Commun. Math. Phys. 273, 1–36 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0230-6
Communications in
Mathematical Physics
The Operator Product Expansion for Perturbative Quantum Field Theory in Curved Spacetime Stefan Hollands Inst. f. Theor. Physik, Georg-August-Universität, D-37077 Göttingen, Germany. E-mail:
[email protected] Received: 11 May 2006 / Accepted: 26 June 2006 Published online: 28 April 2007 – © Springer-Verlag 2007
Abstract: We present an algorithm for constructing the Wilson operator product expansion (OPE) for perturbative interacting quantum field theory in general Lorentzian curved spacetimes, to arbitrary orders in perturbation theory. The remainder in this expansion is shown to go to zero at short distances in the sense of expectation values in arbitrary Hadamard states. We also establish a number of general properties of the OPE coefficients: (a) they only depend (locally and covariantly) upon the spacetime metric and coupling constants, (b) they satisfy an associativity property, (c) they satisfy a renormalization group equation, (d) they satisfy a certain microlocal wave front set condition, (e) they possess a “scaling expansion”. The latter means that each OPE coefficient can be written as a sum of terms, each of which is the product of a curvature polynomial at a spacetime point, times a Lorentz invariant Minkowski distribution in the tangent space of that point. The algorithm is illustrated in an example. 1. Introduction The operator product expansion [37] (OPE, for short) states that a product of n local quantum fields can be expanded at short distances as an asymptotic series, each term of which is given by a model dependent coefficient function of the n spacetime arguments, times a local field at a nearby reference point1 y: Ci1 i2 ...in k (x1 , x2 , . . . , xn ) Ok (y). (1) Oi1 (x1 )Oi2 (x2 ) · · · Oin (xn ) ∼ k
This expansion has been established in perturbative quantum field theory on Minkowski spacetime [39], and is by now a standard tool, for example in the analysis of quantum gauge theories such as QCD. It has also been proven for conformally invariant quantum field theories [35, 31, 32], and has in fact played a major role in the development and 1 In this paper, we will take y = x , but other more symmetric choices are also possible. n
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analysis of such theories [6, 27]. Formal mathematical proofs of the OPE have also been given within various axiomatic settings [15, 2] for quantum field theory on Minkowski spacetime. Given the importance of the OPE in flat spacetime, it is of great interest to construct a corresponding version of the expansion in curved spacetime. In this paper, we present such a construction, within the framework of perturbation theory. It is based on the perturbative construction of quantum field theories on globally hyperbolic Lorentzian spacetimes which was recently achieved in a series of papers [20– 23], which in turn were based upon key results of Brunetti, Fredenhagen, and Köhler [4, 3]. In these papers, the interacting (Heisenberg) quantum fields in curved spacetime are constructed as formal power series in the coupling constant(s), that are valued in a certain *-algebra of quantum observables. The basic idea how to construct the OPE for these interacting fields is as follows: Suppose we have linear functionals yk from the algebra into the complex numbers, that are labelled by an index k enumerating the different composite fields, and a reference spacetime point y, and which form a “dual basis” to the composite quantum fields in the sense that yk (O j (y)) = δ k j . Now apply the functional with label k to the OPE. Then we immediately see that the operator product coefficient in front of the k th term in the sum on the right side of the OPE ought to be given precisely by the c-number distribution obtained by applying that functional to the product of fields on the left side of the OPE. Below, we will give a perturbative construction of such a dual basis of functionals in the context of a scalar, renormalizable field theory model in any 4 dimensional Lorentzian spacetime. In this way, we will obtain the desired perturbative formula for the OPE coefficients, C. While this construction gives a conceptually clean derivation of a perturbative expression for the coefficients C, it does not yet show that the remainder of the OPE expansion defined in this way actually goes to zero (and in what sense) when the points are scaled together. To analyze this question, we apply any Hadamard state to the remainder in the OPE. We then show that the resulting c-number distribution in n points goes to zero in the sense of distributions when the points are scaled together in an arbitrary fashion. Thus, the OPE holds in the sense of an asymptotic expansion of expectation values, to arbitrary order in perturbation theory, and for any Hadamard state. The proof of this statement mainly relies on the known scaling properties of the various terms in the perturbative series for the interacting fields [20, 21, 4]. However, these properties by themselves are not sufficient, for the following reason. The perturbative formulae for the interacting fields at k th order involve an integration over k “interaction points” in the domain of the interaction (which might be the entire spacetime). It turns out that we can control the contributions from these integrations in the OPE if we split the interaction domain into a region “close” to the points x1 , . . . , xn , and a region “far away.” However, the points x1 , . . . , xn themselves are supposed to be scaled, i.e., they move, so the split of the integration domain has to be constantly adapted. We achieve this by dividing the interaction domain into slices of thickness 2− j centered about y, where j = 1, 2, 3, . . . . We find that the contributions from these slices can be controlled individually, and then be summed, if the interaction is renormalizable. Finally, we will derive the following important general properties about the OPE coefficients C. (a) They have a local and covariant dependence upon the spacetime metric, (b) they satisfy an associativity property, (c) they satisfy a renormalization group equation, (d) they satisfy a certain microlocal condition on their so-called wave front set, and (e) they can be expanded in a “scaling expansion”. Let us explain these properties. The local covariance property (a) of the OPE coefficients states that if (M, g) and (M , g ) are globally hyperbolic spacetimes with corresponding OPE coefficients C,
OPE for Perturbative Quantum Field Theory in Curved Spacetime
3
resp. C , and if f : M → M is a causality and orientation preserving isometric embedding, then f ∗ C is equivalent to C at short distances. Thus, in this sense, the OPE coefficients are local functionals of the metric (and the coupling constants), and in particular do not depend upon the large scale structure of spacetime, such as the topology of M. This property can be understood from the fact that, as shown in [22, 4], the interacting fields may be constructed in a local and covariant fashion. More precisely, whenever we have a causality preserving isometric embedding f , there exists a linear map α f from the quantum field algebra associated with (M, g) into the algebra associated with (M , g ) preserving the algebraic relations. Furthermore, the fields are local and covariant in the sense [20, 5] that the image of an interacting field on (M, g) via α f corresponds precisely to the definition of that field on (M , g ). Since the OPE may be interpreted as an asymptotic algebraic relation, and since the local and covariance property as just stated means that algebraic relations only depend on the metric locally and transform covariantly under spacetime embedding, it is natural to expect that also the OPE coefficients depend locally and covariantly upon the metric (and the couplings). This is indeed what we shall prove. The associativity property (b) arises when one studies the different ways in which a configuration of n points can approach the diagonal in M n . For example, in the case n = 3, we may consider a situation in which all three points approach each other at the same rate, or we may alternatively consider a situation in which two points approach each other faster than the third one. The possible ways in which n points can approach each other may be described by corresponding merger trees T which characterize the subsequent mergers [17, 1]. The associativity property states that scaling together the points in an operator product according to a given tree is equivalent to performing subsequently the OPE in the hierarchical order represented by the tree T . We will argue—relying mainly on a general theorem of [24]—that this type of “short distance factorization property” indeed holds in perturbation theory. The associativity may again be understood intuitively from the fact that the OPE coefficients are in some sense the structure “constants” of the abstract associative *-algebra of which the interacting fields are elements. The renormalization property (c) of the OPE coefficients arises from the fact that the algebras of quantum fields satisfy a similar property [22]. Namely, if we rescale the metric by a constant conformal factor λ2 , then this is equivalent (in the sense of giving rise to isomorphic algebras) to redefining the field generators in a particular way, and at the same time letting the coupling constants of the interaction (which enter the structure of the algebra) flow in a particular way dictated by the “renormalization group flow” of the theory [22]. Again, since the OPE coefficients are in a sense the structure constants of the algebra, they can be expected to have a corresponding property. The microlocal property, (d), is a property describing the nature of the singularities in an OPE coefficient C. It states that the wave front set [26] WF(C) has a characteristic form that encodes the positivity of energy momentum in the tangent space. Our condition found for WF(C) is similar in nature to the so-called “microlocal spectrum condition” [3, 34] for the wave front set of correlation functions of linear field theories in curved space. However, our condition differs qualitatively from that proposed in [3, 34] in that the interactions may affect the form of the wave front set in our case. The scaling expansion (e) states that, if we scale n points x1 , x2 , . . . , xn in M together according to a merger tree, then a given OPE coefficient C can be approximated to any desired precision by a finite sum of terms each of which has the form of a polynomial in the mass and curvature tensors at point y = xn , times a Lorentz invariant Minkowski
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distribution in the Riemannian normal coordinates of x1 , . . . , xn−1 relative to y = xn . These Minkowski distributions have, to each order in perturbation theory, a simple homogeneous scaling behavior modified by polynomials in the logarithm. They may be extracted from the given OPE coefficients by taking certain “Mellin moments”, which is an operation defined by first taking the Mellin-transform of a function and then extracting certain residue. The general properties just described (except (e)) are postulated axiomatically in the forthcoming paper [24], and so our present analysis may be viewed as a confirmation of these axioms in perturbation theory. Our plan for this paper is as follows. In Sect. 2, we first recall the general strategy for obtaining the perturbation series for the interacting fields [20, 21, 4]. The OPE is derived in Sect. 3, and its general properties are derived in Sect. 4. An example illustrating our algorithm for computing the OPE coefficients is presented in Sect. 5. As indicated, for simplicity and concreteness, we only consider a single hermitian, scalar field with renormalizable interaction (in 4 spacetime dimensions). While the restriction to a renormalizable interaction seems to be essential, we expect that our algorithm will work for other types of fields with higher spin and renormalizable interaction, in other dimensions. However, the analysis of the OPE in the physically interesting case of Yang-Mills theories would first require an understanding of the renormalization of such theories in curved spacetime, which is considerably complicated by the issue of gauge invariance. This is at present an open problem. 2. Perturbation Theory A single hermitian scalar field φ in 4 dimensions is described classically by the action S= g µν ∇µ φ∇ν φ + (m 2 + α R)φ 2 + 2 (2) κi Oi dµ. M
√
i
Here dµ = ∈ R, the quantities κi ∈ R are coupling parameters parametrizing the strength of the self-interaction of the field, and throughout this paper Oi are polynomials in the field φ and its covariant derivatives, as well as possibly the Riemann tensor and its derivatives. In the above action, they encode the nature of the self-interaction. Later, we will assume that the interaction is renormalizable, but for the moment no such assumption need to be made. The perturbative construction of the quantum field theory associated with this action has been performed in a series of papers [20, 21, 23, 4]. These constructions consist of the following steps. First, one defines an abstract *-algebra [12, 20] F(M, g) containing the quantized field φ, together with its Wick powers Oi , for the corresponding linear theory, which classically corresponds to dropping the self-interaction term κi Oi dµ = L dµ, (3) I = g dx0
∧ · · · ∧ d x 3,
m2, α
M
i
M
from the above action S. From these quantities, one then constructs the corresponding interacting quantum fields (smeared with a compactly supported testfunction) as formal power series in free field quantities via the Bogoliubov formula [18, 28], in Oi (h) I = Rn ( M Oi h dµ, I ⊗n ), h ∈ C0∞ (M), (4) n! n≥0
OPE for Perturbative Quantum Field Theory in Curved Spacetime
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or more formally without smearing, Oi (x) I =
in Rn (Oi (x), I ⊗n ), n!
(5)
n≥0
where the quantities Rn are the so-called retarded products, which are multi-linear maps on the space of local classical action functionals of the form (3), taking values in the underlying free field algebra F(M, g). In order to describe these constructions in more detail, it is first necessary to recall at some length the definition and key features of the linear field algebra F(M, g) and its quantum states, as well as the nature of the retarded products Rn . The definition of F(M, g) can be stated in different equivalent ways [22, 12–14], and we now give a definition that is most suited to our purpose. Let ω2 ∈ D (M × M) be a bidistribution of Hadamard type, meaning (a) that ω2 is a bisolution to the Klein-Gordon equation − m 2 − α R in each entry, (b) that the anti-symmetric part of ω2 is given by ω2 (x1 , x2 ) − ω2 (x2 , x1 ) = i( A (x1 , x2 ) − R (x1 , x2 )) ≡ i (x1 , x2 ),
(6)
where A/R are the unique advanced and retarded propagators of the Klein-Gordon equation [16], and (c) that it has a wave front set [26] of Hadamard type [34]: WF(ω2 ) = {(x1 , k1 ; x2 , k2 ) ∈ T ∗ (M × M) \ 0; k1 = p, k2 = − p, p ∈ V¯+∗ }.
(7)
Here, it is understood that the set can only contain those x1 and x2 that can be joined by a null geodesic, γ , and that p = pµ d x µ is a parallel co-vector field tangent to that null geodesic, meaning that ∇γ˙ p = 0, and V±∗ is the dual of the future/past lightcone. The desired Wick-polynomial algebra F(M, g) is now generated by an identity 11, and the following symbols F: n ⊗n F= : φ :ω u n (x1 , . . . , xn ) dµ(xi ), (8) Mn
i=1
where u n is a symmetric, compactly supported distribution on M n subject to the wave front condition [12] WF(u n ) ∩ [(V¯+∗ )n ∪ (V¯−∗ )n ] = ∅. (9) The relations in the *-algebra F(M, g) are as follows: The *-operation is defined by letting F ∗ be given by the same expression as F, but with u n replaced by its complex conjugate, and the product is defined by : φ ⊗n :ω (x1 , . . . , xn ) : φ ⊗m :ω (xn+1 , . . . , xn+m ) n!m! = ω2 (x pi (1) , x pi (2) ) : φ ⊗n+m−2k :ω ({x j ; j ∈ / |P|}), (n−k)!(m −k)!k! k
p1 ,..., pk ∈P i
(10) where P is the set of all pairs pi ∈ {1, . . . , n} × {n + 1, . . . , n + m}. This formula is identical in nature to the standard Wick theorem for normal ordered quantities (relative to a Gaussian state with 2-point function ω2 ). The wave front conditions on u n and ω2 are needed in order to guarantee that the product between the corresponding integrated quantities as in (8) exists, because the latter involves the pointwise products of distributions [26]. The definition of the algebra F(M, g) superficially seems to depend on the
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particular choice of ω2 , but this is in fact not so: A change of ω2 merely corresponds to a relabeling of the generators, and does not change the definition of F(M, g) as an abstract algebra [20]. The relation between the abstract quantization of the linear Klein-Gordon field just described and more familiar ones is as follows. If f is a smooth test function, then the generator (8) with n = 1 and u 1 = f can be identified with the smeared field φ( f ). Indeed, using the Wick formula (10), and the antisymmetric part of ω2 , Eq. (6), one easily derives the relation [φ( f 1 ), φ( f 2 )] = i ( f 1 , f 2 )11, which is the standard commutation relation for a linear scalar field in curved spacetime. The higher order generators (8) with u n given by the n-fold tensor product f ⊗n = f ⊗ · · · ⊗ f of a smooth test function correspond to a smeared normal ordered product : φ ⊗n :ω ( f ⊗n ), formally related to the field φ itself by :φ
⊗n
1 δn exp iφ( f ) + ω2 ( f, f ) :ω (x1 , . . . , xn ) = n . i δ f (x1 ) · · · δ f (xn ) 2 f =0
(11)
As it stands, the smeared version of the Klein-Gordon equation φ((−m 2 −α R) f ) = 0 is not an algebraic relation. However, this relation could easily be incorporated by factoring F(M, g) by the 2-sided *-ideal J (M, g) consisting of all elements F of the form (8) with u n a distribution in the class (9) which is in the image of this distribution class under the Klein-Gordon operator, such as ( − m 2 − α R) f in the simplest case. The purpose of this paper will be to establish an OPE, and it is technically convenient for this purpose not to factor by the ideal. However, after the OPE has been constructed there is absolutely no problem to factor by this ideal, because it is clear that the OPE will continue to hold on the factor algebra. Quantum states in the algebraic framework are linear expectation functionals : F(M, g) → C that are normalized, meaning (11) = 1, and of positive type, meaning (F ∗ F) ≥ 0. Of particular importance are the so-called Hadamard states on F(M, g). Those states are defined by the fact that their 2-point function 2 (x1 , x2 ) = (φ(x1 )φ(x2 )) satisfies properties (a), (b), and (c) listed above, and that their truncated n-point functions of the field φ for n = 2 are smooth solutions to the Klein-Gordon equation. The key consequence of the Hadamard requirement which we shall need later is [19] that (: φ ⊗n :ω (x1 , . . . , xn )) is smooth. Note that by definition, the n-point functions of a Hadamard state satisfy the Klein-Gordon equation. Consequently, they vanish on the ideal J (M, g) generated by the Klein-Gordon equation and hence induce states on the factor algebra. Later, we want to define an operator product expansion, and for this we will need a notion of what it means for algebra elements to be “close” to each other. For this we now introduce a topology on F(M, g). There are various ways to do that. A particular topology was introduced in [20]. We prefer here to work with a different (weaker) topology, defined by the collection of all linear functionals on F(M, g) with the property that (: φ ⊗n :ω (x1 , . . . , xn )) is smooth. This set includes the Hadamard states as defined above, and we shall, by abuse of notation, sometimes refer to such as “Hadamard” as well. We then introduce a set of seminorms N (F) = | (F)|, labelled by these functionals . We say that a sequence {FN } N ∈N of algebra elements tends to zero if for each , and each ε > 0, there is an N0 such that N (FN ) ≤ ε for all N ≥ N0 . An important feature of the algebra F(M, g) is that it has a local and covariant dependence upon the spacetime. More precisely, if f : M → M is a causality and orientation preserving isometric embedding of a globally hyperbolic spacetime (M, g) into another
OPE for Perturbative Quantum Field Theory in Curved Spacetime
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such spacetime (M , g ), then there exists a continuous, injective *-homomorphism α f : F(M, g) −→ F(M , g ).
(12)
This embedding is most simply described in terms of its action on a smeared field φ(h), where h is a test function on M. If h = f ∗ h is the corresponding pushed forward test function on M , then we define α f [φ(h)] = φ(h ). Furthermore, the action of α f on an arbitrary element in F(M, g) may then be defined by continuity, because the subalgebra generated by expressions of the form φ(h) is dense in F(M, g). The action of α f on the smeared field φ(h) is characteristic for so-called “local covariant fields”. Namely, an algebra valued distribution Oi : C0∞ (M) → F(M, g), h → Oi (h) that is defined for all spacetimes (M, g) is called a (scalar) local and covariant field if α f [Oi (h)] = Oi (h ) h = f ∗ h,
(13)
whenever f is an orientation and causality preserving isometric embedding. Local covariant fields of tensor type are defined in the same way, except that the testfunction h is now a section in the (dual of the) vector bundle Ei corresponding to the tensor type. Thus, the field φ is (by definition) a local and covariant field. On the other hand, the normal ordered n th Wick power of a field defined by putting u n (x1 , . . . , xn ) = f (xn )δ(x1 , . . . , xn ) in Eq. (8) is not a local and covariant field, because it implicitly depends on the choice of the 2-point function ω2 , which is not a local and covariant quantity [20]. The possible definitions of Wick powers giving rise to local and covariant fields (satisfying also various other natural conditions) were classified in [20]. It turns out that the definition of a given classical expression
Oi = ∇ a1 φ∇ a2 φ · · · ∇ an φ, i = {a1 , a2 , . . . , an } ∈ I ≡ Zn≥0 (14) n
as a local, covariant field in F(M, g) is not unique, but contains certain ambiguities. As proven in [20], these ambiguities correspond to the possibility of adding to a given field lower order Wick powers times certain polynomials of the Riemann tensor and its derivatives ∇(µ1 · · · ∇µk ) Rν1 ν2 ν3 ν4 of the same dimension as Oi . Here, the dimension of a Wick power is the map [·] : I → N from the index set labelling the various fields, into the natural numbers defined by n [i] = n + ai . (15) i=1
One definition which is local and covariant (and satisfies also the other natural conditions given in [20]) is the following “local normal ordering prescription”. It is based upon the use of the local Hadamard parametrix H , which is the bidistribution defined on a convex normal neighborhood of the diagonal {(x, x); x ∈ M} of M × M by2 1 v0 n + v ln(σ + i0t). (16) H= σ n+1 σ + i0t 2n n! n≥0
In this expression, σ (x, y) is the signed squared geodesic distance between two points, we have defined t (x, y) = τ (x) − τ (y), where τ is a time-function, and the vn (x, y) are smooth symmetric functions that are determined by requiring that H be a parametrix, 2 The infinite sum is to be understood in the sense of an asymptotic expansion.
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i.e., a solution to the Klein-Gordon equation −m 2 −α R in each entry modulo a smooth remainder. Explicitly v0 = D 1/2 /2π 2 is given in terms of the VanVleck determinant D, defined by 1 |(∇ ⊗ ∇)σ | D=− . (17) 4 |J | Here, J (x, y) ∈ Tx∗ M ⊗ Ty∗ M is the bitensor of parallel transport, (∇ ⊗ ∇)σ (x, y) ∈ Tx∗ M ⊗ Ty∗ M is the bitensor obtained by taking the gradient of σ (x, y) in both x and y, and we are defining a biscalar |B(x, y)| by |B(x, y)| dµ(x) ⊗ dµ(y) =
4
B(x, y)
(18)
from any bitensor B(x, y). The smooth functions vn (x, y) are iteratively defined by the transport equations [10] 2(∇ µ σ )∇µ vn + [(∇ µ σ )∇µ ln D + 4n]vn = −2( − m 2 − α R)vn−1 ,
(19)
where the derivatives act on x. These functions are symmetric in x and y [33, 16], and their germs at the diagonal are locally and covariantly defined in terms of the metric. Where it is well-defined, H has a wave front set of Hadamard type (7). Next, fix a convex normal neighborhood of the diagonal in M n , and in that neighborhood define locally normal ordered products : φ ⊗n :H (x1 , . . . , xn ) by the same formula as (11), but with ω2 in that formula replaced by H , :φ
⊗n
1 δn exp iφ( f ) + H ( f, f ) :H (x1 , . . . , xn ) = n . i δ f (x1 ) · · · δ f (xn ) 2 f =0
(20)
As the expressions (11), their expectation value in any Hadamard state is smooth. Following [20] we define the local covariant n th Wick power of the field as the distribution, valued in F(M, g), given by x (ε)) x(ε) = (expx (εξ1 ), . . . , expx (εξn )) . φ n (x) = lim : φ ⊗n :H ( ε→0
(21)
More generally, fields containing derivatives are defined by first acting with the derivatives on the appropriate tensor factor in : φ ⊗n :H before taking the above “coincidence limit”. That is, if i = (a1 , . . . , an ) ∈ I denotes a collection of natural numbers, then the corresponding local covariant field Oi (x) is defined by applying the partial derivative operator ∂ξa11 · · · ∂ξann prior to the coincidence limit. Note that this definition is covariant, because partial derivatives with respect to Riemannian normal coordinates at x may be expressed in terms of curvature tensors at x and covariant derivatives ∇. Having described the algebra F(M, g) of field observables in the linear field theory associated with the action S without the interaction terms, we now turn to the interacting theory. For this it is technically convenient at an intermediate step to assume that the couplings κi in the action S are not constants, but actually smooth functions of compact support in M, which we assume are locally constant, κi (x) = κi χ (x),
(22)
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where χ ∈ C0∞ (M), and χ (x) = 1 in an open set in M with compact closure. The cutoff functions χ serve as an infra red cutoff and are removed at a later stage. With the introduction of cutoff functions understood, the interacting fields are defined by the Bogoliubov formula (5) in terms of the retarded products Rn . Each retarded product is a continuous, bilinear map Rn : S ×
n S → F(M, g) (I0 , I1 ⊗ · · · ⊗ In ) → Rn (I0 , I1 ⊗ · · · ⊗ In ), (23)
from the tensor powers of the space S of all classical action functionals I j = L j dµ that are local and polynomial in the field φ and whose couplings are compactly supported functions on M. The map is taking values in the algebra F(M, g) associated with the linear field theory, and is symmetric in I1 , . . . , In . Note that the power series expression (5) for the interacting fields is only a formal series, and no statement is made about its convergence. Since each term in these series is an element in F(M, g), the interacting fields are elements of the algebra P ⊗ F(M, g), where P ≡ C[[κ1 , κ2 , . . . ]] =
aα1 ...αk κ1α1 · · · κkαk ; aα1 ...αk ∈ C
(24)
αi ≥0
is the corresponding ring of formal series. All operations, such as multiplication in this ring and in the algebra P ⊗ F(M, g), are defined by simply formally multiplying out the corresponding formal series term by term. Furthermore, this algebra inherits a natural topology from F(M, g): A formal power series converges to another formal power series in the algebra if each coefficient does. We want the zeroth order contribution R0 (Oi (x)) in the Bogoluibov formula (5) to be given by our definition of Wick powers Oi (x) in the linear field theory3 . Thus, we define R0 (Oi (x)) to be equal to the locally normal ordered field Oi (x) given in Eq. (21). The terms with n ≥ 1 in the formal series (5) represent the perturbative corrections coming from the interaction, I . They involve the higher, non-trivial, retarded products. The construction of these retarded products can be reduced4 to the construction of the so-called “time-ordered products,” because there exists a well-known formula of combinatorial nature relating these two quantities, see e.g. the Appendix of [14]. The construction of the time-ordered products in turn has been given in [20, 21, 23], which is based on work of [4]. The strategy in these papers is to first write down a number of functional relations for the time ordered products that are motivated by corresponding properties of the interacting fields defined by the Bogoliubov formula. These properties then dictate to a large extent the construction of the time-ordered (and hence the retarded) products. Since there is a combinatorial formula relating the time ordered products to the retarded products, these relations can equivalently be stated in terms of the retarded products. The relevant relations5 for this paper are as follows: 3 Note that the argument of the retarded product R (O (x)) is a classical action (or density), while the 0 i corresponding Wick power Oi (x) is a distribution valued in the quantum algebra F (M, g). We should strictly speaking distinguish these quantities by introducing a new notation for the Wick power, but we shall not do this for simplicity. 4 Alternatively, it should also be possible to construct the retarded products directly along the lines of [12– 14], by suitably generalizing the arguments of that paper from Minkowski spacetime to curved spacetime. 5 A complete list may be found in [23].
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(r1) Causality: Let F, G, Si ∈ S. Suppose that there is a Cauchy surface such that supp G is in its future and supp F in its past. Then Rn F, G ⊗ (25) Si = 0. i
(r2) GLZ factorization formula [14]: Rn G, Si ⊗ F − Rn F, Si ⊗ G i
i
R|I |,1 F, Si , R|J |,1 G, S j . (26)
=
I ∪J ={1,...,n−1}
i∈I
j∈J
(r3) Expansion: There exist local covariant c-number distributions r near the total diagonal {(x, x, . . . , x); x ∈ M} such that (with l j < i j ) Rn (Oi0 (x), Oi1 (y1 ) · · · Oin (yn )) = l l ...l ri00i11...inn (x, y1 , . . . , yn ) : Ol0 (x)Ol1 (y1 ) . . . Oln (yn ) :H. (27) l0 ,l1 ,...,ln
(r4) Scaling degree: The distributions r have the scaling degree ...ln sd(ril00il11...i )= [i k ] − [lk ] n
(28)
k
at the total diagonal {(x, x, . . . , x); x ∈ M} ⊂ M n+1 . Here, the scaling degree of a distribution u ∈ D (X ) at a submanifold Y ⊂ X is defined as follows [36, 4]. Let Sε : X 0 → X 0 be an injective, smooth map defined on an open neighborhood X 0 of Y with the properties (a) that Sε Y = idY , and (b) that for all y ∈ Y , the map (DSε )(y) : Ty X → Ty X is the identity on Ty Y and scales vectors by ε > 0 on a complementary subspace C y ⊂ Ty X of Ty Y . Then u has scaling degree sd(u) at Y if limε→0 ε D u ◦ Sε = 0 for all D > sd(u) in the sense of D (X 0 ). The definition is independent of the precise choice of Sε . (r5) Locality and covariance: Let f : (M, g) → (M , g ) be a causality and orientation preserving isometric embedding. Then the retarded products satisfy α f Rn F, (29) Si = Rn f ∗ F, f ∗ Si , i
i
where f ∗ denotes the natural push-forward of a local action functional on M to the corresponding action functional on M . (r6) Microlocal condition: The distributions r in the expansion (27) have the following wave front set:
WF(r ) ⊂ (x, k; y1 , l1 ; . . . ; ym , lm ) ∈ T ∗ M m+1 ; there is graph in G1,n such that k=
e:s(e)=x −
pe −
e:t (e)=x
pe , l i =
yi ∈ J (x) i = 1, . . . , m .
e:s(e)=yi
pe −
pe
e:t (e)=yi
(30)
OPE for Perturbative Quantum Field Theory in Curved Spacetime
11
The valence of the vertices yi in the graph is restricted to be less than or equal to the maximum power of φ occurring in the operators Oi1 , . . . , Oin in Eq. (27). In the formulation of the last condition, we are using a graph theoretical notation [3], which will be useful later as well. Most generally, we consider the set Gn,m of embedded, oriented graphs in the spacetime M with n + m vertices. Each such graph has n so-called “external vertices”, x1 , . . . , xn ∈ M, and m so-called “internal” or “interaction vertices” y1 , . . . , ym ∈ M. These vertices are of arbitrary valence, and are joined by edges, e, which are null-geodesic curves γe : (0, 1) → M. It is assumed that an abstract ordering6 < of the vertices is defined, and that the ordering among the external vertices is x1 < · · · < xn , while the ordering of the remaining interaction vertices is unconstrained. If e is an edge joining two vertices, then s(e) (the source) and t (e) (the target) are the two vertices γe (0) and γe (1), where the curve is oriented in such a way that it starts at the smaller vertex relative to the fixed vertex ordering. Each edge carries a future directed, tangent parallel covector field, pe , meaning that ∇γ˙e pe = 0, and pe ∈ ∂ V+∗ . Similar to the case of local covariant Wick products (the case n = 0), the above functional relations (together with other functional relations described in detail in [20, 21]) do not uniquely fix the retarded products: There remains a number of real constants at each order n which parametrize the set of possible definitions of Rn that are compatible with (r1)–(r6). These correspond to the usual “renormalization ambiguities” in perturbative quantum field theory, see [21–23] for details. One finally needs to remove the dependence of the interacting fields on the arbitrary cutoff function χ . For this, one investigates how the interacting field changes when the cutoff function is varied. Assume that χ1 and χ2 are two different cutoff functions, both of which are equal to 1 in an open globally hyperbolic neighborhood U → M. Let I1 and I2 be the corresponding interactions. Note that, as a classical functional, the difference I1 − I2 is supported in a compact region, and vanishes in the neighborhood U where the cutoff functions coincide. The key fact [4], which follows from the above functional relations (r1) and (r2), is now that there exists a unitary operator V ∈ P ⊗ F(M, g), depending upon I1 , I2 , with the property that Oi (x) I1 = V Oi (x) I2 V ∗ , for all x ∈ U , i ∈ I.
(31)
This relation may be interpreted as saying that the algebraic relations between the interacting fields within the region U where the cutoff function is constant do not depend on how the cutoff function is chosen outside this region, and this observation may be used to construct an abstract interacting field algebra associated with the entire spacetime (M, g) that is independent of the choice of cutoff function [4]. However, in the present context, we are actually only interested in a small patch U of spacetime where we want to consider the OPE. Therefore, it will be more convenient for us to simply fix an arbitrary cutoff function that is equal to 1 in the patch U of interest. Since the OPE concerns only local algebraic relations, it is clear from (31) that it should not matter what cutoff function we choose, and this will formally be shown below in Item 2 of Sect. 4. 6 The ordering is not assumed to be related to the causal structure of the manifold at this stage.
12
S. Hollands
3. Operator Product Expansion We will now show that the interacting fields Oi (x) I described in the previous section obey an operator product expansion, Oi1 (x1 ) I Oi2 (x2 ) I · · · Oin (xn ) I ∼ Ci1 i2 ...in k (x1 , x2 , . . . , xn ) I Ok (xn ) I , (32) [k]≤
where C I are certain distributional coefficients depending upon the interaction, I , which are to be determined, and where [k] is the standard dimension function defined above. We mean by the above expression that, as the points x1 , x2 , . . . , xn approach each other, the algebra product of the interacting fields on the left side can be approximated, to the desired precision determined by , by the right side, in the topology on the algebra P ⊗ F(M, g). To make this statement precise, we must, however, take into account that both sides of the OPE are actually distributional, and that a configuration x = (x1 , . . . , xn ) ∈ M n of n mutually distinct points on a manifold may “merge” in qualitatively different manners when n > 2, because the points may approach each other at “different rates”. The appropriate mathematical framework to formalize in a precise manner the possibility of configuration of points to approach each other at different rates is provided by a construction referred to as the “compactification of configuration space,” due to Fulton and MacPherson [17], and Axelrod and Singer [1]. Let M0n = { x = (x1 , x2 , . . . , xn ) ∈ M n ; xi = x j } = { x ∈ Map({1, . . . , n}, M), x(i) = xi ; x injective}
(33) (34)
be the configuration space, i.e., the space of all configurations of n mutually distinct points in M. The union of partial diagonals S ⊂ M n , (35) ∂ M0n = S⊂{1,...,n}
where a partial diagonal is defined by x ∈ Map({1, . . . , n}, M); x S = constant}, S = {
(36)
is the boundary of the configuration space M0n . Configurations of points where some points come close to each other are in some sense close to this boundary. The Fulton-MacPherson compactification M[n] is obtained by attaching a different boundary, ∂ M[n], to M0n , which in addition incorporates the various directions in which ∂ M0n can be approached. This boundary may be characterized as the collection of endpoints of certain curves x(ε), in M[n], which are in M0n for ε > 0, and which end on ∂ M[n] at ε = 0. These curves are labeled by trees T that characterize subsequent mergers of the points in the configuration as ε → 0. A convenient way to describe a tree T (or more generally, the disjoint union of trees, a “forest”) is by a nested set T = {S1 , . . . , Sk } of subsets Si ⊂ {1, . . . , n}. “Nested” means that two sets are either disjoint, or one is a proper subset of the other. We agree that the sets {1}, . . . , {n} are always contained in the tree (or forest). Each set Si in T represents a node of a tree, i.e., the set of vertices Vert(T ) is given by the sets in T , and Si ⊂ S j means that the node corresponding to Si can be reached by moving downward from the node represented by S j . The root(s) of the tree(s) correspond to the maximal elements, i.e., the sets that are not subsets of
OPE for Perturbative Quantum Field Theory in Curved Spacetime
13
any other set. If the set {1, . . . , n} ∈ T , then there is in fact only one tree, while if there are several maximal elements, then there are several trees in the forest, each maximal element corresponding to the root of the respective tree. The leaves correspond to the sets {1}, . . . , {n}, i.e., the minimal elements. The desired curves x(ε) tending to the boundary of M[n] are associated with trees and are constructed as follows. With the root(s) of the tree(s), we associate a point xi ∈ M, where i is a label that runs through the different maximal elements, while with each edge e ∈ Edge(T ) of a given tree (a line joining two nodes), we associate a vector ve ∈ Txi M, where xi is associated with the root of the tree that e belongs to. To describe the definition of this vector it is convenient to identify an edge e ∈ Edge(T ) with the pair e = (S, S ) of nodes that it connects, i.e., an edge defines a relation in T × T . If S ⊂ S , then we write S = t (e) for target, and S = s(e), for the source. We then set ve = ξm(t (e)) − ξm(s(e)) m(S) = max{i; i ∈ S},
(37)
where ξ j denotes the Riemannian normal coordinates (identified with a vector in R4 via a choice of orthonormal tetrad at the corresponding root xi ). We define the desired curve x(ε) by x j (ε) = ve εdepth(t (e)) , (38) e∈ p j
where p j is the unique path connecting the leaf j with the corresponding root, where ve is given in terms of x by Eq. (37), and where depth(S) is the number of edges that connect the node S ∈ T with the root. The following figure illustrates this definition in an example: root = x4
S0
εv1
εv2
S1 ε2 v3 x1 (ε)
x1 (ε) =
εv1 + ε2 v3 ,
S2 ε2 v4
x2 (ε)
ε2 v5
ε2 v6
x4 (ε)
x3 (ε)
T = {S0 , S1 , . . . , S6 } x2 (ε) = εv1 + ε2 v4 , x3 (ε) = εv2 + ε2 v5 , x4 (ε) = εv2 + ε2 v6
For each fixed tree T , and each fixed ε, the above curve defines a map ψT (ε) : M0n → M0n , x → x(ε)
(39)
flowing the point x = x(1) to the point x(ε). For ε = 0, the image of this map may be viewed as a portion of the boundary ∂ M[n] corresponding to the tree. The roots of T correspond to the particular diagonal; in particular, if there is only one tree in T (as we
14
S. Hollands
shall assume from now on) then the configuration x(ε) converges to the total diagonal in M n . It may be checked that the maps ψT (ε) satisfy the composition law ψT (ε) ◦ ψT (ε ) = ψT (εε ).
(40)
Using the maps ψT (ε) we can define an asymptotic equivalence relation ∼δ,T for distributions on M n . Consider distributions u 1 , u 2 defined on M n . For a given tree T and δ > 0, we declare the equivalence relation ∼T ,δ by u 1 ∼T ,δ u 2
:⇐⇒
lim ε−δ (u 1 − u 2 ) ◦ ψT (ε) = 0,
ε→0+
(41)
in the sense of distributions on M n , where we view ψT (ε) as a map M n → M n that is parametrized by ε > 0. Having defined the equivalence relation ∼δ,T we can now state precisely our notion of an OPE. Namely, we require that, for each δ > 0, each given set of operators, and each tree T , there exists a so that the OPE holds in the sense of ∼δ,T . The only issue that we have not yet been quite precise about is that the OPE is not a relation between c-number distributions, but instead distributions valued in the topological algebra P ⊗ F(M, g). This difficulty is simply dealt with by requiring convergence in the equivalence relation (41) (now for algebra valued objects) with respect to the topology in the algebra. Thus, we define the precise sense in which the OPE is supposed to hold to be that for each tree T with one root, and each δ, there exists a ∈ R such that (32) holds in the sense of ∼δ,T as a relation between the corresponding algebra valued distributions. We now come to the actual construction of the operator product coefficients in perturbation theory. As also described in [24], and as originally suggested by Bostelmann [2] and Fredenhagen and Hertel [15] in the context of algebraic quantum field theory on Minkowski spacetime, it is convenient to think of the operator product coefficients as arising via certain “standard functionals” i M,x (. ) I : P ⊗ F(M, g) −→ P ⊗ Ei |x .
(42)
These functionals depend upon the given spacetime (M, g), the index label i ∈ I describing a composite field, a point x ∈ M, and the interaction, as indicated by the subscript “I ”. The functionals take values in the fiber over x in the vector bundle Ei (viewed as a P-module) associated with the tensor character of the field Oi . In our constructions below, the functionals are in fact only defined on the subalgebra P ⊗ F(U, g) corresponding to a convex normal neighborhood U ⊂ M. However, since all of our considerations are entirely local, we may assume without loss of generality and to save writing that U = M. The OPE coefficients are supposed to be given in terms of the above standard functionals by j Ci1 i2 ...in j (x1 , x2 , . . . , xn ) I = xn Oi1 (x1 ) I Oi2 (x2 ) I · · · Oin (xn ) I I .
(43)
We will construct the OPE coefficients in perturbation theory by presenting a suitable set of such standard functionals. We are going to choose these standard functionals as a “dual basis” to the interacting fields, in the sense that we wish them to satisfy xi (O j (x) I ) I = δ i j idEi for all x ∈ M and [i], [ j] < .
(44)
OPE for Perturbative Quantum Field Theory in Curved Spacetime
15
This ansatz is motivated by the following simple consideration. Let us assume that an OPE exists. Let us fix a > 0, carry the OPE out until [k] ≤ , and apply the funcj tionals xn to it, where [ j] ≤ . Using (44), we immediately find that the coefficients in the OPE must be given by (43), up to a remainder term coming from the remainder in the OPE. But this remainder is by assumption small for asymptotically short distances, in the sense of the above equivalence relation, provided we make sufficiently large. It can therefore be ignored. Thus we have argued that if an OPE exists in the sense above, and if standard functionals satisfying (44) have been defined, then the OPE coefficients C ought to be given by (43). Consequently, our first step will be to define the standard functionals as formal power series in the coupling constants κi so that Eq. (44) will be satisfied to arbitrary orders in perturbation theory. To zeroth order in perturbation theory, such standard functionals are defined as follows (see also [24]). Recall that a general algebra element F ∈ F(M, g) can be written as in Eq. (8) in terms of normal ordered generators (11). If we are interested only in elements F so that the corresponding u n in (8) are supported sufficiently close to the diagonal in M n (as we will always assume in the following), then we may rewrite F in terms of the locally normal ordered generators : φ ⊗n :H (x1 , . . . , xn ) given in Eq. (20) instead of the normal ordered generators : φ ⊗n :ω (x1 , . . . , xn ). The action of the zeroth order standard functionals is then declared by xi : φ ⊗m :H (x1 , . . . , xm ) =
δm,n ξ ⊗a1 · · · ξn⊗an i = (a1 , . . . , an ), a1 ! · · · an ! 1
(45)
and extended to all of F(M, g) by linearity. Here, ξi are the Riemannian normal coordinates of xi relative to x, identified with vectors in Tx M. These functionals satisfy the analog of Eq. (44) for the linear fields defined above. Since the interacting fields Oi (x) I are given by formal power series whose zeroth order is the linear field expression (see the Bogoliubov formula (5)), it follows that the action of the linear field functionals on an interacting field is of the form xi (O j (x) I ) = δ i j idEi + Ai j (x),
(46)
where Ai j is the endomorphism in End(E j , Ei ) that arises from the higher perturbative contributions to the interacting field, see (5), and is given by Ai j (x) =
in xi Rn (O j (x); I ⊗n ) . n!
(47)
n≥1
Consequently, using the standard geometric series for the inverse of a linear operator of the form 11 + L and writing out explicitly the above formula for Ai j (x), we find that the functional defined by the following series is a solution to Eq. (44): ∞ im 1 +···+m k (−1)k m ! · · · mk ! m l ≥1 1 k=0 i Rm 1 (O j1 ; I ⊗m 1 ) j1 Rm 2 (O j2 ; I ⊗m 2 ) · · · jk (Rm k O jk+1 ; I ⊗m k ) jk+1 (F). (48) Here, m = m l is the perturbation order of an individual term, and the sums over jl are carried out to order [ jl ] ≤ . Thus, for each fixed m, the sum over k has only a finite
i (F) I =
16
S. Hollands
number of terms, and the resulting expression is a well-defined functional on formal power series, valued in formal power series. Furthermore, all functionals jk and all operators O jk appearing on the right side are taken at a reference point x. We now define the operator product coefficients by formula (43) in terms of the functionals i (. ) I . Writing out all terms explicitly, the interacting OPE-coefficients are thus given by ∞
im 1 +···+m k m ! · · · mk ! m i ≥1 1 k=0 [l j ]≤ j ⊗m 1 l1 ⊗m 2 Rm 1 (Ol1 (xn ); I ) Rm 2 (Ol2 (xn ); I ) · · · lk (Rm k Olk+1 (xn ); I ⊗m k ) n nr i Rn i (Oir (xi ); I ⊗nr ) , (49) × lk+1 nr ! Ci1 ...in j (x1 , . . . , xn ) I ≡
(−1)k
n i ≥0
r =1
where all local functionals l refer to the point xn . In order to make this formula welldefined, it is necessary to assume that the support of the cutoff function χ implicit in I is small enough so that the standard functionals are defined on the corresponding retarded product. However, this is no real restriction, because the OPE is an asymptotic short distance expansion, and we will later show that the coefficients do not depend on the particular choice of χ asymptotically. We claim that the coefficients C I satisfy an OPE: Theorem 1. Let the interaction I = L dµ be renormalizable, i.e., [L] ≤ 4. For a given tree T , δ ≥ 0, and given i 1 , . . . , i n ∈ I, let =δ+
n [i j ] · depth(T ),
(50)
j=1
and define the OPE coefficients Ci1 ...in k by Eq. (49). Then the OPE holds: Oi1 (x1 ) I Oi2 (x2 ) I · · · Oin (xn ) I ∼T ,δ Ci1 i2 ...in k (x1 , x2 , . . . , xn ) I Ok (xn ) I . (51) [k]≤
Remarks. 1) The theorem is false for non-renormalizable interactions. 2) Since the topology on P ⊗ F(M, g) of which the interacting fields are elements is generated by a set of seminorms associated with functionals including the Hadamard states, it follows that the OPE will continue to hold on the factor algebra obtained by dividing by the Klein-Gordon equation, in the sense of expectation values in Hadamard states, to arbitrary orders in perturbation theory. Proof. Let N I be defined as the remainder in the OPE, i.e., the left side of (51) minus the right side. We need to prove that ε−δ (N I ◦ ψT (ε)) tends to 0 in the sense of distributions as ε → 0. The analysis of this limit is easiest in the case when T is the tree T = {S0 , S1 , . . . , Sn } with one root S0 = {1, . . . , n} and n leaves Si = {i}. Then depth(T ) = 1, and ψT (ε) is the map that scales the Riemannian normal coordinates of the points x1 , . . . , xn−1 ∈ U relative to y = xn by ε, where U is a convex normal neighborhood of y. Thus, taking ε = 2−N , we must show that 2δ N (N I (2−N x1 , . . . , 2−N xn )) → 0 as N → ∞,
(52)
OPE for Perturbative Quantum Field Theory in Curved Spacetime
17
in the sense of distributions valued in P, i.e., to any order in perturbation theory. In the above expression, and in the remainder of this proof, points xi have been identified with their Riemannian normal coordinates around y = xn . In order to analyze the above expression, it is necessary to perform several intermediate decompositions of N I , and we now explain how this is done. We first decompose N I into contributions from the different orders in perturbation theory. The ring P of formal power series in the couplings κi contained in the interaction I = Ldµ can be decomposed into a direct sum
P= P(k) , P(k) = Eigenspace of κi d/dκi for eigenvalue k, (53) k
where the k th
summand corresponds to the k th order in perturbation theory. Accordingly, N I may be decomposed as N(k) into contributions from the various orders in pertur C(k) , etc. Using the Bogoliubov formula, the k th bation theory, and likewise C I = order perturbative contribution to N I can be written in the form
N(k) (x1 , . . . , xn ) =
k1 +···+kn =k
−
k
ik Rk j (O(x j ), I ⊗k j ) k1 ! · · · kn ! j
C( p) (x1 , . . . , xn )Rk− p (O(xn ), I ⊗(k− p) ),
(54)
p=0
where the labels on O indicating the field species have been omitted to lighten the notation. Let A(k) be defined as N(k) , but with the k th order OPE coefficient C(k) omitted. Then, using the definition of the OPE-coefficients, it can be seen that (55) Ci1 ...in j (x1 , . . . , xn )(k) = j (Ai1 ...in (x1 , . . . , xn )(k) ) j j and that N(k) = A(k) − [ j]≤ (A(k) )O j , where are the free field reference functionals at point xn and where O j are the free field Wick powers taken at point xn . Thus, the expectation value of the scaled, k th order perturbative contribution to the remainder is given by (N(k) (2−N x1 , . . . , 2−N xn )) = (A(k) (2−N x1 , . . . , 2−N xn )) j (A(k) (2−N x1 , . . . , 2−N xn )) (O j (2−N xn )). − [ j]≤
(56) k k y (F) (Ok (y)),
The right side of this equation is schematically of the form (F)− and for such expressions we will now write down an expression which will be useful to analyze the limit N → ∞ of Eq. (56). To derive this expression, perform a Taylor expansion with remainder about (y, . . . , y) ∈ U m of the m th locally normal ordered product (20), :
m i=1
φ(ξi ) :H −
|α1 |+···+|αm |≤ρ
m 1 : ξiαi ∂ αi φ(0) :H α1 ! · · · αm !
=
1 ρ!
i=1
0
1
ρ+1
(1 − t)ρ ∂t
:
m i=1
φ(tξi ) :H dt.
(57)
18
S. Hollands
Here, the ξi denote Riemannian normal coordinates around y and are identified with points in R4 , αi ∈ N40 is a multiindex, and quantities like |αi | are defined using standard multiindex conventions. As explained above, any element F ∈ F(M, g) supported in U may be written as a linear combination of expressions which consist of distributions u m supported in U m satisfying the wave front condition (9), integrated with locally normal ordered products : φ ⊗m :H . If we apply a Hadamard state to such an expression F, use the above Taylor series with remainder, and use the definition (45) for the standard functionals, then we get the following equation: (F) − yk (F) (Ok (y)) [k]<
=
m |α|= −m+1
× 0
1
1 ( − m)!
Mm
u m (ξ1 , . . . , ξm )ξ1α1 · · · ξmαm
m (1 − t) −m : ∂α1 φ(tξ1 ) · · · ∂αm (tξm ) :H dt dµ(ξi ). (58) i=1
The key point to note about this identity is that there are now factors of ξiαi on the right side, which will work in our advantage when the points ξi are scaled by a small factor. On the other hand, the normal ordered expectation values in the second line are smooth (here we are using the assumption that is Hadamard), and so will not cause any trouble for such a scaling. We will now prove that (52) holds in the sense of distributions by exploiting this identity for F = A(k) in Eq. (56). However, before we efficiently make use of that identity in (56), it is first necessary to rewrite A(k) in a suitable way, and to apply an induction in k. For this, we recall that the interaction Lagrangian density L is confined to the convex normal neighborhood U since we are taking the couplings to be κi (x) = κi χ (x) with χ a smooth cutoff function that is supported in U . We now “slice up” the support of L into contributions from different “shells” in U that are centered around y = xn , and that have thickness 2− j , where j = 1, . . . , N . For this, we choose a compactly supported function ϑ that is 1 on U , and we set ϑ j (x) = ϑ(2 j x).
(59)
Then L may be decomposed as L = ϑN L +
N (ϑ j−1 − ϑ j )L.
(60)
j=1
Each term in the sum is supported in a slice of thickness 2− j , see the figure following Eq. (67). The key step is now to rewrite an interacting field quantity in a way that reflects the subdivision of the interaction region U into these slices. For this, we note that if V j is the unitary in (31) relating the interacting field with interactions I = ϑ j Ldµ and j I j−1 = ϑ j−1 Ldµ, we have O I (2−N x) = V1 V2 · · · VN O I N (2−N x)(V1 V2 · · · VN )−1 , for all x ∈ U . Explicitly, V j is given in terms of the relative S-matrix [4], V j = S ϑ L ρ j L = S ϑ L ρ j L (k) . j
j
k
(61)
(62)
OPE for Perturbative Quantum Field Theory in Curved Spacetime
19
Here, ρ is any smooth function of compact support in U with the property that ρ(x) = 0 for all x ∈ J + (supp(ϑ1 )) and ρ(x) = ϑ0 (x) − ϑ1 (x) for all x ∈ J − (supp(ϑ1 )), and ρ j (x) = ρ(2 j x). Each term in this expansion can in turn be written in terms of retarded products [4]. Substituting Eq. (61) into the formula for the remainder, and expanding in a perturbation expansion, we get the following identity: A(k) (2−N x1 , . . . , 2−N xn ) I = A(k) (2−N x1 , . . . , 2−N xn ) I N +
k−1
p=0 k − p = k1 + · · · + kr 0<α1 ···<αr
S ϑ L ρα j L (k ) αj j
+l1 + · · · + lq
−N
· N( p) (2 x1 , . . . , 2−N xn ) I N ∗ · S ϑ L ρβi L (l ) . 0>β1 ···>βq >N
βi
i
(63)
This complicated identity has the following structure. The sum on the right side is by definition only for p such that p < k, meaning that the terms under the sum only contain the remainder in the OPE up to (k − 1)th order in perturbation theory. This will enable us to use an inductive procedure to estimate the k th order perturbative contribution to the remainder by the lower order contribution. The first term on the right side, A(k) (. . . ) I N , is identical in nature with A(k) (. . . ) I , with the only exception that all the retarded products implicit in its definition are now computed with respect to the interaction I N = ϑ N Ldµ which is supported only in a small ball of radius 2−N around y = xn . This will enable us to use a scaling argument to estimate this term. We now explain more precisely how the decomposition of A(k) given in (63) will make it possible to analyze the scaling behavior of the k th order remainder in the OPE. For this, we take (63), and substitute it into Eq. (56). This gives an expression for (N(k) (. . . ) I ), and to each term in this expression, we can apply Eq. (58). Consider first the term arising from the first term on the right side of Eq. (63). That term makes a contribution to (N(k) (. . . ) I ) of the form (A(k) (2−N x1 , . . . , 2−N xn ) I N ) j (A(k) (2−N x1 , . . . , 2−N xn ) I N ) (O j (2−N xn ) I N ). −
(64)
[ j]≤
We must now substitute the expression for A(k) (. . . ) I N . For this, we use that, on account of the Bogoliubov formula, the scaled interacting field with interaction I N is given by O(2
−N
x) I N =
2−4k N ik k
k!
Uk
k Rk O(2−N x), L(2−N yi ) ϑ0 (yk )dµ N (yk ), i=1
(65) where we have performed the change of integration variables yi → 2 N yi , and where dµ N is 24N -times the pull-back of dµ by the inverse of this map, which is smooth as N → ∞. If we now also use the Wick expansion of the retarded products (r3), along with the scaling degree property (r4) and combine the result with Eq. (58), then we obtain that Eq. (64) scales as 2−N (δ+1) , as desired.
20
S. Hollands
Now we must take the second term on the right side in (63), substitute it into (56), and then analyze its scaling using (58). To do this, we must now proceed iteratively, in the order in perturbation theory k. For k = 0, there is nothing to show. For k > 0, we then inductively know the scaling (52) up to order k − 1, which enables us to estimate the remainders N( p) , p < k in the terms in the sum on the right side of (63). More precisely, we may inductively assume that the p th order remainder ( p < k) has the structure N( p) (2−N x1 , . . . , 2−N xn ) I N =
m
· 0
Mm 1
2−N (4m+ −m+1)
|α|= −m+1
1 ( − m)!
n m (2−N y1 , . . . , 2−N ym , 2−N x1 , . . . , 2−N xn )y1α1 · · · ymαm
(1 − t) −m : ∂α1 φ(t2−N y1 ) · · · ∂αm (t2−N ym ) :H dt
m
dµ N (yi ), (66)
i=1
where n m are the coefficients in a Wick-expansion of N( p) (x1 , . . . , xn ) I (note that we have also performed a change of integration variables yi → 2 N yi ). Using the scaling (r3) and the fact that all terms N( p) (. . . ) I may be written in terms of retarded products by means of the Bogoliubov formula, one can see that 2−N ([ j1 ]+···+[ jn ]+m[L]−m) n m (2−N y1 , . . . , 2−N ym , 2−N x1 , . . . , 2−N xn ) → 0 as N→∞. (67) One now has to take formula (66), and substitute it into the sum on the right side of (63). From the product of N( p) (. . . ) I N with the relative S-matrices there arise terms which blow up as N → ∞, and so these terms have to be carefully controlled. To understand in detail what type of diverging terms can arise, we must write out the explicit formula for the relative S-matrices in terms of retarded products. Then we must write each retarded product in a Wick expansion (r4), and perform the products using Wick’s theorem (10), with ω replaced by H . Then we get a collection of terms, each of which is a product of H , r , n m and a locally normal ordered Wick power. These terms are evaluated on a set of spacetime arguments which are scaled by 2−N , 2−αi , or 2−β j , and which are integrated against the compactly supported smooth functions ϑ or ρ. The arguments scaled by 2−α j arise from points in the interaction domain U within the α th j slice, the arguments −β j scaled by 2 arise from points in the interaction domain U within the β th j slice, and the arguments scaled by 2 N correspond to the scaled arguments 2−N xi . More precisely, when we use Wick’s theorem to perform the products in the second term in (63), there arise “contractions” between points in the αith and β th j slice, indicated by lines in the following figure: Each such contraction is associated with a factor H (2−α j y1 , 2−βi y2 ) (or a derivative thereof), which, using the explicit form of the Hadamard parametrix H , is seen to scale as 22min(αi ,α j ) , in the sense of the scaling degree of a distribution (with a correspondingly larger power when derivatives are present). Furthermore, the scaling of the retarded
OPE for Perturbative Quantum Field Theory in Curved Spacetime
21
supp(ϑ0 ) supp(ϑ1 ) supp(ϑ2 )
supp(ϑ3 ) H=
Interaction domain U
products in a term in Eq. (63) associated with the i th slice may also be controlled. Namely, using the Wick expansion (r3), we see that a retarded product associated with the i th slice contributes factors of r (2−i y1 , . . . , 2−i yl ), the scaling power of which may then be controlled using (r4). Finally, the scaling of n m (2−N x1 , . . . , 2−N ym ) is controlled by Eq. (67). Thus, the rate at which the terms in the sum on the right side of (63) blow up can be controlled. We finally need to substitute each such term into Eq. (56), and use (58). If we carefully keep track of all the scaling powers, then we find that a typical term contributing to N(k) arising from these substitutions has the scaling power 2 N (−δ−4k−1)+[L]
j
k j α j +[L]
i li βi
.
(68)
Using a geometric series, the sum of such terms is estimated by k−1
2 N (−δ−4k−1)+[L]
j
k j α j +[L]
i li βi
p=0 k − p = k1 + · · · + kr 1 < α1 · · · < αr < N
+l1 + · · · + lq 1 < β1 · · · < βq < N
≤ const. 2−N (δ+1−([L]−4)k) ≤ const. 2−N (δ+1) ,
(69)
where we have used in the last step that the interaction is renormalizable, [L] ≤ 4. Thus, the total scaling of the sum of terms in N(k) (2−N x1 , . . . , 2−N xn ) is given by 2−N (δ+1) , which implies the convergence of (52). On the other hand, for non-renormalizable interactions, we would not get convergence. The analysis for a general tree T is in principle not very different from the one just given. For a general tree, the minimum distance between points in a scaled configuration x(2−N ) is of order 2−depth(T )N , and not just 2−N as in the simple tree studied above. This implies that the scaling of the corresponding quantities in the perturbative expansions is different. One now has to go through the above steps again and take that different scaling into account. If this is done, then the result claimed in the theorem is obtained.
22
S. Hollands
4. Properties of the OPE Coefficients We would now like to establish a number of important general properties of the OPE coefficients defined in the previous section. These properties are 1. 2. 3. 4. 5.
Microlocal spectrum condition. Locality and covariance. Renormalization group. Associativity. Scaling expansion.
Except for the last one, these properties were postulated as axioms in the paper [24], so our present work can be viewed as a confirmation of [24]. We first establish the microlocal spectrum condition. With the graph theoretical notation Gn,m introduced in Sect. 2, let us define the following subset of the cotangent space T ∗ Mn:
n,m (M, g) = (x1 , k1 ; . . . ; xn , kn ) ∈ T ∗ M n \ {0}; ∃ decorated graph G( x , y, p) ∈ Gm,n
such that ki =
pe −
e:s(e)=yi
pe for all xi and
e:t (e)=xi
e:s(e)=xi
such that 0 =
pe −
pe for all yi and
e:t (e)=yi
such that yi ∈ J ({x1 , . . . , xn }) ∩ J ({x1 , . . . , xn }) for all 1 ≤ i ≤ m, . −
+
(70) The microlocal spectrum condition for the OPE-coefficients is the statement that WF(C) Un ⊂
m,n ,
(71)
m≥0
where Un is some neighborhood of {(x, x, . . . , x) ∈ M n }, and where “WF” is the wave front set of a distribution [26]. The microlocal condition in the above form (71) is similar in nature to a condition that was obtained by [3] for the n-point correlation functions of Wick powers in Hadamard states in the context of linear field theory. The difference to the above condition is that also interaction vertices are now allowed, which were not considered in [3]. These interaction vertices correspond to the contributions m ≥ 1 in (71) and genuinely weaken the bound on the wave front set relative to the linear case (for n ≥ 4). The interaction vertices arise from the non-linear interactions present in the theory. As we will see, the maximum valence of the interaction vertices allowed in WF(C) is equal to the maximum power of the field φ that appears in the interaction Lagrangian L. Since we restrict ourselves to renormalizable interactions in 4 spacetime dimensions, that maximum valence is equal to 4. Note, however, that the wave front condition (71) is only an upper bound, and does not say whether interaction vertices will actually contribute to WF(C) or not. We have checked this for the OPE-coefficient in front of the identity operator in the expansion of the product φ(x1 ) I · · · φ(x4 ) I of four interacting fields, to first order in perturbation theory in Minkowski space, where a contribution from an interaction vertex in 4,1 would
OPE for Perturbative Quantum Field Theory in Curved Spacetime
23
be allowed according to the above estimate (71). Using our definition (49) of C and using the integrals in [9], we found in this example that such a contribution is actually absent from WF(C). Hence, the estimate (71) is not sharp at this order. Let us now prove the microlocal condition (71). By Eq. (49), the microlocal spectrum condition will follow if we can show that if u n (x1 , . . . , xn ) = j
n
Rnr (Oir (xr ); I ⊗nr ) ,
(72)
r =1
then WF(u n ) ⊂ ∪m n,m . To prove this statement, we expand the retarded products as in Eq. (27), then multiply them using the Wick expansion formula (10), and finally apply the functional j . The result will be the sum of expressions each of which is a product of r ’s, of H ’s and expectation values in j of locally normal ordered expressions, which are integrated over interaction vertices against the smooth test function χ of compact support appearing as an infrared cutoff in the Lagrange density, L. The expectation value in j of any locally normal ordered expression is smooth, the wave front set of the r ’s is given above in (r6), see (30), while the wave front property of H is WF(H ) = {(y1 , k1 ; y2 , k2 ); k1 = p, k2 = − p, p ∈ ∂ V+∗ }, where p is a coparallel, cotangent vector field along a null geodesic (edge) joining y1 , y2 . We now combine these facts using the wave-front set calculus of Hörmander, by which we mean the following theorems about the behavior of the wave front set under the operations of smoothing, and products [26]: Let X, Y be manifolds (in our applications, they are Cartesian powers of M). If K ∈ D (X × Y ) is a distribution and f ∈ D(Y ) a smooth test function, then the distribution u(x) = Y K (x, y) f (y)dµ(y) has wave front set WF (u) ⊂ {(x, k) ∈ T ∗ X ; (x, k; y, 0) ∈ WF(K )}.
(73)
Secondly, let u, v ∈ D (X ) so that [WF(u) + WF(v)] ∩ {0} = ∅. Then the distributional product uv is defined and has wave front set WF(uv) ⊂ {(x, k + p) ∈ T ∗ X ; (x, k) ∈ WF(u) ∪ {0}, (x, p) ∈ WF(v) ∪ {0}}. (74) Applying these rules to the above products of r ’s and H ’s, we essentially obtain that WF(u n ) is a subset of ∪m n,m . For example, the momentum conservation rule in the third line of Eq. (70) follows from the additive and smoothing properties (73), (74) combined with the fact that we are integrating the interaction vertices against the smooth test function, χ , of compact support. Similarly, we obtain the second line from the additive property (74). Finally, we need to prove the support restriction on the interaction vertices in the fourth line of (70). For this, we note that the contribution to the wave front set of u n from the interaction vertices yk arises only from points that are in the support of the interaction, I , i.e., in the support of χ . Let U be an arbitrary small neighborhood of J − ({x1 , . . . , xn }) ∩ J + ({x1 , . . . , xn }), and let χ be a cutoff function supported in U . Then χ − χ is supported outside of the domain of dependence D({x1 , . . . , xn }), and so there exists by (31) a unitary V such that Oik (xk ) I = V Oik (xk ) I V ∗ . (75) k
k
Thus, because of (43), we see that changing χ to χ is equivalent to changing the standard functionals from j (. . . ) to j (V . . . V ∗ ). We claim that this would not, however, change our above wave front argument. Indeed, the only property of the functionals
24
S. Hollands
that was used in the above wave front set argument was that the expectation values of locally normal ordered expressions in j are smooth. This does not change if we change the standard functionals from j (. . . ) to j (V . . . V ∗ ). Consequently, we have shown that contributions to (70) arise only from interaction vertices yk in U . Since U was an arbitrarily small neighborhood of J − ({x1 , . . . , xn }) ∩ J + ({x1 , . . . , xn }), the support restriction in the last line of (70) follows. We next show that the OPE coefficients have the following local and covariance property: Let f : M → M be a causality preserving isometric embedding, let C I respectively C I be the OPE coefficients on the respective spacetimes, and let δ > 0 be given. Finally, assume that there are open neighborhoods U ⊂ M and U ⊂ M with f (U ) ⊂ U , where the cutoff functions χ , respectively χ , implicit in the interactions I and I are equal to 1. Then, supposing that in Eq. (44) is chosen as in Theorem 1, we have on U , f ∗ C I ∼T ,δ C I for all trees T . (76) This condition essentially follows from the fact that the interacting fields are local and covariant, in the sense of (13), which follows in turn from the fact that the individual terms in the perturbation expansion of the interacting fields are local and covariant. However, a complication arises from the fact that the algebra embedding α f in (13) is not simply given in terms of the corresponding free field homomorphism, but is more complicated [22]. Instead of taking into account the more complicated definition of α f at the interacting level, one can also more directly prove (76). For this, we note that, if the cutoff function χ on M were such that f ∗ χ = χ , then we would have equality in (76), because the retarded products and standard functionals which are the ingredients in the definition of C I have a local and covariant dependence simultaneously on both χ and the metric g implicit in I , by property (r5). Thus, it is sufficient to show that C I is essentially independent of the cutoff function χ . In other words, if χ and χ are two cutoff functions (on the same spacetime) which are equal to 1 on U , and if C I and C I are the corresponding OPE coefficients, then we must show that C I ∼δ,T C I
(77)
holds on U . To prove this statement, we simply apply the functionals (V . . . V ∗ ) I to the remainder N I of the OPE formed using the coefficients C I , where V is the unitary in Eq. (31) relating the interactions I and I . Then we find (V N I V ∗ ) I = C I − C I . Since N I is the remainder of the OPE, and since (V . . . V ∗ ) I is a Hadamard functional, it follows that ε−δ (C I − C I ) ◦ ψT (ε) will go to zero by Theorem 1, which is what we needed to show. In [22], it was shown that the perturbative interacting fields obey a “local covariant renormalization group flow”. The construction of this flow involves the consideration of a 1-parameter family λ2 g of conformally rescaled metrics, where λ ∈ R>0 , and states how the interacting fields Oi (x) I change under such a rescaling. In [24], a simple general argument was given that the existence of such a local covariant renormalization group flow implies a corresponding flow of the OPE coefficients if the theory possesses an OPE in the sense described in the previous section. The key assumption on the nature of the RG made in [24] was that there exists a suitable “basis” of functionals which is in some sense “dual” to the fields. This is the case in perturbation theory, on account of (44). Hence, it follows by the argument of [24] that Z (λ)i1 j1 · · · Z (λ)i1 j1 [t Z (λ)−1 ]k l Ci1 ...in k [M, g] I ∼T ,δ C j1 ... jn l [M, λ2 g] I (λ)
(78)
OPE for Perturbative Quantum Field Theory in Curved Spacetime
25
for all trees T . Here, the prefactors are linear maps (whose construction and properties was described in [22]) (79) Z (λ)i j ∈ End(E j , Ei ), where Ei is the vector bundle in which the field Oi lives, and I (λ) is an interaction of the same form as I with suitable “running” couplings κi (λ), whose construction was also described in [22]. An associativity property for the OPE coefficients may be formulated as follows (see [24] for details). Let x(ε) be a curve in configuration space representing the merger of the points according to a tree T , that is, ψT (ε) : x → x(ε). In this situation, we should be able to perform the OPE successively, in the hierarchical order represented by the tree, thus leading to some kind of “asymptotic factorization”. That is, we should be allowed to first perform the OPE for each subtree, and then successively the OPE’s corresponding to the branches relating the subtrees, and so forth. For example, for the tree T given in the figure following Eq. (38), we should be allowed to perform the OPE successively as indicated by the brackets (O1 O2 )(O3 O4 ). To formulate this condition more precisely, we recall the notation s(e), t (e) for the source and targets of an edge, e, in the tree T . Furthermore, for x = (x1 , . . . , xn ) ∈ M n , and for each node S ∈ T of the tree, let us set (80) x S = xm(S) , m(S) = max{i : i ∈ S}. Finally, we consider maps i : T → I which associate with every node S ∈ T of the tree an element i S ∈ I, the index set labelling the fields. With these notations in place, the associativity property can be stated as follows. Let δ > 0, let T be a tree, and let M Sn be the set of all “spacelike configurations” x = (x1 , . . . , xn ), x ∈ M0n ; xi ∈ / J + (x j ) ∪ J − (x j ) forall i, j}. M Sn = {
(81)
Then, on M Sn , we have Ci1 ...in j (x1 , . . . , xn ) I ∼T ,δ C{it (e) ;e such that s(e) = S} i S xt (e) ; e such that s(e) = S I , (82) i S∈T
where the sum is over all i, with the properties that i {k} = i k , k = 1, . . . , n, i {1,...,n} = j, [i t (e) ] < δ S ∀S ∈ T ,
(83) (84)
e:s(e)=S
where δ S > 0 are chosen sufficiently large. Note that it makes sense to consider the relation ∼δ,T with respect to the open subset of spacelike configurations, because a configuration remains spacelike when scaled down by ψT (ε), at least provided the points x ∈ M Sn are in a sufficiently small neighborhood of the total diagonal, which we assume is the case. The technical reason for restricting the OPE to pairwise spacelike related points is that the OPE coefficients C are smooth on M Sn , by the microlocal spectrum condition (see Eq. (71)), and so convergence in the sense of ∼δ,T is more straightforward to study. Furthermore, since the interacting fields commute for spacelike related points, there are no ordering issues when working with the configurations in M Sn . Also, from
26
S. Hollands
a physical viewpoint, the notion of “short distances” is somewhat unclear if lightlike directions are included. In [24], it is shown that associativity in the above sense is an automatic consequence if the theory also possesses a suitable local covariant renormalization group with suitable properties, and if the OPE holds not only for each fixed spacetime and fixed choice of couplings, but instead also uniformly in a suitable sense for smooth families of metrics and couplings (termed “condition (L)” in that paper). As we have already described, the existence of a local covariant renormalization group in perturbation theory was established in [22]. It is a general property of the perturbative renormalization group that Z i j (λ) is given by λ−[i] times a polynomial in ln λ at each finite order in perturbation theory [22], and that the running couplings κi (λ) in I (λ) have a power law dependence λ4−[i] , which is modified by polynomials in ln λ at any given order in perturbation theory. These are essentially the properties for the renormalization group required in (L) at the perturbative level, except that the running of couplings I (λ) is not exactly smooth at λ = 0 as required in (L), but instead contain logarithmic terms at each order in perturbation theory. However, the argument given in [24] is insensitive to such logarithmic corrections. To also establish the desired smooth dependence of the OPE-coefficients under smooth variations of the metric required in (L) at the perturbative level, it is necessary to go through the proof of Theorem 1 for families of spacetimes and corresponding families of states depending smoothly on a parameter in the sense [21] and analyze the behavior of the constructions under variations of the parameter. This can indeed be done, using the smooth dependence of the retarded products under such parameters [21], as well as the techniques and type of arguments employed in the Appendix of [23]. However, even though the repetition of these arguments is in principle straightforward, that analysis is quite lengthy and cumbersome, and not very illuminating. It is therefore omitted. Since a perturbative version of condition (L) holds, Theorem 1 of [24] then implies that the associativity property holds on the space M Sn of pairwise spacelike configurations. In the remainder of this section we will prove that the OPE coefficients themselves can be expanded for asymptotically small distances in terms of curvature terms and Minkowski distributions in the tangent space (for spacelike related configurations in M Sn , to which we shall restrict ourselves in the remainder of this section). The construction of this “scaling expansion” involves the Mellin transform, M[ f, z], of a function f (x) defined on R>0 vanishing near infinity, with at most polynomial type singularity [38] as x → 0. It is defined by ∞ M[ f, z] = x iz−1 f (x) d x, (85) 0
and is an analytic function of z for sufficiently small Im(z) < y0 , where y0 depends upon the strength of the singularity of f . The inverse Mellin transform of F(z) = M[ f, z] is given by +i∞+c 1 M−1 [F, x] = x z F(z) dz, (86) 2π i −i∞+c where the integration contour is to the right of all poles of F(z) in the complex z-plane. The Mellin transform is useful in the context of functions f (x) possessing near x = 0 an asymptotic expansion of the form f (x) ∼
p
l
a p,l x − p (ln x)l ,
(87)
OPE for Perturbative Quantum Field Theory in Curved Spacetime
27
where p is bounded from above, and where the sum over l is finite for any p. It can be seen that the Mellin transform of such a function possesses isolated poles at z = i p in the complex plane, with finite multiplicities. Furthermore, the asymptotic expansion coefficients a p,l are the residues of the Mellin transform, i.e., 1 a p,l = Resz=−i p (z + i p)l M[ f, z] . (88) l! We now define, for each tree T , distributions C IT that give the desired scaling expansion of the OPE coefficient C I relative to the scaling function ψT (ε) : M0n → M0n , x → x(ε) defined above in Eq. (39), by extracting the poles of C I ◦ ψT (ε) in ε using the Mellin transform. In order to do this, let x = (x1 , . . . , xn ) ∈ M Sn be a spacelike configuration of n points, let T be a tree, and let ε → x(ε) be the corresponding curve in M Sn . By the microlocal spectrum property, the OPE coefficients are smooth on M Sn , so we may consider [C I ◦ ψT (ε)]( x ) = C I ( x (ε)) as a smooth function in ε at any fixed value of the argument x. As we will show in the proof of the next theorem, if we fix the parameter δ > 0 in the operator product expansion, this function has an expansion of the form C I ( x (ε)) = a p,l ( x )ε− p (ln ε)l + . . . , (89) p
l
near ε = 0, where the dots stand for a remainder vanishing faster than εδ . Here, p is in the range from −δ to D = depth(T ) · (−[k] + j [i j ]), and the sum over l is finite for each p, at any given order in perturbation theory7 . Consequently, we can define the Mellin transform8 of this function in the variable ε, ∞ T M ( x , z) ≡ M[C I ◦ ψT (ε), z] = C I ( x (ε)) εiz−1 dε, (90) 0
which is now a function of x ∈ M Sn that is analytic in z ∈ C for sufficiently small Im(z). Furthermore, by the above expansion (89), it is meromorphic on a domain including Im(z) ≤ δ, with poles possibly at iδ, i(δ − 1), i(δ − 2), . . . , −iD. Let us now choose a contour C around these points as illustrated in the figure. Im(z)
C
δ δ−1 .. . −D + 1 −D
Re(z)
7 Note however that the range of l increases with the perturbation order. 8 To make this expression well defined, we need to arbitrarily cut off the integral for large ε (where the
map ψT (ε) is not well-defined anyway). How this cutoff is chosen does not affect the following discussion.
28
S. Hollands
1 x )I ≡ MT ( x , z) dz. C ( 2π i C Concerning this function on M Sn , we have the following theorem.
Define
T
(91)
Theorem 2. 1. We have C IT ∼T ,δ C I for spacelike configurations, and therefore CiT1 i2 ...in k (x1 , x2 , . . . , xn ) I Ok (xn ) I Oi1 (x1 ) I Oi2 (x2 ) I · · · Oin (xn ) I ∼T ,δ [k]≤
(92) on M Sn . 2. C IT is local and covariant, i.e., if f : (M, g) → (M , g ) is an orientation, causality preserving isometric embedding then C IT = f ∗ C IT . In particular, C IT does not depend on the choice of cutoff function χ used in the definition of the interacting field. 3. The expression C IT ( x ) is the sum of residue of MT (z, x) corresponding to the poles in the contour C in (91), C IT ( x) = Resz=−i p MT ( x , z) . (93) p≥−δ
These have the following form: x , z) = Pa [∇(α1 · · · ∇αk ) Rµ1 µ2 µ3 µ4 (xn )] W a (ξ1 , . . . , ξn−1 ) , Resz=−i p MT ( a
(94) where Pa is a polynomial in the Riemann tensor and (finitely many) of its covariant derivatives evaluated at xn , valued in some tensor power of the tangent space (Txn M)⊗a , while ξi are the Riemannian normal coordinates of x1 , . . . , xn−1 around xn , identified with vectors in R4 via a tetrad. The sum over a is finite, and each Wa is a Lorentz covariant distribution on R4(n−1) (defined on spacelike configurations), that is valued in (R4 )⊗a (identified with (Tx∗n M)⊗a via the tetrad), depending polynomially on m 2 , α. Thus, for any proper, orthochronous Lorentz transformation , we have Wa (ξ1 , . . . , ξn−1 ) = D b a ()Wb (ξ1 , . . . , ξn−1 ), (95) where D() is the corresponding tensor representation. 4. There are Lorentz invariant distributions Va,l such that N l Wa (ξ (ε)) = ε (ln ε) Va,l (ξ ) , Wa (ξ ) +
(96)
l
where the sum is only over a finite range of l, at any given, but finite order in perturbation theory. Remarks. 1) Equations (93), (94) constitute the claimed scaling expansion. That scaling expansion is similar in nature to a corresponding expansion derived in [21] for the short distance behavior of time ordered products. However, note that the scaling expansion above is more general than that derived in [21], because it involves the consideration of more general scaling functions ψT (ε) corresponding to general trees. Also, the Mellin transformation technique was not used in [21].
OPE for Perturbative Quantum Field Theory in Curved Spacetime
29
2) The restriction to spacelike configurations has mainly been made for technical reasons, to avoid technical issues that could arise when taking the Mellin transformation of a distribution. However, we expect that all constructions and properties summarized in the above theorem hold for all configurations, i.e., in the sense of distributions on M n . Proof. Let us first argue that the claimed meromorphicity of the Mellin transform of C( x (ε)) I holds, or equivalently, that it has an asymptotic expansion as claimed in Eq. (89). For this, we first consider the simple tree T = {S0 , S1 , . . . , Sn } with one root S0 = {1, . . . , n} and n leaves Si = {i}. The depth of this tree is depth(T ) = 1, and ψT (ε) is simply the map which multiplies the Riemann normal coordinates of xi relative to xn by ε. To analyze the corresponding scaling of the OPE coefficients, we note that, by the local and covariance property of the OPE coefficients, a rescaling of the arguments xi is equivalent to changing the metric from g to sε∗ g, where sε : M → M is the diffeomorphism that scales the Riemannian coordinates of a point around xn by ε. Thus, we have C I [M, g] ◦ ψT (ε) ∼T ,δ C I [M, sε∗ g]
(97)
for this tree. Next, we use the fact that in perturbation theory, there exists a local and covariant renormalization group [22]. This implies that, up to terms of order εδ , Z (ε)ij11 · · · Z (ε)ij11 [t Z (ε)−1 ]k l Ci1 ...in k [M, g(ε)] I (ε) = C j1 ... jn l [M, g] I ◦ ψT (ε), (98) where g(ε) = ε2 sε∗ g. The point is now that the metric g(ε) has a smooth dependence upon ε, as may be seen by rewriting it in Riemannian normal coordinates as gµν (εξ )dξ µ dξ ν . The OPE coefficients in turn have a smooth dependence upon smooth variations of the metric, since all the quantities in their definition have this property [21]. The only singular terms (in ε) in the expression on the left side can therefore come from (a) the running couplings in I (ε) and (b) the Z (ε)-factors. However, by the general analysis of the local renormalization given in [22], these quantities are polynomials in ε− p (ln ε)l of finite degree to any given order in perturbation theory. Thus, up to terms vanishing faster than εδ , the quantity C I ( x (ε)) has an expansion of the type (89), as desired. The argument just given may be generalized to arbitrary trees by an induction in the depth of the tree. For trees of depth one we have just proven the statement. Let us inductively assume that we have proven (89) for trees of depth d. To deal with trees of depth d + 1, one notices that a tree of depth ≥ 2 can always be decomposed into a tree S of depth one connected to the root, and trees T1 , . . . , Tr attached to the leaves of S. Thus, we may write T =S∪
r t=1
see the figure for an example with r = 2.
Tt ,
(99)
30
S. Hollands
root = x6 S0 S = {S0 , S1 , S2 } S1
S2
T2 = {S1 , S3 , S4 }
T1 = {S2 , S5 , . . . , S9 } S3
S4
S6
T = {S0 , S1 , . . . , S9 }
S5
S7 S8 S9
The key point is now that the map ψT (ε) factorizes by the inductive nature of its definition (39), while the OPE coefficient C I factorizes by the associativity property (82) under this decomposition of the tree. This gives r jt k k Ci1 ...in ◦ ψT (ε) ∼T ,δ C j1 ... jr · C{i S ; S∈Leaves(Tt )} ◦ ψTt (ε) ◦ ψS (ε), (100) t=1
where we note that, since S has depth one, the map ψS (ε) is given in terms of the diffeomorphism sε : M → M which scales the Riemann normal coordinates around xn of points by ε. On the right side of this expression, we can now apply the induction hypothesis to the expression in brackets, because each of the trees Tt has depth ≤ d. Furthermore, since ψS (ε) is given by the diffeomorphism sε , we may again apply the general covariance and renormalization group property as we did above to convert the action of sε into a smooth change g(ε) in the metric and Z (ε)-factors depending polynomially on ε− p (ln ε)l . This proves Eq. (89). We next come to the actual proof of the theorem. To prove 1), we need to show that ε−δ (C I − C IT ) ◦ ψT (ε) → 0 as ε → 0, to any finite but arbitrary order in perturbation theory. We have, using the definition of C IT and the relation ψT (ε)◦ψT (ε ) = ψT (εε ), ∞ T C I ◦ ψT (ε) = Resz=−i p (C I ◦ ψT (ε )) ◦ ψT (ε) εiz−1 dε p≥−δ
=
0
Resz=−i p eiz ln ε MT ( x , z)
p≥−δ
=
ε− p
p≥−δ
=
p≥−δ
(ln ε)l l
l!
Resz=−i p (z + i p)l MT ( x , z)
ε− p (ln ε)l a p,l ( x ),
(101)
l
where we have performed a change of integration variables in the second step. Comparing with (89), this formula implies that C T ( x (ε)) I differs from C( x (ε)) I by a term vanishing faster than εδ . This proves the assertion 1). To prove 2), we recall that we have already proven above that, at each order in perturbation theory, f ∗ C I [M , g ] ∼δ,T C I [M, g], so the difference between these two
OPE for Perturbative Quantum Field Theory in Curved Spacetime
31
terms vanishes faster than εδ . This difference term will change the Mellin transform MT (z) only by a term that is analytic in a domain including Im(z) ≤ δ, and thus will not contribute to C IT [M, g] respectively C IT [M , g ], because the contour integral of a holomorphic function vanishes. For 3), consider again the metric g(ε) = ε2 sε∗ g. Its components in Riemann nor N ε PN mal coordinates around xn have a Taylor expansion of the form gµν (εξ ) = (∇ k Rα1 α2 α3 α4 (xn ), ξ ρ )µν , where PN are polynomials. Since the C IT are local and covariant by 2), it follows that they can be viewed as functionals of the Riemann tensor and its covariant derivatives at point x (or ξ = 0) which enters via PN . Thus, it follows that the N th ε-derivative is ∂N T N1 ...Nr C [g(ε)]( x )| = W ( ξ ) PNi [∇(α1 · · · ∇αk ) Rµ1 µ2 µ3 µ4 (xn ), ξ νj ], ε=0 I ∂ε N N1 +···+Nr =N
i
(102) where W N1 ...Nr =
∂ r C IT . ∂ PN1 · · · ∂ PNr g=η
(103)
We define W a and Pa by the above relation, i.e., Pa is the appropriate product of the PNi , with the polynomial ξi -dependence taken out and absorbed in the definition of W a . To prove the desired relation (94), we must now show that the ε-derivatives of C IT [g(ε)] at ε = 0 vanish for N sufficiently large. This quantity arises from the quantity C I [g(ε)] ◦ ψT (ε ) by taking a Mellin transform in ε , then taking N derivatives with respect to ε at ε = 0, and finally extracting the residue in z. It follows that we only need to show that the quantity obtained by taking N derivatives with respect to ε of C I [g(ε)] ◦ ψT (ε ) vanishes faster than ε δ , because such a term would not give rise to poles of the Mellin transform in the domain Im(z) ≤ δ. Consider first the case when T has depth one, so that ψT (ε ) is simply a dilation of the Riemann normal coordinates by ε . As above in Eq. (98), by combining the renormalization group and general covariance, the action of such a dilation may be translated into changing g(ε) to g(εε ), along with a suitable set of Z (ε )-factors, and running couplings in I (ε ). The point is now that the N derivatives on ε will produce, when acting on g(εε ), precisely N positive powers of ε . Thus, if N is sufficiently large, then the resulting positive powers will dominate the corresponding negative powers in the Z (ε )-factors, and we get the desired result. The generalization of this argument to arbitrary trees T can be done via an induction argument based upon formula (100), similar to the one given there. To prove the Lorentz invariance of the W a , let us now define, for each Lorentz trans↑ formation ∈ SO(3, 1)0 the diffeomorphism f : ξ → ξ . It defines a causality and orientation preserving isometric embedding between the spacetimes (M, g) and (M, f ∗ g = g ) with the same orientations. Thus, using the local covariance property f ∗ C T [g] = C T [g ], and the transformation property of f ∗ P a = D a b ()P b under this diffeomorphism, it follows that Pa [∇ k Rα1 α2 α3 α4 (xn ), ξiν ] W a (ξ1 , . . . , ξn−1 ) a
=
Pa [∇ k Rα1 α2 α3 α4 (xn ), ξiν ] D a b ()W b (ξ1 , . . . , ξn−1 ). (104)
a,b
However, since this holds for all metrics, Eq. (95) follows.
32
S. Hollands
Using the definition of W a just given, item 4) immediately follows from the fact that C I has an expansion of the form (89), and that C IT ∼δ,T C I . 5. Example We now illustrate our general method for computing the OPE in curved spacetime by an example. Let us summarize again the steps needed in this computation. 1. Fix a desired accuracy, δ, of the OPE, and determine as in Theorem 1. 2. Identify the retarded products in Eq. (49) that are needed to compute the desired OPE coefficient to a given order in perturbation theory and a given accuracy δ, and determine them, using the methods of the papers [20, 21]. 3. Perform a local “Wick expansion (27) of all retarded products. In places in (49) where two retarded products are multiplied, perform the products using Wick’s theorem (10) (with ω2 in that formula replaced by H ). Apply the standard functionals (45) to the resulting expressions in the way indicated in (49). This yields the desired OPE coefficient. If one is interested in the asymptotic behavior of the OPE coefficient (up to order δ in ε) under a rescaling of a point ψT (ε) : x → x(ε) associated with a given tree T , perform the following step: 4. Take the Mellin-transform of C I ( x (ε)) in ε as in Eq. (90), and define C IT ( x ) to be the sum of its residue at the poles iδ, i(δ − 1), . . . , as in Eq. (91). The result then automatically has the form of curvature terms times Minkowski distributions in the relative coordinates as described in item 2) of Theorem 2, and it is equivalent to C I in the sense that C IT ∼T ,δ C I . As an illustration, we now apply this method to the determine the coefficient C I in the triple product of operators φ(x1 ) I φ(x2 ) I φ(x3 ) I = · · · + C(x1 , x2 , x3 ) I φ(x3 ) I + . . . (105) up to first order in perturbation theory in the interaction I = M L dµ = − 4!1 M κφ 4 dµ, and accuracy δ = 0. We then discuss the scaling expansion as in item 2) of Theorem 2. As discussed above, we impose an infrared cutoff by taking κ(x) = κχ (x) at intermediate steps, where χ is a smooth cutoff function, but the final answers will not depend on the choice of χ , see Eq. (77). The different ways of scaling the 3 points together give rise to the different limiting behavior C IT of C I . We choose to investigate the most interesting case when all points are scaled together at the same rate, i.e., under the scaling map ψT (ε) : x → x(ε), where T = {{1, 2, 3}, {1}, {2}, {3}}. If ξi are the Riemannian normal coordinates of the points xi around x3 , this corresponds to ξ (ε) = (εξ1 , εξ2 , εξ3 ) with ξ3 = 0. Consider first the zeroth order perturbative contribution. According to Eq. (49), this is given for a general OPE coefficient of a triple product of operators by Ci1 i2 i3 j (x1 , x2 , x3 )(0) = j (Oi1 (x1 )Oi2 (x2 )Oi3 (x3 )),
(106)
where the reference point for the functional j (see Eq. (45)) is x3 throughout. We are interested in the case Oi1 = Oi2 = Oi3 = O j = φ. In order to determine the action of the functional j in this case, we need to perform the local Wick expansion of the product φ(x1 )φ(x2 )φ(x3 ), which is given by φ(x1 )φ(x2 )φ(x3 ) = : φ ⊗3 :H (x1 , x2 , x3 ) + H (x1 , x2 )φ(x3 ) + cyclic(1, 2, 3). (107)
OPE for Perturbative Quantum Field Theory in Curved Spacetime
33
Applying now the definition of the functional j (with O j chosen to be φ, and reference point x3 ) gives C(x1 , x2 , x3 )(0) = H (x1 , x2 ) + H (x2 , x3 ) + H (x1 , x3 ),
(108)
for any δ. In order to determine the representer C IT to zeroth order in perturbation theory for our choice δ = 0 (see Eq. (91)), we are instructed to compose the above result with the map ψT (ε), then take a Mellin transform in the variable ε, and then extract the residue at the poles 0, −i, −2i in the complex z-plane via the contour integral (91). As explained above, taking a Mellin transform and then extracting those residues is a way to compute the corresponding coefficients of ε0 , ε−1 , ε−2 in the expansion (89) of the distribution C I ◦ ψT (ε) in ε. In the present simple example, it is easier to compute the coefficients directly from Eq. (108), by using the corresponding short distance expansions [10] around x3 of the quantities σ, vn appearing in the Hadamard parametrix H . The result is T
C (x1 , x2 , x3 )(0)
1 1 = 2π 2 (ξi − ξ j )2 i< j
1 R(ξi , ξ j , ξi , ξ j ) 1 R(ξi , ξ j ) + R(ξi , ξi ) + R(ξ j , ξ j ) 1 2 , R ln(ξ − − + − ξ ) i j 3 (ξi − ξ j )4 6 (ξi − ξ j )2 12 (109) where R(X 1 , X 2 , X 3 , X 4 ) stands for the Riemann tensor at y = x3 evaluated on 4 vectors in Ty M, R(X 1 , X 2 ) for the Ricci tensor, and R for the scalar curvature. We are also using the notation X 2 = g(X, X ) for the scalar product in the tangent space Ty M. By item 1) of Theorem 2, we know that C I ∼T ,0 C IT . We next consider the first order perturbative contribution to a general OPE coefficient for a general triple product. By formula (49) this is given by Ci1 i2 i3 j (x1 , x2 , x3 )(1) = i −
j (Oi1 (x1 )Oi2 (x2 )R1 (Oi3 (x3 ), L(y)))+cyclic(1, 2, 3) M
(R1 (Ok (x3 ), L(y)) (Oi1 (x1 )Oi2 (x2 )Oi3 (x3 )) χ (y) dµ(y). j
k
(110)
[k]≤
We are again interested in the case Oi1 = Oi2 = Oi3 = O j = φ. The constant depends on the desired precision of the OPE governed by δ, and is given in Thm. 1. For our choice δ = 0, we have to choose = 3. In this case, it can be seen that only Ok = φ 3 will make a contribution. Thus, the required retarded products and their Wick expansion in Eq. (110) are R1 (φ n (x), φ m (y)) =
4 k=0
n!m! rk (x, y) : φ n−k (x)φ m−k (y) :H (n − k)!(m − k)!
(111)
for n = 1, 3 and m = 4. Consequently, we need to know rk for k = 1, 3. The method [21] for defining the “renormalized” distribution gives r1 = i R , where R is the retarded
34
S. Hollands
propagator, and ln(σ + i0t) 1 3 2 v0 A (−t) 32 σ + i0t n ln2 (σ + i0t) + 21 ln(σ + i0t) σ 3 + v02 vn+1 n A (−t) 2 2 n! σ + i0t n σ n+m + (−t) vm+1 vn+1 n+m 2 n!m! m,n
r3 = −
ln2 (σ + i0t) σk × 3v0 + vk+1 k ln3 (σ + i0t) − h.c., σ + i0t 2 k!
(112)
k
where t (x, y) = τ (x) − τ (y), for any time function τ : M → R. The symbol A stands for the operator9 A = + (∇ µ ln D)∇µ , (113) where D is the VanVleck determinant defined above in Eq. (17). The desired OPE coefficient can now be obtained. The result is C(x1 , x2 , x3 )(1) = iκ [H (x1 , y)H (x2 , y)r1 (x3 , y) M
+ cyclic(1, 2, 3) − r3 (x3 , y)]χ (y) dµ(y),
(114)
where H is the local Hadamard parametrix. To get the desired representer C IT provided by Theorem 2, we again use the definitions Eqs. (90) and (91) corresponding to our choice δ = 0. Thus, we must compose C I with ψT (ε), take a Mellin transform, and extract the residue at 0, −i, −2i in the complex z-plane. To do the Mellin transform (90), we perform first a short distance expansion of the integrand in (114) around x3 , using the corresponding expansions [10] of the quantities σ, vn present in H, ri . This short distance expansion will lead to a sum of terms, each of which is a curvature polynomial at x3 , times a Minkowski distribution in ξ1 , ξ2 and the Riemannian normal coordinates of y. We may also set χ = 1, since the residues are independent of the particular choice of χ , as proven in Thm. 2. Thus, the computation of the Mellin-transform reduces to ordinary Minkowski integrals, times curvature polynomials at x3 . Only a finite number of these terms will give rise to poles at 0, −i, −2i, and so all others can be discarded. Let us consider in detail the pole −2i. For this pole, the Minkowski space integrals that contribute can be reduced to the integral given in [30] using some standard “i0-identities” for distributions: C T (x1 , x2 , x3 )(1) =
3 κ a(ξ1 , ξ2 , ξ3 ) 1 + . . . , (115) Cl2 2 arctan 26 π 4 a(ξ1 , ξ2 , ξ3 ) (ξi+1 − ξi+2 )(ξi+2 − ξi ) i=1
where the dots stand for the contributions from the poles 0, −i, and where ξi = ξi mod 3 . Also, we have set (116) a(ξ1 , ξ2 , ξ3 ) = |δ1 δ2 + δ2 δ3 + δ3 δ1 |, 9 The usefulness of this operator lies in the identity A(σ n ) = 4n(n + 1)σ n−1 .
OPE for Perturbative Quantum Field Theory in Curved Spacetime
35
where δ1 = (ξ2 − ξ3 )(ξ3 − ξ1 ), δ2 = (ξ1 − ξ2 )(ξ2 − ξ3 ), δ3 = (ξ3 − ξ1 )(ξ1 − ξ2 ), (117) and as before it is assumed that the points ξi ∈ R4 are pairwise spacelike, and ξ3 = 0. The function Cl2 is the Clausen function [29]. The arguments of these functions in the are given by twice the angles of a triangle with sides of length above expression (ξ1 − ξ2 )2 , (ξ2 − ξ3 )2 and (ξ3 − ξ1 )2 , and a is the area of that triangle, see the figure. ξ1 (ξ1 − ξ2 )2
ξ2
(ξ3 − ξ1 )2 ξ3
(ξ2 − ξ3 )2 ρ = Area of triangle
Thus, the result for C IT manifestly has the simple form claimed in the scaling expansion in Theorem 2. There are no curvature terms in the terms that we have displayed, but those arise from the other poles 0, −i. These contributions may be obtained in closed form using the Minkowski integrals of [8]. The end result is a sum of terms µνσρ µνσρ (ξ1 , ξ2 , ξ3 ), where Wa are Lorentz invariant distributions in the a Rµνσρ Wa Riemannian normal coordinates ξi of xi around x3 . However, the expressions for these distributions are rather complicated and will therefore be given elsewhere [25]. It should be clear that our method is not confined to the above example, but in principle only limited by the ability to perform complicated Minkowski integrals of Feynman type. Acknowledgements. I have benefitted from conversations with D. Buchholz, N. Nikolov, and R.M. Wald. I would like to thank M. Dütsch for reading an earlier version of the manuscript.
References 1. Axelrod, S., Singer, I.M.: Chern-Simons perturbation theory. 2. J. Diff. Geom. 39, 173 (1994) 2. Bostelmann, H.: Operator product expansions as a consequence of phase space properties. J. Math. Phys. 46, 082304 (2005); Phase space properties and the short distance structure in quantum field theory. J. Math. Phys. 46, 052301 (2005) 3. Brunetti, R., Fredenhagen, K., Köhler, M.: The microlocal spectrum condition and Wick polynomials on curved spacetimes. Commun. Math. Phys. 180, 633–652 (1996) 4. Brunetti, R., Fredenhagen, K.: Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on physical backgrounds. Commun. Math. Phys. 208, 623–661 (2000) 5. Brunetti, R., Fredenhagen, K., Verch, R.: “The generally covariant locality principle: A new paradigm for local quantum physics.” Commun. Math. Phys. 237, 31 (2003); see also K. Fredenhagen, “Locally covariant quantum field theory.” In: XIVth Int. Congress on Mathematical Physics (Lisbon 2003), RiverEdge Co: World Scientific, 2006 6. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite Conformal Symmetry In Two-Dimensional Quantum Field Theory. Nucl. Phys. B 241, 333 (1984) 7. Buchholz, D., Wichmann, E.H.: Causal independence and the energy level density of states in local quantum field theory. Commun. Math. Phys. 106, 321 (1986) 8. Davydychev, A.I.: Some exact results for N -point massive Feynman integrals. J. Math. Phys 32, 1052– 1060 (1991)
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9. Denner, A., Nierste, U., Scharf, R.: A Compact expression for the scalar one loop four point function. Nucl. Phys. B 367, 637 (1991) 10. DeWitt, B.S., Brehme, R.W.: Radiation Damping In: A Gravitational Field. Annals Phys. 9, 220 (1960) 11. Dütsch, M., Fredenhagen, K.: A local (perturbative) construction of observables in gauge theories: the example of QED. Commun. Math. Phys. 203, 71 (1999) 12. Dütsch, M., Fredenhagen, K.: Algebraic quantum field theory, perturbation theory, and the loop expansion. Commun. Math. Phys. 219, 5 (2002); Perturbative algebraic field theory, and deformation quantization. In: Proc. of the Conf. on Mathematical Physics in Mathematics and Physics (Siena 2000), Providence, RI: Amer. Math. Soc., 2001 13. Dütsch, M., Fredenhagen, K.: The Master Ward Identity and Generalized Schwinger-Dyson Equation in Classical Field Theory. Commun. Math. Phys. 243, 275 (2003) 14. Dütsch, M., Fredenhagen, K.: Causal Perturbation Theory in Terms of Retarded Products, and a Proof of the Action Ward Identity. http://arxiv.org/list/2004, hep-th/0403213, to appear in Rev. Math. Phys. 15. Fredenhagen, K., Hertel, J.: Local algebras of observables and point - like localized fields. Commun. Math. Phys. 80, 555 (1981); K. Fredenhagen, M. Jorss, “Conformal Haag-Kastler nets, point - like localized fields and the existence of operator product expansions. Commun. Math. Phys. 176, 541 (1996) 16. Friedlaender, F.G.: The Wave Equation on Curved Space-Time. Cambridge: Cambridge University Press 1975 17. Fulton, W., MacPherson, R.: A compactification of configuration spaces. Ann. Math. 139, 183 (1994) 18. Haag, R.: On Quantum Field Theories. Z. Phys. 141, 217 (1955) [Phil. Mag. 46, 376 (1955)] 19. Hollands, S., Ruan, W.: The state space of perturbative quantum field theory in curved space-times. Annales Henri Poincare 3, 635 (2002) 20. Hollands, S., Wald, R.M.: Local wick polynomials and time ordered products of quantum fields in curved space. Commun. Math. Phys. 223, 289–326 (2001) 21. Hollands, S., Wald, R.M.: Existence of local covariant time-ordered-products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309–345 (2002) 22. Hollands, S., Wald, R.M.: On the Renormalization Group in Curved Spacetime. Commun. Math. Phys. 237, 123–160 (2003) 23. Hollands, S., Wald, R.M.: Conservation of the stress tensor in interacting quantum field theory in curved spacetimes. Rev. Math. Phys. 17, 227 (2005) 24. Hollands, S., Wald, R.M.: On the Operator Product Expansion in Curved Spacetime. In preparation 25. Hollands, S.: in preparation 26. Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Berlin: Springer-Verlag, 1983 27. Kac, V.: Vertex algebras for beginners. AMS University Lecture Series 10, 2nd edition, Providence, R.I. Amer. Math. Soc., 1998; I. Frenkel, J. Lepowsky, A. Meurman. Vertex Operator Algebras And The Monster. Pure and Appl. Math. 134, Boston, MA: Academic Press, 1988 28. Källen, G.: Formal integration of the equations of quantum theory in the Heisenberg representation. Ark. Fysik 2, 371 (1950) 29. Lewin, L.: Polylogarithms and associated Functions, New York: Elsevier North Holland, 1981 30. Lu, H.J., Perez, C.A.: Massless one loop scalar three point integral and associated Clausen, Glaisher and L functions. SLAC-PUB-5809 available at http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub5809.pdf, 1992 31. Luscher, M.: Operator product expansions on the vacuum in conformal quantum field theory in two space-time dimensions. Commun. Math. Phys. 50, 23 (1976) 32. Mack, G.: Convergence of operator product expansions on the vacuum in conformal invariant quantum field theory. Commun. Math. Phys. 53, 155 (1977) 33. Moretti, V.: Comments on the stress-energy tensor operator in curved spacetime. Commun. Math. Phys. 232, 189 (2003) 34. Radzikowski, M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179, 529 (1996) 35. Schroer, B., Swieca, J.A., Volkel, A.H.: Global operator expansions in conformally invariant relativistic quantum field theory. Phys. Rev. D 11, 1509 (1975) 36. Steinmann, O.: Perturbative Expansions in Axiomatic Field Theory. Lecture Notes in Physics 11, Berlin-Heidelberg-New York: Springer 1971 37. Wilson, K.G.: Nonlagrangian models of current algebra. Phys. Rev. 179, 1499 (1969) 38. Szmydt, Z., Ziemian, B.: The Mellin transformation and Fuchsian type partial differential equations. Dordrecht: Kluwer, 1992 39. Zimmermann, W.: Normal products and the short distance expansion in the perturbation theory of renormalizable interactions. Annals Phys. 77, 570 (1973) and [Lect. Notes Phys. 558, 278 (2000)] Communicated by A. Connes
Commun. Math. Phys. 273, 37–65 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0223-5
Communications in
Mathematical Physics
Double Standard Maps Michał Misiurewicz1, , Ana Rodrigues2, 1 Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, IN 46202-3216, USA.
E-mail:
[email protected]
2 Departamento de Matemática Pura, Centro de Matemática , Universidade do Porto, Rua do Campo
Alegre, 687, 4169-007 Porto, Portugal. E-mail:
[email protected] Received: 29 May 2006 / Accepted: 31 August 2006 Published online: 13 March 2007 – © Springer-Verlag 2007
Abstract: We investigate the family of double standard maps of the circle onto itself, given by f a,b (x) = 2x + a + (b/π ) sin(2π x) (mod 1), where the parameters a, b are real and 0 ≤ b ≤ 1. Similarly to the well known family of (Arnold) standard maps of the circle, Aa,b (x) = x + a + (b/(2π )) sin(2π x) (mod 1), any such map has at most one attracting periodic orbit and the set of parameters (a, b) for which such orbit exists is divided into tongues. However, unlike the classical Arnold tongues that begin at the level b = 0, for double standard maps the tongues begin at higher levels, depending on the tongue. Moreover, the order of the tongues is different. For the standard maps it is governed by the continued fraction expansions of rational numbers; for the double standard maps it is governed by their binary expansions. We investigate closer two families of tongues with different behavior.
1. Introduction It is a usual procedure that in order to understand the behavior of a system in higher dimension one investigates first a one-dimensional system that is somewhat similar. The classical example is the Hénon map and similar systems, where serious progress occurred only after unimodal interval maps had been thoroughly understood. Another example of this type was investigation by V. Arnold of the family of standard maps of the circle, given by the formula Aa,b (x) = x + a +
b sin(2π x) (mod 1) 2π
The first author was partially supported by NSF grant DMS 0456526.
(1.1)
The second author was supported by FCT Grant SFRH/BD/18631/2004. CMUP is supported by FCT through POCTI and POSI of Quadro Comunitário de apoio III (2000-2006)
with FEDER and national funding.
38
M. Misiurewicz, A. Rodrigues
(when we write “mod 1,” we mean that both the arguments and the values are taken modulo 1). Those maps, called also Arnold maps, should not be confused with TaylorChirikov maps, which are defined by similar formulas in the annulus, and are also called standard maps. The family (1.1) appeared in [1] and its study was useful in the creation of the KAM Theory. This family has been investigated by various authors since then, see for instance [12, 6] and other papers cited there. Recently, very interesting families of branched covering maps in the plane have been studied, [3–5]. It motivates finding similar (in some sense) one-dimensional maps and studying them. If we consider a branched covering map of the plane that has only one branching point and degree 2, a good choice is to study degree 2 circle maps. To begin with, one should concentrate on some specific family of such maps. Perhaps the most natural choice is the family similar to standard maps, but with the sinusoid added not to the identity but to the doubling map (we also rescale the parameter b in order to keep its critical value at 1). In such a way we get the following family of double standard maps: f a,b (x) = 2x + a +
b sin(2π x) (mod 1). π
(1.2)
There are also other reasons for studying this family. It is a hybrid between the family of standard maps and the family of expanding circle maps (see, e.g., [11]). Both families are of special interest, so it is an important problem to investigate what the result of the cross-breeding may be. Moreover, the circle maps with cubic critical points (this is what we get when we put b = 1 in (1.2)) already proved to be interesting (see, e.g., [8]). A widely accepted method of investigating new dynamical system or their families consists of initial numerical investigation, formulating questions and conjectures based on it, and subsequent attempts to answer the questions and prove the conjectures. We will follow this scheme. In this paper we study this family for the values of b from 0 to 1. In this range, the maps are local homeomorphisms, while for b > 1 they are bimodal circle maps – a class with quite different features, similar to unimodal interval maps. In Sect. 2 we realize the first, easiest part of the plan. Namely, we make computer experiments, look at the pictures and try to understand what we see. In Sect. 3 we develop some tools that will be useful in the next sections. In Sect. 4 we explain the order in which tongues appear as the parameter a increases. In Sect. 5 we look closer at the only tongue for which the direct computations are reasonably simple, namely at the period 1 tongue. We describe its shape and produce explicit estimates for which values of b this is the only tongue. In Sects. 6 and 7 we investigate, for b = 1, two families of attracting periodic orbits with opposite behaviors. One of them consists of orbits which would be very strongly repelling if it did not happen that one point of such an orbit is very close to the critical point. The other one consists of intermittent orbits, for which repelling properties are extremely weak. For both families we estimate the size of the windows in the parameter space and in the phase space. Finally, in Sect. 8 we estimate the size of the tongues in the direction of the parameter b for those two families. 2. Numerical Results In this section we will present several computer generated pictures for the family of double standard maps and describe the apparent features of this family, based on the pictures. The usual pictures produced for the standard (Arnold) family of maps present the situation in the (a, b)-plane and show the parameter values for which there is an attracting
Double Standard Maps
39
periodic orbit (phase locking regions). Those parameter values are grouped in regions called Arnold tongues (see Fig. 2.1). Note that (mod 1) we have A−a,b (−x) = −Aa,b (x), so A−a,b is conjugate to Aa,b via the map x → −x (mod 1). Therefore the picture is symmetric with respect to the line a = 1/2 and therefore we only need to show it for 1/2 ≤ a ≤ 1. The same applies to the maps f a,b replacing Aa,b . Let us describe Fig. 2.1 precisely. The vertical axis is b, from 0 to 1. The horizontal axis is a, from 1/2 to 1. The tongues shown are all tongues of period 5 or less, and their order from left to right is 2, 5, 3, 4, 5, 1. They correspond to the rotation numbers 1/2 < 3/5 < 2/3 < 3/4 < 4/5 < 1/1. Let us compare this picture to the analogous one for the double standard maps (see Fig. 2.2). Here the vertical axis is b, from 1/2 to 1. The horizontal axis is a, from 1/2 to 1. The tongues shown are all tongues of period 5 or less (in fact, almost all, because the last one is so small that it does not show on the picture), and their order from left to right is 1, 5, 5, 4, 5, 5, 4, 3, 5, 5, 4, 5, 5, 4, 3, 5, 5, 2, 5, 5, 4, 5, 3, 5, 4, 5. As we will explain later, they correspond to the rational numbers 0/1 < 1/31 < 2/31 < 1/15 < 3/31 < 4/31 < 2/15 < 1/7 < 5/31 < 6/31 < 3/15 < 7/31 < 8/31 < 4/15 < 2/7 < 9/31 < 10/31 < 1/3 < 11/31 < 12/31 < 6/15 < 13/31 < 3/7 < 14/31 < 7/15 < 15/31 (the denominators are of the form 2n −1, where n is the period). This order is completely different from that for the standard maps. Another big difference is that here the tongues begin not at the level b = 0, like for standard maps, but at much higher levels. The
Fig. 2.1. Arnold tongues for the family of standard maps
Fig. 2.2. Arnold tongues for the family of double standard maps
40
M. Misiurewicz, A. Rodrigues
Fig. 2.3. The (a, x)-plot, with a and x from 0 to 1 and b = 1
lowest tongue tip is at b = 1/2, for the period 1 tongue. There cannot be anything lower, because if 0 < b < 1/2 then the map is expanding. Thus, the natural conjecture is that for the double standard family of maps the phase locking regions come in tongues, whose shapes are similar to the classical Arnold tongues. We have to explain their order. It seems that for a given value of b ∈ [0, 1) there are only finitely many of them (however, we will see in Sect. 8 that this is not true). In particular, only the period 1 tongue begins as low as 1/2. We would like to know the size of the tongues in both a and b directions. The a-size should be measured at the level b = 1. Then, since we fix the value of b, it makes sense to look at the picture in the (a, x)-plane (like the classical pictures for the family of the logistic or real quadratic families of maps). Figure 2.3 presents the global picture, with both a and x varying from 0 to 1. Since 1/2 is the unique critical point of f a,1 and the map has negative Schwarzian derivative, for every a there is at most one attracting periodic orbit (see, e.g., [9]). If such an orbit exists, one of its points must be close to 1/2. Since f a,1 (1/2) = a, there must be a point of such orbit close to a. Figure 2.3 suggests that in order to see well how the attracting periodic orbit varies with a, it is better to look close to the diagonal x = a, rather than close to the line x = 1/2, where the line is very steep unless the period is very small. And indeed, blow-ups at many regions close to the diagonal show a graph of a periodic point as a function of a that is not so steep in its middle part (although of course it has to be vertical at the boundary of the window). This is illustrated on Fig. 2.4, where a and x vary from 0.69053 to 0.69055. However, there are periodic orbits close to the boundaries of tongues of small period, that we can call resonant or intermittent, for which this graph is much steeper. Figure 2.5 shows what happens near the boundary of the period 1 tongue. There a and x vary from 0.61087 to 0.61093. 3. Tools In this section we prove some preliminary results that will serve as tools for more detailed investigation of the family of double standard maps. In most of the paper, f a,b will denote the standard map given by Eq. (1.2) and Fa,b its lifting to the real line, that is, the map given by the same formula, but not considered modulo 1. However, in this
Double Standard Maps
41
Fig. 2.4. The (a, x)-plot, with a and x from 0.69053 to 0.69055 and b = 1
Fig. 2.5. The (a, x)-plot, with a and x from 0.61087 to 0.61093 and b = 1
section (except the very end), we prove some properties of those maps, which do not depend on the precise formula. Therefore at the moment we will only assume that Fa,b are maps from the real line to itself, satisfying the following properties: 1. 2. 3.
Each Fa,b is continuous increasing (as a function of x), Fa,b (x + k) = Fa,b (x) + 2k for every integer k, Fa,b (x) is increasing as a function of a and continuous jointly in x, a, b.
While the fact that local homeomorphisms of the circle of degree 2 are semiconjugate to the doubling map is well known, we need additionally monotonicity properties of the semiconjugacy as the function of a. Therefore we include a simple proof which also gives us this monotonicity. The first lemma establishes semiconjugacy as a certain limit. Lemma 3.1. Under assumptions (1) and (2), the limit Φa,b (x) = lim
n→∞
n (x) Fa,b
2n
(3.1)
42
M. Misiurewicz, A. Rodrigues
exists uniformly in x. The limit Φa,b (x) is a continuous increasing function of x. Moreover, Φa,b (x + k) = Φa,b (x) + k for every integer k and Φa,b (Fa,b (x)) = 2Φa,b (x) for every x, so Φa,b semiconjugates Fa,b with multiplication by 2. Proof. Since Fa,b is continuous and satisfies (2), it has a fixed point xa,b . For this fixed point we know that Fa,b (xa,b ) = xa,b , and therefore by (2), Fa,b (xa,b + k) = xa,b + 2k. n (x n From this by induction we get Fa,b a,b + k) = x a,b + 2 k for any integer k and n ≥ 0. Now, if we take any x and large m, we know that there exists an integer k such that m (x) ≤ x m+n n xa,b + k ≤ Fa,b a,b + k + 1. Then for any n ≥ 0 we get x a,b + 2 k ≤ Fa,b (x) ≤ n xa,b + 2 (k + 1), and therefore m+n (x) Fa,b xa,b k xa,b k+1 + ≤ ≤ m+n + m . m+n m m+n 2 2 2 2 2
This implies that for every r, s ≥ n and every x we have r s (x) Fa,b (x) Fa,b ≤ 1 . − (3.2) 2r 2s 2m F r (x) ∞ Therefore the sequence a,b2r satisfies the uniform Cauchy’s condition, so it conr =1 verges uniformly. F n (x)
Since the limit in (3.1) is uniform and the functions a,b 2n ing, the function Φa,b is also continuous and increasing. Since Φa,b (x + k) = lim
n→∞
= lim
n (x + k) Fa,b
2n n (x + k) Fa,b
n→∞
and Φa (Fa,b (x)) = lim
2n
n (F (x)) Fa,b a
2n also hold.
n→∞
the last two properties of Φa,b
= lim
are continuous and increas-
n (x) + 2n k Fa,b
2n
n→∞
+ k = Φa,b (x) + k
= 2 lim
n→∞
n+1 (x) Fa,b
2n+1
= 2Φa,b (x),
Lemma 3.2. Under assumptions of Lemma 3.1, the map Φa,b is a lifting of a monotone degree one map ϕa,b of the circle to itself, which semiconjugates f a,b with the doubling map D : x → 2x (mod 1). Moreover, if p is a periodic point of f a,b of period n then ϕa,b ( p) is a periodic point of D of period n. Proof. The first statement follows directly from Lemma 3.1. Assume that p is a periodic n ( p) = p, we get D n (ϕ ( p)) = ϕ ( p). Suppose point of f a,b of period n. Since f a,b a,b a,b that the period of ϕa,b ( p) for D is not n. Then it has to be a factor of n and there must be points x = y on the orbit of p which are mapped to the same point under ϕa,b . Since ϕa,b is monotone, a whole arc A joining x with y has to be mapped by ϕa,b to one point. We may assume that this arc goes from x to y in the anticlockwise direction. For every k (x) = q. Then f k (A) contains an point q of the orbit of p there is k such that f a,b a,b arc joining q with its anticlockwise neighbor from the orbit of p, and it is mapped by ϕa,b to one point. This proves that the whole circle is mapped by ϕa,b to one point, a contradiction. Hence, the period of ϕa,b ( p) for D must be n.
Double Standard Maps
43
The next lemma adds monotonicity with respect to a. Lemma 3.3. Under an additional assumption (3), Φa,b (x) is increasing as a function of a and continuous as a function of x, a, b (jointly). Proof. The inequality (3.2) is uniform jointly in x, a, b, so by (3), the limit (3.1) is continuous jointly in x, a, b. Since Fa,b (x) is increasing in a and x, the iterates Fan (x) are increasing in a, and therefore the limit (3.1) is increasing in a. The fourth lemma is of a different nature. It deals with a map close to a saddle-node (in the intermittency regime). While other, in a sense stronger, tools can be used for this purpose (see [10]), this one has a simple proof and gives explicit estimates unavailable otherwise. Lemma 3.4. Let f : R → R be a C 1 orientation preserving diffeomorphism. Choose x0 ∈ R and set xi = f i (x0 ) for i ∈ Z. Assume that x−1 < x0 < · · · < xn < xn+1 . Then: 1. If f is increasing on (x−1 , xn−1 ) then ( f n ) (x0 ) ≥
xn − xn−1 . x0 − x−1
2. If f is decreasing on (x0 , xn ) then ( f n ) (x0 ) ≥
xn+1 − xn . x1 − x0
3. If f is increasing on (x0 , xn ) then ( f n ) (x0 ) ≤
xn+1 − xn . x1 − x0
4. If f is decreasing on (x−1 , xn−1 ) then ( f n ) (x0 ) ≤
xn − xn−1 . x0 − x−1
Proof. We have ( f n )(x0 ) =
n−1
f (xi ).
i=0
By the Mean Value Theorem, for each i there is ξi ∈ (xi−1 , xi ) such that (xi+1 − xi )/ (xi − xi−1 ) = f (ξi ). If f is increasing, we have f (ξi ) ≤ f (xi ), so ( f n ) (x0 ) ≥
n−1 i=0
xi+1 − xi xn − xn−1 = . xi − xi−1 x0 − x−1
If f is decreasing, we have f (ξi ) ≤ f (xi−1 ), so ( f n ) (x0 ) ≥
n−1 i=0
xi+2 − xi+1 xn+1 − xn = . xi+1 − xi x1 − x0
The last two inequalities are proved in the same way.
44
M. Misiurewicz, A. Rodrigues
Let us now return to double standard maps. The way we think of the circle on which the maps f a,b act, this is the circle R/Z. Theorem 3.5. If 0 ≤ b ≤ 1 then the double standard map f a,b , given by (1.2), has at most one attracting or neutral periodic orbit. Proof. We can complexify f a,b by conjugating it via e2πi x . Then we get the map ga,b (z) = e
1 2πia 2 b z− z
z e
,
(3.3)
of the unit circle to itself. This map is the restriction of the map of C \ {0} to itself given by the same formula. By the results in the theory of iterations of complex maps (see [2], Theorem 7), it follows that for a map (3.3) any attracting periodic orbit of ga,b has to attract a critical point. A neutral periodic orbit on the unit circle is parabolic, so it is on a boundary of a periodic Leau domain. Therefore this result applies also to such an orbit. If b < 1 then here is only one pair of critical points, symmetric (in the complex sense) with respect to the unit circle, and the map preserves this symmetry. If b = 1, there is just one critical point, −1. Therefore, there can be at most one attracting or neutral periodic orbit. 4. Order of Tongues Suppose that a double standard map f a,b has an attracting periodic orbit P of period n of both critical points of g n. By Theorem 3.5, the trajectories under ga,b a,b (or of the 2πi p critical point −1 if b = 1) converge to e for some p ∈ P. Let ϕa,b be the semiconjugacy from Lemma 3.2. Then by that lemma, ϕa,b ( p) is a periodic point of period n of the doubling map D. We will denote this point by T (P) and call it the type of the orbit P. For a periodic point T of D we define the tongue of type T as the set of parameter values (a, b) (where we think of a as taken modulo 1 and b is from [0, 1]) for which there exists an attracting periodic orbit of type T . If the period of T is n, we will say that the tongue of type T has period n. Since ga,b and ϕa,b depend continuously on (a, b), each tongue is open. We first investigate some properties of double standard maps with an attracting or neutral periodic orbit. We will use them later in this section in the case b = 1, but we state and prove them in a more general case. Lemma 4.1. Assume that p is an attracting or neutral periodic point of f a,b of period n. Let J be the set of all points x for which ϕa,b (x) = ϕa,b ( p). Then J is either a closed n | is an orientation preserving homeomorinterval (modulo 1) or a singleton and f a,b J n , and one of the following phism of J onto itself. The endpoints of J are fixed points of f a,b four possibilities holds (see Fig. 4.1). In the first three cases J is an interval. 1. The left endpoint of J is neutral, topologically attracting from the right and topologically repelling from the left; the right endpoint of J is repelling; there are no n in J . other fixed points of f a,b 2. The right endpoint of J is neutral, topologically attracting from the left and topologically repelling from the right; the left endpoint of J is repelling; there are no n in J . other fixed points of f a,b
Double Standard Maps
45
Fig. 4.1. Four cases n in the 3. Both endpoints of J are repelling; there is an attracting fixed point of f a,b n interior of J ; there are no other fixed points of f a,b in J . n , repelling from both sides. 4. The set J consists of one neutral fixed point of f a,b
Proof. By Lemma 3.1, ϕa,b is an increasing continuous function. Therefore J is a closed interval or a singleton. Since ϕa,b (J ) is a set consisting of a fixed point of D n and f a,b n | is an orientation is an orientation preserving local homeomorphism, we see that f a,b J preserving homeomorphism of J onto itself. It follows that the endpoints of J are fixed n . None of them can be attracting from the “outside”, because the whole points of f a,b immediate basin of attraction would be contained in J . n is analytic, it has finitely many fixed points in J and if x < y are consecSince f a,b utive fixed points, either x is attracting from the right and y is repelling from the left, or x is repelling from the right and y is attracting from the left (here by attracting and repelling we mean topologically attracting and repelling; such a point can be neutral). By Theorem 3.5, there can be at most one fixed point topologically attracting from one or both sides. All this restricts the possibilities to the four ones listed in the statement of the lemma. n and its derivative depend continuously on (a, b) and a change of (a, b) Since f a,b that strictly increases (respectively decreases) f a,b also strictly increases (respectively n , we get immediately (look at Fig. 4.1 and trace possible changes of the decreases) f a,b graph) the following lemma.
Lemma 4.2. A small change in (a, b) that strictly increases f a,b , applied to Case 1 of Lemma 4.1, or a small change in (a, b) that strictly decreases f a,b , applied to Case 2, results in Case 3, with the periodic point p depending continuously on (a, b). A small change in (a, b) that strictly decreases f a,b , applied to Case 1 of Lemma 4.1, or a small change in (a, b) that strictly increases f a,b , applied to Case 2, results in disappearing of an attracting or neutral periodic point of period n. A small change in (a, b) applied to Case 3, results in Case 3, with the periodic point p depending continuously on (a, b). We are interested in the order of the tongues as we vary a. While Lemma 3.3 gives us monotonicity of ϕa,b with respect to a, we cannot be sure where the point p from the definition of T (P) is located. Fortunately, if b = 1, we know where on the circle the critical point of f a,b is located. Elementary computations show that this point is at 1/2 and that f a,1 has negative Schwarzian derivative. Therefore the whole interval joining n , where n is the period of P. p with 1/2 is attracted to p under the iterates of f a,1 To simplify notation, we will write f a for f a,1 and ϕa for ϕa,1 . Lemma 4.3. If f a has an attracting periodic orbit P then T (P) = ϕa (1/2).
46
M. Misiurewicz, A. Rodrigues
Proof. Let p be the point of P from the definition of T (P) and let n be the period of P. As we observed, the whole interval joining p with 1/2 is attracted to p under the n . Then from the definition of Φ iterates of f a,1 a,b it follows that ϕa ( p) = ϕa (1/2). Thus, T (P) = ϕa (1/2). The next result is a kind of converse to Lemma 4.3. It describes the situation when ϕa (1/2) is a periodic point of D. Proposition 4.4. Let q be a periodic point of D of period n. Then the set of values of a for which ϕa (1/2) = q is a closed interval I (modulo 1). If a ∈ I then f a has an attracting or neutral periodic point p(a) of period n. The set J (a) = ϕa−1 (1/2) is a closed interval (modulo 1). Its interior (together with p(a) if p(a) is an endpoint of J (a)) is the immediate basin of attraction of p(a) and contains 1/2. If a is the left (respectively right) endpoint of I then the left (respectively right) endpoint of J (a) is p(a) and it is neutral; the other endpoint of J (a) is a repelling periodic point of period n and there are no periodic points in J (a) other than these two. If a is in the interior of I , then p(a) is attracting; both endpoints are repelling periodic points of period n and there are no periodic points in J (a) other than these three. Proof. By Lemma 3.3, a → ϕa (1/2) is an increasing continuous function. Therefore I is a closed interval or a singleton. Assume that a ∈ I . By Lemma 4.1, J (a) is a closed interval or a singleton, f an | J (a) is an orientation preserving homeomorphism of J (a) onto itself, the endpoints of J (a) are fixed points of f an , and one of the Cases 1-4 of that lemma holds. By the definition, 1/2 ∈ J (a). If J (a) is a singleton, then 1/2 is periodic for f a , and since f a (1/2) = 0, it is superattracting. On the other hand, by Lemma 4.1, it is neutral, a contradiction. This proves that J (a) is an interval and leaves only Cases 1–3. In all three cases there is an attracting or neutral fixed point p(a) of f an and its immediate basin of attraction is the interior of J (a) (together with p(a) if p(a) is an endpoint of J (a)). Since 1/2 is in the interior of J (a), it belongs to the immediate basin of attraction of p(a). The sets f ak (J (a)) are disjoint from J (a) for k = 1, 2, . . . , n − 1, so the fixed points of f an | J (a) are periodic points of period n of f a . It remains to prove that I is an interval and that the Cases 1, 2 and 3 correspond to a being the left endpoint, right endpoint and an interior point of I , respectively. However, this is a straightforward consequence of Lemma 4.2. Now, Lemma 4.3, Proposition 4.4 and the fact that the function a → ϕa (1/2) is increasing and continuous, imply the main theorem of this section. Theorem 4.5. As a increases, the types of the tongues of f a vary in the order of rational numbers. In particular, this theorem explains the order mentioned in Sect. 2. As a varies from 1/2 to 1, the periodic points of D of period 5 or less are 0/1 < 1/31 < 2/31 < 1/15 < 3/31 < 4/31 < 2/15 < 1/7 < 5/31 < 6/31 < 3/15 < 7/31 < 8/31 < 4/15 < 2/7 < 9/31 < 10/31 < 1/3 < 11/31 < 12/31 < 6/15 < 13/31 < 3/7 < 14/31 < 7/15 < 15/31, and they have periods 1, 5, 5, 4, 5, 5, 4, 3, 5, 5, 4, 5, 5, 4, 3, 5, 5, 2, 5, 5, 4, 5, 3, 5, 4, 5, respectively. If we want to apply the same methods to the tongues at the level b = b0 with b0 < 1, the problems arise already when we want to prove an analogue of Lemma 4.3. Observe that the derivative of f a,b attains its minimum at x = 1/2. It is a natural conjecture that if there is an attracting periodic orbit P then 1/2 is in its immediate basin
Double Standard Maps
47
of attraction, which would prove the desired lemma. While numerical experiments seem to support this conjecture, it is nevertheless false. Let (a0 , b0 ) be the coordinates of a tip of a period 2 tongue in the parameter plane. To be more precise, b0 is the infimum of the values of b for which f a,b has an attracting periodic orbit of period 2 and there is a sequence (an , bn )∞ n=1 convergent to (a0 , b0 ) such that f an ,bn has an attracting periodic point xn of period 2. We may assume that xn → x0 as n → ∞. Then x0 is a periodic point of period 2 of f a0 ,b0 . By Lemma 4.2, for (a0 , b0 ) Case 4 of Lemma 4.1 has to occur. In particular, ( f a2 ,b )
(x0 ) = 0. 0 0 Small change of (a, b) cannot produce a large basin of attraction of a periodic point of period 2, so the length of the immediate basin of attraction of xn shrinks to 0. Therefore, if 1/2 is always in this basin of attraction, we must have x0 = 1/2. We will show that this is impossible. Let f = f a0 ,b0 and assume that x0 = 1/2. Since ( f 2 )
= ( f
◦ f )( f )2 + ( f ◦ f ) f
and ( f 2 )
(1/2) = 0, we get f
( f (1/2))( f (1/2))2 + f ( f (1/2)) f
(1/2) = 0. However, f
(1/2) = 0, so f
( f (1/2))( f (1/2))2 = 0. Since f (1/2) = 0, we get f
( f (1/2)) = 0. The only points at which f
vanishes are 0 and 1/2 (modulo 1), so either f (1/2) = 1/2 or f (1/2) = 0 (modulo 1). In the first case, 1/2 is a fixed point of f , and since b0 > 1/2, this point is attracting, a contradiction. In the second case, since f (0) = a0 and f (1/2) = 1 + a0 (modulo 1), we get 1/2 = a0 = 0 (modulo 1), also a contradiction. This proves that in a period 2 tongue, close to its tip, there must be values of a, b such that 1/2 is not in the immediate basin of attraction of the attracting periodic orbit of period 2. We finish this section with a corollary to Lemma 4.2. Proposition 4.6. Whenever a piece of the boundary of a tongue consists of points for which Case 1 or 2 of Lemma 4.1 holds, it has slope with the absolute value at least π . Proof. Observe that the partial derivative of f a,b (x) with respect to a is 1, while the partial derivative with respect to b is sin(2π x)/π , which has modulus at most 1/π . Therefore any change in (a, b) in the direction of a vector (1, y), where |y| < π , strictly increases f a,b . Similarly, any change in (a, b) in the direction of a vector (−1, y), where |y| < π , strictly decreases f a,b . The statement of the lemma follows from this and Lemma 4.2. The problem with the application of this proposition is that we have to exclude the possibility of pieces of the boundaries of tongues consisting of points for which n (x) = x, Case 4 holds. This would require the solution to three analytic equations: f a,b n n
( f a,b ) (x) = 1 and ( f a,b ) (x) = 0 in the (a, b, x)-space to contain a curve. Generically, this is not a case. However, we do not know how generic the family of double standard maps is. 5. Period 1 Tongue Let us investigate closer the tongue corresponding to period 1. Elementary computations show that its boundary is given by the curves √ √ 1 4b2 − 1 − arctan 4b2 − 1 (5.1) a= ± 2 2π
48
M. Misiurewicz, A. Rodrigues
and the corresponding fixed point is then √ 1 arctan 4b2 − 1 x =− ± . 2 2π Set b = 1/2 + t. Then (5.1) becomes √ √ 1 2 t + t 2 − arctan(2 t + t 2 ) a= ± 2 2π
(5.2)
and the derivative of the right-hand side of (5.2) is √ 2 t + t2 . π(1 + 2t) At t = 0 this is of order t 1/2 , so the tangency of the two lines bounding the period 1 tongue is of order t 3/2 . This tongue begins at the level b = 1/2. We will show that all other tongues begin substantially higher. For the double standard maps we have
f a,b (x) = 2 + 2b cos(2π x).
(5.3)
has one minimum, at x = 1/2 (mod 1), one maximum, at x = 0 (mod 1), Therefore, f a,b is decreasing on (0, 1/2) and increasing on (1/2, 1). This allows us to apply Lemma 3.4
is the to f a,b , or rather, since we use inequalities, to Fa,b . Clearly, the formula for Fa,b
. same as for f a,b k−1 Lemma 5.1. Assume that x ∈ (0, 1), k ≥ 1, Fa,b (x) ≤ 1 and Fa,b (t) > t for k (x)). Then t ∈ (x, Fa,b k
(Fa,b ) (x) ≥
k (x) − F k−1 (x) Fa,b a,b
Fa,b (x) − x
· (Fa,b ) (1/2).
(5.4)
k (x) > 1/2. Then the orbit of x is x < · · · < Proof. Assume first that x < 1/2 and Fa,b 0 xn < y0 < · · · < ym < . . . , where 1/2 is between xn and y0 and n + m + 1 = k (so k (x) = y . Then by Lemma 3.4, Fa,b m k
) (x) ≥ (Fa,b
y0 − xn ym − ym−1 ym − ym−1 · (Fa,b ) (xn ) · ≥ · (Fa,b ) (1/2). x1 − x0 y0 − xn x1 − x0
k (x) ≤ 1/2 (that is, there are no y ’s), then the estimate that we get from If Fa,b i Lemma 3.4, k
) (x) ≥ (Fa,b
Fa,b (xn ) − xn , x1 − x0
so by the Mean Value theorem we get k
(Fa,b ) (x) ≥
Fa,b (xn ) − xn xn − xn−1 xn − xn−1 · ≥ (Fa,b ) (1/2) · . xn − xn−1 x1 − x0 x1 − x0
Double Standard Maps
49
Similarly, if x ≥ 1/2 (that is, there are no xi ’s), then we get k
(Fa,b ) (x) ≥
ym − ym−1 ym − ym−1 . ≥ (Fa,b ) (1/2) · −1 y0 − (Fa,b ) (y0 ) y1 − y0
, if 1/2 < b < 1 then on [0, 1] there are 2 points By our analysis of the derivative Fa,b where it is equal to 1. The first of them is in (0, 1/2):
p=
1 −1 arccos , 2π 2b
(5.5)
and the other one is 1 − p (note that the graph of Fa,b is centrally symmetric about the point (1/2, Fa,b (1/2)), see Fig. 5.1. If Fa,b has no attracting or neutral fixed point then it has a unique fixed point q. If −1/2 < a ≤ 0 then Fa,b (0) = a ≤ 0 and Fa,b (1/2) = 1 + a > 1/2, so 0 ≤ q < 1/2 (see Fig. 5.1).
Fig. 5.1. The graph of Fa,b for a = −0.3, b = 0.7
50
M. Misiurewicz, A. Rodrigues
Lemma 5.2. Assume that −1/2 < a ≤ 0, 1/2 < b < 1, Fa,b (1 − p) ≤ 1, f a,b has no attracting or neutral fixed point, and
−1 1 − Fa,b (1) Fa,b (1/2) > Fa,b ( p) − p. (5.6) Then every periodic orbit of f a,b is repelling. Proof. Any periodic orbit of f a,b of period larger than 1 can be divided into blocks as follows. Any point of the orbit that is in [0, q) forms a block of length 1. The rest of the points of the orbits are divided in a natural way into maximal blocks of the form k−1 (x, f a,b (x), . . . , f a,b ) satisfying the assumptions of Lemma 5.1 (on those blocks we can replace f a,b by Fa,b ). In order to prove the lemma it is enough to show that the derivative of Fa,b along any block is larger than 1. This is true for the blocks of length
is decreasing in [0, q] and is larger than 1 at q. 1 with the point in [0, q), because Fa,b It is also true for the other blocks if the right-hand side of (5.4) is larger than 1.
is larger than 1 at all points of the block, Look at such a block. If x > 1 − p then Fa,b so the product of those derivatives is also larger than 1. Assume now that x ≤ 1 − p. Then Fa,b (x) − x ≤ Fa,b ( p) − p.
is increasing in [1 − p, 1], we get Moreover, since Fa,b (1 − p) ≤ 1 and Fa,b k−1 −1 k (x) − Fa,b (x) ≥ 1 − Fa,b (1). Fa,b
Therefore, the right-hand side of (5.4) is larger than 1 by the inequality (5.6).
We get a similar result also in another situation, not involving intermittency. Lemma 5.3. Assume that −1/2 < a ≤ 0, 1/2 < b < 1, f a,b has no attracting or neutral fixed point, and
(1/2) · Fa,b ( f a,b ( p)) > 1 and Fa,b (1/2) · Fa,b ( f a,b (1 − p)) > 1. Fa,b
(5.7)
Then every periodic orbit of f a,b is repelling. Proof. From (5.7) it follows that Fa,b ( p) > 1 − p and Fa,b (1 − p) < p + 1. Therefore
(x) ≥ F (1/2) and Fa,b ([ p, 1 − p]) ⊂ [1 − p, p + 1]. If x ∈ [ p, 1 − p] then Fa,b a,b
Fa,b (Fa,b (x)) ≥ min Fa,b ( p), Fa,b (1 − p) .
(x) · F (F (x)) > 1. Therefore, we can divide any periodic orbit By (5.7) we get Fa,b a,b a,b of f a,b into blocks of length 1 or 2 (if the point x on the orbit is in [ p, 1 − p] then it is the first point of a block of length 2), and the derivative of f a,b along any block is larger than 1. This completes the proof.
Now we can prove the main theorem of this section. Theorem 5.4. If 0 ≤ b < 0.5 then all periodic orbits of f a,b are repelling. Set b0 = 0.578. If 0.5 ≤ b ≤ b0 then all periodic orbits of f a,b , except perhaps one fixed point, are repelling.
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is larger than 1 everywhere, so all periodic orbits are Proof. If 0 ≤ b < 0.5 then f a,b repelling. Similarly, if b = 0.5 then there cannot be attracting periodic orbits, and the only neutral periodic orbit can have period 1. Assume that 0.5 < b ≤ b0 . If there is an attracting or neutral fixed point, then by Theorem 3.5 there are no other attracting or neutral periodic orbits. Therefore it remains to consider the case when there is no attracting or neutral fixed point. Further reduction can be made in the range of the parameter a. Since we consider our map modulo 1, we may assume that a ∈ [−1, 0]. Moreover, the map x → 1 − x conjugates f a,b with f −1−a,b . Therefore we may assume that a ∈ [−1/2, 0]. If a = −1/2 then the point 1/2 is attracting, so our final assumption on a is that −1/2 < a ≤ 0, the same as in Lemmas 5.2 and 5.3. Set a0 = −0.285. If a ≤ a0 , we would like to apply Lemma 5.2. To do this, we have to check that f a,b (1 − p) ≤ 1 and that (5.6) is satisfied. For a fixed b, the largest value of Fa,b (1 − p) is attained when a = a0 . Thus, we have to check that Fa0 ,b (1 − p) ≤ 1. Since −1 1 = 1− 2, arccos 2b 4b
this inequality is equivalent to 2π(1 + a0 ) ≤ 2 arccos
−1 2 + 4b − 1. 2b
The derivative of the right-hand side of (5.8) is 2 1 4b − , √ b 4b2 − 1
(5.8)
√ so the right-hand side of (5.8) attains its minimum at b = 2/2. The value of this minimum is (3/2)π + 1 > 5.7, while the value of the left-hand side of (5.8) is less than 4.5. This proves that Fa,b (1 − p) ≤ 1. Fix b and consider both sides of (5.6) as functions of a. Since p is independent of a, the derivative of the right-hand side with respect to a is 1. On the left-hand side the
(1/2) is independent of a and smaller than 1. Since F (1 − p) ≤ 1, we factor Fa,b a,b −1
(F −1 (1)) ≥ 1. Therefore, since the derivative of (1) ≥ 1 − p, so Fa,b get that Fa,b a,b Fa,b (x) for any fixed x is 1, and by the Implicit Function Theorem, the absolute value −1 of the derivative of Fa,b (1) with respect to a is smaller than or equal to 1. Therefore the derivative of the left-hand side with respect to a is smaller than 1. This means that we have to check (5.6) only for a = a0 . Now, for the fixed value of a, as b increases, on −1
(1/2) decreases, and on (0, 1/2) (1) decreases, Fa,b (1/2, 1) f a,b decreases, so 1 − Fa,b x → Fa,b (x) − x increases (as a function of b), so Fa,b ( p) − p increases. Therefore we have to check (5.6) only for b = b0 . For those values of a and b the left-hand side of (5.6) is larger than 0.224, while the right-hand side is smaller than 0.224. This shows that (5.6) holds, so by Lemma 5.2, if a ≤ a0 (plus the assumptions of the theorem) then all periodic orbits of f a,b , except perhaps one fixed point, are repelling. Assume now that a > a0 . Then we would like to use Lemma 5.3, so we have to verify that (5.7) holds. Consider the first inequality of (5.7). Since p < 1/2 and b > 1/2, we have Fa,b ( p) > Fa,1/2 ( p). Therefore it is enough to show that
(1/2) · Fa,b (Fa,1/2 ( p)) > 1. Fa,b
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M. Misiurewicz, A. Rodrigues
The value of p decreases as b increases. Therefore −1 1 > 0.4163, arccos 1/2 > p ≥ 2π 2b0 so 1 = f 0,1/2 (1/2) > ( f a,1/2 ( p)) > f a0 ,1/2 (0.4163) > 0.627.
increases. Hence, This shows that Fa,1/2 ( p) is in the region where Fa,b
( f a,1/2 ( p)) > 2 + 2 · 0.578 · cos(2π · 0.627) > 1.19, Fa,b
so
Fa,b (1/2) · f a,b ( f a,1/2 ( p)) > 0.844 · 1.19 > 1.004.
Consider now the second inequality of (5.7). Since p > 0.4, we have 1 − p < 0.6, so Fa,1/2 (1 − p) < 1.2. Moreover, 1 − p > p. The derivative of Fa,b is the same at 1 + t and at 1 − t, so we get
Fa,b (Fa,1/2 (1 − p)) > min(Fa,b (0.627), Fa,b (0.8)) = Fa,b (0.627) > 1.19,
and then we get the same estimate as for the first inequality. This completes the proof. 6. Mostly Repelling Attracting Periodic Orbits In this section we consider again the case b = 1, and we use the same notation as in Sect. 4. We will consider here a special class P of attracting periodic orbits. They are attracting periodic orbits for f a of type 0.0001 ∗ 1 ∗ 1 · · · ∗ 1 (the line over a finite sequence means that it is repeated periodically), where each ∗ can be 0 or 1. There are values of a, al ≈ −0.32221099 and ar ≈ −0.28609229 for which Φal (1/2) = 1/16 and Φar (1/2) = 1/8. We have 1/16 = 0.00010 and 1/8 = 0.0001. The numbers of the form 0.0001 ∗ 1 ∗ 1 · · · ∗ 1 are between those two, so any a for which f a has a periodic orbit of such type is in (al , ar ). Let z(a) be the fixed point of Fa − 1 (see Fig. 6.1). Then Φa (z(a)) = 1. If P ∈ P then z(a) is not in the basin of attraction of P, so the sets Φa−1 (1), as well as Φa−1 (1/2 j ) for j = 1, 2, . . . , consist of one point each. We have Far (1/2) = ar + 1 > 2/3, so Fa2r (2/3) < z(ar ). As a decreases, Fa2 (2/3) decreases, while z(a) increases. Therefore 2/3 < Fa−2 (z(a)) for all a ∈ (al , ar ). The point Fa−2 (z(a)) is the unique point whose image under Φa is 1/4, so the binary expansion of Φa (2/3) starts with 0.00. On the other hand, z(al ) < 4/3, so z(a) < 4/3 for all a ∈ (al , ar ). Since the only points where Fa ≤ 1 are in [1/3, 2/3] and the integer shifts of this interval (note that Fa does not depend on a), we see that there exists a constant λ > 1 such that whenever a ∈ (al , ar ) and the binary expansion of Φa (x) does not start with 0.00, we have Fa (x) > λ. Assume that P ∈ P is an orbit of period n and let p ∈ P be the point for which n 1/2 is in the immediate basin of attraction of p for √ f a (we need p not modulo 1, so we choose p ∈ [0, 1)). Since Fal (1/3) = 2/3 + al + 3/(2π ) > 1/2, there exists a constant
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Fig. 6.1. The map f a for a ∈ (al , ar )
c > 0 such that whenever a ∈ (al , ar ) and x ∈ [1/3, 2/3] then f a ( f a (x)) > c. Thus, if p ≥ 1/3 then f a ( f a ( p)) > c. We cannot easily exclude the case p < 1/3. However, then [1/3, 1/2] is contained in the basin of immediate attraction of p for f an . Then there is ε > 0 such that the length of the basin of immediate attraction of f a ( p) for f an is larger than ε, independently of a. We have to have ελn−2 < 1, so there is N such that if n ≥ N , this is impossible. From now on, we exclude from P the orbits of period less than N . Moreover, we have p ≤ 2/3, since otherwise our periodic orbit would not contain a point with derivative less than 1, so it would not be attracting. To summarize, we get the following structure of an orbit P ∈ P. There is a point p ∈ [1/3, 2/3] ∩ P, such that 1/2 is in the basin of immediate attraction of p for f an . The derivative of f a at f a ( p) is larger than c and at the points of P other than p and f a ( p) is larger than λ. To describe the situation in more geometrical terms, let us look at Fig. 6.1. The interval J consists of points whose image under Φa has binary expansion starting with 0.1. For the intervals I1 and I2 this is respectively 0.01 and 0.11. The point p is in [1/3, 2/3]. Its image under f a2 is in I1 , and further images under the iterates of f a2 are in I1 and I2 . Since f a (I1 ) = f a (I2 ) = J and f a (J ) is the whole circle, we can get periodic orbits that under the iterates of f a2 go through I1 and I2 in the prescribed order. Moreover, the derivative of f a on I1 ∪ I2 ∪ J is larger than λ. We are interested in the sizes in the directions of a and p of the region where our orbit P ∈ P is attracting (we will refer to them as the P-windows in the directions of a and p). Denote those windows by [a1 , a2 ] and [ p1 , p2 ] respectively. Since p depends on a, we will write p(a). Thus, we have pi = p(ai ) for i = 1, 2. We will express those sizes in terms of the exponent of P \ { p(a)},
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M. Misiurewicz, A. Rodrigues
α(a) = ( f an−1 ) ( f a ( p(a)).
(6.1)
We have to choose some specific value of a, and the most natural such value is a0 for which p(a0 ) = 1/2. Theorem 6.1. There exist positive constants K 1 , K 2 , K 3 , K 4 such that if a periodic orbit P belongs to P then for the P-windows [ p1 , p2 ] in the direction of p and [a1 , a2 ] in the direction of a we have
and
K 1 (α(a0 ))−1/2 ≤ p2 − p1 ≤ K 2 (α(a0 ))−1/2
(6.2)
K 3 (α(a0 ))−3/2 ≤ a2 − a1 ≤ K 4 (α(a0 ))−3/2 .
(6.3)
In particular, the size of the P-window in the direction of a is of order of the cube of the size of the P-window in the direction of p. Proof. Let us compute the partial derivatives of the iterates of f a with respect to a. We have to treat f as a function of 2 variables. Use notation f (a, x) = f a (x). Then: ∂ f an+1 (x) ∂ f (a, f an (x)) ∂f ∂f ∂ f n (x) = = (a, f an (x)) + (a, f an (x)) · a . ∂a ∂a ∂a ∂x ∂a Since in our case ∂ f /∂a = 1, we obtain ∂ f an+1 (x) ∂ f n (x) = 1 + f a ( f an (x)) · a . ∂a ∂a Therefore by induction we get ∂ f an (x) i n−i = ( f a ) ( f a (x)). ∂a n−1
(6.4)
i=0
We have f an ( p(a)) = p(a). Differentiate both sides of this equality with respect to a: ∂ f an ( p(a)) + ( f an ) ( p(a)) · p (a) = p (a). ∂a Therefore we get, substituting the formula (6.4), n−1 i n−i (f )(f ( p(a)))
p (a) = i=0 a n a . 1 − ( f a ) ( p(a))
(6.5)
Using notation (6.1), we get ( f an ) ( p(a)) = f a ( p(a)) · α(a). We have n−1
i n−i i=0 ( f a ) ( f a ( p(a)))
α(a)
=
n−1
1
( f an−i ) ( p(a)) i=0
=
n−1
1
( f ai ) ( p(a)) i=0
.
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The term of the last sum above corresponding to i = 0 is equal to 1, so the whole sum is larger than or equal to 1. On the other hand, if i > 0 then we have ( f ai ) ( p(a)) ≥ cλi−1 , so n−1 i=0
∞
1 1 λ ≤ 1 + . =1+ ( f ai ) ( p(a)) cλi−1 c(λ − 1) i=1
Thus, there is a constant C > 0, independent of P ∈ P, such that α(a)
C ≤ n−1
i n−i i=0 ( f a ) ( f a ( p(a)))
≤ 1.
(6.6)
Now we can rewrite the differential equation (6.5) as α(a) 1 da
= n−1 − f a ( p) . i n−i dp α(a) i=0 ( f a ) ( f a ( p))
(6.7)
The right-hand side of (6.5) is positive if a ∈ (a1 , a2 ) and infinite if a = ai , i = 1, 2. Therefore the right-hand side of (6.7) is positive if a ∈ (a1 , a2 ) and zero if a = ai , i = 1, 2. In particular, 1/α(ai ) = f a ( pi ). i Let us estimate the distortion of α as a varies from a1 to a2 . Then p(a) increases and a increases, so f ai ( p(a)) increases for every i ≥ 0. If additionally i ≤ n, then f an ( p1 ) = f an−i ( f ai ( p1 )) ≤ f an−i ( f ai ( p2 )) ≤ f an−i ( f ai ( p2 )) = f an ( p2 ). 1
Since
f an ( p1 ) 1
1
1
1
= p1 + k and
f an ( p2 ) 2
2
2
2
2
= p2 + k for the same integer k, we get
f an−i ( f ai ( p2 )) − f an−i ( f ai ( p1 )) ≤ p2 − p1 < 1. 1
2
1
1
For j = 2, 3, . . . , n −1 the interval [ f ai ( p1 ), f ai ( p2 )], and therefore also its subinterval 1 2 [ f ai ( p1 ), f ai ( p2 )], is in the region where the derivative of f a (which is independent of 1 1
a) is larger than λ; if j = 1, we should replace λ by c. Therefore if 2 ≤ i ≤ n, we get f ai ( p2 ) − f ai ( p1 ) <
1 , λn−i
f a 2 ( p2 ) − f a 1 ( p1 ) <
1 . cλn−1
2
1
and
In those regions the logarithm of the derivative of f a is Lipschitz continuous with some constant L, so by the chain rule we get for any a, b ∈ [a1 , a2 ], ⎞ ⎛
n−1 ∞ 1 1 1 1 ⎠. | log α(a) − log α(b)| ≤ L + ≤ L⎝ + cλn−1 λn−i c λj i=2
j=1
The right-hand side of this inequality is a constant independent of the orbit P ∈ P. Therefore there exists a constant D > 1, independent of the orbit P ∈ P, such that α(a) 1 ≤ ≤D D α(b)
(6.8)
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M. Misiurewicz, A. Rodrigues
for every a, b ∈ [a1 , a2 ]. We have f a
1 +t 2
= 2(1 − cos(2π t)),
so there exist positive constants E 1 , E 2 such that if 1/3 ≤ 1/2 + t ≤ 2/3 then 1 + t ≤ E2 t 2 . E 1 t 2 ≤ f a
2
(6.9)
For i = 1, 2, since 1/α(ai ) = f a ( pi ), we get i
E1
1 − pi 2
2
1 ≤ E2 ≤ α(ai )
1 − pi 2
2 .
From inequalities (6.8) and (6.10) we get 1 1 D 1 1 ≤ − pi ≤ . · · D E 2 α(a0 ) 2 E 1 α(a0 )
(6.10)
(6.11)
Therefore (6.2) holds for some positive constants K 1 , K 2 independent of P ∈ P. By (6.8), we have 1/α(a) − f a ( p) ≤ D/α(a0 ), so from (6.7) and (6.6) we get a2 − a1 ≤
D ( p2 − p1 ). α(a0 )
(6.12)
On the other hand, the right-hand side of (6.7) is non-negative, so in view of (6.6) and since (by (6.8)) 1/α(a) ≥ (1/D)(1/α(a0 )), 1/2+s 1 1 · − f a ( p) dp, a2 − a1 ≥ C (6.13) D α(a0 ) 1/2−s where
s=
1 1 · D E 2 α(a0 )
(note that by (6.11) we have [1/2 − s, 1/2 + s] ⊂ [ p1 , p2 ]). By (6.9) we have 1/2+s 1/2+s 1 2E 2 3 2 s = · √ f a ( p) dp ≤ E 2 t 2 dt = (α(a0 ))−3/2 . 3 3 D D E2 1/2−s 1/2−s Therefore, from (6.13) we get 1 2 1 2s · − · √ a2 − a1 ≥ C (α(a0 ))−3/2 D α(a0 ) 3 D D E 2 1 4 · √ =C (α(a0 ))−3/2 . 3 D D E2 From this, (6.12) and (6.2) we get (6.3) for some positive constants K 3 , K 4 independent of P ∈ P.
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Let us make several comments about Theorem 6.1. The first one is that if instead of looking at the point p of the orbit P for which 1/2 is in its immediate basin of attraction, we look at the next point along the orbit, q = f a ( p) (the one that has a in its basin of attraction), then the scaling of the P-window in the direction of q will be the same as the scaling of the P-window in the direction of a. Indeed, if this window is [q1 , q2 ] then q2 − q1 = Fa2 ( p2 ) − Fa1 ( p1 ) = (Fa1 ( p2 ) − Fa1 ( p1 )) + (a2 − a1 ), and since the map f a1 in [1/3, 2/3] is cubic up to a multiplicative constant (and in view of (6.3)), we get (6.14) K 5 (α(a0 ))−3/2 ≤ q2 − q1 ≤ K 6 (α(a0 ))−3/2 for some positive constants K 5 , K 6 independent of P ∈ P. Therefore we get the following corollary to Theorem 6.1. It is consistent with Figs. 2.3 and 2.4 (remember that when we consider f a , we take a modulo 1). Corollary 6.2. There exist positive constants K 7 , K 8 such that if a periodic orbit P belongs to P then for the P-windows [q1 , q2 ] in the direction of q and [a1 , a2 ] in the direction of a, K7 ≤
q2 − q1 ≤ K8. a2 − a1
The second comment is that since we expressed the sizes of the P-windows in terms of α(a0 ), we have some information how those sizes behave as the period of P ∈ P goes to infinity. Then α(a0 ) grows exponentially with the period n, in the sense that c1 λn−2 ≤ α(a0 ) ≤ c2 Λn−2
(6.15)
and c1 , c2 > 0, Λ ≥ λ > 1 (this follows immediately from the definition of α and our earlier estimates). However, whether (1/n) log α(a0 ) is closer to log λ or log Λ, depends on a concrete orbit P. The third comment is that although the orbits from P are kind of special, there are infinitely many of them. Moreover, the only properties of P that we used were that the growth of the derivatives along the pieces of the orbit P ∈ P not passing through p is exponential in the length of the piece, uniformly in P. Thus, there are many other families similar to P for which the same properties can be proved. 7. Intermittent Periodic Orbits Now we consider periodic orbits with the behavior in a sense opposite to the behavior of the orbits considered in Sect. 6. Again the case is b = 1, so we work with the maps f a and their liftings Fa . Set √ 3 2 aI = − ≈ −0.3910022190. 2π 3 We have Fa I (2/3) = 2/3 and Fa (2/3) = 1. Thus, 2/3 is a neutral fixed point and if a I is slightly larger than a I then we observe intermittency for f a . The trajectories of points in a rather large interval containing 1/2 are increasing and spend a lot of time very close to 2/3.
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M. Misiurewicz, A. Rodrigues
We denote by R the class of attracting periodic orbits for f a such that if p ∈ P ∈ R, 1/2 is in the immediate basin of attraction of p and n is the period of P then p < Fa (P) < Fa2 ( p) < · · · < Fan−1 ( p) and p = Fan ( p) − 1. It follows that Φa ( p) = D n (Φa ( p)) − 1, so Φa ( p) = 1/(2n − 1). Therefore the type of such an orbit is 1/(2n − 1). Since we encounter all those types for the values of a slightly larger than a I , they cannot appear anywhere else, and they are our intermittent ones. The general philosophy for intermittency is that as the period of the attracting periodic orbits increases, we have the same behavior (even quantitatively) in the directions of the variables x and b, while in the direction of a we have scaling depending on the order of tangency of the graph of Fa I to the diagonal. This we will see in Theorems 7.2 and 8.2. We can also observe the repetition of the same behavior on Figs. 7.1 and 7.2. Note that we see there wide windows coming in pairs. Such a pair consists of orbits of types 1/(2n − 1) and 2/(2n+1 − 1). Clearly, our considerations can be applied also to the latter types, as well as to every intermittent family.
Fig. 7.1. The (a, x)-plot, with a from 0.6107 to 0.6111, x from 0 to 1 and b = 1
Fig. 7.2. The (a, x)-plot, with a from 0.6089 to 0.6129, x from 0 to 1 and b = 1
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As in the preceding section, we want to estimate sizes of the P-windows for P ∈ R. This time we will do it in terms of the period of P. We will be using a result of Jonker [7]. Although stated formally for circle homeomorphisms, it is local and applies to any intermittent behavior in one dimension. Let us restate it for our family of maps. Lemma 2.5 of [7] (we skip the dependence of some constants on other constants) gives us the following lemma. Lemma 7.1 ([7]). There are ε, τ > 0 such that if m ≥ 1 then there exist constants K 1 , K 2 > 0 such that K 1 (a − a I )−3/2 <
q
∂ f a (x) < K 2 (a − a I )−3/2 ∂a
whenever a I < a0 < a I + ε, x ∈ [2/3 − τ, Fam (2/3 − τ )) and Fa (x) ∈ [Fa−m (2/3 + τ ), 2/3 + τ ). q
The main result of this section is the following theorem. Theorem 7.2. There exist positive constants M1 , M2 , M3 , M4 such that if Pn is the periodic orbit of R of period n then for the P-windows [ p1 , p2 ] in the direction of p and [a1 , a2 ] in the direction of a, then
and
M 1 ≤ p2 − p1 ≤ M 2
(7.1)
M3 n −3 ≤ a2 − a1 ≤ M4 n −3 .
(7.2)
Moreover, there exist positive constants M5 , M6 such that if cn is the value of the parameter a for which 1/2 ∈ Pn , then M5 n −2 ≤ cn − a I ≤ M6 n −2 .
(7.3)
Proof. We need estimates of the partial derivative with respect to a along our periodic orbit. If we start and end close to 1/2 then we can split a trajectory piece of length n into 3 pieces of lengths k, n − 2k and k, so that Lemma 7.1 applies to the middle piece. As the parameter a approaches a I , the maps f a converge to f a I , so we can find k that will work for all sufficiently large periods. Computations similar to those in Sect. 6 give us the following formulas: ∂ f an−k k ∂f k ∂ f n−2k k ( f a ( p)) = a ( f an−k ( p)) + ( f ak ) ( f an−k ( p)) a ( f a ( p)) ∂a ∂a ∂a
(7.4)
and
∂ f n−k ∂f k ∂ f an ( p) = a ( f ak ( p)) + ( f an−k ) ( f ak ( p)) a ( p). ∂a ∂a ∂a By substituting (7.4) to (7.5) we get ∂ f an ∂ f n−2k k ∂f k ( p) = a ( f an−k ( p)) + ( f ak ) ( f an−k ( p)) a ( f a ( p)) ∂a ∂a ∂a ∂f k +( f an−k ) ( f ak ( p)) a ( p). ∂a
(7.5)
(7.6)
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M. Misiurewicz, A. Rodrigues
Let us now estimate the derivative with respect to x. The point p has 1/2 in its immediate basin of attraction, while other points of the orbit of p do not. Therefore Fa ( p) > 1/2 and p < Fa (1/2). If n is the period of p, then Fa (Fan−1 ( p)) = p + 1 < Fa (1/2) + 1 = Fa (1), so Fan−1 ( p) < 1. This proves that for i = 1, 2, . . . , n the points Fai ( p) belong to the interval (1/2, 1) on which Fa is increasing. Therefore by Lemma 3.4 we get F n ( p) − Fan−1 ( p) (Fan−2 ) (Fa ( p)) ≤ a 2 ≤ (Fan−2 ) (Fa2 ( p)). (7.7) Fa ( p) − Fa ( p) To get estimates from both sides of (Fan−1 ) (Fa ( p)) we need additionally the upper estimate of Fa (Fan−1 ( p)) and the lower estimate of Fa (Fa ( p)). The first one is simple, because the maximal value of the derivative of Fa is 4. The second one requires the proof that Fa ( p) cannot be too close to 1/2. In the same way as (7.7), we get (Fan−3 ) (Fa (x)) ≤
Fan−1 (x) − Fan−2 (x) Fa2 (x) − Fa (x)
for every x ∈ [1/2, Fa ( p)]. Since, as we noticed, Fa ≤ 4, we get (Fan−1 ) (Fa (x)) ≤ 16
Fan−1 (x) − Fan−2 (x) . Fa2 (x) − Fa (x)
(7.8)
Since [1/2, Fa ( p)] ⊂ [1/2, Fa (1/2)], for a sufficiently close to a I the values of Fa2 (x)− Fa (x) are uniformly (in a and x) bounded away from 0. Clearly, Fan−1 (x) − Fan−2 (x) are uniformly bounded from above, so together with (7.8) we get that (Fan−1 ) (Fa (x)) is uniformly bounded from above. Therefore there is δ1 > 0 such that if a is sufficiently close to a I and Fa ( p) < 1/2 + δ1 then (Fan ) (Fa (x)) < 1 for all x ∈ [1/2, Fa ( p)]. This means that 1/2 is in the immediate basin of attraction of Fa ( p), which is impossible. Therefore we must have Fa ( p) ≥ 1/2 + δ1 . Consequently, there is δ2 > 0 such that if a is sufficiently close to a I then Fa (Fa ( p)) > δ2 . This estimate together with Fa (Fan−1 ( p)) ≤ 4 and (7.7) gives us δ2
Fan ( p) − Fan−1 ( p) F n ( p) − Fan−1 ( p) ≤ (Fan−1 ) (Fa ( p)) ≤ 4 a 2 . 2 Fa ( p) − Fa ( p) Fa ( p) − Fa ( p)
(7.9)
The same type of estimates as in the preceding paragraph show that if a is sufficiently close to a I then Fan ( p) − Fan−1 ( p) is uniformly bounded from above and Fa2 ( p) − Fa ( p) is uniformly bounded away from 0. By this and by (7.9) we conclude that there are constants K 3 , K 4 > 0 such that K 3 ≤ (Fan−1 ) (Fa ( p)) ≤ K 4 .
(7.10)
This estimate holds for all a sufficiently close to a I . This leaves out finitely many periods, and for each of them clearly an estimate of this type holds. Therefore, by changing constants K 3 and K 4 , we get (7.10) for all orbits from R. Let us return to (7.6). As we said, Lemma 7.1 applies to the middle piece, so there are constants K 1 , K 2 > 0 such that K 1 (a − a I )−3/2 <
∂ f an−2k k ( f a ( p)) < K 2 (a − a I )−3/2 . ∂a
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As n goes to infinity, a approaches a I , so ∂ f ak /∂a( f an−k ( p)), ( f ak ) ( f an−k ( p)) and ∂ f ak /∂a( p) converge to continuous positive functions of p. Thus, they are bounded from above and bounded away from 0 by constants independent of n and p. Clearly, the considerations that led to (7.10) give the same results for ( f an−k ) ( f ak ( p)). Taking all this into account, we get from (7.6), K 5 + K 6 (a − a I )−3/2 ≤
∂ f an ( p) ≤ K 7 + K 8 (a − a I )−3/2 , ∂a
where K 5 , K 6 , K 7 , K 8 are positive constants. Since (a − a I )−3/2 goes to infinity as a → a I , we conclude that there are constants K 9 , K 10 > 0 such that K 9 (a − a I )−3/2 ≤
∂ f an ( p) ≤ K 10 (a − a I )−3/2 . ∂a
(7.11)
Although this does not appear explicitly in (7.10) and (7.11), the point p depends on a, so we will write p(a) instead of p now. As for the case of mostly repelling attracting orbits, we have ∂ f an ∂a ( p(a))
p (a) = (7.12) 1 − ( f an ) ( p(a)) (this is (6.5) without (6.4) plugged in). As in Sect. 6, denote α(a) = ( f an−1 ) ( f a ( p(a))) and for P ∈ R of period n consider P-windows [a1 , a2 ] and [ p1 , p2 ] in the directions of a and p respectively. In view of (7.10), (6.8) holds in the present case, too, so in the same way as in Sect.6, we get (6.2). Applying (7.10) again, we conclude that there are constants M1 , M2 > 0, independent of n, such that (7.1) holds. In order to estimate the size of the P-window in the direction of a, we have to solve approximately (up to a multiplicative constant) (7.12). Observe that in the region where we want to solve it both the numerator and denominator of the right-hand side of (7.12) are positive. Therefore p is a strictly increasing function of a, and thus we can write a as a function of p, a = a( p). We have p(ai ) = ai for i = 1, 2 and thus p2 a2 n ∂ fa n
( p(a)) da. (7.13) (1 − ( f a( ) ( p)) dp = p) p1 a1 ∂a n ) ( p) ≤ 1, so by (7.1) the left-hand side of (7.13) is bounded from Clearly, 1 − ( f a( p) n ) ( p) = α(a( p)) f
above by M2 . On the other hand, ( f a( a( p) ( p). By (7.10), α(a( p)) p)
is bounded from below by K 3 and from above by K 4 . Moreover, f a( p) ( p) = 2(1 − cos(2π(1/2 − p))). There is s > 0, independent of n, such that | p − 1/2| < s then
1 − cos(2π(1/2 − p)) <
1 . 2K 4
Then n
( f a( p) ) ( p) = α(a( p)) · 2(1 − cos(2π(1/2 − p))) < K 4 ·
2 = 1, 2K 4
so [1/2 − s, 1/2 + s] ⊂ [ p1 , p2 ]. Therefore p2 1/2+s n
(1 − ( f a( p) ) ( p)) dp ≥ K 3 · 2(1 − cos(2π(1/2 − p))) dp. p1
1/2−s
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M. Misiurewicz, A. Rodrigues
The right-hand side of the above equation is a positive constant, call it K 11 , independent of n. Thus, we have p2 n
K 11 ≤ (1 − ( f a( (7.14) p) ) ( p)) dp ≤ M2 . p1
Let us consider the right-hand side of (7.13). By (7.11), we have a2 K 9 (a − a I )−3/2 da 2K 9 ((a1 − a I )−1/2 − (a2 − a I )−1/2 ) = a 1a2 n ∂ fa ( p(a)) da ≤ a1 ∂a a2 ≤ K 10 (a − a I )−3/2 da a1
= 2K 10 ((a1 − a I )−1/2 − (a2 − a I )−1/2 ). Together with (7.13) and (7.14) this gives us the existence of constants K 12 , K 13 > 0, independent of n, such that 1 1 K 12 ≤ √ −√ ≤ K 13 . a1 − a I a2 − a I
(7.15)
Now we have to investigate the dependence between a − a I and n. Recall that Pn is the periodic orbit from R of period n and cn is the value of the parameter a for which 1/2 ∈ Pn . As a moves from cn+1 to cn then Fan moves from Fcnn+1 (1/2) = Fcnn+1 ( p(cn+1 )) to Fcnn (1/2) = 1/2, so the distance it covers is between 1/2 and 1. The estimates that resulted in (7.11) hold also in this situation (maybe with slightly worse constants), so we get K 14 (a − a I )−3/2 ≤
∂ f an (1/2) ≤ K 15 (a − a I )−3/2 ∂a
for all a ∈ [cn+1 , cn ], where the constants K 14 , K 15 > 0 are independent of n. Therefore cn 1 1 −3/2 1≥ K 14 (a − a I ) da = 2K 14 √ −√ cn+1 − a I cn − a I cn+1 and 1 ≤ 2
cn
cn+1
K 15 (a − a I )
−3/2
da = 2K 15
1
1 −√ √ cn+1 − a I cn − a I
.
Thus, 1 1 1 1 ≤√ −√ ≤ . 4K 15 cn+1 − a I cn − a I 2K 14 Summing it from n = 1 to m − 1 we get 1 1 m−1 1 m−1 +√ ≤√ ≤ +√ . 4K 15 c1 − a I cm − a I 2K 14 c1 − a I
(7.16)
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63
Therefore there exist constants K 16 , K 17 > 0, independent of n, such that K 16 n ≤ √
1 ≤ K 17 n. cn − a I
2 and M = 1/K 2 . This gives us (7.3) with M5 = 1/K 17 6 16 Now, by (7.15), (7.3), the identity √ 1 1 √ , x − y = (x y + y x) √ − √ y x
and since cn+1 < a1 < a2 < cn−1 , we conclude that there exist constants M3 , M4 > 0, independent of n, such that (7.2) holds. The scaling we obtained for orbits from R is completely different than the scaling for the orbits from P. In particular, switching from the point p to its image q will not change this scaling. 8. Length of Tongues Despite a clear picture emerging from the numerical experiments, we do not know much about the shape of the tongues, except the tongue of period 1. We do not even know whether the tongues are connected. Therefore, since the bulk of our knowledge concerns the level b = 1, it makes sense to define proper tongues as those components of the tongues that have non-empty intersection with the line b = 1. By Propositions 4.3 and 4.4, the intersection of any tongue with the line b = 1 is connected and nonempty, and therefore there is exactly one proper tongue of each type. For the types considered in Sect. 6, we have enough information to estimate the length of the proper tongues. We measure the length of a tongue in the direction of b. The first result seems to confirm the conjecture that at a given level b < 1 there are only finitely many tongues. Theorem 8.1. Let s, t be periodic points of D with 1/16 < s < t < 1/8. Then there exist constants λ > 1, N > 0 and K 5 > 0 such that any proper tongue of a type between s and t, period n ≥ N , and such that the orbit of this type for some f a belongs to P, has length smaller than K 5 λ−n . Proof. There are as , at such that f as has a periodic attracting orbit of type s and f at has a periodic attracting orbit of type t. If ε is sufficiently small, then for any b ∈ [1 − ε, 1] the map f as ,b has a periodic attracting orbit of type s and f at ,b has a periodic attracting orbit of type t. Since tongues are pairwise disjoint, the proper tongue of any type r ∈ (s, t) intersected with the set [0, 1) × [1 − ε, 1] is contained in (as , at ) × [1 − ε, 1]. If ε is sufficiently small and b ∈ [1 − ε, 1], the maps f a,b are uniformly close to the maps f a . Therefore for the orbits of the types described in the statement of the theorem, the same estimates for the derivatives (with respect to x) as in Sect. 6 hold, perhaps with slightly smaller λ and c. Therefore, using the notation of that section, we get n−1
( f a,b ) ( f a,b ( p)) ≥ cλn−1 .
occurs at 1/2 and is equal to 2 − 2b. If our orbit On the other hand, the minimum of f a,b n−1 is attracting, we get (2 − 2b)cλ < 1, so 1 − b < (λ/(2c))λ−n . Thus, if K 5 λ−n < ε,
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M. Misiurewicz, A. Rodrigues
Fig. 8.1. Tongues of period 50 or less in the intermittent region, 0.6 ≤ a ≤ 0.64, 0.96 ≤ b ≤ 1
where K 5 = λ/(2c), we see that the length of the proper tongue that we consider is smaller than K 5 λ−n . To complete the proof, we note that there exists N such that if n ≥ N then K 5 λ−n < ε. Let us now consider periodic orbits of the types considered in Sect. 7. Here we will see that if b < 1 is sufficiently close to 1 then there are infinitely many tongues at that level. Once we know where they are situated, we can produce a picture showing them (see Fig. 8.1). Let us remark that a straightforward method used to detect attracting periodic orbits does not work well here, since a point that moves only slightly due to intermittency may be mistaken for a fixed point. Theorem 8.2. There exists a constant L > 0 such that any proper tongue such that the orbit of this type for some f a belongs to R, has length larger than L. Proof. As we noticed in Sect. 7, the type of an orbit from the statement of the theorem is 1/(2n − 1) if its period is n. For each value of b and each n there exists a unique value, a(b, n) of a, such that for f a(b,n),b the point 1/2 is periodic and Φa(b,n),b (1/2) = 1/(2n − 1). Clearly, if n is fixed, then a(b, n) depends continuously on b. If b is sufficiently close to 1, then the inequality (7.7) with p replaced by 1/2 and Fa replaced by Fa(b,n),b can be proved in exactly the same way as in Sect. 7. The upper estimate of the derivative of Fa(b,n),b by 4 still holds. Therefore we get an analogue of the estimate from (7.10), n−1 (Fa(b,n),b ) (Fa(b,n),b (1/2)) ≤ K 4
with the value of K 4 possibly changed (but independent of n and of b provided b is
sufficiently close to 1). Now, if b is sufficiently close to 1 then Fa(b,n),b (1/2) < 1/K 4 , so n (Fa(b,n),b ) (Fa(b,n),b (1/2)) ≤ 1.
This proves that (a(b, n), b) belongs to the tongue of type 1/(2n − 1). The estimates on how close b should be to 1 are independent of n and since a(b, n) depends continuously on b, this is the proper tongue. This completes the proof. Acknowledgements. The authors are grateful to Janina Kotus for help in the proof of Theorem 3.5.
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References 1. Arnold, V.I.: Small denominators,. I Mappings of the Circumference onto Itself. Amer. Math. Soc. Translations 46, 213–284 (1965) 2. Bergweiler, W.: Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29, 151–188 (1993) 3. Blokh, A., Cleveland, C., Misiurewicz, M.: Expanding polymodials. In: Modern Dynamical Systems and Applications eds. M. Brin, B. Hasselblatt, Ya. Pesin, Cambridge:Cambridge University Press, 2004, pp. 253–270 4. Blokh, A., Cleveland, C., Misiurewicz, M.: Julia sets of expanding polymodials. Ergodic Theory Dynam. Systems 25, 1691–1718 (2005) 5. Blokh, A., Misiurewicz, M.: Branched derivatives. Nonlinearity 18, 703–715 (2005) b sin(2π x) with any given 6. Epstein, A., Keen, L., Tresser, C.: The set of maps Fa,b : x → x + a + 2π rotation interval is contractible. Commun. Math. Phys. 173, 313–333 (1995) 7. Jonker, L.B.: The scaling of Arnold’s tongues for differentiable homeomorphisms of the circle. Commun. Math. Phys. 129, 1–25 (1990) ´ atek, G.: Universality of critical circle covers. Commun. Math. Phys. 228, 371–399 (2002) 8. Levin, G., Swi¸ 9. de Melo, W., van Strien, S.: One-dimensional dynamics. Berlin: Springer-Verlag, 1993 10. Newhouse, S.E., Palis, J., Takens, F.: Bifurcations and stability of families of diffeomorphisms. Publ. Math. IHES 57, 5–71 (1983) 11. Shub, M., Sullivan, D.: Expanding endomorphisms of the circle revisited. Ergodic Theory Dynam Systems 5, 285–289 (1985) 12. Wenzel, W., Biham, O., Jayaprakash, C.: Periodic orbits in the dissipative standard map. Phys. Rev. A 43, 6550–6557 (1991) Communicated by G. Gallavotti
Commun. Math. Phys. 273, 67–118 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0254-y
Communications in
Mathematical Physics
Functional Integral Construction of the Massive Thirring model: Verification of Axioms and Massless Limit G. Benfatto, P. Falco, V. Mastropietro Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica, I-00133, Roma, Italy. E-mail:
[email protected] Received: 19 June 2006 / Accepted: 19 December 2006 Published online: 18 April 2007 – © Springer-Verlag 2007
Abstract: We present a complete construction of a Quantum Field Theory for the Massive Thirring model by following a functional integral approach. This is done by introducing an ultraviolet and an infrared cutoff and by proving that, if the “bare” parameters are suitably chosen, the Schwinger functions have a well defined limit satisfying the Osterwalder-Schrader axioms, when the cutoffs are removed. Our results, which are restricted to weak coupling, are uniform in the value of the mass. The control of the effective coupling (which is the main ingredient of the proof) is achieved by using the Ward Identities of the massless model, in the approximated form they take in the presence of the cutoffs. As a byproduct, we show that, when the cutoffs are removed, the Ward Identities have anomalies which are not linear in the bare coupling. Moreover, we find for the interacting propagator of the massless theory a closed equation which is different from that usually stated in the physical literature. 1. Introduction and Main Result 1.1. Historical Introduction. Proposed by Thirring [T] half a century ago, the Thirring model is a Quantum Field Theory (QFT) of a spinor field in a two dimensional space time, with a self interaction of the form (λ/4) dx(ψ¯ x γ µ ψx )2 . The interest of such a model, witnessed by the enormous number of papers devoted to it, is mainly due to the fact that it has a non-trivial behavior, similar to that of more realistic models, but at the same time it is simple enough to be in principle accessible to an analytic investigation. Hence the validity of several properties of QFT models, which in general can be verified at most by perturbative expansions, can be checked in principle in the Thirring model at a non-perturbative level. The Thirring model has been studied over the years following different approaches and we will recall here briefly the main achievements. Johnson solution of the massless Thirring model. After a certain number of “solutions” of the model fell into disrepute after inconsistences were encountered (see [W] for a review of such early attempts), Johnson [J] was able to derive, in the massless case, an
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G. Benfatto, P. Falco, V. Mastropietro
exact expression for the two point function T (ψx ψ¯ 0 ). His solution, based on operator techniques, is essentially a self-consistency argument: a number of reasonable requirements on the correlations is assumed from which their explicit expression can be determined. The first assumption is the validity of Ward-Takahashi Identities (WTi) of the form i∂µ T ( jzµ ψx ψ¯ y ) = a[δ(z − x) − δ(z − y)]iT (ψx ψ¯ y ) , i∂µ T ( jzµ,5 ψx ψ¯ y ) = a[δ(z ¯ − x) − δ(z − y)]γ 5 iT (ψx ψ¯ y ) , µ
(1.1)
µ,5
where the current jx and pseudocurrent jx are operators, formally defined respectively as [ψ¯ x γ¯ µ , ψx ]/2 and [ψ¯ x γ¯ µ γ 5 , ψx ]/2, and a, a¯ are introduced to take into account possible quantum anomalies [AB] (in the naive WTi, which one would expect from the classical conservation laws, a = a¯ = 1). The second assumption was the validity of a Schwinger-Dyson equation (formally derived from the equation of motion) for the 2-point function. Combining the WTi (1.1) with the Schwinger-Dyson equation, after some (highly formal) manipulations a closed equation for the 2-point function was found with solution T (ψx ψ¯ 0 ) = i(γ¯ µ ∂µ )−1 (x)(|x|/l0 )−ηz , where ηz =
λ (a − a) ¯ , 4π
(1.2)
γ¯µ are the Minkowski gamma matrices and l0 is an arbitrary constant with the dimension of a length. In order to express a, a¯ in terms of bare coupling λ, Johnson did a similar calculation for the 4-point function and, by a self-consistency argument, found the following explicit values: a −1 = 1 −
λ , 4π
a¯ −1 = 1 +
λ . 4π
(1.3)
The above expressions say that quantum anomalies are present and that ηz = 2(λ/4π )2 [1 − (λ/4π )2 ]−1 . Note also that the quantities a −1 − 1 and a¯ −1 − 1, called anomalies, are linear in the bare coupling, that is no higher order contributions are present; this property is called anomaly non-renormalization or the Adler-Bardeen theorem, and it holds, as a statement valid at all orders in perturbation theory and with suitable regularizations [AB], in Q E D4 . The validity of (1.3) in the Thirring model has been considered [GR] as a non-perturbative verification of the perturbative analysis of [AB] adapted to this case (see also [AF]). Unfortunately the Johnson solution is not satisfactory from a mathematical point of view, as it involves several formal manipulations of diverging quantities; even the meaning of the basic equation (1.1) is unclear as the averages in the l.h.s. and r.h.s. have to be (formally) divided by a vanishing constant, in order to be not identically vanishing. However it contains many deep ideas and all our analysis can be seen as a way to implement the Johnson approach in a rigorous functional integral context. Klaiber solution and axiomatic approach. Later on Klaiber [K] derived an explicit formula for the n-point functions, which depends on a single parameter in terms of which the critical indices are expressed; this formula reduces to the Johnson one for n = 2, 4 and for a suitable choice of that parameter. Klaiber’s solution was obtained by considering the expectation of n operators given by exponentials of bosonic operators, whose form was suggested by the solution of the motion’s equation of the classical Thirring
Functional Integral Construction of the Massive Thirring Model
69
model. An important observation of Klaiber was that that there is not a unique way to relate the critical indices to the bare coupling; for example, he got a relation between the critical index ηz and the bare coupling depending on a free parameter; this property reflects the arbitrariness in the choice of the current regularization. Wightman [W] proposed to construct the massless Thirring model following an axiomatic approach; one can start directly from the explicit expressions of the n-point functions at non-coinciding points derived in [K] (forgetting how they were derived) and try to verify the axioms necessary for the reconstruction theorem. Indeed all axioms can be easily verified from the Klaiber expression except positive definiteness, which was proved later on in [DFZ] and [CRW]; the idea was to define certain field operators, whose expectations verify the positivity property by construction and such that their n-point functions coincide, for a suitable choice of the parameters, with the expressions found in [K]. In this rigorous axiomatic approach, quadratic fermionic operators at coinciding points cannot be considered, hence neither WTi nor Schwinger-Dyson equations can be rigorously derived. Massive Thirring model. The massive Thirring model is much more difficult to study. In [C] clever heuristic arguments were given to suggest that the n-point functions of the currents in the Massive Thirring model, considered as perturbative expansions in the mass µ, should be order by order coinciding with the perturbative expansions of the interaction density n-point √ functions in the sine-Gordon model (a massless boson field with interaction ζ : cos( βϕ) :), if the identifications of parameters ζ = µ and (4π )/β = 1 + λ/π are done. To rigorously prove this conjecture is a very difficult problem and indeed, in [C], the calculations were performed only in the much simpler case where the interaction of the sine-Gordon model and the mass term of the Thirring model are restricted to a finite volume. This more simple problem had also a rigorous treatment in [D] for the special case of λ = 0, where the fermion model is not interacting (hence trivial), corresponding to β = 4π in the sine-Gordon model (which is not trivial even in this case). The relation between the Massive Thirring model (with λ > 0) and the sine-Gordon theory (in the region β < 4π ), both defined in a fixed finite box, was analyzed also in [SU], where rigorous results about the relation between the correlation functions were proved; however the limit of infinite volume could not be done. Another result on boson-fermion equivalence can be found in [FS], where the massive sine-Gordon model for β < 4π was considered; in this case equivalence with the massive SchwingerThirring model (QED2 ) is proved, in a suitable range of the corresponding parameters. A Bethe-ansatz solution for the Massive Thirring model was found in [BT], but as usual the n-point functions cannot be obtained from it. The Massive Thirring model has been extensively studied in the context of the integrable quantum field theories and many remarkable properties are known; in particular the exact S-matrix has been found in [Z] and the form factors in [S]. The Massive Thirring model has been also analyzed by perturbative Renormalization Group methods in [GL], to understand if the n-point functions computed perturbatively in the massive case pass over smoothly in the n-point functions obtained by the exact solution in the massless limit. All the above results provide much interesting information, but it should be remarked that, up to now, no rigorous treatment of the Massive Thirring model (and of the massless sine-Gordon model) has been presented. Bosonic functional integral approach. If the coupling λ is negative (following our definitions), the partition function and the generating functional of the massless (Euclidean) Thirring model can be written as bosonic functional integrals [FGS] by a Hubbard-
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G. Benfatto, P. Falco, V. Mastropietro
Stratonovich transformation; one can then integrate the fermion variables and it turns out that the partition function of the Thirring model can be written as det(γµ [∂µ + Aµ ]) , (1.4) P(d A) det(γµ ∂µ ) where Aµ,x is a two-dimensional Gaussian field with covariance Aµ,x Aν,y = e2 δµ,ν δ(x − y), e2 = −λ/2, and γµ are the Euclidean gamma matrices. A similar expression holds for the generating functional. It is well known [S] that, under suitable regularity conditions over A, log det(γµ ∂µ + γµ Aµ ) − log det(γµ ∂µ ) is quadratic in A; by replacing the determinant with a quadratic exponential, one then gets an explicitly solvable integral, from which the n-point functions can be derived. As stressed in [FGS], in this way one gets in a very simple way the results of the exact approach found in [J] and [K]. In particular the relation (1.2) for the two point critical index ηz is verified and the anomalies (1.3) can be easily computed. If a dimensional regularization is adopted, one finds a = 1 and a¯ −1 − 1 = λ/(2π ), while with a momentum regularization (1.3) holds; in both cases the anomaly non-renormalization holds. Of course in the above derivation an approximation is implicit; the logarithm of the fermionic determinant in (1.4) is given by a quadratic expression only if A is sufficiently regular, but the integral is over all possible fields A, hence one is neglecting the contributions of the irregular fields and there is no guarantee at all that such contribution is negligible. This approximation is usually supported by the fact that one gets in this way the same results found in [J] and [K]. Construction of the Massive Thirring model. We write the generating functional for the Euclidean Massive Thirring model as the following Grassmann integral (see below for a more precise definition) ¯ φx ψ¯ x √ dx[− λ4 (ψ¯ x γ µ ψx )2 +Jµ,x ψ¯ x γµ ψx + √ + φx ψx ] 1 ZN ZN eW (φ,J ) = , (1.5) PN (dψ)e N where N is a normalization constant, φ, J are external fields, Z N is the wave function renormalization, ψx , ψ¯ x are Grassmann variables, PN (dψ) is the fermionic integration corresponding to a fermionic propagator with mass µ N and a (smooth) momentum ultraviolet cut-off γ N , with γ > 1. Note that the average of ψ¯ x γ µ γ 5 ψx can be obtained by the derivatives with respect to Jµ , using the relation ψ¯ x γ µ γ 5 ψx = −iεµ,ν ψ¯ x γ µ ψx , with εµ,ν = −εν,µ and ε1,0 = 1. When J = φ = 0 and µ N = 0, the r.h.s. of (1.5) coincides with (1.4) (if λ is negative) in the limit N → ∞. The ultraviolet cutoff we will choose (for proving all properties except reflection positivity, for which a different choice is more convenient) is a momentum cutoff preserving, in the massless case, the oddness of the propagator under momentum reflection and invariance under the global 5 chiral transformation ψx± → e±α±βγ ψx± ; on the contrary it violates the local chiral 5 symmetry ψ ± → e±αx ±βx γ ψx± . We will show that, by properly choosing the bare wave function renormalization Z N and the bare mass µ N , the Schwinger functions at non-coinciding points obtained from (1.5) converge, for N → ∞, to a set of functions verifying the Osterwalder-Schrader axioms [OS2] for an Euclidean QFT. These functions are represented as convergent expansions, which depend on three parameters, the physical mass, the physical wave function renormalization and the physical coupling, but are independent on the way the ultraviolet cutoff is explicitly realized. On the contrary, the relation between the physical and the bare parameters depends on the details of the ultraviolet cutoff.
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71
The analysis of the functional integral (1.5) is performed by a multiscale analysis using a (Wilsonian) Renormalization Group approach as in [G]. After each iteration step an effective theory with new couplings, mass, wave function and current renormalizations is obtained. The effective parameters obey a recursive equation, called Beta function, and a major technical problem is that this iterative procedure can be controlled only by proving non-trivial cancellations in the Beta function. Such cancellations are established by suitable WTi valid at each scale and reflecting the symmetries of the formal action; contrary to the WTi formally valid when all cutoffs are removed, they have corrections due to the cutoffs introduced for performing the multiscale integration. The crucial role of WTi in the construction of the theory is a feature that the functional integral (1.5) shares with realistic models like Q E D or the Electroweak theory in d = 4, requiring WTi even to prove the perturbative renormalizability, which is absent in the models previously rigorously constructed by functional integral methods, like the massive Yukawa model [Le] or the massive Gross-Neveu model [GK, FMRS]. In this way we have obtained for the first time the construction of a QFT for the Massive Thirring model at weak coupling for any value of the (physical) mass. Our results are uniform in the mass, but the massless case can be investigated in more detail; so we can try to compare the representation we get for the two point function with that obtained via the exact solution. From the functional integral (1.5) we obtain, for N → ∞ and in the massless case, WTi of the same form as those postulated in [J]: ∂µ ψ¯ z γ µ ψz ; ψx ψ¯ y = a[δ(z − x) − δ(z − y)]ψx ψ¯ y , ∂µ ψ¯ z γ µ γ 5 ψz ; ψx ψ¯ y = a[δ(z ¯ − x) − δ(z − y)]γ 5 ψx ψ¯ y ,
(1.6)
2 where ψx ψ¯ y = lim N →∞ ∂ φ¯∂ ∂φ W|0 (similar definitions hold for the other averages); x y however the coefficients in (1.6) are given by the following expressions
a −1 = 1 −
λ λ − c+ λ2 + O(λ3 ) , a¯ −1 = 1 + − c+ λ2 + O(λ3 ) , 4π 4π
(1.7)
where c+ is a non-vanishing constant (its explicit value is calculated in Appendix B). The anomaly coefficients are not linear in the bare coupling (the anomaly nonrenormalization is violated ), contrary to what happens for the values (1.3), found in the exact approach of [J]. In particular, the constant c+ not only is different from 0, but even depends on the way the ultraviolet cutoff is realized. The difference of (1.7) with respect to (1.3) also implies that the approximation in (1.4) of the determinant by a quadratic exponential does not lead to correct results, at least if a momentum regularization is used. Note that the presence of nonlinear terms in the anomalies depends crucially from the fact that a local current-current interaction is assumed in the regularized Grassmann integral (1.5); if one starts from a non-local interaction one gets values a −1 − 1 and a¯ −1 − 1 given by (1.3), see [M], that is the anomaly non-renormalization is verified; the same is true if the local limit is performed after the N → ∞ limit. In (1.5) a bare wave function Z N for the fermionic fields has been introduced, to be fixed so that the “physical” renormalization has a fixed value at the “laboratory scale”; analogously we can introduce a (finite) bare charge also for the current, defining it as ¯ µ ψ. A physically meaningful choice for ξ could be ξ = a −1 , implying that the ξ ψγ current has no anomalies; this choice fixes the renormalization even of the pseudocurrent ¯ µ ψ), which then still has anomalies. ¯ µ γ 5 ψ = iεµ,ν ψγ (remember that ψγ
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Finally we will show that a closed equation for the 2-point function can be indeed obtained from the functional integral (1.5); it is however different with respect to the one postulated in [J] (which was the natural one obtained inserting the WTi in the Schwinger-Dyson equation) for the presence of additional anomalies. As a consequence, we get a relation between the critical index of the two point function and the anomalies different with respect to (1.2), namely ηz =
λ (a − a)[1 ¯ + c0 λ2 + O(λ3 )] , 4π
(1.8)
with c0 > 0 nonvanishing. This additional anomaly says that the closed equation for the 2-point function is not simply obtained by inserting the WTi in the Schwinger-Dyson equation. In the case of local interaction, the additional anomaly is vanishing and (1.2) holds. Note that (1.8) is not in disagreement with [K], where a one parameter family of relations is found between ηz and λ among which there is surely (1.8). In the rest of this section we will define more precisely our regularized functional integral and we state our main results. We will find it more convenient, from the point ± , with ω = ±, such that of view of the notation, to introduce the Weyl spinors ψx,ω − − + , ψ + ); the γ ’s matrices are explicitly ψx = (ψx,+ , ψx,− ), ψ¯ ≡ ψ + γ 0 and ψx+ = (ψx,+ x,− given by 0 1 0 −i 1 0 0 1 5 0 1 γ = , γ = , γ = −iγ γ = . 1 0 i 0 0 −1 1.2. Massive Thirring model with cutoff. We introduce in = [−L/2, L/2] × [−L/2, L/2] a lattice a whose sites are given by the space-time points x = (x, x0 ) = (na, n 0 a), with L/2a integer and n, n 0 = −L/2a, . . . , L/2a − 1. We also consider the set Da of 1 2π space-time momenta k = (k, k0 ), with k = (m + 21 ) 2π L and k0 = (m 0 + 2 ) L and m, m 0 = −L/2a, . . . , L/2a − 1. In order to introduce an ultraviolet and an infrared cutoff, we fix a number γ > 1, a positive integer N and a negative integer h; then we −1 define the function C h,N (k) in the following way; let χ0 ∈ C ∞ (R+ ) be a non-negative, non-increasing smooth function such that de f 1 if 0 ≤ t ≤ 1 χ0 (t) = (1.9) 0 if t ≥ γ0 , for a fixed choice of γ0 : 1 < γ0 ≤ γ ; then we define, for any h ≤ j ≤ N , de f f j (k) = χ0 γ − j |k| − χ0 γ − j+1 |k|
(1.10)
N −1 −1 and C h,N (k) = j=h f j (k); hence C h,N (k) acts as a smooth cutoff for momenta |k| ≥ γ N +1 (ultraviolet region) and |k| ≤ γ h−1 (infrared region). It is useful for technical reasons to choose for χ0 (t) a Gevrey function, for example one of class 2, that is a function such that, for any integer n, |d n χ0 (t)/dt n | ≤ C n (n!)2 ,
(1.11)
where C is a symbol we shall use regularly in the following to denote a generic con [h,N ]σ
stant. With each k ∈ Da we associate four Grassmann variables ψ , σ, ω = ± , k,ω
Functional Integral Construction of the Massive Thirring Model
73
de f −1 to be called field variables; we define D[h,N ] = k ∈ Da : C h,N (k) = 0 . On the finite Grassmannian algebra generated from these variables we define a linear
[h,N ] (the Lebesgue measure), so that, given a monomial Q(ψ) in the functional dψ
[h,N ] Q(ψ) = 0 except in the case Q(ψ) is equal to Q0 (ψ) = field variables, dψ
[h,N ]− ψ
[h,N ]+ or to one of the monomials obtained from Q0 (ψ) by k∈D [h,N ] ω=± ψk,ω k,ω
[h,N ] Q(ψ) is detera permutation of the field variables; in these cases the value of dψ [h,N ]
Q0 (ψ) = 1 and the anticommuting properties of the mined by the condition dψ field variables. We also define a Grassmann field on the lattice a by Fourier transform, according to the following convention: de f
[h,N ]σ ψx,ω =
1 iσ kx [h,N ]σ
ψ e , k,ω L2
x ∈ a .
(1.12)
k∈Da
[h,N ]σ is antiperiodic both in time and in space coordinate. By the definition of Da , ψx,ω The Generating Functional of the Thirring model with cutoff is W(ϕ, J ) = log PZ N (dψ) exp − λV ( Z N ψ) (1.13) [h,N ]+ [h,N ]− + [h,N ]− [h,N ]+ − , + ZN ψx,ω + ψx,ω + ψx,ω ϕx,ω dx Jx,ω ψx,ω dx ϕx,ω
ω
ω
where dx is a short-hand notation for a de f
[h,N ] · PZ N (dψ) = dψ
2
x∈a ,
2 L −4 Z 2N (−|k|2 − µ2N )C h,N (k)
k∈D [h,N ]
⎧ ⎨
1 exp −Z N 2 ⎩ L
Tω,ω (k)
−1 C h,N (k)
ω,ω =± k∈D [h,N ]
de f
Tω,ω (k) =
de f
V (ψ) =
D+ (k) µ N µ N D− (k)
ω,ω
;
[h,N ]−
[h,N ]+ ψ ψ k,ω k,ω
⎫ ⎬ ⎭
−1
,
de f
Dω (k) = − ik0 + ωk1 ,
1 [h,N ]+ [h,N ]− [h,N ]+ [h,N ]−
x,ω ψx,ω ψx,−ω ψx,−ω dx ψ 2 ω=±
·
(1.14)
(1.15)
(1.16)
σ } and {Jx,ω }x,ω are commuting variables, while {ϕx,ω x,ω,σ are anticommuting. {Jx,ω }x,ω σ } and {ϕx,ω are the external field variables. x,ω,σ
Remark. It is immediate to check that (1.13) coincides with (1.5), if the notational √ conventions adopted at the end of § 1.1 are used and up to the trivial rescaling ψ → Z ψ of the Grassmann variables. The above regularization is essentially a continuum one, as the lattice has been introduced just to give a meaning to the functional integral and the continuum limit is taken before the removal of the momentum cut-off. Such a regularization is very suitable to derive WTi and SDe; its main disadvantage is that the positive definiteness property is not automatically ensured. Such a property will be recovered indirectly
74
G. Benfatto, P. Falco, V. Mastropietro
later on by introducing a different regularization preserving positive definiteness and such that, by a proper choice of the bare parameters, the Schwinger functions in the limit of removed cutoffs are coinciding. de f
de f
Setting x = (x1 , . . . , xn ), and y = (y1 , . . . , ym ), for any given choice of the labels de f
de f
de f
σ = (σ1 . . . , σm ), ω = (ω1 . . . , ωn ) and ε = (ε1 . . . , εn ), the Schwinger functions are defined as ∂ n+m W (0, 0) . (1.17) ε ε L→∞ ∂ Jy1 ,σ1 · · · ∂ Jym ,σm ∂ϕx11 ,ω1 · · · ∂ϕxnn ,ωn
,h,a;(m;n) SσN;ω,ε (y; x) = lim de f
We will follow the convention that a missing label means that the corresponding limit has ,h;(m;n) ,h,a;(m;n) been performed, for instance SσN;ω,ε = lima→0 SσN;ω,ε In particular, in order to shorten the notation of the most used Schwinger functions, let: de f
N ,h,a,(0;2)
,h,a (x, y) = Sω,ω,(+,−) (x, y) , G 2,N ω
(1.18)
N ,h,a,(1;2)
,h,a G 2,1,N (z; x, y) = Sω ;ω,ω,+,− (z; x, y) . ω ,ω de f
(1.19)
We define the Fourier transforms so that, for example,
dk −ik(x−y) 2 e G ω (k) , (2π )2 dkdp ip(z−y) −ik(x−y) 2,1 de f G 2,1 e e G ω ,ω (p, k) . ω ,ω (z; x, y) = (2π )4 de f
G 2ω (x, y) =
(1.20) (1.21)
N ,h,a;(m;n)
The presence of the cutoffs makes the Schwinger functions Sσ ;ω,(ε) (y; x) well defined, since the generating functional is simply a polynomial in the external field variables, for any finite L, and the limit L → ∞ gives no problem, if h is finite. Note that the lattice is introduced just to give a meaning to the Grassmann integral and it can be removed safely if h, N are fixed. In § 2 we will prove the following result. Theorem 1.1. Given λ small enough and µ > 0, there exist functions Z N ≡ Z N (λ) µ N ≡ µg N (λ), such that Z N = γ −N ηz 1 + O(λ2 ) , µ N = µγ −N ηµ 1 + O(λ)) ,
(1.22)
with ηz = az λ2 + O(λ4 ), ηµ = −aµ λ + O(λ2 ), az , aµ > 0, and the following is true. 1. The limit lim
N ,−h,a −1 →∞
N ,h,a;(m;n)
Sσ ;ω,ε
(m;n)
(y; x) = Sσ ;ω,ε (y; x) ,
(1.23)
exist at non-coinciding points. (0;2n) 2. The family of functions S2n,ω (x), defined as equal to Sω,ε (x), with εi = +1 for i = 1, . . . , n and εi = −1 for i = n + 1, . . . , 2n, fulfills the OSa.
Functional Integral Construction of the Massive Thirring Model
75
3. The two point Schwinger function verifies the following bound: 2 G ω (x, y) ≤
C e−c |x − y|1+ηz
κµ1+ηµ |x−y|
,
(1.24)
= (1 + η )−1 − 1 = a λ + O(λ2 ). Moreover G 2 (x, y) is singular for x → y with ηµ µ µ ω and it diverges as |x − y|−1−ηz . 4. In the massless case (µ = 0) the two point Schwinger function can be written as
2ω (k) = (1 + f (λ)) G
|k|ηz , −ik0 + ωk
(1.25)
with f (λ) = O(λ) and independent of k. Items 1, 2, except Reflection Positivity (RP), and 3 will be proved in § 2.3, see Theorem 2.2 and the remark following it; Item 4 is proved in § 4. RP follows from Theorem 1.4 below and the remark at the beginning of § 1.5. Note the nontrivial dependence on the ultraviolet cut-off of the bare wave function and mass; the wave function is diverging when the ultraviolet cut-off is removed while the bare mass is vanishing or diverging depending if λ < 0 or λ > 0. The behavior of the two point Schwinger function is deeply modified by the presence of the interaction; at small distances we find a power law divergence with a non-trivial critical index 1 + ηz , with ηz a non-trivial function of λ. For large distances and for small masses µ (µ is the mass at O(1) momentum scales) one can distinguish two different regimes in the a nonasymptotic behavior, discriminated by an intrinsic length O(µ−1−ηµ ), with ηµ trivial critical index. For distances smaller than such scale we have a power law decay
κµ1+ηµ |x−y|
while for greater distances an exponential decay O(e−c second regime our bound is of course not optimal.
) is found; in this
Remark. It is an easy consequence of our proof that the Schwinger functions do not depend on the parameter γ , but are only functions of λ and µ. 1.3. WTi and chiral anomalies. Once the model is constructed and the OSa are verified, we can compute the WTi in the massless limit. We will show that ,h
2,1,N Dω (p)G (p; k) ω,ω ,h ,h ,h
2,N
2,N
2,1,N = δω,ω [G (k − p) − G (k)] + (p; k), ω ω ω,ω
(1.26)
,h
2,1,N (p; k) is a correction term which is formally vanishing if we replace where ω,ω ,h −1
2,1,N is nonvanishing in C h,N (k) by 1. The anomaly manifests itself in the fact that ω,ω the limit N , −h → ∞; we will prove in fact the following theorem.
Theorem 1.2. Under the same conditions of Theorem 1.1, in the massless limit, i.e. µ = 0, it holds that for finite nonvanishing k, k − p, p, ,h ,h ,h +
2,1,N
2,1,N
2,1,N (p; k) = Dω (p) R (p; k) + νh,N Dω (p)G (p; k) ω,ω ω,ω ω,ω −
2,1,N ,h (p; k) +νh,N D−ω (p)G −ω,ω
,
(1.27)
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G. Benfatto, P. Falco, V. Mastropietro
Fig. 1.1. Graphical representation of (1.26); the small circle in the last term represents the function in the r.h.s. of (3.3).
where all the quantities appearing in this identity admit a N , −h → ∞-limit, such that ν− =
λ + O(λ2 ) , 4π
ν+ = c+ λ2 + O(λ3 ) ,
(1.28)
2,1 (p; k)| satisfies the bound (2.63) below, and with c+ < 0, |G ω,ω
2,1 (p; k) = 0 . R ω,ω
(1.29)
It is immediate to check that the above result implies the WTi (1.6), with a −1 = 1 − ν − − ν + and a¯ −1 = 1 + ν − − ν + . Remark. We could write Ward identities involving only the truncated current correlations and, by using the analogue of (1.27), it is possible to prove that all truncated correlations vanish, except that of order two, which is proportional to the free one; this means that the current is indeed gaussian when cut-offs are removed. We will not prove here in detail this statement which is a rather straightforward consequence of our results.
1.4. Closed equation and additional anomaly. It is easy to see (see for instance [BM4]) that the Schwinger functions of (1.13) in the massless limit verify the following SDe:
g N ,h (k) ,h
2,N G (k) = ω − λ
gωN ,h (k) ω ZN
dp ,h
2,1,N χ¯ N (p)G −ω,ω (p; k) , (2π )2
(1.30)
−1 (k)Dω (k)−1 and χ¯ N (p) is a smooth function with support in where
gωN ,h (k) = C h,N |p| ≤ 3γ N +1 , equal to 1 if |p| ≤ 2γ N +1 (we can insert it freely in the SDe, thanks to the support properties of the propagator). Inserting the WTi (1.26) in SDe (1.30) and using (1.27), we get
g N ,h (k) ,h
2,N G (k) = ω − λ
gωN ,h (k) ω ZN gωN ,h (k) − λA+,h,N
dp
2,1,h,N χ¯ h (p)G −ω,ω (p; k) (2π )2
2,N ,h (k − p) dp G (1.31) χ (p) ω 2 (2π ) D−ω (p) Dεω (p) 2,1,N ,h dp +
gωN ,h (k) λAε,h,N χ (p) (p; k) , R 2 (2π ) D−ω (p) εω,ω ε
Functional Integral Construction of the Massive Thirring Model
77
Fig. 1.2. Graphical representation of (1.30); the wiggling line represents the function χ¯ N (p).
where de f
Aε,h,N = 1
ah,N =
− 1 − νh,N
+ − νh,N
,
ah,N − εa¯ h,N , 2 1 a¯ h,N = , − + 1 + νh,N − νh,N
(1.32)
χ¯ h (p) is defined as χ¯ N (p), with h in place of N , and χ (p) = χ¯ N (p) − χ¯ h (p) (so that the support of χ (p) is only for 2γ h+1 ≤ |p| ≤ 3γ N +1 ). The bound (2.63) below implies
2,1,h,N that, if k is fixed to a non-vanishing value, G −ω,ω (p; k) diverges more slowly than −1/8 , as p → 0; hence the second addend in the r.h.s. of (1.31) is vanishing in the |p| limit h = −∞. If the last term in (1.31) were vanishing for −h, N → ∞ (as the second addend),
2ω which is identical to the one postulated in [J]; one would get a closed equation for G it is just the formal Schwinger-Dyson equation combined with the WTi in the limit of removed cutoffs. However this is not what happens; despite when both the WTi and Schwinger-Dyson equations are true in the limit, one cannot simply insert one in the other to obtain a closed equation. The last term is non-vanishing and this is a additional anomaly effect which seems to be unnoticed in the literature. Despite the presence of the additional anomaly, a closed equation (different with respect to the one in [J]) holds, as shown from the following theorem. Theorem 1.3. Under the same conditions of Theorem 1.1, in the massless limit there exist functions αε,h,N , ρε,h,N such that, for non-vanishing k,
dp Dεω (p) 2,1,N ,h χ (p) (p; k) R (2π )2 D−ω (p) εω,ω
g N ,h (k) ,h
2,N
ε4,N ,h (k) , = −αε,h,N ω + αε,h,N + ρε,h,N )G (k) + R ω ZN
gωN ,h (k)
(1.33)
with lim
N ,−h→∞
ε4,N ,h (k) = 0 , R
(1.34)
and, in the limit of removed cutoff, α− = c1 λ + O(λ2 ) , α+ = c3 + O(λ) ,
ρ− = c2 λ + O(λ2 ) , ρ+ = c4 + O(λ) .
(1.35)
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G. Benfatto, P. Falco, V. Mastropietro
The above result says that, up to a vanishing term, the last addend in the r.h.s. of (1.31) can be written in terms of g and G 2 , so that a closed equation still holds in the limit, but different with respect to the one postulated in [J]; in particular one gets a relation between the critical index ηz and the anomalies a, a, ¯ which is different with respect to (1.2), than found in [J]. Corollary 1.1. The critical index of the massless two point Schwinger function (1.25) verifies ηz = with
ε
a − a¯ λ 4π 1 − λ ε Aε (αε + ρε )
(1.36)
Aε (αε + ρε ) = c0 λ + O(λ2 ) with c0 > 0.
1.5. Lattice fermions and positive definiteness. The regularization adopted in (1.13), −1 consisting in replacing the fermionic propagator Tω,ω (k) with a regularized one −1 C h,N (k)Tω,ω (k), is the closest one to the formal continuum limit and preserves several properties valid in the formal theory. If the bare mass µ N is vanishing, the regularized propagator in momentum space is odd under k → −k and the theory is invariant under [h,N ]± [h,N ]± the global symmetry ψx,ω → e±αω ψx,ω ; such properties imply that there is no generation through the interaction of a mass and produce many simplifications in the derivation of WTi and a closed equation for the two point Schwinger function. On the other hand, such regularization does not preserve the Reflection Positivity (RP), which is one of the Osterwalder-Schrader axioms it is necessary to prove, and it is very difficult to prove it directly in the limit. Therefore we prove all the statements in Theorem 1.1 starting from (1.13), except RP; such a property will be instead proved starting from a different regularization, which preserves RP, and showing that in the limit of removed cutoffs the two different regularizations of the Thirring model give the same Schwinger functions, if the “bare” parameters are suitably chosen. The problem of regularizations preserving RP is extensively studied (for a review see [MM]), and a very good solution is the one proposed by Wilson, consisting in approximating the continuum space-time by a lattice and adding a suitable term in the fermionic action to avoid the fermion doubling problem. Indeed, if one simply replaces k, k0 in the propagator with a −1 sin(ka) and a −1 sin(k0 a), the well known fermion doubling problem is encountered, namely that the massless fermion propagator has four poles instead of a single one. In the continuum limit a → 0 this means that there are four fermion states per field component and such extra unwanted fermions influence possibly the physical behavior in a non trivial way. In the Wilson formulation a term is added to the free action, called Wilson term, to cancel the unwanted poles. Then in the Wilson lattice regularization the fermionic integration is given by ⎧ ⎨ Z de f a PZ a (dψ) = exp − 2 ⎩ L ·
ω,ω =± k∈Da
+ dψ
− dψ k,ω k,ω , ¯ Na (k)
k∈Da ω=±
r −1 (k)
ω ,ω
+ ψ
− ψ k,ω k,ω
⎫ ⎬ ⎭
·
(1.37)
Functional Integral Construction of the Massive Thirring Model
79
where the covariance
rω,ω (k) is defined as de f
rω,ω (k) =
1 e− (k) −µa (k) , e+ (k)e− (k) − µa2 (k) −µa (k) e+ (k) ω,ω
(1.38)
with k0 = (m 0 + 1/2)2π/L, k = (m + 1/2)2π/L, n, n 0 = −L/2a, 1, . . . , L/2a − 1, sin(ka) sin(k0 a) +ω , a a 1 − cos(k0 a) 1 − cos(ka) de f µa (k) = µ + + , a a de f
eω (k) = −i
(1.39)
and N¯ a is the normalization. The generating functional is given by
PZ a (dψ) exp − λa Z a2 V (ψ) + νa Z a N (ψ) (1.40)
+ + − − + − ψx,ω + ψx,ω + ψx,ω ϕx,ω , · exp Z a(2) dx Jx,ω ψx,ω dx ϕx,ω
ω
ω
+ ψ− . where N (ψ) = ω=± dx ψx,ω x,−ω The regularization in (1.37) has the crucial property of preserving RP [OSe]; moreover, the fermion doubling problem is solved, as the term (1 − cos(k0 a))/a + (1 − cos(ka))/a has the effect that, in the massless case, only one pole is present. On the other hand, even if the bare mass is vanishing, both the oddness of the propagator and [h,N ]± [h,N ]± the global symmetry ψx,ω → e±αω ψx,ω are not true, and this leads to the generation through the interaction of a mass; to get an interacting massless theory, then a mass counterterm νa has to be introduced. On the other hand the presence of the lattice and of the Wilson term poses no serious extra difficulties to our methods, which do not need any special symmetry and were just developed to study interacting fermions on a lattice [BM1]; in particular the fermionic action in (1.37) is essentially the one appearing in the fermionic representation of two interacting copies of d = 2 classical Ising models, and it was extensively studied (at fixed lattice step) in [Ma] or [GiM]. We will prove in § 5 that the continuum limit in (1.37) can be taken. N ,(m;n) We call Sσ ;ω,ε the Schwinger functions (1.17) (in the limit a = 0 and h = −∞) and S¯σa,(m;n) ;ω,ε the Schwinger functions corresponding to (1.40); in § 5 we shall prove the following theorem.
Theorem 1.4. Given λ small enough and µ > 0, there exist functions λa (λ), νa (λ), µa ≡ µga (λ), Z a ≡ Z a (λ) verifying Z a = a −ηz 1 + O(λ2 ) , µa = µa −ηµ 1 + O(λ) , (1.41) with ηz = az λ2 + O(λ4 ), ηµ = −aµ λ + O(λ2 ), az , aµ > 0 such that the limit lim
L ,a −1 →∞
a,(m;n) (m;n) S¯σ ;ω,ε ≡ S¯σ ;ω,ε ,
(1.42)
exist at non coinciding points. Moreover, given N > 0, let a N = π(4γ N +1 )−1 ; then N ,(m;n)
a ,(m;n)
lim [Sσ ;ω,ε (z; x) − S¯σ N;ω,ε
N →∞
(z; x)] = 0 .
(1.43)
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G. Benfatto, P. Falco, V. Mastropietro
The above result says that in the limit of removed cutoffs the two different regularizations of the Thirring model give the same Schwinger functions, if the “bare” parameters are suitably chosen.
1.6. Contents. The rest of the paper is organized in the following way. In § 2 a multiscale analysis of the functional integral (1.13) is performed; in particular the marginal and relevant terms of the effective interaction are identified. In this way we find an expansion for the Schwinger functions in terms of running coupling constants, which is convergent uniformly in the infrared and ultraviolet cut-off and in the mass, provided that the mass and wave function renormalization are chosen properly and that the quartic running coupling constants are small enough for all scales; this last property follows from the asymptotic vanishing of the Beta function proved in [BM4]. In § 3 we derive the Ward Identities (1.26) and we define a convergent expansion for the last addend in (1.26), from which Theorem 1.2 is derived. In § 4 we analyze the Schwinger-Dyson equation and we define suitable convergent expansions for the last addend in (1.31), from which we derive Theorem 1.3 and Corollary 1.1. Finally in § 5 the lattice construction is performed and the positivity property is achieved. The proof of the above results is based on many technical arguments, some of which were already proved in [BM1–BM4]; hence, in this paper we shall discuss in detail only the arguments not discussed in those papers.
2. Multiscale Analysis for (1.13) 2.1. Renormalization Group analysis. The integration of the generating functional (1.13) is done almost exactly (essentially up to a trivial rescaling) as described in [BM1–BM4]; hence we briefly resume here such procedure to fix notations. It is possible to prove by induction that, for any j : h ≤ j ≤ N , there are a constant E j , two positive functions Z˜ j (k), µ˜ j (k) and two functionals V ( j) and B ( j) , such that, if Z j = maxk Z˜ j (k), eW (ϕ,J ) = e−L
2E
j
PZ˜ j ,µ˜ j ,Ch, j (dψ [h, j] )e−V
( j) (
√
Z j ψ [h, j] )+B( j) (
√
Z j ψ [h, j] ,ϕ,J )
, (2.1)
where: [h, j] ) is the effective Grassmannian measure at scale j, equal to 1. P Z j ,µ˜ j ,Ch, j (dψ
P Z j ,µ˜ j ,Ch, j (dψ ⎧ ⎪ ⎨ 1 · exp − 2 ⎪ ⎩ L ( j)
[h, j]
)
[h, j]+
−1 k:Ch, j (k)>0
C h, j (k) Z˜ j (k)
(2.2)
N j (k)
−1 ω,ω =±1 k:Ch, j (k)>0
[h, j]−
dψ k,ω d ψk,ω
ω±1
[h, j]+ T ( j) (k)ψ
[h, j]− ψ k,ω ω,ω k,ω
with Tω,ω given by (1.15) with µ˜ j (k) replacing µ N , C h, j (k)−1 = N j (k) a suitable normalization constant.
j
r =h
⎫ ⎪ ⎬ ⎪ ⎭
,
fr (k) and
Functional Integral Construction of the Massive Thirring Model
81
2. The effective potential on scale j, V ( j) (ψ), is a sum of monomial of Grassmannian variables multiplied by suitable kernels, i.e. it is of the form $ 2n " 2n # % ∞ σ 1 ( j) ( j)
i V (ψ) = ψ σi ki , ki ,ωi W2n,ω (k1 , ..., k2n−1 )δ L 4n k ,...,k n=1
1 2n ω1 ,...,ω2n
i=1
i=1
(2.3) where σi = + for i = 1, . . . , n, σi = − for i= n+1, . . . , 2n and ω = (ω1 , . . . , ω2n ); 3. The effective source term at scale j, B ( j) ( Z j ψ, ϕ, J ), is a sum of monomials of Grassmannian variables and ϕ ± , J field, with at least one ϕ ± or one J field; we shall write it in the form ( j) ( j) B ( j) ( Z j ψ, ϕ, J ) = Bϕ( j) ( Z j ψ) + B J ( Z j ψ) + W R ( Z j ψ, ϕ, J ) , (2.4) ( j)
( j)
where Bϕ (ψ) and B J (ψ) denote the sums over the terms containing only one ϕ or J field, respectively. B ( j) ( Z j ψ, ϕ, J ) can be written as the sum over monomials
( j) of ψ, ϕ, J multiplied by kernels W . 2n,n ϕ ,n J ,ω
(0) (N ) Of course (2.1) is true for j = N , with Z˜ N (k) = Z N , W R = 0, and V (N ) (ψ), Bϕ , (N )
( j) , V ( j) and B ( j) , j < N , are B J given implicitly by (1.13). The kernels in W functions of µk , Z k and the effective couplings λk (to be defined later), with k ≥ j; the iterative construction below will inductively imply that the dependence on these variables is well defined.
We now begin to describe the iterative construction leading to (2.1). We introduce
( j) in the following way: two operators Pr , r = 0, 1, acting on the kernels W ( j)
∂ W ( j) ( j) ( j)
=W
= P0 W , P1 W µ˜ k (k) . (2.5) µ˜ j ,..µ N =0 ∂ µ˜ k (k) k≥ j,k
µ˜ j ,..µ N =0
( j) in the following We introduce also two operators Lr , r = 0, 1, acting on the kernels W way: 1. If n = 1, ( j)
de f
(k) = L0 W 2,ω
1 4
f 1
( j) (k) de L1 W = 2,ω 4
η,η =±1
η,η =±1
( j) k¯ ηη , W 2,ω
( j) (k¯ ηη ) η k0 L + η k L , W 2,ω π π
(2.6)
where k¯ ηη = η πL , η πL . f
4,ω de = 0 and 2. If n = 2, L1 W f
( j) (k¯ ++ , k¯ ++ , k¯ ++ ) .
( j) (k1 , k2 , k3 )de L0 W =W 4,ω 4,ω ( j) de f
( j) de f
3. If n > 2, L0 W 2n,ω = L1 W2n,ω = 0.
(2.7)
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G. Benfatto, P. Falco, V. Mastropietro
( j) as Given L j , P j , j = 0, 1 as above, we define the action of L on the kernels W 2n,ω follows. 4. If n = 1, then f
( j) de
( j)
( j) LW 2,ω,ω = (L0 + L1 )P0 W2,ω,ω + L0 P1 W2,ω,ω .
(2.8)
f
( j) de
( j) 5. If n = 2, then LW 4,ω = L0 P0 W4,ω .
( j) = 0. 6. If n > 2, then LW 2n,ω
( j) = 0, because of the parity properties (in the exchange k → Note that L0 P0 W 2,ω,ω −k) of the diagonal propagators, whose number is surely odd in each Feynmann graph ( j)
( j) = 0, because there are no contributions of first contributing to W2,ω,ω ; L0 P1 W 2,ω,ω
( j)
( j) = 0, since the only way to get a contribution to W order in µk ; P0 W 2,ω,−ω 2,ω,−ω is to use at least one antidiagonal propagator. Therefore (2.8) reads
( j) = L1 P0 W
( j) , LW 2,ω,ω 2,ω,ω
( j)
( j) LW 2,ω,−ω = L0 P1 W2,ω,−ω .
(2.9)
Note also that L2 V ( j) = LV ( j) . The effect of L on V ( j) is, by definition, to replace on
( j) ; we get
( j) with LW the r.h.s. of (2.3) W 2n,ω 2n,ω [h, j]
LV ( j) (ψ [h, j] ) = z j Fζ
[h, j]
+ s j Fσ[h, j] + l j Fλ
,
(2.10)
where z j , a j and l j are real numbers and 1 [h, j]
[h, j]− ,
[h, j]+ ψ Fζ = 2 Dω (k)ψ k,ω k,ω L ω −1 Fσ[h, j] = [h, j]
Fλ
=
1 L2 ω 1 L8
k:Ch, j (k)>0
[h, j]+ ψ
[h, j]− , ψ k,ω k,−ω
(2.11)
−1 k:Ch, j (k)>0
−1 k1 ,...,k4 :Ch, j (ki )>0
[h, j]+ ψ
[h, j]− ψ
[h, j]+ ψ
[h, j]− δ(k1 − k2 + k3 − k4 ) . ψ k1 ,+ k2 ,+ k3 ,− k4 ,−
Remark. According to dimensional power counting considerations, the quartic terms in the effective interaction are marginal and the quadratic terms are relevant in the RG sense. However we can further split the kernels in a part independent of the mass and the rest which can be proved, by an improvement of the naive dimensional bounds,
( j) only the part of irrelevant in the RG sense; hence it is sufficient to include in LW 2n,ω the kernels independent from the mass. Analogously, we write B ( j) = LB ( j) +RB( j) , R = 1−L, according to the following ( j) ( j) definition. First of all, we put LW R = 0. Let us consider now B J ( Z j ψ). It is easy to see that the field J is equivalent, from the point of view of dimensional considerations, to two ψ fields. Hence, the only terms which need to be renormalized are those of second order in ψ, which are indeed marginal. We shall use for them the definition 1 ( j) ( j,2)
[h, j]+ )( Z j ψ
[h, j]− ) . (2.12) B J ( Z j ψ) = Bω,ω˜ (p, k) J p,ω ( Z j ψ p+k,ω˜ k,ω˜ L4 ω,ω˜
p,k
Functional Integral Construction of the Massive Thirring Model
83
( Z j ψ), in analogy to what we did for the effective potential, by ( j,2) ( j,2) decomposing it as the sum of LB J ( Z j ψ) and RB J ( Z j ψ), where L is defined ( j) through its action on
B (p, k) in the following way: ( j,2)
We regularize B J
ω,ω˜
( j)
L
Bω,ω˜ (p, k) =
1 ( j) δω,ω˜ P0
Bω,ω˜ (0, k¯ η,η ) ; 4
(2.13)
η,η =±1
( j) note that L
Bω,−ω = 0 because of the symmetry property
gω( j) (k∗ ), k = (k, k0 ), k∗ = (−k0 , k) .
gω( j) (k) ≡ f j (k)Dω (k)−1 = −iω
(2.14)
We get ( j,2) LB J ( Z j ψ)
=
Z (2) j ω
Zj
dx Jx,ω
+ Z j ψx,ω
− Z j ψx,ω
,
(2.15)
(2)
which defines the renormalization constant Z j ; we shall extend this definition to j = N (2)
by putting, in agreement with (1.13), Z N = Z N . ( j) Finally we have to define L for Bϕ ( Z j ψ); we want to show that, by a suitable choice of the localization procedure, if j ≤ N − 1, it can be written in the form " N ∂ Q,(i) ( j) + Bϕ ( Z j ψ) = gω,ω V ( j) ( Z j ψ) dxdy ϕx,ω (x − y) + ∂ψy,ω ω,ω i= j+1 # ∂ Q,(i) − + − V ( j) ( Z j ψ)gω,ω (y − x)ϕx,ω (2.16) ∂ψy,ω [h, j]+ ( j+1) 1 − + ( j+1)
(k)
[h, j]− ψ , ϕ +
ϕ (k) ψ Q Q + k,ω k,ω k,ω ω,ω ω,ω k,ω L2 −1 ω,ω
k:Ch, j (k)>0
Q,(i)
where
g ω,ω (k) =
(i)
(i) (k), g ω,ω (k) Q ω
ω ,ω
(i)
gω,ω is the renormalized propagator of
( j) (k) is defined the field on scale j (see (2.22) below for a precise definition) and Q ω,ω inductively by the relations
( j+1) (k) − z j Z j Dω (k)
( j) (k) = Q Q ω,ω ω,ω
N i= j+1
(0) (k) Q ω,ω
=1.
Q,(i)
g ω,ω (k) − s j Z j
N
Q,(i)
g ω,−ω (k) ,
i= j+1
(2.17) ( j)
The L operation for Bϕ is defined by decomposing V ( j) in the r.h.s. of (2.16) as LV ( j) + RV ( j) , LV ( j) being defined by (2.10). After writing V ( j) = LV ( j) + RV ( j) and B ( j) = LB ( j) + RB( j) , the next step is to renormalize the free measure PZ˜ j ,µ˜ j ,Ch, j (dψ [h, j] ), by adding to it part of the r.h.s. of (2.10). We get that (2.1) can be written as √ √ [h, j] [h, j] 2 ˜ ( j) ˜ ( j) e−L t j (2.18) PZ˜ j−1 ,µ˜ j−1 ,Ch, j (dψ [h, j] ) e−V ( Z j ψ )+B ( Z j ψ ) ,
84
G. Benfatto, P. Falco, V. Mastropietro
where, since Z˜ j (k) = Z j ≡ maxk Z˜ j (k) and µ˜ j (k) = µ j ≡ (Z j+1 /Z j )(µ j+1 + s j+1 ), −1 if C h, j (k) = 1, then −1 ˜ j−1 (k) = Z˜ j−1 (k) = Z j [1 + C h, j (k)z j ] , µ
Zj −1 [µ j + C h, j (k)s j ] , ˜ Z j−1 (k)
[h, j] V˜ ( j) (ψ [h, j] ) = V ( j) (ψ [h, j] ) − z j Fζ − s j Fσ[h, j] ,
(2.19) (2.20)
and the factor exp(−L 2 t j ) in (2.18) takes into account the different normalization of the two measures. Moreover ( j) ( j) B˜ ( j) ( Z j ψ [h, j] ) = B˜ϕ( j) ( Z j ψ [h, j] ) + B J ( Z j ψ [h, j] ) + W R , (2.21) ( j)
( j)
where B˜ϕ is obtained from Bϕ by inserting (2.21) in the second line of (2.16) and by absorbing the terms proportional to z j , s j in the terms in the third line of (2.16). If j > h, the r.h.s of (2.18) can be written as 2 PZ˜ j−1 ,µ j−1 ,Ch, j−1 (dψ [h, j−1] ) PZ j−1 ,µ j−1 , f˜−1 (dψ ( j) ) e−L t j j √ √ √ −l j Fλ ( Z j ψ [h, j] )−RV ( Z j ψ [h, j] )+B˜ ( j) ( Z j ψ [h, j] ) ·e , (2.22) where f˜j (k) = f j (k)Z j−1 [ Z˜ j−1 (k)]−1 . The above integration procedure is done till the scale h ∗ = max{h, h¯ ∗ }, where h¯ ∗ is the maximal j such that γ j ≤ µ j . If h¯ ∗ < j ≤ N , by using the Gevray property (1.11) of χ0 , see [DR], we get √ C j ( j) |gω,ω (x, y)| ≤ γ j e−c γ |x−y| , Z j−1 √ µj C ( j) j −c γ j |x−y| γ |gω,−ω (x, y)| ≤ e , (2.23) Z j−1 γ j where C and c are suitable constants; moreover, ¯∗
(≤h ) (x, y)| ≤ |gω,ω (≤h¯ ∗ ) |gω,−ω (x, y)|
≤
C Z h¯ ∗ −1 C Z h¯ ∗ −1
¯∗
γ h e−c
µh¯ ∗ ∗ γ h¯
√
∗
γ h¯ |x−y| ¯∗
γ h e−c
,
√
∗
γ h¯ |x−y|
.
(2.24)
Note that the propagator
gωQ,(i) (k) is equivalent to
gω(i) (k), as concerns the dimensional bounds, since the sum in the r.h.s. of (2.17) contains at most two nonvanishing terms. We now rescale the field so that
( j) ( Z j−1 ψ [h, j] ) , l j Fλ ( Z j ψ [h, j] ) + RV( Z j ψ [h, j] ) = V
( j) ( Z j−1 ψ [h, j] ) ; B˜ ( j) ( Z j ψ [h, j] ) = B (2.25) [h, j]
2
( j) (ψ [h, j] ) = λ j F it follows that LV , where λ j = (Z j Z −1 λ j−1 ) l j ; we shall extend this definition to j = N by putting, in agreement with (1.13), λ N = λ. If we now define √ √ ( j−1) ( Z j−1 ψ [h, j−1] )+B( j−1) ( Z j−1 ψ [h, j−1] )−L 2 E j (2.26) e−V √ √
( j)
( j) Z j−1 [ψ [h, j−1] +ψ ( j) ] +B Z j−1 [ψ [h, j−1] +ψ ( j) ] ( j) −V = PZ j−1 ,µ j−1 , f˜−1 (dψ ) e , j
Functional Integral Construction of the Massive Thirring Model
85
it is easy to see that V ( j−1) and B ( j−1) are of the same form of V ( j) and B ( j) and that the procedure can be iterated. Note that the above procedure allows, in particular, to write λ j , Z j , µ j , for any j such that N > j ≥ h ∗ , in terms of λ j , Z j , µ j , j > j. At the end of the iterative integration procedure, we get W(ϕ, J ) = −L 2 E L +
m ϕ +n J ≥1
(h)
S2m ϕ ,n J (ϕ, J ) ,
(2.27)
(h) where E L is the free energy and S2m ϕ ,n J (ϕ, J ) are suitable functionals, which can be expanded, as well as E L , the effective potentials and the various terms in the r.h.s. of (2.4) and (2.3), in terms of trees. We do not repeat here the analysis leading to the tree expansion, as it is essentially identical to the one for instance in § 3 of [BM1], and (h) we quote the results; it turns out the kernels S2m ϕ ,n J (ϕ, J ) can be written as in formula (102) of [BM2]: (h)
S2m ϕ ,n J (ϕ, J ) =
N −1 ∞
n=0 j0 =h ∗ −1 ω τ ∈T j
P∈P ϕ J 0 ,n,2m ,n |Pv0 |=2m ϕ
dx
ϕ 2m
i=1
J
ϕxσii ,ωi
n r =1
Jx2m ϕ +r ,ω2m ϕ +r S2m ϕ ,n J ,τ,ω (x) ,
(2.28)
where we refer to § 3.4 of [BM2] for the notation. In particular, – T j0 ,n,2m ϕ ,n J is a family of trees (identical to the those defined in § 3.2 of [BM2], up to the (trivial) difference that the maximum scale of the vertices is N + 1 instead of +1), with root at scale j0 , n normal endpoints (i.e. endpoints not associated to ϕ or J fields), n ϕ = 2m ϕ endpoints of type ϕ and n J endpoints of type J . – If v is a vertex of the tree τ , Pv is a set of labels which distinguish the external fields of v, that is the field variables of type ψ which belong to one of the endpoints following v and either are not yet contracted in the vertex v (we shall call Pv(n) the set of these variables) or are contracted with the ψ variable of an endpoint of type ϕ ϕ (n) ϕ through a propagator g Q(h v ) ; note that |Pv | = |Pv | + n v , if n v is the number of endpoints of type ϕ following v. – xv , if v is not an endpoint, is the family of all space-time points associated with one of the endpoints following v. 2.2. Convergence of the RG expansion. In order to control the RG expansion, it is sufficient to show that λ¯ h ≡ maxh≤ j≤N |λ j | stays small if λ = λ N is small enough. This property is surely true if |h − N | is at most of order λ−1 , but to prove that it is true for any h, N is quite nontrivial. In § 4.3, by using WTi and SDe, we shall prove the following theorem, essentially taken from [BM4]. Theorem 2.1. There exists a constant ε1 , independent of N , such that, if |λ| ≤ ε1 , the (2) constants λ j , Z j , Z j and µ j are well defined for any j ≤ N ; moreover there exist suit(2) Z j,
Z and
µ j , defined for j ≤ 0 and independent of N , such that able sequences
λj,
j
86
G. Benfatto, P. Falco, V. Mastropietro
(2) (2) λj =
λ j−N , Z j =
Z j−N , Z j =
Z j−N and µ j =
µ j−N . The sequence
λ j converges, 2 as j → −∞, to a function λ−∞ (λ) = λ + O(λ ), such that
|
λ j − λ−∞ | ≤ Cλ2 γ j/4 .
(2.29)
Finally, there exist ηµ = −aµ λ + O(λ2 ) and ηz = az λ2 + O(λ3 ), with aµ and az strictly (2) (2) Z j−1 /
Z j )−ηz | ≤ Cλ2 γ j/4 , | logγ (
Z j−1 /
Z j )− positive, such that, for any j ≤ 0, | logγ (
ηz | ≤ Cλ2 γ j/4 and | logγ (
µ j−1 /
µ j ) − ηµ | ≤ C|λ|γ j/4 . (2)
Remark. Note that the definitions of λ j , µ j , Z j and Z j are independent of the µ value; however, in the theory with µ = 0, there appear only their values with j ≥ h¯ ∗ . Remark. From (2.29) we see that λ j = λ−∞ + O(λ2 γ −(N − j)/4 ), which means that, at any fixed scale j, the running coupling constant λ j has no flow (it is a constant in j) when the ultraviolet cut-off is removed (N → ∞). On the other hand λ j has a bounded but non-trivial flow from the cut-off scale N to O(1) scales, so that λ−∞ = λ N . Once the dressed coupling λ−∞ is fixed to some value, the bare coupling is a complicated function of λ−∞ , quite sensitive to the details of the regularization; this is in perfect agreement with the general QFT philosophy, in which the bare couplings are unobservable, while the dressed one are fixed by the experiments. The above result implies that we can remove the cutoffs and take the limit N , −h → ∞, by choosing the normalization conditions Z 0 = 1, µ0 = µ . In fact, by using (2.30), it is easy to prove that, if Z N = N and µ N = [ i=1 (µ j−1 /µ j )]−1 , then µ j = µγ −ηµ j F1, j,N (λ),
Z j = γ −ηz j F2, j,N (λ),
(2) ZN
=[
N
(2.30)
i=1 (Z j−1 /Z j )]
−1
−ηz j Z (2) F3, j,N (λ) , j = ζ (λ)γ
(2.31) where ζ (λ) =
0
(2)
Z j−1
Zj
j=−∞
(2)
Zj
Z j−1
(2.32)
and Fi, j,N (λ), i = 1, 2, 3, satisfy the conditions Fi,0,N (λ) = 1,
|Fi, j,N (λ) − 1| ≤ C|λ|2 γ −[N −max{ j,0}]/4 .
(2.33)
Note also that the first of (2.31) implies that, in the limit N , −h → ∞, if [x] denotes the largest integer ≤ x, ' & logγ |µ| h¯ ∗ = . (2.34) 1 + ηµ Moreover, the proof of Theorem 2.1 implies that the critical indices ηz and ηµ are given by tree expansions, such that everywhere the constants λ j and Z j are substituted with λ−∞ and γ −ηz j . In particular ηz is the solution of an equation of the form ηz = az λ2−∞ + λ4−∞ H (λ−∞ , ηz ) ,
(2.35)
which allows to explicitly calculate the perturbative expansion of ηz through an iterative procedure.
Functional Integral Construction of the Massive Thirring Model
87 (2)
Remark. The normalization conditions (2.30) could also include the value of Z j
for
(2) Zj
for j = N , by putting it equal to Z N . j = 0, but we have chosen to fix the value of A different choice would only change the value of ζ (λ) by an arbitrary finite constant. 2.3. The Schwinger functions. Theorem 2.1 allows us to control the expansion of the Schwinger functions, by using the following bound for the kernels appearing in the expansion (2.28): ϕ J dx|S2m ϕ ,n J ,τ,ω (x)| ≤ L 2 C 2m ϕ +n J (C λ¯ j0 )n γ − j0 (−2+m +n ) ·
ϕ 2m
i=1
J
(2)
n γ −h i Z h¯ r (Z h i )1/2 Z h¯ r r =1
v not e.p.
Z hv Z h v −1
|Pv |/2
γ −dv ,
(2.36)
where h i is the scale of the propagator linking the i th endpoint of type ϕ to the tree, h¯ r is the scale of the r th endpoint of type J and dv = −2 + |Pv |/2 + n vJ + z˜ (Pv ) , with
⎧ ⎪ 3/4 ⎪ ⎪ ⎨3/2 z˜ (Pv ) = ⎪ 3/4 ⎪ ⎪ ⎩0
(2.37)
ϕ
if |Pv | = 4, n v = 0, 1, n vJ = 0, ϕ if |Pv | = 2, n v = 0, 1, n vJ = 0, ϕ if |Pv | = 2, n v = 0, n vJ = 1, otherwise.
(2.38)
The above bound has a simple dimensional interpretation; how to prove it rigorously has been explained in detail in the very similar model studied in [BM1] (see also § 3 of [BM2]). We simply remark here that, had we defined L = 0, we would have obtained a bound similar to (2.36) with z˜ (Pv ) = 0 in (2.38). The regularization procedure has the effect that the vertex dimension dv gets an extra z˜ (Pv ), whose value can be understood in the following way. If we apply the regularizing operator 1 − L0 to the kernel associated with the vertex v, the bound improves by a dimensional factor γ h v −h v , if v is the first non-trivial vertex preceding v; if we apply 1 − L0 − L1 , the bound improves by a factor γ 2(h v −h v ) . Moreover, if to a kernel associated with the vertex v the operator 1 − P0 is applied, the bound improves by a factor ∗
¯
∗
|µh v |γ −h v ≤ |µh ∗ ||µh v /µh ∗ |γ −h v ≤ γ h γ cλ j0 (h v −h ) γ −h v =γ
(1−cλ¯ j0 )(h ∗ −h v )
≤γ
3 4 (h v −h v )
(2.39)
; 3
if 1 − P0 − P1 is applied, the bound improves by a factor (|µh v |γ −h v )2 ≤ γ 2 (h v −h v ) . By suitably modifying the analysis leading to the bound (2.36), we can derive a bound for all the Schwinger functions and get a relatively simple tree expansion for their removed cutoffs limit. We shall here consider in detail the Schwinger functions with n J = 0, at fixed non-coinciding points; we shall get a bound sufficient to prove two of the OSa, the boundedness and the cluster property. Since relativistic invariance is obvious by construction, to complete the proof of OSa there will remain to prove only positive definiteness.
88
G. Benfatto, P. Falco, V. Mastropietro
Given a set x = {x1 , . . . , xk } of k (an even integer) space-time points, such that δ ≡ minx =y∈x |x − y| > 0, and a set ω = {ω1 , . . . , ωk } of ω-indices, the k-points Schwinger function Sk,ω (x) is defined as the k th order functional derivative of the generating function (2.28) with respect to ϕx+1 ,ω1 , . . . , ϕx+k/2 ,ωk/2 and ϕx−k/2+1 ,ωk/2+1 , . . . , ϕx−k ,ωk at J = ϕ = 0, see (1.17) and Item 2) in Theorem 1.1. By using (2.28), we can write
Sk,ω (x) =
∞ N −1
lim
|h|,N →∞
π(x,ω) n=0 j0 =h ∗ −1 ω τ ∈T j0 ,n,k,0
Sk,0,τ,ω (x) ,
(2.40)
P∈P |Pv0 |=k
where π(x,ω) denotes the sum over the permutations of the x and ω labels associated with the k/2 endpoints of type ϕ + , as well as those associated with the k/2 endpoints of type ϕ − . We need some extra definitions. Given a tree τ contributing to the r.h.s. of (2.40), we call τ ∗ the tree which is obtained from τ by erasing all the vertices which are not needed to connect the k special endpoints (all of type ϕ). The endpoints of τ ∗ are the k special endpoints of τ , which we denote vi∗ , i = 1, . . . , k; with each of them a space-time point xi is associated. Given a vertex v ∈ τ ∗ , we shall call xv∗ the subset of x made of all points associated with the endpoints following v in τ ∗ ; we shall use also the definition Dv = maxx,y∈xv∗ |x − y|. Moreover, we shall call sv∗ the number of branches following v in τ ∗ , sv∗,1 the number of branches containing only one endpoint and sv∗,2 = sv∗ − sv∗,1 . Note that xv∗ ⊂ xv and sv∗ ≤ sv . The bound of Sk,0,τ,ω (x) can be obtained by slightly modifying the procedure described in detail in § 3 of [BM1], which allowed us to prove the integral estimate (2.36), in order to take into account the fact that the points √ in x are not integrated. First of
all, we note that it is possible to extract a factor e−c γ v Dv for each non trivial (that is with√sv∗ ≥ 2, n.t. in the following) vertex v ∈ τ ∗ , by partially using the decaying factors h
e−c γ |x−y| appearing ( in the bounds (2.23), which are used for the propagators of the spanning tree Tτ = v Tv of τ (see (3.81) of [BM1]); we can indeed use the bound j
e−c
√
γ h |x|
c
≤ e− 2
√
γ h |x|
· e−c
h j=−∞
√
γ j |x|
, c =
2
∞
c
j=0 γ
− j/2
(2.41)
and the remark that, given a n.t. v ∈ τ ∗ , there is a subtree Tv∗ of Tτ , connecting the points in xv∗ (together with a subset of the internal points in xv ), made of propagators of scale j ≥ h v . It follows that, given two points x, y ∈ xv∗ , such that Dv = |x − y|, there is a path connecting x and y, made of propagators in Tv∗ , whose length is at least Dv ; the decomposition of the decaying factors √ in the r.h.s. of (2.41) allows us to extract, for each of these propagators, a factor e−c √
bounded by
e−c
γ h v Dv .
γ h v |x|
and the product of these factors can be
√ j Note that after this operation, there will remain a factor e−(c/2) γ |x−y| for each propagator of Tτ , to be used for the integration over the internal vertices. Moreover, there will be 1 + v∈τ ∗ (sv∗ − 1) = k integrations less to do; by suitably choosing them, ∗ the lacking integrations produce in the bound an extra factor v∈τ ∗ γ 2h v (sv −1) L −2 so
Functional Integral Construction of the Massive Thirring Model
89
that we get " |Sk,0,τ,ω (x)| ≤ C (C λ¯ j0 )n γ − j0 (−2+k/2) k
k γ −h i · (Z h i )1/2 i=1
v not e.p.
Z hv Z h v −1
|Pv |/2
γ
2h v (sv∗ −1) −c
e
√
# γ h v Dv
n.t.v∈τ ∗
γ −dv .
(2.42)
Let E i be the family of trivial vertices belonging to the branch of τ ∗ which connects with the higher non trivial vertex of τ ∗ preceding it; the definition of sv∗,1 and the fact that, by assumption, 1/Z h i ≤ γ h i η , with η ≤ cλ¯ 2j0 , imply that, if E = ∪i E i ,
vi∗
k ∗,1 γ −h i ≤ γ −(1−η/2) γ −h v (1−η/2)sv . 1/2 (Z h i ) ∗ v∈E
i=1
(2.43)
n.t.v∈τ
Let v0∗ be the first vertex following v0 (the vertex immediately following the root of τ , of scale j0 + 1) with sv∗ ≥ 2; then we have, if kv denotes the number of elements in xv∗ (hence kv = k, if v0 ≤ v ≤ v0∗ ),
γ − j0 (−2+k/2 )
γ −dv = γ
−h v ∗ (−2+kv ∗ /2) 0
˜
γ −dv ,
0
v0 ≤v
(2.44)
v0 ≤v
where we used the definition, |Pv | − kv kv ˜ = dv = dv − −2 + ; 2 2
(2.45)
note that d˜v ≥ 1/2, for any v ∈ τ ∗ . By inserting (2.43) and (2.44) in the r.h.s. of (2.42), we get " #" √ k n −c γ h v Dv ¯ e |Sk,0,τ,ω (x)| ≤ C (C λ j0 ) ⎡
⎤
n.t.v∈τ ∗
"
v not e. p.
#⎡
⎢ ⎥ −dv −1+η/2 ⎣ ·⎣ γ −dv ⎦ γ v ∈τ / ∗ v not e. p.
Z hv Z h v −1
|Pv |/2 #
⎤
˜
γ −dv ⎦ · Fτ ,
(2.46)
v0 ≤v
v∈E
where ⎡ Fτ = γ
−h v ∗ (−2+kv ∗ /2) 0
0
n.t.v∈τ ∗
∗
∗,1
γ h v [2(sv −1)−(1−η/2)sv
⎤
⎢ ⎥ γ −dv ⎥ ⎣ ⎦ .
]⎢
(2.47)
v0∗ ≤v∈τ ∗ v ∈E /
Given a n.t. vertex v ∈ τ ∗ , let s = sv∗ , s1 = sv∗,1 , s˜ = s − s1 and v1 , . . . , vs˜ be the s˜ n.t. vertices immediately following v in τ ∗ . Note that kv = s1 + i=1 kvi ; hence, given
90
G. Benfatto, P. Falco, V. Mastropietro
ε > 0, we can write −(−2 + ε + kv /2) + [2(s − 1) − (1 − η/2)s1 ] = 2 − ε − kv /2 + ε(s − 1) + (2 − ε)(s1 + s˜ − 1) − (1 − η/2)s1 ⎞ ⎛ s˜ 1⎝ =− kvi ⎠ + ε(s − 1) + (2 − ε)˜s + s1 (2 − ε − 1 + η/2) s1 + 2 i=1
= ε(s − 1) + s1 (1/2 − ε + η/2) −
s˜ (−2 + ε + kvi /2) .
(2.48)
i=1
This identity, applied to the vertex v0∗ , implies that if v1 , . . . , vs˜ , s˜ = sv∗∗ − sv∗,1 ∗ are the 0 0 ∗ ∗ n.t. vertices immediately following v0 in τ , then γ
∗,1 ∗ −h v ∗ (−2+kv ∗ /2) h v0∗ [2(sv ∗ −1)−(1−η/2)sv ∗ ] 0
εh v ∗
=γ
γ
0
0
γ
αv ∗ h v ∗ 0
s˜
0
⎡
0
0
⎣γ −h vi (−2+ε+kvi /2) ·
⎤ γ −2+ε+kv /2 ⎦ ,
(2.49)
v∈Ci
i=1
where Ci is the path connecting v0∗ with vi in τ ∗ (not including vi ) and we used the definition αv = ε(sv∗ − 1) + sv∗,1 (1/2 − ε + η/2) .
(2.50)
The presence of the factor γ −h vi (−2+ε+kvi /2) for each vertex vi in the r.h.s. of (2.49) implies that an identity similar to (2.49) can be used for each n.t. vertex v ∈ τ ∗ . It is then easy to show that ⎤ " #⎡ εh v ∗ ˜ Fτ = γ 0 γ αv h v ⎣ γ −dv +ε ⎦ . (2.51) n.t.v∈τ ∗
v∈τ ∗ ,v ∈E /
By inserting this equation in (2.46), we get "
n εh v0∗
|Sk,0,τ,ω (x)| ≤ C (C λ¯ j0 ) γ k
" ·
v not
Z h v |Pv |/2 −d¯v ( ) γ Z h v −1 e. p.
#
γ
αv h v −c
√
e
# γ h v Dv
n.t.v∈τ ∗
,
(2.52)
where ⎧ ⎪ d˜ ⎪ ⎪ v ⎨ d˜v − ε d¯v = ⎪ dv + 1 − η/2 ⎪ ⎪ ⎩d v
if v0 ≤ v < v0∗ if v ∈ τ ∗ , v0∗ ≤ v ∈ /E . if v ∈ E otherwise
(2.53)
Functional Integral Construction of the Massive Thirring Model
91
Note that d¯v > 0 for any v ∈ τ , if ε < 1/2; moreover, if this condition is satisfied, αv ≥ ε > 0, for any n.t. vertex v ∈ τ ∗ , uniformly in λ¯ j0 . Moreover, since by hypothesis Dv ≥ δ > 0, there is c0 such that √ c αv √ hv 0 γ αv h v e−c γ Dv ≤ sup x 2αv e−c x δ ≤ αv2αv . (2.54) δ x>0 Note that
1 1 k(1 + η) − ε ≤ k(1 + η) . 2 2
αv =
n.t.v∈τ ∗
(2.55)
Hence, by using (2.52) and (2.55), we get −[k(1+η)/2−ε]+α
∗
v0 |Sk,0,τ,ω (x)| ≤ C k (k!)1+η (C λ¯ j0 )n δ 3 " # h ∗ Z h |Pv |/2 (αv ∗ +ε)h v ∗ −c γ v0 Dx ¯v v − d 0e ·γ 0 γ , Z h v −1 v not e. p.
(2.56)
where Dx denotes the diameter of the set x. Let us now observe that, since the vertex dimensions d¯v are all strictly positive, if we insert the bound (2.56) in the r.h.s of (2.40), we can easily perform all the sums (by using the arguments explained, for instance, in [BM1]), once we have fixed the scale of the vertex v0∗ and the values of sv∗∗ and sv∗,1 ∗ (so that the value of αv ∗ is fixed) and we can 0 0 0 take the limit −h, N → ∞. By using Theorem 2.1 and the remark that the bound (2.56) h ∗ implies that the trees giving the main contribution to Sk,ω (x) are those with γ v0 Dx of order 1, it is easy to prove that the limit can be expressed as an expansion similar to (2.40), with the sum over j0 going from −∞ to +∞, the sum over τ including trees with endpoints of arbitrary scale (satisfying the usual constraints) and the values of Sk,0,τ,ω (x) modified in the following way: 1) in every endpoint there is the same constant λ−∞ in place of λh v ; 2) the constants Z j , and µ j are substituted everywhere by γ −ηz j and µγ −ηµ j , respectively, see (2.31); 3) in the expansion which defines the constants z j and s j needed, respectively, in the definition of Z˜ j−1 (k) and µ j−1 (k), see (2.19), one has to make the same substitutions of Items 1) and 2). The bound (2.56) also implies the following one (valid for Cε |λ| ≤ 1, with Cε → ∞ as ε → 1/2): |Sk,ω ( x )| ≤ C (Cε |λ|) k
k/2−1
(k!)
2+η −[k(1+η)/2−ε]
δ
s k
δ ε(s−1)+s1 (1−2ε+η)/2
s=2 s1 =0
·
+∞
γ [εs+s1 (1−2ε+η)/2]h e−c
√
γ h Dx
h=−∞
≤ C (Cε |λ|) k
k/2−1
(k!)
3+2η −k(1+η)/2
δ
s k δ εs+s1 (1−2ε+η)/2 . Dx s=2 s1 =0
(2.57)
92
G. Benfatto, P. Falco, V. Mastropietro
Since δ/Dx ≤ 1, the sum over s and s1 is bounded by Ck 2 (δ/Dx )2ε ; hence we get the bound |Sk,ω (x)| ≤ C k (Cε |λ|)k/2−1 (k!)3+2η δ −[k(1+η)/2−2ε]
1 , 1 + Dx2ε
(2.58)
which proves both the boundedness and the cluster property, see Appendix A. In conclusion,we have proved the following result. Theorem 2.2. If ε1 is defined as in Theorem 2.1, there exists ε2 ≤ ε1 such that,if the normalization conditions (2.30) are satisfied and |λ| ≤ ε2 , then the Schwinger functions Sk,ω (x) are well defined at non coinciding points and verify all the OS axioms, possibly except RP. The Reflection Positivity is a consequence of Theorem 1.4, to be proved in § 5. Moreover, it is easy to derive from the previous bounds and (2.34) (see for instance [BM4] for the case µ = 0) the bound for the two point Schwinger functions (1.24). The previous arguments can be extended to prove that also the Schwinger functions with n J > 0 are well defined in the limit of removed cutoffs, so completing the proof of Theorem 1.1, except for Eq. (1.25), which will be proved in § 2.5 below. Finally, note that our results are uniform in the mass µ; the radius of convergence of the expansion as a function of the running coupling constants can be chosen independent of the mass, and the running coupling constants are independent of the mass by construction. 2.4. Bounds for the Fourier transform of the Schwinger functions. The main bound (2.36) can be also used to get bounds on the Fourier transform of the Schwinger functions at non-zero external momenta; these bounds are uniform in the cutoffs and allow, in particular, to prove (by some obvious technicality, that we shall skip) that the removed cutoffs limit is well defined. Here we shall only consider, as an example, the function ,h
2,1,N G (p; k) in the massless case. ω,ω By using (2.28), we can write ,h
2,1,N (p; k) = G ω,ω
−1 ∞ N
n=0 j0 =h−1 τ ∈T j0 ,n,2,1
2,1 G τ (p, k) ,
(2.59)
P∈P |Pv0 |=2
2,1 with an obvious definition of G τ (p, k). Let us define, for any k = 0, h k = min{ j : f j (k) = 0} and suppose that p, k, p − k are all different from 0. It follows that, given τ , if h − and h + are the scale indices of the ψ fields belonging to the endpoints associated
2,1 with ϕ + and ϕ − , while h J denotes the scale of the endpoint of type J , G τ (p, k) can be different from 0 only if h − = h k , h k + 1, h + = h k−p , h k−p + 1 and h J ≥ h p − logγ 2. Moreover, if T j0 ,n,p,k denotes the set of trees satisfying the previous conditions and 2,1
2,1 τ ∈ T j0 ,n,p,k , |G τ (p, k)| can be bounded by dzdx|G τ (z; x, y)|. Hence, by using (2.36) and (2.31), we get ,h
2,1,N (p; k)| ≤ Cγ −h k (1−ηz /2) γ −h k−p (1−ηz /2) · |G ω,ω
·
∞ N −1
n=0 j0 =h−1 τ ∈T j0 ,n,p,k
P∈P |Pv0 |=2
(C|λ|)n
v not e.p.
γ −dv .
(2.60)
Functional Integral Construction of the Massive Thirring Model
93
The bound of the r.h.s. of (2.60) could be easily performed by using the procedure described in § 3 of [BM1], if dv were greater than 0 for any v; however, by looking at (2.37), one sees that this is not true. Given τ ∈ T j0 ,n,p,k , let v0∗ be the higher vertex preceding all three special endpoints and v1∗ ≥ v0∗ be the higher vertex preceding either the two endpoints of type ϕ (to be called vϕ,+ and vϕ,− ) or one endpoint of type ϕ and the endpoint of type J (to be called v J ). It turns out that dv > 0, except for the vertices belonging to the path C ∗ connecting v1∗ with v0∗ , where, if |Pv | = 4 and n vJ = 0 or |Pv | = 2 and n vJ = 1, dv = 0. Hence, we can perform as in § 3 of [BM1] the sums over the scale and Pv labels of τ , only if we fix the scale indices h ∗0 and h ∗1 of v0∗ and ∗ ∗ v1∗ , after multiplying by γ −δ(h 1 −h 0 ) the r.h.s. of (2.60), δ being any positive number. Of ∗ ∗ course, we also have to perform the sum over h ∗0 , h ∗1 of γ δ(h 1 −h 0 ) , which is divergent, if we proceed exactly in this way. In order to solve this problem, we note that, if v ∈ / C ∗ , dv − 1/4 > 0. Hence, before performing the sums over the scale and Pv labels, we can extract from each γ −dv factor associated with the vertices belonging to the paths connecting the three special endpoints with v0∗ or v1∗ , a γ −1/4 piece, to be used to perform safely the sums over h ∗0 , h ∗1 in the following way. (1) Let us consider first the family T j0 ,n,p,k of trees such that the two special endpoints ∗ following v1 are vϕ,+ and vϕ,− and let us suppose that |k| ≥ |k − p|. In this case, before doing the sums over the scale and Pv labels, we fix also the scale h J of v J . We get, if h ∗J ≡ max{h p + 2, h ∗0 + 1}: ∞ N −1
n=0 j0 =h−1 τ ∈T (1)
≤C
∗
h1
γ −dv
(2.61)
v not e.p.
P∈P j0 ,n,p,k |Pv0 |=2
h k−p
(C|λ|)n
+∞
∗
∗
∗
1
∗
∗
γ δ(h 1 −h 0 ) γ − 4 [(h k −h 1 )+(h k−p −h 1 )+(h J −h 0 )] ,
h ∗1 =−∞ h ∗0 =−∞ h J =h ∗J
and it is easy to prove that the r.h.s. of (2.61) is bounded by Cγ δ(h k −h p ) , if δ ≤ 1/8. If |k − p| ≥ |k|, we get a similar result, with h k−p in place of h k . (2,+) Let us consider now the family T j0 ,n,p,k of trees such that the two special endpoints following v1∗ are v J and vϕ,+ . We get, if h ∗J ≡ max{h p +2, h ∗1 +1} and h¯ 0 = min{h k−p , h ∗1 }: −1 ∞ N n=0
P∈P j0 =h−1 τ ∈T (1,+) j0 ,n,p,k |Pv0 |=2
≤C
hk
h¯ 0
(C|λ|)n
+∞
γ −dv
(2.62)
v not e.p.
∗
∗
1
∗
∗
∗
γ δ(h 1 −h 0 ) γ − 4 [(h k −h 1 )+(h k−p −h 0 )+(h J −h 1 )] ,
h ∗1 =−∞ h ∗0 =−∞ h J =h ∗J
and it is easy to prove that, if δ ≤ 1/8, the r.h.s. of (2.62) is bounded by Cγ δ(h k −h k−p ) , (2,−) if |k| ≥ |k − p|, by a constant, otherwise. The family T j0 ,n,p,k of trees such that the two special endpoints following v1∗ are v J and vϕ,− can be treated in a similar way and one obtains a bound Cγ δ(h k−p −h k ) , if |k − p| ≥ |k|, or a constant, otherwise.
94
G. Benfatto, P. Falco, V. Mastropietro
By putting together all these bounds, we get, for any positive δ ≤ 1/8: Cδ ,h
2,1,N |G (p; k)| ≤ · ω,ω 1−η |k| z |k − p|1−ηz " δ # |k| δ |k − p| δ |k| |k − p| δ + + + , |p| |p| |k − p| |k|
(2.63)
with Cδ → ∞ as δ → 0.
2ω (k) in the massless case. We want now to discuss the structure 2.5. Calculation of G ,h (k) for µ = 0.
2,N of the limit −h, N → ∞ of the interacting propagator G ω By using (2.28), we can write ,h
2,N (k) = G ω
−1 ∞ N
n=0 j0 =h−1 τ ∈T j0 ,n,2,0
2τ (k) , G
(2.64)
P∈P |Pv0 |=2
2τ (k). with an obvious definition of G Let us define h k as in § 2.4 and suppose that k = 0. It follows that, given τ , if h − and h + are the scale indices of the ψ fields belonging to the endpoints associated with
2τ (k) can be different from 0 only if h ± = h k , h k + 1. Moreover, if T j0 ,n,k ϕ + and ϕ − , G
2τ (k)| can denotes the set of trees satisfying the previous conditions and τ ∈ T j0 ,n,k , |G be bounded by dx|G 2τ (x, y)|. Hence, by using (2.36) and (2.31), we get γ −h k Z hk
,h
2,N |G (k)| ≤ Cγ −(h k − j0 ) ω
·
−1 ∞ N n=0
(C|λ|)n
j0 =h−1 τ ∈T j0 ,n,k P∈P |Pv0 |=2
γ −dv ,
(2.65)
v not e.p.
where dv > 0, except for the vertices belonging to the path connecting the root with v ∗ , the higher vertex preceding both of the two special endpoints, where dv can be equal to 0. These vertices can be regularized by using the factor γ −(h k − j0 ) in the r.h.s. of (2.65); hence, by proceeding as in § 2.4, we can easily perform the sum over the trees with a fixed value of the scale label h ∗ of v ∗ and we get the bound ,h
2,N |G (k)| ≤ C ω
γ −h k Z hk
hk h ∗ =−∞
γ −(h k −h
∗ )/2
≤C
γ −h k . Z hk
(2.66)
By using Theorem 2.1, it is not hard to argue, as in § 2.3, that the removed cutoffs limit
2ω (k) is well defined and is given by an expansion similar to (2.64), with the sum over G
2τ (k) modified by substituting, in every endj0 going from −∞ to +∞ and the quantity G point, λ j with λ−∞ , and, in every propagator, Z j with γ −η j , η ≡ ηz ; this property easily
2ω (k). On the other hand, the symmetries of the model
2ω (γ k) = γ η−1 G implies that G imply that there is a function g(x, λ), defined for x > 0 and λ small enough, such that
2ω (k) = Dω−1 (k)g(|k|, λ); by the previous scaling property, g(γ x, λ) = γ η g(x, λ). G We want to show that g(x, λ) = x η f (λ), with f (λ) independent of x.
Functional Integral Construction of the Massive Thirring Model
95
,h (k) is independent of γ , since the
2,N To prove this claim, first of all note that G ω −1 cutoff function C h,N (k) only depends on γ0 and γ ≥ γ0 , see § 1.2. This property is then
2ω (k), hence for g(x, λ). However, since the expansion heavily depends valid also for G on γ , the value of η is apparently a function of γ ; we want to show that this is not true. Note that for any γ and any integer j, g(γ j , λ) = γ jη(γ ) g(1, λ); it follows that, if j j there exist, given γ1 and γ2 , two integers j1 , j2 , such that γ1 1 = γ2 2 , then η(γ1 ) = η(γ2 ). Hence, given an interval I = [γ0 , γ¯ ] and γ ∈ I , the set {γ ∈ I : η(γ ) = η(γ )} is dense in I , as the set of rational numbers is dense in the interval [logγ γ0 , logγ γ¯ ]. Since η(γ ) is obviously continuous in γ , it follows that it is constant. Let us now put g(x, λ) = x η f (x, λ); we see immediately that f (γ x, λ) = f (x, λ). Hence, by varying γ in the interval [2, 4] and by choosing x = 1/γ , we see that f (1, λ) = f (x, λ), if x ∈ [1/4, 1/2]. By using this equation, by varying x in the interval [1/4, 1/2] and by choosing γ = 2, we get also f (1, λ) = f (x, λ), if x ∈ [1/2, 1]. By proceeding in this way, it is easy to show that f (1, λ) = f (x, λ), for any x > 0. The previous discussion and the fact that in the expansion (2.65), dv > 1/4 for any v > v ∗ , imply also that
|k|η ,h
2,N [ f (λ) + O(|k|γ −N )1/4 ] . G (k) = ω Dω (k)
(2.67)
3. Ward–Takahashi Identities 3.1. Proof of Theorem 1.2. In order to derive WTi in the massless case µ = 0 from the generating functional (1.13) (in the continuum limit a = 0), it is convenient to ε (k)]−1 equivalent to [C −1 as far as the scaling introduce a cutoff function [C h,N h,N (k)] ε (k)]−1 is the whole set D and features are concerned, but such that the support of [C h,N 0 ε −1 −1 limε→0 [C h,N (k)] = [C h,N (k)] ; we refer to [BM2] § 2.2 for its exact definition. ε (k)]−1 in the r.h.s. of (1.13) and perform We then substitute [C h,N (k)]−1 with [C h,N ± ±α ± (equivalent to the usual phase and chiral x,ω the gauge transformation ψx,ω → e ψx,ω transformations). The change in the cutoff function has the effect that the Lebesgue mea [h,N ] is invariant under this transformation and we get the WTi (1.26), where, if sure d ψ 2 < . >h,N denotes the expectation with respect to measure N −1 PZ N (dψ)e−λ N Z N V (ψ) ,h
2,1,N (see (1.14)-(1.16) for the definitions), (p, k) is the Fourier transform of ω,ω ,h − + (x; y, z) = ψy,ω 2,1,N ; ψz,ω ; δTx,ω h,N , ω,ω de f
(3.1)
where −; −; −h,n denotes the truncated expectation with respect to the measure (1.14), de f
δTx,ω =
Z N i(k+ −k− )x ε
−− ,
k++ ,ω ψ e C h,N ;ω (k+ , k− )ψ k ,ω L4 + −
(3.2)
k ,k k+ =k−
and ε + − ε − − ε + + C h,N ;ω (k , k ) = [C h,N (k ) − 1]Dω (k ) − [C h,N (k ) − 1]Dω (k ) .
