...µl (ξ1 , . . . , ξn ) be a Lorentz invariant tensor-valued distribution on R4n \0 which scales almost homogeneously with degree ρ ∈ C under coordinate rescalings, i.e. SρN u = 0

for some natural number N

(170)

where Sρ =

n

ξiµ ∂/∂ξiµ + ρ.

(171)

i=1

Then u has a Lorentz invariant extension, also denoted u, to a distribution on R4n which also scales almost homogeneously with degree ρ under rescalings of the coordinates. Moreover: 1. If ρ ∈ Z, ρ < 4n, then u can be extended by continuity, the extension is unique, and SρN u = 0. 2. If ρ ∈ C\Z then the extension is unique, and SρN u = 0. 3. If ρ ∈ Z, ρ ≥ 4n, then the extension is not unique, and SρN +1 u = 0. Two diﬀerent extensions can diﬀer at most by a distribution of the form Lδ, where L is a Lorentz-invariant partial diﬀerential operator in ξ1 , . . . , ξn containing derivatives of degree ρ − 4n. Proof. The proof of the lemma shows how the desired extension u can be constructed. We will ﬁrst construct an extension that satisﬁes the almost homogeneous scaling property. This extension need not satisfy the Lorentz invariance properties. However, we will show that the extension can be modiﬁed, if necessary, so that the desired Lorentz-invariance property is satisﬁed, while retaining the desired almost homogeneous scaling behavior. The proof of the theorem given here diﬀers from that given in [67], and thereby provides an alternative construction of the extension. A less general result of a similar nature for distributions with an exactly homogeneous scaling has previously been obtained in [76, Theorems 3.2.3 and 3.2.4]. Thus, our theorem generalizes this result to the case of almost homogeneous scaling. To simplify the notation, we set x = (ξ1 , . . . , ξn ) ∈ R4n throughout this proof. The almost homogeneous scaling property of u, Eq. (202), or the equivalent form of this condition (164) implies that u(rx) can be written in the form u(rx) = r

−ρ

N −1 k=0

(log r)k vk (x), k!

r > 0,

(172)

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where vk are the distributions deﬁned on R4n \0 by vk = Sρk u.

(173)

Choose an arbitrary compact (4n − 1)-dimensional surface Σ ⊂ R4n homeomorphic to the sphere S 4n−1 around the origin of R4n that intersects each orbit of the scaling map x → µx transversally and precisely once.h The ﬁrst aim is to show that the distributions vk can be restricted to Σ. To prove this, it is convenient to use the methods of microlocal analysis, in particular the following result [76]: If ϕ is a distribution on a manifold X with a submanifold Y , then ϕ can be restricted to Y if its wave front set (see Appendix C) satisﬁes WF(ϕ)|Y ∩ N ∗ Y = ∅, where N ∗ Y is the “conormal bundle”, deﬁned as N ∗ Y = {(y, k) ∈ Ty∗ X; y ∈ Y, ki wi = 0 ∀w ∈ Ty Y }.

(174)

We would like to apply this result to the situation Σ = Y, R4n \0 = X, and vk = ϕ. To estimate the wave front set of the distributions vk , we use another result from microlocal analysis [76]. Suppose A is a diﬀerential operator on X such that Aϕ is smooth. Then WF(ϕ) ⊂ char(A)\0, where the characteristic set of A is deﬁned by char(A) = {(x, k) ∈ Tx∗ X; a(x, k) = 0}, where a is the principal symbol of A. In our case, we have SρN −k vk = 0, so

N −k ∗ 4n WF(vk ) ⊂ char(Sρ )\0 = (x, k) ∈ T R ; ξi · ki = 0, k = 0 (175) i

# ξi · ki , where we recall because the principal symbol of Sρ is given by s(x, k) = the notation x = (ξ1 , . . . , ξn ), and where we have set k = (k1 , . . . , kn ) ∈ (R4n )∗ . Assume now that (x, k) ∈ N ∗ Σ, and at the same time (x, k) ∈ WF(vk )|Σ . Then, from the ﬁrst condition, we have w · k = 0 for all w ∈ Tx R4n that are tangent to S, while from the second condition, we have x · k = 0 and k = 0. Since Σ is transverse to the scaling orbits, it follows that k = 0, a contradiction. Hence WF(vk )|Σ ∩ N ∗ Σ = ∅, and vk can be restricted to Σ. We denote points x), by the usual abuse of in Σ by x ˆ, and we denote the restriction simply by vk (ˆ notation. Let Σ ⊂ R4n a submanifold of dimension 4n − 1 as above, and deﬁne, for r > 0 x ∈ R4n ; x ˆ ∈ Σ}. Σr = {rˆ

(176)

We let d4n x be the usual 4n-form on R4n with the orientations induced from R4 , i.e. d4n x = d4 ξ1 ∧ · · · ∧ d4 ξn ,

d4 ξ = dξ 0 ∧ · · · ∧ dξ 3 ,

(177)

example, we may choose Σ to be the sphere S 4n−1 deﬁned relative to some auxiliary Euclidean metric on R4n .

h For

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where we have put again x = (ξ1 , . . . , ξn ) to lighten the notation. We also deﬁne the 3-form w on R4 and the 4n − 1 form Ω on R4n by 3

w(ξ) =

$µ ∧ · · · ∧ dξ 3 , ξ µ dξ 1 ∧ · · · ∧ dξ

(178)

d4 ξ1 ∧ · · · ∧ w(ξi ) ∧ · · · ∧ d4 ξn

(179)

µ=0

Ω(x) =

n i=1

where a caret denotes omission. Because we are assuming that the surface Σ is x ∈ R4n \0 transverse to the orbits of dilations in R4n , the map (r, xˆ) ∈ R+ × Σ → rˆ is diﬀeomorphism. If ir : Σr → R4n is the natural inclusion, then we may write d4n x =

dr ∧ i∗r Ω. r

(180)

Now let f be a testfunction of compact support on R4n \0, i.e. f is smooth, vanishes outside a compact set, and vanishes in an open neighborhood of 0. From the equation for d4n x, and from Eq. (172), we then get the following representation for u(f ): u(x)f (x)d4n x u(f ) = R4n ∞

dr u(x)f (x)Ω(x) r 0 Σr ∞ r4n−1 u(rx)f (rx)Ω(x) dr =

=

0

= 0

Σ1 −1 ∞N

r4n−1−ρ

k=0

(log r)k k!

vk (x)f (rx)Ω(x) dr.

(181)

Σ

The terms in the sum may be written as residue using the equality ra =

ak (log r)k k!

k

.

(182)

For this, let fr (ˆ x) be the function on Σ deﬁned by f (rˆ x). Then we may write vk (x)fr (x)Ω(x), (183) vk (fr ) = Σ

and we have u(f ) = Resa=0

N −1 k=0

1 ak+1

∞

ra+4n−1−ρ vk (fr )dr,

(184)

0

This formula is well deﬁned because, since the support of f is bounded away from the origin in R4n , the distribution r → vk (fr ) is in fact a smooth test function on R+ whose support is compact and bounded away from r = 0. We would like to deﬁne

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the desired extension u by generalizing formula (184) to arbitrary test functions f on R4n whose support is not necessarily bounded away from the origin. If f is an arbitrary test function then r → vk (fr ) vanishes for suﬃciently large r > r0 , but it no longer vanishes near r = 0. In that case, it is not obvious that the right-hand side of (184) is still well-deﬁned, and if so, whether it deﬁnes a meromorphic function of a. To show this, we let vk (x)f (rx)Ω(x), (185) hk (r) := vk (fr ) = Σ

and we observe that the distribution u deﬁned by   1 m N −1 j j d 1 r h (0) k  dr u (f ) := Resa=0 ra−ρ+4n−1 hk (r) − ak+1 0 j! drj j=0 k=0

+ Resa=0

N −1 k=0

1 ak+1

∞

ra−ρ+4n−1 hk (r)dr

(186)

1

is well deﬁned for all test functions f if m is chosen to be the largest integer ≤ Re ρ − 4n. Indeed, the ﬁrst integral on the right-hand side is well deﬁned and analytic for Re a > −1, and the second term is well deﬁned and analytic for all a ∈ C. Thus, the terms on the right-hand side of the above equation are linear functionals on the space of test functions that are meromorphic in a. Furthermore, if f has its support away from 0, then hk (r) = 0 in an open neighborhood of r = 0, and we have u (f ) = u(f ). Finally, it can be shown using the methods described in [57, Chap. I, Paragraph 3] that u (f ) is not just a linear functional on the space of test-functions, but deﬁnes in fact a distribution on R4n . Consequently, (186) deﬁnes an extension u of the distribution u. We next need to analyze the scaling behavior of our extension u . A straightforward calculation using Eq. (186) shows that   N −1  ∂N µa N (Sρ u )(f ) = −Resa=0 N k+1   ∂(log µ) k=1 a r=µ  dm    r h (0) m k   r4n−ρ+a hk (0) dr + ···+ ×   4n − ρ + a m!(4n − ρ + a + m)   

4n−ρ+a+m

r=1

.

(187)

µ=1

If we now assume that we are in case (3) of the lemma, i.e. ρ ∈ N0 + 4n, then m = ρ − 4n, and the expression evaluates to dρ−4n h (0) ρ−4n N −1 . (SρN u )(f ) = dr (ρ − 4n)!

(188)

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The terms on the right-hand side can be evaluated as follows using the deﬁnition of hN −1 (r) and vN −1 (x), see Eqs. (185) and (173): dρ−4n α N −1 h (0) = x S u(x)Ω(x) (∂α f )(0), (189) N −1 ρ drρ−4n Σ |α|=ρ−4n

where α = (α1 , . . . , α4n ) ∈ N4n 0 is a multi-index, and we are using the usual multiindex notation ∂ |α| α4n 1 |α| = αi , xα = xα (190) ∂α = α1 α4n , 1 · · · x4n . ∂x1 · · · ∂x4n i Alternatively, we may write SρN u (x) =

cα ∂α δ(x)

(191)

|α|=ρ−4n

in terms of the usual δ-function on R4n concentrated at the origin. The numerical constants cα ∈ C are given by the formula α c = F α (x), (192) Σ

with F

α

the (distributional) (4n − 1)-forms on Σ deﬁned by F α (x) :=

(−1)ρ−4n α N −1 x Sρ u(x) · Ω(x) ∈ D (Σ; ∧4n−1 T ∗ Σ). (ρ − 4n)!

(193)

Since the delta-function is a homogeneous distribution of degree −4n, we have Sρ ∂α δ = ∂α S4n δ = 0, and therefore SρN +1 u = 0 by Eq. (203). Thus our extension u is again an almost homogeneous distribution. One may repeat this argument also for the ﬁrst and second case of the lemma. In those cases, one ﬁnds SρN u = 0. Thus, summarizing, Eq. (186) deﬁnes a distributional extension u of u that is almost homogeneous. To simplify the notation, we will from now on denote this extension again by u. We now investigate the Lorentz transformation properties of u. Our construction of the extension u given above involved a choice of a suitable Σ transverse to the orbits of the dilations. Since no Σ with the above properties exists that is at the same time invariant under the Lorentz group, the extension u just constructed will in general fail to be Lorentz invariant. Restoring the tensor indices on u, we ﬁnd by a calculation using Eq. (186) that for any test function f ∈ C0∞ (R4n ) and any Lorentz transformation, Λ, we have bα (194) uµ1 ···µl (f ) − Λνµ11 · · · Λνµll uν1 ···νl (R(Λ)f ) = µ1 ···µl (Λ)∂α δ(f ), |α|≤ρ−4n −1

bα µ1 ···µl (Λ)

where (R(Λ)f )(x) = f (Λ x) and the are complex constants, which would vanish if and only if the distribution u were Lorentz invariant. We now apply the diﬀerential operator SρN +1 to both sides of the above equation. Since Sρ is

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itself a Lorentz invariant operator, we have R(Λ)Sρ = Sρ R(Λ). Therefore, since SρN +1 u = 0, the operator SρN +1 annihilates the left-hand side of Eq. (194), so we obtain bα 0 = SρN +1 µ1 ··· µl (Λ)∂α δ |α|≤Re(ρ)−4n

=

(ρ − 4n − |α|)N +1 bα µ1 ··· µl (Λ)∂α δ.

(195)

|α|≤Re(ρ)−4n

It follows immediately that bα µ1 ··· µl (Λ) = 0, except possibly when |α| = ρ − 4n, which evidently can only happen when ρ is an integer. Thus, focussing on that case, we have ν ··· ν

(Λ)∂ν1 · · · ∂νρ−4n δ(f ) uµ1 ··· µl (f ) − Λνµ11 · · · Λνµll uν1 ··· νl (R(Λ)f ) = bµ11 ··· µρ−4n l

(196)

for all f and all Lorentz-transformations Λ. Using this equation, one ﬁnds the following transformation property for b(Λ), 0 = b(Λ1 Λ2 ) − b(Λ1 ) − D(Λ1 )b(Λ2 ) ≡ (δb)(Λ1 , Λ2 ),

(197)

where we have now dropped the tensor-indices and where D denotes the tensor representation of the Lorentz-group on the space D = (⊗l R4 )∗ ⊗ (⊗ρ−4n R4 ). This relation is of cohomological nature. To see its relation to cohomology, one deﬁnes the following group-cohomology rings, see e.g. [63]: Deﬁnition 3.1. Let G be a group, D a representation of G on a vector space V , and let cn be the space of functionals ξn : G×n → V . Let δ : cn → cn+1 be deﬁned by (δξn )(g1 , . . . , gn+1 ) = D(g1 )ξn (g2 , . . . , gn+1 ) +

n

(−1)i ξn (g1 , . . . , gi gi+1 , . . . , gn+1 )

i=1

+ (−1)n+1 ξn (g1 , . . . , gn ).

(198)

Then δ 2 = 0. The corresponding cohomology rings are deﬁned as H n (G; D) =

{Kernel δ : cn → cn+1 } . {Image δ : cn−1 → cn }

(199)

According to this deﬁnition, Eq. (197) may be viewed [97] as saying that b ∈ H 1 (SO0 (3, 1); D). It is a classical result of Wigner [118] that this ring is trivial for the Lorentz group and any ﬁnite-dimensional D. It follows that there is an a such that b = δa, or b(Λ) = (δa)(Λ) ≡ a − D(Λ)a

∀Λ,

(200)

where a is an element in H 0 (SO0 (3, 1); D) = D = (⊗l R4 )∗ ⊗ (⊗ρ−4n R4 ). This enables us to deﬁne a modiﬁed extension u by ν ···ν

u µ1 ···µl := uµ1 ···µl − aµ11 ···µρ−4n ∂ν1 · · · ∂νρ−4n δ, l

(201)

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where we have now restored the tensor indices. It is easily checked that u is Lorentz invariant and satisﬁes SρN +1 u = 0. In cases (1) and (2), u actually even satisﬁes SρN u = 0, so the modiﬁed extension (201) even satisﬁes SρN u = 0. We have therefore accomplished the goal of constructing the desired extension of u in cases (1)–(3). The uniqueness statement immediately follows from the fact that the diﬀerence between any two extensions has to be a Lorentz-invariant derivative of the deltafunction, Lδ, such that SρN +1 Lδ = 0. Thus, L can be non-zero only when ρ is an integer, and L must have degree of precisely ρ − 4n. From the proof of the lemma, we get the following interesting proposition: Proposition 1. Let u(x) be a Lorentz invariant (possibly tensor-valued) distribution on R4n \0 which scales almost homogeneously with degree ρ ∈ 4n + N0 under coordinate rescalings, i.e. SρN u(x) = 0

for some natural number N, x = 0.

(202)

Then u has a Lorentz invariant extension, also denoted u, to a distribution on R4n which also scales almost homogeneously with degree ρ under rescalings of the coordinates. We have SρN +1 u = 0, and SρN u(x) = cα ∂α δ(x) (203) |α|=ρ−4n

in terms of the usual δ-function on R4n concentrated at the origin. The numerical constants cα ∈ C are Lorentz-invariants, and are given by the formula F α (x), (204) cα = Σ

where Σ ⊂ R is any closed (4n − 1) submanifold enclosing the origin 0 ∈ R4n which is transverse to the orbits of to the dilations of R4n . Here, the distributional (4n−1)-forms F α ∈ D (Σ; ∧4n−1 T ∗ Σ) on Σ are deﬁned in Eq. (193), and are closed, 4n

dF α = 0.

(205)

Proof. We only need to show that the (4n − 1)-forms F α are closed, and that the cα are Lorentz invariants. We ﬁrst compute dΩ(x) = 4nd4n x

(206)

using the deﬁnition of the (4n − 1)-form Ω, see Eq. (178). By a straightforward computation using the deﬁnition of Ω, we also have d[xα SρN −1 u(x)] ∧ Ω(x) = xα (S0 + |α|)[(S0 − ρ)N −1 u(x)]d4n x.

(207)

Using next the fact that |α| = ρ − 4n, and that SρN u = (S0 − ρ)N u = 0, we ﬁnd d[xα SρN −1 u(x)] ∧ Ω(x) = −4nxα SρN −1 u(x)d4n x

(208)

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so dF α (x) = =

(−1)ρ−4n d[xα SρN −1 u(x)Ω(x)] (ρ − 4n)! (−1)ρ−4n {d[xα SρN −1 u(x)] ∧ Ω(x) + xα SρN −1 u(x)dΩ(x)} = 0. (ρ − 4n)!

(209)

β We would next like to show that cα are Lorentz invariants, in the sense that Λα βc = α c for any Lorentz transformation. We have β β Λα c = Λα β β F (x) Σ

=

Λ∗ Σ

β −1 Λα x) β F (Λ

F α (x)

= Λ∗ Σ

α

=

dF α (x)

F (x) + Σ α

=c .

U

(210)

Here we have used in the ﬁrst step the deﬁnition of cα , in the second step we have used the standard transformation formula of an integral under a diﬀeomorphism, denoting by Λ∗ Σ image of Σ under the natural action of Λ on R4n . In the third step we have used that F α itself is Lorentz invariant, and in the fourth step we have used Stoke’s theorem for the open set U ⊂ R4n such that ∂U = −Σ ∪ Λ∗ Σ, and in the ﬁfth step we used dF α = 0. In summary, we have now described how to construct the time-ordered products Tn (⊗ni=1 φki ) of Wick monomials without derivatives. These constructions can in principle be generalized to time-ordered products of Wick monomials Oi containing derivatives by generalizing the Wick expansion to ﬁelds with derivatives. A nontrivial new renormalization condition now arises from T10, because S0 contains derivatives. This condition is not automatically satisﬁed, but it is not diﬃcult to see that we can change, if necessary, our construction of the time-ordered products, so as to also satisfy T10 [70]. We ﬁnally have to consider condition T11. This condition is satisﬁed by our construction for T1 , but not in general for Tn when n > 1. The operational meaning of this requirement is that “derivatives can be freely pulled through the timeordering symbol”. This identity is a non-trivial requirement because both sides of the equation mean quite diﬀerent things a priori: The ﬁrst expression means the time-ordered product of ﬁelds, one of which contains a total derivative. The second expression denotes the derivative, in the sense of distributions, of the algebra-valued distribution given by the time-ordered product of the ﬁelds without the total derivative. That these two quantities are actually the same is not obvious from the above

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construction, and is therefore an additional renormalization condition, called the “action Ward identity” in [37], and the “Leibniz rule” in [70]. It is shown in these two references how, starting from a prescription that satisﬁes T1–T10 but possibly does not satisfy this renormalization condition, one can get to a prescription which does. The action Ward identity is at odds with conventions often found in standard textbooks on ﬁeld theory in Minkowski spacetime [117], where the derivative is not taken to commute with Tn . To illustrate this diﬀerence in point of view, consider the time-ordered product T2 (φ(x) ⊗ φ(y)). According to condition T11, we have (x − m2 )T2 (φ(x) ⊗ φ(y)) = T2 ((x − m2 )φ(x) ⊗ φ(y)). In our approach, the timeordered products need not vanish when acting on a factor of the wave equation, so this quantity does not need to vanish. In fact, one can see that the time-ordered product under consideration is uniquely determined by the properties T1–T10, and we have T2 ((x − m2 )φ(x) ⊗ φ(y)) = iδ(x, y)1. In standard approaches, on the other hand, it is assumed that the time-ordered product vanishes when acting on (x − m2 )φ(x), because the time-ordering symbol is viewed as on operation acting on on-shell quantized ﬁelds, rather than just classical polynomial expressions in P. On the other hand, in most standard approaches, it is not assumed that derivatives commute with T2 . In this way, one reaches the same conclusion for the example just considered, and both viewpoints are consistent for that example. However, the standard viewpoint gets very awkward in general when considering more complicated time-ordered products of ﬁelds with derivatives, for a discussion see, e.g., [38]. This is because it is in general inconsistent to assume that a timeordered product containing a factor Oφ vanishes, because of possible anomalies. On the other hand, the Leibniz rule can always be satisﬁed, and possible anomalies can thereby be analyzed consistently.

3.4. Examples Here we illustrate the above general construction of the time-ordered product by some simple examples. The simplest non-trivial example of a time-ordered product with one factor is T1 (φ2 (x)) = : φ2 (x) :H . Using the deﬁnition of the locally normalordered product, this may be viewed as a “point-splitting” deﬁnition, see, e.g., [27]. Consider next the time-ordered product T2 (φ2 (x1 ) ⊗ φ2 (x2 )). By T8, it is deﬁned for non-coincident points x1 = x2 by the prescription T2 (φ (x1 ) ⊗ φ (x2 )) = 2

2

: φ2 (x2 ) :H : φ2 (x1 ) :H

when x1 ∈ / J + (x2 );

: φ2 (x1 ) :H : φ2 (x2 ) :H

when x1 ∈ / J − (x2 ).

(211)

In order to extend the deﬁnition to coincident points x1 = x2 , i.e. to make the time-ordered product a well deﬁned distribution on the entire product manifold M 2 , we now use the expansion procedures described in general in the previous section. Using the deﬁnition of the product , and of the locally normal ordered

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products, we have : φ2 (x1 ) :H : φ2 (x2 ) :H = : φ2 (x1 )φ2 (x2 ) :H −2H(x1 , x2 ) : φ(x1 )φ(x2 ) :H +2 H(x1 , x2 )2 1,

(212)

for points x1 , x2 that are suﬃciently close to each other so that the local Hadamard parametrix H(x1 , x2 ) is well deﬁned. Using furthermore the deﬁnition of the local Feynman parametrix HF (see Eq. (519)) and ϑ(T (x) − T (y))H(x, y) + ϑ(T (y) − T (x))H(y, x) = iHF (x, y)

(213)

we can write the time-ordered product under consideration as T2 (φ2 (x1 ) ⊗ φ2 (x2 )) = : φ2 (x1 )φ2 (x2 ) :H +2(/i)HF (x1 , x2 ) : φ(x1 )φ(x2 ) :H + (/i)2 HF (x1 , x2 )2 1,

(214)

for non-coinciding points x1 , x2 . This is the desired local Wick-expansion. Comparing with Eq. (159), we read oﬀ t0,0 (x1 , x2 ) = 1,

t1,1 (x1 , x2 ) = (/i)HF (x1 , x2 ),

t2,2 (x1 , x2 ) = (/i)2 HF (x1 , x2 )2

(215)

for the coeﬃcients in the Wick expansion. The coeﬃcients t0,0 , t1,1 may be extended to coincident points x1 = x2 by continuity, because their scaling degree is 0, respectively, 2, which is less than 4, but the distribution t2,2 has scaling degree 4 and therefore cannot be extended to the diagonal by continuity, but must instead be extended non-trivially. Actually, since t2,2 is the square of the distribution HF with singularities on the lightcone, it is instructive to check explicitly that it is even deﬁned for non-coincident points that are on the lightcone. This can be done using / J + (x2 ), the pair (x1 , k1 ; x2 , k2 ) ∈ T ∗ (M 2 ) is in the the wave front set: For x1 ∈ wave front set of HF (see Appendix C) if and only if x1 and x2 can be joined by ˙ = −k2 , with k1 ∈ V+∗ . a null-geodesic γ : (0, 1) → M , with γ(0) ˙ = k1 and γ(1) − ∗ Similarly, for x1 ∈ / J (x2 ), the pair (x1 , k1 ; x2 , k2 ) ∈ T (M 2 ) is in the wave front set if and only if x1 and x2 can be joined by a null-geodesic γ : (0, 1) → M , with ˙ = −k2 , with k1 ∈ V−∗ . It follows that, when x1 = x2 , elements γ(0) ˙ = k1 and γ(1) (x1 , k1 , x2 , k2 ) ∈ WF(HF ) can never add up to the zero element. Thus, by the general theorems about the wave front set summarized in Appendix C, arbitrary powers HF (x1 , x2 )n exist in the distributional sense, i.e. as distributions on M 2 \∆2 . On the other hand, when x1 = x2 , arbitrary elements of the form (x1 , k, x2 , −k) are in WF(HF ). Thus, for coincident points, the elements in the wave front set can add up to zero, and the product HF (x1 , x2 )n is therefore not deﬁned as a distribution on all of M 2 , i.e. including coincident points. In order to extend t2,2 to a well-deﬁned distribution to all of M 2 , we now need to perform the scaling expansion of t2,2 , which in turn can be obtained from the scaling expansion of HF . The latter can be found using expansions for the

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recursively deﬁned coeﬃcients in the local Hadamard parametrix, see e.g. [27]. Up to numerical prefactors, it is given by (we assume for simplicity that m2 = 0) 1 µν 1 1 ξµξν 2 + Rµν (y) − 2 + η log(ξ + i0) + · · · , HF (expy ξ, y) ∼ 2 ξ + i0 6 ξ + i0 12 (216) where the dots stand for a remainder with scaling degree < 2, where ξ ∈ Ty M has been identiﬁed with a vector in R4 via a tetrad, and where ξ 2 = ηµν ξ µ ξ ν . From this we obtain the ﬁrst terms in the scaling expansion of t2,2 up to numerical prefactors as t2,2 (expy ξ, y) ∼ u(ξ) + Rµν (y)uµν (ξ) + · · ·

(217)

where the dots stand for terms of scaling degree less than 2. The distributions u and uµν are deﬁned on R4 \0 and is given there by u(ξ) =

1 , 2 (ξ + i0)2

uµν (ξ) = −

1 ξµξν 1 η µν log(ξ 2 + i0) . + 3 (ξ 2 + i0)2 6 ξ 2 + i0

(218)

u has scaling degree 4, while uµν has scaling degree 2. Thus, by Lemma 6, we need to extend non-trivially only u, while uµν and the remainder (i.e. the dots in the scaling expansion of t2,2 ) can be extended by continuity. An extension to all of R4 (i.e. including ξ = 0) of u can easily be guessed, but we here prefer to give a systematic method, which is needed anyway in more complicated examples. A constructive method to obtain an extension of u is provided by Lemma 6. However, that is somewhat complicated because it involves a non-Lorentz invariant surface S at intermediate steps, which is awkward in concrete calculations.i Instead we here present a diﬀerent method that is more practical and works in a wide class of examples. That method is based upon the fact that, for complex scaling degree, there is a unique extension of a homogeneous distribution by Lemma 6. The method has also appeared in the context of BPHZ-renormalization in momentum space under the name “analytic renormalization” [105–107]. Consider instead of u the distribution given by 1 , a ∈ C\Z. (219) ua (ξ) = 2 (ξ + i0)2−a By contrast to u, this is well deﬁned on all of R4 , see e.g. [57], and also [91] for a treatment of such so-called “Riesz-distributions”. An extension u of u can now be obtained by taking the residue of the meromorphic function a → ua (f )/a, ua (f ) . (220) a Indeed, if the support of f excludes 0, then u (f ) obviously must coincide with u(f ), because we may then use formula (218) to get u(f ). The almost homogeneous u (f ) = Resa=0

i Note, however, that this is not an obstacle in the corresponding “Euclidean situation”, where one may take S simply to be a Euclidean sphere.

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scaling property of u (f ) under rescalings of f (ξ) → f (µξ) also immediately follows from the deﬁnition. To get a more explicit formula for the extension, we compute the fourier transform of ua , given up to numerical factors by [91] Γ(a) (p2 − i0)−a . (221) Γ(2 − a) We expand this expression around a = 0 using the well-known residue of the Γfunction at 0 and substitute the resulting expression into Eq. (220). We obtain, up to numerical prefactors u ˆa (p) = 4a

u ˆ (p) = ln[l2 (p2 − i0)]

(222)

where l is some constant. Taking an inverse fourier transform then gives the desired extension log[l−2 (ξ 2 + i0)] 1 u (ξ) = − ∂ 2 (223) 2 ξ 2 + i0 where ∂ 2 = η µν ∂ 2 /∂ξ µ ∂ξ ν . Note that the extension has acquired a logarithm, which is a general phenomenon according to Lemma 6. Diﬀerent choices of l change the extension by a term proportional to δ 4 (ξ), and thus correspond to the diﬀerent extensions of u(ξ). Thus, inserting this extension into the scaling expansion of t2,2 , we obtain the desired extension of T2 (φ2 (x1 ) ⊗ φ2 (x2 )). Our last example is the time-ordered product T3 (φ3 (x1 ) ⊗ φ3 (x2 ) ⊗ φ4 (x3 )) with 3 factors. The terms in the Wick expansion of this quantity that need to be extended non-trivially from M 3 \∆3 to M 3 are t3,3,2 (x1 , x2 , x3 ) = t1,1 (x1 , x2 )t1,1 (x2 , x3 )t2,2 (x1 , x3 ),

(224)

t3,3,4 (x1 , x2 , x3 ) = t1,1 (x1 , x2 )t2,2 (x2 , x3 )t2,2 (x1 , x3 ).

(225) 3

All other terms are either already well-deﬁned as distributions on all of M (assuming the corresponding time-ordered products with 2 factors have been deﬁned), or can be extended by continuity. We focus on the last term t3,3,4 . Again, for the sake of illustration of the general construction, we ﬁrst verify explicitly that this / ∆3 . distribution is indeed well deﬁned on M 3 \∆3 . Consider a point (x1 , x2 , x3 ) ∈ Then it must be possible to separate one point from the remaining two points by a Cauchy surface. For deﬁniteness, let us assume that this point is x3 , and that / J + (x3 ). Then (x1 , k1 ; x3 , k3 ) is in the wave front set of t2,2 (x1 , x3 ) if and x1 , x2 ∈ only if k1 ∼ −k3 , and if k1 ∈ V+∗ . Likewise, (x2 , p2 ; x3 , p3 ) is in the wave front set of t2,2 (x2 , x3 ) if and only if p2 ∼ −p3 , and if p2 ∈ V+∗ . Finally (x1 , q1 ; x2 , q2 ) is in the wave front set of t1,1 (x1 , x2 ) if and only if q1 ∼ −q2 and q1 ∈ V±∗ when / J ± (x2 ), or if and only if q1 = −q2 when x1 = x2 . We now add up these wave x1 ∈ front set elements, viewed in the obvious way as elements in Tx∗1 M × Tx∗2 M × Tx∗3 M . We obtain the set S = {(x1 , k1 + q1 ; x2 , p2 + q2 ; x3 , k3 + p3 )}.

(226)

Assume ﬁrst that x1 = x2 . Clearly, if, e.g., k1 + q1 = 0, then q1 ∈ V−∗ , so p2 + q2 = p2 − q1 = 0, because p2 ∈ V+∗ . Thus, S cannot contain the zero element, and the

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product deﬁning t3,3,4 is well deﬁned near (x1 , x2 , x3 ) by Theorem 5. Similarly, if / J − (x2 ), then q2 ∈ V+∗ , and p2 + q2 = 0, and again, S cannot contain the zero x1 ∈ element. The same type of argument can be made for all other conﬁgurations of the points, except the conﬁguration x1 = x2 = x3 . Thus, by the general existence Theorem 5 for products of distributions, t3,3,4 is indeed well deﬁned as a distribution on M 3 \∆3 . We next would like to construct an extension of t3,3,4 along the lines of our general construction. Thus, we must determine the scaling expansion of t3,3,4 . It can be obtained from the expansions of the (extended) distributions t2,2 and of t1,1 that were constructed above. We focus on the terms that require a non-trivial extension (up to numerical prefactors): t3,3,4 (expy ξ1 , expy ξ2 , y) ∼ u(ξ1 , ξ2 ) + Rµν (y)uµν (ξ1 , ξ2 ) + Rµνσρ (y)uµνσρ (ξ1 , ξ2 ) + · · · ,

(227)

where u is the distribution deﬁned on (R4 )2 \0 given by log[l−2 (ξ12 + i0)] log[l−2 (ξ22 + i0)] 1 1 2 u(ξ1 , ξ2 ) = ∂12 ∂ 2 4 ξ12 + i0 ξ22 + i0 (ξ1 − ξ2 )2 + i0 (228) where uµν is the distribution deﬁned on (R4 )2 \0 given by log[l−2 (ξ12 + i0)] 1 1 ξ2µ ξ2ν uµν (ξ1 , ξ2 ) = − ∂12 − 2 ξ12 + i0 3 (ξ22 + i0)2 1 1 η µν log[l−2 (ξ22 + i0)] + (ξ1 ↔ ξ2 ) + 2 6 ξ2 + i0 (ξ1 − ξ2 )2 + i0 and where uµνσρ is the distribution on (R4 )2 \0 deﬁned by log[l−2 (ξ12 + i0)] log[l−2 (ξ22 + i0)] 1 2 uµνσρ (ξ1 , ξ2 ) = ∂12 ∂ 2 4 ξ12 + i0 ξ22 + i0 ξ1µ ξ1σ ξ2ν ξ2ρ 1 η µσ (ξ1ν ξ2ρ + 2ξ1ν ξ1ρ ) 1 − · − 6 [(ξ1 − ξ2 )2 + i0]2 12 (ξ1 − ξ2 )2 + i0 1 + η µσ η νρ log{l−2 [(ξ1 − ξ2 )2 + i0]} + (ξ1 ↔ ξ2 ). 24

(229)

(230)

The dots in Eq. (227) again represent a remainder. This now has scaling degree 6 and can thus be extended by continuity, while the 3 terms in the scaling expansion that are explicitly given have scaling degree 10 for the ﬁrst term respectively 8 for the second and third terms. They must thus be extended non-trivially. The extension of the corresponding distributions u, uµν , uµνσρ now can no longer be found by trial and error, but one must use a constructive method, such as that given in the proof of Lemma 6. We will again not use this method here, but instead

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use a variant of the method given above. For this, we consider the distribution ua,b,c (ξ1 , ξ2 ) =

1 (ξ12

+

i0)2−a (ξ22

+

i0)2−b [(ξ1

− ξ2 )2 + i0]2−c

.

(231)

It can be checked using wave front arguments similar to that given above that this distributional product is well deﬁned on (R4 )2 \0 for a, b, c ∈ C\Z. Furthermore, by Lemma 6, if a + b + c ∈ / Z this distribution has a unique extension to all of (R4 )2 . We deﬁne the desired extension of u by the expression u (f ) = Resc=1 Resb=0 Resa=0

ua,b,c (f ) . ab(c − 1)

(232)

This is an extension, because one can check that u (f ) conicides with u(f ) for any f whose support excludes ξ1 = ξ2 = 0, and it is also clearly Lorentz invariant and has the desired almost homogeneous scaling behavior. To get a more explicit expression for u , we perform a Fourier transformation of ua,b,c using Eq. (221) and Eq. (23) of [26]. This gives, up to numerical factors u ˆa,b,c (p1 , p2 ) =

4a+b+c Ia,b,c (p1 , p2 ) Γ(4 − a − b − c)Γ(2 − a)Γ(2 − b)Γ(2 − c)

(233)

where Ia,b,c (p1 , p2 ) = [(p1 + p2 )2 − i0]2−a−b−c Γ(c)Γ(a + b + c − 2)Γ(2 − a − c)Γ(2 − c − b) p22 p21 , × F4 c, a + b + c − 2, a + c − 1, b + c − 1 (p1 + p2 )2 − i0 (p1 + p2 )2 − i0 + [(p1 + p2 )2 − i0]−a (p22 − i0)2−b−c Γ(a)Γ(2 − b)Γ(2 − a − c)Γ(b + c − 2) p21 p22 , × F4 a, 2 − b, a + c − 1, 3 − b − c (p1 + p2 )2 − i0 (p1 + p2 )2 − i0 + [(p1 + p2 )2 − i0]−b (p21 − i0)2−a−c Γ(b)Γ(2 − a)Γ(a + c − 2)Γ(2 − c − b) p21 p22 , × F4 b, 2 − a, 3 − a − c, b + c − 1 (p1 + p2 )2 − i0 (p1 + p2 )2 − i0 + [(p1 + p2 )2 − i0]c−2 (p21 − i0)2−a−c (p22 − i0)2−b−c × Γ(4 − a − b − c)Γ(2 − c)Γ(a + b − 2)Γ(b + c − 2) × F4 4 − a − b − c, 2 − c, 3 − a − c, 3 − b − c p22 p21 , . × (p1 + p2 )2 − i0 (p1 + p2 )2 − i0

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Here, F4 is the Appell function, deﬁned by F4 (α, β, γ, δ|z1 , z2 ) =

∞ (α)j1 +j2 (β)j1 +j2 j1 j2 z1 z2 , (γ)j1 (δ)j2 j ,j =0 1

(234)

2

with (α)j the Pochhammer symbol. The Fourier transform of the extension is then given by u ˆ (p1 , p2 ) = Resc=1 Resb=0 Resa=0

u ˆa,b,c (p1 , p2 ) , ab(c − 1)

(235)

which may be evaluated readily using the Laurent expansion of the Gammafunction. It is worth noting that the extension u given by expression (232) now implicitly contains third powers of the logarithm, thus again conﬁrming the general theorem that there are logarithmic corrections to the naively expected homogeneous scaling behavior. 3.5. Ghost fields and vector fields The above algebraic construction of Wick-powers and their time-ordered products may be generalized to a multiplet of scalar or tensor ﬁelds satisfying a system of wave equations on M with local covariant coeﬃcients or to Grassmann valued ﬁelds. In the BRST approach to gauge theory, the relevant ﬁelds are (gauge ﬁxed) vector ﬁelds, and ghost ﬁelds. Classical ghost ﬁelds are valued in the Grassmann algebra E. For gauge theory, the relevant ghost ﬁelds are described, at the free level, by the Lagrangian L0 = −idC¯ ∧ ∗dC.

(236)

The ﬁelds C, C¯ are independent and take values in the Grassmann algebra E. In particular, the “bar” over C¯ is a purely conventional notation and is not intended to mean any kind of conjugation. The non-commutative *-algebra W0 corresponding to this classical Lagrangian is described as follows. As above, we consider a bidistribution ω s (x, y) on M × M of Hadamard form (we put a superscript “s” for “scalar”), and we consider distributions u on M n which are anti-symmetric in the variables, and which satisfy the wave front condition (126). With each such distribution, we associate a generator F (u), which we (purely formally) write as ¯ 1 ) · · · C(y ¯ m ) :ω F (u) = u(x1 , . . . , xn ; y1 , . . . , ym ) : C(x1 ) · · · C(xn )C(y × dx1 · · · dxn dy1 · · · dyn .

(237)

We now deﬁne a -product between such generators. This is again deﬁned by Eq. (121), where the derivative operator (124) is now given by δL δL δR δR − ¯ ω s (x, y) ω s (x, y) ¯ dxdy. (238) < D> = −i δC(x) δC(y) δ C(y) δ C(x) Here, as above, it is understood that a functional derivative acting on F (u) is executed by formally treating the ﬁelds in the normal ordered expression as classical

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¯ 1 ) · · · C(y ¯ m ) :ω with the clasﬁelds, i.e. by formally identifying : C(x1 ) · · · C(xn )C(y sical ﬁeld expression. The operation * of conjugation is deﬁned as C(x)∗ = C(x) and ¯ ∗ = C(x). ¯ C(x) This is consistent with the product. It leads to the anti-commutation relations for the ghost ﬁelds, ¯ ¯ = ∆s (x, y)1, C(x) C(y) + C(y) C(x) ¯ ¯ ¯ ¯ + C(y) C(x) = 0, C(x) C(y) + C(y) C(x) = C(x) C(y)

(239) (240)

where we have put a superscript on “s” the scalar causal propagator ∆s to distinguish it from the vector propagator below. The ﬁeld equations may be implemented, as in the scalar case, by dividing W0 by the ideal J0 generated by C(x) and ¯ C(x). Time-ordered products of Grassmann ﬁelds are also deﬁned in the same way as above, the only minor diﬀerence being that they are not symmetric in the tensor factors, but have graded symmetry according to the Grassmann parity of the arguments. For example, T6 reads instead Tn (· · · ⊗ O1 (xj ) ⊗ O2 (xj+1 ) ⊗ · · ·) = (−1)εj εj+1 Tn (· · · ⊗ O2 (xj+1 ) ⊗ O1 (xj ) ⊗ · · ·).

(241)

There are similar signs also in T9. We next consider 1-form (or vector) ﬁelds, A. In the Lorentz gauge, their classical dynamics is described by the Lagrangian L0 =

1 (dA ∧ ∗dA + δA ∧ ∗δA) 2

(242)

where δ = ∗d∗ is the co-diﬀerential (divergence). Their equation of motion is the canonical wave equation for vectors, (dδ + δd)A = 0, or (gµν + Rµν )Aν = 0

(243)

in component notation. It is seen from the component form of the equation that it is hyperbolic in nature, and hence has unique fundamental retarded and advanced solutions, ∆vA and ∆vR , where we have put a superscript “v” in order to distinguish them from their scalar counterparts. To deﬁne the corresponding quantum algebra of observables, we proceed by analogy with the scalar case. For this, we pick an arbitrary distribution ω v taking values in T ∗M × T ∗M of Hadamard form. Thus, ω v (x, y) satisﬁes the vector equations of motion (243) in x and y, its anti-symmetric part is given by i∆v (x, y), where ∆v is the diﬀerence between the fundamental advanced and retarded vector causal propagators, and its wave front set is given by Eq. (119). The algebra W0 is generated by expressions of the form F (u) = u(x1 , . . . , xn ) : A(x1 ) · · · A(xn ) :ω dx1 · · · dxn , (244) where u(x1 , . . . , xn ) is a distribution with wave front set (126), now taking values in the bundle T M × · · · × T M , and the *-operation is declared by A(x)∗ = A(x).

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The -product is again deﬁned by Eq. (121), where the derivative operator (124) is now given by δL δR ω v (x, y) dxdy. (245) < D> = δA(x) δA(y) From this, we can calculate the commutation relations for the ﬁeld A(x) = : A(x) :ω , A(x) A(y) − A(y) A(x) = i∆v (x, y)1.

(246)

The construction of Wick powers and their time-ordered products is completely analogous to the scalar case, the only diﬀerence is that the Hadamard scalar parametrix H must be replaced by a vector Hadamard parametrix, whose construction is described in Appendix C.2. 3.6. Renormalization ambiguities of the time-ordered products In the previous sections, we have described the construction of local and covariant renormalized time-ordered products in globally hyperbolic Lorentzian curved spacetimes. We now address the issue to what extend the time-ordered products are unique. Thus, suppose we are given two prescriptions, called T = {Tn } and Tˆ = {Tˆn }, satisfying the conditions T1–T11. We would like to know how they can diﬀer. To characterize the diﬀerence, we introduce a hierarchy D = {Dn } of linear functionals with the following properties. Each Dn is a linear map Dn : Pk1 (M ) ⊗ · · · ⊗ Pkn (M ) → Pk1 /···/kn (M n )[[]],

(247)

where we denote by Pk1 /···/kn (M n ) the space of all distributional local, covariant functionals of φ and its covariant derivatives ∇k φ, of m2 , of the metric, and of the Riemann tensor and its covariant derivatives ∇k R, which are supported on the total diagonal, and which take values in the bundle k1

T ∗M × · · · ×

kn

T ∗M ⊂

k1 +···+k n

T ∗M n

(248)

of anti-symmetric tensors over M n . Thus, if Oi ∈ Pki (M ), then Dn (⊗i Oi ) ∈ Pk1 /···/kn (M n ), and Dn is a (distributional) polynomial, local, covariant functional of φ, the mass, m2 , and the Riemann tensor and its derivatives taking values in the k1 + · · · + kn forms over M n , which is supported on the total diagonal, i.e. supp Dn (O1 (x1 ) ⊗ · · · ⊗ On (xn )) = {x1 = x2 = · · · = xn } = ∆n .

(249)

It is a k1 -form in the ﬁrst variable x1 , a k2 -form in the second variable x2 , etc. The diﬀerence between two prescriptions T and Tˆ for time-ordered products satisfying T1–T11 may now be expressed in terms of a hierarchy D = {Dn } as follows. Let F = f ∧ O be an integrated local functional O ∈ P(M ), and formally

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combine the time-ordered functionals into a generating functional written T (eF ⊗ ) :=

∞ 1 Tn (F ⊗n ), n! n=0

(250)

where exp⊗ is the standard map from the vector space of local actions to the tensor algebra (i.e. the symmetric Fock space) over the space of local action functionals. We similarly write D(eF ⊗ ) for the corresponding generating functional obtained from D. The diﬀerence between the time-ordered products T and Tˆ may now be expressed in the following way [66]: i[F +D(exp⊗ F )]/ iF/ ) Tˆ(e⊗ ) = T (e⊗

(251)

where D = {Dn } is a hierarchy of functionals of the type just described. Each Dn is a formal power series in , and if each Oi = O(0 ), then it can be shown that Dn (⊗Oi ) = O(), essentially because there are no ambiguities of any kind in the underlying classical theory. The expression D(eF ⊗ ) may be viewed as being equal to the ﬁnite counterterms that characterize the diﬀerence between the two prescriptions for the time-ordered products. Note that in curved space, there is even an ambiguity in deﬁning time-ordered products with one factor (the Wick powers), so even D1 might be non-trivial. The counterterms, i.e. the maps Dn , satisfy a number of properties corresponding to the properties T1–T11 of the time-ordered products [66]. As we have already said, the Dn are supported on the total diagonal, and this corresponds to the causal factorization property T8. The Dn are local and covariant functionals of the ﬁeld φ, the metric, and m2 , in the following sense: Let ψ : M → M be any causality and orientation preserving isometric embedding, i.e. ψ ∗ g = g. If Dn and Dn denote the functionals on M respectively M , then we have that ψ ∗ ◦ Dn = Dn ◦ (ψ ∗ ⊗ · · ·⊗ ψ ∗ ). This follows from T1. It follows from the smoothness and analyticity properties T4, T5 and the scaling property T2 that the Dn depend only polynomially on the Riemann curvature tensor, the mass parameter m2 , and the ﬁeld φ. Since there is no ambiguity in deﬁning the identity operator, 1, or the basic ﬁeld, φ, we must have D1 (1) = D1 (φ) = 0.

(252)

As a consequence of the symmetry of the time-ordered products T6, the maps Dn are symmetric (respectively, graded symmetric when Grassmann valued ﬁelds would be present), and as a consequence the ﬁeld independence property T9, they must satisfy δ Dn (O1 (x1 ) ⊗ · · · ⊗ On (xn )) δφ(y) δOk (xk ) = ⊗ · · · ⊗ On (xn ) . Dn O1 (x1 ) ⊗ · · · ⊗ δφ(y)

(253)

k

In particular, the Dn depend polynomially upon the ﬁeld φ. As a consequence of the scaling property T2 of time-ordered products, the engineering dimension of

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each term appearing in Dn must satisfy the following constraint. As above, let Nr the counter of Riemann curvature tensors, let Nf be the dimension counter for the ﬁelds, and let Nc be the counter for the coupling constant (in this case m2 ), see Eq. (140). Let the dimension counter Nd : P → P be deﬁned as above by Nd = Nc + Nr + Nf . Then we must have (Nd + sd)Dn (O1 (x1 ) ⊗ · · · ⊗ On (xn )) =

n

Dn (O1 (x1 ) ⊗ · · · ⊗ Nd Oi (xi ) ⊗ · · · ⊗ On (xn ))

(254)

i=1

where sd is the scaling degree, see Appendix C. The unitarity requirement T7 on the time-ordered products yields the constraint Dn (O1 (x1 ) ⊗ · · · ⊗ On (xn ))∗ = −Dn (O1 (x1 )∗ ⊗ · · · ⊗ On (xn )∗ )

(255)

and the action Ward identity T11 implies that one can freely pull an exterior derivative di = dxµi ∧ ∂x∂ µ into Dn , i

di Dn (O1 (x1 ) ⊗ · · · ⊗ Oi (xi ) ⊗ · · · ⊗ On (xn )) = Dn (O1 (x1 ) ⊗ · · · ⊗ di Oi (xi ) ⊗ · · · ⊗ On (xn )).

(256)

The meaning of the above restrictions on Dn is maybe best illustrated in some examples. The dimension of the coupling is d(m2 ) = +2, and the dimension of the ﬁeld is d(φ) = +1. Consider the composite ﬁeld φ2 ∈ P. In curved spacetime, there is an ambiguity D1 (φ2 ) in deﬁning T1 (φ2 ), given by Tˆ1 (φ2 ) = T1 (φ2 ) + (/i)T1 (D1 (φ2 )).

(257)

δ By properties (253) and (252), we must have δφ D1 (φ2 ) = 0, so D1 (φ2 ) must be 2 a multiple of the identity operator, so D1 (φ ) = ic1. By the local and covariance property and the dimensional constraint (254), c = aR + bm2 , where a, b are constants that must be real in view of (255). Thus, we have the familiar result that the Wick power T1 (φ2 ) is unique only up to curvature/mass terms. Consider next the ambiguity in deﬁning the time-ordered product of two factors of φ2 , given by

Tˆ2 (φ2 ⊗ φ2 ) = T2 (φ2 ⊗ φ2 ) + (/i)2 T1 (D2 (φ2 ⊗ φ2 ))

(258)

(here we are assuming that D1 (φ2 ) = 0 for simplicity). By the same reasoning as above, this must now be given by D2 (φ2 (x) ⊗ φ2 (y)) = cδ(x, y)

(259)

for some real constant c, because the scaling degree of the delta function in 4 dimensions is +4. If φ2 in this formula would be replaced by φ3 , then the righthand side could be a constant times the wave operator of the delta function, or by a real linear combination of m2 , R and φ2 , times the delta-function.

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We summarize the renormalization ambiguities again in the “main-theorem of renormalization theory”: Theorem 2 [66, 67]. Time-ordered products T with the above properties T1–T11 exist. If T = {Tn } and Tˆ = {Tˆn } are two diﬀerent time-ordered products satisfying conditions T1–T11, then their diﬀerence is given by Tˆn (O1 (x1 ) ⊗ · · · ⊗ On (xn ))  " ! = Tr+1  Oj (xj ) ⊗ (/i)|Ik | D|Ik | Oi (xi )  . (260) I0 ∪I1 ∪···∪Ir ⊂n

j∈I0

k

i∈Ik

Here, the sum runs over all partitions I0 ∪ · · · ∪ Ir = n of n = {1, . . . , n}, and D = {Dn } is a hierarchy of counterterms described above. Conversely, if D is as above, then Tˆ deﬁnes a new hierarchy of time-ordered products with the properties T1–T11. 3.7. Perturbative construction of interacting quantum fields In the previous sections we have given the construction of Wick powers and their time-ordered products in a theory that is classically described by a Lagrangian L0 at most quadratic in the ﬁeld, with associated classical ﬁeld equations of waveequation type. Those quantities may be used to give a deﬁnition of an interacting quantum ﬁeld theory via a perturbation expansion. For deﬁniteness, consider a scalar ﬁeld described by the classical Lagrangian L = L0 + λL1 , L=

1 (dφ ∧ ∗dφ + m2 ∗φ2 ) + λ ∗φN = L0 + λL1 . 2

(261)

We would like to construct quantities in the interacting quantum ﬁeld theory as formal power series in λ. Even in ﬂat spacetime, one may encouter infra-red divergences if one tries to deﬁne the terms in such expansions, but such infra-red divergences are absent if one considers, instead of the interaction I = λL1 , a cutoﬀ interaction, F = λf L1 , where f is a smooth cutoﬀ function of compact support that is one in a globally hyperbolic subregion of the original spacetime (M, g). The perturbative formula for the interacting ﬁelds associated with this interaction is then , iF/ -−1 O(x)F = T e⊗

, iF/+R j∧O δ T e⊗ |j=0 . δj(x)

(262)

This formula is called “Bogoliubov’s formula”, [14]. Each term in the formal power series for O(x)F is a well-deﬁned element in W0 , due to the infra-red cutoﬀ in the interaction F . The subscript “F ” indicates throughout this paper the we mean an “interacting ﬁeld” deﬁned by F , which is an element in the ringj W0 ⊗ C[[λ, ]], as j The fact that, implicit in the notation “C[[]]”, the interacting ﬁeld only contains non-negative powers of , is not so obvious and follows from the fact that Rn itself is of order n , see [41].

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opposed to the classical ﬁeld expression O ∈ P. The expansion coeﬃcients in λ of the interacting ﬁelds deﬁne the so-called “retarded products”, [83] ∞ in iF/ R (O(x); F ⊗n ) =: R(O(x); e⊗ ). O(x)F = n n! n n=0

(263)

The retarded products are maps Rn : P⊗(n+1) → D (M n+1 ) ⊗ W0 with properties similar to the properties T1–T11 of the time-ordered products. The symmetry property only holds with respect to the n-arguments separated by the semicolon. Their deﬁnition in terms of time-ordered products is Rn (Ψ(y); O1 (x1 ) ⊗ · · · ⊗ On (xn ))   = (−1)n+j+1 T|I1 | Ok (xk ) · · · T|Ij | Ψ(y) ⊗ Ok (xk ), I1 ∪···∪Ij =n

k∈I1

k∈Ij

(264) where the sum runs over all partitions I1 ∪ · · · ∪ Ij of n = {1, . . . , n}. An important property of the retarded products is that their support is restricted to the set supp Rn (Ψ(y); O1 (x1) ⊗ · · · ⊗ On (xn )) ⊂ {(y, x1 , . . . , xn ) ∈ M n+1 | xi ∈ J − (y) ∀i}. (265) The support property follows from the causal factorization property of the timeordered products. A useful combinatorial identity for the retarded products is the Glaser–Lehmann–Zimmermann (GLZ) relation, which states that [38] n−1 n−1 Oi (xi ) − Rn Ψ2 (y2 ); Ψ1 (y1 ) ⊗ Oi (xi ) Rn Ψ1 (y1 ); Ψ2 (y2 ) ⊗

=



i=1

R|I|

I∪J=n

Ψ1 (y1 );

i∈I

 Oi (xi ) , R|J| Ψ2 (y2 );

i=1



Oj (xj ) .

(266)

j∈J

The GLZ-relation may be used to express the commutator of two interacting ﬁelds in terms of retarded products as follows: [Ψ1 (y1 )F , Ψ2 (y2 )F ] =

∞ in [Rn+1 (Ψ1 (x1 ); Ψ2 (x2 ) ⊗ F ⊗n ) − (1 ↔ 2)]. n n! n=0

(267)

As a consequence of the GLZ-relation and the support properties of the retarded products, any two interacting ﬁelds located at spacelike separated points commute.k Thus, we have constructed interacting ﬁelds as formal power series in the coupling k In

case when Grassmann valued ﬁelds are present, the commutator is replaced by the graded commutator, and the minus sign on the right-hand side is replaced by −(−1)ε1 ε2 , where εi are the Grassmann parities of Ψi .

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constant via the time-ordered products in the underlying free ﬁeld theory. If one changes the deﬁnition of the time-ordered products along the lines described in the previous subsection, then there is a corresponding change in the interacting theory, aﬀecting the interaction Lagrangian. There is also in general a multiplicative redeﬁnition of the interacting ﬁelds. To describe this in more detail, we introduce the linear map ZF : P(M ) → P(M )[[λ, ]] by ZF (O(x)) := O(x) + D(O(x) ⊗ eF ⊗ ),

(268)

where D = {Dn } is the hierarchy of distributions encoding the diﬀerence between two prescriptions T and Tˆ for time-ordered products. We may introduce a basis in P(M ), and represent this map by its matrix j Zi Oj (x). (269) ZF (Oi (x)) = j

For renormalizable interactions (Nf F ≤ 4), ZF leaves each ﬁnite dimensional subspace of P invariant, but this is no longer the case for non-renormalizable interacˆ F is the deﬁnition of the interacting ﬁeld using the time-ordered tions. Now, if O(x) products Tˆ, and O(x)F that using T , then the two are related by ˆ F = ZF [O(x)]F +D(exp F ) . O(x) ⊗

(270)

We now explain how one can remove the cutoﬀ implemented by the cutoﬀ function f in the interaction F = λf O at the algebraic level. The key identity [21] in this construction is VF1 ,F2 O(x)F2 VF1 ,F2 −1 = O(x)F1

(271)

where F1 , F2 are any two local interactions as above that are equal in an open neighborhood of x, and where VF1 ,F2 ∈ W0 ⊗ C[[, λ]] are unitaries that can be written in terms of retarded products. They satisfy the cocycle condition VF1 ,F2 VF2 ,F3 VF3 ,F1 = 1.

(272)

To construct the limit of the interacting ﬁelds as f → 1, one can proceed as follows. For simplicity, let us assume that M = R × Σ, with Σ compact. The cutoﬀ function may then be chosen to be of compact support in a “time-slice” M2τ = Σ×(−2τ, 2τ ), and to be equal to one in a somewhat smaller time-slice, say Mτ . To indicate the dependence upon the cutoﬀ τ , let us write the cutoﬀ function as fτ . Let Fτ = λfτ L1 and let OFτ be the corresponding interacting ﬁeld, deﬁned using Fτ as the interaction. Finally, let Uτ = VFδ ,Fτ , for some ﬁxed δ. The interacting ﬁelds with respect to the true interaction I = λL1 may now deﬁned as the limit O(x)I = lim Uτ O(x)Fτ Uτ −1 , τ →∞

τ ∈ N.

(273)

The sequence on the right-hand side is trivially convergent, because it only contains a ﬁnite number of terms for each ﬁxed x, by the cocycle condition. More precisely, the terms in the sequence will remain constant once τ has become so

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large that x ∈ Mτ . It is important to note that this would not be the case if we had not inserted the unitary operators under the limit sign. In that case, our notion of interacting ﬁeld would have coincided with the naive “adiabatic limit” which intuitively corresponds to the situation where the interacting ﬁeld is ﬁxed at τ = −∞. By contrast, our limit corresponds intuitively to ﬁxing the ﬁeld during “ﬁnite time interval” Σ × (−δ, δ). One can also see that the deﬁning formula for Uτ and the interacting ﬁeld will still make sense also for spacetimes with non-compact Cauchy surface. We can now deﬁne the algebras of interacting ﬁeld observables as . J0 . FI (M, g) = Alg GI G = g ∧ O

(274)

We note that these are subalgebras of F0 [[λ, ]]. While the embedding of this algebra as a subalgebra of F0 [[λ, ]] depends upon the choice of the cutoﬀ function f , it can be proved [21, 68] that the deﬁnition of FI as an abstract algebra is independent of our choice of the sequence of cutoﬀ functions {fτ }. Another important consequence of our deﬁnition of the interacting ﬁelds is that, if we want to investigate properties of the interacting ﬁeld near a point x, we only have to work in practice with the cutoﬀ interaction F where f is equal to 1 on a suﬃciently large neighborhood containing x. For example, if we want to check whether an interacting current J(x)I is conserved, we only need to check whether dJ(x)F = 0 for any cutoﬀ function f which is equal to 1 in an open neighborhood of x. The eﬀect of changing the renormalization conditions may also be discussed at the level of the interacting ﬁelds OI and the associated interacting ﬁeld algebra FI . For this, consider again two prescriptions T and Tˆ for deﬁning the time-ordered ˆI the respective interacting ﬁelds, and by products, and let us denote by OI and O FI and FˆI the interacting ﬁeld algebras. Let us denote by ZI : P → P[[λ, ]] the limit of the map ZF as the cutoﬀ implicit in F is removed. This limit exists, because all the functionals D = {Dn } in the deﬁning relation (268) for ZF are supported only on the total diagonal. Then one can derive from Eq. (270) that there exists an algebra isomorphism ρ : FˆI → FIˆ,

ˆI ) = ZI (O) ˆ, ρ(O I

(275)

with Iˆ = I + D(eI⊗ ). The algebra isomorphism map ρ is needed in order to comˆτ in the two prescriptions, pensate for the diﬀerence between the unitaries Uτ and U see Eq. (273), and see [68] for details. A particular case of this map again arises when the prescription Tˆ is deﬁned in terms of a change of scale (see T2) from the time-ordered product T . Then we obtain, for each scale µ ∈ R+ , a map ρµ , which depends polynomially on µ and ln µ. This map deﬁnes the renormalization group ﬂow in curved spacetime [68] together with the corresponding “mixing matrices”, i.e. the matrix components Zji (µ) of the maps ZI (µ).

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4. Quantum Yang–Mills Theory 4.1. General outline of construction 4.1.1. Free ﬁelds We now construct quantum Yang–Mills theory along the lines outlined in the introduction. As our starting point, we take the auxiliary theory described classically by the action S with ghosts and anti-ﬁelds, see Eq. (34). Thus, the set of dynamical and background ﬁelds is background ﬁelds spacetime metric g anti-ghost C ‡ , C¯ ‡ anti-vector A‡ anti-auxiliary B ‡

dynamical ﬁelds ghost C, C¯ vector A auxiliary B

We assume that the group G is a direct product of a semi-simple group and U (1)l , and that the dimension of spacetime is 4. We split the action S into a free part S0 containing only expressions at most quadratic in the dynamical ﬁelds, and an interaction part, λS1 + λ2 S2 . The action S0 describes the classical auxiliary theory. Its ﬁeld equations are hyperbolic. As we shall describe in more detail below, we can thus deﬁne an algebra W0 that represents a deformation quantization of the free ﬁeld theory associated with the free auxiliary action S0 , and this algebra contains all local covariant Wick-powers, and their time-ordered products. As in the classical case, the so-obtained auxiliary theory is by itself not equivalent to (free) Yang–Mills theory, because it contains gauge-variant observables and observables with non-zero ghost number. To obtain a quantum theory of (free) Yang–Mills theory, we pass from the algebra of observables, W0 , to the cohomology algebra constructed from the (free) quantum BRST-charge Q0 . For this, we consider ﬁrst the (free) classical BRST-current J0 , which deﬁnes a quantum Wick power T1 (J0 ), which we denote again by J0 by abuse of notation. Let us assume for simplicity that the spacetime (M, g) has a compact Cauchy surface Σ.Then there is a closed compactly supported 1-form γ on M such that M γ ∧ α = Σ α for any closed 3-form α, i.e. [γ] ∈ H01 (M, d) is dual to the cycle [Σ] ∈ H3 (M, ∂). We can then deﬁne the free BRST-charge by γ ∧ J0 . (276) Q0 = M

As we will show below, the local covariant quantum BRST current J0 := T1 (J0 ) can be deﬁned so that it is closed dJ0 = 0 modulo J0 , so evidently Q0 is independent, modulo J0 , of the choice of the representer γ in H 1 (M, d). We will also show that Q0 is nilpotent, Q20 = 0 modulo J0 . It follows from this fact that the linear quotient space Kernel [Q0 , .] ∩ F0 ∩ Kernel Ng Fˆ0 = , Image [Q0 , .] ∩ F0 ∩ Kernel Ng

F0 = W0 /J0

(277)

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is well deﬁned, and that it is again an algebra. Above, we have explained that F0 is a deformation quantization of the classical theory associated with S0 in the sense that, when → 0, the commutator divided by goes over to the Peierls bracket of the classical observables. In particular, the commutator divided by with Q0 goes to the classical BRST-variation, sˆ0 . Furthermore, as we explained above, the cohomology of sˆ0 is in 1-1 correspondence with classical gauge-invariant observables, so that, in the classical limit, the algebra Fˆ0 is the Poisson algebra of physical, gauge-invariant observables. Thus, it is natural to deﬁne Fˆ0 to be the algebra of physical observables also in the quantum case. Consider now a representation π0 of the free algebra F0 on an inner product space H0 . For simplicity, let us denote representer π0 (Q0 ) of the BRST-charge in this representation again by Q0 . We require Q0 to be hermitian with respect to the (necessarily indeﬁnite) inner product. We would like to know under which condition this representation induces a Hilbert-space representation π ˆ0 on the factor algebra Fˆ0 . Following [42], let us suppose that the representation fulﬁlls the following additional Positivity requirement. A representation is called positive if the following hold: (a) if |ψ ∈ Kernel Q0 , then ψ|ψ ≥ 0, and (b) if |ψ ∈ Kernel Q0 , then ψ|ψ = 0 if and only if |ψ ∈ Image Q0 . It is elementary to see that if the positivity requirement is fulﬁlled, then the representation π0 induces a representation π ˆ0 of the physical observables Fˆ0 on the inner product space ˆ 0 = Kernel Q0 , (278) H Image Q0 which is in fact seen to be a pre-Hilbert space, i.e. carries a positive deﬁnite inner product. As we will see below, when G is compact, there do indeed exist representations satisfying the above positivity requirement if we restrict ourselves to the ghost number 0 subalgebra of F0 . As we will also see, in static spacetimes (M, g) ˆ 0 (in the ground state represenor in spacetimes with static regions, the states in H tation) can be put into one-to-one correspondence with ±-helicity particle states of ˆ 0 contains a dense set of Hadamard states. However, the electromagnetic ﬁeld, and H in generic time-dependent spacetimes, such an interpretation in terms of particles states is not possible. When the Cauchy surfaces of M are not compact, the charge Q0 in general cannot be deﬁned as stated. The reason is that the 1-form ﬁeld γ is no longer of compact support, but has non-compact support in spatial directions. Nevertheless, we can see that if we formally consider the graded commutator [Q0 , O(x)] with a local quantum Wick-power, denoted O(x) := T1 (O(x)), then there will be only contributions in the formal integral deﬁning Q0 (see (276)) from the portion of the support of γ that is contained in J + (x) ∪ J − (x). All other contributions vanish due to the (graded) commutativity property, T9. Since the intersection of the support

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of γ and J + (x) ∪ J − (x) is compact for a suitable choice of γ, it follows that the commutator of any local observable in F0 with Q0 is always deﬁned. Thus, while Q0 itself is undeﬁned, the graded commutator still deﬁnes a graded derivation. The deﬁnition of the algebra of gauge invariant observables can then be given in terms of this graded derivation. Unfortunately, the construction of representations explicitly uses (the representer of) Q0 itself, and not just the graded commutator. Thus, it is not straightforward to obtain Hilbert space representations on manifolds with non-compact Cauchy surfaces. 4.1.2. Interacting ﬁelds A similar kind of construction as for free Yang–Mills theory can also be given for interacting Yang–Mills theory. The starting point is now the classical auxiliary interacting ﬁeld theory described by the auxiliary action S = S0 + λS1 + λ2 S2 . Thus, the interaction is (279) I = (λL1 + λ2 L2 ) = λS1 + λ2 S2 . The ﬁrst step is to construct a quantum theory associated with this auxiliary action. For simplicity, we again assume that M has compact Cauchy-surfaces — the general situation can again be treated by complete analogy with the free ﬁeld case as just described. Following the general procedure described in Sec. 3.7, we ﬁrst introduce an infra-red cutoﬀ for the interaction, supported in a compact region of spacetime, and construct the interacting theory in that region. To deﬁne the desired infrared cutoﬀ, we consider a compactly supported cutoﬀ function, f , which is equal to 1 on the submanifold Mτ = (−τ, τ ) × Σ. We deﬁne a cutoﬀ interaction, F , by F = {f λL1 + f 2 λ2 L2 }, and we deﬁne corresponding interacting ﬁelds OF by Bogoliubov’s formula. We then send the cutoﬀ τ to inﬁnity at the algebraic level as described in Sec. 3.7, and get a corresponding algebra FI of interacting ﬁelds OI . This algebra of interacting ﬁelds is not equivalent to quantum Yang–Mills theory, as it contains gauge variant ﬁelds and ﬁelds of non-zero ghost number. As in the free case, we obtain the algebra of physical ﬁeld observables by considering the cohomology of the (now interacting) BRST-charge operator, QI . To deﬁne this object, consider the interacting BRST-current with cutoﬀ interaction, deﬁned by the Bogoliubov formula [see Eq. (263)] R δ iF/+ γ∧J iF/ T (e⊗ )−1 T (e⊗ )|γ=0 J(x)F = δγ(x) 1 i n = Rn (J(x); F ⊗n ). (280) n! n≥0

As in our general deﬁnition of interacting ﬁelds, we can then remove the cutoﬀ at the algebraic level and obtain an interacting current J(x)I . We will show below that the in M , so we can deﬁne a corresponding interacting BRST-current JI is conserved interacting BRST-charge by QI = γ ∧ JI , (compare with Eq. (276)).

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We will furthermore show that the so-deﬁned charge is nil-potent, Q2I = 0. Thus, we can deﬁne the physical observables as in the free ﬁeld theory by the cohomology of the interacting BRST-charge, i.e. the algebras of interacting ﬁelds are deﬁned by Kernel [QI , .] ∩ FI ∩ Kernel Ng . FˆI = Image [QI , .] ∩ FI ∩ Kernel Ng

(281)

Next, one would like to deﬁne representations of the algebra of observables on a Hilbert space. Such representations can be obtained from those of the free theory by a deformation process [42]. For this, consider a state |ψ0 ∈ H0 in a representation π0 of the underlying free theory satisfying the above positivity requirement. Let also |ψ0 ∈ Kernel Q0 . Then, using Q2I = 0, and QI = Q0 + λQ1 + λ2 Q2 + · · · one ﬁrst shows that there exists a formal power series |ψI = |ψ0 + λ|ψ1 + λ2 |ψ2 + · · · ∈ HI = H0 [[λ]]

(282)

such that QI |ψI = 0, where QI has been identiﬁed with its representer in the representation πI that is induced from the representation of the underlying free theory. In order to construct the vectors |ψi , we proceed inductively. We write the condition that |ψI is in the kernel of QI and that Q2I = 0 as 0=

m

Qk |ψm−k ,

0=

k=0

m

Qk Qm−k ,

(283)

k=0

for all m. For m = 0, the ﬁrst equation is certainly satisﬁed, as we are assuming Q0 |ψ0 = 0. Assume now that |ψ0 , |ψ1 , . . . , |ψn−1 have been constructed in such a way that the ﬁrst equation is satisﬁed up to m = n − 1, and put |χm =

n−1

Qm−k |ψk .

