Communications in Mathematical Physics - Volume 243

Commun. Math. Phys. 243, 1–54 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0948-8 Communications in Mathe...

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Commun. Math. Phys. 243, 1–54 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0948-8

Communications in

Mathematical Physics

Banach Lie-Poisson Spaces and Reduction Anatol Odzijewicz1 , Tudor S. Ratiu2 1

Institute of Physics, University of Bialystok, Lipowa 41, 15424 Bialystok, Poland. E-mail: [email protected] 2 Centre Bernoulli, Ecole ´ Polytechnique F´ed´erale de Lausanne, 1015 Lausanne, Switzerland. E-mail: [email protected] Received: 16 November 2002 / Accepted: 17 April 2003 Published online: 17 October 2003 – © Springer-Verlag 2003

Abstract: The category of Banach Lie-Poisson spaces is introduced and studied. It is shown that the category of W ∗ -algebras can be considered as one of its subcategories. Examples and applications of Banach Lie-Poisson spaces to quantization and integration of Hamiltonian systems are given. The relationship between classical and quantum reduction is discussed. Contents 1. 2. 3. 4. 5. 6. 7. 8.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . Banach Poisson Manifolds . . . . . . . . . . . . . . . . Classical Reduction . . . . . . . . . . . . . . . . . . . . Banach Lie-Poisson Spaces . . . . . . . . . . . . . . . . Preduals of W ∗ -Algebras as Banach Lie-Poisson Spaces Quantum Reduction . . . . . . . . . . . . . . . . . . . . Symplectic Leaves and Coadjoint Orbits . . . . . . . . . Momentum Maps and Reduction . . . . . . . . . . . . .

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1 3 8 12 22 27 32 43

1. Introduction This paper investigates the foundations of Banach Poisson differential geometry, including such topics as Banach Lie-Poisson spaces, classical and quantum reduction, integration and quantization of Hamiltonian systems with the aid of the momentum map. We were inspired to study this circle of problems due to the appearance of formal Poisson structures in a large number of works devoted to the integration of infinite dimensional systems and the crucial role played by the momentum map in these approaches. The notion of a Lie-Poisson space is as old as the concept of a Lie algebra and both were introduced simultaneously by Lie [Lie]. A Lie-Poisson space is a Poisson vector

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A. Odzijewicz, T.S. Ratiu

space with the property that its dual is invariant under the Poisson bracket, which is equivalent to the statement that the Poisson bracket is linear. In the finite dimensional case the notions of Lie algebras and Lie-Poisson spaces are equivalent in the sense that for any Lie algebra g its dual g∗ is a Lie-Poisson space and, conversely, given a Lie-Poisson space its dual is a Lie algebra. This is so because finite dimensional vector spaces are reflexive, the operation of taking the dual defines an isomorphism between these two categories. To generalize this to infinite dimensions, it is reasonable to assume that a Lie-Poisson space is a Banach space b endowed with a Poisson bracket {·, ·} such that the bracket of any two linear continuous functions is again a linear continuous function. This implies that (b∗ , {·, ·}) is a Banach Lie algebra. In order to preserve the correspondence between Banach Lie-Poisson spaces and Banach Lie algebras it is necessary to restrict to those Banach Lie algebras (g, [·, ·]) that admit a predual g∗ and satisfy in addition the condition that ad∗g : g∗ → g∗ preserves the predual g∗ . Thus, in the infinite dimensional case, Banach Lie-Poisson spaces form a subcategory of the category of Banach Lie algebras. A crucial example is the Banach space L1 (M) of linear trace class operators on a separable Hilbert space M which is predual to the Banach Lie algebra L∞ (M) of all linear bounded operators on M. As far as we know, the Lie-Poisson structure on L1 (M) was first found by Bona [B]. Momentum maps are an efficient way to encode integrals of motion for a Hamiltonian system. In its modern formulation due to Kostant [Ko1] and Souriau [So1, So2] a momentum map is naturally associated to an infinitesimal Poisson action of a Lie algebra on a Poisson manifold and it maps the phase space to the dual of the Lie algebra of symmetries. It turns out, that in finite dimensions, a momentum map is characterized by the property that it is Poisson, when one endows the dual of the Lie algebra with the Lie-Poisson structure (see, e.g. Marsden and Ratiu [M-R2] for a proof of this fact). In infinite dimensions, due to existence of non-reflexive Banach spaces, we will define a momentum map to be a Poisson map from a Banach Poisson manifold to a Banach Lie-Poisson space, which can always be considered as the predual of the Banach Lie algebra of symmetries. It is shown that the momentum map so defined has all the usual properties, such as being conserved along the flow of any symmetry invariant Hamiltonian vector field (Noether’s theorem). Like in finite dimensions, also in the infinite dimensional case, the notion of momentum map is an important tool in the study of Hamiltonian systems. For example, the knowledge of momentum maps leads to integrals of motion of the considered Hamiltonian system, as will be illustrated here through the example of the infinite Toda lattice. In the special case when one assumes that the momentum map is an injective immersion and its range is linearly dense in the target Banach Lie-Poisson space one discovers that it is the coherent states map in the sense of Odzijewicz [O2]. So, it can be used to quantize the system under consideration. This method of quantization, called Ehrenfest quantization, is a natural unification of the Kostant-Souriau geometric (Kostant [Ko2], Souriau [So1, So2]) and ∗-product quantization; for details see Odzijewicz [O2] and §8. The structure of the paper is as follows. In §2 the notion of a Banach Poisson manifold is introduced modeled on the example of a strong symplectic manifold. Its elementary properties are presented as well as some comments on the compatibility of the Poisson structure with almost complex, complex, and holomorphic structures. Classical reduction for Banach Poisson manifolds is discussed in §3. The Poisson reduction theorem of Marsden and Ratiu [M-R1] and its consequences are generalized to the Banach manifold context.

Banach Lie-Poisson Spaces and Reduction

3

Banach Lie-Poisson spaces and their properties are analyzed in §4. Linear continuous Poisson maps are studied in detail. The realification and complexification of a Banach Lie-Poisson space is also presented. The upshot of this section is the establishment of an isomorphism between a subcategory of Banach Lie-Poisson spaces and a specific subcategory of Banach Lie algebras. The entirety of §5 is devoted to one crucial example: the predual of a W ∗ -algebra and the dual to a C ∗ -algebra are naturally Banach Lie-Poisson spaces. As a consequence it is shown that various spaces related to operator algebras (for example the space of Hermitian trace class operators on a separable Hilbert space) are Banach Lie-Poisson spaces. In §6 we show that the quantum measurement operation in the sense of von Neumann can be considered as a Poisson projection. We shall give examples of other physically important Poisson projections. These examples justify the interpretation of Poisson projections as quantum reduction procedures. The internal structure of Banach Lie-Poisson spaces is presented in §7. If the Lie algebra of a Banach Lie group admits a predual which is invariant under the coadjoint representation it is shown that a large class of coadjoint orbits in the predual, which is naturally a Banach Lie-Poisson space, are symplectic leaves in a weak sense: they are weak symplectic manifolds and are weakly immersed submanifolds (the inclusion is smooth and has injective derivative, but no splitting condition, or even a closed range condition on the derivative, usually imposed in the definition of an immersion, holds). Among these orbits a subclass is determined for which the symplectic form is strong and the orbit is injectively immersed. The section ends with the standard example of a dual pair based on the cotangent bundle (in this case on the precotangent bundle) that illustrates that our definition of a Banach Poisson manifold is violated in this important case and that once one leaves the category of W ∗ -algebras a weakening of this notion will be needed. Section 8 introduces momentum maps as Poisson maps from a Banach Poisson manifold to a Banach Lie-Poisson space. It is shown that the coherent states map is a momentum map with certain special properties. In this way the quantization procedure based on the coherent states map has Banach Poisson geometrical interpretation. The relationship between classical and quantum reduction is explored. Using both procedures of reduction, classical and quantum, one can construct a new momentum map from a given one. The description of the infinite Toda lattice in Banach Poisson geometrical terms is presented. Among others, it is shown that the Flaschka transformation is a momentum map of some Banach weak symplectic space into the Banach Lie-Poisson space of the lower triangular trace class operators.

2. Banach Poisson Manifolds Throughout the paper, given a Banach space b, the notation b∗ will always mean the Banach space dual to b. Given x ∈ b∗ and b ∈ b, we shall denote by x, b the value of x on b. Thus ·, · : b∗ × b → R (or C, depending on whether we work with real or complex Banach spaces and functions) will denote the natural bilinear continuous duality pairing between b and its dual b∗ . A real finite dimensional Poisson manifold is a pair (P , {·, ·}) consisting of a manifold P whose space of Fr´echet smooth functions is endowed with a Lie algebra structure {·, ·} satisfying the Leibniz property in each factor; this bilinear operation {·, ·} is called

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a Poisson bracket. As we shall discuss below, this definition is not appropriate in infinite dimensions and a more stringent condition needs to be imposed. To see this, assume that on the space C ∞ (P ) of smooth functions on the infinite dimensional smooth Banach manifold P there is a Poisson bracket {·, ·}. Due to the Leibniz property, the value of the Poisson bracket at a given point p ∈ P depends only on the differentials df (p), dg(p) ∈ Tp∗ P which implies that there is a smooth section of the vector bundle 2 T ∗∗ P satisfying {f, g} = (df, dg). This means that for each p ∈ P the map p : Tp∗ P × Tp∗ P → R is a continuous bilinear antisymmetric map that depends smoothly on the base point p. In addition, denoting by [·, ·]S the Schouten bracket on skew symmetric contravariant tensors, the equality (see e.g. Marsden and Ratiu [M-R2], §10.6) {{f, g}, h} + {{g, h}, f } + {{h, f }, g} = i[, ]S (df ∧ dg ∧ dh), shows that the Jacobi identity is equivalent to [, ]S = 0, which is an additional differential quadratic condition on . Let : T ∗ P → T ∗∗ P be the bundle map covering the identity defined by p (dh(p)) := (·, dh)(p), that is, p (dh(p))(dg(p)) = {g, h}(p), for any locally defined functions g and h. Denote by b the Banach space modeling the Banach manifold P . Thus Tp P ∼ = b, ∗ Tp P ∼ = b∗ , and Tp∗∗ P ∼ = b∗∗ . If b is not reflexive, that is, b ⊂ b∗∗ and b = b∗∗ , then Xf := (·, df ) = (df ),

or, as a derivation on functions,

Xf = {·, f }

is a smooth section of T ∗∗ P and hence is not, in general, a vector field on P . In analogy with the finite dimensional case, we want Xf to be the Hamiltonian vector field defined by the function f . In order to achieve this, we are forced to make the assumption that the Poisson bracket on P satisfies the condition (T ∗ P ) ⊂ T P ⊂ T ∗∗ P . Thus we give the following definition. Definition 2.1. A Banach Poisson manifold is a pair (P , {·, ·}) consisting of a smooth Banach manifold and a bilinear operation {·, ·} satisfying the following conditions: (i) (C ∞ (P ), {·, ·}) is a Lie algebra; (ii) {·, ·} satisfies the Leibniz identity on each factor; (iii) the vector bundle map : T ∗ P → T ∗∗ P covering the identity satisfies (T ∗ P ) ⊂ TP. Condition (iii) allows one to introduce for any function h ∈ C ∞ (P ) the Hamiltonian vector field by Xh [f ] := df, Xh = {f, h}, where f is an arbitrary smooth locally defined function on P . Given two Banach Poisson manifolds (P1 , { , }1 ) and (P2 , { , }2 ), a smooth map ϕ : P1 → P2 is said to be canonical or a Poisson map if ϕ ∗ {f, g}2 = {ϕ ∗ f, ϕ ∗ g}1

(2.1)

Banach Lie-Poisson Spaces and Reduction

5

for any two smooth locally defined functions f and g on P2 . Condition (iii) in the previous definition implies, like in the finite dimensional case, that (2.1) is equivalent to Xf2 ◦ ϕ = T ϕ ◦ Xf1 ◦ϕ

(2.2)

for any smooth locally defined function f on P2 (for the proof see e.g. Marsden and Ratiu [M-R2], §10.3). Therefore, the flow of a Hamiltonian vector field is a Poisson map and Hamilton’s equations in Poisson bracket formulation are valid. For later applications we shall need the notion of the product of Banach Poisson manifolds. The definition we shall give is the one used in finite dimensions (see, e.g. Weinstein [W1] or Vaisman [V]). However, the proof of the theorem characterizing the product needs some care due to the infinite dimensionality of the manifolds and the additional condition (iii) imposed in Definition 2.1. For this reason we shall sketch it below. Theorem 2.2. Given the Banach Poisson manifolds (P1 , { , }1 ) and (P2 , { , }2 ) there is a unique Banach Poisson structure { , }12 on the product manifold P1 × P2 such that: (i) the canonical projections π1 : P1 × P2 → P1 and π2 : P1 × P2 → P2 are Poisson maps; (ii) π1∗ (C ∞ (P1 )) and π2∗ (C ∞ (P2 )) are Poisson commuting subalgebras of C ∞ (P1 × P2 ). This unique Poisson structure on P1 × P2 is called the product Poisson structure and its bracket is given by the formula {f, g}12 (p1 , p2 ) = {fp2 , gp2 }1 (p1 ) + {fp1 , gp1 }2 (p2 ),

(2.3)

where fp1 , gp1 ∈ C ∞ (P2 ) and fp2 , gp2 ∈ C ∞ (P1 ) are the partial functions given by fp1 (p2 ) = fp2 (p1 ) = f (p1 , p2 ) and similarly for g. Proof. Recall that if f ∈ C ∞ (P1 × P2 ) then the partial exterior derivative d1 f (p1 , p2 ) relative to P1 is defined by d1 f (p1 , p2 ) := dfp2 (p1 ) = (π1∗ dfp2 )(p1 , p2 ) = d(π1∗ fp2 ) (p1 , p2 ) and similarly d2 f (p1 , p2 ) = d(π2∗ fp1 )(p1 , p2 ). Therefore, df (p1 , p2 ) = d1 f (p1 , p2 ) + d2 f (p1 , p2 ) = d(π1∗ fp2 )(p1 , p2 ) + d(π2∗ fp1 )(p1 , p2 ). Thus the functions f and π1∗ fp2 + π2∗ fp1 have the same derivatives at the point (p1 , p2 ) ∈ P1 × P2 . Similarly, g and π1∗ gp2 +π2∗ gp1 have the same derivatives at the point (p1 , p2 ) ∈ P1 ×P2 . Assume that there is a Poisson bracket { , }12 on P1 × P2 satisfying the conditions in the theorem. Since any Poisson bracket depends only on the first derivatives of the functions we necessarily have {f, g}12 (p1 , p2 ) = {π1∗ fp2 + π2∗ fp1 , π1∗ gp2 + π2∗ gp1 }12 (p1 , p2 ) = {π1∗ fp2 , π1∗ gp2 }12 (p1 , p2 ) + {π1∗ fp2 , π2∗ gp1 }12 (p1 , p2 ) +{π2∗ fp1 , π1∗ gp2 }12 (p1 , p2 ) + {π2∗ fp1 , π2∗ gp1 }12 (p1 , p2 ) = (π1∗ {fp2 , gp2 }1 )(p1 , p2 ) + (π2∗ {fp1 , gp1 }2 )(p1 , p2 ) = {fp2 , gp2 }1 (p1 ) + {fp1 , gp1 }2 (p2 ), where condition (ii) and (i) were used in the third equality. This shows that the Poisson bracket, if it exists, is unique and is given by (2.3). Now define { , }12 by (2.3). It remains to show that the axioms in Definition 2.1 hold. It is obvious that this operation satisfies the Leibniz identity, is bilinear, and skew

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symmetric. By Definition 2.1 (iii), one can use Hamiltonian vector fields to express {{f, g}, h}. A direct computation gives {{f, g}, h}12 (p1 , p2 ) = {{fp2 , gp2 }1 , hp2 }1 (p1 ) + {{fp1 , gp1 }2 , hp1 }2 (p2 ) +d1 d2 f (p1 , p2 ) Xh1p (p1 ), Xg2p (p2 ) 1 2 1 2 +d1 d2 f (p1 , p2 ) Xgp (p1 ), Xhp (p2 ) 2 1 1 2 −d1 d2 g(p1 , p2 ) Xhp (p1 ), Xfp (p2 ) 2 1 1 2 −d1 d2 g(p1 , p2 ) Xfp (p1 ), Xhp (p2 ) , 2

1

where d1 d2 f denotes the second mixed partial derivative of f and where Xf1p is the 2 Hamiltonian vector field on P1 corresponding to the function fp2 ∈ C ∞ (P1 ) and similarly for the other ones. Adding the other two terms obtained by circular permutation gives zero since the first two terms summed with their analogues vanish by the Jacobi identity on P1 and P2 respectively and the other terms cancel. Since Hamiltonian vector fields on P1 and P2 exist by Definition 2.1, formula (2.3) shows that the Hamiltonian vector field on P1 × P2 exists and is given by (2.4) Xh12 (p1 , p2 ) = Xh1p (p1 ), Xh2p (p2 ) ∈ Tp1 P1 × Tp2 P2 , 2

1

where condition (iii) in Definition 2.1 was used on P1 and P2 ; we have identified here T(p1 ,p2 ) (P1 × P2 ) with Tp1 P1 × Tp2 P2 . Thus all conditions in Definition 2.1 hold which proves that P1 × P2 is a Banach Poisson manifold.

We remark that (2.3) implies that the product is functorial, that is, if ϕ1 : P1 → P1 and ϕ2 : P2 → P2 are Poisson maps then their product ϕ1 × ϕ2 : P1 × P2 → P1 × P2 is also a Poisson map. Returning to Definition 2.1, it should be noted that the condition (T ∗ P ) ⊂ T P is automatically satisfied in certain cases: • if P is a smooth manifold modeled on a reflexive Banach space, that is b∗∗ = b, or • P is a strong symplectic manifold with symplectic form ω. The first condition holds if P is a Hilbert (and, in particular, a finite dimensional) manifold. Any strong symplectic manifold (P , ω) is a Poisson manifold in the sense of Definition 2.1. Recall that strong means that for each p ∈ P the map vp ∈ Tp P → ω(p)(vp , ·) ∈ Tp∗ P

(2.5)

is a bijective continuous linear map. Therefore, given a smooth function f : P → R there exists a vector field Xf such that df = ω(Xf , ·). The Poisson bracket is defined by {f, g} = ω(Xf , Xg ) = df, Xg , thus df = Xf , so (T ∗ P ) ⊂ T P . On the other hand, a weak symplectic manifold is not a Poisson manifold in the sense of Definition 2.1. Recall that weak means that the map defined by (2.5) is an injective continuous linear map that is, in general, not surjective. Therefore, one cannot construct the map that associates to every differential df of a smooth function f : P → R the Hamiltonian vector field Xf . Since the definition of the Poisson bracket should be {f, g} = ω(Xf , Xg ), one cannot define this operation on functions and hence weak

Banach Lie-Poisson Spaces and Reduction

7

symplectic manifold structures do not define, in general, Poisson manifold structures in the sense of Definition 2.1. There are various ways to deal with this problem. One of them is to restrict the space of functions on which one is working, as is often done in field theory. Another is to deal with densely defined vector fields and invoke the theory of (nonlinear) semigroups; see Chernoff and Marsden [C-M] for this approach. A simple example illustrating the importance of the underlying topology is given by the canonical symplectic structure on b × b∗ , where b is a Banach space. This canonical symplectic structure is in general weak; if b is reflexive then it is strong. In this paper we shall not address these important questions regarding weak symplectic manifolds and their relation to Poisson structures and we shall exclusively consider Banach Poisson manifolds as given by Definition 2.1. Thus, in some sense, the Poisson manifolds considered in this paper are generalizations of strong symplectic manifolds. However, in §7 and §8 we shall give examples illustrating the need for a weakening of Definition 2.1. We shall need in the sequel various notions of Poisson structures defined on almost complex and complex manifolds. We briefly summarize the various possibilities below. Assume that the real Banach manifold P underlying the Poisson structure given by the tensor field has also the structure of an almost complex manifold, that is, there is a smooth vector bundle map I : T P → T P covering the identity which satisfies I 2 = −id. The question then arises: what does it mean for the Poisson and almost complex structures to be compatible? The Poisson structure is said to be compatible with the almost complex structure I if the following diagram commutes:

T ∗P

-

TP

6 I∗

I

T ∗P

? -

TP

that is, I ◦ + ◦ I ∗ = 0.

(2.6)

= (2,0) + (1,1) + (0,2)

(2.7)

The decomposition

induced by the almost complex structure I and the reality of , implies that the compatibility condition (2.6) is equivalent to (1,1) = 0

and

(2,0) = (0,2) .

(2.8)

In view of (2.8), [, ]S = 0 is equivalent to [(2,0) , (2,0) ]S = 0

and

[(2,0) , (2,0) ]S = 0.

(2.9)

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If (2.6) holds, the triple (P , {·, ·}, I ) is called an almost complex Banach Poisson manifold. If I is given by a complex analytic structure PC on P it will be called a complex Banach Poisson manifold. For finite dimensional complex manifolds these structures were introduced and studied by Lichnerowicz [Li]. Denote by O(k,0) (PC ) and O(k,0) (PC ) the space of holomorphic k-forms and k-vector fields respectively. If O(1,0) (PC ) ⊂ O(1,0) (PC ),

(2.10)

that is, the Hamiltonian vector field Xf is holomorphic if f is a holomorphic function, then, in addition to (2.8) and (2.9), one has (2,0) ∈ O(2,0) (PC ). As expected, the compatibility condition (2.10) is stronger than (2.6). Note that (2.10) implies the second condition in (2.9). Thus the compatibility condition (2.10) induces on the underlying complex Banach manifold PC a holomorphic Poisson tensor C := (2,0) . A pair (PC , C ) consisting of an analytic complex manifold PC and a holomorphic skew symmetric contravariant two-tensor field C such that [C , C ]S = 0 and (2.10) holds will be called a holomorphic Banach Poisson manifold. Consider now a holomorphic Poisson manifold (P , ). Denote by PR the underlying real Banach manifold and define the real two-vector field R := Re . It is easy to see that (PR , R ) is a real Poisson manifold compatible with the complex Banach manifold structure of P and (R )C = . Summarizing, we have shown that there are two procedures that are inverses of each other: a holomorphic Poisson manifold corresponds in a bijective manner to a real Poisson manifold whose Poisson tensor is compatible with the underlying complex manifold structure. One can call these constructions the complexification and realification of Poisson structures on complex manifolds.

3. Classical Reduction We shall review in this section the theory of classical Poisson reduction for Banach Poisson manifolds. Let (P , {·, ·}P ) be a real Banach Poisson manifold (in the sense of Definition 2.1), i : N → P be a (locally closed) submanifold, and E ⊂ (T P )|N be a subbundle of the tangent bundle of P restricted to N . For simplicity we make the following topological regularity assumption throughout this section: E ∩ T N is the tangent bundle to a foliation F whose leaves are the fibers of a submersion π : N → M := N/F, that is, one assumes that the quotient topological space N/F admits the quotient manifold structure. The subbundle E is said to be compatible with the Poisson structure provided the following condition holds: if U ⊂ P is any open subset and f, g ∈ C ∞ (U ) are two arbitrary functions whose differentials df and dg vanish on E, then d{f, g}P also vanishes on E. The triple (P , N, E) is said to be reducible, if E is compatible with the Poisson structure on P and the manifold M := N/F carries a Poisson bracket {·, ·}M (in the sense of Definition 2.1) such that for any smooth local functions f¯, g¯ on M and any smooth local extensions f, g of f¯ ◦ π , g¯ ◦ π respectively, satisfying df |E = 0, dg|E = 0, the following relation on the common domain of definition of f and g holds: ¯ M ◦ π. {f, g}P ◦ i = {f¯, g}

(3.1)

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9

If (P , N, E) is a reducible triple then (M = N/F, {·, ·}M ) is called the reduced manifold of P via (N, E). Note that (3.1) guarantees that if the reduced Poisson bracket {·, ·}M on M exists, it is necessarily unique. Given a subbundle E ⊂ T P , its annihilator is defined as the subbundle of T ∗ P given by E ◦ := {α ∈ T ∗ P | α, v = 0 for all v ∈ E}. The following statement generalizing the finite dimensional Poisson reduction theorem of Marsden and Ratiu [M-R1] is central for our purposes. The proof in infinite dimensions is a modification of the original one (see the above mentioned paper or Vaisman [V], §7.2, for the finite dimensional proof). Theorem 3.1. Let P , N , E be as above and assume that E is compatible with the Poisson structure on P . The triple (P , N, E) is reducible if and only if (En◦ ) ⊂ Tn N + En for every n ∈ N . Proof. Assume that (P , N, E) is reducible. Thus M := N/F is a Banach Poisson manifold and (3.1) holds. In addition, recall that N is a (locally closed) submanifold of P and that E ∩ T N is the tangent bundle of a foliation on N . For n ∈ N , choose a chart domain U of n in P with the submanifold property relative to N and such that U ∩ N is foliated. Given αn ∈ En◦ , find a smooth function f on U (shrunk if necessary), such that df (n) = αn and df, E = 0. This is possible since E is a subbundle of T P |N and E ∩ T N is the tangent bundle to a foliation on N . Let f¯ be the smooth function on π(U ∩ N) ⊂ M induced by f , that is, f |N = f¯ ◦ π . Therefore, f : U → R is a local extension of f¯ ◦ π. Next, take an arbitrary βn ∈ (En +Tn N )◦ = En◦ ∩(Tn N )◦ and find a smooth function g on U such that N ∩ U = g −1 (0), dg, E = 0, and dg(n) = βn . Again, the existence of g is insured by the hypothesis that E is a subbundle of T P |N and that E ∩ T N is the tangent bundle to a foliation on N. Thus g : U → R is a local extension of 0 ◦ π , where 0 is the identically zero function on M. Then we have by (3.1), βn , (αn ) = dg(n), Xf (n) = {g, f }P (n) = {0, f }M (π(n)) = 0. This shows that (αn ) ∈ (En + Tn N )◦◦ = En + Tn N , that is, (En◦ ) ⊂ Tn N + En for every n ∈ N. Conversely, assume that (En◦ ) ⊂ Tn N + En for every n ∈ N . For f¯, g¯ locally defined smooth functions on M we need to define their Poisson bracket, show that (3.1) holds, and that all conditions in Definition 2.1 are satisfied. Let f, g be local extensions of f¯ ◦ π and g¯ ◦ π respectively, such that df and dg vanish on E. Since E is compatible with the Poisson bracket on P , d{f, g}P also vanishes on E and thus {f, g}P is constant on the leaves of F thereby inducing a smooth locally defined function on M. We take this function to be the definition of {f¯, g} ¯ M . If we show that this function is well defined, that is, is independent on the extensions chosen, then the axioms of a Poisson bracket (that is, conditions (i) and (ii) in Definition 2.1) are trivially verified and, by construction, (3.1) holds. Since the Poisson bracket is skew symmetric it suffices to show the independence of the extension only for the function f . So let f be another local extension of f¯ ◦ π such that df , E = 0. On the common domain of definition of f and f , we have hence (f − f )|N = 0; in particular, d(f − f ) vanishes on T N . However, since both df and df vanish on E, it follows that d(f − f ) vanishes on E + T N . Let n ∈ N

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be an arbitrary point in the common domain of definition of f and f . By continuity, d(f − f )(n) vanishes on En + Tn N . Since Xg (n) ∈ (En◦ ), using the working hypothesis (En◦ ) ⊂ Tn N + En , we conclude {f − f , g}P (n) = d(f − f )(n), Xg (n) = 0, that is, {f, g}P (n) = {f , g}P (n). ¯ ∗ M) ⊂ T M, where ¯ : It remains to verify condition (iii) of Definition 2.1, that is, (T ∗ ∗∗ T M → T M is the vector bundle map covering the identity defined by ¯ m (d f¯(m)) := ¯ {·, f¯}M (m) for any smooth locally defined function f on M. The idea of the proof below ¯ ¯ is to use (3.1) to show that m (d f (m)) = Tn π Xf (n) ∈ Tm M for every m ∈ M and every locally defined function f¯ around m. To do this, let f¯, g¯ : W → R be two arbitrary smooth functions, where W is a chart domain on M containing the point m. We shall construct now local extensions of f¯ ◦ π and g¯ ◦π adapted to our needs. Since we have already shown that the definition of {·, ·}M is independent on the extensions, we can work only with these extensions and conclude the desired result. Since E ∩ T N is the tangent bundle to a foliation on N , if n ∈ N is such that π(n) = m, there is a foliated chart on N around n whose domain is of the form W × W (after an eventual shrinking of W ), that is, the leaves of the foliation are given by {w} × W for all w ∈ W . Since N is a submanifold of P and since E is defined only along N, there is a chart on P whose domain is of the form W ×W ×V (after shrinking, if necessary, both W and W ). Define the local extension f : W ×W ×V → R of f¯ ◦π by f (w, w , v ) := f¯(w). By condition (iii) of Definition 2.1, n (df )(n) is a vector of the form Xf (n) ∈ Tn P . Let us show that Xf (n) is tangent to N . This is equivalent to proving that for any linear continuous functional βn on the ambient Banach space containing V , we have βn , Xf (n) = 0. However, βn = dk(n), for some smooth function k : W × W × V → R that does not depend on the variables from W and W . But then k is a local extension of 0 ◦ π and, using (3.1), we get βn , Xf (n) = dk(n), Xf (n) = {k, f }P (n) = {0, f }M (π(n)) = 0, which proves the claim. Construct in the same fashion a local extension of g◦π ¯ to the same open neighborhood of n in P . Since dg(n) ◦ Tn i = d g(m) ¯ ◦ Tn π, m = π(n), we have by (3.1), ¯ = {g, ¯ f¯}M (m) = {g, f }P (n) = n (df (n))(dg(n)) ¯ m (d f¯(m))(d g(m)) ¯ Tn π Xf (n) . = dg(n), Xf (n) = d g(m), Since g¯ is an arbitrary smooth function defined on a neighborhood of m, the Hahn-Banach Theorem and the inclusion of the Banach space into its bidual imply that ¯ m (d f¯(m)) = ¯ ∗ M) ⊂ T M. Tn π Xf (n) ∈ Tm M for every m ∈ M , that is (T

The behavior of Poisson maps and Hamiltonian dynamics under reduction is given by the following two theorems whose proofs are identical to the ones in finite dimensions (Marsden and Ratiu [M-R1] or Vaisman [V], §7.4). Theorem 3.2. Let (P1 , N1 , E1 ) and (P2 , N2 , E2 ) be Poisson reducible triples and assume that ϕ : P1 → P2 is a Poisson map satisfying ϕ(N1 ) ⊂ N2 and T ϕ(E1 ) ⊂ E2 . Let Fi be the regular foliation on Ni defined by the subbundle Ei and denote by πi : Ni → Mi := Ni /Fi , i = 1, 2, the reduced Poisson manifolds. Then there is a unique induced Poisson map ϕ : M1 → M2 , called the reduction of ϕ, such that π2 ◦ ϕ = ϕ ◦ π1 .

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11

Proof. The hypotheses imply that T ϕ(E1 ∩ T N1 ) ⊂ E2 ∩ T N2 and hence ϕ maps the leaves of the foliation F1 to those of F2 . Therefore ϕ is a projectable map, that is, there exists a smooth map ϕ : M1 → M2 such that π2 ◦ ϕ = ϕ ◦ π1 . It remains to be shown that ϕ is a Poisson map. Let f¯ and g¯ be two smooth local functions on M2 and let f and g be the local extensions of f¯ ◦ π2 and g¯ ◦ π2 respectively, such that df |E2 = dg|E2 = 0. Since T ϕ(E1 ) ⊂ E2 it follows that d(f ◦ ϕ)|E1 = d(g ◦ ϕ)|E1 = 0. Hence f ◦ ϕ is a smooth local extension of f¯ ◦ π2 ◦ ϕ = f¯ ◦ ϕ¯ ◦ π1 . Similarly, g ◦ ϕ is a smooth local extension of g¯ ◦ π2 ◦ ϕ = g¯ ◦ ϕ¯ ◦ π1 . Definition (3.1) gives then {f¯ ◦ ϕ, ¯ g¯ ◦ ϕ} ¯ M1 ◦ π1 = {f ◦ ϕ, g ◦ ϕ}P1 ◦ i1 = {f, g}P2 ◦ ϕ ◦ i1 = {f, g}P2 ◦ i2 ◦ ϕ = {f¯, g} ¯ M2 ◦ π2 ◦ ϕ = {f¯, g} ¯ M2 ◦ ϕ¯ ◦ π1 , which implies that ϕ¯ is a Poisson map by surjectivity of π1 .

Theorem 3.3. Let (P , N, E) be a Poisson reducible triple and π : N → M be the corresponding Poisson reduced manifold. Assume that h ∈ C ∞ (P ) and the associated flow ϕt of the Hamiltonian vector field Xh satisfies the conditions (i) dh|E = 0, (ii) ϕt (N ) ⊂ N , (iii) T ϕt (E) ⊂ E for all t for which the flow ϕt is defined. Then the reduction ϕ t is the flow of the Hamiltonian vector field on M given by the function h¯ uniquely determined by the condition h¯ ◦ π = h|N . The Hamiltonian vector fields Xh on N ⊂ P and Xh¯ on M are π -related. Proof. The hypotheses guarantee by Theorem 3.2 that the flow ϕt of Xh reduces to a smooth flow ϕ t on the reduced manifold M. The hypothesis on h insures the existence of the smooth function h¯ on M. Let us prove that Xh and Xh¯ are π -related. If this is done, their flows are necessarily π-related and hence, by surjectivity of π , it follows that the flow of Xh¯ is ϕ t . To prove that Xh and Xh¯ are π-related, let f¯ be a smooth locally defined function in a neighborhood of m ∈ M and let f be a smooth local extension of f¯ ◦ π satisfying df |E = 0. Then, since Xh is tangent to N , using the defining identity of the reduced bracket (3.1), for any n ∈ N satisfying π(n) = m, we get ¯ M (m) = {f, h}P (n) = df (n), Xh (n) d f¯(m), Xh¯ (m) = {f¯, h} = d(f¯ ◦ π )(n), Xh (n) = d f¯(m), Tn π (Xh (n)), which proves that Xh¯ ◦ π = T π ◦ Xh .

In this paper we shall not investigate the consistency of Poisson reduction with other structures such as almost complex, complex, and holomorphic structures. For finite dimensional Poisson manifolds the consistency of Poisson reduction with the complex structure was presented in Nunes da Costa [N].

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4. Banach Lie-Poisson Spaces It is well known that the dual of any Lie algebra admits a linear Poisson structure, called the Lie-Poisson structure. In this section we shall extend the definition of this structure to the infinite dimensional case in agreement with Definition 2.1. We shall call such spaces Banach Lie-Poisson spaces and shall investigate their properties. Recall that a Banach Lie algebra (g, [·, ·]) is a Banach space that is also a Lie algebra such that the Lie bracket is a bilinear continuous map g × g → g. Thus the adjoint and coadjoint maps adx : g → g, adx y := [x, y], and ad∗x : g∗ → g∗ are also continuous for each x ∈ g. Definition 4.1. A Banach Lie-Poisson space (b, {·, ·}) is a real or holomorphic Poisson manifold such that b is a Banach space and the dual b∗ ⊂ C ∞ (b) is a Banach Lie algebra under the Poisson bracket operation. Throughout this section we shall treat the real and the holomorphic cases simultaneously. Denote by [·, ·] the restriction of the Poisson bracket {·, ·} from C ∞ (b) to the Lie subalgebra b∗ . For any x, y ∈ b∗ and b ∈ b we have y, ad∗x b = [x, y], b = {x, y}(b) = −{y, x}(b) = −Xx [y](b) = −Dy(b), Xx (b) = −y, Xx (b), where we have used the linearity of y ∈ C ∞ (b) to conclude that the Fr´echet derivative Dy(b) = y. Thus we obtain the following identity in the bidual b∗∗ : Xx (b) = − ad∗x b

for

x ∈ b∗ ,

b ∈ b.

(4.1)

Theorem 4.2. The Banach space b is a Banach Lie-Poisson space (b, {·, ·}) if and only if its dual b∗ is a Banach Lie algebra (b∗ , [·, ·]) satisfying ad∗x b ⊂ b ⊂ b∗∗ for all x ∈ b∗ . Moreover, the Poisson bracket of f, g ∈ C ∞ (b) is given by {f, g}(b) = [Df (b), Dg(b)], b,

(4.2)

where b ∈ b and D denotes the Fr´echet derivative. If h is a smooth function on b, the associated Hamiltonian vector field is given by Xh (b) = − ad∗Dh(b) b.

(4.3)

Proof. Assume that b is a Banach Lie-Poisson space relative to the bracket {·, ·}. By Definition 4.1, its dual b∗ is a Banach Lie algebra relative to the bracket [·, ·] := {·, ·}|b∗ . However, b is also a Poisson manifold and thus, by definition, Xx (b) ∈ b for all x ∈ b∗ and all b ∈ b. Formula (4.1) implies then that ad∗x (b) ∈ b for all x ∈ b∗ and all b ∈ b which is the required condition. Conversely, assume that (b∗ , [·, ·]) is a Banach Lie algebra satisfying ad∗x b ⊂ b ⊂ b∗∗ for all x ∈ b∗ . Define the bracket {f, g} of f, g ∈ C ∞ (b) by (4.2). All properties of the Poisson bracket are trivially satisfied by (4.2) except for the Jacobi identity. For this, we note that from ad∗x b ⊂ b, x ∈ b∗ , one has D{f, g}(b) = [Df (b), Dg(b)], · − D 2 f (b) ad∗Dg(b) b, · +D 2 g(b) ad∗Df (b) b, · (4.4)

Banach Lie-Poisson Spaces and Reduction

13

for f, g ∈ C ∞ (b). Using (4.4) we obtain {{f, g}, h}(b) = [D{f, g}(b), Dh(b)] , b

= [[Df (b), Dg(b)] , Dh(b)] , b + D 2 f (b) ad∗Dg(b) b, ad∗Dh(b) b − D 2 g(b) ad∗Df (b) b, ad∗Dh(b) b .

Taking the two other terms obtained by circular permutation of f , g, and h, using the Jacobi identity for the Lie bracket in the sum of the first three terms and the symmetry of the second derivative in the sum of the remaining terms, proves that (4.2) satisfies the Jacobi identity. Since Df (b), Xh (b) = {f, h}(b) = [Df (b), Dh(b)], b = − Df (b), ad∗Dh(b) b for every f ∈ C ∞ (b) and ad∗x b ⊂ b for every x ∈ b∗ , it follows that the Hamiltonian vector field Xh is given by (4.3).

Example 4.3. Let b be a reflexive Banach Lie algebra, that is, b∗∗ = b. Then its dual b∗ is a Banach Lie-Poisson space. To see this, note that b∗∗ = b is a Banach Lie algebra and that ad∗x (b∗ ) ⊂ b∗ for all x ∈ b, so Theorem 4.2 applies. Example 4.4. Since every finite dimensional Lie algebra is reflexive Example 4.3 yields the following classical result: the dual of any finite dimensional Lie algebra is a LiePoisson space. Definition 4.5. A morphism between two Banach Lie-Poisson spaces b1 and b2 is a continuous linear map φ : b1 → b2 that preserves the Poisson bracket structure, that is, {f ◦ φ, g ◦ φ}1 = {f, g}2 ◦ φ for any f, g ∈ C ∞ (b2 ). Such a map φ is also called a linear Poisson map. We consider now the category B whose objects are the Banach Lie-Poisson spaces and whose morphisms are the linear Poisson maps. Let L denote the category of Banach Lie algebras and continuous Lie algebra homomorphisms. Denote by L0 the following subcategory of L. An object of L0 is a Banach Lie algebra g admitting a predual g∗ , that is, (g∗ )∗ = g, and satisfying ad∗g g∗ ⊂ g∗ , where ad∗ is the coadjoint representation of g on g∗ ; note that g∗ ⊂ g∗ . A morphism in the category L0 is a Banach Lie algebra homomorphism ψ : g1 → g2 such that the dual map ψ ∗ : g∗2 → g∗1 preserves at least one choice of the corresponding preduals, that is, ψ ∗ : (g2 )∗ → (g1 )∗ . Let L0u be the subcategory of L0 whose objects have a unique predual. Theorem 4.6. There is a contravariant functor F : B → L0 defined by F(b) = b∗ and F(φ) = φ ∗ . On the subcategory F−1 (L0u ) this functor is invertible. The inverse of F is given by F−1 (g) = g∗ and F−1 (ψ) = ψ ∗ |(g2 )∗ , where ψ : g1 → g2 . Proof. If b is a Banach Lie-Poisson space, then F(b) = b∗ is a Banach Lie algebra that admits b as a predual and, according to Theorem 4.2, ad∗b∗ b ⊂ b. Thus F(b) is indeed an object in the category L0 . If φ : b1 → b2 is a linear Poisson map let us show that F(φ) = φ ∗ : b∗2 → b∗1 is a Banach Lie algebra homomorphism. First, φ ∗ is a linear

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continuous map between Banach spaces. Second, since the Lie bracket on b∗2 is defined by [x, y]2 = {x, y}2 and similarly for b1 , we get φ ∗ [x, y]2 = φ ∗ {x, y}2 = {φ ∗ x, φ ∗ y}1 = [φ ∗ x, φ ∗ y]1 , which shows that φ ∗ a homomorphism of Banach Lie algebras. Finally, the dual of ∗∗ ∗∗ F(φ), that is, φ ∗∗ : b∗∗ 1 → b2 satisfies φ |b1 = φ. Thus F(φ) is indeed a morphism in the category L0 . Since duality reverses the direction of the arrows and the order of the composition, F is a contravariant functor. Conversely, consider the functor F−1 : L0u → B and let g be an object of L0u . By Theorem 4.2, F−1 (g) = g∗ is a Banach Lie-Poisson space, that is, an object of B. If ψ : g1 → g2 is a morphism in the category L0u , then let us show that F−1 (ψ) = ψ ∗ |(g2 )∗ is a linear Poisson map. Let f, g be smooth functions on (g1 )∗ . From (4.2) and using the fact that ψ is morphism of Banach Lie algebras and that ψ ∗ |(g2 )∗ is a linear map, we get for every b ∈ (g2 )∗ , {f ◦ ψ ∗ |(g2 )∗ , g ◦ ψ ∗ |(g2 )∗ }2 (b) = [D(f ◦ ψ ∗ |(g2 )∗ )(b), D(g ◦ ψ ∗ |(g2 )∗ )(b)]2 , b = [Df (ψ ∗ (b)) ◦ ψ ∗ |(g2 )∗ , Dg(ψ ∗ (b)) ◦ ψ ∗ |(g2 )∗ ]2 , b = [ψ Df (ψ ∗ (b)) , ψ Dg(ψ ∗ (b)) ]2 , b = ψ [Df (ψ ∗ (b)), Dg(ψ ∗ (b))]1 , b = [Df (ψ ∗ (b)), Dg(ψ ∗ (b))]1 , ψ ∗ (b) = {f, g}1 (ψ ∗ |(g2 )∗ (b)), which shows that ψ ∗ |(g2 )∗ : (g2 )∗ → (g1 )∗ is a morphism of Banach Lie-Poisson spaces. The functor F−1 is contravariant since its action on morphisms is given by duality. Finally, it is clear the functors F : F−1 (L0u ) → L0u and F−1 : L0u → F−1 (L0u ) are inverses of each other.

We turn now to the study of the internal structure of morphisms of Banach Lie-Poisson spaces. Proposition 4.7. Let φ : b1 → b2 be a linear Poisson map between Banach Lie-Poisson spaces and assume that im φ is closed in b2 . Then the Banach space b1 / ker φ is predual to b∗2 / ker φ ∗ , that is, (b1 / ker φ)∗ ∼ = b∗2 / ker φ ∗ . In addition, b∗2 / ker φ ∗ is a Banach Lie algebra satisfying the condition ad∗[x] (b1 / ker φ) ⊂ b1 / ker φ for all [x] ∈ b∗2 / ker φ ∗ and b1 / ker φ is a Banach Lie-Poisson space. Moreover, the following properties hold: (i) the quotient map π : b1 → b1 / ker φ is a surjective linear Poisson map; (ii) the map ι : b1 / ker φ → b2 defined by ι([b]) := φ(b), where b ∈ b1 and [b] ∈ b1 / ker φ is an injective linear Poisson map; (iii) the decomposition φ = ι ◦ π into a surjective and an injective linear Poisson map is valid. Proof. We define the pairing ·, · : b∗2 / ker φ ∗ × b1 / ker φ → C (or R) by [x], [b] := x, φ(b)2 = φ ∗ (x), b1 ,

(4.5)

where [x] ∈ b∗2 / ker φ ∗ , [b] ∈ b1 / ker φ, and ·, ·i : b∗i × bi → C (or R), i = 1, 2, are the pairings between the given Banach Lie-Poisson spaces and their duals. This pairing

Banach Lie-Poisson Spaces and Reduction

15

is correctly defined since it does not depend on the choice of the representatives x ∈ b2 and b ∈ b1 . One has |[x], [b]| ≤ φ [x] [b] and if [x], [b] = 0 for each x ∈ b2 (b ∈ b1 ) then [b] = [0] ([x] = [0]). Thus (4.5) defines a continuous weakly non-degenerate pairing and therefore the map [x] ∈ b∗2 / ker φ ∗ → [x], [·] = φ ∗ (x), ·1 = x, φ(·)2 ∈ (b1 / ker φ)∗ is a continuous linear injective map of Banach spaces. To show that this map is surjective, we need to find for a given α ∈ (b1 / ker φ)∗ an [x] ∈ b∗2 / ker φ ∗ such that [x], [b] = φ ∗ (x), b1 = x, φ(b)1 = α([b]) for all b ∈ b1 . Since the range im φ is closed in b2 , it is a Banach subspace and hence the map : [b] ∈ b1 / ker φ → φ(b) ∈ im φ is a Banach space isomorphism. Thus α ◦ −1 ∈ (im φ)∗ . Let x ∈ b∗2 be an extension of α ◦ −1 to b2 . Then we have for any b ∈ b1 , [x], [b] = x, φ(b)2 = α ◦ −1 , φ(b)2 = α([b]). Thus the Banach space b∗2 / ker φ ∗ is isomorphic to the dual of b1 / ker φ. The space b∗2 / ker φ ∗ is a Banach Lie algebra because ker φ ∗ is an ideal in the Banach Lie algebra b∗2 (since φ ∗ : b∗2 → b∗1 is a morphism of Banach Lie algebras). Finally, since φ : b1 → b2 is a linear Poisson map, we have ∗ ∗ (4.6) ad2 x φ(b) = φ ad1 x◦φ b , for any x ∈ b∗2 , b ∈ b1 , and where adi denotes the adjoint operator in the Banach Lie ∗ algebra b∗i , i = 1, 2. Here we have used the fact that ad1 x◦φ b1 ⊂ b1 for any x ∈ b∗2 . ∗ From (4.6) and ad2 x b2 ⊂ b2 for all x ∈ b∗2 we conclude that for all b ∈ b1 we have [y], ad∗[x] [b] = [[x], [y]] , [b] = [[x, y]] , [b] = [x, y] , φ(b)2 ∗

∗ ∗ = y, ad2 x φ(b) = y, φ ad1 x◦φ b = [y], ad1 x◦φ b 2

for each y ∈

b∗2 .

2

This implies that

∗ ad∗[x] [b] = ad1 x◦φ b ∈ b1 / ker φ

for all [x] ∈ b∗2 / ker φ ∗ , [b] ∈ b1 / ker φ, and thus ad∗[x] (b1 / ker φ) ⊂ b1 / ker φ for all [x] ∈ b∗2 / ker φ ∗ . Thus b∗2 / ker φ ∗ is an object in the category L0 . Theorem 4.2 (or Theorem 4.6) guarantees then that the quotient Banach space b1 / ker φ is a Banach Lie-Poisson space. Endow the Banach subspace im φ with the Banach Lie-Poisson structure making the Banach space isomorphism : b1 / ker φ → im φ into a linear Poisson isomorphism. Thus ∗ : (im φ)∗ → (b1 / ker φ)∗ is an isomorphism in the category L0 . Since φ = ◦ π : b1 → im φ is a linear Poisson map by hypothesis, it follows that φ ∗ = π ∗ ◦ ∗ is a morphism in the category L0 which then implies that π ∗ is also a

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morphism in the category L0 . By Theorem 4.6 this is equivalent to the fact that π is a linear Poisson map thereby proving property (i) in the statement of the proposition. Define ι : b1 / ker φ → b2 to be the composition of the inclusion im φ → b2 with the isomorphism : b1 / ker φ → im φ. The definition of is equivalent to the equality φ = ◦ π thought of as a map from b1 to im φ. Composing this identity on the left with the inclusion im φ → b2 yields φ = ι ◦ π which proves property (iii). To prove part (ii), let f, g ∈ C ∞ (b2 ). Then f ◦ ι, g ◦ ι ∈ C ∞ (b1 / ker φ). Since π : b1 → b1 / ker φ is a surjective linear Poisson map and φ = ι ◦ π , the relation {f ◦ ι, g ◦ ι} ◦ π = {f ◦ ι ◦ π, g ◦ ι ◦ π}1 = {f ◦ φ, g ◦ φ}1 = {f, g}2 ◦ φ = {f, g}2 ◦ ι ◦ π implies that {f ◦ ι, g ◦ ι} = {f, g}2 ◦ ι, that is, ι : b1 / ker φ → b2 is an injective linear Poisson map.

Proposition 4.7 reduces the investigation of linear Poisson maps with closed range between Banach Lie-Poisson spaces to the study of surjective and injective linear Poisson maps. Consider therefore the surjective linear continuous map π : b1 → b2 , where b1 is a Banach Lie-Poisson space and b2 is just a Banach space with no additional structure. The dual map π ∗ : b∗2 → b∗1 is therefore an injective continuous linear map of Banach spaces. The space im π ∗ coincides with the Banach subspace of linear continuous functionals on b1 that vanish on ker π, which is closed in b1 . Thus im π ∗ is a closed subspace of b∗1 . Assume next that im π ∗ is also closed under the Lie bracket operation [·, ·]1 of b∗1 . Then π ∗ : b∗2 → im π ∗ is a Banach space isomorphism and, declaring it to be also a Lie algebra morphism, it follows that there is a Banach Lie algebra structure [·, ·]2 on b∗2 and that π ∗ : (b∗2 , [·, ·]2 ) → (b∗1 , [·, ·]1 ) is a Banach Lie algebra morphism. Let x, ˜ y˜ ∈ b∗2 and let π(b) = b˜ ∈ b2 . Then ∗ ˜ y] ˜ 2 , b˜ = [x, ˜ y] ˜ 2 , π(b)2 = π ∗ ([x, ˜ y] ˜ 2 ), b 1 y, ˜ ad2 x˜ b˜ = [x, 2 2 ∗

∗ 1 = π ∗ (x), ˜ π ∗ (y) ˜ 1 , b 1 = π ∗ (y), ˜ ad1 π ∗ (x) b = y, ˜ π ad b ; ∗ ˜ ˜ π (x) 1

∗ ad1 π ∗ (x) ˜ b1

2

b∗2

⊂ b1 for all x˜ ∈ which the last equality is a consequence of the inclusion is insured by the fact that b1 is a Banach Lie-Poisson space. Since this relation holds for any y˜ ∈ b2 , we conclude that ∗ ∗ ad2 x˜ b˜ = π ad1 π ∗ (x) ˜ b ∗

for any x˜ ∈ b∗2 and any b˜ ∈ b2 . This shows that ad2 x˜ b2 ⊂ b2 for any x˜ ∈ b∗2 , and hence, by Theorem 4.2, b2 is a Banach Lie-Poisson space or, equivalently, b∗2 is an object in the category L0 . The map π ∗ is a morphism of Banach Lie algebras. In addition, its dual π ∗∗ : b∗∗ 1 → ∗ ∗∗ b∗∗ 2 has the property π (b1 ) ⊂ b2 . Indeed, for any b1 ∈ b1 and β2 ∈ b2 , the definition of the dual of a linear map gives π ∗∗ (b1 ), β2 2 = b, π ∗ (β2 )1 = π(b1 ), β2 2 , ∗ where ·, ·2 : b∗∗ 2 × b2 → R (or C) is the canonical pairing between a Banach space and its dual and similarly for ·, ·1 : b1 × b∗1 → R (or C) and ·, ·2 : b2 × b∗2 → R (or

Banach Lie-Poisson Spaces and Reduction

17

C). This shows that π ∗∗ (b1 ) = π(b1 ) ∈ b2 . Therefore π ∗ is a morphism in the category L0 and, by Theorem 4.6, π : b1 → b2 is a linear Poisson map. In addition, since π is also surjective, the Banach Lie-Poisson structure on b2 is unique. Therefore, following e.g. Vaisman [V], we shall call this Banach Lie-Poisson structure on b2 coinduced by the surjective mapping π. The above proves the “only if” part of the following proposition; the converse is an easy verification. Proposition 4.8. Let (b1 , {·, ·}) be a Banach Lie-Poisson space and let π : b1 → b2 be a continuous linear surjective map onto the Banach space b2 . Then b2 carries the Banach Lie-Poisson structure coinduced by π if and only if im π ∗ ⊂ b∗1 is closed under the Lie bracket [·, ·]1 of b∗1 . The map π ∗ : b∗2 → b∗1 is a Banach Lie algebra morphism ∗∗ whose dual π ∗∗ : b∗∗ 1 → b2 maps b1 into b2 . Example 4.9. Let (g, [·, ·]) be a complex Banach Lie algebra admitting a predual g∗ satisfying ad∗x g∗ ⊂ g∗ for every x ∈ g. Then, by Theorem 4.2, the predual g∗ admits a holomorphic Banach Lie-Poisson structure, whose holomorphic Poisson tensor is given by (4.2). We shall work with the realification (g∗ R , R ) of (g∗ , ) in the sense of §2. We want to construct a real Banach space gσ∗ with a real Banach Lie-Poisson structure σ such that gσ∗ ⊗ C = g∗ and σ is coinduced from R in the sense of Proposition 4.8. To achieve this, introduce a continuous R-linear map σ : g∗ R → g∗ R satisfying the properties: (i) σ 2 = id; (ii) the dual map σ ∗ : gR → gR defined by σ ∗ z, b = z, σ b

(4.7)

for z ∈ gR , b ∈ g∗ R , and where ·, · is the pairing between the complex Banach spaces g and g∗ , is a homomorphism of the Lie algebra (gR , [·, ·]); (iii) σ ◦ I + I ◦ σ = 0, where I : gR → gR is defined by z, I b := I ∗ z, b := iz, b

(4.8)

for z ∈ gR , b ∈ g∗ R . Consider the projectors R :=

1 (id + σ ) 2

R ∗ :=

1 (id + σ ∗ ) 2

(4.9)

and define gσ∗ := im R, gσ := im R ∗ . Then one has the splittings g∗ R = gσ∗ ⊕ I gσ∗

and gR = gσ ⊕ I gσ

(4.10)

into real Banach subspaces. One can identify canonically the splittings (4.10) with the splittings gσ∗ ⊗R C = gσ∗ ⊗R R ⊕ gσ∗ ⊗R Ri . (4.11) Thus one obtains isomorphisms gσ∗ ⊗R C ∼ = g∗ and gσ ⊗R C ∼ = g of complex Banach spaces. For any x, y ∈ gR one has [R ∗ x, R ∗ y] = R ∗ [x, R ∗ y],

(4.12)

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and thus gσ is a real Banach Lie subalgebra of gR . From Rez, b = R ∗ z, Rb + I ∗ R ∗ I ∗ z, I RI b = R ∗ z, Rb + (1 − R ∗ )z, (1 − R)b

(4.13)

for all z ∈ gR and all b ∈ g∗ R , where for the last equality we used R = 1 + I RI and R ∗ = 1 + I ∗ R ∗ I ∗ , one concludes that the annihilator (gσ∗ )◦ of gσ∗ in gR equals I ∗ g∗ . Therefore gσ∗ is the predual of gσ . Taking into account all of the above facts we conclude from Proposition 4.8 that gσ∗ carries a real Banach Lie-Poisson structure {·, ·}gσ∗ coinduced by R : g∗ R → gσ∗ . According to (4.13), the bracket {·, ·}gσ∗ is given by {f, g}gσ∗ (ρ) = [df (ρ), dg(ρ)], ρ, gσ∗

gσ∗

and the pairing on the right is between where ρ ∈ and real valued functions f, g ∈ C ∞ (gσ∗ ) and any b ∈ g∗ R we have

(4.14) gσ .

In addition, for any

{f ◦ R, g ◦ R}gR (b) = Re[d(f ◦ R)(b), d(g ◦ R)(b)], b = R ∗ [d(f ◦ R)(b), d(g ◦ R)(b)], R(b) +(1 − R ∗ )[d(f ◦ R)(b), d(g ◦ R)(b)], (1 − R)b = R ∗ [R ∗ df (R(b)), R ∗ dg(R(b))], R(b) +(1 − R ∗ )[R ∗ df (R(b)), R ∗ dg(R(b))], (1 − R)b = [df (R(b)), dg(R(b))], R(b) = {f, g}gσ∗ (R(b)), where we have used (4.12). The above computation proves, independently of Proposition 4.8, that R : g∗ R → gσ∗ is a linear Poisson map. Next we investigate the case of injective linear Poisson maps. Proposition 4.10. Let b1 be a Banach space, (b2 , {·, ·}2 ) be a Banach Lie-Poisson space, and ι : b1 → b2 be an injective continuous linear map with closed range. Then b1 carries a unique Banach Lie-Poisson structure such that ι is a linear Poisson map if and only if ker ι∗ is an ideal in the Banach Lie algebra b∗2 . Proof. Assume that ker ι∗ is an ideal in the Banach Lie algebra b∗2 . Denote by [·, ·]2 the Lie bracket of the Banach Lie algebra b∗2 . Since ι : b1 → b2 is an injective linear continuous map, its adjoint ι∗ : b∗2 → b∗1 is a surjective linear continuous map inducing the Banach space isomorphism [ι∗ ] : b∗2 / ker ι∗ −→b ˜ ∗1 . Since ker ι∗ is an ideal in the Banach ∗ ∗ ∗ Lie algebra b2 , it follows that b2 / ker ι is a Banach Lie algebra. The isomorphism [ι∗ ] induces a Banach Lie algebra structure [·, ·]1 on b∗1 . The linear map ι∗ : b∗2 → b∗1 becomes a Banach Lie algebra homomorphism. For each x, y ∈ b∗2 and each b ∈ b1 we have ∗ ∗ y, ι∗∗ ad1 ι∗ (x) b = ι∗ (y), ad1 ι∗ (x) b = [ι∗ (x), ι∗ (y)]1 , b 1 2 1 ∗ ∗ = ι ([x, y]2 ) , b 1 = [x, y]2 , ι(b) = y, ad2 x ι(b) , and hence we get the following identity in b∗∗ 2 : ∗ ∗ ι∗∗ ad1 ι∗ (x) b = ad2 x ι(b) for any x ∈ b∗2 and any b ∈ b1 .

(4.15)

Banach Lie-Poisson Spaces and Reduction

19

Let us now prove that ∗

ad2 b∗ ι(b1 ) ⊂ ι(b1 ), 2

(4.16)

∗

∗ where ad2 denotes the coadjoint action of b∗2 on b∗∗ 2 . We begin by noticing that ker ι = [ι(b1 )]◦ , where [ι(b1 )]◦ is the annihilator of ι(b1 ) in b∗2 . Taking the annihilator of this identity in b2 one obtains

[ker ι∗ ]◦ = [ι(b1 )]◦◦ = ι(b1 ),

(4.17)

where the last equality follows by the closedness of ι(b1 ) in b2 . By the definition of ∗ Banach Lie-Poisson spaces, ad2 b∗ b2 ⊂ b2 . Since ker ι∗ is an ideal in b∗2 it follows that 2

∗

ad2 b∗ [ker ι∗ ]◦ ⊂ [ker ι∗ ]◦ and thus (4.17) implies (4.16). 2

∗

By (4.15) and (4.16), we have ad2 x ι(b) ∈ ι(b1 ) and thus ∗ ι∗∗ ad1 ι∗ (x) b ∈ ι(b1 ) ∗∗ for any x ∈ b∗2 and any b ∈ b1 . The double adjoint ι∗∗ : b∗∗ 1 → b2 is a injective contin∗ ∗ ∗ uous linear map (since ι : b2 → b1 is a surjective continuous linear map) and ι maps 1∗ ∗∗ b1 ⊂ b∗∗ 1 into b2 ⊂ b2 . This shows that ad ι∗ (x) b1 ⊂ b1 . Applying Theorem 4.2 we conclude that b1 is a Banach Lie-Poisson space and that ι : b1 → b2 is an injective linear Poisson map. Uniqueness of the Poisson structure on b1 follows from the injectivity of ι. Conversely, let us assume that b1 is a Banach Lie-Poisson space and that ι : b1 → b2 is a linear Poisson map. Then ι∗ : b∗2 → b∗1 is a homomorphism of Banach Lie algebras and therefore its kernel is an ideal in b∗2 .

Proposition 4.10 allows one to introduce a unique Banach Lie-Poisson structure on b1 relative to which ι is a linear Poisson map. In analogy to the previous case, this Poisson structure on b1 will be said to be the Banach Lie-Poisson structure induced by the mapping ι. Proposition 5.4 in §5 gives an example of an induced Poisson structure. Proposition 4.11. Let b1 be a Banach space, (b2 , {·, ·}2 ) be a Banach Lie-Poisson space, and ι : b1 → b2 be an injective continuous linear map with closed range. Then the equality ∗

ad2 b∗ ι(b1 ) = ι(b1 ) 2

(4.18)

implies that ker ι∗ is an ideal in the Banach Lie algebra b∗2 and thus the map ι : b1 → b2 induces a Banach Lie Poisson structure on b1 . Proof. To show that (4.18) implies that ker ι∗ is an ideal, we prove first the following equality: ∗ ker ι∗ = x ∈ b∗2 | ad2 x ι(b1 ) = 0 . To see this, let x, y ∈ b∗2 and note that ∗ ∗ y, ad2 x ι(b1 ) = − x, ad2 y ι(b1 ) .

20

A. Odzijewicz, T.S. Ratiu ∗

Thus, ad2 x ι(b1 ) = 0 if and only if

∗ x, ad2 y ι(b1 ) = 0

for all

y ∈ b∗2 ,

which, by condition (4.18), is equivalent to 0 = x, ι(b1 ) = ι∗ x, b1 , that is, ι∗ x = 0. Next we prove that

∗

x ∈ b∗2 | ad2 x ι(b1 ) = 0

is an ideal. Indeed if x is in this subspace and y, z ∈ b∗2 are arbitrary, then ∗ ∗ ∗ ∗ ∗ z, ad2 [x,y] ι(b1 ) = z, ad2 y ad2 x ι(b1 ) − z, ad2 x ad2 y ι(b1 ) = 0 ∗

because in the second term ad2 y ι(b1 ) ⊂ ι(b1 ) by condition (4.18) and the element x ∗ satisfies ad2 x ι(b1 ) = 0. These two steps show that ker ι∗ is an ideal in the Banach Lie algebra b∗2 . Therefore, if (4.18) holds, by Proposition 4.10, the space b1 carries a unique Banach Lie-Poisson structure such that ι is a linear Poisson map.

The previous two propositions give an algebraic characterization of linear Poisson maps φ : b1 → b2 between Banach Lie-Poisson spaces analogous to that from linear algebra. We summarize this in the following theorem. Theorem 4.12. The linear continuous map φ : b1 → b2 between the Banach Lie-Poisson spaces b1 and b2 , such that φ(b1 ) is a Banach subspace in b2 , is a linear Poisson map if and only if it has a decomposition φ = ι ◦ π, where (i) π : b1 → b is a linear continuous surjective map of Banach spaces such that im π ∗ ⊂ b∗1 is closed with respect to a Lie bracket of b∗1 ; (ii) ι : b → b2 is a continuous injective linear map of Banach spaces with closed range such that ker ι∗ is an ideal in the Banach Lie algebra b∗2 . We will be interested now in finding the properties of an object in B that correspond to the condition of being an ideal or a subalgebra of the related object in L0 . To do this we shall use Propositions 4.8 and 4.10. Proposition 4.13. Let b ∈ Ob(B) be a Banach Lie-Poisson space and let g ∈ Ob(L0 ) be a Banach Lie algebra such that b∗ = g. Then: (i) There exists a bijective correspondence between the coinduced Banach Lie-Poisson structures from b and the Banach Lie subalgebras of g. If the surjective continuous linear map π : b → c coinduces a Banach Lie-Poisson structure on c, the Banach Lie subalgebra of g given by this correspondence is π ∗ (c∗ ). Conversely, if k ⊂ g is a Banach Lie subalgebra then the Banach Lie-Poisson space given by this correspondence is b/k◦ , where k◦ is the annihilator of k in b, and π : b → b/k◦ is the quotient projection.

Banach Lie-Poisson Spaces and Reduction

21

(ii) There exists a bijective correspondence between the induced Banach Lie-Poisson structures in b (i.e., the Banach Lie-Poisson subspaces of b) and the Banach ideals of g. If the injection ι : c → b with closed range induces a Banach Lie-Poisson structure on c, then the ideal in g given by this correspondence is ker ι∗ . Conversely, if i ⊂ g is a Banach ideal, then the Banach Lie- Poisson subspace of b given by this correspondence is i◦ , where i◦ is the annihilator of i in b and ι : i◦ → b is the inclusion. Proof. (i) If the surjective continuous linear map π : b → c coinduces a Banach Lie-Poisson structure on c, Proposition 4.8 states that π ∗ (c∗ ) is a Banach Lie subalgebra of g. Conversely, if k ⊂ g is a Banach Lie subalgebra then π : b → b/k◦ is a surjective continuous linear map of Banach spaces. Consider the dual map π ∗ : [b/k◦ ]∗ → b∗ . Since b∗ = g and since [b/k◦ ]∗ ∼ = k◦◦ = k, it follows that ∗ im π is a Banach Lie subalgebra of g. Therefore, by Proposition 4.8, there is a unique coinduced Banach Lie-Poisson structure on b/k◦ . (ii) This is a direct consequence of Proposition 4.10.

Consider two Banach Lie-Poisson spaces (b1 , {·, ·}1 ) and (b1 , {·, ·}2 ). According to Theorem 2.2, the product (b1 × b2 , {·, ·}12 ) is a Banach Poisson manifold. The Banach space isomorphism (b1 × b2 )∗ ∼ = b∗1 × b∗2 and formula (2.3) show that (b1 × b2 )∗ is closed under the product Poisson bracket {·, ·}12 which proves that (b1 × b2 , {·, ·}12 ) is also a Banach Lie-Poisson space. As opposed to the general case of Poisson manifolds, the inclusions ik : bk → b1 ×b2 , k = 1, 2, defined by i1 (b1 ) := (b1 , 0) and i2 (b2 ) := (0, b2 ) are Poisson maps. Indeed, by (2.3), we get (i1∗ {f, g}12 )(b1 ) = {f, g}12 (b1 , 0) = {f0 , g0 }1 (b1 ) + {fb1 , gb1 }2 (0) = {i1∗ f, i1∗ g}1 (b1 ), where the term {fb1 , gb1 }2 (0) vanishes because in this case the Poisson bracket is linear. The proof for i2 is similar. Regarding the product construction, the following question arises naturally. When does a Banach Lie-Poisson space (b, {·, ·}) allow a decomposition as a product of two Banach Lie-Poisson spaces (b1 , {·, ·}1 ) and (b2 , {·, ·}2 )? In the category of Banach spaces this means that there is a splitting, i.e., b = b1 ⊕ b2 , for two Banach subspaces b1 and b2 , which is equivalent to b ∼ = b1 × b2 . In view of the previous properties of the product of two Banach Lie-Poisson spaces, this suggests the following definition. Definition 4.14. Let (b, { , }) be a Banach Lie-Poisson space. The splitting b = b1 ⊕ b2 into two Banach subspaces b1 and b2 is called a Poisson splitting if (i) b1 and b2 are Banach Lie Poisson spaces whose brackets shall be denoted by { , }1 and { , }2 respectively; (ii) the projections πk : b → bk and the inclusions ik : bk → b, k = 1, 2, consistent with the above splitting, are Poisson maps; (iii) if f ∈ π1∗ (C ∞ (P1 )) and g ∈ π2∗ (C ∞ (P2 )), then {f, g} = 0. The following proposition gives equivalent conditions for the existence of Poisson splittings.

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Proposition 4.15. The following conditions are equivalent: (i) the Banach Lie-Poisson space (b, {·, ·}) admits a Poisson splitting into the two Banach Lie-Poisson subspaces (b1 , {·, ·}1 ) and (b2 , {·, ·}2 ); (ii) the Banach Lie-Poisson space (b, {·, ·}) is isomorphic to the product Banach LiePoisson space (b1 × b2 , {·, ·}12 ); (iii) the components b∗1 and b∗2 of the dual splitting b∗ = b∗1 ⊕ b∗2 are ideals of the Banach Lie algebra b∗ , where one identifies b∗1 and b∗2 with the annihilators of b2 and b1 in b∗ respectively. Proof. The equivalence of (i) and (ii) is a direct consequence of Theorem 2.2 and the subsequent comments. Conditions (i) and (iii) are equivalent by applying Propositions 4.8 and 4.10.

In this section we established an equivalence between the subcategory of Banach Lie algebras admitting a unique predual and the subcategory of Banach Lie-Poisson spaces that are unique preduals of their dual. The statements proved above give examples how this equivalence can be used in the study of these two categories. For example, the simplicity of a Banach Lie algebra from the category L0u is equivalent to the non-existence of Banach Lie-Poisson subspaces of its predual. 5. Preduals of W ∗ -Algebras as Banach Lie-Poisson Spaces In this section we shall consider the important class of Banach Lie-Poisson spaces related to the category of W ∗ -algebras. Recall that a W ∗ -algebra is a C ∗ -algebra m which possesses a predual Banach space m∗ , i.e. m = (m∗ )∗ ; this predual is unique (Sakai [S]). Since m∗ = (m∗ )∗∗ , the predual Banach space m∗ canonically embeds into the Banach space m∗ dual to m. Thus we shall always think of m∗ as a Banach subspace of m∗ . The existence of m∗ allows the introduction of the σ (m, m∗ )-topology on the W ∗ -algebra m; for simplicity we shall call it the σ -topology in the sequel. Recall that a net {xα }α∈A ⊂ m converges to x ∈ m in the σ -topology if, by definition, limα∈A xα , b = x, b for all b ∈ m∗ . The σ -topology is Hausdorff. Alaoglu’s theorem states that the unit ball of m is compact in the σ -topology. One can characterize the predual space m∗ as the subspace of m∗ consisting of all σ -continuous linear functionals, see Sakai [S]. A theorem of Diximier (see Sakai [S], §1.13) states that a positive linear functional ν ∈ m∗ is σ -continuous if and only if it is normal, i.e. it satisfies ν, l.u.b.xα = l.u.b.ν, xα for every uniformly bounded increasing direct set {xα } of positive elements in m. The normality is determined by the ordering on m only. So, the predual space m∗ and thus the pairing m∗ × m (ν, x) → x, ν := x(ν) ∈ C are defined by the algebraic structure of m in a unique way. Theorem 5.1. Let m be a W ∗ -algebra and m∗ be the predual of m. Then m∗ is a Banach Lie-Poisson space with the Poisson bracket {f, g} of f, g ∈ C ∞ (m∗ ) given by (4.2). The

Banach Lie-Poisson Spaces and Reduction

23

Hamiltonian vector field Xf defined by the smooth function f ∈ C ∞ (m∗ ) is given by (4.3). Proof. We shall prove the theorem by checking the conditions of Theorem 4.2. Since the W ∗ -algebra m is an associative Banach algebra we can define the Lie bracket in m as the commutator [x, y] = xy − yx of x, y ∈ m. Left and right multiplication by a ∈ m define uniformly and σ -continuous maps La : m x → ax ∈ m Ra : m x → xa ∈ m, see Sakai [S]. Let L∗a : m∗ → m∗ and Ra∗ : m∗ → m∗ denote the dual maps of La and Ra respectively. If v ∈ m∗ , then L∗a (v) and Ra∗ (v) are σ -continuous functionals and therefore, by the characterization of the predual m∗ as the subspace of σ -continuous functionals in m∗ , it follows that L∗a (v), Ra∗ (v) ∈ m∗ . One has ada = [a, ·] = La − Ra and thus, ad∗a = L∗a − Ra∗ . We conclude from the above that m is a Banach Lie algebra and ad∗a m∗ ⊂ m∗ for each a ∈ m, which are the conditions of Theorem 4.2.

Corollary 5.2. Let a be a C ∗ -algebra. Then its dual a∗ is a Banach Lie-Poisson space. Proof. The bidual a∗∗ is isomorphic to the universal enveloping von Neumann algebra of a and by the canonical inclusion a → a∗∗ the C ∗ -algebra a can be considered as a C ∗ -subalgebra of a∗∗ (see Sakai [S] §17.1, or Takesaki [T2]). Since a∗ is predual to a∗∗ , Theorem 5.1 guarantees that it is a Banach Lie-Poisson space.

Any σ -closed C ∗ -subalgebra n ⊂ m has the predual given by m∗ /n◦ , where n◦ is the annihilator of n in m∗ . Thus n is a Banach Lie subalgebra of m admitting a predual. By Proposition 4.13 (i), the quotient map π : m∗ → m∗ /n◦ coinduces a Lie-Poisson structure on the quotient Banach space m∗ /n◦ . Therefore, there is a bijective correspondence between W ∗ -subalgebras of m and a subclass of Banach Lie-Poisson spaces coinduced from m∗ . It would be interesting to characterize this subclass in Poisson geometrical terms. If n is a hereditary subalgebra of m, then there exists a projector p ∈ m such that n = im P (see Murphy [M]), where the map P : m → m is defined by P (x) := pxp.

(5.1)

The map P is a σ -continuous projector with P = 1. Thus its dual P ∗ : m∗ → m∗ preserves m∗ . We therefore conclude that P∗ := P ∗ |m∗ : m∗ → m∗ is a projector with P∗ = 1. Thus there is a splitting m∗ = im P∗ ⊕ker P∗ which allows one to canonically identify m∗ /n◦ with im P∗ since ker P∗ = n◦ . Proposition 5.3. Let n be a hereditary W ∗ -subalgebra of m. Then the projector P∗ : m∗ → m∗ coinduces a Banach Lie-Poisson structure on im P∗ . So, a necessary condition for n to be a hereditary subalgebra is that the Lie-Poisson structure on m∗ coinduces one on a Banach subspace of m∗ .

24

A. Odzijewicz, T.S. Ratiu

Let us recall that m∗ and m∗ have natural Banach Lie-Poisson structures according to Theorem 5.1 and Corollary 5.2 respectively. Proposition 5.4. Let m∗ be the predual of the W ∗ -algebra m and ι : m∗ → m∗ be the canonical inclusion. Then ι is an injective linear Poisson map and the Poisson structure induced by it from m∗ coincides with the original Lie-Poisson structure on m∗ . Proof. Since ι(b) = b for b ∈ m∗ , the range of ι : m∗ → m∗ is closed in m∗ . The dual map ι∗ : m∗∗ → m = (m∗ )∗ is a projection of the universal enveloping W ∗ -algebra m∗∗ onto m of norm one. One has the equality ker ι∗ = (m∗ )◦ , where (m∗ )◦ is the annihilator of m∗ in m∗∗ . In addition, L∗x m∗ ⊂ m∗ and Rx∗ m∗ ⊂ m∗ for any x ∈ m∗∗ (see Sakai [S]). Thus, ker ι∗ is a σ (m∗∗ , m∗ )-closed ideal of m∗∗ . Therefore, ker ι∗ is also an ideal in the Banach Lie algebra structure of m∗∗ defined by [x, y] = xy − yx. Proposition 4.10 implies that ι induces a Banach Lie-Poisson structure on m∗ . Since m∗∗ / ker ι∗ is isomorphic to m, this structure coincides with the original Banach Lie-Poisson structure of m∗ defined by Theorem 5.1.

Consider now a σ -closed two sided ideal i ⊂ m. Then it equals im P , where P is given by (5.1), for p a central projector in m (see Sakai [S] §1.10). The projector P⊥ : m → m defined by P⊥ (x) := (1 − p)x(1 − p) is also a σ -continuous linear map with P⊥ = 1 and projects m onto a σ -closed two sided ideal i⊥ . Since P + P⊥ = I , we have the splitting m = i ⊕ i⊥

(5.2)

of m into two sided ideals. The decomposition (5.2) is also a splitting into ideals in the category of Banach Lie algebras. By Proposition 4.15, the direct sum (5.2) induces a Poisson splitting m∗ = i◦ ⊕ i◦⊥ ,

(5.3)

where i◦ and i◦⊥ are the annihilators in m∗ of i and i⊥ respectively. As a special case, one can consider the universal enveloping W ∗ -algebra m∗∗ of the ∗ W -algebra m with predual m∗ . Then m∗ is the predual to m∗∗ and m∗ ⊂ m∗ is a L∗m∗∗ ∗ and Rm ∗∗ invariant Banach subspace. In this case, the splitting (5.3) gives the Poisson splitting m∗ = m∗ ⊕ m⊥ ∗ of m∗ into the normal and singular functionals (see Takesaki [T2] for this terminology). In order to illustrate Theorem 5.1 let us take a complex Hilbert space M . By L1 (M), 2 L (M), and L∞ (M) we shall denote the involutive Banach algebras of the trace class operators, the Hilbert-Schmidt operators, and the bounded operators on M respectively. Recall that L1 (M) and L2 (M) are self adjoint ideals in L∞ (M). Let K(M) ⊂ L∞ (M) denote the ideal of all compact operators on M . Then L1 (M) ⊂ L2 (M) ⊂ K(M) ⊂ L∞ (M),

(5.4)

and the following remarkable dualities hold (see e.g. Murphy [M]): K(M)∗ ∼ = L1 (M),

L2 (M)∗ ∼ = L2 (M),

and

L1 (M)∗ ∼ = L∞ (M).

(5.5)

Banach Lie-Poisson Spaces and Reduction

25

These are implemented by the strongly non-degenerate pairing x, ρ = tr(xρ),

(5.6)

where x ∈ L1 (M), ρ ∈ K(M) for the first isomorphism, ρ, x ∈ L2 (M) for the second isomorphism and x ∈ L∞ (M), ρ ∈ L1 (M) for the third isomorphism. The isomorphism L1 (M)∗ ∼ = L∞ (M) gives the crucial example of the W ∗ -algebra of bounded operators on the complex Hilbert space M . So, we recover the result of Bona [2000] as a corollary of Theorem 5.1. Corollary 5.5. The Banach space L1 (M) of trace class operators on the Hilbert space M is a Banach Lie-Poisson space relative to the Poisson bracket given by {f, g}(ρ) = tr([Df (ρ), Dg(ρ)]ρ),

(5.7)

where ρ ∈ L1 (M) and the bracket [Df (ρ), Dg(ρ)] denotes the commutator of the bounded operators Df (ρ), Dg(ρ) ∈ L∞ (M) ∼ = L1 (M)∗ . The Hamiltonian vector field associated to f ∈ C ∞ (L1 (M)) is given by Xf (ρ) = [Df (ρ), ρ].

(5.8)

Proof. Formula (5.7) follows from (4.2) by using (5.6) for the pairing between L1 (M) and L∞ (M) . In order to obtain (5.8) from (4.3), let us notice that y, − ad∗x ρ = −[x, y], ρ = − tr([x, y]ρ) = tr(y[x, ρ]) = y, [x, ρ]

(5.9)

for ρ ∈ L1 (M) and x, y ∈ L∞ (M). Thus − ad∗x ρ = [x, ρ] ∈ L1 (M), since L1 (M) is an ideal in L∞ (M). (We have identified here {ρ} × L1 (M) with the tangent space Tρ L1 (M).)

The other two isomorphisms in (5.5) also give rise to Banach Lie-Poisson spaces, but as a corollary of Theorem 4.2; Theorem 5.1 cannot be applied because L2 (M) and L1 (M) are not W ∗ -algebras. Example 5.6. The Banach space L2 (M) of Hilbert-Schmidt operators on the Hilbert space M is a Banach Lie-Poisson space. Indeed, we use the isomorphism L2 (M)∗ ∼ = L2 (M) given by the pairing (5.6) and notice that L2 (M) is a reflexive (that is, L2 (M)∗∗= L2 (M)) Banach algebra. The formulas for the Poisson bracket and for the Hamiltonian vector field are (5.7) and (5.8) respectively with ρ ∈ L2 (M). Example 5.7. The Banach space K(M) of compact operators on the Hilbert space M, as a predual of L1 (M), is a Banach Lie-Poisson space. The proof is identical to that of Corollary 5.5. The formulas for the Poisson bracket and for the Hamiltonian vector field are (5.7) and (5.8) respectively with ρ ∈ K(M). Example 5.8. Let L∞ (M, µ) be the W ∗ -algebra of all essentially bounded µ-locally measurable functions on a localizable measure space M. Then its predual is the Banach space L1 (M, µ) of all µ-integrable functions on M. Since L∞ (M, µ) is commutative, the Banach Lie-Poisson structure on L1 (M, µ) is trivial, that is, {f, g} = 0 for all f, g ∈ C ∞ (L1 (M, µ)). ¯ However, one can take the W ∗ -algebra tensor product L∞ (M, µ)⊗m, where m ¯ is is a W ∗ -algebra with predual m∗ . Then (see, e.g. Sakai [S], §1.22) L∞ (M, µ)⊗m isomorphic with the Banach space L∞ (M, µ, m) of all m-valued essentially bounded

26

A. Odzijewicz, T.S. Ratiu

weakly ∗ µ-locally measurable functions on M. Moreover, L∞ (M, µ, m) is a W ∗ -algebra under pointwise multiplication and its predual is the Banach space L1 (M, µ, m∗ ) of all m∗ -valued Bochner µ-integrable functions on M. For details see Sakai [S], Takesaki [T2], or Bourbaki [Bo1] §2.6. The duality pairing between b ∈ L1 (M, µ, m∗ ) and x ∈ L∞ (M, µ, m) is given by b, x = b(m), x(m)mdµ(m), (5.10) M

where ·, ·m is the duality pairing between m∗ and m. Thus, by Theorem 5.1 and formula (4.2) the Lie-Poisson bracket of f, g ∈ C ∞ (L1 (M, µ, m∗ )) is given by δf δg {f, g}(b) = b(m), dµ(m), (5.11) (m), (m) δb δb M m where δf/δb, δg/δb ∈ L∞ (M, µ, m) are the representatives via the pairing (5.10) of the Fr´echet derivatives Df (b) and Dg(b) ∈ L1 (M, µ, m∗ )∗ respectively. Applying to L1 (M, µ, m∗ ) the quantum reduction procedure (see Sect. 6), one obtains the Banach Lie-Poisson space L1 (M, µ, g∗ ), where g∗ is the predual space of the reduced Banach Lie algebra g = R ∗ (m). In the finite dimensional case, for example when m = gl(N, C) and M is a smooth manifold, we will consider the Banach Lie algebra L∞ (M, µ, g) as the Lie algebra of the current group C ∞ (M, G), where G is a Lie group with Lie algebra g and the group structure on C ∞ (M, G) is defined by pointwise multiplication of maps. Usually the Lie algebra of C ∞ (M, G) is taken to be C ∞ (M, g) (see, e.g. Kirillov [K]); in our approach we shall work with the L∞ completion of this Lie algebra. For M = S 1 one has the loop group case. So, we could consider the Banach Lie-Poisson space L1 (M, µ, g∗ ) with the bracket δf δf {f, g}(α) = Cjik αi (m) (m) (m)dµ(m) (5.12) δαj δαk M as one related to the current group. In order to clarify (5.12), let us mention that, since g∗∗ = g, we identified g∗ with g∗ . The scalar functions α1 , . . . , αs , where s = dim g, denote the components of α ∈ L1 (M, µ, g) in a basis of g∗ dual to a given basis of g relative to which the structure constants Cjik , i, j, k = 1, . . . , s are determined. Let us now discuss the realifications mR and m∗ R of the W ∗ -algebras m and its predual m∗ . As was mentioned in §4, m∗ R has a real Banach Lie-Poisson structure. For a fixed Hermitian element η ∈ m satisfying η2 = 1, one defines the involutions σ (b) = −ηb∗ η =: − Ad∗η (b∗ ), σ ∗ (x) = −ηx ∗ η =: − Adη (x ∗ ),

b ∈ m∗ R ,

(5.13)

x ∈ mR

(5.14)

in the sense of Example 4.9, i.e., they satisfy conditions (i), (ii), and (iii) given there. To check them, one uses the defining identity for the conjugation in the predual m∗ : x, b = x ∗ , b∗ , x∗

where and b∗ are the conjugates of Lie algebra mσ := {x ∈ m | σ ∗ x =

(5.15)

x ∈ m and b ∈ m∗ respectively. The real Banach x} = {x ∈ m | ηx ∗ + xη = 0} has underlying

Banach Lie group U (m, η) := {g ∈ m | g ∗ ηg = η}

Banach Lie-Poisson Spaces and Reduction

27

consisting of the set of pseudounitary elements (see Bourbaki [Bo3], Chapter 3, §3.10, Proposition 37). For η = 1 one obtains the group of unitary elements of m. From the considerations presented in Example 4.9, one can conclude that mσ∗ := {b ∈ m∗ | σ (b) = b} has the real Banach Lie-Poisson structure coinduced from mR by the projector R = (id + σ )/2. This structure is given by (4.14) and is Ad∗U (m,η)-invariant. In §7, we will discuss the orbits of this action. The above more general constructions are of course valid if one considers the special case m = L∞ (M) and m∗ = L1 (M). As we have seen, Poisson geometry naturally arises in the theory of operator algebras. The links between these theories established above show the importance of the fact that the category of W ∗ -algebras can be considered as a subcategory of the category of Banach Lie-Poisson spaces. Finally, let us mention that Poisson structures that are fundamental for classical phase spaces appear in a natural way on quantum phase spaces, i.e. duals to C ∗ -algebras. 6. Quantum Reduction Recall that L1 (M) contains the subset of mixed states ρ of the quantum mechanical physical system, i.e., ρ ∗ = ρ ≥ 0 and tr ρ = ρ1 = 1. If the system under consideration is an isolated quantum system, its dynamics is reversible and is described by the Liouville-von Neumann equation ρ˙ = [H, ρ],

(6.1)

which is a Hamiltonian equation on (L1 (M), {·, ·}) with Hamiltonian tr(Hρ). For simplicity, let us assume that H ∗ = H ∈ L∞ (M) is a given (ρ-independent) operator. Therefore the Schr¨odinger flow U (t) = eitH is a Poisson flow on the Banach Lie-Poisson space (L1 (M), {·, ·}). Let us now apply a measurement operation to the system corresponding to the discrete orthonormal decomposition of the unit Pn Pm = δnm Pn ,

∞

Pn = 1.

(6.2)

n=1

For example, this is the case when one measures the physical quantity given by the operator X = ∞ n=1 xn Pn , with xn ∈ R. Then, according to the well known von Neumann projection postulate, the density operator ρ of the state before measurement is transformed by the measurement to the density operator R(ρ) given by R(ρ) :=

∞

Pn ρPn .

(6.3)

n=1

Proposition 6.1. The measurement operator R : L1 (M) → L1 (M) has the following properties: (i) R is a continuous norm one projector, i.e., R 2 = R and R = 1; (ii) it preserves the space of states, i.e., if ρ ∗ = ρ > 0, then R(ρ)∗ = R(ρ ∗ ) = R(ρ) > 0; (iii) the range im R ∗ of its dual R ∗ : L∞ (M) → L∞ (M) is a Banach Lie subalgebra of L∞ (M).

28

A. Odzijewicz, T.S. Ratiu

Proof. Using the natural pairing between L1 (M) and L∞ (M) given by the trace of the product, it follows that R ∗ is given by R ∗ (X) :=

∞

(6.4)

Pn XPn

n=1

for X ∈ L∞ (M). Then, for v ∈ M one concludes R ∗ (X)v2 =

∞ n=1

Pn XPn v2 ≤ X2

∞

Pn v2 = X2 v2

n=1

which proves that R ∗ ≤ 1. Since R ∗ = R ∗ 2 ≤ R ∗ 2 it follows that R ∗ = 1. Now, using the defining identity Tr R(ρ)X = Tr ρR ∗ (X) of R ∗ it follows that R = 1. This proves (i). Property (ii) follows directly from (6.4). Finally, in order to prove (iii) it is enough to remark that R ∗ (X)R ∗ (Y ) = R ∗ (R ∗ (X)R ∗ (Y )).

(6.5)

We conclude from Propositions 6.1 and 4.8 that the quantum measurement procedure gives a Poisson projection R : L1 (M) → im R of L1 (M) on the Banach subspace im R = ker(1 − R) endowed with the Poisson bracket {·, ·}im R coinduced from L1 (M). Clearly, opposite to the U -procedure, i.e., the unitary time evolution U (t), t ∈ R, the Rprocedure is not reversible. However, both the U -procedure and the R-procedure share an essential common feature: they are linear Poisson maps. After this physical introduction, let us now come back to the case when the Banach Lie-Poisson space is the predual space m∗ of a W ∗ -algebra m. In the theory of quantum physical systems (including statistical physics) the W ∗ -algebra is the algebra of observables and the norm one positive elements of m∗ ⊂ m∗ are the normal states of the considered system, see e.g. Bratteli and Robinson [B-R1, B-R2], or Emch [Em]. The norm one map E : m → m which is idempotent (E 2 = E) and maps m onto a C ∗ -subalgebra n is called a conditional expectation. If E is σ -continuous then n is a W ∗ -subalgebra of m. In that case, the adjoint map E ∗ : m∗ → m∗ preserves m∗ ⊂ m∗ and maps m∗ onto n∗ . The conditional expectation is said to be compatible with the state µ ∈ m∗ if E ∗ (µ) = µ. The concept of conditional expectation comes from probability theory where it is very important in martingale theory. The definition of the conditional expectation as the linear map E : m → m on the W ∗ -algebra m with the properties mentioned above is the generalization of the conditional expectation concept in non-commutative probability theory. The role of conditional expectation in the theory of quantum measurement theory and in quantum statistical physics and their remarkable mathematical properties was elucidated in many remarkable publications such as Takesaki [T1], Accardi, Frigerio, and Gorini [A-F-G], and Accardi and von Waldenfels [A-vW]. See Holevo [H] for an extended list of references to publications concerning conditional expectations. Resuming, we see that the restriction R := E ∗ |m∗ : m∗ → m∗ of the map dual to a conditional expectation E : m → m is a continuous projector. Since im R ∗ = im E = n, the range of the projector R ∗ : m → m is a Banach Lie subalgebra (n, [·, ·]) of (m, [·, ·]). So, like in the case of the measurement map (6.3), one can apply Proposition 4.8 in order to coinduce a Banach Lie-Poisson structure on im R.

Banach Lie-Poisson Spaces and Reduction

29

Motivated by the above two examples, we introduce the following definition. Definition 6.2. A quantum reduction map is a continuous projector R : b → b on a Banach Lie-Poisson space (b, {·, ·}) such that the range im R ∗ of the dual map R ∗ : b∗ → b∗ is a Banach Lie subalgebra of b∗ . This immediately implies that R coinduces a Poisson structure on im R (see Proposition 4.8) and, in particular, R : (b, {·, ·}) → (im R, {·, ·}im R ) is a Poisson map. Let us now give some important examples of quantum reduction. Example 6.3. Every self-adjoint projector p in the W ∗ -algebra m defines a uniformly and σ -continuous projector m x → P (x) := pxp ∈ m

(6.6)

of m , see Sakai [S] or Takesaki [T2]. Let P ∗ : m∗ → m∗ be the projector dual to P , i.e. P ∗ µ, x = µ, P x for any µ ∈ m∗ and x ∈ m , where µ, x := µ(x). Since P is σ -continuous, the predual space m∗ ⊂ m∗ is preserved by P ∗ . Let P∗ be the restriction of P ∗ to m∗ . The dual (P∗ )∗ of the projector P∗ is equal to P . The range im P of the projector P is a W ∗ -subalgebra of m (see Sakai [S]). Recalling that ad∗x m∗ ⊂ m∗ for x ∈ m , we see that adx im P∗ ⊂ im P∗ , for x ∈ im P . Summarizing, we have proved the following. Proposition 6.4. The projector P∗ : m∗ → m∗ has the following properties: (i) P∗ = 1; (ii) im(P∗ )∗ is a Banach-Lie algebra; (iii) adx im P∗ ⊂ im P∗ , for x ∈ im P . Therefore P∗ : m∗ → m∗ is a quantum reduction map. If m = L∞ (M) and m∗ = L1 (M) the projector P∗ : L1 (M) → L1 (M) reduces the mixed state ρ of the quantum system to the state pρp = P∗ ρ localized on the subspace L1 (pM ) ⊂ L1 (M). In the quantum mechanical formalism the projector p : M → M represents the so-called elementary observable “proposition” (or “question”) which can have only two alternative outcomes: “yes” or “no”. The measurement of the “proposition” p reduces the state ρ to the state P∗ ρ and the non-negative number tr(P∗ ρ) is the probability of the yes-answer. Since P∗ is a projector, the repetition of the measurement does not change the state P∗ ρ. This is the mathematical expression of the von Neumann reproducing postulate (von Neumann [vN]). Example 6.5. Let m be a W ∗ -algebra and {pα }α∈I ⊂ m, be a family of self-adjoint mutu ally orthogonal projectors (i.e., pα pβ = δαβ pα , and pα∗ = pα ) such that α∈I pα = 1; the index set I is not necessary countable. Define the map R ∗ : m → m by R ∗ (x) :=

pα xpα

α∈I

for x ∈ m, where the summation is taken in the sense of the σ -topology.

(6.7)

30

A. Odzijewicz, T.S. Ratiu

Proposition 6.6. The map R ∗ : m → m is a σ -continuous linear projector with R ∗ = 1. Moreover, im R ∗ is a W ∗ -subalgebra of m and hence a Banach Lie subalgebra of (m, [·, ·]). Additionally, one has R ∗ (R ∗ (x)R ∗ (y)) = R ∗ (x)R ∗ (y)

R ∗ (x ∗ ) = (R ∗ (x))∗

and

(6.8)

for all x, y, ∈ m. Proof. We can always consider m as a von Neumann algebra of operators on the Hilbert space M. If v ∈ M, then 2 pα xpα = pα xpα v2 ≤ x2 pα v2 = x2 v2 R ∗ (x)v2 = α∈I

α∈I

α∈I

which shows that R ∗ (x) ≤ 1. From (6.7) we have 2 pβ pα xpα pβ = δαβ pα xpα δαβ = pα xpα = R ∗ (x); R ∗ (x) = β∈I

α∈I

β∈I α∈I

α∈I

in this computation the σ -continuity of left and right multiplication with an element pβ was used. Thus R ∗ 2 = R ∗ and R ∗ = 1. For any b ∈ m∗ , there is an element ρ ∈ L1 (M) such that x, b = tr(xρ). Thus ∗ ∗ tr(pα xpα ρ) = tr(xpα ρpα ) = tr x pα xpα . R (x), b = tr(R (x)ρ) = α∈I

α∈I

α∈I σ

σ

We want to check that xi → x implies that R ∗ (xi ) → R ∗ (x). To do this, substitute xi in the previous identity to get σ ∗ pα xpα −→ tr x pα xpα = tr(R ∗ (x)b) = R ∗ (x), b R (xi ), b = tr xi α∈I

α∈I

for any b ∈ m∗ . So R ∗ is a σ -continuous linear map. The defining formula for R ∗ shows that for x, y ∈ m one has R ∗ (R ∗ (x)R ∗ (y)) = ∗ R (x)R ∗ (y) and R ∗ (x ∗ ) = (R ∗ (x))∗ . Thus im R ∗ is a W ∗ -subalgebra of m which implies that it is also a Banach Lie subalgebra of (m, [·, ·]).

We conclude from Proposition 6.6 that (R ∗ )∗ : m∗ → m∗ preserves the predual subspace m∗ ⊂ (m∗ )∗∗ = m∗ and hence R := (R ∗ )∗ |m∗ is a quantum reduction. Note that one has the splitting m = im R ∗ ⊕ ker R ∗ . Example 6.7. Take the decomposition of the unit (6.2) and define the map R− : L1 (M) → L1 (M) by R− (ρ) :=

∞ n n=1 m=1

pn ρpm =

∞

pn ρqn ,

(6.9)

n=1

where qn := nm=1 pm . It is clear that R− is a linear projector on L1 (M) whose range is the linear subspace of all “lower triangular” trace class operators L1 (M)− . From

Banach Lie-Poisson Spaces and Reduction

R− (ρ)∗ R− (ρ) =

∞ ∞

31

q ρ ∗ p pn ρqn =

=1 n=1

∞

qn ρ ∗ pn ρqn ≤

n=1

we have tr

∞

ρ ∗ pn ρ = ρ ∗ ρ

n=1

R(ρ)∗ R(ρ) ≤ tr ρ ∗ ρ

which shows that R− (ρ)1 ≤ ρ1 . Thus, R− : L1 (M) → L1 (M) is a continuous ∗ : L∞ (M) → L∞ (M) defined by projector with R− = 1. So, the dual map R− ∗ R− (x)

:=

∞

qn xpn

n=1 ∗ = 1 and projects L∞ (M) onto the “upper triangular” Banach Lie also satisfies R− ∞ subalgebra L (M)+ ⊂ L∞ (M). In this way, R− : L1 (M) → L1 (M) is a quantum ∗ satisfies (6.8). In §8 we will use the quantum reduction R in reduction. Note that R− − the description of the Toda lattice.

The discussion below uses the standard notion of Banach Lie group and its associated Banach Lie algebra. Recall that a (real or complex) Banach Lie group is a (real or complex) smooth Banach manifold G with a group structure such that the multiplication and inversion are smooth maps. As in the finite dimensional case, the tangent space at the identity, Te G, carries a Lie algebra structure which makes it isomorphic to the Lie algebra of left invariant vector fields on G endowed with the usual bracket operation on vector fields. Due to the smoothness of the group operations, this Lie bracket is a continuous bilinear map on the Banach space Te G. Thus Te G is a Banach Lie algebra which will be denoted, as customary, by g; it is called the Lie algebra of G. Let G(m) be the group of invertible elements of a W ∗ -algebra m; it is an open subset (in the norm topology) of m and is therefore a (real or complex) Banach Lie group whose Lie algebra is m relative to the commutator bracket [·, ·] (Bourbaki [Bo3], Chapter III, §3.9). Its exponential map is the usual exponential function on m. Proposition 6.8. Let R : m∗ → m∗ be a quantum reduction as given in Definition 6.2 that also satisfies properties (6.8) and R ∗ (1) = 1. Then the set G(m) ∩ im R ∗ = G(im R ∗ ) is a closed Banach Lie subgroup of G(m) whose Lie algebra is the Banach Lie subalgebra im R ∗ . Proof. We prove that G(im R ∗ ) is a subgroup of G(m). From (6.8) and the fact that R ∗ maps the identity to the identity, it follows that G(im R ∗ ) is closed under multiplication in m and that it contains the identity element. We shall prove now that if R ∗ (x) is invertible, then its inverse is also an element of G(im R ∗ ). To see this, we assume without loss of generality that 1 − x < 1/R ∗ . Since R ∗ (1) = 1 one has R ∗ (1 − x) < 1 and therefore (R ∗ (x))−1 = (1 − R ∗ (1 − x))−1 =

∞

k

R ∗ (1 − x)

k=0

=

∞ k=0

R∗

∞

k

k ∗ ∗ ∗ = R ∗ (R ∗ (x))−1 . R (1 − x) =R R (1 − x)

k=0

Thus G(im R ∗ ) is also closed under inversion and is therefore a subgroup of G(m). This argument also proves the equality in the statement.

32

A. Odzijewicz, T.S. Ratiu

Since im R ∗ is closed in m, it follows that G(im R ∗ ) is a closed subgroup of G(m). As the group of invertible elements of the W ∗ -algebra im R ∗ , G(im R ∗ ) is a Lie group in its own right whose Lie algebra equals im R ∗ . Because im R ∗ splits in m it follows that the inclusion of G(im R ∗ ) into G(m) is an immersion. However, the topologies on G(im R ∗ ) and G(m) are both induced by the norm topology of m and thus the inclusion G(im R ∗ ) → G(m) is a homeomorphism onto its image which shows that this inclusion is an embedding and hence G(im R ∗ ) is a Lie subgroup of G(m).

We shall return to this proposition in §7. Note that both Examples 6.5 and 6.7 satisfy the hypotheses of Proposition 6.8. 7. Symplectic Leaves and Coadjoint Orbits A smooth map f : M → N between finite dimensional manifolds is called an immersion, if for every m ∈ M the derivative Tm f : Tm M → Tf (m) N is injective. In infinite dimensions there are various notions generalizing this concept. Definition 7.1. A smooth map f : M → N between Banach manifolds is called a (i) immersion if for every m ∈ M the tangent map Tm f : Tm M → Tf (m) N is injective with closed split range; (ii) quasi immersion if for every m ∈ M the tangent map Tm f : Tm M → Tf (m) N is injective with closed range; (iii) weak immersion if for every m ∈ M the tangent map Tm f : Tm M → Tf (m) N is injective. An immersion between Banach manifolds has the same properties as an immersion between finite dimensional manifolds. For example, it is characterized by the property that locally it is given by a map of the form u → (u, 0), where the model space of the chart on N necessarily splits. This is the concept widely used in the literature; see e.g. Abraham, Marsden, and Ratiu [A-M-R], or Bourbaki [Bo2]. The notion of quasi immersion is modeled on the concept of a quasi-regular submanifold introduced in Bourbaki [Bo2]. Unfortunately, in the study of Banach Poisson manifolds not even this weaker concept of quasi-immersion is satisfactory and one is forced to work with genuine weak immersions, as we shall see in this section. If (P , {·, ·}P ) is a Banach Poisson manifold (in the sense of Definition 2.1), the vector subspace Sp := {Xf (p) | f ∈ C ∞ (P )} of Tp P is called the characteristic subspace at p. Note that Sp is, in general, not a closed subspace of the Banach space Tp P . The union S := ∪p∈P Sp ⊂ T P is called the characteristic distribution of the Poisson structure on P . Note that even if Sp were closed and split in Tp P for every p ∈ P , S would not necessarily be a subbundle of T P . However, the characteristic distribution S is always smooth, in the sense that for every vp ∈ Sp ⊂ Tp P there is a locally defined smooth vector field (namely some Xf ) whose value at p is vp . Assume that the characteristic distribution is integrable. For finite dimensional manifolds this is automatic by the Stefan-Sussmann theorem (see, e.g. Libermann and Marle [L-M], Appendix 3, Theorem 3.9) which, to our knowledge, is not available in infinite dimensions. Let L be a leaf of the characteristic distribution, that is, • L is a connected smooth Banach manifold, • the inclusion ι : L → P is a weak injective immersion, • Tq ι(Tq L) = Sq for each q ∈ L,

Banach Lie-Poisson Spaces and Reduction

33

• if the inclusion ι : L → P is another weak injective immersion satisfying the three conditions above and L ⊂ L , then necessarily L = L, that is, L is maximal. If we assume in addition that on the leaf L • there is a weak symplectic form ωL consistent with the Poisson structure on P , then L will be called a symplectic leaf. In order to explain what this consistency means, consider from Definition 2.1 the bundle map : T ∗ P → T P associated to the Poisson tensor on P and note that for each p ∈ P , the linear continuous map p : Tp∗ P → Tp P induces a bijective continuous map [p ] : Tp∗ P / ker p → Sp . By definition, ωL is consistent with the Poisson structure on P if (7.1) ωL (q)(uq , vq ) = (ι(q)) ([ι(q) ]−1 ◦ Tq ι)(uq ), ([ι(q) ]−1 ◦ Tq ι)(vq ) . This shows that the weak symplectic form ωL consistent with the Poisson structure on P is unique. For finite dimensional Poisson manifolds, it is known that all leaves are symplectic (see Weinstein [W1]) and so the last assumption above is not necessary. In the infinite dimensional case this question is open, even in the case of a Banach Lie group G whose Lie algebra g has a predual g∗ invariant under the coadjoint action. In this case, g∗ is a Banach Lie-Poisson space and we will characterize a large class of points in g∗ whose coadjoint orbits are all weak symplectic manifolds. Their connected components are therefore symplectic leaves. Among these, we will also identify a class of coadjoint orbits that are strong symplectic manifolds. Proposition 7.2. Let ι : (L, ωL ) → P be a symplectic leaf of the characteristic distribution of P . Then (i) for any f ∈ C ∞ (U ), q ∈ ι−1 (U ) ∩ L, where U is an open subset of P , one has −1 (Xf (ι(q))), · ; (7.2) d (f ◦ ι)|ι−1 (U ) (q) = ωL (q) Tq ι (ii) the subspace ι∗ (C ∞ (P )) of C ∞ (L) consisting of functions that are obtained as restrictions of smooth functions from P is a Poisson algebra relative to the bracket {·, ·}L given by −1 −1 (Xf (ι(q))), Tq ι (Xg (ι(q))) ; (7.3) {f ◦ ι, g ◦ ι}L (q) := ωL (q) Tq ι (iii) ι∗ : C ∞ (P ) → ι∗ (C ∞ (P )) is a homomorphism of Poisson algebras, that is, {f ◦ ι, g ◦ ι}L = {f, g}P ◦ ι

(7.4)

for any f, g ∈ C ∞ (P ). Proof. We begin with the proof of formula (7.2). For any q ∈ L ∩ ι−1 (U ), vq ∈ Tq L, f ∈ C ∞ (U ), U open in P , we have by (7.1) and the definition of , −1 (Xf (ι(q))), vq = (ι(q)) df (ι(q)), ([ι(q) ]−1 ◦ Tq ι(vq ) ωL (q) Tq ι = df (ι(q)), Tq ι(vq ) = d(f ◦ ι)(q)(vq ),

34

A. Odzijewicz, T.S. Ratiu

−1 which proves (7.2). Now replace in the above computation vq by Tq ι (Xg (ι(q))) and get −1 −1 ωL (q) Tq ι (Xf (ι(q))), Tq ι (Xg (ι(q))) = df (ι(q)), Xg (ι(q)) = {f, g}P (ι(q)), which, in view of (7.3), proves (7.4). Finally, (7.4) shows that (7.3) defines a Poisson bracket on L.

Formula (7.2) is remarkable since it guarantees the existence of Hamiltonian vector fields on the weak symplectic manifold L for a large class of functions, namely those that are pull backs to the symplectic leaf. We shall give below a class of Banach Poisson manifolds for which some of the symplectic leaves can be explicitly determined. In what follows G denotes a (real or complex) Banach Lie group, Lg and Rg denote the diffeomorphisms of G given by left and right translations by g ∈ G, and g denotes the (left) Lie algebra of G. Theorem 7.3. Let G be a (real or complex) Banach Lie group with Lie algebra g. Assume that: (i) g admits a predual g∗ , that is, g∗ is a Banach space whose dual is g; (ii) the coadjoint action of G on the dual g∗ leaves the predual g∗ invariant, that is, Ad∗g (g∗ ) ⊂ g∗ , for any g ∈ G; (iii) for a fixed ρ ∈ g∗ the coadjoint isotropy subgroup Gρ := {g ∈ G | Ad∗g ρ = ρ}, which is closed in G, is a Lie subgroup of G (in the sense that it is a submanifold of G and not just injectively immersed). Then the Lie algebra of Gρ equals gρ := {ξ ∈ g | ad∗ξ ρ = 0} and the quotient topological space G/Gρ := {gGρ | g ∈ G} admits a unique smooth (real or complex) Banach manifold structure making the canonical projection π : G → G/Gρ a surjective submersion. The manifold G/Gρ is weakly symplectic relative to the two-form ωρ given by ωρ ([g])(Tg π(Te Lg ξ ), Tg π(Te Lg η)) := ρ, [ξ, η] ,

(7.5)

where ξ, η ∈ g, g ∈ G, [g] := π(g) = gGρ , and ·, · : g∗ × g → R (or C) is the canonical pairing between g∗ and g. Alternatively, this form can be expressed as ωρ ([g])(Tg π(Te Rg ξ ), Tg π(Te Rg η)) := Ad∗g −1 ρ, [ξ, η] . (7.6) The two-form ωρ is invariant under the left action of G on G/Gρ given by g ·[h] := [gh], for g, h ∈ G. Proof. The subgroup Gρ is clearly closed. For Banach Lie groups it is no longer true that closed subgroups are Lie subgroups (for a counterexample see Bourbaki [Bo3], Chapter III, Exercise 8.2). However, as in the finite dimensional case, if Gρ is assumed to be a Lie subgroup of G, then (Bourbaki [Bo3], Chapter III, §6.4, Corollary 1) ξ ∈ g is an element of the Lie algebra of Gρ if and only if exp tξ ∈ Gρ for all t ∈ R (or C depending on whether G is a real or complex Banach Lie group). Thus, since (see, e.g. Marsden and Ratiu [M-R2], Chapter 9) d Ad∗exp tξ ρ = Ad∗exp tξ ad∗ξ ρ, dt

Banach Lie-Poisson Spaces and Reduction

35

it follows that exp tξ ∈ Gρ ⇐⇒ Ad∗exp tξ ρ = ρ ⇐⇒ 0 = ⇐⇒ ad∗ξ ρ = 0 ⇐⇒ ξ ∈ gρ ,

d Ad∗exp tξ ρ = Ad∗exp tξ ad∗ξ ρ dt

which shows that the Lie algebra of Gρ is gρ . Since Gρ is assumed to be a Lie subgroup of G, the set G/Gρ has a unique smooth manifold structure such that the canonical projection π : G → G/Gρ is a submersion. The underlying manifold topology of G/Gρ is the quotient topology. The left action (g, [h]) ∈ G × G/Gρ → g · [h] := [gh] is smooth (see Bourbaki [Bo3], Chapter III, §1.6, Proposition 11, for a proof of these statements). In what follows we shall need the following observation. Condition (ii) implies that ad∗ξ (g∗ ) ⊂ g∗ for any ξ ∈ g. The two-forms defined in formulas (7.5) and (7.6) are equal. Indeed, taking (7.5) as the definition but applying it to tangent vectors of the form Tg π(Te Rg ξ ), Tg π(Te Rg η), we get ωρ ([g])(Tg π(Te Rg ξ ), Tg π(Te Rg η)) = ωρ ([g]) Tg π(Te Lg (Adg −1 ξ )), Tg π(Te Lg (Adg −1 ξ )) = ρ, [Adg −1 ξ, Adg −1 η] = ρ, Adg −1 [ξ, η] = Ad∗g −1 ρ, [ξ, η] , which gives formula (7.6). We shall prove now that the two-form (7.6) is well defined. Indeed, if [g] = [g ] and Tg π(Te Rg ξ ) = Tg π(Te Rg ξ ), then there is some h ∈ Gρ such that g = gh and hence Tg π(Te Rg ξ ) = Tg π(Te Rg ξ ) = Tgh π(Te Rgh ξ ) = Tg (π ◦ Rh )(Te Rg ξ ) = Tg π(Te Rg ξ ), which means that Tg π(Te Rg (ξ − ξ )) = 0. Due to the fact that the fibers of π are of the form gGρ , this is equivalent to Te Rg (ξ − ξ ) ∈ Te Lg (gρ ), that is, ξ − ξ ∈ Adg (gρ ). Thus there is some ζ ∈ gρ such that ξ = ξ + Adg ζ . Similarly, if Tg π(Te Rg η) = Tg π(Te Rg η ) there is some ζ ∈ gρ such that η = η + Adg ζ . Therefore, since ad∗ζ ρ = ad∗ζ ρ = 0, it follows that ωρ ([g ])(Tg π(Te Rg ξ ), Tg π(Te Rg η )) = Ad∗g −1 ρ, [ξ , η ]

= Ad∗(gh)−1 ρ, ξ + Adg ζ, η + Adg ζ

= Ad∗g −1 ρ, ξ + Adg ζ, η + Adg ζ

= Ad∗g −1 ρ, [ξ, η] + Ad∗g −1 ρ, Adg ζ, η

+ Ad∗g −1 ρ, ξ, Adg ζ + Ad∗g −1 ρ, Adg [ζ, ζ ]

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A. Odzijewicz, T.S. Ratiu

= Ad∗g −1 ρ, [ξ, η] + ρ, ζ, Adg −1 η + ρ, Adg −1 ξ, ζ + ρ, [ζ, ζ ] = Ad∗g −1 ρ, [ξ, η] + ad∗ζ ρ, Adg −1 η − ad∗ζ ρ, Adg −1 ξ + ad∗ζ ρ, ζ = Ad∗g −1 ρ, [ξ, η] = ωρ ([g])(Tg π(Te Rg ξ ), Tg π(Te Rg η)). The two-form ωρ is weakly non-degenerate. Indeed if [g] ∈ G/Gρ is given and if ωρ ([g])(Tg π(Te Lg ξ ), Tg π(Te Lg η)) = 0 for all Tg π(Te Lg η) then, by (7.5), ad∗ξ ρ, η = 0 for all η ∈ g and thus ξ ∈ gρ (since ad∗ξ ρ ∈ g∗ ) which is equivalent to Tg π(Te Lg ξ ) = 0. We shall prove that ωρ is a smooth closed two-form on G/Gρ by showing that the smooth one-form on G given by νρ (g)(Te Lg ξ ) := −ρ, ξ satisfies dνρ = π ∗ ωρ . Since π is a surjective submersion this immediately implies that ωρ is smooth and closed. To compute the exterior derivative of νρ , we denote by X, Y the vector fields on G given by X(g) = Te Lg ξ and Y (g) = Te Lg η and note that νρ (X)(g) = −ρ, ξ is constant and [X, Y ](g) = Te Lg [ξ, η]. Therefore, by Cartan’s formula, dνρ (g)(Te Lg ξ, Te Lg η) = dνρ (X, Y )(g) = X[νρ (Y )](g) − Y [νρ (X )](g) − νρ ([X, Y ])(g) = −νρ (g)(Te Lg [ξ, η]) = ρ, [ξ, η] = (π ∗ ωρ )(g)(Te Lg ξ, Te Lg η). To show that ωρ is G-invariant, we note that π is G-equivariant, ωρ = π ∗ νρ , and that νρ is G-invariant.

We shall see a concrete example of a symplectic form ωρ that is weak and not strong after Example 7.9. Next we study the coadjoint orbits of G through points of g∗ . Theorem 7.4. Let the Banach Lie group G and the element ρ ∈ g∗ satisfy the hypotheses of Theorem 7.3. Then the map ι : [g] ∈ G/Gρ → Ad∗g −1 ρ ∈ g∗

(7.7)

is an injective weak immersion of the quotient manifold G/Gρ into the predual space g∗ . Endow the coadjoint orbit O := {Ad∗g ρ | g ∈ G} with the smooth manifold structure making ι into a diffeomorphism. The push forward ι∗ (ωρ ) of the weak symplectic form ωρ ∈ 2 (G/Gρ ) to O has the expression (7.8) ωO (Ad∗g −1 ρ) ad∗Adg ξ Ad∗g −1 ρ, ad∗Adg η Ad∗g −1 ρ = ρ, [ξ, η], for g ∈ G, ξ, η ∈ g, and ρ ∈ g∗ . Relative to this symplectic form the connected components of the coadjoint orbit O are symplectic leaves of the Banach Lie-Poisson space g∗ . Proof. By Theorem 4.6 the predual g∗ is a (real or holomorphic) Banach Lie-Poisson space whose (real or complex) Poisson bracket is given by (4.2). For each ρ ∈ g∗ its characteristic subspace is therefore given by Sρ = {ad∗ξ ρ | ξ ∈ g} ⊂ g∗ since ad∗ξ (g∗ ) ⊂ g∗ for any ξ ∈ g.

Banach Lie-Poisson Spaces and Reduction

37

The map ι : [g] ∈ G/Gρ → Ad∗g −1 ρ ∈ O is a bijection, so we put on O the smooth Banach manifold structure making this bijection into a diffeomorphism. Since the map g ∈ G → Ad∗g −1 ρ ∈ g∗ is continuous, it thus follows that the inclusion O ⊂ g∗ is also continuous. We shall prove now that the map g ∈ G → Ad∗g −1 ρ ∈ g∗ is smooth. Indeed its derivative Te Lg ξ ∈ Tg G → − Ad∗g −1 ad∗ξ ρ = − ad∗Adg ξ Ad∗g −1 ρ ∈ g∗

(7.9)

is a continuous linear map from Tg G to g∗ , that is, the map g ∈ G → Ad∗g −1 ρ ∈ g∗ is

C 1 . Inductively, it follows that it is C ∞ . In addition, the range of the derivative at g is the characteristic subspace SAd∗−1 ρ at Ad∗g −1 ρ. g

Since the map g ∈ G → Ad∗g −1 ρ ∈ g∗ is Gρ -invariant, it follows that ι : [g] ∈ G/Gρ → Ad∗g −1 ρ ∈ g∗ is smooth and that the range of its derivative at [g], given by T[g] ι : Tg π(Te Lg ξ ) ∈ T[g] (G/Gρ ) → − ad∗Adg ξ Ad∗g −1 ρ ∈ g∗ , equals SAd∗−1 ρ . The g

map T[g] ι is injective. Indeed, if 0 = T[g] ι(Tg π(Te Lg ξ )) = − Ad∗g −1 ad∗ξ ρ , then ξ ∈ gρ and hence Tg π(Te Lg ξ ) = 0. This shows that ι is an injective weak immersion. Let us endow the manifold O ⊂ g∗ with the push forward weak symplectic form ωO given by the diffeomorphism [g] ∈ G/Gρ → Ad∗g −1 ρ ∈ O ⊂ g∗ . From the formula for its derivative given by (7.9) and (7.5), it immediately follows that this weak symplectic form on O ⊂ g∗ has the expression ωO (Ad∗g −1 ρ) ad∗Adg ξ Ad∗g −1 ρ, ad∗Adg η Ad∗g −1 ρ = ρ, [ξ, η].

(7.10)

Let now f ∈ C ∞ (g∗ ) and ρ ∈ g∗ . Since Xf (Ad∗g −1 ρ) = − ad∗Df (Ad∗ g

−1 ρ)

Ad∗g −1 ρ ∈ SAd∗−1 ρ , g

and SAd∗−1 ρ is the tangent space at Ad∗g −1 ρ to the orbit O, it follows that this orbit is a g

Poisson submanifold of g∗ . Since

Df (Ad∗g −1 ρ) = Adg D(f ◦ Ad∗g −1 )(ρ) for g ∈ G and ρ ∈ g∗ , it follows that Xf (Ad∗g −1 ρ) = − ad∗

Adg D(f ◦Ad∗ −1 )(ρ) g

Ad∗g −1 ρ ,

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A. Odzijewicz, T.S. Ratiu

and hence for any η ∈ g we have by (7.8), ωO (Ad∗g −1 ρ) Xf (Ad∗g −1 ρ), − ad∗Adg η Ad∗g −1 ρ ∗ ∗ ∗ ∗ ∗ Ad −1 ρ , − ad = ωO (Adg −1 ρ) − ad Adg η Adg −1 ρ ∗ g Adg D(f ◦Ad

g −1

)(ρ)

= −ad∗η ρ, D(f ◦ Ad∗g −1 )(ρ) = −Df (Ad∗g −1 ρ) Ad∗g −1 ad∗η ρ = −Df (Ad∗g −1 ρ) ad∗Adg η Ad∗g −1 ρ , = ρ, [D(f

◦ Ad∗g −1 )(ρ), η]

which shows that the Hamiltonian vector field Xf relative to the Lie-Poisson structure (4.2) computed at a point of the orbit O is also Hamiltonian relative to the weak symplectic form (7.8). Thus the Lie-Poisson structure on g∗ and the weak symplectic form on the orbit are compatible, i.e., the Lie-Poisson structure (4.2) induces the weak symplectic form (7.8) on the orbit. Summarizing, we have shown that each connected component of a coadjoint orbit is a connected smooth Banach manifold, that the inclusion of the orbit in g∗ is a weak injective immersion such that its tangent map has at each point as range the characteristic subspace, and that the Lie-Poisson structure induces the weak symplectic form on the orbit given by the canonical diffeomorphism of the orbit with the quotient G/Gρ . In addition, since the orbits are a partition of g∗ , the maximality condition holds automatically.

Next we analyze a remarkable particular situation that will give much stronger conclusions. Theorem 7.5. Let the Banach Lie group G and the element ρ ∈ g∗ satisfy the hypotheses of Theorem 7.3. The following conditions are equivalent: (i) ι : G/Gρ → g∗ is an injective immersion; (ii) the characteristic subspace Sρ := {ad∗ξ ρ | ξ ∈ g} is closed in g∗ ; (iii) Sρ = g◦ρ , where g◦ρ is the annihilator of gρ in g∗ . Endow the coadjoint orbit O := {Ad∗g ρ | g ∈ G} with the manifold structure making ι a diffeomorphism. Then, under any of the hypotheses (i)–(iii), the two-form given by (7.8) is a strong symplectic form. Proof. It is a general fact that any set is included in its double annihilator, so Sρ ⊂ Sρ◦◦ . We shall prove now that if Sρ is closed, then this inclusion is actually an equality (this is also a general fact for closed subspaces in any Banach space). Indeed, if Sρ = Sρ◦◦ ⊂ g∗ , then closedness of Sρ guarantees by the Hahn-Banach theorem that there exists 0 = ϕ ∈ g∗∗ such that ϕ ∈ Sρ◦ and ϕ ∈ / Sρ◦◦◦ . The inclusion Sρ ⊂ Sρ◦◦ implies Sρ◦◦◦ ⊂ Sρ◦ . Since ◦ it is in general true that Sρ ⊂ Sρ◦◦◦ we get Sρ◦ = Sρ◦◦◦ , which contradicts the existence of ϕ. Thus if Sρ is closed, then Sρ = Sρ◦◦ . Using the identity η, ad∗ξ ρ = −ξ, ad∗η ρ for any ξ, η ∈ g, it follows that Sρ◦ = gρ . Taking the annihilator in this relation and using the equality Sρ = Sρ◦◦ , for Sρ closed, yields Sρ = g◦ρ . Thus Sρ is closed if and only if Sρ = g◦ρ . This proves the equivalence of (ii) and (iii). Assume now that (iii) holds. Since Gρ is a Banach Lie subgroup one has the splitting g = gρ ⊕ gcρ , where gcρ is a closed subspace. This induces the splitting of the dual space ◦ ◦ g∗ = g◦ρ ⊕ gcρ = Sρ ⊕ gcρ . Thus, using the inclusion Sρ ⊂ g∗ we obtain the splitting

◦ g∗ = Sρ ⊕ gcρ ∩ g∗ .