(3.3)
Let us now suppose that p is fixed independently of h and N , as well as k, and that p, k and k − p are all different from 0. This implies, in particular, that the condition
96
G. Benfatto, P. Falco, V. Mastropietro
χ (p) = 1 is satisfied if |h| and N are large enough and χ(p) ˜ is the function appearing in ,h
2,1,N (p, k) with (1.31). Hence we can prove Theorem 1.2 by substituting in (1.27) R ω,ω 2,1,N ,h
(p, k), which is the Fourier transform of χ (p) R ω,ω
∂ ∂2 W | J =ϕ=0 , + ∂ϕ − ∂ Jx,ω ∂ϕy,ω z,ω where eW (J,ϕ) =
de f
dP [h,N ] (ψ) exp
+
ω
· exp
− λN V
ZNψ
+ − + − ψx,ω + ψx,ω ϕx,ω dx ϕx,ω
5
(3.4)
4
T¯0,ω − ν+,N T¯+,ω − ν−,N T¯−,ω
(3.5) J,
ZNψ
6 ,
ω
with ε C h,N 1 1 ;ω (k, k − p) + − ¯ ψk,ω ψk−p,ω ≡ 2 Jp,ω χ (p) Jp,ω δρp,ω , T0,ω (J, ψ) = 4 L Dω (p) L k,p
D±ω (p) + 1
− ψ ψ T¯±,ω (J, ψ) = 4 Jp,ω χ (p) . L Dω (p) k,±ω k−p,±ω
p =0
(3.6)
k,p
The coefficients ν±,N will be fixed by the requirement that (1.29) holds. A crucial role in the analysis is played by the function j) + − (i, ω (k , k ) = ( j)
ε + − C h,N ;ω (k , k )
Dω (k+ − k− )
gω(i) (k+ )
gω( j) (k− ) ,
(3.7)
( j)
gω,ω . By proceeding as in § (4.2.) of [BM2] (where only the case N = 0 where
gω ≡
is considered), one can show that, if p = k+ − k− = 0 and |p| ≥ 2γ h+1 (which is true since χ (p) = 1): ε (k± )]−1 = 1, 1. if h < i, j < N , since [C h,N j) + − (i, ω (k , k ) = 0 ;
(3.8)
2. if h < j ≤ N , , j) + − (k , k ) = (N ω
p S( j) (k+ , k− ) , Dω (p) ω
(3.9)
de f ( j) ( j) ( j) where Sω (k+ , k− ) = Sω,0 (k+ , k− ), Sω,1 (k+ , k− ) is a vector of smooth functions such that ( j)
|∂km++ ∂k−− Sω,i (k+ , k− )| ≤ Cm + +m − m
γ −N (1+m + ) γ − j (1+m − ) ; Z N Z j−1
(3.10)
Functional Integral Construction of the Massive Thirring Model
97
3. if h ≤ i ≤ N , + − −(i−h) |(i,h) ω (k , k )| ≤ Cγ
γ −h−i ; Z i−1 Z N
(3.11)
4. if i = j = h, (h,h) (k+ , k− ) = 0 . ω
(3.12)
Note that, in the r.h.s. of (3.11), there is apparently a Z N /Z h−1 factor missing, but the bound can not be improved; this is a consequence of the fact that Z˜ h−1 (k) = 0 for |k| ≤ γ h−1 , see Eq. (63) of [BM2]. Remark. Equations (3.1) and (3.4) say that the correction to the formal WI can be written ,h , except that the correspondas a functional integral very similar to the one for G 2,1,N ω ing generating function is replaced by the more complicated expression appearing in (3.5). Hence, we can evaluate (3.1) via a multiscale integration similar to the one for ,h , but we have to discuss the effect of the new term appearing in (3.5). The idea G 2,1,N ω is that the operator T¯0,ω can generate, during the multiscale integration, dimensionally marginal terms of the form J ψ + ψ − , whose flow can be controlled by choosing properly the counterterms ν+,N and ν−,N . As regards the dimensionally irrelevant terms, by (3.8) they contain at least a nontrivial integration at the ultraviolet or infrared scale and this implies, as we will see, that such contributions vanish when the cutoffs are removed. The multiscale integration of W has been described in detail in § 4 of [BM2] (of course the scale 0 has to be replaced with the scale N ). After the integration of ψ (N ) we get an expression like (2.1) and the terms linear in J and quadratic in ψ in the exponent (N −1) √ (N −1) (a,N −1) (b,N −1) will be denoted by K J ( Z N −1 ψ [h,N −1] ); we write K J = KJ +K J , (a,N −1) (b,N −1) where K J was obtained by the integration of T¯0 and K J from the integration (a,N −1) as of T¯± . We can write K J (a,N −1)
KJ
+
( Z N −1 ψ) = Z N dx Jx,ω T¯0,ω (J, ψ)
ω˜
(N −1)
(3.13)
ω
4 (N −1) (N −1) − + ψ ] dydz F2,ω,ω˜ (x, y, z) + F1,ω (x, y, z)δω,ω˜ [ψy, ω˜ z,ω˜ , (N −1)
where F2,ω,ω˜ and F1,ω are the analogues of Eq. (132) of [BM2]; they represent the terms in which both or only one of the fields in δρp,+ , respectively, are contracted. Both contributions to the r.h.s. of (3.13) are dimensionally marginal; however, the regulariza(N −1) tion of F1,ω is trivial, as it is of the form (N −1) + − F1,ω (k , k ) =
) + + g (N [C h,N (k− ) − 1]Dω (k− )Z N
ω (k ) − u N (k ) (2) + G ω (k ), + − Dω (k − k ) (3.14)
or the similar one, obtained exchanging k+ with k− ; u N (k) = 0 if |k| ≤ γ N and u N (k) = 1 − f N (k) for |k| ≥ γ N . By the oddness of the propagator in the momentum,
98
G. Benfatto, P. Falco, V. Mastropietro (2)
G ω (0) = 0, hence we can regularize such terms without introducing any local term, by simply rewriting it as (N −1)
F1,ω
+ (2) (k+ , k− ) = [G (2) ω (k ) − G ω (0)] · (N )
[C h,N (k− ) − 1]Dω (k− )Z N
g ω (k+ ) − u N (k+ ) . Dω (k+ − k− )
(3.15)
(N −1)
By using the symmetry property (2.14), F2,ω,ω˜ can be written as (N −1)
F2,ω,ω˜ (k+ , k− ) =
1 p0 A0,ω,ω˜ (k+ , k− ) + p1 A1,ω,ω˜ (k+ , k− ) , Dω (p)
(3.16)
where Ai,ω,ω˜ (k+ , k− ) are functions such that, if we define (N −1)
LF2,ω,ω˜ =
1 p0 A0,ω,ω˜ (0, 0) + p1 A1,ω,ω˜ (0, 0) , Dω (p)
(3.17)
then (N −1)
(N −1)
LF2,ω,ω = Z 3,+ N −1 , LF2,ω,−ω =
D−ω (p) 3,− Z , Dω (p) N −1
(3.18)
3,− where Z 3,+ N −1 and Z N −1 are suitable real constants. Hence the local part of the marginal term in the second line of (3.13) is, by definition, equal to [h,N −1] [h,N −1] ¯ ¯ [Z N Z 3,+ ) + Z N Z 3,− )] . (3.19) N −1 T+,ω (J, ψ N −1 T−,ω (J, ψ ω
The terms linear in J and quadratic in ψ obtained by the integration of T¯± have the form (b,N −1)
KJ
&
1
++ ψ
− ψ ( Z N −1 ψ) = Z N 4 χ (p)Jp,ω k ,ω˜ k+ −p,ω˜ L + k ,p
(N )
· −ν+,N G ω,ω˜ (k+ , k+ − p) − ν−,N
ω,ω˜
' D−ω (p) (N ) + + G (k , k − p) . (3.20) Dω (p) −ω,ω˜ (N )
By using the symmetry property of the propagators, it is easy to show that G ω,−ω (0, 0) = (N ) (N ) 0. Hence, if we regularize (3.20) by subtracting G ω,ω˜ (0, 0) to G ω,ω˜ (k+ , k+ − p), we still get a local term of the form (3.19). Finally by collecting all the local terms linear in J we can write LK JN −1 ( Z N −1 ψ [h,N −1] ) = Z N T¯0,ω J, ψ [h,N −1] (3.21)
−ν+,N −1 T¯+,ω J,
ω
Z N −2 ψ
[h,N −1]
− ν−,N −1 T¯−,ω J, Z N −2 ψ [h,N −1] , (N )
where Z N −2 ν±,N −1 = Z N −1 [ν±,N − Z 3,± N −1 +ν±,N G ±ω,±ω (0, 0)] (our definitions imply that Z N −1 = Z N ). The above integration procedure can be iterated with no important differences up to scale h + 1. In particular, for all the marginal terms such that one of the fields in T¯0,ω in (3.13) is contracted at scale j, we put R = 1; in fact the second
Functional Integral Construction of the Massive Thirring Model
99
Fig. 3.1. Graphical representation of the lowest order contribution to ν − and to ν + ; the small circle represents the function in the r.h.s. of (3.3)
field has to be contracted at scale h and, by (3.11), the extra factor γ h− j has the effect of automatically regularizing such contributions. The above analysis implies that ν+, j gets no contributions from trees with an endpoint of type ν−,k , k > j, and viceversa; moreover, if a tree has an endpoint corresponding to T¯0,ω , this endpoint has scale index N + 1. Hence we can write, for h + 1 ≤ j ≤ N − 1, j
ν±, j−1 = ν±, j + β±,ν (λ j , ν j .., λ N , ν N ) ,
(3.22)
with j
j
β±,ν (λ j , ν j .., λ N , ν N ) = β±,ν (λ j , .., λ N ) +
N
j, j
ν±, j β±,ν (λ j , .., λ N )
(3.23)
j j, j |β±,ν (λ j , .., λ N )| ≤ C λ¯ j γ −2ϑ(N − j) , |β±,ν (λ j , .., λ N )| ≤ C λ¯ 2j γ −2ϑ| j− j | .
(3.24)
j = j
and, given a positive ϑ < 1/4,
We fix ν±,N so that ν±,N = −
N
j
β±,ν (λ j , ν j .., λ N , ν N ) .
(3.25)
j=h+1
By a fixed point argument (see § 4.6 of [BM4]), one can show that, if λ¯ h is small enough, it is possible to choose ν±,N so that |νω, j | ≤ c0 λ¯ h γ −ϑ(N − j) ,
(3.26)
for any h + 1 ≤ j ≤ N . The convergence of ν±,N as |h|, N → ∞ is an easy consequence of the previous conλ siderations. Moreover, from an explicit computation of (3.22), we get ν− = 4π + O(λ2 ) and ν+ = c+ λ2 + O(λ3 ) with c+ < 0. ,h
2,1,N (p; k) was discussed in § 2.4. Hence, to complete the The convergence of G ω,ω ,h
2,1,N (p, k) → 0 , if p, k and k − p proof of Theorem 1.2, we have to prove that χ (p) R ω,ω are all different from 0. In fact, since χ (p) = 1 for p = 0 and |h|, N large enough, this implies (1.29). This result can be obtained by a simple extension of the arguments given in § 2.4 to ,h ,h
2,1,N
2,1,N (p; k) is bounded uniformly in h and N . In fact, χ (p) R (p, k) prove that G ω,ω ω,ω ,h
2,1,N can be written by a sum of trees essentially identical to the ones for G , with the ω,ω only important difference that there are three different special endpoints associated to
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G. Benfatto, P. Falco, V. Mastropietro
the field J , corresponding to the three different terms in (3.21); we call these endpoints of type T0 , T+ , T− respectively. The sum over the trees such that the endpoint is of type T± can be bounded as in (2.60), the only difference being that, thanks to the bound (3.26), one has to multiply the r.h.s. by a factor |λ|γ −ϑ(N −h J ) , which has to be inserted also in the r.h.s. of the bounds (2.61) and (2.62). Hence, it is easy to see that the contributions of these trees vanish as N → ∞. Let us now consider the trees with an endpoint of type T0 . In this case there are two possibilities. The first is that the fields of the T0 endpoint are contracted at scale j, N ; this implies that the sum over h J is missing in the r.h.s. of the bounds (2.61) and (2.62) and h J = N . Hence it is easy to see that the sum over such trees goes to 0 as N → ∞. The second possibility is that the fields of the T0 endpoint are contracted at scale j, h; this implies that the sum over j0 is missing in the r.h.s. of (2.60) and j0 = h. Since dv − 1/4 > 0 for all vertices belonging to the path connecting the root to the vertex v0∗ , we can add a factor γ −( j0 −h))/4 to the r.h.s. of the bounds (2.61) and (2.62), which then go to 0 as h → −∞. 4. Schwinger-Dyson Equations and New Anomalies 4.1. Proof of Theorem 1.3. In this section we study the last addend of the SchwingerDyson equation (1.31), so proving Theorem 1.3; the analysis rests heavily on § 4 of [BM4]. Let us consider a fixed finite k and let us define its scale h k as in § 2.4; then, if −h and N are large enough,
gωN ,h (k) =
gω (k). We start by putting (see § 4.1 of [BM4]): de f ,h 2,N gω (k) G ε,ω (k) =
dp Dεω (p) 2,1,N ,h ∂WT,ε χ (p) , (4.1) Rεω;ω (p, k) = 2 (2π ) D−ω (p) ∂
ϕ +k,ω ∂ J k,ω
where ε = ± and WT,ε is defined (in the infinite volume limit) by the equation: eWT,ε (J,ϕ) =
de f
[h,N ]+ − + ψx,ω ϕx,ω
dPZ N (ψ) exp
+ ZN
(ε) T1
− λ N Z 2N V (ψ)
+ [h,N ]− dx ϕx,ω ψx,ω
− ν+,N T−ε − ν−,N Tε (ψ, J )
4 ,
(4.2)
with
dpdk Cεω (k , k − p)
k,ω
χ (p) J g (k) ω (2π )4 D−ω (p) +
+ ψ
−
k−p,ω ψ , ·ψ k ,εω k −p,εω dpdk de f +
+ ψ
−
k−p,ω ψ T+ (ψ, J ) = χ (p) J k,ω
gω (k)ψ k ,−ω k −p,−ω , (2π )4 dpdk Dω (p) + de f
+ ψ
− ψ ψ χ (p) J k,ω
gω (k) , T− (ψ, J ) = 4 (2π ) D−ω (p) k−p,ω k ,ω k −p,ω
(ε) T1 (ψ,
de f
J) =
and ν±,N are defined as in (3.25).
(4.3)
Functional Integral Construction of the Massive Thirring Model
101
Remark. Equation (4.1) essentially says that the last term in the Schwinger-Dyson equa,h (protion (1.31) can be written as a functional integral very similar to the one for G 2,N ω ceeding as in the derivation of the Schwinger-Dyson equation in the opposite direction), except that the corresponding generating functional is replaced by the more complicated ,h 2,N expression appearing in (4.2). We can then evaluate G ε,ω (k) via a multiscale integra,h , and in the new expansions additional running coupling tion similar to the one for G 2,N ω constants will appear. The expansion is convergent if such new running coupling constants will remain small uniformly in the cut-offs, and this will be proved by showing ,h . that they are close to the ones appearing in the expansion for G 2,N ω ,h 2,N The calculation of G ε,ω (k) is done again via a multiscale expansion, very similar to the one described in § 4 of [BM4]. The main differences are that here we are considering a quantity with two external lines, instead of four, and that the external momenta are on the scale h k , instead of the infrared cutoff scale h. However, the last remark implies that the integration of the fields of scale j > h k + 1 differs from that discussed in [BM4] only ,h 2,N for trivial scaling factors; in particular, there is no contribution to G ε,ω (k) associated with a tree, whose root has scale higher than h k . Let us call V¯ (N −1) (ψ [h,N −1] ) the sum over the terms linear in J , obtained after the integration of the field ψ (N ) ; we put (N −1) (N −1) V¯ (N −1) (ψ [h,N −1] ) = V¯ a,1 (ψ [h,N −1] ) + V¯ a,2 (ψ [h,N −1] ) (N −1) (N −1) V¯ b,1 (ψ [h,N −1] ) + V¯ b,2 (ψ [h,N −1] ) ,
(4.4)
(N −1) (N −1)
+ where V¯ a,1 + V¯ a,2 is the sum of the terms in which the field ψ k−p,ω appearing in (ε) (N −1) (N −1) ¯ ¯ the definition of T (ψ) or T± (ψ) is contracted, V and V denoting the sum 1
a,1
a,2
(N −1) (N −1) over the terms of this type containing a T1 or a T± vertex, respectively; V¯ b,1 + V¯ b,2
+ is the sum of the other terms, that is those where the field ψ k−p,ω is an external field, the index i = 1, 2 having the same meaning as before. (N −1) Let us consider first V¯ a,1 ; we shall still distinguish different group of terms, those
+
− where both fields ψ k ,εω and ψk −p,εω are contracted, those where only one among
+ and them is contracted, and those where none is contracted. If none of the fields ψ k ,εω −
ψ is contracted, we can only have terms with at least four external lines; for the k −p,εω (i, j) − +
properties of εω (see (3.7)), at least one of the fields ψ k ,εω and ψk −p,εω must be contracted at scale h. If one of these terms has four external lines, hence it is marginal, it has the following form:
dpdk +
ω,k−p G (N ) (k − p)
ψ gωN ,h (k − p) 2 4 (2π ) Cεω (k , k − p) +
ψ
− ψ gω (k) χ (p) · J k,ω
k ,εω k −p,εω , D−ω (p)
ZN
(4.5)
) where G (N 2 (k) is a suitable function which can be expressed as a sum of graphs with an (N ) odd number of propagators, hence it vanishes at k = 0. This implies that G 2 (0) = 0, so that we can regularize it without introducing any running coupling.
102
G. Benfatto, P. Falco, V. Mastropietro
Fig. 4.1. Graphical representation of (4.5)
(−1)
Fig. 4.2. Graphical representation of W˜ 4
(−1)
and W˜ 2
(ε)
+ and ψ
− If both ψ k −p,εω in T1 (ψ) are contracted, we get terms of the form (up k ,εω to an integral over the external momenta) n (N −1)
εi , ψ J k,ω gω (k)W˜ n+1 (k, k1 , .., kn )( Z N )n−1 ki
(4.6)
i=1
where n is an odd integer. We want to define an R operation for such terms. There is apparently a problem, as the R operation involves derivatives and any term contributing (N −1) (N ,N ) (p). Hence one can worry about to W˜ n+1 contains the εω and the cutoff function χ the derivatives of the factor χ (p)pD−ω (p)−1 . However, as k is fixed independently from N (and far enough from γ N ) and k − p is fixed at scale N , then |p| ≥ γ N −1 /2, so that we can freely multiply by a smooth cutoff function χ(p) ¯ restricting p to the allowed region; this allows us to pass to coordinate space and shows that the R operation can be defined in the usual way. We define (N −1)
(N −1) (k, k1 , k2 , k3 ) = W˜ 4 (0, .., 0) ,
(4.7)
(N −1) (N −1) (N −1) LW˜ 2 (k) = W˜ 2 (0) + k∂k W˜ 2 (0) .
(4.8)
LW˜ 4
Note that by parity the first term in (4.8) is vanishing; this means that there are only marginal terms.
+ and ψ
− If only one among the fields ψ k −p,εω in T1 (ψ) is contracted, we get terms k ,εω with four external lines of the form (up to an integral over the external momenta): − + +
Z N
gω (k) Jk,ω ψk1 ,ω1 ψk− ,εω ψk− +k−k1 ,ω2 dk+ χ(k ˜ + − k− ) ) + ·
gωN ,h (k − k+ + k− )G (N 4 (k , k1 , k + k− − k1 ) 4 N ,h (k+ ) u N (k+ ) g εω [C h,N (k− ) − 1]Dεω (k− )
− , · D−ω (k+ − k− ) D−ω (k+ − k− )
(4.9)
Functional Integral Construction of the Massive Thirring Model
103
Fig. 4.3. Graphical representation of a single addend in (4.9)
or the similar one with the roles of k+ and k− exchanged. Note that the indices ω1 and ω2 must satisfy the constraint ω1 ω2 = ε. The two terms in (4.9) must be treated differently, as concerns the regularization procedure. The first term is such that one of the external lines is associated with the operator [C h,N (k− ) − 1]Dεω (k− )D−ω (p)−1 . We define R = 1 for such terms; in fact, when such external line is contracted (and this can happen only at scale h), the factor Dεω (k− )D−ω (p)−1 produces an extra factor γ h−N in the bound, with respect to the dimensional one. The second term in (4.9) can be regularized as above, by subtracting the value of the kernel computed at zero external momenta, i.e. for k− = k = k1 = 0. Note that such quantity vanishes, if the four ω-indices are all equal, otherwise it is given by the product of the field variables times u N (k+ ) (N ) −Z N
, (4.10) gω (k) J k,ω dk+ χ(k ˜ + )
g ωN ,h (k+ )G 4 (k+ , 0, 0) D−ω (k+ ) and there is no singularity associated with the factor D−ω (k+ )−1 , thanks to the support on scale N of the propagator
gωN ,h (k+ ). The terms with two external lines can be produced only if ε = +1 and can be treated in a similar way; they have the form (N )
−
ψ dk+ χ˜ (k+ − k)G 1 (k+ ) g (k) J k,ω k,ω ω 5 ε 6 (N ) [C h,N (k) − 1]Dεω (k)
g ω (k+ ) u N (k+ ) − · , (4.11) D−ω (k+ − k) D−ω (k+ − k) (N )
where G 1 (k+ ) is a smooth function of order 0 in λ. However, the first term in the braces ε (k) − 1 = 0, is equal to 0, since we keep k fixed and far from the cutoffs, hence C h,N and the second term can be regularized as above. (N −1) A similar (but simpler) analysis holds for the terms contributing to V¯ a,2 , which contain a vertex of type T+ or T− and are of order λν± . Now, the only thing to analyze carefully is the possible singularities associated with the factors χ(p) ˜ and pD−ω (p)−1 . + N
However, since in these terms the field ψk−p,ω is contracted, |p| ≥ γ −1 /2; hence the regularization procedure can not produce bad dimensional bounds. (ε) (ε) We will define z˜ N −1 and λ˜ N −1 , so that (recall that Z N −1 = Z N ) (N −1) (N −1) + V¯ a,2 ](ψ [h,N −1] ) = −λ˜ (ε) L[V¯ a,1 N −1
Z 2N −2 ZN
F¯λ[h,N −1] (ψ [h,N −1] , J )
Z N −1 [h,N −1]+
ψ Dω (k)
gω (k) J k,ω , Z N k,ω where we used the definition dk1 dk2
+
k+k
− .
k+ ,ω ψ F¯λ (ψ [h,N −1] , J ) = ψ gω (k)ψ Jk,ω
2 1 −k2 ,−ω k1 ,−ω (2π )4 − z˜ (ε) N −1
(4.12)
(4.13)
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G. Benfatto, P. Falco, V. Mastropietro (N −1)
+ Let us consider now the terms contributing to V¯ b,i , that is those where ψ is ¯ k−p,ω not contracted. Such terms can be analyzed exactly as in § 4.3 of [BM4]; it turns out that (N −1)
L[V¯ b,1
(N −1) + V¯ b,2 ](ψ [h,N −1] ) = −ν+,N −1 Z N −2 T−ε (ψ [h,N −1] , J )
− ν−,N −1 Z N −2 Tε (ψ [h,N −1] , J ) ,
(4.14)
ν±,N −1 being exactly the same constants appearing in (3.21). The integration over subsequent scales is performed in a similar way; as described in more detail in § 4.4 of [BM4], it turns out that, if j ≥ h k , the local part of the terms linear in J has the form (coinciding with Eq. (131) of [BM4] for Z N = 1 and ω = ε = −1): (ε)
LV¯ ( j) (ψ [h, j] ) = Z N T1 (ψ [h, j] , J ) − ν+, j Z j−1 T−ε (ψ [h, j] , J ) (ε)
− ν−, j Z j−1 Tε ((ψ [h, j] , J ) − λ˜ j −
N −1 i= j
(ε)
z˜ i
Z 2j−1 ZN
[h, j] F¯λ (ψ [h, j] , J )
Z i [h, j]+
ψ Dω (k)
gω (k) J k,ω . Z N k,ω
(4.15)
If j < h k , LV¯ ( j) has the same structure, but there is indeed no term with two external
[h, j]+ = 0; for a similar reason the term with four external legs is different legs, since ψ k,ω (ε) from 0 only if 3γ j−1 > γ h k . However, the constants λ˜ (ε) j and z˜ i are defined for any j > h and their value is independent of k. Note also that, as in the expansion of a normal Schwinger function, we do not localize the terms with four external legs, containing both a J vertex and a ϕ vertex. ,h 2,N It follows that we can write G ε,ω (k) as a sum of trees with two special endpoints, similar to those described in detail in § 4.5 of [BM4]; they differ from those present in ,h (k), see § 2.5, since one of the special endpoints
2,N the expansion of the function G ω corresponds to one of the addenda in (4.15), to be called of type T , T+ , T− , λ(ε) , z (ε) . By construction the constants ν±, j coincide with those introduced in § 3.1, hence they verify (3.26). Moreover, it was shown in § 4.6) of [BM4], by a fixed point argument, that, if λ¯ h is small enough, it is possible to choose α−,h,N = c1 λ + O(λ2 ) and α+,h,N = c3 + O(λ) so that there exist two positive constants, C and ϑ, independent of h and N , such that, if h + 1 ≤ j ≤ N − 1, |z i αε,h,N − z i(ε) | ≤ C λ¯ h γ −ϑ(N −i) , |λi αε,h,N − λi(ε) | ≤ C λ¯ h γ −ϑ(N −i) .
(4.16)
Then we can write de f T± ,N ,h ,h z,N ,h T,N ,h λ,N ,h 2,N G (k) + Aε,ω (k) , ε,ω (k) = Aε,ω (k) + Aε,ω (k) + Aε,ω
T ,N ,h
(4.17)
± λ,N ,h , A z,N ,h T,N ,h contain respectively one endpoint of type where Aε,ω and Aε,ω ε,ω , Aε,ω λ(ε) , z (ε) , T± , T . T,N ,h , we repeat the analysis in § 4.8 in [BM4]. It follows that it In order to bound Aε,ω is bounded by an expression similar to the r.h.s. of (2.65), with the following differences. T,N ,h , the dimensional bound differs from that of a tree Given a tree τ contributing to Aε,ω 2,N ,h
ω (k) for the following reasons: contributing to G
Functional Integral Construction of the Massive Thirring Model
105
Fig. 4.4. Graphical representation of the leading terms contributing to c1 λ; the other four graphs contributing to c1 λ, as well as the graph contributing to c3 , are vanishing in the limit of short tail (γ0 → 1) of the cutoff function (see Definition (1.9)). The two graphs giving the 0th order expansion in λ of α−,h,N cancel each other by symmetry
(1) there is an extra factor Z h k /Z N , because one external propagator is substituted by (ε) the free one, Z −1 gω (k) (see the definition of T1 ); N
(ε) (2) since there is no external field renormalization for T1 (which is dimensionally equivalent to a term with four external fields), there is an extra factor (Z N /Z jT )2 , if jT is the scale of the endpoint of type T ; (ε) (3) if at least one of fields in T1 ) is contracted on scale h, there is an extra factor Z h /Z N , because of the bound (3.11); (4) because of (3.8), either jT = N + 1 or the root of τ has scale h − 1. T,N ,h can be bounded by an expression equal to the r.h.s. of (2.65), mulHence, Aε,ω
tiplied by a factor Z h Z h k /Z 2jT ≤ γ Cλ [( jT −h)+| jT −h k |] , which takes into account Items (1)–(3) above. This factor can be absorbed in the sum over the scale labels, since all vertices have an “effective” positive dimension (see the remark before (2.66)). Then, by taking into account the remark in Item (4) above, it is easy to show that 2
T,N ,h |Aε,ω (k)| ≤ C
γ −h k −(N −h k )/4 γ + γ −(h k −h)/4 . Z hk
(4.18)
T ,N ,h
± Let us now consider Aε,ω . We still have some extra factors with respect to the bound (2.65), the same factor of Item (1) above and a factor (Z N /Z jT ), due to the partial field renormalization of T± ; the product of these factors can be treated as before. We do not have anymore a condition like Item (4) above, but we have to take into account that the running constant associated with the special vertex of type T± satisfies the bound (3.26). It follows that
T ,N ,h
± |Aε,ω
(k)| ≤ C
γ −h k −(N −h k )/4 γ . Z hk
(4.19)
z,N ,h and let us suppose that |k| = γ h k , so that (see (2.19) and Let us now consider A ε,ω ( j)
( j)
g ω (k) = 0, if j = h k . This (2.23))
g ω (k) = [Dω (k)Z h k −1 ]−1 , if j = h k , while
condition, which greatly simplifies the following discussion, is not really restrictive. In fact, since the external momentum k is fixed in this discussion, one could modify the definition (1.24) of the cutoff functions f j (p), by substituting it with f j (p/ p0 ), p0 being a fixed positive number ≤ γ , to be chosen so that p0−1 |k| = γ h k , for some integer
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G. Benfatto, P. Falco, V. Mastropietro
Fig. 4.5. The first two graphs are the graphical representation of c2 λ; the last is the graph for c4
h k . Since our bounds would be clearly uniform in this new parameter and the removed cutoffs limit is independent of γ (see § 2.5), this procedure can not produce any trouble. By using (4.15), we can write ⎡ ⎤ N −1 1 (k)
g ω (ε) z,N ,h 1, z,N ,h ⎣ A ε,ω (4.20) (k) = Aε,ω (k) − zi Z i ⎦ , ZN Z h k −1 i=h k
1, z,N ,h contains the contributions to A z,N ,h where Aε,ω ε,ω coming from trees with at least one λ endpoint. Since Z j−1 = Z j (1 + z j ) and Z N −1 = Z N ,
Z h k −1 −
N −1
Z jz j = ZN ,
(4.21)
j=h k
hence we can write N −1 i=h k
N −1 (ε) Z z i Zi Z h k −1 i (ε) z i − z i αε,h,N = + αε,h,N −1 . ZN ZN ZN
(4.22)
i=h k
The first term in the r.h.s. of (4.22) can be written as N −1
( z i(ε) − αε,h,N z i )
i=h k
−
h k −1
N −1 Zj Zi = ( z (ε) j − αε,h,N z j ) ZN ZN
(4.23)
j=h
(ε)
( z j − αε,h,N z j )
j=h
Z j de f = − ρε,h,N + Rε2,N ,h (k) , ZN
where ρε,h,N is independent of k and satisfies, by (4.16), the bound |ρε,h,N | ≤ C|λ|
N
¯2
γ −(ϑ−cλh )(N − j) ≤ C|λ| ,
(4.24)
j=h
implying that there exists the limit ρε = lim−h,N →∞ ρε,h,N . By an explicit computation one can show that ρ− = c2 λ+ O(λ2 ) and ρ+ = c4 + O(λ), with c2 and c4 strictly positive constants. On the contrary, Rε2,N ,h (k) is vanishing for −h, N → ∞; in fact |Rε2,N ,h (k)| ≤ C|λ|
hk j=h
¯2
γ −(ϑ−cλh )(N − j) ≤ C|λ|γ −(ϑ/2)(N −h k ) .
(4.25)
Functional Integral Construction of the Massive Thirring Model
107
By collecting all terms we get
gω (k) + (ρε,N ,h + αε,N ,h )
g ω(h k ) (k) ZN −Rε2,N ,h (k)
g ω(h k ) (k) ,
z,N ,h 1, z,N ,h (k) = Aε,ω (k) − αε,N ,h A ε,ω
(4.26)
(h ) with Rε2,N ,h (k)
gω k (k) ≤ C|λ|γ −h k Z h−1 γ −ϑ(N −h k ) . k
We now consider A1, z,N ,h together with Aλ,N ,h . We proceed as in [BM3], formulas (161)–(165); to summarize, given a tree τ ∈ Tλ,n , n ≥ 1, we can associate to it a tree τ ∈ Tz,n+1 , substituting the endpoint v ∗ , on scale j ∗ , of type λ with an endpoint of type λ, and linking the endpoint v ∗ to an endpoint of type z. If we define de f (h k ) ,h
2,N G (k) =
g ω (k) + ω
N j ∗ =h k
2,N ,h λ j ∗ Bω, j ∗ (k) ,
(4.27)
then it is easy to check that N −1 Z h k −1 (ε) 2,N ,h λ j ∗ Bω, j ∗ (k) , ZN ∗ j =h k $ N −1 (ε) % N −1 z i Zi Z h k −1 i=h k 2,N ,h 1, z,N ,h Aε,ω (k) = − λ j∗ Bω, j ∗ (k) . ZN ∗ Z h k −1
,h Aλ,N ε,ω (k) =
(4.28)
j =h k
Using (4.21) and the definitions of ρε,h,N and Rε2,N ,h (k), we get: ∗ λ(ε) j ∗ −λ j
N −1
z (ε) j=h k j Zj Z h k −1
∗ ∗ = ( λ(ε) j ∗ −αε λ j ) + λ j
ZN αε,h,N +ρε,h,N − Rε2,N ,h (k) .
Z h k −1
(4.29) 2,N ,h −h k Z −1 (γ −ϑ| j By the usual arguments, one can see that |Bω, j ∗ (k)| ≤ Cγ hk by summing the two equations in (4.28), we get:
∗ −h
k|
); hence,
,h 1, z,N ,h 2,N ,h (h k )
Aλ,N (k) + A (k) = (α + ρ ) G (k) −
g (k) + Rε3,N ,h (k) , ε,h,N ε,h,N ε,ω ε,ω ω ω (4.30) where de f Rε3,N ,h (k) =
& ' N ZN Z h k −1 2,N ,h (ε) 2,N ,h Bω, j ∗ (k) λ j ∗ − αε,h,N λ j ∗ − λ j ∗ Rε (k) ZN ∗ Z h k −1 j =h k
(4.31) γ −(ϑ/2)(N −h k ) . is bounded by C|λ|γ −h k Z h−1 k
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G. Benfatto, P. Falco, V. Mastropietro
Finally, the summation of all terms in the r.h.s. of (4.17) gives gω (k) ,h ,h 2,N G + (αε,h,N + ρε,h,N )G 2,N (k) + Rε4,N ,h (k) , (4.32) ε,ω (k) = −αε,h,N ω ZN with |Rε4,N ,h (k)| ≤ C|λ|
γ −h k −(ϑ/2)(h k −h) γ + γ −(ϑ/2)(N −h k ) . Z hk
(4.33)
This ends the proof of Theorem 1.3.
4.2. Proof of Corollary 1.1. If we insert the identity (4.32) in the r.h.s. of (1.31) and we take the limit h → −∞, we get
2,N (k − p) G dp + H N ,ω (k) , (4.34) χ¯ N (p) ω 2 (2π ) D−ω (p) where χ¯ N (p) is the function appearing in (1.30), B N ≡ (1 − λ N ε Aε,N aε,N )[1 − λ N ε Aε,N (aε,N + ρε,N )]−1 , b N ≡ λ N A+,N [1 − λ N ε Aε,N (aε,N + ρε,N )]−1 and H N ,ω (k) is a function satisfying the bound BN
2,N G − bN ω (k)Dω (k) = ZN
|H N ,ω (k)| ≤ C|λ|Z h−1 γ −(ϑ/2)(N −h k ) . k
(4.35)
On the other hand, by (2.67), there is a function f (λ), independent of k, such that |k|ηz
2,N FN (k) , G ω (k) = Dω (k)
FN (k) = f (λ) + O(γ −N |k|)ϑ ;
(4.36)
hence, we can rewrite (4.34) as FN (p) dp BN + H N ,ω (k) + bN χ¯ N (p + k)|p|ηz |k|ηz FN (k) = 2 ZN (2π ) D−ω (p + k)Dω (p) (4.37) and, subtracting the equation with k = 0, we obtain ' & χ¯ N (p) χ¯ N (k + p) dp |p|ηz |k|ηz FN (k) = b N F − (p) N (2π )2 Dω (p) D−ω (k + p) D−ω (p) + H N ;ω (k) − H N ,ω (0) . The integral can be written as the sum of two terms dp |p|ηz D−ω (k) FN (p)χ¯ N (k + p) (2π )2 |p|2 D−ω (k + p) dp |p|ηz − FN (p)[χ¯ N (k + p) − χ¯ N (p)] , (2π )2 |p|2
(4.38)
(4.39)
Functional Integral Construction of the Massive Thirring Model
109
and the second addend is vanishing in the N → ∞ limit, as it can be written as dp |p|ηz ηz N γ FN (p)[χ¯ 0 (γ −N k + p) − χ¯ 0 (p)] (4.40) (2π )2 |p|2 and χ¯ 0 (γ −N k + p) − χ¯ 0 (p) is O(γ −N |k|) and with compact support. On the other hand, by (4.36), the integral we obtain, if we substitute FN (p) with f (λ), is vanishing as N → ∞. Hence, in the limit N → ∞ we get the identity: |k|
ηz
= b∞
b∞ dp |p|ηz D−ω (k) = 2 2 (2π ) |p| D−ω (k + p) 2π
∞
0
dρ ρ 1−ηz
2π 0
|k| dϑ , 2π |k| + ρeiϑ (4.41)
that is b∞ 1= 2π
∞ 0
dρ ρ 1−ηz
2π 0
1 dϑ b∞ = 2π 1 + ρeiϑ 2π
1
0
dρ b∞ = , ρ 1−ηz 2π ηz
(4.42)
which proves (1.36).
4.3. Proof of Theorem 2.1. The Beta function equations for the running coupling or renormalization constants are j
λ j−1 = λ j + βλ (λ j , ..., λ N ) , Z j−1 ( j) = 1 + βz (λ j , .., λ N ) , Zj (2)
Z j−1 (2) Zj
( j)
= 1 + βz 2 (λ j , . . . , λ N ) ,
(4.43)
µk µ j−1 = 1+ β ( j,k) (λ j , .., λ N ) , µj µj µ k≥ j
( j)
( j)
( j,k)
with βz , βz 2 , βµ stants,
independent from µ and, if aµ , az , az 2 are suitable positive conβµ( j,k) (λ j , ..λ j ) = aµ λ j δ j,k + O(λ¯ 2j ) , ( j) βz (λ j , .., λ j ) = az λ2j + O(λ¯ 4j ) , ( j) βz 2 (λ j , .., λ j )
=
az 2 λ2j
+
O(λ¯ 4j )
(4.44) .
(4.45)
Moreover, these functions do not depend directly on Z N , but only depend on the ratios Z j−1 /Z j , j ≤ N ; hence the value of λ j is a function of λ N = λ and the number of RG steps needed to reach scale j starting from scale N . It follows that, if we call
λj, j ≤ 0, the constants we get for N = 0, then, for any N > 0 and j ≤ N , λ j =
λ j−N . The problem with N = 0 was studied in detail in [BM4], where it has been proved (see Theorem 2 of that paper) that there exist constants c1 , ε1 (independent of N , h), such that, if |λ| ≤ ε1 , then |λ j | ≤ c1 ε1 for any j. The proof of this statement is based on the
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G. Benfatto, P. Falco, V. Mastropietro
analogue of SDe equation (1.31) for the four point function; if the momenta are calculated at the infrared cut-off scale γ j , a relation is obtained between λ j and λ implying that λ j = λ + O(λ2 ). These properties imply, see (3.48) of [BM3], that |βλ (λ j , ..., λ j )| ≤ C|λ j |2 γ −(N − j)/4 . j
(4.46)
From (4.43) and (4.46) one gets immediately, see § 4.10 of [BM1], the bound (2.29) with λ−∞ (λ) = λ + O(λ2 ) together with | logγ (Z j−1 /Z j ) − ηz | ≤ Cλ2 γ −(N − j)/4 , | logγ (µ j−1 /µ j ) − ηµ | ≤ C|λ|γ −(N − j)/4 ; finally by the WTi (1.26) with momenta (2)
calculated at the infrared cut-off scale γ j one gets, see [BM2], |Z j /Z j − 1| ≤ C|λ|. 5. Lattice Wilson Fermions We prove now Theorem 1.4 for the lattice model (1.37); in this model the momentum k belongs to the two-dimensional torus Da of size 2π/a and we shall denote by |k − k | the corresponding distance. To begin with, we define f¯(k) so that C N−1 (k) + f¯(k) = 1 ,
(5.1)
where C N−1 (k) = Nj=−∞ f j (k), with f j (k) as in (1.10); since C N−1 (k) = 0 for |k| ≥ γ N +1 = π/(4a), the support of the function f¯(k) is given by the set {k : |k − π/a| ≤ 3π/4a}. Therefore, it is possible to decompose the propagator
rω,ω (k), defined in (1.38), as (≤N )
(N +1)
rω,ω (k) =
rω,ω (k) +
rω,ω (k) , (≤N )
(5.2)
(N +1)
with
r ω,ω (k) = C N−1 (k)
rω,ω (k) and
rω,ω (k) = f¯(k)
rω,ω (k). Note that √ N (N +1) |rω,ω (x)| ≤ Cγ N e−c γ |x| ,
(5.3)
since the function f¯(k) is a Gevrais function of class 2, with a compact support of size ˜ −1 = Cγ N on its support. a −2 , and [1 − cos(k0 a) + 1 − cos(ka)]/a ≥ Ca We can therefore write the Generating function (1.40) as (≤N ) (N +1) PZ N +1 ,
)PZ N +1 ,
)· µ N +1 ,C N (dψ µ N +1 , f¯ (dψ
exp − λ N +1 Z 2N +1 V (ψ) + ν N +1 Z N +1 N (ψ) · (5.4)
+ + − − + − · exp Z (2) , ψx,ω + ψx,ω + ψx,ω ϕx,ω dx Jx,ω ψx,ω dx ϕx,ω N +1 ω
(2) where Z N +1 = Z a , Z N +1 ψ = ψ (≤N ) + ψ (N +1) .
ω
=
(2) Za ,
λ N +1 = λa , ν N +1 = νa , µ N +1 (k) = µa (k) and
The integration in (5.4) is performed iteratively in a way very similar to that used for the integration in (1.13); the only difference is that we have one more steps to do, associ(N +1) ). However, thanks to the bound (5.3), this ated with the measure PZ N +1 ,
µ N +1 , f¯ (dψ step gives no trouble, in the sense that the effective potential following from it differs
Functional Integral Construction of the Massive Thirring Model
111
from the local one of (1.13) only for small irrelevant terms, so that the subsequent integration steps are essentially identical, except for the presence in the effective potential of a term proportional to N (ψ) and in the renormalized free measure of a non diagonal term proportional to [1 − cos(k0 a) + 1 − cos(ka)]/a. Hence, after the fields of scale N + 1, N , .., j + 1 are integrated, (5.4) can be written as √ √ −L 2 E¯ j (≤ j) −V¯ ( j) ( Z j ψ (≤ j) )+B¯ ( j) ( Z j ψ (≤ j) ,ϕ,J ) e (dψ )e , (5.5) PZ˜ j ,
µ j ,C j where
−1 (≤ j)
˜ PZ˜ j ,
r ω,ω (k), whose µ j ,C j is the integration with propagator given by Z j (k)
(≤N )
r ω,ω (k) by replacing Z a , µa (k), C N (k) with expression is obtained from that of Z a−1
µ j (k), C j (k); V¯ ( j) , B¯ ( j) are very similar to the analogous quantities in § 2, up Z˜ j (k),
to some obvious modifications that we now discuss. The function
µ j (k), as we shall explain better below, is the sum of a term proportional to [1 − cos(k0 a) + 1 − cos(ka)]/a and another one proportional to µ, that we shall call µ˜ j (k), as it will play the same role of the function with the same name in the continuum model. This allows us to define all localization operators as in § 2, except L1 , which is defined as ' & 1 (h) sin k0 a (h) sin ka ¯
L1 W2,ω,ω (k) = , (5.6) +η W2,ω,ω (kηη ) η 4 sin πa sin πa L L η,η =±1
in order to take into account the lattice structure of the space coordinates; hence the localization procedure is essentially unchanged. However, since the operator P0 does not cancel the non diagonal part of the propagator, which is even in the momentum (while the diagonal one is odd), extra terms are produced by the action of the L operator with respect to the ones in (2.10). It follows that
( j) = L1 P0 W
( j)
( j) , LW
( j)
( j) LW 2,ω,ω 2,ω,ω 2,ω,−ω = L0 P0 W2,ω,−ω + L0 P1 W2,ω,−ω , (5.7) and this implies that we can write [h, j]
LV ( j) (ψ [h, j] ) = z j Fζ
[h, j]
+ (s j + γ j n j )Fσ[h, j] + l j Fλ
,
(5.8)
( j) , while s j = L0 P1 W
( j) , as in (2.9)-(2.10). The proof where γ j n j = L0 P0 W 2,ω,−ω 2,ω,−ω
( j) is given by the sum of graphs of (5.7) follows by induction from the remark that W 2,ω,ω with 1. either an even number of ν vertices, an even number of non diagonal propagators and an odd number of diagonal propagators; 2. or an odd number of ν vertices, an odd number of non diagonal propagators and an odd number of diagonal propagators. ( j)
Moreover W 2,ω,−ω is given by the sum of graphs with 3. either an even number of ν vertices, an odd number of non diagonal propagators and an even number of diagonal propagators; 4. or an odd number of ν vertices, an even number of non diagonal propagators and an even number of diagonal propagators.
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The renormalization of the free measure is done exactly as in § 2, see (2.19), that is we do not put the term proportional to n j in the free measure, but we define a new running
( j) (ψ (≤ j) ) coupling constant ν j = n j (Z j /Z j−1 ). It follows that the rescaled potential V (≤ j) differs from that of (2.26) because its local part contains the term γ j ν j Fσ , that is it is [h, j] (≤ j) equal to λ j Fλ + γ j ν j Fσ . One then performs the integration with respect to ψ ( j) , whose propagator is of the form 1 Z˜ j (k)
de f ( j)
r ω,ω (k) =
f˜j (k) µ j (k) e− (k) −
, µ j (k) e+ (k) ω,ω Z˜ j (k) e+ (k)e− (k) − µ2j (k) −
1
(5.9)
with
µ j (k) = µ˜ j (k) + [Z N +1 / Z˜ j (k)][1 − cos(k0 a) + 1 − cos(ka)]/a, µ˜ j (k) being a function equal to µ for j = N + 1; Z˜ j (k) and µ˜ j (k) satisfy the recursion relations (2.19). In order to control the RG expansion we have to prove that λ¯ h ≡ maxh≤ j≤N +1 |λ j | and ν¯ h ≡ maxh≤ j≤N +1 |ν j | stay small, if λa is small enough and νa is suitably chosen. This can be proved by noting that the propagator (5.9) can be written as ( j) ( j) R,( j) Z˜ j (k)−1
r ω,ω (k) =
g ω,ω (k) +
g ω,ω (k) ,
(5.10)
( j)
where
g ω,ω (k) has exactly the same form as the single scale propagator appearing in the multiscale integration of (1.22), see (2.22). We shall prove below that the flow of the running couplings and the free measure can be controlled as in § 2, if the value of νa is suitably chosen. This implies that there is minimal h ∗ , such that, as far as j > h ∗ , |µ˜ j (k)| ≤ γ j , so that, as it is easy to check, |gω,ω (x, y)| ≤ Cγ −(N − j) γ j e−c R,( j)
√
γ j |x−y|
.
(5.11)
By (5.10) and (5.11) we see that the single scale propagators of the lattice model are equal to those of the continuum model, of course with different Z˜ j (k) and µ˜ j (k) functions, up to a correction which is vanishing in the limit N → ∞. By using the above decomposition, the flow equation for λ j can be written, for j ≤ N + 1, as λ j−1 = λ j + β˜λ (λ N , ..., λ j ) + rλ (λa , λ N , ..., λ j ) j,k + νk β˜λ (λa , νa , λ N , ν N , ..., λ j , ν j ) , j
j
(5.12)
k≥ j
where the functions in the r.h.s. can be represented as sums over trees similar to those of (2.28); in particular, we have included the sum over all trees with at least one ν-endpoint in the last term in the r.h.s. of (5.12) and we have split the sum of all trees with no j j j ( j) ν-endpoint as β˜λ + rλ , where β˜λ contains the trees with propagator gω,ω , j ≤ N (the j
decomposition (5.10) is used), while all other terms are included in rλ . The fact that the contribution of a single tree satisfies a bound similar to that of (2.36), with dv > 0 for any v, easily implies that, if |ν j | ≤ C|λa | for any j, |β˜λ | ≤ C λ¯ j γ −(k− j)/4 , |rλ | ≤ C λ¯ 2j γ −(N − j)/4 . j,k
j
(5.13)
Functional Integral Construction of the Massive Thirring Model
113
j On the other hand the only difference between β˜λ (λ N , ..., λ j ) and the function j βλ (λ N , ..., λ j ) in (4.43) comes from the fact that in the continuum model the delta 2 function of conservation of momenta is L δk,0 δk0 ,0 , while in the lattice model it is 2 L n,m∈Z 2 δk,2π n/a δk0 ,2π m/a . However, the difference between the two delta functions has no effect on the local part LV N , because of the compact support of ψ ≤N and only slightly affects the non local terms. To see that, let us consider a particular tree τ and a vertex v ∈ τ of scale h v with2n external fields of space momenta ki ; the conservation of momentum implies that i εi ki = m 2π a , with m an arbitrary integer. On the other hand, ki is of order γ h v for any i, hence m can be different from 0 only if n is of order γ N −h v . Since the number of endpoints following a vertex with 2n external fields is greater or equal to n − 1 and there is a small factor (of order λ¯ j ) associated with each endpoint, we get an improvement, in the bound of the terms with |m| > 0, with respect to the others, of a factor exp(−Cγ N −h v ). Hence, by using the remark preceding (5.13), it is easy to show that j j
j (λ N , ..., λ j ), β˜λ (λ N , ..., λ j ) = βλ (λ N , ..., λ j ) + β λ
(5.14)
j j where |βλ (λ N , ..., λ j )| ≤ C λ¯ 2j γ −(N − j)/4 for a suitable constant C and βλ (λ N , ..., λ j ) is the beta function for the continuum model, verifying the crucial bound (4.46). In the same way the flow equation for ν j ,
ν j−1 = γ ν j + βν( j) λa , νa ; λ N , ν N ; ...; λ j , ν j ,
(5.15)
can be written, for j ≤ N + 1, as ν j−1 = γ ν j + βν(1, j) (λa , λ N , ..., λ j ) +
νk β˜ν( j,k) λa , νa , λ N , ν N , ..., λ j , ν j ,
k≥ j
(5.16) ( j,1)
where βν is a sum over trees with no endpoints of type ν. By using the decomposition ( j) (5.10), the parity properties of gω,ω (x, y) and the remark preceding (5.13), we get the bounds |βν(1, j) | ≤ C|λa |γ −(N − j)/4 , |β˜ν( j,k) | ≤ C|λa |γ −(k− j)/4 .
(5.17)
From the above properties we can show that it is possible to choose νa = ν N +1 so that ν j = O(λa γ −(N − j)/8 ) and λ j stays close to λa ; the proof is quite standard and it is essentially identical to the one in [BM1] or [GiM], so we just sketch it below. Lemma 5.1. For any given λ N +1 small enough, it is always possible to fix ν N +1 so that, for any j ≤ N + 1, |ν j | ≤ C|λa |γ −(N − j)/8 ,
|λ j − λa | ≤ Cλa2 .
(5.18)
Proof. We consider the Banach space Mξ of sequences ν = {ν j } j≤N +1 such that ||ν||ξ = sup γ (N − j)/8 |ν j | ≤ ξ |λa | , j≤N +1
(5.19)
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with ξ to be fixed later. From (5.12), (5.13) and (4.46) it follows, see § 4 of [BM1] or Appendix 5 of [GiM] for details, that there exists ε0 such that, if both |λa | and ξ |λa | are smaller than ε0 , then, for any ν, ν ∈ Mξ , |λ j ( ν ) − λa | ≤ Cλa2 , |λ j ( ν ) − λ j (ν )| ≤ C|λa |||ν − ν ||ξ .
(5.20)
We want to show that it is possible to choose ν N +1 so that ν ∈ Mξ . Note that ν verifies Eq. (5.15) and, if ν ∈ Mξ , lim j→−∞ ν j = 0; by some simple algebra, this implies that νj = − (5.21) γ k− j−1 βν(k) λa , νa ; λ N , ν N ; ...; λ j , ν j . k≤ j
Hence, we look for a fixed point of the operator T : Mξ → Mξ defined as de f
T( ν ) j = −
γ k− j−1 βν(k) λa , νa , λ N ( ν ), ν N , .., λ j ( ν ), ν j .
(5.22)
k≤ j
By (5.15) and (5.17) we find |T( ν ) j | ≤ C|λa |γ −( j−k) γ −(N −k)/8 ≤ c0 |λa |γ −(N − j)/8 .
(5.23)
k≤ j
Hence the operator T : Mξ → Mξ leaves Mξ invariant, if ξ ≥ c0 and λa is sufficiently small, and it is also a contraction since |T( ν ) j − T(ν ) j | ≤ C|λa |||ν − ν ||ξ . It follows that there is a unique fixed point in Mξ , satisfying the flow equation (5.15). An important consequence of the bound (4.46) is that, if we construct as in § 2 the Schwinger functions, by imposing the normalization conditions (2.30), we get, as N → ∞, exactly the same expansion in terms of trees, containing only λ endpoints with a fixed coupling constant λ˜ −∞ (λa ) = lim j→−∞ λ j ; in fact, the trees containing at least one ν vertex vanish in this limit. By a fixed point argument, one can show that we can fix λa so that λ˜ −∞ (λa ) has the same value as λ−∞ (λ) in the continuum model; this remark completes the proof of Theorem 1.4. A. Osterwalder-Schrader Axioms Osterwalder-Schrader axioms were partially stated in [OS1] and completed in [OS2] by the “linear growth property”. We show here that they are satisfied by the Schwinger functions of our model.
A.1. Linear growth condition and clustering. In order to verify the linear growth prop2k erty, see the bound (4.1) of [OS2], for s = 3, let us consider the space S0 (R ) of the test functions such that, for any m ∈ N, de f (A.1) f m = sup (1 + |x|2 )m/2 D α f (x) < ∞, 2k x∈R |α|≤m
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and which vanish, together with all their partial derivatives, if at least two among the points in the set x = {x1 , . . . , xk } coincide. By (2.58) Sk,ω , f ≤ C k (k!)3+2η
f (x)
dx1 · · · dxk
i< j
|xi − x j |k(1+η)/2−2ε
.
(A.2)
On the other hand, by (A.1), | f ( x )| ≤ f 4k+1 (1 + |x|4k+1 )−1 and, for any i = j, | f ( x )| ≤ 2k [(2k)!]−1 |xi − x j |2k f 2k ; hence, since f 2k ≤ f 4k+1 , | f ( x )| ≤ f 4k+1 (1 + |x|4k+1 )−1 2k [(2k)!]−1 |xi − x j |2k .
(A.3)
It follows that Sk,ω , f ≤ C k (k!)2+2η f 4k+1 .
(A.4)
In order to prove the “cluster property”, any fixed integer p ∈ [1, k − 1], y ∈ R 2k and f ∈ S0 (R ), we first prove that Sk,ω , f p,y goes to 0 as |y| → ∞, if f p,y ( x ) ≡ f (x1 , . . . x p , x p+1 − y, . . . , xk − y). Let us consider the characteristic functions χy ( x ) and χy ( x ) of the set 2
2k M = x ∈ R : max |x j | ≤ |y|/4 , de f
1≤ j≤ p
4 max |x j − y| ≤ |y|/4 ,
p+1≤ j≤k
(A.5)
and of its complementary, respectively. Since Dx ≥ |y|/2 in M, by using (2.58) and (A.3), we see that | S | ≤ [1 + (|y|/2)2ε ]−1 C k (k!)2+2η f 4k+1 , so that , f χ k,ω p,y y Sk,ω , f p,y χy is uniformly bounded and vanishes as |y| → ∞. On the other hand, by (A.3), | Sk,ω , f p,y χy | ≤ C k (k!)2+2η f 4k+1 dx (1 + |x|4k+1 )−1 χ0 ( x ), so that even Sk,ω , f p,y χy is uniformly bounded and vanishes in the limit |y| → ∞, as well as Sk,ω , f p,y . The cluster property E0, defined in § 3 of [OS1], now simply follows, by decomposing the connected Schwinger functions as finite linear combinations of the truncated Schwinger functions.
A.2. Symmetry, Euclidean invariance and Reflection Positivity. From the explicit definition of the generating functional, (1.13), two properties immediately follow. First, since the fields anticommute, the Schwinger functions are antisymmetric in the exchange of their arguments. Moreover, the generating functional (1.13) is Euclidean invariant by construction. Finally the Reflection Positivity E2, defined in § 6 of [OS1], is verified in the lattice regularization (1.40), as proved in [OSe], hence it holds even in the removed cutoffs limit of the regularized model (1.13), which we have shown to be equivalent to the a = 0 limit of the lattice model, see Theorem 1.4.
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B. Lowest Order Computation of ν− and ν+ de f
de f
B.1. Lowest order computation of ν− . Calling
gω,ω (k) =
gω (k) and u 0 (t) = 1 − χ0 (t) the lowest order contibution to the ν−,N , appearing in (3.5), is obtained, from (3.17) and (3.18), by taking the p → 0 limit of the following expression (see the first graph in Fig. 3.1), whose value is independent of the infrared cutoff for any fixed p and |h| large enough: dk C h,N ;ω (k, k − p) (≤N )
g ω (k)
λ g ω(≤N ) (k − p) (2π )2 D−ω (p) Dω (p) dk u 0 γ −N |k − p| χ0 γ −N |k| = −λ D−ω (p) (2π )2 Dω (k − p)Dω (k) −N dk χ0 γ |k| − χ0 γ −N |k − p| , (B.1) +λ (2π )2 Dω (k − p)D−ω (p) where we have used (3.3) and rearranged the terms. In the limit |p| → 0, the first contribution in the r.h.s. of (B.1) vanishes by the symmetry
gω (k) = −iω
gω (k∗ ), k∗ = (−k0 , k). As regards the second term, if we write the first order Taylor expansion in p of the numerator as a linear combination of D−ω (p) and Dω (p), the term proportional to Dω (p) also vanishes, again for the symmetry k → k∗ , so that ∞ λ λ λ dk χ0 (|k|) =− . (B.2) ν− = − dρ χ0 (ρ) = 2 2 (2π ) |k| 4π 1 4π B.2. Lowest order computation of ν+ . If we define dk −N Iω γ k =
g (≤N ) (k )
g ω(≤N ) (k + k) , (2π )2 ω
(B.3)
then the lowest order contribution to the anomaly coefficient ν+,N , appearing in (3.5), is is obtained, from (3.17) and (3.18), by taking the p → 0 limit and, after that, the h → −∞ limit of the following expression (see the second graph in Fig. 3.1): dk C h,N :ω (k, k − p) (≤N ) (≤N ) −N g γ −λ2 (k)g (k − p)I k −ω ω ω (2π )2 Dω (p) −N dk u 0 γ |k − p| χ0 γ −N |k| 2 −N I γ =λ k −ω (2π )2 Dω (k − p)Dω (k) −N γ χ |k| − χ0 γ −N |k − p| dk 0 2 −N I γ −λ k . (B.4) −ω (2π )2 Dω (k − p)Dω (p) In the limit |p| → 0 and h → −∞, we get & ' u 0 (|k|)χ0 (|k|) χ0 (|k|) dk 2 2 I−ω (k)D−ω ν+ = λ − (k) , (2π )2 |k|4 2|k|3
(B.5)
where we are using the symbol I−ω (k) to denote even its h = −∞ limit, which is finite. Note that the term in square brackets is nonnegative; moreover, it is different from 0
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only for 1 ≤ |k| ≤ γ0 (defined in (1.9)). We now fix ω = + for definiteness (the result is ω-independent); then if ik0 + k = yeiφ and ik0 + k = xeiϑ we get: I− (k) = e−2iφ
χ0 (|xe−iϑ + y|) −iϑ d xdϑ χ0 (x) e (xe−iϑ + y) , 2 (2π ) |xe−iϑ + y|2
(B.6)
χ0 (|xe−iϑ + y|) d xdϑ χ (x) (x cos 2ϑ + y cos ϑ) . 0 (2π )2 |xe−iϑ + y|2
(B.7)
so that 2 D− (k)I− (k) = y 2
The integral (B.5) is easily shown to be strictly negative in the limit γ0 → 1; hence by continuity in γ0 , ν+ < 0 for γ0 − 1 small enough. Indeed in the limit γ0 → 1, (B.5) becomes 2π 1 π λ2 χ0 (|xe−iϑ + 1|) d x dϑ (x cos 2ϑ + cos ϑ) ; (B.8) (2π )4 0 |xe−iϑ + 1|2 0 on the other hand, since |xe−iϑ + 1| ≤ 1, cos ϑ < 0 if x > 0 and x cos 2ϑ + cos ϑ = cos ϑ(1 + x cos ϑ) − x sin2 ϑ < 0 if 0 < x < 1; it follows that the integrand of (B.8) is < 0 for x = 0, 1. A numerical calculation also shows that |ν+ | is not constant as a function of γ0 , but is a strictly decreasing function near γ0 = 1. Acknowledgement. We are indebted with G.Gallavotti and K. Gawedzki for enlightening discussions on the Thirring model. P. F. gratefully acknowledges the hospitality and the financial support of the Erwin Schrödinger Institute for Mathematical Physics (Vienna) during the preparation of this work.
References [AB] [AF] [BM1] [BM2] [BM3] [BM4] [BT] [C] [CR] [CRW] [D] [DFZ] [DR]
Adler, S.L., Bardeen, W.A.: Absence of higher order corrections in the anomalous axial vector divergence equation. Phys. Rev. 182, 1517–1536 (1969) Akiyama, A., Futami, Y.: Two-fermion-loop contribution to the axial anomaly in the massive Thirring model. Phys. Rev. D 46, 798–805 (1992) Benfatto, G., Mastropietro, V.: Renormalization group, hidden symmetries and approximate Ward identities in the x yz model. Rev. Math. Phys. 13, 1323–1435 (2001) Benfatto, G., Mastropietro, V.: On the density–density critical indices in interacting Fermi systems. Commun. Math. Phys. 231, 97–134 (2002) Benfatto, G., Mastropietro, V.: Ward identities and vanishing of the Beta function for d = 1 interacting fermi systems. J. Stat. Phys. 115, 143–184 (2004) Benfatto, G., Mastropietro, V.: Ward identities and chiral anomaly in the Luttinger liquid. Commun. Math. Phys. 258, 609–655 (2005) Bergknoff, H., Thacker, H.: Structure and solution of the massive Thirring model. Phys. Rev. D 19, 3666–3679 (1979) Coleman, S.: Quantum sine-Gordon equation as the massive Thirring model. Phys. Rev. D 11, 2088– 2097 (1975) Cooper, A., Rosen, L.: The weakly coupled Yukawa z field theory: cluster expansion and Wightman axioms. Trans. Am. Math. Soc. 234, 1 (1977) Carey, A.L., Ruijsenaars, S.N.M., Wright, J.D.: The massless Thirring model: Positivity of Klaiber’s n-point functions. Commun. Math. Phys. 99, 347–364 (1985) Dimock, J.: Bosonization of Massive Fermions. Commun. Math. Phys. 198, 247–281 (1998) Dell’Antonio, G., Frishman, Y., Zwanziger, D.: Thirring Model in Terms of Currents: Solution and Light–Cone Expansions. Phys. Rev. D 6, 988–1007 (1972) Disertori, M., Rivasseau, V.: Interacting Fermi Liquid in Two Dimensions at Finite Temperature. Commun. Math. Phys. 215, 251–290 (2000)
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Furuya, K., Gamboa Saravi, S., Schaposnik, F.A.: Path integral formulation of chiral invariant fermion models in two dimensions. Nucl. Phys. B 208, 159–181 (1982) [FMRS] Feldman, J., Magnen, J., Rivasseau, V., Sénéor, R.: Massive Gross–Neveu Model: A renormalizable field theory: the massive Gross-Neveu model in two dimensions. Commun. Math. Phys. 103, 67– 103 (1986) [FS] Fröhlich, J., Seiler, E.: The massive Thirring-Schwinger model (QED2): convergence of perturbation theory and particle structure. Helv. Phys. Acta 49, 889–924 (1976) [G] Gallavotti, G.: Renormalization theory and ultraviolet stability for scalar fields via renormalization group methods. Rev. Mod. Phys. 57, 471–562 (1985) [GK] Gawedzki, K., Kupiainen, A.: Gross–Neveu model through convergent perturbation expansions. Commun. Math. Phys. 102, 1–30 (1985) [GL] Gomes, M., Lowenstein, J.H.: Asymptotic scale invariance in a massive Thirring model. Nucl. Phys. B 45, 252–266 (1972) [GR] Georgi, H., Rawls, J.M.: Anomalies of the Axial Vector Current in Two Dimensions. Phys. Rev. D 3, 874–879 (1971) [GiM] Giuliani, A., Mastropietro, V.: Anomalous Universality in the anisotropic Ashkin-Teller model. Commun. Math. Phys. 256, 687–735 (2005) [J] Johnson, K.: Solution of the Equations for the Green’s Functions of a two Dimensional Relativistic Field Theory. Nuovo Cimento 20, 773–790 (1961) [K] Klaiber, B.: The Thirring model. In: Quantum theory and statistical physics, Vol X, A, Barut, A.O., Brittin, W.F., editors. London Gordon and Breach, 1968 [Le] Lesniewski, A.: Effective action for the Yukawa2 quantum field theory. Commun. Math. Phys. 108, 437–467 (1987) [M] Mastropietro, V.: Nonperturbative Adler-Bardeen Theorem. J. Math. Phys. 48, 22302–22332 (2007) [Ma] Mastropietro, V.: Ising models with four spin interaction at criticality. Comm. Math. Phys. 244, 3, 595–642 (2004) [MM] Montvay, I., Münster, G.: Quantum Fields on a Lattice. Cambridge, Cambridge University Press (1994) [OS1] Osterwalder, K., Schrader, R.: Axioms for Euclidean Green’s Functions. Commun. Math. Phys. 31, 83–112 (1973) [OS2] Osterwalder, K., Schrader, R.: Axioms for Euclidean Green’s Functions II. Commun. Math. Phys. 42, 281–305 (1975) [OSe] Osterwalder, K., Seiler, E.: Gauge Field Theories on a Lattice. Ann. Phys. 110, 440–471 (1978) [S] Seiler, E.: Phys. Rev. D 22, 2412–2418 (1980) [Sm] Smirnov, F.A.: “Form factors in completely integrable models of quantum field theory”. Singapore: World Sci., 1992 [SU] Seiler, R., Uhlenbrock, D.A.: On the massive Thirring model. Ann. Physics 105, 81–110 (1977) [T] Thirring, W.: A soluble relativistic field theory. Ann. Phys. 3, 91–112 (1958) [Wi] Wilson, K.G.: Non–Lagrangian Models of Current Algebra. Phys. Rev. 179, 1499–1512 (1969) [W] Wightman, A.S.: Cargese lectures, 1964, New York: Gorden and Beach, [Z] Zamolodchikov Alexander, B., Zamolodchikov Alexey, B.: Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models. Ann. Physics 120, 253–291 (1979) Communicated by G. Gallavotti
Commun. Math. Phys. 273, 119–136 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0245-z
Communications in
Mathematical Physics
Quantization of Symplectic Dynamical r -Matrices and the Quantum Composition Formula Anton Alekseev1 , Damien Calaque2, 1 Section de Mathématiques (UNIGE), 2-4 rue du Lièvre, 1211 Genève, Switzerland.
E-mail:
[email protected]
2 Max Planck Institüt für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany.
E-mail:
[email protected] Received: 20 June 2006 / Accepted: 22 November 2006 Published online: 25 April 2007 – © Springer-Verlag 2007
Abstract: In this paper we quantize symplectic dynamical r -matrices over a possibly nonabelian base. The proof is based on the fact that the existence of a star-product with a nice property (called strong invariance) is sufficient for the existence of a quantization. We also classify such quantizations and prove a quantum analogue of the classical composition formula for coboundary dynamical r -matrices. Introduction Let h ⊂ g be an inclusion of Lie algebras and H ⊂ G the corresponding inclusion of Lie groups. Let U ⊂ h∗ be an invariant open subset and let Z ∈ (∧3 g)g. A (coboundary) dynamical r -matrix is a h-equivariant map r : U → ∧2 g satisfying the (modified) classical dynamical Yang-Baxter equation 1 ∂r [r (λ), r (λ)] − h i ∧ i (λ) = Z (λ ∈ U ) , 2 ∂λ i
where (h i )i and (λi )i are dual bases of h and h∗ , respectively. Then (following [14, 6]) πr := πlin +
∂ − → −−→ ∧ h i + r (λ) , i ∂λ
(1)
i
together with Z , defines a H -invariant (g-)quasi-Poisson structure on M = U × G. Here πlin is the linear Poisson structure on U ⊂ h∗ . Such a dynamical r -matrix is called symplectic if the g-quasi-Poisson manifold (M, πr , Z ) is symplectic, i.e. if πr# : T ∗ M → T M is invertible and Z = 0. Current address: Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France
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By a dynamical twist quantization of r (λ) we mean a h-equivariant map J = 1 ⊗ 1 + O() : U → ⊗2 U g[[]] satisfying the semi-classical limit condition J (λ) − J 2,1 (λ) = r (λ) + O(2 ) (λ ∈ U ) and the modified dynamical twist equation J 12,3 (λ) ∗ P BW J 1,2 (λ + h (3) ) = −1 J 1,23 (λ) ∗ P BW J 2,3 (λ) (λ ∈ U ) , where ∈ (⊗3 U g)g[[]] is an associator quantizing Z , of which we know the existence from [5, Prop. 3.10]. Recall that ∗ P BW is the Poincaré-Birkhoff-Witt star-product given on polynomial functions as the pull-back of the usual product in U (h)1 by the symmetrization map sym : S(h)[[]] → U (h). We also made use of the following notations: J 12,3 (λ) := (∆ ⊗ id)(J (λ)) and k ∂k J J 1,2 (λ + h (3) ) := (λ) ⊗ h i1 · · · h ik . k! ∂λi1 · · · ∂λik k≥0
i 1 ···i k
In this paper we prove the following generalization of [13, Theorem 5.3] to the case of a nonabelian base: Theorem 0.1. Any symplectic dynamical r -matrix admits a dynamical twist quantization (with = 1). Furthermore, two dynamical twist quantizations J1 , J2 : U → ⊗2 U g[[]] of r are said to be gauge equivalent if there exists a h-equivariant map T = 1 + O() : U → U g[[]] such that T 12 (λ) ∗ P BW J1 (λ) = J2 (λ) ∗ P BW T 1 (λ + h (2) ) ∗ P BW T 2 (λ) . In this context we also prove the following generalization of [13, Sect. 6] to the case of a nonabelian base, asserting that dynamical twist quantizations are classified by the second dynamical r -matrix cohomology (see Definition 3.2): Theorem 0.2. Let r be a symplectic dynamical r -matrix. Then the space of gauge equivalence classes of dynamical twist quantizations of r (with = 1) is an affine space modeled on Hr2 (U, g)[[]]. A class of examples of symplectic dynamical r -matrices is given by nondegenerate reductive splittings. A reductive splitting g = h ⊕ m (with [h, m] ⊂ m) is called nondegenerate if there exists λ ∈ h∗ for which ω(λ) ∈ ∧2 m∗ defined by ω(λ)(x, y) =< λ, [x, y]|h > is nondegenerate. Then rhm(λ) := −ω(λ)−1 ∈ ∧2 m ⊂ ∧2 g defines a symplectic dynamical r -matrix on the invariant open subset {λ|detω(λ) = 0} ⊂ h∗ (see [11, Prop. 1] or [14, Theorem 2.3]; see also [7, Prop. 1.1] for a more algebraic proof). Moreover, one can use it to “compose” dynamical r -matrices (see [11, Prop. 1] and [7, Prop. 0.1]): 1 h = h[[]] with bracket [, ] := [, ] . h
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Proposition 0.3 (The composition formula). Assume that h = t ⊕ m is a nondegenerate reductive splitting and let rtm : t∗ ⊃ V → ∧2 h be the corresponding symplectic dynamical r -matrix. If ρ : h∗ ⊃ U → ∧2 g is a dynamical r -matrix with Z ∈ (∧3 g)g, then θρ := rtm + ρ|t∗ : t∗ ⊃ U ∩ V −→ ∧2 g is a dynamical r -matrix with the same Z . We prove that one can “quantize” the map ρ → θρ : Theorem 0.4. With the hypothesis of Proposition 0.3, there exists a map
: {Dynamical twist quantizations of ρ} −→ {Dynamical twist quantizations of θρ } which keeps the associator fixed. The paper is organized as follows. In Sect. 1 we first recall basic facts about quasiPoisson manifolds, compatible quantizations and related results. We then give a sufficient condition for the existence of dynamical twist quantizations. In Sect. 2 we give a very short proof of Theorem 0.1, using a result stating the existence of a quantum momentum map (see also [10, 12]) which is based on Fedosov’s well-known globalization procedure [8, 9]. We start the section with a summary of basic ingredients of Fedosov’s construction. In Sect. 3 we prove Theorem 0.2 using a variant of the well-established classification of star-products on a symplectic manifold by formal series with coefficients in the second De Rham cohomology group of the manifold. In Sect. 4 we prove a quantum analogue of the composition formula for classical dynamical r -matrices. We start with a new proof of the classical composition formula (Proposition 0.3) using (quasi-)Poisson reduction. We then derive its quantum counterpart (Theorem 0.4) using quantum reduction. Notations. We denote by Oh∗ = S(h) polynomial functions on h∗ , OU = C ∞ (U ) ⊃ Oh∗ smooth functions on U ⊂ h∗ , and OG the ring of smooth functions on the (even→ tually formal) Lie group G. For any element x ∈ g we denote by − x (resp. ← x−) the corresponding left (resp. right) invariant vector field on G. 1. A Sufficient Condition for the Existence of a Dynamical Twist Quantization Let r : h∗ ⊃ U → ∧2 g be a coboundary dynamical r -matrix. Denote by πr the bivector field on M = U × G given by (1). 1.1. Quasi-Poisson manifolds and their quantizations. Recall from [1, 2] that a (g-) quasi-Poisson manifold is a manifold X together with a g-action ρ : g → X(X ), an invariant bivector field π ∈ (X, ∧2 T X )g and an element Z ∈ (∧3 g)g such that [π, π ] = ρ(Z ) .
(2)
Let { f, g} :=< π, d f ∧ dg > ( f, g ∈ O X ) be the corresponding quasi-Poisson bracket. Then Eq. (2) is equivalent to {{ f, g}, h} + {{h, f }, g} + {{g, h}, f } =< ρ(Z ), d f ∧ dg ∧ dh >
( f, g, h ∈ O X ) .
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One can define the quasi-Poisson cochain complex of (X, π, Z ) as follows: k-cochains are Cπk (X ) = (X, ∧k T X )g and the differential is dπ = [π, −]. The fact that dπ ◦ dπ = 0 follows from an easy calculation: dπ ◦ dπ (x) = [π, [π, x]] =
1 1 [[π, π ], x] = [ρ(Z ), x] = 0 (x ∈ Cπk (X )) . 2 2
Let us now fix an associator ∈ (⊗3 U g)g[[]] quantizing Z (we know it exists from [5, Prop. 3.10]). Following [6, Definition 4.4], by a quantization of a given quasi-Poisson manifold (X, π, Z ) we mean a series ∗ ∈ Bidiff(X )g[[]] of invariant bidifferential operators such that − f ∗ g = f g + O() for any f, g ∈ O X , − f ∗ g − g ∗ f = { f, g} + O(2 ) for any f, g ∈ O X , and := S ⊗3 (−1 ), then2 − if we write m ∗ ( f ⊗ g) := f ∗ g for f, g ∈ O X , and ) . m ∗ ◦ (m ∗ ⊗ id) = m ∗ ◦ (id ⊗ m ∗ ) ◦ ρ ⊗3 (
(3)
Here, S denotes the antipode of U g. One has a natural notion of gauge transformation for quantizations. It is given by an element Q = id + O() ∈ Diff(X )g[[]] that act on ∗’s in the usual way: f ∗(Q) g := Q −1 (Q( f ) ∗ Q(g)) ( f, g ∈ O X ) . More precisely, if (, ∗) is a quantization of (Z , π ) then (, ∗(Q) ) is also. In this case we say that ∗ and ∗(Q) are gauge equivalent. 1.2. Classical and quantum momentum maps. Let (X, π, Z ) be g-quasi-Poisson manifold and let G be a Lie algebra with Lie group G. A momentum map is a smooth g-invariant map µ : M → G ∗ such that µ∗ π = πlin and for which the corresponding infinitesimal action G → X(X ); x → {µ∗ x, −} integrates to a right action of G. Let us describe the reduction procedure with respect to a given momentum map µ. First of all G acts on µ−1 (0) and hence one can define the reduced space X r ed := µ−1 (0)/G. Let us assume that it is smooth (this is the case when 0 is a regular value and G acts freely); its function algebra is O X r ed = OGX /(OGX ∩ I0 ), where I0 is the ideal generated by im(µ∗ ). Since µ is g-invariant then g acts on µ−1 (0). Moreover the g-action and the G-action commute (because π and µ are g-invariant), consequently g also acts on X r ed . Now observe that OGX = { f ∈ O X |{ f, I0 } ⊂ I0 }, therefore the quasi-Poisson bracket {, } naturally induces a quasi-Poisson bracket (with the same Z ) on O X r ed . In other words, X r ed inherits a structure of a quasi-Poisson manifold from the one of X . Now assume that we are given a quantization (∗, ) of the quasi-Poisson manifold (X, π, Z ). By a quantum momentum map quantizing µ we mean a map of algebras M = µ∗ + O() : (OG ∗ [[]], ∗ P BW ) −→ (O X [[]], ∗) taking its values in g-invariant functions, and such that for any f ∈ O X and any x ∈ G one has [M(x), f ]∗ = {µ∗ x, f }. 2 We thank Pavel Etingof for pointing to us that one has to use instead of in this definition.
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Remark 1.1. One only needs to know M on linear functions x ∈ G. Let us describe the quantum reduction with respect to a given quantum momentum map M. First denote by I the right ideal generated by im(M) in (O X [[]], ∗) and observe h h h that its normalizer is O X [[]]. Therefore O X [[]] ∩ I is a two-sided ideal in O X [[]], h h and we can define the reduced algebra A := O X [[]]/O X [[]] ∩ I. Since I ∼ = I0 [[]] ∼ then A = O X r ed [[]] (as R[[]]-modules). It is easy to see that the induced product on O X r ed [[]], together with , gives a quantization of the quasi-Poisson structure on X r ed . A quantization ∗ of a quasi-Poisson manifold (M, π, Z ) with a momentum map µ : M → G ∗ for which µ∗ itself defines a quantum momentum map (it will be M = U (µ∗ ) ◦ sym) is called strongly (G-)invariant. 1.3. Compatible quantizations. Let us first observe that πr defines a quasi-Poisson structure on M = U × G. Here the action is g x → ← x− ∈ X(M) (it generates ← − − → left translations). Remark that since Z ∈ (∧3 g)g then Z = Z . Moreover, the natural map M → h∗ , (λ, g) → λ is a momentum map and the corresponding right H -action is given by (λ, g) · h := (Ad∗h λ, gh) (following the notation of the previous § we have G = h). Conversely, Proposition 1.2 ([14], Proposition 2.1). A map ρ ∈ C ∞ (U, ∧2 g) is a coboundary classical dynamical r -matrix if and only if π = πlin +
∂ − → −−→ ∧ h i + ρ(λ) i ∂λ
defines a g-quasi-Poisson structure on U × G. Proof. The proof given in [14] is for the case when Z = 0, but it admits a straightforward generalization. Following Ping Xu ([14]), by a compatible quantization of πr we mean a quantization ∗ which is such that for any u, v ∈ Oh∗ and any f ∈ OG , u ∗ v = u ∗ P BW v, f ∗ u = f u and u ∗ f =
k k≥0
k!
i 1 ,...,i k
∂ku − → − → h i1 · · · h ik · f . i i 1 k ∂λ · · · ∂λ
(4)
Proposition 1.3. There is a bijective correspondence between compatible quantizations of πr and dynamical twist quantizations of r . Proof. Let ∗ be a compatible quantization of πr . Since ∗ is G-invariant then for all f, g ∈ OG one has −−→ ( f ∗ g)(λ) = J (λ)( f, g) (λ ∈ h∗ ) with J : U → ⊗2 U g[[]]. Moreover Lemma 1.4. ∗ is strongly h-invariant3 . 3 In particular ∗ is H -invariant. It was not noticed in [14], where the definition of compatible star-products includes this H -invariance property. The lemma claims that it comes for free (like in the classical situation).
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Proof of the lemma. Let f = gu (g ∈ OG and u ∈ Oh∗ ) on U × G. Then for any h ∈ h one has h ∗ f − f ∗ h = h ∗ (gu) − (gu) ∗ h = h ∗ (g ∗ u) − (g ∗ u) ∗ h = (h ∗ g) ∗ u − g ∗ (u ∗ h) ( acts trivialy) − → = (g ∗ h + ( h · g)) ∗ u − g ∗ (u ∗ h) = g ∗ ([h, u]∗ ) + (χh · g) ∗ u = g([h, u]∗ P BW ) + (χh · g)u = (g(χh · u) + (χh · g)u) = (χh · f ). Hence for any f ∈ O M , h ∗ f − f ∗ h = (χh · f ).
Therefore using [14, Proposition 3.2] one obtains that J is H -equivariant. The following lemma ends the first part of the proof: Lemma 1.5. J satisfies the dynamical twist equation. → Proof of the lemma. Let us define L : g x → − x and R : g x → ← x−, and denote ⊗n (n) by m : O M → O M ; f 1 ⊗ · · · ⊗ f n → f 1 · · · f n the standard n-fold product of functions. A computation in [14] emphases the fact that for all f, g, h ∈ OG , one has4 −−−−−−−−−−−−−−−−−−−−−→ m ∗ ◦ (m ∗ ⊗ id)( f ⊗ g ⊗ h) = J 12,3 (λ) ∗ P BW J 1,2 (λ + h (3) )( f ⊗ g ⊗ h) and
−−−−−−−−−−−−−−−→ m ∗ ◦ (id ⊗ m ∗ )( f ⊗ g ⊗ h) = J 1,23 (λ) ∗ P BW J 2,3 (λ)( f ⊗ g ⊗ h) .
)( f ⊗ g ⊗ h) is equal to Therefore, m ∗ ◦ (id ⊗ m ∗ ) ◦ R⊗3 ( )( f ⊗ g ⊗ h ) m (3) L⊗3 (J 1,23 (λ) ∗ P BW J 2,3 (λ))R⊗3 ( )L⊗3 (J 1,23 (λ) ∗ P BW J 2,3 (λ))( f ⊗ g ⊗ h) = m (3) R⊗3 ( ←−−−−−− = S ⊗3 (−1 ) L⊗3 (J 1,23 (λ) ∗ P BW J 2,3 (λ))( f ⊗ g ⊗ h) −−→ = −1 L⊗3 (J 1,23 (λ) ∗ P BW J 2,3 (λ))( f ⊗ g ⊗ h) −−−−−−−−−−−−−−−−−−−→ = −1 J 1,23 (λ) ∗ P BW J 2,3 (λ)( f ⊗ g ⊗ h) , where the equality before the last one follows from the invariance of . This ends the proof of the lemma. Conversely, let J = α f α Aα ⊗ Bα be a dynamical twist quantization of r ( f α ∈ OU [[]] and Aα , Bα ∈ U g). Following [14] we define a G-invariant product ∗ on O M [[]] by g1 ∗ g2 :=
k ∂ k g1 − → − →− → → − f α ∗ P BW ( Aα · i ) ∗ P BW ( Bα h i1 . . . h ik · g2 ) . i 1 k k! ∂λ · · · ∂λ
k≥0,α
i 1 ,...,i k
One can check by direct computations that h-equivariance of J implies strong h-invari ance of ∗ , and that the dynamical twist equation implies Eq. (3). 4 The reader must pay attention to the following important remark: for any P ∈ ⊗n U g we denote by − → ← − P (resp. P ) the corresponding left (resp. right) invariant multidifferential operator, while L⊗n (P) (resp. − → ⊗n R (P)) is an element in ⊗n Diff(G)G le f t (resp. ⊗n Diff(G)G right ). Namely, P = m (n) ◦ L⊗n (P) and ← − P = m (n) ◦ R⊗n (P) .
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Remark 1.6. Since Oh∗ = S(h) is generated as a vector space by h n , h ∈ h and n ∈ N, then one can rewrite condition (4) as h n ∗ f =
n
− → k Cnk ( h k · f )h n−k .
k=0
We saw in Lemma 1.4 that a compatible quantization always satisfies the strongly h-invariance condition. In what follows we show that this condition is actually sufficient for the existence of a compatible quantization. 1.4. A sufficient condition for the existence of a compatible quantization. Proposition 1.7. Assume that we are given a strongly h-invariant quantization ∗ of πr on M. Then there exists a gauge equivalent compatible quantization ∗ of πr . Therefore there exists a dynamical twist quantization J of r . Proof. First observe that h ∗ h − h ∗ h = [h, h ]h = [h, h ]h . Therefore we have an algebra morphism a : U (h) −→ (O M , ∗) . Then define the algebra morphism Q : Oh∗ ×G = S(h) ⊗ OG −→ O M as follows: Q( f u) = f ∗ a(sym(u)) (u ∈ S(h), f ∈ OG ) , where sym : S(h)[[]] −→ U (h) is the isomorphism sending h n to h n for any h ∈ h. Thus Q(h n ⊗ f ) = f ∗ h ∗ · · · ∗ h , and since ∗ can be expressed as a series m 0 + O() n times
of bidifferential operators on M then Q can be expressed as a series id + O() of differential operators on M. Moreover it is obviously g-invariant (since ∗ is), consequently we have a new quantization ∗ of πr , gauge equivalent to ∗, defined as follows: for any f, g ∈ O M , f ∗ g = Q −1 (Q( f ) ∗ Q(g)) . Let us now check that ∗ satisfies all Xu’s properties for compatible quantizations. − For any u, v ∈ S(h), u ∗ v = Q −1 a(sym(u)) ∗ a(sym(v) = Q −1 a(sym(u)sym(v)) = Q −1 a(sym(u ∗ P BW v)) = u ∗ P BW v. − Let u ∈ S(h) and f ∈ OG , then f ∗ u = Q −1 f ∗ a(sym(u)) = f u. Let us now compute u ∗ f ; we can assume that u = h n , h ∈ h, and then · · ∗ h ∗ f u ∗ f = Q −1 a(sym(u)) ∗ f = Q −1 h ∗ · n times
= Q −1
n k=0
=
n k=0
− → Cnk k ( h k · f ) ∗ h ∗ · · · ∗ h
− → k Cnk ( h k · f )h n−k .
n−k times
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− Since ∗ is a H -invariant star-product, then Q is a H -invariant gauge equivalence. Therefore ∗ is also H -invariant. The proposition is proved.
Remark 1.8. The gauge transformation Q constructed above obviously satisfies Q(h) = h for any h ∈ h. 2. Quantization of Symplectic Dynamical r-Matrices In this section we prove Theorem 0.1. We start by recalling Fedosov’s construction of star-products on a symplectic manifold (for more details we refer to [8, 9]). 2.1. Fedosov’s star-products. Let (M, ω) a symplectic manifold and denote by π = ω−1 the corresponding Poisson bivector. Then its tangent bundle T M inherits a Poisson structure π˜ expressed locally as π˜ = π i j (x)
∂ ∂ ∧ , i ∂y ∂y j
where y i ’s are coordinates in the fibers. This Poisson structure is regular and constant on the symplectic leaves which are the fibers Tx M of the bundle. Therefore it is quantized by the series of fiberwize bidifferential operators exp (π˜ ). It defines an associative ˆ ∗ M)[[]] that naturally extends to Ω ∗ (M, W ). The product ◦ on sections of W = S(T ∗ center of (Ω (M, W ), ◦) consists of forms that are constant in the fibers, i.e. lying in Ω ∗ (M)[[]]. By assigning the degree 2k +l to sections of k S m (T ∗ M) there is a natural decreasing filtration W = W0 ⊃ W1 ⊃ · · · ⊃ Wi ⊃ Wi+1 ⊃ · · · ⊃ O M . Now fix (once and for all) a torsion free connection ∇ on M with Christoffel’s symbols ikj . One can assume without loss of generality that it is symplectic (see [9, Sect. 2.5]), which means that ω is parallel w.r.t. ∇. Then consider ∂ : Ω ∗ (M, W ) → Ω ∗+1 (M, W ) its induced covariant derivative. In Darboux local coordinates we have ∂=d+
1 [ , −]◦ ,
where = − 21 i jk y i y j d x k is a local 1-form with values in W ( i jk = ωil ljk ). One has 1 ∂ 2 = − [R, −]◦ , where R = 41 Ri jkl y i y j d x k ∧ d x l , and Ri jkl = ωim R mjkl is the curvature tensor of the symplectic connection ∇.
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Let us consider more general derivations of (Ω ∗ (M, W ), ◦) of the form D =∂ −δ+
1 [r, −]◦ ,
where r ∈ Ω 1 (M, W ) and δ = 1 [ωi j y i d x j , −]◦ . A simple calculation yields that 1 D 2 = − [Ω, −]◦ , where Ω = ω + R + δr − ∂r − 1 r 2 ∈ Ω 2 (M, W ) is called the Weyl curvature of D. In particular D is flat (i.e. D 2 = 0) if and only if Ω ∈ Ω 2 (M)[[]] (i.e. is a central 2-form), and in this case the Bianchi identity for ∇ implies that dΩ = DΩ = 0. In computing D 2 one sees that δ : Ω ∗ (M, Wk ) → Ω ∗+1 (M, Wk−1 ) has square zero and that the torsion freeness of ∇ implies δ∂ + ∂δ = 0. Then we define a homotopy operator κ : Ω ∗ (M, Wk ) → Ω ∗−1 (M, Wk+1 ) on monomials a ∈ Ω p (M, S q (T ∗ M)): if p + q = 0 then κ(a) =
1 y i ∂d x i a p+q
and otherwise κ(a) = 0. One easily check that κ 2 = 0 and δκ + κδ = id − σ where σ : ∗ (M, W ) → C ∞ (M)[[]], a → a|d x i =y i =0 is the projection onto 0-forms constant in the fibers. Theorem 2.1 (Fedosov). For any closed 2-form Ω = ω + O() ∈ Z 2 (M)[[]] there exists a unique r ∈ Ω 1 (M, W3 ) such that κ(r ) = 0 and D =∂ −δ+
1 [r, −]◦
has Weyl curvature Ω and is therefore flat. Proof. First observe that Ω = ω + R + δr − ∂r − 1 r 2 with κ(r ) = 0 if and only if 1 r = κ(Ω − ω − R + ∂r + r 2 ) .
(5)
Since ∂ preserves the filtration and κ raises its degree by 1 then κ(Ω − ω − R) ∈ Ω 1 (M, W3 ) and the sequence (rn )n≥3 defined by the iteration formula 1 rn+1 = r0 + κ(∂rn + rn2 )
(6)
with r3 = κ(Ω − ω − R) converges to a unique element r ∈ Ω 1 (M, W3 ) which is a solution of Eq. (5). We have proved the existence. Conversely, for any solution r ∈ Ω 1 (M, W3 ) of (5) define rk = r mod Wk+1 . Then r3 = κ(Ω − ω − R) and the sequence (rn )n≥3 satisfies (6). Unicity is proved. Such a flat derivation D is called a Fedosov connection (of ∇-type). The previous theorem claims that they are in bijection with series Ω of closed two forms starting with ω.
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Theorem 2.2 (Fedosov). If D is a Fedosov connection then for any f 0 ∈ C ∞ (M)[[]] there exists a unique D-closed section f ∈ (M, W ) such that σ ( f ) = f 0 . Hence σ establishes an isomorphism between Z 0D (W ) and C ∞ (M)[[]]. Proof. Let f 0 ∈ C ∞ (M)[[]]. One has D( f ) = 0 with σ ( f ) = f 0 if and only if f = f 0 + κ(∂ f +
1 [r, f ]◦ ) .
(7)
Like in the proof of Theorem 2.1 we can solve (uniquely) this equation with the help of an iteration formula: f n+1 = f 0 + κ(∂ f n + 1 [r, f n ]◦ ). Then f ∗ g = σ (σ −1 ( f ) ◦ σ −1 (g)) defines a star-product on C ∞ (M)[[]] that quantizes (M, ω). A star-product constructed this way is called a Fedosov star-product (of ∇-type) and is uniquely determined, once ∇ is fixed, by its characteristic 2-form ω :=
1 (Ω − ω) ∈ Z 2 (M)[[]] .
Moreover one can easily prove the following (i) (i) Lemma 2.3 ([4]). Let ω = k>0 k−1 ωk ∈ Z 2 (M)[[]] (i = 1, 2) and denote by (i) (1) (2) ∗i the Fedosov star-product with characteristic two-form ω . If ω = ω + O(k ) then (1)
(2)
∗(1) − ∗(2) = k+1 π # (ωk − ωk ) + O(k+2 ) . 2.2. Fedosov’s construction in the presence of symmetries. Let (M, ω) be a symplectic manifold. Let us prove two results on the compatibility of Fedosov’s construction with group actions and hamiltonian vector fields. Proposition 2.4 (Fedosov). Assume that a group G acts on (M, ω) by symplectomorphisms and is equipped with a G-invariant torsion free connection. Then for any ω ∈ Z 2 (M)G [[]] the corresponding Fedosov star-product is G-invariant. Proof. First observe that starting from a G-invariant torsion free connection ∇ one can assume without loss of generality that it is symplectic (see the proof of Proposition 5.2.2 in [9], where all expressions become obviously G-invariant). Then, being a symplectomorphism of (M, ω), any elements g ∈ G act via its differential dg on (T M, π˜ ) as a Poisson automorphism linear in the fibers. Then its dual map g ∗ : T ∗ M → T ∗ M defined by < g ∗ ξ, x >=< ξ, dg(x) > extends to W as an automorphism. Finally, we need to prove that g ∗ preserves the Fedosov connection with Weyl curvature Ω = ω + ω. On one hand the automorphism g ∗ commutes with ∂ (since ∇ is assumed to be G-invariant) and so g ∗ R = R. On the other hand g ∗ also commutes with δ and κ, thus if r is a solution of Eq. (5) with κ(r ) = 0 then so is g ∗r . By uniqueness g ∗r = r . We are done. Proposition 2.5 (Fedosov). Let H ∈ O M be such that χ = {H, ·} preserves a torsion free connection on M. Then for any ω ∈ Z 2 (M)[[]] such that ιχ ω = 0 the corresponding Fedosov star-product ∗ satisfies H ∗ f − f ∗ H = (χ · f ) for any f ∈ OM .
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Proof. First observe that L χ ω = (dιχ + ιχ d)ω = 0. Therefore, the infinitesimal version of the previous proof ensures us that the Fedosov connection D with Weyl curvature Ω = ω + ω is L X -equivariant. Hence in local Darboux coordinates it is written D = d + 1 [γ , −]◦ with L χ γ = 0, and Ω = −dγ − 1 γ 2 . Let us compute D(H − ιχ γ ) = d H − dιχ γ − = ιχ (Ω + dγ +
1 1 [γ , ιχ γ ]◦ = ιχ Ω + ιχ dγ + [ιχ γ , γ ]◦
1 2 γ ) = 0.
Since σ (H − ιχ γ ) = σ (H ) = H , it means that σ −1 (H ) = H − ιχ γ in local Darboux coordinates. Consequently, for any f ∈ O M [[]], H ∗ f − f ∗ H = σ [H − ιχ γ , σ −1 ( f )]◦ = σ − [ιχ γ , σ −1 ( f )]◦ = σ − ιχ (D − d)σ −1 ( f ) = σ ιχ dσ −1 ( f ) = σ L χ σ −1 ( f ) − dιχ σ −1 ( f ) = L χ ( f ) = (χ · f ). The proposition is proved.
2.3. Proof of Theorem 0.1. Let r : U → ∧2 g be a symplectic dynamical r -matrix. A → basis B of vector fields on M = U × G is given by B = (. . . , ∂λi , . . . , . . . , − ei , . . . ), i ∗ where (λ )i is a base of h and (ei )i is a base of g. Then one defines a torsion free connection ∇ on M as ∇b X =
1 [b, X ] 2
b ∈ B, X ∈ X(M) .
Remark that [χh , b] ∈ spanR B for any b ∈ B. Therefore it follows immediately from the Jacobi identity that ∇ is h-invariant: for all X, Y ∈ X(M) and h ∈ h, [χh , ∇ X Y ] = ∇[χh ,X ] Y + ∇ X [χh , Y ] . Thus from Proposition 2.5 the Fedosov star-product ∗ with the trivial characteristic 2-form is strongly h-invariant. Moreover ∇ is obviously G-invariant, hence Proposition 2.4 implies that ∗ is also G-invariant. Finally, we apply Proposition 1.7 to construct a compatible quantization of πr . We are done.
3. Classification Let r : h∗ ⊃ U → ∧2 g be a dynamical r -matrix. Denote by πr the corresponding H -invariant g-quasi-Poisson structure (1) on M = U × G (together with Z ∈ (∧3 g)g).
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3.1. Strongly invariant equivalences and obstructions. By a strongly invariant equivalence between two strongly h-invariant quantizations of πr we mean a H -invariant equivalence Q (namely, Q = id + O() ∈ Diff(M)G×H [[]]) satisfying Q(h) = h for any h ∈ h ⊂ O M . We will now develop an analogue of the usual obstruction theory in this context. Let us denote by b the Hochschild coboundary operator for cochains on the (commutative) algebra O M . We start with the following result which is a variant of a standard one. Proposition 3.1. Suppose that ∗1 and ∗2 are two strongly invariant quantizations of πr : f ∗i g = k Cki ( f, g) (i = 1, 2) . k≥0
Assume that ∗1 and ∗2 coincide up to order n, i.e. Ck1 = Ck2 if k ≤ n. Then (1) there exists B ∈ (M, ∧2 T M)G×H and E ∈ Diff(M)G×H such that B(h, −) = 0 and E(h) = 0 if h ∈ h ⊂ Oh∗ , [πr , B] = 0, and satisfying 1 2 (Cn+1 − Cn+1 )( f, g) = B( f, g) + (bE)( f, g)
( f, g ∈ O M ) ;
(2) there exists P ∈ Diff(M)G×H such that C1 = πr + b P and P(h) = 0 for h ∈ h; (3) if B = [πr , X ], X ∈ X(M)G×H such that X (h) = 0, then the strongly invariant equivalence Q = 1 + n X + n+1 (E + [X, P]) transforms ∗1 into another strongly invariant star-product which coincides with ∗2 up to order n + 1. Proof. We use a similar argument as in [13, 3, 4]. 1 − C 2 ) = 0. Hence we may write (1) It is well-known that b(Cn+1 n+1 1 2 Cn+1 − Cn+1 = B + b(E 0 ), 1 − C 2 and E ∈ where B ∈ (M, ∧2 T M)G×H is the skew-symmetric part of Cn+1 0 n+1 Diff(M). Moreover, one knows (see e.g. [4]) that [πr , B] = 0, and it follows directly from the strong h-invariance property for ∗1 and ∗2 that B(h, −) = 0 if h ∈ h. Since U × G admits a G × H -invariant connection and b(E 0 ) is obviously G × H invariant, then according to [3, Prop. 2.1] we can assume that E 0 is G × H -invariant. In particular E 0 is G-invariant and hence E 0 ( f ), f ∈ OU , is a function on U only. Thus we can define a H -invariant vector field v on U as follows: < dh, v >= E 0 (h) for any h ∈ h ⊂ OU . Now E := E 0 − v satisfies all the required properties and b(E) = b(E 0 ) − b(v) = b(E 0 ). (2) It is standard that C1 = πr + b(P0 ). By repeating a similar argument as in (1) we can prove that P0 can be chosen so that P0 = P ∈ Diff(M)G×H and satisfies P(h) = 0 for any h ∈ h. The third statement (3) follows from an easy (and standard) calculation.
This proposition means that obstructions to strongly invariant equivalences are in the second cohomology group of the subcomplex C ∞ (U, ∧∗ g)h in the H -invariant quasiPoisson cochain complex of (M, πr , Z ). On such cochains c the (quasi-)Poisson ∂c coboundary operator [πr , −] reduces to dr (c) := h i ∧ ∂λ i + [r, c]. Definition 3.2. The cohomology Hr∗ (U, g) of this cochain complex is called the dynamical r -matrix cohomology associated to r : U → ∧2 g.
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3.2. Classification of strongly invariant star-products. Now assume that the quasi-Poisson manifold (M, πr ) is actually symplectic and denote by ωr the symplectic form; it is G × H -invariant and satisfies ιχh ω = 0 for any h ∈ h ⊂ Oh∗ . The G × H -invariant isomorphism πr# : T ∗ M −→T ˜ M extends to a G × H -invariant isomorphism of cochain complexes ˜ ∧∗ T M), [πr , −]) (Ω ∗ (M), d)−→( (M, which restricts to an isomorphism ∞ ˜ (U, ∧∗ g)h, dr ) , (Ωh∗ (M)G , d)−→(C
where Ωh∗ (M) := {α ∈ Ω ∗ (M) H |ιχh α = 0 , ∀h ∈ h}. Let us fix once and for all a symplectic G × H -invariant connection ∇ on M (we know it exists) and remember from the previous section that for any ω ∈ Ω 2 (M)G×H [[]] such that dω = 0 there exists a (unique) G × H -invariant Fedosov star-product ∗ with characteristic 2-form ω. Moreover, if ω ∈ Ωh2 (M)[[]] then Proposition 2.5 implies that ∗ is strongly h-invariant. Therefore we can associate a strongly invariant quantization of πr (which is actually a Fedosov star-product) to any closed two form ω ∈ Ωh2 (M)G [[]]. In the rest of the section, all Fedosov star-products are assumed to be of ∇-type and G-invariant (since they quantize the g-quasi-Poisson structure πr ). Theorem 3.3. Two strongly invariant Fedosov star-products are equivalent by a strongly invariant equivalence if and only if their characteristic 2-forms lie in the same cohomology class in HhG,2 (M)[[]]. Proof. Let ∗0 and ∗1 be two strongly invariant Fedosov star-products with respective (0) (1) characteristic 2-form ω and ω . First assume that ω(0) = ω(1) + dα for some α = k k α (k) ∈ Ωh1 (M)G [[]], and (0)
define ω(t) = ω + tdα. Let Dt = ∂ − δ + 1 [r (t), −] be the Fedosov differential with Weyl curvature Ω(t) = ω + ω(t). Let H (t) ∈ (M, W ) be the solution of the equation Dt H (t) = −α + r˙ (t) with σ (H (t)) = 0. Then H (t) is G × H -invariant since Dt , α and r (t) are. According to [9, Theorem 5.5.3] the solution of the Heisenberg equation d F(t) + [H (t), F(t)]◦ = 0 , dt
F(0) = f
establishes an isomorphism Z 0D0 (W ) → Z 0D1 (W ), f → F(1) and then the corresponding series of differential operators Q : (O M [[]], ∗0 ) → (O M [[]], ∗1 ) is obviously G × H -invariant. Remember from the proof of Proposition 2.5 that in local Darboux coordinates the Fedosov differential is written Dt = d + 1 [γ (t), −]◦ and σt−1 (h) = h − ιχh γ (t) if
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h ∈ h. Now remark that γ˙ (t) = r˙ (t) and that ιχh r (t) is independent of t. Hence σt−1 (h) does not depend on t and thus Q(h) = h. Conversely, assume that ∗0 and ∗1 are related by a strongly invariant equivalence (0) (1) (i) (i) with [ω ] = [ω ] in HhG,2 (M)[[]]. Write ω = k>0 k−1 ωk (i = 0, 1) and (0)
(1)
denote by l the lowest integer for which [ωl ] = [ωl ] in HhG,2 (M). Let us then define ω(2) =
0
k−1 ωk(0) +
k−1 ωk(1)
k≥l (2)
and ∗2 the strongly invariant Fedosov star-product with characteristic 2-form ω . Since [ω(2) ] = [ω(1) ] ∈ HhG,2 (M)[[]] then ∗2 is equivalent to ∗1 , and hence to ∗0 , by a 2 − C0 = strongly invariant equivalence. Then we deduce from Lemma 2.3 that Cl+1 l+1 (1) (0) i # πr (ωl − ωl ), where Ck ’s (i = 0, 2) are the cochains defining ∗i . Thus it follows (1) (0) from Proposition 3.1 that ωl − ωl is exact and we obtain a contradiction. Theorem 3.4. Any strongly invariant quantization of πr is equivalent to a strongly invariant Fedosov star-product by a strongly invariant equivalence. Therefore the set of strongly invariant star-products quantizing πr up to strongly invariant equivalences is an affine space modeled on HhG,2 (M)[[]] = Hr2 (U, g). Proof. We follow [4, Prop. 4.1]. Let ∗ be an arbitrary strongly invariant quantization of πr . Denote by ∗0 the strongly invariant Fedosov star-product with the trivial characteristic 2-form, which coincides with ∗ up to order 0. Moreover, the skew-symmetric part of the first order term in ∗ − ∗0 vanishes, hence it follows from Proposition 3.1 that there exists a strongly invariant equivalence Q (0) = 1 + Q 0 that transforms ∗ into a new strongly invariant quantization ∗(0) which coincides with ∗0 up to order 1. Now the skew-symmetric part of the second order term in ∗(0) − ∗0 yields a closed form ω1 ∈ Ωh2 (M)G . Denote by ∗1 the strongly invariant Fedosov star-product with characteristic 2-form ω1 . Lemma 2.3 tells us that the skew-symmetric part of the second order term in ∗(0) −∗1 vanishes, hence it follows from Proposition 3.1 that there exists a strongly invariant equivalence Q (1) = 1 + 2 Q 1 that transforms ∗(0) into a new strongly invariant quantization ∗(1) which coincides with ∗1 up to order 2. Repeating this procedure we get a sequence of strongly invariant equivalences Q (k) = 1 + k+1 Q k (k ≥ 0) and a sequence of closed forms ωk (k > 0) such that the strongly invariant quantization ∗(k) obtained from ∗ by applying successively Q (0) , . . . , Q (k) coincides up to order k + 1 with the strongly invariant Fedosov star-product ∗k with characteristic 2-form ω1 + · · · + k−1 ωk . Finally, the strongly invariant equivalence Q := · · · Q (2) Q (1) Q (0) transform ∗ into the strongly invariant Fedosov star-product with characteristic 2-form ω = k>0 k−1 ωk .
3.3. Classification of dynamical twist quantizations. Let T be a gauge equivalence of dynamical twist quantizations J1 and J2 of r . One can view T as an element in Diff(U × G)G×H [[]] such that T (u) = u for any u ∈ Oh∗ . Moreover if we denote by ∗i the
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compatible quantization of πr corresponding to Ji (i = 1, 2) then it follows from an easy calculation that T ( f ∗1 g) = T ( f ) ∗2 T (g) . Conversely, any G × H -invariant gauge equivalence T from ∗1 to ∗2 which is such that T (u) = u for any u ∈ Oh∗ , that we will call from now a compatible equivalence, obviously gives rise to a gauge equivalence of the dynamical twist quantizations J1 and J2 . Therefore, the set of dynamical twist quantization of r up to gauge equivalences is in bijection with the set of compatible quantizations of πr up to compatible equivalences. Remember from Proposition 1.7 and Remark 1.8 that any strongly invariant quantization is equivalent to a compatible one by a strongly invariant equivalence. Moreover the PBW star-product has the following nice property: for any h ∈ h, h ∗ P BW n = h n . Hence any strongly invariant equivalence between two compatible quantizations is actually a compatible equivalence. Consequently: Proposition 3.5. There is a bijection {strongly invariant quantizations of πr } {compatible quantizations of πr } ←→ strongly invariant equivalences compatible equivalences End of the Proof of Theorem 0.2. Assume that the dynamical r -matrix is symplectic. Then Theorem 0.2 follows from Proposition 3.5 and Theorem 3.4. 4. The Quantum Composition Formula In this section we assume that h = t ⊕ m is a nondegenerate reductive splitting and we denote by rtm : t∗ ⊃ V → ∧2 h the corresponding symplectic dynamical r -matrix. Let p : h → t be the t-invariant projection along m. For any function f on h∗ with values in a h-module L we write f |t∗ for the function f ◦ p ∗ on t∗ with values in L viewed as a t-module; in particular if f is h-invariant then f |t∗ is t-invariant. 4.1. The classical composition formula (Proof of Proposition 0.3). Let ρ : U → ∧2 g be a dynamical r -matrix with Z ∈ (∧3 g)g. Then π := πrtm + πρ defines a g-quasi-Poisson structure (with the same Z ) on the manifold X = V × H × U × G which is − H -invariant with respect to left multiplication on H , − H -invariant with respect to the right action on U × G, − T -invariant with respect to the right action on V × H . The right diagonal H -action, given by (τ, x, λ, y) · q = (τ, q −1 x, Adq∗ λ, yq), actually comes from a momentum map: µ : X −→ h∗ (τ, x, λ, y) −→ λ − Ad∗x −1 ( p ∗ τ ) . Consequently we can apply the reduction with respect to µ. The right H -invariant smooth map ψ : X = V × H × U × G −→ M := U ∩ V × G (τ, x, λ, y) −→ (τ, yx)
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restricts to a diffeomorphism µ−1 (0)/H → M with inverse given by (τ, y) −→ (τ, 1, p ∗ τ, y) (τ ∈ U ∩ V , y ∈ G) . Remark 4.1. From an algebraic viewpoint, we have an injective map of commutative h algebras ψ ∗ : O M → O X with values in O X = O X/H and such that, composed with the h h h projection O X → O X /(O X ∩ < im(µ∗ ) >) = Oµ−1 (0)/H , it becomes an isomorphism. Since ψ is obviously left G-invariant then it remains to show that the induced g-quasi-Poisson structure on M is πρ|t ∗ +rtm . Let t, t ∈ t ⊂ Ot∗ and f, g ∈ OG . First of all we have {ψ ∗ t, ψ ∗ t } X = {t, t } X = [t, t ] = ψ ∗ [t, t ] , hence {t, t } M = [t, t ]. Then − → − → − → {ψ ∗ t, ψ ∗ f } X = {t, f (yx)} X = t H · ( f (yx)) = ( t · f )(yx) = ψ ∗ ( t · f ) . − → − → The third equality follows from the left H -invariance of t H . Thus {t, f } M = t · f . Finally −−−→ −−→ {ψ ∗ f, ψ ∗ g} X (τ, x, λ, y) = rtm(τ ) H · ( f (yx), g(yx)) + ρ(λ)G · ( f (yx), g(yx)) −−−−−→ −−−→ = rtm(τ ) · ( f, g) (yx) + ρ(Ad∗x λ) · ( f, g) (yx) . Therefore, when restricting to µ−1 (0) one obtains −−−→ −−−−→ {ψ ∗ f, ψ ∗ g} X (τ, x, Ad∗x −1 ( p ∗ τ ), y) = rtm(τ ) · ( f, g) (yx) + ρ( p ∗ τ ) · ( f, g) (yx) −−−−−−→ = ψ ∗ (rtm + ρ|t∗ ) · ( f, g) . −−−−−−→ Therefore { f, g} M = (rtm + ρ|t∗ ) · ( f, g). This ends the Proof of Proposition 0.3.
4.2. Quantization of the momentum map µ. Let us first consider (V × H, πrtm ). There is a momentum map ν : V × H −→ h∗ (τ, x) −→ −Adx −1 ( p ∗ τ ) with corresponding right H -action on V × H given by (τ, x) · q = (τ, q −1 x). Like in Subsect. 2.3 one has a T -invariant and H -invariant torsion free connexion on V × H , therefore from Proposition 2.5 the corresponding Fedosov star-product ∗ is both strongly h-invariant and strongly t-invariant5 . Then Proposition 1.7 tells us that there exists a strongly t-invariant (and H -invariant) equivalence Q such that ∗ := ∗(Q) is a compatible quantization of πrtm . Consequently we can define the following algebra morphism: N := Q −1 ◦ U (ν ∗ ) ◦ sym : (Oh∗ [[]], ∗ P BW ) → (OV ×H [[]], ∗ ) . 5 Recall that we also have a momentum map V × H → t∗ ; (τ, x) → τ with corresponding right T -action given by (τ, x) · b = (Ad∗b (τ ), xb).
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It is obviously a quantization of the Poisson map ν and, moreover, for any h ∈ h and any f ∈ OV ×H one has [N(h), f ]∗ = Q −1 [ν ∗ h, Q( f )]∗ = Q −1 ({ν ∗ h, Q( f )}) = {ν ∗ h, f } . In other words, N is a quantum momentum map quantizing ν. Let us now assume that we know a dynamical twist quantization J (λ) : U → ⊗2 U g[[]] of ρ(λ) (with some associator ) and denote by ∗ J the corresponding compatible quantization of πρ on U × G. Together with ∗ it induces a quantization ∗J of πrtm + πρ on X (with the same ). Remark 4.2. Actually ∗J is the compatible quantization corresponding to the dynamical twist quantization J(τ, λ) := Jtm(τ )J (λ) : (t ⊕ h)∗ ⊃ V × U −→ ⊗2 U (h ⊕ g)[[]] of the dynamical r -matrix r(τ, λ) := rtm(τ ) + ρ(λ) : V × U −→ ∧2 (h ⊕ g). Here Jtm is the dynamical twist quantizing rtm. For any f ∈ Oh∗ we define M( f ) := (N ⊗ inc) ◦ ∆( f ) ∈ (OV ×H ⊗ OU ×G )[[]] = O X [[]]. Here inc : Oh∗ → OU ×G is the natural inclusion and ∆ : Oh∗ → Oh∗ ⊗Oh∗ = Oh∗ ×h∗ is defined by ∆( f )(λ1 , λ2 ) = f (λ1 + λ2 ). Proposition 4.3. The algebra morphism M : (Oh[[]], ∗ P BW ) −→ (O X [[]], ∗J ) is a quantum momentum map quantizing µ. 4.3. Quantization of the composition formula (Proof of Theorem 0.4). Let us assume that J is a dynamical twist quantization of ρ and keep the notations of the previous subsection. Denote by I the right ideal generated by im(M) in (O X [[]], ∗J ) and consider the h
h
reduced algebra A := O X [[]]/O X [[]] ∩ I. Let = ψ ∗ + O() be the composition h h of ψ ∗ : O M [[]] → O X [[]] with the projection O X [[]] → A ∼ = Oµ−1 (0)/H [[]]. It ∗ is obviously bijective and G-invariant (since ψ is), therefore it defines a quantization ∗˜ of the quasi-Poisson structure πrtm +ρ|t ∗ . We end the proof of Theorem 0.4 using the following proposition: Proposition 4.4. ∗˜ is a compatibe quantization. Proof. First of all for any u, v ∈ Ot∗ one has (ψ ∗ u) ∗J (ψ ∗ v) = u ∗J v = u ∗ P BW v = ψ ∗ (u ∗ P BW v) . Consequently u ∗˜ v = −1 ((u) ·A (v)) = u ∗ P BW v. Then let u ∈ Ot∗ and f ∈ OG . On one hand (ψ ∗ f ) ∗J ψ ∗ u = ( f (yx)) ∗J u = f (yx)u = ψ ∗ ( f u)
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and thus f ∗˜ u = f u. On the other hand for u = t n (t ∈ t) one has ψ ∗ (t n ) ∗J (ψ ∗ f ) = (t n ) ∗J ( f (yx)) =
n
− → k Cnk ( t H )k · ( f (yx)) t n−k
k=0
=
n
k
− → Cnk ( t k
k=0
Therefore t n ∗˜ f =
n
k=0
· f )(yx)t
n−k
=ψ
∗
n
k
− → Cnk ( t k
· f )t
n−k
.
k=0
→k k C k (− n t
· f )t n−k . The proposition is proved.
Acknowledgements. We thank Benjamin Enriquez for pointing out to us the problem of the quantum composition formula and for a very stimulating series of discussions during his short visit in Geneva. A.A. acknowledges the support of the Swiss National Science Foundation. D.C. thanks the Section de Mathématiques de l’Université de Genève, the Institut des Hautes Études Scientifiques and the Max-Planck-Institut für Mathematik Bonn where parts of this work were done.
References 1. Alekseev, A., Kosmann-Schwarzbach, Y.: Manin pairs and momentum maps. J. Diff. Geom. 56(1), 133–165 (2000) 2. Alekseev, A., Kosmann-Schwarzbach, Y., Meinrenken, E.: Quasi-Poisson manifolds. Canad. J. Math. 54(1), 3–29 (2002) 3. Bertelson, M., Bielavsky, P., Gutt, S.: Parametrizing equivalence classes of invariant star-products. Lett. Math. Phys. 46, 339–345 (1998) 4. Bertelson, M., Cahen, M., Gutt, S.: Equivalence of star-products. Classical Quantum Gravity 14, A93– A107 (1997) 5. Drinfeld, V.: Quasi-Hopf algebras. Leningrad Math. J. 1, 1419–1457 (1990) 6. Enriquez, B., Etingof, P.: Quantization of Alekseev-Meinrenken dynamical r -matrices. Trans. Am. Math. Soc. (ser. 2) 210, 81–98 (2003) 7. Enriquez, B., Etingof, P.: Quantization of classical dynamical r -matrices with nonabelian base. Commun. Math. Phys. 254(3), 603–650 (2005) 8. Fedosov, B.: A simple geometric construction of deformation quantization. J. Diff. Geom. 40, 213–238 (1994) 9. Fedosov, B.: Deformation quantization and index theory. Berlin: Akademia Verlag, 1996 10. Fedosov, B.: Nonabelian reduction in deformation quantization. Lett. Math. Phys. 43, 137–154 (1998) 11. Fehér, L., Gábor, A., Pusztai, P.: On dynamical r -matrices obtained from Dirac reduction and their generalizations to affine Lie algebras. J. Phys. A34(36), 7335–7348 (2001) 12. Kravchenko, O.: Deformation quantization of symplectic fibrations. Compos. Math. 123(2), 131–265 (2000) 13. Xu, P.: Triangular dynamical r -matrices and quantization. Adv. Math. 166(1), 1–49 (2002) 14. Xu, P.: Quantum dynamical Yang-Baxter equation over a nonabelian base. Commun. Math. Phys. 226(3), 475–495 (2002) Communicated by L. Takhtajan
Commun. Math. Phys. 273, 137–159 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0244-0
Communications in
Mathematical Physics
Quantum Ergodicity for Graphs Related to Interval Maps G. Berkolaiko1 , J. P. Keating2 , U. Smilansky3 1 Department of Mathematics, Texas A&M University, College Station, TX 77843, USA 2 School of Mathematics, University of Bristol, Bristol BS8 1TW, UK. E-mail:
[email protected] 3 Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
Received: 27 June 2006 / Accepted: 14 September 2006 Published online: 25 April 2007 – © Springer-Verlag 2007
Abstract: We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakónski et al (J. Phys. A, 34, 9303-9317 (2001)). As observables we take the L 2 functions on the interval. The proof is based on the periodic orbit expansion of a majorant of the quantum variance. Specifically, given a one-dimensional, Lebesgue-measure-preserving map of an interval, we consider an increasingly refined sequence of partitions of the interval. To this sequence we associate a sequence of graphs, whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the Perron-Frobenius operator corresponding to the map. We show that, except possibly for subsequences of density 0, the eigenstates of the quantum graphs equidistribute in the limit of large graphs. For a smaller class of observables we also show that the Egorov property, a correspondence between classical and quantum evolution in the semiclassical limit, holds for the quantum graphs in question.
1. Introduction The quantum ergodicity theorem is one of the central results in quantum chaos. Essentially, it asserts that in systems in which the classical dynamics is ergodic, the probability measures associated with the squares of the moduli of the quantum eigenfunctions converge to the classical invariant measure as one approaches the semiclassical limit through almost all sequences of eigenstates (any exceptional subsequences have density zero). This was originally proved for flows [1–5], but it has since been extended to discrete dynamical systems (chaotic maps) of even dimension; see, for example [6–10] (for a very readable introduction to the subject, the reader should consult [11]). The methods of proof typically involve applying Egorov-type theorems, which relate the time evolution of quantum and classical observables in the semiclassical limit.
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Quantum graphs correspond to associating an operator with a graph. For example, this might be the discrete Laplacian acting at the vertices, or the one-dimensional Laplacian acting on functions defined on the edges of a (metric) graph, with matching conditions applied at the vertices. Such systems have recently been the subject of considerable interest [12]. In particular, quantum graphs have emerged as important toy models of quantum chaotic behaviour [13, 14]: if one considers sequences of graphs with increasing numbers of edges then, under certain conditions, the quantum eigenvalue statistics converge to those of random matrix theory [13–20]. However, relatively little attention has been paid to their eigenfunction statistics. For example, quantum ergodicity has not been proved in this context. Even though the classical (Markovian) dynamics on a fixed graph is mixing, the difficulty lies in dealing with sequences of graphs with increasing numbers of bonds. To date, the only examples that have been studied in this limit are the star graphs (in which the bonds are connected at a single central vertex). However, even though any given star graph is classically ergodic, the limit as the number of bonds tends to infinity is not quantum ergodic [21–23]. This is not altogether surprising because the star graphs do not satisfy the condition under which one expects the spectral statistics to coincide with those of random matrix theory (instead, their spectral statistics coincide with those of integrable systems perturbed by a singular scatterer [24, 25]). It turns out that the star graph eigenfunctions are strongly scarred by short periodic orbits (see also [26]). The problem of finding examples of sequences of quantum graphs that are quantum ergodic thus remains open. It is this problem that we address here. We start by discussing how the question of quantum ergodicity on general graphs can be related to the ergodic properties of the eigenvectors of an ensemble of unitary matrices. Each ensemble consists of matrices DS0 , where S0 is a fixed unitary matrix, determined by the corresponding graph, and D is a random diagonal unitary matrix. We then identify a particular sequence of graphs (or matrices S0 ) for which quantum ergodicity can be established. These are graphs obtained from a construction proposed by Pakónski et al [27] involving ergodic one-dimensional maps on an interval. We also prove the analogue of Egorov’s theorem for these graphs. To be explicit, given a one-dimensional, Lebesgue-measure-preserving map S : [0, 1] → [0, 1], we consider an increasingly refined sequence of partitions Mn of the interval [0, 1]. To this sequence we associate a sequence of graphs G n whose directed edges (bonds) correspond to elements of the partitions. The quantum evolution on G n is described by a unitary matrix Un such that the corresponding classical (Markov) dynamics of G n approximates the Perron-Frobenius operator associated with S. To a classical observable φ ∈ L 2 [0, 1] we associate a sequence of quantum observables On (φ) which are defined as operators corresponding to multiplication by the average value of φ on an element of the partition. We prove that there is a sequence of sets Jn ⊂ {1, . . . , |Mn |} such that |Jn | =1 n→∞ |Mn | lim
and for all sequences { jn }∞ n=1 , jn ∈ Jn ,
(n) (n) lim ψ jn , On (φ)ψ jn =
n→∞ (n)
1
φ(x)d x,
(1)
0
where ψ jn is the jnth eigenvector of the graph G n (compare to the corresponding statement for cat maps, [7, 11]). This is the analogue of “quantum ergodicity” for the graphs
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in question. It is equivalent to the decay of the quantum variance, 2 |M | 1 1 n (n) (n) →0 ψ − , O (φ)ψ φ(x)d x n j j |Mn | 0 j=1
in the limit n → ∞. This equivalence follows from a Chebyshev-type inequality and parallels the textbook proof of the statement “for uniformly bounded random variables, convergence in mean square is equivalent to convergence in probability”. If φ is Lipschitz continuous, we also prove the Egorov property, Un On (φ)Un−1 − On (φ ◦ S) = O |Mn |−1 , where Un is the quantum evolution operator corresponding to the graph G n . The existence of the Egorov property provides further justification for the use of the term “quantization” when referring to the sequence Un obtained from a map S. It should be noted that we do not explicitly use the Egorov property (EP) in the proof of quantum ergodicity (QE). Even though the more traditional route of deriving QE from EP is available to us, we feel that the proof in the present form is likely to be more adaptable to other families of quantum graphs. This paper is organized as follows. In Sect. 2 we review some of the main issues relating to the construction of quantum graphs. In Sect. 3 we introduce the construction of Pakónski et al [27] and proceed to discuss some of its properties. In particular, we prove a sufficient condition for a map to be quantizable in the fashion described by [27]. This sufficient condition, although rather restrictive, demonstrates that the class of quantizable maps is sufficiently rich to be interesting. In Sect. 4 we introduce the observables on the quantum graphs obtained from maps. Their quantum variance is analyzed in Sect. 5. In Sects. 6 and 7 we prove quantum ergodicity for these observables by estimating two different contributions to the variance, and in Sect. 8 we prove the Egorov property in this context. Finally, in Sect. 9 we discuss some of the issues arising in the proof of these theorems and the possibility of extending the proofs to larger classes of graphs. 2. Quantum Graphs A quantum graph can be defined in two different, but related, ways. In both constructions we start with a graph G = (V, B), where V is a finite set of vertices (or nodes), and B is the set of bonds (or edges). Each bond b has a non-zero length, denoted L b . The first way to define a quantum graph [14] is to identify each bond b with the interval [0, L b ] of the real line and thus define the L 2 -space of functions on the graph. Then one can consider the eigenproblem −
d2 u b (x) = λ2 u b (x). dx2
(2)
This setup has been studied by mathematicians since the 1980s [28–32] and was used in physical models prior to that [33–35]. To make the operator in (2) self-adjoint we need to impose matching conditions on the behavior of u at the vertices of the graph. One possibility is to impose Kirchhoff
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conditions:1 we require that u is continuous on the vertices, and that the probability current is conserved, i.e. d u b (v) = 0 dx
for all v ∈ V,
(3)
v∈b
where the sum is over all bonds that originate from the vertex v (the bonds are now taken to be undirected) and the derivatives are taken at the vertex v in the outer direction. The admissible boundary conditions were classified in, among other sources, [36, 37]. The second construction considers wave propagation on the graph where each vertex is treated as a scatterer and the propagation along the bonds is free. This construction was first considered in [14] and generalized in [38] to directed graphs. In both constructions one ends up with a unitary matrix S(λ) = eiλL S(0), where L is the diagonal matrix of the bond lengths. This matrix gives the eigenvalues {λn } of (2) via the equation det(I − S(λn )) = 0.
(4)
The dimension of the above matrices is equal to the number B of directed bonds of the graph G. If the bonds were initially undirected, each bond is split into two directed bonds of the same length. In various sources the notion of the “spectrum σ (G) of the graph G” can refer either to the eigenproblem (2) (and thus solutions of (4)) or to the eigenphases of the matrix S(λ) for an arbitrary λ. This is not as confusing as it might seem, since the statistical properties of both versions of the spectrum are conjectured to coincide when averaged over a large interval of λ. Similarly, the “eigenvector” of G can refer to one of three objects: 1. 2. 3.
the function u(x) that solves (2), subject to boundary conditions, for some λ ∈ σ (G), the eigenvector of S(λn ) corresponding to the eigenvalue 1, denoted by φ n , any eigenvector of S(λ) for arbitrary λ, denoted by ψ(λ).
There is a simple correspondence between the first two notions of the eigenvector: the solution u(x) is a superposition of plane waves with coefficients given by the elements of φ n . Below we discuss a heuristic formula which connects the ergodic properties of the second and the third types of eigenvectors. This formula provides an additional motivation for the results in the main body of our paper, where we study the eigenvectors ψ(λ). It should be mentioned that these results are fully rigorous and do not rely on the heuristic connection. To proceed, we need to introduce more notation. By ψ k (λ) we will denote the k th eigenvector of S(λ). Our observables are diagonal matrices O acting in the space of directed bonds. The matrix L, as before, is the diagonal matrix of the bond lengths. The ¯ Quantum ergodicity is the property of average bond length, B −1 Tr L, is denoted by L. almost all eigenvectors to equidistribute. This is equivalent to the vanishing of the variance in some limit. For example, we would like to prove that the variance of φ n |O|φ n (and, correspondingly, ψ k (λ)|O|ψ k (λ)) vanishes. At this point two obvious questions arise: (a) with respect to which ensemble is the variance taken, and (b) in which limit is it expected to vanish? 1 sometimes called “Neumann” conditions
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The mean of both quantities is equal to B −1 Tr O. Taking, without loss of generality, Tr O to be zero, we define two variances V S (, B) =
1
N () λ
φ n |O|φ n 2 ,
n ≤
V U (S(λ), B) =
B 1 ψ k (λ)|O|ψ k (λ)2 , B k=1
where N () = Tr L/2π is the mean number of the eigenvalues in the interval [0, ]. A heuristic calculation presented in Appendix A suggests that, if the bond lengths are rationally independent, φ n |O|φ n 2 1 1 U lim V (S(λ), B)dλ. (5) = lim →∞ N () →∞ 0 φ n |L|φ n / L¯ λn ≤
Thus, if the lengths of the bonds are taken from a narrow distribution (forcing φ n |L|φ n ≈ ¯ the two variances are intimately connected. Moreover, following [39] one can show L), that the limit on the right-hand side coincides with the average of V U (DS(0), B), where D are uniformly distributed random unitary diagonal matrices. Thus Eq. (5) relates the quantum ergodic properties of a graph to the like properties of an ensemble of random matrices. The definition of S(λ) implies that V U (S(λ), B) is an almost periodic function of λ. Therefore, Eq. (5) suggests that one cannot in general expect the variance V S (, B) to vanish in the limit → ∞. It is natural, however, to expect ergodicity in the limit B → ∞ (cf. [7]). A serious associated problem here is the choice of an appropriate sequence of graphs and observables. One sequence of graphs, the quantum star graphs, has been investigated in [21, 22], and it was found, in particular, that the variance V S (, B) does not vanish even when B → ∞. This is not altogether surprising because the star graphs are known to exhibit non-standard spectral statistics [14, 24], corresponding to integrable systems perturbed by a point-scatterer, rather than to chaotic systems [25] (for a review of the quantum fluctuation statistics of star graphs see [23]). This is due to the fact that the spectral gap in their Markov transition matrix closes more quickly as B → ∞ (like 1/B) than is the case for graphs exhibiting truly quantum chaotic behaviour. The lack of quantum ergodicity for the star graphs is related to the existence of strong scarring of the eigenfunctions by short periodic orbits. In the present article we study sequences of graphs generated from 1-dimensional maps of an interval in a fashion suggested in [27]. We prove that for a suitable choice of observables, the variance V U (S(λ), B) converges to 0 for any λ (and independently of the choice of the bond lengths), given that the original 1-dimensional map was ergodic. This is a stronger statement than the convergence when averaged with respect to λ, as suggested by relation (5). 3. Quantum Graphs Obtained from 1d Maps Pakónski et al [27] proposed a procedure to associate a sequence of quantum graphs to a one-dimensional map of an interval. In this section we review their construction and proceed to investigate some of its properties.
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1 2 1 1 4 1 .
2 1 1 .
. . 1 1 1 1 2 2
(a)
1
2
3
4
0 0
1
1 4
2 2 . . . 2 1 1 1 . . . 1 1 1 . . . . . . . . .
. 2 1 . 1 . . .
. . . . . . . . . . . .
(b)
1 2
6 4
1 1 1 1 . . . .
7
5
1 1 1 1 2 2 . . . . 2 2
3
8
Fig. 1. An example of a quantizable map and the corresponding matrices B: (a) |M| = 4, the atoms of the partition are represented by the vertices of the graph; (b) |M| = 8, the atoms of the partition are represented by the edges of the graph.
We consider maps of an interval, which we take to be [0, 1]. A partition M of the interval [0, 1] will be taken to mean a finite collection of open disjoint intervals E j (henceforth called atoms) such that [0, 1] =
M
E j,
j=1
where M = |M| denotes the number of intervals in the partition. We will denote by E(M) the set of endpoints of the partition M. In a slight abuse of the notation we will also denote by M the σ -algebra generated by the atoms of the partition M. When considering sequences {Mn } of partitions, each partition will be a refinement of the previous one, E(Mn ) ⊂ E(Mn+1 ). We will write Mn ⊆ Mn+1 to describe this statement. Condition 1. We consider maps S : [0, 1] → [0, 1] that satisfy the following conditions: (a) the Lebesgue (uniform) measure µ is preserved by the map S: µ(A) = µ S −1 (A) for any measurable set A; (b) there exists a partition M0 of the interval [0, 1] into M0 equal atoms, with S linear on each atom; (c) the set of endpoints E(M0 ) is forward-invariant under the action of S: S(E) ⊂ E. An example of a map S satisfying Condition 1 is shown on Fig. 1. The tent map with slope 2 is another such example. Remark 1. We infer from Condition 1(c) that the image of each atom is a union of several whole atoms together with some endpoints. This implies, in particular, that for any two atoms E and E , either S(E) is disjoint with E or S(E) ⊃ E . Another important consequence is that, since atoms have equal lengths and the map is linear, the slope of the map must be integer. Remark 2. It is possible to generalize the construction to maps that preserve a measure different from Lebesgue, but such a map would have to be topologically conjugate to
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a piecewise linear map satisfying the above properties. To maintain a degree of generality we strive to make explicit the conditions that are imposed on the map S and the measure µ. The Frobenius-Perron operator, reduced to measures constant on each atom of a partition M, can be described by a matrix B of size M = |M|. The entries of the matrix are given by µ E j ∩ S −1 (E k ) 1 = , y ∈ E j, B jk = (6) µ(E j ) |S (y)| and can be described as the answer to the question “what proportion of the set E j gets mapped into E k ”. If we view the interval [0, 1] with the uniform measure as a probabil ity space, we can write B jk = P S(x) ∈ E k |x ∈ E j . An example of a map S and the corresponding matrices B for two different partitions are shown on Fig. 1. Lemma 1. Let the set of endpoints of the partition M be invariant under S. Then the matrix B defined by (6) satisfies the following properties: 1. B is stochastic,
M
B jk = 1.
k=1
2. If the atoms E j of the partition M have equal measure and the map S preserves this measure then B is doubly stochastic, M
B jk = 1.
j=1
3. If the atoms E j of the partition M have equal measure and if the map S is linear with respect to µ on each atom E j (i.e. µ(S(A)) = Cµ(A) for some C and all A ∈ E j ) then k −r E µ jr r =0 S B j0 j1 B j1 j2 · · · B jk−1 jk = . (7) µ E j0 Proof. Part 1 follows directly from (6),
M M µ E j ∩ [0, 1] 1 −1 µ Ej ∩ = 1. B jk = S (E k ) = µ(E j ) µ(E j ) k=1
k=1
Part 2 is similar: if µ(E j ) = m for all j then ⎛⎛ ⎞ ⎞ M M µ S −1 (E k ) µ(E k ) 1 ⎝⎝ −1 µ = = 1. B jk = E j ⎠ ∩ S (E k )⎠ = m m m j=1
j=1
Part 3 is a consequence of the fact that, if µ(S(A)) = C j µ(A) for all A ⊂ E j , then B jk is either 0 or 1/C j . Consider first the case B jr jr +1 = 0 for some r . By definition of B, this means that µ E jr ∩ S −1 (E jr +1 ) = 0. Therefore, µ S −r (E jr ) ∩ S −r −1 (E jr +1 ) = 0,
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and the expression on the right-hand side of (7) evaluates to zero. Now consider the case n = 2 (the case of general n being analogous) with both B j0 j1 and B j1 j2 being different from zero. To simplify the notation, let jr = r . Then µ E 0 ∩ S −1 (E 1 ) µ E 0 ∩ S −1 (E 1 ) B0,1 = = µ(E 0 ) µ(E 1 ) µ E 0 ∩ S −1 E 1 ∩ S −1 (E 2 ) = , µ E 1 ∩ S −1 (E 2 ) where the last equality is true by virtue of linearity of S. Using the definition of B1,2 and the identity S −1 (A ∩ B) = S −1 (A) ∩ S −1 (B), we arrive to µ E 0 ∩ S −1 (E 1 ) ∩ S −2 (E 2 ) , B0,1 B1,2 = µ(E 1 ) which is the sought result, given that µ(E 1 ) = µ(E 0 ).
As mentioned earlier, we are interested in sequences of partitions. Condition 2. We consider sequences of partitions Mn that satisfy (a) the atoms within each partition have equal measure; (b) the sets of endpoints E(Mn ) are forward-invariant under the action of S; (c) the set of the endpoints of Mn contains the j th pre-image of the endpoints of M0 for all j = 1, . . . n. Remark 3. Given a map S satisfying Condition 1 one can always construct a sequence of partitions satisfying Condition 2. Remark 4. Conditions 1(b) and 1(c) imply that the map S is non-contracting, µ(S(A)) ≥ µ(A). If the map is ergodic (see Definition 1 in Sect. 4), Condition 1(a) implies that the preimages of E(M0 ) with respect to S are dense in [0, 1]. This, in turn, implies that the size of the atoms of the partitions Mn tends to zero (or, equivalently, Mn → ∞). The following lemma explains the way in which such sequences of partitions ‘resolve’ the dynamics. Lemma 2. Given a partition Mn satisfying Condition 2, let k0 and kn be such that S n (E k0 ) ⊃ E kn (cf. Remark 1). Then there exists a unique sequence k1 , . . . , kn−1 such that x ∈ E k0 and S n (x) ∈ E kn ⇒ S j (x) ∈ E k j ∀0 ≤ j ≤ n. Proof. Consider an atom E of the partition Mn . Condition 2(c) means that for every j = 0, . . . n, the image S j (E) lies in a single atom of the “primary” partition M0 . Since the map S is one-to-one on each atom of M0 , we conclude, by induction, that S j is one-to-one on E for every j = 1, . . . n + 1. Assume that the statement of the lemma is incorrect: there are two points, x and y, that satisfy, without loss of generality, x, y ∈ E 1 , Sr (x) ∈ E 2 , Sr (y) ∈ E 3 and S n (x), S n (y) ∈ E 4 . Remark 1 implies the following inclusions: Sr (E 1 ) ⊃ E 2 ,
Sr (E 1 ) ⊃ E 3 ,
S n−r (E 2 ) ⊃ E 4 ,
S n−r (E 3 ) ⊃ E 4 .
Thus each z ∈ E 4 has (n − r )-preimages in both sets E 2 and E 3 and, therefore, two distinct n-preimages in E 1 . This contradicts the earlier conclusion that S n is one-to-one on E 1 .
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Remark 5. Obviously, Lemma 2 is valid if, instead of the “position” of S n (x) (i.e. the atom E j such that S n (x) ∈ E j ), we know the position of S m (x) for some m < n: we can still recover positions of all iterates S j (x) for 0 < j < m. In fact, a careful inspection of the proof reveals that the lemma would still be true for m = n + 1. However, if we know only that x ∈ E 1 and S n+2 (x) ∈ E 2 , we would not be able to pinpoint S j (x), 0 < j < n + 2, to any particular atom of the partition Mn . The next lemma exhibits the block structure of the matrix B. Lemma 3. For a partition Mn , n > 0, define an equivalence relation between atoms by setting E j ∼ E k if S(E j ) intersects S(E k ) and then completing by transitivity. Then the maximum number of elements in an equivalence class is uniformly bounded with respect to n. For example, in the partition of Fig. 1, part (b), the atoms E 1 , E 2 , E 3 and E 5 form one equivalence class and the other four atoms form another equivalence class. Note that, if the atoms of a partition are represented by edges of the graph, the equivalence classes correspond to the groups of edges ending in the same vertex. For the map in Fig. 1 the uniform bound on the size of an equivalence class is 4, as will be evident from the proof. Proof. Take an atom E (which is an open interval!) of the primary partition M0 and let (x j , y j ), j = 1, . . . , n be the disjoint intervals forming the pre-image of E with respect to S. By Condition 1(c) these intervals contain no endpoints of M0 , therefore the map S is linear on each interval. As discussed in Remark 1, Condition 1 also implies that all slopes of the map S are integer. Denote the slope of S on the interval (x j , y j ) by s j . To simplify the notation we assume that all s j are positive. Let p be the least common multiple of s j . Condition 2(c) implies that x j and y j are endpoints of the partition Mn . Choose x j ∈ E(Mn ) such that the interval (x j , x j ) contains exactly p/s j atoms of the partition Mn . Since the atoms of Mn have equal length (which we denote by µn ), s j1 (x j1 − x j1 ) = s j2 (x j2 − x j2 ) = pµn ,
for any j1 , j2 .
(8)
Moreover, the selected points x j are the closest to the respective x j to satisfy both condition (8) and x j ∈ E(Mn ). In particular, this implies that x j ≤ y j , since setting x j = y j would also satisfy condition (8). From the above we can conclude that S maps all intervals (x j , x j ) to the same subinterval of E. The atoms of Mn making up the intervals (x j , x j ) thus form an equivalence class of size p/s1 + · · · + p/sk , which is independent of n. We can now repeat this procedure with intervals (x1 , y1 ), . . . , (xk , yk ) and, thereafter, with all atoms E of the partition M0 . If some of the slopes s j are negative, the procedure would still go through with minor variations. Having obtained a sequence of doubly stochastic matrices Bn we define their “quantizations” as unitary matrices Un such that 2 (Bn ) jk = (Un ) jk . (9) The doubly stochastic matrices B for which finding a corresponding U is possible are called unistochastic.
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Condition 3. We assume that the map S is such that all of the corresponding matrices Bn , bar finitely many, are unistochastic. Not all bistochastic matrices are unistochastic. However, formulating general sufficient conditions that ensure unistochasticity is a question of considerable difficulty. The interested reader is referred to [27, 40] and the references therein where some necessary conditions are discussed and where examples of maps satisfying and failing Condition 3 are given. To convince the reader that the class of maps S satisfying Condition 3 is far from empty we state the following sufficient condition. Lemma 4. If the slopes of the map S satisfying Condition 1 are all equal (modulo sign), Condition 3 is also satisfied. Proof. This lemma follows simply from the proof of Lemma 3. Indeed, let s be the absolute value of the slope of S. Then all matrices Bn , n > 0, have a block structure with blocks of the size s × s and elements 1/s. Thus the question is really about finding an s × s unitary matrix with all elements satisfying |U jk |2 = 1/s.√ One example of such matrix is the Fourier matrix with elements U jk = exp{2πi jk/s}/ s. Example 1. An example of a map S which has unequal slopes but is still unistochastic is provided by the map of Fig. 1. Remark 6. An observant reader would notice that, given one unitary U satisfying (9), one can produce infinitely many such matrices. For example, one can multiply a given U by an arbitrary diagonal unitary matrix eiλL . However, the results of our paper do not depend on the precise choice of matrices Un , provided that condition (9) is satisfied. One can associate a graph to the matrices B and U in the following way: the indices of the matrices enumerate the directed edges of the graph; the end of an edge j coincides with the start of the edge k if the matrix element B jk is non-zero. The number of distinct vertices in such a construction should be maximized, then the vertices will correspond to the equivalence classes of Lemma 3. The matrix B defines a Markov chain on the edges of the graph with B jk representing the transition probability from j to k. The matrix U can be viewed as a quantum propagator on the graph. This geometrical interpretation of the two matrices as a graph will be helpful in the later sections when we use trajectories on the graph to describe properties of the eigenvectors of U. It is also possible to associate vertices of a graph to the indices of B, see Fig. 1, part (a). We use directed edges for reasons of tradition, rather than convenience. 4. Quantization of the Observables Having defined the sequences of unitary matrices Un , ergodic properties of whose eigenvectors we are going to study, we need a final ingredient, the observables On . For a general sequence of graphs, it is not obvious how to define a consistent sequence of observables. In our case, however, there is a natural answer. We use the discretizations of functions φ ∈ L 2 [0, 1] as our observables. Fix a partition M (the semi-classical limit corresponds to |M| → ∞). If the function φ is constant on each atom of the partition M, its quantization O = O(φ) is a diagonal matrix with entries O j j = φ(x), where x ∈ E j . If φ is not constant on the atoms of M, we replace
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φ by its local average. More precisely, we introduce the piecewise constant function φ defined by 1 (x) = φ φ(y)dµ(y), where E j x. µ(E j ) E j Then we define O = O(φ) as before, by (E j ) = O j j = O j j (φ) = φ
1 µ(E j )
φ(y)dµ(y).
(10)
Ej
using the notions of probability theory. In probabilistic It is convenient to describe φ is its language, φ is a random variable defined on the probability space [0, 1] and φ = E [ φ| M]. We will also use the notation of expectation to conditional expectation, φ denote the integral over [0, 1]:
1
Eφ =
φ(x)dµ(x).
0
1/2 In particular, φ2 = Eφ 2 . To prove quantum ergodicity, we will rely on the ergodicity of the classical map S. Since our observables are in L 2 , the relevant version of the ergodic theorem is the L 2 ergodic theorem (see, e.g., [41]). Definition 1. A map S : [0, 1] → [0, 1] is ergodic if any set A ⊂ [0, 1] satisfying S −1 (A) = A has either full or zero measure. Theorem 1 (L 2 Ergodic Theorem). If φ ∈ L 2 [0, 1] and S is ergodic, then
def
VT (φ) = E
T 1 φ ◦ S t − Eφ T
2 → 0.
(11)
t=1
Remark 7. One of the classical examples of an ergodic map is the binary shift x → (2xmod1). The binary shift satisfies Condition 1. , Since the ergodic theorem applies to any function from φ ∈ L 2 , it also applies to φ whatever partition M was used to produce it. Unfortunately, a uniform estimate for the rate of convergence in (11) for different hat-versions of the same φ is not known [42]. However, it is easy to see that, for fixed T , ) → VT (φ) V T (φ gets finer. as the partition in the definition of φ
(12)
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5. Quantum Mean and Variance Given a map S and a sequence of partitions Mn we have constructed a sequence of Markov matrices Bn , which, in turn, give rise to unitary matrices Un . On the other hand we are given an observable φ and we have constructed a corresponding sequence of diagonal matrices On , which “quantize” φ. We denote by Mn the number of atoms in the partition Mn . This is also the size of the matrices Bn , Un and On . The semiclassical limit corresponds to Mn → ∞. (n) Let ψ j , where j = 1, . . . , Mn denote the orthonormal eigenvectors of Un . If an eigenvalue is degenerate, the particular choice of the basis for its eigenspace is unimportant. We start by computing the quantum mean of an observable φ. From the unitarity of Un and the definition of On , we get 1 Mn 1 1 (n) (n) ψ j , On ψ j = Tr On = E φ = E(φ) = φ(x)d x, Mn Mn 0 j=1
independently of n. This property is the analogue of the local Weyl law. To show quantum ergodicity it is sufficient to prove that Vn =
M n 2 1 (n) (n) ψ j , On ψ j − E(φ) → 0, Mn
(13)
j=1
as n → ∞. Without loss of generality we can assume that E(φ) = 0. In what follows we will omit the sub- and super-scripts n unless we want to underline the dependence of a quantity on n and on the partition Mn . To obtain an estimate of Vn we employ some standard manipulations. If ψ is an eigenvector of a unitary matrix U, we have, for any matrix O (not necessarily diagonal) and all t ∈ N, (ψ, Oψ) = Ut ψ, OUt ψ = ψ, (U∗ )t OUt ψ . Summing this equality over t = 0, . . . T − 1 we obtain T −1 1 ∗ t t (U ) OU ψ . (ψ, Oψ) = ψ, T
t=0
We introduce the shorthand On,T for the time average of On , On,T =
T −1 1 ∗ t (Un ) On Unt . T t=0
Using Cauchy-Schwarz inequality and orthonormality of {ψ j } we estimate |(ψ, Oψ)|2 = |(ψ, OT ψ)|2 ≤ (OT ψ, OT ψ) = ψ, O∗T OT ψ ,
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and obtain Vn =
Mn 1 ψ j , Oψ j 2 Mn j=1
≤
Mn def 1 1 ψ j , O∗T OT ψ j = Tr O∗T OT = K (n, T ). Mn Mn
(14)
j=1
It is important to note that the above inequality is valid for all values of T . Thus, to show that Vn → 0, we are free to choose an appropriate T = T (n) for each n as long as we can demonstrate that 1 Tr O∗n,T (n) On,T (n) → 0, K (n, T (n)) = Mn as n → ∞. In the following sections we prove that T (n) = n is a suitable choice for this task. For our purposes, it is more convenient to work with the matrices Sn,T defined by Sn,T = UnT On,T =
T −1 1 T −t Un On Unt , T t=0
which is equivalent to working with OT since S∗T ST = (UT OT )∗ (UT OT ) = O∗T OT . Multiplying S∗T S out we obtain M 1 1 Tr S∗T ST = |(ST )s, f |2 . M M s, f =1
We can expand the entries of ST in terms of trajectories on the graph. Using the definition of ST , we obtain (ST )s f
T −1 1 = Ub0 ,b1 · · · UbT −t−1 ,bT −t ObT −t ,bT −t · · · UbT −1 ,bT T t=0 b0 ,...bT
T −1 1 = Ub0 ,b1 · · · UbT −1 ,bT ObT −t ,bT −t , T b0 ,...bT
t=0
where the inner sum in the first line is over all sequences of bonds satisfying b0 = s and bT = f . Such a sequence of bonds we will call a trajectory. Only trajectories compatible with the graph’s geometry (i.e. those for which Ub j b j+1 = 0) contribute to K (n, T ). A trajectory τ = (b0 , . . . , bT ) is said to have length T and amplitude def
Aτ = Ub0 b1 · · · UbT −1 bT . We will denote by τ the average of the observable over the trajectory τ , def
τ =
1 O b1 b1 + . . . + O b T b T . T
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To summarize, we have shown that
2 Mn ∗ 1 1 Vn ≤ K (n, T ) = Tr ST ST = τ Aτ Mn Mn s, f =1 τ :s→ f Mn 1 = Mn
s, f =1 τ1 ,τ2 :s→ f
∗τ1 τ2 A∗τ1 Aτ2 ,
(15)
where the inner sum is over all possible trajectories of length T starting at s and finishing at f . 6. Diagonal Terms Equation (15) is reminiscent of a trace formula expansion of the spectral form factor (i.e. of the Fourier transform of the spectral two-point correlation function)2 , in particular of a graph, see e.g. [14]. Such expansions are notoriously difficult to analyze rigorously as both T and the size of the graph increase. The starting point of any such analysis is the evaluation of the contribution from the diagonal terms, obtained by restricting the last sum in (15) to identical trajectories, τ1 = τ2 . It is usually assumed that the off-diagonal terms sum up to a subdominant contribution, when T and the size of the graph scale appropriately. This idea, called the diagonal approximation was first introduced for a general class of systems in [43]. On graphs it was explored, in particular, in [14, 44]. It is difficult, however, to give an a priori estimate on the size of the off-diagonal contributions and the analysis is usually restricted to evaluating the contributions coming from specific classes of interacting trajectories [15–18]. Our strategy now is to calculate the contribution from the diagonal terms in (15). Then we will show that, in the case of graphs constructed from 1d maps, we can actually estimate the off-diagonal terms by virtue of being able to choose an appropriate T = T (n). To evaluate the diagonal contribution def 1 |τ |2 |Aτ |2 , K (diag) (n, T ) = Mn τ we make two observations. First, by the definition of the amplitude Aτ and the defining property of the matrix U, Eq. (9), we obtain |Aτ |2 = |Ub0 ,b1 |2 · · · |UbT −1 ,bT |2 = Bb0 ,b1 · · · BbT −1 ,bT . Now we recall Lemma 1, part 3, and conclude that T −t (E ) µ S b t t=0 |Aτ |2 = . µ E b0 On the other hand, by definition of τ , τ =
T T T 1 1 1 (E bt ) = ◦ S t S −t (E bt ) , φ φ Obt ,bt = T T T t=1
t=1
t=1
2 To underline this similarity we used in (14) the traditional notation for the form factor, K .
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(E b ) denotes the (constant) value of the function φ on the atom E b . In fact, it is where φ easy to see that τ coincides with the value of the function T def = φ
T 1 ◦ St φ T t=1
T def on the set t=0 S −t (E bt ) = E b0 ,...,bT , if this set is non-empty. If it is empty, the value of τ is of no consequence since the trajectory τ is then incompatible with the graph’s geometry and Aτ = 0. The measure of all atoms E b is assumed to be equal. More precisely, it is equal to 1/Mn , since Mn is the total number of the atoms. Collecting our observations together, we can express the diagonal term as µ(E b0 ,...,bT ) 1 T E b0 ,...,bT 2 φ Mn τ Mn−1
2 1 T 2 1 t ◦ S T (x) d x = E ). φ φ = = V T (φ T 0
K (diag) (n, T ) =
t=1
Thus, by the L 2 ergodic theorem (Theorem 1), K (diag) (n, T ) goes to zero as T → ∞. On the other hand, K (n, T ) is bounded below by a non-negative Vn which is, generically, non-zero for a fixed n. This shows that the diagonal term is a poor approximation to K (n, T ) in the limit T → ∞. Luckily, this is not the limit we have to take. 7. Completion of the Proof of Quantum Ergodicity Lemma 2 has a very important consequence for the inner sum in (15). Lemma 5. The diagonal term K (diag) (n, T ) gives the exact value of K (n, T ) up to time T = n, i.e. ) if T ≤ n, K (n, T ) = K (diag) (n, T ) = VT (φ = E [ φ| Mn ]. where φ Proof. By Lemma 2, for every pair of bonds s and f , there is at most one trajectory going from s to f in T ≤ n steps. Thus, for T ≤ n, 2 Mn 1 K (n, T ) = A τ τ Mn s, f =1 τ :s→ f =
Mn 1 τ (s→ f ) Aτ (s→ f ) 2 = 1 |τ Aτ |2 = K (diag) (n, T ). Mn Mn τ s, f =1
As a consequence, we have the following result.
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Theorem 2 (Quantum Ergodicity). Let the map S and the sequence of partitions {Mn } satisfy Conditions 1, 2 and 3; let {Un } be the corresponding sequence of unitary matrices (n) with eigenvectors ψ j ; and let {On } be a sequence of diagonal matrices corresponding to an observable φ ∈ L 2 [0, 1] with Eφ = 0. If S is ergodic, then M n 1 (n) (n) 2 Vn = ψ j , On ψ j → 0 Mn
as n → ∞.
j=1
Proof. The variance Vn is majorized by K (n, T ) for any T . We combine Lemma 5 with Eq. (12) and we conclude that, for a fixed T , K (n, T ) → VT (φ)
as n → ∞.
Now we use the standard ε/2 argument: for any ε > 0, by Theorem 1 we can find T such that VT (φ) < ε/2. Having fixed this T , we find n(ε, T ) such that |K (n, T ) − VT (φ)| < ε/2 for all n ≥ n(ε, T ). Combining the above, Vn ≤ K (n, T ) < ε/2 + ε/2 as long as n ≥ n(ε, T ). Since ε was arbitrary, we conclude that Vn → 0. Remark 8. One can avoid the ε/2 argument in the following way. Taking the limit n → ∞ of the inequality Vn ≤ K (n, T ) produces 0 ≤ lim sup Vn ≤ lim sup K (n, T ) = VT (φ). n→∞
n→∞
Now taking the T → ∞ limit, we obtain 0 ≤ lim sup Vn = lim sup lim sup Vn ≤ lim sup VT (φ) = 0. n→∞
T →∞
n→∞
T →∞
8. Egorov Property given a In Sect. 4 we defined a procedure to obtain a piecewise constant function φ 2 . function φ ∈ L [0, 1]. It is enlightening to see how φ ◦ S is related to φ By definition, 1 1 dµ(z) φ ◦ S = . φ(S(y))dµ(y) = φ(z) Ej µ(E j ) E j µ(E j ) S(E j ) |S (y)| Since S is linear on E j , its derivative is constant. In fact, it is easy to see that 1/|S (y)| = B jk , where y ∈ E j and S(y) ∈ E k . Thus we have 1 φ ◦ S = B jk φ(z)dµ(z), Ej µ(E j ) E k k:E k ∩S(E j )=∅
where the sum is over the decomposition of the set S(E j ) into atoms E k . Since B jk = 0 whenever E k ∩ S(E j ) is empty and since µ(E j ) is independent of j, we arrive to the following conclusion:
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Lemma 6. If, for a given partition M, the matrices B, O(φ) and O(φ ◦ S) are defined according to (6) and (10) then O j j (φ ◦ S) =
M
B jk Okk (φ),
k=1
where M is the number of atoms in the partition M. Lemma 6 is a rather beautiful manifestation of the inter-consistency between the discretization procedures for maps S and observables φ ∈ L 2 . Namely, the discretization commutes with the action of S on L 2 . In this, Lemma 6 is a classical analogue of the Egorov property, a result which shows that the unitary matrices Un faithfully represent the action of the classical map S. Theorem 3 (Egorov property). Let the map S and the sequence of partitions {Mn } satisfy Conditions 1, 2 and 3; let {Un } be the corresponding sequence of unitary matrices (n) with eigenvectors ψ j ; and let {On } be a sequence of diagonal matrices corresponding to an observable φ. If φ is Lipschitz continuous then Un On (φ)Un−1 − On (φ ◦ S) = O(Mn −1 ), where the norm is the operator norm on the Euclidean space R Mn . Proof. We fix the partition Mn , denote the corresponding UOU−1 by Q and observe that, while O(φ ◦ S) is a diagonal matrix, Q is not necessarily so. First we treat the diagonal elements of Q. Writing them out explicitly we get Q jj =
Mn
U jr Orr Ur−1 j =
r =1
Mn
U jr Orr U jr =
r =1
Mn
|U jr |2 Orr = O j j (φ ◦ S),
r =1
where we used the unitarity of U and its defining property, |U jr |2 = B jr and Lemma 6. For the off-diagonal elements of Q we have Q jk =
Mn
U jr Orr Ur−1 k =
r =0
=
Mn r =0 Mn
U jr (Orr − C)Ukr + C
Mn
U jr Ukr
r =0
U jr (Orr − C)Ukr ,
r =0
where C is any constant and we have used the unitarity of U to conclude that the second sum is zero. We estimate, using Cauchy-Schwarz, |Q jk | ≤ max |Orr − C| r
Mn
≤ max |Orr − C| = r
|U jr Ukr |
r =0
max
x∈S(E j )∩S(E k )
(x) − C|. |φ
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If φ is Lipschitz continuous with def
φLip = sup x= y
|φ(x) − φ(y)| < ∞, µ(x, y)
we can estimate further, by choosing appropriate C, |Q jk | ≤
1 φLip 1 φ µ S(E j ) ∩ S(E k ) ≤ φLip max |S (x)|µ(E j ) ∝ . Lip 2 2 Mn
Since U jr is non-zero only if S(E j )∩Er = ∅ and Ukr is non-zero only if S(E k )∩Er = ∅, we conclude that Q jk = 0 if S(E j ) and S(E k ) are disjoint. Thus the matrix Q is of block-diagonal structure, each block corresponding to an equivalence class as defined by Lemma 3. The norm of Q−O(φ ◦ S) is equal to the maximum of the norms of the blocks. A norm of a block, in turn, is bounded by its dimension times the maximum absolute value of the element of the block. The dimension of a block is uniformly bounded by Lemma 3. Thus we get φLip Q − O(φ ◦ S) ≤ D(S) Mn for some constant D(S) which is independent of φ and n.
Remark 9. If the function φ is only assumed to be continuous on [0, 1], one can prove a weaker property: Un On (φ)Un−1 − On (φ ◦ S) → 0
as Mn → ∞.
9. Discussion We have succeeded in proving quantum ergodicity (QE) for a special class of sequences of quantum graphs. However, we would like to mention that the result is expected to hold for a much broader class of graphs. It is true that, given a finite quantum graph G, one can associate a 1d map to it by reversing the process described in the paper. Thereafter, it is possible to produce a sequence of graphs, one of which will coincide with the original graph G, and answer the question of QE for this sequence. In this sense, each graph corresponds to a 1d map. However, this is not true for every sequence of graphs. In fact, it is not true for most sequences. Examples of such sequences include star graphs with Kirchhoff conditions at the central vertex (for which the question of QE has been answered negatively), the complete (Kirchhoff) graphs, and the star graphs with Fourier central vertex [44], for both of which the QE is expected (but is not known) to hold in some form. It is reassuring that the proof of QE in the present article suggests a direction for possible generalizations: study the diagonal terms and then find an estimate for the off-diagonal ones. However, for the sequences of graphs described above the diagonal approximation ceases to be exact for T > 1 (cf. Lemma 5). This makes estimation of the off-diagonal terms a much more difficult task. Another interesting question to consider is whether quantum unique ergodicity (when the convergence in (1) happens along all sequences of eigenvectors) is true for any quantum graphs. This has been answered in the negative [26] for graphs with Kirchhoff vertices but is unclear for other types if graphs.
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Acknowledgement. We would like to thank Zeev Rudnick for his suggestion to consider proving Egorov property, which we followed with success. We are also grateful to Alexander G. Kachurovskii for enlightening discussions on the speed of convergence in ergodic theorems. One of the authors (GB) wishes to thank the University of Bristol and the Weizmann Institute of Science for the hospitality extended to him. This collaboration was supported by EPSRC Grant GR/T06872/01. JPK is supported by an EPSRC Senior Research Fellowship. GB acknowledges support from NSF award #0604859.
A. Connection Between Variances V S and V U To demonstrate relation (5) we start with summarizing the notation introduced in Sect. 2. Let the unitary B × B matrix S be defined by S = S(λ) = eiλL S(0), where L is the diagonal matrix of the bond lengths of the graph and S(0) is some fixed unitary matrix. Let {λn } be the (real) solutions of the equation det(I − S(λ)) = 0. We assume that the spectrum {λn } is non-degenerate, which is a generic situation [45]. Denote by φ n the normalized eigenvector of S(λn ) corresponding to the eigenvalue 1. By ψ k (λ) we denote the k th normalized eigenvector of S(λ). We further denote by eiθk (λ) the eigenvalues of S(λ), with θk chosen to be continuous (indeed smooth) functions of λ. When λ = λn there is an index k for which θk (λ) = 0 mod 2π . For this index k we also have φ n = ψ k (λ). Let A be a self-adjoint matrix (a generalization of the observable O) with trace 0 (without loss of generality). We are interested in the relationship between two variances, V S (, B) =
1
N () λ
φ n |A|φ n 2 ,
n ≤
and V U (S(λ), B) =
B 1 ψ k (λ)|A|ψ k (λ)2 , B k=1
where N () = Tr L/2π is the mean number of the eigenvalues λn in the interval [0, ]. Introducing the notation Ak = Ak (λ) = ψ k (λ), Aψ k (λ) we observe that Tr ASm (λ) = In particular, Tr A =
B
Ak eimθk (λ) .
k=1
B
Ak = 0. From the theory of distributions we know that
∞ def 1 −mε imθ −imθ
ε (θ ) = e e +e 1+ 2π k=1
m=1
converges, in the limit ε → 0, to def
(θ ) =
∞ r =−∞
δ(θ − 2πr ),
(16)
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where δ is the Dirac delta function. Substituting in the above identity θ = θk , multiplying by Ak and performing the summation over k yields
∞ B 1 −mε
ε (θk )Ak = e Tr ASm + Tr AS−m . (17) 2π m=1
k=1
As ε → 0 this converges to B
(θk )Ak =
∞ Ak (λn ) δ(λ − λn ), |θk (λn )| n=1
k=1
where, given λn , k is chosen to satisfy θk = 0. It is shown in [14] that def θk (λn ) = φ n , Lφ n = ψ k (λn ), Lψ k (λn ) = L k (λn ). Clearly, θk (λn ) > 0 and so we can drop the modulus around θk (λn ) in the previous equation. Now we need the following properties of the approximants to the Dirac delta function, 2π lim ε 2ε (x) = (x) ε→0
and, if x1 = x2 , 2π lim ε ( ε (x − x1 ) + ε (x − x2 ))2 = (x − x1 ) + (x − x2 ). ε→0
Applying these identities gives
B
2π lim ε ε→0
2
ε (θk )Ak
=
k=1
B
(θk )A2k =
∞ A2k (λn ) δ(λ − λn ). L k (λn ) n=1
k=1
Integrating the right-hand side with respect to λ we get ∞ A2 (λn ) def A2k (λn ) 1 1 k S (, B). δ(λ − λn ) dλ = = V L (λ ) L (λ ) N () 0 N () k n k n λn <
n=1
We will use this quantity to approximate V S (, B). It is a good approximation if the bond lengths of the graph are approximately 1 (i.e. the matrix L is approximately unity): S (, B) ≤ V S (, B) ≤ L max V S (, B), L min V where L max and L min are the maximal and minimal bond lengths. Using (17) and expanding the square we obtain S
V (, B) = =
1 N ()
2π lim ε ε→0
0
1 2π N ()
0
lim ε
ε→0
B k=1 ∞
2
ε (θk )Ak
dλ
e−ε(|m 1 |+|m 2 |) Tr ASm 1 Tr ASm 2 dλ.
m 1 ,m 2 =−∞
Quantum Ergodicity for Graphs Related to Interval Maps
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We now take the → ∞ limit of the above expression and interchange it with the ε-limit. Due to the rational independence of the bond lengths, the limit 1 lim Tr ASm 1 Tr ASm 2 dλ →∞ N () 0 is zero whenever m 1 = −m 2 . Indeed, we recall that S = S(λ) = eiλL S(0) and expand the trace Ab0 ,b1 S(0)b1 ,b2 · · · S(0)bm ,b0 exp(iλL p ), Tr ASm = b0 ,...,bm
where L p = L b1 + · · · + L bm . If L p is a sum of m 1 terms and L q is a sum of m 2 = m 1 terms, they cannot be equal. Thus, the only case when the phase factors exp(iλL p ) in a product of two traces can cancel each other is when m 1 = −m 2 . We arrive at ∞ 1 S (, B) = lim lim ε e−2mε Tr ASm Tr AS−m dλ lim V →∞ →∞ π N () 0 ε→0 m=1 B 1 A2k (λ) dλ = lim →∞ 2π N () 0 k=1 1 = lim V U (S(λ), B) dλ. →∞ 2π N () 0 Here we used expansion (16) and the fact that lim ε
ε→0
∞
e−2mε eim(θk1 −θk2 ) = lim
m=1
ε
ε→0 e2ε−i(θk1 −θk2 )
−1
is 1/2 when θk1 = θk2 and 0 otherwise. References 1. Shnirelman, A.I.: Ergodic properties of eigenfunctions. Uspehi Mat. Nauk 29(6(180)), 181–182 (1974) 2. Colin de Verdière, Y.: Ergodicité et fonctions propres du laplacien. Commun. Math. Phys. 102(3), 497– 502 (1985) 3. Helffer, B., Martinez, A., Robert, D.: Ergodicité et limite semi-classique. Commun. Math. Phys. 109(2), 313–326 (1987) 4. Zelditch, S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55(4), 919–941 (1987) 5. Gérard, P., Leichtnam, É.: Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71(2), 559–607 (1993) 6. Degli Esposti, M., Graffi, S., Isola, S.: Classical limit of the quantized hyperbolic toral automorphisms. Commun. Math. Phys. 167(3), 471–507 (1995) 7. Bouzouina, A., De Bièvre, S.: Equipartition of the eigenfunctions of quantized ergodic maps on the torus. Commun. Math. Phys. 178(1), 83–105 (1996) 8. Kurlberg, P., Rudnick, Z.: Hecke theory and equidistribution for the quantization of linear maps of the torus. Duke Math. J. 103(1), 47–77 (2000) 9. Kurlberg, P., Rudnick, Z.: On quantum ergodicity for linear maps of the torus. Commun. Math. Phys. 222(1), 201–227 (2001) 10. Degli Esposti, M., Nonnenmacher, S., Winn, B.: Quantum variance and ergodicity for the baker’s map. Commun. Math. Phys. 263, 325–352 (2006)
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11. De Bièvre, S.: Quantum chaos: a brief first visit. In: Second Summer School in Analysis and Mathematical Physics (Cuernavaca, 2000), Vol. 289 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2001, pp. 161–218 12. Berkolaiko, G., Carlson, R., Fulling, S., Kuchment, P. eds.,: Proceedings of Joint Summer Research Conference on Quantum Graphs and Their Applications, 2005, Contemporary Mathematics, Vol. 415, Providence, RI: Amer. Math. Soc., (2006) 13. Kottos, T., Smilansky, U.: Quantum chaos on graphs. Phys. Rev. Lett. 79, 4794–4797 (1997) 14. Kottos, T., Smilansky, U.: Periodic orbit theory and spectral statistics for quantum graphs. Ann. Phys. 274, 76–124 (1999) 15. Berkolaiko, G., Schanz, H., Whitney, R.S.: Leading off-diagonal correction to the form factor of large graphs. Phys. Rev. Lett. 88(10), 104101 (2002) 16. Berkolaiko, G., Schanz, H., Whitney, R.S.: Form factor for a family of quantum graphs: an expansion to third order. J. Phys. A 36(31), 8373–8392 (2003) 17. Berkolaiko, G.: Form factor for large quantum graphs: evaluating orbits with time reversal. Waves Random Media 14(1), S7–S27 (2004) 18. Berkolaiko, G.: Correlations within the spectrum of a large quantum graph: a diagrammatic approach. In: Proceedings of Joint Summer Research Conference on Quantum Graphs and Their Applications, 2005 (G. Berkolaiko, R. Carlson, S. Fulling, P. Kuchment, eds.), Contemporary Mathematics, Vol. 415, Providence, RI: Amer. Math. Soc., 2006 19. Gnutzmann, S., Altland, A.: Universal spectral statistics in quantum graphs. Phys. Rev. Lett. 93(19), 194101 (2004) 20. Gnutzmann, S., Altland, A.: Spectral correlations of individual quantum graphs. Phys. Rev. E 72(5), 056215 (2005) 21. Berkolaiko, G., Keating, J.P., Winn, B.: Intermediate wave-function statistics. Phys. Rev. Lett. 91, 134103 (2003) 22. Berkolaiko, G., Keating, J.P., Winn, B.: No quantum ergodicity for star graphs. Commun. Math. Phys. 250(2), 259–285 (2004) 23. Keating, J.P.: Fluctuation statistics for quantum star graphs. In: Proceedings of Joint Summer Research Conference on Quantum Graphs and Their Applications, 2005 (G. Berkolaiko, R. Carlson, S. Fulling, P. Kuchment, eds.), Contemporary Mathematics, Vol. 415, Providence, RI: Amer. Math. Soc., 2006 24. Berkolaiko, G., Keating, J.P.: Two-point spectral correlations for star graphs. J. Phys. A 32(45), 7827– 7841 (1999) 25. Berkolaiko, G., Bogomolny, E.B., Keating, J.P.: Star graphs and Šeba billiards. J. Phys. A 34(3), 335– 350 (2001) 26. Schanz, H., Kottos, T.: Scars on quantum networks ignore the lyapunov exponent. Phys. Rev. Lett. 90, 234101 (2003) ˙ 27. Pako´nski, P., Zyczkowski, K., Ku´s, M.: Classical 1D maps, quantum graphs and ensembles of unitary matrices. J. Phys. A 34(43), 9303–9317 (2001) 28. Lumer, G.: Espaces ramifiés, et diffusions sur les réseaux topologiques. C. R. Acad. Sci. Paris Sér. A-B 291(12), A627–A630 (1980) 29. Roth, J.-P.: Le spectre du laplacien sur un graphe. In: Théorie du potentiel (Orsay, 1983), Vol. 1096 of Lecture Notes in Math., Berlin: Springer, 1984, pp. 521–539 30. von Below, J.: A characteristic equation associated to an eigenvalue problem on c2 -networks. Linear Algebra Appl. 71, 309–325 (1985) 31. Nicaise, S.: Spectre des réseaux topologiques finis. Bull. Sci. Math. (2) 111(4), 401–413 (1987) 32. Penkin, O.M., Pokorny˘ı, Y.V.: On a boundary value problem on a graph (in Russian). Differentsial nye Uravneniya 24(4), 701–703, 734–735 (1988) 33. Pauling, L.: The dimagnetic entropy of aromatic molecules. J. Chem. Phys. 4, 673–677 (1936) 34. Griffith, J.: A free-electron theory of conjugated molecules. i. polycyclic hydrocarbons. Trnas. Faraday Soc. 49, 345–351 (1953) 35. Ruedenberg, K., Scherr, C.W.: Free-electron network model for conjugated systems. i. theory. J. Chem. Phys. 21(9), 1565–1581 (1953) 36. Kostrykin, V., Schrader, R.: Kirchhoff’s rule for quantum wires. J. Phys. A 32(4), 595–630 (1999) 37. Harmer, M.: Hermitian symplectic geometry and extension theory. J. Phys. A 33(50), 9193–9203 (2000) 38. Tanner, G.: Spectral statistics for unitary transfer matrices of binary graphs. J. Phys. A 33(18), 3567– 3585 (2000) 39. Barra, F., Gaspard, P.: On the level spacing distribution in quantum graphs. J. Statist. Phys. 101(1–2), 283–319 (2000) ˙ 40. Zyczkowski, K., Ku´s, M., Słomczy´nski, W., Sommers, H.-J.: Random unistochastic matrices. J. Phys. A 36(12), 3425–3450 (2003) 41. Billingsley, P.: Probability and measure. New York: J. Wiley & Sons, 3rd ed., 1995
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42. Kachurovski˘ı, A.G.: Rates of convergence in ergodic theorems. Uspekhi Mat. Nauk. 51(4(310)), 73–124 (1996). Translated in Russ. Math. Surv. 51(4), 653–703 (1996) 43. Berry, M.V.: Semiclassical theory of spectral rigidity. Proc. R. Soc. Lond. A 400, 229–251 (1985) 44. Tanner, G.: Unitary-stochastic matrix ensembles and spectral statistics. J. Phys. A 34(41), 8485– 8500 (2001) 45. Friedlander, L.: Genericity of simple eigenvalues for a metric graph. Israel J. Math. 146, 149–156 (2005) Communicated by P. Sarnak
Commun. Math. Phys. 273, 161–176 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0214-6
Communications in
Mathematical Physics
Interior Regularity Criteria for Suitable Weak Solutions of the Navier-Stokes Equations Stephen Gustafson1 , Kyungkeun Kang2 , Tai-Peng Tsai1 1 Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2.
E-mail:
[email protected];
[email protected]
2 Department of Mathematics, Sungkyunkwan University and Institute of Basic Science, Suwon 440-746,
Republic of Korea. E-mail:
[email protected] Received: 5 July 2006 / Accepted: 10 October 2006 Published online: 6 March 2007 – © Springer-Verlag 2007
Abstract: We present new interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations: a suitable weak solution is regular near an interior point p,q z if either the scaled L x,t -norm of the velocity with 3/ p + 2/q ≤ 2, 1 ≤ q ≤ ∞, or the p,q p,q L x,t -norm of the vorticity with 3/ p + 2/q ≤ 3, 1 ≤ q < ∞, or the L x,t -norm of the gradient of the vorticity with 3/ p + 2/q ≤ 4, 1 ≤ q, 1 ≤ p, is sufficiently small near z. 1. Introduction We continue our study in [10] of the regularity problem for suitable weak solutions (u, p) : × I → R3 × R of the three-dimensional incompressible Navier-Stokes equations (NS) u t − u + (u · ∇)u + ∇ p = f in × I. (1) div u = 0 Here is either a domain in R3 or the 3-dimensional torus T3 , I is a finite time interval, u(x, t) is the velocity field and p(x, t) is the pressure. We also denote the vorticity field by w = curl u. By suitable weak solutions we mean functions which solve (1) in the sense of distributions and satisfy some integrability conditions and the local energy inequality (for details, see Definition 2.1 in Sect. 2). For a point z = (x, t) ∈ R3 × R we denote Bx,r = {y ∈ R3 : |y − x| < r },
Q z,r := Bx,r × (t − r 2 , t).
A solution u is said to be regular at z ∈ × I if u ∈ L ∞ (Q z,r ) for some Q z,r ⊂ × I , r > 0. Otherwise it is singular at z (see [2, p. 780]). Although the existence of weak solutions was proved by Leray and Hopf [17, 11] in R3 and domains, it is not known whether the solution stays regular for all time even if all
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the data are smooth. One type of condition ensuring regularity involves zero-dimensional integrals, u L p,q (×I ) < ∞,
3 2 + = 1, 3 ≤ p ≤ ∞, p q
(2)
where u L p,q (×I ) = u L q L xp (×I ) = u(x, t) L xp () L q (I ) . t t
(3)
These integrals have zero dimension if one assigns the dimensions 1, 2, and −1 to x, t and u. This is related to the scaling property of solutions of (NS): The map {u(x, t), p(x, t)} → {λu(λx, λ2 t), λ2 p(λx, λ2 t)} (λ > 0),
(4)
sends a solution of (NS) to another solution, with a new force λ3 f (λx, λ2 t). The first contributions in this direction, concerning uniqueness and regularity of weak solutions, were made by [20, 31, 32, 15] when 3/ p + 2/q < 1. The borderline cases 3/ p + 2/q = 1, 3 < p ≤ ∞, for different types of domains were later proved by [8, 33, 9, 37]. See [38, 34, 4] for results in the setting of Lorentz spaces. The endpoint case ( p, q) = (3, ∞) was recently resolved [7] (also see the references in [34, 7] for earlier results in subclasses). Similar regularity criteria have been established near the boundary [36, 12, 28]. In a series of papers [21]–[24], Scheffer began to study the partial regularity theory for (NS). His results were further generalized and strengthened in Caffarelli-KohnNirenberg [2], which proved that the set S of possible interior singular points of a suitable weak solution is of one-dimensional parabolic Hausdorff measure zero, i.e. P 1 (S) = 0 (the estimate of the Hausdorff measure was improved by a logarithmic factor in [5]). The key to the analysis in [2] is the following regularity criterion: there is an absolute constant > 0 such that, if u is a suitable weak solution of (NS) in × I and if for an interior point z ∈ × I , 1 |∇u(y, s)|2 dyds ≤ , lim sup (5) r r →0+ Q z,r then u is regular at z. See [18] for a simpler proof and [16] for more details. See [27, 29] for extensions when z lies on a flat or curved boundary. The objective of this paper is to present new sufficient conditions for the regularity of suitable weak solutions to (NS) in the interior, in terms of the smallness of the scaled L p,q -norm of the velocity, vorticity or the gradient of the vorticity. We obtained such results in terms of the velocity either in the interior or on a flat boundary in [10]. We will assume that the force f belongs to a parabolic Morrey space M2,γ , for some γ > 0, equipped with the norm 1 f 2M2,γ (×I ) = | f |2 dz . sup (6) 1+2γ r Q z,r Q z,r ⊂×I, r >0 5
(This space is trivial if γ > 2, and it contains L 2−γ ( × I ) for γ ≤ 2.) Suitable weak solutions will be defined in Definition 2.1 of Sect. 2.
Interior Regularity Criteria for Suitable Weak Solutions of NS Equations
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Theorem 1.1 (Regularity Criteria). Suppose the pair (u, p) is a suitable weak solution of (NS) in × I with force f ∈ M2,γ (× I ) for some γ > 0. Suppose z = (x, t) ∈ × I and Q z,r ⊂ × I . Then u is regular at z if one of the following conditions holds, for a small constant > 0 depending only on p ∗ (or p, p ), q, and γ (but independent of f M2,γ ). p ∗ ,q
(i) Velocity criteria. u ∈ L loc near z and lim sup r
−( p3∗ + q2 −1)
r →0+
where (u)r (s) =
1 |Br |
u − (u)r L p∗ ,q (Q z,r ) ≤ ,
(7)
u(y, s)dy, for some p ∗ , q satisfying
Br
1 ≤ 3/ p ∗ + 2/q ≤ 2, 1 ≤ p ∗ , q ≤ ∞.
(8)
The same result holds if u − (u)r is replaced by u in (7). p,q (ii) Velocity gradient criteria. ∇u ∈ L loc near z and lim sup r
−( 3p + q2 −2)
r →0+
∇u L p,q (Q z,r ) ≤ ,
(9)
for some p, q satisfying 2 ≤ 3/ p + 2/q ≤ 3, 1 ≤ q ≤ ∞.
(10)
p,q
(iii) Vorticity criteria. w = curl u ∈ L loc near z and lim sup r
−( 3p + q2 −2)
r →0+
w L p,q (Q z,r ) ≤ ,
(11)
for some p, q satisfying 2 ≤ 3/ p + 2/q ≤ 3, 1 ≤ q ≤ ∞, ( p, q) = (1, ∞).
(12)
p ,q
(iv) Vorticity gradient criteria. ∇ 2 u ∈ L loc near z and −(
3 2 + −3) p q
∇w L p ,q (Q
≤ ,
(13)
3 ≤ 3/ p + 2/q ≤ 4, 1 ≤ q, 1 ≤ p .
(14)
lim sup r r →0+
z,r )
for some p , q satisfying
Furthermore, for p > 1, ∇w can be replaced by curl w. Comments for Theorem 1.1. 1. The region defined by (8) corresponds to the union of II and III in Fig. 1, including all borderlines. The region defined by (10) corresponds to IV, including all borderlines. The region defined by (12) also corresponds to IV, but without the corner point (1/ p, 1/q) = (1, 0). The region defined by (14) corresponds to V, including all borderlines.
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Fig. 1. Regularity Criteria
2. In (8), the lower bound 1 ≤ 3/ p ∗ + 2/q is only to ensure a non-positive exponent of r in (7). The true limit is the upper bound 3/ p ∗ + 2/q ≤ 2. Similar comments apply to (10), (12) and (14). 3. The quantities in (7), (9), (11) and (13) are zero-dimensional, and are invariant under the scaling (4). Such quantities are useful in the regularity theory for (NS), see e.g. [2]. 4. In [10], the authors obtained Theorem 1.1 (i) only for region II, without the borderline q = 2 (but the result is also valid on a flat boundary of ). Theorem 1.1 (i) extends it to region III, and in particular includes the point (1/ p, 1/q) = (1/3, 1/2). It does not further assume the smallness of the pressure, in contrast to, e.g., Theorem 2.2. Special cases (1/ p, 1/q) = (1/3, 1/3) and (1/2, 0) were obtained in [39] and [30], respectively. 5. Theorem 1.1 (ii) contains the special case ( p, q) = (2, 2) of [2]. 6. Theorem 1.1 (iii) contains the special case ( p, q) = (2, 2) of [39]. Theorem 1.1 implies many known regularity criteria. Some of them are summarized below. For simplicity we assume f = 0. The Lorentz space L ( p,∞) for p < ∞ is defined with the norm v L ( p,∞) = supσ >0 σ |{|v| > σ }|1/ p . Corollary 1.2. Let u be a weak solution of (NS) in × I with f = 0 and Q z 0 ,r0 ⊂ × I for some r0 > 0. Then u is regular at z 0 if one of the following conditions holds. (i) Zero-dimensional integrals of u [8, 33, 9, 37]. If 3 2 + = 1, 3 < p ≤ ∞, p q
u ∈ L p,q (Q z 0 ,r0 ), or u ∈ L 3,∞ (Q z 0 ,r0 ) and u L 3,∞ (Q z
0 ,r0 )
(15)
is sufficiently small.
(ii) Lorentz spaces [38, 13, 34, 4]. If u is in L (q,∞) ((t0 − r 2 , t0 ); L ( p,∞) (Bx0 ,r )) with 3/ p + 2/q = 1, 3 < p < ∞, and u L (q,∞) L ( p,∞) (Q ) is sufficiently small. t
x
z 0 ,r
Interior Regularity Criteria for Suitable Weak Solutions of NS Equations
165
(iii) Zero-dimensional integrals of ∇u [1]. 3 2 + = 2, p q
∇u ∈ L p,q (Q z 0 ,r0 ),
3 < p ≤ ∞, 2
or ∇u ∈ L 3/2,∞ (Q z 0 ,r0 ) and ∇u L 3/2,∞ (Q z ,r ) is sufficiently small. 0 0 (iv) Zero-dimensional integrals of w = curl u [3]. w ∈ L p,q (Q z 0 ,r0 ),
3 2 + = 2, p q
or w ∈ L 3/2,∞ (Q z 0 ,r0 ) and w L 3/2,∞ (Q z
0 ,r0 )
3 < p ≤ ∞, 2
(16)
is sufficiently small.
Comments for Corollary 1.2. 1. To prove Corollary 1.2 using Theorem 1.1, we need to show that u is suitable under the corresponding assumptions. It suffices to show that |u|2 |∇u| ∈ L 1t,x , which justifies the integration by parts and thus one can 2prove the local 2energy inequality. In fact, it is enough to show u ∈ L 4t,x since |u| |∇u|dz ≤ u L 4 ∇u L 2 . 2/q
3/ p
For (i), it follows from u2L 4 ≤ u L p,q u L 2,∞ u L 6,2 . For (ii), since 3 < p < ∞, one can choose p1 ,q1 so that q1 < q,
p1 < p, 1/ p1 + 1/q1 ≤ 1/2, 3/ p1 + 1/q1 ≤ 1.
That is, (1/ p1 , 1/q1 ) lies in region V of Fig. 2 of [10]. By the imbedding of L ( p,∞) ⊂ L p1 and L (q,∞) ⊂ L q1 , we have u ∈ L p1 ,q1 . Interpolating with u ∈ L 2,∞ ∩ L 6,2 , we get u ∈ L 4t,x . For (iii), we have
2.
3.
4.
5.
2/q 3/ p p,q . u 6,2 ∇u L x,t L 2,∞ L x,t x,t
|u|2 |∇u|dz ≤ u
For (iv), since ∇u L p,q (Qr ) ≤ Cw L p,q (Q 2r ) + Cu L p,q (Q 2r ) (see Remark 3.7), it follows from (iii). Strictly speaking, one also needs to show that p ∈ L 3/2 so that (u, p) is suitable. But 5/3 15/14 (Q r ) this has already been done [35, 18]. By [18, Lem. 3.4], one has ∇ p ∈ L t L x for every weak solution in Q r . Let p(x, ˜ t) = p(x, t) − Br p(x, t) d x. The new pair (u, p) ˜ is suitable since the local energy inequality (22) remains the same if one 5/3 replaces p by p, ˜ and p˜ ∈ L t,x (Q r ) by Poincaré inequality. We now complete the proof of Corollary 1.2. For (ii), since 3 < p < ∞, one can (q,∞) ( p,∞) choose q2 < q, p2 < p, and 3/ p2 + 2/q2 = 2. Being small in L t Lx (Q z 0 ,r ) 1 q 2 p2 implies smallness in the scaled norm r L L (Q r ) by imbedding. Then one applies Theorem 1.1. For the rest, one imbeds L p,q to L p2 ,q for some suitable p2 < p. Corollary 1.2 (i) is due to several authors, already quoted above. Theorem 1.1 does not imply the end point case u ∈ L 3,∞ (Q z 0 ,r0 ) without smallness assumption, for which see [7]. (q,∞) p L x (Q r ) For Corollary 1.2 (ii), [38] proved regularity for small u in the classes L t (3,∞) with 3 < p < ∞, [13] in the class L ∞ L (see [14] for improvement), [34] in x t (q,∞) ( p,∞) (q,∞) ( p,∞) the classes L t Lx ( × I ) with 3 < p < ∞, [4] in the classes L t Lx ( p,∞) (q,∞) with 3 < p < ∞ and the classes L x Lt with 3 ≤ p < ∞. It follows from
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these results, in particular, that u is regular at z 0 if it satisfies, for θ ∈ [0, 1] and some = (θ ) > 0, lim ess sup |t − t0 |θ/2 |x − x0 |1−θ |u(x, t)| ≤ .
r →0
(17)
Q z 0 ,r
Our Theorem 1.1 does not cover the endpoint cases p = 3, ∞, except the cases θ = 0, 1 in (17) when suitability is assumed. 6. Corollary 1.2 (iii) was proved in [1] for the cases 3/2 < p < ∞. The endpoints p = 3/2 and p = ∞ were not obtained in [1]. The p = 3/2 case without the smallness assumption follows from [7] and imbedding. 7. Corollary 1.2 (iv) was proved in [3, Prop. 2]. The main result in [3, Th. 1] shows p,q regularity near z 0 assuming only two components of the vorticity belonging to L x,t . Again, the p = 3/2 case without the smallness assumption follows from [7] and Remark 3.7. A major motivation for the study of such regularity criteria is to improve the partial regularity result of [2]. For example, Constantin [6] proved, when = T3 , the existence of suitable weak solutions satisfying (18) ∇w ∈ L 4/3− ( × I ), ∀0 < 1. 4/3− dz has dimension 1 + 3. Combining this estimate Note that the integral |∇w| with Theorem 1.1 (iv), we find that the parabolic Hausdorff dimension of the singular set S of u is at most one. This is slightly weaker than the CKN theorem that the one-dimensional parabolic Hausdorff measure of S is actually zero. Note that Scheffer [25, 26] constructed examples satisfying the local energy inequality and their dimensions of singular sets are arbitrarily close to one. Thus the CKN result is optimal for functions satisfying only the local energy inequality. However, the proof of (18) uses the equation for the vorticity, which may not be satisfied by Scheffer’s examples. Therefore there might be hope to prove other a priori estimates for w and thus improve the partial regularity. The rest of this paper is organized as follows. In Sect. 2 we introduce some scaling invariant functionals, recall the notion of suitable weak solutions and a regularity criterion involving the scaled norms of velocity and pressure. In Sect. 3 we establish some estimates regarding the velocity, pressure and vorticity, and prove Theorem 1.1. We finally correct a typo in Lemma 15 of [10]: In the second line from the bottom of p. 615 and the 1third line3 of p. 616, the function C(r ) should be replaced by ˆ ) = r −2 C(r |u − Qr |Q r | Q r u| dz. 2. Preliminaries In this section we introduce the notation, review suitable weak solutions, and recall a regularity criterion involving scaled norms. We start with the notation. Let be either an open domain in R3 or the 3-dimensional torus T3 , and I be a finite time interval. By N = N (α, β, . . .) we denote a constant depending on the prescribed quantities α, β, . . ., which may change from line to line. For 1 ≤ q ≤ ∞, W k,q () denote the usual Sobolev spaces, i.e. W k,q () = q α q { f ∈ L () : D f ∈ L (), 0 ≤ |α| ≤ k}. We denote by E f the average of f a function f(x, t), E ⊂ and J ⊂ I , we denote on E; i.e., E f = E f /|E|. For f L p,q (E×J ) = f L q L p (E×J ) = f L p (E) L q (J ) .
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Next, we define several scaling-invariant functionals similar to those in [2, 18, 16, 27]. For a suitable weak solution (u, p) and z = (x, t) ∈ × I , let 1 1 2 |u(y, s)| dy, E(r ) := |∇u(y, s)|2 dy ds, A(r ) := sup r Q z,r t−r 2 ≤s
1 1 1 = − . ∗ p p 3
(19)
Recall w = ∇ × u is the vorticity field of u. We define 1 ˜ ) := 1 u(y, s) − (u)r (s) q p∗ , G 1 (r ) := ∇u(y, s) L qs L yp (Q z,r ) , G(r L s L y (Q z,r ) r r W (r ) :=
1 w(y, s) L qs L yp (Q z,r ) . r
When 1 ≤ q ≤ 2, we also define W1 (r ) :=
1 1 ∇w(y, s) q p , W˜ 1 (r ) := curl w(y, s) q p , L s L y (Q z,r ) L s L y (Q z,r ) r r
where p is the number satisfying, for p, q as in (19), 3 2 + = 4, p q
1 1 1 3 = − , 1 ≤ p ≤ . p p 3 2
(20)
We now define suitable weak solutions for the (NS). Definition 2.1. Suppose that f belongs to the parabolic Morrey space M2,γ ( × I ) for some γ ∈ (0, 2]. A pair (u, p) is a suitable weak solution to the Navier-Stokes equations (1) in × I with force f if the following conditions are satisfied: (a) The functions u : × I → R3 and p : × I → R satisfy u ∈ L ∞ (I ; L 2 ()) ∩ L 2 (I ; W 1,2 ()),
3
p ∈ L 2 ( × I ).
(21)
(b) u and p solve (1) in × I in the sense of distributions. (c) u and p satisfy the local energy inequality
≤
|u(x, t)|2 φ(x, t) dx + 2
t t0
∇u(x, t )2 φ(x, t ) dx dt
t |u|2 (∂t φ + φ) + (|u|2 + 2 p)u · ∇φ + 2 f · uφ dx dt t0
for all t ∈ I = (t0 , t1 ) and all nonnegative functions φ ∈ C0∞ ( × I ).
(22)
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In this definition we impose no initial or boundary condition for u. The main difference between suitable weak solutions and Leray-Hopf weak solutions (see [2, p.779]) is the additional condition of the local energy inequality (22). The existence of suitable weak solutions is proved in [22, 2]. Definition 2.1 is the slightly modified version used in [18]. As remarked in [2, p. 823], it is an open question if all weak solutions are suitable. Next we recall a local regularity criterion, which is a refined version of [2, Prop. 1], and is formulated in the present form with f = 0 in [19, 18], and proved with nonzero f ∈ M2,γ in [16, Prop. 2.8]. Theorem 2.2. There exists > 0 depending only on γ > 0 (and independent of f M2,γ ), and r0 > 0 depending on f M2,γ , such that if (u, p) is a suitable weak solution of (NS) with f ∈ M2,γ , then u is regular at z = (x, t) ∈ × I if C(r ) + D(r ) < for some r ∈ (0, r0 ).
(23)
An important feature of (23) is that it requires only one r , not infinitely many r . We will prove our regularity criteria based on this theorem. For our proof in the next section, in order to get (23), it suffices to assume γ > −1. The assumption γ > 0 is made in order to apply Theorem 2.2. 3. Local Interior Regularity In this section, we present the proof of Theorem 1.1. Through the entire section, we assume (u, p) is a suitable weak solution in × I . Without loss of generality, we assume z = (0, 0) and Q r = Q (0,0),r ⊂ × I . By Hölder inequality, it suffices to consider borderline exponents, i.e., those exponents p, p ∗ , p and q satisfying (19) and (20). Denote m γ = f M2,γ . −1/(1+γ )
Lemma 3.1. Suppose Q 2r ⊂ × I and 0 < r ≤ m γ
. Then
A(r ) + E(r ) ≤ N [1 + C(2r ) + D(2r )]. Proof. By choosing suitably localized φ in the local energy inequality (22), we get 2 1 2 +r | f | dz A(r ) + E(r ) ≤ N C 3 (2r ) + C(2r ) + 2 u L 3 (Q 2r ) p 3 L 2 (Q 2r ) r Q 2r which is bounded by N [1 + C(2r ) + D(2r ) + r 2(γ +1) m 2γ ]. Lemma 3.2. Suppose u ∈ L p
∗ ,q
(Q r ) with 3/ p ∗ + 2/q = 2, 1 ≤ q ≤ ∞, then 1
1
˜ ) ≤ N A q (r )E 1− q (r )G(r ˜ ). C(r Proof. Let α = (2 p ∗ − 3)/3 p ∗ and β = 1/ p ∗ . Note 1/3 = α/2 + β/6 + (1 − α − β)/ p ∗ . Using the Hölder inequality and Sobolev imbedding, we obtain β
u − (u)r L 3 (Br ) ≤ N uαL 2 (B ) u − (u)r L 6 (B ) u − (u)r p∗ L (B r
1−α−β r)
r
1
β
≤ N uαL 2 (B ) ∇u L 2 (B ) u − (u)r L3 p∗ (B ) , r
r
r
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where we used 1 − α − β = 1/3. Raising to the third power, integrating in time and dividing both sides by r 2 , we get N 0 3β ˜ u3α ∇u L 2 (B ) u − (u)r L p∗ (Br ) dt C(r ) ≤ 2 L 2 (Br ) r r −r 2 0 3β2 0 q1 N 3α 3α q 2 ∇u L 2 (B ) dt u − (u)r p∗ ≤ 2 r 2 A 2 (r ) dt , L (Br ) r r −r 2 −r 2 1
which equals N A q (r )E
1− q1
˜ ). (r )G(r
Lemma 3.3. Suppose 0 < 2r ≤ ρ and Q ρ ⊂ × I . Then ρ 2 r ˜ C(ρ) + N C(r ) ≤ N C(ρ). ρ r Proof. This follows from the Hölder inequality: ρ 2 r N 3 3 ˜ C(ρ) + N |(u)ρ | + |u − (u)ρ | dz ≤ N C(r ) ≤ 2 C(ρ). r Qr ρ r Lemma 3.4. Suppose 0 < 2r ≤ ρ and Q ρ ⊂ × I . Then ρ 2 3 3 r (γ +1) 2 ˜ 2 D(ρ). (C(ρ) + ρ mγ ) + N D(r ) ≤ N r ρ
(24)
Proof. Let φ(x) ≥ 0 be supported in Bρ with φ = 1 in Bρ/2 . The divergence of (1) gives −p = ∂i ∂ j u i u j − ∇ · f in the sense of distributions. Let
1 p1 (x, t) := ∂i ∂ j (u i − (u i )ρ )(u j − (u j )ρ )φ − ∇ · ( f φ) (y, t)dy 3 |x 4π − y| R and p2 (x, t) := p(x, t) − p1 (x, t). Due to div u = 0, p2 = 0 in Bρ/2 . By the mean value property of harmonic functions, 3 3 3 3 1 Nr Nr Nr 2 2 2 | p2 | d x ≤ 3 | p2 | d x ≤ 3 | p1 | 2 d x. | p| d x + 3 2 r Br ρ ρ ρ Bρ/2 Bρ Bρ By Calderon-Zygmund and potential estimates, 3 Nρ 9/4
3 3 3 r 1 N 2 2 | | | | u − (u) p p d x ≤ d x ≤ + | f |2 d x 4 . 1 1 ρ 3 2 2 2 ρ Bρ r Bρ r Bρ r Bρ Adding these estimates, integrating in time, and using 3/2 Nr −3/2 m γ ρ 3+3γ /2 ,
1 r2
| f |2 d x
we get 3
| p| 2 dz ≤ Qr
ρ 9/4 Bρ −r 2 r 2
0
1 r2
3
3
| p1 | 2 + | p2 | 2 dz ≤ RHS of (24). Qr
3 4
dt ≤
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Now we are ready to prove Theorem 1.1 (i). Proof of Theorem 1.1 (i). It suffices to prove the borderline cases 3/ p ∗ + 2/q = 2 and 1 ≤ q ≤ ∞. The other cases follow by Hölder inequality. Suppose 0 < 4r ≤ ρ. By Lemmas 3.2 and 3.4, and by Lemma 3.3, we get ρ ρ 2 ρ 3 3 r ρ (γ +1) 2 ˜ 2 C(r ) + D(r ) ≤ N C +D +N +ρ mγ C ρ 2 2 r 2 ρ ρ 2 ρ r C +D +N ≤N ρ 2 2 r ρ 3 1 ρ 1 ρ 3 1− q (γ +1) 2 q ˜ 2 E G +ρ × A mγ . 2 2 2 −1/(γ +1)
Suppose ρ ≤ m γ N
ρ 2 r
1
Aq
ρ 2
. By Lemma 3.1,
E
1− q1
ρ
ρ ρ 2 ˜ G˜ ≤N (1 + C(ρ) + D(ρ)) G(ρ). 2 2 r
Combining the above estimates, we obtain r ρ 2 ˜ C(r ) + D(r ) ≤ N2 G(ρ) (C(ρ) + D(ρ)) + ρ r ρ 2 3 3 (γ +1) 2 ˜ 2 + N2 mγ . G(ρ) + ρ r θ 2/3 1/(γ +1) Choose θ ∈ (0, 1/4) so that N2 θ < 1/4. We fix r0 < min{1, m1γ , m1γ ( 8N ) } 2 2
˜ ) < θ min{1, } for all r ≤ r0 , where is the constant in Theorem such that G(r 1+8N2 2.2. Replacing r and ρ by θr and r , respectively, we get 2
C(θr ) + D(θr ) ≤
1 (C(r ) + D(r )) + , ∀r < r0 . 2 4
By iteration, C(θ k r ) + D(θ k r ) ≤
1 (C(r ) + D(r )) + , ∀r < r0 . k 2 2
Thus, for k sufficiently large, C(θ k r ) + D(θ k r ) ≤ , from which z is a regular point due to Theorem 2.2. The last statement of Theorem 1.1 (i), that one can replace u − (u)r by u, is because u − (u)r L p∗ ,q ≤ N u L p∗ ,q . The following modification of Lemma 3.2 is all that is needed to prove Theorem 1.1 (ii). Lemma 3.5. Suppose 0 < 2r ≤ ρ and Q ρ ⊂ × I . Then ˜ ) ≤ N A1/q (r )E 1−1/q (r )G 1 (r ). C(r
(25)
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Proof. The proof is similar to that of Lemma 3.2. When 1 ≤ p < 3, using the same exponents α = 1 − 1/ p and β = 1/ p − 1/3, we have 3β
3(1−α−β) r)
u − (u)r L 6 (B ) u − (u)r p∗ u − (u)r 3L 3 (B ) ≤ N u3α L 2 (B ) L (B r
r
r
2/q
2−2/q
≤ N u L 2 (B ) ∇u L 2 (B ) ∇u L p (Br ) . r
r
If p = 3 (and q = 1), by Gagliardo-Nirenberg and Poincaré inequalities, u − (u)r 3L 3 (B ) ≤ N u − (u)r 2L 2 (B ) ∇u L 3 (Br ) + r
r
N u − (u)r 3L 2 (B ) r r 3/2
≤ N u2L 2 (B ) ∇u L 3 (Br ) . r
Integrating in time and applying the Hölder inequality, we get (25).
Proof of Theorem 1.1 (ii). The proof is the same as that for Theorem 1.1 (i): we only ˜ ) by G 1 (r ). need to replace Lemma 3.2 by Lemma 3.5, and replace the quantity G(r The next lemma shows that the gradient of the velocity can be controlled by the vorticity. This is the key to Theorem 1.1 (iii). p,q
Lemma 3.6. Suppose 0 < 2r ≤ ρ and Q ρ ⊂ × I . Suppose ∇u ∈ L x,t (Q ρ ) with 3 2 p + q = 3 and 1 ≤ q < ∞. Then G 1 (r ) ≤ N
ρ r
3 −1 r p W (ρ) + N G 1 (ρ). ρ
(26)
Furthermore, if p = 3 (so q = 1), then G 1 (r ) ≤ N
ρ r
r W (ρ) + N G 1 (ρ) + g(u; r ), ρ
(27)
where g(u; r ) → 0 as r → 0. Proof. Choose a standard cut off function φ supported in Bρ such that φ = 1 in B3ρ/4 . Define 1 × w(y, t)φ(y)dy, h = u − v. v(x, t) := ∇x 3 4π |x − y| R Note that x h(x, t) = 0 in B3ρ/4 . We give the proof of (26) first. By the mean value property of harmonic functions, for each fixed time t, 3/ p 3/ p r r ∇h L p (Br ) ≤ N ∇h L p (Bρ/2 ) ≤ N ∇u L p (Bρ ) + ∇v L p . ρ ρ On the other hand, due to Calderon-Zygmund estimates, for each fixed time, ∇v L p ≤ N w L p (Bρ ) .
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Combining these estimates, we obtain ∇u L p (Br ) ≤ ∇v L p (Br ) + ∇h L p (Br ) ≤ N w L p (Bρ ) + N
3 r p ∇u L p (Bρ ) . ρ
Taking L q -norm in time and dividing both sides by r , we get (26). To prove (27), set p = 3 (so q = 1), use the above estimate for ∇v, and modify the estimate for ∇h as follows: ∇h L 3 (Br ) ≤ ∇h − (∇h)r L 3 (Br ) + (∇h)r L 3 (Br ) .
(28)
The second term in (28) is just Nr |(∇h)r |. For the first term in (28), use the PoincaréSobolev inequality, the mean-value property, and an interior estimate: 2 r 2 2 ∇h − (∇h)r L 3 (Br ) ≤ N ∇ h 3/2 ≤N ∇ h 3/2 L (Br ) L (Bρ/2 ) ρ 2 2 r r ∇h L 3 (Bρ ) ≤ N ≤N [∇u L 3 (Bρ ) + ∇v L 3 (Bρ ) ] ρ ρ 2 r ∇u L 3 (Bρ ) . ≤N (29) ρ Combine this estimate with the above estimate for ∇v L 3 , divide by r , and integrate in time to get 0 r ρ G 1 (r ) ≤ N W (ρ) + N G 1 (ρ) + N |(∇h)r |dt. (30) r ρ −r 2 Since h (and hence ∇h) is harmonic in B3ρ/4 , (∇h)r = (∇h)ρ/2 , and so |(∇h)r | = |(∇h)ρ/2 | ≤ Thus
0
N ∇h L 3 (Bρ/2 ) . ρ
N g(u; r ) := N |(∇h)r |dt ≤ 2 ρ −r
0
−r 2
∇u L 3 (Bρ ) dt.
Since ∇u ∈ L 3,1 (Q ρ ), we have g(u, r ) → 0 as r → 0, and so (30) yields (27).
p,q
Remark 3.7. By similar argument, if w = curl u ∈ L loc near z, then so is ∇u, since ∇u L p,q (Qr ) ≤ N w L p,q (Q ρ ) + N u L p,q (Q ρ ) if 0 < r < ρ ≤ 2r . Proof of Theorem 1.1 (iii). It suffices to prove the borderline cases 3/ p + 2/q = 3 and 1 < p ≤ 3. The other cases follow by Hölder’s inequality. If p < 3, we use the estimate (26), and if p = 3, we use the refined estimate (27). Choose θ ∈ (0, 1/4) so that if p < 3, then N θ 3/ p−1 < 1/2, where N is the constant in (26), and if p = 3, N θ < 1/2, where N is the constant from (27). Replace r, ρ by θr and r , respectively. Note that G 1 (r ) is finite by Remark 3.7. The estimate (26) ( p < 3) or (27) ( p = 3) then implies 1 N 0 if p < 3 . G 1 (θr ) ≤ W (r ) + G 1 (r ) + N g(u; θr ) if p = 3 θ 2
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θ Choose r0 so that supr
G 1 (θr ) ≤
8N ,
where is the
1 G 1 (r ) + . 2 4
Iterating this estimate, we obtain, for all r ≤ r0 , G 1 (θ k r ) ≤
1 G 1 (r ) + . 2k 2
(31)
Choose an integer k0 ≥ 3 + supθr0
satisfy (19) and (20). If ∇w ∈ L x,t (Q ρ ), then W (r ) ≤ N
ρ r
3 −1 r p W1 (ρ) + N W (ρ). ρ
(32)
Furthermore, if 1 ≤ q < 2, we have G 1 (r ) ≤ N
ρ r
W˜ 1 (ρ) + N
3/ p r G 1 (ρ) + g(u; r ) ρ
(33)
with g(u; r ) → 0 as r → 0. Proof. Statement (32) follows from Sobolev imbedding: w L p (Br ) ≤ (w)ρ L p (B ) + w − (w)ρ L p (B ) r ρ 3 p r w L p (Bρ ) + N ∇w L p (B ) . ≤N ρ ρ Taking the L q norm in time and dividing both sides by r , we get (32). The proof of (33) is similar to that of the second part of Lemma 3.6. Choose a standard cut off function φ supported in Bρ such that φ = 1 in B3ρ/4 . Define 2-tensors V (x, t) :=
R3
∇x
1 (curl w(y, t)) φ(y)dy, 4π |x − y|
H := ∇u − V.
Note that since ∇ · u = 0, −u = curl w, and so x H (x, t) = 0 in B3ρ/4 . Potential estimates give, if p > 1, (i.e. q < 2), V L p ≤ N curl w L p (B ) . ρ
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The same estimate as in (29) gives H − (H )r L p (Br )
3 r p H L p (Bρ ) ≤N ρ 3 r p ∇u L p (Bρ ) + curl w L p (B ) . ≤N ρ ρ 3
3
−1 −1 Using r1 (H )r L p (Br ) = Nr p |(H )r | = Nr p |H |r =0 together with the last two estimates, we find (33) with 0 1/q 3 −1 g(u; r ) := Nr p (|H |r =0 )q dt . −r 2
Now arguing as in the proof of Lemma 3.6, 3
g(u; r ) = Nr p
−1
0
−r 2
1/q |(H )ρ/2 |q dt
3
≤
−1
Nr p ρ 3/ p
0 −r 2
and since ∇u ∈ L p,q (Q ρ ), we have g(u, r ) → 0 as r → 0.
1/q (∇u L p (Bρ ) )q dt
,
Proof of Theorem 1.1 (iv). It suffices to prove the borderline cases 3/ p + 2/q = 4. The other cases follow by Hölder inequality. We also assume z = (0, 0). We first consider 1 < q ≤ 2. Let be the constant in Theorem 1.1 (iii) and we suppose W1 (r ) ≤ /4 for any r < r0 . Our assertion follows a procedure similar to the proof in Theorem 1.1 (iii). Since 3p − 1 > 0 in (32), we replace r and ρ by θr and r , respectively, after choosing θ ∈ (0, 1/2) appropriately, and then iterate (32). This procedure leads to the conclusion that W (θ k r ) < /2 for r < r0 and k sufficiently large, and so W (r ) < for r < θ k r0 . Theorem 1.1 (iii) then implies the regularity. For 1 ≤ q < 2, we can use (33) instead. Arguing just as in the second part of the proof of Theorem 1.1(iii), we conclude that G 1 (r ) can be made small enough to apply Theorem 1.1(ii), provided W˜ 1 (r ) ≤ W1 (r ) can be made arbitrarily small. Acknowledgements. The research of Gustafson and Tsai is partly supported by NSERC grants. The research of Kang is supported partly by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2006-311-C00007). Kang thanks the University of British Columbia and the Pacific Institute of Mathematical Sciences for their hospitality during his visit in February 2006.
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6. Constantin, P.: Navier-Stokes equations and Area of Interfaces. Commun. Math. Phys. 129, 241– 266 (1990) 7. Escauriaza, L., Seregin, G., Šverák, V.: L 3,∞ -solutions of Navier-Stokes equations and backward uniqueness. Usp. Mat. Nauk 58 (2(350)) 3–44 (2003) (in Russian); translation from Russ. Math. Surv. 58(2), 211–250 (2003) 8. Fabes, E.B., Jones, B.F., Riviére, N.M.: The initial value problem for the Navier-Stokes equations with data in L p . Arch. Ration. Mech. Anal. 45, 222–240 (1972) 9. Giga, Y.: Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier-Stokes system. J. Differ Eqs. 62(2), 186–212 (1986) 10. Gustafson, S., Kang, K., Tsai, T.-P.: Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary. J. Differ. Eqs. 226, 594–618 (2006) 11. Hopf, E.: Über die Anfangswertaufgabe f¨r die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951) 12. Kang, K.: On boundary regularity of the Navier-Stokes equations. Comm. Partial Differ. Eqs. 29(7–8), 955–987 (2004) 13. Kozono, H.: Removable singularities of weak solutions to the Navier-Stokes equations. Comm. Partial Differ. Eqs. 23(5–6), 949–966 (1998) 14. Kozono, H.: Weak solutions of the Navier-Stokes equations with test functions in the weak-L n space. Tohoku Math J. (2) 53, 55–79 (2001) 15. Ladyzenskaja, O.A.: Uniqueness and smoothness of generalized solutions of Navier-Stokes equations, (Russian) Zap. Nauˇcn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 5, 169–185 (1967) 16. Ladyzenskaja, O.A., Seregin, G.A.: On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations. J. Math. Fluid Mech. 1, 356–387 (1999) 17. Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 64, 193–248 (1934) 18. Lin, F.-H.: A new proof of the Caffarelli-Kohn-Nirenberg theorem. Comm. Pure Appl. Math. 51(3) 241–257 (1998) 19. Neˇcas, J., R˚užiˇcka, M., Šverák, V.: On Leray’s self-similar solutions of the Navier-Stokes equations. Acta Math. 176(2), 283–294 (1996) 20. Prodi, G.: Un teorema di unicita per le equazioni di Navier-Stokes (Italian). Ann. Mat. Pura Appl. 48(4), 173–182 (1959) 21. Scheffer, V.: Partial regularity of solutions to the Navier-Stokes equations. Pacific J. Math. 66(2), 535–552 (1976) 22. Scheffer, V.: Hausdorff measure and the Navier-Stokes equations. Commun. Math. Phys. 55, 97–112 (1977) 23. Scheffer, V.: The Navier-Stokes equations on a bounded domain. Commun. Math. Phys. 73(1), 1–42 (1980) 24. Scheffer, V.: Boundary regularity for the Navier-Stokes equations in a half-space. Commun. Math. Phys. 85(2), 275–299 (1982) 25. Scheffer, V.: A solution to the Navier-Stokes inequality with an internal singularity. Commun. Math. Phys. 101(1), 47–85 (1985) 26. Scheffer, V.: Nearly one-dimensional singularities of solutions to the Navier-Stokes inequality. Commun. Math. Phys. 110(4), 525–551 (1987) 27. Seregin, G.A.: Local regularity of suitable weak solutions to the Navier-Stokes equations near the boundary. J. Math. Fluid Mech. 4(1), 1–29 (2002) 28. Seregin, G.A.: On Smoothness of L 3,∞ -Solutions to the Navier-Stokes Equations up to Boundary. Math. Ann. 332(1), 219–238 (2005) 29. Seregin, G.A., Shilkin, T.N., Solonnikov, V.A.: Boundary partial regularity for the Navier-Stokes equations. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310 (2004), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 34, 158–190, 228; translation in J. Math. Sci. (N. Y.) 132(3), 339–358 (2006) 30. Seregin, G.A., Sverak, V.: Navier-Stokes equations with lower bounds on the pressure. Ration. Mech. Anal. 163(1), 65–86 (2002) 31. Serrin, J.: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 9, 187–195 (1962) 32. Serrin, J.: The initial value problem for the Navier-Stokes equations. In: 1963 Nonlinear Problems, Proc. Sympos., Madison, Wis., Madison, WI: Univ. of Wisconsin Press, 1963, pp. 69–98 33. Sohr, H.: Zur Regularitätstheorie der instationären Gleichungen von Navier-Stokes. Math. Z. 184(3), 359–375 (1983) 34. Sohr, H.: A regularity class for the Navier-Stokes equations in Lorentz spaces. J. Evol. Equ. 1(4), 441–467 (2001) 35. Sohr, H.: Zur Regularitätstheorie der instationären Gleichungen von Navier-Stokes. Math. Z. 184(3), 359–375 (1983)
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36. Solonnikov, V.A.: Estimates of solutions of the Stokes equations in S. L. Sobolev spaces with a mixed norm. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 288, 204–231 (2002) 37. Struwe, M.: On partial regularity results for the Navier-Stokes equations. Comm. Pure Appl. Math. 41, 437–458 (1988) 38. Takahashi, S.: On interior regularity criteria for weak solutions of the Navier-Stokes equations. Manuscripta Math. 69(3), 237–254 (1990) 39. Tian, G., Xin, Z.: Gradient estimation on Navier-Stokes equations. Comm. Anal. Geom. 7(2), 221–257 (1999) Communicated by P. Constantin
Commun. Math. Phys. 273, 177–202 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0251-1
Communications in
Mathematical Physics
Universality Classes in Burgers Turbulence Govind Menon1 , Robert L. Pego2 1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.
E-mail:
[email protected]
2 Center for Nonlinear Analysis, Department of Mathematical Sciences, Carnegie Mellon University,
Pittsburgh, PA 15213-3890, USA. E-mail:
[email protected] Received: 6 July 2006 / Accepted: 12 November 2006 Published online: 8 May 2007 – © Springer-Verlag 2007
Abstract: We establish necessary and sufficient conditions for the shock statistics to approach self-similar form in Burgers turbulence with Lévy process initial data. The proof relies upon an elegant closure theorem of Bertoin and Carraro and Duchon that reduces the study of shock statistics to Smoluchowski’s coagulation equation with additive kernel, and upon our previous characterization of the domains of attraction of selfsimilar solutions for this equation. 1. Introduction The construction of stochastic processes that are also weak solutions to the equations of fluid mechanics is one approach to rigorous mathematical theories of turbulence. This is poorly understood at present, and we must settle for insights from vastly simplified model problems. We consider the invisicid Burgers equation 2 u = 0, t > 0, x ∈ R, u(x, 0) = u 0 (x), ∂t u + ∂ x (1) 2 with random initial data u 0 . The problem is to determine the statistical properties of the Cole-Hopf (entropy) solution u(x, t) to (1), given the statistical properties of u 0 . There is a large literature on the subject; we refer to Burgers’ book [10] and the more recent survey articles [17, 25, 36]. The problem was proposed by Burgers as a model for turbulence in incompressible fluids, but it has several well-known flaws in this regard. Explicit solutions play a special role in the theory. Burgers studied the case when u 0 is white noise in his monograph [10]. His work remains the foundation for several rigorous results, which culminate with the complete solution by Frachebourg and Martin for the velocity and shock statistics (see [19] and references therein). The case when u 0 is a Brownian motion has attracted much attention since the work of She, Aurell and Frisch [34] and Sinai [35]. An elegant solution to this problem was obtained by
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Bertoin [5] and Carraro and Duchon [12]. More generally, these authors considered initial data that comprise a Lévy process with only downward jumps (i.e., shocks). A Lévy process X x (x ≥ 0) is a continuous-time random walk with stationary and independent increments. It is determined completely by its characteristic exponent , satisfying E(eik X x ) = e−x(k) , via the celebrated Lévy-Khintchine formula σ 2k2 + 1 − eiks + iks1|s|<1 (ds), k ∈ R. (2) (k) = ibk + 2 R The process X is the superposition of three independent processes related to this formula: a Brownian motion with variance σ 2 and drift −b ∈ R, a compound Poisson process with jump measure 1|s|≥1 , and a pure jump martingale with jump measure 1|s|<1 (see [4, Ch.1]). The measure is arbitrary, subject to the condition R (1 ∧ s 2 )(ds) < ∞, where a ∧ b means min(a, b). We say X is spectrally negative if is concentrated on the half-line s < 0. In all that follows we assume 0, x < 0, u 0 (x) = (3) a spectrally negative Lévy process, x ≥ 0. As we show below, we can always reduce to the case where u 0 (x) has zero mean E(u 0 (x)) 0 for all x, and −∞ (|s| ∧ s 2 )(ds) < ∞. It is then more convenient to use the Laplace exponent ∞ −qs σ 2q 2 + ψ(q) = −(−iq) = e − 1 + qs (ds), q > 0, (4) 2 0 where ((s, ∞)) = ((−∞, −s)) for every s > 0, so that E(eq X x ) = e xψ(q) and E(X x ) = xψ (0) = 0. For this class of initial data, Bertoin proved a remarkable closure property for the entropy solution of (1), namely: x → u(x, t) − u(0, t) remains a spectrally negative Lévy process for all t > 0. This closure property was first noted by Carraro and Duchon in connection with their notion of statistical solutions to Burgers equation [11]. That these statistical solutions agree with the Cole-Hopf solution for spectrally negative data was shown by Bertoin [5, Thm. 2]. The closure property fails if u 0 has positive jumps— these positive jumps open into rarefaction waves for t > 0, and this is incompatible with the rigidity of sample paths of Lévy processes. An interesting formal analysis of closure properties of Burgers equation is presented in [13]. Henceforth, we write v(x, t) = u(x, t) − u(0, t) for brevity. The Lévy-Khintchine representation now implies that the law of the Lévy process x → v(x, t) is completely described by a corresponding “Lévy triplet” (bt , σt2 , t ). The mean drift bt = E(v(1, t)) satisfies bt = b0 = 0 for every t ≥ 0. Moreover, for every t > 0, v(·, t) is of bounded variation, thus the variance σt = 0. Consequently, the law of v(·, t) is completely determined by only the jump measure t which contains the shock statistics. It is a striking fact, implicit in [5], that the evolution of t is described by Smoluchowski’s coagulation equation with additive kernel, an equation that arises in entirely different areas such as the analysis of algorithms [14], the kinetics of polymerization [37], and cloud formation from droplets [23] (see [2] for a review). What this means is that mean-field theory is exact for Burgers equation with initial data of the form (3), i.e., random one-sided data with stationary and independent increments. We
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give a precise statement to this effect below in Theorem 2. Several connections between stochastic models of coalescence and Burgers turbulence are reviewed in [8]. Here, we use the closure property as a basis for a rigorous study of universality classes for dynamic scaling in Burgers turbulence. Our motivation is the following. A central theme in studies of homogeneous isotropic turbulence in incompressible fluids is the universality of the Kolmogorov spectrum [30]. A possible rigorous formulation of such universality involves (a) the construction of stochastic processes that mimic a ‘typical turbulent flow’, and (b) a characterization of the domains of attraction of these processes. For Burgers turbulence, step (a) consists of constructing exact solutions for special initial data, say white noise or Brownian motion. In this article, we carry out step (b) for initial data that satisfy (3). Domains of attraction are studied in the classical limit theorems in probability (e.g., the central limit theorem), and their process versions (e.g., Donsker’s invariance principle). For Smoluchowski’s coagulation equation with additive kernel, we characterized all possible domains of attraction in [32], a result akin to the classical limit theorems. In this article we deal with a process version. In all that follows, we consider the processes x → v(x, t) as elements of the space D of right continuous paths R+ → R with left limits (càdlàg paths) equipped with the Skorokhod topology [28, Ch. VI]. The shock statistics determine completely the law of this process (a probability measure on D). Approach to limiting forms will be phrased in terms of weak convergence of probability measures on D. Among the initial data we consider, the stable processes are of particular importance because of their self-similarity. Let X α , α ∈ (1, 2] denote the stable process with Laplace exponent q α (α = 2 corresponds to Brownian motion). The corresponding jump measure (ds) = s −1−α ds/ (−α) for α < 2. There is a one-to-one correspondence between (a) these stable processes, (b) statistically self-similar solutions in Burgers turbulence, and (c) self-similar solutions to Smoluchowski’s coagulation equation. Precisely, this works as follows. Let α ∈ (1, 2], and let T α denote the first-passage process for x → X xα + x, i.e.,
Txα = inf{y X αy + y > x }. (5) The velocity field is obtained from Txα by considering the associated spectrally negative process Vxα = x − Txα , x ≥ 0. (6) Then for the solution to (1) with u 0 (x) = X xα for x ≥ 0, v(x, t) is statistically self-similar, with α−1 L , t, x > 0. (7) v(x, t) = t 1/β−1 Vxtα −1/β , β = α L
Here = means both processes define the same measure on D. The process T α is a pure jump Lévy process with Lévy measure f α (s) ds, where f α is the number density profile of a self-similar solution to Smoluchowski’s coagulation equation [32, Sect. 6]: f α (s) =
∞ 1 (−1)k−1 s kβ−2 (1 + k − kβ) sin π kβ, α ∈ (1, 2]. π k!
(8)
k=1
These solutions are related to classical distributions in probability theory by rescaling. If p(s; α, 2 − α) denotes the density of a maximally-skewed Lévy stable law [18, XVII.7]
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we have [7, 32]
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f α (s) = s β−2 p(s β ; α, 2 − α).
(9)
By the Lévy-Itô decomposition [4, Thm 1.1] and (7) we may conclude that the magnitudes of shocks in u(·, t) form a Poisson point process valued in (0, ∞) whose characteristic measure is (10) αt (ds) = t 1−2/β f α st 1−1/β ds. For 1 < α < 2 the self-similar solutions have algebraic tails, with f α (s) ∼ s −1−α / (−α) as s → ∞. The case α = 2 is particularly important since it corresponds to Brownian initial data. Here we obtain a solution found by Golovin in a model for cloud formation from droplets [23], f 2 (s) = (4π )−1/2 s −3/2 e−s/4 .
(11)
For the corresponding solution to (1), the law of v(x, t) can be recovered from the law of Tx2 , the first-passage time for Brownian motion with unit drift, which is explicitly given as follows (see Sect. 2.6): x1 y>0 (x − y)2 2 dy. (12) P(Tx ∈ (y, y + dy)) = exp − 4y 2 π y3 Considering now arbitrary solutions to (1) with initial data (3), we classify solutions that approach self-similar form as t → ∞ as follows. A rescaled solution au(λx, τ t) is again a solution of Burgers equation if and only if a = τ/λ. If we set t = 1, regard λ as a function of τ , and relabel τ as t, we see it is natural to study the large-t behavior of the processes t x → Vx(t) := v(λ(t)x, t). (13) λ(t) We shall establish necessary and sufficient conditions for convergence of the laws of these rescaled processes in the sense of weak convergence of measures on D. (Since the shocks coalesce, a rescaling λ(t) → ∞ is needed to obtain a non-trivial limit.) L
Convergence to a process V ∗ is written V (t) → V ∗ as in [28]. We say that the process V ∗ is non-zero if Vx∗ is not identically zero with probability one. Recall that a positive function L is said to be slowly varying at ∞ if limt→∞ L(t x)/L(t) = 1 for all x > 0. Theorem 1. Let u 0 be a spectrally negative Lévy process with zero mean E(u 0 (x)), ∞ variance σ02 ≥ 0, and downward jump measure satisfying 0 (s ∧ s 2 )0 (ds) < ∞. 1. Suppose there is a rescaling λ(t) → ∞ as t → ∞ and a non-zero Lévy process V ∗ (t) with zero mean E(V1∗ ) such that the random variables V1 converge to V1∗ in law. Then there exists α ∈ (1, 2] and a function L slowly varying at infinity such that s σ02 + r 2 0 (dr ) ∼ s 2−α L(s) as s → ∞. (14) 0
2. Conversely, assume that there exists α ∈ (1, 2] and a function L slowly varying at infinity such that (14) holds. Then there is a strictly increasing rescaling λ(t) → ∞ L ˜ slowly varying at infinity such that V (t) → V α . Moreover, there is a function L, such that, with β = (α − 1)/α, ˜ λ(t) ∼ t 1/β L(t) as t → ∞.
(15)
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Remark 1. Since V (t) and V ∗ are Lévy processes, we have V (t) → V ∗ if and only if we (t) have convergence in law of the random variables Vx0 for some fixed x0 ∈ (0, ∞) (see (62)-(63) in Sect. 3 below). We take x0 = 1 without loss of generality. Part 2 implies in particular that the only possible limits are statistically self-similar. Remark 2. We say a solution has finite energy if for any finite interval I ⊂ R+ we have E I |v(x, t)|2 d x < ∞. The jump measure t for the solution is related to the energy by (see Sect. 4)
E
v(x, t)2 d x I
∞
= 0
r 2 t (dr )
x d x. I
The integral in (14) is thus a measure of the energy in an interval. If it is initially finite, it is conserved for t > 0, and it remains infinite if it is initially infinite. The only selfsimilar solution with finite energy corresponds to α = 2, and Theorem 1 implies it attracts all solutions with initially finite energy. In this sense, one may say that the finite energy solution is universal. However, Theorem 1 also indicates the delicate dependence of the domains of attraction on the tail behavior of 0 . Heavy-tailed solutions seem to us no less interesting than those with finite energy. Finer results on asymptotics, and a compactness theorem for subsequential limits that builds on Bertoin’s Lévy-Khintchine classification for eternal solutions to Smoluchowski’s equation [7], will be developed elsewhere. Remark 3. The case of zero mean, b0 = 0, is the most interesting. If b0 > 0 or b0 < 0 we can reduce to this case by a change of variables (see Sect. 2.2). If b0 < 0, the solution is defined only for 0 ≤ t < −b0−1 . Theorem 1 then characterizes the approach to self-similarity at the blow-up time. If b0 > 0 then the behavior of the solution as t → ∞ is determined by the zero-mean solution with the same σ02 and 0 at the finite time b0−1 . Remark 4. The Cole-Hopf solution is geometric and Theorem 1 may be a viewed as a limit theorem for statistics of minima. The utility of regular variation in such problems is widely known [33]. If the initial data is white noise, the Cole-Hopf solution is a study of the parabolic hull of Brownian motion. Groeneboom’s work on this problem [24] is the basis for several results on Burgers turbulence (in particular [3, 19, 22]). We have been unable to find a similar reference to the problem we consider in the probability literature ([9] seems the closest). Remark 5. There is a growing literature on intermittence, and the asymptotic self-similarity of Burgers turbulence, see for example [21, 26]. Numerical simulations and heuristic arguments suggest that this is a subtle problem with several distinct regimes. It is hard to obtain rigorous results for general initial data. Theorem 1 tells us that the approach to self-similarity is at least as complex as in the classical limit theorems of probability. The rest of this article is organized as follows. We explain the mapping from Burgers equation to Smoluchowski’s coagulation equation in Sect. 2. This is followed by the proof of Theorem 1 in Sect. 3. Finally, in Sect. 4 we compute a number of statistics of physical interest: energy and dissipation in solutions, the Fourier-Laplace spectrum, and the multifractal spectrum.
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2. Mean Field Theory for Burgers Equation In this section we explain the connection between Burgers equation with spectrally negative Lévy process data and Smoluchowski’s coagulation equation. The main results are due to Bertoin [5] and Carraro and Duchon [12]. We follow Bertoin’s approach, and explain results implicit in [5] and [7]. We think it worthwhile to make this connection widely known in full generality, since the results are of interest to many non-probabilists. Exact solutions of this simplicity are also useful as benchmark problems for numerical calculations. 2.1. Shock coalescence and Smoluchowski’s coagulation equation. Smoluchowski’s coagulation equation is a widely used mean-field model of cluster growth (see [2, 16] for introductions). We begin with a heuristic derivation of the coagulation equation as a mean-field model of shock coalescence. First consider the evolution of a single shock of size s > 0. Let u 0 (x) = −s1x≥0 . Then the solution is s u(x, t) = −s1x≥x1 (t) , x1 (t) = − t. (16) 2
N Shock coalescence is nicely seen as follows. Let u 0 (x) = − k=1 sk (0)1x≥xk (0) , where sk (0) > 0 for k = 1, . . . , N and x1 (0) < . . . x N (0). The solution may be constructed using the method of characteristics and the standard jump condition x˙ =
1 − (u + u + ) 2
(17)
across a shock at x = x(t), where u − and u + denote respectively the left and right limits of u(·, t) at x. At any time t > 0, there are N (t) ≤ N (0) shocks at locations x1 (t) < xk (t) < x N (t) (t) and u(x, t) = −
N k=1
sk (t)1x≥xk (t) , x˙k (t+ ) = −
k−1 j=1
s j (t+ ) −
sk (t+ ) . 2
(18)
The shock sizes sk (t) are constant between collisions, and add upon collision—when shocks k and k + 1 collide, we set sk (t+ ) = sk (t− ) + sk+1 (t− ) and relabel. This yields an appealing sticky particle or ballistic aggregation scenario. We say a system of particles with position, mass and velocity (xk (t), m k (t), vk (t)) undergoes ballistic aggregation if (a) the particles move with constant mass and velocity between collisions, and (b) at collisions, the colliding particles stick to form a single particle, conserving mass and momentum in the process. We map this shock coalescence problem to a sticky particle system by setting m k = sk and vk = x˙k . Suppose particles k and k + 1 meet at time t. Then, with unprimed variables denoting values before collision and primed variables denoting values after, since vk+1 = vk − (m k + m k+1 )/2 we use (18) to obtain m k+1 = m k vk . m k vk + m k+1 vk+1 = (m k + m k+1 ) vk − 2 Thus, the jump condition (17) reflects conservation of momentum. The calculations so far involve no randomness. Suppose now that the shock sizes s j are independent and let f (s, t) ds denote the expected number of shocks per unit length with size in [s, s + ds]. We derive a mean-field rate equation for f as follows. Let I be
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an interval of unit length. The number density changes because of the flux of shocks entering and leaving I and because of shock collisions within I . On average, the velocity difference across I is
∞
M1 (t) =
s f (s, t)ds,
0
therefore the average influx is M1 (t) f (s, t) ds. Next consider the formation of a shock of size s1 + s2 by a collision of shocks of size s1 and s2 as shown in Fig. 1. The relative velocity between these shocks is (s1 + s2 )/2 (see Fig. 1). The expected number of neighboring pairs with sizes in [s1 , s1 + ds1 ], [s2 , s2 + ds2 ] respectively is f (s1 , t) f (s2 , t) ds1 ds2 . The probability that these neighboring shocks are near enough to collide in time dt is 1 2 (s1 + s2 ) dt, thus the number of these shocks that collide in time dt is f (s1 , t) f (s2 , t)
s1 + s2 ds1 ds2 dt. 2
(19)
Summing over all collisions that create shocks of size s = s1 + s2 , and accounting for the loss of shocks of size s (= s1 or s2 ) in collisions with other shocks, we obtain the rate equation ∂t f (s, t) = M1 (t) f + Q( f, f ), where Q( f, f ) denotes the collision operator given by 1 Q( f, f )(s, t) = 2
0
s
s f (s1 , t) f (s − s1 , t) ds1 −
∞
(s + s1 ) f (s, t) f (s1 , t) ds1 .
0
We integrate in s to find M˙ 1 = M12 , therefore the normalized density f /M1 satisfies the equation f f 1 f =Q . (20) ∂t , M1 M1 M1 M1 Up to a change of time scale, this is a fundamental mean-field model of coalescence: Smoluchowski’s coagulation equation with additive kernel. We treat this equation in greater depth below. More precisely, it turns out that the random solution u(x, t) has the structure described in (18) when the initial data u 0 consists of a compound Poisson process with only downward jumps. The mean drift rate at time t is then −M1 (t), and this example shows that the solution blows up at the time M1 (0)−1 . We show below (see (26)) that one may remove the mean drift by a change of scale and slope, yielding ‘sawtooth’ data with a deterministic upward drift that compensates the random downward jumps. For such data we obtain a global solution. Thus, there is no essential distinction between sawtooth data and the decreasing initial data considered above.
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I
s1
s2
Fig. 1. Binary clustering of shocks
2.2. The Cole-Hopf formula. The modern notion of an entropy solution stems from the penetrating analysis by Hopf of the vanishing viscosity limit to (1). His work was based on a change of variables (re)discovered independently by Cole and Hopf [15, 27]. This solution is obtained via minimization of the Cole-Hopf function (x − y)2 H (y, t; x) = + 2t
y
−∞
u 0 (y )dy .
(21)
y The minimum in y is well-defined for all t > 0 provided U (y) = 0 u 0 (y )dy is lower semicontinuous and lim x→±∞ y −2 U (y) = 0. This is a mild assumption and holds for the random data we consider provided that the mean drift is zero. We denote the extreme points where H is minimized by a− (x, t) = inf{z|H (z, t; x) = min H }, a+ (x, t) = sup{z|H (z, t; x) = min H }. y
y
(22) Notice that any z ∈ R such that x = tu 0 (z) + z is a critical point of H , and represents a Lagrangian point that arrives at x at time t. Of these z, the ‘correct’ Lagrangian points are the minimizers of H . If a− (x, t) = a+ (x, t), this point is unique, and we have u(x, t) =
x − a± (x, t) , x ∈ R, t > 0. t
(23)
There is a shock at (x, t) when a− (x, t) = a+ (x, t). In this case, the Lagrangian interval [a− (x, t), a+ (x, t)] is absorbed into the shock and the velocity of the shock is given by the Rankine-Hugoniot condition (conservation of momentum) u(x, t) =
u(x+ , t) + u(x− , t) 1 = 2 a+ (x, t) − a− (x, t)
a+ (x,t)
a− (x,t)
u 0 (y) dy.
(24)
It will be convenient for us to assume that u is right-continuous in x and we call a(x, t) = a+ (x, t) the inverse Lagrangian function. Of course, the speed of shocks are still determined by the right-hand side of (24). In order to deal with non-zero mean drift in initial data, we will use the following interesting invariance of Burgers equation. Assume that u 0 (x) = o(|x|) as |x| → ∞,
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and let u(x, t) be the Cole-Hopf solution with u(x, 0) = u 0 (x), defined for all t ≥ 0. Let c ∈ R and define −1 −c , c < 0, (c) (25) u 0 (x) = u 0 (x) + cx, Tc = +∞, c ≥ 0. (c)
Then the Cole-Hopf solution with initial data u 0 is given by u (c) (x, t) =
1 u 1 + ct
x t , 1 + ct 1 + ct
+
cx , t ∈ [0, Tc ). 1 + ct
(26)
This is seen as follows. An elementary calculation shows that the Cole-Hopf functionals for the different data are related by H
(c)
(y, t; x) = H
x t ; y, 1 + ct 1 + ct
+
cx 2 , 2(1 + ct)
which implies the inverse Lagrangian functions are related by a (c) (x, t) = a
x t , 1 + ct 1 + ct
.
(27)
We now substitute in (23) to obtain (26).
2.3. Solutions with Lévy process initial data. Here we describe how the solution of (1), with initial data of the form (3), is determined in terms of Laplace exponents, essentially following Bertoin’s treatment in [5]. Suppose x → u 0 (x) is an arbitrary spectrally negative Lévy process for x ≥ 0, with Laplace exponent ψ0 having downward jump ∞ measure 0 . We first show that we may assume without loss of generality that 0 (s ∧ s 2 )0 (ds) < ∞. Indeed, if ∞ 1 s0 (ds) = ∞, then u 0 (x)/x → −∞ almost surely as x → ∞. (This follows from the fact that for the compound Poisson process X x with jump measure 0 (ds)1|s|≥1 , one has X x /x → ∞ as x → ∞ by the law of large numbers.) In this case the Cole-Hopf function H (y, t; x) has no minimum for any t > 0, and Eq. ∞(1) has no finite entropy solution for any positive time. Hence, we may suppose that 1 s0 (ds) < ∞. Next, we show that one may assume the mean drift b0 = E(u 0 (1)) is zero. If b0 is nonzero, we have lim x→∞ u 0 (x)/x = b0 a.s. by the strong law of large numbers. If b0 < 0, then by comparison to compression-wave solutions with initial data A + b max(x, 0), we find using the maximum principle that a.s. the solution blows up exactly at time −b0−1 . If b0 > 0 there is a global solution. In either case, we may use the transformation (26) with c = b0 to reduce to the case b0 = 0, replacing u 0 (x) by u 0 (x) − b0 x1x>0 . More precisely, we apply (26) for x ≥ 0 noting that a(0, t) ≥ 0, thus a(x, t) ≥ a(0, t) ≥ 0 for x ≥ 0, so that (27) holds for x ≥ 0. We have: (c)
Lemma 1. If u 0 is a spectrally negative Lévy process with Lévy triplet (c, σ02 , 0 ), the Cole-Hopf solution u (c) (x, t) is determined via (26) for x ≥ 0 and t ∈ [0, Tc ), in terms of a solution u(x, t) having zero mean drift and defined for all t ≥ 0.
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With these reductions, we may restrict ourselves to Laplace exponents ψ0 of the form ∞ −qs σ 2q 2 + e − 1 + qs 0 (ds), q ≥ 0. (28) ψ0 (q) = 0 2 0 We will always assume that a and u are right continuous in x (i.e., a(x, t) = a(x+ , t), compare with (24)). This ensures a is an element of the Skorokhod space D, so that we may use the standard Skorokhod topology to study limiting behavior. For brevity, we write v(x, t) = u(x, t) − u(0, t),
l(x, t) = a(x, t) − a(0, t),
and rewrite (23) as
x − l(x, t) , x ≥ 0, t > 0. (29) t Bertoin has shown that for all t > 0, x → l(x, t) is an increasing Lévy process (a subordinator) with the same law as the first passage process for tu 0 (x) + x. We denote the Laplace exponents of l and v by Φ and ψ respectively: E e−ql(x,t) = e−xΦ(q,t) , E eqv(x,t) = e xψ(q,t) , x, q, t ≥ 0. (30) v(x, t) =
We combine (29) and (30) to obtain ψ(q, t) =
q q −Φ ,t . t t
(31)
Since l is a subordinator, it has the simpler Lévy-Khintchine representation ∞ Φ(q, t) = dt q + (1 − e−qs )µt (ds), q > 0,
(32)
0
where dt ≥ 0 supplies the deterministic ∞ part of the drift, and µt is the Lévy measure of l(·, t), which now must satisfy 0 (1 ∧ s)µt (ds) < ∞ [4]. We see from (29) that v(·, t) is a Lévy process with no Gaussian component, and thus has a Lévy-Khintchine representation ∞ −qs ψ(q, t) = bt q + − 1 + qs t (ds), q ≥ 0, t > 0, (33) e 0
related to (32) by t (ds) = µt (t ds),
∞
bt + 0
st (ds) =
1 − dt . t
(34)
Due to the result that t and the first passage process of tu 0 (x) + x have the same law, a simple functional relation holds between ψ0 and Φ(q, t) [5, Thm. 2]: ψ0 (tΦ(q, t)) + Φ(q, t) = q, q ≥ 0, t > 0.
(35)
The evolution takes a remarkably simple form when we combine Eqs. (31) and (35) to obtain ψ(q, t) = ψ0 (q − tψ(q, t)), q ≥ 0, t > 0. (36)
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But then ψ(q, t) solves the inviscid Burgers equation (in q and t!) ∂t ψ + ψ∂q ψ = 0, ψ(0, q) = ψ0 (q).
(37)
The solution to (37) may be constructed by the method of characteristics and takes the form (36). The remarkable fact that the Laplace exponent is also a solution to Burgers equation was first observed by Carraro and Duchon [12, Thm. 2]. Since ψ0 is analytic and strictly convex, the solution (36) is analytic for all time and unique, and the condition ∂q ψ(0, t) = 0 is preserved for all t > 0. By (31)–(34), we have ∞ bt = 0, dt = 1 − t st (ds), t > 0. (38) 0
Let
M0 = lim ψ0 (q).
(39)
q→∞
We find Φ(q, t) → ∞ as q → ∞ from (35), and differentiate to obtain 1
dt = lim ∂q Φ(q, t) = lim
q→∞ 1 + tψ (tΦ(q, t)) 0
q→∞
=
with the understanding that dt = 0 if M0 = +∞. Then ∞ M(t) := lim ∂q ψ(q, t) = st (ds) = q→∞
0
1 , t > 0, 1 + t M0
M0 , t > 0, 1 + t M0
(40)
(41)
with the understanding that M(t) = t −1 when M0 = ∞. Note M = −M 2 . Below we will characterize the evolution of t differently. 2.4. BV regularity. It is clear from the Cole-Hopf formula that u is locally of bounded variation for every t > 0. We derive a decay estimate that quantifies this. The sample paths of u 0 have unbounded variation if and only if [4, p.15] ∞ σ02 > 0 or s0 (ds) = ∞. (42) 0
Heuristically, this corresponds to the presence of many small jumps (‘dust’). This is reflected in the Laplace exponent as M0 = limq→∞ ψ0 (q) = +∞ in this case. On the ∞ other hand, M0 is finite if and only if u 0 is BV, in which case σ0 = 0 and M0 = 0 s0 (ds) < ∞. The analytic formula (32) has the following probabilistic meaning. If we take a Poisson point process x → m tx (masses of clusters) with jump measure µt we have the representation [4, p.16] m ty . (43) l(x, t) = dt x + 0≤y≤x
The velocity field, and a point process of shock strengths s yt = t −1 m ty are determined from (23), (40) and (43) by v(x, t) = M(t)x −
1 t m y = M(t)x − s yt . t 0≤y≤x
0≤y
(44)
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For every t > 0, v(x, t) is the difference of two increasing functions: a linear drift and a pure jump process. Thus, it is of bounded variation, and by (38) and (43) we have x |∂ y v(y, t)|dy = 2M(t)x, x, t > 0, (45) E because E
0≤y≤x
0
m ty = x M(t) = x(1 − dt ).
2.5. Relation to Smoluchowski’s coagulation equation. We consider a positive measure ντ (ds) interpreted as the number of clusters of mass or size s per unit volume at time τ . Clusters of mass r and s coalesce by binary collisions at a rate governed by a symmetric kernel K (r, s). A weak formulation of Smoluchowski’s coagulation equation can be based on a general moment identity for suitable test functions ζ (see [32]): ∞ ∂τ ζ (s) ντ (ds) = 0 1 ∞ ∞ (46) (ζ (r + s) − ζ (r ) − ζ (s)) K (r, s) ντ (dr ) ντ (ds). 2 0 0 We consider only the additive kernel K (r, s) = r + s. It is classical that (46) can then be solved by the Laplace transform [16]. We denote the initial time by τ0 (to be chosen ∞ below). The minimal (and natural) hypothesis on initial ∞data ντ0 is that the mass sν (ds) is finite. We scale the initial data such that τ 0 0 0 sντ0 = 1. The Laplace exponent ∞
(1 − e−qs )ντ (ds)
(47)
∂τ ϕ − ϕ∂q ϕ = −ϕ, τ > τ0 .
(48)
ϕ(q, τ ) = 0
then satisfies
We showed in [32] that (48) may be used to define unique, global, mass-preserving solutions to (46). In particular, a map τ → ντ from [τ0 , ∞) to the space of positive ∞ Radon measures on (0, ∞), such that 0 sντ (ds) = 1 for all τ ≥ τ0 , is a solution of Smoluchowski’s equation with K (r, s) = r + s (in an appropriate weak sense detailed in [32]) if and only if ϕ satisfies (48). We now connect solutions of the inviscid Burgers equation (1) with Lévy process initial data to solutions of Smoluchowski’s equation through a change of scale. Let u 0 satisfy (3) and assume ∞ as in Subsect. 2.3 that the corresponding downward jump measure 0 satisfies 0 (s ∧ s 2 )0 (ds) < ∞ and the mean drift is zero. Let t be the jump measure of the Cole-Hopf solution. With M0 and M(t) as in (39) and (41), let τ0 = − log M0 if u 0 is of bounded variation, and τ0 = −∞ otherwise, and set τ = − log M(t), ντ (ds) = t (M(t)ds). ∞ From (41) it follows 0 sντ (ds) = 1, and by (47) and (33) we find ϕ(q, τ ) = q − ψ(qeτ , t).
(49)
(50)
We see that ψ solves (37) if and only if ϕ solves (48). Therefore, the rescaled Lévy measure of v(·, t) evolves according to Smoluchowski’s equation. Conversely, given any solution of Smoluchowski’s equation with initial data ν0 at a finite τ0 , we can construct
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a corresponding solution of (1) by choosing u 0 to be a spectrally negative Lévy process with jump measure t0 via (49). Initial data u 0 with unbounded variation are of particular interest. Here we have eternal solutions ντ to (46) defined for all τ ∈ R. We see that eternal solutions are in one-to-one correspondence with initial data u 0 of unbounded variation via (50). A finer correspondence mapping the clustering of shocks to the additive coalescent is found in [6]. To summarize, we have the following correspondence. Theorem 2. Assume u 0 is a spectrally negative Lévy process with Lévy triplet (0, σ02 , 0 ), with the same assumptions as in Theorem 1. Then for all t > 0, v(·, t) is a Lévy process with triplet (0, 0, t ), whose jump measure t determines a solution ντ (ds) to Smoluchowski’s coagulation equation with rate kernel K (r, s) = r + s as described in (49).
2.6. Self-similar solutions. Bertoin’s characterization of eternal solutions is the analogue of the Lévy-Khintchine characterization of infinitely divisible distributions [4, 18]. Among the latter, the stable distributions are of particular interest, and their analogues for Smoluchowski’s equations are obtained by choosing the Laplace exponent ψ0 (q) = q α , α ∈ (1, 2]. For α ∈ (1, 2) the corresponding Lévy measures are (ds) =
s −(1+α) ds. (−α)
The Laplace exponent q 2 corresponds to an atom at the origin. We thereby obtain for α ∈ (1, 2] a family of self-similar solutions to Smoluchowski’s equation with Laplace exponent of the form ϕ(τ, q) = e−βτ ϕα (qeβτ ), where ϕα solves ϕα (q)α + ϕα (q) = q, q > 0.
(51)
The self-similar solutions to Smoluchowski’s coagulation equation are ντ (ds) = e−2τ/β f α (e−τ/β s) ds, β =
α−1 , α ∈ (1, 2], α
(52)
where f α has been defined in (8). An analytic proof that these are the only self-similar solutions to Smoluchowski’s equation may be found in [32]. Each of these solutions corresponds to a self-similar process. Precisely, let X α denote the stable process with Laplace exponent q α , and T α and V α denote the processes Txα = inf{y ≥ 0 : X αy + y > x},
Vxα = x − Txα .
(53)
We have M0 = +∞ and M(t) = t −1 = e−τ in this case, and the Laplace exponent of the process l(·, t) is of the self-similar form Φ(q, t) = ϕ(q, τ ) = t −1/β ϕα qt 1/β , t > 0. (54) The solution processes have the scaling property L
l(x, t) = t 1/β Txtα −1/β ,
L
v(x, t) = t 1/β−1 Vxtα −1/β .
(55)
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The corresponding Lévy measures are obtained from (34), (49) and (52): µαt (ds) = t −2/β f α (t −1/β s) ds, αt (ds) = t 1−2/β f α t 1−1/β s ds. In the important case α = 2, we have 1/β = 2 and ϕ2 (q) = − 21 +
1 4
(56)
+ q, and by Laplace
inversion [1, Ch.29] we obtain the explicit expression in (12) for the distribution of Tx2 . 3. The Convergence Theorem In [32] we proved the following theorem characterizing solutions that approach the selfsimilar form in Smoluchowski’s coagulation equation with additive kernel. To every ∞ solution ντ of (46) with 0 sντ (ds) = 1 we associate the probability distribution function r ντ (dr ). (57) F(s, τ ) = (0,s]
To a self-similar solution f α , α ∈ (1, 2] with β = (α − 1)/α we associate Fα (s) =
s 0
r f α (r ) dr =
∞ (−1)k−1 s kβ k=1
k!
(1 + k − kβ)
sin π kβ . π kβ
(58)
A probability distribution function F ∗ is called nontrivial if F ∗ (s) < 1 for some s > 0; this means the distribution is proper (lims→∞ F ∗ (s) = 1) and not concentrated at 0 (F(s) ≡ 1). Theorem 3. Suppose τ1 ∈ R and ντ , τ ∈ [τ1 , ∞), is a solution to Smoluchowski’s ∞ coagulation equation with additive kernel such that 0 sντ1 (ds) = 1. ˜ ) → ∞ as τ → ∞ and a nontrivial 1. Suppose there is a rescaling function λ(τ probability distribution function F ∗ such that ˜ )s, τ ) = F ∗ (s) lim F(λ(τ
τ →∞
(59)
at all points of continuity of F ∗ . Then there exists α ∈ (1, 2] and a function L slowly varying at infinity such that s r 2 ντ1 (dr ) ∼ s 2−α L(s) as s → ∞. (60) 0
2. Conversely, assume that there exists α ∈ (1, 2] and a function L slowly varying at infinity such that (60) holds. Then there is a strictly increasing rescaling λ˜ (τ ) → ∞ such that lim F(λ˜ (τ )s, τ ) = Fα (s), 0 ≤ s < ∞,
τ →∞
where Fα is a distribution function for a self similar solution as in (58).
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˜ ) in part 2 from [32, (7.12)] corresponds to finding q = eτ /λ˜ The choice of λ˜ = λ(τ to solve 1 − ∂q ϕ(q, τ1 ) = αe−τ . By the Tauberian Lemma 3.3 of [32], (60) implies 1 (3 − α) αe−τ = 1 − ∂q ϕ(q, τ1 ) ∼ q α−1 L as q → 0. q α−1 ˆ τ ) for some It follows q L(1/q)1/(α−1) ∼ cα e−τ/(α−1) , which implies q ∼ e−τ/(α−1) L(e function Lˆ slowly varying at ∞. Hence ˆ τ ). λ˜ (τ ) ∼ eτ/β / L(e
(61)
If L is constant in (60), then λ˜ is asymptotically proportional to eτ/β . We now ∞prove Theorem 1. Let u 0 be a spectrally negative Lévy process with zero mean drift and 0 (s ∧ s 2 )0 (ds) < ∞. To the solution increment v(x, t) = u(x, t) − u(0, t) with downward jump measure t , associate a solution ντ of Smoluchowski’s coagulation equation (46) as in Theorem 2 with Laplace exponent ϕ(q, τ ) given by (50). Let τ1 = τ0 = − log M0 if M0 < ∞, and let τ1 = 0 if M0 = +∞ and τ0 = −∞. We deduce Theorem 1 from Theorem 3 by establishing two equivalences: (a)
(b)
There is a rescaling λ(t) → ∞ as t → ∞ and a non-zero Lévy process V ∗ with L
zero mean drift E(V ∗ ) such that V (t) → V ∗ if and only if there is a rescaling λ˜ (τ ) → ∞ as τ → ∞ and a nontrivial probability distribution function F ∗ such that holds. ∞ (59) 2 ν (ds) < ∞ if and only if ∞ s 2 (ds) < ∞. Moreover, s r 2 ν (dr ) ∼ s τ1 0 τ1 0 0 0 s s 2−α L(s) as s → ∞ if and only if 0 r 2 0 (dr ) ∼ s 2−α L(s) as s → ∞.
Proof of (a). We prove claim (a) by showing each part equivalent to a corresponding convergence statement for rescaled Laplace exponents. First, convergence in law in D for processes with independent increments can be reduced to the convergence of characteristic exponents [28, Cor. VII.4.43, p.440]. In particular, suppose λ(t) → ∞ as t → ∞. Then we have L V (t) → V ∗ , with E(V1∗ ) = 0, (62) (t)
∗
(t)
∗
if and only if E(eikVx ) → E(eikVx ) for all k ∈ R, uniformly for x in compact sets, and E(V1∗ ) = 0. But since we are working with Lévy processes, the Lévy-Khintchine formula shows the dependence on x is trivial, and thus (62) is equivalent to E(eikV1 ) → E(eikV1 ) for all k ∈ R,
and E(V1∗ ) = 0.
(63)
But pointwise convergence of characteristic functions is equivalent to convergence in (t) (t) (t) distribution of the random variables V1 [18, XV.3.2], and since V1 = 1 − T1 ≤ 1, (63) is equivalent to convergence of the Laplace transforms [18, XIII.1.2]: (t)
∗
E(eq V1 ) → E(eq V1 ) for all q > 0, and E(V1∗ ) = 0.
(64)
Taking logarithms and using (13) and (30), (64) is equivalent to λψ (qt/λ, t) → ψ ∗ (q) for all q > 0, and ∂q ψ ∗ (0) = 0, q Vx∗
xψ ∗ (q)
(65)
where E(e ) = e . This expresses the convergence of V (t) in terms of convergence of rescaled Laplace exponents. Note that ψ ∗ (q) ≡ 0 if and only if Vx∗ = 0 for all x ≥ 0 with probability 1.
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˜ ) → ∞ as τ → ∞. Using [18, XIII.1.2] again, the (proper) conNow suppose λ(τ vergence in (59) is equivalent to pointwise convergence of Laplace transforms: (66) η(q, τ ) → η∗ (q) for all q > 0, with η∗ (0) = 1, ∞ −qs ∞ F(λ˜ (τ ) ds, τ ), η∗ (q) := 0 e−qs F ∗ (ds). By (57) and (47), where η(q, τ ) := 0 e we have q η(q, τ ) = (∂q ϕ)(q/λ˜ , τ ), η(r, τ ) dr = λ˜ ϕ(q/λ˜ , τ ). (67) 0
We claim that (66) is equivalent to the statement that (with ϕ ∗ (q) =
q 0
η∗ (r ) dr )
λ˜ ϕ(q/λ˜ , τ ) → ϕ ∗ (q) for all q > 0, and ∂q ϕ ∗ (0) = 1.
(68)
Clearly, since η(·, τ ) is completely monotone and bounded, (66) implies (68). In the other direction, assume (68). For any sequence τ j → ∞ there is a subsequence along which η(q, τ j ) converges for all (rational, hence real) q > 0, to some limit whose integral must be ϕ ∗ . Thus (66) follows. We now finish the proof of claim (a) by observing that due to (50), we have λ˜ ϕ(q/λ˜ , τ ) = q − λ˜ ψ(qeτ /λ˜ , t).