(284)

k=0

Then, using the second equation in (283), we see that 0=

m k=0

Qm−k |χk ,

0=

m

χm |χm−k ,

(285)

k=0

for all m. We now use the inductive assumption that |χm = 0 for m ≤ n − 1, from which we get that Q0 |χn = 0, putting m = n in the ﬁrst equation. Putting m = 2n in the second equation, we get χn |χn = 0. In view of the positivity requirement, we must thus have |χn = −Q0 |ψn for some |ψn . We take this as the deﬁnition of the nth term for the deformed state (282). This then satisﬁes the induction assumption at order n, thus closing the induction loop. Thus, by the above deformation argument, one sees that Kernel QI ⊂ HI is a non-empty subspace. One furthermore shows that the representation πI satisﬁes an

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analog of the positivity requirementl for the interacting theory. Thus, we obtain, as in the free case, a representation π ˆI on the inner product space ˆ I = Kernel QI , H (286) Image QI and this space is again shown to be a pre-Hilbert space. For details of these constructions, see [42, Sec. 4.3]. 4.1.3. Operator product expansions and RG-ﬂow As we have just described, a physical gauge invariant, interacting ﬁeld is an element in the algebra Fˆ0 , i.e. an equivalence class of an interacting ﬁeld operator OI (x) satisfying [QI , OI (x)] = 0 ∀x ∈ M,

(287)

modulo the interacting ﬁelds that can be written as OI (x) = [QI , OI (x)]

∀x ∈ M,

(288)

for some local ﬁeld O (as usual, [, ] means the graded commutator). Our constructions of the interacting BRST-charge do not imply that the action of QI on a local covariant interacting ﬁeld is equivalent to sˆ (cf. Eq. (60)). But it follows from general arguments that q O)I (x) [QI , OI (x)] = (ˆ

∀x ∈ M

(289)

where qˆ is a map qˆ : Pp (M ) → Pp (M )[[]],

qˆ = sˆ + ˆ q1 + 2 qˆ2 + · · · .

(290)

Q2I

= 0, the map qˆ is again a diﬀerential (the “quantum BRSTBecause diﬀerential”), qˆ2 = 0, whose action on general elements in P is diﬀerent from that of sˆ. An exception of this rule are the exactly gauge invariant elements O = Ψ at / zero ghost number, which by Lemma 1 are of the form Ψ = Θsi (F, DF, D2 F, . . .), with Θs invariant polynomials of the Lie-algebra. For such elements, we shall show that we have qˆΨ = sˆΨ = 0. Thus, [QI , ΨI (x)] = 0

∀x ∈ M

(291)

and the corresponding interacting ﬁelds ΨI (x) are always observable. Given n local ﬁelds Oj1 , . . . , Ojn ∈ P, we can construct the operator product expansion of the corresponding interacting quantum ﬁelds, Cjk1 ···jn (x1 , . . . , xn , y)Ok (y)I . (292) Oj1 (x1 )I · · · Ojn (xn )I ∼ k

The operator product expansion is an asymptotic expansion for x1 , . . . , xn → y, see [72], where the construction and properties of the expansion are described. l Since we are working over the ring C[[λ]] of formal power series in λ in the case of interacting Yang–Mills theory, the positivity requirement needs to be formulated appropriately by specifying what it means for a formal power series to be positive. For details, see [42].

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Because the action S of the auxiliary theory has zero ghost number, the OPE coeﬃcients are non-vanishing only when Ng (Ojr ) = Ng (Ok ). (293) r

Now assume that all operators Oj1 , . . . , Ojn are physically observable ﬁelds. Then, since the graded commutator with QI respects the -product, also all local operators Ok appearing on the right-hand side must be in the kernel of QI . By the same argument, if one of the operators on the left-hand side is of the trivial from (288), then it follows that each operator in the expansion on the right-hand side is of that form, too. Thus, we conclude that the OPE closes on gauge invariant operators, and we summarize this important result as a theorem: Theorem 3. Let Oi1 , . . . , Oin ∈ P be in the kernel of sˆ, with vanishing ghost number, as characterized by Theorem 1. Then Cik1 ···in is non-vanishing only for Ok ∈ P of vanishing ghost number that are in the kernel of sˆ. If one Oir is in the image of sˆ, then Cik1 ···in is non-vanishing only for Ok ∈ P of vanishing ghost number that are in the image of sˆ. If one drops the restriction to the 0-ghost number sector, then the same statement is true with sˆ replaced by qˆ. By the same kind of argument, one can also show that the renormalization group ﬂow closes on physical operators. The renormalization ﬂow in curved spacetime was deﬁned in Sec. 3.7 as the behavior of the interacting ﬁelds under a conformal change of the metric, g → µ2 g. In general we have ρµ (Oi (x)I (x)) = Zij (µ) · Oj (x)Iµ for all x ∈ M , where Iµ is the renormalized interaction, and where ρµ : FI (g) → FIµ (µ2 g) is an algebraic isomorphism implementing the conformal change of the metric. Now, in the perturbative quantum ﬁeld theory associated with the auxiliary action S, we have ζi (µ) · Oi (x)Iµ ∀x ∈ M, (294) ρµ (J(x)I ) = Z(µ) · J(x)Iµ + i

for some Z(µ), ζi (µ) ∈ C[[λ, ]], and operators Oi ∈ P3 (M ) of dimension three, not equal to the BRST-current and not equal to 0. If we take the exterior derivative d of this equation and use that the interacting BRST-currents themselves are conserved, # we obtain ζi (µ) · dOi (x)Iµ = 0. Let k be the largest natural number such that ζi (µ) is of order k for all i, and let zi (µ) be the k -contribution to ζi (µ). We can then divide this relation by k , and take the classical limit → 0. Because the classical limit of the interacting ﬁelds gives the corresponding perturbatively deﬁned classical interacting ﬁelds and because Iµ → I as → 0, it follows that # zi (µ) · dOi (x)I = 0 for the corresponding on-shell classical interacting ﬁelds. This means that dOi (x)I = 0 for those i such that zi (µ) = 0. But there are no such 3-form ﬁelds of dimension three at the classical level by the results of [5] except for the zero ﬁeld and the BRST-current. Thus, we have found that zi (µ) = 0 for all i. By repeating this type of argument for the higher orders in in ζi (µ), we can conclude that ζi (µ) = 0 to all orders in .

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Thus, we have found that BRST-current does not mix with other operators under the renormalization group ﬂow, from which it follows that ρµ (QI ) = Z(µ) · QIµ .

(295)

Hence, if [QI , Oi (x)I ] = 0 for all x ∈ M , then, by applying ρµ to this relation, it also follows that Zji (µ)[QIµ , Oi (x)Iµ ] = 0.

(296)

Because Zji (µ) is invertible (it is a formal power series in λ starting with δji ), we thus obtain the following result, which states that the RG-ﬂow does not leave the sector of physical observables: Theorem 4. Let Oi ∈ P be in the kernel of sˆ, with vanishing ghost number, as characterized by Theorem 1. Then Zij (µ) is non-vanishing only for Oj ∈ P of vanishing ghost number that are in the kernel of sˆ. If Oi is in the image of sˆ, then Zij (µ) is non-vanishing only for Oj ∈ P of vanishing ghost number that are in the image of sˆ. If one drops the restriction to the 0-ghost number sector, then the same statement is true with sˆ replaced by qˆ. Remark. An interesting corollary to this theorem arises when one considers the particular case when O is the Yang–Mills Lagrangian. Since it is the only gauge invariant ﬁeld at ghost number 0 of this dimension, it does not mix with other ﬁelds up to QI -exact terms under the renormalization group ﬂow. The corresponding constant ZI (µ) describing the ﬁeld renormalization corresponding to the Yang– Mills Lagrangian then deﬁnes the ﬂow of the coupling constant λ. Since our ﬂow is local and covariant, it follows that this ﬂow automatically must be exactly the same as in Minkowski spacetime! A similar remark would apply to more complicated gauge theories with additional matter ﬁelds, as long as there cannot arise any additional couplings to curvature of engineering dimension 4 (such as, e.g., R Tr Φ2 if the gauge ﬁeld is coupled to a scalar ﬁeld Φ in some representation of the gauge group). Even if there can arise such couplings, the above argument can still be used to directly infer the vanishing of all β-functions in curved spacetimes with R = 0 if the corresponding β-functions vanish in ﬂat spacetime. 4.2. Free gauge theory We now describe in more detail the construction of free gauge theory outlined in the previous Sec. 4.1. As explained, our starting point is the auxiliary theory that is classically described by the free action S0 . The ﬁrst step is to deﬁne a suitable deformation quantization algebra W0 for this theory. The theory contains the dynamical ﬁelds Φ = (AI , B I , C I , C¯ I ), as well as the background ﬁelds Φ‡ = (A‡I , BI‡ , CI‡ , C¯I‡ ). Of the dynamical ﬁelds, B I is only an auxiliary ﬁeld with no kinetic term in S = 0, while the vector ﬁeld AI and the ghost ﬁelds C I , C¯ I were

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quantized above in Sec. 3.5. Thus, the desired W0 is essentially a tensor product of the algebras for the vector and ghost ﬁelds. We now describe the construction in detail. We ﬁrst consider a vector Hadamard 2-point function ω v (x, y), and a scalar Hadamard 2-point function ω s (x, y). These quantities by deﬁnition satisfy the hyperbolic equations (dδ + δd)x ω v (x, y) = 0 = (dδ + δd)y ω v (x, y),

(dδ)x ω s (x, y) = 0 = (dδ)y ω s (x, y), (297)

the commutator property (116), and the wave front condition (119). Below, we will show that we can always choose them so that they additionally satisfy the consistency relation dx ω s (x, y) = −δy ω v (x, y),

dy ω s (x, y) = −δx ω v (x, y),

(298)

where dx = dxµ ∧ ∂x∂ µ , and where δx = ∗dx ∗ is the co-diﬀerential, etc. We deﬁne the desired deformation quantization algebra W0 to be the vector space generated by formal expression of the form ··· kn F (u) = uki11··· im (x1 , . . . , xn ; y1 , . . . , ym ) : Φi1 (y1 ) · · · Φim (ym )Φ‡k1 (x1 ) · · · Φ‡kn (xn ) :ω ,

(299)

where u is a distribution subject to the wave front set condition (126) in the variables y1 , . . . , ym , but not subject to any wave front set condition in the variables x1 , . . . , xn . We deﬁne the -product to be given by the diﬀerential operator δL δR ωjk (x, y) dxdy (300) < D> = δΦk (x) δΦj (y) where j, k = (AI , B I , C I , C¯ I ), and where   −iδy ω v (x, y) 0 0 ω v (x, y)   −iδx ω v (x, y)  0 0 0 . (ωjk (x, y)) = (kIJ ) ⊗   s 0 0 0 iω (x, y)   0 0 0 −iω s (x, y) (301) Our deﬁnitions imply the commutation relations (239), (246) (with obvious modiﬁcation to accommodate the Lie-algebra indices on the ﬁelds AI , C I , C¯ I ), as well as AI (x) B J (y) − B J (y) AI (x) = k IJ δy ∆v (x, y)1.

(302)

The commutators of all other ﬁelds, in particular those involving any of the background ﬁelds A‡I , BI‡ , CI‡ , C¯I‡ , vanish. In this sense the background ﬁelds are Cnumbers, and their product is not deformed. This completes our construction of the quantization algebra W0 of free gauge theory.

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The next step is to deﬁne within W0 the Wick products and time-ordered products satisfying conditions T1–T11. As for the time-ordered products with one factor, we make the same deﬁnition as in the scalar case, with the only diﬀerence that H is replaced by the matrix valued Hadamard parametrix   −iδy H v (x, y) 0 0 H v (x, y)   −iδx H v (x, y)  0 0 0 , (Hjk (x, y)) = (kIJ ) ⊗   s 0 0 0 iH (x, y)   0 0 −iH s (x, y) 0 (303) where j, k = (AI , B I , C I , C¯ I ). Using the Hadamard parametrix, the time-ordered products T1 (O) with one factor O ∈ P are deﬁned by complete analogy with the scalar case, and they satisfy T1–T11. In particular, it follows from the deﬁnition that the Wick product T1 (J0 ) of the free BRST-current (77) is conserved, dT1 (J0 ) = T1 (dJ0 ) = 0 (modulo J0 ). Hence, we can deﬁne a conserved BRST-charge (when the Cauchy surfaces are compact, see above). It also follows directly from the relations in the algebra W0 that Q20 = 0 modulo J0 . Thus, we can deﬁne the algebra of physical observables, Fˆ0 , by the cohomology of Q0 as explained in the previous section. It follows from the Ward identity (c) below that if O ∈ P is a classically gauge / invariant polynomial expression in AI , i.e. O = ∇si dAIi (so that in particular sˆ0 O = 0), then the corresponding Wick power T1 (O) is in the kernel of Q0 under the graded commutator. Thus, at ghost number 0, the algebra contains all local covariant quantum Wick powers of classically gauge invariant observables. Thus, it only remains to prove the existence of Hadamard 2-point functions ω s , ω v satisfying Eq. (298), and to prove that the algebra Fˆ0 has sensible Hilbert space representations. Both statements will now be proved by appealing to a deformation argument, as originally proposed by Fulling, Narcowich and Wald [52], and generalized by Fewster and Pfenning [49] to Maxwell ﬁelds on ultra-static spacetimes. That construction only works for spacetimes M = Σ×R with Σ compact and simply connected (i.e. H 1 (Σ, dΣ ) = 0), which is a physically reasonable assumption in view of the topological censorship theorem [56], and which we shall assume here. Consider, besides the original spacetime, (M, g), an auxiliary deformed asympˆ , gˆ). By this we mean that both spacetimes are identical totically static spacetime (M to the future of some Cauchy surface Σ × {t+ }, and that gˆ is “ultrastatic” to the past of some Cauchy surface Σ × {t− }, meaning that gˆ has the form gˆ = −dt2 + h(dx, dx)

(304)

there, where h = hij dxi dxj is a Riemannian metric on Σ that does not depend upon t. The idea of the deformation argument is now as follows. First, construct ˆ , gˆ). ˆ v ) satisfying the desired Eq. (298) in the ultrastatic part of (M a pair (ˆ ωs , ω Then, because d and δ intertwine the action of the wave operators δd on 0-forms ˆ v ) are bisolutions to the respective wave and dδ + δd on 1-forms, and since (ˆ ωs, ω

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ˆ , gˆ), equations (297), the desired equations (298) must therefore hold on all of (M and not just on the ultrastatic part. Furthermore, one can show [85] using the celebrated “propagation of singularities theorem” [31] (see Appendices C and E) ˆ v ) satisﬁes the desired wave front set condition (119) on all of that the pair (ˆ ωs , ω ˆ , gˆ) if they are satisﬁed in the ultrastatic part. In particular, on the part of (M ˆ , gˆ) (M s v ˆ ) with identical to (M, g), we then have a pair of Hadamard bi-distributions (ˆ ω ,ω ˆ , gˆ) identical with (M, g) ˆ v ) on the part of (M the desired properties. The pair (ˆ ωs, ω may now be propagated to a solution (ω s , ω v ) of the hyperbolic equations (297) on the undeformed spacetime (M, g). By the same arguments as above, this will now have a wave front set of Hadamard form on the undeformed spacetime, and it will satisfy the desired equation (298). ˆ v ) satisfying (298), the Thus, we need only prove the existence of a pair (ˆ ωs, ω Hadamard condition (119), the commutator property, and ﬁeld equations (297) on ˆ , gˆ). This can be shown as follows using the following an ultrastatic spacetime (M construction [49], which in turn builds on results of [82]: On the three-dimensional compact Riemannian spacetime (Σ, h), we consider a complete set of eigenfunctions of the corresponding scalar Laplace-operator ∆h = δΣ dΣ , ∆h ϕk = −ν(S, k)2 ϕk ,

(305)

with positive eigenvalues ν(S, k)2 , labeled by an index k ∈ J(S) in a corresponding ˆ and uk (t, x) = eiν(S,k)t ϕk (x), and one deﬁnes index set. One deﬁnes x = (t, x) ∈ M the “scalar” and “longitudinal” mode 1-forms on M by AS,k (t, x) = uk (t, x)dt AL,k (t, x) =

1 duk (t, x) + iuk (t, x)dt, ν(L, k)

(306) (307)

with ν(L, k) = ν(S, k). These mode functions are smooth by elliptic regularity. One next chooses an orthonormal set of eigenmodes for the Laplacian ∆h = dΣ δΣ +δΣ dΣ on (Σ, h) acting on 1-forms. By the Hodge decomposition theorem (see, e.g., [15]), using H 1 (Σ, dΣ ) = 0, these can be uniquely decomposed into ones in the image of δΣ and those in the image of dΣ . We denote those in the image of δΣ by ξk and their eigenvaluesm by −ν(T, k)2 , where k is now an index from a set J(T ). We deﬁne the corresponding “transversal” mode 1-forms on M by AT,k (t, x) = eiν(T,k)t ξk (x)

(308)

and we deﬁne the vector Hadamard two-point distribution on the ultra-static spacetime by s(λ) Aλ,k (x)Aλ,k (y) (309) ω ˆ v (x, y) = − 2ν(λ, k) λ k∈J(λ)

where s(S) = 1, s(L) = −1 = s(T ), and λ ∈ {S, L, T }. It was proved in [49] that this is of Hadamard form and that it has the desired commutator property. We m Note

that the scalar and transversal eigenvalues need not coincide.

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deﬁne the scalar Hadamard two-point distribution on the ultra-static spacetime by 1 uk (x)uk (y). (310) ω ˆ s (x, y) = 2ν(S, k) k∈J(S)

It was shown by [82] that this is of Hadamard form and that it satisﬁes the desired commutator property. The desired consistency property (298) on the ultrastatic spacetime follows by going through the deﬁnitions. Thus, by the deformation argument, we obtain from this a pair (ω v , ω s ) on the undeformed spacetime satisfying also the desired consistency condition (298). We must ﬁnally construct a Hilbert space representation of the algebra F0 = W0 /J0 that gives rise to a corresponding representation of the algebra of physical ˆ , gˆ), we observables (277) on the factor space (278). On an ultrastatic spacetime (M construct a representation as follows. We let hb be the 1-particle indeﬁnite inner product space spanned by the orthonormal basis elements eI,λ,k , with λ = S, L, T and k ∈ J(λ), with indeﬁnite hermitian inner product deﬁned by (eI,λ,k , eI ,λ ,k ) = s(λ)kII δλλ δkk . We let Fb =

∞ n

hb

(311)

n=0

be the corresponding (indeﬁnite metric) standard bosonic Fock space, with basis vectors 1 eIπ1 λπ1 kπ1 ⊗ · · · ⊗ eIπn λπn kπn (312) |I1 λ1 k1 , . . . , In λn kn = n! π∈Sn

and we let vectors, i.e.

a+ I,λ,k

be the standard creation operators associated with the basis

a+ J,ν,p |I1 λ1 k1 , . . . , In λn kn = |Jνp, I1 λ1 k1 , . . . , In λn kn .

(313)

We let hf be the 1-particle indeﬁnite inner product space spanned by the orthonormal basis elements fI,±,k and k ∈ J(S), with indeﬁnite hermitian inner product deﬁned by (fI,s,k , fI ,s ,k ) = iss kII δkk , where ss is the anti-symmetric tensor in two dimensions. We let ∞ n (314) hf Ff = n=0

be the corresponding (indeﬁnite metric) standard fermionic Fock space, with basis vectors 1 |I1 s1 k1 , . . . , In sn kn = sgn(π)fIπ1 sπ1 kπ1 ⊗ · · · ⊗ fIπn sπn kπn (315) n! π∈Sn

and we let c+ I,s,k be the standard creation operators associated with the basis vectors, i.e. c+ J,r,p |I1 s1 k1 , . . . , In sn kn = |Jrp, I1 s1 k1 , . . . , In sn kn .

(316)

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The (indeﬁnite) metric space H0 is deﬁned as the tensor product H0 = Fb ⊗ Ff . We now deﬁne the representatives of the ﬁelds Φ = (AI , B I , C I , C¯ I ) as linear operators on H0 by 1 2 Aλ,k (x)a+ π0 (AI (x)) = (317) I,λ,k + h.c., 2ν(λ, k) λ k∈J(λ) π0 (C I (x)) = π0 (C¯ I (x)) =

1 2 uk (x)c+ I,+,k + h.c., 2ν(S, k) k∈J(S)

1 2 uk (x)c+ I,−,k + h.c. 2ν(S, k) k∈J(S)

(318)

(319)

We deﬁne the representative π0 (B I (x)) to be −iπ0 (δAI (x)), and we deﬁne the representative of any anti-ﬁeld Φ‡ to be zero. Finally, we deﬁne the representative of any element F (u) of the form (299) by applying a normal ordering on the representatives (all creation operators to the left or all annihilation operators). The two-point functions of the vector- and ghost ﬁelds are then precisely given by ω ˆv, s respectively, by ω ˆ . As in ﬂat spacetime, it may next be checked that, for compact G (i.e. positive deﬁnite Cartan–Killing form kIJ ) and in the ghost number 0 sector, the positivity requirement of Sec. 4.2 is fulﬁlled. Thus, the physical Hilbert space (278) inherits a positive deﬁnite inner product. Furthermore, it follows from the consistency condition (298) that it contains precisely excitation of the longitudinal modes (308). In a general, non-static spacetimes, a similar construction can be applied by promoting the mode functions Aλ,k , uk to solutions of the corresponding wave equation on the spacetime (M, g) by a deformation argument as above. We expect a similar construction to work in the case when H 1 (Σ, dΣ ) = 0, the only diﬀerence being the addition of corresponding zero modes to the mode expansions. We also expect a similar argument to work in spacetimes with noncompact Cauchy-surface, but it appears that this requires more work in general. 4.3. Interacting gauge theory In this section, we describe in detail how the general construction of interacting Yang–Mills theory outlined in Sec. 4.1 is performed. To construct perturbatively the interacting ﬁelds in interacting gauge theory, we need to construct the timeordered products in the free theory considered in the previous subsection. For timeordered products with 1 factor, this was done there. For time-ordered products with n factors, this can be done as described in Sec. 3, and these time-ordered products will satisfy the analog of conditions T1–T11. However, in gauge theory, the time-ordered products must satisfy further constraints related to gauge invariance. As we have argued in Sec. 4.2, in the gauge ﬁxed formalism, we need to be able to deﬁne an interacting BRST-charge operator, QI , and we need that operator to be nilpotent, i.e. Q2I = 0. In order to meaningfully construct QI , we need a conserved interacting BRST-current JI . If our time-ordered

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products only satisfy T1–T11 (with the symmetry property T6 replaced by graded symmetry with respect to the Grassmann parity), then there is in general no guarantee that the interacting BRST-current is conserved, dJI = 0, nor that Q2I = 0, nor that [QI , ΨI ] = 0 for strictly gauge invariant operators Ψ of ghost number 0. We will now formulate a set of Ward identities in the free theory that will guarantee that these conditions are satisﬁed, and which moreover will guarantee (formally) that the S-matrix — when it exists — is BRST-invariant. As argued in the previous section, with such a deﬁnition of time-ordered products, the conditions of gauge invariance of the perturbative interacting quantum ﬁeld theory are then satisﬁed. The Ward identities that we want to propose are to be viewed as an additional normalization condition on the time-ordered product, and are as follows. Consider a local operator O ∈ P, given by an expansion of the form O = O0 + λO1 + · · · + λN ON .

(320)

Let f be a smooth compactly supported test function on M , and let [O0 + λf O1 + · · · + λN f N ON ].

F =

(321)

M

Then the Ward identity that we will consider is iF/

[Q0 , T (e⊗

1 iF/ )] = − T ((S0 + F, S0 + F ) ⊗ e⊗ ) modulo J0 . 2

(322)

Here, Q0 is the free BRST-charge operator, (., .) is the anti-bracket (58), and [., .] is the graded commutator in the algebra W0 . As with all generating type formulae in this work, this is to be understood as a shorthand for the hierarchy of identities that are obtained when the above expression is expanded as a formal power series in λ. We now write out explicitly this hierarchy of identities. For this, it is convenient to introduce some notation. We denote by I = {k1 , . . . , kr } subsets of n = {1, . . . , n}, and we write r = |I| for the number of elements. We set XI = (xk1 , . . . , xkr ), and we put Or (XI ) = r!Or (xk1 )δ(xk1 , . . . , xkr ). With these notations, the Ward identity (322) can be expressed as I1 ∪···∪It =n

= −

t i [Q0 , Tt (O|I1 | (XI1 ) ⊗ · · · ⊗ O|It | (XIt ))]

I1 ∪···∪It =n

t−1 t i (−1)εk Tt (O|I1 | (XI1 ) k=1

⊗ · · · ⊗ sˆ0 O|Ik | (XIk ) ⊗ · · · ⊗ O|It | (XIt ))

(323)

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I1 ∪···∪It =n

t−2 i

1115

(−1)εk εl Tt−1 (O|I1 | (XI1 )

1≤k
⊗ · · · ⊗ (O|Ik | (XIk ), O|Il | (XIl )) ⊗ · · · ⊗ O|It | (XIt ))

(324)

modulo J0 , where εk = ε(O1 ) + · · · + ε(Ok−1 ). We will not prove the above Ward identities for arbitrary operators O in this work, but only for certain special cases, which are relevant for our analysis of gauge invariance. These cases are (T12a) O is given by the interaction Lagrangian, O = λL1 + λ2 L2 , (T12b) O is given by a linear combination of the interaction Lagrangian, and the BRST-current O = λL1 + λ2 L2 + γ ∧ (J0 + λJ1 ) (evaluation of the Ward identity to ﬁrst order in γ ∈ Ω10 (M )). # (T12c) O = λL1 + λ2 L2 + γ ∧ λk Ψk ∈ P4 (M ) is given by a linear combination of the interaction Lagrangian and a strictly gauge invariant operator Ψ = # k p k λ Ψk ∈ P (M ) of ghost number 0, i.e. of the form given by Eq. (47) (M )). (evaluation of the Ward identity to ﬁrst order in γ ∈ Ω4−p 0 It is only for those cases that we will prove the Ward identities (324), and that proof is provided in Sec. 4.4. For convenience, we now give explicitly the form of the Ward identities in the cases (a)–(c). Case (T12a). The Ward identities in that case are given explicitly by t 4 i 3 Q0 , Tt (L|I1 | (XI1 ) ⊗ · · · ⊗ L|It | (XIt )) I1 ∪···∪It =n

=−

I1 ∪···∪It =n

t−1 t i Tt (L|I1 | (XI1 ) k=1

⊗ · · · ⊗ sˆ0 L|Ik | (XIk ) ⊗ · · · ⊗ L|It | (XIt )) t−2 i − Tt−1 (L|I1 | (XI1 ) I1 ∪···∪It =n

1≤j
⊗ · · · ⊗ (L|Ij | (XIj ), L|Ik | (XIk )) ⊗ · · · ⊗ L|It | (XIt )), modulo J0 . Case (T12b). The Ward identities in that case are given explicitly by t−1 3 4 i Q0 , Tt (J|I1 | (y, XI1 ) ⊗ L|I2 | (XI2 ) ⊗ · · · ⊗ L|It | (XIt )) I1 ∪···∪It =n

=

I1 ∪···∪It =n

t−2 t i Tt (J|I1 | (y, XI1 ) ⊗ L|I2 | (XI2 ) i=2

⊗ · · · ⊗ sˆ0 L|Ii | (XIi ) ⊗ · · · ⊗ L|It | (XIt ))

(325)

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−

I1 ∪···∪It =n

+

I1 ∪···∪It =n

t−2 i Tt (ˆ s0 J|I1 | (y, XI1 ) ⊗ L|I2 | (XI2 ) ⊗ · · · ⊗ L|It | (XIt )) t−3 i

Tt−1 (J|I1 | (y, XI1 ) ⊗ L|I2 | (XI2 )

2≤i<j≤t

⊗ · · · ⊗ (L|Ii | (XIi ), L|Ij | (XIj )) ⊗ · · · ⊗ L|It | (XIt )) t−2 i − Tt−1 (L|I2 | (XI2 ) I1 ∪···∪It =n

2≤i≤t

⊗ · · · ⊗ (J|I1 | (y, XI1 ), L|Ii | (XIi )) ⊗ · · · ⊗ L|It | (XIt )),

(326)

modulo J0 . Here J1 (y, x) = J1 (y)δ(x, y). Case (T12c). Let Ψ = Ψ0 + λΨ1 + · · · + λN ΨN be a strictly gauge invariant local ﬁeld polynomial of ghost number zero. Thus, by formula (47), up to local / curvature terms which we may ignore, Ψ = Θsi (F, DF, D2 F, . . .), where Θs are invariant polynomials of the Lie-algebra. The Ward identities in that case are given explicitly by t−1 3 4 i Q0 , Tt (Ψ|I1 | (y, XI1 ) ⊗ L|I2 | (XI2 ) ⊗ · · · ⊗ L|It | (XIt )) I1 ∪··· ∪It =n

=−

I1 ∪··· ∪It =n

t−2 t i Tt (Ψ|I1 | (y, XI1 ) ⊗ L|I2 | (XI2 ) i=2

⊗ · · · ⊗ sˆ0 L|Ii | (XIi ) ⊗ · · · ⊗ L|It | (XIt )) t−2 i − Tt (ˆ s0 Ψ|I1 | (y, XI1 ) ⊗ L|I2 | (XI2 ) ⊗ · · · ⊗ L|It | (XIt )) I1 ∪··· ∪It =n

−

I1 ∪··· ∪It =n

t−3 i

Tt−1 (Ψ|I1 | (y, XI1 ) ⊗ L|I2 | (XI2 )

2≤i<j≤t

⊗ · · · ⊗ (L|Ii | (XIi ), L|Ij | (XIj )) ⊗ · · · ⊗ L|It | (XIt )) t−2 t i Tt−1 (L|I2 | (XI2 ) − i=2 I1 ∪··· ∪It =n

⊗ · · · ⊗ (Ψ|I1 | (y, XI1 ), L|Ii | (XIi )) ⊗ · · · ⊗ L|It | (XIt )),

(327)

modulo J0 . We will give a proof of the Ward identities T12a–T12c in Sec. 4.4. We will then show in Sec. 4.6 that the Ward identities T12a imply the conservation of the interacting BRST-current, dJI = 0. We will prove in Sec. 4.7 that the Ward identities T12b furthermore imply that Q2I = 0 and we will show in Sec. 4.8 that

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the Ward identities T12c imply [QI , ΨI ] = 0 for strictly gauge invariant operators Ψ at ghost number 0. The Ward identity T12a also formally implies the BRSTinvariance of the S-matrix (see Sec. 4.5), provided the latter exists (which is not the case in Minkowski space, and appears even more unlikely in curved spacetime). We will not analyze this existence question here, so in this sense the BRSTinvariance of the S-matrix is not a rigorous result unlike the other results in our paper. As an aside, we note that, the Ward identities T12a, T12b, and T12c are incompatible with the identity [Q0 , Tn (O1 (x1 ) ⊗ · · · ⊗ On (xn ))] n X = i (−1)εi Tn (O1 (x1 ) ⊗ · · · ⊗ sˆ0 Oi (xi ) ⊗ · · · ⊗ On (xn ))

mod J0

(WRONG!),

i=1

(328) unless none of the ﬁelds Oi contains anti-ﬁelds. The above identity has been considered before in the context of ﬂat spacetime in [39], where it has been termed “Master BRST-identity”. It appears that it is impossible to satisfy this identity (even for n = 1) when anti-ﬁelds are present. It would also not imply either the conservation of the interacting BRST current JI nor the nilpotency of the interacting BRST charge in a framework with anti-ﬁelds. Since the use of anti-ﬁelds also appears to be essential in order to derive suﬃciently strong constraints on potential anomalies to the BRST-Ward identities, we believe that Eq. (328) is not a good starting point for the proof of gauge invariance in perturbative Yang–Mills theory. 4.4. Inductive proof of Ward identities T12a, T12b, and T12c We now show that the Ward identities can be satisﬁed together with T1–T11 by making a suitable redeﬁnition of the time-ordered products if necessary. The Ward identity (324) is an identity modulo J0 , that is, it is required to hold only on shell. For the proof of that identity it is actually useful to consider a more stringent “oﬀ-shell” version of the identity. Even though that oﬀ-shell version is more stringent, it will in fact turn out to be easier to prove, as it gives, at the same time, stronger constraints of cohomological nature on the the possible anomalies than the corresponding on-shell version. To set up the oﬀ-shell version of our Ward identity, we ﬁrst recall the deﬁnition sˆ0 = s0 +σ0 of the free Slavnov–Taylor diﬀerential, given above in Eqs. (71) and (72). As it stands, the diﬀerential sˆ0 was deﬁned as a map sˆ0 : P(M ) → P(M ), i.e. it acts on polynomial expressions in the classical ﬁelds Φ, Φ‡ . We will now extend the action of sˆ0 to the non-commutative algebra W0 . For this, we recall that the algebra W0 may be viewed as the closure of the CCR-algebra W00 , which in turn is generated by expressions of the form F1 · · · Fn , where each Fi is given by fi ∧ Oi , with fi smooth and of compact support, and with Oi given by one of the

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“basic ﬁelds” Φ, Φ‡ . To deﬁne the action of sˆ0 on such elements of W00 , we set sˆ0 (O1 (x1 ) · · · Os (xn )) =

n

(−1)εi O1 (x1 ) · · · sˆ0 Oi (xi ) · · · On (xn ),

i=1

(329) where Oi is either a basic ﬁeld Φ, or an anti-ﬁeld Φ‡ . This deﬁnes the Slavnov– Taylor diﬀerential sˆ0 as a graded derivation (denoted by the same symbol) of the algebra W00 . As we have remarked, the subalgebra W00 ⊂ W0 is dense (in the H¨ ormander topology). Thus, we can uniquely extend sˆ0 to a graded derivation on W0 by continuity with respect to this topology. We will again denote this graded derivation sˆ0 : W0 → W0 by the same symbol. Actually, we must still check that the deﬁnition (329) is consistent, i.e. compatible with the algebra relations in W00 . We formulate this result as a lemma: Lemma 7. The formula (329) deﬁnes a graded derivation on W0 . Proof. The basic algebraic relations in W00 are the graded commutation relations [Φi (x), Φj (y)] = i∆ij (x, y)1,

[Φi (x), Φ‡j (y)] = 0 = [Φ‡i (x), Φ‡j (y)],

where ∆ij is the matrix of commutator functions given by  ∆v (x, y) −iδy ∆v (x, y) 0  v −iδx ∆ (x, y) 0 0 (∆jk (x, y)) = (kIJ ) ⊗   0 0 0  s 0 0 −i∆ (x, y)

(330)

0



  , s i∆ (x, y)  0 0

(331) where Φi = (AI , B I , C I , C¯ I ), and where ∆v , ∆s are the advanced minus retarded propagators for vectors and scalars, see Appendix E. To show that the deﬁnition of sˆ0 on W00 is consistent, we next apply the deﬁnition (329) to the above graded commutators and check that we get identities. This follows from the relations dx ∆s (x, y) = −δy ∆v (x, y),

dy ∆s (x, y) = −δx ∆v (x, y),

(332)

which in turn a direct consequence of the ﬁeld equations satisﬁed by the advanced and retarded propagators for scalars and vectors. We are now in a position to formulate the desired oﬀ-shell version of our (anomalous) Ward identity that will eventually enable us to prove T12a, T12b, and T12c. We formulate our result in a proposition: Proposition 2 (Anomalous Ward Identity). For a general prescription for time-ordered products satisfying T1–T11, the identity iF/

sˆ0 T (e⊗

)=

i i iF/ iF/ T ((S0 + F, S0 + F ) ⊗ e⊗ ) + T (A(eF ) ⊗ ) ⊗ e⊗ 2

(333)

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holds. Here F = f ∧ O is any smeared local ﬁeld with O ∈ Pp (M ), f ∈ Ω4−p (M ) 0 F and A(e⊗ ) is the anomaly, given by A(eF ⊗) =

1 An (F ⊗n ), n!

(334)

n≥0

where An : Pk1 (M )⊗· · ·⊗Pkn (M ) → Pk1 /···/kn (M n ) are local functionals supported on the total diagonal. The anomaly satisﬁes the following further properties: (i) A(eF ⊗ ) = O(). (ii) Each An is locally and covariantly constructed out of the metric. (iii) Each An has ghost number one, in the sense that Ng ◦ An − An ◦ Γn Ng = An , where Ng is the number counter for the ghost ﬁelds, see Eq. (50) (with additional terms for the anti-ﬁelds), and Γn Ng =

n

id ⊗ · · · ⊗ Ng ⊗ · · · ⊗ id : P⊗n → P⊗n .

(335)

i=1

(iv) Each An has dimension 0, in the sense that Nd ◦ An − An ◦ Γn Nd = 0, where Nd := Nf +Nr is the dimension counter, which is the sum of the dimensions Nf of the individual ﬁelds and anti-ﬁelds (see the tables above), and the dimensions Nr of the curvature terms. ∗ F∗ (v) The maps An are real in the sense that A(eF ⊗ ) = A(e⊗ ). Before we come to the proof of this key proposition, we note that, in the absence of anomalies, A(eF ⊗ ) = 0, the oﬀ-shell version of our Ward identity becomes iF/

sˆ0 T (e⊗

)=

i iF/ T ((S0 + F, S0 + F ) ⊗ e⊗ ). 2

(336)

The diﬀerence to (322) is that on the left-hand side, we do not have the graded commutator with Q0 , but instead we act with the Slavnov–Taylor map sˆ0 , which is the sum of the standard free BRST-diﬀerential s0 generated by Q0 , and the Koszul–Tate diﬀerential. The addition of the Koszul–Tate diﬀerential is crucial to obtain an identity that holds oﬀ shell, and not just modulo the free ﬁeld equations as Eq. (322). As already indicated, despite being more stringent, the sharpened oﬀ-shell Ward identity (336) is in fact simpler to prove than the corresponding on-shell identity (322), as it also allows one to derive more stringent consistency conditions on the possible anomalies. These consistency conditions rely in an essential way upon the use of the anti-ﬁelds, and this is the principal reason why we have introduced such ﬁelds in our construction. Proof of Proposition 2. The proof of the anomalous Ward identity (333) proceeds by induction in the order n in perturbation theory, noting that the anomalous Ward identity holds at order n if it holds up to order n − 1, modulo a contribution supported on the total diagonal. That contribution is deﬁned to be An . In more detail, consider n local functionals F1 , . . . , Fn with Fi = fi ∧ Oi , with fi a form

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of compact support and form degree complementary to that of Oi ∈ P(M ). For deﬁniteness and simplicity, we assume that all Fi have Grassmann parity 0; in the general case one proceeds similarly. The anomalous Ward identity (333) at order n is then the statement that n Tn (F1 ⊗ · · · ⊗ sˆ0 Fk ⊗ · · · ⊗ Fn ) sˆ0 Tn (F1 ⊗ · · · ⊗ Fn ) = k=0

Tn−1 (F1 ⊗ · · · ⊗ (Fj , Fk ) ⊗ · · · ⊗ Fn ) i k<j t−1 n + i t=1 +

k1 <···
× Tn−t+1 (At (Fk1 ⊗ · · · ⊗ Fkt ) ⊗ Fl1 ⊗ · · · ⊗ Fln−t ).

(337)

We now look at the individual terms in this expression. We decompose sˆ0 = s0 + σ0 into its pure BRST-part s0 and the Koszul–Tate diﬀerential σ0 . Letting εi be the Grassmann parity of fi (equal to that of Oi , since Fi is assumed to be bosonic), we have σ0 Tn (F1 ⊗ · · · ⊗ Fn ) P = σ0 (−1) i<j εi εj f1 (x1 ) · · · fn (xn )Tn (O1 (x1 ) ⊗ · · · ⊗ On (xn ))dx1 · · · dxn P

= (−1)

i<j

εi εj

n

P

(−1)

l
εl

k=1

× [f1 (x1 ) · · · σ0 fk (xk ) · · · fn (xn )] Tn (⊗i Oi (xi ))dx1 · · · dxn n P P εi εj i<j = (−1) (−1) l
δR S0 δL fk (xk ) · · · f × f1 (x1 ) · · · (x ) Tn (⊗i Oi (xi ))dydx1 · · · dxn n n δΦ(y) δΦ† (y) n δL Fk δR S0 Tn F1 ⊗ · · · ⊗ ⊗ · · · ⊗ Fn dy, = (338) δΦ(y) δΦ‡ (y) k=1

and we have n

Tn (F1 ⊗ · · · σ0 Fk ⊗ · · · Fn )

k=1

=

n k=1

δL Fk δR S0 ∧ ⊗ · · · ⊗ Fn dx, Tn F1 ⊗ · · · ⊗ δΦ(x) δΦ‡ (x)

(339)

using the deﬁnition of σ0 , see Eq. (72) and the following table. We may combine these two identities into the following identity for the corresponding generating

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functionals: iF/ σ0 T (e⊗ )

1121

δL F δ R S0 iF/ T ⊗ e⊗ dx δΦ(x) δΦ‡ (x) δR S0 δL F i iF/ ∧ dx. − T ⊗ e⊗ δΦ(x) δΦ‡ (x) (340)

i i iF/ − T (σ0 F ⊗ e⊗ ) =

To manipulate this expression, we now use a proposition formulated and proven ﬁrst in [17] (see Eq. (5.48) in Lemma 11 of this reference). Proposition 3 (“Master Ward Identity”). Let ψ ∈ C0∞ (M ) · P(M ) be arbitrary, i.e. ψ is a local functional of the ﬁelds, times a compactly supported cutoﬀ function. Set δS0 δF ∧ ψ(x), δB F = ∧ ψ(x). (341) B= δΦ(x) δΦ(x) M M Then we have

iF/

T ([F + δB F + ∆B (eF ⊗ )] ⊗ e⊗

)=

δS0 iF/ T (ψ(x) ⊗ e⊗ )dx. δΦ(x)

(342)

# 1 ⊗n ) and each ∆n : Pk1 (M ) ⊗ · · · ⊗ Pkn (M ) → Here ∆B (eF ⊗) = n n! ∆n (F Pk1 /···/kn (M n ) is a linear map that is supported on the total diagonal. If the Fi do not depend on , then the quantity ∆n (F1 ⊗ · · · ⊗ Fn ) is of order O(). We will outline the proof of this proposition at the end of the present proof. We now apply the Master Ward identity to the case when ψ(x) = δL F/δΦ‡ (x). Then we obtain, for the last term in Eq. (340) the expression δR S0 δL F i iF/ ∧ − T ⊗ e⊗ δΦ(x) δΦ‡ (x) δL F i i δ R S0 iF/ iF/ = T (δB F ⊗ e⊗ ) − T ⊗ e ⊗ δΦ(x) δΦ‡ (x) i iF/ + T (∆B (eF ). ⊗ ) ⊗ e⊗ Now, we have, with our choice ψ(x) = δL F/δΦ‡ (x), δL F 1 δR F ∧ = (F, F ). δB F = ‡ δΦ (x) 2 M δΦ(x)

(343)

(344)

Thus, we altogether obtain the identity iF/

σ0 T (e⊗

)−

i i i iF/ iF/ iF/ T (σ0 F ⊗ e⊗ ) = T ((F, F ) ⊗ e⊗ ) + T (∆B (eF ) ⊗ ) ⊗ e⊗ 2 (345)

which is in fact just another equivalent way of expressing the Master Ward Identity. This identity in eﬀect will take care of all terms in Eq. (337) involving the Koszul– Tate diﬀerential. We now look at the terms involving the pure BRST-diﬀerential s0 .

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To deal with these terms, we now use the following identity: Lemma 8. s0 Tn (F1 ⊗ · · · ⊗ Fn ) =

n

Tn (F1 ⊗ · · · ⊗ s0 Fk ⊗ · · · ⊗ Fn )

k=0

+

n

t=1 k1 <···
t−1 i

× Tn−t+1 (δt (Fk1 ⊗ · · · ⊗ Fkt ) ⊗ Fl1 ⊗ · · · ⊗ Fln−t ).

(346)

Here, δn is a map of the same nature as An , i.e. it is supported on the total diagonal, and it is of order O(). A formula generating these identities is iF/

s0 T (e⊗

i i iF/ iF/ ) − T (s0 F ⊗ e⊗ ) = T (δ(eF ). ⊗ ) ⊗ e⊗

(347)

Proof. For n = 1 the identity says that s0 T1 (F ) = T1 (s0 F ) + T1 (δ1 (F )), and we simply deﬁne δ1 (F ) in this way. Since there is no anomaly in the classical limit, it follows that δ1 (F ) is of order . We now proceed inductively to prove the equation for all n. Assume that it has been shown for any number of factors up to n − 1, and the δ1 , . . . , δn−1 have consequently been deﬁned. Take n functionals F1 , . . . , Fn with the property that the support of the ﬁrst l functionals is not in the future of the support of the last n − l functionals, where l is not equal to 0 or n. Deﬁne Mn to be the diﬀerence between the left and right terms in the above equation, with the nth term in the sum (the one containing δn ) omitted. Then, using the causal factorization property of the time-ordered products and the assumed support properties of the Fi , it follows that Mn (F1 ⊗ · · · ⊗ Fn ) = −s0 (Tl (F1 ⊗ · · · ⊗ Fl ) Tn−l (Fl+1 ⊗ · · · ⊗ Fn )) +

l

Tl (F1 ⊗ · · · ⊗ s0 Fk ⊗ · · · ⊗ Fl ) Tn−l (Fl+1 ⊗ · · · ⊗ Fn )

k=0

+

l

Tn (F1 ⊗ · · · ⊗ · · · ⊗ Fl ) Tn−l (Fl+1 ⊗ · · · ⊗ s0 Fk ⊗ · · · ⊗ Fn )

k=l+1

+

l

t=1 k1 <···
t−1 Tl−t+1 (δt (Fk1 ⊗ · · · ⊗ Fkt ) i

⊗ Fl1 ⊗ · · · ⊗ Fll−t ) Tn−l (Fl+1 ⊗ · · · ⊗ Fn ) t−1 n−1 + Tl (F1 ⊗ · · · ⊗ Fl ) i t=1 l
Tn−l−t+1 (δt (Fk1 ⊗ · · · ⊗ Fkt ) ⊗ Fl1 ⊗ · · · ⊗ Fln−l−t ).

(348)

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We now apply the inductive hypothesis that Eq. (346) holds at order n−1, together with the fact that s0 is a graded derivation of W0 (we proved this above for sˆ0 , the proof for s0 is completely analogous). If this is done, then it follows that Mn (F1 ⊗ · · · ⊗ Fn ) = 0 under the assumed support properties for the Fi . Consequently, Mn must be a functional valued in W0 that is supported on the total diagonal. That functional must hence be of the form (/i)n−1 T1 (δn (F1 ⊗ · · · ⊗ Fn )) for some δn , which we hence take as the deﬁnition of δn . We must next show that δn (F ⊗n ) is of order . For this, we pick a quasifree state ω of W0 , and we deﬁne, as described in Appendix B, the “connected time-ordered products” Tωc by the formula , c (F1 ⊗ · · · ⊗ Fn ) := Tn (F1 ⊗ · · · ⊗ Fn ) − : T|J| ⊗j∈J Fj :ω (349) Tn,ω P

J∈P

where P runs over all partitions of {1, . . . , n}, and where J runs through the disjoint sets in the given partition. A generating functional formula can be obtained using the linked cluster theorem, and is given by Eq. (485). The key fact about the connected products is that the nth product is of order O(n−1 ) if the Fi themselves are of order O(1). This will now be used by formulating Eq. (346) in terms of connected products. Using generating functional expression for the connected timeordered products, and using the fact that s0 is a derivation with respect to the Wick product (which follows from Eq. (298)), one can easily see that n−1 n n−1 i i c c s0 Tn,ω (F1 ⊗ · · · ⊗ Fn ) − Tn,ω (F1 ⊗ · · · ⊗ s0 Fk ⊗ · · · ⊗ Fn ) k=0

−

n−1

t=1 k1 <···
n−t i

c × Tn−t+1,ω (δt (Fk1 ⊗ · · · ⊗ Fkt ) ⊗ Fl1 ⊗ · · · ⊗ Fln−t )

= T1 (δn (F1 ⊗ · · · ⊗ Fn )).

(350)

Now, if we inductively assume that δt is of order O() for orders t < n, then it follows that the order of the second sum in the above expression is O(). Furthermore, the ﬁrst two terms on the left-hand side in the above equation precisely cancel up to c /n−1 a term of order O(). This follows from the fact that the limit lim Tn,ω correspond to the “tree diagrams”, and there are no anomalies at tree level [43]. Thus, δn = O(), as we desired to show. We are now in a position to complete the proof. From Eqs. (347) and (345) we get the desired Ward identity (333) with δL F δR S0 F F F ∧ . (351) A(e⊗ ) := ∆B (e⊗ ) + δ(e⊗ ), B = δΦ‡ (x) M δΦ(x) We must ﬁnally show that the maps An have properties analogous to those of the maps Dn in Sec. 3.6, i.e. properties (i)–(vi). The proof is similar as the proof for the

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Dn outlined there. It is again inductive in nature and is based on the expression T1 (An (F1 ⊗ · · · ⊗ Fn )) = sˆ0 Tn (F1 ⊗ · · · ⊗ Fn ) −

n

Tn (F1 ⊗ · · · ⊗ sˆ0 Fk ⊗ · · · ⊗ Fn )

k=0

−

Tn−1 (F1 ⊗ · · · ⊗ (Fj , Fk ) ⊗ · · · ⊗ Fn ) i k<j

−

n−1

t=1 k1 <···
t−1 i

× Tn−t+1 (At (Fk1 ⊗ · · · ⊗ Fkt ) ⊗ Fl1 ⊗ · · · ⊗ Fln−t )

(352)

for the nth order anomaly. We have already shown that An = O(), because this is true for ∆n , δn to all orders. The statement (ii) follows because all quantities on the right-hand side of this equation are locally and covariantly constructed out of the metric. (iii) follows from the fact that sˆ0 increases the ghost number by 1 unit, and because the anti-bracket increases the ghost number by 1 unit. (iv) follows because sˆ0 and the anti-bracket preserve the dimension, and from the known scaling behavior of the time-ordered products, T2. (v) follows because sˆ0 is compatible with the *-operation and because the time-ordered products are unitary, see T7. For more details on such kinds of arguments, see again [66]. To complete the proof, we must still show that Proposition 3 is indeed true. These arguments are given in detail in [17, Theorem 7 and Lemma 11]. For completeness, we here outline a slightly modiﬁed version of these arguments, but we refer the reader to this work for full details.n Proof of Proposition 3. We begin by writing down the nth order part of Eq. (342), given by −T1 (∆n (F1 ⊗ · · · ⊗ Fn )) n i δS0 = Tn+1 F1 ⊗ · · · ⊗ Fn ⊗ ψ(x) ∧ δΦ(x) n−1 n i δFi ⊗ · · · ⊗ Fn Tn F1 ⊗ · · · ⊗ ψ(x) ∧ + δΦ(x) i=1 n i δS0 − Tn (F1 ⊗ · · · ⊗ Fn ⊗ ψ(x)) δΦ(x)

n The

arguments in [17] are given only for the case of ﬂat spacetime, but the key steps generalize to curved manifolds straightforwardly.

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+

n−1 t=1

i

n−t

1125

k1 <···
× Tn−t+1 (∆t (Fk1 ⊗ · · · ⊗ Fkl ) ⊗ Fl1 ⊗ · · · ⊗ Fln−t ). For n = 0, the identity becomes δS0 δS0 −T1 (∆0 ) = T1 ψ(x) ∧ . − T1 (ψ(x)) δΦ(x) δΦ(x)

(353)

(354)

The function ∆0 is trivially local in this case. Because the ﬁrst time-ordered product T1 as well as the -product reduce to the ordinary product in the space of classical local functionals of the ﬁelds when → 0, it follows that ∆0 = O(), as claimed. We now proceed iteratively in n. We assume that the assertion about ∆n in the proposition has already been proved for ∆k up to k = n − 1. In fact, let us assume for simplicity even that ∆k = 0 up to k = n − 1. We deﬁne Mn (F1 ⊗ · · · ⊗ Fn ) to be the right-hand side of Eq. (353). The aim is to prove that this is a local functional valued in W0 . To demonstrate this, consider functionals Fi with the property that n l δFi δFi + supp supp ∪ supp ψ ∩ J =∅ (355) δΦ δΦ i=1 i=l+1

for some l not equal to n. Then Mn can be written as follows using the causal factorization properties of the time-ordered products: Mn (F1 ⊗ · · · ⊗ Fn ) n i δS0 Tl+1 F1 ⊗ · · · ⊗ Fl ⊗ ψ(x) ∧ = Tn−l (Fl+1 ⊗ · · · ⊗ Fn ) δΦ(x) n−1 l i δFi ⊗ · · · ⊗ Fl + Tl F1 ⊗ · · · ⊗ ψ(x) ∧ δΦ(x) i=1 n i Tn (F1 ⊗ · · · ⊗ Fl ⊗ ψ(x)) Tn−l (Fl+1 ⊗ · · · ⊗Fn ) −

δS0 Tn−l (Fl+1 ⊗ · · · ⊗ Fn ), δΦ(x)

(356)

where we have used that, for any G ∈ W0 (of even Grassmann parity), we have the identity G

δS0 δS0 = G, δΦ(x) δΦ(x)

(357)

which in turn follows from the deﬁnition of the star-product given above in Sec. 4.2, Eq. (300), together with the fact that δS0 /δΦi (x) = Dij Φj (x), Dij ω jk (x, y) = 0, with the Dij the matrix of linear partial diﬀerential operators in the ﬁeld equation for the free underlying (gauge ﬁxed) theory with action S0 . Using now the inductive

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assumption in Eq. (356), we conclude that Mn for the Fi with the assumed support properties. It follows from this that Mn can only be supported on the diagonal, which is the desired locality property of ∆n (F1 ⊗ · · · ⊗ Fn ). It remains to be seen that ∆n = O(). For this, we take Eq. (342) and multiply from the left with the anti-time-ordered products (see Eq. (148)), to obtain

δS0 δF iF/ iF/ + ψ(x) ∧ T¯(e⊗ ) T e⊗ ⊗ ψ(x) ∧ dx δΦ(x) δΦ(x) δS0 iF/ iF/ dx = T¯ (e⊗ ) T (e⊗ ⊗ φ(x)) δΦ(x) iF/ iF/ + T¯ (e⊗ ) T (e⊗ ⊗ ∆B (eF ⊗ )).

(358)

Using next the deﬁnition of the retarded products (see Eq. (264)), this may be rewritten in the form δS0 δF iF/ + ψ(x) ∧ ; e⊗ R ψ(x) ∧ dx δΦ(x) δΦ(x) δS0 iF/ iF/ dx + R(∆B (eF ). (359) = R(ψ(x); e⊗ ) ⊗ ); e⊗ δΦ(x) The key point is now the that the retarded products in this equation have a meaningful limit as → 0, as proven in [40], i.e. the above expressions contain no inverse powers of , despite the inverse powers of in the exponentials. This limit is just the classical limit for the interacting ﬁelds as deﬁned by the Bogoliubov formula Eq. (263). Furthermore, the classical limit of is the usual classical product of classical ﬁelds. Thus, the Eq. (359) has a classical limit, the “classical Master Ward Identity” of [17]. It is shown in this reference that this identity in classical ﬁeld theory is indeed true with ∆ = 0. Consequently, ∆ itself must be of order O(), as we desired to show. This concludes our outline of the proof of Proposition 3. Since we have proved Proposition 3, we have proved Proposition 2. We next derive a “consistency condition” on the anomaly. Proposition 4 (“Consistency Condition”). The anomaly satisﬁes the equation 1 F F F (S0 + F, A(eF ⊗ )) − A((S0 + F, S0 + F ) ⊗ e⊗ ) = A(A(e⊗ ) ⊗ e⊗ ). 2

(360)

Proof. We ﬁrst act with sˆ0 on the anomalous Ward identity Eq. (333) and use that sˆ20 = 0. We obtain the equation iF/

0 = sˆ0 T (A(eF ⊗ ) ⊗ e⊗

1 iF/ ) + sˆ0 T ((S0 + F, S0 + F ) ⊗ e⊗ ) = (I) + (II). 2

(361)

The trick is now to apply the anomalous Ward identity one more time to each of the terms on the right-hand side. For simplicity, we assume that F has Grassmann

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parity 0. We can then write the ﬁrst term as , i(F +τ A(eF⊗ )) d sˆ0 T e⊗ |τ =0 i dτ

d 1 i(F +τ A(eF ⊗ ))/ F T ((S0 + F + τ A(eF = ) ⊗ ), S0 + F + τ A(e⊗ )) ⊗ e⊗ dτ 2 F τ A(eF ) + T (A(e⊗ ⊗ ) ⊗ ei(F +τ A(e⊗ ))/ )

(I) =

τ =0

i iF/ iF/ T (A(eF = T ((S0 + F, A(eF )+ ) ⊗ )) ⊗ e⊗ ⊗ ) ⊗ (S0 + F, S0 + F ) ⊗ e⊗ 2 iF/

F − T (A(A(eF ⊗ ) ⊗ e⊗ ) ⊗ e⊗

iF/

F ) + T (A(eF ⊗ ) ⊗ A(e⊗ ) ⊗ e⊗

).

(362)

Since F has Grassmann parity 0, A(eF ⊗ ) has Grassmann parity 1, so by the antisymmetry of the time-ordered products for such elements, see (241), the last term vanishes. Next, we apply the anomalous Ward identity to term (II). We now obtain , i(F +τ (S0 +F,S0 +F ))/ d sˆ0 T e⊗ |τ =0 2i dτ

1 d 1 T ((S0 + F + τ (S0 + F, S0 + F ), S0 + F = 2 dτ 2

(II) =

i(F +τ (S0 +F,S0 +F ))/

+ τ (S0 + F, S0 + F )) ⊗ e⊗ τ (S0 +F,S0 +F )

+ T (A(e⊗

i(F +τ (S0 +F,S0 +F ))/

) ⊗ e⊗

1 iF/ = T ((S0 + F, (S0 + F, S0 + F )) ⊗ e⊗ ) 2 +

)

)

τ =0

i iF/ T ((S0 + F, S0 + F ) ⊗ (S0 + F, S0 + F ) ⊗ e⊗ ) 2 iF/

− T (A((S0 + F, S0 + F ) ⊗ eF ⊗ ) ⊗ e⊗

)

i iF/ F − T (A(e⊗ ) ⊗ (S0 + F, S0 + F ) ⊗ e⊗ ) .

(363)

Now, the ﬁrst term on the right-hand side vanishes due to the graded Jacobi identity (59) for the anti-bracket. The second term vanishes due to the anti-symmetry property of the time-ordered products (241), since (S0 + F, S0 + F ) has Grassmann parity 1. If we now add up terms (I) and (II), we end up with the following identity:

1 iF/ F F T (S0 + F, A(e⊗ )) − A((S0 + F, S0 + F ) ⊗ e⊗ ) ⊗ e⊗ 2 iF/

F = T (A(A(eF ⊗ ) ⊗ e⊗ ) ⊗ e⊗

).

(364)

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Since a time-ordered product T (G⊗ e⊗ consistency condition (360) follows.

) vanishes if and only if G = 0, the desired

Let us summarize what we have shown so far: We ﬁrst demonstrated that the Ward identity (336) always holds with an anomaly term of order , i.e. that Eq. (333) holds. We then showed that the anomaly is not arbitrary, but must obey the consistency condition (360). This condition imposes a strong restriction on the possible anomalies, and we will show in the following subsections using this condition that, when F is as in the cases T12a, T12b, and T12c, then the anomaly A(eF ⊗ ) can in fact be removed by a redeﬁnition of the time-ordered products consistent with T1–T11. Thus, in these cases, we may achieve that the Ward identity (336) holds exactly, without anomaly. To prepare the proof of this statement, we ﬁrst note that, since the anomaly itself is of order , the lowest order in contribution to the “anomaly of the anomaly term” on the right-hand side of Eq. (360) is necessarily of a higher order in than the lowest order contribution left-hand side. An even more stringent consistency condition can therefore be obtained for the lowest order (in ) contribution to the anomaly. For this, we expand A(eF ⊗ ) in powers of the coupling, λ, and , n λ A(eF m (365) Am ⊗) = n (x1 , . . . , xn )f (x1 ) · · · f (xn )dx1 · · · dxn , n! n,m>0 ‡ where Am n is a local, covariant functional of (Φ, Φ ), and the metric that is supported on the total diagonal. Both sums start with positive powers, because the anomaly vanishes in the classical theory (i.e. = 0), and also in the free quantum theory (i.e. λ = 0). An explicit deﬁnition of Am n is given by

Am n (x1 , . . . , xn ) =

δn 1 ∂m A(eF ⊗ )|f =0= . m m! ∂ δf (x1 ) · · · δf (xn )

(366)

F Let Am (eF ⊗ ) now be the lowest order contribution to A(e⊗ ) in the -expansion, that is, m is the smallest integer for which

Am (eF ⊗ ) :=

1 ∂m A(eF ⊗ )|=0 m! ∂m

(367)

is not zero. (Note that the quantity Am is diﬀerent from the quantity An above!) Then, from our consistency condition given in Proposition 4, we get the following version of the consistency condition: Proposition 5 (“-Expanded Consistency Condition”). Let A be the anomaly of the Ward identity in Proposition 2, and let Am be the ﬁrst non-trivial term in the -expansion of A. Then we have 1 m F (S0 + F, Am (eF ⊗ )) − A ((S0 + F, S0 + F ) ⊗ e⊗ ) = 0. 2 Here, (., .) is the anti-bracket (see Eq. (58)).

(368)

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This stronger form of the consistency condition is the key relation that will be used in the proofs of T12a, T12b, and T12c. In those proofs we will actually encounter several quantities like Am n , so it is convenient to use again the notation from Sec. 3.6. As there, (k1 , . . . , kn ) is a set of natural numbers. We denote by Pk1 /···/kn (M n ) the space of all local, covariant functionals of Φ, Φ‡ , and the metric which are supported on the total diagonal, and which take values in the bundle (248) of anti-symmetric tensors over M n . Thus, if Bn ∈ Pk1 /···/kn (M n ), then Bn is a (distributional) polynomial, local, covariant functional of Φ, Φ‡ and the metric taking values in the k1 + · · · + kn forms over M n , which is supported on the total diagonal. It is a k1 -form in the ﬁrst variable x1 , a k2 -form in the second variable x2 , etc. Concerning such quantities, we have a simple lemma that we will use below. Lemma 9. Let Bn ∈ Pk1 /···/kn (M n ), and let fi , i = 1, . . . , n be closed forms on M of degree 4 − ki . Assume that for any such forms, we have fi (xi ) = 0. (369) Bn (x1 , . . . , xn ) ∧ i

Then it is possible to write Bn [Φ, Φ‡ ] =

n

dk Bn/k [Φ, Φ‡ ] + Bn [0, 0],

(370)

k=1

where dk = dxk ∧ (∂/∂xµk ) is the exterior diﬀerential applied to the kth variable. Proof. We ﬁrst consider the case n = 1. If k1 = 4, then the assumptions imply that F = B1 (x)f1 (x) = 0 for any closed 0-form f1 , i.e. for any constant such as f1 (x) = 1. We therefore have δF/δψ(x) = 0, using the abbreviation ψ = (Φ, Φ‡ ). Consider the path ψτ = (τ Φ, τ Φ‡ ) in ﬁeld space. Then d δF [ψτ ] ∂B1 [ψτ ] B1 [ψτ ] = =ψ + dϑ[ψτ ] = dϑ[ψτ ], (∇k ψ) (371) k dτ ∂(∇ ψ) δψ k

for some locally constructed 3-form ϑ. Thus, 1 1 d B1 [ψτ ]dτ = B1 [0, 0] + d ϑ[ψτ ]dτ B1 [Φ, Φ‡ ] = B1 [0, 0] + 0 dτ 0 = B1 [0, 0] + dB1/1 [Φ, Φ‡ ],

(372) (373)

which has the desired form. If k1 = 0, then f1 is a 4-form, which is always closed. Thus, the assumptions of the lemma imply that B1 [Φ, Φ‡ ] = 0, which is again of the desired form. Finally, if 0 < k1 < 4, we may choose f1 = dh1 , implying that dB1 (x) ∧ h1 (x) = 0 for all h1 , and thus that dB1 = 0. The statement now follows from the algebraic Poincare lemma. The proof of the lemma for n > 1 can now be generalized from the case n = 1. Without loss of generality, we may assume Bn [Φ = 0, Φ‡ = 0] = 0, for otherwise, we may simply subtract this quantity. To reduce the situation to n = 1, consider

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the form on M of degree k1 that is obtained by smearing Bn as in (374), but the smearing over the ﬁrst test-form f1 omitted. If k1 < 4, then this form is a closed form that is locally and covariantly constructed from f2 , f2 , . . . , fn and Φ, Φ‡ . This 4-form then by deﬁnition obeys the assumptions of lemma 3, so we may write 0=

Bn (x1 , . . . , xn ) ∧

n

Bn/1 (x1 , . . . , xn ) ∧

fi (xi ) − d1

i=2

n

fi (xi )

(374)

i=2

for some Bn/1 ∈ Pk1 −1/k2 /··· /kn . If k1 = 4 one may argue similarly. We now repeat this argument, now omitting the integration over the second test form f2 . We then get 0=

Bn (x1 , . . . , xn ) ∧

n

fi (xi ) − d1

Bn/1 (x1 , . . . , xn ) ∧

i=3

− d2

Bn/2 (x1 , . . . , xn ) ∧

n

fi (xi )

i=3 n

fi (xi )

(375)

i=3

for some Bn/2 ∈ Pk1 /k2 −1/··· /kn . We may continue this procedure, and thus inductively proceed to construct the remaining Bn/k . 4.4.1. Proof of T12a Up to now, we have shown (Proposition 2) that any prescription for deﬁning timeordered products satisfying properties T1–T11 satisﬁes the Ward identity (333) with anomaly. We shall now prove that we can change the deﬁnition of the timeordered products in such a way that T1-T11 still hold, and such that in addition the anomaly vanishes in the case when F = {λf L1 +λ2 f 2 L2 }, where f ∈ C0∞ (M ). Thus, our new prescription will satisfy (322) (and in fact even Eq. (336)) for this F . This will then enable us to prove that the new prescription for deﬁning time-ordered products will satisfy property T12a. The key tool for proving this statement is the consistency condition on the expanded anomaly given in Proposition 5. To take full advantage of this consistency condition, we would like to put f = 1, for we then have S0 + F = S, and we can take advantage of BRST-invariance of the full action S, see (57). We note that we iF/ cannot simply set f = 1 in T (e⊗ ), for we might encounter infra-red divergences. ‡ However, since the anomaly terms Am n are local, covariant functionals of Φ, Φ that are supported on the total diagonal (taking values in the 4n-forms ∧4n T ∗ M n over M n ), we may without any danger set f = 1 in Eq. (368). As we have already said, in that case we have F = λS1 + λ2 S2 , and consequently S0 + F = S, where S is the full action (57). So from Eq. (368) together with (S, S) = 0 and sˆ = (S, .) we ﬁnd 2

1 +λ sˆAm (eλS ⊗

S2

) = 0.

(376)

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Now, we have m

A

2 1 +λ S2 (eλS ) ⊗

1131

λn = a (x) = am (x), n! M n M n>0 m

(377)

where am ∈ P4 (M ) (and likewise for am n ). Furthermore, from the properties of the anomaly derived in the previous subsection, the dimension of am must be 4, and the ghost number must be +1. Equation (376) may now be viewed as saying s|d, P4 ). From the lemmas given in Sec. 2.2, we have a complete that am ∈ H 1 (ˆ classiﬁcation of all the elements in this ring. In fact, as shown there in Lemma 1, all non-trivial elements in this ring at ghost number +1 and dimension 4 must be even under parity, ε → −ε when the Lie-group has no abelian factors. On the other hand, it follows from the properties of the anomaly A that am is parity odd, i.e. am → −am under parity ε → −ε. Therefore, am must represent the zero element s|d, P4 ), so there are bm ∈ P40 (M ) and cm ∈ P31 (M ) such that in the ring H 1 (ˆ am (x) = sˆbm (x) + dcm (x).

(378)

We expand bm (x) =

λn bm n (x). n! n>0

(379)

We would like to use the coeﬃcients bm n (x) to redeﬁne the time-ordered products Tn (L1 (x1 )⊗· · ·⊗L1 (xn )) containing n factors of the interaction Lagrangian. Recalling that by Theorem 2, the changes in the time-ordered products are parametrized by local, covariant maps Dn : Pp1 (M ) ⊗ · · · ⊗ Ppn (M ) → Pp1 /···/pn (M n ), we deﬁne Dn (L1 (x1 ) ⊗ · · · ⊗ L1 (xn )) := −m bnm (x1 )δ(x1 , . . . , xn ).