Banach Lie-Poisson Spaces and Reduction

The identity

39

T[g] ι T[g] (G/Gρ ) = SAd∗−1 ρ = Ad∗g −1 Sρ , g

(7.11)

and the fact that Ad∗g −1 is a Banach space isomorphism show that ι is a immersion. So (i) holds. Conversely, if (i) is valid then Sρ is closed by definition, so (ii) is satisfied. In order to prove that (7.8) is a strong symplectic form, let us notice that since Sρ is a closed subspace of g∗ , by the Hahn-Banach Theorem, for any f ∈ Sρ∗ there exists an element η ∈ (g∗ )∗ = g such that f (ad∗ξ ρ) = ad∗ξ ρ, η = ρ, [ξ, η] for any ξ ∈ g. So the linear map ad∗η ρ ∈ Sρ → ωO (ρ)(·, ad∗η ρ) ∈ Sρ∗ is surjective. Due to left invariance of ωO this surjectivity will hold at all points of the orbit O. Since ωO is in general a weak symplectic form according to Theorem 7.4, it follows that ωO is a strong symplectic form.

Corollary 7.6. Let the Banach Lie group G and the element ρ ∈ g∗ satisfy the hypotheses of Theorem 7.3. Then ι : G/Gρ → g∗ is a quasi immersion if and only if it is an immersion. We now apply the above theorems to the important class of W ∗ -algebras. From Theorem 5.1 one knows that the predual m∗ of the (complex) W ∗ -algebra m is a holomorphic Banach Lie-Poisson space. Recall that the set G(m) of invertible elements of m is a Banach Lie group who acts on m∗ by the coadjoint action. Corollary 7.7. Let ρ ∈ m∗ be such that G(m)ρ = im R ∗ ∩ G(m), where R ∗ is given by (6.7) (so R := (R ∗ )∗ |m∗ is a quantum reduction). Then the connected components of the coadjoint orbit through ρ are weakly immersed weak symplectic manifolds that are symplectic leaves of the Banach Lie-Poisson space m∗ . Proof. By Proposition 6.8, G(m)ρ is a Lie subgroup of G(m), so the hypotheses of Theorems 7.3 and 7.4 hold and the conclusion follows.

Corollary 7.8. Let ρ ∈ m∗ be such that G(m)ρ = im R ∗ ∩ G(m), where R ∗ is given by (6.7). Then the following conditions are equivalent: (i) Sρ = ker R; (ii) the map ι : G(m)/G(m)ρ → m∗ is an injective immersion. Under any of these conditions, the coadjoint orbit O endowed with the smooth manifold structure making ι a diffeomorphism onto its image is strong symplectic. Proof. If Sρ = ker R, then Sρ is closed and Theorem 7.5 applies thus guaranteeing that ι is an injective immersion. Conversely, if ι is an immersion, Theorem 7.5 states that Sρ = m◦ρ . However, the hypothesis and Proposition 6.8 guarantee that Gρ is a Lie subgroup of G(m) whose Lie algebra is im R ∗ . On the other hand it is clear that the Lie algebra of G(m)ρ is mρ since exp(λx) ∈ G(m)ρ for all x ∈ mρ and all λ ∈ C (see Bourbaki [Bo3], Chapter III, §6.4, Corollary 1). Therefore mρ = im R ∗ . The decomposition m = im R ∗ ⊕ ker R ∗ and the one induced on the dual imply the general identities ◦ ◦ ker R ∗ = im R ∗∗ , im R ∗ = ker R ∗∗ and where the annihilators are taken in m∗ .

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A. Odzijewicz, T.S. Ratiu

m◦ρ

Therefore, using mρ = im R ∗ , we get ker R = ker R ∗∗ ∩ m∗ = (im R ∗ )◦ ∩ m∗ = = Sρ and the equivalence of (i) and (ii) is proved. The last statement follows by directly applying Theorem 7.5.

Example 7.9. Take in the previous considerations m = L∞ (M), m∗ = L1 (M), and the quantum reduction map R : L1 (M) → L1 (M) defined by (6.3), where N k=1 Pk = 1, with N ∈ N or N = ∞. If N = ∞, let ρ=

∞

λk P k ,

λk ∈ C,

λk = λ = 0 for k = ,

rank Pk < ∞ for k ≥ 1,

k=1

(7.12) and if N < ∞ let ρ=

N

λk Pk , λk ∈ C, λk = λ for k = ,

λ1 = 0,

rank Pk < ∞ for k ≥ 2.

k=1

(7.13) Thus ρ ∈ L1 (M). It is easy to check that N Pk XPk | X ∈ L∞ (M) . L∞ (M)ρ = im R ∗ = k=1

So all conclusions of Theorem 7.4 hold, that is, the coadjoint orbit O := {gρg −1 | g ∈ GL∞ (M)} ⊂ L1 (M) through ρ is weakly immersed in L1 (M) and is weakly symplectic relative to the two-form ωO (gρg −1 )([gXg −1 , gρg −1 ], [gY g −1 , gρg −1 ]) = tr(ρ[X, Y ])

(7.14)

for ρ ∈ L1 (M) given by (7.12) or (7.13), g ∈ GL∞ (M), X, Y ∈ L∞ (M). In addition, for M a complex separable Hilbert space, recall that the group GL∞ (M) is path connected (see e.g. Boos and Bleecker [B-B] §I.6) and hence the coadjoint orbit is also connected; thus it is a symplectic leaf of the Banach Lie-Poisson space L1 (M). The characteristic subspace Sρ = {[X, ρ] | X ∈ L∞ (M)} is contained in     Pk XP | X ∈ L∞ (M) = ker R ∗ , ker R   k =

and if N ∈ N one has

Pk XP = [ρ, Y ]

k =

for some Y ∈ L∞ (M) which is related to X through the system of equations Pk XP = (λk − λ )Pk Y P for all k = . Note that if N = ∞, the above system is not solvable for some Y ∈ L∞ (M). Therefore, if N ∈ N, Sρ = ker R and by Theorem 7.5 one concludes that the connected coadjoint orbit is immersed in L1 (M) and that it is strongly symplectic.

Banach Lie-Poisson Spaces and Reduction

41

Remark. The weak symplectic form ωρ given by (7.5) or (7.6) is, in general, not strong since ωρ ([e]) is, in general, not a strong bilinear form on T[e] (G/Gρ ). To begin with, one notices that Te π(ξ ) ∈ T[e] (G/Gρ ) → [ξ ] ∈ g/gρ is a linear continuous bijective map and hence a Banach space isomorphism. Thus ωρ ([e]) can be viewed as a bilinear continuous map ωρ ([e]) : g/gρ × g/gρ → R given by ωρ ([e])([ξ ], [η]) = ρ, [ξ, η]. The map [ξ ] ∈ g/gρ → ωρ ([e])([ξ ], [·]) = ad∗ξ ρ, · ∈ (g/gρ )∗ is clearly linear continuous and injective. Thus, if the symplectic form ωρ were strong, then the Banach spaces g/gρ and (g/gρ )∗ would necessarily be isomorphic. Here is a concrete situation in which this cannot occur. Consider the case described in Example 7.9 for the trace class operator ρ given by (7.12). Then L∞ (M)/L∞ (M)ρ ∼ = ker R ∗ since L∞ (M)ρ = im R ∗ . The map : ker R ∗ X → Tr(ρ[X, ·]) = Tr([ρ, X]·) ∈ (ker R ∗ )∗ given by the defining formula (7.14) of the symplectic form ωO , has the following explicit form ∗

: ker R X →

∞

(λl − λm )Pl XPm ∈ ker R (ker R ∗ )∗ = (im R ∗ )◦ ,

l =m

where the annihilator is taken in (L∞ (M))∗ . This formula shows that : ker R ∗ → (ker R ∗ )∗ is not surjective if ρ is given by (7.12). Hence, in this case, the orbit symplectic form ωO given by (7.14) is not strong. It was shown in Bona [2000] that unitary group coadjoint orbits through Hermitian finite rank operators are always strong symplectic manifolds. We present this case below. Example 7.10. We apply the considerations of this section to the real closed Banach Lie subgroup U ∞ (M) := {U ∈ L∞ (M) | U U ∗ = U ∗ U = I } of unitary elements of GL∞ (M) (Bourbaki [Bo3], Chapter III, §3.10, Corollary 2). Its Lie algebra consists of the skew Hermitian bounded operators u∞ (M) := {X ∈ L∞ (M) | X + X ∗ = 0}; this is a closed split real Banach Lie subalgebra of L∞ (M). To study this case, we place ourselves in the context of Example 4.9, that is, we take g = L∞ (M), g∗ = L1 (M), g∗ R = L1 (M)R , gR = L∞ (M)R (in other words, the Banach spaces L1 (M) and L∞ (M) thought of as real Banach spaces), the continuous R-linear involution σ : L1 (M)R → L1 (M)R given by σρ = −ρ ∗ , for ρ ∈ L1 (M), and the complex structure I given by Iρ = iρ. It is easily verified that the involution σ satisfies the conditions (i), (ii), and (iii) of Example 4.9. Then, by construction, gσ∗ = {ρ ∈ L1 (M) | ρ + ρ ∗ = 0} =: u1 (M) and, as was shown in Example 4.9, u1 (M) is a real Banach Lie-Poisson space and the map R : L1 (M)R → u1 (M) given by R = (id + σ )/2 is a linear Poisson map. The same type of argument as in Example 7.9 shows that one can directly apply Theorems 7.4 and 7.5 to G = U ∞ (M) and g∗ = u1 (M) . The symplectic leaves in this case correspond to the infinite dimensional flag manifolds and the strong symplectic form given by (7.14) (ρ of finite rank and the arguments in the correct spaces) coincides with the imaginary part of the natural K¨ahler metric on these manifolds. A particular example of such an infinite dimensional flag manifold is the projectivized Hilbert space CP(M) thought of as immersed in L1 (M) as the coadjoint orbit through ρ := |ψψ|/ψ|ψ for any |ψ ∈ M. We next discuss the cotangent bundle of a Banach Lie group and introduce a remarkable submanifold, called in the sequel the precotangent bundle. Consider a Banach Lie group G with Banach Lie algebra g admitting a predual g∗ and assume that Ad∗G (g∗ ) ⊂ g∗ . If Lg and Rg denote the left and right translations by g ∈ G respectively, it follows

42

A. Odzijewicz, T.S. Ratiu

that Tg Lg −1 : Tg G → Te G = g and Tg Rg −1 : Tg G → Te G = g are Banach space isomorphisms. Their duals Tg∗ Lg −1 : g∗ → Tg∗ G and Tg∗ Rg −1 : g∗ → Tg∗ G are therefore also Banach space isomorphisms. Define Tg ∗ G := Tg∗ Lg −1 g∗ , T∗ G := ∪g∈G Tg ∗ G, and conclude, as usual, that T∗ G is a vector bundle over G which is also a subbundle of T ∗ G (see, e.g. Abraham, Marsden, Ratiu [A-M-R] for such an argument); it will be called the precotangent bundle of G. This construction could have been equally well done using right translations since Tg∗ Rg −1 Ad∗g −1 g∗ = Tg∗ Lg −1 g∗ and, by hypothesis, Ad∗g g∗ = g∗ for any g ∈ G. The precotangent bundle T∗ G has been constructed using the left trivialization L : T∗ G → G × g∗ , L(ρg ) := (g, Te∗ Lg ρg ) with inverse L−1 (g, ρ) = Tg∗ Lg −1 ρ, for ρg ∈ Tg ∗ G and ρ ∈ g∗ . Completely analogous formulas hold for the right trivialization R : T∗ G → G × g∗ , R(ρg ) := (g, Tg∗ Rg −1 ρg ), R −1 (g, ρ) = Te∗ Rg −1 ρ; L and R are vector bundle isomorphisms covering the identity of G. Denote by π : T ∗ G → G the cotangent bundle projection and use the same letter to denote its restriction to T∗ G. The usual construction of the canonical one-form on the cotangent bundle T ∗ G works also in the case of the precotangent bundle T∗ G. Indeed, define the one form on T ∗ G or on T∗ G by (ρg )(v) := ρg , Tρg π(v)

(7.15)

for any ρg ∈ Tg∗ G (respectively Tg ∗ G), v ∈ Tρg (T ∗ G) (respectively Tρg (T∗ G)) and where the pairing is between Tg∗ G and Tg G (respectively Tg ∗ G and Tg G). Left trivialized, this formula reads L (g, ρ)(ug , µ) := (L∗ )(g, ρ)(ug , υ) = ρ, Tg Lg −1 ug

(7.16)

for g ∈ G, ug ∈ Tg G, and ρ, µ ∈ g∗ (respectively g∗ ), where the pairing is now between g∗ (respectively g∗ ) and g. Define the canonical symplectic form on T ∗ G or T∗ G by := −d and let L := L∗ . A computation identical to the one in the finite dimensional case using the identity L = −dL (see Abraham and Marsden [A-M], §4.4), leads to the expression of the canonical two-form in the left trivialization L (g, ρ) (ug , µ), (vg , ν) := (L∗ )(g, ρ) (ug , µ), (vg , ν) = ν, Tg Lg −1 ug − µ, Tg Lg −1 vg − ρ, [Tg Lg −1 ug , Tg Lg −1 vg ],

(7.17)

where g ∈ G, ug , vg ∈ Tg G, and ρ, µ, ν ∈ g∗ (respectively g∗ ). This formula immediately shows that L and hence is a weak symplectic form on both T ∗ G and T∗ G. We shall see below that it is not strong in general, for different reasons, on both T ∗ G and T∗ G. ∗ To show that L is strong on G×g∗ , one needs to prove that for fixed (g, ρ) ∈ G×g ∗ the linear continuous map (ug , µ) ∈ Tg G × g → L (g, ρ) (ug , µ), (·, ·) ∈ (Tg G × g∗ )∗ ∼ = Tg∗ G × g∗∗ is surjective, that is, given αg ∈ Tg∗ G and ∈ g∗∗ one can find ug ∈ Tg G and µ ∈ g∗ such that ν, Tg Lg −1 ug − µ + ad∗Tg L

g −1 ug

ρ, Tg Lg −1 vg = αg , vg + , ν

for all vg ∈ Tg G, ν ∈ g∗ . If this were possible, a necessary condition is that = Tg Lg −1 ug which is not the case if g is strictly included in g∗∗ . If the Banach space g is reflexive then ∈ g and one can choose µ = − ad∗ ρ − Te∗ Lg αg . Thus, if g is reflexive, the canonical weak symplectic form on T ∗ G is strong. Next we analyze L on G×g∗ . As before, we fix (g, ρ) ∈ G×g∗ and study the linear continuous map (ug , µ) ∈ Tg G × g∗ → L (g, ρ) (ug , µ), (·, ·) ∈ (Tg G × g∗ )∗ ∼ =

Banach Lie-Poisson Spaces and Reduction

43

Tg∗ G × g. To prove its surjectivity one needs to find for given αg ∈ Tg∗ G and ξ ∈ g a vector ug ∈ Tg G and a form µ ∈ g∗ such that ν, Tg Lg −1 ug − µ + ad∗Tg L

g −1 ug

ρ, Tg Lg −1 vg = αg , vg + ν, ξ

for all vg ∈ Tg G, ν ∈ g∗ . This identity implies that ug = Te Lg ξ , which, unlike the previous case, is possible. However, this identity also requires that µ = − ad∗ξ ρ − Te∗ Lg αg which is, in general, impossible to achieve since Te∗ Lg αg ∈ g∗ but is not necessarily an element of g∗ . In spite of this obstruction, there is a Poisson bracket on G × g∗ . Given f, h ∈ C ∞ (G × g∗ ) their Poisson bracket is given by {f, h}(g, ρ) = d1 f (g, ρ), Te Lg d2 h(g, ρ) − d1 h(g, ρ), Te Lg d2 f (g, ρ) − ρ, [d2 f (g, ρ), d2 h(g, ρ)] , (7.18) where d1 f (g, ρ) ∈ Tg∗ G and d2 f (g, ρ) ∈ (g∗ )∗ = g are the first and second partial derivatives of f , the pairing in the first two terms is between Tg∗ G and Tg G, whereas in the third term it is between g∗ and g. Conditions (i) and (ii) in Definition 2.1 are satisfied. The proof of the Jacobi identity is a tedious direct verification. However, condition (iii) does not hold. Indeed formula (7.18) shows that the Hamiltonian vector field of h, if well defined, must have the expression Xh (g, ρ) = Te Lg d2 h(g, ρ), ad∗d2 h(g,ρ) ρ − Te∗ Lg d1 h(g, ρ) (7.19) The same obstruction encountered in the attempted proof of the strongness of L on G × g∗ appears here in the second summand of the second component: the term Te∗ Lg d1 h(g, ρ) is, in general, not an element of g∗ . Thus, unlike T ∗ G, the precotangent bundle T∗ G is naturally endowed with a Poisson bracket, but is not a Poisson manifold in the sense of Definition 2.1. However, before Lie-Poisson reduction, the unreduced space G × g∗ is only weakly symplectic, admits a Poisson bracket, but has no Hamiltonian vector fields. For functions admitting Hamiltonian vector fields, the Poisson bracket is naturally induced by the weak symplectic form which is the pull back of the canonical symplectic form on the cotangent bundle of the group. Finally, the projection G × g∗ → g∗ preserves the Poisson brackets, if one changes the sign of the Lie-Poisson bracket on g∗ . Similar considerations can be carried out for right translations and one obtains, as in finite dimensions, a dual pair (Weinstein [W1], Vaisman [V]) g∗ + ←− T∗ G −→ g∗ − where the signs refer to the Lie-Poisson bracket on g∗ ; the two arrows are the momentum maps for left and right translations (see §8 for a presentation of momentum maps in our setting and Marsden and Ratiu [M-R2] for more information in the finite dimensional case). 8. Momentum Maps and Reduction In this section we shall explore the relationship between the classical theory of reduction for Poisson manifolds discussed in §3 and that of quantum reduction presented

44

A. Odzijewicz, T.S. Ratiu

in § 6. We shall show that this link will be crucial for the integration and quantization of Hamiltonian systems. We shall introduce a definition of the momentum map which is a direct generalization of this concept from finite dimensional Poisson geometry. Definition 8.1. A momentum map is a Poisson map J : P → b from a Banach Poisson manifold P to a Banach Lie-Poisson space b. Recall that b is a Banach space, that C ∞ (b) is endowed with a Poisson bracket {·, ·}, that b∗ is closed under this bracket, and that ad∗b∗ b ⊂ b. Thus b∗ is a Lie algebra and the prescription ξ ∈ b∗ → ξP := Xξ ◦J defines a (left) Lie algebra action on P , that is, [ξP , ηP ] = −[ξ, η]P for any ξ, η ∈ b∗ . Indeed, recalling that the Hamiltonian vector field defined by h ∈ C ∞ (P ) is defined by df (Xh ) = {f, h}, the Jacobi identity for the Poisson bracket is equivalent to [Xf , Xg ] = −X{f,g} . Using this relation and the fact that J is a Poisson map, we conclude [ξP , ηP ] = [Xξ ◦J , Xη◦J ] = −X{ξ ◦J,η◦J } = −X{ξ,η}◦J = −[ξ, η]P , which proves that ξ → ξP is indeed a (left) Lie algebra action. Theorem 8.2 (Noether’s Theorem). If h ∈ C ∞ (P ) is b∗ -invariant, that is, dh, ξP = 0 for all ξ ∈ b∗ , then J is conserved along the flow of the Hamiltonian vector field Xh . Proof. The condition of invariance states that 0 = dh, Xξ ◦J = {h, ξ ◦ J } = −d(ξ ◦ J ), Xh for every ξ ∈ b∗ , which is equivalent to d σh (t)∗ (ξ ◦ J ) = 0 dt for every ξ ∈ b∗ , where σh (t) is the flow of Xh . This in turn means that ξ ◦J ◦σh (t) = ξ ◦J , which says that ξ, (J ◦ σh (t) − J )(b) = 0 for all b ∈ b. Since ξ is arbitrary, one concludes that (J ◦ σh (t) − J )(b) = 0 for every b ∈ b, that is, J ◦ σh (t) = J for all t.

A Hamiltonian system (P , {·, ·}P , h) is called collective if there is a momentum map J : P → b and a function H ∈ C ∞ (b) such that h = H ◦ J . Therefore, the correspondb on P and b respectively are J -related, that ing Hamiltonian vector fields XhP and XH is, b T J ◦ XhP = XH ◦ J,

which is equivalent to the commutation of the respective flows σh (t) and σH (t) of XhP b respectively, that is, and XH σH (t) ◦ J = J ◦ σh (t). Thus the integration of Hamilton’s equations on b given by b˙ = − ad∗DH (b) b,

(8.1)

Banach Lie-Poisson Spaces and Reduction

45

where b ∈ b, leads to the partial integration of Hamilton’s equations f˙ = {f, h}P

(8.2)

on P , where f ∈ C ∞ (P ). If J : P → b is an embedding, then solving (8.1) is equivalent to solving (8.2). In the other extreme case, namely when the trajectory {σH (t)(b) | t ∈ R} is a point b ∈ b, any trajectory σH (t)(p) with p ∈ J −1 (b) remains in the level set J −1 (b). If there is a distribution E covering this level set satisfying the hypotheses of the classical reduction theorem, then the trajectories above drop to the quotient and one is led to the problem of solving a reduced system. Provided this can be integrated, for example, if the reduced system is integrable, then the standard reconstruction method (Abraham and Marsden [A-M], §4.3) gives the solution of the original system on the level set of the momentum map. In the special case when b = L1 (M), Eq. (8.1) assumes the form of the nonlinear Liouville-von Neumann equation ρ˙ = [DH (ρ), ρ]

(8.3)

for ρ ∈ L1 (M). The search for a collective Hamiltonian system on P is equivalent to finding a “Lax representation" on L1 (M). The functions Tk := (tr ρ k )/k, k ∈ N are Casimir functions on L1 (M). Therefore the functions tk := Tk ◦ J , k ∈ N, are, in general, integrals of motion in involution for the system (8.2). If J : P → b is an embedding, then they are Casimirs of the system given by the phase space P . So, the problem of integration of an equation having Lax representation in L1 (M) reduces to a large extent to the integration of Eq. (8.3). For direct investigations of the nonlinear Liouville-von Neumann equation in the physics literature, see for example Leble and Czachor [L-C] and references therein. We shall illustrate the above considerations with the example of the infinite Toda lattice system associated to the Banach Lie-Poisson space L1 (M). Example 8.3. The infinite Toda lattice system. The details for this example can be found in Odzijewicz and Ratiu [O-R]. On the weak symplectic Banach space ∞ × 1 = (1 )∗ × 1 define the Hamiltonian H (q, p) :=

∞

∞

k=1

k=1

1 2 pk + αk λk exp(qk − qk+1 ), 2

(8.4)

where λk = 0, {αk }, {λk } ∈ 1 . Since the Banach space on which this Hamiltonian is defined is only weak symplectic, the existence of the Hamiltonian vector field associated to H is not guaranteed. Formally, this Hamiltonian is that of the Toda lattice. We will also assume that ∞ p k=1 k = 0, which means that the velocity of the center of mass is zero. We also observe that H is invariant relative to the action of R on the q-space by translation. Thus we shall consider H defined on the weak symplectic Banach space (∞ /Rq0 ) × 10 , where ∞ /Rq0 is the quotient Banach space by the closed subspace Rq0 , where q0k = 1 for all k ∈ N, and 10 := ker q0 . Relative to the canonical coordinates xk := qk − qk+1 on ∞ /Rq0 and pk on 10 the weak symplectic form has the expression ∞ ∞ ω = −d pk dxk = dxk ∧ dpk . (8.5) k=1

k=1

46

A. Odzijewicz, T.S. Ratiu

Let M be a separable Hilbert space. Let Pn := |nn| : M → M be the rank one projection onto the span of |n. If ρ ∈ L1 (M) and X ∈ L∞ (M) write ρ=

∞

ρnm |nm|

and

X=

n,m=1

∞

Xnm |nm|,

n,m=1

where ρnm := n|ρ|m and Xnm := n|X|m. From Example 6.7 we know that L1 (M)− is a Banach Lie-Poisson space whose dual is the Banach Lie algebra L∞ (M)+ . The pairing between these two spaces is given by ρ− , X+ := tr(ρ− X+ ) = ρnm Xmn , n≥m

where ρ− ∈ L1 (M)− and X+ ∈ L∞ (M)+ . Using this pairing, a direct verification shows that the coinduced Poisson bracket of L1 (M)− has the expression {f, g}L1 (M)− (ρ− ) = tr([(π−1 )∗ (df (ρ− )), (π−1 )∗ (dg(ρ− ))]ρ− ) $ # ∂f ∂g ∂g ∂f = ρn , − ∂ρnm ∂ρm ∂ρnm ∂ρm

(8.6)

n≥m≥

where π−1 : L1 (M) → L1 (M)− is the projector given by π−1 (ρ) := ρnm |nm|, n≥m

(π−1 )∗ : (L1 (M)− )∗ ∼ = L∞ (M)+ → L∞ (M) is its dual, and the formula (π−1 )∗ (df (ρ− ))mn =

∂f (ρ− ) ∂ρnm

was used in the proof of the second equality. Define the Flaschka transformation J : (∞ /Rq0 ) × 10 → L1 (M)− by J (z) =

∞

pk |kk| + λk exk |k + 1k| .

(8.7)

k=1

One verifies that J is a smooth injective map whose tangent map at every point is a continuous injection. Since the closed two-form ω given by (8.5) is only weak, the Poisson bracket of two functions φ, ψ ∈ C ∞ (∞ /Rq0 × 10 ) cannot be defined, in general. However, for collective functions f ◦ J and g ◦ J , where f , g ∈ C ∞ (L1 (M)− ), one has d(f ◦ J ), Xg◦J = −Xf ◦J [g ◦ J ] = −{f, g}L1 (M)− ◦ J since in this case, the Hamiltonian vector field Xf ◦J =

$ ∞ # ∂(f ◦ J ) ∂ ∂(f ◦ J ) ∂ − ∂pk ∂xk ∂xk ∂pk k=1

Banach Lie-Poisson Spaces and Reduction

47

on the weak symplectic manifold (∞ /Rq0 ×10 , ω) makes sense, as a derivation, if it acts on collective functions g ◦ J . This is so because the sequence {ρk+1,k }k∈N = {λk exk }k∈N belongs to 1 and the four sequences {∂f/∂ρk+1,k }k∈N , {∂g/∂ρk+1,k }k∈N , {∂f/∂ρk,k }k∈N , {∂g/∂ρk,k }k∈N all belong to ∞ . Thus, the Flaschka transformation is a momentum map in a generalized sense. Additionally, let us mention that the Toda lattice Hamiltonian (8.4) is of the form H = h ◦ J , for (8.8) h(ρ− ) = tr(ρ− + a)2 , ∞ where a := n=1 αk |kk + 1| ∈ L1 (M), that is, ∞ n=1 |αk | < ∞. Thus, the Toda lattice is a Hamiltonian system on a Poisson submanifold of L1 (M)− endowed with the bracket (8.6) and relative to the Hamiltonian function (8.8). We shall prove below a version of an involution theorem combining the KostantSymes and the Mischchenko-Fomenko involution theorems for L1 (M)− with the goal to show that the functions hk (ρ− ) := tr(ρ− + a)k /k, k ∈ N, are in involution relative to the Poisson bracket {·, ·}L1 (M)− . The proof turns out to follow the finite dimensional one (see Kostant [Ko3] or Ratiu [R]). Note that f is a Casimir function on L1 (M) if and only if Xf = 0, which by (5.8) is equivalent to [df (ρ), ρ] = 0 for every ρ ∈ L1 (M). Decompose L1 (M) = L1 (M)− ⊕ L1 (M)+ and L∞ (M) = L∞ (M)+ ⊕ L∞ (M)− , where L1 (M)+ := {ρ ∈ L1 (M) | ρnm = 0 for n ≥ m} are the strictly upper triangular linear trace class operators (no diagonal) and L∞ (M)− := {X ∈ L∞ (M) | Xnm = 0 for m ≥ n} are the strictly lower triangular bounded linear operators (no diagonal). Let + π−1 : L1 (M) → L1 (M)− , π 1 : L1 (M) → L1 (M)+ , π+∞ : L∞ (M) → L∞ (M)+ , − and π ∞ : L∞ (M) → L∞ (M)− be the projections associated to the Banach space direct sums L1 (M) = L1 (M)− ⊕ L1 (M)+ and L∞ (M) = L∞ (M)+ ⊕ L∞ (M)− respectively. With this notation the Poisson bracket (8.6) becomes

˜ − )) ρ− , (8.9) {f, g}L1 (M)− (ρ− ) = tr π+∞ (d f˜(ρ− )), π+∞ (d g(ρ where on the right-hand side, f˜ and g˜ are arbitrary extensions of f, g : L1 (M)− → R to L1 (M) respectively. Thus d f˜(ρ− ) ∈ L∞ (M) and π+∞ (d f˜(ρ− )) ∈ L∞ (M)+ and similarly for g. Proposition 8.4. Let a ∈ L1 (M) be a given element satisfying tr(a[L∞ (M)+ , L∞ (M)+ ]) = 0 and

tr(a[L∞ (M)− , L∞ (M)− ]) = 0.

For any two Casimir functions f, g on L1 (M) the functions fa (ρ− ) := f (ρ− + a), ga (ρ− ) := g(ρ− + a) are in involution on L1 (M)− . Proof. Note that in (8.9) one can take f˜a (ρ) = f (ρ + a) for any ρ ∈ L1 (M). Since tr(a[L∞ (M)+ , L∞ (M)+ ]) = 0, it follows {fa , ga }L1 (M)− (ρ− )

= tr π+∞ (d f˜a (ρ− )), π+∞ (d g˜ a (ρ− )) ρ−

= tr π+∞ (df (ρ− + a)), π+∞ (dg(ρ− + a)) (ρ− + a)

= tr π+∞ (df (ρ− + a)) dg(ρ− + a) − π ∞ − (dg(ρ− + a)), ρ− + a

48

A. Odzijewicz, T.S. Ratiu

= − tr π+∞ (df (ρ− + a)) π ∞ − (dg(ρ− + a)), ρ− + a

= − tr π+∞ (df (ρ− + a)), π ∞ − (dg(ρ− + a)) (ρ− + a)

= tr π ∞ − (dg(ρ− + a)), π+∞ (df (ρ− + a)) (ρ− + a)

= tr π ∞ − (dg(ρ− + a)) π+∞ (df (ρ− + a)), ρ− + a

= tr π ∞ − (dg(ρ− + a)) df (ρ− + a) − π ∞ − (df (ρ− + a)), ρ− + a

= − tr π ∞ − (dg(ρ− + a)) π ∞ − (df (ρ− + a)), ρ− + a

= − tr π ∞ − (dg(ρ− + a)), π ∞ − (df (ρ− + a)) (ρ− + a) = 0, because tr(ρ− [L∞ (M)− , L∞ (M)− ]) = 0 and tr(a[L∞ (M)− , L∞ (M)− ]) = 0.

∞ 1 In the case of the Toda lattice one takes a = n=1 αk |k + 1k| ∈ L (M) with ∞ ∞ − ∞ − n=1 |αk | < ∞ and then it immediately follows that tr(a[L (M) , L (M) ]) = 0 ∞ ∞ and tr(a[L (M)+ , L (M)+ ]) = 0. Thus, the hypotheses of Proposition 8.4 are satisfied and we conclude that all the functions hk (ρ− ) := tr(ρ− + a)k /k, k ∈ N, are in involution relative to the Poisson bracket {·, ·}L1 (M)− and hence the relation J ∗ ωO = ω shows that hk ◦ J are commuting conserved quantities for the Toda Hamiltonian H := h2 ◦ J . Next we discuss the Poisson reduction in a Banach Lie-Poisson space b associated to a quantum reduction operator R : b → b. Assume that i : N → b is a (locally closed) submanifold. Since T b|N = N × b, define the subbundle E ⊂ T b|N by Eb := {b} × ker R. Next, make the topological assumption that E ∩ T N is the tangent bundle to a regular foliation F and that the space of leaves M := N/F is a smooth manifold with the projection π : N → M a submersion. Lemma 8.5. The subbundle E is compatible with the Poisson structure of b. Proof. Let U be an open subset of b and f, g ∈ C ∞ (U, C) have the property that df, dg vanish on E, that is, df (b), ker R = 0 and dg(b), ker R = 0 for b ∈ U ∩ N . Therefore, there exist functions f˜, g˜ ∈ C ∞ (R(U ), C) such that f = f˜ ◦ R and g = g˜ ◦ R. Recall from §7 that the quantum reduction R : (b, {·, ·}) → (im R, {·, ·}R ) is a Poisson map. Thus, {f, g} = {f˜◦R, g◦R} ˜ = {f˜, g} ˜ R ◦R whence d{f, g}(b) = d{f˜, g} ˜ R (R(b))◦ R which implies that d{f, g}(b) vanishes on Eb = ker R. The following commutative diagram N

i

-

π

b

R ?

N/F

? J

- im R

summarizes the maps involved in the theorem below.

Banach Lie-Poisson Spaces and Reduction

49

Theorem 8.6. Let i : N → b be a submanifold, R : b → b be a quantum reduction, and E be the distribution on b given at every point by kerR. Assume that: (i) E ∩ T N is the tangent bundle of a regular foliation F on N and the projection π : N → M := N/F is a submersion; (ii) (ker R)◦ ⊂ ker R + Tn N for every n ∈ N . Then M is the reduction of b by (N, E) and is thus a Banach Poisson manifold. The map J : M → im R defined by J ([n]) := (R ◦ i)(n) is Poisson, that is, J is a momentum map. Proof. In view of the previous lemma, by Theorem 3.1, the two hypotheses guarantee that the triple (b, N, E) is reducible. Thus M is a Banach Poisson manifold. Since im R can be regarded as the quotient manifold obtained by collapsing the fibers of R, that is, by dividing with ker R, the inclusion map i is obviously compatible with the equivalence relations on N and on b. Therefore, i induces a smooth map J : M → im R on the quotients (see, for example, Abraham, Marsden, Ratiu [A-M-R] or Bourbaki [Bo2]) given by J ([b]) := R(i(b)). The diagram above commutes by construction. It remains to be shown that J is a Poisson map. Let f, g ∈ C ∞ (im R, C). Then f ◦ R ∈ C ∞ (b, C) is an extension of f ◦ J ◦ π ∈ ∞ C (N, C) and similarly for g. Therefore, by the definition of the reduced bracket on M, since R is a Poisson map, we get {f ◦ J, g ◦ J }M ◦ π = {f ◦ R, g ◦ R} ◦ i = {f, g}R ◦ R ◦ i = {f, g}R ◦ J ◦ π. Since π is a surjective map, this implies that J : M → im R is a Poisson map.

Example 8.7. Averaging. Let i : N → L1 (M) be the inclusion map of a smooth (regular) Banach submanifold in L1 (M). Let G be a compact Lie group and denote by µ(g) the normalized Haar measure on G. Given are: • a smooth left action σ : G → Diff(N ) of G on N , • a smooth Lie group homomorphism U : G → GL∞ (M) such that U (g) is unitary for each g ∈ G. Assume also that the inclusion i : N → L1 (M) is equivariant, that is, i(σ (g)(n)) = U (g)i(n)U (g)−1 , for all n ∈ N and all g ∈ G. The homomorphism U defines the operator R : L1 (M) → L1 (M) by U (g)ρU (g)∗ dµ(g). (8.10) R(ρ) := G

We shall prove below that this R is a quantum reduction operator. We begin by showing that R is a projector. By invariance of the Haar measure under translations, we have for ρ ∈ L1 (M), # $ U (h) U (g)ρU (g)∗ dµ(g) U (h)∗ dµ(h) R 2 (ρ) = G G $ # = U (hg)ρU (hg)∗ dµ(g) dµ(h) G G = R(ρ)dµ(g) = R(ρ). G

50

A. Odzijewicz, T.S. Ratiu

Next, we show that R = 1. Since R is a projector we have R ≥ 1. To prove equality, note first that if ρ ≥ 0, that is, ψ|ρ|ψ ≥ 0 for all |ψ ∈ M, then we also have ψ|U (g)ρU (g)∗ |ψ ≥ 0 for all |ψ ∈ M and integration over G yields R(ρ) ≥ 0. Thus we showed that ρ ≥ 0 implies R(ρ) ≥ 0. Continuity of the trace in the · 1 –norm gives then for ρ ≥ 0, ∗ tr U (g)ρU (g) dµ(g) = tr ρ dµ(g) = tr ρ = ρ1 , R(ρ)1 = tr R(ρ) = G

G

which proves that R = 1. Finally, we need to show that im R ∗ is a Banach Lie subalgebra of L∞ (M). It is easy to see that for any X ∈ L∞ (M), U (g)∗ XU (g)dµ(g). R ∗ (X) = G

Using this formula we find R ∗ (X)R ∗ (Y ) = R ∗ (XR ∗ (Y )) = R ∗ (R ∗ (X)R ∗ (Y )), and the condition that im R ∗ is a Lie subalgebra of L∞ (M) follows immediately. Thus all conditions in the definition of a quantum reduction map are satisfied and hence R given by (8.10) is a quantum reduction operator. One can regard R : L1 (M) → im R as a momentum map. Consider the distribution E on N given at every point n ∈ N by En = {n}×ker R.Assume that E∩T N is the tangent bundle of a regular foliation F on N and that the projection π : N → N/F is a submersion. If (ker R)◦ ⊂ ker R + Tn N for every n ∈ N, the conditions of Theorem 8.6 are satisfied and we obtain a momentum map J : N/F → im R. Note that the tangent spaces to the G-orbits determine a distribution on N that is included in the distribution E ∩ T N. Supposing that the quotient by the group action is regular, that is, that N/G is a smooth manifold with the canonical projection N → N/G a surjective submersion, it follows that there is smooth surjective map between quotient spaces : N/G → N/F and hence a map J ◦ : N/G → im R. If, in addition, N/G is a Poisson manifold and is a Poisson map, then J ◦ : N/G → im R is a momentum map. This is satisfied, for example, if = identity. Certain momentum maps play a special role in the physical description of various systems. An important class of such momentum maps are the coherent states maps. We shall introduce this notion in the context of Banach Poisson manifolds, modeling it on the definition introduced by Odzijewicz [O2] for the case of a canonical map between a finite dimensional symplectic manifold and the projectivization of a complex Hilbert space. Definition 8.8. Let P be a Banach Poisson manifold and b be a Banach Lie-Poisson space. A coherent states map of P into b is a Poisson embedding K : P → b with linearly dense range, that is, the closure of the span of im K equals b. The situation investigated by Odzijewicz [O2] is the case when P is a finite dimensional Poisson manifold, b = h1 (M) is the Banach space of Hermitian trace class operators on a separable complex Hilbert space M, and K(p) is a rank one orthogonal projector for every p ∈ P . In this case, the range of K lies in the projectivization CP(M) of M by identifying a rank one projector with the point in projective space determined

Banach Lie-Poisson Spaces and Reduction

51

by its image. To illustrate this situation, let us recall how K is used in the quantization of a physical system. For the Poisson diffeomorphism σ : P → P , we assume that there is a linear Poisson automorphism : h1 (M) → h1 (M), such that the diagram of canonical maps P

K

- h1 (M)

σ

? P

K

? - h1 (M)

commutes. By a theorem of Wigner, the automorphism is of the form (ρ) = UρU ∗ , where U is a unitary or anti-unitary operator on M. Due to the hypothesis that im K is linearly dense in b, if such an automorphism exists, it is necessarily unique. It is natural to interpret as the quantization of σ . Denote by Aut(h1 (M)) the linear Poisson isomorphisms of h1 (M). The set of all Poisson diffeomorphisms σ for which a as above exists, forms a subgroup Diff K (P , {·, ·}) of the Poisson diffeomorphism group Diff(P , {·, ·}) of P . The map E : Diff K (P , {·, ·}) → Aut(h1 (M)) so defined, is a group homomorphism which will be called, according to Odzijewicz [O2], Ehrenfest quantization. Consider now the flow σt of the Hamiltonian vector field Xh on M and assume that σt ∈ Diff K (P , {·, ·}) for all t. It is known that the set of all Hamiltonian functions satisfying this condition form a Poisson subalgebra in the Poisson algebra of all smooth functions on P . Then E(σt )(ρ) = exp(itH )ρ exp(−itH )

(8.11)

for H a self-adjoint operator (unbounded, in general) whose domain includes the linear span of the set K(P )(M). The correspondence Q : h → H defined above is linear and satisfies the relation Q({h1 , h2 }) = i[Q(h1 ), Q(h2 )], that is, Q is the Lie algebra homomorphism induced by E. For more details on the precise relationship between the coherent states map quantization and the Kostant-Souriau geometric quantization as well as ∗-product quantization, we refer to Odzijewicz [O1, O2, O4]. The traditional coherent states map sends classical states to pure quantum states. Definition 8.8 generalizes this idea by letting the coherent states map send classical states to mixed quantum states. Mathematically, mixed quantum states are in L1 (M), or, more generally, in a Banach Lie-Poisson space. Definition 8.8 further generalizes the usual approach by also allowing in this scheme infinite dimensional classical systems.

52

A. Odzijewicz, T.S. Ratiu

Example 8.9 (See Odzijewicz [O3]). Consider the coherent states map K : P → h1 (M) from a finite dimensional Poisson manifold (P , {·, ·}) into the Banach Lie-Poisson space h1 (M) of Hermitian trace class operators on the separable complex Hilbert space M. Assume that the functions f1 , . . . , fk ∈ C ∞ (P ) are in involution, that is, {fi , fj } = 0, for all i, j = 1, . . . , k, and that their differentials df1 (p), . . . , dfk (p) are linearly independent for all p ∈ N , where ι : N → P is a given submanifold of P , invariant under the Hamiltonian flows σ1 (t), . . . , σk (t), t ∈ R, generated by f1 , . . . , fk respectively. Quantize the flows σ1 (t), . . . , σk (t) using the Ehrenfest quantization procedure (8.11), E : σi (t) → i (t), where it is assumed that the generators Fi , i = 1, . . . , k of the quantum flows i (t) are all self adjoint operators with discrete spectrum, i.e., Fi =

∞

λin Pni ∈ h∞ (M),

(8.12)

n=1

where {Pni }∞ n=1 is an orthonormal decomposition of the unit related to Fi . Consider the orthogonal projector Pλ1 ...λk := Pλ11 . . . Pλkk i1

ik

i1

ik

of M onto the common eigensubspace Mλ1 ...λk := {v ∈ M | F1 v = λ1i1 v, . . . Fk v = λkik v} i1

ik

of the generators F1 , . . . , Fk . According to Example 6.3, the map Rλ1 ...λk : h1 (M) → i1

h1 (M) defined by

ik

Rλ1 ...λk (ρ) := Pλ1 ...λk ρPλ1 ...λk i1

i1

ik

i1

ik

ik

(8.13)

is a quantum reduction. Assume now that conditions (i) and (ii) of Theorem 8.6 hold. In addition, assume that: (iii) the foliation F is given by the Hamiltonian vector fields Xf1 , . . . , Xfk . Then the quotient manifold N/F =: M is a Poisson manifold and the map Iλ1 ...λk := Rλ1 ...λk ◦ ι i1

ik

i1

ik

(8.14)

is a momentum map of M with values in im Rλ1 ...λk . i1

ik

In the special case when N = f −1 (µ), µ ∈ Rk , is the level set of the map f := (f1 , . . . , fk ), conditions (i), (ii), and (iii) imply certain restrictions on µ. For example, it can happen that µ1 = λ1i1 , . . . , µk = λkik . In this case, the existence of the momentum map (8.14), i.e., the existence of the Ehrenfest quantization for the quotient system N/F = M, leads to a discretization (quantization) of f : P → Rk . The above method has been applied to the quantization of the MIC-Kepler system in Odzijewicz and Swietochowski [O-S]. The above examples show the importance of the relation between the classical and quantum reduction procedures for the quantization and the integration of Hamiltonian systems. The present paper raises several important questions regarding these connections to which we shall return in future publications.

Banach Lie-Poisson Spaces and Reduction

53

Acknowledgements. We thank A.B Antonevich, H. Flaschka, J. Huebschmann, A.V. Lebedev J.-P. Ortega, and J. Marsden for several useful discussions that improved the exposition. Special thanks to P. Bona for his interest in our work and his inspired remarks. The first author was partially supported by KBN under grant 2 PO3 A 012 19. The second author was partially supported by the European Commission and the Swiss Federal Government through funding for the Research Training Network Mechanics and Symmetry in Europe (MASIE) as well as the Swiss National Science Foundation.

References [A-M]

Abraham, R., Marsden, J.E.: Foundations of Mechanics. Second Edition, Reading, MA: Addison-Wesley, 1978 [A-M-R] Abraham, R., Marsden, J.E., Ratiu, T.S.: Manifolds, Tensor Analysis, and Applications. Second Edition, Applied Mathematical Sciences 75, New York, NY: Springer-Verlag, 1988 [A-F-G] Accardi, L., Frigerio, A. and Gorini, V.: Quantum probability and Applications. Lecture Notes in Math. 1136, New York, NY: Springer-Verlag, 1984 [A-vW] Accardi, L., von Waldenfels: Quantum probability and Applications III. Lecture Notes in Math. 1396, New York, NY: Springer-Verlag, 1988 [A] Arnold, V.I.: Mathematical Methods of Classical Mechanics. Second Edition, Graduate Texts in Mathematics 60, New York, NY: Springer-Verlag, 1989 [B] Bona, P.: Extended quantum mechanics. Acta Physica Slovaca 50(1), 1198 (2000) [B-B] Boos, B., Bleecker, D.: Topology and Analysis. The Atiyah-Singer Index Formula and GaugeTheoretic Physics. Universitext, New York, NY: Springer-Verlag, 1983 [Bo1] Bourbaki, N.: Int´egration. Chapitre 6, Paris: Hermann, 1959 [Bo2] Bourbaki, N.: Vari´et´es diff´erentielles et analytiques. Fascicule de r´esultats. Paragraphes 1 a` 7. Paris: Hermann, 1967 [Bo3] Bourbaki, N.: Groupes et alg`ebres de Lie. Chapitre 3, Paris: Hermann, 1972 [B-R1] Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I. New York, NY: Springer-Verlag, 1979 [B-R2] Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics II. New York, NY: Springer-Verlag, 1981 [C-M] Chernoff, P.R., Marsden, J.E.: Properties of Infinite Dimensional Hamiltonian Systems. Lecture Notes in Mathematics, 425, New York, NY: Springer-Verlag, 1974 [Eh] Ehrenfest, P.: Z. Physik 45, 455 (1927) [Em] Emch, G.: Algebraic Methods in Statistical Mechanics. New York: Wiley Interscience, 1972 [H] Holevo, A.: Statistical Structure of Quantum theory. Lecture Notes in Physics, Monographs, New York, NY: Springer-Verlag, 2001 [K] Kirillov, A.A.: The orbit method, II: Infinite-dimensional Lie groups and Lie algebras. In: Cont. Math. 145, Representation Theory of Groups and Algebras, Providence, RI: Am. Math. Soc., 1993, pp. 33–63 [K-S] Klauder, J.R., Skagerstem, Bo-Sure: Coherent States – Applications in Physics and Mathematical Physics. Singapore: World Scientific, 1985 [K-N] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. New York: Wiley, 1963 [Ko1] Kostant, B.: Orbits, symplectic structures and representation theory. In: Proc, US-Japan Seminar on Diff. Geom., Kyoto. Tokyo: Nippon Hyronsha, 1966, p. 71 [Ko2] Kostant, B.: Quantization and unitary representations. Lecture Notes in Math. 170, BerlinHeidelberg-New York: Springer-Verlag, 1970, pp. 87–208 [Ko3] Kostant, B.: The solution to a generalized Toda lattice and representation theory. Adv. Math. 34, 195–338 (1979) [L] Landsman, N.P.: Mathematical Topics Between Classical and Quantum Mechanics. Springer Monographs in Mathematics, New York, NY: Springer-Verlag, 1998 [L-C] Leble, S.B., Czachor, M.: Darboux-integrable Liouville-von Neumann equations. Phys. Rev. E 58, 7091 (1998) [L-M] Libermann, P., Marle, C.-M.: Symplectic Geometry and Analytical Mechanics. Dordredut: Kluwer Academic Publishers, 1987 [Li] Lichnerowicz, A.: Vari´et´es de Jacobi et espaces homog`enes de contact complexes. J. Math. Pures et Appl. 67, 131–173 (1988) [Lie] Lie, S.: Theorie der Transformationsgruppen, Zweiter Abschnitt. Leipzig: Teubner, 1890 [M-R1] Marsden, J.E., Ratiu, T.S.: Reduction of Poisson manifolds. Lett. Math. Phys. 11, 161–170 (1986) [M-R2] Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. Texts in Applied Mathematics, 17, Second Edition, Second printing 2003, New York, NY: Springer-Verlag, 1994

54 [M] [N] [O1] [O2] [O3] [O4] [O-R] [O-S] [R] [S] [Sc] [St] [So1] [So2] [T1] [T2] [V] [vN] [W1] [W1]

A. Odzijewicz, T.S. Ratiu Murphy, G.J.: C ∗ -algebras and Operator Theory. San Diego: Academic Press, 1990 Nunes da Costa, J.M.: Reduction of complex Poisson manifolds. Portugaliae Math. 54, 467– 476 (1997) Odzijewicz, A.: On reproducing kernels and quantization of states. Commun. Math. Phys. 114, 577–597 (1988) Odzijewicz, A.: Coherent states and geometric quantization. Commun. Math. Phys. 150, 385– 413 (1992) Odzijewicz, A.: Coherent states for reduced phase spaces. In: Quantization and Coherent States Methods, Ali, S.T., Mladenov, I.M., Odzijewicz, A. (eds.), Singapore: World Sci. Press, 1993, pp. 161–169 Odzijewicz, A.: Covariant and contravariant Berezin symbols of bounded operators. In: Quantization and Infinite- Dimensional Systems, Antoine, J.-P., Ali, S.T., Lisiecki, W., Mladenov, I.M., Odzijewicz, A. (eds.), Singapore: World Sci. Press, 1994, pp. 99–108 Odzijewicz, A., Ratiu, T.S.: The Banach Poisson geometry of the infinite Toda lattice. Preprint, 2003 Odzijewicz, A., Swietochowski, M.: Coherent states map for MIC-Kepler system. J. Math. Phys. 38, 5010–5030, (1997) Ratiu, T.S.: Involution theorems. In: Geometric Methods in Mathematical Physics, Kaiser, G., Marsden, J.E. (eds.), Springer Lecture Notes 775, New York, NY: Springer-Verlag, 1978, pp. 219–257 Sakai, S.: C ∗ -Algebras and W ∗ -Algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete, 60, 1998 (reprint of the 1971 edition) New York, NY: Springer-Verlag, 1971 Schr¨odinger, E.: Naturwissenschaften. 14, 664 (1926) Stefan, P.: Accessible sets, orbits and foliations with singularities. Proc. London Math. Soc. 29, 699–713 (1974) Souriau, J.-M.: Quantification g´eom´etrique. Commun. Math. Phys. 1, 374–398 (1966) Souriau, J.-M.: Quantification g´eom´etrique. Applications. Ann. Inst. H. Poincar´e. 6, 311–341 (1967) Takesaki, M.: Conditional expectations in von Neumann algebra. J. Funct. Anal. 9, 306–321 (1972) Takesaki, M.: Theory of Operator Algebras I. New York, NY: Springer-Verlag, 1979 Vaisman, I.: Lectures on the Geometry of Poisson Manifolds. Progress in Mathematics, 118, Basel: Birkh¨auser Verlag, 1994 von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton, NJ: Princeton Univ. Press, 1955 Weinstein, A.: The local structure of Poisson manifolds. J. Diff. Geom. 18, 523–557 (1983) Weinstein, A.: Poisson geometry. Diff. Geom. Appl. 9, 213–238 (1998)

Communicated by L. Takhtajan

Commun. Math. Phys. 243, 55–91 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0954-x

Communications in

Mathematical Physics

Correlation Decay in Certain Soft Billiards P´eter B´alint1 , Imre P´eter T´oth2 1 2

Alfr´ed R´enyi Institute of the H.A.S., 1053 Re´altanoda u. 13-15, Budapest, Hungary. E-mail: [email protected] Mathematical Institute, Technical University of Budapest, 1111 Egry J´ozsef u. 1. Budapest, Hungary and Research group “Stochastics” of the Hungarian Academy of Sciences, affiliated to TUB. E-mail: [email protected]

Received: 10 February 2003 / Accepted: 12 May 2003 Published online: 21 October 2003 – © Springer-Verlag 2003

Abstract: Motivated by the 2D finite horizon periodic Lorentz gas, soft planar billiard systems with axis-symmetric potentials are studied in this paper. Since Sinai’s celebrated discovery that elastic collisions of a point particle with strictly convex scatterers give rise to hyperbolic, and consequently, nice ergodic behaviour, several authors (most notably Sinai, Kubo, Knauf) have found potentials with analogous properties. These investigations concluded in the work of V. Donnay and C. Liverani who obtained general conditions for a 2-D rotationally symmetric potential to provide ergodic dynamics. Our main aim here is to understand when these potentials lead to stronger stochastic properties, in particular to exponential decay of correlations and the central limit theorem. In the main argument we work with systems in general for which the rotation function satisfies certain conditions. One of these conditions has already been used by Donnay and Liverani to obtain hyperbolicity and ergodicity. What we prove is that if, in addition, the rotation function is regular in a reasonable sense, the rate of mixing is exponential, and, consequently, the central limit theorem applies. Finally, we give examples of specific potentials that fit our assumptions. This way we give a full discussion in the case of constant potentials and show potentials with any kind of power law behaviour at the origin for which the correlations decay exponentially.

1. Introduction Consider the planar motion of a point particle in a periodic array of strictly convex scatterers. Interaction with the scatterers is in the form of elastic collisions, otherwise motion is uniform. This dynamical system, the planar Lorentz process is a paradigm for strongly chaotic behaviour. Among other important properties ergodicity ([Si2, SCh]) and exponential decay of correlations ([Y, Ch]) have been proven for the corresponding billiard system.

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In this paper we consider the following natural modification. The scatterers are no longer hard disks, the point particle may enter them. The particle moves according to some rotation symmetric potential which vanishes identically outside the disks. Even the issue of these softened Lorentz processes has a large amount of literature. Results point in two different directions. On the one hand, for quite general softening of the potential, the chaotic behaviour is no longer present. Stable periodic orbits and islands appear in the phase space. This is generally the case with smooth potentials, see [RT, Do2, Do1] and references therein. However, in many cases, especially when the potential is not C 1 , the chaotic behaviour persists1 . The investigation of such soft billiards dates back to the pioneer works of Sinai ([Si1]) and Kubo et al. ([Ku] and [KM]). There are two different approaches present in the literature to this hyperbolic case. On the one hand, under conditions on the derivatives (up to the second) of the potential the Hamiltonian flow turned out to be equivalent to a geodesic flow on a negatively curved manifold. This point of view is especially suitable for potentials with Coulomb type singularities, see [Kn1] for details. The approach we follow is to study dynamics as a hyperbolic system with singularities. [M] and, especially, [DL] – which is one of our main references – are written in the spirit of this principle. Actually, in most cases it is convenient to study the discrete time dynamics, a naturally defined Poincar´e section map of the Hamiltonian flow – this is the track we are going to take. Hyperbolicity of the system is mainly related to the properties of the so-called rotation function that can be calculated from the potential. Being a bit technical its definition and relevant properties are discussed in the next section. Formulation of our main theorem (Theorem 1) is likewise left to the next section as it is in terms of the rotation function. Nevertheless, it might be useful to point out that – In case the rotation function (and the billiard configuration) satisfies some hyperbolicity condition (see Definition 2), the soft billiard system is hyperbolic and ergodic. Although somewhat otherwise stated, this fact was proved in [DL]. The condition is, essentially, necessary for ergodicity (note however Remark 1). – In this paper we concentrate on decay of correlations. If – in addition to those needed for hyperbolicity – the rotation function satisfies further regularity conditions (see Definition 3), the rate of mixing is proved to be exponential. In most of the paper we think of the rotation function as being fixed with the desired properties. It is only Sect. 5 when we turn to some specific potentials. Nevertheless, two technical conditions supposed to hold throughout the paper are: – In order to be able to define a rotation function at all, we introduce h(r) = r 2 (1 − 2V (r)) (cf. Sect. 5) and require h (r) > 0 for all but finitely many r (this condition ensures the lack of trapping zones, cf. [DL]) – The scattering occurs on rotation symmetric potentials of finite range – that is, potential for every scatterer is concentrated on a circle and depends only on the distance from the center. (Note this is the case in our references like [DL], too.) – The horizon is finite (i.e. the maximum time between two enterings of consecutive potential disks is uniformly bounded above for any trajectory). 1 In [DL] there is a smooth potential example with ergodic behaviour, too; however it is unstable with respect to small perturbations like varying the full energy level.

Correlation Decay in Certain Soft Billiards

57

Proof of our main theorem is based on our second main reference, [Ch]. In this paper, by implementing the techniques of L. S. Young from [Y], N. Chernov showed that given any hyperbolic system with singularities for which one can show the validity of certain technical properties, correlations decay exponentially fast. What we perform below is the proof of these technical properties for our “soft” billiard system. Even though the existence of invariant cone fields is established in [DL], the uniformity of hyperbolicity (Subsect. 3.2) needs detailed investigation. An even more important new difficulty that we have to overcome is the treatment of quantities connected to the second derivative of the dynamics, especially while traveling through the potential. An analysis finer than before – in this sense – of the evolution of fronts is needed. This applies especially to the self-contained proof of curvature and distortion bounds (Subsect. 3.3). It is a key aspect of our method that arguments related to expansion and distortion can be carried out by considering motion inside and outside the potential disks separately. Actually, our choice of the outgoing phase space and the Euclidean metric (see Sect. 2) is related to this point of view and not to the tradition of [Ch]. (Using the Euclidean metric with the phase space of incoming particles instead of outgoing, our distortion bounds would no longer hold.) The splitting of motion into “potential” and “free” intervals is, however, slightly restrictive. Namely, certain soft billiard systems that seem ergodic and exponentially mixing are covered neither in this paper nor in [DL] (see also Sect. 6, especially Remark 1). The paper is organized as follows. In Sect. 2 the dynamical system along with the rotation function and its properties are defined, and our main theorem is formulated. Section 3 gives a detailed geometric analysis of the system. After fixing notations and establishing some basic properties in Subsect. 3.1, Subsect. 3.2 is mainly concerned with uniform hyperbolicity (Proposition 1) and related issues. In Subsect. 3.3 important regularity properties of unstable manifolds are shown. Specifically, curvature bounds, distortion bounds and absolute continuity of the holonomy map are proven (Propositions 2, 3 and 4). As a final bit of the general proof in Sect. 4 we investigate the growth of unstable manifolds. The fact that expansion prevails the harmful effect of singularities is quantificated in the growth formulas of Proposition 5. As a conclusion we refer to Theorem 2.1 from [Ch]: a hyperbolic system with singularities for which Propositions 1, 2, 3, 4 and 5 are valid enjoys exponential decay of correlations. For the reader’s convenience, we formulate the theorem of Chernov in the Appendix. Last but not least, preceding some concluding remarks, in Sect. 5 we turn to the investigation of specific potentials: as corollaries of our main theorem certain soft billiard systems are shown to exhibit exponential decay of correlations. We note that it is not clear how sharp our results are. On the one hand, the conditions for ergodicity – which are part of our conditions – formulated by Donnay and Liverani are more or less sharp (see [Do1]). On the other hand, the conditions formulated for EDC by Chernov are sufficient, but most probably not necessary. So, although we know that Chernov’s conditions (e.g. the bounded curvature assumption and the distortion bounds) are not satisfied when our regularity conditions are not met, it is well possible that EDC still occurs. At some points of the paper we will point out why our regularity conditions are necessary for Chernov’s method to work. Part of the results in this paper and a sketch of the proof can also be found in the proceedings paper [B´aT´o].

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2. Definition and Basic Properties of the System The phase space. Consider finitely many disjoint circles of radius R on the unit twodimensional flat torus T2 . (Thinking of a periodic array of circular disks on the Euclidean plane R2 would not be very much different.) We require that the configuration has finite horizon: there is a certain constant τmax such that any straight segment longer than τmax on R2 intersects at least one of the scatterers. Remark. As the circles are disjoint, the minimum distance between two scatterers is bigger than some positive constant τmin . Let the Hamiltonian motion of our point particle be described by a potential which is identically zero outside and is some rotation symmetric function V (r) inside the circular scatterers (here r is the distance from the center of the scatterer). For simplicity we fix the mass and the full energy of our point particle as m = 1,

E=

1 . 2

This way the free flight velocity has unit length, |v| = 1 (in other words v ∈ S1 , where S1 is the unit circle in R2 ). We assume (cf. Definition 2 and the remarks following it) that the Hamiltonian flow restricted to this surface of constant full energy is ergodic (with respect to Liouville measure). Equivalently one can say that the map corresponding to the naturally defined Poincar´e section of the flow (see below) is ergodic. Our aim is to study the rate of mixing for this map. We work with the Poincar´e section of outgoing velocities (particles that have just left one of the scatterers). Notation. Denote by M the Poincar´e section of outgoing particles. Sometimes we will also use the notation M+ = M to stress that this is the outgoing phase space, to avoid confusion. The phase points are the boundary points of the scatterers, equipped with unit velocities pointing outwards. The phase space M is a finite union of cylinders (each corresponding to one of the circular scatterers). Coordinates for the cylinders are: Notations. s denotes the arclength parameter along the scatterer (starting from a point arbitrarily fixed), describing position of the outgoing particle. ϕ denotes the collision angle, the angle that the outgoing velocity makes with the normal vector of the scatterer in the point s. Clearly ϕ ∈ [− π2 , π2 ]. The position can be equivalently described by another angle parameter ∈ [0, 2π ], for which s = R (here R is the radius of the scatterer). Note that M defined this way is a (finite union of) Riemannian manifold(s). Notation. Let |dx|e =

ds 2 + dϕ 2

(2.1)

denote the Riemannian metric on M, which will be referred to as the Euclidean metric (e-metric). Later on we will introduce another auxiliary metric quantity very common in the billiard literature, the p-metric.

Correlation Decay in Certain Soft Billiards

59

As to dynamics, let T denote the first return map onto M. Notation for the Lebesgue measure on M is m, i.e. dm = ds dϕ. Furthermore, given a curve γ in M we denote the Lebesgue measure on γ with mγ (this is simply the length on γ ). Denote by µ the natural invariant probability measure on M. µ is absolutely continuous w.r.t Lebesgue, and the density is of the form dµ = const. cos(ϕ) dm = const. cos(ϕ) ds dϕ.

(2.2)

It is this latter measure for which T is assumed to be ergodic and K-mixing and this is the one we work with as well. Remark. In a completely similar manner we could consider the Poincar´e section M− of incoming particles. The two coordinates would be the point of income and the angle the incoming velocity makes with the (opposite) normal vector. However, in some key steps of the proof – e.g. the distortion bounds of Subsubsect. 3.3.2 – we heavily use that our phase space is the outgoing, and not the incoming Poincar´e section. With slight abuse of notation we often refer to the incoming Poincar´e coordinates with the same symbols s and ϕ. That should cause no confusion. Rotation function, its basic properties and formulation of the main theorem. To describe the first return map T we decouple the motion into two parts: free flight among the scatterers and flight in the potential of the scatterers. Free flight can be treated completely analogously to the billiard case. The particle leaves one of the scatterers in the point s0 with velocity ϕ0 and reaches some other scatterer in point s (or equivalently, ) with unit incoming velocity that makes an angle ϕ with the (opposite) normal vector n(s) at the point of income. After some inter-potential motion the particle leaves the circle in some point s1 = (R1 ) with outgoing velocity specified by ϕ1 . Out of symmetry reasons ϕ1 = ϕ, thus the only nontrivial quantity is the angle difference = 1 − . Again out of symmetry reasons depends only on the angle ϕ. The role in the map T played by the potential is completely described by the function (ϕ). Definition 1. From here on we will refer to this function (ϕ) as the rotation function. Being mainly interested in the differential aspects of T we introduce one more Notation d(ϕ) κ(ϕ) = . dϕ Below two important properties are defined in terms of which our main theorem is formulated. Definition 2. The soft billiard system satisfies property H in case 1. there is some positive constant c such that |2 + κ(ϕ)| > c for all ϕ; 2. the configuration of scatterers is such that the distance of any two circles is bounded below by τmin , where cos ϕ τmin max −2Rκ(ϕ) . ϕ 2 + κ(ϕ)

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Remarks. – Although a bit otherwise formulated, it was essentially proven in [DL] that soft billiard systems with property H are hyperbolic and ergodic. The mechanism of hyperbolicity is briefly explained in Sect. 3.2. – Note that in case κ > 0 or κ < −2 for all ϕ, the lower bound for τmin turns out to be negative. Thus the second assumption is only restrictive in the opposite case, and the closer κ may get to −2 from above the more restrictive it is. – In case there is some ϕ for which 0 > κ > −2, a positive lower bound on the free path is to be assumed. Thus a planar periodic configuration of circles is needed that has finite horizon and (a possibly great) given τmin simultaneously. At first sight it seems questionable whether such configurations exist at all, nevertheless, as proven in [B¨oTa], this happens with positive probability in a random construction. Definition 3. The rotation function is termed regular in case the following properties hold: 1. (ϕ) is piecewise uniformly H¨older continuous. I.e. there are constants C < ∞ and α > 0, and furthermore, [− π2 , π2 ] can be partitioned into finitely many intervals, such that for any ϕ1 and ϕ2 (from the interiour of one of the intervals): |(ϕ1 ) − (ϕ2 )| ≤ C|ϕ1 − ϕ2 |α . 2. (ϕ) is a piecewise C 2 function of ϕ on the closed interval [− π2 , π2 ], in the above sense. (Note, however, that κ, in contrast to , can happen to have no finite onesided limits at discontinuity points.) 3. There is some finite constant C such that |κ (ϕ)| ≤ C|(2 + κ(ϕ))3 |, where κ (ϕ) is the derivative of κ with respect to ϕ. 4. For the final property consider any discontinuity point ϕ0 , where κ(ϕ) (in contrast to (ϕ)) has no finite limit from the left. Of course, in case there is no finite limit from the right, the analogous property is similarly assumed. Restricted to some interval [ϕ0 − , ϕ0 ); ω(ϕ) = 2+κ(ϕ) cos ϕ is a monotonic function of ϕ. Remark. Note that in case κ is C 1 (or piecewise C 1 with boundedness of itself and of κ ) regularity is automatic. In case the asymptotics of κ near some discontinuity is some power law (ϕ0 − ϕ)−ξ (with ξ > 0), regularity means 21 ≤ ξ < 1. We need two more definitions for the statement of our theorem: Definition 4. Consider a phase space M with a dynamics T and a T -invariant probability measure µ. We say that the dynamical system (M, T , µ) has exponential decay of correlations (EDC) , if for every f, g : M → R H¨older-continuous pair of functions there exist constants C < ∞ and a > 0 such that for every n ∈ N, n f (x)g(T n x)dµ(x) − f (x)dµ(x) g(T x)dµ(x) ≤ Ce−an . M

M

M

Correlation Decay in Certain Soft Billiards

61

Definition 5. We say that (M, T , µ) satisfies the central limit theorem (CLT) (for H¨older continuous functions) if for every η > 0 and every H¨older-continuous function f : M → R with f dµ = 0, there exists a σf ≥ 0 such that 1 distr f ◦ T i −→ N (0, σf ), √ n n−1 i=0

where N (0, σf ) is the Gaussian distribution with variance σf2 . Now we are ready to formulate our main theorem. Theorem 1. Suppose that the soft billiard system (M, T , µ) satisfies property H and the rotation function is regular. Suppose furthermore that there are no corner points and the horizon is finite (0 < τmin , τmax < ∞). Then, dynamics enjoys, in addition to ergodicity and hyperbolicity, exponential decay of correlations and the central limit theorem for H¨older-continuous functions. Proof. Ingredients for the proof are in Sect. 3 and 4. Actually, following tradition (e.g. [Ch]) we modify the dynamical system in several steps (Conventions 1 and 2). We will ¯ which is the original M cut into (countably many) connected use a phase space M, components by singularities and so called “secondary singularities”. We will also use a higher iterate of the dynamics T1 = T m0 with some m0 to be found later. ¯ T1 , µ) for which the conditions for EDC It is the modified dynamical system (M, ¯ T1 , µ) are the and CLT given in [Ch] are checked. Precisely, EDC and CLT for (M, consequence of Propositions 1, 2, 3, 4 and 5 and Theorem 2.1 from [Ch]. Exponential decay of correlations and the central limit theorem for (M, T , µ) follow ¯ T1 , µ). easily from EDC and CLT for (M,

For the reader’s convenience, we give a formulation of Theorem 2.1 from [Ch] in the Appendix. Now we turn to the details of the above proof. Some conventions. Constants that depend only on the map T itself (like τmin , τmax , . . .) will be called global constants. Positive and finite global constants, whose value is otherwise not important, will be often denoted by just c or C (typically c for lower bounds and C for upper). That is, in two different lines of the same section, C can mean two different numbers. Two quantities f and g defined on (the tangent bundle of) M (or on some subset like the unstable cone field, see Subsect. 3.1) will be called equivalent (f ∼ g) if there are some global positive constants c and C such that cf ≤ g ≤ Cf . 2.1. Singularities. Just like in billiards the dynamics T is not smooth at certain one-codimensional submanifolds (curves) of M. Consider the set of tangential reflections: π S0 = (s, ϕ) ∈ M | ϕ = ± . 2 Actually S0 = ∂M (the boundary of the phase space). It is not difficult to see that T is not continuous at S1 = T −1 S0 , i.e. at the preimages of tangential reflections. However, additional singularities appear at Z0 = { (s, ϕ) ∈ M | ϕ = ϕ0 }

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in case ϕ0 is some discontinuity point for (ϕ), κ(ϕ) or κ (ϕ). In such a case we will consider the phase space as if it were cut into two regions, more precisely Z0 is treated as part of the boundary. As κ is not differentiable at Z0 , T is not C 1 at the preimage of this set, at Z1 = T −1 Z0 . Furthermore we introduce the notations S (n) = S1 ∪ T −1 S1 ∪ · · · ∪ T −n+1 S1 and Z (n) , analogously. The nth iterate of the dynamics is not smooth precisely at Z (n) ∪ S (n) . The geometrical structure of Z (n) is much similar to that of S (n) . Indeed, one can think of Z1 as the set of those trajectories that would touch tangentially a smaller disk (one of radius R sin(|ϕ0 |)) at the next collision. The following properties of the singularity set are of crucial importance: – Z (n) ∪ S (n) is a finite union of C 2 curves. – Continuation property. Each endpoint, x0 , of every unextendable smooth curve γ ⊂ Z (n) ∪ S (n) , lies either on the extended boundary Z0 ∪ S0 or on another smooth curve γ ⊂ Z (n) ∪ S (n) that itself does not terminate at x0 . – Complexity property. Let us denote by Kn the complexity of Z (n) ∪ S (n) , i.e. the maximal number of smooth curves in Z (n) ∪ S (n) that intersect or terminate at any point of Z (n) ∪ S (n) . Kn grows sub-exponentially with n. For the proof of these properties in the billiard setting see the literature, especially [ChY], our case is analogous. One more similarity with “hard” billiards is that for technical reasons later on we will introduce countably many secondary singularities parallel to the lines of S (n) . Such secondary singularities are to be introduced parallel to Z (n) as well, in case |κ| is unbounded as ϕ → ϕ0 , at least from one side. We will turn back to this question in Subsubsect. 3.2.6. 3. Fronts, u-Manifolds and Unstable Manifolds 3.1. u-manifolds and their geometric properties. Fronts and their geometric description. Our most important tools in describing hyperbolicity – local orthogonal manifolds or simply fronts – we inherit from billiard theory. A front W is defined in the flow phase space rather than in the Poincar´e section. Definition 6. Take a smooth 1-codim submanifold E of the whole configuration space, and add the unit normal vector v(q) of this submanifold at every point q as a velocity, continuously. Consequently, at every point the velocity points to the same side of the submanifold E. The set W = {(q, v(q))|q ∈ E}, (3.1) where v : E → S1 is continuous (smooth) and v ⊥ E at every point of E, is called a front. Analysis of the time evolution of fronts is the key to almost all the geometric properties of the system that we need. For this reason, we first discuss time evolution of an arbitrary front. Later subsections will deal with special cases. Consider a front with a reference point just before reaching a scatterer, and another “perturbed” point nearby. With the notations introduced before (see also Fig. 1), the

Correlation Decay in Certain Soft Billiards

s

63

(ϕ)

ϕ+

ϕ−

dq− , dv−

dq+ , dv+ Fig. 1. Conventions for notation and signs for fronts

perturbation bringing the reference trajectory into the perturbed one is (dq− , dv− ) just before collision, (ds− , dϕ− ) in the incoming Poincar´e section, (ds+ , dϕ+ ) in the out , dv ) just before the going Poincar´e section, (dq+ , dv+ ) just after collision, and (dq− − next collision. The evolution of the perturbations is: ds− = d− dϕ− d+ dϕ ds+ dq+ dv+

= = = := = = =

dq− , cos ϕ− ds− , R dv− + d− , d− + κdϕ− , dϕ+ = dϕ− , Rd+ , − cos ϕds+ , −d+ − dϕ+ ,

(3.2)

while crossing the potential. For the evolution equations of free flight, we introduce the Notation. τ = τ (x) will denote the length of free flight of the particle before reaching the next scatterer. So, during free flight we have dq− = dq+ + τ dv+ , dv−

= dv+ .

(3.3)

64

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Note that the angles of incidence and reflection are measured in different directions – in order to keep them equal, as they traditionally are, – but dq− and dq+ (just like dv− and dv+ ) are measured in the same direction, unlike usually in billiards. dv Based on these, we can find out about the evolution of the derivative B = dq . Notations. B will denote the derivative of the unit normal vector (velocity) v(q) of a front: dv = Bdq for tangent vectors (dq, dv) of the front. e section. m = dϕ ds will denote the slope of the (trace of the) front in the Poincar´ B is nothing other than the curvature of the submanifold E. Yet we will prefer to call it the second fundamental form (SFF), in order to avoid confusion with other curvatures that are coming up. The term “form” refers to higher dimensional cases when B is a symmetric operator. Equation (3.2) gives 1 m− = cos ϕB− + , R 1 1 = + Rκ, m+ m− 1 cos ϕB+ = m+ + , R

(3.4)

while crossing the potential, which can be summarized in B+ =

2 + κ(ϕ) + (1 + κ(ϕ))R cos ϕB− , R cos ϕ(1 + κ(ϕ) + κ(ϕ)R cos ϕB− )

(3.5)

and (3.3) gives 1 1 = B +τ B− +

(3.6)

during free flight. Notations. dq+ , dq− dq λ2 := − , dq+ λ := λ1 λ2 . λ1 :=

(3.7) (3.8)

These are exactly the expansion factors along the front, for the respective “pieces” of the dynamics. (They are also expansion factors in the Poincar´e section, but in the p-metric to be introduced later.) We have λ1 = 1 + κ + κR cos ϕB− = 1 + κRm− = λ2 = 1 + τ B + =

B+ . B−

m− , m+

(3.9)

(3.10)

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65

To study decay of correlations, we need one more derivative. Notation. D = dB dq . This is exactly the curvature of the front as of a subset of the flow phase space (and not as of a subset of the configuration space – unlike B, cf. (3.1)). To study the evolution of D we need to consider two small pieces of the front, one around the reference point, and one around the perturbed one. Let the change in the SFF be dB−− = D− dq before scattering, and dB++ = D+ dq after scattering. dB−− is not the difference of SFF-s at the points of incidence, because the perturbed point has to travel another dτ− = tan ϕ− dq− to reach the scatterer (dτ can be negative), which changes its SFF according to the rules (3.6) of free flight. Taking that into account, we have 2 2 dB− = dB−− − B− dτ− = dB−− − B− tan ϕ− dq− .

(3.11)

Similarly, for the fronts leaving the potential, 2 2 dB++ = dB+ − B+ dτ+ = dB+ − B+ tan ϕ+ dq+ .

(3.12)

(Note our convention on the signs of dq− , dq+ , ϕ− and ϕ+ .) To follow the evolution of curvature we introduce dm− dm+ dB+ − Notations. D1 = dB dq− (= D− ), K− = ds− , K+ = ds+ , D2 = dq+ and η(ϕ) = With these we get from (3.2), (3.4), (3.7), (3.9), (3.11) and (3.12), 2 , D1 = D− − tan ϕB− 2 K− = cos ϕD1 − sin ϕB− m− ,

1 m− 3 K+ = 3 K1 − R η, λ1 λ1

dκ(ϕ) dϕ .

(3.13)

cos2 ϕD2 = −K+ − sin ϕB+ m+ , 2 D+ = D2 − tan ϕB+ , while crossing the potential, and, from (3.3), (3.6) and (3.8) D− =

during free flight.

1 D+ λ32

(3.14)

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P. B´alint, I. P. T´oth

3.2. Invariance of convex fronts, u-fronts and u-manifolds. In [DL] it is shown – although not explicitly stated in this integrated form – that if Property H (defined in Definition 2) is satisfied, then convex fronts with suitably small SFF-s (the upper bound may be ∞) either remain convex, or focus before reaching the next scatterer, and become convex again, with suitably small SFF. This property is called the “invariance of convex fronts”. In the present work we also require (see Theorem 1) that τ be bounded from below by some τmin > 0 even in the case when [DL] did not (the “no corner points” assumption), and an upper bound τmax (the “finite horizon” assumption). In order to establish estimates that we will need later, we must repeat some steps of the argument in [DL]. We omit details of the calculations; these can be done by the reader or can be found in the above paper. Notations.

cos ϕ τ1 = max 0, max −Rκ(ϕ) ϕ 2 + κ(ϕ) 1 B∗ = (∞ if τ1 = 0). τ1

, (3.15)

From (3.4) we get that if 0 < B− < B ∗ then either m+ > 0 and thus B+ > R1 or < B ∗∗ with some global constants B+ < −B ∗ . This – by (3.6) – implies that c < B− ∗∗ ∗ c > 0 and B < B , assuming that τmin > 2τ1 , which is exactly Property H. All in all, < B ∗∗ . c < B− < B ∗∗ implies c < B−

(3.16)

This motivates our Definition 7. A u-front is a front with c < B− < B ∗∗ . A u-manifold is the trace of a u-front on the Poincar´e phase space. Definition 8. An s-front is a front with c < −B+ < B ∗∗ . An s-manifold is the trace of an s-front on the Poincar´e phase space. As we have seen, u-manifolds remain u-manifolds under time evolution. s-fronts are exactly the u-fronts of the inverse dynamics. The aim of this subsection is to show important properties of u-fronts and u-manifolds, which are stronger than those shown for an arbitrary front in the previous subsection. In Subsect. 3.3 we further restrict to the case of unstable manifolds, which are special kinds of u-manifolds. 3.2.1. Expansion estimates along u-fronts. First we work out estimates for the expansion along a front from one moment of incidence to the next. We will use these estimates later to estimate expansion of our dynamics T in our outgoing Poincar´e phase space M. Consider a u-front with the earlier notations. We start with an easy observation we will often use: from (3.4) and (3.16) we get R1 < m− < R1 + B ∗∗ , which implies m− ∼ 1.

(3.17)

To get the order of magnitude for the expansion factor λ, put the formulas in (3.5) and (3.9) together, and get that cos ϕ R cos ϕB+ λ1 = 1 + (1 + κ(ϕ))RB− . 2 + κ(ϕ) 2 + κ(ϕ)

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The right-hand side is trivially bounded from above since B− is bounded, and so is 1+κ(ϕ) 1 2+κ(ϕ) = 1 − 2+κ(ϕ) . On the other hand, – It is greater than 1 if – If

1+κ(ϕ) 2+κ(ϕ)

1+κ(ϕ) 2+κ(ϕ)

> 0.

≤ 0 (that is, −2 < κ(ϕ) ≤ −1), then cos ϕ cos ϕ ≥ 1 + (1 + κ(ϕ))R B∗ 2 + κ(ϕ) 2 + κ(ϕ) −1 1 2 + κ(ϕ) cos ϕ = ≥ . ≥ 1 + (1 + κ(ϕ))R 2 + κ(ϕ) −Rκ(ϕ) cos ϕ κ(ϕ) 2

1 + (1 + κ(ϕ))RB−

All in all, using λ2 =

B+ B−

∼ B+ (see (3.10) and (3.16)) we have λ ∼ B+ λ1 ∼

2 + κ(ϕ) , cos ϕ

(3.18)

which is one of our key estimates. Notice that the right-hand side cannot be too small due to Property H (Definition 2). We can also get the order of magnitude for λ1 and λ2 separately: (3.9) and (3.17) gives |λ1 | 1 + m2+ = λ21 + m2− ∼ |2 + κ(ϕ)|. (3.19) (The last equivalence is true because both sides are bounded away from zero, and can only be big when they grow linearly with κ.) Notice that λ1 can be very small (even zero), and can even change signs while 2 + κ(ϕ) remains positive. Of course, m+ has to be infinity (and change signs) simultaneously. Putting (3.18) and (3.19) together, we get

|λ2 | 1 + m2+

∼

1 . cos ϕ

(3.20)

This last line can be rewritten as |m+ + R1 | |λ2 | cos ϕ |B+ | cos ϕ 1∼ ∼ = , 1 + m2+ 1 + m2+ 1 + m2+ which implies that there is a global constant c such that m+ + 1 > c. R

(3.21)

3.2.2. Expansivity. To obtain hyperbolicity, we must see that u-manifolds are expanded by the dynamics. In the first round we prove a lemma about the expansion on u-fronts from collision to collision. Lemma 1. There exists a global constant > 1, such that for every u-front, |λ| ≥ .

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Proof. Besides τ > 0 and B− > 0 we will use that τ ≥ 2τ1 +d, where d := τmin −2τ1 > cos ϕ 0, and τ1 ≥ −Rκ(ϕ) 2+κ(ϕ) for every ϕ (see Definition 2 and (3.15)). Altogether: cos ϕ . (3.22) 2 + κ(ϕ) We will also use from (3.15) and Definition 7 that 2 + κ(ϕ) (3.23) 0 < B− ≤ −Rκ(ϕ) cos ϕ whenever the right-hand side is positive, which is the −2 < κ(ϕ) < 0 case. Now we start by putting together (3.9), (3.10) and (3.5) to get τ ≥ d − 2Rκ(ϕ)

λ = (1 + κ(ϕ)Rm− )(1 + τ B+ ) = 1 + κ(ϕ) + κ(ϕ)R cos ϕB− + τ

2 + κ(ϕ) + (1 + κ(ϕ))B− . R cos ϕ

We estimate this taking care of the signs of the particular terms. – If κ(ϕ) ≤ −2 − δ, then λ ≤ 1 + κ(ϕ) ≤ −1 − δ. – If −2 + δ ≤ κ(ϕ) ≤ −1 then both coefficients of B− are negative, so we can use (3.23) to estimate the right-hand side from below. In the next step we find the coefficient of τ positive, so we can use (3.22). What we get is 2 + κ(ϕ) λ ≥ 1 + κ(ϕ) + κ(ϕ)R cos ϕ −Rκ(ϕ) cos ϕ

2 + κ(ϕ) 2 + κ(ϕ) +τ + (1 + κ(ϕ)) R cos ϕ −Rκ(ϕ) cos ϕ 2 + κ(ϕ) = −1 + τ −κ(ϕ)R cos ϕ

cos ϕ 2 + κ(ϕ) ≥ −1 + d − 2Rκ(ϕ) 2 + κ(ϕ) −κ(ϕ)R cos ϕ dδ ≥ 1+ . 2R – If −1 ≤ κ(ϕ) ≤ 0, then the coefficient of τ is positive, so we first use (3.22) to estimate the right-hand side from below. In the next step we find one coefficient of B− positive, so we just use B− > 0, and one coefficient of B− negative, so we can use (3.23). What we get is λ ≥ 1 + κ(ϕ) + κ(ϕ)R cos ϕB−

cos ϕ 2 + κ(ϕ) + d − 2Rκ(ϕ) + (1 + κ(ϕ))B− 2 + κ(ϕ) R cos ϕ 2 + κ(ϕ) κ 2 (ϕ)R cos ϕ + d(1 + κ(ϕ))B− − κ(ϕ) − B− R cos ϕ 2 + κ(ϕ) 2 + κ(ϕ) κ 2 (ϕ)R cos ϕ 2 + κ(ϕ) ≥1+d − κ(ϕ) − R cos ϕ 2 + κ(ϕ) −Rκ(ϕ) cos ϕ d ≥ 1+ . R

= 1+d

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69

– If 0 < κ(ϕ), then λ≥1+

2d . R

3.2.3. Transversality. Lemma 2. We will see that u- and s-manifolds are uniformly transversal. I.e. there is some global constant α0 > 0 such that given any two tangent vectors (in the outgoing Poincar´e phase space) dxs and dxu of an s- and a u-manifold, respectively, we have (dxu , dxs ) > α0 . Proof. To see this, use Definition 8 and (3.4) to get ms+ ∼ −1 for the slope of any smanifold. This way, it is enough to see that the slopes of u- and s-manifolds are bounded away, that is, |mu+ − ms+ | > c. To get this, use – again – Definition 8, Definition 7, the estimates before them and (3.4) to get −

1 1 > ms+ > − − cos ϕB ∗∗ , R R 1 mu+ > 0 or mu+ < − − cos ϕB ∗ , R

(3.24)

so either mu+ − ms+ > R1 or ms+ − mu+ > cos ϕ(B ∗ − B ∗∗ ). This implies the statement when cos ϕ is not too small. However, when cos ϕ is small, we have to use the estimate (3.21) and (3.24) to see also that mu+ > 0 or mu+ < − which completes the proof.

1 − c, R

3.2.4. Hyperbolicity. In what follows we will consider time evolution of vectors tangent to u-manifolds. Notation both in the incoming and the outgoing phase space will be of the type dx = (ds, dϕ). In addition to the e-metric (2.1) we will use one more metric quantity, the p-metric: |dx|p = |ds| cos(ϕ). The p-metric measures distances along the corresponding u-front. It is degenerate on the whole tangent bundle. However, when restricted a u-manifold in the incoming phase space, by (3.17) we have: |dx|p ∼ |dx|e cos(ϕ). According to Lemma 1, u-vectors are expanded uniformly (from collision to collision, that is, in the incoming phase space) in the p-metric: |DT |p = λ ≥ > 1. To obtain expansion in the e-metric and the outgoing phase space, we look at the nth iterate of the outgoing phase space dynamics the following way: – – – – –

switch to p-metric reach the next scatterer do n − 1 steps in the incoming phase space cross the potential switch back to Euclidean metric.

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This way we get

|DT n dx|e =

1 + m2+(n) cos ϕ(n)

λ1(n) λ(n−1) λ(n−2) . . . λ(1) λ2

cos ϕ 1 + m2+

|dx|e ,

where symbols with ()-ed subscripts mean values at the appropriate iterate of the phase point. Using (3.19), (3.20) and Lemma 1 we get |DT n dx|e ∼ λ(n−1) λ(n−2) . . . λ(1)

2 + κ(ϕ(n) ) |dx|e . cos ϕ(n)

(3.25)

This way we have |DT n dx|e > c1 n |dx|e

(3.26)

with some global constant c1 . Again, this is for u-vectors in the outgoing phase space. The transversality of s- and u- vectors, stated in Proposition 1 implies that the product of (length) expansion factors for s- and u- vectors is equivalent to the n-step (Lebesgue) volume expansion factor. Using (2.2), and the T -invariance of µ, we get that if dx is a u-vector and dy is an s-vector, then |DT n dx|e |DT n dy|e cos ϕ ∼ . |dx|e |dy|e cos ϕ(n) Combinig this with (3.25) we get |DT n dy|e ∼

cos ϕ 1 |dy|e , 2 + κ(ϕ(n) ) λ(n−1) λ(n−2) . . . λ(1)

which implies |DT n dy|e <

C1 |dx|e n

(3.27)

with some global constant C1 . Again, this is for s-vectors in the outgoing phase space. Convention 1. We choose a positive integer m0 the following way. First take m1 such that c1 m1 > 1 and Cm11 < 1. This way any high enough power of the dynamics, T m with m > m1 is uniformly expanding along u-manifolds and uniformly contracting along s-manifolds with 1 = m−m1 . Now recall the notion and the basic properties of complexity Kn from Subsect. 2.1. As Kn grows subexponentially we may choose m2 for which we have Km < m−m1 whenever m > m2 . We fix m0 = min(m1 , m2 ) + 1. The advantage of this choice is that the iterate T1 = T m0 is uniformly hyperbolic (see the proposition to come) with constant 1 for which 1 > Km0 + 1. This later fact we only use in Sect. 4. Let us summarize what we have seen so far from the hyperbolic properties in the following

Correlation Decay in Certain Soft Billiards

71

Proposition 1. There exist two families of cones Cs (x) and Cu (x) – called stable and unstable cones – in the tangent space of M such that DT (Cu (x)) ⊂ Cu (T x) and Cs (T x) ⊂ DT (Cs (x)). The stable/unstable cone is uniformly contracting/expanding: |DT1−1 (dx)| ≥ 1 |dx| ∀dx ∈ Cs (x), |DT1 (dx)| ≥ 1 |dx| ∀dx ∈ Cu (x). Furthermore, the two cone fields are uniformly transversal in the sense above. Vectors of the stable/unstable cone are often called s- and u-vectors. Proof. The two cones are formed by the tangent vectors of s- and u-manifolds, respectively. Invariance is the implication (3.16), recalling Definition 7 and 8. Expansion and contraction are (3.26), (3.27) and Convention 1. Transversality is Lemma 2.

We note that so far we have only used that our billiard satisfies property H, which is a property already formulated in [DL], and which is known from [Do1] to be essentially necessary for ergodicity. 3.2.5. Alignment. We need to investigate the relative position of u-manifolds and singularities in order to find out how much of a u-manifold can be “close” to a singularity. Our aim is to prove the following Lemma 3. Take any smooth component Z of T −k Z0 with k ≥ 0, where Z0 = { (s, ϕ) ∈ M | ϕ = ϕ0 } with any ϕ0 ∈ [− π2 , π2 ]. Given some small positive δ let us denote the δ-neighborhood of Z by Z [δ] . There are global constants C < ∞ and α > 0 such that for any u-manifold W we have

(3.28) mW Z [δ] ∩ W ≤ Cδ α , where mW is the Lebesgue measure – the length – on the u-manifold W . Proof. If k > 0, then Z is an s-manifold, and is transversal to our u-manifold W according to Lemma 2, so the statement holds even with α = 1. So take k = 0, then Z is described by mZ = 0. If κ(ϕ) remains bounded near ϕ0 , then for our u-manifold W , 1 1 = + Rκ(ϕ) m+ m− is bounded (see (3.4) and (3.17)), so the two curves are transversal again, we can choose α = 1. The interesting case is k = 0, κ(ϕ) → ∞ as ϕ → ϕ0 . In this case (3.17) ensures that m1− is negligible – say, less than the ε portion – compared to Rκ(ϕ). This – through

(3.4) and the definition m+ =

dϕ+ ds+

=

dϕ ds

(1 − ε)Rκ(ϕ) ≤

– implies that for u-manifolds ds ≤ (1 + ε)Rκ(ϕ). dϕ

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Integrating this with respect to ϕ and using the definition of κ(ϕ), we get (1 − ε)R((ϕ) − (ϕ)) ¯ ≤ s − s¯ ≤ (1 + ε)R((ϕ) − (ϕ)) ¯ which means that, close (enough) to a κ(ϕ) → ∞ singularity, a u-manifold is (arbitrarily) similar to the graph of the rotation function (ϕ). Now the H¨older-continuity of (ϕ) required in the regularity condition (Definition 3) implies the statement of the lemma.

We note that the proof of alignment is the only place where we use our assumption that the rotation function is H¨older-continuous. The above proof shows that H¨older-continuity is indeed a necessary condition for alignment. Alignment is not among the conditions of Chernov’s theorem which our proof is based on, but we will use it in the proof of the growth properties (Proposition 5). At that place it seems to be unavoidable, so we think that H¨older-continuity of the rotation function is needed for Chernov’s method to work. On the other hand, as already pointed out in the introduction, we do not claim that it is a necessary condition for EDC. 3.2.6. Homogeneity strips, secondary singularities and homogeneous u-manifolds. Notation.

2 + κ(ϕ) (3.29) cos ϕ We will see that expansion in the e-metric is unbounded as |ω(ϕ)| → ∞. This certainly happens in the vicinity of ± π2 , nevertheless, there can exist other discontinuity values ϕ0 with the same property. Big expansion comes together with big variations of expansion (i.e. distortion) rates along u-manifolds. For that reason we need to partition the phase space into homogeneity layers in which ω(ϕ) is nearly constant. We fix a large integer k0 (to be specified in Sect. 4) and define for k > k0 the I-strips as (3.30) Ik = (s, ϕ) | k 2 ≤ |ω(ϕ)| < (k + 1)2 . ω(ϕ) :=

Recall from Definition 3 that whenever limϕ→ϕ0 |ω(ϕ)| = ∞, there exists an interval [ϕ0 − , ϕ0 ) restricted to which |ω(ϕ)| is a monotonic function of ϕ. We partition a subinterval of this interval into I-strips, thus k0 is chosen accordingly large. In case there are several discontinuity points of ω(ϕ) (with unbounded one-sided limits) we may (s) construct further I-strips, Ik , analogously. Here the index s labels the finitely many discontinuities of this kind. (u) Furthermore take I0 ; u = 1 · · · U , where the index u labels the finitely many connected components of the complement of all the above layers (that is, the “remaining part” of the phase space). We will use the notations 0 for the countably many boundary components of I-strips. Convention 2. From now on, 0 – just like S0 and Z0 before – is considered as part of ¯ whose the boundary of the phase space. That is, we will use a modified phase space M, (u) connected components are the homogeneity strips Ik (and I0 ). In complete analogy with primary singularities we introduce furthermore the notations 1 and (n) for the corresponding preimages. The geometric properties of these secondary singularity lines are analogous to those of primary ones (for example, (3.28) applies).

Correlation Decay in Certain Soft Billiards

73

Definition 9. We will say that a u-manifold is homogeneous whenever it is contained in (u) one of the homogeneity strips Ik (or I0 ). In Sects. 3.3.2 and 4 we will be concerned with u-manifolds that remain homogeneous for several steps of the dynamics. 3.3. Regularity properties of unstable manifolds. Definition 10. An unstable manifold is a u-manifold for which all past iterates are umanifolds as well. Analogously, a stable manifold is an s-manifold for which all future iterates are s-manifolds as well. From the theory of hyperbolic systems (see [Ch] and references therein) we know that there is a unique unextendable unstable (and similarly a unique unextendable stable) ¯ Thus it makes sense to talk about the manifold through (µ−)almost every point of M. (un)stable manifold through the point. We will also refer to unstable manifolds as “local unstable manifolds” (LUMs), stressing the fact that they are (and all their past iterates as well are) contained in some homogeneity layer Ik . (Remember that our phase space ends on the boundary of Ik , so Ik+1 is already another connected component.) In this subsection we deal with properties of unstable manifolds which are stronger than those proved before for arbitrary u-manifolds in Subsect. 3.2. 3.3.1. Curvature bounds. In what follows we obtain bounds on unstable manifolds that will guarantee that their curvature is uniformly bounded from above. First we look at u-fronts as submanifolds of the flow phase space. Putting the formulas in (3.13) and (3.14) together, we get D− =−

2 2 2 sin ϕB− sin ϕB− η sin ϕB− D− 2 sin ϕB− − + + . + 3 − Rm3− 3 2 3 3 2 λ λ cos ϕ λ2 cos ϕ Rλ cos ϕ Rλ2 cos2 ϕ λ cos2 ϕ

Our key estimate (3.20) implies cos ϕ|λ2 | ∼

1 + m2+ ,

which is bounded from below. So, in the above sum, terms number 2,3,4 and 5 are all bounded in absolute value. The last term is bounded due to our assumption (cf. Definition 3) η(ϕ) (2 + κ(ϕ))3 < C. As a consequence, we have |D− | + C2 , 3 with some global constant C2 and can state |D− |≤

(3.31)

Lemma 4. There is a global constant Dˆ such that for almost any point of the phase space, the front corresponding to the LUM has ˆ |D− | ≤ D.

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Proof. Choose Dˆ =

C2 3 . 3 −1

Now suppose indirectly that there is a set H ⊂ M of positive measure, for the points of which |D− | > Dˆ + ε. Then (3.31) implies that there is a c(ε) > 0 such that |D− | > Dˆ +ε +c on T −1 H . This implies that |D− | > Dˆ +ε +2c on T −2 H , and so on: |D− | > Dˆ + ε + kc on T −k H for all k > 0. But the T −k H -s are all sets of equal positive measure, which contradicts the finiteness of the phase space.

As a consequence, we can give curvature bounds for local unstable manifolds in the incoming and outgoing phase spaces. Since an unstable manifold in the Poincar´e section is the graph of a function ϕ = ϕ(s), its curvature is given by g=

ϕ (s) 1 + (ϕ (s))2

3

=√

K 1 + m2

3

.

We have reached Proposition 2. There is a global constant C such that for almost any point of the phase space, the front corresponding to the LUM has |g+ | ≤ C. Proof. It can be read from (3.13) that |K− | < C,

(3.32)

thus |g− | < C. To find out about g+ , we write 3

 g+ =

K+ 1 + m2+

3

= 

η  K− − R . 3 3 m− 1 + m2+ 1 + ( m1− + Rκ(ϕ))2 m+

This is also bounded in absolute value due to our assumption η(ϕ) (2 + κ(ϕ))3 < C (see Definition 3).

We note that this proof suggests that our condition |κ (ϕ)| ≤ C|(2 + κ(ϕ))3 | is necessary for bounded curvature, and consequently for Chernov’s method to work.

Correlation Decay in Certain Soft Billiards

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3.3.2. Distortion bounds. Length of a u-manifold W is expanded by T n locally with a factor |DT n dx|e , JW,n (x) = |dx|e where dx is the vector tangent to the curve of W at x. The aim of this subsubsection is to prove Proposition 3. Let W be an unstable manifold on which T n is smooth. Assume that Wi = T i W is a homogeneous unstable manifold for each 1 ≤ i ≤ n. Then for all x, x¯ ∈ W , 1 | ln JW,n (x) − ln JW,n (x)| ¯ ≤ C distWn (T n x, T n x) ¯ 5. i Proof. Note that JW,n (x) = n−1 i=0 JWi ,1 (T x). Hence, it is enough to prove the lemma i i for n = 1, because dist(T x, T x) ¯ grows uniformly exponentially in i due to (3.26). So we put n = 1. Denote x = T x and, we will use a to denote quantities related to the point x . Recall from Sect. 2 that the expansion factor is easily calculated in the p-metric. To obtain J := JW,1 (x) we transform |dx|e to |dx|p , take the p-expansion factor from (3.7) and (3.8) and transform back. This way: cos ϕ 1 + m 2 λ 1 λ2 √ . J = cos ϕ 1 + m2 In order to calculate the change in the logarithm of J as we move from x to x, ¯ it is best to write it with the help of (3.29) in the form J = ω(ϕ )J1 J2 with J1 =

1 + m2+ 2 + κ(ϕ)

and J2 =

cos ϕ 1 + m2+

(3.33)

λ1

λ2 .

Equations (3.19) and (3.20) imply |J1 | ∼ |J2 | ∼ 1.

(3.34)

The change in logarithm of the three terms can be calculated independently, moreover, J1 and J2 are expected to change moderately, while ω(ϕ ) can be kept under good control, because it depends only on ϕ . The three terms are investigated in three sublemmas. Thus Proposition 3 is the direct consequence of the three Sublemmas 1, 2 and 3. Of course, the first and third (concerning J1 and ω(ϕ)) have to be applied with -es. When applying Sublemma 3, we use the trivial fact |ϕ − ϕ| ¯ ≤ dist(x, x). ¯

In the arguments below, as usual, quantities with neither + nor − in their index are meant to have a +, that is, in the outgoing phase space.

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Sublemma 1. There exists a global constant C such that when a perturbation of size dx is performed on the base point, we have |d ln J1 | ≤ C|dx|. Proof. In many estimates, we will use – without further mention – that m− and K− are bounded (see (3.17) and (3.32)). With the help of (3.9) we choose the form (1 + κ(ϕ)Rm− )2 + m2− J1 = . 2 + κ(ϕ) When calculating the differential, we use η(ϕ)m+ dκ(ϕ) = η(ϕ)dϕ = dx 1 + m2+ and dm− = K− ds− = K−

ds+ dx K− K− = = dx. λ1 λ1 1 + m 2 2 2 λ + m + − 1

Calculating the differential, we get d ln J1 =

m− + κ(ϕ)R + κ 2 (ϕ)R 2 m− K− dx ((1 + κ(ϕ)Rm− )2 + m2− )3/2 +

2Rm− − 1 − m2− + (2Rm− − 1)Rm− κ(ϕ) η(ϕ)m+ dx. (2 + κ(ϕ))((1 + κ(ϕ)Rm− )2 + m2− ) 1 + m2 +

The coefficient of dx in the first term is obviously bounded since the denominator is one degree higher in κ(ϕ) and is bounded away from zero. In the second term, we use (3.9) and (3.19) to get m+ m 1 − = ∼ , (3.35) 2 + κ(ϕ) 1 + m2+ 1 + m2+ λ1 so, looking again at the degrees of polynomials (in κ) in the numerator and denominator of the second term, we have 2Rm− − 1 − m2− + (2Rm− − 1)Rm− κ(ϕ) η(ϕ)m+ η(ϕ) ≤ C1 . ≤ C

2 3 2 (2 + κ(ϕ)) (2 + κ(ϕ))((1 + κ(ϕ)Rm− ) + m− ) 1 + m2+ Sublemma 2. There exists a global constant C such that when a perturbation of size dx is performed on the base point, we have |d ln J2 | ≤ C|dx | (note the on the right-hand side).

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Proof. With the help of (3.4) and (3.10) we choose the form J2 =

cos ϕ + τ m+ + 1 + m2+

τ R

.

When calculating the differential, we use dϕ =

m+

dx

1 + m2+

and dm+ = K+ ds+ = K+

dx 1 + m2+

= (1 + m2+ )g+ dx.

This way we get d ln J2 =

m+ ( Rτ + cos ϕ) − τ − sin ϕm+ dx + B− dτ − g+ dx. cos ϕλ2 cos ϕλ2 1 + m2+

Due to (3.20), the coefficient of dx in the first term is equivalent to third term to

m ( Rτ +cos ϕ)−τ g+ , − + 1+m2+

− sin ϕm+ , 1+m2+

and in the

both of which are bounded (cf. (3.35)).

We finish by estimating dx and dτ with dx . First, 2 + κ(ϕ ) dx ≥ cdx. dx = J dx ∼ cos ϕ

(3.36)

|. On the one hand, (3.36) Second, the triangle inequality implies |dτ | ≤ |ds| + |ds− implies |ds| ≤ |dx| ≤ C|dx |. On the other hand (3.19) implies,

dx =

1 + m2 + |λ1 |ds− ∼ |2 + κ(ϕ )|ds− ≥ cds− .

These give |dτ | ≤ C|dx |.

Sublemma 3. There exists a global constant C such that if x = (s, ϕ) and x¯ = (¯s , ϕ) ¯ are in the same homogeneity layer Ik = (s, ϕ) | k 2 ≤ |ω(ϕ)| < (k + 1)2 , then |ln |ω(ϕ)| − ln |ω(ϕ)|| ¯ ≤ C|ϕ − ϕ| ¯ 1/5 .

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d Proof. We use the notation ω (ϕ) = dϕ ω(ϕ). It is easy to see that the regularity of κ(ϕ) implies ω (ϕ) ω3 (ϕ) ≤ C.

That is, everywhere inside Jk , d| ln ω(ϕ)| ω (ϕ) 2 4 = ω(ϕ) ≤ C|ω(ϕ)| ≤ 2Ck . dϕ ¯ < (k + 1)2 , implies This, together with the obvious k 2 ≤ |ω(ϕ)|, |ω(ϕ)| ¯ ln(k + 1)2 − ln k 2 | ln |ω(ϕ)| − ln |ω(ϕ)|| ¯ ≤ min 2Ck 4 |ϕ − ϕ|, 2 4 ≤ min 2Ck |ϕ − ϕ|, ¯ . k It is easy to check that for every k and every ξ , 2 4 ≤ 2C 1/5 ξ 1/5 , min 2Ck |ξ |, k which completes the proof.

After proving that the expansion factors vary nicely between nearby points on the same u-manifold, we now investigate their behaviour at points of different u-manifolds that lie on the same s-manifold. This is the absolute continuity property. Just like it was with the distortion bounds, it is important to consider homogeneous manifolds. We introduce the simplified notation Jku (x) and Jks (x) for the k-step length expansion factor at x along the unstable and the stable manifold, respectively. Proposition 4. Let Ws be a small s-manifold, x, x¯ ∈ Ws , and Wu , W¯ u two u-manifolds ¯ respectively. Assume that T k is smooth on Ws and T i Ws is a crossing Ws at x and x, homogeneous s-manifold for each 0 ≤ i ≤ k. Then ¯ ≤ C, | ln Jku (x) − ln Jku (x)| where C is a global constant. Proof. We have bounds on the change in expansion as we move along unstable manifolds. In order to have such bounds as we move along stable manifolds, we wish to use the fact that stable manifolds are turned into unstable ones when we revert time. However, this time reflection symmetry is not complete: we always work in the outgoing Poincar´e section, and reverting time turns this into the incoming one. To deal with the problem, we introduce the map P which is the dynamics through the potential, and which maps from the incoming to the outgoing Poincar´e section. That is, P ((s− , ϕ)) := (s+ , ϕ) = (s− + R(ϕ), ϕ). We can see from (3.4) that if dx− = (ds− , dϕ) is a tangent vector of the incoming phase space, then 1 |DP (dx− )|e = 1 + m2+ |λ1 | |dx− |e . 1 + m2−

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Denote by ν(x) the expansion factor of DP along the unstable manifold at x, that is − )|e ν(x) = |DP|dx(dx , where dx is an unstable vector at x. We can use (3.19) and (3.17) to − |e get ν(x) ∼ |2 + κ(ϕ)|. We also introduce the “turn back” operator, which we will denote by a “−” sign: this turns incoming phase points into outgoing phase points which corresponds to reverting the velocity. “−” is almost the identity function from M− to M+ , only the collision angle is reverted (see our sign convention in Fig. 1): − : M− → M+ , −(s, ϕ− ) := (s+ , ϕ+ ) = (s− , −ϕ− ). With these notations, if x = P (y), the time reflection symmetry implies Jks (x) =

ν(−x) 1 |2 + κ(ϕ)| ∼ u . Jku (−T k y)ν(−T k x) Jk (−T k y) |2 + κ(ϕk )|

(3.37)

The transversality of stable and unstable vectors, stated in Proposition 1 implies that Jku (x)Jks (x) is equivalent to the k-step (Lebesgue) volume expansion factor. Using (2.2), and the T -invariance of µ, we get Jku (x)Jks (x) ∼

cos ϕ . cos ϕk

(3.38)

Putting together (3.37) and (3.38) we get Jku (x) ∼ Jku (−T k y)

2 + κ(ϕk ) cos ϕ ω(ϕk ) = Jku (−T k y) . 2 + κ(ϕ) cos ϕk ω(ϕ)

The same is true for x¯ = P (y), ¯ so we have ω(ϕk ) ω(ϕ) ln +ln +C. | ln Jku (x)−ln Jku (x)| ¯ ≤ | ln Jku (−T k y)−ln Jku (−T k y)|+ ¯ ω(ϕ¯ ) ω(ϕ) ¯ k To see the boundedness of the first term of the right-hand side we can apply Proposition 3, because −T k y and −T k y¯ are on the same local unstable manifold. The second and third term is bounded because Ws and T k Ws are homogeneous, see Sect. 3.2.6. Now the proof of Proposition 4 is complete.

4. Growth Properties of Unstable Manifolds This last section is concerned with the growth properties of LUMs. Our aim is to show that LUMs “grow large and round, on the average”. This is expressed in the formulas of Proposition 5 below. Recall Convention 1. Throughout the section we use the higher iterate of the dynamics, T1 = T m0 . This has singularity set (secondary and primary) = (m0 ) . For the higher iterates of T1 the singularity set is (n) = ∪ T1 −1 ∪ · · · ∪ T −n+1 .

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δ0 -LUM’s. To formulate and prove further important conditions on growth of LUMs we need to recall several notions and notations from [Ch]. Let δ0 > 0. We call W a δ0 -LUM if it is a LUM and diam W ≤ δ0 . For an open subset V ⊂ W and x ∈ V denote by V (x) the connected component of V containing the point x. Let n ≥ 0. We call an open subset V ⊂ W a (δ0 , n)-subset if V ∩ ((n) ) = ∅ (i.e., the map T1n is smooth and homogeneous on V ) and diam T1n V (x) ≤ δ0 for every x ∈ V . Note that T1n V is then a union of δ0 -LUM’s. Define a function rV ,n on V by rV ,n (x) = dT1n V (x) (T1n x, ∂T1n V (x)). Note that rV ,n (x) is the radius of the largest open ball in T1n V (x) centered at T1n x. In particular, rW,0 (x) = dW (x, ∂W ). One further notation we introduce is Uδ (for any δ > 0), the δ-neighborhood of the closed set ∪ S0 ∪ Z0 . The aim of this section is to prove the proposition below. Proposition 5. There are constants α0 ∈ (0, 1) and β0 , D0 , η, χ , ζ > 0 with the following property. For any sufficiently small δ0 , δ > 0 and any δ0 -LUM W there is an open (δ0 , 0)-subset Vδ0 ⊂ W ∩ Uδ and an open (δ0 , 1)-subset Vδ1 ⊂ W \ Uδ (one of these may be empty) such that mW (W \ (Vδ0 ∪ Vδ1 )) = 0 and that ∀ε > 0, mW (rV 1 ,1 < ε) ≤ α0 1 · mW (rW,0 < ε/1 ) + εβ0 δ0−1 mW (W ),

(4.1)

mW (rV 0 ,0 < ε) ≤ D0 δ −η mW (rW,0 < ε)

(4.2)

mW (Vδ0 ) ≤ D0 mW (rW,0 < ζ δ χ ).

(4.3)

δ

δ

and

Proof of this Proposition goes along the lines of the arguments from [Ch]. First let us consider Accumulation of singularity lines. There are two sources of accumulation of the components of the set that can cut LUM’s into arbitrarily many pieces. First, the set 1 consists of countably many curves stretching approximately parallel to some curves in S1 (or Z1 ) and approaching them. So, each set T −1 Ik and k = 0, is a narrow strip with curvilinear boundaries. The expansion of unstable fibers in these strips can be estimated using (3.33), (3.34) and (3.30). More precisely, let W ⊂ T −1 Ik be a LUM, for some k = 0. Then the expansion factor, J u (x), on W satisfies J u (x) ∼ ω(ϕ) ∼ k 2

∀x ∈ W.

(4.4)

Second, there might be multiple intersections of the curves in S1 ∪ Z1 . Recall Kn , the complexity of S (n) ∪ Z (n) and its properties from Subsect. 2.1. Specifically important for us is the choice of the higher iterate T1 = T m0 with its relevant properties, see Convention 1.

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Indexing system. Before proving the proposition we introduce a handy indexing system, cf. [Ch]. Let δ0 > 0 and W be a δ0 -LUM. If δ0 is small enough, then W crosses at most Km0 curves of the set S (m0 ) ∪ Z (m0 ) , so the set W \ (S (m0 ) ∪ Z (m0 ) ) consists of at most Km0 + 1 connected curves, let us call them W1 , . . . , Wp with p ≤ Km0 + 1. On each Wj the map T1 (as a map on M) is smooth, but any Wj may be cut into arbitrary many (countably many) pieces by other curves in , which are the preimages of the boundaries of Ik . Let ⊂ W be a connected component of the set W \ . It can be identified with the (m0 + 1)-tuple (k1 , . . . , km0 ; j ) such that ⊂ Wj and T i ⊂ Iki for 1 ≤ i ≤ m0 . Note that this identification is almost unique. Indeed, given j , (T i ⊂)T i Wj is contained in a strip of the phase space that lies between two horizontal lines: two components of S0 ∪ Z0 . It might happen that expansion factors diverge – and consequently, homogeneity strips have been constructed – at both sides of the strip. Thus given the index ki , we have T i ⊂ Iki , where Iki can be the kith layer from one of the two homogeneity structures. In such a case we use the following convention; the homogeneity layers at the “upper” and “lower” ends of the phase space strip (corresponding to j ) are labelled by odd and even numbers, respectively. This way the indexing system is made unique and (4.4) remains true. All in all, we will write = (k1 , . . . , km0 ; j ). Of course, some strings (k1 , . . . , km0 ; j ) may not correspond to any piece of W , for such strings (k1 , . . . , km0 ; j ) = ∅. Denote by J1u (x) = J u (x) · · · J u (T m0 −1 x) the expansion factor of unstable vectors under DT1 . Let || = m () be the Euclidean length of a LUM . We record two important facts: (a) For every point x ∈ (k1 , . . . , km0 ; j ) we have     ki2 , J1u (x) ≥ Lk1 ,...,km0 := max 1 , C20   ki =0

where C20 is some positive global constant. This follows from (4.4). (b) For each (k1 , . . . , km0 ; j ) we have     ki−2 , |(k1 , . . . , km0 ; j )| ≤ Mk1 ,...,km0 := min |W |, C21   ki =0

−1 |W |max and |W |max is the maximal length of LUMs in M. This where C21 = C20 follows from the previous fact.

Next, put θ0 := 2

∞

k −2 ≤ 4/k0

k=k0

and let us turn to the proof of our growth formulas. Let W be a δ0 -LUM and δ > 0 be small. For each connected component ⊂ W \ put 0 = ∩ Uδ and 1 = int( \ Uδ ) (recall Uδ is the δ-neighborhood of ∪ S0 ∪ Z0 ). Due to the Continuation Property (cf. Subsect. 2.1) and to Alignment (cf. Subsubsect. 3.2.5), the set 0 consists of two subintervals adjacent to the endpoints of (they may overlap and cover , of course). The set 1 is either empty or a subinterval of . We put W 1 = ∪⊂W \ 1 .

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Proof of (4.1). For each 1 the set T1 (1 ∩ {rW 1 ,1 < ε}) is the union of two subintervals of T1 1 of length ε adjacent to the endpoint of T1 1 . Using the above indexing system we get mW (rW 1 ,1 < ε) ≤ 2εL−1 k1 ,...,km 0

k1 ,...,km0 ,j

m0 −1 2 ≤ 2εp −1 1 + C20 (θ0 + θ0 + · · · + θ0 )

−1 ≤ 2ε(Km0 + 1) −1 + C m θ . 20 0 0 1 We now assume that k0 is large enough so that −1 α0 := (Km0 + 1)(−1 1 + C20 m0 θ0 ) < 1

and thus get

mW (rW 1 ,1 < ε) ≤ min{|W |, 2α0 ε}.

The first term on the right-hand side of (4.1) is equal to α0 1 min{|W |, 2ε/1 } = min{α0 1 |W |, 2α0 ε}. Since α0 1 > 1, we get

mW (rW 1 ,1 < ε) ≤ α0 1 · mW rW,0 < ε/1 .

(4.5)

Next, to obtain an open (δ0 , 1)-subset Vδ1 of W 1 , one needs to further subdivide the intervals 1 ⊂ W such that |T1 1 | > δ0 . Each such LUM T1 1 we divide into s equal subintervals of length ≤ δ0 , with s ≤ |T1 1 |/δ0 . If |T1 1 | < δ0 , then we set s = 0 and leave 1 unchanged. Then the union of the preimages under T1 of the above intervals will make Vδ1 . Now we must estimate the measure of the ε-neighborhood of the additional endpoints of the subintervals of T1 1 . This gives mW (rV 1 ,1 < ε) − mW (rW 1 ,1 < ε) ≤ 2s εC22 |1 |/|T1 1 | δ

⊂W \

≤

2C22 ε|1 |/δ0

⊂W \

≤ 2C22 εδ0−1 |W |. 1

5 Here C22 = exp(const · |W |max ) is an upper bound on distortions on LUM’s, see Proposition 3. Combining the above bound with (4.5) completes the proof of (4.1) with β0 = 2C22 . We now prove (4.2). It is enough to consider ε < |W |/2, so that the right-hand side of (4.2) equals 2D0 δ −η ε. We can put Vδ0 = W \ Vδ1 . Then the left-hand side of (4.2) does not exceed 2Jδ ε, where Jδ is the number of nonempty connected components of the set Vδ0 , which is at most the number of connected components of W \ of length > 2δ. Hence, clearly Jδ ≤ |W |/δ ≤ δ0 /δ. This proves (4.2) with η = 1. Finally, we prove the inequality (4.3). Again, let be a connected component of W \ and 0 , 1 be defined as above, with the set 0 consisting of two subintervals adjacent to the endpoints of . By (3.28) – and the analogous property for the secondary

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83

singularities, see Subsubsect. 3.2.5 and 3.2.6 – each of these subintervals has length smaller than Cδ α . Now, the right-hand side of (4.3) equals D0 min{|W |, 2ζ δ χ }. So, it is enough to show that mW (Vδ0 ) ≤ Bδ χ for some B, χ > 0. We have mW (Vδ0 ) ≤

min{2Cδ α , ||}

⊂W \

≤

min{2Cδ α , Mk1 ,...,km0 }

k1 ,...,km0 ,j

≤ const · δ α + const ·

∗ k1 ,...,km0

 

min δ α , 

ki−2

ki =0

  

,

! where ∗ is taken over m0 -tuples that contain at least one nonzero index ki = 0. The following lemma – Lemma 7.2 from [Ch], which was proved in the Appendix of that α paper – completes the proof of (4.3) with χ = 2m . 0 Lemma 5. Let > 0 and m ≥ 1. Then min , (k1 · · · km )−2 ≤ B(m) · 1/2m . k1 ,...,km ≥2

With the help of this lemma Proposition 5, and consequently, Theorem 1 is proved.

5. Specific Potentials In this section we would like to show that, as important corollaries of Theorem 1, exponential decay of correlations can be established for certain specific potentials. To prove such corollaries we need to calculate the rotation function (ϕ) from the potential V (r). As to the detailed description of the Hamiltonian flow in a circularly symmetric potential, we refer to the literature, e.g. [DL] and references therein. Most important is that besides the full energy there is an additional integral of motion, the angular momentum l, that can be calculated for a specific trajectory as l = R sin ϕ, where ϕ is the collision angle at income. For brevity of notation it is worth introducing the function h(r) = (1 − 2V (r))r 2 . By the presence of the angular momentum, motion is completely integrable and is described by the pair of differential equations (recall our convention that the full energy is E = 21 ): r˙ 2 = r −2 (h(r) − l 2 ), ˙ = l. r 2

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P. B´alint, I. P. T´oth

Combining these we get d l =± , dr r h(r) − l 2

(5.1)

where the sign depends on whether r is increasing or decreasing. More precisely, there is a minimum radius h(ˆr ) = l 2 = R 2 sin2 ϕ,

rˆ = rˆ (ϕ) :

down to which r decreases (with negative sign in (5.1)) and from which r increases (with positive sign in (5.1)). This results in R l (ϕ) = 2 dr. (5.2) rˆ r h(r) − l 2 For a generic potential, the dependence of (5.2) on ϕ is rather implicit: ϕ is present both in the integrand (via l) and in the limits (via rˆ ). One possible strategy to follow is to obtain some even more complicated formulas for the derivatives in the general case, and based on those perform estimates that guarantee the desired dynamical properties. This is possible as long as only hyperbolicity and ergodicity is treated – like in [DL] – and thus only the first derivative, κ(ϕ) = (ϕ) is needed. However, for rate of mixing you need one more derivative, κ (ϕ) = (ϕ), cf. Definition 3. Finding good sufficient conditions on the potential V (r) that guarantee the regularity of κ seems to be a very hard task, if possible at all. Thus we have chosen instead to investigate some specific cases where is directly computable from (5.2). Of course, this way we could handle a much narrower class of potentials than [DL], nevertheless, the established dynamical property is stronger. Corollary 1. Consider the case of a constant potential, V (r) = V0 for any r ∈ [0, R). Correlations decay with an exponential rate in case – V0 > 0 and the configuration is arbitrary, – V0 < 0 and the configuration is such that τmin >

√

2R . 1−2V0 −1

Remarks. Actually, the analysis of this constant potential case from the point of ergodicity dates back to the late eighties, to [Kn2] and [Ba]. Rate of mixing is, to our knowledge, discussed for the first time. For potential values V0 > 21 the particle cannot enter the disks, the system is equivalent to the traditional dispersing billiard, thus we consider the opposite case, V0 < 21 . Proof. Let us introduce the quantity ν=

1 − 2V0

(5.3)

which is less or greater than 1 depending on the sign of V0 . Let us consider the case of positive V0 first and introduce furthermore the angle ϕ0 for which: ν = sin ϕ0 . In case |ϕ| > ϕ0 , |l| is greater than the maximum value h(r) can take, which indicates that the particle has too large angular momentum to enter the potential, thus = 0. In

Correlation Decay in Certain Soft Billiards

85

ϕ| the opposite case of |ϕ| < ϕ0 it is easy to obtain rˆ = R| sin and perform the integration ν of (5.2). All in all "

2 arccos sinν ϕ if |ϕ| < ϕ0 , (ϕ) = 0 if |ϕ| > ϕ0 .