(69)
Hence the convergence in (65) is equivalent to that in (68) provided we have λ(t)/t = λ˜ (τ )/eτ ,
(70)
˜ ), since t M(t) → 1 as t → ∞. (Note t M(t) = 1 if M0 = ∞.) or λ(t) = t M(t)λ(τ Moreover, F ∗ is a non-trivial probability measure if and only if ψ ∗ (q) > 0 for q > 0 and ∂q ψ ∗ (0) = 0. Proof of (b). It is only the case M0 = ∞ that requires some work. Indeed, if M0 < ∞ we see from (49) that ντ1 (ds) = 0 (M0 ds). In what follows, we suppose that M0 = ∞. We then have an eternal solution to Smoluchowski’s equation, and t = eτ . We shall compare the tails of ν0 (τ = 0) with that of 0 (t = 0). Claim (b) is a purely analytic fact that follows from Karamata’s Tauberian theorem [18]. We first reformulate it as a statement about Laplace transforms. Let ϕ0 (q) = ϕ(q, 0), ψ0 (q) = ψ(q, 0). For every α ∈ (1, 2] we have s 1 (3 − α) 2 2−α α−1 (71) r ν0 (dr ) ∼ s L(s) ⇐⇒ 1 − ϕ0 (q) ∼ q L q α−1 0 as s → ∞ and q → 0 respectively (see [32, Eq. 7.4]). By the same argument, s 1 (3 − α) , (72) r 2 0 (dr ) ∼ s 2−α L(s) ⇐⇒ ψ0 (q) ∼ q α−1 L q α−1 0 ∞ with the following caveat when α = 2. If 0 r 2 (dr ) = ∞ then (72) holds. On the ∞ 2 other hand, if 0 r 0 (dr ) < ∞ then we must modify the second condition in (72) to ∞ ψ0 (q) ∼ σ 2 + r 2 (dr ) q, q → 0. 0
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We set t = 1 in (50) and differentiate (36) with respect to q to obtain ψ0 (ϕ0 (q)) =
1 − ϕ0 (q) 1 = − 1. ϕ0 (q) ϕ0 (q)
(73)
The functions ψ0 , ϕ0 , and 1/ϕ are strictly increasing. Since ϕ0 (0) = 1 we also have ϕ0 (q) = q(1 + o(1)) as q → 0. A sandwich argument as in [18] may now be used to deduce claim (b). First suppose that (71) holds. Fix b, ε > 0. Then for q sufficiently small we use monotonicity and (73) to obtain 1 − ϕ0 (bq(1 − ε)) ϕ0 (q(1 + ε)) ψ0 (bq) 1 − ϕ0 (bq(1 + ε)) ϕ0 (q(1 − ε)) < < . 1 − ϕ0 (q(1 + ε)) ϕ0 (bq(1 − ε)) ψ0 (q) 1 − ϕ0 (q(1 − ε)) ϕ0 (bq(1 + ε)) Letting first q and then ε → 0, we obtain ψ0 (bq) = bα−1 . q→0 ψ0 (q) lim
Thus, ψ0 is regularly varying with exponent α − 1. Similarly, if we assume that (72) holds, we sandwich ψ0 (b(1 − ε)q) 1 − ϕ0 (bq) ϕ0 (q) ψ0 (b(1 + ε)q) < . < , ψ0 ((1 + ε)q) 1 − ϕ0 (q) ϕ0 (bq) ψ0 ((1 − ε)q) to deduce that 1 − ϕ0 is regularly varying with exponent α − 1. Finally, since ϕ0 (0) = 1 it follows from (73) that limq→0 ψ0 (q)/(1 − ϕ0 (q)) = 1. This finishes the proof of Theorem 1. 4. Energy, Dissipation and Spectra In this section, we compute several statistics of physical interest for the solution increments: mean energy and dissipation, the law of the Fourier-Laplace transform, and the multifractal spectrum. While the computations are routine, some interesting features emerge, namely (i) conservation of energy despite dissipation at shocks, (ii) a simple evolution rule for the Fourier-Laplace spectrum, and (iii) a multifractal spectrum in sharp variance with that of fully developed turbulence. Simple proofs of fine regularity properties (e.g., Hausdorff dimension of the set of Lagrangian regular points) may be found in [5]. 4.1. Energy and dissipation. The energy in any finite interval I ⊂ R+ is computed using the Lévy-Khintchine formula (33) and Fubini’s theorem as follows: 2 2 2 qv(x,t)
∂q E e dx E v(x, t) d x = E v(x, t) d x =
q=0 I I I
= x 2 ∂q ψ(0, t)2 + x∂q2 ψ(0, t) d x ∂q2 e xψ(q,t)
dx = q=0 I I ∞ y 2 t (dy) x d x. (74) = bt2 x 2 d x + I
0
I
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Let us restrict attention to solutions of mean zero, that is bt = 0. Then we have conservation of energy in the sense that 2 2 E v(x, t) d x = E v(x, 0) d x , t ≥ 0. (75) I
I
Indeed, by (74), we see that (75) is equivalent to ∞ s 2 0 (ds) =: M2 , ∂q2 ψ(0, t) = ∂q2 ψ0 = σ02 + 0
with the understanding that ∂q2 ψ(0, t) = ∞ if necessary to differentiate (36) to obtain ∂q ψ(q, t) =
∞ 0
(76)
s 2 0 (ds) is divergent. It is only
ψ0 (q − tψ) 1 , ∂q2 ψ(q, t) = 3 ψ0 (q − tψ), 1 + tψ0 (q − tψ) 1 + tψ0 (q − tψ)
and then take the limit q → 0 to obtain (75). The dissipation at a shock with left and right limits u ± is obtained as follows. The decay of the L 2 norm for solutions to Burgers equations with viscosity ε, u t +uu x = εu x x , is given by d u 2 d x = 2ε u 2x d x. dt R R The right-hand side may be evaluated exactly for traveling waves (viscous shocks) of the form u(x, t) = u ε (x − ct). It is easily seen that for any ε > 0 a traveling wave profile connecting the states u − > u + at ∓∞ is of the form u ε (x − ct) = w((x − ct)/ε), where w satisfies the ordinary differential equation dw u− + u+ 1 2 w − u 2− = , c= . −c (w − u − ) + 2 dξ 2 We therefore have 2 2 2ε u x d x = 2 (w ) dξ = 2 R
= 2(u − − u + )3
0
R 1
u+
u−
w(1 − w) dw =
1 2 2 w − u − dw −c(w − u − ) + 2
(u − − u + )3 . 3
The right-hand side is independent of ε and captures the dissipation of the entropy solution in the limit ε → 0. The dissipation at shocks in any finite interval I ⊂ R+ may now be computed by summing over all shocks in I using (43) and (44): ⎛ ⎛ ⎞ ⎞ 1 ⎝ 1 |I | ∞ 3 E (v(y− , t) − v(y+ , t))3 ⎠ = E ⎝ (s yt )3 ⎠ = s t (ds), (77) 3 3 3 0 y∈I
y∈I
where |I | is the length of I . Conservation of energy in the sense described in (75) is rather surprising ∞in view of the dissipation at shocks. In particular, there are solutions with finite energy ( 0 s 2 t (ds) < ∞ ∞), but infinite dissipation ( 0 s 3 t (ds) = ∞). However, there is no contradiction, since (75) refers to the expected value of the energy in any finite interval I , and the energy dissipated in shocks is compensated by energy input from the endpoints of I .
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4.2. The Fourier-Laplace spectrum. We show that the law of the Fourier transform v(k, ˆ t) of paths x → v(x, t), is determined by a Lévy process with jump measure ¯ t (s) ds, where ¯ t (s) = ∞ t (ds). Here t (ds) denotes the jump measure of s −1 s v(x, t) (see Theorem 4 below). The assertion vˆ ∼ k −1 as k → ∞ for white noise initial data is common in the Burgers turbulence literature (e.g., see [20, 36]). For the present case of Lévy process initial data, we show that v(k, ˆ t) ∼ −i M(t)k −2 as k → ∞. In addition, we find precise corrections under additional assumptions on t (for example, for self-similar solutions). These computations with the laws of the Fourier-Laplace transform should be contrasted with the conventional notion of the power spectrum. Despite its widespread use for wide-sense stationary processes, the power spectrum is of limited utility for the present problem involving stationary increments, as we now show. Fix L > 0 and consider the interval [0, L]. Almost every sample path v(x, t) is bounded on [0, L] and we may define the truncated Fourier transform
L
vˆ L (k, t) =
e−ikx v(x, t) d x.
(78)
0
If the energy is finite (M2 < ∞ in (76)), we may compute a truncated power spectral density SL (k) as follows. We have 1 1 |vˆ L (k, t)|2 = L L
0
L
L
e−ik(x−y) v(x, t)v(y, t) d x d y.
(79)
0
Since v(x, t) is a Lévy process with mean zero, the autocorrelation is E(v(x, t)v(y, t)) = (x ∧ y)M2 .
(80)
We take expectations in (79) to find 2M 1 2 SL (k) := E |vˆ L (k, t)|2 = 2 L k
sin k L 1− , k = 0. kL
(81)
The power spectrum S(k) = lim L→∞ SL (k) = 2M2 /k 2 is now seen to be well-defined, but is unsuitable for distinguishing solutions because all solutions with the same energy (possibly infinite) have identical power spectrum. A well-defined spectrum that distinguishes solutions may be obtained by taking the Fourier-Laplace transform of process paths. For fixed p > 0 we define the random variable ∞ 1 ∞ − px − px Lv( p, t) = e v(x, t) d x = e v(d x, t). (82) p 0 0 The integrals are well-defined because lim x→∞ v(x, t)/x = 0 a.s. by the strong law of large numbers, and v(x, t) is of bounded variation. If s t denotes a point process of shock strengths as in (44) we have p Lv( p, t) =
M(t) − px t − e sx . p 0≤x
(83)
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We determine the law of p Lv( p, t) by computing its Laplace transform via the ‘infinitesimal’ Laplace exponent formula E eqv(d x,t) = eψ(q,t) d x . Due to independence of the increments v(d x, t), we find by a standard discretization argument that ∞ ψ(qe− px , t) d x E eq p Lv( p,t) = exp 0 q 1 1 ψ(q , t) =: exp = exp dq (q, t) , p, q > 0, (84) ψ # p 0 q p − px after the change of variables q = qe ∞ . We now observe that ψ# determines a Laplace ¯ t (s) = exponent as follows. Let s t (ds) denote the tail of the Lévy measure t . ∞ Since 0 (s ∧ s 2 )t (ds) < ∞ we have the bounds ∞ ε ¯ t (s) ≤ ¯ t (ε), s ∈ (0, ε). ¯ t (s) ≤ s r t (dr ), s2 r 2 t (dr ) + s 2 s
0
Therefore, ¯ t (s) = 0, lim s
s→∞
¯ t (s) = 0, lim s 2
s→0
and we may integrate by parts in (33) to obtain ∞ ψ(q , t) ¯ t (s) ds. = (1 − e−q s ) q 0
(85)
Integrating once more in q we find q ∞ ¯ t (s) ψ(q , t) −qs ds. ψ# (q, t) = dq = (e − 1 + qs) q s 0 0 We integrate by parts in (41) to see that ∞ ¯ t (s) ds = 0
∞
st (ds) = M(t) < ∞.
(86)
(87)
0
This enables us to write ψ# (q, t) = M(t)q − Φ# (q, t), Φ# (q, t) = 0
∞
(1 − e−qs )
¯ t (s) ds. s
(88)
¯ t (s) ds satisfies the finiteness conditions for a jump measure, Since (87) ensures s −1 ψ# is a Laplace exponent for a Lévy process with zero mean drift that we denote by Z t . Similarly, Φ# is the Laplace exponent for a subordinator that we denote Y t . We summarize our calculations in the identities t t Z rt = M(t)r − Yrt , E eq Z r = er ψ# (q,t) , E e−qYr = e−r Φ# (q,t) , r, q, t > 0. (89) The result is that the Laplace spectrum of the solution increments is determined by t (90) E eq p Lv( p,t) = E eq Z 1/ p , q > 0, p > 0,
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t which implies that Lv( p, t) has the same law as p −1 Z 1/ p for fixed p > 0. Note that t −1 −1 for fixed r , Z r has the same law as r Lv(r , t), but the latter is not a Lévy process in r . In fact, for a fixed realization, Lv( p, t) is analytic in p. Nevertheless, its law is determined by the Lévy process Z t . We extend this computation to the Fourier spectrum ( p = ik) as follows. The calculations leading to (84) hold for complex q with Re(q) ≥ 0, and in particular for q = iξ , ξ ∈ R. Moreover, Lv( p, t) is a well-defined random variable for every p with Re( p) > 0. Thus, we may analytically continue the identity E(eq p Lv( p,t) ) = exp( p −1 ψ# (q, t)) to all p with Re( p) > 0, and q = iξ . As in (4), let # (ξ, t) = −ψ# (iξ, t) define the characteristic exponent corresponding to the Lévy process Z t . For ε, k > 0 we set p = ε + ik, v(k ˆ − iε, t) = Lv(ε + ik, t) and pass to the limit ε → 0 on both sides of (84) to obtain 1 ˆ lim E eiξ(ik v(k−iε,t)) = exp (91) ψ# (iξ, t) ε↓0 ik t i # (ξ, t) = E eiξ Z 1/k , ξ ∈ R, k > 0. (92) = exp k
Thus, for fixed k > 0, as ε ↓ 0 the random variables ik v(k ˆ − iε, t) converge in law to t . We denote this limit by ik v(k, the (real) random variable Z 1/k ˆ t). As before we do not t , simply the assert that the processes ik v(k ˆ − iε, t) converge in law to the process Z 1/k convergence of random variables for fixed k. We summarize our conclusions as follows. Theorem 4. Let Y t be a subordinator with Laplace exponent Φ# (q, t) from (88), and let Z t be the Lévy process defined by (89). Then for every fixed p > 0 and k > 0 the random t t variables p Lv( p, t) and ik v(k, ˆ t) have the same law as Z 1/ p and Z 1/k , respectively. Due to this result and (89), we always have the upper bound ik 2 v(k, ˆ t) ≤ M(t) a.s. This crude bound may be refined as k → ∞ using information related to the sample path behavior of subordinators (see [4, Ch. III.4]). Corollary 1. For every t > 0, limk→∞ ik 2 v(k, ˆ t) = M(t) in probability. Proof. This follows from the fact that limr ↓0 Yrt /r = 0 in probability, proved as follows. t By (89) we have E(e−qYr /r ) = e−r Φ# (q/r,t) , and since 1 − e−s ≤ 1 ∧ s, by (88) we have that the Laplace exponent ∞ ∞ ¯ t (s) ¯ t (s) ds ≤ ds → 0 r Φ# (q/r, t) = r (1 − e−qs/r ) (r ∧ qs) s s 0 0 as r ↓ 0 for each q > 0. Hence Yrt /r → 0 in law. A similar conclusion holds for the Laplace spectrum as p → ∞. Actually, for the subordinator Yrt , the sample paths have the stronger property that limr ↓0 Yrt /r → 0 a.s. [4, III.4.8]. Under a mild assumption on the integrability of the small jumps, we can strengthen convergence in probability to almost-sure convergence of the Laplace spectrum. Corollary 2. For every t > 0, lim p→∞ p 2 Lv( p, t) = M(t) in probability. If we also 1 ¯ t (s) ds < ∞, then lim p→∞ p 2 Lv( p, t) = M(t) a.s. assume 0 | log s|
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Proof. For notational convenience, we suppress the dependence on t in the proof. Fix ε > 2 Lv( p ) = M 0, and let pm = 2m for positive integers m. We will show that limm→∞ pm m a.s. That is, for every ε > 0, we claim
2
(93) P pm Lv( pm ) − M > ε infinitely often = 0. This is sufficient to establish lim p→∞ p 2 Lv( p) = M a.s. Indeed, since M/ p − p Lv( p) is completely monotone by (83), for p ∈ ( pm , pm+1 ) we have the bounds p 2 2 0 < M(t) − p 2 Lv( p) < M(t) − pm Lv( pm ) < 2 M − pm Lv( pm ) , pm and therefore
2 {M − p 2 Lv( p) > 2ε} ⊂ {M − pm Lv( pm ) > ε}. (94) In order to prove (93) we use the elementary estimate
p
e m
2
E 1 − exp − Y1/ pm P pm Lv( p) − M > ε = P pm Y1/ pm > ε ≤ e−1 ε 1 e 1 e pm pm 1 − exp − Φ# ( ) ≤ = Φ# ( ). e−1 pm ε e − 1 pm ε
∞ −1 Φ ( p /ε) < ∞. The first Borel-Cantelli lemma then We will show that m=1 pm # m implies (93). For clarity, we suppose ε = 1. This causes no essential difference and reveals the main computation. Denote the integrated tail of the Lévy measure for Y t by ∞ ¯ t (s ) t (s) = ds . (95) s s
We integrate by parts and use Tonelli’s theorem to find ∞ ∞ ∞ −1 pm Φ# ( p m ) = e− pm s t (s) ds. m=1
0
(96)
m=1
It is only necessary to check that the integral over s ∈ (0, 1) is finite. Here we use the elementary estimate ∞ ∞ ∞ | log s| + 1 dy 1 m ≤ , e−2 s ≤ exp(−e x log 2 s) d x = e−y log 2 y log 2 0 s m=1
so that
∞ 1
e− pm s t (s) ds ≤
0 m=1
1 log 2
1
(1 + | log s|)t (s) ds.
0
By the definition of t (s) in (95), the last integral is 1 ∞ ¯ ∞ ¯ 1∧r t (r ) t (r ) | log s| | log s| ds dr ds = dr r r 0 s 0 0 ∞ 1 ¯ ¯ t (r ) dr, | log r |t (r ) dr + ≤ 0
which is finite by assumption.
0
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Corrections to the bound ik 2 v(k, ˆ t) ≤ M(t) involve the law of the iterated logarithm [4, III.4]. The following corollary holds for initial data that is not BV (so M0 = +∞ and M(t) = 1/t) with suitably regular small jumps (‘dust’). ¯ 0 (s) = ∞ (dr ) Corollary 3. Assume σ0 = 0 and α = 2, or assume σ0 = 0 and s is regularly varying at zero with exponent −α, where α ∈ (1, 2). Then for every c > 0 and t > 0 we have −1 t − ik 2 v(k, log log k ˆ t) − log P ≤c ∼ , k → ∞, (97) h(k log log k) γ cγ t 1+2γ where γ = 1/(α − 1) and h(k) = k/ψ0 (k). This corollary is a consequence of [4, Lemma III.12] and is associated with the following lemma of independent interest which shows that the evolution preserves the regularity of the dust. ¯ 0 (s) is regularly varying at zero with expoLemma 2. (a) Assume that σ0 = 0 and ¯ t (s) is regularly varying at zero with exponent −1/α nent −α, α ∈ (1, 2). Then for every t > 0. √ ¯ t (s) ∼ (σ0 t)−1 2/(π s) as s → 0, for every t > 0. (b) If σ0 = 0, then t . Combining (31) with (86) Proof. Recall that t −1 − ik 2 v(k, ˆ t) agrees in law with kY1/k we find that the Laplace exponent of the subordinator Y t satisfies q q/t dq dq q ,t Φ = Φ q ,t . (98) Φ# (q, t) = t q q 0 0
We claim that Φ(·, t), and hence Φ# (·, t), is regularly varying at ∞ with exponent αˆ = 1/α ∈ [ 21 , 1). To prove the claim, we integrate by parts in (28) to obtain ∞ ∞ σ02 ψ0 (q) −qs ¯ + 0 (r ) dr ds. = e q2 2 0 s ¯ 0 is regularly varying at zero with exponent −α. Then ψ0 First assume σ0 = 0 and is regularly varying at infinity with exponent α. This follows from [18, XIII.5.3], or may be proved directly. If σ0 = 0, we have limq→∞ ψ0 (q)/q 2 = σ02 /2. The Laplace exponent Φ(q, t) is determined via the functional relation (35). Since α > 1, the map Φ → g0 (Φ) := ψ0 (tΦ)+Φ is regularly varying (in Φ) at ∞ with exponent α. Therefore, the inverse function Φ(q, t) is regularly varying (in q) at ∞ with exponent 1/α. Now let g# (·, t) be the inverse function to Φ# (·, t). Then by [4, Lemma III.4.12] we infer that for every cˆ > 0, t kY1/k ˆ α) ˆ − log P ≤ cˆ ∼ (1 − α)( ˆ α/ ˆ c) ˆ α/(1− log log k, k → ∞, (99) h # (k log log k, t) where h # (k, t) =
k . g# (k, t)
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By (98) and regular variation we have Φ# (q, t) ∼ Φ(q/t, t)/αˆ as q → ∞, and thus by (35) we find that as q → ∞, g# (q, t) ∼ tg0 (αq) ˆ ∼ tψ0 (αtq) ˆ ∼ t 1+α α −α ψ0 (q). Substituting cˆ = ct 1+α α −α into (99) yields Corollary 3. Karamata’s Tauberian theorem and the monotone density theorem now imply that ¯ t (s) is regularly varying at zero with exponent −1/α. If σ0 = 0, we find Φ(q, t) ∼ √ (σ0 t)−1 2q as q → ∞. Assertion (b) of the lemma then follows from the Tauberian theorem. ¯ 0 (s) = s −α /(α(−α)) For the self-similar solutions, ψ0 (q) = q α with α ∈ (1, 2], ¯ t (s) ∼ t −1 (ts)−1/α / (1 − 1/α) as s → 0. for α ∈ (1, 2), and we have 4.3. The multifractal spectrum. The notion of a multifractal spectrum was introduced by Frisch and Parisi to describe the intermittency of velocity fields in fully developed turbulence [20]. The multifractal spectrum d(h) measures the dimension of the set Sh where the velocity field has singularities of order h. There are different mathematical formulations of multifractality, corresponding to different notions of what one means by singularities of order h. Here we follow the treatment by Jaffard, which yields d(h) rather easily [29] (the notation has been changed slightly for consistency with this article). We say a function f : R+ → R, is C r (x0 ) for a point x0 ∈ R+ if there is a polynomial Px0 of degree at most [r ] such that | f (x) − Px0 (x)| ≤ C|x − x0 |r , in a neighborhood of x0 . The Hölder exponent of f at x0 is defined as
h f (x0 ) = sup{r f ∈ C r (x0 ) }. We define Sh to be the set of points where f is of Hölder exponent h. The multifractal spectrum d(h) is the Hausdorff dimension of Sh . If Sh is empty, the convention is d(h) = −∞. We now apply these definitions to v(x, t). As an example, let us compute the multifractal spectrum when the initial data is of bounded variation. Then M0 < ∞ in (44) and there is a finite number of shocks s yt in a finite interval [0, x] with probability 1. Suppose x0 is not a shock location for v(·, t). Then (44) shows v is analytic near x0 and h v (x0 ) = ∞. If x0 is a shock location, then v ∈ C −ε (x0 ) for every ε > 0, so that h v (x0 ) = 0. Thus, we have simply d(0) = 0 and d(h) = −∞ for every h = 0. The multifractal spectrum is more interesting for initial data of unbounded variation, that is, when (42) holds. In this case, the jumps in v are dense. Following Jaffard [29], the multifractal spectrum is computed as follows. We define 2− j log C j (t) ˜ C j (t) = t (ds), βt = max 0, lim sup . (100) j log 2 2− j−1 j→∞ For any t > 0, v(·, t) has no Brownian component. It then follows from [29, Thm. 1] that β˜t h, h ∈ [0, 1/β˜t ], (101) dt (h) = −∞, else.
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Experiments suggest that the multifractal spectrum for fully developed three-dimensional turbulence is a concave curve [31, Fig. 2]. This is in clear contrast with (101). For example, let us compute the multifractal spectrum for the self-similar process V α of index α ∈ (1, 2]. Since αt (s) is a scaled copy of α1 (s), dt (h) is independent of t. We use (8) to obtain the asymptotics as s → 0: α1 (ds) = f α (s) ds ∼
sin πβ β−2 α−1 (2 − β) ds, β = s . π α
We then have β˜t = α −1 , t > 0 and d(h) =
h/α, −∞,
h ∈ [0, α], else.
(102)
In particular, (102) implies that d(α) = 1, that is v(x, t) is C α (x) for a.e x ∈ R+ . For this set a finer characterization of the local variation of v(·, t) may be obtained by using the Fristedt-Pruitt law of the iterated logarithm (see [5, Cor. 1]). However, the multifractal spectrum also describes sets Sh , 0 < h < α, that are not covered by the Fristedt-Pruitt law. Acknowledgement. This material is based upon work supported by the National Science Foundation under grants DMS 03-05985, DMS 04-05343, DMS 06-05006 and DMS 06-04420. G.M. thanks the University of Crete for hospitality during part of this work.
References 1. Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. Vol. 55 of National Bureau of Standards Applied Mathematics Series, Superintendent of Documents, Washington, DC: U.S. Government Printing Office, 1964 2. Aldous, D.J.: Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5, 3–48 (1999) 3. Avallaneda, M., E, W.: Statistical properties of shocks in Burgers turbulence. Commun. Math. Phys. 172, 13–38 (1995) 4. Bertoin, J.: Lévy processes. Vol. 121 of Cambridge Tracts in Mathematics, Cambridge: Cambridge, University Press, 1996 5. Bertoin, J.: The inviscid Burgers equation with Brownian initial velocity. Commun. Math. Phys. 193, 397– 406 (1998) 6. Bertoin, J.: Clustering statistics for sticky particles with Brownian initial velocity. J. Math. Pures Appl. (9) 79, 173–194 (2000) 7. Bertoin, J.: Eternal solutions to Smoluchowski’s coagulation equation with additive kernel and their probabilistic interpretations. Ann. Appl. Probab. 12, 547–564 (2002) 8. Bertoin, J.: Some aspects of additive coalescents. In: Proceedings of the International Congress of Mathematicians, Beijing 2002, Vol. III, Beijing Higher Ed. Press, 2002, pp. 15–23 9. Bingham, N.H.: Maxima of sums of random variables and suprema of stable processes. Z. Wahr. Verw. Geb. 26, 273–296 (1973) 10. Burgers J.M.: The nonlinear diffusion equation. Dordrecht: Reidel, 1974 11. Carraro, L., Duchon, J.: Solutions statistiques intrinsèques de l’équation de Burgers et processus de Lévy. C. R. Acad. Sci. Paris Sér. I Math. 319, 855–858 (1994) 12. Carraro, L., Duchon, J.: Équation de Burgers avec conditions initiales à accroissements indépendants et homogènes. Ann. Inst. H. Poincaré Anal. Non Linéaire 15, 431–458 (1998) 13. Chabanol, M.-L., Duchon, J.: Markovian solutions of inviscid Burgers equation. J. Statist. Phys. 114, 525– 534 (2004) 14. Chassaing, P., Louchard, G.: Phase transition for parking blocks, Brownian excursion and coalescence. Random Structures and Algorithms 21, 76–119 (2002) 15. Cole, J.D.: On a quasi-linear parabolic equation occurring in aerodynamics. Quart. Appl. Math. 9, 225– 236 (1951)
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16. Drake, R.L.: A general mathematical survey of the coagulation equation. In: Topics in Current Aerosol Research, G. M. Hidy and J. R. Brock, eds., No. 2 in International reviews in Aerosol Physics and Chemistry, London: Pergammon, 1972, pp. 201–376 17. E, W., Sina˘ı, Y.G.: New results in mathematical and statistical hydrodynamics. Usp. Mat. Nauk 55, 25– 58 (2000) 18. Feller, W.: An introduction to probability theory and its applications. Vol. II. Second edition, New York: John Wiley & Sons Inc. 1971 19. Frachebourg, L., Martin, P.A.: Exact statistical properties of the Burgers equation. J. Fluid Mech. 417, 323–349 (2000) 20. Frisch, U., Parisi, G.: On the singularity structure of fully developed turbulence. In: Turbulence and predictability in geophysics, M. Ghil, R. Benzi, and R. Parisi, eds., Amsterdam: North-Holland, 1985, pp. 84–87 21. Funaki, T., Surgailis, D., Woyczy´nski, W.A.: Gibbs-Cox random fields and Burgers turbulence. Ann. Appl. Probab. 5, 461–492 (1995) 22. Giraud, C.: Genealogy of shocks in Burgers turbulence with white noise initial velocity. Commun. Math. Phys. 223, 67–86 (2001) 23. Golovin, A.M.: The solution of the coagulating equation for cloud droplets in a rising air current. Izv. Geophys. Ser. 482–487 (1963) 24. Groeneboom, P.: Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81, 79–109 (1989) 25. Gurbatov, S.N., Malakhov, A.N., Saichev, A.I.: Nonlinear random waves and turbulence in nondispersive media: waves, rays, particles. Manchester: Manchester University Press, 1991 26. Gurbatov, S.N., Simdyankin, S.I., Aurell, E., Frisch, U., Tóth, G.: On the decay of Burgers turbulence. J. Fluid Mech. 344, 339–374 (1997) 27. Hopf, E.: The partial differential equation u t +uu x = µx x . Commun. Pure Appl. Math. 3, 201–230 (1950) 28. Jacod, J., Shiryaev, A.N.: Limit theorems for stochastic processes. Vol. 288 of Grundlehren der Mathematischen Wissenschaften, Berlin: Springer-Verlag, Second ed., 2003 29. Jaffard, S.: The multifractal nature of Lévy processes. Probab. Theory Related Fields 114, 207–227 (1999) 30. Kolmogorov, A.N.: Dissipation of energy in the locally isotropic turbulence. Proc. Roy. Soc. London Ser. A 434, 15–17 (1991). Translated from the Russian by V. Levin: Turbulence and stochastic processes: Kolmogorov’s ideas 50 years on. 31. Meneveau, C., Sreenivasan, K.R.: Simple multifractal cascade model for fully developed turbulence. Phys. Rev. Lett. 59, 1424–1427 (1987) 32. Menon, G., Pego, R.: Approach to self-similarity in Smoluchowski’s coagulation equations. Commun. Pure Appl. Math. 57, 1197–1232 (2004) 33. Resnick, S.: Extreme values, regular variation and point processes. New York: Springer-Verlag, 1987 34. She, Z.-S., Aurell, E., Frisch, U.: The inviscid Burgers equation with initial data of Brownian type. Commun. Math. Phys. 148, 623–641 (1992) 35. Sina˘ı, Y.G.: Statistics of shocks in solutions of inviscid Burgers equation. Commun. Math. Phys. 148, 601–621 (1992) 36. Woyczy´nski, W.A.: Burgers-KPZ turbulence, Vol. 1700 of Lecture Notes in Mathematics, Berlin: Springer-Verlag, 1998. 37. Ziff, R.M.: Kinetics of polymerization. J. Statist. Phys. 23, 241–263 (1980) Communicated by A. Kupiainen
Commun. Math. Phys. 273, 203–215 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0249-8
Communications in
Mathematical Physics
Nonexistence of Self-Similar Singularities for the 3D Incompressible Euler Equations Dongho Chae Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea. E-mail:
[email protected] Received: 10 July 2006 / Accepted: 25 October 2006 Published online: 13 April 2007 – © Springer-Verlag 2007
Abstract: We prove that there exists no self-similar finite time blowing up solution to the 3D incompressible Euler equations if the vorticity decays sufficiently fast near infinity in R3 . By a similar method we also show nonexistence of self-similar blowing up solutions to the divergence-free transport equation in Rn . This result has direct applications to the density dependent Euler equations, the Boussinesq system, and the quasi-geostrophic equations, for which we also show nonexistence of self-similar blowing up solutions. 1. The Incompressible Euler Equations We are concerned here with the following Euler equations for the homogeneous incompressible fluid flows in R3 : ⎧ ∂v ⎪ ⎪ + (v · ∇)v = −∇ p, (x, t) ∈ R3 × (0, ∞) ⎪ ⎨ ∂t (E) , div v = 0, (x, t) ∈ R3 × (0, ∞) ⎪ ⎪ ⎪ ⎩ v(x, 0) = v0 (x), x ∈ R3 where v = (v1 , v2 , v3 ), v j = v j (x, t), j = 1, 2, 3, is the velocity of the flow, p = p(x, t) is the scalar pressure, and v0 is the given initial velocity, satisfying div v0 = 0. There are well-known results on the local existence of classical solutions (see e.g. [23, 18, 8] and references therein). The problem of finite time blow-up of the local classical solution is one of the most challenging open problems in mathematical fluid mechanics. On this direction there is a celebrated result on the blow-up criterion by Beale, Kato and Majda ([2]). By geometric type of consideration some of the possible scenarios to the possible singularity have been excluded (see [9, 13, 15]. One of the main purposes of this paper is to exclude the possibility of a self-similar type of singularities for the Euler system. The work was supported partially by the KOSEF Grant no. R01-2005-000-10077-0, and KRF Grant (MOEHRD, Basic Research Promotion Fund).
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The system (E) has scaling property that if (v, p) is a solution of the system (E), then for any λ > 0 and α ∈ R the functions v λ,α (x, t) = λα v(λx, λα+1 t),
p λ,α (x, t) = λ2α p(λx, λα+1 t)
(1.1)
are also solutions of (E) with the initial data v0λ,α (x) = λα v0 (λx). In view of the scaling properties in (1.1), the self-similar blowing up solution v(x, t) of (E) should be of the form, x 1 v(x, t) = (1.2) α V 1 (T∗ − t) α+1 (T∗ − t) α+1 for α = −1 and t sufficiently close to T∗ . Substituting (1.2) into (E), we find that V should be a solution of the system ⎧ ⎨ α V + 1 (x · ∇)V + (V · ∇)V = −∇ P, (S E) α + 1 α+1 ⎩ div V = 0 for some scalar function P, which could be regarded as the Euler version of the Leray equations introduced in [20]. The question of existence of a nontrivial solution to (SE) is equivalent to that of existence of a nontrivial self-similar finite time blowing up solution to the Euler system of the form (1.2). A similar question for the 3D Navier-Stokes equations was raised by J. Leray in [20], and answered negatively by the authors of [24], the result of which was refined later in [28]. Combining the energy conservation with a simple scaling argument, the author of this article showed that if there exists a nontrivial self-similar finite time blowing up solution, then its helicity should be zero ([3], see also [26] for other related discussion). To the author’s knowledge, however, the possibility of self-similar blow-up of the form (1.2) has never been excluded previously. In particular, due to lack of the laplacian term in the right hand side of the first equations of (SE), we cannot apply the argument of the maximum principle, which was crucial in the works in [24] and [28] for the 3D Navier-Stokes equations. Using a completely different argument from those used in [3], or [24], we prove here that there cannot be a self-similar blowing up solution to (E) of the form (1.2), if the vorticity decays sufficiently fast near infinity. Before stating our main theorem we recall the notions of particle trajectory and the back-to-label map, which are used importantly in the recent work of [7]. Given a smooth velocity field v(x, t), the particle trajectory mapping a → X (a, t) is defined by the solution of the system of ordinary differential equations, ∂ X (a, t) = v(X (a, t), t) ; ∂t
X (a, 0) = a.
The inverse A(x, t) := X −1 (x, t) is called the back to label map, which satisfies A(X (a, t), t) = a, and X (A(x, t), t) = x. Theorem 1.1. There exists no finite time blowing up self-similar solution v(x, t) to the 3D Euler equations of the form (1.2) for t ∈ (0, T∗ ) with α = −1, if v and V satisfy the following conditions: (i) For all t ∈ (0, T∗ ) the particle trajectory mapping X (·, t) generated by the classical solution v ∈ C([0, T∗ ); C 1 (R3 ; R3 )) is a C 1 diffeomorphism from R3 onto itself.
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(ii) The vorticity satisfies =curl V = 0, and there exists p1 > 0 such that ∈ L p (R3 ) for all p ∈ (0, p1 ). Remark 1.1. The condition (i), which is equivalent to the existence of the back-to-label map A(·, t) for our smooth velocity v(x, t) for t ∈ (0, T∗ ), is guaranteed if we assume a uniform decay of V (x) near infinity, independent of the decay rate ([6]). 1 (R3 ; R3 ) and there Remark 1.2. Regarding the condition (ii), for example, if ∈ L loc ε2 −ε |x| exist constants R, K and ε1 , ε2 > 0 such that |(x)| ≤ K e 1 for |x| > R, then we have ∈ L p (R3 ; R3 ) for all p ∈ (0, 1). Indeed, for all p ∈ (0, 1), we have p p |(x)| d x = |(x)| d x + |(x)| p d x
R3
|x|≤R
≤ |B R |
|x|>R
1− p |x|≤R
p
|(x)|d x
+Kp
R3
ε2
e− pε1 |x| d x < ∞,
where |B R | is the volume of the ball B R of radius R. Remark 1.3. In the zero vorticity case = 0, from div V = 0 and curl V = 0, we have V = ∇h, where h(x) is a harmonic function in R3 . Hence, we have an easy example of self-similar blow-up, x 1 v(x, t) = , α ∇h 1 (T∗ − t) α+1 (T∗ − t) α+1 in R3 , which is also the case for the 3D Navier-Stokes with α = 1. We do not consider this case in the theorem. Remark 1.4. If we assume that initial vorticity ω0 has compact support, then the nonexistence of self-similar blow-up of the form given by (1.2) is immediate from the well-known formula, ω(X (a, t), t) = ∇a X (a, t)ω0 (a)(see e.g. [23]). The proof of Theorem 1.1 will follow as a corollary of the following more general theorem. Theorem 1.2. Let v ∈ C([0, T ); C 1 (R3 ; R3 )) be a classical solution to the 3D Euler equations generating the particle trajectory mapping X (·, t) which is a C 1 diffeomorphism from R3 onto itself for all t ∈ (0, T ). Suppose we have representation of the vorticity of the solution, by ω(x, t) = (t)( (t)x)
∀t ∈ [0, T ),
(1.3)
where (·) ∈ C([0, T ); (0, ∞)), (·) ∈ C([0, T ); R3×3 ) with det( (t)) = 0 on [0, T ); = curl V for some V , and there exists p1 > 0 such that ∈ L p (R3 ) for all p ∈ (0, p1 ). Then, necessarily either det( (t)) ≡ det( (0)) on [0, T ), or = 0. Proof. By consistency with the initial condition, ω0 (x) = (0)( (0)x), and hence (x) = (0)−1 ω0 ([ (0)]−1 x) for all x ∈ R3 . We can rewrite the representation (1.3) in the form, ∀t ∈ [0, T ), (1.4) ω(x, t) = G(t)ω0 (F(t)x)
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where G(t) = (t)/(0), F(t) = [ (0)]−1 (t). In order to prove the theorem it suffices to show that either det(F(t)) = 1 for all t ∈ [0, T ), or ω0 = 0, since det(F(t))= det( (t))/det( (0)). Taking the curl of the first equation of (E), we obtain the vorticity evolution equation, ∂ω + (v · ∇)ω = (ω · ∇)v. ∂t This, taking the dot product with ω, leads to ∂|ω| + (v · ∇)|ω| = α|ω|, ∂t where α(x, t) is defined as ⎧ 3
⎪ ⎪ ⎨ Si j (x, t)ξi (x, t)ξ j (x, t) α(x, t) = i, j=1 ⎪ ⎪ ⎩ 0 with 1 Si j = 2
∂v j ∂vi + ∂ xi ∂ x j
(1.5)
if ω(x, t) = 0 if ω(x, t) = 0
, and ξ(x, t) =
ω(x, t) . |ω(x, t)|
In terms of the particle trajectory mapping defined by v(x, t), we can rewrite (1.5) as ∂ |ω(X (a, t), t)| = α(X (a, t), t)|ω(X (a, t), t)|. ∂t Integrating (1.6) along the particle trajectories {X (a, t)}, we have t α(X (a, s), s)ds . |ω(X (a, t), t)| = |ω0 (a)| exp
(1.6)
(1.7)
0
Taking into account the simple estimates − ∇v(·, t) L ∞ ≤ α(x, t) ≤ ∇v(·, t) L ∞ ∀x ∈ R3 , we obtain from (1.7) that t ∞ |ω0 (a)| exp − ∇v(·, s) L ds ≤ |ω(X (a, t), t)| 0 t ≤ |ω0 (a)| exp ∇v(·, s) L ∞ ds , 0
which, using the back to label map, can be rewritten as t ∇v(·, s) L ∞ ds ≤ |ω(x, t)| |ω0 (A(x, t))| exp − 0 t ≤ |ω0 (A(x, t))| exp ∇v(·, s) L ∞ ds . 0
(1.8)
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Combining this with the self-similar representation formula in (1.4), we have t ∞ ∇v(·, s) L ds ≤ G(t)|ω0 (F(t)x)| |ω0 (A(x, t))| exp − 0 t ∞ ≤ |ω0 (A(x, t))| exp ∇v(·, s) L ds .
(1.9)
0
Given p ∈ (0, p1 ), computing the L p (R3 ) norm of each side of (1.9), we derive ω0
Lp
t −1 ∞ exp − ∇v(·, s) L ds ≤ G(t)[det(F(t))] p ω0 L p 0 t ≤ ω0 L p exp ∇v(·, s) L ∞ ds ,
(1.10)
0
where we used the fact det(∇ A(x, t)) ≡ 1. Now, suppose = 0, which is equivalent to assuming that ω0 = 0, then we divide (1.10) by ω0 L p to obtain t −1 exp − ∇v(·, s) L ∞ ds ≤ G(t)[det(F(t))] p 0 t ≤ exp ∇v(·, s) L ∞ ds .
(1.11)
0
If there exists t1 ∈ (0, T ) such that det(F(t1 )) = 1, then either det(F(t1 )) > 1 or det(F(t1 )) < 1. In either case, setting t = t1 and passing p 0 in (1.11), we deduce that
t1
∇v(·, s) L ∞ ds = ∞.
0
This contradicts the assumption that the flow is smooth on (0, T ), i.e v ∈ C([0, T ); C 1 (R3 ; R3 )). Proof of Theorem 1.1. We apply Theorem 1.2 with 1
(t) = (T∗ − t)− α+1 I, and (t) = (T∗ − t)−1 , where I is the unit matrix in R3×3 . If α = −1 and t = 0, then 3
3 − α+1
det( (t)) = (T∗ − t)− α+1 = T∗
= det( (0)).
Hence, we conclude that = 0 by Theorem 1.2. In this case, there is no finite time blow-up for v(x, t), since the vorticity is zero.
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2. Divergence-Free Transport Equation The previous argument in the proof of Theorem 1.1 can also be applied to the following transport equations by a divergence-free vector field in Rn , n ≥ 2: ⎧ ∂θ ⎪ ⎪ ⎨ ∂t + (v · ∇)θ = 0, (T E) div v = 0, ⎪ ⎪ ⎩ θ (x, 0) = θ0 (x), where v = (v1 , · · · , vn ) = v(x, t), and θ = θ (x, t). In view of the invariance of the transport equation under the scaling transform, v(x, t) → v λ,α (x, t) = λα v(λx, λα+1 t), θ (x, t) → θ λ,α,β (x, t) = λβ θ (λx, λα+1 t) for all α, β ∈ R and λ > 0, the self-similar blowing up solution is of the form, x 1 v(x, t) = , α V 1 (T∗ − t) α+1 (T∗ − t) α+1 x 1
θ (x, t) = 1 (T∗ − t)β (T∗ − t) α+1
(2.1) (2.2)
for α = −1 and t sufficiently close to T∗ . Substituting (2.1) and (2.2) into the above transport equation, we obtain ⎧ ⎨ β + 1 (x · ∇) + (V · ∇) = 0, (ST ) α+1 ⎩ div V = 0. The question of existence of a suitable nontrivial solution to (ST) is equivalent to that of a existence of nontrivial self-similar finite time blowing up solution to the transport equation. We will establish the following theorem. Theorem 2.1. Let v ∈ C([0, T∗ ); C 1 (Rn ; Rn )) generate a C 1 diffeomorphism from Rn onto itself. Suppose there exist α = −1, β ∈ R and solution (V, ) to the system (ST) with ∈ L p1 (Rn ) ∩ L p2 (Rn ) for some p1 , p2 such that 0 < p1 < p2 ≤ ∞. Then,
= 0. This theorem is a corollary of the following one. Theorem 2.2. Suppose there exists T > 0 such that there exists a representation of the solution θ (x, t) to the system (TE) by θ (x, t) = (t) ( (t)x)
∀t ∈ [0, T ),
(2.3)
where (·) ∈ C([0, T ); (0, ∞)), (·) ∈ C([0, T ); Rn×n ) with det( (t)) = 0 on [0, T ); there exists p1 < p2 with p1 , p2 ∈ (0, ∞] such that ∈ L p1 (Rn ) ∩ L p2 (Rn ). Then, necessarily either det( (t)) ≡ det( (0)) and (t) ≡ (0) on [0, T ), or = 0.
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Proof. Similarly to the proof of Theorem 1.2 the representation (2.3) reduces to the form, θ (x, t) = G(t)θ0 (F(t)x), (2.4) where G(t) = (t)/(0), F(t) = (t)[ (0)]−1 . By standard L p -interpolation and the relation between θ0 and by θ0 (x) = (0) ( (0)x), we have that ∈ L p1 (Rn ) ∩ L p2 (Rn ) implies θ0 ∈ L p (Rn ) for all p ∈ [ p1 , p2 ]. As in the proof of Theorem 1.2 we denote by {X (a, t)} and {A(x, t)} the particle trajectory map and the back to label map respectively, each one of which is defined by v(x, t). As the solution of the first equation of (TE) we have θ (X (a, t), t) = θ0 (a), which can be rewritten as θ (x, t) = θ0 (A(x, t)) in terms of the back to label map. This, combined with (2.4), provides us with the relation θ0 (A(x, t)) = G(t)θ0 (F(t)x).
(2.5)
Using the fact det(∇ A(x, t)) = 1, we compute the L p (Rn ) norm of (2.5) to have θ0 L p = |G(t)||det(F(t))|
− 1p
= |G(t)||det(F(t))|
− 1p
1
|θ (F(t)x)| |det(F(t))|d x p
Rn
p
θ0 L p
(2.6)
for all t ∈ [0, T ) and p ∈ [ p1 , p2 ]. Suppose θ0 = 0, which is equivalent to = 0, then we divide (2.6) by θ0 L p to obtain |G(t)| p = det(F(t)) for all t ∈ [0, T ) and p ∈ [ p1 , p2 ], which is possible only if G(t) = det(F(t)) = 1 for all t ∈ [0, T ). Hence, (t) ≡ (0), and det( (t)) ≡ det( (0)). Proof of Theorem 2.1. We apply Theorem 2.2 with 1
(t) = (T∗ − t)− α+1 I and (t) = (T∗ − t)−β , where I is the unit matrix in Rn×n . Then, n − (α+1)
n
det( (t)) = (T∗ − t)− (α+1) = det( (0)) = T∗ Hence, by Theorem 2.2 we have = 0.
if α = −1, t = 0.
Below we present some examples of fluid mechanics, where we can apply a similar argument to the above to prove nonexistence of nontrivial self-similar blowing up solutions. A. The density-dependent Euler equations. The density-dependent Euler equations in Rn , n ≥ 2, are the following system: ⎧ ∂ρv ⎪ + div (ρv ⊗ v) = −∇ p, ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ∂ρ + v · ∇ρ = 0, (E1 ) ∂t ⎪ ⎪ ⎪ ⎪ div v = 0, ⎪ ⎪ ⎩ v(x, 0) = v0 (x), ρ(x, 0) = ρ0 (x), where v = (v1 , · · · , vn ) = v(x, t) is the velocity, ρ = ρ(x, t) ≥ 0 is the scalar density of the fluid, and p = p(x, t) is the pressure. We refer to Sect. 4.5 in [21] for a more detailed introduction of this system. Here we just note that this system reduces to the
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homogeneous Euler system of the previous section when ρ ≡ 1. The question of finite time blow-up for the system is wide open even in the case of n = 2, although we have local in time existence result of the classical solution and its finite time blow-up criterion (see e.g. [1, 4]). The system (E 1 ) has the scaling property that if (v, ρ, p) is a solution of the system (E 1 ), then for any λ > 0 and α ∈ R the functions v λ,α (x, t) = λα v(λx, λα+1 t), ρ λ,α,β (x, t) = λβ ρ(λx, λα+1 t),
(2.7)
p λ,α,β (x, t) = λ2α+β p(λx, λα+1 t)
(2.8)
are also solutions of (E 1 ) with the initial data λ,α,β
v0λ,α (x) = λα v0 (λx), ρ0
(x) = λβ ρ0 (λx).
In view of the scaling properties in (2.7), we should check if there exists a nontrivial solution (v(x, t), ρ(x, t)) of (E 1 ) of the form, x 1 v(x, t) = , (2.9) α V 1 (T∗ − t) α+1 (T∗ − t) α+1 x 1 R ρ(x, t) = (2.10) 1 (T∗ − t)β (T∗ − t) α+1 for α = −1 and t sufficiently close to T∗ . The solution (v, ρ) of the form (2.9)–(2.10) is called the self-similar blowing up solution of the system (E 1 ). The following theorem establishes the nonexistence of a nontrivial self-similar blowing up solution of the system (E 1 ), which is immediate from Theorem 2.2. Theorem 2.3. Let v generate a particle trajectory, which is a C 1 diffeomorphism from Rn onto itself for all t ∈ (0, T∗ ). Suppose there exist α = −1 and a solution (v, ρ) to the system (E 1 ) of the form (2.9)–(2.10), for which there exists p1 , p2 with 0 < p1 < p2 ≤ ∞ such that R ∈ L p1 (Rn ) ∩ L p2 (Rn ). Then, R = 0. B. The 2D Boussinesq system. The Boussinesq system for the inviscid fluid flows in R2 is given by ⎧ ∂v ⎪ + (v · ∇)v = −∇ p + θ e1 , ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ∂θ + (v · ∇)θ = 0, (B) ∂t ⎪ ⎪ ⎪ ⎪ div v = 0, ⎪ ⎪ ⎩ v(x, 0) = v0 (x), θ (x, 0) = θ0 (x), where v = (v1 , v2 ) = v(x, t) is the velocity, e1 = (1, 0), and p = p(x, t) is the pressure, while θ = θ (x, t) is the temperature function. The local in time existence of the solution and the blow-up criterion of the Beale-Kato-Majda type has been well known (see e.g. [16, 5]). The question of finite time blow-up has been open until now. Here, we exclude the possibility of a self-similar finite time blow-up for the system. The system (B) has scaling property that if (v, θ, p) is a solution of the system (B), then for any λ > 0 and α ∈ R the functions v λ,α (x, t) = λα v(λx, λα+1 t), θ λ,α (x, t) = λ2α+1 θ (λx, λα+1 t),
(2.11)
Nonexistence of Self-Similar Singularities for the 3D Incompressible Euler Equations
p λ,α (x, t) = λ2α p(λx, λα+1 t)
211
(2.12)
are also solutions of (B) with the initial data v0λ,α (x) = λα v0 (λx), θ0λ,α (x) = λ2α+1 θ0 (λx). In view of the scaling properties in (2.11), the self-similar blowing-up solution (v(x, t), θ (x, t)) of (B) should of the form, x 1 v(x, t) = , (2.13) α V 1 (T∗ − t) α+1 (T∗ − t) α+1 x 1
θ (x, t) = , (2.14) 1 (T∗ − t)2α+1 (T∗ − t) α+1 where α = −1. We have the following nonexistence result of such type of solution. Theorem 2.4. Let v generate a particle trajectory, which is a C 1 diffeomorphism from R2 onto itself for all t ∈ (0, T∗ ). There exists no nontrivial solution (v, θ ) of the system (B) of the form (2.13)–(2.14), if there exists p1 , p2 ∈ (0, ∞], p1 < p2 , such that
∈ L p1 (R2 ) ∩ L p2 (R2 ), and V ∈ H m (R2 ), m > 2. Proof. Similarly to the proof of Theorem 2.1, we first conclude = 0, and hence θ (·, t) ≡ 0 on [0, T∗ ). Then, the system (B) reduces to the 2D incompressible Euler equations, for which we have a global in time regular solution for v0 ∈ H m (R2 ), m > 2 (see e.g. [19]). Hence, we should have v(·, t) ≡ 0 on [0, T∗ ). Note added to the proof. A similar proof to the one above leads to the nonexistence of a self-similar blowing up solution to the axisymmetric 3D Euler equations with swirl of the θ m 3 form, (1.2), if = r V
satisfies the condition of Theorem 2.4, and curl V ∈ H (R ), m > 5/2, where r = x12 + x22 , and V θ is the angular component of V . Indeed, applying D Theorem 2.2 to the θ -component of the Euler equations, Dt (r v θ ) = 0, we show that θ v = 0 as in the above proof, and then we use the global regularity result for the 3D axisymmetric Euler equations without swirl ([22, 27]) to conclude that (vr , v 3 ) is also zero. C. The 2D quasi-geostrophic equation. The following 2D quasi-geostrophic equation (QG) models the dynamics of the mixture of cold and hot air, and the fronts between them, ⎧ ∂θ ⎪ + (v · ∇)θ = 0, ⎪ ⎪ ⎪ ⎨ ∂t θ (y, t) (QG) v = −∇ ⊥ (−)− 21 θ = ∇ ⊥ dy , ⎪ ⎪ ⎪ R2 |x − y| ⎪ ⎩ θ (x, 0) = θ0 (x),
where ∇ ⊥ = (−∂2 , ∂1 ). Besides its physical significance, mainly due to its similar structure to the 3D Euler equations, there have been many recent studies on this system (see e.g. [10–12] and references therein). Although the question of finite time singularities is still open, some type of scenarios of singularities have been excluded ([11, 12, 14]).
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Here we exclude the scenario of self-similar singularity. The system (QG) has the scaling property that if θ is a solution of the system, then for any λ > 0 and α ∈ R the functions θ λ,α (x, t) = λα θ (λx, λα+1 t)
(2.15)
are also solutions of (QG) with the initial data θ0λ,α (x) = λα θ0 (λx). Hence, the selfsimilar blowing up solution should be of the form, x 1 θ (x, t) = (2.16) α 1 (T∗ − t) α+1 (T∗ − t) α+1 for t sufficiently close to T∗ and α = −1. Applying the same argument as in the proof of Theorem 2.1, we have the following theorem. Theorem 2.5. Let v generate a particle trajectory, which is a C 1 diffeomorphism from R2 onto itself for all t ∈ (0, T∗ ). There exists no nontrivial solution θ to the system (QG) of the form (2.16), if there exists p1 , p2 ∈ (0, ∞], p1 < p2 , such that
∈ L p1 (R2 ) ∩ L p2 (R2 ). 3. Remarks on the Locally Self-Similar Blow-up The notion of self-similar solutions considered in the previous sections are apparently ‘global’ in the sense that the self-similar representation of the solution in (1.2) should hold for all space points in R3 . For convenience we call the self-similar solutions considered above global self-similar solutions. On the other hand, many physicists have been trying to seek a ‘locally self-similar’ solution of the 3D Euler equations (see e.g. [17, 25] and the references therein). Our aim in this section is to show that the nonexistence of the global self-similar solution as proved in the previous sections implies the nonexistence of the locally self-similar solutions. Thus we exclude the most popular scenario (at least among the physicists) leading to the singularities of the 3D Euler equations. We first formulate the precise definition of the locally self-similar solution of the 3D Euler equations. A similar definition applies obviously to the locally self-similar solutions to the other equations. Definition 1. A solution v(x, t) of the solution to (E) is called a locally self-similar blowing up solution near a space-time point (x∗ , T∗ ) ∈ R3 × (−∞, +∞) if there exist r > 0, α > −1 and a solenoidal vector field V defined on R3 such that the representation x − x∗ 1 v(x, t) = ∀(x, t) ∈ B(x∗ , r ) × (T∗ − r α+1 , T∗ ) α V 1 α+1 (T∗ − t) (T∗ − t) α+1 (3.1) holds true, where B(x∗ , r ) = {x ∈ R3 | |x − x∗ | < r }. The following is our main result in this section. Theorem 3.1. The nonexistence of the globally self-similar solution of the 3D Euler equations implies the nonexistence of the locally self-similar solution. Combining Theorem 3.1 with Theorem 1.1, we have the following corollary.
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Corollary 3.1. Suppose there exists a locally self-similar blowing up solution v of the 3D Euler equations in the form (3.1), which generates a C 1 diffeomorphism on R3 for all time before the blow-up. If there exists p1 > 0 such that = curl V ∈ L p (R3 ) for all p ∈ (0, p1 ), then necessarily = 0. In other words, there exists no nontrivial locally self-similar solution to the 3D Euler equation if the vorticity = 0 satisfies such an integrability condition. Proof of Theorem 3.1. We assume there exists a locally self-similar solution v(x, t) in the sense of Definition 1. The proof of Theorem 3.1 follows if we prove the existence of the global self-similar solution. By translation in space-time variables, we can rewrite the velocity in (3.1) as v(x, t) =
1 α
t α+1
x
V t
1 α+1
for (x, −t) ∈ B(0, r ) × (−r α+1 , 0).
(3.2)
We observe that, under the scaling transform (1.1), we have the invariance of the representation, 1 x (= v(x, t)), v(x, t) → v λ,α (x, t) = λα v(λx, λα+1 t) = α V 1 t α+1 t α+1 while the region of space-time, where the self-similar form of solution is valid, transforms according to B(0, r ) × (−r α+1 , 0) → B(0, r/λ) × −(r/λ)α+1 , 0 . We set λ = 1/n, and define the sequence of locally self-similar solutions {v n (x, t)} 1 by v n (x, t) := v n ,α (x, t) with v 1 (x, t) := v(x, t). In the above we find that v n (x, t) =
1 t
α α+1
x
V t
1 α+1
for (x, −t) ∈ B(0, nr ) × (−(nr )α+1 , 0),
and each v n (x, t) is a solution of the Euler equations for all (x, t) ∈ R3 × (−∞, 0). Let us define v ∞ (x, t) by v ∞ (x, t) =
1 α
t α+1
x
V t
1 α+1
for (x, −t) ∈ R3 × (−∞, 0).
Given a compact set K ⊂ R3 × (−∞, 0), we observe that v n → v ∞ as n → ∞ on K in any strongest possible topology of convergence. Indeed, for sufficiently large N = N (K ), v n (x, t) ≡ v ∞ (x, t) for all (x, t) ∈ K , if n > N . Hence, we find that v ∞ (x, t) is a solution of the Euler equations for all (x, t) ∈ R3 × (−∞, 0), which is a global self-similar blowing up solution, after translation in space and time. We note that the above proof does not depend on the specific form of the Euler equations, and hence obviously works also for the self-similar solutions of the other equations, e.g. for the Leray type of self-similar solutions of the Navier-Stokes equations. We state the result more precisely below.
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Corollary 3.2. Let v be a weak solution of the 3D Navier-Stokes equations. Suppose 2 (R3 ) for some there exist r > 0, (x∗ , T∗ ) ∈ R3 × (−∞, ∞), and V ∈ L p (R3 ) ∩ L loc p ∈ [3, ∞) such that 1 x − x∗ v(x, t) = √ V √ ∀(x, t) ∈ B(x∗ , r ) × (T∗ − r 2 , T∗ ) T∗ − t T∗ − t holds true, then V = 0. The proof is similar to the previous one, where we use the result in [24] for the nonexistence of a weak solution to the Leray system in L 3 (R3 ), while we use the corresponding result in [28] for the case of a weak solution of the Leray system in L p (R3 ), p ∈ (3, ∞). A similar type of nonexistence theorems hold true also for the other equations considered in Sect. 2 with the appropriate integrability conditions. References 1. Beir¯ao da Veiga, H., Valli, A.: Existence of C ∞ solutions of the Euler equations for nonhomogeneous fluids. Comm. P.D.E. 5, 95–107 (1980) 2. Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66 (1984) 3. Chae, D.: Remarks on the blow-up of the Euler Equations and the Related Equations. Commun. Math. Phys. 245(3), 539–550 (2003) 4. Chae, D., Lee, J.: Local existence and blow-up criterion of the inhomogeneous Euler equations. J. Math. Fluid Mech. 5, 144–165 (2003) 5. Chae, D., Nam, H-S.: Local existence and blow-up criterion for the Boussinesq equations. Proc. Roy. Soc. Edinburgh Sect. A 127(5), 935–946 (1997) 6. Constantin, P.: Private communication 7. Constantin, P.: An Eulerian-Lagrangian approach for incompressible fluids: local theory. J. Amer. Math. Soc. 14(2), 263–278 (2001) 8. Constantin, P.: A few results and open problems regarding incompressible fluids. Notices Amer. Math. Soc. 42(6), 658–663 (1995) 9. Constantin, P., Fefferman, C., Majda, A.: Geometric constraints on potential singularity formulation in the 3-D Euler equations. Comm. P.D.E 21(3–4), 559–571 (1996) 10. Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994) 11. Córdoba, D.: Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann. of Math. 148, 1135–1152 (1998) 12. Córdoba, D., Fefferman, C.: Growth of solutions for QG and 2D Euler equations. J. Amer. Math. Soc. 15(3), 665–670 (2002) 13. Córdoba, D., Fefferman, C.: On the collapse of tubes carried by 3D incompressible flows. Commun. Math. Phys. 222(2), 293–298 (2001) 14. Córdoba, D., Fefferman, C., DeLa LLave, R.: On squirt singularities in hydrodynamics. SIAM J. Math. Anal. 36(1), 204–213 (2004) 15. Deng, J., Hou, T.Y., Yu, X.: Geometric and Nonblowup of 3D Incompressible Euler Flow. Comm. P.D.E 30, 225–243 (2005) 16. E, W., Shu, C.: Small scale structures in Boussinesq convection. Phys. Fluids 6, 48–54 (1994) 17. Greene, J.M., Pelz, R.B.: Stability of postulated, self-similar, hydrodynamic blowup solutions. Phys. Rev. E 62(6), 7982–7986 (2000) 18. Kato, T.: Nonstationary flows of viscous and ideal fluids in R3 . J. Funct. Anal. 9, 296–305 (1972) p 19. Kato, T., Ponce, G.: On nonstationary flows of viscous and ideal fluids in L s (R2 ). Duke Math. J. 55, 487–499 (1987) 20. Leray, J.: Essai sur le mouvement d’un fluide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934) 21. Lions, P.L.: Mathematical Topics in Fluid Mechanics: Vol. 1 Incompressible Models. Oxford: Clarendon Press, 1996 22. Majda, A.: Vorticity and the mathematical theory of incompressible fluid flow. Comm. Pure Appl. Math. 39, 187–220 (1986)
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23. Majda, A., Bertozzi, A.: Vorticity and Incompressible Flow. Cambridge: Cambridge Univ. Press, 2002 ˇ ak, 24. Necas, ˇ J., Ruˇz icka, ˇ M., Sver ´ V.: On Leray’s self-similar solutions of the Navier-Stokes equations. Acta Math. 176, 283–294 (1996) 25. Pelz, R.B.: Locally self-similar, finite-time collapse in a high-symmetry vortex filament model. Phys. Rev. E 55, 1617–1626 (1997) 26. Pomeau, Y., Sciamarella, D.: An unfinished tale of nonlinear PDEs: Do solutions of 3D incompressible Euler equations blow-up in finite time?. Physica D 205, 215–221 (2005) 27. Saint Raymond, X.: Remarks on axisymmetric solutions of the incompressible Euler system. Comm. P. D. E. 19, 321–334 (1994) 28. Tsai, T-P.: On Leray’s self-similar solutions of the Navier-Stokes equations satisfying local energy estimates. Arch. Rat. Mech. Anal. 143(1), 29–51 (1998) Communicated by P. Constantin
Commun. Math. Phys. 273, 217–236 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0248-9
Communications in
Mathematical Physics
Intrinsic Definitions of “Relative Velocity” in General Relativity Vicente J. Bolós Dpto. Matemáticas, Facultad de Ciencias, Universidad de Extremadura, Avenida de Elvas s/n, 06071 Badajoz, Spain. E-mail:
[email protected] Received: 10 July 2006 / Accepted: 9 November 2006 Published online: 13 April 2007 – © Springer-Verlag 2007
Abstract: Given two observers, we define the “relative velocity” of one observer with respect to the other in four different ways. All four definitions are given intrinsically, i.e. independently of any coordinate system. Two of them are given in the framework of spacelike simultaneity and, analogously, the other two are given in the framework of observed (lightlike) simultaneity. Properties and physical interpretations are discussed. Finally, we study relations between them in special relativity, and we give some examples in Schwarzschild and Robertson-Walker spacetimes.
1. Introduction The need for a strict definition of “radial velocity” was treated at the General Assembly of the International Astronomical Union (IAU), held in 2000 (see [15, 10]), due to the ambiguity of the classic concepts in general relativity. As a result, they obtained three different concepts of radial velocity: kinematic (which corresponds most closely to the line-of-sight component of space velocity), astrometric (which can be derived from astrometric observations) and spectroscopic (also called barycentric, which can be derived from spectroscopic measurements). The kinematic and astrometric radial velocities were defined using a particular reference system, called Barycentric Celestial Reference System (BCRS). The BCRS is suitable for accurate modelling of motions and events within the solar system, but it has not taken into account the effects produced by gravitational fields outside the solar system, since it describes an asymptotically flat metric at large distances from the Sun. Moreover, from a more theoretical point of view, these concepts can not be defined in an arbitrary space-time since they are not intrinsic, i.e. they only have sense in the framework of the BCRS. So, in this work we are going to define them intrinsically. In fact, we obtain in a natural way four intrinsic definitions of relative velocity (and consequently, radial velocity) of one observer β with respect to another observer β, following the original ideas of the IAU.
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This paper has two big parts: – The first one is formed by Sect. 3 and 4, where all the concepts are defined, trying to make the paper as self-contained as possible. In Sect. 3, we define the kinematic and Fermi relative velocities in the framework of spacelike simultaneity (also called Fermi simultaneity), obtaining some general properties and interpretations. The kinematic relative velocity generalizes the usual concept of relative velocity when the two observers β, β are at the same event. On the other hand, the Fermi relative velocity does not generalize this concept, but it is physically interpreted as the variation of the relative position of β with respect to β along the world line of β. Analogously, in Sect. 4, we define and study the spectroscopic and astrometric relative velocities in the framework of observed (lightlike) simultaneity. – In the second one (Sect. 5 and 6) we give some relations between these concepts in special and general relativity. In Sect. 5 we find general expressions, in special relativity, for the relation between kinematic and Fermi relative velocities, and between spectroscopic and astrometric relative velocities. Finally, in Sect. 6 we show some fundamental examples in Schwarzschild and Robertson-Walker space-times. 2. Preliminaries We work in a 4-dimensional lorentzian space-time manifold (M, g), with c = 1 and ∇ the Levi-Civita connection, using the Landau-Lifshitz Spacelike Convention (LLSC). We suppose that M is a convex normal neighborhood [8]. Thus, given two events p and q in M, there exists a unique geodesic joining p and q and there are not caustics. The parallel transport from p to q along this geodesic will be denoted by τ pq . If β : I → M is a curve with I ⊂ R a real interval, we will identify β with the image β I (that is a subset in M), in order to simplify the notation. If u is a vector, then u ⊥ denotes the orthogonal space of u. The projection of a vector v onto u ⊥ is the projection parallel to u. Moreover, if x is a spacelike vector, then x denotes the modulus of x. Given a pair of vectors u, v, we use g (u, v) instead of u α vα . If X is a vector field, X p will denote the unique vector of X in T p M. In general, we will say that a timelike world line β is an observer. Nevertheless, we will say that a future-pointing timelike unit vector u in T p M is an observer at p, identifying it with its 4-velocity. The relative velocity of an observer with respect to another observer is completely well defined only when these observers are at the same event: given two observers u and u at the same event p, there exists a unique vector v ∈ u ⊥ and a unique positive real number γ such that u = γ (u + v) .
(1)
As consequences, we have 0 ≤ v < 1 and γ := −g u , u = √
1 . 1−v2
We will say
that v is the relative velocity of u observed by u, and γ is the gamma factor corresponding to the velocity v. From (1), we have v=
1 u − u. −g (u , u)
(2)
We will extend this definition of relative velocity in two different ways (kinematic and spectroscopic) for observers at different events. Moreover, we will define another two
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concepts of relative velocity (Fermi and astrometric) that do not extend (2) in general, but they have clear physical sense as the variation of the relative position. A light ray is given by a lightlike geodesic λ and a future-pointing lightlike vector field F defined in λ, tangent to λ and parallelly transported along λ (i.e. ∇ F F = 0), called frequency vector field of λ. Given p ∈ λ and u an observer at p, there exists a unique vector w ∈ u ⊥ and a unique positive real number ν such that F p = ν (u + w) .
(3)
As consequences, we have w = 1 and ν = −g F p , u . We will say that w is the relative velocity of λ observed by u, and ν is the frequency of λ observed by u. In other words, ν is the modulus of the projection of F p onto u ⊥ . A light ray from q to p is a light ray λ such that q, p ∈ λ and expq−1 p is future-pointing. 3. Relative Velocity in the Framework of Spacelike Simultaneity The spacelike simultaneity was introduced by E. Fermi (see [7]), and it was used to define the Fermi coordinates. So, some concepts given in this section are very related to the work of Fermi, as the Fermi surfaces, the Fermi derivative or the Fermi distance. The original Fermi paper and most of the modern discussions of this notion (see [11, 2]) use a coordinate language (Fermi coordinates). On the other hand, in the present work we use a coordinate-free notation that allows us to get a better understanding of the basic concepts of the Fermi work, studying them from an intrinsic point of view and, in the next section, extending them to the framework of lightlike simultaneity. Let u be an observer at p ∈ M and : M → R defined by (q) := g exp−1 q, u . p Then, it is a submersion and the set L p,u := −1 (0) is a regular 3-dimensional submanifold, called Landau submanifold of ( p, u) (see [13, 3]), also known as Fermi surface. In other words, L p,u = exp p u ⊥ . An event q is in L p,u if and only if q is simultaneous with p in the local inertial proper system of u. Definition 1. Given u an observer at p, and a simultaneous event q ∈ L p,u , the relative position of q with respect to u is s := exp−1 p q (see Fig. 1). We can generalize this definition for two observers β and β .
Fig. 1. Scheme in T p M of the relative position s of q with respect to u
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Fig. 2. Scheme in M of the elements that involve the definition of the kinematic relative velocity of u with respect to u
Definition 2. Let β, β be two observers and let U be the 4-velocity of β. The relative position of β with respect to β is the vector field S defined on β such that S p is the relative position of q with respect to U p , where p ∈ β and q is the unique event of β ∩ L p,U p . 3.1. Kinematic relative velocity. We are going to introduce the concept of “kinematic relative velocity” of one observer u with respect to another observer u generalizing the concept of relative velocity given by (2), when the two observers are at different events. Definition 3. Let u, u be two observers at p, q respectively such that q ∈ L p,u . The kinematic relative velocity of u with respect to u is the unique vector vkin ∈ u ⊥ such that τq p u = γ (u + vkin ), where γ is the gamma factor corresponding to the velocity vkin (see Fig. 2). So, it is given by vkin :=
1 τq p u − u. −g τq p u , u
(4)
Let s be the relative position of q with respect to p, the kinematic radial of u velocity s s rad with respect to u is the component of vkin parallel to s, i.e. vkin := g vkin , s s . If rad := v . On the other hand, the kinematic tangential velocity s = 0 (i.e. p = q) then vkin kin tng rad . of u with respect to u is the component of vkin orthogonal to s, i.e. vkin := vkin − vkin
So, the kinematic relative velocity of u with respect to u is the relative velocity of τq p u observed by u, in the sense of expression (2). Note that vkin < 1, since the parallel transported observer τq p u defines an observer at p. We can generalize these definitions for two observers β and β . Definition 4. Let β, β be two observers, and let U , U be the 4-velocities of β, β respectively. The kinematic relative velocity of β with respect to β is the vector field Vkin defined on β such that Vkin p is the kinematic relative velocity of Uq observed by U p (in the sense of Definition 3), where p ∈ β and q is the unique event of β ∩ L p,U p . In the same way, we define the kinematic radial velocity of β with respect to β, denoted rad , and the kinematic tangential velocity of β with respect to β, denoted by V tng . by Vkin kin We will say that β is kinematically comoving with β if Vkin = 0.
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be the kinematic relative velocity of β with respect to β . Then, V Let Vkin kin = 0 if = 0, i.e. the relation “to be kinematically comoving with” is symmetric and only if Vkin and so, we can say that β and β are kinematically comoving (each one with respect to the other). Note that it is not transitive in general.
3.2. Fermi relative velocity. We are going to define the “Fermi relative velocity” as the variation of the relative position. Definition 5. Let β, β be two observers, let U be the 4-velocity of β, and let S be the relative position of β with respect to β. The Fermi relative velocity of β with respect to β is the projection of ∇U S onto U ⊥ , i.e. it is the vector field VFermi := ∇U S + g (∇U S, U ) U
(5)
defined on β. The right-hand side of (5) is known as the Fermi derivative. The Fermi radial velocity of β with respect to β is the component of VFermi parallel to S, i.e. S S rad VFermi := g VFermi , S S if S = 0; if S p = 0 (i.e. β and β intersect at p) then rad VFermi
p
:= VFermi p . On the other hand, the Fermi tangential velocity of β with respect tng
rad . to β is the component of VFermi orthogonal to S, i.e. VFermi := VFermi − VFermi We will say that β is Fermi-comoving with β if VFermi = 0.
It is important to remark that the modulus of the vectors of VFermi is not necessarily smaller than one. Since g (VFermi , S) = g (∇U S, S), if S = 0 we have S S rad VFermi = g ∇U S, . (6) S S The relation “to be Fermi-comoving with” is not symmetric in general. An expression similar to (5) is given by the next proposition, that can be proved easily. Proposition 1. Let β, β be two observers, let U be the 4-velocity of β, let S be the relative position of β with respect to β, and let VFermi be the Fermi relative velocity of β with respect to β. Then VFermi = ∇U S − g (S, ∇U U ) U . Note that if β is geodesic, then ∇U U = 0, and hence VFermi = ∇U S . If S p = 0, i.e. β and β intersect at p, then VFermi p = (∇U S) p . So, it does not coincide in general with the concept of relative velocity given in expression (2). We are going to introduce a concept of distance from the concept of relative position given in Definition 2. This concept of distance was previously introduced by Fermi. Definition 6. Let u be an observer at an event p. Given q, q ∈ L p,u , and s, s the relative positions of q, q with respect to u respectively, distance from q to q the Fermi Fermi q, q := s − s . with respect to u is the modulus of s − s , i.e. du We have that duFermi is symmetric, positive-definite and satisfies the triangular inequality. So, it has all the properties that must verify a topological distance defined on L p,u . As a particular case, if q = p we have 1/2 −1 duFermi (q, p) = s = g exp−1 q, exp q . (7) p p
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The next proposition shows that the concept of Fermi distance is the arclength parameter of a spacelike geodesic, and it can be proved taking into account the properties of the exponential map (see [8]). Proposition 2. Let u be an observer at an event p. Given q ∈ L p,u and α the unique geodesic from p to q, if we parameterize α by its arclength such that α (0) = p, then α duFermi (q, p) = q. Definition 7. Let β, β be two observers and let S be the relative position of β with respect to β. The Fermi distance from β to β with respect to β is the scalar field S defined in β. We are going to characterize the Fermi radial velocity in terms of the Fermi distance. Proposition 3. Let β, β be two observers, let S be the relative position of β with respect to β, and let U be the 4-velocity of β. If S = 0, the Fermi radial velocity of β with rad = U (S) S . respect to β reads VFermi S By Definition 7 and Proposition 3, the Fermi radial velocity of β with respect to β is the rate of change of the Fermi distance from β to β with respect to β. So, if we parameterize β by its proper time τ , the Fermi radial velocity of β with respect to β at S d(S◦β) rad p = β (τ0 ) is given by VFermi (τ0 ) S p . p = dτ p
4. Relative Velocity in the Framework of Lightlike Simultaneity The lightlike (or observed) simultaneity is based on “what an observer is really observing” and it provides an appropriate framework to study optical phenomena and observational cosmology (see [6]). −1 q . Then, it is q, exp Let p ∈ M and ϕ : M → R defined by ϕ (q) := g exp−1 p p a submersion and the set E p := ϕ −1 (0) − { p}
(8)
is a regular 3-dimensional submanifold, called horismos submanifold of p (see [3, 1]). An event q is in E p if and only if q = p and there exists a lightlike geodesic join+ − ing p and q. E p has two connected components, E − p and E p [14]; E p (respectively + E p ) is the past-pointing (respectively future-pointing) horismos submanifold of p, and it is the connected component of (8) in which, for each event q ∈ E − p (respectively q is a past-pointing (respectively future-pointing) lightq ∈ E +p ), the preimage exp−1 p − − + + − like vector. In other words, E p = exp p C p , and E p = exp p C p , where C p and C +p are the past-pointing and the future-pointing light cones of T p M respectively. This section is analogous to Sect. 3, but using E − p instead of L p,u . Definition 8. Given u an observer at p, and an observed event q ∈ E − p ∪{ p}, the relative position of q observed by u (or the observed relative position of q withrespect to u) is ⊥ (see Fig. 3), i.e. s −1 q + g exp−1 q, u u. the projection of exp−1 q onto u := exp obs p p p We can generalize this definition for two observers β and β . Definition 9. Let β, β be two observers and let U be the 4-velocity of β. The relative position of β observed by β is the vector field Sobs defined in β such that Sobs p is the relative position of q observed by U p , where p ∈ β and q is the unique event of β ∩ E − p.
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Fig. 3. Scheme in T p M of the relative position sobs of q observed by u
4.1. Spectroscopic relative velocity. In a previous work (see [4]), we defined a concept of relative velocity of an observer observed by another observer in the framework of lightlike simultaneity. We are going to rename this concept as “spectroscopic relative velocity”, and to review its properties in the context of this work. Definition 10. Let u, u be two observers at p, q respectively such that q ∈ E − p and let observed by u is λ be a light ray from q to p. The spectroscopic relative velocity of u the unique vector vspec ∈ u ⊥ such that τq p u = γ u + vspec , where γ is the gamma factor corresponding to the velocity vspec (see Fig. 4). So, it is given by vspec :=
1 τq p u − u. −g τq p u , u
(9)
We define the spectroscopic radial and tangential velocity of u observed by u analogously to Definition 3, using sobs (see Definition 8) instead of s. So, the spectroscopic relative velocity of u observed by u is the relative velocity of τq p u observed by u, in the sense of expression (2), and vspec < 1.
Fig. 4. Scheme in M of the elements that involve the definition of the spectroscopic relative velocity of u observed by u
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Note that if w is the relative velocity of λ observed by u (see (3)), then w = − ssobs , obs and so rad = g vspec , w w. (10) vspec We can generalize these definitions for two observers β and β . Definition 11. Let β, β be two observers, we define Vspec (the spectroscopic relative velocity of β observed by β) and its radial and tangential components analogously to Definition 4, using E − p instead of L p,U p . We will say that β is spectroscopically comoving with β if Vspec = 0. The relation “to be spectroscopically comoving with” is not symmetric in general. The following result can be found in [4]. Proposition 4. Let λ be a light ray from q to p and let u, u be two observers at p, q respectively. Then (11) ν = γ 1 − g vspec , w ν, where ν, ν are the frequencies of λ observed by u, u respectively, vspec is the spectroscopic relative velocity of u observed by u, w is the relative velocity of λ observed by u, and γ is the gamma factor corresponding to the velocity vspec . Expression (11) is the general expression for Doppler effect (that includes gravitational redshift, see [4]). Therefore, if β is spectroscopically comoving with β , and λ is a light ray from β to β, then, by (11), we have that β and β observe λ with the same frequency. So, if β emits n light rays in a unit of its proper time, then β observes also n light rays in a unit of its proper time. Hence, β observes that β uses the “same clock” as it does. Taking into account (10), expression (11) can be written in the form rad 1 ± vspec ν = ν, 1 − vspec 2
(12)
where we choose “+” if g v , w < 0 (i.e. if u is moving away from u), and we spec choose “−” if g vspec , w > 0 (i.e. if u is getting closer to u). Remark 1. We can not deduce vspec from the shift, ν /ν, unless we make some assumptions (like considering negligible the tangential component of vspec , as we will see in Remark 2). For instance, if ν /ν = 1 then vspec is not necessarily zero. Let us study this particular case: by (11) we have 1 − g vspec , w ν 1= = −→ g vspec , w = 1 − 1 − vspec 2 . ν 1 − vspec 2 Since 1 − 1 − vspec 2 ≥ 0, it is necessary that g vspec , w ≥ 0, i.e. the observed rad = object has to be getting closer to the observer. In this case, by (12) we have vspec 1 − 1 − vspec 2 . So, it is possible that ν /ν = 1 and vspec = 0 if the observed object
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is getting closer to the observer. On the other hand, if the observed object is moving away from the observer then ν /ν = 1 if and only if vspec = 0. That is, for objects moving away, the shift is always redshift; and for objects getting closer, the shift can be blueshift, 1, or redshift. tng
rad = kw with k ∈ ]−1, 1[, then Remark 2. If we suppose that vspec = 0, i.e. vspec = vspec we can deduce vspec from the shift ν /ν: 2 √ ν 1 − 1 − g vspec , w ν ν 1−k 1−k = =√ = √ −→ k = 2 , 2 ν 1+k 1−k 1 − vspec 2 1 + νν
and hence
⎛ vspec
⎜ =⎝
1− 1+
2 ⎞ ν ν
⎛
⎟ ⎜ 2 ⎠ w = − ⎝ ν ν
1− 1+
2 ⎞ ν ν
⎟ sobs . 2 ⎠ sobs ν
(13)
ν
4.2. Astrometric relative velocity. We are going to define the “astrometric relative velocity” as the variation of the observed relative position. Definition 12. Let β, β be two observers, we define Vast (the astrometric relative velocity of β observed by β) and its radial and tangential components analogously to Definition 5, using Sobs (see Definition 9) instead of S. So, Vast := ∇U Sobs + g (∇U Sobs , U ) U,
(14)
where U is the 4-velocity of β. We will say that β is astrometrically comoving with β if Vast = 0. It is important to remark that the modulus of the vectors of Vast is not necessarily smaller than one. Analogously to (6), since g (Vast , Sobs ) = g (∇U Sobs , Sobs ), if Sobs = 0 we have Sobs Sobs rad Vast = g ∇U Sobs , . (15) Sobs Sobs The relation “to be astrometrically comoving with” is not symmetric in general. An expression similar to (14) is given by the next proposition, which proof is analogous to the proof of Proposition 1. Proposition 5. Let β, β be two observers, let U be the 4-velocity of β, let Sobs be the relative position of β observed by β, and let Vast be the astrometric relative velocity of β observed by β. Then Vast = ∇U Sobs − g (Sobs , ∇U U ) U . Note that if β is geodesic, then ∇U U = 0, and hence Vast = ∇U Sobs . If Sobs p = 0, i.e. β and β intersect at p, then Vast p = (∇U Sobs ) p . So, it does not coincide in general with the concept of relative velocity given in (2). We are going to introduce another concept of distance from the concept of observed relative position given in Definition 8. This distance was previously introduced in [9] and studied in [4], and it plays a basic role for the construction of optical coordinates whose relevance for cosmology was stressed in many articles by G. Ellis and his school (see [6]).
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Definition 13. Let u be an observer at an event p. Given q, q ∈ E − p ∪ { p}, and sobs , the relative positions of q, q observed by u respectively, the affine distance from sobs , i.e. d affine q, q := s q to q observed by u is the modulus of sobs − sobs obs − sobs . u We have that duaffine is symmetric, positive-definite and satisfies the triangular inequality. So, it has all the properties that must verify a topological distance defined on E − p ∪{ p}. As a particular case, if q = p we have duaffine (q, p) = sobs = g exp−1 p q, u .
(16)
The next proposition shows that the concept of affine distance is according to the concept of “length” (or “time”) parameter of a lightlike geodesic for an observer, and it is proved in [4]. Proposition 6. Let λ be a light ray from q to p, let u be an observer at p, and let w be the relative velocity of λ observed by u. If we parameterize λ affinely (i.e. the vector field tan. gent toλ is parallelly transported along λ) such that λ (0) = p and λ (0) = − (u + w), then λ duaffine (q, p) = q. Definition 14. Let β, β be two observers and let Sobs be the relative position of β observed by β. The affine distance from β to β observed by β is the scalar field Sobs defined in β. We are going to characterize the astrometric radial velocity in terms of the affine distance. The proof of the next proposition is analogous to the proof of Proposition 3, taking into account expression (15). Proposition 7. Let β, β be two observers, let Sobs be the relative position of β observed by β, and let U be the 4-velocity of β. If Sobs = 0, the astrometric radial velocity of β rad = U (S ) Sobs . observed by β reads Vast obs Sobs By Definition 14 and Proposition 7, the astrometric radial velocity of β observed by β is the rate of change of the affine distance from β to β observed by β. So, if we parameterize β by its proper time τ , the astrometric radial velocity of β observed by β rad = d(Sobs ◦β) (τ ) Sobs p . at p = β (τ0 ) is given by Vast 0 S p dτ obs p
5. Special Relativity In this section, we are going to work in the Minkowski space-time, with β, β two observers, and U the 4-velocity of β. Proposition 8. Let S be the relative position of β with respect to β, and let Vkin , VFermi be the kinematic and Fermi relative velocities of β with respect to β respectively. Then VFermi = (1 + g (S, ∇U U )) Vkin .
(17)
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Fig. 5. Scheme of the proof of Proposition 8
Proof. We are going to consider the observers parameterized by their proper times. Let p = β (τ ) be an event of β, let u (τ ) be the 4-velocity of β at p, and let q = β τ be the event of β such that g (u (τ ) , q − p) = 0 (note that the Minkowski space-time has an affine structure, and q − p denotes the vector which joins p and q). So, q − p is the relative position of q with respect to u (τ ), denoted by s (τ ). Considering the differential diagram given in Fig. 5, where u τ is the 4-velocity of β at q, it is easy to check that . 1 + g s (τ ) , u (τ ) dτ (18) δ= −g (u (τ ) , u (τ + dτ )) for an infinitesimally small dτ (i.e. it holds in quadratic approximation in dτ ). Since s (τ + dτ ) = s (τ ) + u τ δ − u (τ ) dτ , from (18) we have . 1 + g s (τ ) , u (τ ) s (τ + dτ ) − s (τ ) . = u τ − u (τ ) . s (τ ) = lim (19) dτ →0 dτ −g (u (τ ) , u (τ )) Let U , U be the 4-velocities of β and β respectively, and let S be the relative position of β with respect to β. Then, from (19) we have 1 + g S p , (∇U U ) p Uq − U p . (20) (∇U S) p = −g U p , Uq So, by Proposition 1 and expression (20), the Fermi relative velocity VFermi p of β with respect to β at p is given by VFermi p = (∇U S) p − g S p , (∇U U ) p U p ⎞ ⎛ 1 Uq − U p ⎠ . (21) = 1 + g S p , (∇U U ) p ⎝ −g U p , Uq On the other hand, the kinematic relative velocity Vkin reads Vkin
p
=
p
of β with respect to β at p
1 Uq − U p . −g Uq , U p
Hence, from (21) and (22) we obtain (17), concluding the proof.
(22)
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Fig. 6. Scheme of the proof of Proposition 9
So, Vkin and VFermi are proportional. Moreover, if β is geodesic, then VFermi = Vkin . The proof of the next proposition is similar to the proof of Proposition 8 (but a bit more complicated), considering the differential diagram given in Fig. 6. Proposition 9. Let Sobs be the relative position of β observed by β, and let Vspec , Vast be the spectroscopic and astrometric relative velocities of β observed by β respectively. If Sobs = 0 then Vast = Sobs ∇U U +
1
1 + g Vspec , SSobs obs
Vspec .
(23)
So, Vspec and Vast are not proportional unless β is geodesic. If β is geodesic then it is clear that Vspec = Vkin . Moreover, if β is also geodesic then Vspec = Vkin = VFermi . Remark 3. Let us suppose that β and β intersect at p, let u, u be the 4-velocities of β, β at p respectively, and let v be the relative velocity of u observed by u, in the sense of expression (2). Let us study the relations between v, Vkin p , VFermi p , Vspec p and Vast p . It is clear that Vkin p = Vspec p = v, even in general relativity. Moreover, since S p = 0, by (17) we have VFermi p = v. On the other hand, since Sobs p = 0, it is easy 1 to prove that Vast p = 1±v v, where we choose “+” if we consider that β is leaving from β, and we choose “−” if we consider that β is arriving at β. Therefore, if β and β intersect at p, then it is not possible to write Vast p in a unique way in terms of v. Example 1. Using rectangular coordinates (t, x, y, z), let us consider the following τ := observers parameterized by their proper times: β := 0, 0, 0), and β (τ ) (τ, ⎧ ⎨ γ τ , vγ τ , 0, 0 if τ ∈ 0, 1 γv , where v ∈ ]0, 1[ and γ := √ 1 . That is, 1−v 2 ⎩ γ τ , 2 − vγ τ , 0, 0 if τ ∈ 1 , 2 γv γv
Intrinsic Definitions of “Relative Velocity”
229
Fig. 7. Scheme of the observers of Example 1
β is a stationary observer with x = 0, y = 0, z = 0 and β is an observer moving from x = 0, y = 0, z = 0 to x = 1, y = 0, z = 0 with velocity of modulus v and returning (see Fig. 7). It is satisfied that ∂ v ∂ x β(τ ) if τ ∈ 0, v1 , Vkin β(τ ) = −v ∂∂x β(τ ) if τ ∈ v1 , v2 Vspec β(τ ) =
v ∂∂x β(τ ) if τ ∈ 0, 1+v v 2 . −v ∂∂x β(τ ) if τ ∈ 1+v v ,v
Applying (17), we obtain VFermi β(τ ) = Vkin β(τ ) . Moreover vτ ∂ if τ ∈ 0, 1+v 1+v ∂ x β(τ v ) . Sobs β(τ ) = 2−vτ ∂ if τ ∈ 1+v , 2 1−v
Hence, by (23) we have Vast β(τ ) =
∂ x β(τ )
∂ v if 1+v ∂ x β(τ ) ∂ v − 1−v ∂ x β(τ )
v
v
τ ∈ 0, 1+v v 2 . if τ ∈ 1+v v ,v
Consequently, Vast β(τ ) ∈ ]0, 1/2[ if τ ∈ 0, 1+v , i.e. if β is moving away radially. v 1+v 2 On the other hand, Vast β(τ ) ∈ ]0, +∞[ if τ ∈ v , v , i.e. if β is getting closer radially. This corresponds to what β observes.
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6. Examples in General Relativity 6.1. Stationary observers in Schwarzschild. In the Schwarzschild metric with spherical coordinates 1 ds 2 = −a 2 (r ) dt 2 + 2 dr 2 + r 2 dθ 2 + sin2 θ dϕ 2 , a (r ) where a (r ) = 1 − 2m r > 2m, let us consider two equatorial stationary observers, r and β1 (τ ) = a11 τ, r1 , π/2, 0 and β2 (τ ) = a12 τ, r2 , π/2, 0 with τ ∈ R, r2 > r1 > 2m,
a1 := a (r1 ) and a2 := a (r2 ), and let U be the 4-velocity of β2 , i.e. U := a12 ∂t∂ . We are going to study the relative velocities of β1 with respect to and observed by β2 .
6.1.1. Kinematic and Fermi relative velocities Let us consider the vector field X := ∂ a (r ) ∂r . This vector field is spacelike, unit, geodesic, and orthogonal to U . Since 1 ∂ ∇ X a(r ) ∂t = 0, we have that the kinematic relative velocity Vkin of β1 with respect to β2 is given by Vkin = 0. Let α (σ ) = (t0 , αr (σ ) , π/2, 0) be an integral curve of X such that q := α (σ1 ) ∈ β1 and p := α (σ2 ) ∈ β2 , with σ2 > σ1 (i.e. α (σ ) is a spacelike geodesic from q to p, parameterized by its arclength, and its tangent vector at p is X p ). Then, by Proposition 2, the Fermi distance dUFermi (q, p) from q to p with respect to U p is σ2 −σ1 . Since α is an p −1/2 . r2 .r r 2m integral curve of X , we have α (σ ) = 1 − α2m 1 − α (σ ) dσ = r (σ ) . So, r r (σ ) α 1 σ2 − σ1 , and then √ (1 − a1 ) r1 + r 2 a2 − r 1 a1 . (24) p) = 2m ln dUFermi (q, √ p (1 − a2 ) r2 Since (24) does not depend on t0 , the Fermi distance from β1 to β2 with respect to β2 is also given by expression (24). Hence, by (7), the relative position S of β1 with respect to β2 is given by √ ∂ (1 − a2 ) r2 + r 1 a1 − r 2 a2 a2 . S = 2m ln √ ∂r (1 − a1 ) r1 It is easy to prove that ∇U S is proportional to U . Therefore, the Fermi relative velocity VFermi of β1 with respect to β2 reads VFermi = 0. 6.1.2. Spectroscopic and astrometric relative velocities It is easy to prove that the spectroscopic relative velocity Vspec of β1 observed by β2 is radial. Since the gravitational redshift is given by aa21 (see [4]), by (13) we obtain Vspec = −a2
a22 − a12 ∂ . a22 + a12 ∂r
Expression (25) is also obtained in [4]. We have limr1 →2m Vspec = 1.
(25)
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On the other hand, in [4] it is also proved (by using Proposition 6) that the affine 1 distance from β1 to β2 observed by β2 is r2a−r . Hence, by (16), the relative position Sobs 2 of β1 observed by β2 is given by Sobs = (r1 − r2 )
∂ . ∂r
(26)
It is easy to prove that ∇U Sobs is proportional to U . Therefore, the astrometric relative velocity Vast of β1 observed by β2 reads Vast = 0. 6.2. Free-falling observers in Schwarzschild. Let us consider ob a radial free-falling server β1 parameterized by the coordinate time t, β1 (t) = t, β1r (t) , π/2, 0 . Given an event q = (t1 , r1 , π/2, 0) ∈ β1 , the 4-velocity of β1 at q is given by ∂ E ∂ 2 − a2 u1 = 2 − E , (27) 1 ∂r q a1 ∂t q where E is a constant of motion given by E :=
1−2m/r0 1−v02
1/2 , r0 is the radial coordinate
at which the fall begins, v0 is the initial velocity (see [5]), and a1 := a (r1). Moreover, let us consider an equatorial stationary observer β2 (τ ) = a12 τ, r2 , π/2, 0 with τ ∈ R,
r2 > r1 > 2m, a2 := a (r2 ), and U := a12 ∂t∂ its 4-velocity. We are going to study the relative velocities of β1 with respect to and observed by β2 at p, where p will be a determined event of β2 . 6.2.1. Kinematic and Fermi relative velocities Let p = (t1 , r2 , π/2, 0). This is the unique event of β2 such that q ∈ L p,U p , i.e. there exists a spacelike geodesic α (σ ) from . q = α (σ1 ) to p = α (σ2 ) such that the tangent vector α (σ2 ) is orthogonal to U p . We can consider α (σ ) parameterized by its arclength and σ2 > σ1 . So, α (σ ) is an integral curve ∂ of the vector field X = a (r ) ∂r . If we parallelly transport u 1 from q to p along α we a2 E ∂ obtain τq p u 1 = − E 2 − a 2 ∂ . By (4), the kinematic relative velocity a1 a2 ∂t p
Vkin
p
1 ∂r p
a1
of β1 with respect to β2 at p reads
∂ a2 2 2 Vkin p = − E − a1 . E ∂r p So, it is satisfied that limr1 →2m Vkin p = 1. On the other hand, by (24), the relative position S of β1 with respect to β2 is given by √ r ∂ (1 − a2 ) r2 r r r S = 2m ln + β1 (t) a β1 (t) − r2 a2 a2 . ∂r 1 − a β1 (t) β1 (t) By (5), the Fermi relative velocity VFermi of β1 with respect to β2 reads .r
VFermi
1 ∂ Sr ∂ 1 β 1 (t) ∂ ∂ . = = = (∇U S) ∂r a2 ∂t ∂r a2 a β1r (t) ∂r r
232
V. J. Bolós .r
a2
Taking into account (27), we have β 1 (t1 ) = − E1
E 2 − a12 . Hence ∂ a1 2 2 VFermi p = − E − a1 . a2 E ∂r p So, it is satisfied that limr1 →2m VFermi p = 0.
6.2.2. Spectroscopic and astrometric relative velocities Let p be the unique event of β2 such that there exists a light ray λ from q to p, and let us suppose that p = (t2 , r2 , π/2, 0). In [4] it is shown that the spectroscopic relative velocity Vspec p of β1 observed by β2 at p is given by 2 a2 + a12 E 2 − a12 + E a22 − a12 ∂ . (28) Vspec p = −a2 a2 − a2 E 2 − a 2 + E a 2 + a 2 ∂r p 2
1
1
2
1
So, it follows that limr1 →2m Vspec p = 1. On the other hand, it can be checked that r − 2m λ (r ) := t1 + r − r1 + 2m ln , r, π/2, 0 , r ∈ [r1 , r2 ] r1 − 2m is a light ray from q = λ (r1 ) to p = λ (r2 ). So,
t2 = λt (r2 ) = t1 + r2 − r1 + 2m ln
r2 − 2m r1 − 2m
.
Let us define implicitly the function f (t) by the expression r2 − 2m . f (t) := t − r2 − β1r ( f (t)) + 2m ln β1r ( f (t)) − 2m
(29)
(30)
Taking into account (29), f (t) is the coordinate time at which β1 emits a light ray that arrives at β2 at coordinate time t. Applying (26), the relative position Sobs of β1 observed by β2 reads ∂ . Sobs = β1r ( f (t)) − r2 ∂r By (14), the astrometric relative velocity Vast of β1 observed by β2 is given by Vast = (∇U Sobs )r .
r . 1 ∂ Sobs ∂ 1 .r ∂ ∂ = = β 1 ( f (t)) f (t) . ∂r a2 ∂t ∂r a2 ∂r a2
From (30), we have f (t2 ) = 2 2 1 . r . Moreover, taking into account (27), we a1 − a1 −1 β 1 (t1 ) .r a12 2 2 have β 1 (t1 ) = − E E − a1 . Hence 2 E 2 − a12 a1 ∂ Vast p = − , (31) a2 E + a 2 − 1 E 2 − a 2 ∂r p 1 1 and, in consequence, limr1 →2m Vast p =
1 2E 2 a22 1+2E 2
∈ ]0, +∞[.
Intrinsic Definitions of “Relative Velocity”
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6.3. Comoving observers in Robertson-Walker. In a Robertson-Walker metric with cartesian coordinates a 2 (t) ds 2 = −dt 2 + dx 2 + dy 2 + dz 2 , 2 1 + 41 kr 2 where a (t) is the scale factor, k = −1, 0, 1 and r := x 2 + y 2 + z 2 , we consider two comoving (in the classical sense, see [14]) observers β0 (τ ) = (τ, 0, 0, 0) and β1 (τ ) = . (τ, x1 , 0, 0) with τ ∈ R and x1 > 0. Let t0 ∈ R, p := β0 (t0 ) and u := β0 (t0 ) = ∂t∂ p (i.e. the 4-velocity of β0 at p). We are going to study the relative velocities of β1 with respect to and observed by β0 at p. 6.3.1. Kinematic and Fermi relative velocities. The vector field X := −
a02 a0 ∂ 1 2 ∂ + kx − 1 1 + a 2 (t) ∂t a 2 (t) 4 ∂x
is geodesic, spacelike, unit, and X p is orthogonal to u, i.e. it is tangent to the Landau submanifold L p,u . Let β1 (t1 ) =: q be the unique event of β1 ∩ L p,u . We can find t1 for concrete scale factors a (t) taking into account the expression of X , but . we can not find an explicit expression in the general case. If u := β1 (t1 ) = ∂t∂ q , then τq p u = a0 ∂ + 12 − 12 ∂ , where a1 := a (t1 ) (it is well defined because a1 ∂t p
a0 ∂ x p
a1
a0 ≥ a1 > 0). So, by (4), the kinematic relative velocity Vkin at p is given by
Vkin
p
1 = 2 a0
p
of β1 with respect to β0
∂ 2 2 a0 − a1 . ∂x p
Given a concrete scale factor a (t), the Fermi distance d Fermi from β1 to β0 with respect to β0 can be also found, taking into account the expression of X . So, the relative position S of β1 with respect to β0 reads S=d
Fermi
1 + 41 kr 2 ∂ , a (t) ∂ x
because d Fermi = S. Hence, the Fermi relative velocity VFermi β0 at p is given by VFermi
p
=
d dt
p
of β1 with respect to
. a (t0 ) ∂ d Fermi d Fermi p . + a (t) t=t0 ∂x p a02
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6.3.2. Spectroscopic and astrometric relative velocities Let λ be a light ray received by β0 at p and emitted 1 ). Note that t1 can be found from x 1 and t0 taking x from β1 at β1 t(t 0 dt = into account that 0 1 dx 1 2 t1 a(t) . It can be easily proved that the spectroscopic 1+ 4 kx
relative velocity Vspec p of β1 observed by β0 at p is radial (by isotropy). So, by (13) taking into account that the cosmological shift is given by aa01 (see [4]), where a0 := a (t0 ) and a1 := a (t1 ), we have Vspec
p
=
1 a02 − a12 ∂ . a0 a02 + a12 ∂ x p
(32)
Given a concrete scale factor a (t), the affine distance d affine from β1 to β0 observed by β0 can be found. So, the relative position Sobs of β1 observed by β0 is given by Sobs = d
affine
1 + 41 kr 2 ∂ , a (t) ∂ x
because d affine = Sobs . Hence, the astrometric relative velocity Vast by β0 at p reads Vast
p
=
d dt
p
. a (t0 ) ∂ d affine d affine p . + a (t) t=t0 ∂x p a02
of β1 observed
(33)
Let us study these relative velocities in more detail. In cosmology it is usual to consider the scale factor in the form 1 a (t) = a0 1 + H0 (t − t0 ) − q0 H02 (t − t0 )2 + O H03 (t − t0 )3 , 2 .
where t0 ∈ R, a0 = a (t0 ) ..> 0, H (t) = a (t) /a (t) is the Hubble “constant”, H0 = . H (t0 ) > 0, q (t) = −a (t) a (t) /a (t)2 is the deceleration coefficient, and q0 = q (t0 ), with |H0 (t − t0 )| 1 (see [12]). This corresponds to a universe in decelerated expansion and the time scales that we are going to use are relatively small. Let us define . p := β0 (t0 ) and u := β0 (t0 ) = ∂t∂ p . We are going to express the spectroscopic and the astrometric relative velocity of β1 observed by β0 at p in terms of the redshift parameter at t = t0 , defined as z 0 := aa01 − 1, where a1 := a (t1 ). This parameter is very usual in cosmology since it can be measured by spectroscopic observations. By (32), the spectroscopic relative velocity Vspec p of β1 observed by β0 at p is given by Vspec
p
1 a04 − (z 0 + 1)2 ∂ = . a0 a04 + (z 0 + 1)2 ∂ x p
(34)
In [4] it is shown that the affine distance d affine from β1 to β0 observed by β0 reads d affine (t) =
1 z (t) 1 − (3 + q (t)) z (t) + O z 3 (t) , H (t) 2
Intrinsic Definitions of “Relative Velocity”
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where z (t) is the redshift function. So, by (33), the astrometric relative velocity Vast of β1 observed by β0 at p is given by . . ∂ z (t0 ) z 0 z (t0 ) . q0 + 1 − + Vast p = (3 + q0 ) + O z 02 a0 H0 a0 H0 ∂x p
p
.
Hence, if we suppose that z (t0 ) ≈ 0 (i.e., the redshift is constant in our time scale), then ∂ z0 . (35) Vast p ≈ (q0 + 1) + O z 02 a0 ∂x p 7. Discussion and Comments It is usual to consider the spectroscopic relative velocity as a non-acceptable “physical velocity”. However, in this paper we have defined it in a geometric way, showing that it is, in fact, a very plausible physical velocity. – Firstly, in other works (see [3, 4]), we have discussed pros and cons of spacelike and lightlike simultaneities, coming to the conclusion that lightlike simultaneity is physically and mathematically more suitable. Since the spectroscopic relative velocity is the natural generalization (in the framework of lightlike simultaneity) of the usual concept of relative velocity (given by (2)), it might have a lot of importance. – Secondly, there are some good properties suggesting that the spectroscopic relative velocity has a lot of physical sense. For instance, if we work with the spectroscopic relative velocity, it is shown in [4] that gravitational redshift is just a particular case of a generalized Doppler effect. Nevertheless, all four concepts of relative velocity have full physical sense and they must be studied equally. Finally, one can wonder whether the discussed concepts of relative velocity can be actually determined experimentally. A priori, only the spectroscopic and astrometric relative velocities can be measured by direct observation. The shift allows us to find relations between the modulus of the spectroscopic relative velocity and its tangential component, as we show in (12). But, in general, it is not enough information to determine it completely (as we discuss in Remark 1), unless we make some assumptions (see Remark 2) or we use a model for the space-time and apply some expressions like (25), (28), or (34). Finding the astrometric relative velocity is basically the same problem as finding the optical coordinates. It is non-trivial and it has been widely treated, for instance, in [6]. Nevertheless, expressions like (31) or (35) could be very useful in particular situations. Since the measure of these velocities is rather difficult, any expression relating them can be very helpful in order to determine them, as, for example, expression (23) in special relativity. Acknowledgements. I would like to thank E. Minguzzi and P. Sancho for their valuable help and comments.
References 1. Beem, J.K., Ehrlich, P.E.: Global Lorentzian Geometry. Marcel Dekker, New York, 1981 2. Bini, D., Lusanna, L., Mashhoon, B.: Limitations of radar coordinates. Int. J. Mod. Phys. D 14, 1413– 1429 (2005)
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3. Bolós, V.J., Liern, V., Olivert, J.: Relativistic simultaneity and causality. Int. J. Theor. Phys. 41, 1007– 1018 (2002) 4. Bolós, V.J.: Lightlike simultaneity, comoving observers and distances in general relativity. J. Geom. Phys. 56, 813–829 (2006) 5. Crawford, P., Tereno, I.: Generalized observers and velocity measurements in general relativity. Gen. Rel. Grav. 34, 2075–2088 (2002) 6. Ellis, G.F.R., Nel, S.D., Maartens, R., Stoeger, W.R., Whitman, A.P.: Ideal observational cosmology. Phys. Rep. 124, 315–417 (1985) 7. Fermi, E.: Sopra i fenomeni che avvengono in vicinanza di una linea oraria. Atti R. Accad. Naz. Lincei, Rendiconti, Cl. sci. fis. mat & nat. 31(1), 21–23, 51–52, and 101–103 (1922) 8. Helgason, S.: Differential Geometry and Symmetric Spaces. Academic Press, London, 1962 9. Kermack, W.O., McCrea, W.H., Whittacker, E.T.: On properties of null geodesics and their application to the theory of radiation. Proc. R. Soc. Edinburgh 53, 31–47 (1932) 10. Lindegren, L., Dravins, D.: The fundamental definition of “radial velocity”. Astronomy & Astrophysics 401, 1185–1201 (2003) 11. Marzlin, K.P.: The physical meaning of Fermi coordinates. Gen. Rel. Grav. 26, 619–636 (1994) 12. Misner, W., Thorne, K., Wheeler, J.: Gravitation. Freeman, New York, 1973 13. Olivert, J.: On the local simultaneity in general relativity. J. Math. Phys. 21, 1783–1785 (1980) 14. Sachs, R.K., Wu, H.: Relativity for Mathematicians. Springer Verlag, Berlin, 1977 15. Soffel, M., et al.: The IAU 2000 resolutions for astrometry, celestial mechanics and metrology in the relativistic framework: explanatory supplement. The Astronomical Journal 126, 2687–2706 (2003) Communicated by G.W. Gibbons
Commun. Math. Phys. 273, 237–281 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0253-z
Communications in
Mathematical Physics
Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators on Bounded Domains Stathis Filippas1,4 , Luisa Moschini2 , Achilles Tertikas3,4 1 Department of Applied Mathematics, University of Crete, 71409 Heraklion, Greece.
E-mail:
[email protected]
2 Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, University of Rome “La Sapienza”,
00161 Rome, Italy. E-mail:
[email protected]
3 Department of Mathematics, University of Crete, 71409 Heraklion, Greece. E-mail:
[email protected] 4 Institute of Applied and Computational Mathematics, FORTH, 71110 Heraklion, Greece
Received: 10 July 2006 / Accepted: 20 December 2006 Published online: 4 May 2007 – © Springer-Verlag 2007
Abstract: On a smooth bounded domain ⊂ R N we consider the Schrödinger operators − − V , with V being either the critical borderline potential V (x) = (N − 2)2 /4 |x|−2 or V (x) = (1/4) dist(x, ∂)−2 , under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Schrödinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a series of new inequalities such as improved Hardy, logarithmic Hardy Sobolev, Hardy-Moser and weighted Poincaré. As a byproduct of our technique we are able to answer positively to a conjecture of E. B. Davies. 1. Introduction and Main Results Harnack inequalities have been extremely useful in the study of solutions of elliptic and parabolic equations, starting from the pioneering works of De Giorgi [DG], Nash [N] and Moser [Mo1, Mo2]. They are used to prove Hölder continuity of solutions, strong maximum principles, Liouville properties, as well as sharp two-sided heat kernel estimates. In particular, we should mention the influential works of Aronson [A] and Li and Yau [LY] where heat kernel estimates were obtained via parabolic Harnack inequalities. In fact, in certain cases, the parabolic Harnack inequality is equivalent to sharp two– sided heat kernel estimates. This is the case when dealing with second order uniformly elliptic operators in divergence form on R N , or more generally with weighted Laplacians on complete Riemannian manifolds; see the works of Fabes and Stroock [FS], Grigoryan [G1], and Saloff–Coste [SC1]. This equivalence has been also used in order to get sharp two–sided estimates for Schrödinger operators in R N . For instance, the case of a potential that is regular and decays like |x|−2 at infinity was studied by Davies and Simon [DS2], where pointwise upper bounds for the heat kernel were derived. The picture was later completed by Grigoryan [G2] where sharp two sided estimates were
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provided by means of a parabolic Harnack inequality. A recent survey on heat kernels on weighted manifolds can also be found in [G2]. As it was shown in the works of Fabes, Kenig and Serapioni [FKS], and Chiarenza and Serapioni [CS], parabolic Harnack inequalities follow after establishing Poincaré and Sobolev inequalities as well as a doubling volume growth condition. Moreover, on complete Riemannian manifolds parabolic Harnack inequalities are equivalent to Poincaré inequality and a doubling volume growth condition as explained by Grigoryan and Saloff–Coste in [GSC, SC2]. Since the work of Baras and Goldstein [BG], the existence or nonexistence of solutions to the partial differential equation u t = u + V u,
(1.1)
with a potential V involving the inverse square of the distance function have been widely investigated. See [BG], Brezis and Vázquez [BV], Cabré and Martel [CM], as well as Vázquez and Zuazua [VZ], for the case V (x) = c|x|−2 and [CM] for the case V (x) = cd −2 (x) on a bounded domain , where d(x) = dist(x, ∂). Concerning the case where V (x) = c|x|−2 with c < (N − 2)2 /4, sharp two–sided heat kernel estimates have been obtained in R N , see [MT1, MT2] where the approach of [GSC] on complete Riemannian manifolds has been used, after a suitable transformation; see also [MS] for a different method. On the other hand few results are known in the case of incomplete Riemannian manifolds, as it is for example the case of bounded domains in R N . To our knowledge the only sharp two sided estimates in this case, concern the standard Dirichlet Laplacian on a smooth bounded domain ⊂ R N , first studied by Davies and Simon in [D1, D2, DS1], and recently completed by Zhang [Z]. We note that in the case of a bounded domain, the asymptotic of the heat kernel is different for small time than it is for large time. In fact, for the heat kernel h D (t, x, y) of the standard Dirichlet Laplacian and for two positive constants C1 ≤ C2 , we have for small time d(x)d(y) − N −C2 |x−y|2 t t 2e C1 min 1, ≤ h D (t, x, y) t d(x)d(y) − N −C1 |x−y|2 t t 2e , (1.2) ≤ C2 min 1, t whereas for large time C1 d(x) d(y) e−λ1 t ≤ h D (t, x, y) ≤ C2 d(x) d(y) e−λ1 t ,
(1.3)
for all x, y ∈ ; here λ1 is the first Dirichlet eigenvalue. In this work, our main interest is in obtaining sharp two–sided estimates for the heat kernel of the Schrödinger operator − − V under Dirichlet boundary conditions, on a smooth bounded domain ⊂ R N for the following critical borderline potentials: V (x) = ((N − 2)2 /4)|x|−2 or V (x) = (1/4)d −2 (x). Throughout this work is a C 2 bounded domain of R N containing the origin and d(x) = dist(x, ∂). We first consider, for N ≥ 3, the case V (x) = ((N − 2)2 /4)|x|−2 , −2)2 x ∈ and we formally define the operator K by K u = −u − (N4|x| u|∂ = 0. 2 u, More precisely, the Schrödinger operator K is defined in L 2 () as the generator of the symmetric form (N − 2)2 ∇u 1 ∇u 2 − u 1 u 2 d x, K[u 1 , u 2 ] := 4|x|2
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namely, if (N − 2)2 2 D(K ) := u ∈ H () : −u − u ∈ L () , 4|x|2 K u := −u −
(N − 2)2 u for any u ∈ D(K ), 4|x|2
(1.4)
where H () denotes the closure of C0∞ () in the norm u → ||u|| H () :=
|∇u|2 −
21 (N − 2)2 2 dx . u 4|x|2
(1.5)
1,q
Let us recall that H () ⊂ W0 () for any 1 ≤ q < 2, due to the results in Subsect. 4.1 of [VZ]. It follows, using the Hardy inequality, that K is a nonnegative self-adjoint operator 2 −K t has an integral kernel, that is, e−K t u (x) := on 0 L () such that for every t > 0, e k(t, x, y)u (y)dy, where k(t, x, y) is the heat kernel of K . The first Dirichlet eigen0 value of K can be defined by (N −2)2 2 2 dx |∇ϕ| − 4|x|2 ϕ λ1 := , (1.6) inf∞ 2 0=ϕ∈C0 () ϕ dx with λ1 > 0, due to [BV]. Moreover there exists a positive function ϕ1 ∈ H () satisfying −ϕ1 −
(N − 2)2 ϕ1 = λ1 ϕ1 , in , ϕ1 = 0, on ∂, 4|x|2
see for example Davila and Dupaigne [DD]. We then have the following sharp two-sided heat kernel estimate on K for small time Theorem 1.1. Let ⊂ R N , N ≥ 3, be a smooth bounded domain containing the origin. Then there exist positive constants C1 , C2 , with C1 ≤ C2 , and T > 0 depending on such that √ N −2 √ N −2 d(x)d(y)
|x−y|2 2−N N (|x||y|) 2 t − 2 e−C2 t ≤ C1 min (|x| + t) 2 (|y| + t) 2 , t ≤ k(t, x, y) √ N −2 √ N −2 d(x)d(y)
|x−y|2 2−N N (|x||y|) 2 t − 2 e−C1 t , ≤ C2 min (|x| + t) 2 (|y| + t) 2 , t for all x, y ∈ and 0 < t ≤ T . Concerning the large time asymptotic we have: Theorem 1.2. Let ⊂ R N , N ≥ 3, be a smooth bounded domain containing the origin. Then there exist two positive constants C1 , C2 , with C1 ≤ C2 , such that C1 d(x) d(y) (|x||y|)
2−N 2
e−λ1 t ≤ k(t, x, y) ≤ C2 d(x) d(y) (|x||y|)
for all x, y ∈ and t > 0 large enough; here λ1 is defined in (1.6).
2−N 2
e−λ1 t ,
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To prove the above Theorem 1.2 we have shown a new improved Hardy inequality which is of independent interest; see Theorem 3.2. We next consider the case where the Schrödinger operator H has a potential with critical borderline singularity at the boundary H u = −u − 4d 21(x) u, u|∂ = 0; here N ≥ 2 and is a convex domain. More precisely, the Schrödinger operator H is defined in L 2 () as the generator of the symmetric form 1 ∇u 1 ∇u 2 − 2 u 1 u 2 d x, H[u 1 , u 2 ] := 4d (x) namely if
D(H ) := u ∈ W () : −u − H u := −u −
1 2 u ∈ L () , 4d 2 (x)
1 u for any u ∈ D(H ), 4d 2 (x)
(1.7)
where W () denotes the closure of C0∞ () in the norm u → ||u||W ()
|∇u|2 − :=
1 2 1 2 u dx . 2 4d (x)
1,q
Let us recall that W () ⊂ W0 () for any 1 ≤ q < 2, due to Theorem B in [BFT1]. Then, due to the Hardy inequality, H is a nonnegative self-adjoint operator on 2 () such that for every t > 0, e−H t has an integral kernel, that is, e−H t u (x) := L 0 h(t, x, y)u 0 (y)dy; here h(t, x, y) denotes the heat kernel of H . The first Dirichlet eigenvalue of H is defined by 1 2− 2 dx |∇ϕ| ϕ 4d 2 (x) . (1.8) inf∞ λ1 := 2 ϕ dx 0=ϕ∈C0 () It is known that λ1 > −∞ for any bounded domain , and λ1 > 0 if is convex, see [BM]. Moreover there exists a positive function ϕ1 ∈ W () satisfying −ϕ1 −
1
ϕ1 4d 2 (x)
= λ1 ϕ1 , in , ϕ1 = 0, on ∂;
see for example [DD]. We then have the following sharp two-sided heat kernel estimate on H for small time Theorem 1.3. Let ⊂ R N , N ≥ 2, be a smooth bounded and convex domain. Then there exist positive constants C1 , C2 , with C1 ≤ C2 , and T > 0 depending on such that 1 1 d 2 (x)d 2 (y) − N −C2 |x−y|2 t ≤ h(t, x, y) C1 min 1, t 2e 1 t2 1 1 d 2 (x)d 2 (y) − N −C1 |x−y|2 t ≤ C2 min 1, , t 2e 1 t2 for all x, y ∈ and 0 < t ≤ T .
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We next complement this with the large time behavior: Theorem 1.4. Let ⊂ R N , N ≥ 2, be a smooth bounded and convex domain. Then there exist two positive constants C1 , C2 , with C1 ≤ C2 , such that 1
1
1
1
C1 d 2 (x) d 2 (y) e−λ1 t ≤ h(t, x, y) ≤ C2 d 2 (x) d 2 (y) e−λ1 t , for all x, y ∈ and t > 0 large enough; here λ1 is defined in (1.8). The two-sided estimates in Theorems 1.1 and 1.3 are obtained as a consequence of a new parabolic Harnack inequality up to the boundary, for a suitable degenerate elliptic operator. Let us present a model operator in this direction. For this we consider classical solutions of vt =
1 d α (y)
div(d α (y)∇v),
(1.9)
(actually solutions are considered as weak solutions, for the precise formulation we refer to Definition 2.9 with λ = 0 there, note that due to elliptic regularity, any solution is smooth away from the boundary of ). Then, the following Harnack inequality holds true: Theorem 1.5. (Parabolic Harnack inequality up to the boundary). Let N ≥ 2, α ≥ 1 and ⊂ R N be a smooth bounded domain. Then, there exist positive constants C H and R = R() such that for x ∈ , 0 < r < R and for any positive solution v(y, t) of (1.9) in {B(x, r ) ∩ } × (0, r 2 ), the following estimate holds true ess sup
(y,t)∈{B(x, r2 )∩}×( r4 , r2 ) 2
2
v(y, t) ≤ C H ess inf (y,t)∈{B(x, r )∩}×( 3 r 2 ,r 2 ) v(y, t). 2
4
(1.10) Here B(x, r ) denotes roughly speaking an N dimensional cube centered at x and having size r (see Definition 2.1). The restriction on α in Theorem 1.5 is sharp, since in the weakly degenerate case, where 0 < α < 1, even the elliptic Harnack fails. Indeed, let 1 := B(0, 1), then v(y) := |y| (1−s)dsα s N −1 is a positive solution of div(d α (y)∇v) = 0 for 1/2 < |y| < 1, with v(1) = 0. The natural analogue of Theorem 1.5 in the weakly degenerate case, that is 0 < α < 1, is a Harnack inequality for the ratio of any two positive solutions; in the elliptic case this is done in [FKJ] and by a probabilistic approach in [Ga]. To derive heat kernel estimates we define the operator L := − d α1(x) div(d α (x)∇) in 2 L (, d α (x) d x) as the generator of the symmetric form L[v1 , v2 ] := d α (x)∇v1 ∇v2 d x,
namely D(L) := v ∈ H01 (, d α (x) d x) : − Lv := −
1 d α (x)
1 α 2 α div(d (x)∇v) ∈ L (, d (x) d x) , d α (x)
div(d α (x)∇v) for any v ∈ D(L),
(1.11)
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S. Filippas, L. Moschini, A. Tertikas
where H01 (, d α (x) d x) denotes the closure of C0∞ () in the norm 1 2 v → ||v|| Hα1 := d α (x) |∇v|2 + v 2 d x .
(1.12)
We should emphasize that for α ≥ 1, one has H01 (, d α (x) d x) = H 1 (, d α (x) d x), see Theorem 2.11. Let us note that L is a nonnegative self-adjoint operator on L 2 (, d α (y)dy) such that for every t > 0, e−Lt has an integral kernel, that is e−Lt v0 (x) := α l(t, x, y)v0 (y)d (y)dy; the existence of the heat kernel l(t, x, y) can be proved arguing as in [DS1]. Then, arguing as in [GSC], and using the parabolic Harnack inequality up to the boundary (Theorem 1.5), we obtain the following sharp two-sided estimates for the heat kernel generated by L, for small time. The estimate of Theorem 1.3 is a consequence of Theorem 1.5 and corresponds to the extreme value α = 1. We refer to Theorem 2.10 for a more general result that leads to Theorem 1.1. The existence of a uniform upper bound on the size of the admissible “balls” denoted by R(), in Theorem 1.5 is necessary, because otherwise the nonexistence of an upper bound would imply two-sided heat kernel estimates that are the same for small time and large time, which is not the case at least for α = 1, due to Theorems 1.3 and 1.4. Theorem 1.6. Let α ≥ 1, N ≥ 2 and ⊂ R N be a smooth bounded domain. Then there exist positive constants C1 , C2 , with C1 ≤ C2 , and T > 0 depending on such that |x−y|2 N 1 1 t − 2 e−C2 t ≤ l(t, x, y) C1 min α , α α t 2 d 2 (x)d 2 (y) |x−y|2 N 1 1 t − 2 e−C1 t , ≤ C2 min α , α α t 2 d 2 (x)d 2 (y) for all x, y ∈ and 0 < t ≤ T . So far we have considered special potentials V . However as we shall see next we can obtain much more general results. For instance we consider the operator E := − − V , where the potential V is such that V (x) = V1 (x) + V2 (x) ,
(1.13)
where |V1 (x)| ≤
1 N , V2 (x) ∈ L p (), p > . 4d 2 (x) 2
(1.14)
We also suppose that
2 − V ϕ2 d x |∇ϕ| λ1 := > 0, inf 2 0=ϕ∈C0∞ () ϕ dx
(1.15)
and that to λ1 there corresponds a positive eigenfunction ϕ1 satisfying for all x ∈ the following estimate, α
α
c1 d 2 (x) ≤ ϕ1 (x) ≤ c2 d 2 (x), for some and for c1 , c2 two positive constants.
α ≥ 1,
(1.16)
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Then as before it can be shown that E is a well defined nonnegative self-adjoint operator on L 2() such that for every t > 0, e−Et has an integral kernel, that is e−Et u 0 (x) := e(t, x, y)u 0 (y)dy. We consider positive solutions of u t = −Eu;
(1.17)
then our first result reads Corollary 1.7. For N ≥ 2, let ⊂ R N be a smooth bounded domain. Suppose that (1.13), (1.14), (1.15) and (1.16) are satisfied. Then, there exist positive constants C H and R = R() such that for x ∈ , 0 < r < R and for any positive solution u(y, t) of (1.17) in {B(x, r ) ∩ } × (0, r 2 ) we have the estimate ess sup
α
(y,t)∈{B(x, r2 )∩}×( r4 , r2 ) 2
2
u(y, t)d − 2 (y) α
≤ C H ess inf (y,t)∈{B(x, r )∩}×( 3 r 2 ,r 2 ) u(y, t)d − 2 (y). 2
4
Our result in the case α = 2 is basically the local comparison principle by Fabes, Garofalo and Salsa [FGS] in the case of the Schrödinger operator (see Remark 2.16, that covers the uniformly elliptic case). As we have seen before the parabolic Harnack inequality yields sharp two-sided estimates for the heat kernel e(t, x, y), corresponding to the operator E. In particular we have: Corollary 1.8. For N ≥ 2, let ⊂ R N be a smooth bounded domain. Suppose that (1.13), (1.14), (1.15) and (1.16) are satisfied. Then there exist positive constants C1 , C2 , with C1 ≤ C2 , and T > 0 depending on such that for any x, y ∈ and 0 < t ≤ T , α α d 2 (x)d 2 (y) − N −C2 |x−y|2 t C1 min 1, ≤ e(t, x, y) t 2e α t2 α α d 2 (x)d 2 (y) − N −C1 |x−y|2 t ≤ C2 min 1, , t 2e α t2 whereas for t > T we have α
α
α
α
C1 d 2 (x)d 2 (y)e−λ1 t ≤ e(t, x, y) ≤ C2 d 2 (x)d 2 (y)e−λ1 t . As a byproduct of our method we can answer a conjecture by E. B. Davies (Conjecture 7 in [D2]) in the case of the Schrödinger operator (see Sect. 4.4 for a more general case). For this let us introduce the Green function associated to E, that is ∞ e(t, x, y)dt, (1.18) G E (x, y) = 0
then we have Corollary 1.9. For N ≥ 3, let ⊂ R N be a smooth bounded domain. Suppose that (1.13), (1.14), (1.15) and (1.16) are satisfied. Then there exist two positive constants C1 , C2 , with C1 ≤ C2 , such that for any x, y ∈ , α α 1 d 2 (x)d 2 (y) C1 min , ≤ G E (x, y) |x − y| N −2 |x − y| N +α−2 α α 1 d 2 (x)d 2 (y) ≤ C2 min , . |x − y| N −2 |x − y| N +α−2
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The Davies conjecture corresponds to our result in the case α = 2, we should note however that other values of α ≥ 1 are possible. The structure of the paper is as follows. In Sect. 2 we prove the new parabolic Harnack inequality up to the boundary for a doubly degenerate elliptic operator, as well as the two sided small time heat kernel estimates that can be deduced from it. In Sect. 3 we present the proof of the above mentioned results concerning the Schrödinger potential having critical singularity at the origin, while Sect. 4 treats the case of the Schrödinger operator having critical singularity on the boundary.
2. Parabolic Harnack Inequality up to the Boundary for Degenerate Operators In this section we prove a new parabolic Harnack inequality up to the boundary for the doubly degenerate elliptic operator in divergence form L λα := −
1 |x|λ d α (x)
div(|x|λ d α (x)∇),
(2.1)
for any α ≥ 1 and λ ∈ [2 − N , 0]. Our approach is to first obtain a doubling volumegrowth condition (see Corollary 2.4), then a local weighted Poincaré inequality (see Theorem 2.5), and finally a local weighted Moser inequality. In fact we will establish two local weighted Moser inequalities, one will be used near the boundary (Theorem 2.6) and the other one away from the boundary (Theorem 2.13). These three key estimates along with a suitable Moser iteration scheme as in [GSC] or [CS] lead to the small time parabolic, up to the boundary, Harnack inequality, see Theorem 2.10. In order for the Moser iteration to work, a crucial role is played by the density Theorem 2.11. Then, with arguments quite similar to the ones used in [GSC] in the complete Riemannian setting, we deduce from the parabolic Harnack inequality, sharp two-sided heat kernel estimates for small time, see Theorem 2.14. To this end a sharp volume estimate is also needed (see Lemma 2.2). In the sequel we will use the following local representation of the boundary of . There exist a finite number m of coordinate systems (yi , yi N ), yi = (yi1 , · · · , yi N −1 ) and the same number of functions ai = ai (yi ) defined on the closures of the (N − 1) dimensional cubes i := {yi : |yi j | ≤ β for j = 1, · · · , N − 1}, i ∈ {1, · · · , m} so that for each point x ∈ ∂ there is at least one i such that x = (xi , ai (xi )). The functions ai satisfy the Lipschitz condition on i with a constant A > 0 that is |ai (yi ) − ai (z i )| ≤ A|yi − z i | for yi , z i ∈ i ; moreover there exists a positive number β < 1 such that the set Bi defined for any i ∈ {1, · · · , m} by the relation Bi = {(yi , yi N ) : yi ∈ i , ai (yi ) − β < yi N < ai (yi ) + β} satisfy Ui = Bi ∩ = {(yi , yi N ) : yi ∈ i , ai (yi ) − β < yi N < ai (yi )} and i = Bi ∩ ∂ = {(yi , yi N ) : yi ∈ i , yi N = ai (yi )}. Finally let us observe that for any y ∈ Ui one can make use of the following estimate on the distance function (1 + A)−1 (ai (yi ) − yi N ) ≤ d(y) ≤ (ai (yi ) − yi N ) (see Corollary 4.8 in [K] for details) Let us fix from now on a constant γ ∈ (1, 2) and let us define the “balls” we will use in Moser iteration technique. Roughly speaking they will be Euclidean balls if they stay away from the boundary and they will be N dimensional “deformed cubes”, following
Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators
245
the geometry of the boundary, if they are close enough to the boundary or even if they intersect it. More precisely we have Definition 2.1. (i) For any x ∈ and for any 0 < r < β we define the “ball” centered at x and having radius r as follows. B(x, r ) = B(x, r ) the Euclidean ball centered at x and having radius r if d(x) ≥ γ r , while B(x, r ) = {(yi , yi N ) : |yi − xi | ≤ r, ai (yi ) − r − d(x) < yi N < ai (yi ) + r − d(x)} if d(x) < γ r , where i ∈ {1, · · · , m} is uniquely defined by the point x¯ ∈ ∂ such that |x¯ − x| = d(x), that is by the projection of the center x onto ∂. (ii) We also define by V (x, r ) := B(x,r )∩ |y|λ d α (y)dy the volume of the “ball” centered at x and having radius r . We first derive a sharp volume estimate. Lemma 2.2. Let α > 0, N ≥ 2, λ ∈ (−N , 0] and ⊂ R N be a smooth bounded domain containing the origin. Then there exist positive constants c1 , c2 and β such that for any x ∈ and 0 < r < β, we have c1 max{d α (x)(|x| + r )λ , r α }r N ≤ V (x, r ) ≤ c2 max{d α (x)(|x| + r )λ , r α }r N . To this end we make use of the following lemma which can be proved as in [MT2]. Lemma 2.3. Let N ≥ 2, λ ∈ (−N , 0] and ⊂ R N be a smooth bounded domain containing the origin. Then there exist two positive constants d1 , d2 such that for any x ∈ , we have d1 r N (|x| + r )λ ≤ |y|λ dy ≤ d2 r N (|x| + r )λ . (2.2) B(x,r )
Let us accept (2.2) at the moment and let us prove the sharp volume estimate. Proof of Lemma 2.2. Let us first consider the case where d(x) ≥ γ r . Then B(x, r ) = B(x, r ) ⊂ . Due to the fact that any y ∈ B(x, r ) satisfies γ −1 γ +1 d(x) ≤ d(x) − r ≤ d(y) ≤ d(x) + r ≤ d(x) (2.3) γ γ α the claim easily follows making use of Lemma 2.3 with c2 ≥ d2 γγ+1 and c1 ≤ α γ −1 . d1 γ Let us now consider the case where d(x) < γ r and let us denote by L 1 , L 2 two positive constants such that B(0, L 1 ) ⊂ ⊂ B(0, L 2 ) (note that they exist by assumption on ). Then any y ∈ B(x, r ) ∩ satisfies the following estimate L 1 − (γ + 1)β ≤ |y| ≤ L 2 . Indeed, if on the contrary |y| < L 1 − (γ + 1)β, then by definition of L 1 we would have d(y) > (γ + 1)β, and this contradicts our assumption. In fact one obviously has d(y) ≤ d(x) + r < (γ + 1)r < (γ + 1)β and it is not restrictive to suppose from the beginning that the parameter β in the local representation of the boundary of satisfies β < L 1 (γ + 1)−1 . As a consequence we have: ∀ y ∈ B(x, r ) ∩ d3 ≤ |y|λ ≤ d4 . Here d3 := L λ2 and d4 := (L 1 − (γ + 1)β)λ .
(2.4)
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Then for some i ∈ {1, · · · , m}, we have V (x, r ) = |y|λ d α (y)dy B(x,r )∩
∼
|yi −xi |≤r
min{ai (yi ),ai (yi )+r −d(x)}
ai (yi )−r −d(x)
|y|λ (ai (yi ) − yi N )α dyi N dyi .
From now on we omit the subscript i for convenience. Indeed we have V (x, r ) ≤
|y −x |≤r
min{a(y ),a(y )+r −d(x)}
a(y )−r −d(x)
≤ d4 (γ + 1)r (d(x) + r )
α
|y −x |≤r
|y|λ (a(y ) − y N )α dy N dy dy ≤
≤ d4 (γ + 1)α+1 r α+1+N −1 ω N −1 = d4 (γ + 1)α+1 r α+N ω N −1 . On the other hand V (x, r ) ≥ (1 + A)−α
|y −x |≤r
≥ d3 (1 + A)−α
a(y )−r (γ −1)
a(y )−r −d(x)
|y −x |≤r
a(y )−r (γ −1)
a(y )−r −d(x)
≥ d3 (1 + A)−α r α+1 (2 − γ )(γ − 1)α = d3 (1 + = d3 (1 +
|y|λ (a(y ) − y N )α dy N dy ≥
(r (γ − 1))α dy N dy ≥ dy
|y −x |≤r −α α+1+N −1 A) r (2 − γ )(γ − 1)α ω N −1 A)−α r α+N (2 − γ )(γ − 1)α ω N −1 .
=
N Here ω N denotes the standard α of the Euclidean unit ball in R . Thus
the result volume γ −1 −α α follows with c1 := min d1 γ , d3 (1 + A) (2 − γ )(γ − 1) ω N −1 and c2 := α
γ +1 α+1 , d4 (γ + 1) ω N −1 . max d2 γ
Let us now prove estimate (2.2), which is taken from [MT2], we give here the details for the convenience of the reader Proof of Lemma 2.3. (i) Observe that λ λ+N |y| dy = r B(x,r )
where w := rx . Then (2.2) reads d1 (|w| + 1)λ ≤
|w + z|λ dz, B(0,1)
|w + z|λ dz ≤ d2 (|w| + 1)λ . B(0,1)
Since |z| ≤ 1, there holds |w| − 1 ≤ |w| − |z| ≤ |w + z| ≤ |w| + |z| ≤ |w| + 1,
(2.5)
Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators
whence
λ ω N |w| + 1 ≤
247
λ |w + z|λ dz ≤ ω N |w| − 1 .
(2.6)
B(0,1)
Comparing (2.5) and (2.6), we immediately obtain the lower bound in (2.2) with d1 := ω N . (ii) To prove the upper bound in (2.2), observe that three cases are possible: (a) |x| r ≤ |x| 2 , (b) 2 < r < 3|x| and (c) r ≥ 3|x|. In case (a) the claim follows from the right-hand inequality in (2.6), if we exhibit d2 > 0 such that ω N (|w| − 1)λ ≤ d2 (|w| + 1)λ |λ| t+1 for any |w| ≥ 2. It is easily seen that the function F(t) := t−1 (t > 1) is decreas 1 λ ing, thus F(t) ≤ F(2) = 3 for any t ≥ 2. This proves the claim in this case for any d2 ≥ ω N 31λ . To deal with cases (b) − (c), observe first that ω N λ+N |y|λ dy = . r λ +N B(0,r ) In case (b) we have B(x, r ) ⊆ B (0, 4|x|) , whence
B(x,r )
|y|λ dy ≤
In case (c) there holds
(2.7)
ωN ωN (4|x|)λ+N ≤ (8r )λ+N . λ+ N λ+ N
4r B(x, r ) ⊆ B 0, , 3
(2.8)
(2.9)
thus B(x,r )
|y|λ dy ≤
ωN λ+ N
4r 3
λ+N
.
(2.10)
Since r λ+N ≤
r N (|x| + r )λ 3λ
for |x| < 2r
we have B(x,r )
|y|λ dy ≤
ω N 8λ+N N r (|x| + r )λ λ + N 3λ
in cases (b)–(c). Hence the conclusion follows with d2 :=
ω N 8λ+N λ+N 3λ
.
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From Lemma 2.2 one can easily deduce the doubling property which reads as follows: Corollary 2.4. Let α > 0, N ≥ 2, λ ∈ (−N , 0] and ⊂ R N be a smooth bounded domain containing the origin. Then there exist positive constants C D and β such that for any x ∈ and 0 < r < β, we have V (x, 2r ) ≤ C D V (x, r ). Let us state now the local Poincaré inequality. Theorem 2.5. (Local weighted Poincaré inequality) Let α > 0, N ≥ 2, λ ∈ (−N , 0] and ⊂ R N be a smooth bounded domain containing the origin. Then there exist positive constants C P and β such that for any x ∈ and 0 < r < β, we have inf |y|λ d α (y)| f (y) − ξ |2 dy ξ ∈R B(x,r )∩ ≤ CP r2 |y|λ d α (y)|∇ f |2 dy, ∀ f ∈ C 1 (B(x, r ) ∩ ). (2.11) B(x,r )∩
Proof. Let us first consider the case where d(x) ≥ γ r . Then B(x, r ) = B(x, r ) ⊂ . Due to (2.3) the claim corresponds to Theorem 3.1 in [MT2]. We give here the details for the convenience of the reader. (i) As a consequence of the compact embedding of the space H 1 (B(0, 1), |y|λ dy) into L 2 (B(0, 1), |y|λ dy) (e.g. see [KO]) we have that | f − fˆ|2 |y|λ dy ≤ C |∇ f |2 |y|λ dy, ∀ f ∈ C 1 (B(0, 1)); B(0,1)
B(0,1)
−1 λ λ here fˆ := . Then by scaling (2.11) follows B(0,1) f (y)|y| dy B(0,1) |y| dy when x = 0. (ii) Let us now consider the case |x| ≥ 2r and let us define f¯ := ω−1 N B(0,1) f (x + r z)dz. Then as in the proof of Lemma 2.3 we have (|x| + r )λ ≤ |x + r z|λ ≤ 31λ (|x| + r )λ . Hence 2 λ N ¯ | f − f | |y| dy = r | f (x + r z) − f¯|2 |x + r z|λ dz B(x,r )
B(0,1)
rN ≤ λ (|x| + r )λ 3 ≤ Cr
2r
N
3λ
| f (x + r z) − f¯|2 dz ≤ B(0,1)
(|x| + r )
λ
|∇ f |2 (x + r z)dz B(0,1)
rN ≤ Cr 2 λ |∇ f |2 |x + r z|λ dz 3 B(0,1) 1 = Cr 2 λ |∇ f |2 |y|λ dy. 3 B(x,r ) Then (2.11) follows when |x| ≥ 2r .
Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators
249
(iii) For a general x ∈ two cases are possible (a) 0 ≤ |x| < r4 ; (b) |x| ≥ r4 . In case (a) there holds r r ⊆ B 0, ⊆ B(x, r ), B x, 8 2 thus from (i) we have 2 λ inf | f (y) − ξ | |y| dy ≤ inf | f (y) − ξ |2 |y|λ dy ξ ∈R B (x, r ) ξ ∈R B (0, r ) 8 2 C 2 2 λ 2 |∇ f |2 |y|λ dy. ≤ r |∇ f | |y| dy ≤ C P r 4 B (0, r2 ) B(x,r ) This proves (2.11) in case (a) since using a Whitney type covering and arguing as in [SC2] the integration set of the left-hand side which from above is B x, r8 can be increased to cover all B(x, r ).
In case (b) there holds |x| ≥ 2 r8 ; hence from (ii), C inf | f (y) − ξ |2 |y|λ dy ≤ r 2 |∇ f |2 |y|λ dy r ξ ∈R B (x, r ) 64 B x, ( ) 8 8 2 2 λ ≤ CPr |∇ f | |y| dy. B(x,r )
This completes the proof in the case d(x) ≥ γ r . Let us now consider the case where d(x) < γ r . Then for some i ∈ {1, · · · , m} we have d α (y)| f (y) − ξ |2 dy B(x,r )∩
≤
|yi −xi |≤r
min{ai (yi ),ai (yi )+r −d(x)}
ai (yi )−r −d(x)
| f (yi , yi N ) − ξ |2 |y|λ (ai (yi ) − yi N )α dyi N dyi .
From now on we omit the subscript i for convenience. Let us perform then the following change of variables (y , y N ) → (y , z N := a(y ) − y N ) and make use of (2.4); thus the above integral is less than or equal to:
r +d(x)
| f (y , a(y ) − z N ) − ξ |2 z αN dz N dy = |y −x |≤r max{0,d(x)−r } r +d(x) α 2 zN | f (y , a(y ) − z N ) − ξ | dy dz N ≤ = d4 |y −x |≤r max{0,d(x)−r } r +d(x) ∂f ∂ f a(y ) 2 dz N ≤ z αN dy ≤ Cd4 r 2 + ∂ yn ∂ y |y −x |≤r ∂ y max{0,d(x)−r } r +d(x) 2 |∇ f |2 (y , a(y ) − z N )z αN dy dz N ≤ Cd4 r max{0,d(x)−r } |y −x |≤r
d4
≤C
d4 (1 + A)α r 2 |y|λ d α (y)|∇ f |2 dy. d3 B(x,r )∩
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In the above argument we made the following choices ξ = ξ(z N ) := −1 = ω N −1
|y −x |≤r
|z −x |≤1
f (y , a(y ) − z N )dy
r −N +1 ω−1 N −1
f (r z , a(r z ) − z N )dz ,
and C being the Euclidean Poincaré constant on the N − 1 dimensional Euclidean ball of radius one. Since for any ξ¯ ∈ R, | f − ξ¯ |2 ≤ 2| f − ξ(z N )|2 + 2|ξ(z N ) − ξ¯ |2 in order to prove (2.11) in this case, it only remains to estimate the following term:
|y −x |≤r
r +d(x)
max{0,d(x)−r }
|ξ(z N ) − ξ¯ |2 z αN dz N dy =
z α+1 − max{0, d(x) − r }α+1 r +d(x) |ξ(z N ) − ξ¯ |2 N − max{0,d(x)−r } α+1 |y −x |≤r r +d(x) ∂ξ(z N ) α+1 2 z N − max{0, d(x) − r }α+1 dz N . (ξ(z N ) − ξ¯ ) − α + 1 max{0,d(x)−r } ∂z N
dy
=
Thus, choosing ξ¯ := ξ(r + d(x)) above, we obtain by Hölder inequality
|y −x |≤r
2 ≤ α+1 ⎛ ×⎝
r +d(x)
|ξ(z N ) − ξ¯ |2 z αN dz N dy ≤
max{0,d(x)−r }
|y −x |≤r
max{0,d(x)−r }
1 |ξ(z N ) − ξ¯ |2 z αN
dz N dy
2
×
⎞1 2 ∂ξ(z ) 2 α+1 − max{0, d(x) − r }α+1 2 z N α ⎠ dz dy . zN N N ∂z N z αN max{0,d(x)−r }
|y −x |≤r
r +d(x)
r +d(x)
Since ∂ξ(z ) ∂f N ≤ ω−1 (r z , a(r z ) − z N )dz N −1 ∂z N |z −x |≤1 ∂ y N ∂f −N +1 = ω−1 r (y , a(y ) − z N )dy , N −1 |y −x |≤r ∂ y N hence ∂ξ(z ) 2 ∂ f 2 N −1 1−N ≤ ω N −1r (y , a(y ) − z N )dy . ∂z N ∂ y N |y −x |≤r Thus, since d(x) < γ r , we obtain (2.11) with constant C P := 2 dd43 (1+A)α C +
4(γ +1)2 (α+1)2
.
Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators
251
We next prove the following local weighted Moser inequality: Theorem 2.6. (Local weighted Moser inequality) Let α > 0, N ≥ 2 and ⊂ R N be a smooth bounded domain. Then there exist positive constants C M and R = R(α, ) such that for any ν ≥ N + α, x ∈ , 0 < r < R and f ∈ C0∞ (B(x, r )) we have
d α (y)| f (y)|
B(x,r )∩
2
≤ C M r 2 V (x, r )− ν
2 1+ ν2
dy
d α (y)|∇ f |2 dy
B(x,r )∩
B(x,r )∩
d α (y)| f |2 dy
2 ν
.
Proof. Let us first consider the case where d(x) ≥ γ r . By the standard Moser inequality, there exists a positive constant C such that for any x ∈ and any ν ≥ N if N ≥ 3 or any ν > 2 if N = 2, the following holds true B(x,r )
| f (y)|
2 1+ ν2
dy
2 − 2N ν
≤ Cr r
|∇ f | dy
2
2
B(x,r )
ν
| f | dy 2
B(x,r )
, ∀ f ∈ C0∞ (B(x, r ))
(see for example Sect. 2.1.3 in [SC2]). Thus we have B(x,r )
d α (y)| f (y)| α
2 1+ ν2
2 − 2N ν
dy
≤ (d(x) + r ) Cr r ≤ Cr
2
×
d(x) + r d(x) − r
α
|∇ f | dy
B(x,r )
α − ν2
d (y)| f | dy 2
≤ C M r 2 V (x, r )− ν
ν
B(x,r )
(r (d(x) − r ) ) N
2
2 | f | dy 2
2
α
B(x,r )
2
α
B(x,r )
ν
≤
d (y)|∇ f | dy 2
≤
B(x,r )
d α (y)|∇ f |2 dy
B(x,r )
d α (y)| f |2 dy
2 ν
,
α 2α 2 ν γ 2 c2ν and c2 is the constant appearing in the volume where C M := C 1 + γ −1 γ −1 estimate in Lemma 2.2 when λ = 0. Let us now consider the case where d(x) < γ r . Then we claim the following local weighted Sobolev inequality: there exist positive constants C S and R = R(α, ) such that for any x ∈ , 0 < r < R, satisfying d(x) < γ r , and any f ∈ C0∞ (B(x, r )), we have B(x,r )∩
α
d (y)| f (y)|
2(N +α) N +α−2
N +α−2 N +α
dy
≤ CS
B(x,r )∩
d α (y)|∇ f |2 dy. (2.12)
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S. Filippas, L. Moschini, A. Tertikas
If we accept (2.12), then the result follows, with C M = C S by means of Hölder inequality, in fact we have
α
B(x,r )∩
d (y) f
≤
B(x,r )∩
2 1+ N2+α
d α (y) f
dy
2(N +α) N +α−2
N +α−2 N +α
dy
B(x,r )∩
d α (y) f 2 dy
2 N +α
,
as well as for any ν > N + α,
α
B(x,r )∩
d (y) f
≤
2 1+ ν2
α
B(x,r )∩
d (y) f
dy
2 1+ N2+α
2(ν−N −α)
α
B(x,r )∩
d (y) f dy
dy V (x, r )
2
2(ν−N −α) ν(N +α)
ν(N +α)
.
In the sequel we will give the proof of (2.12). We will follow closely the argument of [FMT2]. If V ⊂ R N is any bounded domain and u ∈ C ∞ (V ) then it is well known that S N ||u||
N
L N −1 (V )
≤ ||∇u|| L 1 (V ) + ||u|| L 1 (∂ V ) ,
− 1 1 where S N := N π 2 (1 + N2 ) 2 (see p. 189 in [M]). Let us fix from now on that V := B(x, r )∩, and let us apply the above inequality to u := d a f , for any f ∈ C0∞ (B(x, r )) and any a > 0. Thus we get |∇ f |d a + ad a−1 |∇d|| f | dy. ≤ S N ||d a f || N L N −1 (V )
V
Let us remark at this point that boundary terms on ∂ are zero due to the presence of the weight d a , a > 0. To estimate the last term of the right hand side, we will make use of an integration by parts, noting that ∇d · ∇d = 1 a.e.; that is we have: d a−1 | f |dy = a ∇d · ∇d d a−1 | f |dy = ∇d a · ∇d| f |dy = a V V V a a d d| f |dy − d ∇d · ∇| f |dy + d a ∇d · ν | f | d S. =− V
∂V
V
Under our smoothness assumption on we have that |dd| ≤ c0 δ in δ for δ small, say 0 < δ ≤ δ0 , and for some positive constant c0 independent of δ (δ0 , c0 depending on ). Now, if d(x) + r < δ, that is if r < γ δ+1 , we have that V ⊂ δ and it follows that
d a−1 | f |dy ≤ c0 δ
a V
d a−1 | f |dy + V
d a |∇ f |dy, V
hence
d a−1 | f |dy ≤ (a − c0 δ)−1 V
d a |∇ f |dy. V
(2.13)
Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators
253
Consequently for any r ∈ (0, R(a, )), R(a, ) := γ 1+1 min{δ0 , ca0 } and any a > 0 the following inequality is true a ≤ +1 d a |∇ f |dy. (2.14) S N ||d a f || N L N −1 (V ) a − c0 δ0 V To proceed we will use the following interpolation inequality (cf. Lemma 4.1 of [FMT2]): N (q − 1) a q − N (q − 1) a−1 ||d f || N ||d + f || L 1 (V ) , L N −1 (V ) q q N q −1 ∀1
0. (2.15) N −1 q
||d b f || L q (V ) ≤
From (2.13) and (2.14), we get for any a, b, q as above the following inequality: ||d b f || L q (V ) ≤ C1 ||d a ∇ f || L 1 (V ) , where C1 :=
N (q−1) 1 q SN
a a−c0 δ0
+1 +
q−N (q−1) 1 q a−c0 δ0 to | f |s instead of
Let us now apply inequality (2.16) b := Bs. Due to (2.15) we have a = b+1− q−1 q N = In this way we obtain
d B Q | f | Q dy
Q 1 2 +1 Q
V
Q 2
. f , for s := Q2 + 1, q := Qs , A := B +1− Q−2 2Q N .
BQ 2 + A, where
BQ Q Q +1 d 2 +A | f | 2 |∇ f |dy ≤ 2 V 1 1 2 2 BQ Q 2A 2 ≤ C2 d | f | dy d |∇ f | dy ;
≤ C1
V
where C2 := C1
(2.16)
V
+1 .
After simplifying we see that we have proved the following: there exists R = R( B2Q + A, ) such that for all 0 < r < R and all x ∈ with d(x) < γ r , there holds
2 B(x,r )∩
d
BQ
| f | dy Q
Q
≤C
B(x,r )∩
d 2 A (y)|∇ f |2 dy,
for any N ≥ 2 and any f ∈ C0∞ (B(x, r )) under the following conditions A := B + BQ 2N 1 − Q−2 2Q N , 2 + A > 0, 2 < Q < ∞ if N = 2, 2 < Q ≤ N −2 if N ≥ 3; here C3 = C22 = C3 (N , Q, B, c0 , δ0 ). +α) α Taking A = α2 , Q := 2(N N +α−2 and B := Q we deduce the local weighted Sobolev inequality (2.12) with C S = C S (N , α, c0 , δ0 ) and this completes the proof of Theorem 2.6. Remark 2.7. Note that the upper bound for the length of the “balls” in the local weighted Moser inequality, denoted by R(α, ), goes to zero as α tends to zero.
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Remark 2.8. Let us note that when N = 1, the corresponding analogue of the local weighted Sobolev inequality (2.12) when = (−1, 1) is the following one:
min{1,x+r } max{−1,x−r }
≤ CS r
α
(1 − |y|) | f | (y)dy
α+1 1−α q + 2
q
min{1,x+r }
max{−1,x−r }
q1
α
2
(1 − |y|) | f | (y)dy
21
,
for any f ∈ C0∞ (x − r, x + r ), and any q > 2 if 0 < α ≤ 1 and 2 < q ≤ 2(α+1) α−1 if α > 1. Consequently Theorem 1.5 as well as its consequences can be also stated for N = 1; see [KO]. From the results within this subsection, we will now deduce a new parabolic Harnack inequality up to the boundary for the doubly degenerate elliptic operator L λα defined in (2.1). To this end let us first make precise the notion of a weak solution. Definition 2.9. By a solution v(y, t) to vt = −L λα v in Q := {B(x, r ) ∩ } × (0, r 2 ), we mean a function v ∈ C 1 ((0, r 2 ); L 2 (B(x, r ) ∩ , |y|λ d α (y)dy)) ∩ C 0 ((0, r 2 ); H 1 (B(x, r ) ∩ , |y|λ d α (y)dy)) such that for any ∈ C 0 ((0, r 2 ); C0∞ (B(x, r ) ∩ )) and any 0 < t1 < t2 < r 2 we have t2 {|y|λ d α (y)vt + |y|λ d α (y)∇v∇}dydt = 0. (2.17) t1
B(x,r )∩
Then we have Theorem 2.10. Let α ≥ 1, N ≥ 2, λ ∈ [2 − N , 0] and ⊂ R N be a smooth bounded domain containing the origin. Then there exist positive constants C H and R = R() such that for x ∈ , 0 < r < R and for any positive solution v(y, t) 1 λ α 2 of ∂v ∂t = |y|λ d α (y) div(|y| d (y)∇v) in {B(x, r ) ∩ } × (0, r ), the following estimate holds true: ess sup
(y,t)∈{B(x, r2 )∩}×( r4 , r2 ) 2
2
v(y, t) ≤ C H ess inf (y,t)∈{B(x, r )∩}×( 3 r 2 ,r 2 ) v(y, t). 2
4
In order to prove the parabolic Harnack inequality in Theorem 2.10 we use the Moser iteration technique as adapted to degenerate elliptic operators in [FKS, CS] as well as [GSC]. In this approach one inserts in the weak form of the equation vt = −L λα v suitable test functions . One of the key ideas is to use test functions of the form η2 v q , where v is the weak solution of the equation, η is a cut off function and q ∈ R. To this end one has to check that η2 v q is in the right space of test function. In this direction the following density theorem is crucial. Theorem 2.11. Let N ≥ 2 and ⊂ R N be a smooth bounded domain. Then for any α ≥ 1, H 1 (, d α (y) dy) = H01 (, d α (y) dy). In particular for any α ≥ 1, the set C0∞ () is dense in H 1 (, d α (y)dy).
Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators
Here H 1 (, d α (y)dy) denotes the set {v = v(y) : the corresponding norm being defined in (1.12). We are now ready to prove the density theorem.
d
255
α (y)(v 2
+ |∇v|2 )dy < ∞},
Proof. Let us prove here the result when α = 1. We refer to Proposition 9.10 in [K] for the case α > 1, even though our proof with some minor changes can also cover this range. First of all from Theorem 7.2 in [K] it is known that the set C ∞ () is dense in 1 H (, d(y) dy). Thus for any v ∈ H 1 (, d(y) dy) there exists vm ∈ C ∞ () such that for any > 0 we have ||v − vm || H 1 ≤ if m ≥ m(). Let us choose w := vm() and let 1 us define, for k ≥ 1, the following function: ⎧ ⎪ ⎨0 ϕk (x) = 1 + ⎪ ⎩1
ln(kd(x)) ln(k)
i f d(x) ≤ k12 , i f k12 < d(x) < i f d(x) ≥ k1 .
1 k
,
Then wk := wϕk ∈ C00,1 (), moreover we have ||w − wk || H 1 = ||w(1 − ϕk )|| H 1 1 1 2 ≤ 2 (w + |∇w|2 )(1 − ϕk )2 d(y) dy + 2 w 2 |∇ϕk |2 d(y) dy ≤
≤2
d(y)< k1
(w 2 + |∇w|2 )d(y) dy + 2
1
w2 dy. d(y)(ln(k))2
Now as k → ∞ the right hand side goes to zero, this proves the theorem. The above theorem allows us to take the cut off function η in C0∞ (B(x, r )) instead of taking it as usual in C0∞ (B(x, r ) ∩ ). Clearly the two function spaces differ only if the “ball” intersects the boundary of . To explain what are the appropriate modifications of the standard iteration argument by Moser, we now present in detail the first step, which is the L 2 mean value inequality for any positive local subsolution of the equation vt = −L λα v. Theorem 2.12. Let α ≥ 1, N ≥ 2, λ ∈ [2 − N , 0] and ⊂ R N be a smooth bounded domain containing the origin. Then there exist positive constants C and R() such that for x ∈ , 0 < r < R() and for any positive subsolution v(y, t) of vt − 1 div(|y|λ d α (y)∇v) = 0 in {B(x, r ) ∩ } × (0, r 2 ) we have the estimate |y|λ d α (y) ess sup
2
(y,t)∈{B(x, r2 )∩}×( r2 ,r 2 )
≤
C 2 r V (x, r )
v 2 (y, t)
{B(x,r )∩}×(0,r 2 )
|y|λ d α (y) v 2 (y, t)dydt.
Proof. We will only prove the result in the non-standard case in which the “ball” B(x, r ) intersects the boundary of ; we refer to [MT2] as well as to [GSC] for details in the other case. Similarly to Definition 2.9 we define a subsolution v(y, t) to be a
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S. Filippas, L. Moschini, A. Tertikas
function in C 1 ((0, r 2 ); L 2 (B(x, r ) ∩ , |y|λ d α (y) dy)) ∩ C 0 ((0, r 2 ); H 1 (B(x, r ) ∩ , |y|λ d α (y) dy)) such that the following holds true:
r2
B(x,r )∩ 0
0
{|y|λ d α (y)vt + |y|λ d α (y)∇v∇}dydt ≤ 0,
∀ ∈ C ((0, r 2 ); C0∞ (B(x, r ) ∩ )), ≥ 0. Hence in particular we have also {|y|λ d α (y)vt +|y|λ d α (y)∇v∇}dy ≤ 0, ∀ ∈ C0∞ (B(x, r ) ∩ ), ≥ 0. B(x,r )∩
Let us define for any q, M ≥ 1 the following functions G(z) =√z q if z ≤ M and G(z) = M q + q(z − M)M q−1 if z > M and H (z) ≥ 0 by H (z) = G (z), H (0) = 0; note that G(z) ≤ zG (z) as well as H (z) ≤ z H (z). Due to Theorem 2.11 there exists a sequence of functions vm in C ∞ (B(x, r ) ∩ ) having compact support in such that vm → v in H 1 (B(x, r ) ∩ , d α (y) dy) as m → +∞; whence due to (2.4) also in H 1 (B(x, r ) ∩ , |y|λ d α (y) dy). Hence for any η ∈ C0∞ (B(x, r )) and m ≥ 1 the function := η2 G(vm ) is an admissible test function, that is the following holds true: {|y|λ d α (y)η2 G(vm )vt + |y|λ d α (y)∇v∇(η2 G(vm ))}dy ≤ 0. B(x,r )∩
Passing to the limit as m → +∞ we get {|y|λ d α (y)η2 G(v)vt +|y|λ d α (y)∇v∇(η2 G(v))}dy ≤ 0, ∀ η ∈ C0∞ (B(x, r )). B(x,r )∩
This is the standard starting point in the Moser iteration technique apart from the fact that the cut off function η is not necessarily zero on ∂, this is crucial. Then by the Schwarz inequality we get {|y|λ d α (y)η2 G(v)vt + |y|λ d α (y)|∇v|2 G (v)η2 }dy B(x,r )∩ ≤C |y|λ d α (y)|∇η|2 v 2 G (v)dy, B(x,r )∩
thus also that
B(x,r )∩
{|y|λ d α (y)η2 G(v)vt + |y|λ d α (y)|∇(ηH (v))|2 }dy
≤C
B(x,r )∩
|y|λ d α (y)|∇η|2 v 2 G (v)dy.
For any smooth function χ of the time variable t, we easily get d λ α 2 2 |y| d (y)(ηχ F(v)) dy + χ |y|λ d α (y)|∇(ηH (v))|2 dy ≤ dt B(x,r )∩ B(x,r )∩
≤ Cχ χ ||∇η|| L ∞ (Rn ) + ||χ || L ∞ (R) |y|λ d α (y)v 2 G (v)dy; suppη ∩
Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators
257
here F(z) is such that 2F(z)F (z) = G(z). For 21 ≤ s < s < 1 we choose as usual χ such that 0 ≤ χ ≤ 1, χ = 0 in (−∞, r 2 (1 − s )), χ = 1 in (r 2 (1 − s), ∞), moreover if ξ ∈ C0∞ (0, 1) be a nonnegative non-increasing function such that ξ(z) = 1 if z ≤ s and ξ(z) = 0 if z ≥ s , we define, use oflocal coordinates, the following cut off making |y −x | |a(y )−y N −d(x)| ξ . Then clearly ||∇η|| L ∞ (Rn ) ≤ r (sC −s) function η(y) := ξ r r and ||χ || L ∞ (R) ≤
C . r 2 (s −s)
Integrating our inequality over (0, t), with t ∈ (r 2 (1 − s), r 2 ) we obtain |y|λ d α (y)(ηF(v))2 dy sup t∈J B(x,r )∩ + |y|λ d α (y)|∇(ηH (v))|2 dydt ≤ {B(x,r )∩}×(r 2 (1−s),r 2 ) C ≤ 2 |y|λ d α (y)v 2 G (v)dydt. r (s − s)2 {B(x,s r )∩}×(r 2 (1−s ),r 2 ) Making once again use of Theorem 2.11 we note that we can apply the local weighted Moser inequality in Theorem 2.6 to the function f := ηH (v) thus obtaining 2 1+ N2+α λ α |y| d (y)(ηH (v)) dydt ≤ {B(x,r )∩}×(r 2 (1−s),r 2 )
≤
C 2 r (s − s)2
{B(x,s r )∩}×(r 2 (1−s ),r 2 )
Let us now denote by γ˜ := 1 +
2 N +α ,
{B(x,sr )∩}×(r 2 (1−s),r 2 )
C ≤ 2 r (s − s)2
s s − s
2
1+
2 N +α
.
thus as M tends to infinity we have for p := q + 1,
|y|λ d α (y)v pγ˜ dydt
λ α
{B(x,s r )∩}×(r 2 (1−s ),r 2 )
Thus due to Lemma 2.2 also that −1 2 −1 V (x, sr ) (r s) ≤C
|y|λ d α (y)v 2 G (v)dydt
{B(x,sr )∩}×(r 2 (1−s),r 2 )
V (x, s r )−1 (r 2 s )−1
|y| d (y)v dydt p
γ˜
.
|y|λ d α (y)v pγ˜ dydt ≤
{B(x,s r )∩}×(r 2 (1−s ),r 2 )
|y|λ d α (y)v p dydt
γ˜
.
i+2 then if we denote Take now p = pi := 2γ˜ i , s = θi+1 and s = θi , where θi := 2(i+1) 1 p by I (i) := V (x, θi r )−1 (r 2 θi )−1 {B(x,θi r )∩}×(r 2 (1−θi ),r 2 ) |y|λ d α (y)v pi dydt i the above inequality can be restated as follows I (i +1) ≤ C(i)I (i). Thus since one can show ∞ that the product of C(i) for all i ≥ 0 is finite, we obtain I (∞) ≤ i=0 C(i) I (0); this completes the proof of the proposition. To this end the choice R() := min{β, R(1, )} can be made; here β and R(1, ) are the constants appearing respectively in the local representation of ∂ and in Theorem 2.6 when α := 1.
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Theorem 2.6 corresponds to the local weighted Moser inequality needed in the proof of the parabolic Harnack inequality up to the boundary stated in Theorem 1.5. The local weighted Moser inequality involved in the proof of Theorem 2.10 differs from Theorem 2.6 only if d(x) ≥ γ r , N ≥ 3, λ = 0, and in this case it reads as follows Theorem 2.13. Let N ≥ 3, λ ∈ [2 − N , 0) and ⊂ R N be a smooth bounded domain containing the origin. Then there exist a positive constant C M such that for any ν ≥ N , x ∈ , r > 0 and f ∈ C0∞ (B(x, r )) we have 2 1+ ν2 λ |y| | f (y)| dy B(x,r )
2
≤ C M r 2 (r N (|x| + r )λ )− ν
B(x,r )
|y|λ |∇ f |2 dy
B(x,r )
|y|λ | f |2 dy
2 ν
.
Proof. By Hölder inequality the result easily follows with C M := C S as soon as the following local weighted Sobolev inequality holds true:
2N
B(x,r )
|y|λ | f (y)| N −2 dy
N −2 N
≤ C S (|x| + r )
2|λ| N
B(x,r )
|y|λ |∇ f |2 dy
(2.18)
(we refer to the proof of Theorem 2.6 where a similar argument is used). Let us first prove the above inequality for any λ ∈ (2 − N , 0). As a consequence of the Caffarelli Kohn Nirenberg inequality (e.g. see Corollary 2 in Sect. 2.1.6 of [M]), the following holds true: N −2 N 2N Nλ N −2 N −2 f |y| dy ≤C |∇ f |2 |y|λ dy, ∀ f ∈ C0∞ (B(x, r )), B(x,r )
B(x,r )
and for some positive constant C independent of x and r . Whence also B(x,r )
2N N −2
f
λ
|y| dy
≤C
N −2 N
2|λ| N sup |y|
y∈B(x,r )
B(x,r )
λ
|∇ f | |y| dy ≤ C(|x| + r ) 2
2|λ| N
B(x,r )
|∇ f |2 |y|λ dy.
Let us now prove the result for λ = 2 − N . To this end let us apply Proposition 3.1 to 1 = B(0, 1) with D = e N −2 . Then there exists a positive constant C such that
2N
v N −2 |x|−N X
|∇v|2 |x|2−N d x ≥ C B(0,1)
∀v∈
B(0,1)
C0∞ (B(0, 1));
2(N −1) N −2
N −2 N |x| dx , D
x 1 ∞ here X (t) = 1−ln t , t ∈ (0, 1]. Now let us take v(x) := f R for any f ∈ C 0 (B(0, R)), then from above we have N −2 N −1) 2N |y| 2 2−N −N 2(N N −2 N −2 dy |∇ f | |y| dy ≥ C f |y| X . DR B(0,R) B(0,R)
Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators
259
Then if y ∈ B(x, r ) clearly y ∈ B(0, |x| + r ), thus if we take R := |x| + r and f ∈ C0∞ (B(x, r )) from above we have
B(x,r )
|∇ f |2 |y|2−N dy ≥ C
≥
B(x,r )
f
2N N −2
B(x,r )
f
2N N −2
|y|−N X
N −2 N
|y|2−N dy
inf
y∈B(x,r )
2(N −1) N −2
|y|−2 X
2(N −1) N −2
N −2 N |y| dy ≥ DR N −2 N |y| . DR
Whence the claim easily follows as soon as we prove that sup |y|X
N −1 −N −2
y∈B(x,r )
|y| DR
2(NN−2) ≤ C S (|x| + r )
2(N −2) N
.
This is indeed the case, in fact we have N −1 N −2 N −1 |y| |y| −N −2 ≤ sup |y| 1 − ln sup |y|X = DR DR 0≤|y|≤|x|+r y∈B(x,r )
N −1 (thus using the fact that the function ϕ(t) = t 1 − ln DtR N −2 is an increasing function for t ∈ [0, R] if D and R are as above) N −1 N −2 N −1 |x| + r = (|x| + r ) 1 − ln = (|x| + r )(1 + ln(D)) N −2 DR N −1 N − 1 N −2 = (|x| + r ) . N −2 This completes the proof of Theorem 2.13. To state the heat kernel estimates following from Theorem 2.10 we introduce some notation. The operator L λα is defined for α ≥ 1 and λ ∈ [2−N , 0] in L 2 (, |x|λ d α (x) d x) as the generator of the symmetric form Lλα [v1 , v2 ] := |x|λ d α (x)∇v1 ∇v2 d x,
namely D(L λα )
:= v ∈ H01 (, |x|λ d α (x) d x) :
1 λ α 2 λ α div(|x| d (x)∇v) ∈ L (, |x| d (x) d x) , |x|λ d α (x) 1 div(|x|λ d α (x)∇v) for any v ∈ D(L λα ), L λα v := − λ α |x| d (x) −
where H01 (, |x|λ d α (x) d x) denotes the closure of C0∞ () in the norm v → ||v|| H 1 := α,λ
λ α
|x| d (x) |∇v| + v 2
2
1
2
dx
.
(2.19)
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S. Filippas, L. Moschini, A. Tertikas
Then L λα is a nonnegative self-adjoint operator on L 2 (, |y|λ d α (y)dy) such −L λ t −L λ t := that λ for every t >λ α0, e α has λan integral kernel, that is e α v0 (x) λ lα (t, x, y)v0 (y)|y| d (y)dy; here lα (t, x, y) is called the heat kernel of L α . The existence of lαλ (t, x, y) can be proved arguing as in [DS1]; that is, using a global Sobolev inequality on , which can be easily deduced from its local version (2.12) as well as (2.18), by means of the partition of unity as in [K]. Then, from the parabolic Harnack inequality in Theorem 2.10, the following sharp two-sided heat kernel estimate for small time can be easily deduced: Theorem 2.14. Let α ≥ 1, N ≥ 2, λ ∈ [2 − N , 0] and ⊂ R N be a smooth bounded domain containing the origin. Then there exist positive constants C1 , C2 , with C1 ≤ C2 , and T > 0 depending on such that √ |λ| √ |λ| |x−y|2 N 1 (|x| + t) 2 (|y| + t) 2 C1 min α , t − 2 e−C2 t ≤ lαλ (t, x, y) ≤ α α t2 d 2 (x)d 2 (y) √ |λ| √ |λ| |x−y|2 N 1 (|x| + t) 2 (|y| + t) 2 ≤ C2 min α , t − 2 e−C1 t α α t2 d 2 (x)d 2 (y) for all x, y ∈ and 0 < t ≤ T . Proof of Theorem 2.14. Using the mean value estimate for subsolutions as in Theorem 2.12 and the parabolic Harnack inequality of Theorem 2.10 and arguing as in Theorems 5.2.10, 5.4.10 and 5.4.11 in [SC2] we are lead to the following Li-Yau type estimate: |x−y|2
|x−y|2
C1 e−C2 t C2 e−C1 t λ √ 1 √ 1 ≤ lα (t, x, y) ≤ √ 1 √ 1, V (x, t) 2 V (y, t) 2 V (x, t) 2 V (y, t) 2 for all x, y ∈ and 0 < t ≤ T ; where C1 , C2 are two positive constants with C1 ≤ C2 , and T > 0 depends on . From this the result follows using the volume estimate in Lemma 2.2. Using the machinery we have produced in this section we can handle more general operators than the one in Theorems 2.10 and 2.14. Thus, consider the operator !λ := − L α
N " ∂ 1 ∂ λ α a , (x)|x| d (x) i, j |x|λ d α (x) ∂ xi ∂x j
(2.20)
i, j=1
where ai, j (x) N ×N is a measurable symmetric uniformly elliptic matrix. The operator !λ is defined for α ≥ 1 and λ ∈ [2 − N , 0] in L 2 (, |x|λ d α (x) d x) as the generator of L α the symmetric form !λ [v1 , v2 ] := L α
N " i, j=1
|x|λ d α (x)ai, j (x)
∂v ∂v d x. ∂ xi ∂ x j
Then the existence of a heat kernel l#αλ (t, x, y) follows as in [DS1], and we have
Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators
261
Theorem 2.15. Let α ≥ 1, N ≥ 2, λ ∈ [2 − N , 0] and ⊂ R N be a smooth bounded domain containing the origin. Then there exist positive constants C1 , C2 , with C1 ≤ C2 , and T > 0 depending on such that √ |λ| √ |λ| |x−y|2 N 1 (|x| + t) 2 (|y| + t) 2 C1 min α , t − 2 e−C2 t ≤ l#αλ (t, x, y) ≤ α α t2 d 2 (x)d 2 (y) √ |λ| √ |λ| |x−y|2 N 1 (|x| + t) 2 (|y| + t) 2 ≤ C2 min α , t − 2 e−C1 t α α t2 d 2 (x)d 2 (y) for all x, y ∈ and 0 < t ≤ T . Remark 2.16. A parabolic Harnack inequality up to the boundary similar to the one of Theorem 2.10 can be stated under the same assumptions of Theorem 2.15 for the more !λ . general operator L α 3. Critical Point Singularity In this section we establish a new Improved Hardy inequality (Theorem 3.2) and then we give the proofs of Theorem 1.1 and Theorem 1.2. The structure of this section is as follows. In Subsect. 3.1 we first deduce the improved Hardy inequality and then the global in time pointwise upper bound for the heat kernel of the Schrödinger operator − − ((N − 2)2 /4)|x|−2 , which is sharp when x and y are close to the boundary (see Theorem 3.4); then, due to an argument contained in [D1], we complete the proof of Theorem 1.2 proving the sharp lower bound for time large enough. The proof of Theorem 1.1 is finally completed in Subsect. 3.2, using the parabolic Harnack inequality up to the boundary of Theorem 2.10. 3.1. Boundary upper bounds and complete sharp description of the heat kernel for large values of time. We first recall the following improved Hardy-Sobolev inequality stated in Theorem A in [FT] (see also inequality (3.3) in [BFT2]) Proposition 3.1. For N ≥ 3, let ⊂ R N be a smooth bounded domain containing the origin and D ≥ supx∈ |x|. Then there exists a positive constant C such that
for any v ∈
|∇v|2 |x|2−N d x ≥ C
C0∞ ();
here X (t) =
2N
v N −2 |x|−N X
1 1−ln t ,
2(N −1) N −2
N −2 N |x| dx , D
t ∈ (0, 1].
We next state a new result, the proof of which will be given later on. Theorem 3.2 (Improved Hardy inequality). Let ⊂ R N , N ≥ 3, be a smooth bounded domain containing the origin. Then there exists a constant C = C() ∈ (0, 41 ] such that u2 (N − 2)2 2 |∇u|2 − d x ≥ C() d x, ∀ u ∈ C0∞ (). (3.1) u 2 4|x|2 d (x)
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S. Filippas, L. Moschini, A. Tertikas
The positive constant C() can be taken to be exactly following condition:
1 4
for all domains satisfying the
−div(|x|2−N ∇d(x)) ≥ 0 a.e. in .
(3.2)
For example when ≡ B(0, R), for arbitrary R > 0, condition (3.2) is satisfied. Consequently, in this case the Hardy inequality involving the Schrödinger operator having critical singularity at the origin can be improved exactly by the inverse-square potential having critical singularity at the boundary. As a consequence of Proposition 3.1 and of the improved Hardy inequality of Theorem 3.2, the following logarithmic Sobolev inequality can be easily obtained: Theorem 3.3 (Logarithmic Hardy Sobolev inequality). For N ≥ 3, let ⊂ R N be a smooth bounded domain containing the origin. Then for any u ∈ C0∞ ( \ {0}), u ≥ 0, and any > 0 we have u u 2 log dx 2−N ||u||2 |x| 2 d(x) (N − 2)2 2 N +2 2 |∇u| − u d x + K3 − (3.3) ≤ log ||u||22 ; 4|x|2 4 here K 3 is a positive constant independent of and ||u||2 :=
u
2d x
1 2
.
Then using the Gross theorem of logarithmic Sobolev inequalities, as adapted by Davies and Simon (see Theorem 2.2.7 in [D4]), we will show the following global in time pointwise upper bound for the heat kernel: Theorem 3.4. For N ≥ 3, let ⊂ R N be a smooth bounded domain containing the origin. Then there exists a positive constant C such that k(t, x, y) ≤ C
2−N N d(x)d(y) (|x||y|) 2 t − 2 e−λ1 t , ∀ x, y ∈ , t > 0. t
Let us first prove the logarithmic Hardy Sobolev inequality (3.3). Proof of Theorem 3.3. As a first step we claim that the following logarithmic Hardy Sobolev inequality holds true: (N − 2)2 2 1 2 |∇u| − u (− log d(x)) d x ≤ u d x + K 1 − log ||u||22 , 4|x|2 2 (3.4)
2
for any u ∈ C0∞ (), u ≥ 0, and any > 0; here K 1 is a positive constant independent of . To see this let us first suppose that the nonnegative function u ∈ C0∞ () is such that ||u||2 = 1. We then have 1 1 1 2 2 −2 2 u (− log d(x)) d x = u (log d(x) ) d x ≤ log u dx ≤ 2 2 2 d(x) (N − 2)2 2 1 |∇u|2 − dx ; u ≤ log C −1 2 4|x|2
Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators
263
here we have used first Jensen’s inequality and then the improved Hardy inequality (3.1). For a general nonnegative u ∈ C0∞ () we apply the above inequality to the function u ||u||2 , to get −1 2 C 1 (N − 2) |∇u|2 − u 2 (− log d(x)) d x ≤ ||u||22 log u2 d x . 2 4|x|2 ||u||22 Since log z ≤ z for any z > 0, then also log y ≤ 2C y − log (2C), for any > 0; 1 whence from this we deduce (3.4), with K 1 := 21 log( 2C ). We will next show the following logarithmic Hardy Sobolev inequality: u u 2 log dx 2−N ||u||2 |x| 2 (N − 2)2 2 N 2 |∇u| − u d x + K 2 − log ||u||22 , (3.5) ≤ 4|x|2 4 for any u ∈ C0∞ ( \ {0}), u ≥ 0, and any > 0; here K 2 is a positive constant independent of . By Proposition 3.1 it follows easily that there exists a positive constant C such that
|∇v|2 |x|2−N d x ≥ C
2N
v N −2 |x|2−N d x
N −2 N
,
(3.6)
for any v ∈ C0∞ () (this is inequality (4.12) in [BFT2]). Whence we claim that the following logarithmic Sobolev inequality holds true: v N |x|2−N d x ≤ v 2 log |∇v|2 |x|2−N d x + K 2 − log ||v||22 , ||v||2 4 (3.7) for any v ∈ C0∞ (), v ≥ 0, and any > 0; here K 2 is a positive constant independent of
1 and ||v||2 := v 2 |x|2−N d x 2 . To see this let us first suppose that the nonnegative function v ∈ C0∞ () is such that ||v||2 = 1. We then have 4 N −2 2 2−N v log(v) |x| dx = v 2 log v N −2 |x|2−N d x 4 4 N −2 +2 2−N log v N −2 |x| dx = ≤ 4 N −2 N 2N N N log log C −1 = v N −2 |x|2−N d x ≤ |∇v|2 |x|2−N d x ; 4 4 here we have used first Jensen’s inequality and then the improved Hardy-Sobolev inequality (3.6). For a general nonnegative v ∈ C0∞ () we apply the above inequality to the v function ||v|| , to get 2 C −1 v N 2 2−N 2 2 2−N |x| v log d x ≤ ||v||2 log |∇v| |x| dx . ||v||2 4 ||v||22
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S. Filippas, L. Moschini, A. Tertikas
, for any > 0; Since log z ≤ z for any z > 0, then also log y ≤ 4C y − log 4C N N
N N whence from this we deduce (3.7) with K 2 := 4 log 4C . 2−N
Inequality (3.7) implies (3.5) via the following change of variables u := v|x| 2 . Finally from (3.4) and (3.5), the logarithmic Hardy Sobolev inequality (3.3) easily follows with constant K 3 := K 1 + K 2 + N4+2 log 2. We are now ready to give the proof of Theorem 3.4. Proof of Theorem 3.4. Let us define, as in Sect. 2 of [D2], the operator K˜ := U −1 (K − λ1 )U , U : L 2 (, ϕ12 d x) → L 2 () being the unitary operator U w := ϕ1 w, thus K˜ := − ϕ12 div(ϕ12 ∇). Here ϕ1 > 0 denotes the first eigenfunction and λ1 > 0 the first 1
−2) eigenvalue corresponding to the Dirichlet problem −ϕ1 − (N4|x| 2 ϕ1 = λ1 ϕ1 in , 2 ϕ1 = 0 on ∂, normalized in such a way that ϕ1 (x) d x = 1. Due to the results in Lemma 7 in [DD] and using Theorem 7.1 in [DS1] on one hand and elliptic regularity on the other, there exist two positive constants c1 , c2 such that 2
c1 |x|
2−N 2
d(x) ≤ ϕ1 (x) ≤ c2 |x|
2−N 2
d(x),
∀ x ∈ .
(3.8)
From this and (3.3) we deduce the following logarithmic Sobolev inequality: w N +2 2 2 ˜ w log log ||w||22 , ϕ1 d x ≤ < K w, w > L 2 (,ϕ 2 d x) + K 4 − 1 ||w||2 4 (3.9) for any w ∈ C0∞ ( \ {0}), w ≥ 0, and any > 0; where K 4 := K 3 + λ1 − log c1 and
1 ||w||2 := w 2 ϕ12 d x 2 . Let us remark that only the lower bound in estimate (3.8) was used. From now on one can use the standard approach of [D4] to complete the proof of the theorem. Here are some details for the convenience of the reader. As a first step we claim that the following L p logarithmic Sobolev inequalities holds true: p w p p ϕ12 d x ≤ < K˜ w, w p−1 > L 2 (,ϕ 2 d x) w log 1 2 ||w|| p 2 N +2 p log ||w|| p + K4 − (3.10) 4 for any w ∈ C0∞ ( \ {0}), w ≥ 0, and any > 0, p > 2. To see this we apply inequality p (3.9) to w 2 ; whence due to the fact that p p2 p2 2 2 2 < ∇w, ∇w p−1 > L 2 (,ϕ 2 d x) |∇w | ϕ1 d x = w p−2 |∇w|2 ϕ12 d x = 1 4 4( p − 1) p ≤ < K˜ w, w p−1 > L 2 (,ϕ 2 d x) , 1 2 since
p 2( p−1)
≤ 1 if p ≥ 2; the claim follows.
Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators
265
Let H01 (, ϕ12 d x) be the closure of C0∞ () with respect to the norm ||w|| H 1
0,ϕ12
:=
21 |∇w|2 ϕ12 + w 2 ϕ12 d x ;
as one can easily prove this is also the closure of C0∞ ( \ {0}) with respect to the same norm. Then the operator K˜ defined in the domain D( K˜ ) = {w ∈ H01 (, ϕ12 d x) : K˜ w ∈ L 2 (, ϕ12 d x)} is naturally associated with the bilinear symmetric form defined 2 ˜ 1 , w2 ] :=< K˜ w1 , w2 > 2 as follows K[w L (,ϕ12 d x) = ∇w1 ∇w2 ϕ1 d x, which is a Dirichlet form. Whence Lemma 1.3.4 and Theorems 1.3.2 and 1.3.3 in [D4] imply that ˜ e− K t , which is an analytic contraction semigroup in L 2 (, ϕ12 d x), is also positivity preserving and a contraction semigroup in L p (, ϕ12 d x) for any 1 ≤ p ≤ ∞. As a consequence for any t > 0 and any p ≥ 2, ˜
e− K t [L 2 (, ϕ12 d x) ∩ L ∞ ()]+ ⊂ [H01 (, ϕ12 d x) ∩ L p (, ϕ12 d x) ∩ L ∞ ()]+ ; where we denote by [E]+ the subset of positive functions in the space E. Thus by density argument the L p logarithmic Sobolev inequality (3.10), more gen˜ erally applies to any function in ∪t>0 e− K t [L 2 (, ϕ12 d x) ∩ L ∞ ()]+ . This means that Theorem 2.2.7 in [D4] can be applied, in the same way as in Corollary 2.2.8 in [D4], to the operator K˜ ; whence obtaining that ˜
||e− K t ||2→∞ ≤ Ct −
N +2 4
,
and by duality that ˜
||e− K t ||1→2 ≤ Ct −
N +2 4
,
that is ˜
||e− K t ||1→∞ ≤ Ct −
N +2 2
.
Here we use the following notation: ˜
||e− K t ||q→ p := where || f ||q :=
˜ semigroup e− K t
q 2 | f | ϕ1
˜
||e− K t f (x)|| p , || f (x)||q 0<|| f ||q ≤1 sup
q1
. This implies, by Dunford-Pettis theorem, that the ˜ x, y) is indeed a semigroup of integral operators; that is a heat kernel k(t, dx
˜
associated to the semigroup e− K t is well defined and satisfies the following pointwise ˜ x, y) ≤ C 1 t − N2 , for any x, y ∈ and any t > 0. Theorem 3.4 then upper bound k(t, t follows, due to the upper bound in (3.8) and to the fact that, as a consequence of the ˜ x, y), corresponding respectively unitary operator U , the heat kernels k(t, x, y) and k(t, to K and K˜ , satisfy the following equivalence: ˜ x, y)e−λ1 t . k(t, x, y) ≡ ϕ1 (x)ϕ1 (y) k(t,
(3.11)
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S. Filippas, L. Moschini, A. Tertikas
Remark 3.5. Applying Davies’s method of exponential perturbation to the operator K˜ (see Sect. 2 in [D3] for details), the upper bound in Theorem 3.4 can be improved by |x−y|2
adding a factor cδ e− 4(1+δ)t . Let us now deduce from the upper bound in Theorem 3.4 an analogous lower bound for time large enough, thus completing the proof of Theorem 1.2. We argue as in Theorem 6 of [D1] (see also Prop. 4 of [D2]), we give the details here for the convenience of the reader. Proof of Theorem 1.2. Making use of the same notation as in the proof of Theorem 3.4, ˜ x, y) ≥ C for any the lower bound we want to prove corresponds to the statement k(t, x, y ∈ if t is large enough, C being some positive constant. For any f ∈ L 1 (, ϕ12 d x), we clearly have f =< f, 1 > 1 + g, where < f, 1 >:=< f, 1 > L 2 (,ϕ 2 1
d x) , and <
g, 1 >= 0, since
making use of the fact that by definition K˜ 1 = 0 we have ˜
2 ϕ1 (x)d x
= 1. Thus,
˜
e− K t f =< f, 1 > 1 + e− K t g, ˜
that is the semigroup e−At f := e− K t f − < f, 1 > 1, to whom it is clearly associated ˜ x, y) − 1, is such that for any f ∈ L 1 (, ϕ 2 d x) the heat kernel k(t, 1 ˜
e−At f ≡ e− K t g, where g = g( f ) is a function in L 1 (, ϕ12 d x) such that < g, 1 >= 0. Thus, due to Theorem 3.4, ˜
||e−At ||1→∞ ≤ ||e− K t ||1→∞ ≤ Ct −
N +2 2
,
here C is some positive constant; this is equivalent to say that ˜ x, y) − 1| ≤ Ct − |k(t,
N +2 2
,
from which the claim easily follows for t large enough. In the sequel we will give the proof of Theorem 3.2. We will use the following lemma whose proof will be postponed until the end of this subsection. Lemma 3.6. For N ≥ 3, let ⊂ R N be a smooth bounded domain containing the origin. Then there exists δ0 > 0, such that 2−N |∇ f |2 d x 1 δ |x| inf = , 2 ∞ f 4 2−N f ∈C0 (δ ) dx δ |x| d 2 (x) for all 0 < δ ≤ δ0 ; here δ := {x ∈ : dist(x, ∂) ≤ δ}.
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We are now ready to prove the improved Hardy inequality. Proof of Theorem 3.2. (i) Let us first prove the claim on any domain satisfying condition (3.2). To this end let us define for any u ∈ C0∞ () as a new variable w := |x|
N −2 2
1
d − 2 (x)u, obviously w ∈ H01 (). By direct computations we have ∇u =
thus
1 2−N 1 1 2−N 1 2 − N − N −1 |x| 2 x d 2 w + |x| 2 d − 2 ∇d w + |x| 2 d 2 ∇w, 2 2
|∇u|2 =
(N − 2)2 −N 1 |x| d w 2 + |x|2−N d −1 w 2 + |x|2−N d |∇w|2 d x 4 4 N − 2 −N 2 − |x| + w x ∇d − (N − 2)|x|−N d w x ∇w 2 2−N d x. + ∇d∇w w |x|
Whence (N − 2)2 2 1 2 2 |∇u| − u − 2 u dx 4|x|2 4d N − 2 −N 2 = w x ∇d |∇w|2 d |x|2−N − |x| 2 1 (N − 2) −N 2 2 2−N dx |x| d x ∇w + ∇d ∇w |x| − 2 2 N − 2 −N 2 |∇w|2 d|x|2−N − |x| = w x ∇d 2 1 (N − 2) div(|x|−N d x)w 2 − div(|x|2−N ∇d)w 2 d x + 2 2 1 |∇w|2 d |x|2−N − div(|x|2−N ∇d)w 2 d x ≥ 0, = 2 due to condition (3.2) on . Thus inequality (3.1) is proved with constant C() ≡ 41 in any domain satisfying condition (3.2). (ii) Let us prove indirectly the claim in the remaining case. To this end let us denote by H01 (, |x|2−N d x) the closure of C0∞ () in the norm || f || H 1
2−N
:=
1 (|∇ f | + f )|x| 2
2
2−N
2
dx
.
(3.12)
The improved Hardy inequality (3.1) we are going to prove, in the new variable v := N −2 |x| 2 u reads as follows:
|∇v|2 |x|2−N d x ≥ C
|x|2−N
v2 d x. d2
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Let us suppose that the improved Hardy inequality (3.1) is false; whence let us suppose that the following holds true: inf |x|2−N |∇v|2 d x = 0; {
|x|
2−N v 2 d2
d x = 1}
thus there exists a sequence {v j } j≥0 in H01 (, |x|2−N d x) such that and
|x|
2 2−N v j d2
d x = 1,
|x|2−N |∇v j |2 d x → 0,
as j → ∞.
(3.13)
For any arbitrary function ϕ ∈ C0∞ (), such that ϕ ≡ 1 in a neighborhood of the origin, we also have 2−N 2 |x| |∇(ϕv j )| d x ≤ 2 |x|2−N |∇v j |2 ϕ 2 + |∇ϕ|2 v 2j d x ≤C |x|2−N |∇v j |2 + v 2j d x ≤C |x|2−N |∇v j |2 d x → 0 as j → ∞. (3.14)
Here we use the fact that the following inequality holds true: 2−N 2 |x| f dx ≤ C |x|2−N |∇ f |2 d x, ∀ f ∈ H01 (, |x|2−N d x).
(3.15)
Inequality (3.15) for example follows easily from inequality (3.6) by the Holder inequality. From estimate (3.14) and inequality (3.15) (applied to f := ϕv j ) we easily deduce that |x|2−N ϕ 2 v 2j → 0, as j → ∞,
or similarly (due to the fact that ϕ has compact support inside ) that
|x|2−N ϕ 2
v 2j d2
d x → 0,
as j → ∞.
(3.16)
We then compute v 2j (ϕv j + (1 − ϕ)v j )2 1= |x|2−N 2 d x = |x|2−N dx = d d2 v 2j v 2j v 2j = |x|2−N ϕ 2 2 d x + 2 |x|2−N ϕ(1 − ϕ) 2 d x + |x|2−N (1 − ϕ)2 2 d x. d d d We observe that the first two terms in the last line tend to zero as j tends to infinity and therefore we obtain that v 2j |x|2−N (1 − ϕ)2 2 d x = 1 + o(1), as j → ∞. (3.17) d
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On the other hand we have that |x|2−N |∇[(1 − ϕ)v j ]|2 d x ≤ 2 |x|2−N |∇v j |2 d x + 2 |x|2−N |∇(ϕv j )|2 d x,
both terms in the right-hand side going to zero as j tends to infinity due to (3.13) and (3.14); whence we deduce that
|x|2−N |∇[(1 − ϕ)v j ]|2 d x → 0,
as j → ∞.
(3.18)
Since for any j ≥ 0 the function f := (1 − ϕ)v j is an element of H01 (δ ) for a suitable choice of the function ϕ (take it identically one in a subset containing \ δ ), by means of (3.17) and (3.18) we reach a contradiction with Lemma 3.6, thus proving the improved Hardy inequality. A similar improved Hardy inequality for a potential behaving like ((N − 2)2 /4)|x|−2 near the origin and exactly like (1/4)d −2 (x) near the boundary is also shown without any geometric assumption on the domain (see Theorem 3.10 below). We next prove Lemma 3.6. One can consider it as a consequence of the following more general result. Lemma 3.7. For N ≥ 3, let ⊂ R N be a smooth bounded domain. Then there exists a N positive constant δ0 = δ0 (), such that for any V ∈ L 2 (δ0 ) and 0 < δ ≤ δ0 , we have the following estimate: 1 2 2 |∇u| − 2 u d x ≥ c V u 2 d x, ∀ u ∈ C0∞ (δ ); 4d δ δ here c = c(δ) → ∞ as δ → 0 and δ := {x ∈ : dist(x, ∂) ≤ δ}. −2) Proof of Lemma 3.6. Let us choose V (x) := (N4|x| in Lemma 3.7 above and let us 2 choose δ small enough such that c(δ) ≥ 1 and 0 < δ ≤ δ0 , thus we have 2
δ
|∇u|2 −
(N − 2)2 2 1 2 d x ≥ u u d x, 2 4d 4|x|2 δ 2−N
(3.19)
for any u ∈ C0∞ (δ ). For any f ∈ C0∞ (δ ), u := f |x| 2 will be in C0∞ (δ ), moreover by easy computations we have 1 (N − 2)2 2 1 2 |∇u|2 − 2 u 2 d x = |∇ f |2 + |x|2−N d x, f − f 4d 4|x|2 4d 2 δ δ thus (3.19) can be restated as follows: 1 |∇ f |2 − 2 f 2 |x|2−N d x ≥ 0; 4d δ this proves the claim.
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Whence it only remains to prove Lemma 3.7. Before doing so let us observe that inequality (3.19) simply says that the improved Hardy inequality (3.1) indeed holds true with constant C() = 41 whenever the support of the functions considered is contained in a neighborhood of the boundary. The proof of Lemma 3.7 makes use of the following improved Hardy-Sobolev inequality near the boundary stated in Theorem 3 of [FMT1], we recall it here for the convenience of the reader: Proposition 3.8. For N ≥ 3, let ⊂ R N be a smooth bounded domain. Then there exist positive constants δ0 = δ0 () and C = C(N ), such that N −2 N 2N 1 2 2 |∇u| − 2 u d x ≥ C u N −2 d x , ∀ u ∈ C0∞ (δ ), 4d δ δ
and any 0 < δ ≤ δ0 ; here δ := {x ∈ : dist(x, ∂) ≤ δ}. Let us focus here on the fact that in Proposition 3.8 no convexity assumption on the domain is made; this is due to the fact that we only consider functions whose supports are contained in a neighborhood of the boundary. Proof of Lemma 3.7. By Holder inequality we have
V u dx ≤
δ
≤
2 N
N 2
2
δ
V dx 2
N
N 2
δ
δ
V dx
u
C(N )
2N N −2
−1
N −2 N
dx
≤
1 u2 2 |∇u| − d x, 4 d2 δ
the last step being due to Proposition 3.8. This proves the claim with constant c(δ) :=
δ
N 2
V dx
− 2
N
C(N ),
which tends to infinity as δ tends to zero due to the integrability assumption on V . With some minor changes in the proof of Theorem 3.2 one can indeed prove the following improved Hardy inequality, which does not a priori require the bounded domain to be smooth. Theorem 3.9. For N ≥ 3, let ⊂ R N be a bounded domain containing the origin such that 2 u 2 |∇u| d x ≥ C d x, ∀ u ∈ C0∞ () 2 d and some positive constant C. Then there exists a positive constant C˜ such that 2 u (N − 2)2 2 ˜ |∇u|2 − d x ≥ C u d x, ∀ u ∈ C0∞ (). 2 2 4|x| d We finally mention the following related new Hardy inequality, which we think is of independent interest
Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators
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Theorem 3.10. For N ≥ 3, let ⊂ R N be a smooth bounded domain containing the origin, and define for > 0, (N −2)2 i f {x ∈ : d(x) ≥ } 4|x|2 V (x) = 1 i f {x ∈ : d(x) < }. 4d 2 (x) Then there exists 0 = 0 () such that for all 0 < ≤ 0 and u ∈ C0∞ (), we have 2 |∇u| d x ≥ V (x)u 2 d x.
Proof. We will only sketch it. Let 1 = {x ∈ : d(x) ≥ }. Then using the change of 2−N variable u := |x| 2 v, one can prove the following inequality: 2−N u2 (N − 2)2 2 |∇u|2 − d x ≥ u x · ν d Sx . 2 2 4|x| 2 1 ∂1 |x| 1
1
Similarly using the change of variable u := d 2 (x)X − 2 (d(x))v with X (t) = (1−ln t)−1 one can prove the following inequality: 1 u2 1 |∇u|2 − 2 u 2 d x ≥ − ∇d · ν d Sx , 4d (x) 4 ∂1 d(x) \1 for any 0 < ≤ min{e−1 , 1 }, where 1 > 0 is such that d −1 X (d) + 2d(ln d) ≥ 0 for d ≤ 1 . The result then follows showing that for 0 < ≤ 0 = min{e−1 , 1 , 2NR−3 } we ) 1 have 2(2−N x − d(x) ∇d · ν ≥ 0 since ν := −∇d on ∂1 ; here R denotes a positive |x|2 constant such that B(0, R) ⊂ , which exists due to the assumption on . 3.2. Complete sharp description of the heat kernel for small values of time. In this section we prove the two-sided sharp estimate on the heat kernel k(t, x, y) stated for small time in Theorem 1.1. Proof of Theorem 1.1. Since for any x ∈ and for some positive constants c1 , c2 we λ α λ α have the following estimate c1 |x| 2 d 2 (x) ≤ ϕ1 (x) ≤ c2 |x| 2 d 2 (x) for α = 2 and λ = 2− N , we can apply the result of Theorem 2.15 to the operator K˜ = − ϕ 21(x) div(ϕ12 (x)∇). Hence due to (3.11) the result follows immediately.
1
Let us finally make some remarks concerning Schrödinger operators having potential V (x) = c|x|−2 . Arguing as in Lemma 7 in [DD] one can easily prove that the first Di2 , behaves richlet eigenfunction for the Schrödinger operator − − |x|c 2 , 0 < c < (N −2) 4 $ λ 2 like |x| 2 d(x) on all , where λ := 2 − N + (N − 2) − 4c. Then we have Theorem 3.11. For N ≥ 3, let ⊂ R N be a smooth bounded domain containing the origin. Then there exist positive constants C1 , C2 , with C1 ≤ C2 , and T > 0 depending on such that √ |λ| √ |λ| d(x)d(y)
|x−y|2 λ N (|x||y|) 2 t − 2 e−C2 t ≤ C1 min (|x| + t) 2 (|y| + t) 2 , t √ |λ| √ |λ| d(x)d(y)
|x−y|2 λ N (|x||y|) 2 t − 2 e−C1 t , ≤ kc (t, x, y) ≤ C2 min (|x| + t) 2 (|y| + t) 2 , t
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for all x, y ∈ and 0 < t ≤ T ; here kc (t, x, y) denotes the heat kernel associated to 2 the operator − − |x|c 2 in under Dirichlet boundary conditions for 0 < c < (N −2) , 4 $ 2 and λ := 2 − N + (N − 2) − 4c. Theorem 3.12. For N ≥ 3, let ⊂ R N be a smooth bounded domain containing the origin. Then there exist two positive constants C1 , C2 , with C1 ≤ C2 , such that λ
λ
C1 d(x) d(y) (|x||y|) 2 e−λ1 t ≤ kc (t, x, y) ≤ C2 d(x) d(y) (|x||y|) 2 e−λ1 t , for all x, y ∈ and t > 0 large enough; here kc (t, x, y) denotes the heat kernel associated to the operator − − |x|c 2 in under Dirichlet boundary conditions for 0 < c < $ (N −2)2 , λ1 its (positive) elliptic first eigenvalue and λ := 2 − N + (N − 2)2 − 4c. 4
4. Critical Boundary Singularity In this section we prove Theorems 1.3 and 1.4 as well as a new Hardy-Moser inequality (Theorem 4.3). The structure of this section is as follows. In Subsect. 4.1 we first prove the improved Hardy-Moser inequality. Then in Subsect. 4.2 we get the global in time pointwise upper bound for the heal kernel of the Schrödinger operator − − (1/4)d −2 (x), which is sharp when x and y are close to the boundary (see Theorem 4.4). Then arguing as in [D1], we deduce the sharp heat kernel lower bound for time large enough, thus completing the proof of Theorem 1.4. The proof of Theorem 1.3 is finally completed in Subsect. 4.3, using the parabolic Harnack inequality up to the boundary stated in Theorem 1.5.
4.1. The improved Hardy-Moser inequality. Here we will prove a new improved HardyMoser inequality which we think is of independent interest. The proof is based on an auxiliary Hardy-Sobolev inequality, that we will show here, as well as on the following improved Hardy inequality stated in Theorem A in [BFT1]. Proposition 4.1. For N ≥ 2, let ⊂ R N be a smooth bounded and convex domain. Then there exists D0 positive such that for all D ≥ D0 , |∇u|2 −
1 u2 4d 2 (x)
for any u ∈ C0∞ (); here X (t) :=
dx ≥
1 1−ln t ,
1 4
t ∈ (0, 1].
X2
d(x) D
d 2 (x)
u 2 d x,
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Let us now state the auxiliary Hardy-Sobolev inequality we will use in the sequel. Lemma 4.2. Let α > 0, N ≥ 2 and ⊂ R N be a smooth bounded domain. Then there exist δ0 > 0 and C = C(α, δ0 ) > 0 such that
α
d (x)|∇v| d x +
d
\δ
α−1
(x)|v| d x ≥ C
d
αN N −1
(x)|v|
N N −1
N −1 N
dx
,
for any v ∈ C0∞ () and any 0 < δ ≤ δ0 ; here δ := {x ∈ : dist(x, ∂) ≤ δ}. Proof. We will follow closely the argument of [FMT2]. Our starting point is the following Gagliardo-Nirenberg inequality (see p. 189 in [M]) S N || f ||
N
L N −1 ()
≤ ||∇ f || L 1 () , ∀ f ∈ C0∞ (),
where S N is a positive constant depending only on N . For any v ∈ C0∞ () let us apply the above inequality to the function f := d α v. Hence we obtain S N ||d α v|| N ≤ d α (x)|∇v|d x + α d α−1 (x)|v|d x. L N −1 ()
C0∞ (2δ ),
We next estimate the last term above. Let ϕδ ∈ 0 ≤ ϕδ ≤ 1, be a cut off function which is identically one in δ and identically zero in R N \ 2δ . Clearly v = ϕδ v + (1 − ϕδ )v. Then we have d α−1 (x)|v|d x ≤ α d α−1 (x)|ϕδ v|d x + α d α−1 (x)(1 − ϕδ )|v|d x ≤ α α−1 ≤α d (x)|ϕδ v|d x + α d α−1 (x)|v|d x.
\δ
Concerning the first term on the right-hand side we have d α−1 (x)|ϕδ v|d x = ∇d α · ∇d|ϕδ v|d x α α =− d (x)∇d · ∇|ϕδ v|d x − d α (x)d|ϕδ v|d x ≤ d α (x)|∇(ϕδ v)|d x + c0 δ d α−1 (x)|ϕδ v|d x,
here we used the smoothness assumption on which implies that |dd| ≤ c0 δ in δ for δ small, say 0 < δ ≤ δ0 , and for some positive constant c0 independent of δ (δ0 , c0 depending on ). Thus we have for any 0 < δ ≤ δ0 , α α−1 α d (x)|ϕδ v|d x ≤ d (x)|∇(ϕδ v)|d x ≤ C d α (x)|∇v|ϕδ d x + α α − c0 δ0 C d α (x)|v|d x ≤ C d α (x)|∇v|d x + C d α−1 (x)|v|d x, + δ 2δ \δ 2δ \δ from which the result follows.
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We next state the new improved Hardy-Moser inequality. Theorem 4.3. (Improved Hardy-Moser inequality) For N ≥ 2, let ⊂ R N be a smooth bounded and convex domain. Then there exists a positive constant C such that 2 N 2 1 2 2 2 |∇u| − 2 u dx u dx ≥C u 2(1+ N ) d x, ∀ u ∈ C0∞ (). 4d 1
Proof. Changing variables by v := ud − 2 , we get 2(N +2) αN N N +2 2(N +2) u N dx = d N v N dx = d N −1 (v 2α ) N −1 d x,
with α :=
(N +2)(N −1) . N2
αN
Applying Lemma 4.2 to the function v 2α we have
N
d N −1 (v 2α ) N −1 d x ≤ C
d α |∇v 2α |d x +
\δ
d α−1 v 2α d x
α 2α−1 ≤ C 2α d |∇v||v| dx +
≤C
1 2
2
d(x)|∇v| d x
+
\δ
v2 dx d
\δ
d
v
d
21
≤C
\δ
2α−1 2(2α−1)
dx
d
N N −1
α−1 2α
v dx
N N −1
1 v
2α−1 2(2α−1)
2
dx
d 2α−1 v 2(2α−1) d x
⎫ N 1 ⎬ N −1 2
⎭
N 2(N −1)
N 2(N −1) 1 2 2 × ; |∇v| d − d v d x 2 d ) ≥ 41 X 2 ( Dδ ) if x ∈ \ δ ) and here we used Proposition 4.1 (observe that 41 X 2 ( D standard estimates. Returning to the original variable u, we obtain
u
2(N +2) N
dx ≤ C
u 2(2α−1) d x
N 2(N −1)
N 2(N −1) 1 |∇u|2 − 2 u 2 d x , 4d
that is,
u
2(N +2) N
2(N −1) N
dx
≤C
u
2(2α−1)
dx
1 2 |∇u| − 2 u d x . 4d 2
If N = 2 we have that α = 1, thus the above inequality becomes 1 2 4 2 2 |∇u| − 2 u d x u dx ≤ C u dx 4d
Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators
275
which is the sought for estimate. For N ≥ 3, we use the Hölder inequality to obtain 2(N −1) 2 N −2 N N N 2(N +2) 2(N +2) 2 u N dx ≤C u dx u N dx 1 |∇u|2 − 2 u 2 d x × 4d from which
u
2(N +2) N
2 N 1 2 2 |∇u| − 2 u d x ; u dx 4d
dx ≤ C
2
and this completes the proof of Theorem 4.3.
4.2. Boundary upper bounds and complete sharp description of the heat kernel for large values of time. Here we will first prove the following: Theorem 4.4. For N ≥ 2, let ⊂ R N be a smooth bounded and convex domain. Then there exists a positive constant C such that 1
h(t, x, y) ≤ C
1
d 2 (x)d 2 (y) 1 2
N
t − 2 , ∀ x, y ∈ , t > 0.
t To this end we need the following estimate of [FMT2]:
Proposition 4.5. For N ≥ 2, let ⊂ R N be a smooth bounded and convex domain. Then there exists a positive constant C such that 2 q q 1 2 2 (N −2)−N q 2 |∇u| − 2 u d x ≥ C d (x)|u| d x , (4.1) 4d (x) for any u ∈ C0∞ () and any 2 < q ≤
2N N −2
if N ≥ 3 or any 2 < q < ∞ if N = 2.
Using (4.1) the following logarithmic Sobolev inequality can be easily obtained v 1 N +1 d dx ≤ |∇v|2 d − d v 2 d x + K 1 − log ||v||22 , v 2 log ||v||2 2 4 (4.2)
for all v ∈ C0∞ (), v ≥ 0, and any > 0; here K 1 is a positive constant independent of
1 and ||v||2 := |v|2 d d x 2 . 1
To obtain (4.2) we apply (4.1) to v := ud − 2 to get for any v ∈ C0∞ (), 2 q 1 2 2 q q2 (N −2)−N + q2 |∇v| d − d v d x ≥ C v d dx . 2
Taking q :=
2(N +1) N −1
we have
N −1 N +1 2(N +1) 1 2 2 (N −1) |∇v| d − d v d x ≥ C v d dx . 2
(4.3)
Then arguing in a quite similar way as in the proof of (3.7) in Subsect. 3.1 we obtain
(4.2) with K 1 := N4+1 log N4C+1 .
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Proof of Theorem 4.4. Let H01 (, d d x) be the closure of C0∞ () with respect to the norm 1 2 1 2 2 ||v|| H 1 := |∇v| d + (−d)v d x . 0,d 2 Let H¯ := U −1 HU , U : L 2 (, d d x) → L 2 () being the unitary operator U v := d 2 v, ¯ ¯ thus H¯ := − d1 div(d∇) − 21 d d . To the operator H defined in the domain D( H ) = {v ∈ 1 2 ¯ H0 (, d d x) : H v ∈ L (, d d x)} is naturally associated the bilinear symmetric form ¯ 1 , v2 ] :=< H¯ v1 , v2 > L 2 (,d d x) =< v1 , v2 > 1 defined as follows H[v H0 (,d d x) , which is a Dirichlet form. Whence Lemma 1.3.4 and Theorems 1.3.2 and 1.3.3 in [D4] imply ¯ that e− H t , which is an analytic contraction semigroup in L 2 (, d d x), is also positivity preserving and a contraction semigroup in L p (, d d x) for any 1 ≤ p ≤ ∞. As a consequence for any t > 0 and any p ≥ 2, 1
¯
e− H t [L 2 (, d d x) ∩ L ∞ ()]+ ⊂ [H01 (, d d x) ∩ L p (, d d x) ∩ L ∞ ()]+ ; thus by density argument the L p logarithmic Sobolev inequality, which can be deduced as usual from the L 2 logarithmic Sobolev inequality (4.2) (see Subsect. 3.1 where a simi¯ lar argument is used) more generally applies to any function in ∪t>0 e− H t [L 2 (, d d x)∩ ∞ + L ()] . This means that Theorem 2.2.7 in [D4] can be applied, as in Corollary 2.2.8 in [D4], to the operator H¯ ; whence obtaining that ¯
||e− H t ||2→∞ ≤ Ct −
N +1 4
,
and by duality that ¯
||e− H t ||1→2 ≤ Ct −
N +1 4
,
that is ¯
||e− H t ||1→∞ ≤ Ct −
N +1 2
.
Here we use the following notation: ¯
||e− H t ||q→ p :=
¯
||e− H t f (x)|| p , || f (x)||q 0<|| f ||q ≤1 sup
1 where || f ||q := | f |q d d x q . This implies, by Dunford-Pettis theorem, that the ¯ ¯ x, y) semigroup e− H t is indeed a semigroup of integral operators; that is a heat kernel h(t, ¯t − H associated to the semigroup e is well defined and satisfies the following pointwise 1 − N2 ¯ upper bound h(t, x, y) ≤ C 1 t , for any x, y ∈ and any t > 0. Theorem 4.4 then t2
follows, due to the fact that, as a consequence of the unitary operator U , the heat kernels ¯ x, y), corresponding respectively to H and H¯ , satisfy the following h(t, x, y) and h(t, 1 1 ¯ x, y). equivalence h(t, x, y) ≡ d 2 (x)d 2 (y) h(t, Remark 4.6. Applying Davies’s method of exponential perturbation to the operator H¯ (see Sect. 2 in [D3] for details) the upper bound in Theorem 4.4 can be improved by |x−y|2
adding a factor cδ e− 4(1+δ)t .
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277
Let us now give the sketch of the proof of Theorem 1.4. Proof of Theorem 1.4. Let us first improve by an exponential-decreasing-in-time factor the global in time upper bound stated in Theorem 4.4. To this end let us define, as in Sect. 2 of [D2], the operator H˜ := U −1 (H − λ1 )U , U : L 2 (, ϕ12 d x) → L 2 () being the unitary operator U w := ϕ1 w, thus H˜ := − ϕ12 div(ϕ12 ∇). Here ϕ1 > 0 denotes 1
the first eigenfunction and λ1 > 0 the first eigenvalue corresponding to the Dirichlet problem −ϕ1 − 4d 21(x) ϕ1 = λ1 ϕ1 in , ϕ1 = 0 on ∂, normalized in such a way that 2 ϕ1 (x) d x = 1. Since it is known that there exist two positive constants c1 , c2 such that 1
1
c1 d 2 (x) ≤ ϕ1 (x) ≤ c2 d 2 (x), ∀ x ∈
(4.4)
(as a consequence of Lemma 7 in [DD]), logarithmic Sobolev inequalities analogous to (4.2) also hold true if we replace H¯ by H˜ . Thus as a consequence of the Gross theorem, ˜ x, y) satisexactly as in the proof of Theorem 4.4, the corresponding heat kernel h(t, N ¯ ˜ fies the same pointwise upper bound as h(t, x, y), that is h(t, x, y) ≤ C1 t − 2 for any t2
x, y ∈ and any t > 0. From the definition of U , it follows ˜ x, y)e−λ1 t , h(t, x, y) ≡ ϕ1 (x)ϕ1 (y)h(t, thus we get that h(t, x, y) ≤ C
1 1 d 2 (x)d 2 (y) 1 t2
(4.5)
N
t − 2 e−λ1 t for any x, y ∈ and any t > 0.
Finally arguing as in Theorem 6 of [D1], an analogous lower estimate can be easily deduced (we refer to the proof of Theorem 1.2 where a similar argument is used). 4.3. Complete sharp description of the heat kernel for small values of time. In this section we prove the two-sided sharp estimate on the heat kernel h(t, x, y) stated for small time in Theorem 1.3. Before doing so let us observe that Theorem 1.5 entails also the following parabolic Harnack inequality for the Schrödinger operator having critical singularity at the boundary. Theorem 4.7. For N ≥ 2, let ⊂ R N be a smooth bounded and convex domain. Then there exist positive constants C H and R = R() such that for x ∈ , 0 < r < R and 1 2 for any positive solution u(y, t) of ∂u ∂t = u + 4d 2 (y) u in {B(x, r ) ∩ } × (0, r ) we have the estimate ess sup
1
(y,t)∈{B(x, r2 )∩}×( r4 , r2 ) 2
2
u(y, t)d − 2 (y) 1
≤ C H ess inf (y,t)∈{B(x, r )∩}×( 3 r 2 ,r 2 ) u(y, t)d − 2 (y). 2
4
Proof. As a first step let us observe that if u satisfies u t = −H u then v(y, t) := eλ1 t ϕ1 (y)−1 u(y, t) satisfies vt = − H˜ v. Whence as a consequence of (4.4), due to Remark 2.16, v satisfies Theorem 1.5 for α = 1. From this the claim can be easily deduced. The proof of Corollary 1.7 is similar to the above proof of Theorem 4.7, thus we omit the details. We are now ready to prove Theorem 1.3.
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Proof of Theorem 1.3. Since for any x ∈ and for some positive constants c1 , c2 we α α have the following estimate c1 d 2 (x) ≤ ϕ1 (x) ≤ c2 d 2 (x) for α = 1, we can apply the 1 result of Theorem 2.15 to the operator H˜ = − ϕ 2 (x) div(ϕ12 (x)∇). Hence due to (4.5) 1
the result follows immediately.
The proof of Corollary 1.8 is similar to the above proofs of Theorems 1.3 and 1.4, thus we omit the details. Let us finally make some remarks concerning Schrödinger operators having potential V (x) = cd −2 (x). Arguing as in Lemma 7 in [DD] one can easily prove that the first Dirichlet eigenfunction for the Schrödinger operator − − d 2c(x) , 0 < c < 41 , behaves √ α like d 2 on all , for α := 1 + 1 − 4c. Then we have: Theorem 4.8. For N ≥ 2, let ⊂ R N be a smooth bounded and convex domain. Then there exist positive constants C1 , C2 , with C1 ≤ C2 , and T > 0 depending on such that α α d 2 (x)d 2 (y) − N −C2 |x−y|2 t C1 min 1, ≤ h c (t, x, y) t 2e α t2 α α d 2 (x)d 2 (y) − N −C1 |x−y|2 t ≤ C2 min 1, t 2e α t2 for all x, y ∈ and 0 < t ≤ T ; where h c (t, x, y) denotes the heat kernel associated to the operator − − d 2c(x) in under Dirichlet boundary conditions, for any 0 < c < 41 √ and α := 1 + 1 − 4c. Theorem 4.9. For N ≥ 2, let ⊂ R N be a smooth bounded and convex domain. Then there exist two positive constants C1 , C2 , with C1 ≤ C2 , such that α
α
α
α
C1 d 2 (x) d 2 (y) e−λ1 t ≤ h c (t, x, y) ≤ C2 d 2 (x) d 2 (y) e−λ1 t for all x, y ∈ and t > 0 large enough; where h c (t, x, y) denotes the heat kernel associated to the operator − − d 2c(x) in under Dirichlet boundary conditions, λ1 its √ (positive) elliptic first eigenvalue, for any 0 < c < 41 and α := 1 + 1 − 4c. Remark 4.10. Let us at this point remark that Theorems 1.3 and 4.8 as well as Theorem 4.7 concerning respectively sharp asymptotic for small time and the parabolic Harnack inequality up to the boundary for the Schrödinger operator having potential V (x) = cd −2 (x), hold true also without any convexity assumption on the domain under consideration.
4.4. On Davies conjecture. In this subsection we consider Davies conjecture. For this # denotes the self-adjoint operator associated with the closure of the we suppose that E positive quadratic form ⎛ ⎞ N " ∂ f ∂ f ⎝ ai, j (x) − V f 2 ⎠ d x, Q( f ) = ∂ xi ∂ x j i, j=1
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initially defined on C0∞ (); where (ai, j (x)) N ×N is a measurable symmetric uniformly elliptic matrix such that N "
ai, j (x)ξi ξ j ≥ |ξ |2
i, j=1
and V is a potential on such that V (x) = V1 (x) + V2 (x) ,
(4.6)
where |V1 (x)| ≤
1 4d 2 (x)
, V2 (x) ∈ L p (), p >
N . 2
(4.7)
We also suppose that λ1 :=
inf∞
0=ϕ∈C0 ()
Q(ϕ) > 0, 2 ϕ dx
(4.8)
and that to λ1 there corresponds a positive eigenfunction ϕ1 satisfying for all x ∈ the following estimate, α
α
c1 d 2 (x) ≤ ϕ1 (x) ≤ c2 d 2 (x), for some
α≥1
(4.9)
and for c1 , c2 two positive constants. # is defined on the closure of C ∞ () with respect to the norm defined by Thus E 0 # is a well defined nonnegthe quadratic form Q. Then as before it can be shown that E # 2 ative self-adjoint operator on L () such that for every t > 0, e− Et has an integral # kernel, that is e− Et u 0 (x) := # e(t, x, y)u 0 (y)dy and if N ≥ 3 a Green function ∞ #−1 . e(t, x, y)dt denoting the kernel of E G E#(x, y) = 0 # Theorem 4.11. For N ≥ 3, let ⊂ R N be a smooth bounded domain. Suppose that (4.6), (4.7), (4.8) and (4.9) are satisfied. Then there exist two positive constants C1 , C2 , with C1 ≤ C2 , such that for any x, y ∈ , α α 1 d 2 (x)d 2 (y) C1 min , ≤ G E#(x, y) |x − y| N −2 |x − y| N +α−2 α α 1 d 2 (x)d 2 (y) ≤ C2 min , . |x − y| N −2 |x − y| N +α−2 Davies conjecture is stated under slightly stronger assumptions on V than (4.7) and on ϕ1 (only when α = 2). # − λ1 )U , U : L 2 (, Proof. We note that we have E 1 := − ϕ12 div(ϕ12 ∇) ≡ U −1 ( E 1
ϕ12 d x) → L 2 () being the unitary operator U w := ϕ1 w; hence we have the following relationship between heat kernels: # e(t, x, y) = ϕ1 (x)ϕ1 (y)e1 (t, x, y)e−λ1 t .
(4.10)
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Due to (4.9) we can apply Theorem 2.15 to the operator E 1 . Hence due to (4.10) for two positive constants C1 ≤ C2 , we have for small time α α d 2 (x)d 2 (y) − N −C2 |x−y|2 t C1 min 1, ≤# e(t, x, y) t 2e α t2 α α d 2 (x)d 2 (y) − N −C1 |x−y|2 t ≤ C2 min 1, . (4.11) t 2e α t2 On the other hand for large time α
α
α
α
C1 d 2 (x) d 2 (y) e−λ1 t ≤ # e(t, x, y) ≤ C2 d 2 (x) d 2 (y) e−λ1 t ,
(4.12)
for all x, y ∈ . To obtain this estimate we need to prove a global Sobolev inequality on , which can be easily deduced from its local version (2.12) as well as (2.18) with λ = 0 there, by means of a partition of unity as in [K]. Then the result follows integrating # e(t, x, y) in the time variable. Acknowledgement. This work was largely done whilst the second author was visiting the University of Crete and FORTH in Heraklion, the hospitality of which is acknowledged. This research has been partially supported by the RTN European network Fronts–Singularities, HPRN-CT-2002-00274.
References [A]
Aronson, D.G.: Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73, 890–896 (1967) [BG] Baras, P., Goldstein, J.: The heat equation with a singular potential. Trans. Amer. Math. Soc. 284, 121–139 (1984) [BFT1] Barbatis, G., Filippas, S., Tertikas, A.: A unified approach to improved L p Hardy inequalities with best constants. Trans. Amer. Math. Soc. 356(6), 2169–2196 (2003) [BFT2] Barbatis, G., Filippas, S., Tertikas, A.: Critical heat kernel estimates for Schrödinger operators via Hardy-Sobolev inequalities. J. Funct. Anal. 208, 1–30 (2004) [BM] Brezis, H., Marcus, M.: Hardy’s inequalities revised. Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(1–2), 217–237 (1997) [BV] Brezis, H., Vazquez, J.L.: Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Complut. Madrid 10(2), 443–469 (1997) [CM] Cabré, X., Martel, Y.: Existence versus explosion instantanée pour des équationes de la chaleur linéaires avec potentiel singulier. C.R. Acad. Sci. Paris Ser. I Math. 329(11), 973–978 (1999) [CS] Chiarenza, F.M., Serapioni, R.P.: A remark on a Harnack inequality for degenerate parabolic equations. Rend. Sem. Mat. Univ. Padova 73, 179–190 (1985) [D1] Davies, E.B.: Perturbations of ultracontractive semigroups. Quart. J. Math. Oxford 2(37), 167– 176 (1986) [D2] Davies, E.B.: The equivalence of certain heat kernel and green function bounds. J. Funct. Anal. 71, 88–103 (1987) [D3] Davies, E.B.: Explicit constants for Gaussian upper bounds on heat kernels. Amer. J. of Math. 109, 319–334 (1987) [D4] Davies, E.B.: Heat kernels and spectral theory. Cambridge: Cambridge University Press, 1989 [DS1] Davies, E.B., Simon, B.: Ultracontractivity and the heat kernels for Schrodinger operators and Dirichlet Laplacians. J. Funct. Anal. 59, 335–395 (1984) [DS2] Davies, E.B., Simon, B.: L p norms of non-critical Schrödinger semigroups. J. Funct. Anal. 102, 95–115 (1991) [DD] Dávila, J., Dupaigne, L.: Hardy type inequalities. J. Eur. Math. Soc. 6(3), 335–365 (2004) [DG] De Giorgi, E.: Sulla differenziabilitá e l’analiticitá delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. (3), n. 3, 25–43 (1957) [FGS] Fabes, E.B., Garofalo, N., Salsa, S.: A backward Harnack inequality and Fatou Theorem for nonnegative solutions of parabolic equations. Ill. J. Math. 30(4), 536–565 (1986)
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Fabes, E.B., Kenig, C.E., Jerison, D.: Boundary behavior of solutions to degenerate elliptic equations. Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Belmont, CA: Wadsworth, 1983, pp. 577–589 [FKS] Fabes, E.B., Kenig, C.E., Serapioni, R.P.: The local regularity of solutions of degenerate elliptic equations. Comm. Part. Diff. Eq. 7, 77–116 (1982) [FS] Fabes, E.B., Stroock, D.W.: A new proof of Moser’s parabolic Harnack inequality via the old ideas of Nash. Arch. Rat. Mech. Anal. 96, 327–338 (1986) [FMT1] Filippas, S., Maz’ya, V.G., Tertikas, A.: Sharp Hardy-Sobolev inequalities. C.R. Math. Acad. Sci. Paris 339(7), 483–486 (2004) [FMT2] Filippas, S., Maz’ya, V.G., Tertikas, A.: Critical Hardy-Sobolev inequalities. J. Math. Pures Appl. 87, 37–56 (2007) [FT] Filippas, S., Tertikas, A.: Optimizing Improved Hardy inequalities. J. Funct. Anal. 192, 186– 233 (2002) [Ga] Gao, P.: The boundary Harnack principle for some degenerate elliptic operators. Comm. Partial Differ. Eq. 18(12), 2001–2022 (1993) [G1] Grigoryan, A.: The heat equation on non-compact Riemannian manifolds (in Russian). Matem. Sbornik 182(1), 55–87 (1991). Engl. Transl.: Math. USSR Sb. 72(1), 47–77 (1992) [G2] Grigoryan, A.: Heat kernels on weighted manifolds and applications. Cont. Math. 398, 93–191 (2006) [GSC] Grigoryan, A., Saloff-Coste, L.: Stability results for Harnack inequalities. Ann. Inst. Fourier, Grenoble 55(3), 825–890 (2005) [K] Kufner, A.: Weighted Sobolev spaces. Teubner-Texte zur Mathematik, 31, Stüttgart, Teubner, 1981 [KO] Kufner, A., Opic, B.: Hardy type inequalities. Pitman Research Notes in Math Series 219, London: Pitman, 1990 [LY] Li, P., Yau, S.-T.: On the parabolic kernel of the schrödinger operator. Acta Math. 156(3–4), 153– 201 (1986) [M] Mazya, V.G.: Sobolev spaces. Berlin-Heidelberg. New York: Springer-Verlag, 1985 [MS] Milman, P.D., Semenov, Yu.A.: Global heat kernel bounds via desingularizing weights. J. Funct. Anal. 212, 373–398 (2004) [MT1] Moschini, L., Tesei, A.: Harnack inequality and heat kernel estimates for the Schrödinger operator with Hardy potential. Rend. Mat. Acc. Lincei 16, 171–180 (2005) [MT2] Moschini, L., Tesei, A.: Parabolic Harnack inequality for the heat equation with inverse-square potential. To appear in Forum Mathematicum [Mo1] Moser, J.: On harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 577– 591 (1961) [Mo2] Moser, J.: A Harnack inequality for parabolic differential equations. Comm. Pure. Appl. Math. 17, 101–134 (1964); Correction: 20, 231–236 (1967) [N] Nash, J.: Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80, 931– 954 (1958) [SC1] Saloff-Coste, L.: A note on Poincaré, Sobolev, and Harnack inequalities. Internat. Math. Res. Notices 2, 27–38 (1992) [SC2] Saloff-Coste, L.: Aspects of Sobolev-type inequalities. London Math. Soc. Lecture Notes Series 289, Cambridge: Cambridge University Press, 2002 [VZ] Vázquez, J.L., Zuazua, E.: The hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential. J. Funct. Anal. 173, 103–153 (2000) [Z] Zhang, Qi. S.: The boundary behaviour of heat kernels of Dirichlet Laplacians. J. Diff. Eq. 182, 416– 430 (2002) Communicated by B. Simon
Commun. Math. Phys. 273, 283–304 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0226-2
Communications in
Mathematical Physics
Design of Hyperbolic Billiards Maciej P. Wojtkowski Department of Mathematics and Physics, University of Szczecin, ul. Wielkopolska 15, 70-451 Szczecin, Poland. E-mail: [email protected] Received: 27 February 2006 / Accepted: 8 September 2006 Published online: 11 May 2007 – © M. P. Wojtkowski 2007
Abstract: We formulate a general framework for the construction of hyperbolic billiards. Spherical symmetry is exploited for a simple treatment of billiards with spherical caps and soft billiards in higher dimensions. Other examples include the Papenbrock stadium. 1. Introduction The purpose of this paper is to present a general framework for the construction of hyperbolic billiards, especially with some convex pieces in the boundary, and also “soft” billiards. Most of the examples of hyperbolic billiards that were constructed up to date can be understood in this framework, with the notable exception of systems studied in [W5, W6]. Billiards are a class of dynamical systems with appealingly simple description. A point particle moves with constant velocity in a box of arbitrary dimension (“the billiard table”) and reflects elastically from the boundary (the component of velocity perpendicular to the boundary is reversed and the parallel component is preserved). Mathematically it is a class of hamiltonian systems with collisions defined by symplectic maps on the boundary of the phase space, [W1]. Such systems are also called hybrid systems, being a concatenation of continuous time and discrete time dynamics. The billiard dynamics defines a one parameter group of maps t of the phase space which preserve the Lebesgue measure, and are in general only measurable due to discontinuities. The boundaries of the box are made up of pieces: concave, convex and/or flat. Discontinuities occur in particular at the orbits tangent to concave pieces of the boundary of the box. The orbits hitting two adjacent pieces (“corners”) have two natural continuations, which is another source of discontinuity. These singularities are not too severe so that the flow has well defined Lyapunov exponents and Pesin structural theory Reproduction of the entire article for non-commercial purposes is permitted without charge.
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is applicable, [K-S]. A billiard system is called hyperbolic if it has nonzero Lyapunov exponents almost everywhere (or at least on a subset of positive Lebesgue measure). It is called completely hyperbolic if all of its Lyapunov exponents are nonzero almost everywhere, except for one zero exponent in the direction of the flow. Billiards in smooth strictly convex domains have no singularities, but no such systems are known to be hyperbolic. In dimension 2 Lazutkin showed that near the boundary of such domains the system is near integrable. Applying the KAM theory he proved that for these “grazing orbits” there is always a family of invariant curves with positive total measure in the phase space, and with zero Lyapunov exponents. In general billiards exhibit mixed behavior just like other hamiltonian systems; there are invariant tori intertwined with the “chaotic sea”. In hyperbolic billiards stable behavior is excluded by the choice of the pieces in the boundary of the box, arbitrary concave pieces and special convex ones, and their particular placement, usually separation. Thus hyperbolicity is achieved by design, as in optical instruments. Hyperbolicity is the universal mechanism for random behavior in deterministic dynamical systems. Under additional assumptions it leads to ergodicity, mixing, K-property, Bernoulli property, decay of correlations, central limit theorem, and other stochastic properties. Hyperbolic billiards provide a natural class of examples for which these properties were extensively studied. In this article we restrict ourselves to hyperbolicity itself. The most prominent example of a hyperbolic billiard is the gas of hard spheres. This way of looking at the system was developed in the groundbreaking papers of Sinai, see [Ch-S] for an exhaustive list of references. The excellent collection of papers, [H], contains more up to date information. An important source on hyperbolic billiards is the book by Chernov and Markarian, [Ch-M]. The books by Kozlov and Treschev [K-T], and by Tabachnikov [T] provide broad surveys of billiards from different perspectives. 2. Jacobi Fields and Monotonicity The key to understanding hyperbolicity in billiards lies in two essentially equivalent descriptions of infinitesimal families of trajectories. The basic notion is that of a Jacobi field along a billiard trajectory. Let γ (t, u) be a family of billiard trajectories, where t is time and u is a parameter, |u| < , for some > 0. A Jacobi field J (t) along γ (t) = γ (t, 0) is defined by J (t) = ∂γ ∂u |u=0 . Jacobi fields form a finite dimensional vector space that can be naturally identified with the tangent space of the phase space at any point on the trajectory. Jacobi fields t contain as the flow D if we the same information derivatives of the billiard . Indeed, (t) treat J (0), J (0) and J (t), J (t) as tangent vectors at γ (0), γ (0) and γ (t), γ t respectively then D J (0), J (0) = J (t), J (t) . In particular the Lyapunov exponents are the exponential rates of growth of Jacobi fields. Jacobi fields split naturally into parallel and perpendicular components to the trajectory, each of them a Jacobi field in its own right. The parallel Jacobi field carries the zero Lyapunov exponent. In the following we discuss only the perpendicular Jacobi fields until Sects. 6, 7 and 8, where we are forced to consider general Jacobi fields. They form a codimension 1 subspace in the tangent to the unit tangent bundle, i.e., the phase space restricted by the condition that velocity has length one. Since the billiard trajectories are geodesics of the Euclidean metric the Jacobi fields satisfy between collisions the differential equation J = 0, and hence J (t) = J (0) + t J (0).
(1)
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At a collision a Jacobi field undergoes a change by the map J (tc+ ) = RJ (tc− )
J (tc+ ) = RJ (tc− ) + P ∗ KP J (tc+ ),
(2)
where J (tc− ) and J (tc+ ) are Jacobi fields immediately before and after collision, K is the shape operator of the piece of the boundary (K = ∇n, n is the inside unit normal to the boundary), and P is the projection along the velocity vector from the hyperplane perpendicular to the orbit to the hyperplane tangent to the boundary. Finally R is the orthogonal reflection in the hyperplane tangent to the boundary. To measure the growth/decay of Jacobi fields we introduce form in the a quadratic tangent spaces, or equivalently on Jacobi fields, Q(J, J ) = J, J . Evaluation of Q on a Jacobi field is a function of time Q(t). Definition 1. A billiard trajectory γ (t) is (strictly) monotone on a Jacobi field J , between its two points q0 = γ (0) and q1 = γ (t1 ), or equivalently between time 0 and time t1 , if Q(t1 )(>) ≥ Q(0). A billiard trajectory γ (t) is called (strictly) monotone between its two points q0 = γ (0) and q1 = γ (t1 ) (or between time 0 and time t1 ), if it is (strictly) monotone on any nonzero Jacobi field J (t) between q0 and q1 . A nonzero Jacobi field is called parabolic between time 0 and time t1 if J (0) = 0 and J (t1 ) = 0. A billiard trajectory γ (t) is called parabolic between its two points q0 = γ (0) and q1 = γ (t1 ), if it has a Jacobi field J (t) which is parabolic between time 0 and time t1 (i.e., J (0) = J (t1 ) = 0). It is called completely parabolic if for every Jacobi field if J (0) = 0 then also J (t1 ) = 0. Clearly any trajectory which is strictly monotone between q0 and q1 cannot be parabolic between q0 and q1 . Due to the reversibility of the billiard motion monotonicity is the property of a trajectory in the configuration space with chosen points q0 and q1 . Indeed, if it holds for the trajectory traversed from q0 to q1 then it also holds for the reversed trajectory from q1 to q0 . This is the subject of the following Lemma 2. If a trajectory is (strictly) monotone between its two points q0 and q1 then the reversed trajectory is also (strictly) monotone between q1 and q0 . Proof. Let us consider a nonzero Jacobi field J along the orbit γ (t), 0 ≤ t ≤ T . The orbit γ˜ (t) = γ (T − t) is the reversed orbit and J˜(t) = J (T − t) is a Jacobi field along the orbit γ˜ (t). We get further J˜ (t) = −J (T − t) and the change of Q on J along the orbit γ and on J˜ along the orbit γ˜ are the same. It follows from (1) that if there are no collisions between two points on a trajectory then the trajectory is monotone between the points. Indeed Q(t) − Q(0) = t|J (0)|2 . We have further Lemma 3. If a trajectory is monotone between two noncollision points q0 and q1 then it is also monotone between a point q˜0 earlier than q0 (q˜0 < q0 ) and a point q˜1 later than q1 (q1 < q˜1 ), provided that there are no collisions between q˜0 and q0 , and q1 and q˜1 , respectively. Moreover for two such points q˜0 < q0 and q˜1 > q1 either the orbit is strictly monotone between q˜0 and q˜1 or it is parabolic between the points.
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Proof. The first part of the lemma is obvious. To prove the second part let us consider a nonzero Jacobi field J . If J does not vanish at q0 then the form Q increases strictly on J between q˜0 and q0 . Similarly if J does not vanish at q1 then the form Q increases strictly between q1 and q˜1 . It turns out that a sufficiently long “free flight” ensures monotonicity for most trajectories. Proposition 4. If a trajectory is not parabolic between two noncollision points q0 and q1 then it is strictly monotone between a point q˜0 sufficiently earlier than q0 and a point q˜1 sufficiently later than q1 , provided that there are no collisions between q˜0 and q0 , and q1 and q˜1 , respectively. If a trajectory is completely parabolic between two noncollision points q0 and q1 then it is monotone between a point q˜0 sufficiently earlier than q0 and a point q˜1 sufficiently later than q1 , provided that there are no collisions between q˜0 and q0 , and q1 and q˜1 , respectively. More precisely we extend the segments of the trajectory containing q0 and q1 into rays, which allows us to go arbitrarily far in the past and/or arbitrary far in the future without any new collisions. Note also that in dimension 2 we can apply this result to any trajectory since then any parabolic trajectory is obviously completely parabolic. However for trajectories close to parabolic the necessary interval of free flight may be unbounded. Proof. Let q0 = γ (0) and q1 = γ (t1 ). We are seeking T > 0 so large that Q(t1 + T ) > Q(−T ) for any nonzero Jacobi field J . In other words we want the quadratic form Q(t1 + T ) − Q(−T ) on perpendicular Jacobi fields to be positive definite. We have Q(t1 + T ) − Q(−T ) = Q(t1 + T ) − Q(t1 ) + Q(t1 ) − Q(0) + Q(0) − Q(−T ) = T |J (0)|2 + |J (t1 )|2 + Q(t1 ) − Q(0).
(3)
If a trajectory is not parabolic between q0 and q1 then |J (0)|2 + |J (t1 )|2 is a positive definite quadratic form on perpendicular Jacobi fields. It follows that for sufficiently large T the quadratic form (3) is also positive definite. This proves the first part of the proposition. To prove the second part let us recall that due to the symplectic nature of the billiard dynamics for trajectories completely parabolic between time 0 and time t1 we have J (t1 ) = A J (0), J (t1 ) = A∗−1 J (0) + B J (0), for some linear operators A and B such that A∗ B is symmetric, see for example [W1]. Using (3) we get Q(t1 + T ) − Q(−T ) = T |J (0)|2 + |A J (0)|2 + A∗ B J (0), J (0) . Clearly the quadratic form is positive semidefinite for sufficiently large T > 0, and hence the trajectory is monotone, but not strictly monotone. By (2) the monotonicity of a trajectory at a collision, i.e., Q(tc+ ) ≥ Q(tc− ), is equivalent to the positive semidefiniteness of the shape operator K ≥ 0, it holds for concave pieces of the boundary. Billiards with only concave and flat pieces of the boundary are called semidispersing. If K > 0 at a point of collision then by Lemma 3 we have
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strict monotonicity between a point before the collision and a point after the collision. In semidispersing billiards, where K ≥ 0, K = 0, strict monotonicity may still occur after sufficiently many reflections. Definition 5. We say that a billiard system is eventually strictly monotone (ESM) on a subset X of positive Lebesgue measure in the phase space, if for almost every trajectory beginning in X there is a return time t1 to X such that the trajectory is strictly monotone between time 0 and time t1 . The role of monotonicity is revealed in the following Theorem 6. [W1] If a billiard system is ESM on a subset X of the phase space then for almost every orbit passing through X all Lyapunov exponents are different from zero. Theorem 6 is formulated here for billiard systems. However it can be generalized and applied to other systems, not even hamiltonian (see [W2] for precise formulations, references and the history of this idea). 3. Wave Fronts and Monotonicity There is a geometric formulation of monotonicity (which historically preceded the one given above). Let us consider a local wavefront, i.e., a local hypersurface W (0) perpendicular to a trajectory γ (t) at t = 0. Let us consider further all billiard trajectories perpendicular to W (0). The points on these trajectories at time t form a local hypersurface W (t) perpendicular again to the trajectory (warning: in general at exceptional moments of time the wavefront W (t) is singular). Infinitesimally wavefronts are described by the shape operator U = ∇n, where n is the unit normal field. U is a symmetric operator on the hyperplane tangent to the wavefront (and perpendicular to the trajectory γ (t). The evolution of infinitesimal wavefronts is given by the formulas U (t) = (t I + U (0)−1 )−1 without collisions, U (tc+ ) = RU (tc− )R−1 + P ∗ KP at a collision.
(4)
It follows that between collisions a wavefront that is initially convex (i.e., diverging, or U > 0) will stay convex. Moreover any wavefront after a sufficiently long run without collisions will become convex (after which the normal curvatures of the wavefront will be decreasing). The second part of (4) shows that after a reflection in a strictly concave boundary a convex wavefront becomes strictly convex (and its normal curvatures increase). These properties are equivalent to (strict) monotonicity as formulated in Definition 1. Indeed in the language of Jacobi fields an infinitesimal wavefront represents a linear subspace in the space of perpendicular Jacobi fields (i.e., the tangent space). Moreover it is a Lagrangian subspace with respect to the standard symplectic form. We can follow individual Jacobi fields or whole subspaces of them. It explains the parallel of (1), (2) and (4). The form Q allows the introduction of positive and negative Jacobi fields and positive and negative Lagrangian subspaces. An infinitesimal convex wavefront represents a positive Lagrangian subspace. Monotonicity is equivalent to the property that for every positive Lagrangian subspace at time 0 its image under the derivative of the flow Dt is also positive. It may seem that there is loss of information in formulas (4) compared to (1) and (2). However the symplectic nature of the dynamics makes them actually equivalent, [W1].
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4. Design of Hyperbolic Billiards In view of (4) it seems that a convex piece in the boundary (K < 0) excludes monotonicity. There are two ways around this apparent obstacle to hyperbolicity. First we could change the quadratic form Q at the convex boundary and consider monotonicity with respect to the modified form Q. We follow here another path. We treat convex pieces as “black boxes” and look only at incoming and outgoing trajectories. The first strategy is presented in [W1]. Although the second strategy seems more restrictive, the examples of hyperbolic billiards constructed to date fit the black box scenario with few exceptions, [W5, W6]. To introduce this approach let us consider a billiard table with flat pieces of the boundary and exactly one convex piece. A trajectory in such a billiard experiences visits to the convex piece separated by arbitrary long sequences of collisions in flat pieces, which do not affect the geometry of a wavefront at all. Hence whatever is the geometry of a wavefront emerging from the curved piece it will become convex and very flat by the time it comes back to the curved piece of the boundary again. Hence it follows, at least heuristically, that we must study the complete passage through the convex piece of the boundary, regarding its effect on convex, and especially flat, wavefronts. An important difference between convex and concave pieces is that a trajectory has usually several consecutive collisions in the same convex piece, moreover the number of such collisions is unbounded. A finite billiard trajectory is called complete if it contains reflections in one and the same piece of the boundary, and it is preceded and followed by reflections in other pieces. We can now formulate two principles for the design of hyperbolic billiards. 1. No parabolic trajectories. Convex pieces may have no complete trajectories which are parabolic. 2. Separation. There must be sufficient separation, in space or time, between complete trajectories. The relevance of these two principles can be seen in Proposition 4. For non-parabolic trajectories if there is enough separation we get strict monotonicity and Theorem 6 is applicable. Definition 7. A complete trajectory is (strictly) z-monotone (on a Jacobi field J ) for some z ≥ 0, if it is (strictly) monotone (on the Jacobi field J ) between the point at the distance z before the first reflection and the point at the distance z after the last reflection. A convex piece of the boundary is (strictly) monotone if almost every complete trajectory is (strictly) z-monotone for some z ≥ 0. Additionally the piece of the boundary is called finitely (strictly) monotone if the value of z is uniformly bounded for almost all complete trajectories in the piece. In the language of wavefronts a complete trajectory is z-monotone if every diverging wavefront at a distance at least z from the first reflection becomes diverging at the distance z after the last reflection, or earlier. Clearly a strictly concave piece is strictly monotone. Every complete trajectory has only one reflection and it is strictly -monotone for any small , the property we call 0-monotone. It follows from Theorem 6 that we get a completely hyperbolic billiard if we put together curved strictly monotone pieces of the boundary and some flat pieces, in such a way that for every two consecutive complete trajectories, which are z 1 -monotone and
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z 2 -monotone respectively the distance from the last reflection in the first trajectory to the first reflection in the second one is bigger than z 1 + z 2 . Indeed we can consider the subset X of the phase space containing appropriate midpoints of trajectories leaving one curved piece and hitting another one. We obtain immediately the property ESM on X . This construction seems unlikely to succeed if there is no uniform bound on the distances z at which complete orbits are monotone, so that no separation of the pieces will be sufficient. However in the case of spherical caps, studied by Bunimovich and Rehacek, [B-R], we find a geometric scenario that works without the uniform bound on the value of z. We will discuss it in Sect. 6. 5. Hyperbolic Billiards in Dimension 2 In all of the examples of hyperbolic billiards constructed so far the convex pieces of the boundary have no parabolic complete trajectories. Checking this property is nontrivial due to the unbounded number of reflections in complete trajectories close to tangency. It was accomplished so far only in integrable, or near integrable examples, with one exception described in the following. Billiards in dimension 2 are understood best. First of all there is yet another way of describing infinitesimal families of trajectories. Every infinitesimal family of lines in the plane has a point of focusing (in linear approximation), possibly at infinity. This point of focusing contains the same information as the curvature of the infinitesimal wavefront (it is the center of curvature, rather than curvature itself) and it has the advantage that it does not change in free flight. The change in the focusing point after a reflection is described by the familiar mirror equation of the geometric optics −
1 1 2 + = , f0 f1 d
(5)
where f 0 , f 1 are the signed distances of the points of focusing to the reflection point, d = r cos θ , r is the radius of curvature of the boundary piece (r > 0 for a strictly convex piece) and θ is the angle of incidence. The mirror equation is just the two dimensional version of (4). We can see that f 1 is a fractional linear function of f 0 as it should be, since the focusing point f is a projective coordinate in the projectivization of the two dimensional space of perpendicular Jacobi fields, [W3]. This fractional linear function gives us a mapping of the line extending the billiard segment, to the next one; the lines become topological circles with the addition of the points at infinity. These circles have natural orientations given by the direction of the billiard trajectory. Fractional linear maps given by the mirror equation preserve this orientation. Indeed the perpendicular Jacobi fields form a plane which has the canonical orientation defined by the symplectic form (the form does not vanish on the plane). This orientation, like the symplectic form, is invariant under the dynamics and it induces an orientation of the projectivization (the circle of focusing points with projective coordinate f ). By Proposition 4 in dimension 2 every convex piece is monotone. However in general it is not finitely monotone, i.e., the value of z may be unbounded. To examine this issue let us consider an incoming trajectory before the first collision and the outgoing trajectory after the last collision. The minimal value of z for which this trajectory is z-monotone can be obtained in terms of the linear fractional map that shows the dependence of the focusing point before the first reflection and the focusing point after the last reflection. Let the focusing points before the first reflection and after the last reflection be denoted by f 0 and f 1 respectively. f 0 and f 1 are signed distances to the respective
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reflection points measured in the direction of the motion. There are two cases, parabolic and non-parabolic trajectory. If the complete trajectory is parabolic then for the parabolic Jacobi field J (t) the focusing after the last reflection occurs at infinity and we have f 1 = a f 0 + b, for some b a > 0 and b. We obtain that this parabolic trajectory is z-monotone for z = max{0, a+1 }. Indeed it is the minimal z ≥ 0 such that if f 0 ≤ −z then f 1 ≤ z. In the case of a nonparabolic complete trajectory let the Jacobi field that is initially focused at infinity (J (0) = 0) be focused after the last reflection at finite f 1 = c1 , and the Jacobi field focused at infinity after the last reflection be focused at some finite f 0 = c0 before the first reflection. We get then that f1 =
c1 f 0 + b , c0 c1 + b < 0. f 0 − c0
(6)
Let us note that the condition c0 c1 + b < 0 is equivalent to the orientation preservation c1 +b of the fractional linear map since dd ff01 = − ( cf 0−c > 0. )2 0
0
Lemma 8. The trajectory is z-monotone for 1 c1 − c0 + (c0 + c1 )2 − 4(c0 c1 + b) }. z = max{0, 2 Proof. By direct calculations we obtain two values f ±= 21 (c0−c1 ± (c0 +c1 )2−4(c0 c1+b)) such that if f 0 = f ± then f 1 = − f ± . We have also that f − < c0 < f + and − f + < c1 < − f − . If f − ≤ 0 then the trajectory is z-monotone for z = − f − . If f − > 0 then the trajectory is 0-monotone (in the sense that it is -monotone for arbitrarily small > 0). The problem with the application of these formulas is that in general we cannot get the explicit dependence of f 1 on f 0 for complete trajectories with a large number of reflections. The exception is provided by integrable billiard tables, the disk and the ellipse. Billiard in a disk is integrable due to its rotational symmetry. Let J be a Jacobi field obtained by the rotation of a trajectory. This family of trajectories (“the rotational family”) is focused exactly in the middle between two consecutive reflections (that is where J vanishes). It follows further from the mirror equation (5) that a parallel family of orbits is focused at a distance d2 after the reflection, and any family focusing somewhere between the parallel family and the rotational family will focus at a distance somewhere between d2 and d, not only after the first reflection, but also after an arbitrary long sequence of reflections. Hence any complete trajectory in an arc of a circle is z-monotone, where 2z is the length of a segment of the trajectory, and by Lemma 3 it is strictly z’-monotone, for any z > z. Two arcs of a circle separated by parallel segments form the stadium of Bunimovich, [B1]. Lazutkin showed that billiards in smooth strictly convex domains are near integrable close to the boundary of the table, [L]. Donnay applied Lazutkin’s coordinates to establish that for an arbitrary strictly convex arc the situation near the boundary is similar to that in a circle, i.e., in our language complete near tangent trajectories are z-monotone, where z is of the order of the length of a single segment. This crucial calculation shows that if a strictly convex arc is strictly monotone then any sufficiently small perturbation of this arc is strictly monotone. In view of Proposition 4 the only obstacle to using a convex arc in the boundary of a hyperbolic billiard is the presence of parabolic orbits. Such orbits are not a problem in
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themselves since they are still z-monotone, but nearby orbits may have c0 << −1 and c1 >> 1 which by Lemma 8 will result in a large z value. The case when c0 >> 1 and c1 << −1 is safe, resulting in z = 0. However no examples of such “safe” parabolic orbits are known to us. Bunimovich [B3] introduced the concept of absolutely focusing arcs which in our language means that parabolic trajectories are excluded and additionally for every complete trajectory c0 < 0 and c1 > 0. We do not need the last assumptions for our argument, but it is conceivable that our generalization is superficial, i.e., if there are no parabolic orbits then by necessity c0 < 0 and c1 > 0 for all complete trajectories. The following proposition reformulates in our language the result of Donnay [D]: Proposition 9. Any convex arc without complete parabolic orbits is finitely strictly monotone (and hence can be used in designing completely hyperbolic billiards). Proof. In view of Donnay’s analysis of near tangent orbits we have to consider only compact families of complete trajectories with the number of reflections not exceeding a certain fixed number. Under the assumption of the absence of parabolic complete trajectories, the values of |c0 | and |c1 | are uniformly bounded on these compact families. It follows from Lemma 8 that the z-value is uniformly bounded for these trajectories. It was also observed by Donnay [D] and Markarian [M] that for any strictly convex arc, its sufficiently short piece does not have parabolic orbits. Indeed, if the arc is very short, then any complete trajectory is either near tangent or it has only one reflection. No near tangent trajectory is parabolic and the mirror equation (5) does not allow a trajectory with one reflection to be parabolic. In general checking that parabolic orbits are absent cannot be done by a direct calculation. 2 An arc satisfying dds r2 < 0, where r is the radius of curvature as a function of the arc length s, is strictly monotone, [W3]. Such an arc is called convex scattering. More precisely we have that any complete trajectory in a convex scattering arc is strictly z-monotone with z = max d, maximum of the values of d from the mirror equations (5) for the first and the last segment of the trajectory. This property leads to examples of hyperbolic billiards with one convex piece of the boundary, like the domain bounded by the cardioid. Let us note that the convex scattering property stands out in not being associated with integrability or near integrability. Integrability of the elliptic billiard allows one to establish finite strict monotonicity of the semi-ellipse with endpoints on the longer axis, [W3]. Donnay, [D], √ showed that the other semi-ellipse is also finitely strictly monotone provided that a ≤ 2b, where a ≥ b are the semiaxes. 6. Systems with Local Spherical Symmetry Let us consider two segments of the same orbit lying in one plane R2 ⊂ Rn , with mirror symmetry in the plane R2 reversing the direction of time, Fig. 1. We further assume that there is the center of symmetry which we use as the origin of the coordinate system and that small rotations around it (of Rn ) take the two segments into two segments of another orbit, with the preservation of time. In such a case we say that our system has local spherical symmetry, on the orbit in question. Examples of systems with local spherical symmetry are furnished by billiards with spherical caps, and by soft billiards with spherical scatterers, [B2, W4, B-R , D-L, B-T].
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e2 e1
m0 q0
m1 q1
Fig. 1.
Let γ (t) be the time parameterization of the segments, where q0 = γ (0) belongs to the first segment and q1 = γ (t1 ) belongs to the second segment. It follows from the local spherical symmetry that one parameter groups of rotations produce families of orbits. Let Z ∈ o(n) be an infinitesimal rotation (i.e. Z is an anti-symmetric matrix). We get the family of orbits γ (t, u) = eu Z γ (t) and the respective Jacobi field J Z (t) = Z γ (t), J Z (t) = Z γ (t). This Jacobi field is known to us only on the two segments, but it is enough to check monotonicity between q0 and q1 . We will call such Jacobi fields spherical. We choose an orthonormal basis e1 , e2 in the plane of the orbit so that its axis of symmetry has the direction of e2 , Fig. 1. We will be checking monotonicity of our orbit between γ (0) = q0 = ae1 + be2 and γ (t1 ) = q1 = −ae1 + be2 . We have γ (0) = sin αe1 + cos αe2 with − π2 ≤ α ≤ π2 , and γ (t1 ) = sin αe1 − cos αe2 . It follows that for the spherical Jacobi field J Z , J Z (0) = a Z e1 + bZ e2 , J Z (0) = sin α Z e1 + cos α Z e2 , J Z (t1 ) = −a Z e1 + bZ e2 , J Z (t1 ) = sin α Z e1 − cos α Z e2 .
(7)
We will say that the orbit segments are in general position if their extensions do not contain the origin. Equivalently the segments are in general position if a cos α −b sin α = 0. For orbit segments in general position there are many spherical Jacobi fields. More precisely we have the following Lemma 10. If the orbit segments are in general position then spherical Jacobi fields form a linear subspace of dimension 2n − 3. Nonzero spherical Jacobi fields are not necessarily perpendicular but none of them is parallel. Proof. Let us consider the linear map Z → J Z . It follows from the condition a cos α − b sin α = 0 that the kernel of the map coincides with antisymmetric matrices such that Z e1 = Z e2 = 0. Hence the kernel of the map has the dimension 21 (n − 2)(n − 3) while the space of all matrices Z has the dimension 21 n(n − 1). This gives us the dimension of the space of spherical Jacobi fields. To prove the second part of the lemma let us consider a spherical Jacobi field such that J Z (0) and J Z (0) are parallel to γ (0), and hence linearly dependent. It follows that Z e1 and Z e2 must be perpendicular to e1 and e2 , and further that J Z (0) and J Z (0) are both parallel and perpendicular to γ (0). Only the zero Jacobi field can satisfy it.
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We will consider only orbit segments in general position. It is enough for the study of hyperbolicity because the orbits for which this condition fails form a subset of the phase space of dimension n and can be safely ignored. It follows from (7) and the condition a cos α − b sin α = 0 that by an appropriate choice of a skewsymmetric matrix Z we can get arbitrary vectors perpendicular to the plane spanned by e1 and e2 as the values of J (0) and J (0). Let us call such spherical Jacobi fields transverse. Transverse Jacobi fields form a linear subspace of the space of spherical Jacobi fields of dimension 2n − 4. At this stage we need to invoke the symplectic nature of the dynamics. Jacobi fields form the tangent space to the phase space at any point on the orbit. Hence we get an identification of all of these tangent spaces. This identification amounts to the action of the derivative of the flow. The tangent spaces are equipped with the canonical symplectic form and hence the space of Jacobi symplectic space with fields is a linear the canonical symplectic form ω(J1 , J2 ) = J1 , J2 − J1 , J2 (‘Wronskian’), where the scalar products are evaluated at any point on the orbit segments (in particular we get the same value independent of the point). It follows from this formula that for any Jacobi field J skeworthogonal to the space of transverse Jacobi fields the values of J and J at any point are in the plane spanned by e1 and e2 . We will call such Jacobi fields planar. The space of perpendicular Jacobi fields contains the 2 dimensional subspace of planar Jacobi fields. Further we have the unique splitting of any Jacobi field J = J p + Jt into a planar, J p , and a transverse, Jt , Jacobi fields. Moreover it follows from the definition of the form Q that Q(J ) = Q(J p + Jt ) = Q(J p ) + Q(Jt ). Hence, if we establish monotonicity separately for planar and for transverse Jacobi fields then we get monotonicity for all Jacobi fields. Note that we have used the symplectic formalism to obtain the splitting. In the two classes of examples with spherical symmetry we get it directly from an additional symmetry. Monotonicity between the symmetric points q0 and q1 for spherical Jacobi fields can be analyzed by direct calculation: Q(t1 ) − Q(0) = J Z (t1 ), J Z (t1 ) − J Z (0), J Z (0) = −2a sin α Z e1 , Z e1 (8) −2b cos α Z e2 , Z e2 . Hence we get monotonicity if and only if b cos α ≤ 0 and a sin α ≤ 0. Let us assume that there is monotonicity on all spherical Jacobi fields between the points q0 = γ (0) and q1 = γ (t1 ) and that monotonicity fails on some of these fields between γ () and γ (t1 − ) for arbitrarily small > 0. In the same way as in formula (3) we obtain from (8), Q(t1 − ) − Q() = (−2a sin α − 2 sin2 α) Z e1 , Z e1 +(−2b cos α − 2 cos2 α) Z e2 , Z e2 . Our assumptions lead to the conditions b = 0, a sin α < 0 or a = 0, b cos α < 0. In this way we arrive at two generic configurations, the configuration A where b = 0, a sin α < 0, and the configuration B where a = 0, b cos α < 0. We also have two singular configurations, the configuration As with α = ± π2 , a = 0, and the configuration Bs with α = 0, b = 0, Fig. 2 and Fig. 3. In all cases we get monotonicity between q0 and q1 on all spherical Jacobi fields, but the points q0 and q1 are positioned differently in different configurations. In configuration A and Bs the points q0 and q1 lie on the e1 axis and in configuration B and As they
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q1 q0 e2
q0
e2
q1
e1
e1
Fig. 2. Configurations A (left) and As (right)
e2
q0
e1 q0
e2
q
1
e1
q1
Fig. 3. Configurations B (left) and Bs (right)
coincide geometrically with one point on the axis of symmetry, i.e., the e2 axis. In the singular configuration Bs both segments are vertical and in the singular configuration As both segments lie on the same horizontal line. In any of the configurations the points are optimal in the sense that monotonicity fails for points past q0 and before q1 . The difference between a configuration A and a configuration B is in the location of the point of intersection of the line extensions of the segments. This point lies on the axis of symmetry, above the origin for a configuration A, and below the origin for a configuration B. (The terms ‘below’ and ‘above’ seem arbitrary. To remove this ambiguity we note that the two orbit segments are ordered by time. The first segment in a generic configuration allows us to orient canonically the line of symmetry, which was hidden earlier in the condition that − π2 ≤ α ≤ π2 .) We have thus established that monotonicity on spherical Jacobi fields depends only on the geometry of the incoming and outgoing segment and it is not affected by the dynamics. The generic configurations have no parabolic spherical Jacobi fields but the singular configurations do. However even for the singular configurations we get monotonicity between appropriate points, i.e., the conclusions of Proposition 4 hold, even though we do not have a completely parabolic orbit. It follows from the splitting of an arbitrary perpendicular Jacobi field into a transverse field and a planar one. All transverse fields are spherical and hence are covered by the above analysis. The planar fields form a two
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dimensional invariant subspace and hence the proof of the completely parabolic part of Proposition 4 applies to them. Having monotonicity separately for transverse and planar fields is equivalent to monotonicity. Let us analyze monotonicity on planar Jacobi fields in more detail. We will compare the Jacobi fields at the points γ (0) = m 0 and γ (t1 ) = m 1 which are closest to the origin in the line extensions of the respective orbit segments, Fig. 1. (If the initial segments are too short to contain the points their status is somewhat abstract; they may or may not be actual orbit points. However it will not effect our analysis.) Lemma 11. The fractional linear representation of the dynamics (6) on the planar perpendicular Jacobi fields between m 0 and m 1 has the form f1 =
−c f 0 , f0 − c
where c is the unique value for which there is a planar perpendicular Jacobi field J with J (0) = 0 and J (t1 − c) = 0 (or a field with J (c) = 0 and J (t1 ) = 0). Monotonicity holds for the planar Jacobi fields between γ (−z) and γ (t1 + z) for z = |c| − c. In the limit case of c → ∞ we get f 1 = f 0 and then there is monotonicity for z = 0. Proof. The space of planar perpendicular Jacobi fields is 2 dimensional. The spherical planar Jacobi field J Z generated by the infinitesimal rotation Z with Z e1 = e2 , Z e2 = −e1 is not perpendicular but it has the nonzero perpendicular component which we will denote by Jr . By the choice of the points m 0 and m 1 we have that Jr (0) = 0 and Jr (t1 ) = 0. Moreover introducing compatible orthonormal frames v0 = γ (0),v0⊥ and v1 = γ (0), v1⊥ at m 0 and m 1 respectively, we can calculate that Jr (0), v0⊥ = J Z (0), v0⊥ = 1 and Jr (t1 ), v1⊥ = J Z (t1 ), v1⊥ = 1. Now we use J (0), v0⊥ , J (0), v0⊥ and J (t1 ), v1⊥ , J (t1 ), v1⊥ as coordinates in the 2 dimensional space of planar perpendicular Jacobi fields, at m 0 and m 1 respectively. In these coordinates Jr is the second basic vector both at m 0 and at m 1 . Hence dynamics between m 0 and m 1 the 10 is described in these coordinates by the matrix . Since the focusing distance ∗1
J,v ⊥
f i = − J ,vi⊥ , i = 0, 1, we obtain the lemma from Lemma 8 by direct calculation. i
Note that the effort in the above proof is to show that c0 = −c1 = c in (6). We get it from local spherical symmetry alone. It is quite obvious in the two classes of examples due to the additional reversible symmetry. We have thus established that monotonicity of a complete trajectory in a system with spherical symmetry depends only on the geometry of the incoming and outgoing segments and the value of c from Lemma 11, which is the only information we need to extract from the dynamics. In the case of spherical caps of radius R any complete trajectory lies in a plane passing through the center. Moreover the planar Jacobi fields are just the Jacobi fields of the trajectory in the billiard in the disk of radius R. It was observed in Sect. 5 that in such a case c is always positive and hence z = 0. The analysis of monotonicity in the case of spherical caps is thus complete and can be summarized in the following proposition which was essentially stated in [W4].