(380)

It can be shown that this is within the allowed renormalization freedom for the time-ordered products described in Sec. 3.6: First, the locality and covariance of Dn follows from the corresponding property of bm n . The scaling property (254) follows m from the fact that bn has dimension 4, together with the scaling degree property sdδ = 4(n − 1) for the delta function of n spacetime arguments concentrated on the diagonal in M n . The smooth and analytic dependence of Dn under changes of the spacetime metric again follows from the corresponding properties of bm n, while the symmetry is manifest. The unitarity condition (255) follows from the fact that bm n is real, which in turn follows from the corresponding property of the anomaly A derived in the previous subsection. To satisfy the ﬁeld independence property (253), it is furthermore necessary to also change the time-ordered products of sub-monomials of L1 in order to be consistent with T9. This causes no problems. The identity (256) can be satisﬁed by deﬁning Dn appropriately for entries Oi that are exterior diﬀerentials of L1 . This does not lead to any potential consistency problems, because L1 itself is not the exterior diﬀerential of a locally constructed 3-form. For details of such kinds of arguments see [70], where a very similar situation was treated. Thus, the above Dn (together with the corresponding Dn for sub-Wick

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monomials of L1 and their exterior derivatives) gives a permissible change in the time-ordered products, i.e. the changed time-ordered products Tˆ deﬁned according to (251) with the above D again satisfy T1–T11. With the above deﬁnition of Dn , and F = [λf L1 + λ2 f 2 L2 ], the relevant newly deﬁned time-ordered products Tˆ are given by (see Eq. (251)) , i[F +D(exp⊗ F )]/ , iF/ = T e⊗ , (381) Tˆ e⊗ , , i[F +D(exp⊗ F )]/ iF/ Tˆ (S0 + F, S0 + F ) ⊗ e⊗ = T (S0 + F, S0 + F ) ⊗ e⊗ , (382) , , ˆ F ) + D(A(e ˆ F ) ⊗ eiF/ = T [A(e ˆ F ) ⊗ eF )] ⊗ ei[F +D(exp⊗ F )]/ , Tˆ A(e ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ (383) ˆ F where A(e ⊗ ) is the anomaly (333) in the Ward identity for the modiﬁed timeordered products Tˆ . The second line follows because there are no sub-Wick monomials in L1 that are also contained in (S0 + F, S0 + F ), as one may check explicitly. We would now like to relate the anomaly A(eF ⊗ ) of the “old” time-ordered products ˆ F ) of the “new” time-ordered products Tˆ . We have T to the anomaly A(e ⊗ i , ˆ F ˆ F ) ⊗ eF )] ⊗ ei[F +D(exp⊗ F )]/ T [A(e⊗ ) + D(A(e ⊗ ⊗ ⊗ i ,ˆ F iF/ = Tˆ A(e ⊗ ) ⊗ e⊗ , iF/ i , iF/ − Tˆ (S0 + F, S0 + F ) ⊗ e⊗ = sˆ0 Tˆ e⊗ 2 , i[F +D(exp⊗ F )]/ i , i[F +D(exp⊗ F )]/ − T (S0 + F, S0 + F ) ⊗ e⊗ = sˆ0 T e⊗ 2 i , i[F +D(exp⊗ F )]/ F T (S0 + F + D(eF = ⊗ ), S0 + F + D(e⊗ )) ⊗ e⊗ 2 i , i[F +D(exp⊗ F )]/ − T (S0 + F, S0 + F ) ⊗ e⊗ 2 i , F +D(exp⊗ F ) i[F +D(exp⊗ F )]/ ) ⊗ e⊗ + T A(e⊗ i , i[F +D(exp⊗ F )]/ = T (S0 + F, D(eF ⊗ )) ⊗ e⊗ i , F +D(exp⊗ F ) i[F +D(exp⊗ F )]/ ) ⊗ e⊗ + T A(e⊗ i , i[F +D(exp⊗ F )]/ F . (384) + T (D(eF ⊗ ), D(e⊗ )) ⊗ e⊗ 2 It follows from this equation that , , F ˆ F Aˆ eF ⊗ + D A(e⊗ ) ⊗ e⊗ , F +D(exp⊗ F ) - 1 , , F + D(eF = S0 + F, D(eF ⊗ ) + A e⊗ ⊗ ), D(e⊗ ) . 2

(385)

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m F We now use that, by deﬁnition, D(eF ⊗ ) = O( ), and by assumption A(e⊗ ) = m F m ˆ ⊗ ). Then it follows that, at order , we have O( ) = A(e , , - , m F Aˆm eF (386) e⊗ + S0 + F, Dm (eF ⊗ =A ⊗) . 2

1 +λ S2 ) = Now it follows from our deﬁnition of D and Eq. (378) that sˆDm (eλS ⊗ 2 m λS1 +λ S2 2 2 ). Therefore, if we now put f = 1 in F = [λf L1 + λ f L2 ] in the −A (e⊗ above equation, then we ﬁnd , 1 +λ2 S2 , 1 +λ2 S2 , 1 +λ2 S2 = Am eλS + sˆDm eλS = 0. (387) Aˆm eλS ⊗ ⊗ ⊗

Thus, by our redeﬁnition of the time-ordered products, we have already removed the anomaly for any constant test function f . We will now use this fact to completely remove the anomaly by a further redeﬁnition of the time-ordered products. To simplify the notation, we will now again use the notations T and A for ˆ The the redeﬁned time-ordered products and new anomaly, instead of Tˆ and A. anomaly may be expanded in powers of and λ as in Eq. (365). From Eq. (387) ˆ we then have (remembering that A now denotes A), (388) Am n (x1 , . . . , xn )dx1 · · · dxn = 0, because we can assume at this stage that the anomaly vanishes for constant f . Consequently, by Lemma 9, this quantity must be given by an expression of the form n m dk Cn/k (x1 , . . . , xn ), (389) Am n (x1 , . . . , xn ) = k=1

for some

m Cn/k

∈P

4/···3/···/4

n

(M ). We next deﬁne a set of Dn by the formula

m Dn (L1 (x1 ) ⊗ · · · ⊗ O1 (xk ) ⊗ · · · ⊗ L1 (xn )) := −m Cn/k (x1 , . . . , xn ),

(390)

where O1 ∈ P31 (M ) is the ﬁeld determined by the equation sˆ0 L1 = dO1 ,

(391)

and is given explicitly by 1 O1 := fIJK C I AJ ∧ ∗dAK + fIJK C I C K ∗ dC¯ K . (392) 2 We may again argue that this Dn satisﬁes all the required properties for an allowed redeﬁnition of the time-ordered products, and we denote the new time-ordered ˆ By a calculation similar to products again by Tˆ , and the new anomaly again by A. the one given above, the new anomaly now satisﬁes , F, - 1 , , F F F (393) Aˆ eF ⊗ = A e⊗ + D A(e⊗ ) ⊗ e⊗ + D (S0 + F, S0 + F ) ⊗ e⊗ . 2 Again evaluating this new anomaly at order m , we ﬁnd , , - 1 m, m F Aˆm eF e⊗ + D (S0 + F, S0 + F ) ⊗ eF (394) ⊗ = A ⊗ . 2

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However, our Dn are designed precisely in such a way that Dm ((F, F ) ⊗ eF ⊗) = 0 m F m F ˆ s0 F ⊗ e F ) = −A (e ), so we ﬁnd that A (e ) = 0. and that Dm (ˆ ⊗ ⊗ ⊗ In summary, our subsequent redeﬁnitions of the time-ordered products have m removed the anomaly Am (eF ⊗ ) at order , and to all orders in λ. We now repeat the m+1 F (e⊗ ), i.e. order m+1 , and we can proceed in just the same same argument for A way for any order in . This shows that the anomaly can be removed to arbitrary orders in and λ by a redeﬁnition of the time-ordered products that is compatible with T1–T11. The absence of an anomaly in Eq. (333) for our choice of F implies that T12a is satisﬁed, because Eq. (333) is a generating identity of the identities in T12a. 4.4.2. Proof of T12b The proof that the time-ordered products can be adjusted, if necessary, so that T12b is satisﬁed is very similar in nature as that given above for T12a. We therefore only focus on the essential diﬀerences. Consider the local elements G = γ ∧ (J0 + f λJ1 ) and F = (f λL1 + f 2 λ2 L2 ), where γ is a smooth 1-form of compact support, and f is a smooth scalar function of compact support. The satisfaction of T12b means that the anomaly in , , i , iF/ iF/ iF/ T (S0 + F, S0 + F ) ⊗ G ⊗ e⊗ = T (S0 + F, G) ⊗ e⊗ + sˆ0 T G ⊗ e⊗ 2 , , iF/ iF/ F + T A(eF (395) + T A(G ⊗ e⊗ ) ⊗ e⊗ ⊗ ) ⊗ G ⊗ e⊗ can be removed by a suitable redeﬁnition of the time-ordered products. As above, we write n mλ A(G ⊗ eF ) = Am ⊗ n (x1 , . . . , xn )γ(x1 )f (x1 ) · · · f (xn )dx1 · · · dxn . n! n M m,n>0 (396) Let Am (G ⊗ eF ⊗ ) be the lowest order contribution in to the anomaly. Because the anomaly is of order at least , we have m > 0. We apply the consistency condition (368) to the element F + τ G instead of F in that formula, and we diﬀerentiate with respect to τ and set τ = 0. Then we obtain the consistency condition , , m (S0 + F, G) ⊗ eF S0 + F, Am (G ⊗ eF ⊗) + A ⊗ , 1 − Am (S0 + F, S0 + F ) ⊗ G ⊗ eF ⊗ = 0. 2

(397)

Now, we put f = 1 and we take γ to satisfy dγ = 0. Then, F = λS1 + λ2 S2 , and S0 + F = S, where S is the full action (57) satisfying (S, S) = 0. Furthermore, by sˆJ = dK, γ∧J= γ ∧ dK = − dγ ∧ K = 0. (398) (S0 + F, G) = (S, G) = sˆ M

M

Thus, condition (397) implies the condition , 2 1 +λ S2 =0 sˆAm G ⊗ eλS ⊗

M

(399)

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when γ is closed. Now, we have , 2 1 +λ S2 A G ⊗ eλS = ⊗ m

λn γ ∧ h (x) = γ ∧ hm n (x), n! M M n>0 m

(400)

where hm ∈ P3 (M ) (and likewise for hm n ). Furthermore, from the properties of the anomaly derived in the previous subsection, the dimension of hm must be 3, and the ghost number must be +2. Equation (399), which holds for all closed 1-forms γ in s|d, P3 ). From the the deﬁnition of G, may now be viewed as saying that hm ∈ H 2 (ˆ lemmas given in Sec. 2.2, we again have a complete classiﬁcation of all the elements in this ring. In fact, as shown there in Lemma 1, all non-trivial elements in this ring at ghost number +2 and dimension 3 must be even under parity, ε → −ε when the Lie-group has no abelian factors. On the other hand, it follows again from the properties of the anomaly A that hm is parity odd, i.e. hm → −hm under parity s|d, P3 ), so ε → −ε. Therefore, hm must represent the zero element in the ring H 2 (ˆ there are j m ∈ P31 (M ) and k m ∈ P22 (M ) such that hm (x) = sˆj m (x) + dk m (x).

(401)

We again expand j m in powers of λ j m (x) =

λn jnm (x). n! n>0

(402)

Similar to the proof of T12a, we would like to use the coeﬃcients jnm (x) to redeﬁne the time-ordered products Tn (J1 (x1 ) ⊗ L1 (x2 ) ⊗ · · · ⊗ L1 (xn )) containing n − 1 factors of the interaction Lagrangian and one factor of the free BRST-current. By Theorem 2, the changes in the time-ordered products are parametrized by local, covariant maps Dn : Pp1 (M ) ⊗ · · ·⊗ Ppn (M ) → Pp1 /···/pn (M n ), and we deﬁne Dn (J1 (x1 ) ⊗ L1 (x2 ) ⊗ · · · ⊗ L1 (xn )) := −m jnm (x1 )δ(x1 , . . . , xn ).

(403)

This gives changed time-ordered products via (251), and one may argue as above in T12a that these again satisfy T1–T11. By T12a, we may assume that A(eF ⊗ ) = 0. The above changes in the time-ordered products eﬀected by the maps Dn do not ˆ F invalidate this, i.e. the new time-ordered products Tˆ will also satisfy A(e ⊗ ), where ˆ ˆ A is the anomaly in Eq.(333) for the new time-ordered products T . From Eq. (333), we then have, for F = [λf L1 + λ2 f 2 L2 ] and any G, , , i , iF/ iF/ iF/ sˆ0 T G ⊗ e⊗ = T (S0 + F, G) ⊗ e⊗ + T (S0 + F, S0 + F ) ⊗ G ⊗ e⊗ 2 , iF/ (404) ) ⊗ e + T A(G ⊗ eF ⊗ ⊗

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and likewise for the “hatted” time-ordered products and the hatted anomaly. From this, and the deﬁnition of D above, one ﬁnds, for F, G as above , , ˆ ⊗ eF ) ⊗ eiF/ ˆ ⊗ eF ) ⊗ eF ) ⊗ eiF/ + T A(G T D(A(G ⊗ ⊗ ⊗ ⊗ ⊗ , ˆ ⊗ eF ) ⊗ eiF/ = Tˆ A(G ⊗ ⊗ , , iF/ iF/ = sˆ0 Tˆ G ⊗ e⊗ − Tˆ (S0 + F, G) ⊗ e⊗ i , iF/ − Tˆ (S0 + F, S0 + F ) ⊗ G ⊗ e⊗ 2 , , iF/ iF/ − T (S0 + F, G) ⊗ e⊗ = sˆ0 T [G + D(G ⊗ eF ⊗ )] ⊗ e⊗ i , iF/ − T (S0 + F, S0 + F ) ⊗ [G + D(G ⊗ eF ⊗ )] ⊗ e⊗ 2 , , iF/ iF/ F = T A(G ⊗ eF + T A[D(G ⊗ eF ⊗ ) ⊗ e⊗ ⊗ ) ⊗ e⊗ ] ⊗ e⊗ , iF/ + T (S0 + F, D(G ⊗ eF . ⊗ )) ⊗ e⊗

(405)

It follows that the anomalies for the old and new prescriptions are related by , , - , F F F A G ⊗ eF ⊗ + A D(G ⊗ e⊗ ) ⊗ e⊗ + S0 + F, D(G ⊗ e⊗ ) , , ˆ ⊗ eF ) ⊗ eF + Aˆ G ⊗ eF . = D A(G ⊗ ⊗ ⊗

(406)

F The trick is again to evaluate this at order m , and use that D(G⊗eF ⊗ ) and A(G⊗e⊗ ) m themselves are of order . This gives the equality

, - , , m F ˆm G ⊗ eF Am G ⊗ eF ⊗ + S0 + F, D (G ⊗ e⊗ ) = A ⊗ . Now, if G = that

(407)

γ ∧ J, and if F = λS1 ⊗ λ2 S2 , then it follows from the above equation

, , , 2 2 2 1 +λ S2 1 +λ S2 1 +λ S2 + sˆDm G ⊗ eλS = Aˆm G ⊗ eλS . Am G ⊗ eλS ⊗ ⊗ ⊗

(408)

2

1 +λ S2 Furthermore, it follows from the deﬁnition of D that D(G ⊗ eλS ) is equal ⊗ to − γ ∧ j m. Furthermore, if γ is a closed 1-form, we have shown above that sˆ γ ∧ j m = γ ∧ hm . By Eq. (400) and our deﬁnition of D, we therefore have

, , 2 2 1 +λ S2 1 +λ S2 = −Am G ⊗ eλS . sˆDm G ⊗ eλS ⊗ ⊗

(409)

Consequently, we have shown that , 2 1 +λ S2 Aˆm G ⊗ eλS = 0. ⊗

(410)

Therefore, our redeﬁnition of the time-ordered products has already removed the ˆ ⊗ eF anomaly A(G ⊗ ) in the case when γ is a closed 1-form, and f is a constant. We now drop the caret from our notation for the newly deﬁned time-ordered products and the corresponding anomaly. We may then assume that Eq. (410) holds for Am .

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For the quantities deﬁned in Eq. (411), this means that Am 0= n (x1 , . . . , xn )γ(x1 )dx1 · · · dxn

1137

(411)

Mn

for any closed 1-form γ, and any n. Lemma (9) now implies that we we may write Am n as n m m Am (x , . . . , x ) = d B (x , . . . , x ) + dk Bn/k (x1 , . . . , xn ). (412) 1 n 1 1 n n n/1 k=2

Here, the Bm,n/k are now a local covariant functional of (Φ, Φ‡ ) in the space P2/4/···4 (M n ) for k = 1, and in the space P3/4/···3/···4 (M n ) for k ≥ 2. By elem are supported on the mentary manipulations using δ-functions, using that the Bn/k diagonal and have dimension +2 and ghost number +2, we may shift the derivatives between the terms in the sum if necessary and thereby achieve that m (x1 , . . . , xn ) = const. K1 (x1 )δ(x1 , . . . , xn ), Bn/1

(413)

up to an irrelevant total d1 -derivative. Next, we deﬁne for products with n arguments containing 1 factor of K1 ∈ P22 [see Eq. (76)] and n − 1 factors of L1 ∈ P40 by m Dn (K1 (x1 ) ⊗ L1 (x2 ) ⊗ · · · ⊗ L1 (xn )) := −m Bn/1 (x1 , . . . , xn ).

(414)

We redeﬁne the time-ordered products with n + 1 factors, containing 1 factor of J1 ∈ P31 , one factor of O1 ∈ P31 [see Eq. (391)], and n − 2 factors of L1 ∈ P40 by m Dn (J1 (x1 ) ⊗ L1 (x2 ) ⊗ · · · ⊗ O1 (xk ) ⊗ · · · ⊗ L1 (xn )) := im+1 Bn/k (x1 , . . . , xn ).

(415) By going through the same steps as above in T12a, we ﬁnd that the new anomaly ˆ ⊗ eF A(G ⊗ ) after the above redeﬁnition eﬀected by these D’s is now , , , F F Aˆ G ⊗ eF ⊗ = A G ⊗ e⊗ − D (S0 + F, G) ⊗ e⊗ i , + D (S0 + F, S0 + F ) ⊗ G ⊗ eF (416) ⊗ . 2 Now, it can be seen that, because of the ﬁrst redeﬁnition (414), D((S0 + F, G) ⊗ eF ⊗) λn m (x1 , . . . , xn )γ(x1 )f (x1 ) · · · f (xn )dx1 · · · dxn , = m d1 Bn/1 n!

(417)

n≥0

using (S0 , J1 ) = dK1 + · · · . It follows from the second redeﬁnition (415) that i , D G ⊗ (S0 + F, S0 + F ) ⊗ eF ⊗ 2 n λn m (x1 , . . . , xn )γ(x1 )f (x1 ) · · · f (xk ) · · · f (xn )dx1 · · · dxn , = −m dk Bn/k n! n≥0 k=2

(418) m

using (S0 , L1 ) = dO1 . Thus, taking the O( )-part of Eq. (416), using Eq. (413), we ﬁnd that the new anomaly Aˆm (G ⊗ eF ⊗ ) = 0. Thus, the anomaly for the new

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time-ordered products vanishes at order m and to all orders in λ. We continue this process by redeﬁning the time-ordered products to the next order in , and remove the anomaly Am+1 (G ⊗ eF ⊗ ). Since we can do this for all m, we see that we can satisfy T12b above by a suitable redeﬁnition of the time-ordered products. 4.4.3. Proof of T12c / Let Ψ = Θsi (F, DF, D2 F, . . .) be the gauge-invariant expression of form-degree p under consideration, where Θs are invariant polynomials of the Lie-algebra, so that in particular sˆΨ = 0. Let α be a (4 − p)-form, and let G = α ∧ Ψ. The satisfaction of the Ward identity T12c means that anomaly in Eq. (395) can be removed, where G in that equation is now α ∧ Ψ. As in the proofs of T12a, T12b, one ﬁrst proves the consistency condition , 2 1 +λ S2 = 0, (419) sˆAm G ⊗ eλS ⊗ where m is the ﬁrst order in where the anomaly occurs, and where α is now arbitrary. This condition is again of cohomological nature. As in T12b, it may be used to show that the anomaly can be removed, at nth order in λ, by a redeﬁnition of the time-ordered products with 1 factor of Ψ0 and n factors of L1 , and by the time-ordered products with 1 factor of Ψ0 , 1 factor of O1 (see Eq. (391)) and n − 1 factors of L1 . The details of these arguments are completely analogous to those given above in the proofs of T12a and T12b, so we omit them here. 4.5. Formal BRST-invariance of the S-matrix iF/

We consider the adiabatically switched S-matrix S(F ) = T (e⊗ ) associated with the cutoﬀ interaction F = M {λf L1 + λ2 f 2 L2 }, where f is a smooth switching function of compact support. Let Q0 be the free BRST-charge operator. It follows iF/ from the deﬁnition S(F ) = T (e⊗ ) and the Ward identities T12a (see Eq. (322)) that 1 , iF/ mod J0 . [Q0 , S(F )] = − T (S0 + F, S0 + F ) ⊗ e⊗ (420) 2 Now consider a sequence of cutoﬀ functions such that f → 1 suﬃciently rapidly, i.e. the “adiabatic limit”. Then it follows that S0 + F → S, and consequently that iF/ (S0 + F, S0 + F ) → (S, S) = 0. Thus, formally, T ((S0 + F, S0 + F ) ⊗ e⊗ ) → 0. Furthermore, formally, S(F ) converges to the true S-matrix S. Consequently, assuming that all these limits exist, we would have [Q0 , S] = 0

mod J0

(FORMALLY).

(421)

As we have already said, the adiabatic limit does not appear to exist for pure Yang– Mills theory in Minkowski spacetime, and there is even less reason to believe that it ought to exist in generic curved spacetimes. Therefore, the above statement concerning the BRST-invariance of the S-matrix is most likely only a formal statement,

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unlike the other results in this paper. We have nevertheless mentioned it, because such a condition is often taken to be as the deﬁnition of gauge-invariance at the perturbative level in less rigorous treatments of quantum gauge ﬁeld theories in ﬂat spacetime. 4.6. Proof that dJI = 0

As above, consider the cutoﬀ interaction F = M {λf L1 + λ2 f 2 L2 }, where f is a smooth switching function of compact support, which is equal to one on some time-slice MT = (−T, T ) × Σ. The desired identity dJ(x)I will follow if we can show that, in the sense of formal power series, in 0 = dJ(x)F = (422) R (dJ(x); F ⊗n ), x ∈ MT n n! n n modulo J0 for any such cutoﬀ function f . Expanding the retarded products in terms of time-ordered products gives the equivalent relation iF/

T (dJ(x) ⊗ e⊗

) = 0 mod J0 for all x ∈ MT ,

(423)

which is again to be understood in the sense of formal power series. At the level of classical ﬁelds, we have dJ(x) = (S0 + F, Φ(x)) · (Φ‡ (x), S0 + F )

for all x ∈ MT .

(424)

Hence, (423) is equivalent to the equation ,5 6 , iF/ iF/ = −T sˆ0 Φ(x) · (Φ‡ (x), F ) + sˆ0 Φ‡ (x) · (Φ(x), F ) ⊗ e⊗ T dJ0 (x) ⊗ e⊗ ,5 6 iF/ − T (F, Φ(x)) · (Φ‡ (x), F ) ⊗ e⊗ mod J0 . (425) We claim that this equation can be satisﬁed as a consequence of our Ward identity T12a by a redeﬁnition of the time-ordered products. In fact, we shall now show that our Ward identity T12a can even be used to prove the following stronger identity: t i Tt+1 (dJ0 (y) ⊗ L|I1 | (XI1 ) ⊗ · · · ⊗ L|It | (XIt )) I1 ∪···∪It =n

=−

I1 ∪···∪It =n

t−1 t i Tt (L|I1 | (XI1 ) ⊗ · · · ⊗ {ˆ s0 Φ(y) · (Φ‡ (y), L|Ii | (XIi )) i=1

+ sˆ0 Φ‡ (y) · (Φ(y), L|Ii | (XIi ))} ⊗ · · · ⊗ L|It | (XIt )) t−2 i − Tt−1 (L|I1 | (XI1 ) ⊗ · · · ⊗ (L|Ii | (XIi ), Φ‡ (y)) I1 ∪···∪It =n

1≤i<j≤t

· (Φ(y), L|Ij | (XIj )) ⊗ · · · ⊗ L|It | (XIt ))

(426)

modulo J0 . This identity implies (423) as may be seen by multiplying each term by λn /n!, integrating against f (x1 ), . . . , f (xn ), and summing over n. Thus, it remains

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to be seen that (426) follows from the Ward identity T12a. For n = 0, we get the condition T1 (dJ0 (y)) = 0, which is just the condition of current conservation in the free theory and hence is satisﬁed. For n > 0, we proceed inductively. This shows that, at the order considered, the failure of (426) to be satisﬁed is of the form T1 (αn (y, x1 , . . . , xn )), where αn (y, x1 , . . . , xn ) is a local covariant functional that is supported on the total diagonal. We now show that we can set this quantity to 0. To do this, we pick a testfunction h ∈ C ∞ (M ) with the following properties: h(y) = 1 in an open neighborhood of {x1 , . . . , xn }, h(y) = 0 towards the future of Σ+ , and towards the past of Σ− , where Σ± are Cauchy surfaces in the future/past of {x1 , . . . , xn }. We may thus write dh = γ+ − γ− , where γ± are 1-forms that are supported in the future/past of {x1 , . . . , xn }. Now, from Q0 = M T1 (J0 ) ∧ γ± , and from the causal factorization of the time-ordered products, we have h(y)Tt+1 (dJ0 (y) ⊗ L|I1 | (XI1 ) ⊗ · · · ⊗ L|It | (XIt ))dy M

= [Q0 , Tt (L|I1 | (XI1 ) ⊗ · · · ⊗ L|It | (XIt ))] = iˆ s0 Tt (L|I1 | (XI1 ) ⊗ · · · ⊗ L|It | (XIt )), where the last equation is modulo J0 . We also have h(y)(O(xi ), Φ‡ (y)) · (Φ(y), O(xj ))dy = (O(xi ), O(xj ))

(427)

(428)

M

for any O. It follows from these equations that if we integrate (426) against h(y), then we get an identity follows from the known Ward identity T12a. Stated diﬀerently, because h(y) = 1 in a neighborhood of {x1 , . . . , xn }, and because the failure αn of (426) to hold is supported on the total diagonal, it must satisfy αn (y, x1 , . . . , xn )dy = 0 mod J0 . (429) M

By Lemma 9, it hence follows that there exists a local covariant βn supported on the total diagonal such that dy βn (y, x1 , . . . , xn ) = αn (y, x1 , . . . , xn ), where βn is a 3-form in the y-entry, and a 4-form in each xi -entry, and where dy is the exterior diﬀerential acting on the y-variable. We may now redeﬁne time-ordered products with one factor of J0 (y) and n factors of L1 (xi ), i = 1, . . . , n by taking Dn+1 (J0 (y) ⊗ L1 (x1 ) ⊗ · · · ⊗ L1 (xn )) := βn (y, x1 , . . . , xn ). Then the redeﬁned timeordered products satisfy (426). 4.7. Proof that Q2I = 0 We know from the previous subsection that the interacting BRST-current is conany x in a domain served, dJ(x)I = 0 for any x, or equivalently, dJ(x) F = 0 for 2 2 MT = (−T, T ) × Σ where the function f in F = {λf L1 +λ f L2 } is equal to 1. Thus, the deﬁnition of the interacting BRST-charge QI = γ ∧ JI is independent

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of the choice of the compactly supported closed 1-form γ dual to the Cauchy surface Σ. Using the Bogoliubov formula for the interacting ﬁeld operators, the desired equality Q2I = 0 is equivalent to the equation 2 0 = Q2F = γ(x) ∧ J(x)F =

1 in+m 2 n,m n+m n!m!

[Rn (J(x); F ⊗n ), Rm (J(y); F ⊗m )]γ(x)γ(y)dxdy (430)

modulo J0 , where γ is now chosen to be supported in MT . Note that, as usual, we mean the graded commutator, which is actually the anti-commutator in the above expression. Now, because the interacting BRST-charge QF as deﬁned using the cutoﬀ interaction F is independent upon the choice of the compactly supported closed 1-form in γ dual to Σ, we may write the interacting BRST-charge either as QF = γ (1) ∧ JF , or as QF = γ (2) ∧ JF . We may therefore alternatively write 1 in Q2F = Rn+1 (J(x); J(y) ⊗ F ⊗n )γ (1) (x)γ (2) (y)dxdy + (1 ↔ 2), (431) 2 n n n! where we have also used the GLZ-formula (266). We now make a particular choice for γ (1) and γ (2) that will facilitate the evaluation of this expression. We choose γ (1) = dh(1) + dh(2) , where h(1) and h(2) are smooth scalar functions with the following properties: (a) the support of h(1) is compact, (b) h(1) = 1 on the support of γ (2) , (c) the support of h(2) is contained in the causal past of the support of γ (2) . Due to these support properties and the causal support properties of the retarded products, the above expression can then be written as Q2F = −

1 in Rn+1 (dJ(x); J(y) ⊗ F ⊗n )h(1) (x)γ (2) (y)dxdy. 2 n n n!

(432)

Below, we will show that, for any x, y ∈ MT , the following identity is a consequence of the Ward identity T12b: iF/

R(dJ(x); J(y) ⊗ e⊗

) = iR({(S0 + F, Φ(x)) · (Φ‡ (x), J(y)) + (S0 + F, Φ‡ (x)) · (Φ(x), J(y))}; e⊗

iF/

) mod J0 . (433)

We now apply this identity and use that h(1) = 1 on the support of γ (2) . Then we obtain i iF/ (434) Q2F = R((S, J(x)); e⊗ )γ (2) (x)dx, 2 again, modulo J0 . However, sˆJ = dK, so using T11, the right-hand side vanishes by dγ (2) = 0. Thus, we have proved Q2F = 0 modulo J0 , and it remains to prove Eq. (433). That equation can be written equivalently in terms of time-ordered

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products iF/

T (dJ(x) ⊗ J(y) ⊗ e⊗

) = iT ({(S0 + F, Φ(x)) · (Φ‡ (x), J(y)) + (S0 + F, Φ‡ (x)) · (Φ(x), J(y))} ⊗ e⊗

iF/

mod J0 ,

)

(435) using the formulae relating time-ordered and retarded products given above. We will prove it in this form. Using Eq. (73), Eq. (435) may be written alternatively as iF/

T (dJ0 (x) ⊗ J(y) ⊗ e⊗

)

‡

= − T ({ˆ s0 Φ(x) · (Φ (x), F ) + (Φ ↔ Φ‡ )} ⊗ J(y) ⊗ e⊗

iF/

− T ({(F, Φ(x)) · (Φ‡ (x), F ) + (Φ ↔ Φ‡ )} ⊗ J(y) ⊗

)

iF/ e⊗ )

+ iT ({ˆ s0Φ(x) · (Φ‡ (x), J(y)) + (F, Φ(x)) · (Φ‡ (x), J(y)) + (Φ ↔ Φ‡ )} ⊗ e⊗

iF/

) mod J0 .

(436)

We will now show that this equation can be satisﬁed as a consequence of our Ward identity T12b. To prove this identity, we employ the same technique as in the previous subsection. We ﬁrst formulate a set of stronger identities that will imply (436). This set of conditions is completely analogous to Eq. (426), with the diﬀerence that in Eq. (426), we replace Li (X) everywhere by Li (X) + τ Ji (y, X), and expand the resulting set of equations to ﬁrst order in τ . As in the proof of Eq. (426), the resulting equations are established inductively in n. For n = 0 the identity can be veriﬁed directly using the deﬁnitions made in free gauge theory. Inductively, the resulting equations will then be violated at order n by a potential “anomaly” term of the form T1 (αn (x, y, x1 , . . . , xn )), where αn is now an element of P4/3/4/··· /4 (M n+2 ). As in the treatment of Eq. (426), the Ward identity T12b then implies that αn (x, y, x1 , . . . , xn )dx = 0 (437) M

while the GLZ-identity, together with the fact that dJI = 0 can be seen to imply the relation dy αn (x, y, x1 , . . . , xn )dx1 · · · dxn = 0. (438) M

Equations (437) and (438) can now be used to show that the time-ordered products can be redeﬁned, if necessary, to remove the anomaly αn . By the same argument as in the previous subsection, the ﬁrst identity (437) implies that αn (x, y, x1 , . . . , xn ) = dx δn (x, y, x1 , . . . , xn )

(439)

for some δn ∈ P3/3/4/···/4 (M n+2 ). We would like to redeﬁne the time-ordered products using the quantity Dn (see Sec. 3.6) Dn+2 (J0 (x) ⊗ J0 (y) ⊗ L1 (x1 ) ⊗ · · · ⊗ L1 (xn ) := δn (x, y, x1 , . . . , xn ).

(440)

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In view of Eq. (439), this would remove the anomaly. However, it is not clear that we can make this redeﬁnition, because the time-ordered products with two free BRST-currents at x and y must be anti-symmetric in x and y, and this need not be the case for δn in (439). We will circumvent this problem by using a modiﬁed δˆn in Eq. (440) to redeﬁne the time-ordered products with 2 currents. To construct the modiﬁed δˆn , we consider the quantity (1) (2) β(γ , γ ) = δn (x, y, z1 , . . . , zn )γ (1) (x)γ (2) (y)dxdydz1 · · · dzn + (1 ↔ 2), (441) where γ (1) , γ (2) are now arbitrary 1-forms of compact support. β is evidently closely related to the symmetric part of δn , which we would like to be zero. From Eq. (438), we have β(dh(1) , dh(2) ) = 0 for any pair of compactly supported scalar functions h(1) , h(2) . As we shall show presently, this implies that we can write β(γ (1) , γ (2) ) = C(dγ (1) , γ (2) ) + (1 ↔ 2)

(442)

where C has a distributional kernel C ∈ P2/3 (M 2 ). We now deﬁne δˆn (x, y, z1 , . . . , zn ) = δn (x, y, z1 , . . . , zn ) − dx C(x, y)δ(y, z1 , . . . , zn ) − (x ↔ y), (443) ˆ n in order to redeﬁne which is manifestly anti-symmetric in x, y. We use this new D the time-ordered products with 2 currents as in Eq. (440) instead of the old Dn . Evidently, the new time-ordered product is now anti-symmetric in x, y. Furthermore, as a consequence of Eq. (442), the new anomaly for the redeﬁned time-ordered products α ˆ n satisﬁes α ˆ n (x, y, z1 , . . . , zn )dz1 · · · dzn = 0. (444) It follows from this equation that α ˆn (x, y, z1 , . . . , zn ) =

n

dl δn/l (x, y, z1 , . . . , zn ),

dl = dzl ∧

l=1

∂ ∂zl

(445)

for some δn/l ∈ P4/3/4/··· /3··· /4 (M n+2 ). We use these quantities to make a ﬁnal redeﬁnition of the time-ordered products. We have sˆ0 Φ(x1 ) · (Φ‡ (x1 ), L1 (x2 )) + sˆ0 Φ‡ (x1 ) · (Φ(x1 ), L1 (x2 )) = d1 J1 (x1 )δ(x1 , x2 ) + d2 Σ1 (x1 , x2 )

(446)

for some Σ1 ∈ P3/3 (M 2 ). We redeﬁne the time-ordered products involving these quantities using the quantities (see Sec. 3.6) Dn+1 (J0 (x) ⊗ L1 (z1 ) ⊗ · · · ⊗ Σ1 (y, zl ) ⊗ · · · ⊗ L1 (zn )) := δn/l (x, y, z1 , . . . , zn )). (447) This ﬁnal redeﬁnition then removes the anomaly α ˆn .

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It remains to prove Eq. (442). We formulate this result as a lemma: Lemma 10. Let β ∈ P3/3 (M 2 ) such that β(dh(1) , dh(2) ) = 0 for any pair of compactly supported scalar functions h(1) , h(2) . Then β can be written in the form (442) for some C ∈ P2/3 (M 2 ). Proof. β is of the form

β(γ (1) , γ (2) ) =

dx M

p

β µν1 ··· νm σ γµ(1) ∇ν1 · · · ∇νm γσ(2) ,

(448)

m=0

where β are tensor ﬁelds that are locally constructed out of g, ∇, and Φ, Φ‡ . We claim that the condition β(dh(1) , dh(2) ) = 0 and the symmetry of β implies that β can be put into the form (442). Since the commutator of two derivatives gives a Riemann tensor, we may assume that each tensor β in the sum in (448) is symmetric under the exchange of the indices ν1 , . . . , νm , β µν1 ···νm σ = β µ(ν1 ···νm )σ .

(449)

Now consider the contribution to (448) with the highest number of derivatives, m = p. By varying β(dh(1) , dh(2) ) = 0 with respect to h(1) , h(2) there follows the additional symmetry β (µν1 ···νp σ) = 0.

(450)

Consider now the vector ﬁeld deﬁned by B µ = β µν1 ···νp σ ∇ν1 · · · ∇νp γσ .

(451)

Using the symmetry property (449), this may be rewritten as B µ = β µν1 ··· νp σ ∇ν1 · · · ∇[νp γσ] + β µ(ν1 ··· νp σ) ∇ν1 · · · ∇νp γσ .

(452)

Then, using the symmetry (450), this may further be written as B µ = β µν1 ··· νp σ ∇ν1 · · · ∇[νp γσ] −

2 β σ(µν1 ··· νp ) ∇ν1 · · · ∇[νp γσ] p+2

2(p + 1) ∇ν {β µ(να1 ...αp−1 σ) ∇α1 · · · ∇αp−1 γσ − (µ ↔ ν)} p+2 + terms with (p − 1) derivatives on γσ . −

(2)

(453)

(1)

in this equation, contract both sides with γ , and integrate, Now put γ = γ to obtain an expression for the highest derivative term in β. Using this expression, we ﬁnd that β(γ (1) , γ (2) ) is given by a sum of terms each of which contains either (1) (2) ∇[µ γν] or ∇[µ γν] , or which contains at most derivative terms of order p − 1. Consequently, using the symmetry of β, we can write β(γ (1) , γ (2) ) = C(dγ (1) , γ (2) ) + C(dγ (2) , γ (1) ) + Rp−1 (γ (1) , γ (2) ),

(454)

where Rp−1 stands for a remainder term of the form (448) containing at most p − 1 derivatives, and where C is also of the form (448). If we now take γ (1) = dh(1) , and

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γ (2) = dh(2) in Eq. (454), and use β(dh(1) , dh(2) ) = 0, then we see that Rp−1 again satisﬁes Rp−1 (dh(1) , dh(2) ) = 0. Thus, we may repeat the arguments just given for Rp−1 and conclude that β can be written as in Eq. (454) with a new C, and a remainder Rp−2 containing at most p − 2 derivatives. Thus, further repeating this procedure, we ﬁnd (454) must hold for some C and a remainder of the form that (1) (2) R0 (γ (1) , γ (2) ) = εγµ rµν γν . Now, R0 is symmetric, so r[µν] = 0. Furthermore, we have R0 (dh(1) , dh(2) ) = 0 for all compactly supported h(1) , h(2) . Varying this equation with respect to h(2) , we get 0 = ∇µ (rµν ∇ν h(1) ). Now, pick a point x ∈ M , and choose h(1) so that h(1) (x) = 0. Then it follows that rµν ∇µ ∇ν h(1) = 0 at x. Because ∇µ ∇ν h(1) is an arbitrary symmetric tensor at x, it follows that r(µν) = 0, and therefore that rµν = 0, thus proving the desired decomposition (442). This completes the proof.

4.8. Proof that [QI , ΨI ] = 0 when Ψ is gauge invariant Here we show that the Ward identity T12c implies [QI , ΨI (x)] = 0 modulo J0 , whenever Ψ ∈ P(M ) is a strictly gauge invariant operator of ghost number 0, i.e. / Ψ = Θsi (F, DF, . . . , Dki F ). As in the proof given in the previous subsection, this property will follow from the identity iF/

T (dJ0 (x) ⊗ Ψ(y) ⊗ e⊗

)

‡

= −T ({ˆ s0 Φ(x) · (Φ (x), F ) + (Φ ↔ Φ‡ )} ⊗ Ψ(y) ⊗ e⊗

iF/

− T ({(F, Φ(x)) · (Φ‡ (x), F ) + (Φ ↔ Φ‡ )} ⊗ Ψ(y) ⊗ e⊗

)

iF/

‡

)

‡

+ iT ({ˆ s0Φ(x) · (Φ (x), Ψ(y)) + (F, Φ(x)) · (Φ (x), Ψ(y)) (455) + (Φ ↔ Φ‡ )} ⊗ e⊗ ) mod J0 , where again F = (λf L1 + λ2 f 2 L2 ). One can now formulate a stronger set of local identities analogous to Eq. (426), and one can prove these identities using T12c along the same lines as in the previous subsection, with J(y) there replaced everywhere by Ψ(y). The potential anomaly of the stronger identities (and therefore the possible violation of Eq. (455)) can now be removed by a suitable redeﬁnition of the time-ordered products Tn+2 (J0 (x) ⊗ Ψ0 (y) ⊗ L1 (x1 ) ⊗ L1 (xn )) at nth order in perturbation theory, where Ψ = Ψ0 + λΨ1 + λ2 Ψ2 + · · · . However, contrary to the case in the previous subsection, we now do not have to worry about potential symmetry issues, that had to be dealt with there, because Ψ0 is always distinct from J0 , the latter having ghost number 1. iF/

4.9. Relation to other perturbative formulations of gauge invariance In our approach to interacting quantum gauge theories, the gauge invarince of the theory was incorporated in the conditions that there exists a conserved interacting

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BRST-current operator, and that the corresponding charge operator be nilpotent. As we demonstrated, this follows from our Ward identity (322), the generating identity for T12a, T12b, and T12c. In the literature on perturbative quantum ﬁeld theory in ﬂat spacetime, other notions of gauge invariance of the quantum ﬁeld theory have been suggested, and other conditions have been proposed to ensure those. We now brieﬂy discuss some of these, and explain why these formulations are not suitable in curved spacetime. Diagrammatic Approaches (Dimensional Regularization). Historically, the ﬁrst proofs of gauge invariance of the renormalized perturbation series in gauge theories on ﬂat R4 were performed at the level of Feynman diagrams. The gaugeinvariance of the classical Lagrangian implies certain formal identities between the diagrams at the unrenormalized level. At the renormalized level, these identities in turn would formallyo imply the gauge-invariance of amplitudes. One must thus prove that these identities remain valid at the renormalized level. For this, it is important to have a regularization/renormalization scheme that preserves these identities. Such a scheme was found by ’t Hooft and Veltmann [73–75], namely dimensional regularization. Because that scheme is also very handy for calculations (except for certain calculations involving Dirac-matrices), it has remained the most popular approach among practitioners. Modern presentations of this approach based on the Hopf-algebra structure behind renormalization in the BPHZapproach [24, 25, 87] are [112, 113]. In curved space, scattering amplitudes are not well-deﬁned, because there is no sharp notion of particle in general. At a more formal level, diagrammatic expansions in general are problematic because there does not exist a unique Feynman propagator, so a given Feynman diagram can mean very diﬀerent mathematical expressions depending on one’s choice of Feynman propagator. One may of course expand the theory using any Feynman propagator. However, then the problem arises that the Feynman propagator is not a local covariant functional of the metric, but also depends upon boundary/initial conditions, which are intrinsically non-local. This would interfere with ones ability to reduce the ambiguity to local curvature terms. One might be tempted to take the local Feynman parametrix HF , which is local and covariant. But this has the undesirable property that it is not a solution of the ﬁeld equation, but only a Green’s function modulo a smooth remainder, see Appendix D. This severely complicates the treatment of quantities that vanish due to ﬁeld equations, and of the Ward identities. Finally, in curved space, the Feynman propagator is only well deﬁned as a distribution in position space, while techniques such as dimensional regularization seem to work best in momentum spacetime. Thus, a diagrammatic proof of quantum gauge invariance of the Yang–Mills theory in curved spacetime seems to be diﬃcult and somewhat unnatural.

o We say “formally”, because amplitudes can have additional infra-red divergences, which are very hard to treat in a gauge-invariant manner.

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Zinn–Justin Equation. In many formal approaches to perturbative gauge theory in ﬂat spacetime R4 , gauge invariance of the theory is expressed in terms of an integrated condition involving the so-called “eﬀective action”, Γeﬀ (S) of the theory associated with the classical action S = S0 + λS1 + λ2 S2 . The eﬀective action is a generating functional for the 1-particle irreducible Feynman diagrams of the theory. The condition for perturbative gauge invariance is simply and elegantly encoded in the relation [119] (Γeﬀ (S), Γeﬀ (S)) = 0.

(456)

Condition (456) is referred to as the “Slavnov Taylor identity” in “Zinn–Justin form”. It is closely related to the “master equation” that arises in the Batalin– Vilkovisky formalism [6] (see also [64]), and it reduces to the classical condition (S, S) = 0 for BRST-invariance when one puts = 0. At the formal level, the Slavnov–Taylor identity is most straightforwardly derived from the path integral. It is also in this setting that one can understand relatively easily that it formally implies the absence of (inﬁnite) counterterms to the classical action violating gauge invariance. However, by itself, it does not imply the gauge invariance of physical quantities such as scattering amplitudes, or identities like Q2I = 0. The eﬀective action Γeﬀ (S) is only a formal quantity, since it involves integrations over all of spacetime. These integrations typically lead to infra-red divergences, as is in particular the case also in pure Yang–Mills theory. Therefore, also the Slavnov–Taylor equation (456) is only aformal identity. If the interaction λS1 +λ2 S2 is replaced by a local interaction, F = {λf L1 + λ2 f 2 L2 }, with f a smooth cutoﬀ function of compact support, then the infra-red divergences are avoided, and the eﬀective action Γeﬀ (S0 + F ) is well deﬁned. The precise deﬁnition of Γeﬀ (S0 + F ) within our framework is given in Appendix B. However, for the cutoﬀ-interaction, the Slavnov–Taylor identity no longer holds. Nevertheless, it can be shown that Γeﬀ (S0 + F ) satisﬁes an analogous equation, given by Eq. (490). That equation can be used to formally “derive” Eq. (456), if one could prove that the anomaly in Eq. (490) vanishes. Since the anomaly is closely related to the failure of the interacting BRST-current to be conserved, one might expect to be able to remove the anomaly by an argument similar to our proof of T12a, but this has not been worked out even in ﬂat spacetime. In curved spacetime, we may still deﬁne an eﬀective action, Γeﬀ (S0 + F ), which now depends upon the arbitrary choice of a quasifree Hadamard state ω, see Appendix E. Hence it is deﬁnitely not a quantity that depends locally and covariantly upon the metric, but also on the non-local choice of ω, Therefore, even at the formal level, it is not clear that the Slavnov–Taylor identity can be viewed as a renormalization condition that is compatible with the locality and covariance of the time-ordered products. Also, while the Slavnov–Taylor identity can again be formally derived from our Ward Identity T12a, it does not directly imply the gauge-invariance of physical quantities such as n-point functions, and it also does not prove (even formally) that the OPE closes among physical operators. For these

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reasons, we prefer to work with the Ward-identities T12a, T12b, T12c in this paper, which are rigorous, and have a local and covariant character. Despite the above differences, the Zinn–Justin is probably to be regarded as the closest analogue to our renormalization conditions expressing local gauge invariance. The similarities can be made more explicit using our generating formula (322) (or Eq. (336)) for our Ward identities. Causal Approach. A condition expressing perturbative gauge invariance in ﬂat spacetime that is of a more local nature than (456) has been proposed in a series of papers by D¨ utsch et al. [32–36,103], see also [77–79,60–62]. These works are also related to the “quantum Noether condition” [79]. Let Tn (x1 , . . . , xn ) be the timeordered product of Tn (L1 (x1 ) ⊗ · · · ⊗ L1 (xn )). (In the above papers, the interaction Lagrangian 4-form is here identiﬁed with a scalar by taking the Hodge dual.) Let Q0 be the free BRST-charge. Then it is postulated that there exists a set of timeordered products Tn/l (x1 , . . . , xn ) with the insertionp of some (unspeciﬁed) 3-formvalued ﬁeld in the lth entry such that [Q0 , Tn (x1 , . . . , xn )] = i

n

dl Tn/l (x1 , . . . , xn ) modulo J0

(457)

l=1

for all n > 0, where dl = dxµl ∧ ∂/∂xµl is the exterior derivative acting on the lth entry. The condition is to be viewed as a normalization on the time-ordered products involving n factors of the interaction L1 . Note that there are no explicitq conditions imposed on time-ordered products involving L2 . Note also that the condition is imposed only modulo J0 , that is, on shell. In fact, the authors of the above papers always work in a representation, where the ﬁeld equations automatically hold (see Sec. 3), rather than at the algebraic level, where the ﬁeld equations need not be imposed as a relation. A related diﬀerence is that the above authors do not work with anti-ﬁelds, without which it appears to be very cumbersome to obtain powerful consistency relations for potential anomalies of (457). (Some aspects of this diﬀerence are addressed in [4].) The key motivation for condition (457) is that, as our condition T12a, it formally implies that the S-matrix commutes with Q0 in the “adiabatic limit”, see above. Indeed, if we formally integrate (457) over (R4 )n , then the right-hand side formally vanishes, being a total derivative. This shows that S formally commutes with Q0 . However, unlike our Ward identities, we do not believe that Eq. (457) would imply Q2I = 0 for the interacting BRST-charge, or [QI , ΨI ] = 0 for gauge invariant operators. in particular, Tn/l (x1 , . . . , xn ) should be symmetric in all variables except xl , and it is a 3-form in xl . q As explained in the above papers, however, implicit normalization conditions on time-ordered products with factors of L2 arise from (457). Also, (457) apparently may even be used to determine the form of L1 , which is simply given in our approach. p Thus

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The relation (457) is apparently diﬀerent from our corresponding condition T12a (considered in ﬂat spacetime), so we now brieﬂy outline how they are related. Consider a prescription for the time-ordered products satisfying our Ward identity T12a, so that, in particular, Eq. (457) does not hold for that prescription. However, let us now make the following redeﬁnition of the time-ordered products containing two factors of L1 , that is, T2 (L1 (x1 ) ⊗ L1 (x2 )) → T2 (L1 (x1 ) ⊗ L1 (x2 )) + T1 (L2 (x1 , x2 )),

(458)

where we recall the notation L2 (x1 , x2 ) = 2L2 (x1 )δ(x1 , x2 ). Let us further note that sˆ0 L2 (x1 , x2 ) + (L1 (x1 ), L1 (x2 )) = d1 O2/1 (x1 , x2 ) + d2 O2/2 (x1 , x2 )

(459)

for some ﬁelds O2/1 ∈ P4/3 and O2/2 ∈ P3/4 supported on the diagonal, and s0 Tn modulo J0 , and deﬁning Tn/l by sˆ0 L1 = dO1 . Using that [Q0 , Tn ] = iˆ Tn/l (x1 , . . . , xn ) = Tn−1 (L1 (x1 ) ⊗ · · · ⊗ O2/j (xl+j−1 , xl+j ) ⊗ · · · ⊗ L1 (xn )) j=1,2

+ cycl. perm. + Tn (L1 (x1 ) ⊗ · · · ⊗ O1 (xl ) ⊗ · · · ⊗ L1 (xn )), (460) one can then check that Eq. (457) holds. Thus, our Ward identity implies (457) if a ﬁnite renormalization change is made, and presumably (457) may also be used to deduce our Ward identity T12a. Note, however, that our identities T12b and T12c are conditions that go deﬁnitely beyond the Ward identities (457). 5. Summary and Outlook In this paper, we have given, for the ﬁrst time, a perturbative construction of non-abelian Yang–Mills theory on arbitrary globally hyperbolic curved, Lorentzian spacetime manifolds. Following earlier work on quantum ﬁeld theory in curved spacetime, our strategy was to construct the interacting ﬁeld operators and the algebra that they generate. This was accomplished starting from a gauge ﬁxed version of the theory with ghost and anti-ﬁelds, and then deﬁning the algebra of observables of perturbative Yang–Mills theory as the BRST-cohomology of the corresponding algebra associated with the gauge ﬁxed theory. To implement this strategy it was necessary to ﬁrst ﬁnd a prescription for deﬁning a conserved interacting BRSTcurrent, and for which the corresponding conserved charge is furthermore nilpotent. We were able to characterize such a prescription by a novel set of Ward identities for the time-ordered products in the underlying free theory. We furthermore showed how to ﬁnd a renormalization prescription for which the Ward identities indeed hold. In addition, we showed that our renormalization prescription also satisﬁes other other important properties, notably the condition of general covariance. Altogether, these constructions provide a proof that perturbative Yang–Mills theory

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can be deﬁned as a consistent, local covariant quantum ﬁeld theory (to all orders in perturbation theory), for any globally hyperbolic spacetime. A key feature of our approach is that it is entirely local in nature, in the sense that our renormalization conditions only make reference to local quantities. A local approach is essential in a generic curved spacetime in order ﬁnd the correct renormalization prescription respecting locality and general covariance. But it is also advantageous in ﬂat spacetime in many respects compared to other existing approaches in ﬂat spacetime, such as approaches focused on the scattering matrix, or approaches based on the path-integral. The key advantages of our approach are the following: • Because our approach is completely local, we can completely disentangle the infrared divergences and ultra-violet divergences of the theory. This is mandatory in Yang–Mills theory, where infra-red divergences pose a major problem, even in ﬂat spacetime. • Because our approach is algebraic in nature, the objects of primary interest are the interacting ﬁeld operators, rather than auxiliary quantities such as eﬀective actions or scattering matrices. This makes it easy for us to prove the important result that the operator product expansion of Yang–Mills theory closes among gauge invariant ﬁelds, and that the renormalization group ﬂow does not leave the space of gauge invariant ﬁelds. On the other hand, it tends to be much more complicated to prove such statements in other formalisms even in ﬂat spacetime. • Because our approach is local and covariant, we can directly analyze the dependence of our constructions on the metric. For example, one can directly obtain the following result: If a non-abelian gauge theory has trivial RG-ﬂow in ﬂat spacetime (such as the N = 4 super Yang–Mills theory), then it also must have trivial RG-ﬂow in any spacetime in which possible renormalizable curvature couplings in the Lagrangian (such as a R Tr Φ2 -type term) happen to vanish. Thus, the N = 4 super Yang–Mills theory has trivial RG-ﬂow in any spacetime with vanishing scalar curvature. Note that, unlike in ﬂat spacetime, this does by no means imply that the theory is conformally invariant, because a spacetime with R = 0 will not in general admit any conformal isometries. A weak point of our constructions, as for most other perturbative constructions in quantum ﬁeld theory, is that one does not have any control over the convergence of the perturbation series. This is in particular a problem for quantum states such as bound states that are not expected to have a perturbative description. A partial resolution of this problem is provided by the operator product expansion (see Sec. 4.2), because it allows one to compute n-point correlation functions in terms of OPE-coeﬃcients and 1-point functions (“form factors”), which one may regard as additional phenomenological input. But a full solution would presumably require to go beyond perturbation theory, which seems a distant goal even in ﬂat spacetime.

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Apart from this problem, there remain a couple of technical questions related to the perturbation expansion, of which we list a few: 5.1. Matter fields, anomalies In this paper, we have considered only pure Yang–Mills theory for simplicity. Clearly, one would like to add matter ﬁelds, such as fermion ﬁelds in a representation R of the gauge group G. In that case, the general strategy and methods of our paper can still be applied. But it is no longer clear that the Ward identities formulated in this paper can still be satisﬁed, as there can now be non-trivial solutions to the corresponding consistency conditions in the presence of chiral fermions. If the Ward identities cannot be satisﬁed, one speaks of an anomaly. In our case this would imply that the interacting BRST-current is no longer conserved, and that a conserved BRST-charge cannot be deﬁned, meaning that the theory is inconsistent at the quantum level. In ﬂat space, this can happen if the gauge group contains factors of U (1), for certain representations R. By the general covariance of our construction, the types of anomalies in ﬂat space must then also be present in any curved spacetime. However, in curved space, a new type of anomaly can also arise in the presence of chiral fermions and abelian factors in the gauge group. For example, one can compute that the divergence dJI (exterior diﬀerential) of the quantum BRST current operator is not zero as required by consistency, but it has a contribution to its divergence proportional of the type given in Eq. (63), which cannot be eliminated by ﬁnite renormalization. In particular, one ﬁnds a contribution dJI ∝ AI + · · · at 1-loop order, where A = const. Tr[R(TK )]C K Tr(R ∧ R) (461) K

and where the sum over K is over the abelian generators of the Lie-algebra only. In the standard model, with gauge group G = SU (3)×SU (2)×U (1), the representation of the abelian generator Y (charge assignments of the fermion ﬁelds) is precisely so that A = 0, as also observed by [58, 93]. However, we do not know whether the theory remains free of this kind of anomaly to arbitrary orders in the perturbation series. This would be important to check. It is also important to investigate whether the renormalization conditions considered in this paper can be used to show that a divergence-free interacting stress tensor TIµν can be constructed. Here, one can presumably use the techniques of [70] to show that there is no anomaly for this conservation equation, but it would be important to settle the details. A particularly interesting question in this connection is to see precisely how the expected trace anomaly for this quantity arises in the present framework. 5.2. Other gauge fixing conditions In this paper, we have worked with a speciﬁc gauge ﬁxing condition (the Lorentz gauge). The important feature of this condition for our purposes was that the ﬁeld

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equation for the spin-1 ﬁeld then becomes A + · · · = 0, where the dots represent terms with less derivatives. This was important because only in that case are we able to construct a Hadamard parametrix for the vector ﬁeld, which is a key ingredient in our constructions. However, one may wish to consider other types of gauge ﬁxing conditions, both for practical purposes, as well as a matter of principle. Even if a Hadamard parametrix could still be deﬁned in such cases, it is not a priori clear that the theories deﬁned using diﬀerent gauge ﬁxing conditions are equivalent. In our approach, equivalence would mean that the algebras of observables obtained from diﬀerent gauge ﬁxing conditions are canonically isomorphic. We have not investigated the question whether this is indeed the case.

5.3. Background independence In our constructions (as in all other standard approaches to perturbative Yang–Mills theory), we have split the Yang–Mills connection D = ∇ + iλA into the standard ﬂat, non-dynamical background connection ∇, and a dynamical ﬁeld A. At the level of classical Yang–Mills theory it is evident that it is immaterial how this split is made, i.e. classical Yang–Mills theory is background independent in this sense. In particular, the standard choice ∇ = ∂ in ﬂat spacetime is just one possibility among inﬁnitely many other ones. In the gauge ﬁxed classical theory with ghosts and anti-ﬁelds, diﬀerent choices of the background connection give rise to diﬀerent classical actions. The diﬀerence is, however, only by a BRST-exact term. Since the classical theory is deﬁned as the BRST-cohomology, such a BRST-exact term does not change the brackets between the physical observables, and hence the theory is background independent also in the gauge-ﬁxed formalism. Unfortunately, we do not know whether the same statement is still true in the quantum ﬁeld theory, i.e. we do not know whether the algebras of physical observables associated with diﬀerent choices of the background connection are still isomorphic. The diﬃculty is that, in quantum ﬁeld theory, the background connection ∇ is treated very diﬀerently from the dynamical part A: The background connection would enter the deﬁnition of the propagators, e.g., of the local Hadamard parametrices, while A is a quantum ﬁeld. The question whether one is allowed to shift parts of A into ∇ and vice versa is closely related to the question whether the “principle of perturbative agreement” formulated in [70] can be satisﬁed with respect to the gauge connection. The satisfaction of this principle is equivalent to certain Ward identities at the level of the time-ordered products, but we do not know in the present case whether these Ward identities can be satisﬁed, i.e. whether there are any anomalies. As shown in [70], a potential violation of these identities may be identiﬁed with a certain cohomology class. In our case, when the background structure in question is a gauge connection, the potential violation would be represented by a certain 2-cocycle on the space of all gauge potentials. An anomaly of this sort could arise in theories with chiral

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fermions. Thus, the question of background independence in quantum Yang–Mills theory remains an open problem, which has not been solved, to our knowledge, even in ﬂat spacetime. Acknowledgments I would like to thank F. Brennecke, D. Buchholz, M. D¨ utsch, L. Faddeev, K. Fredenhagen and R. M. Wald for discussions, and I would also like to thank M. Henneaux for discussions at an early stage of this work. Some parts of this work were completed during the 2007 program “Mathematical and Physical Aspects of Perturbative Approaches to Quantum Field Theory” at the Erwin-Schroedinger Institute, Vienna, to which I express my gratitude for its ﬁnancial support and hospitality. Appendix A. U (1)-Gauge Theory Without Vector Potential In the case of a pure U (1)-gauge theory, one may consider a diﬀerent starting point for deﬁning the theory, using as the basic input only the ﬁeld equations for the 2-form ﬁeld strength tensor rather than the action for the gauge potential A. This is because the ﬁeld equations may then be written without reference to the gauge potential as equations for the ﬁeld strength F , viewed now as the dynamical variable. The equations are of course Maxwell’s equations, in diﬀerential forms notation dF = 0 and d ∗ F = 0. On a curved manifold M with nontrivial topology, not every closed form F needs to be exact, so it does not follow from the ﬁeld equation dF = 0 that F can be written in terms of a vector potential as F = dA. Thus, using only Maxwell’s equations as the input deﬁnes a more general theory classically than the action dA ∧ ∗ dA, because cohomologically non-trivial solutions F are possible. In this section, we brieﬂy indicate how one may quantize such a theory. A globally hyperbolic spacetime always has topology M = Σ × R, so closed but non-exact 2-forms F can exist on M if Σ contains any non-contractible 2-cycles, C. Let us cover M by Mi (462) M= i

where each Mi a globally hyperbolic, connected and simply connected spacetime in its own right, which does not contain any non-contractible 2-cycles. Consequently on each Mi , any closed 2-form is exact, and the classical theory deﬁned by Maxwell’s equations dF = 0, d ∗ F is completely equivalent to the theory of a vector potential A with action (34). Thus, by the results of the previous sections, we can construct a corresponding algebra of observables Fˆ0 (Mi ) for each i, containing gauge-invariant observables such as polynomials of the ﬁeld strength. Each Fˆ0 (Mi ) is only given to us as an abstract *-algebra, so we do not a priori know what is the relation between those algebras for diﬀerent i. However, if Mi is

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contained in Mj , then by the general covariance property, there is an embedding of algebras αi,j ≡ αψ(i,j) : Fˆ0 (Mi ) → Fˆ0 (Mj ), where ψ(i, j) : Mi → Mj is the embedding. Thus, following ideas of Fredenhagen and K¨ usk¨ u [50, 51, 90], we may deﬁne an algebra Au (M ) as the universal algebra Au (M ) ≡ ind-limMi Fˆ0 (Mi ).

(463)

The universal algebra is deﬁned as the unique algebra such that there exist *homorphisms αi : Fˆ0 (Mi ) → Au (M ) with the property αj ◦ αj,i = αi . It is characterized by the fact there are no additional relations in Au (M ) apart from the ones in the subalgebras. Thus, Au (M ) is generated by the symbols Fi (f ) where supp f ⊂ Mi , which we think of as smeared ﬁeld strength tensors Fi (f ) = f ∧ F. (464) Mi

Their relations are Fi (f ) = Fj (f ),

supp f ⊂ Mi ∩ Mj ,

if

(465)

and the Fi (f ), with supp f ⊂ Mi satisfy all the relations in Fˆ0 (Mi ), which are [Fi (f ), Fi (h)] = i∆(f, h)1,

Fi (df ) = 0 = Fi (∗df ),

(466)

for any 1-forms f, h of compact support in Mi . Here, ∆ : Ω20 (M ) × Ω20 (M ) → R denotes the advanced minus retarded fundamental solution for the hyperbolic operator δd + dδ acting on 2-forms. For an arbitrary compactly supported 2-form f on M , we may then deﬁne the algebra element F (f ) ∈ Au (M ) as Fi (ψi f ), (467) F (f ) ≡ #

i

where supp ψi ⊂ Mi , and i ψi = 1 on supp f . It is not diﬃcult to show using Eq. (465) that this deﬁnition does not depend upon the particular choice of the covering. From Eq. (466), it then also follows that F (df ) = 0 = F (d∗f ) holds for arbitrary compactly supported forms f in M . One can also easily show that F (f ) F (h) − F (h) F (f ) = 0 for any two test-forms having spacelike related support. Indeed, after splitting f, h using a suitable a partition of unity, we may assume that the supports of f and h are contained in sets Mi and Mj . Since M is assumed to be connected, there exists therefore a globally hyperbolic spacetime N ⊂ Mi ∪ Mj in which every 2-cycle is contractible, and we may assume that N appears in the covering of M . We may then view both F (f ) and F (h) as elements in Fˆ0 (N ), where they commute. Since ∆ is uniquely determined by its action on testfunctions supported in a neighborhood of a Cauchy surface, it then also follows that [F (f ), F (h)] = i∆(f, h)1. The universal algebra contains certain central elements that carry information about the topology of M . They arise as follows. Let C be a 2-cycle in M , and let

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{ψi } be a partition of unity subordinate to the covering {Mi } of M . By Poincare duality, we can ﬁnd a closed 1-form hC on M such that hC ∧ α = α (468) M

C

for any closed 2-form α, and we may arrange hC to have support in a neighborhood of C. The 2-form ψi hC has compact support in Mi , and we may deﬁne Fi (ψi hC ) ∈ Au (M ). (469) Ze [C] = F (hC ) ≡ i

We claim that Ze [C] is independent of the particular choice of hC , and of the partition {Ui , ψi }. Independence of the partition was already shown above for general 2-forms. To show independence of hC , consider another hC with the same properties, and let hC − hC = ω. Then ω is closed, of compact support and, ω ∧ α = 0 for any closed 2-form α. By the well-known fact that the pairing (470) : H 2 (M ) ⊗ H02 (M ) → R is non-degenerate, we therefore must have that [ω] = 0 in H02 (M ), i.e. ω = dβ for some 1-form β of compact support. Independence of Ze [C] on the particular form of hC then follows from F (dβ) = 0. It then also follows that Ze [C] only depends upon the homotopy class of C, i.e. Ze [C] may be viewed as a map Ze : H2 (M ; Z) → Au (M ),

[C] → Ze [C].

(471)

In particular Ze [C] = 0 for any 2-cycle C that can be deformed into a point. Because Ze [C] only depends upon the class [C] of C in H2 (M ), it follows that, given any suﬃciently small compact region K ⊂ M , we may deform C so as to be in the causal complement of K, that is C ⊂ J + (K) ∪ J − (K). By choosing hC to be supported in a suﬃciently small neighborhood of C, it then follows that [Ze [C], F (f )] = 0,

∀f ∈ Ω20 (K).