On the one hand, whatever configuration we have, the system satisfies property H (cf. Definition 2), as either κ = 0 or κ ≤ −2 ν < −2. On the other hand, κ is a piecewise 1

C 1 function of ϕ and it behaves as (ϕ0 − ϕ)− 2 near the discontinuity point ϕ0 . Thus κ is regular (cf. Definition 3 and the remarks following it). This means that the first statement of our Corollary follows from Theorem 1. Now let us turn to the case of V0 < 0 (i.e. ν > 1). It is even simpler to calculate the rotation function (5.2):

sin ϕ (ϕ) = 2 arccos ν

for all ϕ. As ν > 1, this is a C 2 function on the interval [− π2 , π2 ], thus κ is definitely regular. As to property H, we have 0 > κ ≥ − ν2 , where the minimum is obtained at 2R ϕ = 0. Thus the assumption on the configuration from Definition 2 reads as τmin > ν−1 and the second statement of the corollary follows from Theorem 1.

Remark. Note that motion in the constant potential is equivalent to the problem of diffraction from geometric optics. More precisely, we can think of the disks as if they were made of a material optically different from their neighborhood, where the relative diffraction coefficient is ν from (5.3). In case the disks are optically less dense than their neighborhood (i.e. ν < 1, V0 > 0), we may observe the phenomenon of complete reflection that corresponds to the limiting angle ϕ0 . Corollary 2. Given constants A > 0 and β > −2, consider the potential

r β V (r) = A 1 − . R Correlations decay at an exponential rate in case:

(ϕ) 2π

(ϕ) 2π

2π 2π − 2+β

π 2π 2+β

− π2

−ϕ0

0

ϕ0

(a) V (r) = V0 (> 0)

πϕ 2

− π2

0

β (b) V (r) = 21 1 − Rr (β > 0)

Fig. 2. Rotation function for two examples

πϕ 2

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P. B´alint, I. P. T´oth

– A = 21 , 0 > β(> −2) and the configuration is arbitrary, – A = 21 , β > 0 and the configuration is such that τmin > 2R β . Remark. Note that according to our construction the chosen value for the constant A, A = 21 is exactly the full energy. If we had a different value for A, the integration in (5.2) would be much more complicated. In other words, Corollary 2, in contrast to Corollary 1 is unstable with respect to variations of the full energy (see also the discussion below, following the proof). Nevertheless it is nice to have at least one potential with exponential mixing for any kind of power law behaviour (if β ≤ −2, a positive measure set of trajectories is pulled into the center of the disk, cf. [DL]). Proof. By straightforward calculation h(r) =

r 2+β ; Rβ

and

2

rˆ = R | sin ϕ| 2+β .

Then it is not hard to integrate in (5.2): (ϕ) =

4 π −ϕ 2+β 2

for all ϕ = 0. Thus is piecewise linear (in the general case with one discontinuity of the first kind at ϕ = 0) and thus κ=−

4 2+β

identically. Regularity (in terms of Definition 3) is automatic. Let us consider the attracting potentials, β < 0 first. In such a case the potential has a singularity at the center of the disk, resulting in the discontinuity at ϕ = 02 . Nevertheless, κ < −2, thus property H (cf. Definition 2) and consequently the first statement of the corollary follows. Now if β > 0, as A = 21 , the “top” of the potential is equal to the energy. As a consequence, for the initial value ϕ = 0 the flow is not uniquely defined, resulting in the discontinuity for the rotation function. However, in accordance with Definition 2, property H is satisfied if τmin > 2R β . Thus the second statement of the corollary holds.

Discussion. As already mentioned, Corollary 2 is very sensitive to the convention E = 21 . Though very difficult to calculate, it is interesting to guess what happens if one perturbs the constant A (or equivalently, the full energy level). Let us consider the case β > 0 first. With A either increased or decreased from the value 21 , the physical reason for the discontinuity at ϕ = 0 disappears and we expect smooth rotation functions. By continuity of the potential at R, ( π2 ) = 0 seems also reasonable. As to the initial value ϕ = 0 let us have a look at the case A < 21 first. There is no reason for the trajectory to deviate in direction: it slows down, reaches the center and then speeds up following a linear track. Thus (0) = π . This altogether implies on the basis of Lagrange’s mean value theorem that there definitely exists at least one ϕ ∈ (0, π2 ) for which κ(ϕ) = −2. In such a case, however, stable periodic orbits tend to 2 However, in case β = −2(1− 1 ), the left and right limits coincide; this corresponds to the possibility n of regularizing the flow, cf. [DL] and [Kn1].

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87

appear and the system is most likely not even ergodic, cf. [Do1]. One can suspect that a typical repelling potential which has a maximum less than the total energy leads to non-ergodic soft billiards in a similar fashion. In the opposite case of A > 21 the behaviour of trajectories in the vicinity of ϕ = 0 is completely different. As the top of the potential is higher than the full energy, the particle cannot “climb” it thus it should “turn back”. We expect (0) = 0 and a smooth rotation function with κ > −2 for all ϕ. That would mean ergodicity and possibly exponential mixing in case of a suitable configuration (cf. Definition 2). All in all, ergodic and statistical behaviour is very sensitive to perturbation of the full energy level. In the case of β < 0 it is not so easy to guess. Nevertheless, we can say something rather surprising in one particular case that indicates similar sensitivity. Choose β = −1 and A = 1. It is not difficult to obtain h(r) = 2r − r 2 . The integral in (5.2) is a bit more complicated now, nevertheless, it is possible to evaluate: (ϕ) = 2π − 2ϕ

(5.4)

which means κ = −2 identically. This corresponds to the least ergodic behaviour we can have. It is straightforward to obtain that an identically zero potential (V (r) = 0 for all r) would result in (ϕ) = π − 2ϕ. Thus by (5.4) in this particular case of A = 1, β = −1 trajectories evolve as if they passed on freely and were reflected when leaving the disc. Thus if β = −1, we may have exponential mixing (A = 21 ) and stability (A = 1). As to other values of A it is worth mentioning that ergodicity follows from [DL] in case A < 21 . 6. Outlook In this last section we list several possible interesting directions of future research. 1. As to the possibly most direct challenge, we conjecture that there exist rapidly mixing potentials for which the condition |κ + 2| > c (i.e. property H from Definition 2) is not satisfied for nearly tangential trajectories. Thus these systems are not covered by Theorem 1, even more, at least to our knowledge, there is no result in the literature on the ergodicity or hyperbolicity of such soft billiards either. Thus we make the following Remark 1. Note that it is possible that κ tends to −2 as ϕ → π2 , nevertheless, 2+κ | cos ϕ | > c and the system can be hyperbolic (possibly ergodic or exponentially mixing). We will turn back to this question in a separate paper. The difficulty with the treatment of this case is, as already mentioned in Sect. 1, that the separate investigation of motion inside and outside the disks seems not to work at several arguments. 2. Further exciting open questions seem even more difficult. One natural direction of generalization is of course the higher dimensional case. As to softenings of multidimensional dispersing billiards (motivated e.g. by the three dimensional Lorentz process with spherical scatterers) we are not aware of any mathematical result. Even hyperbolicity and ergodicity seem difficult, not to mention decay of correlations, especially in view of the recently observed pathological behaviour of singularity manifolds in multi-dimensional billiards (see [BChSzT]).

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3. Another direction of future research, motivated mainly by applications to physics, could be the further investigation of those systems for which rapid mixing is already established. For example, as mathematical evidence on the existence of diffusion and other transport coefficients is given, it would be interesting to understand the dependence of these on certain parameters like the full energy level. 4. Last but not least, in contrast to the generality of Theorem 1, it is striking how narrow the class of specific potentials is for which we could apply the result in Sect. 5. It would be desirable to establish – at least numerically – our reasonable regularity properties for as wide a class of potentials as possible. Acknowledgements. We are very grateful to our advisor, Domokos Sz´asz for suggesting us this exciting topic and for lots of useful conversations. The financial support of the Hungarian National Foundation for Scientific Research (OTKA), grants T26176, T32022 and TS040719; of the Research Group Stochastics at TUB and the Research Project 2001-122 of the Hungarian Academy of Sciences, both affiliated to the Technical University of Budapest, is also acknowledged.

Appendix Here we provide, for the reader’s convenience, a very short, yet mainly self-contained formulation of Theorem 2.1 from [Ch]. For self-containedness, many notions and notations are repeatedly introduced. First we give the conditions P0 . . . P6 which are required, and then the statement of the theorem. P0. The dynamical system is a map T : M \ → M, where M is an open subset in ¯ and T is a C 2 a C ∞ Riemannian manifold, M¯ is compact. is a closed subset in M, diffeomorphism of its range onto its image. is called the singularity set. P1. Hyperbolicity. We assume there are two families of cone fields Cxu and Cxs in the tangent planes Tx M, x ∈ M¯ and there exists a constant > 1 with the following properties: – DT (Cxu ) ⊂ CTu x and DT (Cxs ) ⊃ CTs x whenever DT exists; – |DT (v)| ≥ |v| ∀v ∈ Cxu ; – |DT −1 (v)| ≥ |v| ∀v ∈ Cxs ; ¯ their axes have the same dimensions – these families of cones are continuous on M, across the entire M¯ which we denote by du and ds , respectively; – du + ds = dim M; – the angles between Cxu and Cxs are uniformly bounded away from zero: ∃ α > 0 such that ∀x ∈ M and for any dw1 ∈ Cxu and dw2 ∈ Cxs one has (dw1 , dw2 ) ≥ α The

Cxu

are called the unstable cones whereas Cxs are called the stable ones.

The property that the angle between stable and unstable cones is uniformly bounded away from zero is called transversality. Some notation and definitions. For any δ > 0 denote by Uδ the δ-neighborhood of the closed set ∪ ∂M. We denote by ρ the Riemannian metric in M and by m the Lebesgue measure (volume) in M. For any submanifold W ⊂ M we denote by ρW the metric on W induced by the Riemannian metric in M, by mW the Lebesgue measure on W generated by ρW , and by diamW the diameter of W in the ρW metric.

Correlation Decay in Certain Soft Billiards

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LUM-s. To be able to formulate the further properties to be checked the reader is kindly reminded of the notion of local unstable manifolds. We call a ball-like submanifold W u ⊂ M a local unstable manifold (LUM) if i) dim W u = du , ii) T −n is defined and smooth on W u for all n ≥ 0, iii) ∀x, y ∈ W u we have ρ(T −n x, T −n y) → 0 exponentially fast as n → ∞. We denote by W u (x) (or just W (x)) a local unstable manifold containing x. Similarly, local stable manifolds (LSM) are defined. P2. SRB measure. The dynamics T has to have an invariant ergodic Sinai-Ruelle-Bowen (SRB) measure µ. That is, there should be an ergodic probability measure µ on M such that for µ-a.e. x ∈ M a LUM W (x) exists, and the conditional measure on W (x) induced by µ is absolutely continuous with respect to mW (x) . Furthermore, the SRB-measure should have nice mixing properties: the system (T n , µ) is ergodic for all finite n ≥ 0. In our case the SRB measure is simply the Liouville-measure defined by (2.2) in Sect. 2. Absolute continuity of µ is straightforward, while the other above required properties (invariance, ergodicity, mixing) are proved in [DL]. P3. Bounded curvature. The tangent plane of an unstable manifold should be a Lipschitz function of the phase point. By this we mean that a base can be chosen in every tangent plane so that every base vector is a Lipschitz function of the phase point. Some notation. Denote by J u (x) = |det(DT |Exu )| the Jacobian of the map T restricted to W (x) at x, i.e. the factor of the volume expansion on the LUM W (x) at the point x. P4. Distortion bounds. Let x, y be in one connected component of W \ (n−1) , which we denote by V . Then log

n−1 i=0

J u (T i x) ≤ ϕ ρT n V (T n x, T n y) , u i J (T y)

where ϕ(·) is some function, independent of W , such that ϕ(s) → 0 as s → 0. P5. Absolute continuity. Let W1 , W2 be two sufficiently small LUM-s, such that any LSM W s intersects each of W1 and W2 in at most one point. Let W1 = {x ∈ W1 : W s (x) ∩ W2 = ∅}. Then we define a map h : W1 → W2 by sliding along stable manifolds. This map is often called a holonomy map. This has to be absolutely continuous with respect to the Lebesgue measures mW1 and mW2 , and its Jacobian (at any density point of W1 ) should be bounded, i.e. 1/C ≤

mW2 (h(W1 )) ≤ C mW1 (W1 )

with some C = C (T ) > 0. A few words are in order to discuss how our Proposition 4 implies property (P5). Let us consider the unique ergodic SRB-measure µ for the dynamical system (in our billiard dynamics this is precisely the Liouville measure defined by (2.2)). We know that the conditional measure on any LUM induced by µ is absolutely continuous with respect to the Lebesgue measure on the unstable manifold. These conditional measures are often

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referred to as u-SRB measures and their density w.r.t. the Lebesgue measure, ρW (x) is given by the following equation (cf. [Ch]): J u (T −i x) ρW (x) = lim . ρW (y) n→∞ J u (T −i y) n

i=1

Actually, what directly follows from Proposition 4 is that if we consider two nearby LUM-s W and W¯ and points x, x¯ on them joint by the holonomy map along an s-manifold, then the ratio of ρW (x) and ρW¯ (x), ¯ the densities for the two u-SRB measures is uniformly bounded. However, taking into account the invariance of µ and the uniform contraction along s-manifolds, we may get the uniform bound on the distortion of Lesbegue measures, i.e. the property we assumed in (P5). Some further notation. Let δ0 > 0. We call W a δ0 -LUM if it is a LUM and diam W ≤ δ0 . For an open subset V ⊂ W and x ∈ V denote by V (x) the connected component of V containing the point x. Let n ≥ 0. We call an open subset V ⊂ W a (δ0 , n)-subset if V ∩ (n) = ∅ (i.e., the map T n is smoothly defined on V ) and diam T n V (x) ≤ δ0 for every x ∈ V . Note that T n V is then a union of δ0 -LUM-s. Define a function rV ,n on V by rV ,n (x) = ρT n V (x) (T n x, ∂T n V (x)). Note that rV ,n (x) is the radius of the largest open ball in T n V (x) centered at T n x. In particular, rW,0 (x) = ρW (x, ∂W ). Now we are able to give the last group of technical properties that have to be verified: P6. Growth of unstable manifolds Let us assume there is a fixed δ0 > 0. Furthermore, there exist constants α0 ∈ (0, 1) and β0 , D0 , κ, σ, ζ > 0 with the following property. For any sufficiently small δ > 0 and any δ0 -LUM W there is an open (δ0 , 0)-subset Vδ0 ⊂ W ∩ Uδ and an open (δ0 , 1)-subset Vδ1 ⊂ W \ Uδ (one of these may be empty) such that the two sets are disjoint, mW (W \ (Vδ0 ∪ Vδ1 )) = 0 and ∀ε > 0, mW (rV 1 ,1 < ε) ≤ α0 · mW (rW,0 < ε/) + εβ0 δ0−1 mW (W ), δ

mW (rV 0 ,0 < ε) ≤ D0 δ −κ mW (rW,0 < ε), δ

and mW (Vδ0 ) ≤ D0 mW (rW,0 < ζ δ σ ). Now we can formulate Theorem 2.1 from [Ch]. Theorem A.1 (Chernov, 1999). Under the conditions P0 . . . P6, the dynamical system enjoys exponential decay of correlations and the central limit theorem for H¨older-continuous functions. The properties stated in the theorem are defined in Definitions 4 and 5.

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References [Ba] Baldwin, P.R.: Soft billiard systems. Physica D 29, 321–342 (1988) [BChSzT] B´alint, P., Chernov, N.I., Sz´asz, D., T´oth, I.P.: Geometry of Multi-dimensional Dispersing Billiards. To appear in Asterisque [B´aT´o] B´alint, P., T´oth, I.P.: Mixing and its rate in “soft” and “hard” billiards motivated by the Lorentz process. To appear in Physica D [B¨oTa] B¨or¨oczky, K., Jr., Tardos, G.: The longest segment in the complement of a packing. Mathematika, to appear [Ch] Chernov, N.: Decay of correlations and dispersing billiards. J. Statist. Phys. 94, 513–556 (1999) [ChY] Chernov, N.,Young, L.S.: Decay of Correlations for Lorentz gases and hard balls. In: Hard ball systems and the Lorentz gas, Encyclopedia Math. Sci. 101, Sz´asz, D. (ed), Berlin: Springer, 2000, pp. 89–120 [Do1] Donnay, V.: Non-ergodicity of two particles interacting via a smooth potential. J. Statist. Phys. 96(5–6), 1021–1048 (1999) [Do2] Donnay, V.: Elliptic islands in generalized Sinai billiards. Ergod. Th. and Dynam. Sys. 16(5), 975–1010 (1997) [DL] Donnay, V., Liverani, C.: Potentials on the two-torus for which the Hamiltonian flow is ergodic. Commun. Math. Phys. 135, 267–302 (1991) [Kn1] Knauf, A.: Ergodic and topological properties of Coulombic periodic potentials. Commun. Math. Phys. 110, 89–112 (1987) [Kn2] Knauf, A.: On soft billiard systems. Physica D 36, 259–262 (1989) [Ku] Kubo, I.: Perturbed billiard systems, I. Nagoya Math. J. 61, 1–57 (1976) [KM] Kubo, I., Murata, H.: Perturbed billiard systems II, Bernoulli properties. Nagoya Math. J. 81, 1–25 (1981) [M] Markarian, R.: Ergodic properties of plane billiards with symmetric potentials. Commun. Math. Phys. 145, 435–446 (1992) [RT] Rom-Kedar, V., Turaev, D.: Big islands in dispersing billiard-like potentials. Physica D 130(3– 4), 187–210 (1999) [Si1] Sinai,Ya.G.: On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics. Dokl. Akad. Nauk SSSR 153, 1262–1264 (1963) [Si2] Sinai, Ya.G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russ. Math. Surv. 25, 137–189 (1970) [SCh] Sinai,Ya.G., Chernov, N.: Ergodic Properties of Certain Systems of 2–D Discs and 3–D Balls. Russ. Math. Surv. 42(3), 181–201 (1987) [Y] Young, L.S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147, 585–650 (1998) Communicated by G. Gallavotti

Commun. Math. Phys. 243, 93–103 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0961-y

Communications in

Mathematical Physics

A Non-Existence Result for Supersonic Travelling Waves in the Gross-Pitaevskii Equation Philippe Gravejat Laboratoire Jacques-Louis Lions, Universit´e Pierre et Marie Curie, Boˆıte Courrier 187, 75252 Paris Cedex 05, France. E-mail: [email protected] Received: 28 April 2003 / Accepted: 14 May 2003 Published online: 17 October 2003 – © Springer-Verlag 2003

Abstract: We prove √ the non-existence of non-constant travelling waves of finite energy and of speed c > 2 in the Gross-Pitaevskii equation in dimension N ≥ 2. Introduction In this paper, we will focus on the Gross-Pitaevskii equation i∂t u = u + u(1 − |u|2 ).

(1)

One of the motivations for this equation is the analysis of Bose-Einstein condensation, which describes the behaviour of interacting bosons near absolute zero. When condensation occurs, Eq. (1) might be used as a model for the Bose condensate (see [4] for more details). In particular, this model is relevant to describe Bose-condensed gases. The model is also sometimes proposed to describe the superfluid state of H elium I I , though in this case the interactions between particles are important and cannot be neglected at temperature different from zero. In order to describe this condensation, E.P. Gross [8] and L.P. Pitaevskii [12] considered a set of N bosons of mass m that fill a volume V : they then assumed almost all bosons are Bose-condensed in the fundamental state of energy. Therefore, they can be described by a macroscopic wave function . They then deduced the Gross-Pitaevskii equation satisfied by the function from a Hartree-Fock approach, 2 i∂t + |(x , t)|2 U (x − x )dx = 0. − 2m V Here, U (x − x ) denotes the interaction between the bosons at positions x and x : this interaction being of very short range, it is often approached by U0 δ(x − x ). Thus, denoting Eb , the average energy level per unit mass of a boson, and, u(t, x) = e

−imEb t

(t, x),

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they computed the equation i∂t u +

2 u + mEb u − U0 u|u|2 = 0. 2m

They finally rescaled the equation by taking the mean density ρ0 = √ 2m2 Eb

as unit length, and,

mEb

mEb U0

as unit,

as unit time, in order to obtain the dimensionless

equation i∂t u + u + u(1 − |u|2 ) = 0. At this point, we can write the hydrodynamic form of this equation by using the Madelung transform ([11]), √ u = ρeiθ , which is only meaningful where ρ does not vanish. Denoting v = 2∇θ, we deduce the equations ∂t ρ + div(ρv) = 0 |∇ρ|2 ρ(∂t v + v.∇v) + ∇ρ 2 = ρ∇( ρ ρ − 2ρ 2 ). Those equations are similar to the Euler equations for a irrotational ideal fluid with as a quantic pressure p(ρ) = ρ 2 : the term of the right member is then considered √ pressure term. Here, we can remark that the sound speed is cs = 2. In this article, we will consider Eq. (1) in the space RN for every integer N ≥ 2: we can notice that this equation is associated to the energy 1 1 E(u) = |∇u|2 + (1 − |u|2 )2 = e(u). 2 RN 4 RN RN We will study the travelling waves of finite energy and of speed c ≥ 0 for this equation i.e. the solutions u which are of the form u(t, x) = v(x1 − ct, . . . , xN ). The simplified equation for v, which we will consider now, is ic∂1 v + v + v(1 − |v|2 ) = 0.

(2)

C.A. Jones, S.J. Putterman and P.H. Roberts [9, 10] first considered formally and numerically those particular solutions because they suppose they play an important role in the long time dynamics of general solutions: they conjectured √that non-constant travelling waves only exist when their speed c is in the interval ]0, 2[ i.e. they all are subsonic. They then noticed the apparition of vortices for those solutions when c tends to 0 in dimension two (two parallel oppositely directed vortices) and in dimension three (a vortex ring). They also gave for each value of c, the asymptotic development at infinity in dimension two, iαx1 v(x) − 1 ∼ |x|→+∞ x 2 + (1 − c2 )x 2 1 2 2

A Non-Existence Result for Supersonic Travelling Waves

95

and in dimension three, v(x) − 1

∼

iαx1

|x|→+∞

(x12

+ (1 −

c2 2 2 )(x2

3

+ x32 )) 2

,

where the constant α is the stretched dipole coefficient. F. B´ethuel and J.C. Saut [1, 2] first studied mathematically those travelling waves: they proved their existence in dimension two when c is small and the apparition of vortices in this case. They also gave a mathematical proof for their limit at infinity. In dimension N ≥ 3, F. B´ethuel, G. Orlandi and D. Smets [3] showed their existence when c is small and the apparition of a vortex ring. In every dimension, A. Farina [5] proved a universal bound for their modulus. Finally, we proved their uniform convergence to a constant of modulus one ([6]) in dimension N ≥ 3, and also studied their decay at infinity ([7]) in dimension N ≥ 2. In this paper, we will complete all those results by the following theorem: √ Theorem 1. In dimension N ≥ 2, a solution of Eq. (2) of finite energy and speed c > 2 is constant. This paper will be organized around the proof of Theorem 1. In the first step, we 2 will write the equation √ satisfied by η = 1 − |v| . Then, we will derive a new integral identity when c > 2: this is the crucial step of the proof of Theorem 1. Finally, we will write the Pohozaev identities in order to prove that the energy E(v) vanishes and that the travelling wave v is constant. 1. Equation Satisfied by η In this part, we will write the equation satisfied by the variable η = 1 − |v|2 for every c ≥ 0: in particular, the results in this section (i.e. Propositions 1, 2 and 3) are valid for every c ≥ 0. We first recall two useful propositions yet mentioned in [6, 7] and based on arguments taken from F. B´ethuel and J.C. Saut [1, 2]. Proposition 1. For every c ≥ 0, if v is a solution of Eq. (2) in L1loc (RN ) of finite energy, then v is regular, bounded and its gradient belongs to all the spaces W k,p (RN ) for k ∈ N and p ∈ [2, +∞]. Thus, a travelling wave is a regular function and a classical solution of Eq. (2), which will simplify the following discussion. Proposition 2. The modulus ρ of v satisfies ρ(x)

→

|x|→+∞

1.

Proof. Indeed, the function η2 is uniformly continuous because v is bounded and lipschitzian by Proposition 1. As RN η2 is finite, η converges uniformly to 0 at infinity, which completes the proof of this proposition.

Thus, the function ρ does not vanish at infinity, and we can define a regular function θ on a neighborhood of infinity such that v can be written v = ρeiθ . Denoting ψ, a regular function from RN to [0, 1] such that ψ = 0 on a neighborhood of Z = {x ∈ RN , ρ(x) = 0}, and ψ = 1 on a neighborhood of infinity, and denoting v = v1 + iv2 , we can write the equation satisfied by the function η:

96

P. Gravejat

Proposition 3. For every c ≥ 0, the function η satisfies the equation 2 2 η − 2η + c2 ∂1,1 η = −F + 2c∂1 div(G),

(3)

where F = 2|∇v|2 + 2η2 + 2c(v1 ∂1 v2 − v2 ∂1 v1 ) − 2c∂1 (ψθ ) and G = v1 ∇v2 − v2 ∇v1 − ∇(ψθ ). Proof. By Eq.(2), we have v1 − c∂1 v2 + v1 (1 − |v|2 ) = 0,

(4)

v2 + c∂1 v1 + v2 (1 − |v| ) = 0.

(5)

2

We then compute 2 2 2 η − 2η + c2 ∂1,1 η = −2|∇v|2 − 2(v.v) − 2η + c2 ∂1,1 η,

and by Eqs. (4)–(5), we have on one hand v.v = v1 v1 + v2 v2 = c(v1 ∂1 v2 − v2 ∂1 v1 ) − |v|2 η, and, on the other hand, c∂1 η = −2c(v1 ∂1 v1 + v2 ∂1 v2 ) = 2(v2 v1 − v1 v2 ) = 2div(∇v2 v1 − ∇v1 v2 ). Therefore, we finally get 2 2 η − 2η + c2 ∂1,1 η = − 2|∇v|2 − 2η2 − 2c(v1 ∂1 v2 − v2 ∂1 v1 )

+ 2c∂1 div(v1 ∇v2 − v2 ∇v1 ) = − (2|∇v|2 + 2η2 + 2c(v1 ∂1 v2 − v2 ∂1 v1 ) − 2c∂1 (ψθ )) + 2c∂1 div(v1 ∇v2 − v2 ∇v1 − ∇(ψθ )) = − F + 2c∂1 div(G), which is the desired equality.

2. A New Integral Relation We have

√ Proposition 4. If c > 2, the travelling wave v satisfies the integral equation 2 (|∇v|2 + η2 ) = c( 2 − 1) (v1 ∂1 v2 − v2 ∂1 v1 − ∂1 (ψθ )). c RN RN √ Remark 1. This is the only point where we use the assumption c > 2. For the proof, we use

Lemma 1. F and G belong to the space W 2,1 (RN ).

(6)

A Non-Existence Result for Supersonic Travelling Waves

97

Proof. Indeed, G is regular and satisfies at infinity G = (ρ 2 − 1)∇θ. By Proposition 1, the functions η and ∇v belong to H 2 (RN ) ∩ W 2,∞ (RN ): since |∇v|2 = |∇ρ|2 + ρ 2 |∇θ|2 , and since ρ uniformly converges to 1 at infinity by Proposition 2, the function ∇θ belongs to H 2 ∩ W 2,∞ on a neighborhood of infinity: thus, the function G belongs to the space W 2,1 (RN ). Since F = 2(|∇v|2 + η2 ) + 2cG1 , the function F also belongs to this space, which completes the proof of Lemma 1.

Proof of Proposition 4. By Proposition 1, the function η belongs to H 4 (RN ), and we can write by taking the Fourier transformation of Eq. (3) (ξ ) − 2c ∀ξ ∈ RN , (|ξ |4 + 2|ξ |2 − c2 ξ12 ) η(ξ ) = |ξ |2 F

N

j (ξ ) := H (ξ ). (7) ξ 1 ξj G

j =1

Consider the set

= {ξ ∈ RN , |ξ |4 + 2|ξ |2 − c2 ξ12 = 0}. √ √ This set is reduced to {0} when c ≤ 2, but, when c > 2, it is a regular hypersurface of codimension 1 except at {0}: in dimension 2, it has the geometry of a bretzel, and in higher dimensions, it has the geometry of two spheres linked at some point. Indeed, is a surface of revolution around axis x1 : in spherical coordinates ξ = (r cos(α), r sin(α) cos(β), . . .), it is described by the equation r 2 = c2 cos2 (α) − 2. In particular, we notice that there are two sequences (xn )n∈N and (yn )n∈N of points of \ {0} which tend to 0 when n tends to +∞ and which satisfy

xn 2 2 2 yn 2 , 1 − 2 , 0, . . . , and, , − 1 − 2 , 0, . . . . → → |xn | n→+∞ c2 c |yn | n→+∞ c2 c (8) Coming back to the study of Eq. (7), we claim that Lemma 2. The function H defined by Eq. (7) is continuous on RN and satisfies H = 0 on . Proof. The first assertion follows from Lemma 1: indeed, since the functions F and G (ξ ) and ξ → ξ1 ξj G j (ξ ) are belong to the space W 2,1 (RN ), the functions ξ → |ξ |2 F N N continuous on R , and therefore, the function H is continuous on R too. In order to prove the second √ assertion, we argue by contradiction and assume there is some point ξ0 ∈ \ {0, ( c2 − 2, 0, . . . , 0)} such that H (ξ0 ) = 0.

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Since the function H is continuous on RN , there is some neighborhood V of the point ξ0 and some strictly positive number A such that ∀ξ ∈ V , |H (ξ )| ≥ A. Hence, we have by Eq. (7) A2 . (|ξ |4 + 2|ξ |2 − c2 ξ12 )2

∀ξ ∈ V \ , | η(ξ )|2 ≥

Integrating this relation and using spherical coordinates, we get dξ | η(ξ )|2 dξ ≥ A2 4 + 2|ξ |2 − c2 ξ 2 )2 (|ξ | V \ V \ 1 s N−2 dsdξ1 ≥ AN 2 2 2 2 2 2 2 V \∩R×R+ ((s + ξ1 ) + 2s + (2 − c )ξ1 ) r N−1 sinN−2 (α)drdα ≥ AN 4 2 2 2 2 V \∩R+ ×[0,π] r (r + 2 − c cos (α)) . Thus, denoting ξ0 = (r0 cos(α0 ), r0 sin(α0 ) cos(β0 ), . . .), there is some real number > 0 such that r0 + α0 + r N−1 sinN−2 (α)drdα | η(ξ )|2 dξ ≥ AN := AN I (α0 , r0 , ). 4 2 2 2 2 V \ r0 − α0 − r (r + 2 − c cos (α)) √ Since ξ0 ∈ \ {0, ( c2 − 2, 0, . . . , 0)}, r0 is different from 0 and α0 is different from 0 and π2 , and so, we can compute for sufficiently small I (α0 , r0 , ) ≥ A(α0 , r0 , )

r0 + r0 −

α0 +

α0 −

(r 2

drdα . + 2 − c2 cos2 (α))2

By doing the change of variable r = c2 cos2 (β) − 2, we know that there is some real number δ > 0 such that α0 +δ α0 +δ dβdα , I (α0 , r0 , ) ≥ A(α0 , r0 , ) 2 2 2 2 2 α0 −δ α0 −δ (c cos (β) − c cos (α)) and finally, by denoting a = α − α0 and b = β − α0 , we get

δ

δ

−δ δ

−δ δ

−δ δ

−δ δ

I (α0 , r0 , ) ≥ A(α0 , r0 , , c) ≥ A(α0 , r0 , , c) ≥ A(α0 , r0 , , c)

−δ

−δ

dadb (cos2 (b + α0 ) − cos2 (a + α0 ))2 dadb (cos(2b + 2α0 ) − cos(2a + 2α0 ))2 dadb . (sin(b − a))2

A Non-Existence Result for Supersonic Travelling Waves

99

1 2 Since the function (a, b) → (sin(b−a)) 2 is not integrable at the origin in R , the integral I (α0 , r0 , ) is not finite and we can conclude that | η(ξ )|2 dξ = +∞. V \

Since the energy of the function v is finite, so is the integral RN η2 , and by the Plancherel theorem, we deduce | η(ξ )|2 dξ < +∞, RN

which leads √ to a contradiction and proves that H is identically equal to 0 on the set \ {0, ( c2 − 2, 0, . . . , 0)}. The second assertion of Lemma 2 then follows from the continuity of the function H .

End of the Proof of Proposition 4. By Lemma 2, we now know that ∀n ∈ N, H (xn ) = 0, which gives by dividing by |xn |2 , (xn ) = 2c ∀n ∈ N, F

N (xn )1 (xn )j j =1

|xn | |xn |

j (xn ). G

and G j , we can take the limit as xn → 0 of this By continuity of the functions F expression and obtain by assertion (8) (0) = 4 G 2 (0). 1 (0) + 2c 1 − 2 G F c c2 Likewise, we know that ∀n ∈ N, H (yn ) = 0, which gives by the same method, 2 4 F (0) = G1 (0) − 2c 1 − 2 G 2 (0). c c Finally, we have (0) = 4 G 1 (0), F c so that,

4 F (x)dx = G1 (x)dx. c RN RN

The conclusion follows from the expressions of the functions F and G.

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3. Pohozaev Identities We now prove for sake of completeness two well-known identities based on the use of Pohozaev multipliers (See [9, 10, 1, 3] for more details). Those estimates do not use the √ fact that c > 2. Proposition 5. Let c ≥ 0. A finite energy solution v to Eq. (2) satisfies the two identities E(v) = |∂1 v|2 , (9) RN c ∀2 ≤ j ≤ N, E(v) = |∂j v|2 + (v2 ∂1 v1 − v1 ∂1 v2 + ∂1 (ψθ )). (10) N 2 R RN Proof. We first fix some real number R > 0 and we multiply Eq. (2) by Pohozaev multiplier x1 ∂1 v on the ball B(0, R), (v.x1 ∂1 v + x1 ∂1 v.v(1 − |v|2 )) = 0. (11) B(0,R)

Integrating by parts, we compute v.x1 ∂1 v = B(0,R)

|∇v|2 − 2

B(0,R)

−

ν1 x 1

|∂1 v| + 2

B(0,R) |∇v|2

2

S(0,R)

S(0,R)

x1 ∂1 v.∂ν v

,

and,

x1 ∂1 v.v(1 − |v|2 ) = B(0,R)

B(0,R)

By Eq. (11), we then get e(v) = B(0,R)

(1 − |v|2 )2 − 4

x 1 ν1 S(0,R)

|∂1 v|2 −

B(0,R)

S(0,R)

x1 ∂1 v.∂ν v +

On one hand, by Proposition 1, we know that e(v) − |∂1 v|2 → B(0,R)

R→+∞

B(0,R)

(1 − |v|2 )2 . 4

ν1 x1 e(v).

(12)

S(0,R)

E(v) −

On the other hand, we have x ∂ v.∂ v − ν x e(v) ≤ AR 1 1 ν 1 1 S(0,R)

RN

S(0,R)

|∂1 v|2 .

e(v).

Since the integral R+ ( S(0,R) e(v))dR is finite, there are some positive real numbers Rn such that Rn → +∞ and n→+∞

∀n ∈ N, Rn

e(v) ≤ S(0,Rn )

1 , ln(Rn )

A Non-Existence Result for Supersonic Travelling Waves

which gives

101

S(0,Rn )

(x1 ∂1 v.∂ν v − ν1 x1 e(v))

and finally, by Eq. (12),

→

n→+∞

0,

E(v) =

RN

|∂1 v|2 .

In order to prove the second identity, we multiply Eq. (2) by Pohozaev multiplier xj ∂j v on the ball B(0, R), (v.xj ∂j v + ic∂1 v.xj ∂j v + xj ∂j v.v(1 − |v|2 )) = 0. (13) B(0,R)

Integrating by parts, we compute v.xj ∂j v = B(0,R)

|∇v|2 − 2

B(0,R)

− S(0,R)

ν j xj

|∂j v| + 2

B(0,R) |∇v|2

2

S(0,R)

xj ∂j v.∂ν v

,

and, B(0,R)

xj ∂j v.v(1 − |v|2 ) =

B(0,R)

(1 − |v|2 )2 − 4

S(0,R)

x j νj

(1 − |v|2 )2 . 4

If R is sufficiently large such as ψ = 1 on S(0, R), we also compute

1 i∂1 v.xj ∂j v = xj (ρ 2 − 1)(νj ∂1 θ − ν1 ∂j θ) 2 S(0,R) B(0,R) − (v2 ∂1 v1 − v1 ∂1 v2 + ∂1 (ψθ )) , B(0,R)

which leads to e(v) = B(0,R)

B(0,R)

|∂j v|2 +

+ S(0,R)

c 2

(v2 ∂1 v1 − v1 ∂1 v2 + ∂1 (ψθ )) B(0,R)

(xj νj e(v) − xj ∂j v.∂ν v − xj (ρ 2 − 1)(νj ∂1 θ − ν1 ∂j θ )). (14)

On one hand, by Proposition 1, we know that c e(v) − |∂j v|2 − (v2 ∂1 v1 − v1 ∂1 v2 + ∂1 (ψθ )) 2 B(0,R) B(0,R) B(0,R) c → E(v) − |∂j v|2 − (v2 ∂1 v1 − v1 ∂1 v2 + ∂1 (ψθ )). R→+∞ 2 RN RN On the other hand, we have 2 ≤ AR (x ν e(v) − x ∂ v.∂ v − x (ρ − 1)(ν ∂ θ − ν ∂ θ)) j j j j ν j j 1 1 j S(0,R)

S(0,R)

e(v).

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P. Gravejat

By using the sequence of positive real numbers Rn constructed for proving equality (9), we get (xj νj e(v) − xj ∂j v.∂ν v − xj (ρ 2 − 1)(νj ∂1 θ − ν1 ∂j θ)) → 0, n→+∞

S(0,Rn )

and finally, by Eq. (14), E(v) =

c |∂j v| + (v2 ∂1 v1 − v1 ∂1 v2 + ∂1 (ψθ )). 2 RN RN 2

4. Conclusion We now complete the proof of Theorem 1. By Proposition 5, we have |∂1 v|2 , E(v) = RN

which gives by denoting ∇⊥ v = (∂2 v, . . . , ∂N v), 1 1 E(v) 1 2 2 |∇⊥ v| + η = |∂1 v|2 = , 2 RN 4 RN 2 RN 2 and

RN

η2 = 2E(v) − 2

RN

|∇⊥ v|2 .

(15)

We then compute

RN

(|∇v|2 + η2 ) = 3E(v) −

and, by Proposition 5, (v1 ∂1 v2 − v2 ∂1 v1 − ∂1 (ψθ )) = c RN

RN

|∇⊥ v|2 ,

2 |∇⊥ v|2 − 2E(v). N − 1 RN

(16)

(17)

Proposition 4 gives

2 (|∇v|2 + η2 ) = c 2 − 1 (v1 ∂1 v2 − v2 ∂1 v1 − ∂1 (ψθ )), c RN RN which leads by Eqs. (16)–(17) to (c2 + 4)(N − 1)E(v) = ((N − 3)c2 + 4) If N = 2, we get

RN

(c2 + 4)E(v) = (4 − c2 )

RN

|∇⊥ v|2 ,

|∇⊥ v|2 .

(18)

A Non-Existence Result for Supersonic Travelling Waves

103

which gives by Eq. (15), c2 + 4 η2 = −2c2 |∇⊥ v|2 = 0. 2 RN RN Finally, we have involving Eq. (15) once more E(v) = 0. If N ≥ 3, since by Eq. (15),

RN

|∇⊥ v|2 ≤ E(v),

Eq. (18) gives (2c2 + 4N − 8)E(v) ≤ 0, and finally, E(v) is also equal to 0 in this case. In conclusion, since E(v) = 0, the function ∇v vanishes on RN and v is a constant (of modulus one since η also vanishes on RN ). Acknowledgement. The author is grateful to F. B´ethuel, G. Orlandi, J.C. Saut and D. Smets for interesting and helpful discussions.

References 1. B´ethuel, F., Saut, J.C.: Travelling waves for the Gross-Pitaevskii equation I. Ann. Inst. Henri Poincar´e. Physique th´eorique 70(2), 147–238 (1999) 2. B´ethuel, F., Saut, J.C.: Travelling waves for the Gross-Pitaevskii equation II. Preprint 3. B´ethuel, F., Orlandi, G., Smets, D.: Vortex rings for the Gross-Pitaevskii equation. J. Eur. Math. Soc., in press 4. Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G.: Photons et atomes. Introduction a` l’´electrodynamique quantique. Interéditions, Editions du CNRS, 1987 5. Farina, A.: From Ginzburg-Landau to Gross-Pitaevskii. Preprint 6. Gravejat, P.: Limit at infinity for travelling waves in the Gross-Pitaevskii equation. C. R. Acad. Sci. Paris, S´er I 336, 147–152 (2003) 7. Gravejat, P.: Decay for travelling waves in the Gross-Pitaevskii equation. Ann. Inst. Henri Poincar´e Analyse non lin´eaire, in press 8. Gross, E.P.: Hydrodynamics of a superfluid condensate. J. Math. Phys. 4(2), 195–207 (1963) 9. Jones, C.A., Roberts, P.H.: Motions in a Bose condensate IV, Axisymmetric solitary waves. J. Phys. A.Math. Gen. 15, 2599–2619 (1982) 10. Jones, C.A., Putterman, S.J., Roberts, P.H.: Motions in a Bose condensate V, Stability of wave solutions of nonlinear Schr¨odinger equations in two and three dimensions. J. Phys. A. Math. Gen. 15, 2991–3011 (1982) 11. Madelung, E.: Quantumtheorie in Hydrodynamische form. Zts. F. Phys. 40, 322–326 (1926) 12. Pitaevskii, L.P.: Vortex lines in an imperfect Bose gas. Soviet Physics JEPT 13(2), 451–454 (1961) Communicated by P. Constantin

Commun. Math. Phys. 243, 105–122 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0956-8

Communications in

Mathematical Physics

Conformal Restriction, Highest-Weight Representations and SLE Roland Friedrich1,2 , Wendelin Werner1,3 1 2 3

Laboratoire de Math´ematiques, Universit´e Paris-Sud, 91405 Orsay cedex, France. E-mail: [email protected] I.H.E.S., 35, route de Chartres, 91440 Bures-sur-Yvette, France. E-mail: [email protected] Institut Universitaire de France

Recieved: 24 February 2003 / Accepted: 27 May 2003 Published online: 14 October 2003 – © Springer-Verlag 2003

Abstract: We show how to relate Schramm-Loewner Evolutions (SLE) to highestweight representations of infinite-dimensional Lie algebras that are singular at level two, using the conformal restriction properties studied by Lawler, Schramm and Werner in [33]. This confirms the prediction from conformal field theory that two-dimensional critical systems are related to degenerate representations. 1. Introduction The goal of this paper is to show how the Schramm-Loewner evolutions (or Stochastic Loewner Evolutions, which is anyway abbreviated by SLE) can be used to interpret in a simple and elementary way some of the starting points of conformal field theory, stated by Belavin-Polyakov-Zamolodchikov in their seminal paper [7]. In particular, we will see how restriction properties studied in [33] can be rephrased in terms of highest-weight representations of the Lie algebra A of vector fields on the unit circle (and its central extension, the Virasoro algebra). The results in this paper were announced in the note [18]. It is probably worthwhile to spend some lines outlining our perception of the history of this subject (see also the recent review paper by Cardy [10]): It has been recognized by physicists some decades ago that two-dimensional systems from statistical physics near their critical temperatures have some universal features. In particular, some quantities (correlation length for instance) obey universal power laws near the critical temperature, and the value of the (critical) exponent in fact depends only on the phenomenological features of the discrete system (for instance, it is the same for the same model, taken on different lattices). In order to identify the value of the exponents, two techniques turned out to be very successful. The first one is the “Coulomb gas approach” (see e.g. [37] and the references therein, as well as the reprinted papers in [21]), which is based on explicit computations for some specific models. The second one (see Polyakov [38], Belavin-Polyakov-Zamolodchikov [7], Cardy [8]) is conformal field theory. Based on

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the analogy with some other problems, it is argued in [7] that two-dimensional critical systems are associated to conformal fields. These fields should then satisfy certain relations, such as the Ward identities, which then allow to make a link with highest-weight representations of the Virasoro algebra. Then the critical exponents can be identified from the corresponding highest weights. We now quote from [22]: “The remarkable link between the theory of highest-weight modules over the Virasoro algebra and conformal field theory and statistical mechanics was discovered by Belavin-Polyakov-Zamolodchikov [6, 7]. Conformal Field Theory has now become a huge field with ramifications to other fields of mathematics and mathematical physics”. We refer for instance to the introduction of [16] and the compilation of papers in [19, 21]. This approach has then been used to develop the related “quantum gravity” method (see e.g. [13]) and the references therein. It is worthwhile to stress some points: The actual mathematical meaning, intuition or definition of these fields (and their properties, such as the Ward identities) in terms of the discrete two-dimensional models was to our knowledge never clarified. Also, the notion of “conformal invariance” itself for these systems remained rather obscure. In the case of critical percolation, Aizenman [2] formulated clearly what it should mean, but for other famous models such as self-avoiding walks, or Ising, the precise conjecture was never stated until recently. In [9], Cardy pointed out that in the case of critical percolation, the arguments from [7, 8] could be used in order to predict the exact formula for asymptotic crossing probabilities of a topological rectangle by a percolation cluster. This prediction was popularized in the mathematical community through the review paper by LanglandsPouliot-StAubin [25], that attracted many mathematicians to this specific problem (including Stas Smirnov). In that paper, the authors also explain how difficult it is for mathematicians to understand Cardy’s arguments. On a rigorous mathematical level, only limited progress towards the understanding of 2D critical phenomena had been made before the late 90’s. In 1999, Oded Schramm [40] defined a one-parameter family of random curves based on Loewner’s differential equation, SLEκ indexed by the positive real parameter κ. These random curves are the only ones which combine conformal invariance and a Markovian-type property (which is usually already satisfied in the discrete setting). Provided that the scaling limit of an interface in a model studied in statistical physics (such as Ising, Potts or percolation) exists and is conformally invariant (and this approach allows one to give a precise meaning to this), then the limiting object must be one of the SLEκ curves. Conformal invariance has now been rigorously shown in some cases (critical site percolation on the triangular lattice has been solved by Stas Smirnov [42], the case of loop-erased random walks and uniform spanning trees is treated in Lawler-Schramm-Werner [31]). For a general discussion of the conjectured relation between the discrete models and SLE, see [39]. See also [32] for self-avoiding walks and self-avoiding polygons. In the SLE setting, the critical exponents simply correspond to principal eigenvalues of some differential operators, see Lawler-Schramm-Werner [27–30]. Recognizing this led to complete mathematical derivations of the values of critical exponents for the models, that have been proved to be conformally invariant, in particular for critical percolation on the triangular lattice (see [43]). In order to establish rigorously the conjectures for the other models, the missing step is to show their conformal invariance. Using the Markovian property (which implies that with “time” the conditional probabilities of macroscopic events are martingales) of SLE and Itˆo’s formula, one readily sees that the probabilities of macroscopic events such as crossing probabilities have to satisfy some second order differential equations [27–29, 41]. This enables one to recover

Conformal Restriction, Highest-Weight Representations and SLE

107

Cardy’s formula in the case of SLE6 , and to generalize it to other models (i.e. for other values of κ). Note that just as observed by Carleson in the case of critical percolation, these crossing probabilities formulae become extremely simple in well-chosen triangles, as pointed out by Dub´edat [11]. It is therefore natural to think that SLE should be related to conformal field theory and to highest-weight representations of the Virasoro Algebra. Bauer-Bernard [3, 4] recently viewed (with a physics approach) SLE as a process living on a “Virasoro group”, which shows such a link and enables them among other things to recover in conformal field theory language, the generalized crossing probabilities mentioned above. Back in 1999, Lawler and Werner [34] had introduced a notion of universality based on a family of conformal restriction measures, that gave a good insight into the fact that the exponents associated to self-avoiding walks, critical percolation and simple random walks were in fact the same (these correspond in CFT language to the models with zero central charge) and pointed out the important role played by these restriction properties (which became also instrumental in the papers [27–29]). In the recent paper [33] by Lawler, Schramm and Werner, closely related (but slightly different) restriction properties are studied. Loosely speaking (and this will be recalled in more precise terms below), one looks for random subsets K of a given set (the upper half-plane, say), joining two boundary points (0 and infinity, say), such that the law of K is invariant under the following operations: For all simply connected subsets H of H, the law of K conditioned on K ⊂ H is equal to the law of (K), where is a conformal map from H onto H preserving the two prescribed boundary points. In some sense, the law of K is “invariant” under perturbation of the boundary. It turns out that one can fully classify these random sets (it is a one-parameter family termed restriction measures, that are indexed by their positive real exponent), and that they can be constructed in different but equivalent ways. For instance, by taking the hull of Brownian excursions (possibly reflected on the boundary of the domain), or by adding to an SLEκ path a certain Poissonian cloud of Brownian loops. This gives an alternative description of the SLE curves, that does not rely on Loewner’s equation and on the Markovian property, but can be interpreted as a variational equation (“how does the law of the SLE change”) with respect to perturbations of the domain. This in turn can be shown to correspond in the geometric setting of CFT to differentiating the partition function with respect to the moduli, which then gives the correlation functions of the stress-energy tensor. In fact, the SLE correlation functions derived below, are those of the stress tensor. This will not be further explained in the present text, but is one of the subjects of the forthcoming paper [17]. The aim of the present paper is to point out that these restriction properties (and their relation to the SLE curves) can be rephrased in a way that exhibits a direct and simple link between the SLE curves (and therefore also the two-dimensional critical systems) and representation theory. In this setting, the Ward identities turn out to be a reformulation of the restriction property. More precisely, we will associate to each restriction measure a highest-weight representation of A (viewed as operators on a properly defined vector space). The degeneracy of the representation corresponds to the Markovian type property of SLE. The density of the Poissonian cloud of Brownian loops that one has to add to the SLEκ is (up to a sign-change) the central charge associated to the representation and the exponent of the restriction measure is its highest-weight. The reader acquainted with conformal field theory will recognize almost all the identities that we will derive as “usual and standard” facts from the CFT perspective, but the point is here to give them a rigorous meaning and interpretation in terms of SLE and discrete models. Also, in the spirit of the conclusion of Cardy’s review paper [10] and as

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already confirmed by [3], the rigorous SLE approach should hopefully become useful and exploited within the theoretical physics community. 2. Background 2.1. Chordal SLE. The chordal SLEκ curve γ is characterized as follows: The conformal maps gt from H \ γ [0, t] onto H such that gt (z) = z + o(1) when z → ∞ solve the ordinary differential equation ∂t gt (z) = 2/(gt (z)−Wt ) (and are started from g0 (z) = z), √ where Wt = κbt (here and in the sequel, (bt , t ≥ 0) is a standard real-valued Brownian motion with b0 = 0). In other words, γt is precisely the point such that gt (γt ) = Wt . See e.g. [27, 39] for the definition and properties of SLE, or [26, 44] for reviews. Note that for any finite set of points, if one defines the function ft (z) = gt (z) − Wt , the Markov property of the Brownian motion b shows that the law of (ft0 +t , t ≥ 0) is identical to that of (ft , t ≥ 0). Then Itˆo’s formula immediately implies that for any set of real points x1 , . . . , xn and any smooth function F : Rn → R, dF (ft (x1 ), . . . , ft (xn )) = −dWt

n j =1

  n n κ  2 ∂j F (ft (x1 ), . . . , ft (xn )) + dt ( ∂j ) ∂j )2 + ( 2 ft (xj )  j =1

j =1

× F (ft (x1 ), . . . , ft (xn )), i.e. if one defines the operators LN := − F (ft (x1 ), . . . , ft (xn )),

n

1+N ∂j , j =1 xj

and the value Ft =

dFt = dWt L−1 Ft + dt (κ/2L2−1 − 2L−2 )F (ft (x1 ), . . . , ft (xn )). From this the chordal crossing probabilities [27, 29] are identified by using the fact that the drift term vanishes iff F is a martingale, i.e. if (κ/2L2−1 − 2L−2 )F = 0. This already enabled [3] to tie a link with conformal field theory. 2.2. Chordal restriction. All the facts recalled in this section are derived in [33]. Let H denote the open upper half-plane. We call H+ (resp. H) the family of simply connected subsets H of H such that: H \ H is bounded and bounded away from R− (resp. from 0). For such an H , we define the conformal map H from H onto H such that H (0) = 0 and H (z) ∼ z when z → ∞. We say that a simply connected set K in H satisfies the “one-sided restriction property” (resp. the two-sided restriction property) if: • It is scale-invariant (the laws of K and of λK are identical for all λ > 0). • For all H ∈ H+ (resp. H ∈ H), the conditional law of H (K) given K ∩(H\H ) = ∅ is identical to the law of K. All such random sets K are classified in [33]. It is not difficult to see that this definition implies that, for all H ∈ H+ (resp. H ∈ H), and for some fixed exponent h > 0, P [K ∩ (H \ H ) = ∅] = H (0)h . This (modulo filling) in fact characterizes the law of the random set K. Conversely, for all h > 0, there exists such a random set K. It can be constructed through three

Conformal Restriction, Highest-Weight Representations and SLE

109

111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 K 000000000 111111111 000000000 111111111 δΚ=:β 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 0 Fig. 1. The set K and its right-boundary β

a priori very different means: By using a variant of SLE8/3 , called SLE(8/3, ρ), by filling certain (reflected) Brownian excursions (see below), or by adding Brownian loops to a certain SLEκ . In the two-sided case, such random sets K only exist when h ≥ 5/8. The only value h corresponding to a simple curve K is h = 5/8 (and this random curve conjecturally corresponds to the scaling limit of half-plane infinite self-avoiding walks, see [32]). Here we will focus mainly on the right boundary of such sets K (which -in the onesided case- is an equivalent way of describing K) that will be denoted by β. It is shown in [33] that this curve is an SLE(8/3, ρ) for some ρ = ρ(h). In particular, the Hausdorff dimension of all these curves β is 4/3. The most important examples of such sets β are: • The SLE8/3 curve itself. In fact, it is the only simple curve satisfying the two-sided restriction property. The corresponding exponent h is 5/8. • If one takes the “right-boundary” of a Brownian excursion from 0 to ∞ in the upperhalf plane (this process is a Markov process that can be loosely described as Brownian motion conditioned never to hit the real line). This corresponds to the exponent h = 1. This last example can in fact be generalized to all h < 1: If one takes the “rightboundary” of a Brownian motion started from the origin that is • Conditioned never to hit the positive half-axis, • Reflected off the negative half-axis with a fixed well-chosen angle θ (h), then, it satisfies the one-sided restriction property with exponent h. See [33] for more details. Also, it is easy to see that if β1 , β2 , . . . , βN are N such independent curves with respective exponents h1 , h2 , . . . , hN , then the right-boundary β of β1 ∪ . . . ∪ βN also satisfies the one-sided restriction property with exponent h1 + · · · + hN . This is simply due to the fact that P [β ∩ (H \ H ) = ∅] =

j =N

P [βj ∩ (H \ H ) = ∅] = H (0)h1 +···+hN

j =1

for all H ∈ H+ . In particular, this shows that any one-sided restriction measure can be constructed using the union of independent (conditioned and reflected) Brownian motions.

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β

1111111111111111111111 0000000000000000000000 0

x1

x2

x3

Fig. 2. The event E = Eε (x1 ) ∩ Eε (x2 ) ∩ Eε (x3 )

3. Boundary Correlation Functions Suppose now that the random simple curve β satisfies the one-sided restriction property. For each real positive x and ε, define the event √ Eε (x) := {β ∩ [x, x + iε 2] = ∅}. The one-sided restriction property of β shows that P [Eε1 (x1 ) ∪ . . . ∪ Eεn (xn )] = 1 − H\∪n

j =1 [xj ,xj +iεj

√

2]

(0)h ,

for all positive xj ’s and εj ’s. These derivatives can (in principle) be determined (−1 is a simple Schwarz-Christoffel transformation, see [1]). This (by a simple inclusionexclusion formula) yields the values of the probabilities f (x1 , ε1 , . . . , xn , εn ) := P [Eε1 (x1 ) ∩ . . . ∩ Eεn (xn )] in terms of x1 , . . . , xn , ε1 , . . . , εn . For example, when n = 1, h

x f (x, ε) = P [Eε (x)] = 1 − √ . x 2 + 2ε 2 (h)

In particular, f (x, ε) ∼ ε2 h/x 2 when ε → 0. We then define B1 (x) = h/x 2 = limε→0 ε −2 f (x, ε). (h) More generally, one can define the functions Bn = Bn as Bn (x1 , . . . , xn ) :=

lim ε −2 . . . εn−2 f (x1 , ε1 , . . . ε1 ,... ,εn →0 1

, xn , εn ).

(1)

An indirect way to justify the existence of the existence of the limit in (1) goes as follows: First, note that when h = 1, the description of β as the right-boundary of a (1) Brownian excursion yields the existence of Bn and the following explicit expression: Bn(1) (x1 , . . . , xn ) =

n−1 (xs(j ) − xs(j −1) )−2 , s∈σn j =1

where σn denotes the group of permutations of {1, . . . , n} and by convention xs(0) = 0. This is due to the fact that β intersects all these slits if and only if the Brownian excursion

Conformal Restriction, Highest-Weight Representations and SLE

111

itself intersects all these slits. One then decomposes this event according to the order with which the excursion actually hits them, and one uses its strong Markov property. Similarly, an analogous reasoning using the Brownian motions reflected on the negative half-axis, and conditioned not to hit the positive half-axis (and its strong Markov property), yields the existence of the limit in (1) for all h < 1. Also, since the right-boundary of the union K1 ∪ . . . ∪ KN of N independent sets satisfying the restriction property with exponents h1 , . . . , hN satisfies the one-sided restriction property with exponent h1 + · · · + hN , we get easily the existence of the limit in (1) for all h (using the existence when h1 , . . . , hN ≤ 1), and the following property of the functions B: For all R : {1, . . . , n} → {1, . . . , N}, write r(j ) = card(R −1 {j }). Then, Bn(h1 +···+hN ) (x1 , . . . , xn ) =

N

(h )

Br(jj) (xR −1 {j } ),

(2)

R j =1

where B0 = 1 and xI denotes the vector with coordinates xk for k ∈ I . This yields a simple explicit formula for B (n) when n is a positive integer. (h) In the general case, one way to compute Bn is to use the following inductive relation (h) (together with the convention B0 ≡ 1): Proposition 1. For all n ∈ N, x, x1 , . . . , xn ∈ R+ , (h)

Bn+1 (x, x1 , x2 , . . . , xn )

n h (h) 1 1 2 = 2 Bn (x1 , . . . , xn ) − + )∂xj − ( x xj − x x (xj − x)2 j =1

× Bn(h) (x1 , . . .

, xn ).

(3)

This relation plays the role of the Ward identities in the CFT formalism. Proof. Suppose now that the real numbers x1 , . . . , xn are fixed and let us focus on the event E = Eε (x1 ) ∩ . . . ∩ Eε (xn ). Let us also √ choose another point x ∈ R and a small δ. Now, either the curve β avoids [x, x + iδ 2] or it does hit it. This additional slit is hit (as well as the n other ones) with a probability A comparable to ε2n δ 2 Bn+1 (x1 , . . . , xn , x) when both√δ and ε vanish. On the other hand, the image of β conditioned to avoid [x, x + iδ 2] under the map ϕ(z) = H\[x,x+iδ √2] = (z − x)2 + 2δ 2 − x 2 + 2δ 2 has the same law as β. In particular, we get immediately that √ A := P[E | β ∩ [x, x + iδ 2] = ∅] n ∼ ε2n |ϕ (xj )|2 B(ϕ(x1 ), . . . , ϕ(xn )) j =1

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when ε → 0 (this square for the derivatives can be interpreted as the fact that the “boundary exponent” for restriction measures is always 2). But when δ vanishes,

1 1 2 + + o(δ 2 ) ϕ(z) = z + δ z−x x and ϕ (z) = 1 −

δ2 + o(δ 2 ). (z − x)2

On the other hand, √ P[E] = A + A P[β ∩ [x, x + iδ 2] = ∅]

(4)

is independent of δ and √ hδ 2 P[β ∩ [x, x + iδ 2] = ∅] = ϕ (0)h = 1 − 2 + o(δ 2 ) x when δ → 0. Looking at the δ 2 term in the δ-expansion of (4), we get (3).

4. Highest-Weight Representations We now define, for all N ∈ Z, the operators LN = {−xj1+N ∂xj − 2(N + 1)xjN } j

acting on functions of the real variables x1 , x2 , . . . . In fact, one should (but we will omit this) make precise the range of j , i.e. define LN on the union over n of the spaces Vn of functions of n variables x1 , . . . , xn . Note that these operators satisfy the commutation relation [LN , LM ] = (N − M)LN+M just as the operators LN do. In other words, the vector space generated by these operators is (isomorphic to) the Witt algebra, i.e. the Lie algebra of vector fields on the unit circle (this is classical, see e.g. [15]). Note also that one can rewrite the Ward identity in terms of these operators as: (h)

Bn+1 (x, x1 , . . . , xn ) =

h (h) Bn (x1 , . . . , xn ) + x N−2 L−N Bn(h) (x1 , . . . , xn ). (5) 2 x N≥1

We are now going to consider vectors w = (w0 , w1 , w2 , . . . ) such that for each n, wn is a function of n variables x1 , . . . , xn . An example of such a vector is (h)

(h)

(h)

B = B (h) = (B0 , B1 , B2 , . . . ), (h)

where B0 is set to be equal to 1. For convenience we will fix h and not always write the (h) superscript.

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For such a vector w, we define for all N ∈ Z the operator lN in such a way that x N−2 (l−N (w))n (x1 , . . . , xn ). wn+1 (x, x1 , . . . , xn ) = N∈Z

In other words, the n-variable component (lN (w))n of lN (w) is the x −N−2 term in the Laurent expansion of wn+1 (x, x1 , . . . , xn ) with respect to x. For example, the Ward identity (5) gives the values of lN (B):  if N > 0  (0, 0, . . . ) if N = 0. lN (B) = (hB0 , hB1 , . . . ) (6)  (L B , L B , . . . ) if N < 0 N 0 N 1 We insist on the fact that lN (B) does not coincide with LN (B) for non-negative N ’s. For instance, L0 (B1 ) = 0 = hB1 = (l0 B)1 . But the identity for negative N ’s can be iterated as follows: Lemma 1. For all k ≥ 1 and negative N1 , . . . , Nk , (lN1 · · · lNk B)n = LN1 . . . LNk Bn .

(7)

Proof of the lemma. This is a rather straightforward consequence of (5). We have just seen that it holds for k = 1. Assume that (7) holds for some given integer k ≥ 1. Then, for all negative N2 , . . . , Nk , (LN2 · · · LNk B)n+1 (x, x1 , . . . , xn ) =u+ x −N−2 LN LN2 . . . LNk Bn (x1 , . . . , xn ), N≤−1

where u is a Laurent series in x such that u(x, x1 , . . . , xn ) = O(x −2 ) when x → ∞. We then apply LN1 (viewed as acting on the space of functions of the n + 1 variables x, x1 , . . . , xn ) to this equation, where N1 < 0. There are two x −N−2 terms in the expansion on the right-hand side: The first one is simply x −N−2 LN1 LN LN2 . . . LNk Bn (x1 , . . . , xn ). The second one comes from the term (LN1 x −N −N1 −2 )LN+N1 LN2 . . . LNk Bn (x1 , . . . , xn ) = (N − N1 )x −N−2 LN+N1 LN2 . . . LNk Bn (x1 , . . . , xn ). The sum of these two contributions is indeed x −N−2 LN LN1 . . . LNk Bn (x1 , . . . , xn ) because of the commutation relation LN1 LN + (N − N1 )LN+N1 = LN LN1 . This proves (7) for k + 1.

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We now define the vector space V generated by the vector B and all vectors lN1 . . . lNk B for negative N1 , . . . , Nk and positive k (we will refer to these vectors as the generating vectors of V ). Then: Proposition 2. For all v ∈ V , for all M, R in Z, lM (v) ∈ V and [lM , lR ]v = (M − R)lM+R v. We insist again on the fact that lN only coincides with LN for negative N . Also, the commutation relation for the lN ’s does not hold for a general vector. The above statement only says that it is valid on this special vector space V . Proof. Note that the commutation relation holds for negative R and M’s because of Lemma 1. Suppose now that N1 , . . . , Nk are negative. Then, LN1 . . . LNk Bn+1 = LNi1 . . . LNir (x −2−N ) N≤0,I

×LNj1 . . . LNjs (lN B)n (x1 , . . . , xn ), where the sum is over all I := {i1 , . . . , ir } ⊂ {1, . . . , k}. One then writes {j1 , . . . js } = {1, . . . , k} \ {i1 , . . . , ir } (and the i’s and j ’s are increasing). We use lN (B)n instead of LN Bn to simplify the expression (otherwise the case N = 0 would have to be treated separately). Since LNi1 . . . LNik (x −2−N ) = (N − 2Nir )(N − Nir − 2Nir−1 ) . . . . . . (N − Nir − . . . − Ni2 − 2Ni1 )x −2−N+Ni1 +···+Nik , it follows immediately that for all integer M, (lM lN1 . . . lNk B)n =

(M + Ni1 + . . . + Nir−1 − Nir ) . . . (M − Ni1 )

I : M+Ni1 +···+Nir ≤0

×LNj1 . . . LNjs (lM+Ni1 +...+Nir B)n .

(8)

This implies that indeed, lM (V ) ⊂ V . When M ≤ 0, then for any i1 , . . . , ir , M + Ni1 + . . . + Nir ≤ 0, so that the sum is over all I . Suppose now that M ≥ 0, R < 0, and consider v = lN1 . . . lNk for some fixed negative N1 , . . . , Nk . We can apply (8) to get the expression of lR+M v, of lM lR v and of lM v. Furthermore, we can use the Lemma to deduce the following expression for lR lM v: (lR lM v)n = (M + Ni1 + . . . + Nir−1 − Nir ) . . . (M − Ni1 ) I : M+Ni1 +···+Nir ≤0

×LR LNj1 . . . LNjs (lM+Ni1 +...+Nir B)n . On the other hand, (lM lR v)n =

(M + Ni0 + . . . + Nir−1 − Nir ) . . . (M − Ni0 )

I0 : M+Ni0 +···+Nir ≤0

×LNj1 . . . LNjs (lM+Ni0 +...+Nir B)n ,

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where this time, the sum is over {i0 , . . . , ir } ⊂ {0, . . . , k}, and we put R = N0 . The difference between these two expressions is due to the terms (in the latter) where i0 = 0: (M + R + Ni1 + . . . + Nir−1 − Nir ) . . . [lM , lR ]v = (M − R) I : M+Ni1 +···+Nir ≤0

. . . (M + R − Ni1 )LNj1 . . . LNjs (lM+R+Ni1 +...+Nir B)n = (M − R)lM+R . This proves the commutation relation for negative R and arbitrary M. Finally, to prove the commutation relation when both R and M are negative and v = lN1 . . . lNk as before, it suffices to use the previously proved commutation relations to write lM v, lR v and lM+R v as the linear combination of the generating vectors of V . Then, one can iterate this procedure to express [lM , lR ]v as a linear combination of the generating vectors of V . Since this formal algebraic calculation is identical to that one would do in the Lie algebra A, one gets indeed [lM , lR ]v = (M − R)lM+R , which therefore also holds for any v ∈ V . To put it differently, to each (one-sided) restriction measure, one can simply associate a highest-weight representation of the Lie algebra A (without central extension) acting on a certain space of function-valued vectors. The value of the highest weight is the exponent of the restriction measure. Note that the right-sided boundary of a simply connected set K satisfying the twosided restriction property satisfies the one-sided restriction property (so that one can also associate a representation to it). In this case, the function Bn also represents the limiting value of √ ε−2n P (K intersects all slits [xj , xj + iε 2], j = 1, . . . , n) even for negative values of some xj ’s. 5. Evolution and Degeneracy 5.1. SLE8/3 . We are now going to see how to combine the previous considerations with a Markovian property. For instance, does there exist a value of κ such that SLEκ satisfies the restriction property? We know from [33] that the answer is yes, that the value of κ is 8/3 and that the corresponding exponent is 5/8. This “boundary exponent” for SLE8/3 has appeared before in the theoretical physics literature (see e.g. [14]) as the boundary exponent for long self-avoiding walks (which is consistent with the conjecture [32] that this SLE is the scaling limit of the half-plane self-avoiding walk). This exponent was identified as the only possible highest-weight of a highest-weight representation of A that isdegenerate at level two. We are now going to see that indeed, the Markovian property of SLE is just a way of 2 (B) are not independent. This shows (without saying that the two vectors l−2 (B) and l−1 using the computations in [33]) why the values κ = 8/3, h = 5/8 pop out. Suppose that β is an SLEκ . Consider the event E := Eε1 (x1 ) ∩ . . . ∩ Eεn (xn ) as (h) in the definition of Bn . If one considers the conditional probability of E given β up to time t, then it√is the probability that an (independent) SLE β˜ hits the (curved) slits ft ([xj , xj + iεj 2]). At first order, this is equivalent to hitting the straight slits √ [ft (xj ), ft (xj ) + iεj 2ft (xj )].

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If the SLE satisfies the restriction property with exponent h, then this means that ft (x1 )−2 . . . ft (xn )−2 Bn(h) (ft (x1 ), . . . , ft (xn )) is a local martingale. Recall that √ ∂t ft (x) = − κdbt +

2 −2ft (x) . and ∂t ft (x) = ft (x) ft (x)2

Hence, since the drift term of the previous local martingale vanishes, Itˆo’s formula yields κ 2 L Bn − 2L−2 Bn = 0 2 −1 for all n ≥ 1. Note that the operators are L’s, and not L’s as in the crossing probability formulae, because of the local scaling properties of the functions B. 2 (B) are collinear and the previously described highestIn other words, l−2 (B) and l−1 weight representation of A must be degenerate at level two. It is elementary to deduce the values of h and κ, using the fact that κ 2 l2 ( l−1 − 2l−2 )B = (3κ − 8)l0 B = 0, 2 which implies that κ = 8/3 and κ 2 κ l1 ( l−1 − 2l−2 )B = (4l−1 l0 B + 2l−1 B) − 6l−1 B = (2κh + κ − 6)l−1 B = 0, 2 2 which then implies that h = 5/8. 5.2. The cloud of bubbles. We are now going to use the description of the “restriction paths” β via SLE curves to which one adds a Poissonian cloud of Brownian bubbles, as explained in [33]. Let us briefly recall how it goes. Consider an SLEκ for κ < 8/3. As we have just seen, it does not satisfy the restriction property. However, if one adds to this curve an appropriate random cloud of Brownian loops, then the obtained set satisfies the two-sided restriction property for a certain exponent h > 5/8 (and its right-boundary β satisfies the one-sided restriction property). More details and properties of the Brownian loop-soup and the procedure of adding loops can be found in [33, 35]. Intuitively this phenomenon can be understood from the case where κ = 2: SLE2 is the scaling limit of the loop-erased random walk excursion (see [31]). Adding Brownian loops to it, one should (in principle) recover the Brownian excursion that satisfies the restriction property with parameter h = 1. More generally, let κ < 8/3 be fixed, and consider an SLEκ curve γ , with its usual time-parametrization. There exists a natural (infinite) measure on Brownian bubbles in H rooted at the origin. This is a measure supported on Brownian paths of finite length in H that start and end at the origin (more generally, we say that a bubble in H rooted at x ∈ ∂H is a path η of finite length T such that η(0, T ) ∈ H and η(0) = η(T ) = x). Consider a Poisson point process of these Brownian bubbles in H, with intensity λ (more precisely, λ times the measure on Brownian bubbles). A realization of this point process is a family (ηˆ t , t ≥ 0) such that for all but a random countable set {tj } of times, ηˆ t = ∅ and for the times tj , ηˆ tj is a (Brownian) bubble in H rooted at the origin. We then define for all t, ηt = ft−1 (ηˆ t ), so that ηt is empty if t ∈ / {tj } and is a bubble in H \ γ [0, tj ]

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rooted at γ (tj ) if t = tj . Another equivalent way to define this random family (ηt , t ≥ 0) via a certain Brownian loop-soup is described in [35]. Define the union of γ and the bubbles ηt , i.e. = ∪t≥0 ({γt } ∪ ηt ). We let Ft denote the σ -field generated by (γs , ηs , s ≤ t). The right outer-boundary β (see [33, 35]) of then satisfies the restriction property (actually satisfies the two-sided restriction property). This is proved in [33] studying the conditional probabilities that avoids a given set A with respect to the filtration generated by γ alone. As observed in [33], the relation between the density λ(κ) of the loops that one has to add to the SLEκ and the exponent h(κ) of the corresponding restriction measure (i.e. h = (6 − κ)/2κ and λ = (8 − 3κ)h) recalls the relation between the central charge and the highest-weight of degenerate highest-weight representations of the Virasoro algebra (which is the central extension of A). We shall try in this subsection (h) to give one way to explain the relation to representations, via the functions Bn , and therefore recover these values of h and λ, just assuming that if one adds the cloud of bubbles with intensity some λ, one obtains a restriction measure. (h) It is worthwhile emphasizing that in this context, the functions Bn are only indirectly related to the SLE curve via this Poissonian cloud of loops. They do for instance not represent the probabilities that the SLE itself does visit the infinitesimal slits, but the probability that some loops that have been attached to this SLE curve do visits the infinitesimal slits. (h) Recall that the functions Bn are related to a highest-weight representation of A, as discussed in the previous section. As in the κ = 8/3 case, we will try to obtain additional information on this representation, using the evolution of the SLE curve. More precisely: How does the (conditional) √ probability with respect to Ft of the event E that β intersects the n slits [xj , xj + iεj 2] for infinitesimal εj ’s evolve with time? Here is a heuristic discussion, that can easily be made rigorous: Consider an infinitesimal time . Let ˜ denote the union of γ [, ∞) and the loops that it does intersect. More precisely, ˜ = ∪t> ({γt } ∪ ηt ). Typically (for very small ), there is no bubble ηt for t ∈ [0, ] that does intersect one of these n slits. In this case, the conditional probability of the event E given F is simply the probability that ˜ does intersect these n slits (given F ). The definition of γ and of the bubbles show that the conditional law of f (˜ ) given F is independent of (in particular, it is the same as for = 0, i.e. the law of ). This shows that (exactly as in the κ = 8/3 case), the conditional probability of E has a drift term due to the distortion of space induced by the SLE (i.e. by f ) of the type κ ( L2−1 Bn − 2L−2 Bn ). 2 But there is an additional term due to the fact that one might in the small time-interval [0, ], have added a Brownian loop √ ηt to the curve that precisely goes through one or several of the n slits [xj , xj + iεj 2]. The probability that one has added a loop that goes through the j th slit is of order λεj2 /xj4 . This fact is due to scale-invariance. Here λ is the (constant) density of loops that is added on top of the SLE curve (we use this

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definition for this density λ in this paper, as in [33]; in other contexts, replacing λ by λ/6 can be more natural). One way to understand the εj2 /xj4 term is that the Brownian bubble has to go from 0 to the slit, which contributes a factor εj2 /xj2 , and then back to the origin, which contributes also 1/xj2 . If such a loop has been added, the conditional probability of E is (at first order) the probability that the SLE+loops hits the remaining n − 1 slits, i.e. fn−1 (x{1,...n}\{j } ) l =j εl2 (here and in the sequel xJ stands for (xj1 , . . . , xjp ) when J = {j1 , . . . , jp }). More generally, define T0 = 0, T1 (x) = 1/x 4 , and for p ≥ 2, Tp (x1 , . . . , xp ) =

1

2 s∈σp xs(1) (xs(2)

− xs(1) )2 . . . (xs(p)

2 − xs(p−1) )2 xs(p)

.

Each s corresponds intuitively to an order of visits of the infinitesimal slits by the loop. For J = {j1 , . . . , jp } ⊂ {1, . . . , n} with |J | = p ≥ 1, the probability to add a loop that goes precisely through the slits near xj for j ∈ J is of the order of εj21 . . . εj2p Tp (xJ )λ. We are therefore naturally led to define the operator U by Tp (xJ ) × fn−p (x{1,...n}\J ). (Uf )n (x1 , . . . , xn ) = J ⊂{1,... ,n}

Then, the fact that P (E|Ft ) is a martingale, shows that the drift term vanishes i.e. that κ 2 − 2l−2 + λU B = 0. (9) l−1 2 Note that the definitions of lN and U show easily that for any w = (w0 , w1 , . . . ) (not only in V ), (lN (T ))p (xJ ) × wn−p (x{1,...n}\J ). ([lN , U ]w)n (x1 , . . . , xn ) = J ⊂{1,... ,n}

In order to compute lN (T )p , one has to look at the Laurent expansion (when x → 0) of Tp+1 (x, x1 , . . . , xp ). Recall that T1 (x) = 1/x 4 and note that for p ≥ 1, Tp+1 (x, x1 , . . . , xp ) = 2x −2 Tp (x1 , . . . , xp ) + o(x −2 )

(10)

(the only terms in the sum that contribute to the leading term are those corresponding to x being visited first or last by the loop). It follows that lN T = 0 if N > 2 and if N = 1 (there are no x −N−2 terms in the expansion). Also, l2 T = (1, 0, 0, . . . ) (the only case where there is an x −4 term is p + 1 = 1). Finally, l0 T = 2T because of (10). Hence,  0 if N > 2   Id if N = 2 . [lN , U ] =   0 if N = 1 2U if N = 0 This enables as before to relate λ to κ and h: 2 /2 − 2l−2 )B = l2 (−λU B) = −λB − λU l2 B = −λB l2 (κl−1

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and

2 l1 (κl−1 /2 − 2l−2 )B = l1 (−λU B) = −λU l1 B = 0. This last relation implies that 6−κ h= , 2κ and the first one then shows that (8 − 3κ)(6 − κ) λ = (8 − 3κ)h = , 2κ which are the formulae appearing in [33]. This relation between h and −λ is indeed that between the highest-weight and the central charge for a representation of the Virasoro algebra that is degenerate at level two. Recall that if l˜n ’s are the generators of the Virasoro Algebra and C its central element, then [l˜2 , l˜−2 ] = 4l˜0 + C/2, so the little two by two linear system leading to the determination of κ and h for a degenerate highest-weight representation of the Virasoro algebra is the same (and therefore leads to the same expression); roughly speaking, l−2 − λU/2 plays the role of l˜−2 . Note that the previous considerations involving the Brownian bubbles is valid only in the range κ ∈ (0, 8/3] and therefore for c ≤ 0. This corresponds to the fact that (h) two-sided restriction measures exist only for h ≥ 5/8. In this case all functions Bn are positive for all (real) values of x1 , . . . , xn .

5.3. Analytic continuation. In the representations that we have just been looking at, we considered simple operators acting on simple rational functions. All the results depend analytically on κ (or h). In other words, for all real κ (even negative!), if one defines (h) the functions Bn recursively, the operators ln , the vector B (h) and the vector space (h) V = V as before, then one obtains a highest-weight representation of A with highest weight h. The values of κ, λ and h are still related by the same formula, but do not correspond necessarily to a quantity that is directly relevant to the SLE curve or the restriction measures. (h) When h ∈ (0, 5/8), the functions Bn can still be interpreted as renormalized probabilities for one-sided restriction measures. They are therefore positive for all positive x1 , . . . , xn but they can become negative for some negative values of the arguments. The “SLE + bubbles” interpretation of the degeneracy (i.e. of the relation (9)) is no longer valid since the “density of bubbles” becomes negative (i.e. the corresponding central charge is positive). In this case, the local martingales measuring the effect of boundary perturbations are no longer bounded (and do not correspond to conditional probabilities anymore). (h) (h) For negative h, the functions Bn can still be defined. This time, the functions Bn n are not (all) positive, even when restricted on (0, ∞) and they do not correspond to any restriction measure. These facts correspond to “negative probabilities” that are often implicit in the physics literature. Note that c (i.e. −λ) cannot take any value: For positive κ, c varies in (−∞, 1] and for negative κ, it varies in [25, ∞). The transformation κ ↔ −κ corresponds to the well-known c ↔ 26 − c duality (e.g. [36]). (h) In other words, the Bn ’s provide the highest-weight representations of A with highest weight h. Each one is related to a highest-weight representation of the Virasoro (h) algebra that is degenerate at level 2. Furthermore, all Bn ’s are related by (2).

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6. Remarks In order to clarify the state of the art seen from a mathematical perspective, let us now try to sum up things: • The interfaces of two-dimensional critical models (such as random cluster interfaces, that are very closely related to Potts models) are believed to be conformally invariant in the scaling limit. In some cases, this is proved (critical percolation, uniform spanning trees). In some other cases (Ising, double-domino tiling), some partial results hold. Anyway, to derive conformal invariance, it seems that one has to work on each specific model separately. • These interfaces can be constructed in a dynamic way, i.e. they have a Markovian type property (at least the critical random cluster interfaces, that have the same correlation functions as the Potts models). Therefore, if conformal invariance holds, their scaling limit must be one of the SLE curves. In general, these limits correspond to the SLE curves with κ > 4 that are not simple curves. The correlation functions of the 2D statistical physics model are related to the fractal properties of the SLE curve, but the knowledge of the SLE curve is a much richer information than just the value of the exponents. • One can understand the dependence of the law of an SLE in a domain with respect to this domain via the restriction properties. This shows that some specific “finitedimensional observables” of the SLE curves satisfy some relations. This can be reformulated in terms of highest-weight representations of the Lie algebra A, and explains the relation between the physics models and these representations. Also, it makes it possible to define conformal fields via SLE. However, and we think that this has to be again stressed, since the initial purpose was to understand the statistical physics models and their behaviour, the SLE itself is a more natural way. Also, one (h) should also again emphasize that in the present paper, the “correlation functions” Bn do correspond only indirectly with the curve γ (via the cloud of Brownian bubbles) when the central charge does not vanish. All functions described in the present paper deal with the boundary (or “surface”) behaviour of the systems. One may want to develop a similar theory for points lying in the inside of the upper half-plane (“in the bulk”). Beffara’s results [5] (for instance in the case κ = 8/3) provide a first step in this direction, and show that the definition of these correlation functions themselves is not an easy task. Acknowledgement. Thanks are of course due to Greg Lawler and Oded Schramm, in particular because of the instrumental role played by the ideas developed in the paper [33]. We have also benefited from very useful discussions with Vincent Beffara and Yves Le Jan. R.F. acknowledges support and hospitality of IHES.

References 1. Ahlfors, L.V.: Complex Analysis. 3rd Ed., New-York: McGraw-Hill, 1978 2. Aizenman, M.: The geometry of critical percolation and conformal invariance. In: StatPhys 19 (Xiamen 1995), River Edg, NJ: World Sci. Publishing, 1996, pp. 104–120 3. Bauer, M., Bernard, D.: SLEκ growth and conformal field theories. Phys. Lett. B543, 135–138 (2002) 4. Bauer, M., Bernard, D.: Conformal Field Theories of Stochastic Loewner Evolutions. Commun. Math. Phys., to appear, 2003 5. Beffara, V.: The dimension of the SLE curves. arxiv:math.PR/0211322, Preprint, 2002 6. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry of critical fluctuations in two dimensions. J. Statist. Phys. 34, 763–774 (1984)

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7. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in twodimensional quantum field theory. Nucl. Phys. B241, 333–380 (1984) 8. Cardy, J.L.: Conformal invariance and surface critical behavior. Nucl. Phys. B240 (FS12), 514–532 (1984) 9. Cardy, J.L.: Critical percolation in finite geometries. J. Phys. A25, L201–206 (1992) 10. Cardy, J.L.: Conformal Invariance in Percolation, Self-Avoiding Walks and Related Problems. condmat/0209638, Preprint, 2003 11. Dub´edat, J.: SLE and triangles. Electr. Comm. Probab. 8, 28–42 (2003) 12. Dub´edat, J.: SLE(κ, ρ) martingales and duality. arxiv:math.PR/0303128, Preprint, 2003 13. Duplantier, B.: Conformally invariant fractals and potential theory. Phys. Rev. Lett. 84, 1363–1367 (2000) 14. Duplantier, B., Saleur, H.: Exact surface and wedge exponents for polymers in two dimensions. Phys. Rev. Lett. 57, 3179–3182 (1986) 15. Feigin, B.L., Fuks, D.B.: Skew-symmetric invariant differential operators on the line and Verma modules over the Virasoro algebra. Funct. Anal. Appl. 16, 114–126 (1982) 16. Frenkel, E., Ben-Zvi, D.: Vertex Algebras and Algebraic curves. A.M.S. monographs 88, Providence, RI: Aus, 2001 17. Friedrich, R., Kalkkinen, J.: On conformal field theory and stochastic Loewner evolution, arxiv:hepth/0308020, Preprint, 2003 18. Friedrich, R., Werner, W.: Conformal fields, restriction properties, degenerate representations and SLE. C.R. Acad. Sci. Paris Ser. I. Math. 335, 947-952 (2002) 19. Goddard, P., Olive, D., ed.: Kac-Moody and Virasoro algebras. A reprint volume for physicists. Advanced Series in Mathematical Physics 3, Singapore: World Scientific, 1988 20. Itzykson, C., Drouffe, J.-M.: Statistical field theory. Vol. 2. Strong coupling, Monte Carlo methods, conformal field theory, and random systems. Cambridge: Cambridge University Press, 1989 21. Itzykson, C., Saleur, H., Zuber, J.-B., ed.: Conformal invariance and applications to statistical mechanics. Singapore: World Scientific, 1988 22. Kac, V.G.: Infinite-dimensional Lie Algebras. 3rd Ed, Combridge: CUP, 1990 23. Kac, V.G., Raina, A.K.: Bombay lectures on highest weight representations of infinite-dimensional Lie algebras. Advanced Series in Mathematical Physics 2, Singapore: World Scientific, 1987 24. Kennedy, T.G.: Monte-Carlo tests of Stochastic Loewner Evolution predictions for the 2D self-avoiding walk. Phys. Rev. Lett. 88, 130601 (2002) 25. Langlands, R., Pouliot, Y., Saint-Aubin, Y.: Conformal invariance in two-dimensional percolation. Bull. A.M.S. 30, 1–61 (1994) 26. Lawler, G.F.: An introduction to the stochastic Loewner evolution, Proceeding of a conference on random walks. ESI Vienna, to appear, 2001 27. Lawler, G.F., Schramm, O., Werner, W.: Values of Brownian intersection exponents I: Half-plane exponents. Acta Math. 187, 237–273 (2001) 28. Lawler, G.F., Schramm, O., Werner, W.: Values of Brownian intersection exponents II: Plane exponents. Acta Math. 187, 275–308 (2001) 29. Lawler, G.F., Schramm, O., Werner, W.: Values of Brownian intersection exponents III: Two sided exponents. Ann. Inst. Henri Poincar´e 38, 109–123 (2002) 30. Lawler, G.F., Schramm, O., Werner, W.: One-arm exponent for critical 2D percolation. Electronic J. Probab. 7(2), (2002) 31. Lawler, G.F., Schramm, O., Werner, W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. arXiv:math.PR/0112234, Ann. Prob., to appear, 2003 32. Lawler, G.F., Schramm, O., Werner, W.: On the scaling limit of planar self-avoiding walks. arXiv:math.PR/0204277. In Fractal geometry and application, A jubilee of Benoit Mandelbrot, AMS Proc. Symp. Pure Math., to appear, 2002 33. Lawler, G.F., Schramm, O., Werner, W.: Conformal restriction. The chordal case. J. Amer. Math. Soc. 16, 917–955 (2003) 34. Lawler, G.F., Werner, W.: Universality for conformally invariant intersection exponents. J. Europ. Math. Soc. 2, 291-328 (2000) 35. Lawler, G.F., Werner, W.: The Brownian loop-soup. Probab. Theor. Relat. Fields, to appear, 2003 36. Neretin, Yu. A.: Representations of Virasoro and affine Lie Algebras. In: Representation theory and non-commutative harmonic analysis I, A.A. Kirillov, (ed.), Berlin-Heidelberg-New York: Springer, 1994, pp. 157–225 37. Nienhuis, B. Coulomb gas description of 2D critical behaviour. J. Stat. Phys. 34, 731–761 (1984) 38. Polyakov, A.M.: A non-Hamiltonian approach to conformal field theory. Sov. Phys. JETP 39, 10–18 (1974) 39. Rohde, S., Schramm, O.: Basic properties of SLE. Ann. Math., to appear, 2003 40. Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288 (2000)

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41. Schramm, O.: A percolation formula, Electr. Comm. Prob. 6, 115–120 (2001) 42. Smirnov, S.: Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris S´er. I Math. 333(3), 239–244 (2001) 43. Smirnov, S., Werner, W.: Critical exponents for two-dimensional percolation. Math. Res. Lett. 8, 729–744 (2001) 44. Werner, W.: Random planar curves and Schramm-Loewner Evolutions. Lecture Notes of the 2002 St-Flour summer school, Berlin-Heidelberg-New York: Springer, to appear 2002 Communicated by A. Kupiainen

Commun. Math. Phys. 243, 123–136 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0951-0

Communications in

Mathematical Physics

Uniqueness of Finite Energy Solutions for Maxwell-Dirac and Maxwell-Klein-Gordon Equations Nader Masmoudi1 , Kenji Nakanishi2 1 2

Courant Institute, New York University, New York, NY 10012, USA Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan

Received: 10 January 2003 / Accepted: 19 June 2003 Published online: 17 October 2003 – © Springer-Verlag 2003

Abstract: We prove uniqueness of solutions to the Maxwell-Dirac system in the energy space, namely C(−T , T ; H 1/2 × H˙ 1 ). We also give a proof for uniqueness of finite energy solutions to the Maxwell-Klein-Gordon equations, which is simpler than that given in [16]. 1. Introduction In this paper, we study uniqueness of solutions with finite energy for the Maxwell-KleinGordon (MKG) and Maxwell-Dirac (MD) systems. The existence and uniqueness with additional restrictions have been obtained by Klainerman and Machedon in [8] for MKG in the energy space C(H 1 ). In [13] the authors obtained a similar result for MD in the energy space C(H 1/2 × H 1 ), where H 1/2 is the space for the spinor field, improving Bournaveas’ result in [2], where regularity strictly above that was necessary, namely C(H 1/2+ε × H 1+ε ). For MKG, unconditional uniqueness was proved essentially by Zhou in [16]. More precisely, he proved the uniqueness for a simplified model which carries only the null quadratic terms, but one can derive the uniqueness for the original MKG by complimenting his argument with that in [4] to treat the elliptic and cubic terms. Their arguments rely on bilinear estimates for the null-forms in the space-time fractional Sobolev-type spaces X3/4+,1/2+ , which are much more complicated than the original null-form estimate in L2t,x used to prove the existence in [8]. In contrast, our uniqueness proof will use only the original null-form estimate, the well-known Strichartz estimate, and their interpolation. It requires even less null-form estimate than [8] in the sense that we do not need it for the Maxwell part. The main idea of our proof is to estimate the difference of two solutions in the spaces with 1/2 less regularity, assuming that one of the solutions has good properties derived The first author is partially supported by an NSF grant and an Alfred Sloan fellowship. The second author is supported by JSPS Postdoctoral Fellowships for Research Abroad (2001–2003)

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in the existence proofs. This idea is very natural when trying to prove a uniqueness result. In the case of MD, we will use the same method as in [13] which consists in rewriting the equation partially in the MKG form depending on the frequency. We will also give a simplified existence proof for the MD system without using the frequency decomposition. However, the uniqueness result requires the frequency decomposition. Our result can be put in a more general set of works concerning uniqueness. Indeed, while trying to prove an existence result for some PDE, one is led to use a fixed point argument, which yields the unique existence of a solution in the space where the fixed point argument is performed. This space is usually smaller than the space where we are looking for uniqueness and one has to come with a different argument for unconditional uniqueness. In some cases this turns out to be a non-trivial problem. In this set of results, we can refer to Planchon, who proved uniqueness for the critical semilinear wave equation [14]. In the same spirit, we mention some results concerning the uniqueness for the Navier-Stokes system in C(LN (RN )) where we have a similar picture ([7, 11]). Now we introduce the Maxwell-Dirac equation as follows. Denote ∂ = (∂0 , . . . , ∂3 ) = (∂t , ∇), Dα = ∂α + iAα , Fαβ = ∂α Aβ − ∂β Aα ,

(1.1) (1.2)

where (t, x) denotes the space-time coordinates, A = (A0 , . . . , A3 ) = (A0 , A ) denotes the electromagnetic potential, D denotes the covariant derivative, and F denotes the electromagnetic field. F is decomposed into the electric field E = (F10 , F20 , F30 ) = −(F01 , F02 , F03 ) and the magnetic field B = (F23 , F31 , F12 ) = −(F32 , F13 , F21 ). In other words, they are given by E = ∇A0 − A˙ ,

B = ∇ × A.

(1.3)

The existence of A satisfying these relations is equivalent to the following equations for E and B: ∇ · B = 0,

B˙ + ∇ × E = 0.

(1.4)

In the sequel, we employ the convention of tacit summation over coupled upper and lower indices, where Greek letters run from 0 to 3 while Latin letters run from 1 to 3. We denote X α = g αβ Xβ with g αβ = diag(−1, 1, 1, 1) for any tensor X. We consider the following Maxwell-Dirac system which describes the evolution of the wave function of a self-interacting relativistic electron: ∂α F αβ = J β = γ 0 u, γ β u, (1.5) iγ α Dα u = mu, where u ∈ C4 denotes the spinor field coupled with F , m ≥ 0 is a constant, J0 = −|u|2 is the charge density, J = (J1 , J2 , J3 ) is the electric current given by Jk = γ 0 u, γ k u for 1 ≤ k ≤ 3, and a, b denotes the real part of the inner product, namely a, b = (ab). The Dirac matrices are given by I2 0 2 02 σ k k 0 γ = , γ = , (1.6) 02 −I2 −σk 02

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125

where 02 is the null 2 × 2 matrix, I2 is the 2 × 2 identity and the Pauli matrices are given by 01 0 −i 1 0 σ1 = , σ2 = , σ3 = . (1.7) 10 i 0 0 −1 This system has a conserved total charge J0 dx = − |u|2 dx and the conserved energy given by E = 2iD0 u, u + |E|2 + |B|2 dx = 2γ j Dj u, iγ 0 u + 2mu, γ 0 u + |E|2 + |B|2 dx, (1.8) A = 0) which does not have a definite sign. The Maxwell-Dirac in the Coulomb gauge (∇·A can be written as  α  iγ ∂α u − mu = γ α Aα u, A = PJJ , −∂t2A + A (1.9)  A0 = −|u|2 , ∇ · A = 0, where P denotes the projection on divergence-free vectors which is given by PJJ = J − ∇ −1 (∇ · J ). We complement (1.9) with the following initial data: (u(0), A (0), A˙ (0)) ∈ H 1/2 (R3 ; C4 ) × (H˙ 1 × L2 )(R3 ; R3 ), ∇ · A (0) = ∇ · A˙ (0) = 0,

(1.10)

where H˙ 1 should be understood as the completion of C0∞ in the space of tempered distributions S with respect to the homogeneous norm ∇ϕ L2 . Also, we define C0 as the completion of C0∞ with respect to the L∞ norm. Thus we have H˙ 1 = {ϕ ∈ L6 (R3 )|∇ϕ ∈ L2 }, C0 = {ϕ ∈ C(R3 )| lim ϕ(x) = 0}. |x|→∞

(1.11)

We have the following existence and uniqueness result. Theorem 1.1. Given any initial data as in (1.10), there exists a unique solution (u, A) to the Maxwell-Dirac in the Coulomb gauge (1.9) on some time interval (−T , T ) satisfying (u, A0 , A , A˙ ) ∈ C(−T , T ; H 1/2 × C0 × H˙ 1 × L2 ).

(1.12)

The uniqueness holds in the class of weakly continuous (for t) functions. The second result concerns the uniqueness of solutions in the energy space for the Maxwell-Klein-Gordon. We recall the system ∂α F αβ = J β = − (uD β u) = iu, D β u, (1.13) Dα D α u = m2 u, where u is a complex scalar field. In the Coulomb gauge, the system can be rewritten as   u − m2 u + 2iAα ∂α u − i A˙ 0 u − Aα Aα u = 0, (1.14) A = PJJ , −∂t2 A + A  A0 = J0 , ∇ · A = 0.

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This system has the conserved charge J0 dx and the conserved energy A|2 + |Dα u|2 + |mu|2 dx, E= |∇A0 |2 + |A˙ |2 + |∇A R3

(1.15)

which is positive definite. We complement (1.14) with the following initial data: (u(0), u(0), ˙ A (0), A˙ (0)) ∈ (H 1 × L2 )(R3 ; C) × (H˙ 1 × L2 )(R3 ; R3 ), ∇ · A (0) = ∇ · A˙ (0) = 0. (1.16) Theorem 1.2. Given any initial data as in (1.16), there exists a unique global solution (u, A) to the Maxwell-Klein-Gordon system in the Coulomb gauge (1.14) satisfying (u, u, ˙ A0 , A , A˙ ) ∈ C(R; H 1 × L2 × C0 × H˙ 1 × L2 ).

(1.17)

The uniqueness holds in the class of functions that are weakly continuous in time. As we have mentioned, the existence part was proved in [8], and the uniqueness was essentially shown in [16], but we will give a different, simpler proof of uniqueness. 2. Preliminaries Here we prepare the different tools and notations that we use in this paper. Let U (±t) = e±i|∇|t

(2.1)

be the free wave propagator. First, we have the Strichartz estimate

U (±t)ϕ Lp1 B˙ s1 ϕ H˙ s , t t q1 ,2 U (±(t − t ))f (t )dt p1 ˙ s1 f Lp2 B˙ s2 , 0

Lt Bq

t

1 ,2

(2.2)

q2 ,2

s and H˙ s denote the homogeneous Besov and Sobolev spaces for x ∈ Rn , see where B˙ q,2 [1] for the definition of these spaces. The exponents should satisfy

2 < pi ≤ ∞,

2 ≤ qi < ∞,

1/pi + 1/qi = 1/2,

si = s − 2/pi ,

(2.3)

and p denotes the H¨older conjugate of p. This estimate is false at the endpoint exponent (p, q) = (2, ∞), otherwise we could derive wellposedness only with the Strichartz estimate for finite energy solutions. We need to complement this lack by the following null-form estimate due to Klainerman and Machedon:

∂i u∂j v − ∂i v∂j u L2 ϕ H˙ 2 ψ H˙ 1 , t,x

(2.4)

where u = U (±t)ϕ, v = U (±t)ψ, and i, j = 1, 2, 3. In order to apply this estimate to the integral equation and especially to interpolate with the Strichartz estimate, it is convenient to introduce the Fourier restriction norms Xs,b : For s, b ∈ R and an interval I ⊂ R, we define

u Xs,b (I ) := ±

inf

u(t)=U (±t)v(t)onI

v H b (R;H s ) , t

x

(2.5)

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127

where H s denotes the inhomogeneous Sobolev space. Schematically, we have s,b X± = U (±t)Htb Hxs .

(2.6)

s,b We will denote X˙ ± when Hxs is replaced with the homogeneous version H˙ xs (but keeps,0 ing Htb inhomogeneous). Since U (±t) is unitary, it is clear that X± = L2t Hxs . We also s,b s,b s,b denote X = X+ + X− . We remark that those spaces do not change if we consider √

the Klein-Gordon propagator Um (±t) = e±it m − instead of U (±t) = U0 (±t), since Um (−t)U0 (t) is a uniformly bounded operator on any Htb (Hxs ) and Htb (H˙ xs ). In other words, we have 2

s,b X± = Um (±t)Htb Hxs ,

(2.7)

where the left-hand side is independent of m. Then it is obvious that we have

Um (±t)ϕ Xs,b (I ) ϕ H s

(2.8)

±

for any m, s, b, I . In what follows, we will assume that I is small enough and to keep ideas clear we assume that |I | < 1. Hence, we have t

Um (±(t − t ))f (t )dt Xs,b (I ) |I |1−b f L2 H s , (2.9) ±

0

t

for 0 ≤ b ≤ 1. Moreover t

Um (±(t − t ))f (t )dt Xs,b (I ) f Xs,b−1 . ±

0

(2.10)

The Strichartz estimate implies the following estimates for X s,b . Let b > 1/2 and assume (2.3). Then, by the Sobolev embedding H b (R) ⊂ L∞ (R) and the Minkowski p p inequality L2t Lt 1 ⊂ Lt 1 L2t , we have

f Lp1 B˙ U (±t)U (∓t )f (t ) Lp1 L∞ B˙ t

t

t

t

t

U (±t)U (∓t )f (t ) Lp1 H b B˙ U (±t)U (∓t )f (t ) H b Lp1 B˙ t

t

U (∓t )f (t ) H b H˙ s f X˙ s,b , t

±

(2.11)

where we abbreviated B˙ = B˙ qs11 ,2 and we should choose the same sign for all ± and the other for ∓. In other words, we have the embedding s,b X˙ ± ⊂ Lp1 B˙ qs11 ,2

(2.12)

for any b > 1/2 when (2.3) is satisfied. We refer the reader to [15] for a different proof of this embedding. Thus, Xs,b may be thought of as carrying all the available Strichartz estimates in it. We can say even a bit more actually—it virtually allows the prohibited L2t L∞ x estimates under interpolation in the following sense, which is quite convenient for our uniqueness argument:

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Lemma 2.1. Assume (2.3). Let b > 1/2, 0 ≤ θ < 1, α, β ∈ R and suppose that 2/θ = p1 ,

(1 − θ )α + θ (β − 1) = s1 ,

s = s1 + θ.

(2.13)

Then we have the embedding ∞ ˙α 2 ˙ 1 ˙ s1 ˙α ˙ [L∞ t H , X± ]θ ⊂ Lt Bq1 ,2 = [Lt H , Lt B∞,2 ]θ , p

β,b

β−1

(2.14)

where the last identity is just a reminder. [·, ·]θ denotes the complex interpolation space. Proof. Since L∞ H s is invariant under U (±t), we have ∞ ˙α b ˙β ˙α ˙ [L∞ t H , X± ]θ = U (±t)[Lt H , Ht H ]θ = U (±t)Ht

bθ,p1

β,b

H˙ s ,

(2.15)

where H s,p denotes the Lp -based Sobolev space, and we used the identity (2.13) and the well-known interpolation property for the mixed Lebesgue-Sobolev spaces (see [1]). bθ,p The Sobolev embedding is inherited by the interpolation space as Ht 1 ⊂ L∞ . Thus, by the same argument as above, we obtain

f Lp1 B˙ U (±t)U (∓t )f (t ) Lp1 L∞ B˙ U (∓t )f (t ) H bθ,p1 H˙ s , t

t

t

t

which, together with the above interpolation property, implies the desired embedding. It is useful also to have some “reverse” embeddings, namely from the Strichartz-type norms into Xs,b spaces, which are possible if we allow some loss in the regularity in time. In fact, we have the following. Lemma 2.2. Let 2 ≤ q < ∞, > 0 and b = 1/2 − 1/q. Then we have ˙ s,−b− (I ) L2t B˙ qs+2b ,2 (I ) ⊂ X±

(2.16)

on any finite interval I . Proof. Let κ > 0 be a small parameter which will be determined later. By the Strichartz estimate (2.11), we have X˙ s,0 ⊂ L2t H˙ s ,

1/(1/2−κ) ˙ s−1+2κ X˙ s,1/2+κ ⊂ Lt B1/κ,2 ,

(2.17)

where the first embedding is trivial. Interpolating them, we obtain 1/(1/2−θκ) ˙ s−1+2/q X˙ s,(1/2+κ)θ ⊂ Lt Bq,2 ,

(2.18)

where we choose θ ∈ [0, 1] such that (1 − θ )/2 + θ κ = 1/q.

(2.19)

As κ → +0, we have (1/2 + κ)θ → 1/2 − 1/q + 0 and 1/(1/2 − θκ) → 2 + 0. Therefore, by the trivial embeddings on the finite interval, we obtain s−1+2/q X˙ s,1/2−1/q+ (I ) ⊂ L2t B˙ q,2 (I ),

from which the desired result follows by the duality.

(2.20)

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The above lemma will be used in the case q = 4, b = 1/4: for any ε > 0, we have −1/2,−1/4−ε

L2 L4/3 (I ) ⊂ X˙ ±

(2.21)

(I ).

We may transfer the bilinear estimate (2.4) to Xs,b in the same manner as for the Strichartz estimate. It works as a substitute of the L2t L∞ x Strichartz in the null quadratic terms. Lemma 2.3. Let b > 1/2, s1 , s2 ≤ 2 ≤ s1 + s2 . Then we have

∂i u∂j v − ∂i v∂j u L2 H˙ s1 +s2 −3 u X˙ s1 ,b v X˙ s2 ,b . t

(2.22)

Proof. First we verify it for free waves u = U (±t)u(0) and v = U (±t)v(0). By interpolation, it suffices to consider the three endpoints (s1 , s2 ) = (2, 2), (2, 0), (0, 2) and the last reduces to the second one by the symmetry between u and v. But the estimate for (2, 2) immediately follows from the (2, 1) case (2.4) by the Leibniz rule, so only the (2, 0) case remains, which can be also derived easily from the (2, 1) case as follows. Now we introduce the Littlewood-Paley decomposition

δ(x) = ϕj , (2.23) j ∈Z

where ϕj has Fourier support in {2j −1 < |ξ | < 2j +1 }. Denote the bilinear form as Q(u, v) = ∂i u∂j v − ∂i v∂j u. Then we have  

 ϕi ∗ Q(ϕj ∗ u, ϕk ∗ v), Q(u, v) =  + + (2.24) i j ∼k

j k∼i

k i∼j

by the Fourier support property of {ϕj }, where the three sums should be taken as disjoint with respect to the index (i, j, k). When we take the L2t H˙ s1 +s2 −3 norm of the above summation, the second sum is dominated by

−i

ϕi ∗ Q ϕj ∗ u, ϕk ∗ v 2 2 2 k∼i j i i Lt,x −i

2 ϕ ∗ u(0)

ϕ ∗ v(0)

(2.25) 1 ˙ j k H 2 H˙ k∼i j i 2 i

u(0) H˙ 2 v(0) L2 , and the third sum is dominated by

−i

ϕi ∗ Q ϕj ∗ u, ϕk ∗ v 2 2 2 j ∼i ki i L t,x

−i 2 ϕk ∗ v 1 ϕj ∗ u(0) ˙ 2 H H˙ 2 j ∼i ki i

u(0) H˙ 2 v(0) L2 ,

(2.26)

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where we used the (2, 1)-estimate in the second inequalities in both cases. For the first sum in (2.24), we use the Strichartz estimate together with the null cancellation. For the space-time Fourier transform, we have

ϕi ∗ Q(ϕj ∗ u, ϕk ∗ v) F i j ∼k

|ξ −η|∼|η|

|(ξ − η) × η| = δ(τ − σ ± |ξ − η|)| u(0, ξ − η)| ×δ(σ ± |η|)| v (0, η)|dσ dη

|ξ |F(u v ),

(2.27)

where Fu and u(0) denote the space-time and space Fourier transforms respectively, and we define u and v by u(0, ξ )|, u := e±i|ξ |t |ξ |3/2 |

v := e±i|ξ |t |ξ |−1/2 | v (0, ξ )|.

(2.28)

Then, using the Strichartz and the Plancherel, we can bound the L2t H˙ −1 norm of the above by

u v L2 u L4 v L4 u(0) H˙ 2 v(0) L2 . t,x

t,x

(2.29)

t,x

Thus we obtain the estimate for free waves in the (2, 0) case, and the remaining cases are covered by interpolation. Next we need to generalize the estimate for X s,b functions. Using the algebraic property of Htb and the Sobolev embedding Htb ⊂ L∞ t , we have

Q(u, v) L2 H˙ s Q(U (t)U (−t )u(t ), U (−t)U (t )v(t )) H b (L2 H˙ s ) t

U (−t )u(t ) H b H˙ xs1 U (t )v(−t ) H b H˙ xs2 t

u X˙ s1 ,b v X˙ s2 ,b , +

−

t

t

x

t

(2.30)

where we fixed the sign for X± spaces for simplicity but there is no difference in the other cases. In particular, we have

|∇|−1 Q(u, v) L2 u X˙ 1,b v X˙ 1,b , t,x

(2.31)

for any b > 1/2.

3. Maxwell-Dirac In this section, we prove Theorem 1.1 relative to the Maxwell-Dirac system in Coulomb gauge. First we give an existence proof, which is simpler than that in [13], since we do not use the frequency decomposition, although we still need that idea for the uniqueness part.

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3.1. Existence. The idea is based on rewriting the Maxwell-Dirac system in a system similar to the MKG, to which we apply those estimates in [8] and we use the “reverse” embedding (2.21) to estimate extra nonlinear terms. We can put u ∈ H 1/2 since the current J is more regular. Indeed, the Maxwell-Dirac system can be rewritten as   u − m2 u + 2iAα ∂α u − i A˙ 0 u − Aα Aα u = 2i γ α γ β Fαβ u, A = PJJ , (3.1) A  A0 = −|u|2 , ∇ · A = 0, where the initial data for u˙ is determined by the Dirac equation as u(0) ˙ = −(γ 0 γ j ∂j + 0 α 0 iγ γ Aα + imγ )u(0). We will just sketch the new estimates we need. We estimate the solution (u, A) in the following spaces : u ∈ X1/2,b , u˙ ∈ X−1/2,b , A ∈ X˙ 1,b and A˙ ∈ X0,b for some b > 1/2. By the standard multiplication estimate in the Besov spaces 3/2 and the H¨older inequality, we obtain the following estimates. Notice that B˙ 2,1 ⊂ C0 by the Sobolev embedding.

A0 L∞ (B˙ 3/2 ) |u|2 L∞ (B˙ −1/2 ) u 2L∞ H 1/2 , 2,1

2,1

A˙ 0 B˙ 1/2 uu

˙ B˙ −3/2 u L∞ H 1/2 u

˙ L∞ H −1/2 , 2,∞

2,∞

A L∞ H˙ 1 + A0 L∞ B˙ 3/2 )2 u L∞ H 1/2 ,

Aα Aα u L∞ H −1/2 ( A

(3.2)

2,1

A0 u

˙ L∞ H −1/2 A0 L∞ B˙ 3/2 u

˙ L∞ H −1/2 , 2,1

PJJ L2 L2 u 2L4 L4 . The null-form estimate Lemma 2.3 yields A · ∇u L2 H −1/2 A A X˙ 1,b u X1/2,b ,

A

(3.3)

since we have A · ∇u = ∂i (∂i −1 Aj )∂j u − ∂j (∂i −1 Aj )∂i u by virtue of ∇ · A = 0. Using the embedding (2.21), we get A)u X−1/2,−1/4−ε (∂A A)u L4 L4/3 ∂A A L∞ L2 u L4 ,

(∂A t,x

(∂A0 )u X−1/2,−1/4−ε (∂A0 )u L4 L4/3 ∂A0 B˙ 1/2 u L∞ L2 ∩L4 . 2,∞

t

x

t,x

(3.4)

A in L2t,x . Then, using the above Thus we can estimate u in X−1/2,−1/4−ε and A estimates as well as a classical iteration argument (which we do not detail here), we deduce the existence of a unique local solution for the Maxwell-Dirac system satisfying u ∈ X1/2,b , u˙ ∈ X −1/2,b , A0 ∈ C(C0 ), A ∈ X˙ 1,b and A˙ ∈ X 0,b for some 3/4 > b > 1/2. In the next subsection, we prove that the uniqueness holds in the energy space. 3.2. Uniqueness. We take two solutions (u, A) and (uw , Aw ) of the Maxwell-Dirac system on some time interval I = (−T , T ) in the energy space, namely (u, A0 , A , A˙ ), (uw , A0w , A w , A˙ w ) ∈ C(I ; H 1/2 × C0 × H˙ 1 × L2 ) with the same initial data u(0) = uw (0),

A (0) = A w (0),

A˙ (0) = A˙ w (0).

(3.5)

We want to prove that (u, A) = (uw , Aw ). Without loss of generality, we can assume that (u, A) is the solution constructed above, namely u ∈ X1/2,b , u˙ ∈ X−1/2,b , A ∈ X˙ 1,b and A˙ ∈ X0,b for some b > 1/2. We

132

N. Masmoudi, K. Nakanishi

denote u = u − uw and A = A − Aw . We want to estimate (u , A ) in the following spaces: u ∈ X0,b , u˙ ∈ X−1,b , A ∈ X˙ 1/2,b and A˙ ∈ X˙ −1/2,b . Indeed, we have

|uw |2 L∞ B˙ −1/2 + PJJ w L∞ B˙ −1/2 uw 2L∞ H 1/2 , 2,1

(3.6)

2,1

3/2 and hence A0w ∈ L∞ B˙ 2,1 , A ∈ X˙ 1/2,b and A˙ ∈ X˙ −1/2,b . Next, using the Dirac equation iγ α ∂α uw − muw = γ α Aαw uw and that Aαw uw ∈ L∞ L2 we deduce that uw , u ∈ X 0,b and u˙ w , u˙ ∈ X −1,b . We introduce the KleinGordon propagator and the Dirac propagator given respectively by

K(t) := ∇−1 sin ∇t, where ∇ =

√

˙ D(t) := K(t) + (γ j ∂j − im)γ 0 K(t),

(3.7)

m2 − . Hence, u satisfies

t

u(t) = D(t)u(0) − i

D(t − s)γ 0 γ α (Aα u)(s)ds

(3.8)

0

and u satisfies u (t) = −i

0

t

D(t − s)γ 0 γ α (A α u + A α u + Aα u )(s)ds.

(3.9)

We want to estimate all the terms appearing in (3.9). We have

A 0 uw L2 L2 |I |1/2 A 0 L∞ H˙ 1 uw L∞ H 1/2 ,

A0 u L2 L2 |I |1/2 A0 L∞ L∞ u L∞ L2 , (3.10) A L4 L4 u L4 L4 A A X˙ 1/2,b u X1/2,b A u L2 L2 A

A 1/3 2/3 A u L2 L2 A A L3 L6 u L6 L3 A A 2/3 A 1/3

A

A

u X0,b u L∞ H 1/2 X˙ 1/2,b L∞ H˙ 1 where we have used the Strichartz estimate (2.11) for the third inequality and the interpolation Lemma 2.1 for the last inequality. For the remaining term A u , we use the idea in [13], namely switching the equation from the Dirac to the Klein-Gordon depending on the frequency of the interacting terms. We write (in what follows the convolutions precede the multiplications in all the formulae)

A u = ϕ ∗ (ϕk ∗ A u ) + ϕ ∗ (SA ϕ ∗ u ), (3.11) k≥−5

where SA := k<−5 ϕk ∗ A denotes the frequencies lower than C 2 of A and ϕ := ϕ−2 + ϕ−1 + ϕ + ϕ+1 + ϕ+2 . The first term appearing in (3.11) is estimated as follows

ϕ ∗ (ϕk ∗ A u ) 0,−1/4−ε ϕ ∗ (ϕk ∗ A u ) 2 1/2 ˙ k≥−5

X

|I |

k≥−5

1/4 2

2

k≥−5

L B4/3,2

ϕ ∗ (ϕk ∗ A u )

4/3

L4t 2 Lx

Uniqueness for Maxwell-Klein-Gordon and Dirac

133

−k k |I |1/4 2 2 2 2 ϕk ∗ A L4x u L2x k≥−5

L4t 2

A L4 B˙ 1/2 u L∞ L2 , |I |1/4 A

(3.12)

4,2

where we used the embedding Lemma 2.2 in the first step. The second term in (3.11) is treated by rewriting it in the Klein-Gordon form. Indeed, integrating by parts, we get for all (see [13, (6.5)(6.7)]) t ϕ ∗ D(t − s)γ 0 γ j (SA j ϕ ∗ u )(s)ds 0 t = ϕ ∗ K(t − s) γ α γ j (∂α SA j )ϕ ∗ u − 2SA · ∇ϕ ∗ u 0 + iγ j SA j γ α ϕ ∗ (A α u + Aα u + A α u) . (3.13) We can estimate the different terms in a straightforward way t ϕ ∗ K(t − s){γ α γ j (∂α S Aj )ϕ ∗ u }(s)ds 0,b X 0 |I |1−b 2− γ α γ j (∂α SA j )ϕ ∗ u 2 2 2 Lt Lx |I |1−b 2− ∂α SA L4x ϕ ∗ u L4x 2 2 |I |1−b ∂α A L4 B˙ −1/2 u L4 B˙ −1/2 t

4,2

t

Lt

4,2

A X˙ 1,b + A˙ X0,b ) u X0,b , |I |1−b ( A t ϕ ∗ K(t − s){SA · ∇ϕ ∗ u }(s)ds 0,b X 0 |I |1−b 2− ϕ ∗ {SA · ∇ϕ ∗ u } 2 2 Lt,x |I |1−b 2− SA X1,b ϕ ∗ u X1,b 2 A X˙ 1,b u X0,b , |I |1−b A

(3.14)

(3.15)

where we used the null-form estimate Lemma 2.4 for the second term. Moreover,

ϕ ∗ {γ j SA j γ α ϕ ∗ (A α u + Aα u + A α u)} ∞ −1 L H˙ A L∞ H˙ 1 A L∞ H˙ 1 u L∞ L2 + A L∞ H˙ 1/2 uw L∞ H 1/2 . (3.16) A Now, let us concentrate on the Maxwell part. We have A = 2Pγ 0 u, γ k u + Pγ 0 u , γ k u . A

(3.17)

Using the embedding (2.21) again, we have

uu X˙ −1/2,−1/4−ε uu L2 L4/3 |I |1/4 u L4 L4 u L∞ L2 ,

u u X˙ −1/2,−1/4−ε |I |1/4 u 2L8 L8/3 |I |1/4 u L∞ H 1/2 u L4 B −1/2 . 4,2

134

N. Masmoudi, K. Nakanishi

Putting these two estimates together, we get A X˙ 1/2,b |I |1/4 ( u X1/2,b + u L∞ H 1/2 ) u X0,b .

A

(3.18)

On the other hand, the Poisson equation giving A 0 yields

A 0 L∞ H˙ 1 ( u L∞ H 1/2 + uw L∞ H 1/2 ) u L∞ L2 . Putting all the above estimates together, we deduce that A X˙ 1/2,b C(|I |1/4 + |I |1−b )( u X0,b + A A X˙ 1/2,b ),

u X0,b + A where C depends here on the two solutions (u, A) and (uw , Aw ). Hence, taking I to be small enough, we deduce that (u, A) = (uw , Aw ) and we can iterate the argument on the whole interval (−T , T ). 4. Maxwell-Klein-Gordon In this section, we prove the uniqueness result relative to the Maxwell-Klein-Gordon system. We take two solutions (u, A) and (uw , Aw ) of the Maxwell-Klein-Gordon system on some time interval I = (−T , T ) in the energy space, namely (u, u, ˙ A , A˙ ), (uw , u˙ w , A w , A˙ w ) ∈ C(I ; H 1 × L2 × H˙ 1 × L2 ) with the same initial data u(0) = uw (0), u(0) ˙ = u˙ w (0),

A (0) = A w (0), A˙ (0) = A˙ w (0).

(4.1)

We want to prove that (u, A) = (uw , Aw ). Without loss of generality, we can assume that (u, A) is the solution constructed by Klainerman and Machedon [8], namely u ∈ X 1,b , u˙ ∈ X0,b , A ∈ X˙ 1,b and A˙ ∈ X0,b for some b > 1/2. We denote u = u − uw and A = A − Aw . We want to estimate (u , A ) in the following spaces: u ∈ X 1/2,b , u˙ ∈ X −1/2,b , A ∈ X˙ 1/2,b and A˙ ∈ X˙ −1/2,b . First we check that they belong to those spaces, just by using the multiplication estimate in the Besov spaces. Indeed, we have −1/2 −3/2 Jαw ∈ L∞ B˙ 2,1 and J˙0w = ∇ · J w ∈ L∞ B˙ 2,1 and hence from the Poisson equation 3/2 1/2 for A0w and the wave equation for A w , we infer that A0w ∈ L∞ B˙ , A˙ 0w ∈ L∞ B˙ ,

A ∈ X˙ 1/2,b and that A˙ ∈ X˙ −1/2,b . Hence, we deduce that

2,1

2iAαw ∂α uw + i A˙ 0w uw − Aαw Aαw uw ∈ L2 H −1/2 , and that uw , u ∈ X1/2,b , u˙ w , u˙ ∈ X−1/2,b . The difference equation reads  u − m2 u + 2iAα ∂ α u + 2iA α ∂ α uw − i A˙ 0 u − i A˙ 0 uw    −Aα Aα u − (Aαw + Aα )A α uw = 0, 2 A0 − |uw   | A 0 = iu , u˙ + iA0 u + iuw , u˙ + iA0 u ,  A = P iu , ∇u + u , A u + iuw , ∇u + uw , A u + (|uw |2A ) . A

2,1

(4.2)

(4.3)

Using an energy estimate for the second equation, we get that

A 0 L∞ H˙ 1 iu , u˙ + iA0 u + iuw , u˙ + iA0 u L∞ H˙ −1 ≤ C( u L∞ H 1/2 + u˙ L∞ H −1/2 ),

(4.4)

where here and below, C denotes any constant depending only on the energy norms of the two solutions.