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Proposition 12. Any complete trajectory, in general position, in a spherical cap is monotone between min(q0 , m 0 ) and max(q1 , m 1 ). (The min and max are understood in the sense of the temporal ordering of the trajectory). Moreover only the trajectories in singular configurations are parabolic. Proof. It remains to analyze parabolic trajectories, i.e., we are looking for a perpendicular Jacobi field such that J (0) = 0, J (t1 ) = 0. We know that there are no such nonzero planar Jacobi fields. It remains to check the transverse (and hence spherical) Jacobi fields. It follows from (7) that if J Z (0) = sin α Z e1 + cos α Z e2 = 0 and J Z (t1 ) = sin α Z e1 − cos α Z e2 = 0 then either α = 0 and the trajectory is in the configuration Bs , or α = ± π2 and the trajectory is in configuration As . We can now apply this analysis to specific examples of billiards with spherical caps, and to soft billiards with spherical scatterers. The first construction of a three dimensional hyperbolic billiard with spherical caps was obtained in [B-R]. We are in a position to recover easily this construction, to see what the obstacles are and how to overcome them. We want to attach spherical caps to a box. We need separation of the caps so that it takes a long time from when an orbit leaves a cup until it reaches another one (or the same one after reflecting in flat pieces). By similarity considerations instead of separating the cups we may fix the rectangular box, and decrease the radius of the sphere. The first observation is that configurations Bs are disastrous, because if they are present then in the same plane we will also have configurations B with the point q0 arbitrarily far away. By elementary geometry we get Proposition 13. If the angle at which a piece S of a sphere is seen from the center is less than π2 then all complete trajectories in S are in configuration A, in particular there are no trajectories in configuration Bs We will call such pieces small spherical caps. One may get the impression that configurations As may pose a similar difficulty because the points q0 and q1 may go to infinity. Indeed if we consider a small spherical cap and a plane of our complete orbit that cuts the edge of the cap then our complete orbit may have the points q0 and q1 far away. What saves the construction is that the points must stay on the line through the center, and hence they have a bounded displacement in one direction. So now the prescription for the design of the billiard with spherical caps is to place the small caps only at the bottom and the top of the box. Such a billiard is equivalent to the billiard between two parallel hyperplanes (the top and the bottom) with small spherical caps attached. It is clear that if the hyperplanes are sufficiently far apart then the configurations A do not pose any difficulty. The exact separation is such that the horizontal hyperplanes through the centers of the spheres of the top spherical caps should be above those for the bottom spherical caps. Clearly more complicated designs can also be produced. One finds several of them in [B-R]. 7. Soft Billiards The analysis in Sect. 6 can be readily applied to soft billiards. These are systems with a point particle moving in a rectangular box, or a torus, with spherical scatterers. However the point particle does not collide elastically with the scatterer, but enters into it and is subjected to a field of force with a spherically symmetric potential. In the 2 dimensional case, after the work of Knauf, [K1, K2] Donnay and Liverani, [D-L], gave general
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conditions on the potential that guarantee, in our present language, that all complete trajectories are z-monotone with uniformly bounded z. A complete trajectory through a scatterer is the piece of a trajectory from entering a scatterer to leaving it. This allowed them to construct a variety of completely hyperbolic soft billiards,. The case of higher dimensions remained open for 15 years. Recently Balint and Toth, [B-T], obtained additional conditions on the potential that guarantee complete hyperbolicity in arbitrary dimension. Our condition that no complete trajectory is parabolic is fully equivalent to those of [B-T]. Moreover our approach results in fairly explicit conditions on the required separation, while such conditions are absent both from [D-L] and [B-T]. The orbit of the point particle inside a scatterer is not in general a straight segment. However we restrict our attention to the incoming and outgoing segments of our trajectory which we denote by γ (t), with γ (0) being a point before the entrance into the scatterer and γ (t1 ) a point after the exit. For a family of trajectories γ (t, u) we consider the Jacobi field J (t) = ∂γ ∂u |u=0 . For Jacobi fields which are not spherical we cannot claim that if J (0) is perpendicular to the trajectory then J (t1 ) is also perpendicular. However since J (t) is perpendicular to the trajectory outside of the scatterer (because the point particle has unit velocity there), then the values of Q(0) and Q(t1 ) depend only on the perpendicular component of J (t). We can then consider these perpendicular components in place of perpendicular fields and the analysis of Sect. 6 is perfectly valid. (What happens here is that an invariant codimension one subspace in the tangent bundle of the phase space is not a priori available and we have to work with a quotient space rather than a subspace. Perpendicular components of Jacobi fields form the quotient space, see [W1, W2].) Definition 14. The halo of a scatterer in a soft billiard is a closed concentric ball of minimal radius, containing the scatterer and such that almost any complete trajectory through the scatterer is monotone between two points outside of the ball. Our goal is to establish the existence of the halo for a given scatterer, and to determine its radius. By Theorem 6 if each scatterer in a soft billiard has a halo and the halos are mutually disjoint then the soft billiard is completely hyperbolic. In dimension 3 and above if the spherical potential V = V (r ) is continuous (i.e. in particular it vanishes at the boundary) and attractive V (r ) > 0 then there is no halo. Indeed let us consider a straight line tangent to the scatterer. Perturbing it we obtain a complete trajectory “grazing” the scatterer. By necessity it is in configuration A with the points q0 and q1 at large distance from the center. Since this distance goes to infinity as the trajectory approaches the tangent line and the points must belong to the halo we conclude that there is no halo for our scatterer. Hence scatterers with continuous attractive potentials are not allowed in the design of a hyperbolic soft billiard in dimension ≥ 3. This was already observed by Balint and Toth, [B-T]. The passage through a scatterer is completely described by the rotation angle = (ϕ), 0 ≤ ϕ ≤ π2 , [B-T]. For a given angle of incidence ϕ the angle is the angular difference between the entrance and the exit points on the complete trajectory that enters the scatterer with the incidence angle ϕ. With a fixed orientation of the circle the value of differs by the sign when we switch the incoming and outgoing lines. For simplicity we consider only the case depicted in Fig. 4, with the counterclockwise orientation of the boundary of the scatterer. It is convenient to introduce the angle − π2 ≤ η ≤ π2 between the perpendicular axis of symmetry of the line of the incoming segment and the axis of symmetry of the configuration, Fig. 4.
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∆ 2
η
m
0
φ Fig. 4.
In configuration A we have 0 < η < π2 and in configuration B we have − π2 < η < 0. Moreover = 2η − 2ϕ + π . To find the radius of the halo of a scatterer we need to find the distance of the points q0 , q1 to the center of symmetry for any complete trajectory passing through the scatterer. By simple geometric considerations we obtain that this distance is equal to R
sin ϕ sin ϕ in configuration A and R , in configuration B, sin η cos η
(9)
where R denotes the radius of the scatterer, and ϕ is the angle of incidence for our complete trajectory. Hence the point q0 is outside of the scatterer when η < ϕ in configuration A, and when η < − π2 + ϕ in configuration B. These formulas allow the direct calculation of the halo of a scatterer in examples. It will be large if for some configurations η is small and positive or close to − π2 . It is guaranteed to be finite if there are no singular configurations (η = 0 or η = ± π2 ). There is also contribution into the halo from the planar Jacobi fields. More specifically we need to calculate the constant c in Lemma 11. It is sufficient to obtain one additional planar Jacobi field, not focused at m 0 (as in the rotational field). For that purpose let us consider the family of trajectories γ (t, ϕ) entering a scatterer at one point γ (0, ϕ) = (R, 0), γ (0, ϕ) = − cos ϕe1 + sin ϕe2 . At the exit time t1 = t1 (ϕ) we get γ (t1 (ϕ), ϕ) = (R cos , R sin ) , γ (t1 (ϕ), ϕ) = cos( + ϕ)e1 + sin( + ϕ)e2 . Hence we get the Jacobi field J , J (0)
J (0) = 0, J (t1 ) = R (− sin e1 + cos e2 ) , = sin ϕe1 + cos ϕe2 , J (t1 ) = ( + 1) (− sin( + ϕ)e1 + cos( + ϕ)e2 ) .
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ϕ By direct calculation we obtain now c = − R cos +2 . Hence by Lemma 11 if + 2 < 0 we get z = 0 and there is no contribution from planar Jacobi fields to the scatterer’s halo. If however + 2 > 0 then z = −2c which translates to the halo of radius −1
+1 . (10) h = R p 2 cos2 ϕ + sin2 ϕ, where p = 2
The application of these formulas, in obtaining explicit separation of scatterers for hyperbolicity, hinges on the representation of the rotation function in terms of the potential, which is somewhat cumbersome. We have nothing new to add on this subject compared to the papers [D-L] and [B-T], where the reader can find a detailed discussion. We will consider here only the simple case of a constant potential V = V0 < E = 21 . In the two dimensional case it was studied by Baldwin, [Ba], and Knauf, [K2], who arrived at sharp conditions for complete hyperbolicity. It was shown in [B-T] that there is always sufficient separation of scatterers that will guarantee complete hyperbolicity also in the multidimensional case, without providing specific bounds. It turns out that the two dimensional conditions are also sufficient in higher dimension. Soft billiards with constant potential are systems where the crossing of scatterers is governed by a version of the law of refraction. One needs to distinguish the case of the positive potential 0 < 2V0 < 1 and the negative potential V0 < 0. We have sin ϕ = arccos , ν = 1 − 2V0 , 2 ν where in the case of a positive potential (ν < 1) the formula is valid for 0 ≤ ϕ ≤ ϕ0 and ϕ0 is such that sin ϕ0 = ν, and in the case of a negative potential (ν > 1) the formula is valid for all 0 ≤ ϕ ≤ π2 . It follows immediately that 1 (ϕ) = −
. 2 2 1 + ν −1 cos2 ϕ
Hence the derivative is a decreasing function in the case of a positive potential (ν < 1) and an increasing function in the case of a negative potential. It follows that in the case of positive potential 2(ϕ) ≤ − ν1 < −1, and there is no contribution into the halo from
the planar Jacobi fields. In the case of a negative potential (ν > 1) 2(ϕ) ≥ − ν1 > −1 νR and we get, using (10), that the halo radius h = ν−1 (the minimal value of h is assumed at ϕ = 0 because p is a decreasing function of ϕ). Further in the positive case we have only configurations B with η < ϕ so that there is no contribution from them into the halo. In the negative case we have only configurations A and using (9) we arrive readily νR at the same halo radius h = ν−1 (it is again assumed at ϕ = 0). The explanation for this coincidence is that the halo is determined by trajectories in the limit of the incidence angle ϕ → 0. In that limit the distinction between planar and transverse Jacobi fields is lost. To summarize, no separation of scatterers is necessary for hyperbolicity in the case of a positive potential and in the case of a negative potential the scatterers should have non√ νR intersecting halos with the radius h = ν−1 , ν = 1 − 2V0 . Baldwin, [Ba], showed that with the violation of these conditions one can construct systems with elliptic periodic orbits. The same can be claimed in higher dimensional systems. Hence our conditions are sharp.
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8. Twisted Cartesian Products We will describe here a construction of higher dimensional hyperbolic systems which generalizes the Papenbrock stadium, [P], and can be understood in the language of monotonicity as developed in this paper. Let us consider two billiard systems, system 1 and system 2, and their cartesian product. Given monotone trajectories γ1 and γ2 in systems 1 and 2 respectively, we address monotonicity of the trajectory (γ1 (t), γ2 (t)) in the cartesian product. We are faced with the basic difficulty that the moments of time between which there is monotonicity may be different for γ1 and γ2 . This difficulty disappears if one of the systems has all trajectories monotone between any two points. The examples of such systems are semidispersing billiards and closely related geodesic flows on manifolds of nonpositive sectional curvature. The simplest example is the motion of a point particle in a segment. We will call such systems universally monotone. Another new element in our construction is monotonicity in the full phase space; so far we have discussed montonicity of systems on one energy level. This restriction was somewhat hidden in the fact that all our Jacobi fields J satisfied J , γ = 0. When we allow all energy levels we have more Jacobi fields and a trajectory could fail to be monotone on some of the additional fields. Monotonicity on all Jacobi fields will be called ambient. In the cartesian product the kinetic energy is split arbitrarily between system 1 and system 2. In other words γ1 may be traversed fast while γ2 is traversed slowly. In a pure cartesian product it is not an issue because both kinetic energies are first integrals of motion. However we are going to modify the cartesian product to obtain a hyperbolic system and such modifications are bound to destroy the first integrals; only the total kinetic energy remains constant. We need to consider each of the systems in all of the phase space and check for the ambient monotonicity. More precisely we need to allow more Jacobi fields by considering families of trajectories γ (t, u) (cf., Sect. 2) in which || ∂t∂ γ (t, u)|| depends on u. It turns out that for billiard systems (and geodesic flows) ambient monotonicity follows automatically from monotonicity. Indeed in such systems the same trajectory can be traversed at different speeds. Hence in particular for any trajectory γ (t) there is a constant a > 0 such that γ (as) is its arc length parameterization. In other words a trajectory on an arbitrary energy level is a reparametrization of a trajectory with unit velocity. We have γ (t, u) = γ (a(u)s, u), J (s) = Y (t) + a sw,
∂ ∂u γ (t, u)|u=0 , J (s) 1 d a a ds J − a w, where w = ∂t∂ γ (t, 0).
Y (t) =
=
=
a
d dt Y
Adopting the convention that is get
d d dt , ds
or
∂ ∂u γ (a(u)s, u)|u=0 , = a(0), a = a (0),
d du , as appropriate, and using
J , w = 0, we
1 a (a )2 J (s), J − J (s), w + 3 s. Q(Y ) = Y, Y = a a a Once we remember that J (s), w does not change along a trajectory (because of the invariant split of Jacobi fields into perpendicular and parallel Jacobi fields) we conclude that Q(Y )(t1 ) − Q(Y )(t0 ) =
1 (a )2 (Q(J )(s1 ) − Q(J )(s0 )) + 3 (s1 − s0 ), where a a
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ti = asi , i = 0, 1.
(11)
We summarize the consequences of (11) in the following Proposition 15. If a trajectory traversed with speed one is monotone between two points then the same trajectory traversed at an arbitrary speed satisfies ambient monotonicity between the points. If a trajectory is strictly monotone in the restricted phase space between two points then in the full phase space the only Jacobi fields Y on which Q is not increased are parallel to the velocity and satisfy Y = 0. Proof. The first part follows immediately from (11). To prove the second part let us observe that by (11) if γ (t) is monotone between γ (t0 ) and γ (t1 ) and there is no increase of Q on a Jacobi field Y then a = 0, which means that the Jacobi field may be obtained from a family of trajectories in one energy level. Now since the trajectory is assumed to be strictly monotone, then the Jacobi field Y must be parallel, i.e., Y = constγ , Y = 0. We are ready to proceed with the construction. We consider a euclidean space with coordinates (x0 , x1 , . . . , xk ) = (x0 , x). Our system 1 is a billiard (or a geodesic flow) in a domain D in the upper halfspace {x0 > 0} with some boundary D0 at {x0 = 0}. We will remove the boundary D0 and allow trajectories to enter freely into the lower halfspace {x0 < 0}. We assume that in the system 1 any trajectory is strictly monotone between two consecutive visits to D0 . Our system 2 is a universally monotone system in the configuration space E which is isometric to D0 . It is clear that the product system has all trajectories monotone between consecutive visits of the first component to D0 , regardless of where the second component is in E. We will now examine the Jacobi fields on which our trajectory is parabolic. Let the euclidean coordinates in E be denoted by y = (y1 , . . . , yk ) and let the coordinate identity map y = x furnish the isometry from D0 to E. Let γ (t) = (γ1 (t), γ2 (t)) ∈ D × E be a trajectory of the cartesian product of our systems and let γ (ti ) ∈ D0 × E, i = 0, 1, be two consecutive visits to D0 × E. We consider a family of trajectories γ (t, u) = (γ1 (t, u), γ2 (t, u)) = (x0 (t, u), x(t, u), y(t, u)) including γ (t) = γ (t, 0) ∂ ∂ and generating the Jacobi field Y = ∂u γ1 (t, 0), ∂u γ2 (t, 0) . Proposition 15 leads immediately to the following Lemma 16. If there is no increase in Q on a Jacobi field Y between t0 and t1 then Y = 0 ∂ and ∂u γ1 (t, 0) = const ∂t∂ γ1 (t, 0). To finish the construction of our system we consider another system in the domain D˜ in the lower halfspace {x0 < 0} such that the part of the boundary of D˜ at {x0 = 0} is the same D0 , and with strict monotonicity between consecutive visits to D0 . The simplest example would be the reflection of our system 1 into the lower halfspace {x0 < 0}. We now glue the domains D × E and D˜ × E along the common boundary D0 × E by the isometric map G(x0 , x, y) = (x0 , −y, x). When a trajectory in D × E reaches {x0 = 0} then it is continued into D˜ × E after the change of position and velocity by the map G. We will call such a system a twisted cartesian product. Theorem 17. A trajectory in the twisted cartesian product is strictly monotone between every second visit to {x0 = 0}. The twisted cartesian product is completely hyperbolic.
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Proof. We can see that the glueing map G preserves the form Q. Hence in the twisted cartesian product trajectories are monotone between visits to {x0 = 0}. We need to examine the Jacobi fields on which there is no increase in the value of the form Q between a first visit to {x0 = 0} and the third. Let Y (t) = (Y0 (t), Y1 (t), Y2 (t)) be such a field with the second visit to {x0 = 0} at t = 0. By Lemma 16 if there is no increase of Q on Y between the first and the second visit then (Y0 (−0), Y1 (−0)) = c1 γ1 (−0), where the values at −0(+0) denote the limits at 0 over negative (positive) t. Further we get that if there is no increase of Q on the Jacobi field Y between the second and third visit then (Y0 (+0), Y1 (+0)) = c2 γ1 (+0). Taking into account the gluing map G applied at t = 0 we have (Y0 (+0), Y1 (+0), Y2 (+0)) = (Y0 (−0), −Y2 (−0), Y1 (−0)) and we can conclude that c1 = c2 = c and Y (t) = cγ (t) both for t < 0 and t > 0, i.e., Y is a parallel Jacobi field in the product system. That means that our trajectory is actually strictly monotone between the first and the third visit. It follows that our twisted cartesian product on one energy level is eventually strictly monotone and hence completely hyperbolic. Indeed if we consider the Poincare section of our flow {x0 = 0, ddtx0 > 0} we see that consecutive visits to the section are separated by exactly one more visit to {x0 = 0}. Let us consider specific examples of systems with the required properties and their twisted cartesian products. Example 1. Let the system 1 be the Sinai billiard with one convex scatterer in a square D with D0 being one of the sides of the square and the system 2 be the motion of a point particle in the segment E = D0 . The resulting twisted cartesian product is a billiard system in a rectangular box in three dimensions with two cylindrical scatterers having perpendicular directions. Such systems were introduced by Simanyi and Szasz, [S-S], in a more general case of cylinders with arbitrary directions. Strictly speaking our analysis of strict monotonicity is incomplete for such a system since in the Sinai billiard the trajectories which do not collide with the scatterer between consecutive visits to D0 are parabolic. However it follows easily from our analysis that every trajectory is strictly monotone if it encounters both scatterers. To establish complete hyperbolicity of such a system it remains to show that the trajectories that encounter at most one scatterer form a set of zero Lebesgue measure. This was established in the paper [S-S], in a more general case. Example 2. Let the system 1 be the billiard in a convex domain D without corners with 2 the curved part of the boundary satisfying the property of convex scattering ( dds r2 < 0), and D0 being the flat part of the boundary (Fig. 5a). We have that any trajectory in D is strictly monotone between consecutive visits to D0 , [W3]. The system 2 is again the motion of a point particle in the segment E = D0 . By Theorem 17 the twisted cartesian product is completely hyperbolic. If instead the system 1 is “half of a Bunimovich stadium” as in Fig. 5b, then the twisted cartesian product is the Papenbrock stadium. The first proof that the Papenbrock stadium is completely hyperbolic was obtained by Bunimovich and del Magno, [B-M]. Example 3. We can take as system 1 a rectangular box in 3 dimensions with a spherical cup on one side and the square D0 on the other, that is essentially the BunimovichRehachek system discussed in Sect. 6, [B-R]. The system 2 is the uniform motion of a point particle in the square E = D0 . Theorem 17 is again not immediately applicable
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E
D0
a
D0
E
b Fig. 5.
because there are orbits in D that visit D0 twice without entering into the spherical cap. However the proof of Theorem 17 gives us strict monotonicity on trajectories that enter both caps, in {x0 > 0} and in {x0 < 0}. The proof that almost all trajectories have this property is straightforward but cumbersome and we omit the details. Let us finally observe that while the billiard domain in Example 1 can be modified with the preservation of strict monotonicity, as shown in [S-S], the billiard systems of Examples 2 and 3 are rigid in the sense that a typical perturbation of the billiard domain destroys the arguments used in Theorem 17. These arguments are based on partial integrability of cartesian products. Thus again we are confronted with the fragility of complete monotonicity in billiards in higher dimensions (≥ 3) with convex pieces of the boundary. It is an open problem to produce a more robust construction, or to explain why it cannot be done. Acknowledgements. Much of the work on this paper was done while the author visited the Institute of Mathematics of the Polish Academy of Sciences in Warsaw and the Institute Henri Poincaré in Paris. I am grateful for the warm hospitality and excellent working conditions that I enjoyed in Warsaw and Paris in the Spring of 2005. Supported in part by the NSF grant DMS-0406074. I am also grateful to Paul Wright and the anonymous referees for their valuable remarks.
References [Ba] [B1] [B2] [B3] [B-D] [B-R] [B-T] [Ch-S] [Ch-M]
Baldwin, P.R.: Soft billiard systems. Pyhysica D. 29, 321–342 (1988) Bunimovich, L.A.: On the ergodic properties of nowhere dispersing billiards. Commun. Math. Phys. 65, 295–312 (1979) Bunimovich, L.A.: Many-dimensional nowhere dispersing billiards with chaotic behavior. Physica D 33, 58–64 (1988) Bunimovich, L.A.: On absolutely focusing mirrors. In: Ergodic Theory and related topics, III, Gustrow 1990, U. Krengel (ed), Lecture Notes in Math. 1514, Berlin-Heidelberg-NewYork: Springer, 1992 pp. 62–82 Bunimovich, L.A., Del Magno, G.: Semi-focusing billiards: hyperbolicity. Commun. Math. Phys. 262, 17–32 (2006) Bunimovich, L.A., Rehacek, J.: How high-dimensional stadia look like. Commun. Math. Phys. 197, 277–301 (1998) Balint, P., Toth, I.P.: Hyperbolicity in multi-dimensional hamiltonian systems with applications to soft billiards. Disc. Cont. Dyn. Syst. 15, 37–59 (2006) Chernov, N.I., Sinai, Ya.G.: Ergodic properties of some systems of 2-dimensional discs and 3dimensional spheres. Russ. Math. Surv. 42, 181–207 (1987) Chernov, N.I., Markarian, R.: Billiards. Providence, RI: Amer. Math. Soc. 2005
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Donnay, V.: Using integrability to produce chaos: billiards with positive entropy. Commun. Math. Phys. 141, 225–257 (1991) Hard ball systems and the Lorentz gas, ed. D. Szasz, Berlin-Heidelberg-New York: SpringerVerlag, 2000 Knauf, A.: Ergodic and topological properties of coulombic periodic potentials. Commun. Math. Phys. 110, 89–112 (1987) Knauf, A.: On soft billiard system. Pyhysica D. 36, 259–262 (1989) Katok, A., Strelcyn, J.-M.: with the collaboration of F. Ledrappier, F. Przytycki: Invariant manifolds, entropy and billiards; smooth maps with singularities. Lecture Notes in Math. 1222, BerlinHeidelberg-New York: Springer-Verlag 1986 Kozlov, V.V., Treschev, D.V.: Billiards. A genetic introduction to the dynamics of systems with impacts. Providence, RI: Amer. Math. Soc. 1990 Lazutkin, V.F.: On the existence of caustics for the billiard ball problem in a convex domain. Math. USSR Izv. 7, 185–215 (1973) Markarian, R.: Billiards with pesin region of measure one. Commun. Math. Phys. 118, 87–97 (1988) Papenbrock, T.: Numerical study of a three dimensional generalized stadium billiard. Phys. Rev. E 61, 4626–4628 (2000) Simanyi, N., Szasz, D.: Nonintegrability of cylindric billiards and transitive lie group actions. Erg. Th. Dyn. Sys 20, 593–610 (2000) Tabachnikov, S.: Billiards. Soc. Math. France 1995 Wojtkowski, M.P.: Systems of classical interacting particles with nonvanishing Lyapunov exponents. In: Lyapunov Exponents, Proceedings, Oberwolfach 1990, L. Arnold, H. Crauel, J.-P. Eckmann (eds.), Lecture Notes in Math. 1486, Berlin-Heidelberg-New York: Springer, 1991, pp. 243–262 Wojtkowski, M.P.: Monotonicity, J-algebra of Potapov and Lyapunov exponents. Smooth Ergodic Theory and Its Applications, Proc. Symp. Pure Math. 69, Providence, RI: Amer. Math. Soc. (2001) pp. 499–521 Wojtkowski, M.P.: Principles for the design of billiards with nonvanishing lyapunov exponents. Commun. Math. Phys. 105, 391–414 (1986) Wojtkowski, M.P.: Linearly stable orbits in 3-dimensional billiards. Commun. Math. Phys. 129, 319–327 (1990) Wojtkowski, M.P.: Hamiltonian systems with linear potential and elastic constraints. Fundamenta Matematicae 157, 305–341 (1998) Wojtkowski, M.P.: Complete hyperbolicity in hamiltonian systems with linear potential and elastic collisions. Rep. Math. Phys. 44, 301–312 (1999)
[H] [K1] [K2] [K-S] [K-T] [L] [M] [P] [S-S] [T] [W1]
[W2] [W3] [W4] [W5] [W6]
Communicated by G. Gallavotti
Commun. Math. Phys. 273, 305–315 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0231-5
Communications in
Mathematical Physics
Counting Regions with Bounded Surface Area P. N. Balister, B. Bollobás Department of Mathematical Science, University of Memphis, Memphis, TN 38152-3240, USA. E-mail: [email protected] Received: 2 March 2006 / Accepted: 28 July 2006 Published online: 12 May 2007 – © Springer-Verlag 2007
Abstract: Define a cubical complex to be a collection of integer-aligned unit cubes in d dimensions. Lebowitz and Mazel (1998) proved that there are between (C1 d)n/2d and (C2 d)64n/d complexes containing a fixed cube with connected boundary of (d − 1)-volume n. In this paper we narrow these bounds to between (C3 d)n/d and (C4 d)2n/d . We also show that there are n n/(2d(d−1))+o(1) connected complexes containing a fixed cube with (not necessarily connected) boundary of volume n. 1. Introduction Define an r -cube C to be an r -dimensional unit cube in Rd with vertices in Zd . In other words, a set of the following form: / I, ai ≤ xi ≤ ai + 1 for i ∈ I }, C = C(a, I ) = {x ∈ Rd : xi = ai for i ∈ where a = (a1 , . . . , ad ) ∈ Zd and I is a subset of {1, . . . , d} of size r . Define an r -dimensional cubical complex (or r -complex) B to be a finite union of r -cubes in Rd . We shall call a complex rooted if it contains the cube Cr = C(0, {1, . . . , r }). Define the volume |B| of B to be the number of r -cubes in B. We shall define the boundary ∂C of a cube C to be the (r − 1)-complex which is the union of the r pairs of faces C((a1 , . . . , ai , . . . , ad ), I \ {i}) and C((a1 , . . . , ai + 1, . . . , ad ), I \ {i}) for i ∈ I . We shall also define the boundary ∂ B of the complex n B = i=1 Ci to be the (r − 1)-complex which contains each (r − 1)-cube that occurs in an odd number of boundaries ∂Ci . (We shall avoid issues of orientation in this paper.) We say B is closed if ∂ B = ∅. Define the surface area of B to be the volume of the boundary |∂ B|. We say that an r -complex B is connected if it is connected via its (r − 1)-dimensional faces. More formally, let G be the graph with vertices equal to the component r -cubes of B and two vertices joined by an edge when these cubes share a common (r − 1)-dimensional face. Then B is connected precisely when G is connected.
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Fig. 1. Examples of contours in 2 dimensions
The number of d-dimensional cubical complexes with a given volume or surface area is interesting in its own right; however it also has applications to the Ising model in d dimensions, where the convergence of the low temperature expansion is dependent on the number of Peierls contours, i.e., the number of connected boundaries of rooted cubical complexes (see Lebowitz and Mazel [3]). Following the notation of [3], we define a contour to be the boundary of some rooted d-complex, provided that this boundary is itself a connected (d −1)-complex. A contour is primitive if it is minimal, i.e., it is not a disjoint union of two non-empty contours. Note that, in general, if ∂ B is a contour, then the cubes of B need only be connected via (d − 2)-dimensional cubes. On the other hand, if we insist that B is itself connected, it does not follow that ∂ B is a contour, since ∂ B may not be connected, and even if it is, it is not necessarily primitive (see Fig. 1 for some examples with d = 2). However, if ∂ B is primitive then B must be connected (since ∂ B is the disjoint union of the boundaries of the components of B). Let Bd be the set of rooted d-complexes in Rd with primitive boundaries, Bd the rooted d-complexes (possibly disconnected) with connected boundary, and Bd the connected rooted d-complexes (possibly with disconnected boundary, see Fig. 1). Write Sd (n) (respectively Sd (n), Sd (n)) for the number of elements of Bd (respectively Bd , Bd ) with surface area n. Write Vd (n) for the number of connected rooted d-complexes with volume n. Note that all these quantities are finite. In this note we shall give upper and lower bounds for all of these quantities. 2. Preliminary Results For two r -complexes, B1 and B2 , define B1 ⊕ B2 to be the complex formed from all r -cubes that are in either B1 or B2 but not both. Note that ∂(B1 ⊕ B2 ) = ∂ B1 ⊕ ∂ B2 . Also, for each complex B and for 1 ≤ i ≤ d, define Bi= to be the subcomplex of all r -cubes of B that have zero extent in dimension i, i.e., that are contained in some hyperplane xi = c. Define Bi⊥ to be the subcomplex consisting of all the r -cubes of B which have positive extent in dimension i, so that B = Bi⊥ ⊕ Bi= . The cubes in B ⊥ will be called vertical cubes, and the cubes in B = will be called horizontal cubes. The following slightly technical lemma will be useful. Lemma 1. Assume B = Bd⊥ and ∂ B ⊆ Rd−1 , where Rd−1 is identified with the hyperplane xd = 0 in Rd . Then B = ∅. Proof. Assume B = ∅ and let a ∈ Z be the maximum integer such that B meets the hyperplane xd = a. Then B contains some r -cube C × [a − 1, a] with C ⊆ Rd−1 . The face C × {a} of this r -cube is a face of precisely two r -cubes in Rd with positive extent
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in dimension d. One of these is C × [a − 1, a], the other is C × [a, a + 1]. Only the first of these is in B, so C × {a} ⊆ ∂ B ⊆ Rd−1 . Hence a = 0 and B ⊆ Rd−1 × (−∞, 0]. A similar argument holds for the minimal a and shows that B ⊆ Rd−1 × [0, ∞). Thus B ⊆ Rd−1 × {0}, contradicting the assumption that every cube of B has positive extent in dimension d.
Lemma 2. An r -complex B is closed if and only if B = ∂ B for some B . Proof. Since each (r − 2)-dimensional subcube of an r -cube C is contained in precisely two faces of C, ∂∂C = ∅. Hence ∂ B is closed for all B. We now prove the converse. For each cube C × {a} in Bd= with a = 0, construct the stack C × [0, a] (or C × [a, 0] if a < 0). The ⊕-sum of all these stacks is a complex E with (∂ E)= d agreeing with Bd= outside Rd−1 . Let F = B ⊕ ∂ E. Then Fd= ⊆ Rd−1 . Now F is closed so ∂(Fd⊥ ) = ∂(Fd= ) ⊆ Rd−1 . Hence, by Lemma 1, Fd⊥ = ∅ and F = Fd= ⊆ Rd−1 is a closed complex in Rd−1 . By induction on d it is equal to ∂ F for some F . Now B = ∂(E ⊕ F ) as required.
= Lemma 3. If B and B are two d-complexes and (∂ B)= d = (∂ B )d , then B = B . ⊥ d−1 , Proof. Let E = B ⊕ B . Then (∂ E)= d = ∅, so ∂ E = (∂ E)d . Also ∂∂ E = ∅ ⊆ R ⊥ so by Lemma 1, ∂ E = ∅. But E d = E since every d-cube has positive extent in dimension d. Hence by Lemma 1 again, E = ∅, and so B = B .
Lemma 3 implies that the boundary ∂ B of a d-complex determines the complex B. Hence counting contours is equivalent to counting elements of B , while counting primitive contours is equivalent to counting elements of B. Following [3], we construct a floor-stack (multi-)graph G of the boundary B of a d-complex as follows. Decompose Bd= as a union of connected components or floors Fi . Decompose Bd⊥ into a union of complexes of the form E j = C × [a, b], where C is an (d − 2)-cube in Rd−1 and a, b ∈ Z with b − a maximal. In other words, we group together the component cubes of Bd⊥ as maximal stacks of cubes in the d th dimension. Since ∂ B = ∅, C × {a} and C × {b} must lie in ∂(Bd= ) = ∂(Bd⊥ ), and hence in some ∂ Fi and ∂ Fi . In other words, E j joins Fi and Fi . Let the vertices of G be the floors Fi and join Fi and Fi whenever there is a stack E j joining Fi and Fi . Lemma 4. If B ∈ Bd then the floor-stack graph of ∂ B is a connected graph. Proof. Let E be the union of the floors Fi and stacks E j in one component of the graph G. Let C be a horizontal (d − 2)-cube of ∂ E. Now C lies in the boundary of four (d − 1)-cubes, two horizontal, and two vertical. If C lies in ∂ E j , then precisely one of these vertical (d −1)-cubes lies in ∂ B. But then one of the horizontal (d −1)-cubes must also lie in ∂ B, since otherwise C would lie in ∂∂ B = ∅. Thus C lies in the boundary of some Fi . But this Fi is then an endvertex of E j , so also lies in the chosen component of G. But then C ∈ / ∂ E, a contradiction. Similarly, if C lies in the boundary of an Fi , then it lies at the end of a stack E j in the same component of G, once again leading to ⊥ d−1 , a contradiction. Hence (∂ E)= d = ∅, and thus ∂ E = (∂ E)d . Since ∂∂ E = ∅ ⊆ R Lemma 1 implies ∂ E = ∅. Thus E is a contour that is contained in ∂ B. Since ∂ B is primitive, E = ∂ B and so G is connected.
Note that the floor-stack graph may be disconnected for non-primitive contours. See Fig. 2 for an example in 3 dimensions.
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Fig. 2. Example of contour with disconnected floor-stack graph
3. Bounds for Vd (n) We start with the easiest quantity to estimate, namely Vd (n), since this illustrates some of the techniques that we shall use for the other quantities. Theorem 5. For all n ≥ 1, d n−1 ≤ Vd (n) ≤
n 1 2d−1 (2ed) .
Proof. The cube Cd is connected to 2d other d-cubes. For each of these choose an affine transformation that maps Cd onto this d-cube. We can construct any connected rooted d-complex by gluing smaller complexes onto Cd at some or all of the adjacent d-cubes via their root cubes using the affine transformations defined above. If we define the polynomial f L (X ) inductively by f 0 (X ) = X and f L+1 (X ) = X (1 + f L (X ))2d , then the coefficient an,L of X n in f L is an upper bound on the number of complexes of volume n that can be constructed by the above process in at most L steps (i.e., complexes for which every cube is within graph-distance L of the root cube). As L increases an,L increases, and for L ≥n, an,L is constant, say an,L = an . Thus f L (X ) increases monotonically to f (X ) = an X n provided X is within the radius of convergence of this limiting series. Hence the number of these of volume nn is bounded above by the coefficient an in the generating function f (X ) = ∞ n=1 an X , where f (X ) satisfies the equation f (X ) = X (1 + f (X ))2d . (1) Rewrite Eq. (1) as X = f (1 + f )−2d and maximize X . If the maximal X = X c occurs at f = f c then one sees inductively that f L (X c ) ≤ f c for all L. Hence the generating function f (X ) converges for all X ≤ X c . By logarithmic differentiation, 1 2d 1 dX = − X df f 1+ f so at f = f c ,
1 fc
=
2d 1+ f c .
Thus f c =
1 2d−1
and
X c = (2d − 1)2d−1 (2d)−2d = (1 +
−(2d−1) 1 (2d)−1 2d−1 )
Therefore Vd (n) ≤
n i=1
ai ≤ f (X c )X c−n ≤
(2ed)n . 2d − 1
≥ (2ed)−1 .
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For the lower bound, note that for each sequence (d2 , . . . , dn ) with di ∈ {1, 2, . . . , d} we can construct a complex by taking a sequence of d-cubes with the i th cube located one step in the positive di th direction from the (i − 1)st cube. This gives d n−1 distinct connected complexes.
4. Bounds for Sd (n) We start with an upper bound for the number of primitive contours with given (d − 1)dimensional volume. Note that this volume is always even since the surface area of each cube is even. Theorem 6. For all d ≥ 2 and even n ≥ 2d, Sd (n) ≤
n (8e2 d 2 )n/d 8d 3
≤ (8d)2n/d .
Proof. Let B = ∂ B be a primitive contour. Then, by Lemma 4, the floor-stack graph G of B is connected. Fix a spanning tree of G. Then we can reconstruct the floors by specifying each floor as a rooted (d − 1)-complex together with connecting stacks. We can obtain an upper bound for the number of primitive contours containing the cube Cd−1 by alternately growing floors and stacks. We shall define a generating function g(X, Y ) = ar,s X r Y s , where ar,s will bound the number of possible spanning trees with total stack size s and total floor volume r . We define g by g(X, Y ) = X (1 + κg(X, Y ))2(d−1) ,
(2)
where κ=
4Y 1−Y
+ 1.
(3)
To see that this gives an upper bound, consider growing a complex starting with Cd−1 . For each of the 2(d − 1) faces of Cd−1 we can either attach nothing, attach the neighboring horizontal (d − 1)-cube (extending the current floor), or attach a stack, together with a horizontal (d − 1)-cube at the other end of the stack. Note that we can never attach two stacks (since together they would form a single stack) or a stack and a horizontal cube (since then the stack would not lie in the boundary of the floor). In the cases when we attach cubes, we continue building the complex from the new horizontal (d − 1)cube. If we attach a stack, then the stack can go in one of two directions (up or down) and the horizonal (d − 1)-cube at the other end of the stack can be attached in one of two positions. The stack can be any positive integral length. Hence we get a factor of 4(Y + Y 2 + Y 3 + . . . ) = 4Y/(1 − Y ). Adding one to include the possibility of extending the floor, we get a factor of κg for each face of Cd−1 that we add something to. If we define g0 (X, Y ) = 0 and g L+1 = X (1 + κg L )2(d−1) then g L is a polynomial in X with each coefficient a polynomial in Y divided by some power of 1 − Y . If 0 < Y < 1 then the coefficients increase and stabilize at the corresponding coefficients of g. Hence, as before, g converges provided X ≤ X c , where X c is the maximum value of g/(1 + κg)2(d−1) . This maximum occurs at g = gc = 1/((2d − 3)κ) with X c = gc (1 + κgc )−2(d−1) = κ −1 (2d − 3)2d−3 (2d − 2)−(2d−2) ≥ (2(d − 1)κe)−1 . (4) Next, we bound the number of contours containing a fixed vertical (d − 1)-cube as the root. In this case, we grow the spanning tree of G starting with a stack. Since the root may
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lie in the middle of a stack, and there are floors at each end of the stack, the generating function for these is bounded by g(X, ˜ Y ) = (Y + 2Y 2 + 3Y 3 + . . . )(2g(X, Y ))2 =
4Y g(X, Y )2 . (1 − Y )2
The term kY k (2g(X, Y ))2 comes from choosing the stack of length k (with k possible choices for the root). We then grow the contour starting with two floors, each of which starts in one of two directions. Let h(X ) be the generating function for the number of primitive contours containing the cube Cd−1 . Fix such a contour B. Then B contributes a term of the form X r Y s to g(X, Y ), where r = |Bd= | and s is the sum of the stack lengths, in particular s ≤ n = |B|. Indeed, B contributes many such terms, one for each spanning tree of the floor-stack graph. Now consider the above construction, except that instead of taking dimension d as vertical, dimension i as vertical for each i = 1, . . . , d. Then B contributes at d take least i=1 X ri Y si to the generating function g(X, Y ) + (d − 1)g(X, ˜ Y ). (The root Cd−1 is vertical in (d − 1) dimensions and horizontal in only one.) Since ri = |Bi= |, ri = n and si ≤ n. Thus the AM-GM inequality and the fact that Y < 1 gives d
X ri Y si ≥ d X n/d Y
si /d
≥ d X n/d Y n .
i=1
Hence, for any 0 < Y < 1 and 0 < X < X c = X c (Y ) we have g(X, Y ) + (d − 1)g(X, ˜ Y ) ≥ d h(X 1/d Y ). We wish to maximize X 1/d Y subject to remaining inside the domain of convergence of d 1 g and g. ˜ A reasonably good choice is Y = Y0 = d+1 and X 0 = 8ed 2 . Then κ = 4d + 1, −1 X c ≥ (2e(d − 1)(4d + 1)) ≥ X 0 , and X 0 Y0d ≥
1 8ed 2 (1 + 1/d)d
≥ X1 =
1 . 8e2 d 2
Now (1 +
κ 2(d−1) ) 8d 2
2
≤ e2(4d+1)(d−1)/8d ≤ e,
so 1 (1 + 8dκ 2 )−2(d−1) 8d 2
≥
1 8ed 2
= X0.
Hence g0 = g(X 0 , Y0 ) ≤ 8d1 2 . Also g˜ 0 = g(X ˜ 0 , Y0 ) = 4d(d + 1)g02 ≤ number of primitive contours of size n containing Cd−1 is at most 1/d
−n/d
h(X 1 )X 1
−n/d
≤ d1 (g0 + (d − 1)g˜ 0 )X 1
≤
1 . 8d 2
Thus the
1 (8e2 d 2 )n/d . 8d 2
Each contour surrounding Cd must contain a vertical translate of Cd−1 at a vertical distance less than n/(2(d − 1)) ≤ n/d below the hyperplane xd = 0. Thus Sd (n) ≤
n (8e2 d 2 )n/d . 8d 3
Counting Regions with Bounded Surface Area
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Finally for d ≥ 2, n 8d 3
3
≤ en/8d ≤ e0.04n/d ,
so n (8e2 d 2 )n/d 8d 3
≤ (8e2.04 d 2 )n/d ≤ (8d)2n/d .
Now, we turn to a lower bound on Sd (n). Theorem 7. For all d ≥ 2 and all even n ≥ 4d 2 we have n−4d 2
Sd (n) ≥ d 2(d−1) ≥ (Cd)n/2d . Proof. Let us use the procedure defined in Theorem 5 that for each sequence (d2 , . . . , dk+1 ) with di ∈ {1, 2, . . . , d} builds a complex by taking a sequence of d-cubes with the i th cube located one step in the positive dith direction from the (i − 1)st cube. This gives d k distinct connected complexes with surface area 2(k + 1)(d − 1) + 2. To get an arbitrary even surface area, add a 2 × j × 1 × · · · × 1 block in one of the negative directions. This increases the surface area by j (4d − 6) + 4 − 2 (the −2 is due to the loss of the joining face). Thus we obtain a complex with surface area 2(k + 1 + 2 j)(d − 1) + 4 − 2 j. If we choose j so that 4 − 2 j ≡ n mod 2(d − 1) then one can solve n = 2(k + 1 + 2 j)(d − 1) + 4 − 2 j for k. We can choose j so that 1 ≤ j ≤ d − 1, so n ≤ 2k(d − 1) + 4(d − 1)2 + 4 ≤ 2k(d − 1) + 4d 2 . The result follows.
Combining the upper and lower bounds we see that (C1 d)n/2d ≤ Sd (n) ≤ (C2 d)2n/d for sufficiently large even n. 5. Bounds for Sd (n) Now we extend the results of the previous section to count all contours, rather than just primitive ones. Theorem 8. For all d ≥ 2 and large even n, Sd (n) ≤
n (8e17/8 d 2 )n/d 8d 3
≤ (9d)2n/d .
Proof. As in Theorem 6, let h(X ) be the generating function for the number of primitive contours of volume n containing Cd−1 . Fix a primitive contour containing n (d − 1)cubes. Then there are a total of (at most) (d − 1)n common (d − 2)-cubes, since each (d − 2) cube occurs as the face of (at least) two (d − 1)-cubes, and each (d − 1)-cube has 2(d − 1) faces. An arbitrary contour can be obtained by attaching a contour to some of the (d −2)-cube boundaries of the component cubes of some primitive contour. The way in which this attachment is done is essentially unique, since there are only two possible other (d −1)-cubes that meet this (d −2)-cube, and both must be in the attached contour. By a suitable ordering of all (d − 1)-cubes in Rd , we can fix one of these as the root of
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the added contour. Thus, the number of contours with volume n is bounded above by the coefficient of X n in f (X ) = h(X (1 + f (X ))d−1 ). From the previous section we know that h((8e2 d 2 )−1/d ) ≤ (8e2 d 2 )−1/d /(1 +
1 d−1 ) 8d 2
= (8e2 d 2 (1 +
1 . 8d 2
1 d(d−1) −1/d ) ) 8d 2
Since
≥ (8e17/8 d 2 )−1/d ,
if we set X 2 = (8e17/8 d 2 )−1/d then f (X 2 ) converges and f (X 2 ) ≤ 8d1 2 . Thus the number of contours of surface area (at most) n containing Cd−1 is bounded by f (X 2 )X 2−n ≤
1 (8e17/8 d 2 )n/d . 8d 2
As before, any contour surrounding Cd must contain one of n/d translates of Cd−1 , so Sd (n) ≤ Finally for d ≥ 2,
n 8d 3
≤ e0.04n/d so
n (8e17/8 d 2 )n/d . 8d 3 n (8e17/8 d 2 )n/d 8d 3
≤ (8e2.165 d 2 )n/d < (9d)2n/d .
Our lower bound on Sd (n) is even easier to prove. Theorem 9. For all d ≥ 2 and even n ≥ 2d 2 we have Sd (n) ≥
d (n−d 2 )/2d 2
≥ (Cd)n/d .
Proof. We can write n = 2dk + 2(d − j) for some k > j, 0 ≤ j ≤ d − 1. Fix a sequence (d1 , . . . , d j ) with di ∈ {1, . . . , d} and a sequence ( p j+1 , . . . , pk ), where each pi is an unordered pair (di,1 , di,2 ), di,s ∈ {1, . . . , d}. Construct a complex by starting with the root Cd and adding cubes so that the (i + 1)st cube is located one step in the positive dith direction from the i th cube when i ≤ j and is one step in both the positive di,1 and di,2 directions when i > j. The boundary of this complex is a contour surrounding Cd with k− j surface area 2d(k + 1) − 2 j = n. There are d j d2 such contours. Thus Sd (n) ≥ d j
d k− j 2
≥
d k− j/2 2
≥
d (n−d 2 )/2d 2
.
Combining the upper and lower bounds we find that (C1 d)n/d ≤ Sd (n) ≤ (C2 d)2n/d for sufficiently large even n. Note that these bounds are ‘closer’ than for Sd (n) since the lower bound is much larger.
Counting Regions with Bounded Surface Area
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6. Bounds for Sd (n) It turns out that this quantity is much larger than Sd (n) or Sd (n), i.e., there are many more connected rooted complexes with a given surface area than there are contours. Theorem 10. For fixed d ≥ 2 and all sufficiently large even n, n
Sd (n) = n 2d(d−1) (1+o(1)) . Proof. For a lower bound, consider the large cube [0, N + 2]d consisting of (N + 2)d d-cubes. Remove k of the N d central cubes which are of the form C((a1 , . . . , ad )) with d d i=1 ai ≡ s mod 3. For some choice of s ∈ {0, 1, 2} there will be at least N /3 choices N d /3 for these cubes, and hence at least k ≥ (N d /3ek)k possible resulting d-complexes. The restrictions on the cubes that are removed ensures that the resulting complex will always be connected, and will have surface area exactly 2d(N + 2)d−1 + 2dk. Assuming N > 2d, we can add cubes of the form C((−1, i, 0, 0, . . . , 0), i = 2, 4, . . . 2 j, increasing the surface area by (2d − 2) j. Assume n is large and even. In particular, assume n ≥ 2d(N + 2)d−1 + 2d 2 . One can choose j and k so that 2dk + (2d − 2) j = n − 2d(N + 2)d−1 . To do this, first choose j so that 2 j ≡ −n mod 2d, 0 ≤ j < d. Then solve for k = k(N ). Finally, choose N so that n ≈ 2d(log N )N d−1 . Then k ≈ (log N − 1)N d−1 n and (N d /3ek)k = n 2d(d−1) (1+o(1)) . This shows that Sd (n) is at least as large as claimed. It is somewhat harder to prove a good upper bound. Note that Rd \ B has precisely one infinite component, and the boundary of this component is a contour ∂ B with B ⊆ B . Thus ∂ B is obtained by fixing a contour ∂ B and then adding some contours inside B . Given n 0 = |∂ B | ≤ n, there are at most (9d)2n 0 /d choices for B , and each such B has volume at most v = (n 0 /2d)d/(d−1) with equality if B is a large cube. We shall add choices k contours, each of which must surround some cube of B . There are v+k−1 k for these root cubes (we allow a root cube to be chosen more than once). The added contours will be of sizesn 1 , . . . , n k , where n 1 + · · · + n k = n − n 0 . Since each n i ≥ 2d, there are n−n 0 −2dk+k−1 choices for the n i , i > 0 (we need to partition the ‘excess’ k−1 r = n − n 0 − 2dk as the sum of k numbers). Now each contour can be chosen in at most (9d)2n i /d ways. Thus n−n 0 −2dk+k−1 Sd (n) ≤ . (9d)2(n 0 +n 1 +···+n k )/d v+k−1 k k−1 k,n 0 :2dk+n 0 ≤n
Now (9d)2(n 0 +···+n k )/d = (9d)2n/d = n o(n) , n 2 = n o(n) choices for (k, n 0 ). Hence Sd (n) ≤
k,n 0 :2dk+n 0 ≤n
m
v+k n−n 0 −2dk+k o(n) n ≤ k k
r
≤ (m/r )r , and there are at most max
2dk+n 0 ≤n
(v+k)(n−n 0 −2dk+k) k2
k
n o(n) .
We bound v = (n 0 /2d)d/(d−1) ≤ n 1/(d−1) n 0 , and n 0 ≤ n − 2dk. Thus v + k ≤ n 1/(d−1) (n − 2dk) + k ≤ n 1/(d−1) (n − 2dk + k) and so (v + k)(n − n 0 − 2dk + k) ≤ n 1/(d−1) (n − 2dk + k)2 .
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This shows that Sd (n) ≤ max
2dk≤n
n 1/(d−1) (n−2dk+k)2 k2
k
n o(n) .
Now we maximize over k. Taking logarithms, we need to maximize 2dk + k) − 2k log k. Differentiating with respect to k gives log n d−1
+ 2 log n−2dk+k − k
2(2d−1)k n−2dk+k
k log n d−1
+ 2k log(n −
− 2.
But k ≤ n − 2dk + k, so for sufficiently large n this is always positive. Hence the maximum is attained for the maximum possible k = n/2d. Substituting this we get Sd (n) ≤ n k/(d−1)+o(n) = n n/(2d(d−1))+o(n) .
7. Polymer Expansion for the Ising Model One application of our bounds is to the convergence of the low temperature expansion of the d-dimensional Ising model in terms of Peierls contours (see [3]). The general result of Kotecký and Preiss [2] (see also Dobrushin [1] and Scott and Sokal [4]) about the convergence of cluster expansion implies the following assertion (see also Lemma 2.1 of [3]). Lemma 11. The polymer expansion constructed for the Ising model in terms of Peierls contours is convergent at inverse temperature β if there exists a positive function a(γ ) such that, for any contour γ , e−β|γ |+a(γ ) ≤ a(γ ), γ
where the sum is taken over all contours γ that intersect γ . Using our results on Sd (n) we can improve considerably the Lebowitz-Mazel bound on β implying the convergence of the polymer expansion. Theorem 12. The polymer expansion constructed for the Ising model in terms of Peierls contours converges at inverse temperature β for all β ≥ d2 log(11d). Proof. Each γ that intersects γ must have some common (d − 2)-cube with γ . Fixing this (d − 2)-cube C, γ is forced to contain at least one of the four (d − 1)-cubes meeting C. Since there are (at most) (d − 1)|γ | (d − 2)-cubes in γ , it is enough to show e(α−β)|γ | ≤ α, 4(d − 1) γ
where we have chosen a(γ ) = α|γ | and the sum is over all contours containing a fixed (d − 1)-cube. In other words, we need to show that 4(d − 1)
∞ n=1
cn e(α−β)n ≤ α,
Counting Regions with Bounded Surface Area
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where cn is the number ofrooted contours with surface area n. If eα−β ≤ X 2 , then from (α−β)n ≤ 1 . Thus we can take α = 1 , provided β the proof of Theorem 8, ∞ n=1 cn e 2d 8d 2 is at least α − log X 2 =
1 2d
+
1 d
log(8e17/8 d 2 ) =
1 d
log(8e21/8 d 2 ) ≤
2 d
log(11d).
Acknowledgements. We are grateful to Roman Kotecký for drawing our attention to the problems discussed in this note.
References 1. Dobrushin, R.L.: Estimates of semi-invariants for the Ising model at low temperatures. In: edited by Topics in Theoretical and Statistical Physics, R.L. Dobrushin, R.A. Minlos, M.A. Shubin and A.M. Vershik, Providence, RI: Amer. Math. Soc., 1996, pp. 59–81 2. Kotecký, R., Preiss, D.: Cluster expansion for abstract polymer models. Commun. Math. Phys. 103, 491– 498 (1986) 3. Lebowitz, J.L., Mazel, A.E.: Improved Peierls argument for high-dimensional Ising models. J. Stat. Phys. 90, 1051–1059 (1998) 4. Scott, A.D., Sokal, A.D.: The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma. J. Stat. Phys. 118, 1151–1261 (2005) Communicated by J.L. Lebowitz
Commun. Math. Phys. 273, 317–355 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0232-4
Communications in
Mathematical Physics
Large N Expansion of q -Deformed Two-Dimensional Yang-Mills Theory and Hecke Algebras Sebastian de Haro1 , Sanjaye Ramgoolam2 , Alessandro Torrielli3 1 Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, 14476 Golm, Germany.
E-mail: [email protected]; [email protected]
2 Department of Physics, Queen Mary, University of London, Mile End Road, London E1 4NS, UK.
E-mail: [email protected]
3 Institut für Physik, Humboldt Universität zu Berlin, Newtonstraße 15, D-12489 Berlin, Germany.
E-mail: [email protected] Received: 21 March 2006 / Accepted: 13 December 2006 Published online: 17 May 2007 – © Springer-Verlag 2007
Abstract: We derive the q-deformation of the chiral Gross-Taylor holomorphic string large N expansion of two dimensional SU (N ) Yang-Mills theory. Delta functions on symmetric group algebras are replaced by the corresponding objects (canonical trace functions) for Hecke algebras. The role of the Schur-Weyl duality between unitary groups and symmetric groups is now played by q-deformed Schur-Weyl duality of quantum groups. The appearance of Euler characters of configuration spaces of Riemann surfaces in the expansion persists. We discuss the geometrical meaning of these formulae.
Contents 1. Introduction and Summary of the Results . . . . . . . . . . . . . 2. Hecke Algebras and the Chiral Expansion of q-Deformed 2dYM 2.1 Review of the Gross-Taylor expansion . . . . . . . . . . . . 2.2 Hecke algebras and Schur-Weyl duality . . . . . . . . . . . 2.3 A Hecke formula for the q-dimension . . . . . . . . . . . . 2.4 Hecke q-generalization of sums over symmetric groups of 2d Yang Mills . . . . . . . . . . . . . . . . . . . . . . . . 3. Manifolds with Boundary . . . . . . . . . . . . . . . . . . . . . 4. Chiral Large N Expansion for Wilson Loops . . . . . . . . . . . 5. On the Role of Quantum Characters in q-Deformed 2d YM . . . 5.1 Consistency of Wilson loops . . . . . . . . . . . . . . . . . 5.2 Gauge invariance of Wilson loops . . . . . . . . . . . . . . 6. Discussion and Outlook . . . . . . . . . . . . . . . . . . . . . . A. Central Elements . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Centrality of q-deformed conjugation sum . . . . . . . . . . A.2 Centrality of q-deformed commutator sum . . . . . . . . . A.3 The elements D and E of Hn (q) . . . . . . . . . . . . . . .
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B. Quantum Dimensions . . . . . . . . . . . . . . . C. Projectors . . . . . . . . . . . . . . . . . . . . . C.1 H3 . . . . . . . . . . . . . . . . . . . . . . . C.2 H4 . . . . . . . . . . . . . . . . . . . . . . . C.3 The construction for Hn . . . . . . . . . . . . D. q-Schur-Weyl Duality and q-Characters . . . . . D.1 Uq (su(2)) conventions . . . . . . . . . . . . D.2 Schur-Weyl duality in spin-one . . . . . . . . D.3 Quantum characters in spin-one representation References . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction and Summary of the Results Two-dimensional Yang-Mills theory, on a Riemann surface of genus G and of area A, can be solved exactly. The partition function is 2 Z YM (G, A) = (dim(R))2−2G e−gYM A C2 (R) . (1.1) R
This result was first obtained using the lattice formulation, followed by a continuum limit [1]. The sum is over all irreducible representations of the gauge group, the cases U (N ) or SU (N ) will be of interest here. Gross and Taylor [2–4] studied the large N expansion of two-dimensional Yang-Mills theory with gauge group U (N ) and SU (N ) and showed that it is equivalent to a string theory. They showed that the large N expansion is given by a non-chiral expansion, which is a sum involving chiral and anti-chiral factors. The chiral expansion of (1.1)1 is given by G ∞ 1 1 + δ 2−2G Z YM (G) = si ti si−1 ti−1 . (1.2) n n! N (2G−2)n n=0
si ,ti ∈Sn
i=1
It is a sum consisting of delta functions over symmetric groups, which count homomorphisms from the fundamental group of punctured Riemann surfaces to the symmetric groups. These homomorphisms are known to count branched covers of G . It was shown in [5, 6] that the chiral sum actually computes an Euler character of moduli spaces of holomorphic maps with fixed target space. This was done by expanding the factors, and recognising that the coefficients in the expansion are Euler characters of configuration spaces of (branch) points on G . Topological string theory constructions were then used to derive a path integral which localizes to an integral of the Euler class on the moduli space of holomorphic maps. For simplicity we are discussing only the chiral part of the partition function here, but there is an analogous expansion for the full partition function. A different string action involving harmonic maps was proposed in [7]. Two-dimensional Yang-Mills has recently found a surprising new application in connection with topological strings on a non-compact Calabi-Yau and black hole entropy [8]. The q-deformation of two-dimensional Yang-Mills has also found an application in this context [9, 10]. The partition function of q-deformed Yang-Mills has been obtained by replacing the scalar dual field of the Yang-Mills field strength by a compact scalar. 1 In this paper we work at zero area. The computations can be generalized to the case of finite area along the lines of [3, 4].
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Such a compact scalar is natural from the point of view of the worldvolume of D4-branes wrapping a 4-cycle of the non-compact Calabi-Yau. New connections with Turaev invariants have also been suggested [11]. The q-deformation of two-dimensional Yang-Mills theory was studied earlier [12, 13] (see also [14]). The q-deformation of the zero area partition function of two-dimensional Yang-Mills is Z q YM (G) = (dimq R)2−2G . (1.3) R
In the context of [10] this is the limit where the degree p of one of the line bundles is zero. In the q-deformed Yang-Mills, the universal enveloping algebra of U (N ) is replaced by Uq (u(N )). The exact partition function for a closed Riemann surface, which is expressed in terms of dimensions of irreducible representations of U (N ), is now expressed in terms of q-dimensions of Uq (u(N )) representations. The same remarks apply to Uq (su(N )). The underlying algebraic relation which leads to the relation between the sum over U (N ) representations in (1.1) and the delta functions over symmetric groups in (1.2) is Schur-Weyl duality, which we describe further in Sect. 2. The q-deformation of the Schur-Weyl duality between U (N ) and Sn is known [15]. In this q-deformation, the role of the group algebra of Sn ( denoted by CSn ) is played by the Hecke algebra Hn (q). In this paper, we show that the large N , chiral Gross-Taylor expansion, in terms of symmetric group data can be q-deformed to give an expansion in terms of Hecke algebra data. In this case we find the following result: Z q YM (G) =
∞ 1 [N ](2−2G)n δ g n=0 si ti ∈Sn G −1 −1 2−2G −l(si )−l(ti ) q h(si )h(ti )h(si )h(ti ) . × D n
(1.4)
i=1
Here, h(s) ∈ Hn is the Hecke algebra element associated to s ∈ Sn . That such an expansion is possible at all in the quantum case is highly non-trivial and very much suggestive of a geometric interpretation in terms of deformations of maps, on which we comment in Sect. 6. The possibility of the expansion (1.4) depends crucially on the existence of suitable central elements of the Hecke algebra (like D and n , to be defined later). These central elements play an important role in that they also determine the data on manifolds with closed boundary: Z (G ; C1 , . . . , C B ) =
⎛ × δ ⎝ D 1−B 2−2G−B n
[N ](2−2G−B)n
R G i=1
1 g st i i
q −l(si )−l(ti ) h(si )h(ti )h(si−1 )h(ti−1 )
B
⎞ Cj⎠ .
(1.5)
j=1
In this formula, the central elements of the Hecke algebra take over the role of the holonomies of the gauge field around the B boundaries of G . We also work out the case of non-intersecting Wilson loops. We develop an analog of the Verlinde formula for the tensor product multiplicity coefficients of SU (N ) in terms of characters of the Hecke algebra. To our knowledge, this formula has not appeared in
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the literature. Expectation values of Wilson loops can now again be written as Hecke delta functions which are natural deformations of the symmetric group delta functions. In four appendices we give some of the facts and proofs about Hecke algebras that we use in the main text. To our knowledge, some of the formulas proven in these appendices are not available in the mathematical literature before. 2. Hecke Algebras and the Chiral Expansion of q-Deformed 2dYM 2.1. Review of the Gross-Taylor expansion. Before we do the q-deformed case, we will review the main tools used in the derivation of the partition function of 2d Yang-Mills as a topological theory counting branched covers of the Riemann surface. For full details we refer to [5]. For simplicity, we discuss the case of zero-area and no Wilson loops in this section. We start writing out the partition function as a sum over Young tableaux: Z 2dYM (G ; A) =
(dim(R))2−2G =
∞
(dim(R(Y )))2−2G ,
(2.1)
n=0 Y ∈YnN
R
where we sum over the set YnN of SU (N ) Young diagrams with n boxes and number of rows less than N . Of course, we also sum over diagrams with arbitrary number of boxes. The chiral expansion is derived by dropping the constraint on the number of rows. Next we use Schur-Weyl duality to derive the following fomula: dim(R) =
Nn χ R (n ) . n!
(2.2)
We are using a notation where R = R(Y ) denotes both the SU (N ) and the Sn representation corresponding to a Young tableau with n boxes, Y . χ R is a character of the symmetric group, and n is a particular central element in CSn given in [3, 4]. The chiral Gross-Taylor expansion is obtained as Z 2dYM (G ; A) = =
∞
N
(2−2G)n
n=0 R ∞ (2−2G)n
N
n=0
dR n!
2−2G
1 χ R (2−2G ) n dR
G 1 −1 −1 2−2G δ n si ti si ti . n! si ,ti ∈Sn
(2.3)
i=1
The fact that n is a central element in the group algebra CSn is important. This is explained in more detail and generalized to the q-deformed case in Sect. (2.3). Another important identity which enters (2.3) is 2 1 n! −1 −1 χ R (sts t ) = , dR dR
(2.4)
s,t∈Sn
where it is easy to see that s,t sts −1 t −1 is a central element of CSn . We find (2.50), which gives the q-deformation of this equation, and we prove related centrality properties for Hn (q) in Appendix A.
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321
2.2. Hecke algebras and Schur-Weyl duality. There is a natural generalization of the previous formulas using Hecke algebras. In this subsection we review basic facts about Hecke algebras and derive some formulas that we will use in what follows. The symmetric group Sn can be defined in terms of generators si (i = 1, . . . , n − 1), which obey relations si2 = 1 for i = 1, . . . , n − 1, si si+1 si = si+1 si si+1 for i = 1, . . . , n − 2, si s j = s j si for |i − j| ≥ 2.
(2.5)
The minimal length of a word in the si which is equal to a permutation σ is called the length of the permutation and is denoted as l(σ ). The Hecke algebra Hn (q) is defined in terms of generators gi which obey [16] gi2 = (q − 1) gi + q for i = 1, . . . , n − 1, gi gi+1 gi = gi+1 gi gi+1 for i = 1, . . . , n − 2, gi g j = g j gi for |i − j| ≥ 2.
(2.6)
The Hecke algebra has, as a vector space, a basis h(σ ) labelled by the elements σ of Sn . This is often called the “standard basis” in the literature. These elements h(σ ) are obtained by expressing the σ as a minimal length word in the si and then replacing the si by gi . These Hecke algebras arise as the algebra of operators on V ⊗n , the n-fold tensor product of the fundamental representation of U (N ) or SU (N ), which commute with the action of Uq (u(N )) or Uq (su(N )), the q-deformation of the universal enveloping algebra of u(N ) or su(N ), respectively. The action of the q-deformed enveloping algebras on V ⊗ V is given by the co-product . This obeys the following relations with respect to the R-matrix: R = R, (P R) = (P R) .
(2.7)
For h ∈ Uq , if we write (h) = h 1 ⊗ h 2 , then (h) = h 2 ⊗ h 1 . P is the permutation ˇ For the R-matrix we will use the convenoperator. P R is also commonly denoted by R. tions of [17]. To make that explicit, we write RFRT . The Hecke algebra is related to the algebra of the Rˇ FRT as: √ √ (2.8) g = q Rˇ FRT qFRT = q . gi corresponds to Rˇ FRT acting in the tensor product Vi ⊗ Vi+1 and is sometimes called a braid operator. Since the centralizer of Uq is the Hecke algebra, we can construct the projectors for irreducible representations of Uq in terms of words in the gi . g1 acts on the product space V1 ⊗ V2 , therefore there are two possible projectors that we can construct [17]: q −1 ˇ = 1 (1 + g), (1 + q R) −1 q +q 1+q q q −1 ˇ P = (1 − q R) = (1 − q −1 g) , −1 q +q 1+q
P
=
(2.9)
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which project onto the totally symmetric and antisymmetric tensor products of the fundamental representation, respectively. Using (2.6), one easily checks that they satisfy PR2 = PR .
(2.10)
The symmetric projector is illustrated in Appendix D in terms of properties of ClebschGordan coefficients of Uq (su(2)). Projectors are useful to compute characters in a particular representation in terms of lower-dimensional representations. For example, taking the trace of the above, q −1 q + q −1 q Tr U = q + q −1
Tr
U =
(trU )2 + q tr ⊗ tr Rˇ (U ⊗ 1)(1 ⊗ U ) ,
(trU )2 − q −1 tr ⊗ tr Rˇ (U ⊗ 1)(1 ⊗ U ) ,
(2.11)
where the traces on the right-hand side are taken in the fundamental representation, Tr = trV = tr. From now on we will indicate such traces by trn = trV ⊗n = tr⊗. . . ⊗tr. The U ’s in (2.11), which are matrix elements of representations of Uq , generate the dual algebra to Uq denoted by Funq (SU (N )) or Funq (U (N )) (see for example [18, 19, 13] ). Using known facts about Hecke algebras and the q-deformation of the Schur-Weyl duality between U (N ) and Sn , we will now derive the generalization for arbitrary irreducible representations: PR =
d R (q) −l(σ ) q χ R (h(σ −1 )) h(σ ) , g σ
(2.12)
where l(σ ) is the length of the permutation, i.e. the number of elements in the minimal presentation of the permutation as a product of simple transpositions. The character is taken in the Hecke algebra Hn . Without danger of confusion, we will denote Hn and Funq (SU (N )) characters with the same symbol. The characters for low values of n can be read off from the tables in [16, 20]. d R (q) is the q-deformation of the dimension of a representation of the symmetric group, and g reduces to n! in the classical limit: li lj (q − 1)(q 2 − 1) . . . (q n − 1) i j (q − q ) d R (q) = m , m(m−1)(m−2) 2 l i 6 i=1 (q − 1)(q − 1) . . . (q − 1) q g=
(1 − q)(1 − q 2 ) . . . (1 − q n ) , (1 − q)n
(2.13)
where li = λi + m − i and λ1 ≥ λ2 ≥ ..λm ≥ 0 are the row lengths of the Young diagram, and m is the number of non-zero λ’s. In order to derive (2.12), recall the familiar relation in the q = 1 case: χ R (U ) =
1 χ R (σ ) trn (σ U ) . n! σ
(2.14)
Here R is both the U (N ) reprsentation corresponding to a Young diagram and the Sn rep corresponding to the same diagram. The trace on the right-hand side is taken in V ⊗n , that is U acts as U ⊗ U ⊗ ... ⊗ U and σ acts by permuting the vectors of the tensor product.
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323
The above is obtained from the fact that, if V is the fundamental representation of U (N ) or the universal enveloping algebra U (u(N )), then V ⊗n can be decomposed on the terms of the product group U (N ) × Sn as U (N )
V ⊗n = ⊕ R V R
⊗ V RSn .
(2.15)
The sum is over Young diagrams of Sn , V RSn is the irrep of Sn corresponding to the Young U (N ) diagram R, while V R is the irrep of U (N ) corresponding to the same Young diagram. Similar relations hold when U (N ) is replaced by SU (N ). An immediate consequence of the above expansion is tr (σ U ) = χ R (σ ) χ R (U ). (2.16) R
Then we can use orthogonality of characters of Sn , χ R (σ ) χ S (σ −1 ) = n! δ R S ,
(2.17)
σ
to obtain (2.14). From (2.15) it also follows that d R χ R (U ) = trn (PR U ) ,
(2.18)
hence we can read off PR =
dR χ R (σ −1 ) σ . n! σ
(2.19)
The decomposition analogous to (2.15) holds for Uq (u(N )), when CSn is replaced by the Hecke algebra Hn (q) [15]: V ⊗n = ⊕ R V R q ⊗ V RHn . U
(2.20)
U
Here V R q is the irrep of Uq (u(N )) corresponding to the Young diagram R and V RHn is the representation of Hn corresponding to the same Young diagram. It follows from (2.20) that χ R (h(σ )) χ R (U ). (2.21) trn (h(σ ) U ) = R
U lives in the deformed algebra of functions on U (N ) denoted as Funq (U (N )). This can be defined as the dual to Uq (U (N )). For further discussion on the duality see for example [18, 19, 13]. In (2.21) U acts as (U ⊗ 1⊗ 1⊗ · · · )(1⊗ U ⊗ 1 ⊗ · · · )(1⊗ 1⊗ U ⊗ 1⊗ · · · ) · · · (1 ⊗ 1 ⊗ · · · ⊗ 1 ⊗ U ). (2.22) This product of n U ’s is dual to the co-product which defines the action of Uq on V ⊗n . As will be explained in Sect. 5 (see also Appendix D), quantum traces contain the u-element associated to the Hopf algebra Uq (su(N )). We get the quantum trace if we take a trace of the action of u U on the left-hand side of (2.20) to get trn (h(σ ) ρn (u U )) = χ R (h(σ )) χ R (u U ). (2.23) R
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Here ρn (u) = u ⊗n and U acts as above. For the case of diagonal U , the formula (2.21) is used in [20]. Multiplying the left- and right-hand side of (2.21) with q −l(σ ) χ S (h(σ −1 )), and using the orthogonality relation [21] for Hecke characters d R (1) δRS , q −l(σ ) χ R (h(σ )) χ S (h(σ −1 )) = g (2.24) d R (q) σ we get
q −l(σ ) χ R (h(σ −1 )) trn (h(σ )U ) = g
σ
d R (1) χ R (U ) . d R (q)
This means that the character can be expressed as 1 d R (q) −l(σ ) χ R (U ) = q χ R (h(σ −1 )) trn (h(σ )U ) . g d R (1) σ
(2.25)
(2.26)
This equation can be interpreted as giving us the projection on a fixed Young diagram from the sum in (2.15). Indeed, note that (2.20) implies, by projecting on a fixed Young diagram: d R (1) χ R (U ) = trn (PR U ) . Comparing with (2.26) we see that the projector is 1 PR = d R (q) q −l(σ ) χ R (h(σ −1 )) h(σ ) , g σ
(2.27)
(2.28)
as claimed above. In the appendix we check that it satisfies (2.10). If we use orthogonality starting from (2.23) rather than (2.21), then we get 1 d R (q) −l(σ ) (q) χ R (U ) ≡ χ R (u U ) = q χ R (h(σ −1 )) trn (h(σ )(u U )) . (2.29) g d R (1) σ Note that u ⊗n commutes with h(σ ). We will specialize to U = 1 in order to get a new formula for the q-dimension in Sect. (2.3). 2.3. A Hecke formula for the q-dimension . Recall that in the case q = 1 there is a very useful formula for the dimension of SU (N ) reps which follows from Schur-Weyl duality [5]. This formula can be obtained by specializing (2.14) to U = 1. To that end we need to compute the trace of a permutation acting on V ⊗n . If σ = 1, we just get N n . If σ = (12)(3)(4)..(n), we get N n−1 . In general we get one factor of N for each cycle in the permutation. If the permutation has cycles of length i occuring with multiplicity ki the power of N is N ki . In the 2d Yang-Mills literature this is also denoted as N K σ . So the useful formula for the dimension in 2d Yang-Mills [3, 4] is 1 dim(R) = χ R (σ ) N K σ n! σ Nn = χ R (σ ) N −n+ i ki (σ ) . (2.30) n! σ =
Nn χ R (n ). n!
(2.31)
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325
The last line defines the element n . It is convenient to write this as a sum over conjugacy classes. Let T be a conjugacy class, which is given by specification of the cycle decomposition of the permutations involved. We will write C T = σ ∈T σ . Note this is a central element of the group algebra CSn , i.e. it commutes with all the elements of Sn . So the above can be rewritten as 1 dim(R) = χ R (C T ) N i ki (T ) . (2.32) n! T
We can now find the q-generalization of this formula by setting U = 1 in (2.29), to obtain
1 d R (q) −l(σ ) dimq (R) = q χ R h(σ −1 ) trn (h(σ )u) . (2.33) g d R (1) σ ∈Sn
We can manipulate the above sum, using cyclicity of the trace and the Hecke relations, to reduce it to a sum over conjugacy classes T in Sn , with the only terms appearing inside trn being the trn (u h(m T )). m T are permutations in the conjugacy class T which have minimal length when expressed in terms of generators. They are the minimal words in [16]. For n = 3, m T are 1, g1 , g1 g2 for the 3 conjugacy classes. We prove in Appendix B (B.7), trn (u h(m T )) = q
N +1 2
l(T )
[N ]
i ki
,
(2.34)
where the q-number is [N ] =
q N /2 − q −N /2 , q 1/2 − q −1/2
(2.35)
and l(T ) is the length of the permutation m T . We will explain below that the Hecke algebra elements C T appearing as the coefficients of q −l(T ) tr(u h(m T )) are central. Hence the formula for the q-dimension becomes dimq (R) =
N −1 1 d R (q) χ R (C T (q)) [N ] i ki (T ) q 2 l(T ) . g d R (1)
(2.36)
T
Examples of this formula are described in Appendix B, along with checks against the standard formula in terms of a product of q-numbers over the cells of the Young diagram. We now explain the centrality property of C T . Starting from the formula for the projector (2.12) we can express it in a reduced form using cyclicity and Hecke relations, where we only have the characters of the minimal words in each conjugacy class: PR =
1 d R (q) χ R (h(m T )) C T . g
(2.37)
T
Here T runs over conjugacy classes, and m T are the minimal words. For the formulas up to n = 4, see Appendix C. We can get the projector to the form (2.37) because, by using cyclicity of χ R and the Hecke relations, the Hecke characters can be expressed in terms of these basic characters [16]. Now for every R, PR is a central element of the Hecke algebra since it is a projector for the irreducible representation R. There are as many conjugacy classes T as irreducible representations R. Hence C T must be central
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elements. When we calculate the q-dimension we get (2.33). When we manipulate the expression to express it in terms of q −l(T ) trn (u h(m T )), we are using the same Hecke relations and cyclicity (of trn this time): dimq (R) =
1 d R (q) −l(T ) q χ R (C T ) trn (h(m T ) u) . g d R (1)
(2.38)
T
This immediately leads to (2.36). Incidentally, (2.37) seems to give a relatively efficient way of calculating the central class elements compared to the ones we are aware of in the mathematical literature. Some interesting papers with explicit formulae for Hecke central elements, which we found useful, are [22, 23].
2.4. Hecke q-generalization of sums over symmetric groups of 2d Yang Mills. The string theory interpretation of 2d Yang Mills at q = 1 is centred on formulae derived from Schur-Weyl duality. The character relations following from Schur-Weyl give rise to a formula for dimensions of SU (N ) reps in terms of Sn reps. Then some group theory manipulations lead to an expression of the chiral partition function in terms of delta functions over the symmetric group. The delta function is defined over the symmetric group or, more generally, over the group algebra of the symmetric group: δ(σ ) = 1 if σ = 1, δ(σ ) = 0 otherwise .
(2.39)
A useful property of this delta function is that it can be expressed in terms of characters, n! δ(σ ) = d R χ R (σ ) . (2.40) R
The expressions arising in the 2d Yang-Mills string take the form δ (σ1 σ2 · · · σk ) ,
(2.41)
and the weights depend on the genus G and on N in precisely such a way that the chiral partition function can be expressed in terms of a sum of Euler characters of moduli spaces of holomorphic maps (see Sect. 7 of [5]). Now we will describe a q-generalization of this story, where the Hecke algebra will replace the group algebra of the symmetric group. A q-analog of the delta function on the symmetric group is known in the theory of Hecke algebras [21]. It is defined as: δ(h(σ )) = 1 if σ = 1, δ(h(σ )) = 0 otherwise .
(2.42)
Our δ(h(σ )) is g1 tr(h(σ )) in the notation of [21] for the canonical trace function tr(h(σ )). This q-deformed delta function reduces exactly to the delta function on the symmetric group defined above when q → 1. It can be expressed as g δ(h(σ )) = d R (q) χ R (h(σ )) , (2.43) R
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327
where R runs over partitions of n or Young diagrams with n boxes, and g is given in (2.13). An important fact we will use in what follows is that for C a central element of the Hecke algebra, and for arbitrary σ ∈ Sn , χ R (C) χ R (h(σ )) = d R (1) χ R (C h(σ )) ,
(2.44)
which follows simply from Schur’s lemma applied to the Hecke algebra. We now have all the elements we need in order to rewrite the quantum dimensions in terms of central elements of the Hecke algebra. Using (2.36), we can write [N ]n d R (q) χ R (n ) . g d R (1)
dimq (R) =
(2.45)
In the quantum case the ’s are expressed as N −1 [N ] K T −n q 2 l(T ) C T n = T
= 1+ ≡
T 1 + n
[N ] K T −n q
N −1 2
l(T )
CT
,
(2.46)
where the unprimed sum runs over the central elements of Hn . The restricted sum (denoted by the prime) runs over all central elements associated with conjugacy classes of Sn which are not the identity. The last line is a definition of n . Making repeated use of (2.44), we find that for a central element we have:
χ R (C) m χ R (C m ) . (2.47) = d R (1) d R (1) It now follows from (2.45) that (dimq (R))m = = =
[N ]n d R (q) g
m
χ R (m ) d R (1)
m ∞ n [N ] d R (q) d(m, ) ⎛
×⎝
g
=0
⎞
C Ti ⎠ [N ]
d R (1) i
(2.48) χR
K Ti −n
q
N −1 2
i
l(Ti )
,
i=1 Ti
(m+1) , and we wrote out the definition of . where d(m, ) = ( +1) (m− +1) Let us develop the q-deformed chiral Gross-Taylor expansion
Z =
∞ 2−2G dimq (R) n=0 R∈Yn
=
∞ n=0 R∈Yn
[N ]
(2−2G)n
d R (q) g
2−2G
1 χ R 2−2G . d R (1)
(2.49)
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Now we can show (see Appendix A) that
2
1 g χ R h(s)h(t)h(s −1 )h(t −1 ) . = q −l(s)−l(t) d R (q) d R (1)
(2.50)
s,t∈Sn
We also show in the appendix that the element q −l(s)−l(t) h(s)h(t)h(s −1 )h(t −1 )
(2.51)
s,t∈Sn
is central in Hn . Hence we have G
2G 1 g −1 −1 − i (l(si )+l(ti )) χR = q h(si )h(ti )h(si )h(ti ) . d R (q) d R (1) s ,t ···s ,t 1 1
i=1
G G
(2.52) Now we employ this equation in (2.49) to get Z =
∞
q−
i (l(si )+l(ti ))
[N ](2−2G)n
n=0 R∈Yn si ti
× χR
G i=1
=
d R (q) g d R (1)
2
h(si )h(ti )h(si−1 )h(ti−1 )
∞
[N ]
(2−2G)n
n=0 R∈Yn si ti
× χ R 2−2G
G
d R (q) g
2
χ R (2−2G ) q−
i (l(si )+l(ti ))
d R (1)
h(si )h(ti )h(si−1 )h(ti−1 ) ,
(2.53)
i=1
where we sum over Sn permutations s1 , t1 , . . . , sG , tG . At this point the manipulations performed in the classical case do not generalize straightforwardly to the quantum case because of the different powers of d R (q) and d R (1). We need to introduce an element D of the Hecke algebra with the property χ R (D) = d R (q) .
(2.54)
The existence of this element is proven in Appendix A, where an explicit expression is given for it in terms of an infinite sum. Let us find it explicitly for low values of n. For n = 2, 3, we can solve the above equation explicitly. We find for n = 2, D=
1 + q2 1 − q + g1 , 1+q 1+q
and for n = 3, D=
1 + q 2 + 2q 3 + q 4 + q 6 (1 − q)(2 + 2q + q 2 + 2q 3 + 2q 4 ) + g1 (1 + q)(1 + q + q 2 ) (1 + q)(1 + q + q 2 ) (1 + q)(1 − q)2 + g1 g2 . 1 + q + q2
(2.55)
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329
We note that D → 1 in the classical limit. Using the form of D in the appendix we can 2 χ R (D) write ddRR(q) (1) = d R (q) d R (1) , which allows us to rewrite (2.53), ∞ G 1 [N ](2−2G)n δ D 2−2G q −l(si )−l(ti ) h(si )h(ti )h(si−1 )h(ti−1 ) . Z = g st n=0
i=1
i i
(2.56) In the last step we used (2.43). This is the q-analog of the Gross-Taylor expansion. We can expand the -factors as follows: Z =
∞
∞ N −1 1 i=1 l(Ti ) [N ](2−2G)n+ i (K (Ti )−n) q 2 g n=0 si ti =0 T1 ...T G −1 −1 ×d (2 − 2G, ) δ D C T1 . . . C T h(si )h(ti )h(si )h(ti ) . (2.57)
q−
i (l(si )+l(ti ))
i=1
As explained in [5], the factor of d(2 − 2G, ) is the Euler character of the configuration space of points on G , denoted as χ (G, ). Hence we can write Z =
∞ n=0 si ti
q−
i (l(si )+l(ti ))
∞ N −1 1 [N ](2−2G)n+ j=1 (K (T j )−n) q 2 g =0 T1 ...T
×χ (G, ) δ D C T1 . . . C T
G
q
j=1 l(T j )
−(l(si )+l(ti ))
h(si )h(ti )h(si−1 )h(ti−1 )
. (2.58)
i=1
3. Manifolds with Boundary We now describe the chiral large [N ] expansion of q-deformed 2d Yang-Mills theory on manifolds with boundary, in terms of Hecke algebras. We recall the classical case first. For a Riemann surface of genus G with B boundaries and boundary holonomies U1 , . . . , U B in SU (N ), the parition function is Z YM (G, B; U1 , . . . , U B ) =
(dim R)2−2G−B χ R (U1 ) χ R (U2 ) . . . χ R (U B ) . (3.1)
R 1 B It is useful in that case to multiply by ( n! ) trn (T1 U1† ) trn (T2 U2† ) . . . tr(TB U B† ) and integrate over the holonomies, where T1 , . . . , Tn are sums of permutations in fixed conjugacy classes in Sn . Then the chiral Gross-Taylor expansion becomes
G 1 N n(2−2G−B) δ T1 . . . TB 2−2G−B si ti si−1 ti−1 . Z YM (G, B; T1 , . . . , TB ) = n n! s ,t i i
i=1
(3.2) This is basically a Fourier transformation, and the derivation is explained in [24].
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For q-deformed 2d Yang-Mills, the holonomies along the boundaries are specified by the quantum characters [13, 12] of Uq (SU (N )): Z q YM (G, B; U1 , . . . , U B ) =
(dimq R)2−2G−B χ R (U1 ) χ R (U2 ) . . . χ R (U B ). (3.3)
R
Now we can insert ( g1 ) B trn (C T1 U1† ) trn (C T2 U2† ) . . . tr(C TB U B† ). In this case, C T1 , . . . , C TB are central elements in Hn (q) which approach the class sums T1 , T2 , . . . , TB in the limit q → 1. They have appeared in the formulae for the q-dimension earlier. We use the expansion tr(C T U † ) =
χ S (C T ) χ S (U † ) ,
(3.4)
S
where χ S (C T ) is the Hecke algebra character in the representation S. Then we integrate the quantum group elements U1 , . . . , U B , and use the orthogonality [13, 12] dU χ R (U ) χ S (U † ) = δ R S .
(3.5)
The result is Z q YM (G, B; C T1 , . . . , C TB ) =
B χ R (C T j ) 2−2G−B dimq R g
R∈Yn
=
R
[N ]
j=1
(2−2G−B)n
d R (q) χ R () g d R (1)
2−2G−B
B χ R (C T j ) j=1
(3.6)
g
1 [N ](2−2G−B)n δ g si ti ⎞ ⎛ B−1 G B E 2−2G−B q −l(si )−l(ti ) h(si )h(ti )h(si−1 )h(ti−1 ) CT j ⎠ . ×⎝ g
=
i=1
j=1
In the second line we used (2.48), and in the last line we employed (2.52). The element E is defined in (A.14). As in manipulations of the partition function we repeatedly used (2.44) to combine products of characters. Finally to obtain the delta function from the Hecke characters, we used (2.43). In the q = 1 limit (3.6) reduces to a delta function over the group algebra of Sn , counting maps with specified conjugacy classes of permutations at the boundaries. There is now some deformation of this geometry, involving central elements of the Hecke algebra Hn (q) associated with the boundaries. It is very intersting that for B = 1 we do not have the Eg factors. Recall also that Eg = 1 in the q = 1 limit. In the q-deformed theory there is a notion of a delta-function over the quantum group -valued holonomies [13]. It is the partition function on the disk, therefore the case G = 0,
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331
B = 1 of the above. We compute directly: δ(U, 1) =
n;σ ∈Sn
=
n;σ ∈Sn
1 [N ]n q −l(σ ) δ(D h(σ −1 )) trn (h(σ ) u U ) g 1 [N ]n Q σ trn (h(σ ) u U ) , g
(3.7)
where we defined D = σ Q σ h(σ ). Using (3.5), we can integrate this expression against any test function to obtain a form that depends purely on the Hecke algebra. In particular, the above gives another expression for the quantum dimensions. Thus, in the q-deformed theory the partition function on a disk of zero area continues to be associated to a flat connection, in the quantum group sense [13].
4. Chiral Large N Expansion for Wilson Loops After having computed the partition function on closed Riemann surfaces and Riemann surfaces with boundaries, we should now discuss the chiral expansion of Wilson loops. For simplicity, we will consider non-intersecting Wilson loops in this section. The basic object we need to take into account are the SU (N ) tensor multiplicity coefficients [13, 12]. Indeed, consider a surface of genus G = G 1 + G 2 with a Wilson loop in representation S, where G 1 and G 2 are the genera of the inner and outer faces of the Wilson loop. The expectation value of this Wilson loop is W S (G) =
1−2G 1 1−2G 2 dimq R2 χ R1 (U ) χ S (U ) χ R2 (U † ) , dU dimq R1
R1 R2
(4.1) where R1 and R2 are the representations of the inner and outer faces, respectively. Since we are discussing the case of q non-root of unity, the result of the above quantum integral is the usual SU (N ) tensor multiplicity coefficients (Littlewood-Richardson coefficients). Thus we are set to compute W S (G) =
(dimq R1 )1−2G 1 (dimq R2 )1−2G 2 N RR12S .
(4.2)
R1 R2
Our next task is to look for an expression for the Littlewood-Richardson coefficients that we can interpret as a deformation of the Riemann surface. Thus, we want to write them as delta functions on the Hecke algebra. We start from the definition: N RR13R2 =
dU χ R1 (U ) χ R2 (U ) χ R3 (U † ),
(4.3)
and observe that the above is a trace of the following operator acting in R1 ⊗ R2 : dU χ R3 (U † ) ρ R1 ⊗R2 (U ) .
(4.4)
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Now R1 can be realized in V ⊗n 1 with multiplicity d R1 (1) when we project on the given Young diagram, and likewise for R2 . It is also useful to note that the above operator is proportional to a projector for the representation R3 , 1 ρ R ⊗R (PR3 ) . (4.5) dU χ R3 (U † ) ρ R1 ⊗R2 (U ) = dimq R3 1 2 Using the expression for the projectors for R1 and R2 in terms of the Hecke algebra, we obtain d R1 (q) d R2 (q) −l(σ1 )−l(σ2 ) 1 N RR13R2 = q χ R1 (h(σ1−1 )) χ R2 (h(σ2−1 )) g1 g2 g3 d R1 (1) d R2 (1) σ σ 1 2
1 × (4.6) trV ⊗n1 ⊗V ⊗n2 (h(σ1 ) · h(σ2 )) PR3 . dimq R3 Here and in what follows we take σi ∈ Sn i for i = 1, 2, 3. Writing out the projector (2.28), we get N RR13R2 =
d R1 (q) d R2 (q) 1 g1 g2 g3 d R1 (1) d R2 (1) × q −l(σ1 )−l(σ2 )−l(σ3 ) χ R1 (h(σ1−1 )) χ R2 (h(σ2−1 )) χ R3 (h(σ3−1 )) σ1 σ2 σ3
×
d R3 (q) trV ⊗n1 ⊗V ⊗n2 (h(σ1 ) · h(σ2 )) h(σ3 ) , dimq R3
(4.7)
and expanding the trace in a basis of Young tableaux with n 1 + n 2 boxes, we get N RR13R2 =
d R1 (q) d R2 (q) 1 g1 g2 g3 d R1 (1) d R2 (1) × q −l(σ1 )−l(σ2 )−l(σ3 ) χ R1 (h(σ1−1 )) χ R2 (h(σ2−1 )) χ R3 (h(σ3−1 )) σ1 σ2 σ3
×
d R3 (q) dimq R3
χ S (h(σ1 ) · h(σ2 )) h(σ3 ) dimq S .
(4.8)
S∈Yn 1 +n 2
If we now use the projector property χ S (PR3 h(σ )) = δ R3 S χ R3 (h(σ ))
(4.9)
and the explicit form of the projector in (2.12) then we have the useful orthogonality relation
d R (q) q −l(σ3 ) 3 χ R3 (h(σ3−1 )) χ S h(σ3 )(h(σ1 ) · h(σ2 )) g3 σ 3
= χ R3 (h(σ1 ) · h(σ2 )) δ R3 S .
(4.10)
This can be used to simplify the expression (4.8) further to 1 d R1 (q) d R2 (q) −l(σ1 )−l(σ2 ) q N RR13,R2 = g1 g2 d R1 (1) d R2 (1) σ σ 1 2
× χ R1 (h(σ1−1 )) χ R2 (h(σ2−1 )) χ R3 (h(σ1 ) · h(σ2 )).
(4.11)
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333
This formula is reminiscent of the Verlinde formula for the fusion coefficients of orbifold conformal field theories [25], or alternatively of Chern-Simons theory with finite groups [26, 27]. It would be interesting to understand the connection. If we go from the character basis to the basis in terms of central elements of the Hecke algebra, and using the above, we get 1 N RR13R2 χ R1 (C1 ) χ R2 (C2 ) χ R3 (C3 ) g1 g2 g3 R1 R2 R3
1 δ(h(σ1−1 )C1 ) δ(h(σ2−1 )C2 ) q −l(σ1 )−l(σ2 ) g1 g2 σ σ 1 2 1 × d R (1) χ R3 (h(σ1 ) · h(σ2 )C3 ) g3 3
=
R3
1 1 δ(h(σ1−1 )C1 ) δ(h(σ2−1 )C2 ) q −l(σ1 )−l(σ2 ) δ E C3 (h(σ1 ) · h(σ2 )) = g1 g2 σ σ g3 1 2
σ σ 1 C1 1 C2 2 δ E C3 (h(σ1 ) · h(σ2 )) . (4.12) = g1 g2 g3 σ σ 1
2
E is the element defined in (A.14) of Appendix A. We have denoted by C σ the coefficients which appear in the expansion of the central element C, C=
C σ h(σ ) ,
(4.13)
σ
and we used the following property of the trace [21]: δ(h(σ )h(σ )) = q l(σ ) if σ σ = 1, δ(h(σ )h(σ )) = 0 otherwise.
(4.14)
Consider now the computation of a simple Wilson loop, in the representation S, separating a region with G 1 handles from another region with G 2 handles, WS =
n 1 ,n 2 R1 R2
(dimq R1 )1−2G 1 (dimq R2 )1−2G 2 N RR12S
d R1 (q) 1−2G 1 = [N ] (χ R1 ())1−2G 1 g1 d R1 (1) R1
d R2 (q) 1−2G 2 n 2 (1−2G 2 ) × [N ] (χ R2 ())1−2G 2 N RR12S . g2 d R2 (1)
n 1 (1−2G 1 )
(4.15)
R2
We now use (4.11) with the fusion coefficient, multiply by the character of some central element C in Hn S (q) and sum over S W (C, G 1 , G 2 ) =
χ S (C) S
gS
WS .
(4.16)
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Collecting all S dependences we have 1 d S (q) χ S (C) χ S (h(σ2−1 )) = δ(C h(σ2−1 )). g S d S (1)
(4.17)
S
Hence we obtain W (C; G 1 , G 2 ) =
1 δn 1 +n S ,n 2 [N ]n 1 (1−2G 1 )+n 2 (1−2G 2 ) q −l(σ1 )−l(σ2 ) g g 1 2 n 1 ,n 2 σ1 σ2
G × δ(C h(σ2−1 )) δ D 1 1 1−2G 1 h(σ1−1 ) (4.18)
× δ 1G 2 1−2G 2 (h(σ1 ) · h(σ2 )) .
The factors of [N ] are as above. We have defined 1G 1 = 1G 2 =
q−
i
l(si )−l(ti )
s1 ,t1 ..sG 1 ,tG 1
G1
h(si )h(ti )h(si−1 )h(ti−1 ),
i=1
q−
i
l(si )−l(ti )
s1 ,t1 ..sG 2 ,tG 2
G2
h(si )h(ti )h(si−1 )h(ti−1 ).
(4.19)
i=1
Expanding C=
C σ h(σ )
σ
P = D 1−2G 1 1G 1 =
P σ h(σ ) ,
(4.20)
σ
we finally get W (C; G 1 , G 2 ) =
∞ n 1 =0
1 [N ]γ P σ C σ δ 1−2G 2 1G 2 (h(σ ) · h(σ )) . g1 g2
(4.21)
σσ
We defined γ = n 1 + n 2 − 2(n 1 G 1 + n 2 G 2 ) = (2 − 2G)n 1 + n S (1 − 2G 2 ), where we used n 2 = n 1 + n S . 5. On the Role of Quantum Characters in q-Deformed 2d YM In this paper we have used quantum Uq (SU (N ), characters rather than classical SU (N ) characters. For the computations in [10] it seemed enough to consider classical SU (N ) characters. So one can ask: does one need to compute with quantum characters, or do the classical ones suffice? In this section we argue that quantum characters are needed in the generic situation; in fact, they are extremely natural and they provide the simplest solution to the problem of crossings and gluing along open lines. Our arguments are consistent with [10], where the dimensions appearing in the partition function (1.3) were quantum dimensions but the characters associated with boundaries and Wilson loops were classical SU (N ) characters. In particular, this paper did not consider crossings on the surface, and gluing constructions involved closed curves only. In the absence
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335
of crossing points, both the classical and the quantum characters lead to a topological invariant theory. It is a well-known fact from Chern-Simons theory that one can do without R-matrices or other quantum group structure as long as one considers simple Wilson loops – for example, toric ones, whose expectation value follows from surgery. In the 2d Yang-Mills case, the basic gluing formula along circles is (3.5), which is valid both for classical and quantum characters, and ensures topological invariance of the gluing construction along circles. More precisely, the need for quantum characters in qYM can be seen: 1) in the presence of Wilson loops with non-trivial crossings; 2) when gluing along open lines. The original definition of qYM is well-known [13, 12] and it involves quantum characters. In the following subsections we collect several arguments that show the need for quantum characters. 5.1. Consistency of Wilson loops. One of the basic consistency conditions to be imposed on a Wilson loop is that, if the charge of the particle is zero, the expectation value of the Wilson loop should be that of the unit operator; in other words, it should give back the partition function of the theory. In our case, if W R (G; C) is the Wilson loop operator in representation R around the curve C on the Riemann surface of genus G, consistency requires W R=ρ (G; C) = 1 = Z q YM (G ),
(5.1)
where ρ is the Weyl vector labeling the trivial representation. Thus, we should reproduce: 1 Wρ (G; C) = (dimq S)2−2g q − 2 A C2 (S) . (5.2) S
We will check whether quantum dimensions and classical characters are consistent with this for a Wilson loop with crossings. Consider the expectation value of the Wilson loop W R (G; C) in Fig. 1. In this case we have A = A1 + A2 + A3 , where A1 is the area of the outer face, which has genus G. We get: W R (G; C) 1 = (dimq (R1 ))1−2g dimq (R2 ) dimq (R3 ) q − 2 (A1 C2 (R1 )+A2 C2 (R2 )+A3 C2 (R3 )) R1 R2 R3
×
dU dV χ R1 (U −1 V −1 ) χ R2 (U ) χ R3 (V )χ R (U V −1 ),
(5.3)
where, since we are dealing with classical characters, dU is the Haar measure. Let us compute this in the trivial case: R = ρ. We can compute the integrals using the character formula χ R2 (V ) . (5.4) dU χ R2 (U )χ R3 (U −1 V ) = δ R2 R3 dim(R2 ) We get (dimq (S))3−2g S
dim(S)
1
q − 2 AC2 (S) ,
(5.5)
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R V U
A3 A2 R3 R2 A1 R1 Fig. 1. A Wilson loop with a crossing.
which disagrees with (5.2). The reason that the dimensions do not come out right is that we were forced to use formula (5.4). We conclude that this procedure is not consistent. On the other hand, the same computation can be carried out with quantum characters, and in that case we do get the quantum dimension in (5.4).
5.2. Gauge invariance of Wilson loops. There is a short proof of gauge invariance for the Wilson loops and boundary elements we have discussed in previous sections. Let U ∈ Funq (SU (N )) (for more details on this see Appendices B and D), and consider the ad-action of Funq (SU (N )) on itself: ad : U → h U S(h) ,
(5.6)
where we are considering Funq (SU (N )) as a Hopf algebra with antipode S [17]. It is easy to see that the quantum trace Tr (u U )
(5.7)
is left invariant under this action (for the definition of the u-element, see Appendix B). We get: Tr (u hU S(h)) = Tr (S 2 (h)uU S(h)) = (S 2 (h))i j (uU ) jk (S(h))ki = (S 2 (h))i j (S(h))ki (uU ) jk = S(h ki (S(h))i j )(uU ) jk = Tr (u U ) ,
(5.8)
where we used (h) = 1, and the fact that u satisfies u x = S 2 (x) u
(5.9)
for any x ∈ Funq (SU (N )). Thus, gauge invariance in Funq (SU (N )) is ensured provided we include the u-element.
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We have proven that the triple
Migdal gluing , quantum dimensions , classical characters
337
(5.10)
is inconsistent in the generic case. To get a consistent theory, we need to modify one of the above. If instead of quantum dimensions we use classical dimensions, we of course get back the usual 2d Yang-Mills. If we want the dimensions to be quantum, we either need quantum characters, or a modification of the gluing rules. The possibility to have quantum characters has been discussed at length in this paper, and it has been shown to be consistent in [13]. In particular, the theory is gauge invariant and independent of the triangulation. We do not exclude that there might be a complicated modification of the gluing rules that would allow to keep quantum dimensions and classical characters even in the presence of crossings. Additional features of the quantum characters are the following. The natural expansion of the quantum dimensions is in terms of quantum characters, which are most easily expressed in terms of a Hecke algebra, as we have shown. This gives a natural deformation of the symmetric group description of covering maps of the Riemann surface. Also in the case with boundaries, the use of quantum characters was essential for this. Finally, q-deformed 2d Yang-Mills computes invariants of knots in Seifert manifolds [28, 11]. This is also expected from open-closed string duality in the A-model with branes. This relation will however only work if on the qYM side we deform the gauge symmetry as well so as to get quantum characters, since only that will give the quantum 6j-symbols that appear in the Reshetikhin-Turaev invariant relevant for knots in Chern-Simons [11]. 6. Discussion and Outlook We have shown that the chiral large N expansion ( note that the q-number [N ] appears as the natural expansion parameter ) for q-deformed Yang-Mills can be described by Hecke algebras. The full large N expansion is expected to be given by a coupled product of chiral and anti-chiral contributions. We expect that techniques of this paper can be extended to give a precise description of this non-chiral expansion in terms of Hecke algebras. The string interpretation of q-deformed 2d Yang-Mills on G has been developed in [9, 10]. The leading order terms in the expansion, obtained by setting the factors to 1, were shown to compute Gromov-Witten invariants of a Calabi-Yau space X which is a direct sum of line bundles L p ⊕ L 2g−2− p fibered over G . The sub-leading terms, due to the factors were intepreted in terms of D-brane insertions at 2G − 2 points. This picture develops the Gross-Taylor interpretation (at q = 1) of the factors in terms of fixed points on the Riemann surface [3, 4]. An alternative interpretation of the factors underlies the topological string theory developed in [5, 6] for q = 1. The latter topological string is different from the standard one. It has been labelled a balanced topological string and has been observed to be an example of a general class of balanced topological field theories naturally related to Euler characters of moduli spaces [29]. It integrates over the moduli space of holomorphic maps the Euler class of the tangent bundle to that moduli space. The concrete connection between Euler characters and the large N expansion of two dimensional Yang-Mills is manifest when one expands the factors and recognizes the binomial coefficients as Euler characters of configuration spaces of points on the Riemann surface G [5]. Our treatment of the factors in the q-deformed case, which
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has expressed it in terms of central elements of the Hecke algebra, naturally lends itself to this interpretation. Euler characters of configuration spaces continue to appear in the expansion for the same reasons as at q = 1. This suggests that a closed topological string interpretation exists for the large N expansion of q-deformed two-dimensional Yang-Mills in terms of a balanced topological string. The simplest proposal along these lines is that the balanced topological string with target space X would give a closed string interpretation for the all orders expansion of q-deformed two-dimensional YangMills. The relation of such a picture to the D-brane insertions of [10] would involve an interesting incarnation of open-string/closed-string duality. Developing these relations requires a clearer understanding of the coupling between holomorphic and anti-holomorphic sectors in the context of the balanced topological string. The connection between the Gross-Taylor expansion and the Gromov-Witten invariants appearing in [9, 10] has also been discussed in [30, 31]. Given the rather simple Hecke q-deformation we have uncovered, of the sums over symmetric group delta functions related to the classical Hurwitz counting of branched covers, it is also natural to speculate that there is an intrinsically two-dimensional picture which would account for the Hecke delta functions, without appealing to the Calabi-Yau X . One possiblity is that we have q-deformed Riemann surfaces and maps between such Riemann surfaces. In fact q-deformed planes, known as Manin planes, have been studied and holomorphy has been discussed ( see for example [32]). One could construct Riemann surfaces which, in some sense, locally look like Manin planes, and consider holomorphic maps between them. As far as we are aware, such a theory of Hurwitz spaces for q-deformed Riemann surfaces has not yet been developed. While Hecke algebras are more familiar to mathematical physicists as centralizers of quantum groups acting in tensor spaces, they have another pure mathematical origin (see for example [33]). Hn (q) is an algebra of double cosets Bn (Fq ) \ G L n (Fq )/Bn (Fq ). Here Fq is the finite field with q elements, where q is a power of a prime p. (If q = p then Fq is just the field of residue classes modulo p.) G L n (Fq ) is the group of n × n matrices with entries in Fq . Bn (Fq ) is the subgroup of the upper triangular matrices. This generalises the fact that Sn appears from double cosets Bn (C) \ G L n (C)/Bn (C). Hence the deformation of CSn to the Hecke algebra Hn (q) corresponds to going from C to Fq . This suggests that, at least for q equal to a power of a prime, our Hecke-q-deformed Hurwitz counting problem might be related to Riemann surfaces over Fq . It is interesting that, in this context, fundamental groups can be defined and they still take the form −1 −1 bG u 1 · · · u B = 1 . a1 b1 a1−1 b1−1 a2 b2 a2−1 b2−1 · · · aG bG aG
(6.1)
There are also results on the moduli spaces of branched covers in this set-up, generalizing properties of classical Hurwitz space [34]. An interesting direction for the future is to determine if there is a relation between Hecke algebras Hn (q) and these moduli spaces, and if such a relation provides the geometrical meaning for the q-deformed Hecke counting problems in (2.56), (2.57). Classical and q-deformed 2d Yang-Mills are closely connected to Chern-Simons theory on Seifert manifolds [35, 11, 10, 36, 37]. On the other hand, some of the formulas in this paper, such as (4.11) , are suggestive of some connection of the chiral large N expansion of q-deformed 2d Yang-Mills and orbifold conformal field theories [25] or Chern-Simons theory for finite gauge groups [26, 27]. It is known that the ChungFukuma-Shapere three-dimensional topological field theory [38] is the absolute value squared of the partition function of the Dijkgraaf-Witten theory. It seems very likely that the chiral expansion in terms of Hecke characters worked out in this paper can
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be formulated in the two-dimensional topological field theory framework of [39, 38] with additional insertions coming from the branch points. It would be interesting to see in detail to what extent the chiral q-deformed 2d Yang-Mills theory is related to the Dijkgraaf-Witten theory. In view of the connection to Chern-Simons theory, it will be interesting to explore the q-deformed chiral as well when q approaches roots of unity. q-Schur Weyl duality at roots of unity has been discussed in [40, 41]. Acknowledgements. We thank Mina Aganagic, Luca Griguolo, Costis Papageorgakis, Gabriele Travaglini, and Ivan Todorov for useful discussions and correspondence. SR is supported by a PPARC Advanced Fellowship. The research of SR and SdH is in part supported by the EC Marie Curie Research Training Network MRTN-CT-2004-512194. The work of AT is supported by DFG (Deutsche Forschungsgemeinschaft) within the “Schwerpunktprogramm Stringtheorie 1096”. We have used the package NCAlgebra for some of our computations with the Hecke algebra.
A. Central Elements A.1. Centrality of q-deformed conjugation sum . We want to show that q −l(s) h(s)h(t)h(s −1 )
(A.1)
s
is central in Hq (n). Since Hq (n) is generated by g1 , ...gn−1 , it suffices to show that the above element commutes with these generators. We will first show it for g1 , and it will be clear the same proof can be repeated for g2 , etc. First recall how this works in the case q = 1. We write s1 sts −1 = (˜s )t s˜ −1 s1 s
s
=
s˜ t s˜ −1 s1 ,
s˜
where we defined s˜ = s1 s. The cancellation only uses a pair of terms at a time. For a fixed s, s1 sts −1 = s˜ t s˜ −1 s1 , s1 s˜ t s˜ −1 = sts −1 s1 , which means that [s1 , sts −1 ] + [s1 , s˜ t s˜ −1 ] = 0 .
(A.2)
It turns out that the same pairwise cancellation works for q = 1. It is instructive to check it explicitly for n = 3, 4. Below we give the general argument. Suppose s is of the form s1 u, where u is a word in the generators. Now recall that before applying the map h to s we must express it in reduced form. This means that if s = s1 u, the leftmost term in u is not s1 . The following can be derived easily h(s) = g1 h(u), l(s) = l(u) + 1, h(s −1 ) = h(u −1 )g1 .
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Then s˜ = s1 s = u. Now we write the pair of elements from (A.1) for the fixed s, s˜ . q −l(s) h(s)h(t)h(s −1 ) = q −l(u)−1 g1 h(u)h(t)h(u −1 )g1 , q −l(˜s ) h(˜s )h(t)h(˜s −1 ) = q −l(u) h(u)h(t)h(u −1 ) .
(A.3)
The commutator with the first term is [g1 , q −l(s) h(s)h(t)h(s −1 )] = q −l(u) h(u)h(t)h(u −1 )g1 + q −l(u)−1 (q − 1)g1 h(u)h(t)h(u −1 )g1 − q −l(u) g1 h(u)h(t)h(u −1 ) − q −l(u)−1 (q − 1)g1 h(u)h(t)h(u −1 )g1 . (A.4) The commutator with the second term in (A.3) is [g1 , q −l(u) h(u)h(t)h(u −1 )] = q −l(u) g1 h(u)h(t)h(u −1 ) − q −l(u) h(u)h(t)h(u −1 )g1 . (A.5) Combining the terms in (A.4) and (A.5) we see that the terms proportional to a power of q cancel between the two equations (as they must for this to work at q = 1). The terms containing a factor q − 1 cancel within (A.4). This proves that the sum (A.1) commutes with g1 . It has been done by decomposing the sum over Sn into a sum over left coset elements by the subgroup S2 generated by s1 , and a sum over representatives in each coset. The vanishing of the commutator with g1 works within the sum over representatives in each coset. To prove that it commutes with g2 · · · gn−1 we similarly decompose with respect to left cosets of s2 , · · · sn−1 . Hence (A.1) is central in Hq (n). It follows that its matrix representation in any irreducible representation must be diagonal. Using the matrices given in [16], we have checked this explicitly up to n = 4. A special case of (A.1) is given by the choice t = 1. Based on evidence described below, we conjecture that its character in an irreducible representation is
q −l(s)
s
χR g d R (1) (h(s −1 )h(s)) = , d R (1) d R (q)
(A.6)
with d R (q) and g as given in (2.13). Since the Hecke element in the character is central ( after summation over s ), it suffices to calculate it on one state in the irrep. We have checked this for general completely symmetric reps and completely antisymmetric reps, as well as for all representations up to n = 4, using the explicit matrices given in [16]. Another check of this formula is to multiply by d R (q)d R (1) and sum over young diagrams R with n boxes. Using (2.43), the LHS becomes gδ
q
−l(s)
h(s
−1
)h(s) .
(A.7)
s −1 l(s) But from [21] δ(h(s )h(s)) = q . Hence the LHS is equal to (g n!). On the RHS we have g R (d R (1))2 = (g n!) . This gives a consistency check of (A.6) for any n.
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Using (A.6) and (2.44), χR χR (h(s)h(t)h(s −1 )) = (h(s)h(s −1 )h(t)) q −l(s) q −l(s) d (1) d (1) R R s s g d R (1) χ R (h(t)) d R (q) d R (1) g = χ R (h(t)) . d R (q) =
Hence
s,t
(A.8)
χR (h(s)h(t)h(s −1 )h(t −1 )) q −l(t)−l(s) d R (1) χR χR (h(s)h(t)h(s −1 )) (h(t −1 )) q −l(t) q −l(s) = d (1) d (1) R R s,t g −l(t) χR (h(t −1 )) = q χ R (h(t)) d R (q) t d R (1) g q −l(t) χ R (h(t))χ R (h(t −1 )) = d R (q)d R (1) t
2 gd R (1) g g = . (A.9) = d R (q)d R (1) d R (q) d R (q)
The last sum over characters was done by using orthogonality (2.24). This shows the desired identity (2.50) A.2. Centrality of q-deformed commutator sum . We prove that the element C≡ q −l(s)−l(t) h(s)h(t)h(s −1 )h(t −1 )
(A.10)
s,t
of the Hecke algebra Hn (q) is central. In the q = 1 limit, this is s,t sts −1 t −1 , a sum of commutators of all group elements. Hence C is a q-deformed sum of commutators. Since Hn (q) is generated by g1 . . . gn−1 it suffices to prove that gi C = Cgi for any gi . We will start with g1 and it will be clear how to generalize to the other generators. Given the centrality of the q-deformed conjugation sum (A.1) we can write 1 ≡ g1 C − Cg1 = q −l(s)−l(t) h(s)h(t)h(s −1 )g1 h(t −1 ) s,t
−
q −l(s)−l(t) h(s)g1 h(t)h(s −1 )h(t −1 ) .
(A.11)
s,t
We want to prove 1 = 0. For q = 1 this can be proved as follows. If we define t = tˆs1 , s = sˆ s1 , we can write ss1 ts −1 t −1 = sˆ tˆsˆ −1 s1 tˆ−1 . (A.12) s,t
sˆ ,tˆ
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This shows that it is useful to think about the sums over Sn in terms of the cosets Sn /S2 , where the S2 is generated by s1 . Let us choose expressions for the elements of Sn in terms of words of minimal length in s1 ..sn . Let S+ be the set of words not ending with s1 on the right, and S− the set of elements of the form sˆ s1 . Clearly sˆ does not end with s1 : if it did s would not be in reduced form. Hence sˆ ∈ S+ . For such s = sˆ s1 , it is easy to see that h(s) = h(ˆs )g1 , l(s) = l(ˆs ) + 1, h(s −1 ) = g1 h(ˆs −1 ).
(A.13)
We can write 1 as
1 = (
+
s∈S+
−(
s∈S+
=
+
)(
s=ˆs s1 ∈S− ; sˆ ∈S+
s=ˆs s1 ∈S− ; sˆ ∈S+
)(
t∈S+
+
q −l(ˆs )−l(t)−1 h(ˆs )g1 h(t)g1 h(ˆs −1 )g1 h(t −1 ) +
q −l(ˆs )−l(tˆ)−2 h(ˆs )g1 h(tˆ)g12 h(ˆs −1 )g12 h(tˆ−1 )
sˆ ,tˆ∈S+
q −l(s)−l(t) h(s)g1 h(t)h(s −1 )h(t −1 ) −
q −l(s)−l(tˆ)−1 h(s)g1 h(t)g1 h(s −1 )g1 h(t −1 )
s,tˆ∈S+
s,t∈S+
−
q −l(s)−l(tˆ)−1 h(s)h(tˆ)g1 h(s −1 )g12 h(tˆ−1 )
s,tˆ∈S+
sˆ ,t∈S+
−
) h(s)g1 h(t)h(s −1 )h(t −1 )q −l(s)−l(t)
t=tˆs1 ∈S− ; tˆ∈S+
q −l(s)−l(t) h(s)h(t)h(s −1 )g1 h(t −1 ) +
s,t∈S+
+
) h(s)h(t)h(s −1 )g1 h(t −1 )q −l(s)−l(t)
t=tˆs1 ∈S− ; tˆ∈S+
t∈S+
+
q −l(ˆs )−l(t)−1 h(ˆs )g12 h(t)g1 h(ˆs −1 )h(t −1 ) −
q −l(ˆs )−l(tˆ)−2 h(ˆs )g12 h(tˆ)g12 h(ˆs −1 )g1 h(t −1 ) .
sˆ ,tˆ∈S+
sˆ ,t∈S+
This can be simplified by using g12 = (q −1)g1 +q. We get terms with powers q −l(s)−l(t) in the summand but without powers of (q − 1), terms proportional to (q − 1) and terms proportional to (q − 1)2 . The terms without powers of (q − 1) cancel pairwise among the 8 terms. The other terms can be written out explicitly, and seen to cancel. This proves that [g1 , C] = 0. When checking for commutation with gi , we organise the sums over Sn according to cosets of the S2 subgroup generated by si . Then the same argument as above applies to show that any of the generating gi commute with C. Hence C is central. A.3. The elements D and E of Hn (q) . Equation (A.6) also allows us to give an expression for D defined in (2.54). Let us write E= q −l(s) h(s −1 )h(s) s
= 1+
q −l(s) h(s −1 )h(s)
s
≡ 1 + E . The primed sum extends over elements in Sn excluding the identity. Then we can write χ R (E) gd R (1) = . d R (1) d R (q)
(A.14)
Large N Expansion of q-Deformed 2-D Yang-Mills Theory and Hecke Algebras
Using that E is central
χ R (E) d R (1)
m
χ R (E m ) = = d R (1)
g d R (1) d R (q)
343
m .
(A.15)
Now let m = −1 to get χ R (E −1 ) =
d R (q) . g
(A.16)
Hence D = g E −1 ∞ =g (−1)k (E )k =g
k=0 ∞
k=0
u 1 ,u 2 ...u k
(−1)k
−1 q −l(u 1 )−l(u 2 )−...−l(u k ) h(u −1 1 )h(u 1 ) · · · h(u k )h(u k )
(A.17)
B. Quantum Dimensions The irreducible representations R of Uq (U (N )) can be realized as subspaces of V ⊗n , where V is the fundamental representation. The matrix elements of the fundamental j representation are denoted by U , with entries Ui (see Appendix D for explicit expressions), and the algebra generated by the U ’s is dual to Uq (U (N )) and is denoted by j Funq (U (N )). The commutation relations of the Ui are given in terms of the R-matrix in the reference we denote as FRT [17] (we are using U for T of this reference). We first derive formula (2.34), used in Sect. 2 to obtain a Hecke formula for the quantum dimensions. Thus we need to compute the trace trn (u h(m T )) that comes from the quantum character expression. The element u is: u=q
2
N N +1 i=1
2
−i E ii
.
(B.1)
The E i j act on the fundamental representation in the usual way E i j vk = δ jk vi .
(B.2)
Now we can use the FRT formula for the R-matrix to show that (tr ⊗ 1)(u ⊗ 1) P R = q N 1 , and tr(u) =
q N −q −N q−q −1
(B.3)
. This means that (tr ⊗ tr)(u ⊗ u) P R = q N
q N − q −N . q − q −1
Going back to the Hecke algebra conventions using (2.8) (q → (tr ⊗ 1)(u ⊗ 1) g1 = q
N +1 2
(tr ⊗ tr)(u ⊗ u) g1 = q
N +1 2
(B.4) √ q ), we get
1, [N ] .
(B.5)
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More generally, tensor products of traces act on uh(m T ) as (tr ⊗ tr ⊗ . . . ⊗ tri )(u ⊗ u ⊗ · · · ⊗ u)(g1 g2 · · · gi−1 ) = q (i−1)
N +1 2
[N ] .
(B.6)
We now need to find out how to built h(m T ) out of the gi ’s. Consider a conjugacy class in Sn , denoted by T , made of permutations which have K i cycles of length i. When expressed in terms of the generators si , the minimal length permutation in this conjugacy class, denoted by m T , has length i (i − 1)K i . The minimal permutations are given in terms of words of the form gi gi+1 ...gi+ j , such as the one appearing in (B.6). For such minimal words, we can use (B.6) to obtain N +1 N +1 trn (u h(m T )) ≡ tr⊗n u ⊗n h(m T ) = q 2 i (i−1)K i [N ] i K i = q 2 l(m T ) [N ] i K i . (B.7) This is the formula (2.34) used in the derivation of the q-dimension formula in Sect. 2. We now show explicitly how formula (2.29) works in some examples, and that it leads to a q-dimension formula in terms of central elements (2.36). For q-traces in V ⊗3 , i.e traces with u ⊗3 inserted, we have trq (1) = [N ]3 , trq (g1 ) = [N ]2 q trq (g1 g2 ) = [N ] q
N +1 2
N +1
, ,
(B.8)
therefore trq (g2 g1 g2 ) = (q − 1) trq (g2 g1 ) + q trq (g1 ) = (q − 1) q N +1 [N ] + q q
N +1 2
[N ]2 .
(B.9)
Now (2.29) gives for the q-dimension N +1 1 d R (q) [N ]3 χ R (1) + 2q −1 q 2 [N ]2 χ R (g1 ) dimq (R) = g d R (1) + q −3 χ R (g1 g2 g1 ) trq (g2 g1 g2 ) +q
−2
χ R (g1 g2 ) q
N +1
[N ] + q
−2
χ R (g2 g1 ) q
N +1
[N ] .
(B.10)
Filling in the above, we finally find N −1 1 d R (q) [N ]3 χ R (1) + q 2 [N ]2 χ R (g1 + g2 + q −1 g1 g2 g1 ) dimq (R) = g d R (1)
+ q N −1 [N ] χ R (g1 g2 + g2 g1 + q −1 (q − 1)g1 g2 g1 )
N −1 1 d R (q) [N ]3 χ R (1) + q 2 [N ]2 χ R (C T (2,1) ) + q N −1 [N ] χ R (C T (3) ) . = g d R (1) (B.11) The final expression contains central elements C T associated to conjugacy classes of Sn . There is the trivial conjugacy class containing the identity element, for which C T (q) = 1. There is C(2,1) (q) = g1 + g2 + q −1 g1 g2 g1 , for the conjugacy class corresponding to a
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345
single transposition. Finally there is C(3) (q) = g1 g2 + g2 g1 + (q−1) q g1 g2 g1 . It is easy to check that these elements commute with g1 , g2 . The above central elements and their generalizations are described in [43, 44]. They approach the correct classical limit of a sum of permutations in the appropriate conjugacy class. Using the Hecke characters given in [16, 20], we have checked that the above is consistent with the standard formula for the q-dimension as a product over the cells of the Young diagram: dimq R =
1≤i j≤N
q (λi −λ j + j−i)/2 − q −(λi −λ j + j−i)/2 , q ( j−i)/2 − q −( j−i)/2
(B.12)
where λ1 , . . . , λ N are the lengths of the rows of the Young tableau, and, for SU (N ), λ N = 0. For n = 2, an easy check gives for the symmetric representation : [N ][N + 1] , [2] and for the antisymmetric representation
(B.13)
:
[N ][N − 1] . [2]
(B.14)
We have also obtained by the above manipulations, explicit formulae for central elements for n = 4 which agree with those given in [22]. We have also checked that our formula for the quantum dimensions (2.36) agrees with the standard formula (B.12) for all representations up to n = 4. C. Projectors Below we give explicit checks that (2.28) indeed defines projectors. We do this for n = 3 and n = 4, that is for the Hecke algebras of H3 and H4 , and outline the method of [21] for the general case. C.1. H3 . We work out the projector for a general representation of H3 . It contains 3! = 6 independent terms, corresponding to the six elements of H3 . Using (B.9), we get
1 1 q −1 1 χ R (1) + χ R (g1 ) (g1 + g2 ) + PR = χ R (g1 g2 ) + 2 χ R (g1 ) g1 g2 g1 cR q q3 q
1 + 2 χ R (g1 g2 ) (g1 g2 + g2 g1 ) , (C.1) q where the term g1 g2 g1 corresponds to the (13) permutation. Notice that in Sn , s1 s2 s1 is in the same conjugacy class as s1 . Indeed, in the classical case where q = 1 the first term in (B.9) is absent and tr(g1 g2 g1 ) = tr(g1 ). In the quantum case, g1 g2 g1 has contributions from both χ (g1 g2 ) and χ (g1 ). This implies that it contributes to two different class elements.
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Using the expressions for the characters in [16], we get for the three H3 representations: =
P P
=
P =
1 c 1 c 1 c
(1 + g1 + g2 + g1 g2 + g2 g1 + g1 g2 g1 ) ,
q −1 1 2+ (g1 + g2 ) − (g1 g2 + g2 g1 ) , q q
1 1 1 1 − (g1 + g2 ) + 2 (g1 g2 + g2 g1 ) − 3 g1 g2 g1 . q q q
(C.2)
We have checked by explicit computation that they satisfy the projection equation (2.10) provided c c
=
q2 + q + 1 , q
= (q + 1)(q 2 + q + 1), c =
(q + 1)(q 2 + q + 1) . q3
(C.3)
This agrees exactly with the values given in [21], Eq. (C.7) below. C.2. H4 . For H4 , the projector contains 4! = 24 independent terms. The projector is: c R PR = a + b(g1 + g2 + g3 ) + c(g1 g2 + g2 g3 + g2 g1 + g3 g2 ) + dg1 g3 + f (g1 g2 g3 + g1 g3 g2 + g2 g1 g3 + g3 g2 g1 ) + h(g1 g2 g1 + g2 g3 g2 ) + k(g1 g2 g1 g3 + g1 g2 g3 g2 + g1 g3 g2 g1 + g2 g3 g2 g1 ) + lg2 g1 g3 g2 + m(g1 g2 g1 g3 g2 + g2 g1 g3 g2 g1 ) + ng1 g2 g3 g2 g1 + pg2 g1 g3 g2 g1 g3 . (C.4) The coefficients a, b, c, d, f, h, k, l, m, n, p depend on the representation. They are characters of q −l(σ ) χ (h(σ −1 )) which can be simplified, using cyclicity and the Hecke relations, to a d h k l
= χ (1) , b = q −1 χ (g1 ) , c = q −2 χ (g1 g2 ), = q −2 χ (g1 g3 ) , f = q −3 χ (g1 g2 g3 ), = q −3 [(q − 1)χ (g1 g2 ) + qχ (g1 )], = q −4 [(q − 1)χ (g1 g2 g3 ) + qχ (g1 g2 )], = q −4 [(q − 1)χ (g1 g2 g3 ) + qχ (g1 g3 )],
m = q −5 [(q 2 − q + 1)χ (g1 g2 g3 ) + q(q − 1)χ (g2 g3 )], n = q −5 [(q − 1)2 χ (g1 g2 g3 ) + 2q(q − 1)χ (g1 g2 ) + q 2 χ (g1 )], p = q −6 [(q − 1)(q 2 + 1)χ (g1 g2 g3 ) + q(q − 1)2 χ (g1 g2 ) + q 2 χ (g1 g3 )] . (C.5) Again, the mixing between different terms comes from using formulas like (B.9) and is related to the contribution of a single term to different central elements. In the limit q = 1, each of the a, d, . . . , p depend on a single character, the one corresponding to the conjugacy class of the element that a, d, . . . , p multiply in the projector.
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Using computer algebra, we have checked that the above are projectors for the five n = 4 representations, provided = (q + 1)(q 2 + q + 1)(q 3 + q 2 + q + 1),
c
(q + 1)(q 3 + q 2 + q + 1) , q (q + 1)2 (q 2 + q + 1) = , q2 (q + 1)(q 3 + q 2 + q + 1) = , q3 =
c c c
c =
(q + 1)(q 2 + q + 1)(q 3 + q 2 + q + 1) . q6
(C.6)
C.3. The construction for Hn . In the general case, the projector contains n! elements. Gyoja has given a formula for the coefficients2 c R : m (q − 1)(q 2 − 1) . . . (q λi +m−i − 1) 1 m(m−1)(m−2) c R = i=1 (q − 1)−n . (C.7) q6 λi +m−i − q λ j +m− j ) 1≤i j≤m (q From (C.3) and (C.6) we easily see that the coefficients satisfy c R (q −1 ) = c R T (q) ,
(C.8)
where R T is the representation with transposed Young tableau. This is indeed a general property of the projectors (C.7) [45]. It is easy to see that in the classical limit, m i=1 li ! c R (q = 1) = , (C.9) 1≤i j≤m (li − l j ) where li = λi + m − i. This is the coefficient of the Young symmetrizer, and is given by the hook formula. Also the quantum coefficients (C.7) can be expressed in terms of a hook formula. For high n, it is tedious to check idempotency of the projector. Also, it relies on having explicit formulas for the characters of the Hecke algebra. Gyoja [21] has given a construction to compute projectors in general without recourse to characters. In this construction, to associate a projector to a particular representation R, we first associate a projector to every state of the representation. Every state is represented by a standard tableau T . A standard tableau is a tableau where the entries (numbered with elements from {1, . . . , n}) are increasing across each row and down each column. The number of states in a given representation is d R (1). Thus, PR will be a sum of d R (1) primitive projectors, which we call E T , where T is the standard tableau they correspond to. The construction proceeds by defining two special tableaux, T+ and T− . These are the tableaux where the entries of the tableaux are numbered from 1 to n successively across the first row (column), then the second, third, etc. I+ and I− are the subgroups of Sn that 2 We corrected a typo in the formula in [21].
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preserve the rows (columns) of T+ (T− ). We associate to them parabolic subgroups W± of Sn and define h(w), e+ = w∈W+
e− =
(−q)−l(w) h(w) .
(C.10)
w∈W−
The primitive projector (up to normalization) associated to T is then −1 E(T ) = h − e− h −1 − h + e+ h + ,
(C.11)
where h + = h + (T ) and h − = h − (T ) are the elements of the Hecke algebra corresponding to the permutation that transforms T+ (resp. T− ) to the standard tableau T . Gyoja showed that the E’s are idempotents. The projector is then the sum of the orthogonal primitive idempotents: PR =
d R (1) 1 E(Ti ) , cR
(C.12)
i=1
where c R was given before3 . We checked the previously constructed projectors for n up to 4 using this construction. The first non-trivial case for n = 3 is the representation . There are two standard tableaux: T+ = [{1, 2}{3}] and T− = [{1, 3}{2}]. The permutation relating both is (23), which is h((23)) = g2 . In this case the parabolic subgroups are W+ = W− = {1, s1 }, and e+ = 1 + g1 , 1 e− = 1 − g1 . q
(C.13)
We further have h + (T+ ) = 1, h − (T+ ) = g2 , therefore E(T+ ) = 1 + g1 +
q −1 1 q −1 1 g2 − g1 g2 + g2 g1 − g1 g2 g1 . q q q q
(C.14)
For E(T− ), h + = g2 and h − = 1, so E(T− ) = 1 −
1 1 g1 − g2 g1 + g1 g2 g1 . q q
(C.15)
The primitive idempotents are automatically orthogonal. We get P
q (E(T+ ) + E(T− )) +q +1
q −1 q 1 2+ (g1 + g2 ) − (g1 g2 + g2 g1 ) , = 2 q +q +1 q q =
q2
(C.16)
3 Gyoja showed that the primitive idempotents are orthogonal using a certain ordering. In order for (C.12) to be a projector, they must be orthogonal independently of the ordering. This can be done defining new primitive idempotents in terms of the old ones, see Theorem 4.5 in [21]. For n up to 4, however, we found that the primitive projectors are automatically orthogonal.
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349
in agreement with the formula obtained earlier. As another example, we do n = 4 for the representation . There are three standard tableaux: T+ = [{1, 2, 3}, {4}], T− = [{1, 3, 4}, {2}], and T3 = [{1, 2, 4}, {3}]. We have I+ = {1, s1 , s2 } and I− = {1, s1 }, so e+ = 1 + g1 + g2 + g1 g2 + g2 g1 + g1 g2 g1 , 1 e− = 1 − g1 . q
(C.17)
In this case h + (T+ ) = 1, h − (T+ ) = g3 g2 . Thus: E(T+ ) = g3 g2 e− (T ) g2−1 g3−1 e+ (T ) ,
(C.18)
which we worked out with the help of computer algebra. In the same way we have h − (T− ) = 1, h + (T− ) = g2 g3 , so E(T− ) = e− (T ) g2 g3 e+ (T ) g3−1 g2−1 .
(C.19)
For T3 , h − (T3 ) = g2 , h + (T3 ) = g3 , hence E(T3 ) = g2 e− (T ) g2−1 g3 e+ (T ) g3−1 .
(C.20)
The projector is the sum of the three, with the appropriate coefficient, and it agrees with the one computed directly. Notice that the primitive idempotents were automatically orthogonal in this case as well. D. q-Schur-Weyl Duality and q-Characters In this appendix, we explain concretely the relation between quantum characters of the q-deformed SU (N ) and the symmetric group, in the special case of SU (2). We will use the quantum group conventions of [46] and [47]. We will use the formulae for matrix elements of spin-one representations from [46] in terms of spin-half representations and show that they are consistent with expressing the characters in spin-one in terms of the characters of spin half, using the Hecke algebra generators, or R-matrices. For the R-matrix we will use the notation of [47]. D.1. Uq (su(2)) conventions. We first summarize some of the formulas of [46, 47] that we will use later. The Uq (su(2)) algebra and coproduct are [46]: H e − eH = 2e, H f − f H = −2 f, q H/2 − q −H/2 e f − f e = 1/2 , q − q −1/2 (e) = e ⊗ q H/4 + q −H/4 ⊗ e .
(D.1)
For later convenience, we note that the map to the notation of [47] is q e f H
→ q, → X +, → X −, → H.
(D.2)
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The universal R-matrix in this basis is [47] R=q
H ⊗H 4
∞ (1 − q −1 )n
[n]!
n=0
(q H/4 X + )n ⊗ (q −H/4 X − )n ,
(D.3)
where [n] is as in (2.35). Together with the action of the generators on spin-half states, 1 1 1 1 e | , − = | , , 2 2 2 2 1 1 1 1 f | , = | , − , 2 2 2 2 1 1 1 1 H | , ± = ±| , , 2 2 2 2
(D.4)
this determines the R-matrix as follows: 1
R 12 R R
, 21
1 2 ,2 1 1 2 ,− 2 1 1 2 ,− 2 1 1 2 ,− 2 − 21 , 12
=R =R
− 12 ,− 12 − 21 ,− 21 − 12 , 21 − 21 , 21
= q 1/4 ,
= q −1/4 ,
= q −1/4 (q 1/2 − q −1/2 ) .
(D.5)
D.2. Schur-Weyl duality in spin-one. As in the classical case, the q-characters in higher representations can be written in terms of q-characters of lower representations. Consider for concreteness the case of spin-one, which is contained in the tensor product of two spin-half representations V . There is a projector (2.9) acting on V ⊗ V that leads to the symmetric representation. In the classical case it is just 21 (1 + P), where P is the permutation of the two tensor factors. In the quantum case P does not commute with the coproduct, but P R ≡ Rˇ does: ˇ Rˇ = R.
(D.6)
When Rˇ acts on the tensor product of two spin half irreps, it satisfies a relation of the form Rˇ 2 = q −1/4 (q 1/2 − q −1/2 ) Rˇ + q −1/2 .
(D.7)
A rescaling g = q 3/4 Rˇ can be done to map to the standard form of the Hecke algebra used in the main text. A matrix element of some element h of Uq (su(2)) in the spin 1 representation can be written in terms of a product of spin half reps by using the Clebsch-Gordan coefficients. Consider now the following matrix element in the spin-one representation: j = 1, n|h| j = 1, m = d nj=1;m (h) = h, d nj=1;m .
(D.8)
d nj;m is the representation matrix in representation j with indices n, m, and d mj;m its trace. In the last equation we have expressed the fact that the matrix elements can be viewed as living in the dual space Uq (su(2)), denoted by Funq (SU (2)). For more details on this duality see for example [18, 19, 13].
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We now express this in terms of matrix elements of the fundamental representation. They generate Funq (SU (2)), the deformed algebra of functions on SU (2). Using the Clebsch-Gordan coefficients, we can rewrite the above as follows: Cnn1 n 2 Cmm 1 m 2 j = 1, n|h| j = 1, m = m 1 ,m 2 ;n 1 ,n 2
1 1 1 1 × j = , n 1 | ⊗ j = , n 2 |(h 1 ⊗h 2 )| j = , m 1 ⊗| j = , m 2 2 2 2 2 Cnn1 n 2 Cmm 1 m 2 d n 1 1 (h 1 ) d n 2 1 (h 2 ) = j= 2 ;m 1
m 1 ,m 2 ;n 1 ,n 2
=
Cnn1 n 2 Cmm 1 m 2 h, d n 1
j= 12 ;m 1
m 1 ,m 2 ;n 1 ,n 2
j= 2 ;m 2
d n2
j= 21 ;m 2
.
(D.9)
In the first equality, the co-product (h) = h 1 ⊗ h 2 gives the action of h on the tensor product V ⊗ V . In the last equality, we used the fact that the dual pairing of a product of two elements in Funq (SU (2)) is given by the co-product. Now we can sum over m and use the identity between Clebsch-Gordan coefficients and projectors (see for example [42])
1 1 , ;1 . (D.10) Cnm1 n 2 Cmm 1 m 2 = Pnm11nm2 2 2 2 m ˇ For j = 1, the projector The projector is a linear combination of the identity and the R. is in the tensor product of two spin-half representations. It has to be a linear combination of 1 and Rˇ since the Hecke algebra generates the centralizer of the quantum group action in the tensor product:
1 1 ˇ P , ; 1 = a + b R, (D.11) 2 2 and for the matrix elements we have d nj=1;m = d n1 m 1 ,m 2 ;n 1 ,n 2
j= 21 ;m 1
d n2
j= 21 ;m 2
Cmm 1 m 2 Cnn1 n 2 .
(D.12)
To compute the character, we want the trace of this equation. Using (D.10), and expanding the projector in terms of the R-matrix as in (D.11), we get: ˇ 1 ⊗ 1)(1 ⊗ d 1 )) , tr1 d = a (tr 1 d) (tr 1 d) + b tr1 ( R(d 2
2
2
which, written out in indices, reads: m m d mj=1;m = a δn 11 δn 22 + b Rnm12nm2 1 d n1 1 m 1 ,m 2 ;n 1 ,n 2
m
(D.13)
2
2
;m 1
d n1 2 2
;m 2
.
(D.14)
We will show that the above equation can indeed be solved for constants a, b. The left-hand side can be calculated to give: √ d mj=1;m = x 2 + (x y + quv) + y 2 m
= x 2 + y 2 + x y(1 + q) − q,
(D.15)
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where we have used Eqs. (36-40) of [46] ( recalling that x, y, u, v are the matrix entries of d in the fundamental representation). For the right-hand side of (D.14) we get √ √ (a + bq 1/4 )(x 2 + y 2 ) + ax y + 2buvq −1/4 + (a + bq −1/4 ( q − 1/ q))yx. (D.16) Using the relations yx = (1 − q) + q x y, uv = q 1/2 (x y − 1),
(D.17)
x 2 , y 2 , x y, 1. Comparing with (D.15) and considering
we can rewrite (D.16) in terms of the coefficient of x 2 + y 2 we immediately see that
a + b q 1/4 = 1 .
(D.18)
With this condition the coefficient of x y becomes (q + 1) as desired. Comparing coefficients of the constant term then determines a=
1 q 3/4 , b= . 1+q 1+q
(D.19)
Putting everything together, and going back to the notation used in the main text, we get:
q 3/4 1 tr U tr U + tr ⊗ tr Rˇ (U ⊗ 1)(1 ⊗ U ) (D.20) tr1 U = 1+q 1+q which is q-Schur-Weyl duality (2.20) for n = 1. By comparing (D.7) and with the first ˇ Then the projector can be read from of (2.6) we can see that we can define g = q 3/4 R. above, P
=
1 (1 + g), 1+q
(D.21)
and agrees with (2.9) and the general form (2.28). D.3. Quantum characters in spin-one representation. The quantum characters can be obtained from the above by including the u-element (B.1) in the trace, which is basically q −H . In fact, we will do a slightly more general computation of the trace with an insertion of q AH . Thus, we consider the matrix element in the spin one representation of hq AH where A is an arbitrary number and h is an arbitrary element o Uq (su(2)) : j = 1, n|h q AH | j = 1, m = q Am d nj=1;m (h).
(D.22)
As before, we now rewrite this in spin-half matrix coefficients using the Clebsch-Gordan coefficients: j = 1, n|hq AH | j = 1, m =
m 1 ,m 2 ;n 1 ,n 2
m m2
Cnn1 n 2 Cm 1
1 1 1 1 × j = , n 1 |⊗ j = , n 2 |(h 1 ⊗h 2 )(q AH ⊗q AH )| j= , m 1 ⊗| j = , m 2 2 2 2 2 m m n1 n Cnn1 n 2 Cm 1 2 d j=1/2;m (h 1 ) d 2 1 (h 2 ) q Am 1 + Am 2 = 1
m 1 ,m 2 ;n 1 ,n 2
=
m 1 ,m 2 ;n 1 ,n 2
j= 2 ;m 2
m m 2 Am 1 + Am 2 n n q h, d 1 1 d 2 . j= 2 ;m 1 j= 21 ;m 2
Cnn1 n 2 Cm 1
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Again, we can sum over m = n to take the trace (notice the presence of q AH so this gives the quantum trace) and use (D.10) to get q Am d mj=1;m = q A(m 1 +m 2 ) a δnm11 δnm22 + b Rnm12nm2 1 d n1 1 d n1 2 . m
2
m 1 ,m 2 ;n 1 ,n 2
;m 1
2
;m 2
(D.23) For comparison to the classical formulae, it is useful to rewrite it as
tr1 (q AH U ) = a tr (q AH U ) tr (q AH U ) + b tr ⊗ tr q AH Rˇ (U ⊗ 1)(1 ⊗ U ) . (D.24) In the q → 1 limit, Rˇ goes to the permutation P and the second term becomes 21 tr(U 2 ). We still need to compute the constants a, b in this case. Writing out the traces using [46], we get: tr1 (q AH U ) = q A x 2 + q −A y 2 + 1 + (q 1/2 + q −1/2 )uv,
tr ⊗ tr q A H ⊗H (U ⊗ 1)(1 ⊗ U ) = (tr (q AH U ))2 = q A x 2 + q −A y 2 + 2 + (q 1/2 + q −1/2 )uv,
tr ⊗ tr q A H ⊗H Rˇ (U ⊗ 1)(1 ⊗ U ) = q 1/4 [q A x 2 + q −A y 2 + 1 − q −1 + (q 1/2 + q −1/2 )uv], (D.25)
where A denotes an arbitrary power. It is now easy to see that the values of a, b (D.19) are still the same, independently of the value of A. From the explicit computation (D.25) we also get the special N = 2 relations, Tr U = 1, Tr
U = (tr U )2 − 1 .
(D.26)
References 1. Migdal, A.A.: Recursion Equations In Gauge Field Theories. Sov. Phys. JETP 42, 413 (1975) [Zh. Eksp. Teor. Fiz. 69, 810 (1975)] 2. Gross, D.J.: Two-dimensional QCD as a string theory. Nucl. Phys. B 400, 161 (1993) 3. Gross, D.J., Taylor, W.I.: Two-dimensional QCD is a string theory. Nucl. Phys. B 400, 181 (1993) 4. Gross, D.J., Taylor, W.I.: Twists and Wilson loops in the string theory of two-dimensional QCD. Nucl. Phys. B 403, 395 (1993) 5. Cordes, S., Moore, G.W., Ramgoolam, S.: Lectures on 2-d Yang-Mills theory, equivariant cohomology and topological field theories. Nucl. Phys. Proc. Suppl. 41, 184 (1995) 6. Cordes, S., Moore, G.W., Ramgoolam, S.: Large N 2-D Yang-Mills theory and topological string theory. Commun. Math. Phys 185, 543 (1997) 7. Horava P.: Topological strings and QCD in two-dimensions. http://arxiv.org/list/hep-th/9311156, 1993 8. Vafa C.: Two dimensional Yang-Mills, black holes and topological strings. http://arxiv.org/list/hepth/0406058, 2004 9. Bryan J., Pandharipande R.: The local Gromov-Witten theory of curves. http://arxiv.org/list/math.ag/ 0411037, 2004 10. Aganagic, M., Ooguri, H., Saulina, N., Vafa, C.: Black holes, q-deformed 2d Yang-Mills, and non-perturbative topological strings. Nucl. Phys. B 715, 304 (2005) 11. Haro, S. de : A note on knot invariants and q-deformed 2d Yang-Mills. Phys. Lett. B 634, 78 (2006) 12. Boulatov, D.V.: q deformed lattice gauge theory and three manifold invariants. Int. J. Mod. Phys. A 8, 3139 (1993) 13. Buffenoir, E., Roche, P.: Two-dimensional lattice gauge theory based on a quantum group. Commun. Math. Phys. 170, 669 (1995)
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14. Klimcik, C.: The formulae of Kontsevich and Verlinde from the perspective of the Drinfeld double. Commun. Math. Phys. 217, 203 (2001) 15. Jimbo, M.: A q-analog of U (gl(N + 1)), Hecke algebras and the Yang-Baxter equation. Lett. Math. Phys. 11, 247–252 (1986) 16. King, R.C., Wybourne, B.G.: Representations and traces of the Hecke algebrasHn (q) of type An−1 . J. Math. Phys. 33(1), 4 (1992) 17. Faddeev, L.D., Reshetikhin, N.Y., Takhtajan, L.A.: Quantization Of Lie Groups And Lie Algebras. Lengingrad Math. J. 1, 193 (1990) 18. Majid, S.: Foundations of quantum group theory. Cambridge: Cambridge Univ. Press, (1995) 19. Coquereaux, R., Schieber, G.E.: Action of a finite quantum group on the algebra of complex N × N matrices. AIP Conf. Proc. 453, Melville, NY: Amer. Inst. of Physics, 1998 pp. 9–23 20. Ram, A.: A Frobenius formula for characters of the Hecke algebra. Invent. Math. 106, 461–488 (1991) 21. Gyoja, A.: A q-analogue of Young Symmetrizer. Osaka J. Math. 23, 841–852 (1986) 22. Francis, A.: The Minimal Basis for the Centre of an Iwahori-Hecke Algebra. J. Algebra 221, 1–28 (1999) 23. Francis, A., Jones, L.: On bases of centres of Iwahori-Hecke algebras of the symmetric group. J. Algebra 289(1), 42–69 (2005) 24. Ramgoolam, S.: Wilson loops in 2-D Yang-Mills: Euler characters and loop equations. Int. J. Mod. Phys. A 11, 3885 (1996) 25. Dijkgraaf, R., Vafa, C., Verlinde, E.P., Verlinde, H.L.: The Operator Algebra Of Orbifold Models. Commun. Math. Phys. 123, 485 (1989) 26. Dijkgraaf, R., Witten, E.: Topological Gauge Theories And Group Cohomology. Commun. Math. Phys. 129, 393 (1990) 27. Freed, D.S., Quinn, F.: Chern-Simons theory with finite gauge group. Commun. Math. Phys. 156, 435 (1993) 28. Haro, S. de: Chern-Simons theory, 2d Yang-Mills, and lie algebra wanderers. Nucl. Phys. B 730, 312 (2005) 29. Dijkgraaf, R., Moore, G.W.: Balanced topological field theories. Commun. Math. Phys. 185, 411 (1997) 30. Caporaso, N., Cirafici, M., Griguolo, L., Pasquetti, S., Seminara, D., Szabo, R.J.: Topological strings and large N phase transitions. I: Nonchiral expansion of q-deformed Yang-Mills theory. JHEP 0601, 035 (2006) 31. Caporaso, N., Cirafici, M., Griguolo, L., Pasquetti, S., Seminara, D., Szabo, R.J.: Topological strings and large N phase transitions. II: Chiral expansion of q-deformed Yang-Mills theory. JHEP 0601, 036 (2006) 32. Brzezinski, T., Dabrowski, H., Rembielinski, J.: On the quantum differential calculus and the quantum holomorphicity. J. Math. Phys. 33(1), 19–24 (1992) 33. Krieg A. Hecke Algebras. Memoirs of the American Mathematical Society, 87, 435, Providence, RI: Amer. Math. Soc, 1990 34. Fulton, W.: Hurwitz schemes and irreducibility of the moduli spaces of algebraic curves. Ann. of Math. (2) 90, 542–575 (1969) 35. Haro, S. de : Chern-Simons theory in lens spaces from 2d Yang-Mills on the cylinder. JHEP 0408, 041 (2004) 36. Beasley, C., Witten, E.: Non-abelian localization for Chern-Simons theory. J. Diff. Geom. 70, 183– 323 (2005) 37. Blau, M., Thompson, G.: Chern-Simons theory on S**1-bundles: Abelianisation and q-deformed YangMills theory. JHEP 0605003 , (2006) 38. Chung, S.W., Fukuma, M., Shapere, A.D.: Structure of topological lattice field theories in three-dimensions. Int. J. Mod. Phys. A 9, 1305 (1994) 39. Fukuma, M., Hosono, S., Kawai, H.: Lattice topological field theory in two-dimensions. Commun. Math. Phys. 161, 157 (1994) 40. Martin P.M.: On Schur-Weyl duality, An Hecke algebras and quantum sl(N ) on n+1 C N . Infinite analysis, Part A, B (Kyoto, 1991) Adv. Ser. Math. Phys. Vol. 16 River Edge. NJ: World Sci. Publ., 1992, pp. 645–673 41. Be˘ılinson, A.A., Lusztig, G., MacPherson, R.D.: Duke Math. J. 61(2), 655–677 (1990) 42. LeClair, A., Ludwig, A., Mussardo, G.: Integrability of coupled conformal field theories. Nucl. Phys. B 512, 523 (1998) 43. Katriel, J., Abdelassam, B., Chakrabarti, A.: The fundamental invariant of the Hecke algebra Hn (q) characterizes the representations of Hn (q), Sn , SUq (N ) and SU (N ). J. Math. Phys. 36, 5139–5158 (1995) 44. Dipper, R., James, G.D.: Blocks and idempotents of Hecke algebras of general linear groups. Proc. Lon. Math. Soc. 3(54), 57 (1987) 45. Ogievetsky O., Pyatov P.: Lecture on Hecke algebra. Based on lectures at the International School “Symmetries and Integrable systems”. (Dubna, 8–11 June, 1999). Dubna: JINR Publ. Dept., 2000
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46. Nomura, M.: Representation functions dmk of U [slq (2)] as wavefunctions of “Quantum symmetric tops” and Relationship to Braiding matrices. J. Phys. Soc. Japan 59(12), 4260–4271 (1990) 47. Kirillov, A.N., Reshetikhin, N.Yu.: Representations of the algebra Uq (sl(2)), q orthogonal polynomials and invariants of links. Adv. Series in Math. Phys. 7, 285–339 (1989) Communicated by M.R. Douglas
Commun. Math. Phys. 273, 357–378 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0237-z
Communications in
Mathematical Physics
Looped Cotangent Virasoro Algebra and Non-Linear Integrable Systems in Dimension 2 + 1 V. Ovsienko, C. Roger Institut Camille Jordan, Université Claude Bernard Lyon 1, 21 Avenue Claude Bernard, 69622 Villeurbanne Cedex, France. E-mail: [email protected]; [email protected] Received: 26 April 2006 / Accepted: 31 October 2006 Published online: 11 May 2007 – © Springer-Verlag 2007
Abstract: We consider a Lie algebra generalizing the Virasoro algebra to the case of two space variables. We study its coadjoint representation and calculate the corresponding Euler equations. In particular, we obtain a bi-Hamiltonian system that leads to an integrable non-linear partial differential equation. This equation is an analogue of the Kadomtsev–Petviashvili (of type B) equation. 1. Introduction and Motivations This work was initiated by the question asked to one of the authors by T. Ratiu: can the Kadomtsev–Petviashvili equation be realized as an Euler equation on some infinitedimensional Lie group? In order to make the above question understandable, let us start with some basic definitions. 1.1. The KP and BKP equations as two generalizations of KdV. The famous Korteweg–de Vries (in short KdV) equation ut = 3 u x u + c u x x x ,
(1.1)
where u(t, x) is a smooth (complex valued) function, is the most classic example of an integrable infinite-dimensional Hamiltonian system. The Kadomtsev–Petviashvili (KP) is a “two space variable” generalization of KdV. A function u(t, x, y) on R3 of time t and two space variables x, y satisfies KP if one has u t = 3 u x u + c1 u x x x + c2 ∂x−1 u yy ,
(1.2)
where, as usual, the partial derivatives are denoted by the corresponding variables as lower indices, c1 , c2 ∈ C are arbitrary constants and ∂x−1 denotes the indefinite integral.
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Of course, there is some ambiguity in this definition, so that one can prefer to use the following form: u t x = 3 u x x u + 3 u 2x + c1 u x x x x + c2 u yy . The constants c1 , c2 are called central charges for reasons that will be clarified below. Let us notice that, if u does not depend explicitly on y then (1.2) reduces to the KdV with c = c1 . Another example is the more recent version of the KP equation, namely the so-called KP of type B (BKP) (see [18, 5]). The dispersionless BKP is of the form u t = α u x u 2 + β u y u + u x ∂x−1 u y + c ∂x−1 u yy ,
(1.3)
where α, β and c are some constants. Equations (1.1)-(1.3) are infinite-dimensional integrable (in a weak algebraic sense, cf. [30]) systems. They correspond to infinite hierarchies of conservation laws and an infinite series of commuting evolution equations on the space of functions. They are interesting both for mathematics and theoretical physics. Remark 1.1. The classic KP equation (1.2) should not be confused with the KP hierarchy that became more popular than the original KP equation itself. The KP hierarchy is a family of integrable P.D.E. obtained by an inductive algebraic construction; the KdV equation appears as the first term in this family while the “classic” KP corresponds to the third term (see [30] and [1]). For more details on the KP and BKP hierarchies see, e.g., [13].
1.2. Euler equations. The notion of Euler equation means in our context some grouptheoretic generalizations of the classic Euler equation for the rigid solid motion. Let G be a Lie group, g the corresponding Lie algebra and g∗ the dual of g equipped with the canonical linear Poisson structure (or the Kirillov-Kostant-Souriau bracket). Consider a linear map I : g → g∗ called the inertia operator. Definition 1.2. The Euler equation is the Hamiltonian vector field on g∗ with the quadratic Hamiltonian function H (m) =
1 −1 I (m), m 2
(1.4)
for m ∈ g∗ and , being the natural pairing between g and g∗ . The well-known formula for this vector field is m t = {H, m} = −ad∗dm H m = −ad∗I −1 (m) m,
(1.5)
where { , } is the canonical Poisson bracket on g∗ and ad∗ is the coadjoint action of g on g∗ and dm H = I −1 (m) is the differential of the function (1.4) (see, e.g., [2, 3]).
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1.3. The role of central extensions. In many cases, one is naturally lead to consider central extensions g of g. A central extension is given by a set of non-trivial 2-cocycles µi ∈ Z 2 (g; C) with i = 1, . . . , k. As a vector space, g∼ = g ⊕ Ck , where the second summand belongs to the center. Consider the dual space g∗ ∼ = g∗ ⊕ Ck and fix arbitrary values (c1 , . . . , ck ) ∈ Ck ; ∗ the affine subspace g(c1 ,...,ck ) ⊂ g∗ is stable with respect to the coadjoint action of g. ∗ Identifying the affine subspace g(c1 ,...,ck ) with g∗ and restricting the coadjoint action of g to g (since the center acts trivially), one obtains a k-parameter family of actions of g on g∗ : ∗x = ad∗x + ad
k
ci Si (x)
for
x ∈ g,
(1.6)
i=1
where Si : g → g∗ are 1-cocycles on g with values in the coadjoint representation. More precisely, the above 1-cocyles Si are related with the 2-cocycles µi via µi (x, y) = Si (x), y − Si (y), x
for
(x, y) ∈ g2 .
Formula (1.6) is due to Kirillov (see [14]) for the details); the 1-cocycles Si are sometimes called the Souriau cocycles associated with µi . With the above modifications of the coadjoint action and the corresponding Poisson structure, the Euler equation becomes xt = −ad ∗I −1 (x) x + ci Si I −1 (x) , (1.7) so that one adds to the original equation, which is usually quadratic, some extra linear terms. 1.4. The main results. Our work consists in two parts: • we introduce an infinite-dimensional Lie algebra which seems to be a nice generalization of the Virasoro algebra with two space variables (the kinematics); • we find the relevant inertia operators and Euler equations (the dynamics), in particular, we are interested in the bi-Hamiltonian Euler equations. We consider the loop algebra over the semidirect product of the Virasoro algebra and its dual and classify its non-trivial central extensions. It turns out that one of these central extensions is related simultaneously to the Virasoro and Kac-Moody algebras. We then study the coadjoint representation of this Lie algebra. We also introduce a Lie superalgebra extending the constructed Lie algebra. This superalgebra is a generalization of the Neveu-Schwarz algebra. We compute several Euler equations corresponding to our Lie algebra. The first example is very similar to the KP equation (1.2), yet different from it. In fact, this equation is nothing but KP with supplementary terms and coupled with another equation. This equation cannot be reduced to KP, however, it reduces to KdV. We do not know if this Hamiltonian system is integrable in any sense. The second Euler equation we obtained in this paper leads to the following differential equation gt = gx ∂x−1 g y − g y g + c ∂x−1 g yy ,
(1.8)
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where g = g(x, y, t) and c ∈ C. To avoid non-local expressions, one can rewrite this equation as a system: gt + gx h − h x g + c h y = 0,
g y + h x = 0.
We prove integrability of (1.8) in the following sense. There exists an infinite hierarchy of vector fields commuting with (1.8) and with commuting flows. The first commuting fields are: gt = gx and gt = g y ; one more higher field is provided by Example 5.9. We use the bi-Hamiltonian technique. More precisely, we obtain Eq. (1.8) coupled together with another differential equation (see formula (5.7)) so that the system of two equations is a bi-Hamiltonian vector field. Equation (1.8) was studied in [8] (see formula (33)) and [9]. It has also been considered in differential geometry [7]. In a more general setting, this equation is the second term of the so-called universal hierarchy [23]. Although Eq. (1.8) resembles KP and especially BKP (1.3), it is different: there are no cubic terms and, foremost, the sign in the quadratic term is different. This may be important, especially if one works over R (rather than over C). 1.5. Historical overview. The most classic case is related to the Lie group G = SO(3) and the inertia operator I given by a symmetric tensor in S 2 g (the usual inertia tensor on the rigid solid). One gets the genuine Euler equation. The first generalization was obtained by V.I. Arnold (1966) for the hydrodynamical Euler equation of an incompressible fluid, hence the name of Euler-Arnold is sometimes granted to these equations. The relevant group in this case is the group SDiff(D) of volume-preserving diffeomorphisms of a domain D ⊂ R2 or R3 (see the books [2, 3]). Some interesting examples, such as the Landau-Lifchitz equation, correspond to the Euler equations on the Kac-Moody groups (see [15]). Another example has already been mentioned: the KdV equation is an Euler equation on the Virasoro-Bott group (see [17]). This group is defined as the unique (up to isomorphism) non-trivial central extension of the group Diff(S 1 ) of all diffeomorphisms of S 1 . The inertia operator is given by the standard L 2 -metric on S 1 . In [24] and [16] different choices of the metric on Diff(S 1 ) (and thus of the inertia operator) were considered in order to get some other equations than KdV, such as the Hunter-Saxon and the Camassa-Holm equations. Let us also mention that there is a huge literature containing different generalizations of KdV (in the super case, matrix versions, etc.) in the dimension 1 + 1. All these generalizations are related to some extensions of the Virasoro algebra. Let us finally stress that the property of an evolution equation to be an Euler equation associated with some Lie group (or Lie algebra) is important for the following reason: it allows one to deal with the equation as with the geodesic flow and to apply a wide spectrum of methods specific for differential geometry (see, e.g., [25]). 2. The Virasoro Algebra and its Loop Algebra In this section we introduce the preliminary examples of infinite-dimensional Lie algebras that we will consider. We define the Virasoro algebra and show how to obtain the KdV equation as an Euler equation on it. We then consider the loop group L Diff(S 1 ) . We classify non-trivial central extensions of the corresponding Lie algebra.
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2.1. Reminder: the Virasoro algebra and KdV equation. The Virasoro algebra is defined as the central extension of Vect(S 1 ) given by the 2-cocycle ∂ ∂ , g(x) = f gx x x d x. (2.1) µ f (x) ∂x ∂x S1 This cocycle was found in [12] and is known as the Gelfand-Fuchs cocycle. The cohomology group H 2 (Vect(S 1 ); C) is one-dimensional so that the cocycle (2.1) defines the unique (up to isomorphism) non-trivial central extension of Vect(S 1 ). Consider a natural family of modules over Vect(S 1 ) (and therefore over the Virasoro algebra with the trivial action of the center). Let Fλ be the space of λ-densities (or the space of tensor densities of degree λ) on S 1 ,
Fλ = ϕ(x) d x λ ϕ(x) ∈ C ∞ (S 1 ) , where λ ∈ C. The action of Vect(S 1 ) on the space Fλ is given by the first-order differential operator L λf ϕ(x) d x λ = ( f ϕx + λ f x ϕ) d x λ (2.2) which is nothing but the Lie derivative along the vector field f = f (x) ∂∂x . Let us calculate the coadjoint action of the Virasoro algebra. The dual space, Vect(S 1 )∗ , corresponds to the space of all distributions on S 1 . Following [15], we will consider only the regular part of this dual space that consists of differentiable 2-densities, that is Vect(S 1 )∗reg = F2 with the natural pairing ∂ , u(x) d x 2 = f (x) f (x) u(x) d x. ∂x S1 The coadjoint action of Vect(S 1 ) coincides with the Vect(S 1 )-action on F2 . The Souriau cocycle on Vect(S 1 ) corresponding to the Gelfand-Fuchs cocycle is ∂ = fx x x d x 2 S f (x) ∂x and one finally obtains the coadjoint action of the Virasoro algebra: 2 ∗ u(x) d x = ( f u x + 2 fx u + c fx x x ) d x 2. (2.3) ad ∂ f (x) ∂x
Consider the simplest quadratic Hamiltonian function on Vect(S 1 )∗ 1 2 H u(x) d x = u(x)2 d x 2 S1 corresponding to the inertia operator I ( f (x) ∂∂x ) = f (x) d x 2 . The following result was obtained in [17]. Proposition 2.1. The Euler equation on the Virasoro algebra corresponding to the Hamiltonian H is precisely the KdV equation (1.1). Proof. Immediately follows from formulæ (1.5) and (2.3).
For more details about the Virasoro algebra, its modules and its cohomology see [10] and [11].
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2.2. The loop group on Diff(S 1 ) and the loop algebra on Vect(S 1 ). We wish to extend the Virasoro algebra to the case of two space variables. A natural way to do this is to consider the loops on it. One defines the loop group on Diff(S 1 ) as follows:
L Diff(S 1 ) = ϕ : S 1 → Diff(S 1 ) | ϕ is differentiable , the group law being given by (ϕ ◦ ψ) (y) = ϕ(y) ◦ ψ(y),
y ∈ S1.
Remark 2.2. Let us stress that there are no difficulties in defining differentiable maps with values in Diff(S 1 ). Indeed, L Diff(S 1 ) is naturally embedded into the space of C ∞ -maps on S 1 × S 1 with values in S 1 . In a similar way, we construct the Lie algebra L Vect(S 1 ) consisting of vector fields on S 1 depending on one more independent variable y ∈ S 1 . The loop variable is thus of S 1 by x. The elements denoted by y and the variable on∂ the “target” copy 1 ∞ of L Vect(S ) are of the form: f (x, y) ∂ x , where f ∈ C (S 1 × S 1 ) and the Lie bracket reads as follows:
∂ ∂ ∂ f (x, y) , g(x, y) = ( f (x, y) gx (x, y) − f x (x, y) g(x, y)) . ∂x ∂x ∂x It is easy to convince oneself that L Vect(S 1 ) is the Lie algebra of L Diff(S 1 ) in the usual weak sense for the infinite-dimensional case; a one-parameter group argumentation gives an identification between the tangent space to L Diff(S 1 ) at the identity and L Vect(S 1 ) , equipped with its Lie bracket. We will now classify non-trivial central extensions of the Lie algebra L Vect(S 1 ) and therefore calculate H 2 (L Vect(S 1 ) ; C). This result can be deduced from a more general one that we will need later. 2.3. Central extensions of tensor products. Following the work of Zusmanovich [32], one can calculate the cohomology group H 2 (g ⊗ A) for a Lie algebra g and a commutative algebra A over a field k, the Lie bracket on g ⊗ A being defined by [x1 ⊗ a1 , x2 ⊗ a2 ] = [x1 , x2 ] ⊗ a1 a2 ,
xi ∈ g, ai ∈ A, i = 1, 2.
From the results of [32], one can easily deduce the following Proposition 2.3. If g = [g, g], then H 2 (g ⊗ A; k) = H 2 (g; k) ⊗ A ⊕ Invg S 2 (g∗ ) ⊗ H C 1 (A) , where H C 1 (A) is the first group of cyclic cohomology of the k-algebra A and Invg S 2 (g∗ ) is the space of g-invariant symmetric bilinear maps from g into k, while A = Homk (A, k) represents the dual of A. One can give explicit formulæ for the cohomology classes.
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• Given µ ∈ Z 2 (g; k) and λ ∈ A , one gets µλ ∈ Z 2 (g ⊗ A; k) defined by µλ (x1 ⊗ a1 , x2 ⊗ a2 ) = µ(x1 , x2 ) λ(a1 a2 ).
(2.4)
• Given K ∈ Invg S 2 (g∗ ) and ∈ H C 1 (A), one gets K ∈ Z 2 (g ⊗ A; k) defined by K (x1 ⊗ a1 , x2 ⊗ a2 ) = K (x1 , x2 ) (a1 da2 ),
(2.5)
where d is the Kähler derivative. For the general results on cyclic homology and cohomology see [19]. Example 2.4. If g is a finite-dimensional semisimple Lie algebra and A = C ∞ (S 1 ), formula (2.5) defines the Kac-Moody cocycle K M(x1 ⊗ a1 , x2 ⊗ a2 ) = K (x1 , x2 )
S1
a1 da2 ,
where K is the Killing form. Remark 2.5. In the general situation, one can call the cohomology classes of the cocycles (2.5) the classes “of Kac-Moody type”. For instance, such a class on the loop algebra over the algebra of pseudodifferential symbols has already been used in [30] in order to obtain the KP equation as a Hamiltonian system. 2.4. Central extensions of L Vect(S 1 ) . In our case, g = Vect(S 1 ) and A = C ∞ (S 1 ), one has the following well-known statement: Invg S 2 (g∗ ) = 0, that is, there is no invariant bilinear symmetric form (“Killing form”) on Vect(S 1 ) (see, e.g., [10]). Proposition 2.3 then implies the following result. Proposition 2.6. One has H 2 (L V ect (S 1 ) ; C) = C ∞ (S 1 ) . To a distribution λ ∈ C ∞ (S 1 ) one associates a 2-cocycle µλ ∈ Z 2 (L Vect(S 1 ) ; C) given by formula (2.4) with µ being the Gelfand-Fuchs cocycle (2.1). Similar results were obtained in [29] in a slightly different context. Proposition 2.6 provides a classification of non-trivial central extensions of the loop algebra L Vect(S 1 ) . This is rather a “negative result” for us since it implies that all these central extensions are of Virasoro type. The KP equation (1.2) contains two different central charges, c1 and c2 , and the second one does not belong to the Virasoro type but to the Kac-Moody one. It is clear then that KP-type (and BKP-type) equations cannot be obtained as Euler equations associated with the group L Diff(S 1 ) . One therefore needs to introduce another group with richer second cohomology. We will discuss further generalizations of L Vect(S 1 ) in the Appendix. However, this Lie algebra is not the one we are interested in.
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3. The Cotangent Virasoro Algebra and its Loop Algebra In this section we introduce the main object of our study. We consider the Lie algebra 1 ) and calculate its cenof loops associated with the cotangent Lie algebra T ∗ Vect(S 1 tral extensions. Unlike the loop Lie algebra L Vect(S ) , the constructed Lie algebra simultaneously has non-trivial central extensions of the Virasoro and the Kac-Moody types. 3.1. General setting: the cotangent group and its Lie algebra. The cotangent space T ∗ G of a Lie group G is naturally identified with the semi-direct product G g∗ , where G acts on g∗ by the coadjoint action Ad∗ . The space T ∗ G is then a Lie group with the product (g1 , u 1 ) · (g2 , u 2 ) = (g1 g2 , u 1 + Ad∗g1 u 2 ). The corresponding Lie algebra T ∗ g is the semi-direct product g g∗ equipped with the commutator (3.1) [(x1 , u 1 ), (x2 , u 2 )] = [x1 , x2 ], ad∗x1 u 2 − ad∗x2 u 1 . The evaluation map gives a natural symmetric bilinear form on T ∗ g, namely K ((x1 , u 1 ), (x2 , u 2 )) = u 1 , x2 + u 2 , x1 .
(3.2)
Furthermore, this form is non-degenerate so that one has a Killing type form on this Lie algebra. Remark 3.1. Let us mention that the semi-direct product g g∗ is the simplest case of so-called Drinfeld’s double which corresponds to the trivial Lie-Poisson structure. 3.2. The cotangent loop group and algebra. We will consider the cotangent group G = T ∗ Diff(S 1 ) and we willbe particularly interested in the associated loop group. We = L T ∗ Diff(S 1 ) for short. One has will use the notation G = L Diff(S 1 ) F2 = L Diff(S 1 ) L(F2 ). G Consider the semidirect product g = Vect(S 1 ) F2 . is The Lie algebra corresponding to the group G g = L(Vect(S 1 ) F2 ) = L Vect(S 1 ) L(F2 ).
(3.3)
An element of g is a couple ( f, u), where f and u are C ∞ -tensor fields on S 1 × S 1 of the following form: ∂ + u(x, y) d x 2 . ∂x The commutator (Lie bracket) is defined accordingly to (3.1). ( f, u) = f (x, y)
Remark 3.2.It is easy to check that, with ∗ the right convention on the duality, one has L(F2 ) = L Vect(S 1 )∗ = L Vect(S 1 ) ; the chosen form L(F2 ) will be more suitable for our computations.
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3.3. Central extensions of g. Let us calculate the cohomology space H 2 ( g, C) using Proposition 2.3. The following result is a classification of central extensions of the Lie algebra g. Theorem 3.3. One has g; C) = C ∞ (S 1 ) ⊕ C. H 2 ( Proof. From the classical results on the cohomology of the Virasoro algebra and its representations, one has H 2 (g, C) = C, where, as above, g = Vect(S 1 ) F2 , cf. [10] and [12]. The non-trivial cohomology class is again generated by the Gelfand-Fuchs cocycle. More precisely, µ (( f, u), (g, v)) = µ( f, g), where µ is as in (2.1). Furthermore, one has Invg S 2 (g∗ ) = C, where the generator is provided by the 2-form (3.2) and it is easy to check that there are no other generators. Finally, H C 1 (A) = C and is generated by the volume form (see, e.g., [19]).
Let us give the explicit formulæ of non-trivial 2-cocycles on g. A distribution λ ∈ C ∞ (S 1 ) corresponds to a 2-cocycle of the first class (2.4) given by f gx x x d x , µλ (( f, u), (g, v)) = λ S1
these are the Virasoro type extensions. For the particular case where λ(a(y)) = S 1 a(y)dy, such a 2-cocycle will be denoted by µ1 so that one has f gx x x d xd y. (3.4) µ1 (( f, u), (g, v)) = S 1 ×S 1
Another non-trivial cohomology class is provided by the 2-cocycle µ2 (( f, u), (g, v)) = f v y − g u y d xd y S 1 ×S 1
(3.5)
of Kac-Moody type (2.5). 3.4. The Lie algebra g. We define the Lie algebra g as the two-dimensional central extension of g given by the cocycles µ1 and µ2 . As a vector space, g = g ⊕ C2 , g. The commutator in g is given by the following where the summand C2 is the center of explicit expression which readily follows from the above formulæ:
∂ ∂ f + u d x 2, g + v dx2 ∂x ∂x ∂ + ( f vx + 2 f x v − g u x − 2 gx u) d x 2 = ( f gx − f x g) (3.6) ∂ x + f v y − g u y d xd y , f gx x x d xd y, × S 1 ×S 1
S 1 ×S 1
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where the last term is an element of the center of g. (Note that we did not write the central elements in the left hand-side since they do not enter into the commutator.) The Lie algebra g and its coadjoint representation is the main object of our study. This algebra is a natural two-dimensional generalization of the Virasoro algebra. Further generalizations will be described in the Appendix. 4. The Coadjoint Representation of g According to the general viewpoint of symplectic geometry and mechanics, the coadjoint representation of a Lie algebra plays a very special role for all sorts of applications. It was observed by Kirillov [15] and Segal [31] that the coadjoint action of the Virasoro group and algebra coincides with the natural action of Diff(S 1 ) and Vect(S 1 ), respectively, on the space of Sturm-Liouville operators. The Casimir functions (i.e., the invariants of the coadjoint action) are then expressed in terms of the monodromy operator. In this section we recall the Kirillov-Segal result and generalize it to the case of the Lie algebra g. 4.1. Computing the coadjoint action. Let us start with the explicit expression for the coadjoint action of g. As usual, in the case of semidirect product of a Lie algebra and its dual, the (regular) dual space g∗ is identified with g; the natural pairing being given by ∂ ∂ 2 2 f (4.1) + u dx , g + v dx = ( f v + g u) d xd y, ∂x ∂x S 1 ×S 1 which is nothing but a specialization of the form (3.2). Furthermore, one immediately obtains the Souriau 1-cocycles on g: ∂ + u d x 2 = fx x x d x 2, S1 f ∂x (4.2) ∂ ∂ S2 f + u dx2 = fy + u y d x 2, ∂x ∂x corresponding to µ1 and µ2 , respectively. We are ready to give the expression of the coadjoint action of the “extended” Lie algebra g on the regular dual space g∗ ∼ g. = Proposition 4.1. The coadjoint action of the Lie algebra g is given by ∂ ∂ ∗( f,u) g + v d x 2 = f gx − f x g + c2 f y ad (4.3) ∂x ∂x + f vx + 2 f x v−u x g−2 u gx + c1 f x x x + c2 u y d x 2 , while the center acts trivially. Proof. According to formula (1.6), one has to calculate the coadjoint action of g, which coincides with the adjoint one (cf. formula (3.6) with no central terms) and then to add the Souriau cocycles.
Our next task is to investigate a “geometric meaning” of the coadjoint action (4.3). However, this result will not be relevant for our main task: computing the Euler equations on g. Sections 4.2–4.6 can be omitted at the first reading.
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4.2. The Virasoro coadjoint action and Sturm-Liouville operators. Let us recall the Kirillov-Segal result on the coadjoint action of the Virasoro algebra (cf. [15, 31, 28]). Consider the space of second-order linear differential operators 2 d A = 2c + u(x), (4.4) dx where u(x) ∈ C ∞ (S 1 ). This is a Sturm-Liouville operator with periodic potential (also called a Hill operator). Define an action of Vect(S 1 ) on this space by the formula 3
−1
L f (A) := L 2f ◦ A − A ◦ L f 2 ,
(4.5)
where f = f (x) ddx and where L λf is the first-order differential operator L λf = f
∂ + λ fx ∂x
corresponding to the Lie derivative on the space Fλ of λ-densities (2.2). An elementary computation shows that the result of the action (4.5) is a differential operator of order 0 (that is, an operator of multiplication by a function). More precisely, one has L f (A) = f vx + 2 f x v + c f x x x . This formula coincides with formula (2.3) of the coadjoint action of the Virasoro algebra. We obtained the following result: the space of operators (4.4) equipped with the Vect(S 1 )-action is isomorphic to the coadjoint representation of the Virasoro algebra. Remark 4.2. The operator A is understood as a differential operator on tensor densities on S 1 , namely A : F− 1 → F 3 . The expression (2.2) is the Lie derivative on the space 2 2 Fλ and thus the action (4.5) is natural. This geometric meaning of the Sturm-Liouville operator has already been known by classics. This remarkable coincidence is just a simple observation, but it has important consequences. It relates the Virasoro algebra with the projective differential geometry (we refer to [28] and references therein for more details). For instance, one can now interpret the monodromy operator associated with A as an invariant of the coadjoint action and eventually obtain a classification of the coadjoint orbits of the Virasoro algebra. Sturm-Liouville operators are also closely related to the KdV hierarchy. Let us mention that another geometric approach to the study of the Virasoro coadjoint orbits can be found in [4]. 4.3. Relations with the Neveu-Schwarz superalgebra. It order to understand the origin of the Sturm-Liouville operator in the context of Virasoro algebra, we will apply Kirillov’s sophisticated method [14] that uses Lie superalgebras. Let us note that this is quite an unsusual situation when superalgebra helps to better understand the usual (“non-super”) situation. Let us recall the definition of the Neveu-Schwarz Lie superalgebra. Consider the direct sum k = Vect(S 1 ) ⊕ F− 1 2
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and define the structure of a Lie superalgebra on the space k by
∂ ∂ ∂ − 12 − 12 f = ( f gx − f x g + ϕψ) + ϕ dx , g + ψ dx ∂x ∂x ∂x 1 1 1 + f ψx − f x ψ − gϕx + gx ϕ d x − 2 , 2 2 which is symmetric on the odd part k1 = F− 1 . The Jacobi (super)identity can be easily 2 checked. Furthermore, the Gelfand-Fuchs cocycle (2.1) can be extended to the superalgebra k: ∂ ∂ − 12 − 12 = µ f (4.6) + ϕ dx , g + ψ dx ( f gx x x + 2 ϕψx x ) d x. ∂x ∂x S1 This 2-cocycle defines a central extension k of k called the Neveu-Schwarz algebra. The dual space of this superalgebra is k ∗ = F2 ⊕ F 3 , 2
since F ∗ 1 ∼ k can be easily calculated: = F 3 . The coadjoint action of − 2
∗ ad f
− 21 ∂ ∂ x +ϕ d x
2
3 1 3 u d x 2 + α d x 2 = f vx + 2 f x v + c f x x x + ϕ αx + ϕx α d x 2 2 2 (4.7) 3 3 f αx + f x α + u ϕ + 2 c ϕx x d x 2 . 2
The operator (4.4) is already present in this formula: it gives the action of the odd part of k on the even part of k∗ , namely ∗ − 1 u d x 2 = A(ϕ), ad ϕ dx
2
where A is as in (4.4). This way, the Sturm-Liouville operator naturally appears in the Virasoro context. The Kirillov-Segal result can now be deduced simply from the Jacobi identity for the superalgebra k. 4.4. Coadjoint action of g and matrix differential operators. It turns out that the coadjoint action of g given by the formula (4.3) can also be realized as an action of the non-extended algebra g on some space of differential operators. We introduce the space of 2 × 2-matrix differential operators ⎛ ⎞ ∂ 1 ∂ −c + g − g 0 2 x ⎜ ⎟ ∂ y ∂x 2 ⎟, A=⎜ (4.8) ⎝ ∂ 2 ∂ 3 ⎠ ∂ +g + gx 2c1 +v −c2 ∂x ∂y ∂x 2 where g and v are functions in (x, y), that is g, v ∈ C ∞ (S 1 × S 1 ). Let us define an action of the Lie algebra g on the space of operators (4.8).
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λ the space of loops in Fλ , i.e., of tensor fields on S 1 × S 1 of Let us now denote by F 1 ⊕F 3 is equipped with a g-action the form ϕ = ϕ(x, y) d x λ . The direct sum F Lf
∂ 2 ∂ x +u d x
1
ϕ dx− 2 3 ψ dx 2
−2
2
1 f ϕx − 21 f x ϕ d x − 2 3 . f ψx + 23 f x ψ + u ϕ d x 2
=
(4.9)
Assume the operator A acting on the above space of tensor densities: 3 → F 1 ⊕F 3 . 1 ⊕F A:F − − 2
2
2
2
Then the g-action is naturally defined by the commutator with the action (4.9): L f ∂ +u d x 2 (A) := L f ∂ +u d x 2 , A . (4.10) ∂x
∂x
The following statement is a generalization of the Kirillov-Segal result. Theorem 4.3. The action (4.10) of g on the space of differential operators (4.8) coincides with the coadjoint action (4.3) of the Lie algebra g. Proof. This statement can be checked by a straightforward computation.
In Sect. 4.6 we will give another, conceptual proof of this theorem. Let us also mention that similar results were obtained in [22, 21] for different examples of Lie algebras generalizing Virasoro. Theorem 4.3 implies that the invariants of the operators (4.8) are now invariants of the coadjoint orbits. It would be interesting to investigate these invariants, for instance, if there is an analogue of the monodromy operator of A. 4.5. Poisson bracket of tensor densities. Many of the explicit formulæ that we calculated in this paper (see for instance, (3.6), (4.9) and (4.7)) use the bilinear operation on tensor densities λ ⊗ F µ → F λ+µ+1 F given by
ϕ d x λ , ψ d x µ = (λ ϕ ψx − µ ϕx ψ) d x λ+µ+1 .
(4.11)
It is easy to check that the bilinear maps (4.11) are invariant. We will call the operation (4.11) the Poisson bracket of tensor densities. Let us rewrite some of the main formulæ using the Poisson bracket. The commutator −1 ⊕ F 2 is simply in g=F [( f, u), (g, v)] = ({ f, g}, { f, v} − {g, u}) ,
(4.12)
3 reads in invariant terms as follows: 1 ⊕F while the formula (4.9) of g-action on F − L ( f, u)
ϕ ψ
2
=
2
{ f ϕ} { f ψ} + u ϕ
.
(4.13)
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4.6. A Lie superalgebra extending g. Although the proof of Theorem 4.3 is, indeed, straightforward, it does not clarify the origin of the operators (4.8). The Kirillov method using the Lie superalgebras proved to be universal (see [27] for the details) and will be useful in our case. The Lie superalgebra we define in this section generalizes the Neveu-Schwarz algebra in the same sense as g generalizes the Virasoro algebra. It can be called the looped cotangent Neveu-Schwarz algebra. 1 ⊕F 3 and denote it by Consider the g-module F g1 . We will define the Lie super−2 2 algebra structure on = G g ⊕ g1 . Since we already know the g-action (4.9) on g1 , it remains to define the symmetric operation g1 → g, g1 ⊗ i.e., the “anticommutator”. Similarly to (4.12) and (4.13), let us set [(ϕ, α), (ψ, β)] = (ϕ ψ, {ϕ, β} + {ψ, α}) .
(4.14)
One immediately obtains the following Proposition 4.4. Formula (4.14) defines a structure of a Lie superalgebra on G. as even cocycles such that The cocycles (3.4) and (3.5) can be extended from g to G µ1 ((ϕ, α), (ψ, β)) = 2 ϕ ψx x d xd y (4.15) S 1 ×S 1
(cf. (4.6)) and µ2 ((ϕ, α), (ψ, β)) =
S 1 ×S 1
ϕ β y + ψ α y d xd y.
(4.16)
This defines a two-dimensional central extension G. First, observe that the (regular) dual space Let us calculate the coadjoint action of G. The corresponding Souriau-type cocycles extending (4.2) to G ∗ is isomorphic to G. G can also be easily calculated: 3
S1 (ϕ, α) = 2 ϕx x d x 2 , 1 3 S2 (ϕ, α) = −ϕ y d x − 2 − α y d x 2 ,
(4.17)
Finally, ∗ of G. so that one can write down the explicit formula of the coadjoint action ad ∗ to the odd part let us consider only the restriction of ad g1 and apply it to the even part ∗ which is of course isomorphic to of G g. More precisely, one obtains the map ∗ : ad g1 → End( g). One obtains the following
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Proposition 4.5. One has ∗ (g, v) = A ad ϕ α
ϕ , α
where A is the operator (4.8). This statement clarifies the origin of the linear differential operators (4.8). Proposition 4.5 also implies Theorem 4.3. The “kinematic” part of our work is complete; we now turn to the “dynamics”. 5. Euler Equations on g∗ In this section we calculate Euler equations associated with the Lie algebra g. We are particularly interested in the Euler equations which are bi-Hamiltonian. 5.1. Hamiltonian formalism on g∗ . Let us recall very briefly the explicit expression of the Hamiltonian vector fields on the dual space of a Lie algebra in the infinite-dimensional functional case (see, e.g., [6, 11] for more details). Given a functional H on g∗ which is a (pseudo)differential polynomial: H (g, v) = h g, v, gx , vx , g y , v y , ∂x−1 g, ∂x−1 v, ∂ y−1 g, ∂ y−1 v, gx y , vx y , . . . S 1 ×S 1
×d xd y,
where h is a polynomial in an infinite set of variables. The Hamiltonian vector field (1.5) with the Hamiltonian H reads ∗ δ H δ H (g, v), (g, v)t = −ad , δv
(5.1)
δg
where δδvH and δδgH are the standard variational derivatives given by the (generalized) Euler-Lagrange equations. For instance, δH ∂ ∂ = hv − h vx − δv ∂x ∂y ∂2 ∂2 + 2 h vx x + ∂x ∂ x∂ y
h v y − ∂x−1 h ∂x−1 v − ∂ y−1 h ∂ y−1 v
∂2 h vx y + 2 h v yy ± · · · , ∂y
where, as usual, h v means the partial derivative
∂h ∂v ,
similarly h vx =
∂h ∂vx .
5.2. An Euler equation on g∗ generalizing KP. Our first example is very close to the classic KP equation (1.2). Consider the following quadratic Hamiltonian on g∗ : H (g, v) =
v2 S 1 ×S 1
2
+ v ∂x−1 g y d xd y.
(5.2)
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The variational derivatives of H can be easily computed ∂ δH (g, v) = v + ∂x−1 g y , δv ∂x δH (g, v) = ∂x−1 v y d x 2 . δg We then use formula (5.1) and apply formula (4.3) of the coadjoint action to obtain the following result. Proposition 5.1. The Euler equation associated with the Lie algebra g and the Hamiltonian H is the following system: gt = vgx − vx g − g y g + gx ∂x−1 g y + c2 v y + c2 ∂x−1 g yy , (5.3) vt = 3vx v + c1 vx x x + c2 ∂x−1 v yy + 2vg y − v y g + vx ∂x−1 g y − 2gx ∂x−1 v y + c1 gx x y , with two indeterminates, g(t, x, y) and v(t, x, y). Note that the second equation in (5.3) can be written as vt = 3vx v + c1 vx x x + c2 ∂x−1 v yy + (linear terms in g), and this is nothing but the KP equation with some extra terms in g. In this sense, one can speak of the system (5.3) as a “generalized KP equation”. Remark 5.2. If one sets g ≡ 0 in (5.3), then the first equation gives v y = 0. The function v is thus independent of y and the second equation coincides with KdV (trivially “looped” by an extra variable y which does not intervene in the derivatives). Unfortunately, we do not know whether Eq. (5.3) is bi-Hamiltonian and have no information regarding its integrability. 5.3. “Bi-Hamiltonian formalism” on the dual of a Lie algebra. The notion of integrability of Hamiltonian systems on infinite-dimensional (functional) spaces can be understood in a number of different ways. A quite popular way to define integrability is related to the notion of bi-Hamiltonian systems that goes back to F. Magri [20]. The best known infinite-dimensional example is the KdV equation. Let us recall the standard way to obtain bi-Hamiltonian vector fields on the dual of a Lie algebra. Given a Lie algebra a, the canonical linear Poisson structure on a∗ is given by {F, G} (m) = [dm F, dm G] , m, where the differentials dm F and dm G at a point m ∈ a∗ are understood as elements of the Lie algebra a ∼ = (a∗ )∗ . Consider a constant Poisson structure: fix a point m 0 of the dual space and set {F, G}0 (m) = [dm F, dm G] , m 0 . It is easy to check that the above Poisson structures are compatible (or form a Poisson pair), i.e., their linear combination { , }λ = { , }0 − λ { , }
(5.4)
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is again a Poisson structure for all λ ∈ C. A function F on a∗ defines now two Hamiltonian vector fields associated with F: m t = ad∗dm F m
and
m t = ad∗dm F m 0
corresponding to the first and the second Poisson structure, respectively. Definition 5.3. A vector field X on a∗ is called bi-Hamiltonian if there are two functions, H and H such that X is a Hamiltonian vector field of H with respect to the Poisson structure { , } and is a Hamiltonian vector field of H with respect to { , }0 . Bi-Hamiltonian vector fields provide a rich source of integrable systems. Let H be a function on a∗ which is a Casimir function of the Poisson structure (5.4). This means, for every function F, one has {H, F}λ = 0.
(5.5)
Assume that H is written in a form of a series H = H0 + λ H1 + λ2 H2 + · · · .
(5.6)
One immediately obtains the following Proposition 5.4. The condition (5.5) is equivalent to the following two conditions: (i) H0 is a Casimir function of { , }0 . (ii) For all k, the Hamiltonian vector field of Hk+1 with respect to { , }0 coincides with the Hamiltonian vector field of Hk with respect to { , }. Furthermore, all the Hamiltonians Hk are in involution with respect to both Poisson structures: {Hk , H } = {Hk , H }0 = 0, and the corresponding Hamiltonian vector fields commute with each other. Indeed, if k ≥ , then one has {Hk , H }0 = {Hk−1 , H } = {Hk−1 , H+1 }0 until one obtains an expression of the form {Hs , Hs } or {Hs , Hs }0 which is identically zero. In practice, to construct an integrable hierarchy, one chooses a Casimir function H0 of the Poisson structure { , }0 and then considers its Hamiltonian vector field with respect to { , }. Thanks to the compatibility condition (5.4), this field is Hamiltonian also with respect to the Poisson structure { , }0 with some Hamiltonian H1 . Then one iterates the procedure. The above method has been successfully applied to the KdV equation viewed as a Hamiltonian field on the dual of the Virasoro algebra. 5.4. Bi-Hamiltonian Euler equation on g∗ . There exists an Euler equation on g∗ which is bi-Hamiltonian. This Euler equation is closely related to the BKP equation (1.3) in the special case α = 0. Theorem 5.5. The following Hamiltonian 1 1 c1 H (g, v) = g ∂x−1 v y − ∂x−1 g y + gx x d xd y, 2 2 c2 S 1 ×S 1 defines a bi-Hamiltonian system on g∗ .
(5.7)
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Proof. Let us fix the following point of g∗ : ∂ , d x 2 , 0, 0 (g, u, c1 , c2 )0 = − ∂x and show that the Euler vector field (5.7) is also Hamiltonian with respect to the constant Poisson structure { , }0 . Consider the following Casimir function of the constant Poisson structure { , }0 : H0 = (v − g) d xd y. S 1 ×S 1
Its Hamiltonian vector field with respect to the linear structure { , } is ! gt = gx vt = 2 gx + vx . The compatibility condition (5.4) guarantees that this vector field is Hamiltonian with respect to the constant structure { , }0 . Its Hamiltonian function can be easily computed: v g d xdy. H1 = S 1 ×S 1
Iterating this procedure, consider its Hamiltonian field with respect to the linear structure { , }: ! gt = c2 g y vt = c1 gx x x + c2 v y . Its Hamiltonian with respect to the constant structure { , }0 is proportional to the function (5.7), namely H2 = c2 H. We proved that the Hamiltonian H belongs to the hierarchy (5.6).
Remark 5.6. Note that the Hamiltonian H1 is nothing but the quadratic form (4.1). In the case of the Lie algebra g this is the Casimir function with identically zero Hamiltonian vector field. This is why, in the case of g, the corresponding Hamiltonian vector field linearly depends on the central charges c1 , c2 . Let us now calculate the explicit formula of the Euler equation. Proposition 5.7. The Euler equation with the Hamiltonian (5.7) is of the form: gt = gx ∂x−1 g y − g y g + c2 ∂x−1 g yy , vt = 2v g y − v y g + vx ∂x−1 g y − 2gx ∂x−1 v y + g y g + 2gx ∂x−1 g y − cc21 (gx x x g + 2gx x gx ) +2c1 gx x y + c2 ∂x−1 v yy + ∂x−1 g yy .
(5.8)
Proof. We compute the variational derivatives of H : ∂ δH (g, v) = ∂x−1 g y , δv ∂x δH c1 (g, v) = ∂x−1 v y − ∂x−1 g y + gx x d x 2 , δg c2 and then use formula (5.1) for the Euler equation together with formula (4.3) for the coadjoint action.
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The first equation in (5.8) is precisely Eq. (1.8) already defined in the Introduction. In the complex case, it is equivalent to BKP (1.3) with α = 0. It is an amazing fact that the KP equation (in the preceding section) and the BKP equation naturally appear in our context on mutually ”dual functions”, namely on v and g. Let us now show how the bi-Hamiltonian technique implies the existence of an infinite hierarchy of commuting flows. 5.5. Integrability of Equation (1.8). A corollary of Theorem 5.5 is the existence of an infinite series of first integrals in involution for the field (5.8). It turns out that the corresponding Hamiltonians are of a particular form. Proposition 5.8. Each Hamiltonian Hk of the constructed hierarchy is an affine functional in v, that is Hk (g, v) = Hk (g, v) + Hk (g) and Hk is linear in v. Hk Proof. Let us show that the variational derivative δδv does not depend on v. By construction, the expression for the Hamiltonian vector field of Hk with respect to the Poisson structure { , } gives δH δH δ Hk k k gt = g + c2 . (5.9) gx − δv δv x δv y
On the other hand, the same vector field is the Hamiltonian field of Hk+1 with respect to the Poisson structure { , }0 . One has δ Hk+1 gt = δv x that expresses the variational derivative proceeds by induction.
δ Hk+1 δv
in terms of the function
δ Hk δv .
One then
It follows that Eqs. (5.9) never depend on v. The flows of these vector fields commute with each other since the corresponding Hamiltonian fields on g commute. Example 5.9. The next vector field of our hierarchy is already quite complicated: gt = g g g y − gx ∂x−1 g y − gx ∂x−1 g g y − gx ∂x−1 g y −c2 g ∂x−1 g yy − gx ∂x−2 g yy + ∂x−1 g g y − gx ∂x−1 g y y +c22 ∂x−2 g yyy .
This is the first higher order equation in the hierarchy of (1.8). We can now give a partial answer to the question of T. Ratiu. The dispersionless BKP equation can be realized as an Euler equation on the dual of the looped cotangent Virasoro algebra. However, the problem remains open in the case of the classic KP equation. Appendix All the Lie algebras considered in this paper are generalizations of the Virasoro algebra with two space variables. However, these algebras themselves have interesting generalizations. These structures seem to be quite rich and deserve further study.
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Natural generalizations of L Vect(S 1 ) . Consider the 2-torus T2 = S 1 × S 1 parameterized with variables x and y. Let Vect(T2 ) be the Lie algebra of tangent vector fields on T2 . The Lie algebra L Vect(S 1 ) is naturally embedded to Vect(T2 ) as the Lie subalgebra of vector fields tangent to the constant field X = ∂∂x . This fact suggests the following generalization. Let V be a compact orientable manifold with a fixed volume form ω and X ∈ Vect(V ) be a non-vanishing vector field on V such that divX = 0. We will denote by A X the Lie algebra of vector fields collinear to X ; the Lie bracket of A X can be written as follows: [ f X, g X ] = ( f L X (g) − g L X ( f )) X. This is clearly a Lie subalgebra of Vect(V ) generalizing L Vect(S 1 ) . Some particular cases, such as the iterated loop were considered in [29]. One can easily construct a generalization of the Gelfand-Fuchs cocycle (2.1) on A X : c( f X, g X ) = f (L X )3 g ω. V
It is easy to check that this cocycle is non-trivial so that H 2 (A X ; C) is not trivial. However, we have no further information about this cohomology group. The geometry of coadjoint orbits of A, as well as possible applications to dynamical systems, also remains an interesting open problem. Remark 5.10. The condition divX = 0 is assumed here mainly for technical reasons (it makes the formulæ nicer); however, one may think of dropping this condition as well as the condition on X to be non-vanishing. A 2-parameter deformation of g. Let us describe a 2-parameter family of Lie algebras which can be obtained as a deformation of g. In [26] we classified the non-central extensions of Vect(S 1 ) by the space of quadratic differentials. The result is as follows. There are exactly two (up to isomorphism) non-trivial extensions of Vect(S 1 ) by F2 defined by the following 2-cocycles: ∂ ∂ ρ1 f ,g = ( f x x x g − f gx x x ) d x 2 , ∂x ∂x ∂ ∂ ρ2 f ,g = ( f x x x gx − f x gx x x ) d x 2 ∂x ∂x from Vect(S 1 ) to F2 . The 2-cocycles ρ1 , ρ2 give rise to the following modification of the Lie algebra law (3.6). We set
∂ ∂ ∂ ∂ f + u d x 2, g + v dx2 + u d x 2, g + v dx2 = f ∂x ∂x ∂x ∂x (κ1 ,κ2 ) ∂ ∂ ∂ ∂ + κ2 ρ2 f , + κ1 ρ1 f ,g ,g ∂x ∂x ∂x ∂x where κ1 κ2 ∈ C are parameters. This deformed commutator satisfies the Jacobi identity and provides an interesting Lie algebra structure.
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Acknowledgements. We are grateful to T. Ratiu for his interest in this work. We also wish to thank B. Khesin and A. Reiman for enlightening discussions.
References 1. Ablowitz, M.J., Clarkson, P.A.: Solitons, nonlinear evolution equations and inverse scattering. London Mathematical Society Lecture Note Series 149, Cambridge: Cambridge University Press, 1991 2. Arnold, V.I.: Mathematical methods of classical mechanics. Third edition, Moscow: Nauka, 1989 3. Arnold, V.I., Khesin, B.A.: Topological methods in hydrodynamics. Applied Mathematical Sciences 125. New York: Springer-Verlag, 1998 4. Balog, J., Fehér, L., Palla, L.: Coadjoint orbits of the Virasoro algebra and the global Liouville equation. Internat. J. Mod. Phys. A 13(2), 315–362 (1998) 5. Bogdanov, L.V., Konopelchenko, B.G.: On dispersionless BKP hierarchy and its reductions. J. Nonlinear Math. Phys. 12, Suppl. 1, 64–73 (2005) 6. Dickey L.A.: Soliton equations and Hamiltonian systems. Second edition, Adv. Ser. in Math. Phys. 26, RivereEdge, NJ World Scientific, 2003 7. Dunajski, M.: A class of Einstein-Weyl spaces associated to an integrable system of hydrodynamic type. J. Geom. Phys. 51(1), 126–137 (2004) 8. Ferapontov, E.V., Khusnutdinova, K.R.: On the integrability of (2 + 1)-dimensional quasilinear systems. Commun. Math. Phys. 248(1), 187–206 (2004) 9. Ferapontov, E.V., Khusnutdinova, K.R.: Hydrodynamic reductions of multi-dimensional dispersionless PDEs: the test for integrability. J. Math. Phys. 45(6), 2365–2377 (2004) 10. Fuks D.B.: Cohomology of infinite-dimensional Lie algebras. New York: Consultants Bureau, 1986 11. Guieu L., Roger C.: L’Algèbre et le Groupe de Virasoro: aspects géométriques et algébriques, généralisations. To appear 12. Gelfand, I.M., Fuks, D.B.: Cohomologies of the Lie algebra of vector fields on the circle. Func. Anal. Appl. 2(4), 92–93 (1968) 13. Hirota R.: The direct method in soliton theory. Cambridge Tracts in Mathematics 155, Cambridge: Cambridge University Press, 2004 14. Kirillov, A.: The orbits of the group of diffeomorphisms of the circle, and local Lie superalgebras. Func. Anal. Appl. 15(2), 75–76 (1981) 15. Kirillov A.: Infinite-dimensional Lie groups: their orbits, invariants and representations. The geometry of moments. Lecture Notes in Math. 970, Berlin: Springer, 1982, pp. 101–123 16. Khesin, B., Misiolek, G.: Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math. 176(1), 116–144 (2003) 17. Khesin, B., Ovsienko, V.: The super Korteweg-de Vries equation as an Euler equation. Func. Anal. Appl. 21(4), 81–82 (1987) 18. Konopelchenko, B., Martinez Alonso, L.: Dispersionless scalar integrable hierarchies, Whitham hierarchy, and the quasiclassical ∂-dressing method. J. Math. Phys. 43(7), 3807–3823 (2002) 19. Loday, J.-L.: Cyclic homology. Second edition. Berlin: Springer-Verlag, 1998 20. Magri, F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19(5), 1156–1162 (1978) 21. Marcel, P.: Generalizations of the Virasoro algebra and matrix Sturm-Liouville operators. J. Geom. Phys. 36(3–4), 211–222 (2000) 22. Marcel, P., Ovsienko, V., Roger, C.: Extension of the Virasoro and Neveu-Schwarz algebras and generalized Sturm-Liouville operators. Lett. Math. Phys. 40(1), 31–39 (1997) 23. Martinez Alonso, L., Shabat, A.B.: Towards a theory of differential constraints of a hydrodynamic hierarchy. J. Nonlinear Math. Phys. 10(2), 229–242 (2003) 24. Misiolek, G.: A shallow water equation as a geodesic flow on the Bott-Virasoro group. J. Geom. Phys. 24(3), 203–208 (1998) 25. Misiolek, G.: Conjugate points in the Bott-Virasoro group and the KdV equation. Proc. Amer. Math. Soc. 125(3), 935–940 (1997) 26. Ovsienko, V., Roger, C.: Generalizations of Virasoro group and Virasoro algebra through extensions by modules of tensor-densities on S 1. Indag. Math. (N.S.) 9(2), 277–288 (1998) 27. Ovsienko V.: Coadjoint representation of Virasoro-type Lie algebras and differential operators on tensor-densities. DMV Sem. 31, Basel: Birkhäuser, 2001, pp. 231–255 28. Ovsienko V., Tabachnikov S.: Projective differential geometry old and new. From the Schwarzian derivative to the cohomology of diffeomorphism groups. Cambridge Tracts in Math. 165, Cambridge: Cambridge University Press, 2005 29. Ramos, E., Sah, C.-H.: R Shrock, Algebras of diffeomorphisms of the N -torus. J. Math. Phys. 31(8), 1805– 1816 (1990)
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30. Reiman, A., Semenov-Tyan-Shanskii M.: Hamiltonian structure of equations of Kadomtsev-Petviashvili type. Differential geometry, Lie groups and mechanics, VI. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 133, 212–227 (1984) 31. Segal, G.: Unitary representations of some infinite-dimensional groups. Commun Math. Phys. 80(3), 301– 342 (1981) 32. Zusmanovich P.: The second homology group of current Lie algebras. In: K -theory (Strasbourg, 1992), Astérisque No. 226(11), 435–452 (1994) Communicated by L. Takhtajan
Commun. Math. Phys. 273, 379–394 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0242-2
Communications in
Mathematical Physics
t 1/3 Superdiffusivity of Finite-Range Asymmetric Exclusion Processes on Z Jeremy Quastel , Benedek Valkó , Departments of Mathematics and Statistics, University of Toronto, Toronto, Ontario MSS 1L2, Canada. E-mail: [email protected]; [email protected] Received: 2 June 2006 / Accepted: 31 October 2006 Published online: 15 May 2007 – © Springer-Verlag 2007
Abstract: We consider finite-range asymmetric exclusion processes on Z with non-zero drift. The diffusivity D(t) is expected be of O(t 1/3 ). We prove that D(t) ≥ Ct 1/3 ∞to−λt in the weak (Tauberian) sense that 0 e t D(t)dt ≥ Cλ−7/3 as λ → 0. The proof employs the resolvent method to make a direct comparison with the totally asymmetric simple exclusion process, for which the result is a consequence of the scaling limit for the two-point function recently obtained by Ferrari and Spohn. In the nearest neighbor case, we show further that t D(t) is monotone, and hence we can conclude that D(t) ≥ Ct 1/3 (log t)−7/3 in the usual sense. 1. Introduction A finite-range exclusion process on the integer lattice Z is a system of continuous time, rate one random walks with finite-range jump law p(·), i.e. p(z) ≥ 0, and p(z) = 0 for z > R for some R < ∞, z p(z) = 1, interacting via exclusion: Attempted jumps to occupied sites are suppressed. We will always assume in this article that p(·) has a non-zero drift, zp(z) = b = 0. (1.1) z
In particular, p(·) is asymmetric and we will refer to the process as the asymmetric exclusion process (AEP). The state space of the process is {0, 1}Z and it is traditional to call configurations η, with ηx ∈ {0, 1} indicating the absence, or presence, of a particle at x ∈ Z. The infinitesimal generator of the process is given by L f (η) = p(z)ηx (1 − ηx+z )( f (η x,x+z ) − f (η)), (1.2) x,z∈Z Supported by the Natural Sciences and Engineering Research Council of Canada.
Partially supported by the Hungarian Scientific Research Fund grants T37685 and K60708.
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where η x,y denotes the configuration obtained from η by interchanging the occupation variables at x and y. The Bernoulli product measures πρ , ρ ∈ [0, 1], with πρ (ηx = 1) = ρ form a one-parameter family of invariant measures for the process. In fact, there exist other invariant measures [BM], but they will not be relevant for our discussion. The process starting from π0 and π1 are trivial and so we consider the stationary process obtained by starting with πρ for some ρ ∈ (0, 1). Let ηx − ρ ηˆ x = √ ηˆ x (1.3) , ηˆ A = ρ(1 − ρ) x∈A for any finite nonempty set A ⊂ Z. The collection {ηˆ A }, where A ranges over a finite subset of Z is an orthonormal basis of L 2 (πρ ) with its natural inner product f, g = f gdπρ . (1.4) {0,1}Z
Then L 2 (πρ ) can naturally be thought of as the direct sum of subspaces H1 , H2 , . . . , where Hn is the linear span of {ηˆ A }, |A| = n. It is natural to think of H1 as being linear functions, H2 as quadratic functions, etc. From a physical point of view, the most basic quantity is the two-point function, S(x, t) = E[(ηx (t) − ρ)(η0 (0) − ρ)].
(1.5)
The expectation is with respect to the stationary process obtained by starting from one of the invariant measures πρ . It is easy to show (see [PS]) that S(x, t) satisfies the sum rules 1 S(x, t) = ρ(1 − ρ) = χ , x S(x, t) = (1 − 2ρ)bt. (1.6) χ x x Note that one should not expect to be able to actually compute S(x, t) but one does hope to find its large scale structure. The next most basic quantity, the diffusivity D(t), is already unknown. It is defined as D(t) = (χ t)−1
(x − (1 − 2ρ)bt)2 S(x, t).
(1.7)
x∈Z
Using coupling (see [L]), the diffusivity can be rewritten in terms of the variance of a second class particle. Suppose one starts with two configurations η and η which are ordered in the sense that η x ≥ ηx for each x ∈ Z. One can couple the two exclusions by having them jump together whenever possible and one observes that at later times the ordering is preserved. If we write η = η + η
, then the “particles” of η
move according to the second class particle dynamics. Among themselves they move with the standard exclusion rule, the other (first class) particles move without noticing them, and if a first class particle attempts to jump to a site occupied by a second class particle, the two exchange positions. Note that χ −1 S(x, t) = P(ηx (t) = 1 | η0 (0) = 1) − P(ηx (t) = 1 | η0 (0) = 0) = P(η
x (t) = 1 | η
(0) = δ0 ) = P(X (t) = x | X (0) = 0)
(1.8)
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are the transition probabilities of a single second class particle X (t) starting at the origin. Here δ0 is the configuration with only one particle at 0 and P is the coupled measure. The diffusivity is then given by D(t) = t −1 V ar (X (t)).
(1.9)
We can alternately write the dynamics as a stochastic differential equation dηˆ x = (∇wx + ηˆ x )dt + dMx ,
(1.10)
where d is a microscopic convective derivative, dηˆ x = d ηˆ x +
(1 − 2ρ) p(z)(ηˆ x+z − ηˆ x−z ), 2 z
∇ and are microscopic analogues of first and second spatial derivatives, ∇wx = χ 1/2 p(z)(ηˆ x,x+z − ηˆ x−z,x ),
(1.11)
(1.12)
z
ηˆ x =
1 p(z)(ηˆ x+z + ηˆ x−z − 2ηˆ x ), 2 z
and Mx (t) are martingales with t t 2 E[( φx dMx ) ] = p(z)(φx+z − φx )2 ds. 0
0
x
(1.13)
(1.14)
x,z
The current wx = τx w, where the specific quadratic function w is given by w= zp(z)ηˆ {0,z} .
(1.15)
z
In this sense, AEP is a natural discretisation of the stochastic Burgers equation, ∂t u = ∂x u 2 + ∂x2 u + ∂x W˙
(1.16)
for a function u(t, x) of x ∈ R and t > 0 where W˙ is a space-time white noise. White noise is supposed to be an invariant measure. Letting ∂x U = u one obtains the Kardar-Parisi-Zhang equation for surface growth, ∂t U = (∂x U )2 + ∂x2 U + W˙ .
(1.17)
We are interested in the large scale behaviour and the only rescalings of u which preserve the initial white noise are u (t, x) = −1/2 u( −z t, −1 x).
(1.18)
The stochastic Burgers equation (1.16) transforms to ∂t u = 2 −z ∂x u 2 + 2−z ∂x2 u + 1− 2 ∂x W˙ , 3
z
(1.19)
which suggests that the dynamical exponent z = 3/2 and that the diffusion and random forcing terms become irrelevant in the limit.
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The exponent z = 3/2 was first predicted for (1.16) by Forster, Nelson and Stephen [FNS], then for AEP by van Beijeren, Kutner and Spohn [BKS] and then for (1.17) by Kardar, Parisi and Zhang [KPZ]. Note that at a rigorous level we are very far from understanding this for either (1.16) or (1.17). At the present time the mathematical problem there is just to make sense of the equation (see [BG]). So it makes sense to consider exclusion processes, which are clearly well defined, yet are supposed to have the same large scale behaviour. The scaling prediction for u suggests that on large scales S(x, t) t −2/3 (t −2/3 (x − (1 − 2ρ)bt))
(1.20)
for some scaling function , and in particular one conjectures that, D(t) Ct 1/3 .
(1.21)
Note that the case of asymmetric exclusion with mean-zero jump law is different and there one has as usual that D(t) → D as t → ∞ (see [V]). The diffusivity can be related to the time integral of current-current correlation functions by the Green-Kubo formula, t s 2 −1 D(t) = z p(z) + 2χ t w, eu L w duds. (1.22) 0
z
0
It uses a special inner product defined for local functions by φ, ψ = φ, τx ψ .
(1.23)
x
Equation (1.22) is proved in [LOY] (in the special case p(1) = 1, but the proof for general AEP is the same.) A useful variant is obtained by taking the Laplace transform, ∞ −λt −2 2 2 e t D(t)dt = λ z p(z) + 2χ |||w|||−1,λ , (1.24) 0
z
where the H−1 norm corresponding to L is defined for local functions by |||φ|||−1,λ = φ, (λ − L)−1 φ 1/2 .
∞
(1.25)
We say that D(t) t ρ , ρ > 0 in the weak (Tauberian) sense if 0 e−λt t D(t)dt λ−(2+ρ) . Hence the weak (Tauberian) version of the conjecture (1.21) is |||w|||2−1,λ λ−1/3 .
(1.26)
One of the key advantages of this resolvent approach is that there is a variational formula (see [LQSY]), |||w|||2−1,λ = sup 2w, f − f, (λ − S) f − A f, (λ − S)−1 A f , (1.27) f
where S = 21 (L + L ∗ ) and A = 21 (L − L ∗ ) are the symmetric and antisymmetric parts of the generator L. S is nothing but the generator of the symmetric exclusion process with p(z) ¯ = 21 ( p(z) + p(−z)). It has the special property that it maps the subspaces Hn
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into themselves, and on each is nothing but the generator of a symmetric random walk. Hence one can hope to obtain non-trivial information from (1.27) by choosing carefully test functions f . This idea was used in [LQSY] to obtain D(t) ≥ Ct 1/4 in d = 1 and D(t) ≥ C(log t)1/2 in d = 2, which was improved to D(t) C(log t)2/3 in [Y]. All of these are in the weak (Tauberian) sense. The special case of jump law p(1) = 1, p(z) = 0, z = 1 is called the totally asymmetric simple exclusion process (TASEP). Simple refers to the nearest-neighbour jumps of the underlying random walk. It is very remarkable that after about 20 years of intense study, TASEP has succumbed to a combination of sophisticated techniques from analysis, combinatorics and random matrix theory (see [FS] and references therein). We now state the main result of Ferrari and Spohn [FS]. Define the height function h t (x) = 2Nt − Mt (x)
(1.28)
t ≥ 0, where Nt counts the number of jumps from site 0 to site 1 up to time t and ⎧ x if x > 0, ⎨ i=1 (2ηi (t) − 1) 0 if x = 0, (1.29) Mt (x) = ⎩ 0 − i=x+1 (2ηi (t) − 1) if x < 0. Note that E[h t (x)] = 2χ t + (1 − 2ρ)x. Let v(x, t) = V ar (h t (x)).
(1.30)
Since h t (x + 1) − h t (x) = 1 − 2ηx+1 (t), it is not hard to check that 8S(x, t) = v(x + 1, t) − 2v(x, t) + v(x − 1, t). See [PS] for a detailed proof. We have V ar (h t (x)) − 4χ |x − (1 − 2ρ)t| D(t) = (4χ t)−1
(1.31)
(1.32)
x∈Z
(see Sect. 4). Now consider a normalised version of h t : hˆ t (x) = χ −2/3 t −1/3 (h t (x) − E[h t (x)]),
(1.33)
and for each fixed t > 0 and ω ∈ R let Fω,t be the cumulative distribution function of −hˆ t ((1 − 2ρ)t + 2ωχ 1/3 t 2/3 ); Fω,t (s) = P(−hˆ t ((1 − 2ρ)t + 2ωχ 1/3 t 2/3 ) ≤ s).
(1.34)
The main result of Ferrari and Spohn concerning TASEP is that d Fω,t converge weakly as probability measures, as t tends to infinity, to d Fω where ∂ FGU E (s + ω2 )g(s + ω2 , ω) , (1.35) Fω (s) = ∂s where FGU E is the Tracy-Widom distribution and g is a scaling function defined through the Airy kernel (see [FS] for details). Note that the convergence stated in [FS] is that for any c1 < c2 , c2 c2 lim Fω (s, t)ds = Fω (s)ds. (1.36) t→∞ c 1
c1
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In fact, this is the same as weak convergence. For by monotonicity, if > 0, s s+ −1 −1 Fω,t (u)du ≤ Fω,t (s) ≤ Fω,t (u)du. s−
(1.37)
s
Taking the limit in t and using (1.36) we see that limt→∞ Fω,t (s) = Fω (s) at any continuity point s of the limit function (in this case all s ∈ R), and this is equivalent to weak convergence. The proof of Ferrari and Spohn is through a direct mapping between TASEP and a particular last passage percolation problem. Such a mapping is not available except for the case of TASEP. So although one expects analogous results for general AEP in one dimension, different techniques will be required. Our main motivation here is to confirm, at least in part, the predicted universality (see Sect. 6 of [PS] for a nice description) by showing that these results for TASEP imply some bounds for general AEP. From (1.32) and (1.36) one expects D T AS E P (t) c T AS E P χ 2/3 t 1/3 , where c
T AS E P
=
dω
s d Fω (s) = 2 2
(1.38)
dω
ds FGU E (s + ω2 )g(s + ω2 , ω).
(1.39)
Here, and throughout this article, we will use the superscript T AS E P to denote the values taken by TASEP of quantities defined for general AEP. Unfortunately, the necessary estimates for the upper bound appear to be missing at this time. However from the weak convergence we have immediately that Corollary 1.
lim inf t −1/3 D T AS E P (t) ≥ c T AS E P χ 2/3 . t→∞
(1.40)
Remark. Another way to see the strict positivity of the left-hand side without computing c T AS E P is that by Schwartz’s inequality and (1.7), 2 −1 −1 |x − (1 − 2ρ)t|S(x, t) . (1.41) χ D(t) ≥ t x∈Z
We have
|x − (1 − 2ρ)t|S(x, t) = 2V ar (h t ((1 − 2ρ)t))
(1.42)
x∈Z
(see Sect. 4) and from the weak convergence we have, lim inf t −2/3 V ar (h t ((1 − 2ρ)t)) ≥ χ 4/3 t→∞
Since by (1.6)
s 2 d Fω (s).
(|x − (1 − 2ρ)t| − |x − (1 − 2ρ)t|)S(x, t) ≤ χ x∈Z
the positive lower bound on lim inf t→∞ t −1/3 D T AS E P (t) follows.
(1.43)
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The main result of the present article is a comparison between the diffusivity of AEP and that of TASEP: Theorem 1. Let D(t) be the diffusivity of a finite range exclusion process in d = 1 with non-zero drift. Let D T AS E P (t) be the diffusivity of the totally asymmetric simple exclusion process. There exist 0 < K , C < ∞ such that ∞ ∞ −1 −λK −1 t T AS E P C e tD (t)dt ≤ e−λt t D(t)dt (1.44) 0 0 ∞ e−λK t t D T AS E P (t)dt. ≤C 0
Combined with (1.40) this gives Theorem 2. For any finite range exclusion process in d = 1 with non-zero drift, D(t) ≥ Ct 1/3 in the weak (Tauberian) sense: There exists C > 0 such that ∞ e−λt t D(t)dt ≥ Cλ−7/3 . (1.45) 0
We now make some comments on obtaining strict versions of the estimates, as opposed to weak (Tauberian) versions. In [LY] it is shown that t t −1 E[ w(s)ds τx w(s)ds] ≤ |||w|||2−1,t −1 , (1.46) t x
0
0
and hence an upper weak (Tauberian) bound implies a strict upper bound in time on the diffusivity. There is no analogous fact for lower bounds. However, it is easy to show the following: Proposition 1. Suppose that v(t) ≥ 0 is a nondecreasing function and β > 0. 1. Suppose there exist c1 < ∞ and λ0 > 0 such that for 0 < λ < λ0 , ∞ e−λt v(t)dt ≤ c1 λ−(1+β) ,
(1.47)
0
then there exist c2 < ∞ and t0 such that for all t > t0 , v(t) ≤ c2 t β .
(1.48)
2. Suppose v(t) ≤ c2 t α for some α ≥ β and t > t0 and for some c3 > 0, for 0 < λ < λ0 , ∞ e−λt v(t)dt ≥ c3 λ−(1+β) . (1.49) 0
Then there exists c4 > 0 and t1 < ∞ such that for t > t1 , c4 t β if α = β; v(t) ≥ β −(1+β) c4 t (log t) if α > β.
(1.50)
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Proof. 1. Since v is monotone nondecreasing we have for t > λ−1 0 , ∞ ∞ e−1 v(t) = e−s v(t)ds ≤ e−s v(ts)ds ≤ c1 t β . 1
(1.51)
0
t 2. Because v(t) is non-decreasing, 0 e−λs v(s)ds ≤ tv(t) and if v(t) ≤ c2 t α we have ∞ −λs v(s)ds ≤ c2 λ−1 e−λt t α for t > t1 . Hence t e c3 λ−(1+β) ≤ tv(t) + c2 λ−1 e−λt t α . Choosing λ = t −1 (1 + (α − β)(log t + c log log t)) gives the result. Note that the bound
∞
e−λt t D(t)dt ≤ Cλ−5/2
(1.52) (1.53)
0
can be derived easily from the variational formula (1.27) (see the proof of Proposition 3 for a similar computation). Certainly one expects t D(t) to be nondecreasing in general. We will show in Lemma 2 that z 2 p(z) − 2ρ z( p(z) − p(−z))E[ X (t)|ηz (0) = 1], (1.54) ∂t (t D(t)) = z
z>0
where
X (t) = X (t) − (1 − 2ρ)bt. (1.55) What one expects is that b E[ X (t) | ηz (0) = 1] ≤ 0. If p(z) ≥ p(−z) for all z > 0, (or for all z < 0) this would imply that t D(t) is increasing. We have only been able to prove this in the special case of the simple (nearest neighbor) exclusion (see Proposition 4). Hence for this class of AEP we can make the following statement: Theorem 3. Let D(t) be the diffusivity of a nearest neighbor ( p(z) = 0, |z| = 1) asymmetric exclusion. 1. There exists c0 > 0 such that D(t) ≥ c0 t 1/3 (log t)−7/3 .
(1.56)
2. Suppose that there exists c1 < ∞ such that D T AS E P (t) ≤ c1 t 1/3 .
(1.57)
Then there exists c2 < ∞ such that c2−1 t 1/3 ≤ D(t) ≤ c2 t 1/3 .
(1.58)
Remarks. 1. Note that in Theorems 1 and 2 we have not made any assumptions about the irreducibility of p(·). Let κ = gcd(y ∈ Z : p(y) > 0).
(1.59)
If κ > 1 then our AEP is the same as κ independent copies of the AEP with jump law p(y) ˜ = p(κ y) on the sublattices κZ + i (i = 0, 1, . . . , κ − 1). Using this simple observation it is easy to extend all our proofs from κ = 1 to κ > 1, so we can assume without loss of generality in the proofs that p(·) is irreducible.
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2. Analogous methods to the ones presented here could in principle be applied to other functionals of AEP. For example, the variance of the occupation time of the origin, t ηs (0)ds, (1.60) 0
O(t 4/3 ).
In [B] a lower bound of the form Ct 5/4 is obtained. This is also expected to be variance is again given by the H−1 norm of a certain function and direct comparisons between its value for TASEP and general AEP can be obtained in a straightforward way. Hence asymptotic order of growth bounds for this variance under TASEP would imply the same for AEP. Unfortunately, at the present time no such bounds are available, though it is plausible they could be derived from the machinery that has been developed for TASEP. 2. Comparison of H−1 Norms The first proposition adapts results of Sethuraman [S] to the present context. Proposition 2. There exist α, β ∈ (0, ∞) depending only on p(·) such that T AS E P T AS E P α −1 |||φ|||−1,β −1 λ ≤ |||φ|||−1,λ ≤ α|||φ|||−1,βλ .
(2.1)
Proof. We can also define H−1 norms based on the standard inner product ·, · : φ−1,λ = φ, (λ − L)−1 φ .
(2.2)
T AS E P T AS E P α −1 φ−1,β −1 λ ≤ φ−1,λ ≤ αφ−1,βλ .
(2.3)
From [S] we have that
From the translation invariance of the generators τx φ, (λ − L)−1 φ |||φ|||2−1,λ = x n n 1 τx φ, (λ − L)−1 τx φ n→∞ 2n x=−n x=−n
= lim
n 1 τx φ2−1,λ . n→∞ 2n x=−n
= lim The proposition follows.
Proposition 3. Let w be the current corresponding to a general AEP as in (1.15) and w T AS E P be the current for TASEP. Then there exists C < ∞ such that for 0 < λ < 1, |||w − bw T AS E P |||−1,λ ≤ C.
(2.4)
Remarks. 1. In the theorem one can use either L or L T AS E P to define ||| · |||−1,λ since the results are equivalent. 2. This is similar to, but not the same as, results in [SX], because of the special norm ||| · |||−1,λ .
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Proof. Since
w − bw T AS E P =
x p(x) ηˆ {0,1} − ηˆ {0,x} ,
x
it is enough to show that
|||ηˆ {0,1} − ηˆ {0,k+1} |||−1,λ ≤ C
(2.5)
for each k > 0, where C is a constant depending on p(·) and k and ||| · |||−1,λ is defined using the generator L T AS E P . Call V = ηˆ {0,1} − ηˆ {0,k+1} . Dropping the third term in the variational formula (1.27) we have |||V |||2−1,λ ≤ V, (λ − S)−1 V .
(2.6)
We now show that the right-hand side is bounded independent of λ. The computation is maps H2 to itself. In particular, if f, g ∈ H2 with done using the fact that S f = x
f (x + z, y + z)g(x, y) =
z x
where
∞
f (x)g(x),
(2.7)
x=0
f (x) =
f (y, y + x + 1),
(2.8)
y
and S f =
x
Sˆ f (x, y)ηˆ {x,y} with
1 f (x, y + 1) + f (x − 1, y) − 2 f (x, y) Sˆ f (x, y) = 2 + 1{y−x>1} ( f (x, y − 1) + f (x + 1, y) − 2 f (x, y)) .
(2.9)
Moreover (2.10) Sˆ f (x) = (S f¯)(x) = f (x + 1) − f (x) + 1{x>0} ( f (x − 1) − f (x)). Our V = x
for some h. Then V, (λ − S)−1 V =
x
h(x, x + 1) −
h(x, x + k + 1)
x
= h(0) − h(k) = (λ − S)−1 V (0) − (λ − S)−1 V (k),
(2.12)
where V (x) = 1{x=0} − 1{x=k} . An explicit computation shows that q(x) :=
γx = (λ − S)−1 1{x=0} (x), λ+1−γ
(2.13)
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where γ = γ (λ) is the solution of the equation λ + 2 = γ −1 + γ
(2.14)
with 0 < γ < 1. This is easy to check: if x > 0 then
(λ − S)q (x) = λ − (γ − 1) − (γ −1 − 1)
γx =0 λ+1−γ
and (λ − S)q (0) = (λ − (γ − 1))
1 = 1. λ+1−γ
A similar calculation shows that if k > 0 then one can find constants c1 , c2 (depending on k and λ) such that 1
(c γ k−x + c2 γ x−k ) if 0 ≤ x < k −1 (λ − S) 1{x=k} (x) = 2 11 x−k if k ≤ x 2 (c1 + c2 ) γ and that there is a C < ∞ such that |ci − λ−1/2 | ≤ C, i = 1, 2.
(2.15)
So (λ − S)−1 V (0) − (λ − S)−1 V (k) =
1 1 − γk + (c1 (1 − γ k ) + c2 (1 − γ −k )). (2.16) λ+1−γ 2
Since γ 1 − λ1/2 as λ → 0, it is not hard to check that the right-hand side is bounded for 0 < λ < 1. 3. Monotonicity of t D(t) Let X (t) be the position of a second class particle at time t started at the origin and X (t) = X (t) − (1 − 2ρ)bt. Lemma 1. For any AEP,
and
(1 − ρ)E[ X (t)|η y (0) = 0] + ρ E[ X (t)|η y (0) = 1] = 0
(3.17)
E[ X (t)|η y (0) = 1] = E[ X (t)|η−y (0) = 1].
(3.18)
Proof. Equation (3.17) is straightforward from E[X (t)] = (1 − 2ρ)tb. To prove (3.18) we first write the difference as x(P(X (t) = x|η y (0) = 1) − P(X (t) = x|η−y (0) = 1)) (3.19) x
=
x
x P(X (t) = x|η y (0) = 1) −
(x + y)P(X (t) = x|η−y (0) = 1) + y. x
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We can write P(X (t) = x|η y (0) = 1) (3.20) = E[ηx (t)|η0 (0) = 1, η y (0) = 1] − E[ηx (t)|η0 (0) = 0, η y (0) = 1], and by the translation invariance P(X (t) = x|η−y (0) = 1) (3.21) = E[ηx (t)|η0 (0) = 1, η−y (0) = 1] − E[ηx (t)|η0 (0) = 0, η−y (0) = 1] = E[ηx+y (t)|η0 (0) = 1, η y (0) = 1] − E[ηx+y (t)|η0 (0) = 1, η y (0) = 0]. Substituting these into the previous equation we get E[ X (t)|η y (0) = 1] − E[ X (t)|η−y (0) = 1] (3.22) = x{E[ηx (t)|η0 (0) = 1, η y (0) = 0] − E[ηx (t)|η0 (0) = 0, η y (0) = 1]} + y x
= χ −1
x E[ηx (t)(η0 (0) − η y (0)] + y
x
=0 by (1.6).
Lemma 2. For any AEP, ∂t (t D(t)) = z 2 p(z) − 2ρ z( p(z) − p(−z))E[ X (t)|ηz (0) = 1]. z
(3.23)
z>0
Proof. We compute ∂t x 2 S(x, t) = x 2 p(z)ηx−z (t)(1−ηx (t))−ηx (t)(1−ηx+z (t)), η0 (0) . (3.24) x
x,z
Summing by parts, using the translation invariance, reversing space and time, we can rewrite (3.24) as (−2x z + z 2 ) p(−z)η0 (0)(1 − ηz (0)), ηx (t) − ρ . (3.25) x,z
Again, by explicit computation η0 (1 − ηz )(0), ηx (t) − ρ is given by
χ ρ(1 − ρ) (e11 − e01 ) + (1 − ρ)2 (e10 − e00 ) − ρ (e11 − e10 ) ,
(3.26)
where ei j = E[ηx (t)|η0 (0) = i, ηz (0) = j].
(3.27)
Equation (3.26) can be rewritten in terms of the second class particle (see (3.21)) as χ ((1 − ρ)P(X (t) = x) − ρ P(X (t) = x − z|η−z (0) = 1)) .
(3.28)
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Substituting this into (3.25) and using (1.1), (1.6): x 2 S(x, t) = χ (1 − ρ) (−2E[X (t)]z + z 2 ) p(−z) ∂t x
z
(−2E[X (t)|η−z (0) = 1]z − z 2 ) p(−z) − χρ =χ
z
z 2 p(z) + 2b2 t (1 − 2ρ)2
z
− 2χρ
(3.29)
E[ X (t)|ηz (0) = 1]zp(z).
z
Using (3.18) and the definition of D(t) completes the proof.
Proposition 4. Suppose that p(z) = 0 for |z| = 1 (nearest neighbor). Then t D(t) is non-decreasing in t. Proof. We can assume p(1) ≥ p(−1). In this case we will show E[ X (t)|η1 (0) = 1] ≤ (1 − ρ).
(3.30)
By the previous lemma, ∂t (t D(t)) ≥ (1 − 2ρ(1 − ρ)) p(1) + (1 + 2ρ(1 − ρ)) p(−1) ≥ 0.
(3.31)
Consider a configuration where at site 1 we have a second class particle, at site 0 we have a third class particle and at all the other sites the distribution of particles is independent Bernoulli with probability ρ. The ordinary particles don’t see the second or third class particles (i.e. they see them as empty sites) and the second class particle doesn’t see the third class particle. Let the process evolve according to the AEP dynamics, and denote the position of the third and second class particle with A(t) and B(t), respectively. It is not hard to see that the law of A(t) is the same as the law of X (t) conditioned on the event {η1 (0) = 1} and we have to prove E[A(t)] ≤ (1 − ρ) + (1 − 2ρ)bt.
(3.32)
Also, the law of B(t) is the same as the law of X (t) + 1 conditioned on the event {η−1 (0) = 0}. By Lemma 1 we have E[ρ A(t) + (1 − ρ)B(t)] = (1 − ρ) + (1 − 2ρ)bt, thus it is enough to prove that E[A(t)] ≤ E[B(t)].
(3.33)
Define the variable Z (t) the following way: Z (t) = 0 if A(t) < B(t) and Z (t) = 1 otherwise. Consider a possible joint trajectory (x1 (t), x2 (t)) for min(A(t), B(t)), max(A(t), B(t)) . Conditioned on {(x1 (s), x2 (s)), 0 ≤ s ≤ t}, Z (t) is a continuous time Markov process on {0, 1} with rate p(−1)1{x2 (t)−x1 (t)=1} for the transition 0 → 1 and p(1)1{x2 (t)−x1 (t)=1} for the transition 1 → 0. This uses the fact that our process is nearest neighbor, and thus Z (t) can change only if the second and third class particles switch places. We
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can now calculate P(Z (t) = 0(x1 (s), x2 (s)), 0 ≤ s ≤ t) explicitly. Let T (t) = |{s : x2 (s) − x1 (s) = 1, 0 ≤ s ≤ t}| be the time spent by the two particles up to time t with distance 1 between them, then (using P(Z (0) = 0) = 1) p(−1)e−T (t)( p(−1)+ p(1)) + p(1) P(Z (t) = 0(x1 (s), x2 (s)), 0 ≤ s ≤ t) = . p(−1) + p(1)
(3.34)
Since p(1) ≥ p(−1), this is always at least 1/2. This means E A(t){(x1 (s), x2 (s)), 0 ≤ s ≤ t} ≤ E B(t){(x1 (s), x2 (s)), 0 ≤ s ≤ t} , from which (3.33) and the proposition follows.
4. Summation by Parts In this section we will prove identities (1.32) and (1.42). They hold for general finite range exclusions, but we only need them in case of the TASEP so we will only give the proofs in that special case. Note b = 1 here. The identities are a simple consequence of (1.31) and summation by parts, once one knows the precise behaviour of v(t, x) as |x| → ∞. They are not new; see, for example [FF] for a proof of (1.42). But we could not find a reference for (1.32), so we include the proof here. For x ∈ Z, t ≥ 0 denote by Nt (x) the number of jumps from site x to site x + 1 up to time t. Lemma 3.
v(x, t) = 4χ |x| + 4Cov(Nt (0), Nt (x)) − 4 sgn(x) Cov Nt (0),
|x| y=−|x|+1
η y (t) .
Proof. We will assume x ≥ 0; the case x < 0 is analogous. Recalling the definition (1.29) of Mt (x) and Nt (x) we have Nt (0) − Nt (x) =
1 (Mt (x) − M0 (x)). 2
(4.35)
It is easy to compute V ar (Mt (x)) = 4χ |x|, and by the definition of v(x, t) we have v(x, t) = V ar (2Nt (0) − Mt (x)) = 4V ar (Nt (0)) + 4χ x − 4Cov(Nt (0), Mt (x)).
(4.36)
Using the identity (4.35) and the translation invariance we get Cov(Nt (0), Mt (x)) = E[Nt (0)Mt (x)] − E[Nt (0)]E[Mt (x)] (4.37) 1 = E[(Nt (x) + (Mt (x) − M0 (x)))Mt (x)] − E[Nt (0)]E[Mt (x)] 2 1 = E[( (Mt (x) − M0 (x)))2 ] + E[Nt (x)Mt (x)] 2 − E[Nt (x)]E[Mt (x)] = E[(Nt (x) − Nt (0))2 ] + E[Nt (x)Mt (x)] − E[Nt (x)]E[Mt (x)] = 2V ar (Nt (0)) − 2Cov(Nt (x), Nt (0)) + Cov(Nt (x), Mt (x)).
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We will substitute this into (4.36) to get v(x, t) = 4χ x + 4Cov(Nt (x), Nt (0)) − 2Cov(Nt (0), Mt (x)) − 2Cov(Nt (x), Mt (x)).
(4.38)
By translation invariance, and because of the sign convention in the definition (1.29) of Mt (x), Cov(Nt (x), Mt (x)) = −Cov(Nt (0), Mt (−x)), and the lemma follows.
(4.39)
Lemma 4. For each t > 0, there exist C1 < ∞ and C2 > 0 such that Cov (Nt (0), ηx (t)) ≤ C1 exp{−C2 |x|}, Cov(Nt (0), Nt (x)) ≤ C1 exp{−C2 |x|}. (4.40) Proof. The lemma is standard, but we could not find an exact reference, so for completeness, we give a sketch of the proof. Consider two copies (η(t), η(t)) ˜ of TASEP, coupled as in the preamble to (1.8), starting with initial data (η y , η˜ y = η y 1{y∈[−x/3,x/3]∪[2x/3,4x/3]} ), y ∈ Z, where η is distributed according to πρ . Discrepancies perform nearest neighbor random walks, and the rate of jumping left or right is always at most 1. Let A = {η0 (s) = η˜ 0 (s) and ηx (s) = η˜ x (s) for all s ∈ [0, t]} .
(4.41)
AC is contained in the event that an initial discrepancy reaches 0 or x during the time interval [0, t]. Because of the preservation of order, there are just 4 candidates and hence P(AC ) ≤ 4P(Poisson(t) > x/3), which is exponentially small in x. On A, Nt (0) = ˜ Both Nt (x) and N˜ t (x) are N˜ t (0) and Nt (x) = N˜ t (x), where N˜ t (·) are the currents in η(t). stochastically dominated by Poisson(t) random variables and hence, for any fixed t, their moments are bounded. Breaking up the respective expectations onto A and AC and applying Schwartz’s inequality we see that both |Cov(Nt (0), ηx (t))−Cov( N˜ t (0), η˜ x (t))| and |Cov(Nt (0), Nt (x)) − Cov( N˜ t (0), N˜ t (x))| are exponentially small in x. Hence it suffices to prove the lemma for the second process. Consider a third process η¯ with the same initial conditions as η, ˜ but disallowing jumps between [2x/3] and [2x/3] − 1. Using the same argument as above, it is enough to prove the lemma for η. ¯ Now N¯ t (0) and η¯ x (t) (and N¯ t (0) and N¯ t (x)) are independent, so the covariances vanish. Once one has Lemma 4, it follows from Lemma 3 that for each fixed t ≥ 0, |v(x, t) − 4χ |x|| ≤ C3 exp{−C4 |x|}
(4.42)
for some C3 < ∞ and C4 > 0. Now (1.32) and (1.42) follow by taking partial summations, applying (1.31) summing by parts, and noting that the boundary terms are exponentially small from (4.42). Acknowledgement. The authors would like to thank the referee for pointing out an error in an earlier version of the manuscript.
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References [B]
Bernardin, C.: Fluctuations in the occupation time of a site in the asymmetric simple exclusion process. Ann. Probab. 32(1B), 855–879 (2004) [BG] Bertini, L., Giacomin, G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183(3), 571–607 (1997) [BKS] Beijeren, H., van Kutner, R., Spohn, H.: Excess noise for driven diffusive systems. Phys. Rev. Lett. 54, 2026–2029 (1985) [BM] Bramson, M., Mountford, T.: Stationary blocking measures for one-dimensional nonzero mean exclusion processes. Ann. Probab. 30(3), 1082–1130 (2002) [FF] Ferrari, P.A., Fontes, L.R.G.: Current fluctuations for the asymmetric simple exclusion process. Ann. Probab. 22(2), 820–832 (1994) [FS] Ferrari, P.L., Spohn, H.: Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Commun. Math. Phys. 265(1), 1–44 (2006) [FNS] Forster, D., Nelson, D., Stephen, M.J.: Large-distance and long time properties of a randomly stirred fluid. Phys. Rev. A 16, 732–749 (1977) [KPZ] Kardar, K., Parisi, G., Zhang, Y.Z.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986) [L] Liggett, T.M.: Interacting particle systems. Grundlehren der Mathematischen Wissenschaften 276. New York: Springer-Verlag, 1985 [LOY] Landim, C., Olla, S., Yau, H.T.: Some properties of the diffusion coefficient for asymmetric simple exclusion processes. Ann. Probab. 24(4), 1779–1808 (1996) [LQSY] Landim, C., Quastel, J., Salmhofer, M., Yau, H.-T.: Superdiffusivity of asymmetric exclusion process in dimensions one and two. Commun. Math. Phys. 244(3), 455–481 (2004) [LY] Landim, C., Yau, H.-T.: Fluctuation-dissipation equation of asymmetric simple exclusion processes, Probab. Theory Related Fields 108(3), 321–356 (1997) [PS] Prähofer, M., Spohn, H.: Current fluctuations for the totally asymmetric simple exclusion process. In: In and out of equilibrium (Mambucaba, 2000), Progr. Probab. 51, Boston, MA: Birkhäuser Boston, 2002, pp. 185–204 [S] Sethuraman, S.: An equivalence of H−1 norms for the simple exclusion process. Ann. Prob. 31(1), 35–62 (2003) [SX] Sethuraman, S., Xu, L.: A central limit theorem for reversible exclusion and zero-range particle systems. Ann. Probab. 24(4), 1842–1870 (1996) [V] Varadhan, S.R.S.: Lectures on hydrodynamic scaling. In: Hydrodynamic limits and related topics (Toronto, ON, 1998), Fields Inst. Commun. 27, Providence, RI: Amer. Math. Soc., 2000, pp. 3–40 [Y] Yau, H.-T.: (log t)2/3 law of the two dimensional asymmetric simple exclusion process. Ann. of Math. (2) 159(1), 377–405 (2004) Communicated by H.-T. Yau
Commun. Math. Phys. 273, 395–414 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0250-2
Communications in
Mathematical Physics
An Algebra of Deformation Quantization for Star-Exponentials on Complex Symplectic Manifolds Giuseppe Dito1 , Pierre Schapira2 1 Institut de Mathématiques de Bourgogne, Université de Bourgogne, B.P. 47870, 21078 Dijon Cedex, France.
E-mail: [email protected]
2 Institut de Mathématiques, Université Pierre et Marie Curie, 175, rue du Chevaleret, 75013 Paris, France.
E-mail: [email protected] Received: 10 July 2006 / Accepted: 4 January 2007 Published online: 4 May 2007 – © Springer-Verlag 2007
Abstract: The cotangent bundle T ∗ X to a complex manifold X is classically endowed with the sheaf of k-algebras WT ∗ X of deformation quantization, where k := W{pt} is a subfield of C[[, −1 ]. Here, we construct a new sheaf of k-algebras WTt ∗ X which contains WT ∗ X as a subalgebra and an extra central parameter t. We give the symbol calculus for this algebra and prove that quantized symplectic transformations operate on it. If P is any section of order zero of WT ∗ X , we show that exp(t−1 P) is well defined in WTt ∗ X . Introduction The cotangent bundle T ∗ X to a complex manifold X is endowed with the sheaf of filtered C-algebras ET ∗ X constructed functorially by Sato-Kashiwara-Kawai in [9] and called the sheaf of microdifferential operators. This sheaf is conic and is associated with the homogeneous symplectic structure of T ∗ X . Another no more conic sheaf of filtered algeT ∗ X and defined over C[[, −1 ], has been constructed bras on T ∗ X , denoted here by W in the framework of formal associative deformations by many authors after [1]. (This construction has been extended to Poisson manifolds in [7].) Its analytic counterpart WT ∗ X is constructed in [8]. The sheaf WT ∗ X is similar to the sheaf ET ∗ X of microdifferential operators of [9], but with an extra central parameter , a substitute to the lack of homogeneity 1 . Here belongs to the field k := W{pt} , a subfield of C[[, −1 ]. (Note that the notation τ = −1 is used in [8].) When X is affine and one denotes by (x; u) a ∗ point of T ∗ X , a section P of this sheaf on an open subset U ⊂ T X is represented by its total symbol σtot (P) = −∞< j≤m p j (x; u)− j , with m ∈ Z, p j ∈ OT ∗ X (U ), the p j ’s satisfying suitable inequalities and the product, denoted here by , being given by the Leibniz formula. 1 In this paper, we write E ∗ and W ∗ instead of the classical notations E and W . X X T X T X
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A fundamental tool for spectral analysis in deformation quantization is the star-exponential of the Hamiltonian H (see [1]): (t−1 H )n exp (t−1 H ) = . n! n≥0
However, at the formal level, the star-exponential does not make sense as a formal series in and −1 . The goal of this article is to construct a new sheaf of algebras on the cotangent bundle T ∗ X to a complex manifold X in which the star-exponential has a meaning and such that quantized symplectic transformations operate on such algebras. More precisely, we construct a new sheaf of k-algebras WTt ∗ X , with an extra central holomorphic parameter t defined in a neighborhood of t = 0, with the property that complex symplectic transformations may be locally quantized as isomorphisms of algeres ι bras and there are natural morphisms of k-algebras WT ∗ X − → WTt ∗ X −→ WT ∗ X whose composition is the identity on WT ∗ X . We give the symbol calculus on WTt ∗ X , which extends naturally that of WT ∗ X (however, now we get series in j with −∞ < j < ∞), and finally we show that, if P is a section of WT ∗ X of order 0, then exp(t−1 P) is well defined in WTt ∗ X . We also briefly discuss the case where T ∗ X is replaced with a general symplectic manifold. Our construction is as follows. First, we add a central holomorphic parameter s ∈ C and consider the sheaf WC×T ∗ X , the subsheaf of WT ∗ (C×X ) consisting of sections not depending on ∂s . Denoting by a : C × T ∗ X − → T ∗ X the projection, we first define an s 1 algebra WT ∗ X := R a! WC×T ∗ X . The algebra structure with respect to the s-variable is given by convolution, as in the case of the space Hc1 (C; OC ). In order to replace this convolution product by an usual product, we define the sheaf WTt ∗ X as the “formal” Laplace transform with respect to the variables s−1 of the algebra WTs ∗ X . In a deformation quantization context, the existence of exp(t−1 P) in WTt ∗ X gives a precise meaning to the star-exponential [1] of P which is heuristically related to the Feynman Path Integral of P. 1. Symbols The fields k and k. We set k := C[[, −1 ]. Hence, an element a ∈ k is a series a= a j − j , a j ∈ C, m ∈ Z. −∞< j≤m
Consider the following condition on a: there exist positive constants C, ε such that |a j | ≤ Cε− j (− j)! for all j < 0.
(1.1)
We denote by k the subfield of k consisting of series satisfying (1.1). Convention. We endow k, hence k, with the filtration associated to ord() = −1.
(1.2)
k(0) and k(0), respectively. The fields k and k are Z-filtered2 and contain the subrings
Note that k(0) = C[[]] and k(0) = k ∩ k(0). Remark 1.1. k is flat over k(0) and k is flat over k(0). 2 In the sequel, we shall say “filtered” instead of “Z-filtered”.
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and O . Let (X, O X ) be a complex manifold. The sheaves O X X is the the sheaf O X [[, −1 ]. In other words, O Definition 1.2.(i) We denote by O X X filtered k-algebra defined as follows: A section f (x, ) of OX of order ≤ m (m ∈ Z) on an open set U of X is a series f j (x)− j , (1.3) f (x, ) = −∞< j≤m
with f j ∈ O X (U ). consisting of sections f (x, ) as (ii) We denote by OX the filtered k-subalgebra of O X above satisfying: for any compact subset K of U there exist positive constants C, ε such that sup | f j | ≤ Cε− j (− j)! for all j < 0. K
(1.4) Note that O (0) ⊗ O X X k(0) k, O X O X (0) ⊗k(0) k.
(1.5)
(To be correct, we should have written k X , the constant sheaf with values in k, instead of k in these formulas, and similarly for k(0), k(0) and k.) Also note that there exist isomorphisms of sheaves (not of algebras) (0) O X ×Cˆ| X ×{0} , O X
(1.6)
OX (0)
(1.7)
O X ×C | X ×{0} ,
where O X ×Cˆ| X ×{0} is the formal completion of O X ×C along the hypersurface X × {0} of X × C and O X ×C | X ×{0} is the restriction of O X ×C to X × {0}. Denoting by t the coordinate on C, the isomorphism (1.7) is given by the map OX (0)
j≤0
f j − j →
j≥0
f− j
tj ∈ O X ×C | X ×{0} . j!
The convolution algebra Hc1 (C; OC ). The results of this subsection are well known and elementary. We recall them for the reader’s convenience. We consider the complex line C endowed with a holomorphic coordinate s. Using this coordinate, we identify the sheaf OC of holomorphic functions on C and the sheaf C of holomorphic forms on C. The space Hc1 (C; OC ) is endowed with a structure of an algebra by → Hc2 (C2 ; OC2 ) Hc1 (C; OC ) × Hc1 (C; OC ) − − → Hc1 (C; OC ), where the first arrow is the cup product and the second arrow is the integration along the fibers of the map C2 − → C, (s, s ) → s + s . When representing the cohomology classes by holomorphic functions, the convolution product is described as follows.
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For a compact subset K of C, we identify the vector space HK1 (C; OC ) with the quotient space (C \ K ; OC )/ (C; OC ) and, if f ∈ (C \ K ; OC ), we still denote by f its image in HK1 (C; OC ) or in Hc1 (C; OC ). Let K and L be compact subsets of C, let f ∈ (C \ K ; OC ) and g ∈ (C \ L; OC ). The convolution product f ∗ g is given by 1 f ∗ g(z) = f (z − w)g(w)dw, (1.8) 2iπ γ where γ is a counter-clockwise oriented circle which contains L and |z| is chosen big enough so that z + K is outside of the disc bounded by γ . It is an easy exercise to show that this definition does not depend on the representatives f and g, and that to interchange the role of f and g in the formula (1.8) modifies the result by a function defined all over C, hence gives the same result in Hc1 (C; OC ). Therefore, we obtain a commutative algebra structure on Hc1 (C; OC ). Example 1.3. 1 z n+1
∗
1 z m+1
=
(n + m)! 1 . n!m! z n+m+1
The sheaf Os, X . From now on, we shall concentrate our study on O X .
Notataion 1.4. We shall often denote by Cs the complex line C endowed with the coordinate s. Lemma 1.5. Let Y be a complex manifold and Z a Stein submanifold of Y . Then H j (Z ; OY (0)| Z ) vanishes for j = 0. Proof. Using the isomorphism (1.7), we may replace the sheaf OY (0) with the sheaf OY ×Ct |t=0 . By a theorem of Siu [11], Z × {0} admits a fundamental system of open Stein neighborhoods in Y × Ct and the result follows. Let X be a complex manifold. The manifold Cs × X is thus endowed with the k-fil tered sheaf OC . Let a : Cs × X − → X denote the projection. s ×X Lemma 1.6. (i) One has the isomorphism R j a! OC R j a! OC (0) ⊗k(0) k. s ×X s ×X (ii) R j a! OC (0) 0 for j = 1. s ×X (iii) Let U ⊂⊂ V ⊂⊂ W be three open subsets of X and assume that W is Stein. Then )− → (U ; R 1 a! OC ) factorizes through the natural morphism (W ; R 1 a! OC s ×X s ×X lim ((Cs \ K ) × V ; OC )/ (Cs × V ; OC ), s ×X s ×X − →
K ⊂Cs
where K ranges over the family of compact subsets of C. L
Proof. (i) follows from the projection formula for sheaves (i.e., Ra ! (F ⊗a −1 G) L
Ra ! F ⊗G) and (1.5), since k is flat over k(0).
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(ii) For x ∈ X , we have H j (Ra ! OC (0))x lim HK (Cs × {x}; OC (0)|Cs ×{x} ). s ×X s ×X − → j
K
Applying the distinguished triangle of functors +1
R K (Cs × {x}; • ) − → R(Cs × {x}; • ) − → R((Cs \ K ) × {x}; • ) −→ to the sheaf OC (0)|Cs ×{x} we get the result by Lemma 1.5 for j > 1 and the case s ×X j = 0 follows from the principle of analytic continuation. (iii) Recall first that if W is a Stein manifold and if W1 ⊂⊂ W is open, there exists a Stein open subset W2 of W with W1 ⊂⊂ W2 ⊂⊂ W . For a compact subset L of X , (L; R 1 a! OC ) (L; R 1 a! OC (0)) ⊗k(0) k. s ×X s ×X Hence, it is enough to prove the result for OC (0). s ×X
By Lemma 1.5, H j (D × U ; OC (0)) vanishes for D open in Cs , U Stein open s ×X
(0)) vanishes for j = 1 and we get in X and j = 0. Therefore, HK ×U (Cs × U ; OC s ×X the exact sequence: j
0− → (Cs × U ; OC (0)) − → ((Cs \ K ) × U ; OC (0)) s ×X s ×X (0)) − → 0. − → HK1 ×U (Cs × U ; OC s ×X
1 Definition 1.7. We set Os, X := R a! OCs ×X . Clearly, Os, X is a sheaf of filtered k-modules. By Lemma 1.6, a section f (s, x, ) of order m of the sheaf Os, X on a Stein open subset W of X may be written on any relatively compact open subset U of W as a series
f (s, x, ) =
f j (s, x)− j ,
−∞< j≤m
where f j (s, x) is a holomorphic function on (Cs \ K 0 ) × U for a compact set K 0 not depending on j and the f j ’s satisfy an estimate (1.4) on each compact subset K of (Cs \ K 0 ) × U . We shall extend the product (1.8) to Os, X as follows. For two sections f (s, x, ) = − j and g(s, x, ) = −∞< j≤m g j (s, x)− j of Os, −∞< j≤m f j (s, x) X , we set:
f (s, x, ) ∗ g(s, x, ) = −∞< j≤m+m h j (s, x)− j , 1 h k (s, x) = i+ j=k 2iπ γ f i (s − w, x)g j (w, x)dw.
(1.9)
Proposition 1.8. The sheaf Os, X has a structure of a filtered commutative k- algebra.
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Proof. It is easily checked that multiplication by −1 induces an isomorphism of sheaves s, ∼ of k-modules Os, that the product of X (m) −→ O X (m + 1). Hence we just need to check two sections of order 0 is a section of order 0. Let f (s, x, ) = −∞ R big enough so that s + K 0 does not meet γ . Then for w ∈ γ and x ∈ K ∩ (Cs \ K 0 ) × U , we have: | f i (s − w, x)g j (w, x)| ≤ C 2 (−k)! i+ j=k,i, j≤0
×
ε−i− j
i+ j=k,i, j≤0
Hence h(s, x, ) =
(−i)!(− j)! ≤ 3C 2 ε−k (−k)!. (−k)!
−∞< j≤0 h k (s, x)
−k
defined by (1.9) is in Os, X (0).
The Laplace transform and the algebra Ot,X. In order to replace the convolution product in the s-variable with the ordinary product, we shall apply a kind of Laplace transform to Os, X . Definition 1.9. On a complex manifold X , we denote by Ot,X the filtered sheaf of k-modules defined as follows. A section f (t, x, ) of Ot,X(m) (i.e., a section of order m) on an open set U of X is a series f j (t, x)− j , f j ∈ (U ; OC×X |t=0 ), (1.10) f (t, x, ) = −∞< j<∞
with the condition that for any compact subset K of U there exists η > 0 such that f j (t, x) is holomorphic in a neighborhood of {|t| ≤ η} × K and satisfies there exist positive constants C, ε such that (1.11) sup | f j (t, x)| ≤ C · ε− j (− j)! for all j < 0, x∈K ,|t|≤η
⎧ ⎨there exist positive constants M and R such that R j−m (1.12) |t| j−m for |t| ≤ η and all j ≥ m. ⎩ sup | f j (t, x)| ≤ M ( j − m)! x∈K Let f (t, x, ) = −∞< j<∞ f j (t, x)− j and g(t, x, ) = −∞< j<∞ g j (t, x)− j be
two sections of Ot,X of order m and m respectively. Define formally h(t, x, ) = h j (t, x)− j , h k (t, x) = f i (t, x)g j (t, x). (1.13) −∞< j<∞
i+ j=k
Lemma 1.10. (i) Multiplication by −1 induces an isomorphism of sheaves of k(0)→ Ot,X(m + 1). modules Ot,X(m) −∼ (ii) The product (1.13) of a section f (t, x, ) ∈ Ot,X(m) and a section g(t, x, ) ∈ Ot,X(m ) is well defined and belongs to Ot,X(m + m ).
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Proof. (i) (a) Let f (t, x, ) = −∞< j<∞ f j (t, x)− j ∈ Ot,X(m). Then −1 f (t, x, ) = −j ˜ ˜ −∞< j<∞ f j (t, x) , with f j = f j−1 . For any integer j < 0, we have: sup
x∈K ,|t|≤η
| f˜j (t, x)| =
sup
x∈K ,|t|≤η
| f j−1 (t, x)| ≤ Cε− j+1 (− j + 1)! ≤ (Cε)(εe)− j (− j)!.
Hence Condition (1.11) is satisfied. For j ≥ m + 1, we have: sup | f˜j (t, x)| = sup | f j−1 (t, x)| ≤ M x∈K
x∈K
R j−m−1 |t| j−m−1 , ( j − m − 1)!
which is simply Condition (1.12) for m + 1 and −1 f (t, x, ) ∈ Ot,X(m + 1). (b) Let f (t, x, ) = −∞< j<∞ f˜j (t, x)− j , with f˜j = f j+1 . For any integer j < −1, we have: sup
x∈K ,|t|≤η
| f˜j (t, x)| =
sup
x∈K ,|t|≤η
| f j+1 (t, x)| ≤ Cε− j−1 (− j − 1)! ≤
C −j ε (− j)!. ε
For j = −1, we have: sup
x∈K ,|t|≤η
| f˜−1 (t, x)| =
sup
x∈K ,|t|≤η
| f 0 (t, x)| = A ≥ 0,
since f 0 (t, x) is holomorphic in a neighborhood of |t| ≤ η × K . Set C = max{ Aε , Cε }, then for all integers j < 0, we have: sup
x∈K ,|t|≤η
| f˜j (t, x)| ≤ C ε− j (− j)!,
and Condition (1.11) is satisfied. For j ≥ m − 1, we have: sup | f˜j (t, x)| = sup | f j+1 (t, x)| ≤ M x∈K
x∈K
R j−m+1 |t| j−m+1 , ( j − m + 1)!
which is Condition (1.12) for m − 1 and f (t, x, ) ∈ Ot,X(m − 1). Therefore, multiplication by −1 induces an isomorphism Ot,X(m) −∼ → Ot,X(m + 1).
(ii) By (i), we may assume m = m = 0. Let f = −∞ 0 −∞< j<∞ g j (t, x) X such that f i (t, x) and g j (t, x) are holomorphic in a neighborhood of {|t| ≤ η} × K . Conditions (1.11) and (1.12) guarantee the existence of the positive constants C1 , ε1 , M1 and R1 for the f i ’s, and C2 , ε2 , M2 and R2 for the g j ’s. We set C = max{C1 , C2 }, ε = max{ε1 , ε2 }, M = max{M1 , M2 } and R = max{R1 , R2 }. We shall show that the product (1.13) is well defined. Let h k (t, x) = i+ j=k f i (t, x) g j (t, x). (a) Consider the case k < 0. The sum defining h k can be divided into three parts: hk = f i gk−i + f i gk−i + f k− j g j . (1.14) k
i≥0
j≥0
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The first sum is finite and defines a holomorphic function in a neighborhood of {|t| ≤ η} × K . In the second sum, k − i is strictly negative and, for each term in this sum, Conditions (1.11) and (1.12) give the following estimates when x ∈ K and |t| ≤ η: (Rη)i i−k Cε (i − k)! i! −k i i −k . ≤ C Mε (−k)!(Rηε) i
| f i (t, x)gk−i (t, x)| ≤ M
Recall that
i≥0 α
i n+i i
=
1 (1−α)n+1
for |α| < 1. When Rηε < 1, (Rηε)i
i−k i
is the
1 2Rε }.
Then the second general term of an absolutely convergent series. Let η˜ = min{η, sum in (1.14) converges uniformly on {|t| ≤ η} ˜ × K. The third sum is handled in a similar way and one gets the estimate: −k j j −k . | f k− j (t, x)g j (t, x)| ≤ C Mε (−k)!(Rηε) j It follows that, for k < 0, h k is a holomorphic function in a neighborhood of {|t| ≤ η}×K ˜ . Let us show that h k satisfies Condition (1.11). For x ∈ K and |t| ≤ η, ˜ the first sum in (1.14) is bounded by: | f i (t, x)gk−i (t, x)| ≤ C 2 ε−i εi−k (−i)!(i − k)! ≤ C 2 ε−k (−k)!. k
k
For the second and third sums we have: f i (t, x)gk−i (t, x)| ≤ C Mε−k (−k)! | i≥0
|
f k− j (t, x)g j (t, x)| ≤ C Mε−k (−k)!
j≥0
1 , (1 − R ηε) ˜ −k+1 1 . (1 − R ηε) ˜ −k+1
ε Let ε˜ = max{ε, (1−R ˜ we find that: ηε) ˜ }. For x ∈ K and |t| ≤ η,
|h k (t, x)| ≤ (C 2 +
2C M )˜ε−k (−k)!. (1 − R ηε) ˜
Hence h k satisfies Condition (1.11). (b) The case k ≥ 0. We again split the sum defining h k into three parts: hk = f i gk−i + f i gk−i + f k− j g j . 0≤i≤k
i<0
(1.15)
j<0
The first sum is a holomorphic function in a neighborhood of {|t| ≤ η} × K . For each term in the second sum, we have the following estimates when x ∈ K and |t| ≤ η: | f i (t, x)gk−i (t, x)| ≤ Cε−i (−i)!M
Rk k R k−i |t|k−i ≤ C M(ε Rη)−i |t| . (k − i)! k!
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Since (ε Rη)−i is the general term of the geometric series, the second sum in (1.15) defines a holomorphic function in a neighborhood of {|t| ≤ η} ˜ × K , where η˜ = 1 min{η, 2Rε }. Similarly, for the third sum we have: | f k− j (t, x)g j (t, x)| ≤ C M(ε Rη)− j
Rk k |t| . k!
Therefore, for k ≥ 0, h k is a holomorphic function in a neighborhood of {|t| ≤ η} ˜ × K. Let us show that h k satisfies Condition (1.12) with m = 0. For x ∈ K and |t| ≤ η, ˜ the first sum in (1.15) is bounded by: k Rk (2R)k k ≤ M2 |t| . | f i (t, x)gk−i (t, x)| ≤ M 2 |t|k i k! k! 0≤i≤k
0≤i≤k
For the second and third sums we find: |
f i (t, x)gk−i (t, x)| ≤ C M
Rk k R ηε ˜ |t| , 1 − R ηε ˜ k!
f k− j (t, x)g j (t, x)| ≤ C M
Rk k R ηε ˜ |t| . 1 − R ηε ˜ k!
i<0
|
j<0
For x ∈ K and |t| ≤ η, ˜ we have: |h k (t, x)| ≤ (M 2 + 2C M
(2R)k k R ηε ˜ ) |t| . 1 − R ηε ˜ k!
Hence h k satisfies Condition (1.12) with m = 0. The product of f ∈ Ot,X(0) and g ∈ Ot,X(0) is well defined and f g ∈ Ot,X(0). Therefore: Proposition 1.11. The sheaf Ot,X is naturally endowed with a structure of a commutative filtered k-algebra. ). We formally Let U be an open subset of X and let f (s, x, ) ∈ ((Cs \ K )×U ; OC s ×X define the Laplace transform L( f ) of f by 1 L( f )(t, x, ) = f (s, x, ) exp(st−1 ) ds, 2iπ γ
where γ is a counter-clockwise oriented circle centered at 0 with radius R 0. Example 1.12. L(s −n−1 ) = −n t n /n!, L(
1 ) = exp(t−1 ). s−1
Lemma 1.13. The Laplace transform induces a k-linear monomorphism s −1 · O X [[s −1 ]][[, −1 ] → O X [[t]][[, −1 ]].
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Proof. One notices that the Laplace transform is given by: an, j t n −n− j , an, j s −n−1 − j → n! −∞< j≤m n≥0
j≤m n≥0
and the result follows. Theorem 1.14. The Laplace transform induces a k-linear isomorphism of filtered k-algebras ∼ t, L : Os, X −→ O X .