(472)

But then this also holds for arbitrary f of compact support, because f may be writ# ten as ψi f , with each supp ψi so small that C and hence supp hC can be deformed so as to lie in the causal complement. Thus, Ze [C] is in the center Z(Au (M )) of Au (M ). By taking the dual of hC in Eq. (469), we may similarly deﬁne Fi (ψi ∗ hC ) ∈ Z(Au (M )), (473) Zm [C] = i

and this quantity has similar properties as Ze [C]. The center-valued quantities Ze [C], Zm [C] correspond to the electric and mag netic ﬂuxes through a 2-cycle C. They are analogous to the classical quantities C F respectively C ∗F and satisfy the same additivity relations under the addition of

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cycles. Other interesting derived quantities may also be deﬁned. For example, let C1 , C2 , . . . be a basis of 2-cycles in H2 (M ; Z), and let (Q−1 )jk = I(Cj , Ck )

(474)

be the matrix of their intersection numbers. Then we may deﬁne qtop =

b2

Qjk Ze [Ci ]Ze [Ck ]

∈ Z(Au (M ))

(475)

j,k

and this is analogous to the classical topological quantity jk F ∧F = Q F qclass = M

j,k

Cj

F

(476)

Ck

by the so-called “Riemann identity” for closed diﬀerential forms. In any factorial representation π : Au (M ) → End(H) on a Hilbert space H, the representers corresponding to Ze [C], Zm [C] are by deﬁnition represented by multiples of the identity, i.e. π(Ze [C]) = ce [C] · I,

π(Zm [C]) = cm [C] · I

(477)

where ce , cm are valued in the complex numbers. By DeRahm’s theorem, they can be represented by 2-forms fe and fm , both of which must be closed. Choosing a basis {ω i } of H 2 (M ), for example dual to a basis of 2-cycles {Ci }, we may thus expand # # fe = i qi ω i , and fm = i gi ω i with numerical constants qi , gi ∈ R depending upon the representation. These constants are then the (canonically normalized) numerical values of the electric and magnetic ﬂux through the respective cycle in the representation π. The above construction of Maxwell theory (without a vector potential) is somewhat abstract, and we now discuss an equivalent description. As above, let {ω i } be a set of closed forms forming a basis of H 2 (M ). Any closed form F may thus be # written uniquely as F = dA + i qi ω i . Substitution into the action S gives 1 S= dA ∧ ∗dA + j ∧ ∗A (478) 2 # where j = qi δω i is considered as an external (conserved) current coupled to A. The quantization of this theory now proceeds along similar lines as for the action S without the external current. We correspondingly get an algebra of observables i Aq (M ), which now depends upon the choice of q ≡ {q i } and {ω } through the external current. The algebra is spanned by generators f ∧ dA, and F (f ) = f ∧ dA + qi (479) ω i ∧ f 1. They satisfy the same relations as the generators F (f ) above in the algebra Au (M ). From this it may be seen that the algebra Aq (M ) only depends upon qi and the equivalence classes [ω i ]. This algebra also has further relations not present

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in Au (M ), because the elements Ze [C] ∈ Aq (M ) deﬁned in the same way as the central elements Ze [C] ∈ Au (M ) above, are now represented by multiples of the identity, namely e [C] = Z qi ω i 1 ∈ Aq (M ), (480) C

while the elements Ze [C] ∈ Au (M ) are only in the center, but not necessarily proportional to the identity. Thus, Au (M ) and Aq (M ) are not isomorphic. Instead, we have ⊕ b2 ∼ dqi Aq (M ). (481) Au (M ) = i=1

m [C], deﬁned as above, are not proportional to By contrast, the magnetic ﬂuxes Z the identity but only elements in the center of Aq (M ). This apparent asymmetry between the electric and magnetic ﬂuxes arises from the fact that we have chosen to quantize the theory starting from a potential for F , rather ∗F , which would also be possible. Then the roles of electric and magnetic ﬂuxes would be reversed. A physically relevant example of a spacetime M with a non-trivial 2-cycle is the Kruskal extension of the Schwarzschild spacetime. It has line element 32M 3 er/2M (−dT 2 + dX 2 ) + r2 (dθ2 + sin2 θdϕ2 ), r > 0, (482) r and topology M = R × R × S 2 , where r is deﬁned through T 2 − X 2 = (1 − r/2M )er/2M . It is a globally hyperbolic spacetime with a non-trivial 2-cycle, homotopic to S 2 . Hence, the universal algebra possesses non-trivial central elements Ze [S 2 ], Zm [S 2 ], and this gives rise to the possibility of having non-trivial electric and magnetic ﬂuxes in that spacetime, as also realized by Ashtekar et al. [3]. We now sketch an argument that arbitrary values of the electric and magnetic charges may be realized in representations π carrying a unitary representation of the time-translation symmetry group. The spacetime is a solution to the vacuum Einstein-equation Rµν = 0, with static timelike Killing ﬁeld K = ∂/∂t, with t = 4M tanh−1 (X/T ). By the standard identity ∇[µ (ενσ]αβ ∇α K β ) = 23 Rαβ K β eα µνσ 1 valid for any Killing ﬁeld K, φµν = 4π ∇[µ Kν] is therefore a static (meaning equations. Given q, g ∈ R, we deﬁne £K φ = 0) solution to the classical Maxwell γp,q : F (f ) → F (f ) + q S2 f ∧ φ1 + g S2 f ∧ ∗φ1. This is an automorphism of Au (M ). Let us assume that there is a factorial vacuum state .0 on Au (M ) invariant under the action of the time-translation isometries (which can presumably be constructed by the techniques of Junker et al. [82]), and let us assume that Ze [S 2 ]0 = 0 = Zm [S 2 ]0 . Then the states .q,g = γq,g (.)0 are also factorial and the corresponding GNS-representation carry a unitary representation of the timetranslation symmetries, with invariant vacuum vector. Furthermore, by S 2 ∗φ = 1, we have ds2 =

πq,g (Ze [S 2 ]) = qI,

πq,g (Zm [S 2 ]) = gI

(483)

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in the corresponding GNS-representations πq,g of these states. Thus, the representations πq,g carry electric ﬂux q and magnetic ﬂux g. In this sense, the numbers q, g may be viewed as superselection charges, as also noted by Ashtekar et al. [3]. Appendix B. Eﬀective Actions in Curved Spacetime We here give the deﬁnition of the eﬀective action in our framework following [16,17] and a derivation of a set of consistency conditions. We also emphasize that the eﬀective action is a state dependent quantity, and therefore, unlike the T -products, does not have a local, covariant dependence upon the metric. In the path integral formulation of quantum ﬁeld theory, the eﬀective action in a scalar ﬁeld theory is formally deﬁned as follows (see e.g., [117]). Let j ∈ C0∞ (M ) be an external current density, and deﬁne, formally, c (484) exp(Z (j)) = [Dφ] exp iS/ + jφ . transforThen the eﬀective action Γeﬀ is deﬁned, again formally, as the Legendre mation of Z c (j): Deﬁne φ through φ = δZ c (j)/δj, and Γeﬀ = jφ − Z c (j). The quantity Γeﬀ is a formal power series in depending on φ (and the action S), and may thus be viewed as an element of F . The above construction is formal in several ways: The quantity Z c (j) is typically viewed as the generating functional for the hierarchy of connected time-ordered n-point functions of the quantum ﬁeld φ. It thus depends upon a choice of state, and the same is consequently true for the eﬀective action. This is obscured in the above functional integral formulation. Here, the choice of state would enter the precise choice of the formal path-integral measure [Dφ]. Also, because the path-integral derivation does not specify the precise deﬁnition of the path-integral measure [Dφ], it necessarily disregards all issues related to renormalization. We therefore now give a precise deﬁnition of the eﬀective action in curved spacetime. For this, we deﬁne, following [16], the quantities Tωc : A⊗n → W0 (A the space of local actions) implicitly by 1 : Tωc (exp⊗ iF/) · · · Tωc (exp⊗ iF/) :ω , (485) T (exp⊗ (iF/)) = n! n≥0

where the nth term has n factors. Unlike T , the quantity Tωc is not local and covariant, but depends upon the global choice of ω. It can be shown that τωc (F ⊗n ) = lim→0 Tωc (F ⊗n )/n−1 ∈ A exist. Next, deﬁne a functional Γω implicitly by iΓω (exp⊗ F )/

τωc (e⊗

iF/

) = Tωc (e⊗

).

(486)

It can be shown that, for F ∈ A Γω (1) = 0,

Γω (F ) = F,

(487)

as well as Γω (eF ⊗ ) = F + O().

(488)

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Given an interaction F ∈ A, we deﬁne an “eﬀective action” (with respect to the state ω) associated with S0 + F by Γeﬀ (S0 + F ) = S0 + Γω (eF ⊗ ) = S0 + F + O().

(489)

Again, the higher order terms in depend upon the state ω, and are not local and covariant. This property makes the eﬀective action in general unsuitable to solve the renormalization problem in curved spacetime, since the local and covariance properties of the renormalization procedure cannot be controlled. The eﬀective action obeys a useful identity that can presumably be used to analyze potential anomalies in the Ward identities (as an alternative to our approach), at least in ﬂat spacetime. To formulate this identity, consider any local ﬁeld poly nomial O, and the modiﬁed action S0 + F → S0 + F + M h ∧ O, where h ∈ Ω0 (M ) is a compactly supported smooth form. Then we have the identity [16] M

δΓeﬀ (S0 + F + h, O) δΓeﬀ (S0 + F + h, O) ∧ δh(x) δφ(x) h=0 δ Γeﬀ (S0 + F + h, Oδ(S0 + F )/δφ + ∆O ) = , M δh(x) h=0

(490)

where ∆O (x) = ∆O (eF ⊗ )(x) ∈ A is the anomaly corresponding to O in the corresponding anomalous “Master Ward Identity” in Sec. 4.4, see also [16, 17]. It is viewed here as a 4-form.

Appendix C. Wave Front Set and Scaling Degree We here recall the basic deﬁnition of the wave front set of a distribution and some of its elementary properties. For details, see [76]. If u is a compactly supported smooth function on Rn , then by standard theorems of distribution theory, its Fourier transform, u ˆ(p) = (2π)−n/2 u(exp(ip.)) is an analytic function on Rn falling oﬀ faster than any inverse power of p, i.e. |ˆ u(tp)| ≤ cN (1 + |t|)−N ,

t∈R

(491)

for some cN not depending upon p, and any N . Conversely, this bound implies that a compactly supported distribution u is in fact smooth. The idea of the wave front set is to use the possible failure of this bound to characterize the non-smoothness of a distribution. For compactly supported distributions u, one deﬁnes the set of singular directions by u(tp) ≥ cN (1 + |t|)−N for some N , all t > 0}. Σ(u) = {p ∈ Rn \0 | |ˆ

(492)

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We deﬁne the wave front set of any distribution at a point x ∈ Rn by 7 WFx (u) = Σ(ψu)

(493)

ψ:x∈supp ψ

where the intersection is over all smooth compactly supported cutoﬀ functions ψ. The wave front set is clearly invariant under dilatation, and therefore a cone, and it only depends on the behavior of u in an arbitrary small neighborhood of x. For distributions u deﬁned on a smooth n-dimensional manifold X one deﬁnes the wave front set as follows. Let κ, U be a coordinate chart covering x. Then, choosing a smooth cutoﬀ function Ψ supported in U that is 1 near x, we can deﬁne κ∗ (ψu), which is now a distribution that is deﬁned on Rn . We deﬁne the wave front set to be the set WFx (u) = (κ−1 )∗ WFκ(x) (κ∗ (ψu)) ⊂ Tx∗ X.

(494)

It can be proved that this deﬁnition does not depend upon the arbitrary choice of κ, ψ, and one deﬁnes WF(u) to be the union of all WFx (u). One relevant application of the wave front set in perturbative quantum ﬁeld theory is the following theorem [76] about the product of distributions. Theorem 5. Let u, v be distributions on X. If 0 ∈ / WFx (u) + WFx (v), then the pointwise product uv is deﬁned in some neighborhood of x, and WFx (uv) ⊂ WFx (u) + WFx (v). Clearly, if the assumption holds for all x ∈ X, then the pointwise product is globally deﬁned on X. Another useful theorem about wave front sets is the following [76]. Let K ⊂ Rn be a convex open cone, and let u(x + iy) be analytic in Rn +iK for |y| < δ and some δ, with the property that |u(x+iy)| ≤ C|y|−N for some N , and all y ∈ K with |y| < δ. Then the boundary value u(x) = B.V.y→0 u(x + iy), with the limit taken for y ∈ K deﬁnes a distribution on Rn . Theorem 6. The wave front set of u(x) = B.V.y→0 u(x + iy) with the limit taken within the cone K, i.e. y ∈ K, is bounded by WF(u) ⊂ Rn × K D ,

(495)

where K D = {k ∈ Rn∗ | k · y < 0 ∀y ∈ K} is the dual cone. In applications, one often deals with distributions that are solutions to a partial diﬀerential equation Au = 0, where A is partial diﬀerential operator on X (or even a pseudo-diﬀerential operator), i.e. A=

N n=0

aµ1 ···µn (x)∇(µ1 · · · ∇µn ) .

(496)

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Under this condition, it can be shown that the wave front set of u must be restricted to the set WF(u) ⊂ {(x, k) | aµ1 ···µN (x)kµ1 · · · kµN = 0}.

(497)

In case when A is the wave operator on a Lorentzian manifold, we hence learn that any distributional solution u of the wave equation can only have vectors of the form (x, k) in the wave front set when k is a null-vector. Another important application of the wave front set for quantum ﬁeld theory in curved spacetime is the propagation of singularities theorem. Consider a distribution u on a spacetime (M, g) that is a solution to the wave equation u = f , with f a smooth source. The wave operator deﬁnes a 1-particle Hamiltonian on “phase space” T ∗ M by h(x, p) = g µν (x)pµ pν , and Hamilton’s equations, deﬁned with respect to the symplectic structure dxµ ∧ dpµ , p˙µ = −2Γνµρ (x)pν pρ , µ

x˙ = 2g

µν

(x)pν ,

(498) (499)

deﬁne a ﬂow in phase space, t → φt , which is just the geodesic ﬂow. The propagation of singularities theorem now states in this example that this ﬂow must leave the wave front set WF(u) invariant, in the sense that φ∗t WF(u) ⊂ WF(u). Thus, the propagation of singularities theorem gives information how singularities propagate along the bicharacteristic ﬂow. The theorem as just stated is in fact just a special case of the celebrated Duistermaat–H¨ ormander propagation of singularities theorem [31], which holds for much more general operators A of real principal type (including, e.g., the massive wave equation). The Hamiltonian is then given simply by h(x, k) = aµ1 ···µN (x)kµ1 · · · kµN in the general case, where N is the degree of the operator. Another useful concept in perturbative quantum ﬁeld theory is that of the scaling degree of a distribution. Let u be a distribution on Rn . The scaling degree, sd0 (u) at the origin of Rn is deﬁned as δ sd0 (u) = inf δ ∈ R lim t u(tx) = 0 (500) t→0+ where the limit is understood in the sense of distributions, i.e. after smearing with a testfunction. One similarly deﬁnes the scaling degree sdx (u) at an arbitrary point x by ﬁrst translating u by x. On a manifold X, the scaling degree is deﬁned by ﬁrst localizing u with a cutoﬀ function and then pulling it back with a coordinate chart, κ∗ (ψu), as in the deﬁnition of the wave front set. One again veriﬁes that the deﬁnition does not depend upon the choice of coordinates. Appendix D. Hadamard Parametrices In this appendix, we review the deﬁnition of the scalar Hadamard parametrix H s , and the vector Hadamard parametrix, H v , as well as the local expressions for the advanced and retarded propagators in curved spacetime.

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D.1. Scalar Hadamard parametrix In a general curved spacetime, it is not possible to ﬁnd a closed form expression for ∆A,R , but it is still possible to present a local expression HA,R involving certain recursively deﬁned coeﬃcients, which locally coincides with ∆A,R modulo C ∞ . The distributions HA,R are called “Hadamard parametrices” for ∆A,R . To construct them, let x, y ∈ M , and consider the length functional b |gµν (γ(t))γ˙ µ (t)γ˙ ν (t)|1/2 dt (501) s(x, y) = a

for C 1 -curves γ : [a, b] → M with the property that γ(a) = x and γ(b) = y, which are either spacelike, timelike, or null (but do not switch from one to the other). The functional s(p, q) is invariant under reparametrizations of the curve, so we may choose a parametrization so that gµν γ˙ µ γ˙ ν = 1 along the curve when γ is either spacelike or timelike (such a parameter is called an “aﬃne parameter”). The Euler–Lagrange equations for the functional are then given by γ˙ µ ∇µ γ˙ ν = 0,

(502)

and curves satisfying this equation are “geodesics”. If γ µ are the components of γ in a local chart, then the geodesic equation reads γ¨ µ + Γµ σν γ˙ σ γ˙ ν = 0.

(503)

Two given points x, y may in general be joined by several geodesics, but one can show [65] that every point in M has a neighborhood U such that any pair of points (x, y) ∈ U × U may be joined by a unique geodesic lying entirely within U . For (x, y) ∈ U × U , we deﬁne σ(x, y) to be the value of the function ±s(x, y)2 evaluated on the unique geodesic joining x and y, where + is chosen for a spacelike, and − is chosen for a timelike geodesic. In Minkowski spacetime, the function σ is equal to the invariant distance between the points x, y. In any spacetime, the function σ has the important property that g µν ∇µ σ∇ν σ = 4σ,

(504)

where the derivative can act on either the ﬁrst or second argument. Now let T : M → R be a time function. By analogy with ﬂat spacetime, we seek Hadamard parametrices for the advanced and retarded propagators by the following ansatz when the dimension of M is 4: HA,R (x, y) =

1 θ(∓t(x, y))[u(x, y)δ(σ(x, y)) − v(x, y)θ(−σ(x, y))]. 2π

(505)

Here, u, v are as yet unknown smooth, symmetric functions on U × U and t(x, y) = T (x) − T (y). This ansatz is consistent with the support properties of the advanced and retarded propagators, and it does not depend on the particular choice of time function. The unknown functions u, v are to be determined imposing in addition

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the Klein–Gordon equation, ( − m2 )x HA,R (x, y) = δ(x, y) modulo C ∞ , ∞

( − m )y HA,R (x, y) = δ(x, y) modulo C . 2

(506) (507)

Using the identity (504) one ﬁnds that HA , HR solve these equations in U × U modulo C ∞ if the following identities hold for u, v: 2∇µ σ∇µ u = (8 − σ)u.

(508)

( − m2 )v = 0,

(509)

as well as

modulo C ∞ , and 2∇µ σ∇µ v + (σ − 4)v = −( − m2 )u,

on ∂J ± (y)

(510)

where the derivative operators act on the point x. One can show that the unique smooth solution to the equation for u is given by u = D1/2 , where D(x, y) is the so-called “VanVleck determinant”, which is deﬁned as follows. Let x, y ∈ U , and let Aµν = (∇µ ⊗ ∇ν )σ, so that Aµν dxµ ⊗ dy ν is a tensor in Tx∗ M ⊗ Ty∗ M . We can consider the 4th anti-symmetric tensor power of this tensor, which may be viewed as a map ∧4 A : ∧4 Tx M → ∧4 Ty∗ M,

(511)

where ∧r Tp M denotes the space of totally anti-symmetric tensors of type (r, 0). Clearly, for r = 4 this space is 1-dimensional (in 4 dimensions), so if we pick a basis element at points x, y, we can identify ∧4 A with a scalar. A choice of the basis element depending only upon the metric (up to a sign) is the Levi–Civita tensor . With this choice, D is deﬁned as the scalar obtained from ∧4 A. In local coordinates, D=

1 ν1 A µ1 Aν2 µ2 Aν3 µ3 Aν4 µ4 εµ1 µ2 µ3 µ4 εν1 ν2 ν3 ν4 4!

(512)

where the ε tensors are evaluated at x and y, respectively. While it is not possible to give a similarly explicit solution to the equation for v, it is possible to obtain a solution v in the form of a convergent power series v=

∞

vn χ(σ/αn )σ n .

(513)

n=0

Here, χ is an arbitrary function of compact support that is equal to 1 in a neighborhood of 0, and {αn } is a sequence growing suﬃciently rapidly so as to enforce the convergence of the series. The coeﬃcients are determined recursively as the solutions of the “transport equations” 2∇µ σ∇µ v0 − (∇µ σ∇µ log D − 4)v0 = −( − m2 )D1/2 ,

(514)

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from Eq. (509) and, for n > 0 1 2∇µ σ∇µ vn − (∇µ σ∇µ log D − 4n − 4)vn = − ( − m2 )vn−1 (515) n from Eq. (510). The solutions to these diﬀerential equations are unique if one assumes, as we have done that vn are smooth (i.e. in particular regular at x = y). These solutions can be given in integral form as 1 1/2 1 ( − m2 )D1/2 2 λ dλ (516) v0 = − D 2 D1/2 0 and, for n > 0 1 vn = − D1/2 2n

1

0

( − m2 )vn−1 2n+2 λ dλ D1/2

(517)

where the integrand is evaluated at the point (x(λ), y), where x(λ) = Expy (λξ), and where ξ ∈ Ty M is chosen so that x(1) = x. Thus, in terms of the Riemannian normal coordinates of x relative to y, then the integrand is thought of as evaluated at the rescaled normal coordinates. Despite the apparent asymmetry in the construction of u, v, it can be shown that these functions are symmetric in x, y [55, 94], and one shows that, indeed, HA,R (x, y) = ∆A,R (x, y)

modulo C ∞

(518)

in U × U . (It can be proved that exact Greens functions ∆A,R exist globally, for which the power series expressions therefore deﬁne local asymptotic expansions.) From the advanced and retarded parametrices one can deﬁne 2 other parametrices HF,D (for “Feynman” and “Dyson”), given by u(x, y) 1 + v(x, y) log(σ ± i0) . (519) HF,D (x, y) = 2 2π σ ± i0 These parametrices are symmetric in x, y. Using the transport equations for u, v, one shows that these, too, are local Green’s functions (with δ-function source) modulo C ∞ . The wave front sets of HA,R,F,D are described by the following theorem: Theorem 7. The wave front set of the 4 Hadamard parametrices are given by WF(HA,R ) = {(x1 , k1 ; x2 , k2 ) | k1 ∼ −k2 , x1 ∈ J ± (x2 )} ∪ {(x, k; x, −k)}, WF(HF,D ) = {(x1 , k1 ; x2 , k2 ) | k1 ∼ −k2 , k1 ∈ ∪ {(x, k; x, −k)}.

(520) V±∗

±

iﬀ x1 ∈ J (x2 )} (521)

The proof of this theorem is similar to that of the next lemma. It can also be proved that the four parametrices HA,R,F,D are uniquely characterized by their wave front properties. In fact, there is a similar classiﬁcation of parametrices for any operator or real principal type, as shown by a profound theorem by Duistermaat and H¨ ormander [31].

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In the body of the paper, we use a combination, H, of the above Hadamard parametrices, which is called simply the “local (scalar) Hadamard parametrix” for the operator − m2. It is the distribution on U × U deﬁned by Eq. (154) in terms of the same coeﬃcients u, v that appear above in the local expressions for the advanced and retarded propagators. From identities like 1 1 1 (log(σ + i0t)) = ε(t)θ(−σ), (522) = ε(t)δ(σ), iπ σ + i0t iπ we get the relations HF − HR = −iH = HA − HD .

(523)

In view of the symmetry of HF,D , there follows the commutator property (531). Furthermore since HA,R,F,D are local Green’s functions modulo C ∞ with a δ-function source, there follow the equations of motion ( − m2 )x H(x, y) = 0

modulo C ∞ ,

( − m2 )y H(x, y) = 0

modulo C ∞ . (524)

The local Hadamard parametrix H is important because it characterizes the short distance behavior of any Hadamard state, see Appendix E. D.2. Vector Hadamard parametrix v (x, y)dxµ ∧ dy ν is constructed by The vector Hadamard parametrix H v (x, y) = Hµν analogy to the scalar case. It now satisﬁes the equations

(dδ + δd)x H v (x, y) = 0

modulo C ∞ ,

(dδ + δd)y H v (x, y) = 0

modulo C ∞ , (525)

where δ = ∗d∗. In component form, the equations of motion are given by the operator (243). The local vector Hadamard parametrix has an expansion similar to that of the scalar Hadamard parametrix: 1 uµν (x, y) v + v (x, y) = (x, y) log(σ + i0t) . (526) Hµν µν 2π 2 σ + i0t The coeﬃcients uµν , vµν have expansions that are analogous to the scalar case. The quantity uµν is given explicitly by uµν = D1/2 Iµν

(527)

where I : Tx M → Ty∗ M is the holonomy of the Levi–Civita connection along the unique geodesic connecting x, y (“bitensor of parallel transport”). The expansion coeﬃcients of vµν as in Eq. (513) are again determined by transport equations. The solutions to these equations take exactly the same form as in the scalar case, Eq. (517), with the only diﬀerence that the scalar Klein–Gordon operator − m2 in those expressions is replaced by the vector wave-operator gµν + Rµν .

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Appendix E. Hadamard States In the body of the paper, Hadamard 2-point functions play a key role. They were introduced in Sec. 3.1 as bidistributions that are solutions to the wave equation in both entries, that satisfy the commutator property, and that have a certain wave front set. Here we show that these conditions allow one to identify the short distance behavior of any Hadamard 2-point function with that of the local parametrix H introduced in the previous subsection. Lemma 11. Let ω(x, y) be a 2-point function of Hadamard form, i.e. the wave front set WF(ω) is given by (119). Then locally (i.e. where H is deﬁned), ω − H is smooth, i.e. u(x, y) 1 + v(x, y) log(σ + it0) + (smooth function in x, y). ω(x, y) = 2π 2 σ + it0 (528) Furthermore, any two Hadamard states can at most diﬀer by a globally smooth function in x, y. Proof. We ﬁrst show that, where it is deﬁned, H has a wave front set WF(H) of Hadamard form, i.e. is given by Eq. (119). Since vi are smooth functions on a convex normal neighborhood, it suﬃces to prove that WF([σ +i0t]−1 ) and WF(log[σ +i0t]) have the desired form. To determine the wave front set of such distributions, we use the above Theorem 6. We apply this theorem to the distributions in question as follows. First, we pick a local coordinate system (ψ, U ) in a convex normal neighborhood U . Within U , we pick a tetrad e0 , . . . , e3 which we use to identify each Tx M with R4 via the map sending ξ = (ξ 0 , . . . , ξ 3 ) in R4 to the point ex (ξ) = ξ 0 e0 |x + · · · + ξ 3 e3 |x in Tx M . For each given x ∈ U , we can then write a point y ∈ U uniquely as y = expx ex (ξ) for some ξ ∈ R4 . The mapping (x, y) ∈ U ×U → (ψ(x), ξ) thus deﬁnes a local coordinate chart in M × M , which we call again ψ. Evidently, it then follows that the pull-back of (σ + i0t)−1 under ψ is given by the distribution 1 1 = B. V. , (529) (y + i0e)2 η∈V + ,η→0 (ξ + iη)2 where e = (1, 0, 0, 0), which is of the form to which we can apply our lemma. Using that the dual cone of the open future lightcone V + in Minkowski spacetime is the closure of the past lightcone V¯ − , it follows WF([σ + i0t]−1 ) ⊂ ψ ∗ [(R4 × 0) × (R4 × V¯ − )]. (530) From this, the desired wave front set follows. The logarithmic term is treated in exactly the same fashion. Consider now the distribution d = ω − H. The antisymmetric part of ω is given by i∆, and the anti-symmetric part of H is given by H(x, y) − H(y, x) = iε(t){u(x, y)δ(σ) + v(x, y)θ(σ)},

(531)

where ε(t) = 1 for t > 0, and ε(t) = −1 for t ≤ 0. It can be shown that the righthand side of the equation is equal to i∆ modulo a smooth function. Thus, d(x, y) is

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symmetric in x, y modulo a smooth remainder. On the other hand, since we know that H has the same wave front set as ω, we know that WF(d) ⊂ {(x1 , k1 , x2 , k2 ) ∈ T ∗ M × T ∗ M ; x1 and x2 can be joined by null-geodesic γ ˙ and k2 = −γ(1), ˙ and k1 ∈ V¯ + } k1 = γ(0)

(532) (533)

which is evidently not a symmetric set. Thus, the only possibility is that, in fact, WF(d) = ∅, meaning that d ∈ C ∞ , or equivalently, that ω = H modulo smooth. This proves the lemma. Another proposition about Hadamard 2-point function underlying the “deformation argument construction” of Hadamard states given in Sec. 4.2 is the following: Theorem 8. Let ω be a positive deﬁnite distributional bi-solution such that WF(ω) has the Hadamard wave front property in an open neighborhood of Σ × Σ, where Σ is a Cauchy surface. Then WF(ω) has the Hadamard form globally on M × M . The proof of the theorem is a simple application of the propagation of singularities theorem for solutions of the Klein–Gordon equation described in the previous subsection. A (quasifree) Hadamard state is a 2-point function that is in addition positive deﬁnite, ω(f¯, f ) ≥ 0 for any testfunction. The positivity implies an even stronger “local-to-global theorem” than the one given above [99]: Theorem 9. Let ω be a bi-solution to the Klein–Gordon equation in both entries, with anti-symmetric part i∆, and with the property that any point x ∈ M has a globally hyperbolic neighborhood N such that WF(ω) is of Hadamard form in N × N . Then WF(ω) has the Hadamard form globally in M × M . References [1] A. Abrahams, A. Anderson, Y. Choquet-Bruhat and J. W. York, Einstein and Yang– Mills theories in hyperbolic form without gauge ﬁxing, Phys. Rev. Lett. 75 (1995) 3377–3381; arXiv:gr-qc/9506072. [2] L. Alvarez-Gaume and E. Witten, Gravitational anomalies, Nucl. Phys. B 234 (1984) 269. [3] A. Ashtekar and A. Sen, On the role of space-time topology in quantum phenomena: Superselection of charge and emergence of nontrivial vacua, J. Math. Phys. 21 (1980) 526. [4] G. Barnich, M. Henneaux, T. Hurth and K. Skenderis, Cohomological analysis of gauge-ﬁxed theories, hep-th/9910201. [5] G. Barnich, F. Brandt and M. Henneaux, Local BRST cohomology in gauge theories, Phys. Rep. 338 (2000) 439–569. [6] I. A. Batalin and G. A. Vilkovisky, Gauge algebra and quantization, Phys. Lett. B 102 (1981) 27–31. [7] I. A. Batalin and G. A. Vilkovisky, Closure of the gauge algebra, generalized Lie equations and Feynman rules, Nucl. Phys. B 234 (1984) 106–124.

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[99] M. J. Radzikowski, A local to global singularity theorem for quantum ﬁeld theory on curved space-time, Comm. Math. Phys. 180 (1996) 1–22. [100] M. Sato, T. Kawai and M. Kashiwara, Hyperfunctions and Pseudo-Diﬀerential Equations, Lecture Notes in Mathematics, Vol. 287 (Springer-Verlag, 1973), pp. 265–529. [101] M. Sato, Theory of hyperfunctions. I, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1959) 139–193; MR 22:4951. [102] M. Sato, Theory of hyperfunctions. II, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960) 387–437; MR 24A:2237. [103] G. Scharf, Quantum Gauge Theories: A True Ghost Story (Wiley, New York, 2001). [104] G. Scharf, Finite Quantum Electrodynamics (Springer, 1989 and 1995). [105] E. R. Speer, Analytic renormalization, J. Math. Phys. 9 (1968) 1404–1410. [106] E. R. Speer, On the structure of analytic renormalization, Comm. Math. Phys. 31 (1971) 23–36. [107] E. R. Speer, Analytic renormalization using many space-time dimensions, Comm. Math. Phys. 37 (1974) 83–92. [108] O. Steinmann, Perturbative QED and Axiomatic Field Theory (Springer Verlag, 2000). [109] R. Stora, Pedagogical experiments in renormalized perturbation theory, Contribution to the Conference Theory of Renormalization and Regularization, Hesselberg, Germany (2002); http://wwwthep.physik.uni-mainz.de/scheck/Hessbg02.html. [110] R. Stora, Continuum gauge theories, New Developments in Quantum Field Theory and Statistical Physics, eds. M. Levy and P. Mitter, 1976 Cargese Lectures, NATO ASI Series B26 (Plenum, 1977), pp. 201–224. [111] R. Stora, Algebraic structure and topological origin of anomalies, in Progress in Gauge Field Theory, eds. ’t Hooft et al. (Plenum Press, 1984), pp. 218–223. [112] W. D. van Suijlekom, Renormalization of gauge ﬁelds: A Hopf algebra approach, arXiv:hep-th/0610137. [113] W. D. van Suijlekom, The Hopf algebra of Feynman graphs in quantum electrodynamics, Lett. Math. Phys. 77 (2006) 265–281. [114] R. Verch, A spin-statistics theorem for quantum ﬁelds on curved spacetime manifolds in a generally covariant framework, Comm. Math. Phys. 223 (2001) 261–288; arXiv:math-ph/0102035. [115] R. M. Wald, Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics (University Press, Chicago, 1994). [116] R. M. Wald, On identically closed forms locally constructed from a ﬁeld, J. Math. Phys. 31 (1990) 2378–2384. [117] S. Weinberg, The Quantum Theory of Fields, Volume II: Modern Applications (Cambridge University Press, 1996). [118] E. Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. Math. 40 (1939) 149–204. [119] J. Zinn-Justin, Renormalization of gauge theories, in Trends in Elementary Particle Theory — International Summer Institute on Theoretical Physics in Bonn 1974 (Springer-Verlag, Berlin, 1975).

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Reviews in Mathematical Physics Vol. 20, No. 10 (2008) 1173–1190 c World Scientiﬁc Publishing Company

MACROSCOPIC OBSERVABLES AND THE BORN RULE, I. LONG RUN FREQUENCIES

N. P. LANDSMAN Institute for Mathematics, Astrophysics, and Particle Physics, Faculty of Science, Radboud University Nijmegen, Toernooiveld 1, 6525 Ed Nijmegen, The Netherlands [email protected] Received 30 April 2008 Revised 8 July 2008 Dedicated to the Memory of Bernd Kuckert (1968–2008) We clarify the role of the Born rule in the Copenhagen Interpretation of quantum mechanics by deriving it from Bohr’s doctrine of classical concepts, translated into the following mathematical statement: a quantum system described by a noncommutative C ∗ -algebra of observables is empirically accessible only through associated commutative C ∗ -algebras. The Born probabilities emerge as the relative frequencies of outcomes in long runs of measurements on a quantum system; it is not necessary to adopt the frequency interpretation of single-case probabilities (which will be the subject of a sequel paper). Our derivation of the Born rule uses ideas from a program begun by Finkelstein [17] and Hartle [21], intending to remove the Born rule as a separate postulate of quantum mechanics. Mathematically speaking, our approach reﬁnes previous elaborations of this program — notably the one due to Farhi, Goldstone, and Gutmann [15] as completed by Van Wesep [50] — in replacing inﬁnite tensor products of Hilbert spaces by continuous ﬁelds of C ∗ -algebras. Furthermore, instead of relying on the controversial eigenstate-eigenvalue link in quantum theory, our derivation just assumes that pure states in classical physics have the usual interpretation as truthmakers that assign sharp values to observables. Keywords: Quantum theory; Born rule; C ∗ -algebras. Mathematics Subject Classiﬁcation 2000: 81P15, 47L40

1. Introduction In its simplest formulation, the Born rule says that if A is some quantummechanical observable with nondegenerate discrete spectrum σ(A), then the probability Pψ (A = λi ) that a measurement of A in a state |ψ yields the result λi ∈ σ(A) is given by Pψ (A = λi ) = |ei , ψ|2 , 1173

(1)

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where |ei is a normalized eigenvector of A with eigenvector λi . In other words, if |ψ = i ci |ei with i |ci |2 = 1, then Pψ (A = λi ) = |ci |2 . More generally, if A is a self-adjoint operator on a Hilbert space H with associated spectral measure ∆ → E(∆), then the probability Pψ (A ∈ ∆) that the proposition A ∈ ∆ comes out to be true if A is measured in a state |ψ equals Pψ (A ∈ ∆) = ψ|E(∆)|ψ.

(2)

The Born rule provides the key link between the mathematical formalism of quantum physics and experiment, and as such is responsible for most predictions of quantum theory. In the history and philosophy of science, the Born rule (on par with the Heisenberg uncertainty relations) is often seen as a turning point where indeterminism entered fundamental physics.a Of course, classical physics is full of random phenomena as well. But in all known cases, their apparent random character may be retraced to ignorance about the initial state or about microscopic degrees of freedom or time scales; see, e.g., [13, 40]. In contrast, the type of randomness to which quantum mechanics gives rise via the Born rule is generally felt to be “irreducible” (in the sense of not being reducible to ignorance, not even about the Laws of Nature).b Even the assumption that quantum mechanics is a correct and fundamental theory by no means implies that this feeling is correct. Indeed, although among rival interpretations the Copenhagen Interpretation is the one that arguably puts most emphasis on both the fundamental and the probabilistic character of quantum theory, a mature work by one of its founders actually contains the following passage: “One may call these uncertainties objective, in that they are simply a consequence of the fact that we describe the experiment in terms of classical physics; they do not depend in detail on the observer. One may call them subjective, in that they reﬂect our incomplete knowledge of the world.” (Heisenberg, [22, pp. 53–54].) This claim is in tune with one of the two main principles of the Copenhagen Interpretation,c namely Bohr’s doctrine of classical concepts. A mature and well-known expression of this doctrine is as follows: “However far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms. a The Born rule was ﬁrst stated by Max Born in the context of scattering theory [2], following a slightly earlier paper in which he famously omitted the absolute value squared signs (though he corrected this in a footnote added in proof). The well-known application to the position operator is due to Pauli [39]. The general formulation (2) is due to von Neumann [36, §iii.1]. See [33] for a detailed reconstruction of the historical origin of the Born rule within the context of quantum mechanics, as well as [40] for a briefer historical treatment in the more general setting of the emergence of modern probability theory and probabilistic thinking. b Notable exceptions are Einstein [12, pp. 129–130] and ’t Hooft [25]. c The other one, the Principle of Complementarity, plays no role in this paper.

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(· · ·) The argument is simply that by the word experiment we refer to a situation where we can tell others what we have done and what we have learned and that, therefore, the account of the experimental arrangements and of the results of the observations must be expressed in unambiguous language with suitable application of the terminology of classical physics.” (Bohr, [1, p. 209].) Elsewhere, Bohr time and again stresses that measurement devices must be described classically “if these are to serve their purpose”. We take this to mean that, although such devices are ontologically quantum-mechanical by nature, they become a tool (in fact, the only tool) for the description of quantum phenomena as soon as they are epistemically treated as if they were classical. Thus the so-called Heisenberg cut, i.e. the borderline between the part of the world that is described classically and the part that is described quantum-mechanically, is epistemic or (inter)subjective in nature and hence movable; see also [43, 44]. In our opinion, this ideology provides an attractive qualitative basis for the understanding of randomness in Nature, for it preserves the fundamental diﬀerence between random phenomena in classical and in quantum physics (the given explanation of quantum probabilities as arising from the classical description of some part of the world would not make any sense if applied to classical probabilities), while discarding the notion of strictly “irreducible” randomness (which is only deﬁned through negation and quite possibly makes no philosophical sense at all). However, little (if any) work has been done in relating this ideology to the Born rule, which in the Copenhagen Interpretation simply seems to be taken for granted as a mathematical recipe that requires no explanation. It is the purpose of the present paper to ﬁll this gap: as we shall see, the Born rule can actually be derived from a particular instance of Bohr’s doctrine of classical concepts, provided one identiﬁes the Born probabilities with the relative frequencies of outcomes in long runs of measurements on a quantum system.d As always, the mathematical implementation of Bohr’s philosophical ideas is ambiguous; as far as his doctrine of classical concepts is concerned, we read it as saying that a quantum system described by a noncommutative algebra A of observables is empirically accessible only through commutative algebras associated with A.e For convenience, and in line with the modern mathematical description of quantum theory [47, 3, 19, 45, 31, 32], we assume that these algebras are in fact (unital) C ∗ -algebras. The simplest kind of commutative algebras associated with A are its (unital) commutative d With

this limited goal it is not even necessary to mention single-case probabilities, let alone interpret them; doing so requires a far deeper analysis, which will be the subject of a sequel paper, based on [24]. e Apart from leading to the Born rule, this reading also gives rise to a very pretty description of complementarity through the mathematical framework of topos theory; see [24]. Cf. Scheibe [43] and Howard [26] for diﬀerent readings of the doctrine of classical concepts.

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C ∗ -subalgebras; in this paper we need a more subtle limiting procedure to “extract” a commutative C ∗ -algebra of macroscopic observables. Our derivation of the Born rule relies on certain ideas that were originally proposed by Finkelstein [17] and Hartle [21], whose work was continued by Ochs [38], Bugajski and Motyka [5], Farhi, Goldstone and Gutmann [15], and Van Wesep [50].f We review this development in Sec. 2, aware of the critique that has been issued by a number of authors, including Cassinello and S´ anchez-G´omez [6] and Caves and Schack [7], and its refutation by Van Wesep [50]. Our objection against the derivations just cited is quite diﬀerent from the criticism in [6, 7]; we take issue with their reliance on the so-called eigenstate-eigenvalue link. This terminology, which may sound like a tautology in mathematics, is often used in the philosophy of physics; see, e.g., [4, 8]. The link in question is the postulate that a measurement of an observable A in a state |ψ yields the result λ with certainty if and only if |ψ is an eigenstate of A with eigenvalue λ.g Plausible as this may sound, the eigenstate-eigenvalue link is the source of the measurement problem in quantum mechanics and hence is held to be unsound by most contemporary specialists in the foundations of quantum mechanics (see [4, 8] and references therein). Indeed, the eigenstate-eigenvalue link cannot be found in the writings of Bohr and Heisenberg; it was ﬁrst postulated by Dirac [9]. We will show how the program of deriving the Born rule from “ﬁrst principles” can nonetheless be carried out if, instead, it is underwritten by Bohr’s doctrine of classical concepts (construed as above).h The mathematical formalism needed to accomplish this starts from the modern algebraic approach to the quantum theory of large systems [47, 3, 19, 23, 35, 45], which, however, we need to reformulate in order to incorporate Bohr’s doctrine in an optimal way. This reformulation is based on the uniﬁed picture provided by continuous ﬁelds of C ∗ -algebras [10, 29] in the description of the classical limit of quantum mechanics. This limit actually has (at least) two guises, namely the limit → 0 of Planck’s constant going to zero, and the limit N → ∞ of a system size going to inﬁnity. Both can be brought under the umbrella of continuous ﬁelds of C ∗ -algebras; for → 0 this was done in [31], and for N → ∞ it was announced in [32] and will be completed in the present paper, where essential use is made of ideas of Raggio and Werner [41] and Duﬃeld and Werner [11]. In fact, once the appropriate framework has been set up in Sec. 3, the derivation of the Born rule in Sec. 4 will turn out to be almost trivial. This paper is part of a larger research programme, whose goal it is to interpret quantum mechanics entirely in terms of its classical limit. This is meant as f Some

of these papers were not quite written in support of the Copenhagen Interpretation but rather against it, usually defending the Everettian stance. g Philosophical realists adhering to the eigenstate-eigenvalue link would simply say that A has the value λ in the eigenstate |ψ, but precisely among realists it has become fashionable to deny the eigenstate-eigenvalue link for the reasons mentioned in the main text [4, 8]. For example, in Bohmian mechanics position always has a sharp value, whereas in the modal interpretation of quantum mechanics the link is dropped in a more ﬂexible way. h A completely diﬀerent type of derivation of the Born rule may be found in [42, 51].

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a technical implementation of the Copenhagen Interpretation as originally formulated by Bohr and Heisenberg (cf. [27]), whose goal was expressed quite well by Landau and Lifshitz [30, p. 3]: “Thus quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formulation.” 2. The Strong Law of Large Numbers in Quantum Theory Let us ﬁrst review what has been achieved mathematically in [17, 21, 38, 5, 15, 50]. For simplicity, we restrict ourselves to the simple situation of repeated measurements on a two-level (or, in current parlance, one-qubit) system, i.e. with Hilbert space C2 . Suppose we have an observable A (i.e. a hermitian 2 × 2 matrix) with eigenvalues 0 and 1 and corresponding orthogonal eigenstates |0 and |1. A long series of measurements of A in a given initial state |ψ ∈ C2 (prepared anew for each subsequent measurement) will produce a sequence x = (x1 , x2 , . . .), where xi = 0 or 1. We idealize a long series of measurements as an inﬁnite one, so that x ∈ 2N , with 2 = {0, 1} and the space of inﬁnite binary sequences is denoted by 2N = {x : N → 2}. We deﬁne p ∈ [0, 1] as the Born probability p = |1|ψ|2 ,

(3)

so that |0|ψ|2 = 1 − p. We ﬁrst review the classical strong law of large numbers relevant to 2N , seen as a measure space with Borel structure generated by the sets ()

Bk = {x : N → 2 | xk = },

(4)

where k ∈ N and ∈ 2. For any p ∈ [0, 1], consider the probability measure µp on 2 deﬁned by µp (0) = 1 − p and µp (1) = p. This deﬁnes a probability measure µ∞ p

∞ on 2N for which µ∞ p (Bk ) = p and µp (Bk ) = 1 − p for all k. Let N 1 Lp = x ∈ 2N lim xk = p ⊂ 2N . N →∞ N (1)

(0)

(5)

k=1

This is a Borel set. The strong law of large numbers states that µ∞ p (Lp ) = 1.

(6)

A measure theorist will read this as is stands: Lp has measure one with respect N to µ∞ p . A probability theorist deﬁnes functions fk : 2 → 2 by fk (x) = xk , notes that the fk are i.i.d. random variables, and says that the sequence of functions N (1/N ) k=1 fk on 2N converges pointwise to p with probability one (or almost surely) with respect to µ∞ p . A physicist deﬁnes an elementary proposition (or “yesno question”) χLp (i.e. the characteristic function of Lp ) on the “phase space” 2N , which is answered by yes in a pure state x if limN →∞ N1 N k=1 xk = p, and by no

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N otherwise. The probability measure µ∞ p deﬁnes a mixed state on 2 , and (6) gives the state-proposition pairing in the case at hand as

µ∞ p , χLp = 1.

(7)

If, for a proper yes-no question Q and state ρ, one initially interprets ρ, Q as the probability of obtaining a positive answer to Q in the state ρ (or, more generally, interprets ρ, f as the expectation value of an observable f in a state ρ), then one still has to expand this interpretation by stipulating what notion of probability one is using [18, 34]. Even if a probability equals one, as in (7), one still has to declare whether or not one adopts the so-called Necessity Thesis [34] (stating that probability one implies certainty). These questions cannot be answered by the mathematical formalism. The papers just cited attempt to extend the strong law of large numbers to the quantum case, and, not always sensitive to the last remark, draw certain conclusion about quantum mechanics from such an extension. A correct way of proceeding at least mathematically emerges from a combinination of results in [15, 50], as follows. be the Let (C2 )⊗N ∼ = C2N be the N -fold tensor product of C2 , and let (C2 )⊗∞ ψ separable component of the inﬁnite tensor product (C2 )⊗∞ Hilbert space (in the sense of von Neumann [37]) of C2 that contains |ψ⊗∞ , where |ψ ∈ C2 is a given ⊂ (C2 )⊗∞ , seen as a state, is (unit) vector.i The unit vector |ψ⊗∞ ∈ (C2 )⊗∞ ψ ∞ the quantum analogue of the probability measure µp in the classical situation just reviewed. The quantum analogue of the proposition χLp is a projection P(Lp ) on (1) on (C2 )⊗∞ (C2 )⊗∞ ψ , deﬁned as follows. For each k ∈ N, deﬁne a projection Pk ψ (1)

by Pk = 1 ⊗ · · · |11| ⊗ 1 · · · , where the projection |11| on C2 acts on the kth copy of C2 in the inﬁnite tensor product and all other entries are unit matri(0) ces on C2 . Similarly, Pk is deﬁned by replacing |11| by |00|. The projections (0) (1) {Pk , Pk }k∈N commute, and generate a complete Boolean algebra Pψ of projections on (C2 )⊗∞ ψ . Let B2N be the (countably complete) Boolean algebra of Borel N sets in 2 . By [50, Theorem 1], there is a unique homomorphism P : B2N → Pψ of Boolean algebras that satisﬁes (0)

(0)

(1)

(1)

P(Bk ) = Pk ; P(Bk ) = Pk ,

(8)

for each k ∈ N. The projection P(Lp ), then, is what its notation says, i.e. the image of the Borel set Lp ∈ B2N under P. Interpreted as a yes-no question, it asks if a given measurement outcome x has mean p. i Apart from the original source [37], this formalism is also explained in e.g. [15] or [14, §6.2]. The details are not relevant here, as we will replace the use of inﬁnite tensor products of Hilbert is to spaces by a diﬀerent formalism later on. In any case, the simplest way to deﬁne (C2 )⊗∞ NψN regard it as the Hilbert space of the GNS-representation of the inﬁnite tensor product M2 (C) (cf. [28, §11.4]) induced by the vector state |ψ⊗∞ ; see Sec. 4.

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Let p be as in (3). It is easy to show [50] that (6) implies P(Lp )|ψ⊗∞ = |ψ⊗∞ .

(9)

Regarding the unit vector |ψ⊗∞ as a state ψ ⊗∞ (in the algebraic sense) on any von Neumann algebra of operators on (C2 )⊗∞ containing P(Lp ), we can rewrite ψ (9) in the form of the classical pairing (7), i.e. ψ ⊗∞ , P(Lp ) = 1.

(10)

Indeed, since P(Lp ) is a projection, Eqs. (9) and (10) are equivalent. Furthermore, let us deﬁne the frequency operator fN on C2N by stipulating that its eigenstates are |x1 · · · |xN (where xi = 0 or 1), with eigenvalues fN |x1 · · · |xN =

N 1 xk |x1 · · · |xN . N

(11)

k=1

In words, fN is the relative frequency of the entry 1 in the list (x1 , . . . , xN ). Clearly, ψ on (C2 )⊗∞ by fN can be extended to an operator fN ψ ψ fN =

N 1 (1) Pk . N

(12)

k=1

It then follows from (9) that lim f ψ |ψ⊗∞ N →∞ N

= p|ψ⊗∞ ,

(13)

with p given by (3). In fact, deﬁning ψ ψ f∞ = s-lim fN , N →∞

(14)

where s-lim denotes the limit in the strong operator topology on (C2 )⊗∞ ψ , it can even be shown that ψ = p · 1, f∞

(15)

where 1 is the unit operator on (C2 )⊗∞ ψ . Results of this type can be derived quite easily from the modern algebraic approach to the quantum theory of large systems [47, 3, 19, 32]; see below. For the moment, we discuss the interpretation of (15) and especially of its corollary (13). Authors of papers like [17, 21, 15, 50] argue that, in view of (3) and the deﬁniψ , Eq. (13) provides a derivation of the Born rule from the eigenstatetion (12) of fN eigenvalue link (in the “if” direction). To assess this claim, we make three points: ψ has a (1) Applying the eigenstate-eigenvalue link to conclude from (13) that f∞ ⊗∞ ψ sharp value p in the state |ψ , is inconsistent with measuring f∞ by measur(1) ing each of its components Pk in (12) separately. For each such measurement will disturb the state; neither |ψ⊗∞ nor any |ψ⊗N is an eigenstate of any

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Pk , not to mention all of them.j Hence f is to be measured directly. Although according to [15] this can be done in some cases, it precludes any inference of single-case Born probabilities from (13). ψ were to take place by computing the limit (14) (2) Even if a measurement of f∞ from an inﬁnite list x of single-case measurements, interpreting Born probabilities as limiting frequencies would face all the usual objections to the frequency interpretation of probability [16, 18, 20, 34]. (3) Although (15) suggests that the eigenstate-eigenvalue link is being applied to an observable that is a multiple of the identity (in which case the said link would be uncontroversial and indeed nonprobabilistic), in actual fact the unit operator 1 in (15) is the one on (C2 )⊗∞ ψ , rather than on the entire (nonseparable) state space. Thus, despite appearances, the eigenstate-eigenvalue link is applied to a proper projection and in principle suﬀers from the drawbacks mentioned in the introduction. The conclusion that in practice (and, indeed, in [17, 21, 15, 50]) its application turns out to be correct in the case at hand follows, in our opinion, only from our analysis below. The ﬁrst two points were also made in [6, 7], but despite our adding the third objection, we do not follow the authors of these papers in concluding that the program of deriving the Born probabilities from properties of the frequency operator is “ﬂawed at every step” [7]. Indeed, by changing both the conceptual and the mathematical setting we will see that each of these objections can be met: ψ from a strong operator (1) Changing the deﬁnition of the frequency operator f∞ limit on a Hilbert space (which even depends on the state |ψ) to an element f∞ of a commutative (i.e. classical) C ∗ -algebra of macroscopic observables (which is independent of |ψ) “stabilizes” f against perturbations. Thus, without jeopardizing our derivation and interpretation of the Born rule, the frequency operator can be measured either directly (as suggested in [15]), or in terms of repeated measurements of the underlying observable A in the state |ψ. The latter proce(1) dure determines the possible values (0 or 1) of each Pk for k = 1, . . . , N < ∞, upon which one takes the limit N → ∞. This seems to correspond to experimental practice. (2) The second objection is obviated if one simply interprets the possible values of the frequency operator f∞ according to its deﬁnition, i.e. as limiting frequencies of either a single experiment on a large number of sites or a long run of individual experiments on single sites. In particular, one should refrain from making any statement about single-case probabilities. On this view, the Born rule simply says nothing about individual experiments on single sites.k

fact, any component of the complete von Neumann tensor product (C2 )⊗∞ containing at least (1) one simultaneous eigenstate of all Pk is orthogonal in its entirety to (C2 )⊗∞ ψ . k Except in an empty way, as in Popper’s so-called propensity interpretation of probability [18, 34]. j In

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(3) Instead of relying on the controversial eigenstate-eigenvalue link in quantum theory, our derivation will just assume that pure states in classical physics have the usual interpretation as “truthmakers” that assign sharp values to observables. 3. Large Quantum Systems and the Born Rule 3.1. Continuous fields of C ∗ -algebras In experimental physics, theoretical predictions based on the Born rule are typically checked by performing N identical experiments on a given quantum system in a given state |ψ, where N is large. This situation is idealized by taking the limit N → ∞. We describe this limit in a way that reorganizes the well-known algebraic description of inﬁnite quantum systems by quasilocal C ∗ -algebras [47, 3, 19] and macroscopic observables [23, 35, 41, 11, 45] into the tool of choice in the mathematical analysis of classical behavior in quantum theory [31, 32], namely continuous ﬁelds of C ∗ -algebras. For the reader’s convenience, we recall the latter notion, replacing the original deﬁnition of Dixmier [10] by the equivalent formulation of Kirchberg and Wassermann [29]. By a morphism we mean a ∗ -homomorphism. Definition 1. A continuous field of C ∗ -algebras over a locally compact Hausdorﬀ space X consists of a C ∗ -algebra A, a collection of C ∗ -algebras {Ax }x∈X , and a surjective morphism ϕx : A → Ax for each x ∈ X, such that: (1) The function x → ϕx (A) x is in C0 (X) for each A ∈ A (where · x is the norm in Ax ). (2) The norm of A ∈ A is A = supx∈X ϕx (A) . (3) The C ∗ -algebra A is a C0 (X) module in the sense that for any f ∈ C0 (X) and A ∈ A, there is an element f A ∈ A for which ϕx (f A) = f (x)ϕx (A) for all x ∈ X. A continuous section of the ﬁeld is a map x → Ax ∈ Ax for which there is an A ∈ A such that Ax = ϕx (A) for all x ∈ X. It follows that the C ∗ -algebra A may actually be identiﬁed with the space of continuous sections of the ﬁeld: if we do so, the morphism ϕx is just the evaluation map at x. The general idea is that the family (Ax )x∈X of C ∗ -algebras is glued together by specifying a topology on the bundle x∈X Ax (disjoint union). This topology is deﬁned indirectly via the speciﬁcation of the space of continuous sections of the bundle (cf. the Serre–Swan Theorem for vector bundles). The third condition makes A a C0 (X)-module in the sense that there exists a nondegenerate morphism from C0 (X) to the center of the multiplier algebra of A. This seemingly technical deﬁnition turns out to provide an attractive framework for the study of the classical limit of quantum mechanics. In the scenario → 0, the parameter space X is typically X = [0, 1], and A0 is the commutative C ∗ -algebra of C0 -functions on some classical phase space. For each > 0, one then constructs A

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as the algebra of quantum observables for varying (it may or may not be the case that the A are isomorphic for diﬀerent values of ). Continuous sections of the ﬁeld then describe quantization and the classical limit of observables at one go [31]. More generally, the classical theory is “glued” to the corresponding quantum theories via the continuous ﬁeld structure. 3.2. Macroscopic and quasilocal observables To describe large quantum systems and their possible classical behavior, we use the ˙ This is homeomorphic to {0} ∪ 1/N ⊂ R in the one-point compactiﬁcation X = N. relative topology borrowed from R, viz. under the map n → 1/n and ∞ → 0 (where ∞ is the compactiﬁcation point added to N). To derive the Born rule, we need N copies of a single quantum system with unital algebra of observables A1 (e.g., A1 = M2 (C) as above). From the single C ∗ -algebra A1 , we are going to construct two quite diﬀerent continuous ﬁelds of ˙ called A(c) and A(q) . These ﬁelds coincide as far as their ﬁbers C ∗ -algebras over N, above N ∈ N are concerned, which are given by AN = AN = A⊗N 1 . (c)

(q)

(16)

Here ⊗ is the spatial tensor product; see, e.g., [28, Chap. 11]. However, the two ﬁelds completely diﬀer in their respective ﬁbers above the limit point ∞, given by A(c) ∞ = C(S(A1 ));

(17)

limN A⊗N 1 .

(18)

A(q) ∞

=

Here S(A1 ) is the state space of A1 (equipped with the weak∗ -topology),l and the C ∗ -algebra in the right-hand side of (18) is the inductive limit with respect to the +1 → AN given by AN → AN ⊗ 1 (see below for an explicit inclusion maps A⊗N 1 1 description).m (c) In order to deﬁne the continuous sections of A∞ , we deﬁne, for M ≤ N , sym→ A⊗N by metrization maps jN M : A⊗M 1 1 jNM (AM ) = SN (AM ⊗ 1 ⊗ · · · ⊗ 1),

(19)

where one has N − M copies of the unit 1 ∈ A1 so as to obtain an element of ⊗N → A⊗N is given by (linear and A⊗N 1 . The symmetrization operator SN : A1 1 example, the state space of A1 = S(M2 (C)) is isomorphic as a compact convex set to the three-ball B 3 = {(x, y, z) ∈ R3 | x2 + y 2 + z 2 ≤ 1}: describing a state as a density matrix ρ on C2 , the corresponding point (x, y, z) ∈ B 3 is given by the well-known parametrization ρ(x, y, z) = “ x − iy ” 1 1+z . 2 x + iy 1−z

l For

often writes ∪N∈N AN for the inductive limit limN AN , where the N bar denotes norm comcorresponds to pletion. In the notation of [28, §11.4], our limN A⊗N 1 a∈A Aa with A = N and Aa = A1 for all a.

m One

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continuous) extension of SN (B1 ⊗ · · · ⊗ BN ) =

1 Bσ(1) ⊗ · · · ⊗ Bσ(N ) , N!

(20)

σ∈SN

where SN is the permutation group (i.e. symmetric group) on N elements and is given by Bi ∈ A1 for all i = 1, . . . , N . For example, jN 1 : A1 → A⊗N 1 jN 1 (B) = B

(N )

N 1 = 1 ⊗ · · · ⊗ Bk ⊗ 1 ⊗ · · · ⊗ 1, N

(21)

k=1

where Bk is B seen as an element of the kth copy of A1 in A⊗N 1 . In particular, for deﬁned by (11) is of this form, A1 = M2 (C) the frequency operator fN in A⊗N 1 since from (12) we infer that fN = jN 1 (|11|).

(22)

More generally, for A1 = B(H) (the algebra of all bounded operators on a Hilbert space H), the operator that counts the frequency of the eigenstate |λ ∈ H of some observable A upon N measurements of A is given by fN = jN 1 (|λλ|).

(23)

Definition 2. We say that a sequence (A1 , A2 , . . .) with AN ∈ A⊗N is symmetric 1 when AN = jN M (AM )

(24)

for some ﬁxed M and all N ≥ M . We call (A1 , A2 , . . .) quasisymmetric if for any ε > 0 there is an Nε and a symmetric sequence (A1 , A2 , . . .) such that AN −AN < ε for all N ≥ Nε . Physically speaking, the tail of a symmetric sequence entirely consists of “averaged” or “intensive” observables. which become macroscopic in the limit N → ∞. Quasisymmetric sequences have the important property that they mutually commute in the limit N → ∞; more precisely, if (A1 , A2 , . . .) and (A1 , A2 , . . .) are quasisymmetric sequences, then lim AN AN − AN AN = 0.

N →∞

(25)

Hence we see that in the limit N → ∞ the quasisymmetric sequences organize themselves in a commutative C ∗ -algebra, which we call the C ∗ -algebra of macroscopic observables of the given large system. To see that this limit algebra of macroscopic observables is isomorphic to C(S(A1 )), we complete the deﬁnition of the continuous (c) ﬁeld A∞ by deﬁning its continuous sections.

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Theorem 1. For any unital C ∗ -algebra A1 , the fibers AN = A⊗N 1 ; (c) (c)

A∞ = C(S(A1 ))

(26)

˙ if the C ∗ -algebra A(c) of continuous form a continuous field of C ∗ -algebra s over N sections is defined as follows: a section A : N → AN ∈ A⊗N 1 ; : ∞ → A∞ ∈ C(S(A1 ))

(27)

of the above field is declared to be continuous if the sequence (A1 , A2 , . . .) is quasisymmetric, and A∞ (ω) = lim ω ⊗N (AN ). N →∞

(28)

Here ω ∈ S(A1 ) and ω ⊗N ∈ S(A⊗N 1 ) is the tensor product of N copies of ω, deﬁned by (linear and continuous) extension of ω ⊗N (B1 ⊗ · · · ⊗ BN ) = ω(B1 ) · · · ω(BN ); cf. [28, Proposition 11.4.6]. The limit (28) then exists by deﬁnition of an approximately symmetric sequence: if (A1 , A2 , . . .) is symmetric with (24), one has ω ⊗N (AN ) = ω ⊗M (AM ) for N > M , so that the tail of the sequence (ω ⊗N (AN )) is even independent of N . In the approximately symmetric case, one easily proves that (ω ⊗N (AN )) is a Cauchy sequence. (c)

Proof. To prove that A∞ is a continuous ﬁeld, the main point is to show that lim AN = A∞ ,

N →∞

(29)

if (A1 , A2 , . . .) is quasisymmetric and A∞ is given by (28). This is easy to show for symmetric sequences: assume (24), so that AN = jNM (AM ) for N ≥ M . By the C ∗ -axiom A∗ A = A2 , it suﬃces to prove (29) for A∗∞ = A∞ , which implies A∗M = AM and hence A∗N = AN for all N ≥ M . One then has

AN = sup{|ρ(AN )|, ρ ∈ S(A⊗N 1 )}. Because of the special form of AN , one may ⊗N by the supremum replace the supremum over the set S(A⊗N 1 ) of all states on A1 ⊗N p over the set S (A1 ) of all symmetric states (see Deﬁnition 5 below), which in turn may be replaced by the supremum over the extreme boundary ∂S p (A⊗N 1 ) of ⊗N p ⊗N [46], so that AN = S (A1 ). The latter consists of all states of the form ρ = ω sup{|ω ⊗N (AN )|, ω ∈ S(A1 )}. This is actually equal to AM = sup {|ω ⊗M (AM )|}. (c) Now the norm in A∞ is A∞ = sup {|A∞ (ω)|, ω ∈ S(A1 )}, and by deﬁnition of A∞ one has A∞ (ω) = ω ⊗M (AM ). Hence (29) follows. Given (29), the theorem follows from [31, Proposition II.1.2.3] and the fact that the set of functions A∞ on S(A1 ) arising in the said way are dense in C(S(A1 )) (equipped with the supremum-norm). This follows from the Stone–Weierstrass theorem, from which one infers that the functions in question exhaust S(A1 ). We now turn to the continuous ﬁeld A(q) deﬁned by the quasilocal observables.

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Definition 3. A sequence (A1 , A2 , . . .) (where AN ∈ A⊗N 1 , as before) is called local when for some ﬁxed M and all N ≥ M one has AN = AM ⊗ 1 ⊗ · · · ⊗ 1 (where one has N − M copies of the unit 1 ∈ A1 ), and quasilocal when for any ε > 0 there is an Nε and a local sequence (A1 , A2 , . . .) such that AN − AN < ε for all N ≥ Nε . The inductive limit C ∗ -algebra limN A⊗N then simply consists of all equivalence 1 classes [A1 , A2 , . . .] of quasilocal sequences (A1 , A2 , . . .) under the equivalence relation (A1 , A2 , . . .) ∼ (B1 , B2 , . . .) when limN →∞ AN − BN = 0. The C ∗ -algebraic (q) structure on A∞ is inherited from the quasilocal sequences in the obvious (pointwise) way, except for the norm, which is given by

[A1 , A2 , . . .] = lim AN . N →∞

(30)

Each A⊗N is contained in A∞ as a C ∗ -subalgebra by identifying AN ∈ A⊗N with 1 1 the equivalence class [0, . . . , 0, AN ⊗1, AN ⊗1⊗1, . . .] (where the zero’s are irrelevant, of course; any entry could have been chosen). (q)

Theorem 2. For any unital C ∗ -algebra A1 , the fibers AN = A⊗N 1 ; (q)

A∞ = limN A⊗N 1 (q)

(31)

˙ if the C ∗ -algebra A(q) of continuous form a continuous field of C ∗ -algebras over N sections is defined as follows: a section A : N → AN ∈ A⊗N 1 ; ∞ → A∞ ∈ limN A⊗N 1

(32)

of the above field is declared to be continuous if the sequence (A1 , A2 , . . .) is quasilocal, and A∞ = [A1 , A2 , . . .].

(33)

Proof. This time, the ﬁrst property in Deﬁnition 1 is immediate from (30). The other properties are either trivial or follow from [31, Proposition II.1.2.3]. 4. The Born Rule To see the relevance of the above considerations to the Born rule, we ﬁrst rederive (15). For any A1 , we note that a state ρ on A1 deﬁnes a state ρ⊗∞ on limN AN by ρ⊗∞ ([A1 , A2 , . . . , ]) = lim ρ⊗N (AN ), N →∞

(34)

which limit is easily seen to exist by ﬁrst approximating a quasilocal sequence by a local one. We take A1 = B(H) and pick a unit vector |ψ ∈ H with associated pure state ψ on A1 . As in (34), the inﬁnite tensor product ψ ⊗∞ deﬁnes a state

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on A∞ , which is pure by [28, Proposition 11.4.7]. Hence the associated GNS(q) representation πψ⊗∞ (A∞ ) is irreducible; we may identify its carrier space Hψ⊗∞ with the separable component Hψ⊗∞ of von Neumann’s inﬁnite tensor product H ⊗∞ that contains |ψ⊗∞ as the cyclic vector Ωψ⊗∞ of the GNS-construction (cf. [14, Chap. 6] and also see Sec. 2 above). (q) and hence in A∞ . Although Each operator fN deﬁned by (23) lies in A⊗N 1 (q) limN →∞ fN does not exist within A∞ , one may consider a possible limit limN →∞ πψ⊗∞ (fN ) as an operator on Hψ⊗∞ . This limit indeed exists in the strong (q) operator topology, and commutes with all elements of πψ⊗∞ (A∞ ) (this is easily checked to be the case for any quasisymmetric sequence). Since πψ⊗∞ is irreducible, the limit operator must be a multiple of the unit, and using (34) and (23) one computes the constant as s-lim πψ⊗∞ (fN ) = |λ|ψ|2 · 1. N →∞

(35)

This generalizes (15), and also, to our mind, gives an impeccable derivation of it. The type of derivation of the Born rule reviewed in Sec. 2 is based on (35), but despite the fact that its mathematical status has now been clariﬁed, it faces the conceptual problems listed in that section. To solve these problems, we use the continuous ﬁeld A(c) instead of A(q) , again with A1 = B(H). Identifying a density matrix ρ on H with a state on B(H) in the usual way by ρ(B) = Tr ρB, for a symmetric sequence with AN = jN 1 (B) (see (21)) one easily ﬁnds A∞ (ρ) = Tr ρB.

(36)

Our key application then arises from the frequency operator (23), which amounts to the choice B = |λλ|. In that case (36) becomes f∞ (ρ) = λ|ρ|λ.

(37)

In particular, if |ψ ∈ H is a unit vector and ρ = |ψψ|, deﬁning a vector state ψ on B(H) by ψ(A) = ψ|A|ψ, one has f∞ (ψ) = |λ|ψ|2 .

(38)

This is the Born rule, at least formally. To understand why this identiﬁcation is correct also conceptually, at least in the context of the Copenhagen Interpretation, one has to realize the following. Unlike its counterpart in (35), the limit operator f∞ in (38) is by construction an element of a commutative algebra, namely the C ∗ -algebra C(S(A1 )) of macroscopic observables attached to the N -fold duplication of A1 for N → ∞. According to Bohr’s doctrine of classical concepts (cf. the Introduction), any statement about the quantum system described by A1 has to be made through commutative C ∗ -algebras C associated to A1 , and has to use “the terminology of classical physics”. This terminology includes the role of pure states as “truthmakers”, in the sense that if f : M → R is a classical observable deﬁned

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as a real-valued function on some phase space M , then a point ρ ∈ M validates the proposition f = λ for λ = f (ρ) with certainty. This is precisely what happens in (38), which uses C = C(S(A1 )), hence M = S(A1 ), and states that in the classical state ψ, the observable f∞ simply has the (sharp) value |λ|ψ|2 . Thus one has a non-probabilistic statement in classical physics, which expresses a probabilistic observation about quantum physics. The speciﬁc way in which fN converges to f∞ as a continuous section of A(c) , as well as its relationship to (35), is clariﬁed by the following device [31, 32]. Definition 4. A continuous field of states on a continuous ﬁeld of C ∗ -algebras (A, {Ax }x∈X , {ϕx }x∈X ) over X is a family of states ωx on Ax , deﬁned for each x ∈ X, such that x → ωx (Ax ) is continuous on X for each A ∈ A (i.e. for each continuous section x → ϕx (A) ≡ Ax of the ﬁeld of C ∗ -algebras). ˙ this only imposes the condition In the case at hand, where X = N, ω∞ (A∞ ) = lim ωN (AN ), N →∞

(39)

for each continuous section A of the ﬁeld in question, which we take to be either A(c) or A(q) . Indeed, the relationship between these two continuous ﬁelds of C ∗ -algebras is most easily studied through their respective continuous ﬁelds of states. (q) Any state ω on A∞ trivially deﬁnes a continuous ﬁelds of states on A(q) by ⊂ limN →∞ A⊗N explained just above Theorestriction, using the inclusion A⊗N 1 1 ⊗N rem 2. The ensuing family of states ωN on A1 does not necessarily extend to a continuous ﬁeld on A(c) , and — especially in the context of the Born rule — it is interesting to ﬁnd examples when they do. is symmetric when each of its restrictions Definition 5. A state ω on limN A⊗N 1 is invariant under the natural action of the symmetric group SN on A⊗N to A⊗N 1 1 (under which σ ∈ SN maps an elementary tensor AN = B1 ⊗ · · · ⊗ BN ∈ A⊗N to 1 Bσ(1) ⊗ · · · ⊗ Bσ(N ) ). Such states were analyzed by Størmer [46], who proved a noncommutative version of De Finetti’s well-known representation theorem in classical probability: any has a unique decomposition symmetric state ω on limN A⊗N 1 dµ(ρ)ρ⊗∞ , (40) ω= S(A1 )

where µ is a probability measure on S(A1 ), and ρ⊗∞ is deﬁned as in (34). Theorem 3. Let ω be a symmetric state on limN A⊗N with decomposition (40), and 1 (c) . Define a state ω let ωN be the restriction of ω to A⊗N ∞ on A∞ = C(S(A1 )) by 1 dµ(ρ)f (ρ). (41) ω∞ (f ) = µ(f ) ≡ S(A1 )

Then the family of states {ωN , ω∞ }N ∈N satisfies (39) for any A ∈ A(c) and hence defines a continuous family of states on A(c) .