Uniqueness for Maxwell-Klein-Gordon and Dirac

135

Now, let us estimate the different terms appearing on the right hand side of the third equation in (4.3). We have

Piu , ∇u L4 L4/3 u L4 L4 u L∞ H 1 ≤ C u X1/2,b , A L∞ H˙ 1 u L∞ H 1 .

Pu , A u L∞ H˙ −1/2 u L∞ H 1/2 A

(4.5)

The third term can be rewritten as Piuw , ∇u = −Pi∇uw , u , and it can be estimated as the first one. The fourth term can be estimated as the second one. For the fifth term, we have A L∞ H˙ 1/2 .

P(|uw |2 A ) L∞ H˙ −1/2 ≤ C uw 2L∞ H 1 A

(4.6)

Putting all these estimates together, we deduce that A X˙ 1/2,b + A˙ X˙ −1/2,b ≤ C|I |1/4 ( u X1/2,b + A A X˙ 1/2,b ),

A

(4.7)

where we used (2.21) to deal with (4.5). Using that J˙0 = ∇ · J we deduce that

A˙ 0 L∞ L2 ≤ C JJ L∞ H˙ −1 A L∞ H˙ 1/2 ). ≤ C( u L∞ H 1/2 + A

(4.8)

Notice here that the regularity of J is worse than the one of PJJ . Indeed, if we estimate the third term of J without the projection, we only get

iuw , ∇u L∞ H˙ −1 uw L∞ H 1 u L∞ H 1/2 .

(4.9)

Now, let us concentrate on the terms appearing in the first equation of (4.3). We have the following estimates. We use the null-form estimate only in the first one. A · ∇u L2 H −1/2

A A · ∇uw L4 L4/3

A

Aα Aα u L2 H −1/2

Aαw A α uw L2 H −1/2

A0 u˙ L∞ H −1/2

A X˙ 1,b u X1/2,b ,

A A X˙ 1/2,b uw L∞ H 1 ,

A

A 2L∞ H 1 u L∞ H 1/2 ,

Aw L∞ H˙ 1 uw L∞ H 1 A L∞ H˙ 1/2 ,

A0 L∞ B˙ 3/2 u˙ L∞ H −1/2 ,

(4.10)

2,1

A 0 u˙ w L∞ H −1/2 A 0 L∞ H˙ 1 u˙ w L∞ L2 ,

A˙ 0 u L∞ H −1/2 A˙ 0 L∞ H˙ 1/2 u L∞ H 1/2 ,

A˙ 0 uw L∞ H −1/2 A˙ 0 L∞ L2 uw L∞ H 1 .

Putting all these estimates together, and using the first equation of (4.3), we deduce that

u X1/2,b + u˙ X−1/2,b A X˙ 1/2,b + A˙ X˙ −1/2,b ), ≤ C|I |1/4 ( u X1/2,b + u˙ X−1/2,b + A

(4.11)

where we have also used (4.4) and (4.8) to estimate A 0 and A˙ 0 . Combining (4.7) and (4.11) and choosing I to be small enough, we get that u = 0 and A = 0 on I . Iterating the argument, we deduce the uniqueness result.

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N. Masmoudi, K. Nakanishi

References 1. Bergh, J., L¨ofstr¨om, J.: Interpolation spaces. An introduction. Grundlehren Math. Wiss. 223, Berlin-Heidelberg-New York: Springer, 1976 2. Bournaveas, N.: Local existence for the Maxwell-Dirac equations in three space dimensions. Commun. Partial Diff. Eqs. 21(5–6), 693–720 (1996) 3. Bournaveas, N.: Local existence of energy class solutions for the Dirac-Klein-Gordon equations. Comm. Partial Differ. Eqs. 24(7–8), 1167–1193 (1999) 4. Cuccagna, S.: On the local existence for the Maxwell-Klein-Gordon system in R 3+1 . Comm. Partial Differ. Eqs. 24(5–6), 851–867 (1999) 5. Dirac, P.A.M.: Principles of Quantum Mechanics. 4th ed., London: Oxford University Press, 1958 6. Esteban, M.J., Georgiev, V., S´er´e, E.: Stationary solutions of the Maxwell-Dirac and the KleinGordon-Dirac equations, Calc. Var. 4, 265–281 (1996) 7. Furioli, G., Lemari´e-Rieusset, P.G., Terraneo, E.: Unicit´e dans L3 (R3 ) et d’autres espaces fonctionnels limites pour Navier-Stokes. Rev. Mat. Iberoamericana 16(3), 605–667 (2000) 8. Klainerman, S., Machedon, M.: On the Maxwell-Klein-Gordon equation with finite energy. Duke Math. J. 74(1), 19–44 (1994) 9. Klainerman, S., Machedon, M.: Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46(9), 1221–1268 (1993) 10. Klainerman, S., Machedon, M.: Smoothing estimates for null forms and applications. Internat. Math. Res. Notices 9, 383–390 (1994) 11. Lions, P.L., Masmoudi, N.: Uniqueness of mild solutions of the Navier-Stokes system in LN . Comm. Partial Differ. Eqs. 26(11-12), 2211–2226 (2001) 12. Masmoudi, N., Mauser, N.: The selfconsistent Pauli equation and its semiclassical/nonrelativistic limits. Monatsh. Math. 132(1), 19–24 (2001) 13. Masmoudi, N., Nakanishi, K.: Nonrelativistic limit from Maxwell-Klein-Gordon and Maxwell-Dirac to Poisson-Schr¨odinger. In. Math. Res. No. 13, 697–734 (2003) 14. Planchon, F.: On uniqueness for semilinear wave equations. To appear in Math. Z. 15. Tataru, D.: The Xθs spaces and unique continuation for solutions to the semilinear wave equation. Comm. Partial Differ. Eqs. 21(5-6), 841–887 (1996) 16. Zhou, Y.: Uniqueness of generalized solutions to nonlinear wave equations. Am. J. Math. 122(5), 939–965 (2000) Communicated by H.-T. Yau

Commun. Math. Phys. 243, 137–162 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0959-5

Communications in

Mathematical Physics

On the Existence of a Maximizer for the Strichartz Inequality Markus Kunze Universit¨at Essen, FB 6 – Mathematik, 45117 Essen, Germany. E-mail: [email protected] Received: 24 April 2003 / Accepted: 19 June 2003 Published online: 17 October 2003 – © Springer-Verlag 2003

Abstract: It is shown that a maximizing function u∗ ∈ L2 does exist for the Strichartz 2 inequality eit∂x uL6 (L6 ) ≤ SuL2 , with S > 0 being the sharp constant. t

x

1. Introduction and Main Result The L6t (L6x )-Strichartz inequality in one spatial dimension states that there is a constant C > 0 such that 2

eit∂x uL6 (L6 ) ≤ CuL2 t

for all

x

u ∈ L2 = L2 (R; C),

see [12] for instance. The sharp (or best) constant for this estimate is S := sup

eit∂x2 u

L6t (L6x )

uL2

: u ∈ L , u = 0 2

2 = sup eit∂x uL6 (L6 ) : u ∈ L2 , uL2 = 1 . t

x

it∂x2

denotes the evolution operator of the free Schr¨odinger equation, so that Here e 2 u(t, x) = (eit∂x u0 )(x) by definition solves iut + uxx = 0,

u(0, x) = u0 (x).

(1.1)

The purpose of this paper is to verify that a maximizing function u∗ ∈ L2 does exist, i.e., 2 u∗ gives equality in the estimate eit∂x uL6 (L6 ) ≤ SuL2 . Stated differently, ϕ(u∗ ) = t x S 6 for some u∗ ∈ L2 with u∗ L2 = 1, where 6 2 2 ϕ(u) = eit∂x uL6t (L6x ) = |(eit∂x u)(x)|6 dxdt. (1.2) R R

138

M. Kunze

The main difficulty of this problem results from the many invariances of ϕ: it is not hard to show that ϕ(u(·+x0 )) = ϕ(u) for x0 ∈ R, ϕ(eixξ0 u) = ϕ(u) for ξ0 ∈ R, and moreover ϕ(uλ ) = ϕ(u) for λ > 0, where uλ (x) = λ1/2 u(λx); see Corollary 2.3 below. Since all these invariances preserve the L2 -norm, it is in particular not true that every maximizing sequence (uj ) for ϕ under the constraint uj L2 = 1 converges strongly in L2 . An outline of the proof in this paper that a maximizing function does exist is as follows. First the concentration compactness principle is applied to the sequence (uˆ j ); the observation that it can be helpful to use this principle for the Fourier transforms rather than for the maximizing sequence itself seems to be new. It found a first application in [8], where a variational problem from non-linear fiber optics was studied, although it turned out later that the proof could be simplified a lot and this idea in fact did not have to be used. The concentration compactness principle asserts that basically there are three possible alternatives for a L2 -bounded sequence of functions: either it is tight (in the sense of measures), or it is “vanishing” (it tends to zero uniformly on every interval of fixed length), or it is “splitting” (into at least two parts with supports widely separated). The first issue will be to rule out vanishing and splitting. The former is achieved by suitably shifting the uˆ j (corresponding to multiplication of uj with an appropriate eixξj ) and by rescaling the uˆ j (corresponding to a rescaling of the uj ) such that sup

ξ0 +1

ξ0 ∈R ξ0 −1

|uˆ j |2 dξ =

1 −1

|uˆ j |2 dξ =

1 , 2

j ∈ N,

(1.3)

is satisfied. Note that so far already two of the three invariances have been used. Next it has to be discussed why splitting of the sequence (uˆ j ) cannot occur. For this we assume that uˆ j ∼ vˆj + wˆ j , with vˆj L2 ∼ γˆ ∈]0, 1[, wˆ j L2 ∼ (1 − γˆ ), and the supports of vˆj and wˆ j widely separated, say supp(vˆj ) ⊂ {ξ : |ξ | ≤ a} and supp(wˆ j ) ⊂ {ξ : |ξ | ≥ b}. From standard applications of the concentration compactness principle it is known that due to homogeneity properties of a functional a contradiction would be obtained if ϕ(uj ) ∼ ϕ(vj ) + ϕ(wj ) could be shown. Using uj ∼ vj + wj , it can be seen from the definition of ϕ that roughly ϕ(uj ) − ϕ(vj ) − ϕ(wj ) ∼

R R it∂x2

(e

2

2

|eit∂x vj |3 |eit∂x wj |3 dxdt 2

2

2

vj )2 (eit∂x wj )L2 (eit∂x vj )(eit∂x wj )2 L2 tx

tx

(1.4)

holds. At this point it is helpful to recall that there are quite recent multilinear refinements of the Strichartz estimate, in particular of the kind which deal with functions whose Fourier supports are contained in different sets. The usefulness of such estimates has been recognized in [1] (in the case of two space dimensions), and later on a large number of variants and applications have been developed, also in different function spaces or for evolution equations different from the Schr¨odinger equation; see for instance [4, 7] and very many other papers. From [6, Lemma 3.1] we recall the particular multilinear 2 2 2 estimate (eit∂x u)(eit∂x v)(eit∂x w)L2 ≤ Cu 1 v − 1 wL2 which is appropriate tx H4 H 4 for our purposes. From (1.4) and vˆj L2 , wˆ j L2 ≤ 1 thus ϕ(uj ) − ϕ(vj ) − ϕ(wj ) vj 2 1 wj 2 H4

1

H−4

a 1/2 b−1/2 .

On the Existence of a Maximizer for the Strichartz Inequality

139

It appears to not have been noticed before that the concentration compactness principle can be (slightly) refined in such a way that in the case of a splitting sequence the two parts can be moved arbitrarily far apart; see Lemma 3.1 below. Hence b a can be achieved, and the multilinear estimate works together perfectly with the concentration compactness principle to imply that the sequence (uˆ j ) cannot be splitting. Having now excluded two alternatives, it follows that the sequence of measures µj = |uˆ j |2 dξ is tight, i.e., roughly speaking localized by cutting off the high frequencies (which leads to an L2 -small remainder term). In particular, “almost” (uj ) ⊂ H 1 holds and if µj ∗ µ as j → ∞ (along a subsequence) in the sense of measures, then R dµ = 1. However, this improvement is still not sufficient to ensure the needed strong convergence, since the shift invariance of ϕ has not been used yet. This observation leads to the idea to apply, in a next step, the concentration compactness principle also to (uj ). As soon as it is known that both vanishing and splitting is impossible for (uj ) the strong convergence would follow (if the uj are shifted appropriately). Indeed, in this case the sequence is also almost localized in x-space to some interval ] − M, M[, and it can be used that the embedding H 1 (] − M, M[) ⊂ L2 (] − M, M[) is compact. The fact that splitting cannot occur for (uj ) is proved by using the above-mentioned refinement of the splitting alternative: if uj ∼ vj + wj , vj L2 ∼ γ ∈]0, 1[, and wj L2 ∼ (1 − γ ) are satisfied, and if supp(vj ) ⊂ {x : |x| ≤ a} and supp(wj ) ⊂ {x : |x| ≥ b}, then it can be shown that 1/6 1/6 ϕ(uj ) − ϕ(vj ) − ϕ(wj ) vj H 1 + wj H 1 (1 + a)1/2 (b − a)−1/12 (1 + a)1/2 (b − a)−1/12 . The latter estimate holds, since we now have H 1 -bounds also for vj and wj . By moving the two splitting components far enough apart (choosing b a + (1 + a)6 for instance), the right-hand side can be made as small as necessary to verify that splitting of (uj ) is impossible. Therefore it remains to be seen that also vanishing of (uj ) cannot happen. It should be remarked that in general it is easy to construct (by shifting and scaling of any maximizing sequence) a maximizing sequence (u˜ j ) such that (uˆ˜ j ) is tight and (u˜ j ) is vanishing. Typically such a sequence will be concentrating at one or several points in ξ , i.e., µ = al δξl , but we additionally dispose of the normalization (1.3) which will show that concentration to a point is impossible for the special maximizing sequence (uj ). It is quite technical to make this argument rigorous in the case of the Strichartz estimate, since ϕ is a highly non-local functional. Stated differently, for test functions χ = χ (x) there is no obvious rela 2 tion between ϕ(χ u) and R R χ 6 |eit∂x u|6 dxdt. To advance at this point it turns out to be helpful to consider ϕ as a functional of uˆ rather than as a functional of u, 2 2 i.e., we introduce ψ(v) = ϕ(v). ˇ Using (eit∂x uj )(x) = C R ei(xξ −tξ ) uˆ j (ξ ) dξ and integrating out completely the functions δ0 (ξ1 − ξ2 + ξ3 − ξ4 + ξ5 − ξ6 )δ0 (ξ12 − ξ22 + ξ32 − ξ42 + ξ52 − ξ62 ) which then appear, it is possible to derive an explicit form ψ(uˆ j ) = R fj (ξ ) dξ of ψ, where in our case the fj are non-negative functions which are related to uj . From the of (uˆ j ) and the vanishing of (uj ) one can then tightness 6 deduce that ψ(χuj ) ∼ R χ fj dξ as j → ∞ for test functions χ = χ (ξ ). To use this let νj = fj dξ . First, tightness of (uˆ j ) shows that the sequence of measures (νj ) is tight as well, hence νj ∗ ν as j → ∞ (along a subsequence) in the sense of measures for a ν such that R dν = limj →∞ R dνj = limj →∞ R fj dξ = limj →∞ ψ(uˆ j ) = limj →∞ ϕ(uj ) = S 6 = S 6 R dµ. Second, for a test function

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χ = χ (ξ ) one gets χ 6 dν ∼ χ 6 dνj ∼ ψ(χ uˆ j ) = ϕ(χˇ ∗ uj ) ≤ S 6 χˇ ∗ uj 6L2 R

R

= S6

R

3

χ 2 |uˆ j |2 dξ

∼ S6

R

3 χ 2 dµ

as j → ∞ by Strichartz’inequality. Therefore a reversed H¨older-type inequality relating dν and dµ has been derived. In this situation a well-known result on the concentration of measures (see [11, Lemma I.2, p. 161]) can be applied to yield µ = δξ∗ for some ξ∗ ∈ R, which gives a contradiction to (1.3). Filling in the necessary details we obtain the following main result of this paper. 2

Theorem 1.1. There exists a function u∗ ∈ L2 such that S is attained, i.e., eit∂x u∗ L6 (L6 ) t x = S for some u∗ ∈ L2 with R |u∗ |2 dx = 1. It is to be expected that the technique of proof described above can also be applied successfully to other Strichartz-type inequalities to yield the existence of sharp constants, as soon as a multilinear refinement of the inequality in question is available. In connection with Theorem 1.1 an interesting open problem is to determine the numerical values of the sharp constants and to find (up to the invariances) the maximizing function(s); the associated Euler-Lagrange equation seems not to be very helpful in this respect. In the mathematical literature there are several examples of inequalities which are preserved by shift and scaling, for instance the Hardy-Littlewood-Sobolev inequality and the Sobolev inequality. The first general method to prove the existence of sharp constants for such problems has been developed by Lieb in [9] (see also the related [2]). In [9] it has also been possible in many cases to evaluate explicitly the sharp constants and the maximizing functions. The approach taken in [9] rests upon the fact that the left-hand side of the inequality does not decrease, whereas the right-hand side does not increase, if u is replaced by u∗ , its symmetric decreasing rearrangement; this is the case for the Hardy-Littlewood-Sobolev inequality and for the Sobolev inequality. However, for the Strichartz inequality such rearrangement arguments do not appear to lead very far. A second general method which is closer to the approach used in the present paper can be found in [11], but in this way also no information on sharp constants or maximizing functions could be obtained. The paper is organized as follows. In Sect. 2 some auxiliary results and estimates are collected. Section 3 discusses certain aspects related to concentration compactness, and the proof of Theorem 1.1 is developed in Sect. 4. Concerning notation, we write Lp = Lp (R; C) and H s = H s (R; C), with norms · Lp and · H s , respectively. The inner product on L2 is (u, v)L2 = R uv¯ dx, whereas the (spatial) Fourier transform of u ∈ L2 is u(ξ ˆ ) = (2π )−1/2 R e−iξ x u(x) dx with inverse u. ˇ We mostly make explicit the (2π)-factors in our formulas, since some of them could perhaps be useful later to determine the sharp constant explicitly. For + s, b ∈ R we denote Xs,b the closure of S(R2 ) under the norm u2X+ = s,b

where

R R

s

b

(1 + ξ 2 ) (1 + |τ + ξ 2 |) |Fu(τ, ξ )|2 dξ dτ,

On the Existence of a Maximizer for the Strichartz Inequality

Fu(τ, ξ ) = (2π )−1

R R

141

e−i(τ t+ξ x) u(t, x) dxdt = (2π )−1/2

R

e−itτ u(t, ˆ ξ ) dt

is the space-time Fourier transform of u = u(t, x). In the particular case of u(t, x) = 2 2 ˆ ξ ) = e−itξ u(ξ ˆ ) that Fu(τ, ξ ) = (2π )1/2 δ0 (τ +ξ 2 )u(ξ ˆ ), (eit∂x u)(x) it follows from u(t, whence

1/2 2 it∂x2 1/2 2 s e uX+ = (2π) (1 + ξ ) |u(ξ ˆ )| dξ ∼ uH s . (1.5) R

s,b

By C we denote unimportant positive numerical constants which may change from line to line. 2. Some Preliminaries and Technical Lemmas It will be useful to consider along with ϕ from (1.2) also its multilinear version , which is defined as 2 2 2 2

(u1 , u2 , u3 , u4 , u5 , u6 ) = (eit∂x u1 )(x)(eit∂x u2 )(x)(eit∂x u3 )(x)(eit∂x u4 )(x) R R

2

2

(2.1)

ϕ(u) = (u, u, u, u, u, u).

(2.2)

× (eit∂x u5 )(x)(eit∂x u6 )(x) dxdt for u1 , u2 , u3 , u4 , u5 , u6 ∈ L2 . Then

To derive certain estimates related to ϕ and the special case s = 1/4 of the trilinear estimate from [6, Lemma 3.1] is recalled, which reads uvwL2 ≤ CuX+ vX+ tx

1 4 ,b

− 41 ,b

wX+ , 0,b

where b > 1/2 and C > 0 depends only on b. In view of (1.5) this specializes to 2

2

2

(eit∂x u)(eit∂x v)(eit∂x w)L2 ≤ Cu tx

1

H4

v

1

H−4

wL2 .

(2.3)

First we need to study the basic properties of ϕ and in more detail. Lemma 2.1. The following assertions hold. (a) The estimate | (u1 , u2 , u3 , u4 , u5 , u6 )| ≤ S 6

6

ui L2

i=1

is satisfied for u1 , u2 , u3 , u4 , u5 , u6 ∈

L2 .

(b) For u1 , . . . , u6 , v1 , . . . , v6 ∈ L2 we have | (u1 , u2 , u3 , u4 , u5 , u6 ) − (v1 , v2 , v3 , v4 , v5 , v6 )| 5 ≤ C max {ui L2 , vi L2 } max ui − vi L2 , 1≤i≤6

1≤i≤6

in particular, by (2.2), 5 |ϕ(u) − ϕ(v)| ≤ C max{uL2 , vL2 } u − vL2 ,

u, v ∈ L2 .

(2.4)

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M. Kunze

(c) Assume u = v + w. If vL2 , wL2 ≤ 1, then |ϕ(u) − ϕ(v) − ϕ(w)| ≤ C v 1 w

1 H−4

H4

+ v2 1 w2

1

H−4

H4

.

(2.5)

(d) Assume u = v + w, a < b, supp(v) ⊂ {x ∈ R : |x| ≤ a}, and supp(w) ⊂ {x ∈ R : |x| ≥ b}. If vL2 , wL2 ≤ 1, then 1/6 1/6 |ϕ(u) − ϕ(v) − ϕ(w)| ≤ C vH 1 + wH 1 (1 + a)1/2 (b − a)−1/12 . (2.6) Proof. (a) From H¨older’s inequality in t, x with 6 factors and by definition of S we obtain | (u1 , u2 , u3 , u4 , u5 , u6 )| ≤

6

ϕ(ui )1/6 ≤ S 6

i=1

6

ui L2 .

i=1

(b) Due to the multilinearity of we have | (u1 , u2 , u3 , u4 , u5 , u6 ) − (v1 , v2 , v3 , v4 , v5 , v6 )|

=

(u1 − v1 , u2 , u3 , u4 , u5 , u6 ) + (v1 , u2 − v2 , u3 , u4 , u5 , u6 ) + (v1 , v2 , u3 − v3 , u4 , u5 , u6 ) + (v1 , v2 , v3 , u4 − v4 , u5 , u6 )

+ (v1 , v2 , v3 , v4 , u5 − v5 , u6 ) + (v1 , v2 , v3 , v4 , v5 , u6 − v6 ) , whence (a) applies. 2 2 (c) Writing v(t, x) = (eit∂x v)(x) and w(t, x) = (eit∂x w)(x) we get |ϕ(u) −ϕ(v) ϕ(w)| − ≤C |v|5 |w| + |v|4 |w|2 + |v|3 |w|3 + |v|2 |w|4 + |v||w|5 dxdt (2.7) R R

from u = v + w. For the first term on the right-hand side we write |v|5 |w| = |v|3 (|v|2 |w|) and apply H¨older’s inequality in t, x with p = 2, then Strichartz’ inequality. This way the bound (. . . ) ≤ Cv3L2 v 2 wL2 ≤ Cv 2 wL2 is obtx tx tained. The fifth term is treated analogously, and for the third term we write |v|3 |w|3 = (|v|2 |w|)(|v||w|2 ) and use again H¨older’s inequality in t, x with p = 2. From (2.3) it then follows that 2 |ϕ(u) − ϕ(v) − ϕ(w)| ≤ C v 2 wL2 + v 2 wL2tx + v 2 wL2 vw 2 L2 tx tx tx 2 2 2 +vw L2tx + vw L2 tx 2 ≤ C v 1 w − 1 + v 1 w2 − 1 . H4

H

4

H4

H

4

(d) We continue to use the notation from (c). As before, we start with (2.7). To estimate the first term on the right-hand side, we write now |v|5 |w| = |v|4 (|v||w|) and apply H¨older’s inequality in t, x with p = p = 3. Strichartz’ inequality and and 3/2 3 vL2 ≤ 1 then yield the bound C( R R |v| |w|3 dxdt)1/3 . The second term can be bounded by the same expression, since |v|4 |w|2 = |v|3 |w|(|v||w|), and H¨older’s

On the Existence of a Maximizer for the Strichartz Inequality

143

inequality with p1 = 2, p2 = 6, and p3 = 3 can be used. The other terms are handled similarly, resulting in

1/3 3 3 |ϕ(u) − ϕ(v) − ϕ(w)| ≤ C |v| |w| dxdt +C |v|3 |w|3 dxdt R R

≤C

R R

R R

1/3

|v|3 |w|3 dxdt

(2.8)

.

Next we fix t0 > 0 and split R R |v|3 |w|3 dxdt = |t|≤t0 R |v|3 |w|3 dxdt + 3 3 |t|>t0 R |v| |w| dxdt =: (I ) + (I I ). For (I ) let = {x ∈ R : |x| ≤ (a + b)/2}. Then 3 3 |v| |w| dxdt + |v|3 |w|3 dxdt (I ) = |t|≤t0

≤

|v|10 dx

dt

c

|t|≤t0

≤C

|t|≤t0

sup

c

1/2

2

3

2

2

eit∂x vL5t (L10 eit∂x wL5t (L10 x ) x )

|v|3 |w|3 dxdt

|t|≤t0

|v| |w| dxdt

+

1/5 |w|10 dx

3

|w(t)|2 dx

sup

1/2

3

|t|≤t0

c

|w|2 dx

≤

3/10

|t|≤t0

+

|t|≤t0

|w(t)|2 dx

1/2

+

|t|≤t0

c

|v|3 |w|3 dxdt,

by the general Strichartz inequality, cf. [3, Thm. 3.2.5(i)], since the pair (q, r) = (5, 10) is admissible in one spatial dimension. For the c dx-part one can argue in the same way, exchanging the roles of v and w. This yields

1/2

1/2 2 2 (I ) ≤ C sup |w(t)| dx + C sup |v(t)| dx . (2.9) |t|≤t0

|t|≤t0 c

In order to bound the right-hand side further, we use a well-known argument. We take a function β ∈ C0∞ (R) with β(x) ∈ [0, 1] such that β(x) = 1 for |x| ≤ (a + b)/2 and β(x) = 0 2for |x| ≥ (a + 3b)/4. Then β L∞ ≤ C(b − −1 a) . Defining J (t) = R |w(t)| β(x) dx we have J (0) = 0. From (1.1) we get d dw dw −1 J (t)| = |(−2) Im R w(t) ¯ | dt dx (t)β dx| ≤ C(b − a) w(t)L2 dx (t)L2 = −1 C(b − a)−1 wL2 dw dx L2 ≤ C(b − a) wH 1 . Thus for |t| ≤ t0 , 2 2 |w(t)| dx = |w(t)| β(x) dx ≤ J (t) ≤ C(b − a)−1 wH 1 |t| ≤ C(b − a)−1 wH 1 t0 . In an analogous manner c |v(t)|2 dx can be bounded, whence (2.9) gives 1/2 (2.10) (I ) ≤ C(b − a)−1/2 vH 1 + wH 1 t0 .

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Concerning (I I ), we recall the pseudo-conformal estimate v(t)L6 ≤ C(vL2 + xvL2 )|t|−1/3 ; see [3, Cor. 3.3.4(ii)] with r = 6. This yields (I I ) = |v|3 |w|3 dxdt |t|>t0

≤

R

|t|>t0

≤C

1/2

|t|>t0

R

|v|6 dxdt 1/2

v(t)6L6

−1/2

R R

≤ C(1 + xvL2 )

3

dt

≤ C(1 + xvL2 )3 t0

1/2 |w|6 dxdt

|t|>t0

|t|

−2

1/2 dt

.

Summarizing this bound and (2.10) it follows that for every t0 > 0, 1/2 |v|3 |w|3 dxdt ≤ C(b − a)−1/2 vH 1 + wH 1 t0 R R

−1/2

+ C(1 + xvL2 )3 t0

.

One can then optimize this estimate with respect to t0 to obtain 1/2 1/2 |v|3 |w|3 dxdt ≤ C(1 + xvL2 )3/2 vH 1 + wH 1 (b − a)−1/4 . R R

Hence going back to (2.8) and noting xvL2 ≤ avL2 ≤ a, (2.6) is seen to hold.

The next lemma states a more explicit representation of . Lemma 2.2. For θ, ϑ ∈ [0, 2π] we define 2 3 1 1 1 a1 (θ, ϑ) = − √ cos θ + √ sin θ + sin ϑ, a2 (θ, ϑ) = √ cos ϑ + sin ϑ, 3 2 2 6 2 1 1 2 3 1 a3 (θ, ϑ) = √ cos θ + √ sin θ + sin ϑ, a4 (θ, ϑ) = − √ cos ϑ + sin ϑ, 3 2 2 6 2 2 a5 (θ, ϑ) = (− sin θ + sin ϑ). 3 Then

(u1 , u2 , u3 , u4 , u5 , u6 ) = (4π)

−1

R

∞

dξ

dr r 0

2π

2π

dθ 0

dϑ 0

uˆ 1 (a1 (θ, ϑ)r + ξ ) u¯ˆ 2 (a2 (θ, ϑ)r + ξ ) uˆ 3 (a3 (θ, ϑ)r + ξ ) (2.11) × u¯ˆ 4 (a4 (θ, ϑ)r + ξ ) uˆ 5 (a5 (θ, ϑ)r + ξ ) u¯ˆ 6 (ξ ) =: (uˆ 1 , uˆ 2 , uˆ 3 , uˆ 4 , uˆ 5 , uˆ 6 ). Proof. For 1 ≤ j ≤ 6 we write (eit∂x uj )(x) = (2π)−1/2 2

R

(2.12)

ei(xξ −tξ ) uˆ j (ξ ) dξ 2

(2.13)

On the Existence of a Maximizer for the Strichartz Inequality

145

and insert these expressions with variable ξj for uˆ j into the definition of , cf. (2.1). After integrating out R R dxdt, we arrive at

(u1 , u2 , u3 , u4 , u5 , u6 ) = (2π )−1 . . . dξ1 . . . dξ6 uˆ 1 (ξ1 )u¯ˆ 2 (ξ2 )uˆ 3 (ξ3 )u¯ˆ 4 (ξ4 )uˆ 5 (ξ5 )u¯ˆ 6 (ξ6 ) R

R

× δ0 (−ξ1 + ξ2 − ξ3 + ξ4 − ξ5 + ξ6 )δ0 (ξ12 − ξ22 + ξ32 − ξ42 + ξ52 − ξ62 ) . . . dξ1 . . . dξ5 uˆ 1 (ξ1 )u¯ˆ 2 (ξ2 )uˆ 3 (ξ3 )u¯ˆ 4 (ξ4 )uˆ 5 (ξ5 ) = (2π )−1 R R ¯ (2.14) × uˆ 6 (ξ1 − ξ2 + ξ3 − ξ4 + ξ5 )δ0 α(ξ1 , ξ2 , ξ3 , ξ4 , ξ5 ) , where α(ξ1 , ξ2 , ξ3 , ξ4 , ξ5 ) = (−2) (ξ3 + ξ5 − ξ2 − ξ4 )(ξ1 − ξ2 ) + (ξ5 − ξ4 )(ξ3 − ξ4 ) .

(2.15)

Next the transformation ξ = Az is introduced, with the orthogonal matrix A ∈ R5×5 given by   3 − √1 √1 √1 0 − 23 10 2 6 5     1 1 3 √ √  0  0 10  2  5   1 1 1 2 3  A= (2.16)  √2 √6 √5 0 − 3 10  .     3 0 √1 − √1  0 10  2 5   1 2 2 3 0 −√ √ 0 − 3 10 6

5

Since α(Az) = z12 + z22 − z42 − 5z52 , this leads to

(u1 , u2 , u3 , u4 , u5 , u6 ) 1 1 2 3 1 −1 . . . dz1 . . . dz5 uˆ 1 − √ z1 + √ z2 + √ z3 − = (2π ) z5 3 10 2 6 5 R R 1 3 1 ¯ × uˆ 2 √ z3 + √ z4 + z5 10 2 5 1 1 1 2 3 × uˆ 3 √ z1 + √ z2 + √ z3 − z5 3 10 2 6 5 1 3 1 1 2 3 2 ¯ × uˆ 4 √ z3 − √ z4 + z5 uˆ 5 − √ z2 + √ z3 − z5 10 3 10 2 6 5 5 1 3 × u¯ˆ 6 √ z3 − 4 z5 δ0 (z12 + z22 − z42 − 5z52 ). (2.17) 10 5 Then we pass to polar coordinates (z1 , z2 ) = r(cos θ, sin θ) and ∞ (z4, z∞5 ) = s(cos ϑ, √1 sin ϑ), and we set z˜ 3 = √1 z3 . If we also observe the relation 0 dr 0 ds rs 5 5 ∞ f (r, s)δ0 (r 2 − s 2 ) = 21 0 dr rf (r, r), (2.17) may be rewritten as

146

M. Kunze

(u1 , u2 , u3 , u4 , u5 , u6 ) ∞ 2π dr r dθ = (4π )−1

2π

d z˜ 3 1 1 2 3 × uˆ 1 − √ cos θ + √ sin θ − sin ϑ r + z˜ 3 15 2 2 6 1 1 3 × u¯ˆ 2 √ cos ϑ + sin ϑ r + z˜ 3 5 2 2 1 1 2 3 × uˆ 3 √ cos θ + √ sin θ − sin ϑ r + z˜ 3 15 2 2 6 1 1 3 × u¯ˆ 4 − √ cos ϑ + sin ϑ r + z˜ 3 5 2 2 4 3 2 2 3 × uˆ 5 − √ sin θ − sin ϑ r + z˜ 3 u¯ˆ 6 − sin ϑ r + z˜ 3 . 15 2 5 2 6 Hence it remains to make the transformation ξ = [− 45 23 sin ϑ]r + z˜ 3 , dξ = d z˜ 3 , to get (2.11). 0

0

dϑ

0

R

With from (2.12) let ψ(v) := (v, v, v, v, v, v). Then ψ(u) ˆ = (u, ˆ u, ˆ u, ˆ u, ˆ u, ˆ u) ˆ = (u, u, u, u, u, u) = ϕ(u)

(2.18)

by Lemma 2.2 and (2.2). In particular, Strichartz’ inequality yields ˇ 6L2 = S 6 v6L2 , |ψ(v)| = ϕ(v) ˇ ≤ S 6 v

v ∈ L2 .

(2.19)

Due to (2.4) from Lemma 2.1(b), moreover 5 |ψ(v) − ψ(w)| ≤ C max{vL2 , wL2 } v − wL2 ,

v, w ∈ L2 .

(2.20)

We note some useful consequences. Corollary 2.3. The functional ϕ has the following invariances: (a) ϕ(u(· + x0 )) = ϕ(u) for x0 ∈ R; (b) ϕ(eixξ0 u) = ϕ(u) for ξ0 ∈ R; (c) ϕ(uλ ) = ϕ(u) for λ > 0, where uλ (x) = λ1/2 u(λx). 2

2

Proof. (a) Note that (eit∂x u(·+x0 ))(x) = (eit∂x u)(x +x0 ), since both sides have Fourier 2 ˆ ). Hence (a) follows from the definition of ϕ, cf. (1.2). (b) From transform eix0 ξ e−itξ u(ξ ixξ 0 e u(ξ ) = u(ξ ˆ −ξ0 ) one gets ϕ(eixξ0 u) = ψ(u(·−ξ ˆ ˆ ˆ 0 )) = (u(·−ξ 0 ), . . . , u(·−ξ 0 )), see (2.18). But (2.11) implies that the latter equals (u, ˆ . . . , u) ˆ = ψ(u) ˆ = ϕ(u), as the transformation ξ˜ = ξ − ξ0 , d ξ˜ = dξ , in the R dξ -integral can be made. (c) We 2 2 2 have (eit∂x uλ )(x) = λ1/2 (eiλ t∂x u)(λx), whence (1.2) together with the substitution 2 3 (y, s) = (λx, λ t), dyds = λ dxdt, show that ϕ(uλ ) = ϕ(u).

On the Existence of a Maximizer for the Strichartz Inequality

147

Corollary 2.4. Let u ∈ L2 . Then ϕ(u) ≤ ϕ(|u| ˆ ˇ ). Proof. According to (2.18) the claim is equivalent to ψ(u) ˆ ≤ ψ(|u|). ˆ But (2.11) yields ψ(u) ˆ = (u, ˆ . . . , u) ˆ ≤ (|u|, ˆ . . . , |u|) ˆ = ψ(|u|). ˆ

For u ∈ L2 we moreover introduce ∞ 2π fu (ξ ) = (4π )−1 dr r dθ 0

0

2π

¯ˆ 2 (θ, ϑ)r + ξ ) dϑ u(a ˆ 1 (θ, ϑ)r + ξ ) u(a

0

¯ˆ 4 (θ, ϑ)r + ξ ) u(a ¯ˆ ), × u(a ˆ 3 (θ, ϑ)r + ξ )u(a ˆ 5 (θ, ϑ)r + ξ ) u(ξ

(2.21)

so that R R

fu (ξ ) dξ = ψ(u) ˆ = ϕ(u)

and

χ (ξ )fu (ξ ) dξ = (u, ˆ u, ˆ u, ˆ u, ˆ u, ˆ χ u) ˆ = (u, u, u, u, u, χˇ ∗ u)

(2.22)

for χ real and bounded, by Lemma 2.2 and (2.18). Lemma 2.5. Let χ ∈ C0∞ (R), uˆ ∈ L1 ∩ L2 , and δ > 0. Then

χ (ξ )6 fu (ξ ) dξ ˆ −

ψ(χ u) R 16 2 ≤ C δχ 5L∞ χ L∞ u6L2 + δ −1/3 χ 6L∞ uL32 u ˆ L3 1 , with C > 0 independent of χ , u, and δ. Proof. By definition we have ψ(χ u) ˆ − χ 6 (ξ )fu (ξ ) dξ R

= (4π )−1

∞ 0

2π

dr r 0

2π

dθ

dϑ 0

R

dξ χ (a1 r + ξ )u(a ˆ 1r + ξ )

¯ˆ 2 r + ξ )χ (a3 r + ξ )u(a ¯ˆ 4 r + ξ ) × χ (a2 r + ξ )u(a ˆ 3 r + ξ ) χ (a4 r + ξ )u(a ¯ˆ ) × [χ (a5 r + ξ ) − χ (ξ )]u(a ˆ 5 r + ξ ) χ (ξ )u(ξ

¯ˆ 2 r + ξ ) χ (a3 r + ξ )u(a + χ(a1 r + ξ )u(a ˆ 1 r + ξ ) χ (a2 r + ξ )u(a ˆ 3r + ξ ) ¯ ¯ × [χ (a4 r + ξ ) − χ (ξ )]u(a ˆ 4 r + ξ ) χ (ξ )u(a ˆ 5 r + ξ ) χ (ξ )u(ξ ˆ ) ¯ˆ 2 r + ξ ) [χ (a3 r + ξ ) − χ (ξ )] + χ (a1 r + ξ )u(a ˆ 1 r + ξ ) χ (a2 r + ξ )u(a ¯ˆ 4 r + ξ ) χ (ξ )u(a ¯ˆ ) × u(a ˆ 3 r + ξ )χ (ξ )u(a ˆ 5 r + ξ ) χ (ξ )u(ξ

¯ˆ 2 r + ξ ) χ (ξ )u(a + χ (a1 r + ξ )u(a ˆ 1 r + ξ ) [χ (a2 r + ξ ) − χ (ξ )]u(a ˆ 3r + ξ ) ¯ ¯ × χ (ξ )u(a ˆ 4 r + ξ ) χ (ξ )u(a ˆ 5 r + ξ ) χ (ξ )u(ξ ˆ ) ¯ˆ 2 r + ξ ) χ (ξ )u(a + [χ (a1 r + ξ ) − χ (ξ )]u(a ˆ 1 r + ξ ) χ (ξ )u(a ˆ 3r + ξ ) ¯ ¯ × χ (ξ )u(a ˆ 4 r + ξ ) χ (ξ )u(a ˆ 5 r + ξ ) χ (ξ )u(ξ ˆ ) .

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∞ δ ∞ Now we split 0 dr = 0 dr + δ dr. Noting |ai (θ, ϑ)| ≤ 3 for 1 ≤ i ≤ 5 and all θ, ϑ ∈ [0, 2π ], we can estimate |χ (ai r + ξ ) − χ (ξ )| ≤ 3χ L∞ r ≤ 3χ L∞ δ in the first part (I ) of the above integral where r ∈ [0, δ]. Hence (I ) ≤ ≤

Cδχ 5L∞ χ L∞

∞

2π

dr r

2π

dθ

dϑ

dξ

R 0 0 0 × |u(a ˆ 1 r + ξ )| |u(a ˆ 2 r + ξ )| |u(a ˆ 3 r + ξ )||u(a ˆ 4 r + ξ )| |u(a ˆ 5r 5 5 6 CδχL∞ χ L∞ ψ(|u|) ˆ ≤ Cδχ L∞ χ L∞ uL2 ,

+ ξ )| |u(ξ ˆ )| (2.23)

cf. (2.19). The second part (I I ) where r ∈]δ, ∞[ is bounded by (I I ) ≤

Cχ6L∞

∞

dr r

0 ˆ 2r × |u(a ˆ 1 r + ξ )| |u(a δ

2π

2π

dθ

dϑ 0

R

dξ

+ ξ )| |u(a ˆ 3 r + ξ )| |u(a ˆ 4 r + ξ )| |u(a ˆ 5 r + ξ )| |u(ξ ˆ )|.

To take advantage of the fact that r > δ, we will undo the transformations from Lemma 2.2 which led to (2.11). Going back the steps to (2.17), it hence follows with v := |u| ˆ that (I I ) ≤ Cχ 6L∞ . . . dz1 . . . dz5 1M (z1 , . . . , z5 ) R R 1 1 1 2 3 × v − √ z1 + √ z2 + √ z3 − z5 3 10 2 6 5 1 3 1 ×v √ z3 + √ z4 + z5 10 2 5 1 1 1 2 3 ×v √ z1 + √ z2 + √ z3 − z5 3 10 2 6 5 1 2 3 1 1 2 3 ×v √ z3 − √ z4 + z5 v − √ z2 + √ z3 − z5 10 3 10 2 6 5 5 1 3 ×v √ z3 − 4 (2.24) z5 δ0 (z12 + z22 − z42 − 5z52 ), 10 5 where M = {(z1 , . . . , z5 ) : z12 + z22 ≥ δ 2 , z42 + 5z52 ≥ δ 2 }; observe that z12 + z22 = r 2 and z42 + 5z52 = s 2 . Next we set z = A−1 ξ = At ξ ∈ R5 , with A from (2.16). Since 1 z1 = √ (ξ3 − ξ1 ), 2

1 z2 = √ (ξ1 + ξ3 − 2ξ5 ), 6

it follows that A(M) ⊂ N (1) ∪ N (2) , where N (1) = {(ξ1 , . . . , ξ5 ) : |ξ3 − ξ1 | ≥ δ/2},

N (2) = {(ξ1 , . . . , ξ5 ) : |ξ5 − ξ1 | ≥ δ/2}.

Indeed, otherwise for some ξ = Az ∈ A(M) we would have |ξ3 − ξ1 | < δ/2 as well as |ξ5 − ξ1 | < δ/2, whence z12 + z22 = 21 (ξ3 − ξ1 )2 + 16 ([ξ3 − ξ1 ] + 2[ξ1 − ξ5 ])2 ≤

On the Existence of a Maximizer for the Strichartz Inequality

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1 2 1 2 2 (δ /4) + 6 ((δ/2) + 2(δ/2))

= δ 2 /2, in contradiction to z ∈ M. Thus if the transformation ξ = Az in (2.24) is introduced, we get (I I ) ≤

Cχ6L∞

2 i=1

R

...

R

dξ1 . . . dξ5 1N (i) (ξ1 , . . . , ξ5 ) v(ξ1 )v(ξ2 )v(ξ3 )v(ξ4 )

× v(ξ5 )v(ξ1 − ξ2 + ξ3 − ξ4 + ξ5 )δ0 α(ξ1 , ξ2 , ξ3 , ξ4 , ξ5 ) = Cχ 6L∞

2 i=1

R

...

R

dξ1 . . . dξ6 1N (i) (ξ1 , . . . , ξ5 ) v(ξ1 )v(ξ2 )v(ξ3 )v(ξ4 )v(ξ5 )

× v(ξ6 )δ0 (−ξ1 + ξ2 − ξ3 + ξ4 − ξ5 + ξ6 )δ0 (ξ12 − ξ22 + ξ32 − ξ42 + ξ52 − ξ62 ), cf. (2.14) and (2.15). Since the term with 1N (2) is transformed into the term with 1N (1) when ξ3 and ξ5 are exchanged, it suffices to consider the contribution with 1N (1) . Hence, 6 . . . dξ1 . . . dξ6 1N (1) (ξ1 , . . . , ξ5 ) v(ξ1 )v(ξ2 )v(ξ3 )v(ξ4 )v(ξ5 ) (I I ) ≤ Cχ L∞ R

R

× v(ξ6 )δ0 (−ξ1 + ξ2 − ξ3 + ξ4 − ξ5 + ξ6 )δ0 (ξ12 − ξ22 + ξ32 − ξ42 + ξ52 − ξ62 ) 6 1{|ξ3 −ξ1 |≥δ/2} v(ξ1 )v(ξ3 ) G(−ξ12 − ξ32 , ξ1 + ξ3 ) dξ1 dξ3 , (2.25) = Cχ L∞ R R

where

G(τ, ξ ) =

R R R R

dξ2 dξ4 dξ5 dξ6 v(ξ2 )v(ξ4 )v(ξ5 )v(ξ6 )

× δ0 (−ξ + ξ2 + ξ4 − ξ5 + ξ6 )δ0 (−τ − ξ22 − ξ42 + ξ52 − ξ62 ). Basically we can now follow a standard proof of the Strichartz inequality to obtain the desired estimate.Using (2.13) it is straightforward to verify that G(τ, ξ ) = 2π 2 2 F |eit∂x v| ˇ 2 (eit∂x v) ˇ 2 (−τ, −ξ ). Thus the Hausdorff-Young inequality in conjunction with Strichartz’ inequality imply 2 2 GL3 ≤ C |eit∂x v| ˇ 2 (eit∂x v) ˇ 2 3/2 ≤ Cv ˇ 4L2 = Cu4L2 , (2.26) τξ

Ltx

recall v = |u|. ˆ Therefore we can continue in (2.25), v(ξ1 )v(ξ3 ) 6 (I I ) ≤ Cχ L∞ 1{|ξ3 −ξ1 |≥δ/2} |ξ − ξ1 |1/3 1/3 3 |ξ 3 − ξ1 | R R × G(−ξ12 − ξ32 , ξ1 + ξ3 ) dξ1 dξ3

2/3 v(ξ1 )3/2 v(ξ3 )3/2 ≤ Cχ 6L∞ 1{|ξ3 −ξ1 |≥δ/2} dξ dξ 1 3 |ξ3 − ξ1 |1/2 R R 1/3

3 2 2 |ξ3 − ξ1 | G(−ξ1 − ξ3 , ξ1 + ξ3 ) dξ1 dξ3 × R R

≤ Cδ −1/3 χ 6L∞

R R

2/3 v(ξ1 )3/2 v(ξ3 )3/2 dξ1 dξ3

GL3 , τξ

(2.27)

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the latter through the substitution τ = −ξ12 − ξ32 , ξ = ξ1 + ξ3 , which has dτ dξ = 2|ξ3 − ξ1 |dξ1 dξ3 . Hence it follows from (2.26) that 2

4

ˆ L3 1 u ˆ L3 2 . (I I ) ≤ Cδ −1/3 χ 6L∞ u4L2 v2 3 ≤ Cδ −1/3 χ 6L∞ u4L2 u L2

Summarizing this estimate and (2.23), the proof of the lemma is complete.

We remark that using |ξ3 − ξ1 |1/2 ≥ C|ξ3 − ξ1 |1/2−κ δ κ and the Hardy-LittlewoodSobolev inequality in (2.27), the bound on (I I ) can be improved to Cδ −2κ/3 χ 6L∞ 6− 4κ

4κ

uL2 3 u ˆ L31 for κ ∈]0, 21 ], but this fact will be of no relevance to us here. For a proof of the following lemma see [8, Lemma 2.6]. Lemma 2.6. Let (uj ) ⊂ L2 be bounded and such that for any A > 0, lim sup

j →∞ x0 ∈R

x0 +A

x0 −A

|uj |2 dx = 0,

i.e., (uj ) is “vanishing”. With φ ∈ S(R) (Schwartz functions) we define uj = (φ uˆ j )ˇ = (l) φˇ ∗ uj . Then (u ) is also vanishing. (l)

j

3. Concentration Compactness This terminology is used for the fact that basically there are three possibilities for a (L2 or H 1 -) bounded sequence of functions: either it is tight (in the sense of measures), or it is “vanishing” (it tends to zero uniformly on every interval of fixed length), or it is “splitting” (into at least two parts with supports widely separated). This principle, see [10], has turned out to be very helpful in a number of variational problems. For our purposes here we need a small refinement which relies on the observation that in the case of a splitting sequence the two parts can be moved arbitrarily far apart, see (3.1) below. Since we will need a very explicit form of alternative (3), we include some details. Lemma 3.1. Let (fj ) ⊂ L2 be a sequence such that fj L2 = 1 for j ∈ N. Then there is a subsequence (not relabelled) such that exactly one of the following three possibilities occurs. (1) There exists a sequence (zj ) ⊂ R such that for every ε > 0 there is R = Rε > 0 with the property that

zj +R

zj −R

|fj |2 dz ≥ 1 − ε,

j ∈ N.

(2) For every A > 0 we have lim sup

j →∞ z0 ∈R

z0 +A

z0 −A

|fj |2 dz = 0.

On the Existence of a Maximizer for the Strichartz Inequality

151

(3) There is γ ∈]0, 1[ with the following property. For every δ ∈]0, γ [ there exist j0 = j0 (δ) ∈ N and z1∗ , z2∗ > 0 with z2∗ ≥ z1∗ + 4δ −1 + δ −6 (z1∗ + 2δ −1 )6 such that

γ − δ < sup

z0 ∈R

z0 +z2∗ z0 −z2∗

|fj |2 dz < γ + δ,

j ≥ j0 ,

and for every j ≥ j0 we may select zj ∈ R satisfying zj +z∗ 1 γ −δ < |fj |2 dz < γ + δ. zj −z1∗

(3.1)

(3.2)

(3.3)

In particular, if we fix functions ρ, η ∈ C0∞ (R) with values in [0, 1] which satisfy ρ(z) = 1 for |z| ≤ z1∗ , ρ(z) = 0 for |z| ≥ z1∗ + 2δ −1 , η(z) = 0 for |z| ≤ z2∗ − 2δ −1 , and η(z) = 1 for |z| ≥ z2∗ , then defining vj (z) = ρ(z − zj )fj (z) and wj (z) = η(z − zj )fj (z) one obtains for j ≥ j0 the estimates

fj − (vj + wj )2L2 ≤ 2δ, vj 2L2 − γ ≤ 3δ, and

wj 2L2 − (1 − γ ) ≤ 9δ. Proof. The argument relies on the L´evy concentration functions j (z) = z +z supz0 ∈R z00−z |fj (y)|2 dy, z ≥ 0. Then 0 ≤ j (z) ≤ 1 and j is non-decreasing. Hence there exists a subsequence of (fj ), a countable set E ⊂ [0, ∞[, and a nonnegative and non-decreasing function such that j (z) → (z) as j → ∞ for every z ∈ [0, ∞[\E. With γ := limz→∞ (z) ∈ [0, 1], there are three possibilities: the cases γ = 1 or γ = 0 lead to alternative (1) or (2), respectively, cf. [3, Lemma 8.3.8] (here it is not needed that (fj ) is bounded in H 1 ). So it remains to show that γ ∈]0, 1[ implies (3). To see this, we fix δ ∈]0, γ [ and choose z∗ > 0 such that γ − δ < (z) ≤ γ for z ≥ z∗ . Then we take two widely separated points where j converges to , i.e., we fix some z1∗ ∈ ([0, ∞[\E) ∩ [z∗ , ∞[ and then choose z2∗ ∈ ([0, ∞[\E) ∩ [z∗ , ∞[ satisfying z2∗ ≥ z1∗ + 4δ −1 + δ −6 (z1∗ + 2δ −1 )6 . Note that then in particular (z2∗ − 2δ −1 ) − (z1∗ + 2δ −1 ) > 0, whence the supports of ρ and η are separated. Since z1∗ and z2∗ are convergence points, we also find j0 ∈ N with γ − δ < j (z1∗ ) ≤ j (z2∗ ) < γ + δ for j ≥ j0 . By definition of j , this yields (3.2), and moreover for every j ≥ j0 we find zj ∈ R such that (3.3) holds. With ρ and η as in (3), we then define vj and wj . In view of (3.2) and (3.3) we have zj +z∗ zj +z∗ 2 1 |fj (z)|2 dz = |fj |2 dz − |fj |2 dz z1∗ ≤|z−zj |≤z2∗

zj −z2∗

zj −z1∗

≤ γ + δ − (γ − δ) = 2δ.

(3.4)

Due to the support properties of vj and wj therefore fj − (vj + wj )2L2 = (1 − ρ(z − zj ) − η(z − zj ))2 |fj (z)|2 dz z1∗ ≤|z−zj |≤z2∗

≤

z1∗ ≤|z−zj |≤z2∗

|fj (z)|2 dz ≤ 2δ.

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M. Kunze

In addition, (3.3), z1∗ + 2δ −1 ≤ z2∗ , and (3.4) imply

vj 2L2 − γ =

zj +z1∗

zj −z1∗

≤δ+

|fj (z)| dz − γ + 2

z1∗ ≤|z−zj |≤z2∗

ρ(z − zj ) |fj (z)| dz

−1 2

z1∗ ≤|z−zj |≤z1∗ +2δ

2

|fj (z)|2 dz ≤ 3δ.

Finally, from this and fj L2 = 1 we get

wj 2L2 − (1 − γ ) ≤ 3δ + wj 2L2 + vj 2L2 − fj 2L2

= 3δ +

|fj (z)|2 dz |z−zj |>z2∗

+

z2∗ −2δ −1 ≤|z−zj |≤z2∗

η(z − zj )2 |fj (z)|2 dz

+ +

|z−zj |
|fj (z)|2 dz

z1∗ ≤|z−zj |≤z1∗ +2δ −1

ρ(z − zj )2 |fj (z)|2 dz −

≤ 3δ + +

z1∗ ≤|z−zj |≤z2∗

|fj (z)|2 dz +

z1∗ ≤|z−zj |≤z1∗ +2δ −1

R

|fj (z)|2 dz

z2∗ −2δ −1 ≤|z−zj |≤z2∗

|fj (z)|2 dz

|fj (z)|2 dz ≤ 9δ,

where once more z1∗ + 2δ −1 ≤ z2∗ and (3.4) have been used.

The following result examines what can be said in the case that a reversed H¨oldertype estimate holds for measures. It is a special case of [11, Lemma I.2, p. 161], see also [5, p. 13]. Lemma 3.2. Let ν, µ be finite non-negative measures on R such that for some 1 < p < q < ∞ and S > 0 the estimate

1/q

1/p q p |χ | dν ≤S |χ | dµ , χ ∈ C0∞ (R), R

R

holds. If ν(R)1/q ≥ Sµ(R)1/p , then ν = γ δξ∗ and µ = γ p/q S −p δξ∗ for some γ ≥ 0 and ξ∗ ∈ R. 4. Proof of Theorem 1.1 We consider a maximizing sequence, i.e., (Uj ) ⊂ L2 is such that Uj L2 = 1 for j ∈ N and ϕ(Uj ) → S 6 as j → ∞. Then we introduce Vj = |Uˆ j |ˇ and note that Vˆj = |Uˆ j | ≥ 0 as well as Vˆj ∈ L2 . Hence we can select wj ∈ S(R) satisfying wj ≥ 0 and wj − Vˆj L2 ≤ 1/j . With these definitions we let uj = wj −1 wˇ . Then L2 j

On the Existence of a Maximizer for the Strichartz Inequality

153

uj ∈ S(R), uj L2 = 1, and uˆ j ≥ 0. Therefore Strichartz’inequality yields ϕ(uj ) ≤ S 6 . Since Vˆj L2 = Uj L2 = 1, we also have |wj L2 −1| ≤ wj − Vˆj L2 ≤ 1/j . Hence e.g. wj L2 ≤ 2, and it follows from Corollary 2.4 and with (2.4) from Lemma 2.1(b) that 0 ≤ S 6 − ϕ(uj ) ≤ |S 6 − ϕ(Uj )| + ϕ(Uj ) − ϕ(uj ) ≤ |S 6 − ϕ(Uj )| + ϕ(|Uˆ j |ˇ ) − ϕ(uj ) = |S 6 − ϕ(Uj )| + ϕ(Vj ) − wj −6 ϕ(wˇ j ) L2

≤ |S 6 − ϕ(Uj )| + |ϕ(Vj ) − ϕ(wˇ j )| + wj −6 − 1

ϕ(wˇ j ) 2 L

≤ |S 6 − ϕ(Uj )| + CVj − wˇ j L2 + C wj −6 − 1

→ 0, L2

j → ∞.

Thus ϕ(uj ) → S 6 , i.e., (uj ) is a maximizing sequence with the additional properties that uj ∈ S(R) and uˆ j ≥ 0. We need to modify the uj in other respects, too. For this we fix ξ +z j ∈ N and examine the concentration function ˆ j (z) = supξ0 ∈R ξ00−z |uˆ j |2 dξ of uˆ j . Since limz→∞ ˆ j (z) = 1 we may select λj > 0 such that ˆ j (λ−1 j ) = 1/2. In addition, ξ0 +λ−1 j ˆ j |2 dξ is continuous and tending to zero as ξ0 → ±∞. the function ξ0 → −1 |u ξ0 −λj

Hence we also find ξ0,j ∈ R with

ξ0,j +λ−1 j ξ0,j −λ−1 j

|uˆ j |2 dξ = supξ0 ∈R (...) = ˆ j (λ−1 j ) = 1/2. 1/2

We then define ξj = −λj ξ0,j and u˜ j (x) = eixξj λj uj (λj x). It follows that u˜ j L2 = 1, −1/2 uˆ j (λ−1 (ξ − ξj )) ≥ 0. From Corollary 2.3(b) and (c) we u˜ j ∈ S(R), and uˆ˜ j (ξ ) = λ j

j

deduce that ϕ(u˜ j ) = ϕ(uj ), and we calculate ξ0 +1 λ−1 (ξ0 +1−ξj ) j sup |uˆ˜ j (ξ )|2 dξ = sup |uˆ j (ξ )|2 dξ = sup ξ0 ∈R λ−1 j (ξ0 −1−ξj )

ξ0 ∈R ξ0 −1

ξ0 +λ−1 j

ξ0 ∈R ξ0 −λ−1 j

|uˆ j (ξ )|2 dξ

= ˆ j (λ−1 j ) = 1/2. Moreover, 1 −1

|uˆ˜ j (ξ )|2 dξ =

λ−1 j (1−ξj )

λ−1 j (−1−ξj )

|uˆ j (ξ )| dξ = 2

ξ0,j +λ−1 j

ξ0,j −λ−1 j

|uˆ j (ξ )|2 dξ =

1 . 2

Renaming u˜ j to uj , we can summarize the foregoing modifications as follows. There exists (uj ) ⊂ S(R) such that uj L2 = 1, uˆ j ≥ 0, and ϕ(uj ) → S 6 as j → ∞, and also ξ0 +1 1 1 2 sup (4.1) |uˆ j (ξ )| dξ = |uˆ j (ξ )|2 dξ = , j ∈ N, 2 −1 ξ0 ∈R ξ0 −1 is satisfied. We are going to work with this special improved maximizing sequence in the sequel. Since also uˆ j L2 = 1 for j ∈ N, Lemma 3.1 can be applied to the sequence (fj ) = (uˆ j ). The following two Subsects. 4.1 and 4.2 deal with the two possibilities which then may occur (for a subsequence which is not relabelled) according to Lemma 3.1; note that the Fourier transforms ξ +1cannot vanish in the sense of alternative (2) in Lemma 3.1, as limj →∞ supξ0 ∈R ξ00−1 |uˆ j |2 dξ = 1/2 = 0 by (4.1).

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M. Kunze

4.1. The Fourier transforms cannot be splitting. In this section we suppose that alternative (3) from Lemma 3.1 is satisfied for (uˆ j ). Then we have γˆ ∈]0, 1[ for γˆ = ˆ ), the function ˆ being the pointwise limit (outside a countable set) of the limξ →∞ (ξ ξ +ξ concentration functions ˆ j (ξ ) := supξ0 ∈R ξ00−ξ |uˆ j |2 dζ of the uˆ j . We fix δ ∈]0, γˆ [ and select j0 ∈ N, ξ1∗ = z1∗ > 0, ξ2∗ = z2∗ > 0, ξj = zj for j ≥ j0 , and moreover the functions ρ and η as stated in (3) of Lemma 3.1; all these quantities depend on δ. With aj (ξ ) = ρ(ξ − ξj )uˆ j (ξ ) and bj (ξ ) = η(ξ − ξj )uˆ j (ξ ), we then have aj L2 ≤ 1, bj L2 ≤ 1, and in addition for j ≥ j0 the estimates uˆ j − (aj + bj )2L2 ≤ 2δ, |aj 2L2 − γˆ | ≤ 3δ, as well as |bj 2L2 − (1 − γˆ )| ≤ 9δ. Setting vj = aˇ j and wj = bˇj , this leads to vj L2 ≤ 1, wj L2 ≤ 1, and also

uj − (vj + wj )2L2 ≤ 2δ, vj 2L2 − γˆ ≤ 3δ, and

(4.2)

wj 2L2 − (1 − γˆ ) ≤ 9δ for j ≥ j0 . Then (2.4) from Lemma 2.1(b), (4.2), and Corollary 2.3(b) imply the estimate |ϕ(uj ) − ϕ(vj ) − ϕ(wj )| ≤ |ϕ(uj ) − ϕ(vj + wj )| + |ϕ(vj + wj ) − ϕ(vj ) − ϕ(wj )| 5 ≤ C max{uj L2 , vj + wj L2 } uj − (vj + wj )L2 + |ϕ(vj + wj ) − ϕ(vj ) − ϕ(wj )| ≤ Cδ 1/2 + |ϕ(v˜j + w˜ j ) − ϕ(v˜j ) − ϕ(w˜ j )|, where v˜j (x) := e−ixξj vj (x) and w˜ j (x) := e−ixξj wj (x). Since v˜j L2 = vj L2 ≤ 1 and w˜ j L2 = wj L2 ≤ 1 we can then apply (2.5) from Lemma 2.1 to get |ϕ(uj ) − ϕ(vj ) − ϕ(wj )| ≤ Cδ 1/2 + C v˜j

H

1 4

w˜ j

H

− 41

+ v˜j 2 1 w˜ j 2 H4

1 H−4

.

(4.3)

The second term on the right-hand side can be handled by means of the following observation, where the notation from Lemma 3.1 is used. Lemma 4.1. We have the bound v˜j

1

H4

w˜ j

1

H−4

≤ 21/8 δ 1/4 .

Proof. Due to vˆ˜ j (ξ ) = vˆj (ξ + ξj ) = aj (ξ + ξj ) = ρ(ξ )uˆ j (ξ + ξj ) we find 2 1/4 1/4 (1 + ξ 2 ) |vˆ˜ j (ξ )| dξ = (1 + ξ 2 ) ρ(ξ )2 |uˆ j (ξ + ξj )|2 dξ. v˜j 2 1 = H4

R

R

By definition of ρ we have ρ(ξ ) = 0 for |ξ | ≥ ξ1∗ + 2δ −1 = z1∗ + 2δ −1 . Therefore ρ(ξ ) ∈ [0, 1] yields 1/4 1/4 v˜j 2 1 ≤ (1 + [z1∗ + 2δ −1 ]2 ) |uˆ j (ξ + ξj )|2 dξ = (1 + [z1∗ + 2δ −1 ]2 ) . R

H4

ˆ˜ j (ξ ) = η(ξ )uˆ j (ξ + ξj ), thus Similarly, we have w −1/4 (1 + ξ 2 ) η(ξ )2 |uˆ j (ξ + ξj )|2 dξ. w˜ j 2 − 1 = H

4

R

On the Existence of a Maximizer for the Strichartz Inequality

155

Recalling that η(ξ ) = 0 for |ξ | ≤ ξ2∗ − 2δ −1 = z2∗ − 2δ −1 and η(ξ ) ∈ [0, 1], we obtain −1/4 2 ∗ −1 2 −1/4 |uˆ j (ξ + ξj )|2 dξ = (1 + [z2∗ − 2δ −1 ]2 ) . w˜ j − 1 ≤ (1 + [z2 − 2δ ] ) H

R

4

Therefore we arrive at v˜j

1

H4

w˜ j

1/8

1

H−4

≤ (1 + [z1∗ + 2δ −1 ]2 ) 1/4

≤ 21/8 (z1∗ + 2δ −1 )

−1/8

(1 + [z2∗ − 2δ −1 ]2 ) −1/4

(z2∗ − 2δ −1 )

,

where δ < γˆ < 1 ≤ 2 has been used, hence 1 ≤ z1∗ + 2δ −1 . Due to (3.1) we have z2∗ ≥ z1∗ + 4δ −1 + δ −6 (z1∗ + 2δ −1 )6 ≥ 2δ −1 + δ −1 (z1∗ + 2δ −1 ). This yields the estimate −1 (z1∗ + 2δ −1 )(z2∗ − 2δ −1 ) ≤ δ, hence the claim holds. By Lemma 4.1 we can continue from (4.3) as |ϕ(uj ) − ϕ(vj ) − ϕ(wj )| ≤ Cδ 1/2 + Cδ 1/4 ≤ Cδ 1/4 ,

j ≥ j0 .

(4.4)

Using Strichartz’ inequality, it follows that ϕ(uj ) ≤ Cδ 1/4 + ϕ(vj ) + ϕ(wj ) ≤ Cδ 1/4 + S 6 vj 6L2 + S 6 wj 6L2 ≤ Cδ 1/4 + S 6 (3δ + γˆ )3 + S 6 (9δ + (1 − γˆ ))3 . At the beginning of the argument δ ∈]0, γˆ [ has been fixed, and we have found that the latter estimate holds for all j ≥ j0 = j0 (δ). Since (uj ) is a maximizing sequence, as j → ∞ this yields S 6 ≤ Cδ 1/4 + S 6 (3δ + γˆ )3 + S 6 (9δ + (1 − γˆ ))3 . Taking the limit δ → 0 we finally arrive at 1 ≤ γˆ 3 + (1 + γˆ )3 , contradicting γˆ ∈]0, 1[. Hence it is not possible that the Fourier transforms are splitting. 4.2. The Fourier transforms are tight. So far we have shown that the alternatives (2) and (3) from Lemma 3.1 cannot hold for the sequence (uˆ j ). Therefore (1) has to be satisfied, i.e., there exists a sequence (ξj ) ⊂ R such that for every ε > 0 there is R = Rε > 0 with ξ +R the property that ξjj−R |uˆ j |2 dξ ≥ 1 − ε for j ∈ N. It is well-known that in this kind of argument, in view of (4.1), one can assume ξj = 0 for every j ∈ N by replacing Rε with 2Rε + 1; see e.g. [13, p. 48]. Indeed, if ε < 1/2, then we choose a corresponding Rε and note that Ij, ε = [ξj − Rε , ξj + Rε ] ∩ [−1, 1] = ∅, since otherwise by (4.1) the con ξ +R 1 tradiction 1 = R |uˆ j |2 dξ ≥ ξjj−Rεε |uˆ j |2 dξ + −1 |uˆ j |2 dξ ≥ 1 − ε + 1/2 = 3/2 − ε would be obtained. But Ij, ε = ∅ implies [ξj − Rε , ξj + Rε ] ⊂ [−(2Rε + 1), 2Rε + 1], 2Rε +1 ξ +R whence −(2R |uˆ j |2 dξ ≥ ξjj−Rεε |uˆ j |2 dξ ≥ 1 − ε. Hence we can assume that ε +1) ∀ ε ∈]0, 1/2[ ∃ R = Rε > 0 : is satisfied.

R

−R

|uˆ j |2 dξ ≥ 1 − ε,

j ∈ N,

(4.5)

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M. Kunze

Lemma 4.2. For every j ∈ N let the measures νj and µj on R be defined as νj = fuj dξ and µj = |uˆ j |2 dξ , where fuj is given by (2.21). Then νj and µj are non-negative measures, and the sequences (νj ) and (µj ) are tight. There exist non-negative measures ν and µ on R such that, possibly after selecting subsequences, νj ∗ ν as well as µj ∗ µ as j → ∞ in the sense of measures. In addition, 6 dν = S and dµ = 1. (4.6) R

R

Proof. By construction we have uˆ j ≥ 0, whence (2.21) shows that fuj ≥ 0. Moreover, (2.22) implies R dνj = R fuj dξ = ϕ(uj ) → S 6 as j → ∞. Using the fact that uˆ j L2 = 1, we also get R dµj = R |uˆ j |2 dξ = 1 for j ∈ N. From (4.5) we deduce that (µj ) is tight. Hence, by passing to a subsequence if necessary, µj ∗ µ as j → ∞ in the sense of measures (i.e., R χ dµj → R χ dµ for all bounded χ ∈ C(R)), by Prochorov’s compactness theorem. In view of the Portmanteau theorem this in turn implies R dµ = 1. Therefore it remains to be verified that (νj ) is tight, too. For this we note that, due to the second relation from (2.22) and Lemma 2.1(a), dνj = 1|ξ |≥R fuj dξ = (uj , uj , uj , uj , uj , (1|ξ |≥R )ˇ ∗ uj ) R 6

|ξ |≥R

≤ S (1|ξ |≥R )ˇ ∗ uj L2

1/2 6 2 = S 1− |uˆ j | dξ ≤ S 6 ε 1/2 , |ξ |
j ∈ N,

the latter provided that ε ∈]0, 1/2[ is prescribed and R = Rε is chosen according to (4.5). Hence as before to obtain νj ∗ ν along a subsequence (νj ) is tight andwe may argue 6 and R dν = limj →∞ R dνj = S . The next step consists in the application of Lemma 3.1 to (fj ) = (uj ), yielding a further subsequence of (uj ), which however will be not relabelled for notational convenience. The argument is then divided once more according to which one of the possibilities (1), (2), or (3) from Lemma 3.1 arises. 4.2.1. The case that the sequence (uj ) is vanishing. Throughout this subsection we assume that (2) in Lemma 3.1 occurs for (uj ), i.e., lim sup

j →∞ x0 ∈R

x0 +A

x0 −A

|uj |2 dx = 0

(4.7)

is satisfied for every A > 0. Our aim is to derive a contradiction from this, whence in fact (uj ) cannot be vanishing. Lemma 4.3. With ν and µ from Lemma 4.2 the estimate

1/6 6

R

holds.

χ dν

≤S

1/2 χ dµ , 2

R

χ ∈ C0∞ (R),

On the Existence of a Maximizer for the Strichartz Inequality

157

Before we go on to the proof of Lemma 4.3, let us first argue why this leads to a contradiction. From (4.6) in Lemma 4.2 we know that ν(R)1/6 = S = Sµ(R)1/2 . Hence we can apply Lemma 3.2 with q = 6 and p = 2 to find that ν = γ δξ∗ and µ = γ 1/3 S −2 δξ∗ for some γ ≥ 0 and ξ∗ ∈ R. Then ν(R) = S 6 yields γ = S 6 and µ = δξ∗ . Therefore 1 = ξ +1 µ(]ξ∗ − 1, ξ∗ + 1[) ≤ lim inf j →∞ µj (]ξ∗ − 1, ξ∗ + 1[) = lim inf j →∞ ξ∗∗−1 |uˆ j |2 dξ ≤ ξ +1 lim inf j →∞ supξ0 ∈R ξ00−1 |uˆ j |2 dξ ≤ 1/2 by (4.1), which is a contradiction. Proof of Lemma 4.3. Using (4.5) we will first split every uj into a regular low-frequency part and a small high-frequency part as follows. We choose φ ∈ C0∞ (R) with values in [0, 1] such that φ(ξ ) = 1 for |ξ | ≤ 1 and φ(ξ ) = 0 for |ξ | ≥ 2, and for R > 0 we define φR (ξ ) = φ(ξ/R). With a fixed sequence (εk ) ⊂]0, 1/2[, εk → 0 as k → ∞, we select Rk := Rεk corresponding to εk via (4.5) for k ∈ N. Setting φk = φRk , we decompose every uj as (l)

(h)

uj = uj,k + uj,k ,

j, k ∈ N,

(4.8)

where uj,k = (φk uˆ j )ˇ = φˇ k ∗ uj (l)

and uj,k = ((1 − φk )uˆ j )ˇ = (1 − φk )ˇ ∗ uj . (h)

(l)

(4.9)

(l)

Since uj ∈ S(R) and uˆ j ≥ 0 by construction, we get uj,k ∈ S(R) and uˆ j,k = φk uˆ j ≥ 0. For all j, k ∈ N, (l) 2 duj,k = u(l) 2 = |ξ |2 |φk uˆ j |2 dξ j,k H˙ 1 dx 2 R L |ξ |2 |φk uˆ j |2 dξ ≤ CRk2 uˆ j 2L2 = CRk2 , = |ξ |≤2Rk

(4.10)

and also, by (4.8) and (4.5), (l) 2 L2

uj − uj,k

(h) 2 = uj,k 2 = |(1 − φk )uˆ j |2 dξ L |ξ |≥R k ≤ |uˆ j |2 dξ = 1 − |uˆ j |2 dξ ≤ εk . |ξ |≥Rk

|ξ |
(4.11)

In addition, (l)

uj,k

L2

≤ uˆ j L2 = 1.

(4.12)

In the following we fix χ ∈ C0∞ (R). Then by definition of νj , the second relation in (2.22), Lemma 2.1(b), (4.12), and (4.11), 6 6 6 6 χ dνj = χ fuj dξ = χ fu(l) dξ + χ fuj dξ − χ 6 fu(l) dξ j,k j,k R R R R R = χ 6 fu(l) dξ + uj , uj , uj , uj , uj , (χ 6 )ˇ ∗ uj R j,k (l) (l) (l) (l) (l) (l) − uj,k , uj,k , uj,k , uj,k , uj,k , (χ 6 )ˇ ∗ uj,k ≤ χ 6 fu(l) dξ R

j,k

158

M. Kunze

5 (l) (l) +C max uj L2 , (χ 6 )ˇ ∗ uj L2 , uj,k 2 , (χ 6 )ˇ ∗ uj,k 2 L L 6ˇ (l) (l) 6ˇ × max uj − uj,k 2 , (χ ) ∗ uj − (χ ) ∗ uj,k 2 L L

1/2 (l) (l) 6 ≤ ψ χ uˆ j,k + χ fu(l) dξ − ψ χ uˆ j,k + Cεk , j, k ∈ N, R

j,k

where here and in the sequel C > 0 may depend on χ, but not on k or j . Next we can 1/2 (l) (l) invoke Lemma 2.5 with u = uj,k satisfying uˆ = uˆ j,k ∈ S(R) ⊂ L1 ∩ L2 and δ = εk to obtain from (2.20), Strichartz’ inequality, and (4.11), χ 6 dνj R

16 2 1/2 −1/6 (l) (l) 6 (l) (l) ≤ ψ χ uˆ j,k + C εk χ 5L∞ χ L∞ uj,k 2 + εk χ 6L∞ uj,k 32 uˆ j,k 3 1 L

L

1/2 + Cεk

L

2

1/2 −1/6 (l) (l) ≤ ψ χ uˆ j,k − ψ(χ uˆ j ) + ϕ χˇ ∗ uj + Cεk + Cεk uˆ j,k 3 1 L 5 (l) (l) 6 ≤ C max χ uˆ j,k 2 , χ uˆ j L2 χ uˆ j,k − χ uˆ j 2 + S 6 χˇ ∗ uj L2 L

1/2 + Cεk

≤S

6

R

= S6

L

2

−1/6 (l) + Cεk uˆ j,k 3 1 L 3 1/2 χ 2 |uˆ j |2 dξ + Cεk

R

3

1/2

+ Cεk

χ 2 dµj

−1/6

+ Cεk

−1/6

+ Cεk

(l)

2

uˆ j,k 3 1 L

2

(l)

uˆ j,k 3 1 .

(4.13)

L

Here C = C(χ), and this estimate holds for all j, k ∈ N. We claim that for fixed k ∈ N, (l)

uˆ j,k

L1

→0

as j → ∞.

(4.14)

We remark that in order to verify (4.14) it is sufficient to prove that there exists a sub(l) sequence (j ) ⊂ (j ) such that limj →∞ uˆ j ,k 1 = 0, since the following argument L

(l)

can also be used if we start with a subsequence of the original sequence (uˆ j,k ) . To j ∈N ∞ finally invoke (4.7), we apply Lemma 2.6 to φ = φk ∈ C0 (R). This yields, cf. (4.9), x +A (l) (l) limj →∞ supx0 ∈R x00−A |uj,k |2 dx = 0 for every A > 0. Hence in particular uj,k → 0 as j → ∞ in L2loc . Then a diagonal argument implies that there is a subsequence (l) (l) (j ) ⊂ (j ) such that uj ,k (x) → 0 as j → ∞ for a.e. x ∈ R. Recalling uj ,k ∈ S(R) (l)

(l)

and (4.10), it follows that |uj ,k (x) − uj ,k (y)| ≤ CRk |x − y|1/2 . Thus we must in fact

have uj ,k (x) → 0 as j → ∞ for every x ∈ R. Therefore uˆ j ,k ≥ 0 leads to (l)

(l)

(l)

uˆ j ,k

L1

=

R

(l)

(l)

uˆ j ,k (ξ ) dξ = (2π)1/2 uj ,k (0) → 0

as

j → ∞.

This completes the proof of the claim (4.14). Then going back to (4.13) we fix k ∈ N and take the limit j → ∞. Since νj ∗ ν and µj ∗ µ as j → ∞ in the sense of

On the Existence of a Maximizer for the Strichartz Inequality

159

measures, cf. Lemma 4.2, it follows from (4.14) that

3 1/2 |χ |6 dν ≤ S 6 |χ |2 dµ + Cεk . R

R

As this estimate holds for every k ∈ N, we can pass to the limit k → ∞ to finish the proof of Lemma 4.3. 4.2.2. The case that the sequence (uj ) is splitting. In this subsection we suppose that (3) in Lemma 3.1 is satisfied for (uj ), and again our goal is to show that this is impossible. Since (4.5) holds, we can once more use the decomposition of the uj in low and high frequencies, recall (4.8)–(4.12). Furthermore, we have γ ∈]0, 1[, where γ = x +x limx→∞ (x). Here (x) = limj →∞ j (x) = limj →∞ supx0 ∈R x00−x |uj |2 dy is the pointwise (outside a countable set) limit of the concentration functions corresponding to (uj ). We now fix δ ∈]0, γ [ and choose j0 ∈ N, x1∗ = z1∗ > 0, x2∗ = z2∗ > 0, xj = zj for j ≥ j0 , and moreover the functions ρ and η as described in (3) of Lemma 3.1; once again, all these quantities depend on δ. Defining vj (x) = ρ(x − xj )uj (x) and wj (x) = η(x − xj )uj (x), we recall that then vj L2 ≤ 1, wj L2 ≤ 1, and also

uj − (vj + wj )2L2 ≤ 2δ, vj 2L2 − γ ≤ 3δ, and wj 2L2 − (1 − γ ) ≤ 9δ (4.15) holds for j ≥ j0 . Next we transfer these estimates for every k ∈ N to the functions (l) obtained in an analogous way from the low-frequency parts uj,k of uj , cf. (4.9). To this end, we introduce (l)

(l)

vj,k (x) = ρ(x − xj )uj,k (x) and wj,k (x) = η(x − xj )uj,k (x).

(4.16)

Since ρ and η attain their values in [0, 1], it follows from (4.12) that (l)

vj,k L2 ≤ uj,k

L2

≤1

(l)

and wj,k L2 ≤ uj,k

L2

≤ 1.

(4.17)

Moreover, vj,k ∈ H 1 and wj,k ∈ H 1 . Due to ρ L∞ ∼ δ ≤ 1 and η L∞ ∼ δ ≤ 1, we obtain, using (4.10), the bounds vj,k H 1 + wj,k H 1 ≤ CRk .

(4.18)

The estimates from (4.15) are modified to

1/2 1/2 uj − (vj,k + wj,k )L2 ≤ 2(δ 1/2 + εk ), vj,k 2L2 − γ ≤ 3(δ + εk ), (4.19)

1/2 (4.20) and wj,k 2L2 − (1 − γ ) ≤ 9(δ + εk ) for j ≥ j0 and k ∈ N. Indeed, since ρ(x) ∈ [0, 1], (4.8) and (4.11) imply (l) vj − vj,k 2L2 = ρ(x − xj )2 |uj (x) − uj,k (x)|2 dx R

(l) 2 L2

≤ uj − uj,k

(h) 2 L2

= uj,k

≤ εk ,

160

M. Kunze

and in the same way wj − wj,k 2L2 ≤ εk follows, whence (4.19) and (4.20) are obtained. The following argument is similar to that given in Sect. 4.1. By (2.4) in Lemma 2.1(b), (4.17), the first estimate from (4.19), and Corollary 2.3(a), |ϕ(uj ) − ϕ(vj,k ) − ϕ(wj,k )| ≤ |ϕ(uj ) − ϕ(vj,k + wj,k )| + |ϕ(vj,k + wj,k ) − ϕ(vj,k ) − ϕ(wj,k )| 5 ≤ C max{uj L2 , vj,k + wj,k L2 } uj − (vj,k + wj,k )L2 + |ϕ(vj,k + wj,k ) − ϕ(vj,k ) − ϕ(wj,k )| 1/2

≤ C(δ 1/2 + εk ) + |ϕ(v˜j,k + w˜ j,k ) − ϕ(v˜j,k ) − ϕ(w˜ j,k )|,

(4.21)

(l)

where v˜j,k (x) := vj,k (x + xj ) = ρ(x)uj,k (x + xj ) and w˜ j,k (x) := wj,k (x + xj ) = (l)

η(x)uj,k (x + xj ), cf. (4.16). We recall from Lemma 3.1 that supp(ρ) ⊂ {x ∈ R : |x| ≤ x1∗ + 2δ −1 = z1∗ + 2δ −1 } and supp(η) ⊂ {x ∈ R : |x| ≥ x2∗ − 2δ −1 = z2∗ − 2δ −1 }. Since v˜j,k L2 = vj,k L2 ≤ 1 and w˜ j,k L2 = wj,k L2 ≤ 1 by (4.17), (2.6) from Lemma 2.1(d) can thus be applied with a = z1∗ + 2δ −1 and b = z2∗ − 2δ −1 . Using (4.18) we obtain |ϕ(v˜j,k + w˜ j,k ) − ϕ(v˜j,k ) − ϕ(w˜ j,k )|

≤ C v˜j,k H 1 + w˜ j,k H 1 (1 + a)1/2 (b − a)−1/12 1/6 1/6 = C vj,k H 1 + wj,k H 1 (1 + a)1/2 (b − a)−1/12 1/6

1/6

≤ CRk (1 + a)1/2 (b − a)−1/12 . 1/6

(4.22)

Lemma 4.4. With a = z1∗ + 2δ −1 and b = z2∗ − 2δ −1 the estimate (1 + a)1/2 (b − a)−1/12 ≤ 21/2 δ 1/2 holds. Proof. By choice of z2∗ , see (3.1), z2∗ ≥ z1∗ + 4δ −1 + δ −6 (z1∗ + 2δ −1 )6 which is equivalent to (z1∗ + 2δ −1 )(z2∗ − z1∗ − 4δ −1 )−1/6 ≤ δ. From δ < γ < 1 ≤ 2 and z1∗ > 0 1/2 we get 1 ≤ z1∗ + 2δ −1 , hence (1 + a)1/2 (b − a)−1/12 ≤ 21/2 (z1∗ + 2δ −1 ) (z2∗ − z1∗ − 4δ −1 )−1/12 ≤ 21/2 δ 1/2 . From (4.21), (4.22), and Lemma 4.4, 1/2

1/6

|ϕ(uj ) − ϕ(vj,k ) − ϕ(wj,k )| ≤ C(δ 1/2 + εk ) + CRk δ 1/2 . This estimate holds for all k ∈ N, δ ∈]0, γ [, and j ≥ j0 (δ), with however vj,k and wj,k depending on δ, j , and k. Thus Strichartz’ inequality, the second bound from (4.19), and (4.20) yield 1/2

1/6

1/2

1/6

ϕ(uj ) ≤ C(δ 1/2 + εk ) + CRk δ 1/2 + ϕ(vj,k ) + ϕ(wj,k ) ≤ C(δ 1/2 + εk ) + CRk δ 1/2 + S 6 vj,k 6L2 + S 6 wj,k 6L2 3 1/2 1/6 1/2 ≤ C(δ 1/2 + εk ) + CRk δ 1/2 + S 6 3(δ + εk ) + γ 3 1/2 +S 6 9(δ + εk ) + (1 − γ ) .

On the Existence of a Maximizer for the Strichartz Inequality

161

Since we have got rid of the functions vj,k and wj,k , we may take the limit j → ∞ to find 3 1/2 1/6 1/2 S 6 ≤ C(δ 1/2 + εk ) + CRk δ 1/2 + S 6 3(δ + εk ) + γ 3 1/2 +S 6 9(δ + εk ) + (1 − γ ) . This estimate is satisfied for all δ ∈]0, γ [ and all k ∈ N. Hence we can pass successively to the limits first δ → 0 and then k → ∞ to arrive at S 6 ≤ S 6 γ 3 + S 6 (1 − γ )3 , which is a contradiction to γ ∈]0, 1[. 4.2.3. The case that the sequence (uj ) is tight. Summarizing the results of the preceding sections, we have constructed a maximizing sequence (uj ) which satisfies (4.5) and then applied the concentration compactness principle to (uj ). Since the alternatives (2) and (3) do not occur by Sects. 4.2.1 and 4.2.2, it follows that (1) holds, i.e., there exists a sequence (xj ) ⊂ R such that for every δ > 0 there is M = Mδ > 0 with xj +M 2 ˜ j (x) = uj (x + xj ) and claim that xj −M |uj | dx ≥ 1 − δ for all j ∈ N. Finally we let u (u˜ j ) is a maximizing sequence which (along a subsequence) strongly converges in L2 . Indeed, since u˜ j L2 = uj L2 = 1 and, by Corollary 2.3(a), ϕ(u˜ j ) = ϕ(uj ) → S 6 as j → ∞, (u˜ j ) is a maximizing sequence. It follows from uˆ˜ j (ξ ) = eixj ξ uˆ j (ξ ) that (4.5) is satisfied for (u˜ j ), too. In addition, ∀δ > 0

∃ M = Mδ > 0 :

M

−M

|u˜ j |2 dx ≥ 1 − δ,

j ∈ N.

(4.23)

Passing to a subsequence (which is not relabelled) we can suppose that u˜ j u∗ in L2 as j → ∞ for some u∗ ∈ L2 with u∗ L2 ≤ 1. Again we fix a sequence εk → 0, (l) (h) choose Rk according to (4.5), and define u˜ j,k = (φk uˆ˜ j )ˇ and u˜ j,k = ((1 − φk )uˆ˜ j )ˇ. ∞ Here φk (ξ ) = φ(ξ/Rk ), with φ ∈ C0 (R) taking values in [0, 1] such that φ(ξ ) = 1 for |ξ | ≤ 1 and φ(ξ ) = 0 for |ξ | ≥ 2. Then again (l) u˜ j − u˜ j,k 2 L

(h) u˜ j,k 2 L

1/2 εk ,

(l) u˜ j,k 2 L

(l)

d u˜ j,k dx L2

(l)

= u˜ j,k

H˙ 1

≤ CRk ,

= ≤ and ≤ 1 are satisfied, see (4.10), (4.11), and (4.12). Now we fix k ∈ N. Then there exists a subsequence (j ) ⊂ (j ), v˜k ∈ H 1 , and (l) (h) w˜ k ∈ L2 such that u˜ j ,k v˜k in H 1 and u˜ j ,k w˜ k in L2 as j → ∞. In particular, (h)

w˜ k L2 ≤ lim inf u˜ j ,k j →∞

(l)

L2

1/2

≤ εk ,

(4.24)

(h)

and also u∗ = v˜k + w˜ k due to u˜ j = u˜ j,k + u˜ j,k . Next we fix δ > 0 and choose M = Mδ according to (4.23). By compactness of the embedding H 1 ⊂ L2 (] − M, M[), (l) u˜ j ,k → v˜k strongly in L2 (] − M, M[) as j → ∞. Therefore, using (4.24), the above bounds, and (4.23), u∗ L2 ≥ u∗ L2 (]−M,M[) = v˜k + w˜ k L2 (]−M,M[) ≥ v˜k L2 (]−M,M[) − w˜ k L2 (l)

≥ lim u˜ j ,k j →∞

L2 (]−M,M[)

1/2

− εk

(h)

= lim u˜ j − u˜ j ,k j →∞

L2 (]−M,M[)

1/2

− εk

162

M. Kunze

1/2 (h) ≥ lim sup u˜ j L2 (]−M,M[) − u˜ j ,k 2 − εk L

j →∞

1/2

≥ lim sup u˜ j L2 (]−M,M[) − 2εk j →∞

1/2

≥ (1 − δ)1/2 − 2εk . 1/2

This estimate holds for every δ > 0, whence u∗ L2 ≥ 1 − 2εk for every k ∈ N. Passing to the limit k → ∞ thus u∗ L2 = 1 = limj →∞ u˜ j L2 . Since also u˜ j u∗ in L2 , we obtain u˜ j → u∗ strongly in L2 . By continuity of ϕ : L2 → R, cf. (2.4), this finally yields ϕ(u∗ ) = limj →∞ ϕ(u˜ j ) = S 6 , i.e., u∗ is a maximizing function and the proof of Theorem 1.1 is completed. Acknowledgement. I am grateful to V. Zharnitsky for ongoing discussions. I am also indebted to the referee of [8] who suggested to study the problem in the present paper.

References 1. Bourgain, J.: Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity. Internat. Math. Res. Notices 5, 253–283 (1998) 2. Carlen, E.A., Loss, M.: Extremals of functionals with competing symmetries. J. Funct. Anal. 88, 437–456 (1990) 3. Cazenave, Th.: An Introduction to Nonlinear Schr¨odinger Equations. 3rd edition, Instituto de Mathematica – UFJR, Rio de Janeiro, RJ 1996; available at http://www.ann.jussieu.fr/∼ cazenave/List Art Tele.html 4. Colliander, J.E., Delort, J.-M., Kenig, C.E., Staffilani, G.: Bilinear estimates and applications to 2D NLS. Trans. Am. Math. Soc. 353, 3307–3325 (2001) 5. Evans, L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations. Providence, RI: American Mathematical Society, 1990 6. Gr¨unrock, A.: Some local wellposedness results for nonlinear Schr¨odinger equations below L2 . Preprint arXiv:math.AP/0011157 v2, 2001 7. Kenig, C.E., Ponce, G., Vega, L.: Quadratic forms for the 1-D semilinear Schr¨odinger equation. Trans. Am. Math. Soc. 348, 3323–3353 (1996) 8. Kunze, M.: A variational problem with lack of compactness related to the Strichartz inequality. To appear in Calc. Var. Partial Differential Equations 9. Lieb, E.H.: Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. Math. 118, 349–374 (1983) 10. Lions, P.-L.: The concentration compactness principle in the calculus of variations. The locally compact case. Ann. Inst. H. Poincar´e Anal. Non Lineair´e 1, I.: 109–145 (1984) and II.: 223–283 (1984) 11. Lions, P.-L.: The concentration compactness principle in the calculus of variations. The limit case, Rev. Mat. Iberoamericana 1, I.:(1), 145–201 (1985) and II.:(2), 45–121 (1985) 12. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton: Princeton University Press, 1993 13. Struwe, M.: Variational Methods. 2nd edition, Berlin-New York: Springer, 1996 Communicated by B. Simon

Commun. Math. Phys. 243, 163–191 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0960-z

Communications in

Mathematical Physics

Universality for Eigenvalue Correlations at the Origin of the Spectrum A.B.J. Kuijlaars , M. Vanlessen Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium. E-mail: [email protected]; [email protected] Received: 25 April 2003 / Accepted: 24 June 2003 Published online: 14 October 2003 – © Springer-Verlag 2003

Abstract: We establish universality of local eigenvalue correlations in unitary random matrix ensembles Z1n | det M|2α e−ntr V (M) dM near the origin of the spectrum. If V is even, and if the recurrence coefficients of the orthogonal polynomials associated with |x|2α e−nV (x) have a regular limiting behavior, then it is known from work of Akemann et al., and Kanzieper and Freilikher that the local eigenvalue correlations have universal behavior described in terms of Bessel functions. We extend this to a much wider class of confining potentials V . Our approach is based on the steepest descent method of Deift and Zhou for the asymptotic analysis of Riemann-Hilbert problems. This method was used by Deift et al. to establish universality in the bulk of the spectrum. A main part of the present work is devoted to the analysis of a local Riemann-Hilbert problem near the origin. 1. Introduction In the present paper we consider the following unitary ensemble of random matrices, cf. [2, 3] 1 | det M|2α e−ntr V (M) dM, Zn

α > −1/2.

(1.1)

The matrices M are n × n Hermitian and dM is the associated flat Lebesgue measure on the space of n × n Hermitian matrices, and Zn is a normalizing constant (partition function). The confining potential V in (1.1) is a real valued function with enough increase at infinity, for example a polynomial of even degree with positive leading coefficient. Random matrix ensembles are important in many branches of mathematics and physics, Supported by FWO research project G.0176.02 and by INTAS project 00-272 and by the Ministry of Science and Technology (MCYT) of Spain, project code BFM2001-3878-C02-02 Research Assistant of the Fund for Scientific Research – Flanders (Belgium)

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see the recent survey paper [19]. The specific ensemble (1.1) is relevant in three-dimensional quantum chromodynamics [40]. The ensemble (1.1) induces a probability density function on the n eigenvalues x1 , . . . , xn of M, given by P (n) (x1 , . . . , xn ) =

n 1 wn (xj ) |xi − xj |2 , Zˆ n j =1

i<j

where Zˆ n is the normalizing constant, and where wn is the following varying weight on the real line, wn (x) = |x|2α e−nV (x) ,

for x ∈ R.

(1.2)

Of particular interest are the local correlations between the eigenvalues of the ensemble (1.1) of random matrices, when their size tends to infinity, at the origin of the spectrum, see [2, 22, 32]. The correlations between eigenvalues can be expressed in terms of the orthonormal polynomials pk,n (x) = γk,n x k + · · · with γk,n > 0 with respect to wn , that is pk,n (x)pj,n (x)|x|2α e−nV (x) dx = δj k . Namely, for 1 ≤ m ≤ n − 1, the m-point correlation function Rn,m (y1 , . . . , ym ) ∞ ∞ n! ... P (n) (y1 , . . . , ym , xm+1 , . . . , xn )dxm+1 . . . dxn , = (n − m)! −∞ −∞ n−m

satisfies, by a well-known computation of Gaudin and Mehta [29], Rn,m (y1 , . . . , ym ) = det(Kn (yi , yj ))1≤i,j ≤m , where Kn (x, y) =

n−1 wn (x) wn (y) pj,n (x)pj,n (y) j =0

=

γn−1,n pn,n (x)pn−1,n (y) − pn−1,n (x)pn,n (y) , (1.3) wn (x) wn (y) γn,n x−y

which gives the connection with orthogonal polynomials. The second equality in (1.3) follows from the Christoffel-Darboux formula [36]. Akemann et al. [2] showed that the local eigenvalue correlations at the origin of the spectrum have a universal behavior, described in terms of the following Bessel kernel: √ √ Jα+ 21 (π u)Jα− 21 (π v) − Jα− 21 (π u)Jα+ 21 (π v) Jαo (u, v) = π u v , 2(u − v)

(1.4)

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where Jα± 1 denotes the usual Bessel function of order α ± 21 . In [2] it was assumed 2 that the parameter α is a non-negative integer, that the potential V is even, and that the coefficients ck,n in the recurrence relation xpk,n (x) = ck+1,n pk+1,n + ck,n pk−1,n satisfied by the orthonormal polynomials have a limiting behavior in the sense that the limit ck,n exists whenever k, n → ∞ such that k/n → t for some t > 0. The restriction that α is a non-negative integer was removed by Kanzieper and Freilikher [22], but they still required the assumption that V is even and that the recurrence coefficients have a limiting behavior. In fact, their method of proof (which they call Shohat’s method) relies heavily on these recurrence coefficients. It is the goal of this paper to establish the universality of the Bessel kernel (1.4) at the origin of the spectrum without any assumption on the recurrence coefficients. We can also allow V to be quite arbitrary. We assume the following: V : R → R is real analytic, V (x) lim = +∞, |x|→∞ log(x 2 + 1) ψ(0) > 0,

(1.5) (1.6) (1.7)

where ψ is the density of the equilibrium measure in the presence of the external field V , [12, 34]. Let us explain the condition (1.7). Denote the space of all probability measures on R by M1 (R), and consider the following minimization problem: inf

µ∈M1 (R)

log

1 dµ(s)dµ(t) + |s − t|

V (t)dµ(t) .

(1.8)

Under the assumptions (1.5) and (1.6) it is known that the infimum is achieved [10, 34] uniquely at the equilibrium measure µV ∈ M1 (R) for V . The measure µV has compact support, and since V is real analytic, it is supported on a finite union of intervals. In addition it is absolutely continuous with respect to the Lebesgue measure, i.e. dµV (x) = ψ(x)dx, and ψ is real analytic on the interior of the support of µV , see [12, 13]. The importance of the equilibrium measure lies in the fact that ψ is the limiting (as n → ∞) mean eigenvalue density of the matrix ensemble (1.1), cf. [10, 13]. The condition (1.7) then says that the mean eigenvalue density ψ should be strictly positive there. If the origin belongs to the interior of the support of µV but the mean eigenvalue density vanishes there, then the potential is called multicritical, see [3, 4, 21]. This case will not be treated in this paper. The regular behavior of the recurrence coefficients assumed in [2, 22] is probably satisfied if V is even and if the support of µV consists of one single interval. Note that we make no assumptions on the nature of the support of µV . It can consist of any (finite) number of intervals.

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Our main result is the following. Theorem 1.1. Assume that the conditions (1.5)–(1.7) are satisfied. Let wn be the varying weight (1.2), let Kn be the kernel (1.3) associated with wn , and let ψ be the density of the equilibrium measure for V . Then, for u, v ∈ (0, ∞),

α α

u v u v 1 Kn , = Jαo (u, v) + O , as n → ∞, (1.9) nψ(0) nψ(0) nψ(0) n where Jαo is the Bessel kernel given by (1.4). The error term in (1.9) is uniform for u, v in bounded subsets of (0, ∞). Other types of universal correlations have been established in the bulk [9, 13, 22, 33], at the soft edge of the spectrum [8, 18, 22, 30, 37], and at the hard edge [18, 28, 31, 38]. The universality at the hard edge is also described in terms of a Bessel kernel, which we have denoted in [28] by Jα , namely √ √ √ √ √ √ Jα ( u) vJα ( v) − Jα ( v) uJα ( u) Jα (u, v) = . 2(u − v) To distinguish this Bessel kernel, we use Jαo to denote the Bessel kernel (1.4) relevant at the origin of the spectrum. Remark 1.2. The universality (1.9) is restricted to u, v > 0. It can be extended to arbitrary real u and v in the following way. For u, v ∈ R, we have that

1 u 1 v Kn , = u−α v −α Jαo (u, v) + O , |u|−α |v|−α nψ(0) nψ(0) nψ(0) n as n → ∞, (1.10) and the error term holds uniformly for u, v in compact subsets of R. We will restrict ourselves to proving (1.9), but the same methods allow us to establish (1.10). Our proof of Theorem 1.1 is based on the characterization of the orthogonal polynomials via a Riemann-Hilbert problem (RH problem) for 2 × 2 matrix valued functions, due to Fokas, Its and Kitaev [17], and on an application of the steepest descent method of Deift and Zhou [15]. See [10, 24] for an introduction. The Riemann-Hilbert approach gives asymptotics for the orthogonal polynomials in all regions of the complex plane, and it has been applied before on orthogonal polynomials by a number of authors, see for example [5, 13, 14, 23, 26, 27, 39]. Bleher and Its [6] and Deift et al. [13] were the first to apply Riemann-Hilbert problems to universality results in random matrix theory. Later developments include [5, 7, 35, 28]. In this paper we use many of the ideas of [13]. That paper deals with the varying weights e−nV (x) with V satisfying (1.5) and (1.6). The steepest descent method for Riemann-Hilbert problems is used to establish universality of the sine kernel in the bulk of the spectrum for the associated unitary matrix ensembles. In our case the general scheme of the analysis is the same, and we refer to [13, 14] for some of the details and motivations. The extra factor |x|2α in our weights |x|2α e−nV (x) gives rise to two important technical differences. The first difference lies in the construction of the socalled parametrix for the outside region. To compensate for the factor |x|2α we need to construct a Szeg˝o function on multiple intervals associated to |x|2α . The second and most important difference lies in the fact that we have to do a local analysis near the

Universality for Eigenvalue Correlations at the Origin of the Spectrum

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origin. This is where the Bessel functions Jα± 1 come in. The construction of the local 2 parametrix near the origin is analogous to the construction of the parametrix near the algebraic singularities of the generalized Jacobi weight, recently done by one of us in [39]. The local parametrix determines the asymptotics of the orthonormal polynomials near the origin, and thus also governs the universality at the origin of the spectrum. The rest of the paper is organized as follows. In Sect. 2.1 we characterize the orthogonal polynomials via a RH problem, due to Fokas, Its and Kitaev [17]. Via a series of transformations, we perform the asymptotic analysis of the RH problem as in [13, 14]. The first transformation will be done in Sect. 2.2, the second transformation in Sect. 3. Next, we construct the parametrices for the outside region and near the origin in Sect. 4 and 5, respectively. The final transformation will be done in Sect. 6. Then we have all the ingredients to prove Theorem 1.1 in Sect. 7. Here we use some techniques from [28]. 2. Associated RH Problem and First Transformation Y → T In this section we will characterize the orthonormal polynomials pk,n with respect to the weight (1.2) as a solution of a RH problem for a 2 × 2 matrix valued function Y (z) = Y (z; n, w), due to Fokas, Its and Kitaev [17], and do the first transformation in the asymptotic analysis of this RH problem. 2.1. Associated RH problem. We seek a 2 × 2 matrix valued function Y that satisfies the following RH problem. RH problem for Y . (a) Y : C \ R → C2×2 is analytic. (b) Y possesses continuous boundary values for x ∈ R \ {0} 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 |x|2α e−nV (x) , for x ∈ R \ {0}. (2.1) Y+ (x) = Y− (x) 0 1 (c) Y (z) has the following asymptotic behavior at infinity:

n

1 z 0 Y (z) = I + O , as z → ∞. 0 z−n z (d) Y (z) has the following behavior near z = 0:

 1 |z|2α    O 1 |z|2α , if α < 0,

Y (z) =    O 1 1 , if α > 0, 11

(2.2)

(2.3)

as z → 0, z ∈ C \ R. Compared with the case of no singularity at the origin, see [13], we now have an extra condition (2.3) near the origin. This condition is used to control the behavior near the origin, see also [27, 39].

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Remark 2.1. The O-terms in (2.3) are to be taken entrywise. So for example Y (z) = 1 |z|2α O means that Y11 (z) = O(1), Y12 (z) = O(|z|2α ), etc. 1 |z|2α The unique solution of the RH problem is given by 

1 1 2πi γn,n

1 γn,n pn,n (z)

Y (z) =  −2πiγn−1,n pn−1,n (z) −γn−1,n

pn,n (x)wn (x) dx x−z pn−1,n (x)wn (x) dx x−z

 ,

(2.4)

where pk,n is the k th degree orthonormal polynomial with respect to the varying weight wn , and where γk,n is the leading coefficient of the orthonormal polynomial pk,n . The solution (2.4) is due to Fokas, Its and Kitaev [17], see also [10, 13, 14]. See [24, 27] for the condition (2.3). Note that (2.4) contains the orthonormal polynomials of degrees n − 1 and n. By (1.3) it is then possible to write Kn in terms of the first column of Y . So in order to prove Theorem 1.1 an asymptotic analysis of the RH problem for Y is necessary. Via a series of transformations Y → T → S → R we want to obtain 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, as n → ∞. Then R is also uniformly close to the identity matrix, as n → ∞. Unfolding the series of transformations, we obtain the asymptotics of Y . In particular, we need the asymptotic behavior of Y near the origin, which follows from the parametrix near the origin. 2.2. First transformation Y → T . We first need some properties of the equilibrium mea sure µV for V . Its support is a finite union of disjoint intervals, say N+1 j =1 [bj −1 , aj ]. So the support consists of N + 1 intervals and we refer to these as the bands. The complementary N intervals (aj , bj ) are the gaps. Following [13], we define J =

N+1

(bj −1 , aj )

j =1

so that J is the interior of the support. The density ψ of µV has the form [13] ψ(x) =

1 1/2 R (x)h(x), 2πi +

for x ∈ J ,

(2.5)

where R(z) =

N+1

(z − bj −1 )(z − aj ),

(2.6)

j =1

√ and where h is real analytic on R. In this paper we use R 1/2 to denote the branch of R which behaves like zN+1 as z → ∞ and which is defined and analytic on C \ J¯. In (2.5) 1/2 we have that R+ denotes the boundary value of R 1/2 on J from above. There exists an explicit expression for h in terms of V , see [12], but we will not need that here.

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The equilibrium measure minimizes the weighted energy (1.8). The associated Euler-Lagrange variational conditions state that there exists a constant ∈ R such that 2 log |x − s|ψ(s)ds − V (x) = , for x ∈ J¯, (2.7) 2

log |x − s|ψ(s)ds − V (x) ≤ ,

for x ∈ R \ J¯.

(2.8)

The external field V is called regular if the inequality in (2.8) is strict for every x ∈ R\ J¯, and if h(x) = 0 for every x ∈ J¯. Otherwise, V is called singular. The regular case holds generically [25]. In the singular case there are a finite number of singular points. Singular points in J¯ are such that h vanishes there. Singular points in R \ J¯ are such that equality holds in (2.8). In order to do the first transformation, we introduce the so-called g-function [13, Sect. 3.2] g(z) = log(z − s)ψ(s)ds, for z ∈ C \ (−∞, aN+1 ], (2.9) where ψ(s)ds is the equilibrium measure for V . In (2.9) we take the principal branch of the logarithm, so that g is analytic on C \ (−∞, aN+1 ]. We now give properties of g which are crucial in the following, [13, Sect. 3.2]. From the Euler-Lagrange conditions (2.7) and (2.8) it follows that g+ (x) + g− (x) − V (x) − = 0,

for x ∈ J¯,

(2.10)

g+ (x) + g− (x) − V (x) − ≤ 0,

for x ∈ R \ J¯.

(2.11)

for x ∈ (−∞, aN+1 ),

(2.12)

A second crucial property is that aN +1 g+ (x) − g− (x) = 2πi dµV (s), x

so that g+ (x) − g− (x) is purely imaginary for all x ∈ R and constant in each of the gaps, namely  for x < b0 ,   2πi, aN +1 g+ (x) − g− (x) = 2πi bj dµV (s) =: 2πij , for x ∈ (aj , bj ), j = 1 . . . N,   0, for x > aN+1 . (2.13) From (2.13) we see that j is the total µV -mass of the N + 1 − j largest bands. These constants all belong to (0, 1). Note that j was defined with an extra factor 2π in [13]. As in [13, Sect. 3.3], we define the matrix valued function T as T (z) = e− 2 σ3 Y (z)e 2 σ3 e−ng(z)σ3 , for z ∈ C \ R, (2.14) 1 0 where σ3 = 0 −1 is the Pauli matrix. Then T is the unique solution of the following equivalent RH problem. n

n

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RH problem for T . (a) T : C \ R → C2×2 is analytic. (b) T satisfies the following jump relations on R:

−n(g (x)−g (x)) + − e |x|2α T+ (x) = T− (x) , 0 en(g+ (x)−g− (x))

for x ∈ J¯ \ {0}, (2.15)

e−2πinj |x|2α en(g+ (x)+g− (x)−V (x)−) T+ (x) = T− (x) , 0 e2πinj for x ∈ (aj , bj ), j = 1 . . . , N,

1 |x|2α en(g+ (x)+g− (x)−V (x)−) , 0 1

(2.16)

T+ (x) = T− (x)

for x < b0 or x > aN+1 . (2.17)

(c) T (z) = I + O(1/z), as z → ∞. (d) T (z) has the same behavior as Y (z) as z → 0, given by (2.3). 3. Second Transformation T → S In this section we transform the oscillatory diagonal entries of the jump matrix in (2.15) into exponentially decaying off-diagonal entries. This lies at the heart of the steepest descent method for RH problems of Deift and Zhou [15], and this step is often referred to as the opening of the lens. For every z ∈ C \ R lying in the region of analyticity of h, we define 1 aN +1 1/2 φ(z) = R (s)h(s)ds, 2 z b where the path of integration does not cross the real axis. Since akk R 1/2 (s)h(s)ds = 0 for every k = 1, . . . , N, (this follows easily from the formulas in [13, Sect. 3.1 and a 1/2 3.2]), and bjN +1 R+ (s)h(s)ds = 2πij , we find that for every j , 2φ(z) =

aj

R 1/2 (s)h(s)ds + 2πij ,

if Im z > 0,

(3.1)

R 1/2 (s)h(s)ds − 2πij ,

if Im z < 0.

(3.2)

z

2φ(z) =

aj

z

Note that in [13] a function G is defined which is analytic through the bands. We found it more convenient to have a function with branch cuts along the bands, see also [10]. The functions G and φ also differ by a factor ±2. The point of the function φ is that φ+ and φ− are purely imaginary on the bands, and that 2φ+ = −2φ− = g+ − g− .

(3.3)

Universality for Eigenvalue Correlations at the Origin of the Spectrum

171

0

bj-1

aj

bj

aj+1

bj+1

aj+2

Fig. 1. Part of the contour

This means that 2φ and −2φ provide analytic extensions of g+ − g− into the upper half-plane and lower half-plane, respectively. We also have that for z in a neighborhood of a regular point x ∈ J , (see [13, Sect. 3.3] for details) that Re φ(z) > 0,

if Im z = 0.

(3.4)

We will now discuss the opening of the lens in the regular case. In the singular case we need to modify the opening of the lens somewhat, since we have to take into account the singular points. We do not open the lens around singular points that belong to J , see [13, Sect. 4] for details. For V regular, there is a suitable neighborhood U of J such that the inequality in (3.4) holds for every z ∈ U . The opening of the lens is based on the factorization of the jump matrix (2.15) into the following product of three matrices, see also (3.3), −n(g (x)−g (x))

−2nφ (x)

+ − + e e |x|2α |x|2α = 0 en(g+ (x)−g− (x)) 0 e−2nφ− (x)

1 0 1 0 0 |x|2α . (3.5) = |x|−2α e−2nφ− (x) 1 |x|−2α e−2nφ+ (x) 1 −|x|−2α 0 As in [39] we take an analytic continuation of the factor |x|2α by defining (−z)2α , if Re z < 0, ω(z) = 2α z , if Re z > 0,

(3.6)

with principal branches of powers. In contrast to the situation in [13], here we have to open the lens also going through the origin, cf. [39]. This follows from the fact that |x|2α does not have an analytic continuation to a full neighborhood of the origin. We thus transform the RH problem for T into a RH problem for S with jumps on the oriented contour , shown in Fig. 1. The precise form of the lens is not yet defined, but it will be contained in U . Define the piecewise analytic matrix valued function S as  T (z), for z outside the lens,     

  1 0 T (z) , for z in the upper parts of the lens, S(z) = (3.7) −ω(z)−1 e−2nφ(z) 1  

  1 0    T (z) ω(z)−1 e−2nφ(z) 1 , for z in the lower parts of the lens. Then, S is the unique solution of the following equivalent RH problem. In (3.8), C+ and C− are used to denote the upper half-plane {Im z > 0} and the lower half-plane {Im z < 0}, respectively.

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RH problem for S. (a) S : C \ → C2×2 is analytic. (b) S satisfies the following jump relations on :

1 0 , S+ (z) = S− (z) ω(z)−1 e−2nφ(z) 1

for z ∈ ∩ C± ,

(3.8)

0 |x|2α , S+ (x) = S− (x) −|x|−2α 0

for x ∈ J \ {0},

−2πin j |x|2α e n(g+ (x)+g− (x)−V (x)−) e , S+ (x) = S− (x) 0 e2πinj for x ∈ (aj , bj ), j = 1 . . . N,

1 |x|2α en(g+ (x)+g− (x)−V (x)−) , 0 1

(3.9)

(3.10)

S+ (x) = S− (x)

for x < b0 or x > aN+1 . (3.11)

(c) S(z) = I + O(1/z), as z → ∞. (d) For α < 0, the matrix function S(z) has the following behavior as z → 0:

1 |z|2α , as z → 0, z ∈ C \ . S(z) = O 1 |z|2α For α > 0, the matrix function S(z) has the following behavior as z → 0:

 11   O , as z → 0 from outside the lens,  11 −2α

S(z) =    O |z|−2α 1 , as z → 0 from inside the lens. |z| 1

(3.12)

(3.13)

(e) S remains bounded near each of the endpoints ai , bj . By (3.4) the factor e−2nφ(z) in (3.8) is exponentially decaying for z ∈ ∩ C± as n → ∞. This implies that the jump matrix for S converges exponentially fast to the identity matrix as n → ∞, on the lips of the lens. Since V is regular, we have the strict inequality g+ (x) + g− (x) − V (x) − < 0,

for x ∈ R \ J¯,

(3.14)

so that the factor en(g+ (x)+g− (x)−V (x)−) in (3.10) and (3.11) is also exponentially decaying as n → ∞. 4. Parametrix for the Outside Region From the discussion at the end of the previous section we expect that the leading order asymptotics are determined by the solution of the following RH problem.

Universality for Eigenvalue Correlations at the Origin of the Spectrum

173

RH problem for P (∞) . (a) P (∞) : C \ [b0 , aN+1 ] → C2×2 is analytic. (b) P (∞) satisfies the following jump relations:

0 |x|2α (∞) (∞) P+ (x) = P− (x) , −|x|−2α 0 (∞)

(∞)

P+ (x) = P− (x)

for x ∈ J \ {0},

−2πin j e 0 , 0 e2πinj

(4.1)

for x ∈ (aj , bj ), j = 1 . . . N. (4.2)

(c) P (∞) (z) = I + O (1/z), as z → ∞. The solution of this RH problem is referred to as the parametrix for the outside region, and will be constructed using the so-called Szeg˝o function on the union of disjoint intervals J , associated to |x|2α . The importance of the Szeg˝ is that it transforms o0 function 1 this RH problem into a RH problem with jump matrix −1 0 on J . 4.1. The Szeg˝o function. We seek a scalar function D : C \ [b0 , aN+1 ] → C that solves the following RH problem. RH problem for D. (a) D is non-zero and analytic on C \ [b0 , aN+1 ]. (b) D satisfies the following jump relations: D+ (x)D− (x) = |x|2α , D+ (x) = e

2πiξj

D− (x),

for x ∈ J \ {0}, for x ∈ (aj , bj ), j = 1, . . . N,

(4.3) (4.4)

for certain unknown constants ξ1 , . . . , ξN ∈ R. The selection of ξ1 , . . . , ξN is part of the problem. We should choose them such that it is possible to construct D. (c) D and D −1 remain bounded near the endpoints ai , bj of J , and D∞ := lim D(z) z→∞

(4.5)

exists and is non-zero. We seek D in the form D(z) = exp (z). Then the problem is reduced to constructing a scalar function , analytic on C \ [b0 , aN+1 ], remaining bounded near the endpoints ai , bj of J and at infinity, and having the following jumps: + (x) + − (x) = 2α log |x|, + (x) = − (x) + 2πiξj ,

for x ∈ J \ {0}, for x ∈ (aj , bj ), j = 1, . . . , N.

(4.6) (4.7)

We can easily check, using Cauchy’s formula, the Sokhotskii-Plemelj formula [20], 1/2 1/2 and the fact that R− (x) = −R+ (x) for x ∈ J , see (2.6), that defined by   bj N 2α log |x| dx 1 1 dx  + , (4.8) ξj (z) = R 1/2 (z)  1/2 1/2 (x) x − z 2πi J R+ x − z R aj (x) j =1

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satisfies the jump conditions (4.6) and (4.7). We note that is analytic on C \ [b0 , aN+1 ] and remains bounded near the endpoints ai , bj of J . We use the freedom we have in choosing the constants ξ1 , . . . , ξN to ensure that remains bounded at infinity. Since R 1/2 (z) behaves like zN+1 as z → ∞, and since

N−1 xk 1 1 , as z → ∞, + O =− x−z zk+1 zN+1 k=0

we have to choose ξ1 , . . . , ξN such that the N conditions bj N 2α log |x| k x k dx 1 = 0, dx + ξ x j 1/2 1/2 (x) 2π i J R+ aj R (x) j =1

k = 0, . . . , N − 1, (4.9)

are satisfied. Note that (4.9) represents a system of N linear equations with coefficient matrix  b2 dx bN dx  b1 dx a1 R 1/2 (x) a2 R 1/2 (x) · · · aN R 1/2 (x)     b  1 xdx b2 xdx · · · bN xdx  1/2  a1 R 1/2 (x) a2 R 1/2 (x) aN R (x)    (4.10) A= . .. .. .. ..   .   . . .     b1 x N −1 dx b2 x N −1 dx bN x N −1 dx · · · a1 R 1/2 (x) a2 R 1/2 (x) aN R 1/2 (x) By the multilinearity of the determinant, we have  1 1 ··· 1   x1 x 2 · · · xN bN b1   ... det  . det A = .. . . .  . a1 aN . ..  . .  =

b1

bN

... a1

aN

j
    dx dxN 1  · · · 1/2  1/2  R (x1 ) R (xN )  

N−1 x1N−1 x2N−1 · · · xN

(xk − xj )

dx1 dxN · · · 1/2 . R 1/2 (x1 ) R (xN )

(4.11)

Since the gaps (aj , bj ) are disjoint, we have xj < xk for j < k, so that j 0. Using the fact that R 1/2 does not change sign on each of the gaps (aj , bj ), it then follows that the integrand in (4.11) has a constant sign in the region of integration, so that det A = 0 and A is invertible. We then define ξ1 , . . . , ξN as follows:   2α log |x| 1 2πi J R 1/2 (x) dx +      1 2α log |x|  ξ1    2πi J R+1/2 (x) xdx   ξ2     .  = −A−1  (4.12) . ..  ..    .     ξN  2α log |x| N−1  1 x dx 1/2 2πi J R+ (x)

Universality for Eigenvalue Correlations at the Origin of the Spectrum

175

z

bj

d

aj+1

0

gd Fig. 2. The oriented contour γδ

Note that R 1/2 is real on each of the gaps, so that by (4.10) all entries of A are real. This 1/2 implies, from (4.12) and the fact that R+ is purely imaginary on J , that the constants ξ1 , . . . , ξN are real. We proved the following Theorem 4.1. The scalar function D(z) = exp (z), where is given by (4.8), and the constants ξ1 , . . . , ξN by (4.12), solves the RH problem for D. For later use we state the following lemma. Lemma 4.2. We have that z−α D(z) and zα D(z)−1 remain bounded near the origin. Proof. For definiteness, suppose that the origin lies on the band (bj , aj +1 ) with j ∈ {0, . . . , N}. Since D(z) = exp (z), with given by (4.8), it is sufficient to prove that aj +1 1 2α log |x| dx α log z = 1/2 + F (z), (4.13) 1/2 2πi bj x − z R (z) R+ (x) with F analytic near the origin. Fix z near the origin with Im z = 0, and let γδ be the oriented contour shown in Fig. 2, with δ > 0 small. Cauchy’s formula implies log ζ dζ log z 1 = 1/2 . 1/2 2πi γδ R (ζ ) ζ − z R (z) 1/2

1/2

Letting δ → 0, we then have, since R+ (x) = −R− (x) for x ∈ (bj , aj +1 ), 0 aj +1 1 log |x| + iπ dx log |x| dx 1 ˜ F (z) + + 1/2 1/2 2π i bj R+ (x) x − z 2πi 0 R+ (x) x − z −

1 2π i

0

aj +1

log |x|

1 dx − 1/2 R+ (x) x − z 2πi

0

bj

log |x| − iπ dx log z = 1/2 , 1/2 R (z) R+ (x) x − z

with F˜ analytic near the origin. Hence aj +1 1 1 log z 1 log |x| dx = − F˜ (z), 1/2 1/2 (z) 2πi bj x − z 2 R 2 R+ (x) so that (4.13) holds with F (z) = −α F˜ (z), which proves the lemma.