(1.16)
Proof. (i) By Lemma 1.10, it is enough to check that L induces an isomorphism t, ∼ Os, X (0) −→ O X (0). (ii) Let W be a Stein open subset of X and let U be a relatively compact open subset of W . Let us develop a section f (s, x, ) of (W ; R 1 a! OC (0)) with respect to s −1 for s ×X s > R. We get f j (s, x)− j f˜(s, x, ) = −∞< j≤0
=
f j,n (x)s −n−1 − j
−∞< j≤0 n≥0
with the following Cauchy estimates: for any compact subset K of U there exist positive constants C, ε, R such that sup | f j,n (x)| ≤ Cε− j (− j)!R n . x∈K
tn Applying the Laplace transform to f˜(s, x, ) means to replace s −n−1 with −n . Hence, n! we find tn f j (t, x)− j = f j,n (x) − j−n , L( f˜)(t, x, ) = n! −∞< j<∞
−∞< j≤0 n≥0
where f j (t, x) =
f j−n,n (x)
j≤n,0≤n
tn n!
satisfies | f j (t, x)| ≤ C
j≤n,0≤n
εn− j
(n − j)! (|t|R)n . n!
Let η < (ε R)−1 . It follows that f j (t, x) is holomorphic in a neighborhood of {|t| ≤ η} × K . Assume j < 0, |t| ≤ η and x ∈ K . We get (n − j)! ε C | f j (t, x)| ≤ Cε− j (− j)! (ηε R)n ≤ ( )− j (− j)!. (− j)!n! 1 − ηε R 1 − ηε R 0≤n
Hence Condition (1.11) is satisfied.
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Assume j ≥ 0. We get for |t| ≤ η and x ∈ K , | f j (t, x)| ≤ C
Rj j ε− j j!(n − j)! C (|t|Rε)n ≤ |t| . j! n! 1 − ηε R j! j≤n
Hence Condition (1.12) for m = 0 is satisfied and L( f )(t, x, ) is in Ot,X(0). (iii) Conversely, let f (t, x, ) be a section of Ot,X(0). We develop f as f (t, x, ) =
f j (t, x)− j =
−∞< j<∞
n! f j,n (x)
−∞< j<∞ n≥0
t n −n − j+n . n!
(1.17)
For any compact set K , there exists η > 0 such that f j (t, x) is holomorphic in a neighborhood of {|t| ≤ η} × K . Conditions (1.11) and (1.12) give the Cauchy estimates | f j,n (x)| ≤ Cε− j (− j)!η−n for j < 0, | f j,n (x)| ≤ M
R j j−n η for j ≥ 0. j!
Notice that Condition (1.12) for j > 0 implies that f j (0, x) =
∂ j−1 f j ∂fj (0, x) = · · · = (0, x) = 0, ∂t ∂t j−1
(1.18)
or f j,n (x) = 0 for 0 ≤ n ≤ j − 1.
tn The inverse Laplace transform consists formally in replacing −n by s −n−1 in n! (1.17). We then get L−1 ( f )(s, x, ) = f˜(s, x, ) = n! f j+n,n (x)s −n−1 − j . −∞< j<∞ n≥0
−j ˜ ˜ Writing f˜(s, x, ) = −∞< j<∞ f j (s, x) , (1.18) implies that f j (s, x) = 0 for j ≥ 1. Let R1 > R be large enough so that (η R1 )−1 ≤ 1. We shall check that the sum f˜j (s, x) = n≥0 n! f j+n,n (x)s −n−1 defines a holomorphic function in a neighborhood of {|s| ≥ R1 } × K for any j ≤ 0. For j ≤ 0, let us split the sum f˜j (s, x) as n! f j+n,n (x)s −n−1 + n! f j+n,n (x)s −n−1 . (1.19) f˜j (s, x) = n≥− j
0≤n<− j
In the first sum we have n + j ≤ 0 and for |s| ≥ R1 and x ∈ K , we get from the Cauchy estimates R n+ j j −n−1 M n R −n−1 j η |s| |n! f j+n,n (x)s | ≤ n!M ≤ (η R) (− j)! ( )n . (n + j)! R1 − j R1 (1.20) The right-hand side is the general term of a convergent series since R1 > R and we get the result by noticing that the second sum in (1.19) is finite.
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Finally we shall show that f˜j (s, x) satisfies the required estimates. From (1.20), the first sum in (1.19) is bounded by n R M −n−1 j ( )n | n! f j+n,n (x)s |≤ (η R) (− j)! − j R1 R1 n≥− j
n≥− j
− j 1 M ≤ (− j)!. R1 − R η(R1 − R) Similarly, for the second sum we have |
n! f j+n,n (x)s −n−1 | ≤
0≤n<− j
C −j ε (− j)! R1
0≤n<− j
1 n 1 2C − j ) ≤ ε (− j)!,
− j ( η R1 R1
where the last inequality follows from (η R1 )−1 ≤ 1 and
n
0≤n<− j
Combining these estimates we get for j ≤ 0,
1
(−n j )
≤ 2.
| f˜j (s, x)| ≤ C˜ ε˜ − j (− j)!, 1 with C˜ = max{ R1M−R , 2C R1 } and ε˜ = max{ε, η(R1 −R) }. ˜ Therefore f˜(s, x, ) = j≤0 f˜j (s, x)− j is a section of Os, X (0) and L( f )(t, x, ) = f (t, x, ). (iv) The fact that L is a morphism of algebras follows easily from Example 1.3.
The ring gr Ot,X. If A is a filtered sheaf of rings, we denote as usual by gr A the associated graded ring. Let Cu be the complex line endowed with the coordinate u and denote by b : X × Cu − → X the projection. exp u
Definition 1.15. (i) One denotes by O X the subsheaf of C-algebras on X of the sheaf b∗ O X ×Cu whose sections on an open set U ⊂ X are the holomorphic functions f (x, u) on U × Cu satisfying: for any compact subset K of U there exist positive constants C, R such that sup | f (x, u)| ≤ C exp(R|u|). x∈K exp t −1
(ii) One sets O X
exp t −1
[, −1 ] = O X
⊗C C[, −1 ].
Proposition 1.16. There is a natural isomorphism of graded sheaves of rings exp t −1
gr Ot,X O X
[, −1 ].
Proof. First note the isomorphism s, 1 Os, X (0)/O X (−1) R a! OCs ×X ,
from which we deduce the isomorphism 1 −1 gr Os, X R a! OCs ×X ⊗C C[, ].
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The classical Paley-Wiener theorem says that the Laplace transform induces an isomorphism between Hc1 (C; OC ) and the space of entire functions of exponential type. An extension of this result with holomorphic parameters provides an isomorphism exp t −1 L : R 1 a! OCs ×X −∼ , → OX
and the result follows. in the preceding constructions The formal case. It is possible to replace OX with O X and to set s, := R 1 a! O O Cs ×X . X
(1.21)
s, does not seem to have an easy description. However the Laplace transform of O X Indeed, its sections are no longer germs of holomorphic functions with respect to t as shown in the next example. Example 1.17. Consider a sequence {c j } j≤0 of complex numbers and the section f of s, given by O X cj − j . f (s, ) = (s − 1) j≤0
Then, formally, the Laplace transform of f is given by L( f )(t, ) =
j≤0 n≥0
and the coefficient of 0 is
tn
n≥0 c−n n! ,
cj
t n −n− j , n!
which does not belong to OCt |t=0 in general.
2. The Algebra WT ∗ X Let (X, O X ) be a complex manifold. The cotangent bundle T ∗ X is a homogeneous symplectic manifold endowed with the C× -conic sheaf of rings ET ∗ X of finite-order microdifferential operators. This ring is filtered and contains in particular the subring ET ∗ X (0) of operators of order ≤ 0. This ring is constructed in [9] and we assume that the reader is familiar with this theory, referring to [5] or [10] for an exposition. On the symplectic manifold T ∗ X there exists another (no more conic) useful sheaf of rings constructed as follows (see [8]). Let C be the complex line endowed with the coordinate t and (t; τ ) the associated coordinates on T ∗ C. Set T{τ∗ =0} (X × C) = {(x, t; ξ, τ ); τ = 0} and consider the map ρ : T{τ∗ =0} (X × C) − → T ∗ X, (x, t; ξ, τ ) → (x; ξ/τ ).
(2.1)
ET ∗ (X ×C),t = {P ∈ ET ∗ (X ×C) ; [P, ∂/∂t ] = 0}.
(2.2)
Set
The ring WT ∗ X on T ∗ X is given by WT ∗ X := ρ∗ (ET ∗ (X ×C),t ).
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In the sequel we set := τ −1 .
(2.3)
The ring WT ∗ X is filtered and we denote by WT ∗ X ( j) the subsheaf of WT ∗ X consisting of sections of order less than or equal to j. The following result was obtained in [8]. Theorem 2.1. (i) The sheaf WT ∗ X is naturally endowed with a structure of a filtered k-algebra and gr WT ∗ X OT ∗ X [, −1 ]. (ii) Consider two complex manifolds X and Y , two open subsets U X ⊂ T ∗ X and UY ⊂ → UY . Then, locally, ψ may be T ∗ Y and a symplectic isomorphism ψ : U X −∼ → WT ∗ Y such that quantized as an isomorphism of filtered k-algebras : WT ∗ X −∼ the isomorphism induced on the graded algebras coincides with the isomorphism OT ∗ X [, −1 ] −∼ → OT ∗ Y [, −1 ] induced by ψ. Total symbols. Assume that X is affine of dimension n, that is, X is open in some C-vector space V of dimension n. Theorem 2.2. Assume X is affine. There is an isomorphism of filtered sheaves of k-modules (not of algebras), called the “total symbol” morphism: σtot : WT ∗ X −∼ → OT∗ X .
(2.4)
The total symbol of a product is given by the Leibniz formula. Denote by (x) a local coordinate system on X and denote by (x, u) the associated local symplectic coordinate system on T ∗ X . If Q is an operator of total symbol σtot (Q), then σtot (P ◦ Q) =
|α| ∂uα σtot (P) · ∂xα σtot (Q). α! n
(2.5)
α∈N
The total symbol of a section P ∈ WT ∗ X (U ) is thus written as a formal series: σtot (P) =
p j (x; u)− j , m ∈ Z,
p j ∈ OT ∗ X (U ),
(2.6)
−∞≤ j≤m
with the condition (1.4). Note that (2.5) does not depend of the choice of a local coordinate system on X but only on the affine structure of V . Indeed, (2.5) may be rewritten as σtot (P ◦ Q) = (exp(du , d y )σtot (P)(x, u)σtot (Q)(y, v))|x=y,u=v , where du , d y =
n
i=1 ∂u i ∂ yi
does not depend on the affine coordinate system.
Remark 2.3. Let us identify X with the zero section of T ∗ X . Then the sheaf OX (see Def. 1.2) is isomorphic to the left coherent WT ∗ X -module obtained as the quotient of WT ∗ X by the left ideal generated by the vector fields on X .
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3. The Algebra WTs ∗ X . Operations on W. Let S be a complex manifold of complex dimension d S . One defines the sheaf W S×T ∗ X on S × T ∗ X as the subsheaf of WT ∗ (S×X ) consisting of sections which commute with the holomorphic functions on S. Heuristically, W S×T ∗ X is the sheaf WT ∗ X with holomorphic parameters on S. For a morphism of complex manifolds f: S − → Z we shall still denote by f the map S × X − → Z × X , as well as the map S × T∗X − → Z × T ∗ X . One denotes as usual by S the sheaf of holomorphic forms of maximal degree and one sets for short: (d )
S W S×T ∗ X = W S×T ∗ X ⊗O S . S
(3.1)
Let us recall well-known operations of the theory of microdifferential operators. Although these results do not seem to be explicitly written in the literature, their proofs are straightforward and will not be given here. Let f : S − → Z be a morphism of complex manifolds. The usual operations of inverse image f ∗ : f −1 O Z − → O S and of direct image f : R f ! S [d S ] − → Z [d Z ] extend to W S×T ∗ X . More precisely, there exist morphisms of sheaves of k-modules (the second morphism holds in the derived category Db (k Z ×T ∗ X )): → W S×T ∗ X , f ∗ : f −1 W Z ×T ∗ X − (d S ) Z) : R f ! (W S×T → W Z(d×T ∗ X [d S ]) − ∗ X [d Z ],
(3.2) (3.3)
f
these morphisms having the following properties: • they are functorial with respect to f , that is, for a morphism of complex manifolds g: Z − → W , one has (g ◦ f )∗ f ∗ ◦ g ∗ and g◦ f = g ◦ f , and moreover the inverse (resp. direct) image of the identity morphism is the identity, • when X is affine, f ∗ and f commute with the total symbol morphism (2.4). As a convention, we choose the morphism in (3.3) so that the integral of Hc1 (Cs ; Cs ) is 1. In other words, a
1 1 = s 2iπ
γ
ds ∈ s
ds , s
where γ is a counter-clockwise oriented circle around the origin. The algebra WTs ∗ X . Denote by → T∗X a : Cs × T ∗ X −
(3.4)
the projection. Then, after identifying the sheaves OCs and Cs by f (s) → f (s)ds, the sheaf R 1 a! WCs ×T ∗ X is endowed with a structure of a filtered k-algebra by Hc1 (Cs × T ∗ X ; WCs ×T ∗ X ) × Hc1 (Cs × T ∗ X ; WCs ×T ∗ X ) − → Hc2 (C2s,s × T ∗ X ; WC2
s,s
×T ∗ X )
− → Hc1 (Cs ; WCs ×T ∗ X ), where the first arrow is the cup product and the second arrow is the integration along → C, (s, s ) → s + s . the fibers of the map C2 −
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Definition 3.1. The sheaf WTs ∗ X of k-modules on T ∗ X is given by WTs ∗ X = R 1 a! (WCs ×T ∗ X ).
(3.5)
1 with the cohomology class it defines s in Hc1 (Cs ; OCs ), we define the morphism of sheaves After identifying the holomorphic function
1 P. s Clearly, the morphism (3.6) is a monomorphism of sheaves of k-algebras. We define the morphism of sheaves ι : WT ∗ X − → WTs ∗ X ,
P →
res : WTs ∗ X − → WT ∗ X
(3.6)
(3.7)
by the integration morphism (3.3) associated to the map (3.4). Clearly, the morphism (3.7) is a morphism of sheaves of k-algebras. Hence: Theorem 3.2. (i) The sheaf WTs ∗ X is naturally endowed with a structure of a filtered k-algebra and gr WTs ∗ X R 1 a! OCs ×T ∗ X [, −1 ]. (ii) The monomorphism ι in (3.6) is a morphism of filtered k-algebras, the integration morphism res in (3.7) is a morphism of filtered k-algebras and the composition res ◦ ι : WT ∗ X − → WTs ∗ X − → WT ∗ X is the identity. (iii) Consider two complex manifolds X and Y , two open subsets U X ⊂ T ∗ X and UY ⊂ → UY . Then, locally, ψ may be T ∗ Y and a symplectic isomorphism ψ : U X −∼ → WTs ∗ Y such that quantized as an isomorphism of filtered k-algebras : WTs ∗ X −∼ the isomorphism induced on the graded algebras coincides with the isomorphism R 1 a! OCs ×T ∗ X [, −1 ] −∼ → R 1 a! OCs ×T ∗ Y [, −1 ] induced by ψ. (iv) Assume X is affine. There is an isomorphism of filtered sheaves of k-modules (not of algebras), called the “total symbol” morphism: σ : W s ∗ −∼ → Os,∗ . (3.8) tot
T X
T X
The total symbol of a product is given by the Leibniz formula with a convolution product in the s variable (see (3.10)). Proof. These results follow immediately from Theorem 2.1. Assume that X is affine. For each Stein open subset W of T ∗ X and each relatively compact open subset U ⊂⊂ W , a section P of WTs ∗ X on W admits a total symbol σtot (P)(s, x, u) = p j (s, x; u)− j , m ∈ Z, (3.9) −∞< j≤m
where p j belongs to ((Cs \ K 0 ) × U ; OCs ×T ∗ X ), for a compact subset K 0 of Cs which depends only on P and U , and the p j ’s satisfy an estimate as in (1.4) on each compact subset K of (Cs \ K 0 ) × U . Consider now two sections P and Q of WTs ∗ X on a Stein open set W with total symbols as in (3.9) (replacing p j with q j and m with m for Q). Then the total symbol of P ◦ Q is given by the Leibniz formula: |α| ∂ α σtot (P) ∗ ∂xα σtot (Q), σtot (P ◦ Q) = (3.10) α! u n α∈N
where, setting f (s, x, u) = ∂uα σtot (P)(s, x; u) and g(s, x, u) = ∂xα σtot (Q)(s, x; u), the product f ∗ g is given by (1.9).
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4. The Laplace Transform and the Algebra WTt ∗ X The filtered k-algebra WTt ∗ X on T ∗ X is the algebra WTs ∗ X , but with a different symbol calculus. Definition 4.1. We set WTt ∗ X := WTs ∗ X . For X affine, the total symbol morphism of k-modules (not of algebras) → OTt, σtot : WTt ∗ X −∼ ∗X
(4.1)
∼ is the composition WTs ∗ X −− → OTs,∗X −∼ → OTt, ∗X. σtot
L
For P a section of WTt ∗ X on a Stein open subset V of T ∗ X and an open subset U ⊂⊂ V , σtot (P) is written as a series p j (t, x, u)− j , p j ∈ OC×T ∗ X |t=0 (U ) σtot (P)(t, x, u, ) = −∞< j<∞
satisfying (1.11) and (1.12). Applying Theorem 3.2, we get: exp t −1
Theorem 4.2. (i) WTt ∗ X is a filtered k-algebra and gr WTt ∗ X OT ∗ X [, −1 ] (see Definition 1.15). (ii) The morphism ι in (3.6) induces a monomorphism of filtered k-algebras ι : WT ∗ X → WTt ∗ X , the morphism res in (3.7) induces a morphism of filtered k-algebras res : → WT ∗ X and the composition WT ∗ X − → WTt ∗ X − → WT ∗ X is the identity. WTt ∗ X − (iii) Consider two complex manifolds X and Y , two open subsets U X ⊂ T ∗ X and UY ⊂ → UY . Then, locally, ψ may be T ∗ Y and a symplectic isomorphism ψ : U X −∼ → WTt ∗ Y such that quantized as an isomorphism of filtered k-algebras : WTt ∗ X −∼ the isomorphism induced on the graded algebras coincides with the isomorphism exp t −1 exp t −1 OT ∗ X [, −1 ] −∼ → OT ∗ Y [, −1 ] induced by ψ. (iv) Assume X is affine. There is an isomorphism of filtered sheaves of k-modules (not of algebras), called the “total symbol” morphism: σtot : WTt ∗ X −∼ → OTt, ∗X.
(4.2)
The total symbol of a product is given by the Leibniz formula. For P and Q two sections of WTt ∗ X on an open subset U of T ∗ X , with X affine, the total symbol of P ◦ Q is thus given by the formula: σtot (P ◦ Q) =
|α| ∂uα σtot (P) · ∂xα σtot (Q), α! n
(4.3)
α∈N
where the product ∂uα σtot (P) · ∂xα σtot (Q) is given by the usual commutative algebra structure of OTt, ∗ X of Lemma 1.10. Remark 4.3. In Theorem 4.2, the monomorphism WT ∗ X − → WTt ∗ X is given on symbols t → WT ∗ X is given on symbols by by σtot (P) → σtot (P) and the morphism WT ∗ X − σtot (P)(t, x; u, ) → σtot (P)(0, x; u, ).
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The formal case. The above constructions also work when replacing the sheaf WT ∗ X T ∗ X . Let us briefly explain it. with its formal counterpart, the sheaf W Let X be a complex manifold, as above. Replacing the sheaf of rings ET ∗ X on T ∗ X with the sheaf of rings ET ∗ X of formal microdifferential operators and proceeding as for T ∗ X of finite-order formal WKB-operators on T ∗ X . WT ∗ X , we get the sheaf of rings W It is defined by T ∗ X := ρ∗ (ET ∗ (X ×C),t ). W When X is affine of dimension n, the total symbol morphism induces an isomorphism of k-modules T ∗ X −∼ ∗ , σtot : W →O T X and the symbol σtot (P ◦ Q) is given by the Leibniz formula (2.5). Then by a similar s ∗ . Namely, construction as for WTs ∗ X we construct the filtered sheaf of k-algebras W T X we set C×T ∗ X . s ∗ := R 1 a! W W T X If X is affine, the total symbol morphism induces an isomorphism of k-modules s ∗ −∼ s,∗ and the product is again given by the Leibniz formula (3.10). W → O T X T X However, as already noticed, the Laplace transform does not seem to behave as well for the formal case as for the analytic case, and we shall not construct the Laplace s,∗ . transform of O T X s on a Symplectic Manifold X 5. Remark: The Algebra WX
The complex case. Consider a complex symplectic manifold X. There exists an open covering X = i Ui and complex symplectic isomorphisms ϕi : Ui −∼ → Vi where the Vi ’s are open in some cotangent bundles T ∗ X i of complex manifolds X i . Set WUi := ϕi−1 WT ∗ X i |Vi . In general, the WUi ’s do not glue in order to give a globally defined sheaf of algebras WX on X. However the prestack S on X (roughly speaking, a prestack is a sheaf of categories) Ui → Mod(WUi ) is a stack and the category Mod(WX) := S(X) is well defined. Moreover, one can give a precise meaning to WX by replacing the notion of a sheaf of algebras with that of an algebroid. We refer to [4] for the construction of (an analogue of) this stack in the contact complex case and to [6] in the symplectic complex X and for the definition of an algebroid. See also [8] for a construction of case for W WX (by a different method). By adapting the construction of [8], one easily constructs s associated with the locally defined sheaves of algebras W s . Details the algebroid WX Ui are left to the reader. The real case. Let M be a real analytic manifold, X a complexification of M and denote by ω X the canonical 2-form on T ∗ X . The conormal bundle TM∗ X is Lagrangian for Re ω X and symplectic for Im ω X . In particular, the real manifold TM∗ X is symplectic. For an open subset U of TM∗ X , we set WU := WT ∗ X |U . Now, consider a real analytic symplectic manifold M. It is well known that it is possible to construct a globally defined sheaf of algebras WM on M such that:
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• there exists an open covering M = i∈I Ui and real symplectic isomorphisms ϕi : Ui −∼ → Vi where the Vi ’s are open in the conormal bundles TM∗ i X i for some real manifolds Mi with complexification X i , • WM|Vi ϕi−1 WVi for all i ∈ I . Replacing M with Cs × M, one easily constructs the sheaf of algebras WCs ×M of sections with holomorphic parameter s ∈ Cs . Setting s WM := R 1 a! WCs ×M
we get a filtered k-algebra similar to the algebra WTs ∗ X of Definition 3.1. Then, if P 1 s . is well defined in WM belongs to WM and has order 0, the section s−P 6. Applications As an application, let us construct the exponential of sections of order 0 of WT ∗ X . Consider a section P of WT ∗ X (0) on an open subset U of T ∗ X . For each compact subset K of U , there exists R > 0 such that the section s − P of WTs ∗ X defined on Cs ×U is invertible on (Cs \ D(0, R)) × K , where D(0, R) denotes the closed disc centered at 0 1 defines an element of Hc1 (Cs × U ; WCs ×T ∗ X ), hence, with radius R. Therefore s−P 1 . an element of (U ; WTs ∗ X ). We still denote this section of WTs ∗ X on U by s−P n 1 P By developing as n≥0 n+1 and applying the Laplace transform, we get s−P s 1 formally: L( s−P ) = exp(t−1 P). Notataion 6.1. We denote by exp(t−1 P) the image in WTt ∗ X of the section WTs ∗ X .
1 of s−P
Proposition 6.2. For P ∈ WT ∗ X (0), there is a section exp(t−1 P) ∈ WTt ∗ X such that, when X is affine: σtot (exp(t−1 P)) =
(t−1 σtot (P))n n≥0
n!
,
where the star-product f n means the product given by the Leibniz formula (2.5). Remark 6.3. The Leibniz formula (2.5) is nothing but the standard or normal or Wick star-product and Proposition 6.2 tells us that the star-exponential [1] of P makes sense in WTt ∗ X . In a holomorphic deformation quantization context, the star-exponential of P is heuristically related to the Feynman Path Integral FPI(P) of P. Indeed, the Feynman Path Integral of a Hamiltonian H is the symbol of the evolution operator associated to H , the precise relation being given (see [2]) by exp(−xu−1 )FPI(P) = σtot (exp(t−1 P)).
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Example 6.4. As a simple example, take X = C and P ∈ WT ∗ X (0) with σtot (P) = p0 (t, x; u) = θ xu, θ ∈ C. Up to a change of holomorphic symplectic coordinates, σtot (P) represents the Hamiltonian of the harmonic oscillator in the holomorphic representation. Clearly P is in WC (0), and the total symbol of exp(t−1 P) is easily computed: ∂ σtot (exp(t−1 P)) = σtot (−1 P ◦ exp(t−1 P)) ∂t
= −1 σtot (P)σtot (exp(t−1 P)) ∂ ∂ + σtot (P) σtot (exp(t−1 P)) ∂u ∂x ∂ −1 = θ uxσtot (exp(t−1 P)) + θ x σtot (exp(t−1 P)). ∂x Since σtot (exp(t−1 P))|t=0 = 1, the solution to the preceding equation is:
σtot (exp(t−1 P)) = exp (exp(θ t) − 1)xu−1 . The Feynman Path Integral for the harmonic oscillator is well known in the Physics literature and is given by exp exp(θ t)xu−1 [3]. Acknowledgement. We would like to thank Masaki Kashiwara for extremely useful conversations and helpful insights. The first named author thanks Yoshiaki Maeda for warm hospitality at Keio University where this work was finalized, and the JSPS for financial support.
References 1. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization I,II, Ann. Phys. 111:61–110, 111–151 (1978) 2. Dito, J.: Star product approach to quantum field theory: The free scalar field. Lett. Math. Phys. 20, 125– 134 (1990) 3. Faddeev, L.D., Slanov, A.A.: Gauge Fields. Introduction to Quantum Theory. Reading MA: Benjamin Cummings Publishing, 1980 4. Kashiwara, M.: Quantization of contact manifolds. Publ. RIMS, Kyoto Univ. 32, 1–5 (1996) 5. Kashiwara, M.: D-modules and Microlocal Calculus, Translations of Mathematical Monographs 217, Providence, RI: American Math. Soc., 2003 6. Kontsevich, M.: Deformation quantization of algebraic varieties, In: EuroConférence Moshé Flato, Part III (Dijon, 2000). Lett. Math. Phys. 56, 271–294 (2001) 7. Kontsevich, M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66, 157–216 (2003) 8. Polesello, P., Schapira, P.: Stacks of quantization-deformation modules over complex symplectic manifolds. Int. Math. Res. Notices 49, 2637–2664 (2004) 9. Sato, M., Kawai, T., Kashiwara, M.: Microfunctions and pseudo-differential equations. In: Komatsu, H. (ed.), Hyperfunctions and pseudo-differential equations. Proceedings, Katata 1971. Lecture Notes in Math. 287. New-York: Springer-Verlag, 1973, pp. 265–529 10. Schapira, P.: Microdifferential Systems in the Complex Domain. Grundlehren der Math. Wiss. 269. Berlin: Springer-Verlag, 1985 11. Siu, Y.T.: Every Stein subvariety admits a Stein neighborhood. Invent. Math. 38, 89–100 (1976/77) Communicated by L. Takhtajan
Commun. Math. Phys. 273, 415–443 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0194-6
Communications in
Mathematical Physics
Heat Kernel and Number Theory on NC-Torus V. Gayral1 , B. Iochum2,3 , D. V. Vassilevich 4,5 1 Matematisk Afdeling, Københavns Universitet, 2100 Kobenhavn, Denmark.
E-mail: [email protected]
2 UMR 6207, Unité Mixte de Recherche du CNRS et des Universités Aix-Marseille I, Aix-Marseille II et de
l’Université du Sud Toulon-Var, Laboratoire affilié à la FRUMAM – FR 2291, Luming Case 907, F-13288 Marseille Cedex, France. E-mail: [email protected] 3 Université de Provence, Marseille, France 4 Institut für Theoretische Physik, Universität Leipzig, Posttach 100920, D-04099 Leipzig, Germany. E-mail: [email protected] 5 V. A. Fock Institute of Physics, St. Petersburg University, St. Petersburg 198504, Russia Received: 12 July 2006 / Accepted: 11 September 2006 Published online: 13 March 2007 – © Springer-Verlag 2007
Abstract: The heat trace asymptotics on the noncommutative torus, where generalized Laplacians are made out of left and right regular representations, is fully determined. It turns out that this question is very sensitive to the number-theoretical aspect of the deformation parameters. The central condition we use is of a Diophantine type. More generally, the importance of number theory is made explicit in a few examples. We apply the results to the spectral action computation and revisit the UV/IR mixing phenomenon for a scalar theory. Although we find non-local counterterms in the NC φ 4 theory on T4 , we show that this theory can be made renormalizable at least at one loop, and maybe even beyond. 1. Introduction The importance of heat kernel techniques in spectral analysis (see [26, 32]) or in quantum field theory has been known for a long time (see for instance the references in [40]). This type of expansion is particularly very useful for the control of anomalies and loop divergences. Naturally, its extension to noncommutative theories using for instance the Moyal product instead of the pointwise one, was also begun a long time ago (see reviews [18, 37, 39] and also [10]). The idea, originally due to Heisenberg, behind this generalization is that it could help to suppress some divergences. Unfortunately, a consequence of this idea is that the situation is as difficult as in the classical setting, or even worse since some UV/IR mixing can occur, except in some peculiar cases where the renormalisability of the model is proved [29]. Meanwhile, the noncommutative geometry (NCG) pioneered by Alain Connes [12] has shown its capacity to cover isospectral deformations like the deformation of a classical torus into the celebrated noncommutative torus (nc-torus). While many physical ideas coming from string theory have justified a systematic study of noncommutative quantum field theory, the interest of NCG stems also from its mathematical roots. In particular, the spectral action introduced by Chamseddine–Connes refers to a spectral triple (A, H, D) of an algebra A acting on a Hilbert space H and a
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given Dirac operator D which generates the inner fluctuations corresponding to gauge potentials. This spectral action is simply Tr (D A /) , where is a positive even function, is a mass scale parameter, D A = D + A and A is a one-form. This torus depends on parameters through a deformation matrix and it appears that the heat asymptotics are very sensitive to it. In particular, some Diophantine conditions, see (14), (16), are necessary to control the number-theoretic deviation from rational numbers. We also investigate a situation beyond this condition which yields precise aspects of number theory. To control the heat trace asymptotic we apply a two-step procedure. First, we define a trace (cf. (11)), which is indeed proportional to the Dixmier trace, and calculate it through the Fourier coefficients (cf. (18)). Then we prove that being expressed in terms of this trace the heat trace asymptotics for generalized Laplacians look precisely the same as in the commutative case. We first apply the control of the heat trace asymptotic to the spectral action computation. This action has been partially computed in [24], but here we pay attention to the natural existing real structure J of the triple ([13]): now D A = D + A + J A J −1 , so we are in the most difficult situation where simultaneously left and right regular representations exist (see also [27] for the further physical motivations). We show here that this full spectral action is the expected one for a 4-dimensional nc-torus. The amazing fact is that unlike for arbitrary generalized Laplacians, where non-standard terms appear in the heat kernel expansion (typically of the form ‘product of traces’), for the square of the covariant Dirac operator, such weird terms are absent. Thus the formula (57) we obtain is the expected one, up to some numerical coefficients. Then, we apply our results to the study of a scalar field on an nc-4-torus. We show that the divergent part of the effective action does not reproduce the structure of the classical one, that is, the divergences cannot be cancelled by a proper couplings re-definition. Nevertheless, the theory can be made renormalizable at one loop by adding to the classical action a non-local term, which perfectly fits in with the philosophy of [29]. We conjecture that the modified theory is renormalizable to all orders in perturbation theory. The paper is organized as follows: we recall in Sect. 2 some useful facts on nc-tori and study a trace, in fact a Dixmier trace, applied to operators like L(a)R(b), a, b ∈ A, where L (resp. R) is the left (resp. right) multiplication, giving a full asymptotic of Tr L(a)R(b)e−t P as t → 0 for a generalized Laplacian P. Section 3 touches on toric noncommutative manifolds, not necessarily compact. The spectral action is computed in Sect. 4 and the last section is devoted to the study of divergences of a scalar field theory. Since the proof of the asymptotics of the heat trace is technical, it is postponed to Appendix A while some consequences of number theory in this setting are developed in Appendix B. 2. Heat Trace Asymptotic on NC-Torus 2.1. Traces and number theory. Let C ∞ (Tn ) be the smooth noncommutative n-torus associated to a skewsymmetric deformation matrix ∈ Mn (R) (see [11, 35]). This means that C ∞ (Tn ) is the algebra generated by n unitaries u i , i = 1, . . . , n subject to the relations u i u j = eii j u j u i ,
(1)
we denote k.q := and with Schwartz coefficients. In the following, for k, q ∈ gµν k µ q ν and |k|2 := k.k, with gµν a constant metric on Tn . Naturally, the matrix satisfies k.q = −k.q. Zn ,
Heat Kernel and Number Theory on NC-Torus
417 i
Using the Weyl elements Uk := e− 2 k.χ k u k11 · · · u knn , k ∈ Zn , where χ is the matrix restriction of to its upper triangular part, the relation (1) reads i
Uk Uq = e− 2 k.q Uk+q ,
(2)
where χ is the matrix restriction of to its upper triangular part. Thus unitary operators Uk satisfy Uk∗ = U−k and a typical element a ∈ C ∞ (Tn ) can be written as a = (2π )−n/2 k∈Zn ak Uk , where {ak } ∈ S(Zn ). We use this non-standard normalization in order to simplify upcoming formulas. Let τ be the (unique) normalized faithful trace on C ∞ (Tn ) defined by τ a := (2π )−n/2 a0 and Hτ be the GNS Hilbert space obtained by completion of C ∞ (Tn ) with respect to the norm induced by the scalar product a, b := τ (a ∗ b). On Hτ , we consider the left and right regular representations of C ∞ (Tn ) by bounded operators, that we denote respectively by L(.) and R(.). An easy consequence of the associativity of the algebra is the commutativity of these two representations, namely L(a)R(b) = R(b)L(a), for all a, b ∈ C ∞ (Tn ). Let also δµ , µ = 1, . . . , n be the n (pairwise commuting) canonical derivations, defined by δµ (Uk ) := ikµ Uk .
(3)
They extend to unbounded operators on Hτ (with suitable domain), and let := −g µν δµ δν ≥ 0 be the associated Laplacian. (with constant metric). There is (at least) one analogous description of C ∞ (Tn ) given in terms of Rieffel n n ∞ n star-product ∞ n [36]. If α denotes the (periodic) action of R on T , then C (T ) C (T ), , where the star-product is defined by the following oscillatory integrals: f α−z h , f, g ∈ C ∞ (Tn ), (4) d n y d n z e−i yz α 1 f h := (2π )−n 2 y
R n ×R n
yielding relation (2) on the Fourier modes, i
eikx eiq x = e− 2 k.q ei(k+q)x . A counterpart of Eq. (3) reads ∂µ eikx = ikµ eikx . In this description, the above defined Laplacian is nothing else but the ordinary one (associated to the constant metric g) and the trace τ is the normalized integral, √ τ ( f ) := (2π )−n d n x f (x) = (vol Tn )−1 d n x g f (x), f ∈ C ∞ (Tn ). Tn
Tn
We can consider a non-flat metric, but we need (for later use) a severe restriction on it: the Rn -action must be isometric. Thus only constant metrics are allowed.1 1 There are very few attempts to deal with metrics which are not constant in the non-commutative directions. So far, one was able to obtain expressions for the heat trace asymptotics as formal power series in deviations of the metric from the flat one only [42]. Similar difficulties appear if the metric is matrix-valued [2].
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The purpose of this section is to establish the small-t asymptotics of the function t → Tr L(l)R(r )e−t P , where P is a generalized Laplacian, i.e. P has the form P := −(g µν ∇µ ∇ν + E),
(5)
∇µ := δµ + ωµ := δµ + L(λµ ) − R(ρµ ) , E := L(l1 ) − R(r1 ) + L(l2 )R(r2 ) ,
(6) (7)
where
and l, r, λµ , ρµ , li , ri ∈ C ∞ (Tn ). The arbitrary choice of the sign −R will be justified in (49). One can also take more general forms of E and ω. For example, ωµ can contain a term like R(ρµ )L(l ) with some smooth ρµ and l . Such modifications change very little in our considerations below. For that, the central asymptotic to compute is the one of the function t → Tr L(l) R(r ) e−t , l, r ∈ C ∞ (Tn ). (8) Indeed, after an expansion of the semi-group e−t P , viewed as an unbounded perturbation of the heat operator e−t , the only other asymptotics we need are t → Tr L(l) e−t , t → Tr R(r ) e−t , (9) but we have shown in [24, 41] that they have the same asymptotics as their commutative ( = 0) counterparts. Note that the heat semi-group e−t is trace-class, since it is diagonal in the ortho2 normal basis {Uk }k∈Zn , with eigenvalues e−t |k| . Of course, the same property holds −t P for e , as shown in the next lemma, based on a simple application of the Duhamel expansion. Lemma 2.1. For any r, l, λµ , ρµ , li , ri ∈ C ∞ (T ), the operator e−t P is trace-class for t > 0. Proof. We are going to use Duhamel’s expansion for the semi-group generated by P, viewed as an unbounded perturbation of . We write P = − B, where B = 2g µν ωµ δν + C with C = g µν (ωµ ων + ων,µ ) + E and ων,µ = L(δµ λν ) − R(δµ ρν ). From the Duhamel principle 1 e−st (A+B) B e−(1−s)t A ds, e−t (A+B) = e−t A − t 0
we first formally write e−t P =
∞ (−t) j E j (t), j=0
(10)
Heat Kernel and Number Theory on NC-Torus
where E 0 (t) := e
−t
419
and E j (t) :=
e−s1 t B e−(s2 −s1 )t · · · B e−(1−s j )t d j s.
j
Here j denotes the ordinary j-simplex: j := {s ∈ R ; 0 ≤ s1 ≤ · · · ≤ s j ≤ 1} {s ∈ R j
j+1
; si ≥ 0,
j
si = 1}.
i=0
We prove convergence of the expansion (10) in the trace-norm and for reasonably small t: from the Hölder inequality for Schatten classes, we have
E j (t) 1 ≤
e−s0 t s −1 B e−s1 t s −1 · · · B e−s j t s −1 d j s, j
0
1
j
where B = 2ωµ δµ + C and ωµ , C are bounded. By functional calculus,
δµ e−si t s −1 ≤ δµ e−si t /2 e−si t /2 s −1 i
i
≤ c(g) (esi t)−1/2 (Tr e−t /2 )si , using the inequality || f (δ1 , · · · , δn )||op ≤ || f ||∞ , where f (x) = xµ e−x.x which follows from f (δ1 , · · · , δn )Uk = f (ik)Uk . So
E j (t) 1 ≤ (Tr e−t )s0 C (Tr e−t )s1 +2c(g) ωµ (es1 t)−1/2 (Tr e−t /2 )s1 · · · j
µ
· · · C (Tr e−t )s j + 2c(g)
ωµ (es j t)−1/2 (Tr e−t /2 )s j d j s ≤ Tr e−t /2
j
· · · C + 2c(g)
µ
C + 2c(g)
ωµ (es1 t)−1/2 · · ·
µ
ω (es j t)−1/2 d j s. µ
µ
Using
j
j i=1
−1/2
si
d j s ≤ 2 j−1 ,
the last expression can be estimated for t ≤ e−1 (since s j ≤ 1) by j−1 t −( j−1)/2 e−( j−1)/2 2 j−1 C + 2c(g)
ωµ Tr e−t /2 . µ
Thus ∞ ∞ √ j−1 √2 ( C + 2c(g) (−t) j E j (t) ≤ t Tr e−t /2
ω
) t , µ e j=0
1
j=0
µ
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which is finite for 0
e := t0 . 4( C + 2c(g) µ ωµ )2
Finally, for t0 ≤ t ≤ 2t0 , note that
e−t P 1 ≤ e−(t−t0 )P 1 e−t0 P , and the result follows inductively. One can probably prove this lemma also by using some estimates involving Sobolev spaces, cf. [41]. However, the Duhamel principle is quite important in its own right for physical applications. We shall use (10) below in Sect. 5. The convergence of the Duhamel expansion is necessary to construct the covariant perturbation series in the approach of Barvinsky and Vilkovisky [4]. Note also that the Duhamel expansion has been used to compute one-loop divergences in a more general framework of NCQFT, namely on the Moyal plane with degenerate but non-constant [23]. Let us define the functional Sp on L(Hτ ), given for a bounded operator A by (11) Sp(A) := lim (4π t)n/2 Tr A e−t . t→0+
Note that this definition is not sensitive to the kernel of (which is CU0 ), so we may change in + 1 or assume that is invertible. We will show that Sp L(.)R(.) , as a functional on C ∞ (Tn ) × C ∞ (Tn ), is indeed a finite and faithful trace in each argument, namely it vanishes whenever one of its arguments is a commutator, see Lemma 19. This should not be surprising once one knows that Sp(A) is a multiple of the Dixmier trace of A(1 + )−n/2 . Indeed, from the knowledge of the eigenvalues of and theboundedness of A, one has N µk A(1 + )−n/2 = O(ln N ), where µk (X ) A(1 + )−n/2 ∈ L(1,∞) (Hτ ), i.e., k=1 are the ordered singular values of X . It follows then (see [14, p. 236] with immediate modifications) that Tr ω A(1 + )−n/2 =
1 1 lim t n/2 Tr(A e−t ) = Sp(A). n/2 (n/2 + 1) t→0 (4π ) (n/2 + 1)
However, we leave this feature now since our goal is to find an algorithm to obtain analytic expressions for the heat coefficients. For that, Dixmier-trace technology is not so helpful. From relations (1) and using the orthonormal basis {Uk }k∈Zn to compute the trace, we find for l = (2π )−n/2 q1 lq1 Uq1 , r = (2π )−n/2 q2 rq2 Uq2 in C ∞ (Tn ), 2 τ Uk∗ L(l) R(r ) e−t Uk = e−t|k| τ Uk∗ l Uk r Tr L(l) R(r ) e−t = k∈Zn
= (2π )−n
k∈Zn
e
−t|k|2
lq1 rq2 τ U−k Uq1 Uk Uq2
k, q1 , q2 ∈Zn
= (2π )−n
q, k∈Zn
lq r−q e−ik.q e−t|k| , 2
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421
which after Poisson resummation reads √ 2 lq r−q e−|q−2π k| /4t Tr L(l) R(r ) e−t = g (4π t)−n/2 =
√
g
q, k∈Zn
lq r−q K (t, q − 2π k),
(12)
q, k∈Zn
where K (t, x) := (4π t)−n/2 e−|x| /4t is the heat kernel of Rn with metric g µν . To proceed further, we need to impose some restrictions on the matrix . Whereas it does not for the asymptotics of (9), the number-theoretical aspect of has huge consequences for the asymptotics of (8). To explain what this is about,
let us review what happens in the (nondegenerate) two0 −1 , with θ an arbitrary real number. (Actually, dimensional case, where = θ 1 0 up to an isomorphism, the range of θ can be reduced to the interval [0, 21 ].) In this case, there are two distinct situations to consider [21]. When θ is rational (relative to 2π ), then in the sum (12), only terms with a q multiple of the θ -denominator, will contribute to the small-t asymptotics. When θ is irrational (again relative to 2π ),one can guess that only the zero-mode will contribute to the small-t asymptotics of Sp L(l)R(r ) . This is indeed true, provided one has a control on the sum 2
k∈Z2
0=q∈Z2
e−|θq−2π k| /4t , (4π t)n/2 2
lq r−q
(13)
i.e., provided one can measure how far from rationals θ is, since |θq − 2π k| could be in principle arbitrarily close to 0. This control is precisely given by a Diophantine condition. Definition 2.2. A number θ is said to satisfy a Diophantine condition (relative to 2π ) if there exist two constants C > 0, β ≥ 0 such that for all q ∈ Z∗ ,
θq T := inf |θq − 2π k| ≥ k∈Z
C 2C 2 ⇐⇒ |1 − cos(θq)| ≥ . 2 |q|1+β |q|1+2β+β
(14)
In fact, inf p∈Z |θq − p| ≤ | sin(π θq)| : actually, for a given q, their exists an integer p0 such that |θq − p0 | ≤ 21 and since we work under modulus, we may assume that 0 ≤ θq − p0 ≤ 21 . Since (sin x)/x is decreasing on [0, π2 ], we get sin(π(θq − p0 )) ≥ 2(θq − p0 ) and the above equivalence by taking the square of the inequality with 1 sin2 ( θq 2 ) = 2 (1 − cos(θq)). In other words, this condition states that the inverse torus norm of θq is a temperate distribution over Z∗ . This is exactly what we need since in Eq. (13), the complex coefficients lq , rq are of Schwartz-class by the smoothness assumption. Note that this condition is not too restrictive since the set of irrational numbers satisfying a Diophantine condition is of full Lebesgue measure. In the general Poisson case, one can always change the coordinates on Tn , y = Bx with a constant matrix B, so that the new coordinates y are 2π -periodic again, and the ¯ = B T B becomes Poisson matrix
0 1 0 1 ¯ = 0l , (15) θi θj −1 0 −1 0, {i : θi ∈2π Q∗ }
{ j : θ j ∈R\2π Q}
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where 0l is the zero matrix of size l and n = l +m 1 +m 2 . Here m 1 is the size of the rational part (with θi = 2π pi /qi , i = 1, · · · , m 1 ) and m 2 of the irrational one. We define then Z = Zl × q1 Z × · · · × qm 1 Z × {(0, · · · , 0) ∈ Zm 2 }. The rest of Zn is denoted by K and is split into two parts, Kper {(k1 , · · · , km 1 ) ∈ Nm 1 , 1 ≤ ki ≤ qi − 1, i = 1, · · · , m 1 } and Kinf Zm 2 \ {0}. Kper is a finite set which lays in an R-linear space generated by Z. We shall omit the bar over in what follows. One can avoid the use of a particular coordinate system. Then Z is defined as the set of q ∈ Zn such that (2π )−1 q ∈ Zn . This definition reveals the actual meaning of this set. One can also figure out how to give coordinate independent definitions of Kper and Kinf . Now we are ready to formulate our restriction on . We assume the following: The matrix satisfies a Diophantine condition with respect to Kinf , i.e., there are two positive constants C and β such that inf |q − 2π k| ≥
k∈Zn
C for all q ∈ Kinf . |q|1+β
(16)
In the standard definition of a Diophantine condition for an l-tuple of real numbers, one assumes that 1 + β ≥ l. Our consideration is valid in a more general case for any positive 1 + β, so we do not need this restriction as far as such exist; note, for instance, that there exists a set of full Lebesgue measure of l-tuples of Roth type, that is of α ∈ Rl such that for all > 0, there exists C with inf k∈Zl |αq − 2π k| ≥ |q|Cn+ , see [31]. µν We remark that it does not matter whether one uses the metric g or the normalized diagonal metric in norm-values of (16) since it can be absorbed in the constant C. With this restriction on we can prove the following formula which governs the asymptotic behavior of the trace at t → 0+ : Tr L(l)R(r )e−t =
√
g lq r−q + e.s.t., (4π t)n/2
(17)
q∈Z
where e.s.t. denotes some exponentially small terms in t, i.e., the terms which vanish faster than any power of t as t → 0+ . The proof, which is somewhat technical, is postponed to Appendix A. Equation (17) immediately yields the following theorem. Theorem 2.3. Assume satisfies condition (16). Then for any l, r ∈ C ∞ (Tn ), √ lq r−q . Sp L(l)R(r ) = g
(18)
q∈Z
The explicit expression (18) makes it possible to show rather directly that (11) indeed defines a trace in each variable: it vanishes whenever l or r is a commutator: Corollary 2.4. Let l, r, s ∈ C ∞ (Tn ). Then, Sp L(l)R([r, s]) = 0.
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Proof. From commutation relations (2), we find rk sq sin( 21 k.q) Uk+q , [r, s] = −2i(2π )−n k, q∈Zn
and thus [r, s]k = 2i(2π )−n/2
q∈Zn
rk−q sq sin( 21 q.k),
which is zero whenever k ∈ Z since it is equivalent to (2π )−1 k ∈ Zn . Remark 2.5. We have the following relations between the functional Sp and the trace τ : Sp L(a) = Sp R(a) = (vol Tn ) τ (a), for all a ∈ C ∞ (Tn ) and any , and when Z = {0} (i.e. pure Diophantine case), then Sp L(l) R(r ) = (vol Tn ) τ (l) τ (r ), which makes transparent the statement of the corollary. This completes our study of the trace (11). Below we present several relations similar to (17) which will be used in the next section. One can show (see Appendix A) that Tr [L(l)R(r )]µ δµ e−t = 0 + e.s.t., (19) where the notation [L(l)R(r )]µ1 ...µm means that the vector indices are distributed between l and r . For higher derivatives we have ∗ Tr [L(l)R(r )]µ1 ...µm δµ1 . . . δµm e−t = τ Uk [L(l)R(r )]µ1 ...µm δµ1 . . . δµm e−t Uk k∈Zn
= im =:
kµ1 . . . kµmτ Uk∗ [L(l)R(r )]µ1 ...µme−t Uk
k∈Zn (m) i m Gµ Tr [L(l)R(r )]µ1 ...µm 1 ...µm
e−t . (20)
One can calculate the tensors G (m) by varying (17) or (19) with respect to the metric g µν (this is a standard way to include derivatives in the heat trace expansion, cf. [6].) All G (2 j+1) are exponentially small and can be neglected in our analysis. For even m, corresponding tensors G (m) are obtained from the following recursion relation: 1 δ p+2) G (2 p) , (21) G (2 µνµ1 ...µ2 p = − t δg µν µ1 ...µ2 p √ with G (0) = g. One has to take into account that g µν is symmetric, so that not all of the components are indeed independent, and δ 1 gρσ = − (gµρ gνσ + gµσ gνρ ). (22) δg µν 2 For example, √ g (2) gµν , G µν = 2t (23) √ g (4) G µνρσ = 2 (gµν gρσ + gµρ gνσ + gµσ gνρ ). 4t
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2.2. Heat trace asymptotics for generalized Laplacians. From expression (5) of generalized Laplacian P = −g µν ∇µ ∇ν − E, we need also to define the associated curvature: L R
µν := ∇µ ∇ν − ∇ν ∇µ = L(µν ) − R(µν ),
as repeated commutators with ∇. For and higher “covariant derivatives” of E and example, E ;µ := [∇µ , E], E ;µν := [∇ν , E ;µ ]. Actually, there are two covariant derivatives, ∇µL := δµ + L(λµ ) and ∇µR := δµ − R(ρµ ) and two gauge symmetries in the problem. One of these symmetries acts on “left” fields λµ , l1 , l2 , the other acts on “right” fields ρµ , r1 , r2 . The gauge group has a direct product structure. Explicit expressions for the symmetry transformations can be found
and derivatives is gauge invariant. in [43]. The functional Sp of any polynomial of E, Full invariance will mean that also all vector indices are contracted in pairs. This is precisely the class of invariants which will appear in the heat trace asymptotics. Due to the product structure of the gauge group there are gauge which do not belong invariants to the class we have just described. For example, Sp L(l1 ) is such an invariant. We will see that in the spectral action picture, there is only one gauge group, namely the automorphism group of the algebra but lifted to the spinor bundle via the charge conjugation operator. In such an application, the representation really looks like the adjoint one (see Sect. 4). We shall need the notion of canonical mass dimension. We assign canonical mass
, and canonical mass dimension 1 to each derivative. Canonical dimension 2 to E and mass dimension of any monomial is the sum of canonical mass dimensions of all factors. The heat trace asymptotic is then given by the following Theorem 2.6. Let P be as defined above. Then i) There is a full asymptotic expansion of the heat trace ∞ Tr L(l)R(r )e−t P ∼+ ak (l, r ; P) t (k−n)/2 . t→0
(24)
k=0
ii) The coefficients ak can be expressed as ak (l, r ; P) = bα Sp L(l)R(r )Aα ,
(25)
α
where Aα are independent invariant free polynomials of canonical mass dimension k of
and their covariant derivatives. The numbers bα are constants. The odd-numbered E, coefficients a2 j+1 vanish. iii) Values of bα are uniquely defined by considering “pure left” (ρµ = r1 = r2 = 0, r = 1) or “pure right” (λµ = l1 = l2 = 0, l = 1) cases. In particular, (26) a0 (l, r, P) = (4π )−n/2 Sp L(l)R(r ) , −n/2 a2 (l, r, P) = (4π ) Sp L(l)R(r )E , (27)
µν ) .
µν a4 (l, r, P) = (4π )−n/2 1 Sp L(l)R(r )(6E 2 + 2E ;µ µ + (28) 12
Proof. Existence of the trace follows from Lemma 2.1. The proof of the second statement (below) can be considered as a constructive proof for the existence of the asymptotic expansion.
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To evaluate the asymptotic behavior of the trace on the left-hand side of (24) we use the canonical basis {Uk } of Hτ . This is a standard procedure used in quantum field theory for a long time (cf. [33]), which was recently applied to noncommutative theories 2 [41, 43]: one first factors out a global e−t|k| term, Tr L(l)R(r )e−t P = τ Uk∗ L(l)R(r )e−t P Uk k
=
2 µ µ µ e−t|k| τ Uk∗ L(l)R(r )et ((∇ −ik )(∇µ −ikµ )+2ik (∇µ −ikµ )+E) Uk .
k
(29) Then, one expands the exponential in (29) as a power series in E and (∇ − ik). As a result, one gets a sum of monomials of the form 2 (30) e−t|k| kµ1 · · · kµm τ Uk∗ L(l) R(r ) F(E, (∇ − ik))µ1 ···µm Uk . k
We stress that it is important that each ∇ appear in the combination (∇ − ik). We take in F all (∇ − ik) one by one starting with the rightmost (∇ − ik) and push them to the right. Being commuted through an operator of the type L( f )R(h), (∇ − ik) replaces the functions f, h in this operator by their derivatives, e.g., (∇ − ik)L( f ) = L( f )(∇ − ik) + L(∇ L f ), where only the Leibniz rule satisfied by the derivations δµ has been used. When (∇ − ik) hits Uk , it becomes an operator of (left and right) multiplication by the connection, i.e., (∇µ − ikµ ) Uk = L(λµ ) − R(ρµ ) Uk . In this way, one can remove all derivative operators (which are replaced by left and right multiplication operators) and all momenta k from the expression inside the NC-torus trace in (30). Therefore, one can apply (20) to obtain instead of (30) the following expression 2 (m) Gµ (31) e−t|k| τ Uk∗ L(l) R(r ) F(E, (∇ − ik))µ1 ···µm Uk . 1 ...µm k
Let us now evaluate the power of t corresponding to each monomial. If F contains an N Eth power of E, an N Pth power of (∇ − ik)2 and an N Kth power of 2ik µ (∇µ − ikµ ), then F itself contains t N E +N P +N K . Explicit multipliers kµ which we put in front of F in (30) come from 2ik µ (∇µ − ikµ ) only. Consequently, m = N K . For odd m, the tensors G (m) vanish up to exponentially small terms, while for an even m the tensor G (m) is proportional to t −m/2 . The sum over k brings another t −n/2 . Altogether, we have t N E +N P +N K /2−n/2 . This means that such monomials contribute to the coefficient a p (l, r, P) with p = 2N E + 2N P + N K , which are precisely the canonical mass dimensions of the monomials as defined above. Besides, N K should be even. Consequently, odd numbered heat kernel coefficients a2 j+1 vanish. Now we have to prove that the heat kernel coefficients are of the form declared in
µν and their the theorem, i.e., that they are invariant polynomials constructed from E, derivatives. We have already proved that the expression inside the trace τ in (31) is in fact a multiplication operator (i.e., a combination of left and right regular representation operators) which does not contain k. Together with gauge invariance of the heat trace this could have been enough to get the statement. However, since the gauge group has a product structure, there are more gauge invariants than we expect to find in the heat trace asymptotics. Let us collect all monomials Fa (E, (∇ − ik))µ1 ...µm of a given (even)
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canonical mass dimension p which appear in the expansion of the exponential of (29), and consider the sum
∗ (m) µ1 ...µm τ Uk L(l) R(r ) (32) G µ1 ...µm Fa (E, (∇ − ik)) Uk . a
As we have demonstrated above, Fa are free polynomials of E, ω and their derivatives δµ E, δµ ων , etc. The same procedure as above can be carried out for an arbitrary Laplace type operator P¯ acting on smooth sections of an arbitrary (non-abelian) vector bun¯ the dle over the commutative torus Tn . The free polynomials of the endomorphism E, ¯ and their derivatives are in one-to-one corresponconnection ω¯ µ , which characterize P, dence with the polynomials in (32). In the case of P¯ we know that all terms can be recombined into covariant derivatives and field strengths thus giving standard heat kernel coefficients. This is a purely combinatorial statement, which does not depend on the ¯ Therefore, the same recombination can be done also ¯ and E (or E). nature of ∇ (or ∇) in the noncommutative case considered here. This completes the proof of the second assertion. The third point is easy, it simply means that independent invariants remain independent when reduced to “pure left” or “pure right” cases. The coefficients in front of these invariants can therefore be read off from “pure left” heat kernel coefficients [41] on the torus. This includes (26), (27), (28) and even a6 which is not given explicitly in the present work. The interested reader can calculate also higher terms in the heat trace asymptotics by using the expressions for a8 [1] and a10 [38] obtained in the commutative case. 3. Toward the Asymptotics for Toric Noncommutative Manifolds This section is devoted to the study of the asymptotic (17) in a more general setting of noncommutative spaces. We will concentrate here on toric noncommutative manifolds, C ∞ (M ), also called periodic isospectral deformations (the aperiodic case [25], akin to the Moyal plane, will be studied elsewhere). This class of quantum spaces can be thought of as a curved space generalization of NC-tori. They were originally defined by Connes and Landi [15] (from cohomological considerations) within a twisted product approach (that we will follow here) and later by Connes and Dubois-Violette [16] in a more intrinsic way via fixed-point algebra techniques. We first recall the definition of Cc∞ (M ): let (M, g) be a Riemannian (compact or not) n-dimensional manifold without boundary. Consider α : Tl → Isom(M, g), a smooth isometric action of a l-torus on M, typically given by the maximal abelian subgroup of the isometry group of the manifold (the interesting class is l ≥ 2). This action induces a spectral (Peter–Weyl) decomposition of any smooth function with compact support f ∈ Cc∞ (M), f = fr , such that αz ( fr ) = e−ir.z fr , ∀z ∈ Tl , r ∈Zl
where the action by automorphism of the l-torus on Cc∞ (M) (also denoted α) is given by (αz f )( p) := f (α−z ( p)). It is important to notice that this expansion is convergent in the sup-norm . ∞ (in fact fr ∞ is a Schwartz sequence for f ∈ Cc∞ (M).)
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By analogy with the noncommutative torus, given a skewsymmetric l × l matrix , one can deform the algebra Cc∞ (M) to a noncommutative one Cc∞ (M ), defining the following twisted product on pairs of homogeneous elements i
fr gs = e− 2 r.s fr .gs .
(33)
It should be clear that this product can also be realized via the Rieffel star-product (4) associated to the action α [36]. In this setting, the natural trace τ of this algebra is the integral with Riemannian volume form µg τ (.) = (.) µg , (34) M
and all first-order differential operators which commute with the action α form a Lie algebra of derivations. On the Hilbert space H of square integrable functions on M with Riemannian volume form µg , one defines left and right twisted multiplication operators L(l), R(r ), l, r ∈ Cc∞ (M), by L(l)ψ = l ψ,
R(r )ψ = ψ r, for all ψ ∈ H.
Because the Peter–Weyl expansion is sup-norm convergent, those operators are bounded for (at least) smooth compactly supported functions. For any isometric Rl -action on a Riemannian manifold, from the expression of the kernel of the operators L(l), R(r ) (see for instance [21]) and using the heat kernel K t ( p, p ) of the scalar Laplacian associated with the metric g and its volume form µg , one can compute Tr L(l) R(r ) e−t = (2π )−l µg ( p) d l y d l z l( p) r (αz ( p)) K t α−y ( p), p . M
After a Peter–Weyl expansion of the functions l, r , this reads µg ( p) lq ( p) r−q ( p) K t αq ( p), p . Tr L(l) R(r ) e−t = q∈Zl
(35)
M
For a large class of manifolds (see [17, 25] for a review of sufficient conditions), the behavior of the off-diagonal heat kernel is controlled by the geodesic distance function: 2 2 1 1 e−dg ( p, p )/4t ≤ K t ( p, p ) ≤ C e−dg ( p, p )/4(1+c)t , (4π t)n/2 (4π t)n/2
(36)
where dg is the geodesic distance and C, c are positive constants. This estimate and the fact that the metric on the orbits of the torus action is constant (since Tl -action is isometric) show that the previous discussion on the role of the arithmetic nature of the deformation parameters applies also in this setting (since the geodesic distance on the orbits is the torus one). But in this framework this is not the end of the story. Indeed, such an action is not necessarily free (for example it is for NC-torus but not for Connes–Landi spheres), that is, there may exist fixed or rather singular points for the action. For instance, on a neighborhood of a fixed point, we see that in the integral
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(35), we are left with the heat kernel on the diagonal (i.e., the damping factor of the exponential of geodesic distance disappears). This has certainly some consequences for the power-t expansion. In view of (36), note that the lack of freedom for the action can be rephrased in terms of non-local integrability of the function p → dg−2 (α y ( p), p) for certain 0 = y ∈ Tl , in the neighborhood of singular points. At this level of generality, we are only able to treat the free torus-action case, where one can easily derive the asymptotic of (35). From previous techniques, the estimate (36) and under the Diophantine assumption (16), one gets 1 Tr L(l)R(r )e−t = µg ( p) lq ( p) r−q ( p) kt ( p, p) + e.s.t. (4π t)n/2 M =
1 (4π t)n/2
q∈Z ∞ k=0
tk
q∈Z
M
µg ( p) lq ( p) r−q ( p) a2k ( p) + e.s.t.,
(37) ( p) are the local heat kernel coefficients for the scalar Riemannian Laplacian. where a2k For a non-free torus action, it seems to be difficult to outstrip the qualitative level in general, i.e., to express the asymptotic of (35) in terms of geometric invariants. We will instead treat the (quite simple but non-trivial) example of the ambient space of Connes–Landi 3-sphere S3θ [15]. One standard way to construct this ambient space goes as follow. One parameterizes R4 (with standard metric) in spherical (φ1 , φ2 , ψ), φi ∈ T, ψ ∈ [0, π/2] (with nontrivial boundary conditions) and radial R ∈ [0, +∞[ coordinates. That is to say, in terms of Cartesian coordinates:
x1 = R cos ψ cos φ1 , x2 = R cos ψ sin φ1 , x3 = R sin ψ cos φ2 , x4 = R sin ψ sin φ2 . Then, one twists the product via the T2 -action y.(R, ψ, φ1 , φ2 ) = (R, ψ, φ1 + y1 mod 2π, φ2 + y2 mod 2π ),
y ∈ R2 .
In other words, we are mapping the commutative generators u i = e2iπ φi , i = 1, 2, to those of the noncommutative 2-torus (of course S3θ is obtained by imposing the sphere relation on the generators). For the question of the asymptotic of (35), it is more convenient to move to another coordinate system. It allows to identify the ambient space of S3θ with the ambient space of T2θ . This is achieved by setting r1 = R cos ψ, r2 = R sin ψ, which leads to a parameterization of R4 in double polar coordinates (r1 , φ1 ; r2 , φ2 ). Thus, it corresponds to twist the product of the commutative algebra S(R4 ) via the action of T2 given by the two S O(2)-rotations (which generate the maximal compact Abelian subgroup of the isometry group of R4 ). In such a case, the only interesting 0 1 , with θ irrational. situation is when = θ −1 0
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We make the Fourier transform in the two angular directions and leave two radial coordinates r1 , r2 as they are. From the expression of the heat kernel of R4 parameterized by (r1 cos φ1 , r1 sin φ1 , r2 cos φ2 , r2 sin φ2 ) and action (φ1 , φ2 ).(r, φ1 , φ2 ) := (r, φ1 + φ1 mod 2π, φ2 + φ2 mod 2π ), we have 1 − r12 (1−cos θq2 )+r22 (1−cos θq1 ) /2t K t α−q ( p), p = e , (4π t)2 and we obtain, up to an irrelevant numerical constant c, for (35) c 2 2 Tr L(l)R(r )e−t = 2 d 2 r r1 r2 lq (r) r−q (r)e− r1 (1−cos θq2 )+r2 (1−cos θq1 ) /2t , t R + ×R + 2 q∈Z
(38) where lq (r) (resp. rq (r)) is the r-dependent coefficients in the Fourier expansion of l (resp. r ) in terms of ei(φ1 q1 +φ2 q2 ) as opposed to the terms in the spectral decomposition of l (resp r ). The term with q = 0 in (38) gives our “standard” result. Other terms give (in the asymptotics) contributions from the “singularities” in r1 = 0, r2 = 0. There is a relation between oscillations of a smooth function in the angular directions and its behavior near the origin of the coordinate system r = 0. Consider a smooth complex function ψ on R2 . Let us restrict it to the unit disc in R2 . We are going to expand ψ in a series of eigenfunctions of the Laplace operator on the disc. Consider a polar coordinate system (r, φ) centered at the origin. Let ψ 0 be a restriction of ψ to 0 the boundary of the disc, and ψ (φ) = l ψl0 eilφ . Then ψr0 (φ) := l ψl0 eilφ r|l| is ˜ a (smooth) zero mode of the Laplacian, and ψ(r, φ) := ψ(r, φ) − ψr0 (φ) is a smooth function satisfying Dirichlet boundary conditions on the boundary of the disc. ψ˜ can be expanded in a sum of non-zero eigenfunctions of the Laplacian, which are eilφ J|l| (rλ), where J|l| is the Bessel function, and the eigenvalues λ are defined by the boundary condition J|l| (λ) = 0. The Taylor expansion for J|l| around r = 0 starts with r|l| and contains the powers r|l|+2k , k ∈ N0 . Since ψ is smooth, its harmonic expansion is rapidly convergent, and we can conclude that each Fourier mode ψl behaves near r = 0 as (0)
ψl (r) = r|l| (ψl
(1)
+ r2 ψl
+ . . . ).
(39)
Let us see in detail what happens near r1 = 0. For that we have to look at the sum over q1 = 0, q2 = 0. The corresponding terms in (38) read c 2 f q2 (r) e−r1 (1−cos θq2 )/2t , (40) d 2 r r1 r2 2 t q2 =0
where f q2 (r) := l0,q2 (r) r0,−q2 (r). Fix 1 > 0. It is easy to see that the integral gives an exponentially small term. For r1 ≤ 1 we use the Taylor expansion of (0)
∞ 1
dr1
(1)
f 0,q2 (r) = f 0,q2 (r2 ) + r12 f 0,q2 (r2 ) + . . . . Then (40) takes the form (up to higher order terms): (0) 2t 4t 2 c 1 (1) f 0,q2 (r2 ) + f 0,q2 (r2 ) + ··· . dr2 r2 t2 2 1 − cos θq2 (1 − cos θq2 )2 q2 =0
(41)
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The sum and the integral are convergent if θ satisfies a Diophantine condition. Recall the control of (1 − cos θqi )−1 by (14). It is interesting to note that already the 1/t term receives a contribution from the singularity. ∞ ∞ Next, let q1 = 0, q2 = 0. The contributions from the integrals 1 dr1 and 2 dr2 1 2 are exponentially small. Therefore, we restrict ourselves to the integral 0 0 dr1 dr2 , where we use again the Taylor expansion in the radii. The Taylor expansion of any smooth q q 2q 2q function starts with r11 r22 and the Taylor expansion for lq · r−q starts with r1 1 r2 2 . The corresponding terms contribute to the heat kernel coefficients with
q1 +1
q2 +1 2t 2t 1 (42) t 2 1 − cos θq2 1 − cos θq1 so that the modifications start with t 2 . One can easily evaluate corresponding terms (which describe the effect of the singularity at r1 = r2 = 0). The asymptotic we obtain strongly depends on the functions l and r through their Taylor coefficients in a neighborhood of singular points. This is typical for the heat trace asymptotics if boundaries or singularities are present, cf. [26, 40]. 4. Spectral Action for NC-Tori Within noncommutative geometry, the spectral action introduced by Chamseddine– Connes plays an important role [8]. More precisely, given a spectral triple (A, H, D), where A is an algebra acting on the Hilbert space H and D is a Dirac-like operator (see [12, 28]), they proposed a physical action depending only on the spectrum of the covariant Dirac operator D A := D + A + J A J −1 , where A is a one-form represented on H, i.e. it is of the form A= ai [D, bi ],
(43)
(44)
i
where ai , bi ∈ A, J is a real structure on the triple corresponding to charge conjugation and ∈ { 1, −1 } depending on the dimension of this triple and comes from the commutation relation J D = D J.
(45)
S(D A , ) := Tr (D2A /2 ) ,
(46)
This action is
where is any positive function viewed as a cut-off which could be replaced by a step function up to some mathematical difficulties surmounted in [20]. This means that counts the spectral values of |D A | less than the mass scale (note that the resolvent of D A is compact since, by assumption, the same is true for D). In [24], the spectral action on NC-tori has been computed only for operators of the form D + A. Thanks to our previous result, we can fill the gap and compute (46) in full generality.
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431
We need to fix notations: Let A := C ∞ (Tn ) acting on H := Hτ ⊗ C2 with n = 2m or n = 2m + 1 (i.e., m = n2 is the integer part of n2 ), the square integrable sections of the trivial spin bundle over Tn . Each element of A is represented on H as L(a) ⊗ 12m . The Tomita conjugation m
J0 (a) := a ∗ satisfies [J0 , ∂µ ] = 0 since J0 ∂µ Uk = J0 (ikµ )Uk = −ikµ U−k = ∂µ U−k = ∂µ J0 Uk . Besides, it induces the analogous operator on H, J := J0 ⊗ C0 , m
where C0 is an operator on C2 . The Dirac operator is defined by D := −i eaµ δµ ⊗ γ α = −i δµ ⊗ γ µ , where we use hermitian Dirac matrices γ . This implies C0 γ µ = −γ µ C0 since J D = (J0 ⊗ C0 )(−i∂µ ⊗ γ µ ) = J0 (−i)∂µ ⊗ C0 γ µ = i∂µ J0 ⊗ γ µ C0 , which by (45) is equal to (−i∂µ )J0 ⊗ γ µ C0 . Moreover, C02 = ±12m depending on the parity of m. Finally, one introduces the chirality (which in the even case is χ := id ⊗ (−i)m γ 1 · · · γ n ) and this yields that (A , H, D, J, χ ) satisfies all axioms of a spectral triple, see [12, 28]. The unitary elements u of A (or of its generated C∗ -algebra) play an important role since they reflect the inner automorphisms of A . For instance Uu := (u ⊗ 12m )J (u ⊗ 12m )J −1 is a unitary on H (with Uu∗ = (u ∗ ⊗ 12m )J (u ∗ ⊗ 12m )J −1 ) such that Uu DUu∗ = D + u ⊗ 12m [D, u ∗ ⊗ 12m ] + J u ⊗ 12m [D, u ∗ ⊗ 12m ]J −1 , explaining the construction of one-forms A in (44) thus satisfying Uu AUu∗ = (u ⊗ 12m )Au ∗ ⊗ 12m . These properties follow from the axioms: for all a, b ∈ A , [a ⊗ 12m , J b ⊗ 12m J −1 ] = 0 and [D, a ⊗ 12m ], J b ⊗ 12m J −1 = 0. In conclusion, the fact that the perturbed Dirac operator must satisfy condition (45) (which is equivalent to H being endowed with a structure of A -bimodule: for a, b ∈ A and ψ ∈ Hτ , a.ψ.b := (a ⊗ 12m )J (b∗ ⊗ 12m )J −1 ψ), yields the necessity of a symmetrized covariant Dirac operator: D A = D + A + J A J −1 . Note that for a ∈ A , using J0 L(a)J0−1 = R(a ∗ ), J L(a) ⊗ γ µ J −1 = R(a ∗ ) ⊗ C0 γ µ C0−1 = −R(a ∗ ) ⊗ γ µ , (47) and that the representation L and the antirepresentation R are C-linear, commute and satisfy [δµ , L(a)] = L(δµ a), [δµ , R(a)] = R(δµ a). Choosing an arbitrary selfadjoint one-form A, it can be written as A = L(−i Aµ ) ⊗ γ µ , Aµ = −A∗µ ∈ A ,
(48)
D A = −i δµ + L(Aµ ) − R(Aµ ) ⊗ γ µ .
(49)
and using (47)
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Defining A˜ µ := L(Aµ ) − R(Aµ ), we get D2A = −g µν (δµ + A˜ µ )(δν + A˜ ν ) ⊗ 12m − 21 µν ⊗ γ µν , where γ µν := 21 (γ µ γ ν − γ ν γ µ ) and µν := [δµ + A˜ µ , δν + A˜ ν ] = L(Fµν ) − R(Fµν ), where Fµν := δµ (Aν ) − δν (Aµ ) + [Aµ , Aν ]. Gathering all results, D2A = −g µν δµ + L(Aµ ) − R(Aµ ) δν + L(Aν ) − R(Aν ) ⊗ 12m −
1 L(Fµν ) − R(Fµν ) ⊗ γ µν . 2
(50)
Now, comparing (50) and (5), we can apply the previous result with the following replacement in (6) and (7)
or in Theorem 2.6,
⎧ L(λµ ) → L( Aµ ) ⊗ 12m , ⎪ ⎪ ⎪ ⎪ ⎪ R(ρµ ) → R( Aµ ) ⊗ 12m , ⎪ ⎪ ⎨ L(l ) → − 1 L(F ) ⊗ γ µν , 1 µν 2 1 µν ⎪ R(r ) → − R(F 1 µν ) ⊗ γ , ⎪ 2 ⎪ ⎪ ⎪ L(l2 ) → 0, ⎪ ⎪ ⎩ R(r2 ) → 0,
(51)
⎧ ∇µ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨E
µν ⎪ ⎪ ⎪ L(l) ⎪ ⎪ ⎩ R(r )
(52)
→ → → → →
δµ + L(Aµ ) − R(Aµ ) ⊗ 12m , − 1 L(Fµν ) − R(Fµν ) ⊗ γ µν , 2 L(Fµν ) − R(Fµν ) ⊗ 12m , 1, 1.
It is interesting to note that, in the commutative case when = 0, L = R thus D A = D for any selfadjoint one-form A: the Dirac operator does not fluctuate. We will derive the spectral action by Laplace transform techniques such as in [34], see [44] for details on Laplace transform (alternatively one can follow [20]). We assume that the function has the following property: ∈ C ∞ (R+ ) is the Laplace transform of ψˆ ∈ S(R+ ) := { g ∈ S(R) : g(x) = 0, x ≤ 0}. (53)
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Thus, any function with this property has necessarily an analytic extension on the right complex plane and is a Laplace transform. Consequently, any m-differentiable function ψ such that ψ (m) = is the Laplace transform of a function ψˆ and by differentiation, it satisfies ∞ ˆ (z) = ψ (m) (z) = (−1)m dt, z > 0. e−t z t m ψ(t) 0
One can invoke dominated convergence (see [24]), to obtain: ∞ 2 2 ˆ dt Tr e−t D A / t m ψ(t) Tr (D2A /2 ) = (−1)m 0
m ∞
= (−1)m 0
=
m
ˆ n−2k a˜ 2k t m+k−n/2 ψ(t) dt + O(n−2(m+1) )
k=0
n−2k 2k a˜ 2k + O(n−2(m+1) ),
k=0
where 2k is defined by
2k := (−1)
∞
m
ˆ dt, t m+k−n/2 ψ(t)
(54)
0
and a˜ 2k := a2k (1, 1; D2A ),
(55)
obtained from (25) with replacement (51). When n = 2m is even, 2k has the more familiar form: ∞ 1 m−1−k dt, for k = 0, · · · , m − 1, 0 (t) t (m−k) 2k = for k = m, · · · , n. (−1)k (k−m) (0),
(56)
For n odd, the coefficients 2k have less explicit forms because they involve fractional derivatives of , so in this case, it is better to stick with definition (54). Let us summarize: Theorem 4.1. Let D A = D + A + J A J −1 and A = L(−i Aµ ) ⊗ γ µ a hermitian oneform where A∗µ = −Aµ ∈ A = C ∞ (Tn ). Let ∈ C ∞ (R+ ) be a positive function satisfying condition (53). Then the following expansion of the spectral action holds: S(D A , ) =
n/2
n−2k 2k a˜ 2k + O(n−2(n/2+1) ),
k=0
where the 2k are defined in (54) or (56) depending on the dimension and the a˜ 2k are defined in (55) with replacement (51). More precisely, a˜ 0 = 2n/2 π n/2 , a˜ 2 = 0, a˜ 4 = (4π )−n/2 21 Sp(E 2 ) +
1
µν 12 Sp(
Moreover, all terms in a˜ 2k linear in l1 , r1 are zero.
µν ) .
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V. Gayral, B. Iochum, D. V. Vassilevich
Proof. The last assertions follow from a˜ 0 = (4π )−n/2 Sp(1⊗12m) , a˜ 2 = (4π )−n/2 Sp(E), 1 1 1 −n/2 2 µν E
µν ) and Tr(γ νµ ) = 0, so all
µν a˜ 4 = (4π ) ;µν ) + 12 Sp( 2 Sp(E ) + 6 Sp(g linear terms in E are of zero trace. Now comes an amazing fact: in four dimensions, the non-standard terms (those with products of traces) simply disappear. Indeed, when n = 4, with g = diag(1, 1, 1, 1) and a direct sum of two “Diophantine” matrices, we find, up to negative powers of , S(D A , )
µν µν µν = (4π )−n/2 4 0 4 Sp(1) − (0) 6 Sp L(F Fµν ) + R(F Fµν ) − 2L(F )R(Fµν ) µν µν = 4π n/2 0 4 − (0) F ) − τ (F ) τ (F ) . τ (F µν µν 3
But F µν is a sum of derivatives plus a commutator, so is of zero trace. Thus, the spectral action has the standard form: µν −2 S(D A , ) = 4π n/2 0 4 − (0) (57) 3 τ (F Fµν ) + O( ). The only difference appears in the numerical value of the coefficients. What happens for generic compact toric NC manifolds (we add the assumption that M carries a spin structure as well)? Even lacking a trace asymptotic expansion of the semi-group generated by a generalized Laplacian (i.e., an analogue of Theorem 2.6), we are able to finish in the 4-d pure Diophantine case, using the symmetries we have at disposal. First, it should be clear from examples treated in previous section that the supplementary terms coming from the ‘singular points’ actually do not appear. Indeed, they should appear only in the sub-leading order of a given term, but here the only one we have is the ‘Yang–Mills’ one, which is already the last with non-negative power of (i.e., such correction terms appears in 4-d with negative powers of ). In summary, the torus action being free or not has no serious implication for the structure of the spectral action in dimension four or less. One should emphasize that this is no longer true in higher dimensions. What previous examples also do show is that, up to sub-leading order terms, −t D 2 t ( p, p), Tr L(l) R(r ) e = µg ( p) l0 ( p) r0 ( p) K M
t is the on-diagonal kernel of e−t D . However, a0 = 0 whenever a is either a where K commutator or a derivative (with respect to the action α) in C ∞ (M ). Thus the same conclusion holds, namely, that the asymptotics of Tr[ (D+ A+ J A J −1 )2 /2 ] can be easily derived from those of Tr[ (D + A)2 /2 ], in 4-d with Diophantine deformation matrix. Note that the latter is easily computable from the classical asymptotics of the t ( p, p). kernel K To compute the spectral action, it can also be convenient to use the full force of [9]. 2
5. NC-QFT: Structure of Divergences for a Scalar Field Theory Let us consider a real scalar field φ in a four-dimensional NC torus with the classical action √ λ S[φ] = d 4 x g 21 (∂µ φ)2 + 21 m 2 φ 2 + 24 φ φ φ φ , (58)
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where λ is a coupling constant. Here we change to the notations which are more common in quantum field theory and write the star products (4) or (33) and partial derivatives. To calculate the effective action in this theory, we split φ = ϕ − δφ into a background part ϕ and quantum fluctuations δφ. Then one expands S[φ] about the background. The first term, S[ϕ], simply gives the classical approximation to the effective action, The second term, which is proportional to the first derivative of S[ϕ] is canceled by external sources. The quadratic term can be rewritten as √ (2) 1 S [ϕ, δφ] = 2 d 4 x g (δφ)P(δφ) , where (cf. [22]) P = −g µν ∂µ ∂ν +
λ 6
[R(ϕ ϕ) + L(ϕ ϕ) + L(ϕ) R(ϕ)] + m 2 .
(59)
Note that P > 0 for λ > 0, since −g µν ∂µ ∂ν ≥ 0 and ϕ ∗ = ϕ so that R(ϕ ϕ) + L(ϕ ϕ) + L(ϕ) R(ϕ) ∗ = 21 L(ϕ) + R(ϕ) L(ϕ) + R(ϕ) + 21 L(ϕ)∗ L(ϕ) + 21 R(ϕ)∗ R(ϕ). This operator corresponds to the following choice in (5–7): λµ = ρµ = 0 , l1 = −r1 = − λ6 ϕ ϕ −
m2 2
, l2 = −r2 =
λ 6
ϕ.
Formally, the one-loop effective action reads W =
1 2
ln det P .
This expression has to be regularized. We shall use the zeta-function regularization [19, 30]. The zeta function can then be defined as an L 2 -trace, ζ (s, P) = Tr(P −s ) . The pole structure of the zeta function is determined by the asymptotic properties of heat trace at t → 0 (see, e.g. [26]). Due to Theorem 2.6, the zeta function of P has only simple poles and is regular at s = 0. There is a useful relation ak (P) = Ress= n−k (s) ζ (s, P) . 2
In particular, an = ζ (0, P). The regularized effective action is defined as W (s) = − 21 µ2s (s) ζ (s, P) . The regularization is removed in the limit s → 0. µ is a dimensional constant introduced to keep the regularized effective action dimensionless. Because of a pole of the -function, W (s) is divergent at s → 0. The divergent part of the effective action reads 1 1 ζ (0, P) = − 2s a4 (P). Wdiv (s) = − 2s
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Let us assume that satisfies a Diophantine condition. Then, by Eq. (28), 1 λm 2 √ 4 4 √ 2 −1 vm − 2 d x g ϕ + v ( d 4 x g ϕ)2 a4 (P) = 2 32π 3 λ2 √ √ √ √ 2 d 4 x g ϕ 4 + 3v −1 ( d 4 x g ϕ 2 )2 + 4v −1 d 4 x g ϕ 3 d 4 x g ϕ . + 36 (60) √ Here v = vol T4 = (2π )4 g, ϕ k denotes the k th star-power of ϕ. For example, ϕ 3 := ϕ ϕ ϕ. The theory is called form-renormalizable if all divergences in the effective action can be compensated by redefinitions of couplings in the classical action, i.e., if the divergent part of the effective action repeats the structure of the classical action. The term m 4 in (60) causes no problem as it does not depend on the field (it is said that such terms are removed by a renormalization of the cosmological constant). The terms with m 2 ϕ 2 and ϕ 4 can be removed by suitable renormalization of m 2 and λ in (58). The remaining non-local terms cannot be renormalized away. Therefore, the model (58) is not form-renormalizable. It is instructive to consider an infinite-volume limit of (60). Let us introduce normala x µ , where ea is a constant vierbein, ea ea = g , and let us ized coordinates y a = eµ µν µ µ ν a assume that ϕ(y ) is kept constant as v → ∞. Then all nonlocal terms vanish in the limit v → ∞. This is consistent with the conclusion of [43] that the counterterms for φ 4 on R4 are local in the zeta function regularization if is nondegenerate. Note that, for a degenerate , this is no longer true [22]. This is because the IR divergence developed in the non-planar sector of the 2-point function (proportional to |ξ |−2 in momentum space), turns out to be non-locally integrable and thus the associated Green function does not define a tempered distribution. Of course, one can add several terms to the classical action which repeat the structure of the one-loop divergences. However, such terms change, in general, also the divergent part of the effective action and bring up new structures. Can this process be closed after several steps? Below we show that, at least in one-loop approximation, the answer is affirmative. First of all, we impose antiperiodic conditions in one of the coordinates, say x 1 on the field φ (and also on the background field ϕ, and on quantum fluctuations δφ): φ(x 1 , x 2 , x 3 , x 4 ) = −φ(x 1 + 2π, x 2 , x 3 , x 4 ).
(61)
In the language of NC-QFT, this corresponds to considering a field theory on a finite projective module over the noncommutative torus. This Hermitian module is well defined. One first lifts the torus action on the commutative vector bundle and then defines the module structure on the smooth sections via a star product made out of the lifted torus action (see [16]). anti-periodic condition cancels all terms with linear integral of the field, This d 4 xϕ = 0. By condition (61) also the Laplacian spectrum is changed, ∼ pµ p µ , p ∈ Zn(1/2) := Zn + (1/2, 0, 0, 0). However, as t → 0, k∈Z
e−(k+1/2) t = 2
π t
+ e.s.t.
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and all asymptotic relations derived above remain true after obvious modifications. Since quantum fluctuations are anti-periodic, one has to take in (12) k ∈ Zn(1/2) . The fields l and r are powers of the background field ϕ. Therefore, r and l may be both periodic and anti-periodic. Thus, q ∈ Zn ∪ Zn(1/2) in that equation (but only half of the Fourier coefficients may be non-zero in each particular case). The Diophantine condition also should be understood with respect to k ∈ Zn(1/2) and q ∈ Zn ∪ Zn(1/2) . To deal with the remaining non-local terms in (60) we add to the classical action a non-local part 2 λ˜ 4 √ 2 δS[φ] = √ , (62) d x gφ 2 g where λ˜ is a new coupling constant. Renormalization of λ˜ allows to remove all existing one-loop divergences. It is important to make sure that no new types of divergences appear. Because of (62), the quadratic form of the action receives the contribution 2 λ˜ 2λ˜ √ √ (2) 4 √ δS [ϕ, δφ] = √ d 4 x g (δφ)2 . d x g ϕ · (δφ) + √ d 4 x g ϕ2 g g (63) The second term in (63) is harmless. Due to that, the term m 2 in (59) is replaced by a background-dependent but still coordinate-independent mass term: 2 2 2 ˜ m → m¯ = m + 2λ d 4 x ϕ 2 . (64) Let us assume λ˜ > 0 so that after this replacement the spectrumof P remains positive. Next we substitute m¯ 2 for m 2 in (60) and take into account that d 4 x ϕ = 0 to see that the divergent part of the effective action does not receive any new structure beyond those which are already present in S + δS. The first term on the right hand side of (63) does not contribute to the divergences at all. Let us denote by P¯ the operator (59) with the replacement (64). Then, the operator acting on quantum fluctuations reads P = P¯ + B,
˜ −1/2 |ϕϕ|, B = 4λg
where |ϕϕ| is a rank one operator (with suggestive notations). B being proportional to a projector to a one-dimensional subspace of the Hilbert space, it is clear that the insertions of B improve ultraviolet behavior of quantum amplitudes. To make this argument more precise we use the Duhamel principle (10): ¯
e−t ( P+B) =
∞ (−t) j β j . j=0
For j ≥ 1,
Trβ j =
j
¯ ¯ d j s Tr Be−(s2 −s1 )t P B . . . e−(1−s j +s1 )t P
˜ j (4λ) = j/2 g
j
¯
¯
d j s ϕ|e−(s2 −s1 )t P |ϕ . . . ϕ|e−(1−s j +s1 )t P |ϕ .
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V. Gayral, B. Iochum, D. V. Vassilevich ¯
Since under our assumptions the operator P¯ is positive, |ϕ|e−s P |ϕ| ≤ ϕ 2L 2 for s ≥ 0. Thus |Trβ j | ≤
˜ j ϕ 2 j2 |4λ| L j!g j/2
,
and this means that the series expansion Tr(e
¯ −t ( P+B)
) − Tr(e
−t P¯
∞ )= (−t) j Trβ j j=1
is absolutely convergent in the UV regime, and ¯
¯
Tr(e−t ( P+B) ) − Tr(e−t P ) = O(t) , i.e., the operator B does not contribute to the coefficient a4 ( P¯ + B) or to the one-loop divergences. The action S + δS with anti-periodic boundary conditions is indeed renormalizable at one loop. The canonical mass dimension of the coupling constant e˜ is +4. Standard power-counting arguments show that the insertions of the interactions with e˜ can only improve the ultraviolet behavior of corresponding Feynman diagrams (since in the momentum cut-off regularization each positive power of e˜ should be accompanied by negative powers of the cut-off momentum). Although this power-counting may break down for noncommutative theories, it is nevertheless natural to conjecture that the theory with S + δS will remain renormalizable also at higher orders of the loop expansion. The need to add non-standard terms to the NC action in order to achieve renormalizability is not really surprising (see [29]). Note that in the approach of [29], the Diophantine condition does not play any role. The difference probably comes from the fact that we are working with compact noncommutative directions right from the beginning, while in [29] the “compactification” appears dynamically due to the presence of an oscillator potential. Physical consequences of adding the non-local term (62) to the action are still unclear to us. Since we consider the case of a compact Euclidean manifold (a torus) there are no immediate problems with causality (note that on Rn , no terms like (62) are required for the one-loop renormalization [43]). Alternatively, noncommutative theories may be viewed as effective low-energy theories, so that renormalizability is not required. In this case one needs a self-consistent subtraction scheme only. An example of such a scheme is given by the large-mass subtraction [5], which also uses the asymptotic expansion of the heat trace. The technical tools developed in the previous sections are sufficient to analyze other fields (spinors, gauge fields, etc). In this paper, we restricted ourselves to abelian gauge fields. An extension to non-abelian gauge fields can be done rather straightforwardly. Probably, even an extension to superfields can be achieved since the technique used in superspace [3] is similar to the one presented here. Acknowledgements. We thank Yann Bugeaud and Christian Mauduit for their help with number theory and Harald Grosse for helpful comments and discussions. We are indebted to José Gracia–Bondía and Joseph Várilly for philological advice and constructive comments. The work of DV was supported in part by the DFG project BO 1112/13-1 (Germany) and by the grant RNP 2.1.1.1112 (Russia).
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Appendix A. Proof of Asymptotic Formulae Here we prove Eq. (17). Starting from (12), |q−2π k|2 √ n 4t Tr L(l) R(r ) e−t = g (4π t)− 2 lq r−q e− , q∈Zn k∈Zn
we split the sum over q as
=
+
q∈Z
q∈Zn
.
(65)
q∈K
Consider the sum over Z first. Since for q ∈ Z the vector q/2π belongs to Zn , one can shift k in the subsequent sum by q/2π . This yields π 2 |k|2 n n √ lq r−q + = (4π t)− 2 g (4π t)− 2 lq r−q e− t lq r−q + e.s.t. q∈Z
q∈Z
0=k∈Zn
since π 2 |k|2 π 2 (|k|2 +2k) π2 π2 t lq r−q e− t ≤ 2 e− t |lq r−q | e− ≤ C e− t , q∈Z
q∈Z
0=k∈Zn
k∈Nn
2 2 because k∈Nn e−π (|k| +2k)/t is uniformly bounded in t, and {lq r−q } is a Schwartz sequence. Next we have to consider the second sum in (65). For each q ∈ K we choose k0 (q) ∈ Zn which minimizes the distance to q/(2π ) (if there are several such k, we can take any one of them.) For k = k0 , one can estimate |q −2π k|2 ≥ 4π 2 (c1 +c2 |k −k0 |2 ). From now on, ci ’s denote some positive constants. Therefore, the terms with k = k0 give only exponentially small terms in the second sum of (65). It remains to evaluate the sum 2 √ lq r−q e−|q−2π k0 (q)| /4t . (66) S = g (4π t)−n/2 q∈K
Note that q − 2π k0 (q) = 0 in (66). We split the sum in (66) into the sums over Kper and over Kinf . The first sum consists of a finite number of exponentially small terms. Consequently, it is exponentially small itself. For q ∈ Kinf we use Diophantine condition (16) to obtain 2 2(1+β) t √ |S| ≤ g (4π t)−n/2 |lq r−q |e−C /4|q| + e.s.t. q∈Kinf
Let us again divide the sum into two parts. The first one (which we denote S≤ ) is taken over a cube |qµ | ≤ Q except for q = 0. The rest is denoted S> . We estimate S> first. We would like to show that as t → 0 this sum vanishes faster n than t p− 2 for arbitrary positive p (for convenience n2 is extracted explicitly). Since l and r are smooth, their Fourier coefficients are of rapid decay. This means that the partial
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sums of |lq r−q | outside of the cube vanish faster than any power of Q, i.e., if Q is sufficiently large, |lq r−q | ≤ c3 Q −m (67) |qµ |>Q
for any chosen m. For reasons which will become clear later, we choose Q = c4 t −1/(4(1+β)) , m = 4(1 + β) p + 1. The inequality (67) holds for a sufficiently large c4 and a sufficiently small t. Now we estimate n
1
n
|S> | t − p+ 2 ≤ c4 (4π )− 2 t − p Q −m = c5 t 4(1+β) so that the left hand side vanishes for any p as t → 0. We now turn to S≤ . This is a finite sum with at most Q n terms. The Fourier coefficients entering this sum are bounded by a constant, |l−q rq | ≤ c6 . We also have −
C2 C2 c7 ≤ − = − 1/2 . t 4|q|2(1+β) t 4|Q|2(1+β) t
Therefore, n
n
|S≤ | t − p+ 2 ≤ c6 (4π )− 2 t − p (2Q)n e−c7 /t
1/2
.
This expression vanishes at t → 0 because of the exponential damping. This completes the proof of (17). The proof of (19) goes in the same way. The main observation is that the sum over K \ {0} remains exponentially small, while the “main” terms (produced by q ∈ Z) are zero since the first derivative of the geodesic distance vanishes in the coincidence limit. B. Beyond the Diophantine Condition This section is an attempt to understand what happens if is ‘in between’ rational numbers and “Diophantine numbers”. Consider the simplest case: T2 with
0 −1 µν , =θ 1 0 and g = diag(1, 1). To proceed, we need some results from number theory [7]. Let f : R≥1 → R>0 be a continuous function such that x → x 2 f (x) is non-increasing. Consider the set F( f ) := { θ ∈ R : |θq − p| < q f (q) for infinitely many rational numbers
p q
}.
The elements of F( f ) are termed f -approximable. Note that we cannot expect the above estimate to be valid for all rational numbers qp since for all irrational numbers θ , the set of fractional values of (θq)q≥1 is dense in [0, 1]. Then, there exists an uncountable set of real numbers θ/(2π ) which are f -approximable but not c f -approximable for any 0 < c < 1, see [7, Exercise 1.5]. Let us choose f (x) = (2π x)−1 e−c8 x ,
(68)
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c8 > 1, and fix a constant c9 < 1. Let us pick a θ which is f -approximable, but not c9 f -approximable. We restrict our attention to the functions l and r which do not depend on the second coordinate x 2 and lq r−q = l−q rq . Then, according to (12), 1 2 l0 r0 + 2 lq1 ,0 r−q1 ,0 e−(θq1 −2π k2 ) /4t Tr L(l)R(r )e−t = 4π t q1 >1
k2 ∈Z
up to exponentially small terms. The first term in parentheses is our standard result. Let (0) us consider the “correction” term only. Let k2 (q1 ) be an integer which minimizes the (0) distance to θq1 /2π . The sum over k2 = k2 (q1 ) is exponentially small. Consequently, the correction term becomes (0) 1 2 T (t) := lq1 ,0 r−q1 ,0 e−(θq1 +2π k2 (q1 )) /4t + e.s.t. (69) 2π t q1 >1
For a Diophantine θ , the whole correction term (69) is exponentially small. For θ/(2π ) ∈ Q this term is O(1/t). Below we work out two explicit examples and show that for the values of θ which we consider in this section the correction term is, in general, neither one nor the other. Example 1. Let us take lq1 ,0 r−q1 ,0 = e−α|q1 | . According to our assumption, θ/(2π ) is not c9 f -approximable. Consequently, for all but a finite number of q1 ∈ N, |θq1 − 2π k2(0) (q1 )| > c9 e−c8 q1 . Then we can estimate (69) as ∞ c92 e−2c8 q1 1 −α|q1 | T (t) ≤ e exp − + e.s.t. 2π t 4t q1 =0
(adding or removing any finite number of terms in this sum does not change it up to e.s.t.). Now we use the Euler–Maclaurin formula to transform this sum to an integral (with exponentially small correction terms): ∞ c92 e−2c8 q 1 dq e−αq e− 4t . 2π t 0 This integral can be easily evaluated: − cα
T (t) ≤ (2π c8 )−1 ( 2cα8 )c9
8
·t
−1+ 2cα
8
+ e.s.t.
(70)
Example 2. In our second example we shall obtain a lower bound on T (t), though for a different choice of l and r . According to our assumption, θ/(2π ) is f -approximable with f given by (68). Therefore, for an infinite set of q j , j = 1, 2, . . . , 0 < |θq j − 2π k (0) (q j )| < e−c8 q j .
(71)
We suppose that the q j are ordered, q j < q j+1 . Consequently j ≤ q j and (71) yields |θq j − 2π k (0) (q j )| < e−c8 j .
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Let us now take lq,0 r−q,0 = δq,q j e−α j . By repeating the arguments from the previous example, one obtains T (t) ≥ (2π c8 )−1 ( 2cα8 ) · t
−1+ 2cα
8
+ e.s.t.
(72)
The two estimates (70), (72) suggest that for non-Diophantine irrational θ/(2π ) one has to expect power-law corrections to the asymptotics (17). These power-law corrections are unstable in the sense that they crucially depend on the asymptotic behavior of lq r−q for large q. References 1. Avramidi, I.G.: (1998) A Covariant technique for the calculation of the one-loop effective action. Nucl. Phys. B 355, 712 (1991); Erratum-ibid. B 509 557 2. Avramidi, I.G.: Matrix general relativity: A new look at old problems. Class. Quant. Grav. 21, 103 (2004) 3. Azorkina, O.D., Banin, A.T., Buchbinder, I.L.: Pletnev, N.G.: One-loop effective potential in N = 1/2 generic chiral superfield model. Phys. Lett. B 635, 50–55 (2006) 4. Barvinsky, A.O., Vilkovisky, G.A.: Beyond the Schwinger-Dewitt technique: Converting loops into trees and in-in currents. Nucl. Phys. B 282, 163–188 (1987) 5. Bordag, M., Kirsten, K., Vassilevich, D.: On the ground state energy for a penetrable sphere and for a dielectric ball. Phys. Rev. D 59, 085011 (1999) 6. Branson T.P., Gilkey P.B., Vassilevich D.V.: (2000) Vacuum expectation value asymptotics for second order differential operators on manifolds with boundary. J. Math. Phys. 39, 1040 (1998); Erratum-ibid. 41:3301 7. Bugeaud, Y.: Approximation by Algebraic Numbers. Cambridge: Cambridge Univ. Press 2004 8. Chamseddine, A., Connes, A.: The spectral action principle. Commun. Math. Phys. 186, 731–750 (1997) 9. Chamseddine, A., Connes, A.: Inner fluctuations of the spectral action. J. Geom. Phys. 57, 1-21 (2006) 10. Connes, A., Douglas, M.R., Schwarz, A.: Noncommutative geometry and Matrix theory: Compactification on tori. J. High Energy Phys. 02, 003 (1998) 11. Connes, A.: C ∗ -algèbres et géométrie différentielle. C.R. Acad. Sci. Paris 290, 599–604 (1980) 12. Connes, A.: Noncommutative Geometry. London and San Diego: Academic Press, 1994 13. Connes, A.: Noncommutative geometry and reality. J. Math. Phys. 36, 6194–6231 (1995) 14. Connes, A., Moscovici, H.: The local index formula in noncommutative geometry. Geom. Funct. Anal. 5, 174–243 (1995) 15. Connes, A., Landi, G.: Noncommutative manifolds, the instanton algebra and isospectral deformations. Commun. Math. Phys. 221, 141–159 (2001) 16. Connes, A., Dubois-Violette, M.: Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples. Commun. Math. Phys. 230, 539–579 (2002) 17. Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge, Cambridge University Press, 1989 18. Douglas, M.R. Nekrasov, N.A.: Noncommutative field theory. Rev. Mod. Phys. 73, 977–1029 (2001) 19. Dowker, J.S., Critchley, R.: Effective Lagrangian and energy momentum tensor in de Sitter space. Phys. Rev. D 13, 3224 (1976) 20. Estrada, R., Gracia-Bondía, J.M., Várilly, J.C.: On summability of distributions and spectral geometry. Commun. Math. Phys. 191, 219–248 (1998) 21. Gayral, V.: Heat-kernel approach to UV/IR mixing on isospectral deformation manifolds. Ann. Henri Poincaré, 6, 991–1023 (2005) 22. Gayral, V., Gracia-Bondía, J.M., Ruiz, F.R.: Trouble with space-like noncommutative field theory. Phys. Lett. B 610, 141–146 (2005) 23. Gayral, V., Gracia-Bondía, J.M., Ruiz, F.R.: Position-dependent noncommutative products: Classical construction and field theory. Nucl. Phys. B 727, 513–536 (2005) 24. Gayral, V., Iochum, B.: The spectral action for Moyal plane. J. Math. Phys. 46(4), 043503 (2005) 25. Gayral, V., Iochum, B., Várilly, J.C.: Dixmier traces on noncompact isospectral deformations. J. Funct. Anal. 237, 507–539 (2006) 26. Gilkey P.B.: Asymptotic Formulae in Spectral Geometry. Boca Raton, FL: Chapman & Hall/CRC, 2004 27. Gracia-Bondía, J.M., Iochum, B., Schücker, T.: The standard model in noncommutative geometry and fermion doubling. Phys. Lett. B 416, 123–128 (1998)
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Commun. Math. Phys. 273, 445–471 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0246-y
Communications in
Mathematical Physics
Contour Dynamics of Incompressible 3-D Fluids in a Porous Medium with Different Densities Diego Córdoba, Francisco Gancedo Instituto de Matemáticas y Fisica Fundamental, Consejo Superior de Investigaciones Cientificas, Serrano 123, 28006 Madrid, Spain. E-mail: [email protected] Received: 14 July 2006 / Accepted: 1 November 2006 Published online: 25 April 2007 – © Springer-Verlag 2007
Abstract: We consider the problem of the evolution of the interface given by two incompressible fluids through a porous medium, which is known as the Muskat problem and in two dimensions it is mathematically analogous to the two-phase Hele–Shaw cell. We focus on a fluid interface given by a jump of densities, being the equation of the evolution obtained using Darcy’s law. We prove local well-posedness when the smaller density is above (stable case) and in the unstable case we show ill-posedness. 1. Introduction The evolution of a fluid in a porous medium is an important and interesting topic of fluid mechanics (see [3]). This phenomena is based on an experimental physical principle given by H. Darcy in 1856. Darcy’s law for a 3-D fluid is given by the momentum equation µ v = −∇ p − (0, 0, g ρ), κ where v is the incompressible velocity, p is the pressure, µ is the dynamic viscosity, κ is the permeability of the medium, ρ is the liquid density and g is the acceleration due to gravity. A different problem is the motion of a 2-D fluid in a Hele–Shaw cell (see [12]). In this case the fluid is set between two fixed parallel plates. These plates are close enough in such a way that the mean velocity is described by 12µ v = −∇ p − (0, g ρ), b2 where b denotes the distance between the plates. Considering that the fluid in the porous medium only moves in two directions suppressing one of the variables in the horizontal plane, these two different physical phenomena of fluid dynamics become nevertheless mathematically analogous if we identify the permeability of the medium κ and the constant b2 /12.
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The Muskat problem (see [14]) and the two-phase Hele-Shaw flow (see [17]) model the evolution of an interface between two fluids (in a porous medium and in a Hele– Shaw cell respectively) with different viscosities and densities. A lot of information can be found in the literature about both problems (see references in [9] and [13]). These free boundary problems are considered with surface tension using the Laplace– Young condition and also without surface tension in which case the pressures are equal on the interface. With surface tension, in the two dimensional case it has been proven that the problems have classical solutions (see [11]). Without surface tension, Siegel, Caflisch and Howison [18] proved ill-posedness in an unstable 2-D case, namely when the higher-viscosity fluid contracts, and they show global-in-time existence of small initial data in the stable case when the higher-viscosity fluid expands. The results rely on the assumption that the Atwood number Aµ =
µ1 − µ2 µ1 + µ2
is nonzero where µ1 and µ2 are the viscosities of the fluids. In the same year, Ambrose [1] treated the 2-D problem with an initial data fulfilling (ρ2 − ρ1 )g cos(θ (α, 0)) + 2 Aµ U (α, 0) > 0, and the following condition (x(α, 0) − x(α , 0))2 + (y(α, 0) − y(α , 0))2 > 0, (α − α )2
(1)
where the interface is the curve (x(α, t), y(α, t)), ρ1 and ρ2 are the densities of the fluids, θ is the angle that the tangent to the curve forms with the horizontal and U is the normal velocity (given by the Birkhoff-Rott integral). We are interested in the case Aµ = 0 that presents the evolution of the interface for different densities. This case, for example, models moist and dry regions in a porous medium. Meanwhile the work of Ambrose is based on the arclength and the tangent angle formulation used by Hou, Lowengrub and Shelley [13], due to the particular form of the vorticity in the case Aµ = 0, we get to parameterize the curve in the two dimensional problem getting the condition (1) for any time (see Eq. (16)). The free boundary problems given by fluids with different densities have been intensely studied. Notice the classical paper of Taylor [22] and the works of Wu [23] and [24] where the full water wave problem is solved considering the water with positive density and the air with zero density. A study of the two-dimensional case can be found in [2] due to Ambrose and Masmoudi. In order to simplify the notation, we consider µ/k = 12µ/b2 = 1 and g = 1. Thus, the 3-D system is written as v(x1 , x2 , x3 , t) = −∇ p(x1 , x2 , x3 , t) − (0, 0, ρ(x1 , x2 , x3 , t)),
(2)
where (x1 , x2 , x3 ) ∈ R3 are the spatial variables and t ≥ 0 denotes the time. Here ρ is defined by ρ1 in 1 (t) ρ(x1 , x2 , x3 , t) = ρ2 in 2 (t), with ρ1 , ρ2 ≥ 0 constants and ρ1 = ρ2 .
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Ω (t) 1
X3
Ω2(t)
X2
X1
We show in Sect. 2 that in this case it is not necessary to assume any condition on the pressure along the interface to obtain the contour equation. Furthermore, we illustrate below that the solutions to this model are weak solutions to the following conservation of mass equation: Dρ = ρt + v · ∇ρ = 0, Dt
(3)
where div v = 0. We notice the similarity with the 2-D vortex patch problem given by the twodimensional Euler equation where the vorticity is conserved along trajectories in a weak sense. The vorticity is considered to be a characteristic function of a domain. Chemin [8] proved global-in-time existence using paradifferential calculus. A simpler proof can be found in [4] due to Bertozzi and Constantin. In Sect. 2 we show that due to (2) the velocity can be determined from the density by singular integral operators (see [21]). It makes the equation more singular than the 2-D vortex patch problem where the velocity is given by the Biot-Savart law. A singular problem, more analogous to (3), is the evolution of the 2-D quasigeostrophic (QG) equation for sharp fronts. The QG equation models the dynamics of cold and hot air and the formation of fronts. Here the temperature θ is conserved along particle trajectories and the velocity is given by singular integral operators in the following form: v = (−R2 θ, R1 θ ), where R1 and R2 are the Riesz transforms (see [10] for more details of the QG equation). Rodrigo [19] proposed the contour equation of the sharp fronts where the temperature is concentrated in a domain and proved local existence and uniqueness. The paper is organized as follows. In Sect. 2 we derive the contour equation. We show that this equation fulfills the conservation of mass equation in Sect. 3. In Sect. 4 we prove local existence and uniqueness of the stable case. In Sect. 5 we get a family of global solutions of the 2-D stable case with small initial data. Finally, as a consequence of the previous section, in Sect. 6 we prove ill-posedness for the 3-D unstable case.
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2. The Contour Equation We consider the equation with (x1 , x2 , x3 ) ∈ R3 where the fluid has different densities, that is ρ is represented by ρ1 , {x3 > f (x1 , x2 , t)} ρ(x1 , x2 , x3 , t) = (4) ρ2 , {x3 < f (x1 , x2 , t)}, f being the interface. Using Darcy’s Law we get curl curl v = (−∂x1 ∂x3 ρ, −∂x2 ∂x3 ρ, (∂x21 + ∂x22 )ρ). Since div v = 0 we have curl curl v = −v, therefore it follows v = (∂x1 −1 ∂x3 ρ, ∂x2 −1 ∂x3 ρ, −(∂x21 + ∂x22 )−1 ρ).
(5)
The integral operators ∂x1 −1 and ∂x2 −1 are given by the kernels K 1 (x1 , x2 , x3 ) =
x1 1 , 4π (x12 + x22 + x32 )3/2
K 2 (x1 , x2 , x3 ) =
x2 1 , 4π (x12 + x22 + x32 )3/2
respectively, thus the velocity can be expressed by v = (K 1 ∗ ∂x3 ρ, K 2 ∗ ∂x3 ρ, −K 1 ∗ ∂x1 ρ − K 2 ∗ ∂x2 ρ).
(6)
Since ρ satisfies (4) we have ∇ρ = (ρ2 − ρ1 )(∂x1 f (x1 , x2 , t), ∂x2 f (x1 , x2 , t), −1)δ(x3 − f (x1 , x2 , t)), where δ is the Dirac distribution. Using (6) we obtain (y1 , y2 , ∇ f (x − y, t) · y) ρ2 − ρ1 PV v(x1 , x2 , x3 , t) = − dy, 2 2 3/2 4π R2 [|y| + (x 3 − f (x − y, t)) ]
(7)
(8)
where we note x = (x1 , x2 ), y = (y1 , y2 ) and ∇ f (x − y, t) · y = ∂x1 f (x − y, t)y1 + ∂x2 f (x − y, t)y2 . In (8) x3 = f (x, t) and the principal value is taken at infinity (see [21]). When x3 approaches f (x, t) in the normal direction, we get a discontinuity on the velocity due to the fact that the vorticity is concentrated on the interface. Thus, for ε > 0 we define v 1 (x, f (x, t), t) = lim v(x1 − ε∂x1 f (x, t), x2 − ε∂x2 f (x, t), f (x, t) + ε, t), ε→0
and v 2 (x, f (x, t), t) = lim v(x1 + ε∂x1 f (x, t), x2 + ε∂x2 f (x, t), f (x, t) − ε, t). ε→0
It follows
(y1 , y2 , ∇ f (x − y, t) · y) ρ2 − ρ1 PV dy 2 + ( f (x, t) − f (x − y, t))2 ]3/2 2 4π [|y| R ρ2 − ρ1 ∂x1 f (x, t)(1, 0, ∂x1 f (x, t)) (9) + 2 1 + (∂x1 f (x, t))2 + (∂x2 f (x, t))2 ρ2 − ρ1 ∂x2 f (x, t)(0, 1, ∂x2 f (x, t)) + , 2 1 + (∂x1 f (x, t))2 + (∂x2 f (x, t))2
v 1 (x, f (x, t), t) = −
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(y1 , y2 , ∇ f (x − y, t) · y) ρ2 − ρ1 PV dy 2 2 3/2 4π R2 [|y| + ( f (x, t) − f (x − y, t)) ] ρ2 − ρ1 ∂x1 f (x, t)(1, 0, ∂x1 f (x, t)) − (10) 2 1 + (∂x1 f (x, t))2 + (∂x2 f (x, t))2 ρ2 − ρ1 ∂x2 f (x, t)(0, 1, ∂x2 f (x, t)) − . 2 1 + (∂x1 f (x, t))2 + (∂x2 f (x, t))2
v 2 (x, f (x, t), t) = −
The velocity in the tangential directions only moves the particles on the surface f (x, t); i.e., if we rewrite the velocity in the tangential directions, we only make a change on the parametrization and do not alter the shape of the interface. Thus, it follows that ρ2 − ρ1 (y1 , y2 , ∇ f (x − y, t) · y) v(x, f (x, t), t) = − dy, (11) PV 2 + ( f (x, t) − f (x − y, t))2 ]3/2 2 4π [|y| R due to the fact that the terms ρ2 − ρ1 ∂x1 f (x, t)(1, 0, ∂x1 f (x, t)) , 2 1 + (∂x1 f (x, t))2 + (∂x2 f (x, t))2 ρ2 − ρ1 ∂x2 f (x, t)(0, 1, ∂x2 f (x, t)) ± , 2 1 + (∂x1 f (x, t))2 + (∂x2 f (x, t))2 ±
are in the tangential directions. Moreover, if we add the following tangential terms to (11): y1 ρ2 − ρ1 PV dy(1, 0, ∂x1 f (x, t)), 2 2 [|y| + ( f (x, t) − f (x − y, t))2 ]3/2 4π R y2 ρ2 − ρ1 PV dy(0, 1, ∂x2 f (x, t)), 2 2 3/2 4π R2 [|y| + ( f (x, t) − f (x − y, t)) ] we obtain v(x, f (x, t), t) =
ρ2 − ρ1 (∇ f (x, t) − ∇ f (x − y, t)) · y (0, 0, P V dy). 2 2 3/2 4π R2 [|y| + ( f (x, t) − f (x − y, t)) ] (12)
Finally we have the contour equation given by df ρ2 − ρ1 (∇ f (x, t) − ∇ f (x − y, t)) · y (x, t) = PV dy, 2 2 3/2 dt 4π R2 [|y| + ( f (x, t) − f (x − y, t)) ] f (x, 0) = f 0 (x).
(13)
In the periodic case, we can obtain an equivalent equation to (13) due to the integral operators ∂x1 −1 and ∂x2 −1 can be presented by the kernels x1 1 ( L(x1 , x2 , x3 ) + M(x1 , x2 , x3 )), 4π (x12 + x22 + x32 )3/2 1 x2 p K 2 (x1 , x2 , x3 ) = L(x1 , x2 , x3 ) + M(x1 , x2 , x3 )), ( 4π (x12 + x22 + x32 )3/2 p
K 1 (x1 , x2 , x3 ) =
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respectively for (x1 , x2 , x3 ) ∈ T2 × R with T2 = [−π, π ]2 and the functions L , M ∈ C ∞ (T2 × R) (see [20] for the kernel of the Riesz potentials on the n-torus). Adding an p appropiate function to the singular part of K 1 and K 2P , we can choose L ∈ Cc∞ (T2 × R),
L ≥ 0, supp L ⊂ {x12 + x22 + x32 ≤ 4},
(14)
L = 1 in {x12 + x22 + x32 ≤ 1} and L(−x1 , −x2 , −x3 ) = L(x1 , x2 , x3 ). The function M belongs to Cb∞ (T2 × R) and M(0, 0, 0) = 0. The velocity can be expressed by p
p
p
p
v = (K 1 ∗ ∂x3 ρ, K 2 ∗ ∂x3 ρ, −K 1 ∗ ∂x1 ρ − K 2 ∗ ∂x2 ρ), and due to (7) it follows (suppressing the dependence on t) (y1 , y2 , ∇ f (x − y) · y) ρ2 − ρ1 PV v(x1 , x2 , x3 ) =− L(y, x3 − f (x − y))dy 2 + (x − f (x − y))2 ]3/2 2 4π [|y| 3 T ρ2 − ρ1 (1, 1, ∇ f (x − y) · (1, 1))M(y, x3 − f (x − y))dy, − 4π T2 if x3 = f (x). Adding a term in the tangential direction we obtain (∇ f (x)−∇ f (x − y)) · y ρ2 − ρ1 0, 0, L(y, f (x)− f (x − y))dy v(x, f (x)) = 2 +( f (x)− f (x − y))2 ]3/2 2 4π [|y| T + (∇ f (x)−∇ f (x − y)) · (1, 1))M(y, f (x)− f (x − y))dy . T2
Finally we have the contour equation in the periodic case given by ρ2 − ρ1 (∇ f (x, t) − ∇ f (x − y, t)) · y df (x, t) = 2 + ( f (x, t) − f (x − y, t))2 ]3/2 2 dt 4π [|y| T × L(y, f (x, t) − f (x − y, t))dy ρ2 − ρ1 + (∇ f (x, t) − ∇ f (x − y, t)) · (1, 1)) 4π T2 × M(y, f (x, t) − f (x − y, t))dy, f (x, 0) = f 0 (x).
(15)
We use both formulations throughout the paper. Suppose that the function f (x) only depends on x1 in Eq. (13). Then the contour equation in the 2-D case (with a 1-D interface) follows: ρ2 − ρ1 (∂x f (x, t) − ∂x f (x − α, t))α df (x, t) = PV dα, 2 + ( f (x, t) − f (x − α, t))2 dt 2π α (16) R f (x, 0) = f 0 (x); x ∈ R. This equation can be obtained in a similar way to (13) using the stream function. Performing a two-dimensional analysis using the stream function, we obtain an equivalent
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equation to (16) in the two dimensional periodic case as follows: ρ2 − ρ1 (∂x f (x, t) − ∂x f (x − α, t))α df (x, t) = P(α, f (x, t)− f (x −α, t))dα 2 + ( f (x, t) − f (x − α, t))2 dt 2π α T ρ2 − ρ1 (∂x f (x, t) − ∂x f (x − α, t))Q(α, f (x, t)− f (x −α, t))dα, + 2π T f (x, 0) = f 0 (x), (17) with P(x1 , x2 ) ∈ Cc∞ (T × R), P ≥ 0, supp P ⊂ {x12 + x22 ≤ 4}, P = 1 in {x12 + x22 ≤ 1}
and P(−x1 , −x2 ) = P(x1 , x2 ).
The function Q(x1 , x2 ) belongs to Cb∞ (T × R) and Q(0, 0) = 0. If we consider the linearized equation of the motion, we obtain a dissipative equation when ρ1 < ρ2 (the greater density is below) and an unstable equation when ρ1 > ρ2 . The unstable linearized equation presents an instability similar to Kelvin-Helmholtz’s (see [6]). As usual, we note the Riesz transforms in R2 (see [21]) y1 1 P.V. f (x − y)dy, R1 f (x) = 3 2 2π |y| R y2 1 P.V. R2 f (x) = f (x − y)dy, 3 2 2π |y| R s f (ξ ) = |ξ |s and the operator s f defined by the fourier transform f (ξ ). Suppose that f (x) is uniformly small and we can neglect the terms of order greater than one in (13), then it reduces to the following linear equation:
ρ1 − ρ2 ρ1 − ρ2 (R1 ∂x1 f + R2 ∂x2 f ) = f, 2 2 f (x, 0) = f 0 (x).
ft =
(18)
Applying the Fourier transform we get fˆ(ξ ) = fˆ0 (ξ )e
ρ1 −ρ2 2 |ξ |t
,
and therefore (18) is a dissipative equation when ρ1 < ρ2 and an ill posed problem in the case ρ1 > ρ2 with a general initial data in the Schwartz class. We need an analytic initial data in order to get a well posed problem for ρ1 > ρ2 . 3. The Conservation of Mass Equation We show that if ρ is defined by (4) and f (x, t) is convected by the velocity (12) then ρ is a weak solution of the conservation of mass Eq. (3) and conversely. From now on, is equal to R2 or T2 and x = (x1 , x2 , x3 ).
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Definition 3.1. The density ρ is a weak solution of the conservation of mass equation if for any ϕ ∈ C ∞ ( × R × (0, T )), ϕ with compact support in the real case and periodic in (x1 , x2 ) otherwise, we have T (ρ( x , t)∂t ϕ( x , t) + v( x , t)ρ( x , t)∇ϕ( x , t))d x dt = 0, (19) R
0
where the incompressible velocity v is given by Darcy’s law. Then Proposition 3.2. If f (x, t) satisfies (13) and ρ( x , t) is defined by (4), then ρ is a weak solution of the conservation of mass equation. Furthermore, if ρ is a weak solution of the conservation of mass equation given by (4), then f (x, t) satisfies (13). Proof. Let ρ be a weak solution of (3) defined by (4). Integrating by parts we have T T T I = ρ∂t ϕd x dt = ρ1 ∂t ϕd x dt + ρ2 ∂t ϕd x dt R
0
T
= (ρ1 − ρ2 ) 0
0
{x3 > f }
ϕ(x, f (x, t), t)∂t f (x, t)d xdt.
On the other hand, due to (9) and (10) we obtain T T J= ρv ∇ ϕ d x dt = ρ1 v∇ϕd x dt + ρ2 0
=
T 0
R
{x3 < f }
0
0
{x3 > f }
0
T
{x3 < f }
v∇ϕd x dt
ϕ(x, f (x, t), t)(ρ1 v 1 (x, f (x, t), t)
2
− ρ2 v (x, f (x, t), t))·(∂x1 f (x, t), ∂x2 f (x, t), −1)d xdt T = (ρ1 −ρ2 ) ϕ(x, f (x, t), t)v(x, f (x, t), t) · (∂x1 f (x, t), ∂x2 f (x, t),−1)d xdt, 0
where v(x, f (x, t), t) is given by (11). We get (ρ1 − ρ2 )2 T J = ϕ(x, f (x, t), t) 4π 0 (∇ f (x, t) − ∇ f (x − y, t)) · y ×P V d yd xdt. 2 2 3/2 R2 [|y| + ( f (x, t) − f (x − y, t)) ] Then I + J = 0 due to (19). Thus, if we choose ϕ( x , t) = ϕ(x, t) for x3 ∈ [− f L ∞ , f L ∞ ] it follows that f (x, t) fulfills (13). Following the same arguments it is easy to check that if f (x, t) satisfies (13), then ρ is a weak solution given by (4).
Remark 3.3. Note that due to (5), the velocity satisfies v = (R1 (R3 ρ), R2 (R3 ρ), −(R12 + R22 )(ρ)), where the operators R1 , R2 and R3 are the Riesz transforms in three dimensions (see [21]). Since ρ ∈ L ∞ ( × R) then v belongs to B M O (bounded mean oscillation) and therefore v is in L 2 ( × R) locally (see [21] for the definitions and properties of the B M O space).
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4. Local Well-Posedness for the Stable Case In this section we prove local existence and uniqueness for the stable case using energy estimates. First we study the case = R2 and at the end of the section we give the main differences with the periodic domain. Denote the Sobolev spaces by H k , the Hölder spaces by C k,δ with 0 ≤ δ < 1 the Hölder continuity and the hessian matrix of a function f (x) by ∇ 2 f (x). The norms of H k and C k,δ are defined as follows: f 2H k = f 2L 2 + k f 2L 2 , j
f C k,δ = f C k + max max
j
|∂xi 1 ∂x2 f (x) − ∂xi 1 ∂x2 f (y)| |x − y|δ
i+ j=k x= y
.
4.1. Case = R2 . The main theorem in this section is the following Theorem 4.1. Let f 0 (x) ∈ H k (R2 ) for k ≥ 4 and ρ2 > ρ1 . Then there exists a time T > 0 so that there is a unique solution to (13) in C 1 ([0, T ]; H k (R2 )) with f (x, 0) = f 0 (x). We choose ρ2 − ρ1 = 4π without loss of generality, then (∇ f (x, t) − ∇ f (x − y, t)) · y df (x, t) = P V dy, 2 2 3/2 dt R2 [|y| + ( f (x, t) − f (x − y, t)) ] f (x, 0) = f 0 (x).
(20)
We show the proof with k = 4 being analogous for k > 4. We apply energy methods (see [5] for more details). Then 1 d (∇ f (x) − ∇ f (x − y)) · y f 2L 2 (t) = f (x)P V d yd x 2 2 3/2 2 2 dt R2 [|y| + ( f (x) − f (x − y)) ] R (∇ f (x) − ∇ f (x − y)) · y = f (x) d yd x 2 + ( f (x) − f (x − y))2 ]3/2 2 [|y| R |y|<1 ∇ f (x) · y + f (x)P V d yd x 2 2 3/2 2 R |y|>1 [|y| + ( f (x) − f (x − y)) ] ∇ f (x − y) · y − f (x)P V d yd x 2 2 3/2 R2 |y|>1 [|y| + ( f (x) − f (x − y)) ] = I1 + I2 + I3 . The identity
∂xi f (x) − ∂xi f (x − y) =
yields
0
1
∇∂xi f (x + (s − 1)y) · y ds,
| f (x)||∇ 2 f (x + (s − 1)y)| d xd y 2 −2 3/2 0 |y|<1 R2 [1 + (( f (x) − f (x − y)) |y| ] 1 j ds |y|−1 dy f L 2 ∂xi 1 ∂x2 f L 2 ≤ C f 2H 2 . ≤C
I1 ≤ C
1
|y|−1
ds
0
|y|<1
i+ j=2
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Integrating by parts, the term I2 is written 3 I2 = 2 ≤C ≤
R2
|y|>1
|y|>1
| f (x)|2
|y|−3
R2
( f (x) − f (x − y))(∇ f (x) − ∇ f (x − y)) · y d xd y [|y|2 + (( f (x) − f (x − y))2 ]5/2
| f (x)|2
| f (x) − f (x − y)||y|−1 |∇ f (x) − ∇ f (x − y)| d xd y [1 + (( f (x) − f (x − y))2 |y|−2 ]5/2
C f L ∞ f 2H 1 .
Integrating by parts in I3 , it follows
|y|2 − 2( f (x) − f (x − y))2 d xd y [|y|2 + ( f (x) − f (x − y))2 ]5/2 |y|>1 R2 ( f (x) − f (x − y))∇ f (x − y) · y +3 f (x) f (x − y) d xd y [|y|2 + ( f (x) − f (x − y))2 ]5/2 |y|>1 R2 |y|2 f (x) f (x − y) d xdσ (y) − 2 [|y| + ( f (x) − f (x − y))2 ]3/2 |y|=1 R2
I3 =
f (x) f (x − y)
≤ C( f L ∞ + 1) f 2H 1 . Using Sobolev inequalities, we get finally d f 2L 2 (t) ≤ C( f 3H 2 (t) + 1). dt We consider the quantity 1 d 4 ∂ f 2 2 (t) = I4 + I5 + I6 + I7 + I8 , 2 dt x1 L where
(∇∂x41 f (x) − ∇∂x41 f (x − y)) · y ∂x41 f (x)P V d yd x, 2 2 3/2 R2 R2 [|y| + ( f (x) − f (x − y)) ] =4 ∂x41 f (x) (∇∂x31 f (x) − ∇∂x31 f (x − y)) · y ∂x1 A(x, y)d yd x, R2 R2 4 =6 ∂x1 f (x) (∇∂x21 f (x) − ∇∂x21 f (x − y)) · y ∂x21 A(x, y)d yd x, R2 R2 =4 ∂x41 f (x) (∇∂x1 f (x) − ∇∂x1 f (x − y)) · y ∂x31 A(x, y)d yd x, 2 2 R R 4 = ∂x1 f (x) (∇ f (x) − ∇ f (x − y)) · y ∂x41 A(x, y)d yd x,
I4 = I5 I6 I7 I8
R2
R2
and A(x, y) = [|y|2 + ( f (x) − f (x − y))2 ]−3/2 .
(21)
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The most singular term is I4 . In order to estimate it ∇∂x41 f (x, t) · y I4 = ∂x41 f (x)P V d yd x 2 2 3/2 R2 R2 [|y| + ( f (x, t) − f (x − y, t)) ] ∇∂x41 f (y, t) · (x − y) 4 − ∂x1 f (x)P V d yd x 2 2 3/2 R2 R2 [|x − y| + ( f (x, t) − f (y, t)) ] = J1 + J2 . Integrating by parts ( f (x) − f (x − y))(∇ f (x) − ∇ f (x − y)) · y 3 J1 = |∂x41 f (x)|2 P V d yd x 2 2 R2 [|y|2 + (( f (x) − f (x − y))2 ]5/2 R 3 = |∂ 4 f (x)|2 ( dy + P V dy)d x 2 R2 x 1 |y|>1 |y|<1 3 3 ≤ f C 1 ∂x41 f 2L 2 + M( f ) ∂x41 f 2L 2 , 2 2 (22) where
M( f ) = max P V
( f (x) − f (x − y))(∇ f (x) − ∇ f (x − y)) · y dy . [|y|2 + (( f (x) − f (x − y))2 ]5/2 |y|<1
x
We estimate this maximum in the following form: ( f (x) − f (x − y) − ∇ f (x) · y)(∇ f (x) − ∇ f (x − y)) · y M( f ) ≤ max dy x [|y|2 + (( f (x) − f (x − y))2 ]5/2 |y|<1 (∇ f (x) · y)((∇ f (x) − ∇ f (x − y)) · y − y · ∇ 2 f (x) · y) + max dy 2 2 5/2 x [|y| + (( f (x) − f (x − y)) ] |y|<1 + max (∇ f (x) · y)(y · ∇ 2 f (x) · y)(B(x, y) − C(x, y))dy x
|y|<1
+ max P V x
(∇ f (x) · y)(y · ∇ 2 f (x) · y) dy , 2 2 5/2 |y|<1 [|y| + (∇ f (x) · y) ] (23)
where B(x, y) = [|y|2 + (( f (x) − f (x − y))2 ]−5/2 ,
C(x, y) = [|y|2 + (∇ f (x) · y)2 ]−5/2 .
Making the change of variables y = −z, we obtain that the last integral in (23) is null, then we can estimate M( f ) by |y|−1 2 M( f ) ≤ f C max dy 2 −1 )2 ]5/2 x [1 + (( f (x) − f (x − y))|y| |y|<1 2 2 −1 + f C 1 f C 2,δ |y|−2+δ dy + f C f |y| dy 1 C2 |y|<1
2 2 2 ≤ C( f C 2 + f C 1 f C 2,δ + f C 1 f C 2 ),
|y|<1
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with 0 < δ < 1, having finally 4 4 2 J1 ≤ C( f C 2,δ + 1) ∂ x 1 f L 2 .
(24)
In order to estimate J2 , we integrate by parts getting ∇ y (∂x41 f (x) − ∂x41 f (y)) · (x − y) ∂x41 f (x)P V d yd x J2 = 2 2 3/2 R2 R2 [|x − y| + ( f (x) − f (y)) ] = K1 + K2, with K1 = − and
R2
∂x41
f (x)P V
K2 =
R2
∂x41 f (x) − ∂x41 f (y) [|x − y|2 + ( f (x) − f (y))2 ]3/2
d yd x,
∂x41 f (x)
R2
(∂x41 f (x) − ∂x41 f (y))
R2
3( f (x) − f (y))( f (x) − f (y) − ∇ f (y) · (x − y)) × d yd x. [|x − y|2 + ( f (x) − f (y))2 ]5/2 Making a change of variables we can obtain ∂x41 f (x) − ∂x41 f (y) ∂x41 f (x) d yd x K 1 = −P V [|x − y|2 + ( f (x) − f (y))2 ]3/2 R2 R2 ∂x41 f (x) − ∂x41 f (y) = PV ∂x41 f (y) d yd x [|x − y|2 + ( f (x) − f (y))2 ]3/2 R2 R2 (∂x41 f (x) − ∂x41 f (y))2 1 =− d yd x 2 R2 R2 [|x − y|2 + ( f (x) − f (y))2 ]3/2 ≤ 0. Here we observe the main difference with the unstable case in which we obtain the opposite sign. Now we consider K2 = L 1 + L 2 + L 3, being L1 = 3
R2
|∂x41 f (x)|2 P V
( f (x) − f (y))( f (x) − f (y) − ∇ f (y) · (x − y)) d yd x, [|x − y|2 + ( f (x) − f (y))2 ]5/2 R2
3 L2 = − P V ∂x41 f (x)∂x41 f (y) 4 R2 R2 ( f (x) − f (y))(x − y) · (∇ 2 f (x) + ∇ 2 f (y)) · (x − y) × d yd x, [|x − y|2 + ( f (x) − f (y))2 ]5/2 L 3 = −3
R2 R2
∂x41 f (x)∂x41 f (y)( f (x) − f (y))D(x, y)d yd x,
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with D(x, y) =
( f (x)− f (y)−∇ f (y) · (x − y)− 41 (x − y) · (∇ 2 f (x)+∇ 2 f (y)) · (x − y)) . [|x − y|2 +( f (x)− f (y))2 ]5/2
The term L 1 can be estimated like J1 in (22) and one finds that 4 4 2 L 1 ≤ C(1 + f C 2,δ ) ∂ x 1 f L 2 .
Exchanging x for y we obtain that L 2 = 0. For the last term L 3 , it follows 4 2 |∂x1 f (x)| | f (x) − f (y)||D(x, y)|d yd x L3 ≤ C R2 R2 +C |∂x41 f (y)|2 | f (x) − f (y)||D(x, y)|d xd y 2 2 R R ≤C |∂x41 f (x)|2 | f (x) − f (x − y)||D(x, x − y)|d yd x R2 R2 +C |∂x41 f (y)|2 | f (x + y) − f (y)||D(x + y, y)|d xd y R2 R2 |∂x41 f (x)|2 d x dy + dy ≤C R2 |y|<1 |y|>1 4 2 |∂x1 f (y)| dy dx + dx + R2
≤
C f C 1 f C 2,δ ∂x41
|x|<1 f 2L 2 .
|x|>1
Finally 4 4 2 J2 = K 1 + K 2 ≤ K 2 = L 1 + L 2 + L 3 = L 1 + L 3 ≤ C( f C 2,δ + 1) ∂ x 1 f L 2 ,
and due to (24) we obtain 4 4 2 I4 ≤ C( f C 2,δ + 1) ∂ x 1 f L 2 .
Now we estimate the integral I5 . We have I5 = J3 + J4 where J3 = 4 ∂x41 f (x)∇∂x31 f (x) · R2 ( f (x) − f (x − y))(∂x1 f (x) − ∂x1 f (x − y)) ×P V y d yd x, [|y|2 + ( f (x) − f (x − y))2 ]5/2 R2 and
J4 = − 4P V
R2 R2
∂x41 f (x)∇∂x31 f (y) · (x − y)
( f (x) − f (y))(∂x1 f (x) − ∂x1 f (y)) d yd x [|x − y|2 + ( f (x) − f (y))2 ]5/2 = − 4P V ∂x41 f (x)∂x2 ∂x31 f (y)(x2 − y2 ) ×
R2 R2
( f (x) − f (y))(∂x1 f (x) − ∂x1 f (y)) × d yd x. [|x − y|2 + ( f (x) − f (y))2 ]5/2
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The way to estimate J3 is similar to the term J1 in (22), and we find that 4 2 J3 ≤ C(1 + f C 2,δ ) f H 4 .
We decompose the term J4 = K 3 + K 4 + K 5 + K 6 as follows: ∂x41 f (x)∂x2 ∂x31 f (x − y)y2 E(x, y)d yd x, K 3 = −4P V R2 R2 K 4 = −4P V ∂x41 f (x)∂x2 ∂x31 f (x − y)y2 F(x, y)d yd x, R2 R2
K 5 = −4P V
∂x41 f (x)∂x2 ∂x31 f (x − y)y2 (∇ f (x) · y)(∇∂x1 f (x) · y)(B(x, y)
R2 |y|<1
−C(x, y))d yd x and
K 6 = −4P V
R2
|y|<1
∂x41 f (x)∂x2 ∂x31 f (x − y)y2
(∇ f (x) · y)(∇∂x1 f (x) · y) d yd x, [|y|2 + (∇ f (x) · y)2 ]5/2
where E(x, y) =
( f (x) − f (x − y) − ∇ f (x) · y)(∂x1 f (x) − ∂x1 f (x − y)) , [|y|2 + ( f (x) − f (x − y))2 ]5/2
and F(x, y) =
(∇ f (x) · y)(∂x1 f (x) − ∂x1 f (x − y) − ∇∂x1 f (x) · yX{|y|<1} ) . [|y|2 + ( f (x) − f (x − y))2 ]5/2
The terms K 3 , K 4 , and K 5 are estimated as J1 . Then we find that 2 2 K 3 ≤ C(1 + f C 2 ) f H 4 ,
K 4 ≤ C(1 + f C 1 f C 2,δ ) f 2H 4 , and 2 2 2 K 5 ≤ C(1 + f C 1 f C 2 ) f H 4 .
We rewrite K 6 and we get
K 6 = −4P V
R2
with the operator S defined by
S(g)(x) = P V and (x, y) =
∂x41 f (x)S(∂x2 ∂x31 f )(x)d x,
|y|<1
(x, y) g(x − y)dy, |y|2
y y y2 (∇ f (x) · |y| )(∇∂x1 f (x) · |y| ) . y 2 5/2 |y| [1 + (∇ f (x) · |y| ) ]
(26)
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The function (x, y) satisfies that (i) (x, λy) = (x, y), ∀ λ > 0, (ii) (x, −y) = −(x, y), (iii) sup |(x, y)| ≤ ∇∂x1 f L ∞ , x
and therefore S is a bounded linear map on L p (R2 ) for 1 < p < ∞ and S p ≤ C ∇∂x1 f L ∞ (see [21] and references therein for more details). Then K 6 ≤ C f C 2 ∂x41 f L 2 ∂x2 ∂x31 f L 2 . We obtain finally 4 2 I5 ≤ C(1 + f C 2,δ ) f H 4 .
In order to estimate the term I6 we take 1 ds dy ∂x41 f (x) y · (∇ 2 ∂x21 f (x + (s − 1)y)) · y ∂x21 A(x, y)d x I6 = 6 R2 R2 0 1 ≤ ds dy + dy |y|<1 |y|>1 0 |∂x41 f (x)||∇ 2 ∂x21 f (x + (s − 1)y)||∂x21 A(x, y)||y|2 d x × R2 −2+δ 4 2 ≤ C( |y| dy + |y|−3 dy)(1 + f C 2,δ ) f H 4 . |y|<1
|y|>1
The most singular term of I7 is K 7 , K 7 = −12 ∂x41 f (x)(∇∂x1 f (x) − ∇∂x1 f (x − y)) · y G(x, y)d yd x, R2 R2
where G(x, y) =
( f (x) − f (x − y))(∂x31 f (x) − ∂x31 f (x − y)) [|y|2 + ( f (x) − f (x − y))2 ]5/2
.
Due to |∇∂x1 f (x) − ∇∂x1 f (x − y)| ≤ f C 2,δ |y|δ and writing ∂x31
f (x) − ∂x31
f (x − y) = 0
1
∇∂x31 f (x + (s − 1)y) · y ds,
2 2 4 2 we obtain K 7 ≤ C( f C 2,δ + 1) f H 4 and I7 ≤ C(1 + f C 2,δ ) f H 4 . The most singular term of I8 is K 8 , K 8 = −12 ∂x41 f (x)(∂x41 f (x) − ∂x41 f (x − y)) H (x, y)d yd x,
R2 R2
where H (x, y) =
( f (x) − f (x − y))(∇ f (x) − ∇ f (x − y)) · y . [|y|2 + ( f (x) − f (x − y))2 ]5/2
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Then
K 8 = −12
R2
|∂x41 f (x)|2 P V
R2
( f (x) − f (x − y))(∇ f (x) − ∇ f (x − y)) · y d yd x, [|y|2 + ( f (x) − f (x − y))2 ]5/2
4 ) ∂ 4 f 2 and I ≤ C(1 + is controlled as before. We obtain K 8 ≤ C(1 + f C 8 2,δ x1 L2 4 2 f C 2,δ ) f H 4 . Finally, we have
d 4 4 2 ∂ f 2 2 (t) ≤ C(1 + f C 2,δ (t)) f H 4 (t), dt x1 L and using Sobolev inequalities we get d 4 ∂ f 2 2 (t) ≤ C( f 6H 4 (t) + 1). dt x1 L
(27)
In a similar way we obtain d 4 ∂ f 2 2 (t) ≤ C( f 6H 4 (t) + 1), dt x2 L
(28)
and since we can define f 2H 4 = f 2L 2 + ∂x41 f 2L 2 + ∂x42 f 2L 2 , due to (21), (27) and (28) it follows d f H 4 (t) ≤ C( f 5H 4 (t) + 1). dt Using Gronwall’s inequality we get that the quantity f H 4 is bounded up to a time T = T ( f 0 H 4 ). Then, applying energy methods the local existence result follows. Let the functions f 1 (x, t), f 2 (x, t) be two solutions of Eq. (13) with f 1 (x, 0) = f 2 (x, 0) = f 0 (x), and f = f 1 − f 2 . Then d f 2L 2 (t) = I9 + I10 + I11 , dt with
I9 =
R2
f (x)∇ f (x) · P V
I10 = −
R2
f (x)P V
and
R2
R2
y[|y|2 + ( f 1 (x) − f 1 (x − y))2 ]−3/2 d yd x,
∇ f (y, t) · (x − y)[|x − y|2 + ( f 1 (x) − f 1 (y))2 ]−3/2 d yd x,
I11 =
R2
f (x)P V
R2
(∇ f 2 (x) − ∇ f 2 (x − y)) · y N (x, y)d yd x,
with N (x, y) = [|y|2 + ( f 1 (x) − f 1 (x − y))2 ]−3/2 − [|y|2 + ( f 2 (x) − f 2 (x − y))2 ]−3/2 .
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Integrating by part in I9 we have I9 ≤ C( f 1 H 4 ) f 2L 2 , and I11 ≤ C( f 1 H 4 , f 2 H 4 ) f 2L 2 . The term I10 = −
R2
f (x)P V
R2
∇ y ( f (y) − f (x)) · (x − y)
−3/2 × |x − y|2 + ( f 1 (x) − f 1 (y))2 d yd x = −P V f (x)( f (x) − f (y))[|x − y|2 + ( f 1 (x) − f 1 (y))2 ]−3/2 d yd x R2 R2 +P V f (x)( f (x) − f (y)) R2 R2
3( f 1 (x) − f 1 (y))( f 1 (x) − f 1 (y) − ∇ f 1 (x)(x − y)) × d yd x. [|x − y|2 + ( f 1 (x) − f 1 (y))2 ]5/2 Then we have that I10 ≤ J5 + J6 , where 3( f 1 (x) − f 1 (y))( f 1 (x) − f 1 (y) − ∇ f 1 (x)(x − y)) J5 = | f (x)|2 P V d yd x, [|x − y|2 + ( f 1 (x) − f 1 (y))2 ]5/2 R2 R2 and J6 = − ×
R2
f (x)P V
R2
f (x − y)
3( f 1 (x) − f 1 (x − y))( f 1 (x) − f 1 (x − y) − ∇ f 1 (x) · y) d yd x. [|y|2 + ( f 1 (x) − f 1 (x − y))2 ]5/2
The term J5 is estimated as J1 obtaining J5 ≤ C( f 1 H 4 ) f 2L 2 . The term J6 can be expressed as J6 = K 9 + K 10 with ( f 1 (x) − f 1 (x − y))G(x, y) f (x) f (x − y) d yd x, K 9 = −3 2 + ( f (x) − f (x − y))2 ]5/2 2 2 [|y| 1 1 R R where the function G(x, y) is given by G(x, y) = f 1 (x) − f 1 (x − y) − ∇ f 1 (x) · y −
1 y · (∇ 2 f 1 (x) + ∇ 2 f 1 (x − y)) · y. 4
One finds that the principal value K 10 is null. Therefore, we obtain finally J6 ≤ C( f 1 H 4 ) f 2L 2 . Applying Gronwall’s inequality we get uniqueness. 4.2. Case = T2 . In the periodic case we give the theorem of local well-posedness and the differences with = R2 . Theorem 4.2. Let f 0 (x) ∈ H k (T2 ) for k ≥ 4 and ρ2 > ρ1 . Then there exists a time T > 0 so that there is a unique solution to (15) in C 1 ([0, T ]; H k (T2 )) with f (x, 0) = f 0 (x).
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The proof is similar to Theorem 4.1 but we must use the properties of the function L in (14). We consider without loss of generality ρ2 − ρ1 = 4π . In order to control the evolution of the quantity ∂x41 f L 2 , the most singular term is (∇∂x41 f (x) − ∇∂x41 f (x − y)) · y 4 ∂x1 f (x) L(y, f (x) − f (x − y))d yd x I = 2 2 3/2 2 T2 [|y| + ( f (x) − f (x − y)) ] T L(y, f (x) − f (x − y)) ∂x41 f (x)∇∂x41 f (x) · P V y d yd x = 2 + ( f (x) − f (x − y))2 ]3/2 2 2 [|y| T T ∇∂x41 f (y) · (x − y) − ∂x41 f (x)P V L(x − y, f (x)− f (y))d yd x 2 2 3/2 T2 T2 [|x − y| + ( f (x) − f (y)) ] = J1 + J2 . Integrating by parts 3 |∂x41 f (x)|2 P V A(x, y)L(y, f (x) − f (x − y))d yd x J1 = 2 T2 T2 1 L x3 (y, f (x)− f (x − y))(∇ f (x)−∇ f (x − y)) · y − |∂x41 f (x)|2 P V d yd x 2 T2 [|y|2 +( f (x)− f (x − y))2 ]3/2 T2 = K1 + K2, (29) where A(x, y) =
( f (x) − f (x − y))(∇ f (x) − ∇ f (x − y)) · y [|y|2 + (( f (x) − f (x − y))2 ]5/2
and L x3 (x1 , x2 , x3 ) = ∂x3 L(x1 , x2 , x3 ). Due to |L(x1 , x2 , x3 ) − 1| ≤ C|(x1 , x2 , x3 )| we have (∇ f (x) · y)(y · ∇ 2 f (x) · y) 4 2 4 2 V ) f +C ∂ f max dy K 1 ≤ C(1+ f C P 2,δ x1 H4 L2 x [|y|2 +(∇ f (x) · y))2 ]5/2 T2 4 2 ≤ C(1 + f C 2,δ ) f H 4 .
Using that | f (x) − f (x − y)| ≤ f C 1 |y| and L x3 = 0 in {x12 + x22 + x32 ≤ 4}, we have that 1 K2 = − |∂ 4 f (x)|2 2 T2 x1 L x3 (y, f (x) − f (x − y))(∇ f (x) − ∇ f (x − y)) · y × d yd x 2 [|y|2 + ( f (x) − f (x − y))2 ]3/2 |y|> 1+ f C1
≤
4 C( f C 2,δ
+ 1) f 2H 4 .
In order to estimate J2 , we integrate by parts getting ∇ y (∂x41 f (y)−∂x41 f (x)) · (x − y) J2 =− ∂x41 f (x)P V L(x − y, f (x)− f (y))d yd x 2 2 3/2 T2 T2 [|x − y| +( f (x)− f (y)) ] = K3 + K4 + K5 + K6
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with K3 = −
T2
∂x41
f (x)P V
K4 = K5 =
T2
∂x41 f (x)−∂x41 f (y) T2
[|x − y|2 +( f (x)− f (y))2 ]3/2
L(x − y, f (x)− f (y))d yd x,
∂x41 f (x)
(∂x41 f (x) − ∂x41 f (y))B(x, y)L(x − y, f (x) − f (y))d yd x,
T2
T2
T2
∂x41 f (x)(∂x41 f (y) − ∂x41 f (x))C(x, y)L x3 (x − y, f (x) − f (y))d yd x,
K6 =
T2
T2
∂x41 f (x)(∂x41 f (y) − ∂x41 f (x))D(x, y)d yd x,
and B(x, y) =
3( f (x) − f (y))( f (x) − f (y) − ∇ f (y) · (x − y)) , [|x − y|2 + ( f (x) − f (y))2 ]5/2
C(x, y) = −
D(x, y) = −
∇ f (y) · (x − y) , [|x − y|2 + ( f (x) − f (y))2 ]3/2
L x1 (x − y, f (x) − f (y))(x1 − y1 ) + L x2 (x − y, f (x) − f (y))(x2 − y2 ) , [|x − y|2 + ( f (x) − f (y))2 ]3/2
L x1 (x1 , x2 , x3 ) = ∂x1 L(x1 , x2 , x3 ),
L x2 (x1 , x2 , x3 ) = ∂x2 L(x1 , x2 , x3 ).
Exchanging the variables x and y we can obtain K 3 ≤ 0. The terms K 4 , K 5 and K 6 can be estimated in a similar way as K 1 . Therefore, we obtain d 4 ∂ f 2 (t) ≤ C( f 5H 4 (t) + 1), dt x1 L and analogously d 4 ∂ f 2 (t) ≤ C( f 5H 4 (t) + 1). dt x2 L This proves local existence. The proof of the uniqueness is similar to the case = R2 . 4.3. 2-D case. Using Eq. (16) in R and Eq. (17) in the periodic case we obtain the following theorem Theorem 4.3. Let f 0 (x) ∈ H k for k ≥ 3 and ρ2 > ρ1 . Then there exists a time T > 0 so that there is a unique solution to (16) in C 1 ([0, T ]; H k ) with f (x, 0) = f 0 (x). The proof is similar to that of Theorems 4.1 and 4.2.
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5. Global Solution for the 2-D Stable Case In this section we obtain a family of global solutions for the 2-D stable case with a small initial data with respect to a fixed norm. Indeed, we can get the result with an initial data with the property f 0 H s = ∞ for s > 3/2. In this section we consider x ∈ R and f a = | fˆ(k)|ea|k| . For a > 0, if f a < ∞, then the function f can be extended analytically on the strip |z| < a. Furthermore f b , (30) ∂x f a ≤ C b−a for b > a. The main result of this section is Theorem 5.1. Let f 0 (x) be a function such that T f 0 (x) d x = 0, ∂x f 0 0 ≤ ε for ε small enough and ∂x2 f 0 b(t) ≤ εeb(t) (1 + |b(t)|γ −1 ), (31) with 0 < γ < 1, b(t) = a − (ρ2 − ρ1 )t/2, ρ2 > ρ1 and a ≤ (ρ2 − ρ1 )t/2. Then, there exists a unique solution of (16) with f (x, 0) = f 0 (x) and ρ2 > ρ1 satisfying ∂x f a (t) ≤ C(ε) exp((2σ a − (ρ2 − ρ1 )t)/4),
(32)
and ∂x2 f a (t) ≤ C(ε)(1 + |σ a − for a ≤
ρ2 −ρ1 2σ t,
ρ2 − ρ1 γ −1 t| ) exp((2σ a − (ρ2 − ρ1 )t)/4), 2
(33)
σ = 1 + δ and 0 < δ < 1.
The condition (31) can be satisfied for example if 1+γ f 0 0 < ε and fˆ0 (0) = fˆ0 (1) = fˆ0 (−1) = 0 since ∂x2 f 0 b(t) ≤ eb(t) 1+γ f 0 0 max |k|1−γ eb(t)(|k|−1) . k≥2
In order to prove the theorem, we use the Cauchy-Kowalewski method (see [15] and [16]) in a similar way as Caflisch and Orellana [7] and Siegel, Caflisch and Howison [18]. We show the proof with ρ2 − ρ1 = 2 without loss of generality. Let g(x, t) and h(x, t) be functions satisfying gt = − g, g(x, 0) = f 0 (x), h t = − h + T (g + h), h(x, 0) = 0, with T ( f ) = −π
−1
f (x)− f (x−α) 2 ∂x f (x) − ∂x f (x − α) α f (x)− f (x−α) 2 dα. α R 1+ α
(34)
Then the function f (x, t) = g(x, t) + h(x, t) is a solution of (16). First, we show some properties of the nonlinear operator T .
(35)
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Lemma 5.2. If ∂x f a , ∂x g a < 1 for a ≥ 0 then T ( f )(0) = 0,
(36)
∂x T ( f ) a ≤ C1 ∂x2 f a ∂x f a ,
(37)
and ∂x T ( f ) − ∂x T (g) a ≤ C2 ( ∂x2 f a + ∂x2 g a ) ∂x f − ∂x g a
(38)
+ C2 ( ∂x f a + ∂x g a ) ∂x2 f − ∂x2 g a , with C1 = 4(1 − ∂x f a2 )−2 and C2 = 4(1 − ∂x f a2 )−2 + (1 − ∂x g a2 )−2 .
Proof of the lemma. Due to the inequality |∂x f (x)| ≤ ∂x f a < 1 and by (35) we obtain T ( f ) = π −1
∂x f (x) − ∂x f (x − α) f (x) − f (x − α) 2n (−1)n dα, (39) α α R n≥1
and T ( f ) = π −1 ∂x
(−1)n f (x) − f (x − α) 2n+1 dα. 2n + 1 R α n≥1
Thus T ( f )(0) = 0. Using (39) T ( f )(k) = π −1
n≥1
=
(−1)
n
R k ,...,k 0 2n
δ(
2n
k j , k)ik0
j=0
2n j=0
−iαk j
1−e fˆ(k j ) α
dα
2n 2n (−1)n δ( k j , k) Mn (k0 , . . . , k2n ) ik0 fˆ(k j ), n≥1
k0 ,...,k2n
j=0
j=0
where Mn (k0 , . . . , k2n ) = π −1
2n 1 − e−iαk j dα. α R j=0
We get Mn (k0 , . . . , k2n ) = (−1)n m n (k0 , . . . , k2n )
2n j=1
kj,
(40)
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D. Córdoba, F. Gancedo
with m n (k0 , . . . , k2n ) = π −1
1
=π
0
1
0
1
=i
2n dα ds2n P V ex p iα (s j − 1)k j α R j=1
ds1 . . .
0
1
1
ds1 . . .
0
2n dα ds2n P V ex p − iαk0 +iα (s j − 1)k j α R
1
0
2n
1
ds1 . . .
0
− π −1
1 − e−iαk0 ex p iα (s j − 1)k j dα α R j=1
ds2n
0 −1
1
ds1 . . .
j=1
ds2n (sing A − sing B),
0
and A=
2n (s j − 1)k j ,
B = −k +
j=1
2n
sjkj.
j=1
It follows T ( f )(k) =
δ(
n≥1 k0 ,...,kn
2n
k j , k) m n (k0 , . . . , k2n )
j=0
2n
k j fˆ(k j ),
j=0
with |m n (k0 , . . . , k2n )| ≤ 2. We have
ea|k| |k||T ( f )(k)| ≤ 2
ea|k| |k|δ(
k n≥1 k0 ,...,kn
k
≤2
2n
k j , k)
2n
j=0
|k j || fˆ(k j )|
j=0
2n (2n + 1) ea|k0 | |k0 |2 | fˆ(k0 )| ea|k j | |k j || fˆ(k j )|, k0 ,...,kn
n≥1
j=1
and therefore ∂x T ( f ) a ≤ 2 ∂x2 f a
3 ∂x f a3 − ∂x f a4 (2n + 1) ∂x f a2n = 2 ∂x2 f a . (1 − ∂x f a2 )2 n≥1
We get (37) for ∂x f a < 1. In a similar way we obtain (38).
From (34) g can be expressed as follows: g(k, ˆ t) = e−|k|t fˆ0 (k), and by the hypothesis of the initial data we have ∂x g a (t) ≤ εea−t , ∂x2 g a (t)
≤ εe
a−t
(1 + (t − a)
(41) γ −1
),
(42)
Contour Dynamics of Incompressible 3-D Fluids in a Porous Medium
467
for t ≥ a. We will prove the existence of h by an induction argument on the iterative equation: ∂t h n+1 = − h n+1 + T (g + h n ), h n+1 (x, 0) = 0, h 0 = 0, or h n+1 (k, t) =
t
e−|k|(t−s) (T (g + h n ))(k, s)ds,
0
h 0 = 0. For h 1 we obtain the following estimates: t 1 ∂x h a (t) ≤ T (g) a+s−t (s)ds = 0
0
t−a
e−s (1 + s γ −1 )ds ≤
0
By (41) and (42) we have I2 ≤ C
t
t−a
= I1 + I2 .
t−a
t−a 0
≤ Cε2 ea−t
t
+ 0
Using (37), (41) and (42) we get t−a I1 ≤ ea−t es ∂x T (g) 0 (s)ds ≤ Cea−t
t−a
es ∂x2 g 0 (s) ∂x g 0 (s)ds
Cε2 (1 + 2γ ) a−t e . γ
∂x2 g a+s−t (s) ∂x g a+s−t (s)ds
≤ Cε2 e2(a−t) a(1 + (t − a)γ −1 ) ≤
2Cε2 a−t e , δ
due to the inequalities (aδ)γ −1 > (t − a)γ −1 and aea−t ≤ δ −1 for σ a < t. Then ∂x h 1 a (t) ≤
5Cε2 a−t e . δγ
Choosing b = a + s − t + t−a 2 we have t t t ∂x T (g) b (s) ∂x T (g) b ds ≤ 2 ∂x2 h 1 a (t) ≤ ∂x2 T (g) a+s−t (s)ds ≤ b − (a + s − t) t −a 0 0 0 t t−a 2 ≤ = I3 + I4 , + 0
t−a 2
where a−t t−a a−t t−a 2 2 2Ce 2 2Cε2 e 2 s 2 e ∂x g 0 (s) ∂x g 0 (s)ds ≤ e−s (1 + s γ −1 )ds I3 ≤ t −a 0 t −a 0 2Cε2 a−t e 2 (1 + (t − a)γ −1 ), ≤ γ
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D. Córdoba, F. Gancedo
and I4 ≤
2C t −a
t t−a 2
∂x2 g b (s) ∂x g b (s)ds ≤
2Cε2 a−t t − a γ −1 t a e (1 + ( ) )( + ) t −a 2 2 2
3Cε2 a−t e (1 + (t − a)γ −1 ). δ Therefore ≤
5Cε2 a−t e , δγ 5Cε2 a−t e 2 (1 + (t − a)γ −1 ). ∂x2 h 1 a (t) ≤ δγ ∂x h 1 a (t) ≤
(43) (44)
Define r n+1 = h n+1 − h n , Rn =
sup 0≤a<∞ σa < t
∂x r n a +
t−σ a ∂x2 r n a e 2 , 1 + (t − σ a)γ −1
t−σ a ∂x2 h n a e 2 . 1 + (t − σ a)γ −1
and Mn =
sup 0≤a<∞ σa < t
∂x h n a +
ε0 ε0 Take M1 = R1 ≤ 5Cε δγ ≤ 2 and suppose that M j , R j ≤ 2 for any j = 2, . . . , n, then t t−a t n+1 n n−1 ∂x r a ≤ ∂x T (g + h ) − ∂x T (g + h ) a+s−t (s)ds = + = I7 + I8 . 2
0
0
t−a
Using (38) we have t−a I7 ≤ Cea−t es ( ∂x r n 0 (s)( ∂x2 g + ∂x2 h n 0 (s) + ∂x2 g + ∂x2 h n−1 0 (s)) 0 t−a + Cea−t es ( ∂x2 r n 0 (s)( ∂x g + ∂x h n 0 (s) + ∂x g + ∂x h n−1 0 (s))ds 0 t−a σ a−t 2Cε0 Rn e 2 , ≤ 2Cε0 Rn ea−t (1 + s γ −1 )ds ≤ γ 0 and
I8 ≤ C
t
( ∂x r n a+s−t (s)( ∂x2 g + ∂x2 h n a+s−t (s) + ∂x2 g + ∂x2 h n−1 a+s−t (s))ds
t−a t
( ∂x2 r n a+s−t (s)( ∂x g + ∂x h n a+s−t (s) + ∂x g + ∂x h n−1 a+s−t (s))ds t eδs−σ (t−a) (1 + (σ (t − a) − δs)γ −1 )ds ≤ 2Cε0 Rn +C
t−a
2Cε0 Rn ≤ δ
t−a t−a
t−σ a
e−x (1 + x γ −1 )d x ≤
6Cε0 Rn eσ a−t . γδ
Contour Dynamics of Incompressible 3-D Fluids in a Porous Medium
469
We obtain for b = a + s − t + σ (t−a)−δs , 2σ t ∂x2 r n+1 a (t) ≤ ∂x2 (T (g + h n ) − T (g + h n−1 )) a+s−t (s)ds 0
+ h n ) − T (g + h n−1 ) b (s) ds b − (a + s − t) 0 t ∂x (T (g + h n ) − T (g + h n−1 ) b ≤ 2σ σ (t − a) − δs 0 t σσ+1 (t−a) ≤ = I9 + I10 . + ≤
t ∂
x (T (g
σ σ +1 (t−a)
0
We have σ (t − a) − δs > σ2σ+1 (t − a) for 0 ≤ s ≤ σσ+1 (t − a) and therefore we obtain σ (σ + 1)C σ +1 (t−a) b e ( ∂x r n 0 (s)( ∂x2 g + ∂x2 h n 0 (s) + ∂x2 g + ∂x2 h n−1 0 (s)) I9 ≤ t −a 0 σ (σ + 1)C σ +1 (t−a) b 2 n e ( ∂x r 0 (s)( ∂x g + ∂x h n 0 (s) + ∂x g + ∂x h n−1 0 (s))ds + t −a 0 σ (t−a) σ +1 a−t 4σ Cε0 2(σ + 1)Cε0 eb−s (1 + s γ −1 )ds ≤ ≤ Rn Rn e 2 (1 + (t − a)γ −1 ). t −a γ 0 Using (38) and the induction hypothesis we get t (1 + (s − σ b)γ −1 ) ds eσ b−s I10 ≤ 4σ Cε0 Rn σ σ (t − a) − δs σ +1 (t−a) t σ (t−a)−δs γ −1 ) δs−σ (t−a) 1 + ( 2 ds ≤ 4σ Cε0 Rn e 2 σ σ (t − a) − δs σ +1 (t−a) σ (t−a) σ +1 σ a−t 4σ Cε0 8σ Cε0 Rn Rn e 2 (1 + (σ a − t)γ −1 ). ≤ e−x (x −1 + x γ −2 )d x ≤ t−σ a δ δ(1 − γ ) 2 Due to the estimates for I7 , I8 , I9 and I10 we obtain Cσ ε0 Rn . δγ (γ − 1)
Rn+1 ≤
(45)
Choosing ε0 small enough we get Rn+1 ≤
1 1 ε0 Rn ≤ . . . ≤ n R1 ≤ n+1 , 2 2 2
and Mn+1 ≤
n+1
Rn+1 ≤ ε0 .
j=1
Therefore, we obtain the function h = lim h n satisfying n→∞
∂x h a (t) ≤
n
Rn e
σ a−t 2
≤ ε0 e
σ a−t 2
.
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D. Córdoba, F. Gancedo
Taking f (x, t) = g(x, t) + h(x, t), we get (32) for ρ2 − ρ1 = 2. In order to show the uniqueness, we write Eq. (13) for ρ2 − ρ1 = 2 in the following form: f t = − f + T ( f ), f (x, 0) = f 0 (x). Suppose that there exist two solutions f 1 and f 2 with f 1 (x, 0) = f 2 (x, 0). Define R by R=
sup 0≤a<∞ σa < t
It follows that R ≤ f1 = f2 .
∂ 2 f 1 − ∂x2 f 2 a t−σ a ∂x f 1 − ∂x f 2 a + x e 2 . 1 + (t − σ a)γ −1
C(ε)σ δγ (γ −1) R
and for ε small enough it yields
C(ε)σ δγ (γ −1)
< 1 and therefore
6. Ill-Posedness for the Unstable Case Here we show ill-posedness for the unstable case ρ1 > ρ2 . We use the global solution for the 2-D stable case f (x1 , t) satisfying (32) with 1+γ f 0 0 < C and 1+γ +ζ f 0 0 = ∞ for γ , ζ > 0. Making a change of variables, we define fλ (x1 , t) = λ−1 f (λx1 , −λt+λ1/2 ) obtaining { f λ }λ>0 a family of solutions to the unstable case. Using (32) follows 3
3
3
f λ H s (0) = |λ|s− 2 f H s (λ1/2 ) ≤ C|λ|s− 2 f 1 (λ1/2 ) ≤ C|λ|s− 2 e− and 3
3
f λ H s (λ−1/2 ) = |λ|s− 2 f H s (0) ≥ |λ|s− 2 C
|ρ2 −ρ1 | 1/2 λ 4
,
|k|1+γ +ζ | fˆ0 (k)| = ∞,
k
for s > 3/2 and γ , ζ small enough. We obtain an ill posed problem for s > 3/2. Theorem 6.1. Let s > 3/2, then for any ε > 0 there exists a solution f of (16) with ρ1 > ρ2 and 0 < δ < ε such that f H s (0) ≤ ε and f H s (δ) = ∞. Remark 6.2. If one considers a solution of the 3-D problem satisfying f (x1 , x2 , t) = f (x1 , t), from Eq. (13) one obtains a solution of (16). This shows that solutions of the 2-D case are solutions of the 3-D problem and therefore, using the above theorem, one obtains ill-posedness for the 3-D case with ρ1 > ρ2 . References 1. Ambrose, D.: Well-posedness of Two-phase Hele-Shaw Flow without Surface Tension. Euro. J. Appl. Math. 15, 597–607 (2004) 2. Ambrose, D., Masmoudi, N.: The zero surface tension limit of two-dimensional water waves. Commun. Pure Appl. Math. 58, 1287–1315 (2005) 3. Bear, J.: Dynamics of Fluids in Porous Media. New York: American Elsevier, 1972 4. Bertozzi, A.L., Constantin, P.: Global regularity for vortex patches. Commun. Math. Phys. 152(1), 19– 28 (1993) 5. Bertozzi, A.L., Majda, A.J.: Vorticity and the Mathematical Theory of Incompresible Fluid Flow. Cambridge: Cambridge Press, 2002
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6. Birkhoff, G.: Helmholtz and Taylor instability. In: Hydrodynamics Instability, Proc. Symp. Appl. Math. XII, Providence, RI: Amer. Math. Soc., 55–76, 1962, pp. 55–76 7. Caflisch, R., Orellana, O.: Singular solutions and ill-posedness for the evolution of vortex sheets. SIAM J. Math. Anal. 20(2), 293–307 (1989) 8. Chemin, J.Y.: Persistence of geometric structures in two-dimensional incompressible fluids. Ann. Sci. Ecole. Norm. Sup. 26(4), 517–542 (1993) 9. Constantin, P., Dupont, T.F., Goldstein, R.E., Kadanoff, L.P., Shelley, M.J., Zhou, S.M.: Droplet breakup in a model of the Hele-Shaw cell. Physical Review E 47, 4169–4181 (1993) 10. Constantin, P., Majda, A.J., Tabak, E.: Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994) 11. Escher, J., Simonett, G.: Classical solutions for Hele-Shaw models with surface tension. Adv. Differ. Eqs. 2, 619–642 (1997) 12. Hele-Shaw, H.S.: Nature 58, 34 (1898) 13. Hou, T.Y., Lowengrub, J.S., Shelley, M.J.: Removing the Stiffness from Interfacial Flows with Surface Tension. J. Comput. Phys. 114, 312–338 (1994) 14. Muskat, M.: The flow of homogeneous fluids through porous media. New York:Springer, 1982 15. Nirenberg, L.: An abstract form of the nonlinear Cauchy-Kowalewski theorem. J. Differ. Geom. 6, 561– 576 (1972) 16. Nishida, T.: A note on a theorem of Nirenberg. J. Differ. Geom. 12, 629–633 (1977) 17. Saffman, P.G., Taylor, G.: The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. London, Ser. A 245, 312–329 (1958) 18. Siegel, M., Caflisch, R., Howison, S.: Global Existence, Singular Solutions, and Ill-Posedness for the Muskat Problem. Comm. Pure and Appl. Math. 57, 1374–1411 (2004) 19. Rodrigo, J.L.: On the Evolution of Sharp Fronts for the Quasi-Geostrophic Equation. Comm. Pure and Appl. Math. 58, 0821–0866 (2005) 20. Stein, E., Weiss, G.: Introduction to Fourier Analysis on Euclidean spaces. Princeton, NJ: Princeton University Press, 1971 21. Stein, E.: Harmonic Analysis. Princeton, NJ: Princeton University Press, 1993 22. Taylor, G.: The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. Roy. Soc. London. Ser. A. 201, 192–196 (1950) 23. Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130, 39– 72 (1997) 24. Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Amer. Math. Soc. 12, 445–495 (1999) Communicated by P. Constantin
Commun. Math. Phys. 273, 473–498 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0189-3
Communications in
Mathematical Physics
One-and-a-Half Quantum de Finetti Theorems Matthias Christandl, Robert König, Graeme Mitchison, Renato Renner Centre for Quantum Computation, DAMTP, University of Cambridge, Cambridge CB3 0WA, UK. E-mail: [email protected]; [email protected]; [email protected]; [email protected] Received: 19 July 2006 / Accepted: 20 September 2006 Published online: 13 March 2007 – © Springer-Verlag 2007
Abstract: When n − k systems of an n-partite permutation-invariant state are traced out, the resulting state can be approximated by a convex combination of tensor product states. This is the quantum de Finetti theorem. In this paper, we show that an upper bound 2 on the trace distance of this approximation is given by 2 kdn , where d is the dimension of the individual system, thereby improving previously known bounds. Our result follows from a more general approximation theorem for representations of the unitary group. Consider a pure state that lies in the irreducible representation Uµ+ν ⊂ Uµ ⊗ Uν of the unitary group U(d), for highest weights µ, ν and µ + ν. Let ξµ be the state obtained by tracing out Uν . Then ξµ is close to a convex combination of the coherent states Uµ (g)|vµ , where g ∈ U(d) and |vµ is the highest weight vector in Uµ . For the class of symmetric Werner states, which are invariant under both the permutation and unitary groups, we give a second de Finetti-style theorem (our “half” theorem). It arises from a combinatorial formula for the distance of certain special symmetric Werner states to states of fixed spectrum, making a connection to the recently defined shifted Schur functions [1]. This formula also provides us with useful examples that allow us to conclude that finite quantum de Finetti theorems (unlike their classical counterparts) must depend on the dimension d. The last part of this paper analyses the structure of the set of symmetric Werner states and shows that the product states in this set do not form a polytope in general.
I. Introduction There is a famous theorem about classical probability distributions, the de Finetti theorem [2], whose quantum analogue has stirred up some interest recently. The original (k) theorem states that a symmetric probability distribution of k random variables, PX 1 ···X k , that is infinitely exchangeable, i.e. can be extended to an n-partite symmetric distribution for all n > k, can be written as a convex combination of identical product distributions,
474
M. Christandl, R. König, G. Mitchison, R. Renner
i.e. for all x1 , . . . , xk , PX 1 ···X k (x1 , . . . , xk ) =
PX (x1 ) · · · PX (xk )dµ(PX ),
(1)
where µ is a measure on the set of probability distributions, PX , of one variable. In the quantum analogue [3–8] a state ρ k on H⊗k is said to be infinitely exchangeable if it is symmetric (or permutation-invariant), i.e. πρ k π † = ρ k for all π ∈ Sk and, for all n > k, there is a symmetric state ρ n on H⊗n with ρ k = trn−k ρ n . The theorem then states that ρ k = σ ⊗k dm(σ ) (2) for a measure m on the set of states on H. However, the versions of this theorem that have the greatest promise for applications relax the strong assumption of infinite exchangeability [9, 10]. For instance, one can assume that ρ k is n-exchangeable for some specific n > k, viz. that ρ k = trn−k ρ n for some symmetric state ρ n . In that case, the exact statement in Eq. (2) is replaced by an approximation ρ k ≈ σ ⊗k dm(σ ), (3) as proved in [9], where it was shown that the error is bounded by an expression propor6 tional to √kd . n−k Our paper is structured as follows. In Sect. II we derive an approximation theorem for states in spaces of irreducible representations of the unitary group. Our main application of this theorem is an improvement of the error bound in the approximation in (3) to kd 2 2 kd n for Bose-symmetric states and to 2 n for arbitrary permutation-invariant states. The last step from Bose-symmetry to permutation-invariance is achieved by embedding permutation-invariant states into the symmetric subspace, a technique which might be of independent interest. We conclude this section with a discussion of the optimality of our bounds and explain how our results can be generalised to permutation-symmetry with respect to an additional system. In Sect. III, we prove the “half” theorem of our title. This refers to a de Finetti theorem for a particular class of states, the symmetric Werner states [11], which are invariant under the action on the tensor product space of both the unitary and symmetric groups. In order to prove our result we derive an exact combinatorial expression for the distance of extremal n-exchangeable Werner states to product states of fixed spectrum. This has some mathematical interest because of the connection it makes with shifted Schur functions [1]. It also provides us with a rich supply of examples that can be used to test the tightness of the bounds of the error in Eq. (3) and, in Sect. IV, to explore the structure of the set of convex combinations of tensor product states. II. On Coherent States and the de Finetti theorem A. Approximation by coherent states. In order to state our result we need to introduce some notation from Lie group theory [12]. Let U(d) be the unitary group and fix a basis d−1 {|i}i=0 of Cd in order to distinguish the diagonal matrices with respect to this basis as the Cartan subgroup H(d) of U(d). A weight vector with weight λ = (λ1 , . . . , λd ),
One-and-a-Half Quantum de Finetti Theorems
475
where each λi is an integer, is a vector |v in the representation U of U(d) satisfying U (h)|v = h iλi |v, where h 1 , . . . , h d are the diagonal entries of h ∈ H(d). We can equip the set of weights with an ordering: λ is said to be (lexicographically) higher than λ if λi > λi for the smallest i with λi = λi . It is a fundamental fact of representation theory that every irreducible representation of U(d) has a unique highest weight vector (up to scaling); the corresponding weights must be dominant, i.e. λi ≥ λi+1 . Two irreducible representations are equivalent if and only if they have identical highest weights. It is therefore convenient to label irreducible representations by their highest weights and write Uλ for the irreducible representation of U(d) with highest weight λ. It will also be convenient to choose the normalisation of the highest weight vector |vλ to be
vλ |vλ = 1 in order to be able to view |vλ vλ | as a quantum state. Given two irreducible representations Uµ and Uν with corresponding spaces Uµ and Uν we can define the tensor product representation Uµ ⊗ Uν acting on Uµ ⊗ Uν by (Uµ ⊗ Uν )(g) = Uµ (g) ⊗ Uν (g), for any g ∈ U(d). In general this representation is reducible and decomposes as λ Uµ ⊗ Uν ∼ cµν Uλ . = λ
λ are known as Littlewood-Richardson coefficients. It follows from The multiplicities cµν the definition of the tensor product that |vµ ⊗ |vν is a vector of weight µ + ν, where (µ + ν)i = µi + νi . By the ordering of the weights, µ + ν is the highest weight in Uµ ⊗Uν and |vµ ⊗ |vν is the only vector with this weight. We therefore identify |vµ+ν with |vµ ⊗ |vν and remark that Uµ+ν appears exactly once in Uµ ⊗ Uν . Our first result is an approximation theorem for states in the spaces of irreducible representations of U(d). Consider a normalised vector | in the space Uµ+ν of the irreducible representation Uµ+ν . By the above discussion we can embed Uµ+ν uniquely into the tensor product representation Uµ ⊗ Uν . This allows us to define the reduced state of | on Uµ by ξµ = trν | |. We shall prove that the reduced state on Uµ is approximated by convex combinations of rotated highest weight states:
rotated highest weight Definition II.1. For g ∈ U(d), let |vµ := Uµ (g)|v µ be the g g vector in Uµ . Let Pµ (Cd ) be the set of states of the form |vµ vµ |dm(g), where m is a probability measure on U(d). g
g
Here, the states |vµ , with g ∈ U(d), are coherent states in the sense of [13]. For d = 2 and µ = (k, 0) ≡ (k), these states are the well-known SU(2)-coherent states. In the following theorem, we use the trace distance, which is induced by the trace norm A := 21 tr|A| on the set of hermitian operators. Theorem II.2 (Approximation by coherent states). Let | be in Uµ+ν which we consider to be embedded into Uµ ⊗ Uν as described above. Then ξµ = trν | | is ε-close to Pµ (Cd ), where ε := 2(1 −
dim Uνd d ). That is, there exists a probability measure m dim Uµ+ν
U(d) such that
ξµ −
|vµg vµg |dm(g) ≤ ε .
on
476
M. Christandl, R. König, G. Mitchison, R. Renner g
g
g
g
Proof. By the definition of |vτ and Schur’s lemma, the operators E τ := dim Uτ |vτ vτ |, g ∈ U(d) together with the normalised uniform (Haar) measure dg on U (d) form a POVM on Uτ , i.e., (4) E τg dg = 11Uτ . This allows us to write
ξµ =
wg ξµg dg ,
g
(5) g
where ξµ is the residual state on Uµ obtained when applying {E ν } to |, i.e., wg ξµg = trν ((11Uµ ⊗ E νg )| |) , where wg dg determines the probability of outcomes. g We claim that ξµ is close to a convex combination of the states |vµ , with coefficients g corresponding to the outcome probabilities when measuring | with {E µ+ν }. That is, we show that the probability measure m on U(d) in the statement of the theorem can be g defined as dm(g) := tr(E µ+ν | |)dg. Our goal is thus to estimate g
ξµ − tr(E µ+ν | |)|vµg vµg |dg = S − δ , where, using (5),
dim Uν g tr(E µ+ν | |)|vµg vµg |dg , dim Uµ+ν dim Uν g δ := 1 − tr(E µ+ν | |)|vµg vµg |dg . dim Uµ+ν
S :=
wg ξµg −
dim Uν 3 Because δ = 21 (1 − dim Uµ+ν ), it suffices to show that S ≤ 2 (1 − Uµ+ν ⊂ Uµ ⊗ Uν and |vµ ⊗ |vν = |vµ+ν , we have
dim Uν dim Uµ+ν ).
Since
dim Uν g tr(E µ+ν | |) = vµg |trν ((11µ ⊗ E νg )| |)|vµg dim Uµ+ν = wg vµg |ξµg |vµg . So
S=
wg ξµg − |vµg vµg |ξµg |vµg vµg | dg
Now, for all operators A, B, we have A − B AB = (A − B A) + (A − AB) − (11 − B)A(11 − B) , g
g
g
so putting A = ξµ and B = |vµ vµ | in (6), we have S = α + β − γ,
(6)
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477
where α := β :=
γ :=
wg (ξµg − |vµg vµg |ξµg )dg, wg (ξµg − ξµg |vµg vµg |)dg, wg (11Uµ − |vµg vµg |)ξµg (11Uµ − |vµg vµg |)dg.
g g g g g g Combining wg |vµ vµ |ξµ = trν (|vµ vµ | ⊗ E ν )| | = with (4) and (5), we get dim Uν trν | | . α = 1− dim Uµ+ν
g dim Uν dim Uµ+ν tr ν E µ+ν | |
Similarly, dim Uν trν | | , β = 1− dim Uµ+ν and hence
α = β =
1 dim Uν . 1− 2 dim Uµ+ν
Note that for a projector P and a state ξ on H, we have 1 tr(Pξ ) = Pξ P , 2 as√a consequence of the cyclicity of the trace and the fact that the operator Pξ P = √ ( ξ P)† ( ξ P) is nonnegative. This identity together with the convexity of the trace g g distance applied to the projectors 11Uµ − |vµ vµ | gives
γ ≤
wg (11Uµ − |vµg vµg |)ξµg (11Uµ − |vµg vµg |) dg 1 = tr wg (ξµg − |vµg vµg |ξµg )dg 2 = α .
This concludes the proof because g
ξµ − tr(E µ+ν | |)|vµg vµg |dg ≤ S + δ ≤ α + β + γ + δ , and each of the quantities in the sum on the r.h.s. is upper bounded by 21 (1 −
dim Uν dim Uµ+ν ).
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An important special case of Theorem II.2 is the case where µ = (k) ≡ (k, 0, . . . , 0) d−1
and µ+ν = (n). In this case, U(k) ∼ = Symk (Cd ) (and likewise for U(n) ) is the symmetric d ⊗k subspace of (C ) . Its importance stems from the fact that any n-exchangeable density operator has a symmetric purification, and this leads to a new de Finetti theorem for general mixed symmetric states (cf. Sect. IIB). Corollary II.3. Let | ∈ Symn (Cd ) be a symmetric state and let ξ k := trn−k | |, k ≤ n, be the state obtained by tracing out n − k systems. Then ξ k is ε-close to P(k) (Cd ), where ε := 2 dk n . Equivalently, there exists a probability measure m on pure states on d C such that k
ξ − |ϕ ϕ|⊗k dm(ϕ) ≤ ε . Proof. Put µ = (k), ν = (n − k) in Theorem II.2. Then | ∈ U(n) = Symn (Cd ) is a symmetric state, the highest weight vector of Uµ is just the product |0⊗k , and tracing out Uν corresponds to tracing out (Cd )⊗n−k . Since Uµ (g)|v µ = (g|0)⊗k , an arbitrary state |ϕ ∈ Cd can be written as g|0 for some g ∈ U(d). , so the error in the theorem For the symmetric representation U(l) , dim U(l) = l+d−1 l (n−k+d−1 n−k ) is ε := 2(1 − n+d−1 ), and ( n ) n−k+d−1
(n − k + d − 1)! n!(d − 1)! (n − k)!(d − 1)! (n + d − 1)! n−k+d −1 n−k+1 ··· = n+1 n+d −1 d−1 n−k+1 ≥ n+1 d−1 k = 1− n+1 (d − 1)k ≥1− n+1 dk . ≥1− n
n−k n+d−1 = n
n+ j n+i The first inequality here follows from n+k+i ≤ n+k+ j , which holds for all i ≤ j, and the second to last inequality is also known as the ‘union bound’ in probability theory.
Example II.4. To get some feel for the more general case, where Uµ+ν is not the symmetric representation, let 1 ≤ p ≤ d and consider µ = ( j p ) ≡ ( j, . . . , j), ν = ((m − j) p ) p
and µ + ν = (m p ). We can consider the representation Uµ+ν given by the Weyl tensorial construction [14], with the tableau numbering running from 1 to p down the first column, p + 1 to 2 p down the second, and so on. Then the embedding Uµ+ν ⊂ Uµ ⊗ Uν
One-and-a-Half Quantum de Finetti Theorems
479
corresponds to the factoring of tensors in (Cd )⊗n = (Cd )⊗k ⊗ (Cd )⊗n−k , where k = j p and n = mp. The fact that the Young projector is obtained by symmetrising over rows and antisymmetrising over columns implies that
p Uµ+ν ⊂ Symm ( (Cd )), where p is the antisymmetric subspace on p systems corresponding to a column in the diagram. States in Uµ+ν can thus be regarded as symmetric states of m systems of dimension q = dim p (Cd ), and one can apply Corollary II.3 to deduce that ξµ is close to P( j) (Cq ). However, Theorem II.2 makes the assertion that ξµ is close to Pµ (Cd ). This statement is stronger in certain cases. For instance, when p = 2, the highest weight vector |vµ is ( √1 |01−10)⊗k and The2
orem II.2 says that ξµ is close to a convex combination of states |ϕ ϕ|⊗k/2 , where |ϕ is of the form (g ⊗ g) √1 |01 − 10 with g ∈ U(d). Note that the single-system reduced den2 sity operator of every such |ϕ has rank 2. By contrast, Corollary II.3 allows the |ϕ’s to lie in 2 (Cd ), i.e. in the span of the basis elements √1 |i 1 i 2 − i 2 i 1 , for 1 ≤ i 1 < i 2 ≤ d. 2 This includes |ϕ’s whose reduced density operator has rank larger than 2, if d > 3. B. Symmetry and purification. We now show how the symmetric-state version of our de Finetti theorem, Corollary II.3, can be generalised to prove a de Finetti theorem for arbitrary (not necessarily pure) n-exchangeable states ρ k on H⊗k . We say a (mixed) state ξ n on H⊗n is permutation-invariant or symmetric if π ξ n π † = ξ n , for any permutation π ∈ Sn . Here, the symmetric group Sn acts on H⊗n by permuting the n subsystems, i.e. every permutation π ∈ Sn gives a unitary π on H⊗n defined by π |ei1 ⊗ · · · ⊗ |ein = |eiπ −1 (1) ⊗ · · · ⊗ |eiπ −1 (n)
(7)
d of H. Note that, as a unitary operator, π † corresponds for an orthonormal basis {|ei }i=1 to the action of π −1 ∈ Sn .
Lemma II.5. Let ξ be a permutation-invariant state on H⊗n . Then there exists a purification of ξ in Symn (K ⊗ H) with K ∼ = H. Proof. Let A be the set of eigenvalues of ξ and let Ha , for a ∈ A, be the eigenspace of ξ , so ξ |φ = a|φ, for any |φ ∈ Ha . Because ξ is invariant under permutations, we have π † ξ π |φ = a|φ, for any |φ ∈ Ha and π ∈ Sn . Applying the unitary operation π to both sides of this equality gives ξ π |φ = aπ |φ; so π |φ ∈ Ha . This proves that √ the eigenspaces Ha of ξ are√invariant under permutations. Since the eigenspaces of ξ are identical to those of ξ , ξ is invariant under permutations, too. We now show how this symmetry carries over to the vector |ξ := 11 ⊗ ξ |, d where | = ( i |ei ⊗ |ei )⊗n ∈ (K ⊗ H)⊗n for an orthonormal basis {|ei }i=1 of K∼ H. Observe that | is invariant under permutations, i.e. (π ⊗ π )| = |. Using = √ this fact and the permutation invariance of ξ we find
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(π ⊗ π )(11 ⊗
ξ )| = 11 ⊗ π ξ π † (π ⊗ π )| = (11 ⊗ ξ )|,
so |ξ is invariant under permutations, and hence an element of Symn (K ⊗ H). Computing the partial trace over K⊗n gives † trK⊗n (11 ⊗ ξ )| |(11 ⊗ ξ )† = ξ 11 ξ = ξ, which shows that |ξ is a symmetric purification of ξ .
Definition II.6. Let P k = P k (H) be the set of states of the form m is a probability measure on the set of (mixed) states on H.
σ ⊗k dm(σ ), where
Theorem II.7 (Approximation of symmetric states by product states). Let ξ n be a permutation-invariant density operator on (Cd )⊗n and k ≤ n. Then ξ k := trn−k (ξ n ) is 2 ε-close to P k (Cd ) for ε := 2 dn k . Proof. By Lemma II.5, there is a purification | ∈ Symn (Cd ⊗ Cd ) of ξ n , and the 2 partial trace trn−k | | is ε-close to P(k) (Cd ) by Corollary II.3. The claim then is a consequence of the fact that the trace-distance does not increase when systems are traced out. We close this section by looking at a stronger notion of symmetry than permutationinvariance. This is Bose-symmetry, defined by the condition that π ξ n = ξ n for every π ∈ Sn . Bose-exchangeability is then defined in the obvious way. In the course of their paper proving an infinite-exchangeability de Finetti theorem, Hudson and Moody [4] also showed that if ξ k is infinitely Bose-exchangeable, then ξ k is in P(k) (Cd ). We now show that this result holds (approximately) for Bose-n-exchangeable states. Theorem II.8 (Approximation of Bose symmetric states by pure product states). Let ξ n be a Bose-symmetric state on (Cd )⊗n , and let ξ k := trn−k (ξ n ), k ≤ n. Then ξ k is ε-close to P(k) (Cd ), for ε := 2 dk n . Proof. We can decompose ξ n as ξn =
ai |ψi ψi |,
i
where |ψi is a set of orthonormal eigenvectors of ξ n with strictly positive eigenvalues ai . For all π ∈ Sn we have π |ψi =
1 1 π ξ n |ψi = ξ n |ψi = |ψi , ai ai
making use of the assumption π ξ n = ξ n . This shows that all |ψi are elements of Symn (Cd ). By Corollary II.3, every ξψk i = trn−k |ψi ψi | is -close to a state σψk i that is in P(k) (Cd ). This leads to ai ξψk i − ai σψk i ≤ ai ξψk i − σψk i ≤ , i
and concludes the proof.
i
i
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481
C. Optimality. The error bound we obtain in Theorem II.7 is of size d6k
d2k n ,
which is
tighter than the √n−k bound obtained in [9]. Is there scope for further improvement? For classical probability distributions, Diaconis and Freedman [15] showed that, for n-exchangeable distributions, the error, measured by the trace distance, is bounded by k(k−1) k(k−1) min{ dk n , 2n }, where d is the alphabet size. This implies that there is a bound, 2n , that is independent of d. The following example shows that there cannot be an analogous dimension-independent bound for a quantum de Finetti theorem. Example II.9. Suppose n = d, and define a permutation-invariant state on (Cn )⊗n by 1 ξn = sign(π )sign(π )π |12 · · · n 12 · · · n|π † , n! π,π
n where {|i}i=1 is an orthonormal basis of Cn . n n (C ). Tracing out n − 2 systems gives the
ξ2 =
2 n(n − 1)
This is just the normalised projector onto projector onto 2 (Cn ), i.e. the state |i j − ji i j − ji|,
(8)
1≤i< j≤n
which has trace distance at least 1/2 from P 2 (Cn ), as will be shown by Corollary III.9 and Example IV.3. We must therefore expect our quantum de Finetti error bound to depend on d, as is 2 indeed the case for the error term kdn in Theorem II.7. By generalising this example, we d (1 − d12 ). will show in Lemma III.9 that the error term must be at least 2n This example shows that some aspects of the de Finetti theorem cannot be carried over from probability distributions to quantum states. The following argument shows that probability distributions can, however, be used to find lower bounds for the quantum case. Given an n-partite probability distribution PX = PX 1 ···X n on X n , define a state | := PX (x)|x1 ⊗ · · · ⊗ |xn ∈ H⊗n , x∈X n
where {|x}x∈X is an orthonormal basis of H. Applying the von Neumann measurement M defined by this basis to every system of ξ k := trn−k (| |) gives a measurement k outcome distributed according to M⊗k (ξ )⊗k= PX . If m is a normalised measure on the setof states on H, then measuring σ dm(σ ) gives a distribution of the form M⊗k ( σ ⊗k dm(σ )) = PXk dµ(PX ). Because the trace distance of the distributions obtained by applying the same measurement is a lower bound on the distance between two states, this implies that inf PX 1 ···X k − PXk dµ(PX ) ≤ ξ k − σ ⊗k dm(σ ) , (9) µ
where the infimum is over all normalised measures µ on the set of probability distributions on X . If PX is permutation-invariant, that is, if PX (x1 , . . . , xn ) = PX (xπ −1 (1) , . . . , xπ −1 (n) ) for all (x1 , . . . , xn ) ∈ X n and π ∈ Sn , then | ∈ Symn (H). Applying this to a distribution PX studied by Diaconis and Freedman [15], and using their lower bound on the quantity on the l.h.s. of (9) gives the following result.
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M. Christandl, R. König, G. Mitchison, R. Renner
Theorem II.10. There is a state | ∈ Symn (C2 ) such that the distance of ξ k = trn−k | | to P k is lower bounded by 1 k k · + o( ) if n → ∞ and k = o(n), √ n 2π e n φ(α) + o(1) if n → ∞ and k/n → α ∈]0, 1/2[, where φ(α) :=
√1 2 2π
1
|1 − (1 − α) 2 eαu
2 /2
|e−u
2 /2
du.
For a fixed dimension and up to a multiplicative factor, the dependence on k and n in Corollary II.3 and Theorem II.7 is therefore tight. D. De Finetti representations relative to an additional system. A state ξ An on H A ⊗H⊗n is called permutation-invariant or symmetric relative to H A if (11 A ⊗ π )ξ An (11 A ⊗ π † ) = ξ An , for any permutation π ∈ Sn (see [16, 17, 9]). This property is strictly stronger than symmetry of the partial state ξ n := tr A (ξ An ), since symmetry of ξ n does not necessarily imply symmetry of ξ An relative to H A , as the pure state √1 (|001 + |110) ∈ C2 ⊗ (C2 )⊗2 illustrates. Taking a broader view where ξ n is part of 2 a state on a larger Hilbert space thus gives rise to additional structure. As we shall see, this stronger notion of symmetry also yields stronger de Finetti style statements. These are useful in applications, for instance those related to separability problems (cf. [18] and [19], where an alternative extended de Finetti-type theorem has been proposed). More precisely, symmetry of a state ξ An on H A ⊗ H⊗n relative to H A implies that the partial state ξ Ak := trn−k (ξ An ) is close to a convex combination of states where the part on H⊗k has product form and, in addition, is independent of the part on H A . In particular, ξ Ak is close to being separable with respect to the bipartition H A versus H⊗n . This property is formalised by the following definition which generalises Definition II.6. Definition II.6 . Let P k (H A , H) be the set of states of the form ξσA ⊗ σ ⊗k dm(σ ), where, m is a probability measure on the set of (mixed) states on H and where {ξσA }σ is a family of states on H A parameterised by states on H. The main results of Sect. IIB can be extended as follows. Theorem II.7 (Approximation of symmetric states by product states). Let ξ An be a density operator on H A ⊗ (Cd )⊗n which is symmetric relative to H A and let k ≤ n. 2 Then ξ Ak := trn−k (ξ An ) is ε-close to P k (H A , Cd ) for ε := 2 dn k . ξ n,
A state ξ An on H A ⊗H⊗n is called Bose-symmetric relative to H A if (11 A ⊗ π )ξ An = for any π ∈ Sn .
Theorem II.8 (Approximation of Bose symmetric states by product states). Let ξ An be a state on H A ⊗ (Cd )⊗n which is Bose-symmetric relative to H A , and let ξ Ak := trn−k (ξ An ), k ≤ n. Then ξ Ak is ε-close to P k (H A ⊗ Cd ), for ε := 2 dk n . The proofs of these theorems are obtained by a simple modification of the arguments used for the derivation of the corresponding statements of Sect. IIB. The main ingredient are straightforward generalisations of Theorem II.2 and Lemma II.5.
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483
Theorem II.2 (Approximation by coherent states). Let | be in H A ⊗ Uµ+ν and define ξµ := trν | |. Then there exists a probability measure m on U(d) and a family {τg }g∈U(d) of states on H A such that dim Uνd g g
ξµ − τg ⊗ |vµ vµ |dm(g) ≤ 2 1 − . d dim Uµ+ν Lemma II.5 . Let ξ be a state on H A ⊗ H⊗n which is permutation-invariant relative to H A . Then there exists a purification of ξ in H A ⊗ K A ⊗ Symn (K ⊗ H) with H A ∼ = KA and K ∼ = H. III. On Werner States and the de Finetti theorem A. Symmetric Werner states. We now consider a more restricted class of states, the Werner states [11]. Their defining property is that they are invariant under the action of the unitary group given by Eq. (11). Werner states are an interesting class of states because they exhibit many types of phenomena, for example different kinds of entanglement, but have a simple structure that makes them easy to analyse. One reason for narrowing our focus to these special states is that a de Finetti theorem can be proved for them using entirely different methods from the proof of Theorem II.2. We also obtain a rich supply of examples that give insight into the structure of exchangeable states and provide us with an O( dn ) lower bound for Theorem II.7. Schur-Weyl duality gives a decomposition (Cd )⊗k ∼ Uλd ⊗ Vλ , (10) = λ∈Par(k,d)
with respect to the action of the symmetric group Sk given by (7) and the action of the unitary group U(d) on (Cd )⊗k given by g|ψ = g ⊗k |ψ,
(11)
for g ∈ U(d) and |ψ ∈ (Cd )⊗k . Here Par(k, d) denotes the set of Young diagrams with k boxes and at most d rows, Uλd is the irreducible representation of U(d) with highest weight λ, and Vλ is the corresponding irreducible representation of Sk . Let ρ k be a symmetric Werner state on (Cd )⊗k . Schur’s lemma tells us that ρ k must be proportional to the identity on each irreducible component Uλd ⊗ Vλ , so ρk = wλ ρλk , (12) λ
where ρλk =Pλ /(dim Uλd dim Vλ ), with Pλ the projector onto Uλd ⊗ Vλ , and wλ ≥ 0 for all λ, with wλ = 1. Let Tk (ρ k ) denote the state obtained by “twirling” a state ρ k on (Cd )⊗k , i.e., k k T (ρ ) := g ⊗k ρ k (g ⊗k )† dg, where the Haar measure on U(d) with normalisation dg = 1 is used. A state of the form Tk (σ ⊗k ) is a symmetric Werner state since its product structure ensures symmetry
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and twirling makes it invariant under unitary action. We call such a state a “twirled product state”. Any two states with the same spectra are equivalent under twirling, so σ → Tk (σ ⊗k ) defines a map fk : Specd → W k , where Specd is the set of possible d-dimensional spectra and W k the set of symmetric Werner states on (Cd )⊗k . The map fk can be characterised as follows: Lemma III.1. Given r = (r1 , . . . , rd ) ∈ Specd , the twirled product state fk (r ) on (Cd )⊗k satisfies fk (r ) =
λ∈Par(k,d)
wλ (r )ρλk ,
where wλ (r ) = dim Vλ sλ (r ) and sλ (r ) is the Schur function (cf. Eq. (16)). Proof. Since fk (r ) is a symmetric Werner state, Eq. (12) shows that it has the required form and it remains to compute the coefficients wλ (r ). Since the states ρλk are supported on orthogonal subspaces, wλ (r ) = tr Pλ fk (r ) , where Pλ is the projector onto the component Uλd ⊗ Vλ of the Schur-Weyl decomposition of (Cd )⊗k . Let σ = diag(r ) be a state with spectrum r . By the linearity and cyclicity of the trace, tr(PTk (Q)) = tr(Tk (P)Q)
(13)
for all operators P and Q on (Cd )⊗k , hence we obtain wλ (r ) = tr Pλ Tk (σ ⊗k ) = tr Tk (Pλ )σ ⊗k = tr Pλ σ ⊗k . In the last step, we used the fact that Pλ is invariant under the action (11). Note that Pλ projects onto the isotypic subspace of the irreducible representation Uλd in the k-fold tensor product representation of U(d). On the one hand, this shows that tr Pλ σ ⊗k is the character of the representation σ˜ → Pλ σ˜ ⊗k Pλ , evaluated at σ˜ = σ . On the other hand this representation is equivalent to dim Vλ copies of Uλd , whose character equals sλ (r ). Hence, wλ (r ) = dim Vλ sλ (r ). B. A combinatorial formula. We know from Eq. (12) that the states ρλn with λ ∈ Par(n, d) are the extreme points of the set of symmetric Werner states. A de Finetti theorem for the n-exchangeable states
One-and-a-Half Quantum de Finetti Theorems
trn−k ρλn ,
485
for λ ∈ Par(n, d) ,
(14)
therefore implies a de Finetti theorem for arbitrary n-exchangeable Werner states by the convexity of the trace distance. Note further that a de Finetti-type statement about all states of the form (14) applies to general n-exchangeable Werner states, that is, to states ρ k ∈ W k such that there is some symmetric state τ n on (Cd )⊗n with ρ k = trn−k τ n . This is because we can assume that τ n is a Werner state as ρ k = trn−k Tn (τ n ) and Tn (τ n ) ∈ W n . Our main step in the derivation of a de Finetti theorem for symmetric Werner states is a combinatorial formula for the distance of trn−k ρλn and the symmetric Werner state fk (r ). Note that for every r ∈ Specd , the state fk (r ) is a convex combination of k-fold product states with spectrum r , since (15) fk (r ) = (g diag(r ) g † )⊗k dg . In order to present our formula for trn−k ρλn − fk (r ) , we need to introduce the well-known Schur functions and also the more recently defined shifted Schur functions. We first recall the combinatorial description of the Schur function sµ by sµ (λ1 , . . . , λd ) = λT (α) , (16) T α∈µ
where the sum is over all semi-standard tableaux T of shape µ with entries between 1 and d. A semi-standard (Young) tableau of shape µ is a Young frame filled with numbers weakly increasing to the right and strictly increasing downwards. The product is over all boxes α of µ and T (α) denotes the entry of box α in tableaux T . Note that sµ (λ) is homogeneous of degree k, where k is the number of boxes in µ. It is easy to see that the sum over semi-standard tableaux in (16) can be replaced by a sum over all reverse tableaux T of shape µ, where, in a reverse tableau, the entries decrease left to right along each row (weakly) and down each column (strictly). In the sequel, all the sums will be over reverse tableaux. The shifted Schur functions are given by the following combinatorial formula [1, Theorem (11.1)]: sµ∗ (λ1 , . . . , λd ) = (λT (α) − c(α)) , (17) T α∈µ
where c(α) is independent of T and is defined by c(α) = j − i if α = (i, j) is the box in the i th row and j th column of µ. Theorem III.2 (Distance to a twirled product state). Let λ ∈ Par(n, d) and r ∈ Specd . Let fk (r ) be the twirled product state defined in (15). The distance between the partial trace trn−k ρλn of the symmetric Werner state ρλn and fk (r ) is given by
trn−k ρλn − fk (r ) =
1 2
dim Vµ |
µ∈Par(k,d)
sµ (λ) (n k)
− sµ (r )| ,
where the falling factorial (n k) is defined to be n(n − 1) · · · (n − k + 1) if k > 0 and 1 if k = 0.
(18)
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M. Christandl, R. König, G. Mitchison, R. Renner
In order to prove the theorem we will need a number of lemmas. Our first step is to express the coefficients in trn−k ρλn in terms of Littlewood-Richardson coefficients. Lemma III.3. Let λ ∈ Par(n, d) and let Pλ be the projector onto Uλd ⊗ Vλ embedded in (Cd )⊗n . Then λ tr((Pµ ⊗ Pν )Pλ ) = cµν dim Uλd dim Vµ dim Vν λ is the Littlewood-Richardson for all µ ∈ Par(k, d) and ν ∈ Par(n − k, d), where cµν coefficient. λ is the multiplicity of the irreducible Proof. The Littlewood-Richardson coefficient cµν d representation Uλ in the decomposition of the tensor product representation Uµd ⊗ Uνd of U(d), i.e., λ Uµd ⊗ Uνd ∼ cµν Uλd . (19) = λ
This implies that the image of Pµ ⊗ Pν in (Cd )⊗n is isomorphic to ⎛ ⎞ λ cµν ⎜ ⎟ d Uλ,i ⊗ (Vµ,i ⊗ Vν,i )⎠ , ⎝ λ
i=1
as a representation of U(d) × Sn where, for each λ, the underbraced part consists of λ dim V dim V copies of U d and is contained in the component U d ⊗ V of the cµν µ ν λ λ λ Schur-Weyl decomposition of (Cd )⊗n . The conclusion follows from this. Lemma III.3 allows us to compute the partial trace of the projector Pλ . Lemma III.4. Let λ ∈ Par(n, d) and let Pλ be the projector onto Uλd ⊗ Vλ embedded in (Cd )⊗n . Then λ dim Vν trn−k Pλ = dim Uλd cµν Pµ , dim Uµd µν where the sum extends over all µ ∈ Par(k, d) and ν ∈ Par(n − k, d). Proof. Since trn−k Pλ is symmetric and invariant under the action of U(d), it has the form (cf. (12)) αµ Pµ . trn−k Pλ = µ
The claim then immediately follows from dim Uµd dim Vµ αµ = tr(Pµ trn−k Pλ ) = tr((Pµ ⊗ 11⊗n−k )Pλ ) = tr (Pµ ⊗ Pν )Pλ ν
and Lemma III.3.
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487
In the special case where n = k + 1 we obtain a statement that has recently been derived by Audenaert [20, Prop. 4]. We now show how the expression for tr n−k Pλ in Lemma III.4 can be rewritten in terms of shifted Schur functions. To do so we use the following result expressing dim λ/µ, the number of standard tableaux of shape λ/µ, in terms of shifted Schur functions. Theorem III.5 ([1, Theorem 8.1]). Let λ ∈ Par(n, d), µ ∈ Par(k, d) be such that µi ≤ λi for all i. Then sµ (λ) dim λ/µ . = dim Vλ (n k) Okounkov and Olshanski give a number of proofs for this theorem, the second of which only uses elementary representation theory. The shifted Schur functions allow us to express partial traces of Werner states in a form analogous to Lemma III.1. Lemma III.6. Let λ ∈ Par(n, d). The partial trace of the symmetric Werner state ρλn on (Cd )⊗n satisfies trn−k ρλn = αµλ ρµk , µ∈Par(k,d)
where αµλ = dim Vµ
sµ (λ) (n k)
.
Proof. Lemma III.4 gives αµλ = dim Vµ
ν∈Par(n−k,d)
λ cµν
dim Vν . dim Vλ
λ = 0 (by the Littlewood-Richardson rule) and s (λ) = 0 (by [1, TheoNote that cµν µ rem 3.1]) unless µi ≤ λi for all i. The claim therefore follows from Theorem III.5 and the identity (see [21, p. 67]) λ dim λ/µ = cµν dim Vν . ν∈Par(n−k,d)
We are now ready to give the proof of the combinatorial formula. Proof of Theorem III.2. This is an immediate consequence of Lemmas III.1 and III.6, since
trn−k ρλn − fk (r ) = αµλ ρµk − wµ (r )ρµk µ µ =
1 λ |α − wµ (r )| , 2 µ µ
where we used the fact that the support of the ρµk ’s is orthogonal.
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C. A de Finetti theorem for Werner states. The following de Finetti style theorem is a consequence of Theorem III.2. We call it “half a theorem” as it is a quantum de Finetti theorem for a restricted class of quantum states, the Werner states. Theorem III.7 (Approximation by twirled products). Let λ ∈ Par(n, d) and define ¯ be defined as in (15). Then the partial trace λ¯ := ( λn1 , . . . , λnd ) ∈ Specd . Let fk (λ) n trn−k ρλ of the symmetric Werner state ρλn satisfies k4 3 k(k − 1) n k ¯ ||trn−k ρλ − f (λ)|| ≤ · +O , 4 λ λ2 where λ is the smallest non-zero row of λ. The dimension d does not appear explicitly in this bound, nor in the order term O(·). Proof. First note that we can restrict the sum to diagrams µ with no more than rows, since by definition of , λq = 0 for q > , and sµ (λ1 , . . . , λ , 0, . . . , 0) = sµ (λ1 , . . . , λ , 0, . . . , 0) = 0 for µ+1 > 0. Furthermore, Schur as well as shifted Schur functions satisfy the stability condition [1] sµ (λ1 , . . . , λ , 0, . . . , 0) = sµ (λ1 , . . . , λ ), sµ (λ1 , . . . , λ , 0, . . . , 0) = sµ (λ1 , . . . , λ ) , so that we can safely assume that λ has (non-vanishing) rows and that the tableaux are numbered from 1 to only. Note that 4 1 k(k − 1) k −k 1+ , (20) =n +O (n k) 2n n2 and n −k sµ (λ) = =
T
α
T
β
λ¯ T (α) −
c(α) n
c(α) 1 c(α)c(α ) λ¯ T (β) 1 − + + ··· , λT (α) 2 λT (α) λT (α ) α α=α
where we have made use of (17) in the first line. Using (16), the bound |c(α)| ≤ k − 1 and the fact that α enumerates k boxes, we find the bounds 4 k(k − 1) k + O( 2 ) . |n −k sµ (λ) − sµ (λ¯ )| ≤ sµ (λ¯ ) λ λ Combining this with the estimate (20) we obtain k4 sµ (λ) 1 3 k(k − 1) −1 ≤ +O 2 , 2 (n k)sµ (λ¯ ) 4 λ λ
(21)
where we have used λ ≤ n. Since trn−k ρλn − fk (λ¯ ) is a convex combination with ¯ = dim Vµ sµ (λ) ¯ of the terms on the l.h.s. of (21), this concludes the weights tr(Pµ fk (λ)) proof.
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Example III.8. Three special cases may be noted: • Fix λ¯ and consider λ = n λ¯ for an integer n. The bound then turns into O
k2 n
just as in the classical case. Thus when one restricts attention to a particular diagram ¯ one obtains the same type of dimension-independent bound as Diaconis and shape λ, Freedman [15]. (This does not contradict Example II.9 where we focus on a single diagram with 1. The bound of Theorem III.7 gives no information here.) √ λ = √ • For λ = ( n, . . . , n) we have an error of order O
k2 √ n
.
• Finally, λ = (n): In this case, trn−k ρλn = fk (1, 0, . . . , 0) which means that trn−k ρλn has a product form and an application of Theorem III.7 is not needed. Note that in Theorem III.7 we only kept the dependence on the last nonzero row λ of λ. For specific applications (or for cases such as λ = (λ1 , . . . , λ−1 , 1)) one may want to derive bounds that depend on more details of λ. By the (infinite) quantum de Finetti theorem, convex combinations of tensor product states are the same thing as infinitely exchangeable states. In this light, a finite de Finetti theorem says how close n-exchangeable states are to ∞-exchangeable states, and one can generalise the notion of a de Finetti theorem, and ask How well can n-exchangeable states be approximated by m-exchangeable states, where m ≥ n? In the realm of symmetric Werner states, this amounts to bounding the distance
trn−k ρnnλ¯ − trm−k ρmmλ¯ , which is 1 2
µ∈Par(k,d)
dim Vµ
¯ sµ (n λ) (n k)
−
¯ sµ (m λ) (m k)
.
A straightforward calculation very similar to the proof of Theorem III.7 leads to an interpolation between the trivial case where m equals n and the case where m → ∞ which we have considered in Theorem III.7. D. Necessity of d-dependence. We end this section with a lower bound, which is a direct corollary to Theorem III.2. Corollary III.9. Let k < d and let λ = (m d ) be the diagram consisting of d rows of d length m. Then the distance of trn−k ρλn to P k is lower bounded by 2(n−1) (1 − d12 ), where n = md. Note that this bound can be seen as a generalisation of Example II.9, where we set d = n. It implies that any quantum de Finetti theorem can only give an interesting statement if d is small compared to n.
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¯ and sµ (λ) take a particularly simple form for Proof. Note first that the functions sµ (λ) the diagram λ under consideration. From Eq. (16) sµ (λ¯ ) = d −k dim Uµk ,
(22)
since dim Uµk is equal to the number of semi-standard tableaux T of shape µ, and from Eq. (17), c(α)d −k −k k n sµ (λ) = d dim Uµ 1− . (23) n α Because the trace distance does not increase when tracing out systems, and trk−2 τ k ∈ P 2 for every τ k ∈ P k , we can bound the distance of trn−k ρλn to P k as follows: min trn−k ρλn − τ k ≥ min trn−2 ρλn − τ 2 . τ 2 ∈P 2
τ k ∈P k
Let µ = (12 ). We show below that max sµ (r ) = sµ (λ¯ ) ,
(24)
r
where the maximisation ranges over all spectra. With dim Vµ = 1, this gives for every τ 2 ∈ P 2,
trn−2 ρλn − τ 2 ≥ tr(Pµ (trn−2 ρλn − τ 2 ))
≥ tr(Pµ trn−2 ρλn ) − max tr(Pµ σ ⊗2 ) ≥
sµ (λ) (n 2)
σ
− sµ (λ¯ ) ,
by Lemma III.6 and Lemma III.1. Equation (23) implies d . n −2 sµ (λ) = d −2 dim Uµ2 1 + n We thus obtain sµ (λ) (n 2)
¯ =d − sµ (λ)
−2
dim Uµ2
= dim Uµ2
1 1−
1 n
(25)
d 1+ −1 n
d +1 (n − 1)d 2
(26)
by (22) and (25). The claim then immediately follows from dim Uµ2 = d2 . It remains to prove (24). According to definition (16), for µ = (12 ), sµ (r1 , . . . , rd ) = ri 1 ri 2 , i 1
where the sum is over all indices i 1 , i 2 ∈ {1, . . . , d}. We claim that r1 + r2 r1 + r2 sµ (r1 , . . . , rd ) ≤ sµ , , r3 , . . . , rd . 2 2
(27)
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This follows from the fact that we can write sµ (r ) = r1r2 + (r1 + r2 )
ri +
i≥3
ri 1 ri 2
3≤i 1
and the inequality √ r1 + r2 r1 r2 ≤ 2 relating the geometric and the arithmetic mean of r1 , r2 . Inequality (27) and the symmetry of sµ imply (24). IV. The structure of P k for Werner states We focus now on the set Tk (P k ) = P k ∩ W k of Werner states that are convex combinations of product states. Theorem III.2 approximates elements of W k by elements of P k ∩ W k . We can ask whether it is possible for a symmetric Werner state to be closer to P k than to the set P k ∩ W k . The negative answer is given by the following lemma. Lemma IV.1. The closest state τ k ∈ P k to a symmetric Werner state ρ k ∈ W k is itself a Werner state, i.e., an element of P k ∩ W k . Proof. Suppose τ k ∈ P k is the nearest (not necessarily Werner) product state, so
ρ k − τ k is minimal. Then, using the convexity of the distance
ρ k − Tk (τ k ) = Tk (ρ k − τ k ) ≤ g ⊗k (ρ k − τ k )(g † )⊗k dg = ρ k − τ k , so the Werner state Tk (τ k ) is at least as close to ρ k as τ k (and in fact the triangle inequality is strict unless τ is U(d)-invariant). This means that the closest state τ k is an element of P k ∩ W k . A symmetric Werner state ρλk , for λ ∈ Par(k, d), has the following optimality property: Lemma IV.2. Let λ ∈ Par(k, d). The state ρλk ∈ W k is closer to P k than any other state ρ k with support on Uλd ⊗ Vλ . Proof. Let ρ k be a state with support on Uλd ⊗ Vλ , and let τ k ∈ P k be the state that is closest to ρ k . By Schur’s Lemma, ρλk = Tk ((ρ k )), where (ρ k ) := k!1 π ∈Sk πρ k π † . Thus, using the triangle inequality and the unitary invariance of the trace norm,
ρλk − Tk (τ k ) = Tk ((ρ k )) − Tk ((τ k )) ≤ ρ k − τ k .
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Fig. 1. This shows schematically the maps underlying Theorem III.7. From the Young diagram λ on n systems, one can go to W k by taking the state ρλn and tracing out n − k systems, or one can go to (d) by normalising the row lengths of λ. From the latter space, the map fk takes one to W k , and the two routes approximately end up at the same point
The set Tk (P k ) is the convex hull of all twirled tensor products Tk (σ ⊗k ), which is the convex hull of fk (Specd ). Since fk (πr ) = fk (r ), for any permutation π of r1 , .., rd , we can restrict fk to the simplex (d) = Specd /Sd . The vertices of (d) are the points x q ∈ Specd whose first q coordinates are 1/q and the remainder zero, for q = 1, . . . ., d. Thus fk (x 1 ) is just the twirl of |0 0|⊗k , which is the projector onto Symk (H), and fk (x d ) = (11/d)⊗k is the fully mixed state. The set of Werner states in P k is thus the convex hull of fk ((d)) (see Fig. 1). What does this set look like? Example IV.3. Let us look first at the case where k = 2 and d is arbitrary. By Lemma III.1, the point r in (d) is mapped to s(2) (r )ρ(2) + s(12 ) (r )ρ(12 ) , and it is easy to check that s(12 ) (r ) = i< j ri r j is maximised by r = x d , giving s(12 ) (r ) = 1/2(1 − 1/d). The states in f2 ((d)) are therefore those of the form aρ(2) + bρ(12 ) with b ≤ 1/2(1 − 1/d). Thus P 2 ∩ W 2 has a rather trivial polytope structure, being a line segment. It follows that the state aρ(2) + bρ(12 ) lies at a distance max(0, b − 1/2(1 − 1/d)) from P 2 ∩ W 2 . By Lemma IV.1, this is also the minimum distance to P 2 . This result implies that the state ξ 2 = ρ(12 ) considered in Example II.9, Eq. (8), has distance at least 1/2 from P 2 , showing the impossibility of a dimension-free bound on the error of a quantum de Finetti theorem (see remarks following Corollary III.9). In fact, Lemma IV.2 implies that any symmetric state with support on 2 (Cd ) has distance at least 21 to P 2 . Example IV.4. Consider next the case d = 3, k = 3. We will henceforth regard the set of Werner states W 3 as a subset of R3 by identifying a state ρ = λ vλ ρλ with the vector v = (v(3) , v(2,1) , v(13 ) ). If σ = r1 |1 1| + r2 |2 2| + r3 |3 3|, Lemma III.1 tells us that f3 (r ) = (s(3) (r ), 2s(2,1) (r ), s(13 ) (r )) ⎞ ⎛ ri3 + ri2 r j + r1r2 r3 , 2 ri2 r j + 4r1 r2 r3 , r1r2 r3 ⎠ . =⎝ i= j
i= j
(28)
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Fig. 2. Left: the image fk ((d)) for d = 3, k = 3, projected onto the coordinates ρ(13 ) and ρ(2,1) . The convex hull of fk ((d)) is a polytope (see Example IV.4). Right: the image fk ((d)) for d = 3, k = 4 projected onto ρ(4) and ρ(2,2) . The convex hull of this figure is not a polytope; since it is equal to the projection of the convex hull of fk ((d)), the latter set, P 4 ∩ W 4 , cannot be a polytope
The vertices of (d) are mapped to f3 (x 1 ) = (1, 0, 0), 1 1 , ,0 , f3 (x 2 ) = 2 2 10 16 1 , f3 (x 3 ) = , , 27 27 27 and comparison of Eq. (28) and the coordinates of the vertices gives f3 (r ) = ri3 − ri2 r j + 3r1r2 r3 f3 (x 1 ) + 4 ri2 r j − 24r1r2 r3 f3 (x 2 ) + (27r1 r2 r3 )f3 (x 3 ), and one can show that the polynomial coefficients are positive. So f3 (r ) lies in the convex span of {f3 (x 1 ), f3 (x 2 ), f3 (x 3 )}. Note that f3 ((d)) is a subset of the set of triseparable Werner states studied in [22]. Thus for d = 3, P 3 ∩ W 3 is a polytope (see Fig. 2), as in the previous example. However, if the number of diagrams with a given value of k and d exceeds d, the situation is different: Theorem IV.5. Let k, d be such that |Par(k, d)| > d. Then the set Tk (P k ) is not a polytope. Proof. Let X denote the subspace spanned by fk (x q ) for q = 1, . . . , d, where we identify W k with a subset of R|Par(k,d)| , as in Example IV.4. Since |Par(k, d)| > d, there is a non-zero vector v in R|Par(k,d)| that is orthogonal to X with respect to the Euclidean scalar product in R|Par(k,d)| . Suppose fk (r ) lies in X for all r ∈ (d). Then fk (r ).v = 0, for all r , so from Lemma III.1 we have for all r ∈ (d), (vλ dim Vλ )sλ (r ) = 0 , (29)
λ∈Par(k,d)
where v = λ vλ ρλ . Since the Schur polynomials are homogeneous, Eq. (29) extends from (d) to all r with non-negative components, and therefore all derivatives of the
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polynomial on the l.h.s. of this equation are zero at the origin. Since every coefficient of this polynomial is proportional to one of these derivatives, it must be identically zero. But the Schur functions sλ form a basis for the space of homogeneous symmetric polynomials of degree k in d variables, and therefore no such relationship can hold. Therefore Tk (P k ) includes a point outside X . If Tk (P k ) is a polytope, it has a vertex w not in X . Since Tk (P k ) is the convex hull of fk ((d)), w has the form w = fk (a). As w not in X , a is not a vertex of (d), which implies that there is a line segment in (d) passing through a. Because fk is smooth, the image under fk of the line segment t → a +tξ has a tangent vector at the vertex w. If this tangent vector does not vanish, then we have a contradiction, since then the curve must contain points outside the polytope Tk (P k ) in any neighbourhood of w, however small. It remains to show that, for any point a ∈ (d) that is not a vertex, there is a vector ξ ∈ Rd such that 1. the line segment t → a + tξ lies within (d) for sufficiently small absolute values of the real parameter t, and 2. the derivative of fk in the direction ξ at the point a has non-vanishing tangent vector, k ) i.e. ∂f (a+tξ |t=0 = 0. ∂t It is enough to show that the component of this tangent vector in some direction τ ∈ R|Par(k,d)| is non-vanishing, i.e. that ∂(τ.fk (a + tξ )) ∂t
t=0
= ξ.(∇r (τ.fk (r )))
r =a
= 0 .
(30)
We choose ξ as follows: Suppose a lies in the convex hull of the h vertices x q1 , . . . , x qh of (d), arranged in increasing size of the index qi , with 2 ≤ h ≤ d. Thus a=
h
u i x qi , with 0 < u i < 1 for 1 ≤ i ≤ h.
(31)
q1 q2 q1 x − x q2 , q2 − q1
(32)
i=1
Define ξ=
= (1, . . . , 1, β, . . . , β , 0, . . . , 0) ∈ Rd , q2 −q1
q1
1 where β = q−q . Then a + tξ lies within the convex hull of x q1 , . . . , x qh , and hence in 2 −q1 (d), for small enough values of |t|. To define τ , we use the fact the monomial symmetric functions m λ , for λ ∈ Par(k, d), also form a basis of the homogeneous symmetric polynomials of degree k in d variables. In particular, m (d) (r ) = rid = κλ,(d) sλ (r ) ,
λ
where the coefficients κλµ constitute the transition matrix, which is given by the inverse of the matrix of Kostka numbers [23]. We now take κλ,(d) τ= ρλ , dim Vλ λ
One-and-a-Half Quantum de Finetti Theorems
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which implies that τ.fk (r ) =
rid .
From (30) and (32) therefore ξ.(∇r (τ.fk (r )))|r =a =
ξi
i
=d
q1
∂
d j rj
∂ri aid−1 −
i=1
r =a
q2 dq1 aid−1 q2 − q1 i=q1 +1
>0, the last inequality holding because Eq. (31) implies a1 = · · · = aq1 > aq1 +1 = · · · = aq2 . The tangent vector at a in the direction ξ is therefore non-vanishing, which completes the proof. Figure 2 shows an example where d = 3, k = 4 and |Par(k, d)| = 4 > d. One might wonder whether Theorem IV.5 is tight, in the sense that, for |Par(k, d)| ≤ d, the set Tk (P k ) is a polytope. For k = 3, d = 3, where |Par(k, d)| = d, we have seen that this is true. However, for k = 4, d = 5, which also gives |Par(k, d)| = d, empirical evidence suggests that Tk (P k ) is not a polytope, having a convex boundary. This is shown in Fig. 3, which also plots the images of traced-out states trn−k ρλn with n = 10 and n = 60 and shows how the approximation to Tk (P k ) improves as more systems are traced out; it also reveals some intriguing striations in the case n = 60, corresponding to diagrams whose top rows are the same length. Thus the characterisation of the set P k ∩ W k seems to be quite subtle, and Werner states again uphold their reputation for exhibiting an interesting variety of phenomena.
V. Conclusions Although the quantum de Finetti theorem is usually thought of as a theorem about symmetric states, the unitary group shares the limelight in the results described here. Our highest weight version of the de Finetti theorem (Theorem II.2) generalises the usual symmetric-state version, but the extra generality almost comes free; indeed, one could argue that the structure of the proof is made clearer by taking the broader viewpoint. One can regard a highest weight vector as the state in a representation that is as unentangled as possible; this point of view has been taken by Klyachko [24]. It is therefore natural to regard highest weight vectors as analogues of product states, which is the role they have in our theorem. In the special case of symmetric states, our Theorem II.7 gives bounds for the distance between the n-exchangeable state ρ k and the set P k of convex combinations of products σ ⊗k ; these bounds are optimal in their dependence on n and k, the theorem giving an upper bound of order k/n and there being examples of states that achieve this bound
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Fig. 3. The figures show the image fk ((d)) (shaded region) for d = 5, k = 4, projected onto the coordinates ρ(4) and ρ(2,2) . The image has a smooth convex boundary, so P 4 ∩ W 4 cannot be a polytope. Also shown are the points obtained by tracing out n − k systems from states in W n . Each point corresponds to a diagram with n = 10 boxes (top figure), n = 20 (centre figure) and n = 60 boxes (bottom figure); the line segments demarcate the convex hull of all the points. As expected, fk ((d)) is approximated more closely as n increases
(see Theorem II.10). The dependence of the bound on the dimension d is less clear, the theorem giving a factor of d 2 whereas in the classical case Diaconis and Freedman [15] obtained a bound with a dimension factor of order d.
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Diaconis and Freedman also obtained a bound, k(k−1) 2n , that is independent of the dimension. No such bound can exist for quantum states, as Example II.9 shows; one can find a state ρ n with the property that ρ 2 , obtained by tracing out all but two of the systems, lies at a distance at least 1/2 from P 2 . This example is a Werner state, in fact the fully antisymmetric state on d = n systems, and it is an illustration of the usefulness of this family of states in giving information about P k . Lemma III.6 shows that the shifted Schur functions [1] are closely connected with partial traces of Werner states. The meaning of this connection needs to be further explored: does the algebra of shifted symmetric functions have a quantum-informational significance? Another intriguing connection is with the theorem of Keyl and Werner [25]. They show that the spectrum of a state ρ can be measured by carrying out a von Neumann measurement of ρ ⊗n on the subspaces Uλ ⊗ Vλ in the Schur-Weyl decomposition of (Cd )⊗n (Eq. (10)); if λ is obtained, then λ¯ = ( λn1 , . . . , λnd ) approximates the spectrum of ρ. Our theorem tells us that ρ k = trn−k ρλn can be approximated by the twirled product σ ⊗k , where σ has spectrum λ¯ . By the Keyl-Werner theorem, the state trn−k ρλn must therefore project predominantly into subspaces Uµ ⊗ Vµ with µ close to λ in shape (but rescaled by k/n). In this sense, tracing out a Werner state approximately ‘preserves the shape’ of its diagram. We can get an intuition for why this should be by iterating the special case of Lemma III.4 where one box is removed (cf. [20, Prop. 4]). This shows that tracing out is approximately equivalent, for large n, to a process that selects a row of a diagram with probability proportional to the length of that row and then removes a box from the end of the row. There have been many applications of the de Finetti theorem to topics including foundational issues [7, 26], mathematical physics [17, 27] and quantum information theory [10, 18, 28–31]; there have also been various generalisations [3–7, 9, 10, 15–17]. We have taken one-and-a-half footsteps along this route. Acknowledgements. We thank Aram Harrow and Andreas Winter for helpful discussions, and Ignacio Cirac and Frank Verstraete for raising the question of how to approximate n-exchangeable states by m-exchangeable states (see end of Sect. IIIC). We also thank the anonymous reviewers for their helpful comments. This work was supported by the EU project RESQ (IST-2001-37559) and the European Commission through the FP6-FET Integrated Project SCALA, CT-015714. MC acknowledges the support of an EPSRC Postdoctoral Fellowship and a Nevile Research Fellowship, which he holds at Magdalene College Cambridge. GM acknowledges support from the project PROSECCO (IST-2001-39227) of the IST-FET programme of the EC. RR was supported by Hewlett Packard Labs, Bristol.
References 1. Okounkov, A., Olshanski, G.: Alg. i Anal. 9, no. 2, 13–146 (1997) (Russian); Eng. in st. Petersburg Math. J 9, no. 2 (1998) 2. de Finetti, B.: Ann. Inst. H. Poincaré 7, 1 (1937) 3. Størmer, E.: J. Funct. Anal. 3, 48 (1969) 4. Hudson, R.L., Moody, G.R., Wahrschein, Z.: Verw. Geb. 33, 343 (1976) 5. Petz, D.: Prob. Th. Rel. Fields. 85, 1–11 (1990) 6. Caves, C.M., Fuchs, C.A., Schack, R.: J. Math. Phys. 43, 4537 (2002) 7. Fuchs CA, Schack R: In: Quantum Estimation Theory, M.G.A. Paris, J. Rehaeck (eds), Berlin: Springer, 2004 8. Fuchs, C.A., Schack, R., Scudo, P.F.: Phys. Rev. A 69, 062305 (2004) 9. König, R., Renner, R.: J. Math. Phys. 46, 122108 (2005) 10. Renner, R.: Security of Quantum Key Distribution. PhD thesis, ETH Zurich, 2005, available at http://axiv.org/list/quant-ph/0512258, 2005 11. Werner, R.F.: Phys. Rev. A 40, 4277 (1989)
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12. Carter, R., Segal, G., MacDonald, I.: Lectures on Lie Groups and Lie Algebras. London Mathematical Society Student Texts vol. 32, 1st ed, Cambridge: Cambridge Univ. Press, 1995 13. Perelomov, A.: Generalized coherent states and their application. Texts and Monographs in Physics, Berlin: Springer-Verlag, 1986 14. Weyl, H.: The Theory of Groups and Quantum Mechanics. New York: Dover Publications, Inc., 1950 15. Diaconis, P., Freedman, D.: The Annals of Probability 8, 745 (1980) 16. Fannes, M., Lewis, J.T., Verbeure, A.: Lett. Math. Phys. 15, 255 (1988) 17. Raggio, G.A., Werner, R.F.: Helv. Phys. Acta 62, 980 (1989) 18. Ioannou, L.M.: Deterministic computational complexity of the quantum separability problem. http://arxiv.org/list/quant-ph/0603199; 2006, to appear in QIP, 2006 19. Doherty, A.: Personal communication, 2006 20. Audenaert, K.: Available at http://qols.ph.ic.ac.uk/ ∼ kauden/QITNotes_files/irreps.pdf, 2004 21. Fulton, W.F.: Young Tableaux. Cambridge: Cambridge University Press, 1997 22. Eggeling, T., Werner, R.F.: Phys. Rev. A 63, 04211 (2001) 23. Macdonald, I.G.: Symmetric functions and Hall polynomials. Oxford: Clarendon Press, 1979 24. Klyachko, A.: http:/laxiv.org/list/quant-ph/0206012, 2002 25. Keyl, M., Werner, R.F.: Phys. Rev. A 64, 052311 (2001) 26. Hudson, R.L.: Found. Phys. 11, 805 (1981) 27. Fannes, M., Spohn, H., Verbeure, A.: J. Math. Phys. 21, 355 (1980) 28. Brun, T.A., Caves, C.M., Schack, R.: Phys. Rev. A 63, 042309 (2001) 29. Doherty, A.C., Parillo, P.A., Spedalieri, F.M.: Phys. Rev. A 69, 022308 (2004) 30. Audenaert KMR.: In: Proceedings of MTNS2004 (2004), available at http://arxiv.org/list/quant-ph/ 0402076, 2004 31. Terhal, B.M., Doherty, A.C., Schwab, D.: Phys. Rev. Lett 90, 157903 (2003) Communicated by M.B. Ruskai
Commun. Math. Phys. 273, 499–532 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0256-9
Communications in
Mathematical Physics
Universality of a Double Scaling Limit near Singular Edge Points in Random Matrix Models T. Claeys, M. Vanlessen Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium. E-mail: [email protected] Received: 21 July 2006 / Accepted: 9 November 2006 Published online: 8 May 2007 – © Springer-Verlag 2007 −1 −n tr Vs,t (M) Abstract: We consider unitary random matrix ensembles Z n,s,t e d M on the space of Hermitian n × n matrices M, where the confining potential Vs,t is such that the limiting mean density of eigenvalues (as n → ∞ and s, t → 0) vanishes like a power 5/2 at a (singular) endpoint of its support. The main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit. The limiting kernel is built out of functions associated with a special solution of the PI2 equation, which is a fourth order analogue of the Painlevé I equation. In order to prove our result, we use the well-known connection between the eigenvalue correlation kernel and the Riemann-Hilbert (RH) problem for orthogonal polynomials, together with the Deift/Zhou steepest descent method to analyze the RH problem asymptotically. The key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint, for which we use the model RH problem for the special solution of the PI2 equation. In addition, the RH method allows us to determine the asymptotics (in a double scaling limit) of the recurrence coefficients of the orthogonal polynomials with respect to the varying weights e−nVs,t on R. The special solution of the PI2 equation pops up in the n −2/7 -term of the asymptotics.
1. Introduction and Statement of Results 1.1. Unitary random matrix ensembles. On the space Hn of Hermitian n × n matrices M, we consider for n ∈ N and s, t ∈ R the unitary random matrix ensemble, 1 Z n,s,t
e−n tr Vs,t (M) d M.
(1.1)
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Here, Z n,s,t is a normalization constant and the confining potential Vs,t is a real analytic function, depending on two parameters s, t ∈ R, satisfying the asymptotic condition, lim
x→±∞
Vs,t (x) = +∞, uniformly for s, t ∈ [−δ0 , δ0 ] for some δ0 > 0. log(x 2 + 1)
(1.2)
Then, Z n,s,t =
Hn
e−n tr Vs,t (M) d M
is convergent as n → ∞ so that the random matrix model is well- defined. It is well-known, see e.g. [25], that an important role in the study of the unitary random matrix ensemble (1.1) is played by the following scalar 2-point (correlation) kernel, n
n
K n(s,t) (x, y) = e− 2 Vs,t (x) e− 2 Vs,t (y)
n−1
(n,s,t)
pk
(n,s,t)
(x) pk
(y),
(1.3)
k=0
constructed out of the orthonormal polynomials (n,s,t)
pk
(n,s,t) k
(x) = κk
x + ··· ,
(n,s,t)
κk
> 0,
with respect to the varying weights e−nVs,t on R. Indeed, the correlations between the eigenvalues of M can be written in terms of the correlation kernel. More precisely, the m-point correlation function R(s,t) n,m satisfies [25], (s,t) R(s,t) (x , . . . , x ) = det K (x , x ) . (1.4) m i j n,m 1 n 1≤i, j≤m
Further, the limiting mean eigenvalue distribution µs,t has a density ρs,t which can be retrieved from the correlation kernel as follows: ρs,t (x) = lim
n→∞
1 (s,t) K (x, x). n n
(1.5)
The limiting mean eigenvalue distribution µs,t equals [10] the equilibrium measure in external field Vs,t . This is the unique measure minimizing the logarithmic energy [28] 1 dµ(x)dµ(y) + Vs,t (y)dµ(y), log (1.6) I Vs,t (µ) = |x − y| among all probability measures µ on R. Furthermore, there exists a real analytic function qs,t , such that [9], 1 − ρs,t (x) = qs,t (x), (1.7) π − + − q − , with q ± ≥ 0 and denotes the negative part of qs,t , i.e. qs,t = qs,t where qs,t s,t s,t − + q qs,t s,t = 0. Due to condition (1.2) we have that qs,t (x) → +∞ as x → ±∞, so that µs,t is supported on a finite union of intervals, which we denote by Ss,t . It is known
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[28] that the equilibrium measure µs,t satisfies the following Euler-Lagrange variational conditions: there exists a constant κs,t ∈ R such that for x ∈ Ss,t , (1.8) 2 log |x − u|dµs,t (u) − Vs,t (x) = κs,t , 2
log |x − u|dµs,t (u) − Vs,t (x) ≤ κs,t ,
for x ∈ R \ Ss,t .
(1.9)
The external field Vs,t is called regular if strict inequality in (1.9) holds, if the density ρs,t does not vanish in the interior of the support Ss,t , and if qs,t has a simple zero at each of the endpoints of the support Ss,t . If one of these conditions is not valid, Vs,t is called singular. The singular points x ∗ are classified as follows, see [10, 21]: (i) x ∗ ∈ R \ Ss,t is a type I singular point if equality in (1.9) holds. Then, x ∗ is a + of multiplicity 4m with m ∈ N. zero of qs,t ∗ (ii) x ∈ Ss,t is a type II singular point if it is an interior point of Ss,t where the − equilibrium density ρs,t vanishes. Then, x ∗ is a zero of qs,t of multiplicity 4m. ∗ (iii) x is a type III singular point if it is an endpoint of the support Ss,t and a zero of qs,t of multiplicity larger than one. Then, x ∗ is a zero of qs,t of multiplicity 4m + 1, which means that ρs,t (x) ∼ c|x − x ∗ |(4m+1)/2 . In this paper, we consider external fields Vs,t which are such that in the critical case s = t = 0, V0 = V0,0 has a type III singular (edge) point x ∗ with m = 1, i.e. ρ0,0 (x) ∼ c|x − x ∗ |5/2 ,
as x → x ∗ .
(1.10)
Further, we take Vs,t of the special form, Vs,t = V0 + sV1 + t V2 ,
(1.11)
where V1 is an arbitrary real analytic function, while V2 is real analytic and in addition satisfies some critical condition which we will specify in Sect. 1.4 below. 1.2. Universality in random matrix theory. Consider for now unitary random matrix ensembles Z n−1 e−n tr V (M) d M on the space of Hermitian n × n matrices M. Scaling limits of the associated correlation kernel K n show universal behavior. Near regular points, universality results have been established in [1, 8, 10, 11, 27]. For example, if x ∗ lies in the bulk of the spectrum (i.e. x ∗ is such that it lies in the interior of the support S of the equilibrium measure in external field V , and such that the equilibrium density ρ does not vanish at x ∗ ) there is a constant c such that 1 u v sin π(u − v) Kn x ∗ + , x ∗ + = . (1.12) lim n→∞ cn cn cn π(u − v) On the other hand, if x ∗ is a regular edge point of the spectrum (i.e. x ∗ is an endpoint of S and ρ vanishes like a square root at x ∗ ), there is a constant c such that 1 u v Ai (u)Ai (v) − Ai (v)Ai (u) , K n x ∗ + 2/3 , x ∗ + 2/3 = 2/3 n→∞ cn cn cn u−v lim
where Ai is the Airy function.
(1.13)
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Near singular points, similar results hold. In those singular cases it is interesting to consider double scaling limits where the external field V depends on additional parameters. In [2, 5, 6, 29], an external field V was considered such that there is a type II singular (interior) point x ∗ with m = 1, i.e. ρ(x) ∼ c(x − x ∗ )2 ,
as x → x ∗ .
If an additional parameter is included in the external field, Vt = V /t, one observes for t close to 1 the transition where two intervals in the support of the limiting mean density of eigenvalues merge to one interval through the critical case of a type II singular point. In the double scaling limit where n → ∞ and t → 1 in such a way that c0 n 2/3 (t −1) → s ∈ R for some appropriately chosen constant c0 , there exists a constant c such that (for the associated correlation kernel K n,t ), lim
1 u v K n,t x ∗ + 1/3 , x ∗ + 1/3 = K crit,II (u, v; s). 1/3 cn cn cn
Here, K crit,II (u, v; s) is built out of functions associated with the Hastings-McLeod solution [18] of the second Painlevé equation. The main purpose of this paper is to obtain, for the random matrix models in Sect. 1.1 above, a similar result near the type III singular (edge) point of V0 with m = 1. We take a double scaling limit (n → ∞ and s, t → 0), and the limiting kernel K crit,III will be built out of functions which are associated with a special solution of the fourth order analogue of the Painlevé I equation. The case of a type III singular (edge) point was also studied in the physics literature [3, 4]. In addition, the techniques that we use to prove this allow us to determine the asymptotics (in a double scaling limit) of the recurrence coefficients in the three-term (n,s,t) recurrence relation satisfied by the orthogonal polynomials pk with respect to the varying weights e−nVs,t on R. 1.3. -functions associated with a special solution of the PI2 equation. We consider the following differential equation for y = y(s, t), which we denote as the PI2 equation, 1 1 3 1 2 (1.14) y + (ys + 2yyss ) + yssss . s = ty − 6 24 240 For t = 0, this equation is the second member in the Painlevé I hierarchy [20, 23]. The PI2 equation has been studied for example in [4, 19, 26] (for t = 0) and [7, 15] (for general t). The Lax pair for the PI2 equation is the linear system of differential equations ∂ = U , ∂ζ
∂ = W , ∂s
(1.15)
where
1 −4ys ζ − (12yys + ysss ) 8ζ 2 + 8yζ + (12y 2 + 2yss − 120t) , U= U21 4ys ζ + (12yys + ysss ) 240
(1.16)
U21 = 8ζ 3 − 8yζ 2 − (4y 2 + 2yss + 120t)ζ + (16y 3 − 2ys2 + 4yyss + 240s), (1.17)
Universality of Eigen value Correlation Kernel in Double Scaling Limit
and
W =
0 ζ − 2y
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1 . 0
(1.18)
The system of differential equations (1.15)–(1.18) can only be solvable if y = y(s, t) is a solution to the PI2 equation (1.14). For different solutions y, we have different Lax pairs. We are interested in the special solution y which was studied in [4, 7, 15]. This solution y = y(s, t) is characterized by the vanishing of its Stokes multipliers s1 , s2 , s5 , and s6 , see [19] for details. It was shown in [7] that y has no poles for real s and t, and that it has, for fixed t ∈ R, the following asymptotic behavior: 1 y(s, t) = ∓(6|s|)1/3 ∓ 62/3 t|s|−1/3 + O(|s|−1 ), 3
as s → ±∞.
(1.19)
It has been shown in [26, App. A] that for t = 0, y is uniquely determined by realness and asymptotic condition (1.19). For general t we are not aware of a similar result although it is supported by a conjecture of Dubrovin [15] that this should hold for general t. For s, t ∈ R,the Lax pair (1.15)–(1.18) associated with this special choice of y has a unique 1 solution for which the following limit holds, see [7, 19], 2 1 1 1 (ζ ; s, t) θ(ζ ;s,t) 1 −→ √ e e− 4 πi , 2 (ζ ; s, t) −1 2 as ζ → ∞ with 0 < Arg ζ < 6π/7, 1 0 where σ3 = 0 −1 denotes the third Pauli-matrix, and where θ is given by 1
ζ 4 σ3
θ (ζ ; s, t) =
1 7/2 1 3/2 ζ − tζ + sζ 1/2 . 105 3
(1.20)
(1.21)
The functions 1 and 2 will appear below in the universal limiting correlation kernel near type III singular (edge) points of V0 with m = 1. 1.4. Statement of results. We work under the following assumptions. Assumptions 1.1. (i) We consider external fields Vs,t of the form Vs,t = V0 + sV1 + t V2 ,
(1.22)
where V0 , V1 , and V2 are real analytic and are such that there exists a δ0 > 0 such that the following holds lim
|x|→∞
Vs,t (x) = +∞, log(x 2 + 1)
uniformly for s, t ∈ [−δ0 , δ0 ].
(1.23)
(ii) V0 is such that the equilibrium measure ν0 in external field V0 is supported on one single interval [a, b] ⊂ R, and b is a type III singular (edge) point of V0 with m = 1. Then, ν0 is of the form [9], dν0 (x) =
1 h 0 (x) (b − x)(x − a) χ[a,b] (x)d x, 2π
(1.24)
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with χ[a,b] the indicator function of the set [a, b], and with h 0 real analytic and satisfying, h 0 (b) = h 0 (b) = 0,
and
h 0 (b) > 0.
(1.25)
Furthermore, we assume that V0 has no other singular points besides b. In particular, a is a regular (edge) point and we then have that h 0 (a) > 0.
(1.26)
(iii) V2 is such that it satisfies the critical condition b u−a V (u)du = 0. b−u 2 a
(1.27)
Throughout the rest of this paper we let V be the neighborhood of the real line where V0 , V1 , V2 , and h 0 are analytic. Example 1.2. The assumptions above are valid for the particular example where V0 , V1 , and V2 are given by V0 (x) =
1 4 4 1 8 x − x 3 + x 2 + x, 20 15 5 5
V1 (x) = x,
V2 (x) = x 3 − 6x.
(1.28)
Then, the equilibrium measure ν0 is supported on the interval [−2, 2] and given by dν0 (x) =
1 (x + 2)1/2 (x − 2)5/2 χ[−2,2] (x)d x. 10π
(1.29)
It should be noted that a type III singular (edge) point cannot occur when V0 is a polynomial of degree lower than 4. Example 1.3. In the continuum limit of the Toda lattice [12], an external field of the form Vt1 ,t2 (x) = (1 + t1 )(V0 (x) + t2 x) has been studied. This deformation of V0 can be written in the form (1.22) (so that it is included in the class of external fields studied in this paper). Indeed, if we let V1 (x) = x and V2 (x) = V0 (x) + cx, with c some constant chosen such that the critical condition (1.27) holds, then Vt1 ,t2 = V0 + sV1 + t V2 , with s = t2 + t1 t2 − ct1 and t = t1 . Remark 1.4. In Sect. 2 we will show that assumption (iii) is equivalent to the vanishing of the equilibrium density dνd2x(x) at the right endpoint b, where ν2 is the unique measure which minimizes I V2 (ν), see (1.6), among all signed measures ν, supported on [a, b] and having zero mass, ν([a, b]) = 0. Remark 1.5. The case where the left (instead of the right) endpoint of the support is singular can be transformed to our case by considering the external field Vs,t (−x).
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Remark 1.6. Without giving any mathematical details, we now describe the transitions that can occur for s and t near 0. First, if we let t = 0 and s vary around 0, one typically observes the transition from the regular one-interval case to the singular case and back to the regular one-interval case. Next, for s = 0 and t around 0, we can observe the transition from the regular one-interval case to the regular two-interval case. Finally, letting both s and t vary around 0, we can observe one of the above described transitions, or the critical transition where a type II singular point moves to the endpoint b, where it becomes a type III singular point before moving on as a type I singular point. Further, to describe our results, we have to introduce constants c, c1 , and c2 , 2/7 h (b) h 1 (b) 15 √ h 0 (b) b − a > 0, c1 = 1/2 , c2 = − 3/2 2 , c= 1/2 2 c (b − a) c (b − a)1/2 (1.30) where h 0 is the real analytic function appearing in (1.24), and where the functions h 1 and h 2 are defined as, 1 b
du , for x ∈ [a, b] and j = 1, 2. (b − u)(u − a)V j (u) h j (x) = − π a u−x (1.31) 1.4.1. Universality of the double scaling limit. Our main result is the following. Theorem 1.7. Let Vs,t = V0 +sV1 +t V2 be such that Assumptions 1.1 above are satisfied. We take a double scaling limit where we let n → ∞ and at the same time s, t → 0, in such a way that lim n 6/7 s and lim n 4/7 t exists, and put s0 = c1 · lim n 6/7 s ∈ R,
t0 = c2 · lim n 4/7 t ∈ R,
(1.32) (s,t)
where the constants c1 and c2 are defined by (1.30). Then, the 2-point kernel K n satisfies the following universality result: u v 1 (1.33) lim 2/7 K n(s,t) b + 2/7 , b + 2/7 = K crit,III (u, v; s0 , t0 ), cn cn cn uniformly for u, v in compact subsets of R. Here, K crit,III is built out of the functions 1 and 2 defined in Sect. 1.3, K crit,III (u, v; s, t) =
1 (u; s, t)2 (v; s, t) − 1 (v; s, t)2 (u; s, t) . −2πi(u − v)
(1.34)
Remark 1.8. Since y(s, t) has no poles [7] for s, t ∈ R, the kernel K crit,III (u, v; s, t) exists for all real u, v, s, and t. Furthermore, using a similar argument as in [7, Lemma 2.3 (ii)], one can show that eπi/4 1 and eπi/4 2 are real. It then follows that K crit,III (u, v; s, t) is real for real u, v, s, and t. crit,III . Using the fact that Remark 1 1.9. It is possible to give an integral formula for K 2 satisfies the second differential equation of the Lax pair (1.15), we have that
∂1 (ζ ; s, t) = 2 (ζ ; s, t), ∂s
and
∂2 (ζ ; s, t) = (ζ − 2y(s, t))1 (ζ ; s, t). ∂s
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Using (1.34) this yields, 1 ∂ K crit,III (u, v; s, t) = 1 (u; s, t)1 (v; s, t). ∂s 2πi Now, since lims→−∞ K crit,III (u, v; s, t) = 0, which can be shown using a Deift/Zhou steepest descent method argument [13], it then follows that K crit,III has the following integral formula, s 1 crit,III K (u, v; s, t) = 1 (u; σ, t)1 (v; σ, t)dσ. (1.35) 2πi −∞ Remark 1.10. Theorem 1.7 can be generalized to the case where the support of ν0 (the equilibrium measure in external field V0 ) consists of more than one interval. Then, the proof becomes much more technical, although the main ideas remain the same. We comment in Remark 3.8 on the modifications that have to be made in the multi-interval case. 1.4.2. Recurrence coefficients for orthogonal polynomials. It is well-known [30] that the orthonormal polynomials pk = pk(n,s,t) satisfy a three-term recurrence relation of the form, x pk (x) = ak+1 pk+1 (x) + bk pk (x) + ak pk−1 (x),
(1.36)
where ak = ak(n,s,t) > 0 and bk = bk(n,s,t) ∈ R (we suppress the s and t dependence for brevity). In the generic case where V0 has no singular points, the recurrence coefficients for s = t = 0 have the following asymptotics, see e.g. [2, 8]: an(n,0,0) =
b−a + O(n −1 ), 4
bn(n,0,0) =
b+a + O(n −1 ), 2
as n → ∞. (1.37)
For singular potentials V0 , the constant terms in the expansions (1.37) remain the same, but the error terms behave differently [2, 6]. In our case of interest, where we have a type III singular (edge) point of V0 with m = 1, the error term is of order O(n −2/7 ), and the coefficient of the n −2/7 term is expressed in terms of the special solution y of the PI2 equation discussed in Sect. 1.3. Theorem 1.11. Let Vs,t be such that Assumptions 1.1 above are satisfied. Consider the three-term recurrence relation (1.36) satisfied by the orthonormal polynomials pk = pk(n,s,t) with respect to the weight function e−nVs,t . Then, in the double scaling limit where n → ∞ and s, t → 0, in such a way that lim n 6/7 s and lim n 4/7 t exists, and put s0 = c1 · lim n 6/7 s ∈ R,
t0 = c2 · lim n 4/7 t ∈ R,
(1.38)
with c1 and c2 given by (1.30), we have 1 b−a + y(c1 n 6/7 s, c2 n 4/7 t)n −2/7 + O(n −3/7 ), 4 2c 1 b−a + y(s0 , t0 )n −2/7 (1 + o(1)), = 4 2c
an(n,s,t) =
(1.39)
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and b+a 1 + y(c1 n 6/7 s, c2 n 4/7 t)n −2/7 + O(n −3/7 ) 2 c b+a 1 = + y(s0 , t0 )n −2/7 (1 + o(1)), 2 c
bn(n,s,t) =
(1.40)
where the constant c is given by (1.30), and where y is the special solution of the PI2 equation discussed in Sect. 1.3. Remark 1.12. Note that the expansions of the recurrence coefficients are of the same form as the conjectured (by Dubrovin [15, Main Conjecture, Part 3], see also [14]) expansions for solutions of perturbed hyperbolic equations. Here, the perturbation parameter plays the role of 1/n in our context. Remark 1.13. For polynomials which are orthogonal on certain complex contours, it can occur that the equilibrium density vanishes like a power 3/2. Asymptotics of the recurrence coefficients in this case were obtained in [16]. Here, a special solution of the Painlevé I equation occurs instead of a solution of the PI2 equation and the asymptotics are in powers of n −1/5 . Observe further that in [16] there is no term of order n −1/5 in the asymptotics. In (1.39) and (1.40) we see that there is no term of order n −1/7 . In the proof of Theorem 1.11 this term will drop out in a similar way as the n −1/5 -term in [16]. 1.5. Outline of the rest of the paper. We prove our results by characterizing the orthogonal polynomials via the well-known 2 × 2 matrix valued Fokas-Its-Kitaev RiemannHilbert (RH) problem [17] and applying the Deift/Zhou steepest descent method [13] to analyze this RH problem asymptotically. This approach has been used many times before, see e.g. [5, 6, 8, 10, 11, 16, 22, 31, 32]. An important step in the Deift/Zhou steepest descent method is the construction of so-called g-functions associated with equilibrium measures. Those equilibrium measures will be constructed in Sect. 2. In order to deal with the deformations Vs,t of V0 , we use modified equilibrium problems where we allow the measures to be negative, which was also done in [5, 6, 16]. Another modification of the equilibrium problem is that we choose the support of the equilibrium measure fixed, instead of allowing it to choose its own support. In Sect. 3, we perform the Deift/Zhou steepest descent analysis to the RH problem Y for orthogonal polynomials. Via a series of transformations Y → T → S → R we want to arrive at a RH problem for R which is normalized at infinity (i.e. R(z) → I as z → ∞) and with jumps uniformly close to the identity matrix. Then, R itself is close to the identity matrix. By unfolding the series of transformations we then get the asymptotics of Y . The key step in this method will be the local analysis near the endpoints a and b. Near the regular endpoint a, we construct (in Sect. 3.5) a parametrix built out of Airy functions. Due to the modified equilibrium measures, which have a fixed support, we also need to make a technical modification in the construction of the Airy parametrix, compared with the parametrix as used e.g. in [8]. To construct the local parametrix near the singular endpoint b (in Sect. 3.6) we use a model RH problem associated with the special solution y of the PI2 equation as discussed in Sect. 1.3. The results of Sect. 3 will be used in Sect. 4 to prove the universality result for the correlation kernel (see Theorem 1.7) and in Sect. 5 to determine the asymptotics of the recurrence coefficients (see Theorem 1.11).
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2. Equilibrium Measures We consider external fields Vs,t = V0 + sV1 + t V2 which satisfy Assumptions 1.1 in the beginning of Sect. 1.4. In order to perform the Deift/Zhou steepest descent analysis to the RH problem for orthogonal polynomials one would expect to use the equilibrium measure µs,t in external field Vs,t minimizing I Vs,t (µ), see (1.6), among all probability measures µ on R. However, as in [5, 6, 16] it will be more convenient to use modified equilibrium measures νs,t which we allow to be negative. Furthermore, unlike in [5, 6, 16], we take the support of the measures νs,t to be fixed instead of letting it depend on s and t. The aim of this section is to find measures νs,t (depending on the parameters s, t ∈ R) supported on the interval [a, b] ⊂ R (where [a, b] is the support of the equilibrium measure ν0 in external field V0 ), such that νs,t ([a, b]) = 1, and such that they satisfy the following condition: there exist s,t ∈ R such that for every δ > 0 there are ε, κ > 0 sufficiently small such that for s, t ∈ [−ε, ε], 2 log |x − u|dνs,t (u) − Vs,t (x) = s,t , for x ∈ [a, b], (2.1) 2 log |x − u|dνs,t (u) − Vs,t (x) < s,t − κ, for x ∈ R \ [a − δ, b + δ]. (2.2) We seek νs,t in the following form: νs,t = ν0 + sν1 + tν2 ,
(2.3)
where ν0 is the equilibrium measure in external field V0 minimizing I V0 (ν), see (1.6), among all probability measures ν on R. From Assumption 1.1 (ii) we know that ν0 can be written as follows: dν0 (x) = ψ0,+ (x)χ[a,b] (x)d x,
(2.4)
where χ[a,b] is the indicator function of the set [a, b], and where ψ0,+ is the +boundary value of the function ψ0 (z) =
1 R(z)h 0 (z), 2πi
for z ∈ V \ [a, b],
with h 0 analytic in the neighborhood V of the real line, and with 1/2 R(z) = (z − a)(z − b) , for z ∈ C \ [a, b].
(2.5)
(2.6)
Here, we take the principal branch of the square root so that R is analytic in C \ [a, b]. Further, since a is a regular (edge) point and since b is a type III singular (edge) point with m = 1, we have, cf. (1.25) and (1.26), h 0 (a) > 0,
h 0 (b) = h 0 (b) = 0,
and
h 0 (b) > 0.
(2.7)
Since V0 is assumed to have no other singular points besides b, we know (cf. (1.8) and (1.9)) that ν0 satisfies the following condition: there exists 0 ∈ R such that 2 log |x − u|dν0 (u) − V0 (x) = 0 , for x ∈ [a, b], (2.8) 2 log |x − u|dν0 (u) − V0 (x) < 0 , for x ∈ R \ [a, b]. (2.9)
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We will now construct the two measures ν1 and ν2 . In order to do this we introduce the following auxiliary (analytic) functions: 1 dξ , for z ∈ V and j = 1, 2, (2.10) R(ξ )V j (ξ ) h j (z) = 2πi γ ξ −z where γ is a positively oriented contour in V with [a, b] and z in its interior, and where R is given by (2.6). Observe that, using the fractional residue theorem, one has, b 1 du h j (x) = − , for x ∈ [a, b], (2.11) R+ (u)V j (u) πi a u−x where the integral is a Cauchy principal value integral. So, h j is real on [a, b]. Observe that by Assumption 1.1 (iii) and (2.11), h 2 (b) = 0.
(2.12)
Lemma 2.1. Define two signed measures ν1 and ν2 supported on [a, b] as dν j (x) = ψ j,+ (x)χ[a,b] d x,
j = 1, 2,
(2.13)
where χ[a,b] is the indicator function of the set [a, b], and where ψ j,+ is the +boundary value of the function ψ j (z) =
1 h j (z) , 2πi R(z)
for z ∈ V \ [a, b].
(2.14)
Here, h j is given by (2.10), see also (2.11) for its expression on [a, b], and R is given by (2.6). Then, ν j has zero mass, i.e. b ψ j,+ (u)du = 0, (2.15) ν j ([a, b]) = a
and there exist constants j ∈ R such that 2 log |x − u|dν j (u) − V j (x) = j , Proof. Define, for j = 1, 2, the auxiliary functions b 1 du F j (z) = , R+ (u)V j (u) 2πi R(z) a u−z
for x ∈ [a, b].
(2.16)
for z ∈ C \ [a, b],
(2.17)
which, by standard techniques and by (2.10) and (2.14), are equal to 1 1 dξ F j (z) = V j (z) − R(ξ )V j (ξ ) 2 4πi R(z) γ ξ −z 1 for z ∈ V \ [a, b], = V j (z) − πiψ j (z), 2 where γ is a positively oriented contour in V with [a, b] and z in its interior. This, together with the fact that ψ j,+ = −ψ j,− on (a, b), yields F j,+ (x) − F j,− (x) = −2πiψ j,+ (x), F j,+ (x) + F j,− (x) =
V j (x),
for x ∈ [a, b],
(2.18)
for x ∈ [a, b].
(2.19)
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Since F j is analytic in C \ [a, b] and since, by (2.17), F j (z) = O(z −2 ) as z → ∞, a standard complex analysis argument shows that
1 2πi
b
a
F j,+ (u) − F j,− (u) ds = F j (z), u−z
for z ∈ C \ [a, b].
By (2.18), this yields, F j (z) = −
b
a
ψ j,+ (u) du = −z −1 u−z
b
ψ j,+ (u)du + O(z −2 ),
as z → ∞.
a
Comparing this with the fact that F j (z) = O(z −2 ) as z → ∞, we obtain
b a ψ j,+ (u)du = 0, so that (2.15) is proven. It remains to prove (2.16). It is straightforward to check that, F j (z) = − a
b
ψ j,+ (u) 1 du = −πiψ j (z) + u−z 2
γ
ψ j (ξ ) dξ, ξ −z
for z ∈ V \ [a, b],
so that, using the fractional residue theorem, F j,± (x) = −πiψ j,± (x) − a
b
ψ j,+ (u) du, u−x
for x ∈ [a, b].
From (2.19) and the fact that ψ j,+ + ψ j,− = 0 on [a, b] this yields, d dx
2 log |x − u|dν j (u) + V j (x) = 2
b
a
This proves (2.16).
ψ j,+ (u) du + F j,+ (x) + F j,− (x) = 0. u−x (2.20)
Corollary 2.2. Let νs,t = ν0 + sν1 + tνt . Then, dνs,t (x) = ψs,t,+ (x)χ[a,b] d x, where ψs,t = ψ0 + sψ1 + tψ2 ,
on V \ [a, b],
(2.21)
with ψ0 given by (2.5) and ψ1 and ψ2 given by (2.14). So, νs,t is supported on [a, b] and has mass one, i.e. νs,t ([a, b]) = 1. Further, there exist constants s,t ∈ R such that for any δ > 0 there are ε, κ > 0 sufficiently small such that for s, t ∈ [−ε, ε] the conditions (2.1) and (2.2) are satisfied. Proof. Since νs,t = ν0 + sν1 + tνt , from (2.15), and from the fact that ν0 ([a, b]) = 1 it is clear that νs,t ([a, b]) = 1. Next, with s,t = 0 + s1 + t2 , we have (2.22) 2 log |x − u|dνs,t (u) − Vs,t (x) − s,t = I0 (x) + s I1 (x) + t I2 (x), where
I j (x) = 2
log |x − u|dν j (u) − V j (x) − j ,
j = 1, 2, 3.
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Then, condition (2.1) follows from (2.8) and (2.16). Now, by using (2.9) and the fact that I0 (x) → −∞ as |x| → ∞, there exists κ > 0 such that 3 I0 < − κ, 2
on R \ [a − δ, b + δ].
(2.23)
Further, one can check that I1 and I2 are bounded on R \ [a − δ, b + δ], and thus there exists ε > 0 such that for s, t ∈ [−ε, ε], s I1 + t I2 <
1 κ, 2
on R \ [a − δ, b + δ].
Inserting (2.23) and (2.24) into (2.22) we obtain condition (2.2).