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Proof. This is immediate from (28) and (40).n We now see that the state ω = ψ ⊗∞ used at the beginning of this section is an example of Deﬁnition 5, for which the associated measure µ in (40) and (41) are, is the Dirac measure δψ concentrated at ψ ∈ S(A1 ). The states ωN on A⊗N 1 of course, given by ωN = ψ ⊗N , and the function N → ωN (fN ) has constant value |λ|ψ|2 . Hence one recovers the limit (38) either from (39) or from (41), since δψ (f∞ ) = f∞ (ψ); the fact that these computations coincide is an illustration of Theorem 3. References [1] N. Bohr, Discussion with Einstein on epistemological problems in atomic physics, in Albert Einstein: Philosopher-Scientist, ed. P. A. Schilpp (Open Court, La Salle, 1949), pp. 201–241. [2] M. Born, Quantenmechanik der Stoßvorg¨ ange, Z. Phys. 38 (1926) 819–827. [3] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol. II: Equilibrium States, Models in Statistical Mechanics (Springer, Berlin, 1981). [4] J. Bub, Interpreting the Quantum World (Cambridge University Press, Cambridge, 1999). [5] S. Bugajski and Z. Motyka, Generalized Borel law and quantum probabilities, Int. J. Theor. Physics 20 (1981) 262–268. [6] A. Cassinello and J. L. S´ anchez-G´ omez, On the probabilistic postulate of quantum mechanics, Found. Phys. 26 (1996) 1357–1374. [7] C. Caves and R. Schack, Properties of the frequency operator do not imply the quantum probability postulate, Ann. Phys. (N.Y.) 315 (2005) 123–146. [8] M. Dickson, Non-relativistic quantum mechanics, in Handbook of the Philosophy of Science, Vol. 2: Philosophy of Physics, eds. J. Butterﬁeld and J. Earman (NorthHolland, Elsevier, Amsterdam 2007), pp. 275–415. [9] P. A. M. Dirac, The Principles of Quantum Mechanics (Clarendon Press, Oxford, 1930). [10] J. Dixmier, C ∗ -Algebras (North-Holland, Amsterdam, 1977). [11] N. G. Duﬃeld and R. F. Werner, Local dynamics of mean-ﬁeld quantum systems, Helv. Phys. Acta 65 (1992) 1016–1054. [12] A. Einstein and M. Born, Briefwechsel 1916–1955 (Langen M¨ uller, M¨ unchen, 2005). [13] E. M. R. A. Engel, A Road to Randomness in Physical Systems, Lecture Notes in Statistics, Vol. 71 (Springer-Verlag, Berlin, 1992). [14] D. E. Evans and Y. Kawahigashi, Quantum Symmetries on Operator Algebras (Oxford University Press, Oxford, 1998). [15] E. Farhi, J. Goldstone and S. Gutmann, How probability arises in quantum mechanics, Ann. Phys. (N.Y.) 192 (1989) 368–382. [16] T. L. Fine, Theories of Probability (Academic Press, New York, 1978). [17] D. Finkelstein, The logic of quantum physics, Trans. New York Acad. Sci. 25 (1965) 621–637. [18] D. Gillies, Philosophical Theories of Probability (Cambridge University Press, Cambridge, 2000). n Analogous

results appear in the work of Unnerstall [48, 49].

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[19] R. Haag, Local Quantum Physics: Fields, Particles, Algebras (Springer-Verlag, Heidelberg, 1992). [20] A. H´ ajek, Interpretations of probability, in The Stanford Encyclopedia of Philosophy, ed. E. N. Zalta (The Metaphysics Research Lab, summer 2003) (electronic); http://www.science.uva.nl/∼seop/ entries/ probability-interpret/. [21] J. B. Hartle, Quantum mechanics of individual systems, Am. J. Phys. 36 (1968) 704–712. [22] W. Heisenberg, Physics and Philosophy: The Revolution in Modern Science (Allen and Unwin, London, 1958). [23] K. Hepp, Quantum theory of measurement and macroscopic observables, Helv. Phys. Acta 45 (1972) 237–248. [24] C. Heunen, N. P. Landsman and B. Spitters, A topos for algebraic quantum theory, arXiv:0709.4364v2. [25] G. ’t Hooft, The mathematical basis for deterministic quantum mechanics, arxiv: quant-ph/0604008. [26] D. Howard, What makes a classical concept classical? Towards a reconstruction of Niels Bohr’s philosophy of physics, in Niels Bohr and Contemporary Philosophy, eds. J. Faye and H. Folse (Kluwer Academic Publishers, Dordrecht, 1994), pp. 201–229. [27] D. Howard, Who invented the Copenhagen Interpretation? Philos. Sci. 71 (2004) 669–682. [28] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. 2: Advanced Theory (Academic Press, New York, 1986). [29] E. Kirchberg and S. Wassermann, Operations on continuous bundles of C ∗ -algebras, Math. Ann. 303 (1995) 677–697. [30] L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd edn. (Pergamon Press, Oxford, 1977). [31] N. P. Landsman, Mathematical Topics Between Classical and Quantum Mechanics (Springer-Verlag, New York, 1998). [32] N. P. Landsman, Between classical and quantum, in Handbook of the Philosophy of Science, Vol. 2: Philosophy of Physics, eds. J. Butterﬁeld and J. Earman (NorthHolland, Elsevier, Amsterdam 2007), pp. 417–554. [33] J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory, Vol. 6: The Completion of Quantum Mechanics 1926–1941. Part 1: The Probabilistic Interpretation and the Empirical and Mathematical Foundation of Quantum Mechanics. 1926–1936 (Springer-Verlag, New York, 2000). [34] D. H. Mellor, Probability: A Philosophical Introduction (Routledge, London, 2005). [35] G. Morchio and F. Strocchi, Mathematical structures for long-range dynamics and symmetry breaking, J. Math. Phys. 28 (1987) 622–635. [36] J. von Neumann, Mathematische Grundlagen der Quantenmechanik (Springer-Verlag, Berlin, 1932); English translation: Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955). [37] J. von Neumann, On inﬁnite direct products, Compos. Math. 6 (1938) 1–77. [38] W. Ochs, On the strong law of large numbers in quantum probability theory, J. Philos. Logic 6 (1977) 473–480. ¨ [39] W. Pauli, Uber Gasentartung und Paramagnetismus, Z. Phys. 41 (1927) 81–102. [40] J. von Plato, Creating Modern Probability (Cambridge University Press, Cambridge, 1994). [41] G. A. Raggio and R. F. Werner, Quantum statistical mechanics of general mean ﬁeld systems, Helv. Phys. Acta 62 (1989) 980–1003.

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[42] S. Saunders, Derivation of the Born rule from operational assumptions, Proc. Roy. Soc. Lond. A 460 (2004) 1771–1788. [43] E. Scheibe, The Logical Analysis of Quantum Mechanics (Pergamon Press, Oxford, 1973). [44] M. Schlosshauer and K. Camilleri, The quantum-to-classical transition: Bohr’s doctrine of classical concepts, emergent classicality, and decoherence, arxiv:0804.1609v1. [45] G. L. Sewell, Quantum Mechanics and its Emergent Macrophysics (Princeton University Press, Princeton, 2002). [46] E. Størmer, Symmetric states of inﬁnite tensor products of C ∗ -algebras, J. Funct. Anal. 3 (1969) 48–68. [47] W. Thirring, A Course in Mathematical Physics. Vol. 4: Quantum Mechanics of Large Systems (Springer-Verlag, New York, 1983). [48] T. Unnerstall, Phase-spaces and dynamical descriptions of inﬁnite mean-ﬁeld quantum systems, J. Math. Phys. 31 (1990) 680–688. [49] T. Unnerstall, Schr¨ odinger dynamics and physical folia of inﬁnite mean-ﬁeld quantum systems, Comm. Math. Phys. 130 (1990) 237–255. [50] R. A. Van Wesep, Many worlds and the appearance of probability in quantum mechanics, Ann. Phys. (N. Y.) 321 (2006) 2438–2452. [51] D. Wallace, Everettian Rationality: Defending Deutsch’s approach to probability in the Everett interpretation, Stud. Hist. Phil. Mod. Phys. 34 (2003) 415–438.

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Reviews in Mathematical Physics Vol. 20, No. 10 (2008) 1191–1208 c World Scientiﬁc Publishing Company

GEODESIC FLOW ON EXTENDED BOTT–VIRASORO GROUP AND GENERALIZED TWO-COMPONENT PEAKON TYPE DUAL SYSTEMS

PARTHA GUHA Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, D-04103 Leipzig, Germany and S. N. Bose National Center for Basic Sciences, JD Block, Sector-3, Salt Lake, Calcutta-700098, India [email protected] [email protected] Received 31 December 2007 Revised 26 July 2008 This paper discusses an algorithmic way of constructing integrable evolution equations based on Lie algebraic structure. We derive, in a pedagogical style, a large class of two-component peakon type dual systems from their two-component soliton equations counter part. We study the essential aspects of Hamiltonian ﬂows on coadjoint orbits 1 ) C ∞ (S 1 ) to give a sysof the centrally extended semidirect product group Diﬀ(S tematic derivation of the dual counter parts of various two-component of integrable systems, viz., the dispersive water wave equation, the Kaup–Boussinesq system and the Broer–Kaup system, using moment of inertia operators method and the (frozen) Lie–Poisson structure. This paper essentially gives Lie algebraic explanation of Olver– Rosenau’s paper [31]. Keywords: Geodesic ﬂow; diﬀeomorphism; Virasoro orbit; Sobolev norm; dual equation; frozen Lie–Poisson structure. Mathematics Subject Classiﬁcations 2000: 53A07, 53B50

1. Introduction Recently a 2-component generalization of the Camassa–Holm equation has drawn a lot of interest among scientists. The integrable system group at SISSA, Dubrovin and his coworkers have been working on multi-component analogues, using reciprocal transformations and studying their eﬀect on the Hamiltonian structures, [9, 10, 27]. They show that the 2-component system cited above admits peakons, albeit of a diﬀerent shape owing to the diﬀerence in the corresponding Green’s functions. It has been shown [22] that a 2-component generalization of the Camassa– Holm equation and its supersymmetric analogue also follow from the geodesic motion with respect to the H 1 metric on the extended semidirect product space 1191

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) C ∞ (S 1 ) and its supergroup, respectively. In fact it is known that numerDiﬀ(S 1 ous coupled KdV equations [19–21] follow from the geodesic ﬂows of the right invari) C ∞ (S 1 ) [1,28]. ant L2 metric on the extended semidirect product group Diﬀ(S 1 Since the 80’s, the coupled KdV systems are considered to be important mathematical models. These set of equations are used in various physical phenomena. In 1981, Fuchssteiner [15] made a detailed study of four coupled KdV equation and formulated the bihamiltonian structure of them. Later, Antonowicz and Fordy [2] gave ﬁrst systematic derivations of a large number of coupled KdV systems. About ten years ago, Rosenau [33], introduced a class of solitary waves with compact support as solutions of certain wave equations with nonlinear dispersion. It was found that the solutions of such systems unchanged from collision and were thus called compactons. Later, Olver and Rosenau showed [31] that a simple scaling argument shows that most integrable bihamiltonian systems are governed by triHamiltonian structures. They formulated a method of “tri-Hamiltonian duality”, in which a recombination of the Hamiltonian operators leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons or peakons. A related construction can be found in the contemporaneous paper of Fuchssteiner [16]. The spirit of the Olver and Rosenau method, i.e. algorithmic ways to derive integrable generalizations of the standard integrable systems was given in [12, 17] in the early 1980’s. However, it was not until these models reappeared in physical problems, and their novel solutions such as compactons and peakons were discovered, that the method achieved recognition. It should be emphasized that a large class of Hamiltonian structures was obtain in [13] using a Backlund transformation, from which it is immediately obvious that the relevant systems are tri-Hamiltonian (without the need for a scaling argument). Among the equations obtained in [13] is the so called Camassa–Holm equation (this is the reason that several authors refer to this equation as the Fuchssteiner–Fokas–Camassa–Holm (FFCH) equation). Furthermore, the “tri-Hamiltonian” approach was used in [11] when in addition to the FFCH, similar analogues for the NLS and sG were derived. In fact, in a more recent paper Fokas et al. [14] discusses several algorithmic ways of constructing integrable evolution equations based on the use of multi-Hamiltonain structures. The tri-Hamiltonian formalism can be best described through examples. The Korteweg–deVries equation ut = uxxx + 3uux,

(1)

can be written in bihamiltonian form ut = J1 dH2 = J2 dH1 using the two compatible Hamiltonian operators J2 = D3 + uD + Du

J1 = D, and H1 =

1 2

u2 dx,

H2 =

1 2

where D ≡

d dx

(−u2x + u3 ) dx.

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The tri-Hamiltonian duality construction is implemented as follows: • A simple scaling argument shows that J2 is in fact the sum of two compatible Hamiltonian operators, namely K2 = D3 and K3 = uD + Du, so that K1 = J1 , K2 , K3 form a triple of mutually compatible Hamiltonian operators. • Thus, when we can recombine the Hamiltonian triple as transfer the leading term D3 from J2 to J1 , thereby constructing the Hamiltonian pairs J1 = K2 ± K1 = D3 ± D. The resulting self-adjoint operator S = 1 ± D2 is used to deﬁne the new ﬁeld variable ρ = Su = u ± uxx . • Finally, the second Hamiltonian structure is constructed by replacing u by ρ in the remaining part of the original Hamiltonian operator K3 , so that J2 = ρD + Dρ. Note that this change of variables does not aﬀect J1 . As a result of this procedure, we recover the tri-Hamiltonian dual of the KdV equation 2 1 δH δH = J2 , ρt = J1 δρ δρ where 1 = 1 H 2

1 uρ dx = 2

2

(u ∓

u2x ) dx,

2 = 1 H 2

(2)

(u3 ∓ uu2x) dx.

In this case, (2) reduces to the celebrated Camassa–Holm equation [4, 5]: 1 2 ut ± uxxt = 3uux ± uuxx + ux . 2 x

(3)

Thus the Camassa–Holm (CH) equation is dual to the KdV equation. It is known that the KdV and the CH equations have a geometric derivation and both of them are models of shallow water waves, the two equations have quite diﬀerent structural properties. Similarly one can also study the two-component KdV equation, one prototypical example is the Ito equation [23], ut = uxxx + 3uux + vvx , vt = (uv)x ,

(4)

which is a protypical example of a two-component KdV equation. The triHamiltonian dual of Ito equation follows from (2) where the ﬁrst and second Hamiltonian operators for the new equation are given by D ± D3 0 J1 = , 0 D ρ D + Dρ vD J2 = Dv 0

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with 1 = 1 H 2

2

(uρ + v ) dx

2 = 1 H 2

(u3 + uv 2 ∓ uu2x ) dx.

The dual system (2) takes the explicit form

1 1 2 , ut ± uxxt = 3uux + vvx + uuxx + uuxx + ux 2 2 x vt = (uv)x .

(5)

1.1. Result and organization This paper elucidates the Lie algebraic structure of a well known approach for constructing integrable PDEs and employs this approach for constructing a certain two-component integrable system. In this article we construct such dual systems from the Lie–Poisson method. We study this rearrangement of Hamiltonian operators via the construction of moment of inertia operator. This moment of inertia operator is tacitly connected to H 1 -Sobolev norm, at least for the Ito equation this is readily observable. Using the moment of inertia operators, we compute dual variables. We give a systematic method to derive such operators from the frozen Lie–Poisson structure. Using these operators, we obtain the dual system for various two-component tri-Hamiltonian systems. In these algorithmic ways, we can derive various new dual extension of known integrable systems, which cannot be derived via traditional H 1 -metric approach. Certainly our method can be thought of a Lie theoretic interpretation of the Olver–Rosenau paper [31]. In other words, the duality method of Olver–Rosenau can be manifested in terms of moment of inertial operator related to coadjoint orbit ) C ∞ (S 1 ). The whole of the centrally extended semidirect product group Diﬀ(S 1 paper discusses algorithmic ways of constructing dual systems. The paper is organized as follows. At ﬁrst, we recapitulate the basic deﬁnitions of semidirect product and extension of the Bott–Virasoro group. We compute the coadjoint orbit and the Hamiltonian operator in Sec. 2. We introduce the frozen Lie–Poisson the structure, moment of inertial operator in Sec. 3. In Sec. 4, we construct the generalized two-component peakon type systems using the method of moment of inertia operator. We also give several examples of dual equations. 2. Semidirect Product and Extended Bott–Virasoro Group Let ρ : G → Aut(V ) denotes a Lie group (left) representation of G in the vector space V , and ρ˜ : g → End(V ) is the induced Lie algebra representation. Let us denote G V the semidirect product group of G with V by ρ with multiplication [8, 30] (g1 , v1 )(g2 , v2 ) = (g1 g2 , v1 + ρ(g1 )v2 ) .

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Let g V be the Lie algebra of G V . The Lie bracket on g V is given by [(ξ1 , u1 ), (ξ2 , u2 )] = ([ξ1 , ξ1 ], ρ˜(ξ1 )u2 − ρ˜(ξ2 )u1 ) . An example of a semidirect product structure is when g is the Lie algebra so(3) associated with the rotation group SO(3) and u is R3 . Their semidirect product is the algebra of the 6-parameter Galilean group of rotations and translations. We can build the Lie–Poisson brackets from these algebras. The ± Lie–Poisson bracket of f, g : (g V )∗ → R is given as δf δg δf δg δg δf {f, g}± (µ, a) = ± µ, , ± a, ρ˜ · ∓ a, ρ˜ · , δµ δµ δµ δa δµ δa where

δf δµ

∈ g and

δf δa

∈ V , dual of µ under the pairing <, >: h∗ × h → R.

2.1. Extension of the Bott–Virasoro group The Lie algebra of Diﬀ(S 1 ) C ∞ (S 1 ) is the semidirect product Lie algebra g = Vect(S 1 ) C ∞ (S 1 ). An element of g is a pair d d ∈ Vect(S 1 ), f (x) , a(x) , where f (x) dx dx

and a(x) ∈ C ∞ (S 1 ).

It is known that this algebra has a three-dimensional central extension given by the non-trivial cocycles d d = ω1 f (x) , a(x) , g , b f (x)g (x) dx , dx dx S1 d d = [f (x)b(x) − g (x)a(x)] d x, (6) ω2 f (x) , a(x) , g , b dx dx S1 d d ω3 f (x) , a(x) , g , b =2 a(x)b (x) dx. dx dx S1 The ﬁrst cocycle ω1 is the well-known Gelfand–Fuchs cocycle. The Virasoro algebra Vir = Vect(S 1 ) R is the unique non-trivial central extension of Vect(S 1 ) based on the Gelfand–Fuchs

cocycle. The space C ∞ (S 1 ) R is identiﬁed as regular part of the dual space to the Virasoro algebra. Since the topological dual of the Fr´echet space Vect(S 1 ) is too big, we restrict our attention to the regular dual g∗ , the subspace of Vect(S 1 )∗ deﬁned by linear functionals of the form 2π d ∈ Vect(S 1 ) , F (u) = u(x), f (x) = u(x)f (x) dx, f (x) dx 0 for some function u(x) ∈ C ∞ (S 1 ). The regular part of the dual space Vect(S 1 )∗ is therefore isomorphic to C ∞ (S 1 ) via the L2 inner product. We say that a smooth

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real-valued function F is a regular function if there exists a smooth map [6, 7] δF : C ∞ (S 1 ) → C ∞ (S 1 ) such that dF (µ)u = u · δF (µ) dx, µ, u ∈ C ∞ (S 1 ). S1

In other words, the Fr´echet derivative dF (µ) belongs to the regular dual Vect∗(S 1 ) and the mapping µ → δF (µ) is smooth. Here δF (µ) stands for variational derivative. For any functional f : g∗ → R one can deﬁne its variational δf : derivative δµ δf d f (µ + sw)|s=0 , w, = δµ ds

µ, w ∈ g∗ .

δf is always a vector on In the ﬁnite-dimensional situation, variational derivative δµ the Lie algebra g. In the inﬁnite-dimensional case this is not always true, as not any linear operator on the dual algebra g∗ can be represented by a smooth vector ﬁeld from the Lie algebra g. Other examples are nonlinear polynomial functionals [25] N (u) dx, F (u) = S1

where N is a polynomial in derivatives of u up to an order n and the corresponding variational derivative is given by δF di = (−1)i i δu dx i=0 n

∂N (u) , ∂Xi

where Xi are vector ﬁelds generated by the Sobolev H k -metric and the operator n ∂ is also known as the Euler operator. E = i=0 (−D)i ∂X i The pairing between this space and the Virasoro algebra is given by: d (u(x), α), f (x) , a = u(x)f (x) dx + aα . dx S1 Similarly we consider the following extension of g, R3 . g = Vects (S 1 ) C ∞ (S 1 ) The commutation relation in g is given by d d d := (f g − f g) , f b − ga , ω f , a, α , g , b, β dx dx dx

(7)

(8)

where α = (α1 , α2 , α3 ), β = (β1 , β2 , β3 ) ∈ R3 , and where ω = (ω1 , ω2 , ω3 ) are the cocycles.

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Let ∗ = C ∞ (S 1 ) greg

C ∞ (S 1 )

1197

R3

denote the regular part of the dual space g ∗ to the Lie algebra g, under the following pairing: [f (x)u(x) + a(x)v(x)] dx + α · γ, (9) u, f = S1

d ∗ , f = (f dx , a, α) ∈ g. Of particular interest are the where u = (u(x), v, γ) ∈ greg ∗ coadjoint orbits in greg . In this case, Gelfand, Vershik and Graev, [18], have constructed some of the corresponding representations.

2.2. Computation of Hamiltonian operator and coadjoint representation g Let us introduce H 1 inner product on the algebra f, gH 1 = [f (x)g(x) + a(x)b(x) + ∂x f (x)∂x g(x)] dx + α · β,

(10)

S1

where f =

d f , a, α , dx

d g = g , b, β . dx

Now we compute: Lemma 2.1. The coadjoint operator with respect to the H 1 inner product is given by

u g ad∗fb v   (1 − ∂ 2 )−1 [2f (x)(1 − ∂x2 )u(x) + f (x)(1 − ∂x2 )u (x) + a v(x)]   . + c1 f + c2 a =   f v(x) + f (x)v (x) − c2 f (x) + 2c3 a (x) (11) Proof. Since we have identiﬁed g with g∗ , it follows from the deﬁnition that ad∗fb, u u, [f, g]H 1 , gH 1 = =− [(f g − f g)u − (f b − ga )v − ∂x (f g − f g)∂x u] dx. S1

After computing all the terms by integrating by parts and using the fact that the functions f (x), g(x), u(x) and a(x), b(x), v(x) are periodic, the right-hand side can be expressed as above.

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Let us compute now the left-hand side: ad∗fb

u [(ad∗fbu)g + (ad∗fbu) g + (ad∗fbv)b] dx = v S1 = [[(1 − ∂ 2 )ad∗fbu]g + (ad∗fbv)b] dx = ((1 − ∂ 2 )ad∗fbu(ad∗fbv)), (g, b). S1

Thus by equating the right- and left-hand sides, we obtain the desired formula. Using standard technique of integrable systems [3] we extract the Hamiltonain operator from the coadjoint action (11). Proposition 2.2. The Hamiltonian operator associated to extended Bott–Virasoro orbit with respect to H 1 -metric is given by c1 D3 + Dρ + ρD vD + c2 D2 OH 1 = , (12) Dv − c2 D2 2c3 D where ρ = (1 − ∂x2 )u. Corollary 2.3. The Hamiltonian operator with respect to right invariant L2 metric is given by c1 D3 + Du + uD vD + c2 D2 . (13) OL2 = Dv − c2 D2 2c3 D Subsequently, we have to restrict on speciﬁc hyperplanes for the construction of various types of peakon systems.

2.3. Modified Gelfand–Fuchs cocycle Consider the following “modiﬁed” Gelfand–Fuchs cocycle on Vect(S 1 ): d d (af g + bf g) dx. ωmGF f (x) , g(x) = dx dx S1

(14)

This cocycle is cohomologues to the Gelfand–Fuchs cocycle, hence, the corresponding central-extension is isomorphic to the Virasoro algebra. The additional term in (14) is a coboundary term. It is easy to check that the functional 1 f g dx = (f g − f g ) dx 2 S1 S1 d d , g dx ∈ Vect(S 1 ). depends on the commutator of f dx

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The Gelfand–Fuchs theorem states that H 2 (Vect(S 1 )) = R, and therefore, every nontrivial cocycle is proportional to the Gelfand–Fuchs cocycle upto a coboundary. Thus one has ω ˜ 1 = λω1 + b, where b is a coboundary

d d b f ,g = u, [f, g] dx dx

for some u belongs to space quadratic diﬀerential form or dual of Vect(S 1 ). The new coboundary term modiﬁed the original Lie–Poisson structure on Vect∗ (S 1 ). Thus the new bivector is an aﬃne perturbation of the canonical Lie– Poisson structure on Vir∗ . It is given by Λ = Λ0 + Λ1 , where Λ0 is the canonical or unperturbed Poisson bivector. The perturbed (constant) bivector Λ1 is itself a Poisson bivector since it satisﬁes automatically the Schouten–Nijenhuis condition [Λ1 , Λ1 ] = 0. 2.4. Modified Hamiltonian structure We now compute the modiﬁed Hamiltonian structure corresponding to modiﬁed Lie–Poisson structure on the extended Bott–Virasoro group. It is clear that Vect(S 1 ) C ∞ (S 1 ) algebra is extended by the non-trivial three 2-cocycles

(˜ ω1 , ω2 , ω3 ). Let us compute the coadjoint action of Vect(S 1 ) C ∞ (S 1 ) R3 on

its dual C ∞ (S 1 ) C ∞ (S 1 ) R3 and is given by ad∗fˆuˆ



 = 

(2f (x)u(x) + f (x)u (x) + a v(x) − c1 (af + bf ) + c2 a f v(x) + f (x)v (x) − c2 f (x) + 2c3 a (x)

  . 

0 Thus the modiﬁed Hamiltonian structure associated with the coadjoint action in presence of modiﬁed cocycle is given by −c1 (aD3 + bD) + 2uD + ux vD + c2 D2 L2 = O vx + vD − c2 D2 2c3 D c1 D3 + c4 D + 2uD + ux vD + c2 D2 ≡ , (15) vx + vD − c2 D2 2c3 D where c4 is a new constant.

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3. Duality, Moment of Inertia and Equations In this section, we give an algorithmic construction of the generalized multicomponent Camassa–Holm equation. This method depends directly on the frozen Lie– Poisson structure. We brieﬂy recapitulate frozen Lie–Poisson in the next section. 3.1. Frozen Lie–Poisson structure Consider the dual of the Lie algebra of g∗ with a Poisson structure given by the “frozen” Lie–Poisson structure. In otherwords, we ﬁx some point µ0 ∈ g∗ and deﬁne a Poisson structure given by {f, g}0 (µ) := [df (µ), dg(µ)], µ0 . It was shown by Khesin and Misiolek [24] that Proposition 3.1. The brackets {·, ·}LP and {·, ·}0 are compatible for every “freezing” point µ0 . Proof. Let us take any linear combination {·, ·}λ := {·, ·}LP + λ{·, ·}0 . is again a Poisson bracket, it is just the translation of the Lie–Poisson bracket from the origin to the point −λµ0 . Let us proceed to compute frozen brackets on the dual space of the extended semi-direct product space Vect(S 1 ) C ∞ (S 1 ) . In general, given ∗ (u0 , v0 , c) ∈ Vect(S 1 ) C ∞ (S 1 ) C ∞ (S 1 ) C ∞ (S 1 ) R3 , the frozen bracket is given by

{f, g}(u, v, c) = (u0 , v0 , c),

δf δg , , δ(u, v, c) δ(u, v, c) δf ∗ . = −ad δf (u0 , v0 , c), δ(u,v,c) δ(u, v, c)

Furthermore, recall the corresponding Euler–Poincar´e equations of motions are given by d (u, v, c) = −ad∗ δf (u0 , v0 , c). δ(u,v,c) dt We compute the generalized frozen Hamiltonian structure from Eq. (15). Lemma 3.2. We consider the frozen Poisson structure at u0 = µ, v0 = λ, c1 = a, c2 = c and c3 = d, where µ, λ, c and d are some constants. It is given by aD3 + (c4 + µ)D λD + cD2 aD3 + bD λD + cD2 OgFrozen = ≡ (16) λD − cD2 2dD λD − cD2 2dD where we assume (c4 + µ) = b.

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Remark on coboundary operator and frozen structure. Every 2-cocycle Γ deﬁnes a Lie–Poisson structure on g∗ . The vanishing of Schouten–Nijenhuis bracket for Poisson bivector can be recast as a cocycle condition ∂Γ = 0, where ∂ : ∧k g∗ → ∧k+1 g∗ . A special case of Lie–Poisson structure is given by a 2-cocycle Γ which is a coboundary [6, 7]. If Γ = ∂µ0 for some µ0 ∈ g∗ , the expression {f, g}0 (µ) = µ0 ([df (µ), dg(µ)]) considered to be Lie–Poisson bracket which has been “frozen” at a point µ0 ∈ g∗ . 3.2. Frozen Hamiltonian structure and moment of inertia operator Let G be an arbitrary Lie group, g be its Lie algebra, and g∗ be the corresponding dual algebra. Let I : g → g∗ be a positive deﬁnite symmetric operator, known as moment of inertia operator, deﬁning a scalar product on the Lie algebra. The moment of inertia operator deﬁnes a left- or right-invariant inertia operator IG on the group. This deﬁnes a left- or right-invariant metric or inner product. Consider this inner product on the Lie algebra of vector ﬁelds Vect(S 1 ) on S 1 . If this inner product is local, it is deﬁned via the moment of inertia operator I d d d d ∈ Vect(S 1 ). ηIβ dx, η , β η ,β = dx dx dx dx 1 S We deﬁne a quadratic functional, Hamiltonian function 1 uI−1 (u) H(u) = 2 S1 on the regular dual Vect∗ (S 1 ). If the metric is left-invariant, then geodesics of this metric are described by the Euler–Poincar´e equation u˙ = adI−1 u u,

u ∈ g∗ .

Let us generalized this to semidirect product algebra Vect(S 1 ) C ∞ (S 1 ). It must be noticed that in the inﬁnite dimensional case the operator I is invertible only on a regular part of the dual algebra (Vect(S 1 ) C ∞ (S 1 ))∗ . Definition 3.3. The generalized moment of inertia I maps   d u m(dx)2 I :  dx  → , p(x) v(x) given by

m(x) p(x)

=I

u(x) v(x)

(17)

.

(18)

Now we state the algorithmic method to compute the generalized moment of inertia operator. This follows directly from the Lie–Poisson structure.

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Recipe to compute generalized moment of inertia operator. The generalized moment of inertia is obtained from the frozen Poisson structure Ofrozen . It is given by Ofrozen = IDI,

(19)

where I is the identity matrix, and it is given by aD2 + b λ + cD I2×2 = . λ − cD 2d

(20)

This is a symmetric operator, and Eq. (20) is the most generalized form of moment of inertia associated to the coadjoint orbit of ˆg. 3.2.1. Examples of moment of inertia operators We illustrate this phenomena by examples. Let us start with a few special cases: 1. The moment of inertia operator for the KdV equation is a trivial operator. 2. In the case of the two-component Camassa–Holm equation, we choose a = −b. The moment of inertia operator for the Camassa–Holm equation is I = (1 − ∂ 2 ). 3. The moment of inertia operator for the two-component Camassa–Holm equation is b(1 − D2 ) λ + cD , (21) I2×2 = λ − cD 2d where we assume a = −b in Eq. (20). Lemma 3.4. The moment of inertia operator relates (u, v) pair to (m, p) pair for two-component Camassa–Holm equation as m(x) = b(u − uxx ) + λv + cvx , p(x) = λu − cux + 2dv.

(22)

Proof. Using Eq. (18) we obtain m and p. Let ˆ g = Vect(S 1 ) C ∞ (S 1 ) of inertia operator

R3 . The inner product is deﬁned via the moment g→ g∗ . I2×2 :

Thus the moment of inertia I2×2 plays the role of a metric — it allows us to build a quadratic form from two elements of g∗ . Thus the Hamiltonian is deﬁned as

m(x) m(x) m(x) u(x) −1 H(m, p) = , I2×2 = , p(x) p(x) p(x) v(x) = (mu + pv) dx. (23) S1

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4. Moment of Inertia, Boussinesq System and Duality Equation Let us compute the dual equation associated to the generalized Hamiltonian operator associated to coadjoint orbit of the extended Bott–Virasoro group ) C ∞ (S 1 ). Diﬀ(S 1 4.1. Construction of generalized dual equation associated to the Bott–Virasoro group The frozen Hamiltonian structure is computed straight away from Eq. (16) and it is given by c1 D3 + αD λD + c2 D2 Ofrozen = , (24) −c2 D2 + λD 2c3 D where α = c4 + u0 . Thus the generalized moment of inertia operator associated to g is given by c1 D 2 + α λ + c2 D . (25) Igen = 2c3 −c2 D + λ Therefore, the dual variables are m(x) = c1 uxx + αu + λv + c2 vx , p(x) = −c2 ux + λu + 2c3 v. The dual variables induces the modiﬁed Hamiltonian structure Dm + mD Dp Ogen = . Dv 0

(26)

(27)

The Euler–Poincar´e ﬂow on dual space of Lie algebra g∗ can be written in the form   δH1   m gen  δm  = −O (28)  p t δH1  δp where the Hamiltonian is given by 1 H= 2

mu + pv dx. S1

Proposition 4.1. The Euler–Poincar´e flow associated to the modified Hamiltonian operator yields a flow on g∗ , given by c1 α mt + mu + u2x + u2 + λuv + c3 v 2 = 0, 2 2 x pt + (pu)x = 0, (29) where m and p is defined as (26).

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Proof. By direct computation. Equation (29) is the most general form of peakon version of coupled KdV equation. All other two-component peakon type equations are reductions of this equation. Hence, we call this equation as the peakon/compacton (or dual) version of the Antonowicz–Fordy equation. Example. We give a prototypical example of the two-component Camassa–Holm equation. The Euler–Poincar´e ﬂow of the dual equation yields the generalized twocomponent Camassa–Holm equation (see for example, [10]) mt + mx u + 2mux + pvx = 0, pt + (pu)x = 0,

(30)

where m and p satisfy Eq. (22). 4.2. Dual equation of the Boussinesq system Let us consider the Kupershmidt’s version [26] of the Boussinesq system ut = uux + vx − vxx , vt = (uv)x + vxx . The Hamiltonian structure of the Boussinesq system uD + Du 2D2 + vD O= 2D −2D2 + Dv

(31)

(32)

is associated to hyperplane at c1 = 0, c2 = 2, c3 = 1 in the coadjoint orbit of ˆg and corresponding Hamiltonian is given by 1 (u2 + v 2 ) dx. H= 4 S1 The ﬁrst Hamiltonian operator is just O1 = DI, where I is the 2 × 2 identity matrix. We start with the frozen Hamiltonian structure associated to (31). We compute the Hamiltonian structure at u0 = µ and v0 = λ. Here λ and µ are constants. Thus the frozen Hamiltonian structure is µD 2D2 + λD . (33) Ofrozen = 2D −2D2 + λD Thus the moment of inertia operator of the Boussinesq system becomes µ λ + 2D IBoussinesq = . λ − 2D 2

(34)

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Let us ﬁx µ = λ = 1. Then IBoussinesq yields m(x) u(x) 2u(x) + v − 2vx = IBoussinesq = . p(x) v(x) u(x) + 2ux Once again the moment of inertia transforms the primitive pair (u, v) to a newer pair (m, p), given by m(x) = v + 2vx , p(x) = u − 2ux + 2v.

(35)

Equation (35) transforms the Hamiltonian operator to uD + Du vD mD + Dm pD → . Dv 0 Dp 0 The Hamiltonian function becomes =1 H (mu + pv) dx. 2 S1

(36)

Proposition 4.2. The Euler–Poincar´e flow with respect to new Hamiltonian and Poisson structure yields the dual equation of Boussinesq system mt + (mv)x = 0, pt + (vp + u2 + uv)x = 0.

(37)

4.3. Dual equation of various dispersive water wave equations We narrate our construction with another prototypical example, the dispersive water waves equation, given by ut = vxxx + 2(uv)x , vt = ux + 2vvx .

(38)

This is a geodesic ﬂow on the extension of the Bott–Virasoro group. It is cong and the nected to a hyperplane c1 = 1, c2 = 0, c3 = 12 in the coadjoint orbit of ﬂow is given by   δH ut D3 + Du + uD vD  δu   , H = uv dx. (39) = vt Dv D  δH  S1 δv This equation is also known as the Kaup–Boussinesq system. The KB system has a natural two wave structure, which enables one to capture the eﬀects of interaction of unmodular bores or rarefaction waves arising in the decay of a jump discontinuity.

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4.4. Dual equation of the Kaup–Boussinesq system The frozen Hamiltonian structure at u0 = µ and v0 = λ of the Kaup–Boussinesq system is given by D3 + µD λD . (40) Ofrozen = λD D It is ready to see that the moment of inertia operator of the KB system IKB yields m(x) = uxx + µu + λv,

p(x) = λu + v.

Once again we can normalize µ = λ = 2. The modiﬁed Hamiltonian structure of the dual equation is given by Dm + mD pD = O (41) Dp 0 yields the dual equation for the Kaup–Boussinesq system 1 mt + mu + u2x + u2 + 2uv = 0, 2 x pt + (pu)x = 0 ˆ = 1 (mu + pv) dx. for H S

(42)

4.5. Dual equation of the Broer–Kaup system Let us study the Broer–Kaup system ut = −uxx + 2(uv)x + uux ,

vt = vxx + 2vvx − 2ux

is a geodesic ﬂow associated to the hyperplane c1 = 0, c2 = −1, c3 = −1. Hence, the Hamiltonian structure is uD + Du −D2 + vD , with H = uv dx. OBK = −2D D2 + Dv S1 The moment of inertia operator of the Broer–Kaup system computed from the frozen Hamiltonian structure at u0 = v0 = 1 and given by 1 1−D IBK = . D+1 −2 Therefore, IBK yields m(x) = u + v − vx ,

p(x) = u + ux − 2v.

Thus the dual equation for the Kaup–Boussinesq system is mt + (mu + uv − v 2 )x = 0,

ˆ = for H

1 2

S1

pt + (pu)x = 0 (mu + pv) dx.

(43)

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Hence in this paper we present a more generalized formalism to construct twocomponent type Camassa–Holm type equations. We have demonstrated our method with several examples. All these systems obtained appears to be bi-Hamiltonian ﬂow on the coadjoint orbit of Diﬀ(S 1 ) C ∞ (S 1 ). 5. Conclusion and Outlook In this paper, we have derived various two-component generalization of the Camassa–Holm type systems using extended Bott–Virasoro algebra. Our method is closely related to Olver–Rosenau method. We have given a more Lie algebraic illustration of their construction. It would be rather interesting to generalize this method to supersymmetric dual integrable systems. This would involve extended superconformal group. Acknowledgment The author wish to express his deepest gratitude to Professor Peter Olver who has given time to explain the problem with invaluable insights. He is also grateful to Professors Valentin Ovsienko and Andy Hone for various stimulating discussions. He would like to thank Professor J¨ urgen Jost at MPI-MIS where the work has been done in a stimulating atmosphere. References [1] E. Arbarello, C. De Concini, V. G. Kac and C. Procesi, Moduli space of curves and representation theory, Comm. Math. Phys. 117 (1988) 1–36. [2] M. Antonowicz and A. Fordy, Coupled KdV equation with multi-Hamiltonian structures, Phys. D 28 (1987) 345–357. [3] O. Babelon, D. Bernard and M. Talon, Introduction to Classical Integrable Systems, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 2003), xii + 602 pp. [4] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993) 1661–1664. [5] R. Camassa, D. D. Holm and J. M. Hyman, A new integrable shallow water equation, Adv. Appl. Mech. 31 (1994) 1–33. [6] A. Constantin and B. Kolev, Integrability of invariant metrics on the Virasoro group, Phys. Lett. A 350(1–2) (2006) 75–80. [7] A. Constantin and B. Kolev, Integrability of invariant metrics on the diﬀeomorphism group of the circle, J. Nonlinear Sci. 16(2) (2006) 109–122. [8] D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler–Poincar´e equations and semidirect products with applications to continuum theories, Adv. Math. 137 (1998) 1–81. [9] M. Chen, S.-Q. Liu and Y. Zhang, A 2-component generalization of the Camassa– Holm equation and its solutions, preprint, nlin.SI/0501028 (2005). [10] G. Falqui, On a Camassa–Holm type equation with two dependent variables, preprint, nlin.SI/0505059 (2005). [11] A. S. Fokas, On a class of physically important integrable equations. The nonlinear Schr¨ odinger equation, Phys. D 87 (1995) 145–150.

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[12] A. S. Fokas and B. Fuchssteiner, On the structure of symplectic operators and hereditary symmetries, Lett. Nuovo Cimento 28(2) (1980) 299–303. [13] B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Baecklund transformations and hereditary symmetries, Phys. D 4 (1981) 47–66. [14] A. S. Fokas, P. J. Olver and P. Rosenau, A plethora of integrable bi-Hamiltonian equations, in Algebraic Aspects of Integrable System: In Memory of Irene Dorfman, eds. A. S. Fokas and I. M. Gel’fand, Progress in Nonlinear Equations, Vol. 26 (Birkhauser, 1996), pp. 93–101. [15] B. Fuchssteiner, The Lie algebra structure of nonlinear evolution equations admitting inﬁnite-dimensional abelian symmetry groups, Progr. Theoret. Phys. 65(3) (1981) 861–876. [16] B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa–Holm equation, Phys. D 95 (1996) 229–243. [17] I. M. Gelfand and I. Ya. Dorfman, Schouten bracket and Hamiltonian operators, Funct. Anal. Appl. 14(3) (1980) 71–74. [18] I. M. Gelfand, I. M. Graev and A. M. Vershik, Models of representations of current groups, in Representations of Lie groups and Lie Algebras (Akad. Kiad, Budapest, 1985), pp. 121–179. [19] P. Guha, Integrable geodesic ﬂows on the (super)extension of the Bott–Virasoro group, Lett. Math. Phys. 52 (2000) 311–328. [20] P. Guha, Geodesic ﬂows, bihamiltonian structure and coupled KdV type systems, J. Math. Anal. Appl. 310 (2005) 45–56. [21] P. Guha, Euler–Poincar´e formalism of coupled KdV type systems and diﬀeomorphism group on S 1 , J. Appl. Anal. 11 (2005) 261–282. [22] P. Guha and P. J. Olver, Geodesic ﬂow and two (super) component analog of the Camassa–Holm equation, SIGMA Symmetry Integrability Geom. Methods Appl. 2 (2006) Paper 054, 9 pp. (electronic). [23] M. Ito, Symmetries and conservation laws of a coupled nonlinear wave equation, Phys. Lett. A 91 (1982) 335–338. [24] B. Khesin and G. Misiolek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math. 176(1) (2003) 116–144. [25] B. Kolev, Bi-Hamiltonian systems on the dual of the Lie algebra of vector ﬁelds of the circle and periodic shallow water equations, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 365 (2007) no. 1858, 2333–2357. [26] B. A. Kupershmidt, A multicomponent water wave equation, J. Phys. A 18(18) (1985) L1119–L1122. [27] S.-Q. Liu and Y. Zhang, Deformations of semisimple bihamiltonian structures of hydrodynamic type, J. Geom. Phys. 54 (2005) 427–453. [28] P. Marcel, V. Ovsienko and C. Roger, Extension of the Virasoro and Neveu–Schwartz algebras and generalized Sturm–Liouville operators, Lett. Math. Phys. 40 (1997) 31–39. [29] G. Misiolek, A shallow water equation as a geodesic ﬂow on the Bott–Virasoro group, J. Geom. Phys. 24 (1998) 203–208. [30] J. Marsden, T. Ratiu and A. Weinstein, Semidirect products and reduction in mechanics, Trans. Amer. Math. Soc. 281(1) (1984) 147–177. [31] P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E 53 (1996) 1900–1906. [32] V. Yu. Ovsienko and B. A. Khesin, KdV super equation as an Euler equation, Funct. Anal. Appl. 21 (1987) 329–331. [33] P. Rosenau, Nonlinear dispersion and compact structures, Phys. Rev. Lett. 73 (1994) 1737–1741.

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Reviews in Mathematical Physics Vol. 20, No. 10 (2008) 1209–1248 c World Scientiﬁc Publishing Company

ABELIAN TODA SOLITONS REVISITED

KH. S. NIROV Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect 7a, 117312 Moscow, Russia [email protected] A. V. RAZUMOV Institute for High Energy Physics, 142281 Protvino, Moscow Region, Russia [email protected] Received 20 April 2008 Revised 4 September 2008 We present a systematic and detailed review of the application of the method of Hirota and the rational dressing method to abelian Toda systems associated with the untwisted loop groups of complex general linear groups. Emphasizing the rational dressing method, we compare the soliton solutions constructed within these two approaches, and show that the solutions obtained by the Hirota’s method are a subset of those obtained by the rational dressing method. Keywords: Loop Toda equations; Hirota’s and rational dressing methods; soliton solutions. Mathematics Subject Classiﬁcation 2000: 22E67, 35Q51, 37K10, 37K15

1. Introduction Two-dimensional Toda equations associated with loop groupsa are very interesting examples of completely integrable systems, see, for example, the monographs [1, 2]. They possess soliton solutions having a nice physical interpretation as interacting extended objects. Actually there is no clear deﬁnition of a soliton solution. In the present paper, we call a solution of equations an n-soliton solution, if it depends on n linear combinations of independent variables. Soliton solutions for Toda equations can be constructed with the help of various methods. As far as we know, ﬁrst explicit solutions of Toda equations associated a Sometimes,

one deals with Toda equations associated with aﬃne groups being central extensions of loop groups. Usually, it is possible to construct solutions of the equations associated with aﬃne groups starting from solutions of the equations associated with loop groups. 1209

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with loop groups were found by Mikhailov [3]. He used the rational dressing method being a version of the inverse scattering method [4]. Note that in general the solutions obtained by Mikhailov are not soliton solutions. Besides, they are described by a redundant set of parameters. Another method used here is the Hirota’s one. Its essence [5] is a change of the dependent variables which introduces the so called τ -functions. Here the ﬁnal goal is to come to some special bilinear partial diﬀerential equations which are solved then perturbatively. The soliton solutions arise when the perturbation series truncates at some ﬁnite order. This method was applied to aﬃne Toda systems, for example, in the papers [6–10]. The main disadvantage of the Hirota’s method is that there is no regular method to ﬁnd the desired transformation from the initial dependent variables to τ -functions. Therefore, sometimes it is used in combination with other methods that helps to obtain a desired ansatz, see, for example, the papers [11,12]. There are also two additional approaches to the problem, being a development of the Leznov–Saveliev method [13–15], and of the B¨ acklund–Darboux transformation [16–20]. These methods give the same soliton solutions as the Hirota’s one and are not in the scope of the present paper. The basic purpose of our review is to reproduce, in a possibly systematic and detailed way, the application of the Hirota’s and rational dressing methods to Toda systems associated with the untwisted loop groups of complex general linear groups, making an emphasis on the rational dressing method, and compare the soliton solutions constructed along these approaches. We show that all soliton solutions obtained by the Hirota’s method are contained among the solutions obtained by the rational dressing method. 2. Equations 2.1. Zero-curvature representation of Toda equations It is well known that Toda equations can be formulated as the zero-curvature condition for a connection of a special form on the trivial ﬁber bundle R2 × G → R2 , where G is a Lie group with the Lie algebra G. The connection under consideration can be identiﬁed with a G-valued 1-form O on R2 . One can decompose such a connection over basis 1-forms as O = O− dz − + O+ dz + , where z − , z + are the standard coordinates on the base manifold R2 , and the components O− , O+ are G-valued functions on it. Let us assume that the connection O is ﬂat that means that its curvature is zero. This condition in terms of the components has the formb ∂− O+ − ∂+ O− + [O− , O+ ] = 0. b We

use the usual notation ∂− = ∂/∂z − and ∂+ = ∂/∂z + .

(2.1)

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One can consider this relation as a system of partial diﬀerential equations. In a sense, this system is trivial, and its general solution is well known. It is given by the relations O− = Φ−1 ∂− Φ,

O+ = Φ−1 ∂+ Φ,

where Φ is an arbitrary mapping of R2 to G. Actually the triviality of the zero-curvature condition is due to its gauge invariance. That means that if a connection O satisﬁes (2.1) then for an arbitrary mapping Ψ of R2 to G the gauge transformed connection OΨ = Ψ−1 OΨ + Ψ−1 dΨ,

(2.2)

satisﬁes (2.1) as well. To obtain a nontrivial integrable system out of the zero-curvature condition one imposes on the connection O some restriction destroying the gauge invariance. For the case of Toda equations they are the grading and gauge ﬁxing conditions which are introduced as follows. Suppose that the Lie algebra G is endowed with a Z-gradation, Gk , [Gk , Gl ] ⊂ Gk+l , G= k∈Z

and that a positive integer L is such that the grading subspaces Gk , where 0 < |k| < L, are trivial.c The grading condition states that the components of O have the form O− = O−0 + O−L ,

O+ = O+0 + O+L ,

(2.3)

where O−0 and O+0 take values in G0 , while O−L and O+L take values in G−L and G+L , respectively. There is a residual gauge invariance. Indeed, the gauge transformation (2.2) with Ψ taking values in the Lie subgroup G0 corresponding to the subalgebra G0 does not violate the validity of the grading condition (2.3). Therefore, one imposes an additional condition, called the gauge ﬁxing condition, of the form O+0 = 0. After that one can show that the components of the connection O can be represented as O− = Ξ−1 ∂− Ξ + F− ,

O+ = Ξ−1 F+ Ξ,

(2.4)

where Ξ is a mapping of R2 to G0 , F− and F+ are some mappings of R2 to G−L and G+L . One can easily get convinced that the zero-curvature condition is equivalent c It

can be shown that by rejecting this restriction, we do not come to new Toda equations, see [21] for details.

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to the equalityd ∂+ (Ξ−1 ∂− Ξ) = [F− , Ξ−1 F+ Ξ]

(2.5)

and the relations ∂+ F− = 0,

∂− F+ = 0.

(2.6)

One supposes that the mappings F− and F+ are ﬁxed and considers (2.5) as an equation for Ξ called the Toda equation. When the group G0 is abelian the corresponding Toda equations are called abelian. Thus, a Toda equation associated with a Lie group G is speciﬁed by a choice of a Z-gradation of the Lie algebra G of G and mappings F− , F+ satisfying the conditions (2.6). To classify the Toda equations associated with a Lie group G one should classify Z-gradations of the Lie algebra G of G. Two remarks are in order. First, let Σ be an isomorphism from a Z-graded Lie algebra G to a Lie algebra H. One can consider H as a Z-graded Lie algebra with grading subspaces Hk = Σ(Gk ). In such situation, one says that Z-gradations of G and H are conjugated by Σ. Now let the Lie algebra G of the Lie group G be supplied with a Z-gradation, and σ be an isomorphism from G to a Lie group H. Denote by Σ the isomorphism from the Lie algebra G to the Lie algebra H of the Lie group H induced by the isomorphism σ. It is clear that if Ξ is a solution of the Toda equation (2.5), then the mapping Ξ = σ ◦ Ξ

(2.7)

satisﬁes the Toda equation (2.5) with the mappings F− , F+ replaced by the mappings = Σ ◦ F− , F−

F+ = Σ ◦ F+ .

(2.8)

In other words, the solutions to two Toda equations under consideration are connected via the isomorphism σ, and in this sense these equations are equivalent. Thus, to describe really diﬀerent Toda equations it suﬃces to consider Lie groups and Z-gradations of their Lie algebras up to isomorphisms. Secondly, let Θ− and Θ+ be some mappings of R2 to G0 which satisfy the conditions ∂+ Θ− = 0,

∂− Θ+ = 0.

If a mapping Ξ satisﬁes the Toda equation (2.5), then the mapping Ξ = Θ−1 + ΞΘ−

(2.9)

assume for simplicity that G is a subgroup of the group formed by invertible elements of some unital associative algebra A. In this case, G can be considered as a subalgebra of the Lie algebra associated with A. Actually, one can generalize our consideration to the case of an arbitrary Lie group G.

d We

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satisﬁes the Toda equation (2.5) where the mappings F− , F+ are replaced by the mappings = Θ−1 F− − F− Θ − ,

F+ = Θ−1 + F+ Θ + .

(2.10)

Again, the Toda equations for Ξ and Ξ are not actually diﬀerent, and it is natural to use the transformations (2.10) to make the mappings F− , F+ as simple as possible. If the mappings Θ− and Θ+ are such that Θ−1 − F− Θ − = F− ,

Θ−1 + F+ Θ + = F+ ,

then the mapping Ξ satisﬁes the same Toda equation as the mapping Ξ. Hence, in this case the transformation described by relation (2.9) is a symmetry transformation for the Toda equation under consideration. 2.2. Toda equations associated with loop groups of complex simple Lie groups Let g be a ﬁnite dimensional real or complex Lie algebra. The loop Lie algebra of g, denoted L(g), is deﬁned alternatively either as the linear space C ∞ (S 1 , g) of ∞ (R, g) of smooth smooth mappings of the circle S 1 to g, or as the linear space C2π 2π-periodic mappings of the real line R to g with the Lie algebra operation deﬁned in both cases pointwise, see, for example, [22–24]. In this paper, we adopt the second deﬁnition and think of the circle S 1 as consisting of complex numbers of modulus one. There is a convenient way to supply L(g) with the structure of a Fr´echet space, so that the Lie algebra operation becomes a continuous mapping, see, for example, [23–25]. Now, let G be a ﬁnite dimensional Lie group with the Lie algebra g. We deﬁne the loop group of G, denoted L(G), as the set C ∞ (S 1 , G) of smooth mappings of S 1 to G with the group law deﬁned pointwise. We assume that L(G) is supplied with the structure of a Fr´echet manifold modeled on L(g) in such a way that it becomes a Lie group, see, for example, [23–25]. The Lie algebra of the Lie group L(G) is naturally identiﬁed with the loop Lie algebra L(g). Let A be an automorphism of a ﬁnite dimensional Lie algebra g satisfying the relation AM = idg for some positive integer M .e The twisted loop Lie algebra LA,M (g) is a subalgebra of the loop Lie algebra L(g) formed by the elements ξ that satisfy the equality ξ(M p¯) = A(ξ(¯ p)), where M = e2πi/M is the M th principal root of unity. Similarly, given an automorphism a of a Lie group G that satisﬁes the relation aM = idG , we deﬁne the e Note that we do not assume that M is the order of the automorphism A. It can be an arbitrary multiple of the order.

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twisted loop group La,M (G) as the subgroup of the loop group L(G) formed by the elements χ satisfying the equality p)). χ(M p¯) = a(χ(¯ The Lie algebra of a twisted loop group La,M (G) is naturally identiﬁed with the twisted loop Lie algebra LA,M (g), where we denote by A the automorphism of the Lie algebra g corresponding to the automorphism a of the Lie group G. It is clear that loop groups and loop Lie algebras are partial cases of twisted loop groups and twisted loop Lie algebras, respectively. Therefore, below by a loop group we mean either a usual loop group or a twisted loop group, and by a loop Lie algebra we mean either a usual loop Lie algebra or a twisted loop Lie algebra. Now let us discuss the form of the Toda equations associated with a loop group La,M (G). First of all, note that the group La,M (G) and its Lie algebra LA,M (g) are inﬁnite dimensional manifolds. It appears that it is convenient to reformulate the zero curvature representation of the Toda equations associated with La,M (G) in terms of ﬁnite dimensional manifolds. To this end we use the so-called exponential law, see, for example, [26, 27]. Let M, N , P be three ﬁnite dimensional manifolds, and N be compact. Consider a smooth mapping F of M to C ∞ (N , P). This mapping induces a mapping f of M × N to P deﬁned by the equality f (m, ¯ n ¯ ) = (F (m))(¯ ¯ n). It can be proved that the mapping f is smooth. Conversely, if one has a smooth mapping of M × N to P, reversing the above equality one deﬁnes a mapping of M to C ∞ (N , P), and this mapping is also smooth. Thus, we have the following canonical identiﬁcation C ∞ (M, C ∞ (N , P)) = C ∞ (M × N , P). It is this equality that is called the exponential law. In the case under consideration the connection components O− and O+ entering the equality (2.1) are mappings of R2 to the loop Lie algebra LA,M (g). We will denote the corresponding mappings of R2 × S 1 to g by ω− and ω+ , and call them also the connection components. The mapping Φ generating the connection is a mapping of R2 to La,M (G). Denoting the corresponding mapping of R2 × S 1 by ϕ we write ϕ−1 ∂− ϕ = ω− ,

ϕ−1 ∂+ ϕ = ω+ .

(2.11)

Having in mind that the mapping ϕ uniquely determines the mapping Φ, we say that the mapping ϕ also generates the connection under consideration. The relations (2.4) are equivalent to the equalities ω− = γ −1 ∂− γ + f− ,

ω+ = γ −1 f+ γ,

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where γ is a smooth mapping of R2 × S 1 to G corresponding to the mapping Ξ, f− and f+ are smooth mappings of R2 × S 1 to the Lie algebra g of G corresponding to the mappings F− and F+ . The mappings f− and f+ satisfy the conditions ∂+ f− = 0,

∂− f+ = 0,

(2.12)

which follow from the conditions (2.6). The Toda equation (2.5) in the case under consideration is equivalent to the equation ∂+ (γ −1 ∂− γ) = [f− , γ −1 f+ γ].

(2.13)

To classify Toda equations associated with loop groups one has to classify Zgradations of the corresponding loop Lie algebras. This problem was partially solved in the paper [24], see also [21]. In these papers, the case of loop Lie algebras of complex simple Lie algebras was considered and for this case a wide class of the socalled integrable Z-gradations [24] with ﬁnite dimensional grading subspaces was described. Actually, it was shown that when g is a complex simple Lie algebra any integrable Z-gradation of a loop Lie algebra LA,M (g) with ﬁnite dimensional grading subspaces is conjugated by an isomorphism to the standard gradation of another loop Lie algebra LA ,M (g), where the automorphisms A and A diﬀer by an inner automorphism of g. In particular, if A is an inner (outer) automorphism of g, then A is also an inner (outer) automorphism of g. Assume now that G is a ﬁnite dimensional complex simple Lie group, then its Lie algebra g is a complex simple Lie algebra. Consider a Toda equation associated with a loop group La,M (G). The corresponding Z-gradation of LA,M (g) is conjugated by an isomorphism to the standard Z-gradation of an appropriate loop Lie algebra LA ,M (g). Since the automorphisms A and A diﬀer by an inner automorphism of g, the automorphism A can be lifted to an automorphism a of G, and the isomorphism from LA,M (g) to LA ,M (g) under consideration can be lifted to an isomorphism from La,M (G) to La ,M (G). Actually this means that the initial Toda equation associated with La,M (G) is equivalent to a Toda equation associated with La ,M (G) arising when we supply LA ,M (g) with the standard Z-gradation. The grading subspaces for the standard Z-gradation of a loop Lie algebra LA,M (g) are LA,M (g)k = {ξ ∈ LA,M (g) | ξ = λk x, A(x) = kM x}, where by λ we denote the restriction of the standard coordinate on C to S 1 . It is very useful to realize that every automorphism A of the Lie algebra g satisfying the relation AM = idg induces a ZM -gradation of g with the grading subspacesf g[k]M = {x ∈ g | A(x) = kM x},

k = 0, . . . , M − 1.

Vice versa, any ZM -gradation of g deﬁnes in an evident way an automorphism A of g satisfying the relation AM = idg . A ZM -gradation of g is called an inner or outer f We

denote by [k]M the element of the ring ZM corresponding to the integer k.

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type gradation, if the associated automorphism A of g is of inner or outer type respectively. In terms of the corresponding ZM -gradation the grading subspaces for the standard Z-gradation of a loop Lie algebra LA,M (g) are LA,M (g)k = {ξ ∈ LA,M (g) | ξ = λk x, x ∈ g[k]M }. It is evident that for the standard Z-gradation the subalgebra LA,M (g)0 is isomorphic to the subalgebra g[0]M of g, and the Lie group La,M (G)0 is isomorphic to the connected Lie subgroup G0 of G corresponding to the Lie algebra g[0]M . Hence, the mapping γ is actually a mapping of R2 to G0 . The mappings f− and f+ are given by the relation ¯ p¯) = p¯−L c− (m), ¯ f− (m,

f+ (m, ¯ p¯) = p¯L c+ (m), ¯

m ¯ ∈ R2 ,

p¯ ∈ S 1 ,

where c− and c+ are mappings of R2 to g−[L]M and g+[L]M , respectively. For the connection components ω− and ω+ , we have ω− = γ −1 ∂− γ + λ−L c− ,

ω+ = λL γ −1 c+ γ,

(2.14)

and the Toda equation (2.13) can be written as ∂+ (γ −1 ∂− γ) = [c− , γ −1 c+ γ].

(2.15)

The conditions (2.12) imply that ∂+ c− = 0,

∂− c+ = 0.

(2.16)

It is natural to call an equation of the form (2.15) also a Toda equation. Let b be an automorphism of the Lie group G and B be the corresponding automorphism of the Lie algebra g. The mapping σ deﬁned by the equality σ(χ) = b ◦ χ,

χ ∈ La,M (G),

is an isomorphism from La,M (G) to La ,M (G), where a is an automorphism of G deﬁned as a = bab−1 . It is clear that the mapping Σ deﬁned by the equality Σ(ξ) = B ◦ ξ,

ξ ∈ LA,M (g),

is an isomorphism from LA,M (g) to LA ,M (g), where A is an automorphism of g corresponding to the automorphism a of G. With such isomorphism in the case under consideration the transformations (2.7) and (2.8) take the form γ = b ◦ γ, c− = B ◦ c− ,

c+ = B ◦ c+ .

If a mapping γ satisﬁes the Toda equation (2.15), then the mapping γ satisﬁes the Toda equation (2.15) where the mappings c− , c+ are replaced by the mappings

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c− and c+ . Actually, this means that Toda equations of the form (2.15) deﬁned by means of conjugated ZM -gradations are equivalent. The transformations (2.9) and (2.10) take now the forms −1 γη− , γ = η+

c−

=

−1 η− c− η− ,

c+

=

(2.17) −1 η+ c+ η+ ,

(2.18)

where η− and η+ are some mappings of R2 × S 1 to G0 that satisfy the conditions ∂+ η− = 0,

∂− η+ = 0.

(2.19)

Again, if a mapping γ satisﬁes the Toda equation (2.15), then the mapping γ satisﬁes the Toda equation (2.15) where the mappings c− , c+ are replaced by the mappings c− and c+ . If the mappings η− and η+ are such that −1 c− η− = c− , η−

−1 η+ c+ η+ = c+ ,

(2.20)

then the transformation (2.17) is a symmetry transformation for the Toda equation under consideration. Thus, if G is a ﬁnite dimensional complex simple Lie group, then the Toda equation associated with a loop group La,M (G) and deﬁned with the help of an integrable Z-gradation of LA,M (g) with ﬁnite dimensional grading subspaces is equivalent to an equation of the form (2.15). To describe all nonequivalent Toda equations of this type one has to classify ﬁnite order automorphisms of the Lie algebra g or, equivalently, its ZM -gradations up to conjugations.g This problem was solved quite a long time ago, see, for example, [28, 29]. However, it appeared that the classiﬁcation described in [28, 29] is not convenient for classiﬁcation of Toda equations. Restricting to the case of loop Lie algebras of complex classical Lie algebras one can use another classiﬁcation based on a convenient block matrix representation of the grading subspaces [30, 21]. Let us describe the main points of the resulting classiﬁcation of Toda equations. Each element x of the complex classical Lie algebra g under consideration is considered as a p × p block matrix (xαβ ), where xαβ is an nα × nβ matrix. Certainly, the sum of the positive integers nα is the size n of the matrices representing the elements of g. For the case of Toda systems associated with the loop groups La,M (GLn (C)), where a is an inner automorphism of GLn (C), the integers nα are arbitrary. For the other cases, they should satisfy some restrictions dictated by the structure of the Lie algebra g. The mapping γ has the block diagonal form   Γ1   Γ2 . γ=   Γp g Strictly speaking, we have to consider only conjugations by automorphisms of g which can be lifted to automorphisms of G.

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For each α = 1, . . . , p the mapping Γα is a mapping of R2 to the Lie group GLnα (C). For the case of Toda systems associated with the loop groups La,M (GLn (C)), where a is an inner automorphism of GLn (C), the mappings Γα are arbitrary. For the other cases, they satisfy some additional restrictions. The mapping c+ has the following block matrix structure:   0 C+1     0    , c+ =     0 C+(p−1)  C+0 0 where for each α = 1, . . . , p − 1 the mapping C+α is a mapping of R2 to the space of nα × nα+1 complex matrices, and C+0 is a mapping of R2 to the space of np × n1 complex matrices. The mapping c− has a similar block matrix structure:   0 C−0     C−1 0   ,  c− =       0 C−(p−1) 0 where for each α = 1, . . . , p − 1 the mapping C−α is a mapping of M to the space of nα+1 × nα complex matrices, and C−0 is a mapping of M to the space of n1 × np complex matrices. The conditions (2.16) imply ∂+ C−α = 0,

∂− C+α = 0,

α = 0, 1, . . . , p − 1.

For the case of Toda systems associated with the loop groups La,M (GLn (C)), where a is an inner automorphism of GLn (C), the mappings C±α are arbitrary. For the other cases they should satisfy some additional restrictions. It is not diﬃcult to show that the Toda equation (2.15) is equivalent to the following system of equations for the mappings Γα : −1 −1 ∂+ (Γ−1 1 ∂− Γ1 ) = −Γ1 C+1 Γ2 C−1 + C−0 Γp C+0 Γ1 , −1 −1 ∂+ (Γ−1 2 ∂− Γ2 ) = −Γ2 C+2 Γ3 C−2 + C−1 Γ1 C+1 Γ2 ,

.. .

(2.21)

−1 −1 ∂+ (Γ−1 p−1 ∂− Γp−1 ) = −Γp−1 C+(p−1) Γp C−(p−1) + C−(p−2) Γp−2 C+(p−2) Γp−1 , −1 −1 ∂+ (Γ−1 p ∂− Γp ) = −Γp C+0 Γ1 C−0 + C−(p−1) Γp−1 C+(p−1) Γp .

It appears that in the case under consideration without any loss of generality one can assume that the positive integer L, entering the construction of Toda equations, is equal to 1. Note also that if any of the mappings C+α or C−α is a zero

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mapping, then Eqs. (2.21) are actually equivalent to a Toda equation associated with a ﬁnite dimensional group or to a set of two such equations. Indeed, this is a direct consequence of the classiﬁcation of the loop Toda equations [30, 31]. 2.3. Abelian Toda equations associated with loop groups of complex general linear Lie groups There are three types of abelian Toda equations associated with La,M (GLn (C)). 2.3.1. First type The abelian Toda equations of the ﬁrst type arise when the automorphism A is deﬁned by the equality A(x) = hxh−1 ,

x ∈ gln (C),

where h is a diagonal matrix with the diagonal matrix elements hkk = n−k+1 , n

k = 1, . . . , n.

(2.22)

The corresponding automorphism a of GLn (C) is deﬁned by the equality a(g) = hgh−1 ,

g ∈ GLn (C),

(2.23)

where again h is a diagonal matrix determined by the relation (2.22). Here the integer M is equal to n, and A is an inner automorphism which generates a Zn gradation of gln (C). The block matrix structure related to this gradation is the matrix structure itself. In other words, all blocks are of size one by one. The mappings Γα are mappings of R2 to the Lie group GL1 (C) which is isomorphic to the Lie group C× = C\{0}. The mappings C±α are just complex functions on R2 . The Toda equations under consideration have the form (2.21) with p = n. Let us describe the action of the transformations (2.17) and (2.18) on Eqs. (2.21) in the case under consideration. The mappings η− and η+ have a diagonal form and we denote (η− )αα = H−α ,

(η+ )αα = H+α ,

α = 1, . . . , n.

The functions H−α and H+α satisfy the relations ∂+ H−α = 0,

∂− H+α = 0,

which follow from the relations (2.19). In terms of the functions C−α and C+α the transformations (2.17) and (2.18) look as −1 Γα = H+α Γα H−α , C−α

=

−1 H−(α+1) C−α H−α ,

C+α

=

−1 H+α C+α H+(α+1) .

(2.24) (2.25)

Assume that the functions C−α and C+α have no zeros. Let us show that in this = C− and case the functions H−α and H+α can be chosen in such a way that C−α

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C+α = C+ for some functions C− and C+ which have no zeros and are subject to the conditions

∂+ C− = 0,

∂− C+ = 0.

(2.26)

Indeed, let C− and C+ be some functions which satisfy the equalities n = C−

n

C−α ,

n C+ =

α=1

n

C+α .

α=1

One can verify that the transformations (2.25) with H−α =

n C− , C−β

H+α =

β=α C−α

n C+β C+

β=α C+α

give the desired result, = C− and = C+ . The methods to ﬁnd soliton solutions described below work for arbitrary functions C− and C+ . However, to simplify formulas we will only consider the case when C− = m, and C+ = m, where m is a nonzero constant. In other words, we will assume that     0 1 0 1  1 0    0      , c+ = m  . (2.27) c− = m          0 0 1 1 0 0 1 The equations under consideration take in this case the form −1 2 −1 ∂+ (Γ−1 1 ∂− Γ1 ) = −m (Γ1 Γ2 − Γp Γ1 ), −1 −1 2 ∂+ (Γ−1 2 ∂− Γ2 ) = −m (Γ2 Γ3 − Γ1 Γ2 ),

.. .

(2.28)

−1 −1 2 ∂+ (Γ−1 n−1 ∂− Γn−1 ) = −m (Γn−1 Γn − Γn−2 Γn−1 ), −1 2 −1 ∂+ (Γ−1 n ∂− Γn ) = −m (Γn Γ1 − Γn−1 Γn ).

It is worth to note that when the functions C− and C+ are real, one can come to the Toda equations with C− = m and C+ = m by an appropriate change of the coordinates z − and z + . The symmetry transformations (2.17) for the system under consideration are described by the relation (2.24) where H−α = H− and H+α = H+ for some functions H− and H+ satisfying the conditions ∂+ H− = 0,

∂− H+ = 0.