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A.B.J. Kuijlaars, M. Vanlessen

4.2. Construction of P (∞) . We now use the Szeg˝o function D from the previous sub(∞) into a RH problem with jump matrix to transform the RH problem for P section, 0 1 on J . We seek P (∞) in the form, cf. [27, 39], −1 0 σ3 ˜ (∞) P (∞) (z) = D∞ P (z)D(z)−σ3 ,

for z ∈ C \ [b0 , aN+1 ].

(4.14)

Then, by (4.1)–(4.4) the problem is reduced to constructing a solution of the following RH problem. RH problem for P˜ (∞) . (a) (b)

P˜ (∞) : C \ [b0 , aN+1 ] → C2×2 is analytic. P˜ (∞) satisfies the following jump relations:

0 1 (∞) (∞) ˜ ˜ P+ (x) = P− (x) , −1 0

for x ∈ J \ {0},

−2πin 2πiξ je j e 0 (∞) (∞) ˜ ˜ P+ (x) = P− (x) , 0 e2πinj e−2πiξj for x ∈ (aj , bj ), j = 1, . . . , N.

(4.15)

(4.16)

(c) P˜ (∞) (z) = I + O(1/z), as z → ∞. This corresponds to the RH problem [13, (4.24)–(4.26)], which has been solved there using Riemann theta functions. Note that, in contrast to the RH problem [13, (4.24)– (4.26)], the jump matrix in (4.16) contains extra factors exp(±2π iξj ) in the diagonal entries, which come from the Szeg˝o function D. However, this does not create any problems. In order to formulate the solution of the RH problem for P˜ (∞) we need to introduce some additional notations. Here we closely follow [13], see also [11]. Let J˜ = R \ J¯ be the complement of J¯, and a0 ≡ aN+1 . Letting the point ∞ lie on the interval (a0 , b0 ), J˜ can be displayed as a union of intervals on the √ Riemann sphere. Let X be the two-sheeted Riemann surface of genus N associated to R(z), obtained by gluing together two copies of the slit plane C \ J˜ along J˜. We draw cycles Aj winding once, in the negative direction, around the slit (aj , bj ) in the first sheet, and cycles Bj starting from a point on the slit (aj , bj ) going on the first sheet through a point on the slit (a0 , b0 ), and returning on the second sheet to the original point, as indicated in Fig. 3. The cycles {Ai , Bj }1≤i,j ≤N form a canonical homology basis for X, see [16]. Let ω = (ω1 , . . . , ωN ) be the basis of holomorphic one-forms on X dual to the canonical homology basis, that is ωi = δij , 1 ≤ i, j ≤ N. (4.17) Aj

The associated Riemann matrix of B periods, denoted by τ and with entries τij = ωi , 1 ≤ i, j ≤ N, Bj

(4.18)

Universality for Eigenvalue Correlations at the Origin of the Spectrum

Aj

A1 a0

b0

a1

b1

177

…

B1

aj

bj

Bj

AN …

aN

bN

BN

Fig. 3. The canonical homology basis {Ai , Bj }1≤i,j ≤N for X. The full lines denote paths on the first sheet, while the dotted lines denote paths on the second sheet

is symmetric with positive definite imaginary part, see [16]. The associated Riemann theta function is defined by

1 exp 2πi m, z + m, τ m , (4.19) z ∈ CN , θ (z) = 2 N m∈Z

where ·, · is the real scalar product, which defines an analytic function on CN . The Riemann theta function has the periodicity properties [16] with respect to the lattice ZN + τ ZN , θ (z + ej ) = θ (z),

θ (z ± τj ) = e∓2πizj −πiτjj θ(z),

(4.20)

where z = (z1 , . . . , zN ) and ej is the j th unit vector in CN with 1 on the j th entry and zeros elsewhere, and where τj is the j th column vector of τ . Define the scalar function 1/4 N z − b i z − b0

, γ (z) = z − ai z − aN+1

(4.21)

i=1

which is analytic on C \ J˜, with γ (z) ∼ 1 as z → ∞, z ∈ C+ . It is known [13, Lemma 4.1] that γ has the following properties: γ + γ −1 possesses N roots {zj }N j =1 with zj

on the − side of (aj , bj ),

γ − γ −1 possesses N roots {zj }N j =1 with zj

on the + side of (aj , bj ).

(−)

(−)

(+)

(+)

Fix the base point for the Riemann surface X to be aN+1 = a0 , let K be the associated vector of Riemann constants [16], and define the multivalued function z u(z) = ω. (4.22) aN +1

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A.B.J. Kuijlaars, M. Vanlessen

Here, we take the integral along any path from aN+1 to z on the first sheet. Since the integral is taken on the first sheet, u(z) is uniquely defined in CN /ZN because of (4.17). Let d be defined as N z(−) j ω, (4.23) d = −K − j =1 aN +1

where again the integrals are taken on the first sheet. We now have introduced the necessary ingredients to formulate the solution of the RH problem for P˜ (∞) . Together with (4.14) this gives the parametrix P (∞) for the outside region. The solution of the RH problem for P˜ (∞) is given by, see [13, Lemma 4.3], θ(u+ (∞)+d) + (∞)+d) , P˜ (∞) (z) = diag θ(u+θ(u (∞)−n+ξ +d) θ(−u+ (∞)−n+ξ −d) 



γ +γ −1 θ(u(z)−n+ξ +d) γ −γ −1 θ(−u(z)−n+ξ +d) θ(u(z)+d) −2i θ(−u(z)+d)  2 

× for z ∈ C+ , and P˜ (∞) (z) = diag

γ −γ −1 θ(u(z)−n+ξ −d) γ +γ −1 θ(−u(z)−n+ξ −d) 2i θ(u(z)−d) 2 θ(u(z)+d)

θ(u+ (∞)+d) θ(u+ (∞)+d) θ(u+ (∞)−n+ξ +d) , θ(−u+ (∞)−n+ξ −d)



,



γ −γ −1 θ(−u(z)−n+ξ +d) θ(−u(z)+d)  −2i

− γ +γ2

−1

θ(u(z)−n+ξ +d) θ(u(z)+d) 

γ +γ −1 θ(−u(z)−n+ξ −d) 2 θ(u(z)+d)

− γ −γ 2i

−1

θ(u(z)−n+ξ −d) θ(u(z)−d)

×

(4.24)

,

(4.25)

for z ∈ C− . Here, = (1 , . . . , N ) and ξ = (ξ1 , . . . , ξN ). Remark 4.3. In contrast to [13] we have an extra term ξ in the Riemann theta functions. This comes from the slightly different jump matrix in (4.2) due to the Szeg˝o function, as noted before. If ξ ∈ ZN the factors e±2πiξj in (4.16) disappear and the RH problem for P˜ (∞) is exactly the same as the RH problem [13, (4.24)–(4.26)]. Since the Riemann theta functions possess the periodicity properties (4.20), the term ξ in (4.24) and (4.25) disappears in this case. This is in agreement with [13, Lemma 4.3]. For later use, we need P (∞) to be invertible. In [13, Sect. 4.2] it has been shown that det P˜ (∞) ≡ 1, so that by (4.14) det P (∞) ≡ 1.

(4.26)

5. Parametrix Near the Origin In this section we construct the parametrix near the origin. As noted in the introduction, it is similar to the construction of the parametrix near the algebraic singularities of the generalized Jacobi weight [39], and we skip some details and motivations. We surround the origin by a disk Uδ with radius δ > 0. We assume that δ is small, so that in any case, we have that [−δ, δ] ⊂ J . We seek a matrix valued function P that satisfies the following RH problem.

Universality for Eigenvalue Correlations at the Origin of the Spectrum

179

RH problem for P . (a) P (z) is defined and analytic for z ∈ Uδ0 \ for some δ0 > δ. (b) On ∩ Uδ , P satisfies the same jump relations as S, that is,

1 0 , for z ∈ ∩ (Uδ ∩ C± ), P+ (z) = P− (z) ω(z)−1 e−2nφ(z) 1

0 |x|2α P+ (x) = P− (x) , for x ∈ (−δ, δ) \ {0}. −|x|−2α 0 (c) On ∂Uδ we have, as n → ∞,

−1 1 P (z) P (∞) (z) = I + O , n

uniformly for z ∈ ∂Uδ \ .

(d) For α < 0, the matrix function P (z) has the following behavior as z → 0:

1 |z|2α P (z) = O , as z → 0. 1 |z|2α For α > 0, the matrix function P (z) has the following behavior as z → 0:

   O 1 1 , as z → 0 from outside the lens,   11 P (z) =

  |z|−2α 1  O , as z → 0 from inside the lens. |z|−2α 1

(5.1) (5.2)

(5.3)

(5.4)

(5.5)

We construct P as follows. First, we focus on conditions (a), (b) and (d). We transform the RH problem for P into a RH problem for P (1) with constant jump matrices, and solve the latter RH problem explicitly. Afterwards, we also consider the matching condition (c) of the RH problem. We start with the following map f defined on a neighborhood of the origin iφ(z) − iφ+ (0), if Im z > 0, f (z) = (5.6) −iφ(z) − iφ+ (0), if Im z < 0. Since φ+ = −φ− , we have that f is analytic for z in a neighborhood of the origin. An easy calculation, based on the fact that 2φ+ (x) = g+ (x) − g− (x) and on (2.9) and (5.6), shows that x f (x) = π ψ(s)ds, for x ∈ (−δ, δ), (5.7) 0

which implies that f (0) = πψ(0) > 0. So, the behavior of f near the origin is given by f (z) = πψ(0)z + O z2 , as z → 0. (5.8) So if we choose δ > 0 sufficiently small, ζ = f (z) is a conformal mapping on Uδ onto a convex neighborhood of 0 in the complex ζ -plane. We also note that f (x) is real and positive (negative) for x ∈ Uδ positive (negative), which follows from (5.7).

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A.B.J. Kuijlaars, M. Vanlessen III

G4 -

IV

G5 V

II

G3

G2

+ +

+

0

+ -

+

p 4

-

I + -

- +

G6

-

G1

+

VIII

G8

G7 VI

VII

Fig. 4. The contour

S3 S4 S5

Ud S2

G4

f

S6

G2 0

G5

S1

0

f (Ud)

G3

S8

G1

G6

G8

S7

G7

Fig. 5. The conformal mapping f . Every k is mapped onto the part of the corresponding ray k in f (Uδ )

Let j , j = 1, . . . , 8 be the infinite ray π j = {ζ ∈ C | arg ζ = (j − 1) }. 4 These rays divide the ζ -plane into eight sectors I–VIII as shown in Fig. 4. We define the contours j , j = 1, 2, . . . , 8 as the preimages under the mapping ζ = f (z) of the part of the corresponding rays j in f (Uδ ), see Fig. 5. We have some freedom in the selection of the contour . We now specify that we open the lens in such a way that ∩ Uδ =

j .

j =1,2,4,5,6,8

As a consequence we have that f maps to part of the union of rays

j

j .

Universality for Eigenvalue Correlations at the Origin of the Spectrum

181

In order to transform to constant jumps we use a piecewise analytic function W corresponding to the analytic continuation of |x|2α . For z ∈ Uδ , we define zα , if π/2 < | arg f (z)| < π , W (z) = (5.9) α if 0 < | arg f (z)| < π/2, (−z) , with principal branches of powers. Then W is defined and analytic in Uδ \ (1 ∪ 3 ∪ 5 ∪ 7 ). We seek P in the form P (z) = En (z)P (1) (z)W (z)−σ3 e−nφ(z)σ3 .

(5.10)

Here the matrix valued function En is analytic in a neighborhood of Uδ , and En will be determined below so that the matching condition (c) of the RH problem for P is satisfied. Similar considerations as in [39] show that P (1) should satisfy the following RH problem, with jumps on the system of contours 8i=1 i , oriented as in the left part o of Fig. 5. In (5.11)–(5.14), i is used to denote i without the origin. RH problem for P (1) . (a) P (1) (z) is defined and analytic for z ∈ Uδ0 \ ( ∪ ) for some δ0 > δ. (b) P (1) satisfies the following jump relations on Uδ ∩ ( ∪ ):

0 1 (1) (1) P+ (x) = P− (x) , for x ∈ 1o ∪ 5o , −1 0 (1)

(1)

(1)

(1)

1

P+ (z) = P− (z)

e−2πiα

0 , 1

P+ (z) = P− (z)eπiασ3 , (1) P+ (z)

=

(1) P− (z)

1 e2πiα

for z ∈ 2o ∪ 6o ,

for z ∈ 3o ∪ 7o ,

0 , 1

for z ∈ 4o ∪ 8o .

(c) For α < 0, P (1) (z) has the following behavior as z → 0: α α

|z| |z| P (1) (z) = O , as z → 0. |z|α |z|α For α > 0, P (1) (z) has the following behavior as z → 0:  α −α

|z| |z|     O |z|α |z|−α , as z → 0 from outside the lens, P (1) (z) = −α −α

  |z| |z|  O , as z → 0 from inside the lens. |z|−α |z|−α

(5.11)

(5.12) (5.13) (5.14)

(5.15)

(5.16)

Next we construct an explicit solution of the RH problem for P (1) . This is based on a model RH problem for α in the ζ -plane, see [39]. We denote by the contour 8 j =1 j oriented as shown in Fig. 4.

182

A.B.J. Kuijlaars, M. Vanlessen

RH problem for α . (a) α : C \ → C2×2 is analytic. (b) α satisfies the following jump relations on :

0 1 , for ζ ∈ 1 ∪ 5 , α,+ (ζ ) = α,− (ζ ) −1 0

1

α,+ (ζ ) = α,− (ζ )

e−2πiα

0 , 1

α,+ (ζ ) = α,− (ζ )eπiασ3 , α,+ (ζ ) = α,− (ζ )

1 e2πiα

(5.17)

for ζ ∈ 2 ∪ 6 ,

(5.18)

for ζ ∈ 3 ∪ 7 ,

0 , 1

(5.19)

for ζ ∈ 4 ∪ 8 .

(5.20)

(c) For α < 0 the matrix function α (ζ ) has the following behavior as ζ → 0:

α |ζ | |ζ |α , as ζ → 0. (5.21) α (ζ ) = O |ζ |α |ζ |α For α > 0 the matrix function α (ζ ) has the following behavior as ζ → 0:

 α |ζ | |ζ |−α     O |ζ |α |ζ |−α , as ζ → 0 with ζ ∈ II, III, VI, VII, α (ζ ) = (5.22)

−α   |ζ | |ζ |−α  O , as ζ → 0 with ζ ∈ I, IV, V, VIII. |ζ |−α |ζ |−α This RH problem was solved in [39, formulas (4.26)–(4.33)]. It is built out of the (1) (2) modified Bessel functions Iα± 1 , Kα± 1 and out of the Hankel functions H 1 , H 1 . 2

2

α± 2

α± 2

For our purpose here, it suffices to know the explicit formula for α in sector I. There we have   (2) (1) (ζ ) −iH (ζ ) H 1 1 α+ 2 1 √ 1/2  α+ 2  −(α+ 41 )πiσ3 π ζ  (2) , for 0 < arg ζ < π4 . α (ζ ) = e (1) 2 H 1 (ζ ) −iH 1 (ζ ) α− 2

α− 2

(5.23) Starting from (5.23) we can find the solution in the other sectors by following the jumps (5.17)–(5.20). See [39] for explicit expressions. Now we define P (1) (z) = α (nf (z)),

(5.24)

and P (1) will solve the RH problem for P (1) . This ends the construction of P (1) . So far, we have proven that for every matrix valued function En analytic in a neighborhood of Uδ , the matrix valued function P given by P (z) = En (z)α (nf (z))W (z)−σ3 e−nφ(z)σ3 ,

(5.25)

Universality for Eigenvalue Correlations at the Origin of the Spectrum

183

satisfies conditions (a), (b) and (d) of the RH problem for P . We now use the freedom we have in choosing En to ensure that P , given by (5.25), also satisfies the matching condition (c) of the RH problem for P . To this end, we use the asymptotic behavior of α at infinity, see [39, (4.43)–(4.46)]. Similar calculations as in [39] show that we have to define En as

1i nφ+ (0)σ3 − π4i σ3 1 , (5.26) e En (z) = E(z)e √ 2 i 1 where the matrix valued function E is given by 1

E(z) = P (∞) (z)W (z)σ3 e 2 απiσ3 , 1

E(z) = P (∞) (z)W (z)σ3 e− 2 απiσ3 ,

for z ∈ f −1 (I ∪ I I ), for z ∈ f −1 (I I I ∪ I V ),

E(z) = P

(∞)

E(z) = P

(∞)

(z)W (z)

σ3

(z)W (z)

σ3

0 1 − 21 πiασ3 e , −1 0

0 1 21 πiασ3 e , −1 0

(5.27) (5.28)

for z ∈ f −1 (V ∪ V I ), (5.29) for z ∈ f −1 (V I I ∪ V I I I ). (5.30)

Following the proof of [39, Prop. 4.5], we obtain that E is analytic in a full neighborhood of Uδ . Here we need the fact that D(z)/W (z) and W (z)/D(z) remain bounded as z → 0, which follows from (5.9) and Lemma 4.2. Then we see from (5.26) that En is also analytic in a neighborhood of Uδ . This completes the construction of the parametrix near the origin. Remark 5.1. Note that, in contrast to the case of the generalized Jacobi weight [39], here E depends on n. This follows from the fact that the parametrix P (∞) for the outside region in our case depends on n. For later use we state, since E is analytic in Uδ and from the explicit form of P (∞) , d cf. [13], that E(z) and dz E(z) are uniformly bounded for z ∈ Uδ , as n → ∞. 6. Third Transformation S → R At each of the endpoints ai , bj of J , we have to do a local analysis as well as at each of the singular points (if any). The endpoints and singular points are surrounded by small disks, say of radius δ, that do not overlap and that also do not overlap with the disk Uδ around the region. Within each disk we construct a parametrix P which satisfies a local RH problem: RH problem for P near x0 where x0 is an endpoint or a singular point. (a) P (z) is defined and analytic for z ∈ {|z − x0 | < δ} \ for some δ0 > δ. (b) P satisfies the same jump relations as S does on ∩ {|z − x0 | < δ}. (c) There is κ > 0 such that we have as n → ∞:

−1 1 (∞) P (z) P , uniformly for |z − x0 | = δ. (z) = I + O nκ

(6.1)

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A.B.J. Kuijlaars, M. Vanlessen

z1

bj-1

z2

aj

bj

0

aj+1

bj+1

Fig. 6. Part of the contour R . The singular point z1 corresponds to a point where h vanishes at the interior of J , the singular point z2 corresponds to a point where we obtain equality in (3.14)

The local RH problem near the regular endpoints ai , bj of J is similar to the situation in [13]. Here however, we have extra factors |x|±2α and ω(z)−1 in the jump matrices. These factors can easily be removed via an appropriate transformation, and the local RH problem is then solved as in [13, Sects. 4.3–4.5] with the use of Airy functions. For our purpose, we do not need the explicit formulas for the parametrix near the endpoints. It suffices to know that P exists. For regular endpoints we can take κ = 1 in (6.1). Near the singular points we can follow the analysis of [13, Sect. 5]. Here we do not construct an explicit parametrix out of special functions, only existence of the local RH problem is obtained. For singular points we have κ < 1 in (6.1), see [13]. So the singular points lead to error terms with decay slower than for the regular points (κ = 1). However, this has no influence on the universality result at the origin (not even in the error term), since that only depends on the leading order asymptotics. We are now ready to do the final transformation. As noted before, we surround the endpoints ai , bj of J , the origin, and the singular points of the potential V by nonoverlapping small disks. Using the parametrix P (∞) for the outside region and the parametrix P defined inside each of the disks, we define the matrix valued function R as −1 (z), for z outside the disks, S(z) P (∞) (6.2) R(z) = S(z)P −1 (z), for z inside the disks. Remark 6.1. It is known that the inverses of the parametrices P (∞) and P exist, since all matrices have determinant one. For P (∞) , see (4.26). For P within the disks around the endpoints ai , bj of J , as well as within the disks around the singular points of V we refer to [13]. For P within the disk around the origin we refer to [39, Sect. 4]. Note that P (∞) and S have the same jumps on J \ {0}, and that P and S have the same jumps on the lens within the disks. This implies that R is analytic on the entire plane, except for jumps on the reduced system of contours R , as shown in Fig. 6, cf. [13], and except for a possible isolated singularity at the origin. Yet, as in [27, 39], it follows easily from the behavior of S and P near the origin, given by (3.12) and (3.13), and by (5.4) and (5.5), respectively, that the isolated singularity of R at the origin is removable. Therefore R is analytic on C \ R . Recall that the matrix valued functions S and P (∞) are normalized at infinity. Since det P (∞) ≡ 1, this implies, by (6.2), that also R is normalized at infinity. Let vR be the jump matrix for R. It can be calculated explicitly for each component of R . However, all that we require are the following estimates, cf. [13]: vR (z) = I + O(e−cn|z| ), vR (z) = I + O(1/n ), κ

as n → ∞, z ∈ R \ circles, as n → ∞, z ∈ circles,

Universality for Eigenvalue Correlations at the Origin of the Spectrum

185

for some c > 0 and 0 < κ ≤ 1, and where · is any matrix norm. We note that the extra factor |x|2α , which we will meet in vR , does not cause any difficulties to obtain this behavior. These estimates then imply that vR is uniformly close to the identity matrix as n → ∞, and, since R is normalized at infinity, we then find uniformly for z ∈ R \ R , R(z) = I + O(1/nκ ),

as n → ∞.

(6.3)

d R(z) is uniformly bounded So, R is uniformly bounded as n → ∞. We also have that dz as n → ∞. Another useful property is det R ≡ 1, which follows from (6.2) and the fact that S, P (∞) and P all have determinant 1.

7. Proof of Theorem 1.1 We now have all the ingredients necessary to prove Theorem 1.1. We point out that the general scheme of this proof is the same as the proof of [28, Theorem 1.1(c)]. We replace in the kernel Kn , given by (1.3), the orthonormal polynomials pn−1,n and pn,n , together with their leading coefficients γn−1,n and γn,n , by the appropriate entries of Y , given by (2.4), and find Kn (x, y) = −

1 Y11 (x)Y21 (y) − Y21 (x)Y11 (y) wn (x) wn (y) . 2πi x−y

(7.1)

This means that the kernel Kn can be expressed in terms of the first column of Y . Hence, we want to know the asymptotic behavior of Y near the origin. This will be determined in the following lemma. Lemma 7.1. For x ∈ (0, δ), " !

(nf (x))1/2 Jα+ 1 (nf (x)) n π Y11 (x) − π4i σ 2 3 , e 2 M+ (x) =e Y21 (x) (nf (x))1/2 Jα− 1 (nf (x)) wn (x)

(7.2)

2

with M(z) given by

πi 1 1i M(z) = R(z)E(z)enφ+ (0)σ3 e− 4 σ3 √ , 2 i 1

(7.3)

where R is the result of the transformations Y → T → S → R of the RH problem, and the matrix valued function E is given by (5.27)–(5.30). The matrix valued function d M(z) uniformly bounded for z ∈ Uδ as n → ∞. M is analytic in Uδ with M(z) and dz Furthermore, det M(z) ≡ 1.

(7.4)

Proof. We use the series of transformations Y → T → S → R and unfold them for z inside the disk Uδ and in the right upper part of the lens, so that z ∈ f −1 (I ). Since ω(z) = z2α and W (z) = zα e−πiα for our choice of z, see (3.6) and (5.9), we have by (2.14), (3.7), (5.25) and (6.2), Y (z) = e 2 σ3 R(z)En (z)α (nf (z))e−nφ(z)σ3 z−ασ3 eπiασ3

1 0 − n2 σ3 ng(z)σ3 e × −2α −2nφ(z) e . 1 z e n

(7.5)

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A.B.J. Kuijlaars, M. Vanlessen

We then get for the first column of Y ,

Y11 (z) −α n(g(z)−φ(z)− 2 ) n σ3 πiασ3 1 2 . e R(z)En (z)α (nf (z))e =z e 1 Y21 (z)

(7.6)

Since z is in the right upper part of the lens and inside the disk Uδ , we have 0 < arg nf (z) < π/4, cf. Fig. 5, and we thus use (5.23) to evaluate α (nf (z)). Using the formulas 9.1.3 and 9.1.4 of [1] which connect the Hankel functions with the usual J -Bessel functions, we find " !

(nf (z))1/2 Jα+ 1 (nf (z)) πiασ3 1 − π4i √ 2 α (nf (z))e . (7.7) =e π 1 (nf (z))1/2 Jα− 1 (nf (z)) 2

By (5.26) and (7.3) we have R(z)En (z) = M(z). Inserting this and (7.7) into (7.6) we get " !

(nf (z))1/2 Jα+ 1 (nf (z)) Y11 (z) − π4i √ −α n(g(z)−φ(z)− 2 ) n σ 2 . πz e e 2 3 M(z) =e Y21 (z) (nf (z))1/2 Jα− 1 (nf (z)) 2 (7.8) Letting z → x ∈ (0, δ), and noting that 1

1

x −α en(g+ (x)−φ+ (x)− 2 ) = x −α e 2 n(g+ (x)+g− (x)−) = x −α e 2 nV (x) = wn (x)−1/2 , (7.9)

which follows from the fact that 2φ+ (x) = g+ (x) − g− (x), see Sect. 3, and from (1.2) and (2.10), we obtain (7.2). The matrix valued function M is analytic in the disk Uδ since both R and E are analytic in this disk. So, we may write M(x) instead of M+ (x) in (7.2). d d We recall that R(z), dz R(z), E(z) and dz E(z) are uniformly bounded for z ∈ Uδ as n → ∞, see Sect. 5 and 6. If we also use that |enφ+ (0) | = 1, which follows from the d fact that φ+ is purely imaginary on J , we have from (7.3) that M(z) and dz M(z) are uniformly bounded for z ∈ Uδ as n → ∞. Since M is a product of five matrices all with determinant one, (7.4) is true.

Lemma 7.2. Let u ∈ (0, ∞), un =

u nψ(0)

u˜ n = πu + O

and u˜ n = nf (un ). Then

u2 n

, !

Jα+ 1 (u˜ n ) = Jα+ 1 (π u) + O 2

2

! Jα− 1 (u˜ n ) = Jα− 1 (π u) + O 2

2

as n → ∞,

3

uα+ 2 n

1

uα+ 2 n

(7.10)

" ,

as n → ∞,

(7.11)

,

as n → ∞,

(7.12)

"

where the error terms hold uniformly for u in bounded subsets of (0, ∞).

Universality for Eigenvalue Correlations at the Origin of the Spectrum

187

Proof. Since, see (5.8) as x → 0,

f (x) = πψ(0)x + O(x 2 ),

we have, uniformly for u in bounded subsets of (0, ∞),

2

u u u , as n → ∞, =π +O f nψ(0) n n2 which proves (7.10). 1 We note [1, Formula 9.1.10] that Jα+ 1 (z) = zα+ 2 H (z), with H an entire function. It 2 then follows from (7.10) that, as n → ∞, uniformly for u in bounded subsets of (0, ∞), " # ! 2 $ 3 u uα+ 2 α+ 21 Jα+ 1 (u˜ n ) = (π u) H (π u) + O +O 2 n n ! = Jα+ 1 (π u) + O 2

3

uα+ 2 n

" ,

so that Eq. (7.11) is proved. Similarly, we can prove (7.12).

We are now able to prove Theorem 1.1. Proof of Theorem 1.1. Let u, v ∈ (0, ∞) and define un =

u , nψ(0)

vn =

v , nψ(0)

u˜ n = nf (un ),

v˜n = nf (vn ).

We put Kˆ n (u, v) =

1 Kn (un , vn ). nψ(0)

From (7.1) and (7.2) we then have ! n √ " n √ e− 2 wn (un )Y11 (un ) e− 2 wn (vn )Y11 (vn ) 1 det Kˆ n (u, v) = − n √ n √ 2π i(u − v) e 2 wn (un )Y21 (un ) e 2 wn (vn )Y21 (vn ) 

 =

1/2

u˜ n Jα+ 1 (u˜ n ) 0







1/2

0 v˜n Jα+ 1 (v˜n )

1 2 2  +M(vn )   . det M(un )  1/2 1/2 2(u − v) u˜ n Jα− 1 (u˜ n ) 0 0 v˜n Jα− 1 (v˜n ) 2

2

The matrix in the determinant can be written as   1/2 1/2 u˜ n Jα+ 1 (u˜ n ) v˜n Jα+ 1 (v˜n ) 2 2  M(vn )  1/2 1/2 u˜ n Jα− 1 (u˜ n ) v˜n Jα− 1 (v˜n ) 2

2



1/2

u˜ n Jα+ 1 (u˜ n ) 0



2  . + M(vn )−1 (M(un ) − M(vn ))  1/2 u˜ n Jα− 1 (u˜ n ) 0 2

(7.13)

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A.B.J. Kuijlaars, M. Vanlessen

We will now determine the asymptotics of the second term in (7.13). Since det M(vn ) = 1 and since M(z) is uniformly bounded for z ∈ Uδ , see Lemma 7.1, the entries of M(vn )−1 d are uniformly bounded. By Lemma 7.1 we also have that dz M(z) is uniformly bounded for z ∈ Uδ , so that from the mean value theorem M(un ) − M(vn ) = O u−v n . From 1/2 1/2 Lemma 7.2 it follows that u˜ n Jα+ 1 (u˜ n ) = O(uα+1 ) and u˜ n Jα− 1 (u˜ n ) = O(uα ) 2 2 uniformly for u in bounded subsets of (0, ∞) as n → ∞. Hence we have, uniformly for u, v in bounded subsets of (0, ∞),   ! " 1/2 u˜ n Jα+ 1 (u˜ n ) 0 O u−v uα 0 n −1 2 = M(vn ) (M(un ) − M(vn ))  1/2 . α 0 O u−v u˜ n Jα− 1 (u˜ n ) 0 n u 2

1/2

Inserting this into (7.13), using the fact that det M(vn ) = 1, and that v˜n Jα± 1 (v˜n ) = 2 O(v α ) as n → ∞, we then find uniformly for u, v in bounded subsets of (0, ∞),   u−v α 1/2 1/2 J ( u ˜ ) + O u J ( v ˜ ) v ˜ u ˜ 1 1 n n n n 1 α+ 2 α+ 2 n  Kˆ n (u, v) = det  1/2 u−v α 1/2 2(u − v) u˜ n Jα− 1 (u˜ n ) + O u v˜n Jα− 1 (v˜n ) 2

n

2

  1/2 1/2 α α

u˜ n Jα+ 1 (u˜ n ) v˜n Jα+ 1 (v˜n ) 1 u v 2 2   = det . +O 1/2 1/2 2(u − v) n u˜ n Jα− 1 (u˜ n ) v˜n Jα− 1 (v˜n ) 2

(7.14)

2

We note, from Lemma 7.2, that we can replace in the determinant, u˜ n by π u and v˜n by π v. We then make an error which can be estimated by Lemma 7.2. However, since this estimate is not uniform for u − v close to zero, we have to be more careful. We insert a factor u−α in the first column of the determinant, and a factor v −α in the second. Then we subtract the second column from the first to obtain Kˆ n (u, v) =

uα v α 2(u −v)  1/2 1/2 1/2 u−α u˜ n Jα+ 1 (u˜ n ) − v −α v˜n Jα+ 1 (v˜n ) v −α v˜n Jα+ 1 (v˜n ) 2 2 2  × det  1/2 1/2 1/2 u−α u˜ n Jα− 1 (u˜ n ) − v −α v˜n Jα− 1 (v˜n ) v −α v˜n Jα− 1 (v˜n ) 2 2 2 α α

u v +O . (7.15) n

One can check, using (7.10), (7.11) and the facts that J and

d dx x˜ n

=π

+ O( xn ),

where we have put x˜n =

α+ 21

(x˜n ) = J

x nf ( nψ(0) ),

α+ 21

(π x)+O( x

α+ 21

n

)

that

* x d ) −α 1/2 x x˜n Jα+ 1 (x˜n ) − x −α (π x)1/2 Jα+ 1 (π x) = O , 2 2 dx n

as n → ∞,

uniformly for x in bounded subsets of (0, ∞). It then follows that the (1,1)–entry in the determinant of (7.15) is equal to

u−v . u−α (π u)1/2 Jα+ 1 (π u) − v −α (π v)1/2 Jα+ 1 (π v) + O 2 2 n

Universality for Eigenvalue Correlations at the Origin of the Spectrum

189

Similarly, we have from

* 1 d ) −α 1/2 −α 1/2 x x˜n Jα− 1 (x˜n ) − x (π x) Jα− 1 (π x) = O , 2 2 dx n

as n → ∞,

that the (2,1)–entry in the determinant of (7.15) is equal to u−α (π u)1/2 Jα− 1 (π u) − v −α (π v)1/2 Jα− 1 (π v) + O 2

2

u−v n

.

From Lemma 7.2 it also follows that v˜n Jα± 1 (v˜n ) = (π v)Jα± 1 (π v) + O(1/n). There2 2 fore, uniformly for u, v in bounded subsets of (0, ∞), Kˆ n (u, v) =

uα v α 2(u − v)

  −α u−α (π u)1/2 Jα+ 1 (π u)−v −α (πv)1/2 Jα+ 1 (πv)+O u−v v (π v)1/2 Jα+ 1 (π v)+O n1 n 2 2 2  × det  −α u−α (π u)1/2 Jα− 1 (π u)−v −α (πv)1/2 Jα− 1 (πv)+O u−v v (π v)1/2 Jα− 1 (π v)+O n1 n 2

2

2

α α

u v +O n   α vα u−α (πu)1/2 Jα+ 1 (πu) − v −α (πv)1/2 Jα+ 1 (πv) O n1 u 2 2  = Joα (u, v) + det  2(u − v) u−α (πu)1/2 Jα− 1 (πu) − v −α (πv)1/2 Jα− 1 (πv) O n1 2

α α

u v +O . n

2

(7.16)

1

Since z−α+ 2 Jα± 1 (z) is an entire function we have by the mean value theorem that 2

u−α (π u)1/2 Jα± 1 (π u) − v −α (π v)1/2 Jα± 1 (π v) 2

2

u−v is bounded for u, v in bounded subsets of (0, ∞). From (7.16) we then have α α

u v o ˆ , Kn (u, v) = Jα (u, v) + O n uniformly for u, v in bounded subsets of (0, ∞), which completes the proof of Theorem 1.1.

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Commun. Math. Phys. 243, 193–240 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0934-1

Communications in

Mathematical Physics

Differential Systems for Biorthogonal Polynomials Appearing in 2-Matrix Models and the Associated Riemann–Hilbert Problem M. Bertola1,2 , B. Eynard2,3 , J. Harnad1,2 1

Centre de recherches math´ematiques, Universit´e de Montr´eal, C. P. 6128, succ. centre ville, Montr´eal, Qu´ebec, Canada H3C 3J7. E-mail: [email protected]; [email protected] 2 Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke W., Montr´eal, Qu´ebec, Canada H4B 1R6 3 Service de Physique Th´eorique, CEA/Saclay, Orme des Merisiers, 91191 Gif-sur-Yvette Cedex, France. E-mail: [email protected] Received: 10 September 2002 / Accepted: 9 April 2003 Published online: 5 November 2003 – © Springer-Verlag 2003

Abstract: We consider biorthogonal polynomials that arise in the study of a generalization of two–matrix Hermitian models with two polynomial potentials V1 (x), V2 (y) of any degree, with arbitrary complex coefficients. Finite consecutive subsequences of biorthogonal polynomials (“windows”), of lengths equal to the degrees of the potentials V1 and V2 , satisfy systems of ODE’s with polynomial coefficients as well as PDE’s (deformation equations) with respect to the coefficients of the potentials and recursion relations connecting consecutive windows. A compatible sequence of fundamental systems of solutions is constructed for these equations. The (Stokes) sectorial asymptotics of these fundamental systems are derived through saddle-point integration and the Riemann-Hilbert problem characterizing the differential equations is deduced.

1. Introduction In [2, 3] the differential systems satisfied by sequences of biorthogonal polynomials associated to 2-matrix models were studied, together with the deformations induced by changes in the coefficients of the potentials determining the orthogonality measure. For ensembles consisting of pairs of N × N hermitian matrices M1 and M2 , the U (N ) invariant probability measure is taken to be of the form: 1 1 1 dµ(M1 , M2 ) := exp tr (−V1 (M1 ) − V2 (M2 ) + M1 M2 ) dM1 dM2 , (1.0.1) τN τN where dM1 dM2 is the standard Lebesgue measure for pairs of Hermitian matrices and the potentials V1 and V2 are chosen to be polynomials of degrees d1 + 1, d2 + 1 respectively, with real coefficients. The overall positive small parameter in the exponential Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds FCAR du Qu´ebec

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is taken of order N −1 when considering the large N limit, but in the present context it will just play the role of Planck’s constant in the string equation. Using the Harish-Chandra-Itzykson-Zuber formula, one can reduce the computation of the corresponding partition function to an integral over only the eigenvalues of the two matrices N 1 N dµ(M1 , M2 ) ∝ dxi dyi (x)(y)e− j =1 V1 (xj )+V2 (yj )−xj yj , τN := i=1

(1.0.2) and then express all spectral statistics in terms of the associated biorthogonal polynomials, in the same spirit as orthogonal polynomials are used in the spectral statistics of one-matrix models [24]. In this context, what is meant by biorthogonal polynomials is a pair of sequences of monic polynomials πn (x) = x n + · · · ,

σn (y) = y n + · · · ,

n ∈ N,

(1.0.3)

which are mutually dual with respect to the associated coupled measure on the product space 1

R R

dx dy πn (x)σm (y)e− (V1 (x)+V2 (y)−xy) = hn δmn .

(1.0.4)

In this work, we use essentially the same definition of orthogonality, but extend it to the case of polynomials V1 and V2 with arbitrary (possibly complex) coefficients, with the contours of integration no longer restricted to the real axis, but taken as curves in the complex plane starting and ending at ∞, chosen so that the integrals are convergent. The orthogonality relations determine the two families uniquely, if they exist [13, 3]. It was shown in [2, 3] that the finite consecutive subsequences of lengths d2 + 1 and d1 + 1 respectively, within the sequences of dual quasi-polynomials: 1 1 1 1 ψn (x) = √ πn (x)e− V1 (x) , φn (y) = √ σn (y)e− V2 (y) , hn hn

(1.0.5)

beginning (or ending) at the points n = N , satisfy compatible over-determined systems of first order differential equations with polynomial coefficients of degrees d1 and d2 , respectively, as well as recursion relations for consecutive values of N . In fact, certain quadruples of Differential–Deformation–Difference equations (DDD for short) were derived for these “windows”, as well as for their Fourier Laplace transforms, in which the deformation parameters were taken to be the coefficients of the potentials V1 and V2 . It was shown in [2, 3] that these systems are Frobenius compatible and hence admit joint fundamental systems of solutions. In the present work we explicitly construct such fundamental systems in terms of certain integral transforms applied to the biorthogonal polynomials. The main purpose is to derive the Riemann–Hilbert problem characterizing the sectorial asymptotic behavior at x = ∞ or y = ∞. The ultimate purpose of this analysis is to deduce in a rigorous way the double–scaling limits N → ∞, N = O(1) of the partition function and spectral statistics, for which the corresponding large N asymptotics of the biorthogonal polynomials are required. (See [12, 22] and references therein for further background on 2-matrix models, and [9, 14–16, 13] for other more recent developments.) The study of the large N limit

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of matrix integrals is of considerable interest in physics, since many physical systems having a large number of strongly correlated degrees of freedom (quantum chaos, mesoscopic conductors, . . .) share the statistical properties of the spectra of random matrices. Also, the large N expansion of a random matrix integral (if it exists) is expected to be the generating functional of discretized surfaces, and therefore random matrices provide a powerful tool for studying statistical physics on a random surface. (The 2-matrix model was first introduced in this context, as the Ising model on a random surface [22].) It has been understood for some time that the 1-matrix model is not general enough, since it cannot represent all models of statistical physics (e.g., it contains only the (p, 2) conformal minimal models). In order to recover the missing conformal models ((p, q) with p and q integers), it is necessary to introduce at least a two-matrix model [9]. The 1-matrix models are actually included in the 2-matrix models, since if one takes d2 = 1, and integrates over the Gaussian matrix M2 , one sees that the 1-matrix model follows, and hence may be seen as a particular case. It should also be mentioned that most of the results about the 2-matrix model (in particular those derived in the present work) can easily be extended to multi-matrix models without major modifications (see the appendix of [2]). Indeed, the multi-matrix model is not expected to be very different from the 2-matrix case [9]. (In particular, it contains the same conformal models.) The present paper is organized as follows: in Sect. 2, we present the required formalism for biorthogonal polynomials, beginning with the systems of differential and recursion relations they satisfy, and recalling the main definitions and results of [2]. We then derive the fundamental systems of solutions to the over-determined systems for the “windows” of biorthogonal polynomials in two ways: one, by exploiting the recursion matrices Q, P for the biorthogonal polynomials, which satisfy the string equation, and another by giving explicit integral formulas for solutions and showing their independence when taken over a suitably defined homology basis of inequivalent integration paths. In Sect. 3 we use saddle-point integration methods to deduce the asymptotic form of these fundamental systems of solutions within the various Stokes sectors, and from these, to deduce the Stokes matrices and jump discontinuities at ∞. The full formulation of the matrix Riemann-Hilbert problem characterizing these solutions is given in Theorem 3.1. 2. Differential Systems for Biorthogonal Polynomials and Fundamental Solutions 2.1. Biorthogonal polynomials, recursion relations and differential systems. The definitions and notation to be used generally follow [2], with some minor modifications, and will be recalled here. Let us fix two polynomials which will be referred to as the “potentials”, V1 (x) = u0 +

d 1 +1 K=1

uK K x , K

V2 (y) = v0 +

d 2 +1 J =1

vJ J y . J

(2.1.1)

In terms of these potentials we define a bimoment functional [5], i.e., a bilinear pairing between polynomials π(x) and σ (y) by means of the following formula: 1 π, σ := dxdy e− (V1 (x)+V2 (y)−xy) π(x)σ (y) . (2.1.2) (x)

(y)

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The contours of integration (x) , (y) in the complex x and y planes remain to be specified. They will generally be chosen as differentiable contours on the respective Riemann spheres that begin and end at the point at ∞, approaching it in a direction that assures convergence. More generally, linear combinations of such integrals along various inequivalent contours may also be chosen. In fact there are precisely d1 (homologically) independent choices for the individual contours in the x plane and d2 in the y-plane, due to the choice of the integrands [5]. The convergence of these integrals is guaranteed by choosing the contours in such a way that they are differentiable in the compact topology of the respective Riemann spheres and such that their tangent line at ∞ satisfies (V1 (x)) −→ +∞, x→∞

(V2 (y)) −→ +∞.

(2.1.3)

y→∞

This amounts to requiring that the asymptotic directions at infinity lie within certain sectors defined below. To simplify even further we could choose these contours to be straight lines in a neighborhood of ∞. For k = 0, . . . 2d1 +1 (mod 2d1 +2) and l = 0, . . . 2d2 +1 (mod 2d2 + 2), let us define the sectors

(2k − 1)π (2k + 1)π (x) , ϑy + , (2.1.4) Sk := x ∈ C , arg(x) ∈ ϑx + 2(d1 + 1) 2(d1 + 1) ϑx := − arg(ud1 +1)/(d1 + 1) ,

(2l − 1)π (2l + 1)π (y) Sl := y ∈ C , arg(y) ∈ ϑy + , ϑy + , (2.1.5) 2(d2 + 1) 2(d2 + 1) ϑy := − arg(vd2 +1 )/(d2 + 1) . (x)

Definition 2.1 (See Fig. 1). The wedge contours in the x-plane (resp. y-plane) i (y) (x) (resp. j ) for i = 0, . . . d1 (resp. j = 0, . . . d2 ), and the anti-wedge contours ˜ i (y) (resp. ˜ ) for i = 0, . . . d1 (resp. j = 0, . . . d2 ) are defined as follows: j

(x)

• i

(y)

(x)

(y)

(resp. j ) comes from ∞ within the sector S2i−2 (resp. S2j −2 ) and returns (x)

(y)

to infinity in the sector S2i (resp. S2j ). (y) (y) (x) (x) • ˜ i (resp. ˜ j ) comes from ∞ within the sector S2i−1 (resp. S2j −1 ) and returns (x)

(y)

to infinity in the sector S2i+1 (resp. S2j +1 ). (x)

They have the property that V1 (x) → ∞ on the wedge contours i and V1 (x) → (x) −∞ on the anti-wedge contours i as |x| → ∞ (and similarly for V2 (y)). Note that since there are no singularities in the finite region of the x- or y-planes for (y) (x) the integrands we are considering, we can deform the contours j and k arbitrarily in the finite part of the x and y planes. We could therefore also take such contours as straight rays coming from the (2k −2)th sector combined with another ray going to infinity within the 2k th sector. Moreover, by the Cauchy theorem, we have the homological1 relation 1 The reference to “homology” in this context is coined by analogy. What is meant is simply that, considering these contours as associated to linear functionals over the differentials of the form π(x)e−V1 (x) dx or σ (y)e−V2 (y) dy (π and σ arbitrary polynomials) they satisfy the linear relation (2.1.6) and therefore they do not represent linearly independent linear functionals.

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Fig. 1. An example of wedge (solid) and anti-wedge (line-dot-line) contours in the x-plane for a case with d1 = 7 and real positive leading coefficient in V1 . The white sectors are the even numbered ones and the gray sectors the odd-numbered ones. (Only the first few are labelled)

d1

(x)

i

d2

≡ 0,

(y)

j

≡ 0 , homologically .

(2.1.6)

j =0

i=0

(y)

There are therefore only d2 homologically independent contours k and d1 of type (x) k . (Similar remarks apply to the anti-wedge contours which will be needed later in this article.) In general, for any complex d1 × d2 matrix with elements {κ i,j } i=1,...d1 , define the j =1,...d2

homology class of contours κ to be κ :=

(x)

κ i,j i

(y)

× j .

(2.1.7)

ij

For brevity, we will denote the corresponding integral operator as follows: d1 d2 i=1 j =1

κ i,j

(x)

(y)

i × j

=:

κ

.

(2.1.8)

For a given κ, we denote the corresponding bilinear pairing as: π, σ κ :=

κ

1

dx dy π(x)e− (V1 (x)+V2 (y)−xy) σ (y) ,

(2.1.9)

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M. Bertola, B. Eynard, J.Harnad

and define two sequences of monic polynomials πn (x), σn (y) of degree n such that they are biorthogonal with respect to this pairing (2.1.10) πn , σm κ = hn δnm , πn (x) = x n + ... , σn (y) = y n + ... .

(2.1.11)

The matrix κ i,j must be chosen such that the N × N finite submatrices of the matrix of bimoments (2.1.12) {Bij := x i , y j }i,j =0,...,N −1 are nonsingular for all N ; i.e., the nondegeneracy condition that ensures the existence of the biorthogonal polynomials is given by N (κ) := det x i , y j κ = 0 ∀N ∈ N . (2.1.13) i,j =0,...,N

It can be proved that these expressions are not identically zero (see [13]) for the case of real potentials or [14] using Heine’s formula, or it may be seen to follow from the linear independence of the corresponding moment functionals [5]). Since these N (κ)’s form a denumerable sequence of homogeneous polynomials in the coefficients κ i,j , this will hold on the complement of a denumerable set of hypersurfaces, and in this sense the permissible choices of κ’s are “generic”, whatever the choice of potentials V1 and V2 . We introduce the corresponding normalized quasipolynomials and combine them into semi-infinite column vectors (“wave-vectors”) (x), (y) as follows: ∞

∞

1 1 1 1 ψn (x) := √ πn (x)e− V1 (x) , φn (y) := √ σn (y)e− V2 (y) , hn hn (2.1.14) (x) := [ψ0 (x), ..., ψn (x), ...]t , (y) := [φ0 (y), ..., φn (y), ...]t .

∞

∞

In matrix notation the biorthogonality reads xy dxdy e (x) t (y) = 1 , ∞

κ

∞

(2.1.15)

where 1 denotes the semi-infinite unit matrix. We denote the Fourier-Laplace transforms of the wave-vectors along the wedge contours by xy dx e t (x) , i = 0, . . . , d1 , (2.1.16) (i) (y) := ∞

(j ) (x) := ∞

∞

(x)

i

(y) j

xy

dy e t (y) , j = 0, . . . , d2 , ∞

(2.1.17)

) (x) respectively. (y) and φ (j viewed as row vectors, with components denoted ψ (i) n n Because of Eq. (2.1.6) only d1 (or d2 ) are linearly independent d1 i=0

(i) (y) = ∞

d2 j =0

(j ) (x) ≡ 0. ∞

(2.1.18)

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The recursion relations for the biorthogonal polynomials are expressed by the following matrix equations for the wave-vectors [2]: x (x) = Q (x) ,

∂x (x) = −P (x) ,

y (y) = P t (y) ,

∂y (y) =

∞

∞

∞

∞

∞

∞

∞ −Qt (y) ∞

(2.1.19) ,

(2.1.20)

where the matrices Q and P have finite band sizes d2 + 1 and d1 + 1 respectively, with Q having only one nonzero diagonal above the principal one and P only one below 

α0 (0) α1 (1) .. .

γ (0) α0 (1) .. .

0 γ (1) .. .

0 0 .. .

··· ··· .. .



      Q :=  ,  α (d ) α   d2 2 d2 −1 (d2 ) · · · α0 (d2 ) γ (d2 )  .. .. .. .. . . . . 0   β0 (0) β1 (1) · · · βd1 (d1 ) ···   ..  γ (0) β0 (1) β1 (2) . βd1 (d1 + 1)      .. ..  . . . (2) 0 γ (1) β P :=  0     . .  0  . 0 γ (2) β0 (3)   .. .. .. .. .. . . . . .

(2.1.21)

(2.1.22)

These semi-infinite matrices satisfy the string equation [P , Q] = 1 .

(2.1.23)

(See [2, 12, 14] for proofs of these assertions.) For the dual sequences of Fourier–Laplace transforms, a simple integration by parts in Eqs. (2.1.19), (2.1.20) gives x (j ) (x) = (j ) (x)Q ,

∂x (j ) (x) = (j ) (x)P , j = 0, . . . , d2 ,

(2.1.24)

y (k) (y) = (k) (y)P t ,

∂y (k) (y) = (k) (y)Qt , k = 0, . . . , d1 .

(2.1.25)

∞

∞

∞

∞

∞

∞

∞

∞

Notice that integration by parts is allowed due to the exponential decay of the integrand along the chosen contours. We recall from [2] the definition of dual sequences of windows. Definition 2.2. Call a window of size d1 + 1 or d2 + 1 any subset of d1 + 1 or d2 + 1 consecutive elements of type ψn , φ n , φn or ψ n , with the notations := [ψN−d2 , . . . , ψN ]t ,

N ≥ d2 ,

:= [φN−d1 , . . . , φN ]t ,

N ≥ d1 ,

N

N

N

:= [ψN−1 , . . . , ψN+d1 −1 ]t , N

t

:= [φN−1 , . . . , φN+d2 −1 ] ,

N ≥ 0, N ≥ 1,

(2.1.26)

(2.1.27)

200

M. Bertola, B. Eynard, J.Harnad ) ) (j ) := [ψ (j , . . . , ψ (j ] N−d N

N ≥ d2 ,

, . . . , φ (k) ], (k) := [φ (k) N−d N

N ≥ d1 ,

2

N

1

N

N

) ) (j ) := [ψ (j , . . . , ψ (j N−1 N+d

1 −1

N ≥ 1,

],

N

(k) := [φ (k) , . . . , φ (k) N−1 N+d

2

(2.1.28)

(2.1.29)

N ≥1.

], −1

Note the difference in positioning and size of the windows in the various cases, and the fact that the barred quantities are defined as row vectors while the unbarred ones are column vectors. The notation used here differs slightly from that used in [2], N+1

N

what is denoted, e.g., here would be denoted in [2]. N

N

Definition 2.3. Each of the two pairs of windows ( , ) and ( , ) of dimensions N

N

d2 + 1 and d1 + 1 respectively, will be called dual windows (and these are defined for N ≥ d2 and N ≥ d1 respectively). We now recall the notion of folding, as defined in [2]. We give here only the main statements without proofs and refer the reader to ref. [2] for further details. Let us introduce N

the sequence of companion–like ladder matrices a (x) and a(x) of size d2 + 1, N



0

1

  a (x) :=  N 

0 0

0 0 ···

    a(x) :=    

N

−αd2 (N) γ (N) x−α0 (N) γ (N−1)

−α1 (N+1) γ (N−1)

.. . −αd2 (N+d2 ) γ (N−1)

0 .. . 0

0



0 1

   , 

−α1 (N) (x−α0 (N)) γ (N) γ (N)

1 0 · 0 ··

0

N ≥ d2 ,

(2.1.30)



 0   , N ≥ 1,  0 0 1  0 0 0

(2.1.31)

N

and also the analogous sequence of matrices b (y) and b (y) of size d1 + 1, N



0

1

  b (y) :=   N

0 0

0 0 ···

    b (y) :=    

N

−βd1 (N) γ (N) y−β0 (N) γ (N−1)

−β1 (N+1) γ (N−1)

.. . −βd1 (N+d1 ) γ (N−1)

0 .. . 0

0



0 1

   , 

−β1 (N) (y−β0 (N )) γ (N) γ (N)

1 0 · 0 ··

0

N ≥ d1 ,

(2.1.32)



 0   , N ≥ 1.  0 0 1  0 0 0

(2.1.33)

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The first equations in (2.1.19), (2.1.20) and (2.1.24), (2.1.25) imply the following N N

Lemma 2.1. The sequences of matrices a , b and a, b implement the shift N → N + 1 N N

and N → N − 1 in the windows of quasi-polynomials and Fourier–Laplace transforms in the sense that N

N+1

N

a (x) = (x) ,

(x) = (x) a ,

b (y) = (y) ,

(y) = (y) b ,

N N

N+1

N N

N+1

N

N+1

(2.1.34)

N

(2.1.35)

and in general =

N+j

=

N+j N

N

N+j N+j −1

a

··· a ,

=

b

···b ,

=

N+j −1

N+j −1

N N

N N

N

N

··· a ,

a

N+j N+j −1

b

(2.1.36)

N

···b ,

(2.1.37)

N

where , here denote windows in the Fourier–Laplace transforms defined in Eqs. (2.1.16), (2.1.17). (In this lemma, and in what follows, if no superscript distinguishing the integration N

N

path in the Fourier-Laplace transform in Eqs. (2.1.16), (2.1.17) is present in , , this means that the result being discussed holds for all cases.) Equations (2.1.34), (2.1.35) will henceforth be referred to as the “ladder relations”. We refer to the process of expressing any ψn (x) or φ n (x) by means of linear combinations of elements in a specific window with polynomial coefficients, as folding onto N

N

the specified window. We also recall that matrices a (y) and a(y), b (y) and b (y) are N

N

invertible (see [2]). The opposite shifts are therefore implemented by the inverse matrices and the folding may take place in either direction with respect to polynomial degrees. Lemma 2.2 (From [2], with the adapted notation.). The windows of quasi-polynomials N

N

(x), (y) and Fourier–Laplace transforms (x), (y) satisfy the following differenN

N

tial systems: N ∂ (x) = D1 (x) (x) , N ≥ d2 + 1 , ∂x N N N N ∂ N (x) = (x)D 1 (x) , N ≥ d2 + 1 , ∂x

−

N ∂ (y) = D2 (y) (y) , N ≥ d1 + 1 , ∂y N N N N ∂ N (y) = (y)D 2 (y) , N ≥ d1 + 1 , ∂y

−

(2.1.38) (2.1.39)

(2.1.40) (2.1.41)

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M. Bertola, B. Eynard, J.Harnad

where N

N

−1 + N−1

D1 (x) :=β −1 a N

N

N

N

β0 +

d1

N

βj

j =1

N

D 1 (x) := a −1 β −1 + β 0 +

a

a

N+j −1 N+j −2

· · · a ∈ gld2 +1 [x] , (2.1.42) N

d1 N−j N N −1 N−2 a a · · · a β j ∈ gld2 +1 [x] ,

(2.1.43)

j =1

N β j := diag βj (N + j − d2 ), βj (N + j − d2 + 1), . . . , βj (N + j ) , j = 0, . . . d1 , (2.1.44) N β −1 := diag γ (N − d2 − 1), . . . , γ (N − 1) , (2.1.45) N β j := diag βj (N − 1), βj (N), . . . βj (N + d2 − 1) , j = 0, . . . , d1 , (2.1.46) N

β −1 := diag (γ (N − 1), γ (N), . . . , γ (N + d2 − 1)) , N

N

N

D2 (y) :=α −1 b −1 + α0 + N−1

N

N

N

d2 j =1

N

D 2 (y) := b −1 α −1 + α 0 +

N

αj

b

b

N+j −1 N+j −2

(2.1.47)

· · · b ∈ gld1 +1 [y] , (2.1.48) N

d2 N−j N N −1 N−2 b b · · · b α j ∈ gld1 +1 [y] ,

(2.1.49)

j =1

N α j := diag αj (N + j − d1 ), αj (N + j − d1 + 1), . . . , αj (N + j ) , j = 0, . . . d2 , (2.1.50) N α −1 := diag γ (N − d1 − 1), . . . , γ (N − 1) , (2.1.51) N α j := diag αj (N − 1), αj (N), . . . αj (N + d1 − 1) , j = 0, . . . , d2 , (2.1.52) N α −1 := diag (γ (N − 1), γ (N ), . . . , γ (N + d1 − 1)) , (2.1.53) Moreover the systems (2.1.38)–(2.1.41) are compatible with the ladder relations (2.1.34), (2.1.35) since N+1 D1 (x) N+1

N

= a (x) D1 (x)a −1 (x) − a (x)a −1 (x), N

N

N

N

N

N

N

N

(2.1.54)

N

D 1 (x) = a(x) D 1 (x) a−1 (x) − a (x) a−1 (x) ,

(2.1.55)

N+1 D2 (y)

(2.1.56)

N+1

N

= b (y) D2 (y) b −1 (y) − b (y) b −1 (y), N N

N

N

N

N

N

N

N

D 2 (y) = b (y) D 2 (y) b−1 (y) − b (y) b−1 (y) .

(2.1.57)

Remark 2.1. We stress that the windows taken from any wave-vector solution to Eqs. (2.1.19), (2.1.20) and Eqs. (2.1.24), (2.1.25) will automatically satisfy the systems specified in Lemma 2.2 (differential and difference equations).

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203

In the following, we will only consider the fundamental systems of solutions to the sequence of Eqs. (2.1.38), (2.1.39), (2.1.54), (2.1.55), since the corresponding dual N

sequence of equations for the quantities (y), (y) may be treated analogously by just N

making the appropriate interchange of notations x ↔ y, ↔ , d1 ↔ d2 , etc., and the interchange of integration contours in the x and y planes. In [2] it was shown that there exists a natural nondegenerate Christoffel–Darboux N

(x) to Eqs. (2.1.34) and (2.1.39) such (x) and pairing between any pair of solutions N

that

N

N

, N

N

(x) A (x) , :=

(2.1.58)

N

N

N

is constant both in x and N , where the invertible matrix A defining the pairing is  −γ (N − 1) 0 0 0 0   αd2 (N ) ··· α2 (N ) α1 (N ) 0 N   αd2 (N + 1) ··· α2 (N + 1) 0 (2.1.59) A :=  0   0 0 αd2 (N + 2) ··· 0 0 0 0 αd2 (N + d2 − 1) 0 

N

N

This follows from the fact that the matrices D1 and D 1 are conjugate to each other by N

means of the matrix A (Theorem 4.1 in [2]) N N

N

N

A D1 (x) = D 1 (x) A ,

(2.1.60) N

together with the shift relation (2.1.54). The (d2 + 1) × (d2 + 1) matrix A is the only N−1

N−1

0

0

, Q , where denotes the projector onto the

nonzero block in the commutator

first N basis elements (a “canonical” projector). In the following it is convenient to use N

the same notation A both for the finite matrix and the semi-infinite one. It was also proved in [2] (Proposition 3.3) that one can choose the windows N

N

as joint solutions of the PDE’s following from the infinitesimal changes of the and coefficients {uJ } and {vK } of the potentials (deformation equations). For this choice, the pairing also becomes independent of these deformation parameters.

2.2. Fundamental solutions of the D1 and D 1 systems. In this section we explicitly N

N

construct solutions of the pairs of dual ODE’s defined by the matrices D1 (x) and D 1 (x). In fact these solutions will simultaneously satisfy the deformation equations and the ladder recursion relations in N. It will be left to the reader to formulate the correspondN

N

ing statements for the other pairs (D2 (y), D 2 (y)), which are essentially the same, mutatis mutandis.

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As mentioned in Remark 2.1, taking windows within any wave-vector solution to Eqs. (2.1.19), (2.1.24) one obtains solutions to the difference-differential equations in Lemma 2.2. Therefore we could try to construct d2 + 1 such wave-vector solutions in order to obtain a fundamental system for Eqs. (2.1.39). On the other hand it will be shown in Proposition 2.1 below that the relations (2.1.24) have precisely d2 linearly independent wave-vector solutions (given by the d2 Fourier–Laplace transforms (2.1.17)) while relations (2.1.19) have only one solution (given by the quasipolynomials). Therefore we will look for solutions to slightly modified recursion relations differing from (2.1.19), (2.1.25) by changing the initial terms2 . The solutions to these modified recursion relations {ψ˜ n (x)}n≥0 or {ψ˜n (x)}n≥0 will still satisfy the unmodified recursions relations, but only for n large enough. This will provide us with the required solutions of the difference-differential equations of Lemma 2.2. We start by proving that the Fourier–Laplace transforms and the quasipolynomials are the only wave-vector solutions to Eqs. (2.1.19), (2.1.24). Notational remark. Throughout the remainder of this section, (x) and (x) will be ∞

∞

used generally to denote arbitrary wave-vectors consisting of arbitrary solutions and not just the previously defined quasipolynomials and their Fourier–Laplace transforms. Proposition 2.1. The semi-infinite systems x (x) = Q (x) ∞

∞

∂x (x) = −P (x) ∞

,

(2.2.1)

∞

  x (x) = (x)Q ∞

∞

(2.2.2)

 ∂x (x) = (x)P ∞

∞

have 1 and d2 linearly independent solutions, respectively, for the wave-vectors (x) ∞

and (x). (A similar statement holds for , , as solutions to Eqs. (2.1.20) and (2.1.25) ∞

∞

with d2 → d1 ).

∞

Proof. The compatibility is guaranteed by the string equation (2.1.23). Recalling [2] that = Q − V2 (P ) =0, (2.2.3) P − V1 (Q) ≥0

we have

P − V1 (Q)

≤0

= −ψ0 (x) − V1 (x)ψ0 (x),

(2.2.4)

0 = Q − V2 (P ) = xφ 0 (x) − V2 (∂x )φ 0 (x) .

(2.2.5)

0=

∞ 0

∞

0

Thus we have only one solution of the first equation (up to normalization) and d2 independent solutions for the second. Using the x recursion relations for the ψ 2 This is not new in the context of orthogonal polynomials, where there exist, besides the orthogonal polynomials, solutions of the second kind. (See e.g. [8, 25]).

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sequence, we can build the rest of the sequence by starting from the first term ψ0 (x), since the matrix Q has only one nonzero diagonal above the principal one. On the other hand, using the ∂x recursion relations we can build the rest of the φ n sequence starting from any given solutions φ 0 of (2.2.5) because the matrix P has only one diagonal below the main one. The solutions in Proposition 2.1 are, up to multiplicative constants, those given by the quasipolynomials for the wave-vector and the different possible ’s corresponding ∞

∞

to the different solutions of Eq. (2.2.5), which are just the d2 different Fourier–Laplace transforms of the quasipolynomials φn (y). Indeed, the solutions φ 0 (x) of Eq. (2.2.5) can be expressed by 1 φ 0 (x) ∝ dy e (xy−V2 (y)) , (2.2.6)

(y) k

where is any of the contours or a linear combination of them. As indicated in the introduction to this section, in order to find a fundamental system of solutions to the Differential-Difference equations of Lemma 2.2, we consider systems for the wave-vectors modified by the addition of terms that only change the equations involving the first few entries of the wave-vectors. The recursion relations will therefore remain unchanged for n sufficiently large. Proposition 2.2. The semi-infinite systems: x (x) = Q (x) − W2 (−∂x )F (x) ∞

∞

,

∂x (x) = −P (x) + F (x) ∞

(2.2.7)

∞

  x (x) = (x)Q + U (x) ∞

∞

 ∂x (x) = (x)P + U (x)W1 (x) ∞

,

(2.2.8)

∞

where V1 (Q) − V1 (x) , Q−x F (x) := [f (x), 0, 0, · · ·]t ,

W1 (x) :=

V2 (P ) − V2 (∂x ) , (2.2.9) P − ∂x U (x) := [u(x), 0, 0, · · ·] , (2.2.10) W2 (∂x ) :=

both have d2 + 1 linearly independent solutions for the extended unknown wave-vectors f (x), (x) and u(x), (x) . ∞

∞

Remark 2.2. The terms in the RHS of Eqs. (2.2.7), (2.2.8) which are not present in Eqs. (2.2.1), (2.2.2) are semi-infinite vectors with at most d2 nonzero entries for (2.2.7) or d1 for (2.2.8). Indeed the matrices W1 and W2 (which are polynomials in Q and P of degrees d1 − 1 and d2 − 1 respectively) have finite band sizes, and hence when acting on the vectors U (x) and F (x), which have only a first nonzero entry, produce a finite number of nonzero entries. As a result, windows taken from any solution of the modified wave-vector systems (2.2.7), (2.2.8) will provide solutions of the Differential-Difference systems of Lemma 2.2, but only beyond a minimal N value (d1 or d2 respectively).

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Proof. The compatibility of these systems is not obvious. We have: = [∂x , x] = ∂x Q −W2 (−∂x )F − x −P +F (2.2.11) ∞ ∞ ∞ ∞ = Q −P +F − ∂x W2 (−∂x )F ∞ +P Q −W2 (−∂x )F + xF (2.2.12) ∞

V (P ) − V2 (−∂x ) F (x) + xF (2.2.13) = [P , Q] +QF − (∂x + P ) 2 ∞ P + ∂x = + Q − V2 (P ) F + x − V2 (−∂x ) F. (2.2.14) ∞

Since only the first entry of F is nonzero and Q − V2 (P ) is strictly upper-triangular [2], the second term vanishes. The last term gives the following ODE for the first entry of F (x): V2 (−∂x )f (x) = xf (x). (2.2.15) The solutions of Eq. (2.2.15) are easily written as Fourier–Laplace integrals, and give a compatible system (see Eq. (2.3.16), where we also fix the most convenient normalization for them). We denote them by f (α) with α = 0, . . . , d2 , with f (0) ≡ 0 the trivial solution, corresponding to the unmodified system (2.2.1). Fixing any such solution f (x), we can now solve for . First of all, one can prove by induction that Qk − x k Qk = x k + (2.2.16) W2 (−∂x )F. ∞ ∞ Q−x Next we compute as for the previous proposition 0 = P − V1 (Q) ∞ 0 V1 (Q) − V1 (x) W2 (−∂x )F = −∂x +F − V1 (x) − (2.2.17) ∞ ∞ Q−x 0 = −ψ0 (x) − V1 (x)ψ0 (x) + [1 − W1 (x)W2 (−∂x )]00 f (x). (2.2.18) Thus, ψ0 (x) must solve this first order system of ODE’s. Since there are d2 + 1 choices for the function f = f (α) , we correspondingly obtain d2 + 1 independent solutions (α) (x) to the system. Now consider the second system (2.2.8). The compatibility gives = [∂x , x] = ∂x Q + U − x P + U W1 (x) (2.2.19) ∞ ∞ ∞ ∞ = P + U W1 (x) Q + ∂x U − Q + U P − U W1 (x)x (2.2.20) ∞

∞

V (Q) − V1 (x) (Q − x) + ∂x U = [P , Q] − U P + U 1 Q−x ∞ = −U P − V1 (Q) + ∂x − V1 (x) U (x). ∞

(2.2.21) (2.2.22)

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207

Since the first entry of U is nonzero and P − V1 (Q) is strictly lower-triangular, the second term vanishes. The last term gives the following ODE for the first entry of U (x): 1

u (x) = V1 (x)u(x) ⇒ u(x) = ce V1 (x) or u(x) ≡ 0 .

(2.2.23)

We next consider the solutions . By a computation similar to the previous one, we have ∞

0 = Q − V2 (P ) 0 ∞ = x −U ∞

← − V (P ) − V2 ( ∂ x ) − V2 (∂x ) +U W1 (x) 2 ← − ∞ P −∂ x

← − = x − V2 (∂x ) φ0 (x) − u(x) 1 − W1 (x)W2 ( ∂ x )

00

(2.2.24) 0

(2.2.25)

.

This is a d2th order inhomogeneous ODE for the function φ 0 (x). Choosing u(x) ≡ 0 as solution to (2.2.25) we have d2 independent solutions corresponding to the “unmodified” system (2.2.2). The choice 1

u(x) = ce V1 (x)

(2.2.26)

leads to one more independent solution of the inhomogeneous equation.

We denote the solutions to system (2.2.8) in general as (α) , α = 0, . . . d2 , where ∞

α = 0 corresponds to the inhomogeneous solutions and α = 1, . . . d2 correspond to the independent solutions to the homogeneous case (U (x) ≡ 0). We will give explicit integral representations for the d2 + 1 solutions in the next section. With the solutions (α) , (β) , α, β = 0, . . . , d2 , we can construct the (d2 + 1) × (d2 + 1) modified kernels

∞

∞

N (α,β)

K 11

(x, x ) :=

N−1

N−1

φ (α) (x)ψn(β) (x ) = (α) (x) (β) (x ) , n ∞

n=0

0

∞

(2.2.27)

and also obtain the following modified Christoffel–Darboux formulae (cf. [2]). Proposition 2.3 (Christoffel–Darboux Kernels). N (α,β)

(x − x ) K 11 (x, x ) N−1 N−1 (α) = (x)Q + δα0 U (x) (β) (x ) − (α) (x) ∞ 0 0 ∞ ∞ × Q (β) (x ) − W2 (−∂x )F (β) (x ) ∞ N−1 (β) (α) = − (x) , Q (β) (x ) + δα0 u(x)ψ0 (x ) ∞

0 N−1 0

(2.2.29)

∞

+ (α) (x) W2 (−∂x )F (β) (x ) . ∞

(2.2.28)

(2.2.30)

208

M. Bertola, B. Eynard, J.Harnad N−1

Recall that the commutator of the finite band matrix Q with the projector gives 0

a finite-rank semi-infinite matrix that corresponds to the Christoffel–Darboux kernel N

matrix A. As a corollary, we obtain the pairing between the solutions of the systems (2.1.39) by setting x = x . Corollary 2.1. N N N N−1 (β) (β) (α) (α) (α) := (x) A (x) = (x) , Q (β) (x) , N

∞

N

N

0

∞

N−1

(β)

= δα0 u(x)ψ0 (x) + (α) (x) W2 (−∂x )F (β) (x) . (2.2.31) ∞

0

As shown in [2], this is a constant (in x) if N > d2 . But if N > d2 the projector in the second term is irrelevant because the vector W2 (−∂x )F (β) (x) has only its first d2 + 1 entries nonvanishing, and hence we have, for N > d2 , N V (P ) − V2 (−∂x ) (β) (β) (α) (β) , = δα0 u(x)ψ0 (x) + (α) (x) 2 F (x) . (2.2.32) P + ∂x N ∞ N If α = 0 then (α) (x)P = ∂x (α) (x) , ∞

and hence we have, for N > d2 , N

∞

(2.2.33)

α = 0,

← − − → V2 ( ∂ x ) − V2 (− ∂ x ) (β) F (x) ← − − → ∞ ∂ x +∂ x V (∂x ) − V2 (−∂x ) (α) (β) = 2 φ 0 (x )f (x) . ∂x + ∂x x =x

(α) (x) A (β) (x) = (α) (x) ∞

∞

(2.2.34)

In this expression φ 0 and f are kernel solutions of the pair of adjoint differential equations (V2 (∂x ) − x)φ (α) (x) = 0 , (V2 (−∂x ) − x)f (β) (x) = 0 , 0

(2.2.35)

and the last expression in Eq. (2.2.34) is just the bilinear concomitant of the pair (which is a constant in our case). For β = 0 we have N 1 1 1 c (0) (α) (x) A (0) (x) = δα0 u(x)ψ0 (x) = δα0 ce V1 (x) √ e− V1 (x) = δα0 √ . ∞ h0 h0 ∞ (2.2.36)

For completeness we recall that the matrices P , Q satisfy the following deformation equations: (2.2.37) ∂uK Q = − Q, UK , ∂vJ Q = Q, VJ , ∂uK P = − P , UK , ∂vJ P = P , VJ , (2.2.38)

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209

where

1 K 1 K 1 J 1 J J , V := − . + + U := − Q Q P P >0 0 <0 0 K 2 J 2 (2.2.39) (Note that in [2] the matrix that here is denoted by P was denoted −P .) Here the subscripts < 0 (resp. > 0) mean the part of the matrix with non-zero entries only below (resp., above) the principal diagonal and the subscript 0 denotes the diagonal. K

Proposition 2.4. The equations in Proposition 2.2 are compatible with the deformation equations   ∂uK (x) = UK (x) ∞ ∞ , (2.2.40)  ∂ F (x) = UK F (x) uK  1 P J − (−∂x )J   F (x)  ∂vJ (x) = −VJ (x) − ∞ ∞ J P + ∂x , (2.2.41) (−∂x )J  J   ∂vJ F (x) = + V00 F (x) J  1 QK − x K    ∂uK (x) = − (x)UK + U (x) Q−x , ∞ K ∞ (2.2.42) xK  K   ∂uK U (x) = + U00 U (x) K ∂vJ (x) = (x)VJ ∞ ∞ (2.2.43) ∂vJ U (x) = U (x)VJ . Proof. We only sketch the proof, which is quite straightforward but rather long. Compatibility of the deformation equations amongst themselves follows from the zero curvature equations

[∂uK + UK , ∂uK + UK ] = [∂vJ − VJ , ∂vJ − VJ ] = [∂uK + UK , ∂vJ − VJ ] = 0 (2.2.44) (which are established in the general theory [26, 2]), together with the fact that both vectors U (x) and F (x) have, by assumption, nonzero entries only in the first position. Compatibility with Eqs. (2.2.7), (2.2.8) requires some more computation. We first prove the compatibility between Eq. (2.2.8) and the two Eqs. (2.2.42), (2.2.43). Compatibility between the equations involving multiplication by x and application of ∂vJ : x∂vJ (x) = (x)QVJ + U (x)VJ ∞

(2.2.45)

∞

∂vJ x (x) = (x)VJ Q + (x)[Q, VJ ] + U (x)VJ = RHS of (2.2.45) . (2.2.46) ∞

∞

∞

Compatibility between the equation involving multiplication by x and application of ∂uK : x∂uK (x) = − (x)QUK − U (x)UK + ∞

∞

QK − x K 1 U (x) x K Q−x

(2.2.47)

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M. Bertola, B. Eynard, J.Harnad

∂uK x (x) = ∂uK (x)Q − (x)[Q, UK ] + ∂uK U (x) ∞

∞

∞

1 QK − x K U (x) Q K Q−x ∞ K x K − (x)[Q, UK ] + U (x) + U00 K ∞ 1 = − (x)QUK + U (x)(QK − x K ) K ∞ K 1 QK − x K x K + U (x) U (x) x+ + U00 K Q−x K

(2.2.48)

= − (x)UK Q +

()

= RHS of (2.2.47),

(2.2.49)

(2.2.50) (2.2.51)

where in the step (), we have used the fact that (since only the first entry of U (x) is nonzero) 1 K U (x) = −U (x)UK . (2.2.52) U (x)QK + U00 K Compatibility between the ∂x and ∂uK equations is a bit more involved: 2 ∂x ∂vJ (x) = ∂x (x)VJ = (x)P VJ + U (x)W1 (x)VJ ∞ ∞ ∞ 2 ∂vJ ∂x (x) = ∂vJ (x)P + U (x)W1 (x) ∞

(2.2.53)

∞

= (x)VJ P + (x)[P , VJ ] + U (x)VJ W1 (x) ∞

∞

+U (x)[W1 (x), VJ ] = RHS of (2.2.53) .

(2.2.54)

Finally we have the ∂x , ∂vJ compatibility: QK − x K 1 2 ∂x ∂vJ (x) = ∂x − (x)UK + U (x) K ∞ ∞ Q−x = − (x)P + U (x)W1 (x) UK ∞

QK − x K 1 QK − x K 1 + ∂x U (x) + U (x)∂x K Q−x Q−x K K = − (x)P + U (x)W1 (x) U ∞

1 QK − x K + (∂x − V1 (x))U (x) K Q−x lower tri. QK − x K 1 + U (x) V1 (x) − V1 (Q) + V1 (Q) − P Q−x K 1 QK − x K QK − x K 1 U (x)P + U (x)∂x K Q−x K Q−x 1 () = − (x)P + U (x)W1 (x) UK − U (x)W1 (x)(QK − x K ) K ∞ QK − x K 1 P + U (x) K Q−x +

Differential Systems for Biorthogonal Polynomials

211

1 QK − x K (∂x − V1 (x))U (x) K Q−x QK − x K 1 + U (x)(∂x + ∂Q ) K Q−x xK 1 K = − (x)P − U (x)W1 (x) Q + UK + U (x)W1 (x) K K ∞ 1 QK − x K + U (x) P K Q−x QK − x K 1 QK−1 − x K−1 + (∂x − V1 (x))U (x) + U (x) K Q − x Q−x K 1 K x = − (x)P − U (x) Q + UK W1 (x) + U (x)W1 (x) K K ∞ K K 1 Q −x + U (x) P K Q−x 1 QK − x K QK−1 − x K−1 + (∂x − V1 (x))U (x) + U (x) K Q−x Q−x K U ] −U (x)[W1 (x), K x 1 QK − x K () K = − (x)P + + U00 U (x)W1 (x) + U (x) P K K Q−x ∞ 1 QK − x K QK−1 − x K−1 + (∂x − V1 (x))U (x) + U (x) K Q−x Q−x −U (x)[W1 (x), UK ] . (2.2.55) +

In step () we have used the fact that the commutator of P with a function of Q is equivalent to the (formal) derivative ∂Q , while in step ( ) we have used Eq. (2.2.52). Computing the derivatives in the opposite order we obtain

∂vJ ∂x (x) = ∂vJ (x)P + U (x)W1 (x) ∞ K ∞ QK − x K x 1 K K P+ + U00 U (x)W1 (x) = − (x)U + U (x) K Q−x K ∞ x K−1 − QK−1 +U (x) − U (x)[W1 (x), UK ]. (2.2.56) x−Q 2

Subtracting Eq. (2.2.55) from Eq. (2.2.56), we are left with 0 = RHS of (2.2.55) – RHS of (2.2.56) QK − x K 1 ⇒ ∂x − V1 (x) U (x) = 0 , (2.2.57) = (∂x − V1 (x))U (x) K Q−x where the implication follows from the fact that only the first entry of U (x) is nonzero. This is precisely the same compatibility condition that we found in Prop. 2.2. The other compatibility checks are also rather long but routine and are left to the reader to fill in.

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2.3. Explicit integral representations for the dual wave-vectors. Proposition 2.5. The d2 + 1 semi-infinite wave-vectors (α) and functions u(α) (x), α = ∞

0, . . . d2 defined by ∞

(0)

1 V1 (x)

(x) = e

" 1 e− (V1 (s)−sy) t ds dy (y) , u(0) (x) := h0 e V1 (x) (2.3.1) x−s κ

(k) (x) = ∞

1

xy

(y) k

dy e t (y) , u(k) (x) ≡ 0,

k = 1, . . . , d2 ,

(2.3.2)

are independent solutions of the modified system (2.2.8). Proof. There is nothing to prove for (k) , since for k > 0 these are just the Fourier∞

Laplace transforms, which satisfy the corresponding unmodified system (2.2.1), which is the same as the modified system with u(x) ≡ 0. Let us therefore consider (0) . We ∞

first check the x recurrence relations. − 1 (V1 (s)−sy) − 1 (V1 (s)−sy) 1 (x − s)e e x (0) (x) = e V1 (x) ds dy s t (y) t (y) + ∞ ∞ x−s x−s ∞ κ 1 "

1 1 e− (V1 (s)+sy) = h0 e V1 (x) , 0, . . . − e V1 (x) ds dy ∂y t (y) ∞ x−s κ ()

= U (0) (x) + (0) (x)Q , ∞

where in () we have used Eq. (2.1.20) and defined the semi-infinite vector "

1 U (0) (x) := h0 e V1 (x) , 0, . . . .

(2.3.3)

It is easy to prove by induction in k, starting from the first part of Eq. (2.2.8), that x k (0) (x) = U (0) (x) ∞

x k − Qk + (0) (x)Qk . x−Q ∞

(2.3.4)

We now consider the ∂x differential recursion. After shifting the derivative in x to one in s inside the integral in Eq. (2.3.1), and integrating by parts, using Eq. (2.3.4), we get ∂x (0) (x) = U (0) (x) ∞

V1 (x) − V1 (Q) + (0) (x)P . x−Q ∞

(2.3.5)

This proves that the given integral expression is indeed the additional solution to Eq. (2.2.8). Remark 2.3. We would also have solutions for any other choice of admissible contours (or linear combinations of them) in Eq. (2.3.2). This arbitrariness will be used later.

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213

Remark 2.4. The functions φ (0) (x) are piecewise analytic functions in each connected n #d1 (x) component of Cx \ j =1 j , but can be analytically continued from each such connected component to entire functions by deforming the contours of integration. With regard to the deformation equations studied in Proposition 2.4, we have: Lemma 2.3. The wave-vectors (0) (x) (i.e. the quasipolynomials) and (k) (x), k = ∞

∞

1, . . . , d2 (i.e. the Fourier–Laplace transform of the quasipolynomials φn (y)) satisfy the following deformation equations: ∂uK (0) (x) = UK (0) (x) , ∞

∂vJ (0) (x) = ∞

∞ −VJ (0) (x) ∞ K (k)

∂uK (k) (x) = − ∞

∞ (k)

(2.3.6) ,

(2.3.7)

(x)U ,

(2.3.8)

∂vJ (k) (x) = (x)VJ . ∞

∞

(2.3.9)

Proof. This follows straightforwardly from the two equations (2.2.37), (2.2.38) together with the fact that taking a Fourier–Laplace transform commutes with differentiation with respect to the deformation parameters The given integral expressions also satisfy the modified deformation equations given by the following proposition. Proposition 2.6. The dual wave-vectors defined in Proposition 2.5 satisfy the following deformation equations: ∂uK (0) (x) = − (0) (x)UK + ∞ (0)

∂vJ ∞

(x) = ∞

∞ (0)

1 (0) x K − QK U (x) , K x−Q

(x)VJ ,

(2.3.10) (2.3.11)

where the semi-infinite vector U (0) (x) is defined in Eq. (2.3.3). Proof. As for the {vJ } deformation equations for (0) , the proof follows from ∞

Lemma 2.3 and from the fact that the integral transform defining this wave-vector does not depend on the vJ ’s. On the other hand, the uK deformations give (again using Lemma 2.3): ∂uK (0) (x) = − (0) (x)UK + ∞

∞

% $ 1 (0) (x) x K − QK , K∞

from which the proof follows using Eq. (2.3.4).

(2.3.12)

214

M. Bertola, B. Eynard, J.Harnad

Remark 2.5. Note that from the definition of the vector U (x), one can verify directly that it satisfies the deformation equations (2.2.42), (2.2.43) by using the equations for the normalization constant h0 : 1 dxdy e− (V1 (x)+V2 (y)−xy) , (2.3.13) h0 = κ

K J , ∂vJ ln(h0 ) = V00 . ∂uK ln(h0 ) = U00 2 2

(2.3.14)

Remark 2.6. In particular, the sequences of functions defined in Proposition 2.5 satisfy identical deformation equations for N large enough (N > d1 ) and hence the relevant windows satisfy the full DDD equations specified in [2] for the barred (dual) quantities. We conclude with a discussion regarding the construction of an integral representation for the wave-vectors . For the unmodified system (2.2.1) the solution is explicitly given ∞

by the quasipolynomials (0) (x). The other solutions of the wave-equations (2.2.7) are ∞

in 1 − 1 correspondence with the d2 solutions of Eq. (2.2.15) via Eq. (2.2.18). We could try to define an integral representation of the form

∞

(k)

(x) =

(y) k

1 dy 1 (V2 (y)−yx) e− (V2 (t)−zt) dz dt e (z) , ∞ 2iπ t −y κ

(2.3.15)

corresponding to the following solution of Eq. (2.2.15) √

f

(k)

h0 (x) := 2iπ

(y) k

1

dy e (V2 (y)−xy) ,

k = 1, . . . , d2 .

(2.3.16)

However formula (2.3.15) is not well-defined, since the inner integral defines only a (y) piecewise analytic function with jump discontinuities along the k ’s, and each of the (y) (y) anti-wedge contours k crosses at least one of the contours j . One could alternatively take the analytic continuation of the inner double integral from one connected #2 (y) component of Cy \ dk=1 k to an entire function, which amounts to deforming the contours in κ to avoid any jump discontinuities. But then the outer integral is in general divergent, due to the behavior of the integrand as y → ∞. Thus, as it stands, Eq. (2.3.15) may only be viewed as a formal expression which at best defines a divergent integral. However, treating it naively (as though it were uniformly convergent with respect to the parameters {uJ , vK }, as well as x) we would find that it indeed satisfies Eqs. (2.2.7), (2.2.40) and (2.2.41). Nonetheless we know that there are solutions (k) of Eq. (2.2.7), (2.2.40) corresponding to the f (k) (x) defined in ∞

(2.3.16), determined recursively from the solutions of Eq. (2.2.18). Such solutions are uniquely defined by the choice of a particular solution to Eq. (2.2.15), up to the addition of a solution of the homogeneous equation (that is, the quasipolynomials) forming (0) . ∞

There does exist a way to arrive at convergent integral representations, however, by forming suitable linear combinations of Fourier-Laplace transforms of a piecewise analytic function defined along appropriate contours. This was done for the case of cubic potentials by Kapaev in [21]. In Appendix A, it will be indicated how the procedure

Differential Systems for Biorthogonal Polynomials

215

used there may be extended to polynomial potentials of higher degree. For present purposes, however, we may view the solutions as just defined by the recursion relations (2.2.7), together with Eq. (2.2.18). It follows from compatibility that these satisfy the deformation equations as well. As a further remark, we note that the solutions (k) ( mod C (0) ) associated to the ∞

∞

f (k) s defined by Eq. (2.3.16), may be associated to any linear combination of the antiwedge contours, used as contours of integration in (2.3.16). This freedom will be used in the next section. 2.4. Diagonalization of the Christoffel–Darboux pairing. We know from Proposition 2.6 that the windows constructed from the integral representations in Proposition 2.5 provide solutions to the DDD equations for the barred system (for N > d1 ). On the other hand, we are guaranteed by Theorem 4.1 and Corollary 4.1 of [2] that the Christoffel– Darboux pairing between such solutions and any solution of the unbarred DDD system does not depend on x, N, or the deformation parameters determining the two potentials. It is of interest therefore to compute this pairing for the explicit solutions at hand. The solutions of the unbarred DDD equations are obtained by taking suitable windows in the wave-vector solutions of Eq. (2.2.7), consisting of the quasipolynomials (0) (x) ∞

and the solutions (k) (x) associated to the f (k) (Eq. 2.3.16). It should be clear that the ∞

diagonalization of the pairing depends on a careful choice of the contours of integration in Eqs. (2.3.2), (2.3.16). Here we prove that the choice of contours that diagonalizes the pairing (2.1.58) is linked to the notion of dual steepest descent-ascent contours, whose definition is given here and will be needed again in the following section. Definition 2.4. The steepest descent contours (SDC’s) and the dual steepest ascent contours (SAC’s) for integrals of the form 1 1 I (x) := dy e− (V2 (y)−xy) H (y) , I(x) := dy e (V2 (y)−xy) H (y) , (2.4.1)

respectively, passing through any saddle point yk (x), k = 0 . . . d2 − 1 with H (y) at most of exponential type, are the contours γk and γ˜k , respectively, uniquely defined by     γk := y ∈ C; (V2 (y) − xy) = (V2 (yk (x)) − xyk (x)) , (V2 (y)) −→ +∞ , y→∞   y ∈ γk

(2.4.2)    γk := y ∈ C; (V2 (y) − xy) = (V2 (yk (x)) − xyk (x)) , (V2 (y)) −→ −∞ . y→∞       

y∈ γk

(2.4.3) Here yk (x) denotes one of the d2 branches of the solution to the algebraic equation V2 (y) = x ,

(2.4.4)

which behaves like yk (x) ∼ (vd2 +1 ) x→∞

− d1

2

1

2iπ

ωk x d2 , ω := e d2 ,

(2.4.5)

216

M. Bertola, B. Eynard, J.Harnad (y)

(y)

S7

(y)

S6

(y)

S5

(y)

S4

S8 S S

(y)

(y)

S3

9

(y)

(y)

S2

10

3

2

4

(y)

S 11

(y)

S1

1 5 0

(y)

S 12

(y)

S0

6 10 (y)

(y)

S 13

S 23

7 9

8 (y)

(y)

S 14

S 22 (y)

(y)

S 15

S 21 (y)

(y)

S 16

(y)

S 17

(y)

S 18

(y)

S 20

S 19

Fig. 2. Example of the asymptotic (as |x| → ∞) SDC’s γ (solid) and SAC’s γ˜ (line-dot-line) in the y-plane for a potential of degree d2 + 1 = 12 with positive leading coefficient (vd2 +1 > 0) and for the non-Stokes line arg(x) = 0. The numbers at the vertices of the endecagon label the d2 = 11 critical points and the pair of dual SDC and SAC passing through each 1

where x d2 denotes the principal d2th root of x. (Note that the homology class of the SDC’s and SAC’s becomes a constant for |x| sufficiently large along a generic ray (see Fig. 2). It will be proved in Sect. 3.2 that the homology class is also locally constant with respect to the angle of the ray. For the following, we also define the sectors

(2k − 1)π (2k + 1)π Sk := x ∈ C , arg(x) ∈ −ϑy + , −ϑy + , for d2 odd, 2(d2 + 1) 2(d2 + 1)

kπ (k + 1)π Sk := x ∈ C , arg(x) ∈ −ϑy + , −ϑy + , for d2 even, (d2 + 1) (d2 + 1) (2.4.6) k = 0, . . . , 2d2 + 1 , (where ϑy was defined in Eq. (2.1.5)), and denote by Rk , k = 0, . . . 2d2 + 1 the rays separating them, counting counterclockwise, starting from S0 . These will be shown in the following section to be interpretable as Stokes’ rays, defining a part of the Stokes’ sectors in our Riemann–Hilbert problem. It will be shown in Sect. 3.1 that if x approaches ∞ along a ray distinct from the Rk ’s, then the homology class of the SDC’s and SAC’s in Def. 2.4 is well defined. We then have

Differential Systems for Biorthogonal Polynomials

217

Proposition 2.7. Fix any (non-Stokes) ray {arg(x) = α = const.} = Rk and the corresponding limiting (for large |x|) homology class of the dual steepest descent-ascent contours γk and γ˜k , k = 1, . . . d2 of Def. 2.4. With this choice of the contours of integration, use the SDC’s to redefine the Fourier-Laplace transform (2.3.2), and the SAC’s to redefine the f (k) s of Eq. (2.3.16), and the associated wave-vectors (k) mod C (0) . ∞

Then  N N  C αβ := (α) (x) A (β) (x) =  N

1

0

∞



 = 1d2 +1 , N > d2 , 0 1d2 

(2.4.7)

up to the addition of a suitable multiple of (0) (x) to the solutions (k) (x). ∞

∞

Proof. For β = 0 the statement follows from Eq. (2.2.36) (where the constant c equals √ h0 ). For α = 0 = β we use the fact that the solutions (k) (x) associated to f (k) ∞

are defined only up to the addition of a multiple of (0) (x). Indeed, we know from the ∞

general theory that the pairing of the two solutions is constant, and we may use this freedom to normalize the constant to zero by adding a suitable multiple of the homogeneous solution of Eq. (2.2.7) (0) (x) (i.e., the quasipolynomials), which is “orthogonal” to the ∞

solutions (α) (x), α = 0, as follows again from Eq. (2.2.36). ∞

Finally for k = α = 0 = β = j , C kj equals the bilinear concomitant of the corresponding functions (Eq. 2.2.34). Let us consider x belonging to a fixed ray, and choose a basis of steepest descent contours γk and steepest ascent contours γk (whose homology does not change as x → ∞ on the fixed ray for |x| big enough). We use formula (2.2.34) with √ 1 1 h0 1 f (j ) (x) := dy e (V2 (y)−xy) , ϕ (k) (x) := dy e− (V2 (y)−xy) . √ 0 2iπ h γj 0 γk (2.4.8) The respective asymptotic behaviors for large |x|, computed by the saddle–point method, are ) √ h0 1 (V2 (yj (x))−xyj (x)) −2π (j ) 1 + O(λ−1 ) , f (x) (2.4.9) e 2iπ V2 (yj (x)) ) 1 2π 1 ϕ (k) (x) √ e− (V2 (yk (x))−xyk (x)) 1 + O(λ−1 ) , (2.4.10) 0 V2 (yk (x)) h0 where yk (x) are as in Def. 2.4. The bilinear concomitant is a constant, so it must vanish for j = k since the exponential parts of the asymptotic forms in Eqs. (2.4.9), (2.4.10) cannot give a nonzero constant when multiplied together. For j = k, the bilinear concomitant is given by the integral V2 (∂x ) − V2 (−∂x ) (k) (k) ϕ 0 (x )f (x) ∂x + ∂x

x =x

218

M. Bertola, B. Eynard, J.Harnad

1 V (y) − V2 (y ) dy dy e (V2 (y)−V2 (y )−x(y−y )) 2 y−y γk ×γk ) 1 2π 1 e (V2 (yk (x))−xyk (x)) 2iπ V2 (yk (x)) ) 1 −2π ×e− (V2 (yk (x))−xyk (x)) 1 + O(λ−1 ) V2 (yk (x)) = 1 . (2.4.11) V2 (yk (x))

=

1 2iπ

This concludes the proof.

(y) (y) Had we chosen the contours as k and k rather than the steepest descent contours, we would have had a constant matrix for C αβ . Notice that the pairing of these integrals is also independent of the deformation parameters determining V1 , V2 , as well as of the choice of the integer N defining the window. As shown in [2], this can always be accomplished through a suitable choice of basis. But here we have explicitly shown how this occurs for the particular normalizations chosen in the integrals.

3. Asymptotic Behavior at Infinity and Riemann–Hilbert Problem 3.1. Stokes sectors and sectorial asymptotics. Given the duality between the ODE’s involving D1N and D N 1 implied by Eq. (2.1.60), it is clear that the Stokes matrices around the irregular singularity at x = ∞ for systems of the form (2.1.38) and (2.1.39) are related. Using the explicit integral representations of the fundamental solutions, we can determine the asymptotic behavior from these integral representations by saddle N

point methods. The solutions for the D 1 system are simpler to analyze and do not involve the problems of divergent integrals discussed in Sect. 2.3 (and the Appendix). But in principle one could consider the asymptotic behavior of the solutions to the system (2.1.38) directly, by taking suitable windows in the wave-vector solutions of Eq. (2.2.7). Remark 3.1. We have seen in Sect. 2.4 that one can choose the wave-vector solutions to Eqs. (2.2.7), (2.2.8) so that the Christoffel-Darboux pairing between the dual windows of fundamental solutions implies the identity N

N

(x) A (x) = 1d2 +1 , N

(3.1.1)

N > d2 . N

Therefore, if we formulate a Riemann–Hilbert problem for (x), we can immediately derive the corresponding one for (x). Since the asymptotic forms are mutual inverses, N

the Stokes (and jump) matrices for the one must be the inverses of those for the other. N

(In our conventions, the Stokes matrices for (x) act on the left, while for (x) they N

act on the right.) The formulation for the dual wave-functions is considerably easier, so this is what we analyze here in detail. N

From the general theory of ODE’s, since the matrix D 1 (x) is of degree d1 , one would expect d1 + 1 Stokes sectors. However, this is true only if the leading term of the matrix

Differential Systems for Biorthogonal Polynomials

219

has a nondegenerate spectrum. In the case at hand, however, we have   1 0 N   D 1 (x) ∼ x d1  + O(x d1 −1 ) . 0 0 

(3.1.2)

Since the spectrum of the leading term has a d2 -fold degeneracy, we have more complicated asymptotic behavior and the occurrence of more Stokes sectors. A few more preparatory remarks are required. From the discussion in Sects. 2.2 and 2.3 we obtain that the fundamental system for the differential-difference equations specified in Lemma 2.2 is provided by   N (0)  (x)   N (1)  N  (x)    , (x) :=  (3.1.3) N ≥ d1 + 1,  ..   .   N

(d2 ) (x) N

where { (k) (x)}k=0,...d2 are windows constructed from the wave-vectors defined in Proposition 2.5. Given these integral representations of the solutions, the asymptotic behavior and the Riemann–Hilbert problem can be determined by means of the steepest descent method. N

N

While { (k) (x)}k=1,...d2 are entire functions, the integral defining the window (0) (x) defines in fact a piecewise analytic vector function. Its domains of analyticity are the (x) d1 + 1 connected components in which the x-plane is partitioned by the contours j , (x)

j = 1, . . . d1 . We denote by Dj the connected domain to the right of the contours j for j = 1, . . . , d1 and by D0 the domain to the left of all contours. (See Fig. 3 for the case d1 = 7.) The fundamental piecewise-analytic solution N (x) defined in Eq. (3.1.3) then satisfies the jump equations   1 2iπκ µ,1 2iπκ µ,2 · · · 2iπ κ µ,d2 1 0 ··· 0 0  N  N 0 0 1 · · · 0   (x) , + (x) =   − ..   . 0 0 0 0 0 0 ··· 1 (x) x ∈ µ , µ = 1, . . . , d1 , (3.1.4) where the subscripts + , − denote the limiting values from the right or the left, respectively, with respect to the orientation of the contour. Since we can arbitrarily deform the contours in the finite part of the x-plane, we can, by retracting the d1 contours to the origin arrange that the d1 + 1 regions all become wedge-shaped sectors. Using this (x) freedom in the choice of the contours j , denote by Lµ µ = 0, . . . , d1 the oriented (x)

rays starting from the origin and going to infinity in the sectors S2µ defined in (2.1.4). (x)

From this point on we choose the contours j (x)

j

as follows:

:= Lj − Lj −1 , j = 1, . . . , d1 .

(3.1.5)

220

M. Bertola, B. Eynard, J.Harnad L2 L1 L3 D3 D

4

D2

Γ

(X)

3

Γ

Γ

(X)

2 (X)

Γ1

(X)

4

Γ5

Γ

(X)

6

D6

D

1

3

D

2

D

D0

(X)

D5

D

D

4

L0

1

(X)

Γ

L4

7

D

D

5

D7

0

D

6

D7

L7 L5 L6

Fig. 3. Example of the two possible choices for the domains of definition of φ (0) (in the x-plane) in the n case d1 = 7

In this way all the regions, including D0 , become wedge-shaped sectors (see Fig. 3). The corresponding (equivalent) jump discontinuities are   1 2iπ Jµ,1 2iπJµ,2 · · · 2iπ Jµ,d2 1 0 ··· 0 0  N N 0 N 0 1 ··· 0  (x), + (x) = Gµ − (x) :=    − .. x∈Lµ 0  . 0 0 0 0 0 ··· 1 Jµ,j := κ µ,j − κ µ+1,j , µ = 0, . . . , d1 , j = 1, . . . , d2 , κ 0,j := κ d1 +1,j := 0. (3.1.6) In order to formulate the complete RH problem, we need to supplement this discontinuity data with the sectorial asymptotics around the irregular singularity at x = ∞ and the Stokes matrices. In doing so one should be careful that the lines Lµ for which the discontinuities are defined do not coincide with any of the Stokes’ lines. We can always (x) arrange this by perturbing the rays Lµ within the same sector S2µ . N

It should be clear that each of the piecewise analytic functions (x) can alternatively (x) be analytically continued to entire functions, since the contours j can be deformed arbitrarily in the finite part of the x-plane. Therefore the “discontinuities” in the definition of the Hilbert integral are just apparent and have an intrinsic meaning only when studying the asymptotic behavior at infinity. Indeed in the final formulation of our Riemann–Hilbert problem we prefer to let the lines Lµ also play the roles of Stokes’ lines. Proposition 3.1 (Sectorial Asymptotics). In each of the sectors around x = ∞ with boundaries given by the Stokes’ lines Lµ , µ = 0, . . . , d1 and Rk , k = 0, . . . 2d2 + 1 (defined right after (2.4.6)) the system

N N d N (x) = (x) D 1 (x) dx

(3.1.7)

Differential Systems for Biorthogonal Polynomials

221

possesses a solution whose leading asymptotic form at x = ∞ coincides within this sector with the following formal asymptotic expansion: 1

1

T (x) W x G Y (x d2 ), N f orm (x) ∼ e

(3.1.8)

where Y = Y0 + O(x −1/d2 ) is a matrix–valued function analytic at infinity, Y0 is a diagonal invertible matrix (specified in the proof) and d2

d2 +1−j d2 tj x d2 d2 +1−j + V1 (x)E, d −j +1 j =0 2   1 0 ··· 0 0  0 ω ω 2 · · · ω d2    2 4 2d2   W :=  0 ω ω · · · ω  , ..   ..  . .

T (x) :=

(3.1.9)

(3.1.10)

2

0 ωd2 ω2d2 · · · ωd2 N + 21 − d22 N + 23 − , G := diag −N, d2 d2

d2 2

,...,

N−

1 2

d2

+

d2 2

, (3.1.11)

2iπ

:= diag(0, ω, ω2 , . . . , ωd2 −1 , ωd2 ) , ω := e d2 , E := diag(1, 0, . . . , 0), 1 vd2 −1 t0 := (vd2 +1 ) d2 , t1 := − , d2 vd2 +1 j −1 1 tj := res V2 (y) d2 dy j = 2, . . . , d2 . j − 1 y=∞

(3.1.12) (3.1.13) (3.1.14) (3.1.15)

Proof. In any given sector Sk bounded by the lines Rk−1 and Rk (Eq. 2.4.6) we can choose a basis of steepest descent contours γk , k = 1, . . . , d2 . The reason why the Sk ’s are Stokes sectors and the proper construction of the steepest descent contours is delayed to the discussion of the Stokes’ matrices in Sect. 3.2. We Fourier-Laplace transform the quasipolynomials φn (y) along these contours in order to obtain the functions φ n (x). (x)’s Notice that they are not necessarily the same as the previously introduced φ (k) n (y)

since the steepest descent contours do not necessarily coincide with the contours k defined previously. However they are suitable linear combinations with integer coefficients of such φ (k) (x)’s since the choice of the steepest descent contours is just a different n basis in the homology space of the y-plane. Now consider the asymptotic expansions for −1 (k) dy e− (V2 (y)−xy) φn (y) . (3.1.16) ϕ n (x) := γk

Here we use the notation ϕ rather than φ to stress that these are Fourier–Laplace transforms along contours of a homology class equivalent to SDC’s. The leading asymptotic term in the sector Sk is given at the critical point of the exponent V2 (y) − xy corresponding to the steepest descent contour γk . That is, we must compute V2 (y) − xy near a solution to: (3.1.17) V2 (y) − x = 0

222

M. Bertola, B. Eynard, J.Harnad

asymptotically as x → ∞ within the specified sector. Let us solve Eq. (3.1.17) in a series expansion in the local parameter at ∞ given by one determination, λ, of the d2th root of x: vd2 +1 y d2 + vd2 y d2 −1 + . . . = V2 (y) = x := λd2 , ∞ tj λ−j . y(λ) = λ

(3.1.18) (3.1.19)

j =0 1

We then have the formulae (recalling λ = (V2 (y)) d2 ) t0 = (vd2 +1 )

− d1

2

,

V (y) dλ 1 1 vd2 res y 2 , = dy = − λ=∞ λ d2 y=∞ V2 (y) d2 vd2 +1 j −1 λj −1 1 tj = res y (λ)dλ = res V2 (y) d2 dy , j = 2 . . . ∞ . λ=∞ j − 1 j − 1 y=∞

t1 = res y(λ)

(3.1.20)

As before, denote by {yk (x)}k=0,...d2 +1 the d2 solutions of the equation V2 (y) = x, which are solved in (3.1.20) by a series in the d2th root of x. We then have in a neighborhood of x = ∞, V2 (yk (x)) − xyk (x) = − =−

x

∞ j =0

yk (x )dx = −d2

λk

y(λ )λ d2 −1 dλ

d2 tj d +1−j λ2 − ck , d2 + 1 − j k

(3.1.21)

where λk := ωk λ and ck is a constant depending only on the coefficients of V2 and the branch of the solution yk (x). This formula is proved by taking the derivative (with respect to x) of both sides and using the defining equation for yk (x). Notice that there is no logarithmic contribution since td2 +1 = 0 as follows immediately from the residue formula (3.1.20). The different saddle points are computed by replacing λ with λk := ωk λ. Substituting into the integral representation of the functions ϕ (k) (x), we get n 1 (x) := √ ϕ (k) n hn

1

dy e− (V2 (y)−xy) σn (y)

(3.1.22)

γk

ck 1 e 1 *x 2 √ e yk (x )dx σn (y(λk )) e− 2 V2 (y(λk ))t dt hn R ) ck * x 2π e . = √ e yk (x )dx σn (y(λk )) V2 (y(λk )) hn

(3.1.23) (3.1.24)

Differentiating V2 (y(λk )) = λk d2 implicitly we obtain the relation V2 (y(λk )) =

d2 λk d2 −1 , y (λk )

(3.1.25)

Differential Systems for Biorthogonal Polynomials

223

where y (λ) means differentiation with respect to λ. Therefore we obtain ) 2π n− d2 −1 − 2n−1 2d2 2 v λk d2 +1 hn + ,. ) 2π σn (yk (x)) −1 ϕ (k) (x) ∼ e− (V2 (yk (x))−xyk (x)) √ (y (x)) n V hn 2 k ) 2π −−1 (V2 (yk (x))−xyk (x)) n− d2 −1 − 2n−1 2d2 2 v e λk (3.1.26) ∼ d2 +1 d2 h n ) 2πy (λk ) 1 1 * x yk (x )dx σn (y(λk )) (3.1.27) = √ e hn d2 λk d2 −1   d2 − n+1 %% $ $ 1−d2 d t vd2 +1 d2 1 2 j = Ck √ , λk n+ 2 exp  λk d2 +1−j  1 + O λ−1 d2 + 1 − j hn j =0

(3.1.28) where

/ 0 0 Ck := e 1

2π

ck

(3.1.29)

.

1

vd2 +1 d2 d2 Note that the full series (3.1.21) which should appear in the exponent of (3.1.28) has been truncated to j ≤ d2 because the terms corresponding to j > d2 + 1 contribute to negative powers in the exponential and hence give a 1+O(λ−1 ) term and, as remarked above, there is no term for j = d2 + 1 since td2 +1 = 0. (0) On the other hand the functions φn (x) have the following asymptotic expansion as x → ∞ within the Dµ sectors:

1

φn(0) (x) := e V1 (x) e =

1 V1 (x)

" hn x

ds dy

κ

∞

x −k−1

e−

−1 (V (s)−sy) 1

φn (y)

(x − s)

κ k=0 $ −n−1 −1 V1 (x)

ds dy s k e− $

1 + O x −1

e

−1 V (s) 1

%%

φ n (y) (3.1.30)

.

N

Therefore the matrix of leading terms of (x) as defined in Eq. (3.1.3) is given by   N +d2 N 2 −N  hN−1 vd2 +1 x  d N − 21 − 22  1 T (x)  λ1 1 d2  diag 1, C1 , · · · Cd2 e  λ2 N − 2 − 2  ..  .  d

1

d2

λd2 N− 2 − 2

2 · · · hN+d2 −1 vd2 +1 x −N−d2   d 1 2  ··· λ1 N− 2 + 2  d · N− 21 + 22 ··· λ2   ..  ··· .  d N− 21 + 22 ··· λd2 d

224

M. Bertola, B. Eynard, J.Harnad



N +d

− dN 2 vd2 +1

 ·diag  √

− d 2 vd2 +12

hN−1

,..., "

hN+d2 −1

   .

(3.1.31)

The determinant of the Vandermonde–like matrix is very simple: by computing it along the first row one realizes that only the first and last minors are not zero. Indeed for all other minors the corresponding submatrix has the first and last column proportional. The first minor is a constant in x while the last is of order x −d2 . Therefore we can write         

N

N +d2 d2 d N− 21 + 22 1 d N− 21 + 22 2

hN−1 vd2 +1 d2 x −N · · · hN+d2 −1 vd2 +1 1

d2

1

d2

λ1 N− 2 − 2

λ2 N− 2 − 2 .. .

···

λ

···

λ

.. .

···

d N− 21 − 22

x −N −d2

1

        

d2

λ ··· λd2 N− 2 + 2 d2 N hN−1 vd2 +1 d2 x −N 0 ··· 0  d d N + 21 − 22 N− 21 + 22  0 λ · · · λ 1 1  1 d2 1 d2  = 0 λ2 N − 2 − 2 · · · λ2 N − 2 + 2  .. ..   . ··· .      =   

d2

1

N

hN−1 vd2 +1 d2 x −N 0 0 .. . 0



1 0  0 ωN+ 21 − d22   1 d2 2 N+ 2 − 2 =  0 (ω ) .  .. 0

1

d2

λd2 N− 2 − 2 · · · λd2 N− 2 + 2

0

···

0

      (1 + O(x −1 ))    

0

1 d2 1 d2  λ1 N + 2 − 2 · · · λ1 N− 2 + 2   1 d2 1 d2  λ2 N − 2 − 2 · · · λ2 N − 2 + 2  (1 + O(x −1 ))  ..   ··· . d2

1

(3.1.32)

1

(3.1.33)

d2

λd2 N− 2 − 2 · · · λd2 N− 2 + 2  ··· 0 d 1 2 · · · ωN− 2 + 2   d  N− 21 + 22  2 · · · (ω )   ..  ··· . ··· 1

1 N 1 d2 1 d2 −N N+ − N− + d ×diag hN−1 vd2 +1 2 x , λ 2 2 , . . . , λ 2 2 (1 + O(x −1 )). (3.1.34) When inserting this into the asymptotic form, we see that, up to factoring the constant invertible (diagonal) matrix on the left 1

d2

diag(1, C1 ωN+ 2 − 2 , C2 ω2(N+1/2−d2 /2) , · · · , Cd2 ) ,

(3.1.35)

which is irrelevant for the asymptotics and depends on N in a rather trivial manner, we obtain a solution with the asymptotic form

Differential Systems for Biorthogonal Polynomials

225

  d2 d2 tj λd2 +1−j d2 +1−j 1 W f orm (x) exp EV1 (x) + d2 − j + 1 j =0 N 1 d2 1 d2 ×diag hN−1 vd2 +1 d2 x −N , λN+ 2 − 2 , . . . , λN− 2 + 2   N +d − dN − d 2 2 2 vd +1 vd +1  (1 + O(λ−1 )) ×diag  √2 , . . . , "2 hN−1 hN+d2 −1  − Nd+1 N+ 1 − d2 " 2 λ 2 2 1 T (x) −N vd2 +1  =e W diag hN−1 x , ,..., √ hN  N +d2 1 d2 − vd2 +1 d2 λN− 2 + 2  " × (1 + O(λ−1 )) hN+d2 −1   N +d − Nd+1 − d 2 " 2 2 1 v v d +1 d +1  = exp , . . . , "2 W T (x)W −1 x G diag  hN−1 , 2√ hN hN+d2 −1 N

×(1 + O(λ−1 )) ,

(3.1.36)

where W is the matrix defined in Eq. (3.1.10). Note that W −1 W is just the permutation matrix (in the subblock).

3.2. Stokes matrices for the Fourier–Laplace transforms. The fundamental solution of N

the system D 1 is formed from d2 Fourier-Laplace transforms (k) (x), k = 1, . . . d2 , and N

one Hilbert-Fourier-Laplace transform (0) (x). The asymptotic behavior of the d2 FL transforms is analyzed by means of the steepest descent method in each of the sectors Sk , k = 0, . . . 2d2 + 1 (2.4.6), while the behavior of the (piecewise) analytic function N

(0) (x) is obtained (in each Dµ ) from Eq. (3.1.30). The computation is achieved by (y) (y) expressing the change of homology basis from the 1 , . . . , d2 contours to the steepest descent contours. In order to simplify the analysis of the Stokes matrices we point out that there is no essential loss of generality in assuming V2 (y) = y d2 +1 d21+1 . Indeed, we are concerned with just the homology classes of the SDC’s, and as x → ∞ the d2 solutions of the equation V2 (y) = x entering Def. 2.4 are distinct and asymptotic to the d2 roots of x (up to a nonzero factor). The particular choice of the leading coefficient is also essentially irrelevant. If we choose a different coefficient, we must just appropriately rotate by ϑ = arg(vd2 +1 )/(d2 + 1) counterclockwise in the pictures to follow, but without any essential difference. Therefore, we proceed with vd2 +1 set equal to unity. With these simplifications, the Stokes phenomenon can be studied directly on the integrals

dy e where

− 1

y d+1 d+1 −xy

1

= |x| d

d+1 z dz exp − , − eiα z d +1

(3.2.1)

226

M. Bertola, B. Eynard, J.Harnad

1 d+1 |x| d , α := arg(x) , (3.2.2) and, in order to avoid too many subscripts in the formulae to follow, we have here, and 1 for the remainder of this section, set d = d2 . Disregarding the positive factor |x| d , which is inessential for these considerations, the integrals in (3.2.2) can be written as d+1 z dz dz exp − = ds e−s , − eiα z (3.2.3) d +1 ds :=

where z = Z(s) is the D + 1-valued inverse to s = S(z) :=

zd+1 − eiα z , z = Z(s). d +1

(3.2.4)

It defines a (d +1)-fold covering of the s-plane branching around the points (z, s) whose projection on the s-plane are the d critical values (j )

(j )

scr = scr (α) = −

(d+1)α 2iπ d ei d ωj , ω := e d , j = 0, . . . , d − 1. d +1

(3.2.5)

In realizing this d + 1-fold covering, we take the branch cuts on the s-plane to be the (j ) rays (s) = (scr ) =const, extending to (s) = +∞. As → +∞ the integrals (3.2.2) have leading asymptotic behavior that depends only on the critical values of the map s(z) and on the homology class of the contour. We now return to the computation of the Stokes’ lines. By the definition of the SDC’s γk (Def. 2.4, with V2 (y) now taken as just y d+1 /(d + 1)), their image in the s-plane consists of contours which come from (s) = +∞ on one side of the branch-cut (and on the appropriate sheet) and go back to (s) = +∞ on the other side of the branch-cut, on the same sheet. (For the SAC’s, we choose cuts extending to (s) = −∞.) The cuts on the s-plane may overlap only for those values of α = arg(x) for which the (j ) (i) imaginary parts of two different critical values scr (α) and scr (α) coincide.A straightforward computation, with our simplifying assumptions on V2 (y), yields the lines separating the sectors Sk to be those defined in Eqs. (2.4.6) (with d2 replaced by d and ϑ = 0). We also need the following. Definition 3.1. For a given sector S, of width A < π, centered around a ray arg(x) = α0 , the dual sector S ∨ is the sector centered around the ray arg(x) = −α0 + π , with width π − A. For x → ∞ in each sector the SDC’s are constant integral linear combinations of (y) the contours k ’s. When x crosses the Stokes line between two adjacent sectors, the homology of the SDC’s changes discontinuously. We denote the SDC’s relative to the (k) sector Sk by γj , j = 0 . . . d − 1, and denote the column vector with these as entries denote the column vector with entries { (y) }j =1...d . Denoting γ (k) , and similarly, let j the matrix of change of basis by Ck , we have , Ck ∈ GL(d, Z) . γ (k) = Ck

(3.2.6)

Our first objective is to compute these matrices Ck . For each fixed generic α (i.e. away from the Stokes’ lines), we can construct a diagram (essentially a Hurwitz

Differential Systems for Biorthogonal Polynomials

227

diagram) which describes the sheet structure of the inverse map z = Z(s). We draw d + 1 identical ordered d-gons each representing a copy of the s-plane and whose (la(j ) beled) vertices represent the projections of the d critical values scr . Two vertices with the same label of two different d-gons are joined by a segment if the two sheets are glued together along a horizontal branch-cut originating at the corresponding critical value and going to (s) = +∞. Since all the branch-points of the inverse map are of order 2, there are at most two sheets glued along each cut. Furthermore we give an orientation to the segments (represented by an arrow) with the understanding that this gives an orientation to the corresponding SDC. The convention is that an arrow going from sheet j to sheet k means that the SDC runs on sheet j coming from (s) = +∞ below the cut and goes back above the same cut (or, what is “homologically” the same, the contour runs on sheet k coming from +∞ above the cut and returns to +∞ below it). The diagram can be uniquely associated to a matrix Qk of size d × (d + 1), in which each row corresponds to a SDC and each column to a sheet. The matrix element (Qk )ij is taken to equal: −1 if the i th SDC points to the j th sheet, 1 if the i th SDC originates on the j th sheet, 0 otherwise. Hence each row of the matrix has exactly one +1 and one −1 entry. In Fig. 4, we indicate, for the case d = 11, α = 0, the SDS contours and sheet structure represented by the extended Hurwitz diagram. The matrix corresponding to this diagram is: 0 1 1 −1 0 1 0 0  0 0  0 0  Q0 :=  0 1 1 0  0 0  0 0 0 0 

2 0 −1 1 0 1 0 0 0 0 0 1 0 0

3 0 0 −1 1 0 0 0 0 0 0 0

4 0 0 0 −1 0 0 0 0 0 0 0

5 0 0 0 0 −1 0 0 0 0 0 0

6 0 0 0 0 0 −1 0 0 0 0 0

7 0 0 0 0 0 0 −1 0 0 0 0

8 9 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 −1 1 0 0 −1 0 0 0

11  0 0  0   0   0   0   0   1   0  1 

−1

0 1 2 3 4 5 , 6 7 8 9 10

(3.2.7)

where the boldface numbers labeling the columns are the corresponding labels for the sheets, while the numbers labeling the rows are the labels of the SDC’s. Since α = arg(x) ranges within a fixed sector Sk , the diagram does not change topology and the corresponding matrix Qk remains unchanged. We can now describe how a given diagram changes when α = arg(x) crosses the line between two adjacent sectors Sk and Sk+1 (counterclockwise). These lines correspond precisely to the values of α for which two distinct critical values have the same imaginary parts (so that the cuts may overlap if they are on the same sheet). We leave to the reader to check that these lines are precisely the boundaries of the Stokes sectors Sk defined in Eq. (2.4.6). As α increases by π/(d + 1) from Sk to Sk+1 , the d-gons rotate by π/d. In this process the connections between the sheets change according to the following rule: if the branch-point Pj on sheet r crosses the cut originating from a different branch-point Ph on the same sheet (on the left of Pj on sheet r, and

M. Bertola, B. Eynard, J.Harnad

α+ π 12 5 6

1

α

11 9

10

7

4

3

7

8

10

2

1 11

2

5

0

5 4 0

0

6

11 10

3 1

2

9 0

10

9

α

4

5

8

8

7

6

α+ π 12

7

6

5

8

54

7

7

9

43

8

10

9

1

10

2

9

8 1

6

10

9

32

3

7

6

11

10 1

2

0

1

8

4

3

0

7

10

9

0

6

8

1

9

7

6

3

4

5

5

2

11

8

6

7

5

6

5

4

0

98

3

2 3

4

2

0 10

1

2

0

1

4

3

228

Fig. 4. Example of the sheet structure for the case d = 11, α = 0. The contours depicted in the figure are the SDC’s in the z plane. The second image represents the contours after incrementing α by π/6, i.e., after crossing two Stokes lines. The labeling of the contours (smaller numbers in the top figures) is by the α (k) subscripts of the γk ’s and γk ’s passing through the critical points zcr := ei d ωk . This also corresponds to (k)

d+1

d e d α ωk . the numbering of the vertices of the endecagons representing the critical values scr := − d+1 The different sheets of the s-plane are mapped onto the connected components in which the z-plane is cut by the SDC’s. These are the images in the z-plane of the cuts in the s-plane. The labeling of each sheet is given by the boldface numbers and corresponds to the labels inside the endecagons appearing in the extended Hurwitz diagrams below the corresponding z-planes. The intermediate Hurwitz diagram represents the gluing of the sheets after crossing the intermediate Stokes line, i.e., after incrementing α by π/12

Differential Systems for Biorthogonal Polynomials

229

Ph

Ph

Pj Sheet r

Sheet r

Jum

ps o

f she

et Pj

Ph

Ph

Sheet s

Sheet s

Before crossing

After crossing

Fig. 5. The crossing of two critical values and the corresponding cuts

hence Pj crosses the cut from below as it moves upwards), then the Pj (and its cut) jumps to the sheet s which is glued to sheet r along the cut originating at Ph (see Fig. 5). Diagrammatically, the tip (or the tail) of the corresponding arrow moves from one d-gon to another one connected along the vertex h. In terms of the matrix Qk , the j th row reflects along the hyperplane orthogonal to the hth row. The homology class of (k) the SDC γh will then change because the branch-cut attached to Pj which “emerges” (k) from the branch cut attached to Ph “extracts” a contribution proportional to γj . The proportionality factor is ±1 depending on the relative orientations. The corresponding (k) (k) (k+1) (k+1) and γh by the relations SDC’s γj and γh are related to the SDC’s γj (k+1)

= γj

(k+1)

= γh + hj γj

γj

γh

(k) (k)

(3.2.8)

(k) (k)

(k)

where the incidence number hj = j h is 1 if the SDC’s γj , γh have the opposite orientation and −1 if they have the same orientation. Alternatively, the incidence number is just the (standard) inner product of the corresponding rows h, j of the matrix Qk . We denote by Mk the d ×d matrix which expresses the changes in the SDC’s described by Eq. (3.2.8) through the relation γ (k+1) = Mk γ (k) .