(2.24)
Remark 2.3. The measure ν1 (ν2 ) is the equilibrium measure that minimizes I V1 (ν) (I V2 (ν)) among all signed measures ν, supported on [a, b] with ν([a, b]) = 0. The measures νs,t on the other hand minimize I Vs,t (ν) among all signed measures supported on [a, b] with ν([a, b]) = 1. Observe that since ν0 has a strictly positive density on (a, b) (since ν0 has no type II singular points) we have for any δ > 0 that νs,t is positive on (a + δ, b − δ) for s, t sufficiently small. 3. Riemann-Hilbert Analysis 3.1. RH problem for orthogonal polynomials. For each fixed n, s, and t, we consider the Fokas-Its-Kitaev Riemann-Hilbert problem [17] characterizing the orthogonal polyno(n,s,t) mials pk with respect to the weight functions e−nVs,t . We seek a 2 × 2 matrix-valued function Y (z) = Y (z; n, s, t) (we suppress the n, s, and t dependence for brevity) that satisfies the following conditions. RH problem for Y . (a) Y : C \ R → C2×2 is analytic. (b) Y possesses continuous boundary values for x ∈ R denoted by Y+ (x) and Y− (x), where Y+ (x) and Y− (x) denote the limiting values of Y (z ) as z approaches x from above and below, respectively, and 1 e−nVs,t (x) Y+ (x) = Y− (x) , for x ∈ R. (3.1) 0 1 (c) Y has the following asymptotic behavior at infinity: n z 0 , as z → ∞. Y (z) = I + O(z −1 ) 0 z −n The unique solution of the RH problem is given by ⎛ ⎞ pn (u)e−nVs,t (u) κn−1 −1 p (z) κ du n n ⎜ ⎟ 2πi R u−z ⎟, Y (z) = ⎜ −nV (u) s,t ⎝ ⎠ pn−1 (u)e −2πiκn−1 pn−1 (z) −κn−1 du u−z R
(3.2)
for z ∈ C \ R, (3.3)
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where pk = pk is the k th degree orthonormal polynomial with respect to the varying (n,s,t) > 0 is the leading coefficient of pk . The solution weight e−nVs,t , and where κk = κk (3.3) is due to Fokas, Its, and Kitaev [17], see also [8, 10, 11]. (s,t) It is now possible to write the 2-point kernel K n , see (1.3), in terms of Y . Indeed using the Christoffel-Darboux formula for orthogonal polynomials and the fact that det Y ≡ 1 (which follows easily from (3.1), (3.2), and Liouville’s theorem), we get K n(s,t) (x,
y) = e
− n2 Vs,t (x) − n2 Vs,t (y)
e
−1 1 1 0 1 Y± (y)Y± (x) . 0 2πi(x − y)
(3.4)
So, in order to prove Theorem 1.7, we need to analyze the RH problem for Y asymptotically. We do this by applying the Deift/Zhou steepest descent method [13] to this RH problem. 3.2. Normalization of the RH problem at infinity: Y → T . In order to normalize the RH problem for Y at infinity, the equilibrium measures νs,t , introduced in Sect. 2 play a key role. Consider the log-transform gs,t of νs,t , gs,t (z) =
b
log(z − u)dνs,t (u),
for z ∈ C \ (−∞, b].
(3.5)
a
Here, we take the principal branch of the logarithm so that gs,t is analytic in C\(−∞, b]. We now give properties of gs,t which are crucial in the following. From (3.5) and condition (2.1) it follows that gs,t,+ (x) + gs,t,− (x) − Vs,t (x) − s,t = 0,
for x ∈ [a, b].
(3.6)
for x ∈ R,
(3.7)
Another crucial property is that
b
gs,t,+ (x) − gs,t,− (x) = 2πi
dνs,t (u),
x
so that since νs,t is supported on [a, b] and has mass one (see Corollary 2.2), gs,t,+ (x) − gs,t,− (x) =
2πi, for x < a, 0, for x > b.
(3.8)
Now, we are ready to perform the first transformation Y → T . Define the matrix valued function T as 1
1
T (z) = e− 2 ns,t σ3 Y (z)e−ngs,t (z)σ3 e 2 ns,t σ3 ,
for z ∈ C \ R,
(3.9)
that appears in the variational conditions (2.1) and (2.2), and where s,t isthe constant 0 denotes the third Pauli-matrix. Using (3.6), (3.8), the RH conditions where σ3 = 01 −1 for Y , and the fact that gs,t (z) = log z + O(1/z) as z → ∞, it is straightforward to check that T is a solution to the following RH problem.
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RH problem for T . (a) T : C \ R → C2×2 is analytic. (b) T+ (x) = T− (x)vT (x) for x ∈ R, with ⎧ ⎪ e−n(gs,t,+ −gs,t,− ) 1 ⎪ ⎪ , on (a, b), ⎪ ⎪ ⎨ 0 en(gs,t,+ −gs,t,− ) vT = ⎪ n(gs,t,+ +gs,t,− −Vs,t −s,t ) ⎪ 1 e ⎪ ⎪ , on R \ (a, b). ⎪ ⎩ 0 1 (c) T (z) = I + O(1/z),
(3.10)
as z → ∞.
Remark 3.1. From (3.7) we see that the diagonal entries of vT on (a, b) are rapidly oscillating for large n. Further, using condition (2.2) and (3.5), we see that vT − I decays exponentially on R \ [a − δ, b + δ]. 3.3. Opening of the lens: T → S. Here, we will transform the oscillatory diagonal entries of the jump matrix vT on (a, b) into exponentially decaying off-diagonal entries. This step is referred to as the opening of the lens. Introduce a scalar function φs,t as,
b
φs,t (z) = −πi
ψs,t (ξ )dξ,
for z ∈ V \ (−∞, b],
(3.11)
z
where the path of integration does not cross the real line, and where ψs,t is defined by (2.21). The important feature of the function φs,t is that by (3.7), φs,t,+ and φs,t,− are purely imaginary on (a, b) and satisfy, b −2φs,t,+ (x) = 2φs,t,− (x) = 2πi dνs,t (u) = gs,t,+ (x) − gs,t,− (x), for x ∈ (a, b), x
(3.12) which means that −2φs,t and 2φs,t provide analytic extensions of gs,t,+ − gs,t,− into the upper half-plane and lower half-plane, respectively. Further, 2gs,t + 2φs,t − Vs,t − s,t is analytic in V \ (−∞, b] and satisfies by (3.12) and (3.6), 2gs,t,± + 2φs,t,± − Vs,t − s,t = gs,t,+ + gs,t,− − Vs,t − s,t = 0,
on (a, b),
so that by the identity theorem, 2gs,t − Vs,t − s,t = −2φs,t ,
on V \ (−∞, a].
(3.13)
Using (3.8) this yields, gs,t,+ + gs,t,− − Vs,t − s,t = 2gs,t,− − Vs,t − s,t + (gs,t,+ − gs,t,− ) = −2φs,t,− + 2πi,
on (−∞, a). (3.14)
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Inserting (3.12), (3.13), and (3.14) into (3.10), the jump matrix for T can be written in terms of φs,t as ⎧ 2nφs,t,+ ⎪ e 1 ⎪ ⎪ , on (a, b), ⎪ ⎪ ⎨ 0 e2nφs,t,− vT = ⎪ −2nφs,t,− ⎪ ⎪ 1e ⎪ , on R \ (a, b). ⎪ ⎩ 0 1
(3.15)
It is straightforward to check, using the fact that φs,t,+ + φs,t,− = 0 on (a, b), that vT has on the interval (a, b) the following factorization: vT =
1
e2nφs,t,−
0 1
0 1 −1 0
1
e2nφs,t,+
0 , 1
on (a, b),
(3.16)
and the opening of the lens is based on this factorization. Observe that, since
b Re φs,t,± (x) = 0 and Im φs,t,± (x) = ∓ x dνs,t (u) for x ∈ (a, b) (see (3.12)), and since νs,t is positive on (a + δ, b − δ) for δ > 0 and s, t sufficiently small (see Remark 2.3), it follows (as in [8]) from the Cauchy-Riemann conditions that Re φs,t (z) < 0,
for |Im z| = 0 small and a + δ < Re z < b − δ.
(3.17)
We deform the RH problem for T into a RH problem for S by opening a lens as shown in Fig. 1, so that we obtain a contour . For now, we choose the lens to be contained in V, but we will specify later how we choose the lens exactly. Let ⎧ ⎪ T (z), for z outside the lens, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 0 ⎪ ⎨T (z) , for z in the upper part of the lens, −e2nφs,t (z) 1 S(z) = ⎪ ⎪ ⎪ ⎪ ⎪ 1 0 ⎪ ⎪ for z in the lower part of the lens. ⎪ ⎩T (z) e2nφs,t (z) 1 ,
(3.18)
Then, using (3.16) and the RH conditions for T , one can check that S is the unique solution of the following RH problem:
Fig. 1. The lens
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RH problem for S. (a) S : C \ → C2×2 is analytic. (b) S+ (z) = S− (z)v S (z) for z ∈ , with ⎧ ⎪ 0 1 ⎪ ⎪ , on (a, b), ⎪ ⎪ ⎪ −1 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 0 vS = , on ∩ C± , ⎪ e2nφs,t 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 e−2nφs,t,− ⎪ ⎪ , on R \ (a, b). ⎪ ⎩ 0 1 (c) S(z) = I + O(1/z),
(3.19)
as z → ∞.
Remark 3.2. On the lips of the lens (away from a and b) and on R \ [a − δ, b + δ], it follows from (3.17) and (2.2) that the jump matrix for S converges exponentially fast to the identity matrix as n → ∞. This convergence is uniform as long as we stay away from small disks surrounding the endpoints a and b. Near these endpoints we have to construct local parametrices. 3.4. Parametrix P (∞) for the outside region. From Remark 3.2, we expect that the leading order asymptotics of Y will be determined by a solution P (∞) of the following RH problem: RH problem for P (∞) . (a) P (∞) : C \ [a, b] → C2×2 isanalytic. 0 1 (∞) (∞) , for x ∈ (a, b). (b) P+ (x) = P− (x) −1 0 (c) P (∞) (z) = I + O(1/z), as z → ∞. It is well known, see for example [8, 11], that P (∞) given by P
(∞)
z − b σ3 /4 1 1 −1 1 1 (z) = , i −i i −i z−a
for z ∈ C \ [a, b],
(3.20)
is a solution to the above RH problem. Note that P (∞) is independent of the parameters s, t and n. 3.5. Parametrix P (a) near the regular endpoint a. Here, we do the local analysis near the regular endpoint a. Let Uδ,a = {z ∈ C : |z − a| < δ} be a small disk with center a and radius δ > 0 sufficiently small such that the disk lies in V. We seek a 2 × 2 matrix valued function P (a) (depending on the parameters n, s, and t) in the disk Uδ,a with the same jumps as S and which matches with P (∞) on the boundary ∂Uδ,a of the disk. We thus seek a 2 × 2 matrix valued function that satisfies the following RH problem:
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RH problem for P (a) . (a) P (a) : Uδ,a \ → C2×2 is analytic. (a) (a) (b) P+ (z) = P− (z)v S (z) for z ∈ ∩ Uδ,a , where v S is given by (3.19). (c) P (a) satisfies the matching condition P (a) (z)(P (∞) )(−1) (z) = I + O(n −1/7 ),
(3.21)
as n → ∞ and s, t → 0 such that (1.32) holds, uniformly for z ∈ ∂Uδ,a \ . 3.5.1. Airy model RH problem. We will construct P (a) by introducing an auxiliary 2 × 2 matrix valued function A(ζ ; r ) with jumps (in the variable ζ ) on an oriented contour = j j , shown in Fig. 2, consisting of four straight rays 1 : arg ζ = 0,
2 : arg ζ =
6π , 7
3 : arg ζ = π,
4 : arg ζ = −
6π . 7
These four rays divide the complex plane into four regions I , II , III , and IV , also shown in Fig. 2. Put y j = y j (ζ ; r ) = ω j Ai (ω j (ζ + r )), with ω = e
2πi 3
and with Ai the Airy function, and let, ⎧ ⎪ y0 −y2 ⎪ ⎪ , ⎪ ⎪ y0 −y2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −y1 −y2 ⎪ ⎪ , ⎪ ⎪ ⎨ −y1 −y2 √ − πi A(ζ ; r ) = 2π e 4 × ⎪ ⎪ −y y 2 1 ⎪ ⎪ , ⎪ ⎪ −y2 y1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y0 y1 ⎪ ⎪ ⎩ , y0 y1
j = 0, 1, 2,
for ζ ∈ I , for ζ ∈ II , (3.22) for ζ ∈ III , for ζ ∈ IV .
With y j we mean the derivative of y j with respect to ζ . It is well-known, see e.g. [8, 11], that A satisfies the following RH problem:
Fig. 2. The oriented contour . The four straight rays 1 , . . . , 4 divide the complex plane into four regions I,II,III and IV.
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RH problem for A. (a) A is analytic for ζ ∈ C \ and for r in C. (b) A satisfies the following jump relations on , A+ (ζ ) = A− (ζ ) A+ (ζ ) = A− (ζ ) A+ (ζ ) = A− (ζ )
0 −1 1 0 1 1
1 , 0 1 , 1 0 , 1
for ζ ∈ 3 ,
(3.23)
for ζ ∈ 1 ,
(3.24)
for ζ ∈ 2 ∪ 4 .
(3.25)
(c) A has the following asymptotic behavior at infinity, A(ζ ; r ) = (ζ + r )− =ζ
−
σ3 4
σ3 4
2 3/2 N I + O (ζ + r )−3/2 e− 3 (ζ +r ) σ3
1 1 N I − r 2 ζ −1/2 σ3 + r 4 ζ −1 I + O(r 6 ζ −3/2 ) + O(r ζ −1 ) 4 32 2 3/2 +r ζ 1/2 )σ3
× e−( 3 ζ
,
(3.26)
as ζ → ∞, uniformly for r such that sgn Im (ζ + r ) = sgn Im ζ ,
and
|r | < |ζ |1/4 .
(3.27)
In (3.26), N is given by 1 1 1 − 14 πiσ3 e N=√ . 2 −1 1
(3.28)
3.5.2. Construction of P (a) . We seek P (a) in the following form P (a) (z) = E (a) (z)σ3 A n 2/3 f a (z); n 2/3 rs,t (z) σ3 enφs,t (z)σ3 ,
(3.29)
where E (a) is an invertible 2 × 2 matrix valued function analytic on Uδ,a and where f a and rs,t are (scalar) analytic functions on Uδ,a which are real on (a −δ, a +δ). In addition we take f a to be a conformal map from Uδ,a onto a convex neighborhood f a (Uδ,a ) of 0 such that f a (a) = 0 and f a (a) < 0. If those conditions are all satisfied, and if we open the lens (recall that the lens was not yet fully specified) such that f a ( ∩ (Uδ,a ∩ C+ )) = 4 ∩ f a (Uδ,a ), and f a ( ∩ (Uδ,a ∩ C− )) = 2 ∩ f a (Uδ,a ), then it is straightforward to verify, using (3.19) and (3.23)–(3.25), that P (a) defined by (3.29) satisfies conditions (a) and (b) of the RH problem for P (a) .
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Let 2/3 a 3 −3/2 −πi f a (z) = ψ0 (ξ )dξ (a − z) (a − z) 2 z 2/3 √ 1 (z − a) + O((z − a)2 ), =− h 0 (a) b − a 2
as z → a,
where we have used (2.5), and let a (sψ1 (ξ ) + tψ2 (ξ ))dξ f a (z)−1/2 . rs,t (z) = −πi
(3.30)
(3.31)
z
Then, f a is analytic with f a (a) = 0 and f a (a) < 0, it is real on (a − δ, a + δ), and it is a conformal mapping on Uδ,a provided δ > 0 is sufficiently small. Further, it is straightforward to check that rs,t is analytic on Uδ,a and real on (a − δ, a + δ), as well. Thus, f a and rs,t satisfy the above conditions, so that P (a) defined by (3.29), with E (a) any invertible analytic matrix valued function, satisfies conditions (a) and (b) of the RH problem for P (a) . Remark 3.3. We can use any functions f a and rs,t , satisfying the conditions stated under Eq. (3.29), to construct the parametrix P (a) . However, we have to choose them so as to compensate for the factor enφs,t σ3 in (3.29). Using (2.21), (3.11), and the fact that
b a ψs,t+ (u)du = 1 we have e
−n
2 3/2 +r (z) f (z)1/2 s,t a 3 f a (z)
σ3
= (−1)n e−nφs,t (z)σ3 ,
for z ∈ Uδ,a \ [a, a + δ). (3.32)
From this and (3.26) it is clear that our choice of f a and rs,t will do the job. It now remains to determine E (a) such that the matching condition (c) holds as well. In order to do this we make use of the following result. Proposition 3.4. Let n → ∞ and s, t → 0 such that (1.32) holds. Then, −σ3 /4 P (a) (z) = (−1)n E (a) (z) n 2/3 f a (z) σ3 N σ3 (n 4/7rs,t (z))2 (n 4/7rs,t (z))4 −1/7 −2/7 −3/7 In × I− σ3 n + + O(n ) . 4 f a (z)1/2 32 f a (z) (3.33) Proof. We will use the asymptotics (3.26) of A. In order to do this we have to check that condition (3.27) is satisfied for our choice of ζ = n 2/3 f a (z) and r = n 2/3rs,t (z). Obviously rs,t (z) = O(n −4/7 ) as n → ∞ and s, t → 0 such that (1.32) holds, uniformly for z ∈ ∂Uδ,a . Then, it is straightforward to check that there exists n 0 ∈ N sufficiently large, and κ1 , κ2 > 0 sufficiently small, such that |n 2/3rs,t (z)| < |n 2/3 f a (z)|1/4 ,
(3.34)
for z ∈ ∂Uδ,a (for a possible smaller δ), for n ≥ n 0 , and for s and t such that |c1 n 6/7 s − s0 | ≤ κ1 and |c1 n 4/7 t − t0 | ≤ κ2 .
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Further, since f a and rs,t are analytic near a and real valued on (a − δ, a + δ) one can check that Im f a (z) = f a (Re z)Im z + O((Im z)2 ), (Re z) + O((Im z)2 ), Im rs,t (z) = rs,t
as z → a, as z → a.
(Re z) = O(n −4/7 ) uniformly for z ∈ U Now, since f a (a) = 0 and rs,t δ,a one then can find a constant C > 0 such that,
|Im rs,t (z)| < C|Im z| < |Im f a (z)|, for z ∈ ∂Uδ,a \ (a − δ, a + δ), for n ≥ n 0 , and for s and t such that |c1 n 6/7 s − s0 | ≤ κ1 and |c1 n 4/7 t − t0 | ≤ κ2 (for a possible smaller δ, κ1 and κ2 , and for a possible larger n 0 ). This yields sgn(Im ( f a (z) + rs,t (z))) = sgn(Im f a (z)).
(3.35)
We now have shown that condition (3.27) is satisfied so that we can use the asymptotic behavior (3.26) of A. Using (3.29), (3.26), (3.32), and the fact that rs,t (z) = O(n −4/7 ) we obtain (3.33). From (3.33) and the fact that rs,t = O(n −4/7 ) it is clear that (in order that the matching condition (c) is satisfied) we have to define E (a) by E (a) = (−1)n P (∞) σ3 N −1 σ3 (n 2/3 f a )σ3 /4 .
(3.36)
E (a)
Obviously, is well-defined and analytic in Uδ,a \ (a, a + δ). Further, using condition 1/4 1/4 (b) of the RH problem for P (∞) , Eq. (3.28), and the fact that f a,− = i f a,+ on (a, a + δ), it is straightforward to check that E (a) has no jump on (a, a + δ). We then have that E (a) is analytic in Uδ,a except for a possible isolated singularity at a. However, E (a) has at most a square root singularity at a and hence it has to be a removable singularity. Further, since det P (∞) ≡ 1 and det N = 1 it is clear that det E (a) ≡ 1 and thus E (a) is invertible. This ends the construction of the parametrix near the regular endpoint. 3.6. Parametrix P (b) near the critical endpoint b. Here, we do the local analysis near the critical endpoint b. Let Uδ,b = {z ∈ C : |z − b| < δ} be a small disk with center b and radius δ > 0 sufficiently small such that Uδ,b lies in V and such that the disks Uδ,a and Uδ,b do not intersect. We seek a 2 × 2 matrix valued function P (b) (depending on n, s and t) in the disk Uδ,b with the same jumps as S and with matches with P (∞) on the boundary ∂Uδ,b of the disk. We thus seek a 2 × 2 matrix valued function that satisfies the following RH problem: RH problem for P (b) . (a) P (b) : Uδ,b \ → C2×2 is analytic. (b) P+(b) (z) = P−(b) (z)v S (z) for z ∈ Uδ,b ∩ , where v S is given by (3.19). (c) P (b) satisfies the matching condition P (b) (z)(P (∞) )−1 (z) = I + O(n −1/7 ),
(3.37)
as n → ∞ and s, t → 0 such that (1.32) holds, uniformly for z ∈ ∂Uδ,b \ . Due to the singular behavior of the equilibrium measure dν0 (x) near b, see Assumptions 1.1 (ii), the Airy parametrix does not fit near b. Instead we use a different model RH problem associated with the PI2 equation (1.14).
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3.6.1. Model RH problem for the PI2 equation. We construct P (b) by introducing the following model RH problem for the special solution y of the PI2 equation (1.14) as discussed in Sect. 1.3. This RH problem depends on two complex parameters s, t and has jumps on the oriented contour as defined in Sect. 3.5, see Fig. 2. We seek a 2 × 2 matrix valued function (ζ ) = (ζ ; s, t) satisfying the following conditions: RH problem for . (a) is analytic for ζ ∈ C \ . (b) satisfies the following jump relations on , 0 1 , + (ζ ) = − (ζ ) −1 0 1 1 + (ζ ) = − (ζ ) , 0 1 1 0 + (ζ ) = − (ζ ) , 1 1
for ζ ∈ 3 ,
(3.38)
for ζ ∈ 1 ,
(3.39)
for ζ ∈ 2 ∪ 4 .
(3.40)
(c) has the following behavior at infinity, 1
(ζ ) = ζ − 4 σ3 N
I − hσ3 ζ −1/2 +
1 h2 i y −1 −3/2 ζ + O(ζ ) e−θ(ζ ;s,t)σ3 , 2 −i y h 2 (3.41)
where y = y(s, t) is the special solution of the PI2 equation (1.14) as discussed in Sect. 1.3, where ∂h ∂s = −y, where N is given by (3.28), and where θ is given by (1.21). Remark 3.5. Note that the only difference between the model RH problem for Airy functions and the one for PI2 lies in the asymptotic condition (c). In particular, in θ we have an extra factor ζ 7/2 . If we fix s0 , t0 ∈ R, it was proven in [7, Lemma 2.3 and Prop. 2.5] that there exists a neighborhood U of s0 and a neighborhood W of t0 such that the RH problem for is (uniquely) solvable for all (s, t) ∈ U × W. Furthermore, for (s, t) ∈ U × W, is analytic both in s and t, and condition (c) holds uniformly for (s, t) in compact subsets of U × W. In [7, Sect. 2.3], the authors have shown that the solution of the RH problem for satisfies the Lax pair (1.15)–(1.18). From (1.20), (3.41), and (3.40) we then obtain ⎧ ⎪ (ζ ; s, t) ⎪ 11 ⎪ , ⎪ ⎪ ⎨ 21 (ζ ; s, t) 1 (ζ ; s, t) = 2 (ζ ; s, t) ⎪ ⎪ 1 ⎪ 11 (ζ ; s, t) ⎪ ⎪ ⎩ 21 (ζ ; s, t) 1
for 0 < Arg ζ < 6π/7, 0 , 1
(3.42) for 6π/7 < Arg ζ < π .
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3.6.2. Construction of P (b) . We seek P (b) in the following form: P (b) (z) = E (b) (z) n 2/7 f b (z); n 6/7 s f 1 (z), n 4/7 t f 2 (z) enφs,t (z)σ3 ,
(3.43)
where E (b) is an invertible 2 × 2 matrix valued function analytic on Uδ,b and where f b , f 1 , and f 2 are (scalar) analytic functions on Uδ,b which are real on (b − δ, b + δ). We take f 1 and f 2 to be such that f 1 (b) = c1 and f 2 (b) = c2 (where c1 and c2 are given by (1.30)). Then it is clear from (1.32) that for n sufficiently large and s and t sufficiently small, n 6/7 s f 1 (z) ∈ U, and n 4/7 t f 2 (z) ∈ W, for z ∈ Uδ,b , where U and W are the neighborhoods of s0 and t0 where exists. In addition we take f b to be a conformal map from Uδ,b onto a convex neighborhood f b (Uδ,b ) of 0 such that f b (b) = 0 and f b (b) > 0. If those conditions are all satisfied, and if we open the lens (recall that the lens was not yet fully specified near b) such that f b ( ∩ (Uδ,b ∩ C+ )) = 2 ∩ f b (Uδ,b ), and f b ( ∩ (Uδ,b ∩ C− )) = 4 ∩ f b (Uδ,a ), then it is straightforward to verify, using (3.19) and (3.38)–(3.40), that P (b) defined by (3.43) satisfies conditions (a) and (b) of the RH problem for P (b) . Let 2/7 b −7/2 f b (z) = 105 −πi ψ0 (ξ )dξ (z − b) (z − b) = c(z − b) + O(z − b)2 , z
(3.44) as z → 0, where
c=
15 √ h (b) b − a 2 0
2/7 .
To get the expansion of f b near b we have used (2.5) and the facts that h 0 (b) = h 0 (b) = 0 (see (2.7)). Further since h 0 (b) > 0 we have that c > 0. So, we have defined an analytic function f b with f b (b) = 0 and f b (b) = c > 0, which is real on (b − δ, b + δ), and which is a conformal mapping on Uδ,b provided δ > 0 is sufficiently small. Next, let f 1 and f 2 be defined by b b −1/2 f 1 (z) = −πi ψ1 (ξ )dξ f b (z) , f 2 (z) = −3 −πi ψ2 (ξ )dξ f b (z)−3/2 . z
z
(3.45) Since f b is a conformal mapping in Uδ,b it is clear from (2.14) and (2.6) that f 1 is analytic in Uδ,b . To see that f 2 is analytic in Uδ,b as well, we also need to use the extra condition h 2 (b) = 0 (see (2.12)). Further, f 1 and f 2 are real on (b − δ, b + δ) and one can check that, f 1 (b) =
h 1 (b) c1/2 (b − a)1/2
= c1 ,
f 2 (b) = −
h 2 (b) 3/2 c (b − a)1/2
= c2 .
(3.46)
Thus, f b , f 1 , and f 2 satisfy the above conditions, so that P (b) defined by (3.43), with E (b) any invertible analytic matrix valued function, satisfies conditions (a) and (b) of the RH problem for P (b) .
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Remark 3.6. As in Remark 3.3 we note that we could have also used different functions f b , f 1 , and f 2 . However, we have to choose them so as to compensate for the factor enφs,t σ3 in (3.43). Using (1.21), (3.44), (3.45), (2.21), and (3.11) we have θ (n 2/7 f b (z); n 6/7 s f 1 (z), n 4/7 t f 2 (z)) = nφs,t (z),
for z ∈ Uδ,b \ (b − δ, b]. (3.47)
From this and (3.41) it is clear that our choice of f b , f 1 , and f 2 will do the job. It now remains to determine E (b) such that the matching condition (c) holds as well. In order to do this we make use of the following proposition (the analogon of Proposition 3.4). Proposition 3.7. Let n → ∞ and s, t → 0 such that (1.32) holds. Then, −σ3 /4 P (b) (z) = E (b) (z) n 2/7 f b (z) N 1 h2 i y −1 −2/7 −3/7 (z) n + O(n ) , × I − h f b (z)−1/2 σ3 n −1/7 + f b 2 −i y h 2 (3.48) where we have used for brevity the notation h = h(n 6/7 s f 1 (z), n 4/7 t f 2 (z)),
and
y = y(n 6/7 s f 1 (z), n 4/7 t f 2 (z)).
Proof. This follows easily from (3.43), (3.41), and (3.47)
From (3.48) it is clear that (in order that the matching condition (c) is satisfied) we have to define E (b) by, σ3 /4 E (b) = P (∞) N −1 n 2/7 f b ,
(3.49)
where N is given by (3.28) and where P (∞) is the parametrix for the outside region, given by (3.20). Similarly as we have proven that E (a) is an invertible analytic matrix valued function in Uδ,a , we can check that E (b) is invertible and analytic in Uδ,b . This completes the construction of the parametrix near the singular endpoint. 3.7. Final transformation: S → R. Having the parametrix P (∞) for the outside region and the parametrices P (a) and P (b) near the endpoints a and b, we have all the ingredients to perform the final transformation of the RH problem. Define ⎧ (a) −1 ⎪ ⎨ S(z) P (z), for z ∈ Uδ,a \ , R(z) = S(z) P (b) −1 (z), for z ∈ Uδ,b \ , ⎪ −1 ⎩ S(z) P (∞) (z), for z ∈ C \ ( ∪ Uδ,a ∪ Uδ,b ).
(3.50)
Then, by construction of the parametrices, R has only jumps on the reduced system of contours R shown in Fig. 3, and R satisfies the following RH problem. The circles around a and b are oriented clockwise.
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523
Fig. 3. The contour R after the third and final transformation
RH problem for R. (a) R : C \ R → C2×2 is analytic. (b) R+ (z) = R− (z)v R (z) for z ∈ R , with ⎧ (∞) −1 (a) ⎪ on ∂Uδ,a , ⎨P P , v R = P (b) P (∞) −1 , on ∂Uδ,b , ⎪ ⎩ (∞) (∞) −1 P vS P , on the rest of R .
(3.51)
(c) R(z) = I + O(1/z), as z → ∞. (d) R remains bounded near the intersection points of R . As n → ∞ and s, t → 0 such that (1.32) holds, we have by construction of the parametrices that the jump matrix for R is close to the identity matrix, both in L 2 and L ∞ - sense on R , I + O(n −1/7 ), on ∂Uδ,a ∪ ∂Uδ,b , v R (z) = (3.52) I + O(e−γ n ), on the rest of R , with γ > 0 some fixed constant. Then, arguments as in [10, 11] guarantee that R itself is close to the identity matrix, R(z) = I + O(n −1/7 ),
uniformly for z ∈ C \ R ,
(3.53)
as n → ∞ and s, t → 0 such that (1.32) holds. This completes the Deift/Zhou steepest descent analysis. Remark 3.8. The Deift/Zhou steepest descent method can be generalized to the case where the support of ν0 consists of more than one interval. However, there are two (technical) differences. First, in the multi-interval case, the equilibrium measures ν1 and ν2 have densities which are more complicated than in the one-interval case, but it remains possible to give explicit formulae. Consequently, condition (1.27), which expresses the requirement that the density of ν2 vanishes at the singular endpoint, has to be modified. Further, the construction of the outside parametrix P (∞) is more complicated, since it uses -functions as in [10, Lemma 4.3]. With these modifications the asymptotic analysis can be carried through in the multi-interval case. 3.8. Asymptotics of R. For the purpose of proving the universality result for the kernel K n(s,t) (Theorem 1.7) it is enough to unfold the series of transformations Y → T → S → R and to use (3.53). This will be done in the next section. However, in order to (n,s,t) (n,s,t) and bn (Theorem determine the asymptotics of the recurrence coefficients an −1/7 1.11) we need to expand the O(n ) term in (3.53).
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We show that the jump matrix v R for R has an expansion of the form, v R (z) = I +
1 (z) 2 (z) + 2/7 + O(n −3/7 ), n 1/7 n
(3.54)
as n → ∞ and s, t → 0 such that (1.32) holds, uniformly for z ∈ R , and we will explicitly determine 1 and 2 . On R \(∂Uδ,a ∪∂Uδ,b ), the jump matrix is the identity matrix plus an exponentially small term, so that 1 (z) = 0,
2 (z) = 0,
for z ∈ R \ (∂Uδ,a ∪ ∂Uδ,b ).
(3.55)
Now, from (3.51), (3.33), and (3.48) we obtain (3.54) with, 2 1 4/7 n rs,t (z) f a (z)−1/2 P (∞) (z)σ3 P (∞) (z)−1 , for z ∈ ∂Uδ,a , 4 1 (z) = −h f 0 (z)−1/2 P (∞) (z)σ3 P (∞) (z)−1 , for z ∈ ∂Uδ,b ,
1 (z) = −
(3.56) (3.57)
and 4 1 4/7 n rs,t (z) f a (z)−1 I, 32 2 1 h iy −1 , 2 (z) = f 0 (z) −i y h 2 2 2 (z) =
for z ∈ ∂Uδ,a ,
(3.58)
for z ∈ ∂Uδ,b ,
(3.59)
where we have used for brevity h = h(n 6/7 s f 1 (z), n 4/7 t f 2 (z)),
y = y(n 6/7 s f 1 (z), n 4/7 t f 2 (z)).
Observe that 1 and 2 have an extension to an analytic function in a punctured neighborhood of a and a punctured neighborhood of b with simple poles at a and b. As in [11, Theorem 7.10] we obtain from (3.54) that R satisfies, R(z) = I +
R (1) (z) R (2) (z) + + O(n −3/7 ), n 1/7 n 2/7
(3.60)
as n → ∞ and s, t → 0 such that (1.32) holds, which is valid uniformly for z ∈ C \ (∂Uδ,a ∪ ∂Uδ,b ). We have that R (1) and R (2) are analytic on C \ (∂Uδ,a ∪ ∂Uδ,b ), R
(1)
(z) = O(1/z),
R
(2)
(z) = O(1/z),
We will now compute the functions R (1) and R (2) explicitly.
as z → ∞.
(3.61) (3.62)
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525
Determination of R (1) . Expanding the jump relation R+ = R− v R using (3.54) and (3.60), and collecting the terms with n −1/7 we find (1)
(1)
R+ (z) = R− (z) + 1 (z),
for z ∈ ∂Uδ,a ∪ ∂Uδ,b .
This together with (3.61) and (3.62) gives an additive RH problem for R (1) . Recall that 1 is analytic in a neighborhood of z = a and z = b except for simple poles at a and b. So, 1 (z) =
A(1) + O(1), z−a
as z → a,
1 (z) =
B (1) + O(1), z−b
as z → b,
for certain matrices A(1) and B (1) . We then see by inspection that ⎧ (1) A B (1) ⎪ ⎪ + for z ∈ C \ (U δ,a ∪ U δ,b ), ⎪ ⎨z − a z − b, R (1) (z) = ⎪ (1) ⎪ B (1) ⎪ ⎩ A + − 1 (z), for z ∈ Uδ,a ∪ Uδ,b , z−a z−b
(3.63)
solves the additive RH problem for R (1) . It now remains to determine A(1) and B (1) . This can be done by expanding the formulas (3.56) and (3.57) near z = a and z = b, respectively. We then find after a straightforward calculation (using also the fact that f 1 (b) = c1 and f 2 (b) = c2 , see (3.46), 1√ 1 i , b − a (n 4/7rs,t (a))2 (− f a (a))−1/2 i −1 8 1 √ −1 i , = h b − a f b (b)−1/2 i 1 2
A(1) =
(3.64)
B (1)
(3.65)
where we used h to denote h(c1 n 6/7 s, c2 n 4/7 t) for brevity. Determination of R (2) . Next, expanding the jump relation R+ = R− v R using (3.54) and (3.60), and collecting the terms with n −2/7 we find (2)
(2)
(1)
R+ (z) = R− (z) + R− (z)1 (z) + 2 (z),
for z ∈ ∂Uδ,a ∪ ∂Uδ,b .
(1) This together with (3.61) and (3.62) gives an additive RH problem for R (2) . Since R− is the boundary value of the restriction of R (1) to the disks Uδ,a and Uδ,b and since 1 and 2 are analytic in a neighborhood of a and b, except for simple poles at a and b, we have
A(2) + O(1), z−a B (2) + O(1), R (1) (z)1 (z) + 2 (z) = z−b
R (1) (z)1 (z) + 2 (z) =
as z → a, as z → b,
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T. Claeys, M. Vanlessen
for certain matrices A(2) and B (2) . As in the determination of R (1) we then see by inspection that ⎧ (2) A B (2) ⎪ ⎪ for z ∈ C \ (U δ,a ∪ U δ,b ), ⎪ ⎨z − a + z − b, (2) R (z) = (3.66) ⎪ A(2) (2) ⎪ B ⎪ (1) ⎩ + − R (z)1 (z) − 2 (z), for z ∈ Uδ,a ∪ Uδ,b , z−a z−b solves the additive RH problem for R (2) . The determination of A(2) and B (2) is more complicated than the determination of A(1) and B (1) . It involves R (1) (a) and R (1) (b) for which we need to determine also the constant terms in the expansions of 1 near z = a and z = b. After a straightforward (but rather long calculation) we find, (n 4/7rs,t (a))2 h (n 4/7rs,t (a))4 0 i 1 i , (3.67) + A(2) = 32(− f a (a)) −i 0 8(− f a (a))1/2 f b (b)1/2 −i 1 B
(2)
y + h2 0 = 2 f b (b) −i
(n 4/7rs,t (a))2 h −1 i + 1/2 1/2 −i 0 8(− f a (a)) f b (b)
i , −1
(3.68)
where we used h and y to denote h(c1 n 6/7 s, c2 n 4/7 t) and y(c1 n 6/7 s, c2 n 4/7 t) for brevity. 4. Universality of the Double Scaling Limit Here, we will prove the universality result for the 2-point correlation kernel K n(s,t) . We (s,t) do this by using the expression (3.4) for K n in terms of Y and by unfolding the series of transformations Y → T → S → R. Proof of Theorem 1.11. From Eqs. (3.4), (3.9), and (3.13), the reader can verify that the (s,t) 2-point kernel K n can be written as, cf. [5, 6], −1 1 1 0 1 T+ (y)T+ (x) , for x, y ∈ R. K n(s,t) (x, y) = e−nφs,t,+ (x) e−nφs,t,+ (y) 0 2πi(x − y) From (3.18) and the fact that S+ = R P+(b) on (b − δ, b + δ), see (3.50), we have ⎧ ⎪ R P (b) , on (b, b + δ), ⎪ ⎨ + T+ = 1 0 (b) ⎪ , on (b − δ, b). ⎪ ⎩ R P+ 2nφ s,t,+ e 1 (s,t)
Inserting this in the previous equation for K n
we arrive at,
1 01 2πi(x − y) 1 , × P −1 (y)R −1 (y)R(x) P(x) 0
K n(s,t) (x, y) = e−nφs,t,+ (x) e−nφs,t,+ (y)
(4.1)
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527
for x ∈ (b − δ, b + δ), where ⎧ (b) ⎪ ⎨ P+ , on (b, b + δ), P= 1 0 (b) ⎪ , on (b − δ, b). ⎩ P+ e2nφs,t,+ 1
(4.2)
Further, we define ⎧ ⎪ ⎨+ = 1 ⎪ ⎩+ 1
on R+ , (4.3)
0 , on R− , 1
where is the solution of the RH problem for , see Sect. 3.6. By (3.42), we have that 11 = 1 and 21 = 2 . Using (3.43), (4.2), and (4.3) a straightforward calculation yields, P(x) = E (b) (x) n 2/7 f b (x); n 6/7 s f 1 (x), n 4/7 t f 2 (x) enφs,t,+ (x)σ3 , for x ∈ (b − δ, b + δ). Inserting this into (4.1) we then obtain, −1 2/7 1 01 n f b (y); n 6/7 s f 1 (y), n 4/7 t f 2 (y) 2πi(x − y) 1 ×(E (b) )−1 (y)R −1 (y)R(x)E (b) (x) n 2/7 f b (x); n 6/7 s f 1 (x), n 4/7 t f 2 (y) , 0 (4.4)
K n(s,t) (x, y) =
for x ∈ (b − δ, b + δ). Now, we introduce for the sake of brevity some notation. Let un = b +
2/7 √ u v , and vn = b + 2/7 , with c = f b (b) = 15 h 0 (b) b − a . 2 2/7 cn cn (4.5)
We then have, lim n 2/7 f b (u n ) = u,
n→∞
and
lim n 2/7 f b (vn ) = v.
n→∞
(4.6)
Furthermore, since f 1 (b) = c1 and f 2 (b) = c2 (see (3.46)) we have in the limit as n → ∞ and s, t → 0 such that (1.32) holds, lim n 6/7 s f 1 (u n ) = s0 , lim n
4/7
t f 2 (u n ) = t0 ,
lim n 6/7 s f 1 (vn ) = s0 ,
(4.7)
t f 2 (vn ) = t0 .
(4.8)
lim n
4/7
Now, a similar argument as in [24] shows that lim E b−1 (vn )R(vn )−1 R(u n )E b (u n ) = I.
(4.9)
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Inserting (4.6)–(4.9) into (4.4) and using the fact that 11 = 1 and 21 = 2 it is then straightforward to obtain lim
1 K (s,t) (u n , vn ) cn 2/7 n −1 1 1 0 1 (v; s0 , t0 )(u; s0 , t0 ) = 0 2πi(u − v) 1 = (1 (u; s0 , t0 )2 (v; s0 , t0 ) − 1 (v; s0 , t0 )2 (u; s0 , t0 )) , (4.10) −2πi(u − v)
where we take the limit n → ∞ and s, t → 0 such that (1.32) holds. This completes the proof of Theorem 1.7. 5. Asymptotics of the Recurrence Coefficients We will now determine the asymptotics of an(n,s,t) and bn(n,s,t) as n → ∞ and s, t → 0 such that (1.32) holds. In order to do this, we make use of the following result, see e.g. [8, 11]. Let Y be the unique solution of the RH problem for Y . There exist 2 × 2 constant (independent of z but depending on n, s and t) matrices Y1 and Y2 such that Y (z)
z −n 0 0 zn
=I+
Y1 Y2 + 2 + O(1/z 3 ), z z
as z → ∞,
(5.1)
and an(n,s,t) =
(Y1 )12 (Y1 )21 ,
bn(n,s,t) = (Y1 )11 +
(Y2 )12 . (Y1 )12
(5.2)
We need to determine the constant matrices Y1 and Y2 . For large |z| it follows from (3.9), (3.18), and (3.50), that 1
1
Y (z) = e 2 ns,t σ3 R(z)P (∞) (z)engs,t (z)σ3 e− 2 ns,t σ3 .
(5.3)
So, in order to determine Y1 and Y2 we need the asymptotic behavior of P (∞) (z), engs,t (z)σ3 , and R(z) as z → ∞. Asymptotic behavior of P (∞) (z) as z → ∞. Expanding the factor ((z − b)/(z − a))σ3 /4 in (3.20) at z = ∞ it is clear that,
P (∞) (z) = I +
(∞)
P1 z
(∞)
+
P2 z
+ O(1/z 3 ),
as z → ∞,
(5.4)
with P1(∞)
i 0 1 , = (b − a) −1 0 4
P2(∞)
i 2 ∗ 1 2 . = (b − a ) −1 ∗ 8
(5.5)
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Asymptotic behavior of engs,t (z)σ3 as z → ∞. By (3.5) we have −n G1 G2 0 z + 2 + O(1/z 3 ), =I+ as z → ∞, engs,t (z)σ3 0 zn z z with
G 1 = −n
b
a
1 0 udνs,t (u) , 0 −1
∗0 G2 = . 0∗
(5.6)
(5.7)
Asymptotic behavior of R(z) as z → ∞. As in [11] the matrix valued function R has the following asymptotic behavior at infinity: R(z) = I +
R1 R2 + 2 + O(1/z 3 ), z z
as z → ∞.
The compatibility with (3.60), (3.63), and (3.66) yields that R1 = A(1) + B (1) n −1/7 + A(2) + B (2) n −2/7 + O(n −3/7 ), R2 = a A(1) + bB (1) n −1/7 + a A(2) + bB (2) n −2/7 + O(n −3/7 ),
(5.8)
(5.9) (5.10)
as n → ∞ and s, t → 0 such that (1.32) holds. Here, A(1) , B (1) , A(2) , and B (2) are given by (3.64), (3.65), (3.67), and (3.68), respectively. Now, we are ready to determine the asymptotics of the recurrence coefficients. Proof of Theorem 1.11. Note that by (5.3), (5.4), (5.6) and (5.8), 1 1 Y1 = e 2 ns,t σ3 P1(∞) + G 1 + R1 e− 2 ns,t σ3
(5.11)
and
1 1 (∞) (∞) (∞) Y2 = e 2 ns,t σ3 P2 + G 2 + R2 + R1 P1 + P1 + R1 G 1 e− 2 ns,t σ3 .
(5.12)
(n,s,t)
. Inserting (5.11) into (5.2), and using We start with the recurrence coefficient an (∞) (∞) the facts that (P1 )12 = −(P1 )21 = i(b −a)/4 (by (5.5)) and (G 1 )12 = (G 1 )21 = 0 (by (5.7)), we obtain ! "1/2 b−a 2 b−a (n,s,t) = +i . (5.13) an ((R1 )21 − (R1 )12 ) + (R1 )12 (R1 )21 4 4 Now, from the formula (5.9) for R1 and the formulas (3.64), (3.65), (3.67), and (3.68) for A(1) , B (1) , A(2) , and B (2) , we have ! 4/7 2 " y (n rs,t (a))2 h + (R1 )21 − (R1 )12 = −i + n −2/7 + O(n −3/7 ), f b (b) 4(− f a (a))1/2 f b (b) and (R1 )12 (R1 )21
b−a =− 4
(n 4/7rs,t (a))2 h + 1/2 4(− f a (a)) f b (b)
2
n −2/7 + O(n −3/7 ).
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Note that we have used y to denote y(c1 n 6/7 s, c2 n 4/7 t) for brevity. Inserting the latter two equations into (5.13) and using the fact that f b (b) = c (by (3.44)) we then obtain (1.39). We will now consider the recurrence coefficient bn(n,s,t) . Inserting (5.11) and (5.12) (∞) (∞) into (5.2), and using the facts that (P1 )11 = (P1 )22 = 0, (G 1 )12 = (G 2 )12 = 0, (G 1 )11 + (G 1 )22 = 0, and R1 = O(n −1/7 ), we obtain bn(n,s,t) = (R1 )11 +
(∞)
(P2 !
= (R1 )11 +
(∞)
)12 + (R1 )11(P1 (∞) (P1 )12
(P2
(∞)
)12
(P1
(∞)
)12
⎡ × ⎣1 −
(P1(∞) )12
(∞)
+
+ (R1 )12
+ (R1 )11 +
(R1 )12
)12 + (R2 )12
(R1 )12
(R2 )12 (∞)
(P1
"
)12 ⎤
2
(P1(∞) )12
+ O(n −3/7 )⎦ .
(∞)
Since (P1 )12 = i(b − a)/4, (P2 )12 = i(b2 − a 2 )/8, R1 = O(n −1/7 ), and R2 = O(n −1/7 ) we then obtain after a straightforward calculation and combining terms, bn(n,s,t) =
4i b+a b+a 4i + 2(R1 )11 + 2i (R1 )12 − (R2 )12 (R1 )12 1+ 2 b−a b−a b−a 4i (R1 )11 (R1 )12 + O(n −3/7 ). − (5.14) b−a
Now, from (5.9), (5.10), (3.64), (3.65), (3.67), and (3.68) we have b+a 4i (2) (2) (R1 )12 − (R2 )12 = 2i A12 − B12 n −2/7 + O(n −3/7 ) b−a b−a h2 (n 4/7rs,t (a))4 y + − n −2/7 + O(n −3/7 ), = f b (b) f b (b) 16(− f a (a))
2(R1 )11 + 2i
and (R1 )11 (R1 )12 = −i = −i
(1) 2 A12
b−a 4
−
(1) 2 B12
n −2/7 + O(n −3/7 )
h2 (n 4/7rs,t (a))4 − f b (b) 16(− f a (a))
n −2/7 + O(n −3/7 ).
Inserting the latter two equations into (5.14) and using the facts that (R1 )12 = O(n −1/7 ) and f b (b) = c we obtain (1.40). So, the theorem is proven.
Universality of Eigen value Correlation Kernel in Double Scaling Limit
531
Acknowledgements. We thank Arno Kuijlaars for careful reading and stimulating discussions. The authors are supported by FWO research project G.0455.04, by K.U.Leuven research grant OT/04/24, and by INTAS Research Network NeCCA 03-51-6637. The second author is Postdoctoral Fellow of the Fund for Scientific Research - Flanders (Belgium).
References 1. Bleher, P., Its, A.: Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. Math. 150, 185–266 (1999) 2. Bleher, P., Its, A.: Double scaling limit in the random matrix model: the Riemann-Hilbert approach. Comm. Pure Appl. Math. 56, 433–516 (2003) 3. Bowick, M.J., Brézin, E.: Universal scaling of the tail of the density of eigenvalues in random matrix models, Phys. Lett. B 268(1), 21–28 (1991) 4. Brézin, E., Marinari, E., Parisi, G.: A non-perturbative ambiguity free solution of a string model. Phys. Lett. B 242(1), 35–38 (1990) 5. Claeys, T., Kuijlaars, A.B.J.: Universality of the double scaling limit in random matrix models. Comm. Pure Appl. Math. 59, 1573–1603 (2006) 6. Claeys, T., Kuijlaars, A.B.J., Vanlessen, M.: Multi- critical unitary random matrix ensembles and the general Painlevé II equation. http://arxiv.org/list/math-ph/0508062, to appear in Ann. Math. 7. Claeys, T., Vanlessen, M.: The existence of a real pole- free solution of the fourth order analogue of the Painlevé I equation. Nonlinearity 20, 1163–1184 (2007) 8. Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann- Hilbert Approach. Courant Lecture Notes 3, New York: New York University, 1999 9. Deift, P., Kriecherbauer, T., McLaughlin, K.T-R.: New results on the equilibrium measure for logarithmic potentials in the presence of an external field. J. Approx. Theory 95, 388–475 (1998) 10. Deift, P., Kriecherbauer, T., McLaughlin, K.T-R., Venakides, S., Zhou, X.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52, 1335–1425 (1999) 11. Deift, P., Kriecherbauer, T., McLaughlin, K.T-R., Venakides, S., Zhou, X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52, 1491–1552 (1999) 12. Deift, P., McLaughlin, K.T-R.: A continuum limit of the Toda lattice. Vol. 131, Memoirs of the Amer. Math. Soc. 624, Providence, RI: Amer. Math. Soc., 1998 13. Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. 137, 295–368 (1993) 14. Dubrovin, B., Liu, S.-Q., Zhang, Y.: On Hamiltonian perturbations of hyperbolic systems of conservation laws I: quasi-triviality of bi-Hamiltonian perturbations. Comm. Pure Appl. Math. 59(4), 559–615 (2006) 15. Dubrovin, B.: On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour. Commun. Math. Phys. 267, 117–139 (2006) 16. Duits, M., Kuijlaars, A.B.J.: Painlevé I asymptotics for orthogonal polynomials with respect to a varying quartic weight. Nonlinearity 19, 2211–2245 (2006) 17. Fokas, A.S., Its, A.R., Kitaev, A.V.: The isomonodromy approach to matrix models in 2D quantum gravity. Commun. Math. Phys. 147, 395–430 (1992) 18. Hastings, S.P., McLeod, J.B.: A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation. Arch. Rat. Mech. Anal. 73, 31–51 (1980) 19. Kapaev, A.A.: Weakly nonlinear solutions of equation PI2 . J. Math. Sc. 73(4), 468–481 (1995) 20. Kawai, T., Koike, T., Nishikawa, Y., Takei, Y.: On the Stokes geometry of higher order Painlevé equations. Analyse Complexe, systèmes Dynamiques, sommabilité Des séries Divergentes Et théories Galoisiennes. II. Astérisque 297, 117–166 (2004) 21. Kuijlaars, A.B.J., McLaughlin, K.T-R.: Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields. Comm. Pure Appl. Math. 53, 736–785 (2000) 22. Kuijlaars, A.B.J., McLaughlin, K.T-R., Van Assche, W., Vanlessen, M.: The Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials. Adv. Math. 188(2), 337–398 (2004) 23. Kudryashov, N.A., Soukharev, M.B.: Uniformization and transcendence of solutions for the first and second Painlevé hierarchies, Phys. Lett. A 237(4–5), 206–216 (1998) 24. Kuijlaars, A.B.J., Vanlessen, M.: Universality for eigenvalue correlations at the origin of the spectrum. Commun. Math. Phys. 243, 163–191 (2003) 25. Mehta, M.L.: Random Matrices. 2nd. ed. Boston: Academic Press, 1991 26. Moore, G.: Geometry of the string equations. Commun. Math. Phys. 133(2), 261–304 (1990) 27. Pastur, L., Shcherbina, M.: Universality of the local eigennvalue statistics for a class of unitary invariant random matrix ensembles. J. Stat. Phys. 86(1–2), 109–147 (1997)
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28. Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. New York: Springer-Verlag, 1997 29. Shcherbina, M.: Double scaling limit for matrix models with non analytic potentials. http:// arxiv./org/list/cond-math/0511161, 2005 30. Szeg˝o, G.: “Orthogonal polynomials”. 3r d ed., Providence, RI: Amer. Math. Soc. 1974 31. Vanlessen, M.: Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight. J. Approx. Theory 125, 198–237 (2003) 32. Vanlessen, M.: Strong asymptotics of Laguerre-type orthogonal polynomials and applications in random matrix theory. http://arxiv.org/list/math.CA/0504604, 2005, to appear in Constr. Approx Communicated by B. Simon
Commun. Math. Phys. 273, 533–559 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0220-8
Communications in
Mathematical Physics
Scattering Solutions in Networks of Thin Fibers: Small Diameter Asymptotics S. Molchanov, B. Vainberg Dept. of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA. E-mail: [email protected] Received: 24 July 2006 / Accepted: 5 October 2006 Published online: 13 March 2007 – © Springer-Verlag 2007
Abstract: Small diameter asymptotics is obtained for scattering solutions in a network of thin fibers. The asymptotics is expressed in terms of solutions of related problems on the limiting quantum graph . We calculate the Lagrangian gluing conditions at vertices v ∈ for the problems on the limiting graph. If the frequency of the incident wave is above the bottom of the absolutely continuous spectrum, the gluing conditions are formulated in terms of the scattering data for each individual junction of the network. 1. Formulation of the Problem and Statement of the Results The paper concerns the asymptotic analysis of wave propagation through a system of wave guides when the thickness ε of the wave guides is very small and the wave length is comparable to ε. The problem is described by the stationary wave (Helmholtz) equation −ε2 u = λu,
x ∈ ε ,
(1)
in a domain ε ⊂ R d , d ≥ 2, with infinitely smooth boundary (for simplicity) which has the following structure: ε is a union of a finite number of cylinders C j,ε (which we shall call channels), 1 ≤ j ≤ N , of lengths l j with the diameters of cross-sections of order O (ε) and domains J1,ε , . . . , J M,ε (which we shall call junctions) connecting the channels into a network. It is assumed that the junctions have diameters of the same order O(ε). Let m channels have infinite length. We start the numeration of C j,ε with the infinite channels. So, l j = ∞ for 1 ≤ j ≤ m. The axes of the channels form edges j of the limiting (ε → 0) metric graph . The vertices v j ∈ V of the graph correspond to the junctions J j,ε . The Helmholtz equation in ε must be complemented by the boundary conditions (BC) on ∂ε . In some cases (for instance, when studying heat transport in ε ) the Neumann BC is natural. In fact, the Neumann BC presents the simplest case due to The authors were supported partially by the NSF grant DMS-0405927.
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J3,ε
C1,ε
C4,ε
C6,ε J1,ε
J2,ε C8,ε
C5,ε C7,ε C2,ε
J4,ε C3,ε
Γ
Fig. 1. An example of a domain ε with four junctions, four unbounded channels and four bounded channels.
the existence of a simple ground state (a constant) of the problem in ε . However, in many applications, the Dirichlet, Robin or impedance BC are more important. We shall consider (apart from a general discussion) only the Dirichlet BC, but all the arguments and results can be modified to be applied to the problem with other BC. An important class of domains ε are self-similar domains with only one junction and all the channels being infinite. We will call them spider domains. Thus, if ε is a spider domain, then there exist a point x = x(ε) and an ε-independent domain such that ε = {( x + εx) : x ∈ }.
(2)
Thus, ε is the ε-contraction of = 1 . For any ε , let J j (v),ε be the junction which corresponds to a vertex v ∈ V of the limiting graph . Consider a junction J j (v),ε and all adjacent to J j (v),ε channels. If some of these channels have a finite length, we extend them to infinity. We assume that, for each v ∈ V, the resulting domain v,ε which consists of a junction J j (v),ε and emanating from it semi-infinite channels is a spider domain (i.e., v,ε is self-similar). This assumption can be weakened. For example, one can consider some type of “curved” channels, and the final results (with some changes) will remain valid. Simple equations on the limiting graph in this case will be replaced by more complicated equations with variable coefficients. However, even small deviation from the assumption on the selfsimilarity of v,ε would make the statement of the results and the proofs much more technical. So, we consider only domains ε for which v,ε , v ∈ V, are self-similar. Hence, the cross sections ω j,ε of channels C j,ε are ε−homothety of bounded domains ω j ∈ R d−1 . Let λ j,0 < λ j,1 ≤ λ j,2 . . . be eigenvalues of the negative Laplacian −d−1 in ω j with the Dirichlet boundary condition on ∂ω j , and let {ϕ j,n } be the set of corresponding orthonormal eigenfunctions. The eigenvalues λ j,n coincide with the eigenvalues of −ε2 d−1 in ω j,ε . In the presence of infinite channels, the spectrum of the operator −ε2 in ε with the Dirichlet boundary condition on ∂ε has an absolutely continuous component which coincides with the semi-bounded interval [λ0 , ∞), where λ0 = min λ j,0 . 1≤ j≤m
(3)
Equation (1) is considered under the assumption that λ ≥ λ0 , when propagation of waves is possible. There are two very different cases: λ → λ0 as ε → 0, i.e. the frequency is at the edge (or bottom) of the absolutely continuous spectrum, or λ → λ > λ0 , i.e. the
Scattering Solutions in Networks of Thin Fibers: Small Diameter Asymptotics
535
frequency is above the bottom of the absolutely continuous spectrum. There are many results about the first case, the references will be given later. This paper concerns the asymptotic analysis of the scattering solutions for the Dirichlet problem in ε when λ is close to λ > λ0 . If ε → 0, one can expect that the solution u ε of (1) in ε can be described in terms of the solution ς = ςε (t) of a much simpler problem on the graph . For example, if λ j,0 < λ < λ j,1 for all j, then ς satisfies the following equation on each edge of the graph −
ε2 d 2 ς (t) = (λ − λ j,0 )ς (t) , dt 2
(4)
where t is the length parameter on the edges. One has to add appropriate gluing conditions (GC) at the vertices v of . These gluing conditions give basic information on the propagation of waves through the junctions. They define the solution ς of the problem (4) on the limiting graph. The ordinary differential equation (4), the GC, and the solution ς depend on ε. However, we shall often call the corresponding problem on the graph the limiting problem, since it enables one to find the main term of the asymptotics as ε → 0 for the solution u = u ε of the problem (1) in ε . One of the main difficulties in the problem under investigation was to find the GC, in particular, since the GC differ dramatically from those which were known in the case of λ close to the bottom of the spectrum. Let us define the scattering solutions for the Dirichlet problem in ε . We introduce local coordinates (t, y) in each channel C j,ε with t axis parallel to the cylinder C j,ε , 0 < t < l j , and y ∈ R n−1 being Euclidean coordinates in the plane perpendicular to the t axis. The coordinate y is chosen in such a way that ω j,ε = {(εy) : y ∈ ω j ∈ R n−1 }. 1−d For each j, the set {ε 2 ϕ j,n ( εy )} is the orthonormal basis in L 2 (ω j,ε ) consisting of eigenfunctions of the operator −ε2 d−1 . Let l be a bounded closed interval of the real axis which does not contain the points λ j,n , j ≤ N . Thus, there exist m j ≥ 1 such that λ j,m j < λ < λ j,m j +1 for all λ ∈ l. As will be seen from the definitions below, m j + 1 is the number of waves which may propagate in each √ direction in the channel C j,ε without loss of energy and with frequencies less than λ, λ ∈ l. We put m j = −1, thus {λ j,n , 0 ≤ n ≤ m j } is the empty set if λ j,0 > λ for λ ∈ l. Consider the non-homogeneous Dirichlet problem (−ε2 − λ)u = f, x ∈ ε ;
u = 0 on ∂ε .
(5)
Definition 1. Let f ∈ L 2com (ε ) have a compact support, and λ ∈ l. A solution u of (5) is called an outgoing solution if it has the following asymptotic behavior at infinity in each infinite channel C j,ε , 1 ≤ j ≤ m: u=
mj
√ a j,n e
i
λ−λ j,n ε
t
ϕ j,n (y/ε) + O(e−γ t ), γ = γ (ε) > 0,
(6)
n=0 (ε)
Definition 2. A function = s,k , 1 ≤ s ≤ m, 0 ≤ k ≤ m j , is called a solution of the scattering problem in ε if (−ε2 − λ) = 0, x ∈ ε ;
= 0 on ∂ε ,
(7)
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and has the following asymptotic behavior at infinity in each infinite channel C j,ε , 1 ≤ j ≤ m: (ε) s,k
= δs, j e
−i
√
λ−λs,k ε
t
ϕs,k (y/ε) +
mj
√ t j,n e
i
λ−λ j,n ε
t
ϕ j,n (y/ε) + O(e−γ t ),
(8)
n=0
where γ = γ (ε) > 0, and δs, j is the Kronecker symbol, i.e. δs, j = 1 if s = j, δs, j = 0 if s = j. The first term in (8) corresponds to the incident wave, and all other terms describe the transmitted waves. The incident wave depends on s and k, where s determines the channel, and s and k together determine the frequency of the incident wave. The transmission coefficients t j,n also depend on s and k (i.e. on the choice of the incident wave), so sometimes we will denote them by t s,k j,n . We introduce an order in the set of incident waves and corresponding scattering solutions and the same order in the set of transmitted waves. Namely, we number the incident waves in the channel C1,ε taking them in the order of increase of absolute values of their frequencies, then we number all the solutions in the channel C2,ε , and so on. With this order taken into account, the transmission coefficients for a particular scattering solution form a column vector with M=
m (m j + 1)
(9)
j=1
entries. Together, they form an M × M scattering matrix T = {t s,k j,n },
(10)
where s, k define the column of T and j, n define the row. We denote by D the diagonal M × M matrix with elements λ − λ j,n on the diagonal taken in the same order as above. The following statement can be useful in some applications, and will be proved in the next section (although it will not be used in this paper). Theorem 3. The matrix D 1/2 T D −1/2 is unitary and symmetric. The operator H = −ε2 with the Dirichlet boundary conditions on ∂ε is non-negative, and therefore the resolvent Rλ = (−ε2 − λ)−1 : L 2 (ε ) → L 2 (ε )
(11)
is analytic in the complex λ plane outside the positive semi-axis λ ≥ 0. Hence, the operator Rk 2 is analytic in k in the half plane Imk > 0. We are going to consider an analytic extension of the operator Rk 2 onto the real axis and in the lower half plane. Such an extension does not exist if Rk 2 is considered as an operator in L 2 (ε ) since Rk 2 is an unbounded operator when λ = k 2 belongs to the spectrum of the operator Rλ . However, one can extend Rk 2 analytically if it is considered as an operator in the following spaces (with a smaller domain and a larger range): 2 Rk 2 : L 2com (ε ) → L loc (ε ).
(12)
Scattering Solutions in Networks of Thin Fibers: Small Diameter Asymptotics
537
Theorem 4. (1) The spectrum of the operator H = −ε2 in ε with the Dirichlet boundary conditions on ∂ε consists of the absolutely continuous component [λ0 , ∞), where λ0 > 0 is given by (3) and, possibly, a discrete set of positive eigenvalues {λ j,ε } with the only possible limiting point at infinity. The multiplicity of the a.c. spectrum changes at points λ = λ j,n , and at any point λ, it is equal to the number of points λ j,n , 1 ≤ j ≤ m, located below λ. The eigenvalues λ j,ε = λ j for spider domains ε do not depend on ε. (2) The operator (12) admits a meromorphic extension from the upper half plane Imk > 0 into lower half plane Imk < 0 with the branch points at k = ± λ j,n of the √ second order and the real poles at k = ± λs,εand, perhaps, at some of the branch points. The resolvent (12) has a pole at k = ± λ j,n if and only if the homogeneous problem (5) with λ = λ j,n has a nontrivial solution u such that u= a j,n ϕ j,n (y/ε) + (e−γ t ), x ∈ C j,ε , t → ∞, 1 ≤ j ≤ m. (13) j,n:λ j,n =λ
√ (3) If f ∈ L 2com (ε ), and k = λ is real and is not a pole or a branch point of the operator (12), and λ > λ0 , then the problem (5), (6) is uniquely solvable and the 2 ( ) limit outgoing solution u can be found as the L loc ε u = Rλ+i0 f.
(14)
(4) There exist √ exactly M (see (9)) different scattering solutions for values of λ > λ0 such that k = λ is not a pole or a branch point of the operator (12), and the scattering solution is defined uniquely after the incident wave is chosen. Remark 1. Operator H = −ε2 and its domain depend on ε. One could use the term “family of operators” when referring to H . We prefer to drop the word “family”, but one must always keep in mind that H depends on ε. 2. Existence of a pole of the operator (12) at a branch point meansthat Rk 2 has a pole at z = 0 if this operator function is considered as a function of z = k 2 − λ j,n . 3. One can not identify poles of the resolvent and eigenvalues of the operator based only on general theorems of functional analysis since we deal with the poles of the modified resolvent (12) which belong to the absolutely continuous spectrum of the operator. 4. The eigenvalues λ j,ε of the operator H can be embedded into the absolutely continuous spectrum, and can be located below the absolutely continuous spectrum. In particular, from the minimax principle it follows that H necessarily has a non-empty discrete spectrum below λ0 if at least one of the junctions is wide enough. For example, non-empty discrete spectrum below λ0 exists if a junction contains a ball Bρ of the radius ρ = r ε such that the negative Dirichlet Laplacian in the ball Br has an eigenvalue below λ0 . (ε)
Let us describe the asymptotic behavior of scattering solutions = s,k as ε → 0, λ ∈ l. Note that an arbitrary solution u of Eq. (1) in a channel C j,ε can be represented as a series with respect to the orthogonal basis {ϕ j,n (y/ε)} of the eigenfunctions of the Laplacian in the cross-section of C j,ε . Thus it can be represented as a linear combination of the travelling waves √ e±i
λ−λ j.n ε
t
ϕ j,n (y/ε), 1 ≤ n ≤ m j ,
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S. Molchanov, B. Vainberg
and functions which grow or decay exponentially along the axis of C j,ε . The main term of small ε asymptotics of scattering solutions contains only travelling waves, i.e. on each channel C j,ε , any function has the form =
(ε) s,k
=
mj
√ (α j,n e
i
λ−λ j.n ε
√ t
+ β j,n e
−i
λ−λ j.n ε
t
ε )ϕ j,n (y/ε) + rs,k ,
(15)
n=0
where ε |rs,k | ≤ Ce−
γ d(t) ε
, γ > 0, and d(t) = min(t, l j − t).
The constants α j,n and β j,n depend also on s, k and ε. The formula (15) can be written in a shorter form as follows: =
(ε) s,k
=
mj
ε ς j · ϕ j + rs,k ,
ε |rs,k | ≤ Ce−
γ d(t) ε
,
n=0
where ϕ j = ϕ j (y/ε) is the vector with components ϕ j,n (y/ε), 0 ≤ n ≤ m j , and ς j = ς j (t) is a (m j + 1)-vector whose components ς j,n are linear combinations of the corresponding oscillating exponents in t, i.e. ς j satisfies the following equation: d2 + D 2j )ς j = 0, 0 < t < l j , (16) dt 2 where D j is the diagonal matrix with elements λ − λ j.n , 0 ≤ n ≤ m j , on the diagonal. In order to complete the description of the main term of the asymptotic expansion (15), we need to provide the choice of constants in the representation of ς j,n as linear combinations of the exponents. Thus, 2(m j + 1) constants must be chosen for each channel C j,ε . We consider the limiting graph , whose edges j are the axes of the channels C j,ε . Let ς be the vector valued function on which is equal to ς j on j . The vector ς has a different number of coordinates on different edges j of the graph . We specify ς by imposing conditions at infinity and gluing conditions (GC) at each vertex v of the graph . Let V = {v} be the set of vertices v of the limiting graph . These vertices correspond to the junctions in ε . The conditions at infinity concern only the infinite channel C j,ε , j ≤ m. They depend on the choice of the incident wave and have the form: 1 if ( j, n) = (s, k) , 1 ≤ j ≤ m. (17) β j,n = 0 if ( j, n) = (s, k) (ε2
The GC at vertices v of the graph are universal for all incident waves and depend on λ. In order to state the GC at a vertex v, we choose the parametrization on in such a way that t = 0 at v for all edges adjacent to this particular vertex. The origin (t = 0) on all other edges can be chosen at any of the end points of the edge. Consider auxiliary scattering problems for the spider type domain v,ε formed by the individual junction, which corresponds to the vertex v, and all channels with an end at this junction, where the channels are extended to infinity if they have a finite length. We denote by v the limiting graph which is defined by v,ε . Definitions 1, 2 and Theorem 4 remain valid for the domain v,ε . In particular, one can define the scattering matrix T = Tv for the problem (1) in the domain v,ε . Let v1 , v2 , . . . vl , l = l(v), be indices of channels
Scattering Solutions in Networks of Thin Fibers: Small Diameter Asymptotics
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in ε which correspond to channels in v,ε . Let us form a vector ς (v) by writing the coordinates of all vectors ςvs in one column, starting with coordinates of ςv1 , then coordinates ofςv2 , and so on. Let us denote by Dv (λ) the diagonal matrix with the diagonal elements λ − λvs,k written in the same order as the coordinates of the vector ς (v) . Let Iv be the unit matrix of the same size as the size of the matrix Dv (λ). The GC at the vertex v has the form ε[Iv + Tv ]Dv−1 (λ)
d (v) ς (t) + i[Iv − Tv ]ς (v) (t) = 0, dt
t = 0.
(18)
The GC (18) has the following form in the coordinate representation. Let Z = Z (v) be the set of indices ( j, n), where j are the indices of the edges of ending at v and 0 ≤ n ≤ m j . Then s,k s,k s,k −1/2 d ε δ j,n + t s,k = 0 at v, ς (v) (λ − λ ) + i δ − t (v) ς j,n j,n j,n j,n j,n j,n dt ( j,n)∈Z
(s, k) ∈ Z , where t s,k j,n (v) are the transmission coefficients of the auxiliary problem in the spider s,k domain v,ε (i.e. t s,k j,n (v) are the elements of Tv ), and δ j,n = 1 if (s, k) = ( j, n),
δ s,k j,n = 0 if (s, k) = ( j, n). Definition 5. A family of subsets l(ε) of a bounded closed interval l ⊂ R 1 will be called thin if, for any δ > 0, there exist constants β > 0 and c1 , independent of δ and ε, and c2 = c2 (δ), such that l(ε) can be covered by c1 intervals of length δ together with c2 ε−1 intervals of length c2 e−β/ε . Note that |l(ε)| → 0 as ε → 0. Theorem 6. Let l be a bounded closed interval of the λ-axis which does not contain points λ j,n . Then there exists γ = γ (ω j , l) > 0 and a thin family of sets l(ε) such that the asymptotic expansion (15) holds on all (finite and infinite) channels C j,ε uniformly in λ ∈ l \ l(ε) and x in any bounded region of R d . The function ς in (15) is a vector function on the limiting graph which satisfies Eq. (16), conditions (17) at infinity, and the GC (18). Remark 1) It will be shown in the proof of Lemma 11 that for spider domains the estimate of the remainder is uniform for all x ∈ R d . For general domains, we provide the estimate of the remainder only in bounded regions of R d in order not to complicate the exposition. 2) The arguments, used to justify the asymptotic behavior of the scattering solutions and prove Theorem 6, can be applied to study the asymptotic behavior of the outgoing solutions of the non-homogeneous problem (5) as ε → 0, λ > λ0 . The asymptotics will be expressed in terms of solutions of the corresponding non-homogeneous equation on the limiting graph. One can easily show that the GC can not be chosen independently of f even if we consider only functions f with compact support. However, if the support of f is separated from the junctions then the solution of the non-homogeneous equation on the limiting graph satisfies the same universal GC (18) that appear when scattering solutions are studied. The latter is related to the following fact: the outgoing solution in a narrow channel behaves as a combination of plane waves plus a term which decays exponentially outside of the support of f when ε → 0.
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Note that the GC for the function ς on the limiting graph depend on λ. In fact, there exists an effective matrix potential on which is independent of λ, and allows one to single out the scattering solutions ς on with the same scattering data as for the original problem in ε . These results will be published elsewhere. The convergence of the spectrum of the problem in ε to the spectrum of a problem on the limiting graph has been extensively discussed in the physical and mathematical literature (e.g., [4–7, 9, 12, 13, 16, 18] and references therein). What makes our paper different is the following: all the publications that we are aware of, are devoted to the convergence of the spectra (or resolvents) only in a small (in fact, shrinking with ε → 0) neighborhood of λ0 (bottom of the absolutely continuous spectrum), or below λ0 . Usually, the Neumann BC on ∂ε is assumed. We deal with asymptotic behavior of solutions of the scattering problem in ε when λ is close to λ > λ0 , and the BC on ∂ε can be arbitrary. In particular, papers [5, 12, 13, 18] contain the gluing conditions and the justification of the limiting procedure ε → 0 near the bottom of the spectrum λ0 under assumption that the Neumann BC is imposed at the boundary of ε . Note that λ0 = 0 for the Neumann BC. Typically, the GC in this case are: the continuity of ς (s) at each vertex v and dj=1 ς j (v) = 0, i.e. the continuity of both the field and the flow. These GC are called Kirchhoff’s GC. In the case when the shrinkage rate of the volume of the junction neighborhoods is lower than the one of the area of the cross-sections of the guides, more complex energy dependent or decoupling conditions can arise (see [9, 13, 7] for details). Let us stress again that this is the situation near the bottom λ0 = 0 of the absolutely continuous spectrum. As follows from Theorem 3, the GC and the small ε asymptotics are different when λ > λ0 . Both assumptions (λ → λ0 , and the fact that the BC is the Neumann condition) in the papers above are very essential. The Dirichlet Laplacian near the bottom of the absolutely continuous spectrum λ0 > 0 was studied in a recent paper [16] under the condition that the junctions are more narrow than the tubes. It is assumed there that the domain ε is bounded. Therefore, the spectrum of the operator (1) is discrete. It is proved that the eigenvalues of the operator (1) in the O(ε2 )-neighborhood of λ0 behave asymptotically, when ε → 0, as eigenvalues of the problem in the disconnected domain that one gets by omitting the junctions, separating the channels in ε , and adding the Dirichlet conditions on the bottoms of the channels. This result indicates that the waves do not propagate through the narrow junctions when λ is close to the bottom of the absolutely continuous spectrum. A similar result was obtained in [3] for the Schrödinger operator with a potential having a deep strict minimum on the graph, when the width of the walls shrinks to zero. We also studied the Dirichlet problem for general domains ε without special assumptions on the geometry of the junctions when, simultaneously, ε → 0, λ → λ0 , and the diameters of the guides and junctions have the same order O(ε). Our conclusion is that, generically, waves do not propagate through the junctions when the frequency is close to the bottom of the absolutely continuous spectrum. Let us stress that this is true both in the case when the diameters of the junctions are smaller than the diameters of the guides, and in the case when they are larger. Some special conditions must be satisfied for waves to propagate if λ → λ0 . An infinite cylinder, which can be considered as two half-infinite tubes with the junction of the same shape, can be considered as an example of a domain where the propagation of waves at λ = λ0 is not suppressed. Less trivial examples will be given in our next paper. We do not deal with the problem near the bottom of the absolutely continuous spectrum in this publication. A detailed analysis of
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this problem will be published elsewhere. However, we show here that the GC on the limiting graph with λ > λ0 , generically, have a limit as ε → 0, λ → λ0 , and the limiting conditions are the Dirichlet conditions. To be more exact, the following statement will be proved. √ λ0 . Then Theorem 7. 1) Assume that the resolvent (12) does not have a pole at k = √ the scattering matrix (10), defined for λ > λ0 , admits an analytic in z = λ − λ0 extension to a neighborhood of the point z = 0 and is equal to −I at z = 0, where I is the (m 0 × m 0 )-identity matrix and m 0 is the number of infinite channels C j,ε with λ j,0 = λ0 . 2) Assume that the resolvent √ of the auxiliary problem in the spider type domain ν,ε does not have a pole at k = λ0 . Then the GC (18) have a limit as λ → λ0 of the form εTv
d (v) ς (t) + 2iς (v) (t) = 0, dt
t = 0,
d Tv . The GC also have a limit when ε → 0, λ → λ0 independently. This where Tv = dz limit is the Dirichlet condition ς (v) (0) = 0.
A simple version of the results presented in this paper (for models admitting the separation of variables) was published in our paper [14]. The next section contains the proofs of Theorem 4 and 3. The statements of these theorems mostly concern problems with a fixed value of ε. Without loss of generality, one can assume that ε = 1 there. The last section is devoted to the proof of Theorem 6 on asymptotic behavior of the scattering solutions as ε → 0. Here the dependence of all objects on ε is essential. At the end of the last section, one can find a proof and a short discussion of Theorem 7. 2. Analytic Properties of the Resolvent Rλ We denote by (a) ε the following bounded part of ε : (a) ε = ε \ ∪ (C j,ε ∩ {t > a}).
(19)
j≤m
The next lemma will be needed later. Lemma 8. If the homogeneous problem (5), (6) with a real λ > 0 has a non-trivial √ solution u, then either λ is an eigenvalue of −ε2 and u decays exponentially at infinity, or λ ∈ {λ j,n } and (13) holds. (a)
Proof. From the Green formula for u and u in the domain ε , a > 0, it follows that
∂u Im ud S = 0, (a) ∂ν ∂ε (a)
where ν is the unit normal to ∂ε and d S is an element of the surface area. Using the boundary condition (5) we arrive at
Im u t udy = 0. (20) (a)
∂ε \∂ε
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This, (6), and the orthogonality of the functions ϕ j,n imply, for a → ∞, λ − λ j,n |a j,n |2 + O(e−γ a ) = 0, j,n:λ j,n <λ
which justifies the lemma after taking the limit as a → ∞. This completes the proof. Let C j,ε be the channel C j,ε extended along the whole t axis, C j,ε = {(t, εy) : t ∈ R, y ∈ ω j ⊂ R n−1 }. ( j)
We denote by Rλ the resolvent (11) of the operator −ε2 in the extended channel C j,ε . Let L a2 (C j,ε ) be the set of functions from L 2 (C j,ε ) with the support in the region |t| ≤ a, and let H 2 (C bj,ε ) be the Sobolev space of functions in the domain C j,ε ∩{b < |t| < b+1}. Consider the operator ( j)
2 Rλ : L 2com (C j,ε ) → L loc (C j,ε ).
(21)
The following lemma can be easily proved using the method of separation of variables. Lemma 9. (1) The operator (21) admits an analytic continuation from the upper half plane Imλ > 0 onto the real axis with the branch points at λ = λ j,n , n = 0, 1, . . . . ( j) (2) If λ j,m j < λ < λ j,m j +1 and h ∈ L 2com (C j ) then Rλ h has the following behavior as t → ±∞, √ mj λ−λ j,n ( j) ± i |t| ε c j,n e ϕ j,n (y/ε) + O(e−γ (ε)|t| ), γ > 0, (22) Rλ h = n=1
where c±j,n
=
c±j,n (h)
ε−d = 2i λ − λ j,n
ω j,ε
√
∞ −∞
e
∓i
λ−λ j,n ε
τ
ϕ j,n (y/ε)h(τ, y)dτ dy.
(23)
(3) Let λ ∈ l, where l is a bounded closed interval of the real axis such that λ j,m j < λ < λ j,m j +1 for all λ ∈ l. Let h ∈ L 23ε (C j,ε ) and b ≥ 0. Then there exist positive constants c = c(l) and γ = γ (l) which are independent of λ ∈ l, ε and h, and such that the remainder term r in the right-hand side of (22) has the estimate ||r || H 2 (C b
j,ε )
≤ ce−γ b/ε ||h|| L 2
3ε (C j,ε )
.
Proof of Theorem 4. The statements of the theorem mostly concern the problem with a fixed value of ε. Without loss of generality, we can assume that ε = 1, and we omit ε in the notations of all objects (ε , C j,ε , and so on). The dependence on ε will be restored in some parts of the proof, when this dependence on ε is essential. Step 1. Construction of the resolvent. Let us introduce the following partition of unity on m j=0
φ j = 1.
(24)
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We fix arbitrary functions φ j ∈ C ∞ (), 1 ≤ j ≤ m, such that φ j = 1 in the (infinite) channel C j for t ≥ 2, φ j = 0 in C j for t ≤ 1 and outside of C j . The function φ0 is defined as follows: φ0 = 1 − j≤m φ j . We also need functions ψ j that are equal to one on the supports of ϕ j , which will allow us to smoothly extend functions defined only on infinite channels or only in a bounded part of onto the whole domain . We fix functions ψ j ∈ C ∞ (), 1 ≤ j ≤ m, such that ψ j = 1 in the infinite channel C j for t ≥ 1 (i. e. on the support of φ j ), ψ j = 0 outside of C j . Let ψ0 ∈ C ∞ () be a function such that ψ0 = 1 on the support of φ0 , and ψ0 = 0 in all infinite channels C j when t ≥ 3. Note that ψ j φ j = φ j , 0 ≤ j ≤ m.
(25)
We construct the parametrix (almost resolvent) for the problem (5) in the form Pλ : L 2 () → L 2 (),
Pλ f = ψ0 Rλ (φ0 f ) +
m
( j)
ψ j Rλ (φ j f ),
(26)
j=1
where Rλ is the resolvent (11) of the operator in with a fixed λ = iσ, σ > 0, ( j) which will be chosen later, and Rλ are resolvents of the negative Dirichlet Laplacians 2 in C j . If f ∈ L () then φ j f = 0 outside C j , and we consider φ j f as an element of ( j) ( j) L 2 (C j ). Then the operator Rλ can be applied to φ j f and Rλ (φ j f ) ∈ L 2 (C j ). Since ( j) ψ j = 0 at the bottom of C j and outside of C j , we consider ψ j Rλ (φ j f ) as an element of L 2 () that is equal to zero outside of C j . In this way, the operator Pλ is well defined for λ ∈ / [0, ∞). Let us look for a solution u ∈ L 2 () of the problem (5) with λ ∈ / [0, ∞) in the form of u = Pλ h with unknown h ∈ L 2 (). Obviously, u satisfies the Dirichlet boundary condition since each term in (26), applied to any h, satisfies the Dirichlet boundary condition. The substitution of Pλ h for u in Eq. (5) with λ ∈ / [0, ∞) (and ε = 1) leads to (− − λ)Pλ h = −(ψ0 )[Rλ (φ0 h)] − 2∇ψ0 · ∇[Rλ (φ0 h)] −ψ0 ( + λ )[Rλ (φ0 h)] − (λ − λ )ψ0 [Rλ (φ0 h)] m ( j) ( j) ( j) (ψ j )[Rλ (φ j f )] + 2∇ψ j · ∇ Rλ (φ j h) + ψ j ( + λ)[Rλ (φ j h)] = f. − j=1
Using (25), (24), the last relation can be rewritten in the form h + Fλ h = f,
(27)
where Fλ h = −[( + λ − λ )ψ0 ][Rλ (φ0 h)] − 2∇ψ0 · ∇[Rλ (φ0 h)] m ( j) ( j) (ψ j )[Rλ (φ j h)] + 2∇ψ j · ∇ Rλ (φ j h) . −
(28)
j=1
Let us show that the operator Fλ : L 2 () → L 2 (),
λ∈ / [λ0 , ∞),
(29)
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is compact and depends analytically on λ. Indeed, the resolvents Rλ and Rλ map any function f ∈ L 2 into the solution of the problem (5) in the domains , C j , respectively. Thus, these operators are bounded as operators from L 2 into the Sobolev spaces H 2 . Since the formula (28) contains at most first derivatives of the resolvents, the operator Fλ , λ ∈ / [λ0 , ∞), is bounded if it is considered as an operator from L 2 () into the Sobolev space H 1 (). Since ∇ψ0 = ∇ψ j = 0 at points x ∈ C j with t > 3, from (28) it follows that, for any infinite channel C j , Fλ h = 0, x ∈ C j ∩ {t > 3}.
(30)
Hence, the Sobolev imbedding theorem implies that the operator (29) is compact. The ( j) analyticity of the operator (29) is obvious since the operators Rλ depend analytically on λ, and Rλ does not depend on λ. Now we put λ = λ = iσ and show that ||Fiσ || → 0 as σ → ∞. In fact, since the norm of the resolvent does not exceed the inverse distance from the spectrum, we have that ( j)
||Rλ ||, ||Rλ || ≤ 1/σ,
(31)
where the first norm is considered in the space L 2 () and the second one is in the space L 2 (C j ). Multiplying Eq. (5), considered in the domain or C j , by u and integrating over ( j) the domain, we get the following relation for the functions u = Rλ f and u = Rλ f, respectively:
||∇u||2L 2 − iσ ||u||2L 2 = u f d x, which implies that
||∇u||2L 2
≤|
u f d x| ≤ ||u|| L 2 || f || L 2 .
Thus, ( j)
||Rλ f || H 1 () , ||Rλ f || H 1 (C j ) ≤ Cσ −1/2 || f || L 2 .
(32)
Since the formula (28) contains at most first derivatives of the resolvents, estimates (31), (32) imply that ||Fiσ || → 0 as σ → ∞. We fix λ = iσ in (26) in such a way that ||Fλ || < 1. Then from the analytic Fredholm theorem it follows that the operator (E + Fλ )−1 : L 2 () → L 2 (), λ ∈ / [λ0 , ∞),
(33)
exists and depends meromorphically on λ. From here, (26) and ( 27) the representation for the resolvent follows: Rλ = Pλ (E + Fλ )−1 , λ ∈ / [λ0 , ∞).
(34)
Step 2. Analytic continuation of the resolvent. In order to extend the operator (12) meromorphically into the lower half plane Imk < 0 we need to repeat the arguments used to justify (34). Consider the space L a2 () of functions f ∈ L 2 () with supports in (a) (see (19)), i.e. f = 0 in the infinite channels C j when t > a. Let f ∈ L a2 (). Without
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loss of generality, one can assume that a > 3. Then (27) and (30) imply that h is also supported in (a) , i.e. Fλ can be considered as an operator in L a2 () : Fλ : L a2 () → L a2 (), λ ∈ / [0, ∞). Let χ = χa (t) be a function equal to one when t ≤ a and zero when t > a. From Lemma 9 it follows that the operators ( j)
χ Rk 2 : L a2 (C j ) → L a2 (C j ), Imk > 0, admit an analytic continuation into the lower half plane with the branch points at ( j) k = ± λ j,n . Further, u = Rk 2 f satisfies Eq. (5) with λ = k 2 for all complex k ∈ C, and therefore the operators ( j)
( j)
χ Rk 2 , χ ∇ Rk 2 : L a2 (C j ) → L a2 (C j ),
k ∈ C,
are compact and analytic in the complex plane C. Since χ = 1 on the supports of ∇ψ j , ( j) 0 ≤ j ≤ m, we can insert the factor χ on the left of all the resolvents Rλ in ( 28). From here it follows that the operator Fk 2 : L a2 () → L a2 (),
k ∈ C, is compact and analytic with branch points at k = ± λ j,n . Hence, the operator (35) (E + Fk 2 )−1 : L a2 () → L a2 (), k ∈ C, is meromorphic with the branch points at k = ± λ j,n . Together with (26), (34) and ( j)
2 (C ), k ∈ C, this implies that the analyticity of the operators Rk 2 : L a2 (C j ) → L loc j the operator (12) admitsa meromorphic continuation to the lower half plane with the branch points at k = ± λ j,n and poles determined by the poles of the operator (35). Obviously, the poles of the operator (35) may have a limiting point only at λ = ∞. Step 3. Spectral analysis. First of all note that the existence of the meromorphic extension of the operator (12) together with the Stone formula immediately imply that the operator H = − does not have singular spectrum. The proof of this fact can be found in [17] (see Theorem XIII.20). In order to prove the part of statement (1) of the theorem concerning the absolutely continuous spectrum of the operator H = −, we split the domain into pieces by introducing cuts along the bases t = 0 of all infinite channels. We denote the new (not connected) domain by , and denote the negative Dirichlet Laplacian in by H , i. e. H is obtained from H by introducing additional Dirichlet boundary conditions on the cuts. Obviously, the operator H has the absolutely continuous spectrum described in statement (1) of the theorem. Thus, it remains to show that the wave operators for the couple H, H exist and are complete. The justification of the existence and completeness of the wave operators can be found in [1]. Another option is to derive the latter fact independently using the Birman theorem stating that the validity of the inclusion
(H − λ)−n − (H − λ)−n ∈ J1
(36)
for some λ and n ≥ 1 implies the existence and completeness of the wave operators. Here J1 is the space of operators of the trace class. The inclusion (36) can be derived
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from (34) and a similar formula for the resolvent of the operator H . This completes the proof of the statement about the absolutely continuous spectrum. The discreteness of the set {λ j,ε } of eigenvalues follows from the fact that the operator (12) is meromorphic in λ and has poles at {λ j,ε }. The existence of the poles at {λ j,ε } can be derived from the Stone formula. Another proof will be given below. Let us prove the part of statement (1) concerning the spider domains. If ε is a spider domain, then there exists a point x (ε) and an ε-independent domain such that the transformation (see (2)) Lε : x → x (ε) + εx,
(37)
maps into ε . In order to stress the fact that the operator H = −ε2 in the domain ε depends on ε, we shall denote it by H (ε) . The operator − in the domain shall be denoted by H (1) . Obviously, H (ε) = L ε H (1) L −1 ε ,
(38)
and this implies the independence of the eigenvalues of the operator H (ε) of ε. This completes the proof of statement (1). Step 4. Real poles of the resolvent. The first part of statement (2) about the existence of the analytic extension of the resolvent was justified in Step 2 of the proof. Now we are going to prove the second part of that statement concerning the set of real poles of the operator (12 ). We denote this set of poles by K . Let us assume that either u is an eigenfunction of the operator H = − with an eigenvalue λ = λ > 0 or u is a non-trivial solution of the homogeneous problem (5), (13)√with λ = λ > 0 (recall that we assume that ε = 1). We are going to show that k = ± λ ∈ K . Consider the restrictions u j of u to the cylinders C j , 1 ≤ j ≤ m. Let v j ∈ L 2 (C j ) be the solution of the problem (− − λ)v j = 0,
x ∈ Cj;
v j = 0 on ∂ C j ,
v j = u j when t = 0,
where λ ∈ / [0, ∞), and ∂ C j is the lateral boundary of C j . The solution v j ∈ L 2 (C j ) of this problem is unique and can be found by separation of variables. The function u j satisfies the same equation with the fixed λ = λ and the same boundary conditions. It is also defined uniquely by its values at t = 0 and can be found by separation of variables. This implies that v j converges to u as λ → λ + i0. Since u is a solution of a homogeneous elliptic problem, u ∈ C ∞ . Thus, u j is infinitely smooth when t = 0, and the convergence v j → u j takes place, for example, in the Sobolev space H 2 on the part of the cylinder C j where 0 ≤ t ≤ 2. Let v=
m
φ j v j + φ0 u ∈ L 2 (C j ), λ ∈ / [0, ∞),
j=1
where {φ j } is the partition of unity which was introduced above. The function u can not be equal to zero identically on \ ∪ C j due to the uniqueness of the solution of the Cauchy problem for the operator − − λ . Thus ||v|| L 2 (\∪C j ) = ||u|| L 2 (\∪C j ) = c0 > 0.
(39)
On the other hand, (− − λ)v = −
m j=1
[(φ j )v j + 2∇φ j · ∇v j ] − (λ − λ )φ0 u − (φ0 )u − 2∇φ0 · ∇u.
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Thus, (− − λ)v ∈ L a2 (). From the convergence v j → u and (24) it follows that (− − λ)v tends to zero in L a2 () as λ → λ + i0. √ provides the √ Together with (39) this existence of the pole of the operator √(12) at k = λ . The pole at k = − λ exists due to the relation Rλ = Rλ . Hence, ± λ ∈ K . √ to K . The relation Now let us assume that at least one of the points ± λ belongs √ Rλ = Rλ implies that the second point also belongs to K , i.e. λ ∈ K , and there exist a > 0 and f ∈ L a2 () such that w := Rλ f =
u(x) v(x, λ) + ; ||v|| L 2 ((a+2) ) ≤ c, n (λ − λ ) (λ − λ )n−1
λ → λ + i0, (40)
where n ≥ 1 and u does not vanish identically. In fact, n can not exceed one, but it is not important for us now. Obviously, (− − λ )u = 0, x ∈ ;
u = 0 on ∂.
(41)
From here and Lemma 8 it follows that in order to complete the proof of the second statement of the theorem it is sufficient to show that the asymptotic expansion (6) holds for the function u. Note that (41) implies that u ∈ C ∞ . Since f = 0 in all infinite channels C j when t > a, from relation (40) it follows that (− − λ)v = (λ − λ )u, x ∈ C j ∩ {t > a};
v = 0 on ∂.
From here, the estimate in (40), and standard local a priori estimates for solutions of elliptic problems it follows that for any vector α, |
1 3 ∂αv | ≤ c(α), x ∈ C j ∩ {a + > t > a + }, λ → λ + i0, ∂xα 2 2
and therefore ∂ α [(λ − λ )n w] ∂αu → ∂xα ∂xα
(42)
uniformly on C j ∩ {t = a + 1} as λ → λ + i0. We restrict the functions (λ − λ )n w and u to C j ∩ {t = a + 1} and expand the restrictions with respect to the basis. {ϕ j,n } 0 be the of the operator − in the cross section of the channel C j . Let γ j,n (λ) and γ j,n coefficients of these expansions. Then (42) implies that for any β, 0 |γ j,n (λ) − γ j,n | < cβ n −β , λ → λ + i0.
(43)
The function w := (λ − λ )n w satisfies the following relations in C j ∩ {t ≥ a + 1} : |t=a+1 = γ j,n (λ)ϕ j,n (y), (− − λ) w = 0, w = 0 for x ∈ ∂C j ∩ {t > a + 1}, w n
where λ ∈ / [0, ∞). One can find the solution w ∈ L 2 of this problem by the method of separation of variables and then pass to the limit as λ → λ + i0 using (43). This leads to the asymptotic expansion (6) for u and completes the proof of the second statement of Theorem 4.
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Step 5. The proof of the last two statements of the theorem. If k = pole or a branch point of Rk 2 then w := Rλ f = u(x) + (λ − λ )v(x, λ); ||v|| L 2 ((a) ) ≤ c(a),
√
λ , λ > 0, is not a
λ → λ + i0,
where a > 0 is arbitrary and (− − λ )u = f, x ∈ ;
u = 0 on ∂.
In order to prove the third statement of the theorem, we need only to show that the asymptotic expansion (6) holds for u. It can be done exactly in the same way as it was done for function u in (40) by representing u in C j ∩ {t > a + 1} as the limit of functions w as λ → λ + i0. In order to prove the last statement of the theorem one can look for the solution = s,k of the scattering problem in the form √ = φs e−i λ−λs,k t ϕs,k (y) + u, where φs is the function from the partition of unity (24). This reduces problem (7), (8) to the uniquely solvable problem (5), (6) for u. This completes the proof of Theorem 4. Proposition 10. Let s,k and s ,k be two scattering solutions, and let a s,k j,n be the transmission coefficients for the scattering solution s,k . Then 1) The following energy conservation law is valid:
2 λ − λ j,n |a s,k j,n | =
mj m
2 λ − λ j,n |a s,k j,n | =
λ − λs,k .
j=1 n=0
j,n
2) If these solutions correspond to different incident waves ((s, k) = (s , k )), then s ,k λ − λ j,n a s,k j,n a j,n = 0. j,n
Proof. Since the statement concerns the problem with a fixed value of ε, one can put ε = 1 and omit ε in the notations ε , (a) ε . Green’s formula for s,k and s ,k in the (a) domain implies, similarly to (20), that
[( s,k )t s ,k − s,k ( s ,k )t ]dy = 0. ∂(a) \∂
From here, (8), and the orthogonality of the functions ϕ j,n it follows that √ s ,k s,k −2i λ−λs,k a λ − λ j,n a s,k j,n a j,n − λ − λs,k a j,n e j,n
√ + λ − λs ,k a sj,n,k e2i λ−λs ,k a − λ − λs,k δ + O(e−γ a ) = 0, a → ∞,
s s , and δ = 0 otherwise. We take the average with respect where δ = 1 if = k k to a ∈ (A, 2 A) and pass to the limit as A → ∞. Then we get s ,k λ − λ j,n a s,k j,n a j,n = λ − λs,k δ, j,n
which justifies both statements of the proposition. This completes the proof.
Scattering Solutions in Networks of Thin Fibers: Small Diameter Asymptotics
549
Proof of Theorem 3. Proposition 10 is equivalent to the relation A∗ A = I for A = D 1/2 T D −1/2 , which provides the unitarity of the matrix A. If one applies Green’s formula to the scattering solutions s,k and s ,k , then the arguments used in the proof of Proposition 10 lead to the symmetry of D 1/2 T D −1/2 . This completes the proof of Theorem 3. 3. Asymptotic Behavior of Scattering Solutions as ε → 0 We start with a study of scattering solutions in spider domains ε . Lemma 11. Theorem 6 is valid for spider domains. Proof. The transformation L −1 ε , see (37), maps the spider domain ε into the ε−independent domain with the channels C j , 1 ≤ j ≤ m. The coordinates ( t, y) in C j are related to coordinates (t, y) in C j,ε via the formulas t = t/ε, y = y/ε.
(44)
= s,k of the problem in has the form similar to (8): The scattering solution s,k = δs, j e−i
√
t λ−λs,k
ϕs,k ( y) +
mj
t j,n ei
√
t λ−λ j,n
ϕ j,n ( y) + O(e−γt ),
n=0
t → ∞. x ∈ Cj, s,k is a smooth function, the remainder term Since r in the formula above can be estimated for all values of t: | r | ≤ Ce−γt , x ∈ C j . Since the scattering solutions in the domains ε and are related via the formula s,k (L −1 s,k (x) = ε x), it follows that √ mj √ λ−λ j,n λ−λs,k (ε) −i t i t ε ε ϕs,k (y/ε) + t j,n e ϕ j,n (y/ε) + r (ε) , s,k = δs, j e n=0
|r
(ε)
| ≤ Ce
−γ t/ε
, x ∈ Cj.
(45)
Thus, the asymptotic expansion (15), (17) is valid, and it only remains to show that the GC (18) holds for vectors ς = ςs,k determined by (45) (the definition of these vectors is given in the paragraph above formula (18)). We form the matrix = (t) with columns ςs,k taking them in the same order as the order chosen for elements in each of these vectors (first we put columns with s = 1 and k = 1, 2, . . . , m 1 , then columns with s = 2, and so on). From (45) it follows that (0) = I + T, (0) =
i D(−I + T ), ε
where T is the scattering matrix, I is the identity matrix of the same size, and D is the diagonal matrix of the same size with elements λ − λ j,n on the diagonal. Hence, ε(I + T )D −1 (0) + i(I − T )(0) = 0 and GC (18) holds for the columns of the matrix . This completes the proof of the lemma.
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The following two lemmas about spider domains will be needed in order to prove Theorem 6 for general domains. Let Rλ be the resolvent of the operator H = −ε2 in λ be the resolvent of the similar operator H = − in the a spider domain ε , and let R domain which is the image of ε under the map L −1 ε , see (37). Note that the operλ is ε-independent. ator Rλ and its domain, L 2 (ε ), depend on ε, while the operator R Formula (38) implies Lemma 12. The following relation holds λ L −1 Rλ = L ε R ε . Let us fix m constants t j > 0, 1 ≤ j ≤ m. Let ε be a spider domain with the channels C j,ε , 1 ≤ j ≤ m. Consider slices D j,ε of C j,ε defined by the inequalities |t − t j | ≤ 3ε. Let ε be a bounded domain which is obtained from ε by cutting off the infinite parts of channels C j,ε on which t ≥ 43 t j. Let a function h ∈ L 2 (ε ) be supported in one of the domains D j,ε , for example, with j = s. Below, when the resolvent Rλ of the operator H = −ε2 in ε is considered with λ belonging to the continuous spectrum of the operator, Rλ is understood in the sense of the analytic extension described in Theorem 4. We denote the Sobolev spaces of functions which are square integrable together with their derivatives of up to the second order by H 2 ( ε ) and H 2 (D j,ε ). Lemma 13. Let ε be a spider domain. Let l be a bounded closed interval of the λ-axis that does not contain points λ j,n , and let a function h ∈ L 2 (ε ) be supported in the domain Ds,ε . Then there exists γ = γ (ω j , l) > 0 such that (1)
Rλ h =
ms
cs.k s,k + r0 in ε , |r0 (x)| ≤
k=0
Ce−γ /ε ||h|| L 2 (ε ) , λ j ∈l |λ − λ j |
(46)
− (h) are given by (23), and where s,k are scattering solutions, the coefficients cs.k = cs.k λ j are eigenvalues of the operator H in ε (see statement (1) of Theorem 4); √ mj ms λ−λ j,n (s) s,k i t ε cs.k t j,n e ϕ j,n (y/ε) + r j in D j,ε , (47) (2) Rλ h = δs, j Rλ h + k=0
n=0
where ||r j || H 2 (D j,ε ) ≤
Ce−γ /ε ||h|| L 2 (ε ) . λ j ∈l |λ − λ j | (s)
Here δs, j is the Kronecker symbol (δs, j = 1 if s = j, δs, j = 0 if s = j), Rλ is the (channel C resolvent of − in the extended channel Cs,ε s,ε extended to −∞ along the t s,k − axis), cs.k = cs.k (h), and t j,n are the transmission coefficients (see the remark following Definition 2). Proof. Let a function α ∈ C ∞ (ε ) have the form: α = 1 in Cs,ε when t > 78 t j + ε, α = 0 in ε \Cs,ε , and α = 0 in Cs,ε when t < 78 t j . Consider the function (s)
u = α Rλ h +
ms k=0
cs.k [ s,k − αe−i
√
λ−λs,k t ε
ϕs,k (y/ε)].
(48)
Scattering Solutions in Networks of Thin Fibers: Small Diameter Asymptotics
551
Obviously, u = 0 on ∂ε , since each term in the right hand side above satisfies the Dirichlet boundary condition. Furthermore, −ε2 u − λu = h − ε2 [∇α · ∇ Rλ(s) h + (α)Rλ(s) h − ∇α · ∇g − (α)g],
(49)
where g=
ms
cs.k e−i
√
λ−λs,k t ε
ϕs,k (y/ε).
k=0
The right hand side in (49) has the form h + h 1 , where h 1 is supported in the slice 7 7 8 ts ≤ t ≤ 8 ts + ε of C s,ε . From Lemma 9 it follows that ||h 1 || L 2 (ε ) ≤ Ce−γ /ε ||h|| L 2 (ε ) . It is also clear that the behavior of the function u at infinity is described by (6). Hence, u = Rλ (h + h 1 ) due to statement (3) of Theorem 4. From here and (48) it follows that Rλ h =
α Rλ(s) h
+
ms
cs.k ( s,k − αe−i
√
λ−λs,k t ε
ϕs,k (y/ε)) − Rλ h 1 .
(50)
k=0
This implies equality (46) with r0 = −Rλ h 1 . Let ε be obtained from ε by cutting off the parts of channels C j,ε where t ≥ 78 t j . Since operator (12) is meromorphic (due to Theorem 4) and has poles of at most first order due to the Stone formula, ||r0 || L 2 ( ε ) can be estimated by the right-hand side of inequality (46). Since (−ε2 − λ)r0 = 0 in ε and r0 = 0 on the lateral side of ∂ ε , standard a priori estimates for elliptic equations lead to the estimates on r0 in Sobolev norms in ε . These estimates, together with Sobolev imbedding theorems, justify the estimate (46). Similarly, Eq. (47) follows from (50) and Lemma 12. This completes the proof of the lemma. We need two more auxiliary statements in order to prove Theorem 6. Lemma 14. Let a real-valued function f belong to C n+1 (R 1 ) and || f ||C n+1 =
n+1 k=0
n
sup | f (k) | = A+ < ∞,
(51)
x
| f (k) (x)| ≥ A− > 0,
x ∈ R1.
(52)
k=0
Then for any σ ≤ A− /2, the set σ = {x : | f (x)| ≤ σ } has the following structure. There exists a constant c which depends only on A± and n and such that, for any bounded interval ⊂ R 1 , a) the number of connected components of σ in is finite and does not exceed c(|| + 1), b) the measure of each connected component of σ in does not exceed cσ 1/n . Remark The last estimate can not be improved. In fact, if f (x) = sinn x then σ ∩ [− π2 , − π2 ] ∼ 2σ 1/n .
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Proof. We shall denote by c j different constants which depend on A± and n but not on f. If x ∈ σ then (52) implies that n
| f (k) (x)| ≥ A− /2,
k=1
and therefore, | f (k) (x)| ≥ A− /2n for the chosen x and some k = k(x), 1 ≤ k ≤ n. Since | f (k+1) | ≤ A+ , x ∈ R 1 , there exists an interval x such that x ∈ x , | f (k) (x)| ≥ A− /4n − on x , and |x | = c0 = 4nAA+ . The set of intervals x covers σ ∩ . Hence, one can select a finite number of intervals x covering σ ∩ . Then one can omit some of them in such a way that the remaining intervals still cover σ ∩ with multiplicity at most two. This leaves us with at most 2( || c0 + 1) ≤ c1 (|| + 1) intervals x covering σ ∩ . Thus, it is enough to prove the lemma for an individual interval (one of the intervals x ) such that | | = c0 and | f (k) (x)| ≥ c2 on for some fixed value of k, 1 ≤ k ≤ n. Equations f (x) = ±σ have at most k solutions on . In fact, if there exist k + 1 points where f (x) = σ then there are k intermediate points where f (x) = 0. Thus, there are k − 1 points where f (x) = 0, and so on. Finally, there has to be a point where f (k) (x) = 0. This contradicts the assumption that | f (k) (x)| ≥ c2 on . Hence, the set σ ∩ consists of at most k + 1 intervals. It remains only to show that the length of these intervals does not exceed cσ 1/k . In order to estimate this length, we assume that there is an interval [x1 , x1 + h], where | f (x)| ≤ σ, | f (k) (x)| ≥ c2 . Put h = h/k and consider the k th difference
k k k = f (x1 ) − f (x1 + h ) + f (x1 + 2h ) − · · · + (−1)k f (x1 + kh ). (53) 1 2 There exists a point ξk ∈ [x1 , x1 + h] such that k = (h )k f (k) (ξk ). Thus, |k | ≥ c2 h k = c3 h k . On the other hand, from (53) and the estimate | f (x)| ≤ σ it follows kk that |k | ≤ σ 2k . Hence, c3 h k ≤ σ 2k , i.e. h ≤ cσ 1/k . This completes the proof of the lemma. Lemma 15. Let a set of functions f ε = f ε (λ), ε → 0, on a closed interval l ⊂ R 1 , have the form fε =
M
C j (λ)ei
g j (λ) ε
,
(54)
j=1
where functions C j (λ) are real valued, functions g j (λ) are analytic, there are no two functions g j (λ) whose difference is a constant, and M
|C j (λ)| ≥ 1.
(55)
j=1
Then, for any η > 0, the set η (ε) = {λ : | f ε (λ)| ≤ e−η/ε } is thin (see the definition in the introduction).
(56)
Scattering Solutions in Networks of Thin Fibers: Small Diameter Asymptotics
553
Proof. Consider the set 0 , where gi (λ) = g j (λ) for some i = j. Due to the analyticity of the functions g j (λ), this set consists of a finite number of points. Let us denote the number of points in 0 by c1 . Let δ be the δ/2-neighborhood of 0 . Then |gi (λ) − g j (λ)| ≥ a(δ) > 0,
i = j,
λ ∈ l \ δ .
(57)
Consider the functions f ε (µ) =
M
C j (εµ)ei
g j (εµ) ε
,
εµ ∈ l \ δ .
j=1
For any k, g (εµ) dk k i jε (µ) = [g (εµ)] C (εµ)e + O(ε). f ε j j dµk
M
(58)
j=1
We move the remainders to the left hand side and consider (58) with 1 ≤ k ≤ M as g j (εµ)
equations for unknowns C j (εµ)ei ε . The matrix of this system of equations with the elements a j,k = [g j (εµ)]k is a Vandermond matrix, and its determinant is bounded from below due to (57). This and (55) imply that M dk | k f ε (µ)| ≥ A− (δ) > 0 dµ j=1
if ε is small enough. It also follows from (58) that M+1 j=1
|
dk f ε (µ)| ≤ A+ . dµk
f ε (µ) on Hence, Lemma 14 is applicable to at least one of the functions Re f ε (µ) or Im each connected interval of the set l \ δ stretched by a factor of ε−1 . Since we have at most c1 +1 of those intervals, this implies that the set {λ : | f ε (λ)| ≤ σ } can be covered by δ and c2 (δ)ε−1 intervals of length c2 (δ)σ 1/M . We take σ = e−η/ε , and this completes the proof of the lemma. Proof of Theorem 6. The proof is based on a representation of the resolvent Rλ of the problem in ε through the resolvents Rv,λ of the operators H = −ε2 in the spider domains v,ε , formed by an individual junction, which corresponds to a vertex v, and all the channels with an end at this junction, where the channels are extended to infinity if they have finite length. Let us consider the slices D j,ε of the finite channels C j,ε , j > m, defined by the conditions t j ≤ t ≤ t j + 3ε where t j = 4l j /5. We construct the following partition of the unity ε : φv = 1, v∈V
where V is the set of all the vertices v of the limiting graph , φv ∈ C ∞ (ε ), and is defined as follows. The function φv is equal to one on the junction Jv , which corresponds to the vertex v; on the infinite channels adjacent to Jv and on the parts of the
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S. Molchanov, B. Vainberg
finite channels adjacent to Jv , where t ≤ t j + ε. The function φv is equal to zero on the parts of finite channels adjacent to Jv , where t ≥ t j + 2ε, and also on all the other junctions and channels which are not adjacent to Jv . Let ψv ∈ C ∞ (ε ), ψv = 1 on the support of φv , ψv = 0 on the parts of finite channels adjacent to Jv , where t ≥ t j + 3ε, and also on all other junctions and channels which are not adjacent to Jv . We fix a vertex v = v . Let J v be the corresponding junction of ε . We choose the parametrization on in such a way that the value t = 0 on all the edges adjacent to v corresponds to v . The origin (t = 0) on all the other edges can be chosen at any end of the edge. We are going to justify the asymptotic expansion (15) in the domain C( v ) consisting of the infinite channels adjacent to J v and the parts t < 3l j /5 of the finite channels C j,ε adjacent to J . Moreover, it will be shown that the function ς in v the asymptotic expansion satisfies Eq. (16), conditions (17) at infinity, and the GC (18). Since v is arbitrary and the union of all domains C( v ), v ∈ V , covers all the channels, the validity of (15) in C( v ) justifies the statements of Theorem 6. (ε) Let us show that the asymptotic expansion (15) in C( v ) for any scattering solution s,k follows from a similar expansion for functions of the form u = Rλ f, where f ∈ L 2 (ε ) (ε) is supported in ∪D j,ε . In fact, let u = ψv1 s,k,v1 , where the vertex v1 = v1 (s) corre(ε)
sponds to the first junction Jv1 encountered by the incident wave, s,k,v1 is the solution of the scattering problem in the spider domain v1 ,ε , and the function u is considered as a function in ε which is equal to zero outside of the support of ψv1 . Then (−ε2 − λ)u = f,
(ε)
(ε)
f := −ε2 [∇ψv1 · ∇ s,k,v1 + (ψv1 ) s,k,v1 ].
Obviously, f ∈ L 2 (ε ) and f is supported in ∪D j,ε . From statement (3) of Theorem 4 it follows that there exists the unique outgoing solution v = Rλ f of the equation (−ε2 − λ)v = f, λ ∈ l \{λ j }. Then (ε)
(ε)
s,k = ψv1 s,k,v1 − Rλ f, since this function satisfies (7) and (8). From here and Lemma 11 it follows that the asymptotic expansion (15) and the properties of ς mentioned in Theorem 6 hold for (ε) s,k in C( v ) if the corresponding properties are valid for Rλ f in C( v ). Hence, the proof of the theorem will be complete as soon as we show that, for any f ∈ L 2 (ε ) with the support in ∪D j,ε , the function u = Rλ f has expansion (15) in C( v ) with β j,n = 0 and ς satisfying the GC (18). Consider the operator Pλ defined by the formula ψv Rv,λ (φv h), λ ∈ l, (59) Pλ h = v∈V
L 2 (ε )
is supported in ∪D j,ε , l is defined in the statement of Theorem 6, where h ∈ and the resolvents Rv,λ for real λ ∈ l are understood in the sense of Theorem 4. We look for u = Rλ f in the form of Pλ h with an unknown h ∈ L 2 (ε ). This leads to the equation (compare with (27), (28)) h + Fλ h = f, Fλ h = −ε2 [2∇ψv · ∇ Rv,λ (φv h) + (ψv )Rv,λ (φv h)]. (60) v∈V
From here, similarly to (34), it follows that Rλ f = Pλ (I + Fλ )−1 f
(61)
Scattering Solutions in Networks of Thin Fibers: Small Diameter Asymptotics
555
for Imλ > 0. Similarly to (35), one can show that the operator (I + Fλ )−1 : L a2 (ε ) → L a2 (ε )
(62)
admits a meromorphic extension into the lower half plane Imλ ≤ 0 with the branch points at λ = λ j,n . The only difference is that now we use operators Rv,λ instead of ( j) ( j) Rλ , and Rv,λ depend meromorphically on λ, while Rλ are analytic in λ. So, one needs to refer to a version of an analytic Fredholm theorem where the operator may have poles (with residues of finite ranks). This version of the theorem can be found in [2], and applications of this theorem to operators similar to (62) can be found in [19, 20]. Hence, formula (61) is established for all complex λ. All the operators in (61) depend on ε. The function (I + Fλ )−1 f is meromorphic in λ, and its poles depend on ε. In order to find a set on the interval l where the operator (I + Fλ )−1 exists and is bounded uniformly in λ we shall use the following reduction of Eq. (60) to a system of equations where the domains of the operators do not depend on ε. Recall that f is supported in ∪D j,ε . Formula (60 ) for Fλ implies that the function Fλ h is also supported in ∪D j,ε . Thus, the support of any solution h of (60) belongs to ∪D j,ε . We shall identify functions f and h with vector functions whose components are the restrictions of f and h, respectively, to individual domains D j,ε , m + 1 ≤ j ≤ N . Furthermore, we map D j,ε onto the ε-independent domain D j by the transformation L ε defined by formulas t − t j = ε t,
y = ε y.
(63)
This transformation differs from (44) by a shift in t (compare with (37)). The vector functions of variables ( t, y) with components from L 2 (D j ) defined by f, h will be denoted by f and h , respectively. Then Eq. (60) can be written in the form h + Fλ h = f , where Fλ is the [(N − m) × (N − m)]-matrix operator which corresponds to the operator Fλ . Here N − m is the number of finite channels in ε . Recall that the entries of the vectors f and h are functions with the domains D j , which do not depend on ε (and λ). Our next goal is to describe how the entries of the matrix Fλ depend on ε and λ. It will be done using (60), where each resolvent Rv,λ can be specified using (47). The first term in the right hand side of (47), (s)
Rλ : L 2 (Cs,ε ) → L 2 (Cs,ε ), onto the ε-independent cylinder C . depends on ε. The transformation (63) maps Cs,ε s The operator
(s) := L ε R (s) (L ε )−1 : L 2 (Cs ) → L 2 (Cs ) R λ λ does not depend on ε (Lemma 12), and it depends meromorphically on λ. Thus, the contributions from the first term in the right-hand side of (47) to the entries of the matrix Fλ are operators which are ε-independent and meromorphic in λ. The rest of the terms in the right-hand side of (47) (other than the remainder) are operators of finite ranks. − Due to Lemma 13 (see also the formula (23) for cs,k = cs,k ), the contributions of these
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S. Molchanov, B. Vainberg
terms to the entries of Fλ are ε-independent operators which are analytic in λ and are of the rank one, multiplied by functions qv; j,n,s,k of the form √ √ qv; j,n,s,k (λ, ε) = ei
αj
λ−λ j,n +βs
λ−λs,k
ε
.
(64)
Here α j = t j or α j = l j − t j (independently, βs = ts or βs = ls − ts ). Formula (47) leads to α j = t j , βs = ts if 1) the channels C j,ε and Cs,ε are adjacent to a common junction, which corresponds to the vertex v, and 2) the parameter t on both channels C j,ε and Cs,ε is introduced in such a way that t = 0 at the vertex v. Other options in the choice of α j and βs correspond to opposite parametrization of the channels C j,ε , Cs,ε , or both. If C j,ε and Cs,ε do not have a common junction which corresponds to the vertex v then qv; j,n,s,k = 0. Thus, the matrix operator Fλ can be represented in the form ⎡ ⎤ j,n,s,k ⎦ Fλ = Fλ0 + ⎣ qv; j,n,s,k (λ, ε)Fλ + R, (65) v;n,k
j,s>m
j,n,s,k
where Fλ0 , Fλ are ε-independent operators, Fλ0 is meromorphic in λ, operators j,n,s,k Fλ are analytic in λ and have rank one, the summation extends over all the vertices v and over n ∈ [0, m j ], k ∈ [0, m s ]. The operator R = R(ε, λ) corresponds to the remainder term in (47) and has the following estimate ||R|| ≤
Ce−γ /ε . λ j ∈l |λ − λ j |
Since the analytic Fredholm theorem [2] is applicable to the operator I + Fλ , from (65) it follows that it is also applicable to the operator I + Fλ0 . Let l δ be the δ/2neighborhood of the set consisting of both the poles λ j of the operator (I + Fλ0 )−1 j located inside l and the points λ ∈ l. Then ||(I + Fλ0 )−1 || ≤ C(δ),
||R || ≤ C(δ)e−γ /ε ,
λ ∈ l \ lδ,
(66)
where R = R(I + Fλ0 )−1 . Formula (65) implies, for λ ∈ l \ l δ , −1 (I + Fλ )−1 = (I + Fλ0 )−1 I + qG + R , where qG is the matrix operator with matrix elements N − m < j, s ≤ N . Here j,n,s,k
Gλ j,n,s,k
The operators G λ The equation
j,n,s,k
= Fλ
v;n,k
(67) j,n,s,k
qv; j,n,s,k (λ, ε)G λ
,
(I + Fλ0 )−1 .
are meromorphic in λ and have rank one. (I + qG)x = g
(68)
Scattering Solutions in Networks of Thin Fibers: Small Diameter Asymptotics
557
for x can be reduced to an equation in the finite dimensional space S spanned by the j,n,s,k ranges of the operators G λ . We fix a basis in S, reduce Eq. (68) to the algebraic system A x = g for coordinates of the projection of x on S, and solve the system using the Kramer rule. Since functions qv; j,n,s,k are bounded when λ ∈ l, the procedure described above allows us to estimate the norm of the operator (I + qG)−1 through | det −1 A|. Hence, there exists a polynomial P = P(qv; j,n,s,k ) of variables qv; j,n,s,k which has the following properties. Its coefficients are meromorphic in λ (with poles belonging to the set { λ j } ∪ {λ j }), the polynomial is linear with respect to each variable qv; j,n,s,k , and is such that ||(I + qG)−1 || ≤ C| f ε (λ)|−1 ,
λ ∈ l \ lδ,
f ε (λ) := 1 + P(qv; j,n,s,k (λ, ε)).
The function f ε (λ) here has the form (54) with one of g j identically equal to zero, and the corresponding coefficient C j equal to one. The latter implies (55). Thus, Lemma 15 can be applied to the function f ε (λ) above on each connected interval of the set l\l δ . There are only finitely many such intervals. Thus, for any η > 0, there exists a thin set η (ε) such that ||(I + qG)−1 || ≤ Ceη/ε ,
λ ∈ l\η (ε).
We choose η < γ , where γ is defined in (66). Then ||(I + qG + R )−1 || ≤ Ceγ /2ε ,
λ ∈ l\η (ε),
when ε is small enough,. A similar estimate holds for operator (67): ||(I + Fλ )−1 || ≤ Ceγ /2ε ,
λ ∈ l\η (ε).
Hence, the same estimate is valid for the operator (I + Fλ )−1 , and from (59), (61) it follows that Rλ f = ψv Rv,λ (φv h), λ ∈ l , (69) v∈V
where h ∈ L 2 (ε ) is supported in ∪D j,ε , j > m, and ||h|| L 2 (ε ) ≤ Ceγ /2ε || f || L 2 (ε ) , λ ∈ l\η (ε).
(70)
Relations (69), (70), (46), and Lemma 11 together provide the asymptotic expansion (15) for Rλ f needed to complete the proof of Theorem 6. The last result, which we are going to discuss now, concerns the limiting behavior of the GC as λ approaches λ0 , the bottom of the absolutely √ continuous spectrum. We assume that the resolvent (12) does not have a pole at k = λ0 . Obviously this assumption holds for generic domains ε . Theorem 4 implies that this assumption is equivalent to the absence of bounded solutions of the homogeneous problem (5) with λ = λ0 . Recall that the scattering matrix (10) depends on λ > λ0 and the GC (18) depend on both λ > λ0 and ε > 0.
558
S. Molchanov, B. Vainberg (ε)
Proof of Theorem 7. Consider an infinite channel C s , for which λs,0 = λ0 . Let s,0 be the scattering solution which corresponds to the incident wave ψinc = e−i
√
λ−λs,0 t ε
ϕs,0 (y/ε).
Let φs ∈ C ∞ (ε ), φs = 1 in the channel Cs for t ≥ 2, φs = 0 in Cs for t ≤ 1 and (ε) outside of Cs . We represent s,0 in the form (ε) s,0 = φs ψinc + u, λ > λ0 .
Then u is the outgoing solution of the problem (−ε2 − λ)u = f, x ∈ ε ;
u = 0 on ∂ε ,
where f = −ε2 (φs )ψinc − 2ε2 ∇φs ∇ψinc has a compact support. Hence, u = R √λ f. From here, the second statement of Theorem 4, and the absence of a pole at k√= λ0 2 ( ), is analytic in z = it follows that u, if considered as an element of L loc λ − λ0 ε in a neighborhood of the point z = 0. Then from standard local a priori estimates for solutions of elliptic problems it follows that u is analytic, if considered as an element of any Sobolev space of functions on any bounded part of ε . Hence, the restrictions u j of u to cross-sections t = 2 of infinite channels C j are analytic in z. Thus, for any infinite channel C j , u is an outgoing solution of the problem (−ε2 − λ)u = 0, x ∈ ε ∩ {t > 2}; u = 0 on ∂ε ∩ {t > 2}; u|t=2 = u j . (71) √ Since u j is analytic in z = λ − λ0 in a neighborhood of the point z = 0, the coefficients a j,n in the asymptotic expansion (6) for the solution u of (71) are analytic in z. This proves the analyticity of the scattering matrix. (ε) From analyticity of u in z and (71) it also follows that the scattering solution s,0 , when z = 0, is a solution of the homogeneous problem (5) with λ = λ0 , and satisfies (ε) (13). Thus s,0 ≡ 0 when z = 0 due to Theorem 4. This implies that T = −I and completes the proof of the first statement. The second statement of the theorem is an obvious consequence of the analyticity of Tv and (18). This completes the proof of the theorem. Remarks concerning Theorem 7. 1) Consider a bounded domain ε with one junction and several channels of finite length. Let ε be a spider type domain which one gets by extending the channels of ε to infinity. The spectrum of the problem (5) in ε is discrete, and there exists a sequence of eigenvalues which approach λ0 as ε → 0. Each of these eigenvalues has the form λn (ε) = λ0 + O(ε2 ).
(72)
Theorem 7, concerning the problem in ε , can be used to specify the asymptotic behavior (72) of the eigenvalues λn (ε). The last statement of the theorem and (72) indicate that, for generic domains ε , the asymptotic behavior of λn (ε) as ε → 0 (when n is fixed or n → ∞ not very fast) is the same as for eigenvalues of the corresponding Dirichlet problem on the limiting graph with the Dirichlet GC at the vertex. This result will be discussed in more detail elsewhere. 2) Our paper [14] contains a mistake in the statement of Theorem 5.1 (which is a simplified version of Theorem 7 above) about the form of the GC at the bottom of the
Scattering Solutions in Networks of Thin Fibers: Small Diameter Asymptotics
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absolutely continuous spectrum: k → 0 has to be replaced there by k → 0, ε → 0. The arguments in the last 5 lines of the proof are wrong, but can be easily corrected with the additional assumption that ε → 0. (Also, the index of summation in the formulas (5.2), (5.4), (5.6) of that paper must be n, not j.) References 1. Birman, M.S.: Perturbations of the continuous spectrum of a singular elliptic operator by varying the boundary and the boundary conditions (Russian, English summary). Vestnik Leningrad. Univ. 17(1), 22–55 (1962) 2. Blekher, P.M.: On operators that depend meromorphically on a parameter. Moscow Univ. Math. Bull. 24(5–6), 21–26 (1972) 3. Dell’Antonio, G., Tenuta, L.: Quantum graphs as holonomic constraints. J. Math. Phys. 47, 072102:1–21 (2006) 4. Duclos, P., Exner, P.: Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7, 73–102 (1995) 5. Freidlin, M., Wentzel, A.: Diffusion processes on graphs and averaging principle. Ann. Probab. 21(4), 2215–2245 (1993) 6. Exner, P., Seba, P.: Electrons in semiconductor microstructures: a challengee to operator theorists. In: Schrödinger Operators, Standard and Nonstandard (Dubna 1988), Singapure: World Scientific, 1989 pp. 79–100 7. Exner, P., Post, O.: Convergence of spectra of graph-like thin manifolds. J. Geom. Phys. 54, 77–115 (2005) 8. Kostrykin, V., Schrader, R.: Kirchhoff’s rule for quantum waves. J. Phys. A: Mathematical and General 32, 595–630 (1999) 9. Kuchment, P.: Graph models of wave propagation in thin structures. Waves in Random Media 12, 1– 24 (2002) 10. Kuchment, P.: Quantum graphs. I. Some basic structures, Waves in Random Media 14(1), 107–128 (2004) 11. Kuchment, P.: Quantum graphs. II. Some spectral properties of quantum and combinatorial graphs. J. Phys. A: Mathematical and General 38(22), 4887–4900 (2005) 12. Kuchment, P., Zeng, H.: Convergence of spectra of mesoscopic systems collapsing onto a graph. J. Math. Anal. Appl. 258, 671–700 (2001) 13. Kuchment, P., Zeng, H.: Asymptotics of spectra of Neumann Laplacians in thin domains. In: Advances in Differential Equations and mathematical Physics, Yu. Karpeshina etc. (editors), Contemporary Mathematics 387, Providence, RI: Amer. Math. Sec., 2003 pp. 199–213 14. Molchanov, S., Vainberg, B.: Transition from a network of thin fibers to quantum graph: an explicitly solvable model. Contemporary Mathematics 115, Providence, RI: Amer. Math. Sec., 2006 pp. 227–239 15. Novikov, S.: Schrödinger operators on graphs and symplectic geometry. The Arnold fest., Fields Inst. Commun. 24, Providence, RI: Amer. Math. Sec., 1999 pp. 397–413 16. Post, O.: Branched quantum wave guides with Dirichle BC: the decoupling case. J. Phys. A: Mathematical and General 38(22), 4917–4932 (2005) 17. Reed, M., Simon, B.: Methods of modern mathematical Physics, IV: Analysis of operators, New York: Acadamic Press, A Subsidiary of Harcourt Brace Jovnnovich, Publishers, 1978 18. Rubinstein, J., Schatzman, M.: Variational problems on myltiply connected thin strips. I. Basic estimates and convergence of the Laplacian spectrum. Arch. Ration. Mech. Anal. 160(4), 293–306 (2001) 19. Vainberg, B.: On analytic properties of the resolvent for a class of sheaves of operators, Math. USSR-Sb. 6, 241–273 (1968) 20. Vainberg, B.: On short wave asymptotic behavior of solutions of stationary problems and the asymptotic behavior as t → ∞ of solutions of non-stationary problems. Russ. Math. Surv. 30(2), 1–58 (1975) Communicated by B. Simon
Commun. Math. Phys. 273, 561–599 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0270-y
Communications in
Mathematical Physics
On Absolute Moments of Characteristic Polynomials of a Certain Class of Complex Random Matrices Yan V. Fyodorov1, , Boris A. Khoruzhenko2, 1 School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK.
E-mail: [email protected]
2 School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK.
E-mail: [email protected] Received: 31 January 2006 / Accepted: 8 March 2007 Published online: 31 May 2007 – © Springer-Verlag 2007
Abstract: The integer moments of the spectral determinant | det(z I − W )|2 of complex random matrices W are obtained in terms of the characteristic polynomial of the positive-semidefinite matrix W W † for the class of matrices W = AU , where A is a given matrix and U is random unitary. This work is motivated by studies of complex eigenvalues of random matrices and potential applications of the obtained results are discussed in this context. 1. Introduction The characteristic polynomials of random matrices have recently attracted considerable interest in the mathematical physics literature. Initially, the interest was stimulated by applications in number theory [35, 36], quantum chaos [3, 21, 27] and quantum chromodynamics (QCD) [43, 46, 29, 22], but with the emerging connections to integrable systems [39, 47], combinatorics [16, 44], representation theory [9, 10, 12, 15] and analysis [5] it has become apparent that the characteristic polynomials of random matrices are also of independent interest. In this paper we are concerned with the integer moments of the squared modulus of the characteristic polynomial of complex random matrices in a rather general class of matrices W = AU , where A ≥ 0 is fixed and U is a random unitary matrix distributed uniformly over the unitary group. In the particular case when A is an identity matrix, the matrix W is unitary and its eigenvalues lie on the unit circle. Various moments of the characteristic polynomial for this class of matrices were obtained recently, see [35, 36, 13–15]. In the general case, the eigenvalues of W = AU will be distributed in a region in the complex plane. Eigenvalue The research in Nottingham was supported by EPSRC grant EP/C515056/1: “Random Matrices and Polynomials: a tool to understand complexity”. Part of this work was carried out during the Newton Institute programme on Random Matrix Approaches in Number Theory.
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statistics of such complex eigenvalues, and in particular the mean eigenvalue density, are of interest for physics of open chaotic systems, see, e.g. [24, 25], and in QCD, see, e.g. [47] and references therein, and are difficult to study analytically. In this context moments of the squared modulus of the characteristic polynomial frequently provide a very useful tool. Indeed, in a variety of random matrix ensembles the mean eigenvalue density, ρ(x, y) = tr δ(z I − W )W , z = x + i y, (1.1) can be expressed in terms of the mean-square-modulus of the characteristic polynomial in a closely related random matrix ensemble. In (1.1) the angle brackets stand for the average over the matrix distribution, and I is the identity matrix. An obvious example is served by the Ginibre ensemble of complex matrices [31]. In this ensemble the matrix distribution has density Const. exp(− tr W W † ), where W † is the complex conjugate transpose of W . The mean density ρn (x, y) of eigenvalues of the Ginibre matrices of size n × n is given by 1 −|z|2 |z|2k e . π k! n−1
ρn (x, y) =
(1.2)
k=0
One can arrive at (1.2) in various ways. Ginibre computed the joint probability density function of the eigenvalues and then applied the method of orthogonal polynomials. Another way is to use the method of dimensional reduction, see e.g. [45, 17, 18] which gives the following relation: e−|z| |det(z In−1 − Wn−1 )|2 Wn−1 . π(n − 1)! 2
ρn (x, y) =
(1.3)
Here the angle brackets stand for averaging over the Ginibre ensemble of complex matrices of size (n − 1) × (n − 1). The mean-square on the right-hand side in (1.3) can be easily computed giving again (1.2). A less obvious example, which in fact provided the initial impetus for the present study, is the so-called ensemble of ‘random contractions’ [25]. In its simplest variant of rank-one deviations from the unitarity, these are random n × n matrices satisfying the constraint 1−γ 0 † Wn Wn = , 0 < γ < 1. (1.4) 0 In−1 In the ‘polar’ coordinates, Wn = G n Un , where Un is a CUE √n matrix, i.e. a matrix drawn at random from the unitary group U (n), and G n = diag ( 1 − γ , 1, . . . , 1). The mean density of eigenvalues1 of Wn can be expressed as the mean-square-modulus of the characteristic polynomial of (n − 1) × (n − 1) matrices G˜ n−1 Un−1 , n − 1 γ˜ n−2 ρn (x, y) = | det(z In−1 − G˜ n−1 Un−1 )|2 Un−1 , 1 − γ < |z|2 < 1, π γ |z|2 γ (1.5) where now the angle brackets stand for averaging over the unitary group U (n − 1) with respect to the normalized Haar measure, and G˜ n−1 = diag( 1 − γ˜ , 1, . . . , 1),
γ˜ =
|z|2 + γ − 1 . |z|2
1 Note that constraint (1.4) implies that the eigenvalues of W lie in the annulus 1 − γ ≤ |z|2 ≤ 1.
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Another example is provided by finite-rank deviations from Hermiticity [23]. We only consider the simplest but still non-trivial case of rank-one deviations. Let Wn = Hn + iΓn ,
(1.6)
where Hn is a GUEn matrix, i.e. random Hermitian matrix of size n × n with probability distribution β
d Pβ,n (H ) = Const.e− 2 tr H
2
n j=1
d Hjj
1≤ j
d Hjjd Hjj , 2
β > 0,
and Γn = diag(γ , 0, . . . , 0), γ > 0. For the mean eigenvalue density ρn (x, y) of matrices (1.6) we have ρn (x, y) = rβ,n (x, y) | det(z In−1 − (Hn−1 + i Γ˜n−1 )|2 Hn−1 ,
0 < y < γ,
(1.7)
where βx 2
β n (γ − y)n−2 e− 2 −β(γ −y)y rβ,n (x, y) = , √ 4 2πβ γ n−1 (n − 2)!
Γ˜n−1 = diag(γ − y, 0, . . . , 0),
and the angle brackets stand for averaging with respect to the distribution d Pβ,n−1 (H ). We derive (1.5) and (1.7) in Sect. 6. The above formulas relating the mean eigenvalue density and the mean-squaremodulus of the characteristic polynomial are specific to the considered matrix distributions. In the general situation, the mean density of eigenvalues can be determined from the fractional moments of | det(z I − W )|2 = det(z I − W )(z I − W )† (e.g., by way of the logarithmic potential of the eigenvalue distribution), or from averages of ratios of det[(z I − W )(z I − W )† + ε2 I ], see e.g. [25]. Getting explicit formulas for that kind of objects outside the classes of Hermitian and unitary matrices is, however, a considerable challenge. Although it is known that the fractional moments of | det(z I − W )|2 can be written in terms of a hypergeometric function of matrix argument W W † [39], the corresponding series are hard to deal with in the limit of the infinite matrix dimension. Our main result, Theorem 1, expresses ±m U , det(z I − AU )(z I − AU )†
m = 1, 2, . . . ,
where the integration is over the unitary matrices U with respect to the Haar measure, as an m-fold integral of powers of the characteristic polynomial of A A† . This integral can be written as an m × m determinant with entries given by a certain integral transform of the characteristic polynomial of A A† , see (2.11)-(2.12). In particular, this result implies that for the ensembles of random complex matrices W with unitary invariant matrix distribution (e.g., for the Feinberg-Zee ensemble [19]) our formulas effectively reduce the original non-Hermitian problem to a Hermitian one, albeit on the level of characteristic polynomials. This, as explained in more detail at the end of the next section, has a clear computational advantage, as one can then use various formulas for the averages of products and ratios of the characteristic polynomials of Hermitian matrices which have been obtained recently, see [11, 26, 5]. In contrast, with the exception of essentially Gaussian weights [1, 2], no such formulas are known for complex matrices.
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We also express
1 det[(z I − AU )(z I − AU )† + ε2 I ]
U
as a two-fold integral of the inverse spectral determinant of A A† , see Theorem 2. Again, the non-Hermitian problem is reduced to a Hermitian one. This regularized inverse spectral determinant can be useful as an indicator of the domain of the distribution of complex eigenvalues. 2. Statement of Main Results and Discussion Let n and m be positive integers. Define dµn (t1 , . . . , tm ) =
m 1 2 ∆ (t1 , . . . , tm ) (1 + t j )−n−2m dt1 . . . dtm , cn
t j ≥ 0, (2.1)
j=1
and, for n ≥ 2m, dνn (t1 , . . . , tm ) =
m 1 2 ∆ (t1 , . . . , tm ) (1 − t j )n−2m dt1 . . . dtm , kn
0 ≤ t j ≤ 1,
j=1
(2.2)
m− j m ∆(t1 , . . . , tm ) = det ti
where
i, j=1
=
(ti − t j )
(2.3)
1≤i< j≤m
is the Vardermonde determinant, and cn =
m−1 j=0
j!( j + 1)!(n + j)! (n + m + j)!
and kn =
m−1 j=0
j!( j + 1)!(n − m − j − 1)! (n − j − 1)!
(2.4)
are the normalization constants. The Selberg Integral, see e.g. [38], asserts that dµn and dνn are unit mass measures, ∞ ∞ 1 1 . . . dµn (t1 , . . . , tm ) = . . . dνn (t1 , . . . , tm ) = 1. 0
0
0
0
The measures dµn and dνn define probability distributions which have the following random matrix interpretation. Consider two families of matrix distributions on the space
m of m × m complex matrices Z = x jk + i y jk j,k=1 : m 1 1 d x jk dy jk , d µˆ n (Z ) = cˆn det n+2m (Im + Z Z † ) j,k=1
and d νˆ n (Z ) =
m 1 det n−2m (Im − Z Z † ) d x jk dy jk , kˆn j,k=1
n ≥ 0,
(2.5)
n ≥ 2m.
(2.6)
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The measures d νˆ n (Z ) are defined on the matrix ball Z Z † < Im and the constants cˆn and kˆn are determined by the normalization condition d µˆ n (Z ) = d νˆ n (Z ) = 1. Z Z † ≥0
0≤Z Z † ≤Im
A standard calculation, see e.g. [30], shows that if s(Z Z † ) is a symmetric function of the eigenvalues t1 , . . . , tm of Z Z † , i.e. s(Z Z † ) = s(t1 , . . . , tm ), then
+∞ +∞ s(Z Z ) d µˆ n (Z ) = · · · s(t1 , . . . , tm ) dµn (t1 , . . . tm ), †
Z Z † ≥0
0≤Z Z † ≤Im
0
0
1
1
s(Z Z † ) d νˆ n (Z ) =
··· 0
(2.7)
s(t1 , . . . , tm ) dνn (t1 , . . . tm ).
(2.8)
0
Theorem 1 below, which we state in slightly more generality than required for spectral determinants, tells how to integrate moments of determinantsover the unitary group equipped with the Haar measure dU fixed by the normalization U (n) dU = 1. Theorem 1. Let A, B, C, D be complex matrices of size n × n. (i) For any positive integer m,
∞ ∞ m det [(AU + C)(BU + D) ]dU = . . . det(C D † + t j AB † ) dµn (t1 , . . . , tm ). m
†
U (n)
0
0
j=1
(2.9) (ii) If A A† < CC † and B B † < D D † then for any positive integer m such that 2m ≤ n, U (n)
dU = m det [(AU + C)(BU + D)† ]
1 1 ... 0
m
0
dνn (t1 , . . . , tm )
.
(2.10)
det(C D † − t j AB † )
j=1
Remark. Identities (2.9) – (2.10) may be written in yet another form by making use of the well-known identity, see e.g. [40] Part 2, Problem 68,
m
m
m . . . det p j (ti ) i, j=1 det q j (ti ) i, j=1 dt1 . . . dtm = m! det pi (t)q j (t)dt
.
i, j=1
We have U (n)
det m [(AU + C)(BU + D)† ]dU m! det = cn
0
+∞
det(C D † + t AB † ) t i+ j dt (1 + t)n+2m
m−1 (2.11) i, j=0
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Y. V. Fyodorov, B. A. Khoruzhenko
and U (n)
dU m! det = det m [(AU + C)(BU + D)† ] kn
1 0
(1 − t)n−2m t i+ j dt det(C D † − t AB † )
m−1 , i, j=0
(2.12) where the constants cn and kn are given in (2.4). Obviously, by letting C = D = z I in (2.9) and (2.10) one obtains formulas for the integer moments of the spectral determinants | det(z I − AU )|2 . In particular,
∞ | det(z I − AU )| dU = (n + 1) 2
U (n)
and, provided n ≥ 2,
0
det(|z|2 I + t A A† ) dt (1 + t)n+2
(2.13)
⎧ 1 ⎪ ⎪ (n − 1)(1 − t)n−2 ⎪ ⎪ dt, if |z|2 < λmin (A A† ); ⎪ ⎪ ⎨ det(A A† − t|z|2 I )
dU = 01 ⎪ | det(z I − AU )|2 ⎪ (n − 1)(1 − t)n−2 ⎪ ⎪ U (n) 2 † ⎪ ⎪ det(|z|2 I − t A A† ) dt, if |z| > λmax (A A ), ⎩
(2.14)
0
where λmin (A A† ) and λmax (A A† ) are respectively the smallest and the largest eigenvalues of A A† . If λmin (A A† ) ≤ |z|2 ≤ λmax (A A† ), then the integral on the left-hand side in (2.14) should be handled with care. One way to do this is to regularize the integrand. For positive ε, define dU † Rz,ε (A, A ) =
† . U (n) 1 1 2 det ε I + I − z AU I − z AU The integral on the right-hand side is, in fact, a function of A A† and our next theorem evaluates this function in terms of the eigenvalues of A A† . Theorem 2. Let ε > 0, and assume that n ≥ 2. Then for any n × n matrix A and any non-zero complex z, Rz,ε (A, A† ) +∞ 1 n−1 1 dx n−2 = (1 − t) dt .
√ 1 2πi 0 † + ε 2 − t I − iε t x + 1 I −∞ x det A A 2 x |z| If the eigenvalues a 2j of A A† are all distinct then for any z in the annulus λmin (A A† ) < |z|2 < λmax (A A† ) we have lim
ε→0
n Rz,ε (A, A† ) 1 2 2 2 n−2 2 2 (|z| − a ) θ (|z| − a ) , (2.15) = (n − 1)|z| j j 2 2 ln(1/ε ) a − a 2j j=1 k= j k
where θ is the Heaviside step function.
Absolute Moments of Characteristic Polynomials of Complex Random Matrices
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We prove Theorems 1 and 2 in Sects. 4 and 5, respectively, by making use of two techniques, which to a certain extent are equivalent. One is based on the expansion of moments of the spectral determinants in characters of the unitary group and subsequent use of the orthogonality of the characters. In this way, Theorem 1 is equivalent to two combinatorial identities (3.18) and (3.19), one of which is a particular case of the Selberg integral in the form of Kaneko [34] and Kadell [33]. We prove (3.18) and (3.19) in Sect. 3. These combinatorial identities can be stated in the form of matrix integrals (3.27) and (3.28) and are of independent interest. They lead to evaluation of some nontrivial matrix integrals, as discussed at the end of Sect. 3. The other technique is based on the so-called color-flavor transformation, due to Zirnbauer [49]. This transformation has many uses, and in the random matrix context it provides a very convenient tool to handle moments of spectral determinants. As an application of Theorem 1, let us consider random matrices (1.4). In the limit n → ∞ the eigenvalues of Wn get closer and closer to the unit circle. Let Nn (a, b) be the number of eigenvalues of Wn in the annulus 2a 2b 2 ≤ 1 − |z| ≤ , 0 < a < b. Da,b = z : n n By (1.5), Nn (a, b)U (n) ⎛ n−2 ⎜ n − 1 γ˜ = ⎝ π γ |z|2 γ
⎞ ⎟ | det(z In−1 − G˜ n−1 Un−1 )|2 dUn−1 ⎠ d xd y.
U (n−1)
Da,b
Making use of (2.13),
∞
| det(z In−1 − G˜ n−1 Un−1 )|2 dUn−1 = n
U (n−1)
0
(|z|2 + t (1 − γ˜ ))(|z|2 + t)n−2 dt (1 + t)n+1
and 1 Nn (a, b)U (n) = π n where f n (q) =
n−1 πγ q
1− 2a n
1− 2b n
f n (q)dq,
n−2 ∞ γ˜ [q + t (1 − γ˜ )](q + t)n−2 dt γ (1 + t)n+1 0
with γ˜ = (|z|2 + γ − 1)/|z|2 . Letting n → ∞, we obtain, after simple manipulations, a(γ − 2) sinh b b(γ − 2) 1 sinh a exp − exp , lim Nn (a, b)U (n) = n→∞ n a γ b γ recovering one of the formulas of [25], in which, using a different method requiring knowledge of the joint probability distribution of eigenvalues, they found the mean density of eigenvalues and higher order correlation functions for the general case of finite-rank deviation from the CUE. Note that when γ = 1 the nonzero eigenvalues of
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G n Un coincide with those of the (n − 1) × (n − 1) matrix obtained from Un by removing its first row and column, see [50] for more information about eigenvalue statistics of truncated unitary matrices. Now, we would like to elaborate on the point made at the end of the Introduction. Consider random complex matrices W of the size n × n with unitary invariant matrix distribution. Then, by making use of the unitary invariance and Theorem 1, ∞ pn (|z|2 t) 2 2 | det(z I − W )| = | det(z I − W U )| dU = dt, (2.16) W (1 + t)n+2 U (n) 0 W where pn (x) = det(x I + W W † )W . A similar formula holds for higher order moments of | det(z I − W )|2 . Thus, Theorem 1 reduces the original non-Hermitian problem to a Hermitian one. The integral on the right-hand side in (2.16) can be evaluated, in the limit of the infinite matrix dimension, in terms of the limiting eigenvalue distribution of W W † . To this end, consider, for example, the complex n × n matrices W with the matrix distribution characterized by the Feinberg-Zee density Const. e−n tr V (W W ) , †
(2.17)
where V (r ) is a polynomial in r , V (r ) = am r m + . . ., am > 0. Then pn (x) = en
ln(x+λ)dw(λ)
(1 + o(1)),
where dw(λ) is the limiting normalized eigenvalue counting measure of W W † , and it follows that 1 lim ln | det(z I − W )|2 = (x, y), n→∞ n W where
⎧ 2 if |z| > m 1 = λdw(λ), ⎪ ⎪ln |z| ⎪ ⎪ ⎪ −1 ⎪ ⎨ ∞ −1 ln λdw(λ) if |z| < 1/m −1 = , λ dw(λ)
(x, y) = ⎪ 0 ⎪ ⎪ ⎪ ∞ ⎪ λ + t0 ⎪ ⎩|z|2 + ln 2 dw(λ) if 1/m −1 < |z| < m 1 , |z| + t0 0 (2.18) where t0 is the unique non-negative solution of ∞ 1 dw(λ) = 2 . λ+t |z| + t 0 1 The function (x, y) is subharmonic and, hence, defines a measure dν = 4π ∆ in the complex plane. Here ∆ is the Laplacian in variables x and y. For the Ginibre ensemble of random matrices this measure can be found explicitly. In this case V (r ) = r and W W † is the Wishart ensemble √ of random matrices. Its limiting eigenvalue distribution 1 dw(λ) is given by dw(λ) = 2π (4 − λ)/λ, 0 < λ < 4, with m 1 = 1 and m −1 = ∞.
Absolute Moments of Characteristic Polynomials of Complex Random Matrices
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A straightforward but tedious calculation shows that (x, y) = |z|2 − 1 inside the unit disk |z|2 < 1. Therefore dν is the uniform distribution on the unit disk, which is the same as the limiting eigenvalue distribution in the Ginibre ensemble of random matrices, and, hence, for this ensemble 1 ln | det(z I − W )|2 n→∞ n lim
W
1 ln | det(z I − W )|2 n→∞ n
= lim
W
,
(2.19)
so that the operations of taking the logarithm and average commute in the limit n → ∞. A similar relation is known to hold for Wigner ensembles of Hermitian matrices [6]. It would be interesting to investigate conditions on random matrix distributions which guarantee (2.19). As the left hand-side in (2.19) is the logarithmic potential of the limiting eigenvalue distribution of W , this together with our Theorem 1 would give a useful tool for calculating eigenvalue distributions in the complex plane. There are indications that the range of matrix distributions for which (2.19) holds is quite wide and contains the invariant ensembles (2.17). Indeed, (x, y) of Eq. (2.18) reproduces the density of eigenvalue distribution in ensembles (2.17) which was obtained in [19, 20] with the help of the method of Hermitization2 . In this context we would like to mention calculation of Brown’s measure for R-diagonal elements in finite von Neumann algebras [28], see also [8]. A matrix model for such elements is provided by random matrices RU , where U is random unitary and R is positive-definite, and Brown’s measure is in a way a regularized version of the eigenvalue distribution. Again, (x, y) of Eq. (2.18) reproduces Brown’s measure found in [28]. 3. Combinatorial Identities Schur functions. In order to make our paper self-contained we recall below the required facts from the theory of symmetric polynomials. A partition is a finite sequence λ = (λ1 , λ2 , . . . , λn ) of integers, called parts, such that λ1!≥ λ2 ≥ · · · ≥ λn ≥ 0. The weight of a partition, |λ|, is the sum of its parts |λ| = j λ j , and the length, l(λ), is the number of its non-zero parts. No distinction is made between partitions which differ merely by the number of zero parts, and different partitions of weight r represent different ways to write r as a sum of natural numbers. Partitions can be viewed as Young diagrams. The Young diagram of λ is a rectangular array of boxes (or dots), with λ j boxes in the j th row, the rows being lined up on the left. By transposing the diagram of λ (i.e. interchanging the rows and columns) one obtains another partition. This partition is called the conjugate of λ and denoted by λ . For example the conjugate of the partition (r ) of length one is the partition (1, . . . , 1) ≡ (1r ) of length r . Obviously, l(λ ) = λ1 and |λ| = |λ |. For any partition λ of length l(λ) ≤ n,
λ +n− j n det xi j i, j=1 sλ (x1 , . . . , xn ) = (3.1)
n− j n det xi i, j=1
is a symmetric polynomial in x1 , . . . , xn , homogeneous of degree |λ|. These polynomials are known as the Schur functions. By convention, sλ (x1 , . . . , xn ) = 0 if l(λ) > n. 2 This method has a hidden regularization procedure which has to be justified to satisfy mathematical rigor.
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This convention is in agreement with the apparent identities sλ (x1 , . . . , xn−1 , 0) = sλ (x1 , . . . , xn−1 ) =0
if l(λ) ≤ n − 1, if l(λ) > n − 1.
(3.2) (3.3)
For partitions of length one, λ = (r ), the Schur functions sλ are the complete symmetric functions h r , s(r ) (x1 , . . . , xn ) = h r (x1 , . . . , xn ) = x i 1 x i 2 . . . x ir , (3.4) 1≤i 1 ≤i 2 ≤...≤ir ≤n
and sλ are the elementary symmetric functions er , s(1r ) (x1 , . . . , xn ) = er (x1 , . . . , xn ) =
x i 1 x i 2 . . . x ir .
(3.5)
1≤i 1
More generally, see e.g. [37] p. 41, the Jacobi-Trudi identity asserts that for any n ≥ l(λ), n n
sλ = det h λi −i+ j i, j=1 , sλ = det eλi −i+ j i, j=1 , (3.6) where, by convention, er = h r = 0 if r < 0. We shall also need the Schur functions of matrix argument. If M is an n × n matrix then sλ (M) = sλ (x1 , . . . , xn ), where x1 , . . . , xn are the eigenvalues of M. Thus sλ (M) is a symmetric polynomial in the eigenvalues of M. In view of (3.6), it is also a polynomial in the matrix entries of M. The Schur functions of the matrix argument are the characters of irreducible representations of the general linear group and its unitary subgroup and, as a consequence, have an important property of orthogonality. If λ and µ are two partitions and A and B are two n × n matrices then, see e.g. [37] p. 445, sλ (AB † ) sλ (AU )sµ (BU )dU = δλ,µ , (3.7) dλ U (n) and
U (n)
sλ (AU BU † )dU =
sλ (A)sλ (B) , dλ
(3.8)
where dλ is the dimension of the irreducible representations of U (n) with signature λ, dλ = sλ (In ) = sλ (1n ). We use the notation (1n ) for the n-tuple (1, . . . , 1). If λ is a partition of length 1, λ = (r ), then (n + r − 1)! n +r −1 , sλ (1n ) = h r (1n ) = = r r !(n − 1)! and sλ (1n ) = er (1n ) =
n! n = . r r !(n − r )!
(3.9)
(3.10)
Absolute Moments of Characteristic Polynomials of Complex Random Matrices
571
In general, explicit expressions are known for sλ (1n ) and sλ (1n ) in terms of the λ j ’s. If l(λ) ≤ n then for any m ≥ l(λ),
sλ (1n ) =
m
(n + λ j − j)! . (m + λ j − j)!(n − j)!
(3.11)
(n + j − 1)! . (n + j − 1 − λ j )!(m + λ j − j)!
(3.12)
(λi − i − λ j + j)
1≤i< j≤m
j=1
If l(λ ) ≤ n then for any m ≥ l(λ), sλ (1n ) =
(λi − i − λ j + j)
1≤i< j≤m
m j=1
Both identities can be derived by evaluating the binomial determinants in (3.6). We shall also need the Cauchy identities for Schur functions, see, e.g., [37], pp. 63, 65. Let X be an n × n matrix. Then m
det(In + ti X ) =
sλ (t1 , . . . , tm )sλ (X ),
(3.13)
1 = sλ (t1 , . . . , tm )sλ (X ). det(In − ti X )
(3.14)
i=1 m
λ
i=1
λ
The summation in (3.13) is over all partitions such that l(λ) ≤ m and l(λ ) ≤ n and is finite. The summation in (3.14) is over all partitions such that l(λ) ≤ min(m, n) and is infinite. The corresponding series converges absolutely if X X † < In . Beta-function determinants. When m = 1 identities (3.13) and (3.14) take the familiar form of the expansion of the characteristic polynomial and its reciprocal in terms of the elementary symmetric functions and complete symmetric functions, respectively. In this case Theorem 1 is a straightforward consequence of the orthogonality property of the Schur functions (3.7) and the Euler integral 1 +∞ t p−1 Γ ( p)Γ (q) = B( p, q), Re p, q > 0, t p−1 (1−t)q−1 dt = dt = (1 + t) p+q Γ ( p + q) 0 0 (3.15) where Γ ( p) and B( p, q) are the Gamma and Beta functions respectively. Indeed, for example, n er (AB † ) det(I + AU ) det(I + BU )† dU = er (1n ) U (n) r =0 ∞ dt det(I + t AB † ) , = (n + 1) (1 + t)n+2 0 where we have used (3.15) in the form 1 = (n + 1) er (1n )
+∞ 0
tr dt. (1 + t)n+2
(3.16)
Similarly,
1 1 = (n − 1) t r (1 − t)n−2 dt, h r (1n ) 0 and this identity does the trick for the reciprocal characteristic polynomials.
(3.17)
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Our proof of Theorem 1 uses the following generalization of (3.16) – (3.17) to the multivariate setting. Lemma 3. Let m and n be nonnegative integers.
(a) For any partition λ such that l(λ) ≤ m and l(λ ) ≤ n, ∞ ∞ m . . . sλ (t1 , . . . tm )∆2 (t1 , . . . , tm )
sλ2 (1m ) 1 = sλ (1n ) cn
0
j=1
0
dt j . (1 + t j )n+2m
(3.18)
(b) If 2m ≤ n then for any partition λ such that l(λ) ≤ m, sλ2 (1m ) 1 = sλ (1n ) kn
1 1 m . . . sλ (t1 , . . . tm )∆2 (t1 , . . . , tm ) (1 − t j )n−2m dt j . 0
(3.19)
j=1
0
The normalization constants cn and kn are given in (2.4), and ∆(t1 , . . . , tm ) is the Vandermonde determinant (2.3). Remark. Identity (3.19) can be inferred from a generalization of the Selberg Integral due to Kaneko [34] and Kadell [33], of which (3.19) is a particular case. However, we are not aware about any generalization of the Selberg Integral leading to (3.18). Below, we give an elementary proof of (3.18) and (3.19) based on evaluation of a determinant consisting of Beta functions. Our proof has limited scope and does not extend to the generality of Kaneko and Kadell formulas.
Proof. Let f j = m + λ j − j, j = 1, 2, . . . , m. If l(λ) ≤ m ≤ n and l(λ ) ≤ n then by (3.11) – (3.12), m−1 m sλ2 (1m ) 1 = ∆( f 1 , . . . , f m ) f j !(n + m − 1 − f j )!, sλ (1n ) j!2 (n + j)!
where ∆( f 1 , . . . , f m ) =
j=0
j=1
m
( f i − f j ) = det f jm−i
1≤i< j≤m
i, j=1
.
By adding rows in the Vandermonde determinant det f jm−i ,
m ∆( f 1 , . . . , f m ) = det pm−i ( f j ) i, j=1 , where pk (x) = (x + 1)(x + 2) . . . (x + k). Hence
m f 1 ! f 2 ! . . . f m !∆( f 1 , . . . , f m ) = det ( f j + m − i)! i, j=1 ,
and m−1 (n + m + j)!
m sλ2 (1m ) = det B( f j + m − i + 1, n + m − f j ) i, j=1 , sλ (1n ) ( j!)2 (n + j)! j=0
(3.20)
Absolute Moments of Characteristic Polynomials of Complex Random Matrices
573
where B is the Beta function. By making use of Proposition 4 below,
m det B( f j + m − i + 1, n + m − f j ) i, j=1
m = det B( f j + m − i + 1, n + m − f j + i − 1) i, j=1 m +∞ f j m−i t t dt = det . (1 + t)n+2m i, j=1 0 It is apparent that ⎛ +∞ ⎞ +∞ +∞ m
f j t m−i dt tim−i dti t m− j ⎝ ⎠ det = · · · s (t , . . . t ) det t λ 1 m i (1 + t)n+2m (1 + ti )n+2m 0
0
=
1 m!
i=1
0 +∞
+∞
0
0
m
m− j · · · sλ (t1 , . . . tm )det 2 ti i=1
dti , (1 + ti )n+2m
and (3.18) follows. Similarly, if 2m ≤ n and l(λ) ≤ m then by (3.11), m m−1 (n − j − 1)! sλ2 (1m ) f j! = ∆( f 1 , . . . , f m ) , 2 sλ (1n ) j! (n − m + f j )! j=0
j=1
and, in view of (3.20), m−1 sλ2 (1m ) (n − j − 1)! = sλ (1n ) ( j!)2 (n − m − j − 1)! j=0
m × det B( f j + m − i + 1, n − 2m + i) i, j=1 .
(3.21)
By Proposition 4 below,
m det B( f j + m − i + 1, n − 2m + i) i, j=1
m = det B( f j + m − i + 1, n − 2m + 1) i, j=1 m 1 f j m−i n−2m t t (1 − t) dt , = det 0
and (3.19) follows.
(3.22) (3.23)
i, j=1
Proposition 4. For any p1 , p2 , . . . , pm and q1 , q2 , . . . , qm such that Re p j > m and Re q j > −1 we have
m
m det B( p j − i, q j + i) i, j=1 = det B( p j − i, q j + 1) i, j=1 .
(3.24)
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Y. V. Fyodorov, B. A. Khoruzhenko
Proof. We shall use the identity B( p, q − 1) + B( p − 1, q) = B( p − 1, q − 1)
(3.25)
and the operation of addition of columns to transform the determinant on the left in (3.24) to the one on the right. It is convenient to write determinants by showing their columns. With this convention, |B( p − 1, q + 1), B( p − 2, q + 2), . . . , B( p − m + 1, q + m − 1), B( p − m, q + m)| represent the determinant on the left-hand side in (3.24). Let us label its columns by numbers 1, . . . , m from left to right (so that the leftmost column is column 1). Note a particular property of columns in this determinant. As we move from column j to column j + 1 the first argument of the Beta function decreases by 1, the second argument increases by 1. To be able to refer to this property, we say that columns 1, 2, . . . , m are balanced. Observing that column 1 has the desired form already, let us perform the following operation on columns 2, 3, . . . , m. Starting at column m and working backwards, let us add to each column the one that precedes it. In view of (3.25) and the above mentioned property of columns, this operation yields |B( p − 1, q + 1), B( p − 2, q + 1), B( p − 3, q + 2), . . . , B( p − m, q + m − 1)|. Observing that columns 1 and 2 have the desired form now, and that columns 2, . . . , m remain balanced, we apply our operation again, now on columns 3, . . . , m. This yields the determinant |B( p − 1, q + 1), B( p − 2, q + 1), B( p − 3, q + 1), . . . , B( p − m, q + m − 2)|, where columns 1, 2, 3 have the desired form and columns 4, . . . , m are balanced. It is clear that repeated application of our operation will yield the determinant |B( p − 1, q + 1), B( p − 2, q + 1), B( p − 3, q + 1), . . . , B( p − m, q + 1)| after the final step. This is exactly the determinant on the right in (3.25).
Applications to matrix integrals. In view of the integration formulas (2.7) and (2.8), identities (3.18) and (3.19) can be rewritten as: s 2 (Im ) s 2 (Im ) , , (3.26) sλ (Z Z † ) d µˆ n (Z ) = λ sλ (Z Z † ) d νˆ n (Z ) = λ sλ (In ) sλ (In ) Z Z † ≥0
Z Z † ≤Im
where Z are complex m × m matrices, and Im and In are identity matrices of sizes m × m and n × n, respectively. The first identity holds for any non-negative integer n and any partition λ such that l(λ ) ≤ n. The second one holds for any integer n ≥ 2m and any λ. Since the above identities become trivial (both sides vanish) for partitions of length > m we drop the restriction l(λ) ≤ m. These two identities lead to several useful matrix integrals. Let M be an m × m matrix. Then, for any non-negative integer n, sλ (M)sλ (Im ) , (3.27) sλ (M Z Z † ) d µˆ n (Z ) = sλ (In ) Z Z † ≥0
Absolute Moments of Characteristic Polynomials of Complex Random Matrices
575
provided l(λ ) ≤ n, and if n ≥ 2m then sλ (M)sλ (Im ) sλ (M Z Z † ) d νˆ n (Z ) = † sλ (In ) Z Z ≤Im
(3.28)
for any λ. These two integrals follow from (3.26) and (3.8) and the unitary invariance of d µˆ n (Z ) and d νˆ n (Z ). If L and M are two m × m matrices then for any non-negative integer n, sλ (L M † ) , (3.29) sλ (L Z )sµ (M Z ) d µˆ n (Z ) = δλ,µ sλ (In ) Z Z † ≥0 provided l(λ ) ≤ n and l(µ ) ≤ n, and if n ≥ 2m then sλ (L M † ) . sλ (L Z )sµ (M Z ) d νˆ n (Z ) = δλ,µ sλ (In ) Z Z † ≤Im
(3.30)
These orthogonality relations follow from (3.7) and (3.27) – (3.28), and, in turn, lead to Berezin-Hua integrals [30, 7] det n (Im + L Z )det n (Im + M Z )† d µˆ n (Z ) = det n (Im + L M † ) Z Z † ≥0 d νˆ n (Z ) 1 , n ≥ 2m. = n n n † det (Im − L M † ) Z Z † ≤Im det (Im − L Z )det (Im − M Z ) One only has to recall the Cauchy identites (3.13) and (3.14). If P and Q are two n × m matrices and n ≥ 2m then it follows from (3.7) and (3.30) that sλ (P Q † U )sµ (P Q † U ) dU = sλ (P † P Z )sµ (Q † Q Z ) d νˆ n (Z ) (3.31) U (n)
Z Z † ≤Im
for any λ and µ. Identity (3.31) implies that † † † etr(P Q U +U Q P ) dU = U (n)
Z Z † ≤Im
etr(P
† P Z +Z † Q † Q)
d νˆ n (Z ).
(3.32)
The duality relation (3.32) is a particular case of Zirnbauer’s color-flavor transformation [49]. It can be easily obtained from (3.31) by making use of the expansion etr A = cλ sλ (A). (3.33) λ
! In fact, (3.32) extends to any series g(A) = λ cλ sλ (A), |g(P Q † U )|2 dU = g(P † P Z )g(Q † Q Z ) d νˆ n (Z ). U (n)
Z Z † ≤Im
It follows from (3.33) and (3.7) that the integral over the unitary group on the lefthand side in (3.32) is a function of Q † Q P † P. This function can be evaluated explicitly in terms of the eigenvalues of Q † Q P † P. We would like to demonstrate this in a slightly more general setting.
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Y. V. Fyodorov, B. A. Khoruzhenko
For square matrices A and B of size n × n define † † etr(AU +U B ) dU. Fn (AB † ) = U (n)
(3.34)
If the eigenvalues z 12 , . . . , z n2 of the matrix AB † are all distinct then [42] n
Const. j−1 Fn (AB † ) = I (z ) , det z j−1 i i i, j=1 ∆(z 12 , . . . , z n2 ) where Ik is the modified Bessel function, Ik (z) =
z 2 j+k
∞
2
j!( j + k)!
j=0
.
For our purposes, we want to know Fn (AB † ) for matrices AB † of low rank, e.g. when AB † is rank one. 2 and 2m ≤ Lemma 5. Suppose that AB † has m distinct non-zero eigenvalues z 12 , . . . , z m n. Then
m 1 1 det g(t z 2 ) m m i (n − j)! j i, j=1 Fn (AB † ) = ... tim−i (1 − ti )n−2m dti , 2, . . . , z2 ) (n − m − j)! ∆(z m 1 j=1 i=1 0
0
√ where g(x) = I0 2 x . In particular, if AB † is rank one and z 2 is its non-zero eigenvalue then 1 Fn (AB † ) = (n − 1) I0 2 t z 2 (1 − t)n−2 dt. (3.35) 0
Proof. It follows from (3.33) and (3.7) that Fn (AB † ) =
λ
cλ2 sλ (AB † ). sλ (1n )
The coefficients cλ are given by m m (m − j)! 1 cλ = det = sλ (1m ) (λ j − j + i)! i, j=1 (m + λ j − j)! j=1
see, e.g., [4] and references therein, and Fn (AB ) = †
m−1
j!2
s 2 (1m )
sλ (AB † ) , sλ (1n ) f 1 !2 · . . . · f m !2 λ
λ
j=0
where as before f j = m + λ j − j. Note that the summation is over all partitions λ of length ≤ m, or, equivalently, over all f 1 > f 2 > . . . > f m ≥ 0. It follows now from (3.21) – (3.23) that Fn (AB ) = †
m j=1
(n − j)! (n − m − j)!
1 1 m g(t1 , . . . tm ) m−i ... ti (1 − ti )n−2m dti , 2) ∆(z 12 , . . . , z m i=1 0
0
Absolute Moments of Characteristic Polynomials of Complex Random Matrices
where
f m det ti j
g(t1 , . . . tm ) =
2 f m det z i j
i, j=1 f 1 !2 . . . f m !2
f 1 > f 2 >...> f m ≥0
577
i, j=1
.
To complete the proof, recall ! the following generalization of the Cauchy-Binet formula, see e.g. [30] p. 22. If g(x) = f ≥0 γ f x f is an analytic function in the complex x-plane then
f m
f m m
γ f1 . . . γ fm det ti j det xi j . det g(ti x j ) i, j=1 = i, j=1
f 1 > f 2 >...> f m ≥0
i, j=1
By making use of this formula, m
" g(t1 , . . . , tm ) = det I0 2 ti z 2j
i, j=1
and the lemma follows.
,
4. Proof of Theorem 1 After all the preparatory work of the previous section, Theorem 1 becomes almost evident. We first prove (2.9) – (2.10) for C = D = I . With Lemma 3 in hand, this becomes a routine calculation. Expanding powers of determinants in the Schur functions as in (3.13) – (3.14) and integrating over the unitary group with the help of (3.7), one gets det m [(I + AU )(I + BU )† ]dU =
s 2 (1m ) λ
λ
U (n)
and
U (n)
det m [(I
sλ (1n )
sλ (AB † )
(4.1)
s 2 (1m ) dU λ = sλ (AB † ). † − AU )(I − BU ) ] sλ (1n )
(4.2)
λ
The sum in (4.1) is finite and the sum in (4.2) is absolutely converging for any A A† < I and B B † < I . Now, by making use of (3.18) and (3.19), and then (3.7) again, one arrives at
∞ ∞ m det [(I + AU )(I + BU ) ]dU = . . . det(I + t j AB † ) dµn (t1 , . . . , tm ) (4.3) m
†
U (n)
0
0
j=1
and U (n)
dU = m det [(I − AU )(I − BU )† ]
1 1 ... 0
dνn (t1 , . . . , tm )
m
0 j=1
det(I − t j AB † )
.
(4.4)
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Y. V. Fyodorov, B. A. Khoruzhenko
Extending (4.3) and (4.4) to the generality of (2.9) and (2.10) is straightforward. If C and D are not degenerate, then det m [(AU + C)(BU + D)† ] dU U (n) = det m (C D † ) det m [(I + C −1 AU )(I + D −1 BU )† ] dU U (n)
and (2.9) follows from (4.3). The assumption that C and D are not degenerate can be removed by the continuity argument.
5. Regularization of the Inverse Determinant In this section we employ another approach to the problem of evaluation of negative moments of spectral determinants. This approach is to write the determinants as Gaussian integrals and then perform the integration over the unitary group with the help of the color-flavor transformation. Such approach is not new. It was pioneered by Zirnbauer in the context of unitary random matrix ensembles. The new element here is that we apply it in the general context of complex matrices. We shall write the spectral determinant det[(I − AU )(I − AU )† ] of n × n matrices as an 2n × 2n block determinant # # # 0 i(U † − A) ## † † † † # det[(I − AU )(I − AU ) ] = det[(U − A)(U − A) ] = # # i(U † − A)† 0 and more generally # # # εI i(U † − A) ## # det[ε I + (I − AU )(I − AU ) ] = # #. εI i(U † − A)† 2
†
Proposition 6. Suppose that Re λ j > 0, j = 1, 2. Then for any complex n × n matrix we have # # # λ1 I i #−1 # # † # i λ2 I # $ % 1 = n d 2v d 2w exp −[λ1 v † v + λ2 w† w + i(w † † v + v † w)] . π Cn Cn
(5.1)
The integral on the right-hand side converges absolutely. Remark. In this section, we shall use letters in boldface to represent column vectors in Cn . The symbol d 2v will denote the volume element of v in Cn , d 2v =
n j=1
d 2η j =
n j=1
d Re v j d Im v j .
Absolute Moments of Characteristic Polynomials of Complex Random Matrices
579
Proof. Note that
λ I i λ1 v v + λ2 w w + i(w v + v w) = (v , w ) 1 † i λ2 I †
†
†
†
†
†
†
v w
.
In view of the singular value decomposition = U † ωV , † λ1 I i U 0 λ1 I iω U 0 , = 0 V iω λ2 I i† λ2 I 0 V† where ω is diagonal matrix of singular values of , ω = diag(ω1 , . . . , ωn ), and U and V are unitary matrices. Introducing f = U v and g = V w, λ I i f v λ I iω = ( f †, g†) 1 (v † , w † ) 1 † iω λ2 I g w i λ2 I n fj λ1 iω j = . ( f¯j , g¯ j ) iω j λ2 gj j=1
Since U and V are unitary, d 2v = d 2 f and d 2w = d 2g. Changing the variables of integration in (5.1) from v and w to f and g breaks this 2n-fold integral into the product of the 2-fold integrals 1 exp −λ1 | f j |2 − λ2 |g j |2 − iω j ( f j g¯ j + f¯j g j ) d 2 f j d 2g j = (λ1 λ2 + ω2j )−1 . π C2 Thus, the integral on the right-hand side in (5.1) equals j=1 (λ1 λ2 + ω2j )−1 which is obviously the same as the determinant on the left-hand side. Proof of Theorem 2. Obviously, without loss of generality we can put z = 1. Let dU , ε > 0. (5.2) Rε (A, A† ) = 2 † U (n) det[ε I + (I − AU )(I − AU ) ] The integral on the right-hand side converges for any n × n matrix A. It follows from Proposition 6 that 1 † † † † † Rε (A, A† ) = n d 2v d 2w e−[ε(v v +w w)−i(w A v +v Aw)] f n (v † vw† w), π Cn Cn
(5.3)
where, cf. (3.34), f n (v † vw † w) =
U (n)
† † † ei(w U v +v U w) dU =
By Lemma 5,
1
f n (v vw w) = †
†
U (n)
ei tr(vw
† U +U † wv † )
J0 2 t v † v w † w dσn (t),
0
where dσn (t) = (n − 1)(1 − t)n−2 dt
dU.
(5.4)
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Y. V. Fyodorov, B. A. Khoruzhenko
and J0 is the Bessel function J0 (z) =
∞ j=0
i z 2 j 2
j!2
= I0 (i z).
We have | f n (v † vw† w)| ≤ 1 for all v and w. This is because |J0 (z)| ≤ 1 for all z. Therefore we can interchange the order of integrations on replacing f n in (5.3) by the integral of (5.4). This yields Rε (A, A† ) 1
1 † † † † † = dσn (t) n d 2v d 2w e−ε(v v +w w)+i(w A v +v Aw) J0 2 t v † v w† w . (5.5) π Cn Cn 0 In order to perform the integration in variables v and w we shall make use of the integral representation +∞ d x i( px+ q ) 1 √ x e J0 (2 p q) = (5.6) 2πi −∞ x which holds any p > 0 and q > 0 and is a particular case of Eq. 3.871.1 in [32]. The integral in (5.6) converges because of the oscillations of the exponential function, however the convergence is not absolute. We have +∞
d x i √t(x v † v + w† w ) 1 † † x J0 2 t v v w w = e . (5.7) 2πi −∞ x Note that by Proposition 6, √ † 1 2 2 −ε(v † v +w † w )+i(w † A† v +v † Aw ) i t(x v † v + w x w ) d v d w e e π n Cn Cn #−1 # # (ε − i √t x)I −A√ ## # =# # . # −A† (ε − i x t )I # Therefore, on replacing J0 in (5.5) by the integral of (5.7) and reversing the order of integrations one arrives at 1 +∞ 1 dx 1 † , dσn (t) Rε (A, A ) = √ † 2 2πi x det[A A + (ε − t)I − iε t(x + x1 )I ] 0 −∞ (5.8) which is the identity claimed in Theorem 2. It remains to justify reversing the order of integrations with respect to x and v, w. Firstly, we will show that the integral on the right-hand side in (5.8) is well-defined. Proposition 7. For any ε > 0 and n ≥ 2 the integral in (5.8) converges absolutely (and uniformly in A). Proof. Let a 2j be the eigenvalues of A A† so that n 1 1 1 × = w(a j , t), √ x x det[A A† + (ε2 − t)I − iε t(x + x1 )I ] j=1
Absolute Moments of Characteristic Polynomials of Complex Random Matrices
581
where w(a, t) = Since |z| ≥ | Im z|, we have
1
a2
+ ε2
. √ − t − iε t(x + x1 )
# # #1 # # w(a1 , t)# ≤ √ 1 #x # ε t(1 + x 2 ) .
Also, for all 0 ≤ t ≤ ε2 /2 we have |w(a j , t)| ≤
|ε2
1 2 ≤ 2 2 ε − t + aj|
and for all ε2 /2 ≤ t ≤ 1 we have |w(a j , t)| ≤
1 1 1 ≤ √ ≤√ . √ 2ε t |ε t(x + x1 )| 2ε2
Therefore the absolute value of the integrand in (5.8) is majorated by the function n−1 2 1 , √ 2 ε t(1 + x ) ε2 which is obviously integrable with respect to dσn (t) × d x.
We can now turn to justification of reversing the order of integrations in v, w and x in the integral 1 † † † † † 2 I= dσn (t) d v d 2w e−ε(v v +w w)+i(w A v +v Aw) n n C C 0 +∞ d x i √t(x v † v + w† w ) x e × . −∞ x The corresponding calculation is routine but tedious. First we restrict the x-integration to the finite interval δ ≤ |x| ≤ 1/δ, δ > 0, reverse the order of integrations, and then show that the corresponding tail integrals are negligible in the limit δ → 0. Let
1 Iδ =
dσn (t) 0
d v d w 2
Cn
2
Cn
δ≤|x|≤1/δ
d x −ε(v † v +w† w)+i(w† A† v +v † Aw) e x ×ei
√
†
t(x v † v + w x w ).
The absolute value of the integrand is majorated by the integrable function 1 −ε(v † v +w † w ) , and therefore we can reverse the order of integrations and then per|x| e form the integration in v, w. This yields 1 1 dx . Iδ = dσn (t) √ † 2 x det[A A + (ε − t)I − iε t(x + x1 )I ] δ≤|x|≤1/δ 0
582
Y. V. Fyodorov, B. A. Khoruzhenko
It follows from this, in view of Proposition 7, that 1 +∞ 1 dx + o(1) dσn (t) Iδ = √ 1 † 2 0 −∞ x det[A A + (ε − t)I − iε t(x + x )I ] in the limit δ → 0. It only remains to show that the tail integrals 1 † † † † † Iδ = dσn (t) d 2v d 2w e−ε(v v +w w)+i(w A v +v Aw) Cn Cn 0 d x i √t(x v † v + w† w ) x × e |x|≥1/δ x and Iδ =
1
dσn (t)
d 2v
Cn
0
† † † † † d 2w e−ε(v v +w w)+i(w A v +v Aw) d x i √t(x v † v + w† w ) x e × |x|≤δ x
Cn
vanish in the limit δ → 0. For real r, p and q define g L (r, p, q) =
+∞
d x ir ( px+ q ) x = e x
L
1/L
0
d x ir (q x+ p ) x . e x
(5.9)
By integration by parts, 1 ir ( pL+ q ) 1 L + e g L (t; p, q) = − i pr L i pr
+∞ L
q
eir ( px+ x ) p dx + 2 x q
+∞ L
q
eir ( px+ x ) d x, x3
and, therefore, for L > 0 we have |g L (r, p, q)| ≤ Obviously, |x|≥1/δ
|q| 2 + . | p||r |L 2| p|L 2
(5.10)
√ √ d x i √t(x v † v + w† w ) x e = g 1 ( t, v † v, w † w) − g 1 (− t, v † v, w † w), δ δ x
and, by (5.10), # # # # Therefore # # #I # ≤ δ
1 0
|x|≤δ
# 4δ d x i √t(x v † v + w† w ) ## w† wδ 2 x ≤ † √ + e . # x v†v v v t
dσn (t)
d 2v
Cn
† † d 2w e−ε(v v +w w)
Cn
4δ w† wδ 2 + √ v†v v†v t
.
As the function v1† v is locally integrable with respect to d 2v for n ≥ 2, we conclude that Iδ = O(δ)
when δ → 0.
(5.11)
Absolute Moments of Characteristic Polynomials of Complex Random Matrices
Similarly |x|≤δ
583
√ √ d x i √t(x v † v + w† w ) x = g 1 ( t, w † w, v † v) − g 1 (− t, w † w, v † v), e δ δ x
and repeating the above argument one obtains that Iδ = O(δ)
when δ → 0,
so that both Iδ and Iδ vanish in the limit δ → 0. This completes our proof of the first part of Theorem 2. It is worth mentioning another formula for the regularized average of the inverse spectral determinant, Rε (A, A ) = (n − 1) †
|z|2 ≤1
(1 − |z|2 )n−2 d 2z , det[ε2 I + (1 − z¯ )I + (1 − z)A A† ]
which is almost an immediate corollary of (3.32) and the representation 1 det[ε2 In + (In − AU )(In − AU )† ] 1 † 2 † † † † † † = n e−[v (1+ε )v +v A A v ] etr(vv AU +U A vv ) d 2v. π Cn This formula, however, does not seem to be easy to handle in the limit ε → 0. We now turn to the integral in (5.8) and evaluate it in the limit ε → 0 under the assumption that A A† has no repeated eigenvalues. The limit in (2.15) follows immediately from the asymptotic relation (5.19) which is the end-product of our calculation. The following identities, which can be obtained from the Lagrange interpolation formula, see, e.g., [41] Part VI, Problem 67, will be useful for our purposes. Proposition 8. Suppose that x1 , . . . , xn are pairwise distinct. Then n j=1
n 1 1 1 = , xj − t xj − t xk − x j
(5.12)
k= j
j=1
and, for non-negative integer r , n
x rj
j=1
k= j
& 1 0 if r ≤ n − 2, = h r −n+1 (x1 , . . . , xn ) if r ≥ n − 1, xk − x j
(5.13)
where h r , r = 0, 1, 2, . . . , are the complete symmetric functions. It follows from (5.12) that 1 det[A A†
+ (ε2
√
− t)I − iε t(x +
1 x )I ]
=
n j=1
w(a j , t, x)
a2 k= j k
1 , − a 2j
(5.14)
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Y. V. Fyodorov, B. A. Khoruzhenko
where a1 , . . . , an are the eigenvalues of A A† and w(a, t, x) =
1 . √ a 2 + ε2 − t − iε t(x + x1 )
By the calculus of residues, +∞ 1 dx 1 , w(a, t, x) = 2 2πi −∞ x (a − t − ε2 )2 + 4ε2 a 2
(5.15)
and putting (5.14) and (5.15) into (5.8) we arrive at the following expression of Rε (A, A† ) in terms of the eigenvalues of A A† : Rε (A, A† ) =
n
Fε (a j )
j=1
where Fε (a) =
0
1
dσn (t) , 2 (a − t − ε2 )2 + 4ε2 a 2
a2 k= j k
1 , − a 2j
dσn (t) = (n − 1)(1 − t)n−2 dt.
(5.16)
(5.17)
This formula is convenient for finding Rε (A, A† ) in the limit ε → 0. If A A† > I , by letting → 0 in (5.16) and recalling (5.12) we immediately obtain 1 dσn (t) , lim Rε (A, A† ) = ε→0 det(A A† − t I ) 0 thus reproducing the corresponding formula of Theorem 1, part (ii). If A A† < I or if A A† has eigenvalues on each side of a 2 = 1, evaluation of the right-hand side in (5.16) in the limit ε → 0 requires some work. The integral in (5.17) is standard. There are different methods available to evaluate it. None seems to give an explicit expression for all parameter values. However, we are only interested in ε → 0, and in this regime Fε (a) = (n − 1)(1 − a 2 )n−2 γn−2 sgn(a 2 − 1) + L 0 (ε, a) + qn−2 (a 2 ) + O(ε), (5.18) where
⎧ 1−a 2 2 ⎪ ⎨ln ε2 if a < 1, 2 L 0 (ε, a) = ln 2a if a 2 > 1, ⎪ ⎩ a2 −1 ln ε if a 2 = 1;
γn−2 is the partial sum of the harmonic series, γn−2 =
n−2 1 , j j=1
sgn is the sign function, sgn(x) takes value 1 if x > 0, -1 if x < 0 and 0 if x = 0, and qn−2 (a 2 ) is a polynomial of degree n − 2 in a 2 with coefficients which do not depend on ε. Details of the derivation of (5.18) are given in Appendix A.
Absolute Moments of Characteristic Polynomials of Complex Random Matrices
585
Let us now put (5.18) into (5.16). In view of (5.13) the polynomial qn−2 gives no contribution. After rearranging the remaining terms we obtain, Rε (A, A† ) = α(A A† ) ln
1 + β(A A† ) + O(ε), ε2
(5.19)
with the coefficients α and β given by α(A A† ) = (n − 1)
n
(1 − a 2j )n−2 θ (1 − a 2j )
j=1
β(A A† ) = (n − 1)
n
(1 − a 2j )n−2 ψ(a 2j )
j=1
1 , − a 2j
a2 k= j k
a2 k= j k
1 , − a 2j
where θ is Heaviside’s step function, ⎧ ⎪ ⎨1 if x > 1, θ (x) = 21 if x = 1, ⎪ ⎩0 if x < 1, and ⎧ 2 2 2 ⎪ ⎨γn−2 + ln a − ln(a − 1) if a > 1, 2 2 ψ(a ) = −γn−2 + ln(1 − a ) if a 2 < 1, ⎪ ⎩ln 2 if a 2 = 1. As one would expect, the coefficient α vanishes if A A† < I or A A† > I . This follows from identity (5.13). If A A† > I then the constant γn−2 gives no contribution, again by (5.13) and lim Rε (A, A† )
ε→0
= (n − 1)
n
(1 − a 2j )n−2 ln
j=1
a 2j a 2j
−1
k= j
1 = ak2 − a 2j
1
dσn (t) . det(A A† − t I )
1
dσn (t) . det(I − t A A† )
0
Similarly, if A A† < I then lim Rε (A, A† )
ε→0
= (n − 1)
n j=1
(1 − a 2j )n−2 ln(1 − a 2j )
k= j
1 = ak2 − a 2j
0
Thus, (5.8) indeed reproduces formulas of Theorem 1 part (ii), and our proof of Theorem 1 is now complete.
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Y. V. Fyodorov, B. A. Khoruzhenko
6. Rank-One Deviations form CUE and GUE In this section we express the mean eigenvalue density for the random matrix ensembles (1.4) and (1.6) in terms of the spectral determinants. Our calculation is inspired by similar calculations in [17, 18] and makes use of a process known as eigenvalue deflation which was introduced in the context of random matrices in [45]. We need to recall a few facts about elementary unitary Hermitian matrices [48]. Let v be a column-vector in Cn . The matrix Rv = In − 2vv † /|v|2 ,
|v|2 = v † v,
where In is the identity matrix, defines a linear transformation which is a reflection across the hyperplane through the origin with normal v/|v|. It is straightforward to verify that Rv is unitary and Hermitian, Rv = Rv†
and
Rv Rv† = Rv2 = In .
In the context of numerical linear algebra the matrices Rv are known as Householder reflections. Any matrix can be brought to triangular form by a succession of Householder reflections. We only need the first step of this process which we now describe. Let Wn be an n × n matrix and z and x = (x1 , . . . , xn )T be an eigenvalue and eigenvector of Wn , so that Wn x = zx. Without loss of generality we may assume that x1 ≥ 0 and |x|2 = x † x = 1. Let e1 = (1, 0, . . . , 0)T and x + e1 x + e1 =√ . (6.1) v= |x + e1 | 2(1 + x1 ) Since the vector v bisects the angle of x and e1 , we have Rv x = −e1 and Rv e1 = −x. Therefore Rv Wn Rv e1 = ze1 and (recall that Rv2 = In ) z w† Rv , (6.2) Wn = R v 0 Wn−1 for some Wn−1 and w. Note that Wn−1 is (n − 1) × (n − 1) and w † is 1 × (n − 1). Obviously, applying this procedure again (to the matrix Wn−1 ) and again, one can reduce Wn to triangular form by means of unitary transformations. Such factorization is known as Schur decomposition. It is convenient to write v = (v1 , q)T , where q = (v2 , . . . , vn )T√. Since v is a unit vector, v12 + |q|2 = 1. Note that the first equation in (6.1) reads v1 = (1 + x1 )/2. Since 0 ≤ x1 ≤ 1, we must have 1/2 ≤ v1 ≤ 1. Therefore, " 1 ≤ |q|2 ≤ 1. (6.3) v1 = 1 − |q|2 and 2 In terms of q the matrix Rv is given by 2−1 2|q| −2 1 − |q|2 q † 1 − 2v12 −2v1 q † = . Rv = −2v1 q In−1 − 2qq † −2 1 − |q|2 q In−1 − 2qq †
(6.4)
The incomplete Schur decomposition (6.2) gives rise to a new coordinate system in the space of complex matrices, the new (complex) coordinates being z, w, q and
Absolute Moments of Characteristic Polynomials of Complex Random Matrices
587
the matrix entries of Wn−1 . There are no restrictions on the range of variation of z, w and Wn−1 , and, in view of (6.3), the vector q is restricted to the spherical segment 1 2 2 ≤ |q| ≤ 1. The Jacobian of the transformation from (Wn, jk ) to this new system of coordinates,3 n d(Wn ) jk d(Wn ) jk 2
j,k=1
= J (z, w, q, Wn−1 )
n−1 n−1 dzdz dw j dw j dq j dq j 2 2 2 j=1
j=1
n−1 j,k=1
d(Wn−1 ) jk d(Wn−1 ) jk , 2
is given by (cf. Lemma 3.2 in [18]) J (z, w, q, Wn−1 ) = 22n−2 | det(z In−1 − Wn−1 )|2 (1 − |q|2 )n−2 (2|q|2 − 1).
(6.5)
We derive (6.5) in Appendix B. Suppose that we have a probability distribution d P(Wn ) = p(Wn )
n d(Wn )i j d(Wn )i j 2
(6.6)
i, j=1
on the space of complex n × n matrices. Then, following the argument of [18], see their Lemma 3.1, the mean eigenvalue density ρn (x, y), z = x + i y, of Wn is given by ρn (x, y) z w† 2 2 2 Rv , (6.7) d Wn−1 d w d q J (z, w, q, Wn−1 ) p Rv = 0 Wn−1 C2(n−1)2
Cn−1
1 2 2 ≤|q | ≤1
where d 2 Wn−1 =
n−1 j,k=1
n−1 n−1 dq j dq j dw j dw j d(Wn−1 ) jk d(Wn−1 ) jk , d2q = , d 2w = . 2 2 2 j=1
j=1
Since we integrate in the q-space over the spherical segment 1/2 ≤ |q|2 ≤ 1, it is convenient to introduce spherical coordinates, √ q = tσ , t = |q|2 , σ = q/|q|. The element of volume in the q-space is then d2q =
1 n−2 t dt d S(σ ), 2
where d S(σ ) is the element of area of the sphere |σ |2 = 1. The range of variation of t is 1/2 ≤ t ≤ 1. Next, on making the substitution √ 1 1 1+ r 2 r = (2t − 1) , (2t − 1)dt = dr, (1 − t)t = (1 − r ), t = , 4 4 2 3 For Jacobian computations it is convenient to consider z and z as functionally independent variables, so that d 2 z ≡ d Re zd Im z = dzdz/2.
588
Y. V. Fyodorov, B. A. Khoruzhenko
the expression for the Jacobian becomes simpler, √ J (z, w, q, Wn−1 )d 2 q = J (z, w, tσ , Wn−1 )d 2 q = 22n−3 | det(z In−1 − Wn−1 )|2 [(1 − t)t]n−2 (2t − 1)dtd S(σ ) 1 = | det(z In−1 − Wn−1 )|2 (1 − r )n−2 dr d S(σ ). 2 Substituting this into (6.7), we arrive at the desired formula for the mean density of eigenvalues in the ensemble with matrix distribution (6.6): ρn (x, y) 1 1 z w† 2 2 n−2 2 R . = d Wn−1 d w d S(σ ) (1−r ) dr | det(z In−1 −Wn−1 )| p R 0 Wn−1 2 C2(n−1)2
Cn−1
|σ |2 =1
0
(6.8) Here
⎛
⎞ √ 1 − r σ† ⎠. R=⎝ √ √ † − 1 − r σ In−1 − (1 + r )σ σ √ r
(6.9)
We shall now apply this result to express the mean density of eigenvalues in terms of the absolute square modulus of characteristic polynomials for two ensembles of random matrices. Rank-one deviations from unitarity. Let Un be an n × n unitary matrix and √ 1−γ 0 , 0 ≤ γ ≤ 1. (6.10) Gn = 0 In−1 The Haar measure on U (n) induces a measure on the manifold Wn† Wn = G 2n in the space of complex n × n matrices via the correspondence Wn = Un G n . The corresponding matrix distribution is uniform and can be conveniently described via matrix delta-function d P(Wn ) =
cn δ(Wn† Wn
−
G 2n )
n d(Wn )i j d(Wn )i j . 2
(6.11)
i, j=1
For Hermitian H , we define δ(H ) as δ(H ) = δ(H j j ) δ(H jk )δ(H jk ). j
(6.12)
j
The normalization constant cn can be easily computed by changing to the matrix ‘polar’ coordinates, 1!2! · · · (n − 1)! 2n cn = = . (6.13) Vol(U (n)) π n(n+1)/2 Note that the eigenvalues of G n Un and Un G n coincide, and, therefore, for the purpose of calculating the eigenvalue statistics the ensembles G n Un and Un G n are equivalent. In view of (6.2) it is more convenient to work with matrices Un G n .
Absolute Moments of Characteristic Polynomials of Complex Random Matrices
589
Changing the coordinate system to z, w, r , σ , Wn−1 , ' ( ( ' z w† z 0 cn 2 δ −RG n R d P(Wn ) = † w Wn−1 2 0 Wn−1 × | det(z In−1 −Wn−1 )|2 d 2 z (1 − r )n−2 dr d S(σ ) d 2 w d 2 Wn−1 , (6.14) where we have used the unitary invariance of the matrix delta-function. The matrix inside the delta-function in (6.14) is ⎛ ⎞ √ |z|2 − 1 + γ r zw† − γ (1 − r )r σ † ⎝ ⎠ √ † zw − γ (1 − r )r σ Wn−1 Wn−1 − In−1 + ww † + γ (1 − r )σ σ † and the delta-function factorizes into the product of the three delta-functions, correspondingly. On substituting this into (6.8) we obtain cn ρn (x, y) = | det(z In−1 − Wn−1 )|2 f (Wn−1 ) d 2 Wn−1 , (6.15) 2 C2(n−1)2
where f (Wn−1 ) 1
= d S(σ ) (1 − r )n−2 dr δ |z|2 − 1 + γ r |σ |2 =1
×
0
† d 2 w δ Wn−1 Wn−1 − In−1 + ww† + γ (1 − r )σ σ † δ zw − γ (1 − r )r σ .
Cn−1
Note that
δ zw − γ (1 − r )r σ =
1 |z|2(n−1)
δ w−
γ
√ (1 − r )r σ . z
Therefore the integral over w yields 1 |z|2(n−1)
γ (1 − r )(γ r + |z|2 ) † † δ Wn−1 Wn−1 − In−1 + σ σ |z|2
and f (Wn−1 ) 1
1 d S(σ ) (1 − r )n−2 dr δ |z|2 − 1 + γ r = 2(n−1) |z| |σ |2 =1 0 γ (1 − r )(γ r + |z|2 ) † † . Wn−1 − In−1 + σ σ ×δ Wn−1 |z|2
590
Y. V. Fyodorov, B. A. Khoruzhenko
It is apparent that if |z|2 < 1 − γ or |z|2 > 1 then the integral over r vanishes and f (Wn−1 ) = 0. Therefore ρn (x, y) = 0 for such values of z. If 1 − γ < |z|2 < 1 then the integral over r produces a non-trivial contribution and f (Wn−1 ) =
1 γ |z|2(n−1)
n−2 1−|z|2 γ − 1 + |z|2 † † d S(σ ) 1− δ Wn−1 Wn−1 − In−1 + σσ . γ |z|2
|σ |2 =1
Introducing γ˜ =
γ − 1 + |z|2 , |z|2
we can rewrite the above expression in a shorter form, n−2
γ˜ 1 † † . f (Wn−1 ) = d S(σ ) δ W W − I + γ ˜ σ σ n−1 n−1 n−1 γ |z|2 γ |σ |2 =1
On substituting this into (6.15), we arrive at n−2 γ˜ cn ρn (x, y) = d 2 Wn−1 | det(z In−1 − Wn−1 )|2 2 2γ |z| γ C2(n−1)2
×
† d S(σ ) δ Wn−1 Wn−1 − In−1 + γ˜ σ σ † .
|σ |2 =1
Since the matrix σ σ † is unitary equivalent to the matrix 1 0 , 0 0n−1 the integration over σ can be easily performed yielding cn Vol(S 2n−3 ) 2γ |z|2 n−2
γ˜ † × | det(z In−1 − Wn−1 )|2 δ Wn−1 Wn−1 − G˜ 2n−1 d 2 Wn−1 , γ
ρn (x, y) =
C2(n−1)2
where G˜ n−1 is the (n − 1) × (n − 1) matrix (cf. (6.10)) 1 − γ˜ 0 ˜ G n−1 = 0 In−2 and Vol(S 2n−3 ) is the area of the unit sphere in R2(n−1) , Vol(S 2n−3 ) =
2π n−1 . (n − 2)!
(6.16)
Absolute Moments of Characteristic Polynomials of Complex Random Matrices
591
It follows from (6.13) that cn /cn−1 = (n − 1)!/π n . Hence cn Vol(S 2n−3 ) n−1 = cn−1 , 2 π and finally ρn (x, y) n−1 = π γ |z|2
n−2 γ˜ cn−1 γ
† | det(z In−1 − Wn−1 )|2 δ Wn−1 Wn−1 − G˜ 2n−1 d 2 Wn−1
C2(n−1)2
#
#2 n − 1 γ˜ n−2 # ˜ n−1 ## dUn−1 det z I = − U G # n−1 n−1 π γ |z|2 γ U (n−1) as claimed in (1.5). Rank-one deviations from hermiticity. Let Hn be a GUEn matrix, i.e. random Hermitian matrix of size n × n with probability distribution cβ,n e
− β2 tr Hn2
n
d(Hn ) j j
j=1
1≤ j
d(Hn ) j j d(Hn ) j j , 2
β > 0,
and Γn be the n × n matrix 1 0 , 0 0n−1
Γn = γ
γ > 0.
Consider the random matrices Wn = Hn + iΓn . Obviously, Re Wn :=
Wn + Wn† = Hn , 2
and
Im Wn :=
Wn − Wn† = Γn . 2i
The matrices Wn are complex and their probability distribution is given by β
d P(Wn ) = cβ,n e− 2 tr(Re Wn ) δ(Im Wn − Γn ) 2
n d(Wn )i j d(Wn )i j , 2
(6.17)
i, j=1
where cβ,n is the normalization constant, cβ,n =
n/2 n 2 /2 β 1 2 π
and δ is the matrix delta-function (6.12).
(6.18)
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Y. V. Fyodorov, B. A. Khoruzhenko
Changing the coordinate system to z, w, r , σ and Wn−1 , d P(Wn ) β 1 2 β 2 β 2 = cβ,n e− 2 tr(Re Wn−1 ) − 4 |w| − 2 (Re z) δ 2
'' Im z
w† 2i
w − 2i Im Wn−1
(
( − RΓn R
×| det(z In−1 − Wn−1 )|2 d 2 z (1 − r )n−2 dr d S(σ ) d 2 w d 2 Wn−1 , where we have used the unitary invariance of the matrix delta-function. The matrix inside the delta-function in (6.14) is '
( γ√ w† † Im z − γ r 2i + 2 (1 − r )r σ , √ † − w2i + γ2 (1 − r )r σ † Im Wn−1 − γ (1 − r )σ σ †
and the delta-function factorizes into the product of the three delta-functions correspondingly. On substituting this into (6.8) we obtain ρn (x, y) =
βx 2 β 1 2 cβ,n e− 2 | det(z In−1 − Wn−1 )|2 f (Wn−1 )e− 2 tr(Re Wn−1 ) d 2 Wn−1 , 2 2 C2(n−1) (6.19)
where f (Wn−1 ) 1
= d S(σ ) (1 − r )n−2 dr δ (y − γ r ) δ Im Wn−1 − γ (1 − r )σ σ † |σ |2 =1
0
×
The integral over w yields
1 4
e−βγ
d we 2
Cn−1
2 (1−r )r
− β|w| 4
2
δ
w† γ † + (1 − r )r σ . 2i 2
, and we arrive at
f (Wn−1 ) 1
1 2 = d S(σ ) dr (1 − r )n−2 e−βγ (1−r )r δ (y − γ r ) δ Im Wn−1 − γ (1 − r )σ σ † . 4 |σ |2 =1 0 It is apparent that if y < 0 or y > γ then the integral over r vanishes. Therefore ρn (x, y) = 0 if y < 0 or y > γ . If 0 < y < γ , then the integration over r produces the factor γ1 and the constraint r = γy , so that (γ − y)n−2 e−β(γ −y)y f (Wn−1 ) = 4γ n−1
d S(σ ) δ Im Wn−1 − (γ − y)σ σ † .
|σ |2 =1
Absolute Moments of Characteristic Polynomials of Complex Random Matrices
593
On substituting the obtained expression for f (Wn−1 ) into (6.19), we arrive at ρn (x, y) cβ,n (γ − y)n−2 e− = 8γ n−1
βx 2 2 −β(γ −y)y
β
d 2 Wn−1 | det(z In−1 −Wn−1 )|2 e− 2 tr(Re(Wn−1 )
C2(n−1)2
×
2
d S(σ ) δ Im Wn−1 −(γ − y)σ σ † .
|σ |2 =1
The integral over σ yields
Vol(S 2n−3 ) δ Im Wn−1 − Γ˜n−1 , where Γ˜n−1 is the (n − 1) × (n − 1) matrix
1 0 Γ˜n−1 = (γ − y) 0 0n−2
and Vol(S 2n−3 ) is the area of the unit sphere in R2(n−1) (6.16). We have that cβ,n cβ,n−1
n 1/2 π β = , π 2β
and it now follows that ρn (x, y) βx 2
1 β n (γ − y)n−2 e− 2 −β(γ −y)y = √ cβ,n−1 γ n−1 4 2πβ (n − 2)!
β 2 × d 2 Wn−1 | det(z In−1 − Wn−1 )|2 e− 2 tr(Re(Wn−1 ) δ Im Wn−1 − Γ˜n−1 , C2(n−1)2
as claimed in (1.7). Acknowledgement. We would like to thank A. Gamburd for useful discussions and, in particular, for bringing reference [45] to our attention and for pointing us towards the link between our Lemma 3 and the Selberg Integral. We are also grateful to Ph. Biane for bringing reference [28] to our attention.
A. Appendix In this appendix we evaluate the integral 1 (1 − t)k dt Ik (ε2 , a 2 ) = 0 (t − a 2 + ε2 )2 + 4ε2 a 2 in the limit ε → 0.
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We shall use the following fact from calculus. If P(t) is a polynomial of degree k then (integrate by parts) " dt P(t) dt = Q(t) t 2 + pt + q + λ , (A.1) 2 2 t + pt + q t + pt + q where Q is a polynomial of degree k − 1 and λ is a constant. For Q and λ one has the equation (differentiate (A.1)) P(t) = Q (t)(t 2 + pt + q) +
1 Q(t)(t 2 + pt + q) + λ. 2
It follows from this that t=1 Ik (ε2 , a 2 ) = Q ε,a (t) (t − a 2 + ε2 )2 + 4ε2 a 2 + λε,a I0 (ε, a) t=0
= Q ε,a (1) (1 − a 2 + ε2 )2 + 4ε2 a 2 − Q ε,a (0)(ε2 + a 2 ) + λε,a I0 (ε, a), and the equation for Q(t) and λ is (1 − t)k = Q ε,a (t) (t − a 2 + ε2 )2 + 4ε2 a 2 + Q ε,a (t)(t − a 2 + ε2 ) + λε,a .
(A.2)
It is apparent from (A.2) that Q(t) and λ must be polynomials in a 2 and ε2 and, therefore, in the limit ε → 0, Q ε,a (t) = Q a (t) + O(ε2 ) and λε,a = λa + O(ε2 ), and (1 − t)k = Q a (t)(t − a 2 )2 + Q a (t)(t − a 2 ) + λa . This equation for Q a and λa can be explicitly solved, the solution being λa = (1 − a )
2 k
k (−1)l k (t − a 2 )l−1 (1 − a 2 )k−l . and Q a (t) = l l l=1
Note that at Q a (0) is a polynomial in a 2 of degree k − 1, and Q a (1) = (1 − a )
2 k−1
k (−1)l k = −(1 − a 2 )k−1 γk , l l l=1
where γk is the partial sum of the harmonic series, γk = 1 +
1 1 + ··· + . 2 k
We now turn to I0 (ε2 , a 2 ). Recalling the table integral dt = ln |t + t 2 + α 2 |, √ t 2 + α2
Absolute Moments of Characteristic Polynomials of Complex Random Matrices
we have I0 (ε , a ) = ln 2
2
1 − a 2 + ε2 +
At a 2 = 1, I0 (ε , 1) = ln 2
ε2 +
595
(1 − a 2 + ε2 )2 + 4ε2 a 2 . 2ε2
√
ε4 + 4ε2 1 = ln + O(ε). 2ε2 ε
For a 2 = 1, and therefore
(1 − a 2 + ε2 )2 + 4ε2 a 2 = |1 − a 2 | +
ε2 (1 + a 2 ) + O(ε4 ), |1 − a 2 |
⎧ 1 − a2 ⎪ ⎪ ⎨ln + O(ε2 ) if a 2 < 1, ε2 I0 (ε2 , a 2 ) = a2 ⎪ ⎪ ⎩ln + O(ε2 ) if a 2 > 1. a2 − 1
After collecting all relevant terms we arrive at the desired formula Ik (ε2 , a 2 ) = (1 − a 2 )k θ (1 − a 2 ) ln
1 + β(a 2 ) + qk (a 2 ) + O(ε), ε2
where θ is the Heaviside step-function,
⎧ ⎪ ⎨1 if x > 0, θ (x) = 21 if x = 0, ⎪ ⎩0 if x < 0,
qk is a polynomial of degree k and β(a 2 ) = sgn(a 2 − 1)(1 − a 2 )k (γk − ln |1 − a 2 |) + θ (a 2 − 1) ln a 2 . We use the convention according to which the sign function, sgn(x), vanishes at x = 0. B. Appendix In this appendix we derive Eq. (6.5). Let †q − 1 2q −2 1 − q † qq † z w† R, R= . Wn = R 0 Wn−1 −2 1 − q † qq In−1 − 2qq † When z, w, η and Wn−1 get infinitesimal increments dz, dw, dq and dWn−1 the matrix Wn gets increment dz dw † z w† z w† R+R R+R d R. dWn = d R 0 Wn−1 0 dWn−1 0 Wn−1
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Since R is unitary Hermitian, the matrix Rd R is skew-Hermitian, so that d f −d h† dT = Rd R = dσ dTn−1 for some f , h and Tn−1 . Also Rd R = −(d R)R, and it follows that R(dWn ) = dT =
z w† z w† dz dw † − dT + 0 Wn−1 0 Wn−1 0 dWn−1
w† d h w † d f − zd h† − d h† Wn−1 + w† dTn−1 dz dw † + . 0 dWn−1 (z I − Wn−1 )d h d hw† + dTn−1 Wn−1 − Wn−1 d Sn−1 (B.1)
Let d M = R(dWn )R. It is apparent that n
n
d(Wn ) jk d(Wn ) jk =
j,k=1
d M jk d M jk .
(B.2)
j,k=1
On the other hand, it follows from (B.1) that n
d M jk d M jk
j,k=1
= | det(z I − Wn−1 )|2 dzdz
n−1
dw j dw j
j=1
n−1 j=1
dh j dh j
n−1
d(Wn−1 ) jk d(Wn−1 ) jk .
j,k=1
(B.3) To complete our derivation we now compute the Jacobian of the transformation from h to q. Recall that d h is the (2,1)-entry of the matrix dT = Rd R. A straightforward computation yields d h = (2a + b)(dq † )qq + a(dq) + bqq † (dq), where " a = −2 1 − q † q,
1 − 2q † q b= . 1 − q†q
Equation (B.4) can be written as d h = (a I + bqq † )(dq) + (2a + b)qq T (dq), and, therefore, a I + bqq † (2a + b)qq T dq dq = . dq dq (2a + b)qq † a I + bqq T
(B.4)
Absolute Moments of Characteristic Polynomials of Complex Random Matrices
It now follows that
n−1 j=1
dh j dh j = det(a I + L)
n−1
dq j dq j ,
597
(B.5)
j=1
where L is the 2(n − 1) × 2(n − 1) matrix (2a + b)qq T bqq † . (2a + b)qq † bqq T If we find the eigenvalues of L, we shall know det(a I + L). To solve the eigenvalue problem for L, we observe that if ( f , g)T is an eigenvector of L then b(q † f )q + (2a + b)(q T g)q = λ f (2a + b)(q † f )q + b(q T g)q = λg for some λ. If λ = 0 we must have f = c1 q and g = c2 q for some c1 and c2 , and b(q † q)c1 + (2a + b)(q † q)c2 = λc1 . (2a + b)(q † q)c1 + b(q † q)c2 = λc2 This reduced eigenvalue problem yields the two non-zero eigenvalues of L, λ1 = −2aq † q and λ2 = 2(a + b)q † q. It is now apparent that λ = 0 is an eigenvalue of L of multiplicity 2(n − 2). This fact can be verified independently of the eigenvalue count by observing that for any vector u which is orthogonal to q, u 0 = 0. L = 0 and L u 0 It follows now that det(a I + L) = a 2(n−2) (a + λ1 )(a + λ2 ) = (−2)2n−2 (1 − q † q)n−2 (1 − 2q † q). (B.6) Collecting (B.2)–(B.3) and (B.5)–(B.6), one arrives at (6.5). References 1. Akemann, G., Vernizzi, G.: Characteristic Polynomials of Complex Random Matrix Models. Nucl. Phys. B 660, 532–556 (2003) 2. Akemann, G., Pottier, A.: Ratios of characteristic polynomials in complex matrix models. J. Phys. A: Math and General 37, L453–L460 (2004) 3. Andreev, A.V., Simons, B.D.: Correlators of spectral determinants in quantum chaos. Phys. Rev. Lett. 75, 2304–2307 (1995) 4. Balantekin, A.B.: Character expansions, Itzykson-Zuber integrals, and the QCD partition function. Phys. Rev. D(3) 62, 085017–085023 (2000) 5. Baik, J., Deift, P., Strahov, E.: Products and ratios of characteristic polynomials of random Hermitian matrices. J. Math. Phys. 44, 3657–3670 (2003) 6. Berezin, F.A.: Some remarks on the Wigner distribution (in Russian). Teor. Mat. Fiz. 17, 305–318 (1973). English translation: Theoret. and Math. Phys. 17(3), 1163–1171 (1974) 7. Berezin, F.A.: Quantization in complex symmetric spaces (in Russian). Izv Akad Nauk SSSR, Ser Math 39, 363–402 (1975); English translation: Math USSR-Izv 9(2), 341–379 (1976)
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8. Biane, Ph., Lehner, F.: Computation of some examples of Brown’s spectral measure in free probability. Colloq. Math. 90, 181–211 (2001) 9. Borodin, A., Olshanski, G., Strahov, E.: Giambelli compatible point processes. Adv. in Appl. Math. 37(2), 209–248 (2006) 10. Borodin, A., Strahov, E.: Averages of characteristic polynomials in Random Matrix Theory. Commun. Pure and Applied Math. 59(2), 161–253 (2006) 11. Brezin, E., Hikami, S.: Characteristic polynomials of random matrices. Commun. Math. Phys. 214, 111– 135 (2000) 12. Bump, D., Gamburd, A.: On the average of characteristic polynomials from classical groups. Commun. Math. Phys. 265, 227–274 (2006) 13. Conrey, J.B., Farmer, D.W., Keating, J.P., Rubinstein, M.O., Snaith, N.C.: Autocorrelation of random matrix polynomials. Commun. Math. Phys. 237, 365–395 (2003) 14. Conrey, J.B., Forrester, P.J., Snaith, N.C.: Averages of ratios of characteristic polynomials for the compact classical groups. Int. Math. Res. Not. 7, 397–431 (2005) 15. Conrey, J.B., Farmer, D.W., Zirnbauer, M.R.: Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groups U(N). http://arxiv.org/list/math-ph/0511024, 2005 16. Diaconis, P., Gamburd, A.: Random matrices, magic squares and matching polynomials. Electron. J. Combin. 11(2), Research Paper 2, 26 pp. (2004/05) 17. Edelman, A.: The probability that a random real gaussian matrix has k real eigenvalues, related distributions, and the Cirular law. J. Multiv. Anal. 60, 203–232 (1997) 18. Edelman, A., Kostlan, E., Shub, M.: How many eigenvalues of a random matrix are real?. J. Amer. Math. Soc. 7, 247–267 (1994) 19. Feinberg, J., Zee, A.: Non-Gaussian Non-Hermitean Random Matrix Theory: phase transitions and addition formalism. Nucl. Phys. B 501, 643–669 (1997) 20. Feinberg, J., Scalettar, R., Zee, A.: “Single Ring Theorem” and the Disk-Annulus Phase Transition. J. Math. Phys. 42, 5718–5740 (2001) 21. Fyodorov, Y.V.: Negative moments of characteristic polynomials of random matrices: Ingham-Siegel integral as an alternative to Hubbard-Stratonovich transformation. Nucl. Phys. B 621, 643–674 (2002) 22. Fyodorov, Y.V., Akemann, G.: On the supersymmetric partition function in QCD-inspired random matrix models. JETP Lett. 77, 438–441 (2003) 23. Fyodorov, Y.V., Khoruzhenko, B.A.: Systematic analytical approach to correlation functions of resonances in quantum chaotic scattering. Phys. Rev. Let. 83, 65–68 (1999) 24. Fyodorov, Y.V., Sommers, H.-J.: Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering: Random matrix approach for systems with broken time-reversal invariance. J. Math. Phys. 38, 1918–1981 (1997) 25. Fyodorov, Y.V., Sommers, H.-J.: Random matrices close to Hermitian or unitary: overview of methods and results. J. Phys. A 36, 3303–3347 (2003) 26. Fyodorov, Y.V., Strahov, E.: An exact formula for general spectral correlation function of random Hermitian matrices. J. Phys. A: Maths and General 36, 3203–3213 (2003) 27. Fyodorov, Y.V., Strahov, E.: Characteristic polynomials of random Hermitian matrices and DuistermaatHeckman localisation on non-compact Kähler Manifolds. Nucl. Phys. B 630, 453–491 (2002) 28. Haagerup, U., Larsen, F.: Brown’s spectral distribution measure for R-diagonal elements in finite von Neumann algebras. J. Funct. Anal. 176, 331–367 (2000) 29. Halasz, M.A., Jackson, A.D., Verbaarschot, J.J.M.: Fermion determinants in matrix models of QCD at nonzero chemical potential. Phys. Rev. D 56, 5140–5152 (1997) 30. Hua, L.K.: Harmonic Analysis of Functions of Several Complex variables in the Classical Domains. Providence, RI: Amer. Math. Soc., 1963 31. Ginibre, J.: Statistical Ensembles of Complex, Quaternion, and Real Matrices. J. Math. Phys. 6, 440– 449 (1964) 32. Gradshtein, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 5th ed., A. Jeffrey, ed. New York: Academic Press, 1994 33. Kadell, K.W.J.: The Selberg-Jack symmetric functions. Adv. Math. 130, 33–102 (1997) 34. Kaneko, J.: Selberg integrals and hypergeometric functions associated with Jack polynomials. SIAM J. Math. Anal. 24, 1086–1110 (1993) 35. Keating, J.P., Snaith, N.C.: Random matrix theory and ζ (1/2 + it). Commun. Math. Phys. 214, 57– 89 (2000) 36. Keating, J.P., Snaith, N.C.: Random matrix theory and L-functions at s = 1/2. Commun. Math. Phys. 214, 91–110 (2000) 37. Macdonald, I.G.: Symmetric Functions and Hall Polynomials. 2nd ed. Oxford: Clarendon Press, Oxford, 1995 38. Mehta, M.L.: Random Matrices. 3rd ed. Amsterdam: Elsevier/Academic Press, 2004)
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39. Orlov, A.Yu.: New Solvable Matrix Integrals. In: Proceedings of 6th International Workshop on Conformal Field Theory and Integrable Models. Internat. J. Modern Phys. A 19, May, suppl., 276–293 (2004) 40. Pólya, G., Szegö, G.: Problems and Theorems in Analysis. Vol. I, Berlin-Heidelberg-New York: SpringerVerlag, 1972 41. Pólya, G., Szegö, G.: Problems and Theorems in Analysis. Vol. II, Berlin-Heidelberg-New York: SpringerVerlag, 1976 42. Schlittgen, B., Wettig, T.: Generalizations of some integrals over the unitary group. J. Phys. A: Math and General 36, 3195–3202 (2003) 43. Shuryak, E.V., Verbaarschot, J.J.M.: Random matrix theory and spectral sum rules for the Dirac operator in QCD. Nucl. Phys. A 560, 306–320 (1993) 44. Strahov, E.: Moments of characteristic polynomials enumerate two-rowed lexicographic arrays. Electron. J. Combin. 10, Research paper 24, 8 pp. (2003) 45. Trotter, H.F.: Eigenvalue distributions of large Hermitian matrices: Wigner semicircle and a theorem of Kac, Murdock, and Szego. Adv. Math. 54, 67–82 (1984) 46. Verbaarschot, J.J.M.: Spectrum of the QCD Dirac Operator and Chiral Random Matrix Theory. Phys. Rev. Lett. 72, 2531–2533 (1994) 47. Verbaarschot, J.J.M.: QCD, chiral random matrix theory and integrability. In: Applications of random matrices in physics, NATO Sci. Ser. II Math. Phys. Chem. 221, Dordrecht: Springer, 2006, pp. 163–217 48. Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Oxford: Clarendon Press, 1965 49. Zirnbauer, M.R.: Supersymmetry for systems with unitary disorder: circular ensembles. J. Phys. A: Math and General 29, 7113–7136 (1996) 50. Zyczkowski, K., Sommers, H.-J.: Truncations of random unitary matrices. J. Phys. A: Math. and General 33, 2045–2058 (2000) Communicated by P. Sarnak
Commun. Math. Phys. 273, 601–618 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0252-0
Communications in
Mathematical Physics
Upper Bounds On Wavepacket Spreading For Random Jacobi Matrices Svetlana Jitomirskaya1 , Hermann Schulz-Baldes2 1 Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA 2 Mathematisches Institut, Universität Erlangen-Nürnberg, Erlangen, Germany.
E-mail: [email protected] Received: 10 July 2006 / Accepted: 9 November 2006 Published online: 28 April 2007 – © Springer-Verlag 2007
Abstract: A method is presented for proving upper bounds on the moments of the position operator when the dynamics of quantum wavepackets is governed by a random (possibly correlated) Jacobi matrix. As an application, one obtains sharp upper bounds on the diffusion exponents for random polymer models, coinciding with the lower bounds obtained in a prior work. The second application is an elementary argument (not using multiscale analysis or the Aizenman-Molchanov method) showing that under the condition of uniformly positive Lyapunov exponents, the moments of the position operator grow at most logarithmically in time. 1. Introduction One of the fundamental questions of quantum mechanics concerns the spreading of an initially localized wave packet φ under the time evolution e−ıt H associated to a Schrödinger operator H . If the physical space is Rd or Zd and the position operator is denoted by X , the spreading can be quantified using the time-averaged moments of X (or equivalently the moments of the associated classical probability distribution): ∞ dt − 2t q MT = e T φ| eı H t |X |q e−ı H t |φ , q >0. (1) T /2 0 It is well known that for short-range operators H , the moments cannot grow faster than q ballistically, that is MT ≤ C(q) T q . The growth actually is ballistic in typical scattering situations and for periodic operators H describing Bloch electrons. On the other hand, if q the moments MT are bounded uniformly in time, one speaks of dynamical localization. This can be proven in the regime of Anderson localization for random operators, but also for certain almost-periodic operators (see [Jit] for a review). There are many models q q where the moments MT exhibit some non-trivial power law behavior. If MT ∼ T q/2 , the quantum motion is diffusive, and any other asymptotic growth behavior is called
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S. Jitomirskaya, H. Schulz-Baldes
anomalous diffusion. In order to distinguish various anomalous diffusive motions, one defines the diffusion exponents q
βq+ = lim sup T →∞
log(MT ) , log(T q )
βq− = lim inf T →∞
q
log(MT ) , log(T q )
q > 0.
(2)
If the limit exists, we write βq = βq+ = βq− . The diffusion exponents correspond to the Levy-Khinchin classification of Levy flights in classical probability, however, we stress that the quantum anomalous diffusion does not result from a probabilistic dynamics, but rather from a Hamiltonian one. It is due to delicate quantum interference phenomena. The ballistic bound implies 0 ≤ βq± ≤ 1 and convexity inequalities show that βq± is non-decreasing in q. In the regime of dynamical localization βq = 0 and for quantum diffusion βq = 21 . Anomalous diffusion corresponds to all other values of βq± . Typically βq± is then also varying with q which reflects a rich multiscale behavior of the wave packet spreading. Such anomalous diffusion was exhibited numerically in several almost-periodic Jacobi matrices having singular continuous spectra (e.g. Fibonacci and critical Harper operator), and also some random and sparse Jacobi matrices. It is a challenging problem of mathematical physics to calculate the diffusion exponents for a given Schrödinger operator, in particular, when the quantum motion is anomalous diffusive. In this work we accomplish this for the so-called random polymer models, a random Jacobi matrix described in detail in the next section, and show that 1 βq = max 0, 1 − , q > 0, (3) 2q see Theorem 3. This result confirms the heuristics and numerical results of Dunlap, Wu and Phillips [DWP] for the random dimer model, the prototype of a random polymer. The latter model was introduced and analyzed in our prior work in collaboration with G. 1 Stolz [JSS], which already contained a rigorous proof of the lower bound βq− ≥ 1 − 2q . In this work we hence focus on the upper bound, which amounts to proving quantitative localization estimates. Next let us discuss this result in the context of prior rigorous work on other onedimensional models (Jacobi matrices) exhibiting anomalous diffusion. First of all, the Guarneri bound [Gua] and its subsequent improvements [Com, Las, GS2, GS3, KL, BGT] allow to estimate the diffusion exponents from below in terms of various fractal dimensions of the spectral measure. However, those results do not allow to prove the lower bound in (3) because, as was shown by de Bievre and Germinet [BG], the random dimer model has pure-point spectrum so that the Hausdorff dimension vanishes and the Guarneri bound is empty. In fact, the argument in [JSS] is based on a large deviation estimate on the localization length of the eigenstates near the so-called critical energies at which the Lyapunov exponent vanishes. Upper bounds on anomalous quantum diffusion were first proven for Jacobi matrices with self-similar spectra [GS1, BS], and these bounds are even optimal for the so-called Julia matrices. Kiselev, Killip and Last [KKL] presented a technique based on subordinacy theory allowing to control the spread of a certain portion of the wave packet (not the fastest one and therefore not the moments), and applied it to the Fibonacci model. Tcheremchantsev [Tch] proved tight upper bounds on growing sparse potential Hamiltonians introduced in [JL] and further analyzed in [CM]. Recently, Damanik and Tcheremchantsev [DT] developed a transfer matrix based method that allows to prove upper bounds on the diffusion exponents and also applied it to the Fibonacci model.
Upper Bounds On Wavepacket Spreading For Random Jacobi Matrices
603
Another way to achieve upper bounds on the diffusion exponents in terms of properties of the finite size approximants (Thouless widths and eigenvalue clustering) was recently developed and applied to the Fibonacci operator by Breuer, Last, and Strauss [BLS]. In the models considered in [GS1, BS, DT, BLS, Tch] the anomalous diffusion is closely linked to dimensional properties of the spectral measures, even though the (generalized) eigenfunctions have to be controlled as well. As a result, the transport slows down as the fractal dimension of the spectral measure decreases. The origin of the anomalous transport in the random polymer model is of a different nature. In fact, in the random polymer model only a few, but very extended localized states near the critical energies lead to the growth of the moments. Hence this model illustrates that spectral theory may be of little use for the calculation of the diffusion exponents. This statement is even more true if the dimension of physical space is higher. There are examples of three-dimensional operators with absolutely continuous spectral measures, but subdiffusive quantum diffusion with diffusion exponents as low as 13 [BeS]. The strategy for proving upper bounds advocated in [DT] appears to be more efficient in the present context than prior techniques [GS1, KKL]. We refine and generalize the relevant part in Sect. 3, see in particular Proposition 2. It allows to give a rather simple proof of our second result, namely Theorem 1, which establishes a logarithmic q bound MT ≤ log(T )βq , β > 2, under the condition of a uniformly positive Lyapunov exponent. This result is neither new nor fully optimal. However, in the generality we have (any condition on the distribution of randomness, including e.g. Bernoulli) the Aizenman-Molchanov method [AM] cannot be applied, and the only technique previously available to obtain this statement was the multi-scale analysis of [CKM] (see also [BG, DSS]). Thus our proof is significantly simpler. Moreover, one can argue that it captures the physically relevant effect of localization. Indeed, it was shown by Gordon [Gor] and del Rio, Makarov and Simon [DMS] that a generic rank one perturbation of q a model in the regime of strict dynamical localization (MT bounded) leads to singular continuous spectrum, and therefore, by the RAGE theorem, growth of the moments. However, it was shown in [DJLS] that this growth can be at most logarithmic, just as proven in Theorem 1. The proof of Theorem 1 constitutes essentially a part of the proof of Theorem 3. In the next section we present our models and results with technical details. Section 3 contains a general (non-random) strategy for proving the upper bounds. Section 4 provides the proof of Theorem 1 as well as some statements that are used in Sect. 6. In Sect. 5 we obtain probabilistic bounds on the transfer matrices near a critical energy based on the large deviation estimate of [JSS]. Section 6 contains the proof of Theorem 3, that is the identity (3). 2. Models and Results A Jacobi matrix is an operator Hω on 2 (Z) associated to the data ω = (tn , vn )n∈Z of positive numbers tn and real numbers vn which we suppose to be both bounded by a constant C, and tn bounded away from 0. Using the Dirac notation |n for the canonical basis in 2 (Z), Hω is given by Hω |n = tn+1 |n + 1 + vn |n + tn |n − 1.
(4)
Each ω is called a configuration. The set of all configurations is contained in = ([−C, C]×2 )×Z . The left shift S is naturally defined on . A stochastic Jacobi matrix is a family (Hω )ω∈ of Jacobi matrices drawn according to a probability measure P on
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which is invariant and ergodic w.r.t. S. Furthermore, we speak of a random Jacobi matrix if P has at most finite distance correlations, namely there exists a finite correlation length L ∈ N such that (tn , vn ) and (tm , vm ) are independent whenever |n −m| ≥ L. The most prominent example of a random Jacobi matrix is the one-dimensional Anderson model for which tn = 1 and the vn are independent and identically distributed so that L = 1. Random polymer models as studied in [DWP, JSS] and described in more detail below provide an example of a random Jacobi matrix with finite distance correlations (the second crucial feature of these models is that the (tn , vn ) only take a finite number of values). We will consider the disorder and time averaged moments of the position operator q X on 2 (Z), denoted by MT as in (1): ∞ dt − 2t q e T E 0| eı Hω t |X |q e−ı Hω t |0 , MT = q >0. (5) T /2 0 Here E denotes the average over ω w.r.t. P. (Note that upper bounds on the expectation w.r.t. P yield upper bounds almost surely.) One may replace |0 by any other localized initial state (at least in 1 (Z)). As discussed in the introduction, it is well-known that the Anderson model exhibits q dynamical localization, that is MT ≤ C(q) < ∞ uniformly in T for all q > 0. A byproduct of our analysis of the random polymer model discussed below is a simple proof q of the weaker result that MT grows at most logarithmically in T whenever the Lyapunov exponent is strictly positive. In order to define the latter, let us introduce as usual in the analysis of one-dimensional systems the transfer matrices at a complex energy z by (z − vn )tn−1 −tn z Tωz (n, m) = Tn−1 · . . . · Tmz , n > m, Tnz = . tn−1 0 Furthermore Tωz (n, m) = Tωz (m, n)−1 for n < m and Tωz (n, n) = 1. Then the Lyapunov exponent is γ (z) =
lim
N →∞
1 E log Tωz (N , 0) . N
Because the tn and vn are uniformly bounded, one shows by estimating the norm of a product of matrices by the product of their norms that for every bounded set U ⊂ C, Tωz (n, m) ≤ eγ1 |n−m| ,
z ∈ U,
(6)
where the γ1 depends on U . This implies that γ (z) ≤ γ1 for z ∈ U . For a random Jacobi matrix, uniform lower bounds on γ (z) can be proven by the Furstenberg theorem (e.g. [PF]). This applies in particular to the Anderson model (also with a Bernoulli potential), and more generally to random Jacobi matrices with correlation length equal to 1. Theorem 1. Consider a random Jacobi matrix. Let the (non-random) spectrum be σ (H ) ⊂ (E 0 , E 1 ) Suppose that the Lyapunov exponent is strictly positive: γ (z) ≥ γ0 > 0,
z ∈ (E 0 , E 1 ).
(7)
Then for any β > 2 there exists a constant C(β, q) such that q
MT ≤ (log T )qβ + C(β, q) .
(8)
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Another self-averaging quantity associated to ergodic Jacobi matrices is the integrated density of states (IDS) N (E) which can be defined by N (d E) f (E) = E 0| f (Hω )|0, f ∈ C0 (R). The Thouless formula (e.g. [PF]) links the Lyapunov exponent to the IDS. It implies that, if (7) holds for real E ∈ (E 0 , E 1 ), then one also has γ (z) ≥ γ0 for all z with e(z) ∈ (E 0 , E 1 ). As shown by Theorem 1 and the remark just before it, it is necessary for a random q Jacobi matrix to have a correlation length larger than 1 for the moments MT to grow faster than logarithmically, so that the diffusion exponents βq do not vanish. That this actually happens for the random dimer model was discovered by Dunlap, Wu and Phillips [DWP]. Next let us describe in more detail the more general random polymer model considered in [JSS]. Given are two finite sequences tˆ± = (tˆ± (0), . . . , tˆ± (L ± − 1)) and vˆ± = (vˆ± (0), . . . , vˆ± (L ± − 1)) of real numbers, satisfying tˆ± (l) > 0 for all l = 0, . . . , L ± − 1, L ± ≥ 1. The associated random polymer model is the random Jacobi matrix constructed by random juxtaposition of these sequences and randomizing the origin. More precisely, configurations ω ∈ can be identified with the data of a sequence of signs (σn )n∈Z and an integer 0 ≤ l ≥ L σ1 − 1, via the correspondence (tn ) = (. . . , tˆσ1 (l), . . . , tˆσ1 (L σ1 −1), tˆσ2 (0), . . . , tˆσ2 (L σ2 −1), tˆσ3 (0), . . .) and similarly for (vn ), with choice of origin t0 = tˆσ1 (l) and v0 = vˆσ1 (l). The shift is as usual, and the probability P is the Bernoulli measure with probabilities p+ and p− = 1 − p+ combined with a randomization for l (cf. [JSS] for details). The correlation length in this model is L = max{L + , L − }. It is now natural and convenient to consider the polymer transfer matrices (E − v)t −1 −t z z z z . where Tv,t = T± = Tvˆ (L −1),tˆ (L −1) · . . . · Tvˆ (0),tˆ (0) , ± ± ± ± ± ± 0 t −1 (9) Definition 1. An energy E c ∈ R is called critical for the random polymer model (Hω )ω∈ if the polymer transfer matrices T±E c are elliptic (i.e. |Tr(T±E c )| < 2) or equal to ±1 and commute [T−E c , T+E c ] = 0 .
(10)
If L ± = 1, the model reduces to the Bernoulli-Anderson model and there are no critical energies. The most studied [DWP, Bov, BG] example is the random dimer model for which L + = L − = 2 and tˆ± (0) = tˆ± (1) = 1, vˆ+ (0) = vˆ+ (1) = λ and vˆ− (0) = vˆ− (1) = −λ for some λ ∈ R. This model has two critical energies E c = λ and E c = −λ as long as λ < 1. For further examples we refer to [JSS]. It follows from the definition that a simultaneous change of coordinates reduces both T+ and T− to rotations by angles that are denoted by η+ and η− . It is immediate from (10) that the Lyapunov exponent vanishes at a critical energy. Because the transfer matrices T±z are analytic in z, it follows that there is a constant c0 such that for all ∈ C with | | < 0 one has for n, m ∈ N, E c +
(11) Tω (n, m) ≤ ec0 | | |n−m| .
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In particular, |γ (E c + )| ≤ c0 | |. However, the correct asymptotics for the Lyapunov exponent is γ (E c + ) = O( 2 ). This was first shown (non-rigorously) by Bovier for the case of the random dimer model [Bov], but heuristics were already given in [DWP]. The rigorous result about the Lyapunov exponent and also the integrated density of states N are combined in the following theorem. Theorem 2 [JSS]. Suppose that E(e2ıησ ) = 1 and E(e4ıησ ) = 1. Then for ∈ R and some D ≥ 0, the Lyapunov exponent of a random polymer model satisfies γ (E c + ) = D 2 + O( 3 ),
(12)
in the vicinity of a critical energy E c . If [T−E c + , T+E c + ] ≥ C for some C > 0 and small , one has D > 0. Moreover, the IDS N satisfies N (E c + ) − N (E c − ) = D + O( 2 ),
(13)
for some constant D > 0. Furthermore [JSS] contains explicit formulas for D and D . The bound D > 0 is not explicitly contained in [JSS], but can be efficiently checked using Proposition 1 in [SSS]. The statement about the IDS only requires E(e2ıησ ) = 1. Let us remark that the hypothesis E(e4ıησ ) = √ 1 does not hold, for example, in the special case of a random dimer model if λ = 1/ 2. In this situation, one is confronted with an anomaly. Nevertheless, the asymptotics is as in (12) and again one can calculate D explicitly [Sch]. As we did not perform the large deviation analysis of [JSS] in the case of an anomaly, we retain the hypothesis of Theorem 2 throughout. The following is the main result of this work. Theorem 3. Suppose that E(e2ıησ ) = 1 and E(e4ıησ ) = 1, and that the random polymer model has a critical energy at which (12) holds with D > 0. Then for q > 0, 1 βq = max 0, 1 − . 2q As already pointed out, the lower bound βq− ≥ 1 −
1 2q
was proven in [JSS].
3. A Strategy for Proving Upper Bounds on Dynamics In this section it is not necessary for the Jacobi matrix to be random or ergodic; hence the index ω and the average E are suppressed. Let the notation for the Green’s function be G z (n, m) = n|
1 |m , H −z
(14)
where n, m ∈ Z and z ∈ C is not in the spectrum σ (H ) of H . The starting point of the analysis is to express the time averaged moments (5) in terms of the Green’s function ı dE q MT = |n|q (15) |G E+ T (0, n)|2 . R πT n∈Z
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Using the spectral theorem, this well known identity can be checked immediately by a contour integration. In order to decompose the expression on the r.h.s., let us introduce for 0 < α0 < α1 and E 0 ≤ E 1 , q,α0 ,α1
MT
(E 0 , E 1 ) =
T α0 <|n|≤T α1
|n|q
E1 E0
ı dE |G E+ T (0, n)|2 , πT
(16)
and MT (E 0 , E 1 ) is defined similarly with the sum running over 0 ≤ |n| ≤ T α , α > 0. The following result also holds for higher dimensional models. q,0,α
Proposition 1. Suppose σ (H ) ⊂ (E 0 , E 1 ), = dist({E 0 , E 1 }, σ (H )) > 0, and α > 1. Then there exists a constant C1 = C1 (α, , q) such that
C1
q q,0,α .
MT − MT (E 0 , E 1 ) ≤ T The proof is based on the following Combes-Thomas estimate. Even though standard, its proof is sufficiently short and beautiful to reproduce it. Lemma 1. Let (z) = dist(z, σ (H )) and C2 = (4 t ∞ )−1 where t ∞ = supn∈Z tn . Then | G z (n, m) | ≤
2 exp ( − arcsinh(C2 (z)) |n − m|) .
(z)
Proof. For η ∈ R, set Hη = eηX H e−ηX . A short calculation shows that Hη − H ≤ t ∞ |eη − e−η |. Hence one has −1 −1 −1 −1 − z) ≤ − z) − H − H . (H (Hη η Since (H − z)−1 ≤ (z)−1 , the choice |η| = arcsinh ( (z)/(4 t ∞ )) hence gives (Hη − z)−1 ≤ 2/ (z). The bound now follows from n|(H − z)−1 |m = eη(m−n) n|(Hη − z)−1 |m. The following estimate will be used not only for the proof of Proposition 1, but at several reprises below. Lemma 2. Let , α > 0, q ≥ 0 and N ∈ N. Let p = [ q+1 α ] where [ · ] stands for the integer part. Then n>N
n q e− n
α
α
≤
p e− N 2 p! N + −1 . α
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Proof. Bounding the sum by the integral on each interval of monotonicity and then extending this integral to the entire range [N , ∞), we obtain ∞ α p 1 ∞ e− N p! α α (N ) j . n q e− n ≤ 2 d x x q e− x ≤ dy y p e− y = α Nα α 1+ p j! N n>N
j=0
Bounding the sum over j by p! (N + 1) p completes the proof.
Proof of Proposition 1. We first consider the energies above the spectrum and set = dist(E 1 , σ (H )). Due to the previous two lemmata and if p = [q + 1], ∞ dE 2 q,0,∞ exp (−arcsinh(C2 ( + E))|n|) (E 1 , ∞) ≤ |n|q MT π T
+ E 0 |n|≥1 ∞ d E 8 p! arcsinh(C2 ( + E))−( p+1) . ≤ π T + E 0 q,0,∞
Since arcsinh(y) ≥ ln(y) for large y, this shows that MT (E 1 , ∞) ≤ C/T for some q,0,∞ constant C = C(q, ). A similar bound holds for MT (−∞, E 0 ). Now using the imaginary part of the energy in Lemma 1 and the bound arcsinh(y) ≥ y for sufficiently small y ≥ 0, we obtain E1 2 q,α,∞ (E 0 , E 1 ) ≤ dE |n|q exp (−arcsinh(C2 /T )|n|) MT π E0 α |n|>T
α−1
4 (E 1 − E 0 ) p! e−C2 T ≤ π C2 /T
T α + T /C2
p
.
For α > 1, this decreases faster than any power of T . Combining these estimates implies the proposition. According to Proposition 1 and Eq. (16), one now needs a good bound on the decay (in n) of the Green’s function for complex energies in the vicinity of the spectrum. As shown by Damanik and Tcheremchantsev [DT], such bounds can be obtained for Jacobi matrices in a very efficient way in terms of the transfer matrices. Here we give a streamlined proof of this statement which also works for arbitrary Jacobi matrices (the kinetic part is not necessarily the discrete Lapalacian) and does not contain energy dependent constants as in [DT]. It will be convenient to first consider such bounds for the halfline problem, which is operator (4) on 2 (N) with Dirichlet boundary conditions. This operator and its Green’s function are denoted by Hˆ and Gˆ z (n, m). Proposition 2. Set τ = max{ t 2∞ , 1, t −1 2∞ } and z = E +
ı T
. One has the bounds
|Gˆ z (0, n)|2 ≤
4 τ3 T4 , max0≤n≤N T z (n, 0) 2
|G z (0, n)|2 ≤
16 τ 4 T 6 . max0≤|n|≤N T z (n, 0) 2
n>N
and, for T ≥ 1, |n|>N
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Proof. Let N be the decoupling operator at N defined by N = t N +1 (|N N + 1| + |N + 1N |) and set Gˆ zN = ( Hˆ − z − N )−1 and Gˆ z = ( Hˆ − z)−1 . The resolvent identity reads Gˆ z = Gˆ zN − Gˆ zN N Gˆ z . Thus, with the notation Gˆ zN (n, m) = n|Gˆ zN |m, |Gˆ z (0, n)|2 = |Gˆ zN N Gˆ z (0, n)|2 n>N
n>N
= |Gˆ zN (0, N )t N +1 |2
|Gˆ z (N + 1, n)|2 ≤ t 2∞ T 2 |Gˆ zN (0, N )|2 ,
n>N
since T −1 = m(z) and Gˆ z ≤ T. As the l.h.s. is decreasing in N , we therefore have |Gˆ z (0, n)|2 ≤ t 2∞ T 2 min |Gˆ nz (0, n)|2 . (17) 0≤n≤N
n>N
N Now let N = n=0 |nn| be the projection on the states on the first N + 1 sites ˆ and set H N = N H N . Then Gˆ zN (n, m), 0 ≤ n, m ≤ N , are the matrix elements of the inverse of an (N + 1) × (N + 1) matrix Hˆ N − z. The matrix elements are closely linked to the transfer matrix ab z , T (N + 1, 0) = cd N since a, b, c, d when multiplied by the n=1 tn , are the determinants of certain minors of z − Hˆ N . Namely by Cramer’s rule (or, alternatively, by the Stieltjes continued fraction expansion and geometric resolvent identity) the following identities hold for z ∈ / σ ( Hˆ N ): 1 b Gˆ zN (0, 0) = 2 , t0 a
c Gˆ zN (N , N ) = − , a
1 d Gˆ zN −1 (0, 0) = 2 , t0 c
and 1 1 Gˆ zN (0, N ) = − , t0 a
1 1 Gˆ zN −1 (0, N − 1) = − . t0 t N c
Therefore |b| ≤ t02 T |a|, |c| ≤ T |a| and |d| ≤ t02 T |c|. As the matrix norm is bounded by the Hilbert-Schmidt norm, it follows that T z (N + 1, 0) 2 ≤
4 T2 τ2 . min{|Gˆ zN (0, N )|2 , |Gˆ zN −1 (0, N − 1)|2 }
By (17) this proves the first inequality. The second one follows from the first one (coupled with the same statement for the left half-line) by observing that the resolvent identity gives G z (0, n) = Gˆ z (0, n) − G z (0, −1) t0 Gˆ z (0, n). Therefore |G z (0, n)| ≤ (1 + T t0 ) |Gˆ z (0, n)| which implies the second bound.
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4. Logarithmic Bounds in the Localization Phase In this section we provide the proof of Theorem 1 and hence suppose throughout that the stated hypothesis hold. The main idea is to use the given positivity of the Lyapunov exponent (7), combine it with the given uniform upper bound (6) in order to deduce good probabilistic estimates on the growth of the transfer matrices. This growth in turn allows to bound the Green’s function due to Proposition 2 which then readily leads to the logarithmic upper bound on the moments. Let us set U = {z ∈ C | E 0 ≤ e(z) ≤ E 1 , |m(z)| ≤ 1 }. Lemma 3. For z ∈ U and N ∈ N, the set
N (z) = ω ∈ Tωz (N , 0) 2 ≥ eγ0 N satisfies P( N (z)) ≥
γ0 . 2 γ1 − γ0
Proof. Let us set P = P( N (z)). Due to (7), the subadditivity of the transfer-matrix cocycle and the bound (6), it follows that 1 1 E log( Tωz (N , 0) ) ≤ (1 − P) γ0 + P γ1 , N 2
γ0 ≤
with γ1 defined by (6) using U as above. This directly implies the result.
Lemma 4. Let z ∈ U and N ∈ N. Then there is a constant C3 = C3 (γ0 , γ1 ) such that the set
1
ˆ N (z) = ω ∈ max Tωz (n, 0) 2 ≥ eC3 N 2
0≤n≤N
satisfies 1 2
ˆ N (z)) ≥ 1 − e− C3 N . P( Proof. Let us split N into NN0 pieces of length N0 (here and in the sections below, we suppose without giving further details that there is an integer number of pieces and that the boundary terms are treated separately). By the stationarity, on each piece [ j N0 + 1, ( j + 1)N0 ), Lemma 3 with N = N0 applies. As the pieces are independent, we deduce
N
z 2 γ0 N0
≤ (1 − p0 ) N0 , P ω ∈ max Tω (( j + 1)N0 , j N0 + 1) ≤ e 0≤ j≤N /N0 where p0 = γ0 /(2 γ1 − γ0 ). Furthermore Tωz (( j + 1)N0 , j N0 + 1) = Tωz (( j + 1)N0 , 0) 1 1 Tωz ( j N0 , 0)−1 . As A = BC implies either B ≥ A 2 or C ≥ A 2 for arbitrary matrices, and A−1 = A for A ∈ SL(2, C), it therefore follows that
N
1 ≥ 1 − (1 − p0 ) N0 . P ω ∈
max Tωz ( j N0 , 0) 2 ≥ e 2 γ0 N0 0≤ j≤N /N0 1
Choosing N0 = cN 2 with adequate c concludes the proof.
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Since the above lemma applies equally well to negative integers N , it is a direct corollary of Lemma 4 and Proposition 2 that, for T sufficiently large, 1 ı 2 E |G ωz (0, n)|2 ≤ 32 τ 4 T 6 e− C3 N , z = E + ∈ U. (18) T |n|>N
The following lemma, holding for arbitrary ergodic families of Jacobi matrices, is useful for bounding a sum similar to the one in (18), but over |n| ≤ N . Let Bµ (z) = µ(d E) (z − E)−1 denote the Borel transform of a measure µ. Lemma 5. For any E ∈ R, T > 0 and N ≥ 1, one has |n|q Nq E |G ωz (0, n)|2 ≤ m BN (z), πT π
z = E +
0≤|n|≤N
Furthermore, for any E 0 < E 1 , |n|q
E1 E0
0≤|n|≤N
(19)
dE E+ ı E |G ω T (0, n)|2 ≤ N q . πT
Proof. One has |n|q 1 1 E |G ωz (0, n)|2 ≤ N q E 0| πT πT Hω − E − n∈Z
0≤|n|
ı . T
≤ Nq E
1 1 0| πT (Hω − E)2 +
ı T
1 T2
which by the spectral theorem gives (19). Using E1 dE dE 1 m BN (E + ı T −1 ) ≤ N (de) π (e − E)2 + E0 R πT R the inequality (20) follows upon integrating (19).
(20)
1 Hω − E +
|n n|
ı T
|0
|0, 1 T2
=
R
N (de) = 1,
q,0,α
Proof of Theorem 1. According to Proposition 1 it remains to bound E MT for α > 1. We further split this into two contributions: E1 dE E+ ı q E |G ω T (0, n)|2 , |n| MT (1) = E0 π T β
(E 0 , E 1 )
0≤|n|≤(log T )
and MT (2) corresponding to the sum over (log T )β < |n| ≤ T α . The bound (20) with N = (log T )β immediately gives MT (1) ≤ (log T )βq . The second contribution can be bounded using (18): MT (2) ≤
64 τ 4 T 5 π
1 2
n q e− C3 n ,
n≥(log T )β
which is bounded by a constant C(β, q) due to Lemma 2 as long as β > 2.
This above proof applies equally well to the half-line problem with arbitrary boundary condition, the only difference being that the IDS N has to be replaced by the E(µω (d E)), where µω is the spectral measure of |0 and Hˆ .
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5. Probabilistic Estimates Near a Critical Energy In this section, we first derive more quantitative versions of Lemmata 3 and 4 by replacing the input (6) and (7) by the estimates (11) and (12). However, these estimates are not sufficient for the proof of Theorem 3. In fact, one can further improve the lemmata by replacing the uniform upper bound (11) by a probabilistic one, deduced from a large deviation estimate from [JSS] recalled below and showing that the transfer matrices grow no more than given by the Lyapunov exponent with high probability. For sake of notational simplicity, we suppose that E c = 0. Furthermore, according to (12) and the Thouless formula we may choose positive d < D and 0 such that γ (z) ≥ d 2 ,
for z = + ıδ with | | < 0 .
(21)
In order to further simplify notation, we also assume that δ, > 0 even though all estimates hold with |δ| and | |. Lemma 6. For z = + ıδ with δ ≤ < 0 introduce the set
2
N (z) = ω ∈ Tωz (N , 0) 2 ≥ e d N .
(22)
Then P( N (z)) ≥ c1 . Proof. This is an immediate corollary of Lemma 3 with c1 = d/(2c0 ), where c0 is introduced in (11). Lemma 7. Let z = + ıδ with δ ≤ < 0 . Then there exists a constant c2 such that the set
z 2
3 N
ˆ N (z) = ω ∈ max Tω (n, 0) ≥ e 0≤n≤N
satisfies ˆ N (z)) ≥ 1 − e− c2 N . P( Proof. Let us split N into NN0 pieces of length N0 and follow the proof of Lemma 4 invoking Lemma 6 instead of Lemma 3, giving
N
1 2N −c N z 2 d
0
≥ 1 − (1 − c1 ) N0 ≥ 1 − e 1 N0 . P ω ∈ max Tω (n, 0) ≥ e 2 0≤n≤N
Choosing N0 =
2
d
shows that one may take c2 = c1 d/2.
Certainly other choices of N0 are possible in the previous proof, but the present one leading to Lemma 7 implies the following estimate, which is sufficient in order to deal with one of the terms in the next section (a boundary term of energies close to 0 ). Corollary 1. Let z = + ıδ with δ ≤ < 0 . There is a constant c3 such that 1 3 ≤ e−c3 N . E z 2 max0≤|n|≤N T (n, 0) Now we turn to the refined statements and start by recalling the following
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Theorem 4. [JSS] Suppose that E(e2ıησ ) = 1 and 0 < α ≤ 21 . Then there exist constants c4 , c5 , c6 , c7 such that the set
1 αN = ω ∈
max Tω +ıδ (n, m) ≤ ec4 ∀ |δ| ≤ c5 N −1 , | | ≤ N − 2 −α , 0≤n,m≤N
satisfies α
P(αN ) ≥ 1 − c6 e−c7 N . For a fixed energy z, this can be extended to length scales N beyond the localization length (inverse Lyapunov exponent). 1
Lemma 8. For z = + ıδ with δ ≤ c5 2 and ≥ N − 2 −α , the set
α z c4 N 2(1−2α)
, N (z) = ω ∈ max Tω (n, m) ≤ e 0≤n,m≤N
satisfies −α
P(αN (z)) ≥ 1 − c6 N e−c7 . 2
Proof. We split N into NN0 pieces of length N0 = − 2α+1 . The condition ≥ N − 2 −α insures that N0 ≤ N . By the stationarity, on each piece we may apply Theorem 4 (with 1
− 1 −α
N0 instead of N ) because ≤ N0 2 . For the j th piece, denote the set appearing in α, j α, j Theorem 4 by N . For any ω ∈ ∩ j=1,..., N N one then has for δ ≤ c5 N0−1 (and hence also δ ≤ c5 2 ) the estimate max
0≤n,m≤N
Therefore ∩ j=1,...,
N N0
N0
Tωz (n, m) ≤ e
c4
N N0
≤ e c4 N
2(1−2α)
.
α, j
N ⊂ αN (z). We hence deduce from Theorem 4 that
P(αN (z)c ) ≤ c6
N −c7 N α −α 0 ≤ c N e −c7
e , 6 N0
which is precisely the statement of the lemma.
This last lemma can now be used in order to improve Lemma 6 in the range δ < c 2 . 1
Lemma 9. Let z = + ıδ with δ ≤ c5 2 and N − 2 −α ≤ ≤ N −α , 0 < α ≤ 21 . Then the set N (z) defined in (22) satisfies for some constant c8 , P( N (z)) ≥ c8 4α . Proof. We argue as in the proof of Lemma 3. Let us set again P = P( N (z)), and estimate separately the contribution from the complement of N (z), αN (z) and its complement. Due to (21), Lemma 8 and the a priori bound (11) (used on the complement of αN (z)), it follows that d 2 ≤ (1 − P)
1 −α d 2 + P c4 2(1−2α) + 2 c0 c6 N 2 e−c7 . 2
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Hence P ≥
4α
d − 4 c0 c6 N e−c7
2 c4 − d 4α
−α
.
The hypothesis ≤ N −α implies the result (it would be enough to assume ≤ log(N ) p for some p). As the final preparatory step for the next section, we improve Lemma 7 in the range δ < c 2 , by invoking Lemma 9 in its proof. 1
Lemma 10. Let z = + ıδ with δ ≤ c5 2 and N − 2 −α ≤ ≤ N −α , 0 < α ≤ 21 . Then the set
ˆ αN (z) = ω ∈ max Tωz (n, 0) 2 ≥ e N 1−α 2(1+2α)
0≤n≤N
satisfies for some constant c9 α
ˆ αN (z)) ≥ 1 − e− c9 N . P( Proof. Splitting N into NN0 pieces of length N0 and arguing exactly as in Lemma 4 invoking Lemma 9 instead of Lemma 3, we obtain
N
1 z 2 d 2 N0
2 P ω ∈ max Tω (n, 0) ≥ e ≥ 1 − (1 − c8 4α ) N0 . 0≤n≤N
Choosing N0 = (2N 1−α 4α )/d shows that one may take c9 = c8 d/2.
Lemma 10 implies the following estimate, which is the main result of this section and will be used in the next one. 1
Corollary 2. Let z = + ıδ. If δ ≤ c5 2 and N − 2 −α ≤ ≤ N −α , 0 < α ≤ 21 , one has 1 1−α 2(1+2α) α E ≤ e−N
+ e − c9 N . max0≤|n|≤N T z (n, 0) 2 6. Proof of Upper Bound for Random Polymer Models In this section we complete the proof of Theorem 3. For this purpose, we follow the q,α ,α strategy discussed in Sect. 3 and consider MT 0 1 (E 0 , E 1 ) defined as in (16), but with q,0,1+α a disorder average E. By Proposition 1 it is sufficient to bound MT (E 0 , E 1 ) for α > 0 if (E 0 , E 1 ) contains the spectrum. Moreover, energies bounded away from critical energies have a strictly positive Lyapunov exponent [BG]. By the results of Sect. 4, these energies hence lead at most to logarithmic growth in time, and therefore give no contribution to the diffusion exponents βq . Thus we are left to deal with energy intervals around the critical energies. All of them are treated the same way, so we focus on one of them. We suppose that E c = 0 and consider only the energy interval [0, 0 ] with 0 chosen as in (21); the other side [− 0 , 0] is again treated similarly. Furthermore we split the contribution as follows: q,0,1+α
MT
q,0,1+α
(0, 0 ) = MT
(0, T −η ) + MT
q,0,1+α
(T −η , T −α ) + MT
q,0,1+α
(T −α , 0 ),
Upper Bounds On Wavepacket Spreading For Random Jacobi Matrices
615
where η = min{q, 21 }. This is a good choice due to the following lemma, showing that q,0,1+α the contribution MT (0, T −η ) is bounded by the diffusion exponent as given in Theorem 3. Lemma 11. For some constant C1 , one has q,0,1+α
MT
(0, T −η ) ≤ C1 T q−η+αq .
Proof. By (19) we have q,0,1+α
MT
(0, T −η ) ≤ T q(1+α)
T −η
0
d m BN ( + ı T −1 ).
The estimate now follows from (13) and Proposition 3 in the Appendix, with 0 = T −η . q,0,1+α
Next let us consider the boundary term MT
(T −α , 0 ).
Lemma 12. For some constant C2 = C2 (α), one has q,0,1+α
MT
(T −α , 0 ) ≤ T 4 α q + C2 .
Proof. Let us split the sum over n appearing in the definition of MT (T −α , 0 ) into 4α 4α one over |n| ≤ T and the other over |n| > T . The first one can be bounded by T 4αq using (20). In the second one we apply Proposition 2 combined with Corollary 1 in order q,0,1+α to bound the Green’s function. This shows MT (T −α , 0 ) is bounded above by 0 d
3 T4αq + |n|q 16 τ 4 T 6 e−c3 |n| . −α π T T 4α q,0,1+α
|n|> T
In the second term, the sum over n is bounded by Lemma 2. As > T −α , it follows that α p −c T the second term is bounded by C T e 3 for some C, p > 0. Hence this term gives the second contribution in the bound. Lemma 13. For η = min{q, 21 } and some constants C3 = C3 (α) and C4 , one has ⎧ 1 ⎨ T q− 2 q ≥ 21 , q,0,∞ −η −α 6qα (T , T ) ≤ C3 + C4 T · MT ⎩ α(2q−1) T q ≤ 21 . Proof. We first split MT (T −η , T −α ) into two contributions MT (1) and MT (2), the first containing all the summands with |n| smaller than the (energy dependent) localization length: Tα d
+ ı MT (1) = |n|q E |G ω T (0, n)|2 , T −η π T −2 4α q,0,∞
1≤|n|≤
T
and the second MT (2) containing the sum over |n| > −2 T 4α corresponding to the summands beyond the localization length. MT (1) is bounded using (19): T −α d −2 q 1
MT (1) ≤ T 4 α q . N (d E) πT (E − )2 + T −2 T −η
616
S. Jitomirskaya, H. Schulz-Baldes
In order to bound the factor −2q , let us split the integral over into η−α
MT (1) ≤ T
4αq
α
N (d E)
T 2q(η−( j−1)α)
j=1 η−α
=T
4αq
α
T
T −η+ jα
2q(η−( j−1)α)
T −η+( j−1)α
j=1
T −η+ jα T −η+( j−1)α
η−α α
pieces:
1 d
π T (E − )2 + T −2
d m BN ( + ı T −1 ).
Using Proposition 3 we obtain η−α
MT (1) ≤ C T
4αq
α
η−α
T
2q(η−( j−1)α)
T
−η+ jα
= CT
6qα
T
(2q−1)η
j=1
α
T (1−2q)α j .
j=1
For q ≥ 1/2, we use the bound T (1−2q)α j ≤ 1 showing that the sum is bounded by η−α (1−2q)α j ≤ T (1−2q)(η−α) . This gives α . For q ≤ 1/2 we bound each summand by T the second contribution in the lemma. It remains to show that MT (2) ≤ C3 . Due to Proposition 2 and Corollary 2, MT (2) ≤
T −α T −η
d
πT
2(1+2α) 1−α α |n| |n|q 16 τ 4 T 6 e−
+ e−c9 |n| .
|n|> −2 T 4 α
Using Lemma 2 it is now elementary to bound MT (2) by a constant.
Combining Lemmata 11, 12 and 13, and recalling that α can be taken arbitrary close to 0, proves Theorem 3. Appendix: Estimates on the Borel Transform In Sect. 6 we used well-known estimates on the Borel transform Bµ (z) = µ(d E) (z − E)−1 of a measure µ. For sake of completeness we provide a short proof. Proposition 3. If a measure µ satisfies at some E the bound µ([E − , E + ]) < C
for all > 0, then for any finite positive δ and 0 , 0 π C, m Bµ (E + ıδ) < d m Bµ (E + + ıδ) < π 2 C 0 . 2 0 Proof. One has, uniformly in δ, m Bµ (E + ıδ) = µ(de)
δ (e − E)2 + δ 2
1
1 δ2
− δ2 =δ dt µ e ∈ R |e − E| <
t 0 √ ∞ 1 1 x δ2 − δ2 < C dt dx ,
Upper Bounds On Wavepacket Spreading For Random Jacobi Matrices
617
which gives the first inequality. For the second one, let us bound the indicator function χ[− 0 , 0 ] ( ) above by obtain
0
0
2 02 .
2 + 02
Then one can use the stability of Cauchy distribution to
2 02
δ (E + − e)2 + δ 2 δ + 0 = 2 π 0 , µ(de) (E − e)2 + (δ + 0 )2 = 2 π 0 m Bµ (E + ı(δ + 0 )),
d m Bµ (E + + ıδ) <
d
2
+ 02
µ(de)
and thus the second inequality follows from the first one.
Acknowledgement. This work would have been impossible without [JSS]. We are very thankful to G. Stolz for this collaboration. We also thank the anonymous referees for comments that improved the paper. The work of S. J. was supported in part by the NSF, grant DMS-0300974, and Grant No. 2002068 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. H. S.-B. acknowledges support by the DFG.