In particular, multiplication of all Γα by the same constant is a symmetry transformation. Deﬁning Γ = Γ1 Γ2 · · · Γn ,

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one can easily see that ∂+ (Γ

−1

n

∂− Γ) =

∂+ (Γ−1 α ∂− Γα ).

α=1

Equations (2.28) give ∂+ (Γ−1 ∂− Γ) = 0, therefore, Γ = Γ+ Γ−1 − , for some functions Γ− and Γ+ which satisfy the relations ∂+ Γ− = 0,

∂− Γ+ = 0.

Thus, if we perform the symmetry transformation (2.24) with H−α and H+α given by 1/n

H−α = Γ− ,

1/n

H+α = Γ+ ,

we will obtain functions Γα which satisfy the Toda equations (2.28) and obey the equality Γ = Γ1 Γ2 · · · Γn = 1. Actually this means that via appropriate symmetry transformations we can reduce solutions of the abelian Toda equations associated with the loop group of GLn (C) under consideration to solutions of the corresponding Toda equations associated with the loop group of SLn (C). Suppose now that we have a solution of Eqs. (2.28) with Γ = 1. The mappings Γα for α = 1, . . . , n − 1 are independent. Introduce a new set of n − 1 independent mappings Φα , α = 1, . . . , n − 1, deﬁned as Φα =

α

Γβ .

β=1

It is easy to show that the inverse transition to the mappings Γα is described by the equalities Γ 1 = Φ1 ,

−1 Γ2 = Φ−1 1 Φ2 , . . . , Γn−1 = Φn−2 Φn−1 ,

Γn = Φ−1 n−1 ,

and that the mappings Φα satisfy the equations −2 2 ∂+ (Φ−1 1 ∂− Φ1 ) = −m (Φ1 Φ2 − Φn−1 Φ1 ), −2 2 ∂+ (Φ−1 2 ∂− Φ2 ) = −m (Φ1 Φ2 Φ3 − Φn−1 Φ1 ),

.. . −2 2 ∂+ (Φ−1 n−2 ∂− Φn−2 ) = −m (Φn−3 Φn−2 Φn−1 − Φn−1 Φ1 ), −2 2 ∂+ (Φ−1 n−1 ∂− Φn−1 ) = −m (Φn−2 Φn−1 − Φn−1 Φ1 ).

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This system can be written in a more symmetric form. To this end, one introduces an additional mapping ∆0 , which satisﬁes the equation 2 ∂+ (∆−1 0 ∂− ∆0 ) = −m Φn−1 Φ1 ,

and denotes ∆α = ∆0 Φα ,

α = 1, . . . , n − 1.

It is easy to see that the mappings ∆α , α = 0, 1, . . . , n − 1, satisfy the equations −2 2 ∂+ (∆−1 0 ∂− ∆0 ) = −m ∆n−1 ∆0 ∆1 , −2 2 ∂+ (∆−1 1 ∂− ∆1 ) = −m ∆0 ∆1 ∆2 ,

.. .

(2.29)

−2 2 ∂+ (∆−1 n−2 ∂− ∆n−2 ) = −m ∆n−3 ∆n−2 ∆n−1 , −2 2 ∂+ (∆−1 n−1 ∂− ∆n−1 ) = −m ∆n−2 ∆n−1 ∆0 ,

which can be written as 2 ∂+ (∆−1 α ∂− ∆α ) = −m

n−1 β=0

−aαβ

∆β

,

(2.30)

where aαβ are the matrix elements of the Cartan matrix of an aﬃne Lie algebra of (1) type An−1 : 

 2 −1 0 ··· 0 0 −1   2 −1 · · · 0 0 0 −1    0 −1 2 ··· 0 0 0    .  . . . . . .  .. .. .. .. .. ..  (aαβ ) =  .. .    0 0 0 ··· 2 −1 0     0 0 · · · −1 2 −1  0 −1

0

0 ···

0 −1

(2.31)

2

Equation (2.30) is of the standard form of the Toda equations associated with the (1) An−1 aﬃne Lie group. 2.3.2. Second type The abelian Toda equations of the second and third types arise when we use outer automorphisms of gln (C). For the equations of the second type, n is odd, and for the equations of the third type, n is even.

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Consider ﬁrst the case of an odd n = 2s − 1, s ≥ 2. In this case, an abelian Toda equation arises when the automorphism A is deﬁned by the equality A(x) = −h(B −1txB)h−1 ,

(2.32)

where tx means the transpose of x, h is a diagonal matrix with the diagonal matrix elements 2s−k = 4s−2 , hkk = n−k+1 2n

and B is an n × n matrix of the form  1    B=  



−1

1   .  

1

−1 The order of the automorphism A is 2n = 4s − 2 and the integer p is 2s − 1. The mapping γ is a diagonal matrix, and the mappings Γα are mappings of R2 to C× subject to the constraints Γ1 = 1,

Γ2s−α+1 = Γ−1 α ,

α = 2, . . . , s.

The mappings C±α are complex functions satisfying the equality C±0 = C±1 ,

(2.33)

and for s > 2 the equalities C±(2s−α) = −C±α ,

α = 2, . . . , s − 1.

(2.34)

Let us choose the mappings Γα , α = 2, . . . , s, as a complete set of mappings parametrizing the mapping γ. Taking into account the equalities (2.33) and (2.34), we come to the following set of independent equations equivalent to the Toda equation under consideration −1 ∂+ (Γ−1 2 ∂− Γ2 ) = −C+2 C−2 Γ2 Γ3 + C+1 C−1 Γ2 , −1 −1 ∂+ (Γ−1 3 ∂− Γ3 ) = −C+3 C−3 Γ3 Γ4 + C+2 C−2 Γ2 Γ3 ,

.. .

(2.35)

−1 −1 ∂+ (Γ−1 s−1 ∂− Γs−1 ) = −C+(s−1) C−(s−1) Γs−1 Γs + C+(s−2) C−(s−2) Γs−2 Γs−1 , −1 −2 ∂+ (Γ−1 s ∂− Γs ) = −C+s C−s Γs + C+(s−1) C−(s−1) Γs−1 Γs .

As above, in the case when the functions C−α and C+α have no zeros, using the transformations (2.17) and (2.18), one can show that Eqs. (2.35) are equivalent to the same equations, where C−α = C− and C+α = C+ for some functions C− and C+ which have no zeros and are subject to the conditions (2.26). If these functions

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are real, then with the help of an appropriate change of the coordinates z − and z + we can come to the equations −1 2 ∂+ (Γ−1 2 ∂− Γ2 ) = −m (Γ2 Γ3 − Γ2 ), −1 −1 2 ∂+ (Γ−1 3 ∂− Γ3 ) = −m (Γ3 Γ4 − Γ2 Γ3 ), .. . −1 −1 2 ∂+ (Γ−1 s−1 ∂− Γs−1 ) = −m (Γs−1 Γs − Γs−2 Γs−1 ), −1 2 −2 ∂+ (Γ−1 s ∂− Γs ) = −m (Γs − Γs−1 Γs ),

where m is a nonzero constant, see also papers [3, 33]. For s = 2, denoting Γ2 by Γ we have the equation ∂+ (Γ−1 ∂− Γ) = −m2 (Γ−2 − Γ). Putting Γ = exp(F ), we obtain ∂+ ∂− F = −m2 [exp(−2F ) − exp(F )]. This is the Tzitz´eica equation [34] which is now usually called the Dodd–Bullough– Mikhailov equation [35, 3]. Let us show how the above equations are related to the Toda equations associ(2) ated with the A2s−2 aﬃne Lie group. Assume that s > 2. Introduce an additional mapping ∆0 which satisﬁes the equation ∂+ (∆−1 0 ∂− ∆0 ) = −

m2 Γ2 2

(2.36)

and denote ∆α = 2−α ∆20

α+1

Γβ ,

α = 1, . . . , s − 2,

β=2

∆s−1 = 2−s+2 ∆20

s

Γβ .

β=2

Now one can get convinced that the mappings ∆α , α = 0, 1, . . . , s − 1, satisfy the equations of the form (2.30) where n = s and aαβ are the matrix elements of the (2) Cartan matrix of an aﬃne Lie algebra of type A2s−2 :   2 −1 0 ··· 0 0 0   2 −1 · · · 0 0 0 −2    0 −1 2 ··· 0 0 0     . . . . . . .   . . . . . . . (aαβ ) =  . . . . . . . .    0 0 0 ··· 2 −1 0     0 0 · · · −1 2 −1  0 0 0 0 ··· 0 −2 2 In the case of s = 2, we again deﬁne the mapping ∆0 with the help of the relation (2.36) and denote ∆1 = 2−1/3 ∆20 Γ2 .

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After an appropriate rescaling of the coordinates z − and z + we come to Eq. (2.30) where n = 2 and aαβ are the matrix elements of the Cartan matrix of an aﬃne Lie (2) algebra of type A2 :

(aαβ ) =

2 −4

−1 2

.

2.3.3. Third type In the case of an even n = 2s, s ≥ 2, to come to an abelian Toda system one should use again the automorphism A deﬁned by the relation (2.32) where now   1  1  1     B=  1     −1   −1 and h is a diagonal matrix with the diagonal matrix elements 2s−1 h11 = n−1 2n−2 = 4s−2 = −1,

2s−i+1 hii = n−i+1 2n−2 = 4s−2 ,

i = 2, . . . , n.

The number p characterizing the block structure is equal to n − 1 = 2s − 1, n1 = 2, and nα = 1 for α = 2, . . . , 2s − 1. The mapping Γ1 is a mapping of R2 to the Lie group SO2 (C) which is isomorphic to C× . Actually Γ1 is a 2 × 2 complex matrix valued function satisfying the relation J2−1 t Γ1 J2 = Γ−1 1 , where

J2 =

0 1 . 1 0

It is easy to show that Γ1 has the form

(Γ1 )11 Γ1 = 0

0 (Γ1 )−1 11

,

where (Γ1 )11 is a mapping of R2 to C× . The mappings Γα , α = 2, . . . , 2s − 1, are mappings of R2 to C× satisfying the relations Γ2s−α+1 = Γ−1 α .

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The mappings C−1 , C+0 are complex 1×2 matrix valued functions, the mappings C−0 , C+1 are complex 2 × 1 matrix valued functions. Here one has C−0 = J2−1 tC−1 ,

C+0 = tC+1 J2 .

(2.37)

The mappings C±α , α = 2, . . . , p − 1 = 2s − 2, are just complex functions, satisfying for s > 2 the equalities C±(2s−α) = −C±α ,

α = 2, . . . , s − 1.

(2.38)

The mappings (Γ1 )11 and Γα , α = 2, . . . , s, form a complete set of mappings parametrizing the mapping γ. Taking into account the equalities (2.37) and (2.38), we come to the following set of independent equations equivalent to the Toda equation under consideration: −1 ∂+ ((Γ1 )−1 11 ∂− (Γ1 )11 ) = −(C+1 )11 (C−1 )11 (Γ1 )11 Γ2 + (C+1 )21 (C−1 )12 Γ2 (Γ1 )11 , −1 ∂+ (Γ−1 2 ∂− Γ2 ) = −C+2 C−2 Γ2 Γ3

+ (C+1 )11 (C−1 )11 (Γ1 )−1 11 Γ2 + (C+1 )21 (C−1 )12 Γ2 (Γ1 )11 , −1 −1 ∂+ (Γ−1 3 ∂− Γ3 ) = −C+3 C−3 Γ3 Γ4 + C+2 C−2 Γ2 Γ3 ,

.. . −1 −1 ∂+ (Γ−1 s−1 ∂− Γs−1 ) = −C+(s−1) C−(s−1) Γs−1 Γs + C+(s−2) C−(s−2) Γs−2 Γs−1 , −1 −2 ∂+ (Γ−1 s ∂− Γs ) = −C+s C−s Γs + C+(s−1) C−(s−1) Γs−1 Γs .

As well as for the ﬁrst two types, under appropriate conditions these equations can be reduced to the equations with C−α = m, C+α = m√for α = 2, . . . , s, √ (C−1 )11 = (C−1 )12 = m/ 2 and (C+1 )11 = (C+1 )21 = m/ 2, where m is a nonzero constant.h Thus, we come to the equations m2 −1 (Γ1 − Γ1 )Γ2 , 2 m2 −1 2 −1 (Γ1 + Γ1 )Γ2 , ∂+ (Γ−1 2 ∂− Γ2 ) = −m Γ2 Γ3 + 2

∂+ (Γ−1 1 ∂− Γ1 ) = −

−1 −1 2 ∂+ (Γ−1 3 ∂− Γ3 ) = −m (Γ3 Γ4 − Γ2 Γ3 ), .. . −1 −1 2 ∂+ (Γ−1 s−1 ∂− Γs−1 ) = −m (Γs−1 Γs − Γs−2 Γs−1 ), −1 2 −2 ∂+ (Γ−1 s ∂− Γs ) = −m (Γs − Γs−1 Γs ),

where slightly abusing notation we denote (Γ1 )11 by Γ1 . Introduce now an additional mapping ∆0 which satisﬁes the equation ∂+ (∆−1 0 ∂− ∆0 ) = − h This

m2 Γ1 Γ2 2

choice is convenient for applications of the rational dressing method.

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and denote ∆1 = ∆0 Γ1 ,

∆α = 2α(α−1)/(2s−1)−α+1 ∆20

α

Γβ ,

α = 2, . . . , s.

β=1

The mappings ∆α , α = 0, 1, . . . , s, satisfy the equations which, after an appropriate rescaling of the coordinates z − and z + , take the form (2.30), where now n = s + 1 and aαβ are the matrix elements of the Cartan matrix of an aﬃne Lie algebra of (2) type A2s−1 :   2 0 −1 · · · 0 0 0   2 −1 · · · 0 0 0  0   −1 −1 2 ··· 0 0 0    .  . . . . . .  .. .. .. .. .. ..  (aαβ ) =  .. .    0 0 0 ··· 2 −1 0     0 0 · · · −1 2 −1  0 0 0 0 ··· 0 −2 2 3. Soliton Solutions In this section, we compare two methods used to construct soliton solutions of the abelian Toda systems associated with the loop groups of the complex general linear groups. We restrict ourselves to the abelian Toda equations of the ﬁrst type which have the form (2.28). 3.1. Hirota’s method It is convenient to treat the system (2.28) as an inﬁnite system −1 2 −1 ∂+ (Γ−1 α ∂− Γα ) = −m (Γα Γα+1 − Γα−1 Γα ),

(3.1)

where the functions Γα are deﬁned for arbitrary integer values of the index α in the periodic way, Γα+n = Γα .

(3.2)

Following the Hirota’s approach [6–10], one introduces τ -functions connected with Γα by the relation Γα = τα /τα−1 ,

(3.3)

where we assume that the τ -functions are deﬁned for all integer values of the index α. This change of variables is the essence of the Hirota’s method. The periodicity condition (3.2) in terms of the τ -functions takes the form τα+n /τα = τα−1+n /τα−1 .

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It means that the ratio τα+n /τα does not depend on α. Noting that Γ = Γ1 Γ2 · · · Γn = τn /τ0 , we write τα+n = Γτα . As was explained in the previous section, with the appropriate symmetry transformation, one can make Γ = 1. We will assume that the corresponding symmetry transformation was performed, and, therefore, τα+n = τα .

(3.4)

Equation (3.1) in terms of the τ -functions looks as −1 −2 ∂− τα−1 ) = −m2 (τα−1 τα−2 τα+1 − τα−2 τα−1 τα ). ∂+ (τα−1 ∂− τα ) − ∂+ (τα−1

Consider the following decoupling of the above equations ∂+ (τα−1 ∂− τα ) = m2 (1 − τα−1 τα−2 τα+1 ).

(3.5)

It is evident that if the τ -functions satisfy these equations, then the functions Γα deﬁned by (3.3) satisfy the system (2.28). Moreover, it is easy to show that in this case the functions ∆α = exp(−m2 z + z − )τα satisfy the system (2.29). It is convenient to rewrite Eq. (3.5) in the form τα ∂+ ∂− τα − ∂+ τα ∂− τα = m2 (τα2 − τα−1 τα+1 ).

(3.6)

These equations are of the Hirota bilinear type. Their solutions, leading to multisoliton solutions of the system (2.28), can be found perturbatively in the following way. Consider a series expansion of the functions τα in some parameter ε which will be set to one at the ﬁnal step of the construction. So we represent the functions τα in the form τα = τα(0) + ετα(1) + ε2 τα(2) + · · · , and assume that

(0) τα

(3.7)

are constants. The periodicity condition (3.4) gives (k)

τα+n = τα(k) ,

k = 0, 1, . . . .

Let us try to solve Eq. (3.6) order by order in ε. Actually our goal is to ﬁnd solutions for which the series (3.7) truncates at some ﬁnite order in ε. In such a case we have an exact solution. Using the expansion (3.7), one obtains τα ∂+ ∂− τα =

∞ k=0

εk

k =0

τα(k−) ∂+ ∂− τα() .

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Similarly one has ∂+ τα ∂− τα =

∞

εk

k=0

k

∂+ τα(k−) ∂− τα() .

=0

Now, using the equality τα2 − τα−1 τα+1 =

∞

εk

k=0

k

()

(k−)

(τα() τα(k−) − τα−1 τα+1 ),

=0

we see that Eq. (3.6) are equivalent to the equations k

(τα(k−) ∂+ ∂− τα() − ∂+ τα(k−) ∂− τα() ) = m2

=0

k

()

(k−)

(τα() τα(k−) − τα−1 τα+1 ),

(3.8)

=0

which can be solved step by step starting from k = 0. For k = 0, one has (0)

(0)

τα−1 τα+1 − τα(0) τα(0) = 0, that can be rewritten as (0)

(0)

τα+1 /τα(0) = τα(0) /τα−1 . It is clear that the general solution to this relation is τα(0) = τ0 dα , (0)

(3.9)

where d is an arbitrary constant. Recall that the Toda equation (3.1) is invariant with respect to the multiplication of all Γα by the same constant. From the point of view of the τ -functions, this is equivalent to the multiplication of the function τα by the αth power of the constant. Hence, diﬀerent values of the constant d in the relation (3.9) correspond to the functions Γα connected by a rescaling. Moreover, dividing all τ -functions by the same constant we do not change the functions Γα . Therefore, actually without any loss of generality, one can put τα(0) = 1.

(3.10)

Using this equality, we rewrite (3.8) as ∂+ ∂− τα(k) − m2

n−1 β=0

(k)

aαβ τβ

=−

k−1

(τα(k−) ∂+ ∂− τα() − ∂+ τα(k−) ∂− τα() )

=1

+ m2

k−1

()

(k−)

(τα() τα(k−) − τα−1 τα+1 ),

(3.11)

=1

where aαβ are the matrix elements of the Cartan matrix (2.31) of an aﬃne Lie (1) algebra of type An−1 . Thus, we see that at each step we should solve a system of linear diﬀerential equations.

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In particular, for k = 1, one has to solve the system of equations ∂+ ∂− τα(1) − m2

n−1 β=0

(1)

aαβ τβ

= 0.

(3.12)

It is easy to ﬁnd solutions of these equations using the eigenvectors θρ of the Cartan matrix (aαβ ) which are given by (θρ )α = n(α+1)ρ ,

ρ = 0, 1, . . . , n − 1.

(3.13)

Here the corresponding eigenvalues are 2 κ2ρ = 2 − ρn − −ρ n = 4 sin (πρ/n). (1)

Let us assume that the functions τα are of the form τα(1)

=

r

Eαi ,

(3.14)

i=1

where i Eαi = (α+1)ρ exp[mκρi (ζi−1 z − + ζi z + ) + δi ]. n

(3.15)

Here ρi is an integer from the interval from 1 to n − 1, ζi and δi are arbitrary complex numbers. Note that the choice ρi = 0 is excluded because it gives a con(0) stant contribution to the τ -functions which can be included into τα . Then after a corresponding rescaling one can satisfy the normalization (3.10). For deﬁniteness we assume that −ρ/2 ) = 2 sin(πρ/n). κρ = −i(ρ/2 n − n

(3.16)

Certainly, the ansatz (3.14) does not give a general solution to the equations (3.12) but it ensures truncation of the expansion (3.7). For k = 2, Eqs. (3.11) have the form ∂+ ∂− τα(2) − m2

n−1 β=0

(2)

aαβ τβ

= −τα(1) ∂+ ∂− τα(1) + ∂+ τα(1) ∂− τα(1) (1)

(1)

+ m2 (τα(1) τα(1) − τα−1 τα+1 ). Using the equalities ∂− Eαi = mκρi ζi−1 Eαi ,

∂+ Eαi = mκρi ζi Eαi ,

one obtains (1)

(1)

−τα(1) ∂+ ∂− τα(1) + ∂+ τα(1) ∂− τα(1) + m2 (τα(1) τα(1) − τα−1 τα+1 ) =

r m2 [κρ κρ (ζi ζ −1 + ζi−1 ζi2 ) − κ2ρi1 − κ2ρi2 + κ2ρi1 −ρi2 ]Eαi1Eαi2 . 1 2 i ,i =1 i1 i2 1 i2 1

2

(3.17)

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Note that in the case of r = 1, we have (1)

(1)

−τα(1) ∂+ ∂− τα(1) + ∂+ τα(1) ∂− τα(1) + m2 (τα(1) τα(1) − τα−1 τα+1 ) = 0 (2)

and we can put τα = 0. This gives the one-soliton solutions ρ(α+1)

Γα =

exp[mκρ (ζ −1 z − + ζz + ) + δ] 1 + n . ρα 1 + n exp[mκρ (ζ −1 z − + ζz + ) + δ]

(3.18) (2)

In the case when r > 1, the equality (3.17) suggests to look for τα of the form r

τα(2) =

1 ηi i Eαi1Eαi2 . 2 i ,i =1 1 2 1

2

Here one can easily ﬁnd that ∂+ ∂− τα(2) − m2

n−1 β=0

(2)

aαβ τβ

=

r m2 [κρ κρ (ζi ζ −1 + ζi−1 ζi2 ) + κ2ρi1 + κ2ρi2 1 2 i ,i =1 i1 i2 1 i2 1

2

− κ2ρi1 +ρi2 ]ηi1 i2 Eαi1Eαi2 and, therefore, ηi1 i2 =

+ ζi−1 ζi2 ) − κ2ρi1 − κ2ρi2 + κ2ρi1 −ρi2 κρi1κρi2 (ζi1 ζi−1 2 1 κρi1κρi2 (ζi1 ζi−1 + ζi−1 ζi2 ) + κ2ρi1 + κ2ρi2 − κ2ρi1 +ρi2 2 1

,

that can be written as ηi1 i2 =

(ζi1 ζi−1 + ζi−1 ζi2 ) − 2 cos[π(ρi1 − ρi2 )/n] 2 1 (ζi1 ζi−1 + ζi−1 ζi2 ) − 2 cos[π(ρi1 + ρi2 )/n] 2 1

.

(3.19)

The quantities ηi1 i2 are symmetric with respect to the indices i1 , i2 and they turn to zero when i1 = i2 . Hence one can write τα(2) = ηi1 i2 Eαi1Eαi2 . 1≤i1
It can be shown that when r = 2 one can choose τα = 0. In general, it can be shown that for ≤ r, one can choose    ηij ik  Eαi1 Eαi2 · · · Eαi τα() = 1≤i1
1≤j
()

and τi = 0 for > r. In other words, the equations (3.6) have the following solutions     r r   Eαi + ηij ik  Eαi1 Eαi2 · · · Eαi  . τα = 1 + i=1

=2

1≤i1
1≤j
(3.20)

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3.2. Rational dressing Since for any m ¯ ∈ R2 , the matrices c− (m) ¯ and c+ (m) ¯ commute,i it is obvious that γ = In ,

(3.21)

where In is the n × n unit matrix, is a solution to the Toda equation (2.15). Denote a mapping of R2 × S 1 to GLn (C) which generates the corresponding connection by ϕ. Using the equalities (2.11) and (2.14) and remembering that in our case L = 1, we write ϕ−1 ∂− ϕ = λ−1 c− ,

ϕ−1 ∂+ ϕ = λc+ ,

where the matrices c− and c+ are deﬁned by the relation (2.27). To construct more interesting solutions to the Toda equations we will look for a mapping ψ, such that the mapping ϕ = ϕψ

(3.22)

would generate a connection satisfying the grading condition and the gauge-ﬁxing constraint ω+0 = 0. p) = ψ(m, ¯ p¯), For any m ¯ ∈ R2 , the mapping ψ˜m deﬁned by the equality ψ˜m (¯ 1 p¯ ∈ S , is a smooth mapping of S 1 to GLn (C). Recall that we treat S 1 as a subset of the complex plane which, in turn, will be treated as a subset of the Riemann sphere. Assume that it is possible to extend analytically each mapping ψ˜m to all of the Riemann sphere. As the result we get a mapping of the direct product of R2 and the Riemann sphere to GLn (C) which we also denote by ψ. Suppose that for any m ¯ ∈ R2 the analytic extension of ψ˜m results in a rational mapping regular at the points 0 and ∞, hence the name rational dressing. Below, for each point p¯ of the Riemann sphere we denote by ψp the mapping of R2 to GLn (C) deﬁned by the ¯ = ψ(m, ¯ p¯). equality ψp (m) Since we deal with the Toda equations described in Sec. 2.3.1, that is, the mapping ψ is generated by a mapping of R2 to the loop group La,n (GLn (C)) with the automorphism a deﬁned by the relations (2.23) and (2.22), for any m ¯ ∈ R2 and 1 p¯ ∈ S we should have ¯ p¯)h−1 , ψ(m, ¯ n p¯) = hψ(m,

(3.23)

where h is a diagonal matrix described by the relation (2.22). The equality (3.23) means that two rational mappings coincide on S 1 , therefore, they must coincide on the entire Riemann sphere. A mapping, satisfying the equality (3.23), can be constructed by the following procedure. Let χ be an arbitrary mapping of the direct product of R2 and the ˆ be a linear operator acting on χ as Riemann sphere to GLn (C). Let a ¯)h−1 . a ˆχ(m, ¯ p¯) = hχ(m, ¯ −1 n p i Actually

in the case under consideration c− and c+ are constant mappings.

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It is easy to get convinced that the mapping ψ=

n

a ˆk χ

k=1

satisﬁes the relation a ˆψ = ψ which is equivalent to the equality (3.23). Note that a ˆn χ = χ. To construct a rational mapping satisfying (3.23) we start with a rational mapping regular at the points 0 and ∞ and having poles at r diﬀerent nonzero points µi , i = 1, . . . , r. Concretely speaking, we consider a mapping χ of the form

r λ χ = In + n Pi χ0 , λ − µi i=1 where Pi are some smooth mappings of R2 to the algebra Matn (C) of n× n complex matrices and χ0 is a mapping of R2 to the Lie subgroup of GLn (C) formed by the elements g ∈ GLn (C) satisfying the equality hgh−1 = g.

(3.24)

Actually this subgroup coincides with the subgroup G0 . The averaging procedure leads to the mapping

r n λ hk Pi h−k ψ0 , (3.25) ψ = In + kµ λ − i n i=1 k=1

where ψ0 = nχ0 . It is convenient to assume that µni = µnj for all i = j. Denote by ψ −1 the mapping of R2 × S 1 to GLn (C) deﬁned by the relation ψ −1 (m, ¯ p¯) = (ψ(m, ¯ p¯))−1 . −1 Suppose that for any ﬁxed m ¯ ∈ R2 the mapping ψ˜m of S 1 to GLn (C) can be extended analytically to a mapping of the Riemann sphere to GLn (C) and as the result we obtain a rational mapping of the same structure as the mapping ψ,

r n λ −1 −1 k −k ψ = ψ0 h Qi h In + , (3.26) λ − kn νi i=1 k=1

with the pole positions satisfying the conditions νi = 0, νin = νjn for all i = j, and additionally νin = µnj for any i and j. We will denote the mapping of the direct product of R2 and the Riemann sphere to GLn (C) again by ψ −1 . By deﬁnition, the equality ψ −1 ψ = In

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is valid at all points of the direct product of R2 and S 1 . Since ψ −1 ψ is a rational mapping, the above equality is valid at all points of the direct product of R2 and the Riemann sphere. Hence, the residues of ψ −1 ψ at the points νi and µi should be equal to zero. Explicitly we have   r n ν i (3.27) hk Pj h−k  = 0, Qi In + kµ ν − i j n j=1 k=1



In +

r n j=1 k=1

 µi k −k  Pi = 0. h Qj h µi − kn νj

(3.28)

In this case due to the relation (3.23) the residues of ψ −1 ψ at the points kn µi and kn νi vanish for k = 1, . . . , n. We will discuss later how to satisfy these relations, and now let us consider what connection is generated by the mapping ϕ deﬁned by (3.22) with the mapping ψ possessing the prescribed properties. Using the representation (3.22), we obtain for the components of the connection generated by ϕ the expressions ω− = ψ −1 ∂− ψ + λ−1 ψ −1 c− ψ, ω+ = ψ

−1

∂+ ψ + λψ

−1

(3.29)

c+ ψ.

(3.30)

We see that the component ω− is a rational mapping which has simple poles at the points µi , νi and zero.j Similarly, the component ω+ is a rational mapping which has simple poles at the points µi , νi and inﬁnity. We are looking for a connection which satisﬁes the grading and gauge-ﬁxing conditions. The grading condition in our case is the requirement that for each point of R2 the component ω− is rational and has the only simple pole at zero, while the component ω+ is rational and has the only simple pole at inﬁnity. Hence, we demand that the residues of ω− and ω+ at the points µi and νi should vanish. In this case, as above, due to the relation (3.23) the residues of ω− and ω+ at the points kn µi and kn νi vanish for k = 1, . . . , n. The residues of ω− and ω+ at the points νi are equal to zero if and only if   r n ν i (3.31) hk Pj h−k  = 0, (∂− Qi − νi−1 Qi c− ) In + kµ ν − i j n j=1 

(∂+ Qi − νi Qi c+ ) In +

k=1

r n j=1 k=1



νi hk Pj h−k  = 0, νi − kn µj

(3.32)

j Here and below discussing the holomorphic properties of mappings and functions we assume that the point of the space R2 is arbitrary but ﬁxed.

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respectively. Similarly, the requirement of vanishing of the residues at the points µi gives the relations   r n µ i In + hk Qj h−k  (∂− Pi + µ−1 (3.33) i c− Pi ) = 0, kν µ − i j n j=1 k=1



In +

n r j=1 k=1

 µi hk Qj h−k  (∂+ Pi + µi c+ Pi ) = 0. µi − kn νj

(3.34)

To obtain the relations (3.31)–(3.34) we made use of the equalities (3.27), (3.28). Suppose that we have succeeded in satisfying the relations (3.27), (3.28) and (3.31)–(3.34). In such a case, from the equalities (3.29) and (3.30), it follows that the connection under consideration satisﬁes the grading condition. It is easy to see from (3.30) that ω+ (m, ¯ 0) = ψ0−1 (m)∂ ¯ + ψ0 (m). ¯ Taking into account that ω+0 (m) ¯ = ω+ (m, ¯ 0), we conclude that the gauge-ﬁxing constraint ω+0 = 0 is equivalent to the relation ∂+ ψ0 = 0.

(3.35)

Assuming that this relation is satisﬁed, we come to a connection satisfying both the grading condition and the gauge-ﬁxing condition. Recall that if a ﬂat connection ω satisﬁes the grading and gauge-ﬁxing conditions, then there exist a mapping γ of R2 to G and mappings c− and c+ of R2 to g−1 and g+1 , respectively, such that the representation (2.14) for the components ω− and ω+ is valid. In general, the mappings c− and c+ parametrizing the connection components may be diﬀerent from the mappings c− and c+ which determine the mapping ϕ. Let us denote the mappings corresponding to the connection under consideration by γ , c− and c+ . Thus, we have ψ −1 ∂− ψ + λ−1 ψ −1 c− ψ = γ −1 ∂− γ + λ−1 c− , ψ −1 ∂+ ψ + λψ −1 c+ ψ = λγ −1 c+ γ .

(3.36) (3.37)

Note that ψ∞ is a mapping of R2 to the Lie subgroup of GLn (C) deﬁned by the relation (3.24). Recall that this subgroup coincides with G0 , and denote ψ∞ by γ. From the relation (3.36) we obtain the equality γ −1 ∂− γ = γ −1 ∂− γ. The same relation (3.36) gives ψ0−1 c− ψ0 = c− . Impose the condition ψ0 = In , which is consistent with (3.35). Here we have c− = c− .

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Finally, from (3.37) we obtain γ −1 c+ γ = γ −1 c+ γ. We see that if we impose the condition ψ0 = In , then the components of the connection under consideration have the form given by (2.14) where γ = ψ∞ . Thus, to ﬁnd solutions to Toda equations under consideration, we can use the following procedure. Fix 2r complex numbers µi and νi . Find matrix-valued functions Pi and Qi satisfying the relations (3.27), (3.28) and (3.31)–(3.34). With the help of (3.25), (3.26), assuming that ψ0 = In , construct the mappings ψ and ψ −1 . Then, the mapping (3.38)

γ = ψ∞

satisﬁes the Toda equation (2.15). Let us return to the relations (3.27), (3.28). It can be shown that, if we suppose that the matrices Pi and Qi are of maximum rank, then we get the trivial solution of the Toda equation given by (3.21). Hence, we will assume that Pi and Qi are not of maximum rank. The simplest case here is given by matrices of rank one which can be represented as Pi = ui twi ,

Qi = xi tyi ,

where u, w, x and y are n-dimensional column vectors. This representation allows one to write the relations (3.27) and (3.28) as t

yi +

r n j=1 k=1

ui +

νi (tyi hk uj )twj h−k = 0, νi − kn µj

r n j=1 k=1

µi hk xj (tyj h−k ui ) = 0. µi − kn νj

(3.39)

(3.40)

Using the identity n−1 k=0

z n−|j|n z−jk n =n n , k z − n z −1

(3.41)

where |j|n is the residue of division of j by n, we can rewrite (3.39) in terms of the components, yik + n

r

(Rk )ij wjk = 0.

j=1

Here the r × r matrices Rk are deﬁned as (Rk )ij =

n 1 n−|−k|n |−k|n νi µj yi uj . νin − µnj =1

(3.42)

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The same identity (3.41) allows one to write component form of (3.40) as uik + n

r

xjk (Sk )ji = 0,

(3.43)

j=1

where (Sk )ji = −

n 1 |k−|n n−|k−|n νj µi yj ui . νjn − µni =1

With the help of the equality n − 1 − |i − 1|n = | − i|n it is straightforward to demonstrate that (Sk )ji = −

µi (Rk+1 )ji . νj

Consequently, we can write Eq. (3.43) as uik − nµi

r

xjk

j=1

1 (Rk+1 )ji = 0. νj

(3.44)

We use Eqs. (3.42) and (3.44) to express the vectors wi and xi via the vectors ui and yi , r

wik = −

r

1 −1 (R )ij yjk , n j=1 k

xik =

1 1 −1 ujk (Rk+1 )ji νi . n j=1 µj

As the result, we come to the following solution of the relations (3.27) and (3.28): r 1 (R−1 )ij yj , (Pi )k = − uik n j=1

r

(Qi )k =

1 1 −1 ujk (Rk+1 )ji νi yi . n j=1 µj

Further, it follows from (3.39) and (3.40) that, to fulﬁll also (3.31)–(3.34), it is suﬃcient to satisfy the equations ∂− yi = νi−1 tc− yi , ∂− ui = −µ−1 i c− u i ,

∂+ yi = νi tc+ yi ,

(3.45)

∂+ ui = −µi c+ ui .

(3.46)

The n-dimensional column vectors θρ , deﬁned by the relation (3.13), are eigenvectors of the matrices tc− , tc+ , c− and c+ , t

c− θρ = mρn θρ ,

t

c+ θρ = m−ρ n θρ ,

c− θρ = m−ρ n θρ ,

c+ θρ = mρn θρ ,

and form a basis in the space Cn . Hence, the general solution of Eqs. (3.45) and (3.46) can be written in the form uik =

n ρ=1

−Zρ (µi ) ciρ kρ , n e

yik =

n ρ=1

Z−ρ (νi ) diρ kρ , n e

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where ciρ , diρ are arbitrary constants and −1 − Zρ (µ) = m(−ρ z + ρn µz + ). n µ

Thus, we see that it is possible to satisfy (3.27), (3.28) and (3.31)–(3.34). This gives us solutions of the Toda equations (2.28). Let us show that they can be written in a simple determinant form. Using (3.38) and (3.25), one gets γ = ψ∞ = In +

r n

hk Pi h−k .

i=1 k=1

For the matrix elements of γ, this gives the expression  

r r γk = δk 1 + n (Pi )kk = δk 1 − uik (Rk−1 )ij yjk  . i=1

i,j=1

Hence, we have Γα = 1 −

r

−1 uiα (Rα )ij yjα .

i,j=1

To this expression can also be given the form −1 Γα = 1 − tuα Rα yα ,

where uα and yα are r-dimensional column vectors with the components uiα and yiα , respectively. We assume for convenience that the functions uiα and yiα are deﬁned for arbitrary integral values of α and ui,α+n = uiα ,

yi,α+n = yiα .

The periodicity of Rα in the index α follows from the deﬁnition. It appears that it α deﬁned as is more appropriate to use quasi-periodic quantities u α , yα and R u α = M α uα ,

yα = N −α yα ,

α = N −α Rα M α , R where N and M are diagonal r × r matrices given by Nij = νi δij ,

Mij = µi δij .

For these quantities one has quasi-periodicity conditions u α+n = M n u α ,

yα+n = N −n yα ,

α+n = N −n R α M n . R

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α through the functions The expression of the matrix elements of the matrices R iα has a remarkably simple form [3] yiα and u   α−1 n 1 α )ij = µnj (R yiβ u jβ + νin yiβ u jβ  . (3.47) νin − µnj β=1

β=α

In terms of the quasi-periodic quantities, for the functions Γα we have −1 α Γα = 1 − tu α R yα ,

and it can be shown that Γα =

α − yα tu α ) det(R . det Rα

α , one comes to the equality Using the explicit form of R α+1 = R α − yα tu R α . Therefore, one can write [3] Γα =

α+1 det R . α det R

(3.48)

3.3. Solitons through the rational dressing α are ordinary To obtain a one-soliton solution one puts r = 1. In this case R functions for which one has the expression α = R

n 1 cρ dσ e−Zρ (µ)+Z−σ (ν) ν n − µn ρ,σ=1



× µn

α−1

µβ ν −β (ρ+σ)β + νn n

β=1

n

 . µβ ν −β (ρ+σ)β n

β=α

It is not diﬃcult to verify that µn

α−1

µβ ν −β (ρ+σ)β + νn n

β=1

n

µβ ν −β (ρ+σ)β = (ν n − µn )µα ν −α n

β=α

(ρ+σ)α

n . 1 − µν −1 ρ+σ n

α : Thus one obtains the following expression for R α = µα ν −α R

n ρ,σ=1

cρ dσ e−Zρ (µ)+Z−σ (ν)

(ρ+σ)α

n . 1 − µν −1 ρ+σ n

To obtain a solution which depends on only one combination of z − and z + we suppose that cρ is diﬀerent from zero for only one value of ρ which we denote

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by I, and that dσ is diﬀerent from zero for only two values of σ which we denote α , that is by J and K. In this case we arrive at a simpliﬁed version of R (I+J)α (I+K)α n n α −α −ZI (µ) Z−J (ν) Z−K (ν) Rα = µ ν cI e + dK e dJ e , 1 − µν −1 I+J 1 − µν −1 I+K n n and the corresponding solution can be written as Γα = µν −1 I+J n

(K−J)(α+1) Z−K (ν)−Z−J (ν)

1 + dn 1+

e

(K−J)α Z−K (ν)−Z−J (ν) dn e

,

where d=

dK (1 − µν −1 I+J ) n . dJ (1 − µν −1 I+K ) n

Making use of the freedom in multiplying a solution by a constant, we can write the obtained solution as (3.18), where ρ = K − J, κρ is deﬁned by (3.16), ζ = −(K+J)/2 ν, and δ is a constant introduced by the relation exp δ = d. Thus we −in arrive at the one-soliton solution obtained before by the Hirota’s method. In the case of r > 1 (multi-soliton construction) we suppose that for any i the coeﬃcients ciρ are diﬀerent from zero for only one value of ρ which we denote by Ii , and that the coeﬃcients diσ are diﬀerent from zero for only two values of σ which we denote by Ji and Ki . This leads to the following expression for the matrix elements α : of the matrices R 1 −α Ji α Z−Ji (νi ) (Rα )ij = νi n dJi e I +J 1 − µj νi−1 nj i (K −J )α n i i dKi Z−K (νi )−Z−J (νi ) −ZIj (µj ) Ij α i i + e . µα j n cIj e I +K dJi 1 − µj νi−1 nj i Immediately we see from (3.48) that the solution in question has the form r α+1 det R Γα = µi νi−1 Ini +Ji , (3.49) det R α i=1 are deﬁned by where the matrices R α α )ij = (R

(K −J )α

1 I +Ji

1 − µj νi−1 nj

+

n i i dKi Z−K (νi )−Z−J (νi ) i i e . I +K dJi 1 − µj νi−1 nj i

α Using the matrices D deﬁned in Appendix A, we rewrite the expression for R in the form

)ij = Dij (ν−J , µI ) + (R α n n

dKi (Ki −Ji )α Z−K (νi )−Z−J (νi ) I i i e Dij (ν−K n , µn ). dJi n

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one can use in the relation (3.49) the matrices R It is clear that instead of R i i deﬁned as r dKi (Ki −Ji )α Z−K (νi )−Z−J (νi ) −1 I −J I α i i )ij = δij + n e Dik (ν−K (R n , µn )Dkj (νn , µn ). dJi k=1

Using the equality (A.3), one comes to the expression νi α (R )ij =

r

i (νi −J − ν −J n n )

=1 =i r

Jni

−

µ In )

j (νj −J − µ In ) n

=1

(Tα )ij

i (νi −J n =1

r

Jnj

,

r

j νj (νj −J n =1 =j

−

(3.50)

ν −J n )

where i −Ji )α Z−Ki (νi )−Z−Ji (νi ) (Tα )ij = δij + di (K e n

i i νi −K − νi −J n n

−Jj

i νi −K − νj n n

and, with a slight abuse of notation, dKi Jni di = i dJi K n

r

i (νi −K − ν −J n n )

=1 =i r

r

i (νi −J − µ In ) n

=1

i (νi −J n =1 =i

−

ν −J n )

.

r

i (νi −K n =1

−

µ In )

Utilizing the expression (3.50) and having in mind the freedom in multiplying a solution by a constant, we write the solution under consideration as follows: Γα =

det Tα+1 . det Tα

−(K +J )/2

Deﬁning ρi = Ki − Ji , ζi = −in i i the relations exp δi = di , one can write

νi and introducing constants δi satisfying

(Tα )ij = δij + ρni α exp[mκρi (ζi−1 z − + ζi z + ) + δi ] It is proved in Appendix B that  r r  det Tα+1 = 1 + Eαi + i=1

=2

1≤i1
 

−ρi /2

n

−ρi /2

n

ρ /2

ζi − ni ζi ρ /2

ζi − nj ζj 

.

(3.51)



ηij ikEαi1 Eαi2 · · · Eαi,

1≤j
(3.52) where the functions Eαi and ηij ik are deﬁned by the relations (3.15) and (3.19), respectively. Thus, we come to the multi-soliton solutions which coincide with those

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obtained by the Hirota’s method. The Hirota’s τ -functions (3.20) are given by the equality τα = det Tα+1 . It is clear that the quantities ηij ik here make the same sense as do the normal ordering coeﬃcients eﬀectively describing the interaction between solitons in the vertex operators approach of Olive, Turok and Underwood [14, 15]. We refer the reader to the papers [10,36,37] for some more speciﬁc properties of such coeﬃcients. 4. Conclusion In this paper, we have considered abelian Toda systems associated with the loop groups of the complex general linear groups. We have reviewed two diﬀerent approaches to construct soliton solutions to these equations in the untwisted case, namely, the Hirota’s and rational dressing methods. Subsequently, basic ingredients representing soliton solutions within the frameworks of these methods have been explicitly related. As we have seen in Sec. 3.2, the rational dressing method allows one to construct solutions to the loop Toda equations, presenting them as the ratio of the determinants of speciﬁc matrices (3.47), (3.48), and they actually represent a class of solutions being wider than that formed by the soliton solutions of the Hirota’s method in Sec. 3.1: By setting the initial-data coeﬃcients arising in the rational dressing method to some speciﬁc values we have shown in Sec. 3.3 that the Hirota’s soliton solutions are contained among the solutions constructed by the rational dressing approach. Note also that the reduction to the systems based on the loop groups of the complex special linear groups can easily be performed. Our consideration can be generalized to Toda systems based on other loop groups, such as twisted loop groups of the complex general linear groups, twisted and untwisted loop groups of the complex orthogonal and symplectic groups. However, one should take into account that the change of ﬁeld variables in the Hirota’s method is more tricky there, and besides, when applying the rational dressing to obtain soliton solutions, one faces that the pole positions of the dressing meromorphic mappings and their inverse ones are to be related, just due to the group conditions. These circumstances make part of the formulae more intricate than in the general linear case considered in the present paper. We will address to this problem and present our results in some future publications. Appendix A. Some Properties of the Matrices D In this appendix, we investigate r × r matrices D(f, g) with matrix elements given by the equality Dij (f, g) =

1 fi = . −1 f − gj 1 − fi gj i

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Let us show that for the matrix elements of the inverse matrix D−1 (f, g) one has the representation r

(f − gi )

=1 =j r

−1 (f, g) = Dij

fj

r

(fj − g )

=1 r

(g − gi )

=1 =i

.

(A.1)

= δij .

(A.2)

(fj − f )

=1 =j

To prove the above equality one has to demonstrate that fi r k=1

r

(f − gk )

=1 =j

r

(fj − g )

=1 r

fj (fi − gk )

(g − gk )

=1 =k

r

(fj − f )

=1 =j

Consider the set of meromorphic functions of z deﬁned as r

(f − z)

=1 =j

Fij (f, g, z) =

(fi − z)

r

. (g − z)

=1

The residue of Fij (f, g, z) at inﬁnity is equal to zero, therefore, the sum of the residues at the point fi and at the points g , = 1, . . . , r, is also zero. Hence we have the following equality r r k=1

r

(f − gk )

=1 =j r

(fi − gk )

= − (g − gk )

(f − fi )

=1 =j r

,

(g − fi )

=1 =k

=1

and, therefore, fi r k=1

r

(f − gk )

=1 =j

fj (fi − gk )

r

(fj − g )

fi

=1 r =1 =k

(g − gk )

r

= (fj − f )

=1 =j

fj

r

(fi − f )

=1 =j r

r

(fj − g )

=1

(fi − g )

=1

r

. (fj − f )

=1 =j

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Now, taking into account the identity r r (fi − f ) (fj − f ) = δij , =1 =j

=1 =j

we see that the relation (A.2) is true. Thus the equivalent relation (A.1) is also true. In a similar way, one can prove the validity of the equality fi r k=1

−1 Dik (f, g)Dkj (f, g) =

fj

r

(fi − f )

=1 =j r

r

(fj − g )

=1

(fi − g )

=1

r

.

(A.3)

(fj − f )

=1 =j

Appendix B. Proof of Relation (3.52) Proceeding from the relation (3.51), one obtains (Tα+1 )ij = δij + Eαi

f˜i − fi , f˜i − fj

where i /2 fi = −ρ ζi , n

fi = ρni /2 ζi .

(B.1)

and the functions Eαi are deﬁned by the relation (3.15). Then it is not diﬃcult to get convinced that   r r  Eαi + ηi1 i2 ···i Eαi1 Eαi2 · · · Eαi  , (B.2) det Tα+1 = 1 + i=1

=2

1≤i1
where ηi1 ···i =

π∈S

sgn(π)

fij − fij . fi − fi

j=1

j

π(j)

As is customary, we denote by S the symmetric group on the set {1, 2, . . . , } and by sgn(π) the signature of the permutation π. For = 2, one has ηi1 i2 = 1 −

(fi1 − fi2 )(fi2 − fi1 ) (fi1 − fi1 )(fi2 − fi2 ) = . (fi1 − fi2 )(fi2 − fi1 ) (fi1 − fi2 )(fi2 − fi1 )

Using the deﬁnition (B.1) of fi and fi , we see that the quantities ηi1 i2 coincide with the coeﬃcients ηi1 i2 deﬁned by the relation (3.19).

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Let us prove by induction that

ηi1 i2 ···i =

ηij ik .

(B.3)

1≤j
Certainly, for = 2 the equality (B.3) is valid. Suppose that it is valid up to some ﬁxed value of and show that it is valid for its value incremented by one. The group S can be identiﬁed with a subgroup of S+1 formed by the permutations π ∈ S+1 satisfying the condition π(+1) = +1. Denote by πm , m = 1, . . . , , the transposition exchanging m and + 1 and represent the group S+1 as the union of the right cosets S πm . This allows us to write ηi1 ···i i+1 = ηi1 ···i −

sgn(π)

m=1 π∈S

fij − fij

+1 j=1

fij − fiπ(πm (j))

.

(B.4)

It is not diﬃcult to realize that

sgn(π)

π∈S

+1 j=1

fij − fij

fij − fiπ(πm (j))

=

(fim − fim )(fi+1 − fi+1 ) ηi1 ···i |fei =fei , m +1 (fim − fi+1 )(fi+1 − fim )

and that ηi1 ···i |fei

e m =fi+1

=

(fi+1 − fij )(fim − fij ) ηi1 ···i . j=1 (fim − fij )(fi+1 − fij )

j=m

Using these equalities in (B.4), we obtain ηi1 ···i i+1 = ηi1 ···i

(fi+1 − fi+1 ) + ηi1 ···i

(fi+1 − fij )

j=1

m=1

(fi+1 − fij )

(fim − fij )

j=1

(fim − fi+1 )

j=1

+1

.

(fim − fij )

j=1 j=m

(B.5) Now consider a meromorphic function of z deﬁned as

F (f, f, z) =

(z − fij )

j=1

(z − fi+1 )

+1 j=1

. (z − fij )

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The equality of the sum of the residues of F (f, f, z) to zero gives the relation m=1

(fim − fij )

j=1 +1

(fim − fi+1 )

(fim − fij )

j=1 j=m

=−

j=1 +1

(fi+1 − fij ) − (fi+1

− fij )

(fil+1 − fij )

j=1

(fi+1 − fi+1 )

j=1

. (fi+1 − fij )

j=1

Using it in (B.5), we come to the equality ηi1 ···i i+1 = ηi1 ···i

(fij − fi+1 )(fi+1 − fij ) = ηi1 ···i ηij i+1 j=1 (fij − fi+1 )(fi+1 − fij ) j=1

that gives (B.3). It is clear that (B.2) and (B.3) prove the validity of (3.52). Acknowledgment This work was supported in part by the Russian Foundation for Basic Research under grant #07–01–00234. References [1] A. N. Leznov and M. V. Saveliev, Group-Theoretical Methods for the Integration of Nonlinear Dynamical Systems (Birkh¨ auser, Basel, 1992). [2] A. V. Razumov and M. V. Saveliev, Lie Algebras, Geometry and Toda-Type Systems (Cambridge University Press, Cambridge, 1997). [3] A. V. Mikhailov, The reduction problem and the inverse scattering method, Phys. D 3 (1981) 73–117. [4] V. E. Zakharov and A. B. Shabat, Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II, Funct. Anal. Appl. 13 (1979) 166–174. [5] R. Hirota, The Direct Method in Soliton Theory (Cambridge University Press, Cambridge, 2004). [6] T. Hollowood, Solitons in aﬃne Toda ﬁeld theories, Nucl. Phys. B 384 (1992) 523– 540. [7] C. P. Constantinidis, L. A. Ferreira, J. F. Gomes and A. H. Zimerman, Connection between aﬃne and conformal aﬃne Toda models and their Hirota’s solution, Phys. Lett. B 298 (1993) 88–94; arXiv:hep-th/9207061. [8] N. J. MacKay and W. A. McGhee, Aﬃne Toda solutions and automorphisms of Dynkin diagrams, Int. J. Mod. Phys. A 8 (1993) 2791–2807; Erratum, ibid. 8 (1993) 3830; arXiv:hep-th/9208057.

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[9] H. Aratyn, C. P. Constantinidis, L. A. Ferreira, J. F. Gomes and A. H. Zimerman, Hirota’s solitons in the aﬃne and the conformal aﬃne Toda models, Nucl. Phys. B 406 (1993) 727–770; arXiv:hep-th/9212086. [10] Z. Zhu and D. G. Caldi, Multi-soliton solutions of aﬃne Toda models, Nucl. Phys. B 436 (1995) 659–680; arXiv:hep-th/9307175. [11] A. G. Bueno, L. A. Ferreira and A. V. Razumov, Conﬁnement and soliton solutions in the SL(3) Toda model coupled to matter ﬁelds, Nucl. Phys. B 626 (2002) 463–499 (2002); arXiv:hep-th/0105078. [12] P. E. G. Assis and L. A. Ferreira, The Bullough–Dodd model coupled to matter ﬁelds, arXiv:0708.1342. [13] D. I. Olive, M. V. Saveliev and J. W. R. Underwood, On a solitonic specialisation for the general solutions of some two-dimensional completly integrable systems, Phys. Lett. B 311 (1993) 117–122; arXiv:hep-th/9212123. [14] D. I. Olive, N. Turok and J. W. R. Underwood, Solitons and the energy-momentum tensor for aﬃne Toda theory, Nucl. Phys. B 401 (1993) 663–697. [15] D. I. Olive, N. Turok and J. W. R. Underwood, Aﬃne Toda solitons and vertex operators, Nucl. Phys. B 409 (1993) 509–546; arXiv:hep-th/9305160. [16] A. P. Fordy and J. Gibbons, Integrable nonlinear Klein–Gordon equations and Toda lattices, Commun. Math. Phys. 77 (1980) 21–30. [17] V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons (Heidelberg, Springer, 1991). [18] H. Ch. Liao, D. I. Olive and N. Turok, Topological solitons in Ar Aﬃne Toda theory, Phys. Lett. B 298 (1993) 95–102. [19] C. Rogers and W. K. Schief, B¨ acklund and Darboux Transformations, Geometry and Modern Applications in Soliton Theory (Cambridge, Cambridge University Press, 2002). (1) [20] Zi-Xiang Zhou, Darboux transformations and exact solutions of two-dimensional Cl (2)

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[30] Kh. S. Nirov and A. V. Razumov, Toda equations associated with loop groups of complex classical Lie groups, Nucl. Phys. B 782 (2007) 241–275; arXiv:math-ph/0612054. [31] A. V. Razumov, M. V. Saveliev and A. B. Zuevsky, Non-abelian Toda equations associated with classical Lie groups, in Symmetries and Integrable Systems, Proc. of the Seminar, ed. A. N. Sissakian (JINR, Dubna, Russia, 1999), pp. 190–203; arXiv: math-ph/9909008. [32] Kh. S. Nirov and A. V. Razumov, On classiﬁcation of non-abelian Toda systems, in Geometrical and Topologicl Ideas in Modern Physics, ed. V. A. Petrov (IHEP, Protvino, Russia, 2002), pp. 213–221; arXiv:nlin.SI/0305023. [33] A. V. Mikhailov, M. A. Olshanetsky and A. M. Perelomov, Two-dimensional generalized Toda lattice, Commun. Math. Phys. 79 (1981) 473–488. [34] G. Tzitz´eica, Sur une nouvelle classe de surfaces, Rend. Circ. Mat. Palermo 25 (1908) 180–187. [35] R. K. Dodd and R. K. Bullough, Polynomial conserved densities for the sine-Gordon equations, Proc. R. Soc. Lond. A 352 (1977) 481–503. [36] E. J. Beggs and P. R. Johnson, Inverse scattering and solitons in An−1 aﬃne Toda ﬁeld theories, Nucl. Phys. B 484 (1997) 653–681; arXiv:hep-th/9610104. [37] E. J. Beggs and P. R. Johnson, Inverse scattering and solitons in An−1 aﬃne Toda ﬁeld theories. II, Nucl. Phys. B 529 (1998) 567–587; arXiv:hep-th/9803248.

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Reviews in Mathematical Physics Vol. 20, No. 10 (2008) 1249–1282 c World Scientiﬁc Publishing Company

SEMICLASSICAL ANALYSIS FOR SPECTRAL SHIFT FUNCTIONS IN MAGNETIC SCATTERING BY TWO SOLENOIDAL FIELDS

HIDEO TAMURA Department of Mathematics, Okayama University, Okayama, 700-8530, Japan [email protected] Received 18 May 2008 Revised 16 September 2008 We study the Aharonov–Bohm eﬀect through the semiclassical analysis for the spectral shift function and its derivative in magnetic scattering by two solenoidal ﬁelds in two dimensions, assuming that the total magnetic ﬂux vanishes. The corresponding classical system has a trajectory oscillating between the centers of two solenoidal ﬁelds. The emphasis is placed on analyzing how the trapping eﬀect is reﬂected in the semiclassical asymptotic formula. We also make a comment on the case of scattering by a ﬁnite number of solenoidal ﬁelds and discuss the relation between the Aharonov–Bohm eﬀect from quantum mechanics and the trapping eﬀect from classical mechanics. Keywords: Aharonov–Bohm eﬀect; Birman–Krein trace formula; magnetic scattering; semiclassical limit; spectral shift function. Mathematics Subject Classiﬁcation 2000: 81U99, 35P25, 35Q40, 81Q70

1. Introduction We study the semiclassical asymptotic behavior of the spectral shift function and of its derivative in magnetic scattering by two solenoidal ﬁelds in two dimensions under the assumption that the total magnetic ﬂux vanishes. The system has a trajectory oscillating between the centers of two solenoidal ﬁelds. We place the special emphasis on analyzing how the trapping eﬀect caused by the oscillating trajectory is reﬂected in the semiclassical asymptotic formula. We work in the two-dimensional space R2 with generic point x = (x1 , x2 ) throughout the entire discussion and write ∂j for ∂/∂xj . We deﬁne Λ(x) by Λ(x) = (−x2 /|x|2 , x1 /|x|2 ) = (−∂2 log |x|, ∂1 log |x|). The potential Λ : R2 → R2 deﬁnes the solenoidal ﬁeld ∇ × Λ = (∂12 + ∂22 ) log |x| = ∆ log |x| = 2πδ(x) 1249

(1.1)

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with center at the origin, and it is often called the Aharonov–Bohm potential in physics literatures. A quantum particle moving in two solenoidal ﬁelds with centers odinger operator e± is governed by the magnetic Schr¨ Hh = (−ih∇ − A)2 =

2

(−ih∂j − aj )2 ,

0 < h 1,

(1.2)

j=1

where the potential A = (a1 , a2 ) : R2 → R2 takes the form A(x) = αΛ(x − e+ ) − αΛ(x − e− ),

e+ = e− .

The real number α ∈ R is called the ﬂux of the ﬁeld 2παδ(x). The operator Hh formally deﬁned above is not necessarily essentially self-adjoint in C0∞ (R2 \{e+ , e− }) because of a strong singularity at e± of A(x). We have to impose the boundary condition lim

|x−e± |→0

|u(x)| < ∞

(1.3)

at center e± to obtain the self-adjoint realization (Friedrichs extension) in L2 = L2 (R2 ). We denote by the same notation Hh this self-adjoint realization. The spectral shift function ξh (λ) is deﬁned by the Birman–Krein theory [4, 28]. Let H0h = −h2 ∆ be the free Hamiltonian. The total ﬂux of A(x) vanishes, and the line integral C A(x) · dx = 0 along closed curves in the region {|x| > M } with M 1 large enough. This allows us to construct a smooth real function g(x) falling at inﬁnity such that A = ∇g over the above region. Hence the original operator Hh is unitarily equivalent to ˜ h = exp(−ig/h)Hh exp(ig/h) = (−ih∇ − (A − ∇g))2 H with potential A − ∇g compactly supported, so that the diﬀerence between two ˜ h − i)−1 is of trace class. Then, by the Birman–Krein resolvents (H0h − i)−1 and (H theory, there exists a unique locally integrable function ξh (λ) ∈ L1loc (R) such that ξh (λ) vanishes away from the spectral support of Hh and satisﬁes the trace formula ˜ h ) − f (H0h )] = f (λ)ξh (λ) dλ Tr[f (H for f ∈ C0∞ (R), where the integration without the domain attached is taken over the whole space. We often use this abbreviation throughout the discussion in the sequel. We use the notation tr[G1 − G2 ] = (G1 (x, x) − G2 (x, x)) dx for two integral operators Gj with kernels Gj (x, y). If G1 − G2 is of trace class, then this coincides with the usual trace Tr[G1 − G2 ]. However, the above integral is well deﬁned even for G1 − G2 not necessarily belonging to trace class. For example,

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˜ h ) with f ∈ C0∞ (R). According to tr[G1 − G2 ] = 0 for G1 = f (Hh ) and G2 = f (H this notation, the trace formula takes the form (1.4) tr[f (Hh ) − f (H0h )] = f (λ)ξh (λ) dλ, f ∈ C0∞ (R), for the pair (H0h , Hh ). The function ξh (λ) is called the spectral shift function. The function ξh (λ) with λ > 0 is related to the scattering matrix Sh (λ) at energy ˜ h be as above. Then both the pairs (H0h , Hh ) λ > 0 for the pair (H0h , Hh ). Let H ˜ and (H0h , Hh ) deﬁne the same scattering matrix Sh (λ) as a unitary operator acting on L2 (S 1 ), S 1 being the unit circle. Since the perturbation A − ∇g is of compact support, Sh (λ) takes the form Sh (λ) = Id + Th (λ) with operator Th (λ) of trace class, where Id denotes the identity operator. Hence det Sh (λ) is well deﬁned and is related to ξh (λ) through det Sh (λ) = exp(−2πiξh (λ)). For this reason, ξh (λ) is often called the scattering phase. The function ξh (λ) is also known to be smooth over (0, ∞), and ξh (λ) is calculated as ξh (λ) = −(2πi)−1 Tr[Sh (λ)∗ (dSh (λ)/dλ)]

(1.5)

by the well-known formula (see [7, p. 163] for example). The operator −iSh (λ)∗ Sh (λ) is called the Eisenbud–Wigner time delay operator in physics literatures and its trace describes the time delay for a monoenergetic beam at energy λ (see [3] for the physical background). We introduce a basic cut-oﬀ function χ ∈ C ∞ [0, ∞) such that 0 ≤ χ ≤ 1,

supp χ ⊂ [0, 2),

χ = 1 on [0, 1].

(1.6)

The function χ is often used without further references. We denote by E(λ; H) the spectral resolution associated with self-adjoint operator H = λdE(λ; H). Then both the operators χL E (λ; H0h )χL and χL E (λ; Hh )χL are of trace class for χL = χ(|x|/L), and hence we have Tr[χL (f (Hh ) − f (H0h ))χL ] = f (λ) Tr[χL (E (λ; Hh ) − E (λ; H0h ))χL ] dλ. This, together with (1.4), implies that ξh (λ) = − lim Tr[χL (E (λ; Hh ) − E (λ; H0 ))χL ] L→∞

(1.7)

exists in D (0, ∞). We will prove in Sec. 3 that the convergence makes meaning pointwise as well as in the sense of distribution. The singularity at e± of potential A(x) in (1.2) makes it diﬃcult for us to control ξh (λ) through (1.5). The direct representation (1.7) without using the scattering matrix is better to see the relation between the semiclassical asymptotic behavior of ξh (λ) and the trajectory oscillating between two centers e− and e+ . The derivation of (1.7) relies on the idea due to Bruneau and Petkov [5].

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The asymptotic behavior as h → 0 of ξh (λ) and of ξh (λ) is described in terms of the scattering amplitude by single solenoidal ﬁeld, which has been explicitly calculated in the early works [1, 2, 20]. We consider the operator H±h = (−ih∇ ∓ αΛ)2 under the boundary condition (1.3) at the origin. We denote by f±h (ω → θ; λ) the amplitude for the scattering from incident direction ω ∈ S 1 to ﬁnal one θ at energy λ > 0 for the pair (H0h , H±h ). We often identify ω ∈ S 1 with the azimuth angle from the positive x1 axis. The scattering amplitude is known to have the representation f±h = (2i/π)1/2 λ−1/4 h1/2 sin(±απ/h) exp(i[±α/h](θ − ω))F0 (θ − ω),

(1.8)

where the Gauss notation [α/h] denotes the greatest integer not exceeding α/h and F0 (s) is deﬁned by F0 (s) = eis /(1 − eis ) for s = 0. In particular, the backward amplitude takes the simple form f±h (ω → −ω; λ) = −(i/2π)1/2 λ−1/4 h1/2 (−1)[α/h] sin(απ/h) and also the backward amplitude f±h (ω → −ω; λ, e± ) by the ﬁeld ±2παδ(x − e± ) with center e± is shown to be represented as f±h (ω → −ω; λ, e± ) = exp(i2h−1 λ1/2 e± · ω)f±h (ω → −ω; λ),

(1.9)

where the notation · denotes the scalar product in two dimensions. We are going to discuss the scattering by single ﬁeld in some detail in Sec. 5. We will prove the above relation there. We note that the spectral shift function cannot be necessarily deﬁned for the scattering by a single solenoidal ﬁeld, because the Aharonov–Bohm potential Λ(x) does not fall oﬀ rapidly at inﬁnity. We are now in a position to mention the two main theorems. Theorem 1.1. Let e = e+ − e− = 0 and let eˆ = e/|e| ∈ S 1 . Write f±h (λ) = f±h (±ˆ e → ∓ˆ e; λ, e± ) and define ξ0 (λ; h) = f+h (λ)f−h (λ)h−1 = (i/2π)λ−1/2 sin2 (κπ) exp(i2λ1/2 |e|/h), where κ = α/h − [α/h]. Then ξh (λ) obeys ξh (λ) = −π −1 λ−1/2 Re(ξ0 (λ; h)) + O(h1/3−δ ),

h → 0,

locally uniformly in λ > 0 for any δ, 0 < δ < 1/3. Theorem 1.2. Let κ be as above. As h → 0, ξh (λ) obeys ξh (λ) = κ(1 − κ) − 2(2π)−2 λ−1/2 sin2 (κπ) cos(2λ1/2 |e|/h)|e|−1 h + o(h) locally uniformly in λ > 0.

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In quantum mechanics, a vector potential is known to have a direct signiﬁcance to particles moving in magnetic ﬁelds. This quantum phenomenon is called the Aharonov–Bohm eﬀect (A–B eﬀect) [2]. The leading term κ(1−κ) in the asymptotic formula of ξh (λ) seems to describe this quantum eﬀect, while the second term highly oscillating describes the trapping eﬀect from trajectory oscillating between two centers. We prove Theorem 1.1 in Sec. 2 by reducing the proof to two basic lemmas after formulating the problem as the scattering by two solenoidal ﬁelds with centers at large separation. The two lemmas are proved in Secs. 3–5. Theorem 1.2 is veriﬁed in Sec. 6 by combining Theorem 1.1 with trace formula (1.4). The method developed in the paper applies not only to the special case of two solenoidal ﬁelds but also to the general case of a ﬁnite number of solenoidal ﬁelds. We make only a brief comment on the possible extension without proofs in the last section (Sec. 7). The result heavily depends on the location of centers. If, in particular, centers are placed in a collinear way, then the A–B eﬀect is strongly reﬂected in the asymptotic formula. We have studied the A–B eﬀect in magnetic scattering by two solenoidal ﬁelds through the semiclassical analysis for amplitudes and total cross sections in the previous works [12, 24, 25]. The present paper is thought of as a continuation of these works. We also refer to [22, 23] for related subjects. The spectral shift function is one of important physical quantities in scattering theory, and it plays an important role in the study of the location of resonances in various scattering problems. In his work [17], Melrose has studied how the location of resonances is reﬂected in the asymptotic behavior at high energies of spectral shift function in obstacle scattering through the trace formula (1.4). Since then, a lot of studies have been made in this direction. We refer to [5, 6, 13, 18, 19, 21] and references cited there for comprehensive information on related subjects. Among them, the literature [21] by Sj¨ ostrand is an excellent survey on the relation between the location of resonances near the real axis and classical trapped trajectories. Theorem 1.1 suggests that ξh (λ) remains bounded for Im λ > −M h with M 1 ﬁxed arbitrarily. This implies that for any M 1, there exists hM such that λ with Im λ > −M h is not a resonance for 0 < h < hM . It makes a complement to the result due to Martinez [16], which says that for any M 1, there exists hM such that λ with Im λ > −M h log h−1 is not a resonance for 0 < h < hM in the nontrapping energy range. The spectral shift function is also used for studying the integrated density of states for random Schr¨ odinger operators (see [26] and the references therein).