(3.2.9)

These will be seen presently to be precisely the Stokes matrices for the passage between these sectors. One can check that the matrices Qk and Qk+1 are related by Qk+1 = Mkt

−1

Qk .

(3.2.10)

Therefore we can reconstruct the matrices Mk once we have an initial diagram representing the sheet structure. Before constructing the initial diagram, we note that when α increases by 2π/(d + 1) (i.e. when we cross two Stokes lines), the steepest descent (unoriented) contours are like the original ones, but rotated by the same amount clockwise. That is, the unoriented diagram (i.e. forgetting the orientation of the SDC’s) is the same, up to cyclically permuting the labels of the (d + 1) sheets and the d cuts. Note also that the critical points 2π rotate by d(d+1) and the critical values by 2π d counterclockwise. Indeed we have

230

M. Bertola, B. Eynard, J.Harnad

2π s z; α + d +1

=

2π 2π zd+1 − eiα ei d+1 z = s(ei d+1 z; α) . d +1

(3.2.11)

As for the orientation of the SDC’s relative to the new labeling, the d th SDC passing (d−1) reverses its orientation relative to the (oriented) SDC’s obtained by just through zcr rotating the initial SDC’s (see Fig. 4). This implies that the matrices Qk representing the diagrams in the various sectors and the Stokes’ matrices Mk satisfy the recursion relations Qk+2 = p −1 Qk Sd+1 , Mk+2 = p · Mk · p −1 ,    0 0 ··· 0 1 0 1 0 ··· 0 0 1 0 1 ··· 0 0   ∈ GL(d + 1, Z) , p :=  0 Sd+1 =      0 0 . . . 0 ...  0 0 0 ··· 1 0 0

0 ··· 0 0 ··· 0 1 ··· 0 . 0 .. 0



(3.2.12)

−1 0  0   ∈ GL(d, Z)  0 

0 ··· 1 0

It is therefore only necessary to compute Q0 , Q1 and M0 , M1 . (y) Notice also that the wedge contours j , j = 0, . . . , d are in 1 − 1 correspondence with the sheets of the map Z(s). Indeed, each connected component to the right of the (y) wedge contours contains only one sector S2k+1 in the z-plane which corresponds to the sector (s) < 0 in the s-plane. Thus, the same argument used to arrive at the recursion (3.2.12) proves that Ck+2 = p Ck P ,  −1 −1 · · · −1  1 0 ··· 0  0 1 ··· 0 P :=    0 0 ... 0 0

0 ··· 1

(3.2.13)

 −1 0  0   ∈ GL(d, Z) ,  0  0

where both the matrix Sd+1 in Eq. (3.2.12) and P in Eq. (3.2.13) generate a representation of the cyclic group Zd+1 (although P is of size d × d). The matrix P is just the generator of Zd+1 on the “wedge” contours; i.e. with the additional constraint d (y) i=0 i = 0. We now describe how to derive the initial diagram - for example, how Fig. 4 is (y) obtained. We start by noting that the “wedge” contours k enclose the sector S2k−1 of width π/(d +1). It is to see that the corresponding integrals are (more than) exponentially decreasing in the dual sector, since

1

e− (y)

y d+1 /(d+1)−xy

dy ≤ exp (|x|M)

k

where M is the supremum of

1 |x| (xy)

(y) k

1

e− y

d+1 /(d+1)

(y)

dy ,

(3.2.14) (y)

as y goes along k . Now the contour k can (y)

be deformed so as to approach the sector S2k−1 as closely as we wish. Then the constant (y) ∨

M is finite and negative if x lies within the dual sector Sk

.

Differential Systems for Biorthogonal Polynomials

231

We now consider the case of odd d, leaving the easy generalization to even d’s to the reader. (The only difference is that for odd d, α = arg(x) = 0 is not a Stokes’ line. To study the even d case, one should choose a convenient initial value of α (e.g. α = 1 or α = π/2(d + 1), which corresponds to an anti–Stokes line). Let us start with the (non-Stokes) value α = 0 and focus on any of the SDC’s attached to a critical value lying in the right s-plane for this value of α (remembering that d is assumed to be odd). Since the real part of such a critical value is positive, d+1 d the corresponding integrals decrease as exp(− d+1 ωj ), ( := 1 |x| d ) i.e., they are (more than) exponentially suppressed on the line x ∈ R+ . As we increase α, the d-gons rotate by d+1 d α counterclockwise. It should be clear (from the previous description of the change of homology of the SDC’s) that the SDC’s attached to such critical values do not change homology class while they remain in the right half of the s-plane. This is so because there are no branch points to their right which can pass through the branch-cut through the given saddle point (which extends to the right) as α is increased. Therefore the corresponding integrals are exponentially suppressed as long as α ranges in a d 1 corresponding sector of width d+1 π = π − d+1 π . This is precisely the width of the (y)

π dual sector to a sector of width d+1 ; i.e., the width of any of the S2k+1 ’s. By careful inspection of the anti-Stokes lines for these integrals3 , which correspond to the value of α for which the corresponding critical value is real and positive, one concludes that they (y) coincide with an appropriate k in the left half of the z-plane. This argument proves (y) that all “wedge” contours k lying in the left plane are homological to SDC’s. (These are the SDC’s in the left z-plane in Fig. 4.) We then take the first critical value lying in the left half of the s-plane (in Fig. 4, the SDC number 9). As we increase α so as to move this critical value to the right half of the s-plane, the corresponding SDC can acquire a contribution only from the first SDC in the left half-plane (number 8 in our example). As a consequence the correspondπ ing integral is exponentially suppressed in a sector of width π − 2 d+1 . Considering its anti-Stokes line and its linear independence from the previously identified SDC’s, we conclude that it must enclose two odd-numbered sectors S2k+1 and S2k+3 (in our example this is contour 9). Proceeding this way we can easily identify the homology classes of all SDC’s for α = 0. The labeling of the sheets is largely arbitrary. (The choice we have made in Fig. 4 is (y) just for “aesthetic” reasons). The fixed basis of contours i , i = 1, . . . d and the bases (k) γj , j = 0, . . . d − 1 k = 0, . . . 2d + 1 are related by

= Mk−1 · · · M0 C0 , γ (k) = Ck (0) γ = C0 .

(3.2.15) (3.2.16)

Therefore only the matrices Mk and the first change of basis matrix C0 are needed. The matrix C0 can easily be constructed from the initial sheet structure. For our present purposes it is not actually necessary to have its general form, since the objects of primary interest are the Stokes matrices, which will be seen to be just the Mk ’s. But to illustrate by example the form that C0 takes, within the basis we have chosen, for the case d = 11, it is: 3 The anti-Stokes’ line for exponential integrals of the type of Def. 2.4 are the lines along which the integrals are most rapidly decreasing as |x| → ∞.

232

M. Bertola, B. Eynard, J.Harnad



1 0 0  0  0  C0 :=  0 0  0  0 0

1 1 0 0 0 0 0 0 0 0 00

1 1 1 0 0 0 0 0 0 0 0

1 1 1 1 0 0 0 0 0 0 0

1 1 0 0 1 0 0 0 0 0 0

1 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 1 0 0 1

0 0 0 0 0 0 0 0 1 1 1

0 0 0 0 0 0 0 0 0 1 1

 0 0 0  0  0  0 . 0  0  0 0

(3.2.17)

1

It is not difficult to give an explicit description of the matrices Q0 , Q1 and M0 , M1 but, for the sake of brevity, we will not give it here. It consists of a lengthy but straightforward calculation, which leads to the matrices in Table 1 (listed for the cases d = 2, . . . , 11), where one can clearly extrapolate to the correct rule. We now turn our attention to the Stokes matrices. Consider the subblock of the fundamental system corresponding to solutions of the D 1 ODE given by the Fourier-Laplace transforms. If we denote by Y (x) the d2 × d2 such block whose rows are the integrals of N

ϕ (y) on the contours k(y) , and by Yk (x) the analogous matrix obtained by integrating over the SDC’s, we have (3.2.18) Yk (x) = Ck Y (x). The Stokes matrices are then given by Sk := Yk+1 Yk −1 = Ck+1 Ck −1 = Mk .

(3.2.19)

These are just the matrices expressing the relative change of homology basis of the SDC’s corresponding to two consecutive Stokes’ sectors. In order to complete the description of the Riemann–Hilbert problem, we need to also consider the extra solution given by a Hilbert–Fourier-Laplace transform. In doing so, we extend the previously computed Stokes matrices Mk to the full fundamental system of d2 + 1 solutions by means of 1 0 8 . (3.2.20) Mk := 0 Mk Summarizing the whole discussion, we have proved the following theorem. Theorem 3.1 (Riemann–Hilbert Problem). There exists a fundamental system of soluN

tions, (x) which is analytic where defined, with the properties: 1. In each of the sectors around x = ∞ defined by removing the lines Lµ , µ = 0, . . . d1 N

and Rk , k = 0, . . . , 2d2 + 1, (x) exists, is analytic and invertible, and it can be normalized on the left by a constant matrix to have the same asymptotic behavior as N

(x) defined in Proposition 3.1. f orm

2. Crossing the lines Lµ , the corresponding Stokes matrix is given by Gµ as defined in Eq. (3.1.6).

Differential Systems for Biorthogonal Polynomials

233

Table 1. The first two Stokes matrices for degrees d2 = 2 . . . 11. The remaining ones are obtained using Eq. (3.2.12)

1 0  1 =0 0  1 0 = 0 0  1 0  =0 0 0  1 0  0  = 0 0 0 1 0 0  = 0 0  0 0 1 0  0  0 = 0  0  0 0 1 0 0  0  = 0 0  0  0 0 1 0 0  0  0 = 0  0  0  0 0  1 0 0  0  0  = 0 0  0  0  0 0

M0 = M0

M0

M0

M0

M0

M0

M0

M0

M0

0 1  −1 0 1 0 0 1  −1 0 0 1 0 0 0 1 0 0 1 1  0 −1 0 0 1 0 0 0  0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

0 1 1  −1 0 0 0 0 0 0 0  1 0 0 0 0 1 0 0  0 0 1 0 0 1 0 1 0 −1 0 0 0  −1 0 0 0 0  1 0 0 0 0  0 1 0 0 0  0 0 1 0 0  0 0 0 1 0 0 0 1 0 1 0 −1 0 0 0 0  −1 0 0 0 0 0  1 0 0 0 0 0  0 1 0 0 0 0  0 0 1 0 0 0  0 0 0 1 0 0  0 0 0 1 1 0 0 0 1 0 0 1 0 0 −1 0 0 0 0  0 −1 0 0 0 0 0  1 0 0 0 0 0 0  0 1 0 0 0 0 0  0 0 1 0 0 0 0  0 0 0 1 0 0 0  0 0 0 0 1 0 0  0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 −1 0 0 0 0 0  0 −1 0 0 0 0 0 0  1 0 0 0 0 0 0 0  0 1 0 0 0 0 0 0  0 0 1 0 0 0 0 0  0 0 0 1 0 0 0 0  0 0 0 0 1 0 0 0  0 0 0 0 0 1 0 0  0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1  0 0 0 −1 0 0 0 0 0 0 0 −1 0 0 0 0 0 0  1 −1 0 0 0 0 0 0 0   0 1 0 0 0 0 0 0 0  0 0 1 0 0 0 0 0 0  0 0 0 1 0 0 0 0 0  0 0 0 0 1 0 0 0 0  0 0 0 0 0 1 0 0 0  0 0 0 0 0 0 1 0 0  0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1

1 0 1 1   1 0 0 M1 =  0 1 0  0 1 1   1 0 0 0 0 1 0 0 M1 =  0 0 1 0 0 1 0 1   1 −1 0 0 0 0 1 0 0 0   M1 =  0 0 1 0 0  0 0 0 1 0 M1 =

0 0 1 0 1   1 −1 0 0 0 0 0 1 0 0 0 0   0 0 1 0 0 0 M1 =   0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 1  1 0 −1 0 0 0 0  0  1 0 0 0 0 0 0 0 1 0 0 0 0    M1 =  0 0 0 1 0 0 0 0 0 0 0 1 0 0   0 0 0 0 1 1 0 0 0 0 1 0 0 1  1 0 −1 0 0 0 0 0  0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0   0 0 0 1 0 0 0 0  M1 =  0 0 0 0 1 0 0 0   0 0 0 0 0 1 0 0   0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1  1 0 0 −1 0 0 0 0 0   0 1 −1 0 0 0 0 0 0  0 0 1 0 0 0 0 0 0   0 0 0 1 0 0 0 0 0    0 0 0 0 1 0 0 0 0 M1 =    0 0 0 0 0 1 0 0 0   0 0 0 0 0 0 1 0 0   0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1  1 0 0 −1 0 0 0 0 0 0   0 1 −1 0 0 0 0 0 0 0  0 0 1 0 0 0 0 0 0 0   0 0 0 1 0 0 0 0 0 0   0 0 0 0 1 0 0 0 0 0   M1 =   0 0 0 0 0 1 0 0 0 0   0 0 0 0 0 0 1 0 0 0   0 0 0 0 0 0 1 1 0 0   0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1   1 0 0 0 −1 0 0 0 0 0 0  0 1 0 −1 0 0 0 0 0 0 0  0 0 1 0 0 0 0 0 0 0 0   0 0 0 1 0 0 0 0 0 0 0   0 0 0 0 1 0 0 0 0 0 0    M1 =  0 0 0 0 0 1 0 0 0 0 0   0 0 0 0 0 0 1 0 0 0 0   0 0 0 0 0 0 0 1 0 0 0   0 0 0 0 0 0 0 1 1 0 0   0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1

d=2 d=3

d=4

d=5

d=6

d=7

d=8

d=9

d = 10

d = 11 .

234

M. Bertola, B. Eynard, J.Harnad L2 S7

S6

S5 S4

S8

L3

L1

S3

S9

S2

S 10

S1

S 11

L4

S0

L0

S 12

S 23

S 13

S 14

S 22 S 21

L5

S 15

L7 S 20

S 16 S 17

S 18

S 19

L6

Fig. 6. Example of the structure of Stokes sectors and discontinuity lines Lµ (which can equivalently be viewed as Stokes lines) in the x-plane for the case d1 = 7 and d2 = 11 and both leading coefficients of the potentials real and positive

8k in Eq. (3.2.20), 3. Crossing the lines Rk the corresponding Stokes matrix is given by M constructed according to the algorithm described in Sect. 3.2. For an example of all Stokes’ sectors and lines see Fig. 6 and for examples of explicit Stokes’ matrices M0 , M1 from which all the others are computed easily, see Table 1. We conclude this section with the remark that in the formulation of Theorem 3.1 the lines Lµ were viewed as Stokes’ lines. However, we could alternatively have formulated an equivalent Riemann–Hilbert problem, in which the lines Lµ define discontinuities, according to Eq. (3.1.6), by retaining a fixed choice of the contours of integration along the boundaries of the wedge sectors, thereby giving rise to genuine jump discontinuities across the boundaries. 4. Summary and Comments on Large N Asymptotics in Multi-Matrix Models A complete formulation of the Riemann–Hilbert problem characterizing fundamental systems of solutions to the differential-recursion equations satisfied by biorthogonal polynomials associated to 2-matrix models with polynomial potentials is provided in Theorem 3.1. The approach derived here can also be extended quite straightforwardly to the case of biorthogonal polynomials associated to a finite chain of coupled matrices with polynomial potentials, following the lines indicated in the appendix of [2].

Differential Systems for Biorthogonal Polynomials

235

The Riemann-Hilbert data, consisting essentially of the Stokes matrices at ∞, are independent of both the integer parameter N corresponding to the matrix size and the deformation parameters determining the potentials; that is, the fundamental solutions constructed are solutions simultaneously of a differential-difference generalized isomonodromic deformation problem. This provides the first main step towards a rigorous analysis of the N → ∞, N = O(1) limit of the partition function and the Fredholm kernels determining the spectral statistics of coupled random matrices (essential, e.g., to the question of universality in 2-matrix models). Such an analysis should follow similar lines to those previously successfully applied to ordinary orthogonal polynomials in the 1-matrix case [6, 17, 19, 20, 10, 11]. The main difference in the 2-(or more) matrix case is that in the double–scaling limit the functional dependence of the free energy on the eigenvalue distributions is not as explicit as in the 1-matrix models [23, 18]. It is also clear that the hyperelliptic spectral curve that arises in the solution of the one-matrix model must be replaced by a more general algebraic curve, which arises naturally in the spectral duality of the spectral curves of [2] (cf. [14]). In order to determine the large N asymptotics with the help of the data defining the Riemann-Hilbert problem, one should begin with an ansatz that can be checked a posteriori against the given case. In the 1-matrix case [10, 11], this was provided by means of hyperelliptic -functions. The physical heuristics and the basic tools for generating such an ansatz were also given in [7, 14, 23, 18]. Much of this can be extended to the 2-matrix case [4], and this will be the subject of a subsequent work [1]. A. Appendix: Convergent Integral Representations of Ψ (x) ∞

In this appendix, we indicate how to overcome the problem in the formal definition of Eq. (2.3.15), illustrated through an example. The idea follows that used in [21] for the case of cubic potentials4 . As discussed in section (0) (x), a natural approach to finding solutions of Eq. (2.2.7) ∞ would be to take the inverse Fourier-Laplace transform of the function

1

(0) (y) := e V2 (y) ∞

1

κ

dx dt

e− (V1 (x)−xt) t (x) , y−t

(A-1)

along the anti-wedge contours k . Equation (A-1) defines a piecewise analytic function in the domains d2 d2 9 9 (y) Cy \ k = Dν , (A-2) (y)

ν=0

k=1 (y)

with jumps across the k given by

∞

(0)

(y+ ) −

∞

(0)

(y− ) = 2iπ

d1 j =1

κ j,k (j ) (y) , ∞

(y)

y ∈ k .

(A-3)

4 We wish to thank A. Kapaev for pointing out the problem of divergent integrals appearing in an earlier version of this paper, and for helpful discussions. The approach outlined here, for potentials of arbitrary degrees, will be the subject of further work, and the results detailed in a subsequent, joint publication.

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However, either retaining this definition or modifying it through analytic continuation leads to a divergent integral when the Fourier-Laplace transform is applied. In order to resolve this difficulty, the idea is to consider a suitable contour approaching y = ∞ y within the odd-numbered sectors S(2k+1) and define a locally analytic function in a neighborhood of this contour so as to render the Fourier–Laplace transform convergent. A general approach resolving this problem is not needed for the results presented in this paper since, as pointed out, the Riemann-Hilbert problem for (x) is more simply N

obtained by means of duality (Remark 3.1). Therefore, we just give an indication of how to proceed by means of an example. A.1. An Example: d1 = 7, d2 = 4. The example we consider consists of two potentials V1 (x) and V2 (y) of degrees 8 and 5 respectively, both with positive real leading coef(y) ficients. We refer to Fig. 7 as “the figure” throughout this section. The Sk sectors are

(x)

(x)

S3

S5

S1(y) (x)

S7

(x)

Γ3

(y)

S5(y)

lL

∼ Γ

S1

Γ

(y) 1

L

Γ1

(x)

lR

R

(x)

S 15 (x)

S9

S9(y) (x)

(x)

S 11

S 13

Fig. 7. The figure illustrating the definition of (Lef t,Right) (y). The contour must approach ∞ within (y)

∞

the sectors S2k+1 ’s (lighter shaded) and asymptotic to the two rays L , R . The terms added to (0) in (x)

∞

the definition of (Lef t) ((Right) ) must correspond to sectors S2j +1 which fall in the dual sector L ∞ ∞ (R) indicated by the dashed (dash-dotted) arc

Differential Systems for Biorthogonal Polynomials (x)

the light-shaded ones while the Sj

237

are the darker-shaded ones. Fix an asymptotic ray (y)

L with arg(y) = α0 (= π) within an odd-numbered sector, in our example S5 , and an sector around it. Let L be the dual sector, i.e., the sector centered around π − α0 (= 0) (x) with width π − . Any of the functions (j ) (y) whose wedge contours j lie in the ∞

sector L decay faster than any exponential e−My on the given ray L . These are the (x) sectors S(1,3,13,15) in Fig. 7. Fix another ray R within a different odd-numbered sector (y)

S2k+1 in such a way that the dual sector to the sector around R (denoted by R) (x) contains all but one of the remaining sectors S2j +1 . In the choice shown in the figure R (y) (x) lies in S9 and the sector R contains the sectors S7,9,11 . Finally, fix a contour which goes to infinity asymptotically to the two rays L , R (as in the figure). With these choices, redefine the duality pairing according to the following rule: (x)

(i) In the x-plane use as a basis only the contours j

which fall within one or the (x)

other of the sectors L, R. In our example this leaves out contour 3 , which has not been drawn in the figure. (y) (ii) In the y-plane use a basis in which the contours intersect only one of the k s. (y) (y) In our example this would leave out either 3 or 0 and we have chosen the latter. Note that we can accomplish an equivalent duality pairing using such bases by exploiting the homological equations d1

(x)

j

j =0

=

d2

(y)

k

= 0.

(A-1-1)

k=0

Indeed, using (A-1-1), one can avoid one contour and redefine the κ’s: (y) (y) (x) (x) κ i,j i × j = κ i,j i × j , κ = i=0,j =0

(A-1-2)

i=3,j =0

where κ i,j in this example would be κ 0,j := −κ 3,j . κ i,j := κ i,j − κ 3,j , i = 3, 0; Now define the wave-vector Lef t (y) := (0),t (y) − 2iπ ∞

∞

=:

∞

(0),t

(y) − 2iπ

κ i,3 (i),t (y) ∞

i=0,1,2,7 i,3

κ

i∈IL

(A-1-3)

(i),t

∞

(y) ,

(A-1-4)

(y) for y belonging to the part of inside contour 3 . (Note the transposition of row vectors to column vectors.) In Eq. (A-1-4) in general IL would be the set of indices j (x) corresponding to the contours j contained in sector L. Note that this function decays

on the ray L because L is in the dual sector of all the (j ) entering Eq. (A-1-4). ∞

Moreover the piecewise analytic function

∞

(0)

(y)

decays exponentially in each sector Sk .

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M. Bertola, B. Eynard, J.Harnad

(y) As we cross the contour 3 along , by virtue of the jump condition (A-3), the Lef t function (y) is analytically continued to the function ∞

Right (y) := (0),t (y) + 2iπ ∞

∞

:=

κ j,3 (j ),t (y) ∞

j ∈L (0),t

∞

(y) + 2iπ

κ j,3 (j ),t (y) ,

(A-1-5)

∞

j =4,5,6

which, due to the choices made above, also decays along the asymptotic direction R . The two functions Lef t,Right (y) together define a function (y) which is (locally) ∞

∞

analytic in a neighborhood of the contour and decays sufficiently fast in either direction to allow us to take an inverse Fourier-Laplace transform using xy ( ) dy e− ( ) (y) . (A-1-6) (x) := ∞

∞

This wave-vector is a solution of Eqs. (2.2.8), (2.2.40) and (2.2.41) as we now show. The equations "

t 1 y ( ) (y) = P ( ) (y) + h0 e V2 (y) , 0, . . . , (A-1-7) ∞ ∞ "

t 1 ∂y ( ) (y) = Q( ) (y) + W2 (y) h0 e V2 (y) , 0, . . . , (A-1-8) ∞

∞

( )

∂uK ∞

(y) = UK ( ) (y), ∞

∂vJ ( ) (y) = −VJ ( ) (y) − ∞

∞

(A-1-9)

t 1 1 P J − y J " h0 e V2 (y) , 0, . . . , (A-1-10) J P −y

are satisfied as a consequence of the similar equations for (0) (y) and (k) (y), k = ∞

∞

1, . . . d1 . From these, Eqs. (2.2.8), (2.2.40) and (2.2.41) for ( ) (x) follow from standard ∞

manipulations of the inverse Fourier–Laplace transform which are now made rigorous by the convergence of the integral. In particular this solution of (2.2.8) is associated with the function 1 ( ) f (x) := dy e (V2 (y)−xy) . (A-1-11)

One should repeat the scheme outlined here for other contours as well, until they (y) span the same homology space spanned by the anti-wedge contours { j }, so as to obtain a basis of solutions to Eq. (2.2.7). Just to complete the given example we indicate how to choose the other contours and corresponding bases: 1. The contour asymptotic to the rays arg(y) = −π and arg(y) = π/10 + . The basis on which to re-define the coefficients κ (which we now denote by κ ) is given by (y) (x) i × j , i = 6, j = 1 and the wave vectors are given by κ i,3 (i),t (y), (A-1-12) Lef t (y) := (0),t (y) − 2iπ ∞

∞

Right (y) := (0),t (y) + 2iπ ∞

∞

i=0,1,2,7

j =3,4,5

∞

κ j,3 (j ),t (y) . ∞

(A-1-13)

Differential Systems for Biorthogonal Polynomials

239

2. The contour asymptotic to the rays arg(y) = π/2 + and arg(y) = 3π/2 − : The (y) (x) basis on which to re-define the coefficients κ is given by i × j , i = 3, j = 4: Lef t (y) := (0),t (y) − 2iπ ∞

∞

∞

κ i,2 (i),t (y),

(A-1-14)

κ j,2 (j ),t (y) .

(A-1-15)

i=0,1,4

Right (y) := (0),t (y) + 2iπ ∞

∞

∞

j =5,6,7,0

3. The contour asymptotic to the rays arg(y) = π/2 + and arg(y) = 17π/10 + . The (y) (x) basis on which to re-define the coefficients κ is given by i × j , i = 0, j = 0: Lef t (y) := (0),t (y) − 2iπ ∞

∞

Right (y) := (0),t (y) + 2iπ ∞

∞

κ i,2 (i),t (y),

(A-1-16)

κ j,2 (j ),t (y) .

(A-1-17)

i=1,2,3,4

j =5,6,7

∞

∞

We leave the details of the general case to subsequent work. Acknowledgement. The authors would like to thank A. Kapaev for helpful comments concerning nonsingular integral representations of the fundamental systems of solutions to Eq. (2.1.38), which resulted in the addition of Appendix A and the discussion at the end of Sect. 2.3.

References 1. Bertola, M., Eynard, B., Harnad, J.: An ansatz for the solution of the Riemann–Hilbert problem for biorthogonal polynomials. In preparation 2. Bertola, M., Eynard, B., Harnad, J.: Duality: Biorthogonal Polynomials and Multi–Matrix Models. Commun. Math. Phys. 229(1), 73–120 (2002) 3. Bertola, M., Eynard, B., Harnad, J.: Duality of spectral curves arising in two-matrix models. Theor. Math. Phys 134(1), (2003) 4. Bertola, M., Eynard, B., Harnad, J.: Genus zero large n asymptotics of bi-orthogonal polynomials involved in the random 2-matrix model. Presentation by B.E. at AMS Northeastern Regional Meeting, Montreal, May 3–5, 2002 5. Bertola, M.: Bilinear semi–classical moment functionals and their integral representation. J. App. Theory 121, 71–99 (2003) 6. Bleher, P., Its, A.: Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. Math. (2) 150(1), 185–266 (1999) 7. Bonnet, G., David, F., Eynard, B.: Breakdown of universality in multi-cut matrix models. J. Phys. A 33, 6739–6768 (2000) 8. Chihara, T.S.: An introduction to orthogonal polynomials. Mathematics and its Applications. Vol. 13 New York-London-Paris: Gordon and Breach Science Publishers, 1978 9. Daul, J.M., Kazakov, V., Kostov, I.K.: Rational Theories of 2D Gravity from the Two-Matrix Model. Nucl. Phys B409, 311–338 (1993), hep-th/9303093. 10. Deift, P., Kriecherbauer, T., McLaughlin, K.T.R., Venakides, S., Zhou, Z.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commun. Pure Appl. Math 52, 1335–1425 (1999) 11. Deift, P., Kriecherbauer, T., McLaughlin, K.T.R., Venakides, S., Zhou, Z.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math 52, 1491–1552 (1999) 12. Di Francesco, P., Ginsparg, P., Zinn-Justin, J.: 2D Gravity and Random Matrices. Phys. Rep. 254, 1 (1995) 13. Ercolani, N.M., McLaughlin, K.T.-R.: Asymptotics and integrable structures for biorthogonal polynomials associated to a random two-matrix model. Physica D 152–153, 232–268 (2001)

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14. Eynard, B.: Eigenvalue distribution of large random matrices, from one matrix to several coupled matrices. Nucl. Phys. B 506, 633 (1997), cond-mat/9707005 15. Eynard, B.: Correlation functions of eigenvalues of multi-matrix models, and the limit of a time dependent matrix. J. Phys. A: Math. Gen 31, 8081 (1998), cond-mat/9801075 16. Eynard, B., Mehta, M.L.: Matrices coupled in a chain: eigenvalue correlations. J. Phys. A: Math. Gen 31, 4449 (1998), cond-mat/9710230 17. Fokas, A., Its, A., Kitaev, A.: The isomonodromy approach to matrix models in 2D quantum gravity. Commun. Math. Phys 147, 395–430 (1992) 18. Guionnet, A., Zeitouni A.: Large deviations asymptotics for spherical integrals. J. F. A. 188, 461–515 (2002) 19. Its, A.R., Kitaev, A.V., Fokas, A.S.: Matrix models of two-dimensional quantum gravity and isomonodromy solutions of discrete Painleve equations. Zap. Nauch. Sem. LOMI 187, 3–30 (1991) (Russian), translation in J. Math. Sci. 73(4), 415–429 (1995) 20. Its, A.R., Kitaev, A.V., Fokas, A.S.: An isomonodromic Approach in the Theory of Two-Dimensional Quantum Gravity. Usp. Matem. Nauk, 45(6), 135–136, 276 (1990) (Russian), translation in Russ Math. Surveys 45(6), 155–157 (1990) 21. Kapaev, A.A.: The Riemann–Hilbert problem for the bi-orthogonal polynomials. nlin.SI/0207036 22. Kazakov, V.A.: Ising model on a dynamical planar random lattice: exact solution. Phys Lett. A119, 140–144 (1986) 23. Matytsin, A.: On the large N limit of the Itzykson Zuber Integral. Nuc. Phys B411, 805 (1994), hep-th/9306077 24. Mehta, M.L.: Random Matrices. Second edition. New York: Academic Press, 1991 25. Szeg¨o, G.: Orthogonal Polynomials. Providence, Rhode Island: AMS, 1939 26. Ueno, K., Takasaki, K.: Toda Lattice Hierarchy. Adv. Studies Pure Math. 4, 1–95 (1984) Communicated by L. Takhtajan

Commun. Math. Phys. 243, 241–260 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0958-6

Communications in

Mathematical Physics

Non-Perturbative Mass and Charge Renormalization in Relativistic No-Photon Quantum Electrodynamics, Christian Hainzl∗∗∗ , Heinz Siedentop Mathematik, Theresienstrasse 39, 80333 M¨unchen, Germany. E-mail: [email protected] Received: 18 March 2003 / Accepted: 1 May 2003 Published online: 5 November 2003 – © Christian Hainzl and Heinz Siedentop 2003

Abstract: Starting from a formal Hamiltonian as found in the physics literature – omitting photons – we define a renormalized Hamiltonian through charge and mass renormalization. We show that the restriction to the one-electron subspace is well-defined. Our construction is non-perturbative and does not use a cut-off. The Hamiltonian is relevant for the description of the Lamb shift in muonic atoms.

1. Introduction According to Dirac’s hole theory the vacuum consists of electrons which occupy the negative energy states of the free Dirac operator (Dirac sea). Dirac postulated that their charge is not measurable. However, if one introduces an external electric field, e.g., the field of a nucleus, these electrons should rearrange, occupying the negative energy states of the Dirac operator with the external electric field. Physically speaking, the nucleus polarizes the vaccuum. (This rearrangement may be interpreted as the creation of virtual electron-positron pairs when expressed in terms of the free Dirac operator.) In other words, the vacuum is polarized. Dirac [2] indicates that these polarization effects result in a logarithmically divergent charge density, which cannot be neglected. As a solution, he suggested that a momentum cut-off must be introduced, since he expected that the Dirac equation would fail for energies higher than 137 mc2 . In [3] he changed this train of thought and suggested that the infinities occurring should be absorbed by a procedure c 2003 The authors. Reproduction of this article for non-commercial purposes by any means is permitted. C.H. has been supported by a Marie Curie Fellowship of the European Community program “Improving Human Research Potential and the Socio-economic Knowledge Base” under contract number HPMFCT-2000-00660. Both authors acknowledge partial support through the European Union’s IHP network Analysis & Quantum HPRN-CT-2002-00277 *** Current address: CEREMADE, Universit´e Paris-Dauphine, Place du Mar´echal de Tassigny, F-75775 Paris Cedex 16, France. E-mail: [email protected]

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which is now called charge renormalization. A similar step was independently undertaken by Furry and Oppenheimer in [8] who circumvented the hole theory by introducing annihilation and creation operators. Heisenberg [11] clarified Dirac’s picture and generalized his approach extracting the physically relevant terms by subtraction of an unambiguous infinite constant, at least to first order in α. Serber [26] and Uehling [29] gave detailed calculations (in first order of α). Uehling demonstrated that the vacuum polarization alters the Coulomb potential of a charged particle resulting in the electron being slightly more bound in the s-states (angular momentum 0) of hydrogenic atoms. Later Weisskopf [31] gave a thorough discussion of the physics involved in charge renormalization. After the experiments of Lamb and Retherford [18] in 1947, which gave a much higher discrepancy concerning the hyperfine structure of hydrogen, in addition to a different sign, than Uehling’s calculation showed, and the first explanation by Bethe [1], the insight into quantum electrodynamics (QED) changed and the interaction with the radiation field turned out to be the dominating part in describing the splitting of the energy levels of hydrogenic atoms beyond the Dirac equation. Similar to vacuum polarization, which was now treated together with the radiative corrections, the photon interaction caused fundamental problems such as infinities, which were “removed” – at least in first order of α – by mass renormalization by Tomonaga, Schwinger, and Feynman. Eventually, Dyson “succeeded” with the renormalization program to every order in α. Since then, QED has proven to be of extraordinary predictive power. (We refer the reader interested in more historical details to Schweber [24].) But despite the predictive power of quantum electrodynamics, the description in terms of perturbation theory causes great uneasiness among mathematicians; a mathematically consistent formulation of QED is still unknown; in fact Dyson [5] indicated that the perturbation theory is divergent. A self-adjoint Hamiltonian for QED is not known. In the present paper we address a particular kind of singularities arising in QED, namely those stemming from the vacuum polarization. As opposed to the prevalent physics literature we will not use any Feynman diagram but will rather construct a Hamiltonian (in Coulomb gauge). This we have in common with the above cited early works in the field. However, the fact that we start from a formal Hamiltonian and renormalize it non-perturbatively distinguishes us from those authors. Although our approach is rigorous, the resulting renormalization is far from being of academic interest only. In fact, the restriction Dren of the fully renormalized Hamiltonian H to the one-particle electron sector accounts already for a precise description of the low energy levels of µ-mesonic atoms where the vacuum polarization effect dominates the radiative corrections by far, since the Bohr orbits traverse the support of the polarization potential in this case. (See, e.g., Peterman and Yamaguchi [21], Glauber et al. [9], Milonni [19], Weinberg [30], and Greiner et al [10].) Eventually we would note explicitly that we do not address the further renormalization of the completely normal ordered pair-interaction. 2. Model In relativistic QED the quantized electron-positron field (x), which is an operator valued spinor, is written formally as (x) = a(x) + b∗ (x),

(1)

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243

where a(x) annihilates an electron at x and b∗ (x) creates a positron at x. (We use the notation that x = (x, σ ) ∈ = R3 × {1, 2, 3, 4}, where σ is the spin index and dx denotes integration over R3 and a summation over σ .) The underlying Hilbert space is given by H = L2 (). The definition of a one-electron, respectively one-positron, state will correspond to the positive, respectively negative, energy solutions of the Dirac operator 1 D ϕ = α · ∇ + β − αϕ i

(2)

in which α, β denote the 4 × 4 Dirac matrices. The constant α is a positive real number, the Sommerfeld fine structure constant which is approximately 1/137. (We have picked units in which the electron mass is equal to one.) We will not assume that the nucleus is a point particle; we rather associate with it a density n ∈ L1 (R) whose integral gives the atomic Z number of the atom under consideration. For technical convenience we assume n to be a spherically symmetric Schwartz function whose Fourier transform has compact support. We remark: it is an experimental fact that the nucleus is not a point particle but an extended object. In fact the numerical calculations of the Lamb shift depend on the size of the support, which actually limits the accuracy of numerical value of the calculation of the Lamb shift because of the experimental uncertainty of the radius of the nucleus (Weinberg [30], p. 593). (A point nucleus leads also to mathematical difficulties, since the renormalized potential is more singular than the Coulomb potential, i.e., it could not be controlled by the kinetic energy (see Uehling [29] and Subsect. 3.5 of this paper). The electric potential of the nucleus is given as ϕ = | · |−1 ∗ n.

(3)

An application of the Young inequality shows that the nuclear potential ϕ ∈ L3+ (R3 ) ∩ L∞ (R3 )

(4)

under our assumption on the nuclear density n for any positive . Moreover, by Newton’s theorem 0 ≤ ϕ(x) ≤ Z/|x|.

(5)

For completeness we note the following fact whose proof is obvious: Lemma 1. Fix αZ ∈ [0, 1) and assume n to be a non-negative Schwartz function with n = Z and ϕ = | · |−1 ∗ n. Then, D ϕ is selfadjoint with domain H 1 (), i.e., the same domain as the free Dirac operator D0 . Moreover, if n is spherically symmetric, D ϕ has no negative discrete spectrum and its k th eigenvalue is bounded from below by the k th eigenvalue of the Coulomb Dirac operator D Z/|·| . We remark that Lemma 1 implies that the lowest eigenvalue is positive. ϕ ϕ Now, we can specify the electron and positron state spaces H+ and CH− respecϕ ϕ ϕ tively: the orthogonal projection on H+ and CH− are defined as P+ := χ[0,∞) (D ϕ ) ϕ ϕ and P− := 1 − P+ . The (anti-unitary) charge conjugation operator is given on H by ˆ ˆ = iβα2 ψ(−p). (Here (Cψ)(x) = iβα2 ψ(x). In momentum space it acts as (Cˆψ)(p) and in the following we follow the notation of Thaller [28]; see also [12].)

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Note that the Hilbert space can be written as the orthogonal sum ϕ

ϕ

H = H+ ⊕ H− .

(6)

Correspondingly a ∗ (f ) creates an electron in the state P+ f , whereas b∗ (g) creates a ϕ positron with wave function CP− g. Note that the definition of the operator a and b depends explicitly on the choice of the potential ϕ. The Hamiltonian for the non-interacting electron-positron field is given by (7) Dϕ = dx : ∗ (x)D ϕ (x) :, ϕ

where : : denotes normal ordering, i.e., anti-commuting of all creation operators to the left of all annihilation operators ignoring the anti-commutators. Note that for our renormalization procedure the choice of the electron-positron subspaces as the positive and negative eigenspaces of D ϕ is crucial, in fact it is a choice already proposed by Dirac [2]. The creation and annihilation operators fulfill the canonical anti-commutation relations {a(f ), a(g)} = {a ∗ (f ), a ∗ (g)} = {a(f ), b(g)} = {a ∗ (f ), b∗ (g)} = {a ∗ (f ), b(g)} = {a(f ), b∗ (g)} = 0,

(8)

{a(f ), a ∗ (g)} = (f, P+ g), {b∗ (f ), b(g)} = (f, P− g).

(9)

and ϕ

ϕ

Formally this is equivalent to {a(x), a(y)} = {a ∗ (x), a ∗ (y)} = {a(x), b(y)} = {a ∗ (x), b∗ (y)} = {a ∗ (x), b(y)} = {a(x), b∗ (y)} = 0,

(10)

{a(x), a ∗ (y)} = P+ (x, y), {b∗ (x), b(y)} = P− (x, y),

(11)

and ϕ

ϕ

ϕ

ϕ

ϕ

ϕ

where P+ (x, y), P− (x, y) are the integral kernels of the projectors P+ , P− . If there is no external potential there should be no polarization effects present. It is therefore expected that the difference Qϕ of one-particle density matrices of the perturbed and unperturbed vacua ϕ

ϕ

Qϕ := P+ − P+0 = −P− + P−0

(12)

plays a central role in defining the renormalized Hamiltonian. Using Cauchy’s formula we can express the Qϕ in terms of the respective resolvents (Kato [14], Sect. VI,5, Lemma 5.6) 1 ∞ 1 1 dη − . (13) Qϕ = 2π −∞ D ϕ + iη D 0 + iη The difference of Qϕ and of the first order resolvent expansion, i.e., α ∞ 1 1 ϕ Q − dη 0 ϕ 0 2π −∞ D + iη D + iη

(14)

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will contribute to the renormalized operator as follows: one interprets its spin summed diagonal as density. The corresponding electric potential should be added to the one particle operator. To avoid any unnecessary difficulties defining the operator we split (14) again in three summands motivated by iterating the resolvent equation: α 2 Q2 + α 3 Q3 + α 4 Q4 , where the indices 2, 3, 4 indicate the number of ϕ’s in the expression, i.e., 1 ∞ 1 1 1 dη 0 ϕ 0 ϕ 0 , Q2 := 2π −∞ D + iη D + iη D + iη ∞ 1 1 1 1 1 Q3 := dη 0 ϕ 0 ϕ 0 ϕ 0 , 2π −∞ D + iη D + iη D + iη D + iη 1 ∞ 1 1 1 1 1 Q4 := dη 0 ϕ 0 ϕ ϕ ϕ 0 ϕ 0 . 2π −∞ D + iη D + iη D + iη D + iη D + iη

(15)

(16)

We can immediately remark that the density corresponding to Q2 vanishes: the terms linear in the Dirac matrices vanish after summation over σ , since the Dirac matrices are traceless; the remaining terms are odd in η and vanish after integration over η. Consequently, we can disregard this term in defining the operator. We now define the density ρ3 (x) := (2π)

−3

R3

dp

R3

dq

4

ˆ 3 (p, σ ; q, σ ), ei(p−q)·x Q

(17)

σ =1

where

∞ ˆ 3 (p, q) = 1 dη dp1 dp2 (Dp + iη)−1 ◦ ϕ(p ˆ − p1 ) ◦ (Dp1 + iη)−1 Q 2π −∞ R3 R3 ◦ϕ(p ˆ 1 − p2 ) ◦ (Dp2 + iη)−1 ◦ ϕ(p ˆ 2 − q) ◦ (Dq + iη)−1 (18)

with Dr := α · r + β. The corresponding electric potential is P3 := ρ3 ∗ | · |−1 .

(19)

The quadratic form defining P4 is given by

(ψ, P4 ψ) := tr(χ Q4 ),

where χ (x) := dy|ψ(y)|2 /|y − x|. It will be useful to introduce the function C, 1 1 dx(1 − x 2 ) log[1 + k2 (1 − x 2 )/4] C(k) = k2 2 0 1 + 4/k2 + 1 1 2 2 4 5 4 = k (1 − 2 ) 1 + 2 log − + 3 k k 3 1 + 4/k2 − 1 k2

(20)

(21)

as already done by Serber [26] and Uehling [29] and later by Pauli and Rose [20], Jauch and Rohrlich [13], Schwinger [25], and Klaus and Scharf [16]. The vacuum polarization potential U , also known as Uehling potential, is defined via its Fourier transform C(k) Uˆ (k) = ϕ(k) ˆ . π|k|2

(22)

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The renormalized one-particle operator is 1 Dren := α · ∇ + β − αϕ − α 2 U + α 2 X − α 4 P3 − α 5 P4 , i

(23)

where X is the renormalized operator with integral kernel αX(x, y) :=

Qϕ (x, y) . |x − y|

(24)

To introduce the operator Dren might appear unmotivated at this point. However, it has a solid physical motivation: it emerges through mass and charge renormalization from the canonical formal textbook Hamiltonian (see, e.g., Milonni [19], p. 385, Formula (11.25)) for the interaction of electrons when there are no photons present. In turn the charge and mass renormalization originates in three physical principles W1, W2, and W3 as we will explain in Sect. 3. ϕ ϕ Moreover, and this is our main mathematical result, the operator P+ Dren on H+ turns out to be well defined and self-adjoint on the same form domain as the free Dirac operator: Theorem 1. Assume αZ ∈ [0, 1) and n with spherically symmetric Schwartz function with compact support in Fourier space, and ϕ = n ∗ | · |−1 . Then the quadratic forms ϕ (ψ, U ψ), (ψ, Xψ), (ψ, P3 ψ), (ψ, P4 ψ) on P+ (H 1 ()) are relatively form bounded with respect to (ψ, |D 0 |ψ) with form bound zero. This has the following consequence: Corollary 1. There exists a unique self-adjoint operator D+ fulfilling (ψ, D+ ψ) = ϕ (ψ, (D ϕ + α 2 X − α 2 U − α 4 P3 − α 5 P4 )ψ) for all ψ ∈ P+ (H 1 ()) with form domain ϕ P+ (H 1/2 ()). Furthermore it is bounded from below. √ √ Proof. Define c := 2 and M := 2 ϕ ∞ . Then, obviously 0 ≤ (c2 − 2)(D 0 )2 − 2ϕ 2 + M 2 implying (D 0 )2 ≤ c2 (D02 − 2(ϕD 0 ) + ϕ 2 ) + M 2 + 2Mc|D ϕ |. Rewriting this and taking the square root – which is an operator monotone function – yields |D 0 | ≤ c|D ϕ | + M.

(25)

This means that infinitesimal form boundedness with respect to |D 0 | implies also infinitesimal form boundedness with respect to |D ϕ |. Moreover, we have (ψ, |D ϕ |ψ) = ϕ (ψ, D ϕ ψ) for ψ ∈ P+ (H 1 ()). Thus, according to the KLMN theorem (Reed and Simon [23], Theorem X.17) there ϕ exists a unique self-adjoint operator D+ on H+ whose form domain is the form domain ϕ ϕ of D ϕ (which – since ϕ is bounded – is the form domain of P+ D 0 , i.e., P+ H 1/2 ()). Moreover, the quadratic form of D+ fulfills (ψ, D+ ψ) = (ψ, (D ϕ + α 2 X − α 2 U − α 4 P3 − α 5 P4 )ψ).

(26)

As already mentioned in the introduction, the Hamiltonian (23) can be used to describe µ-mesonic atoms where the interaction with the photon field is negligible as indicated experimentally by Peterman and Yamaguchi [21] and theoretically by Glauber et al [9].

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3. Physical “Derivation” of the Renormalized Hamiltonian We start with the formal expression for the interaction of electrons when no photons are present (Kroll and Lamb [17], French and Weisskopf [7]) found also in textbooks (see, e.g., Milonni [19]): ∗ (x)(x) ∗ (y)(y) 1 Wur = dx dy . (27) 2 |x − y| The Hamiltonian describing our system is formally given by Hur = Dϕur + αWur , where

Dϕur

:=

dx ∗ (x)D ϕ (x).

(28)

(29)

It is well known that this expression contains several singular terms. In particular it does not even contain any normal ordering. The remaining part of this section can be viewed as manipulating on it a physical allowed way and transforming it to a physically equivalent expression that is mathematically meaningful, namely the renormalized Hamiltonian. We emphasize that none of the steps taken is mathematically justified, i.e., the eventual justification of the renormalized Hamiltonian is its successful predictive power. We use three guiding principles to transform expressions for the energy into other physically equivalent ones as formulated and justified by Weisskopf [31], p. 6: “The following three properties of the vacuum electrons are assumed to be irrelevant: W1: The energy of the vacuum electrons in field free space. W2: The charge and current density of the vacuum electrons in field free space. W3: A field independent electric and magnetic polarizability that is constant in space and time.” Similar procedures have been suggested by Heisenberg [11], French and Weisskopf [7], Kroll and Lamb [17], and Dyson [4]. Exploiting the canonical anti-commutation relations (10) we can rewrite (28). For the one-particle part we have ϕ (30) Dϕur = Dϕ + dx(D ϕ P− )(x, x). The last summand is a – although infinite – constant which we drop, since it does not influence energy differences. For the two-particle part we get : ∗ (x)(x) ∗ (y)(y) : 1 Wur = dx dy 2 |x − y| ϕ ϕ P (x, y) − P− (x, y) 1 ∗ + dx dy : (x)(y) : + 2 |x − y| ϕ P (y, y) − + dx dy : ∗ (x)(x) : |x − y| ϕ ϕ ϕ ϕ P+ (x, y)P− (x, y) 1 P (x, x)P− (y, y) 1 + dx dy dx dy − + . (31) 2 2 |x − y| |x − y|

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The last two terms are again constants which we drop. The first term is the normal ordered two-particle interaction which has finite expectation in states of finite kinetic energy. We will denote it by : ∗ (x)(x) ∗ (y)(y) : 1 dx dy . (32) W= 2 |x − y| The remaining two other terms are one-particle operators of particular interest to us. Both terms, the classical electrostatic interaction energy of the electron with the polarized Dirac sea called the “non-exchange energy” Pur := −

dx

dy : ∗ (x)(x) :

ϕ

P− (y, y) , |x − y|

(33)

and the exchange energy 1 Xur := 2

dx

ϕ

ϕ

P (x, y) − P− (x, y) dy : (x)(y) : + |x − y| ∗

(34)

are not well defined. (For curiosity we remark that the latter is logarithmically divergent ˆ in , if one introduces a cut-off by (x) = |p|≤ (p, σ )e−ip·x dp). To renormalize the exchange energy we introduce the operators P+0 , P−0 which are the projectors on the positive and negative subspace of the free Dirac operator D 0 . (Note that we can interpret P−0 as the one-particle density matrix of the free Dirac sea.) 3.1. The renormalization of the exchange energy. To renormalize Xur we subtract the exchange interaction energy of the electron with the free Dirac sea using Principle W1, i.e., ϕ ϕ [(P+ − P+0 ) − (P− − P−0 )](x, y) 1 αX := dx dy : ∗ (x)(y) : 2 |x − y| ∗ = α dx dy : (x)(y) : X(x, y)

(35)

with X as defined in (24). (In physical language this subtraction of an undefined operator – known as “counter term” – is called “mass renormalization”. We refer to French and Weisskopf [7], Eq. (30), for the motivation of this terminology.) From now on, we will assume that the external potential ϕ is so weak that all there are only positive eigenvalues in the gap of D ϕ . Then, ∞ 1 1 1 ϕ P+ = + (36) dη 2 2π −∞ D ϕ + iη as well as P+0

1 1 = + 2 2π

∞

−∞

dη

D0

1 + iη

(37)

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249

(Kato [14], Sect. VI,5, Lemma 5.6). Thus, Qϕ =

α 2π

∞ −∞

dη

1 α2 1 ϕ 0 + 0 D + iη D + iη 2π

∞ −∞

dη

1 1 1 ϕ ϕ , D 0 + iη D ϕ + iη D 0 + iη (38)

where the first summand of the right-hand side is denoted by αQ1 and the second summand is treated in (14) through (16). Furthermore, since 2 ˆ p2 |ϕ(p)| <∞ dp 1 + |p| R3 the potential ϕ is regular in the sense of Klaus and Scharf [15], namely the operator Qϕ ∈ S2 (H), i.e., Qϕ is an Hilbert-Schmidt operator. (See also [12], Theorem 4.) This allows to show the finiteness of the exchange energy between the one-particle density matrix of the electron-positron field and the difference of the density matrices of the polarized Dirac sea and the free Dirac sea. To formulate the next lemma we fix the following notation: let Cp,q be the optimal constant in the generalized Young inequality, i.e., f ∗ g r ≤ Cp,q f p g q , 1 < p, q, r < ∞, r −1 + 1 = p −1 + q −1 . Lemma 2. Let ψ ∈ L3 () ∩ L2 (). Then

ψ(x)Qϕ (x, y)ψ(y)

dx dy

≤ C3/2,3/2 1/| · |2 3/2,w Qϕ 2 ψ 23 ,

|x − y| and for every > 0 there exists a constant C > 0 such that

ψ(x)Qϕ (x, y)ψ(y)

dx dy

≤ ψ 23 + C ψ 22 .

|x − y|

(39)

(40)

Proof. Since Qϕ is a Hilbert-Schmidt operator we get using the Schwarz inequality

ψ(x)Qϕ (x, y)ψ(y)

L := dx dy

|x − y| 1/2 1/2 |ψ(x)|2 |ψ(y)|2 ϕ 2 dx dy|Q (x, y)| . (41) ≤ dx dy |x − y|2 The second factor of the right-hand side is the Hilbert-Schmidt norm Qϕ 2 of Qϕ . To estimate the first factor we decompose the kernel into two functions f (x) := χBR (0) (x)/|x|2 and the rest g, i.e., 1/|x|2 = f (x) + g(x). Thus, using inequality (41) we get

L ≤ (|ψ|2 ∗ f, |ψ|2 )1/2 + (|ψ|2 ∗ g, |ψ|2 )1/2 Qϕ 2 . (42) We estimate the first and second summand of the first factor on the right-hand side separately from above.

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C. Hainzl, H. Siedentop

The first summand yields using the H¨older inequality followed by the generalized Young inequality (see, e.g., Reed and Simon [22], p. 32) (|ψ|2 ∗ f, |ψ|2 ) ≤ C3/2,3/2 ψ 2 23/2 f 3/2,w ,

(43)

where w indicates the weak-norm. Picking the radius R = ∞, i.e., g = 0, yields immediately (39). To prove (40) we also use (43) but pick the radius R > 0 sufficiently small: in this case we need to bound also the second summand containing g; we use again H¨older’s inequality now followed by using Young’s inequality (|ψ|2 ∗ g, |ψ|2 ) ≤ ψ 2 21 g ∞ .

(44)

Thus, the first factor on the right hand side of (42) is bounded by 1/2 1/2 C3/2,3/2 ψ 2 3/2 f 3/2,w + ψ 2 1 g ∞ . Since f 3/2,w tends to zero as R tends to zero, the claimed inequality follows.

3.2. Electrostatic vacuum polarization energy (non-exchange energy). In the expression for the electrostatic vacuum polarization energy we replace the density of the polarized sea by the difference of this density and the free Dirac sea using Principle W2: ϕ P (y, y) − P−0 (y, y) P˜ = − dx dy : ∗ (x)(x) : − |x − y| Qϕ (y, y) ∗ = dx dy : (x)(x) : |x − y| tr 4 Qϕ (y, y) ∗ dy : (x)(x) : C . (45) = dx |x − y| R3 (Here and in the following we will denote by Qϕ (x, y) the 4 × 4 matrix with entries (Qϕ (x, σ ; y, τ ))4σ,τ =1 .) However, the integral kernel of Qϕ is always singular on the diagonal except for vanishing potential as can be seen from (51) implying that P˜ is not well defined; one more renormalization is necessary. The question how to extract the physical relevant information from tr C4 Qϕ (y, y) was already asked by Dirac [2] and partially answered by Dirac [3], Heisenberg [11], Serber [26], Uehling [29], Weisskopf [31], Schwinger [25], Dyson [4], Klaus and Scharf [16], and others. The proposed solution amounted to a perturbative renormalization according to Principle W3. — One of our main results is that this renormalization can be done non-perturbatively: subtracting ϕ the zeroth order expansion Q1 of the difference Qϕ of P+ and P+0 will turn tr C4 Qϕ (y, y) into well defined quantities given in (19) and (20). Recall that 1 1 ∞ 1 ϕ 0 . (46) dη 0 Q1 = 2π −∞ D + iη D + iη Thus, in momentum space Q1 is given by ∞ α · p + β − iη α · q + β − iη −5/2 ˆ dη 2 ϕ(p ˆ − q) 2 , Q1 (p, q) = (2π) 2 p +1+η q + 1 + η2 −∞

(47)

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251

which leads to p · q + 1 − E(p)E(q) ˆ 1 (p, q) = 2−1/2 π −3/2 ϕ(p (48) ˆ − q) tr C4 Q E(p)E(q)(E(p) + E(q)) by a straightforward calculation with E(p) = p2 + 1. In configuration space we obtain ˆ 1 (r, q)e−iq·y dr dqeir·x tr C4 Q tr C4 Q1 (x, y) = (2π)−3 3 3 R R ˆ 1 (p − k/2, p + k/2)eip·(x−y) e−ik·(x+y)/2 = (2π)−3 dpdktr C4 Q ˜ x − y, x + y =: Q (49) 2 after introducing new variables of integration r = p − k/2 and q = p + k/2. Defin˜ x) ing ξ := x − y we remark that the “limits” limy→x tr C4 Q1 (x, y) and limξ →0 Q(ξ, ˜ are formally the same. The corresponding expression P in the electrostatic energy (45) becomes formally : ∗ (x)(x) : tr C4 Q1 (y, y) 4π ˆ˜ dx dy k), (50) = dk : ∗ :(k) 2 Q(0, |x − y| k ˆ˜ ˜ with respect to the second variable for fixed where Q(ξ, ·) is the Fourier transform of Q ξ = 0, i.e., formally ˆ −3/2 ˆ 1 (p − k/2, p + k/2)eip·ξ ˜ dp tr C4 Q Q(ξ, k) = (2π ) 3 R 1 p2 − k2 /4 + 1 − E(p − k/2)E(p + k/2) eip·ξ . = ϕ(k) ˆ dp 3 4π E(p − k/2)E(p + k/2) (E(p − k/2) + E(p + k/2)) R3 (51) We note that the integral (51) is logarithmically divergent at ξ = 0 independently of the form of the external potential ϕ. This shows – as already remarked above – that the limit limy→x tr C4 Qϕ (x, y) only exists, if ϕ vanishes. ˆ˜ , k) has been intensively studied in the literature (see, e.g., The expression Q(ξ Heisenberg [11], Serber [26], Pauli and Rose [20], Weisskopf [31], and Klaus and Scharf [16]). We will follow mainly the calculations of Pauli and Rose. (Since their treatment is time-dependent one has to set k0 = 0 to translate to our situation.) According to [20], ˆ˜ into two terms Eqs. (5) – (9), we can separate Q ˆ˜ , k) = F (ξ , k) + ϕ(k)k 2 ˆ F0 (ξ ), Q(ξ 1 with F0 (ξ ) = −

eip·ξ 1 p2 cos2 θ dp 1 − , 3 2 16π R3 1+p (1 + p2 )3/2

(52)

(53)

θ being the angle between ξ and p. With this definition of F0 the function F1 is finite for ξ = 0 and has there the value ρ vac (k) := F1 (0, k)

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C. Hainzl, H. Siedentop

=

2 p2 − k4 + 1 − E(p − k2 )E(p + k2 ) ϕ(k) ˆ 4π 3 R3 E(p − k )E(p + k ) E(p − k ) + E(p + k ) 2 2 2 2

+k2

p2 sin2 θ + 1 1 dp = ϕ(k)C(k), ˆ 4π 2 4E(p)5

(54)

where C is the function defined in (21). While each of the summands in the latter formula decreases like |p|−3 for large values of |p| and therefore the corresponding parts of the integral are logarithmically divergent, the difference in the integrand decreases like |p|−5 and is therefore convergent. Pauli and Rose [20] obtain the following asymptotic behavior for C: C(k)/k = 2

1 2 2 15 k + o(k ) 2 5 3 log(|k|) − 9

|k| → 0 . + o(1) |k| → ∞

(55)

We note that the second summand on the right-hand side of (52) can be written as 4π n(k)F ˆ 0 (ξ ), i.e., this depends only on the density of the nucleus. This implies that it can be dropped according to W3. This means that ρvac as defined in (54) can be considered as the physically relevant density of the polarized vacuum.

3.3. The fully renormalized Hamiltonian. The potential of the vacuum polarization density is U = | · |−1 ∗ ρvac ;

(56)

the corresponding term of the electrostatic interaction of the electron-positron field with the vacuum reads −α

2

∗

dx : (x)(x) : U (x) = −α

2

= −α 2

dx : ∗ (x)(x) : | · |−1 ∗ ρvac (x) dk : ∗ :(k)

4π ρ vac (k), k2

(57)

√ where we used F(| · |−1 ) = 2/π| · |−2 . According to (45), (38), (17), and (19) the second quantized renormalized polarization energy becomes P=

dx : ∗ (x)(x) : U (x) + α 2 P3 (x) + α 3 P4 (x) .

(58)

Consequently our fully renormalized Hamiltonian is H = Dϕ + αW + α 2 (−P + X) .

(59)

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253

3.4. Physical interpretation of the renormalization procedure of the vacuum polarization. In physics literature the subtraction of the singular part of the diagonal term of the one-particle density matrix of the Dirac sea, i.e., the dropping of the second summand (52)) is called charge renormalization for the following reason: the term subtracted from P˜ to obtain P is 2 α 4πF0 (ξ ) dx : ∗ (x)(x) : ϕ(x). (60) Formally this result can also be obtained by replacing the square of the (bare) charge e2 = α in D ϕ , e2 → e2 (1 + 4πe2 F0 (ξ )).

(61)

Note that F0 (ξ ) = log(|ξ |) + o(1) for small |ξ | which leads to a well known formula in the literature (see, e.g., Milonni [19], p. 417). It is interesting that a change in the effective charge due to the polarization of the vacuum was already suggested by Furry and Oppenheimer [8]. The word polarization is due to the following picture: according to Dirac the electrostatic field causes a redistribution of charge in the Dirac sea, i.e., it polarizes the vacuum. In particular the nucleus polarizes the vacuum in its vicinity causing a screening lowering the effective charge for an observer at a distance. Another reason for the fact that the infinity of the diagonal part of the density matrix of the Dirac sea in (52) is invisible in experiments is the following: in first order the factor in front of F0 (0) changes α in D ϕ by an (infinite) constant only, which does not effect the degeneracy of the eigenvalues of D ϕ , i.e., it does not cause a splitting of degenerated eigenvalues, a fact that is confirmed by experiment.

3.5. The vacuum polarization potential of the Coulomb potential. Recall that the nuclear 2 . Thus the Fourier transform ˆ = 4π n(k)/k ˆ potential is ϕ = | · |−1 ∗ n; consequently ϕ(k) of the vacuum polarization potential using (54) gives ϕ(k)C(k) ˆ n(k)C(k) ˆ Uˆ (k) = =4 , 2 π|k| |k|4

(62)

which is spherically symmetric and compactly supported under our assumptions on the charge distributions of the nucleus. If we assume that the nuclear density is a spherically symmetric Schwartz function – as we do in the mathematical part of this paper – this implies that the vacuum polarization potential U is bounded continuous and decreases exponentially at infinity. However, if we assume that we have a point nucleus as assumed by Uehling [29], this is no longer the case. To relate to Uehling’s work we will discuss this case as well although the corresponding potential is no longer form bounded with respect to the kinetic energy. From (62) we have 2 −1 C U =Z . (63) F π3 | · |4

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According to Uehling [29] and Schwinger [25], Eq. (2.53), this is 2 Z ∞ −2|x|s 1 (s 2 − 1)1/2 U (x) = 1+ 2 e ds, 3π |x| 1 2s s2 which means asymptotically  − 2 Z|x|−1 log |x| + 5 + γ + O(1) |x| → 0 3π 6 U (x) = , Z −2|x| −5/2  √ e |x| (1 + O(1/|x|)) |x| → ∞ 4 π

(64)

(65)

where γ ≈ 0.5772 is Euler’s constant. Consequently one obtains an effective potential 5 Z 2 2 2 Z ϕeff (x) = −αϕ − α U = −α +α log |x| + + γ + O(1) (66) |x| 3π |x| 6 close to the nucleus. Obviously, due to vacuum polarization, the effective potential becomes more singular than the Coulomb potential. This implies that the energy |p| − α 2

2Z 1 log 3π|x| |x|

is unbounded from below for all positive values of α and Z and (66) is no longer relatively form bounded with respect to the relativistic kinetic energy operator. This suggests to avoid the mathematical idealization of a point nucleus and to take the experimental fact that the nuclei are extended into account. 3.6. Splitting of the bound state energies. The effect of the vacuum polarization potential to lowest order in α is given by the effective one-particle operator ϕ ϕ P+ D ϕ − α 2 U (x) P+ . Therefore, the energy eigenvalues are shifted in lowest order of α by 2 δE = −α dxU (x)|ψ(x)|2 , R3 Dϕ .

(67)

where ψ(x) denotes an eigenstate of To get a rough heuristic estimate on the numerical effect Uehling [29] assumes the nucleus to be a point particle, i.e., its density is n(x) = Zδ(x), and takes the corresponding Schr¨odinger eigenstates ψn,l , where n is the principal quantum number and l the orbital-angular-momentum quantum number: δEn,l = −α

2

R3

dxU (x)|ψn,l (x)|2 ≈ −

4Zα 2 4Z 4 α 5 |ψn,l (0)|2 = − δ0,l . 15 15π n3

(68)

Concerning the first excited eigenvalue n = 2 this indicates an energy level splitting (of the 2s and 2p state) of δE2,0 ∼ − the Uehling effect [29], Eq. (29).

Z4α5 , 30

(69)

Vacuum Polarization

255

The vacuum polarization (69) accounts for only one percent of the 2s1/2 − 2p1/2 Lamb shift of hydrogen, since the Bohr radius is much bigger than the range of the vacuum polarization potential. However, the Bohr radii of muonic atoms are much smaller because of the large effective mass which means that the vacuum polarization of muonic helium accounts for 90 percent of the Lamb shift (Peterman andYamaguchi [21], Glauber et al. [9], see also Greiner et al. [10], p. 413). 4. Proof of the Self-Adjointness of the Renormalized Hamiltonian In the following we are going to show that the quadratic form associated to the operator ϕ H of the electron-positron field restricted to the one-electron sector H+ defines a selfadjoint operator which is bounded from below. This means – among other things – that higher order renormalizations, as introduced in perturbation theory by Dyson [4], are unnecessary. We will show infinitesimal form boundedness of all perturbations, namely U , P3 , ϕ ϕ and P4 relative to P+ D ϕ P+ . For X this has been already shown in Lemma 2. U: According to (62) and the remark thereafter U is bounded and therefore trivially infinitesimally relatively form bounded. P4 :We will show that for any positive there exists a C such that for all ψ ∈ D D ϕ ⊂ L3 () ∩ L2 (), (ψ, P4 ψ) ≤ ψ 23 + C ψ 2 , which implies the infinitesimal form boundedness by Sobolev’s inequality for Lemma 3. Assume αZ ∈ [0, 1), χ ∈

L5 ().

(70) √

−.

Then

| tr(χ Q4 )| ≤ χ Q4 1 ≤

Cϕ,4

ϕ 45 χ 5 3π 2

(71)

with Cϕ,4 := 1 + α ϕ

1

. Dϕ

(72)

Proof. We have 1 1 1 1 1 ∞ 1 ϕ ϕ ϕ ϕ dη χ 0 0 ϕ 0 0 2π −∞ D + iη D + iη D + iη D + iη D + iη 1 3 1 1 1 ∞ 1 ϕ ϕ . ≤ dη χ (73) 0 0 ϕ 2π −∞ D + iη 5 D + iη 5 D + iη 5 √ (We use the standard notation A p = p tr |A|p .) We will estimate the right-hand side of the above inequality which will also show that χ Q(4) ∈ S1 (H). To this end we estimate the factor containing the perturbed resolvent: 1 1 1 0 ϕ = ϕ D ϕ + iη D 0 + iη (D + iη) D ϕ + iη 5 5 0 1 1 ϕ . ≤ (D + iη) (74) ϕ 0 D + iη ∞ D + iη 5 | tr(χ Q4 )| ≤

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The first factor on the right side is finite: 0 1 1 (D + iη) ≤ 1 + α ϕ ≤ 1 + α ϕ 1 < ∞, D ϕ + iη Dϕ D ϕ + iη ∞ ∞ ∞ where we use that D ϕ is invertible because of Lemma 1 and that ϕ is bounded (see (4)). (Note the boundedness would also hold if ϕ = Zα/| · |, since 1/| · | is relatively bounded √ with respect to −.) Since

f (x)g(−i∇) 5 ≤ (2π)−3/5 f 5 g 5 ,

(75)

the norm being the one of the trace ideal S5 (L2 (R3 )) (Simon [27], Theorem 4.1), we can estimate the other factors occurring in (73): 1 1 2 + 1 + η2 χ ≤

χ

1/ | · | (76) 5 D 0 + iη 1/5 3/5 2 π 5 5 (norm in S5 (H) on the left-hand side and in S5 (L2 (R3 )) on the right-hand side) and a similar expression of the term containing ϕ. Using (74) and (76) allows us to continue (73) as ∞ Cϕ,4 1 | tr(χ Q4 )| ≤ 4 χ 5 ϕ 45 dη dp 2 3 4π (p + 1 + η2 )5/2 −∞ R ∞ Cϕ,4 Cϕ,4 1 1 ≤ 4 dη dp χ 5 ϕ 45 ≤

χ 5 ϕ 45 . 4π −∞ (η2 + 1) R3 (1 + p2 )5/2 3π 2

This has the Corollary 2. The perturbation P4 is relatively form bounded with respect to |D 0 | with form bound zero. Proof. We pick χ = |ψ|2 ∗ | · |−1 in Lemma 3 with ψ ∈ H 1/2 (). Using Young’s inequality followed by Sobolev’s inequality yields the desired result.

P3 : Unfortunately, Simon’s elegant trace inequality used in (73) does not suffice to handle the Q3 containing only four resolvents. In that case we estimate directly: Lemma 4. Denote the electric potential of a state ψ ∈ H 1/2 () of finite kinetic energy by χ(x) := dy|ψ(y)|2 /|x − y| and assume p > 3. Then there exists a constant Cϕ,p such that |(ψ, P3 ψ)| ≤ Cϕ,p χ p . Proof. We have

(ψ, P3 ψ) =

R3

(77)

dxχ (x)ρ3 (x) =

= (2π)−3/2

R3

dp1

R3

R3

dpχˆ (p)ρ3 (p)

dp2

4 σ =1

ˆ 3 (p1 , σ ; p2 , σ ), χˆ (p1 − p2 )Q

(78)

Vacuum Polarization

257

where we use Definition (17) of ρ3 . The “eigenfunctions” of the free Dirac operator in momentum space are   σ · p e  τ 1  τ = 1, 2,   N+ (p) −(1 − E(p))eτ uτ (p) := (79)  1 σ · p eτ   τ = 3, 4   N− (p) −(1 + E(p))eτ with eτ := (1, 0)t for τ = 1, 3 and eτ := (0, 1)t for τ = 2, 4 and N+ (p) = 2E(p)(E(p) − 1), N− (p) = 2E(p)(E(p) + 1).

(80)

The indices 1 and 2 refer to positive “eigenvalue” E(p) and the indices 3 and 4 to negative −E(p). (See, e.g., Evans et al. [6].) Using Plancherel’s theorem we get (ψ, P3 ψ) =

4 1 dp1 dp2 dp3 dp4 7 (2π ) R3 R3 R3 R3 τ1 ,τ2 ,τ3 ,τ4 =1 ˆ 2 − p3 )ϕ(p ˆ 3 − p4 )ϕ(p ˆ 4 − p1 ) ×χˆ (p1 − p2 )ϕ(p ×uτ1 (p1 )|uτ2 (p2 )uτ2 (p2 )|uτ3 (p3 )uτ3 (p3 )|uτ4 (p4 )uτ4 (p4 )|uτ1 (p1 ) ∞ 1 , (81) dη × (iaτ1 E(p1 ) − η)(iaτ2 E(p2 ) − η)(iaτ3 E(p3 ) − η)(iaτ4 E(p4 ) − η) −∞

with aτ = 1 for τ = 1, 2 and aτ = −1 for τ = 3, 4. The integral over η is seen to vanish by Cauchy’s theorem, if all four aτj have the same sign. In fact we have to distinguish only two cases, namely three of the aτj are equal and two of the aτj are equal. Therefore in the following we will only treat two different cases. The others then work analogously. We begin with aτ1 = −1, aτ2 = aτ3 = aτ4 = 1.

(82)

In that case the first factor in (81) reads uτ1 (p1 )|uτ2 (p2 ) uτ2 (p2 )|uτ3 (p3 ) τ1 =3,4

×

τ2 =1,2

uτ3 (p3 )|uτ4 (p4 )

τ3 =1,2

uτ4 (p4 )|uτ1 (p1 )

τ4 =1,2

σ · p σ · p + (1 + E(p ))(1 − E(p )) 2 1 2 1 = tr C2 2 N (p )2 N (p )2 N (p )2 N (p ) − 1 + 2 + 3 + 4 × σ · p2 σ · p3 + (1 − E(p2 ))(1 − E(p3 )) × σ · p3 σ · p4 + (1 − E(p3 ))(1 − E(p4 )) × σ · p4 σ · p1 + (1 − E(p4 ))(1 + E(p1 )) . We estimate the modulus of (83) and obtain tr 2 |σ · p4 σ · p1 + (1 − E(p4 ))(1 + E(p1 ))| |(83)| ≤ c C N− (p1 )N+ (p4 ) |p4 · p1 − (E(p4 ) − 1)(1 + E(p1 ))| + |p4 ∧ p1 | . ≤c N− (p1 )N+ (p4 )

(83)

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(Here and in the following c is a generic positive constant.) Since 1 ∞ 1 − dη 2π −∞ (−iE(p1 ) − η)(iE(p2 ) − η)(iE(p3 ) − η)(iE(p4 ) − η) 1 , = (E(p2 ) + E(p1 ))(E(p3 ) + E(p1 ))(E(p4 ) + E(p1 )) our term of interest (81) is bounded by constant times dp1 dp2 dp3 dp4 |χˆ (p1 − p2 )ϕ(p ˆ 2 − p3 )ϕ(p ˆ 3 − p4 )ϕ(p ˆ 4 − p1 )| R3

R3

R3

R3

|p4 · p1 − (E(p4 ) − 1)(E(p1 ) + 1)| + |p4 ∧ p1 | × . N− (p1 )N+ (p4 )(E(p2 ) + E(p1 ))(E(p3 ) + E(p1 ))(E(p4 ) + E(p1 ))

(84)

Substituting p2 → p1 + p2 turns (84) into ˆ 3 − p4 )ϕ(p ˆ 4 − p1 )| |ϕ(p ˆ 2 + p1 − p3 )ϕ(p dp1 dp2 dp3 dp4 |χˆ (−p2 )| 3 3 3 3 N (p )N (p ) − 1 + 4 R R R R |p4 · p1 − (E(p4 ) − 1)(E(p1 ) + 1)| + |p4 ∧ p1 | × (E(p2 + p1 ) + E(p1 ))(E(p1 ) + E(p3 ))(E(p4 ) + E(p1 )) =

R3

dp2 |χ(−p ˆ 2 )|f (p2 ),

(85)

where we introduce f to be the remaining integrand. We will now estimate f . Substituting p1 → p1 + p4 , p3 → p3 + p4 we get f (p2 ) = dp1 dp3 dp4 |ϕ(p ˆ 2 + p1 − p3 )ϕ(p ˆ 3 )ϕ(p ˆ 1 )| R3

R3

R3

|p4 · (p1 + p4 ) − (E(p4 ) − 1)(1 + E(p1 + p4 ))| + |p4 ∧ p1 | N− (p1 + p4 )N+ (p4 )(E(p2 + p1 + p4 ) + E(p4 + p1 )) 1 . × (E(p1 + p4 ) + E(p3 + p4 ))(E(p4 + p1 ) + E(p4 ))

×

Since E(p4 + p1 ) = E(p4 ) +

(p4 + µp1 ) · p1 E(p4 + µp1 )

for some µ ∈ [0, 1], we see that |p4 · (p1 + p4 ) − (E(p4 ) − 1)(E(p1 + p4 ) + 1)| + |p4 ∧ p1 | ≤ 4|p1 ||p4 |. Notice, we can bound dp4 R3

|p4 | ≤c E(p1 + p4 )2 E(p4 )N+ (p4 )N− (p1 + p4 )

independent of p1 and p3 . Therefore, f (p2 ) ≤ c dp1 dp3 |ϕ(p ˆ 2 + p1 − p3 )ϕ(p ˆ 3 )ϕ(p ˆ 1 )||p1 |. R3

R3

2 , we have that f (0) is finite; since n ˆ is compactly supported, Since ϕ(k) ˆ = 4π n(k)/k ˆ p1 and p2 are bounded. We conclude that f has also compact support.

Vacuum Polarization

Consequently, since

259

R3

dp2 |χˆ (−p2 )|f (p2 ) ≤ χˆ q f p ,

we see using the Hausdorff-Young inequality that for all p ≥ 2, dp2 |χ (−p2 )|f (p2 ) ≤ cp,ϕ χ p R3

(86)

(87)

for constant cp,ϕ depending on p and ϕ. Next, we take a peek at the case aτ1 = aτ2 = 1 and aτ3 = aτ4 = −1. The corresponding integral over η gives 1 ∞ 1 dη 2π −∞ (iE(p1 ) − η)(iE(p2 ) − η)(−iE(p3 ) − η)(−iE(p4 ) − η) 1 = (E(p2 ) − E(p1 ))(E(p1 ) + E(p3 ))(E(p1 ) + E(p4 )) 1 + . (−E(p2 ) + E(p1 ))(E(p2 ) + E(p3 ))(E(p2 ) + E(p4 )) |p2 ||p3 | Observe now that the corresponding first factor in (81) can be bounded by c·4 N+ (p . 2 )N− (p3 ) Now, we do similar variable transforms as above and arrive at an analogue of (87).

Again this has the Corollary 3. The perturbation P3 is relatively form bounded with respect to |D 0 | with form bound zero. The above result was the final step in showing that all four perturbations, U , P3 , P4 , ϕ ϕ and X, are form bounded with respect to P+ D ϕ P+ . References 1. Bethe, H.A.: The electromagnetic shift of energy levels. Phys. Rev., Minneapolis, II. Ser. 72, 339–341 (1947) 2. Dirac, P.-A.-M.: Th´eorie du positron. In: Cockcroft, J. Chadwick, F. Joliot, J. Joliot, N. Bohr, G. Gamov, P.A.M. Dirac, W. Heisenberg, (eds.), Structure et propri´et´es des noyaux atomiques. Rapports et discussions du septieme conseil de physique tenu a` Bruxelles du 22 au 29 octobre 1933 sous les auspices de l’institut international de physique Solvay. Publies par la commission administrative de l’institut. Paris: Gauthier-Villars. XXV, 353 S., 1934, pp. 203–212 3. Dirac, P.A.M.: Discussion of the infinite distribution of electrons in the theory of the positron. Proc. Camb. Philos. Soc. 30, 150–163 (1934) 4. Dyson, F.J.: The radiation theories of Tomonaga, Schwinger, and Feynman. Phys. Rev. (2) 75, 486–502 (1949) 5. Dyson, F.J.: Divergence of perturbation theory in quantum electrodynamics. Phys. Rev. (2) 85, 631–632 (1952) 6. Evans, W.D., Perry, P., Siedentop, H.: The spectrum of relativistic one-electron atoms according to Bethe and Salpeter. Commun. Math. Phys. 178(3), 733–746 (1996) 7. French, J.D., Weisskopf, V.F.: The Electromagnetic Shift of Energy Levels. Phys. Rev. II. Ser. 75, 1240–1248 (1949) 8. Furry, W.H., Oppenheimer, J.R.: On the Theory of the Electron and Positive. Phys. Rev. II. Ser. 45, 245–262 (1934) 9. Glauber, R., Rarita, W., Schwed, P.: Vacuum polarization effects on energy levels in µ-mesonic atoms. Phys. Rev. 120(2), 609–613 (1960)

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10. Greiner, W., M¨uller, B., Rafelski, J.: Quantum Electrodynamics of Strong Fields. Texts and Mongraphs in Physics. Berlin-Heidelberg-New York-Tokyo: Springer-Verlag, 1985 11. Heisenberg, W.: Bemerkungen zur Diracschen Theorie des Positrons. Z. Phys. 90, 209–231 (1934) 12. Helffer, B., Siedentop, H.: Form perturbations of the second quantized Dirac field. Math. Phys. Electron. J. 4(4), 16 (electronic), (1998) 13. Jauch, J.M., Rohrlich, F.: The theory of photons and electrons. The relativistic quantum field theory of charged particles with spin one-half. Cambridge, MA: Addison-Wesley Publishing Company, Inc., 1955 14. Kato, T.: Perturbation Theory for Linear Operators, Volume 132 of Grundlehren der mathematischen Wissenschaften. Berlin: Springer-Verlag, 1966 15. Klaus, M., Scharf, G.: The regular external field problem in quantum electrodynamics. Helv. Phys. Acta 50(6), 779–802 (1977) 16. Klaus, M., Scharf, G.: Vacuum polarization in Fock space. Helv. Phys. Acta 50(6), 803–814 (1977) 17. Kroll, N.M., Lamb jun, W.E.: On the Self-Energy of a Bound Electron. Phys. Rev. II. Ser. 75, 388–398 (1949) 18. Lamb, W.E., Retherford., R.C.: Fine structure of the hydrogen atom by a microwave method. Phys. Rev. 72(3), 241–243 (1947) 19. Milonni, P.W.: The Quantum Vacuum: An Introduction to Quantum Electrodynamics. 1st ed., Boston: Academic Press, Inc., 1994 20. Pauli, W., Rose, M.E.: Remarks on the Polarization Effects in the Positron Theory. Phys. Rev., II. Ser. 49, 462–465 (1936) 21. Peterman, A., Yamaguchi., Y.: Corrections to the 3d-2p transitions in µ-mesonic phosphorus and the mass of the muon. Phys. Rev. Lett. 2(8), 359–361 (1959) 22. Reed, M., Simon, B.: Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1975 23. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Volume 4, Analysis of Operators. 1st ed., New York: Academic Press, 1978 24. Schweber, S.S.: QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga. Princeton Series in Physics. Princeton, NJ: Princeton University Press, 1994 25. Schwinger., J.: Quantum Electrodynamics II. Vacuum Polarization and Self-Energy. Phys. Rev., II. Ser. 75, 651–679 (1949) 26. Serber, R.: Linear modifications in the Maxwell field equations. Phys. Rev., II. Ser. 48, 49–54 (1935) 27. Simon, B.: Trace Ideals and their Applications, Volume 35 of London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press, 1979 28. Thaller, B.: The Dirac Equation. 1st ed., Texts and Monographs in Physics. Berlin: Springer-Verlag, 1992 29. Uehling, E.A.: Polarization effects in the positron theory. Phys. Rev., II. Ser. 48, 55–63 (1935) 30. Weinberg, S.: The quantum theory of fields. Vol. I. Foundations, Corrected reprint of the 1995 original. Cambridge: Cambridge Univ. Press, 1996 ¨ 31. Weisskopf, V.: Uber die Elektrodynamik des Vakuums auf Grund der Quantentheorie des Elektrons. Math.-Fys. Medd., Danske Vid. Selsk. 16(6), 1–39 (1936) Communicated by B. Simon

Commun. Math. Phys. 243, 261–273 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0963-9

Communications in

Mathematical Physics

Nonlinear Instability in Two Dimensional Ideal Fluids: The Case of a Dominant Eigenvalue Misha Vishik1 , Susan Friedlander2 1 2

Department of Mathematics, University of Texas, Austin, TX 78712, USA Department of Mathematics, Statistics and Computer Science (M/C 249), University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607-7045, USA

Received: 7 March 2003 / Accepted: 7 April 2003 Published online: 5 November 2003 – © Springer-Verlag 2003

Abstract: It is proved that any steady 2 dimensional ideal fluid flow is nonlinearly unstable with respect to L2 growth in the velocity, provided there exists an eigenvalue λ for the linearised Euler equation with Reλ > . Here is the maximal Lyapunov exponent of the steady flow. Introduction The instability of a flow of an inviscid, incompressible fluid governed by the Euler equations is a classical problem investigated, for example, by Rayleigh [R], Arnold [A]. There are a number of explicit examples of steady flows U0 that have been shown to be linearly unstable. In these examples the instability may arise from unstable eigenvalues in the spectrum of the generator L of the linearised Euler equations, e.g. plane parallel shear flow with a sinusoidal profile [FSV-1]. However the instability may also arise from the existence of an unstable essential spectrum for the evolution operator eLt . Vishik [V] obtained an expression for the essential spectral radius ress (eLt ) in terms of a geometric quantity which in 2 dimensions is the maximal Lyapunov exponent of the flow U0 (and in 3 dimensions it is bounded from below by /2). Hence any steady Euler flow with a positive Lyapunov exponent is linearly unstable. From this result it follows, for example, that all Euler flows with a hyperbolic point (in 2 or 3 dimensions) are linearly unstable. Since the full Euler equations are a system of nonlinear PDEs, it is natural to ask if those flows that are unstable in the context of the linearised problem, are also unstable to perturbations governed by the full nonlinear equations. For the Euler equations the answer is not obvious because the dynamics of an ideal fluid are in no sense finite dimensional. Partial answers to this question have been given in several papers. In [FSV-1] it is proved that linear instability in L2 implies nonlinear instability in H s , s > n/2 + 1 (n being the space dimension), provided that the spectrum σ (eLt ) has a suitable gap. However this condition is difficult to verify for specific flows. The main application

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M. Vishik, S. Friedlander

is to those cases where = 0 and there exist unstable eigenvalues (e.g. shear flows with sinusoidal profiles). A stronger result was proved by Grenier [G] for classes of 2 dimensional flows in which = 0, namely shear flows or flows with no fixed points. For such flows Grenier proved that spectral instability (in the sense that there exists a linear exponentially growing mode) implies nonlinear instability with growth in L2 and L∞ . Koch [K] proved that for Euler flows in 2 dimensions nonlinear stability in C 1,α requires that all flows lines are periodic with a single period hence, in particular, all shear flows are unstable in C 1,α . Another partial result for 2 dimensional flows is proved by Bardos, Guo and Strauss [BGS], namely a steady flow U0 for which L has an unstable eigenvalue λ with Reλ > is nonlinearly unstable with respect to growth of the vorticity in L2 . In this present paper we give a further partial result concerning nonlinear instability of 2D Euler flows. We prove that steady flows U0 for which there exist an unstable eigenvalue with Reλ > are nonlinearly unstable with respect to L2 growth of the velocity (i.e. the energy norm). We remark that although it might be conceptually desirable to consider nonlinear instability in the same space for which it is possible to exhibit examples of nonlinearly stable flows, to date there are no satisfactory procedures for proving nonlinear stability for the Euler equations, even in 2D. The celebrated stability theorems of Arnold [A] are valid in the H 1 norm of the velocity. However the well posedness of the Cauchy problem for the incompressible Euler system is currently completely open in this space due to fundamental difficulties in the question of uniqueness. We claim that the L2 energy norm is the natural norm in which to consider the issue of stability or instability as opposed to the norms used in the partial results for nonlinear instability cited in the paragraph above. For example, Yudovich [Y] remarks that the H s , s > n/2 + 1, definition of instability used in [FSV-1] is probably too strong to appropriately represent hydrodynamic transitions from stability to instability. In this context L2 instability in velocity seems physically more reasonable. As is generally the case in fluid dynamics, the issue of nonlinear instability is much more difficult in 3 dimensions than in 2 dimensions. Although some of the constructions that we will develop work in both 2 and 3 dimensions, several crucial ideas utilize strongly the simpler structure of the equations in 2 dimensions. In particular, bounds on the nonlinear term in the velocity equation are estimated via the evolution equation for the vorticity which is considerably more tractable in 2 dimensions. The method of proof in this paper is by contradiction and utilises a bootstrap argument of the type employed in [BGS]. The initial condition for the nonlinear equation is taken to be the eigenfunction with Reλ > . We will address the general case of instability in the spectrum of the generator L in the energy space in the next paper.

1. Notation and Formulation 1.1. We introduce some notation and formulate the results of the paper. We treat the case when the dimension n = 2, and although some intermediate statements are valid for n = 3 too, we will not make an attempt to study this case in the present paper. Throughout the paper, T 2 = R2 /2πZ2 will denote the 2 dimensional torus. For any p ∈ [1, ∞), Lp (T 2 ) = Lp will denote the usual Lebesgue spaces. For s ∈ [−1, ∞), p ∈ (1, ∞), 1+s,p

W 1+s,p (T 2 ) = Lp (T 2 ) ∩ Wloc

(R2 )

Nonlinear Instability in Two Dimensional Ideal Fluids

263

denotes a Sobolev space. Here a function (distribution) on T 2 is viewed as a periodic 1+s,p function on R2 . Also Wsol = (W 1+s,p )2 ∩ {v|div v = 0}. The subscript sol denotes solenoidal, i.e. divergence free, functions. In particular, L2sol = L2 (T 2 )2 ∩ {v|divv = 0}. Let u(x, t), x ∈ T 2 , t ∈ R denote a solution to the Euler equations on T 2 :  ∂t u(x, t) = −(u, ∂x )u − ∂x p(x, t)    div u=0 (1.1)   = u(0).  u t=0

To discuss stability of (1.1) we introduce an equilibrium U0 (x), x ∈ T 2 , −(U0 , ∂x )U0 − ∂x P0 (x) = 0 div U0 = 0.

(1.2)

There are many examples of equilibrium solutions in the literature, see e.g., [L]. Linearizing (1.1) about U0 , yields  ∂t v = −(U0 , ∂x )v − (v, ∂x )U0 − ∂x q ≡ Lv    div v = 0 (1.3)   = v(0).  v t=0

Assuming U0 ∈ C ∞ (T 2 ), (1.3) is known to define a strongly continuous semigroup s,p (group) in every Sobolev space Wsol with generator L. This group will be denoted by exp{Lt}, v(t) = exp{Lt}v(0),

s,p

v(0) ∈ Wsol .

(1.4)

We introduce the spectrum σL2 (exp{Lt}) ⊂ C sol

and a partition disc σL2 (exp{Lt}) = σLess 2 (exp{Lt}) ∪ σL2 (exp{Lt}) sol

sol

as in [V]. A complex number z ∈

sol

σLdisc 2 (exp{Lt}) if and only if the operator exp{Lt} has sol

an eigenvalue z of finite multiplicity and

L2sol = V ⊕ W,

where V and W are closed invariant subspaces; dim V < ∞, (exp{Lt} − z)

is a sum W

of nilpotent Jordan cells; (exp{Lt} − z) is an isomorphism W → W . Any other point in σL2 (exp{Lt}) belongs to σLess 2 (exp{Lt}). sol

sol

We now define a suitable version of Lyapunov stability for the fluid equations (1.1). 1+s,p Let X = Wsol , where s > p2 ; Z = L2sol and U0 ∈ C ∞ (T 2 ) be an equilibrium solution as in (1.2).

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Remark 1. It follows immediately from definitions that X → Z. Definition. An equilibrium solution U0 is called (X, Z) Lyapunov stable if ∀ ρ > 0 ∃ δ > 0 so that for any u(0) ∈ X such that u(0) − U0 Z < δ the unique solution u(t) to (1.1) satisfies u(t) − U0 Z < ρ,

t ∈ [0, ∞).

An equilibrium U0 that is not (X, Z)-Lyapunov stable is called (X, Z) Lyapunov unstable. We will drop the reference to (X, Z) where it does not lead to confusion. Remark 2. It is well known that the Euler equations (1.1) are globally in time well posed in X. Throughout the paper C denotes a generic constant and ci , for some integer i, denotes a specific constant that has the same value each time it occurs. 1.2. We now can state the main result of this paper. ∞ (T 2 ) be a solution to (1.2). Denote by the maximal LyapuTheorem 1. Let U0 ∈ Csol nov exponent to the ODE x˙ = U0 (x):

1 |g∗t (x)η)|, log sup t→∞ t 2 2 x∈T ,η∈R ,|η|=1

= lim

(1.5)

where g∗t is the differential of the flow map g t : x → x(t). Assume there is a λ ∈ σL2 (L) sol such that Reλ > .

(1.6) 1+s,p

Then, U0 is (X, Z)-Lyapunov unstable for Z = L2sol , X = Wsol

,s >

2 p.

Remark 3. It follows from the result of [V] (see also [FSV-1]) that (1.6) is equivalent to the following condition. For any t > 0 there is a z ∈ σLdisc 2 (exp{tL}) such that sol

σLess 2 (exp{Lt}) sol

⊂ {w ∈ C|w| < |z|}.

(1.7)

∞ (T 2 ) Remark 4. As the proof below shows it is possible to relax the condition U0 ∈ Csol 1+s,p 2,∞ 2 2 2 to U0 ∈ Wsol (T ) ∩ Wsol (T ) for given s, p; s > p .

2. Proof of Theorem 1 in Case of a Smooth Eigenfunction 2.1. We first present the proof with an additional assumption that we state below. Let ψ be a normalized eigenfunction, corresponding to λ: Lψ = λψ,

ψ ∈ L2sol (T 2 ),

ψL2 = 1.

(2.1)

We first assume that ψ ∈ C ∞ (T 2 ). Without restricting the generality of our proof we may, and will, assume that λ in (1.6) has the maximal real part amongst the eigenvalues of L satisfying (1.6). Indeed, the discrete spectrum of L exponentiates, and for any κ > 0

Nonlinear Instability in Two Dimensional Ideal Fluids

265

there are only a finite number of eigenvalues z in σLdisc 2 (exp{Lt}) (with multiplicities) sol

such that

σLess 2 (exp{Lt}) ∈ {w ∈ C |w| < |z| − κ}. sol

The deeper result of [LV] that the growth bound is equal to the spectral bound is not used here. The Lyapunov stability of U0 is equivalent to the Lyapunov stability of the trivial solution v = 0 to the problem  ∂t v = −(U0 , ∂x )v − (v, ∂x )U0 − (v, ∂x )v − ∂x q    divv = 0 (2.2)   = v(0).  v t=0

We study a family of solutions v (x, t) to (2.2) with the initial condition v(0) = v (x, 0) = ψ(x).

(2.3)

Such a solution exists and is unique (see Remark 2) with v(·) ∈ L∞ loc ([0, ∞), X). Theorem 1 follows from the proposition below. Proposition 1. Assume there is a ψ ∈ C ∞ (T 2 ) satisfying (2.1). There exists a constant c0 > 0 and a constant 0 > 0 with the following property: for every ∈ (0, 0 ] there is a time T > 0 such that v ( ) (T )L2 ≥ c0 .

(2.4)

Indeed, given Proposition 1, and choosing ρ = c20 in the definition of Lyapunov stability, we get a contradiction with (2.4) at t = T , for sufficiently small. We need the following lemma. Lemma 1. Let α = Reλ, λ is the eigenvalue of L with maximal real part satisfying (1.6). Then there is a constant C > 0 and M0 > 0 such that exp{Lt}L2 ≤ C(1 + t)M0 eαt ,

t ≥ 0.

(2.5)

Proof. Fix t > 0 and consider the spectrum of exp{Lt} in L2sol . We have L2sol = Q ⊕ W , where Q is a spectral subspace corresponding to a finite number of eigenvalues zk , k = 1, . . . , N (with multiplicities) satisfying |zk | = eαt ,

k = 1, . . . , N.

On the complimentary closed subspace W we have exp{Lt} L(W ) ≤ Ce(α−κ)t W

for some κ > 0. Considering a finite number of Jordan cells in Q we arrive at (2.5) where, for example, M0 = dim Q − 1.

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Proof of Proposition 1. We consider the solution v(x, t) = v (x, t) to the problem (2.2), (2.3). We denote curl v by ω, where ω(x, t) ≡ ∂1 v2 (x, t) − ∂2 v1 (x, t). The solution v(x, t) to the Euler equations belongs to any space L∞ ). loc ([0, ∞); Wsol We choose p ∈ (2, ∞). Choose c1 = max(C, curl ψLp ), where C ≥ 1 is a sufficiently large constant to be chosen later. For a constant R ∈ ( 43 , 2), to be specified later, we consider the maximal time interval [0, T ] for which both of the following inequalities are satisfied: 1+s,p

v(t)L2 ≤ Reαt , t ∈ [0, T ]; ω(t)Lp ≤ c1 Reαt , t ∈ [0, T ].

(2.6) (2.7)

T can be equal to ∞. Because of our choice of c1 and because of the strong continuity of t → v(t), t → ω(t) in L2 we have T > 0. The first part of the proof will establish that T has to be sufficiently large for small > 0. To see this we proceed as follows, for any t ∈ [0, T ] using (2.2) and treating the nonlinear term F (v) = −(v, ∂x )v − ∂x q as a perturbation,

t

v(t) = exp{tL}v(0) +

exp{(t − τ )L}F (v(τ ))dτ.

(2.8)

0

Therefore, using (2.3), (2.1), (2.5)

(2.8 )

v(t)L2 ≤ e

αt

t

+C 0

(1 + (t − τ ))M0 eα(t−τ ) F (v(τ ))L2 dτ.

To estimate F (v)L2 we use the following simple lemma: Lemma 2. The following inequality holds: F (v)L2 ≤ C(vL2 + ωLp )ωLp ,

p ∈ (2, ∞).

(2.9)

Proof. Since the Leray projector is of norm 1 in L2 , F (v)L2 = − (v, ∂x )v − ∂x qL2 ≤ (v, ∂x )vL2 ≤ v 2p ∂x vLp ≤ C(vL2 + ωLp )ωLp . L p−2

Here we have used the Sobolev embedding theorem and the boundedness of the Calder´on-Zygmund operator ω → ∂x v in Lp . From (2.8 ), (2.9), (2.6), (2.7), v(t)L2 ≤ eαt + C ≤

t

(1 + (t − τ ))M0 eα(t−τ ) c12 2 R 2 e2ατ dτ

0

eαt + c2 c12 2 R 2 e2αt ,

since (1 + (t − τ ))M0 ≤ Ceα(t−τ )/2 for an appropriate constant C > 0.

(2.10)

Nonlinear Instability in Two Dimensional Ideal Fluids

267

We next use the equation for the vorticity ω(t): ∂t ω(x, t) = −(U0 + v, ∂x )ω − (v, ∂x )0 ,

(2.11)

where 0 = curl U0 . By treating the last term on the right side of (2.11) as a perturbation we get t (v(τ ), ∂x )0 Lp dτ ω(t)Lp ≤ ω(0)Lp + t 0 ≤ c1 + c3 v(τ )Lp dτ,

(2.12)

0

where c3 = ∂x 0 L∞ . We use compactness of the embedding Lp → W 1,p and a well known argument to conclude that for every ζ > 0 there is C(ζ ) > 0 so that vLp ≤ C(ζ )vL2 + ζ ωLp .

(2.13)

From (2.12), (2.13), (2.6), (2.7), ω(t)Lp ≤ c1 + CC(ζ ) Reαt + (1/α)c3 ζ c1 Reαt . We choose ζ =

α 8c3 ,

(2.14)

thus from (2.14),

R αt ω(t)Lp ≤ c1 1 + e + C Reαt 8 R αt ≤ c1 1 + e < c1 Reαt , t ∈ [0, T ], 4

(2.15)

provided that c1 is sufficiently large: namely c1 > 8C,

(2.16)

where C appears in the right side of the first line of (2.15). We now go back to the definition of T . Assuming T < ∞, at least one of the inequalities (2.6), (2.7) has to be an equality by a continuity argument; it cannot be (2.7) because of (2.15). Therefore T < ∞ ⇒ v(T )L2 = ReαT .

(2.17)

If this is the case, (2.10) implies R − 1 ≤ c2 c12 R 2 eαT , i.e.,

eαT ≥

R−1 . c2 c12 R 2

This means T ≥

R − 1 def 1 = T , log α

c2 c12 R 2

(2.18)

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M. Vishik, S. Friedlander

which is growing logarithmically as → 0. Therefore, both inequalities (2.6) and (2.7) are valid over the time interval [0, T ] where T is given by the right side of (2.18). The same conclusion is trivially valid in the opposite case when T = ∞. Using once again (2.8) we get t (1 + (t − τ ))M0 eα(t−τ ) F (v(τ ))L2 dτ, t ∈ [0, T ]. v(t)L2 ≥ eαt − C 0

(2.19) Applying (2.9), (2.6), (2.7) to (2.19) we get v(t)L2 ≥ eαt − c2 c12 2 R 2 e2αt ,

t ∈ [0, T ].

(2.20)

Choosing t = T , we arrive at v(T )L2 ≥ eαT − c2 c12 2 R 2 e2αT =

2 R−1 2 2 (R − 1) − c c R 2 1 c2 c12 R 2 c22 c14 R 4

=

R−1 (R − 1)2 1 − = (R − 1)(2 − R) = c0 > 0. 2 2 2 2 c2 c 1 R c2 c 1 R c2 c12 R 2

This completes the proof of the Proposition 1.*

3. Proof of Theorem 1: General Case 3.1. In this section we prove Proposition 1 under the less restrictive assumption ψ ∈ L2sol (T 2 ) without assuming smoothness. Remark 5. We note that although some regularity of the eigenfunction can be deduced from the equation it satisfies (cf. the result in Friedlander, et al [FSV-2] proving that all ∞ eigenfunctions

are C smooth when = 0), this regularity decreases with the quantity (Reλ − ) . Proposition 2. Assume there is a ψ ∈ L2sol (T 2 ) that satisfies (2.1). Then the same conclusion as in Proposition 1 holds. This proposition implies Theorem 1 in the way explained in Sect. 2. One additional difficulty that appears in the general case ψ ∈ L2sol (T 2 ) compared to Sect. 2 is the following. For the proof in Sect. 2 we used a linearised perturbation that grows asymptotically as eαt as t → ∞ and which is smooth in x. In the general case this cannot be guaranteed. Here is a replacement for the properties just described. Proposition 3. Assume there is a ψ ∈ L2sol (T 2 ) that satisfies (2.1). Then there is a func∞ (T 2 ), ψ tion ψ0 ∈ Csol 0 L2 = 1, real numbers c4 ≥ c5 > 0, and an integer M ∈ Z+ so that the following inequality holds true: c5 (1 + t)M eαt ≤ exp{Lt}ψ0 L2 ≤ c4 (1 + t)M eαt , t ∈ [0, ∞). *

In case T < ∞ we don’t need to use the lower bound (2.19) to get v(T )L2 = (R−1) > 0. 2 c2 c1 R

(3.1)

Nonlinear Instability in Two Dimensional Ideal Fluids

269

Proof. We partition the spectrum of L into the following 2 parts. Let Q be the spectral subspace corresponding to a finite number of eigenvalues λk , k = 1, . . . , N, with Reλk = α. As usual the eigenvalues are counted with multiplicity. Let W be a complementary spectral subspace, so that L2sol = Q ⊕ W.

(3.2)

The part of σL2 (L) corresponding to W belongs to a half-plane Reλ < α − κ for an sol appropriate κ > 0. This easily follows from the results of [V] and the general fact that the discrete part of the spectrum exponentiates. As a consequence we have etL L(W ) ≤ Ce(α−κ)t ,

t ∈ [0, ∞)

(3.3)

for some, possibly smaller, κ > 0. Let PQ and PW be the spectral projectors onto Q and W respectively; they commute with the group exp{tL} and PQ + PW = id; PQ PW = PW PQ = 0. Consider the ball Bγψ = f ∈ L2sol f − ψL2 < γ . sol

∞ (T 2 ) is dense in L2 it is possible to select a ψ ∈ B ∩ C ∞ (T 2 ) for every Since Csol 0 γ sol γ > 0. We now claim that for sufficiently small γ such a vector ψ0 satisfies (3.1) for an appropriate c4 , c5 , and for some M ∈ [0, N − 1] ∩ Z. Indeed, ψ

PQ ψ0 − ψL2 ≤ PQ γ . sol

Thus, by the normalization condition (2.1), and for γ ∈ (0, PQ −1 ), PQ ψ0 = 0.

(3.4)

Let PQ ψ0 =

N0

(3.5)

ψk ,

k=1

where ψk is in a spectral subspace Qk of L corresponding to distinct λk ∈ σL2 (L), sol Reλk = α, k = 1, . . . N0 ; N0 ≤ N. By construction we have Q=

N0

(3.6)

Qk .

k=1

By the theorem of Jordan, L

= λk id + Nk , Qk

where each Nk is nilpotent and commutes with L

. Therefore, Qk

270

M. Vishik, S. Friedlander

etL ψ0 = etL PQ ψ0 + etL PW ψ0 N0

= etL ψk + etL PW ψ0 k=1

=

N0

eλk t

k=1

N −1

=0

1 t Nk ψk + etL PW ψ0 . !

(3.7)

Let M ∈ [0, N − 1] be the maximal in the right side of (3.7) for which there is a k0 ∈ [1, N0 ] such that Nk0 ψk0 = 0. This choice of can be made because of (3.4)(3.6). We claim that with M constructed above (3.1) holds. To prove an upper bound we estimate the terms on the right side of (3.7), making use of (3.3): etL ψ0 L2 ≤ eαt sol

N0 M

1 t Nk ψk L2 + Ce(α−κ)t PW ψ0 L2 sol sol ! k=1 =0

≤ Ceαt (1 + t)M + Ce(α−κ)t ≤ c4 eαt (1 + t)M , t ∈ [0, ∞).

(3.8)

As for the lower bound, we again use (3.7) plus the fact that (3.6) is a direct sum to conclude etL ψ0 L2 ≥

N0

sol

M

1 t Nk ψk L2 − etL PW ψ0 L2 sol sol !

eλk t

k=1

=0

≥ C −1 eαt

M

1 t Nk0 ψk0 − Ce(α−κ)t ! =0

≥ C −1 eαt (1 + t)M

(3.9)

for some C > 0 and for t ∈ [c6 , ∞), where c6 > 0 is an appropriate constant. To recover the values t ∈ [0, c6 ] we note that etL ψ0 L2 ≥ c7 > 0, sol

t ∈ [0, c6 ].

(3.10)

Therefore, etL ψ0 L2 ≥ c5 eαt (1 + t)M sol

(3.11)

for an appropriate c5 > 0 and for all t ∈ [0, ∞). Combining (3.8), (3.11) leads to (3.1). Normalizing ψ0 produces a vector satisfying all the claimed properties. This concludes the proof of Proposition 3. Proof of Proposition 2. Following the procedure of Sect. 2.1, we consider the family v(x, t) = v ( ) (x, t) of solutions to problem (2.2), satisfying the initial conditions v(x, 0) = v ( ) (x, 0) = ψ0 (x),

(3.12)

Nonlinear Instability in Two Dimensional Ideal Fluids

271

when the function ψ0 (x) was constructed in Sect.3.1. This function will be fixed for the duration of this proof and that the constants c4 , c5 (and M0 ) are fixed and independent of . We choose c1 = max(C, c4 , curlψ0 Lp ) (3.12 ) as above, C ≥ 1 being a sufficiently large constant to be chosen below. We next c5 −1 c5 −1 choose R ∈ 1 − 4c , 1 + cc45 . We note that 1 − 4c < 1 + cc45 , since cc45 ∈ 4 4

(0, 1]. In particular, R > 1. The solution v (x, t) defined above belongs to any space 1+s,p L∞ , s > p2 . We choose p ∈ (2, ∞). loc ([0, ∞); Wsol We define T ≥ 0 so that [0, T ] is the maximal time interval for which we have both estimates v(t)L2 ≤ c4 Reαt (1 + t)M ,

t ∈ [0, T ];

(3.13)

ω(t)Lp ≤ c1 Re (1 + t) ,

t ∈ [0, T ].

(3.14)

αt

M

It is possible that T = ∞. From the definition of c1 , c4 and from the strong continuity of v(t), ω(t) with values in L2sol , Lp , we conclude T > 0. To prove that T is logarithmically growing as → 0 we obtain an upper bound for v(t)L2 , ω(t)Lp . First, for t ∈ [0, T ], using (2.5), (2.8), (3.1) we get t αt M v(t)L2 ≤ c4 e (1 + t) + C (1 + (t − τ ))M0 eα(t−τ ) F (v(τ ))L2 dτ. (3.15) 0

The estimates (2.9), (3.13), (3.14) applied to (3.15) yield t v(t)L2 ≤ c4 eαt (1 + t)M + C (1 + (t − τ ))M0 eα(t−τ ) c12 2 R 2 e2ατ (1 + τ )2M dτ ≤ c4 e (1 + t) αt

M

0 2 2 2 2αt + c8 c1 R e (1 + t)2M ,

(3.16)

since 1 + (t − τ )M0 ≤ Ceα(t−τ )/2 . We next turn to Eq. (2.11) for ω(t). Using the definition of c1 we get, as in (2.12), t ω(t)Lp ≤ c1 + c3 v(τ )Lp dτ, (3.17) 0

with c3 = ∂x ω0 L∞ . From (3.17), (2.13), (3.13), (3.14), ω(t)Lp ≤ c1 + CC(ζ ) Reαt (1 + t)M + (C/α)c3 ζ c1 Reαt (1 + t)M , (3.18) for any t ∈ [0, T ], ζ ∈ (0, ∞). We now choose ζ = ω(t)Lp

c5 α 8c3 c4 C

to obtain from (3.18),

Rc5 αt e (1 + t)M + c9 Reαt (1 + t)M . ≤ c1 1 + 8c4

For c1 sufficiently large (see (3.12 )), e.g., c1 > 8c9 cc45 this yields R c5 αt ω(t)Lp ≤ c1 1 + e (1 + t)M . 4 c4

272

But 1 +

M. Vishik, S. Friedlander Rc5 4c4

< R for R > 1 −

c5 −1 , 4c4

thus

ω(t)Lp < c1 Reαt (1 + t)M ,

t ∈ [0, T ].

(3.19)

We now can conclude that T is growing at least logarithmically as → 0. Indeed, since T is a maximal time for (3.13), (3.14) to be valid, using the continuity argument and inequality (3.19) for t = T yields the statement T < ∞ ⇒ v(T )L2 = c4 ReαT (1 + T )M .

(3.20)

From (3.20), (3.16) T < ∞ implies c4 (R − 1) ≤ c8 c12 R 2 eαT (1 + T )M , i.e.

eαT (1 + T )M ≥

c4 (R − 1) . c8 c12 R 2

(3.21)

We denote by T the unique positive solution to the equation

eαT (1 + T )M =

c4 (R − 1) , c8 c12 R 2

∈ (0, 0 ),

(3.22)

where 0 is sufficiently small. Then, from (3.21), (3.22), T ≥ T . Therefore, both (3.13), (3.14) are valid over the time interval [0, T ]. To obtain a lower bound on v(t)L2 we use (2.8), (3.1), v(t)L2 ≥ c5 eαt (1 + t)M − c8 c12 2 R 2 e2αt (1 + t)2M .

(3.23)

For t = T this yields, (see (3.22)), v(T )L2 ≥

c5 c4 (R − 1) c8 c12 R 2 (R − 1)2 c42 − c8 c12 R 2 c82 c14 R 4

c5 c4 (R − 1) (R − 1)2 c42 − = c8 c12 R 2 c8 c12 R 2 c4 (R − 1)(c5 − (R − 1)c4 ) = = c0 > 0 R2 c8 c12

=

(3.24)

In the last step we used the fact that (R − 1)(c5 − (R − 1)c4 ) > 0, since R ∈ 1 − c5 −1 , 1 + cc45 . The constant c0 > 0 is obviously independent of as → 0. This 4c4 concludes the proof of Proposition 2. Acknowledgements. The authors thank Roman Shvydkoy for many useful comments. The work of Misha Vishik is partially supported by NSF grants DMS-9876947, and DMS-0301531. The work of Susan Friedlander is partially supported by NSF grant DMS-0202767.

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Arnold, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, Vol. 60, Berlin-Heidelberg-New York: Springer, 1980 [BGS] Bardos, C., Guo, Y., Strauss, W.: Stable and unstable ideal plane flows. To appear [FSV-1] Friedlander, S., Strauss, W., Vishik, M.: Nonlinear instability in an ideal fluid. Ann. Inst. Henri. Poincar´e, Analyse Nonlin´eaire 14(2), 187–209 (1997) [FSV-2] Friedlander, S., Strauss, W., Vishik, M.: Robustness of instability for the two dimensional Euler equations. SIAM J. Math. Anal. 30(6), 1343–1355 (1999) [G] Grenier, E.: On the nonlinear instability of the Euler and Prandtl equations. Comm. Pure Appl. Math. 53(9), 1067–1091 (2000) [K] Koch, H.: Instability for incompressible and inviscid fluids. In: Partial Differential Equations, Praha, 1998, CRC Res. Notes Math. 406, London: Chapman Hall, 1999, pp. 240–247 [L] Lamb, H.: Hydrodynamics. 6th Ed., New York: Dover, 1945 [LV] Latushkin, Y., Vishik, M.: Linear stability in an ideal incompressible fluid. Commun. Math. Phys. 233, 439–461 (2003) [R] Lord Rayleigh: On the stability or instability of certain fluid motions. Proc. Lond. Math. Soc. 9, 57–70 (1880) [V] Vishik, M.: Spectrum of small oscillations of an ideal fluid and Lyapunov exponents. J. Math. Pures et. Appl. 75, 531–557 (1996) [Y] Yudovich, V.: On the loss of smoothness of the solutions of the Euler equations and the inherent instability of flows of an ideal fluid. Chaos. 10(3), 705–719 (2001) Communicated by P. Constantin

Commun. Math. Phys. 243, 275–314 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0968-4

Communications in

Mathematical Physics

The Master Ward Identity and Generalized Schwinger-Dyson Equation in Classical Field Theory 1, , Klaus Fredenhagen2 Michael Dutsch ¨ 1 2

Institut f¨ur Theoretische Physik, 37073 G¨ottingen, Germany. E-mail: [email protected] Institut f¨ur Theoretische Physik, 22761 Hamburg, Germany. E-mail: [email protected]

Received: 26 November 2002 / Accepted: 26 June 2003 Published online: 5 November 2003 – © Springer-Verlag 2003

Abstract: In the framework of perturbative quantum field theory a new, universal renormalization condition (called Master Ward Identity) was recently proposed by one of us (M.D.) in a joint paper with F.-M. Boas. The main aim of the present paper is to get a better understanding of the Master Ward Identity by analyzing its meaning in classical field theory. It turns out that it is the most general identity for classical local fields which follows from the field equations. It is equivalent to a generalization of the Schwinger-Dyson Equation and is closely related to the Quantum Action Principle of Lowenstein and Lam. As a byproduct we give a self-contained treatment of Peierls’ manifestly covariant definition of the Poisson bracket. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Classical Field Theory for Localized Interactions . . . . . . . . . 2.1 Retarded product and Poisson bracket . . . . . . . . . . . . 2.2 Higher order retarded products and perturbation theory . . . 2.3 Elimination of derivative couplings . . . . . . . . . . . . . . 3. The Master Ward Identity . . . . . . . . . . . . . . . . . . . . . . 3.1 Generalized Schwinger-Dyson Equation . . . . . . . . . . . 3.2 Definition of a map σ from free fields to unrestricted fields . 3.3 The Master Ward Identity . . . . . . . . . . . . . . . . . . . 4. Quantization: Defining Properties of Perturbative Quantum Fields 5. Application to BRS-Symmetry . . . . . . . . . . . . . . . . . . . 5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The free BRS-transformation . . . . . . . . . . . . . . . . . 5.3 Admissible interaction . . . . . . . . . . . . . . . . . . . .

Work supported by the Deutsche Forschungsgemeinschaft.

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. . . . . . . . . . . . . .

276 279 279 284 289 290 290 292 293 296 299 299 300 303

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6. Appendix A: Construction of the Map σ . . . . . . . . . . . . . . . . . . 6.1 Particular solution for σ . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Uniqueness of σ for a single real Klein-Gordon field . . . . . . . . 7. Appendix B: Formulation of BRS-Invariance of the Interaction Used in [7]

. . . .

307 307 308 309

1. Introduction The hard question in the renormalization of a perturbative quantum field theory (QFT) is whether the symmetries of the underlying classical theory can be maintained in the process of renormalization. The difficulties are connected with the singular character of quantized fields which forbids a straightforward transfer of the arguments valid for the classical theory. Typically the various symmetries which one wants to be present in the quantized theories are implied by certain identities (the Ward identities) which one imposes as renormalization conditions. Traditionally, the impact of symmetries of the classical theory on the structure of quantum theory was analyzed in terms of the functional formulation of QFT (see [19–22] and, e.g. [27]). In order to avoid infrared problems, it is, however, preferable to focus, in the spirit of algebraic quantum field theory, on the algebra of interacting fields. Actually, this becomes mandatory for quantum field theories on generic curved spacetimes (see, e.g. [3] and [16]). In the functional approach, the algebraic properties of the interacting fields are not immediately visible. In causal perturbation theory a` la Bogoliubov-EpsteinGlaser [2, 12], on the other hand, the local algebras of observables of the interacting theory can be constructed directly [3] and, hence, we work with this method. In causal perturbation theory a general treatment of symmetries is the Quantum Noether Condition (QNC) of Hurth and Skenderis [17]. It addresses the problem: given a free classical theory with a symmetry, find a deformed classical Lagrangian which possesses a deformed symmetry, and extend the symmetry to the quantized theory. In case of the BRS-current the QNC is closely related to the “perturbative gauge invariance” of [9], see the last Remark in Sect. 4.5.2 of [7] (published version). The main motivation for our works [5, 7] and to a certain extent for this paper is to give a construction of the local algebras of observables in quantum gauge theories, i.e. the elimination of the unphysical fields and the construction of physical states in the presence of an adiabatically switched off interaction. In [5] this construction was performed under certain conditions which were shown to be satisfied in QED. However, it turned out that in non-Abelian gauge theories additional relations, beyond QNC and perturbative gauge invariance, had to be fulfilled, in order to allow a local construction of the algebra of observables. The Master Ward Identity (MWI) (postulated in [7]) is a universal formulation of symmetries. In [7] it was shown that the MWI implies field equations, energy momentum conservation, charge conservation and a rigorous substitute for equal-time commutation relations of quark currents. Application of the MWI to the ghost- and to the BRS-current of non-Abelian gauge theories yields ghost number conservation and the “Master BRST Identity” [7]. These symmetries contain the information which is needed for our local construction of non-Abelian gauge theories. Reference [7] addresses the following problem: (I) Given a free theory, find a mapping from free fields to interacting fields, as a function of an interaction.

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The MWI was obtained there in the following way: the difference between different orders of differentiation and time-ordering, ∂xν1 T˜ (W1 , . . . , Wn )(x1 , . . . , xn ) − T˜ (∂ ν W1 , . . . , Wn )(x1 , . . . , xn )

(1)

(T˜ (W1 , . . . , Wn )(x1 , . . . , xn ) denotes the time-ordered product of the Wick polynomials W1 (x1 ), . . . , Wn (xn ) in free fields), is formally computed by means of the Feynman rules and the causal Wick expansion (see Sect. 4 of [12]) (or equivalently the normalization condition (N3) [5]). The MWI requires then that renormalization has to be done in such a way that this heuristically derived result is preserved. The main motivations for imposing this condition were, on the one hand, the many, important and far-reaching consequences of the MWI, and on the other hand, the experience that the MWI can nearly always be fulfilled. In this paper we give a further important argument in favor of the MWI: it is the straightforward generalization to QFT of the most general classical identity for local fields which can be obtained from the field equations and the fact that classical fields may be multiplied point-wise (see (11)). Since quantum fields are distributions which cannot, in general, be multiplied point-wise, the derivation of the MWI in classical field theory is not transferable to quantum field theory. There, the MWI is a highly non-trivial normalization condition which contains much more information than merely the field equations. In order for the MWI to be a well-defined and solvable renormalization condition also in models with anomalies, it must be generalized and a certain deviation from classical field theory must be admitted. This has been worked out in Sect. 5 of [7]. The present paper is mainly dedicated to the problem: (II) Given a classical theory, find an appropriate dictionary, relating classical fields to quantum fields. We formulate identities in classical field theory in such a way that they remain meaningful in quantum field theory; concerning symmetries these identities are classical versions of the Schwinger-Dyson equations. As far as we know, our formulation is new (see however the papers [4, 23]). As just indicated we start our study of the MWI with another equation, which will turn out to be equivalent to the MWI. Namely we first formulate the most general identity which follows in classical field theory from the field equations. Due to formal similarity we call it the Generalized Schwinger-Dyson Equation (GSDE). In this form it does not depend on a splitting of the Lagrangian into a free and an interaction part. We then introduce such a splitting and obtain the perturbative version of the GSDE which can be imposed as renormalization condition. It turns out that it is appropriate to use an off shell formalism where the entries of time-ordered products are classical fields not subject to any field equation, as advocated by Stora [32]. The MWI then gives a formula for time-ordered products of fields where one of the entries vanishes if the free field equations are imposed. In the traditional version of causal perturbation theory all calculations are done in terms of Wick products of free fields. There the same identity becomes visible as a non-commutativity of differentiation and time-ordering (1). In order to understand the connection between both formalisms we introduce a map σ which associates free fields to general (off shell) fields. The time-ordered products T of off shell fields in this paper are then related to the time-ordered products T˜ (1) of on shell fields by T˜ (W1 (x1 ), . . . , Wn (xn )) = T (σ (W1 )(x1 ), . . . , σ (Wn )(xn )) .