References [AM] [BGT] [BS] [BeS] [BG] [BL] [Bov] [BLS] [CKM] [Com] [CM] [DSS] [DT] [DJLS] [DMS] [DWP]
Aizenman, M., Molchanov, S.: Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys. 157, 245–278 (1993) Barbaroux, J.M., Germinet, F., Tcheremchantsev, S.: Fractal dimensions and the phenomenon of intermittency in quantum dynamics. Duke Math. J. 110, 161–193 (2001) Barbaroux, J.-M., Schulz-Baldes, H.: Anomalous transport in presence of self-similar spectra. Ann. I.H.P. Phys. Théo. 71, 539–559 (1999) Bellissard, J., Schulz-Baldes, H.: Subdiffusive quantum transport for 3d-hamiltonians with absolutely continuous spectra. J. Stat. Phys. 99, 587–594 (2000) de Bièvre, S., Germinet, F.: Dynamical localization for the random dimer schrödinger operator. J. Stat. Phys. 98, 1135–1148 (2000) Bougerol, P., Lacroix, J.: Products of Random Matrices with Applications to Schrödinger Operators. Boston, Birkhäuser, 1985 Bovier, A.: Perturbation theory for the random dimer model. J. Phys. A: Math. Gen. 25, 1021–1029 (1992) Breuer, J., Last, Y., Strauss, Y.: Upper bounds on the dynamical spreading of wavepackets. In: preparation Carmona, R., Klein, A., Martinelli, F.: Anderson localization for bernoulli and other singular potentials. Commun. Math. Phys. 108, 41–66 (1987) Combes, J.-M.: Connections between quantum dynamics and spectral properties of time- evolution operators. In: Differential Equations with Applications to Mathematical Physics, Ames, W.F., Harell, E.M., Herod, J.V., eds. Boston: Academic Press, 1993 Combes, J.-M., Mantica, G.: Fractal dimensions and quantum evolution associated with sparse potential Jacobi matrices. In: Proceedings of the Bologna APTEX International Conference, World Scientific. Ser. Concr. Appl. Math. 1, River Edge, NJ: world Scientific, pp. 107–123 (2001) Damanik, D., Sims, R., Stolz, G.: Localization for discrete one-dimensional random word models. J. Funct. Anal. 208, 423–445 (2004) Damanik, D., Tcheremchantsev, S.: Upper bounds in quantum dynamics. To appear in J. Amer. Math. Soc., electronically posted on Nov. 30, 2006 del Rio, R., Jitomirskaya, S., Last, Y., Simon, B.: Operators with singular continuous spectrum: iv, hausdorff dimension, rank-one perturbations and localization. J. d’Analyse Math. 69, 153–200 (1996) del Rio, R., Makarov, N., Simon, B.: Operators with singular continuous spectrum: ii, rank one operators. Commun. Math. Phys. 165, 59–67 (1994) Dunlap, D.H., Wu, H.-L., Phillips, P.W.: Absence of localization in random-dimer model. Phys. Rev. Lett. 65, 88–91 (1990)
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Gordon, A.: Pure point spectrum under 1-parameter perturbations and instability of Anderson localization. Commun. Math. Phys. 164, 489–505 (1994) Guarneri, I.: Spectral properties of quantum diffusion on discrete lattices. Europhys. Lett. 10, 95–100 (1989); On an estimate concerning quantum diffusion in the presence of a fractal spectrum. Europhys. Lett. 21, 729–733 (1993) Guarneri, I., Schulz-Baldes, H.: Upper bounds for quantum dynamics governed by jacobi matrices with self-similar spectra. Rev. Math. Phys. 11, 1249–1268 (1999) Guarneri, I., Schulz-Baldes, H.: Lower bounds on wave packet propagation by packing dimensions of spectral measures. Math. Phys. Elect. J. 5(1), 16 pages (1999) Guarneri, I., Schulz-Baldes, H.: Intermittent lower bound on quantum diffusion. Lett. Math. Phys. 49, 317–324 (1999) Jitomirskaya, S.: Ergodic Schrödinger operators (on one foot). Preprint 2006, to appear in Barry Simon Festschrift Jitomirskaya, S., Last, Y.: Power-law subordinacy and singular spectra, i. half line operators. Acta Math. 183(2), 171–189 (1999) Jitomirskaya, S., Schulz-Baldes, H., Stolz, G.: Delocalization in random polymer chains. Commun. Math. Phys. 233, 27–48 (2003) Killip, R., Kiselev, A., Last, Y.: Dynamical upper bounds on wavepacket spreading. Amer. J. Math. 125, 1165–1198 (2003) Kiselev, A., Last, Y.: Solutions, spectrum, and dynamics for schrödinger operators on infinite domains. Duke Math. J. 102, 125–150 (2000) Last, Y.: Quantum dynamics and decomposition of singular continuous spectra. J. Funct. Anal. 142, 402–445 (1996) Pastur, L., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Berlin:Springer, 1992 Schrader, R., Schulz-Baldes, H., Sedrakyan, A.: Perturbative test of single parameter scaling for 1d random media. Ann. H. Poincare 5, 1159–1180 (2004) Schulz-Baldes, H.: Lyapunov exponents at anomalies of SL(2, R) actions, to appear in Operator Theory: Advances and Applications. Basel:Birkhäuser, 2006 Tcheremchantsev, S.: Dynamical analysis of schrödinger operators with growing sparse potentials. Commun. Math. Phys. 253, 221–252 (2005)
Communicated by B. Simon
Commun. Math. Phys. 273, 619–636 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0221-7
Communications in
Mathematical Physics
On the Distinguishability of Random Quantum States Ashley Montanaro Department of Computer Science, University of Bristol, Woodland Road, Bristol, BS8 1UB, UK. E-mail: [email protected] Received: 13 July 2006 / Accepted: 16 October 2006 Published online: 6 March 2007 – © Springer-Verlag 2007
Abstract: We develop two analytic lower bounds on the probability of success p of identifying a state picked from a known ensemble of pure states: a bound based on the pairwise inner products of the states, and a bound based on the eigenvalues of their Gram matrix. We use the latter, and results from random matrix theory, to lower bound the asymptotic distinguishability of ensembles of n random quantum states in d dimensions, where n/d approaches a constant. In particular, for almost all ensembles of n states in n dimensions, p > 0.72. An application to distinguishing Boolean functions (the “oracle identification problem”) in quantum computation is given. 1. Introduction A fundamental property of quantum mechanics is that non-orthogonal pure quantum states may not be distinguished perfectly. This leads to the following quantum detection problem: given an unknown quantum state |ψ? , picked from a known set E with known a priori probabilities, find the “optimal” measurement M opt to determine |ψ? . Several different criteria for optimality may be considered [7, 8, 14]; here we only concern ourselves with optimising the probability of success P opt , and in particular the related state distinguishability problem of finding P opt without necessarily finding M opt . Efficient optimisation techniques can be used to estimate P opt numerically [9]; however, the problem of finding an analytic expression for P opt seems intractable. We are therefore led to attempting to produce bounds on P opt . This note derives two lower bounds on P opt ; one based on the pairwise distinguishability of the states in E, and one based on the eigenvalues of their Gram matrix. We use the latter, and a powerful result from random matrix theory (the Marˇcenko-Pastur law [19]), to bound the probability of distinguishing a set of random quantum states, for a quite general notion of randomness. This has an application to quantum computation in the so-called oracle identification problem introduced by Ambainis et al. [1], where we are given an n-bit Boolean function f picked from a known set of N functions, and
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must identify f with the minimum number of queries to f . We show that, for all but an exponentially small fraction of sets with N = 2n , a quantum computer can perform this task successfully in a constant number of queries (with arbitrarily high probability), whereas classical computation requires n queries for all such sets. As showing that a set of quantum states are quite distinguishable forms an essential part of proofs in many areas of quantum information theory, we hope that these results will find application elsewhere. The organisation of the paper is as follows. Section 2 introduces notation and our main tool, the so-called “pretty good measurement”, before moving on to give the lower bounds on P opt . An extension of the lower bounds to mixed states is considered. Section 3 applies the bounds to a specific family of ensembles (those where all the states have constant inner product). Section 4 describes the random matrix theory we will be using, and applies it to the distinguishability of random quantum states. Section 5 gives the application to the oracle identification problem, and the paper closes with some discussion in Sect. 6.
2. Bounds on the Distinguishability of Quantum States We consider an ensemble E containing n d-dimensional pure states |ψi with their a priori probabilities pi . We will use {|ψi } to denote the √ set containing the same states, renormalised to reflect their probabilities (i.e. |ψi = pi |ψi ). Given an unknown state |ψ? , picked in accordance with these probabilities, the quantity we are interested in is the average probability of success for a given generalised measurement to distinguish which state we were given. For a measurement M (given by a set of positive operators {Mi } summing to the identity), let this probability be denoted by P M (E). Then we have P M (E) =
ψi |Mi |ψi = pi ψi |Mi |ψi . i
(1)
i
M opt (E) will denote the measurement with the optimal probability of success, and in an abuse of notation P opt (E) will denote this optimal probability. We call this the optimal probability of distinguishing the states in E. √ We use three matrix norms: the Euclidean (Frobenius) norm A2 = tr A† A = √ 2 † i, j |Ai j | , the trace norm A1 = tr A A = i σi (A), where σi (A) denotes the th i singular value of A, and the l1 norm i, j |Ai j |. We will often use the d ×n state matrix S = S(E) = (|ψ1 , . . . , |ψn ) whose i th column is the state |ψi . Then G = S † S gives the n × n Gram matrix [16] encoding all the inner products between the renormalised states in E. If n < d, G will have d − n zero eigenvalues. Note that every rectangular matrix M with M2 = 1 is a state matrix. ρ will represent the density matrix of the ensemble: ρ=
n
|ψi ψi |.
i=1
It is well-known [17] that G and ρ have the same non-zero eigenvalues.
(2)
On the Distinguishability of Random Quantum States
621
2.1. Use of the “pretty good measurement”. We will use a specific measurement to provide bounds on P opt (E), which is “canonical” in the sense that it performs reasonably well for any ensemble E. This is the so-called pretty good measurement (PGM), which was independently identified by several authors (e.g. [11, 12]) and has a number of useful properties. It is usually defined as a set of projectors {|νi νi |} onto “measurement vectors” |νi , where |νi = ρ −1/2 |ψi (the inverse only being taken on the support of ρ). However, it may also be defined implicitly, which brings out its “canonical” nature. To this end, consider an arbitrary measurement M for E that consists of a set of n rank 1 projectors onto unnormalised measurement vectors |µi , where each measurement vector corresponds to a state |ψi in the ensemble. (In fact, it turns out that the optimal measurement for an ensemble of pure states always falls into this category [9].) The probability of getting measurement outcome i and receiving state j is then |µi |ψ j |2 , n and the overall probability of success of this measurement is i=1 |µi |ψi |2 . We may thus encode all the inner products (and hence the probabilities) in a matrix P, where Pi j = µi |ψ j ; and rather than looking for an optimal measurement M, we can rephrase our task as looking for an optimal matrix P that corresponds to a valid measurement. We have the following requirement on P, from the fact that M must be a valid POVM: n n † ψi |µk µk |ψ j = ψi | |µk µk | |ψ j = G i j = (S † S)i j . (3) (P P)i j = k=1
k=1
A natural way √ to produce a matrix P that satisfies this condition from any given S is to take P = G, the positive semidefinite square root of G. The PGM turns out to be a measurement corresponding to this matrix P, for, if Pi j = νi |ψ j , then (P 2 )i j =
n ψi |ρ −1/2 |ψk ψk |ρ −1/2 |ψ j k=1
=
ψi |
ρ
−1/2
n
(4)
|ψk ψk |ρ −1/2
|ψ j = G i j .
(5)
k=1
n √ 2 ( G)ii . The probability of success for the PGM is thus given by P pgm (E) = i=1 Barnum and Knill have proved [4] that the PGM has the further property that it is almost optimal in the following sense. Theorem 1 (Barnum, Knill) [4]. P pgm (E) ≥ P opt (E)2 . So there is the overall relationship P opt (E)2 ≤ P pgm (E) ≤ P opt (E). For completeness, we include (in Appendix A) a simplified proof of Barnum and Knill’s result in the case of pure states. 2.2. Bounds from the pairwise inner products. A set of states that are pairwise almost orthogonal are pairwise almost distinguishable. It thus seems intuitively clear that, given such a set, the probability of success in distinguishing one state from all the others must also be high. However, this intuition is wrong. This was noted by Jozsa and Schlienz [17], who showed that the inner products of an ensemble of states may all be reduced, while simultaneously reducing the von Neumann entropy of the ensemble (which gives a measure of overall distinguishability). This effect also manifests itself in quantum
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fingerprinting [6]. Here, d-dimensional states are “compressed” to log d-dimensional “fingerprint” states that can be distinguished pairwise. However, given such a fingerprint the corresponding original state may not be identified, as this would violate Holevo’s theorem [15]. Nevertheless, for certain ensembles the pairwise inner products can give a good lower bound on the overall distinguishability, as noted by several authors [11, 4]. In this section we derive such a bound. Our approach is based on that of Hausladen et al. [11], who found a parabola forming a lower bound on the square root function, which is useful because of the following lemma. √ Lemma 1. If the function x is bounded below by f (x) = ax + bx 2 for x ≥ 0, then √ n ( G)ii ≥ aG ii + b j=1 |G i j |2 . Proof. G is a positive semidefinite matrix and thus may be diagonalised: G = U DU † , where D =√diag({λi }) and U = (u i j ) is√unitary. Working out the matrix algebra √ shows that ( G)ii = nk=1 λk |u ik |2 , so ( G)ii ≥ nk=1 f (λk )|u ik |2 = f (G)ii . But n f (G)ii = (aG + bG 2 )ii = aG ii + b j=1 G i j G ji = aG ii + b nj=1 |G i j |2 . Our goal will be to find a and b to parametrise f such that aG ii + b nj=1 |G i j |2 is √ maximised. It is clear that, for this to be maximised, f (r ) must equal r for √ some r (or we could just increase a or b). So we will pick a and b such that f (r ) = r and 1 (i.e. the curves are tangent at this point). This leads to the simultaneous f (r ) = 2√ r equations ar + br 2 =
√ 1 r , a + 2br = √ . 2 r
(6)
Solving for a and b gives the optimal values 3 1 a = √ , b = − 3/2 . 2r 2 r
(7)
√ To see that f (x) actually is a lower bound for x for any positive value of r (with these 2 values for a and b), note that the only solutions to the related equation √ f (x) = x are x = 0, x = r , or x = 4r . As f (4r ) is negative, we have that f (x)√= x if and only if x = 0 or x = r . So the only remaining possibility is that f (x) > x for all 0 < x < r . Substituting a suitable value of x (e.g. r/2) shows that this is not the case. The expression aG ii + b nj=1 |G i j |2 may now be expressed solely in terms of r . Optimising this for r gives that the maximum is found at the point n r=
j=1 |G i j |
G ii
2
.
(8)
Returning to the original inequality, we have n √ √ G3 3 1 |G i j |2 ⇒ ( G)ii2 ≥ n ii . ( G)ii ≥ √ G ii − 3/2 2 2r 2 r j=1 |G i j | j=1
(9)
On the Distinguishability of Random Quantum States
623
We thus have the following bound on the probability of distinguishing the states in E: P pgm (E) ≥
n i=1
ψi |ψi 3 pi2 = . n 2 2 j=1 |ψi |ψ j | j=1 p j |ψi |ψ j | n
n
(10)
i=1
If all the states have equal a priori probabilities, the bound simplifies further to P pgm (E) ≥
n 1 1 n . 2 n j=1 |ψi |ψ j |
(11)
i=1
Unlike previous bounds obtained by other authors for the probability of success n of the PGM [11, 4], the bound (10) is always positive and greater than or equal to i=1 pi2 , thus showing that the PGM always does at least as well as the “non-measurement” of guessing which state was received in accordance with their a priori probabilities. 2.3. Bounds from eigenvalues. The eigenvalues of a Hermitian matrix are closely related to its diagonal elements; indeed, the former majorises the latter √ [16]. With this in mind, we look for a bound on the unknown diagonal elements of G in terms of the known eigenvalues {λi } of G. n √ 2 Lemma 2. P pgm (E) ≥ n1 = n1 S21 . i=1 λi Proof. By the fact that the trace of a matrix is the sum of its eigenvalues, we have n √ n λi ( G)ii = i=1
n √ ⇒ ( G)ii
n 2 = λi
i=1
i=1
⇒P
(13)
i=1
n 2 n √ ( G)ii2 ≥ λi ⇒n pgm
(12)
i=1 2
(14)
i=1
1 (E) ≥ n
2 n λi ,
(15)
i=1
where in (14) we used a Cauchy-Schwarz inequality, showing that equality can only be √ attained in step (14) when all the ( G)ii are equal. Interestingly, this bound is the same as the fidelity of G with the maximally mixed state
2 I /n, where the fidelity F(ρ, σ ) is defined as tr ρ 1/2 σ ρ 1/2 [20]. It is worth noting that no upper bound on the success probability in terms of the eigenvalues alone can be found, for the following reason. Any set of eigenvalues {λi } summing to 1 can give rise to a Gram matrix G, where G ii = λi , and G i j = 0 (for i = j). Such matrices correspond to an ensemble E of perfectly distinguishable states where P pgm (E) = 1. As future work, it would be interesting to determine whether an upper bound (or an improved lower bound) could be produced by considering the diagonal entries of G as well as its eigenvalues.
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2.4. Distinguishing mixed states. It is natural to ask to what extent these lower bounds hold for the generalised problem of distinguishing an ensemble E consisting of mixed states {ρi }. The following theorem allows the problem to be related to that of distinguishing pure states. Theorem 2. Let E be an ensemble of nd-dimensional mixed states {ρi } with a priori probabilities { pi }, and having spectral decompositions ρi = dk=1 λik |vik vik |. Let F be an ensemble of the nd pure states given by the eigenvectors {|vik } with a priori probabilities { pi λik }. Then P pgm (E) ≥ P pgm (F). Proof. For mixed states, the PGM is defined by the following measurement operators {Mi }: Mi =
ρ −1/2 ρi ρ −1/2 ,
where
ρi
= pi ρi and ρ =
n
ρi .
(16)
i=1
So the probability of success √ can be bounded as follows, where we use the renormalised = √p eigenvectors |vik i λik |vik : P pgm (E) =
n
tr ρ −1/2 ρi ρ −1/2 ρi i=1
=
n
tr ρ −1/2
i=1
=
d
(17)
|vik vik | ρ −1/2
k=1
d
|vil vil |
tr ρ −1/2 |vik vik |ρ −1/2 |vil vil |
n d
(18)
l=1
(19)
i=1 k,l=1
=
n d
|vik |ρ −1/2 |vil |2 ≥
i=1 k,l=1
n d
2 |vik |ρ −1/2 |vik | = P pgm (F).
i=1 k=1
3. The Distinguishability of States with Constant Inner Product An illustrative case to apply these bounds to is that of equiprobable states where the pairwise inner products are all equal, so the states are all equally distinguishable from each other. Consider an ensemble E with Gram matrix G, where G ii = 1/n and G i j = p/n for i = j (and p is a positive real constant). In this case, the inner product bound of Sect. 2.2 gives the bound P pgm (E) ≥
1 = O(1/n). 1 + p 2 (n − 1)
(20)
The eigenvalue bound, however, gives much better results. The symmetry of G shows immediately that it has an eigenvector (1, 1, . . . , 1); the corresponding eigenvalue is λ1 = p + (1 − p)/n. The set of eigenvectors may be completed by taking any n − 1
On the Distinguishability of Random Quantum States
625
vectors orthogonal to (1, 1, . . . , 1), which will be eigenvectors with eigenvalues λ2...n = (1 − p)/n. We therefore have
P
pgm
2 1− p 1− p + (n − 1) p+ n n
2(1 − p) (1 − p) 1 (n − 1)2 ≥ (1 − p) − ≥ n n n
1 (E) ≥ n
(21) (22)
so the probability of distinguishing these states approaches a constant as n → ∞. In fact, one can show that inequality (21) is actually an equality giving the precise probability √ of success P pgm (E) (this follows from showing that the diagonal entries of G are all equal). Such an ensemble therefore provides a kind of converse to the ensemble of states used in quantum fingerprinting [6]: in this case, no matter how many states there are in the ensemble, their joint distinguishability is of the same order as their pairwise distinguishability. We will see below that this behaviour is not typical; however, it is perhaps not surprising, because E can only be realised in n dimensions. To see this, note that G is non-singular, so the states in E must be linearly independent. 4. The Distinguishability of Random Quantum States We will use Lemma 2 and some results from the theory of random matrices to put a lower bound on the probability of distinguishing random quantum states. We will find that it is possible to give strong lower bounds on the distinguishability of n random states in d dimensions, in the regime where n/d is constant.
4.1. A little random matrix theory. In this section, we will calculate the expected value of the trace norm of a random matrix. The distribution of the trace norm (i.e. the sum of singular values) of a matrix M is clearly related to that of the eigenvalues of the matrix M M † , which is known to statisticians as a sample covariance matrix. The asymptotic distribution of the eigenvalues of such a matrix is given by the Marˇcenko-Pastur law [19], which is stated in the form we need in [3]. Theorem 3 (Marˇcenko-Pastur law) [19]. Let Rr be a family of d × n matrices with n ≥ d and d/n → r ∈ (0, 1] as n, d → ∞, where the entries of Rr are i.i.d. complex random variables with mean 0 and variance 1. Then, as n, d → ∞, the eigenvalues of the rescaled matrix n1 Rr Rr† tend almost surely to a limiting distribution with density (x − A2 )(B 2 − x) pr (x) = 2πr x for A2 ≤ x ≤ B 2 (where A = 1 −
(23)
√ √ r , B = 1 + r ), and density 0 elsewhere.
We will translate this to a similar statement about the singular values of Rr . The following lemma is straightforward.
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Lemma 3. Let Rr be a family of d × n matrices with k/m → r ∈ (0, 1] as n, d → ∞, where k = min(n, d) and m = max(n, d), and the entries of Rr are i.i.d. complex random√variables with mean 0 and variance 1. Then, as n, d → ∞, the singular values of Rr / m tend almost surely to a limiting distribution with density (y 2 − A2 )(B 2 − y 2 ) pr (y) = (24) πr y √ √ for A ≤ y ≤ B (where A = 1 − r , B = 1 + r ), and density 0 elsewhere. √ Proof. The lemma follows from Theorem 3 for n ≥ d by substituting y = x. For n ≤ d, note that the singular values of R are the same as those of R T , so the roles of n and d need merely be interchanged. Lemma 4. Let Rr be a family of d × n matrices with k/m → r ∈ (0, 1] as n, d → ∞, where k = min(n, d) and m = max(n, d), and the entries of Rr are i.i.d. complex random variables with mean 0 and variance 1. Then, as n, d → ∞, the expected trace norm of Rr tends almost surely to m 3/2 B E(Rr 1 ) = (y 2 − A2 )(B 2 − y 2 ) dy, (25) π A √ √ where A = 1 − r , B = 1 + r . Proof. With probability 1, Rr will have k non-zero singular values. Let σi (Rr ) denote the value of the i th (unsorted) singular value of Rr , for arbitrary i between 1 and k. We have B √ √ √ E(Rr 1 ) = (k m) E(σi (Rr / m)) = k m y pr (y) dy (26) A
and using Lemma 3 gives the desired result.
This turns out to be an elliptic integral which cannot be expressed in terms of elementary functions [10]. However, it is possible to produce a good lower bound, which is tight in the case r = 1: Lemma 5. Let Rr , k, m be defined as in Lemma 4. Then, as k, m → ∞, the expected trace norm of Rr
√ 64 (27) E(Rr 1 ) ≥ k m 1 − r 1 − 9π 2 with equality when r = 1. Proof. See Appendix B.
As these are asymptotic results, it is important to bound the rate of convergence of this expected value to that given by Lemma 5. This can be done using a theorem of Bai [2], who has shown that the Kolmogorov distance between the (rescaled) expected empirical spectral distribution of an m × k matrix (with m ≥ k) and the asymptotic distribution given by the Marˇcenko-Pastur law is O(m −5/48 ). After some algebra, this may be used with Lemma 5 to give
√ 64 −5/48 E(Rr 1 ) ≥ k m − O(m 1−r 1− ) (28) 9π 2 for a finite-dimensional m × k matrix Rr .
On the Distinguishability of Random Quantum States
627
4.2. Random quantum states. We can apply this result, and the lower bound of Lemma 2, to estimate the distinguishability of random quantum states uniformly distributed on the complex unit sphere in d dimensions. In fact, we may exploit the concentration of measure effects characteristic of high-dimensional spaces to show lower bounds on the distinguishability of almost all ensembles of quantum states. A uniformly random quantum state may be produced by creating a vector v, each of whose components are complex Gaussians (say vi ∼ N˜ (0, 1/d)), and normalising the result. The intuition that this normalisation step is “almost unnecessary” [22] can be formalised as follows. It is straightforward to see that E(v) = 1. In order to get an explicit expression for the concentration around this expectation the following result from Appendix A of [5] can be used. Lemma 6 (Norm concentration of Gaussian vectors) [5]. Let v be a d-dimensional random vector, each of whose components vi ∼ N˜ (0, 1/d). Then, for any , Pr[|v22 − 1| ≥ ] < 2e−d
2 /12
.
(29)
Similarly, the state matrix of an ensemble E of n equiprobable d-dimensional uniformly random quantum states is given by a d × n matrix S whose columns √ are uniformly random quantum states renormalised so that each column √ has norm 1/ n. Let S denote the matrix produced by rescaling each column by 1/ n, rather than normalising them. We will show that S and S are close with high probability. Consider an arbitrary column of S and the same column in S , denoted v and v respectively. Lemma 6 allows a bound to be put on the probability of v and v being far apart, as √ 1 √ v − v 22 = ( nv 2 − 1)v22 = ( nv 2 − 1)2 . n
(30)
We may therefore obtain √ Pr[v − v 22 ≥ ] = Pr[( nv 2 − 1)2 ≥ n] ≤ Pr[|nv 22 − 1| ≥ n] ≤ 2e−n
2 d 2 /12
.
(31)
Considering all the columns in the matrices S and S , and using the union bound, we have Pr[S − S 22 ≥ ] ≤ 2ne−d
2 /12
.
(32)
In order to convert this to a statement about the “distinguishability” function f (S) = n1 S21 that we are interested in, we need the following lemma, which is proved in Appendix C. Lemma 7. Let S be an n × d matrix with S2 ≤ l, and define f (S) = n1 S21 . Then the Lipschitz constant η of f , η = supx,y | f (x) − f (y)|/x − y2 , satisfies η ≤ 2l. Lemma 7 implies the following relationship, for any l > 0: √ Pr |(S 21 − S21 )/n| ≥ 2l ≤ Pr[S 2 ≥ l] + Pr[S − S 22 ≥ ] ≤ 2e−nd(l
2 −1)2 /12
+ 2ne−d
The final result we will need is the following concentration lemma.
2 /12
.
(33)
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Lemma 8 (Concentration of Gaussian measure) [18]. Let p be a point in Rd picked in accordance with standard Gaussian measure. Then Pr[| f ( p) − E( f )| ≥ ] ≤ 2e−
2 /2η2
,
(34)
where η is the Lipschitz constant of f , η = supx,y | f (x) − f (y)|/x − y2 . We now have all the required ingredients to prove a lower bound on the distinguishability of almost all quantum states. Theorem 4. Let E be an ensemble of n equiprobable d-dimensional quantum states
64 −5/48 ) if n ≥ d, and picked uniformly at random. Set p = r1 1 − r1 1 − 9π − O(n 2
64 −5/48 p = 1 − r 1 − 9π 2 − O(d ) otherwise. Then, for any ≤ p/2,
4 2 Pr[P pgm (E) ≤ p − 2] ≤ 2 (n + 1)e−d /K + e−nd /5 ,
(35)
where K is a constant ≤ 300. Proof. As before, let S be the state matrix of E, and let S be the matrix produced by rescaling the√ vectors of Gaussians which would produce S if they were normalised. The matrix R = nd S fulfills the criteria for the Marˇcenko-Pastur law (Theorem 3), as its entries are complex random variables with mean 0 and variance 1. We therefore have E
1 2 S 1 n
≥
1 1 E(S 1 )2 = 2 E(R1 )2 ≥ p n n d
(36)
using the lower bound on the expected trace norm of R from Lemma 5 and the convergence result of Bai [2]. We will show that this implies a bound on n1 S21 , and hence (by Lemma 2) a bound on P pgm (E). From Lemma 8 (identifying Cd with R2d ) and Eqn. (33), we have for any l, Pr S21 /n ≤ p − 2
≤ Pr |(S 21 − S21 )/n| ≥ + Pr |S 21 /n − E S 21 /n | ≥
nd 2 d 4 nd(l 2 − 1)2 + exp − + n exp − ≤ 2 exp − 12 192 l 4 4 l2
nd 4 d 4 nd 2 ≤ 2 exp − + n exp − + exp − , 12 300 5 where, in the last line, we pick l 2 = 1 + 2 and note that ≤ 1/2.
(37) (38) (39) (40)
Despite the large constants that appear in these expressions, Fig. 1 shows numerical evidence that ensembles E of quantum states picked uniformly at random in fact appear to have a value of P pgm (E) close to the asymptotic lower bound, even when the states are (relatively) low-dimensional.
On the Distinguishability of Random Quantum States
629
1 Asymptotic lower bound Numerical results 0.9
P pgm(E)
0.8 0.7 0.6 0.5 0.4 0
0.5
1 r
1.5
2
(a) 0 ≤ r ≤ 2 1 Asymptotic lower bound Numerical results
0.9 0.8
P
pgm
(E)
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
2
4
6
8
10
r
(b) 0 ≤ r ≤ 10 Fig. 1. Asymptotic bound on P pgm (E) vs. numerical results (averaged over 10 runs) for ensembles of n = 50r 50-dimensional uniformly random states.
5. Application to Oracle Identification The oracle identification problem may be defined as follows [1]. Given an unknown n-bit Boolean function f : {0, 1}n → {0, 1} (the oracle), picked uniformly at random from a known set F of functions, identify f with the minimum number of uses of f . Set N = |F| and D = 2n . Clearly, classical computation cannot identify f with certainty in fewer than log2 N queries (as each query may reduce the search space by at most half). However, quantum computation can sometimes do better. On a quantum computer, we can encode the oracle as an n qubit unitary operator U f , defined by the action 1 2n −1 U f |x → (−1) f (x) |x. Now if the uniform superposition 2n/2 x=0 |x is input to the oracle, the following oracle state will be produced: |ψ f =
1 2n/2
n −1 2
(−1) f (x) |x.
(41)
x=0
In some cases, a single quantum query to U f may be enough to identify f with certainty. This will be the case if ψ f |ψg = 0 for all f = g (although this is not a necessary
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A. Montanaro
condition). The satisfaction of this orthogonality condition may be expected to be a rare event, and is certainly impossible when N > D. However, if we are content with a small probability of error, the situation is better: we will show here that, in particular, almost all sets of N = D oracles may be distinguished almost certainly in a constant number of quantum queries. The oracle identification problem was introduced and studied by Ambainis et al. [1], who (among other results) developed a hybrid quantum-classical algorithm for the random oracle case with which we concern ourselves here. However, the upper bound they obtained in the case where N = D is only O(log2 N ) queries, which can be shown to be no better than classical computation. Indeed, consider a set of k random classical queries to the unknown function f . The probability that any two of the set F of random functions agree on all k queries is 2−k , so by the union bound they will all differ on at least one of the queries with probability ≥ 1 − N2 2−k . Setting k = 3 log2 N makes this success probability approach 1 for large N , showing that almost all oracles can be identified with O(log2 N ) classical queries. Lemma 9. Let E be an ensemble of N D-dimensional oracle states corresponding to Boolean functions picked√ uniformly at random (call these random oracle states). Then the rescaled state matrix N D S(E) defines a point picked uniformly at random on the N D-dimensional hypercube {−1, 1} N D . √ Proof. Each component of each state will be ±1/ N D, with equal probability of each. √ N D S(E) therefore meets the required conditions for the Marˇcenko-Pastur law (Theorem 3), so we may say immediately Lemma 10. Let E be an ensemble of N D-dimensional random oracle states, and set r = N /D. Then ⎧
⎨ 1 1 − 1 1 − 642 − O(N −5/48 ) if N ≥ D r r 9π pgm
(42) E(P (E)) ≥ ⎩ 1 − r 1 − 642 − O(D −5/48 ) otherwise 9π and in particular E(P pgm (E)) ≥ 0.720 − O(D −5/48 ) when N ≤ D. Like the sphere, the high-dimensional hypercube exhibits the concentration of measure phenomenon, and we can write down a similar result to Levy’s Lemma [18]: Lemma 11 (Concentration of measure on the cube) [18]. Given a function f : {−1, 1}d → R defined on a d-dimensional hypercube, and a point p on the hypercube chosen uniformly at random,
−2 2 , (43) Pr[| f ( p) − E( f )| ≥ ] ≤ 2 exp dη2 where η is the Lipschitz constant of f with respect to the Hamming distance, η = supx,y | f (x) − f (y)|/d(x, y). Lemma 12. Let H be a point on the nd-dimensional hypercube written down as an n ×d {−1, 1}-matrix, and let f (H ) = n 12 d H 21 . Then the Lipschitz constant η of f satisfies η ≤ 4/nd.
On the Distinguishability of Random Quantum States
Proof. See Appendix C.
631
Inserting this value of η into Lemma 11 gives Theorem 5. Let E be an
ensemble of N D-dimensional random oracle states. Set 64 −5/48 ) if N ≥ D, and p = 1 − r 1 − 64 p = r1 1 − r1 1 − 9π − O(N − 2 9π 2 O(D −5/48 ) otherwise, where r = N /D. Then
Pr[P pgm (E) ≤ p − ] ≤ 2 exp
−2N D 2 16
.
(44)
We have our desired result: with 1 query, all but an exponentially small fraction of the possible sets of N N -dimensional random oracle states may be distinguished with a constant probability bounded away from 1/2 (in fact, to get a probability of success greater than 1/2, we may take r = N /D to be as high as ∼ 1.66). A constant number of repetitions allows this probability to be boosted to be arbitrarily high. 6. Discussion This work can be seen as part of an overall programme of understanding the behaviour of random quantum states [13, 21–23]. There is a fundamental correspondence between the mixed state obtained from an equal mixture of uniformly random pure states, and that produced by starting with a larger system in a uniformly random pure state, and tracing out part of the system. Consider a d-dimensional state ρn,d =
n 1 |ψi ψi |, n
(45)
i=1
where each state in the set E = {|ψi } is picked uniformly at random. We can think of ρn,d as being produced from the following dn-dimensional state (which we consider to live in a Hilbert space Hd ⊗ Hn ) by tracing out the second subsystem: n−1 n−1 d−1 1 1 |υk |k = √ αkl |l|k |υ = √ n n k=0
(46)
k=0 l=0
for some coefficients αkl . As mentioned previously, the αkl will be approximately normally distributed as N˜ (0, 1/d). So, because of the normalisation factor at the front of the sum, the overall state |υ has coefficients which are normally distributed and scaled as N˜ (0, 1/dn). Therefore, this state is approximately picked from the uniform distribution on the unit sphere in Cdn . Popescu, Short and Winter [21] obtained an upper bound on the expected trace distance of such a state ρn,d from the maximally mixed state I /d, and used this to show that for n d, ρ ≈ I /d. Because the non-zero eigenvalues of the Gram matrix of (rescaled) states in E are the same as the eigenvalues of ρn,d [17], this paper can be seen as obtaining a similar result to [21] for the fidelity of ρn,d with the maximally mixed state, via quite different methods. However, the bound is tighter for n close to d, and the notion of “randomness” of the states {|ψi } is more general (which is simply a side-effect of relying on the powerful Marˇcenko-Pastur law).
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Acknowledgements. I would like to thank Richard Jozsa for careful reading of this manuscript, Tony Short for helpful discussions, and Aram Harrow for many helpful comments and suggestions. I would also like to thank Jon Tyson for pointing out an error in Appendix A, and a referee for comments that greatly improved the paper. This work was supported in part by the UK Engineering and Physical Sciences Research Council QIP-IRC grant.
Appendices A. The PGM is Close to Optimal Theorem 1 (Barnum, Knill) [4]. P pgm (E) ≥ P opt (E)2 . Proof. Consider an arbitrary POVM R consisting of measurement operators {Ri }, and an arbitrary ensemble E of renormalised states {|ψi }, with a priori probabilities pi , where √ n as before |ψi = pi |ψi and ρ = i=1 |ψi ψi |. Assume wlog that Ri = |µi µi | for some vectors |µi , as the optimal measurement will always be of this form [9]. Then P R (E) = ≤
n i=1 n
ψi |Ri |ψi =
n
|ψi |µi |2 =
i=1
n
|ψi |ρ −1/4 ρ 1/4 |µi |2
(47)
i=1
ψi |ρ −1/2 |ψi µi |ρ 1/2 |µi
(48)
i=1
⎞ ⎛ n n ψi |ρ −1/2 |ψi 2 ⎝ µ j |ρ 1/2 |µ j 2 ⎠ ≤ i=1
(49)
j=1
n ≤ ψi |ρ −1/2 |ψi 2 = P pgm (E).
(50)
i=1
The first and second inequalities are Cauchy-Schwarz inequalities, and the third follows because the vectors {ρ 1/2 |µi } can easily be seen to define an ensemble with density matrix ρ: n n 1/2 1/2 1/2 ρ |µi µi |ρ =ρ |µi µi | ρ 1/2 = ρ, (51) i=1
n
i=1
1/2 |µ 2 i i=1 µi |ρ
≤ 1, as this is the probability of success of and we therefore have the measurement R applied to this ensemble. B. Proof of Lemma 5 In this appendix we will prove a lemma which immediately implies Lemma 5. See [10] for the facts used about elliptic integrals and hypergeometric series. √ √ Lemma 13. Let 0 ≤ r ≤ 1 and A = 1 − r , B = 1 + r . Then
B 64 2 2 2 2 (52) (y − A )(B − y ) dy ≥ r π 1 − r 1 − 9π 2 A with equality at r = 0, r = 1.
On the Distinguishability of Random Quantum States
633
Proof. We have
B
(y 2 − A2 )(B 2 − y 2 ) dy √ √ B 2 − A2 B 2 − A2 B 2 2 2 = − 2A K (A + B )E 3 B2 B2 √ 1/4
1/4 √ 2r 2r 2(1 + r ) = , (1 + r )E √ − (1 − r )2 K √ 3 1+ r 1+ r
f (r ) =
(53)
A
(54) (55)
where K (r ) and E(r ) are the complete elliptic integrals of the first and second kind, respectively: 1√ 1 dx 1 − r2x2 , E(r ) = d x. (56) K (r ) = √ 1 − x2 0 0 (1 − x 2 )(1 − r 2 x 2 ) Note that f (r ) may be evaluated explicitly for r = 0 and r = 1, giving 0 and 8/3 respectively. Now we may apply a standard change of variables (Landen’s transformation) to both elliptic integrals, giving √
√ √ √ 2 √ √ 2(1 + r ) 1 + r f (r ) = √ 2E( r ) − (1 − r )K ( r ) −(1− r ) (1 + r )K ( r ) 3 1+ r √ √ 4 (1 + r )E( r ) − (1 − r )K ( r ) . (57) = 3 We now move to the representation of K (r ) and E(r ) as hypergeometric series, which are defined as follows (using the notation a n¯ = a(a + 1) · · · (a + n − 1)): 2 F1 (a, b; c; r ) =
∞ a n¯ bn¯ n=0
cn¯ n!
rn,
(58)
K (r ) = (π/2) 2 F1 (1/2, 1/2; 1; r 2 ) , E(r ) = (π/2) 2 F1 (−1/2, 1/2; 1; r 2 ). (59) This has the advantage that, by a transformation rule due to Gauss, we can rewrite f (r ) as a single hypergeometric series, 2π ((1 + r ) 2 F1 (−1/2, 1/2; 1; r ) − (1 − r ) 2 F1 (1/2, 1/2; 1; r )) 3 = πr 2 F1 (−1/2, 1/2; 2; r ).
f (r ) =
(60) (61)
Returning to the original inequality, our task has been simplified to showing that
64 . (62) g(r ) = 2 F1 (−1/2, 1/2; 2; r )2 ≥ 1 − r 1 − 9π 2 Evaluating g(r ) at 0 and 1 makes it clear that this is equivalent to showing that g(r ) is concave for 0 ≤ r ≤ 1, which would follow from showing the second derivative g (r )
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to be negative in this region. From the rules governing differentiation of hypergeometric series, it is easy to show that g (r ) =
1 2 2 F1 (1/2, 3/2; 3; r ) − 2 2 F1 (−1/2, 1/2; 2; r )2 F1 (3/2, 5/2; 4; r ) . 32 (63)
The following hypergeometric transformation allows this to be simplified. 2 F1 (a, b; c; r )
= (1 − r )c−a−b 2 F1 (c − a, c − b; c; r ) 1 (1 − r )2 2 F1 (5/2, 3/2; 3; r )2 ⇒ g (r ) = 32 − 2(1 − r )2 2 F1 (5/2, 3/2; 2; r ) 2 F1 (3/2, 5/2; 4; r ) .
(64) (65) (66)
We will show that 2 F1 (5/2, 3/2; 3; r )2 ≤ 2 F1 (5/2, 3/2; 2; r ) 2 F1 (5/2, 3/2; 4; r ) for all positive r , implying that g (r ) is negative in this region. We write out the two hypergeometric series explicitly: 2 2 F1 (5/2, 3/2; 3; r )
=
∞ km kn (5/2)n¯ (3/2)n¯ n r , , where k = n 3m¯ 3n¯ n!
(67)
∞ km kn 4m¯ 2n¯
(68)
m,n=0
2 F1 (5/2, 3/2; 2; r ) 2 F1 (5/2, 3/2; 4; r ) =
=
m,n=0 ∞ m,n=0
km kn 3m¯ 3n¯
3 3+m
2+n 2
∞ ∞ 2 6 + 3m km km kn 3(2 + n) 3(2 + m) + + = 3m¯ 3m¯ 6 + 2m 3m¯ 3n¯ 2(3 + m) 2(3 + n) m=0
≥
(69)
(70)
m,n=0 m>n
∞ ∞ 2 km 2km kn + = 2 F1 (5/2, 3/2; 3; r )2 , 3m¯ 3m¯ 3m¯ 3n¯
m=0
(71)
m,n=0 m>n
where elementary methods can be used to show that the bracketed last term in Eq. (70) is at least 2 for any non-negative m and n. This completes the proof of the lemma.
C. Lipschitz Constants This appendix contains derivations of the Lipschitz constants of the functions used for the concentration of measure results. Lemma 7. Let S be an n × d matrix with S2 ≤ l, and define f (S) = n1 S21 . Then the Lipschitz constant η of f satisfies η ≤ 2l.
On the Distinguishability of Random Quantum States
635
Error 0.012
0.01
0.008
0.006
0.004
0.002
r 0.2
0.4
0.6
0.8
1
Fig. 2. Error in approximation to elliptic integral (52) for 0 ≤ r ≤ 1.
Proof. Let k = min(n, d). We have | S21 − T 21 | | f (S) − f (T )| = sup S − T 2 nS − T 2 S,T S,T
S1 + T 1 | S1 − T 1 | = sup n S − T 2 S,T
S1 + T 1 S − T 1 ≤ sup n S − T 2 S,T √ k (S1 + T 1 ) ≤ 2kl/n ≤ 2l. ≤ sup n S,T
η = sup
(72) (73) (74) (75)
The first inequality is a triangle inequality, and the second two are derived from
S1 =
k i=1
k √ σi (S) ≤ k σi2 (S) ≤ kS2 ,
(76)
i=1
which in turn uses a Cauchy-Schwarz inequality.
Lemma 12. Let S be a point on the nd-dimensional hypercube written down as an n × d {−1, 1}-matrix, and let f (S) = n 12 d S21 . Then the Lipschitz constant η of f (with respect to the Hamming distance) satisfies η ≤ 4/nd.
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Proof. The proof is very similar to that of Lemma 7. As before, let k = min(n, d). We have | f (S) − f (T )| 1 | S21 − T 21 | = sup 2 d(S, T ) d(S, T ) S,T S,T n d
S − T 1 S1 + T 1 ≤ sup 1 2 n d S,T i, j |S − T |i j 2 √ 2 k (S1 + T 1 ) ≤ 4k/n 2 d ≤ 4/nd, ≤ sup n2d S,T √ √ where, extending inequality (76), we use S1 ≤ kS2 ≤ k i, j |S|i j . η = sup
(77) (78) (79)
References 1. Ambainis, A., Iwama, K., Kawachi, A., Masuda, H., Putra, R.H., Yamashita, S.: Quantum identification of boolean oracles. Proc. STACS ’04, LNCS 2996, Berlin-Heidelberg: Springer, 2004, pp. 105–116 2. Bai, Z.D.: Convergence rate of expected spectral distributions of large random matrices. Part II. Sample covariance matrices. Ann. Prob. 21, 649–672 (1993) 3. Bai, Z.D.: Methodologies in spectral analysis of large dimensional random matrices. a review. Statist. Sinica 9, 611–677 (1999) 4. Barnum, H., Knill, E.: Reversing quantum dynamics with near-optimal quantum and classical fidelity. J. Math. Phys. 43, 2097–2106 (2002) 5. Bennett, C.H., Hayden, P., Leung, D., Shor, P., Winter, A.: Remote preparation of quantum states. IEEE Trans. Inform. Theory 51, 56–74 (2003) 6. Buhrman, H., Cleve, R., Watrous, J., de Wolf, R.: Quantum fingerprinting. Phys. Rev. Lett. 87, 167902 (2001) 7. Davies, E.B.: Information and quantum measurement. IEEE Trans. Inform. Theory 24, 596–599 (1978) 8. Eldar, Y.C., Forney, G.D. Jr.: On quantum detection and the square-root measurement. IEEE Trans. Inform. Theory 47, 858–872 (2001) 9. Eldar, Y.C., Megretski, A., Verghese, G.: Designing optimal quantum detectors via semidefinite programming. IEEE Trans. Inform. Theory 49, 1007–1012 (2003) 10. Gradshteyn I.S., Ryzhik I.M.: Table of integrals, series and products. New York, Academic Press (1980) 11. Hausladen, P., Jozsa, R., Schumacher, B., Westmoreland, M., Wootters, W.: Classical information capacity of a quantum channel. Phys. Rev. A 54, 1869–1876 (1996) 12. Hausladen, P., Wootters, W.: A “pretty good” measurement for distinguishing quantum states. J. Mod. Opt. 41, 2385 (1994) 13. Hayden, P., Leung, D.W., Winter, A.: Aspects of generic entanglement. Commun. Math. Phys. 265, 95– 117 (2006) 14. Helstrom C.W.: Quantum Detection and Estimation Theory. New York, Academic Press (1976) 15. Holevo, A.S.: Bounds for the quantity of information transmittable by a quantum communications channel. Problemy Peredachi Informatsii 9, no. 3, 3–11 (1993) English translation: Problems of Information Transmission 9, 177–183 (1973) 16. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge, University Press (1985) 17. Jozsa, R., Schlienz, J.: Distinguishability of states and von Neumann entropy. Phys. Rev. A 62, 012301 (2000) 18. Ledoux, M.: The concentration of measure phenomenon. AMS Mathematical Surveys and Monographs 89, Providence, RI: Amer. Math. Soc. 2001 19. Marˇcenko V.A., J.Pastur, L.A.: Distributions of eigenvalues of some sets of random matrices. Math. USSR-Sb. 1, 507–536 (1967) 20. Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge, Cambridge University Press (2000) 21. Popescu, S., Short, A.J., Winter, A.: Entanglement and the foundations of statistical mechanics. http:// arXiv.org/list/quant-ph/0511225 (2005) 22. Wootters, W.K.: Random quantum states. Found. Phys. 20, 1365 (1990) 23. Zyczkowski, K., Sommers H.: Average fidelity between random quantum states. Phys. Rev. A 71, 032313 (2005) Communicated by M.B. Ruskai
Commun. Math. Phys. 273, 637–650 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0247-x
Communications in
Mathematical Physics
Regularity of the Diffusion Coefficient Matrix for Lattice Gas Reversible under Gibbs Measures with Mixing Condition Yukio Nagahata Department of Mathematical Science, Graduate School of Engineering Science, Osaka University, Toyonaka, 560–8531, Japan. E-mail: [email protected] Received: 20 July 2006 / Accepted: 11 December 2006 Published online: 11 April 2007 – © Springer-Verlag 2007
Abstract: In this paper we obtain that the diffusion coefficient matrix for lattice gas reversible under Gibbs measures with mixing condition is continuously differentiable with respect to order parameter.
1. Introduction In [10], Varadhan and Yau proved the hydrodynamic limit of a stochastic lattice gas reversible under Gibbs measures which satisfy a certain mixing condition. They comment that the uniqueness of weak solution to the Cauchy problem for the limiting diffusion equation is quite subtle and give two sufficient conditions for the uniqueness to hold good: if the diffusion coefficient is either diagonal, or Lipshitz continuous, then the uniqueness is valid. They also give a sufficient condition to be satisfied by the jump rate in order that the coefficient becomes diagonal. In this paper we prove that the coefficient is continuously differentiable, and accordingly complete the proof of the hydrodynamic limit of the model whether the coefficient is diagonal or not. The smoothness of the bulk-diffusion coefficient is proved by Bernardin [1] for a lattice gas reversible under the Bernoulli measures, by Sued [9] for a mean zero exclusion process and by the present author [6, 7] both for a lattice gas with energy and for the generalized exclusion process. Previous to these works the smoothness of the self-diffusion coefficient of the symmetric simple exclusion process has been proved by Landim, Olla, and Varadhan [4]. According to [4], we have only to prove the differentiability of the central limit theorem variance of current which appears in the Green-Kubo formula (see Sect. 5). But it seems difficult to adapt to our model the method which is introduced by [4] and developed in [1, 9], since we do not have any suitable orthonormal basis (with respect to invariant measures) of functions on the configuration space. In this paper we use a certain basis, which, although not orthonormal, is natural: by means of it, our process of particles is transformed to a dual process which may be viewed as a family of random
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Y. Nagahata
walks on the square lattices of various dimensions and leads to an inductive Poisson equation (inductive resolvent equation), which is rather tractable. By getting some tail estimate (estimate of the rate of convergence to zero of the tail) of the solution of the inductive Poisson equation, we prove the differentiability of the central limit theorem variance of the current. This paper is organized as follows: In Sect. 2 we state the model and results. In Sect. 3 we introduce a basis of continuous functions on the configuration space and dual operator. In Sect. 4 by using the inductive Poisson equation (inductive resolvent equation), we get a coefficient of a function which solves the resolvent equation and order estimate of it. In Sect. 5 we prove the main result. 2. Model and Result Let N be a cube in Zd with width 2N + 1, centered at the origin and X N := {0, 1} N d (or X := {0, 1}Z ) be the state space of lattice gas. Let η = (ηx )x∈ N (or η = (ηx )x∈Zd ) stand for a generic element of X N (resp. X ), so that for each x, ηx is equal to 0 or 1. Each element of X N or X represents a configuration of particles on understanding that ηx = 1 if there exists a particle at x and ηx = 0 if x is vacant. We define shift operators τx ( x ∈ Zd ), which act on A ⊂ Zd and local functions f as well as configurations η, by τx A := x + A, τx f (η) := f (τx η), (τx η)z := ηz−x and a family of functions { A } A , A ⊂ Zd , by A := 1{ηx =1} .
(1)
x∈A
Let J = {J A } A⊂Z be a family of real numbers, called a potential, and suppose that it is of finite range and invariant under shift, namely there exists r = r (J ) such that if diam A > r , then J A = 0, and if A = τx B, then J A = J B . We use the letter ω to denote a configuration in X N +r (J ) as representing a boundary condition. Let us define the Hamiltonian in N with potential {J A } and boundary condition ω ∈ X N +r (J ) by J A A (η ∪ ω), H N ,ω (η) := A;A∩=∅
where η∪ω ∈ X N +r (J ) such that (η∪ω)x = ηx if x ∈ N and (η∪ω)x = ωx if x ∈ / N . We define the Gibbs measure µ N ,ω in N with potential {J A }, chemical potential λ and boundary condition ω ∈ X N +r (J ) by 1 exp[−H N ,ω (η) − λ ηx ]. µ N ,ω,λ (η) := Z N ,ω,λ x∈ N
Assumption 1. The potential {J A } A satisfies Dobrushin’s condition. A useful sufficient condition for this assumption is found in [2, Example 8.9 (p. 144)]. By Assumption 1, there is a unique infinite volume Gibbs measure, Pρ say, corresponding to each density ρ ∈ [0, 1]: Pρ [{η : η0 = 1}] = ρ. Let E ρ denote the expectation with respect to Pρ . According to [2, Corollary 8.37 (p. 162)], it holds that if f is a local function on X , then E ρ [ f ] is a continuously
Regularity of the Diffusion Coefficient Matrix for Lattice Gas
639
differentiable function of ρ. Furthermore, if f is a local function, then we have d 1 (E ρ [ f {x} ] − ρ E ρ [ f ]), Eρ [ f ] = dρ χ (ρ) x where χ (ρ) = x E ρ [({x} − ρ)({0} − ρ)]. For any function f on X N (or X ) we define π x,y by π x,y f (η) := f (η x,y ) − f (η), where η x,y is a configuration defined by ⎧ ⎨ η y if z = x, (η x,y )z := ηx if z = y, ⎩ η otherwise. z d Assumption 2. There exists a set of local functions {ci }i=1 that satisfies the detailed balance condition
ci (η) exp[−H N ,ω (η)] = ci (η0,ei ) exp[−H N ,ω (η0,ei )] for some large N and all ω ∈ X N +r (J ) , where ei is the usual unit vector along the i th coordinate axis, and that if η0 = ηei , then ci (η) = 0 and if η0 = ηei , then ci (η) > 0. d Given local functions {ci }i=1 as in Assumption 2 we define the operator L N (resp. L) by
L N f (η) :=
d
τx ci (η)π x,x+ei f (η),
i=1 x;x,x+ei ∈ N
L f (η) :=
d
τx ci (η)π x,x+ei f (η),
i=1 x∈Zd
where f is a function on X N (or a local function on X ). In a standard way one can show that there exists a unique closed extension of L in the Banach space of continuous functions on X equipped with supremum norm (cf. [5]). We denote by et L the semigroup generated by the closed extension. According to [10], Assumptions 1 and 2 together guarantee that the diffusion coefficient matrix for the limit non-linear diffusion equation is the symmetric d × d matrix given by variational formula as follows: χ (ρ) := E ρ [({0} − ρ)({x} − ρ)], x∈Zd
α, D(ρ)α :=
d
αi Di, j α j
(2)
i=1
=
d 1 inf E ρ [ ci (η)(αi ({ei } − {0} ) − π 0,ei τx g)2 ], 2χ (ρ) g d i=1
where α =
t (α , . . . , α ) 1 d
x∈Z
and the infimum is taken over all local functions.
Theorem 1. Under Assumptions 1 and 2, the diffusion coefficient matrix given by (2) is a continuously differentiable function of ρ.
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Y. Nagahata
3. Basis of C(X) and Dual Operator Let C(X ) denote the space of continuous functions on X which we equip with the supremum norm. We recall the definition of { A } A given by (1), A := 1{ηx =1} . x∈A
It is easy to see that { A } A is a basis of local functions and the coefficient fˆ in the expansion of a local function f with respect to { A } A is given by fˆ(A) = (−1)#(A\B) f (η B ), B⊂A
where η B is a special configuration defined by 1 if x ∈ B (η B )x := 0 if x ∈ / B. Note that if a function f (η) depends only on {ηx |x ∈ A}, then fˆ(B) = 0 if B ∩ Ac = ∅ as well as that for every pair of A and B, A B = A∪B .
(3)
Let us define A x,y ⊂ Zd by ⎧ / A, ⎨ A \ {x} ∪ {y} if x ∈ A and y ∈ / A, A x,y := A \ {y} ∪ {x} if y ∈ A and x ∈ ⎩ A otherwise. It is easy to see that π x,y A (η) = A (η x,y ) − A (η) = A x,y (η) − A (η), and that if = A then with (4) we have A x,y
L A =
π x,y
d
A
(4)
= 0. Making expansion of ci and using (3) together
τx ci (η)π x,x+ei A
i=1 x∈Zd
=
d i=1 x∈Zd D⊂Zd
=
d
cˆi (τ−x D) D ( A x,x+ei − A )
cˆi (τ−x D) A x,x+ei
i=1 x:A x,x+ei = A D⊂A x,x+ei
+ −
E⊂A x,x+ei
F⊂Zd \A x,x+ei :F=∅
cˆi (τ−x D) A
D⊂A
−
cˆi (τ−x (E ∪ F)) A x,x+ei ∪F
E⊂A F⊂Zd \A:F=∅
cˆi (τ−x (E ∪ F)) A∪F .
(5)
Regularity of the Diffusion Coefficient Matrix for Lattice Gas
641
In this formula we regard the first and third terms as main terms and denote the remainder by A , namely A :=
d
cˆi (τ−x (E ∪ F)) A x,x+ei ∪F
i=1 x:A x,x+ei = A E⊂A x,x+ei F⊂Zd \A x,x+ei :F=∅
−
cˆi (τ−x (E ∪ F)) A∪F .
E⊂A F⊂Zd \A:F=∅
Since ci are local functions, there exists r such that if # A > r then cˆi (A) = 0. Therefore
A (B), which is the coefficient of A , satisfies that if # B > # A + r , then
A (B) = 0.
A (B) also satisfies that if # B ≤ # A, then By the definition of A , the coefficient
A (B) = 0. Let us define c(A, B) by
c(A, B) :=
⎧ ⎨ τx ci (η A ) = cˆi (τ−x D) if A = B x,x+ei = B, ⎩
D⊂A
0
otherwise.
Then the main term in (5) is rewritten as d {c(A x,x+ei , A) A x,x+ei − c(A, A x,x+ei ) A }. i=1
x
By using a summation by parts formula we have Lf = L
fˆ(A) A
A
=
fˆ(A)[
A
=
d {c(A x,x+ei , A) A x,x+ei − c(A, A x,x+ei ) A } + A ] i=1
d A i=1
x
c(A, A x,x+ei )( fˆ(A x,x+ei ) − fˆ(A)) A +
x
fˆ(A) A
(6)
A
for any local function f . We define L by L fˆ(A) :=
d i=1
c(A, A x,x+ei )( fˆ(A x,x+ei ) − fˆ(A)).
x
Let Ys be a Markov process on P(Zd ) generated by L and PA a distribution of the Markov process starting from A. Then this process is equivalent to the original process which starts from the configuration η A . We call L the dual operator of L.
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Y. Nagahata
4. The Resolvent Equation and the Tail Estimate At the end of the preceding section, we remarked that the Markov process generated by L starting from A is equivalent to the original process starting from η A . Let us decompose the power set P(Zd ) into Pn = Pn (Zd ) := {A ⊂ Zd ; # A = n}. It is easy to see that Pn are ergodic classes of the Markov process generated by L. First we suppose that d = 1. On considering the position of the left-most particle and the distances of particles, there is a bijection from Pn to Z × Nn−1 : for A = {x1 , x2 , . . . , xn }, where xi < x j if i < j, the bijection φ is given by φ(A) = φ({x1 , x2 , . . . , xn }) := (x1 , x2 − x1 , x3 − x2 , . . . , xn − xn−1 ). Therefore for each n the ergodic class Pn may be identified with Z × Nn−1 , and the Markov process generated by L can be regarded as a (continuous time) random walk on Z × Nn−1 . By the definition of L each particle in the original process moves to one of two nearest neighbor sites subject to the exclusion rule. Corresponding to this transition rule the random walk moves from φ(A) to φ(A) ± ji , i = 1, . . . , n, with the reflecting boundary condition such that it suppresses the transition when the walker attempts to move out of the space Z × Nn−1 . Here ji are n dimensional vectors defined by ( ji )i = 1, ( ji )i+1 = −1, ( ji )l = 0 for l = i, i + 1. Let us define a discrete measure m(A) whose mass is given by m(A) := E 1/2 [η A ]. If we consider a family of discrete measures {m ρ (A)} whose mass is given by m ρ (A) := E ρ [η A ], they are essentially equivalent to one another on each ergodic class of the Markov process in the sense that m ρ1 and m ρ2 are absolutely continuous to each other and the Radon-Nikodym derivative is a constant which depends only on ρ1 , ρ2 and the ergodic class. By Assumption 2, it is easy to see that m is a reversible measure of L on each ergodic class. Since the Markov process on Pn is a random walk on Z × Nn−1 reversible under m, it is not difficult to obtain an estimate of the rate of convergence to zero at infinity of the Green function G of L, which we call the tail estimate of G. Furthermore since the random walk has reflecting boundary condition, it is not difficult to estimate the tail of the difference of G. Now suppose that d ≥ 2. The state space of the Markov process then fails to have linear order structure and we cannot follow the argument made in the 1-dimensional case. But in general (including the case in d = 1), we can still get the tail estimate of the Green function of L similar to that obtained in the case d = 1 by matching an element of Pn with a set of n! elements of (Zd )n , as follows: Pick A = {x1 , x2 , . . . xn } ∈ Pn . Since xi = x j if i = j, there are n! permutations, say ϕ 1 (A) = (x11 , x21 , . . . , xn1 ), . . . , ϕ n! (A) = (x1n! , x2n! , . . . , xnn! ). Then An := {ϕ i (A)|A ∈ Pn , i = 1, 2, . . . , n!} is the same as (Zd )n except for the diagonal part. We define ϕ −1 : An → Pn by ϕ −1 (x1 , x2 , . . . , xn ) := {x1 , x2 , x3 , . . . , xn }.
Regularity of the Diffusion Coefficient Matrix for Lattice Gas
643
We consider a Markov generator L˜ on An by L˜ f (x1 , . . . , xn ) =
n d
c(ϕ −1 (x1 , . . . , xn ), ϕ −1 ((x1 , . . . , xn ) + lei, j ))
j=1 i=1 l=±1
×{ f ((x1 , . . . , xn ) + lei, j ) − f (x1 , . . . , xn )}, where ei, j is an element of (Zd )n whose i th component is e j ∈ Zd and the others are 0. Here it sometimes happens that (x1 , . . . , xn ) + lei, j ∈ / An . In such cases, we regard (x1 , . . . , xn ) + lei, j as (x1 , . . . , xn ). Suppose that G = G A (B) and G˜ = G˜ A (B) are the Green functions of L and L˜ ˜ we have respectively. Then by the definition of L and L, G A (B) =
n!
G˜ ϕ 1 (A) (ϕ i (B)).
i=1
Since the Markov process generated by L˜ is essentially a (continuous time) random walk ˜ Furtheron Zdn , it is not difficult to get the decay estimate of the Green function of L. more by the definition of ϕ i , we can find the symmetry relative to planes. Therefore we can adapt the reflection principle. We state these as a proposition, but we omit the details of the proof. Proposition 1. The resolvent kernel G λ , (λ > 0) is well-defined and vanishes exponentially fast, namely there exists G λ = G λA (B) for A, B ∈ P and # A = # B such that G λ solves λG λA (B) − LG λA (B) = δ A (B), where δ A (B) = 0 or 1 according to A = B or A = B and there exist constants λ > 0 and C which may depend on λ such that |G λA (B)| ≤ C exp(−λ d(ϕ(A), ϕ(B))), where d(·, ·) is an Euclidean distance on Zd# A . If d# A ≥ 3, then the Green function G is also well-defined and given by the limit of G λ and vanishes polynomially fast, namely there exists limλ→0 G λ = G = G A (B) for A, B ∈ P with # A = # B ≥ 3/d such that G solves −LG A (B) = δ A (B) and there exists a constant C such that |G A (B)| ≤ Cd(ϕ(A), ϕ(B))−(d# A−2) . Furthermore for all d ≥ 1, there exists limλ→0 (G λA /m(A) − G λB /m(B)) = G A,B = G A,B (C) for A, B, C ∈ P and # A = # B = #C such that G solves −LG A,B (C) =
1 1 δ A (C) − δ B (C), m(A) m(B)
and there exists a constant D which may depend on A, B such that |G A,B (C)| ≤ Dd(ϕ(A), ϕ(C))−d# A .
(7)
644
Y. Nagahata
Let us define currents we by we (η) := ce (η)({e} − {0} ). Denote by wˆ e (A) the coefficient of the current, i.e., we (η) = A wˆ e (A) A (η) and by
A (B) the coefficient of the remainder term A in the formula (6). From now on, we simply write w for we and wˆ for wˆ e . By using Proposition 1, we have the following lemma. Lemma 1. There is a function gˆ λ : P → R such that gˆ λ solves the resolvent equation λgˆ λ (A) − Lgˆ λ (A) − gˆ λ (B) ˆ (8) B (A) = w(A). It also satisfies that gˆ λ vanishes exponentially fast, namely there exists C and λ > 0 such that |gˆ λ (A)| ≤ C exp(−λ diam(A ∪ {0})). Furthermore gλ (η) := A gˆ λ A (η) is well-defined and solves the resolvent equation λgλ − Lgλ = w. We also have a function gˆ 0 : P → R such that gˆ 0 = limλ→0 gˆ λ and solves the Poisson equation −Lgˆ 0 (A) − gˆ 0 (B) ˆ (9) B (A) = w(A). Furthermore gˆ 0 vanishes polynomially fast: there exists a constant C such that |gˆ 0 (A)| ≤ Cdiam(A ∪ {0})−d# A . Proof. We will construct gˆ 0 by using the Green function in Proposition 1. Construction of gˆ λ can be given in a similar way. We rewrite (9) by −Lgˆ 0 (A) = w(A) ˆ + gˆ 0 (B) (10) B (A), B;# B<# A
since B (A) = 0 if # A ≤ # B. Therefore we call (10) an inductive Poisson equation and can give gˆ 0 (A) inductively on # A, as follows: First, suppose that # A = 0, then A = ∅. Then B (A) = 0 for all B, since 0 = # A ≤ # B for all B. By the definition of L, L f (∅) = 0 for all f . Therefore if w(∅) ˆ = 0, then we get gˆ 0 (∅) = 0 (to simplify). Second, suppose that # A = 1, i.e., A ∈ P1 . Since if 1 = # A ≤ # B then B (A) = 0 and gˆ 0 (∅) = 0, the right-hand side in (10) for A ∈ P1 is also w(A). ˆ Suppose that d ≥ 3, then by using the Green function G B (A) in Proposition 1, we have for A ∈ P1 , w(B)G ˆ gˆ 0 (A) = B (A). B∈P1
ˆ = 0. Since w is a local function, there exists a finite Suppose that A∈P1 w(A)m(A) sequence of sets {Ai1 ; 1 ≤ i ≤ m} ⊂ P1 such that if B ∈ P1 \ {Ai1 ; 1 ≤ i ≤ m} then
Regularity of the Diffusion Coefficient Matrix for Lattice Gas
w(B) ˆ = 0. Put w˜ i1 := see that w(A) ˆ =
i
ˆ i1 )m(Ai1 ) j=1 w(A
m−1
w˜ i1 {
i=1
645
for 1 ≤ i ≤ m − 1, then we can easily
1 1 δ A1 (A) − δ 1 (A)}. 1 1 ) Ai+1 m(Ai ) i m(Ai+1
Therefore by using the Green function G B,C (A) in Proposition 1, we have for A ∈ P1 , gˆ 0 (A) =
m−1 B∈P1 i=1
w˜ i1 G A1 ,A1 (A). i
i+1
Third, suppose that gˆ 0 (B) for # B ≤ n − 1 is given. Since B (A) = 0 for all A such that # B ≥ # A, for A ∈ P the right-hand side in (10) is equal to w(A) ˆ + n B;# B≤n−1 gˆ 0 (B) B (A) and well-defined. If dn = d# A ≥ 3, then we have
gˆ 0 (A) =
{w(B) ˆ +
B∈Pn
gˆ 0 (C) C (B)}G B (A)
C;#C≤n−1
ˆ + B;# B≤n−1 gˆ 0 (B) for A ∈ Pn . If B (A)) = 0 then we can find A∈Pn (w(A) i n n a pair of sequences {Ai ; 1 ≤ i ≤ m n } and {w˜ i }i , that is w˜ in := ˆ in ) + j=1 {w(A n B;# B≤n−1 gˆ 0 (B) B (A )}, such that i
w(A) ˆ +
gˆ 0 (B) B (A) =
w˜ in {
i
B;# B≤n−1
Therefore we have gˆ 0 (A) =
B
1 1 n δ An (A) − n ) δ Ai+1 (A)}. m(Ain ) i m(Ai+1
n (A). w˜ in G Ain ,Ai+1
i
Finally, we conclude that if we can prove the following condition for w(A) ˆ then the existence of gˆ 0 is proved; for d ≥ 3, w(∅) ˆ =0, for d = 2, w(∅) ˆ = 0 and A∈P1 w(A)m(A) ˆ = 0, and for d = 3, w(∅) ˆ = 0, A∈P1 w(A)m(A) ˆ = 0 and {w(A) ˆ + B;# B≤1 gˆ 0 (B) B (A)}m(A) = 0. We note that the last condition A∈ P 2 ˆ + B;# B≤1 gˆ 0 (B) B (A)}m(A) = 0 makes sense if wˆ satisfies the other A∈P2 {w(A) conditions. Note that gˆ λ (A) for λ > 0 is given without any condition. We note that there exists s ∈ R such that E[w|Fs ] = 0, where Fs is a σ -algebra generated by x∈s ηx and {ηx ; x ∈ / s }. Note that w depend on {ηx ; x ∈ s }. There fore if we pick A0 = {η; x∈s η y = 0, ηz = 0 for all z ∈ / s }, A1 = {η; x∈s η y = 1, ηz = 0 if z ∈ / s } and A2 = {η; x∈s η y = 2, ηz = 0 if z ∈ / s }. Then we have 0 = E[w|A0 ] = w(∅), ˆ 0 = E[w|A1 ] = w(B)m(B)Z ˆ ˆ A1 + w(∅), B∈P1
0 = E[w|A2 ] = (
B∈P2
w(B)m(B) ˆ +
B∈P1 x∈s \B
(11) w(B)m(B ˆ ∪ {x}))Z A2 + w(∅), ˆ
646
Y. Nagahata
respectively. Here Z Ai are normalizing constants. By using (11), we get w(∅) ˆ = 0, and w(B)m(B) ˆ =0
(12)
B∈P1
for all d. Suppose that d = 1 and we consider L on P1 . Then L is discrete Laplacian on Z+ , and m(A) = 1/2 for all A ∈ P1 . Since w is a local function there exists a finite set B ⊂ P1 such that w(B) ˆ = 0 if B ∈ / B. Therefore we conclude that there exist {gˆ 0 (B)} B∈P1 and a finite set B ⊂ P1 such that gˆ 0 solves the Poisson equation (10) and 0 if B ∈ / B . According to the binomial expansion 0 = (1 − 1)# A = gˆ 0 (B) = #(A\B) if A = ∅, in general we have (−1) B⊂A ⎞ ⎛ (−1)#G ⎝ fˆ(E ∪ F) = fˆ(E)⎠ (13) E⊂A
G⊂F
E⊂(A∪F)\G
if F = ∅. Suppose F = ∅ and B ∩ F = ∅. Put A := B ∪ F. First we decompose B (A) into B (A) :=
d
i,x B (A),
i=1 x;A x,x+ei = A
cˆi (τ−x (E ∪ F)). i,x B (A) := E⊂A
By using (13), we have
⎞ ⎛ i,x B (A) = (−1)#G ⎝ cˆi (τ−x E)⎠ G⊂F
=
E⊂A\G
(−1)
#G
c(A \ G, A x,x+ei \ G).
G⊂F
Similarly we have x,x+ei )= B (A
(−1)#G c(A x,x+ei \ G, A \ G).
G⊂F
It is easy to see that if neither B ⊂ A nor B ⊂ A x,x+ei , then B (A) = 0. We conclude that the coefficient of B is rewritten as B (A) =
d i=1
x
i,x B (A),
⎧ ⎪ − (−1)#G c(A \ G, A x,x+ei \ G) ⎪ ⎪ ⎪ ⎪ G⊂A\B ⎪ ⎪ ⎪ if B ⊂ A and B \ A x,x+ei = ∅ ⎨ i,x #G x,x+e i \ G, A \ G) . (A) = B (−1) c(A ⎪ ⎪ ⎪ G⊂A x,x+ei \B ⎪ ⎪ ⎪ ⎪ if B ⊂ A x,x+ei and B \ A = ∅ ⎪ ⎩ 0 otherwise
Regularity of the Diffusion Coefficient Matrix for Lattice Gas
647
We note that the summation in the above formula is only a finite summation since if i,x A = A x,x+ei then B (A) = 0. By using this we have
gˆ 0 (B)
B∈P1
B (A)m(A) =
A∈P2
gˆ 0 (B)
B∈P1
d A∈P2 i=1
i,x B (A)m(A).
x
Since B ∈ P1 , A ∈ P2 and if neither B ⊂ A nor B ⊂ A x,x+ei , then B (A) = 0, the right-hand side above is rewritten as i,x
d B∈P1 i=1
×
gˆ 0 (B)
x
[{c(B x,x+ei ∪ {y}, B ∪ {y}) − c(B x,x+ei , B)}m(B x,x+ei ∪ {y}) y
− {c(B ∪ {y}, B x,x+ei ∪ {y}) − c(B, B x,x+ei )}m(B ∪ {y})]. By Assumption 2 we have c(A, B)m(A) = c(B, A)m(B), in general. Therefore we arrive at d
gˆ 0 (B)
B∈P1 i=1 x x,x+ei
+c(B, B
[−c(B x,x+ei , B)m(B x,x+ei ∪ {y})
y
)m(B ∪ {y})].
By using summation by parts formula, we have d [−c(B, B x,x+ei )(gˆ 0 (B x,x+ei ) − gˆ 0 (B))m(B ∪ {y})]. B∈P1 i=1
x
y
d x,x+ei )(gˆ (B x,x+ei ) − gˆ (B))] = −Lgˆ (B) and gˆ (B) solves Since i=1 0 0 0 0 x [−c(B, B the Poisson equation (10), we have B∈P1
w(B)m(B ˆ ∪ {y}).
y
By using (11), we have
w(A)m(A) ˆ +
A∈P2
=
A∈P2
gˆ 0 (B)
B∈P1
w(A)m(A) ˆ +
B∈P1
B (A)m(A)
A∈P2
w(B)m(B ˆ ∪ {y}) = 0.
y
Therefore we conclude that there exists a solution of the inductive Poisson equation (10).
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Y. Nagahata
i,x . The proof is In order to get the tail estimate, we also use the notation B i,x (A) + also inductive on the cardinality of A. It is not so difficult to see that g(B) ˆ B i,xx,x+e (A) is dominated by a constant multiple of diam(A ∪ {0})−d# A , due g(B ˆ x,x+ei ) i B to cancellation of the first order term. It is also not so difficult to see that i,x (A) + g(B i,xx,x+e (A)}m(A) {g(B) ˆ ˆ x,x+ei ) i B B i,xx,x+e (A x,x+ei )}m(A x,x+ei ) = 0. i,x (A x,x+ei ) + g(B ˆ x,x+ei ) +{g(B) ˆ i B B Adapting the argument for the estimate (7) in Proposition 1, we get that |gˆ 0 (A)| ≤ Cdima(A ∪ {0})−d# A . 5. Proof of the Theorem Proof of Theorem 1. According to [8, Prop. II.2.2 (p.180)], the diffusion coefficient matrix defined by the variational formula (2) coincides with the diffusion coefficient matrix defined by the Green-Kubo formula based on the current-current correlation function: ¯ α, D(ρ)α :=
1 Eρ [ ci (η)(αi ({ei } − {0} ))2 ] 2χ (ρ) i=1 1 ∞ − E ρ [wα τx et L wα ]dt . 2 0 x d
Since ci and {x} are local functions, we have only to prove the differentiability of the second term. According to [3], we have ∞ E ρ [wτx et L w]dt = lim E ρ [wτx gλ ], 0
λ→0
x
x
where gλ is a solution of the resolvent equation λgλ = Lgλ = w. We note that there exists s ∈ Z such that E ρ [w|Fs ] = 0. Therefore it is easy to see that if A ∩ s = ∅, then we have E ρ [w A ] = 0. Furthermore if x ∈ s , x = 0, ei and B ∩ s = ∅, then by using Assumption 2, we also have E ρ [w{x} B ] = 0. By substituting A gˆ λ (A) (A) for gλ , where gˆ λ is given by Lemma 1 and using these equalities, we have E ρ [wτx gλ ] = E ρ [wτx gˆ λ (A) A ] = E ρ [w gˆ λ (A)τ−x A ], x
x
A⊂Zd
x
A∈Bx
where Bx := {A : #(τ−x A ∩ s ) ≥ 2 or τ−x A ∩ {0, ei } = ∅}. In order to apply Fubini’s theorem (and Lebesgue’s convergence theorem), we estimate |gˆ λ (A)|E ρ [|wτ−x A |], x
A∈Bx
Regularity of the Diffusion Coefficient Matrix for Lattice Gas
649
for λ ≥ 0. We infer that E ρ [|w A |] ≤ w∞ E ρ [ A ] ≤ w∞ θ # A , where θ = θ (ρ) is defined by θ :=
E ρ [{0} |F] < 1.
sup d \{0}
F∈σ ({0,1}Z
)
Therefore we have x
|gˆ λ (A)|E ρ [|wτ−x A |] ≤
∞
A∈Bx
|gˆ λ (A)|w∞ θ n .
x n=1 A∈Bx :# A=n
By using Lemma 1, it is not difficult to see that there exists C which may depend on λ such that for all n, |gˆ λ (A)| ≤ Cn. x
A∈Bx :# A=n
Since θ < 1, we conclude that x
|gˆ λ (A)|E ρ [|wτ−x A |] < ∞.
A∈Bx
By using Fubini’s theorem, we have
E ρ [w
gˆ λ (A)τ−x A ] =
A∈Bx
x
∞ n=1 x
gˆ λ (A)E ρ [wτ−x A ].
A∈Bx :# A=n
By using Lebesgue’s convergence theorem and Lemma 1, we have lim
λ→0
E ρ [wτx gλ ] =
x
∞ n=1 x
gˆ 0 (A)E ρ [wτ−x A ].
A∈Bx :# A=n
d We know that dρ E ρ [ f ] = x E ρ [ f ({x} − ρ)]/χ (ρ) and it is not difficult to see that there exists a constant C such that x E ρ [ A ({x} − ρ)] ≤ Cθ # A . Therefore we can justify the exchange of the order of summation and differentiation. Remark 1. Let us define a sequence of local functions {gn }n by gˆ 0 (A) A . gn := A⊂n
Then it is not difficult to see that lim E ρ [
n→∞
d i=1
= lim E ρ [ λ→0
d i=1
ci (η)(αi ({ei } − {0} ) −
π 0,ei τx gn )2 ]
x∈Zd
ci (η)(αi ({ei } − {0} ))2 ] −
1 E ρ [wα τx gλ ] . 2 x
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Y. Nagahata
Namely, {gn }n is a minimizing sequence of variational formula (2), which does not depend on the density ρ. Furthermore the left-hand side above and differential (with respect to ρ) of it converge uniformly in ρ. Acknowledgement. The author would like to thank Professor K.Uchiyama for helping him with valuable suggestions.
References 1. Bernardin, C.: Regularity of the diffusion coefficient for lattice gas reversible under Bernoulli measures. Stochastic Process. Appl. 101(1), 43–68 (2002) 2. Georgii, H.O.: Gibbs Measures and Phase Transitions. Berlin: Walter de Gruyter & Co., 1988 3. Kipnis, C., Varadhan, S.R.S.: Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion. Commun. Math. Phys. 104(1), 1–19 (1986) 4. Landim, C., Olla, S., Varadhan, S.R.S.: Symmetric simple exclusion process: regularity of the selfdiffusion coefficient. Commun. Math. Phys. 224(1), 307–321 (2001) 5. Liggett, T.M.: Interacting particle systems. Berlin-Heidelberg-New York: Springer, 1985 6. Nagahata, Y.: Regularity of the diffusion coefficient matrix for the lattice gas with energy. Ann. Inst. H. Poincare Probab. Statist. 41, 45–67 (2005) 7. Nagahata, Y.: Regularity of the diffusion coefficient matrix for generalized exclusion process. Preprint 8. Spohn, H.: Large Scale Dynamics of Interacting Particles. Berlin-Heidelberg-New York: Springer, 1991 9. Sued, M.: Regularity properties of the diffusion coefficient for a mean zero exclusion process. Ann. Inst. H. Poincare Probab. Statist. 41, 1–33 (2005) 10. Varadhan, S.R.S., Yau, H.T.: Diffusive limit of lattice gases with mixing condition. Asian J. Math. 1, 623–678 (1997) Communicated by H.-T. Yau
Commun. Math. Phys. 273, 651–675 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0198-2
Communications in
Mathematical Physics
Adiabatic Theorems for Quantum Resonances Walid K. Abou Salem , Jürg Fröhlich Institute for Theoretical Physics, ETH Zurich, CH-8093 Zurich, Switzerland. E-mail: [email protected]; [email protected] Received: 25 July 2006 / Accepted: 3 October 2006 Published online: 17 March 2007 – © Springer-Verlag 2007
Abstract: We study the adiabatic time evolution of quantum resonances over time scales which are small compared to the lifetime of the resonances. We consider three typical examples of resonances: The first one is that of shape resonances corresponding, for example, to the state of a quantum-mechanical particle in a potential well whose shape changes over time scales small compared to the escape time of the particle from the well. Our approach to studying the adiabatic evolution of shape resonances is based on a precise form of the time-energy uncertainty relation and the usual adiabatic theorem in quantum mechanics. The second example concerns resonances that appear as isolated complex eigenvalues of spectrally deformed Hamiltonians, such as those encountered in the N-body Stark effect. Our approach to study such resonances is based on the Balslev-Combes theory of dilatation-analytic Hamiltonians and an adiabatic theorem for nonnormal generators of time evolution. Our third example concerns resonances arising from eigenvalues embedded in the continuous spectrum when a perturbation is turned on, such as those encountered when a small system is coupled to an infinitely extended, dispersive medium. Our approach to this class of examples is based on an extension of adiabatic theorems without a spectral gap condition. We finally comment on resonance crossings, which can be studied using the last approach. 1. Introduction There are many physically interesting examples of quantum resonances in atomic physics and quantum optics. To mention one, the state of a cold gas of atoms localized in a trap may be metastable, since the trap may be not strictly confining. In typical Bose-Einstein condensation experiments, the shape of the trap usually varies slowly over time scales small compared to the lifetime of the metastable state, yet larger than a typical relaxation time (see for example [1]). This is an example of an adiabatic evolution of shape resonances. While there has been much progress in a time-independent theory of quantum Current address: Department of Mathematics, University of Toronto, M5S 2E4 Toronto, ON, Canada
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W. K. Abou Salem, J. Fröhlich
resonances (see [2–7]), there has been relatively little work on a time-dependent theory of quantum resonances (see [8–11]). Surprisingly, and in spite of its relevance to the interpretation of many experiments and phenomena in atomic physics, the problem of adiabatic evolution of quantum resonances received very litte attention, so far (but see [12]). In this paper, we study the adiabatic evolution of three general types of quantum resonances. This is a first step towards a rigorous understanding of resonance- and metastability phenomena, such as hysteresis in magnets and Sisyphus cooling of atomic gases (see for example [13–15]). We first consider the adiabatic evolution of so-called shape resonances. More specifically, we consider a quantum-mechanical particle in a potential well, say that of a quantum dot or a locally harmonic trap, with the property that the shape of the potential well changes over time scales which are small compared to the time needed for the particle to escape from the well. The analysis of this problem is based on a precise form of the time-energy uncertainty relation, see [8], and the standard adiabatic theorem in quantum mechanics, [16]. In our approach, we obtain an explicit estimate on the distance between the true state of the system and an instantaneous metastable state. Our approach can also be applied to study the time evolution of the state of an electron in a H e+ ion moving in a time-dependent magnetic field which changes over time scales that are small compared to the ionization time of the ion; (see [8] for a discussion of this example in the time-independent situation). The second class of examples concerns quantum resonances that appear as isolated complex eigenvalues of spectrally deformed Hamiltonians, such as the N-body Stark effect (see for example [4, 6]).1 Our analysis is based on Balslev-Combes theory for dilatation analytic Hamiltonians, [17], and on an adiabatic theorem for generators of evolution that are not necessarily normal or bounded, [18]. This approach, too, yields explicit estimates on the distance between the true state of the system and an instantaneous metastable state. The third class of examples concerns resonances that emerge from eigenvalues of an unperturbed Hamiltonian embedded in the continuous spectrum after a perturbation has been added to the Hamiltonian. Typical examples of such resonances arise when a small quantum-mechanical system, say an impurity spin, is coupled to an infinite, dispersive medium, such as magnons (see for example [22–24] for relevant physical models). Our approach to such examples is based on an extension of adiabatic theorems without a spectral gap condition, [25–29]. Our results also cover the case of resonance crossings. Further details of applications where our assumptions are explicitly verified for various physical models will appear in [30]. 2. Adiabatic Evolution of Shape Resonances In this section, we study the time evolution of the state of a quantum-mechanical particle moving in Rd under the influence of a potential well which is not strictly confining. The potential well is described by a time-dependent function on Rd , x (1) vθτ (x, t) ≡ θ 2 v( , s), θ where τ is the adiabatic time scale, t is the time, s = τt is rescaled time, θ ≥ 1 is a parameter characterizing the width and height of the well, and v(x, s) is a function on 1 For the sake of simplicity, we consider nondegenerate resonances. However, our analysis can be extended to the case of degenerate resonances; (see [10, 11] for a discussion of the latter in the time-independent case).
Adiabatic Theorems for Quantum Resonances
653
Rd × R that is twice differentiable in s ∈ R and smooth in x ∈ Rd ; see below for precise assumptions on the potential. We assume that τ is small compared to the escape time of the particle from the well.2 By introducing an auxiliary adiabatic evolution, we obtain precise estimates on the difference between the true state of the particle and an instantaneous metastable state. Our analysis is based on the generalized time-energy uncertainty relation, as derived in [8], and on the usual adiabatic theorem in quantum mechanics, [16]. The Hilbert space of the system is H := L 2 (Rd , d d x). Its dynamics is generated by the time-dependent Hamiltonian H τ (t) := −/2 + vθτ (x, t),
(2)
where is the d-dimensional Laplacian.3 We make the following assumptions on the potential vθ (x, s), for s ∈ I, where I is an arbitrary, but fixed compact interval of R. (A1) The origin x = 0 is a local minimum of v(x, s), for all s ∈ I, and, without loss of generality, v(0, s) = 0 for s ∈ I. (A2) The Hessian of v(·, s) at x = 0 is positive-definite, with eigenvalues i2 (s) > 20 , i = 1, . . . , d, and 0 > 0 is a constant independent of s. (A3) Consider a smooth function g(x) with the properties that g(x) = 1 for |x| < 21 d 2 and g(x) = 0 for |x| > 1, where |x| := i=1 x i . For > 0, we define the rescaled function g,θ by x . (3) g,θ (x) := g (θ )1/3 We assume that, for all > 0, 1 max g,θ (x)|vθ (x, s) − x2 (s)x| ≤ c, d 2 x∈R
(4)
uniformly in s ∈ I , where 2 (s) is the Hessian of v(·, s) at x = 0 and c is a finite constant independent of s ∈ I. (A4) v(x, s) is smooth, polynomially bounded in x ∈ Rd , and bounded from below, uniformly in s ∈ I. Moreover, v(x, s) is twice differentiable in s ∈ I. We also assume that H (s1 ) − H (s2 ) ≤ C, ∀s1 , s2 ∈ I, where C is a finite constant. Note that under these assumptions, vθ is a potential well of diameter of order O(θ ) and height O(θ 2 ). Let 1 H0 (s) := −/2 + x2 (s)x, (5) 2 and (6) H1 (s) := H0 (s) + w,θ (x, s), where4
1 w,θ (x, s) := g,θ (x)[vθ (x, s) − x2 (s)x]. 2
(7)
2 The escape time of the particle from the well, which is related to θ, will be estimated later in this section. 3 We work in units where the mass of the particle m = 1, and Planck’s constant = 1. 4 H (s) depends on the parameters θ and , but we drop the explicit dependence to simplify notation. 1
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W. K. Abou Salem, J. Fröhlich
Note that H (s) = H1 (s) + δv,θ (x, s),
(8)
where
1 δv,θ (x, s) := (1 − g,θ (x))[vθ (x, s) − x2 (s)x]. 2 It follows from Assumptions (A3) and (A4), and (9) that 0, |x| ≤ 21 (θ )1/3 max |δv,θ (x, s)| ≤ , s∈I θ 2 P(x/θ ), |x| ≥ 21 (θ )1/3
(9)
(10)
uniformly in s ∈ I, for some polynomial P(x) of x. Denote by P1n (s), n ∈ N, the projection onto the eigenstates of H1 (s) corresponding to the n th eigenvalue of H1 (s). It follows from Assumptions (A3) and (A4) that, for small enough, P1n (s) is twice differentiable in s as a bounded operator for s ∈ [0, 1]. Denote by U τ (s, s ) the propagator generated by H (s), which solves the equation5 ∂s U τ (s, s ) = −iτ H (s)U τ (s, s ), U τ (s, s) = 1.
(11)
Suppose that the initial state of the system is given by a density matrix ρ0 , which is a positive trace-class operator with unit trace. Then the state of the particle at time t = τ s is given by the density matrix ρs , which satisfies the Liouville equation ρ˙s = −iτ [H (s), ρs ]
(12)
and ρs=0 = ρ0 . The solution of (12) is given by ρs = U τ (s, 0)ρ0 U τ (0, s).
(13)
Let P be an orthogonal projection onto a reference subspace PH, and let ps denote the probability of finding the state of the particle in the reference subspace PH at time t = τ s. This probability is given by ps := T r (ρs P).
(14)
We are interested in studying the adiabatic evolution of a state of a particle which initially, at time t = 0, is localized inside the well. Such a state may be approximated by a superposition of eigenstates of H1 (0) (defined in (6)). The initial state of the particle is chosen to be given by N ρ0 = cn P1n (0), (15) n=0
P1n (0)
where are the eigenprojections onto the states corresponding to the eigenvalues N cn = 1, for some finite integer N . E n of H1 (0), cn ≥ 0, with n=1 We let U1 be the propagator of the auxiliary evolution generated by H1 (s). It is given as the solution of the equation ∂s U1 (s, s ) = −iτ H1 (s)U1 (s, s ), U1 (s, s) = 1.
(16)
5 Assumptions (A1)–(A4) are sufficient to show that U τ exists as a unique unitary operator with domain D, a common dense core of H (s), s ∈ I.
Adiabatic Theorems for Quantum Resonances
Moreover, let
655
W (s, 0) := U1 (0, s)U τ (s, 0).
(17)
Then W (s, 0) solves the equation
where
(s)W (s, 0), W (0, 0) = 1, ∂s W (s, 0) = −iτ H
(18)
(s) = U1∗ (s, 0)δv,θ (s)U1 (s, 0), H
(19)
as follows from (11), (6), (16) and (17). Then ps = T r (ρs P) =
cn psn ,
(20)
n
where We define
psn := T r (PU τ (s, 0)P1n (0)U τ (0, s)).
(21)
1n (s) := U1 (s, 0)P1n (0)U1 (0, s). P
(22)
We have the following proposition. Proposition 2.1. Suppose Assumptions (A1)–(A4) hold. Then s ≤ n (s)) ± 2τ (s )), psn sin2∗ (ar csin T r (P P ds f (P1n (0), H 1 ≥ 0 (s) in (19), P n (s) in (22), for s ≥ 0, where psn is defined in (21), H 1 ⎧ ⎪ ⎨0, x < 0 sin ∗ (x) := sin(x), 0 ≤ x ≤ π2 , ⎪ ⎩1, x > π 2 and f (P, A) :=
T r (P A∗ (1 − P)A).
(23)
(24)
(25)
The proof of Proposition 2.1 is given in the Appendix, and it is based on the generalized time-energy uncertainty relation derived in [8]. Before stating an adiabatic theorem for shape resonances, we want to estimate the time needed for the quantum-mechanical particle to escape from the potential well if its initial state is given by (15). Note that, for each fixed value of s ∈ I, the spectrum of H0 (s), σ (H0 (s)), is formed of the eigenvalues E ls =
d i=1
1 i (s)(li + ), 2
(26)
where l = (l1 , . . . , ld ) ∈ Nd , with corresponding eigenfunctions φls (x) =
d i=1
1/4 i (s)h li ( i (s)xi ),
(27)
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W. K. Abou Salem, J. Fröhlich
where h li are Hermite functions normalized such that d xh l (x)h k (x) = δlk . Recall that the Hermite functions decay like a Gaussian away from the origin, 1
|h l (x)| ≤ cl,δ e−( 2 −δ)x , 2
(28)
for an arbitrary δ > 0 and a finite constant cl,δ ; (see for example [36]). It follows from analytic perturbation theory (Lemma A.1 in the Appendix) that the eigenstates of H1 (s) decay like a Gaussian away from the origin. Moreover, it follows from Assumption (A3) that δv,θ is supported outside a ball of radius 21 (θ )1/3 . Let π1n (x, y; s) denote the kernel of U1 (s, 0)P1n (0)U1 (0, s), whose modulus decays like a Gaussian away from the origin for arbitrary finite τ ; see Lemma A.1 in the Appendix. For each fixed s ∈ I, the following estimate follows from Lemma A.1, Assumptions (A3)–(A4) and (19). (s))2 = |T r (P1n (0) H (s)2 − P1n (0) H (s)P1n (0) H (s))| f (P1n (0), H (s)]2 )| = |T r ([P1n (0), H = |T r ([U1 (s, 0)P1n (0)U1 (0, s), δv,θ (s)]2 )| = d xd y|π1n (x, y; s)|2 (δv,θ (x, s) − δv,θ (y, s))2 ≤ C,n e−µ θ
2/3
,
(29)
where µ is proportional to 2/3 , C,n is a finite constant independent of s ∈ I (for finite n appearing in (15) and fixed ). Let τl ∼ eµ θ
2/3 /2
,
(30)
which, by (29) and (15), is a lower bound for the time needed for the particle to escape from the well.6 We now introduce the generator of the adiabatic time evolution for each eigenprojection, i Han (s) := H1 (s) + [ P˙1n (s), P1n (s)], (31) τ and the corresponding propagator Uan (s, s ) which satisfies ∂s Uan (s, s ) = −iτ Han (s)Uan (s, s ); Uan (s, s) = 1.
(32)
By Assumptions (A1)–(A4), it follows that (32) has a unique solution, Uan (s, s ), which is a unitary operator. From the standard adiabatic theorem in quantum mechanics [16], 6 In other words, the particle spends an exponentially large time in θ inside the well. Note that one may also directly use time-dependent perturbation theory to estimate the time needed for the particle to escape from the well, see [8].
Adiabatic Theorems for Quantum Resonances
657
we know that Uan (s, s )P1n (s )Uan (s , s) = P1n (s), sup s∈[0,1]
Uan (s, 0) − U1 (s, 0)
= O(τ
−1
(33) ),
(34)
for τ 1.7 For 1 τ τl , where τl is given in (30), it follows from (23), (22), (33) and (34) that psn = T r (PP1n (s)) + O(max(1/τ, τ/τl )). (35) Let ρ s :=
N
cn P1n (s),
(36)
n=1
the instantaneous metastable state of the particle inside the well. By (15), (35) and (36), we have that, for 1 τ τl , sup | ps − T r (P ρs )| ≤ A/τ + Bτ/τl ,
(37) (38)
s∈[0,1]
where A and B are finite constants. This proves the following theorem for the adiabatic evolution of shape resonances. Theorem 2.2. (Adiabatic evolution of shape resonances). Suppose Assumptions (A1)–(A4) hold for some τ satisfying (37). Then 1 τ ps = T r (P ρ (s)) + O(max( , )). τ τl
(39)
In other words, over time scales that are small compared to the escape time τl , given in (30), of the particle from the potential well, the true state of the particle which is initially localized inside the well, as given by the choice (15), is approximately equal to the instantaneous metastable state given in (36). We remark that a similar analysis can be applied to study the adiabatic evolution of the metastable state of the electron of an H e+ ion moving in a time-dependent magnetic field (see [8] for a discussion of this model in the time-independent case). 3. Isolated Eigenvalues of Spectrally Deformed Hamiltonians In this section, we discuss the adiabatic evolution of quantum resonances which appear as isolated eigenvalues of spectrally deformed Hamiltonians. Examples of such resonances include ones of the Stark effect and the N-body Stark effect (see for example [4, 6, 2, 3, 12]). Our analysis is based on Balslev-Combes theory for dilatation analytic Hamiltonians and on an adiabatic theorem for nonnormal and unbounded generators of evolution. The main result of this section is Theorem 3.3, which gives an estimate on the distance between the true state and an instantaneous metastable state when the adiabatic time scale is much smaller than the lifetime of the metastable state. 7 We work in units where a microscopic relaxation time is of order unity.
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Fig. 1.
3.1. Approximate metastable states. Consider a quantum mechanical system with Hilbert space H and a family of selfadjoint Hamiltonians {Hgτ (t)}t∈R , which are given by Hgτ (t) = Hg (s),
(40)
with fixed dense domain of definition, where Hg (s) = H0 (s) + gV (s),
(41)
and H0 (s) is the (generally time-dependent) unperturbed Hamiltonian, while gV (s) is a perturbation bounded relative to H0 (s), unless specified otherwise; see the footnote after assumption (B1) below. Here, s = t/τ ∈ [0, 1] is the rescaled time. Let U (θ ), θ ∈ R, denote the one-parameter unitary group of dilatations. For fixed g, we assume that there exists a positive β, independent of s ∈ [0, 1], such that Hg (s, θ ) := U (θ )Hg (s)U (−θ ),
(42)
extends from real values of θ to an analytic family in a strip |I mθ | < β, for all s ∈ [0, 1]. The spectrum of Hg (s, θ ) is assumed to lie in the closed lower half-plane for I mθ ∈ (0, β). The relation (43) Hg (s, θ )∗ = Hg (s, θ ) holds for real θ and extends by analyticity to the strip |I mθ | < β. We make the following assumptions: (B1) λ0 (s) is an isolated or embedded simple eigenvalue of H0 (s) with eigenprojection P0 (s).8 We assume that, for each fixed s ∈ [0, 1] and I mθ ∈ (0, β), λ0 (s) is separated from the essential spectrum of H0 (s, θ ). We also assume that the corresponding eigenprojection P0 (s, θ ) is analytic in θ for I mθ ∈ (0, β) and strongly continuous in θ for I mθ ∈ [0, β). g (s, θ ) = Pg (s, θ )Hg (s, θ )Pg (s, θ ) denote the reduced (B2) For 0 < I mθ < β, let H Hamiltonian acting on Ran(Pg (s, θ )), and let λg (s) be its corresponding eigenvalue. Then g→0
λg (s) −→ λ0 (s). We assume that λg (s) is differentiable in s ∈ [0, 1]. 8 The Stark effect for discrete eigenvalues of Coulumb systems is an example where isolated eigenvalues of the unperturbed Hamiltonian become resonances once the unbounded perturbation is turned on [32, 33].
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(B3) For each fixed s ∈ [0, 1] and fixed θ with I mθ ∈ (0, β), there is an annulus N (s, θ ) ⊂ C centered at λ0 (s) such that the resolvent, Rg (s, θ ; z) := (z − Hg (s, θ ))−1 ,
(44)
exists for each z ∈ N (s, θ ) and 0 ≤ g < g0 (z). (B4) Let γ (s) be an arbitrary contour in N (s, θ ) enclosing λ0 (s) and λg (s), for I mθ ∈ (0, β). Then, for 0 ≤ g < g(γ (s)), the spectral projection dz Rg (s, θ ; z) (45) Pg (s, θ ) := 2πi γ (s) satisfies
lim Pg (s, θ ) − P0 (s, θ ) = 0.
g→0
(46)
We assume that Pg (s, θ ) is twice differentiable in s ∈ [0, 1] as a bounded operator, for fixed θ, I mθ ∈ (0, β). (B5) RS (Rayleigh-Schrödinger) Expansion. The perturbation V (s, θ ), for |I mθ | < β, is densely defined and closed, and V (s, θ )∗ = V (s, θ ). We define Hg (s, θ ) := H0 (s, θ ) + gV (s, θ ) on a core of Hg (s, θ ). For I mθ = 0, z ∈ N (s, θ ) and g small enough, the iterated resolvent equation is Rg (s, θ ; z)P0 (s, θ ) =
N −1
g n R0 (s, θ ; z)An (s, θ ; z) + g N Rg (s, θ ; z)A N (s, θ ; z),
n=0
for N ≥ 1 (depending on the model), where An (s, θ ; z) := (V (s, θ )R0 (s, θ ; z))n P0 (s, θ ).
(47) (48)
We assume that the individual terms in (47) are well-defined, and that An (s, θ ; z) defined in (48) are analytic in θ in the strip I mθ ∈ (0, β), for n = 1, . . . , N , and z ∈ N (s, θ ), and strongly continuous in I mθ ∈ [0, β). This assumption is satisfied for N = 1 in dilatation-analytic systems where V (s, θ ) is bounded relative to H0 (s, θ ), I mθ ∈ [0, β); see, e.g., [2, 3]. Moreover, this assumption holds for arbitrary N ≥ 1, if λ0 (s) is an isolated eigenvalue of the unperturbed Hamiltonian H0 (s), as in the case of discrete eigenvalues of Coulumb systems, with V (s, θ ) a perturbation describing the Stark effect, [32, 33]. The RS-expansion for Pg (s, θ ) implies that, for I mθ ∈ (0, β), Pg (s, θ ) = PgN (s, θ ) + O(g N ),
(49)
where PgN (s, θ ) is analytic in the strip I mθ ∈ (0, β), and strongly continuous in I mθ ∈ [0, β). In other words, the spectral projection onto the resonance state is only defined up to a certain order N in the coupling constant g. This is to be expected since resonance states decay with time. We now show that, for each fixed s ∈ [0, 1], the projections PgN (s) can be regarded as projections onto approximate metastable states, up to an error of order O(g N ).
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Denote by ψ0 (s) the eigenstate of H0 (s) with corresponding eigenvector λ0 (s), and let ψgN (s) =
1 PgN (s)ψ0 (s)
PgN (s)ψ0 (s).
(50)
We have the following proposition for approximate metastable states, for each fixed s ∈ [0, 1]; see [4]. Proposition 3.1. (Approximate metastable states). Assume that (B1)–(B5) hold, and fix s ∈ [0, 1]. Let ξ ∈ C0∞ (R) be supported close to λ0 (s) with ξ = 1 in some open interval containing λ0 (s). Then ψgN (s), e−i Hg (s)t ξ(Hg (s))ψgN (s) = agN (s)e−iλg (s)t + bgN (t),
(51)
for small g, where agN (s) = ψgN (s, θ ), Pg (s, θ )ψgN (s, θ ) = 1 + O(g 2N ), I mθ ∈ (0, β), and
bgN (t) ≤ g 2N Cm (1 + t)−m ,
for m > 0, where Cm is a finite constant, independent of s ∈ [0, 1]. Although the proof of Proposition 3.1 is a straightforward extension of the results in [4], it is sketched in the Appendix to make the presentation self-contained. Choosing t = 0 in (51) gives ψgN (s), (1 − ξ(Hg (s)))ψgN (s) = O(g 2N ).
(52)
In particular, for 0 < ξ ≤ 1, ψgN (s), e−i Hg (s)t ψgN (s) = e−iλg (s)t + O(g 2N ).
(53)
This motivates considering ψgN (s) as approximate instantaneous metastable states, up to an error term of order O(g 2N ). In the next subsection, we recall a general adiabatic theorem proven in [18]. 3.2. A general adiabatic theorem. Consider a family of closed operators {A(t)}t∈R acting on a Hilbert space H, with common dense domain of definition D. Let U (t) be the propagator given by ∂t U (t)ψ = −A(t)U (t)ψ , U (t = 0) = 1 ,
(54)
for t ≥ 0; ψ ∈ D. We make the following assumptions, which will be verified in the application we consider later in this section. (C1) U (t) is a bounded semigroup, for t ∈ R+ , i.e., U (t) ≤ M, where M is a finite constant. (C2) For z ∈ ρ(A(t)), the resolvent set of A(t), let R(z, t) := (z − A(t))−1 . Assume that R(−1, t) is bounded and differentiable as a bounded operator on H, and that ˙ A(t) R(−1, t) is bounded, where the (˙) stands for differentiation with respect to t.
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Assume that A(t) ≡ A(0) for t ≤ 0, and that it is perturbed slowly over a time scale τ such that A(τ ) (t) ≡ A(s), where s := τt ∈ [0, 1] is the rescaled time. The following two assumptions are needed to prove an adiabatic theorem. (C3) The eigenvalue λ(s) ∈ σ (A(s)) is isolated and simple, with dist (λ(s), σ (A(s))\{λ(s)}) > δ, where δ > 0 is a constant independent of s ∈ [0, 1], and λ(s) is continuously differentiable in s ∈ [0, 1]. (C4) The projection onto λ(s), 1 Pλ (s) := R(z, s)dz, (55) 2πi γλ (s) where γλ (s) is a contour enclosing λ(s) only, is twice differentiable as a bounded operator. Note that, since λ(s) is simple, the resolvent of A(s) in a neighborhood N of λ(s), contained in a ball B(λ(s), r ) centered at λ(s) with radius r < δ, is R(z, s) =
Pλ (s) + Ranalytic (z, s), z − λ(s)
(56)
where Ranalytic (z, s) is analytic in N . We now discuss our general adiabatic theorem. Let Uτ (s, s ) be the propagator given by ∂s Uτ (s, s ) = −τ A(s)Uτ (s, s ) , Uτ (s, s) = 1, (57) for s ≥ s . Moreover, define the generator of the adiabatic time evolution, Aa (s) := A(s) −
1 ˙ [ Pλ (s), Pλ (s)], τ
(58)
with the corresponding propagator Ua (s, s ), which is given by ∂s Ua (s, s ) = −τ Aa (s)Ua (s, s ) ; Ua (s, s) = 1,
(59)
for s ≥ s . It follows from Assumption (C4) that sup [ P˙λ (s), Pλ (s)] ≤ C,
s∈[0,1]
for some finite constant C, and hence by perturbation theory for semigroups, [35] Chap. IX, and Assumption (C1), Ua defined on the domain D exists and is unique, and Ua (s, s ) < M for s ≥ s , where M = MeC . We are in a position to state our adiabatic theorem. Theorem 3.2. (A general adiabatic theorem). Assume (C1)–(C4). Then the following holds: (i)
Pλ (s)Ua (s, 0) = Ua (s, 0)Pλ (0) , for s ≥ 0 (the intertwining property).
(60)
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(ii) sup Uτ (s, 0) − Ua (s, 0) ≤
s∈[0,1]
C , 1+τ
for τ > 0 and C a finite constant. In particular, sup Uτ (s, 0) − Ua (s, 0) = O(τ −1 ),
s∈[0,1]
for τ 1. We refer the reader to [18] for a proof of Theorem 3.2. Remark. Assumption (C1) can be relaxed, but the result of Theorem 3.2 will be weakened. Suppose A(t) generates a quasi-bounded semigroup, i.e., there exist finite positive constants M and γ such that U (t) ≤ Meγ t , t ∈ R+ , then (ii) in Theorem 3.2 becomes sup Uτ (s, 0) − Ua (s, 0) ≤ C
s∈[0,1]
eτ γ , τ
for 1 τ γ −1 . 3.3. Adiabatic evolution of resonances that appear as isolated eigenvalues of spectrally deformed Hamiltonians. We consider a quantum mechanical system satisfying Assumptions (B1)–(B5), Subsect. 3.1. Denote by Uτ (s, s , θ ) the propagator corresponding to the deformed time evolution, which is given by ∂s Uτ (s, s , θ ) = −iτ Hg (s, θ )Uτ (s, s , θ ), Uτ (s, s, θ ) = 1,
(61)
for 0 ≤ s ≤ s ≤ 1 and I mθ ∈ [0, β). We make the following assumption on the existence of the deformed time evolution, which can be shown to hold in specific physical models; see [30, 4, 32] and [35], Chap. IX. (B6) For fixed θ with I mθ ∈ (0, β), Uτ (s, s , θ ), 0 ≤ s ≤ s ≤ 1, exists and is unique as a bounded semigroup with some dense domain of definition D.9 In particular, there exists a finite constant M such that Uτ (s, s , θ ) ≤ M, 0 ≤ s ≤ s ≤ 1. The generator of the deformed adiabatic time evolution is given by Ha (s, θ ) := Hg (s, θ ) +
i ˙ [ Pg (s, θ ), Pg (s, θ )], τ
(62)
and it generates the propagator ∂s Ua (s, s , θ ) = −iτ Ha (s, θ )Ua (s, s , θ ), Ua (s, s, θ ) = 1, for 0 ≤ s ≤ s ≤ 1 and fixed θ with I mθ ∈ (0, β). 9 We remark later how this assumption can be relaxed.
(63)
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For fixed θ with I mθ ∈ (0, β), Assumptions (B4) and (B6) and perturbation theory for semigroups, [35], imply that Ua (s, s , θ ), s ≥ s , exists and [ P˙g (s, θ ), Pg (s, θ )] < ∞, I mθ ∈ (0, β) Ua (s, s , θ ) ≤ M ,
(64) (65)
where M is a finite constant independent of s, s ∈ [0, 1]. Assumptions (B1)–(B6) in Subsect. 3.1 imply Assumptions (C1)–(C4) in Subsect. 3.2, with the identification Hg (s, θ ) ↔ −i A(s), λg (s) ↔ −iλ(s), Pg (s, θ ) ↔ i Pλ (s), for fixed θ with I mθ ∈ (0, β). We consider a reference subspace corresponding to a projection P which is dilatation analytic, i.e., P(θ ) = U (θ )PU (−θ ) extends from real values of θ to a family in a strip |I mθ | < β, β > 0. Moreover, we assume that the initial state of the quantum mechanical system is ρ0 = |ψgN (0)ψgN (0)|, (66) where ψgN (s) has been defined in (50). We are interested in estimating the difference between the true state of the system and the instantaneous metastable state defined in (50) when Hg varies over a time scale smaller than the lifetime of the metastable state, τl = min (I mλg (s))−1 ∼ g −2 . s∈[0,1]
More precisely, we are interested in comparing pτ s := T r (PUτ (s, 0)ρ0 Uτ∗ (s, 0))
= T r (PUτ (s, 0)|ψgN (0)ψgN (0)|Uτ∗ (s, 0))
to
pτ s := T r (P|ψgN (s)ψgN (s)|).
(67) (68)
This is given in the following theorem. Theorem 3.3. (Adiabatic evolution of isolated resonances). Suppose Assumptions (B1)–(B6) hold. Then, for g small enough and for 1 τ τl ∼ g −2 , pτ s | = O(max(1/τ, g N τ, τ/τl (g))). | pτ s −
(69)
Proof. This result is a consequence of Theorem 3.2. Since Assumptions (B1)–(B6) hold, we know that, for fixed θ with I mθ ∈ (0, β), Ua (s, 0, θ )PgN (0, θ ) = PgN (s, θ )Ua (s, 0, θ ) + O(g N τ ), C sup Ua (s, 0, θ ) − Uτ (s, 0, θ ) ≤ , τ s∈[0,1]
(70) (71)
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for τ 1, where C is a finite constant. For 1 τ g −2 , and I mθ ∈ (0, β), we have pτ s = Uτ (s, 0)ψgN (0), PUτ (s, 0)ψgN (0) = Uτ (s, 0, θ )ψgN (0, θ ), P(θ )Uτ (s, 0, θ )ψgN (0, θ ) = Ua (s, 0, θ )ψgN (0, θ ), P(θ )Ua (s, 0, θ )ψgN (0, θ ) + O(1/τ ) = ψgN (s, θ ), P(θ )ψgN (s, θ ) + O(max(τ/τl (g), 1/τ, g N τ )) = pτ s + O(max(1/τ, g N τ, τ/τl (g))). Remarks. (1) To estimate the survival probability of the true state of the system, choose P = |ψgN (0)ψgN (0)|, where ψgN (s) is defined in (50). (2) One may also estimate the difference between the true expectation value of a bounded operator A and its expectation value in the instantaneous metastable state, provided the operator A is dilatation analytic. Similar to the proof of Theorem 3.3, one can show that ψgN (0), Uτ (s, 0)∗ AUτ (s, 0)ψgN (0) = ψgN (s), AψgN (s) + O(max(1/τ, g N τ, τ/τl (g))), for 1 τ g −2 . (3) The results of this section can be extended to study the quasi-static evolution of equilibrium and nonequilibrium steady states of quantum mechanical systems at positive temperatures, e.g., when one or more thermal reservoirs are coupled to a small system with a finite dimensional Hilbert space; see [18, 29] for further details. In these applications, the generator of time evolution is deformed using complex translations instead of complex dilatations. (4) Assumption (B6) can be relaxed. Fix θ with I mθ ∈ (0, β). Suppose that Hg (s, θ ) generates a quasi-bounded semigroup,
Uτ (s, s , θ ) ≤ Me gατ (s−s ) ,
(72)
where M and α are positive constants and g is the coupling constant. It follows from Assumption (B4) that 1 C sup [ P˙g (s, θ )Pg (s, θ )] ≤ τ s∈[0,1] τ for finite C. Together with (72), this implies that
Ua (s, s , θ ) ≤ M e gατ (s−s ) , where M is a finite constant. Then, under Assumptions (B1)–(B6), the result of Theorem 3.3 becomes | pτ s − pτ s | = O(max(e gατ /τ, g N τ, τ/τl (g))), for 1 τ g −2 .
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(5) The results of this section can be extended to study “superadiabatic” evolution of quantum resonances.10 In the last decade, there has been a lot of progress in studying superadiabatic processes (see for example [19] and references therein). Depending on the smoothness of the generator of the time evolution, superadiabatic theorems give improved estimates of the difference between the true time evolution and the adiabatic one. Very recently, and after the submission of this paper, superadiabatic theorems with a gap condition have been extended to evolutions generated by nonselfadjoint operators [20]. Using superadiabatic theorems and methods developed in [21], the results of this section can be extended to longer time scales under additional regularity assumptions on the Hamiltonian. Further details will appear in [30].
4. General Resonances In this section, we study the case of resonances which emerge from eigenvalues of an unperturbed Hamiltonian embedded in the continuous spectrum after a perturbation has been added to the Hamiltonian. Such resonances arise, for example, when a small system, say a toy atom or impurity spin, is coupled to a quantized field, e.g. to magnons or the electromagnetic field. The main result of this section is Theorem 4.1, which is based on an extension of the adiabatic theorem without a spectral gap; see for example [25, 27, 29]. The results of this section are more general than Sect. 3, since the perturbation is not restricted to be dilatation analytic. Consider a quantum mechanical system with a Hilbert space H and a family of time-dependent selfadjoint Hamiltonians {Hg (t)}t∈R such that Hg (t) = H0 (t) + gV (t), where H0 (t) is the unperturbed Hamiltonian with fixed common dense domain of definition D, ∀t ∈ R, and V (t) is a perturbation which is bounded relative to H0 (t) in the sense of Kato[35]. We assume that the variation of the true Hamiltonian, Hgτ (t), in time is given by Hgτ (t) ≡ Hg (s), where s ∈ [0, 1] is the rescaled time. We make the following assumptions on the model. (D1) Hg (s) is a generator of a contraction semigroup for s ∈ [0, 1] with fixed dense core. Let Rg (z, s) := (z − Hg (s))−1 for z ∈ ρ(Hg (s)), the resolvent set of Hg (s). We assume that Rg (i, s) is differentiable in s as a bounded operator, and Hg (s) R˙ g (i, s) is bounded uniformly in s ∈ [0, 1]. This assumption is sufficient to show that the unitary propagator generated by Hg (s) exists and is unique. (D2) λ0 (s) is a simple eigenvalue of H0 (s) which is embedded in the continuous spectrum of H0 (s), with corresponding eigenvector φ(s), H0 (s)φ(s) = λ0 (s)φ(s). Furthermore, the eigenprojection P0 (s) corresponding to λ0 (s) is twice differentiable in s as a bounded operator for almost all s ∈ [0, 1], and is continuous in s, s ∈ [0, 1], as a bounded operator. 10 We are grateful to an anonymous referee for indicating this possibility to us.
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(D3) Let P 0 (s) := 1 − P0 (s), and, for a given operator A on H, denote by Aˆ s its restriction to the range of P 0 (s), Aˆ s := P 0 (s)A P 0 (s). Let F(z, s) := φ(s), V (s)P 0 (s)(z − Hˆ 0 (s)s )−1 P 0 (s)V (s)φ(s).
(73)
For each s ∈ [0, 1], we have I m F(λ0 (s) + i0, s) ≤ 0, (Fer mi s Golden Rule).
(74)
We note that P0 (s)Hg (s) = λ0 (s)P0 (s) + O(g), Hg (s)P0 (s) = λ0 (s)P0 (s) + O(g). (D4) Instantaneous metastable states. Let ξ ∈ C0∞ (R) be supported in a neighborhood of λ0 (s). For each fixed s ∈ [0, 1], we have φ(s), e−it Hg (s) ξ(Hg (s))φ(s) = ag (s)e−itλg (s) + bg (t), t ≥ 0,
(75)
where λg (s) = λ0 (s) + gφ(s), V (s)φ(s) + g 2 F(λ0 (s) − i0, s) + o(g 2 ), and |ag (s) − 1| ≤ Cg 2 , |bg (t)| ≤ Cg 2 (1 + t)−n , C is a finite constant independent of s ∈ [0, 1], for some n ≥ 1. Note that I mλg (s) ≤ 0. Equation (75) uniquely defines the instantaneous resonance state, up to an error O(g 4 ).11 11 The latter assumption is satisfied if the following holds, for each fixed s ∈ [0, 1]; see [7] for a proof of this claim in the s-independent case: (1) There exists a selfadjoint operator As such that
eit As D ⊂ D, for each fixed s ∈ [0, 1] and t ∈ R. This implies that D ∩ D(As ) is a core of H0 (s). j j−1 (2) Denote by ad A (·) := [As , ad A ], ad 1A (·) := [As , ·]. For some integer m ≥ n + 6, where n appears s
s
s
in (D4), the multiple commutators ad iA (H0 (s)) and ad iA (V (s)), i = 1, . . . , m, exist as H0 (s)-bounded s s operators in the sense of Kato. [35] (3) Mourre’s inequality holds for some open interval s λ0 (s), E s (H0 (s))i[H0 (s), As ]E s (H0 (s)) ≥ θ E s (H0 (s)) + K , where E s (H0 (s)) is the spectral projection of H0 (s) onto s , θ is a positive constant, and K is a compact operator.
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A physical example where Assumptions (D1)–(D4) may be satisfied is a small system interacting with a field of noninteracting bosons or fermions, for example, a spin system coupled to a time-dependent magnetic field; see [22–24, 30] for further details on the relevant model of a toy atom interacting with the electromagnetic radiation field. We are interested in the adiabatic evolution of the quantum resonance over time scales which are much smaller than the lifetime of the resonance. We will prove an adiabatic theorem without a spectral gap condition for quantum resonances for weak coupling g (see [25, 27–29]). Let Uτ (s, s ) be the propagator given by ∂s Uτ (s, s ) = −iτ Hg (s)Uτ (s, s ), Uτ (s, s) = 1,
(76)
with some dense domain of definition D. Existence of Uτ as a unique unitary operator follows from Assumption (D1) and Theorem X.70 in [36]. Moreover, we introduce the generator of the adiabatic time evolution Ha0 (s) := Hg (s) +
i ˙ [ P0 (s), P0 (s)]. τ
(77)
The propagator corresponding to the approximate adiabatic evolution is given by ∂s Ua0 (s, s ) = −iτ Ha0 (s)Ua0 (s, s ), Ua0 (s, s) = 1,
(78)
with domain of definition D. Note that Ua exists as a unique unitary operator due to Assumptions (D1) and (D2). We have the following theorem, which is an extension of the results in [25, 27, 29]. Theorem 4.1. (Adiabatic theorem for embedded resonances). Suppose Assumptions (D1)–(D4) hold. Then, for small enough coupling g and large enough τ, Ua0 (s, 0)P0 (0)Ua0 (0, s) = P0 (s) + O(τ g), and sup Uτ (s, 0) − Ua0 (s, 0) ≤
s∈[0,1]
A τ 1/2
+ Bgτ 1/4 + C(τ −1/4 ),
(79)
(80)
where A and B are finite constants, and C(x) is a positive function of x ∈ R such that lim x→0 C(x) = 0. In particular, choosing τ ∼ g −2/3 gives sup Uτ (s, 0)P0 (0) − P0 (s) ≤ Ag 1/3 + C(g 1/6 ).
(81)
h(s, s ) := Ua0 (s, s )P0 (s )Ua0 (s , 0).
(82)
s∈[0,1]
Proof. Let Then
∂s h(s, s ) = iτ Ua0 (s, s ){Ha0 (s )P0 (s ) − P0 (s )Ha0 (s )}Ua0 (s , 0) i = iτ Ua0 (s, s ){λ0 (s )P0 (s ) + P˙0 (s )P0 (s ) − λ0 (s )P0 (s ) τ i + P0 (s ) P˙0 (s ) + O(g)}Ua0 (s , 0) τ = O(gτ ),
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where we have used the definition of the generator of the adiabatic evolution and the property that P˙0 (s)P0 (s) + P0 (s) P˙0 (s) = 0. It follows that h(s, 0) = h(s, s), which is claim (79). Moreover, we are interested in estimating the difference between the true evolution and the adiabatic time evolution. For ψ ∈ D, we have that s (Uτ (s, 0) − Ua0 (s, 0))ψ = − ds ∂s (Uτ (s, s )Ua0 (s , 0))ψ 0 s = −iτ ds Uτ (s, s )[Hg (s ) − Ha0 (s )]Ua0 (s , 0)ψ 0 s =− ds Uτ (s, s )[ P˙0 (s ), P0 (s )]Ua0 (s , 0)ψ. 0
Since the domain of definition D is dense in H, it follows that s 0 Uτ (s, 0) − Ua (s, 0) = ds Uτ (s, s )[ P˙0 (s ), P0 (s )]Ua0 (s , 0).
(83)
0
We will now use a variant of Kato’s commutator method to express the integrand as a total derivative plus a remainder term, see [25]. Let X (s) := Rg (λ0 (s) + i, s) P˙0 (s)P0 (s) + P0 (s) P˙0 (s)Rg (λ0 (s) − i, s).
(84)
Note that [Hg (s), X (s)] = [Hg (s) − λ0 (s) − i, Rg (λ0 (s) + i, s) P˙0 (s)P0 (s)] + [Hg (s) − λ0 (s) + i, P0 (s) P˙0 (s)Rg (λ0 (s) − i, s)] = [ P˙0 (s), P0 (s)] + i X (s) + O(g/). Furthermore, ∂s (Uτ (s, s )X (s )Ua0 (s , 0)) = iτ Uτ (s, s )[Hg (s ), X (s )]Ua0 (s , 0) + Uτ (s, s )X (s )[ P˙0 (s ), P0 (s )]Ua0 (s , 0) + Uτ (s, s ) X˙ (s )Ua0 (s , 0). Therefore, s 1 ds Uτ (s, s )[ P˙0 (s ), P0 (s )]Ua0 (s , 0) ≤ sup { [X (s)(1 + 2 P˙0 (s)P0 (s)) τ 0 s∈[0,1] + X˙ (s)] + X (s)} + Cg/, (85) where C is a finite constant independent of s ∈ [0, 1]. We claim that the following estimates are true for small enough and g: (i) X (s) < C/, (ii) X˙ (s) < C/ 2 , (iii) X (s) < B() + Cg/, where lim→0 B() = 0, and C is a finite constant, uniformly in s ∈ [0, 1].
(86) (87) (88)
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Estimates (i) and (ii) follow from our knowledge of the spectrum of Hg (s) and the resolvent identity. To prove estimate (iii), we compare the LHS of (88) to the case when g = 0. Let (89) X (s) := R0 (λ0 (s) + i, s) P˙0 (s)P0 (s) + P0 (s) P˙0 (s)R0 (λ0 (s) − i, s). Then, by the second resolvent identity, X (s) + Cg/ 2 , X (s) ≤ uniformly in s, for some finite constant C. We claim that X (s)2 = 0. lim 2 →0
(90)
Consider φ ∈ D, then ψ(s) = P˙0 (s)P0 (s)φ ∈ K er (P0 (s)). Using the spectral theorem for H0 (s), we have the following result: lim 2 R0 (λ0 (s) + i, s) P˙0 (s)P0 (s)φ2 = lim 2 ψ(s), R0 (λ0 (s) − i, s)R0 (λ0 (s) →0
→0
+i, s)ψ(s) = lim 2 dµψ(s) (λ)
1 (λ − λ0 (s))2 + 2 = µ(ψ(s) ∈ Ran(P0 (s))) = 0, →0
and hence claim (90). Therefore, sup Uτ (s, 0) − Ua0 (s, 0) ≤
s∈[0,1]
C1 C2 g + C(), + τ 2
(91)
where C1,2 are finite constants, and lim→0 C() = 0. Choosing = τ −1/4 gives (80). By choosing τ ∼ g −2/3 , (81) follows from Assumption (D4), (79) and (80). Remarks. (1) We note that, using an argument due to Kato, [16], the case of finitely many resonance crossings is already covered by Theorem 4.1, since the latter holds for P0 (s) twice differentiable as a bounded operator for almost all s ∈ [0, 1] and continuous as a bounded operator for s ∈ [0, 1]. Suppose that at time s0 ∈ [0, 1], a crossing of λ0 (s) with an eigenvalue of H0 (s) happens. It follows from continuity of P0 (s) that, for small > 0, Ran(P0 (s0 − )) and Ran(P0 (s0 + )) are close up to an error which is arbitrarily small in , and hence our claim follows. (2) Further knowledge of the spectrum of H0 (s) will yield a better estimate of the convergence of X to zero as → 0. For example, it is shown in [25–27] that if the spectral measure µφ(s) , φ(s) ∈ Ran(P0 (s)), is α-Hölder continuous, for α ∈ [0, 1], uniformly in s ∈ [0, 1], then12 sup R0 (λ0 (s) + i, s) P˙0 (s)P0 (s) ≤ A α/2 , (92) s∈[0,1]
for small enough, where A is a finite constant, and hence estimate (81) becomes sup Uτ (s, 0)P0 (0) − P0 (s) = O(g α/12 )
(93)
s∈[0,1]
for g small enough. Acknowledgements. WAS is grateful to an anonymous referee for pointing out references [12, 19–21]. 12 A measure µ is α-Hölder continuous, α ∈ [0, 1], if there exists a finite constant C such that, for every set with Lebesgue measure || < 1, µ() < C||α , see, e.g., [35].
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5. Appendix Proof of Proposition 2.1, Sect. 2. Proof of Proposition 2.1 . This proposition effectively follows by integrating the Liouville equation and applying the Cauchy-Schwarz inequality. It is a special case of the generalized time-energy uncertainty relations derived in [8]. Consider an orthogonal projection P and selfadjoint operators A and B acting on a Hilbert space H. Then it follows from a direct application of the Cauchy-Schwarz inequality that T r (P[A, B])2 ≤ 4T r (P A2 − P A P A)T r (P B 2 − P B P B),
(94)
with equality when there exist a, b ∈ R\{0} such that [a A + ibB, P]P = 0.
(95)
We use inequality (94) to derive upper and lower bounds for psn . Let n n ps,s := T r (PUτ (s, s )P1 (0)Uτ (s , s)).
(96)
Then n n |∂s ps,s | = |iτ T r (PUτ (s, s )[H (s ), P1 (0)]Uτ (s , s))|
= |τ T r (P1n (0)[Uτ (s , s)PUτ (s, s ), H (s )])| ≤ 2τ T r (Uτ (s, s )P1n (0)Uτ (s , s)P2 − Uτ (s, s )P1n (0)Uτ (s , s)PUτ (s, s )P1n (0)Uτ (s , s)P)1/2 × T r (P1n (0)H (s )2 − P1n (0)H (s )P1n (0)H (s ))1/2 n − ( p n )2 f (P n (0), H (s )), ≤ 2τ ps,s 1 s,s √ where f (P, A) := T r (P A∗ (1 − P)A). It follows that s n ∂s ps,s n n | − ar csin | ds | = |ar csin ps,0 ps,s n − ( p n )2 0 ps,s s,s s ≤ 2τ ds f (P1n (0), H (s )), 0
and hence psn
=
n ≤ ps,0 sin 2 ≥ ∗
n ar csin T r (PP1 (0)) ± 2τ
s
ds
0
f (P1n (0)),
H (s )
.
(97)
We note that psn = T r (PUτ (s, 0)P1n (0)Uτ (0, s)) = T r (U1 (0, s)PU1 (s, 0)W (s, 0)P1n (0)W (0, s)). (98) Together with (97), and the identification P ↔ U1 (0, s)PU1 (s, 0) (s), H (s) ↔ H
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(s) is the generator of the auxiliary propagator W , as defined in (19), we have where H s n≤ 2 n (s ))). ds f (P1n (0), H ps sin ∗ (ar csin T r (PU1 (s, 0)P1 (0)U1 (0, s)) ± 2τ ≥ 0 (99) Proof that the eigenstates of H1 , Sect. 2, decay like a Gaussian in space. It follows from Assumption (A3) that w,θ (x, s) defined in (7) is uniformly bounded by c, for s ∈ I. Therefore, the spectrum of H1 (s), for each fixed s ∈ I, can be computed by applying analytic perturbation theory (see, e.g., [35, 36]). Also using analytic perturbation theory, one can show that the eigenstates of H1 (s), for each fixed s, decay like a Gaussian away from the origin (see [31, 32]). To prove the last claim, choose E > 0. There exist finitely many sequences l(1) , . . . , l(k E ) , such that E ls( j) < E, j = 1, . . . , k E ,
(100)
where
E ), 0 A is a finite geometrical constant, 0 appears in Assumption (A2), and E ls( j) is given in (26). Let |l| := max li . Then |l ( j) | < E0 for j = 1, . . . , k E . Choose a contour γ E in the complex plane surrounding σ (H0 (s)) ∩ [0, E), such that k E ≤ A(
1 min(E s(k +1) − E ls(k E ) ) > 0. 2 s∈I l E For each fixed time s ∈ I , we define the spectral projection of H1 (s), 1 dz(z − H1 (s))−1 . PEθ, (s) := 2πi γ E d E := min dist[γ E , σ (H0 (s))] = s∈I
(101)
(102)
Let PE0 (s) be the orthogonal projection of H0 (s) onto the subspace H E(s) spanned by the eigenfunctions {φl(1) , . . . , φl(k E ) }, and choose such that c <
d E2 , 3(E + 0 )
(103)
where c is a finite constant appearing in Assumption (A3). It follows from analytic perturbation theory, with satisfying (103), that T r (PEθ, (s)) = T r (PE0 (s)) = k E ,
(104)
PEθ, (s) − PE0 (s) < 1.
(105)
and We have the following lemma.
Lemma A.1. Suppose Assumptions (A2) and (A3) hold. Choose satisfying (103), and fix s ∈ I. Furthermore, suppose that ψ s ∈ Ran PEθ, (s). Then there exist finite constants C > 1 and α > 0 (depending on ) such that, for sufficiently small α, eα|x| ψ s ≤ Cψ s . 2
Furthermore,
(106)
eα|x| U1 (s, s )ψ s ≤ Cψ s , for τ < ∞ and α small enough. 2
(107)
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Proof. It follows from (105) that there exists φ s ∈ H E(s) , the subspace spanned by the eigenfunctions {φl(1) , . . . , φl(k E ) }, such that
and hence
ψ s := PEθ, φ s ,
(108)
ψ s ≤ Cφ s ,
(109)
for some finite constant C. Moreover, it follows from (102) and (108) that dz α|x|2 2 2 α|x|2 s e ψ = (z − H1 (s))−1 e−α|x| eα|x| φ s . e γ E(s) 2πi
(110)
For α small enough, we know from (27) and (28) that eα|x| φ s ≤ C φ s , 2
(111)
for some finite constant C . Moreover, for z ∈ γ E , it follows from analytic perturbation theory, [35], that eα|x| (z − H1 (s))−1 e−α|x| = (z − H 1 (s))−1 ) < ∞, 2
2
(112)
for α small enough (depending on ), where H 1 (s) := H1 (s) + 2αd − 4α 2 |x|2 + 4αx · ∇. The claim (106) follows from (110), (111) and (112). Now,
eα|x| U1 (s, s )ψ s = U 1 (s, s )eα|x| ψ s , 2
2
where U 1 = eα|x| U1 (s, s )e−α|x| is the propagator generated by H 1 (s). By applying analytic perturbation theory, it follows that 2
2
U 1 (s, s )eα|x| ψ s ≤ e M(α)τ eα|x| ψ s , 2
2
where M(α) is a positive constant such that M(α) → 0 as α → 0. Together with (106), this implies (107) for α small enough. Proof of Proposition 3.1, Sect. 3. Proof of Proposition 3.1 . Fix θ, with 0 < I mθ < β. By Assumptions (B1) and (B3), there exists an open interval I ⊂ N (s, θ ) ∩ R, with λ0 (s) ∈ I. Choose ξ ∈ C0∞ (I ). Then F(s, t) := ψgN (s), e−i Hg (s)t ξ(Hg (s))ψgN (s) dz −i zt = lim ξ(z)ψgN (s), (Rg (s, z − i) e →0 I 2πi − Rg (s, z + i))ψgN (s). Let f (θ, s, t) :=
1 2πi
I
dze−i zt ξ(z)ψgN (s, θ ), Rg (s, θ ; z)ψgN (s, θ ),
(113)
(114)
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where ψgN (s, θ ) := U (θ )ψgN (s). Then F(s, t) = f (θ, s, t) − f (θ, s, t). The resolvent in N (s, θ ) can be decomposed into a singular and regular part, Rg (s, θ ; z) = analytic
where Rg
Pg (s, θ ) analytic (s, θ ; z), + Rg z − λg (s)
(115)
(s, θ ; z) is analytic in z. Note that analytic
Rg
analytic
(s, θ ; z)Pg (s, θ ) = Pg (s, θ )Rg
(s, θ ; z) = 0.
(116)
Using (116), the contribution of the regular part to f (θ, s, t) defined in (114) is 1 analytic N u g (s, θ ), dze−i zt ξ(z)Rg (s, θ ; z)u gN (s, θ ), 2πi I where u gN (s, θ ) :=
1 PgN (s)ψ0 (s)
[PgN (s, θ ) − Pg (s, θ )]ψ0 (s, θ ),
is of order g N . Since ξ ∈ C0∞ (I ), the last integral is bounded by Cm t −m for any m ≥ 0, and hence the contribution of the regular part is bounded by g 2N Cm t −m . The contribution of the singular part of the resolvent to F(s, t) is 1 1 −i zt −1 N N ag (s) e ξ(z)(z − λg (s)) − ag (s) dze−i zt ξ(z)(z − λg (s))−1 . 2πi I 2πi I (117) Using the fact that ξ = 1 in some open interval I0 λ0 , one may deform the path I into two contours, C0 and C1 , in the lower complex half-plane, as shown in Fig. 2. The term in (117) corresponding to the path C0 picks the residue agN (s)e−iλg (s)t . It follows from the identity PgN (s, θ )Pg (s, θ )PgN (s, θ ) = (PgN (s, θ ))2 + [PgN (s, θ ) − Pg (s, θ )][Pg (s, θ ) − 1][PgN (s, θ ) − Pg (s, θ )], and from the fact that PgN (s, θ ) − Pg (s, θ ) = O(g N ), that
agN (s) = 1 + O(g 2N ).
Fig. 2.
(118)
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Using (118), one may write the remainder term in (117) due to the path C1 as dz −i zt e ξ(z)(z − λg (s))−1 (z − λg (s))−1 I mλg (s) C1 πi + O(g 2N ) dze−i zt (z − λg (s))−1 + C1 dze−i zt (z − λg (s))−1 , + O(g 2N ) C1
which is of order O(g 2N ). References 1. Dalfovo, F., Giorgini, S., Pitaevskii, L.P.: Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys. 71, 463–512 (1999) 2. Simon, B.: Resonances in N-body quantum systems with dilation analytic potentials and the foundations of time-dependent perturbation theory. Ann. Math. 97, 247 (1973) 3. Simon, B.: Resonances and complex scaling: a rigorous overview. Int. J. Quant. Chem. 14, 529 (1978) 4. Hunziker, W.: Resonances, metastable states and exponential decay laws in perturbation theory. Commun. Math. Phys. 132, 177 (1990) 5. Orth, A.: Quantum mechanical resonances and limiting absorption: the many body problem. Commun. Math. Phys. 126, 559 (1990) 6. Herbst, I.: Exponential decay in the Stark effect. Commun. Math. Phys. 87, 429 (1982/3) 7. Cattaneo, L., Graf, G.M., Hunziker, W.: A general resonance theory based on Mourre’s inequality. Annales Henri Poincare 7, 583–601 (2006) 8. Pfeifer, P., Fröhlich, J.: Generalized time-energy uncertainty relations and bounds on lifetimes of resonances. Rev. Mod. Phys. 67, 759 (1995) 9. Soffer, A., Weinstein, M.I.: Time-dependent resonance theory. Geom. Funct. Anal. 8, 1086 (1998) 10. Merkli, M., Sigal, I.M.: A time-dependent theory of quantum resonances. Commun. Math. Phys. 201, 549 (1999) 11. Jensen, A., Nenciu, G.: On the Fermi Golden Rule: Degenerate Eigenvalues. http://mpej.unige.ch/ mp-arc/html/html/c/06/06-157.pdf,2006 12. Davies, E.B.: An adiabatic theorem applicable to the Stark effect. Commun. Math. Phys. 89, 329–339 (1983) 13. Chakrabarti, B.K., Ascharyya, M.: Dynamical Transitions and Hysteresis. Rev. Mod. Phys. 71, 847–859 (1999) 14. Cohen-Tannoudji, C.: Manipulating atoms with photons. Rev. Mod. Phys. 70, 707–719 (1998) 15. Phillips, W.D.: Laser cooling and trapping of neutral atoms. Rev. Mod. Phys. 70, 721–741 (1998) 16. Kato, T.: On the adiabatic theorem of quantum mechanics. Phys. Soc. Jap. 5, 435–439 (1958) 17. Balslev, E., Combes, J.M.: Spectral properties of Schrödinger operators with dilation analytic interactions. Commun. Math. Phys. 22, 280 (1971) 18. Abou Salem, W.: On the quasi-static evolution of nonequilibrium steady states. http://arxiv.org/list/mathph/060104, 2006. To appear in Annales Henri Poincare (2007) 19. Hagedorn, G., Joye, A.: Elementary exponential error estimates for the adiabatic approximation. J. Math. Anal. Appl. 267, 235–246 (2002) 20. Joye, A.: General adiabatic evolution with a gap condition. http://arxiv.org/list/math-ph/0608059, 2006 21. Nenciu, G.: Linear adiabatic theory, exponential estimates. Commun. Math. Phys. 152, 479–496 (1993) 22. Bach, V., Fröhlich, J., Sigal, I.M.: Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field. Commun. Math. Phys. 207, 249–290 (1999) 23. Bach, V., Fröhlich, J., Sigal, I.M.: Mathematical Theory of nonrelativistic matter and radiation. Lett. Math. Phys. 34, 183–201 (1995) 24. Bach, V., Fröhlich, J., Sigal, I.M.: Quantum electrodynamics of confined nonrelativistic particles. Adv. in Math. 137, 299–395 (1998) 25. Teufel, S.: A note on the adiabatic theorem without a gap condition. Lett. Math. Phys. 58, 261–266 (2002) 26. Teufel, S.: Adiabatic perturbation theory in quantum dynamics. Lecture Notes in Mathematics 1821, Heidelberg, New York: Springer-Verlag (2003) 27. Avron, J.E., Elgart, A.: Adiabatic theorem without a gap condition, Commun. Math. Phys. 203, 445– 463 (1999)
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28. Avron, J.E., Elgart, A.: An adiabatic theorem without a spectral gap. In: Mathematical results in quantum mechanics (Prague 1998), Oper. Theory Adv. Appl. 108, Basel: Birkhäuser, (1999), pp. 3–12 29. Abou Salem, W., Fröhlich, J.: Adiabatic theorems and reversible isothermal processes. Lett. Math. Phys. 72, 153–163 (2005) 30. Abou Salem, W., Fröhlich, J.: In preparation 31. Combes, J.M., Thomas, L.: Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators. Commun. Math. Phys. 34, 251 (1973) 32. Hunziker, W.: Notes on asymptotic perturbation theory for Schrödinger eigenvalue problems. Helv. Phys. Acta. 61, 257–304 (1988) 33. Herbst, I., Simon, B.: Dilation analyticity in constant electric field II. Commun. Math. Phys. 80, 181– 216 (1981) 34. Aguilar, J., Combes, J.M.: A class of analytic perturbations for one-body Schrödinger Hamiltonians. Commun. Math. Phys. 22, 269–279 (1971) 35. Kato, T.: Perturbation theory for linear operators, Berlin: Springer (1980) 36. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol. I (Functional Analysis), Vol. II (Fourier Analysis, Self-Adjointness). New York: Academic Press (1975) Communicated by B. Simon
Commun. Math. Phys. 273, 677–704 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0229-z
Communications in
Mathematical Physics
Spectral Analysis and Zeta Determinant on the Deformed Spheres M. Spreafico1, , S. Zerbini2 1 ICMC-Universidade de São Paulo, São Carlos, Brazil. E-mail: [email protected] 2 Dipartimento di Fisica, Universitá di Trento, Gruppo Collegato di Trento, Sezione INFN di Padova,
Padova, Italy. E-mail: [email protected] Received: 25 July 2006 / Accepted: 17 October 2006 Published online: 13 March 2007 – © Springer-Verlag 2007
Abstract: We consider a class of singular Riemannian manifolds, the deformed spheres SkN , defined as the classical spheres with a one parameter family g[k] of singular Riemannian structures, that reduces for k = 1 to the classical metric. After giving explicit formulas for the eigenvalues and eigenfunctions of the metric Laplacian S N , k we study the associated zeta functions ζ (s, S N ). We introduce a general method to deal k with some classes of simple and double abstract zeta functions, generalizing the ones appearing in ζ (s, S N ). An application of this method allows to obtain the main zeta k invariants for these zeta functions in all dimensions, and in particular ζ (0, S N ) and k ζ (0, S N ). We give explicit formulas for the zeta regularized determinant in the low k dimensional cases, N = 2, 3, thus generalizing a result of Dowker [25], and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter k. 1. Introduction In the last decades there has been a (continuously increasing) interest in the problem of obtaining explicit information on the zeta regularized determinant of differential operators [2, 49, 37, 50, 43, 61]. Despite the lack of a general method, a lot of results are available in the literature for various particular cases or by means of some kind of approximation. Moreover, quite complete results have been obtained for the geometric case of the metric Laplacian on a Riemannian compact manifold for some classes of simple spaces: spheres [18, 11], projective spaces [54], balls [5], orbifolded spheres [25], compact (and non-compact) hyperbolic manifolds [20, 9, 10] or in particular cases: Sturm operators on a line segment [8, 45], cone on a circle [56]. In particular, many works in the recent physical literature applied this zeta function regularization process to study the modifications induced at quantum level by some Partially supported by FAPESP: 2005/04363-4
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kind of deformation of the background space geometry of physical models [40, 24, 22, 52]. In this context, a full class of deformed spaces, called deformed spheres, has been introduced in [52], where the perturbation of the heat kernel expansion has been studied. This is a particularly interesting class of spaces in Einstein theory of gravitation and in cosmology, since the appearance of a non-trivial deformation produces a symmetry breaking of the space. In fact, the deformed sphere may be considered as the Euclidean version of the a deformed de Sitter space, which is particularly relevant in modern cosmology, since it represents the inflationary as well as the recent accelerated phase. It is well known that the quantum effective action is related to the regularized functional determinant of Laplace type operators (see, for example [30] and references therein). As a consequence, an expansion of the functional determinant with respect to the deformation parameter around its spherical symmetric value describes the effects of such geometric symmetry breaking. It is therefore a natural question to see if the explicit calculation of the zeta determinant for the Laplace operator on this class of spaces is possible. In this work we give a positive answer to this question, establishing a general method that permits to compute the zeta regularized determinant on a deformed sphere of any dimension. Actually, for a particular discrete set of values of the deformation parameter k, the N -dimensional deformed sphere turns out to be isometric to the so-called orbifolded sphere, the quotient space S N / , of the standard N -sphere by a finite subgroup of the rotation group O N (R). Determinants on these spaces have been studied by J.S. Dowker in a series of works [25–27], where results are also obtained for different couplings. Under this point of view, the present work is a generalization of the results of Dowker to the continuous range of variation of the deformation parameter k, and in fact the results are consistent (see Sect. 4). The main motivation of the present work, beside the particular result, is that the method introduced has the advantage of being completely general and not related to this specific problem. In particular, we show how it can be applied to obtain the main zeta invariants of some classes of abstract simple and double zeta functions (Sect. 4.2 and 4.3). In order to give the explicit form for the zeta function on the deformed spheres, we produce an explicit description of the spectrum and of the eigenfunctions of the associated Laplace operator in any dimension (Proposition 3.2). In particular, the 2 dimensional case turns out to be very interesting, from the point of view of geometry: in fact the 2 dimensional deformed sphere is a space with singularities of conical type. This class of singular spaces was introduced and studied by Cheeger [17] and although since then became a subject of deep interest and investigation, there are in fact relatively few occasions where explicit results can be obtained. 2. The Geometry of the Deformed Spheres In this section we provide the definition of the N dimensional deformed sphere SkN , where k is the deformation parameter, and we study its geometry. This produces a particular interesting relation with elliptic function and conical singularity, at least in the 2 dimensional case. The deformed N -sphere is defined as the standard N -sphere with a singular Riemannian structure. When N = 2, we have an isometry with the surface immersed in R3 that can be obtained by rotating around an axis a curve described by an elliptic integral function. The surface obtained presents two singular points of conical type, as considered by Brüning and Seeley in [7] generalizing the definition of metric cone of Cheeger [17]. Thus, the 2 dimensional deformed sphere is a space with singularities of
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conical type, and due to the great interest in this kind of singular space, both from the point of view of differential geometry and zeta function analysis (see for example [23, 34, 19, 6, 68, 21]), its study is of particular interest (compare also with [56]). Consider the immersion of the N + 1 dimensional sphere S N +1 in R N +2 , ⎧ x = sin θ0 sin θ1 . . . sin θ N , ⎪ ⎨ 0 x1 = sin θ0 sin θ1 . . . cos θ N , ⎪ ⎩... x N +1 = cos θ0 , and the induced metric (in local coordinates) g S N +1 = (dθ0 )2 + sin2 θ0 g S N . We deform this metric as follows. Let k be a real parameter with 0 < k ≤ 1, and consider the family g S N +1 [k] = (dθ0 )2 + sin2 θ0 g S N [k], g S 1 [k] = k 2 (dθ0 )2 . This is a one parameter family of singular Riemannian metric on S N +1 . We call the singular Riemannian manifolds (S N +1 , g S N +1 [k]) the deformed spheres of dimension N + 1 and we use the notation SkN +1 . By direct inspection, we see that the locus of the singular points of the metric in dimension N + 1 is a sub-manifold isomorphic to two disjoint copies of S N −1 . In particular, in the 2 dimensional case we have g S 2 [k] = (dθ0 )2 + k 2 sin2 θ0 (dθ1 )2 , that shows that the deformed 2-sphere Sk2 is a space with singularities of conical type as defined in [7]. Proceeding as in [7] Sect. 7, we will show in the next subsection that the singularity is generated by rotation of a curve in the plane. Observe that, in a different language, Sk2 is a periodic lune, that is to say it can be pictured by taking a segment of the standard 2-sphere (a lune) and identifying the sides. This situation generalizes to higher dimensions [1], and when the angle of the lune is πn , n ∈ Z, we obtain a spherical orbifold S N / , as pointed out in the introduction. Note also that, by direct verification on the local description of the metric g S N [k], the non-compact Riemannian manifold obtained by subtracting the singular subspace of the metric from SkN is a space of constant curvature and locally symmetric. It is not symmetric, as it is clear from the geometry of the low dimensional cases, or observing that it is not simply connected (see Corollary 8.3.13 of [65]). On the other side, the classical sphere S1N is a symmetric space; for example, the 2 dimensional one having the maximum number N (N2 +1) of global isometries, namely the 3 spatial rotations. Therefore, the variation of the parameter k away from the trivial value produces a breaking of the global symmetric type of the space. In particular for example on the 2 sphere it breaks two continuous rotations in one discrete symmetry, namely the reflection through the horizontal plane. We conclude this subsection with the explicit expression for the Laplace operator. With a = k1 , the (negative) of the induced Laplace operator on the deformed sphere SkN +1 is S N +1 = −dθ20 − N k
cos θ0 1 dθ0 + N. sin θ0 sin2 θ0 Sk
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2.1. Elliptic integrals and the deformed 2-sphere. The geometry of the 2 dimensional case is particularly interesting and this subsection is dedicated to its study. The ellipse 2 x 2 + by2 = 1 can be given parametrically in the first quadrant by the formula x = t,√ y = b 1 − t 2, where 0 ≤ t ≤ 1. If we assume b ≤ 1, the arc length is t 1 − k2s2 ds, l(t) = 1 − s2 0
√ where k = 1 − b2 . With the new variables t = sin θ , s = sin ψ, we obtain x = sin θ, y = b cos θ, with 0 ≤ θ ≤
π 2,
and the arc length is θ E(θ, k) = l(sin θ ) = 1 − k 2 sin2 ψdψ, 0
that is the elliptic integral of the second kind in Legendre normal form [35] 8.110.2 (see [48] or [64] for elliptic functions and integrals). Note that we cannot find a parameterization of the curve by the arc length reversing the above equation using Jacobi elliptic functions. Consider now the curve f (sin θ ) = E(θ, k). This is a smooth curve in the interval 0 ≤ t ≤ 1, with f (0) = 0 and f (1) = E( π2 , k). We can rotate this function around the horizontal axis getting a surface with a geometric singularity at the origin. For further use, it is more convenient to place the surface in the upper half space. Thus, we consider the function 2 1− t 2
t f (t) = E(arccos , k) = k
k
0
1 − k2s2 ds, 1 − s2
with 0 ≤ t ≤ k, and the curve: x = t, y = f (t). We reparametrize this curve by its arc length t t 1 t 2 θ = l(t) = 1 + (y (s)) ds = ds = arcsin , √ 2 2 k k −s 0
obtaining
0
x = k sin θ, y = f (k sin θ ) = E( π2 − θ, k) =
with 0 ≤ θ ≤
π 2
π 2
0
−θ
1 − k 2 sin2 ψdψ,
(as before θ is the angle from the vertical axis).
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Let us now consider the surface Yk+ obtained by rotating the above curve along the vertical axis. We have the parameterization ⎧ ⎨ x = k sin θ cos φ, Yk+ : y = k sin θ sin φ, ⎩ z = E( π − θ, k), 2 where 0 ≤ φ ≤ 2π . This is clearly a smooth surface except at the possible singular point (0, 0, E(π/2, k))), with the circle Ck : x 2 + y 2 = k 2 , z = 0 of radius k as boundary. Moreover, since the coordinate line tangent vectors on the boundary are vφ = − sin φex +cos φe y and vθ = ez , the tangent space is vertical and hence we can glue smoothly Yk+ with the surface Yk− obtained by reflecting through the horizontal plane. We call the surface obtained Yk2 = Yk+ ∪Yk− , and the parameter k deformation parameter. The surface X k obtained from Yk2 by removing the poles (0, 0, ±k) is clearly a smooth (non-compact) surface. The Riemannian metric induced on X k from the immersion in R3 is gY 2 (θ, φ) = (dθ )2 + k 2 sin2 θ (dφ)2 . k
It is clear that the local map f : (θ, φ) → (θ, φ), extends to a diffeomorphism f : Yk2 → S 2 , and since f ∗ g S 2 [k] = gY 2 , it follows that f is an isometry between k
Sk2 = (S 2 , g S 2 [k]) and (Yk2 , gY 2 ). k
3. Spectral Analysis In this section we give the eigenvalues and eigenfunctions of the Laplace operator on the deformed sphere. As observed in Sect. 2, the two dimensional case is of particular interest, since it represents an instance of a space with singularities of conical type that can be solved explicitly. Therefore we spend a few words to describe the concrete operator appearing in that case, using the language of spectral analysis for spaces with conical singularities [17, 7]. With a = k1 , the (negative) of the induced Laplace operator on the deformed sphere Sk2 is S 2 = −∂θ2 − 1/a
cos θ a2 2 ∂θ − ∂φ , sin θ sin2 θ
2 ). With the Liouville transform u = Ev, with E(θ ) = on L 2 (S1/a operator
1 a2 2 1 . 1 + L a = −∂θ2 − − ∂ φ 4 sin2 θ sin2 θ
This is a regular singular operator as defined in [7], L a = −dθ2 + where A(θ ) =
θ2 sin2 θ
1 A(θ ), θ2
1 1 −a 2 ∂φ2 − − θ 2, 4 4
√1 , sin θ
we obtain the
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is a family of operators on the section of the cone, that radius 1. It is is the circle of clear that the operator −∂φ2 has the complete system µm = m 2 , eimφ , with m ∈ Z, where all the eigenvalues are double up to the null one that is simple with the unique eigenfunction given by the constant map. Since the problem decomposes spectrally on this system, we reduce to study the family of singular Sturm operators Tam = −dθ2 +
a2m2 − sin2 θ
1 4
1 − . 4
In order to define an appropriate self adjoint extension, we introduce the following boundary conditions at the singular points: √ 1 1 am BC0 : am + f (θ ) − θ f (θ ) = 0, lim θ θ→0 2 θ and BCπ :
lim (θ − π )
am
θ→π
√ 1 1 am + f (θ − π ) − θ − π f (θ − π ) = 0. 2 θ −π
These are the natural generalizations of the classical Dirichlet boundary conditions (compare with [62] 8.4) and were first considered in [7]. In particular, it was proved in [7], Sect. 7, that the self adjoint extension defined by these conditions is the Friedrich extension. The eigenvalues equation associated with the operators Tam , can be more easily studied going back to the original Hilbert space. This equation was in fact already studied by Gromes [36], who found a complete solution. Generalizing the standard approach used for the standard sphere (see for example [39]), we can prove that in fact this solution provide a complete set of eigenvalues and eigenfunctions for the metric Laplacian, as stated in the following lemma. Lemma 3.1. The operator S 2 , has the complete system: 1/a
λn,m = (a|m| + n)(a|m| + n + 1), n ∈ N, m ∈ Z, µ
where all the eigenvalues with m = 0 are double with eigenfunctions (where the Pν are the associated Legendre functions) −am −am eiamφ Pam+n , e−iamφ Pam+n ,
while the eigenvalues n(n + 1) are simple with eigenfunctions the functions Pn . Next, we pass to the higher dimensions. The (negative) of the induced Laplace operator on the deformed sphere SkN +1 is S N +1 = −dθ20 − N k
cos θ0 1 dθ + N sin θ0 0 sin2 θ0 Sk
on L 2 (SkN +1 ). Projecting on the spectrum of S N , we obtain the differential equation −dθ20
k
λS N cos θ0 k −N dθ + u = λ S N +1 u. k sin θ0 0 sin2 θ0
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Following [52], we make the substitutions u(θ0 ) = sinb θ0 v(θ0 ), 1 (cos θ0 + 1), 2 where b = 21 1 − N + (N − 1)2 + 4λ S N . This gives the hypergeometric equation z=
k
[35] 9.151, z(1 − z)v + [γ − (α + β + 1)z]v − αβv = 0, with α=
1 2b + N ∓ N 2 + 4λ S N +1 , k 2
β=
1 2b + N ± N 2 + 4λ S N +1 , k 2
1 (2b + N + 1). 2 Boundary conditions give the equation 2n + 2b + N = ± N 2 + 4λ S N +1 , γ =
k
where n ∈ N, that, in turn, gives the recurrence relation 1 1 − N + (1 − N )2 + 4λ S N + 2λ S N . λ S N +1 = n 2 + 1 + (1 − N )2 + 4λ S N n + k k k k 2 We can prove that this recurrence relation is satisfied by the numbers
N − 1 2 (N − 1)2 λ S N = am + n 1 + · · · + n N −1 + , − k 2 4 where n i ∈ N, must be a positive integer. We have obtained b = am + n 1 + · · · + n N −1 , α = −n N , β = 2(am + n 1 + · · · + n N −1 ) + n N + N , N +1 , 2 and the family of solutions for the eigenvalues equation (up to a constant) γ = am + n 1 + · · · + n N −1 +
u n N (cos θ0 ) = sin
1−N 2
θ0 P
−am−n 1 −···−n N −1 − N 2−1
am+n 1 +···+n N −1 + N 2−1 +n N
(cos θ0 ).
Using standard argument, we can then prove the following result.
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Lemma 3.2. The operator S N +1 , has the complete system: 1/a
λm,n 1 ,...,n N = (a|m| + n 1 + · · · + n N )(a|m| + n 1 + · · · + n N + N ), n i ∈ N, m ∈ Z, where all the eigenvalues with m = 0 are double with eigenfunctions (up to normalization) eiamθ N
N −1
sin
1−N + j 2
(θ j )P
j=0
e−iamθ N
N −1
sin
1−N + j 2
j −am−n 1 −···−n N −1− j − N −1− 2
j am+n 1 +···+n N − j + N −1− 2
(θ j )P
(cos θ j ),
j −am−n 1 −···−n N −1− j − N −1− 2
j am+n 1 +···+n N − j + N −1− 2
j=0
(cos θ j ),
while the eigenvalues with m = 0 are simple with eigenfunctions N −1
sin
1−N + j 2
(θ j )P
j=0
j −n 1 −···−n N −1− j − N −1− 2
j n 1 +···+n N − j + N −1− 2
(cos θ j ).
4. Zeta Regularized Determinants In this section we study the zeta function associated to the Laplace operator on the deformed sphere SkN +1 . For this, we introduce two quite general classes of zeta functions and we compute the main zeta invariants of them. This allows us to define a general technique to obtain the zeta regularized determinant of the Laplace operator on SkN +1 as a function of the deformation parameter. We apply this technique to the lower cases, N = 1 and 2, giving explicit formulas. Our last result is the computation of the coefficients in the expansions of the zeta determinants in powers of the deformation parameter. By Lemma 3.2, the zeta function on SkN +1 is the function defined by the series ζ (s, S N +1 ) = 1/a
n∈N0N
[n(n + N )]−s + 2
∞
[(am + n)(am + n + N )]−s ,
m=1 n∈N N
when Re(s) > N +1, and by analytic continuation elsewhere. Here n is a positive integer vector n = (n 1 , . . . , n N ), and the notation N0N means N × · · · × N − {0, . . . , 0}. Multidimensional gamma and zeta functions, namely zeta functions where the general term is of the form (n T an + b T n + c)−s , where a is a real symmetric matrix of rank k ≥ 1, b a vector in Rk , c a real number and n an integer vector in Zk , were originally introduced by Barnes [3, 4] and Epstein [32, 33] as natural generalizations of the Euler gamma function. Whenever the sum is on the integers (i.e. n ∈ Zk ), there is a large symmetry that allows one to express the zeta function by a theta series. Multidimensional theta series have been deeply studied in the literature, and by a generalization of the Poisson summation formula (see for example [16] XI.2, 3) it is possible to compute the main zeta invariants for multiple series of this type (see [63, 47, 30, 31 , and 15] and references there in). The main problem in the present case is that the zeta functions are associated to series of Dirichlet type, namely the sums are over Nk0 . We lose then
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many symmetries and in particular a formula of Poisson type. Consequently, it is more difficult to find general results, and different techniques have been introduced to deal with the specific cases (see for example [12, 13, 18, 29, 54, 55] for simple series or series that can be reduced to simple series or [15, 46] for multiple linear series). Note in particular that the case of a double (k = 2) homogeneous quadratic series of Dirichlet type is much harder. The zeta functions of this type (with integer coefficients) appear when dealing with the zeta functions of a narrow ideal class for a real quadratic field as shown by Zagier in [66 and 67], where he also computes the values at non-positive integers (see also [51, 28, 14, 15], and in particular [58] for the derivative). Beside, we can overcome this difficulty in the case under study first by reducing the multi-dimensional zeta functions to a sum of 2 dimensional linear and quadratic zeta functions, and then studying the quadratic one by means of a general method introduced in [59] in order to deal with non homogeneous zeta functions. Note that, for particular values of the deformation parameter, the zeta function can be reduced to a sum of zeta functions of Barnes type [3], and this allows a direct computation of the main zeta invariants [25, 26]. This approach does not work for generic values of the deformation parameter, and therefore the more sophisticated technique introduced here is necessary. We present in the next subsection some generalizations of some results of [59] necessary in order to treat the present case, and we give in the following subsections some applications to the case of some general classes of abstract simple and double zeta functions. As explained hereafter, by means of these two classes of zeta functions, we can in principle calculate the zeta invariants for the deformed sphere in any dimensions. Eventually in the last subsections we apply the method to obtain the main zeta invariants for the zeta functions on the 2 and 3 dimensional deformed spheres. By the following lemma (see [60 or 59]), we can reduce ζ (s, S N +1 ) to a sum of k simple and double zeta functions. Lemma 4.1. Let f (z) be a regular function of z. Then
f (n) =
n∈N N +1
∞ n+N n=0
N
f (n),
n∈N0N +1
f (n) =
∞ n+N n=1
N
f (n).
Proposition 4.2. The zeta function associated with the Laplace operator on the N + 1 dimensional deformed sphere is (N ≥ 1) ζ (s, S N +1 ) = 1/a
∞ ∞ ∞ n+ N −1 n+ N −1 [n(n + N )]−s + 2 N −1 N −1 n=1
m=1 n=0
[(am + n)(am + n + N )]−s . −1 Since n+N N −1 = PN (n) is a polynomial of order N in n, and since given any polynomial PN (n) we have a polynomial Q N (n + x) for any given x, such that PN (n) = Q N (n + x) (and we can find explicitly the coefficients of Q as functions on those of P and x), it is sufficient to consider the two classes of zeta functions z(s; α, 2, x, p) =
∞ n=1
(n + x)α [(n + x)2 + p]−s ,
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and Z (s; α, a, x, p) =
∞ ∞
n α [(n + am + x)2 + p]−s .
m=1 n=1
This will be done in Subsects. 4.2 and 4.3, but first, the next subsection is dedicated to recall and generalize some results on sequences of spectral type and associated zeta functions introduced in [59], necessary in the following.
4.1. Sequences of spectral type and zeta invariants. In this subsection we will use some concepts and results developed in [59], that briefly we recall here. We refer to that work for further details and complete proofs. Let T = {λn }∞ n=1 be a sequence of positive numbers with unique accumulation point at infinite, finite exponent s0 and genus q. We associate to T , the heat function f (t, T ) = 1 +
∞
e−λn t ,
n=1
the logarithmic Fredholm determinant ∞ z 1+ e log F(z, T ) = log λn
q j=1
(−1) j z j j j λn
,
n=1
and the zeta function ζ (s, T ) =
∞
λ−s n .
n=1
The sequence T is called of spectral type if there exists an asymptotic expansion of the associated heat function for small t in powers of t and powers of t times positive integer powers of log t. In particular it is said to be a simply regular sequence of spectral type if the associated zeta function has at most simple poles (see [59] pp. 4 and 9). Formulas to deal with the zeta invariants for sequences of spectral type are given in [59]. In particular, there are considered non-homogeneous sequences as well. We generalize the concept of non-homogeneous sequence here, by considering, for any given sequence ∞ of spectral type T0 = {λn }∞ n=1 , the shifted sequence Td = {λn + d}n=1 , where d is a parameter, subject to the unique condition that Re(λn + d) is always positive. We can prove the following results for a shifted sequence (see [59] Proposition 2.9 and Corollary 2.10 for details). Lemma 4.3. Let T0 = {λn }∞ n=1 be a sequence of finite exponent s0 and genus q, then the associated shifted sequence Td = {λn + d}∞ n=1 , with d such that Re(λn + d) > 0 for all n, is a sequence of finite exponent s0 and genus q. Moreover, T0 is of spectral type if and only if Td is of spectral type. If T0 is simply regular, so is Td .
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Proposition 4.4. Let T0 = {λn }∞ n=1 be a simply regular sequence of spectral type with finite exponent s0 and genus q, and Td = {λn + d}∞ n=1 , with d such that Re(λn + d) > 0 for all n, an associated shifted sequence. Then, ζ (0, Td ) = ζ (0, T0 ) +
q (−1) j
j
j=1
Res1 (ζ (s, T0 ), s = j)d j ,
ζ (0, Td ) = ζ (0, T0 ) − log F(d, T0 ) + +
q (−1) j Res0 (ζ (s, T0 ), s = j) + (γ + ψ( j))Res1 (ζ (s, T0 ), s = j) d j . j j=1
Proposition 4.5. Let T0 = {λn }∞ n=1 be a simply regular sequence of spectral type with finite exponent s0 and genus q. Let L 0 = {λ2n }∞ n=1 , and d such that Re(λn + d) > 0 for all n. Then, 1 ζ (0, L d 2 ) = ζ (0, Tid ) + ζ (0, T−id , 2 p
ζ (0, L d 2 ) = ζ (0, Tid ) + ζ (0, T−id )−
j 2 (−1) j j=1
j
k=1
1 Res1 (ζ (s, T0 ), s = 2 j)d 2 j . 2k − 1
Remark 4.6. Note that the numbers λn in the sequence need not to be different, i.e. the cases with multiplicity are covered by Propositions 4.4 and 4.5. In particular, assume the sequence is T0 = {λn }∞ n=1 , each λn having multiplicity ρn (we cover the case of a general abstract multiplicity, given by any positive real number). Then, the unique difficulty can be in defining the exponent of convergence of the sequence. But actually for our purpose it is sufficient to know the genus, and this can be obtained whenever we know the asymptotic of λn and ρn for large n. In fact, if λn ∼ n b and ρn ∼ n a , then the general term of the associated zeta function behaves as n a−bs , and therefore the genus is q = a+1 (the integer part). b Some more remarks on these results are in order. First, note that the approach of considering some general class of abstract sequences and of studying the analytic properties of the associated spectral functions has been developed by various authors, and in particular instances of Proposition 4.4 can be found in the literature. The original idea is probably due to Voros [61], while a good reference for a rigorous and very general setting is the work of Jorgenson and Lang [41]. However, for our purpose here, the simpler setting of [59] is more convenient. Second, observe that Proposition 4.5 was originally proved by Choi and Quine in [18], and also obtained in [25], Eq. (25). In particular, the reader can see the proof given in [59], as the more rapid route to this result suggested in [25]. 4.2. A class of simple zeta functions. We consider the following class of simple zeta functions (compare with [57]) z(s; α, β, x, p) =
∞ n=1
(n + x)α [(n + x)β + p]−s ,
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for Re(s) > 1+α β , where α and β are real positive numbers, and x and p are real numbers subject to the conditions that n + x > 0 and (n + x)β + p > 0 for all n. Note that different equivalent techniques could be applied to deal with this case; namely one could use the Plana theorem as in [54], a regularized product like in [18], a complex integral representation as in [55], or heat-kernel techniques [30, 31]. Proposition 4.7. The function z(s; α, β, x, p) has a regular analytic continuation in the whole complex s-plane up to simple poles at s = 1+α β − j, j = 0, 1, 2, . . . , whenever these values are not 0, −1, −2, . . . . The origin is a regular point and if positive integer
1+α β
is not a
z(0; α, β, x, p) = ζ H (−α, x + 1), and
z (0; α, β, x, p) = βζ H (−α, x + 1) +
α+1 β
(−1) j ζ H (β j − α, x + 1) p j + j j=1
∞ 1+ − log n=1
while if
1+α β
p (n + x)β
(n+x)α e
(n+x)α
α+1 β j=1
pj (−1) j j (n+x)β j
,
is a positive integer α+1
(−1) β α+1 p β , z(0; α, β, x, p) = ζ H (−α, x + 1) + α+1 and z (0; α, β, x, p) = βζ H (−α, x + 1) +
α+1 β −1
(−1) j ζ H (β j − α, x + 1) p j + j j=1
+
(−1)
α+1 β
α+1 β
α+1 1+α 1 −(x + 1) + γ + p β + β β
∞ 1+ − log n=1
p (n + x)β
(n+x)α e
(n+x)α
α+1 β j=1
pj (−1) j j (n+x)β j
.
Proof. The result follows applying Proposition 4.4. First, note that the unshifted β α sequences is T0 = {(n + x) }, with multiplicity (n + x) . By the Remark 4.6, the α+1 sequence has genus q = β . The associated zeta function is z(s; α, β, x, 0) = ζ H (βs − α, x + 1), and this clearly shows that T0 is a simply regular sequence of spectral type, and so is T p by Lemma 4.3. The unique pole is at s = 1+α β and
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1+α 1 = , Res1 z(0; α, β, x, 0), s = β β
1+α Res0 z(0; α, β, x, 0), s = = −ψ(x + 1). β The associated Fredholm determinant is (n+x)α ∞ q j zj z (n+x)α j=1 (−1) j (n+x)β j . F(z, T0 ) = 1+ e (n + x)β n=1
Next, using the expression given in the proof of Proposition 4.4 we have ∞ −s ζ H (β(s + j) − α) p j , z(s; α, β, x, p) = j j=0
thus we have poles when β(s + j) − α = 1, i.e. s = 1+α β − j, j = 0, 1, 2, . . . , when ever these values are not 0, −1, −2, . . . , and the residua are easily computed. To obtain the value at s = 0, it is useful to distinguish two cases (see [57]). In fact, from the above expression, when s = 0 the unique term that is singular is the one with α+1 β j − α = 1, i.e. j = α+1 β , that is necessarily a positive integer since α ≥ 0. Now, if β = α+1 is not a positive integer, then we have no integer poles, q = α+1 β β , and hence z(0; α, β, x, p) = z(0; α, β, x, 0), and since Res0 (z(s; α, β, x, 0), s = j) = z( j; α, β, x, 0) = ζ H (β j − α, x + 1), z (0; α, β, x, p) = z (0; α, β, x, 0)
α+1 β
(−1) j Res0 (z(s; α, β, x, 0), s = j) p j − log F( p, T0 ). j j=1 α+1 = α+1 If α+1 β is a positive integer, we have a pole, q = β β , and we need to take in account also the residuum. As we have seen, since the Hurwitz zeta function has only one pole at s = 1 with residuum 1, all the terms up to the ones with j = 0 and the one with j = α+1 β have vanishing residuum, and we obtain +
z(0; α, β, x, p) = z(0; α, β, x, 0) +
(−1)
α+1 β
α+1 β
α+1 1+α p β , Res1 z(s; α, β, x, 0), s = β
and α+1 β −1
(−1) j Res0 (z(s; α, β, x, 0), s = j) p j + j j=1 α+1
(−1) β 1+α + α+1 + Res0 z(s; α, β, x, 0), s =
β β
α+1 1+α 1+α Res1 z(s; α, β, x, 0), s = p β − log F( p, T0 ), + γ + β β that gives the formula stated in the thesis. z (0; α, β, x, p) = z (0; α, β, x, 0) +
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4.3. A class of double zeta functions. Consider the following class of double zeta functions ∞
Z (s; α, a, x, p) =
n α [(am + n + x)2 + p]−s ,
m,n=1
for Re(s) > 1 + α, and where x and p are real constants subject to the conditions that am + n + x > 0 and (am + n + x)2 + p > 0 for all n and m, and α is a non-negative integer (the case where α is any real number can be treated by similar methods, but is much more complicated, see [56]). Remark 4.8. In the more general case ∞
Z (s; α, β, a, x, p) =
n α [(am + n + x)β + p]−s ,
m,n=1
for Re(s) > 2(1+α) β , and where x and p are real constants subject to the conditions that am + n + x > 0 and (am + n + x)β + p > 0 for all n and m, we would have genus by Remark 4.6 since the leading term behaves like n α n −βs/2 , but we would q = 2(1+α) β not be able to prove that these are regular sequences of spectral type as in the following proof of Lemma 4.9. The sequences appearing in these zeta functions are: S0 = {λm,n = (am + n + ∞ x)2 }∞ m,n=1 and the associated shifted sequence S p = {λm,n + p}m,n=1 , both with multiα plicity n . These are sequences with finite exponent and genus q = [1 + α] by Remark 4.6. We first show that S p is a simply regular sequence of spectral type. Lemma 4.9. The sequence S p = {(am + n + x)2 + p}∞ m,n=1 is a simply regular sequence of spectral type. Proof. By Lemma 4.3, we need to show that there exists an expansion of the desired type for the heat function f (t, S0 ) = 1 +
∞
n α e−(am+n+x) t . 2
m,n=1 α Consider the sequence L = {am +n +x}∞ m,n=1 , with multiplicity n , of finite exponent ab
and genus 2 (since m a + n b ≤ (mn) a+b ). The associated heat function is f (t, L) = 1 +
∞
n α e−(am+bn+c)t ,
m,n=1
and the associated Fredholm determinant is F(z, L) =
∞ m,n=1
1+
n α z e am + bn + c
2 j=1
(−1) j nα z j j (am+bn+c) j
.
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Since f (t, L) = 1 +
∞
n α e−(am+bn+c)t = 1 + e−ct
m,n=1
∞
e−amt
m=1
∞
n α e−bnt ,
n=1
and we have an expansion of each factor in powers of t (see [57] Sect. 3.1 for the last sum), it is clear that we have an expansion of the form f (t, L) =
∞
e j tδj .
j=0
By Lemma 2.5 of [57], L is simply regular, and hence the unique logarithmic terms in the expansion of F(z, L) are of the form z k log z, with integer k ≤ 2. Now, consider the product ∞ 1+ F(i z, L)F(−i z, L) = m,n=1
×e
2 j=1
iz am + bn + c
n α (i z) j (−1) j j (am+bn+c) j
2
e
j=1
n α 1−
iz am + bn + c
(−1) j n α (−i z) j j (am+bn+c) j
n α ×
.
Since i j + (−i) j = 0 for odd j, and −2 when j = 2, this gives ∞ 1+ F(i z, L)F(−i z, L) = m,n=1
z2 (am + bn + c)2
n α
1
nα z2
e 2 (am+bn+c)2 = F(z 2 , S0 ),
and we obtain a decomposition of the Fredholm determinant associated to the sequence S0 . This means that log F(z, S0 ) has an expansion with unique logarithmic terms of the form z k log z, with integer k ≤ 1, and therefore S0 is a simply regular sequence of spectral type by Lemma 2.5 of [59]. Lemma 4.9 shows that the sequence appearing in the definition of the function Z (s; α, a, x, p) = ζ (s, S p ) are such that we can apply Proposition 4.4 in order to obtain all the desired zeta invariants. For we need explicit knowledge of the zeta invariants of the sequence S0 . This is in the next lemma. Lemma 4.10. The function χ (s; α, a, x) defined for real a and x such that am+n+x > 0, for all m, n ∈ N0 , and α a non-negative integer, by the sum χ (s; α, a, x) =
∞
n α (am + n + x)−s ,
m,n=1
when Re(s) > 2(α + 1), can be continued analytically to the whole complex plane up to a finite set of simple poles at s = 1, 2, . . . , α + 2, by means of the following formula:
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χ (s; α, a, x) = +
α
α(α − 1) . . . (α − j + 1) a j+1−s ζ H (s − j − 1, (x + 1)/a + 1) + (s − 1)(s − 2) . . . (s − j − 1)
j=1
+ia
−s
∞ 0
1 −s a 1−s a ζ H (s, (x + 1)/a + 1) + ζ H (s − 1, (x + 1)/a + 1) + 2 s−1
(1 + i y)α ζ H (s, (x + 1 + i y)/a + 1) − (1 − i y)α ζ H (s, (x + 1 − i y)/a + 1) dy. e2π y − 1
In particular, this shows that the point s = 0 is a regular point. Proof. We apply the Plana theorem as in [54]. Since the general term behaves as n α n −s/2 , we assume Re(s) > 2(α + 1), ∞ ∞ 1 χ (s; α, a, x) = (am + x + 1)−s + t α (am + t + x)−s dt + 2 ∞
m=1
m=1 1
∞ (1 + i y)α (am + x + 1 + i y)−s − (1 − i y)α (am + x + 1 − i y)−s +i dy. e2π y − 1 ∞
m=1 0
Recall that α is a non-negative integer, then we can integrate recursively the middle term obtaining, for α > 0, ∞
α
−s
t (am + t + x) dt =
α j=0
1
α(α − 1) . . . (α − j + 1) (am + x + 1) j+1−s ; (s − 1)(s − 2) . . . (s − j − 1)
this gives χ (s; α, a, x) =
∞ a 1−s 1 −s a (m + (x + 1)/a)1−s + (m + (x + 1)/a)−s + 2 s−1 m=1
+
α j=1
∞ α(α − 1) . . . (α − j + 1) a j+1−s (m + (x + 1)/a) j+1−s + (s − 1)(s − 2) . . . (s − j − 1) m=1
∞ (1 + i y)α (m + (x + 1 + i y)/a)−s − (1 − i y)α (m + (x + 1 − i y)/a)−s dy, e2π y − 1 ∞
+ia
−s
m=1 0
and, due to uniform convergence of the integral, concludes the proof. Remark 4.11. We could deal with this kind of double zeta function by applying the classical integral formula of Hermite as in the case of the Riemann zeta function. This approach confirms the above results, but it would not give a tractable expression for the singular part.
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We can now obtain the zeta invariants of the zeta function Z (s; α, a, x, p) for all the acceptable values of the parameters. This allows us to compute the regularized determinant of the deformed sphere of any dimension, as pointed out at the beginning of this section. Besides, we will give explicit formulas and results for the low dimensional cases in the next subsections. 4.4. Zeta determinant on the deformed 2 sphere. By Proposition 4.2, the zeta function associated to the operator S 2 is the function defined by the series 1/a
ζ (s, S 2 ) = 1/a
∞ [n(n + 1)]−s + 2 n=1
∞
[(am + n)(am + n + 1)]−s ,
m=1,n=0
when Re(s) > 2, and by analytic continuation elsewhere. The aim of this section is to study this zeta function and in particular to obtain a formula for the values of ζ (0, S 2 ) and ζ (0, S 2 ). When a = 1, this reduces to the zeta function on the 1/a 1/a ∞ 2 −s [18, 54, 55]. The zeta function 2-sphere: ζ (s, S 2 ) = n=1 (2n + 1)(n + n) 1/a=1 ζ (s, SS 2 ) decomposes as 1/a
ζ (s, S 2 ) = z(s; 0, 2, 1/2, −1/4) + 2Z (s; 0, a, −1/2, −1/4), 1/a
and we can easily check that the values of the parameters satisfy the condition of definition of these functions. We provide two equivalent formulas for the zeta determinant on the deformed 2-sphere, Theorems 4.15 and 4.16. The first is obtained applying Proposition 4.4, the second applying Proposition 4.5. Computations are given in the proofs of the following lemmas. The first lemma follows by a direct application of Proposition 4.7 and properties of special functions. Lemma 4.12. z(0; 0, 2, 1/2, −1/4) = −1, z (0; 0, 2, 1/2, −1/4) = − log 2π.
Lemma 4.13. 1 a + , 12 12a
1 1 Z (0; 0, a, −1/2, −1/4) = − a log a+ 6 2a Z (0; 0, a, −1/2, −1/4) =
+ζ H (0, 1/(2a) + 1) − 2aζ H (−1, 1/(2a) + 1) − 2aζ H (−1, 1/(2a) + 1)+ ∞ +2i 0
ζ H (0, (1/2 + i y)/a + 1) − ζ H (0, (1/2 − i y)/a + 1) dy+ e2π y − 1
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+
1 1 ζ H (2, 1/(2a) + 1) − ((1/(2a) + 1) + 1 + log a) + 8a 2 4a
i + 2 4a
∞ 0
ζ H (2, (1/2 + i y)/a + 1) − ζ H (2, (1/2 − i y)/a + 1) dy+ e2π y − 1 ∞ + 1− m,n=1
1 4(am + n − 1/2)2
1
e 4(am+n−1/2)2 .
Proof. The function Z (s; 0, a, −1/2, −1/4) is the zeta function associated with the sequence S−1/4 = {(am +n −1/2)2 −1/4}, all terms with multiplicity 1. By Lemma 4.9, S−1/4 is a simply regular sequence of spectral type. In order to apply Proposition 4.4, we need to study the unshifted sequence S0 = {(am + n − 1/2)2 }. This sequence has genus 1, the associate Fredholm determinant is ∞ z z − F(z, S0 ) = 1+ e (am+n−1/2)2 , 2 (am + n − 1/2) m,n=1
and the associated zeta function is ζ (s, S0 ) = χ (2s; 0, a, −1/2). By Proposition 4.4 and since the genus is 1, we have that 1 Z (0; 0, a, −1/2, −1/4) = χ (0; 0, a, −1/2) + Res1 (χ (2s; 0, a, −1/2), s = 1), 4 and that Z (0; 0, a, −1/2, −1/4) = χ (2s; 0, a, −1/2)|s=0 1 + Res0 (χ (2s; 0, a, −1/2), s = 1) − log F(−1/4, S0 ), 4 and hence we need to compute the values at s = 0 of ζ (s, S0 ) = χ (2s; 0, 1, −1/2), and the residua at s = 1. For, we use the formula provided in Lemma 4.10, namely χ (2s; 0, a, −1/2) =
+ia
−2s
∞ 0
1 −2s 1 a ζ H (2s, 1/(2a) + 1) + a 1−2s ζ H (2s −1, 1/(2a) + 1)+ 2 2s − 1
ζ H (2s, (1/2 + i y)/a + 1) − ζ H (2s, (1/2 − i y)/a + 1) dy. e2π y − 1
We obtain χ (0; 0, a, −1/2) = ∞ +i 0
1 ζ H (0, 1/(2a) + 1) − aζ H (−1, 1/(2a) + 1)+ 2
a 1 ζ H (0, (1/2 + i y)/a + 1) − ζ H (0, (1/2 − i y)/a + 1) dy = − , e2π y − 1 12 24a
(1)
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where we have used [35] 9.531 and 9.611.1. Next, we use Eq. (1) to compute the residua at the pole s = 1. The unique singular term is the middle one, so we expand the different factors in it near s = 1, using [35] 9.533.2, a −2s 1 1 ζ H (2s − 1, 1 + 1/(2a)) = 2s − 1 2a s − 1 1 − ((1 + 1/(2a)) + 1 + log a) + O(s − 1). a This gives a
Res1 (χ (2s; 0, a, −1/2), s = 1) =
1 , 2a
and Res0 (χ (2s; 0, a, −1/2), s = 1) = i 1 − (1/(2a) + 1) + 2 a a
∞ 0
1 1 ζ H (2, 1/(2a) + 1) − (1 + log a) + 2 2a a
ζ H (2, (1/2 + i y)/a + 1) − ζ H (2, (1/2 − i y)/a + 1) dy. e2π y − 1
Last, we compute the derivative: χ (0; 0, a, −1/2) = −2χ (0; 0, a, −1/2) log a+ +ζ H (0, 1/(2a) + 1) − 2aζ H (−1, 1/(2a) + 1) − 2aζ H (−1, 1/(2a) + 1)+ ∞ +2i 0
=
1 6
ζ H (0, (1/2 + i y)/a + 1) − ζ H (0, (1/2 − i y)/a + 1) dy = e2π y − 1
1 1 a 1 1 −a log a + (1/(2a)+1)− log 2π + + + −2aζ H (−1, 1/(2a) + 1)+ 2a 2 6 4a 2 ∞ +2i
log 0
dy ((1/2 + i y)/a + 1) dy. 2π ((1/2 − i y)/a + 1) e y − 1
Collecting, we obtain the thesis. Lemma 4.14.
1 a Z (0; 0, a, −1/2, −1/4) = − + 6 6a
+
log a −
1 1 1 log 2π + log (1 + )+ 2 2 a
a 1 3 1 + + − aζ R (−1) − aζ H (−1, 1 + )+ 6 2 4a a ∞
+i
log 0
(1 + i ay )(1 + (1 − i ay )(1 +
1 a 1 a
+ i ay )
dy . −1
− i ay ) e2π y
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Proof. In the language of Proposition 4.5, we have L 0 = {(am + n + x)2 },
L b2 = {(am + n + x)2 + b2 },
S0 = {am + n + x},
Sib = {am + n + x + ib},
where the genus of S0 is p = 2. Therefore, by Proposition 4.5, Z (0; 0, a, x, b2 ) = ζ (0, L b2 ) = ζ (0, Sib ) + ζ (0, S−ib )) − Res1 (ζ (s, S0 ), s = 2)b2 . Also, we have that ∞
ζ (s, Sib ) =
(am + n + x + ib)−s = χ (s; 0, a, x + ib),
m,n=1
and therefore, we need information on χ . Use Lemma 4.10. We have, with z = x ± ib, χ (s; 0, a, z) =
+ia
−s
a 1−s z+1 z+1 1 −s a ζ H (s, + 1) + ζ (s − 1, + 1)+ 2 a s−1 a
∞
ζ H (s,
z+1+i y a
0
y + 1) − ζ H (s, z+1−i + 1) a dy. 2π y e −1
This gives Res1 (χ (s; 0, a, z), s = 2) =
χ (0; 0, a, z) =
1 , a
1 a 1 z2 z z + + + + + , 4 12 12a 2a 2a 2
1 z+1 z+1 χ (0; 0, a, z) = −χ (0; 0, a, z) log a + ζ H (0, + 1) − aζ H (−1, + 1)+ 2 a a
−aζ H (−1,
z+1 + 1) + i a
∞ log 0
(1 + (1 +
z+i y+1 dy a ) . z−i y+1 e2π y − 1 ) a
Using the decomposition at the beginning of this subsection and the results in Lemmas 4.12, 4.13 and 4.14 respectively, we can prove the following theorems. Theorem 4.15. 1 a + , 6 6a
1 1 a a+ log a + 2 log (1/(2a) + 1)+ ζ (0, S 2 ) = −2 log 2π + 1 + − 1/a 3 3 a ζ (0, S 2 ) = −1 + 1/a
Spectral Analysis and Zeta Determinant on the Deformed Spheres
+
697
1 1 ζ H (2, 1/(2a) + 1) − (1/(2a) + 1) − 4aζ H (−1, 1/(2a) + 1)+ 2 4a 2a ∞
dy ((1/2 + i y)/a + 1) + ((1/2 − i y)/a + 1) e2π y − 1
log
+4i 0
i + 2 2a
∞ 0
ζ H (2, (1/2 + i y)/a + 1) − ζ H (2, (1/2 − i y)/a + 1) dy+ e2π y − 1
∞ 1− −2 log m,n=1
1 4(am + n − 1/2)2
1
e 4(am+n−1/2)2 .
Theorem 4.16. ζ (0, S 2 ) = −
1/a
1 a + 3 3a
−2aζ R (−1) − 2aζ H (−1, 1 +
log a − 2 log 2π +
1 ) + 2i a
∞ log 0
a 3 1 +1+ + log (1 + )+ 3 2a a
(1 + i ay )(1 + (1 − i ay )(1 +
1 a 1 a
+ i ay )
dy . −1
− i ay ) e2π y
Observe that, although the formula given in Theorem 4.16 looks nicer, it is in fact less useful than the one given in Theorem 4.15, since convergence of the integral is much lower than convergence of the infinite product. Note also that the analytic formulas obtained in the previous theorems, provide a rigorous answer to the problem studied in [27], where an attempt to obtain such formulas was performed. In particular, we can compare the graphs given in [27] Sect. XI (where observe the opposite sign), with the following one, where ζ (0, S 2 ) is plotted using the formula given in Theorem 4.15, 1/a
and the relation with the lune angle ω is a =
π ω.
0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 –1
2
4
a
6
8
10
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4.5. The zeta determinant on the deformed 3 sphere. On the deformed 3-sphere we have N = 2 and ζ (s, S 3 ) = 1/a
∞ ∞ ∞ (n + 1)[n(n + 2)]−s + 2 (n + 1)[(am + n)(am + n + 2)]−s . n=1
m=1 n=0
We can check that this reduces to the usual zeta function on the 3-sphere ζ (s, S 3 ) = 1 ∞ 2 −s [54, 18], and we can decompose it as follows n=1 (n + 1) [n(n + 2)] ζ (s, S 3 ) = z(s; 1, 2, 1, −1) + 2Z (s; 1, a, 0, −1). 1/a
As in the previous subsection, we apply Propositions 4.4 and 4.5 and properties of special functions to prove the following lemmas. Observe that, in this case, an application of Proposition 4.5 gives a simpler formula for z (0; 1, 2, 1, −1), we thank the referee for pointing out this fact. Lemma 4.17. z(0; 1, 2, 1, −1) = −1, (−1, 2) − log z (0; 1, 2, 1, −1) = γ − 1 + 2ζ H
∞ 1 n 1 (−1, 2) + log 2 − 1. 1− 2 e n = 2ζ H n
n=2
Remark 4.18. The above result allows to obtain the following interesting formulas for the Barnes G-function G(z) and the double sine function S(z) (see [3, 53 or 59] for the definition of the G-function, and [44 or 59] for the multiple sine function): lim
z→1
lim
z→1
4 G(1 − z) = , 1−z e
π S(π(1 − z)) = . 1−z e
The proofs of the next lemmas are the same as for Lemmas 4.13 and 4.14. Besides the increasing difficulty of the calculation and the fact that now the multiplicity is not trivial (α = 1), the main difference is that a new singular term appears in the unshifted zeta function, namely applying Lemma 4.10, we obtain the expression χ (2s; 1, a, 0) =
+
+ia
−2s
∞ 0
a −2s a 1−2s ζ H (2s, 1/a + 1) + ζ H (2s − 1, 1/a + 1)+ 2 2s − 1
a 2−2s ζ H (2s − 2, 1/a + 1)+ (2s − 1)(2s − 2)
(1 + i y)ζ H (2s, (1 + i y)/a + 1) − (1 − i y)ζ H (2s, (1 − i y)/a + 1) dy e2π y − 1
instead of formula (1).
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Lemma 4.19. Z (0; 1, a, 0, −1) = − Z (0; 1, a, 0, −1) =
+
5 , 24
3 a 1 5 11 − + + log a − log 2π + 2 log (1/a + 1)+ 4 12 2a 12 12
1 1 ζ H (2, 1/a + 1) − (1 + 1/a) − 2aζ H (−1, 1/a + 1) + a 2 ζ H (−2, 1/a + 1)+ 2a 2 a ∞
+2i 0
dy ((1 + i y)/a + 1) −2 log ((1 − i y)/a + 1) e2π y − 1 ∞
i + 2 a
0
−
1 a2
y log |((1 + i y)/a + 1)|2 0
dy + e2π y − 1
ζ H (2, (1 + i y)/a + 1) − ζ H (2, (1 − i y)/a + 1) dy+ e2π y − 1
∞ y 0
∞
ζ H (2, (1 + i y)/a + 1) + ζ H (2, (1 − i y)/a + 1) dy+ e2π y − 1
− log
∞ 1− m,n=1
1 (am + n)2
n
n
e (am+n)2 .
Lemma 4.20. Z (0; 1, a, 0, −1) =
a 5 1 2 5 log a − 1 − − log 2π + log ( + 1)+ 12 12 12 2 a
a2 2 2 ζ R (−2) + ζ H (−2, + 1) + −a ζ R (−1) + ζ H (−1, + 1) + a 2 a ∞ +i
log 0
∞ −
y log 0
(1 + i ay )(1 + (1 − i ay )(1 +
2+i y dy a ) + 2−i y e2π y − 1 a )
2 − iy dy πy 2 + iy )(1 + ) 2π y . (1 + a a e −1 ash πay
Using the decomposition at the beginning of this subsection and the results in Lemmas 4.17, 4.19 and 4.20 we can prove the following theorems.
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Theorem 4.21. ζ (0, S 3 ) = −1, 1/a
ζ (0, S 3 ) = γ − 1 + 2ζ H (−1, 2) − log 1/a
∞ 1 n 1 1 − 2 en + n
n=2
11 3 a 1 5 log 2π + 4 log (1/a + 1)+ + − + + log a − 2 6 a 6 6 +
1 2 ζ H (2, 1/a + 1) − (1 + 1/a) − 4aζ H (−1, 1/a + 1) + 2a 2 ζ H (−2, 1/a + 1)+ a2 a ∞ +4i 0
2i + 2 a
(1 + i y) log ((1 + i y)/a + 1) − (1 − i y) log ((1 − i y)/a + 1) dy+ e2π y − 1
∞ 0
(1 + i y)ζ H (2, (1 + i y)/a + 1) − (1 − i y)ζ H (2, (1 − i y)/a + 1) dy+ e2π y − 1 ∞ −2 log 1− m,n=1
1 (am + n)2
n
n
e (am+n)2 .
Theorem 4.22. a 5 2 5 log a − 2 − − log 2π + log ( + 1)+ 6 6 6 a
2 2 −2a ζ R (−1) + ζ H (−1, + 1) + a 2 ζ R (−2) + ζ H (−2, + 1) + a a
ζ (0, S 3 ) = log 2 + 2ζ R (−1) − 1 + 1/a
∞ +2i 0
(1+i ay )(1+ 2+ia y )
dy −2 log y 2−i y e2π y −1 (1−i )(1+ ) a
a
∞ y log 0
y π y(1+ 2+ia y )(1+ 2−i a )
ash πay
dy . e2π y −1
4.6. Expansions. In this subsection we give explicit formulas and numerical values of the first coefficients appearing in the expansions of the determinants of the Laplace operator on the 2 and 3 dimensional deformed sphere SkN for small deformations of the parameter k = 1 − δ, with small positive δ. We first state a lemma that allows to deal with the expansion of the values of the zeta function, and thus justify the formal series expansion of all the functions appearing in Theorems 4.15 and 4.21 up to the infinite products, but the last can be treated directly. The proof of Lemma 4.23 follows by the same argument as the one used in the proof of Proposition 4.4. Lemma 4.23. Let x, q and δ be real with 0 ≤ δ ≤ 1, then for all Re(s) > −2 we have the expansion ζ H (s, 1 + x + qδ)) = ζ H (s, 1 + x) − sζ H (s + 1, 1 + x)qδ +
s(s + 1) ζ H (s + 2, 1 + x)q 2 δ 2 + O(δ 3 ), 2
Spectral Analysis and Zeta Determinant on the Deformed Spheres
701
and ζ H (s, 1 + x + qδ)) = ζ H (s, 1 + x) − ζ H (s + 1, 1 + x) + sζ H (s + 1, 1 + x) qδ+
1 s(s + 1) + s+ ζ H (s + 2, 1 + x) + ζ H (s + 2, 1 + x) q 2 δ 2 + O(δ 3 ), 2 2 where note that the coefficients of the second and third term in the second formula are defined as limits. Proposition 4.24. For a = 1 + δ + O(δ 2 ), ζ (0, S 2 ) = ζ (0, S 2 ) + Z 2 δ + O(δ 2 ), 1−δ
1
where ζ (0, S 2 ) = 4ζ H (−1) − 1
∞
2 π γ i = − log 2π − + + − 4ζ H (−1, 1/2) + 3 8 2 2
0
∞ +4i
1 = 2
(3/2 + i y) − (3/2 − i y) dy+ e2π y − 1
∞ 1 1 dy (3/2 + i y) 4(m+n+1/2)2 1 − e − 2 log log (3/2 − i y) e2π y − 1 4(m + n + 1/2)2 m,n=1
0
= −1.161684575, π2 7 1 γ + ζ R (3) − 4ζ H (−1, 1/2) + 2π Z2 = − + − 3 2 8 4
∞ 0
∞ +4 0
i − 4
1 (1/2 + i y) + (1/2 − i y) dy + y e2π y − 1 2
∞ 0
∞ y 0
tanh π y dy+ e2π y − 1
(3/2 + i y) + (3/2 − i y) dy+ e2π y − 1
∞ ∞ (3/2 + i y) − (3/2 − i y) 1 m dy − 4 = e2π y − 1 4j (m + n + 1/2)2 j+1 j=2
m,n=1
= 0.7116523492. Corollary 4.25. det S 2
1−δ
= det S 2 − Z 2 det S 2 δ + O(δ 2 ) = 3.195311305 − 2.273950797δ + O(δ 2 ). 1
1
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Proposition 4.26. For a = 1 + δ + O(δ 2 ), ζ (0, S 3 ) = ζ (0, S 3 ) + Z 3 δ + O(δ 2 ), 1−δ
1
where ζ (0, S 3 ) = 2ζ R (−2) + 2ζ R (0) + log 2 = 1
∞ 1 n 1 π2 11 5 1 − 2 en + log(2π ) + + 2ζ R (−2) − log = 3γ − − 2ζ R (−1) − 3 6 6 n n=2
∞ log
+4i ∞ +2i 0
0
dy (2 + i y) −4 (2 − i y) e2π y − 1
∞ y log |(2 + i y)|2 0
ζ H (2, 2 + i y) − ζ H (2, 2 − i y) dy − 2 e2π y − 1 ∞ = −2 log 1− m,n=1
1 (m + n)2
∞ y 0
n
dy + e2π y − 1
ζ H (2, 2 + i y) + ζ H (2, 2 − i y) dy+ e2π y − 1 n
e (m+n)2 = −1.205626800,
1 Z 3 = − + 2γ + 2ζ R (3) − 8ζ R (−1) − 2 log(2π ) + 4ζ R (−2)+ 2 ∞ (1 + i y)2 (2 + i y) − (1 − i y)2 (2 − i y) −4i dy+ e2π y − 1 0
∞
−4i ∞ −2i
0
(1 + i y) (2 + i y) − (1 − i y) (2 − i y) dy+ e2π y − 1
(1+i y)2 (2 + i y) − (2 − i y)2 (2−i y) e2π y − 1
0
dy +
3 π2 2 − = 0.6666666661 = . 2 9 3
Corollary 4.27. det S 3
1−δ
= det S 3 − Z 3 det S 3 δ + O(δ 2 ) = 3.338845845 − 2.225897228δ + O(δ 2 ). 1
1
Acknowledgements. We would like to thank an anonymous referee for useful remarks and suggestions. One of the authors, M. S., thanks the Departments of Mathematics and Physics of their University of Trento, and the INFN for their nice hospitality. S. Z. thanks V. Moretti for discussions.
References 1. Apps, J.S., Dowker, J.S.: The C2 heat-kernel coefficient in the presence of boundary discontinuities. Class. Quant. Grav. 15, 1121–1139 (1998) 2. Atiyah, M., Bott, R., Patodi, V.K.: On the Heat Equation and the Index Theorem. Invent. Math. 19, 279– 330 (1973) 3. Barnes, E.W.: The theory of the multiple Gamma function. Trans. Cambridge Phil. Soc. 19, 374– 425 (1904) 4. Barnes, E.W.: The theory of the G function. Quart. J. Math. 31, 264–314 (1899)
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Commun. Math. Phys. 273, 705–754 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0255-x
Communications in
Mathematical Physics
Geometrical (2+1)-Gravity and the Chern-Simons Formulation: Grafting, Dehn Twists, Wilson Loop Observables and the Cosmological Constant C. Meusburger Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada. E-mail: [email protected] Received: 27 July 2006 / Accepted: 8 November 2006 Published online: 15 May 2007 – © Springer-Verlag 2007
Abstract: We relate the geometrical and the Chern-Simons description of (2+1)-dimensional gravity for spacetimes of topology R × Sg , where Sg is an oriented two-surface of genus g > 1, for Lorentzian signature and general cosmological constant and the Euclidean case with negative cosmological constant. We show how the variables parametrising the phase space in the Chern-Simons formalism are obtained from the geometrical description and how the geometrical construction of (2+1)-spacetimes via grafting along closed, simple geodesics gives rise to transformations on the phase space. We demonstrate that these transformations are generated via the Poisson bracket by one of the two canonical Wilson loop observables associated to the geodesic, while the other acts as the Hamiltonian for infinitesimal Dehn twists. For spacetimes with Lorentzian signature, we discuss the role of the cosmological constant as a deformation parameter in the geometrical and the Chern-Simons formulation of the theory. In particular, we show that the Lie algebras of the Chern-Simons gauge groups can be identified with the (2+1)-dimensional Lorentz algebra over a commutative ring, characterised by a formal parameter whose square is minus the cosmological constant. In this framework, the Wilson loop observables that generate grafting and Dehn twists are obtained as the real and the -component of a Wilson loop observable with values in the ring, and the grafting transformations can be viewed as infinitesimal Dehn twists with the parameter . 1. Introduction The quantisation of Einstein’s theory of gravity is often viewed as the problem of constructing a quantum theory of geometry. In particular, a physically meaningful quantum theory of gravity should allow one to recover spacetime geometry from the gauge theory-like formulations used in most quantisation approaches. While the quantisation of gravity in (3+1) dimensions is far from complete, the (2+1)-dimensional version of the theory has been used successfully as a testing ground for various quantisation formalisms [1, 2]. As in the (3+1)-dimensional case, most of these formalisms are based on gauge
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theoretical descriptions of the theory. To apply these results to concrete physics questions, it would therefore be necessary to recover their geometrical interpretation. Yet the relation between the phase space variables used in these approaches and spacetime geometry is not fully clarified even in the classical theory. The simplifications in (2+1)-dimensional gravity compared to the (3+1)-dimensional case are due to the absence of local gravitational degrees of freedom and the finitedimensionality of its phase space. In the geometrical formulation of the theory, this manifests itself in the fact that vacuum solutions of Einstein’s equations are flat or of constant curvature. They are therefore locally isometric to certain model spacetimes, into which any simply connected region of the spacetime can be embedded. The physical degrees of freedom are purely topological and encoded in transition functions, which take values in the isometry group of the model spacetime and relate the embedding of different spacetime regions. From a gauge theoretical perspective, the absence of local gravitational degrees of freedom in (2+1)-dimensional gravity results in its formulation as a Chern-Simons gauge theory with the isometry group of the associated model spacetime as the gauge group [3, 4]. The Einstein equations then take the form of a flatness condition on the gauge field, and their solutions can be locally trivialised, i. e. written as pure gauge. The physical degrees of freedom are then encoded in a set of elements of the gauge group which relate the trivialisations on different regions of the spacetime manifold. The advantage of the Chern-Simons formulation of (2+1)-dimensional gravity is that it allows one to apply gauge theoretical concepts and methods to achieve an explicit parametrisation of the phase space that serves as a starting point for quantisation. As gauge fields solving the equations of motions are flat, physical states can be characterised in terms of the holonomies along closed curves in the spacetime manifold. Conjugation invariant functions of such holonomies then define a complete set of gauge invariant Wilson loop observables, which were first investigated in the context of (2+1)-dimensional gravity in [5–7, 9, 8, 10, 11]. Moreover, by parametrising the phase space in terms of the holonomies along a set of generators of the fundamental group, one obtains an efficient description of the Poisson structure [12, 13]. These descriptions were used in [14] to investigate the classical phase space of theory and are the basis of Alekseev, Grosse and Schomerus combinatorial quantisation formalism [15, 16] and the related approaches in [17, 18]. The drawback of the Chern-Simons formulation is that it complicates the physical interpretation of the theory by obscuring the underlying spacetime geometry. Except for particularly simple spacetimes such as static spacetimes and the torus universe, it is in general difficult to reconstruct spacetime geometry from the gauge theoretical variables that parametrise the phase space. In a geometrical framework, the relation between holonomies and geometry was first investigated by Mess [19], who shows how the holonomies determine the geometry of the spacetime. More recent results on this problem were obtained by Benedetti and Guadagnini [20] and by Benedetti and Bonsante [21, 22], who focus on the construction of (2+1)-dimensional spacetimes via grafting and relate the resulting spacetimes for different values of the cosmological constant. However, despite these results, the relation between spacetime geometry and the description of the phase space of (2+1)-dimensional gravity in the Chern-Simons formalism is still not fully clarified. While the results in [19–22] establish a relation between holonomies and geometry in the geometrical formulation of the theory, they do not relate these variables to the quantities encoding the physical degrees of freedom in the Chern-Simons formalism. In particular, it is not clear how the embedding of spacetime regions into model spacetimes and the associated transition functions are related to
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the corresponding concepts in Chern-Simons theory, the trivialisation of the gauge field and the gauge group elements linking the trivialisations on different regions. Moreover, a full understanding of the relation between spacetime geometry and the Chern-Simons formulation should clarify the role of phase space and Poisson structure. This includes a geometrical interpretation of the phase space transformations generated by the Wilson loop observables as well as the question of how constructions that change the geometry of a spacetime such as grafting and Dehn twists manifest themselves on the phase space of the theory. These questions concerning the relation between geometrical and Chern-Simons formulation in (2+1)-dimensional gravity are the subject of the present paper, in which we consider vacuum spacetimes of topology R × Sg , where Sg is an orientable two-surface of general genus g > 1. Our results are valid for spacetimes of Lorentzian signature and with general cosmological constant and for the Euclidean case with negative cosmological constant. They can be summarised as follows. 1. Embedding and trivialisation. We relate the embedding of spacetime regions into model manifolds in the geometrical formulation and the trivialisation of the gauge field in the Chern-Simons formalism and derive explicit formulas linking the variables which encode the physical degrees of freedom in the two approaches. 2. Grafting transformations on phase space. We show how the geometrical construction of (2+1)-spacetimes by grafting along closed, simple geodesics gives rise to a transformation on the phase space in the Chern-Simons formulation and derive explicit expressions for the action of this transformation on the holonomies along a set of generators of the fundamental group. 3. The transformations generated by Wilson loop observables. We investigate the two basic Wilson loop observables associated to a closed, simple curve on Sg and to the two linearly independent Ad-invariant, symmetric bilinear forms on the Lie algebra of the gauge group. We derive explicit expressions for the phase space transformations these observables generate via the Poisson bracket and show that one of these observables acts as a Hamiltonian for the grafting transformations, while the other generates infinitesimal Dehn twists. 4. Relation between grafting and Dehn twists. We demonstrate that the phase space transformations representing grafting and Dehn twists are closely related for all values of the cosmological constant and that this relation is reflected in a general symmetry relation for the corresponding Wilson loop observables. We show that grafting can be viewed as a Dehn twist with a formal parameter whose square is identified with minus the cosmological constant. 5. The cosmological constant as a deformation parameter. We establish a unified description for spacetimes of Lorentzian signature in which the cosmological constant plays the role of a deformation parameter. In the geometrical description, its square root appears as a parameter relating the embedding into the different model spacetimes and the action of the associated isometry groups. In the Chern-Simons formulation, it plays the role of a deformation parameter in the gauge group and the associated Lie algebra. More precisely, we demonstrate that the Lie algebra of the gauge group can be viewed as the (2+1)-dimensional Lorentz algebra over a commutative ring with a multiplication law that depends on the cosmological constant. Results similar to 1 to 4 were obtained in an earlier paper [23] for the case of Lorentzian (2+1)-spacetimes with vanishing cosmological constant. Although the general approach in [23] is similar, the reasoning and many proofs in [23] make use of specific simplifications resulting from the properties of Minkowski space and the (2+1)-
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dimensional Poincaré group. The inclusion of these spacetimes in the present paper allows one to see how these results arise from a general pattern present for all values of the cosmological constant and to investigate the role of the cosmological constant as a deformation parameter. The paper is structured as follows: In Sect. 2 we introduce definitions and notations for the Lie groups and Lie algebras considered in this paper and summarise some facts from hyperbolic geometry used in the geometrical description of (2+1)-spacetimes. Section 3 gives an overview of the geometrical description of (2+1)-dimensional spacetimes of topology R × Sg for Lorentzian signature and general cosmological constant and for the Euclidean case with negative cosmological constant. We start by introducing the relevant model spacetimes which are (2+1)-dimensional Minkowski space, anti de Sitter space and de Sitter space, respectively, for Lorentzian signature and vanishing, negative and positive cosmological constant and the three-dimensional hyperbolic space for the Euclidean case with negative cosmological constant. We then review the description of (2+1)-spacetimes of topology R × Sg which are obtained as the quotients of regions in the model spacetimes by certain actions of a cocompact Fuchsian group. After summarising the description of static universes, we describe the construction of evolving universes via grafting along closed, simple geodesics following the presentation in [21, 22]. In Sect. 4 we review the formulation of (2+1)-dimensional gravity as a Hamiltonian Chern-Simons gauge theory, where the gauge group is the isometry group of the associated model spacetime, the (2+1)-dimensional Poincaré group P SU (1, 1) R2 ∼ = P S L(2, R) R3 for Lorentzian signature and vanishing cosmological constant, the group P SU (1, 1)× P SU (1, 1) ∼ = P S L(2, R)× P S L(2, R) for Lorentzian signature and negative cosmological constant and S L(2, C)/Z2 for Lorentzian signature and positive cosmological constant and for the Euclidean case. We discuss how the local trivialisation of the gauge field gives rise to a parametrisation of the phase space in terms of the holonomies along a set of generators of the fundamental group π1 (Sg ) and introduce Fock and Rosly’s description of the Poisson structure [12]. Section 5 relates the geometry of (2+1)-spacetimes to their description in the ChernSimons formalism. We discuss the relation between the variables encoding the physical degrees of freedom in the geometrical and in the Chern-Simons approach and show how the embedding into the model spacetimes is obtained from the trivialisation of the gauge field in the Chern-Simons formalism. In Sect. 6 we demonstrate how the construction of evolving (2+1)-spacetimes via grafting along closed, simple geodesics in [21, 22] is implemented in the Chern-Simons formalism and show that gives rise to a transformation on phase space, given explicitly by its action on the holonomies along a set of generators of the fundamental group π1 (Sg ). In Sect. 7, we relate this transformation to the Poisson structure and to the Wilson loop observables. We show that the phase space transformation obtained by grafting along a closed, simple geodesic η is generated via the Poisson bracket by one of the two basic Wilson loop observables associated to η, while the other observable acts as the Hamiltonian for Dehn twists. We discuss the properties of the grafting transformations and their relation to Dehn twists, which manifests itself in a general symmetry relation for the Poisson brackets of the associated observables. Section 8 investigates the role of the cosmological constant in spacetimes of Lorentzian signature. Using the results by Benedetti and Bonsante [21, 22], we show
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that its square root can be viewed as a deformation parameter in the geometrical description of both static and grafted (2+1)-spacetimes. For the Chern-Simons formulation, we establish a common framework relating the different gauge groups by identifying their Lie algebras with the (2+1)-dimensional Lorentz algebra over a commutative ring. The cosmological constant then appears in the ring’s multiplication law and can be implemented by introducing a formal parameter whose square is minus the cosmological constant. We show that the grafting transformations can be viewed as Dehn twists with this parameter . Section 9 contains a discussion of our results and conclusions. 2. Definitions and Notation 2.1. Lie groups and Lie algebras. Throughout the paper we employ Einstein’s summation convention. Indices are raised and lowered either with the three-dimensional Minkowski metric x · y = η L (x, y) = ηab x a y b = −x0 y0 + x1 y1 + x2 y2
(1)
or with the three-dimensional Euclidean metric x · y = η E (x, y) = δab x a y b = x0 y0 + x1 y1 + x2 y2 .
(2)
To avoid confusion, we denote the signature of the spacetime by a variable S and write S = L for Lorentzian and S = E for Euclidean signature. In the following we consider a set of six-dimensional Lie algebras h,S over R whose generators we denote by JaS , PaS , a = 0, 1, 2. For Lorentzian signature, the Lie algebras h,S depend on a parameter ∈ R, and their Lie brackets are given by [JaL , JbL ] = abc JcL
[JaL , PbL ] = abc PcL
[PaL , PbL ] = abc JcL ,
(3)
where indices are raised and lowered with the three-dimensional Minkowski metric (1) and abc is the three-dimensional antisymmetric tensor satisfying 012 = 1. For Euclidean signature, we consider parameters < 0, and the Lie algebra h,E has the bracket [JaE , JbE ] = ab c JcE ,
[JaE , PbE ] = ab c PcE , [PaE , PbE ] = ab c JcE ,
< 0, (4)
where indices are raised with the Euclidean metric (2)1 . The generators JaE in (4) span the real Lie algebra su(2) and can be represented by the matrices i 0 0 i 0 −1 , J1E = 21 , J2E = 21 . (5) J0E = 21 0 −i i 0 1 0 Similarly, the bracket of the generators JaL in (3) is the Lie bracket of the three-dimensional Lorentz algebra so(2, 1) ∼ = sl(2, R) ∼ = su(1, 1). A set of sl(2, R)-matrices representing these generators is given by 0 −1 1 0 0 1 J0 = 21 , J1 = 21 , J2 = 21 . (6) 1 0 0 −1 1 0 1 Note that the parameter in (3), denoted by λ in [4], is not equal to the cosmological constant but to minus the cosmological constant for Lorentzian signature, while its Euclidean analogue in (4) agrees with the cosmological constant. See also the discussion at the beginning of Sect. 3.1.
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However, in the following we will mostly work with the Lie algebra su(1, 1), which is conjugate to sl(2, R) in sl(2, C) via 1 1 1 i 1 −i · sl(2, R) · √ . (7) su(1, 1) = √ 2 i 1 2 −i 1 The su(1, 1) matrices associated to the generators (6) are given by i 0 L L 1 1 0 −i 1 0 1 L J1 = 2 J2 = 2 , J0 = 2 0 −i i 0 1 0
(8)
and by exponentiating linear combinations of these matrices over R, one obtains the Lie group a b 2 2 ∼ SU (1, 1) = | a, b ∈ C, |a| − |b| = 1 (9) = S L(2, R). b¯ a¯ The group SU (1, 1) ∼ = S L(2, R) is the double cover of the proper orthochronous Lorentz group in three dimensions S O(2, 1)+ = P S L(2, R) ∼ = P SU (1, 1) = SU (1, 1)/Z2 . In the following, we will often parametrise elements of SU (1, 1) and P SU (1, 1) via the a L exponential map which in both cases we denote by exp : pa JaL → e p Ja . Using expressions (8) for the generators of su(1, 1), we find that the parametrisation of SU (1, 1) in terms of a vector p ∈ R3 is given by exp : su(1, 1) → SU (1, 1),
⎧ | p| | p| ⎪ cosh 2 1 + 2 sinh 2 pˆ a JaL for p2 > 0 ⎪ ⎨ a L pa JaL → e p Ja = 1 + pa JaL
for p2 = 0
⎪ ⎪ ⎩cos | p| 1 + 2 sin | p| pˆ a J L for p2 < 0 a 2 2
pˆ = √ 1 2 p. |p | (10)
a
L
Elements u = e p Ja ∈ SU (1, 1) are called elliptic, parabolic and hyperbolic, respectively, for p2 < 0, p2 = 0 and p2 > 0. It follows directly from expression (10) that the exponential map for SU (1, 1) is neither surjective nor injective. The exponential map exp : su(1, 1) → P SU (1, 1) ∼ = S O(2, 1)+ is surjective, but again not injective, aJL p 2 2 since e a = 1 for p = −(2π n) , n ∈ Z. However, in the following we will mainly consider hyperbolic elements of P SU (1, 1), for which the parametrisation in terms of a vector p = ( p 0 , p 1 , p 2 ) ∈ R3 is unique. For = 0, the six-dimensional real Lie algebra (3) is the three-dimensional Poincaré algebra h0,L = iso(2, 1) = su(1, 1) ⊕ R3 , and the associated Lie group obtained by exponentiation is the semidirect product SU (1, 1) R3 ∼ = S L(2, R) R3 , where 3 ∼ SU (1, 1) acts on R = su(1, 1) via the adjoint action (u 1 , a1 ) · (u 2 , a2 ) = (u 1 u 2 , a1 + Ad(u 1 )a2 ), u 1 , u 2 ∈ SU (1, 1), a1 , a2 ∈ R3 . (11) For > 0, one can introduce an alternative set of generators Ja± , in terms of which the Lie bracket (3) takes the form of a direct sum Ja± = 21 (JaL ±
√1 PaL )
⇒
c ± [Ja± , Jb± ]>0 = ab Jc
[Ja± , Jb∓ ]>0 = 0. (12)
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Hence, for > 0, the Lie algebra h>0,L = su(1, 1) ⊕ su(1, 1) is the direct sum of two copies of su(1, 1) and the associated Lie group is SU (1, 1) × SU (1, 1), whose elements we will parametrise using an index + for the first and − for the second component (u + , u − ) · (v+ , v− ) = (u + v+ , u − v− )
u ± , v± ∈ SU (1, 1).
(13)
For the Lie algebras h<0,S a set of matrices representing the generators JaS , PaS in (3), (4) √ is obtained by setting PaS = i ||JaS . This implies that the Lie algebras h<0,L , h<0,E are both isomorphic to sl(2, C). In the first case sl(2, C) is realised as the complexification h<0,L = sl(2, C) = sl(2, R) ⊕ i sl(2, R) of its normal real form sl(2, R). In the second case it is given as the complexification h<0,E = sl(2, C) = su(2) ⊕ i su(2) of its compact real form su(2). Hence, depending on the signature S and on the parameter , the Lie algebras h,S and the associated Lie groups H,S are given by ⎧ 3 ⎪ = 0, S = L ⎨su(1, 1) ⊕ R h,S = su(1, 1) ⊕ su(1, 1) > 0, S = L , ⎪ ⎩sl(2, C) < 0, S = L , E ⎧ 3 ⎪ = 0, S = L ⎨ SU (1, 1) R H,S = SU (1, 1) × SU (1, 1) > 0, S = L ⎪ ⎩ S L(2, C) < 0, S = L , E. For all signatures and all values of the parameter , the three-dimensional Lorentz algebra su(1, 1) ∼ = sl(2, R) is a subalgebra of the Lie algebra h,S . The corresponding embedding of the group SU (1, 1) into the groups H,S is given by ⎧ 3 ⎪ = 0, S = L ⎨(v, 0) ∈ SU (1, 1) R ı,S (v) = (v, v) ∈ SU (1, 1) × SU (1, 1) > 0, S = L (14) ⎪ ⎩v ∈ S L(2, C) < 0, S = L , E. The embedding ı,S (±1) induces an action of Z2 on H,S . The quotients of H,S by this action are the (2+1)-dimensional Poincaré group H0,L /Z2 = P SU (1, 1)R3 , the group H>0,L /Z2 = SU (1, 1) × SU (1, 1)/Z2 and the proper orthochronous Lorentz group in (3+1)-dimensions H<0,L /Z2 = H<0,E /Z2 = S L(2, C)/Z2 = S O(3, 1)+ . In addition to these groups, we will need to consider the group P SU (1, 1) × P SU (1, 1) = H>0,L /(Z2 × Z2 ), whose elements we parametrise as in (13). The embedding (14) then induces an embedding of P SU (1, 1) into the quotients H,S /Z2 and into the group P SU (1, 1) × P SU (1, 1), which we will also denote by ı,S . In the following we will sometimes parametrise elements of the groups H,S , H,S /Z2 and of P SU (1, 1) × P SU (1, 1) via the exponential map, for which in all cases we use the symbol exp,S . Depending on the value of the parameter and the signature, these exponential maps are given by ⎧ pa J L (e a√, −T (− p)k) √ ⎪ ⎪ ⎪ ⎨(e( pa + k a )JaL , e( pa − k a )JaL ) √ exp,S ( pa JaS + k a PaS ) = ( pa +i ||k a )JaL ⎪ e ⎪ ⎪ ⎩ ( pa +i √k a )J E a e
= 0, > 0, < 0, < 0,
Lorentzian Lorentzian (15) Lorentzian Euclidean,
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C. Meusburger a
S
where expressions of the form e p Ja denote the image of the exponential map (10) or a a S the associated exponential map for P SU (1, 1), expressions e( p +iq )Ja the image of the 3 exponential map for S L(2, C) and S L(2, C)/Z2 and T ( p) : R → R3 , p ∈ R3 is a bijective linear map given via the identification R3 ∼ = su(1, 1) by (T ( p)k)
a
JaL
=
∞ adn (k a JaL ) pa J L a
n=0
(n + 1)!
= k a JaL + 21 [ p b JbL , k a JaL ]
+ 16 [ p c JcL , [ p b JbL , k a JaL ]] + . . . .
(16)
Note that for all values of and all signatures under consideration, the exponential map exp,S : h,S → H,S is neither surjective nor injective, which follows from the corresponding statement for the group SU (1, 1). For the groups P SU (1, 1) R3 , S L(2, C)/Z2 and P SU (1, 1) × P SU (1, 1) = SU (1, 1) × SU (1, 1)/(Z2 × Z2 ) the exponential maps are surjective but again not injective. 2.2. Hyperbolic geometry. In this section we summarise some facts and definitions from hyperbolic geometry used in this paper. For a general reference, we refer the reader to the book [24] by Benedetti and Petronio, for a specialised treatment focusing on Fuchsian groups to the book [25] by Katok. In the following, we denote by Hdk the d-dimensional hyperbolic space of curvature −|k|, realised as the hyperboloid 1 Hdk = {x = (x 0 , x 1 , . . . , x d ) ∈ Rd+1 | − x02 + x12 + . . . + xd2 = − |k| , x 0 > 0}
(17)
with the metric induced by the (d + 1)-dimensional Minkowski metric. In the twodimensional case, we also work with the disc model, in which H2 = H21 is realised as the unit disc D = {z ∈ C | |z| < 1}
ds 2 =
4|dz|2 , (1 − |z|2 )2
(18)
and which is related to the two-dimensional hyperboloid model (17) via a map z ∈ D → x(z) ∈ H2k , x0 (z) =
2 √1 1+|z| 2 |k| 1−|z|
x1 (z) =
√1 2Re(z)2 |k| 1−|z|
x 2 (z) =
√1 2Im(z)2 |k| 1−|z|
∀z ∈ D.
(19)
In the hyperboloid model, the geodesics of Hdk are obtained as the intersection of Hdk with d-dimensional hyperplanes through the origin. In the two-dimensional disc model, the geodesics are the diameters of the disc and arcs of circles orthogonal to its boundary. The isometry group Isom(H2k ) = Isom(D, ds 2 ) is the proper orthochronous Lorentz group P S L(2, R) ∼ = P SU (1, 1) = SU (1, 1)/Z2 , which acts on the hyperboloid H2k via its canonical action on Minkowski space and whose action on the disc D is given by az + b ab . (20) ∈ SU (1, 1) : z → ¯ + a¯ b¯ a¯ bz The uniformization theorem states that every orientable two-surface of genus g > 1 with a metric of constant curvature −|k| is isometric to a quotient H2k / of H2k by the action of a cocompact Fuchsian group with 2g hyperbolic generators −1 = v A1 , v B1 , . . . , v Ag , v Bg | [v Bg , v −1 A g ] · · · [v B1 , v A1 ] = 1 ⊂ P SU (1, 1). (21)
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The group induces a tessellation of H2k by geodesic arc 4g-gons, which are mapped into each other by the elements of . Hence, for each polygon in the tessellation, there exist 4g elements of which map this polygon into its 4g neighbours and identify its sides pairwise. The surface H2k / is obtained by glueing these pairs of sides of a polygon in the tessellation. In particular, there exists a polygon, in the following referred to as a fundamental polygon and denoted by P , which is mapped into its 4g neighbours by a fixed set of generators of and their inverses. If we label the sides of P as in Fig. 5, the generators v Ai , v Bi in (21) identify the sides of this fundamental polygon P according to v Ai : ai → ai
v Bi : bi → bi .
(22)
The geodesics on the surface H2k / are obtained by projecting the geodesics on H2k . In particular, closed geodesics η : [0, 1] → H2k / , η(0) = η(1) on H2 / arise as the projections of geodesics cη : [0, 1] → H2k for which there exists an element of that maps these geodesics to itself: cη (1) = vη cη (0),
a
vη = en η Ja ∈ .
(23)
In the following we will refer to this element as the translation element of η and to 2 the associated vectors nη and nˆ η = nη / |nη | as the translation vector and unit translation vector of η. Closed geodesics on the surface H2k / are therefore in one-to-one correspondence with elements of the cocompact Fuchsian group , which is isomorphic to the surface’s fundamental group π1 (H2k / ) ∼ = . In the following we will often not distinguish notationally between such geodesics, their homotopy equivalence classes in π1 (H2k / ) and general curves on the surface which represent these homotopy equivalence classes. 3. (2+1)-Dimensional Gravity: The Geometrical Formulation 3.1. Model spacetimes. In this section, we summarise the geometrical description of (2+1)-spacetimes as quotients of certain model spacetimes. A general reference for (2+1)-spacetimes is the book [1] by Carlip. A more specific treatment focusing on the construction of (2+1)-spacetimes via grafting is given in the papers by Benedetti and Bonsante [21, 22]. (2+1)-dimensional gravity is a theory without local gravitational degrees of freedom. As the curvature tensor of a three-dimensional manifold is determined completely by its Ricci tensor, vacuum solutions of the (2+1)-dimensional Einstein equations are flat or of constant curvature. This implies that they are locally isometric to a three-dimensional model spacetime. In this paper, we consider Lorentzian (2+1)-gravity with general cosmological constant and the Euclidean case with negative cosmological constant. In the following, we work with a parameter ∈ R which is identified with minus the cosmological constant for Lorentzian spacetimes and agrees with the cosmological constant in the Euclidean case. The choice of this convention is motivated by the conventions in the Chern-Simons formulation of the theory and leads to notational simplifications there. The model spacetimes for Lorentzian signature are then three-dimensional Anti de Sitter space AdS , three-dimensional Minkowski space M3 , three-dimensional de Sitter space dS , respectively, for > 0 (negative cosmological constant), = 0 (vanishing cosmological constant) and < 0 (positive cosmological constant). The
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model spacetime for Euclidean signature and negative cosmological constant ( < 0) is three-dimensional hyperbolic space H3 . In the following, we parametrise these spacetimes in terms of matrices, which is convenient for establishing a link with their description in the Chern-Simons formalism. For the Lorentzian case with vanishing cosmological constant, the relevant model spacetime is (2+1)-dimensional Minkowski space X0,L = M3 and the group of orientation and time orientation preserving isometries is the (2+1)-dimensional Poincaré group Isom(X0,L ) = P SU (1, 1) R3 = H0,L /Z2 . In the canonical identification of Minkowski space with the set of su(1, 1)-matrices i x0 −i(x 1 + i x 2 ) ∈ su(1, 1), M3 x = (x 0 , x 1 , x 2 ) → X = 2x a JaL = −i x 0 i(x 1 − i x 2 ) (24) the (2+1)-dimensional Minkowski metric agrees with the Killing form of su(1, 1), −x0 y0 + x1 y1 + x2 y2 = 21 Tr (X · Y ) ,
(25)
and the action of Isom(X0,L ) = P SU (1, 1) R3 is given by (u, a) ∈ P SU (1, 1) R3 : X → u Xu −1 + 2a b JbL
∀X ∈ su(1, 1).
(26)
The model spacetime for negative cosmological constant ( > 0) and Lorentzian signature is three-dimensional Anti de Sitter space AdS . We adopt the conventions of [21, 22] in which Anti de Sitter space is realised as a quotient of the universal cover AdS by the action of Z2 . The universal cover AdS is the manifold AdS = {(t1 , t2 , x1 , x2 ) ∈ R4 | t12 + t22 − x12 − x22 = X>0,L =
1 ,
},
d x 2 = −(dt1 )2 − (dt2 )2 + (d x1 )2 + (d x2 )2 .
(27)
Via the map AdS x = (t1 , t2 , x1 , x2 ) → X =
t1 + it2 −i(x1 + i x2 ) i(x1 − i x2 ) t1 − it2
∈ SU (1, 1), (28)
it can be identified with the group SU (1, 1) such that its metric is given by minus the determinant AdS = { √1 A | A ∈ SU (1, 1)}
d x 2 = − det ( d X ).
(29)
The group of orientation and time orientation preserving isometries of AdS is the group SU (1, 1) × SU (1, 1)/Z2 = H>0,L /Z2 , whose action is given by the action of SU (1, 1) × SU (1, 1) via (G + , G − ) ∈ SU (1, 1) × SU (1, 1) : X → G + X G −1 −
∀X ∈ SU (1, 1).
(30)
AdS by the action of the elements (±1, ∓1) Anti de Sitter space AdS is the quotient of via (30) and its isometry group is the quotient P SU (1, 1) × P SU (1, 1)/Z2 , AdS /Z2 , Isom(AdS ) = P SU (1, 1) × P SU (1, 1). (31) X>0,L = AdS =
Geometrical (2+1)-Gravity and the Chern-Simons Formulation
715
For positive cosmological constant ( < 0) and Lorentzian signature, the model spacetime is three-dimensional de Sitter space X<0,L = dS = {x = (x 0 , x 1 , x 2 , x 3 ) ∈ M4 | − x02 + x12 + x22 + x32 =
1 || },
(32)
and for the Euclidean case with negative cosmological constant ( < 0), it is threedimensional hyperbolic space 1 , x 0 > 0}. X<0,E = H3 = {x = (x 0 , x 1 , x 2 , x 3 ) ∈ M4 | − x02 + x12 + x22 + x32 = − ||
(33) In both cases the metric is the one induced by the four-dimensional Minkowski metric, and the group of orientation and, in the de Sitter case, time orientation preserving isometries is the proper orthochronous Lorentz group in three dimensions Isom(X<0,L ) = Isom(X<0,E ) = S O(3, 1)+ = S L(2, C)/Z2 . The parametrisation of these spacetimes in terms of matrices is obtained by identifying vectors in four-dimensional Minkowski space with certain sets of matrices in G L(2, C). The first identification is the standard identification of Minkowski space with the set of hermitian G L(2, C) matrices 0 x + x3 x1 + i x2 4 0 1 2 3 0 i , (34) M x = (x , x , x , x ) → X = x 1 + x σi = x1 − i x2 x0 − x3 in which the four-dimensional Minkowski metric takes the form − det ( X ) = −x02 + x12 + x22 + x32 .
(35)
The action of the isometry group S O(3, 1)+ ∼ = S L(2, C)/Z2 on the set of hermitian 2 × 2-matrices is given by the action of S L(2, C) via G ∈ S L(2, C) : X → G X G †
∀X ∈ G L(2, C), X † = X.
(36)
As this action has kernel ±1 and leaves the determinant invariant, it induces an action of S O(3, 1)+ which preserves the metric (35). Hence, three-dimensional hyperbolic space H3 can be identified with the set of hermitian S L(2, C) matrices with metric (35) and the action of the isometry group Isom(H3 ) = S O(3, 1)+ is given by (36), 1 H3 = { √|| A | A ∈ S L(2, C), A† = A}.
(37)
The matrix representation of dS is similar, but instead of the identification (34), one uses the identification x3 + x0 x1 + i x2 , (38) M4 x = (x 0 , x 1 , x 2 , x 3 ) → X = −(x 1 − i x 2 ) x 3 − x 0 which assigns to each vector in Minkowski space a matrix in the set C L(2, C) = {A ∈ G L(2, C) | A◦ = A} ◦ † a¯ −c¯ −i 0 a b i 0 a b , = = 0 i c d 0 −i c d −b¯ d¯
(39) (40)
such that the four-dimensional Minkowski metric is given by the determinant det (X) = −x02 + x12 + x22 + x32 .
(41)
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C. Meusburger
As the map ◦ : G L(2, C) → G L(2, C) satisfies (A◦ )◦ = A, (AB)◦ = B ◦ A◦ , det(A◦ ) = det A, one obtains an action of the group S L(2, C) on C L(2, C) via G ∈ S L(2, C) : M → G M G ◦
(42)
which has kernel ±1, preserves the determinant, and thus induces an action of S O(3, 1)+ = S L(2, C)/Z2 which preserves the metric (41). Hence, de Sitter space dS can be realised as the set of S L(2, C) matrices invariant under the operation ◦ with metric (41), and with isometry group S L(2, C)/Z2 whose action is given by (42) 1 A | A ∈ S L(2, C), A◦ = A}. dS = { √||
(43)
Hence, depending on the cosmological constant and the signature, the model spacetimes X,S can be identified with the sets of matrices ⎧ ⎪ M3 ∼ = 0, S = L = su(1, 1) ⎪ ⎪ ⎪ 1 ⎨AdS ∼ > 0, S = L = √ P SU (1, 1) X,S = (44) 1 ◦ ∼ √ dS = || {A ∈ S L(2, C) : A = A } < 0, S = L ⎪ ⎪ ⎪ ⎪ 1 ⎩H3 ∼ {A ∈ S L(2, C) : A = A† } < 0, S = E = √|| with metrics given by (25), (29), (35), (41) and their groups of (orientation and time orientation) preserving isometries ⎧ ⎪ P SU (1, 1) R3 = 0, S = L ⎪ ⎪ ⎨ P SU (1, 1) × P SU (1, 1) > 0, S = L (45) Isom(X,S ) = ⎪ S L(2, C)/Z2 < 0, S = L ⎪ ⎪ ⎩ > 0, S = E S L(2, C)/Z2 act via (26), (30), (36), (42). 3.2. Static universes and the embedding of hyperbolic space H2 . The defining characteristic of the model spacetimes introduced in the last subsection is that their topology is trivial. (2+1)-spacetimes with nontrivial topology are obtained as the quotients of domains U,S ⊂ X,S in the model spacetimes by the action of certain subgroups of the isometry groups Isom(X,S ). In this paper we restrict attention to spacetimes for which these subgroups are cocompact Fuchsian groups with 2g > 2 generators and act via group homomorphisms h ,S : → Isom(X,S ). The resulting spacetimes have topology R × Sg , where Sg is an oriented two-surface of genus g > 1. The simplest such spacetimes are the static spacetimes associated to a cocompact Fuchsian group , for a detailed discussion see for example [1]. For Lorentzian signature, the associated domain U,L ⊂ X,L in the model spacetime is the interior of a forward lightcone, i. e. the set of points connected to a given point x ,L ∈ X,L by timelike geodesics. In the Euclidean case, it is the whole model spacetime H3 . In each model spacetime X,S , this domain is foliated by two-surfaces U,S (T ) of constant cosmological time T , i. e. surfaces of constant geodesic distance T from a given point x ,S , which represents a singularity of the spacetime. For all values of the cosmological constant and all signatures under consideration, the surfaces U,S (T ) are surfaces of constant curvature and can be identified with copies of two-dimensional hyperbolic space. The
Geometrical (2+1)-Gravity and the Chern-Simons Formulation
717
action of the (2+1)-dimensional Lorentz group P SU (1, 1) via its canonical embedding ı,S : P SU (1, 1) → Isom(X,S ) preserves the surfaces U,S (T ) and agrees with the action induced by (20). This induces an action of the cocompact Fuchsian group and a tessellation of each surface U,S (T ) by geodesic arc 4g-gons as described in Sect. 2.2. st The static spacetimes M,,S associated to are then obtained by identifying on each surface of constant cosmological time the points related by this action of , st = U,S / . M,,S
(46)
To obtain explicit expressions for the static domains U,S ⊂ X,S and their foliation by copies of hyperbolic space, we consider timelike geodesics c,L in the Lorentzian model spacetimes X,L and an associated geodesic c,E in H3 , ⎧ ⎪ 2T J0L√ , T ∈ (0, ∞) = 0, Lorentzian ⎪ ⎨ √ L 1 T J 0 √ e , T ∈ (0, π/ ) > 0, Lorentzian c,S (T ) = (47) √ ⎪ ⎪ ⎩ √1 e−i ||T J0L , T ∈ (0, ∞) < 0, Lorentzian and Euclidean, ||
which are parametrised by arclength and based at the identity. Furthermore we introduce a map 1 1 z . (48) g : H2 → SU (1, 1), z → g(z) = 1 − |z|2 z¯ 1 A brief calculation shows that - up to right-multiplication with a phase - the action of SU (1, 1) on the disc via (20) corresponds to left-multiplication of the image g(z), g(M z) = M · g(z) · eψ(M,z)J0 , L
ψ(M, z) ∈ R
∀M ∈ P SU (1, 1), z ∈ H2 . (49)
As the phase commutes with J0L and is mapped to its inverse by the operations ◦ , †, one finds the map ,S : H2 → X,S defined by T ⎧ ⎪ g(z)c0,L (T )g(z)−1 = 0, Lorentzian ⎪ ⎪ ⎨ −1 > 0, Lorentzian g(z)c>0,L (T )g(z) (50) ,S T (z) = ı,S ◦ g(z) c,S (T ) = ⎪g(z)c ◦ (T )g(z) < 0, Lorentzian <0,L ⎪ ⎪ ⎩g(z)c † < 0, Euclidean <0,E (T )g(z) satisfies the covariance condition
⎧ −1 ⎪ M ,L ⎪ T (z)M ⎪ ⎨ M ,L (z)M −1 ,S T ,S T (M z) = ı,S (M) T (z) = ⎪ ◦ M ,L ⎪ T (z)M ⎪ ⎩ † M ,E T (z)M
= 0, > 0, < 0, < 0,
Lorentzian Lorentzian Lorentzian Euclidean.
(51)
The action of the (2+1)-dimensional Lorentz group P SU (1, 1) via its canonical 2 embedding into Isom(X,S ) therefore preserves the images ,S T (H ) and agrees with its action induced by its action on the Poincaré disc via (20). Furthermore, as the geode2 sics (47) are parametrised by arclength, all points in the image ,S T (H ) have constant geodesic distance T from the initial singularity c,S (0). Hence, one obtains a foliation
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C. Meusburger
of the forward lightcone or, for Euclidean signature, of H3 by surfaces of constant cosmological time ⎧ = 0, Lorentzian ⎪ T ∈(0,∞) U0,L (T ) ⎪ ⎪ ⎨ √ U ,L (T ) > 0, Lorentzian U,S = T ∈(0,π/ ) U,S (T ) = φT,S (H2 ). (52) ⎪ U (T ) < 0, Lorentzian ⎪ ⎪T ∈(0,∞) ,L ⎩ < 0, Euclidean T ∈(0,∞) U,E (T ) To obtain concrete expressions for the matrices in (50), one evaluates (47) using expression (10) for the exponential map. For Lorentzian signature and vanishing cosmological constant, this yields ⎛ ⎞ 1+|z|2 2i z −T 1−|z| i T 1−|z| 2 2 0,L ⎠. T (z) = ⎝ (53) 1+|z|2 2i z¯ T 1−|z| −i T 1−|z| 2 2 By comparing with (24), we recover the formula (19) which relates the disc model of hyperbolic space to the hyperboloids H21/T 2 of curvature 1/T 2 . For Lorentzian signature
and > 0, we consider the associated geodesic in the double cover AdS and find that the parameters in (28) and the metric (27) take the form √
√
√ √ 1+|z|2 2Re(z) 2Im(z) , x1 = sin(√ T ) , x2 = sin(√ T ) , 2 2 1−|z| 1−|z| 1−|z|2 (54) √ 2 2 4 sin ( T ) |dz| . d x 2 = −(dt1 )2 − (dt2 )2 + (d x1 )2 + (d x2 )2 = −dT 2 + (1 − |z|2 )2
t1 = cos(√ T ) , t2 = sin(√ T )
The surfaces U>0,L (T ) ⊂ X>0,L = AdS therefore have constant curvature √ 2 −/sin ( T ). For Lorentzian signature and < 0, the coordinates parametrising d S in (38) and the metric are given by √ √ 1 + |z|2 ||T ) 2Re(z) ||T ) 2Im(z) , x 1 = − sinh(√|| , x 2 = − sinh(√|| , 2 2 1 − |z| 1 − |z| 1 − |z|2 (55) √ 2 2 √ 4 sinh ( ||T ) |dz| ||T ) x 3 = cosh(√|| , d x 2 = −dT 2 + , || (1 − |z|2 )2 √ and the surfaces U<0,L (T ) have constant curvature −||/sinh2 ( ||T ). For √ Euclid2 ean signature and < 0, the curvature of the surfaces U<0,E (T ) is −/cosh ( T ), since the parameters in (34) and the metric (35) take the form
x0 =
√ sinh(√ ||T ) ||
√ 1 + |z|2 ||T ) 2Re(z) , x 1 = cosh(√|| , 2 1 − |z| 1 − |z|2 √ √ ||T ) 2Im(z) sinh(√ ||T ) 3 x 2 = cosh(√|| , x = , || 1 − |z|2 √ cosh2 ( ||T ) |dz|2 d x 2 = dT 2 + 4 . || (1 − |z|2 )2
x0 =
√ cosh( √ ||T ) ||
(56)
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719
The cocompact Fuchsian group ⊂ P SU (1, 1) acts on the domains U,S freely and properly discontinuously via the canonical inclusion ı,S : P SU (1, 1) → Isom(X,S ) induced by (14). It follows from the identity (51) that this action preserves the surfaces U,S (T ) of constant cosmological time and agrees with the action induced by the idenst tification of these surfaces with hyperbolic space. The static (2+1)-spacetimes M,S, associated to are given as the quotients of the domains U,S by this action of , st M,S, = U,S / =
U,S (T )/ .
(57)
T
3.3. The construction of evolving (2+1)-spacetimes via grafting. After discussing the static spacetimes associated to a cocompact Fuchsian group , we will now summarise the construction of evolving (2+1)-spacetimes via grafting following the presentation in [21, 22]. Grafting along measured geodesic laminations is a method for constructing twosurfaces. The simplest case are geodesic laminations which are sets of non-intersecting closed, simple geodesics on a two-surface. Grafting along closed, simple geodesics was first investigated in the context of complex projective structures and Teichmüller theory [26–28]. General geodesic laminations were first considered by Thurston [29, 30], for historical remarks see for instance [31]. The role of geodesic laminations in (2+1)-dimensional gravity was first explored by Mess [19] who investigated the characterisation of (2+1)-dimensional spacetimes in terms of holonomies. More recent work on grafting in the context of (2+1)-dimensional gravity are the papers by Benedetti and Bonsante [21, 22], which relate the construction of (2+1)-spacetimes via grafting for different values of the cosmological constant. The ingredients of the grafting construction are a cocompact Fuchsian group and a measured geodesic lamination on the associated two-surface H2k / . In the following, we restrict attention to the case where this geodesic lamination is a weighted multicurve on H2k / , i. e. a set of non-intersecting closed, simple geodesics ηi on H2k / , each equipped with a weight wi > 0. G = {(ηi , wi ) | i ∈ I }.
(58)
Geometrically, grafting along the multicurve (58) amounts to cutting the surface H2k / along each geodesic ci , and inserting a strip of width wi as shown in Fig. 1. c
w Fig. 1. Grafting along a closed simple geodesic c with weight w on a genus 2 surface
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C. Meusburger
In the construction of (2+1)-spacetimes via grafting, the grafting procedure is applied to each two-surface U,S (T )/ in (57). The construction is performed on their universal covers, i. e. the constant cosmological time surfaces U,S (T ), which are identified with copies of hyperbolic space via (50) and foliate the static domains U,S ⊂ X,S as in (52). The first step in the grafting construction is to lift each geodesic ηi in the multicurve (58) to a geodesic cηi on the universal cover H2k . By acting on these geodesics with the cocompact Fuchsian group , one obtains a -invariant multicurve on H2k , G Hk = {(vcηi , wi ) | i ∈ I, v ∈ }, 2
(59)
i. e. a set of non-intersecting geodesics on H2k with associated weights wi > 0, which are mapped into each other by the elements of . Via the maps ,S : H2 → U,S (T ) ⊂ T X,S in (50), which identify hyperbolic space with the constant cosmological time surfaces U,S (T ), one then obtains a -invariant set of non-intersecting geodesics on each surface U,S (T ). Grafting along the multicurve (58) assigns to each surface U,S (T ) a deformed surG (T ) constructed as follows. One selects a basepoint q ∈ H2 outside of the face U,S 0
geodesics in the multicurve (58) and considers the images ,S T (q0 ) on the surfaces U,S (T ). One then cuts each surface U,S (T ) along the images ,S T (vcηi ), i ∈ I , v ∈ of the geodesics in the multicurve (59) on U,S (T ). The resulting pieces which do not contain the images of the basepoint are then shifted away from the basepoint in the direction determined by the geodesics’ unit translation vectors and by a distance given by the geodesic’s weight. Finally, one inserts strips, which connect the shifted pieces of each constant cosmological time surface U,S (T ), and thus obtains a connected deformed G (T ) surface U,S The union of these deformed surfaces for all values of the cosmological time T then forms a simply connected regular domain in X,S : ⎧ G = 0, Lorentzian ⎪ T ∈(0,∞) U0,L (T ) ⎪ ⎪ ⎨ √ U G (T ) > 0, Lorentzian G (60) = T ∈(0,π/ )G ,L U,S ⎪ U,L (T ) < 0, Lorentzian ⎪ T ∈(0,∞) ⎪ ⎩ G < 0, Euclidean. T ∈(0,∞) U,E (T )
Under the grafting construction, the initial singularity of the static domains U,S is G (T ) mapped to a graph in X,S . It is shown in [21, 22] that the deformed surfaces U,S are surfaces of constant geodesic distance T from this graph and therefore again surfaces of constant cosmological time T . It is discussed in [21, 22] that the cocompact Fuchsian group acts on the grafted G (T ) via a group homomorphism h G : → Isom(X domain U,S ,S ). This action is ,S G (T ). Hence, by taking free and properly discontinuous and preserves each surface U,S G (T )/ h G () of the deformed constant cosmological time surfaces by the quotient U,S ,S ,G this action of one obtains a two-surface of genus g. The grafted spacetimes M,S associated to the cocompact Fuchsian group and the multicurve (58) on H2k / are then given as the union of these surfaces for all values of the cosmological time or, G by this action of , equivalently, as the quotient of the regular domains U,S ,G G G G G M,S = U,S / h ,S () = U,S (T )/ h ,S (). (61) T
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w
Fig. 2. Grafting along a geodesic with weight w in hyperbolic space
The procedure is most easily visualised in Lorentzian (2+1)-gravity with vanishing cosmological constant, where the surfaces of constant cosmological time are the hyperboloids H21/T 2 which foliate the interior of the forward lightcone. Geodesics on the hyperboloids H21/T 2 are given as the intersection of H21/T 2 with planes through the origin, whose unit normal vector is the unit translation vector of the geodesic given in (23). Cutting each surface U0,L (T ) along these geodesics therefore amounts to cutting the interior of the forward lightcone along the associated planes. The resulting pieces are then shifted away from the basepoint in the direction of the plane’s normal vector by a distance given by the weight of the associated geodesic as shown in Fig.2. The strips connecting the different pieces of a surface U0,L (T ) are obtained by connecting the points of the different pieces of U0,L (T ) which correspond to a single point on a geodesic by straight lines. For the other model spacetimes the construction is similar but its description is more involved. As we will not need the details of the construction, we refer the reader to the papers [21, 22], which give an explicit parametrisation of the resulting surfaces and relate these surfaces for different values of the cosmological constant. In the following, we will only make use of a formula for the translation of the images ,S T (x) ∈ U,S (T ) of points outside of the geodesics in the multicurve (59). The relative shift of such points under the grafting construction is determined by their position relative to the geodesics in (59) and given by a map BG,,S : H2 × H2 → Isom(X,S ). To determine the value of BG,,S ( p, q) for two points p, q ∈ H2 outside the geodesics in (59), one connects them with a geodesic a pq on H2 oriented towards q. One then determines the geodesics in the multicurve (59) which intersect this geodesic as well as the associated oriented intersection numbers. It is shown in [21, 22], see in particular Sects. 4.2.1, 4.4.1, 4.6.1 and 4.7.2, that if these geodesics are labelled by ci , i = 1, . . . , m, such that the intersection point of a pq with ci occurs before the one with c j for i < j and if i are the associated oriented intersection numbers with the convention i = 1 if ci crosses a pq from the left to the right, then the relative shift BG,,S ( p, q) is given by2 √
√ , || are not present in [21, 22], where only spacetimes with cosmological constant ∈ {0, ±1} are considered. However, this normalisation is suggested by the fact that the associated spacelike geodesics should be parametrised by arc length. 2 The factors
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BG,0,L ( p, q) = BG,>0,L ( p, q) =
m
i wi nˆ i ∈ R3 ⊂ P SU (1, 1) R3 ,
i=1 + (BG,>0,L ( p, q),
(62)
− BG,>0,L ( p, q)) ∈ P SU (1, 1) × P SU (1, 1) √
± ( p, q) = e± BG,>0,L
BG,<0,S ( p, q) = BG,<0,E ( p, q) = ei ∈ S L(2, C)/Z2 ,
√
√ 1 w1 nˆ a1 JaL ± 2 w2 nˆ a2 JaL
√
e
||1 w1 nˆ a1 JaL
ei
√
||2 w2 nˆ a2 JaL
· · · e±
n wn nˆ am JaL
· · · ei
||n wn nˆ am JaL
√
,
where wi is the weight of the geodesic ci and nˆ i its unit translation vector as defined in (23). It is shown in [21, 22] that map BG,,S : H2 × H2 → Isom(X,S ) satisfies the identities BG,,S ( p, q) · BG,,S (q, r ) = BG,,S ( p, r ) BG,,S (vp, vq) = ı,S (v) · BG,,S ( p, q) · ı,S
∀ p, q, r ∈ H2 ,
(v)−1
(63)
∀ p, q ∈ H , v ∈ P SU (1, 1), 2
which reflect the geometrical properties of the grafting procedure. This allows one to G : → Isom(X define a group homomorphism h ,S ,S ) from the cocompact Fuchsian group into the isometry group of the model spacetime by setting G h ,S (v) = BG,,S (q0 , vq0 ) · ı,S (v)
∀v ∈ ,
(64)
where ı,S : P SU (1, 1) → Isom(X,S ) is the canonical embedding of P SU (1, 1) into the isometry group of the model spacetime given by (14) and q0 ∈ H2 the basepoint. It is discussed in [21, 22] that this group homomorphism defines a free and properly G which maps each discontinuous action of the group on the grafted domains U,S G (T ) to itself. Furthermore, for any two points (x), (x ) ∈ U surface U,S ,S (T ) T T outside the geodesics which are related by the canonical action (51) of an element v ∈ , G (T ) are related by the action of v the corresponding points on the grafted surface U,S via (64) T T (x ) = ı,S (v) · ,S (x) ,S
⇒
BG,,S (q0 , x
T ) ,S (x )
=
G h ,S (v) ·
(65) T BG,,S (q0 , x) ,S (x).
G ⊂ X The quotient (61) of the domains U,S ,S by this action of is therefore welldefined and gives rise to a spacetime of topology R × Sg .
4. (2+1)-Dimensional Gravity: The Chern-Simons Formulation 4.1. (2+1)-dimensional gravity as a Chern-Simons gauge theory. The absence of local gravitational degrees of freedom in (2+1)-dimensional gravity allows one to formulate the theory as a Chern-Simons gauge theory [3, 4]. The Chern-Simons formulation of (2+1)-dimensional gravity is derived from Cartan’s description, in which a spacetime manifold M is characterised in terms of a dreibein of one forms ea , a = 0, 1, 2, and spin connection one-forms ωa , a = 0, 1, 2, on M. The metric on M is given by the dreibein g = ηab S ea ⊗ eb ,
(66)
Geometrical (2+1)-Gravity and the Chern-Simons Formulation
723
where ηab S denotes the Minkowski metric (1) or the Euclidean metric (2), while the one-forms ωa are the coefficients of the spin connection ω = ωa JaS . Einstein’s equations of motion then take the form of the requirements of vanishing torsion and constant curvature T a = dea + abc ωb ec = 0
Fωa = dωa + 21 abc ωb ∧ ωc = − 2 abc eb ∧ ec .
(67)
To obtain the Chern-Simons formulation of (2+1)-dimensional gravity, one combines dreibein and spin connection into the Cartan connection [32] or Chern-Simons gauge field, A = ωa JaS + ea PaS ,
(68)
where JaS , PaS , a = 0, 1, 2, denote the generators of the six-dimensional Lie algebras h,S with bracket (3),(4). Hence, depending on the signature and the cosmological constant, the Chern-Simons gauge field is a one-form on M with values in the Lie algebra h,S . The choice of the Lie algebra determines the gauge group of the associated ChernSimons theory up to coverings, and in the following, we will take the isometry groups Isom(X,S ) of the associated model spacetimes as the gauge groups. When expressed in terms of the one-form (68), the Einstein-Hilbert action in Cartan’s formulation of the theory takes the form of a Chern-Simons action SC S [A] = A ∧ d A + 23 A ∧ A ∧ A , (69) M
where , is an Ad-invariant, non-degenerate bilinear form on the Lie algebra h,S given by S JaS , PbS = ηab
JaS , JbS = PaS , PbS = 0.
(70)
The equations of motion derived from (69) are a flatness condition on the gauge field F = d A + A ∧ A = 0,
(71)
which combines the requirements (67) of vanishing torsion and constant curvature F = T a PaS + (Fωa +
b a 2 bc e
∧ ec ) JaS .
(72)
The Chern-Simons action (69) is invariant under Chern-Simons gauge transformations A → γ Aγ −1 + γ dγ −1
γ : M → Isom(X,S ).
(73)
It has been shown by Witten [4] that infinitesimal Chern-Simons gauge transformations are on-shell equivalent to infinitesimal diffeomorphisms. The space of metrics solving Einstein’s equation modulo infinitesimally generated diffeomorphisms is therefore isomorphic to the space of flat Chern-Simons gauge fields modulo infinitesimally generated Chern-Simons gauge transformations. Note, however, that some caution should be applied when identifying the phase space of (2+1)-dimensional gravity in its geometrical formulation with the phase space of the associated Chern-Simons theory. First, the equivalence between diffeomorphisms and Chern-Simons gauge transformations does not hold for large diffeomorphisms, which are not infinitesimally generated, and for the large gauge transformations arising in Chern-Simons theory with non-simply connected gauge groups. Second, in order to
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C. Meusburger
define a metric of Lorentzian or Euclidean signature via (66), the dreibein ea has to be non-degenerate, which is not required in the Chern-Simons formalism. It is discussed in [33] for the Lorentzian case with vanishing cosmological constant and spacetimes containing particles that this leads to differences in the global structure of the phase spaces. A similar result for a spacetime with three particles is derived in [22], Sect. 4.9, where it is argued that such problems arise generically when the spacetime is of topology R × S, where S is a non-compact two-surface. However, as this paper restricts attention to spacetimes with compact spatial surfaces and is mainly concerned with the local properties of the phase space, we will not address these issues in the following. On the spacetime manifolds of topology M ∼ = R × Sg considered in this paper, it is possible to give a Hamiltonian formulation of the theory. For this, one introduces coordinates x 0 , x 1 , x 2 on M ≈ R × Sg such that x 0 parametrises R and x 1 , x 2 are coordinates on Sg and splits the gauge field (68) as A = A0 d x 0 + A S ,
(74)
where A0 : R × Sg → h,S is a function with values in the Lie algebras h,S and A S a gauge field on Sg . The Chern-Simons action (69) on M then takes the form 1 S[A S , A0 ] = dx0 (75) 2 ∂0 A S ∧ A S + A0 , FS , R
Sg
where FS is the curvature of the spatial gauge field A S , FS = d S A S + A S ∧ A S ,
(76)
with d S denoting differentiation on the surface Sg . The function A0 plays the role of a Lagrange multiplier. Varying it leads to the flatness constraint FS = 0,
(77)
while variation of A S results in the evolution equation ∂0 A S = d S A0 + [A S , A0 ].
(78) Isom(X
)
,S of flat The phase space of the theory is therefore the moduli space Mg Isom(X,S )-connections A S modulo gauge transformations on the spatial surface Sg .
4.2. Trivialisation and holonomies. As discussed in Sect. 3, the absence of local gravitational degrees of freedom in (2+1)-dimensional gravity implies that each (2+1)-spacetime is locally isometric to one of the model spacetimes X,S . In the Chern-Simons formalism, this absence of local degrees of freedom manifests itself in the fact that gauge fields solving the equations of motions are flat and can be trivialised, i. e. written as pure gauge on any simply connected region R ⊂ R × Sg , A = γ dγ −1 ,
γ : R → Isom(X,S ).
(79)
Given a function γ : R → Isom(X,S ) which trivialises a flat gauge field A on R, the associated functions γ (x 0 , ·) trivialise the corresponding flat spatial gauge fields As for all values of x 0 , A S (x 0 , ·) = γ (x 0 , ·)d S γ −1 (x 0 , ·).
(80)
Geometrical (2+1)-Gravity and the Chern-Simons Formulation
725
bj
bj
aj
bj
aj
bj
aj
aj
bj Fig. 3. Cutting the surface Sg along the generators of π1 (Sg )
To simplify notation, we will often neglect the dependence on the parameter x 0 in the following and denote this function also by γ . A maximal simply connected region in R × Sg is obtained by cutting the spatial surface Sg along a set of generators of the fundamental group π1 (Sg ) as in Fig. 3. As discussed in Sect. 2.2, the fundamental group of a genus g surface Sg is isomorphic to a cocompact Fuchsian group with 2g generators, which are subject to a single defining relation, π1 (Sg ) = a1 , b1 , . . . , ag , bg ; [bg , ag−1 ] · · · [b1 , a1−1 ] = 1 , [bi , ai−1 ] = bi ◦ ai−1 ◦ bi−1 ◦ ai .
(81)
Throughout the paper, we work with a fixed system of generators ai , bi , i = 1, . . . , g, which are the homotopy equivalence classes of two loops around each handle and based at a point p ∈ Sg as shown in Fig. 4. Cutting the surface along each of the curves representing these generators results in a 4g-gon Pg pictured in Fig. 5. As discussed by Alekseev and Malkin [13], a function γ : Pg → Isom(X,S ) on Pg defines a flat gauge field on Sg if and only if it satisfies an overlap condition relating its value on the two sides which correspond to a given generator of the fundamental group. For any y ∈ {a1 , b1 , . . . , ag , bg }, y ∈ {a1 , b1 , . . . , ag , bg } one must have A S | y = γ d S γ −1 | y = γ d S γ −1 | y = A S | y ,
(82)
which is the case if and only if there exist constant elements NY ∈ Isom(X,S ) such that γ −1 | y = NY γ −1 | y .
(83)
The elements NY , Y ∈ {A1 , B1 , . . . , A g , Bg } are the Chern-Simons analogue of the group isomorphisms (64) in the geometrical formulation. They contain all information
726
C. Meusburger j
i
i−1 i−2
g−1
bj
ai
ai
bj g
h i−2
2
1
p
Fig. 4. Generators and dual generators of the fundamental group π1 (Sg )
p
p4g−3 p4g−2
4g−4
ag
bg
p
8
p
p7
4g−1
p =p 4g
0
1
6
1
a1 p
p
NA
b2
NB1
b
a2
1
p2
p
5
p p
4
3
Fig. 5. The polygon Pg
about the physical state and are closely related to the holonomies Ai , Bi along the generators ai , bi of the fundamental group. These holonomies are given by the value of the trivialising function on the corners of the polygon Pg [13], Ai = γ ( p4i−3 )γ ( p4i−4 )−1= γ ( p4i−2 )γ ( p4i−1 )−1 Bi = γ ( p4i−3 )γ ( p4i−2 )−1= γ ( p4i )γ ( p4i−1 )−1 ,
(84)
and satisfy a single relation arising from the defining relation of the fundamental group π1 (Sg ), −1 [Bg , A−1 g ] · · · [B1 , A1 ] ≈ 1
[Bi , Ai−1 ] = Bi · Ai −1 · Bi −1 · Ai .
(85)
Geometrical (2+1)-Gravity and the Chern-Simons Formulation
727
Via the overlap condition (82), one can relate the value of the trivialising function γ at the corners of the polygon Pg to its value at a given corner p0 , −1 −1 γ −1 ( p4i ) = N Hi N Hi−1 · · · N H1 γ −1 ( p0 ) = γ −1 ( p0 )H1−1 · · · Hi−1 Hi ,
(86)
−1 −1 −1 Hi Ai+1 , γ −1 ( p4i+1 ) = N A−1 N B−1 N Ai+1 N Hi · · · N H1 γ −1 ( p0 ) = γ −1 ( p0 )H1−1 · · · Hi−1 i+1 i+1 −1 −1 −1 Hi Ai+1 Bi+1 , γ −1 ( p4i+2 ) = N B−1 N Ai+1 N Hi · · · N H1 γ −1 ( p0 ) = γ −1 ( p0 )H1−1 · · · Hi−1 i+1 −1 −1 −1 −1 −1 −1 −1 γ ( p4i+3 ) = N Ai+1 N Hi · · · N H1 γ ( p0 ) = γ ( p0 )H1 · · · Hi−1 Hi Ai+1 Bi+1 Ai+1 , N Hi = [N Bi , N A−1 ], Hi = [Bi , Ai−1 ] i
which allows one to express the holonomies Ai , Bi along the generators ai , bi ∈ π1 (Sg ) in terms of the Poincaré elements N Ai , N Bi in the overlap condition (82) and vice versa −1 −1 −1 · · · NH NH · N Bi · N Hi−1 · · · N H1 γ −1 ( p0 ), Ai = γ ( p0 )N H 1 i−1 i
(87)
−1 −1 −1 Bi = γ ( p0 )N H · · · NH NH · N Ai · N Hi−1 · · · N H1 γ −1 ( p0 ), 1 i−1 i −1 −1 N Ai =γ −1 ( p0 )H1−1 · · ·Hi−1 Hi Bi Hi−1 · · ·H1 γ ( p0 ), −1 −1 −1 −1 N Bi =γ ( p0 )H1 · · ·Hi−1 Hi Ai Hi−1 · · ·H1 γ ( p0 ).
(88)
Up to conjugation with the value γ ( p0 ) of the trivialising function at the basepoint, the expressions (87) and (88) relating the holonomies Ai , Bi and the group elements N Ai , N Bi are of the same form. This reflects the fact that, up to conjugation with γ ( p0 ), the elements N Ai , N Bi are the holonomies along another system of generators a i , bi ∈ π1 (Sg ) pictured in Fig. 4 and given by −1 a i = h −1 1 ◦ . . . ◦ h i ◦ bi ◦ h i−1 ◦ . . . ◦ h 1 −1 bi = h −1 1 ◦ . . . ◦ h i ◦ ai ◦ h i−1 ◦ . . . ◦ h 1 .