2. Reduction to Main Lemmas and Proof of Theorem 1.1 In this section we prove Theorem 1.1 by reduction to two main lemmas (Lemmas 2.1 and 2.2) after restating the theorems in the previous section under the formulation as the scattering by solenoidal ﬁelds with two centers at large separation. We begin by introducing the standard notation in scattering theory. We denote by W± (H, K)

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the wave operator W± (H, K) = s − lim exp(itH) exp(−itK) : L2 → L2 t→±∞

and by S(H, K) the scattering operator S(H, K) = W+ (H, K)∗ W− (H, K) : L2 → L2 for two given self-adjoint operators H and K acting on L2 = L2 (R2 ). Let ϕ0 (x; λ, ω) = exp(iλ1/2 x · ω),

λ > 0,

ω ∈ S1,

be the generalized eigenfunction of the free Hamiltonian H0 = −∆. We deﬁne the unitary mapping F : L2 → L2 (0, ∞) ⊗ L2 (S 1 ) by (F u)(λ, ω) = 2−1/2 (2π)−1 ϕ¯0 (x; λ, ω)u(x) dx = 2−1/2 uˆ(λ1/2 ω) (2.1) and Fh by (Fh u)(λ, ω) = 2−1/2 (2πh)−1

ϕ¯0 (x/h; λ, ω)u(x) dx,

(2.2)

where uˆ(ξ) is the Fourier transform of u. Let Hh be deﬁned by (1.2). According to the results obtained by [10, Sec. 7], Hh admits the self-adjoint realization in L2 with domain D = u ∈ L2 : (−ih∇ − A)2 u ∈ L2 , lim |u(x)| < ∞ , |x−e± |→0

where (−ih∇ − A)2 u is understood in D (R2 \{e+ , e− }). We know that Hh has no bound states and its spectrum is absolutely continuous. Moreover it has been shown that the wave operator W± (Hh , H0h ) exists and is asymptotically complete Ran(W+ (Hh , H0h )) = Ran(W− (Hh , H0h )) = L2 . Hence the scattering operator S(Hh , H0h ) : L2 → L2 can be deﬁned as a unitary operator. The mapping Fh deﬁned by (2.2) decomposes S(Hh , H0h ) into the direct integral ∞ ⊕ Sh (λ) dλ, (2.3) S(Hh , H0h ) Fh S(Hh , H0h )Fh∗ 0

where the ﬁber Sh (λ) : L2 (S 1 ) → L2 (S 1 ) is called the scattering matrix at energy λ > 0 and it acts as (Sh (λ)(Fh u)(λ, · ))(ω) = (Fh S(Hh , H0h )u)(λ, ω) on u ∈ L2 .

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We denote by γ(x; ω) the azimuth angle from ω ∈ S 1 to x ˆ = x/|x|. The Aharonov–Bohm potential Λ(x) deﬁned by (1.1) is related to γ(x; ω) through the relation Λ(x) = (−x2 /|x|2 , x1 /|x|2 ) = ∇γ(x; ω).

(2.4)

We deﬁne the two unitary operators (U1 f )(x) = h−1 f (h−1 x),

(U2 f )(x) = exp(ig0 (x))f (x)

(2.5)

acting on L2 , where g0 (x) = [α/h]γ(x − d+ ; eˆ) − [α/h]γ(x − d− ; eˆ),

d± = e± /h.

The function g0 (x) satisﬁes ∇g0 = [α/h]Λ(x − d+ ) − [α/h]Λ(x − d− ) by (2.4), and exp(ig0 (x)) is well deﬁned as a single valued function. Hence Hh is unitarily transformed to Kd := (U1 U2 )∗ Hh (U1 U2 ) = (−i∇ − Bd )2 ,

(2.6)

where Bd (x) = κΛ(x − d+ ) − κΛ(x − d− ) with κ = α/h − [α/h]. The operator Kd deﬁned above is self-adjoint with domain lim |u(x)| < ∞ D(Kd ) = u ∈ L2 : (−i∇ − Bd )2 u ∈ L2 , |x−d± |→0

and enjoys the same spectral properties as Hh . The mapping F deﬁned by (2.1) decomposes the scattering operator S(Kd , H0 ) for the pair (H0 , Kd ) into the direct integral as in (2.3). We assert that S(Kd , H0 ) = U1∗ S(Hh , Hh0 )U1 .

(2.7)

To see this, we represent the propagators exp(−itH0 ) and exp(−itKd ) as exp(−itH0 ) = U1∗ exp(−itH0h )U1 ,

exp(−itKd ) = (U1 U2 )∗ exp(−itHh )U2 U1 .

Since g0 (x) in (2.5) falls oﬀ at inﬁnity, we have W± (Kd , H0 ) = (U1 U2 )∗ W± (Hh , H0h )U1 , and hence (2.7) follows. A simple computation yields F = Fh U1 . This, together with (2.7), implies that the pair (H0 , Kd ) deﬁnes the same spectral shift function ξh (λ) as (H0h , Hh ). Thus Theorems 1.1 and 1.2 are reformulated as the asymptotic behavior as the distance |d| = |d+ − d− | = |e+ − e− |/h = |e|/h → ∞ between centers d− and d+ of two solenoidal ﬁelds obtained from potential Bd (x) goes to inﬁnity.

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Theorem 2.1. Let d = e/h be as above. Then ξh (λ) = 2(2π)−2 λ−1 sin2 (κπ) sin(2λ1/2 |d|) + O(|d|−1/3+δ ),

|d| → ∞,

locally uniformly in λ > 0 for any δ, 0 < δ < 1/3. Theorem 2.2. As |d| → ∞, one has ξh (λ) = κ(1 − κ) − 2(2π)−2 λ−1/2 sin2 (κπ) cos(2λ1/2 |d|)|d|−1 + o(|d|−1 ) locally uniformly in λ > 0. The asymptotic behavior of the spectral shift function has been studied by Kostrykin and Schrader [14, 15] in the case of scattering by potentials with two compact supports at large separation. We make a brief review on the results obtained in these works. They have considered the operator Hd = H0 + V1 (x) + V2 (x − d), H0 = −∆, with potentials Vj rapidly falling oﬀ at inﬁnity, Vj being not necessarily assumed to be compactly supported. In [14], they have shown that the spectral shift function ξ(λ, d) for the pair (H0 , Hd ) obeys ξ(λ, d) ∼ ξ1 (λ) + ξ2 (λ), where ξj (λ) is the spectral shift function for the pair (H0 , Hj ) with Hj = H0 + Vj . In the second work [15], they have established the improved asymptotic formula with the second term, which is described in terms of backward amplitudes as in Theorem 2.2. However the situation is diﬀerent in magnetic scattering, in particular, in two dimensions. This comes from the fact that vector potentials corresponding to magnetic ﬁelds with compact supports at large separation can not necessarily have separate support due to the topological feature of dimension two. We denoteby R(z; H) = (H − z)−1 , Im z = 0, the resolvent of self-adjoint operator H = λdE(λ; H). The derivative E (λ; H) is known to be represented by the formula E (λ; H) = dE(λ; H)/dλ = (2πi)−1 (R(λ + i0; H) − R(λ − i0; H)),

(2.8)

where R(λ ± i0; H) = lim R(λ ± iε; H) as ε ↓ 0. By the principle of limiting absorption, the boundary values R(λ ± i0; Kd) = lim R(λ ± iε; Kd) ε↓0

to the positive real axis exist as a bounded operator from L2s to L2−s for s > 1/2 (see [10, Sec. 7]), where L2s = L2s (R2 ) denotes the weighted L2 space L2 (R2 ; x2s dx) with x = (1 + |x|2 )1/2 . By (1.7), we have ξh (λ) = − lim Tr[χL (E (λ; Kd ) − E (λ; H0 ))χL ] L→∞

in D (0, ∞), where χL = χ(|x|/L). We are now in a position to formulate two main lemmas to which the proof of Theorem 2.1 is reduced. We complete the proof of the theorem, accepting these lemmas as proved. We prove the ﬁrst lemma in Sec. 3 and the second one in Secs. 4 and 5.

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Lemma 2.1. Let χ∞ (x) = 1 − χ(|x|/M |d|) for M 1 fixed large enough. Then the limit lim Tr[χL χ∞ (E (λ; Kd ) − E (λ; H0 ))χ∞ χL ]

L→∞

exists pointwise as well as in the sense of distribution, and it obeys the bound O(|d|−N ) for any N 1. Lemma 2.2. Let χ0 (x) = χ(|x|/M |d|) for M 1 as in Lemma 2.1. Then Tr[χ0 (E (λ; Kd ) − E (λ; H0 ))χ0 ] = −2(2π)−2 λ−1 sin2 (κπ) sin(2λ1/2 |d|) + O(|d|−1/3+δ ) locally uniformly in λ > 0. Proof of Theorem 2.1. Let χ0 and χ∞ be as in the lemmas above. We may assume that χ20 + χ2∞ = 1. Then ξh (λ) is decomposed into −Tr[χ0 (E (λ; Kd ) − E (λ; H0 ))χ0 ] − lim Tr[χL χ∞ (E (λ; Kd ) − E (λ; H0 ))χ∞ χL ]. L→∞

We apply Lemma 2.2 to the ﬁrst term and Lemma 2.1 to the second one. If we take account of the cyclic property of trace, then the theorem is obtained at once.

3. Proof of Lemma 2.1 In this section we prove Lemma 2.1. We use the notation H(B) to denote the magnetic Schr¨ odinger operator H(B) = (−i∇ − B)2

(3.1)

with potential B(x) : R2 → R2 . We also denote by Tr the trace norm of bounded operators acting on L2 . The proof of Lemma 2.1 uses the two lemmas below. The ﬁrst lemma has been already established as [10, Lemma 3.2] or [11, Theorem 4.1]. We prove Lemma 3.2 after completing the proof of Lemma 2.1. Lemma 3.1. There exists k > 0 large enough such that x−k R(λ + i0; Kd )x−k = O(|d|k ) locally uniformly in λ > 0. Lemma 3.2. Let q(x) be a bounded function with support in {|x| < c|d|} for some c > 1. Assume that qM ∈ C ∞ (R2 ) has support in {|x| > M |d|, |ˆ x − ω| < a},

x ˆ = x/|x|,

M 1,

0 < a < 1,

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for ω ∈ S 1 and that |∂ l qM | = O(|x|−|l| ) as |x| → ∞. Then we can take M so large that the following statements hold true: (1) If p+ ∈ C0∞ (R2 ) has support in {λ/3 < |ξ|2 < 3λ, ξˆ · ω > −1/2}, then qR(λ + i0; Kd)qM p+ (Dx )xN Tr = O(|d|−N ) for any N 1. (2) If p− ∈ C0∞ (R2 ) has support in {λ/3 < |ξ|2 < 3λ, ξˆ · ω < 1/2}, then qR(λ − i0; Kd)qM p− (Dx )xN Tr = O(|d|−N ). (3) If p ∈ C ∞ (R2 ) is supported away from {λ/2 < |ξ|2 < 2λ} and satisfies |∂ l p| = O(|ξ|−|l| ) as |ξ| → ∞, then qR(λ ± i0; Kd)qM p(Dx )xN Tr = O(|d|−N ). Proof of Lemma 2.1. We set TL = Tr[χL χ∞ (E (λ; Kd ) − E (λ; H0 ))χ∞ χL ]. According to notation (3.1), we write Kd = H(Bd ). The total ﬂux of the ﬁeld deﬁned from Bd vanishes, and hence there exists a smooth real function ζ ∈ C ∞ (R2 ) such that Bd = ∇ζ over {|x| > c|d|} for some c > 0. We deﬁne K0 = exp(iζ)H0 exp(−iζ) = H(∇ζ). The operator K0 has smooth bounded coeﬃcients and satisﬁes the relation Tr[χL χ∞ E (λ; H0 )χ∞ χL ] = Tr[χL χ∞ E (λ; K0 )χ∞ χL ] and it follows from (2.8) that TL = π −1 Im(Tr[χL χ∞ (R(λ + i0; Kd) − R(λ + i0; K0 ))χ∞ χL ]). We set v0 = 1 − χ(|x|/c|d|)

(3.2)

with c > 0 ﬁxed above. Then Kd = K0 on the support of v0 . We calculate R(λ + i0; Kd)v0 − v0 R(λ + i0; K0) = R(λ + i0; Kd)(v0 K0 − Kd v0 )R(λ + i0; K0) and write v0 K0 − Kd v0 = v0 K0 − K0 v0 = [v0 , K0 ]. The coeﬃcients of commutator [v0 , K0 ] are bounded uniformly in |d| and have support in {c|d| < |x| < 2c|d|}. We take q0 ∈ C0∞ (R2 ) such that q0 = 1 there. Since

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χ∞ v0 = χ∞ for M 1, we have the relation TL = π −1 Im(Tr[[v0 , K0 ]R(λ + i0; K0 )(χ∞ χL )2 R(λ + i0; Kd)q0 ]) by the cyclic property of trace. Let qM and {p+ , p− , p} be as in Lemma 3.2 for some ω ∈ S 1 . We write p− for the operator p− (Dx ) and consider the trace T−L = Im(Tr[[v0 , K0 ]R(λ + i0; K0 )(χ∞ χL )2 p− qM R(λ + i0; Kd)q0 ]). If we write xN p− qM R(λ + i0; Kd)q0 = (q0 R(λ − i0; Kd)qM p− xN )∗ , then it follows from Lemmas 3.1 and 3.2 that the limit lim T−L exists as L → ∞ and obeys the bound O(|d|−N ). A similar result holds true for p+ (Dx ) and p(Dx ). Thus we can show limL→∞ TL = O(|d|−N ) by dividing {|x| > M |d|} into a ﬁnite number of conic regions. This completes the proof. Proof of Lemma 3.2. (1) Let K0 = H(∇ζ) be as above and let v0 be deﬁned by (3.2). We may assume that qv0 = 0. Since v0 qM = qM for M 1, we have qR(λ + i0; Kd)qM = qR(λ + i0; Kd)[v0 , K0 ]R(λ + i0; K0 )qM . We can take M 1 so large that the free particle starting from supp qM with momentum ξ ∈ supp p+ at time t = 0 never passes over supp ∇v0 for t > 0. This implies that [v0 , K0 ]R(λ + i0; K0 )qM p+ xN Tr = O(|d|−N ). Thus (1) follows from Lemma 3.1. (2) This is veriﬁed in exactly the same way as (1). We have only to note that the free particle starting from supp qM with momentum ξ ∈ supp p− at time t = 0 never passes over supp ∇v0 for t < 0, provided that M 1 is taken large enough. (3) This is also easy to prove. We use the calculus of pseudodiﬀerential operators to construct the representation for the operator qR(λ ± i0; Kd )qM p in question. The operator Kd equals K0 = H(∇ζ) on the support of qM , and the symbol (|ξ|2 − λ) has the bounded inverse on the support of p. Moreover the supports of q and qM does not intersect with each other for M 1. Thus the operator takes the form qR(λ ± i0; Kd )qM p = qR(λ ± i0; Kd)RN , where RN satisﬁes xN RN Tr = O(|d|−N ) for any N 1. This, together with Lemma 3.1, yields the desired result. We make repeated use of the argument in the proof of Lemma 3.2 at many stages in the course of the proof of Lemma 2.2 also.

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4. Preliminary to Proof of Lemma 2.2 The present and next sections are devoted to proving Lemma 2.2. As the ﬁrst step, we here prove the following lemma. Lemma 4.1. Let q± (x) be defined by q± = χ(|x − d± |/|d|1/3 ). Then Tr[q± (E (λ; Kd ) − E (λ; H0 ))q± ] = O(|d|−1/3+δ ) locally uniformly in λ > 0. We deﬁne the three Hamiltonians K± = H(±κΛ± ) = (−i∇ ∓ κΛ± )2 ,

κ = α/h − [α/h],

(4.1)

and Hβ = H(βΛ), where Λ± = Λ(x−d± ). These operators are all self-adjoint under boundary condition (1.3) at the center of the ﬁeld. The lemma is obtained as an immediate consequence of the two lemmas below. Lemma 4.2. Let qσ (x) be defined by qσ = χ(r/|d|σ ), r = |x|, for 0 < σ ≤ 1. Then Tr[qσ (E (λ; Hβ ) − E (λ; H0 ))qσ ] = O(|d|−σ ). Lemma 4.3. Let q± be as in Lemma 4.1. Then Tr[q± (E (λ; Kd ) − E (λ; K± ))q± ] = O(|d|−1/3+δ ). Proof of Lemma 4.1. We prove the lemma for q+ only. The trace in question is decomposed into the sum Tr[q+ (E (λ; Kd ) − E (λ; K+ ))q+ ] + Tr[q+ (E (λ; K+ ) − E (λ; H0 ))q+ ]. We apply Lemma 4.3 to the ﬁrst term and Lemma 4.2 with σ = 1/3 to the second one. Then the desired bound is obtained and the proof is complete. The proof of Lemma 4.2 uses the formulae of Bessel functions: ∞

Jl (z)2 = 1,

(4.2)

l=−∞

d/dz{z 2(Jµ (az)2 − Jµ+1 (az)Jµ−1 (az))} = 2zJµ (az)2 , a > 0, z ∞ Jµ+l (z)2 = 2µ Jµ (z)2 z −1 dz, µ > 0, Jµ (z)2 + 2

0

(4.4)

0

l=1

µ

(4.3)

∞

Jµ (z)2 z −1 dz = 1/2,

µ > 0.

(4.5)

We refer to [27, pp. 31, 135, 152, 405] for (4.2)–(4.5), respectively. Moreover, Jµ (z) is known to behave like Jµ (z) = (2/πz)1/2 (Aµ (z) cos(z − (2µ + 1)π/4) − Bµ (z) sin(z − (2µ + 1)π/4))

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as z → ∞, where Aµ (z) and Bµ (z) are asymptotically expanded as Aµ = 1 +

N −1

aµn z

−2n

+ O(z

−2N

),

Bµ = z

−1

N −1

n=1

bµn z

−2n

+ O(z

−2N

) .

n=0

Lemma 4.4. Let qσ (r) be as in Lemma 4.2. Define e(r) = r

∞

Jµ (ar)2 ,

µ = |l − β|.

l=−∞

for a > 0 fixed. Then

∞

qσ (r)e(r) dr =

0

∞

0

qσ (r)r dr + O(|d|−σ ).

Proof. If β = 0, then the relation follows immediately from (4.2). Assume that 0 ≤ β < 1, and set ρ = 1 − β. We make use of (4.4) to calculate e(r) as follows: 2

e(r) = (r/2) Jβ (ar) + 2

∞

2

Jβ+l (ar)

+ rJβ (ar)2 /2

l=1

+ (r/2)(Jρ (ar)2 + 2

∞

Jρ+l (ar)2 ) + rJρ (ar)2 /2

l=1

ar

= βr 0

Jβ (t)2 t−1 dt + rJβ (ar)2 /2 + ρr

0

ar

Jρ (t)2 t−1 dt + rJρ (ar)2 /2.

We deﬁne

∞

eβ (r) = −βr

Jβ (t)2 t−1 dt + rJβ (ar)2 /2,

Iβ = 2

ar

0

∞

qσ (r)eβ (r) dr,

and similarly for eρ (r) and Iρ . Then e(r) = r + eβ (r) + eρ (r) by (4.5), and we have

∞

0

qσ (r)e(r) dr =

0

∞

qσ (r)r dr + (Iβ + Iρ )/2.

The integration by parts yields Iβ = β

0

∞

qσ (r)r2

∞

ar

Jβ (t)2 t−1 dt dr + (1 − β)

0

∞

qσ (r)rJβ (ar)2 dr.

(4.6)

|d|σ < r < 2|d|σ on the support of qσ , such an integral as Since ∞ qσ (r)r−n cos ar dr decreases rapidly as |d| → ∞. If we take account of the 0

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asymptotic form at inﬁnity of the Bessel function Jβ (t), then we see that the ﬁrst integral on the right-hand side of (4.6) behaves like ∞ ∞ ∞ 2 2 −1 qσ (r)r Jβ (t) t dt dr = (1/πa) qσ (r)r dr + O(|d|−σ ). 0

0

ar

To see the behavior of the second integral, we use (4.3). Then we have the relation ∞ ∞ 2 −1 qσ (r)rJβ (ar) dr = −2 qσ (r)r2 (Jβ (ar)2 − Jβ+1 (ar)Jβ−1 (ar)) dr 0

0

again by partial integration. By the asymptotic formula, Jβ±1 (ar) takes the form (2/πar)1/2 (±Aβ±1 (ar) sin(ar − (2β + 1)π/4) ± Bβ±1 (ar) cos(ar − (2β + 1)π/4)), and hence the integral obeys ∞ qσ (r)rJβ (ar)2 dr = −(1/πa) 0

0

Thus we have

Iβ = (1/πa)(β − (1 − β))

∞ 0

∞

qσ (r)r dr + O(|d|−σ ).

qσ (r)r dr + O(|d|−σ ).

The other term Iρ with ρ = 1−β takes a similar asymptotic form. Hence the leading term of the sum Iβ + Iρ vanishes. This completes the proof. Proof of Lemma 4.2. The operator Hβ admits the partial wave expansion Hβ =

∞

⊕hβl ,

hβl = −∂ 2 /∂ 2 r + (µ2 − 1/4)/r2 ,

µ = |l − β|,

l=−∞

where hβl is self-adjoint in L2 (0, ∞) with boundary condition lim r−1/2 |u(r)| < ∞ as r → 0. Since the system of eigenfunctions {ψβl },

ψβl (r, λ) = (r/2)1/2 Jµ (λ1/2 r),

hβl ψβl = λψβl ,

associated with hβl is complete in L2 (0, ∞), we have ∞ ∞ 2 1/2 2 qσ (r) rJµ (λ r) /2 dr. Tr[qσ E (λ; Hβ )qσ ] = 0

l=−∞

On the other hand, it follows from (4.2) that ∞ ∞ 2 1/2 2 qσ (r) rJl (λ r) /2 dr = Tr[qσ E (λ; H0 )qσ ] = 0

l=−∞

0

∞

qσ (r)2 r/2 dr.

Hence the lemma follows from Lemma 4.4. The proof of Lemma 4.3 uses the following two lemmas. The ﬁrst lemma is well known by the principle of limiting absorption, and the second one has been veriﬁed as [10, Lemma 3.3] or [11, Theorem 4.1].

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Lemma 4.5. The operator R(λ ± i0; Hβ ) : L2s → L2−s ,

s > 1/2,

is bounded locally uniformly in λ > 0. Lemma 4.6. Let χ± (x) be defined by χ± = χ(|x − d± |/|d|δ ) for δ > 0 fixed arbitrarily but small enough. Then there exists c > 0 independent of δ such that χ± R(λ + i0; Kd)χ± = O(|d|cδ ),

χ± R(λ + i0; Kd )χ∓ = O(|d|−1/2+cδ ),

where denotes the norm of bounded operators acting on L2 . Let δ > 0 be ﬁxed arbitrarily but small enough and let η ∈ C ∞ (R) be a real periodic function with period 2π such that η has support in (ε, 2π − ε) and η(s) = s

on [2ε, 2π − 2ε]

(4.7)

for ε > 0 small enough. Then we deﬁne the function ζ± (x) by ˆ ζ± = ±κη(γ(x − d± ; ±d)) on |x − d± | ≥ ε|d|δ ,

ζ± = 0

on |x − d± | ≤ ε|d|δ /2

˜ ± by and the operator K ˜ ± = exp(iζ∓ )K± exp(−iζ∓ ) = H(±κΛ± + ∇ζ∓ ), K where γ(x; ω) again denotes the azimuth angle from ω ∈ S to x ˆ = x/|x|. By (2.4), ∇ζ± = ±κΛ± on ˆ ≤ 2π − 2ε}, D± = {x : |x − d± | > ε|d|δ , 2ε ≤ γ(x − d± ; ±d)

(4.8)

˜ ± = Kd there. We set and hence K w± (x) = 1 − χ(|x − d± |/M |d|δ ),

M 1,

(4.9)

and calculate ˜ ±) R(λ + i0; Kd)w∓ − w∓ R(λ + i0; K ˜ ± − Kd w∓ )R(λ + i0; K ˜ ±) = R(λ + i0; Kd )(w∓ K ˜ ± ), = R(λ + i0; Kd )(W∓ + R∓ )R(λ + i0; K

(4.10)

˜∓ − K ˜ ∓ ] = w± K ˜ ∓ w± and R± = (K ˜ ∓ − Kd )w± . The coeﬃwhere W± = [w± , K cients of diﬀerential operator R± vanish over ˆ < 2π − 2ε}. {x : |x − d± | > M |d|δ , 2ε < γ(x − d± ; ±d) Proof of Lemma 4.3. We prove the lemma for K+ only. We consider the diﬀerence ˜ + ))q+ , q+ (R(λ + i0; Kd ) − R(λ + i0; K

q+ = χ(|x − d+ |/|d|1/3 ).

Since w− q+ = q+ , it equals ˜ +)q+ q+ R(λ + i0; Kd)(W− + R− )R(λ + i0; K

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by (4.10). As stated above, the coeﬃcients of R− have support in a conic neighborhood around direction −dˆ with d− as a vertex. We can take M 1 so large that ˜ +)q+ Tr = O(|d|−N ). q+ R(λ + i0; Kd )R− R(λ + i0; K This is shown by almost the same argument as in the proof of Lemma 3.2. Hence Im(Tr[q+ (R(λ + i0; Kd) − R(λ + i0; K+))q+ ]) ˜ + ))q+ ]) + O(|d|−N ). = Im(Tr[q+ (R(λ + i0; Kd )W− R(λ + i0; K The three lemmas below completes the proof. Lemma 4.7. Let χ− be as in Lemma 4.6 and let HS denote the Hilbert–Schmidt norm of bounded operators. Then χ− R(λ + i0; H0 )q+ HS + χ− ∇R(λ + i0; H0 )q+ HS = O(|d|−1/6+δ ). Lemma 4.8. There exists c > 0 such that χ− R(λ + i0; K+)q+ HS + χ− ∇R(λ + i0; K+ )q+ HS = O(|d|−1/6+cδ ). Lemma 4.9. There exists c > 0 such that q+ R(λ + i0; Kd )χ− HS = O(|d|−1/6+cδ ). Completion of proof of Lemma 4.3. By Lemmas 4.8 and 4.9, we have ˜ + ))q+ ]) = O(|d|−1/3+cδ ) Im(Tr[q+ (R(λ + i0; Kd )W− R(λ + i0; K for some c > 0. This completes the proof. (1)

Proof of Lemma 4.7. We denote by H0 (z) the Hankel function of ﬁrst kind and order zero. Then the kernel G0 (x, y; λ) of R(λ + i0; H0 ) is given by (1)

G0 (x, y; λ) = (i/4)H0 (λ1/2 |x − y|) and it behaves like G0 (x, y; λ) = (ic(λ)/4π) exp(iλ1/2 |x − y|)|x − y|−1/2 (1 + O(|x − y|−1 ))

(4.11)

as |x − y| → ∞, where c(λ) = (2π)1/2 e−iπ/4 λ−1/4 . If x ∈ supp χ− and y ∈ supp q+ , then |x − y| > |d|/2. Hence the lemma is easily obtained. ˜ 0 by Proof of Lemma 4.8. Let ζ+ be as above. We deﬁne K ˜ 0 = exp(iζ+ )H0 exp(−iζ+ ) = H(∇ζ+ ). K ˜ 0 coincides with K+ over the domain D+ deﬁned by (4.8). If we set The operator K v+ (x) = 1 − χ(|x − d+ |/M |d|1/3 )

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for M 1, then χ− v+ = χ− and v+ q+ = 0, so that we have the relation ˜ 0 )(V ∗ + R ˜ ∗ )R(λ + i0; K+ )q+ χ− R(λ + i0; K+)q+ = χ− R(λ + i0; K + +

(4.12)

˜ 0 ] and R ˜+ = in almost the same way as used to derive (4.10), where V+ = [v+ , K ˜ (K0 − K+ )v+ . We again follow the same argument as in the proof of Lemma 3.2 to obtain that ˜ 0 )R ˜ ∗ R(λ + i0; K+ )q+ Tr = O(|d|−N ). χ− R(λ + i0; K + The coeﬃcients of V+ have support in {M |d|1/3 /2 < |x − d+ | < 2M |d|1/3 } and obeys the bound O(|d|−1/3 ) there. Hence, by elliptic estimate, it follows from Lemma 4.5 that V+∗ R(λ + i0; K+ )q+ = O(|d|cδ ). Thus (4.12), together with Lemma 4.7, completes the proof. Proof of Lemma 4.9. The proof is done in almost the same way as in the proof of Lemma 4.8. We have the relation ∗ ˜ + )(W−∗ + R− )R(λ + i0; Kd)χ− . q+ R(λ + i0; Kd)χ− = q+ R(λ + i0; K

Then the lemma follows from Lemmas 4.6 and 4.8. 5. Completion of Proof of Lemma 2.2 In this section we complete the proof of Lemma 2.2. Throughout the argument in the section, δ > 0 and ε > 0 are ﬁxed arbitrarily but small enough. We deﬁne ˆ < 2ε, |(x ˆ < 2ε}, − d− ) − d| − d+ ) + d| D0 = {|x − d± | > |d|1/3 /2, |(x ˆ < ε, |(x ˆ < ε} ⊂ D0 , D1 = {|x − d± | > |d|1/3 , |(x − d− ) − d| − d+ ) + d| where (x − d± ) = (x − d± )/|x − d± |. The proof is completed by combining Lemma 4.1 with the two lemmas below. Lemma 5.1. Assume that b ∈ R2 fulfills |b| < 2M |d|,

|b − d± | > |d|1/3 /2,

b ∈ D1 .

Define ψb (x) = χ(|x − b|/|d| ). Then δ

Tr[ψb (E (λ; Kd ) − E (λ; H0 ))ψb ] = O(|d|−N ),

N 1,

uniformly in b. Lemma 5.2. Let ψ0 ∈ C0∞ (R2 ) be a real smooth function such that ψ0 has support in D0 and ψ0 = 1 on D1 . Then Tr[ψ0 (E (λ; Kd ) − E (λ; H0 ))ψ0 ] = −2(2π)−2 λ−1 sin2 (κπ) sin(2λ1/2 |d|) + O(|d|−1/3+δ ) locally uniformly in λ > 0.

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Proof of Lemma 2.2. We divide the region {|x| < 2M |d|} by cut oﬀ functions q± , ψb and ψ0 as in Lemmas 4.1, 5.1 and 5.2, respectively. Then the lemma follows from these lemmas. 5.1. Proof of Lemma 5.1 We shall prove Lemma 5.1. Let η ∈ C ∞ (R) be as in (4.7). We deﬁne the function ζb (x) by ζb = κη(γ(x − d+ ; ˆb+ )) − κη(γ(x − d− ; ˆb− )),

ˆb± = (d± − b)/|d± − b|,

on {|x − d− | ≥ ε|d|δ } ∩ {|x − d+ | ≥ ε|d|δ } and by ζb = 0 on {|x − d− | ≤ ε|d|δ /2} ∪ {|x − d+ | ≤ ε|d|δ /2}. We also deﬁne the operator K0 by K0 = exp(iζb )H0 exp(−iζb ) = H(∇ζb ). By deﬁnition, K0 coincides with Kd on the outside of a conic neighborhood around ˆb± with d± as a vertex. Proof of Lemma 5.1. We set u0 (x) = 1 − χ(|x − d− |/|d|δ ) − χ(|x − d+ |/|d|δ ) and calculate R(λ + i0; Kd)u0 − u0 R(λ + i0; K0) = R(λ + i0; Kd)(U0 + R)R(λ + i0; K0 ), where U0 = [u0 , K0 ] and R = (K0 − Kd )u0 . Since ψb u0 = ψb , we have Im(Tr[ψb (R(λ + i0; Kd) − R(λ + i0; H0 ))ψb ]) = Im(Tr[ψb (R(λ + i0; Kd) − R(λ + i0; K0))ψb ]) = Im(Tr[ψb (R(λ + i0; Kd)U0 R(λ + i0; K0 ))ψb ]) + O(|d|−N ). The last relation is obtained in the same way as in the proof of Lemma 3.2. We decompose U0 into the sum U0 = U+ + U− ,

U± = [u± , K0 ],

u± (x) = 1 − χ(|x − d± |/|d|δ ).

Then we further have Im(Tr[ψb (R(λ + i0; Kd) − R(λ + i0; H0 ))ψb ]) = I− + I+ + O(|d|−N ), where I± = Im(Tr[ψb R(λ + i0; Kd)U± R(λ + i0; K0 )ψb ]).

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We evaluate I− only. A similar argument applies to I+ also. We deﬁne w0 (x) = 1 − χ(|x − d− |/M |d|δ ) − χ(|x − d+ |/M |d|δ ),

M 1,

and set W0 = [w0 , K0 ] = W− + W+ , where W± = [w± , K0 ] and w± is deﬁned by (4.9). We represent ψb R(λ + i0; Kd)U− by use of relation w0 R(λ + i0; Kd) − R(λ + i0; K0 )w0 = R(λ + i0; K0 )(K0 w0 − w0 Kd )R(λ + i0; Kd ). Since w0 ψb = ψb and w0 U− = 0 for M 1, we have ψb R(λ + i0; Kd )U− = ψb R(λ + i0; K0)(K0 w0 − w0 Kd )R(λ + i0; Kd)U− . We again repeat the same argument as in the proof of Lemma 3.2. Then we can choose M so large that I− takes the form of Im(Tr[ψb R(λ + i0; K0 )W0∗ R(λ + i0; Kd )U− R(λ + i0; K0)ψb ]) + O(|d|−N ). We assert that the kernel G± (y, z) of the operator G± = U− R(λ + i0; K0 )ψb2 R(λ + i0; K0 )W± obeys the bound |G± (y, z)| = O(|d|−N ). Then, by the cyclic property of trace, the lemma follows from Lemma 4.6. The kernel of R(λ + i0; H0 ) takes the asymptotic form (4.11). If |y − d− | < 2|d|δ and |z − d+ | < 2M |d|δ and if x ∈ supp ψb , then

x−z y − x

|∇x (|x − z| + |y − x|)| =

− > c > 0. |x − z| |y − x|

Hence a repeated use of partial integration proves the bound for G+ (y, z). A similar argument applies to G− (y, z) also. Thus the proof of the lemma is complete.

5.2. Proof of Lemma 5.2 We shall prove Lemma 5.2. We use the functions u0 , u± and w0 , w± with the same meanings as ascribed in the proof of Lemma 5.1. Proof of Lemma 5.2. The proof is divided into several steps. The auxiliary lemmas used in the course of the proof are all veriﬁed after the completion of this lemma. (1) We ﬁx the notation. Let ψ0 (x) be as in the lemma. We may assume that ψ02 2 2 takes the form ψ02 = ψ− + ψ+ , where ψ± has support in D0 ∩ {|x − d± | < 2|d|/3}.

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The trace in the lemma equals π −1 Im(Tr[ψ0 (R(λ + i0; Kd) − R(λ + i0; H0 ))ψ0 ]) and admits the decomposition Tr[ψ0 (E (λ; Kd ) − E (λ; H0 ))ψ0 ] = π −1 (Ψ− + Ψ+ ),

(5.1)

where Ψ± = Im(Tr[ψ± (R(λ + i0; Kd ) − R(λ + i0; H0 ))ψ± ]). Let ζ± be as in Sec. 4. We set ˜ 0 = exp(iζ0 )H0 exp(−iζ0 ) = H(∇ζ0 ), K

ζ0 = ζ− + ζ+ ,

˜ ± again by and deﬁne K ˜ ± = exp(iζ∓ )K± exp(−iζ∓ ) = H(±κΛ± + ∇ζ∓ ), K

K± = H(±κΛ± ).

We further write ˜ 0), R0 (λ) = R(λ + i0; K

˜ ± ), R± (λ) = R(λ + i0; K

Rd (λ) = R(λ + i0; Kd ).

(2) We analyze the behavior as |d| → ∞ of Ψ− only. We make use of the relation ψ− u0 = ψ− to calculate ˜ 0 − Kd u0 )R0 (λ)ψ− . ψ− (Rd (λ) − R0 (λ))ψ− = ψ− Rd (λ)(u0 K Then we obtain Ψ− = J− + J+ + O(|d|−N )

(5.2)

in the same way as in the proof of Lemma 3.2, where ˜± R0 (λ)ψ− ]), J± = Im(Tr[ψ− Rd (λ)U

˜± = [u± , K ˜ 0 ]. U

We make repeated use of the same argument as in the proof of Lemma 3.2 without ˜− to analyze the behavior further references. We consider the operator ψ− Rd (λ)U of J− . Since ˜ − − Kd u+ )R− (λ) Rd (λ)u+ − u+ R− (λ) = Rd (λ)(u+ K ˜− = U ˜− , we see that J− takes the asymptotic form and since ψ− u+ = ψ− and u+ U J− = Im(Tr[ψ− (R− (λ) + Rd (λ)V˜+ R− (λ))U˜− R0 (λ)ψ− ]) + O(|d|−N ), where ˜ − ]. V˜+ = [u+ , K Lemma 5.3. One has ˜− R0 (λ)ψ− ]) = O(|d|−N ) Im(Tr[ψ− R− (λ)U ˜ ∗ HS = O(|d|−N ), where W ˜ ± = [w± , K ˜ 0 ]. ˜− R0 (λ)ψ 2 R0 (λ)W and U − −

(5.3)

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We represent V˜+ R− (λ)U˜− by use of the relation ˜ −∗ + w− (K ˜0 − K ˜ − ))R− (λ). w− R− (λ) − R0 (λ)w− = R0 (λ)(W ˜− w− = 0, it follows from Lemma 5.3 that Since V˜+ w− = V˜+ and U ˜ −∗ R− (λ)U˜− R0 (λ)ψ− ]) + O(|d|−N ). J− = Im(Tr[ψ− Rd (λ)V˜+ R0 (λ)W

(5.4)

We look at the operator ψ− Rd (λ)V˜+ in (5.4). Since ˜ 0 w0 − w0 Kd )Rd (λ) w0 Rd (λ) − R0 (λ)w0 = R0 (λ)(K and since ψ− w0 = ψ− and w0 V˜+ = 0, we see again from Lemma 5.3 that ˜ ∗ Rd (λ)V˜+ R0 (λ)W ˜ ∗ R− (λ)U ˜− R0 (λ)ψ− ]) + O(|d|−N ). J− = Im(Tr[ψ− R0 (λ)W + − Lemma 5.4. There exists c > 0 such that ˜ ∗ (Rd (λ) − R+ (λ))V˜+ = O(|d|−1+cδ ). W + We can easily show that ˜ ∗ HS = O(|d|1/2+δ ), ψ− R0 (λ)W +

˜ ∗ HS = O(|d|−1/2+2δ ) V˜+ R0 (λ)W −

(5.5)

˜− R0 (λ)ψ− = O(|d|1/2+cδ ). In fact, the ﬁrst two bounds follow from the and U asymptotic form (4.11) of the kernel G0 (x, y; λ) of R(λ + i0; H0 ), because the distance between the supports of two functions ψ− and w+ satisﬁes dist(supp ψ− , supp w+ ) ≥ c |d| for some c > 0. The third bound is a consequence of the principle of limiting absorption. Thus Lemmas 4.5 and 5.4, together with these bounds, imply that ˜ ∗ R+ (λ)V˜+ R0 (λ)W ˜ ∗ R− (λ)U ˜− R0 (λ)ψ− ]) + O(|d|−1/2+cδ ) J− = Im(Tr[ψ− R0 (λ)W + − for some c > 0 independent of δ. (3) We denote by ( , ) the L2 scalar product. The argument in this step is based on the following two lemmas. √ Lemma 5.5. Let ϕ0 (x; ω) = ϕ0 (x; λ, ω) = exp(i λx · ω) and let c(λ) = (2π)1/2 e−iπ/4 λ−1/4 .

(5.6)

be as in (4.11). Then ˜ ∗ = (ic(λ)/4π)|d|−1/2 (V˜+ (eiζ0 Π+ e−iζ0 )W ˜ ∗ + OHS (|d|−1+cδ )) V˜+ R0 (λ)W − − for some c > 0, where Π± acts as ˆ 0 (x; ±d) ˆ = (Π± u)(x) = (u, ϕ0 (·; ±d))ϕ

ˆ dy ϕ0 (x; ±d) ˆ u(y)ϕ¯0 (y; ±d)

on u(x), and the remainder OHS (|d|ν ) denotes an operator the Hilbert–Schmidt norm of which obeys the bound O(|d|ν ).

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2 ˜− R0 (λ)ψ− ˜ +∗ takes the Lemma 5.6. Let Π± be as in Lemma 5.5. Then U R0 (λ)W form

˜ ∗ ) + OHS (|d|−1/3+cδ )) ˜− eiζ0 Π− e−iζ0 W (iλ−1/2 /2)(ic(λ)/4π)τ− |d|−1/2 ((U + 2 ˆ dt. for some c > 0, where τ± = τ± (d) = ψ± (td) By the cyclic property of trace, it follows from Lemma 5.5 that J− = |d|−1/2 Im(Tr[T0 ]) + O(|d|−1/2+cδ ) where ˜− R0 (λ)ψ 2 R0 (λ)W ˜ ∗ R+ (λ)V˜+ (eiζ0 Π+ e−iζ0 )W ˜ ∗ R− (λ). T0 = (ic(λ)/4π)U − + − Since τ± (d) = O(|d|), Lemma 5.6 implies that J− = 2−1 λ−1/2 τ− |d|−1 Re(Tr[T1 ]) + O(|d|−1/3+cδ ) where ˜ +∗ R+ (λ)V˜+ (eiζ0 Π+ e−iζ0 )W ˜ −∗ R− (λ)U ˜− . T1 = (ic(λ)/4π)2 (eiζ0 Π− e−iζ0 )W (4) We complete the proof of the lemma in this step. Let f± (ω → θ) denote the amplitude for the scattering from incident direction ω to ﬁnal one θ at energy λ by the solenoidal ﬁeld ±κδ(x − d± ). Lemma 5.7. One has the relations ˆ W ˆ = f− (−dˆ → d) ˆ + O(|d|−N ), ˜ − eiζ0 ϕ0 (·; d)) (ic(λ)/4π)(R− (λ)U˜− eiζ0 ϕ0 (·; −d), ˆ W ˆ = f+ (dˆ → −d) ˆ + O(|d|−N ). ˜ + eiζ0 ϕ0 (·; −d)) (ic(λ)/4π)(R+ (λ)V˜+ eiζ0 ϕ0 (·; d), By this lemma, we have ˆ + (dˆ → −d)) ˆ + O(|d|−1/3+cδ ) J− = 2−1 λ−1/2 τ− |d|−1 Re(f− (−dˆ → d)f and similarly for J+ . Thus we have ˆ + (dˆ → −d)) ˆ + O(|d|−1/3+cδ ) Ψ− = λ−1/2 τ− |d|−1 Re(f− (−dˆ → d)f by (5.2). We also have ˆ + (dˆ → −d)) ˆ + O(|d|−1/3+cδ ). Ψ+ = λ−1/2 τ+ |d|−1 Re(f− (−dˆ → d)f Since

τ− + τ+ =

ˆ 2 dt + ψ− (td)

ˆ 2 dt = ψ+ (td)

ˆ 2 dt = |d|(1 + O(|d|−2/3 )), ψ0 (td)

it follows from (5.1) that the trace in the lemma behaves like ˆ + (dˆ → −d)) ˆ + O(|d|−1/3+cδ ). π −1 λ−1/2 Re(f− (−dˆ → d)f The amplitude is explicitly calculated as ˆ = −(i/2π)1/2 λ−1/4 sin(κπ) exp(±i2λ1/2 d± · d) ˆ f± (±dˆ → ∓d)

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by (1.8) and (1.9) with h = 1. This yields the desired relation, and the proof of the lemma is now complete. 5.3. Proof of Lemmas 5.3–5.6 We prove Lemmas 5.3–5.6. Proof of Lemma 5.3. We prove the ﬁrst relation. It is easy to see that the operator ˜− = 0, we use the relation under consideration is of trace class. Since w− U ˜ − )R− (λ) ˜ 0 w− − w− K w− R− (λ) − R0 (λ)w− = R0 (λ)(K to obtain ˜ − )R− (λ)U ˜ 0 w− − w− K ˜− . ψ− R− (λ)U˜− = ψ− R0 (λ)(K Hence the trace in the lemma obeys ˜ ∗ R− (λ)U ˜− R0 (λ)ψ− ]) + O(|d|−N ). Im(Tr[ψ− R0 (λ)W − If we take account of asymptotic form (4.11) of the kernel of R(λ + i0; H0 ) and of the cyclic property of trace, an argument similar to that used in the proof of Lemma 5.1 yields the bound O(|d|−N ) on the ﬁrst term. The second relation is also veriﬁed in a similar way. Thus the lemma is obtained. Proof of Lemma 5.4. Since ˜ + − Kd u− )R+ (λ), Rd (λ)u− − u− R+ (λ) = Rd (λ)(u− K we have ˜ ∗ (Rd (λ) − R+ (λ))V˜+ = W ˜ ∗ Rd (λ)(V˜ ∗ + (K ˜ + − Kd )u− )R+ (λ)V˜+ , W + + − ˜ + ]. By elliptic estimate, the lemma follows from Lemma 4.6. where V˜− = [u− , K Proof of Lemma 5.5. By deﬁnition, R0 (λ) = exp(iζ0 )R(λ + i0; H0 ) exp(−iζ0 ). The kernel G0 (x, y; λ) of R(λ+i0; H0 ) obeys (4.11). If |x−d+ | < |d|δ and |y −d− | < M |d|δ , then |x − y| = (x − y) · dˆ + O(|d|−1+cδ ) for some c > 0, and hence we have √ √ √ ˆ exp(−i λy · d)(1 ˆ + O(|d|−1+cδ )). exp(i λ|x − y|) = exp(i λx · d) This yields the desired relation. Proof of Lemma 5.6. The proof uses the stationary phase method ([9, Theorem 7.7.5]). We write 2 2 ˜ +∗ ˜− R0 (λ)ψ− ˜ +∗ = U ˜− eiζ0 R(λ + i0; H0 )ψ− U R0 (λ)W R(λ + i0; H0 )e−iζ0 W

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and analyze the behavior of the integral I(y, z) = G0 (y, x; λ)ψ− (x)2 G0 (x; z; λ) dx when y ∈ supp ∇u− and z ∈ supp ∇w+ . To do this, we take d± as d− = (0, 0) and d+ = (|d|, 0), and we work in the coordinates x = (x1 , x2 ). If x ∈ supp ψ− , then |d|1/3 /c < x1 < 2|d|/3 and |x2 | < c x1 for some c > 0. We represent the integral as I(y, z) =

G0 (y, x; λ)ψ− (x)2 G0 (x; z; λ) dx2 dx1

and apply the stationary phase method to the integral in brackets after making change of variable x2 = x1 s. We look at the phase function. If we take account of asymptotic form (4.11), then we can write the phase function as follows: iλ1/2 (|y − x| + |x − z|) = iλ1/2 (|x| + |x − d+ | − |x1 − |d||) + iλ1/2 ν(x, y, z), where ν = ν(x1 , x2 , y, z) is deﬁned by ν = (|x − y| − |x|) + (|x − z| − |x − d+ |) + |x1 − |d||. We further make change of variable x2 = x1 s to see that the ﬁrst term on the right-hand side takes the form iλ1/2 x1 g(x1 , s), where g(x1 , s) = (1 + s2 )1/2 + x1 s2 (|x − d+ | + |x1 − |d||)−1 with x = (x1 , x1 s). A simple computation shows that s = 0 is the only stationary point, g (x1 , 0) = 0, and g (x1 , 0) = 1 + x1 /(|d| − x1 ) = |d|/(|d| − x1 ). We get exp(iλ1/2 x1 g(x1 , 0)) = exp(iλ1/2 x1 ) and −1/2

(λ1/2 x1 g (x1 , 0)/2πi)−1/2 = ic(λ)x1

((|d| − x1 )/|d|)1/2 ,

where c(λ) is deﬁned by (5.6). We also obtain ν(x1 , 0, y, z) = ((x1 − y1 )2 + y22 )1/2 − x1 + ((x1 − z1 )2 + z22 )1/2 = −y1 + (z1 − x1 ) + O(|d|−1/3+2δ ). We make use of (4.11) to calculate the leading term of the integral x1

G0 (y, x; λ)ψ− (x)2 G0 (x; z; λ) ds,

x = (x1 , x1 s),

|s| < c.

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Since (ic(λ)/4π)2 = (iλ−1/2 /2)(4π)−1 , we take account of all the above relations to obtain that the integral behaves like 1/2

(iλ−1/2 /2)(ic(λ)/4π)|d|−1/2 ψ− (x1 , 0)2 e−iλ

y1 iλ1/2 z1

e

(1 + O(|d|−1/3+2δ )).

Thus the proof is complete. 5.4. Scattering by single solenoidal field Before proving Lemma 5.7, we begin by a quick review on the scattering by a single solenoidal ﬁeld without detailed proof. The amplitude is known to have the explicit representation for such a scattering system. We refer to [1, 2, 20] for the earlier works, as stated in Sec. 1. We consider the Schr¨ odinger operator Hβ = H(βΛ) = (−i∇ − βΛ)2 ,

0 ≤ β < 1,

which is self-adjoint under the boundary condition (1.3) at the origin and admits the partial wave expansion ⊕hlβ , hlβ = −∂r2 + (µ2 − 1/4)r−2 , µ = |l − β|. Hβ l∈Z

We denote by ϕ+ (x; λ, ω), Hβ ϕ+ = λϕ+ , the outgoing eigenfunction with incident direction ω. According to the partial wave expansion, ϕ+ (x; λ, ω) is given by √ exp(−iµπ/2) exp(ilγ(x; −ω))Jµ ( λ|x|), ϕ+ = l∈Z

where γ(x; ω) again denotes the azimuth angle from ω to xˆ = x/|x|. If, in particular, β = 0, then this yields the well-known expansion formula for the free eigenfunction 1/2 ϕ0 (x; λ, ω) = eiλ x·ω in terms of Bessel functions. The eigenfunction ϕ+ converges to ϕ0 (x; λ, ω) as |x| → ∞ along direction −ω and it is decomposed as the sum ϕ+ = ϕin (x; λ, ω) + ϕsc (x; λ, ω), where ϕin = exp(iβ(γ(x; ω) − π))ϕ0 (x; λ, ω) and 1/2 ϕsc = −(sin(βπ)/π) eiλ |x| cosh t

e−βt −t e + eiσ

dt eiσ

with σ(x; ω) = γ(x; ω) − π. We apply the stationary phase method to the integral to see that ϕsc takes the asymptotic form ˆ; λ) exp(iλ1/2 |x|)|x|−1/2 + o(|x|−1/2 ), ϕsc = gβ (ω → x

|x| → ∞,

xˆ = ω,

and hence ϕ+ (x; λ, ω) behaves like 1/2

ϕ+ = eiβ(γ(x;ω)−π) eiλ

x·ω

1/2

+ gβ (ω → x ˆ; λ)eiλ

|x|

|x|−1/2 (1 + o(1))

(5.7)

as |x| → ∞ along direction x ˆ = x/|x|. The ﬁrst term on the right-hand side describes the wave incident from direction ω and the second one describes the wave scattered

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into direction x ˆ. The scattering amplitude gβ (ω → θ; λ) is explicitly represented as gβ (ω → θ; λ) = (2i/π)1/2 λ−1/4 sin(βπ) F0 (θ − ω− ), where F0 (θ) is deﬁned by F0 (θ) = eiθ /(1 − eiθ ) under the identiﬁcation of θ ∈ S 1 with the azimuth angle from the positive x1 axis. We add a comment to the√incident wave ϕin which takes a form diﬀerent from the usual plane wave exp(i λx · ω). The modiﬁed factor eiβ(γ(x;ω)−π) is due to the long-range property of the potential βΛ(x). Since Λ(x) = ∇γ(x; ω) by (2.4), β(γ(x; ω) − π) is represented as the integral Λ(y) · dy

β(γ(x; ω) − π) = β l

along the line l = {y = x + tω : t < 0}. Thus the modiﬁed factor may be interpreted as the change of phase generated by the potential βΛ to the free motion. We represent gβ (ω → θ; λ) in terms of R(E + i0; Hβ ). The next lemma has been veriﬁed as [10, Lemma 3.2]. Lemma 5.8. Let u(x) = 1 − χ(|x|/|d|δ ) and let j(x; ω) ∈ C ∞ (R2 → R) be a smooth function with support in a conic neighborhood around −ω such that j(x; ω) = γ(x; ω)

on {|x| > ε|d|δ , |ˆ x + ω| < ε}

and ∂xm j = O(|x|−|m| ). If θ = ω, then gβ (ω → θ; λ) = (ic(λ)/4π)(R(λ + i0; Hβ )Q− ϕ0 (ω), Q+ ϕ0 (θ)) + O(|d|−N ) for any N 1, where we write ϕ0 (ω) for exp(iλ1/2 x · ω) and Q− = exp(iβj(x; ω))[u, H0 ],

Q+ = exp(iβj(x; −θ))[u, H0 ].

We add some comments. If we denote by gβ (ω → θ; λ, p) the amplitude for the scattering by the ﬁeld 2πβδ(x − p) with center p ∈ R2 , it is easily seen from (5.7) that gβ (ω → θ; λ, p) = exp(−iλ1/2 p · (θ − ω))gβ (ω → θ; λ),

(5.8)

because |x − p| = |x| − p · θ + O(|x|−1 ) as |x| → ∞ along direction θ. We further denote by g−β (ω → θ; λ) the scattering amplitude by the ﬁeld −2πβδ(x). The

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operator H−β = H(−βΛ) is unitarily equivalent to H1−β = H((1 − β)Λ) = exp(iγ(x))H−β exp(−iγ(x)), where γ(x) stands for the azimuth angle from the positive x1 axis. Hence it follows that g−β (ω → θ; λ) = exp(−i(θ − ω))g1−β (ω → θ; λ). Thus Lemma 5.8 allows us to represent the amplitude g−β (ω → θ; λ) as ˜ − ϕ0 (ω), Q ˜ + ϕ0 (θ)) + O(|d|−N ), g−β = (ic(λ)/4π)(R(λ + i0; H−β )Q

(5.9)

where ˜ − = exp(−iβj(x; ω))[u, H0 ], Q

˜ + = exp(−iβj(x; −θ))[u, H0 ]. Q

The same relation g−β (ω → θ; λ, p) = exp(−iλ1/2 p · (θ − ω))g−β (ω → θ; λ)

(5.10)

as in (5.8) also remains true for the amplitude g−β (ω → θ; λ, p) in scattering by the ﬁeld −2πβδ(x − p). Proof of Lemma 5.7. According to the notation applied to K± = H(±κΛ± ), 0 ≤ κ < 1, we have ˆ = g−κ (−dˆ → d; ˆ λ, d− ), f− (−dˆ → d)

ˆ = gκ (dˆ → −d; ˆ λ, d+ ). f+ (dˆ → −d)

We write ˆ W ˆ ˜ − eiζ0 ϕ0 (d)), A− = (ic(λ)/4π)(R− (λ)U˜− eiζ0 ϕ0 (−d), iζ iζ 0 0 ˆ W ˆ ˜ + e ϕ0 (−d)) A+ = (ic(λ)/4π)(R+ (λ)V˜+ e ϕ0 (d), for the scalar products on the left-hand side of the relations in the lemma. By deﬁnition, ˜ 0 ] = exp(iζ0 )[u− , H0 ] exp(−iζ0 ), ˜− = [u− , K U

˜ − = exp(iζ0 )[w− , H0 ] exp(−iζ0 ) W

and R− (λ) = exp(iζ+ )R(λ + i0; K− ) exp(−iζ+ ). We insert these relations into the scalar product A− . We note that ˆ ζ0 − ζ+ = ζ− = −κη(γ(x − d− ; −d)), ˆ in a where η ∈ C ∞ (R) is deﬁned by (4.7). Thus ζ0 − ζ+ equals −κγ(x − d− ; −d) ˆ conic neighborhood around d with d− as a vertex. If we make change of variables from x − d− to x, then it follows from (5.9) and (5.10) that ˆ λ, d− ) + O(|d|−N ). A− = g−κ (−dˆ → d; ˜ − ] by (5.3). Since K− = H(−κΛ− ), we have Recall V˜+ = [u+ , K V˜+ = exp(iζ+ )[u+ , K− ] exp(−iζ+ ) = exp(iζ0 )[u+ , H0 ] exp(−iζ0 )

(5.11)

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on |x − d+ | < |d|/2. This enables us to repeat the same argument as used to prove (5.11), and we obtain ˆ λ, d+ ) + O(|d|−N ). A+ = gκ (dˆ → −d; Thus the proof is complete. 6. Proof of Theorem 1.2 In this section we prove Theorem 2.2 (and hence, Theorem 1.2). The proof is based on the two lemmas below. We prove the ﬁrst lemma after completing the proof of the theorem. The second lemma has been already established as [25, Theorem 1.5]. Lemma 6.1. Assume that f ∈ C0∞ (R) is a smooth function such that f is supported away from the origin and obeys f (k) (λ) = O(|d|kρ ) for some 0 < ρ < 1. Then tr(f (Kd ) − f (H0 )) = |supp f | × f ∞ O(|d|−1 ) + o(|d|−1 ), where |supp f | denotes the size of supp f . Lemma 6.2. Assume that f ∈ C0∞ (R) obeys f (k) (λ) = O(1) uniformly in d and that f (λ) vanishes around the origin. Then tr(f (Kd ) − f (H0 )) = −κ(1 − κ)f (0) + o(|d|−1 ), where κ = α/h − [α/h]. Proof of Theorem 2.2. We deﬁne η0 (λ; h) = −2(2π)−2 λ−1/2 sin2 (κπ) cos(2λ1/2 |d|)|e|−1 ,

|d| = |e|/h.

Then it follows from Theorem 2.1 that η0 (λ; h)h and ξh (λ) have the same leading term as |d| → ∞. We ﬁx E > 0 arbitrarily and take ρ, 2/3 < ρ < 1, close enough to 1. Let g ∈ C ∞ (R) be a smooth real function such that 0 ≤ g ≤ 1,

g=0

on (−∞, E − 2|d|−ρ ],

Then ξh (E) is represented as E g(λ)ξh (λ) dλ + ξh (E) = −∞

g=1

E

−∞

on [E − |d|−ρ , ∞).

g (λ)ξh (λ) dλ.

We apply Theorem 2.1 to the ﬁrst integral on the right-hand side to obtain that E g(λ)ξh (λ) dλ = η0 (E; h)h + o(|d|−1 ). −∞

On the other hand, the behavior of the second integral is controlled by the trace −ρ formula. If we set f (λ) = g(λ)−1, then f (λ) = g (λ) and f (λ) = 0 for λ > E−|d| , so that the integral equals f (λ)ξh (λ) dλ. We decompose f (λ) into the sum f =

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f1 + f2 , where f1 ∈ C0∞ (R) has support in (E − 2ε, E − |d|−ρ ) and f2 ∈ C ∞ (R) has support in (−∞, E − ε) for ε > 0 ﬁxed arbitrarily but small enough. We may assume that g(λ) obeys g (k) (λ) = O(|d|kρ ), and hence f1 fulﬁlls the assumption in Lemma 6.1. Thus we have f1 (λ)ξh (λ) dλ = tr(f1 (Kd ) − f1 (H0 )) = ε O(|d|−1 ) + o(|d|−1 ). Since ξh (λ) vanishes for λ < 0 and f2 (0) = −1 at the origin, it follows from Lemma 6.2 that f2 (λ)ξh (λ) dλ = tr(f2 (Kd ) − f2 (H0 )) = κ(1 − κ) + o(|d|−1 ). Thus we sum up all the above integrals to obtain the desired asymptotic formula and the proof is complete. We proceed with proving Lemma 6.1 which remains unproved. To formulate the auxiliary lemma, we consider a triplet {v0 , v1 , v2 } of smooth real functions with the following properties: (v.0) (v.1) (v.2) (v.3)

vj , ∇ vj and ∇∇ vj are bounded uniformly in d. v0 v1 = v0 and v1 v2 = v1 . dist(supp vj , supp ∇v2 ) ≥ c0 |d| for some c0 > 0, j = 0, 1. ∇vj has support in a bounded domain {|d|/c < |x| < c|d|}, c > 1.

These functions depend on d, but we skip the dependence. By (v.1), we have the inclusion relations supp v0 ⊂ supp v1 ⊂ supp v2 and v1 = 1 on supp v0 ,

v2 = 1

on supp v1 .

We do not necessarily assume vj to be of compact support. Lemma 6.3. Let {v0 , v1 , v2 } be as above. Consider a self-adjoint operator K = H(B) = (−i∇ − B)2 . Assume that the potential B satisfies B = ∇g on supp v2 for some smooth real function g defined over R2 . Set K0 = H(∇g). Then v1 ((K − z)−1 − (K0 − z)−1 )v0 Tr = |Im z|−N −4 O(|d|−N ) for any N 1. Proof. We calculate v1 ((K − z)−1 − (K0 − z)−1 )v0 as v1 (K − z)−1 (v2 K0 − Kv2 )(K0 − z)−1 v0 = v1 (K − z)−1 [v2 , K0 ](K0 − z)−1 v0 . By a simple calculus of pseudodiﬀerential operators, it follows from (v.2) and (v.3) that [v2 , K0 ](K0 − z)−1 v0 HS = |Im z|−N −2 O(|d|−N ). This completes the proof.

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Proof of Lemma 6.1. The proof uses the Helﬀer–Sj¨ ostrand calculus for selfadjoint operators [8]. According to the calculus, we have f (Kd ) = (i/2π) ∂¯z f˜(z)(Kd − z)−1 dz d z¯ for f ∈ C0∞ (R) as in the lemma, where f˜ ∈ C0∞ (C) is an almost analytic extension of f such that f˜ fulﬁlls f˜ = f on R and obeys |∂¯zm f˜(z)| = |Im z|N O(|d|N ρ ),

m ≥ 1,

(6.1)

for any N 1. We introduce a smooth nonnegative partition of unity {w− , w+ , w∞ , w1 , . . . , wm },

2 w−

+

2 w+

+

2 w∞

+

m

wk2 = 1,

k=1 2

over R , where m is independent of d and each function has the following property: supp w± ⊂ {|x − d± | < 2ε|d|},

supp w∞ ⊂ {|x| > M |d|}

for 0 < ε 1 small enough and M 1 large enough, and supp wk ⊂ {|x − bk | < ε|d|},

dist(bk , supp w± ) > ε|d|/2

for some bk ∈ R2 . We assert that Tr[wk (f (Kd ) − f (H0 ))wk ] = O(|d|−N ), −N

tr[w∞ (f (Kd ) − f (H0 ))w∞ ] = O(|d|

)

(6.2) (6.3)

for any N 1 and that Tr[w± (f (Kd ) − f (H0 ))w± ] = |supp f | × f ∞ O(|d|−1 ) + o(|d|−1 ).

(6.4)

Then the lemma is obtained. We begin by proving (6.2). To prove this, we note that Kd is represented as Kd = H(Bd ) = exp(igk )H0 exp(−igk ) for some real smooth function gk over the support of wk . In fact, the ﬁeld ∇ × Bd has support only at two centers d− and d+ . If we denote by K0 the operator on the right side, then it follows from Lemma 6.3 that wk ((Kd − z)−1 − (K0 − z)−1 )wk Tr = |Im z|−N −4 O(|d|−N ). Since ρ < 1 strictly in (6.1) by assumption, the Helﬀer–Sj¨ ostrand formula implies (6.2). A similar argument applies to (6.3) also. The proof of (6.4) uses Lemma 4.1. We consider the + case only. We take ˜+ w+ = w+ . Then there exists a real smooth w ˜+ ∈ C0∞ (R2 ) in such a way that w

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function g− such that Kd = exp(ig− )K+ exp(−ig− ) ˜ + the operator on the right side. Then we have over supp w ˜+ . We denote by K ˜ + − z)−1 )w+ = w+ (Kd − z)−1 [w ˜ + ](K ˜ + − z)−1 w+ . w+ ((Kd − z)−1 − (K ˜+ , K ˜+ , the operator on the right side further Since w+ vanishes over the support of ∇w equals ˜ + ](K ˜ + − z)−1 [w+ , K ˜ + ](K ˜ + − z)−1 w w+ (Kd − z)−1 [w ˜+ , K ˜+ . We may assume that dist(supp ∇w+ , supp ∇w ˜+ ) ≥ c |d| for some c > 0. We apply ˜ + − z)−1 (∇w+ ) to obtain that Lemma 6.3 to (∇w˜+ )(K Tr(w+ ((Kd − z)−1 − (K+ − z)−1 )w+ ) = |Im z|−N −4 O(|d|−N ). Hence the Helﬀer–Sj¨ostrand formula yields Tr(w+ (f (Kd ) − f (H0 ))w+ ) = Tr(w+ (f (K+ ) − f (H0 ))w+ ) + o(|d|−1 ). Since f is supported away from the origin, Lemma 4.2 with σ = 1 implies that Tr(w+ (E (λ; K+ ) − E (λ; H0 ))w+ ) = O(|d|−1 ) uniformly in λ ∈ supp f . Thus (6.4) is obtained and the proof is complete. 7. Concluding Remark: A Finite Number of Solenoidal Fields We conclude the paper by making comments on the possible generalization to the case of scattering by a ﬁnite number of solenoidal ﬁelds. We consider the magnetic Schr¨odinger operator 2

Hh = (−ih∇ − A) ,

A=

n

αj Λ(x − ej ).

j=1

The potential A(x) deﬁnes the n solenoidal ﬁelds with ﬂux αj ∈ R and center ej ∈ R2 , and the operator Hh becomes self-adjoint under the boundary condition (1.3) at each center ej . We assume that n

αj = 0.

(7.1)

j=1

Then the spectral shift function ξh (λ) at energy λ > 0 is deﬁned for the pair (H0h , Hh ). We denote by fjh (ω → −ω; λ, ej ) = exp(i2h−1 λ1/2 ej · ω)fjh (ω → −ω; λ), fjh (ω → −ω; λ) = −(i/2π)1/2 λ−1/4 h1/2 (−1)[αj /h] sin(αj π/h),

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the backward amplitude in the scattering by 2παj δ(x−ej ) and by 2παj δ(x), respectively. For pair a = (j, k) with 1 ≤ j < k ≤ n, we deﬁne ea → eˆa ; λ, ej )fkh (ˆ ea → −ˆ ea ; λ, ek )h−1 ξa (λ; h) = fjh (−ˆ = exp(i2λ1/2 |ea |/h)fjh (−ˆ ea → eˆa ; λ)fkh (ˆ ea → −ˆ ea ; λ)h−1 in the same way as ξ0 (λ; h) in Theorem 1.1, where eˆa = ea /|ea | with ea = ek − ej . The quantity ξa (λ; h) is associated with the trajectory oscillating between ej and ek . We also deﬁne ηa (λ; h) by ηa = −2(2π)−2 (−1)[αj /h]+[αk /h] sin(αj π/h) sin(αk π/h) cos(2λ1/2 |ea |/h)|ea |−1 . By deﬁnition, we have ηa (λ; h)h = −π −1 λ−1/2 Re(ξa (λ; h)) + O(h). We make the following assumption on the location of centers: For any pair a = (j, k), there are no other centers on the segment joining ej and ek .

(7.2)

Under assumptions (7.1) and (7.2), we can establish ξh (λ) =

n

κj (1 − κj )/2 + h

ηa (λ; h) + o(h)

a=(j,k), 1≤j
j=1

locally uniformly in λ > 0, where κj = αj /h − [αj /h]. The situation is more delicate when (7.2) is violated. For example, such a case occurs when centers are placed in a collinear way. We now assume that the three centers e1 , e2 and e3 are located along the x1 axis with e2 as a middle point. Then the quantity ηa (λ; h) associated with a = (1, 2) or (2, 3) does not undergo any change, but ηb (λ; h) with b = (1, 3) requires a modiﬁcation, because the magnetic potential α2 Λ(x − e2 ) has a direct inﬂuence on the quantum particle going from e1 to e3 or from e3 to e1 by the Aharonov–Bohm eﬀect. If the particle goes from e1 to e3 , then we distinguish the trajectory l+ passing over the upper half plane {x2 > 0} from l− passing over the lower half plane {x2 < 0}. The change of phase caused by the potential is given by the line integral α2 Λ(y − e2 ) · dy = α2 ∇γ(y − e2 ) · dy = ∓α2 π, l±

l±

where γ(x) again denotes the azimuth angle from the positive x1 axis. Hence the two kinds of trajectories give rises to the factor (exp(−iα2 π/h) + exp(iα2 π/h))/2 = cos(α2 π/h). We have the same factor for the trajectory from e3 to e1 . Thus the asymptotic formula takes the form   3 κj (1 − κj )/2 + h ηa (λ; h) + cos2 (κ2 π)ηb (λ; h) + o(h), ξh (λ) = j=1

a =b

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where b = (1, 3). We have developed the asymptotic analysis for amplitudes in scattering by a chain of solenoidal ﬁelds in the earlier work [11]. References [1] G. N. Afanasiev, Topological Eﬀects in Quantum Mechanics (Kluwer Academic Publishers, 1999). [2] Y. Aharonov and D. Bohm, Signiﬁcance of electromagnetic potential in the quantum theory, Phys. Rev. 115 (1959) 485–491. [3] W. O. Amrein and K. B. Sinha, Time delay and resonances in potential scattering, J. Phys. A 39 (2006) 9231–9254. [4] M. Sh. Birman and D. Yafaev, The spectral shift function, The papers of M. G. Krein and their further development, St. Petersburg Math. J. 4 (1993) 833–870. [5] V. Bruneau and V. Petkov, Representation of the spectral shift function and spectral asymptotics for trapping perturbations, Commun. Partial Diﬀerential Equations 26 (2001) 2081–2019. [6] M. Dimassi, Spectral shift function and resonances for slowly varying perturbations of periodic Schr¨ odinger operators, J. Funct. Anal. 225 (2005) 193–228. [7] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs, Vol. 18 (American Mathematical Society, 1969). ´ [8] B. Helﬀer and J. Sj¨ ostrand, Equation de Schr¨ odinger avec champ magn´etique et ´equation de Harper in Schr¨ odinger Operators (S∅nderborg, 1988), eds. A. Jensen and H. Holden, Lecture Notes in Physics, Vol. 345 (Springer, 1989), pp. 118–197. [9] L. H¨ ormander, The Analysis of Linear Partial Diﬀerential Operators I (Springer Verlag, 1983). [10] H. T. Ito and H. Tamura, Aharonov–Bohm eﬀect in scattering by point-like magnetic ﬁelds at large separation, Ann. Henri Poincar´e 2 (2001) 309–359. [11] H. T. Ito and H. Tamura, Aharonov–Bohm eﬀect in scattering by a chain of point-like magnetic ﬁelds, Asymptot. Anal. 34 (2003) 199–240. [12] H. T. Ito and H. Tamura, Semiclassical analysis for magnetic scattering by two solenoidal ﬁelds, J. London Math. Soc. 74 (2006) 695–716. [13] A. Khochman, Resonances and spectral shift function for the semiclassical Dirac operators, Rev. Math. Phys. 19 (2007) 1071–1115. [14] V. Kostrykin and R. Schrader, Cluster properties of one particle Schr¨ odinger operators, Rev. Math. Phys. 6 (1994) 833–853. [15] V. Kostrykin and R. Schrader, Cluster properties of one particle Schr¨ odinger operators, II, Rev. Math. Phys. 10 (1998) 627–683. [16] A. Martinez, Resonance free domains for non globally analytic potentials, Ann. Henri Poincar´e 3 (2002) 739–756; Erratum, ibid. 8 (2007) 1425–1431. [17] R. Melrose, Weyl asymptotics for the phase in obstacle scattering, Commun. Partial Diﬀerential Equations 13 (1988) 1431–1439. [18] S. Nakamura, Spectral shift function for trapping energies in the semi-classical limit, Commun. Math. Phys. 208 (1999) 173–193. [19] D. Robert, Relative time-delay for perturbations of elliptic operators and semiclassical asymptotics, J. Funct. Anal. 126 (1994) 36–82. [20] S. N. M. Ruijsenaars, The Aharonov–Bohm eﬀect and scattering theory, Ann. Physics 146 (1983) 1–34. [21] J. Sj¨ ostrand, Quantum resonances and trapped trajectories; in Long Time Behavior of Classical and Quantum Systems (Bologna, 1999), Proc. Bologna APTEX Int. Conf.