(2)

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In contrast to T˜ , there is no reason why T should not commute with derivatives. Therefore, we adopt here the proposal of Stora [32] and postulate that T can be freely commuted with derivatives. Stora calls this the Action Ward Identity (AWI), because it means that the interacting fields as well as the S-matrix depend on the interaction Lagrangian only via its contribution to the action.1 With that the non-commutativity of T˜ with derivatives is traced back to the noncommutativity of σ with derivatives. So, the MWI of [7] can be formulated in terms of time ordered products where one of the entries is of the form [∂µ , σ ](W ). The latter expression vanishes if the free field equations are imposed, hence the MWI of [7] is a special case of the MWI proposed in this paper. Actually, under a natural condition on the choice of σ , the two formulations are even equivalent. A puzzling observation was that the MWI of [7] seemed to provide renormalization conditions already on the tree level, involving, in general, free parameters, in spite of the fact that the classical theory is unique. The solution to this puzzle is that the map σ is non-unique: the freedom in the choice of parameters in the Feynman propagators of derivated fields (see [9] and [7]) is converted in the present formalism into the freedom in the choice of σ . Formula (2) can be interpreted in the following way: a fixed choice of σ gives a solution of problem (I) in terms of a solution of problem (II). The use of off shell fields has another advantage: it facilitates the introduction of auxiliary fields which in the presence of derivative couplings or in the definition of the BRS transformation may lead to a more elegant formulation. On the other hand, the use of auxiliary fields introduces more free parameters in the choice of σ . Our analysis might be compared with the formulation of the Quantum Action Principle of Lowenstein [21, 22] and Lam [19, 20]. These authors showed in the framework of BPHZ renormalization how classical symmetries can be transferred into renormalized perturbation theory. In contrast to these works, we emphasize the structural similarity of classical and quantum perturbative field theory. As a consequence, our arguments do not rely on the rather involved combinatorics of BPHZ renormalization. However, we did not yet investigate the structure of anomalies of the MWI. Another difference is that the formalism of Lam seems to be inconsistent for vertices containing higher than first derivatives of the basic fields, see Sect. V of [19]. We overcome these difficulties by means of the map σ (cf. the Example (112)). To the best of our knowledge, a general prescription for the renormalization of terms with higher order derivatives does not seem to exist in the framework of causal perturbation theory (prior to [7], cf. the remark by Stora in [32]).2 The paper is organized as follows: in Sect. 2.1 we study the canonical structure of classical field theory. We use Peierls’ covariant definition of the Poisson bracket [26] which does not rely on a Hamiltonian formalism. In Sect. 2.2 we determine the perturbative expansion of the classical fields as formal power series. The coefficients of this expansion are the retarded products. We prove that they satisfy the GLZ relations [13]. We briefly discuss the possibility of eliminating derivative couplings by introducing auxiliary fields in Sect. 2.3. In Sect. 3 we formulate the GSDE and the MWI, introduce the map σ and discuss the relation to the formulation of the MWI in [7]. 1 It might be that Lemma 1 in [22] or Lemma 1 in [19] actually implies the AWI, but due to the rather different formalisms this conjecture could not yet be verified. 2 The treatment of vertices containing higher than first derivatives is not an academic problem, e.g. the free BRS-current contains second derivatives of the gauge fields.

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In Sect. 4 we introduce the perturbative expansion of the interacting quantum fields (as formal power series) by the principle that as much as possible of the classical structure is maintained in the process of quantization, in particular the GLZ relation, the AWI and the MWI. All these conditions are formulated in terms of the retarded products, because in this formulation the conditions have the same form in the classical theory as well as in the quantum theory. In quantum theory, on the other hand, one can equivalently formulate everything in terms of the more familiar time-ordered products. The resulting formalism is not completely equivalent to the one given in [7]. We clarify the significance of the difference. In Sect. 5 we derive the “Master BRST Identity” [7] (which results from the application of the MWI and AWI to the free BRS-current) in the formalism of this paper. We also use the MWI and AWI to determine the admissible interaction of a BRS-invariant local gauge theory. By a modification of this procedure one can derive the conditions which are used in [7] to express BRS-invariance of the interaction from more fundamental principles. This is done in Appendix B. As a byproduct this will clarify the relation to perturbative gauge invariance (in the sense of [9]). Appendix A gives an explicit formula for the map σ and shows its uniqueness in a particular framework.

2. Classical Field Theory for Localized Interactions 2.1. Retarded product and Poisson bracket. To keep the notations simple we consider only one real scalar field ϕ (on the d-dimensional Minkowski space M, d > 2) and Lagrangians L = L0 + Lint ,

(3)

where the free part L0 is fixed. We will vary the interaction part def

Lint = −gP (ϕ, ∂µ ϕ) = : −gLint ,

(4)

which is a polynomial P = Lint in ϕ and ∂µ ϕ (later we will also allow for higher derivatives of ϕ) multiplied by a test function g ∈ D(M). The latter is interpreted as a space-time dependent coupling constant. We assume that the Cauchy problem is well posed for all Lagrangians in our class. For simplicity we restrict our formalism to smooth solutions. In non-linear theories, classical fields which are initially smooth may get singularities. But, in this paper, we are mainly interested in perturbation theory. It follows from the analysis in Sect. 2.2 that there is a unique smooth perturbative solution, if the given incoming free solution is smooth. Let CL be the set of smooth solutions f : M → R of the Euler-Lagrange equation ∂µ

∂L ∂L = , ∂(∂µ ϕ) ∂ϕ

(5)

with compactly supported Cauchy data. We consider CL as the classical phase space. (This is equivalent to the traditional point of view in which an element of the phase space is the set of the corresponding Cauchy data, e.g. the functions (f, f˙), f ∈ CL , restricted to the time x 0 = 0.)

280

M. D¨utsch, K. Fredenhagen def

We interpret the field ϕ as the evaluation functional on C = C ∞ (M, R): 3 def

ϕ(x)(f ) = f (x),

f ∈ C ∞ (M, R).

(6)

Functionals of the field F (ϕ) ≡

N

dx1 · · · dxn ϕ(x1 ) . . . ϕ(xn )tn (x1 , . . . , xn ),

N < ∞,

(7)

n=0

then lead in a natural way to functionals on C, def

F (ϕ)(f ) = F (f ),

f ∈ C.

(8)

Here t0 ∈ C and the tn are suitable test functions where we admit also certain distributions with compact support, in particular δ d(n−1) (x1 − xn , . . . , xn−1 − xn )f (xn ), f ∈ D(M). More precisely, we admit all distributions with compact support whose Fourier transform decays rapidly outside of the hyperplane {(k1 , . . . , kn ), i ki = 0}. We denote the algebra of functionals of the form (7) by F(C) and, when restricted to CL , by F(CL ). The field ϕL which satisfies the field equation (5) is obtained as the restriction of ϕ to CL , def

ϕL = ϕ|CL .

(9)

This restriction induces a homomorphism of the algebras of functionals F , F (ϕ) → F (ϕ)L = F (ϕL ).

(10)

In particular, the factorization property (AB)L (x) = AL (x)BL (x)

(11)

holds for polynomials A, B in ϕ and its partial derivatives. (The algebra of these polynomials will be denoted by P, P= {∂ a ϕ, a ∈ Nd0 } . ) (12) It is a main difficulty of quantum field theory that the factorization property (11) is no longer valid. To compare theories with different Lagrangians we use the fact that by the assumed uniqueness of the solution of the Cauchy problem there exists to each f2 ∈ CL2 precisely one f1 ∈ CL1 which coincides with f2 outside of the future of the region where the respective Lagrangians differ, f1 (x) = f2 (x) ∀x ∈ (supp (L1 − L2 ) + V + ).4 We denote the corresponding map (the “wave operator”) by rL1 ,L2 : rL1 ,L2 : CL2 → CL1 , f2 → f1 .

(13)

3 A complex scalar field is the analogous evaluation functional on C ∞ (M, C) and we define ϕ ∗ (x)(f ) ≡ f ∗ (x). 4 V denote as usual the forward and backward light-cones, respectively, and V their closures. ± ±

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Obviously it holds rL1 ,L2 ◦ rL2 ,L3 = rL1 ,L3 . Analogously we define aL1 ,L2 : CL2 → CL1 , f2 → f1 by requiring that f1 and f2 agree in the distant future. This bijection between the spaces of solutions can be used to express the interacting fields as functionals on the space of free solutions. We call def

Aret Lint (x) = A(x) ◦ rL0 +Lint ,L0 : CL0 → R

(14)

for A ∈ P the “retarded field”. The retarded field is a functional on the free solutions which solves the interacting field equation. We will define the perturbation expansion of classical interacting fields as the Taylor series of the retarded fields as functionals of the interaction Lagrangian, Aret Lint (x) =

∞ 1 Rn,1 (L⊗n int , A(x)) . n!

(15)

n=0

The retarded products Rn,1 will be constructed in Sect. 2.2. Note µ ret ∂xµ Aret Lint (x) = (∂ A)Lint (x)

(16)

and that the factorization property (11) holds for the retarded fields, too. Besides the commutative and associative product (11) there is a second product for classical fields: the Poisson bracket. Peierls [26] has given a definition of the Poisson bracket without recourse to a Hamiltonian formalism. We now review his procedure. It is convenient to generalize our formalism somewhat: we admit also non-localinteractions, i.e. the interaction part of the action S does not need to be of the form Sint = dx Lint (x), but may be replaced by an arbitrary functional F ∈ F(C). The field equations are still obtained by the principle of least action, but in contrast to (5) they may involve non-local terms. The classical phase space CL , the field ϕL , F (ϕ)L and the maps rL1 ,L2 , aL1 ,L2 (13) are defined in the same way as before, but now denoted by CS , ϕS , F (ϕ)S and rS1 ,S2 , aS1 ,S2 . Equation (10) and the factorization (11) still hold true. We will not discuss whether solutions of the general Cauchy problem for these non-local actions exist. It is sufficient for our purpose that perturbative solutions always exist and are unique. Let F ≡ F (ϕ) and G ≡ G(ϕ) be functionals from F(C). We introduce the retarded product RS (F, G) and the advanced product AS (F, G), d |λ=0 G ◦ rS+λF,S , dλ def d AS (F, G) = |λ=0 G ◦ aS+λF,S dλ def

RS (F, G) =

(17) (18)

which are functionals on CS . Note that the entries of the retarded and advanced products are unrestricted functionals of the field. In general it is not possible to replace them by their restriction to the space of solutions, as the following example shows: let S = dx L0 (x) with L0 = 21 ∂µ ϕ∂ µ ϕ, F = dx g(x)ϕ(x) and G = ϕ(y). Then ϕ(y) ◦ rS+λF,S (f ) = f (y) + λg(y) and hence RS (F, ϕ(y)) = g(y), but FS = 0. The retarded and advanced products have the following important properties [4, 23]: Proposition 1. (a) The retarded product can be expressed in terms of the retarded Green function, δG δF RS (F, G) = − dx dy ret (x, y) . (19) δϕ(x) S δϕ(y) S

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(b) The advanced and retarded products are related by AS (F, G) = RS (G, F ) .

(20)

2

δ S Here ret S is the unique retarded Green function of δϕ(x)δϕ(y) considered as an integral operator, i.e. it satisfies the equations δ2 S δ2 S ret dy S (x, y) = δ(x − z) = dy ret (y, z), (21) δϕ(y)ϕ(z) δϕ(x)ϕ(y) S

and (in case S is local) supp ret S ⊂ {(x, y) | x ∈ y + V + }.

(22)

For non-local interactions with compact support we may construct the retarded Green function (and also the retarded product) in the sense of formal power series. δ2 S is symmetric, it follows that Since δϕ(x)ϕ(y) def

ret adv S (x, y) = S (y, x)

(23)

is the advanced Green function. Similarly to the proof of (a) one finds that the advanced adv product AS (F, G) fulfills (19) with ret S replaced by S . This and (23) immediately imply (b). Example. The abstract formalism may be illustrated by the example of a real Klein Gordon field with a polynomial interaction, S = dx 21 [∂ µ ϕ∂µ ϕ − m2 ϕ 2 − gP (ϕ)]. We obtain

δ2 S δϕ(x)δϕ(y) (f )

= −( + m2 + g(x)P (f (x)))δ(x − y), and

ret ret S (x, y)(f ) = − (x, y; gP (f )),

(24)

is the unique retarded Green function of the Klein-Gordon operator with a potential gP (f ), where f is the classical field configuration on which the functionals are evaluated. We will use the formula (19) as definition of the retarded product outside of the space of solutions f ∈ CS (and analogously for the advanced products), as it was done by Marolf [23]. Proof. It remains to prove (a). Let f ∈ CS and rS+λF,S (f ) = f + λh + O(λ2 ). Then d δ(S + λF ) δF δ2 S 0= |λ=0 (f + λh) = (f ) + dz (f )h(z) (25) dλ δϕ(y) δϕ(y) δϕ(y)ϕ(z) and (in the case of a local action S) h(z) = 0 if z ∈ supp (F ) + V¯+ . Hence, δF RS (F, ϕ(x))(f ) = h(x) = − dy ret (f ), S (x, y) δϕ(y) and by means of d RS (F, G)(f ) = |λ=0 G(ϕ)(f + λh) = dλ

dx

δG (f )h(x), δϕ(x)

(26)

(27)

we obtain the assertion (19). (In the case of a non-local action the condition on the support of h has to be appropriately modified.)

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Definition 1. The Peierls bracket associated to an action S is a product on F(C) with values in F(CS ) defined by def

{F, G}S = RS (F, G) − RS (G, F ) = RS (F, G) − AS (F, G).

(28)

The Peierls bracket depends only on the restriction of the functionals to the space of solutions, {F, G}S = {F , G}S if FS = FS .

(29)

Namely, let F be a functional which vanishes on the space of solutions. This is the case if F is of the form5 δS . (30) F = dx G(x) δϕ(x) Then the retarded product with a functional H is RS (F, H ) = dx dy dz

δG(x) δS δH δ2 S + G(x) ret . S (y, z) δϕ(y) δϕ(x) δϕ(x)δϕ(y) δϕ(z)

The first term vanishes on the space of solutions, therefore we obtain δH . RS (F, H ) = dz G(z) δϕ(z)

(31)

(32)

The same expression is obtained for the advanced product, thus the Peierls bracket of F and H vanishes. We may therefore define the Peierls bracket for functionals on the space of solutions by {FS , GS } = {F, G}S . It is easy to see that for the example of the Klein Gordon field with a polynomial interaction without derivatives the Peierls bracket coincides with the Poisson bracket obtained from the Hamiltonian formalism. The Peierls bracket, however is defined also for derivative couplings and even for non-local interactions where the Hamiltonian formalism has problems. Moreover, it is manifestly covariant and does not use a splitting of space-time into space and time. We now want to show that the Peierls bracket fulfills in general the usual properties of a Poisson bracket. Antisymmetry, linearity and the Leibniz rule are obvious 6 , but the Jacobi identity is non-trivial (actually it is not discussed in the paper of Peierls, however in [4, 23]). We will see that the Jacobi identity follows from the fact that rS2 ,S1 commutes with the Peierls bracket (hence it is a canonical transformation). 5 We tacitly assume here that the ideal J generated by the field equation (i.e. the set of functionals of S the form (30)) is identical with the set of functionals which vanish on the space of solutions. This seems to be true in relevant cases. Otherwise, we have to replace the restriction map F → FS by the quotient map with respect to JS . 6 Linearity and the Leibniz rule hold already for the retarded and advanced products.

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Proposition 2. (a) The retarded wave operator rS2 ,S1 preserves the Peierls bracket (28), {F ◦ rS2 ,S1 , G ◦ rS2 ,S1 }S1 = {F, G}S2 ◦ rS2 ,S1

(33)

and the same statement holds for aS2 ,S1 . (b) The Peierls bracket (28) satisfies the conditions which are required for a Poisson bracket, in particular the Jacobi identity {FS , {HS , GS }} + {GS , {FS , HS }} + {HS , {GS , FS }} = 0 .

(34)

Proof. It suffices to prove (33) for an infinitesimal change of the interaction: setting S1 = S and S2 = S + λH , the infinitesimal version of (33) reads {RS (H, F ), GS } + {FS , RS (H, G)} d |λ=0 (RS+λH (F, G) − AS+λH (F, G)), = RS (H, {F, G}) + dλ

(35)

where we used in the last term the extended definition of the retarded and advanced products outside of the space of solutions introduced after Proposition 1. We now insert the formula for the retarded product of Proposition 1 everywhere in δ (35). Applying δϕ to (21) we find δ adv (x, y) = − δϕ(z) S

dv dw adv S (x, v)

δ3S adv (w, y), δϕ(v)δϕ(w)δϕ(z) S

(36)

and analogously d |λ=0 adv S+λH (x, y) = − dλ

dv dw adv S (x, v)

δ2 H adv (w, y). δϕ(v)δϕ(w) S

(37)

With that (35) can be verified by a straightforward calculation. By an analogous calculation we prove that the advanced transformation aS2 ,S1 is a canonical transformation. The infinitesimal version is (35) where in the first three terms RS is replaced by AS and where the last term, the term with the λ-derivative, is unchanged. Hence, considering the difference of these two versions of (35), the latter term drops out, and we obtain the Jacobi identity (34). 2.2. Higher order retarded products and perturbation theory. In analogy to Eq. (17) we define the higher order retarded products by RS (F ⊗n , G) =

dn |λ=0 G ◦ rS+λF,S . dλn

(38)

They have a unique extension to (n+1)-linear functionals on F(C) which are symmetric in the first n variables. With that the perturbative expansion of G ◦ rS+λF,S in λ reads G ◦ rS+λF,S

∞ λn n=0

n!

λF RS (F ⊗n , G) ≡: RS (e⊗ , G)

(39)

in the sense of formal power series. If S is the free part of the action, this is the perturbative expansion of the retarded fields (14) in terms of free fields.

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In the first paper of [6], Eq. (71), we gave an explicit formula for R in the case where all functionals are local and where F and G do not contain derivatives. It took the form of a retarded multi-Poisson bracket. In case of derivative couplings this can no longer be true, as we already know from the discussion of the retarded product of two factors, cf. the counterexample after Eq. (18). The general case where S, F and G might be non-local can be obtained from the formula d λF λF , G) = RS (e⊗ , RS+λF (F, G)) RS (e⊗ dλ

(40)

which is the Taylor series expansion of d G ◦ rS+λF,S = RS+λF (F, G) ◦ rS+λF,S , dλ

(41)

the latter identity following directly from the definition of the retarded products. On the r.h.s. of (40) we understand RS+λF (F, G) as an unrestricted functional, i.e. RS+λF (F, G) ∈ F(C). Comparing the coefficients on both sides of (40) yields a recursion relation: n n δF δG adv (n−l) RS (F ⊗(n+1) , G) = − RS F ⊗l , dx dy S+λF (x, y) , (42) l δϕ(x) δϕ(y) l=0

where k def d adv (k) S+λF (x, y) = k |λ=0 adv S+λF (x, y) dλ = (−1)k k! dv1 · · · dvk dz1 · · · dzk adv S (x, v1 )

·

δ2 F δ2 F adv adv (zk , y) . S (z1 , v2 ) · · · δϕ(v1 )δϕ(z1 ) δϕ(vk )δϕ(zk ) S

This shows the existence of solutions in the sense of formal power series. Peierls’ formula for the Poisson bracket together with the fact that the retarded wave operators rS,S0 are canonical transformations lead to an interesting relation between the higher order retarded products. Let F, G and S1 be functionals from F(C) and let S = S0 + λS1 . Then we have {F ◦ rS,S0 , G ◦ rS,S0 }S0 = (RS (F, G) − RS (G, F )) ◦ rS,S0 .

(43)

According to the definition of the retarded products it holds RS (F, G) ◦ rS,S0 =

d |µ=0 G ◦ rS+µF,S0 , dµ

where we used the composition property of the wave operators. If we now take the nth derivative with respect to λ on both sides of (43) we find {RS0 (⊗i∈I Hi , F ), RS0 (⊗j ∈I Hj , G)}S0 I ⊂{1,... ,n}

= RS0 (⊗ni=1 Hi ⊗ F, G) − RS0 (⊗ni=1 Hi ⊗ G, F )

(44)

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M. D¨utsch, K. Fredenhagen

with Hi ∈ F(C). This relation is in quantum field theory known as the GLZ-Relation (see below). It plays an important role in renormalization. In case S and F are local, we can find an elegant expression for the retarded products. We introduce the following differential operator on the space of functionals F(C) δF δ R(x) := − dy ret . (45) S (y, x) δϕ(x) δϕ(y) Note that R(x) is smooth in x since ret S maps smooth functions with compact support onto smooth functions. According to (19) we have RS (F, G) = dx (R(x)G)S . (46) The nth order case looks quite similar: Proposition 3. The nth order retarded product is given by the formula ⊗n RS (F , G) = n! dx1 . . . dxn (R(x1 ) · · · R(xn )G)S . x10 ≤...≤xn0

(47)

Proof. We first show that the power series defined by the r.h.s. of formula (47) G → G(λ) = RS0 (exp⊗ λF, G)

(48)

defines a homomorphism on the algebra of functionals G ∈ F(C). This means that for two functionals G and H we have the factorization RS0 (F ⊗n , GH )

=

n n k=0

k

RS0 (F ⊗k , G)RS0 (F ⊗n−k , H ) .

(49)

We use the fact that the operators R(x) are functional derivatives of first order. Hence from Leibniz’ rule we get R(x1 ) · · · R(xn )GH =

(

I ⊂{1,... ,n} i∈I

R(xi )G)(

R(xj )H ) .

(50)

j ∈I

It remains to check the time ordering prescription. For any n-tupel of times t = (x10 , . . . , xn0 ) we choose a permutation πt with xπ0t (1) ≤ . . . ≤ xπ0t (n) . We obtain RS0 (F ⊗n , G) =

d n x(R(xπt (1) ) . . . R(xπt (n) )G)S .

(51)

If we insert (50) into (51) we see that the permutation πt restricted to I as well as to the complement of I yields the correct time ordering. The n-fold integral factorizes, and the integrals over (xi , i ∈ I ) and (xj , j ∈ I ) do not depend on the choice of I , but only on the cardinality of I . This proves (49). We now show that the formal power series for the retarded field (48) with acδ (S + λF ) = 0 and thus coincides with tion S + λF satisfies the field equation δϕ(x)

Master Ward Identity in Classical Field Theory

287

RS (exp⊗ λF, G) because of (49) and the uniqueness of the retarded solutions. We have ) δ to show RS0 (exp⊗ λF, δ(S+λF δϕ(x) ) = 0. So we insert G = δϕ(y) S into (48), use δ δF R(x) S=− δϕ(y) δϕ(x)

dz ret S (z, x)

δ2 S δF =− δ(x − y) , δϕ(x)δϕ(z) δϕ(x)

(52)

δF = 0 if y and obtain the wanted field equation where we exploit the fact that R(y) δϕ(x) is not in the past of x.

Since we are mainly interested in local functionals, we change our point of view somewhat. Let P denote as before the set of polynomials of ϕ and its derivatives (12). Each field A ∈ P defines a distribution with values in F(C), A(f ) = dx A(x)f (x) , f ∈ D(M) . We fix a local action S which later will be the free action. We now define the retarded products of fields as F(CS ) valued distributions in several variables by Rn,1 (A 1 ⊗ · · · ⊗ An , B)(f1 ⊗ · · · ⊗ fn , g) ≡

d(x, y) Rn,1 (A1 (x1 ), . . . , An (xn ); B(y))f1 (x1 ) · · · fn (xn )g(y)

def

= RS (A1 (f1 ) ⊗ · · · ⊗ An (fn ), B(g)).

(53)

The retarded products Rn,1 are multi-linear functionals on P with values in the space of F(CS ) valued distributions. We may equivalently consider them as distributions on the space of P ⊗n+1 valued test functions D(Mn+1 , P ⊗n+1 ) = (P ⊗ D(M))⊗n+1 , i.e. we sometimes write RS (A1 f1 ⊗ · · · ⊗ An fn , Bg) In Sect. 4 we will define perturbative quantum fields by the principle that as much as possible of the structure of perturbative classical fields is maintained in the process of quantization. For this purpose we are going to work out main properties of the retarded products Rn,1 (53). • The causality of the retarded fields, BS+A(f ) (x) = BS (x),

if

supp f ∩ (x + V− ) = ∅

(54)

(where f ∈ D(M), A, B ∈ P) translates into the support property: supp Rn,1 ⊂ {(x1 , . . . , xn , x)|xl ∈ x + V− , ∀l = 1, . . . , n}.

(55)

• A deep property of the retarded products Rn,1 is the GLZ-Relation. In (44) we formulated it for general retarded products. For the retarded products of local fields the GLZ-Relation reads {R|I |,1 (⊗i∈I fi ; f ), R|I c |,1 (⊗k∈I c fk ; g)} I ⊂{1,... ,n}

= Rn+1,1 (f1 ⊗ · · · ⊗ fn ⊗ f ; g) − Rn+1,1 (f1 ⊗ · · · ⊗ fn ⊗ g; f ) (56)

for f1 , . . . , fn , f, g ∈ P ⊗ D(M).

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Glaser, Lehmann and Zimmermann (GLZ) [13] found this formula in the framework of non-perturbative QFT for the retarded products introduced by Lehmann, Symanzik and Zimmermann [25]. In causal perturbation theory [12] the GLZ-relation is a consequence of Bogoliubov’s definition of interacting fields [2], see Proposition 2 in [5]. The important point is that the retarded products on the l.h.s. in (56) are of lower orders, |I |, |I c | < n + 1. • From their definition (53) it is evident that the retarded products Rn,1 commute with partial derivatives ∂xµl Rn,1 (. . . , Al (xl ), . . . ) = Rn,1 (. . . , ∂ µ Al , . . . ) , l = 1, . . . , n + 1 ,

(57)

where A1 , . . . , An+1 ∈ P. Note that the kernel of the linear map P ⊗ D(M) −→ F(C) : A ⊗ g → A(g)

(58)

is precisely the linear span of {∂ µ A ⊗ g + A ⊗ ∂ µ g | A ∈ P, g ∈ D(M)}. Equation (57) expresses the fact that the retarded products Rn,1 depend on the functionals (i.e. the images of the map (58)) only. This can be interpreted in physical terms: Lagrangians which give the same action yield the same physics. This is the motivation for Raymond Stora to require (57) for the retarded (or equivalently: time ordered) products of QFT, and he calls this the Action Ward Identity (AWI) [32], see Sect. 4. • We now assume that S is at most quadratic in the fields. Then the second derivative is independent of the fields, and therefore also the Green functions. We set def

adv S (x, y) = − ret S (x, y) + S (x, y) .

(59)

We look at the Poisson bracket of a retarded product with a free field. Let A, B ∈ P and f, g, h ∈ D(M). We are interested in {Rn,1 (A⊗n , B)(f ⊗n , g), ϕ(h)}S .

(60)

By definition of the retarded products of local fields this is equal to {RS (A(f )⊗n , B(g)), ϕ(h)}S .

(61)

We apply Proposition 1 to compute the Poisson bracket. From Proposition 3 it follows that RS commutes with functional derivatives if S is a quadratic functional. Hence we obtain δA(f ) ⊗ A(f )⊗n−1 , B(g)) dx dy RS (n δϕ(x) δB(g) +RS (A(f )⊗n , ) S (x, y)h(y). (62) δϕ(x) Using the formula ∂A δA(f ) dx k(x) = (x)f (x)∂ a k(x) dx a ϕ) δϕ(x) ∂(∂ a

(63)

for the functional derivative of a local functional, we finally arrive at the formula

Master Ward Identity in Classical Field Theory

{Rn,1 (f1 , . . . , fn+1 ), ϕ(h)}S n+1 ∂fk = (Rn,1 (f1 , . . . , ∂ a S h, . . . , fn+1 ), a ϕ) ∂(∂ a

289

(64)

k=1

where f1 , . . . , fn+1 ∈ D(M, P), h ∈ D(M) and where S was considered as an integral operator acting on h. The requirement that this relation holds also in perturbative QFT plays an important role in the inductive construction of perturbative quantum fields (see Sect. 4). • Symmetries. There are natural automorphic actions αL and βL of the Poincar´e group ↑ (L ∈ P+ ) on (P ⊗ D(M))⊗n+1 and on F(C), respectively. The retarded products are Poincar´e covariant: Rn,1 ◦ αL = βL ◦ Rn,1 , provided S is invariant. A universal formulation of all symmetries which can be derived from the field equations in classical field theory is given by the MWI, see Sect. 3. ret ret • The factorization (AB)ret Lint (x) = ALint (x)BLint (x) and the Leibniz rule for the retarded product of two factors yield, in general, ill-defined expressions in QFT. It is the Master Ward Identity which allows to implement the consequences of the factorization property of classical field theory into quantum field theory.

2.3. Elimination of derivative couplings. Interaction Lagrangians containing derivatives of fields usually cause complications in the canonical formalism. They also change relations between different fields, as may be seen by the non-linear term in the formula which expresses the field strength F µν of a Yang-Mills theory in terms of the vector potential Aµ . A convenient way to deal with these complications is the introduction of auxiliary fields. As an example we consider the Lagrangian L(ϕ, ∂ µ ϕ) =

1 µ m2 2 ∂ ϕ∂µ ϕ − ϕ + Lint (ϕ, ∂ µ ϕ) 2 2

(65)

of a real scalar field ϕ with the Euler-Lagrange equation ( + m2 )ϕ =

∂Lint (ϕ, ∂ µ ϕ) ∂Lint (ϕ, ∂ µ ϕ) − ∂ν . ∂ϕ ∂ ∂νϕ

(66)

To eliminate ∂ µ ϕ in Lint we introduce a vector field ϕ µ and a Lagrangian 1 m2 2 L(ϕ, ∂ µ ϕ, ϕ µ ) = − ϕ µ ϕµ + ϕµ ∂ µ ϕ − ϕ + Lint (ϕ, ϕ µ ), 2 2

(67)

with the Euler-Lagrange equations: ∂Lint (ϕ, ϕ µ ) , ∂ϕ µ ∂Lint (ϕ, ϕ µ ) ∂µ ϕ µ = −m2 ϕ + , ∂ϕ

0 = −ϕµ + ∂µ ϕ +

(68) (69)

which are equivalent to (66). We see that precisely in the case when the interaction Lagrangian depends on ∂ µ ϕ the interacting field ϕ µ differs from ∂ µ ϕ.

290

M. D¨utsch, K. Fredenhagen

Example. By explicit calculation we are going to show that the retarded products µ RL0 (ϕ ν (y), ∂ µ ϕ(x)) and RL0 (ϕ ν (y), ϕ µ (x)) are different, although ∂ µ ϕL0 = ϕL0 (where L0 is the free part of the Lagrangian (67)), so we see again that the entries of the retarded products must not be replaced by their restriction to the space of solutions. The fastest way to compute these retarded products is to use Proposition 1(a), as in (24). However, we find it more instructive to go back to Peierls’ definition of retarded products (17): by definition rL0 +λδy ϕ ν ,L0 (f0 , h0 ) is the solution (f, h) of (68)–(69) with Lint (z) = λδ(z − y)ϕ ν (z) which agrees in the distant past with (f0 , h0 ). We obtain ϕ(x) ◦ rL0 +λδy ϕ ν ,L0 (f0 , h0 ) = f (x) = f0 (x) − λ∂ ν ret (x − y), ϕ µ (x) ◦ rL0 +λδy ϕ ν ,L0 (f0 , h0 ) = hµ (x) µ = h0 (x) − λ(∂ µ ∂ ν ret (x − y) − g µν δ(x − y)).

(70)

RL0 (ϕ ν (y), ∂ µ ϕ(x)) = ∂xµ RL0 (ϕ ν (y), ϕ(x)) = −∂ ν ∂ µ ret (x − y),

(71)

RL0 (ϕ ν (y), ϕ µ (x)) = −(∂ ν ∂ µ ret (x − y) − g νµ δ(x − y)).

(72)

Hence,

but

3. The Master Ward Identity 3.1. Generalized Equation. It is an immediate consequence of the Schwinger-Dyson δS field equations δϕ = 0 for a given local action S that all functionals of the form S

δS A (h) δϕ

(73)

δS (by which we mean the point-wise product of an arbitrary classical field A ∈ P with δϕ smeared out with the test function h) vanish on the space of solutions CS . If we set S = S0 + λS1 , and differentiate with respect to λ at λ = 0 we obtain the identity δS0 δS1 RS0 S1 , A = 0. (74) (h) + A (h) S0 δϕ δϕ

For A ≡ 1 this equation looks similar to the Schwinger-Dyson equation i δS1 δS0 T S1 + = 0, δϕ δϕ

(75)

which holds for the vacuum expectation values of time ordered products for a quantum field theory with action S0 (see, e.g. [15]). (Note that the factor i is absorbed in (74) in the retarded part of the Poisson bracket.) For this reason we call (74) the retarded Schwinger-Dyson Equation and the vanishing of (73) the generalized SchwingerDyson Equation (GSDE). Note that the retarded Schwinger-Dyson Equation has the same form in classical physics as in quantum physics. In the retarded Schwinger-Dyson equation we may permute the two entries in the retarded product. Namely, the difference is just the Poisson bracket which vanishes if one of the entries vanishes on the space of solutions.

Master Ward Identity in Classical Field Theory

291

For the retarded products of local fields we obtain the perturbative version of the generalized Schwinger-Dyson Equation, n δS0 δfl Rn,1 f1 , . . . , fn , h Rn−1,1 f1 , . . . , fl−1 , fl+1 , . . . , fn , h + =0 δϕ δϕ l=1

(76) with fi , h ∈ D(M, P), i = 1, . . . , n, and where the functional derivative of f ∈ D(M, P) is defined by δ dy f (y) δf (x) = , δϕ δϕ(x)

(77)

i.e. δf ∂f (−1)|a| ∂ a = . δϕ ∂(∂ a ϕ) a

(78)

Proceeding by induction on n and using the GLZ-Relation (56), we obtain an equiv0 alent formula for the case that the term h δS δϕ is one of the first n entries of Rn,1 , namely δS0 Rn,1 f1 , . . . , fn−1 , h , fn δϕ n δfl + Rn−1,1 f1 , . . . , fl−1 , h , fl+1 , . . . , fn = 0. δϕ

(79)

l=1

Equations (76) and (79) remain meaningful in perturbative QFT. Also there they are equivalent since the GLZ-Relation still holds. We require either of them as a renormalization condition (see Sect. 4). Equivalently, an analogous identity may be postulated for the time ordered products (where the two versions above coincide in view of the symmetry of time ordered products). It is a generalization of the condition (N4) in [5].7 But there, following the tradition in causal perturbation theory, we considered the entries of time ordered or retarded products as Wick polynomials of the free field. Therefore, time ordering and partial derivatives did not commute, and the formulation of identities involving derivatives of fields contained many free parameters. A consistent choice of these parameters was made possible by the MWI proposed in [7]. The QFT version of (76) or (79) seems to correspond to the “broomstick identity” of Lam given in Fig. 8 of [20]. We think that Lam is unable to write down this identity as an equation because the arguments of his time-ordered products are on shell fields; and compared to [7] (which uses also an on shell formalism, cf. Sect. 4) he is not equipped with the “external derivative”. We will see that the MWI is equivalent to the generalized Schwinger-Dyson Equation (76). Therefore, the MWI can be interpreted as a quantum version of all identities for local fields which follow in classical field theory from the field equations. 7 A first generalization (in the framework of causal perturbation theory) of the renormalization condition (N4) was given in an unpublished preversion of [28].

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M. D¨utsch, K. Fredenhagen

3.2. Definition of a map σ from free fields to unrestricted fields. To keep the formulas simple, we consider the case of one real scalar field ϕ. The procedure, however, applies to a general model. Let J denote the ideal in the algebra P of polynomials of fields which is generated from the free field equation, J ={ Aa ∂ a ( + m2 )ϕ, Aa ∈ P, a ∈ Nd0 } , (80) a

let P0 = P/J be the algebra of free fields and let π : P → P0 be the canonical surjection. Since J is translation invariant, we may define derivatives with respect to def

space-time coordinates in P0 by ∂ µ π(B) = π(∂ µ B), and in this sense the free field equation holds true for πϕ. The wanted map σ is a section σ : P0 → P. In contrast to the surjection π, the section σ is not canonically given. We restrict its choice by the following requirements: π ◦ σ = id, i.e. σ is a section. σ is an algebra homomorphism.8 The Lorentz transformations commute with σ π . σ π(P1 ) ⊂ P1 , where P1 is the subspace of fields which are linear in ∂ a ϕ. σ π does not increase the mass dimension of the fields, i.e. σ π(B) is a sum of terms with mass dimension ≤ dim (B). In particular we find σ π(ϕ) = ϕ. (vi) P is generated by fields in the image of σ and their derivatives. In the present case (one real scalar field), this condition is automatically satisfied.

(i) (ii) (iii) (iv) (v)

By (i) σ π : P → P is a projection: σ πσ π = σ π . The linearity of σ and condition (i) imply ker σ π = ker π = J , and hence σ π ϕ = −m2 σ πϕ .

(81)

We are now looking for the most general explicit formula for σ π which satisfies the above requirements. Due to (ii) it suffices to determine σ π(∂ a ϕ). By definition of π and σ , σ π(A) − A ∈ J

∀A ∈ P

(82)

cba ∂ b ( + m2 )ϕ

(83)

must hold, and hence σ π∂ a ϕ = ∂ a ϕ +

b

with constants cba ∈ R. The determination of an admissible section σ satisfying conditions (i) to (v) is equivdef

alent to the determination of a complementary subspace K = σ π(P1 ) of J1 = J ∩ P1 which is Lorentz invariant and satisfies the condition that already the subspaces with mass dimension ≤ n are complementary, (n)

K (n) + J1

(n)

= P1 .

(84)

Since the finite dimensional representations of the Lorentz group are completely reducible; this is always possible. Namely, for the lowest mass dimension n0 = (d − 2)/2 8

In case of complex fields we additionally require σ (A∗ ) = σ (A)∗ .

Master Ward Identity in Classical Field Theory

293 (n )

the subspace generated by the field equation is zero, thus K (n0 ) = P1 0 = Rϕ. From this the existence of K (n+1) may be proved by induction on n. One just has to choose a (n+1) (n+1) in P1 and to Lorentz invariant complementary subspace L(n+1) of K (n) + J1 (n+1) (n) (n+1) set K =K +L . The arbitrariness in the choice of L(n) depends on the multiplicity in which the irreducible subrepresentations of the Lorentz group occur in the respective subspaces. In the present case it turns out that σ is unique (see the second part of Appendix A). In case one introduces the auxiliary field ϕ µ , the choice of σ involves free parameters. A special choice for σ is given in the first part of Appendix A. For the lowest derivatives we obtain the following general solution of the requirements (i)-(vi): σ π(ϕ) = ϕ, σ π(∂ µ ϕ) = σ π(ϕ µ ) = γ ϕ µ + (1 − γ )∂ µ ϕ, γ ∈ R \ {0}, σ π(∂ µ ∂ ν ϕ) = σ π(∂ ν ϕ µ ) = (1 + 2α)∂ µ ∂ ν ϕ 1 + 2α µν 1 2α µν σ −α(∂ µ ϕ ν + ∂ ν ϕ µ ) − g ϕ − g µν m2 ϕ + g ∂ ϕσ , d d d

(85) (86) (87)

where γ and α ∈ R are free parameters. The condition γ = 0 is necessary and sufficient for (vi) provided (i)-(v) are satisfied. A preferred choice is γ = 1. 3.3. The Master Ward Identity. Let A be a functional of the field which vanishes according to the field equation derived from the action S0 , i.e. AS0 = 0. A is of the form (cf. the remark in footnote 3) δS0 , (88) A = dx G(x) δϕ(x) with G(x) ∈ F(C). (This formula states that A is an arbitrary element of the ideal JS0 generated by the field equation belonging to S0 .) We may introduce the derivation δ (89) δA = dx G(x) δϕ(x) on F(C). The GSDE imply (MWI)

S1 S1 , A) = −RS0 (e⊗ , δA (S1 )) RS0 (e⊗

∀S1 ∈ F(C)

(90)

and, by using the GLZ equation S1 S1 S1 ⊗ A, B) = −RS0 (e⊗ ⊗ δA (S1 ), B) − RS0 (e⊗ , δA (B)) RS0 (e⊗

∀S1 , B ∈ F(C) . (91)

This equation (in a different form, see (110) below) was proposed by Boas and D¨utsch [7] under the name Master Ward Identity (MWI) as a universal renormalization condition in perturbative QFT. In the present formulation it is evidently equivalent to the GSDE. Note that up to now (in contrast to [7]) the formulation of the MWI makes sense also in the case that S0 is not a quadratic functional of the field. Similarly to the GSDE, the MWI holds also non-perturbatively: (92) AS0 +S1 = − δA (S1 ) S +S , A ∈ JS0 , S1 ∈ F(C). 0

1

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M. D¨utsch, K. Fredenhagen

Example. As a typical application let us look at the free complex scalar field with the conserved current jµ = Let A = ∂j, g ≡

and hence

1 ∗ (ϕ ∂µ ϕ − ϕ∂µ ϕ ∗ ) . i

dx ∂ µ jµ (x)g(x) with g ∈ D(M). We have δS0 1 ∗ δS0 A= gϕ , ∗ − gϕ, , i δϕ δϕ 1 δ δ ∗ gϕ , ∗ − gϕ, . δA = i δϕ δϕ

(93)

(94)

(95)

If g ≡ 1 on the localization region of F ∈ F(C), δA F is the infinitesimal gauge transformation of F , and inserting A into the MWI yields the well known Ward identity of the model. A problem with the MWI in the form presented above is the non-uniqueness of the derivation δA for a given A, e.g. for A = dx h(x)(( + m2 )ϕ ∗ (x))( + m2 )ϕ(x) (96) in case of the complex scalar field. We therefore turn now to the free field case and use the techniques and conventions developed in the preceding section. We will give a unique prescription to write any A with B ∈ J ⊂ P = dx h(x)B(x) and h ∈ D(M) (i.e. A ∈ JS0 ) in the form A = dx h(x) j Bj (x)bj (x) with bj ∈ P1 ∩ J . Then we may set δA = dx h(x) Bj (x)δbj (x) (97) j

with δbj (x) (F ) = −RS0 (bj (x), F ) ,

F ∈ F(C) ,

(98)

where we used (31) and the fact that for terms which are linear in the field the first term on the right-hand side vanishes, such that (32) holds everywhere, not only on the space of solutions. To give the mentioned prescription let B ∈ J . Then B = p(B), where p = 1 − σ π is a projection from P onto J . Since σ π is an algebraic homomorphism we find p(B1 B2 ) = B1 p(B2 ) + p(B1 )σ π(B2 ) . Hence, we may write every B ∈ J in the form Bχ p(χ ), B= χ∈G

(99)

(100)

Master Ward Identity in Classical Field Theory

295

where we introduced the set G of generators of the field algebra P, def

G = {∂ a ϕ | a ∈ Nd0 }

(101)

which is a vector space basis of P1 . The coefficients Bχ can be found in the following way: Any B ∈ P is a polynomial P in finitely many different elements χ1 , . . . , χn ∈ G, B = P (χ ), χ = (χ1 , . . . , χn ). Since B ∈ J we have B = B − σ π(B) = P (χ ) − P (σ π(χ )) 1 d dλ P (λχ + (1 − λ)σ π(χ )) = dλ 0 n = Pi (χ , σ π(χ ))p(χi )

(102) (103) (104)

i=1

with

1

Pi (χ , σ π(χ )) =

dλ∂i P (λχ + (1 − λ)σ π(χ )).

(105)

0

Hence we may set Bχi = Pi (χ , σ π(χ )), i = 1, . . . , n and Bχ = 0 if χ ∈ {χ1 , . . . χn }. Example. Let ϕ be a real scalar field. For χ = ϕ, ∂ µ ϕ the expression p(χ ) vanishes. µν µν 0 But for χ = ∂ µ ∂ ν ϕ we obtain p(χ ) = gd ( + m2 )ϕ = gd δS δϕ and hence δp(χ)(x) (F ) =

g µν δF . d δϕ

(106)

In many applications one wants to compare derivatives in the free theory with those in the interacting theory. One is therefore interested in expressions of the form (107) A = dx h(x)(∂µ σ π(B)(x) − σ π(∂µ B)(x)) with B ∈ P and h ∈ D(M). To get a more general identity we even admit h ∈ D(M, P). Clearly, [∂µ , σ π ](B) ∈ J , hence A ∈ JS0 . Using the Leibniz’ rule ∂ µ B = χ∈G ∂B ∂χ ∂µ χ we find [∂µ , σ π](B) =

∂B )[∂µ , σ π ](χ ) ∂χ

(108)

∂B )(x)δ[∂µ ,σ π](χ)(x) . ∂χ

(109)

σ π(

χ∈G

and get the formula δA =

dx h(x)

χ∈G

σ π(

In terms of the retarded fields (14) we end up with ∂B µ ∂Lint (MWI ) δχ,ψ h([∂ µ , σ π ]B) L = h σπ , int ∂χ ∂ψ Lint χ,ψ∈G

(110)

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M. D¨utsch, K. Fredenhagen µ

where the differential operator δχ,ψ is defined by µ

δχ,ψ f (x) = −

dy RS0 [∂ µ , σ π ](χ )(x), ψ(y) f (y) , f ∈ D(M, P) .

(111)

Note that RS0 [∂ µ , σ π](χ )(x), ψ(y) is a linear combination of partial derivatives of δ(x − y). µν

Example. Let χ = ∂ ν ϕ. Then [∂ µ , σ π ](χ ) = gd ( + m2 )ϕ. Hence δ[∂ µ ,σ π](χ) = g µν δ d δϕ , therefore one obtains for the difference between derivatives of free or interacting fields g µν δS1 ∂ µ (∂ ν ϕ)S0 +S1 − (σ π ∂ µ ∂ ν ϕ))S0 +S1 = . (112) d δϕ S0 +S1 We think that the non-vanishing of [∂ µ , σ π ](∂ ν ϕ) compared to [∂ µ , σ π ](ϕ) = 0 explains why the formalism of Lam becomes inconsistent for vertices containing higher than first derivatives of the basic fields (cf. Sect. V of [19]). We derived (110) as a consequence of the MWI (92). It is even equivalent if J ∩ P1 is spanned by fields of the form [∂ µ , σ π](ψ) and their derivatives, with ψ ∈ P1 , since then the l.h.s. of the MWI (90) can be written as a linear combination of fields of the form of the l.h.s. of (110). In the case of the free scalar field this condition is clearly fulfilled, since [∂ µ , σ π ](∂µ ϕ) = ( + m2 )ϕ. In the enlarged model with the auxiliary field ϕµ we find [∂ µ , σ π ](ϕ) = γ (∂ µ ϕ − ϕ µ ) and [∂ µ , σ π ](ϕµ ) = (∂ µ ϕµ + m2 ϕ) + (γ − 1)∂ µ (ϕµ − ∂µ ϕ), hence here it follows from γ = 0. Also in general it follows from condition (vi) on σ in Sect. 3.2. Namely, let χ ∈ J ∩ P1 . According to (vi) and (iv), χ is of the form χ= ∂ a σ π(ψk,a ) ψk,a ∈ P1 . k,a

Hence, χ = χ − σ (πχ ) =

[∂ a , σ π ]σ π(ψk,a ) . k,a

The result now follows from the derivation property of the commutator [∂µ1 · · · ∂µn , σ π] = ∂µ1 · · · ∂µk−1 [∂µk , σ π ]∂µk+1 · · · ∂µn . k

We point out that the MWI (in the original form (90) or (91) as well as in the second form (110)) is well defined in perturbative QFT, too. This is a main ingredient of the next section. 4. Quantization: Defining Properties of Perturbative Quantum Fields The structure of perturbative classical field theory which was analyzed in this paper can to a large degree be preserved during quantization. The main change is the replacement

Master Ward Identity in Classical Field Theory

297

of the commutative product of functionals F ∈ F(C) by a -dependent non-commutative associative product, and by the replacement of the Poisson bracket by i1 times the commutator. The definition of the product can be read off from Wick’s Theorem, F ∗ G =

∞ n n=0

n!

δnG δnF + (xi − yi ) , δϕ(x1 ) · · · δϕ(xn ) δϕ(y1 ) · · · δϕ(yn ) n

d(x, y)

i=1

(113) where + denotes the positive frequency fundamental solution of the Klein Gordon equation. This abstract algebra (which we still denote by F(C)) can be represented on Fock space by Wick polynomials 1 δnF π(F ) = (114) d nx |ϕ=0 : ϕ(x1 ) · · · ϕ(xn ) : , n! δϕ(x1 ) · · · δϕ(xn ) n the kernel of this representation being the set of functionals F vanishing on CL0 , cf. [6]. A direct construction of solutions of the field equations in the case of local interactions is, in general, not possible because of ultraviolet divergences. One may, however, start from the ansatz ∞ 1 BLint (f ) = Rn,1 (L⊗n int , Bf ) n!

(115)

n=0

in analogy to (15) and try to determine the retarded products Rn,1 as polynomials in such that they satisfy the following properties: They are (n + 1)-linear continuous functionals on D(M, P) with values in F(C)L0 which are symmetric in the first n entries, have retarded support (55) and satisfy the GLZ-Equation (56). Moreover they have to fulfill the unitarity condition Rn,1 (f1 ⊗ · · · ⊗ fn , f )∗ = Rn,1 (f1∗ ⊗ · · · ⊗ fn∗ , f ∗ ).

(116)

It turns out that already by these properties, the retarded products Rn,1 are uniquely determined outside of the total diagonal x1 = . . . = xn+1 in terms of the lower order retarded products where the lowest order is defined by R0,1 (Bf ) = B(f )|CL0 . Renormalization then means the extension of the retarded products to the diagonal. This is a variant of the Bogoliubov-Epstein-Glaser renormalization method and, in the adiabatic limit g ≡ 1, it has been worked out by Steinmann [30]. A modernized and local version of the procedure will be presented in [8]. The main work which remains to be done is the so-called finite renormalization, i.e., the analysis of the ambiguities in the extension process. In a first step, condition (64) (condition (N3) of [5]) can be used to reduce the extension problem to a problem for numerical distributions. By requiring translation invariance these numerical distributions depend only on the differences of coordinates, thus one has to study the mathematical problem of extending a distribution which is defined outside of the origin to an everywhere defined distribution. The possible extensions can be classified in terms of Steinmann’s [30] scaling degree, and it is a natural requirement that the scaling degree should not increase during the extension process. In addition one can show that the extension can always be done such that the retarded products are Lorentz covariant.

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The steps described above can always be performed and leave, for every numerical distribution, a finite set of parameters undefined. The proposal is now to add two further conditions. One is the Action Ward Identity (57) proposed by Stora [32]. A proof that it can always be satisfied and is compatible with the other normalization conditions will be given in [8]. We require then the Master Ward Identity in the form (90) or (91), or, equivalently, the generalized Schwinger-Dyson equation in the form (76) or (79). Here anomalies may occur, and one has to check in a given model whether these identities can be satisfied. Fortunately, for a typical application, one needs these identities only for special cases which may be characterized by the polynomial degree of the fields and the number of derivatives which are involved. The formalism given here is not completely equivalent to the one of [7]. The algebras of symbols P and P0 given in [7] may be identified with our algebras P and P0 of classical fields. The main difference is the absence of the Action Ward Identity, thus derivatives could not freely be shifted from fields to test functions. Therefore, in [7] an “external derivative” ∂˜ µ on P0 was introduced which generates new symbols ∂˜ a A (A ∈ P0 , a ∈ Nd0 ). The argument of the retarded product of [7] (we denote it here by R˜ n,1 ) is an element of (P˜ 0 ⊗ D(M))⊗(n+1)

where

def P˜ 0 =

{∂˜ a A | A ∈ P0 , a ∈ Nd0 }

(117)

(for details see [7]). The retarded products R˜ n,1 of symbols with external derivative(s) can be defined in terms of retarded products without external derivative (normalization ˜ The MWI is then expressed by a further normalization condition (N), condition (N)). ˜ In particular (N) ˜ and (N) imply combined with (N). R˜ n,1 (W1 g1 , . . . , (∂˜ ν Wl )gl + Wl ∂ ν gl , . . . , Wn+1 gn+1 ) = 0 , W1 , . . . Wn+1 ∈ P0 . (118) To clarify the relation of the two formalisms we extend σ to a map σ˜ : P˜ 0 → P by setting σ˜ (∂˜ a A) = ∂ a σ (A), def

A ∈ P0 ,

(119)

and requiring that σ˜ is an algebra ∗-homomorphism, similarly to [7]. The defining property (vi) of σ means that σ˜ is surjective. But σ˜ is not injective, even if the auxiliary field ϕ µ is introduced. So, from the retarded product Rn,1 of this paper, we can construct a retarded product R˜ n,1 in the sense of [7], by defining def R˜ n,1 (W1 g1 , . . . , Wn+1 gn+1 ) = Rn,1 (σ˜ (W1 )g1 , . . . , σ˜ (Wn+1 )gn+1 ).

(120)

But, in general, it might happen that R˜ n,1 does not vanish if one entry is in (ker σ˜ ), and then it would be impossible to construct Rn,1 from R˜ n,1 . If the Rl,1 , l ≤ n, satisfy all defining properties given here (including the AWI and the MWI), then the corresponding R˜ l,1 , l ≤ n, (120) fulfill the requirements on a retarded product given in [7], in particular ˜ and (N). (N) It seems that the normalization conditions on R˜ n,1 are weaker than the normalization conditions on Rn,1 . However, the formalism given here is the natural one when departing from classical field theory.

Master Ward Identity in Classical Field Theory

299

Remark. In the formalism of [7] and in [9] the Feynman (or retarded) propagators of perturbative QFT contain undetermined parameters, if there are at least two derivatives present (in d = 4 dimensions). On the classical side (retarded) fields and their perturbative expansion are unique. The non-uniqueness is located in the choice of the map σ : the free parameters in σ can be identified with the free parameters in the Feynman propagators of QFT. This is obvious from ˜ ret ˜ (χ0 )(y), σ˜ (ϕ0 )(x)). ϕ0 ,χ0 (x − y) = R1,1 (χ0 (y); ϕ0 (x)) = R1,1 (σ def

(121)

In the non-enlarged formalism (without ϕ µ ), in which σ is unique, a particular choice of the parameters in the Feynman propagators of [7] is done. 5. Application to BRS-Symmetry 5.1. Motivation. The canonical formalism as developed in this paper cannot directly be applied to gauge theories because there the Cauchy problem is ill posed due to the existence of time dependent gauge transformations. As usual, one may add a gauge fixing term as well as a coupling to ghost and antighost fields to the Lagrangian such that the Cauchy problem becomes well posed. The algebra of observables is then obtained as the cohomology of the BRS transformation s [1] which is a graded derivation which is implemented by the BRS charge Q. In QFT one finally constructs the space of physical states as the cohomology of Q (see e.g. Sects. 4.1-4.2 of [5]). The implementation of this program in the case of perturbative gauge field theory meets the problem that in general the BRS operator Q is changed due to the interaction [5]. It is a major problem to exhibit the corresponding Ward identities which generalize the Slavnov Taylor identities to the case of couplings of compact support. In the case of a purely massive theory one may adopt a formalism due to Kugo and Ojima [18] who use the fact that in these theories the BRS charge Q can be identified with the incomimg (free) BRS charge, which we denote by Q0 . For the S-matrix to be a well defined operator on the physical Hilbert space of the free theory one then has to require [Q0 , T ((gL)⊗n )]|kerQ0 → 0

(122)

in the adiabatic limit g → 1, see e.g. [11, 14]. This is the motivation to require “perturbative gauge invariance” [9, 10, 29], which is a somewhat stronger condition than (122) but has the advantage that it is well defined independent of the adiabtic limit. The condition (122) (or perturbative gauge invariance) can be satisfied if additional scalar fields (corresponding to Higgs fields) are included. Unfortunately, in the massless case, it is unlikely that the adiabatic limit exists.9 So, in the general case an S-matrix formalism is problematic. One should better rely on the construction of local observables in terms of couplings with compact support. But then Q is a formal power series with zeroth order term Q0 , and it is not obvious which conditions one should put on the retarded (or time ordered) products. The difficulty is that one has to formulate symmetry conditions for the perturbed fields which themselves are deformed due to the interaction. But using the formalism of the present paper we can disentangle these two problems. Namely, we first use the MWI together with the AWI to compute the commutator of the free BRS charge Q0 9 To motivate perturbative gauge invariance in that case one can derive it from a suitable form of conservation of the BRS-current. To lowest orders this results (as a byproduct) from Appendix B.

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with the retarded (or time ordered) products. The resulting family of identities is called the Master BRST Identity and may be used as a renormalization condition in its own right. One then can formulate conditions on the interaction which ensure that the Master BRST Identity implies BRS-invariance of the interacting theory. 5.2. The free BRS-transformation. We illustrate the general ideas on the example of µ N massless gauge fields Aa , a = 1, . . . , N, each of them accompanied by a pair of fermionic ghost fields u˜ a , ua . We may also introduce auxiliary fields Ba (the NakanishiLautrup fields [24]). We work in Feynman gauge, in which the free field equations read Aµ a = 0,

ua = 0 = u˜ a ,

∀a

(123)

together with the equation for the auxiliary field ∂µ Aµ a = Ba .

(124)

We omit in what follows the colour index a by using matrix notation. The free BRS-current def

j µ = B∂ µ u − (∂ µ B)u

(125)

is conserved due to the free field equations, ∂µ = Bu − uB. The corresponding charge def Q0 = d 3 xj 0 (x) , (126) jµ

x 0 =const.

is nilpotent, i.e.

Q20

S0

= 0 and {Q0 , Q0 }S0 = 0,

(127)

where we introduced a grading into our Poisson bracket corresponding to ghost number. Using current conservation as well as the GLZ relation we find for the Poisson bracket of Q0 with a retarded product {Q0 , RS0 (F1 , . . . , Fn )}S0 = −RS0 (∂j, h, F1 , . . . , Fn ) ,

(128)

where h ≡ 1 on a causally complete open region O containing the localization regions of all Fi ∈ F(C), i = 1, . . . , n (cf. the analogous argument for time ordered products in [6]).10 To avoid signs which are due to fermionic permutations, we assume that all 10 It is instructive to derive (128) in terms of retarded products: let ∂ µ h = bµ − a µ with supp bµ ∩ (V + + O) = ∅ and supp a µ ∩ (V − + O) = ∅. Since RS0 (F1 , . . . , Fn ) is localized in O, we may vary bµ in the spacelike complement of O without affecting {jµ , bµ S0 , RS0 (F1 , . . . , Fn )}. In this way and by using (∂ µ jµ (x))S0 = 0 we find

{Q0 , RS0 (F1 , . . . , Fn )}S0 = {jµ , bµ S0 , RS0 (F1 , . . . , Fn )}.

(129)

By means of the support property (55) of the retarded products and the GLZ-Relation (56) we obtain for the r.h.s. of (129) = {RS0 ((Fl )l∈I , j, b), RS0 ((Fk )k∈I c , Fn )} I ⊂{1,... ,n−1}

= RS0 (j, b, F1 , . . . , Fn ) − RS0 (F1 , . . . , Fn , j, b) = RS0 (j, b, F1 , . . . , Fn ) = −RS0 (∂j, h, F1 , . . . , Fn ) .

(130)

Master Ward Identity in Classical Field Theory

301

(local) functionals F1 , . . . , Fn are bosonic, i.e. a field polynomial with an odd ghost number is smeared out with a Grassmann valued test function. The free field equations are derived from the Lagrangian L0 =

1 1 ∂µ Aν (∂ ν Aµ − ∂ µ Aν ) + ∂µ u˜ ∂ µ u − B∂µ Aµ + B 2 , 2 2

(131)

hence u = − with S0 =

δS0 δS0 , B = ∂µ δ u˜ δAµ

(132)

L0 . Thus we obtain δ∂j (x) = −B(x)

δ def δ = s˜0 (x) , − u(x)∂µ δ u(x) ˜ δAµ (x)

hence

δ∂j,h =

def

dx h(x)˜s0 (x) = s0

(133)

(134)

on fields localized in the region where h ≡ 1, i.e. we obtain the free BRS transformation ˜ = −B , s0 (B) = 0 , s0 (Aµ ) = ∂µ u , s0 (u) = 0 , s0 (u)

(135)

which is obviously nilpotent. Note that the “local” free BRS-transformation s˜0 (x) dy f (y) (where f ∈ P ⊗ D(M)) differs from s0 f (x) by a sum of divergences. We now apply the MWI to (128) and find the identity {Q0 , RS0 (F1 , . . . , Fn )}S0 =

n

RS0 (F1 , . . . , s0 (Fk ), . . . , Fn ) .

(136)

k=1

In [7] this identity (in a somewhat different form, see (142) and (145) below) is called “Master BRST Identity”. We may ask how the Master BRST Identity (136) changes if one eliminates the Nakanishi-Lautrup field B by using the field equation B = ∂A. The problem is that the ideal JB generated from B − ∂A in the algebra PB of all polynomials in the fields and their derivatives is not stable under s0 . We discuss two possibilities. • The quotient algebra P = PB /JB may be identified with the subalgebra of polynomials in PB which do not contain B or its derivatives. Let σB : P → PB denote this identification (i.e. σB (∂ a B + JB ) = ∂ a ∂ν Aν ) and πB : PB → P : X → X + JB the canonical homomorphism. Then we set t = π B s0 σ B ,

(137)

t (u) ˜ = ∂A , t (Aµ ) = ∂µ u , t (u) = 0 .

(138)

i.e.

This choice has the disadvantage that t 2 = 0. On the other hand, it has the advantage that it commutes with derivatives. Another advantage is that the arising form of the Master BRST identity is equally simple as in the model with the auxiliary field B. Namely, let F be a functional of the fields Aµ , u, u. ˜ The image under s0 might depend on B, but

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M. D¨utsch, K. Fredenhagen

s0 (F ) − t (F ) =

dx (B(x) − ∂A(x))

δF δ u(x) ˜

(139)

0 and B − ∂A = δS δB . Hence if we replace in the Master BRST identity s0 by t the correction terms due to the MWI involve derivatives with respect to B. Hence if none of the functionals Fi depends on B, we obtain the Master BRST Identity (second form)

{Q0 , RS0 (F1 , . . . , Fn )}S0 =

n

RS0 (F1 , . . . , t (Fk ), . . . , Fn ),

(140)

k=1

where now the field B has been eliminated. • Another possibility is to use the fact that on the algebra P0 = P/J , where J is the ideal of P which is generated by the free field equations, the BRS transformation sˆ0 is well defined e.g. by the adjoint action of Q0 (w.r.t. the Poisson bracket). Using the section σ : P0 → P of Sect. 3.2 11 we set def tˆ = σ sˆ0 π = σ πt,

(141)

π denoting the canonical homomorphism P → P0 (as in Sect. 3.2). tˆ has vanishing square but does not commute with derivatives. It naturally occurs if one considers the entries of retarded (and time-ordered) products as functionals of the free fields, as traditionally done in causal perturbation theory. The price to be paid is a more complicated form of the Master BRST Identity. Namely, (t − tˆ)(F ) = (1 − σ π )t (F ) ∈ JS0 , hence from the MWI we find the Master BRST identity (third form) {Q0 , RS0 (F1 , . . . , Fn )}S0 n = RS0 (F1 , . . . , tˆ(Fk ), . . . , Fn ) − RS0 ((Fi )i<max(l,k) , δ(t−tˆ)Fk Fl , k=l

k=1

(Fj )j >max(l,k) ) .

(142)

In many applications tˆ(P ), P ∈ P, can be written as the divergence of another field polynomial P ν ∈ P by using the free field equations, i.e. π tˆ(P ) ≡ π t (P ) = π(∂ν P ν ), or equivalently tˆ(P ) = σ π(∂ν P ν ).

(143)

In the next subsect. we will see that an admissible interaction Lint must fulfill this property. So let us assume that in (142) Fk = fk Pk

with

tˆ(Pk ) = σ π(∂ν Pk ν ) ,

fk ∈ D(M),

k = 1, . . . , n. (144)

In RS0 (F1 , . . . , fk σ π(∂ν Pk ν ), . . . , Fn ) we would then like to move the derivative to the test function fk (i.e. outside of the unsmeared retarded product). This produces corrections which can directly be read off from the second formulation of the MWI (110): 11 The avoidance of the field B contradicts the requirement (vi) on σ , but this is no harm. To do so we choose σ π(∂ν Aν ) = ∂ν Aν , cf. (86).

Master Ward Identity in Classical Field Theory

303

{Q0 , RS0 (f1 P1 , . . . , fn Pn )}S0 n =− RS0 (f1 P1 , . . . , (∂ν fk )σ π(Pk ν ), . . . , fn Pn ) +

k=1

RS0 ((fi Pi )i<max(l,k) , G((Pk , Pk )fk , Pl fl ), (fj Pj )j >max(l,k) ) , (145)

k=l

where G((P1 , P1 )f1 , P2 f2 ) = −δ(t−tˆ)f1 P1 (f2 P2 ) ∂P ∂(f2 P2 ) 1ν ν f1 σ π δχ,ψ . − ∂χ ∂ψ def

(146)

χ,ψ∈G

Obviously, there is also a mixed formula in which the step from (136) (or (140), or (142)) to (145) is done for some of the factors Fk only (not for all). In the forms (142) and (145) the Master BRST Identity was found in [7] with δ(t−tˆ)Fk Fl corresponding to the terms G(1) (Fk , Fl ). (G(. . . ) (146) denotes the same terms.) Note that the Master BRST Identity (in either form) is independent of the choice of an interaction and is therefore well suited for the formalism of causal perturbation theory where one aims at finding the retarded (or time ordered) products not only for the interaction Lagrangian itself but for a whole class of fields. Given the free (quantum) gauge fields, requirements on the interaction are formulated in [7], in particular a suitable form of BRS-invariance. These requirements determine the interaction to a far extent [31, 10, 29]. Then it is demonstrated that for such an interaction the validity of particular cases of the Master BRST Identity and of ghost number conservation (which is another consequence of the MWI) suffices for a construction of the net of local algebras of observables. This construction yields also a space of physical states and an explicit formula for the computation of the BRS-transformation of an arbitrary quantum field. In that reference the requirements expressing BRS-invariance of the interaction have been motivated by the particular case of purely massive gauge models (122) (in which the adiabatic limit exists, see e.g. [11, 14, 9]), and by what has been used in the construction and holds true in the most important examples. However, it is desirable to derive these conditions from more fundamental principles without using the adiabatic limit. The MWI and AWI are well suited tools for such a derivation, as it is demonstrated in Appendix B. However, in the next subsection we determine the admissible interaction of a local gauge theory independently of the corresponding procedure in [7]. This is a further important application of the MWI and AWI. 5.3. Admissible interaction. By an admissible interaction we understand an interaction for which a deformed BRS charge Q exists.12 Let the free action S0 and the free BRScurrent be given. We make the ansatz Sint = Sn λn (147) n≥1 12 For simplicity we do not investigate the existence of Q as an operator (which is used for the construction of physical states in [5] and [7]). This existence involves an infrared problem which can be avoided by a spatial compactification [5]. Here, we only require that Q implements a nilpotent (graded) derivation on the interacting fields (150)–(151), which is a deformation of the free BRS-transformation s0 (134).

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M. D¨utsch, K. Fredenhagen

with Sn = dxg(x)n Ln (x), Ln ∈ P, g ∈ D(M), g ≡ 1 on some causally complete open region O1 . We want to find a conserved current of the interacting theory jµ = jµ(n) λn , (148) n≥0 (0)

where jµ is the free BRS current. The BRS-transformation s : P → P, will be constructed in the form s= dx s˜n (x)λn (149) n≥0

with s˜n (x) a local (graded) derivation, and s˜0 (x) given by (133). In addition we require that s is nilpotent on the space of solutions and fulfills (s(F ))S = sˆ (FS ) ,

(150)

where sˆ is defined in terms of the BRS-current j (148) by sˆ (FS ) = {Q, FS }

def

, Q = j, bS

, ∀ local F with supp F ⊂ O

(151)

(with S = S0 + Sint ). Thereby, O is a causally complete open region with O ⊂ O1 . Due to current conservation there is a rather large freedom in the choice of b = (bµ ). It only needs to be a smooth version of a delta function on a Cauchy surface of O with supp b ⊂ O1 . For a later purpose we choose b in the following way: let h ∈ D(O1 ) with h ≡ 1 on O. Then the bµ which we will use later on is obtained from ∂µ h by the same causal splitting as in (129). Note that for an arbitrary given local F ∈ F(C) the regions O, O1 as well as h and b can be suitably adjusted. We require current conservation within the region O1 (where g is constant) only, Sint RS0 (e⊗ , ∂j (x)) = 0 , ∀x ∈ O1 .

(152)

To zeroth order in λ this is simply the condition that the free BRS current is conserved in the free theory ∂j (0) ≡: G0 ∈ J

(153)

and apply the MWI, Sint 0 = RS0 (e⊗ , s˜0 (x)Sint −

∂j (n) (x)λn ) ,

x ∈ O1 ,

(154)

n≥1

where s˜0 (x) = δG0 (x) (133). To first order we find the requirement s˜0 (x)S1 − ∂j (1) (x) ≡: −G1 (x) ∈ J . Therefore we can apply again the MWI and obtain Sint 0 = RS0 (e⊗ , s˜1 (x)Sint + s˜0 (x) Sn λn − ∂j (n) (x)λn ) , n≥2

n≥2

(155)

x ∈ O1 ,

(156)

Master Ward Identity in Classical Field Theory

305

with s˜1 (x) = δG1 (x) . Iterating the procedure we obtain the conditions s˜0 (x)Sn + s˜1 (x)Sn−1 + . . . + s˜n−1 (x)S1 − ∂j (n) (x) ≡: −Gn (x) ∈ J ,

(157)

and set s˜n (x) := δGn (x) . We see that at every order, Sn must be chosen such that n−1

s˜k (x)Sn−k ∈ Pdiv + J ,

(158)

k=0

where Pdiv = {∂ µ fµ , fµ ∈ P}. This inductive determination of s and Sint by requiring ∂j = 0 has some similarity with the procedure in [17], cf. Appendix B. Since Gn (x) = δGn (x) S0 = s˜n (x)S0 the relation (157) and current conservation imply “local” BRS-invariance of the action S = S0 + Sint within O1 : s˜ (x)S =

λn

n

n≥0

k=0

s˜k (x)Sn−k = ∂j (x) ,

x ∈ O1 ,

(159)

and hence s˜ (x)S

S

=0,

∀x ∈ O1 .

(160)

It remains to verify that the so constructed BRS-transformation s (149) satisfies (150) and the nilpotency 0 = (s 2 (F ))S = sˆ 2 (FS )

∀F .

(161)

To prove the first property we choose h and b as in (129). Analogously to (130) we find {Q, FS } = −RS (∂j, h, F ) .

(162)

Because ∂j, h ∈ JS we can apply the MWI: {Q, FS } = (δ∂j,h F )S = (s(F ))S .

(163)

In the last step we have used (159) as well as supp h ⊂ O1 and h ≡ 1 on supp F . Finally, we want to check the nilpotency of sˆ . Using the Jacobi identity we find (s 2 (F ))S = =

1

1 Q(b), Q(b ), FS + Q(b ), Q(b), FS 2 2

Q(b ), Q(b) , FS

(164)

for all admissible test functions b, b (depending on the support of F ). We may now choose b, b such that b satisfies the conditions also with respect to the support of b (and of course supp b ⊂ O1 ). Then (s 2 (F ))S = {s(j, b)S , FS } .

(165)

s(jµ ) = ∂ ν Cµν + Hµ

(166)

We now assume that

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M. D¨utsch, K. Fredenhagen

with an antisymmetric tensor field Cµν ∈ P and Hµ ∈ JS . Then (s 2 (F ))S = {∂ ν Cµν , bµ S , FS } =

1 {Cµν , ∂ µ bν − ∂ ν bµ S , FS } = 0 2

(167)

since the support of (∂ ν bµ − ∂ µ bν ) is spacelike to the support of F .13 For massless gauge fields without matter fields (i.e. the model studied in the preceding subsect.) the usual expression for the BRS-current 1 j µ = B · D µ u − ∂ µ B · u + ∂ µ u˜ · (u × u) 2

(168)

µ

(where (D µ u)a = ∂ µ ua + fabc Ab uc ) is BRS-invariant: s(j µ ) = 0. So the assumption (166) is trivially satisfied. In cases where the condition (166) cannot be directly checked one may use a perturbative formulation. Set def

H = s(j ) − ∂C

(169)

for some choice of C = (Cµν ). The condition H ∈ JS means that Sint RS0 (e⊗ ,H) = 0

(170)

for all λ. In zeroth order we find that H (0) ∈ JS0 . Set K (0) = −H (0) . We apply the MWI and get Sint RS0 (e⊗ , δK (0) Sint + H (n≥1) ) = 0 .

(171)

In lowest order this implies def

K (1) = − δK (0) S1 − H (1) ∈ JS0 .

(172)

We now define recursively def

K (n) = −

n

δK (n−k) Sk − H (n)

(173)

δK (n−k) Sl≥k + H (l≥n) ) = 0 .

(174)

k=1

and prove by induction that Sint , RS0 (e⊗

n k=1

The lowest order term of (174) is (−K (n) )S0 , hence K (n) ∈ JS0 . The recursion problem can be solved if for every n there exists an antisymmetric tensor field C (n) and a vector (n) field Kµ ∈ JS0 such that n k=1 13

δK (n−k) Sk +

n

sn−k (j (k) ) = ∂C (n) − K (n) .

k=0

In supp F + V¯± we have b = 0 or bµ = ∂ µ h.

(175)

Master Ward Identity in Classical Field Theory

307

In the given derivation we have used various cases of the MWI. In QFT it may therefore happen that the appearance of anomalies restricts the set of admissible interactions further, e.g. models with (non-compensated) axial anomalies must be excluded. The conditions on a gauge interaction found here differ somewhat from the corresponding conditions in [7] (cf. Appendix B) or in [29]. For example: to ensure renormalizability it is required Sl = 0 for all l ≥ 3 in [7]. And the condition (190) (which is the input for the derivation of the conditions of [7] given in Appendix B) is stronger than (152). However, for the class of renormalizable (by power counting) interactions we expect that the requirements derived here and the ones given in [7] have precisely the same solutions. 6. Appendix A: Construction of the Map σ We work in d = 4 dimensions. In the first part we construct recursively a particular σ in the enlarged model (i.e. with the field ϕ µ ). This construction applies also to the non-enlarged model. For the latter we prove that σ is unique (in the second part of this appendix). 6.1. Particular solution for σ . We define µ1 ···µs Hs,n = n ∂ µ1 · · · ∂ µs ϕ − (−m2 )n σ π(∂ µ1 · · · ∂ µs ϕ), def

(176)

which is obviously an element of the ideal J (80) and totally symmetrical in µ1 , . . . , µs . Hence, these properties hold true also for ν1 ···νr Fr,n =

(−1)n Nr,n ···g

1≤j1 <···<j2n ≤r π∈S2n

νjπ(2n−1) νjπ 2n

1 νj νj g π1 π2 · · · 2n n!

ν ···jˆ ···jˆ2n ···νr

1 1 Hr−2n,n

,

n = 1, . . . ,

r

(the hat means that the corresponding index is omitted, and [r/2] = [r/2] = r−1 2 if r odd) where n

Nr,n =

2 r 2

,

(177)

if r even and

(2 − 2l + 2r).

(178)

l=1

We now claim that r

σ π(∂

ν1

· · · ∂ ϕ) = σ π(∂ νr

ν1

···∂

ϕ )=∂

νr−1 νr

ν1

···∂ ϕ + νr

[2]

ν1 ···νr Fr,n

(179)

n=1

≥ 2 by recursion with respect to r. Together with the formulas (85)-(86) for r = 0, 1 this determines a map σ completely. yields a particular solution for σ π(∂ ν1

· · · ∂ νr ϕ) for r

Proof. The only non-trivial point is to verify σ π(∂ a ∂µ ϕ µ ) = −m2 σ π(∂ a ϕ). Because the r.h.s. of (179) is totally symmetrical in ν1 , . . . , νr , it suffices to show gν1 ν2 σ π(∂ ν1 · · · ∂ νr ϕ) = −m2 σ π(∂ ν3 · · · ∂ νr ϕ).

(180)

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M. D¨utsch, K. Fredenhagen def

For this purpose we set Nr,0 = 1 and def Gνr,s3 ···νr =

1 νj νj g π1 π2 · · · 2s s!

3≤j1 <···<j2s ≤r π∈S2s ν ν ν3 ···jˆ1 ···jˆ2s ···νr · · · g jπ(2s−1) jπ 2s Hr−2(s+1),s+1

(181)

def

for 0 ≤ 2s ≤ r − 2, and Gr,s = 0 for 2s > r − 2. By inserting the definitions we find that the identity r (−1)n ν3 ···νr (−1)n ν3 ···νr ν1 ···νr gν1 ν2 Fr,n = Gr,n−1 + Gr,n , n = 1, . . . , , (182) Nr,n−1 Nr,n 2 implies the assertion (180). ν1 ···νr in the following form: To prove (182) we write Fr,n n (−1) ν1 ···νr 3 ···νr = g ν1 ν2 Gνr,n−1 Fr,n Nr,n−1 (2 − 2n + 2r) (g ν1 νl g ν2 νk + g ν1 νk g ν2 νl ) g ··· · · · Hr−2n,n + 3≤l
+

g ν1 νl

ν2 ··· g ··· · · · Hr−2n,n + (ν1 ↔ ν2 )

3≤l≤r

+

(−1)n ··· ν1 ν2 ··· g · · · Hr−2n,n . Nr,n

(183)

3 ···νr Contraction of the second (third respectively) line with gν1 ν2 gives 2(n − 1)Gνr,n−1 (and ν3 ···νr 2(r − 2n)Gr,n−1 resp.). In the last line we use

µ ···µ

µ1 ···µs s 3 = Hs−2,n+1 , gµ1 µ2 Hs,n

(184)

and end up with ν1 ···νr gν1 ν2 Fr,n =

(−1)n 3 ···νr (4 + 2(n − 1) + 2(r − 2n))Gνr,n−1 Nr,n−1 (2 − 2n + 2r) (−1)n ν3 ···νr + G . (185) Nr,n r,n

6.2. Uniqueness of σ for a single real Klein-Gordon field. We will prove that the map σ is unique for P and J given by (12) and (80). Obviously, the relations σ π(ϕ) = ϕ, σ π(∂ µ ϕ) = ∂ µ ϕ and the recursive formula (179) give a solution for σ which we denote by σ0 . Analogously to (83) and by the requirements (iii) and (v) on σ we conclude that the most general solution for σ is of the form: σ π(∂ ν1 · · · ∂ νr ϕ) = σ0 π(∂ ν1 · · · ∂ νr ϕ) + M ν1 ···νr , M ν1 ···νr =

def

[r/2] l 1 al,j g νπ 1 νπ 2 · · · g νπ(2l−1) νπ 2l r! π∈Sr l=1 j =1

× ∂ νπ(2l+1) · · · ∂ νπ r (l−j ) ( + m2 )ϕ,

(186)

Master Ward Identity in Classical Field Theory

309

where al,j ∈ R is a constant. An -tensor is excluded in M ν1 ···νr by the total symmetry in ν1 , . . . , νr . We are now going to show that gνr−1 νr M ν1 ···νr = 0 (see (180)) yields al,j = 0, ∀l, j . We use the equation gνr−1 νr

1 νπ 1 νπ 2 g · · · g νπ(2l−1) νπ 2l ∂ νπ(2l+1) · · · ∂ νπ r r! π∈Sr

1 = {Nr,l g νπ 1 νπ 2 · · · g νπ(2l−3) νπ(2l−2) ∂ νπ(2l−1) · · · ∂ νπ(r−2) (r − 2)! π∈Sr−2

+Mr,l g νπ 1 νπ 2 · · · g νπ(2l−1) νπ 2l ∂ νπ(2l+1) · · · ∂ νπ(r−2) },

(187)

where Nr,l and Mr,l are non-vanishing combinatorical factors, except Mr,[r/2] = 0. The requirement gνr−1 νr M ν1 ···νr = 0 gives the following chains of equations: as,s Nr,s = 0,

al,s Mr,l + al+1,s Nr,l+1 = 0

∀l ∈ {s, s + 1, . . . , [r/2] − 1}, (188)

where the chains are labeled by s = 1, 2, . . . , [r/2]. We find indeed al,j = 0, ∀l, j .

7. Appendix B: Formulation of BRS-Invariance of the Interaction Used in [7] To derive the conditions which are used in [7] to express BRS-invariance of the interaction14 we only require current conservation (152) but generalized to all x ∈ M. The latter makes possible an integration over x ∈ M, which removes ∂j (n) from (157) and replaces s˜0 (x) by s0 . This generalization is done by admitting that current conservation at x is violated by a term of the form Mν (x)∂ ν g(x). Since we want to obtain conditions on the interaction which are expressable in terms of free fields, we require that Mν is in the range of σ : Mν = σ π(Kν ) for some Kν . In detail our input is the requirement that there exist an interaction Sint (147), a BRS-current jµ (148) (with compact support15 of (n) jµ ∀n ≥ 1) and a formal power series Kν (x) = λn Kν(n) (x), Kν(n) ∈ P ⊗ C(M) , (189) n≥1

such that Sint , ∂j (x) − σ π(Kν )(x)∂ ν g(x)) = 0 , ∀x ∈ M , ∀g ∈ D(M) . RS0 (e⊗

(190)

So, in a formalism with auxiliary fields, Sint and j depend on the choice of σ , but this dependence drops out when the auxiliary fields are eliminated.16 The condition (190) corresponds to the “Quantum Noether condition in terms of interacting fields” [17], however the formalisms are quite different. In working out the consequences of (190) we will frequently use the AWI and the MWI (in particular various formulations of the Master BRST Identity) without mentioning it. As in [7] we use a formalism without B-field; so we may replace s0 by t 14 We shall not obtain that conditions in full generality. However, as far as we know, the particular version which we shall obtain determines the interaction to the same extent, see below. 15 Usually we expect supp j (n) ⊂ supp g for n ≥ 1. µ 16 We explain this for the 4-gluon interaction in the formalism of [7]. For a σ corresponding to C = − 1 A 2 this coupling is generated by S1 , but for CA = 0 it appears in S2 , cf. the Remark at the end of Sect. 4.

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due to (139). Similarly to (157) our requirement (190) is equivalent to the sequence of conditions s˜0 (x)Sn + s˜1 (x)Sn−1 + . . . + s˜n−1 (x)S1 − ∂j (n) (x) + σ π(Kν(n) )(x)∂ ν g(x) ≡: −Gn (x) ∈ J , n ≥ 1 , (191) def

and (153) for n = 0, where s˜n (x) = δGn (x) . We will proceed in the following way: first we insert formulas which are inductively known into s˜1 (x)Sn−1 + . . . + s˜n−1 (x)S1 . Then we integrate the resulting equation over x ∈ M. Thereby, we take into account that j (n) has compact support. • To first order we obtain (tS1 )S0 = (∂ ν σ π(Kν(1) ))(g)S0 = σ π(∂ ν Kν(1) )(g)S0 .

(192) (1)

Since g is arbitrary we end up with the condition that there must exist L1 , Kν ∈ P with tˆL1 = σ π(∂ ν Kν(1) ) .

(193)

• To second order n = 2 we multiply Eq. (191) by 2. Then we insert once (˜s1 (x)S1 )S0 = −RS0 (G1 (x), S1 ) = − RS0 (∂j (1) (x), S1 ) + RS0 (˜s0 (x)S1 , S1 ) + RS0 (σ π(Kν(1) )(x)∂ ν g(x), S1 )

(194)

and once (˜s1 (x)S1 )S0 = −RS0 (S1 , G1 (x)) = . . . . Integration yields {Q0 , RSN0 (S1 , S1 )} = −RSN0 (σ π(Kν(1) )∂ ν g, S1 ) − RSN0 (S1 , σ π(Kν(1) )∂ ν g) , (195) where def

RSN0 (S1 , S1 ) = RS0 (S1 , S1 ) + 2(S2 )S0

(196)

and def

(RSN0 − RS0 )(σ π(Kν(1) )f, S1 ) : = σ π(Kν(2) )(f )S0 def

= : (RSN0 − RS0 )(S1 , σ π(Kν(1) )f )

(197)

are shorthand notations which we will interpret below. We find the additional require(2) ment that there must exist L2 ∈ P, Kν ∈ P ⊗ C(M) with tˆL2 (g 2 ) + σ π G((L1 , K (1) )g, L1 g) + σ π(Kν(2) )(∂ ν g) = 0 .

(198)

• To third order n = 3 we multiply Eq. (191) by 3! = 6. Then we insert twice (˜s1 (x)S2 )S0 = −RS0 (S2 , G1 (x)) = · · · (cf. (194)) and (˜s2 (x)S1 )S0 = −RS0 (G2 (x), S1 ) = −RS0 (∂j (2) (x), S1 ) + RS0 (˜s0 (x)S2 , S1 ) + RS0 (δG1 (x) S1 , S1 ) (199) + RS0 (σ π(Kν(2) )(x)∂ ν g(x), S1 ) ,

Master Ward Identity in Classical Field Theory

311

as well as four times (˜s1 (x)S2 )S0 = −RS0 (G1 (x), S2 ) = · · · and (˜s2 (x)S1 )S0 = −RS0 (S1 , G2 (x)) = · · · . Next we use RS0 (δG1 (x) S1 , S1 ) + RS0 (S1 , δG1 (x) S1 ) = −RS0 (G1 (x), S1 , S1 ) = −RS0 (∂j (1) (x), S1 , S1 ) + RS0 (˜s0 (x)S1 , S1 , S1 ) +RS0 (σ π(Kν(1) )(x)∂ ν g(x), S1 , S1 )

(200)

and RS0 (S1 , δG1 (x) S1 ) = − 21 RS0 (S1 , S1 , G1 (x)) = · · · . By integration we obtain {Q0 , RSN0 (S1 , S1 , S1 )} = −2RSN0 (σ π(Kν(1) )∂ ν g, S1 , S1 ) −RSN0 (S1 , S1 , σ π(Kν(1) )∂ ν g) ,

(201)

where def

RSN0 (S1 , S1 , S1 ) = RS0 (S1 , S1 , S1 ) + 2RS0 (S2 , S1 ) +4RS0 (S1 , S2 ) + 6(S3 )S0

(202)

and (RSN0 − RS0 )(σ π(Kν(1) )f, S1 , S1 ) def

: = 2RS0 (σ π(Kν(1) )f, S2 ) + RS0 (σ π(Kν(2) )f, S1 ) + RS0 (S1 , σ π(Kν(2) )f ) + 2σ π(Kν(3) )(f ), N (RS0 − RS0 )(S1 , S1 , σ π(Kν(1) )f ) def

: = 2RS0 × (S2 , σ π(Kν(1) )f ) + 2RS0 (S1 , σ π(Kν(2) )f ) + 2σ π(Kν(3) )(f ). (203) def

(2)

With H = tS2 + G((L1 , K (1) )g, L1 g) + σ π(Kν )(∂ ν g) and by using also a mixed version of the Master BRST Identity, {Q0 , RS0 (S1 , S2 )} = −RS0 (σ π(Kν(1) )∂ ν g, S2 ) + RS0 (S1 , tS2 ) +G((L1 , K (1) )g, S2 )S0 ,

(204)

we find the following additional requirement: there must exist L3 , K (3) with (tL3 (g 3 ))S0 + G((L1 , K (1) )g, L2 g 2 )S0 + σ π(Kν(3) )(∂ ν g)S0 − (δH S1 )S0 = 0, (205) where we have used H ∈ JS0 (198). For the models studied in [7] it holds G((L1 , K (1) )g, L2 g) ∈ ran σ and it is possible to fulfill (205) with L3 = 0 = K (3) . With that we may write H = (t − tˆ)S2 by using (198), and the condition (205) reduces to −π δ(t−tˆ)S2 S1 + π G((L1 , K (1) )g, L2 g 2 ) = 0 .

(206)

• In general it may happen that the higher orders n ≥ 4 of (190) restrict L1 , L2 and L3 further and non-vanishing expressions for Lj and K (j ) are possible for arbitrary high j . We do not work this out here. Because for the models treated in [7] any solution (L1 , L2 , L3 = 0, K (1) , K (2) , K (3) = 0) of (193), (198) and (206) fulfills (190) to all higher orders n ≥ 4 with Ll = 0 = K (l) , ∀l ≥ 4 (up to anomalies, i.e. violations of the MWI and the AWI).

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In [7] generalizations of (193), (198) and (206) have been used to determine the interaction, namely (141)17 , (209) and (216) in the published version. However, at least for the models studied in [7] it seems that any solution of (193), (198) and (206) with L3 = 0 = K (3) satisfies also these generalizations. So far this Appendix applies to classical field theory and QFT. The following disdef

(1)

cussions are restricted to QFT. The definition RSN0 (h) = RS0 (h) for h = S1 , σ π(Kν )f (zeroth order), and the above given first order (196)–(197) and second order definitions (202)–(203) of RSN0 are compatible with the main properties of a retarded product. By the latter we mean linearity, symmetry of Rn,1 in the first n entries, causal support (55) and in particular the recursion given by the GLZ-relation. The continuation (by analogy) of our inductive evaluation of (190) yields the definitions of RSN0 (S1⊗n , S1 ),

RSN0 (S1⊗n , σ π(Kν )f ) and RSN0 (σ π(Kν )f ⊗ S1⊗n−1 , S1 ) for all n ≥ 3. We expect that these whole sequences are compatible with the (just mentioned) main properties of a retarded product. If this holds true we can proceed similarly to formula (214) in [7] (which is a particular simple case of Theorem 3.1 in [28]): to all orders we extend RSN0 to arbitrary factors such that RSN0 agrees with RS0 as far as possible and that RS0 −→ RSN0 is an admissible finite renormalization (i.e. the main properties of a retarded product are preserved in this replacement). However, in general the RSN0 violate some normalization conditions, in particular (64) (i.e. (N3)), the AWI and the MWI, see [7]. With RSN0 being an admissible retarded product, Eq. (195) and (201) are perturbative gauge invariance (in the sense of [9, 10, 29]) to second and third order. So, we have shown to lowest orders that our condition (190) implies that the interaction is such that perturbative gauge invariance can be fulfilled by admissible finite renormalizations. From the requirement that the latter property holds true one can derive the interaction (including the Lie-algebraic structure [31]) of massless and massive spin-1 gauge fields and the couplings of spin-2 gauge theories (for an overview see [29]). To interpret the RSN0 -products in terms of simple equations we go over to the corresponding time-ordered products18 T N . As far as they are determined by (196)–(197) and (202)–(203) they fulfill (1)

Sint ⊗ T (e⊗

(1)

Sint S1 T (e⊗ ) = T N (e⊗ ), S1 λn σ π(Kν(n) )f ) = T N (e⊗ ⊗ σ π(Kν(1) )f )

(207)

n≥1

as formal power series in λ, and we expect that this holds true also for the higher orders. But understood as series with respect to the order n of Tn and TnN , the Eq. (207) express (l) a reordering: the contributions to Tn of the terms Sl , σ π(Kν ) with l ≥ 2, appear N on the r.h.s. in time ordered products Tm of higher orders: (m − n) ≥ (l − 1). The (1) RSN0 (S1⊗n , σ π(Kν )f ) satisfy Sint RS0 (e⊗ ,

S1 λn σ π(Kν(n) )f ) = RSN0 (e⊗ , σ π(Kν(1) )f ) ,

(208)

n≥1 17

µν

More precisely we mean here (141) and the antisymmetry of the L2 which appears in that formula. We are not aware of a good notion of “time-ordered products” in classical field theory. But in QFT the retarded products {Rn,1 | 0 ≤ n ≤ N} determine uniquely the corresponding time-ordered products {Tn | 1 ≤ n ≤ N + 1} and vice versa, see e.g. [12]. 18

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313

however the corresponding relations for RSN0 (S1⊗n , S1 ) and RSN0 (σ π(Kν )f ⊗S1⊗n−1 , S1 ) are more involved. (1)

Acknowledgements. We profitted from several, very interesting and detailed letters from Raymond Stora. We also thank Karl-Henning Rehren for valuable comments, and Tobias Hurth and Kostas Skenderis for discussions.

References 1. Becchi, C., Rouet, A., Stora, R.: Renormalization of the abelian Higgs-Kibble model. Commun. Math. Phys. 42, 127 (1975); Becchi, C., Rouet, A., Stora, R.: Renormalization of gauge theories. Ann. Phys. (N.Y.) 98, 287 (1976) 2. Bogoliubov, N.N., Shirkov, D.V.: Introduction to the Theory of Quantized Fields. New York, 1959 3. Brunetti, R., Fredenhagen, K.: Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds. Commun. Math. Phys. 208, 623 (2000) 4. DeWitt, B.S.: The spacetime approach to quantum field theory. In: Relativity, Groups, and Topology II: Les Houches 1983, B.S. DeWitt and R. Stora, (eds.), Part 2, New York: North-Holland, 1984, pp. 381–738 5. D¨utsch, M., Fredenhagen, K.: A local (perturbative) construction of observables in gauge theories: The example of QED. Commun. Math. Phys. 203, 71 (1999); D¨utsch, M., and Fredenhagen, K.: Deformation stability of BRST-quantization. Preprint: hep-th/9807215, DESY 98-098, Proceedings of the conference ’Particles, Fields and Gravitation’, Lodz, Poland, 1998 6. D¨utsch, M., Fredenhagen, K.: Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion. Commun. Math. Phys. 219, 5 (2001); D¨utsch, M., Fredenhagen, K.: Perturbative Algebraic Field Theory, and Deformation Quantization. Fields Inst. Commun. 30, 151 (2001), hepth/0101079 7. D¨utsch, M., Boas, F.-M.: The Master Ward Identity. Rev. Math. Phys 14, 977–1049 (2002) 8. D¨utsch, M., Fredenhagen, K.: Causal perturbation theory in terms of retarded products, and a local formulation of the renormalization group. Work in progress 9. D¨utsch, M., Hurth, T., Krahe, K., Scharf, G.: Causal construction of Yang-Mills theories. I. N. Cimento A 106, 1029 (1993); D¨utsch, M., Hurth, T., Krahe, K., Scharf, G.: Causal construction of Yang-Mills theories. II. N. Cimento A 107, 375 (1994) 10. D¨utsch, M., Scharf, G.: Perturbative Gauge Invariance: The Electroweak Theory. Ann. Phys. (Leipzig), 8, (1999) 359; Aste, A., D¨utsch, M., Scharf, G.: Perturbative Gauge Invariance: The Electroweak Theory II. Ann. Phys. (Leipzig), 8 (1999) 389 11. D¨utsch, M., Schroer, B.: Massive Vector Mesons and Gauge Theory. J. Phys. A 33, 4317 (2000) 12. Epstein, H., Glaser, V.: The role of locality in perturbation theory. Ann. Inst. H. Poincar´e A 19, 211 (1973) 13. Glaser, V., Lehmann, H., Zimmermann, W.: Field Operators and Retarded Functions. Nuovo Cimen. 6, 1122 (1957) 14. Grigore, D.R.: On the uniqueness of the nonabelian gauge theories in Epstein-Glaser approach to renormalisation theory. Romanian J. Phys. 44, 853 (1999), hep-th/9806244 15. Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. Princeton, NJ: Princeton University Press, 1992 16. Hollands, S., Wald, R. M.: Local Wick Polynomials and Time-Ordered-Products of Quantum Fields in Curved Spacetime. Commun. Math. Phys. 223, (2001) 289; Hollands, S., Wald, R. M.: Existence of Local Covariant Time-Ordered-Products of Quantum Fields in Curved Spacetime. Commun. Math. Phys., (2002) 17. Hurth, T., Skenderis, K.: The quantum Noether condition in terms of interacting fields. In: New Developments in Quantum Field Theory, P. Breitenlohner, D. Maison, J.Wess, (eds.), Lect. Notes Phys. 558, Berlin-Heidelberg-New York: Springer, 2000, p. 86 18. Kugo, T., Ojima, I.: Local covariant operator formalism of nonabelian gauge theories and quark confinement problem. Suppl. Progr. Theor. Phys. 66, 1 (1979) 19. Lam, Y.-M.P.: Perturbation Lagrangian Theory for Scalar Fields - Ward-Takahashi Identity and Current Algebra. Phys. Rev. D6, 2145 (1972) 20. Lam, Y.-M.P.: Equivalence Theorem on Bogoliubov-Parasiuk-Hepp-Zimmermann - Renormalized Lagrangian Field Theories. Phys. Rev. D7, 2943 (1973) 21. Lowenstein, J.H.: Differential vertex operations in Lagrangian field theory. Commun. Math. Phys. 24, 1 (1971)

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22. Lowenstein, J.H.: Normal-Product Quantization of Currents in Lagrangian Field Theory. Phys. Rev. D 4, 2281 (1971) 23. Marolf, D.M.: The Generalized Peierls Bracket. Ann. Phys. (N.Y.) 236, 392 (1994) 24. Nakanishi, N.: Prog. Theor. Phys. 35, 1111 (1966); Lautrup, B.: Kgl. Danske Videnskab. Selskab. Mat.-fys. Medd. 35 (11), 1 (1967) 25. Lehmann, H., Symanzik, K., Zimmermann, W.: On the formulation of quantized field theories II. Nuovo Cimen. 6, 319 (1957) 26. Peierls, R.: The commutation laws of relativistic field theory. Proc. Roy. Soc. (London) A 214, 143 (1952) 27. Piguet, O., Sorella, S.: Algebraic renormalization: Perturbative renormalization, symmetries and anomalies. Berlin: Springer, Lecture notes in Physics, 1995 28. Pinter, G.: Finite Renormalizations in the Epstein-Glaser Framework and Renormalization of the S-Matrix of φ 4 -Theory. Ann. Phys. (Leipzig) 10, 333 (2001) 29. Scharf, G.: Quantum Gauge Theories - A True Ghost Story. New York: John Wiley and Sons, 2001 30. Steinmann, O.: Perturbation expansions in axiomatic field theory. Lecture Notes in Physics 11, Berlin-Heidelberg-New York: Springer-Verlag, 1971 31. Stora, R.: Local gauge groups in quantum field theory: Perturbative gauge theories. Talk given at the workshop ‘Local Quantum Physics’ at the Erwin-Schroedinger-Institute, Vienna, 1997 32. Stora, R.: Pedagogical Experiments in Renormalized Perturbation Theory. Contribution to the conference ‘Theory of Renormalization and Regularization’, Hesselberg, Germany (2002), http://www.thep.physik.uni-mainz.de/ scheck/Hessbg02.html; and private communication Communicated by J.Z. Imbrie

Commun. Math. Phys. 243, 315–328 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0972-8

Communications in

Mathematical Physics

Periodic Solutions of Nonlinear Wave Equations with General Nonlinearities Massimiliano Berti1 , Philippe Bolle2 1 2

S.I.S.S.A., Via Beirut 2-4, 34014 Trieste, Italy. E-mail: [email protected] Universit´e d’Avignon, 33, rue Louis Pasteur, 84000 Avignon, France. E-mail: [email protected]

Received: 12 December 2002 / Accepted: 9 July 2003 Published online: 5 November 2003 – © Springer-Verlag 2003

Abstract: We present a variational principle for small amplitude periodic solutions, with fixed frequency, of a completely resonant nonlinear wave equation. Existence and multiplicity results follow by min-max variational arguments. 1. Introduction We consider the completely resonant nonlinear wave equation utt − uxx + f (u) = 0, u(t, 0) = u(t, π ) = 0,

(1)

where f (0) = f (0) = 0. We present a variational principle for finding small amplitude periodic solutions of (1), with fixed frequency, “branching off” from the infinite dimensional space of solutions of the linearized equation at 0,

utt − uxx = 0,

u(t, 0) = u(t, π ) = 0.

(2)

Any solution v = j ≥1 ξj cos(j t + θj ) sin(j x) of (2) is 2π -periodic in time. For proving the existence of small amplitude periodic solutions of completely resonant Hamiltonian PDEs like (1) two main difficulties must be overcome. The first (i) is a “small denominators” problem and the second (ii) is the presence of an infinite dimensional space of periodic solutions of the linearized equation with commensurable periods (infinitely many resonances among the linear frequencies of small oscillations): which solutions of the linearized equation (2) can be continued to solutions of the non linear equation (1)? Problem (i) can be tackled via KAM techniques (see Kuksin [14] and references therein) or Nash-Moser Implicit Function Theorems (Craig-Wayne [11], Bourgain [9]).

Supported by M.U.R.S.T. Variational Methods and Nonlinear Differential Equations.

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Recently Bambusi [3], see also [5, 7], has introduced a strong non resonance condition for the frequency which allows to use, for second order equations, the standard Contraction mapping Theorem. Concerning Problem (ii) no general results are yet available. In [5, 7] and [4] a method based on the Implicit Function Theorem, is introduced: non-degenerate critical points of a suitable functional can be continued, into periodic solutions of the nonlinear equation. Clearly the main difficulty in this method is to check, if ever true, such nondegeneracy condition (it is verified in [5, 6] for f (u) = u3 + o(u3 )). In [12] some results are obtained for partially resonant cases, i.e. when only finitely many linear resonances are present, generalizing the variational principle of Weinstein [16] and Moser [15] for periodic solutions with fixed energy, see also [10]. The main drawback of this approach for infinite dimensional systems is that the frequency of the periodic solution appears as a “Lagrange multiplier” of a suitable action functional constrained on a sphere-like manifold and it is difficult to control these multipliers. On the other hand, for solving the small denominators problem (i), one has to fix the frequency in some good nonresonant set. Some control on the frequency is recovered requiring suitable non-degeneracy assumptions on the nonlinearity and this leads to existence results. In this paper we introduce a variational principle, never used before for PDEs, for getting solutions with fixed frequency: we impose the frequency to satisfy a strong irrationality property for overcoming the small divisor problem (i) and we solve problem (ii) exploiting min-max variational arguments. More precisely, by the classical Lyapunov-Schmidt procedure one has to solve two equations: the (P ) equation, where the small denominators problem (i) appear, and the (Q) equation which is the “bifurcation equation” on the infinite dimensional space of solutions of (2), problem (ii). The (P ) equation is solved by an Implicit Function Theorem. In order to focus our attention on problem (ii), we simplify the small denominators problem (i) imposing on the frequency ω the same strong irrationality condition as in [5] and using the Contraction mapping Theorem. Once the (P ) equation is solved it remains the infinite dimensional (Q) equation: we solve it via a variational principle noting that it is the Euler-Lagrange equation of a “reduced action functional” ω defined in (7). Non-trivial critical points of ω can be obtained by min-max variational arguments. We underline that our variational approach for solving the (Q) equation keeps its validity if the (P ) equation can be solved for more general frequencies with a Nash-Moser Implicit Function Theorem. We illustrate in Sect. 3 the potentialities of our approach showing the existence of periodic solutions of Eq. (1) for the nonlinearities f (u) = aup (a = 0), where p is either an odd or an even integer. Much sharper results can be obtained, see Sect. 4, and are proved in [8]: “optimal” multiplicity results, information on minimal periods and regularity of the solutions. For finite dimensional Hamiltonian systems we quote the results of Fadell and Rabinowitz who proved in [13] existence and multiplicity of periodic solutions with fixed period through a minimax argument which exploits the natural S 1 symmetry of the problem (induced by time translations).

2. The Variational Principle 2.1. The variational Lyapunov-Schmidt reduction. We look for periodic solutions of (1) with frequency ω as u(t, x) = q(ωt, x), where q(·, x) is 2π periodic. After this

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317

normalization of the period we have to solve the problem ω2 utt − uxx + f (u) = 0,

u(t, 0) = u(t, π ) = 0,

We look for solutions of (3) u : → space X := u ∈ H 1 (, R) ∩ L∞ (, R)

u(t + 2π, x) = u(t, x).

(3)

R, where := R/2π Z × (0, π ), in the Banach u(t, 0) = u(t, π ) = 0, u(−t, x) = u(t, x)

endowed with the norm ||u||X := |u|∞ + ||u||, || || being the norm on

H 1 ()

associated to the scalar product dt dx ut wt + ux wx . (u, w) :=

We can restrict to the space X of functions even in time because Eq. (1) is reversible. Any u ∈ X can be developed in Fourier series as ulj cos lt sin j x, u(t, x) = l≥0,j ≥1

and (u, v) =

π2 2

ulj wlj (l 2 + j 2 )

∀u, w ∈ X.

l≥0,j ≥1

We denote by |u|L2 := ( |u|2 )1/2 the L2 -norm of u ∈ X and by (u, w)L2 := uw the L2 -scalar product. The space of the solutions in H01 (, R) of the linear equation vtt − vxx = 0 that are even in time is ξj cos(j t) sin j x ξj ∈ R, j 2 ξj2 < +∞ . V := v(t, x) = j ≥1

j ≥1

The space V can also be written as V := v(t, x) = η(t + x) − η(t − x) η(·) ∈ H 1 (T), η odd , where T := R/2πZ. 1 On V the norm || ||X is equivalent to the H -norm || || since, for v = ξj cos(j t) |ξj | ≤ C||v|| by the Cauchy-Schwarz inequality. sin(j x) ∈ V , we have |v|∞ ≤ Moreover the embedding (V , || · ||) → (V , | · |∞ ) is compact. For the sequel we endow V with the H 1 norm. Remark 2.1. We do not use the same norm as in [5] on X because we want V to be endowed with the H 1 norm: it appears in the quadratic part at 0 of the reduced action functional ω , see (10). We do not work just in H 1 () because Lemma 2.3 requires a space of bounded functions to work.

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We next consider the Lagrangian action functional : X → R, 2π π ω2 2 1 2 dt dx u − ux − F (u),

(u) := 2 t 2 0 0 u where F (u) := 0 f (s) ds. It is easy to check that ∈ C 1 (X, R) and 2π π dt dx ω2 ut ht − ux hx − f (u)h, ∀h ∈ X. D (u)[h] = 0

0

Critical points of are weak solutions of (3). In order to find critical points of we perform a Lyapunov-Schmidt reduction. The space X can be decomposed as X = V ⊕ W , where1 W := w ∈ X (w, v)L2 = 0, ∀ v ∈ V = wlj cos(lt) sin j x ∈ X wjj = 0 ∀j ≥ 1 . l≥0,j ≥1

W is also the H 1 -orthogonal of V in X. Setting u := v + w with v ∈ V and w ∈ W , (3) is equivalent to the following system of two equations (called resp. the (Q) and the (P ) equations) (Q)

− ω2 vtt + vxx = V f (v + w),

(P )

− ω2 wtt + wxx = W f (v + w).

Note that w solves the (P ) equation (in a weak sense) if and only if it is a critical point of the restricted functional w → (v + w) ∈ R, i.e. if and only if 2π π dt dxω2 wt ht − wx hx − f (v + w)h = 0, ∀h ∈ W. (4) D (v + w)[h] = 0

0

We first solve the (P ) equation by an Implicit Function Theorem. Small denominators appear here. As already said, we will bypass this aspect of the problem assuming that ω ∈ W := ∪γ >0 Wγ , where Wγ is the set of strongly non-resonant frequencies introduced in [5] γ Wγ := ω ∈ R |ωl − j | ≥ , ∀j = l . l As proved in Remark 2.4 of [5], for γ < 1/3, the set Wγ is uncountable, has zero measure and accumulates to ω = 1 both from the left and from the right. Lemma 2.1. For ω ∈ Wγ the operator Lω := −ω2 ∂tt + ∂xx : D(Lω ) ⊂ W → W has a bounded inverse L−1 ω : W → W satisfying, for a positive constant C1 independent of −1 : W → W be the inverse operator of −∂ + ∂ . γ and ω, ||L−1 || ≤ (C 1 /γ ). Let L tt xx ω There exists C2 > 0 such that ∀r, s ∈ X,

|ω − 1| −1 r(t, x) L−1 − L s(t, x)) dt dx (5) ( ||r||X ||s||X . ≤ C2 W ω γ 1 ulj cos(lt) sin(j x) ∈ X, the H 1 norm, and so the L∞ norm, of V (u) := If u = ujj cos(j t) sin(j x) is finite and V (u) ∈ X. As a consequence W (u) := u − V (u) ∈ X. Moreover the projectors V : X → V and W : X → W are continuous.

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319

∞ Proof. W.r.t. to Lemma 4.6 of [5] we have to prove that L−1 ω W ⊂ L () and (5). It is done in the Appendix.

Fixed points of the nonlinear operator G : W → W defined by G(w) := L−1 ω W f (v+ w) are solutions of the (P ) equation. In order to prove that G is a contraction for v sufficiently small, we need the following standard lemma on the Nemytski operator induced by f . Lemma 2.2. The Nemitski operator u → f (u) is in C 1 (X, X) and its derivative at u is Df (u)[h] = f (u)h. Moreover, if f (0) = f (0) = . . . = f (p−1) (0) = 0, there is ρ0 > 0 and positive constants C3 , C4 , depending only on f , such that, ∀u ∈ X with ||u||X < ρ0 , p−1

p

||f (u)||X ≤ C3 |u|∞ ||u||X ≤ C3 ||u||X ,

||f (u)h||X ≤ C4 |u|∞ ||u||X ||h||X p−2

p−1

≤ C4 ||u||X ||h||X .

(6)

The next lemma, based on the Contraction mapping Theorem, is similar to Lemma 4.6 of [5]. Lemma 2.3. Let f (0) = f (0) = . . . = f (p−1) (0) = 0 and assume that ω ∈ Wγ . There exists ρ > 0 such that, ∀v ∈ D := {v ∈ V | ||v||p−1 /γ < ρ} there exists a unique w(v) ∈ W which solves the (P ) equation. Moreover • i) ||w(v)||X = O(||v||p /γ );

2 )||v||2p−1 ||r|| f (v)) r| = O (1/γ ; • ii) ∀r ∈ X, | (w(v) − L−1 W X ω • iii) the map v → w(v) is in C 1 (D, W ); • iv) w(−v)(t, x) = w(v)(t + π, π − x). Once the (P ) equation is solved by w(v) ∈ W , there remains the infinite dimensional (Q) equation −ω2 vtt + vxx = V f (v + w(v)). We claim that such an equation is the Euler-Lagrange equation of the reduced Lagrangian action functional ω : D → R defined by 2π π ω2 dt dx (vt + (w(v))t )2 ω (v) := (v + w(v)) = 2 0 0 1 − (vx + (w(v))x )2 − F (v + w(v)). (7) 2 Indeed, by Lemma 2.3-iii) ω is in C 1 (D, R) and ∀h ∈ V , Dω (v)[h] = D (v + w(v)) h + dw(v)[h] = D (v + w(v))[h],

(8)

since dw(v)h ∈ W and w(v) solves the (P ) equation (recall (4)). By (8), for v, h ∈ V , dt dx ω2 vt ht − vx hx − V f (v + w(v))h Dω (v)[h] = = ε(v, h) − dt dx V f (v + w(v))h, (9)

where ε := (ω2 − 1)/2. By (9), a critical point v ∈ D ⊂ V of ω is a solution of the (Q)-equation:

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Theorem 2.1. If v ∈ D ⊂ V is a critical point of the reduced action functional ω : D → R then u = v + w(v) is a weak solution of (3). Remark 2.2. This reduction gives also a necessary condition: any critical point u of ω , sufficiently close to 0, can be written as u = v + w, where v ∈ V is a critical point of ω and w = w(v), see [1]. We have reduced the problem of finding non-trivial solutions of the infinite dimensional (Q)-equation to the problem of finding non trivial critical points of the reduced action functional ω (by (9) v = 0 is a critical point of ω for all ω). By (7) and formula (4) with h = w(v), the reduced action functional can be developed as ω2 2 vx2 1 ω (v) = dt dx vt − − F (v + w(v)) + f (v + w(v))w(v), 2 2 2 and since |vt |2L2 = |vx |2L2 = ||v||2 /2, ε ω (v) = ||v||2 + 2

dt dx

1 2

f (v + w(v))w(v) − F (v + w(v)) ,

(10)

where ε := (ω2 − 1)/2. By (10) and the bounds of Lemma 2.3, ω possesses a local minimum at the origin and we shall show that ω satisfies the geometrical hypotheses of the Mountain Pass Theorem [2]. However we can not apply directly this theorem since ω is defined only on a neighborhood of the origin. In the next Subsect. 2.2, we prove an abstract theorem which can be applied directly to our problem. 2.2. Existence of critical points: An abstract result. Let : Br ⊂ E → R be a C 1 functional defined on the ball Br := {v ∈ E | ||v|| < r} of a Hilbert space E with scalar product (·, ·), of the form (v) =

ε ||v||2 − G(v) + R(v), 2

(11)

where G ≡ 0 and • (H 1) G ∈ C 1 (E, R) is homogeneous of degree q + 1 with q > 1, i.e. G(λv) = λq+1 G(v) ∀λ ∈ R+ ; • (H 2) ∇G : E → E is compact; • (H 3) R ∈ C 1 (Br , R), R(0) = 0 and for any r ∈ (0, r), ∇R maps Br into a compact subset of E. Theorem 2.2. Let G satisfy (H 1), (H 2) and suppose that G(v) > 0 for some v ∈ E (resp. G(v) < 0). There is α > 0 (depending only on G) and ε0 > 0 (depending on r) such that, for all R ∈ C 1 (Br , R) satisfying (H 3) and |(∇R(v), v)| ≤ α||v||q+1 , ∀v ∈ Br

(12)

for all ε ∈ (0, ε0 ) (resp. ∈ (−ε0 , 0)), has a non-trivial critical point v ∈ Br satisfying ||v|| = O(ε1/(q−1) ).

Periodic Solutions of Nonlinear Wave Equations

321

Remark 2.3. The key difference with the approach of [5] is the following: instead of trying to show that the functional (ε/2)||v||2 − G(v) possesses non-degenerate critical points (if ever true), and then continuing them through the Implicit Function Theorem as critical points of , we find critical points of showing that the nonlinear perturbation term R does not affect the mountain pass geometry of the functional (ε/2)||v||2 − G(v). Actually, in [5], non-degenerate critical points of G constrained on S are continued. This is equivalent to what was said before since any critical point v of G constrained to S gives rise, by homogeneity, to a critical point (ε/(q + 1)G( v ))1/q−1 v ∈ E of the functional (ε/2)||v||2 − G(v), provided that ε and G( v ) have the same sign. Proof of Theorem 2.2. For definiteness we assume that G(v) > 0 for some v ∈ V and we take ε > 0. The steps of the proof are the following: 1) Define on the whole space E a new

which is an extension of |V for some neighborhood V of 0, in such a way functional

that possesses the Mountain-Pass geometry, see (15). 2) Derive by the Mountain-Pass

, see (16). 3) Prove that (vn ) Theorem the existence of a “Palais-Smale” sequence for

converges to some critical point v in an open ball where and coincide. Step 1. Let us consider v ∈ S := {v ∈ E | ||v|| = 1} such that m := G(v) > 0. The function t → (ε/2)||tv||2 − G(tv) = (ε/2)t 2 − t q+1 m attains its maximum at rε :=

ε (q + 1)m

1/(q−1)

with maximum value ((1/2) − 1/(q + 1))εrε2 . Let λ = [0, +∞) → R be a smooth cut-off non-increasing function such that λ(s) = 1 if s ∈ [0, 4]

and λ(s) = 0 if s ∈ [16, +∞).

(13)

Choose ε0 > 0 small enough, such that 4rε0 < r. For all 0 < ε < ε0

ε : E → R by we define a functional R 2

ε (v) := λ ||v|| R(v) if v ∈ Br

ε (v) := 0 if v ∈ R and R / Br . rε2

|B2r = R|B2r and, by (12)-(13), there is a constant C depending on

ε ∈ C 1 (E, R), R R λ and q only, such that ∀v ∈ E

ε (v)| ≤ Cα||v||q+1 |R

and

ε (v), v)| + |R

ε (v)| ≤ Cαrεq+1 . |(∇ R (14)

on the whole E as Then we can define

(v) :=

ε

ε (v). ||v||2 − G(v) + R 2

(0) = 0 and

possesses the Mountain pass geometry: ∃δ > 0 and w ∈ Rv with ||w|| > δ, such that

(v) > 0, (i) inf v∈∂Bδ

(w) < 0. (ii)

(15)

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M. Berti, P. Bolle

(Rv) = −∞. Equation (15)-(ii) holds for w = Rv with R large enough since limR→∞ For Eq. (15)-(i), observe first that by the compactness of ∇G : E → E, G maps the sphere S into a bounded set. Hence there is a constant K > 0 such that |G(v)| ≤ K||v||q+1 , and by (14), any δ ∈ (0, R) such that (ε/2)δ 2 − (K + Cα)δ q+1 > 0 is suitable. Step 2. Define the Mountain pass paths = γ ∈ C([0, 1], E) γ (0) = 0, γ (1) = w and the Mountain-pass level

(γ (s)). cε = inf max γ ∈ s∈[0,1]

By (15)-(i), cε > 0. By the Mountain-Pass Theorem [2] there exists a sequence (vn ) such that

(vn ) → 0, ∇

(vn ) → cε ,

(16)

(Palais-Smale sequence). Step 3. We shall prove that for n large enough vn lies in a ball Bh for some h < 2rε . For this we need an estimate of the level cε . By the definition of cε and (14), ε

(sRv) ≤ max ||tv||2 − (m − Cα)||tv||q+1 . cε ≤ max s∈[0,1] t∈[0,R] 2 Computing the maximum in the right-hand side, we find the estimate 2/(q−1) 1 1 m rε2 . − cε ≤ ε 2 q +1 m − Cα

(17)

We claim that lim supn→+∞ ||vn || < 2rε . If not, up to a subsequence, limn→∞ ||vn || :=

(v), v) = ε||v||2 − (q + 1)G(v)+ ν ∈ [2rε , +∞]. By the homogeneity of G, (∇

ε (v), v) and, by (16), (∇ R

ε (vn ), vn ) = µn ||vn ||

(vn ), vn ) = ε||vn ||2 − (q + 1)G(vn ) + (∇ R (∇ with limn→∞ µn = 0. This implies 1 µn 1

ε (vn ) + 1 (∇ R

ε (vn ), vn ).

(vn ) − ε − ||vn ||2 = ||vn || − R 2 q +1 q +1 q +1

(vn ) → cε and using (14), we derive that the sequence (||vn ||) is bounded and Since so ν < ∞. Taking limits as n → ∞, we obtain 1 α 1 2 ε (18) − ν ≤ cε + Cαrεq+1 ≤ cε + C εrε2 , 2 q +1 m by the definition of rε and for some positive constant C . Since ν ≥ 2rε , (18) contradicts estimate (17), provided α has been chosen small enough (depending on q and m only).

on B2rε , Thus vn ∈ Bh (for n large), for some h < 2rε , and since ≡

(vn ) = ∇(vn ) = εvn − ∇G(vn ) + ∇R(vn ) → 0. ∇ Since (vn ) is bounded, by the compactness assumptions (H 2) and (H 3), (vn ) converges in B2rε to some non-trivial critical point v of at the critical level cε > 0.

Periodic Solutions of Nonlinear Wave Equations

323

To complete this section, we note that when is invariant under some symmetry group (e.g. is even), multiplicity of non-trivial critical points can be obtained, as in the symmetric version of the Mountain pass Theorem [2]. We remark that the reduced action functional ω defined in Subsect. 2.1 is even2 . Indeed defining the linear operator I : X → X by (Iu)(t, x) := u(t + π, π − x),

◦ I = , and, by Lemma 2.3-iv) and since −v = Iv, ω (−v) = (−v + w(−v)) = (I(v + w(v))) = (v + w(v)) = ω (v).

(19)

We can prove that the number Nω of non-trivial critical points of ω in D increases to +∞ as the frequency ω tends to 1. These and other results are presented in Sect. 4 and proved in [8]. 3. Applications to Nonlinear Wave Equations As an illustration of our method we prove existence of periodic solutions of the nonlinear wave equation (1) when f (u) = aup (a = 0) for p ≥ 2 odd and p even integer. Here F (u) := aup+1 /(p + 1). 3.1. Case I: p odd. Lemma 3.1. Let f (u) = aup for an odd integer p. Then the reduced action functional ω : D → R defined in (7) has the form (11) with ε = (ω2 − 1)/2, v p+1 F (v) = a −F (v + w(v)) + F (v) and R(v) : = G(v) := p+1 1 + f (v + w(v))w(v). 2 Moreover (∇R(v), v) = O(||v||2p ).

Proof. We find, by (9), (∇R(v), v) = (f (v) − f (v + w(v)))v and so, by Lemma 2.3 and |v|∞ ≤ C||v||, p |f (v) − f (v + w(v)| |v| ≤ |w(v)|∞ |v|∞ = O(||v||2p ). |(∇R(v), v)| ≤

We have to check the compactness properties (H 2) and (H 3). Lemma 3.2. G and R satisfy assumptions (H 2) and (H 3). Proof. We have (∇G(v), h) = av p h and (H 2) stems from the compactness of the embedding H 1 () → L2p (). Now let (vk ) be some sequence in Br , with r < r, Br ⊂ D. Then, up to a subsequence, vk v ∈ V weakly for the H 1 topology and vk → v in |·|∞ . Moreover since wk := w(vk ) too is bounded we can also assume that wk w weakly in H 1 and wk → w in | · |Lq norm for all q < ∞. We claim that ∇R(vk ) → R, where (R, h) = f (v + w)h − av p h. Indeed, since wk → w in | · |Lq , it converges (up to a subsequence) also a.e. We can deduce, by the Lebesgue dominated convergence theorem, that f (vk + wk ) → f (v + w) in L2 since f (vk + wk ) → f (v + w) a.e. and p (f (vk + wk )) is bounded in L∞ . Hence, since (∇R(vk ), h) = f (vk + wk )h − avk h, ∇R(vk ) → R. 2 Not restricting to the space X of functions even in time the reduced functional would inherit the natural S 1 invariance symmetry defined by time translations.