(89)
These generators are investigated in detail in [34], where it is shown that their representatives can be viewed as a dual graph for the curves representing ai , bi ∈ π1 (Sg ) and that they can be used to determine the intersection points of a general embedded curve on Sg with the generators ai , bi . In the following we will therefore refer to the generators a i , bi as the dual generators. As the elements N Ai , N Bi ∈ Isom(X,S ) in the overlap condition or, equivalently, the holonomies Ai , Bi ∈ Isom(X,S ) contain all information about the physical state, they can be used to parametrise the phase space of the theory. Taking into account that these variables are subject to a constraint (85) and that gauge transformations on the surface Sg act on the holonomies Ai , Bi by simultaneous conjugation, one finds that the moduli space of flat Isom(X,S )-connections on Sg is given as the quotient Isom(X,S )
Mg
= {(A1 , B1 , . . . , A g , Bg )
−1 ∈ Isom(X,S )2g | [Bg , A−1 g ] · · · [B1 , A1 ] = 1}/Isom(X,S ).
(90)
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C. Meusburger
4.3. Phase space and Poisson structure. The advantage of the Chern-Simons formulation of (2+1)-dimensional gravity is that it allows one to give a rather simple description of the Poisson structure on the phase space, which is based on its parametrisation (90) in terms of the holonomies along a set of generators ai , bi ∈ π1 (Sg ) [13, 12]. In the following we will use the formalism by Fock and Rosly [12], which is defined for ChernSimons theory with a general gauge group H and parametrises the Poisson structure on the moduli space MgH in terms of an auxiliary Poisson structure on the manifold H 2g . The description is summarised in the following theorem. Theorem 4.1 (Fock,Rosly [12]). Consider Chern-Simons theory with gauge group H on a manifold R × Sg . Denote by Ta , a = 1, . . . , dim H , a basis of the Lie algebra h = Lie H and by tab the matrix representing the Ad-invariant symmetric bilinear form in the Chern-Simons action (69) with respect to this basis tab = Ta , Tb
t ab tbc = δca .
(91)
Let r = r ab Ta ⊗ Tb be a classical r -matrix for the gauge group H , i. e. an element r ∈ h ⊗ h which satisfies the classical Yang Baxter equation (CYBE) [[r, r ]] = [r12 , r13 ] + [r12 , r23 ] + [r13 , r23 ] = 0, (92) r12 = r ab Ta ⊗ Tb ⊗ 1, r13 = r ab Ta ⊗ 1 ⊗ Tb , r23 = r ab 1 ⊗ Ta ⊗ Tb , and whose symmetric part is dual to the bilinear form (91) r = r(s) + r(a) , r(a) = 21 (r ab − r ba )Ta ⊗ Tb , r(s) = 21 (r ab + r ba )Ta ⊗ Tb = 21 t ab Ta ⊗ Tb .
(93)
Consider the manifold H 2g , where the different copies of H are identified with the holonomies Ai , Bi ∈ H along a set of generators of the fundamental group π1 (Sg ) and denote by RaX , L aX , X ∈ {A1 , B1 , . . . , A g , Bg } the left-and right-invariant vector fields associated to a basis of h and the different components of H 2g , d |t=0 f (. . . , e−t Ta · X, . . .) dt d RaX f (A1 , . . . , Bg ) = |t=0 f (. . . , X · et Ta , . . .). dt L aX f (A1 , . . . , Bg ) =
(94)
Then, the bivector B ∈ Vec(H ) ⊗ Vec(H ), ⎛ ⎞ ⎛ ⎞ g g A A B B A A B B ab⎝ Ra j + L a j + Ra j + L a j ⎠ ⊗ ⎝ Rb j + L b j + Rb j + L b j ⎠ B = r(a) j=1
+ 21 t ab
g
j=1 A
A
B
B
(RaAi + L aAi + RaBi + L aBi ) ∧ (Rb j + L b j + Rb j + L b j )
i, j=1, i< j g 1 ab +2t RaAi i=1
∧ (RbBi + L bAi + L bBi ) + RaBi ∧ (L bAi + L bBi ) + L aAi ∧ L bBi ,
(95)
defines a Poisson structure on H 2g . After imposing the constraint (85) and dividing by the associated gauge transformations, which act by simultaneous conjugation of all
Geometrical (2+1)-Gravity and the Chern-Simons Formulation
729
components with H , this Poisson structure agrees with the canonical Poisson structure on the moduli space −1 MgH = {(A1 , B1 , . . . , A g , Bg ) ∈ H 2g | [Bg , A−1 g ] · · · [B1 , A1 ] = 1}/H.
(96)
In Fock and Rosly’s formalism, physical observables are given by functions on the manifold H 2g which are invariant under simultaneous conjugation of all components with the gauge group H . Note that the Poisson bracket of such observables with a general function g ∈ C ∞ (H 2g ) does not depend on the particular choice of the classical r -matrix but only on the matrix t ab representing the Ad-invariant, symmetric bilinear form in the Chern-Simons action. As the component of the bivector (95) which depends g on the antisymmetric component r(a) is proportional to terms of the form i=1 RaAi + L aAi + RaBi + L aBi , this contribution vanishes if one of the functions is invariant under simultaneous conjugation of its arguments with H . A particular set of physical observables, in the following referred to as Wilson loop observables, are conjugation invariant functions of the holonomies along closed curves on Sg . As the equations of motion are a flatness condition on the gauge field, these observables do not depend on the curve itself but only on its homotopy equivalence class in π1 (Sg ) and are invariant under a change of the basepoint. In Fock and Rosly’s formalism, these observables are described by expressing the holonomy along an element η ∈ π1 (Sg ) as a product in the holonomies along the generators ai , bi ∈ π1 (Sg ), η = xrαr · · · x1α1 , xk ∈ {a1 , . . . , bg }, αk ∈ {±1}
⇒
Hη = X rαr · · · X 1α1 .
(97)
The Wilson loop observable f η ∈ C ∞ (H 2g ) associated to η ∈ π1 (Sg ) and a general conjugation invariant function f ∈ C ∞ (H ) on the gauge group is then given by f η : (A1 , . . . , Bg ) → f (Hη )
(98)
with Hη given as a product in the elements Ai , Bi and their inverses as in (97). It follows directly that the Wilson loop observables are invariant under simultaneous conjugation of all holonomies Ai , Bi with elements of H and satisfy f τ ◦η◦τ −1 = f η
∀η, τ ∈ π1 (Sg ).
(99)
In order to apply Fock and Rosly’s description [12] of phase space and Poisson structure to the Chern-Simons formulation of (2+1)-dimensional gravity, one needs classical r -matrices for the Lie algebras h,S such that the symmetric components of these r -matrices agree with the Ad-invariant, symmetric bilinear forms (70). For Lorentzian signature, such a classical r -matrix is given by r = PaL ⊗ JLa + n a abc JbL ⊗ JcL
(100)
with a constant vector n = (n 0 , n 1 , n 2 ) ∈ R3 satisfying η L (n, n) = . The corresponding r -matrix for the Euclidean case with < 0 has the form r E = PaE ⊗ J Ea + n a abc JbE ⊗ JcE
(101)
with a constant vector n = (n 0 , n 1 , n 2 ) ∈ R3 satisfying η E (n, n) = ||. Note that the choice of the classical r -matrix and hence the Poisson structure (95) is not necessarily unique - for a list of classical r -matrices for the (2+1)-dimensional Poincaré algebra see [35]. However, in the following we will only consider Poisson brackets where at least one of the functions is a Wilson loop observable so that our results do not depend on this choice.
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5. Trivialisation and Embedding As discussed in Sect. 3 and Sect. 4.2, the absence of local gravitational degrees of freedom manifests itself in the geometrical and the Chern-Simons description of (2+1)spacetimes in, respectively, the embedding of simply connected regions into the model spacetimes X,S and the trivialisation of the Chern-Simons gauge field. In this section, we discuss the relation between these concepts and show how the embedding of spacetime regions can be constructed from the function trivialising the Chern-Simons gauge field. We consider a simply connected region R ⊂ R × Sg in the spacetime manifold with a metric g and a flat Chern-Simons gauge field A, related to the metric via (66). We denote by X ,S : R → X,S the embedding into the model spacetime X,S and by γ : R → Isom(X,S ) a function which trivialises the gauge field as in (79). The decomposition (68) of the gauge field in terms of the generators JaS , PaS then implies that the dreibein ea and the spin connection ωa are given by ea = γ dγ −1 , JaS
ωa = γ dγ −1 , PaS ,
(102)
where JaS , PaS denote the generators of the Lie algebras (3), (4) and , the Ad-invariant bilinear form (70) in the Chern-Simons action. The expression (66) for the metric in terms of the dreibein relating then implies that the metric g on R takes the form S ea ⊗ eb = ηab γ dγ −1 g = ηab , JaS γ dγ −1, JbS = −4 det(η S ) det( γ dγ −1 , JSa JaS ). S (103)
On the other hand, the metric g must agree with the pull-back of the metric in the model spacetime X,S via the embedding X ,S . To relate the trivialising function γ : R → Isom(X,S ) to the embedding X ,S : R → X,S , one therefore has to construct a function ,S : Isom(X,S ) → X,S from the isometry group into the model spacetime such that the pull-back of the metric in the model spacetime via ,S ◦ γ −1 agrees with the metric (103), −1 S −1 S d(,S ◦ γ −1 )2 = ηab S γ dγ , Ja γ dγ , Jb .
(104)
Furthermore, as the embedding of the region R into the model spacetime X,S is only defined up to a global action of the isometry group Isom(X,S ), two embeddings related by such an action of the isometry group should correspond to the same gauge field on R. This the case if and only if the action of Isom(X,S ) on X,S corresponds to leftmultiplication of the trivialising function γ −1 → N γ −1 , N ∈ Isom(X,S ), i. e. if the function ,S : Isom(X,S ) → X,S satisfies the condition ,S (N γ −1 ) = N ,S (γ −1 )
∀N ∈ Isom(X,S ).
(105)
This suggests that the functions ,S : Isom(X,S ) → X,S should be defined as 0,L (v, x) = x >0,L (v+ , v− ) = <0,L (v) = <0,E (v) =
∀(v, x) ∈ P SU (1, 1) R3 , −1 √1 v+ v− ◦
√1 vv || √1 vv † ||
∀(v+ , v− ) ∈ P SU (1, 1) × P SU (1, 1), ∀v ∈ S L(2, C)/Z2 , ∀v ∈ S L(2, C)/Z2 ,
(106)
Geometrical (2+1)-Gravity and the Chern-Simons Formulation
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since this ensures that identity (105) is satisfied. It remains to show that these functions yield the right metric on R, i. e. that identity (104) holds for each value of the cosmological constant and each signature of the spacetime. The simplest case is the one with Lorentzian signature and vanishing cosmological constant, which is investigated in [14, 23]. Parametrising the trivialising function as γ −1 = (v, x) with v : R → P SU (1, 1), x : R → R3 and using the group multiplication law (11), one finds γ dγ −1 = ωa JaL + ea PaL = v −1 dv + v −1 d x
ea PaL = v −1 d x, ωa JaL = v −1 dv. (107)
The Lorentz invariance of the (2+1)-dimensional Minkowski metric then implies that the metric given by (103) agrees with the pull-back of the (2+1)-dimensional Minkowski metric via 0,L ◦ γ −1 , −1 −1 2 2 2 g = ηab L (v d x)a (v d x)b = −d x 0 + d x 1 + d x 2 .
(108)
For Lorentzian signature and > 0, the trivialising function can be parametrised as γ −1 = (γ+ , γ− ) : R → P SU (1, 1) × P SU (1, 1). Using the relation between the generators Ja , Pa of the Lie algebra (3) and the alternative generators Ja± defined by (12), we find that the gauge field is given by γ dγ −1 = (ωa + ea )Ja+ + (ωa − ea )Ja− = γ+−1 dγ+ + γ−−1 dγ− ,
(109)
and the expression for the dreibein in terms of γ takes the form ea =
1 √ γ −1 dγ+ 2 +
− γ−−1 dγ− , JaL = γ−−1 · (γ+ γ−−1 )−1 d(γ+ γ−−1 ) · γ− , JaL . (110)
To prove that the metric defined by this dreibein via (103) agrees with the one induced by the metric on the model spacetime X,S , we use the following lemma, which can be proved by direct calculation. Lemma 5.1. For general M ∈ SU (1, 1) parametrised as in (9), we have M −1 d M = 2ea JaL
⇒
det d M = det(M −1 d M) = e02 − e12 − e22 = |da|2 − |db|2 . (111)
Applying this lemma to γ+ γ−−1 together with expression (110) for the dreibein and using the Ad-invariance of the determinant, we find that the metric defined by (103) agrees with the pull-back of the AdS-metric via the embedding <0,L ◦ γ −1 , g = − 1 det(d(γ+ γ−−1 )) = ηab γ dγ −1 , JaL γ dγ −1 , JbL .
(112)
For Lorentzian and Euclidean signature and < 0, the gauge field takes the form
(113) γ dγ −1 = ωa JaS + ea PaS = (ωa + i ||ea )JaS . To prove that the metric defined via (103) agrees with the pull-back of the metric on the model spacetime via ,S ◦ γ −1 we apply the following lemma to M = γ −1 (γ −1 )◦ and M = γ −1 (γ −1 )† .
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Lemma 5.2. For general M ∈ S L(2, C) we have
1 M −1 d M = (ωa + i ||ea )JaL ⇒ −e02 + e12 + e22 = || det(d(M M ◦ )), (114)
1 M −1 d M = (ωa + i ||ea )JaE ⇒ e02 + e12 + e22 = − || det(d(M M † )). (115) Proof. The proof is a straightforward calculation. Parametrising the matrix M ∈ S L(2, C) as u v uz − vw = 1, (116) M= w z we find that the S L(2, C) matrices M M ◦ and M M † are given by
|u|2 − |v|2 v z¯ − u w¯ MM = −(vz ¯ − uw) ¯ |z|2 −|w|2 ◦
MM = †
|u|2 + |v|2 v z¯ + u w¯ vz ¯ + uw ¯ |z|2 + |w|2
, (117)
and, after some computation det(d(M M ◦ )) = |wdu|2 + |udw|2 + |zdv|2 + |vdz|2 − 2|udz − vdw|2 +2Re(dudz − dvdw − v z¯ d vd ¯ z¯ − u wd ¯ udw), ¯ † 2 2 2 2 det(d(M M )) = −(|wdu| + |udw| + |zdv| + |vdz| ) − 2|udz − vdw|2 +2Re(dudz − dvdw + v z¯ d vd ¯ z¯ + u wd ¯ udw). ¯
(118) (119)
On the other hand, expanding M −1 d M = (ωa + iea )JaL yields
||e0 = 2Re(udz −wdv),
||e1 = −Re(udw−wdu)+Re(zdv−vdz),
||e2 = Im(udw−wdu)+Im(zdv−vdz),
(120)
while the corresponding expressions for the Euclidean case are given by
||e0 = −2Re(udz −wdv),
||e1 = Re(udw−wdu)+Re(zdv−vdz),
||e2 = −Im(udw−wdu)+Im(zdv−vdz).
(121)
After some further computation using udz − wdv = vdw − zdu we obtain (114), (115). Hence, we have shown for all values of the cosmological constant and all signatures under consideration that the maps ,S : Isom(X,S ) → X,S defined in (106) satisfy the identities (104) and (105). The embedding into the model spacetimes X,S characterised by these conditions is thus given by composing these maps with the trivialising function γ −1 : R → Isom(X,S ), X ,S = ,S ◦ γ −1 : R → X,S .
(122)
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6. Grafting in the Chern-Simons Formalism 6.1. Embedding into the regular domain and action of the group . After deriving explicit expressions which relate the embedding of a spacetime region into the model spacetimes X,S to the function trivialising the Chern-Simons gauge field, we will now apply these results to investigate the construction of (2+1)-spacetimes via grafting from the Chern-Simons viewpoint. The reasoning is similar to the one in [23] but does not make use of the simplifications specific to Lorentzian spacetimes with vanishing cosmological constant. To see how grafting manifests itself on the phase space of the associated Chern-Simons theory, one needs to determine how the variables parametrising the phase space, the holonomies Ai , Bi along a set of generators of the fundamental group π1 (Sg ), transform under the grafting construction. This requires relating these holonomies to the variables which encode the physical degrees of freedom in the geometrical description. For this, we recall that in the geometrical formulation of (2+1)-dimensional gravity, spacetimes are given as quotients of regular domains U,S ⊂ X,S by the action of G : → Isom(X,S ). This a cocompact Fuchsian group via a homomorphism h ,S action leaves the surfaces U,S (T ) of constant cosmological time T invariant, and the spacetime is given by identifying on each surface U,S (T ) the points related by this action of . The physical degrees of freedom are therefore encoded in the cocompact G : → Isom(X Fuchsian group and the group homomorphism h ,S ,S ). In the Chern-Simons formalism, the physical degrees of freedom are given by the holonomies Ai , Bi ∈ Isom(X,S ) or, equivalently, the elements N Ai , N Bi ∈ Isom(X,S ) which arise in the overlap condition (83). By cutting the manifold along the representatives of the generators of the fundamental group and trivialising the gauge field on the resulting region, one obtains a set of functions γ (x 0 , ·) : Pg → Isom(X,S ) on a 4g-gon Pg . The values of γ −1 at the two sides corresponding to a given generator are related by left-multiplication with the elements N Ai , N Bi , and it is shown in Sect. 5 that the left multiplication of the trivialising function γ −1 with elements of the isometry group Isom(X,S ) corresponds to the action of this group on the model space time X,S . This suggests identifying the parameter x 0 in the splitting (74) of the Chern-Simons gauge field with the cosmological time T , identifying the generators ai , bi ∈ π1 (Sg ) with the projection of the geodesics on U,S (T ) which are identified by the action of the generators v Ai , v Bi ∈ and to take the corner p0 of the resulting polygon Pg as the basepoint for the grafting. With these identifications, the embedding constructed from the trivialising function γ (x 0 , ·) : Pg → Isom(X,S ) for constant x 0 = T then maps the polygon Pg into the surface U,S (T ) of constant cosmological time T . As the sides of the embedded polygon in U,S (T ) are identified pairwise by the action h ,S (vY ), Y ∈ {A1 , B1 , . . . , A g , Bg }, of the generators of , this implies that the group elements in the overlap condition (82) must agree with the image of these generators under the group homomorphism h ,S : → Isom(X,S ), NY = h ,S (vY )
∀Y ∈ {A1 , B1 , . . . , A g , Bg }.
(123)
Using the formula (86) which expresses the value of the trivialising function at the corners pi of the polygon Pg in terms of the elements N Ai , N Bi and the value of γ at a point p0 ∈ Pg we can then determine the holonomies Ai , Bi and find that they are given by −1 −1 Ai = γ ( p0 ) · h ,S (v −1 H1 · · · v Hi v Bi v Hi−1 · · · v H1 ) · γ ( p0 ) , −1 −1 Bi = γ ( p0 ) · h ,S (v −1 H1 · · · v Hi v Ai v Hi−1 · · · v H1 ) · γ ( p0 ) .
(124)
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6.2. The transformation of the holonomies under grafting. We will now use the relation between the phase space variables N Ai , N Bi in the Chern-Simons formalism and the group homomorphism h ,S : → Isom(X,S ) in the geometrical description to derive the transformation of the holonomies Ai , Bi along a set of generators of the fundamental group under grafting. We start by considering the static universes associated to the cocompact Fuchsian group . As discussed in Sect. 3.2, the surfaces of constant cosmological time are then copies of hyperbolic space H2 embedded into the model spacetime via the maps : H2 → X,S defined in (50). The polygon Pg obtained by cutting the spatial sur ,S T T (P ) in the tessellation face Sg is embedded into the image fundamental polygon ,S of U,S (T ) induced by T X st ,S (T, ·) : Pg → ,S (P ) ⊂ U,S (T ).
(125)
The group homomorphism h st ,S : → Isom(X,S ) is given by the canonical embedding of P SU (1, 1) into the isometry group of the model spacetime h st ,S (v) = ı,S (v). Hence, using the identities (123) and (124), we find that, up to conjugation with the value of γ at the basepoint p0 ∈ Sg , all holonomies Ai , Bi are purely Lorentzian −1 −1 Ai = γ ( p0 ) · ı,S (v −1 H1 · · · v Hi v Bi v Hi−1 · · · v H1 ) · γ ( p0 ) ,
(126)
−1 −1 Bi = γ ( p0 ) · ı,S (v −1 H1 · · · v Hi v Ai v Hi−1 · · · v H1 ) · γ ( p0 ) .
We now consider the spacetimes obtained from these static spacetimes by grafting along a closed, simple geodesic η on H2k / with weight w. Again, the identification of the parameter x 0 in (74) with the cosmological time implies that for each value of the parameη ter x 0 = T the polygon Pg is embedded into a surface U,S (T ) of constant cosmological time. However, these surfaces are no longer copies of hyperbolic space but the deformed surfaces obtained by inserting a strip along each geodesic in the -invariant multicurve on U,S (T ) associated to η. The cocompact Fuchsian group acts on these surfaces η of constant cosmological time via the group homomorphism h ,S : → Isom(X,S ) defined by (62),(64), and the group elements in the overlap condition are given by η NY = h ,S (vY ). To derive a formula for the transformation of the holonomies Ai , Bi along the generators of the fundamental group, we consider a generic side y of the polygon Pg with starting point and endpoint piY , p Yf ∈ { p0 , . . . p4g }. Denoting by viY , v Yf ∈ the elements of the cocompact Fuchsian group that relate the value of the trivialising function γ −1 at the points piY , p Yf to its value at p0 in the static case γst−1 ( piY ) = ı,S (viY )γst−1 ( p0 )
γst−1 ( p Yf ) = ı,S (v Yf )γst−1 ( p0 ),
(127)
we can express the holonomy along y as η
η
Y = γ ( p Yf )γ −1 ( piY ) = γ ( p0 )h ,S (v Yf )−1 h ,S (viY )γ ( p0 )−1 .
(128)
η
Using identity (64) for the group homomorphism h ,S and the identities (63) we then obtain Y = γ ( p0 )ı,S (v Yf )−1 Bη,,S (q0 , v Yf q0 )−1 Bη,,S (q0 , viY q0 )ı,S (viY )γ ( p0 )−1 (129) = γst ( p Yf )Bη,,S (viY q0 , v Yf q0 )−1 γst ( piY )−1 = Yst · γst ( piY )Bη,,S (viY q0 , v Yf q0 )−1 γst ( piY )−1 ,
Geometrical (2+1)-Gravity and the Chern-Simons Formulation
735
where γst , Yst denote, respectively, the trivialising function and the holonomy along y in the static universe associated to , q0 ∈ H2 the basepoint for the grafting, which we took to coincide with the embedding of the corner p0 and Bη,,S is given by (62). Setting piAi = p4(i−1) , p Af i = p4i−3 and piBi = p4i−2 , p Bf i = p4i−3 in (129) and using expression (86) for the value of the trivialising function γ at the corners of the polygon Pg , we find that the group elements viAi , viBi , v Af i , v Bf i are given by viAi = v Hi−1 · · · v H1 , viBi = v −1 Bi v Ai v Hi−1 · · · v H1 ,
−1 v Af i = v −1 Ai v Bi v Ai v Hi−1 · · · v H1 , −1 v Bf i = v −1 Ai v Bi v Ai v Hi−1 · · · v H1 ,
(130)
and that the transformation of the holonomies Ai , Bi under grafting along η takes the form st st Aist → Aist ·(Hi−1 · · · H1st ) γ ( p0 )Bη,,S (viAi q0 , v Af i q0 )−1 γ −1 ( p0 ) (Hi−1 · · · H1st )−1 ,
(131) st · · · H1st ) γ ( p0 )Bη,,S (viBi q0 , v Bf i q0 )−1 γ −1 ( p0 ) Bist → Bist · (Bist −1 Aist Hi−1 st ×(B st −1 Aist Hi−1 · · · H1st )−1 .
We will now evaluate this formula for the case of a simple, closed geodesic η with weight w on H2k / using the concrete expression (62) for the map Bη,,S : H2 × H2 → Isom(X,S ). As discussed in Sect. 3, the geodesic η lifts to a -invariant multicurve on H2k , H2
k = {Ad(v)cη | v ∈ }, G η,
(132)
where cη : R → H2k , cη (0) ∈ P is the lift of η with basepoint in the fundamental polygon P ⊂ H2k and with translation element a
cη (1) = vη cη (0) = en η Ja ,
cη (t) = vcη (0) ∀v ∈ , t ∈ (0, 1).
(133)
In order to evaluate the formula (62) for the multicurve (132), we need to determine which geodesics in (132) intersect a given side of the fundamental polygon P and to derive their translation vectors. For this, we note that a geodesic c = vcη , v ∈ , intersects a side of P if and only if cη intersects the corresponding side of the polygon P = v −1 P . Hence, the intersection points of geodesics in the multicurve (132) with the sides of P are in one-to-one correspondence with intersection points of cη |[0,1) : [0, 1) → H2k with polygons in the tessellation of H2k induced by . Furthermore, these intersection points are labelled by the factors in the expression of the translation element (133) as a product in the generators v Ai , v Bi of , which can be seen as follows. Because cη is the lift of closed, simple geodesic η on H2k / , it traverses a sequence of polygons in the tessellation of H2k induced by P1 = P ,
P2 = vr P ,
P3 = vr −1 vr P , . . . ,
Pr +1 = v1 · · · vr P = vη P , (134)
which are mapped into each other by group elements vi ∈ , until it reaches the point cη (1) = v1 · · · vr cη (0) = vη cη (0) ∈ Pr +1 identified with cη (0). As the elements of the Fuchsian group which map the polygon P into its neighbours are the generators v Ai , v Bi ∈ and their inverses, we find that the group element vr is of the form
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C. Meusburger
vr = v αXrr , with v X r ∈ {v A1 , . . . , v Bg }, αr ∈ {±1}. Similarly, for a general polygon P = v P the elements of which map this polygon into its neighbours are given by ±1 −1 −1 vv ±1 Ai v , vv Bi v , which implies that the group elements vk in (134) are of the form r vk = v αXrr · · · v αXk+1 v αk v −αk+1 · · · v −α Xr k+1 X k X k+1
(135)
with v X i ∈ {v A1 , . . . v Bg }, αi =∈ {±1}. In particular, the translation element vη in (133) and the associated translation vector nη are given by vη = v1 · · · vr = v αXrr · · · v αX11 = en η Ja . a
(136) H2
k Hence, intersection points of the geodesics in the multicurve G η, with a given side y ∈ {a1 , b1 , . . . , ag , bg } of P are in one-to-one correspondence with factors v αXkk , X k = Y , in the expression (136) of the translation element vη in terms of the generators of . The geodesics in (132) which intersect the fundamental polygon P are therefore given by
α
c1 = cη , c2 = v αX11 cη , c3 = v αX22 v αX11 cη , . . . , cr = v Xrr−1 · · · v αX11 cη , (137) −1 and the associated translation vectors take the form α
nck = Ad(v Xk−1 · · · v αX11 )nη . k−1
(138)
Furthermore, we note that the geodesic in (137) which intersects the side y = xk is ck if αk = 1 and ck+1 if αk = −1. Taking into account the orientation of the sides ai , bi in the polygon Pg , see Fig. 5, we find that intersections with sides ai have positive intersection number for αk = 1 and negative intersection number for αk = −1, while the intersection numbers for sides bi are positive and negative, respectively, for αk = −1 and αk = 1. With the definition (Y ) = 1 for Y = Ai , (Y ) = −1 for Y = Bi and α Ad(v Xk−1 · · · v αX11 )nη for αk = 1 k−1 (139) nk = αk for αk = −1, Ad(v X k · · · v αX11 )nη we can express the group elements Bη,,S (viY q0 , v Yf q0 ) in (131) as
Bη,>0,L (viY q0 , v Yf q0 ) = (
X k =Y
e
αk nˆ k , X k =Y √ −w (Y )αk nˆ ak JaL
Bη,=0,L (viY q0 , v Yf q0 ) = −w(Y )
(140)
,
ew
√
(Y )αk nˆ ak JaL
X k =Y
Bη,<0,L (viY q0 , v Yf q0 ) = Bη,<0,E (viY q0 , v Yf q0 ) =
e−i
√
),
||w(Y )αk nˆ ak JaL
,
X k =Y
where the factors are ordered from the left to the right in the order in which the intersection points occur on the generator y ∈ {a1 , . . . , bg }. Inserting formula (140) into (131), then yields an expression for the transformation of the holonomies under grafting along η in terms of the translation vector of cη . We obtain the following theorem.
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Theorem 6.1. Consider a closed, simple geodesic η : [0, 1] → H2k / with weight w > 0 and its lift cη : [0, 1] → H2k with basepoint cη (0) ∈ P . Let vη ∈ be the translation element of cη defined as in (133) and given in terms of the generators v Ai , v Bi ∈ by (136) and denote by vηk the associated cyclic permutations of (136), α v k−1 · · · v α1 v αr · · · v αXkk αk = 1 k n ak Ja (141) = αXkk−1 α1 X 1αr X r αk+1 vη = e v X k · · · v X 1 v X r · · · v X k+1 αk = −1. Then the transformation of the holonomies Ai , Bi under grafting along η with weight w is given by ⎛ ⎞ −wα w st k Grη,,S : Aist → Aist · (Hi−1 · · · H1st )γ ( p0 ) ⎝ F (vηk )⎠ X k =Ai
st ×γ −1 ( p0 )(Hi−1 · · · H1st )−1 ,
,S
⎛
st · · · H1st )γ ( p0 ) ⎝ Bist → Bist · (Bist −1 Aist Hi−1
X k =Bi
×γ
−1
st ( p0 )(Bist −1 Aist Hi−1 ···
⎞
(142)
wαk k ⎠ F,S (vη )
H1st )−1 ,
where the factors are ordered from the right to the left in the order in which the associated intersection point occurs on the generators ai , bi and Fw : P SU (1, 1) → Isom(X,S ) is given by ⎧ ⎪ ˆ n) = 0, Lorentzian ⎨(1, w √ a L √ a L w n a JaL w J −w n ˆ J n ˆ a a ) = (e F,S (e (143) ,e ) > 0, Lorentzian ⎪ ⎩ i √||wnˆ a JaL e < 0, Lorentzian and Euclidean. nˆ = √ 1 2 n ∈ R3 . |n | 7. Grafting and Poisson Structure 7.1. The transformations generated by the physical observables. Theorem 6.1 gives a formula for the transformation of the holonomies Ai , Bi for a static spacetime under grafting along a closed simple geodesic η on H2k / with weight w. In this section, we will demonstrate that the associated one-parameter group of diffeomorphisms on the phase space is generated via the Poisson bracket by a gauge invariant Hamiltonian. We find that for all values of the cosmological constant and all signatures under consideration this Hamiltonian is a Wilson loop observable associated to η and constructed from an Ad-invariant, symmetric bilinear form on the Lie algebra h,S . In the Chern-Simons formulation of (2+1)-dimensional gravity, the Poisson brackets of Wilson loop observables were first investigated in the work of Regge and Nelson [5–7, 9, 8] and by Ashtekar, Husain, Rovelli and Smolin [10], for the case of a punctured disc see also Martin [11]. In a mathematical context, the Poisson brackets of Wilson loop observables and the associated flows on phase space were first derived in the classical paper [36] by Goldman, who considers the moduli space of flat H -connections on surfaces Sg and for general groups H . However, the formulation in [36] is rather abstract
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C. Meusburger
and does not characterise these flows in terms of the holonomies of a set of generators of the fundamental group π1 (Sg ). It is shown in [34] that Fock and Rosly’s description of the phase space [12] allows one to obtain concrete expressions for the transformations of these holonomies under the flow generated by the Wilson loop observables associated to a general simple curve on Sg and a general conjugation invariant function of the gauge group. The results are valid for Chern-Simons theory with a general gauge group H and can be summarised as follows. Theorem 7.1 ([34], see also [36]). Consider Fock and Rosly’s description of the moduli space MgH of flat H -connections on Sg with the notation introduced in Theorem 4.1. Let η, λ be closed, simple curves on Sg , whose homotopy equivalence classes are given, respectively, as a product of the dual generators a i , bi defined in (89) and as a product in the generators ai , bi and their inverses α
r −1 ◦ . . . ◦ x α1 1 , η = x rαr ◦ x r −1
λ=
ysβs
◦
βs−1 ys−1
◦ ... ◦
x k ∈ {a 1 , b1 , . . . , a g , b g }, αk ∈ {±1},
β y1 1 ,
(144)
yk ∈ {a1 , . . . , bg }, βk ∈ {±1}.
Denote by ηk , λk the cyclic permutations of the products in (144) ηk = λk =
α
k−1 x k−1 ◦ . . . ◦ x α1 1 ◦ x rαr ◦ . . . ◦ x αk k αk = 1 αk k+1 x k ◦ . . . ◦ x α1 1 ◦ x rαr ◦ . . . ◦ x αk+1 αk = −1
β
β
β
(145)
β
k−1 ◦ . . . ◦ y1 1 ◦ ys s ◦ . . . ◦ yk k βk = 1 yk−1 β β β βk+1 yk k ◦ . . . ◦ y1 1 ◦ ys s ◦ . . . ◦ yk+1 βk = −1
(146)
and by Hηk , Hλk the associated holonomies. Then, the intersection points of η and λ, respectively, with curves representing the generators ai , bi ∈ π1 (Sg ) and a i , bi are in one-to-one correspondence with factors x k = a i , x k = bi in (145) and factors yk = ai , yk = bi in (146) and the exponents αk , βk determine the associated oriented intersection numbers. Furthermore, let f η , h λ be the Wilson loop observables associated to η, λ and to conjugation invariant functions f, h ∈ C ∞ (H ) of the gauge group. Then, the following statements hold. 1. The Poisson bracket of the Wilson loop observables f η , h λ is given by {h λ , f η }(A1 , . . . , Bg ) =
g
−1 αk βl g f (Hηk ), gh (Hi−1 · · · H1 Hλl H1−1 · · · Hi−1 )
i=1 x k=a i ,yl=ai
−
g
−1 −1 αk βl g f (Hηk ), gh (Bi−1 Ai Hi−1 · · · H1 Hλl H1−1 · · · Hi−1 Ai Bi )) ,
i=1 x k=bi ,yl =bi
(147) where g f , gh : H → h are defined by the action of the left invariant vector fields Ra ∈ Vec(H ) on f, h ∈ C ∞ (H ), g f (u), Ta = Ra f (u),
gh (u), Ta = Ra h(u)
∀u ∈ H.
(148)
Geometrical (2+1)-Gravity and the Chern-Simons Formulation
739
t : H 2g → H 2g generated by the 2. The one-parameter group of diffeomorphisms T f,η Wilson loop observable f η via the Poisson bracket acts on the holonomies Ai , Bi according to
d t |t=0 h ◦ T f,η ∀h ∈ C ∞ (H 2g ), dt −tα : Ai → Ai · (Hi−1 · · · H1 ) · G f k (Hηk ) · (Hi−1 · · · H1 )−1 ,
{ f η , h} = t T f,η
x k =a i
Bi → Bi ·
(Bi−1 Ai Hi−1 · · ·
H1 ) ·
k G −tα (Hηk ) · (Bi−1 Ai Hi−1 · · · H1 )−1 , f
x k =bi
(149) where the factors are ordered from the right to the left in the order in which the associated intersection points occur on the curves representing the generators ai , bi and the function G tf : H → H is obtained by exponentiating the function g f : H → h, G tf (u) = etg f (u)
∀u ∈ H.
(150)
A particular set of Wilson loop observables which will be relevant in the following are the observables associated to Ad-invariant, symmetric bilinear forms κ ∈ h∗ ⊗ h∗ on the Lie algebra of the gauge group. These observables are constructed using the parametrisation via the exponential map exp : h → H by setting κ(e ˜ k
aT a
) = 21 κ(k a Ta , k a Ta ) = 21 κab k a k b .
(151)
As it is supposed in [34] that the exponential map is surjective, but not necessarily bijective, one has to restrict the range of admissible vectors k ∈ Rdim H to obtain a well-defined expression. The resulting function κ˜ : H → R is then not necessarily continuous everywhere. However, in the following we are interested in the local properties of the phase space. As the exponential map is locally bijective, any element u ∈ H has a neighbourhood in which the Wilson loop observable κ˜ : H → R is a C ∞ -function. It has been shown by Goldman [36] that the associated functions gκ˜ : H → h, G tκ˜ : H → H defined in (148), (150) then take the form gκ˜ (ek
aT a
) = l κ (k a Ta )
G tκ˜ (ek
aT a
) = et l κ (k
aT ) a
(152)
with a linear map l κ : h → h given by l κ (x), y = x, l κ ( y) = κ(x, y)
∀x, y ∈ h.
(153)
In particular, one obtains a generic set of observables constructed from the Ad-invariant, non-degenerate, symmetric bilinear form , in the Chern-Simons action with associated conjugation invariant function t˜ ∈ C ∞ (H ), t˜(ek
aT a
) = 21 k a Ta , k a Ta = 21 tab k a k b .
(154)
In this case, the linear map l , : h → h is the identity l , = idh and the associated functions gt˜, G tt˜ in (148), (150) are given by gt˜(e x ) = x
G tt˜(e x ) = et x
∀x ∈ h.
(155)
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C. Meusburger
The transformations (149) the associated observables t˜λ generate via the Poisson bracket are investigated in [34], for the case of semidirect product gauge groups G g∗ , see also [37], where it is shown that they have the interpretation of infinitesimal Dehn twists along λ. We will discuss these observables and the associated flows on phase space in more detail in Sect. 7.3 and Sect. 8.3, where we investigate the relation between grafting and infinitesimal Dehn twists. 7.2. Hamiltonians for grafting. Using the results from [34, 36], summarised in the last subsection, we can now investigate the transformations on phase space generated by observables associated to certain Ad-invariant, symmetric bilinear forms in the ChernSimons formulation of (2+1)-dimensional gravity and show that they agree with the grafting transformation (142). The first step is to identify the Ad-invariant bilinear forms on the Lie algebras h,S . It is shown in [4] that for all values of the cosmological constant and all signatures under consideration, the space of Ad-invariant, symmetric bilinear forms on h,S is two-dimensional. Besides the pairing , in the Chern-Simons action given by (70), the Lie algebras h,S admit another Ad-invariant symmetric bilinear form, which is given in terms of the generators JaS , PaS by S κ(JaS , JbS ) = ηab
κ(JaS , PbS ) = 0
S κ(PaS , PbS ) = ηab .
(156)
It follows that the associated linear maps l κ : h,S → h,S defined in (153) take the form l κ (JaS ) = PaS l κ (PaS ) = JaS .
(157)
: Isom(X,S ) → Isom(X,S ) defined in (150) is obtained The associated map by exponentiating this map l κ : h,S → h,S as in (152). As the exponential maps exp,S : h,S → Isom(X,S ) are surjective and therefore locally bijective for all isometry groups Isom(X,S ), see the remark after (16), we obtain a one-parameter group of transformations G tκ,,S : Isom(X,S ) → Isom(X,S ) given by ˜ G tκ,,S ˜
(exp,S ( pa JaS + k a PaS )) G tκ,,S ˜ = exp,S (t ( pa Pas + k a JaS )) ⎧ (1, t p) ⎪ ⎪ ⎪ t √( pa +√k a )J L −t √( pa −√k a )J L ⎨ a ,e a ) (e √ = a +i √||k a )J L it ||( p a ⎪ e ⎪ ⎪ ⎩ it √( pa +i √k a )J E a e
(158) = 0, > 0, < 0, < 0,
Lorentzian Lorentzian Lorentzian Euclidean,
: Isom(X,S ) → Isom(X,S ) as and can formally express the map G tκ,,S ˜ G tκ,=0,L (e p ˜
aJL a
, a) = (1, t p)
√ √ t −t (u , u ) = (u , u ) G tκ,>0,L + − − + ˜ √ (u) = G t<0,E (u) = u it || G tκ,<0,L ˜
∀ p, a ∈ R3 ,
(159)
∀u ± ∈ P SU (1, 1), ∀u ∈ S L(2, C)/Z2 .
The fact that the exponential map exp,S : h,S → Isom(X,S ) is locally, but not globally bijective, implies that the maps (158), (159) are only defined locally. In order
Geometrical (2+1)-Gravity and the Chern-Simons Formulation
741
to obtain a unique parametrisation of elements of Isom(X,S ) in terms of elements of h,S , one has to restrict the range of the vectors p, k ∈ R3 appropriately, which implies that the Wilson loop observable κ˜ and the map G tκ,,S : Isom(X,S ) → Isom(X,S ) ˜ are not necessarily continuous everywhere. However, as the exponential map is locally bijective, every element of Isom(X,S ) has a neighbourhood in which the Wilson loop observable κ˜ is C ∞ and the map G tκ,,S a one-parameter group of diffeomorphisms. ˜ In particular, the parametrisation via the exponential map is unique for elements of the group P SU (1, 1) embedded into Isom(X,S ) via (14), see the remark after (10), which is the form the holonomies take for the static universes. Hence, the parametrisation via the exponential map is well-defined and C ∞ in this case, and we can insert expressions (158), (159) into the general formula (149). We obtain the following theorem, which generalises Theorem 5.2. in [23]. Theorem 7.2. Let η be a closed simple curve on Sg whose homotopy equivalence class is given as a product in the dual generators (89) and their inverses as in (144) and κ the Ad-invariant, symmetric bilinear form on h,S defined in (156). Then, the one-parameter t group of diffeomorphisms Tκ,η,,S : Isom(X,S )2g → Isom(X,S )2g generated by the ˜ Wilson loop observable κ˜ η is given by d 2g t |t=0 g ◦ Tκ,η,,S ∀g ∈ C ∞ (H ), ˜ dt −tα k : Ai → Ai · (Hi−1 · · · H1 ) · G κ,,S (Hηk ) · (Hi−1 · · · H1 )−1 , ˜
{κ˜ η , g} = t Tκ,η,,S ˜
x k =a i
Bi →
Bi · (Bi−1 Ai Hi−1 · · ·
H1 ) ·
(160)
−tαk G κ,,S (Hηk ) · (Bi−1 Ai Hi−1 · · · H1 )−1 , ˜
x k =bi
where Hηk : Isom(X,S )2g → Isom(X,S ) represents the holonomy along ηk defined by (145), G tκ,,S : Isom(X,S ) → Isom(X,S ) is given by (158), (159) and the factors ˜ are ordered from the right to the left in the order in which the corresponding intersection points occur on the generators ai , bi . t : We will now demonstrate that the one-parameter group of transformations Tκ,η,,S ˜ 2g 2g Isom(X,S ) → Isom(X,S ) is related to the transformation (142) of the holonomies Ai , Bi under grafting along a closed simple geodesic η on the surface H2k / with homotopy equivalence class (144). For this, we evaluate (160) for the static case, where the group elements N Ai , N Bi in the overlap condition (83) are the images of the generators v Ai , v Bi ∈ under the canonical embedding (14) of P SU (1, 1) into the isometry group of the model spacetime N Ai = ı,S (v Ai ), N Bi = ı,S (v Bi ). From expressions (124) for the holonomies Ai , Bi in terms of N Ai , N Bi it then follows that the holonomies along the elements ηk defined in (145) take the form
Hηk = γ ( p0 )vηk γ −1 ( p0 ) = γ ( p 0 )en k Ja γ −1 ( p0 ) a
L
(161)
: with vηk , nk given by (141). As it is shown in [36, 34] that the functions G tκ,,S ˜ Isom(X,S ) → Isom(X,S ) satisfy the covariance condition G tκ,,S (gug −1 ) = g · G t,S (u) · g −1 ˜
∀u, g ∈ Isom(X,S )
(162)
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C. Meusburger
we obtain an expression in terms of the translation vectors nk defined in (139) ⎧ −1 = 0, Lorentzian ⎪ 0) ⎨γ ( p0 )(1,√t nk )γ ( p√ a aJ L tαk t n J − tn −1 a a k k G κ,,S (Hηk ) = γ ( p0 )(e ,e )γ ( p0 ) > 0, Lorentzian ˜ √ ⎪ a L ⎩ γ ( p0 )eit ||n k Ja γ −1 ( p0 ) < 0, Lorentzian or Euclidean. By inserting this expression into (160) and comparing with the transformation of the holonomies under grafting given by (142), (143), we find that these transformations agree up to normalisation. Theorem 7.3. Consider a static spacetime, where the holonomies Ai , Bi along the generators of the fundamental group are given by −1 −1 Ai = γ ( p0 ) · ı,S (v −1 H1 · · · v Hi v Bi v Hi−1 · · · v H1 ) · γ ( p0 ) , −1 −1 Bi = γ ( p0 ) · ı,S (v −1 H1 · · · v Hi v Ai v Hi−1 · · · v H1 ) · γ ( p0 ) .
(163)
Then, the transformation (142) of the holonomies under grafting along a closed simple geodesic η on H2k / agrees with the transformation (160) generated by the observable κ˜ η if the parameter t in (160) is related to the weight w by t = w|nη |, w t Grη,,S (A1 , . . . , Bg ) = Tκ,η,,S (A1 , . . . , Bg ) ˜
for t = w |κ(nη , nη )| = w|nη |. (164)
The fact that the transformation of the holonomies Ai , Bi under grafting is generated by a gauge invariant observable allows us to directly deduce some of its properties, which are summarised in the following corollary, for the Lorentzian case with vanishing cosmological constant see also [23]. Corollary 7.4. t act by Poisson isomorphisms 1. The grafting transformations Tκ,η,,S ˜ t t t , g ◦ Tκ,η,,S } = { f, g} ◦ Tκ,η,,S { f ◦ Tκ,η,,S ˜ ˜ ˜
∀ f, g ∈ C ∞ (Isom(X,S )2g ). (165)
t 2. The grafting transformation Tκ,η,,S leaves the constraint (85) invariant and com˜ mutes with the associated gauge transformations by simultaneous conjugation with Isom(X,S ), t t (g A1 g −1 , . . . , g Bg g −1 ) = gTκ,η,,S (A1 , . . . , Bg )g −1 ∀g ∈ Isom(X,S ). Tκ,η,,S ˜ ˜
(166) 3. In the Lorentzian case with = 0, all grafting transformations commute. Proof. The first two statements are a direct consequence of the fact that the transfort are generated by the gauge invariant observable κ˜ η . The last statement mations Tκ,η,,S ˜ follows immediately from the formula (158).
Geometrical (2+1)-Gravity and the Chern-Simons Formulation
743
c
c 2 πw
Fig. 6. A Dehn twist with parameter w around a closed, simple geodesic c on a genus 2 surface
7.3. Grafting and Dehn twists. After demonstrating that the transformation of the holonomies under grafting along a closed, simple curve η on Sg is generated by the Wilson loop observable κ˜ η , we will now investigate the relation between this grafting transformation and the action of infinitesimal Dehn twists along η. Geometrically, an infinitesimal Dehn twist along a closed geodesic η on a two-surface H2k / with parameter w amounts to cutting the surface along η and rotating the edges of the cut by an angle 2π w as shown in Fig. 6. In Chern-Simons theory, infinitesimal Dehn twists along closed, simple curves on the spatial surface are present generically for any gauge group H and give rise to a transformation on the moduli space of flat H -connections. The action of (infinitesimal) Dehn twists in Fock and Rosly’s description [12] of the moduli space is investigated in [34], for the case of semidirect product groups G g∗ see also [37], where it is shown that they are generated by the gauge invariant observable constructed from the Ad-invariant bilinear form , in the Chern-Simons action. The results can be summarised as follows. Theorem 7.5 ([34, 37]). Consider Fock and Rosly’s description of the moduli space MgH in terms of the auxiliary Poisson structure (95) on H 2g . Let η be a closed, simple curve on Sg , whose homotopy equivalence class is given as a product in the dual generators (89) and their inverses as in (144). Denote by ηk the cyclic permutations of this expression as in (145) and by Hηk the associated holonomies. Then the transformation Tt˜t,η : H 2g → H 2g generated by the Wilson loop observable t˜η associated to η and the Ad-invariant symmetric bilinear form in the Chern-Simons action as in (154) is given by −1 k · (H Tt˜t,η : Ai → Ai · (Hi−1 · · · H1 ) · Hη−tα i−1 · · · H1 ) , k x k =a i
Bi →
Bi · (Bi−1 Ai Hi−1 · · ·
H1 ) ·
−1 k · (B −1 A H Hη−tα i i−1 · · · H1 ) , i k
(167)
x k =bi
where the elements ηk ∈ π1 (Sg ) with holonomies Hηk are defined as in (145) and the products are ordered from the right to the left in the order in which the associated intersection points occur on the generators ai , bi ∈ π1 (Sg ). For flow parameter t = −1, this transformation agrees with the transformation of the holonomies Ai , Bi under a Dehn twist around η, = Dη , Tt˜−1 ,η
(168)
744
C. Meusburger
where Dη : H 2g → H 2g is the diffeomorphism induced by the action of the Dehn twist along η on the generators of the fundamental group π1 (Sg ). Using the results from [34, 37] summarised in Theorem 7.5, we can now compare t the transformation of the holonomies Ai , Bi under the grafting transformations Tκ,η,,S ˜ in (160) with their transformations (167) under infinitesimal Dehn twists. For this, we note that the formulas (160), (167) differ only in the fact that instead of the factor Hηtαk k in (167), the grafting transformation (160) contains a factor tαk (Hηk ) = G 1κ,,S (Hηtαk k ), G κ,,S ˜ ˜
(169)
with the map G tκ,,S : Isom(X,S ) → Isom(X,S ) given by (159). For the Lorentz˜ ian case with = 0, the map G tκ,0,L assigns to each element of the gauge group ˜ 3 P SU (1, 1) R the translation vector associated to its Lorentz component G tκ,0,L (exp=0,L (( pa JaL + k a PaL ))) = G tκ,0,L ((e p ˜ ˜ = (1, t p) = exp0,L (t p
a
PaL ).
aJ a
, T ( p)k)) (170)
For static universes the holonomies Hηk in (160) and (167) are conjugated to elements of the (2+1)-dimensional Lorentz group P SU (1, 1) and grafting acts on the holonomies Ai , Bi by right-multiplication with the associated elements of the Lie algebra su(1, 1). Hence, for the Lorentzian case with vanishing cosmological constant, grafting along a closed, simple curve η can be viewed as the derivative or first-order approximation of a Dehn twist along η. For Lorentzian signature with > 0 and gauge group tαk P SU (1, 1)× P SU (1, 1), identity (159) allows one to express the factors G κ,>0,L (Hηk ) ˜ in (160) as √
tαk G κ,>0,L (Hηk ) = ((Hηαkk )+t ˜
, (Hηαkk )−t −
√
)
where Hηαkk = ((Hηαkk )+ , (Hηαkk )− ).
(171) √ Hence, we find that the grafting transformation (160) with √ parameter t along η acts on the first component as an infinitesimal Dehn twist along η with parameter t √ and as an infinitesimal Dehn twist with parameter −t on the second component of P SU (1, 1) × P SU (1, 1). In Lorentzian and Euclidean (2+1)-gravity with < 0, the factor G tκ,,S (Hηαkk ) in the expression (160) for the transformation of the holonomies ˜ under grafting is (Hηαkk ) = Hηitk G tκ,,S ˜
√
||αk
,
(172)
t with parameter t given by (160) can therefore and the grafting transformation Tκ,η,,S ˜ √ be viewed as a Dehn twist (167) with parameter it ||. We will come back to this relation between grafting and Dehn twist in Sect. 8.3, where we discuss the role of the cosmological constant as a deformation parameter. These relations between the transformation of the holonomies under grafting and Dehn twists for the different values of the cosmological constant and different signatures are mirrored in a relation for the Poisson brackets of the Wilson loop observables t˜η , κ˜ η associated to closed, simple curves η on Sg . By inserting the maps gt˜, gκ˜ : Isom(X,S ) → h,S into the formula (147) for the Poisson brackets of the Wilson loop observables and using the identity l 2κ = idh,S , we obtain the following theorem which generalises Theorem 5.4. in [23] for the case of Lorentzian signature with = 0.
Geometrical (2+1)-Gravity and the Chern-Simons Formulation
745
Theorem 7.6 (Symmetry relation for the observables). For any two closed, simple curves λ, η on Sg , the associated Wilson loop observables t˜λ , κ˜ λ and t˜η , κ˜ η satisfy the symmetry relations {t˜η , κ˜ λ } = {κ˜ η , t˜λ }
{κ˜ η , κ˜ λ } = {t˜η , t˜λ }.
(173)
Theorem 7.6 establishes a relation between the transformation of the gauge invariant observables t˜η , κ˜ η associated to a closed, simple curve η on Sg under infinitesimal Dehn twist and grafting along another closed simple curve λ. The first identities in (173) imply that, infinitesimally, the transformation of the observable t˜η under grafting along λ is the same as the transformation of κ˜ η under a Dehn twist along λ. The second identity states that the transformation of the observable κ˜ η under infinitesimal grafting along λ corresponds to the transformation of t˜η under an infinitesimal Dehn twist along λ, rescaled by a factor . 8. The Cosmological Constant as a Deformation Parameter 8.1. The geometrical description. In this section we restrict attention to (2+1)-spacetimes with Lorentzian signature and investigate the role of the cosmological constant as a deformation parameter in both the geometrical and the Chern-Simons formulation of (2+1)-dimensional gravity. In the geometrical formulation, the cosmological constant appears as a parameter in G ⊂ X the model spacetime X,L , the domains U,L ,L and in the group isomorphism G h ,L : → Isom(X,L ) which determines the action of the cocompact Fuchsian group G . The dependence of the domains U G on the sign of the cosmo on the domain U,L ,S logical constant is investigated in detail in the papers by Benedetti and Bonsante [21, 22] who show that Lorentzian spacetimes with ∈ {0, ±1} and the Euclidean case with = −1 can be related by rescalings and a Wick rotation compatible with the associated actions of the cocompact Fuchsian group . In contrast, our focus is on the role of the cosmological constant as a continuous parameter deforming these domains, the associated model spacetimes and group isomorphisms. We start by considering the static spacetimes associated to . As discussed in Sect. 3.2, these spacetimes are given as quotients of the interior of the forward lightcone U,L ⊂ X,L by the canonical action of via the embedding ı,S : P SU (1, 1) → Isom(X,S ) into the isometry group of the model spacetime. The parametrisation of these lightcones in terms of matrices and their foliation by copies of hyperbolic space are given by expressions (53), (54) and (55), respectively, for = 0, > 0 and < 0. For vanishing cosmological constant, the entries in the su(1, 1) matrix (24) parametrising the forward lightcone are given by the identification (19) of the disc model of hyperbolic space with the hyperboloids. For > 0, the entries in the P SU (1, 1)-matrices (28) which √ parametrise √ the model spacetime are given by Eq. (54) and involve the functions sin( T ), cos( T ). Using the identities √ √ 1 (174) √lim cos( T ) = 1 √lim √ sin( T ) = T, →0
→0
√ one finds that in the limit → 0, these parameters behave according to 2
√lim t2 (T, z) →0
1+|z| L = T 1−|z| 2 = x 0 (T, z),
√lim x 2 (T, z) →0
Re(z) L = T 1−|z| 2 = x 2 (T, z),
Re(z) √lim x 1 (T, z) = T 1−|z|2 →0 √lim t1 (T, z) →0
= ∞,
= x1L (T, z),
(175)
746
C. Meusburger
where x0L (T, z), x1L (T, z), x2L (T, z) denote the coordinates in the identification of the unit disc with the hyperboloid H21/T 2 given by (19). For < 0, the entries in the parametrisation (38) of the forward lightcone is given by (55). Using the identities
1 (176) √ lim cosh( ||T ) = 1 √ lim √|| sinh( ||T ) = T, ||→0
||→0
in (55), we obtain √ lim x 0 (T, z) ||→0
2
1+|z| L = T 1−|z| 2 = x 0 (T, z),
√ lim x 1 (T, z) ||→0
Re(z) L = T 1−|z| 2 = x 1 (T, z),
(177) √ lim x 2 (T, z) ||→0
=T
Re(z) 1−|z|2
=
x2L (T, z),
√ lim x 3 (T, z) ||→0
→ ∞.
√ Hence, for both positive and negative cosmological constant, in the limit || → 0 one coordinate in the parametrisation of the forward lightcone in the model spacetime X,L tends to infinity while the other coordinates converge to the corresponding coordinates parametrising the forward lightcone in Minkowski space. To determine the role of the cosmological constant as a deformation parameter of G ⊂X the domains U,L ,L associated to grafted spacetimes, we make use of a result by G on Benedetti and Bonsante [22] concerning the dependence of the grafted domains U,L the weight of the multicurve G. In [22], Proposition 4.7.1, they consider the multicurve t G on H2k obtained by multiplying all weights in a multicurve G with a factor t and the G , U tG ⊂ X associated domains U,L ,L for cosmological constant ∈ {0, ±1}. They ,L t G are rescaled by a factor 1/t, they converge to the show that if the grafted domains U,L G in Minkowski space in the limit t → 0, corresponding domain U0,L tG lim 1 U,L t→0 t
G = U0,L ,
∈ {0, ±1},
(178)
where the limit is understood in terms of the coordinates in a certain parametrisation of these domains given in [22]. Although they do not consider cosmological constants = 0, ±1, this result can be applied to determine the dependence of the grafted domains G on the cosmological constant . For this, it is sufficient to recall from Sect. 3.1 that U,L the parametrisation of the model spacetimes X,L for || = 0, 1 is√obtained by rescaling the associated matrices parametrising X±1,L with a factor 1/ ||. Furthermore, we found in Sect. 3.3, see in particular √ Eq. (62), that the weight of the multicurves for || = 1 had to be rescaled with a factor || to ensure that the associated geodesics are G ⊂X parametrised by arclength. The domains U,L ,L for non-vanishing cosmological G constant are therefore related to the associated domains U±1,L via the identity G = U,L
√ ||G √1 U sgn(),L ||
= 0,
(179)
and Eq. (178) implies G √ lim U,L ||→0
G = U0,L .
(180)
Hence, with an appropriate identification of the coordinates parametrising the domains √ G ⊂ X U,L in the model spacetimes, the limit || → 0 is defined and yields the ,L
Geometrical (2+1)-Gravity and the Chern-Simons Formulation
747
G in Minkowski space for both positive and coordinates parametrising the domain U0,L negative cosmological constant. Finally, we consider the role of the cosmological constant as a deformation paramG eter of the group homomorphism h ,L : → Isom(X,L ) by which the cocompact G Fuchsian group acts on the domains U,L ⊂ X,L . For this we note that the map 2 2 BG,,L : H × H → Isom(X,L ) in formula (62) satisfies
BG,,L ( p, q) = B√||G , sgn() , M ( p, q)
∀ p, q ∈ H2 , = 0,
(181)
which implies d √ √ | d =0
G G G h >0,L (v) = (h 0,L (v), −h 0,L (v)) ∈ su(1, 1) ⊕ su(1, 1)
√d |√ d || ||=0
G h <0,L (v)
∀v ∈ , (182)
=
G i h 0,L (v)
∈ sl(2, C)
∀v ∈ . (183)
Hence, one finds that in the geometrical description√of grafted (2+1)-spacetimes given by Benedetti and Bonsante [22], the square root || of the cosmological constant G ⊂ X plays the role of a deformation parameter for both the domains U,L ,L and the G isomorphisms h ,L : → Isom(X,L ) which determine how the group acts on these domains. In the former, it appears as a parameter in the coordinates parametrising the √ G in the model spacetimes X domains U,L ,L , and the limit || → 0 relates these coorG in Minkowski space. In the dinates to the parametrisation of the associated domain U0,L G : → Isom(X latter, it appears as a parameter in the group homomorphisms h ,S ,S ), G G and one finds that the group homomorphism h 0,L is given by the derivatives of h >0,L √ G and h <0,L with respect to ||. 8.2. The Chern-Simons description. In the Chern-Simons description of (2+1)spacetimes, the cosmological constant appears as a parameter in the Lie algebras h,L and the associated gauge groups Isom(X,L ). To clarify in what sense it can be viewed as a deformation parameter, one has to introduce a common framework which encompasses the Lie algebras (3) for all signs of the cosmological constant. Such a framework can be realised by interpreting the Lie algebras (3) as the (2+1)-dimensional Lorentz algebra over a commutative ring R with a -dependent multiplication law. Lemma 8.1. Consider the vector space R2 with the usual vector addition and a multiplication operation · : R2 × R2 → R2 which depends on a real parameter and is given by (a, b) · (c, d) = (ac + bd, ad + bc)
∀a, b, c, d ∈ R.
(184)
Then, (184) equips R2 with the structure of a commutative ring R = (R2 , +, ·) with the unit elements for the addition + and the multiplication · given by, respectively, (0, 0), (1, 0).
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In the following we will express elements of the ring R in terms of a formal parameter and write a + b for the element (a, b). Formally, the product of two elements (a, b), (c, d) ∈ R is then given by (a + b ) · (c + d ) = ac + (ad + bc) + bd2 ,
(185)
and to obtain agreement with the multiplication law (184), one must set 2 = , and the parameter can therefore be viewed as a formal square root of . For < 0, the commutative ring R is isomorphic to the field C and the formal √ √ parameter is the complex number = = i ||. For = 0, the formal parameter satisfies 20 = 0 like the one occurring in supersymmetry and corresponds to the formal parameter θ used to parametrise the (2+1)-dimensional Poincaré group in [11], for a more detailed discussion see also [23]. This implies that the commutative ring R0 is not a field, since elements of the form b, b ∈ R \ {0} are zero divisors b · c = 0
∀b, c ∈ R.
(186)
For > 0, the formal parameter satisfies 2 = > 0. Again, R is not a field √ and has zero divisors a ± a, a ∈ R \ {0}, which satisfy √ √ √ √ √ ( a + a)( a − a) = 0 ( a ± a)2 = 2a ( a ± a). (187) The ring R allows one to identify all Lie algebras h,L with brackets (3) with the (2+1)dimensional Lorentz algebra, only that now this Lie algebra is no longer considered as a Lie algebra over R but as a Lie algebra over the commutative ring3 R . This identification of the Lie algebras h,L with the (2+1)-dimensional Lorentz algebra over R generalises the concept of complexification of real Lie algebras and in the case of negative cosmological constant yields the complexification sl(2, C) = sl(2, R) ⊕ i sl(2, R). We consider the (2+1)-dimensional Lorentz algebra with generators Ja , a = 0, 1, 2, and bracket [Ja , Jb ] = abc Jc .
(188)
By identifying the generators Ja with the sl(2, R) matrices in (6), we obtain a R mod3 into the set sl(2, R ) of traceless two-by-two matrices with ule isomorphism from R 2 with entries in R or, equivalently, the set of endomorphisms of the R -module R vanishing trace form, x + y ∈ R2 →(x a + y a )Ja (189) 1 1 1 0 1 1 0 2 2 − 2 (x + y ) 2 (x + y ) + 2 (x + y ) . = 1 1 1 − 21 (x 0 + y 0 ) + 21 (x 2 + y 2 ) 2 (x + y ) The commutator of two sl(2, R ) matrices then agrees with the bracket obtained by extending (188) bilinearly in R , and with the identification Pa = Ja
(190)
one recovers the Lie bracket (3) of the real Lie algebras h,L . 3 Definitions and properties concerning Lie algebras over commutative rings can be found for instance in [38], Chapter 1, but in the following we will make only use of some basic concepts.
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Moreover, the identification of the Lie algebras g,L with sl(2, R ) allows one to relate the Ad-invariant, symmetric bilinear forms , and κ on h,L defined by (70), (156) to the Killing form of the (2+1)-dimensional Lorentz algebra. For this one extends the Killing form g K on sl(2, R) L g K (Ja , Jb ) = 21 ηab = Tr(Ja · Jb )
(191)
bilinearly to an Ad-invariant symmetric R -bilinear form g K : sl(2, R )×sl(2, R ) → R . Using the parametrisation (189) and comparing the resulting expressions with (70), (156) one obtains g K (( p + k), (q + m)) L L = 21 ( pa q b + k a m b )ηab + 21 ( pa m b + k a q b )ηab =
a 1 2 κ( p Ja
+ k Pa , q Jb + m a
b
b
Pb ) + 21 pa Ja
(192)
+ k Pa , q Jb + m Pb . a
b
b
Hence, the Ad-invariant symmetric forms κ and , on h,L are realised as the projections of the Killing form on sl(2, R ) on, respectively, the first and second component of the ring R = (R2 , +, ·). This generalises the situation for < 0, where these forms can be identified with the real and imaginary part of the Killing form on sl(2, C). Moreover, it sheds some light on the distinguished role played by the Ad-invariant symmetric bilinear forms , and κ on the Lie algebra h,L . While any linear combination of these two forms is again an Ad-invariant, symmetric bilinear form on h,L , the forms , and κ are the only ones that arise canonically from the Killing form on sl(2, R ). We will now demonstrate how the identification of the Lie algebra h,L with the (2+1)-dimensional Lorentz algebra over the commutative ring R gives rise to the associated matrix groups H,L . Although in general the exponential of Lie algebras over commutative rings cannot be defined in a straightforward manner, the particularly simple structure of the ring R allows us to obtain the groups H>0,L = SU (1, 1)×SU (1, 1) ∼ = S L(2, R)×S L(2, R), H0,L = SU (1, 1)R3 ∼ = S L(2, R)R3 and H<0,L = S L(2, C) by exponentiating matrices in sl(2, R ). For this, we consider the formal expression exp (x + y) =
∞ ((x a + y a )Ja )n n!
x + y ∈ sl(2, R ),
(193)
n=0
where ((x a + y a )Ja )n stands for the n th power of√the matrices (189). In the case of negative cosmological constant, we have = i || and expression (193) is the exponential map exp<0,L : sl(2, C) → S L(2, C). The case of vanishing cosmological constant is investigated in [23]. Using identity 20 = 0, one can express (193) as exp=0 ((x a + 0 y a )Ja ) =
∞ (x a Ja )n n=0
n!
+ 0
∞ n−1 (x a Ja ) L (y b Jb )(x c Jc )n−m−1 . n! n=0 m=0 (194)
To simplify this expression further, we move the factors y b Jb in (194) to the left and evaluate the resulting commutators using the formulas L m adkx a Ja (y b Jb ) · (x c Jc )m−k , [(x Ja ) , y Jb ] = k a
L
b
k=1
n−1 n m , = k+1 k
m=k
(195)
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C. Meusburger
which can be proved by induction. After some further computation, this yields exp=0 ((x a + 0 y a )Ja ) = (1 + 0 (T (x) y)b Jb ) · e x
aJ a
,
(196)
x a Ja
stands for the exponential of the sl(2, R)-matrix x a Ja given by (10) and where e the linear map T (x) : R3 → R3 is the one defined in (16). By comparing with the parametrisation of the (2+1)-dimensional Poincaré group SU (1, 1) R3 introduced in Sect. 2, we find that (196) agrees with (15) if we identify (u, y) ∈ SU (1, 1) R3 ∼ ∀u ∈ S L(2, R), y ∈ R3 , (197) = (1 + 0 y a Ja ) · u and we recover the multiplication law (11) (1 + 0 x a Ja )u · (1 + 0 y b Jb )v = (1 + 0 (x a Ja + Ad(u)y b Jb )uv ×∀u, v ∈ S L(2, R), x, y ∈ R3 . Hence, by exponentiating the Lie algebra sl(2, R ) for cosmological constant = 0, one obtains the parametrisation of the (2+1)-dimensional Poincaré group in Sect. 2 whose description in terms of a formal supersymmetry parameter 0 satisfying 20 = 0 was first introduced in the context of (2+1)-dimensional gravity by Martin [11]. For > 0, the exponential (193) can be evaluated by introducing the generators defined in (12) Ja± = 21 (1 ±
√ )J , a
(198)
in terms of which the argument of (193) takes the form √ √ (x a + y a )Ja = (x a + y a )Ja+ + (x a − y a )Ja− .
(199)
Using identity (187), we recover the splitting of h>0,L into the direct sum sl(2, R) ⊕ sl(2, R), Ja+ · Jb− = Ja− · Jb+ = 0
Ja± · Jb± = 21 (1 ±
1 √ )( η 4 ab
+ 21 abc )Jc ,
(200)
and by applying these identities to (193) we obtain √ √ ∞ ∞ ((x a + y a )Ja+ )n ((x a − y a )Ja− )n a a + exp>0 ((x + y )Ja ) = n! n! n=0 √ (x a + y a )Ja 1 )e = 2 (1 + √
+
n=0 √ (x a − y a )Ja 1 √ (1 − )e . 2
(201) With the identification (u + , u − ) ∼ = 21 (1 +
√ )u +
+ 21 (1 −
√ )u −
∀u ± ∈ S L(2, R)
(202)
we then recover formula (15) for the exponential map exp : sl(2, R) ⊕ sl(2, R) → S L(2, R) × S L(2, R) and the group multiplication law (13). Hence, for all values of the cosmological constant, the group H,L and the exponential map exp,L : h,L → H,L can be obtained from the identification of the Lie algebra h,L with sl(2, R ). The cosmological constant can therefore be implemented in the Chern-Simons formulation of (2+1)-dimensional gravity by interpreting it as a parameter in the multiplication law (184) of a commutative ring R . By parametrising the elements of this ring in terms of a formal parameter satisfying 2 = one then obtains a unified description of the Lie algebras h,L and the associated Lie groups H,L , which can be identified, respectively, with the Lie algebra sl(2, R ) and the associated matrix groups obtained by exponentiation.
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8.3. Grafting transformations as deformed Dehn twists. We will now apply these results to demonstrate that the parameter which can be viewed as a formal square root of appears as a deformation parameter relating the Dehn twist and grafting transformations (167) and (160) for all values of the cosmological constant . For this we recall the discussion from Sect. 7.2 and Sect. 7.3, where it is shown that the grafting transformat tion Tκ,η,,L : Isom(X,S )2g → Isom(X,S )2g associated to a closed, simple curve η ˜ on Sg and the corresponding Dehn twist Tt˜t,η,,L : Isom(X,S )2g → Isom(X,S )2g are generated, respectively, by Wilson loop observables κ˜ η and t˜η constructed from bilinear forms κ and , on h,S . The fact that the bilinear forms κ and , on h,L appear as projections on, respectively, the real and the -component of Killing form g K on sl(2, R ) allows one to interpret the Wilson loop observables κ˜ η , t˜η as projections on ∞ 2g real and component of a Wilson loop observable g K η ∈ C (Isom(X,L ) ) which takes values in the Ring R , 1 1 g K η (A1 , . . . , Bg ) = 2 κ˜ η (A1 , . . . , Bg ) + 2 t˜η (A1 , . . . , Bg ).
Moreover, we found in Sect. 7.3 that the grafting and Dehn twist transformations are of a similar form. Both act on the holonomies Ai , Bi by right-multiplication with functions G tt˜,,L , G tκ,,L : Isom(X,L ) → Isom(X,L ) of the holonomies of certain curves con˜ jugated to η, which are obtained as exponentials of linear maps l t , l κ : h,L → h,L . For all values of the cosmological constant, the linear map for the Dehn twist is the identity l t = idh,L , and the associated map G tt˜,,L takes the form G tt˜,,L (u) = u t
∀u ∈ Isom(X,L ).
(203)
The linear map l κ and the associated one parameter group of diffeomorphisms G tκ,,L ˜ for the grafting transformation are given by, respectively, (157) and (158). Unlike the corresponding maps for the Dehn twists, they show an explicit dependence on the cosmological constant . The identification of the Lie algebra h,L with sl(2, R ) allows us to relate these maps for the different values of the cosmological constant and to establish a link with the corresponding maps for Dehn twists. For this we note that for all values of the cosmological constant the linear map l κ on h,L given by (157) can be identified with a linear map on the Lie algebra sl(2, R ), which acts by multiplication with , l κ (x + y) = (x + y)
∀x + y ∈ sl(2, R ).
(204)
The discussion in the previous subsection then allows us to express the associated oneparameter group of transformations G tκ,,L : Isom(X,L ) → Isom(X,L ) via the expo˜ nential map (193), G tκ,,L (exp (x + y)) = exp (t (x + y)) ˜
∀x + y ∈ sl(2, R ). (205)
√ Evaluating this expression by setting = i || for < 0 and by using expressions (196) and (201) for = 0 and > 0, we recover expression (158). Hence, the oneparameter group of transformations G tκ,,L , G tt˜,,L : Isom(X,L ) → Isom(X,L ) are ˜ formally related by the identity = G tt G tκ,,L ˜ ˜,,L .
(206)
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After inserting this identity in the expressions (167), (160) for the Dehn twist and the grafting transformation, we find that formally, the transformation of the holonomies Ai , Bi under grafting along a closed, simple curve η on Sg with parameter t can be expressed as a Dehn twist with parameter t, t = Tt˜t . Tκ,η,,L ˜ ,η,,L
(207)
By interpreting the Lie algebra h,L of the gauge group in Chern-Simons formulation as a (2+1)-dimensional Lorentz algebra over the commutative ring R , we therefore establish a common pattern which relates the grafting and Dehn twist transformations for all values of the cosmological constant. The dependence on the cosmological constant is encoded in the formal parameter satisfying 2 = which plays the role of a deformation parameter. In this formalism, the two Wilson loop observables κ˜ η , t˜η associated to a closed, simple curve η on Sg arise canonically as the projection on the -component and on the real component of the R -valued Wilson loop observable (203) constructed from the Killing form on sl(2, R ). Via the Poisson bracket, these two canonical observables generate the two basic geometry changing transformations on the phase space. The former acts as the Hamiltonian for the Dehn twist transformations (167), while the latter is the Hamiltonian for the grafting transformation (160). Viewed as transformations over the ring R , these two phase space transformations exhibit a similar structure and can be transformed into each other by substituting t → t . 9. Conclusions In this paper we clarified the relation between the geometrical and the Chern-Simons description of (2+1)-dimensional spacetimes of topology R × Sg for Lorentzian signature and general cosmological constant and for the Euclidean case with negative cosmological constant. We showed how the fact that such spacetimes are obtained as quotients of the model spacetimes X,S corresponds to the trivialisation of the gauge field in the Chern-Simons formalism. This allowed us to relate the variables encoding the physical degrees of freedom in the two approaches, the group homomorphism G h , : → Isom(X,S ) in the geometric formulation and the holonomies along a set of generators of the fundamental group π1 (Sg ) in the Chern-Simons description. We demonstrated how the construction of evolving (2+1)-spacetimes via grafting along closed, simple geodesics η gives rise to a transformation on the phase space of the associated Chern-Simons theory. After deriving an explicit expression for the transformation of the holonomies, we showed that this transformation is generated via the Poisson bracket by one of the two canonical Wilson loop observables associated to η, while the other observable generates Dehn twists. We found a close relation between the action of these transformations on the phase space which is reflected in a general symmetry relation for the associated Wilson loop observables. Finally, we investigated the role of the cosmological constant in the geometrical and the Chern-Simons formulation of the theory with Lorentzian signature. We found that the square root of minus the cosmological constant can be viewed as a deformation G ⊂X parameter in the parametrisation of the domains U,L ,L and in the group homoG morphisms h ,L : → Isom(X,L ). In the Chern-Simons formulation, we obtained a unified description for the different signs of the cosmological constant by identifying the Lie algebras of the gauge groups with the (2+1)-dimensional Lorentz algebra sl(2, R ) over a commutative ring R . In this framework, the cosmological constant
Geometrical (2+1)-Gravity and the Chern-Simons Formulation
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arises as a parameter in the ring’s multiplication law and can be implemented via a formal parameter satisfying 2 = . By extending the Killing form on the (2+1)-dimensional Lorentz algebra to an Ad-invariant, bilinear form on sl(2, R ) and considering the associated Wilson loop observables with values in R , we found that the Wilson loop observables generating grafting and Dehn twists arise canonically as the projections on the real and the component of this R -valued observable. Moreover, we found that a grafting transformation with weight w associated to a closed, simple curve η on Sg can be viewed as a Dehn twist around η with parameter w. These results clarify the relation between spacetime geometry and the description of the phase space in the Chern-Simons formalism and provide a geometrical interpretation of the Wilson loop observables. Moreover, we obtained explicit expressions for the action of grafting and Dehn twists in Fock and Rosly’s description of the phase space [12], which is the basis of the combinatorial quantisation formalism [15, 16] and the related approaches [17] and [18] for the group S L(2, C) and semidirect product groups G g∗ such as the (2+1)-dimensional Poincaré group. It would therefore be interesting to see how these results can be applied to the quantised theory and to use them to investigate concrete physics questions in quantum (2+1)-gravity. Acknowledgements. I thank Bernd Schroers for comments on the draft of this paper. Research at Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MEDT.
References 1. Carlip, S.: Quantum gravity in 2+1 dimensions. Cambridge University Press, Cambridge (1998) 2. Carlip, S.: Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe. Living Rev.Rel. 8, 1 (2005) 3. Achucarro, A., Townsend, P.: A Chern–Simons action for three-dimensional anti-de Sitter supergravity theories. Phys. Lett. B 180, 85–100 (1986) 4. Witten, E.: 2+1 dimensional gravity as an exactly soluble system. Nucl. Phys. B 311, 46–78 (1988), Nucl. Phys. B 339, 516–32 (1988) 5. Nelson, J.E., Regge, T.: Homotopy groups and (2+1)-dimensional quantum gravity. Nucl. Phys. B 328, 190–202 (1989) 6. Nelson, J.E., Regge, T.: (2+1) Gravity for genus > 1. Commun. Math. Phys. 141, 211–23 (1991) 7. Nelson, J.E., Regge, T.: (2+1) Gravity for higher genus. Class Quant Grav. 9, 187–96 (1992) 8. Nelson, J.E., Regge, T.: The mapping class group for genus 2. Int. J. Mod. Phys. B 6, 1847–1856 (1992) 9. Nelson, J.E., Regge, T.: Invariants of 2+1 quantum gravity. Commun. Math. Phys. 155, 561–568 (1993) 10. Ashtekar, A., Husain, V., Rovelli, C., Samuel, J., Smolin, L.: (2+1) quantum gravity as a toy model for the (3+1) theory. Class. Quant. Grav. 6, L185–L193 (1989) 11. Martin, S.P.: Observables in 2+1 dimensional gravity. Nucl. Phys. B 327, 178–204 (1989) 12. Fock, V.V., Rosly, A.A.: Poisson structure on moduli of flat connections on Riemann surfaces and r -matrices. Am. Math. Soc. Transl. 191, 67–86 (1999) 13. Alekseev, A.Y., Malkin, A.Z.: Symplectic structure of the moduli space of flat connections on a Riemann surface. Commun. Math. Phys. 169, 99–119 (1995) 14. Meusburger, C., Schroers, B.J.: Poisson structure and symmetry in the Chern-Simons formulation of (2+1)-dimensional gravity. Class. Quant. Grav. 20, 2193–2234 (2003) 15. Alekseev, A.Y., Grosse, H., Schomerus, V.: Combinatorial quantization of the Hamiltonian Chern-Simons Theory. Commun. Math. Phys. 172, 317–58 (1995) 16. Alekseev, A.Y., Grosse, H., Schomerus, V.: Combinatorial quantization of the Hamiltonian Chern-Simons Theory II. Commun. Math. Phys. 174, 561–604 (1995) 17. Buffenoir, E., Noui, K., Roche, P.: Hamiltonian Quantization of Chern-Simons theory with S L(2, C) Group. Class. Quant. Grav. 19, 4953–5016 (2002) 18. Meusburger, C., Schroers, B.J.: The quantisation of Poisson structures arising in Chern-Simons theory with gauge group G g∗ . Adv. Theor. Math. Phys. 7, 1003–1043 (2004) 19. Mess, G.: Lorentz spacetimes of constant curvature. Preprint IHES/M/90/28, Avril 1990 20. Benedetti, R., Guadgnini, E.: Cosmological time in (2+1)-gravity. Nucl. Phys. B 613, 330–352 (2001)
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21. Benedetti, R., Bonsante, F.: Wick rotations in 3D gravity: ML(H2 ) spacetimes. http://arxiv./org/list/ math.DG/0412470, 2004 22. Benedetti, R., Bonsante, F.: Canonical Wick Rotations in 3-dimensional gravity. http://arxiv./org/list/ math.DG/0508485, 2004 23. Meusburger, C.: Grafting and Poisson structure in (2+1)-gravity with vanishing cosmological constant. Commun. Math. Phys. 266, 735–775 (2006) 24. Benedetti, R., Petronio, C.: Lectures on Hyperbolic Geometry. Berlin-Heidelberg: Springer Verlag, 1992 25. Katok, S.: Fuchsian Groups. Chicago: The University of Chicago Press, 1992 26. Goldman, W.M.: Projective structures with Fuchsian holonomy. J. Diff. Geom. 25, 297–326 (1987) 27. Hejhal, D.A.: Monodromy groups and linearly polymorphic functions. Acta. Math. 135, 1–55 (1975) 28. Maskit, B.: On a class of Kleinian groups. Ann. Acad. Sci. Fenn. Ser. A 442, 1–8 (1969) 29. Thurston, W.P.: Geometry and Topology of Three-Manifolds. Lecture notes, Princeton, NJ: Princeton University, 1979 30. Thurston, W.P.: Earthquakes in two-dimensional hyperbolic geometry. In: Epstein, D.B. (ed.), Low dimensional topology and Kleinian groups. Cambridge: Cambridge University Press, 1987, pp. 91–112 31. McMullen, C.: Complex Earthquakes and Teichmüller theory. J. Amer. Math. Soc. 11, 283–320 (1998) 32. Sharpe, R.W.: Differential Geometry. Springer Verlag, New York (1996) 33. Matschull, H.-J.: On the relation between (2+1) Einstein gravity and Chern-Simons Theory. Class. Quant. Grav. 16, 2599–609 (1999) 34. Meusburger, C.: Dual generators of the fundamental group and the moduli space of flat connections. J. Phys. A: Math. Gen. 39, 14781–14832 (2006) 35. Stachura, P.: Poisson-Lie structures on Poincaré and Euclidean groups in three dimensions. J. Phys. A 31, 4555–4564 (1998) 36. Goldman, W.M.: Invariant functions on Lie groups and Hamiltonian flows of surface group representations. Invent. Math. 85, 263–302 (1986) 37. Meusburger, C., Schroers, B.J.: Mapping class group actions in Chern-Simons theory with gauge group G g∗ . Nucl. Phys. B 706, 569–597 (2005) 38. Bourbaki, N. (Pseud.): Elements of Mathematics, Lie groups and Lie algebras, Part I: Chapters 1–3. Paris: Hermann, 1975 Communicated by G.W. Gibbons
Commun. Math. Phys. 273, 755–783 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0265-8
Communications in
Mathematical Physics
The Parameter Planes of λz m exp(z) for m ≥ 2 Núria Fagella1 , Antonio Garijo2 1 Dep. de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via de les Corts Catalanes,
585, 08005 Barcelona, Spain. E-mail: [email protected]
2 Dep. d’Eng. Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans, 26,
43007 Tarragona, Spain. E-mail: [email protected] Received: 3 August 2006 / Accepted: 24 November 2006 Published online: 31 May 2007 – © Springer-Verlag 2007
Abstract: We consider the families of entire transcendental maps given by Fλ,m (z) = λz m exp(z), where m ≥ 2. All functions Fλ,m have a superattracting fixed point at z = 0, and a critical point at z = −m. In the parameter planes we focus on the capture zones, i.e., λ values for which the critical point belongs to the basin of attraction of z = 0, denoted by A(0). In particular, we study the main capture zone (parameter values for which the critical point lies in the immediate basin, A∗ (0)) and prove that is bounded, connected and simply connected. All other capture zones are unbounded and simply connected. For each parameter λ in the main capture zone, A(0) consists of a single connected component with non-locally connected boundary. For all remaining values of λ, A∗ (0) is a quasidisk. On a different approach, we introduce some families of holomorphic maps of C∗ which serve as a model for Fλ,m , in the sense that they are related by means of quasiconformal surgery to Fλ,m . 1. Introduction and Results One of the central topics in complex dynamics is the study of the dynamics of the quadratic polynomial Q c (z) = z 2 + c. The dynamical behavior of the map Q c is determined by the orbit of the unique critical point z = 0. These maps have been thoroughly studied by many authors (see for example [DH1, DH2, CG, M1, L]). In analogy with the quadratic family of polynomials Q c , the exponential map E λ (z) = λ exp(z), with a unique asymptotic value at v = 0, is the simplest example of an entire transcendental map with rich and interesting dynamics. The systematic study of cubic polynomials began with the work of Branner and Hubbard ([BH1]), who considered the two parameter family of monic and centered cubic polynomials which, after a suitable normalization, is given by Ca,b (z) = z 3 − 3a 2 z + b. Both authors were supported by MTM2005-02139/Consolider (including a FEDER contribution) and CIRIT 2005 SGR01028. The first author was also supported by MTM2006-05849/Consolider (including a FEDER contribution).
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Notice that any cubic polynomial is affine conjugate to one in this family. The dynamics of monic centered cubic polynomials is determined by the orbits of the two critical points located at ±a. Moreover, they proved that the cubic connectedness locus, which is a subset of C2 , consisting of all the parameters (a, b) ∈ C × C such that J (Ca,b ) is connected, is compact and connected. Many authors have investigated subfamilies, or slices, of the family of cubic polynomials (among others see [M, Fau, BH2, R, Z, BuHe]). Milnor studied the one parameter family of cubic polynomials having a superattracting fixed point ([M]). These polynomials are given by Ma (z) = z 3 −
3 2 az . 2
(1)
It is easy to see that Ma has a superattracting fixed point at z = 0, and a free critical point at z = a. When z = a belongs to the basin of attraction of the superattracting fixed point z = 0 we say that the critical point z = a has been captured. The connected components of the parameter space for which this phenomenon occurs are called capture zones. We also define the main capture zone, as the set of parameter values a for which the critical point z = a belongs to the immediate basin of z = 0. The original parametrization of the Milnor cubic polynomials was C˜ a (z) = z 3 − 3a 2 z + 2a 3 + a, but both families are equivalent since they are conjugate under an affine change of coordinates. Milnor ([M]) suggested two questions about the family of cubic polynomials Ma , one in the dynamical plane and another one in the parameter plane. The first one was to investigate whether for all parameter values a, the boundary of the immediate basin of attraction of z = 0 is a Jordan curve. The second one was to investigate whether the boundary of the main capture zone is a Jordan curve. Both questions were answered by Faught ([Fau]) using a modification of Yoccoz’s puzzle for a rational like mapping (see [R]). Faught proved that for all parameter values, a ∈ C, the immediate basin of attraction of z = 0 is a Jordan domain and also the boundary of the main capture zone is a Jordan curve. Roesch ([R]) generalized this result, in the dynamical plane, for an extension family of the Milnor cubic polynomial. More precisely, we can consider the family of polynomials Mm,a (z) = z m+1 −
m+1 m az m
(2)
as a generalized family of the Milnor cubic polynomials. For each m ≥ 2 the point z = 0 is a superattracting fixed point of multiplicity m, and z = a is a free critical point (when m = 2 we find exactly the Milnor cubic polynomial Ma ). It is proven ([R]) that for every value of m ≥ 2 and for all parameters a ∈ C, the boundary of the immediate basin of attraction of the superattracting fixed point z = 0 is a Jordan curve. Our goal in this work is to study some dynamical aspects of the families of entire transcendental maps Fλ,m (z) = λz m exp(z), m ≥ 2.
(3)
All functions of the form Fλ,m , with m ≥ 2, have a superattracting fixed point at z = 0 of multiplicity m, which is also an asymptotic value. The only other critical point is z = −m. The coexistence of a superattracting fixed point and a free critical point makes this family an entire transcendental analogue of the generalized Milnor polynomials (Eq. (2)).
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Some functions in the family Fλ,m = λz m exp(z) for m ≥ 2 have been used in the literature as examples of certain dynamical phenomena (see for example [Be], for a Baker domain at a positive distance from any singular orbit for a lift of a certain member Fλ,m ). We also notice that fixed points of Fλ,m appear in a different mathematical context. More precisely, Fλ,m (z) = z is the characteristic equation of the following delay differential equation d m−1 x 1 = x(t − 1). dt m−1 λ If we search for some value z 0 such that x(t) = c e z 0 t is a solution, we obtain the characteristic equation λz 0m exp(z 0 ) = z 0 . In ([FG]) we made an initial study of the discrete dynamical system generated by the map Fλ,m . We focussed our attention in a description of the dynamical planes, and specially on the basin of attraction of the superattracting fixed point at z = 0. In this paper we turn our attention to the parameter planes of the family of functions Fλ,m . As usual in complex dynamics as A. Douady said: “you first plow in the dynamical plane and then harvest in the parameter space”. As we mentioned, the origin is a superattracting fixed point of the function Fλ,m , for all m ≥ 2 and λ ∈ C. We denote by A(0) = Aλ,m (0) the basin of attraction of the origin, given by ◦n A(0) = Aλ,m (0) = {z ∈ C, Fλ,m (z) → 0 as n → ∞}.
(4)
The immediate basin of attraction of z = 0 is the connected component of A(0) containing z = 0, and we denote it by A∗ (0) = A∗λ,m (0). One of the main objectives of this work is the study of Aλ,m (0). We would like to answer the following questions: How many connected components does Aλ,m (0) have? Are they simply connected? Are they bounded? When is the boundary of A∗λ,m (0) locally connected? For some parameter values, the free critical point z = −m belongs to the basin of attraction of z = 0, in which case we say that it has been captured. The connected components of parameter space for which this phenomenon occurs are called capture zones, and they clearly do not exist for members of the family Fλ,m with m < 2, i.e., for the exponential family. We will study the capture zones given by n n Hm = {λ ∈ C | Fλ,m (−m) ∈ A∗λ,m (0)and n is the smallest number with this property}. (5) 0 , as the set of parameter values As a special case, we define the main capture zone, Hm λ for which the critical point z = −m itself belongs to the immediate basin of 0. That is, 0 Hm = {λ ∈ C | − m ∈ A∗λ,m (0)}.
(6)
We shall see that this is a quite special capture zone since its boundary separates the parameter values for which F(Fλ,m ) has one connected component from those for which it has infinitely many. n connected? In the parameter plane we will answer the following questions: Is Hm n simply connected? Are they bounded? How does Are the connected components of Hm the boundary of A∗λ,m (0) depend on λ? Is ∂ A∗λ,m (0) locally connected when λ belongs n? to Hm
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In order to answer all of these questions we divide our study into two parts. In the first one, we study directly the family of functions Fλ,m = λz m exp(z) using standard tools in complex dynamics. In the second one, we relate it to a new family of maps given by G α,β,m (z) = exp(iα)z m exp(β/2(z − 1/z)) ,
(7)
where α and β are real numbers and m ≥ 2. The family of functions G α,β,2 have been investigated as real maps on the unit circle by M. Misiurewicz and A. Rodrigues ([MR]). Using quasiconformal surgery, we relate members of G α,β,m to those of Fλ,m , and use this correspondence to prove some results for the original maps. We first concentrate on the dynamical plane and especially in the basin of attraction Aλ,m (0). More precisely, we prove the following result related to the topology of the connected components of Aλ,m (0). Proposition A. Let λ ∈ C, m ∈ N, m ≥ 2 and Fλ,m (z) = λz m exp(z). Let Aλ,m (0) and A∗λ,m (0) be the basin and the immediate basin of attraction of z = 0 for the map Fλ,m , respectively. The following statements hold. a) All connected components of the Fatou set of Fλ,m are simply connected. b) Aλ,m (0) has either one or infinitely many connected components. c) All the connected components of Aλ,m (0) different from A∗λ,m (0) are unbounded. Further we describe the main features of the parameter planes of the functions Fλ,m and, in particular, the structure of the capture zones. We summarize some of these facts in the following theorems. In the first one we study the topology of the capture zones. In the second one we investigate the local connectivity of the boundary of A∗λ,m (0). In 0. the third we study the complement of the closure of the main capture zone Hm n , H0 be the capture zones as in Theorem B. For all parameters m ∈ N, m ≥ 2, let Hm m (5) and (6), respectively. The following statements hold:
a) The critical point −m belongs to A∗λ,m (0) if and only if the critical value Fλ,m (−m) 1 = ∅. belongs to A∗λ,m (0). Hence Hm 0 ⊂ D , b) There exist ρ = ρ(m), ρ = ρ (m) verifying 0 < ρ < ρ such that Dρ ⊂ Hm ρ where Dr = {z ∈ C | |z| < r }. 0 is connected and simply connected. c) The main capture zone Hm n are simply connected and d) Let n ≥ 2. All the connected components of Hm unbounded. n , H0 be the capture zones as in (5) and Theorem C. Let λ ∈ C, m ∈ N, m ≥ 2. Let Hm m (6), respectively. The following statements hold: ∗ 0 then A 0 then A / Hm a) If λ ∈ Hm λ,m (0) = Aλ,m (0). However if λ ∈ λ,m (0) has infinitely many connected components. 0 the boundary of A∗ (0) (which is equal to the Julia set) is a Cantor b) If λ ∈ Hm λ,m bouquet not locally connected. 0 . If λ ∈ U , then the c) Let Um be the unbounded connected component of C \ Hm m n for any n ≥ 2 the boundary of A∗λ,m (0) is a quasicircle. In particular, if λ ∈ Hm boundary of A∗λ,m (0) is a quasicircle.
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0 . The following Theorem D. Let Um be the unbounded connected component of C \ Hm statements hold: 0 = ∂U . a) ∂Hm m 0 , then U , H0 and V b) If there exists a bounded connected component V of C \ Hm m m are lakes of Wada, i.e., they have a common boundary.
There is another question which will remain unanswered in this work and which we state as a conjecture. 0 is a Jordan curve. Conjecture. The boundary of Hm Finally, we take a second approach, using quasiconformal surgery, to further describe the maps at hand. More precisely,
G α,β,m (z) = exp(iα)z m exp(β/2(z − 1/z)), where α and β are real numbers and m ≥ 2, which we relate to the original one by means of quasiconformal surgery. Roughly speaking quasiconformal surgery is a technique to construct holomorphic maps with some prescribed dynamics. In our case, we combine two dynamical systems acting in different parts of the plane to construct a new system that combines the dynamics of both. In this process we use quasiconformal mappings to glue different behaviors. The key ingredient of this technique is to use the Measurable Riemann Mapping Theorem ([Ah, LV]) in order to assure that the corresponding mapping is a holomorphic map. For our construction, it will play a fundamental role the fact that G α,β preserves the unit circle S 1 . More precisely, G α,β induces a one dimensional mapping on the unit circle α,β,m : θ → α + mθ + β sin(θ ) G
mod (2π ) ,
θ ∈ R/2π Z.
We are interested in the set of parameters α,β,m is quasisymmetrically conjugate to θ → mθ } Wm = {α, β | G α,β,m is an expanding map on the unit which in particular includes all those for which G circle ([SS]). This is summarized in the following theorem. 0 such Theorem E. For any (α, β) ∈ Wm , there exists a λ in the complement of Hm that Fλ,m is quasiconformally conjugate on the complement of A∗λ,m (0) to G α,β on the complement of the closed unit disc. For this value of λ the boundary of A∗λ,m (0) is a quasicircle.
The rest of the paper is organized as follows. In Sect. 2 we present some previous results concerning the basin of attraction of the origin. In Sect. 3 we summarize some tools which we will use in this paper. Finally, Sects. 4 and 5 are devoted to prove the main results of this work. Experts can read directly Sect. 4.
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Fig. 1. Sketch of some sets included in the basin of attraction of z = 0. Precisely, D 0 ⊂ A∗λ,m (0), k )⊂ H Fλ,m (H|λ|,m ) ⊂ D 0 and Fλ,m (Sλ,m |λ|,m
2. Preliminaries The systematic study of the functions Fλ,m was started in [FG]. In this section we recall some results from that work that will be useful later on. Theorem 2.1 (Skeleton of Aλ,m (0), see Fig. 1). Let λ ∈ C, m ∈ N, m ≥ 2 and Fλ,m (z) = λz m exp(z). Let A∗λ,m (0) be the immediate basin of attraction of z = 0 for the map Fλ,m . The following statements hold: a) For λ = 0, if we define 0 = 0 (|λ|, m) > 0 as the unique positive solution of x m−1 e x = 1/|λ|; then A∗λ,m (0) contains the disk D 0 = {z ∈ C ; |z| < 0 }. b) There exist x0 = x0 (|λ|, m) < 0 and a continuous (decreasing) function x → C(x) > 0 defined for x < x0 such that the open set x ∈ (−∞, x0 ) H|λ|,m = z = x + yi y ∈ (−C(x), C(x)) satisfies Fλ,m (H|λ|,m ) ⊂ D 0 . k , which are preimages of H c) There exist infinitely many strips, denoted by Sλ,m |λ|,m . These horizontal strips extend to +∞, and they have asymptotic width equal to π . The skeleton of the main components of Aλ,m (0) is needed to study later the parameter planes. In the first statement of Theorem 2.1 we give an estimate of the size of the immediate basin of attraction of z = 0. Since z = 0 is a superattracting fixed point, there exists 0 > 0 such that the open disk D 0 = {z ∈ C ; |z| < 0 } is contained in the immediate basin of attraction of z = 0. In the second statement we find the first preimage D 0 , which contains an unbounded open set in C extending to the left and containing an unbounded interval (−∞, x0 ) for some real value x0 . In the third statement we find the second preimage of D 0 , which contains countably many horizontal strips extending to +∞. In the following auxiliary result we find a lower bound for 0 , which will be used later on. 1
1 m−1 Lemma 2.2. The value of 0 is always larger than or equal to min{1, ( |λ|e ) }.
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3. Tools In this section we present well known tools in complex dynamics which we will use in this paper. We also present applications of some of them to our particular case. The first tool is a classical result related to the behavior of holomorphic maps near a superattracting fixed point ([Bo]), which we apply to make a detailed description of the superattracting basin of z = 0 for Fλ,m . The second section is related to the extension of a Holomorphic motion, established by Słodkowski ([Sl]). In the third one, we recall shortly the relevant definitions and results relative to quasiconformal mappings ([Ah, LV]). Finally, in the miscellanea section we provide precise definitions of several concepts related to circle maps. 3.1. Böttcher coordinates near a superattracting fixed point. Theorem 3.1. Suppose that f is an holomorphic map, defined in some neighborhood U of 0, having a superattracting fixed point at 0, i.e., f (z) = am z m + am+1 z m+1 + · · · where m ≥ 2, and am = 0. Then, there exists a local conformal change of coordinate w = ϕ(z), called Böttcher coordinate at 0 (or Böttcher map), such that ϕ ◦ f ◦ϕ −1 is the map w → w m throughout some neighborhood of ϕ(0) = 0. Furthermore, ϕ is unique up to multiplication by an (m − 1)st root of unity. In practice, it is customary to make a linear change of coordinates so that the map f is monic, i.e., so that am = 1. When f is monic we obtain a unique Böttcher coordinate such that lim z→0 ϕ(z) z = 1. Also it is natural to extend ϕ to a maximal domain using the functional relation ϕ( f (z)) = ϕ(z)m (see, [DH1, DH2] or [BuHe] for details). One might hope that the change of coordinates z → ϕ(z) extends throughout the entire immediate basin of attraction of the superattractive point as a holomorphic mapping. However, this is not always possible. Such an extension involves computing expressions of the form z → m ϕ( f (z)), and this does not work in general since the n th root cannot be defined as a single valued function. For example, when some other point in the basin maps exactly onto the superattracting point, or when the basin is not simply connected. Using the Böttcher map we can define a useful polar coordinate near 0. We define the dynamical ray of argument θ , where θ ∈ R/Z, to be the image under the inverse of the Böttcher map of the half line through 0 with argument θ turns, i.e. 2π θ radians, R0 (θ ) = ϕ −1 ({se2πiθ | s ≥ 0}). We say that the dynamical ray R0 (θ ) lands if and only if there exist lim ϕ −1 (se2πiθ ).
s→1
When a dynamical ray R0 (θ ) lands we call the limit the landing point of the ray R0 (θ ). We define the dynamical equipotential of level s, where 0 < s < 1, to be the image under the inverse of the Böttcher map of the circle of radius s and centered at 0, E 0 (s) = ϕ −1 ({se2πiθ | 0 ≤ θ < 1}).
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Since ϕ conjugates f to w → wm , the dynamics under f is easy to compute on these dynamical objects (rays and equipotentials). Precisely, we have f (R0 (θ )) = R0 (mθ )
and
f (E 0 (s)) = E 0 (s m ).
As we already mentioned, the Böttcher map verifies the functional equation or Böttcher equation ϕ( f (z)) = ϕ(z)m . On the other hand, there exists an explicit form of the Böttcher map, given by ϕ(z) = lim ( f ◦n (z))1/m . n
n→∞
In order to remove the ambiguity of the root, we write the sequence in the following form: ϕ(z) = z ·
f (z) zm
1/m ◦2 1/m 2 1/m n f (z) f ◦n (z) · . . . .... ( f (z))m ( f ◦(n−1) (z))m
(8)
For the general term, we have
f ◦n (z) ( f ◦(n−1) (z))m
1/m n
1/m n = 1 + O( f ◦(n−1) (z)) .
Hence, in a neighborhood of the superattracting fixed point z = 0, we can define the root by the binomial formula: (1 + u)α =
∞
α(α − 1) · · · (α − n + 1) n=0
n!
u n when |u| < 1.
It is not difficult to see that the product converges uniformly. In our case, z = 0 is a superattracting fixed point of Fλ,m = λz m exp(z). Using a suitable linear change of variables we obtain a new family of entire transcendental maps, so that near the superattracting fixed point z = 0, the functions can be written as z m + O(z m+1 ), and thus have a preferred Böttcher coordinate in this region. More precisely, we consider the following auxiliary family of entire transcendental maps: L a,m (z) = z m e z/a , a ∈ C \ {0}, and m ∈ N, m ≥ 2.
(9)
In the next lemma we prove some fundamental properties of the Böttcher coordinate near z = 0 for the map L a,m . In particular, we obtain an explicit expression of the Böttcher map and we see that it extends to the whole immediate basin of attraction of z = 0. Lemma 3.2. Consider L a,m (z) = z m exp(z/a) for a = 0 and m ≥ 2. Then, the Böttcher coordinate ϕa extends to the whole immediate basin of attraction of the superattracting fixed point z = 0.
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Proof. The map L a,m is affine conjugate to Fλ,m with λ = a m−1 through the map ca (z) = az. In other words, if we choose two parameter values λ0 and a0 such that λ0 = a0m−1 , then Fλ0 ,m and L a0 ,m are conformally conjugate, i.e. L a0 ,m (z) = (ca−1 ◦ Fλ0 ,m ◦ ca0 )(z) 0
∀z ∈ C.
For each a = 0, and when z is small enough we can write the Böttcher coordinate ϕa (z) using the auxiliary expression (8). More precisely, we have
L a,m (z) ϕa,m (z) = z · zm
1/m 2 1/m n 1/m ◦2 (z) ◦n (z) L a,m L a,m · ... .... ◦(n−1) (L a,m (z))m (L a,m (z))m
For the general term, we have
◦n (z) L a,m ◦(n−1)
(L a,m
1/m n
(z))m
◦(n−1) L a,m (z) = exp . a mn
Hence, in a neighborhood of the superattracting fixed point z = 0, we obtain ∞
L ◦n (z) . ϕa (z) = z exp a m n+1
(10)
n=0
Finally, we observe that this holomorphic map is well defined (the series converges) in the whole immediate basin of attraction of z = 0. 3.2. Holomorphic motions. ˆ we say that a map Definition. Let X ⊂ C, ˆ : D× X → D×C (c, z) → (c, z) = (c, c (z)) = (c, z (c)) is a holomorphic motion of X parameterized by D if (a) 0 (z) = z for all z ∈ X . (b) c (z) is injective for all fixed c ∈ D. (c) For all z ∈ X , the map z : D → C is holomorphic. There are two important theorems studying the extension of a holomorphic motion. The first one is the λ Lemma ([MSS]) and it extends a holomorphic motion of X to the closure of X . The second one is Słodkowski Lemma ([Sl]) and it extends a holomorphic motion parameterized in D to the whole Riemann sphere. We only recall the Słodkowski Lemma, since it is a generalization of the λ Lemma. ˆ be a holomorphic Theorem 3.3 (Słodkowski Lemma, [Sl]). Let : D × X → D × C ˆ ˆ Moreover, motion. Then, we can extend to a holomorphic motion : D× C → D× C. ˆ ˆ c : C → C is a quasiconformal homeomorphism for every parameter c ∈ D, the map 1+|c| . whose dilatation ratio K c is bounded by 1−|c|
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In the following lemma we prove that the holomorphic motion of a quasidisk is also a quasidisk. This property will play a fundamental role to prove Theorem C. Lemma 3.4. Let U be a quasidisk, i.e., assume that there exists a quasiconformal mapping h : C → C so that U = h(D). Let : D × U → D × C be a holomorphic motion of U. Then for all c ∈ D we have that c (U) is also a quasidisk. Proof. Applying the Słodkowski Lemma (Theorem 3.3) we can extend to a holoˆ → D×C ˆ such that for every parameter c ∈ D, the map : D×C morphic motion ˆ →C ˆ is a quasiconformal mapping. If we denote by Uc := { (c, z) | z ∈ U}, we c : C c ◦ h : C → C is a quasiconformal mapping and Uc = c ◦ h(D). have that 3.3. Quasiconformal surgery. Definition. A quasiconformal map of C is a homeomorphism ϕ such that small infinitesimal circles are mapped onto small infinitesimal ellipses of bounded axes ratio. The analytic formulation of this condition is that ϕ(x + i y) is absolutely continuous in x for almost every y and in y for almost every x and that the partial derivatives are locally square integrable and satisfy the Beltrami differential equation ∂ϕ ∂ϕ = µ(z) for almost all z ∈ C, ∂z ∂z where µ is a complex measurable function with |µ(z)| ≤ κ < 1 for z ∈ C. In this case we say that ϕ is κ–quasiconformal. An almost complex structure σ on C is a measurable field of ellipses (E z )z∈C , equivalently defined by a measurable Beltrami form µ on C, µ=u
d z¯ . dz
The correspondence between Beltrami forms and complex structures is as follows: the argument of u(z) is twice the argument of the major axis of E z , and |u(z)| = KK−1 +1 , where K ≥ 1 is the ratio of the lengths of the axes. The standard complex structure σ0 is defined by circles or by the Beltrami form µ0 = 0. Suppose that ϕ : C → C is a quasiconformal homeomorphism. Then ϕ gives rise to an almost complex structure σ on C. For almost every z ∈ C, ϕ is differentiable and the R−linear tangent map Tϕ : Tz C → Tϕ(z) C defines, up to multiplication by a positive factor, an ellipse E z in Tz C: E z = (Tz ϕ)−1 (S 1 ). Moreover, there exists a constant K > 1 such that the ratio of the axes of E z is bounded by K for almost every z ∈ C. The smallest bound is called the dilatation ratio of ϕ. Equivalently, ϕ defines a measurable Beltrami form on C µ=
∂ϕ = ∂ϕ
∂ϕ ∂ z¯ ∂ϕ ∂z
d z¯ d z¯ = u(z) . dz dz
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An almost complex structure is quasiconformally equivalent to the standard structure if it is defined by a measurable field of ellipses with bounded dilatation ratio. Given ϕ : C → C a quasiconformal homeomorphism, an almost complex structure σ on C can be pulled back into an almost complex structure ϕ ∗ σ on C. If σ is defined by an infinitesimal field of ellipses (E z )z∈C , then ϕ ∗ σ is defined by (E z )z∈C , where E z = (Tz ϕ)−1 E ϕ(z) whenever defined. To integrate an almost complex structure σ means to find a quasiconformal homeomorphism ϕ such that (Tz ϕ)−1 (S 1 ) = ρ(z)E z for almost every z ∈ C. Informally, we will say that σ is transported to σ0 by σ . Surgery techniques are based on the following result: Theorem 3.5 (Measurable Riemann mapping Theorem, [Ah, LV]). Let σµ be any almost complex structure on C given by the Beltrami form µ=u
d z¯ dz
with bounded dilatation ratio, i.e., ||µ||∞ := sup |u(z)| < m < 1. Then σµ is integrable, i.e., there exists a quasiconformal homeomorphism ϕ such that µ=
∂ϕ , ∂ϕ
or equivalently ϕ ∗ σ0 = σµ . Moreover, ϕ : C → C is unique up to composition with an affine map. Remarks 3.6. The application of Ahlfors-Bers’ theorem to complex dynamics is the following. Let f and σµ be, a quasiregular mapping of C and an almost complex structure with bounded dilatation ratio, such that f ∗ σµ = σµ . If we apply Theorem 3.5 to integrate σµ , we obtain a quasiconformal mapping ϕ such that ϕ ∗ σ0 = σµ . Then g = ϕ ◦ f ◦ ϕ −1 verifies g ∗ σ0 = σ0 , and hence g is a holomorphic map of C. Moreover, f and g are quasiconformally conjugate, i.e., they have the same dynamics. 3.4. Miscellanea. Our goal in this subsection is to make precise definitions of expanding maps ([dMvS]), and the quasiconformal extension of a quasisymmetric map on the circle ([Pom]). We also need the concept of growth order of a continuous function. Definition. We say that a C 1 map f : T → T is expanding if there exist real constants C > 0 and µ > 1 such that |D( f ◦n (x))| > Cµn for all n ∈ N and all x ∈ T. We observe that a sufficient condition to assure that f is expanding is given by min{| f (x)| , x ∈ T} > 1. The following theorem states that any two expanding maps of the same degree are quasisymmetrically conjugate.
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Theorem 3.7 (Shub and Sullivan, [SS]). Let f, g : T → T be expanding and C 1+δ , with δ ∈ (0, 1), maps of degree m. Then there exist a quasisymmetric conjugacy ϕ : T → T such that f = h −1 ◦ g ◦ h. Quasisymmetry is precisely the property that allows a circle map to be extendable to a quasiconformal map of the disc, as shown by the following theorem. Theorem 3.8 (Beurling and Ahlfors [BA], Douady and Earle [DE]). Let h : T → T be an orientation preserving quasisymmetric map. We can extend h to a quasiconformal
: D → D. Moreover, if σ, τ ∈ M ob(D) map H ¨ then the extension of σ ◦ h ◦ τ is given
◦ τ. by σ ◦ H Finally we will need the definition of the growth order of a continuous function. Definition. Let f : C → C be a continuous function. We define M(r, f ) := max|z|=r | f (z)| and the growth order ρ( f ) by ρ( f ) := lim sup r →∞
log+ log+ M(r, f ) , log r
where log+ (t) = log(max(1, t)), 4. Transcendental Part When we consider a holomorphic map f : C → C with an essential singularity at infinity, this point plays a crucial role. For instance, the little Picard Theorem says that an entire function assumes every value in the complex plane with at most one exception, in any neighborhood of infinity. Thus, in general, iteration of entire transcendental maps is more complicated than rational maps. As an example, there are transcendental maps presenting wandering domains ([B1, B2]) and/or Baker domains ([F]), also called “parabolic domains at ∞”. We concentrate on the class of entire transcendental maps of finite type, that is S = { f : C → C, f trans. entire with only finitely many critical and asymptotic values}. Dynamically, entire maps of finite type share some of the properties of polynomials since their Fatou sets cannot include wandering or Baker domains, nor Herman rings ([EL2, GK]). Observe that the family of functions Fλ,m (z) = λz m exp(z) belongs to S. The function Fλ,m has two critical values at 0 and at λ(−m)m ex p(−m), since the critical points are located at z = 0 and z = −m. It has also an asymptotic value at v = 0, since the function tends to 0 as z tends to ∞ along R− . If f ∈ S, there exists a characterization of the Julia set ([EL1]), namely as the closure of the set of points whose orbits tend to ∞. Using the characterization above we can plot an approximation of J (Fλ,m ). Generally, orbits tend to ∞ in specific directions. ◦n (z)| = +∞ , then we have lim ◦n In our case, if limn→∞ |Fλ,m n→∞ Re(Fλ,m (z)) = +∞. Thus, an approximation of the Julia set is given by the set of points whose orbit contains a point with real part greater than, say, 90. Observe that filled black regions are due to numerics, since the Julia set contains no open set.
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767
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 2. The Julia set for Fλ,2
In Fig. 2, we display the Julia set of Fλ,m for different values of λ and m. The immediate basin of attraction of z = 0 is shown 1 in blue, while the other components of Aλ,m (0) \ A∗λ,m (0) are shown in red. The components of the Fatou set different from Aλ,m (0) are shown in orange. Points in the Julia set are shown in black. We show the dynamical plane of the function Fλ,2 = λz 2 exp(z) , for three different values of λ and different ranges. As we proved in [FG] the basin of 0 contains an infinite number of horizontal strips, that extend to +∞ as their real parts tend to +∞. Between these strips we find the well known structures, named Cantor Bouquets which are invariant sets of curves governed by some symbolic dynamics. This kind of structures in the Julia set are typical for critically finite entire transcendental functions ([DT]). Also, as we change the parameter λ we observe that the relative position of these bands also changes, but not their width. Finally, we can see the existence of an unbounded region that extends to −∞ contained in Aλ,m (0). In the zoom plates of Fig. 2, range (−1, 1) × (−1, 1), we can see the dynamical plane near the origin. It seems that the immediate basin of attraction of z = 0 is a Jordan domain for λ = −8 and λ = 6.9. The orbit of the free critical point z = −m, determines in large measure the dynamics of Fλ,m . Indeed, the functions Fλ,m (z) = λz m exp(z) are entire maps with a finite number of critical and asymptotic values, hence we know that if the orbit of z = −m tends to ∞ no other Fatou components can exist besides those that belong to Aλ,m (0). Hence the Fatou set must coincide with the basin of 0, i.e., F(Fλ,m ) = Aλ,m (0). The set Bm is defined as 1 Color plots are available in the online version of this paper. Otherwise, blue is darker than red and orange is light.
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(a)
(b)
(c) Fig. 3. Parameter plane for Fλ,2 . Color codes are explained in the text ◦n Bm = {λ ∈ C | Fλ,m (−m) ∞}.
In each of these sets, we may also distinguish between two different behaviors: those parameter values for which −m ∈ Aλ,m (0) and those for which this does not occur. Let int(Bm ) denote the interior of Bm . Definition. Let U be a connected component of int(Bm ). We say that U is a capture ◦n (−m) = 0, or in other words, zone if for all λ in U we have that limn→+∞ Fλ,m −m ∈ Aλ,m (0). We then say that the orbit of the critical point is captured by the basin of attraction of the superattracting fixed point z = 0. In Fig. 3, we show a numerical approximation of the set B2 . The main capture zone is shown in blue, while other capture zones are shown in red. All other components of B2 are shown in orange. The parameter values for which the orbit of the free critical point tends to ∞ are shown in black. In these sets we can see a countable quantity of 0. horizontal strips. In Fig. 3 (c) we can see the main capture zone Hm 4.1. Dynamical plane: Proof of Proposition A. The first assertion of this theorem, i.e. that all connected components of the Fatou set are simply connected, is a general result for all functions in class S ([B]), which we have included here for completeness. To see that the number of connected components of Aλ,m (0) is either 1 or ∞, we observe that the basin of z = 0, Aλ,m (0), consists of the immediate basin A∗λ,m (0) and all its preimages. For all connected components of Aλ,m (0) other than A∗λ,m (0) there i exists a number i > 0 such that Fλ,m (U ) ⊂ A∗λ,m (0), where i is the smallest number with this property. Suppose that there exist a finite number of connected components, say A∗λ,m (0) , U1 , U2 , . . . , U N . By assumption, for each Uk there exist a number i k
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ik such that Fλ,m (Uk ) ⊂ A∗λ,m (0), for 1 ≤ k ≤ N . Let il be the maximum of the indexes −1 (z) belong i 1 , · · · , i N . Consider z ∈ Ul such that is not exceptional; then, points in Fλ,m ∗ to Aλ,m (0), but not to Aλ,m (0) ∪ U1 ∪ · · · ∪ U N , which is a contradiction. It remains to prove that all connected components of Aλ,m (0) are unbounded except, maybe, A∗λ,m (0). To this end, suppose that U is a connected component of Aλ,m (0) differi (U ) ⊂ A∗λ,m (0). ent from A∗λ,m (0), and let i > 0 be the smallest number such that Fλ,m i (z) and 0. The Let z ∈ U , and denote by γ a simple path in A∗λ,m (0) that joins Fλ,m preimage of γ in U must include a path γ1 that joins z and ∞, since 0 is an asymptotical value with no other finite preimage than itself. Thus we conclude that U is unbounded. This concludes the proof of Proposition A.
4.2. Parameter plane: Proof of Theorem B. In this section we describe some properties n . We are mainly interested in their topological properties. For of the capture zones Hm clarity’s sake we prove each of the statements in a different proposition. Proposition 4.1. The critical point −m belongs to A∗λ,m (0) if and only if the critical 1 = ∅. value Fλ,m (−m) belongs to A∗λ,m (0). Hence Hm Proof. Suppose that Fλ,m (−m) ∈ A∗λ,m (0). Let γ be a simple path in A∗λ,m (0) that joins Fλ,m (−m) and 0. The set of preimages of γ must include a path γ1 that joins −∞ with −m, and also a path γ2 that joins −m and 0 (since −m is a critical point and 0 is a fixed point and asymptotic value). Hence γ1 ∪ γ2 ⊂ A∗λ,m (0) and so does −m. Conversely, if −m ∈ A∗λ,m (0) we have that Fλ,m (−m) ∈ A∗λ,m (0). We define ρ = min{ 1e , ( me )m }, i.e., ρ = 1/e for m = 2, 3 and ρ = ( me )m for m ≥ 4. e m−1 ) . We also define ρ = ( m−1 0 ⊂ D . Proposition 4.2. Dρ ⊂ Hm ρ 0 . For λ ∈ D , we will prove Proof. First we prove that Dρ = {λ ∈ C ; |λ| < ρ} ⊂ Hm ρ that Fλ,m (−m) lies in D 0 which we know belongs to A∗λ,m (0). In order to do so, we 1 1/(m−1) ) ) (Lemma 2.2). If λ ∈ Dρ , then |λ| < 1e , and hence use that 0 ≥ min(1, ( |λ|e 0 ≥ 1. The condition λ ∈ Dρ also implies that |λ| < ( me )m . Hence m m |Fλ,m (−m)| = |λ||(−m)m e−m | = |λ| < 1 ≤ 0 , e
and Fλ,m (−m) lies in A∗λ,m (0). 0 ⊂ D . We will prove that −m ∈ Second we prove that Hm / A∗λ,m (0) for all λ ∈ C ρ e m−1 ) . Let D be the disk centered at 0 of radius m −1. If we calculate such that |λ| > ( m−1 the modulus of the image of its boundary, {|z| = m − 1}, we obtain |Fλ,m (z)| = |λ||z|m e Re(z) ≥ |λ|(m − 1)m e−(m−1) > m − 1, e m−1 ) . This shows that D ⊂ Fλ,m (D). where the inequality is obtained using |λ| > ( m−1 −1 Let W be the component of Fλ,m (D) that contains the origin. It is clear that W ⊂ D and A∗λ,m (0) ⊂ W . Moreover, Fλ,m is a proper function of degree m from W onto
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Fig. 4. Fλ,m is a polynomial-like mapping of degree m near the origin
D, (see Fig. 4). In the terminology of polynomial-like mappings, developed by Douady and Hubbard ([DH3]), the triple (Fλ,m ; W, D) is a polynomial-like mapping of degree m. By the Straightening Theorem ([DH3]), there exists a quasiconformal mapping, φ, that conjugates Fλ,m to a polynomial P of degree m, on the set W . That is (φ −1 ◦ Fλ,m ◦ φ)(z) = P(z) for all z ∈ W . Since z = 0 is superattracting for Fλ,m and φ is a conjugacy, we have that z = 0 is superattracting for P. Hence, after perhaps a holomorphic change of variables, we may assume that P(z) = z m . 0 is bounded. Hence A∗λ,m (0) ⊂ D. Since −m ∈ / D we conclude that Hm 0 is connected and simply connected. Proposition 4.3. The main capture zone Hm 0 is conformally a disk. Since F Proof. We prove that Hm λ,m (z) has a superattracting fixed point at z = 0, we can use the Böttcher coordinate near the origin (see Sect. 3.1) to define a suitable biholomorphic map in the main capture zone. Using a suitable linear change of variables we obtain a new family of entire transcendental maps, so that near the superattracting fixed point z = 0, the functions can be written as z m + O(z m+1 ), and thus having a preferred Böttcher coordinate in this region (see Sect. 3.1). We consider
L a,m (z) = z m e z/a , a ∈ C \ {0}, and m ∈ N, m ≥ 2.
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Under this map, the superattracting fixed point z = 0 is still at z = 0, and the free critical point (located at z = −m for Fλ,m ) is now at ca,m = −m a for L a,m . We now define the following auxiliary set for the family of maps L a,m which is closely related to the main capture zone, more precisely 0 0 Hˆ m = {a ∈ C, such that a m−1 ∈ Hm }. 0 → a m−1 ∈ H0 is a (m − 1)–fold branched covering. By construction, a ∈ Hˆ m m We consider the following mapping 0 : Hˆ m →D a → ϕa,m (ca,m ),
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where ϕa is the Böttcher coordinate defined in the immediate basin of attraction of z = 0 (Sect. 3.1 and Lemma 3.2). We claim that the map is well defined and, in fact, is a ˆ0 conformal isomorphism which is tangent to a → −m e a at the origin. If a ∈ Hm \ {0},
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the Böttcher map extends until the critical point ca,m = −ma, and using Eq. (10) we have that ∞
L ◦n (−ma) ϕa,m (ca,m ) = (−ma) exp . a m n+1 n=0
◦n
We see inductively that L (−ma) is a holomorphic function of a m−1 . Indeed, using a m z/a the definition of L a,m (z) = z e , we have ◦0 (−ma) L a,m = −m a
Assuming then that see that ◦(n+1) L a,m (−ma)
a
=
L ◦n (−ma) a
◦1 (−ma) L a,m = a
and
−m e
m a m−1 .
= R(a m−1 ), where R(w) is a holomorphic map on w, we
◦n (−ma) L a,m
m
exp a
L ◦n
a,m (−ma)
a
= a m−1 [R(a m−1 )]m exp[R(a m−1 )]
◦n
proving thus that L (−ma) is a holomorphic function of a m−1 . a As a → 0, a brief computation shows that ϕa,m (ca,m ) = − me a η(a m−1 ), where η(w) is a holomorphic mapping so that η(0) = 1. Hence the apparent singularity at a = 0 is removable. Since the correspondence a → ϕa,m (cm,a ) (Eq. 12) is well defined and holomorphic, it suffices to show that ϕa,m (ca,m ) is a proper map of degree one from Hm onto D. To this end, we first consider a boundary point a0 ∈ ∂H0,m . Then, as noted earlier, the Böttcher mapping from the immediate basin Aa∗0 ,m (0) onto the unit disc has no critical points, and in fact is a conformal diffeomorphism. In particular, ϕa0 ,m can be defined as a single valued function on the disc of radius 1 − , for any > 0. This last property must be preserved under any small perturbation of a0 , and it follows that |ϕa,m (−m a)| > 1 − for any a ∈ H0,m sufficiently close to a0 . Thus is a proper map from H0,m onto D. Since −1 (0) is the single point 0, with (0) = − me = 0, it follows that is a conformal diffeomorphism. 0 to D using the construction We can define now the conformal mapping from Hm 0 → D, is written as (a) = −e a η(a m−1 ), above. Since the conformal mapping : Hˆ m m 2π it follows that −1 sends a sector S = {z ∈ D, 0 ≤ arg(z) ≤ m−1 } into a sector 0 with an amplitude equal to 2π . We can see that S ∼ D. Hence we −1 (S) ⊂ Hˆ m = m−1 0 defined as obtain a conformal mapping from S to Hm 0 : S → Hm m−1 z → −1 (z) .
(13)
n are unbounded. Proposition 4.4. For all n, m ≥ 2, the connected components of Hm
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0 . We assume Proof. Let U be a connected component of a capture zone different from Hm that U is bounded, then
sup |λ| = M1 < +∞.
λ∈∂U
0 , we observe that 0 ∈ Since λ = 0 belongs to Hm / U. We claim that there exist 1 (m) > 0 such that for all λ ∈ ∂U , we have that ◦k (−m)| ≥ (m) for all k ≥ 0. To see this, we only need to prove that for all |Fλ,m 1 λ ∈ ∂U we can find 1 > 0 such that D(0, 1 ) ⊂ A∗λ,m (0). For all λ ∈ C there exists 0 > 0, depending on |λ| and m, (Theorem 2.1) such that D(0, 0 ) ⊂ A∗λ,m (0). We also know (see Lemma 2.2) that 1 1 m−1 0 (|λ|, m) ≥ min 1, . |λ|e
If λ belongs to ∂U , then |λ| ≤ M1 , and we have
1 |λ| e
Hence, we define 1 = min{1,
1 M1 e
1 m−1
≥
1 m−1
1 M1 e
1 m−1
.
} and this proves the claim.
Let λ0 ∈ U . Since U is a capture zone, by definition we have that Fλ◦k0 ,m (−m) → 0 as k → ∞. Let k0 ≥ 0 be such that, for all k ≥ k0 , we have |Fλ◦k0 ,m (−m)| < 1 /2. We consider now the mapping, ◦k0 Fλ,m (−m) : U → C ◦k0 λ → Fλ,m (−m).
On the one hand, this is a holomorphic function of λ. On the other hand, since 0 ∈ / U, ◦k0 we have that Fλ,m (−m) = 0 for all λ ∈ U (the only preimage of z = 0 under Fλ,m (z) ◦k0 (−m), we have is z = 0). If we apply the minimum principle to Fλ,m
1 ◦k0 ◦k0 0 ≥ |Fλ◦k0 ,m (−m)| ≥ inf |Fλ,m (−m)| = inf |Fλ,m (−m)| ≥ 1 , λ∈U λ∈∂U 2 obtaining thus a contradiction.
n are simply Proposition 4.5. For all n, m ≥ 2, the connected components of Hm connected.
Proof. The proof uses a surgery construction (see Sect. 3.3 for preliminaries on this n where m, n ≥ 2. We consider the technique). Let U be a connected component of Hm following mapping: U : U → D \ {0} ◦k+1 (−m)), λ → ϕλ (Fλ,m where ϕλ denotes the Bötcher coordinate near the origin. As in the previous proposition the map U is a proper mapping and we will prove that it is a local homeomorphism.
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Let λ0 ∈ U and z 0 = U (λ0 ). The idea of this surgery construction is the following: k+1 (−m) has Böttcher coordifor z near z 0 we can build a map Fλ(z),m such that Fλ(z),m nate z. We denote by Wλ0 the connected component of Aλ0 ,m (0) containing Fλ◦n0 ,m (−m), preimage of A∗λ0 ,m (0). Let Cλ0 be an small open neighborhood of Fλ◦n+1 (−m) contained 0 ,m in A∗λ0 ,m (0), and Bλ0 ⊂ Wλ0 be the preimage of Cλ0 containing Fλ◦n0 ,m (−m). For any 0 < < min{|z 0 |, 1 − |z 0 |} and any z ∈ D(z 0 , ), we choose a diffeomorphism δz : Bλ0 → Cλ0 with the following properties: • δz 0 = Fλ0 ,m ; • δz coincides with Fλ0 ,m in a neighborhood of ∂ Bλ0 for any z; (z). • δz (Fλ◦k0 ,m (−m)) = ϕλ−1 0 We consider, for any z ∈ D(z 0 , ), the following mapping G z : C → C: if x ∈ Bλ0 δ (x) G z (x) = z Fλ0 ,m (x) if x ∈ / Bλ0 . We proceed to construct an invariant almost complex structure, σz , with bounded dilatation ratio. Let σ0 be the standard complex structure of C. We define a new almost complex structure σz in C, ⎧ ∗ on Bλ0 ⎨ (δz ) σ0 n ∗ σ on F −n (B ) for all n ≥ 1 (F ) σz := . λ0 ,m λ0 ⎩ λ0 ,m on C \ n≥1 Fλ−n (B ) σ0 λ 0 ,m 0 By construction σ is G z -invariant, i.e., (G z )∗ σ = σ , and it has bounded distortion since δz is a diffeomorphism and Fλ0 is holomorphic. If we apply the Measurable Riemann Mapping Theorem (see Sect. 3.3 and Remark 3.6) we obtain a quasiconformal map τz : C → C such that τz integrates the complex structure σz , i.e., (τz )∗ σ = σ0 , normalized so that τ (0) = 0 and τ (−m) = −m. Finally, we define Rz = τz ◦ G z ◦ τz−1 , which is analytic, hence an entire function. We claim that this resulting mapping is Rz (x) = λx m exp(x), for some λ. Indeed, the map Rz : C → C is an entire map (∞ is an essential singularity) with a superattracting fixed point at the origin. Moreover, Rz has a critical point at z = −m. Thus Rz (x) = νx m exp(h 1 (x)). It is easy to show that the growth order of Fλ,m is equal to 1, hence Rz has the same growth order. We know that the composition of a function of finite growth order by a quasiconformal function can only change the growth order by a real factor ([G]). We can conclude that Rz has finite growth order, hence h 1 (x) is a polynomial of degree d. Then there are d directions where Re(h 1 (x)) → +∞ and I m(h 1 (x)) is bounded for x → ∞, separated by d directions where Re(h 1 (x)) → −∞ and I m(h 1 (x)) is bounded. Thus there are d directions where Rz → ∞ separated by d directions where Rz → 0. This behavior is invariant under topological conjugation. Since Fλ0 ,m has only one direction (along the positive real axis) where Fλ0 ,m → ∞ and one (the negative real axis) where Fλ0 ,m → 0, we conclude that d = 1 and Rz (x) = νx m exp(a0 + a1 x). If we use that −m is a critical point, then a1 must be equal to 1. Finally, if we define λ = ν exp(a0 ), we can conclude that Rz (x) = λx m exp(x). By construction, τz 0 is the identity for z = z 0 , then there exists a continuous function z ∈ D(z 0 , ) → λ(z) ∈ U such that λ(z 0 ) = z 0 and Fλ(z),m = τz ◦ G z ◦ τz−1 .
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Moreover, τz is holomorphic on A∗λ0 ,m (0) conjugating Fλ0 ,m and Fλ(z),m . Hence, observing the following commutative diagram: D ⏐ ϕλ0 ⏐
z2
−−−−→
D ⏐ ⏐ ϕλ 0
Fλ0 ,m
A∗λ0 ,m (0) −−−−→ A∗λ0 ,m (0) ⏐τ z ⏐ τz ⏐ ⏐ Fλ(z),m
A∗λ(z),m (0) −−−−→ A∗λ(z),m (0) we have that ϕλ(z) = ϕλ0 ◦ τz−1 is the Böttcher coordinate of A∗λ(z),m (0). Finally we conclude that ◦n+1 (−m)) = z, U (λ(z)) = ϕλ(z) (Fλ(z),m ◦n+1 (−m) = τ ◦G ◦n+1 ◦τ −1 (−m) = τ ◦G ◦n+1 (−m) = τ ◦G (F ◦n (−m)) = since Fλ(z),m z z z z λ0 ,m z z z −1 −1 τz ◦ ϕλ−1 (z) = τz ◦ τz−1 ◦ ϕλ(z) (z) = ϕλ(z) (z). 0
4.3. Parameter Plane: Proof of Theorem C. ∗ 0 then A 0 then Proposition 4.6. If λ ∈ Hm / Hm λ,m (0) = Aλ,m (0). Otherwise if λ ∈ Aλ,m (0) has infinitely many connected components. 0 . As in Proposition 4.1, let γ be a simple path in A∗ (0) that joins Proof. Let λ ∈ Hm λ,m Fλ,m (−m) and 0. The preimage of γ must include a path γ˜ contained in A∗λ,m (0) that joins −∞ with 0 passing through −m (γ˜ maps 2-1 to γ ). Since H|λ|,m intersects γ˜ , it follows that H|λ|,m ⊂ A∗λ,m (0). We recall that H|λ|,m is a preimage of a small disk of radius 0 (see Sect. 2). All preimages of γ˜ , are contained in A∗λ,m (0) as well, since they all intersect H|λ|,m . In fact, we have that Aλ,m (0) = A∗λ,m (0) since any preimage of D 0 must contain points of H|λ|,m . Hence Aλ,m (0) has a unique connected component. 0 . From Proposition A(b) we have that A Now assume λ ∈ / Hm λ,m (0) has either one or infinitely many connected components. If we suppose that Aλ,m (0) has only one connected component, then Aλ,m (0) is a completely invariant component of the Fatou set. Then, all the critical values of Fλ,m are in Aλ,m (0) (see [B2]), and hence we conclude 0. that −m belongs to Aλ,m (0). However, this is impossible if λ ∈ / Hm 0 , the boundary of A∗ (0) (which is equal to the Julia set) Proposition 4.7. If λ ∈ Hm λ,m is a Cantor bouquet and it is not locally connected. 0 , we have Proof. Using the proposition above, if λ belongs to the main capture zone, Hm ∗ that Aλ,m (0) = Aλ,m (0). Hence, the Fatou set contains a totally invariant component. In fact, from [BD], it follows that the Julia set has an uncountable number of connected components and it is not locally connected at any point. Using standard techniques analogous to [DT] one can show that the Julia set contains a Cantor Bouquet tending to ∞ in the direction of the positive real axis. Indeed,
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it is sufficient to construct a hyperbolic exponential tract on which Fλ,m has asymptotic direction θ ∗ . To this end, let Br be an open disk containing Fλ,m (−m). The preimage of this set is an open set similar to H|λ|,m (see Sect. 2). Let D be the complement of this set. We have that Fλ,m maps D onto the exterior of Br , then D is an exponential tract for Fλ,m . We may choose the negative real axis to define the fundamental domains in D. More precisely, we can find the preimage of the negative real axis under the function Fλ,m . Hereafter, we denote by Arg(.) ∈ (−π, π ] the principal argument. Using the definition of Fλ,m it is easy to see that Arg(Fλ,m (z)) = Arg(λ) + m Arg(z) + I m(z)
mod (2π ).
Finding the preimages of R− is equivalent to solving Arg(Fλ,m (z)) = π. The equation above is equivalent to Arg(λ) + mα + rsin(α) = (2k + 1)π
k ∈ Z,
where r = |z| and α = Arg(z). Hence, we obtain r = ρ(α) =
(2k + 1)π − mα − Arg(λ) sin(α)
α ∈ (−π, π ).
We denote each of these curves by σk = σk (λ, m), where the possible values of the argument depend on k. As their real parts tend to +∞, the σk ’s are asymptotic to the lines I m(z) = (2k + 1)π − Arg(λ). Since the curves σk for k ∈ Z are mapped by Fλ,m onto the negative real axis, it follows that D has asymptotic direction θ ∗ = 0. Furthermore, since Fλ,m (z) = λz m exp(z), one may check readily that D is a hyperbolic exponential tract. Before proving assertion (c) of Theorem C we prove the following auxiliary lemma. e m−1 Lemma 4.8. If |λ| > ( m−1 ) , then the boundary of A∗λ,m (0) is a quasicircle. 0 be such that |λ| > ( e )m−1 . By using the same arguments of Proof. Let λ ∈ / Hm m−1 Proposition 4.2 we have that Fλ,m is polynomial-like of degree m near the origin. From this construction we obtain that ∂ A∗λ,m (0) = φ(T), and the lemma follows. e m−1 Remarks 4.9. The reason to ask for |λ| > ( m−1 ) as a condition is as follows. We want to find a value K > 0 such that if |z| = K then |Fλ,m (z)| > K . This condition is equivalent to
|Fλ,m (z)| ≥ |λ||z|m e−|z| = |λ|(K )m e−K > K or equivalently |λ| > K 1−m e K .
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We want to use this argument for the largest possible region of values of λ. Hence, we choose K > 0, such that K 1−m e K is minimum. This minimum value is reached exactly at K = m − 1. 0 . If λ ∈ U , Proposition 4.10. Let Um be the unbounded connected component of C\Hm m then the boundary of A∗λ,m (0) is a quasicircle. 0 . Since U is unbounded let Proof. Let Um be the unbounded component of C \ Hm m λ0 ∈ Um be such that ∂ A∗λ0 ,m (0) is a quasicircle (see Lemma 4.8), and hence A∗λ0 ,m (0) is a quasidisk. On the other hand, since Um is an open and simply connected set, let ψ : D → Um be the Riemann mapping such that ψ(0) = λ0 . We claim that for all λ ∈ Um , the Böttcher mapping ϕλ conjugating Fλ,m to z → λz m extends to the whole immediate basin of attraction A∗λ,m (0) (see Sect. 3.1). To see the claim we only need to observe that, when λ ∈ Um the critical point −m does not belong to A∗λ,m (0), hence no other critical point than z = 0 belongs to A∗λ,m (0). It follows that for all λ ∈ Um , the Böttcher coordinate
ϕλ : A∗λ,m (0) → D, is a conformal mapping. We can define now a holomorphic motion of A∗λ0 ,m (0) (see Sect. 3.2). We use as main ingredients the Böttcher map, ϕλ , and the conformal Riemann mapping ψ. More precisely, we consider the following map: : D × A∗λ0 ,m (0) → D × C −1 (c, z) → (c, c (z)) = (c, z (c)) = (c, ϕψ(c) ◦ ϕλ0 (z))
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We can check that is a holomorphic motion. By construction, we have that 0 (z) = −1 ◦ ϕλ0 (z) = z. If we fix the parameter c we must see that the map c (z) is injective. ϕψ(0) This is immediate, since the Bötcher mapping ϕλ is conformal. Finally, if we fix a point z ∈ A∗λ0 ,m (0) we must see that z : D → C is a holomorphic map. In this case the map z is a composition of holomorphic maps, since the Böttcher map depends analytically on parameters (see Fig. 5). Geometrically, if we fix λ ∈ Um , the map z → ψ −1 (λ) (z) sends points in A∗λ0 ,m (0) to points in A∗λ,m (0) according to the Böttcher coordinates. Finally, we apply Lemma (3.4) to the holomorphic motion , which roughly speaking, says that a holomorphic motion of a quasidisk is also a quasidisk. The final assertion of Theorem C(c), follows directly from the fact that all the sets n are unbounded and hence belong to U Hm m ˆ of A∗ (0), we have that Remarks 4.11. Since extends to a holomorphic motion λ0 ,m ˆ c (z 2 ) for all z 1 , z 2 ∈ A∗ (0). In other words, if we take ˆ c (z 1 ) = for all c ∈ D, λ0 ,m z 1 and z 2 in the boundary of A∗λ0 ,m (0), the property above proves that two internal rays never land at a common point. 4.4. Parameter Plane: Proof of Theorem D. From Theorem B, statements b) and c), 0 is bounded, connected and simply connected. As we mentioned in we know that Hm 0 is a topological disc. If this were the case then the introduction, we conjecture that Hm
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Fig. 5. Sketch of the Holomorphic motion c (z), where λ = ψ(c). Geometrically, c (z) sends equipotentials and rays from A∗λ ,m (0) to A∗λ,m (0) according to Bötcher coordinates 0
0 would consist of only one connected component which would be unbounded. But C\Hm 0 might have other connected as long as this result it is not proven, the complement of Hm components different from the unbounded one, which we denote by Um . In Theorem D we study the topological relation between these sets.
Proof. of Theorem D. In this proof we use the monic family of functions L a,m = z m exp(z/a) (see Sect. 3.1 and Proposition 4.3). We recall that L a,m (z) is conformally conjugate to Fλ,m (z). We introduce this new family of maps in order to obtain a preferred Böttcher coordinate near z = 0. We also recall that the free critical point for the family L a,m (z) is at the point ca,m = −ma. We denoted by ϕa the Böttcher coordinate defined in the whole immediate basin of attraction of z = 0 (Lemma 3.2) and by (a) = ϕa (ca,m ) (Eq. 12) the uniformization mapping of the main capture zone (Proposition 4.3). 0 and U have a common boundary. Since U is the unWe want to show that Hm m m 0 we have that ∂U ⊂ ∂H0 . Now, we will prove that bounded component of C \ Hm m m 0 ⊂ ∂U and thus statement a) follows. In order to do this, we first observe that the ∂Hm m n , for n ≥ 2, are contained in U since they are unbounded rest of the capture zones Hm m 0 0 , the sequence of and disjoint from Hm . Second, notice that for any point a0 in ∂Hm n {L a,m (ca,m )}n≥0 is not a normal family in any neighborhood of a0 . 0 meets Hn for Third, we claim that any arbitrary neighborhood of any point in ∂Hm m 0 some n ≥ 2. To see the claim, let a0 be a point in ∂Hm , let W be a neighborhood of n = ∅ for some n ≥ 2. We also consider α = 1/2 and a0 . We must show that W ∩ Hm β1 , · · · , βm are complex numbers such that (βi )m = α . Set √ m 0 0 K = {a ∈ Hm | | (a)| > | α |} and P = C \ {a ∈ Hm | | (a)| ≤ | m α |}. By shrinking W , if necessary, we can assume that W ⊂ P. Define functions α(a) = ϕa−1 (α ) and βi (a) = ϕa−1 (βi ) for i = 1, · · · , m. See Fig. 6 for a sketch of the relevant 0 \ {0} then objects of this construction. Notice that by construction of ϕa , if a ∈ Hm −1 the forward orbit of the free critical point ca,m = −ma is contained in ϕa (D| (a)| ). In n (c particular, if a ∈ K and L a,m a,m ) = α(a), then n−1 L a,m (ca,m ) ∈ {β1 (a), . . . , βm (a)}.
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Fig. 6. Sketch of the relevant objects in proof of Theorem D
Now, let xa0 be a preimage of α(a0 ), that is not equal to βi (a0 ) for any 1 ≤ i ≤ m, n is ∞ to 1. We cannot have c n notice that L a,m a,m = x a0 because then L a,m (ca,m ) would be normal in a neighborhood of a0 . From the implicit function theorem, we know that there exists a holomorphic function x(a) such that L a,m (x(a)) = α(a) in some neighborhood of a0 , which we can suppose is W by shrinking it, if necessary. Again by shrinking W , we can suppose that x(a) = βi (a) for all a ∈ W , for i = 1, . . . , m. By lack of normality, n (c the iterates L a,m a,m ) do not avoid 0, ∞ and x(a). So, there exist a ∈ W and n ≥ 0 such that
L an ,m (ca ,m ) = x(a ). n for some n ≥ 0. We finally claim that n > 0. If n = 0, then It follows that a ∈ Hm +1 n a ∈ K , and L a ,m (ca ,m ) = α(a ); this would mean that L an ,m (ca,m ) = βi (a ) for some i, a contradiction. To prove the second statement of Theorem D, let V be a bounded connected compo0 . Hence, we have that ∂V ⊂ ∂H0 and ∂V ⊂ ∂U , since, by statement nent of C \ Hm m m 0 . Then, V has a common boundary with H0 and U . a), ∂Um = ∂Hm m m
5. A Model for Fλ,m For each natural value m ∈ N, m ≥ 2 and α, β ∈ R we define the two-parameter family of maps G α,β,m (z) = eiα z m eβ/2(z−1/z) . It is easy to check that, G α,β,m preserves the unit circle, S 1 , and on this circle we have the following dynamical system: α,β,m : θ → α + mθ + β sin(θ ) G
mod (2π ) ,
θ ∈ R/2π Z.
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When β < 1 and m = 1 this family of circle diffeomorphisms is known as the standard family or Arnold family and its parameter space contains the well known Arnold α,β,m is an m to 1 Tongues ([A]). When m ≥ 2 the situation is very different because G map of T and hence not a circle diffeomorphism. For each parameter value α and β the map G α,β,m is a holomorphic function defined on the punctured plane, C∗ , with 0 and ∞ as essential singularities. We denote by P this class of functions. Maps of this type, are studied in [Ke, Ko1, Ko2] and [Mak] among others. Let f be a holomorphic self-mapping of C∗ . The usual definitions of Fatou and Julia sets apply for functions in class P, although in this case, the Julia set can by characterized by the closure of the set of points whose orbits tend to 0 or to ∞ under iteration. Using the above characterization we can plot the Julia set of G α,β,m for different values of α, β and m. Sullivan’s Theorem of nonwandering domains has been extended to the class P ∩ S by many authors ([Ke, Ko2]). Also for this kind of functions it is proved ([EL]) that they do not have Baker domains. Hence the classification of Fatou components is exactly the same as in the rational case. The map G α,β,m is of finite type, because it has only two critical points in C∗ . Indeed, if we compute G α,β,m we obtain G α,β,m (z) =
1 iα m−2 β/2(z−1/z) e z e (βz 2 + 2mz + β), 2
and hence the two critical points z + (β) and z − (β) are given by, −m ± m 2 − β 2 . z± = β In the case where α and β are real parameters we have that G α,β,m is symmetric with respect to the unit circle which is also invariant. This condition is equivalent to τ ◦ G α,β,m = G α,β,m ◦ τ, where τ (z) = 1/z. When |β| < m the critical points z ± have the same dynamical behavior since τ (z − ) = z + . Also, it is easy to check that z + belongs to D and hence z − ∈ C \ D. In Fig. 7 we display the parameter plane of G α,β,m for m = 2, 3 and 4. We distinguish between two different behaviors of the free critical points z ± . Parameter values α and β for which the critical points tend to infinity or to zero are plotted in color, depending on the rate of escape. Parameter values α and β for which this does not occur are plotted in black. Black shapes that look like chess figures consist of parameter regions (shaped as Arnold tongues) where the attracting periodic orbit is contained in the unit circle and parameter regions (shaped as Mandelbrot sets) where the attracting periodic orbit is disjoint from the unit circle. An exhaustive analysis of these Arnold tongues can be found in [MR]. In Fig. 8 we display the dynamical plane of G α,β,m for m = 2 and different values of α and β. Points tending to z = 0 and z = ∞ are shown in color, depending on the rate of escape, while points for which this does not occur are shown in black. We also plot the unit circle in blue. The following is the main idea of our surgery construction. First, we consider two α,β,m is quasisymmetrically conjugate to θ → mθ real parameters α and β such that G
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(b)
(a)
(c)
Fig. 7. Parameter planes of G α,β,m for m = 2, 3 and m = 4
(a)
(b)
(c)
Fig. 8. Dynamical plane of G α,β,m for m = 2. Range (−2, 2) × (−2, 2)
on the unit circle. Under this condition we can change the behavior of G α,β,m on the unit disk. More precisely, we quasiconformally paste the superattracting behavior of z → z m inside the unit disk. The corresponding map acts like G α,β,m outside on the complement of D and acts like z → z m on D. Second, applying the Measurable Riemann Mapping Theorem ([Ah, LV]) we can obtain a holomorphic mapping with this dynamical behavior, and finally we will prove that this map is precisely Fλ,m (z) = λz m exp z for some parameter λ. We obtain thus that Fλ,m is quasiconformally conjugate on the complement of A∗ (0) to G α,β,m on the complement of the closed unit disc. 5.1. The connection: Proof of Theorem E. Before proving Theorem E we can prove that Wm contains an open set of parameters. We recall that Wm is given by α,β,m is quasisymmetrically conjugate to θ → mθ }. Wm = {α, β | G Lemma 5.1. {(α, β) ∈ R2 | |β| < m − 1} ⊂ Wm . α,β,m is quasisymmetrically conjugate to Proof. From Theorem 3.7 we can prove that G θ → mθ if we are able to prove that G α,β,m is an expanding map. In order to do so, a sufficient condition is to impose that min{|G α,β,m (θ )|, θ ∈ T} > 1. From the definition α,β,m (θ ) = θ → α + mθ + β sin θ we have that of G α,β,m (θ ) = m + β cos θ. G
The Parameter Planes of λz m exp(z) for m ≥ 2
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Hence, it is easy to see that when |β| < m − 1 we obtain that min{|G α,β,m (θ )|, θ ∈ T} > 1. Proof of Theorem E. Let α and β be in Wm . Let h = h α,β,m be the quasisymmetric α,β,m = h −1 ◦ g ◦ h, where g(θ ) = mθ . conjugacy, defined on the unit circle, such that G
Consider H = Hα,β,m : D → D to be the Douady-Earle quasiconformal extension of h
(0) = 0. such that H We now define a new function R = Rα,β,m : C → C as follows: G (z) z∈ /D R(z) := α,β,m
(z))m ) z ∈ D . H −1 (( H This map is equal to G α,β,m outside D and it has the desired superattracting dynamics in D, but is not holomorphic on D. We proceed to construct an invariant almost complex structure, σ = σα,β,m , with bounded dilatation ratio. Let σ0 be the standard complex structure of C. We define a new almost complex structure σ in C, ⎧
)∗ σ0 on D ⎨ (H (D) for all n ≥ 1 . σ := (R n )∗ σ on R −n ⎩σ on C \ n≥1 R −n (D) 0 By construction σ is R-invariant, i.e., (R)∗ σ = σ , and it has bounded distortion
is quasiconformal and R is holomorphic outside D. If we apply the Measurable since H Riemann Mapping Theorem we obtain a quasiconformal map ϕ = ϕα,β,m : C → C such that ϕ integrates the complex structure σ , i.e., (ϕ)∗ σ = σ0 , normalized so that ϕ(0) = 0 and ϕ(z − ) = −m. Finally, we define R˜ = R˜ α,β,m = ϕ ◦ R ◦ ϕ −1 , which is analytic, hence an entire function. Our goal now is to show that there exists a complex ˜ value λ such that R(z) = λz m exp(z). ˜ The map R : C → C is an entire map (∞ is an essential singularity) with a superattracting fixed point at the origin. Near the origin R˜ is conjugate to the map z → z m . Moreover, R˜ has a critical point at z = −m, since the map R has one critical point at z − ∈ C \ D and ϕ(z − ) = −m. The other critical point of G α,β,m is at z + and it has been erased by the quasiconformal surgery construction because it belonged to D. Thus ˜ R(z) = νz m exp(h 1 (z)). By using the same arguments as in Proposition 4.5 we can ˜ conclude that R(z) = Fλ,m (z) = λz m exp(z) for a suitable value of λ. By construction, the boundary of A∗λ,m (0) is a quasicircle, since A∗ (0) is the quasiconformal image of the unit disk, obtaining thus a value λ ∈ C such that ∂ A∗λ,m (0) is a quasicircle. Acknowledgements. We would like to thank Christian Henriksen for many discussions and in particular for providing the idea of the proof of Theorem D. We would also like to thank Adrien Douady for all his valuable suggestions.
References [Ah] [A]
Ahlfors, L.: Lectures on quasiconformal mappings. New York: Wadswoth & Brooks/Cole Mathematics Series, 1966 Arnold, V.: Small denominators i, on the mappings of the circumference into itself. Amer. Math. Soc. Transl. (2) 46, 213–284 (1965)
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[B] [B1] [B2] [BD] [BA] [Be] [Bo] [BH1] [BH2] [BuHe] [CG] [DE] [DT] [dMvS] [DH1] [DH2] [DH3] [EL] [EL1] [EL2] [FG] [F] [Fau] [G] [GK] [Ke] [Ko1] [Ko2] [L] [LV] [Mak] [MSS]
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Baker, I.N.: The domains of normality of an entire function. Ann. Acad. Sci. Fenn. Ser. A I Math. 1, 277–283 (1975) Baker, I.N.: An entire function which has wandering domains. J. Austral. Math. Soc. 22, 173– 176 (1976) Baker, I.N.: Wandering domains in the iteration of entire functions. Proc. London Math. Soc. 49, 563– 576 (1984) Baker, I.N., Dominguez, P.: Some connectedness properties of julia sets. Complex Variables Theory Appl. 41, 371–389 (2000) Beurling, A., Ahlfors, L.V.: The boundary correspondence under quasiconformal mappings. Acta Math. 96, 125–142 (1956) Bergweiler, W.: Invariant domains and singularities. Math. Proc. Camb. Phil. Soc. 117, 525– 532 (1995) Böttcher, L.E.: The principal laws of convergence of iterates and their application to analysis (russian). Izv. Kazan. Fiz.-Mat. Obshch. 14, 155–234 (1904) Branner, B., Hubbard, J.H.: The iteration of cubic polynomials part i: the global toology of parameter space. Acta Math. 169, 143–206 (1992) Branner, B., Hubbard, J.H.: The iteration of cubic polynomials part ii: patterns and parapatterns. Acta Math. 169, 229–325 (1992) Buff, X., Henriksen, C.: Julia sets in parameter spaces. Commun. Math. Phys. 220, 333–375 (2001) Carleson, L., Gamelin, Th.: Complex Dynamics. Berlin-Heidelberg-New York: Springer, 1993 Douady, A., Earle, C.J.: Conformally natural extension of homeomorphism of the circle. Acta Math. 157, 23–48 (1986) Devaney, R.L., Tangerman, F.: Dynamics of entire functions near the essential singularity. Ergodic Theory Dynam. Systems 6, 489–503 (1986) de Melo, W., van Strien, S.: One-Dimensional dynamics. Berlin-Heidelberg-New York: SpringerVerlag, 1993 Douady, A., Hubbard, J.H.: Étude dynamique des polynômes complexes. Part I. Publ. math. d’Orsay, 1984 Douady, A., Hubbard, J.H.: Étude dynamique des polynômes complexes. Part II. Publ. math. d’Orsay, 1985 Douady, A., Hubbard, J.H.: On the dynamics of polynomial-like mappings. Ann. Scient. Ec. Norm. Sup. 18, 287–343 (1985) Eremenko, A.E., Lyubich, M.Yu.: Iterates of entire functions. Soviet Math. Dokl. 30, 592–594 (1984); translation from Dokl. Akad. Nauk. SSSR 279, 25–27 (1984) Eremenko, A.E., Lyubich, M.Yu.: The dynamics of analytic transforms. Leningrad. Math. J. 1, 563– 634 (1990) Eremenko, A.E., Lyubich, M.Yu.: Dynamical properties of some classes of entire functions. Ann. Inst. Fourier 42, 989–1020 (1992) Fagella, N., Garijo, A.: Capture zones of the family of functions f λ,m (z) = λz m exp(z). Inter. J. of Bif. and Chaos (3) 9, 2623–2640 (2003) Fatou, P.: Sur l’iterátion des fonctions transcendentes entières. Acta Math. 47, 337–370 (1926) Faught, D.: Local connectivity in a family of cubic polynomials. Ph.D Thesis, Cornell University, 1992 Geyer, L.: Siegel discs, herman rings and the arnold family. Trans. Amer. Math. Soc. 353, 3661– 3683 (2001) Goldberg, L.R., Keen, L.: A finiteness theorem for a dynamical class of entire functions. Ergodic Th. Dynam. Sys. 6, 183–192 (1986) Keen, L.: Dynamics of holomorphic self-maps of C∗ . Proc. Workshop of Holomorphic Functions and Moduli, Berlin-Heidelberg-New York: Springer-Verlag 1988, pp. 9–30 Kotus, J.: Iterated holomorphic maps on the punctered plane. In: Dynamical Systems, Kurzhanski, A.B., Sigmund, K.J. eds. 287, Berline-Heidelber-New York: Springer Verlag, 1987, pp. 10–29 Kotus, J.: The domains of normality of holomorphic self-maps of C∗ . Ann. Acad. Sci. Fenn. (Ser. A, I. Math) 15, 329–340 (1990) Lei, T.: The Mandelbrot set, theme and variations. London Math. Soc. Lecture Note Ser. 274, Cambridge: Cambridge Univ. Press, 2000 Letho, O., Virtanen, K.I.: Quasiconformal mappings in the plane. Berlin-Heidelberg-New York: Springer-Verlag, 1973 Makienko, P.: Iteration of analytic functions of C∗ (Russian). Dokl. Akad. Nauk. SSRR 297, 35–37 (1987); Translation in Sov. Math. Dokl 36, 418–420 (1988) Mañé, R., Sad, P., Sullivan, D.: On the dynamics of rational maps. Ann. Sci. École Norm. Sup. 16, 193–217 (1983)
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Milnor, J.: On cubic polynomials with periodic critical point. Stony Brook Institute for Mathematical Sciences. http://www.math.sunysb.edu/dynamics/surveys.html, 1991 Milnor, J.: Dynamics in one complex variable: Introductory lectures. Weshaden: Vieweg, 1999 Misiurewicz, M., Rodrigues, A.: Double standard maps. Preprint. http://www.math.iupui.edu/ mmisiure/publlist.html Pommrenke, Ch.: Boundary Behavior of Conformal Maps, Berlin-Heidelberg-New York: SpringerVerlag, 1991 Roesch, P.: Puzzles de yoccoz pour les applications à allure rationnelle. Enseign. Math. (2) 45(1–2), 133–168 (1999) Shub, M., Sullivan, D.: Expanding endomorphims of the circle revisited. Ergodic Theory Dynamical Systems 5, 285–289 (1985) Słodkowski, Z.: Holomorphic motions and polynomials hulls. Proc. Amer. Math. Soc. 111, 347– 355 (1991) Zakeri, S.: Dynamics of cubic siegel polynomials. Commun. Math. Physics. 206, 185–233 (1999)
Communicated by G. Gallavotti
Commun. Math. Phys. 273, 785–801 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0259-6
Communications in
Mathematical Physics
Partial Regularity of Solutions to the Four-Dimensional Navier-Stokes Equations at the First Blow-up Time Hongjie Dong1,2, , Dapeng Du3 1 Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA 2 School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA.
E-mail: [email protected]
3 School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China.
E-mail: [email protected] Received: 16 August 2006 / Accepted: 8 December 2006 Published online: 15 May 2007 – © Springer-Verlag 2007
Abstract: The solutions of incompressible Navier-Stokes equations in four spatial dimensions are considered. We prove that the two-dimensional Hausdorff measure of the set of singular points at the first blow-up time is equal to zero. 1. Introduction In this paper we consider both the Cauchy problem and the initial-boundary value problem for incompressible Navier-Stokes equations in four spatial dimensions with unit viscosity and zero external force: u t + u∇u − u + ∇ p = 0, div u = 0
(1.1)
in a smooth domain Q T = × (0, T ) ⊂ Rd × R. Boundary condition u|×[0,T ] = 0 is imposed if = Rd . Here d = 4 and the initial data a is in the closure of {u ∈ C0∞ () ; div u = 0} in L d () if is bounded, or is in the closure of {u ∈ C ∞ () ; div u = 0} in L d () ∩ L 2 () if = Rd . The local well-posedness of such problems is well-known (see, for example, [9] and [5]). The solution u is locally smooth in both spacial and time variables. We are interested in the partial regularity of u at the first blow-up time T . Many authors have studied the partial regularity of solutions (in particular, weak solutions) of the Navier-Stokes equations, especially when d is equal to three. V. Scheffer studied partial regularity in a series of papers [17, 18, 20]. In three space dimensions, he established various partial regularity results for weak solutions satisfying the so-called local energy inequality. For d = 3, the notion of suitable weak solutions was first introduced in a celebrated paper [1] by L. Caffarelli, R. Kohn and L. Nirenberg. They called Hongjie Dong was partially supported by the National Science Foundation under agreement No. DMS0111298. Dapeng Du was partially supported by a postdoctoral grant from School of Mathematical Sciences at Fudan University.
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a pair consisting of velocity u and pressure p a suitable weak solution if u has finite energy norm, p belongs to the Lebesgue space L 5/4 , u and p are weak solutions to the Navier-Stokes equations and satisfy a local energy inequality. After proving some criteria for local boundedness of solutions, they established partial regularity of solutions and estimated the Hausdorff dimension of the singular set. They proved that, for any suitable weak solution u, p, there is an open subset where the velocity field u is Hölder continuous and they showed that the 1-D Hausdorff measure of the complement of this subset is equal to zero. In [15], with zero external force, F. Lin gave a more direct and sketched proof of Caffarelli, Kohn and Nirenberg’s result. A detailed treatment was later given by O. Ladyzhenskaya and G. A. Seregin in [13]. Very recently, some extended results are obtained in Seregin [16] and Gustafson, Kang and Tsai [6]. For d = 4, V. Scheffer proved in [19] that there exists a weak solution u in R4 × R+ such that u is continuous outside a locally closed set of R4 × R+ whose 3-D Hausdorff measure is finite. Although Scheffer’s paper is not recent, it appears to us that this is the only published results on the partial regularity of 4-D Navier-Stokes equations. Remark 1.1. The weak solution considered in [19] doesn’t verify the local energy estimate. The existence of a weak solution satisfying the local energy estimate is still an open problem. Now let’s state our result. Instead of dealing with weak solutions, we work on classical solutions of 4-D Navier-Stokes equations, which are regular before they blow up. Our first result is that the singular set at the first blowup time is a compact set with zero 2-D Hausdorff measure. We show this after two partial regularity criterions are obtained. Our proof is conceptually similar to Lin’s in [15], but the problem is technically harder. The main difficulty is the lack of certain compactness. We overcome it by a novel use of the backward heat kernel (see the proof of Lemma 2.12) and by the use of two appropriate scaled norms of the pressure. It is possible because the nonlinear term is controlled by using the Sobolev embedding theorem, although we don’t have a compact embedding here. Remark 1.2. In the setting of classical solutions, our result is the 4-D version of Caffarelli, Kohn and Nirenberg’s theorem in [1]. As an application of one of the partial regularity criterions derived in the proof of the first result we get our second result: in case = R4 if a solution blows up, it must blow up at a finite time. Remark 1.3. We can prove a similar result in 3-D by using the same argument. Detailed discussions on the long-time behavior of solutions to 3-D Navier-Stokes can be found in J. Heywood [7] and M. Wiegner [24] (and references therein). It seems that we need some new method to deal with the five or higher dimensional case. To the authors’ best knowledge all the existing methods on partial regularity for the Navier-Stokes equations share the following prerequisite condition: in the energy inequality the nonlinear term should be controlled by the energy norm under the Sobolev imbedding theorem. Actually, four is the highest dimension in which we have such condition. In five or higher dimensions, such condition fails. Therefore, we cannot hope the existing methods work in the five or higher dimensional case. The article is organized as follows. Our main theorems (Theorem 2.1-2.5) are given in the following section. Some auxiliary estimates are proved in Sects. 3 and 4, among which Lemma 2.12 plays a crucial role. We give the proof of our main theorems in the last section.
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To conclude this Introduction, we explain some notation used in what follows: Rd is a d-dimensional Euclidean space with a fixed orthonormal basis. A typical point in Rd is denoted by x = (x1 , x2 , . . . , xd ). As usual the summation convention over repeated indices is enforced. And x · y = xi yi is the inner product for x, y ∈ Rd . For t > 0, we denote Ht = R4 × (0, t) and space points are denoted by z = (x, t). Various constants are denoted by N in general and the expression N = N (· · · ) means that the given constant N depends only on the contents of the parentheses. 2. Setting and Main Results We shall use the notation in [13]. Let ω be a domain in some finite-dimensional space. Denote L p (ω; Rn ) and W pk (ω; Rn ) to be the usual Lebesgue and Sobolev spaces of functions from ω into Rn . Denote the norm of the spaces L p (ω; Rn ) and W pk (ω; Rn ) by · L p ,ω and · W pk ,ω respectively. As usual, for any measurable function u = u(x, t) and any p, q ∈ [1, +∞], we define u(x, t) L tp L qx := u(x, t) L qx L p . t
For summable functions p, u = (u i ) and τ = (τi j ), we use the following differential operators ∂u ∂u , u ,i = , ∇ p = ( p,i ), ∇u = (u i, j ), ∂t ∂ xi div u = u i,i , div τ = (τi j, j ), u = div∇u,
∂t u = u t =
which are understood in the sense of distributions. We use the notation of spheres, balls and parabolic cylinders, S(x0 , r ) = {x ∈ R4 ||x − x0 | = r }, S(r ) = S(0, r ), S = S(1); B(x0 , r ) = {x ∈ R4 ||x − x0 | < r },
B(r ) = B(0, r ),
B = B(1);
Q(z 0 , r ) = B(x0 , r ) × (t0 − r 2 , t0 ),
Q(r ) = Q(0, r ),
Q = Q(1).
Also we denote mean values of summable functions as follows: 1 [u]x0 ,r (t) = u(x, t) d x, |B(r )| B(x0 ,r ) (u)z 0 ,r (t) =
1 |Q(r )|
Q(z 0 ,r )
u dz.
In case = Rd , in a well-known paper [9] Kato proved that the problem is locally well-posed. By known local regularity theory for Navier-Stokes equations it can be proved that Kato’s (also known as mild) solutions are smooth in Rd × (0, T∗ ] for some T∗ > 0. Meanwhile, for bounded , it is also known (see [5]) that there exists a unique solution u of (1.1) satisfying (1) (2) (3)
u ∈ C([0, T∗ ]; L d ), u(0) = a for some T∗ > 0; u ∈ C((0, T∗ ]; D(Aα )) for any 0 < α < 1; Aα u(t) = o(t −α ) as t → 0.
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Here A is the stokes operator. Moreover, such solution is smooth in × (0, T∗ ]. In both cases, let T = sup T∗ be the first blow-up time. Then u is a smooth function in Q T . Let η(x) be a smooth function on R4 supported in the unit ball B(1), 0 ≤ η ≤ 1 and ¯ η ≡ 1 on B(2/3). Let z 0 be a given point in × (0, T ] and r > 0 a real number such that Q(z 0 , r ) ⊂ Q T . It’s known that for a.e. t ∈ (t0 − r 2 , t0 ), in the sense of distribution, one has ∂2 ui u j ∂ xi ∂ x j ∂2 = (u i − [u i ]x0 ,r )(u j − [u j ]x0 ,r ) in B(x0 , r ). ∂ xi ∂ x j
p =
For these t, we consider the decomposition p = p˜ x0 ,r + h x0 ,r in B(x0 , r ), where p˜ x0 ,r is the Newtonian potential of (u i − [u i ]x0 ,r )(u j − [u j ]x0 ,r )η(x/r ). Then h x0 ,r is harmonic in B(x0 , r/2). In the sequel, we omit the indices of p˜ and h whenever there is no confusion. The following notation will be used throughout the article: 1 A(r ) = A(r, z 0 ) = ess supt0 −r 2 ≤t≤t0 2 |u(x, t)|2 d x, r B(x0 ,r ) 1 E(r ) = E(r, z 0 ) = 2 |∇u|2 dz, r Q(z 0 ,r ) 1 C(r ) = C(r, z 0 ) = 3 |u|3 dz, r Q(z 0 ,r ) 1 D(r ) = D(r, z 0 ) = 3 | p − [h]x0 ,r |3/2 dz, r Q(z 0 ,r ) 1 2α 1 t0 F(r ) = F(r, z 0 ) = 2 | p − [h]x0 ,r |1+α d x 2α dt 1+α , r t0 −r 2 B(x0 ,r ) where α ∈ (0, 1) is a number to be specified later. Notice that these objects are all invariant under the natural scaling. Here are our main results: Theorem 2.1. Let be a smooth bounded set or the whole space R4 and let (u, p) be the solution of (1.1). There is a positive number ε0 satisfying the following property. Assume that for a point z 0 ∈ × T the inequality lim sup E(r ) ≤ ε0 r ↓0
(2.1)
holds. Then z 0 is a regular point. Theorem 2.2. Let be a smooth bounded set or the whole space R4 and (u, p) be the solution of (1.1). There is a positive number ε0 satisfying the following property. Assume that for a point z 0 ∈ Q T and for some ρ0 > 0 such that Q(z 0 , ρ0 ) ⊂ Q T and C(ρ0 ) + D(ρ0 ) + F(ρ0 ) ≤ ε0 . Then z 0 is a regular point.
(2.2)
Partial Regularity of Solutions to the NSE
789
Remark 2.3. It is worth noting that the object under estimation in condition (2.1) involves the gradient of u while the objects in condition (2.2) involve only u and p themselves. However, by using condition (2.1) one can obtain a better estimate of the Hausdorff dimension of the set of all singular points. Theorem 2.4. Let be a smooth bounded set or the whole space R4 and (u, p) be the solution of (1.1). Then the 2-D Hausdorff measure of the set of singular points in × T is equal to zero. Theorem 2.5. Assume is the whole space R4 . Let (u, p) be the solution of (1.1). If the solution does not blow up in finite time, then u is bounded and smooth in R4 × (0, +∞). In the sequel, we shall make use of the following well-known interpolation inequality. Lemma 2.6. For any functions u ∈ W21 (R4 ) and real numbers q ∈ [2, 4] and r > 0, q−2 2−q/2 |u|q d x ≤ N (q) |∇u|2 d x |u|2 d x Br
Br
+r −2(q−2)
Br
|u|2 d x
q/2 .
(2.3)
Br
Let (u, p) be the solution of the Navier-Stokes equation (1.1). Lemma 2.7. (i) We have u ∈ L ∞ (0, T ; L 2 (; R4 )) ∩ L 2 (0, T ; W21 (; R4 )) ∩ L 3 (Q T ),
(2.4)
p ∈ L 3/2 (Q T ).
(2.5)
(ii) For 0 < t ≤ T and for all non-negative functions ψ ∈ C0∞ ( × (0, ∞)), the following generalized energy inequality is satisfied: 2 ess sup0<s≤t |u(x, s)| ψ(x, s) d x + 2 |∇u|2 ψ d xds Qt ≤ {|u|2 (ψt + ψ) + (|u|2 + 2 p)u · ∇ψ} d x ds.
(2.6)
Qt
Sketch of the proof. To prove u ∈ L ∞ (0, T ; L 2 (; R4 )) ∩ L 2 (0, T ; W21 (; R4 )), it suffices to multiply the first equation of (1.1) by u and integrate by parts. By using Lemma 2.6 with q = 3 and integrating in t, we obtain u ∈ L 3 (Q T ). Since in Q T it holds that p =
∂2 ui u j , ∂ xi ∂ x j
(2.5) follows from the Calderón-Zygmund estimate. Next, let’s prove part (ii). First, notice that for 0 < t < T , (2.6) can be obtained by multiplying the first equation of
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(1.1) by uψ, integrating by parts and integrating with respect to t. For the case when t = T , due to part (i) and Hölder’s inequality both sides of (2.6) are finite. Then it remains to let t → T and take the limit on both sides. We shall prove Theorem 2.4 in three steps. First, we want to control A, C, D, F in a smaller ball by their values in a larger ball under the assumption that E is sufficiently small. Similar results can be found in [13] or [15] in the case when the space dimension is three. Lemma 2.8. Suppose γ ∈ (0, 1), ρ > 0 are constants and Q(z 0 , ρ) ⊂ Q T . Then we have C(γρ) ≤ N γ −3 A1/2 (ρ)E(ρ) + γ −9/2 A3/4 (ρ)E 3/4 (ρ) + γ C(ρ) , (2.7) where N is a constant independent of γ , ρ and z 0 . Lemma 2.9. Suppose α ∈ (0, 1/2], γ ∈ (0, 1/3], ρ > 0 are constants and Q(z 0 , ρ) ⊂ Q T . Then we have 1−α 2α 3−α F(γρ) ≤ N (α) γ −2 A 1+α (ρ)E 1+α (ρ) + γ 1+α F(ρ) , (2.8) where N (α) is a constant independent of γ , ρ and z 0 . In particular, for α = 1/2 we have, D(γρ) ≤ N γ −3 A1/2 (ρ)E(ρ) + γ 5/2 D(ρ) . (2.9) Moreover, it holds that
D(γρ) ≤ N (α) γ −3 (A(ρ) + E(ρ))3/2 + γ (9−3α)/(2+2α) F 3/2 (ρ) .
(2.10)
Lemma 2.10. Suppose θ ∈ (0, 1/2], ρ > 0 are constants and Q(z 0 , ρ) ⊂ Q T . Then we have A(θρ) + E(θρ) ≤ N θ −2 C 2/3 (ρ) + C(ρ) + C 1/3 (ρ)D 2/3 (ρ) . In particular, when θ = 1/2 we have A(ρ/2) + E(ρ/2) ≤ N [C 2/3 (ρ) + C(ρ) + C 1/3 (ρ)D 2/3 (ρ)].
(2.11)
As a conclusion, we obtain Proposition 2.11. For any ε0 > 0, there exists ε1 > 0 small such that for any z 0 ∈ Q T ∪ (R4 × {T }) satisfying lim sup E(r ) ≤ ε1 , (2.12) r →0
we can find ρ0 sufficiently small such that A(ρ0 ) + E(ρ0 ) + C(ρ0 ) + D(ρ0 ) + F(ρ0 ) ≤ ε0 .
(2.13)
In the second step, our goal is to estimate the values of A, E, C and F in a smaller ball by the values of themselves in a larger ball. Lemma 2.12. Suppose ρ > 0, θ ∈ (0, 1/3] are constants and Q(z 1 , ρ) ⊂ Q T . Then we have A(θρ) + E(θρ) ≤ N θ 2 A(ρ) + N θ −3 [A(ρ) + E(ρ) + F(ρ)]3/2 , where N is a constant independent of ρ, θ and z 1 .
(2.14)
Partial Regularity of Solutions to the NSE
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Lemma 2.13. Suppose ρ > 0 is constant and Q(z 1 , ρ) ⊂ Q T . Then we can find θ1 > 0 small such that A(θ1 ρ) + E(θ1 ρ) + F(θ1 ρ) ≤
1 A(ρ) + E(ρ) + F(ρ) 2
3/2 +N (θ1 ) A(ρ) + E(ρ) + F(ρ) ,
(2.15)
where N is a constant independent of ρ and z 1 . Proposition 2.14. For any ε2 > 0, there exists ε0 > 0 small such that: if for some z 0 ∈ Q T ∪ (R4 × {T }) and ρ0 > 0 satisfying Q(z 0 , ρ0 ) ⊂ Q T and C(ρ0 ) + D(ρ0 ) + F(ρ0 ) ≤ ε0 ,
(2.16)
then for any ρ ∈ (0, ρ0 /4) and z 1 ∈ Q(z 0 , ρ/4) we have A(ρ, z 1 ) + C(ρ, z 1 ) + E(ρ, z 1 ) + F(ρ, z 1 ) ≤ ε2 .
(2.17)
Finally, we apply Schoen’s trick to prove the main theorems. 3. Proof of Proposition 2.11 We will prove these lemma briefly. For more detail, we refer the reader to [13]. Proof of Lemma 2.8. Denote r = γρ. We have, by using Poincaré’s inequality and Cauchy’s inequality, 2 |u| − [|u|2 ]x0 ,ρ d x + |u|2 d x = [|u|2 ]x0 ,ρ d x B(x0 ,r ) B(x0 ,r ) B(x0 ,r ) r 4 ≤ Nρ |∇u||u| d x + |u|2 d x ρ B(x0 ,ρ) B(x0 ,ρ) 1/2 1/2 2 |∇u| d x |u|2 d x ≤ Nρ B(x0 ,ρ) B(x0 ,ρ) r 4 2 + |u| d x ρ B(x0 ,ρ) 1/2 |∇u|2 d x ≤ Nρ 2 A1/2 (ρ) B(x0 ,ρ) 2/3 r 4 + |u|3 d x ρ 4/3 . ρ B(x0 ,ρ) Owing to Lemma 2.6 with q = 3 and using the inequality above, one gets B(x0 ,r )
|∇u|2 d x ρ A1/2 (ρ) B(x0 ,r ) 3/4 r 4 3 −2 3/4 2 + ρ r A (ρ) |∇u| d x + |u|3 d x . ρ B(x0 ,ρ) B(x0 ,ρ)
|u| d x ≤ N 3
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By integrating with respect to t on (t0 − r 2 , t0 ) and applying Hölder’s inequality, we get Q(z 0 ,r )
|∇u|2 dz ρ A1/2 (ρ) Q(z 0 ,ρ) 3/4 r
4 3 −3/2 3/4 A (ρ) |∇u|2 dz + |u|3 dz . +ρ r ρ Q(z 0 ,ρ) Q(z 0 ,r )
|u| dz ≤ N 3
The conclusion of Lemma 2.8 follows immediately. Proof of Lemma 2.9. Denote r = γρ. Recall the decomposition of p introduced in Sect. 2. By using the Calderón-Zygmund estimate, Lemma 2.6 with q = 2(1 + α) and the Poincaré inequality, one has | p˜ x0 ,r (x, t)|1+α d x B(x0 ,r ) ≤N |u − [u]x0 ,r |2(1+α) d x B(x0 ,r ) 2α 1−α ≤N |∇u|2 d x |u − [u]x0 ,r |2 d x B(x0 ,r ) B(x0 ,r ) 1+α −4α + Nr |u − [u]x0 ,r |2 d x B(x0 ,r ) 2α 1−α ≤N |∇u|2 d x |u|2 d x . (3.1) B(x0 ,r )
B(x0 ,r )
Here we also use the inequality |u − [u]x0 ,r |2 d x ≤ B(x0 ,r )
Similarly, B(x0 ,ρ)
| p˜ x0 ,ρ |1+α d x ≤ N
B(x0 ,ρ)
B(x0 ,r )
|∇u|2 d x
|u|2 d x.
2α B(x0 ,ρ)
|u|2 d x
1−α
.
(3.2)
Since h x0 ,ρ is harmonic in B(x0 , ρ/2), any Sobolev norm of h x0 ,ρ in a smaller ball can be estimated by any of its L p norm in B(x0 , ρ/2). Thus, by using the Poincaré inequality one can obtain |h x0 ,ρ − [h x0 ,ρ ]x0 ,r |1+α d x B(x0 ,r ) ≤ Nr 1+α |∇h x0 ,ρ |1+α d x B(x0 ,r )
≤ Nr 5+α sup |∇h x0 ,ρ |1+α B(x0 ,r )
≤N
r 5+α
|h x0 ,ρ (x, t) − [h x0 ,ρ ]x0 ,ρ |1+α d x ρ B(x0 ,ρ/2)
r 5+α ≤N | p(x, t) − [h x0 ,ρ ]x0 ,ρ |1+α + | p˜ x0 ,ρ (x, t)|1+α d x . ρ B(x0 ,ρ)
(3.3)
Partial Regularity of Solutions to the NSE
793
Combining (3.2) and (3.3) together yields, | p(x, t) − [h x0 ,ρ ]x0 ,r |1+α d x B(x0 ,r ) 2α 1−α 2 ≤N |∇u(x, t)| d x |u(x, t)|2 d x B(x0 ,ρ) B(x0 ,ρ) r 5+α +N | p(x, t) − [h x0 ,ρ ]x0 ,ρ |1+α d x. ρ B(x0 ,ρ)
(3.4)
Since p˜ x0 ,r + h x0 ,r = p = p˜ x0 ,ρ + h x0 ,ρ in B(x0 , r ), by Hölder’s inequality, |[h x0 ,ρ ]x0 ,r − [h x0 ,r ]x0 ,r |1+α d x B(x0 ,r )
= Nr 4 |[h x0 ,ρ ]x0 ,r − [h x0 ,r ]x0 ,r |1+α = Nr 4 |[ p˜ x0 ,ρ ]x0 ,r − [ p˜ x0 ,r ]x0 ,r |1+α ≤N | p˜ x0 ,ρ |1+α + | p˜ x0 ,r |1+α d x.
(3.5)
B(x0 ,r )
From (3.1), (3.2), (3.4) and (3.5) we get | p(x, t) − [h x0 ,r ]x0 ,r |1+α d x B(x0 ,r ) 2α 1−α 2 ≤N |∇u(x, t)| d x |u(x, t)|2 d x B(x0 ,ρ) B(x0 ,ρ) r 5+α +N | p(x, t) − [h x0 ,ρ ]x0 ,ρ |1+α d x. ρ B(x0 ,ρ)
(3.6)
Raising to the power 1/(2α) and integrating with respect to t in (t0 − r 2 , t0 ) completes the proof of (2.8) and also (2.9). To prove (2.10), we use a slightly different estimate from (3.3). Again, since h is harmonic in B(x0 , ρ/2), we have |h x0 ,ρ − [h x0 ,ρ ]x0 ,r |3/2 d x B(x0 ,r ) ≤ Nr 3/2 |∇h x0 ,ρ |3/2 d x B(x0 ,r )
≤ Nr ≤N ≤N
11/2
sup |∇h x0 ,ρ |3/2
B(x0 ,r )
r 11/2 ρ 3/2+6/(1+α) r 11/2
B(x0 ,ρ)
|h x0 ,ρ/2 (x, t) − [h x0 ,ρ ]x0 ,ρ |1+α d x | p(x, t) − [h x0 ,ρ ]x0 ,ρ |1+α d x
ρ 3/2+6/(1+α) B(x0 ,ρ)
3 2(1+α) . + | p˜ x0 ,ρ (x, t)|1+α d x B(x0 ,ρ)
3 2(1+α)
3 2(1+α)
(3.7)
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Similar to (3.6), we obtain | p(x, t) − [h x0 ,r ]x0 ,r |3/2 d x B(x0 ,r ) ≤N |∇u(x, t)|2 d x B(x0 ,ρ) r 11/2
B(x0 ,ρ)
|u(x, t)|2 d x
1/2
3 + N 3/2+6/(1+α) [ | p(x, t) − [h]x0 ,ρ |1+α d x 2(1+α) ρ B(x0 ,ρ) 3α 3(1−α) 1+α 2(1+α) . + |∇u(x, t)|2 d x |u(x, t)|2 d x B(x0 ,ρ)
B(x0 ,ρ)
(3.8)
Integrating with respect to t in (t0 −r 2 , t0 ) and applying Hölder’s inequality completes the proof of (2.10). Proof of Lemma 2.10. Let r = θρ. In the energy inequality (2.6), we put t = t0 and choose a suitable smooth cut-off function φ such that ψ ≡ 0 in Q t0 \ Q(z 0 , ρ), 0 ≤ ψ ≤ 1 in Q T , ψ ≡ 1 in Q(z 0 , r ), |∇ψ| < Nρ −1 , |∂t ψ| + |∇ 2 ψ| < Nρ −2 in Q t0 . By using (2.6) and because u is divergence free, we get A(r ) + 2E(r )
1 N 1 |u|2 dz (|u|2 + 2| p − [h]x0 ,ρ |)|u| dz . ≤ 2 2 r ρ Q(z 0 ,ρ) ρ Q(z 0 ,ρ) Due to Hölder’s inequality, one can obtain 2/3 |u|2 dz ≤ |u|3 dz Q(z 0 ,ρ)
Q(z 0 ,ρ)
Q(z 0 ,ρ)
dz
1/3
≤ ρ 4 C 2/3 (ρ),
Q(z 0 ,ρ)
≤
| p − [h]x0 ,ρ ||u| dz
Q(z 0 ,ρ)
| p − [h]x0 ,ρ |
3/2
dz
2/3
Q(z 0 ,ρ)
|u|3 dz
1/3
≤ Nρ 3 D 2/3 (ρ)C 1/3 (ρ). Then the conclusion of Lemma 2.10 follows immediately. Proof of Proposition 2.11. Let’s prove first (2.13) without the presence of F on the left-hand side. For a given point z 0 = (x0 , t0 ) ∈ Q T ∪ (R4 × {T }) satisfying (2.12), choose ρ0 > 0 such that Q(z 0 , ρ0 ) ⊂ Q T . Then for any ρ ∈ (0, ρ0 ] and γ ∈ (0, 1/6), by using (2.11), A(γρ) + E(γρ) ≤ N [C 2/3 (2γρ) + C(2γρ) + D(2γρ)].
Partial Regularity of Solutions to the NSE
795
This estimate, (2.7) and (2.9) together with Young’s inequality imply A(γρ) + E(γρ) + C(γρ) + D(γρ) ≤ N [γ 2/3 C 2/3 (ρ) + γ 5/2 D(ρ) + γ C(ρ) + γ A(ρ)] + N γ −100 (E(ρ) + E 3 (ρ)) ≤ N γ 2/3 [A(ρ) + E(ρ) + C(ρ) + D(ρ)] + N γ 2/3 + N γ −100 (E(ρ) + E 3 (ρ)).
(3.9)
It is easy to see that for any ε3 > 0, there are sufficiently small real numbers γ ≤ 1/(2N )3/2 and ε1 such that if (2.12) holds then for all small ρ we have N γ 2/3 + N γ −100 (E(ρ) + E 3 (ρ)) < ε3 /2. By using (3.9) we reach A(ρ1 ) + C(ρ1 ) + D(ρ1 ) ≤ ε3 for some ρ1 > 0 small enough. To include F in the estimate, it suffices to use (2.8). 4. Proof of Proposition 2.14 Proof of Lemma 2.12. Let r = θρ. Define the backward heat kernel as 2
(t, x) =
|x−x | − 2 1 1 2(r +t1 −t) . e 4π 2 (r 2 + t1 − t)2
In the energy inequality (2.6) we put t = t1 and choose ψ = φ := φ1 (x)φ2 (t), where φ1 , φ2 are suitable smooth cut-off functions satisfying φ1 ≡ 0 in R4 \ B(x1 , ρ), 0 ≤ φ1 ≤ 1 in R4 , φ1 ≡ 1 in B(x1 , ρ/2) φ2 ≡ 0 in (−∞, t1 − ρ 2 ) ∪ (t1 + ρ 2 , +∞), 0 ≤ φ2 ≤ 1 in R, φ2 ≡ 1 in (t1 − ρ 2 /4, t1 + ρ 2 /4), |φ2 | ≤ Nρ −2 in R, |∇φ1 | < Nρ −1 , |∇ 2 φ1 | < Nρ −2 in R4 .
(4.1)
By using the equality + t = 0, we have
|u(x, t)| (t, x)φ(x, t) d x + 2 2
B(x0 ,ρ)
≤
Q(z 0 ,ρ) 2
Q(z 0 ,ρ)
|∇u|2 φ dz
{|u|2 (φt + φ + 2∇φ∇)
+ (|u| + 2 p)u · (∇φ + φ∇)} dz.
(4.2)
After some straightforward computations, it is easy to see the following three properties:
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¯ 1 , r ) it holds that For some constant c > 0, on Q(z φ = ≥ cr −4 .
(ii)
For any z ∈ Q(z 1 , ρ), we have |φ(z)∇(z)| + |∇φ(z)(z)| ≤ Nr −5 .
(iii)
For any z ∈ Q(z 1 , ρ) \ Q(z 1 , r ), we have |(z)φt (z)| + |(z)φ(z)| + |∇φ∇| ≤ Nρ −6 .
These properties together with (4.2) and (4.1) yield A(r ) + E(r ) ≤ N [θ 2 A(ρ) + θ −3 (C(ρ) + D(ρ))].
(4.3)
Owing to Lemma 2.6 with q = 3, one easily gets C(ρ/3) ≤ N C(ρ) ≤ N [A(ρ) + E(ρ)]3/2 .
(4.4)
By using (2.10) with γ = 1/3, we have D(ρ/3) ≤ N [A(ρ) + E(ρ) + F(ρ)]3/2 .
(4.5)
Upon combining (4.3) (with ρ/3 in place of ρ), (4.4) and (4.5) together, the lemma is proved. Proof of Lemma 2.13. Due to (2.8) and (2.14), for any γ , θ ∈ (0, 1/3], we have F(γ θρ) ≤ N γ −2 (A(θρ) + E(θρ)) + γ (3−α)/(1+α) F(θρ) ≤ N γ −2 θ 2 A(ρ) + γ (3−α)/(1+α) θ −2 F(ρ) 3/2 + N γ −2 θ −3 A(ρ) + E(ρ) + F(ρ) A(γ θρ) + E(γ θρ) ≤ (γ θ )2 A(ρ) + (γ θ )−3 [A(ρ) + E(ρ) + F(ρ)]3/2 .
(4.6) (4.7)
Now we put α = 1/27 such that (3 − α)/(1 + α) = 20/7 > 2. In Sect. 5, we will give more explanation why we choose α = 1/27. Now one can choose and fix γ and θ sufficiently small such that N [γ −2 θ 2 + γ 20/7 θ −2 + (γ θ )2 ] ≤ 1/2. Upon adding (4.6) and (4.7), we obtain A(γ θρ) + E(γ θρ) + F(γ θρ) ≤
1 A(ρ) + N [A(ρ) + E(ρ) + F(ρ)]3/2 , 2
where N depends only on θ and γ . After putting θ1 = γ θ , the lemma is proved.
Partial Regularity of Solutions to the NSE
797
Proof of Proposition 2.14. Take the constant θ1 from Lemma 2.13. Due to Lemma 2.10, we may choose ε0 , ε > 0 small enough such that A(ρ0 /2) + E(ρ0 /2) + C(ρ0 /2) + D(ρ0 /2) + F(ρ0 /2) ≤ ε . 2ε + 8N (θ1 )ε3/2 ≤ min(4ε , θ12 ε2 ),
(4.8)
where the constant N (θ1 ) is the same one as in (2.15). Since z 1 ∈ Q(z 0 , ρ/4), we have Q(z 1 , ρ0 /4) ⊂ Q(z 0 , ρ0 /2) ⊂ Q T , A(ρ0 /4, z 1 ) + E(ρ0 /4, z 1 ) + F(ρ0 /4, z 1 ) ≤ 4ε . By using (4.8) and (2.15), one obtains inductively for k = 1, 2, · · · , A(θ1k ρ0 /4, z 1 ) + E(θ1k ρ0 /4, z 1 ) + F(θ1k ρ0 /4, z 1 ) ≤ min{θ12 ε2 , 4ε }. Thus, for any ρ ∈ (0, ρ0 /4], it holds that A(ρ, z 1 ) + E(ρ, z 1 ) + F(ρ, z 1 ) ≤ ε2 . To include the term C(ρ, z 1 ) in the estimate, it suffices to use (4.4). The proposition is proved. 5. Proof of Theorems 2.1-2.5 Proof of Theorems 2.1 and 2.2. Let z 0 ∈ Q T ∪ (R4 × {T }) be a given point. Proposition 2.11 and 2.14 imply that for any ε2 > 0 there exist small numbers ε1 , ε0 , ρ0 > 0 such that either (5.1) lim sup E(r, z 0 ) ≤ ε1 r →0
or C(ρ0 ) + D(ρ0 ) + F(ρ0 ) ≤ ε0
(5.2)
holds true, we can find ρ1 > 0 so that Q(z 0 , ρ1 ) ⊂ Q T and for any z 1 ∈ Q(z 0 , ρ1 /2), ρ ∈ (0, ρ1 /2) we have (5.3) C(ρ, z 1 ) + F(ρ, z 1 ) ≤ ε2 . Let δ ∈ (0, ρ12 /4) be a number and denote Mδ =
max
¯ 0 ,ρ1 /2)∩ Q¯ T −δ Q(z
d(z)|u(z)|,
where d(z) = min[dist(x, ∂), (t + ρ12 /4 − T )1/2 ]. Lemma 5.1. If (5.3) holds true for a sufficiently small ε2 , then sup
Q(z 0 ,ρ1 /4)
|u(z)| < +∞.
(5.4)
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Proof. If for all δ ∈ (0, ρ12 /4) we have Mδ ≤ 2, then there’s nothing to prove. Otherwise, ¯ 0 , ρ1 /2) ∩ Q¯ T −δ , suppose for some δ and z 1 ∈ Q(z M := Mδ = |u(z 1 )|d(z 1 ) > 2. Let r1 = d(z 1 )/M < d(z 1 )/2. We make the scaling as follows: u(y, ¯ s) = r1 u(r1 y + x1 , r12 s + t1 ), p(y, ¯ s) = r1 p(r1 y + x1 , r12 s + t1 ). It’s known that the pair (u, ¯ p) ¯ satisfies the Navier-Stokes equations (1.1) in Q(0, 1). Obviously, sup |u| ¯ ≤ 2, |u(0, ¯ 0)| = 1. (5.5) Q(0,1)
Due to the scaling-invariant property of our objects A, E, C, D and F, in what follows we look at them as objects associated to (u, ¯ p) ¯ at the origin. For any ρ ∈ (0, 1], we have (5.6) C(ρ) + F(ρ) ≤ ε2 . Recall what we did before in the proof of Lemma 2.9. Since u¯ is bounded in Q(0, 1), we have | p˜¯ 0,1 |14 dz ≤ | p˜¯ 0,1 |14 dz ≤ N , (5.7) Q(0,1/3)
Q(0,1/2)
|h¯ 0,1 (z) − [h¯ 0,1 ]0,1/3 |14 d x B(0,1/3)
sup |∇ h¯ 0,1 (x, t)|14
≤N
B(0,1/3)
≤N
|h¯ 0,1 − [h¯ 0,1 ]0,1/2 |28/27 d x
27/2
,
B(0,1/2)
and
|h¯ 0,1 (z) − [h¯ 0,1 ]0,1/3 |14 dz Q(0,1/3) 0
≤N
−1/9
|h¯ 0,1 − [h¯ 0,1 ]0,1/2 |28/27 d x
27/2
dt
B(0,1/2)
≤ N (1 + F 14 (1)). Estimates (5.7) and (5.8) yield
| p(z) ¯ − [h¯ 0,1 ]0,1/3 |14 dz ≤ N .
Q(0,1/3)
Because (u, ¯ p) ¯ satisfies the equation ¯ − ∇( p¯ − [h¯ 0,1 ]0,1/3 ) u¯ t − u¯ = div(u¯ ⊗ u)
(5.8)
(5.9)
Partial Regularity of Solutions to the NSE
799
in Q(0, 1). Owing to (5.5), (5.9) and the classical Sobolev space theory of the parabolic equation, we have 1,1/2
u¯ ∈ W14
(Q(0, 1/4)), u ¯ W 1,1/2 (Q(0,1/4)) ≤ N .
(5.10)
14
Since 1/2 − 6/14 = 1/14 > 0, owing to the Sobolev embedding theorem (see [11]), we obtain ¯ C 1/14 (Q(0,1/5)) ≤ N , u¯ ∈ C 1/14 (Q(0, 1/5)), u where N is a universal constant independent of ε1 and ε2 . Therefore, we can find δ1 < 1/5 independent of ε1 , ε2 such that |u(x, ¯ t)| ≥ 1/2 in Q(0, δ1 ).
(5.11)
Now we choose ε2 small enough which makes (5.11) and (5.6) a contradiction. The lemma is proved. Theorem 2.1 and 2.2 follow immediately from Lemma 5.1. Proof of Theorem 2.4. Take the number ε1 in Lemma 5.1. Denote ∗ := {z ∈ × {T } | lim sup E(r, z) ≤ ε1 }. r ↓0
It is well known that the 2-D Hausdorff measure of \ ∗ is zero. By using Lemma 5.1, for any z ∈ ∗ we can find ρ > 0 such that u is bounded in Q(z, ρ). Then there’s no blow-up at z and z is a regular point. The theorem is proved. Proof of Theorem 2.5. For any α0 ∈ [0, 1], due to Lemma 2.7 (i), the interpolation inequality (2.8) with r = +∞, q = 2(1 + α0 ) and Hölder’s inequality, one can easily get u L t
x 4 + (1+α0 )/α0 L 2(1+α0 ) (R ×R )
< +∞.
(5.12)
Since (u, p) satisfies ∂2 (u i u j ) in R4 × R+ , ∂ xi ∂ x j
p =
due to the Calderón-Zygmund estimate, we have p L t
x 4 + (1+α0 )/(2α0 ) L 1+α0 (R ×R )
< +∞.
(5.13)
Because of (5.12) with α0 = 1/2 and (5.13) with α0 = 1/2, α, and again by the Calderón-Zygmund estimate, for any ε4 ∈ (0, 1) we can find R ≥ 1 sufficiently large such that for any z 0 ∈ R4 × (R, +∞) it holds that C(1, z 0 ) + p L t
x 3/2 L 3/2 (Q(z 0 ,1))
p˜ z 0 ,1 L t
x 3/2 L 3/2 (Q(z 0 ,1))
+ p L t
x (1+α0 )/(2α0 ) L 1+α0 (Q(z 0 ,1))
+ p˜ z 0 ,1 L t
≤ ε4 ,
x (1+α0 )/(2α0 ) L 1+α0 (Q(z 0 ,1))
≤ ε4 .
(5.14) (5.15)
Thus, h z 0 ,1 L t
x 3/2 L 3/2 (Q(z 0 ,1))
+ h z 0 ,1 L t
x (1+α0 )/(2α0 ) L 1+α0 (Q(z 0 ,1))
≤ 2ε4 .
(5.16)
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H. Dong, D. Du
After combining (5.14) and (5.16) together, it is clear by using Hölder’s inequality that C(1, z 0 ) + D(1, z 0 ) + F(1, z 0 ) ≤ N ε4 , where N is independent of ε4 . Then owing to Proposition 2.14 and Lemma 5.1, for sufficiently small ε4 we can find a uniform upper bound M0 > 0 such that for any z 0 ∈ R4 × (R, +∞), sup
z∈Q(z 0 ,1/4)
|u(z)| ≤ M0 .
Therefore, u will not blow up as t goes to infinity, and Theorem 2.5 is proved. Acknowledgement. The authors would like to express their sincere gratitude to Prof. V. Sverak for pointing out this problem and giving many useful comments for improvement. The authors would also thank to Prof. N.V. Krylov for helpful discussions and the referee for his careful review of the article.
References 1. Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier-stokes equations. Comm. Pure Appl. Math. 35, 771–831 (1982) 2. Cannone, M.: A generalization of a theorem by Kato on Navier-Stokes equations. R. Mat. Iberoam 13, 515–541 (1997) 3. Galdi, G.P.: An introduction to the mathematical theory of Navier-Stokes equations, I,II. New York: Springer-Verlag, 1994 4. Giga, Y., Miyakawa, T.: Navier-Stokes flow in R3 with measures as initial vorticity and Morrey spaces. Commun. Part. Differ. Eqs. 14, 577–618 (1989) 5. Giga, Y., Miyakawa, T.: Solution in L r of the Navier-Stokes initial value problem. Arch. Rat. Mech. Anal. 89, 267–281 (1985) 6. Gustafson, S., Kang, K., Tsai, T.: Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations. Commun. Math. Phys. DOI 10.1007/s00220-007-0214-6, 2007 7. Heywood, J.G.: The Navier-Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29, 639–681 (1980) 8. Iftimie, D.: The resolution of the Navier-Stokes equations in anisotropic spaces. Rev. Mat. Iberoam 15, 1–36 (1999) 9. Kato, T.: Strong L p -solutions of the Navier-Stokes equation in Rm with applications to weak solutions. Math. Z. 187, 471–480 (1984) 10. Koch, H., Tataru, D.: Well-posedness for the Navier-Stokes equations. Adv. Math. 157(1), 22–35 (2001) 11. Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and quasi-Linear equations of parabolic type. Moscow: Nauka, 1967 (in Russian); English translation: Providence, RI: Amer. Math. Soc., 1968 12. Ladyzhenskaya, O.: The Mathematical Theory of Viscous Incompressible Flows. 2nd edition, New York: Gordon and Breach, 1969 13. Ladyzhenskaya, O., Seregin, G.A.: On partial regularity of suitable weak solutions to the three-dimensional Navier–Stokes equations. J. Math. Fluid Mech. 1, 356–387 (1999) 14. Leray, J.: Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933) 15. Lin, F.: A new proof of the Caffarelli-Kohn-Nirenberg theorem. Comm. Pure Appl. Math. 51, 241–257 (1998) 16. Seregin, G.: Regularity for Suitable Weak Solutions to the Navier-Stokes Equations in Critical Morrey Spaces. Preprint, http://arxiv.org/list/math.AP/0607537, 2006 17. Scheffer, V.: Partial regularity of solutions to the Navier-Stokes equations. Pacific J. Math. 66, 535–552 (1976) 18. Scheffer, V.: Hausdorff measure and the Navier-Stokes equations. Commun. Math. Phys. 55, 97–112 (1977) 19. Scheffer, V.: The Navier-Stokes equations in space dimension four. Commun. Math. Phys. 61, 41–68 (1978) 20. Scheffer, V.: The Navier-Stokes equations on a bounded domain. Commun. Math. Phys. 73, 1–42 (1980)
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21. Serrin, J.: On the interior regularity of weak solutions of Navier-Stokes equations. Arch. Rat. Mech. Anal. 9, 187–195 (1962) 22. Solonikov, V.A.: Estimates of solutions to the linearized systems of the Navier-Stokes equations. Trudy Steklov Math. Inst. LXX, 213–317 (1964) 23. Taylor, M.: Analysis on Morrey spaces and applications to Navier-Stokes equation. Comm. Part. Differ. Eqs. 17, 1407–1456 (1992) 24. Wiegner, M.: Higher order estimates in further dimensions for the solutions of Navier-Stokes equations. Evolution equations (Warsaw, 2001), Banach Center Publ. 60, Warsaw: Polish Acad. Sci., 2003, pp 81–84 Communicated by P. Constantin
Commun. Math. Phys. 273, 803–827 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0213-7
Communications in
Mathematical Physics
Obstructions to the Existence of Sasaki–Einstein Metrics Jerome P. Gauntlett1,2 , Dario Martelli3 , James Sparks4,5 , Shing-Tung Yau4 1 2 3 4
Blackett Laboratory, Imperial College, London SW7 2AZ, U.K. The Institute for Mathematical Sciences, Imperial College, London SW7 2PG, U.K. Department of Physics, CERN Theory Unit, 1211 Geneva 23, Switzerland Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, U.S.A. E-mail: [email protected] 5 Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, U.S.A. Received: 17 August 2006 / Accepted: 5 October 2006 Published online: 4 May 2007 – © Springer-Verlag 2007
Abstract: We describe two simple obstructions to the existence of Ricci–flat Kähler cone metrics on isolated Gorenstein singularities or, equivalently, to the existence of Sasaki–Einstein metrics on the links of these singularities. In particular, this also leads to new obstructions for Kähler–Einstein metrics on Fano orbifolds. We present several families of hypersurface singularities that are obstructed, including 3–fold and 4–fold singularities of ADE type that have been studied previously in the physics literature. We show that the AdS/CFT dual of one obstruction is that the R–charge of a gauge invariant chiral primary operator violates the unitarity bound. Contents 1. Introduction . . . . . . . . . . . . . . 2. The Obstructions . . . . . . . . . . . . 2.1 The Bishop obstruction . . . . . . 2.2 The Lichnerowicz obstruction . . 2.3 Smooth Fanos . . . . . . . . . . . 2.4 AdS/CFT interpretation . . . . . . 3. Isolated Hypersurface Singularities . . 3.1 The Bishop obstruction . . . . . . 3.2 The Lichnerowicz obstruction . . 3.3 Sufficient conditions for existence 4. A Class of 3–Fold Examples . . . . . . 4.1 Obstructions . . . . . . . . . . . . 4.2 Cohomogeneity one metrics . . . 4.3 Field theory . . . . . . . . . . . . 5. Other Examples . . . . . . . . . . . . 5.1 ADE 4–fold singularities . . . . . 5.2 Weighted actions on Cn . . . . . .
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804 806 807 808 809 811 813 815 816 816 817 817 818 818 820 820 821
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J. P. Gauntlett, D. Martelli, J. Sparks, S.-T. Yau
6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822 A. Cohomogeneity One Metrics . . . . . . . . . . . . . . . . . . . . . . . . . 823 1. Introduction The study of string theory and M–theory on singular manifolds is a very rich subject that has led to many important insights. For geometries that develop an isolated singularity, one can model the local behaviour using a non–compact manifold. In this case, a natural geometric boundary condition is for the metric to asymptote to a cone away from the singularity. This means that one studies a family of metrics that asymptotically approach the conical form g X = dr 2 + r 2 g L
(1.1)
with (L , g L ) a compact Riemannian manifold. The dynamics of string theory or M–theory on special holonomy manifolds that are developing an isolated conical singularity X , with metric (1.1), has proved to be an extremely intricate subject. A particularly interesting setting is in the context of the AdS/CFT correspondence [1]. The worldvolume theory of a large number of D3–branes placed at an isolated conical Calabi–Yau 3–fold singularity is expected to flow, at low energies, to a four–dimensional N = 1 superconformal field theory. In this case, the AdS/CFT conjecture states that this theory is dual to type IIB string theory on AdS5 × L [2–5]. Similar remarks apply to M–theory on conical eight–dimensional singularities with special holonomy, which lead to superconformal theories in three dimensions that are dual to AdS4 × L, although far less is known about this situation. The focus of this paper will be on conical Calabi–Yau singularities, by which we mean Ricci–flat Kähler metrics of the conical form (1.1). This gives, by definition, a Sasaki–Einstein metric on the base of the cone L. We tacitly assume that L is simply– connected which, although not entirely necessary, always ensures the existence of a globally defined Killing spinor on L. A central role is played by the Reeb vector field ∂ ξ=J r , (1.2) ∂r where J denotes the complex structure tensor on the cone X . ξ is holomorphic, Killing, and has constant norm on the link L = {r = 1} of the singularity at r = 0. If the orbits of ξ all close then L has a U (1) isometry, which necessarily acts locally freely, and the Sasakian structure is said to be either regular or quasi–regular if this action is free or not, respectively. The orbit space is in general a positively curved Kähler–Einstein orbifold (V, gV ), which is a smooth manifold in the regular case. More generally, the generic orbits of ξ need not close, in which case the Sasakian structure is said to be irregular. The AdS/CFT correspondence maps the symmetry generated by the Reeb vector field to the R–symmetry of the dual CFT. Thus for the quasi–regular case the CFT has a U (1) R–symmetry, whereas for the irregular case it has a non–compact R R–symmetry. Given a Sasaki–Einstein manifold (L , g L ), the cone X , as a complex variety, is an isolated Gorenstein singularity. If X 0 denotes X with the singular point removed, we have X 0 = R+ × L with r > 0 a coordinate on R+ . X being Gorenstein means simply that there exists a nowhere zero holomorphic (n, 0)–form on X 0 . One may then turn things around and ask which isolated Gorenstein singularities admit Sasaki–Einstein metrics on their links. This is a question in algebraic geometry, and it is an extremely
Obstructions to the Existence of Sasaki–Einstein Metrics
805
difficult one. To give some idea of how difficult this question is, let us focus on the quasi–regular case. Thus, suppose that X has a holomorphic C∗ action, with orbit space being a Fano1 manifold, or Fano orbifold, V . Then existence of a Ricci–flat Kähler cone metric on X , with conical symmetry generated by R+ ⊂ C∗ , is well known to be equivalent to finding a Kähler–Einstein metric on V – for a review, see [6]. Existence of Kähler–Einstein metrics on Fanos is a very subtle problem that is still unsolved. That is, a set of necessary and sufficient algebraic conditions on V are not known in general. There are two well–known holomorphic obstructions, due to Matsushima [7] and Futaki [8]. The latter was related to Sasakian geometry in [9] and is not in fact an obstruction from the Sasaki–Einstein point of view. Specifically, it is possible to have a Fano V that has non–zero Futaki invariant and thus does not admit a Kähler–Einstein metric, but nevertheless the link of the total space of the canonical bundle over V can admit a Sasaki–Einstein metric – the point is simply that the Reeb vector field is not2 the one associated with the canonical bundle over V . It is also known that vanishing of these two obstructions is, in general, insufficient for there to exist a Kähler–Einstein metric on V . It has been conjectured in [11] that V admits a Kähler–Einstein metric if and only if it is stable; proving this conjecture is currently a major research programme in geometry – see, for example, [12]. Thus, one also expects the existence of Ricci–flat Kähler cone metrics on an isolated Gorenstein singularity X to be a subtle problem. This issue has been overlooked in some of the physics literature, and it has sometimes been incorrectly assumed, or stated, that such conical Calabi–Yau metrics exist on particular singularities, as we shall discuss later. The Reeb vector field contains a significant amount of information about the metric. For a fixed X 0 = R+ × L, the Reeb vector field ξ for a Sasaki–Einstein metric on L satisfies a variational problem that depends only on the complex structure of X [13, 9]. This is the geometric analogue of a–maximisation [14] in four dimensional superconformal field theories. This allows one, in principle and often in practice, to obtain ξ , and hence in particular the volume, of a Sasaki–Einstein metric on L – assuming that this metric exists. Now, for any (2n−1)–dimensional Einstein manifold (L , g L ) with Ric = 2(n−1)g L , Bishop’s theorem [15] (see also [16]) implies that the volume of L is bounded from above by that of the round unit radius sphere. Thus we are immediately led to what we will call the Bishop obstruction to the existence of Sasaki–Einstein metrics: If the volume of the putative Sasaki–Einstein manifold, calculated using the results of [13, 9], is greater than that of the round sphere, then the metric cannot exist. It is not immediately obvious that this can ever happen, but we shall see later that this remarkably simple fact can often serve as a powerful obstruction. We will also discuss the AdS/CFT interpretation of this result. The Reeb vector field ξ also leads to a second possible obstruction. Given ξ for a putative Sasaki–Einstein metric, it is a simple matter to show that holomorphic functions f on the corresponding cone X with definite charge λ > 0, Lξ f = λi f
(1.3)
1 We define a Fano orbifold V to be a compact Kähler orbifold, such that the cohomology class of the Ricci–form in H 2 (V ; R) is represented by a positive (1, 1)–form on V . 2 This happens, for example, when V = F – the first del Pezzo surface. In this case both the Matsushima 1 and Futaki theorems obstruct existence of a Kähler–Einstein metric on V , but there is nevertheless an irregular Sasaki–Einstein metric on the link in the total space of the canonical bundle over V [10].
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J. P. Gauntlett, D. Martelli, J. Sparks, S.-T. Yau
give rise to eigenfunctions of the Laplacian on the Sasaki–Einstein manifold with eigenvalue λ(λ + 2n − 2). Lichnerowicz’s theorem [17] states that the smallest eigenvalue of this Laplacian is bounded from below by the dimension of the manifold, and this leads to the restriction λ ≥ 1. Thus we have what we will call the Lichnerowicz obstruction: If one can demonstrate the existence of a holomorphic function on X with positive charge λ < 1 with respect to the putative Reeb vector field ξ , one concludes that no Sasaki–Einstein metric can exist with this Reeb vector field. Again, it is not immediately obvious that this can ever happen. Indeed, if ξ is regular, so that the orbit space V is a Fano manifold, we show that this cannot happen. Nevertheless, there are infinitely many examples of simple hypersurface singularities with non–regular Reeb vector fields that violate Lichnerowicz’s bound. We shall show that for Calabi–Yau 3–folds and 4–folds, the Lichnerowicz obstruction has a beautiful AdS/CFT interpretation: holomorphic functions on the cone X are dual to chiral primary operators in the dual superconformal field theory. The Lichnerowicz bound then translates into the unitarity bound for the dimensions of the operators. The plan of the rest of the paper is as follows. In Sect. 2 we discuss the two obstructions in a little more detail. In Sect. 3 we investigate the obstructions in the context of isolated quasi–homogeneous hypersurface singularities. We also compare our results with the sufficient conditions reviewed in [6] for existence of Sasaki–Einstein metrics on links of such singularities. In Sect. 4 we show that some 3–fold examples discussed in [18] do not admit Ricci–flat Kähler cone metrics. We briefly discuss the implications for the dual field theory. The results of this section leave open the possibility3 of a single new cohomogeneity one Sasaki–Einstein metric on S 5 and we present some details of the relevant ODE that needs to be solved in an appendix. In Sect. 5 we show that some of the 4–fold examples discussed in [20] also do not admit Ricci–flat Kähler cone metrics. Section 6 briefly concludes. 2. The Obstructions In this section we describe two obstructions to the existence of a putative Sasaki–Einstein metric on the link of an isolated Gorenstein singularity X with Reeb vector field ξ . These are based on Bishop’s theorem [15] and Lichnerowicz’s theorem [17], respectively. We prove that the case when ξ generates a freely acting circle action, with orbit space a Fano manifold V , is never obstructed by Lichnerowicz. We also give an interpretation of Lichnerowicz’s bound in terms of the unitarity bound in field theory, via the AdS/CFT correspondence. Let X be an isolated Gorenstein singularity, and X 0 be the smooth part of X . We take X 0 to be diffeomorphic as a real manifold to R+ × L, where L is compact, and let r be a coordinate on R+ with r > 0, so that r = 0 is the isolated singular point of X . We shall refer to L as the link of the singularity. Since X is Gorenstein, by definition there exists a nowhere zero holomorphic (n, 0)–form on X 0 . Suppose that X admits a Kähler metric that is a cone with respect to a homothetic vector field r ∂/∂r , as in (1.1). This in particular means that L is the orbit space of r ∂/∂r and g L is a Sasakian metric. The Reeb vector field is defined to be ∂ . (2.1) ξ=J r ∂r 3 Recently reference [19] appeared. The conclusions of the latter imply that this solution does not in fact exist.
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807
In the special case that the Kähler metric on X is Ricci–flat, the case of central interest, (L , g L ) is Sasaki–Einstein and we have Lξ = ni
(2.2)
since is homogeneous of degree n under r ∂/∂r . This fixes the normalisation of ξ . 2.1. The Bishop obstruction. The volume vol(L , g L ) of a Sasakian metric on the link L depends only on the Reeb vector field [9]. Thus, specifying a Reeb vector field ξ for a putative Sasaki–Einstein metric on L is sufficient to specify the volume, assuming that the metric in fact exists. We define the normalised volume as V (ξ ) =
vol(L , g L ) , vol(S 2n−1 )
(2.3)
where vol(S 2n−1 ) is the volume of the round sphere. Since Bishop’s theorem [15] (see also [16]) implies that for any (2n − 1)–dimensional Einstein manifold (L , g L ) with Ric = 2(n − 1)g L vol(L , g L ) ≤ vol(S 2n−1 )
(2.4)
we immediately have Bishop obstruction. Let (X, ) be an isolated Gorenstein singularity with link L and putative Reeb vector field ξ . If V (ξ ) > 1 then X admits no Ricci–flat Kähler cone metric with Reeb vector field ξ . In particular L does not admit a Sasaki–Einstein metric with this Reeb vector field. There are a number of methods for computing the normalised volume V (ξ ). For quasi– regular ξ , the volume V (ξ ) is essentially just a Chern number, which makes it clear that V (ξ ) is a holomorphic invariant. In general, one can compute V (ξ ) as a function of ξ , and a number of different formulae have been derived in [13, 9]. In [9] a general formula for the normalised volume V (ξ ) was given that involves (partially) resolving the singularity X and applying localisation. For toric Sasakian manifolds there is a simpler formula [13], giving the volume in terms of the toric data defining the singularity. In this paper we shall instead exploit the fact that the volume V (ξ ) can be extracted from a limit of a certain index–character [9]; this is easily computed algebraically for isolated hypersurface singularities, which shall constitute our main set of examples in this paper. We briefly recall some of the details from [9]. Suppose we have a holomorphic (C∗ )r action on X . We may define the character C(q, X ) = Tr q
(2.5)
as the trace4 of the action of q ∈ (C∗ )r on the holomorphic functions on X . Holomorphic functions f on X that are eigenvectors of the induced (C∗ )r action (C∗ )r : f → qm f,
(2.6)
4 As in [9], we don’t worry about where this trace converges, since we are mainly interested in the behaviour near a certain pole.
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J. P. Gauntlett, D. Martelli, J. Sparks, S.-T. Yau
with eigenvalue qm = ra=1 qam a form a vector space over C of dimension n m . Each eigenvalue then contributes n m qm to the trace (2.5). Let ζa form a basis for the Lie algebra of U (1)r ⊂ (C∗ )r , and write the Reeb vector field as ξ=
r
ba ζa .
(2.7)
a=1
Then the volume of a Sasakian metric on L with Reeb vector field ξ , relative to that of the round sphere, is given by V (ξ ) = lim t n C(qa = exp(−tba ), X ). t→0
(2.8)
In general, the right-hand side of this formula may be computed by partially resolving X and using localisation. However, for isolated quasi–homogeneous hypersurface singularities it is straightforward to compute this algebraically. In addition, it was shown in [9] that the Reeb vector field for a Sasaki–Einstein metric on L extremises V as a function of the ba , subject to the constraint (2.2). This is a geometric analogue of a–maximisation [14] in superconformal field theories.
2.2. The Lichnerowicz obstruction. Let f be a holomorphic function on X with Lξ f = λi f,
(2.9)
where R λ > 0, and we refer to λ as the charge of f under ξ . Since f is holomorphic, this immediately implies that f = r λ f˜,
(2.10)
where f˜ is homogeneous degree zero under r ∂/∂r – that is, f˜ is the pull–back to X of a function on the link L. Moreover, since (X, g X ) is Kähler, ∇ X2 f = 0,
(2.11)
where −∇ X2 is the Laplacian on (X, g X ). For a metric cone, this is related to the Laplacian on the link (L , g L ) at r = 1 by ∂ 1 1 ∂ r 2n−1 . (2.12) ∇ X2 = 2 ∇ L2 + 2n−1 r r ∂r ∂r From this, one sees that −∇ L2 f˜ = E f˜,
(2.13)
E = λ[λ + (2n − 2)].
(2.14)
where
Thus any holomorphic function f of definite charge under ξ , or equivalently degree under r ∂/∂r , corresponds to an eigenfunction of the Laplacian on the link. The charge λ is then related simply to the eigenvalue E by the above formula (2.14).
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By assumption, (X, g X ) is Ricci–flat Kähler, which implies that (L , g L ) is Einstein with Ricci curvature 2n − 2. The first non–zero eigenvalue E 1 > 0 of −∇ L2 is bounded from below: E 1 ≥ 2n − 1.
(2.15)
This is Lichnerowicz’s theorem [17]. Moreover, equality holds if and only if (L , g L ) is isometric to the round sphere S 2n−1 [21]. This is important as we shall find examples of links, that are not even diffeomorphic to the sphere, which hit this bound. From (2.14), we immediately see that Lichnerowicz’s bound becomes λ ≥ 1. This leads to a potential holomorphic obstruction to the existence of Sasaki–Einstein metrics: Lichnerowicz obstruction. Let (X, ) be an isolated Gorenstein singularity with link L and putative Reeb vector field ξ . Suppose that there exists a holomorphic function f on X of positive charge λ < 1 under ξ . Then X admits no Ricci–flat Kähler cone metric with Reeb vector field ξ . In particular L does not admit a Sasaki–Einstein metric with this Reeb vector field. As we stated earlier, it is not immediately clear that this can ever happen. In fact, there are examples of hypersurface singularities where this serves as the only obvious simple obstruction, as we explain later. However, in the next subsection we treat a situation where Lichnerowicz never obstructs. Before concluding this subsection we note that the volume of a Sasakian metric on L with Reeb vector field ξ is also related to holomorphic functions on X of definite charge, as we briefly reviewed in the previous subsection. In fact we may write (2.8) as V (ξ ) = lim t n Tr exp(−tLr ∂/∂r ), t→0
(2.16)
where r ∂/∂r = −J (ξ ). Here the trace denotes a trace of the action of Lr ∂/∂r on the holomorphic functions on X . Thus a holomorphic function f of charge λ under ξ contributes exp(−tλ) to the trace. That (2.16) agrees with (2.8) follows from the fact that we can write λ = (b, m). Given our earlier discussion relating λ to eigenvalues of the Laplacian on L, the above trace very much resembles the trace of the heat kernel, also known as the partition function, on L. In fact, since it is a sum over only holomorphic eigenvalues, we propose to call it the holomorphic partition function. The fact that the volume of a Riemannian manifold appears as a pole in the heat kernel is well known [22], and (2.16) can be considered a holomorphic Sasakian analogue. Notice then that the Lichnerowicz obstruction involves holomorphic functions on X of small charge with respect to ξ , whereas the Bishop obstruction is a statement about the volume, which is determined by the asymptotic growth of holomorphic functions on X . 2.3. Smooth Fanos. Let V be a smooth Fano Kähler manifold. Let K denote the canonical line bundle over V . By definition, K −1 is an ample holomorphic line bundle, which thus specifies a positive class c1 (K −1 ) = −c1 (K ) ∈ H 2 (V ; Z) ∩ H 1,1 (V ; R) ∼ = Pic(V ).
(2.17)
Recall here that Pic(V ) is the group of holomorphic line bundles on V . Let I (V ) denote the largest positive integer such that c1 (K −1 )/I (V ) is an integral class in Pic(V ). I (V ) is called the Fano index of V . For example, I (CP2 ) = 3, I (CP1 ×CP1 ) = 2, I (F1 ) = 1.
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Let L be the holomorphic line bundle L = K 1/I (V ) , which is primitive in Pic(V ) by construction. Denote the total space of the unit circle bundle in L by L – this is our link. We thus have a circle bundle S 1 → L → V,
(2.18)
where L is the associated line bundle. If V is simply–connected then L is also simply– connected, as follows from the Gysin sequence of the fibration (2.18). Note that V admits a Kähler–Einstein metric if and only if L admits a regular Sasaki–Einstein metric with Reeb vector field that rotates the S 1 fibre of (2.18). X is obtained from the total space of L by collapsing (or deleting, to obtain X 0 ) the zero section. Holomorphic functions on X of definite charge are then in 1–1 correspondence with global sections of L−k , which are elements of the group H 0 (O(L−k )). Let ζ be the holomorphic vector field on X that rotates the fibre of L with weight one. That is, if s ∈ H 0 (O(L−1 )) is a holomorphic section of the ample line bundle L−1 , viewed as a holomorphic function on X , then Lζ s = is.
(2.19)
L I (V )
is the canonical bundle of V , it follows that the correctly normalised Since K = Reeb vector field is (see, for example, [9]) n ζ. (2.20) ξ= I (V ) We briefly recall why this is true. Let ψ be a local coordinate such that ξ = ∂/∂ψ. Then nψ/I (V ) is a local coordinate on the circle fibre of (2.18) with period 2π . This follows since locally the contact one–form of the Sasakian manifold is η = dψ − A, where A/n is a connection on the canonical bundle of V . Holomorphic functions of smallest positive charge obviously correspond to k = 1. Any section s ∈ H 0 (O(L−1 )) then has charge n (2.21) λ= I (V ) under ξ . However, it is well known (see, for example, [23], p. 245) that for smooth Fanos V we have I (V ) ≤ n, with I (V ) = n if and only if V = CPn−1 . Thus, in this situation, we always have λ ≥ 1 and Lichnerowicz never obstructs. Lichnerowicz’s theorem can only obstruct for non–regular Reeb vector fields. We expect a similar statement to be true for the Bishop bound. For a regular Sasaki– Einstein manifold with Reeb vector field ξ and orbit space a Fano manifold V , Bishop’s bound may be written I (V ) c1 (V )n−1 ≤ n c1 (CPn−1 )n−1 = n n . (2.22) V
CPn−1
It seems reasonable to expect the topological statement (2.22) to be true for any Fano manifold V , so that Bishop never obstructs in the regular case, although we are unaware of any proof. Interestingly, this is closely related to a standard conjecture in algebraic geometry, that bounds V c1 (V )n−1 from above by n n−1 for any Fano manifold V , with equality if and only if V = CPn−1 . In general, this stronger statement is false (see [23], p. 251), although it is believed to be true in the special case that V has Picard number one, i.e. rank(Pic(V )) = 1. This has recently been proven up to dimension n = 5 [24]. It would be interesting to investigate (2.22) further.
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2.4. AdS/CFT interpretation. In this section we show that the Lichnerowicz obstruction has a very natural interpretation in the AdS/CFT dual field theory, in terms of a unitarity bound. We also briefly discuss the Bishop bound. Recall that every superconformal field theory possesses a supergroup of symmetries and that the AdS/CFT duality maps this to the superisometries of the dual geometry. In particular, in the context of Sasaki–Einstein geometry, it maps the R–symmetry in the field theory to the isometry generated by the Reeb vector field ξ , and the R–charges of operators in the field theory are proportional to the weights under ξ . Generically, Kaluza–Klein excitations in the geometry correspond to gauge invariant operators in the field theory. These operators are characterised by their scaling dimensions . The supersymmetry algebra then implies that a general operator satisfies a BPS bound relating the dimension to the R–charge R: ≥ (d − 1)R/2. When this bound is saturated the corresponding BPS operators belong to short representations of the supersymmetry algebra, and in particular are chiral. Here we will only consider scalar gauge invariant operators which are chiral. It is well known that for any conformal field theory, in arbitrary dimension d, the scaling dimensions of all operators are bounded as a consequence of unitarity. In particular, for scalar operators, we have d −2 . (2.23) 2 In Sect. 2.2 we have argued that a necessary condition for the existence of a Sasaki– Einstein metric is that the charge λ > 0 of any holomorphic function on the corresponding Calabi–Yau cone must satisfy the bound
≥
λ ≥ 1.
(2.24)
In the following, we will show that these two bounds coincide. We start with a gauge theory realised on the world–volume of a large number of D3 branes, placed at a 3–fold Gorenstein singularity X . The affine variety X can then be thought of as (part of) the moduli space of vacua of this gauge theory. In particular, the holomorphic functions, defining the coordinate ring of X , correspond to (scalar) elements of the chiral ring of the gauge theory [25]. Recalling that an AdS4/5 × L 7/5 solution arises as the near–horizon limit of a large number of branes at a Calabi–Yau 4–fold/3–fold conical singularity, it is clear that the weights λ of these holomorphic functions under the action of r ∂/∂r must be proportional to the scaling dimensions
of the dual operators, corresponding to excitations in AdS space. We now make this relation more precise. According to the AdS/CFT dictionary [26, 27], a generic scalar excitation in AdS obeying (AdSd+1 − m 2 ) = 0
(2.25)
and which behaves like ρ − near the boundary of AdS (ρ → ∞), is dual to an operator in the dual CFT with scaling dimension d2 d m 2 = ( − d) ⇒ ± = ± + m2. (2.26) 2 4 More precisely, for m 2 ≥ −d 2 /4 + 1 the dimension of the operator is given by + . However, for −d 2 /4 < m 2 < −d 2 /4 + 1 one can take either ± and these will correspond
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to inequivalent CFTs [28, 29]. Notice that + is always well above the bound implied by unitarity. On the other hand, − saturates this bound for m 2 = −d 2 /4 + 1. The values for m 2 can be obtained from the eigenvalues E of the scalar Laplacian −∇ L2 on the internal manifold L by performing a Kaluza–Klein analysis. The modes corresponding to the chiral primary operators have been identified in the literature in the context of a more general analysis for Einstein manifolds; see [28, 30] for type IIB supergravity compactified on L 5 , and [31, 32] for M–theory compactified on L 7 . Consider first d = 4 (i.e. n = 3). The supergravity modes dual to chiral operators are a mixture of the trace mode of the internal metric and the RR four–form and lead to [33, 28, 30] √ m 2 = E + 16 − 8 E + 4. (2.27) Combining this with (2.14) it follows that
± = 2 ± |λ − 2|
(2.28)
so that = λ, providing that we take − for λ < 2 and + for λ ≥ 2. Notice that for λ = 2, + = − , and this corresponds to the Breitenlohner–Freedman bound m 2 (λ = 2) = −4 for stability in AdS5 . The case d = 3 (i.e. n = 4), relevant for AdS4 × L 7 geometries, is similar. The scalar supergravity modes corresponding to chiral primaries [31, 32] are again a mixture of the metric trace and the three–form potential [34], and fall into short N = 2 multiplets. Their masses are given by5 [34, 35, 32] √ E + 9 − 3 E + 9. 4 Combing this with (2.14) it follows that m2 =
± =
1 (3 ± |λ − 3|) 2
(2.29)
(2.30)
so that = 21 λ, providing that we take − for λ < 3 and + for λ ≥ 3. Once again the switching of the two branches occurs at the Breitenlohner–Freedman bound m 2 (λ = 3) = −9/4 for stability in AdS4 . In summary, we have shown that ⎧ for d = 4 ⎨ λ . (2.31)
= 1 ⎩ λ for d = 3 2 Thus in both cases relevant for AdS/CFT the Lichnerowicz bound λ ≥ 1 is equivalent to the unitarity bound (2.23). The Bishop bound also has a direct interpretation in field theory. Recall that the volume of the Einstein 5–manifold (L , g L ) is related to the exact a central charge of the dual four dimensional conformal field theory via [36] (see also [37]) a(L) =
π3N2 , 4vol(L , g L )
(2.32)
5 Note that the mass formulae in [35] are relative to the operator Ad S4 − 32. Moreover, the factor of four mismatch between their m 2 and ours is simply due to the fact that it is actually m 2 R 2 that enters in (2.26),
and the radius of AdS4 is 1/2 that of AdS5 .
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where N is the number of D3–branes. The Bishop bound then implies that a(L) ≥
N2 = a(N = 4), 4
(2.33)
where N 2 /4 is the central charge of N = 4 super Yang–Mills theory. One can give a heuristic argument for this inequality, as follows6 . By appropriately Higgsing the dual field theory, and then integrating out the massive fields, one expects to be able to flow to N = 4 super Yang–Mills theory. This is because the Higgsing corresponds to moving the D3–branes away from the singular point to a smooth point of the cone, at which the near horizon geometry becomes Ad S5 × S 5 . Since the number of massless degrees of freedom is expected to decrease in such a process, we also expect the a central charge to decrease. This would then explain the inequality (2.33). 3. Isolated Hypersurface Singularities In this section we describe links of isolated quasi–homogeneous hypersurface singularities. These provide many simple examples of both obstructions. Let wi ∈ Z+ , i = 1, . . . , n + 1, be a set of positive weights. We denote these by a vector w ∈ (Z+ )n+1 . This defines an action of C∗ on Cn+1 via (z 1 , . . . , z n+1 ) → (q w1 z 1 , . . . , q wn+1 z n+1 ),
(3.1)
C∗ .
Without loss of generality one can take the set {wi } to have no common where q ∈ factor. This ensures that the above C∗ action is effective. However, for the most part, this is unnecessary for our purposes and we shall not always do this. Let F : Cn+1 → C
(3.2)
be a quasi–homogeneous polynomial on Cn+1 with respect to w. This means that F has definite degree d under the above C∗ action: F(q w1 z 1 , . . . , q wn+1 z n+1 ) = q d F(z 1 , . . . , z n+1 ).
(3.3)
Moreover we assume that the affine algebraic variety X = {F = 0} ⊂ Cn+1
(3.4)
is smooth everywhere except at the origin (0, 0, . . . , 0). For obvious reasons, such X are called isolated quasi–homogeneous hypersurface singularities. The corresponding link L is the intersection of X with the unit sphere in Cn+1 : n+1
|z i |2 = 1.
(3.5)
i=1
A particularly nice set of such singularities are provided by so–called Brieskorn– Pham singularities. These take the particular form F=
n+1 i=1
6 We thank Ken Intriligator for this argument.
z iai
(3.6)
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J. P. Gauntlett, D. Martelli, J. Sparks, S.-T. Yau
with a ∈ (Z+ )n+1 . Thus the weights of the C∗ action are given by wi = d/ai . The corresponding hypersurface singularities X are always isolated, as is easily checked. Moreover, the topology of the links L are also extremely well understood – see [39] for a complete description of the homology groups of L. In particular, L is known to be (n − 2)–connected, meaning that the homotopy groups are πa (L) = 0 for all a = 1, . . . , n − 2. Returning to the general case, we may define a nowhere zero holomorphic (n, 0)– form on the smooth part of X by =
dz 1 ∧ · · · ∧ dz n . ∂ F/∂z n+1
(3.7)
This defines on the patch where ∂ F/∂z n+1 = 0. One has similar expressions on patches where ∂ F/∂z i = 0 for each i, and it is simple to check that these glue together into a nowhere zero form . Thus all such X are Gorenstein, and moreover they come equipped with a holomorphic C∗ action by construction. The orbit space of this C∗ action, or equivalently the orbit space of U (1) ⊂ C∗ on the link, is a complex orbifold V . In fact, V is the weighted variety defined by {F = 0} in the weighted projective space WCPn[w1 ,w2 ,...,wn+1 ] . The latter is the quotient of the non–zero vectors in Cn+1 by the weighted C∗ action WCPn[w1 ,w2 ,...,wn+1 ] = Cn+1 \ {(0, 0, . . . , 0)} /C∗ (3.8) and is a complex orbifold with a natural Kähler orbifold metric, up to scale, induced from Kähler reduction of the flat metric on Cn+1 . It is not difficult to show that V is a Fano orbifold if and only if |w| − d > 0,
(3.9)
Lζ z j = w j i z j
(3.10)
Lζ = (|w| − d)i.
(3.11)
n+1 where |w| = i=1 wi . To see this, first notice that |w| − d is the charge of under ∗ U (1) ⊂ C . To be precise, if ζ denotes the holomorphic vector field on X with
for each j = 1, . . . , n + 1, then
Positivity of this charge |w| − d then implies [9] that the cohomology class of the natural Ricci–form induced on V is represented by a positive (1, 1)–form, which is the definition that V is Fano. If there exists a Ricci–flat Kähler metric on X which is a cone under R+ ⊂ C∗ , then the correctly normalised Reeb vector field is thus ξ=
n ζ. |w| − d
(3.12)
We emphasise that here we will focus on the possible (non–)existence of a Sasaki– Einstein metric on the link L which has this canonical vector field as its Reeb vector field. It is possible that such metrics are obstructed, but that there exists a Sasaki–Einstein metric on L with a different Reeb vector field. This may be investigated using the results of [9]. In particular we shall come back to this point for a class of 3–fold examples in Sect. 4.
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3.1. The Bishop obstruction. A general formula for the volume of a Sasaki–Einstein metric on the link of an isolated quasi–homogeneous hypersurface singularity was given in [40]. Strictly speaking, this formula was proven only when the Fano V is well–formed. This means that the orbifold loci of V are at least complex codimension two. When V is not well–formed, the singular sets of V considered as an orbifold and as an algebraic variety are in fact different. A simple example is the weighted projective space WCP1[ p,q] , where hcf( p, q) = 1. As an orbifold, this is topologically a 2–sphere with conical singularities at the north and south poles of polar angle 2π/ p and 2π/q, respectively. As an algebraic variety, this weighted projective space is just CP1 since C/Z p = C. In fact, as a manifold it is diffeomorphic to S 2 , for the same reason. When we say Kähler–Einstein orbifold metric, we must keep track of this complex codimension one orbifold data in the non–well–formed case. For further details, the reader is directed to the review [6]. Assuming that there exists a Sasaki–Einstein metric with Reeb action U (1) ⊂ C∗ , then the volume of this link when V is well–formed is given by [40] π(|w| − d) n 2d vol(L) = . (3.13) w(n − 1)! n n+1 Here w = i=1 wi denotes the product of the weights. Using the earlier formula (2.8), we may now give an alternative derivation of this formula. The advantage of this approach is that, in contrast to [40], we never descend to the orbifold V . This allows us to dispense with the well–formed condition, and show that (3.13) holds in general. The authors of [40] noted that their formula seemed to apply to the general case. Let us apply (2.8) to isolated quasi–homogeneous hypersurface singularities. Let q ∈ C∗ denote the weighted action on X . We may compute the character C(q, X ) rather easily, since holomorphic functions on X descend from holomorphic functions on Cn+1 , and the trace over the latter is simple to compute. A discussion of precisely this problem may be found in [41]. According to the latter reference, the character is simply 1 − qd . C(q, X ) = n+1 wi i=1 (1 − q )
(3.14)
The limit (2.8) is straightforward to take, giving the normalised volume V (ξ ) =
d , wbn
(3.15)
where, as above, ξ = bζ,
(3.16)
and ζ generates the U (1) ⊂ C∗ action. Thus, from our earlier discussion on the charge of , we have b=
n , |w| − d
(3.17)
giving vol(L) =
d (|w| − d)n vol(S 2n−1 ). wn n
(3.18)
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Restoring vol(S 2n−1 ) =
2π n (n − 1)!
(3.19)
we thus obtain the result (3.13). Bishop’s theorem then requires, for existence of a Sasaki–Einstein metric on L with Reeb vector field ξ generating the canonical U (1) action, d (|w| − d)n ≤ wn n .
(3.20)
We shall see that infinitely many isolated quasi–homogeneous hypersurface singularities with Fano V violate this inequality. 3.2. The Lichnerowicz obstruction. As we already mentioned, holomorphic functions on X are simply restrictions of holomorphic functions on Cn+1 . Thus the smallest positive charge holomorphic function is z m , where m ∈ {1, . . . , n + 1} is such that wm = min{wi , i = 1, . . . , n + 1}.
(3.21)
Of course, m might not be unique, but this is irrelevant since all such z m have the same charge in any case. This charge is λ=
nwm |w| − d
(3.22)
and thus the Lichnerowicz obstruction becomes |w| − d ≤ nwm .
(3.23)
Moreover, this bound can be saturated if and only if X is Cn with its flat metric. It is again clearly trivial to construct many examples of isolated hypersurface singularities that violate this bound. 3.3. Sufficient conditions for existence. In a series of works by Boyer, Galicki and collaborators, many examples of Sasaki–Einstein metrics have been shown to exist on links of isolated quasi–homogeneous hypersurface singularities of the form (3.6). Weighted homogeneous perturbations of these singularities can lead to continuous families of Sasaki–Einstein metrics. For a recent review of this work, we refer the reader to [6] and references therein. Existence of these metrics is proven using the continuity method. One of the sufficient (but far from necessary) conditions for there to exist a Sasaki–Einstein metric is that the weights satisfy the condition [6] |w| − d <
n wm . (n − 1)
(3.24)
In particular, for n > 2 this implies that |w| − d < nwm ,
(3.25)
which is precisely Lichnerowicz’s bound. Curiously, for n = 2 the Lichnerowicz bound and (3.24) are the same, although this case is rather trivial.
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4. A Class of 3–Fold Examples Our first set of examples are given by the 3–fold singularities with weights w = (k, k, k, 2) and polynomial F=
3
z i2 + z 4k ,
(4.1)
i=1
where k is a positive integer. The corresponding isolated hypersurface singularities X k = {F = 0} are of Brieskorn–Pham type. Notice that X 1 = C3 and X 2 is the ordinary double point singularity, better known to physicists as the conifold. Clearly, both of these admit Ricci–flat Kähler cone metrics and moreover, the Sasaki–Einstein metrics are homogeneous. The differential topology of the links L k can be deduced using the results of [39], together with Smale’s theorem for 5–manifolds. In particular, for k odd, the link L k is diffeomorphic to S 5 . For k = 2 p even, one can show that L 2 p ∼ = S 2 × S 3 (alternatively, see Lemma 7.1 of [42]). The Fanos Vk are not well–formed for k > 2. In fact, the subvariety z 4 = 0 in Vk is a copy of CP1 , which is a locus of Zk orbifold singularities for k odd, and Zk/2 orbifold singularities for k even. As algebraic varieties, all the odd k are equivalent to CP2 , and all the even k are equivalent to CP1 × CP1 . As orbifolds, they are clearly all distinct. 4.1. Obstructions. These singularities have appeared in the physics literature [18] where it was assumed that all X k admitted conical Ricci–flat Kähler metrics, with Reeb action corresponding to the canonical U (1) action. In fact, it is trivial to show that the Bishop bound (3.20) is violated for all k > 20. Moreover, the Lichnerowicz bound (3.23) is even sharper: for k ≥ 2, z 4 has smallest7 charge under ξ , namely 6 , (4.2) k+2 which immediately rules out all k > 4. For k = 4 we have λ = 1. Recall that, according to [21], this can happen if and only if L 4 is the round sphere. But we already argued that L 4 = S 2 × S 3 , which rules out k = 4 also. Thus the only link that might possibly admit a Sasaki–Einstein metric with this U (1) Reeb action, apart from k = 1, 2, is k = 3. We shall return to the k = 3 case in the next subsection. Given the contradiction, one might think that perhaps the canonical C∗ action is not the critical one, in the sense of [9]. Writing λ=
F = z 12 + uv + z 4k ,
(4.3)
there is clearly a (C∗ )2 action generated by weights (k, k, k, 2) and (0, 1, −1, 0) on (z 1 , u, v, z 4 ), respectively. The second U (1) ⊂ C∗ is the maximal torus of S O(3) acting on the z i , i = 1, 2, 3, in the vector representation. It is then straightforward to compute the volume of the link as a function of the Reeb vector field ξ=
2
ba ζa
(4.4)
a=1 7 For k = 1, z , i = 1, 2, 3 have the smallest charge λ = 1. This is consistent with the fact that k = 1 i corresponds to the link L 1 = S 5 with its round metric.
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using the character formula in [41] and taking the limit as in (2.8). We obtain V (b1 , b2 ) =
1 2kb1 = . 2b1 · kb1 (kb1 + b2 )(kb1 − b2 ) b1 (kb1 + b2 )(kb1 − b2 )
(4.5)
The first component b1 is fixed by the charge of , as above, to be b1 =
3 . k+2
(4.6)
According to [9], the critical Reeb vector field for the putative Sasaki–Einstein metric is obtained by setting to zero the derivative of V (b1 , b2 ), with respect to b2 . This immediately gives b2 = 0. Thus the original weighted C∗ action is indeed a critical point of the Sasakian–Einstein–Hilbert action on the link, in this 2–dimensional space of Reeb vector fields. We could have anticipated this result without computing anything. According to [9], the critical Reeb vector field could not have mixed with any vector field in the Lie algebra of a U (1) subgroup of S O(3), since the latter group is semi–simple. 4.2. Cohomogeneity one metrics. It is interesting to observe that any conical Ricci–flat Kähler metric on X k would necessarily have U (1) × S O(3) isometry (the global form of the effectively acting isometry group will depend on k mod 2). This statement follows from Matsushima’s theorem [7]. Specifically, Matsushima’s theorem says that the isometry group of a Kähler–Einstein manifold8 (V, gV ) is a maximal compact subgroup of the group of complex automorphisms of V . Quotienting L k by the U (1) action, one would thus have Kähler–Einstein orbifold metrics on Vk with an S O(3) isometry, whose generic orbit is three–dimensional. In other words, these metrics, when they exist, can be constructed using standard cohomogeneity one techniques. In fact this type of construction is very well motivated since demanding a local SU (2) × U (1)2 isometry is one way in which the Sasaki–Einstein metrics of [43] can be constructed (in fact they were actually found much more indirectly via M–theory [44]). However, apart from the k = 3 case, and of course the k = 1 and k = 2 cases, we have already shown that any such construction must fail. For k = 3, the relevant ODEs that need to be solved have actually been written down in [45]. In Appendix A we record these equations, as well as the boundary conditions that need to be imposed. We have been unable to integrate these equations, so the question of existence of a Sasaki–Einstein metric on L 3 remains open. 4.3. Field theory. In [18] a family of supersymmetric quiver gauge theories were studied whose classical vacuum moduli space reproduces the affine varieties X 2 p . These theories were argued to flow for large N in the IR to a superconformal fixed point, AdS/CFT dual to a Sasaki–Einstein metric on the link L 2 p for all p. Indeed, the R–charges of fields may be computed using a–maximisation [14], and agree with the naive geometric computations, assuming that the Sasaki–Einstein metrics on L 2 p exist. However, as we have already seen, these metrics cannot exist for any p > 1. We argued in Sect. 2 that this bound, coming from Lichnerowicz’s theorem, is equivalent to the unitarity bound in the CFT. We indeed show that a gauge invariant chiral primary operator, dual to the holomorphic function z 4 that provides the geometric obstruction, violates the unitarity bound for p > 1. 8 The generalisation to orbifolds is straightforward.
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A
X
Y
B Fig. 1. Quiver diagram of the A1 orbifold gauge theory.
Before we recall the field theories for k = 2 p even, let us make a remark on the X k singularities when k is odd. In the latter case, it is not difficult to prove that X k admits no crepant resolution9 . That is, there is no blow–up of X k to a smooth manifold X˜ with trivial canonical bundle. In such cases the field theories might be quite exotic, and in particular not take the form of quiver gauge theories. In contrast, the X 2 p singularities are resolved by blowing up a single exceptional CP1 [47], which leads to a very simple class of gauge theories. Consider the quiver diagram for the N = 2 A1 orbifold, depicted in Fig. 1. The two nodes represent two U (N ) gauge groups. There are 6 matter fields: an adjoint for each gauge group, that we denote by X and Y , and two sets of bifundamental fields A I and B I , where I = 1, 2 are SU (2) flavour indices. Here the A I are in the (N , N¯ ) representation of U (N ) × U (N ), and the B I are in the ( N¯ , N ) representation. This is the quiver for N D3–branes at the C × (C2 /Z2 ) singularity, where C2 /Z2 is the A1 surface singularity. However, for our field theories indexed by p, the superpotential is given by W = Tr X p+1 + (−1) p Y p+1 + X (A1 B1 + A2 B2 ) + Y (B1 A1 + B2 A2 ) . (4.7) It is straightforward to verify that the classical vacuum moduli space of this gauge theory gives rise to the X 2 p singularities. In fact these gauge theories were also studied in detail in [48], and we refer the reader to this reference for further details. The SU (2) flavour symmetry corresponds to the S O(3) automorphism of X 2 p . It is a simple matter to perform a–maximisation for this theory, taken at face value. Recall this requires one to assign trial R–charges to each field, and impose the constraints that W has R–charge 2, and that the β–functions of each gauge group vanish. One then locally maximises the a–function 3N 2 a= 3(R(X i ) − 1)3 − (R(X i ) − 1) (4.8) 2+ 32 i
subject to these constraints, where the sum is taken over all R–charges of fields X i . One finds the results, as in [18], 2 , R(X ) = R(Y ) = p+1 p R(A I ) = R(B I ) = , (4.9) p+1 9 Since the link L = S 5 for all odd k, it follows that Pic(X \ {r = 0}) is trivial, and hence X is factorial. k k k The isolated singularity at r = 0 is terminal for all k. These two facts, together with Corollary 4.11 of [46], imply that X k has no crepant resolution.
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and central charge a(L 2 p ) =
27 p 2 N 2 . 8( p + 1)3
(4.10)
This corresponds, under the AdS/CFT relation (2.32), to a Sasaki–Einstein volume vol(L 2 p ) =
2π 3 ( p + 1)3 , 27 p 2
(4.11)
which agrees with the general formula (3.13). Thus an initial reaction [18] is that one has found agreement between geometric and field theory results. However, the results of this paper imply that the Sasaki–Einstein metrics on L 2 p do not exist for p > 1. In fact, it is clear that, upon closer inspection, the gauge invariant chiral primary operator Tr X (or TrY ) has R–charge 2/( p + 1), which violates the unitarity bound for p > 1. In fact, when one computes the vacuum moduli space for a single D3–brane N = 1, Tr X is identified with the holomorphic function z 4 , and the unitarity bound and Lichnerowicz bound are identical, as we argued to be generally true in Sect. 2. The superpotential (4.7) can be regarded as a deformation of the A1 orbifold theory. For p > 2, using a–maximisation (and assuming an a–theorem), this was argued in [48] to be an irrelevant deformation (rather than a “dangerously irrelevant” operator). This is therefore consistent with our geometric results. The case p = 2 is interesting since it appears to be marginal. If it is exactly marginal, we expect a one parameter family of solutions with fluxes that interpolates between the A1 orbifold with link S 5 /Z2 and the X 4 singularity with flux. 5. Other Examples In this section we present some further obstructed examples. In particular we examine ADE 4–fold singularities, studied in [20]. All of these, with the exception of D4 and the obvious cases of A0 and A1 , do not admit Ricci–flat Kähler cone metrics with the canonical weighted C∗ action. We also examine weighted C∗ actions on Cn . 5.1. ADE 4–fold singularities. Consider the polynomials H = z 1k + z 22 + z 32 Ak−1 , H = z 1k + z 1 z 22 + z 32 Dk+1 , H = z 13 + z 24 + z 32 E 6 , H = z 13 + z 1 z 23 + z 32 E 7 , H = z 13 + z 25 + z 32 E 8 .
(5.1)
The hypersurfaces {H = 0} ⊂ C3 are known as the ADE surface singularities. Their links L AD E are precisely S 3 / , where ⊂ SU (2) are the finite ADE subgroups of SU (2) acting on C2 in the vector representation. Thus these Gorenstein singularities are both hypersurface singularities and quotient singularities. Clearly, the links admit Sasaki–Einstein metrics – they are just the quotient of the round metric on S 3 by the group .
Obstructions to the Existence of Sasaki–Einstein Metrics
821
The 3–fold singularities of the previous section are obtained from the polynomial H for Ak−1 by simply adding an additional term z 42 (and relabelling). More generally, we may define the ADE n–fold singularities as the zero loci X = {F = 0} of F=H+
n+1
z i2 .
(5.2)
i=4
Let us consider the particular case n = 4. The C∗ actions, for the above cases, are generated by the weight vectors w = (2, k, k, k, k) w = (2, k − 1, k, k, k) w = (4, 3, 6, 6, 6) w = (6, 4, 9, 9, 9) w = (10, 6, 15, 15, 15)
d d d d d
= 2k = 2k = 12 = 18 = 30
Ak−1 , Dk+1 , E6, E7, E8.
(5.3)
It is then straightforward to verify that for all the Ak−1 singularities with k > 3 the holomorphic function z 1 on X violates the Lichnerowicz bound (3.23). The case k = 3 saturates the bound, but since the link is not10 diffeomorphic to S 7 Obata’s result [21] again rules this out. For all the exceptional singularities the holomorphic function z 2 on X violates (3.23). The Dk+1 singularities are a little more involved. The holomorphic function z 1 rules out all k > 3. On the other hand the function z 2 rules out k = 2, but the Lichnerowicz bound is unable to rule out k = 3. To summarise, the only ADE 4–fold singularity that might possibly admit a Ricci–flat Kähler cone metric with the canonical C∗ action above, apart from the obvious cases of A0 and A1 , is D4 . Existence of a Sasaki–Einstein metric on the link of this singularity is therefore left open. It would be interesting to investigate whether or not there exist Ricci–flat Kähler metrics that are cones with a different Reeb action. In light of our results on the non–existence of the above Sasaki–Einstein metrics, it would also be interesting to revisit the field theory analysis of [20]. 5.2. Weighted actions on Cn . Consider X = Cn , with a weighted C∗ action with weights v ∈ (Z+ )n . The orbit space of non–zero vectors is the weighted projective n space WCPn−1 [v1 ,...,vn ] . Existence of a Ricci–flat Kähler cone metric on C , with the coni∗ cal symmetry generated by this C action, is equivalent to existence of a Kähler–Einstein orbifold metric on the weighted projective space. In fact, it is well known that no such metric exists: the Futaki invariant of the weighted projective space is non–zero. In fact, one can see this also from the Sasakian perspective through the results of [13, 9]. The diagonal action with weights v = (1, 1, . . . , 1) is clearly a critical point of the Sasakian– Einstein–Hilbert action, and this critical point was shown to be unique in the space of toric Sasakian metrics. Nonetheless, in this subsection we show that Lichnerowicz’s bound and Bishop’s bound both obstruct existence of these metrics. The holomorphic (n, 0)–form on X = Cn has charge |v| under the weighted C∗ action, which implies that the correctly normalised Reeb vector field is ξ=
n ζ |v|
10 One can easily show that H (L; Z) = Z for the links of the A 3 k k−1 4–fold singularities.
(5.4)
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with notation as before, so that ζ is the vector field that generates U (1) ⊂ C∗ . The Lichnerowicz bound is therefore nvm ≥ 1, (5.5) |v| where vm is the (or a particular) smallest weight. However, clearly |v| ≥ nvm , with equality if and only if v is proportional to (1, 1, . . . , 1). Thus in fact nvm ≤1 |v|
(5.6)
with equality only in the diagonal case, which is just Cn with the canonical Reeb vector field. Thus our Lichnerowicz bound obstructs Kähler–Einstein orbifold metrics on all weighted projective spaces, apart from CPn−1 of course. For the Bishop bound, notice that a Kähler–Einstein orbifold metric on WCPn−1 [v1 ,...,vn ] would give rise to a Sasaki–Einstein metric on S 2n−1 with a weighted Reeb action. The volume of this metric, relative to the round sphere, would be V =
|v|n , nn v
(5.7)
n where v = i=1 vi denotes the product of the weights. This may either be derived using the methods described earlier, or using the toric methods of [13]. Amusingly, (5.7) is precisely the arithmetic mean of the weights v divided by their geometric mean, all to the n th power. Thus the usual arithmetic mean–geometric mean inequality gives V ≥ 1 with equality if and only if v is proportional to (1, 1, . . . , 1). This is precisely opposite to Bishop’s bound, thus again ruling out all weighted projective spaces, apart from CPn−1 . Thus Kähler–Einstein orbifold metrics on weighted projective spaces are obstructed by the Futaki invariant, the Bishop obstruction, and the Lichnerowicz obstruction. In some sense, these Fano orbifolds couldn’t have more wrong with them. 6. Conclusions The problem of existence of conical Ricci–flat Kähler metrics on a Gorenstein n–fold singularity X is a subtle one; a set of necessary and sufficient algebraic conditions is unknown. This is to be contrasted with the case of compact Calabi–Yau manifolds, where Yau’s theorem guarantees the existence of a unique Ricci–flat Kähler metric in a given Kähler class. In this paper we have presented two simple necessary conditions for existence of a Ricci–flat Kähler cone metric on a given isolated Gorenstein singularity X with specified Reeb vector field. The latter is in many ways similar to specifying a “Kähler class”, or polarisation. These necessary conditions are based on the classical results of Bishop and Lichnerowicz, that bound the volume and the smallest eigenvalue of the Laplacian on Einstien manifolds, respectively. The key point that allows us to use these as obstructions is that, in both cases, fixing a putative Reeb vector field ξ for the Sasaki–Einstein metric is sufficient to determine both the volume and the “holomorphic” eigenvalues using only the holomorphic data of X . Note that any such vector field ξ must also be a critical point of the Sasakian–Einstein–Hilbert action of [13, 9], which in Kähler–Einstein terms means that the transverse Futaki invariant is zero. We emphasize, however, that the possible obstructions presented here may be analysed independently of this, the weighted
Obstructions to the Existence of Sasaki–Einstein Metrics
823
projective spaces at the end of Sect. 5 being examples that are obstructed by more than one obstruction. To demonstrate the utility of these criteria, we have provided many explicit examples of Gorenstein singularities that do not admit Sasaki–Einstein metrics on their links, for a particular choice of Reeb vector field. The examples include various quasi–homogeneous hypersurface singularities, previously studied in the physics literature, that have been erroneously assumed to admit such Ricci–flat Kähler cone metrics. We expect that in the particular case that the singularity is toric, neither Lichnerowicz nor Bishop’s bound will obstruct for the critical Reeb vector field b∗ of [13]. This is certainly true for all cases that have been analysed in the literature. In this case both bounds reduce to simple geometrical statements on the polyhedral cone C ∗ and its associated semi–group SC = C ∗ ∩ Zn . For instance, given the critical Reeb vector field b∗ , the Lichnerowicz bound implies that (b∗ , m) ≥ 1 for all m ∈ SC . It would be interesting to try to prove that this automatically follows from the extremal problem in [13], for any toric Gorenstein singularity. We have also explained the relevance of these bounds to the AdS/CFT correspondence. We have shown that the Lichnerowicz bound is equivalent to the unitarity bound on the scaling dimensions of BPS chiral operators of the dual field theories. In particular, we analysed a class of obstructed 3–fold singularities, parameterised by a positive integer k, for which, in the case that k is even, the field theory dual is known and has been extensively studied in the literature. The fact that the links L k do not admit Sasaki–Einstein metrics for any k > 3 supports the field theory arguments of [48]. It would be interesting to know whether a Sasaki–Einstein metric exists on L 3 ; if it does exist, it might be dual to an exotic type of field theory since the corresponding Calabi– Yau cone does not admit a crepant resolution. For the 4–folds studied in [20], it will be interesting to analyse the implications of our results for the field theories. Acknowledgements. We would like to thank O. Mac Conamhna and especially D. Waldram for collaboration in the early stages of this work. We would also like to thank G. Dall’Agata, M. Haskins, N. Hitchin, K. Intriligator, P. Li, R. Thomas, C. Vafa, N. Warner, and S. S.–T. Yau for discussions. We particularly thank R. Thomas for comments on a draft version of this paper. J. F. S. is supported by NSF grants DMS–0244464, DMS–0074329 and DMS–9803347. S.–T. Y. is supported in part by NSF grants DMS–0306600 and DMS– 0074329.
A. Cohomogeneity One Metrics Here we discuss the equations that need to be solved to obtain a Kähler–Einstein orbifold metric on the Fano orbifold Vk of Sect. 4, which we recall is a hypersurface F = z 12 + z 22 + z 32 + z 4k = 0 in the weighted projective space WCP3[k,k,k,2] . The group S O(3) acts on z i , i = 1, 2, 3, in the vector representation, and then Matsushima’s theorem [7] implies that this acts isometrically on any Kähler–Einstein metric. The generic orbit is three–dimensional, and hence these metrics are cohomogeneity one. The Kähler–Einstein condition then reduces to a set of ordinary differential equations in a rather standard way. The ODEs for a local Kähler–Einstein 4–metric with cohomogeneity one SU (2) action have been written down in [45]. The metric may be written as ds 2 = dt 2 + a 2 (t)σ12 + b2 (t)σ22 + c2 (t)σ32 ,
(A.1)
where σi , i = 1, 2, 3, are (locally) left–invariant one–forms on SU (2), and t is a coordinate transverse to the principal orbit. The ODEs are then [45]
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J. P. Gauntlett, D. Martelli, J. Sparks, S.-T. Yau
1 a˙ =− (b2 + c2 − a 2 ), a 2abc 1 b˙ =− (a 2 + c2 − b2 ), b 2abc c˙ 1 ab =− (a 2 + b2 − c2 ) + , c 2abc c
(A.2)
where is the Einstein constant, which is = 6 in the normalisation relevant for Sasaki–Einstein metrics on L k . The key question, given these local equations, is what the boundary conditions are. For a complete metric on Vk , the parameter t must take values in a finite interval, which without loss of generality we may take to be [0, t∗ ] for some t∗ . At the endpoints, the principal orbit collapses smoothly (in an orbifold sense) to a special orbit. It is not difficult to work out the details for Vk , given its embedding in WCP3[k,k,k,2] . One must separate k = 2 p even and k odd. For k odd, the principal orbit is S O(3)/Z2 . This collapses to the two special orbits Bt=0 = (S O(3)/Z2 ) /U (1)1 = RP2 , Bt=t∗ = (S O(3)/Z2 ) /U (1)3 = CP1 ,
(A.3)
where the circle subgroups U (1)1 , U (1)3 ⊂ S O(3) are rotations about the planes transverse to the 1–axis and the 3–axis, respectively, thinking of S O(3) acting on R3 in the usual way. Thus the two U (1) subgroups are related by a conjugation. For k = 2 p even, the principal orbit is instead simply S O(3) = RP3 . The two special orbits are Bt=0 = S O(3)/U (1)1 = S 2 , Bt=t∗ = S O(3)/U (1)3 = CP1 .
(A.4)
Of course, these are diffeomorphic, but the notation indicates that the second orbit is embedded as a complex curve in V2 p , whereas S 2 is embedded as a real submanifold of V2 p . In both bases, with k odd or k = 2 p even, the bolts are the real section of Vk , and the subvariety z 4 = 0, respectively. The latter is the image of the conic in CP2 ⊂ WCP3[k,k,k,2] at z 4 = 0, and is a locus of orbifold singularities. This is the only singular set on Vk . The boundary conditions at t = 0 are then, in all cases, a(t) = β + O(t), b(t) = β + O(t), 2 c(t) = t + O(t 2 ), k
(A.5)
where β2 =
k+2 . 6k
(A.6)
Obstructions to the Existence of Sasaki–Einstein Metrics
825
At t = t∗ , one simply requires that a collapses to zero a(t∗ ) = 0, with b(t∗ ) = c(t∗ ) positive and finite. The metric functions should remain strictly positive on the open interval (0, t∗ ). The system of first order ODEs (A.2) may be reduced to a single second order ODE as follows. The change of variables dr/dt = 1/c allows one to find the integral a (r ) = − coth(r ), b
(A.7)
where an integration constant can be reabsorbed by a shift of r . Defining f (r ) = ab, one obtains d log dr
df f dr
= 2 [ f + coth(2r )] .
(A.8)
Any solution of this equation gives rise to a solution of (A.2), using the fact that c2 = −
df . dr
(A.9)
For k = 1, k = 2, one can write down explicit solutions to these equations and boundary conditions, corresponding to the standard metrics on CP2 and CP1 × CP1 , respectively. For k = 1 we have π , a(t) = cos t + 4
π b(t) = sin t + , 4
c(t) = sin(2t),
(A.10)
where the range of t is 0 ≤ t ≤ π/4. Correspondingly, 1 f (r ) = − tanh(2r ) 2
(A.11)
with tan(t) = exp (2r ), so that −∞ ≤ r ≤ 0. For k = 2 we instead have √ 1 a(t) = √ cos( 3t), 3
1 b(t) = √ , 3
√ 1 c(t) = √ sin( 3t), 3
(A.12)
√ where the range of t is 0 ≤ t ≤ π/(2 3). Correspondingly, 1 f (r ) = − tanh(r ) 3
(A.13)
√ with tan( 3t/2) = exp (r ) and −∞ ≤ r ≤ 0. For all k > 3, this paper implies that there do not exist any solutions. This still leaves the case k = 3. We have neither been able to integrate the equations explicitly, nor have our preliminary numerical investigations been conclusive. We leave the issue of existence of this solution open.
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