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Ser. Concr. Appl. Math., Vol. 1, eds. S. Graﬃ and A. Martinez (World Sci. Publ., River Edge, NJ, 2001), pp. 33–61. P. Stovicek, Scattering matrix for the two-solenoid Aharonov–Bohm eﬀect, Phys. Lett. A 161 (1991) 13–20. P. Stovicek, Scattering on two solenoids, Phys. Rev. A 48 (1993) 3987–3990. H. Tamura, Semiclassical analysis for magnetic scattering by two solenoidal ﬁelds: Total cross sections, Ann. Henri Poincar´e 8 (2007) 1071–1114. H. Tamura, Time delay in scattering by potentials and by magnetic ﬁelds with two supports at large separation, J. Funct. Anal. 254 (2008) 1735–1775. I. Veseli´c, Existence and Regularity Properties of the Integrated Density of States of Random Schr¨ odinger Operators, Lec. Notes in Math., Vol. 1917 (Springer, 2008). G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd edn. (Cambridge University Press, 1995). D. Yafaev, Scattering Theory: Some Old and New Problems, Lecture Notes in Math., Vol. 1735 (Springer, 2000).

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Reviews in Mathematical Physics Vol. 20, No. 10 (2008) 1283–1307 c World Scientific Publishing Company

A NONLINEAR MODEL FOR RELATIVISTIC ELECTRONS AT POSITIVE TEMPERATURE

CHRISTIAN HAINZL∗ , MATHIEU LEWIN† and ROBERT SEIRINGER‡ ∗Department

of Mathematics, UAB, Birmingham, AL 35294-1170, USA [email protected]

†CNRS

and Department of Mathematics (CNRS UMR8088), University of Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France [email protected]

‡Department

of Physics, Jadwin Hall, Princeton University, P. O. Box 708, Princeton, New Jersey 08544, USA [email protected] Received 10 March 2008 Revised 22 September 2008

We study the relativistic electron-positron field at positive temperature in the Hartree– Fock approximation. We consider both the case with and without exchange terms, and investigate the existence and properties of minimizers. Our approach is non-perturbative in the sense that the relevant electron subspace is determined in a self-consistent way. The present work is an extension of previous work by Hainzl, Lewin, S´er´ e and Solovej where the case of zero temperature was considered. Keywords: QED; Dirac vacuum; positive temperature. Mathematics Subject Classification 2000: 81Q99, 46T99

0. Introduction In Coulomb gauge and when photons are neglected, the Hamiltonian of Quantum Electrodynamics (QED) reads formally [2, 13, 14, 16] as α ρ(x)ρ(y) φ ∗ 0 H = Ψ (x)D Ψ(x)dx − φ(x)ρ(x)dx + dx dy. (1) 2 |x − y| Here Ψ(x) is the second-quantized ﬁeld operator satisfying the usual anticommutation relations, and ρ(x) is the density operator ρ(x) =

4 4 1 ∗ 1 ∗ [Ψ (x)σ , Ψ(x)σ ] = {Ψ (x)σ Ψ(x)σ − Ψ(x)σ Ψ∗ (x)σ }, 2 σ=1 2 σ=1

1283

(2)

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where σ is the spin variable. In (1), D0 = −iα · ∇ + β is the usual free Dirac operator, α is the bare Sommerfeld ﬁne structure constant and φ is the external potential. The matrices α = (α1 , α2 , α3 ) and β are the usual 4 × 4 anti-commuting Dirac matrices. We have chosen a system of units such that = c = m = 1. In QED, one main issue is the minimization of the Hamiltonian (1). However, even if we implement a UV-cutoﬀ, the Hamiltonian is unbounded from below, since the particle number can be arbitrary. In a formal sense, this problem was ﬁrst overcome by Dirac, who suggested that the vacuum is ﬁlled with inﬁnitely many particles occupying the negative energy states of the free Dirac operator D0 . With this axiom, Dirac was able to conjecture the existence of holes in the Dirac sea which he interpreted as antielectrons or positrons. His prediction was veriﬁed by Anderson in 1932. Dirac also predicted [6,7] the phenomenon of vacuum polarization: in the presence of an electric ﬁeld, the virtual electrons are displaced and the vacuum acquires a non-uniform charge density. In Quantum Electrodynamics, Dirac’s assumption is sometimes implemented via normal ordering which essentially consists of subtracting the kinetic energy of the negative free Dirac sea, in such a way that the kinetic energy of electrons as well as positrons (holes) becomes positive. With this procedure, the distinction between electrons and positrons is put in by hand. It was pointed out in [14] (see also the review [13]), however, that normal ordering is probably not well suited to the case α = 0 of interacting particles (the interaction is the last term of (1)). Instead, a procedure was presented where the distinction between electrons and positrons is not an input but rather a consequence of the theory. The approach of [14] is rigorous and fully non-perturbative, but so far it was only applied to the mean-ﬁeld (Hartree–Fock) approximation, with the photon ﬁeld neglected. It allowed to justify the use of the Bogoliubov–Dirac–Fock model (BDF) [4], studied previously in [10–12]. The purpose of the present paper is to extend these results to the nonzero temperature case. The methodology of [14] is a two steps procedure. First, the free vacuum is constructed by minimizing the Hamiltonian (1) over Hartree–Fock (or quasi-free) states in a box with an ultraviolet cutoﬀ, and then taking the thermodynamic limit 0 when the size of the box goes to inﬁnity. The limit is a Hartree–Fock state P− describing the (Hartree–Fock) free vacuum [13, 14]. It has an inﬁnite energy, since it contains inﬁnitely many virtual particles forming the (self-consistent) Dirac sea. We remark that this state is not the usual sea of negative electrons of the free Dirac operator because all interactions between particles are taken into account, but it corresponds to ﬁlling negative energies of an eﬀective mean-ﬁeld translation invariant operator. The second step of [14] consists of constructing an energy functional that is bounded from below in the presence of an external ﬁeld, by subtracting the (inﬁnite) energy of the free self-consistent Dirac sea. The key observation is that the diﬀerence of the energy of a general state P minus the (inﬁnite) energy of the free vacuum

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0 P− can be represented by an eﬀective functional (called Bogoliubov–Dirac–Fock 0 , describing the variations with (BDF) [4]) which only depends on Q = P − P− respect to the free Dirac sea. The BDF energy was studied in [10–12]. The existence of ground states was shown for the vacuum case in [10, 11] and in charge sectors in [12]. For a detailed review of all these results, we refer to [13]. An associated timedependent evolution equation, which is in the spirit of Dirac’s original paper [6], was studied in [15]. Let us now turn to the case of a nonzero temperature T = 1/β > 0. We consider a Hartree–Fock state with one-particle density matrix 0 ≤ P ≤ 1. Because of the deﬁnition of the Hamiltonian (1) and the anti-commutator in (2), it is more convenient to consider as variable the renormalized density matrix γ = P − 1/2. We remark that the anti-commutator in (2) is a kind of renormalization which does not depend on any reference as normal ordering does (it just corresponds to subtracting the identity divided by 2). The anticommutator of (2) is due to Heisenberg [16] (see also [18, Eq. (96)]) and it is necessary for a covariant formulation of QED, see [22, Eq. (1.14)] and [8, Eq. (38)]. Computing the free energy of our Hartree–Fock state using (1) (and ignoring inﬁnite constant terms) one arrives at the following free energy functional [3, 14] α ργ (x)ργ (y) QED 0 FT (γ) = tr(D γ) − α ϕ(x)ργ (x) + 2 |x − y| α trC4 |γ(x, y)|2 − TS (γ) (3) − 2 |x − y|

where the entropy is given by the formula 1 1 1 1 S(γ) = −tr + γ ln +γ − tr − γ ln −γ . 2 2 2 2

(4)

The (matrix-valued) function γ(x, y) is the formal integral kernel of the operator γ and ργ (x) := trC4 γ(x, x) is the associated charge density. The above formulas are purely formal; they only make sense in a ﬁnite box with an ultraviolet cutoﬀ, in general. Note that we only consider Hartree–Fock states, i.e. quasi-free states having no pairing [3]. Indeed, using [14, Formula (2.14)], one sees that a quasi-free state with pairing density matrix a(x, y) has a Hartree–Fock energy equal to (3) plus a pairing energy (α/2) |a(x, y)|2 |x−y|−1 dx dy. Hence, the Coulomb potential being of positive type, discarding the pairing term always decreases the energy. This shows that the pairing density matrix must vanish for minimizers (in a ﬁnite box with an ultraviolet cutoﬀ) and that we may restrict the whole study to Hartree–Fock states. As in [14] the ﬁrst step is to deﬁne the free vacuum at temperature T , which is the formal minimizer of (3) when φ = 0. Following [14], one can ﬁrst conﬁne the system to a box, then study the limit as the size of the box goes to inﬁnity and identify the free vacuum as the limit of the sequence of ground states. Alternatively, it was

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proved in [14] that the free vacuum can also be obtained as the unique minimizer of the free energy per unit volume. In the nonzero temperature case, this energy reads 1 TT (γ) = trC4 [D0 (p)γ(p)]dp (2π)3 B(0,Λ) α trC4 [γ(p)γ(q)] − dp dq (2π)5 |p − q|2 2 B(0,Λ) 1 1 T 4 + γ(p) ln + γ(p) trC + (2π)3 B(0,Λ) 2 2 1 1 + − γ(p) ln − γ(p) dp 2 2 and it is deﬁned for translation-invariant states γ = γ(p) only, under the constraint −1/2 ≤ γ ≤ 1/2. Here, B(0, Λ) denotes the ball of radius Λ centered at the origin. The real number Λ > 0 is the ultraviolet cutoﬀ. We shall prove in Theorem 4 that the above energy has a unique minimizer γ˜ 0 , and prove several interesting properties of it. In particular, we shall see that it satisﬁes a nonlinear equation of the form 1 1 1 γ˜ 0 = (5) − 2 1 + eβDγ˜ 0 1 + e−βDγ˜ 0 i.e. it is the Fermi–Dirac distribution of a (self-consistent) free Dirac operator, deﬁned as Dγ˜ 0 = D0 − α

γ˜ 0 (x, y) . |x − y|

(The last term stands for the operator having this integral kernel.) This extends results of [14] to the T > 0 case. The next step is to formally subtract the (inﬁnite) energy of γ˜ 0 from the energy of any state γ. In this way one obtains a Bogoliubov–Dirac–Fock free energy at temperature T = 1/β which can be formally written as γ 0 )” FT (γ) = “FTQED (γ) − FTQED (˜ ρ[γ−˜γ 0 ] (x)ρ[γ−˜γ 0 ] (y) α 0 = TH (γ, γ˜ ) − α ϕ(x)ρ[γ−˜γ 0 ] (x) + 2 |x − y| α trC4 |(γ − γ˜ 0 )(x, y)|2 − 2 |x − y|

(6)

where H is the relative entropy formally deﬁned as TH (γ, γ˜ 0 ) = “tr(Dγ˜ 0 (γ − γ˜ 0 )) − TS (γ) + TS (˜ γ 0 )”.

(7)

1 , where ν represents the We shall consider external ﬁelds of the form ϕ = ν ∗ |x| density distribution of the external particles, like nuclei, or molecules.

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In Sec. 2.2, we show how to give a correct mathematical meaning to the previous formulas and we prove that the BDF free energy is bounded from below. An important tool is the following inequality TH (γ, γ˜ 0 ) ≥ tr[|Dγ˜ 0 |(γ − γ˜ 0 )2 ] ≥ tr[|D0 |(γ − γ˜ 0 )2 ].

(8)

This implies that the relative entropy can control the exchange term and enables us to show that FT is bounded from below. Unfortunately, like for the T = 0 case, the free BDF energy is not convex, which makes it a diﬃcult task to prove the existence of a minimizer. Although we leave this question open, we derive some properties for a potential minimizer in Sec. 2.2. In particular we prove that any minimizer γ satisﬁes the following nonlinear equation 1 1 1 − (9) γ= 2 1 + eβDγ 1 + e−βDγ where the (self-consistent) Dirac operator reads Dγ = D0 + αργ ∗ | · |−1 − αφ − α

γ 0 (x, y) . |x − y|

Compared with the zero temperature case, the main diﬃculty in proving the existence of a minimizer comes from localization issues of the relative entropy which are more involved than in the zero temperature case. As a slight simpliﬁcation, we thoroughly study the reduced Hartree–Fock case for T > 0, where the exchange term (the ﬁrst term of the second line of (3)) is neglected. In the zero-temperature case, this model was already studied in detail in [9, 11]. The corresponding free vacuum is now simple: it is the Fermi–Dirac distribution corresponding to the usual free Dirac operator D0 , 1 1 1 γ0 = − . 2 1 + eβD0 1 + e−βD0 The reduced Bogoliubov–Dirac–Fock free energy is obtained in the same way as before by subtracting the inﬁnite energy of the free Dirac see γ 0 to the (reduced) Hartree–Fock energy. It is given by ρ[γ−γ 0 ] (x)ρ[γ−γ 0 ] (y) α red 0 dx dy, FT (γ) = T H(γ, γ ) − α ϕρ[γ−γ 0 ] + 2 |x − y| H(γ, γ 0 ) being deﬁned similarly as before. As this functional is now convex, we can prove in Theorem 2 that it has a unique minimizer γ¯, which satisﬁes the selfconsistent equation 1 1 1 − γ¯ = 2 1 + eβDγ¯ 1 + e−βDγ¯ where Dγ¯ := D0 + αργ¯ −γ 0 ∗ | · |−1 − αφ in this case.

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Additionally we show in Theorem 3 that this minimizer has two interesting properties. First, γ¯ − γ 0 is a trace-class operator. In the zero temperature case, on the other hand, it was proved in [11] that the minimizer is never trace-class for α > 0. This was indeed the source of complications concerning the deﬁnition of the trace (and hence of the charge) of Hartree–Fock states [10] when T = 0. This is related to the issue of renormalization [9, 11, 14]. Although we do not minimize in the trace-class in the case T = 0 but rather in the Hilbert–Schmidt class because the free energy is only coercive for the Hilbert–Schmidt norm, it turns out that the minimizer is trace-class nevertheless. The second (and related) interesting property shown in Theorem 3 below is that the total electrostatic potential created by the density ν and the polarized Dirac sea decays very fast. More precisely we prove that ργ¯ − ν ∈ L1 (R3 ) and (ργ¯ − ν) ∗

1 ∈ L1 (R3 ). |x|

Necessarily, the charge of ργ¯ and the charge of the external sources have to be equal. More precisely the eﬀective potential has a much faster decay at inﬁnity than 1/|x|, which shows that the eﬀective potential is screened. In other words due to the positive temperature, the particles occupying the Dirac-sea have enough freedom to rearrange in such a way that the external sources are totally shielded. Within non-relativistic fermionic plasma this eﬀect is known as Debye-screening. Let us emphasize that in order to recover such a screening, it is essential to calculate the Gibbs-state in a self-consistent way. These two properties of the minimizer of the reduced theory probably also hold for the full BDF model with exchange term. However, like for the case T = 0, the generalization does not seem to be straightforward. The paper is organized as follows. The ﬁrst section is devoted to the presentation of our results for the reduced model which is simpler and for which we can prove much more than for the general case. In the second section, we consider the original Hartree–Fock model with exchange term. We prove the existence and uniqueness of the free Hartree–Fock vacuum, deﬁne the BDF free energy in the presence of an external ﬁeld and provide some interesting properties of potential minimizers. In the last section, we provide some details of proofs which are a too lengthy to be put in the main text. 1. The Reduced Bogoliubov–Dirac–Fock Free Energy 1.1. Relative entropy Throughout this paper, we shall denote by Sp (H) the usual Schatten class of operators Q acting on a Hilbert space H and such that tr(|Q|p ) < ∞. The UV cutoﬀ is implemented like in [10–12, 14] in Fourier space by considering the Hilbert space HΛ := {ψ ∈ L2 (R3 , C4 ) | supp ψˆ ⊂ B(0, Λ)},

(10)

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with B(0, Λ) denoting the ball of radius Λ centered at the origin. We denote by “tr” the usual trace functional on S1 (HΛ ). Within the reduced theory, the free vacuum at temperature T = β −1 > 0 is the self-adjoint operator acting on HΛ deﬁned by γ0 =

1 2

1 1 − 1 + eβD0 1 + e−βD0

.

(11)

Notice when T → 0 (β → ∞), we recover the usual formula [9–11] γ 0 = −D0 /2|D0 |. We assume that T > 0 henceforth. Notice that thanks to the cutoﬀ in Fourier space and the gap in the spectrum of D0 , the spectrum of γ 0 does not include 0 or ±1/2. In fact, it is given by

e−βE(Λ) e−β 1 1 + σ(γ ) = − + , − 2 1 + e−βE(Λ) 2 1 + e−β 1 e−β e−βE(Λ) 1 − , − ∪ 2 1 + e−β 2 1 + e−βE(Λ)

0

(12)

√ where E(Λ) = 1 + Λ2 . Also the charge density of the free vacuum γ 0 at temperature T vanishes: 1 1 1 trC4 − ργ 0 = dk = 0. (13) 2(2π)3 B(0,Λ) 1 + eβD0 (k) 1 + e−βD0 (k) We shall denote the class of Hilbert–Schmidt perturbations of γ 0 by K: K :=

γ ∈ B(HΛ ) | γ ∗ = γ, −

1 1 ≤ γ ≤ , γ − γ 0 ∈ S2 (HΛ ) . 2 2

(14)

The relative entropy reads 1 1 1 0 H(γ, γ ) = tr +γ ln + γ − ln +γ 2 2 2 1 1 1 0 + −γ ln − γ − ln −γ . 2 2 2

0

(15)

Note that since γ ∈ K is a compact perturbation of γ 0 , we always have σess (γ) = σess (γ 0 ). Hence σ(γ) only contains eigenvalues of ﬁnite multiplicity in the neighborhood of ±1/2. Using the integral formula ln a − ln b = − 0

∞

∞ 1 1 1 1 − (a − b) dt, dt = a+t b+t a + t b + t 0

(16)

we easily see that Eq. (15) is well deﬁned as soon as γ ∈ K, γ − γ 0 ∈ S1 (HΛ ), since the spectrum of γ 0 does not contain ±1/2.

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When γ − γ 0 ∈ K is merely Hilbert–Schmidt, we may deﬁne the relative entropy by the integral formula H(γ, γ 0 ) = tr

1

−1

2 1 0 1 − |u| 0 (γ − γ (γ − γ ) ) du . 1 + 2uγ 0 1 + 2uγ 1 + 2uγ 0

(17)

It is clear that this provides a well deﬁned object in K as one has ∀γ ∈ K, ∀u ∈ [−1, 1],

0≤

1 − |u| 1 1 ≤ 1 and 0 ≤ ≤ 1 + 2uγ 1 + 2uγ 0

for some > 0, by (12). It is not diﬃcult to see that (17) and (15) coincide when γ −γ 0 ∈ S1 (HΛ ). We shall discuss this in the Appendix. But (17) has the advantage of being well deﬁned for all γ ∈ K, and hence we use (17) for a deﬁnition of H henceforth. We remark that (17) also formally coincides with the formula (7) we gave in the introduction (with γ˜ 0 replaced by γ 0 ). More precisely, they coincide when all the terms are well deﬁned, for instance when the system is restricted to a box with periodic boundary conditions and an ultraviolet cutoﬀ. Our ﬁrst result is the Theorem 1 (Properties of Relative Entropy). The functional γ → H(γ, γ 0 ) deﬁned in (17) is strongly continuous on K for the topology of S2 (HΛ ). It is convex, hence weakly lower semi-continuous (wlsc). Moreover, it is coercive on K for the Hilbert–Schmidt norm: ∀γ ∈ K,

TH (γ, γ 0 ) ≥ tr(|D0 |(γ − γ 0 )2 )

(18)

where we recall that T = β −1 is the temperature. Coercive in this context means that H(γ, γ 0 ) → ∞ if γ − γ 0 S2 (HΛ ) → ∞. This follows from (18) since |D0 | ≥ 1. Proof of Theorem 1. First, we prove that H(·, γ 0 ) is strongly continuous for the S2 (HΛ ) topology. This is indeed a consequence of the following Lemma 1. Let γ, γ ∈ K. Then we have for some constant C (depending on Λ) and all 0 ≤ η ≤ 1, |H(γ, γ 0 ) − H(γ , γ 0 )| ≤

C γ − γ S2 (HΛ ) + Cη(γ − γ 0 2S2 (HΛ ) η + γ − γ 0 2S2 (HΛ ) ).

(19)

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Proof. We use formula (17) and split the integrals as follows: 1 −1+η 1−η 1 = + + . −1

−1

−1+η

1−η

We estimate

1−η

tr du

1 1 − |u| 1 (γ − γ 0 ) (γ − γ 0 ) 1 + 2uγ 0 1 + 2uγ 1 + 2uγ 0

C 1 1 0 1 − |u| 0

≤ γ − γ S (H ) − (γ − γ ) (γ − γ ) 2 Λ 0 0 1 + 2uγ 1 + 2uγ 1 + 2uγ

η −1+η

using in particular 1 − |u| 1 1 − |u| 1 − |u| = 2u − (γ − γ) 1 + 2uγ 1 + 2uγ 1 + 2uγ 1 + 2uγ and 0 ≤ (1 + 2uγ )−1 ≤ η −1 as γ ∈ K and −1 + η ≤ u ≤ 1 − η. Similarly

1

1 1 0 1 − |u| 0

tr

≤ Cηγ − γ 0 2 (γ − γ du (γ − γ ) ) S2 (HΛ ) .

0 0 1 + 2uγ 1 + 2uγ 1 + 2uγ

1−η The other terms are treated in the same way. Convexity of γ → H(γ, γ0 ) is a simple consequence of the integral representation (17). In fact, the integrand is convex for any ﬁxed u ∈ [−1, 1], since 1 (γ − γ 0 ) 1 + 2uγ 1 1 = (1 + 2uγ0 ) 2uγ − 1 − 4uγ0 + (1 + 2uγ0 ) (2u)2 1 + 2uγ

γ → (γ − γ 0 )

is clearly convex. Finally, we prove formula (18). Consider the following function 1 1 1 +x ln + x − ln +y f (x, y) = 2 2 2 1 1 1 + −x ln − x − ln −y 2 2 2 deﬁned on (−1/2, 1/2)2. Minimizing 1/2+y over x for ﬁxed y, one ﬁnds that f (x, y) ≥ 2 (x − y) C(y) where C(y) = ln 1/2−y /(2y). If we write y as 1 1 1 − , (20) y= 2 1 + eh 1 + e−h we obtain C(y) = h tanh(h/2)−1 ≥ max(|h|, 2). Hence if y takes the form (20), we deduce f (x, y) ≥ max{(x − y)2 |h|, 2(x − y)2 }.

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Assume now that X and Y are self-adjoint operators acting on a Hilbert space H, with −1/2 ≤ X, Y ≤ 1/2 and 1 1 1 − Y = 2 1 + eH 1 + e−H for some H. By Klein’s inequality [23, p. 330], one also has H(X, Y ) = trf (X, Y ) ≥ max{tr(X − Y )2 |H|, 2 tr(X − Y )2 }.

(21)

This gives (18), taking X = γ and Y = γ 0 . 1.2. Existence of a minimizer and Debye screening Now we are able to deﬁne the reduced Bogoliubov–Dirac–Fock energy at temperature T = β −1 . For this purpose, we introduce the Coulomb space C := {ρ ∈ S (R3 ) | D(ρ, ρ) < ∞} where

D(f, g) = 4π R3

|k|−2 f (k)g(k)dk.

(22)

(23)

We remark that the Fourier transform of Q = γ −γ 0 in an L2 -function with support in B(0, Λ) × B(0, Λ). Hence Q(x, y) is a smooth kernel and ρQ (x) = trC4 (Q(x, x)) is a well deﬁned function. Indeed, the map γ ∈ K → ργ−γ 0 ∈ L2 (R3 ) is continuous for the topology of S2 (HΛ ). It is easy to see that the Fourier transform of ργ−γ 0 is given by the formula 1 0 (k) = trC4 (γ (24) ρ − γ 0 )(p + k/2, p − k/2) dp. γ−γ |p+k/2|≤Λ 3/2 (2π) |p−k/2|≤Λ

We also deﬁne our variational set by KC := γ ∈ K | ργ−γ 0 ∈ C .

(25)

The reduced Bogoliubov–Dirac–Fock energy reads FTred (γ) = T H(γ, γ 0) − αD(ν, ργ−γ 0 ) +

α D(ργ−γ 0 , ργ−γ 0 ) 2

(26)

and it is well deﬁned on KC by Theorem 1. In (26), ν ∈ C is an external density creating an electrostatic potential −ν ∗1/|x|. The number α > 0 is the ﬁne structure constant . The following is an easy consequence on Theorem 1: Theorem 2 (Existence of a Minimizer). Assume T > 0, α ≥ 0 and ν ∈ C. Then FTred satisﬁes ∀γ ∈ KC ,

α FTred (γ) ≥ − D(ν, ν) 2

(27)

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hence it is bounded below on KC . It has a unique minimizer γ¯ on KC . The operator γ¯ satisﬁes the self-consistent equation  1 1  γ¯ = 1 − , 2 1 + eβDγ¯ 1 + e−βDγ¯ (28)  D := D0 + α(ρ −1 0 − ν) ∗ | · | . γ ¯

γ ¯ −γ

Remark 1. When T = 0, a similar result was proved in [11, Theorem 3], but there might be no uniqueness in this case. Remark 2. If there is no external ﬁeld, ν = 0, we recover that the optimal state is γ − γ 0 = 0, and its energy is zero, by (27). Proof of Theorem 2. Equation (27) is an obvious consequence of positivity of the relative entropy H and positive deﬁniteness of D(·, ·). The existence of a minimizer is obtained by noticing that FTred is weakly lower semi-continuous for the topology of S2 (HΛ ) and C, by Theorem 1. As FTred is convexa and strictly convex with respect to ργ−γ 0 , we deduce that all the minimizers share the same density. Next we notice that ±1/2 ∈ / σ(¯ γ ) since the derivative of the relative entropy with respect to variations of an eigenvalue is inﬁnite at these two points. Hence γ¯ does not saturate the constraint and it is a solution of Eq. (28). This a fortiori proves that γ¯ is unique, since Dγ¯ depends only on the density ργ¯−γ 0 . Now we provide some interesting properties of any solution of Eq. (28), thus in particular of our minimizer γ¯ . Theorem 3 (Debye Screening). Assume T > 0, α > 0 and ν ∈ C ∩ L1 (R3 ). Any γ ∈ K that solves Eq. (28) is a trace-class perturbation of γ 0 , i.e. γ − γ 0 ∈ S1 (HΛ ). Its charge density ργ−γ 0 is an L1 (R3 ) function which satisﬁes 1 ∈ L1 (R3 ). ργ−γ 0 = ν and (ργ−γ 0 − ν) ∗ (29) |x| 3 3 R R This result implies that the particles arrange themselves such that the total eﬀective potential (ργ−γ 0 − ν) ∗ 1/|x| has a decay much faster than 1/|x|. This implies that the nuclear charge of the external sources is completely screened. The proof of Theorem 3 is lengthy and is given later in Sec. 3.1. 2. The Bogoliubov–Dirac–Fock Free Energy 2.1. Definition of the free vacuum When the exchange term is not neglected, the free vacuum is no longer described by the operator γ 0 introduced in the previous section. Instead it is another translation-invariant operator γ˜ 0 that solves a self-consistent equation. Following a It

can indeed be proved that H(·, γ 0 ) is strictly convex but we do not need that here.

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ideas from [14], we deﬁne in this section γ˜ 0 as the (unique) minimizer of the free energy per unit volume. We consider translation-invariant operators γ = γ(p) acting on HΛ and such that −1/2 ≤ γ ≤ 1/2 which is obviously equivalent to −1/2 ≤ γ(p) ≤ 1/2, for a.e. p ∈ B(0, Λ), in the sense of C4 × C4 hermitian matrices. The free energy per unit volume of such a translation-invariant operator γ at temperature T is given by [14] 1 trC4 [D0 (p)γ(p)]dp TT (γ) = (2π)3 B(0,Λ) trC4 [γ(p)γ(q)] α − dp dq − T S(γ) (30) (2π)2 |p − q|2 B(0,Λ)2 where the entropy is deﬁned as 1 1 + γ(p) ln + γ(p) S(γ) = − trC4 2 2 B(0,Λ) 1 1 + − γ(p) ln − γ(p) dp. 2 2 The free energy is deﬁned on the convex set of matrix-valued functions, such that, for all p ∈ B(0, Λ), γ(p) is a hermitian 4 × 4 matrix, i.e. A := {γ : B(0, Λ) → M 4 | γ(p)∗ = γ(p), − 1/2 ≤ γ(p) ≤ 1/2 for all p ∈ B(0, Λ)}.

(31)

Theorem 4 (The Free Vacuum at Temperature T ). For all T > 0 and all 0 ≤ α < 4/π, the free energy per unit volume TT in (30) has a unique minimizer γ˜ 0 on A. It is a solution of the self-consistent equation  1 1 1  0  γ ˜ = −   2 1 + eβDγ˜ 0 1 + e−βDγ˜ 0 (32)  γ˜ 0 (x, y)  0  . Dγ˜ 0 = D − α |x − y| Furthermore, γ˜ 0 has the form γ˜ 0 (p) = f1 (|p|)α · p + f0 (|p|)β

(33)

with f0 , f1 ≤ 0 a.e. on B(0, Λ) and Dγ˜ 0 satisﬁes |Dγ˜ 0 | ≥ |D0 |.

(34)

Here and in the following, we shall identify operators with their integral kernels for simplicity of the notation. That is, the last term in the second line of (32) γ 0 )(x,y) denotes the operator with integral kernel given by (γ−˜ , where (γ − γ˜ 0 )(x, y) |x−y| is the integral kernel of the translation-invariant operator γ − γ˜ 0 (it is a function of x − y).

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Remark 3. The assumption α < 4/π guarantees that the functional (30) is bounded from below, independently of the UV cutoﬀ Λ, which is arbitrary in this paper. This is a consequence of Kato’s inequality. For α > 4/π, this is not the case [5, 17]. For comparison, we note that in the non-interacting case α = 0, the functions f1 (|p|) and f0 (|p|) appearing in Theorem 4 are given by 1 1 1 f1 (|p|) = f0 (|p|) = − . 2E(p) 1 + eβE(p) 1 + e−βE(p) A result similar to Theorem 4 was proved in the zero temperature case in [14]. As in [14], it is also possible to justify the introduction of TT by a thermodynamic limit procedure. Namely, the free energy (3) can properly be deﬁned in a box of size L with periodic boundary conditions and an ultraviolet cutoﬀ. Next using the properties of γ˜0 , one can prove that for L large enough there is a unique minimizer in the box, which is translation invariant and converges to γ˜ 0 as L → ∞. This justiﬁes that the free HF vacuum in the whole space is described by γ˜ 0 . For shortness, we shall not write here the proof of this result which follows that of [14]. Like for the reduced case, we have that 1 1 σ(˜ γ 0 ) ⊂ − + , − ∪ , − 2 2 for some > 0. This can be seen from (34) and the fact that Dγ˜ 0 is a bounded operator on HΛ due to the presence of the ultraviolet cutoﬀ. Notice also that we have formally ργ˜ 0 ≡ 0 by (33), as in (13). The proof of Theorem 4 is given in Sec. 3.3. 2.2. The external field case As in Sec. 1.2, one can consider the Bogoliubov–Dirac–Fock energy with an external ﬁeld. It is formally obtained by subtracting the inﬁnite free energy of the free vacuum at temperature T > 0 from the free energy of our state γ. This procedure can be justiﬁed like in [14] by a thermodynamic limit procedure. Using the same notation as in Sec. 1.2, the Bogoliubov–Dirac–Fock free energy reads α FT (γ) = TH (γ, γ˜ 0 ) − αD(ν, ργ−˜γ 0 ) + D(ργ−˜γ 0 , ργ−˜γ 0 ) 2 α trC4 |(γ − γ˜ 0 )(x, y)|2 dx dy, (35) − 2 |x − y| where H is the relative entropy deﬁned like in Sec. 1.1. Like for the reduced case, we see that the functional FT is well deﬁned on the following convex set ˜ C := γ ∈ B(HΛ ) | γ ∗ = γ, − 1 ≤ γ ≤ 1 , γ − γ˜ 0 ∈ S2 (HΛ ), ργ−˜γ 0 ∈ C . K 2 2 (36)

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Note that although the function γ → H(γ, γ˜ 0 ) is convex, FT is not a convex functional because of the presence of the exchange term. This is of course a great obstacle in proving the existence of a minimizer, and we have to leave this as an open problem. Following the method of Theorem 1, we shall show that ˜C , ∀γ ∈ K

TH (γ, γ˜ 0 ) ≥ tr(|Dγ˜ 0 |(γ − γ˜ 0 )2 ).

(37)

With the aid of this inequality, we can prove the Theorem 5 (Minimizer in External Field). Assume that 0 ≤ α < 4/π and that T > 0. We have ˜C , ∀γ ∈ K

α FT (γ) ≥ − D(ν, ν) 2

(38)

˜C . and hence FT is bounded below on K ˜ C is a minimizer of FT . Then it satisﬁes the self-consistent Assume that γ ∈ K equation  1 1  γ = 1 − ,   2 1 + eβDγ 1 + e−βDγ (39)  (γ − γ˜ 0 )(x, y)  −1  Dγ := Dγ˜ 0 + α(ργ−γ 0 − ν) ∗ | · | − α |x − y| with Dγ˜ 0 deﬁned in (32). It is unique when π 0≤α 4

−1 π α/2 1/6 11/6 1/2 +π 2 1−α ≤ 1. D(ν, ν) 2 1 − απ/4

(40)

The proof of Theorem 5 is provided in Sec. 3.4. Remark 4. If there is no external ﬁeld, ν = 0, we recover that the unique minimizer ˜ C is γ = γ˜ 0 , its energy being zero by (38). This is usually referred to as of FT on K stability of the free vacuum γ˜0 (under Hilbert–Schmidt perturbations) [1,4,5,10,14]. 3. Proofs 3.1. Proof of Theorem 3 Let γ be a solution of  1 1  γ = 1 − , 2 1 + eβDγ 1 + e−βDγ  D := D0 + α(ρ −1 . γ γ−γ 0 − ν) ∗ | · |

(41)

For the sake of simplicity, we deﬁne ρ := ργ−γ 0 − ν and V = α(ργ−γ 0 − ν) ∗ | · |−1 . Note that ∇V ∈ L2 (R3 ) as ρ ∈ C, hence V ∈ L6 (R3 ). Following [11, p. 4495], we

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may use the Kato–Seiler–Simon inequality (see [20] and [21, Theorem 4.1]) ∀p ≥ 2, to obtain

V

f (−i∇)g(x)Sp (L2 (R3 )) ≤ (2π)−3/p gLp(R3 ) f Lp(R3 )

(42)

1 1 ≤ V ≤ C V L6 (R3 ) ≤ CρC . |D0 | S∞ (HΛ ) |D0 | S6 (HΛ )

This shows that |Dγ | ≤ (1 + αCρC )|D0 |. Thanks to the cutoﬀ in Fourier space, we deduce that Dγ is a bounded operator or HΛ . Recall Duhamel’s formula 1 0 0 etβDγ V e(1−t)βD dt. (43) eβDγ = eβD + β Denoting K := β

1 0

0

e

tβDγ

Ve

(1−t)βD0

dt and using (43), we have

K = K0 + K 1 0 0 := β dt etβD V e(1−t)βD + β 2 0

1

t

dt

0

0

0

ds esβDγ V e(t−s)βD V e(1−t)βD .

0

We obtain for the self-consistent solution 1 1 γ − γ0 = − 1 + eβDγ 1 + eβD0 1 1 1 1 =− K0 − K 1 + eβD0 1 + eβD0 1 + eβD0 1 + eβD0 1 1 1 + K K 1 + eβD0 1 + eβDγ 1 + eβD0 which we write as γ − γ 0 = A + B where A=−

1 1 K0 = −β 1 + eβD0 1 + eβD0

0

1

0

(44)

0

e(1−t)βD etβD dt. 0 V βD 1+e 1 + eβD0

As V ∈ L6 (R3 ) and Dγ is bounded, using the cutoﬀ in Fourier space and the Kato–Seiler–Simon inequality (42), we have K ∈ S6 (HΛ ). Hence we obtain that K ∈ S3 (HΛ ) and B ∈ S3 (HΛ ). The next step is to compute the density of A. The kernel of A is given by 1 0 0 e(1−t)βD (q) etβD (p)

q) = −β(2π)−3/2 V (p − q) A(p, dt. βD0 (p) 1 + eβD0 (q) 0 1+e Using (24), we obtain ρ A (k) = − where β C(|k|) := 2π 2

|p+k/2|≤Λ |p−k/2|≤Λ

dp 0

αC(|k|) ρ (k) |k|2

1

dt trC4

0 0 etβD (p+k/2) e(1−t)βD (p−k/2) . 1 + eβD0 (p+k/2) 1 + eβD0 (p−k/2) (45)

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Inserting this into the self-consistent Eq. (44) gives ρ (k) = − ν (k) −

αC(|k|) ρ (k) + ρ B (k) |k|2

(46)

or, equivalently, ν (k) + ρ ρ (k) = b 1 (k)(− B (k)),

(47)

V (k) = 4π b 2 (k)(− ν (k) + ρ B (k)),

(48)

and

where b1 := F −1

|k|2 2 |k| + αC(|k|)

and b2 := F −1

1 2 |k| + αC(|k|)

,

(49)

with F −1 denoting the inverse Fourier transform. Our main tool will be the following Proposition 1 (Properties of b1 , b2 ). The two functions b1 (x) and b2 (x), deﬁned in (49) and (45), belong to L1 (R3 ). We postpone the proof of Proposition 1 to Sec. 3.2 and ﬁrst complete the proof of Theorem 3. First we claim that ρB ∈ L3 (R3 ). To see this, we take a function ξ ∈ L3/2 (R3 ) ∩ C0∞ (R3 ) and compute |tr(Bξ)| = |tr(B1B(0,Λ) (p)ξ1B(0,Λ) (p))| ≤ BS3 (HΛ ) 1B(0,Λ) (p)ξ1B(0,Λ) (p)S3/2 (HΛ ) . Writing ξ = |ξ|1/2 sgn(ξ)|ξ|1/2 and using the Kato–Seiler–Simon inequality (42) twice in S3 (HΛ ), we obtain |tr(Bξ)| ≤ CBS3 (HΛ ) ξL3/2 (R3 ) where C depends on the cutoﬀ Λ. This proves by duality that ρB ∈ L3 (R3 ). Next we use a boot-strap argument. As ν ∈ L1 (R3 ) and ρB ∈ L3 (R3 ), we get from (48) and Proposition 1 that V ∈ L3 (R3 ). Inserting in the deﬁnition of K and using (42) once more, we obtain that K ∈ S3/2 (HΛ ), hence B ∈ S2 (R3 ) and ρB ∈ L2 (R3 ). Using again (48) and Proposition 1, we get that V ∈ L2 (R3 ), hence B ∈ S1 (HΛ ) and ρB ∈ L1 (R3 ). This ﬁnishes the proof of Theorem 3, by (47), (48) and Proposition 1.

3.2. Proof of Proposition 1 The proof proceeds along the same lines as in [9, Appendix]. In the following, we shall denote by P0+ and P0− the projection onto the positive and negative spectral

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subspace of D0 , respectively. As multiplication operators in momentum space, 1 1 α·p+β α·p+β + − P0 (p) = 1+ , P0 (p) = 1− . 2 E(p) 2 E(p) The function C in (45) can be written as β dp C(|k|) = 2 π |p+k/2|≤Λ |p−k/2|≤Λ

tβE(p+k/2) (1−t)βE(p−k/2) e e + + dt trC4 P (p + k/2)P (p − k/2) 0 1 + eβE(p+k/2) 1 + eβE(p−k/2) 0 0 tβE(p+k/2) −(1−t)βE(p−k/2) e e + − + trC4 P (p + k/2)P (p − k/2) . 0 1 + eβE(p+k/2) 1 + e−βE(p−k/2) 0 (50)

1

×

Hence 1 C(|k|) = 2 π

1 eβE(p+k/2) − eβE(p−k/2) 1 + eβE(p+k/2) E(p + k/2) − E(p − k/2) 1 + eβE(p−k/2) 1

|p+k/2|≤Λ |p−k/2|≤Λ

(p + k/2) · (p − k/2) + 1 × 1+ dp E(p + k/2)E(p − k/2) 1 1 eβE(p+k/2) − e−βE(p−k/2) 1 + 2 π |p+k/2|≤Λ 1 + eβE(p+k/2) E(p + k/2) + E(p − k/2) 1 + e−βE(p−k/2) |p−k/2|≤Λ

(p + k/2) · (p − k/2) + 1 × 1− dp. E(p + k/2)E(p − k/2) For the sake of clarity, we denote by C1 (|k|) (respectively, C2 (|k|)) the ﬁrst (respectively, second) integral of the previous formula. By the monotonicity of the exponential function, it is easily seen that C1 (|k|) ≥ 0 and C2 (|k|) ≥ 0. The next step is to simplify the above integral formula. We follow a method of Pauli and Rose [19] which was recently used in [9, Appendix]. After two changes of variables, we end up with ZΛ (|k|) |k| 2 z sinh(βv) 8 eβw(k,z) dz dv C1 (|k|) = π|k| 0 v 1 + eβ(w(k,z)+v) 0 z w(k, z)−1 1 + eβ(w(k,z)−v) (1 − z 2 )3 |k| z 2 ZΛ (|k|) 1 sinh(βv) 8 eβ(E(Λ)−z) + dz dv β(E(Λ)−z+v) β(E(Λ)−z−v) π|k| 0 v 1 + e 1 + e 0 2 |k| × (E(Λ) − z)2 − , (51) 4 ×

1

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8 C2 (|k|) = π|k|

ZΛ (|k|)

dz 0

|k| 2 z

dv

0

eβv 1+

eβ(w(k,z)+v)

sinh(βw) 1+

|k|2 4 (1

− z 2)

|k|2 z − v2 4 1 − z2 1 + eβ(v−w(k,z)) z |k| 2 ZΛ (|k|) sinh(β(E(Λ) − z)) 8 eβv + dz dv β(E(Λ)−z+v) π|k| 0 E(Λ) − z 1 + e 0 2 1 |k| × − v2 . β(v+z−E(Λ)) 4 1+e ×

1

(52)

In the above formulas we have used the notation (as in [9]) √ 1 + Λ2 − 1 + (Λ − r)2 . ZΛ (r) = r Note that ZΛ is a decreasing C ∞ function on [0, 2Λ] satisfying ZΛ (0) = Λ/E(Λ), ZΛ (2Λ) = 0. We have also used the shorthand notation 1 + |k|2 (1 − z 2 )/4 . 1 − z2

w(k, z) =

All integrands of the above formulas are real analytic functions of r = |k| on a neighborhood of [0, 2Λ]. Also all the integrals vanish at k = 0. We deduce that C1 and C2 are smooth functions on [0, 2Λ]. Using ZΛ (2Λ) = 0, one also sees that C1 (2Λ) = C1 (2Λ) = C2 (2Λ) = C2 (2Λ) = 0. A Taylor expansion of the ﬁrst integral of C1 yields E(Λ) 4 t2 dt > 0. C1 (0) = β −βt π (1 + e )(1 + eβt ) 1 The end of the proof of Proposition 1 is then the same as in [9, Proposition 17]. First, we notice that as C(r) is bounded and has a compact support, b1 and b2 are in L∞ (R3 ). We now prove that they decay at least like |x|−4 at inﬁnity meaning that they also belong to L1 (R3 ). To this end we write for b = b1 or = b2 the inverse Fourier transform in radial coordinates: 2Λ 1 (r b(r)) sin(r|x|)dr. (53) ∀x ∈ R3 \{0}, b(x) = √ 2π|x| 0 Integrating by parts and using b(2Λ) = b (2Λ) = 0 yields 1 3 2Λ b (2Λ) cos(2Λ|x|) − 2 b (0) ∀x ∈ R \{0}, b(x) = √ 2π|x|4 2Λ (r b)(3) (r) cos(r|x|)dr . − 0

This completes the proof of Proposition 1.

(54)

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3.3. Proof of Theorem 4 The proof is inspired by ideas from [14]. We denote I := inf γ∈A TT (γ). We start by introducing the following auxiliary minimization problem J = inf TT (γ)

(55)

B := {γ ∈ A, γ(p) = f1 (|p|)α · p + f0 (|p|)β, f0 , f1 ≤ 0} .

(56)

γ∈B

where B ⊂ A is given by

Lemma 2. There exists a minimizer γ˜ 0 ∈ B for (55). Proof. The functional TT is weakly lower semi-continuous for the weak-∗ topology of L∞ (B(0, Λ)). This is because −S is convex and the exchange term is continuous for the weak topology of L2 (B(0, Λ)) as shown in [14]. Also B is a bounded closed convex subset of L∞ (B(0, Λ)). Hence there exists a minimum. Lemma 3. Let γ˜ 0 ∈ B be a minimizer of (55). Then there exists an > 0 such that |˜ γ 0 | ≤ 1/2 − . Proof. For x ∈ [1/2, 1/2], 1 1 1 1 s(x) := + x ln +x + − x ln −x 2 2 2 2 is an even function of x. Because of the special form of γ˜ 0 , we have γ˜ 0 (p)2 = ˜ γ 0 (p)2 IC4 for all p ∈ B(0, Λ), where · denotes the matrix norm. Hence s(γ(p))dp. ∀γ ∈ B, S(γ) = −4 B(0,Λ)

The derivative of s is inﬁnite at x = 1/2 and the derivative of the terms of the ﬁrst line of (30) stays bounded. It is therefore clear that {p ∈ B(0, Λ) | 1/2 − ≤ ˜ γ 0 (p) ≤ 1/2} has zero measure for small enough. Let us now write the ﬁrst order condition satisﬁed by γ˜ 0 . Since ˜ γ 0 (p) ≤ 1/2− for some small enough, we can consider a perturbation of the form γ(p) = γ˜ 0 (p) + t (g1 (|p|)α · p + g0 (|p|)β) with g0 , g1 ≤ 0 and t > 0 small enough. We obtain 1/2 + γ˜ 0 (p) 4 0 g1 (|p|)α · p + g0 (|p|)β dp ≥ 0 trC Dγ˜ (p) + T ln 1/2 − γ˜ 0 (p) B(0,Λ) (57) for all g1 , g0 ≤ 0.

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1/2+x is odd, hence We notice that the function x → ln 1/2−x 1/2 + γ(p) 1/2 + γ(p) ∀γ ∈ B, ln = sgn(γ) ln , 1/2 − γ(p) 1/2 − γ(p)

(58)

with sgn(γ) = γ/|γ|. We obtain that 1/2 + γ˜ 0 (p) γ 0 (p)) ln = γ˜ 0 (p)F (˜ 1/2 − γ˜ 0 (p) 1/2+x where F (x) = ln 1/2−x /x. On the other hand, we can write Dγ˜ 0 = d1 (|p|)α · p + d0 (|p|)β where d1 and d0 are given by [14, Eqs. (72) and (73)]. Using f1 , f0 ≤ 0, we immediately see that d1 (|p|) ≥ 1 and d0 (|p|) ≥ 1,

(59)

which in particular proves that Dγ˜ 0 (p) ≥ D0 (p) ≥ |p|.

(60)

All this gives Dγ˜ 0 + T ln

1/2 + γ˜ 0 = (d1 (|p|) + T f1 (|p|)F (˜ γ 0 (p)))α · p 1/2 − γ˜ 0 γ 0 (p)))β. + (d0 (|p|) + T f0(|p|)F (˜

Inserting this in (57), we obtain the ﬁrst order conditions d1 (|p|) + T f1 (|p|)F (˜ γ 0 (p)) ≤ 0, γ 0 (p)) ≤ 0. d0 (|p|) + T f0 (|p|)F (˜ In particular, because of (59) we infer that γ 0 (p)) ≤ −1/T, f1 (|p|)F (˜ γ 0 (p)) ≤ −1/T. f0 (|p|)F (˜

(61)

(62)

(63)

As F (˜ γ 0 (p)) ≥ 0 and f0 ≥ −˜ γ 0 (p), we obtain from (63) the inequality |γ(p)|F (γ(p)) ≥ 1/T . Hence γ(p) ≥

e1/T − 1 . 2(1 + e1/T )

(64)

This inequality means that f0 and f1 cannot vanish simultaneously. But we can indeed prove that each of them cannot vanish, as expressed in: Lemma 4. Let γ˜ 0 (p) = f1 (|p|)α · p + f0 (|p|)β be a minimizer of (55). Then there exists an > 0 such that f0 ≤ −

and

f1 ≤ −.

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Proof. By Lemma 3 we know that ˜ γ 0 (p) ≤ 1/2 − for some > 0. By (63) and 1 for k = 0, 1. the monotonicity of F we obtain fk ≤ − T F (1/2−) Lemma 5. Let γ˜ 0 ∈ B be a minimizer of (55). Then it solves the self-consistent equation 1 1 1 0 − . (65) γ˜ = 2 1 + eβDγ˜ 0 1 + e−βDγ˜ 0 Proof. As the constraints are not saturated by Lemmas 3 and 4, we obtain that the derivative vanishes, i.e. γ 0 (p)) = 0, d1 (|p|) + T f1 (|p|)F (˜ (66) d0 (|p|) + T f0 (|p|)F (˜ γ 0 (p)) = 0, which means that Dγ˜ 0 + T ln

1/2 + γ˜ 0 = 0. 1/2 − γ˜ 0

Hence γ˜ 0 solves (65). Now we prove that the operator γ˜ 0 deﬁned in the previous step is the unique minimizer of TT on the full space A deﬁned in (31), not merely on the subset B in (56). We have γ0) TT (γ) − TT (˜ = TH (γ, γ˜ 0 ) −

α (2π)5

B(0,Λ)2

trC4 [(γ − γ˜ 0 )(p)(γ − γ˜ 0 )(q)] dp dq |p − q|2

where H is the relative entropy per unit volume 1 1 1 +γ ln + γ − ln + γ˜ 0 trC4 H(γ, γ˜ 0 ) = (2π)−3 2 2 2 B(0,Λ) 1 1 1 + −γ ln − γ − ln − γ˜ 0 dp. (67) 2 2 2 We shall use the important Lemma 6. For H in (67) the inequality trC4 |Dγ˜ 0 (p)|(γ(p) − γ˜ 0 (p))2 dp TH (γ, γ˜ 0 ) ≥ (2π)−3

(68)

B(0,Λ)

holds for all γ ∈ A. Proof. This is a simple application of (21), taking X = γ(p), Y = γ˜ 0 (p) and integrating over the ball B(0, Λ).

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Using Lemma 6 and the formula α (2π)2

B(0,Λ)2

trC4 [γ(p)γ(q)] α dp dq = 2 |p − q| 2

R3

trC4 |ˇ γ (x)|2 dx |x|

where γˇ(x) is the Fourier inverse of the function γ(p), we ﬁnd −3

TT (γ) − TT (˜ γ ) ≥ (2π) 0

−

α 2

R3

trC4 |Dγ˜ 0 (p)|(γ(p) − γ˜ 0 (p))2 dp

B(0,Λ)

trC4 |(ˇ γ − γˇ 0 )(x)|2 dx |x|

(69)

for all γ ∈ A. We now use ideas of [1,10,14]. Kato’s inequality |x|−1 ≤ π/2|∇| gives α 2

R3

απ trC4 |(ˇ γ − γˇ 0 )(x)|2 dx ≤ |x| 4

B(0,Λ)

trC4 |p|(γ(p) − γ˜ 0 (p))2 dp.

By (60), we deduce TT (γ) − TT (˜ γ 0 ) ≥ (1 − πα/4)(2π)−3

B(0,Λ)

trC4 |Dγ˜ 0 (p)|(γ(p) − γ˜ 0 (p))2 dp.

Hence γ˜ 0 is the unique minimizer of TT on A when 0 ≤ α < 4/π. This completes the proof of Theorem 4.

3.4. Proof of Theorem 5 The lower bound (38) is obtained by following an argument of [1, 10]. By (37) we ˜C have for all γ ∈ K FT (γ) ≥ tr(|Dγ˜ 0 |(γ − γ˜ 0 )2 ) − +

α 2

α D(ργ−˜γ 0 − ν, ργ−˜γ 0 2

trC4 |(γ − γ˜ 0 )(x, y)|2 dx dy |x − y| α − ν) − D(ν, ν). 2

(70)

By (34) together with Kato’s inequality |x|−1 ≤ (π/2)|∇| ≤ (π/2)|D0 |,

π trC4 |(γ − γ˜ 0 )(x, y)|2 dx dy ≤ tr(|Dγ˜ 0 |(γ − γ˜ 0 )2 ) |x − y| 2

which yields (38) when 0 ≤ α < 4/π. ˜ C . The proof that it satisﬁes Assume now that γ is a minimizer of FT on K the self-consistent equation (39) is the same as in the case of the reduced BDF

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functional in Sec. 1.2. Note that because of (70) and inf K˜ C FT ≤ 0, we obtain 2 α α α trC4 |(γ − γ˜ 0 )(x, y)|2 − dx dy + D(ργ−˜γ 0 , ργ−˜γ 0 ) ≤ D(ν, ν). π 2 |x − y| 2 2 It was proved in [11, p. 4495] that this implies (under the condition (40)) that |Dγ | ≥ d−1 |D0 | with

d=

(71)

−1 α/2 π + π 1/6 211/6 D(ν, ν)1/2 1−α . 2 1 − απ/4

˜ C and use that Next we ﬁx some γ ∈ K H(γ , γ˜ 0 ) = H(γ, γ˜ 0 ) + H(γ , γ)   

   1 1 +γ + γ˜ 0       + tr(γ − γ) ln  2  .  − ln 21 1 0 −γ − γ˜ 2 2

(72)

˜C Inserting Eq. (39) for our minimizer γ and Eq. (32) for γ˜ 0 , we obtain for any γ ∈ K the formula α FT (γ ) = FT (γ) + TH (γ , γ) + D(ργ −γ , ργ −γ ) 2 α trC4 |(γ − γ˜ 0 )(x, y)|2 dx dy. (73) − 2 |x − y| We may use one more time (16) and the self-consistent equation (39) to obtain TH (γ , γ) ≥ tr(|Dγ |(γ − γ)2 ). By (71) and Kato’s inequality as before, we eventually get α FT (γ ) ≥ FT (γ) + D(ργ −γ , ργ −γ ) 2 2 α trC4 |(γ − γ˜ 0 )(x, y)|2 − dx dy. + πd 2 |x − y| Hence we obtain that any minimizer is unique when απd/4 ≤ 1, as stated. Let us remark that the expression in last term of (72) is indeed a trace-class operator. It would, however, have been suﬃcient to choose γ as trace class perturbation of γ 0 and conclude the rest by a density argument. Acknowledgments M. L. acknowledges support from the ANR project “ACCQUAREL” of the French ministry of research. R. S. was partially supported by US NSF grant PHY-0652356 and by an A. P. Sloan fellowship. The authors are thankful to the Erwin Schr¨ odinger Institute in Vienna, where this work was started.

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Appendix. Integral Representation of Relative Entropy Here we shall prove the claim made in Sec. 1.1 that (15) and (17) coincide as long as γ − γ 0 ∈ S1 (HΛ ) and the spectrum of γ 0 does not contain ±1/2. From the integral representation (16), we have ∞ a 1 (a − b) dt. a(ln a − ln b) = a + t b + t 0 We split the ﬁrst factor as a b t 1 = + (a − b) a+t b+t b+t a+t and obtain

a(ln a − ln b) = 0

∞

+ 0

1 1 t (a − b) (a − b) dt b+t a+t b+t ∞

1 b (a − b) dt. b+t b+t

Now if a − b is trace class and the spectrum of b is contained in (0, ∞), then ∞ ∞ 1 b b tr (a − b) dt = tr (a − b) dt = tr(a − b). b+t b+t (b + t)2 0 0 If we apply this reasoning to a = 1/2 + γ, b = 1/2 + γ 0 and to a = 1/2 − γ and b = 1/2 − γ 0 , respectively, we thus obtain  ∞ t 1 1  0 H(γ, γ ) = tr (γ − γ 0 ) (γ − γ 0 ) dt 1 1 1 0 +γ+t + γ0 + t + γ0 + t 2 2 2  ∞ t 1 1  (γ − γ 0 ) + (γ − γ 0 ) dt . 1 1 1 0 0 0 −γ +t −γ +t −γ +t 2 2 2 Changing variables from t to u = 1/(1 + 2t) in the ﬁrst integral and u = −1/(1 + 2t) in the second integral, respectively, we arrive at the integral representation (17). References [1] V. Bach, J.-M. Barbaroux, B. Helﬀer and H. Siedentop, On the stability of the relativistic electron-positron ﬁeld, Comm. Math. Phys. 201 (1999) 445–460. [2] J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields (McGraw-Hill, New YorkToronto-London-Sydney, 1965). [3] V. Bach, E. H. Lieb and J.-P. Solovej, Generalized Hartree–Fock Theory and the Hubbard model, J. Stat. Phys. 76(1–2) (1994) 3–89. [4] P. Chaix and D. Iracane, From quantum electrodynamics to mean ﬁeld theory: I. The Bogoliubov–Dirac–Fock formalism, J. Phys. B 22 (1989) 3791–3814.

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[5] P. Chaix, D. Iracane and P. L. Lions, From quantum electrodynamics to mean-ﬁeld theory. II. Variational stability of the vacuum of quantum electrodynamics in the mean-ﬁeld approximation, J. Phys. B 22 (1989) 3815–3828. [6] P. A. M. Dirac, Th´eorie du Positron, Solvay Report, Vol. 25 (Gauthier-Villars, 1934), pp. 203–212. [7] P. A. M. Dirac, Discussion of the inﬁnite distribution of electrons in the theory of the positron, Proc. Camb. Philos. Soc. 30 (1934) 150–163. [8] F. J. Dyson, The radiation theories of Tomonaga, Schwinger, and Feynman, Phys. Rev. 75(3) (1949) 486–502. ´ S´er´e, Ground state and charge renormalization in a [9] P. Gravejat, M. Lewin and E. nonlinear model of relativistic atoms, to appear in Comm. Math. Phys. ´ S´er´e, Existence of a stable polarized vacuum in the [10] C. Hainzl, M. Lewin and E. Bogoliubov–Dirac–Fock approximation, Comm. Math. Phys. 257(3) (2005) 515–562. ´ S´er´e, Self-consistent solution for the polarized vacuum [11] C. Hainzl, M. Lewin and E. in a no-photon QED model, J. Phys. A 38 (2005) 4483–4499. ´ S´er´e, Existence of atoms and molecules in the mean-ﬁeld [12] C. Hainzl, M. Lewin and E. approximation of no-photon quantum electrodynamics, to appear in Arch. Ration. Mech. Anal.; doi: 10.1007/s00205-008-0144-2. [13] C. Hainzl, M. Lewin, E. S´er´e and J. P. Solovej, A minimization method for relativistic electrons in a mean-ﬁeld approximation of quantum electrodynamics, Phys. Rev. A 76 (2007) 052104. [14] C. Hainzl, M. Lewin and J. P. Solovej, The mean-ﬁeld approximation in Quantum Electrodynamics. The no-photon case, Comm. Pure Appl. Math. 60(4) (2007) 546– 596. [15] C. Hainzl, M. Lewin and C. Sparber, Existence of global-in-time solutions to a generalized Dirac–Fock type evolution equation, Lett. Math. Phys. 72 (2005) 99–113. [16] W. Heisenberg, Bemerkungen zur Diracschen Theorie des Positrons, Z. Phys. 90 (1934) 209–223. [17] D. Hundertmark, N. R¨ ohrl and H. Siedentop, The sharp bound on the stability of the relativistic electron-positron ﬁeld in Hartree–Fock approximation, Comm. Math. Phys. 211 (2000) 629–642. [18] W. Pauli, Relativistic ﬁeld theories of elementary particles, Rev. Mod. Phys. 13 (1941) 203–232. [19] W. Pauli and M. E. Rose, Remarks on the polarization eﬀects in the positron theory, Phys. Rev. II 49 (1936) 462–465. [20] E. Seiler and B. Simon, Bounds in the Yukawa quantum ﬁeld theory: Upper bound on the pressure, Hamiltonian bound and linear lower bound, Comm. Math. Phys. 45 (1975) 99–114. [21] B. Simon, Trace Ideals and Their Applications, London Mathematical Society Lecture Notes Series, Vol. 35 (Cambridge University Press, 1979). [22] J. Schwinger, Quantum electrodynamics I. A covariant formulation, Phys. Rev. 74(10) (1948) 1439–1461. [23] W. Thirring, Quantum Mathematical Physics. Atoms, Molecules and Large Systems, 2nd edn. (Springer-Verlag, Berlin, 2002). [24] A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50(2) (1978) 221–260.

November 7, 2008 11:44 WSPC/148-RMP J070-00356

Reviews in Mathematical Physics Vol. 20, No. 10 (2008) 1309–1311 c World Scientiﬁc Publishing Company

REVIEWS IN MATHEMATICAL PHYSICS Author Index Volume 20 (2008)

Arai, A. & Matsuzawa, Y., Time operators of a Hamiltonian with purely discrete spectrum Besbes, A., Uniform ergodic theorems on aperiodic linearly repetitive tilings and applications Boumaza, H., H¨ older continuity of the integrated density of states for matrix-valued Anderson models Brennecke, F. & D¨ utsch, M., Removal of violations of the Master Ward Identity in perturbative QFT Br¨ uning, J., Geyler, V. & Pankrashkin, K., Spectra of self-adjoint extensions and applications to solvable Schr¨ odinger operators Costin, O., Lebowitz, J. L. & Stucchio, C., Ionization in a 1-dimensional dipole model D¸ abrowski, L., see D’Andrea, F. D’Andrea, F., D¸ abrowski, L. & Landi, G.,

The noncommutative geometry of the quantum projective plane Dappiaggi, C., On the Lagrangian and Hamiltonian formulation of a scalar free ﬁeld theory at null inﬁnity Ding, Y. & Wei, J., Stationary states of nonlinear Dirac equations with general potentials Dobrev, V. K., Invariant diﬀerential operators for non-compact Lie groups: Parabolic subalgebras Duclos, P., Soccorsi, E., ˇˇtov´ıˇ S cek, P. & Vittot, M., On the stability of periodically time-dependent quantum systems D¨ utsch, M., see Brennecke, F. Feldman, J. & Salmhofer, M., Singular Fermi surfaces I. General power counting and higher dimensional cases Feldman, J. & Salmhofer, M., Singular Fermi surfaces II. The

8 (2008) 951

5 (2008) 597

7 (2008) 873

2 (2008) 119

1 (2008) 1

7 (2008) 835 8 (2008) 979

1309

8 (2008) 979

7 (2008) 801

8 (2008) 1007

4 (2008) 407

6 (2008) 725 2 (2008) 119

3 (2008) 233

November 7, 2008 11:44 WSPC/148-RMP J070-00356

1310

Author Index

two-dimensional case Geyler, V., see Br¨ uning, J. Guha, P., Geodesic ﬂow on extended Bott–Virasoro group and generalized two-component peakon type dual systems Hainzl, C., Lewin, M. & Seiringer, R., A nonlinear model for relativistic electrons at positive temperature Hasler, D. & Herbst, I., On the self-adjointness and domain of Pauli–Fierz type Hamiltonians Herbst, I., see Hasler, D. Hiai, F., Mosonyi, M., Ohno, H. & Petz, D., Free energy density for mean ﬁeld perturbation of states of a one-dimensional spin chain Hislop, P. D. & Soccorsi, E., Edge currents for quantum Hall systems, I. One-edge, unbounded geometries Hollands, S., Renormalized quantum Yang–Mills ﬁelds in curved spacetime Isobe, T., On a minimizing property of the Hopf soliton in the Faddeev–Skyrme model Kachmar, A., Weyl asymptotics for magnetic Schr¨ odinger operators and de Gennes’ boundary condition Kargol, A., Kondratiev, Y. & Kozitsky, Y.,

3 (2008) 275 1 (2008) 1

10 (2008) 1191

10 (2008) 1283

7 (2008) 787 7 (2008) 787

3 (2008) 335

1 (2008) 71

9 (2008) 1033

7 (2008) 765

8 (2008) 901

Phase transitions and quantum stabilization in quantum anharmonic crystals Keyl, M., Matsui, T., Schlingemann, D. & Werner, R. F., On Haag duality for pure states of quantum spin chains Kondratiev, Y., Minlos, R. & Zhizhina, E., Self-organizing birth-and-death stochastic systems in continuum Kondratiev, Y., see Kargol, A. Kozitsky, Y., see Kargol, A. Kuniba, A. & Sakamoto, R., Combinatorial Bethe ansatz and generalized periodic box-ball system Landi, G., see D’Andrea, F. Landsman, N. P., Macroscopic observables and the Born rule, I. Long run frequencies Lebowitz, J. L., see Costin, O. Lewin, M., see Hainzl, C. Linhares, C. A., Malbouisson, A. P. C. & Roditi, I., Existence of asymptotic expansions in noncommutative quantum ﬁeld theories Lled´ o, F. & Post, O., Existence of spectral gaps, covering manifolds and residually ﬁnite groups Malbouisson, A. P. C., see Linhares, C. A. Matsui, T., see Keyl, M. Matsuzawa, Y., see Arai, A.

5 (2008) 529

6 (2008) 707

4 (2008) 451 5 (2008) 529 5 (2008) 529

5 (2008) 493 8 (2008) 979

10 (2008) 1173 7 (2008) 835 10 (2008) 1283

8 (2008) 933

2 (2008) 199 8 (2008) 933 6 (2008) 707 8 (2008) 951

November 7, 2008 11:44 WSPC/148-RMP

J070-00356

Author Index Minlos, R., see Kondratiev, Y. Molev, A. I. & Ragoucy, E., Symmetries and invariants of twisted quantum algebras and associated Poisson algebras Morosi, C. & Pizzocchero, L., On approximate solutions of semilinear evolution equations II. Generalizations, and applications to Navier–Stokes equations Mosonyi, M., see Hiai, F. Nirov, Kh. S. & Razumov, A. V., Abelian Toda solitons revisited Ohno, H., see Hiai, F. Pankrashkin, K., see Br¨ uning, J. Petz, D., see Hiai, F. Pizzocchero, L., see Morosi, C. Post, O., see Lled´ o, F. Ragoucy, E., see Molev, A. I. Razumov, A. V., see Nirov, Kh. S. Roditi, I., see Linhares, C. A.

4 (2008) 451

2 (2008) 173

6 (2008) 625 3 (2008) 335

10 (2008) 1209 3 (2008) 335 1 (2008) 1 3 (2008) 335 6 (2008) 625 2 (2008) 199 2 (2008) 173 10 (2008) 1209 8 (2008) 933

Sakamoto, R., see Kuniba, A. Salmhofer, M., see Feldman, J. Salmhofer, M., see Feldman, J. Schlingemann, D., see Keyl, M. Seiringer, R., see Hainzl, C. Soccorsi, E., see Duclos, P. Soccorsi, E., see Hislop, P. D. ˇˇtov´ıˇ S cek, P., see Duclos, P. Stucchio, C., see Costin, O. Tamura, H. Semiclassical analysis for spectral shift functions in magnetic scattering by two solenoidal ﬁelds Vittot, M., see Duclos, P. Wei, J., see Ding, Y. Werner, R. F., see Keyl, M. Zenk, H., Ionization by quantized electromagnetic ﬁelds: The photoelectric eﬀect Zhizhina, E., see Kondratiev, Y.

1311

5 (2008) 493 3 (2008) 275 3 (2008) 233 6 (2008) 707 10 (2008) 1283 6 (2008) 725 1 (2008) 71 6 (2008) 725 7 (2008) 835

10 (2008) 1249 6 (2008) 725 8 (2008) 1007 6 (2008) 707

4 (2008) 367 4 (2008) 451

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