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G(v) is homogeneous of order p +1 and for a > 0, G(v) := a v p+1 > 0 ∀v = 0. By Lemma 3.1 we can choose r small enough so that ∀v ∈ Br (∇R(v), v) ≤ α||v||p+1 , where α is defined in (12). Applying Theorem 2.2 we obtain : Theorem 3.1. Let f (u) = aup (a = 0) for an odd integer p ≥ 3. There exists a positive constant C5 := C5 (f ) such that, ∀ω ∈ Wγ satisfying |ω − 1| ≤ C5 and ω > 1 if a > 0 (resp. ω < 1 if a < 0), Eq. (1) possesses at least one 2π/ω-periodic, even in time solution. 3.2. Case II: p even. The case = aup with p even integer requires more attention f (u) p+1 since, by Lemma 3.4 below, v ≡ 0. Lemma 3.3. Let m : R2 → R be 2π-periodic w.r.t. both variables. Then 2π π 1 2π 2π m(t + x, t − x) dt dx = m(s1 , s2 ) ds1 ds2 . 2 0 0 0 0 Proof. Make the change of variables (s1 , s2 ) = (t + x, t − x) and use the periodicity of m. Lemma 3.4. If v ∈ V then v p ∈ W . In particular v p+1 = 0. Proof. For all v(t, x) = η(t + x) − η(t − x), u(t, x) = q(t + x) − q(t − x) ∈ V , by Lemma 3.3, 2π π v 2p (t, x)u(t, x) dx dt 0

0

1 = 2

2π

0

2π

2p

η(s1 ) − η(s2 )

(q(s1 ) − q(s2 )) ds1 ds2 = 0,

0

because (s1 , s2 ) → (η(s1 ) − η(s2 ))2p (q(s1 ) − q(s2 )) is an odd function.

We have to look for the dominant nonquadratic term in the reduced functional ω . Lemma 3.5. Let f (u) = aup with p ≥ 2 an even integer. Then ω : D → R defined in (7) has the form (11) with ε := (ω2 − 1)/2, a2 G(v) := v p L−1 v p and 2 1 a2 R(v) := f (v + w(v))w(v) − F (v + w(v)) − v p L−1 v p . 2 2 Moreover (∇R(v), v) = O(||v||3p−1 + |ε| ||v||2p ). Proof. We find, by (9), (∇R(v), v) = f (v + w(v))v − pa 2 v p L−1 v p . Developing in Taylor series, using Lemma 2.3-i)-ii) and v p+1 = 0, we obtain (∇R(v), v) = f (v)v + f (v)v w(v) + O(||v||3p−1 ) − pa 2 v p L−1 v p p 3p−1 = pa 2 v p L−1 ) − pa 2 v p L−1 v p ω v + O(||v||

Periodic Solutions of Nonlinear Wave Equations

=

325

p −1 p + O(||v||3p−1 ) pa 2 v p L−1 v − L v ω

= O(|ε| ||v||2p + ||v||3p−1 ), by (5).

The next lemma, proved in the Appendix, ensures that G ≡ 0. Lemma 3.6. G(v) := (a 2 /2) v p L−1 v p < 0, ∀v = 0. G is homogeneous of degree 2p (property (H 1)); ∇G, ∇R still satisfy assumptions (H 2) and (H 3). By Lemma 3.5, choosing first r then ε small enough, we can apply Theorem 2.2 and we get: Theorem 3.2. Let f (u) = aup (a = 0) and p be an even integer. There is C6 := C6 (f ) > 0 such that, ∀ω ∈ Wγ , ω < 1, with |ω − 1| ≤ C6 , Eq. (1) possesses at least one 2π/ω-periodic, even in time solution. 4. Further Results Much stronger results than Theorems 3.1 and 3.2 can be obtained. For any smooth nonlinearity f (u) = O(u2 ) with some f p (0) = 0, for ω ∈ W in a right or left neighborhood of 1, we prove in [8] the existence of a large number Nω of 2π/ω-periodic √ classical C 2 () solutions u1 , . . . , un , . . . , uNω with Nω → +∞ as ω → 1 (Nω ≈ γ τ /|ω − 1| for some τ ∈ [1, 2]). Moreover the minimal period of the nth solution un is proved to be 2π/nω. The following theorems are proved in [8]. For ω ∈ W := ∪γ >0 Wγ define γω := max{γ | ω ∈ Wγ }. Theorem 4.1. Let f (u) = aup + h.o.t. (a = 0) for an odd integer p ≥ 3. Then there exists a positive constant C7 := C7 (f ) such that, ∀ω ∈ W and ∀n ∈ N\{0} satisfying |ω − 1|n2 ≤ C7 γω and ω > 1 if a > 0 (resp. ω < 1 if a < 0), Eq. (1) possesses at least one pair of even periodic in time classical C 2 solutions with minimal period 2π/(nω). Theorem 4.2. Let f (u) = aup (a = 0) for some even integer p. Then there exists a positive constant C8 depending only on f such that, ∀ω ∈ W with ω < 1, ∀n ≥ 2 such that (|ω − 1|n2 )1/2 ≤ C8 , γω Equation (1) possesses at least one pair of even periodic in time classical C 2 solutions with minimal period 2π/(nω). If p = 2 the existence result holds true for n = 1 as well. When f (u) = aup +o(up ), p even, two other cases have to be considered, according to the behaviour of the higher order terms of the nonlinearity f . We shall not enter into the details, see [8].

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5. Appendix Proof of Lemma 2.1. Writing w(t, x) = l≥0,j ≥1,j =l wl,j cos(lt) sin(j x) ∈ W we claim that

C |wl,j | 1 |L−1 =: S ≤ C |w|L2 + ||w|| ≤ ||w||. ω w|∞ ≤ |ωl − j |(ωl + j ) γ γ l≥0,j ≥1,j =l

For l ∈ N, let e(l) ∈ N be defined by |e(l) − ωl| = minj ∈N |j − ωl|. Since ω is not rational, e(l) is the only integer e such that |e − ωl| < 1/2. S = S1 + S2 , where

S1 :=

l≥0,j ≥1,j =l,j =e(l)

|wl,j | and |ωl − j |(ωl + j )

S2 :=

l≥0,e(l)=l

|wl,e(l) | . |ωl − e(l)|(ωl + e(l))

We first find an upper bound for S1 . For j = e(l) we have that |j − ωl| ≥ |j − e(l)| − |e(l) − ωl| ≥ |j − e(l)| − 1/2 ≥ |j − e(l)|/2. Moreover, since |e(l) − ωl| < 1/2, it is easy to see (remember that ω ≥ 1/2) that e(l) + l ≤ 4ωl, and hence |j − e(l)| + l ≤ j + e(l) + l ≤ 4(j + ωl). Defining wl,j by wl,j = 0 if j ≤ 0 or j = l, we deduce S1 ≤

l≥0,j ∈Z,j =e(l)

8|wl,j | . |j − e(l)|(|j − e(l)| + l)

Hence, by the Cauchy-Schwarz inequality, S1 ≤ 8R1 |w|L2 , where

R12 =

1

l≥0,j ∈Z,j =e(l)

(j

− e(l))2 (|j

− e(l)| + l)2

=

l≥0,j ∈Z,j =0

≤

l≥0,j ∈Z,j =0

1 j 2 (|j | + l)2 1 j 2 (1 + l)2

< ∞.

We now find an upper bound for S2 . Since ω ∈ Wγ , for l = e(l), |ωl − e(l)||ωl + e(l)| ≥γ l −1 (ωl + e(l)) ≥ γ . Hence, still by the Cauchy-Schwarz inequality, S2 ≤ (1/γ ) e(l)=l |wl,e(l) | ≤ (C/γ )||w||. Finally we prove (5). Writing r = l≥0,j ≥1 rlj cos(lt) sin(j x), s = l≥0,j ≥1 slj cos(lt) sin(j x), slj slj L−1 W s = cos(lt) sin(j x) , L−1 cos(lt) sin(j x), ω s = 2 2 2 2 l −j ω l − j2 j =l

j =l

−1 2 r(L−1 ω − L )(W s) dt dx = π

j =l

slj rlj (1 − ω2 )l 2 . (ω2 l 2 − j 2 )(l 2 − j 2 )

(20)

By (20), since ω2 l 2 − j 2 ≥ ωγ and l 2 − j 2 ≥ 1, |ω − 1| |ω − 1| −1 ||r||X ||s||X , r(L−1 −L )( s) dt dx |sl,j ||rl,j | l 2 ≤ C ≤C W ω γ γ j =l

which proves (5).

Periodic Solutions of Nonlinear Wave Equations

327

Proof of Lemma 3.6. Write v(t, x) = η(t + x) − η(t − x) ∈ V and v p (t, x) = m(t + x, t − x) with m(s1 , s2 ) = (η(s1 ) − η(s2 ))p . Define, for s2 ≤ s1 ≤ s2 + 2π , 1 M(s1 , s2 ) := − m(ξ1 , ξ2 ) dξ1 dξ2 , (21) 8 Qs1 ,s2 where Qs1 ,s2 := {(ξ1 , ξ2 ) ∈ R2 | s1 ≤ ξ1 ≤ s2 + 2π, s2 ≤ ξ2 ≤ s1 }. The partial derivative of M w.r.t. s1 is given by 1 s2 +2π 1 s1 m(ξ1 , s1 ) dξ1 + m(s1 , ξ2 ) dξ2 . ∂s1 M(s1 , s2 ) = − 8 s1 8 s2 Differentiating w.r.t. s2 , remembering that m(s1 , s2 ) = (η(s1 ) − η(s2 ))p with p even, and that η is 2π-periodic, we obtain 1 1 1 ∂s2 ∂s1 M = − m(s2 + 2π, s1 ) − m(s1 , s2 ) = − m(s1 , s2 ). 8 8 4 Moreover M(s1 , s1 ) = M(s1 , s1 − 2π) = 0 and M(s1 + 2π, s2 + 2π ) = M(s1 , s2 ). This implies that z(t, x) = M(t + x, t − x) satisfies −ztt + zxx = v p and z(t, 0) = z(t, π) = 0, z(t + 2π, x) = z(t, x). As a consequence, z ∈ L−1 (v p ) + V and G(v) = (a 2 /2) v p z. By (21), since (p being even) m ≥ 0, M and z ≤ 0. Hence G(v) ≤ 0. If G(v) = 0, then v p (t, x)z(t, x) vanishes everywhere, which implies that m(s1 , s2 ) = 0 or M(s1 , s2 ) = 0 for all s2 ≤ s1 ≤ s2 + 2π . If M(s1 , s2 ) = 0, then m(ξ1 , ξ2 ) = 0 for all (ξ1 , ξ2 ) ∈ Qs1 ,s2 and so m(s1 , s2 ) = 0. In any case m(s1 , s2 ) = 0, and so v = 0. We can conclude that G(v) < 0 for all v = 0. Acknowledgements. The authors thank A. Ambrosetti, D. Bambusi and M. Procesi for useful discussions. Part of this paper was written when the second author was visiting S.I.S.S.A. in Trieste.

References 1. Ambrosetti,A., Badiale, M.: Homoclinics: Poincar´e-Melnikov type results via a variational approach. Annales I. H. P. - Analyse nonlin. 15(2), 233–252 (1998) 2. Ambrosetti, A., Rabinowitz, P.: Dual Variational Methods in Critical Point Theory and Applications. J. Func. Anal. 14, 349–381 (1973) 3. Bambusi, D.: Lyapunov Center Theorems for some nonlinear PDEs: A simple proof. Ann. Sc. Norm. Sup. di Pisa, Ser. IV XXIX, fasc. 4, (2000) 4. Bambusi, D.: Families of periodic solutions of reversible PDEs. Preprint available at http://www.math.utexas.edu/mp arc 5. Bambusi, D., Paleari, S.: Families of periodic solutions of resonant PDEs. J. Nonlinear Sci. 11, 69–87 (2001) 6. Bambusi, D., Cacciatori, S., Paleari, S.: Normal form and exponential stability for some nonlinear string equations. Z. Angew. Math. Phys. 52(6), 1033–1052 (2001) 7. Bambusi, D., Paleari, S.: Families of periodic orbits for some PDE’s in higher dimensions. Comm. Pure and Appl. Analysis 1(4) (2002) 8. Berti, M., Bolle, P.: Multiplicity of periodic solutions of nonlinear wave equations, to appear in Nonlinear Analysis: Theory, Methods, & Applications 9. Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schr¨odinger equations. Ann. Math. 148, 363–439 (1998) 10. Craig, W.: Probl`emes de petits diviseurs dans les e´ quations aux d´eriv´ees partielles. Panoramas et Synth`eses, 9, Paris: Soci´et´e Math´ematique de France, 2000 11. Craig, W., Wayne, E.: Newton’s method and periodic solutions of nonlinear wave equation. Commun. Pure and Appl. Math. XLVI, 1409–1498 (1993)

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12. Craig, W., Wayne, E.: Nonlinear waves and the 1 : 1 : 2 resonance. In: Singular limits of dispersive waves (Lyon, 1991), NATO Adv. Sci. Inst. Ser. B Phys., 320, New York: Plenum, 1994, pp. 297–313 13. Fadell, E.R., Rabinowitz, P.: Generalized cohomological index theories for the group actions with an application to bifurcations question for Hamiltonian systems. Inv. Math. 45, 139–174 (1978) 14. Kuksin, S.B.: Perturbation of conditionally periodic solutions of infinite-dimensional Hamiltonian systems. Izv. Akad. Nauk SSSR, Ser. Mat. 52(1), 41–63 (1988) 15. Moser, J.: Periodic orbits near an Equilibrium and a Theorem by Alan Weinstein. Commun. Pure Appl. Math. XXIX (1976) 16. Weinstein, A.: Normal modes for Nonlinear Hamiltonian Systems. Inv. Math. 20, 47–57 (1973) Communicated by G. Gallavotti

Commun. Math. Phys. 243, 329–342 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0962-x

Communications in

Mathematical Physics

Spin in the q-Deformed Poincar´e Algebra Christian Blohmann1,2 1

Ludwig-Maximilians-Universit¨at M¨unchen, Sektion Physik, Lehrstuhl Prof. Wess, Theresienstr. 37, 80333 M¨unchen, Germany. E-mail: [email protected] 2 Max-Planck-Institut f¨ ur Physik, F¨ohringer Ring 6, 80805 M¨unchen, Germany Received: 30 December 2001 / Accepted: 16 July 2003 Published online: 14 October 2003 – © Springer-Verlag 2003

Abstract: We investigate spin as algebraic structure within the q-deformed Poincar´e algebra, proceeding in the same manner as in the undeformed case. The q-Pauli-Lubanski vector, the q-spin Casimir, and the q-little algebras for the massless and the massive case are constructed explicitly. 1. Introduction From the beginnings of quantum field theory it has been argued that the pathological ultraviolet divergences should be remedied by limiting the precision of position measurements by a fundamental length [1–3]. In view of how position-momentum uncertainty enters into quantum mechanics, a natural way to integrate such a position uncertainty in quantum theory would have been to replace the commutative algebra of space observables with a non-commutative one [4]. However, deforming the space alone will in general break the symmetry of spacetime. In order to preserve a background symmetry, the symmetry group must be deformed together with the space it acts on. This reasoning led to the discovery of quantum groups [5], that is, generic methods to continuously deform Lie algebras [6, 7] and matrix groups [8–10] within the category of Hopf algebras. Starting from the non-commutative plane [11], the q-deformations of a series of objects, differential calculi on non-commutative spaces [12], Euclidean space [9], Minkowski space [13], the Lorentz group and the Lorentz algebra [14–17], led to the q-deformed Poincar´e algebra [18, 19]. Describing the symmetry of flat spacetime, the Poincar´e group or, equivalently, its enveloping algebra is sufficient to construct special relativity and even a considerable part of relativistic quantum theory. More precisely, Wigner has shown that a free elementary particle can be identified with an irreducible Hilbert space representation of the Poincar´e group [20]. These representations are constructed using the method of induced representations, which reduces the representation theoretic problem to a structural analysis of the Poincar´e algebra. In mathematical terms: We start from a representation of the

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inhomogeneous part of the algebra, determine the stabilizer (little algebra) of this representation, construct the irreducible representations of the stabilizer, and, finally, induce these representations to representations of the entire algebra, yielding all irreducible representations of the Poincar´e algebra. Seemingly abstract, each step in this construction has a clear physical interpretation: The representation of the inhomogeneous part is the description of a momentum eigenspace. The stabilizer is the spin symmetry lifting a possible momentum degeneracy. The representation of the stabilizer defines the transformations of the spin degrees of freedom, the canonical example being a massive spin particle at rest carrying a representation of SU(2). Finally, the induction is the boosting of a rest state to arbitrary momentum. We see that this procedure is not only the mathematical means to construct the wanted representations, but provides insight in the physical nature of spin. It tells us that there is spin, because in general momentum is not sufficient to characterize a particle uniquely. It tells us what the symmetry structure of the spin degrees of freedom is, SU(2) in the massive case but ISO(2) in the massless case. And it tells us that momentum and spin are all possible exterior degrees of freedom of a particle. The physical line of thought described in the last paragraph relies on the sole assumption that the Poincar´e algebra describes the basic symmetry of spacetime. The q-deformed Poincar´e algebra has been constructed to describe the basic symmetry of q-deformed space time. Therefore, we can proceed in exactly the same manner to find out what q-deformed spin is. In Sect. 2 we review the q-Poincar´e algebra with focus on its general structure. In Sect. 3 we define the key properties of a useful q-deformed Pauli-Lubanski vector and present such a vector in Theorem 1. Its square yields the spin Casimir. Section 4 uses this q-Pauli-Lubanski vector to compute the q-little algebras for both the massive and the massless case. Throughout this article, it is assumed that q is a real number q > 1. We will frequently j −q −j use the abbreviations λ = q − q −1 and [j ] = qq−q −1 for a real number j , in particular [2] = q + q −1 . The lower case Greek letters µ, ν, σ , τ denote 4-vector indices running through {0, −, +, 3}. The upper case Roman letters A, B, C denote 3-vector indices running through {−1, 0, +1} = {−, 3, +}. 2. The q-Deformed Poincar´e Algebra The q-Poincar´e algebra can be defined very explicitly by listing its generators and the commutation relations between them. This has been done in Appendix A. Here, we give an overview of the more general algebraic structure. The q-Lorentz algebra H = Uq (sl2 (C)) is a Hopf ∗-algebra, with coproduct , counit ε, and antipode S. We will also use the Sweedler notation (h) = h(1) ⊗ h(2) . Several forms of the q-Lorentz algebra can be found in the literature, which are essentially equivalent [21–23]. Here, it is natural to use the form, where H is described as Drinfeld double of Uq (su2 ) with the opposite of its Hopf dual SUq (2)op [14], H = Uq (su2 )a SUq (2)op ,

(1)

that is, the Hopf ∗-algebra generated by the algebra of rotations Uq (su2 ) and the algebra of boosts SUq (2)op with cross commutation relations bl = l(1) , b(1) l(2) b(2) S(l(3) ), b(3)

(2)

Spin in the q-Deformed Poincar´e Algebra

331

for all l ∈ Uq (su2 ), b ∈ SUq (2)op , where l, b denotes the skew pairing. In addition to the Drinfeld-Jimbo generators E, F , K = q H of Uq (su2 ) we will also use the Casimir operator W and the 3-vector {JA } = {J− , J+ , J3 } of angular momentum as defined in Eqs. (45) and (44). The generators a, b, c, d of boosts form a multiplicative quantum matrix ( ac db ). H possesses two universal R-matrices, RI and RII , the first of which is ∗⊗∗ = R−1 = RII 21 . We often write in a Sweedler antireal R∗⊗∗ I I , the second is real RII like notation R = R[1] ⊗ R[2] . The q-Minkowski space algebra X = R1,3 q is generated by the 4-momentum vector {Pµ } = {P0 , P− , P+ , P3 } with relations νµ στ

Pµ Pν RI

= Pσ Pτ ⇔ Pµ Pν (RI−1 )µνσ τ = Pτ Pσ ,

(3)

µν

where the R-matrix RI σ τ = (RI[1] )µ σ (RI[2] )ν τ is the 4-vector representation of RI . The 4-momentum vector is the basis of this 4-vector representation of H, h Pν ≡ Pµ (h)µ ν , where is the representation map. Relations (3) are the only homogeneous commutation relations of X , which are consistent with this representation and which have the right commutative limit. Consistency means that X is a left H-module ∗-algebra, that is, h xx = (h(1) x)(h(2) x ) ,

(h x)∗ = (Sh)∗ x ∗

(4)

for all h ∈ H, x ∈ X . The q-Poincar´e algebra A is the Hopf semidirect product A = X H,

(5)

the ∗-algebra generated by the ∗-algebras X and H with cross commutation relations hx = (h(1) x)h(2) . More accurately, we have the following Definition 1. Let H be a Hopf ∗-algebra and X a left H-module ∗-algebra. The semidirect product X H is the ∗-algebra defined as the vector space X ⊗H with multiplication (x ⊗ h)(x ⊗ h ) := x(h(1) x ) ⊗ h(2) h

(6)

and ∗-structure (x ⊗ h)∗ = (1 ⊗ h∗ )(x ∗ ⊗ 1). We often abbreviate x ≡ x ⊗ 1 and h ≡ 1 ⊗ h. There is a left and a right Hopf adjoint action of H on A defined as adL h a := h(1) aS(h(2) ) ,

a adR h := S(h(1) )ah(2) .

(7)

The commutation relations (6) are precisely such that the left Hopf adjoint action of H on X equals the module action adL h x = h x. Let ρ be a finite representation of the q-Lorentz algebra H. We call a set of operators {Ti } a left or a right ρ-tensor operator if adL h Tj = Ti ρ(h)i j

or

Tj adL h = Ti ρ(S −1 h)i j

(8)

holds, respectively, for all h ∈ H. By definition of the q-Poincar´e algebra, the momenta Pµ form a left -tensor operator, that is, a left 4-vector operator.

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3. The q-Pauli-Lubanski Vector and the Spin Casimir 3.1. Defining properties of the Pauli-Lubanski vector. In the undeformed case one defines the Pauli-Lubanski (pseudo) 4-vector operator 1 Wµq=1 := − εµνσ τ Lνσ P τ , 2

(9)

where ε is the totally antisymmetric tensor, Lνσ the matrix of Lorentz generators, q=1 and P τ the momentum 4-vector. It is useful because each component of Wµ comν mutes with each component of P , from which follows that the 4-vector square W 2 = q=1 q=1 ηµν Wµ Wν is a Casimir operator. The eigenvalues of this Casimir operator are q=1 2 −m s(s + 1) where s is the spin. Therefore, Wµ can be viewed as square root of the spin Casimir. In the q-deformed case we can try to define Wµ by Eq. (9), as well, with the qdeformed versions of the epsilon tensor, of the matrix of Lorentz generators, and of the momenta. By construction, this definition yields a left 4-vector operator. But the square of this 4-vector does not commute with the momenta and, hence, it is not the searched-for spin Casimir. In general, the assumption that Wµ and Pν commute is not consistent with both, Wµ and Pν , being left 4-vector operators. Otherwise, the expression adL h [Wµ , Pν ] would have to vanish for all h ∈ H, that is,

! (10) Wµ Pν (h(1) )µ µ (h(2) )ν ν − (h(2) )µ µ (h(1) )ν ν = 0 . Since the coproduct is not cocommutative as in the undeformed case, this seems only possible for degenerate forms of Wµ . We can avoid this problem if we assume that the q-Pauli-Lubanski vector is a right 4-vector operator, making the following general observation: Proposition 1. Let a ∈ A = X H commute with X , [a, x] = 0 for all x ∈ X . Then a adR h also commutes with X for any h ∈ H. In other words, the centralizer of X is invariant under the right Hopf adjoint action of H. Proof. Let x ∈ X be any element of the quantum space. Then (a adR h) x = S(h(1) )ah(2) x = S(h(1) )a(h(2) x)h(3) = S(h(1) )(h(2) x)ah(3) = (S(h(1) )(1) h(2) x)S(h(1) )(2) ah(3) = (S(h(2) )h(3) x)S(h(1) )ah(4) = x (a adR h) for any h ∈ H.

(11)

This means that if a single component of a right vector operator commutes with all momenta, then the other components commute with all momenta, as well. Hence, the requirement that Pµ and Wν commute does no longer generate linear dependencies of type (10). We come to the following Definition 2. A set of operators Wµ ∈ A with the properties (PL1) Wµ is a right 4-vector operator, (PL2) each component Wµ commutes with all translations Pν ,

Spin in the q-Deformed Poincar´e Algebra q=1

(PL3) limq→1 Wµ = Wµ

333

as defined in (9),

is called a q-Pauli-Lubanski vector. Obviously, (PL1)-(PL3) do not determine Wµ uniquely. For example, we could multiply it by any q-polynomial which evaluates to 1 at q = 1. Property (PL1) tells us that the square of Wµ is a q-Lorentz scalar, that is, commutes with all h ∈ H. As a consequence of (PL2) this square commutes with all momenta. Therefore, (PL1) and (PL2) together guarantee that the square of a q-Pauli-Lubanski vector is a Casimir operator. The additional property (PL3) is the obvious requirement that Wµ be a q-deformation of the undeformed Pauli-Lubanski vector.

3.2. Constructing the q-Pauli-Lubanski vector. The only 4-vector operator we know so far is the 4-momentum Pµ . By construction, it is a left 4-vector operator. Being given a universal R-matrix, there is a generic way to construct a right tensor operator form of a given left tensor operator: Proposition 2. Let Tj be a left ρ-tensor operator, that is, adL h Tj = Ti ρ(h)i j for all h ∈ H, where ρ is a finite representation of H. Let R be a universal R-matrix of H. The set of operators

R (Tj ) := S 2 (R[1] )Ti ρ(R[2] )i j

(12)

is a right ρ-tensor operator. Proof. Abbreviating adL h Tj ≡ h Tj , we have

R (Tj ) adR h = S(h(1) )S 2 (R[1] )(R[2] Tj )h(2) = S(h(1) )S 2 (R[1] )h(3) S −1 (h(2) )R[2] Tj = S S(R[1] )h(1) h(3) S −1 (S(R[2] )h(2) ) Tj −1 = S h(2) S(R[1] ) h(3) S (h(1) S(R[2] )) Tj = S 2 (R[1] )S(h(2) )h(3) R[2] S −1 (h(1) ) Tj = S 2 (R[1] )(R[2] S −1 h Tj ) = R (Ti )ρ(S −1 h)i j . According to Eq. (8), R (Tj ) is indeed a right ρ-tensor operator.

(13)

This proposition tells us in particular, that R (Pµ ) satisfies (PL1). The next proposition takes care of (PL2). Proposition 3. Let Pµ be the momentum 4-vector, RI the antireal universal R-matrix of the q-Lorentz algebra, and be defined as in Proposition 2. Then [ RI (Pµ ), Pν ] = 0 , for all µ, ν.

[ R−1 (Pµ ), Pν ] = 0 I 21

(14)

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Proof. We denote the 4-vector representation by adL h Pν = h Pν = Pµ (h)µ ν . Recall that the commutation relations of the momenta can be written as νµ

Pµ Pν RI

στ

= Pσ Pτ

⇔

Pµ Pν (RI−1 )µν σ τ = Pτ Pσ ,

(15)

µν

where the R-matrix RI σ τ = (RI[1] )µ σ (RI[2] )ν τ is the 4-vector representation of the antireal universal R-matrix RI . Using the commutation relations between tensor operators and the Hopf algebra, we find Pσ RI (Pτ ) = Pσ S 2 (RI[1] )Pν (RI[2] )ν τ = S 2 (RI[1 ] ) S(RI[1] ) Pσ Pν (RI[2] RI[2 ] )ν τ −1 ν µ τ

= S 2 (RI[1 ] )Pµ (R−1 I[1] ) σ Pν (RI[2] ) τ (RI[2 ] ) τ

= S 2 (RI[1 ] ) Pµ Pν (RI−1 )µν σ τ (RI[2 ] )τ τ

= S 2 (RI[1 ] )Pτ Pσ (RI[2 ] )τ τ = RI (Pτ ) Pσ .

(16)

On the second and third line we have used (57), (R[1] ) ⊗ R[2] = R[1] ⊗ R[1 ] ⊗ −1 −1 R[2] R[2 ] , and S(R[1] ) ⊗ R[2] = R−1 [1] ⊗ R[2] . The calculations for RI → RI 21 are completely analogous. Propositions 2 and 3 tell us that both RI (Pµ ) and R−1 (Pµ ) satisfy properties I 21 (PL1) and (PL2), respectively. In order to check (PL3) we must find explicit expressions for RI (Pµ ) and R−1 (Pµ ). This amounts to calculating the L-matrices I 21

µ µ (L I+ ) ν := RI[1] (RI[2] ) ν ,

−1 −1 µ µ (L I− ) ν := RI[2] (RI[1] ) ν .

For the 4-vector of these L-matrices we find   1 0 0 0 1 1 0 b2 q 2 [2] 2 ab  a2   µ 1 1 (LI+ ) ν =  , c2 d2 q 2 [2] 2 cd  0 1 1 1 1 0 q 2 [2] 2 ac q 2 [2] 2 bd (1 + [2]bc)   W λK −1 J− λK −1 J+ W − K −1 −q −1 λJ+ 1 0 −q −1 λJ+  µ ,  (L I− ) ν =  −qλJ 0 1 −qλJ−  − λJ3 −λK −1 J− −λK −1 J+ λJ3 + K −1

(17)

(18a)

(18b)

A with respect to the basis {0, −, +, 3}. Observe that (L I+ ) B is the 3-dimensional corepop resentation matrix of SUq (2) . With a linear combination of these two L-matrices we can satisfy (PL3).

Theorem 1. The set of operators Wν := λ−1 R−1 (Pν ) − RI (Pν ) = λ−1 S 2 (L )µ ν − (L ) µ ν Pµ I− I+ I 21 is a q-Pauli-Lubanski vector in the sense of Definition 2.

(19)

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335

Proof. Properties (PL1) and (PL2) have been shown in Propositions 2 and 3, respectively. It remains to show (PL3). We note that the undeformed limit of the (left and right) Hopf adjoint action is the ordinary adjoint action. Hence, the limit q → 1 preserves tensor operators. Since a 4-vector is an irreducible tensor operator it is sufficient to examine the limit of one component only. The limits of the other components follow by application of the adjoint action. We choose the zero component for which we have to show that q→1

W0 = λ−1 (W − 1)P0 + JA PB g AB −→ W0

q=1

= JA PB g AB .

(20)

λ−1 (W − 1) = λ−1 [2]−1 [q −1 K + qK −1 + λ2 EF ] − 1 = λ−1 [2]−1 q −1 K + qK −1 − [2] + λ[2]−1 EF

(21)

All there is to show is that

vanishes for q → 1. Clearly, the λ[2]−1 EF term of the last line vanishes. Using K = q H we get for the other terms λ−1 (q −1 K + qK −1 − [2]) =

∞ (q + (−1)n q −1 )(ln q)n

λn!

n=1

Hn

[2](ln q)2 2 (ln q)3 3 H + H λ2! λ3! [2](ln q)4 4 + H + ... , λ4!

= ln q H +

which vanishes for q → 1, since limq→1 λ−1 (ln q)n = 0 for n > 1.

(22)

3.3. The Spin Casimir. We proceed to calculate the spin Casimir W τ Wτ = η τ ν Wν W τ

σ µ µ 2 σ = λ−2 ητ ν S 2 (L I− ) ν − (LI+ ) ν Pµ S (LI− ) τ − (LI+ ) τ Pσ σ µ µ σ = λ−2 ητ ν S 2 (L I− ) ν − (LI+ ) ν (LI− ) τ − (LI+ ) τ Pσ Pµ µ σ µ σ = λ−2 ητ ν S 2 (L I− ) ν (LI− ) τ + (LI+ ) ν (LI+ ) τ µ σ µ σ − (L I− ) ν (LI+ ) τ − (LI+ ) ν (LI− ) τ Pσ Pµ .

(23)

This can be further simplified. We first note that the commutation relations of the L-matrices are such that µσ µ

σ

ντ (LI+ ) ν (LI− ) τ RI σ µ Pσ Pµ µ σ ητ ν (L (24) I+ ) ν (LI− ) τ Pσ Pµ ,

−1 τ τν µ σ ητ ν (L I− ) ν (LI+ ) τ Pσ Pµ = η (RI )

=

ν

where in the second step we have used the commutation relations (3) of the momenta

and that (RI−1 )τ ν ντ ητ ν = ητ ν . Moreover, using Eq. (56) one can see that µ σ µσ . ητ ν (L I± ) ν (LI± ) τ = η

With the last two results Eq. (23) becomes µ σ W τ Wτ = 2λ−2 S 2 ηµσ − ητ ν (L I+ ) ν (LI− ) τ Pσ Pµ .

(25)

(26)

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4. The Little Algebras 4.1. Little algebras in the q-deformed setting. In classical relativistic mechanics the state of motion of a free particle is completely determined by its 4-momentum. In quantum mechanics particles can have an additional degree of freedom called spin: Let us assume we have a free relativistic particle described by an irreducible representation of the Poincar´e algebra. We pick all states with a given momentum, Lp := {|ψ ∈ L : Pµ |ψ = pµ |ψ} ,

(27)

where L is the Hilbert space of the particle and p = (pµ ) is the 4-vector of momentum eigenvalues. If the state of the particle is not uniquely determined by the eigenvalues of the momentum, then the eigenspace Lp will be degenerate. In that case we need, besides the momentum eigenvalues, an additional quantity to label the basis of our Hilbert space uniquely. This additional degree of freedom is spin. The spin symmetry is then the set of Lorentz transformations that leave the momentum eigenvalues invariant and, hence, act on the spin degrees of freedom only, Kp := {h ∈ H : Pµ h|ψ = pµ h|ψ for all |ψ ∈ Lp } ,

(28)

where H is the enveloping Lorentz algebra. In mathematical terms, Kp is the stabilizer of Lp . Clearly, Kp is an algebra, called the little algebra. A priori, there are a lot of different little algebras for each representation and each vector p of momentum eigenvalues. In the undeformed case it turns out that for the physically relevant representations (real mass) there are (up to isomorphism) only two little algebras, depending on the mass being either positive or zero [20]. For positive mass we get the algebra of rotations, U(su2 ), for zero mass an algebra which is isomorphic to the algebra of rotations and translations of the 2-dimensional plane denoted by U(iso2 ). The proof that Kp does not depend on the particular representation but on the mass does not generalize to the q-deformed case: If we define for representations of the q-Poincar´e algebra the little algebra as in Eq. (28), Kp for a spin- 21 particle will not be the same as for spin-1. We will therefore define the q-little algebras differently. In the undeformed case there is an alternative but equivalent definition of the little algebras. Kp is the algebra generated by the components of the q-Pauli-Lubanski vector as defined in Eq. (9) with the momentum generators replaced by their eigenvalues. Let us formalize this to see why this definition works and how it is generalized to the q-deformed case. Let χp be the map that maps the momentum generators to the eigenvalues, χp (Pµ ) = pµ . Being the restriction of a representation, χp must extend to a one dimensional ∗representation of the momentum algebra χp : X → C, a non-trivial condition only in the q-deformed case. Noting that every a ∈ A = X H can be uniquely written as a = i hi xi , where hi ∈ H and xi ∈ X , we can extend χp to a linear map on all of A by defining χˆ p : A → H as hi xi := hi χp (xi ). (29) χˆ p i

i

The little algebra can now be alternatively defined as the unital algebra generated by the images of the q-Pauli-Lubanski vector under χˆ p , Kp := Cχˆ p (Wµ ) .

(30)

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337

Why is this a reasonable definition? By construction the action of every element of A on Lp is the same as of its image under χˆ p . For any |ψ ∈ Lp this means Pµ χˆ p (Wν )|ψ = χˆ p (Pµ Wν )|ψ = χˆ p (Wν Pµ )|ψ = pµ χˆ p (Wν )|ψ ,

(31)

which shows that Kp ⊂ Kp . It still could happen that Kp is strictly smaller than Kp . In the undeformed case there are theorems [24, 25] telling us that this cannot happen, so we really have Kp = Kp . For the q-deformed case no such theorem is known [26]. However, if there were more generators in the stabilizer of some momentum eigenspace they would have to vanish for q → 1. In this sense Eq. (30) with the q-deformed Pauli-Lubanski vector can be considered to define the q-deformed little algebras.

4.2. Computation of the q-little algebras. To begin the explicit calculation of the q-deformed little algebras, we need to figure out if there are eigenstates of q-momentum at all. That is, we want to determine the one-dimensional ∗-representations of X = R1,3 q , that is the homomorphisms of ∗-Algebras χ : X → C. Let us again denote the eigenvalues of the generators by lower case letters pµ := χ (Pµ ). According to Eq. (52), we ∗ = −qp for χ to be a ∗-map. To find the conditions for must have p0 , p3 real and p+ − χ to be a homomorphism of algebras, we apply χ to the relations (51) of X , yielding pA (p0 − p3 ) = 0. There are two cases. The first is p0 = p3 , which immediately leads to pA = 0, and p0 = ±m. The second case is p0 = p3 , leading to m2 = −|p− |2 − |p+ |2 , where, if the mass m is to be real, we must have p± = 0. In summary, for real mass m we have a massive and a massless type of momentum eigenstate with eigenvalues given by (p0 , p− , p+ , p3 ) =

(±m, 0, 0, 0), m > 0 . (k, 0, 0, k), m = 0, k ∈ R

(32)

According to Eq. (30) we now have to replace the momenta in the definition (19) of the q-Pauli-Lubanski vector with these eigenvalues. For the massive case we get χˆ p (W0 ) = λ−1 (W − 1)m, χˆ p (W− ) = J− K −1 m, χˆ p (W+ ) = J+ K −1 m, χˆ p (W3 ) = λ−1 (W − K −1 )m ,

(33)

so the set of generators of the little algebra is essentially {W, K −1 , J± K −1 }. Since K −1 stabilizes the momentum eigenspace, so does its inverse K. Hence, it is safe to add K to the little algebra which would exist, anyway, within the h-adic extension. We thus get K(m,0,0,0) = Uq (su2 ) , completely analogous to the undeformed case.

(34)

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The massless case is more interesting. Replacing the momentum generators with (P0 , P− , P+ , P3 ) → (k, 0, 0, k) we get χˆ p (W0 ) = λ−1 (K − 1)k, 3

1

χˆ p (W− ) = −λ−1 q − 2 [2] 2 ack, 5

1

χˆ p (W+ ) = −λ−1 q 2 [2] 2 bdk, χˆ p (W3 ) = λ−1 K − (1 + [2]bc) k .

(35)

The set of generators of this little algebra is essentially {K, ac, bd, bc}. The commutation relations of these generators can be written more conveniently in terms of K and 3 NA := (L I+ ) A , that is 1

1

1

N− = q 2 [2] 2 ac ,

1

N+ = q 2 [2] 2 bd ,

N3 = 1 + [2]bc .

(36)

KNA = q −2A NA K ,

(37)

The commutation relations are NB NA ε AB C = −λNC ,

NA NB g BA = 1 ,

with conjugation NA∗ = NB g BA , K ∗ = K. In words: The NA generate the opposite op algebra of a unit quantum sphere, Sq∞ [27]. K, the generator of Uq (u1 ), acts on NA as on a right 3-vector operator. In total we have op

K(k,0,0,k) = Uq (u1 ) Sq∞ .

(38)

As opposed to the massive case, this is no Hopf algebra. However, since L-matrices are µ ν µ multiplicative, that is, [(L I+ ) σ ] = (LI+ ) ν ⊗ (LI+ ) σ , we have A (NB ) = NA ⊗ (L I+ ) B ,

(39)

hence, K(k,0,0,k) is a right coideal. 5. Conclusion We have determined the algebraic structures of the q-Poincar´e algebra which are important for the description of spin: the q-Pauli-Lubanski vector, the spin Casimir, the characters of the momentum subalgebra and their stabilizers, the q-deformed little algebras. The spin Casimir is particularly useful for the construction of irreducible spin representations of the q-Poincar´e algebra. In [22] massive spin representations were constructed. Within these representations the values of the q-deformed spin Casimir take the form W τ Wτ = −2[2]−1 m2 [s + 1][s]

(40)

for half integer s. This shows that our purely algebraic construction of the q-spin Casimir as a central element indeed yields a reasonable q-deformation of the spin Casimir operator of a representation. The construction of massive representations by induction is also in [22], while the construction of massless representations is work in progress.

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The q-deformed little algebras have proven their value for determining free wave equations on q-Minkowski space. In a representation theoretic approach a free q-relativistic wave equation must be such that the space of solutions is an irreducible representation of the q-Poincar´e algebra. The q-little algebras describe the symmetry of the restriction of q-wave equations to momentum eigenspaces as defined by characters of the momentum algebra. In [28] this approach was used to calculate the q-Dirac equation (including q-gamma matrices which satisfy a q-Clifford algebra), the q-Weyl equations, and the q-Maxwell equations. One intriguing point of the results obtained here is the form of the q-little algebra op for the massless case. By construction the algebraic limit q → 1 of Uq (u1 ) Sq∞ is the undeformed little algebra U(iso2 ), if the generators are identified appropriately. op However, from the viewpoint of representation theory Uq (u1 ) Sq∞ is compact – all irreducible ∗-representations are finite dimensional [22] – while its undeformed counterpart is non-compact. As a consequence, the unphysical continuous spin representations of massless particles are no longer present in the q-deformed setting. A. Appendix: The q-Poincar´e Algebra The Hopf ∗-algebra generated by E, F , K, and K −1 with relations KK −1 = 1 = K −1 K , KEK −1 = q 2 E , KF K −1 = q −2 F , [E, F ] = λ−1 (K − K −1 ) ,

(41)

(E) = E ⊗ K + 1 ⊗ E , (F ) = F ⊗ 1 + K −1 ⊗ F , (K) = K ⊗ K , ε(E) = 0 = ε(F ) , ε(K) = 1 , S(E) = −EK −1 , S(F ) = −KF , S(K) = K −1 ,

(42)

Hopf structure

and ∗-structure E ∗ = F K , F ∗ = K −1 E , K ∗ = K

(43)

is called Uq (su2 ), the q-deformation of the enveloping algebra U(su2 ). The set of generators {JA } = {J− , J3 , J+ } of Uq (su2 ) defined as 1

J− := q[2]− 2 KF, J3 := [2]−1 (q −1 EF − qF E), 1 J+ := −[2]− 2 E,

(44)

is the left 3-vector operator of angular momentum. The center of Uq (su2 ) is generated by W := K − λJ3 = K − λ[2]−1 (q −1 EF − qF E) ,

(45)

the Casimir operator of angular momentum. W is related to JA by W 2 − 1 = λ2 (J32 − q −1 J− J+ − qJ+ J− ) = λ2 JA JB g AB ,

(46)

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thus defining the 3-metric g AB, by which we raise 3-vector indices J A = g AB JB . It is also useful to define an ε-tensor ε−3 − = q −1 ,

ε 3− − = −q,

ε−+ 3 = 1,

ε +− 3 = −1,

ε3+ + = q −1 ,

ε +3 + = −q ,

ε 33 3 = −λ. (47)

The Hopf ∗-algebra generated by the 2 × 2-matrix of generators B i j = ( ac db ) with relations ba = qab, ca = qac, db = qbd, dc = qcd, bc = cb, da − ad = (q − q −1 )bc, da − qbc = 1 , (B i

coproduct ∗-structure

k)

ab S cd

=

Bi

j

⊗ Bj

k

(summation over j), counit

d −qb = , −q −1 c a

ab cd

∗

ε(B i

j)

=

δji ,

(48) antipode and

d −q −1 c , = −qb a

(49)

is SUq (2)op , the opposite algebra of the quantum group SUq (2). The Hopf ∗-algebra generated by the Hopf ∗-sub-algebras Uq (su2 ) and SUq (2)op with cross commutation relations 3 q −1 Eb ab qEa − q 2 b E= , 3 3 1 cd qEc + q 2 Ka − q 2 d q −1 Ed + q − 2 Kb 1 1 1 qF a + q − 2 c qF b − q − 2 K −1 a + q − 2 d ab F = , 5 cd q −1 F c q −1 F d − q − 2 K −1 c ab a q −2 b , (50) K=K 2 cd q c d which is the Drinfeld double of Uq (su2 ) and SUq (2)op , is the q-Lorentz algebra H = Uq (sl2 (C)) [14]. The ∗-algebra generated by P0 , P− , P+ , P3 with commutation relations P0 PA = PA P0 ,

PA PB ε AB C = −λP0 PC

(51)

and ∗-structure P0∗ = P0 ,

P−∗ = −q −1 P+ ,

P+∗ = −q,

P− ,

P3∗ = P3

(52)

1,3 is the q-Minkowski space algebra X = R1,3 q . The center of Rq is generated by the mass Casimir

m2 := Pµ Pν ηµν = P02 + q −1 P− P+ + qP+ P− − P32 ,

(53)

thus defining the 4-metric ηµν . It is related to the 3-metric by ηAB = −g AB for A, B ∈ {−, +, 3}. The commutation relations of X = R1,3 q are consistent with the 4-vector action h Pν = Pµ (h)ν µ of H on X . is defined on the generators of rotations as 0 0 0 ρ (JA ) 0 (JA ) = , (54) = 0 εA B C 0 ρ 1 (JA )

Spin in the q-Deformed Poincar´e Algebra

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where ρ 0 and ρ 1 are the spin-0 and the spin-1 representations of Uq (su2 ), respectively. On the boost generators is given by   [4][2]−2 0 0 qλ[2]−1 0 10 0   , (55a) (a) =  0 01 0  q −1 λ[2]−1 0 0 2[2]−1   0 −1 0 0 1 1 0 0 0 0 (b) = q − 2 λ[2]− 2  , (55b) 1 0 0 1 0 1 00   0 0 −1 0 1 1 1 0 0 1 (c) = −q 2 λ[2]− 2  , (55c) 0 0 0 0 00 1 0   −2 [4][2] 0 0 −q −1 λ[2]−1 0 10 0   (d) =  (55d)  0 01 0 −qλ[2]−1 0 0 2[2]−1 with respect to the {0, −, +, 3} basis. It has the property

ηνν (h)µ ν ηµ µ = (Sh)ν µ

(56)

for all h ∈ H. Finally, the q-Poincar´e algebra is the ∗-algebra generated by the q-Lorentz algebra H = Uq (sl2 (C)) and the q-Minkowski algebra X = R1,3 q with cross commutation relations µ (57) hPν = Pµ (h(1) )µ ν h(2) ⇔ Pν h = h(2) Pµ S −1 (h(1) ) ν . More details and mathematical background information has been compiled in [22]. Acknowledgement. This work was supported by the Studienstiftung des deutschen Volkes.

References 1. Born, M.: On the Quantum Theory of the Electromagnetic Field. Proc. Roy. Soc. London A143, 410 (1933) 2. March, A.: Die Geometrie kleinster R¨aume. Z. Phys. 104, 93 (1936) ¨ 3. Heisenberg, W.: Uber die in der Theorie der Elementarteilchen auftretende universelle L¨ange. Ann. Phys. 32, 20 (1938) 4. Snyder, H.S.: Quantized space-time. Phys. Rev. 71, 38 (1947) 5. Drinfeld, V.G.: Quantum groups. In: Proceedings of the International Congress of Mathematicians, A. M. Gleason, (ed.), Providence, RI: Amer. Math. Soc., 1986, pp. 798–820 6. Drinfeld, V.G.: Hopf algebras and the quantum Yang-Baxter equation. Sov. Math. Dokl. 32, 254 (1985) 7. Jimbo, M.: A q-analogue of U (g) and the Yang-Baxter equation. Lett. Math. Phys. 10, 63 (1985) 8. Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys. 111, 613 (1987) 9. Faddeev, L.D., Reshetikhin, N.Y., Takhtajan, L.A.: Quantization of Lie Groups and Lie Algebras. Leningrad Math. J. 1, 193 (1990) 10. Takeuchi, M.: Matrix Bialgebras and Quantum Groups. Israel J. Math. 72, 232 (1990)

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11. Manin, Y.I.: Quantum Groups and Non-Commutative Geometry. Montr´eal: Centre de Recherche Math´ematiques, 1988 12. Wess, J., Zumino, B.: Covariant Differential Calculus on the Quantum Hyperplane. Nucl. Phys. Proc. Suppl. 18B, 302 (1991) 13. Carow-Watamura, U., Schlieker, M., Scholl, M., Watamura, S.: Tensor Representation of the Quantum Group SLq (2) and Quantum Minkowski Space. Z. Phys. C48, 159 (1990) 14. Podles, P., Woronowicz, S.L.: Quantum deformation of Lorentz group. Commun. Math. Phys. 130, 381 (1990) 15. Carow-Watamura, U., Schlieker, M., Scholl, M., Watamura, S.: Quantum Lorentz group. Int. J. Mod. Phys. A6, 3081 (1991) 16. Schmidke, W.B., Wess, J., Zumino, B.: A q-deformed Lorentz algebra. Z. Phys. C52, 471 (1991) 17. Ogievetskii, O., Schmidke, W.B., Wess, J., Zumino, B.: Six generator q-deformed Lorentz algebra. Lett. Math. Phys. 23, 233 (1991) 18. Ogievetskii, O., Schmidke, W.B., Wess, J., Zumino, B.: q-Deformed Poincar´e algebra. Commun. Math. Phys. 150, 495 (1992) 19. Majid, S.: Braided momentum in the q-Poincar´e group. J. Math. Phys. 34, 2045 (1993), hepth/9210141 20. Wigner, E.P.: On Unitary Representations of the Inhomogeneous Lorentz Group. Annals. Math. 40, 149 (1939) 21. Rohregger, M., Wess, J.: q-deformed Lorentz-algebra in Minkowski phase space. Eur. Phys. J. C7, 177 (1999) 22. Blohmann, C.: Spin Representations of the q-Poincar´e Algebra. PhD thesis, Ludwig-MaximiliansUniversit¨at M¨unchen, 2001, math.qa/0110219 23. Kr¨ahmer, U.: The FRT-dual U (O(Gq ) O(Gq )) and the q-Lorentz Algebra. math.qa/0109157 24. Blattner, R.J.: Induced and Produced Representations of Lie Algebras. Trans. Am. Math. Soc. 144, 457 (1969) 25. Dixmier, J.: Enveloping Algebras. North-Holland Mathematical Library Vol. 14, Amsterdam: NorthHolland, 1977 26. Schneider, H.-J.: On Inner Actions of Hopf Algebras and Stabilizers of Representations. J. Algebra 165, 138 (1994) 27. Podles, P.: Quantum Spheres: Lett. Math. Phys. 14, 193 (1987) 28. Blohmann, C.: Free q-Deformed Relativistic Wave Equations by Representation Theory. Eur. Phys. J. C30, 435–445 (2003) Communicated by H. Araki

Commun. Math. Phys. 243, 343–387 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0964-8

Communications in

Mathematical Physics

Instanton Expansion of Noncommutative Gauge Theory in Two Dimensions L.D. Paniak1 , R.J. Szabo2 1 2

Michigan Center for Theoretical Physics, University of Michigan, Ann Arbor, MI 48109-1120, USA. E-mail: [email protected] Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK. E-mail: [email protected]

Received: 3 April 2002 / Accepted: 23 July 2003 Published online: 21 October 2003 – © Springer-Verlag 2003

Abstract: We show that noncommutative gauge theory in two dimensions is an exactly solvable model. A cohomological formulation of gauge theory defined on the noncommutative torus is used to show that its quantum partition function can be written as a sum over contributions from classical solutions. We derive an explicit formula for the partition function of Yang-Mills theory defined on a projective module for an arbitrary noncommutativity parameter θ which is manifestly invariant under gauge Morita equivalence. The energy observables are shown to be smooth functions of θ. The construction of noncommutative instanton contributions to the path integral is described in some detail. In general, there are infinitely many gauge inequivalent contributions of fixed topological charge, along with a finite number of quantum fluctuations about each instanton. The associated moduli spaces are combinations of symmetric products of an ordinary two-torus whose orbifold singularities are not resolved by noncommutativity. In particular, the weak coupling limit of the gauge theory is independent of θ and computes the symplectic volume of the moduli space of constant curvature connections on the noncommutative torus. Contents 1. Introduction and Summary . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Outline and summary of results . . . . . . . . . . . . . . . . . . . 2. Noncommutative Gauge Theory in Two Dimensions . . . . . . . . . . . 2.1 The noncommutative torus . . . . . . . . . . . . . . . . . . . . . 2.2 Gauge theory on the noncommutative torus . . . . . . . . . . . . 2.3 Gauge symmetry and area preserving diffeomorphisms . . . . . . 3. Localization of the Partition Function . . . . . . . . . . . . . . . . . . 3.1 Symplectic structure . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Hamiltonian structure . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Cohomological formulation of noncommutative Yang-Mills theory

. . . . . . . . . .

. . . . . . . . . .

344 346 348 348 350 353 354 355 356 358

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4. Classification of Instanton Contributions . . . . . . . . . . . . . . . . . . . 4.1 Heisenberg modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Stationary points of noncommutative gauge theory . . . . . . . . . . 5. Yang-Mills Theory on a Commutative Torus . . . . . . . . . . . . . . . . . 6. Yang-Mills Theory on a Noncommutative Torus: Rational Case . . . . . . . 6.1 Gauge Morita equivalence . . . . . . . . . . . . . . . . . . . . . . . 6.2 The partition function for rational θ . . . . . . . . . . . . . . . . . . 6.3 Relation between commutative and rational noncommutative gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Yang-Mills Theory on a Noncommutative Torus: Irrational Case . . . . . . 8. Smoothness in θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Graphical determination of classical solutions . . . . . . . . . . . . . 8.2 Proof of θ-smoothness . . . . . . . . . . . . . . . . . . . . . . . . . 9. Instanton Moduli Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Weak coupling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Instanton partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

361 362 364 367 370 371 372 373 373 375 376 377 378 379 381 382

1. Introduction and Summary Quantum field theories on noncommutative spacetimes provide field theoretical contexts in which to study the dynamics of D-branes, while at the same time retaining the nonlocality inherent in string theory (see [1–3] for reviews). Recent studies of these field theories have raised many questions regarding their existence and properties, and even after extensive study there remain numerous questions concerning the new phenomena they exhibit even in the simplest cases. Of particular interest is Yang-Mills theory defined on a noncommutative torus which serves as an effective description of open strings propagating in flat backgrounds. In particular, noncommutative gauge theory on a two-dimensional torus describes codimension two vortex bound states of D-branes inside D-branes. In this paper we will show that this quantum field theory is exactly solvable and explicitly evaluate its partition function. Various non-trivial aspects of noncommutative gauge theories in two dimensions may be found in [4–10] The commutative version of this theory has a well-known history as an exactly solvable model, which gives the first example of a confining gauge theory whose infrared limit can be reformulated analytically as a string theory (see [11, 12] for reviews). The key feature of two dimensions is that there are no gluons and the theory must be investigated on spacetimes of non-trivial topology or with Wilson loops in order to see any degrees of freedom. This suppression of degrees of freedom owes to the fact that the group of local symmetries of two-dimensional Yang-Mills theory contains not only local gauge invariance, but also invariance under area-preserving diffeomorphisms. Of the several different methods for solving this quantum field theory, a particularly fruitful approach is provided by the lattice formulation [13]. Using the area-preserving diffeomorphism invariance, the heat kernel expansion of the disk amplitude may be interpreted as a wavefunction for a plaquette. The fusion rules for group characters allow one to glue together disconnected plaquettes. The basic plaquette Boltzmann weight in this way turns out to be renormalization group invariant [14], so that the lattice gauge theory reproduces exactly the continuum answer. While a lattice formulation of noncommutative Yang-Mills theory does exist [15], it does not exhibit an obvious self-similarity property as its commutative counterpart

Instanton Expansion of Noncommutative Gauge Theory in Two Dimensions

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does. The non-locality of the star-product mixes the link variables in the lattice action and the theory no longer has the nice Gaussian form that its commutative limit does. While under certain circumstances Morita equivalence can be used to disentangle the lattice star-product by mapping the noncommutative lattice gauge theory onto a commutative one, the continuum limit always requires a complicated double scaling limit to be performed with small lattice spacing and large commutative gauge group rank N , in order that the scale of noncommutativity θ remain finite in the continuum limit. A similar approach to solving gauge theory on the noncommutative plane has been advocated recently in [10]. Nevertheless, the lattice theory at finite N can be solved explicitly by mapping it onto a unitary two-matrix model [16], whose path integral can be reduced to a well-defined sum over integers [17]. This proves that the lattice model is exactly solvable, and thereby gives a strong indication that noncommutative gauge theory in two dimensions is a topological field theory (with no propagating degrees of freedom). However, such an approach, like canonical quantization in the commutative case, is based almost entirely on the representation theory of the gauge group. This group is a somewhat mysterious object in noncommutative gauge theory whose full properties have not yet been unveiled. This infinite-dimensional Lie group is analyzed in [18–27] and it involves a non-trivial mixing of colour degrees of freedom with spacetime diffeomorphisms. A related difficulty arises in the diagonalization approach which requires fixing a gauge symmetry locally [28]. The resulting Faddeev-Popov functional determinants are difficult to analyze in the noncommutative setting. Hamiltonian methods are likewise undesirable because of problems associated with non-localities in time. An approach which doesn’t rely on the (unknown) features of the noncommutative gauge group is thereby desired. We will see, however, that the basic geometric structure underlying this gauge group implies that the noncommutative theory is still invariant under area-preserving diffeomorphisms of the spacetime (though in a much stronger manner) and is thereby an exactly solvable model. As we shall demonstrate, one technique of solving commutative U (N ) Yang-Mills theory which continues to be useful in the noncommutative case is that of nonAbelian localization [29]. This method takes advantage of the fact that in two dimensions a gauge fixed Yang-Mills theory is essentially a cohomological quantum field theory. A judicious deformation of the action by cohomologically exact terms allows one to reduce the quantum path integral defining the partition function to a sum over a discrete set of points which are in one-to-one correspondence with the critical points of the Yang-Mills action. Of course, these critical points are given by gauge field configurations which solve the classical equations of motion. Even though these solutions may be unstable, we will refer to any such configuration as an instanton. As a consequence, the quantum partition function can be evaluated as a sum over all instanton configurations of the gauge theory. In other words, the semi-classical approximation to this field theory is exact, provided that one sums over all critical points of the action. The feature which makes this approach work is the interpretation of noncommutative Yang-Mills theory as ordinary Yang-Mills theory (on a noncommutative space) with its infinite dimensional gauge symmetry group that is formally some sort of large N limit of U (N ). In what follows we will derive an exact, nonperturbative expression for the partition function of quantumYang-Mills theory defined on a projective module over the noncommutative two-torus. Using a combination of localization techniques and Morita duality, we are able to give an explicit formula written as the sum of contributions from the vicinity of instantons. The instantons themselves are parameterized by a collection of lists of pairs of integers (p , q ) ≡ (pk , qk ) which arise from partitions of the topological k≥1

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numbers (p, q) of the projective module on which the gauge theory is defined. The result for the partition function Zp,q is then given as a sum, over all partitions, of terms involving the Boltzmann weights of the noncommutative Yang-Mills action S(p , q ; θ) evaluated at its extrema, along with prefactors W (p , q ; θ) which describe the quantum fluctuations about each instanton configuration. Schematically, we have Zp,q = W (p , q ; θ ) e −S(p ,q ;θ) . (1.1) partitions

We will show that the full expression (1.1) is explicitly invariant under gauge Morita equivalence and that it is a smooth function of the noncommutativity parameter θ. The formalism which we develop in this paper gives the tools necessary to explore and answer all questions about two-dimensional noncommutative Yang-Mills theory, and it gives a model which should capture some features of the more physical higherdimensional theories, but within a much simplified setting. For example, the techniques developed here can be used to learn more about the observables of Yang-Mills theory on the noncommutative torus. The evaluation of the partition function as a sum of contributions from instantons is of course familiar from commutative Yang-Mills theory [30–33]. In that case there exists an equivalent expression via Poisson resummation which is interpreted as a sum over irreducible representations of the gauge group. For Yang-Mills theory on a noncommutative torus we have not been able to find an analogous group theoretical expansion though we believe it would give great insight into the representation theory of the noncommutative gauge group on the two-dimensional torus. The Yang-Mills action can be thought of as defining invariants of the star-gauge group, and the discrete sums over instantons as labelling its representations. The discrete nature of the action is necessary for it to be a Morse function and hence a candidate for the localization formalism [34], and it suggests that the noncommutative gauge group is compact. We expect to report on progress in understanding the details of the noncommutative gauge group on the torus in the near future. 1.1. Outline and summary of results. In the next section we shall begin with a review of the construction of gauge connections and Yang-Mills theory on the two-dimensional noncommutative torus. We include a brief discussion on the area-preserving nature of the noncommutative gauge symmetry which suggests that there are no local degrees of freedom in the noncommutative gauge theory, only global ones as in the commutative case. In Sect. 3 we give an overview of non-Abelian localization and how it applies to the evaluation of the quantum partition function of two-dimensional Yang-Mills theory on the noncommutative torus. We pay particular attention to rewriting the formalism in a manner which does not rely on the details of the noncommutative gauge group. We show in detail how the Yang-Mills action defines a system of Hamiltonian flows which coincide with the Lie algebra action of the group of noncommutative gauge transformations. This compatibility allows us to formally reduce the path integral defining the quantum partition function to a discrete sum. The procedure is applicable to Yang-Mills theory defined on a noncommutative torus with any value of the noncommutativity parameter θ , including vanishing, rational or irrational θ . The localization of the path integral is onto gauge field configurations which are solutions of the classical equations of motion and provide critical points of the YangMills action. In order to characterize these solutions and the spaces in which they are defined, in Sect. 4 we begin by giving a brief description of finitely-generated projective

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(Heisenberg) modules over the noncommutative torus. We characterize all such classical solutions of Yang-Mills theory defined on a projective module for any value of the noncommutativity parameter in terms of partitions of the topological numbers of the projective module. These results serve to bridge previous constructions of classical solutions for two-dimensional Yang-Mills theory in the commutative case [33, 35] and in the noncommutative case for irrational θ [36]. In order to obtain explicit results for the partition function, in Sect. 5 we revisit YangMills theory on the commutative torus and re-interpret the well-known evaluation of the quantum partition function in this case in terms of projective modules. In doing so we will find it necessary to make a distinction between the commonly known “physical” definition of two-dimensional Yang-Mills theory and a “module” definition where we restrict gauge field configurations to have a particular Chern (twist) number. The physical theory can then be recovered by summing over all such cohomological sectors. Given the partition function of ordinary Yang-Mills theory on the torus written in terms of projective modules, in Sect. 6 we use Morita equivalence to construct a mapping from the commutative theory to one with rational values of the noncommutativity parameter θ . Discarding the scaffolding of Morita equivalence, the result is an explicit expression for the quantum partition function of noncommutative Yang-Mills theory defined on a projective module with rational θ purely by the topological numbers of the module. Our construction also provides a more transparent interpretation of the Morita equivalence of Yang-Mills theories on commutative tori and ones with rational values of θ. By exploiting the fact that the localization arguments hold irrespective of the particular value of θ, in Sect. 7 we propose a formula for the partition function at irrational values of θ by natural extension of the rational case. We give strong arguments in favour of this conjecture. The two independent constructions of this formula come from Morita equivalence, whereby the Morita invariant commutative partition function determines exactly the rational noncommutative one, and localization theory, which proves that the partition function is given by a sum over classical solutions for any θ. Further support for this proposal is provided by rational approximations to the irrational noncommutative gauge theory. We will find that the schematic expression (1.1) may be written explicitly as 3 −νa /2 (−1)νa g 2 A Zp,q = p − q θ a a νa ! 2π 2 partitions a≥1   2 2 q 2π q k  , (1.2) − × exp − 2 (pk − qk θ ) g A pk − q k θ p − qθ k≥1

where g is the Yang-Mills coupling constant and A is the area of the torus. The integer νa is the number of partition components (pk , qk ) which have the same distinct values of the quantity pa − qa θ . The sign factor in (1.2) is determined by a Morse index which measures the overall contribution from unstable modes in a given instanton configuration (p , q ). The exponential prefactors are the Gaussian fluctuation determinants, weighted with the appropriate permutation symmetry factors νa ! associated with a partition. From (1.2) we see that the area dependence of the noncommutative gauge theory is similar to that of the commutative case. If A → ∞ for fixed g and θ , then the theory is exponentially dominated by trivial instanton configurations. Essentially the energy of electric flux in the noncommutative theory is still proportional to the length of the flux line, and so the overall details of the dynamics (or lack thereof) are the same as in commutative

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Yang-Mills theory. Thus, in direct analogy to the commutative situation, the gauge theory on the noncommutative plane is essentially trivial. In Sect. 8 we develop a graphical method of analyzing the instanton contributions to Yang-Mills theory which works for θ irrational, rational or vanishing. This graphical approach is applied to the universal expression (1.2) for the partition function to show that the vacuum energy, along with a certain class of topological observables, of YangMills theory on the noncommutative torus are smooth functions of θ. Finally, in Sect. 9 we end with a description of the moduli spaces of classical solutions of Yang-Mills theory on the noncommutative torus. The partition function in the weak coupling limit agrees with that of the commutative gauge theory, except that now it formally computes the symplectic volume of the moduli space of all (not necessarily flat) constant curvature gauge connections on the torus. The rearrangement of the series (1.2) into distinct gauge inequivalent instanton configurations is described. They are determined by rearranging the critical partition components (pk , qk ) into distinct relatively prime pairs (pa , qa ) of topological numbers with (pa , qa ) = Na (pa , qa ). We will see that the moduli space of such gauge orbits is given by ˜2 , Mp,q = SymNa T (1.3) a≥1

˜ 2 . This ˜ 2 is the symmetric product of a certain dual, ordinary two-torus T where SymNa T N 2 ˜ generalizes the moduli space Sym T of flat gauge connections in commutative U (N ) gauge theory. The instanton moduli space (1.3) has a natural physical interpretation in terms of that for a collection of distinct configurations of Na free indistinguishable D0-branes in codimension two. In particular, the point-like instanton singularities are not resolved by noncommutativity. We will show how the orbifold singularities of (1.3) can be used to systematically construct the gauge inequivalent contributions to YangMills theory. Such an explicit classification is only possible within the noncommutative setting. We shall find that, like for the instanton contributions to ordinary Yang-Mills theory, there are a finite number of quantum fluctuations about each gauge inequivalent classical solution. In contrast to the commutative case, however, for irrational θ there are infinitely many distinct instanton contributions to the path integral for fixed quantum numbers (p, q). 2. Noncommutative Gauge Theory in Two Dimensions To set notation and conventions, we will start by reviewing some well-known facts about Yang-Mills theory on a noncommutative two-torus [1, 37, 38]. Our presentation will exhibit the interplay between the physical, quantum field theoretical approach and the mathematical approach within the framework of noncommutative geometry, as both descriptions will be fruitful for our subsequent analysis in later sections. We will also give the first indication that this theory is exactly solvable. For simplicity, we consider a square torus of radii R. 2.1. The noncommutative torus. The noncommutative two-torus may be defined as the abstract, noncommutative, associative unital ∗-algebra generated by two unitary operators Zˆ 1 and Zˆ 2 with the commutation relation Zˆ 1 Zˆ 2 = e 2π i θ Zˆ 2 Zˆ 1 ,

(2.1)

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where θ is the real-valued, dimensionless noncommutativity parameter. Unless otherwise specified, we will assume that θ ∈ (0, 1) is an irrational number. The “smooth” completion Aθ of the algebra generated by Zˆ 1 and Zˆ 2 consists of the power series fˆ =

∞

∞

m1 =−∞ m2 =−∞

f(m1 ,m2 ) e π i θ m1 m2 Zˆ 1m1 Zˆ 2m2 ,

(2.2)

where the coefficients f(m1 ,m2 ) are Schwartz functions of (m1 , m2 ) ∈ Z2 , i.e. f(m1 ,m2 ) → 0 faster than any power of |m1 | + |m2 | as |m1 | + |m2 | → ∞. The phase factor in (2.2) is inserted to symmetrically order the operator product. There are natural, anti-Hermitian linear derivations ∂ˆ1 and ∂ˆ2 of the algebra Aθ which are defined by the commutation relations

(2.3) ∂ˆ1 , ∂ˆ2 = i · I ,

i ∂ˆi , Zˆ j = δij Zˆ j , i, j = 1, 2 , (2.4) R where ∈ R can be interpreted as a background magnetic flux and I is the unit of Aθ . From (2.3) it follows that the Heisenberg Lie algebra L acts on Aθ by infinitesimal ˆ automorphisms. This action defines a Lie algebra homomorphism X → ∂X , X ∈ L ,

i.e. ∂ˆX , ∂ˆY = ∂ˆ[X,Y ] , yielding a linear map ∂ˆ : Aθ −→ Aθ ⊗ L∗ .

(2.5)

The unique normalized trace on Aθ is given by projection onto zero modes as Tr fˆ = f(0,0) ,

(2.6)

which defines a positive linear functional Aθ → C, i.e. Tr fˆ† fˆ ≥ 0 for any fˆ ∈ Aθ . The trace (2.6) satisfies Tr fˆ† = Tr fˆ, and it is invariant under the action of the Lie algebra L of automorphisms of Aθ , i.e.

(2.7) Tr ∂ˆi , fˆ = 0 . The conventional field theoretic approach employs a “dual” description to this analytic one in terms of functions on an ordinary torus T2 . Let x 1 , x 2 ∈ [0, 2π R] be the coordinates of T2 . Then given any element fˆ ∈ Aθ with series expansion of the form (2.2), we can use the Schwartz sequence f(m1 ,m2 ) to define a smooth function on the torus by the Fourier series f (x) =

∞

∞

m1 =−∞ m2 =−∞

f(m1 ,m2 ) e

i mi x i /R

.

(2.8)

This establishes a one-to-one correspondence between elements of the abstract algebra Aθ and elements of the algebra C ∞ (T2 ) of smooth functions on the torus. Under this correspondence, the noncommutativity of Aθ is encoded in the multiplication relation fˆ gˆ = f g ,

(2.9)

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where the star-product is given by (f g)(x) =

∞

−π i R 2 θ

n=0

n n r=0

(−1)r r n−r ∂1 ∂2 f (x) ∂1n−r ∂2r g(x) (2.10) (n − r)! r!

with ∂i = ∂/∂x i . In addition, the actions of the derivations (2.4) correspond to ordinary differentiation of functions,

(2.11) ∂ˆi , fˆ = ∂ if , while the canonical normalized trace (2.6) can be represented in terms of the classical average of functions over the torus, 1 Tr fˆ = d2 x f (x) . (2.12) 4π 2 R 2 Integration by parts also shows that d2 x (f g)(x) = d2 x f (x) g(x) .

(2.13)

Here and in the following, unless specified otherwise, all coordinate integrations extend over T2 .

2.2. Gauge theory on the noncommutative torus. In the noncommutative setting, the generalizations of vector bundles are provided by projective modules, which are vector spaces on which the algebra is represented. Let E be a finitely-generated projective module over the algebra Aθ . We consider only right modules in the following. The free ˆ ˆ ˆ module AM θ = Aθ ⊕ · · · ⊕ Aθ consists of M-tuples ξ = (f1 , . . . , fM ) of elements fˆa ∈ Aθ . It is the analog of a trivial vector bundle. Let P ∈ MM (Aθ ) be a projector with 2 † E = P AM θ , P =P=P ,

(2.14)

where MM (Aθ ) = Aθ ⊗ MM is the algebra of M × M matrices with entries in the algebra Aθ , whose multiplication is the tensor product of the multiplication in Aθ with ordinary matrix multiplication. Alternatively, we may consider E as the subspace of ˆ ˆ elements ξˆ ∈ AM θ with P ξ = ξ . The endomorphism algebra EndAθ (E) = E ∗ ⊗Aθ E of the module E is the algebra of linear maps E → E that commute with the right action of Aθ on E. It is isomorphic ˆ This to the subalgebra of Aθ -valued matrices Aˆ ∈ MM (Aθ ) which obey P Aˆ P = A. means that the identity operator on E can be identified with the projector, IE = P. To simplify some of the formulas which follow, we shall frequently refrain from writing IE explicitly. Let N ≤ M be the largest integer such that the module E can be represented as a direct sum E = E ⊕ · · · ⊕ E of N isomorphic Aθ -modules. Then EndAθ (E ) ∼ = Aθ is also a noncommutative torus [37], where θ is the dual noncommutativity parameter which depends on θ and the projective module E, so that EndAθ (E) ∼ = MN (Aθ ) .

(2.15)

Instanton Expansion of Noncommutative Gauge Theory in Two Dimensions

The derivations ∂ˆi naturally extend to operators on AM θ via the definition

∂ˆi , ξˆ = ∂ˆi , fˆ1 , . . . , ∂ˆi , fˆM

351

(2.16)

ˆ for ξˆ = (fˆ1 , . . . , fˆM ) ∈ AM θ . Then P ◦ ∂i ◦ P is a linear derivation on E. The trace Tr on Aθ also naturally extends to a trace on EndAθ (E) defined by Tr E = Tr ⊗ tr M ,

(2.17)

where tr M is the usual M ×M matrix trace. On E there is a natural Aθ -valued inner product which is compatible with the Aθ -module structure of E and is defined on M-tuples ξˆ = (fˆ1 , . . . , fˆM ) and ηˆ = (gˆ 1 , . . . , gˆ M ) by

The object

ξˆ , ηˆ

Aθ

=

M

fˆa† gˆ a .

(2.18)

a=1

ξˆ , ηˆ = Tr ξˆ , ηˆ

(2.19)

Aθ

then defines an ordinary Hermitian scalar product E × E → C. This turns E into a separable Hilbert space. We will present the explicit classification of the projective modules over the noncommutative torus in Sect. 4.1. We now define a connection on a module E over the noncommutative torus to be a pair of linear operators ∇ˆ 1 , ∇ˆ 2 : E → E satisfying

i ∇ˆ i , Zˆ j = δij Zˆ j , i, j = 1, 2 , (2.20) R where in this equation the Zˆ j are regarded as operators E → E representing the right action on E of the corresponding generators of Aθ . When acting on elements of E, the requirement (2.20) is just the usual Leibnitz rule with respect to the derivations ∂ˆ1 and ∂ˆ2 . In an analogous way to these operators, there is a linear map X → ∇ˆ X , X ∈ L , which defines a vector space homomorphism ∇ˆ : E −→ E ⊗C L∗ .

(2.21)

This definition makes use of the bimodule structure on Aθ ⊗ L∗ . From the definitions (2.4) and (2.20) it follows that an arbitrary connection ∇ˆ i can be expressed in the form ∇ˆ i = ∂ˆi + Aˆ i ,

(2.22)

where Aˆ i ∈ EndAθ (E) are N × NAθ -valued matrices which we will refer to as gauge fields. We stress that here, and below, the quantity ∂ˆi is implicitly understood as the operator P ◦ ∂ˆi ◦ P on AM θ → E. The same is true of similarly defined objects. In the following we shall work only with connections which are compatible with the inner product (2.18), i.e. those which satisfy ∇ˆ i ξˆ , ηˆ (2.23) + ξˆ , ∇ˆ i ηˆ = ∂ˆi , ξˆ , ηˆ Aθ

Aθ

Aθ

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for any ξˆ , ηˆ ∈ E. The compatibility condition (2.23) implies that ∇ˆ i is an anti-Hermitian operator

with respect to the scalar product (2.19). It also implies that its curvature ˆ ˆ ∇1 , ∇2 , which is a two-form on the Heisenberg algebra L with values in the space of linear operators on E, commutes with the action of Aθ on E, and hence takes values in the 1 space EndH Aθ (E) of anti-Hermitian endomorphisms of E. The space of all compatible connections on a module E will be denoted by C(E). From (2.20) and (2.23) it follows that C(E) is an affine space over the vector space of linear maps L → EndH Aθ (E). In this paper we will be interested in evaluating the partition function of two dimensional quantum Yang-Mills theory on the noncommutative torus, which is defined formally by the infinite-dimensional integral 1 ˆ 2 Z(g , θ, , E) = DAˆ e −S A , (2.24) vol G(E) C(E )

where the Yang-Mills action on C(E) is defined for an arbitrary connection (2.22) by

2π 2 R 2

2 ˆ 1 , ∇ˆ 2 , S Aˆ = S ∇ˆ = Tr ∇ E g2

(2.25)

with g the Yang-Mills coupling constant of unit mass dimension. The area factor 4π 2 R 2 is inserted to make the action dimensionless. Here G(E) is the group of gauge transformations, which will be described in the next subsection, and vol G(E) is its volume. The ˆ and also the volume vol G(E), will be defined more precisely in Sect. 3. measure DA, By using the operator-field correspondence of the previous subsection, we can express (2.24) in a more standard quantum field theoretical form as the Euclidean Feynman path integral 1 Z(g 2 , θ, , E) = DA e −S[A] , (2.26) vol G(E) C(E )

where

2 d2 x tr N FA (x) + · IE

(2.27)

FA = ∂1 A2 − ∂2 A1 + A1 A2 − A2 A1 ,

(2.28)

S[A] =

1 2g 2

with

the noncommutative field strength of the anti-Hermitian U (N ) gauge field Ai . The multiplication in (2.28) is the tensor product of the associative star-product (2.10), defined with θ replaced by its dual θ , and ordinary matrix multiplication. This extended starproduct is still associative. 1 Usually one would define the curvature to be a measure of the deviation of the mapping X → ∇ ˆX from being a homomorphism of the Lie algebra (2.3) of automorphisms of Aθ . This means that the curvature should be defined as ∇ˆ 1 , ∇ˆ 2 − · IE . However, later on we will wish to work with an action which is explicitly invariant under Morita duality, which can only be accomplished with the definition of curvature given in the text. This change of convention is mathematically harmless since it corresponds to a shift of the curvature by the central element of the Heisenberg algebra. Physically, it will only add constants to the usual gauge theory action and so will not affect any local dynamics, only topological aspects.

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2.3. Gauge symmetry and area preserving diffeomorphisms. Let us now describe the symmetries of the noncommutative Yang-Mills action (2.25). It is invariant under any covariant transformation of the gauge connection of the form ∇ˆ i −→ Uˆ ∇ˆ i Uˆ † ,

(2.29)

where Uˆ ∈ EndAθ (E) is a unitary endomorphism of the projective module E, Uˆ † Uˆ = Uˆ Uˆ † = IE ,

(2.30)

which determines an inner automorphism of the right action of Aθ on E. In other words, Uˆ ∈ UN (Aθ ), where UN (Aθ ) is the group of unitary elements of the algebra MN (Aθ ). These gauge transformations comprise operators of the form Uˆ = IE + Kˆ ,

(2.31)

where Kˆ lies in an appropriate completion of the algebra of finite rank endomorphisms of ˆ ηˆ ∈ E, let 25]. These latter endomorphisms are defined as follows. For any η, E [24, ηˆ ηˆ be the operator defined by ηˆ ηˆ ξˆ = ηˆ ηˆ , ξˆ (2.32) Aθ

for ξˆ ∈ E, with adjoint ηˆ ηˆ . The Aθ -linear span of endomorphisms of the form (2.32) forms a self-adjoint two-sided ideal in EndAθ (E). Since, as mentioned before, E is a separable Hilbert space, this ideal is isomorphic to the infinite-dimensional algebra M∞ of finite rank matrices. Its operator norm closure is the algebra End∞ Aθ (E) of compact endomorphisms of the module E. The Schwartz restriction on the expansion (2.2) implies that elements fˆ ∈ Aθ act as compact operators on E [26]. Therefore, in (2.29) we should restrict to those unitary endomorphisms (2.31) with Kˆ ∈ End∞ Aθ (E). We denote this infinite dimensional Lie group by U ∞ (E). It is the operator norm completion of the infinite unitary group U (∞) obtained by taking Kˆ to be a finite rank endomorphism. 2 By Palais’ theorem [39], these two unitary groups have the same homotopy type, and their homotopy groups are determined by Bott periodicity as Z , k odd , ∞ πk U (E) = πk U (∞) = (2.33) 0 , k even . In particular, the gauge symmetry group is connected. It should be pointed out here that this is only a local description of the full gauge group of noncommutativeYang-Mills theory. The group of connected components of G(E) acts on the gauge orbit space, obtained by quotienting C(E) by the action of the group G0 (E) of smooth maps T2 → U ∞ (E), as a global symmetry group [1, 25, 27]. By using (2.22), one finds that the infinitesimal form of the gauge transformation rule (2.29) is Aˆ i → Aˆ i + δλˆ Aˆ i , where

(2.34) δλˆ Aˆ i = − ∂ˆi , λˆ + λˆ , Aˆ i 2 The gauge group can also be chosen to be smaller than U ∞ (E ) by completing U (∞) in other Schatten norms [25–27]. The various choices all have the same topology and group theory, and so we shall work for definiteness with only the compact unitaries defined above.

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and λˆ is an anti-Hermitian compact operator on E. In terms of gauge potentials on the ordinary torus T2 this reads δλ Ai = −∂i λ + λ Ai − Ai λ ,

(2.35)

where λ(x) is a smooth, anti-Hermitian N × N matrix-valued field on T2 . The noncommutative gauge transformations (2.35) mix internal, U (N ) gauge degrees of freedom with general coordinate transformations of the torus. Their geometrical significance has been elucidated in [26] by exploiting the relationship between appropriate completions of U (∞) and canonical transformations. The Lie algebra of noncommutative gauge transformations (2.35) is equivalent to the Fairlie-Fletcher-Zachos trigonometric deformation [40] of the algebra w∞ (T2 ) of area-preserving diffeomorphisms of T2 . Therefore, the gauge symmetry group of noncommutative Yang-Mills theory in twodimensions consists of area-preserving diffeomorphisms, which “almost” makes it a topological field theory. Its gauge symmetry “almost” coincides with general covariance, thereby killing most of its degrees of freedom. From this feature we would expect the theory to contain no local propagating degrees of freedom, and hence to be exactly solvable. This reasoning is further supported by the Seiberg-Witten map [41] and the exact solvability of ordinary, commutative Yang-Mills gauge theory in two dimensions. Note however that the topological nature here is quite different than that of the commutative case, because in the noncommutative setting it arises due to the gauge symmetry of the theory, i.e. an inner automorphism of the algebra of functions, while in the commutative case it corresponds to an outer automorphism which preserves the local area element 4π 2 R 2 d2 x. For this reason, the partition function will only depend on the dimensionless combination 4π 2 g 2 R 2 of the Yang-Mills coupling constant and the area of the surface. This fact makes it difficult to make sense of the theory on a non-compact surface. In contrast, this argument breaks down for noncommutative tori of dimension larger than two. In any even dimension the transformations (2.35) generate symplectic diffeomorphisms [26], i.e. coordinate transformations which leave the symplectic two-form of the torus invariant. These are the diffeomorphisms which preserve the Poisson bivector defining the star-product in (2.10). In general, these transformations generate a group that is much smaller than the group of volume-preserving diffeomorphisms. In the D-brane interpretation, this latter group would be the natural worldvolume symmetry group of a static brane. However, particular to the two-dimensional case is the fact that canonical transformations and area-preserving diffeomorphisms are the same. 3. Localization of the Partition Function The path integral (2.24) for quantum Yang-Mills theory on the noncommutative torus has several features in common with non-Abelian gauge theory defined on an ordinary, commutative torus. Formally, it can be regarded as a certain “large N limit” of ordinary U (N) Yang-Mills where we have generalized the gauge fields to measurable operators. In this section we will exploit these similarities to show how one may compute exactly the partition function for noncommutative gauge theory on the torus via the technique of non-Abelian localization [29]. The first key observation we shall make is that the integration measure DAˆ in (2.24) may be naturally identified with the gauge invariant Liouville measure induced on the infinite dimensional operator space of compatible connections C(E) by a symplectic two-form ω[·, ·]. Moreover, the volume of the gauge group vol G(E) is determined formally from the volume form on G(E) associated with

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the metric λˆ , λˆ = Tr E λˆ 2 on EndH,∞ Aθ (E). This metric also induces an invariant qua∗ dratic form (·, ·) on the dual Lie algebra EndH,∞ Aθ (E) , such that the noncommutative Yang-Mills action (2.25) is proportional to the square of the moment map µ corresponding to the symplectic action of G(E) on C(E). Equivalently, the Lie algebra action of the group of gauge transformations G(E) coincides with a system of Hamiltonian flows defined by the Yang-Mills action. In particular, this implies that the action (2.25) is a gauge-equivariant Morse function on C(E) [35, 36]. Consequently, the partition function of Yang-Mills theory defined on the noncommutative torus can be expressed formally as an infinite-dimensional statistical mechanics model 1 β 2 Z(g , θ, , E) = µ, µ , (3.1) exp ω − vol G(E) 2 C(E )

where β=

4π 2 R 2 . g2

(3.2)

As shown in [29], path integrals of the form (3.1) are formally calculable through a generalized non-Abelian localization technique. Here “localization” refers to the fact that the path integral (3.1) is given exactly by the sum over contributions from neighbourhoods of stationary points of the Yang-Mills action (2.25). If we denote the discrete set of all such critical points by P(θ, E), then

β ˆ cl ˆ cl Z(g 2 , θ, , E) = W Aˆ cl e − 2 µ[A ] , µ[A ] , (3.3) Aˆ cl ∈P (θ,E )

where the function W gives the contributions due to the quantum fluctuations about the stationary points. In the remainder of this section we will derive all of these properties in some detail. 3.1. Symplectic structure. Let E be a finitely-generated projective module over the noncommutative torus, and consider the space C(E) of compatible connections on E introduced in Sect. 2.2. The group of gauge transformations G(E) acts on C(E) and it has Lie algebra EndH,∞ Aθ (E) consisting of anti-Hermitian compact operators on E. On this Lie algebra we introduce a natural invariant, non-degenerate quadratic form by ˆ λˆ ∈ EndH,∞ (E) . λˆ , λˆ = Tr E λˆ λˆ , λ, (3.4) Aθ The infinitesimal gauge transformations (2.34) define a group action on C(E) because δλˆ , δλˆ Aˆ i = δλˆ , λˆ Aˆ i . (3.5) Consider the representation (2.3) of the Heisenberg algebra L in the Lie algebra of derivations of Aθ , and let (L∗ ) = n≥0 n (L∗ ) be the Z+ -graded exterior algebra of L . To this representation there corresponds the graded differential algebra ∗ n (E) = (3.6) n (E) , n (E) = EndH,∞ Aθ (E) ⊗C L n≥0

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of left-invariant differential forms on exp(L ) with coefficients in EndH,∞ Aθ (E). For

instance, the curvature ∇ˆ 1 , ∇ˆ 2 = FˆAˆ + · IE ∈ 2 (E) with G(E) acting infinitesimally through

(3.7) δλˆ FˆAˆ = λˆ , FˆAˆ with the usual product on the differential algebra (3.6) implicitly understood. Functional differentiation at a point Aˆ ∈ C(E) is then defined through d ˆ δ , aˆ ∈ 1 (E) . (3.8) f aˆ ≡ f A + t aˆ dt δ Aˆ t=0 As mentioned in Sect. 2.2, C(E) is an affine space over the vector space EndH,∞ Aθ (E)⊗C

L∗ of linear maps L → EndH,∞ Aθ (E), whose tangent space can be identified with the H,∞ cotangent space EndAθ (E) ⊗C 1 (L∗ ) = 1 (E). A natural symplectic structure may then be defined on C(E) by the two-form ˆ aˆ ∈ 1 (E) , (3.9) ω a, ˆ aˆ = Tr E aˆ ∧ aˆ , a, where3 aˆ ∧ aˆ = aˆ 1 aˆ 2 − aˆ 2 aˆ 1 .

(3.10)

Since (3.9) is independent of the point Aˆ ∈ C(E) at which it is evaluated, it is closed, i.e. δω/δ Aˆ = 0, and it is also clearly non-degenerate. In fact, because of the identities (2.12), (2.13) and (2.17), the symplectic two-form (3.9) coincides with the canonical, commutative one that is usually introduced in ordinary two-dimensional U (N ) YangMills theory [35]. Its main characteristic is that it is invariant under the infinitesimal action

δλˆ aˆ i = λˆ , aˆ i (3.11) of the gauge group G(E) on C(E), ˆ aˆ = 0 . ω a, ˆ δλˆ aˆ + ω δλˆ a,

(3.12)

3.2. Hamiltonian structure. Since C(E) is contractible and G(E) acts symplectically on C(E) with respect to the symplectic structure (3.9), there exists a moment map ∗ µ : C(E) −→ EndH,∞ (3.13) Aθ (E) which naturally generates a system of Hamiltonians Hλˆ : C(E) → R by

µ Aˆ , λˆ = Hλˆ Aˆ .

(3.14)

3 Note that components associated with the central elements of the Heisenberg algebra L trivially drop out of all formulas such as (3.10), and hence will not be explicitly written in what follows.

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To determine the moment map explicitly in the present case, we use the Hamiltonian flow condition

δ Hλˆ aˆ = −ω δλˆ Aˆ , aˆ , aˆ ∈ 1 (E) , δ Aˆ

(3.15)

which is equivalent to the G(E)-invariance (3.12). Using (2.34) and (3.9), the condition (3.15) reads

δ Hλˆ aˆ = − Tr E ∇ˆ , λˆ ∧ aˆ . δ Aˆ

(3.16)

Since the trace is invariant under the natural action of the connection on EndAθ (E), i.e.

Tr E ∇ˆ i , λˆ = 0 ,

(3.17)

using the Leibnitz rule we can write (3.16) equivalently as

δ Hλˆ aˆ = Tr E ∇ˆ ∧, aˆ λˆ . δ Aˆ

(3.18)

Let us now compare (3.18) with the first order perturbation of the shifted field strength in a neighbourhood of a point Aˆ ∈ C(E), which is easily computed to be

ˆ ˆ ∧ ˆ + O t2 . FˆA+t ˆ aˆ + · IE = FAˆ + · IE + t ∇ , a

(3.19)

By using (3.8) we may thereby write (3.18) as δ δ Hλˆ aˆ = Tr E FˆAˆ + · IE λˆ , δ Aˆ δ Aˆ

(3.20)

which is equivalent to

Hλˆ Aˆ = FˆAˆ + · IE , λˆ

(3.21)

in the quadratic form (3.4). Comparing with (3.14) we see that the moment map for the action of the noncommutative gauge group on the space C(E) is the shifted noncommutative field strength,

µ Aˆ = FˆAˆ + · IE .

(3.22)

Since π2 U ∞ (E) = 0, the map λˆ → Hλˆ determines a homomorphism from the Lie algebra EndH,∞ Aθ (E) to the infinite-dimensional Poisson algebra induced on the space of functions C(E) → R by the symplectic two-form (3.9).

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3.3. Cohomological formulation of noncommutative Yang-Mills theory. The fact that noncommutative gauge theory is so naturally a Hamiltonian system leads immediately to the localization of the path integral (2.24) onto the critical points of the action (2.25). We will now sketch the argument. First of all, the integration measure appearing in (2.24) is defined to be the Liouville measure corresponding to the symplectic two-form (3.9),4 ˆ ˆ (3.23) DAˆ = dAˆ dψˆ e − i ω ψ , ψ , 1 (E )

where dAˆ is the “ordinary” Feynman measure which may be defined by using the identification (2.15) and the operator-field correspondence as dAˆ =

N N

ab dAab 1 (x) dA2 (x) .

(3.24)

a=1 b=1 x∈T2

In (3.23), denotes the parity reversion operator, ψˆ are the odd generators of functions on the infinite dimensional superspace (E) = C(E) ⊕ 1 (E) ,

(3.25)

and dAˆ dψˆ is the corresponding functional Berezin measure. The result of the previous subsection shows that the noncommutative Yang-Mills action (2.25) is proportional to the square of the moment map, in the quadratic form (3.4), for the symplectic action of the gauge group G(E) on C(E),

2π 2 R 2 ˆ , µ Aˆ µ A . S Aˆ = g2

(3.26)

We can linearize the action (3.26) in µ via a functional Gaussian integration over an auxiliary field φˆ ∈ 0 (E), and by using (3.14) we can write the partition function (2.24) as 1 − 1 φˆ , φˆ dφˆ e 2β Z(g 2 , θ, , E) = vol G(E) 0 (E ) ˆ ˆ ψ]−H ˆ − i ω[ψ, ˆ [A] φ ˆ ˆ , (3.27) × dA dψ e (E )

with the measure dφˆ defined analogously to (3.24). Note that the operator φˆ appears only quadratically in (3.27) and thereby essentially corresponds to a commutative field. Because the functional integration measures in (3.27) are the same as those which occur in the corresponding commutative case, the only place that noncommutativity is present 4 To prove this formula, one needs to carefully study the gauge-fixed path integral measure. Since the gauge-fixed quantum action in two dimensions is Gaussian in the Faddeev-Popov ghost fields, and (3.9) coincides with the symplectic structure of the commutative case, the same arguments as in the commutative case [14] apply here and (3.23) is indeed the appropriate gauge-invariant measure to use on C(E ).

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is in the field strength which appears in the moment map (3.22). Indeed, it is essentially this feature that leads to the exact solvability of the model, in parallel with its commutative limit. The representation (3.27) of noncommutative gauge theory is the crux of the matter. Notice first that in the weak coupling limit g 2 = 0 (β = ∞), the noncommutative field

φˆ appears linearly in (3.27), and its integration yields the constraint µ Aˆ = 0 which localizes the path integral onto gauge connections Aˆ of constant curvature FˆAˆ = −·IE . The partition function in this limit then formally computes the symplectic volume of the moduli space of constant curvature connections modulo noncommutative gauge

trans ˆ formations. The key property which enables this localization is that A → µ Aˆ is a complete Nicolai map which trivializes the integration over C(E). In this respect, the g 2 = 0 limit of (3.27) is a topological gauge theory, and indeed it coincides with a noncommutative version of BF theory in two dimensions [42]. What is remarkable though is that the same Nicolai map appears to trivialize to the full theory (3.27) at g 2 = 0 to a Gaussian integral over C(E). This works up to the points in C(E) where this map has singularities, which coincide with the solutions of the classical equations of motion of noncommutative gauge theory. Thus in the generic case the partition function receives only contributions from the classical noncommutative gauge field configurations. To make these arguments precise, we first observe that the integral over the superspace (E) in (3.27) is formally the partition function of an infinite dimensional statistical mechanics system, and, in the present situation whereby there is a symplectic group

action generated by the Hamiltonian Hφˆ Aˆ , it is known that such integrals can be typically reduced to finite dimensional integrals, or sums, determined by the critical points

of Hφˆ Aˆ [34]. The main difference here is that there is no temperature parameter in front of the Hamiltonian through which to expand, but rather the noncommutative field ˆ The argument for localization can nonetheless be carried through by adapting the φ. non-Abelian localization principle [29] to the present noncommutative setting. This is achieved through a study of the cohomology of the infinite dimensional operator

δ δ ˆ ˆ ˆ (3.28) + ∇i , φ Qφˆ = Tr E ψi δ ψˆ i δ Aˆ i which is defined on the space G(E ) = Sym 0 (E) ⊗ C(E) ⊕ (E) ,

(3.29)

where Sym 0 (E) is the algebra of gauge-covariant polynomial functions on EndH,∞ Aθ (E). The linear derivation (3.28) acts on the basic multiplet Aˆ i , ψˆ i , φˆ of the noncommutative quantum field theory (3.27) through the transformation laws

Qφˆ , Aˆ i = ψˆ i , Qφˆ , ψˆ i = ∇ˆ i , φˆ ,

Qφˆ , φˆ = 0 , (3.30)

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and its square coincides with the generator of an infinitesimal gauge transformation with ˆ gauge parameter φ, 2 Qφˆ = δφˆ . (3.31) The key property of the operator (3.28) is that the Boltzmann weight over (E) in (3.27) is annihilated by it,

Qφˆ Tr E ψˆ ∧ ψˆ − Hφˆ Aˆ = 0 , (3.32) where we have used the Hamiltonian flow equation (3.16). Via integration by parts over the superspace (E), this implies that the partition function (3.27) is unchanged under multiplication of the Boltzmann factor by Qφˆ α for any gauge-invariant α ∈ G(E ) , i.e. Qφˆ

2

α=0.

(3.33)

In particular, we may write (3.27) in the form 1 1 φˆ , φˆ − 2β 2 ˆ dφ e Z(g , θ, , E) = vol G(E) ×

0 (E )

−i dAˆ dψˆ e

ˆ ˆ ˆ ˆ ψ]−H ˆ ω[ψ, φˆ [A]−t Qφˆ α[A,ψ]

.

(3.34)

(E )

That the right-hand side of (3.34) is independent of the parameter t ∈ R for gauge invariant α follows by noting that its derivative with respect to t vanishes upon integrating by parts over (E), and using (3.32) and (3.33) along with the Leibnitz rule for the functional derivative operator (3.28). This will be true so long as the perturbation by Qφˆ α yields an effective action which has a nondegenerate kinetic energy term, and that it does not allow any new Qφˆ fixed points to flow in from infinity in field space. The t = 0 limit of (3.34) coincides with the original partition function of noncommutative gauge theory, while its t → ∞ limit yields the desired reduction for appropriately chosen α. At this stage we will choose

α Aˆ , ψˆ = 4π 2 R 2 Tr E ψˆ i ∇ˆ i , µ Aˆ . (3.35) Substituting (3.35) into (3.34) using (3.28), performing the Gaussian integral over φˆ ∈ 0 (E), and taking the large t limit, we arrive at Z(g 2 , θ, , E) 1 ˆ 2 ˆ ˆ β = dAˆ dψˆ e − Tr E i ψ∧ψ+ 2 µ[A] vol G(E) (E )

2

(4π 2 R 2 )3 2 i ˆ ˆ × lim exp − t Tr E ∇ , ∇i , µ Aˆ t→∞ 2g 2 × exp 4π 2 i R 2 t Tr E µ Aˆ ψˆ i , ψˆ i − ∇ˆ i , ψˆ i ∇ˆ ∧, ψˆ , (3.36)

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where we have further applied the Leibnitz rule along with (3.17), and also dropped overall constants for ease of notation. The ψˆ integrations in (3.36) produce polynomial functions of the parameter t, and the Aˆ integration is therefore suppressed by the Gaussian term in t as t → ∞. Nondegeneracy of the quadratic form (3.4) implies that the functional integral thereby becomes localized near the solutions of the equation

∇ˆ i , ∇ˆ i , µ Aˆ =0, (3.37) and it can be written as a sum over contributions which depend only on local data near the solutions of (3.37). Along with (3.17) and the Leibnitz rule, Eq. (3.37) implies

2 0 = Tr E µ Aˆ ∇ˆ i , ∇ˆ i , µ Aˆ

2

= − Tr E ∇ˆ i , µ Aˆ , (3.38) which again by the non-degeneracy of (3.4) is equivalent to

∇ˆ i , µ Aˆ = 0 .

(3.39)

Since µ Aˆ = ∇ˆ 1 , ∇ˆ 2 , Eqs. (3.39) coincide with the classical equations of motion of

the action (2.25), i.e. δS Aˆ /δ Aˆ = 0. This establishes the localization of the partition function (2.24) of noncommutative gauge theory in two dimensions onto the space of solutions of the noncommutative Yang-Mills equations. This space will be studied in detail in the next section. Although the above technique leads to a formal proof of the localization of the partition function onto classical gauge field configurations, it does not yield any immediate useful information as to the precise form of the function W in (3.3) encoding the quantum fluctuations about the classical solutions. The infinite-dimensional determinants that arise from (3.36) have very large symmetries and are difficult to evaluate. The fluctuation determinants W will be determined later on by another technique. From a mathematical perspective, the action in (3.27) over (E) is the G(E)-equivariant extension of the moment map on C(E), the integration over (E) defines an equivariant differential form, and the integral over φˆ ∈ 0 (E) defines equivariant integration of such forms. The operator (3.28) is the Cartan differential for the G(E)-equivariant cohomology of C(E) [34]. The localization may then also be understood via a mapping onto a purely cohomological noncommutative gauge theory in the limit t → ∞. These aspects will not be developed any further here. 4. Classification of Instanton Contributions In the previous section we proved that the partition function is given by a sum over contributions localized at the classical solutions of the noncommutative gauge theory. In this section we will classify the instantons of two-dimensional gauge theory on the noncommutative torus, and later on explicitly evaluate their contribution to the partition function. By an “instanton” here we mean a solution Ai = Acl i of the classical noncommutative field equations ∂i FA + Ai FA − FA Ai = 0

(4.1)

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which is not a gauge transformation of the trivial solution Ai = 0. Here FA is the noncommutative field strength (2.28). Note that this definition also includes the unstable modes. In the commutative case, instanton contributions have a well-known geometrical classification based on the fundamental group of the spacetime [35]. In the noncommutative setting, however, the role of homotopy groups is played by the K-theory of the algebra and one must resort to an algebraic characterization of the contributing projective modules. For irrational values of the noncommutativity parameter θ, an elegant classification of the stationary points of noncommutative Yang-Mills theory has been given in [36]. In what follows we shall modify this construction somewhat to more properly suit our purposes. 4.1. Heisenberg modules. In order to classify the instanton solutions of gauge theory on the noncommutative torus, we need to specify the topological structures involved. This requirement leads us into the explicit classification of the projective modules over the algebra Aθ [1, 38]. They are classified by the K-theory group [43] (4.2) K0 (Aθ ) = π1 U∞ (Aθ ) = Z ⊕ Z . The cohomologically invariant trace Tr : Aθ → C induces an isomorphism K0 (Aθ ) → Z + Z θ ⊂ R of ordered groups. To each pair of integers (p, q) ∈ K0 (Aθ ) there corresponds a virtual projector Pp,q with Tr ⊗ tr M Pp,q = p − qθ . However, given a projective module E determined by a Hermitian projector P, positivity of the trace implies dim E = Tr E IE = Tr ⊗ tr M P = Tr ⊗ tr M P P† ≥ 0 ,

(4.3)

and so the stable (rather than virtual) projective modules are classified by the positive cone of K0 (Aθ ). Thus to each pair of integers (p, q) we can associate a Heisenberg module Ep,q [44] of positive Murray-von Neumann dimension dim Ep,q = p − qθ > 0 .

(4.4)

Such pairs of integers parameterize the connected components of the infinite dimensional manifold Gr θ of Hermitian projectors of the algebra Aθ . In what follows we will be interested in studying the critical points of the noncommutative Yang-Mills action within a given homotopy class of Gr θ . The integer q=

1 Tr ⊗ tr M Pp,q ∂ˆ , Pp,q ∧ ∂ˆ , Pp,q 2π i

(4.5)

is the Chern number (or magnetic flux) of the corresponding gauge bundle [45]. In the case of irrational θ , any finitely generated projective module over the noncommutative torus is either a free module or it is isomorphic to a Heisenberg module [46]. We will view free modules as special instances of Heisenberg modules obtained by setting q = 0. Any two projective modules representing the same element of K-theory are isomorphic. The main property of Heisenberg modules that we will exploit in the following is that they always admit a constant curvature connection ∇ˆ c ∈ Cp,q = C(Ep,q ),

(4.6) ∇ˆ 1c , ∇ˆ 2c = i f · IEp,q ,

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where f ∈ R is a constant. In this subsection we shall set = 0, as the background magnetic flux can be reinstated afterwards by the shift f → f + . In the presence of supersymmetry, such a field configuration gives rise to a BPS state [47, 48]. It also leads to an explicit representation of the Heisenberg module Ep,q as the separable Hilbert space [38] q

Ep,q = L2 (R) ⊗ C , q = 0 .

(4.7)

L2 (R)

The Hilbert space is the Schr¨odinger representation of the Heisenberg commutation relations (4.6). By the Stone-von Neumann theorem, it is the unique irreducible q representation. The factor C defines the q × q representation of the Weyl-’t Hooft algebra in two dimensions, 1 2 = e 2π i p/q 2 1 ,

(4.8)

which may be solved explicitly by SU (q) shift and clock matrices. The generators of the noncommutative torus are then represented on (4.7) as Zˆ i = e

i f −1 ∇ˆ ic /R

⊗ i ,

(4.9)

and computing (2.1) using (4.6), (4.8) and the Baker-Campbell-Hausdorff formula thereby leads to a relation between the noncommutativity parameter θ and the constant flux f through θ =−

p 1 + . 2πR 2 f q

The Aθ -valued inner product on Ep,q is given by  ∞ ∞ ∞  ξˆ , ηˆ = ds 1m1 2m2 ξ s − Aθ

m1 =−∞ m2 =−∞

−∞

(4.10)

m1 2π R 2 f

× Zˆ 1m1 Zˆ 2m2 .

†

 η(s) e 2π i m2  (4.11)

For q = 0 we define Ep,0 to be the free module of rank p, i.e. p

Ep,0 = L2 (T2 ) ⊗ C .

(4.12)

The Heisenberg module Ep,q so constructed coincides, in the D-brane picture, with the Hilbert space of ground states of open strings stretching between a single Dr-brane and p Dr-branes carrying q units of D(r − 2)-brane charge [41]. It is irreducible if and only if the integers p and q are relatively prime. The Weyl-’t Hooft algebra (4.8) has a unique irreducible representation (up to SU (q) equivalence) of dimension q/gcd(p, q) [49, 50], and so the rank N of the resulting gauge theory as defined in Sect. 2.2 is given by N = gcd(p, q) .

(4.13)

Furthermore, the commutant MN (Aθ ) of Aθ in EndAθ (Ep,q ) is Morita equivalent to the noncommutative torus with dual noncommutativity parameter θ determined by the SL(2, Z) transformation [38] n − sθ N, (4.14) θ = p − qθ where n and s are integers which solve the Diophantine equation ps − qn = N .

(4.15)

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4.2. Stationary points of noncommutative gauge theory. We will now describe the critical points of the noncommutative Yang-Mills action (2.25). Let us fix a Heisenberg module Ep,q over the noncommutative torus, which is labelled by a pair of integers (p, q) obeying the constraint (4.4). From (4.6) and (4.10) it follows that this projective module is characterized by a connection ∇ˆ c ∈ Cp,q of constant curvature FˆAˆ c =

q 1 · IEp,q . 2πR 2 p − qθ

(4.16)

Such constant curvature connections are of fundamental importance in finding solutions of the noncommutative Yang-Mills equations because they not only solve (4.1), but they moreover yield the absolute minimum value of the Yang-Mills action on the module Ep,q [15, 37]. This follows by using (3.19) to compute the infinitesimal variation FˆAˆ c +t aˆ about a constant curvature connection to get 2

2π 2 R 2 ˆ ˆc Tr + · I F S ∇ˆ c + t aˆ = E p,q Ep,q A +t aˆ g2

2π 2 R 2 t 2

2 ˆ c ∧, aˆ + O t 4 . = S ∇ˆ c + Tr ∇ Ep,q g2

(4.17)

The cross terms of order t in (4.17) vanish due to the property (3.17) and the fact that the field strength (4.16) is proportional to the identity operator

on Ep,q . Since the quac dratic term in t is positive definite, we have S ∇ˆ + aˆ ≥ S ∇ˆ c ∀aˆ ∈ 1 (Ep,q ). To establish that ∇ˆ c is a global minimum, we can exploit the freedom of choice of the background flux (see Sect. 6.1) to identify it with the constant curvature (4.16). Then

c ˆ S ∇ = 0, and since (2.25) is a positive functional, the claimed property follows. This shifting of the curvature will be used explicitly below. In addition to yielding the minimum of the Yang-Mills action, constant curvature connections can also be used to construct all solutions of the classical equations of motion [36]. The main observation is that insofar as solutions of theYang-Mills equations are concerned, the module Ep,q may be considered to be a direct sum of submodules [35]. To see this, we note that the equations of motion (3.39) imply that, at the critical points

cl ∇ˆ = ∇ˆ , the moment map µ Aˆ is invariant under the induced action of the Heisenberg algebra L of automorphisms on the algebra EndAθ (Ep,q ). In particular, it corresponds

to the central element of the Heisenberg Lie algebra generated by ∇ˆ cl , ∇ˆ cl and µ Aˆ cl . 1

2

This feature provides a natural direct sum decomposition of the module Ep,q through the adjoint action of the moment map on p,q = (Ep,q ). For this, we consider the self-adjoint linear operators ∇ˆ : p,q → p,q defined for each connection ∇ˆ ∈ Cp,q by

∇ˆ (α) ˆ = µ Aˆ , αˆ , αˆ ∈ p,q . (4.18) From the equations of motion (3.39) it follows that the Aθ -valued eigenvalues cˆk of ∇ˆ are constant in the vicinity of a critical point ∇ˆ = ∇ˆ cl , and so there is a natural direct sum decomposition of the module Ep,q into projective submodules Epk ,qk , Epk ,qk , (4.19) Ep,q = k≥1

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corresponding to the eigenspace decomposition p,q = k≥1 pk ,qk with respect to ∇ˆ .5 On each pk ,qk the operator ∇ˆ acts as multiplication by a fixed scalar ck . Since

µ Aˆ cl commutes with ∇ˆ cl , the connection ∇ˆ cl is also a linear operator on each Epk ,qk →

c = ∇ ˆ cl Epk ,qk , and its restriction ∇ˆ (k) has constant curvature µ Aˆ cl . Epk ,qk

Epk ,qk

Given such a direct sum decomposition6 of the module Ep,q , we can define a connection ∇ˆ on Ep,q by taking the sum of connections on each of the submodules, ˆ ∇ˆ = k≥1 ∇(k) . The noncommutative Yang-Mills action is additive with respect to this decomposition,   (4.20) ∇ˆ (k)  = S ∇ˆ (k) . S k≥1

k≥1

c on each It follows that for the particular choice of constant curvature connections ∇ˆ (k) of the submodules Epk ,qk , the Yang-Mills action has a critical point c ∇ˆ cl = ∇ˆ (k) (4.21) k≥1

on Ep,q . Moreover, from the above arguments it also follows that every Yang-Mills critical point on Ep,q is of this form. This construction thereby exhausts all possible critical points, and is essentially the noncommutative version of the bundle splitting method of constructing classical solutions to ordinary, commutative gauge theory in two dimensions [35]. While there are many possibilities for the decomposition (4.19) of the given module Ep,q into submodules, there are two important constraints that must be taken into account. First of all, the (positive) Murray-von Neumann dimension of the module is additive with respect to the decomposition (4.19), dim Ep,q = dim Epk ,qk = (4.22) (pk − qk θ) . k≥1

k≥1

Secondly, since a module over the noncommutative torus is completely and uniquely determined (up to isomorphism) by two integers, we need an additional constraint. This is the requirement that the Chern number of the module be equal to the total magnetic flux of the direct sum decomposition. For the module Ep,q this gives the relation q= qk . (4.23) k≥1

5 It should be stressed that (4.19) is not the statement that the given Heisenberg module is reducible. It simply reflects the behaviour of connections near a stationary point of the noncommutative Yang-Mills action, in which one may interpret the eigenspaces pk ,qk = (Epk ,qk ) as the differential algebras of submodules Epk ,qk ⊂ Ep,q . For more technical details of the decomposition (4.19) as an Aθ -module, we refer to [36]. Notice also that here we abuse notation by making no distinction between ∇ˆ c acting on p,q or Ep,q . Only the latter operator will be pertinent in what follows. 6 In Sect. 7 we will give an elementary proof that such decompositions necessarily contain only a finite number of direct summands. See [36] for a functional analytic proof.

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For irrational values of the noncommutativity parameter θ it is clear from (4.4) that the constraint (4.23) follows from (4.22). This is not the case for rational θ and the constraint on the Chern class makes necessary a distinction between physical noncommutative Yang-Mills theory and Yang-Mills theory defined on a particular projective module Ep,q . The latter field theory imposes #a K-theory charge conservation law for submodule decompositions (4.19), (p, q) = k≥1 (pk , qk ). This distinction will be discussed further when the partition function for Yang-Mills theory on the noncommutative torus is calculated explicitly. We can now summarize the classification of the critical points of the noncommutative Yang-Mills action as follows. For any value of the noncommutativity parameter θ , any solution of the classical equations of motion of Yang-Mills theory defined on the Heisenberg module Ep,q is completely characterized by a collection of pairs of integers (pk , qk )

k≥1

obeying the constraints

pk − q k θ > 0 , (pk − qk θ ) = p − qθ ,

k≥1

qk = q .

(4.24)

k≥1 7 We will call such a collection of integers a “partition” and will denote it by (p , q ) ≡ (pk , qk ) . In order to avoid overcounting partitions which will contribute to the k≥1

Yang-Mills partition function, we also need to introduce a partial ordering for submodules in a given partition based on the dimension of each submodule, 0 < p1 − q1 θ ≤ p2 − q2 θ ≤ p3 − q3 θ ≤ . . . .

(4.25)

Any number of partitions which are identical after such an ordering will be regarded as equivalent presentations of the same partition. The set of all distinct partitions associated with the Heisenberg module Ep,q will be denoted Pp,q (θ ) = P(θ, Ep,q ). It remains to evaluate the Yang-Mills action at a solution of the classical equations of motion, which, by the arguments of the previous section, is one of the key ingredients in the computation of the partition function of noncommutative gauge theory. At a critical point, i.e. a partition (p , q ), according to (4.20) it is just the sum of contributions from constant curvature connections on each of the submodules of the partition,   c  S(p , q ; θ ) = S  ∇ˆ (k) k≥1

2 q qk 1 − , = 2 2 (pk − qk θ ) 2g R pk − q k θ p − qθ

(4.26)

k≥1

7 This definition of partition is more general than that of [36]. It is the one that is the most useful for the computation of the noncommutative gauge theory partition function in the following. In particular, it contains contributions from reducible connections, as these will also turn out to contribute to the Yang-Mills partition function. These connections are in fact responsible for the orbifold singularities that appear in the instanton moduli spaces. These points, as well as how to avoid the overcounting of critical points through combinatoric factors in the partition function, will be described in detail in Sect. 9.

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where we have used Tr Ep ,q IEpk ,qk = pk − qk θ and fixed the value of the background k k field to be [15] = p,q = −

q . 2R 2 (p − qθ)

(4.27)

This value ensures that the constant curvature connection ∇ˆ c on the module Ep,q , which corresponds to a solution of the equations of motion parameterized by the trivial partition (p, q), gives a vanishing global minimum of the action, S(p, q; θ) = 0. This is the natural boundary condition, and generally the inclusion of ensures that the classical action is invariant under Morita duality [15, 41, 48, 51, 52]. 5. Yang-Mills Theory on a Commutative Torus As we mentioned in Sect. 3, while we can prove that noncommutative gauge theory on a two-dimensional torus is given exactly by a sum over classical solutions (instantons), evaluating directly the fluctuation factors, which multiply the Boltzmann weights of the corresponding critical action values computed in the previous section, is a difficult task. We will therefore proceed as follows. We start with the well-known exact solution for Yang-Mills theory on a commutative torus and identify quantities which are invariant under gauge Morita equivalence. This will yield the partition function of noncommutative Yang-Mills theory for any rational value of the noncommutativity parameter θ. From this expression we will then be able to deduce the corresponding expression for Yang-Mills theory defined on a noncommutative torus with arbitrary θ . In this section we will analyze the instanton contributions to commutative Yang-Mills theory in order to set up this construction. The physical Hilbert space Hphys of ordinary U (p) quantum gauge theory defined on a (commutative) two-torus is the space of class functions Ad U (p) 2 Hphys = L U (p) (5.1) in the invariant Haar measure on the U (p) gauge group. By the Peter-Weyl theorem, it has a natural basis |R determined by the unitary irreducible representations R of the unitary Lie group U (p). The Hamiltonian is essentially the Laplacian on the group manifold of U (p), and so the corresponding vacuum amplitude has the well-known heat kernel expansion [14, 53, 54] 2 2 2 Z(g 2 , p) = e −2π R g C2 (R) , (5.2) R

where the Boltzmann weight contains the quadratic Casimir invariant C2 (R) of the representation R. This concise form does not have a direct interpretation in terms of a sum over contributions from critical points of the classical action that we expect from the arguments of Sect. 3. In order to find a more appropriate form, it is useful to make explicit the sum over irreducible representations as a sum over integers and perform a Poisson resummation of (5.2) [33]. The representations R of U (p) can be labelled by sets of p integers +∞ > n1 > n2 > · · · > np > −∞

(5.3)

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which give the lengths of the rows of the corresponding Young tableaux. In terms of these integers the Casimir operator is given by p p 2 p−1 2 C2 (R) = C2 (n1 , . . . , np ) = p −1 + na − , 12 2

(5.4)

a=1

and by using its symmetry under permutations of the integers na we can write (5.2) as Z(g 2 , p) =

1 p!

e −2π

2 R 2 g 2 C (n ,...,n ) p 2 1

.

(5.5)

n1 =···=np

One can extend the sums in (5.5) over all integers n1 , . . . , np by inserting the determinant det

1≤a,b≤p

p sgn(σ ) δna ,nσ (a) , δna ,nb =

σ ∈p

(5.6)

a=1

where p is the group of permutations on p objects. The permutation symmetry of (5.4) implies that all elements in the same conjugacy class of p yield the same contribution to the partition function. The sum over permutations (5.6) thereby truncates to a sum over conjugacy classes of p . They are labelled by the sets of p integers 0 ≤ νa ≤ [p/a], each giving the number of elementary cycles of length a in the usual cycle decomposition of elements of p , and which define a partition of p, i.e. ν1 + 2ν2 + · · · + pνp = p .

(5.7)

The parity of the elements of a conjugacy class C[ν ] = C[ν1 , . . . , νp ] is #

sgn C[ν ] = (−1)

a ν2a

(5.8)

and it contains C[ν ] =

p! p

(5.9)

νa

a νa !

a=1

elements. The sum over the na ’s in (5.5) then yields a theta-function, and the corresponding Jacobi inversion formula can be derived in the usual way by means of the Poisson resummation formula ∞ n=−∞

f (n) =

∞

∞

ds f (s) e 2π i qs .

(5.10)

q=−∞ −∞

The Fourier transformations required in (5.10) are all Gaussian integrals in the present case, and after some algebra the partition function (5.5) can be expressed as a sum over dual integers qk as [33] 8 8

We have corrected here a few typographical errors appearing in Eqs. (13)–(15) of [33].

Instanton Expansion of Noncommutative Gauge Theory in Two Dimensions

Z(g 2 , p) = e −

×

π 2 g2 R2 6

p a=1

p(p2 −1)

ν :

#

a

∞

···

aνa =p q1 =−∞

3 −νa /2

2g 2 R 2 a νa !

∞

e

iπ

369

#

a ν2a +(p−1)

#

k qk

q|ν | =−∞

e −Sν (q1 ,... ,q|ν | ) .

(5.11)

Here |ν | = ν1 + ν2 + · · · + νp

(5.12)

is the total number of cycles contained in the elements of the conjugacy class C[ν ] of p , and the action is given by  ν1 ν ν1 +ν 1 +ν2 2 +ν3 qk21 qk22 qk23 1 Sν (q1 , . . . , q|ν | ) = 2 2  + + 2g R 1 2 3 k1 =1 k2 =ν1 +1 k3 =ν1 +ν2 +1  |ν | qk2p  . + ··· + (5.13) p kp =ν1 +···+νp−1 +1

It is understood here that if some νa = 0, then qν1 +···+νa−1 +1 = · · · = qν1 +...νa−1 +νa = 0. The remarkable feature of this rewriting is that the action (5.13) is precisely of the ∞ 2 general form (4.26). Since K0 C (T ) = Z ⊕ Z, any finitely-generated projective module E = Ep,q over the algebra A0 = C ∞ (T2 ) is determined (up to isomorphism) by a pair of relatively prime integers (p, q) ∈ Z+ ⊕ Z with dimension given by p and constant curvature q/p [44]. Geometrically, Ep,q is the space of sections of a vector bundle over the torus T2 of rank p, topological charge q, and with structure group U (p). Consider a direct sum decomposition (4.19) of this module. We will enumerate submodules in a partition according to increasing dimension. Let νa be the number of submodules of dimension a, corresponding to the splitting of the bundle into sub-bundles of rank a, so that dim Ep,q = ν1 + 2ν2 + · · · + pνp .

(5.14)

This condition is simply the constraint (4.22) on the total dimension of the sum of sub# modules in this case, i.e. p = k≥1 pk with 1 ≤ pk ≤ p. Therefore, the expression (5.11) is nothing but the localization of the partition function of commutativeYang-Mills theory onto its classical solutions. Note that here the magnetic charges qk are dual to the lengths of the rows of the Young tableaux of the unitary group U (p). There are, however, two important differences here. First of all, the action (5.13) is evaluated for a topologically trivial bundle, i.e. q = 0, which yields a vanishing background flux p,q . Consequently, (5.11) is not the most general result. Secondly, and most importantly, the sum over Chern numbers q1 , . . . , q|ν | in the partition function is #not constrained to satisfy (4.23), which in view of our first point is the restriction k qk = 0. In fact, the partition function (5.11) for physical U (p) Yang-Mills gauge theory on the commutative torus is a sum of contributions from topologically distinct bundles (of different Chern numbers) over the torus. In order to generalize the calculation of the partition function to the case of Yang-Mills theory defined on a projective module, we need to separate out of (5.11) the terms which are well-defined on a particular isomorphism class Ep,q of modules.

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In order to facilitate the identification of such a module definition of Yang-Mills theory, we write the partition function (5.11) in terms of the topological numbers of the module Ep,q . We will first enforce the constraint (4.23) on the magnetic charges. # It is also useful for further calculations to re-interpret the parity factors (−1) a ν2a in terms of the rank p and the total number |ν | of submodules in a given partition |ν | (p , q ) = (pk , qk ) labelling a critical point of the action. If p is odd (even) then k=1 there is an odd (even) number of submodules Epk ,qk with pk odd. By considering all possible cases one can show that [p/2]

ν2a = p + |ν | (mod 2) .

(5.15)

a =1

With these adjustments we are led to the module Yang-Mills theory with partition function Zp,q which is well-defined on Ep,q , Z(g 2 , p) = e −

π 2 g2 R2 6

p(p2 −1)

∞

(−1)(p−1)q+p Zp,q (g 2 , θ = 0) ,

(5.16)

q=−∞

where the module partition function is given by a sum over partitions associated with the module Ep,q , Zp,q (g 2 , θ = 0) = Z(g 2 , θ = 0, p,q , Ep,q ) =

(p , q )∈Pp,q (θ=0)

(−1)|ν |

p 2 2 3 −νa /2 2g R a e −S(p ,q ;θ=0) . νa !

a=1

(5.17) Note that the critical points of the action are defined by partitions obeying the constraints (4.24), including a restriction to submodules with total Chern number q. We have also generalized to the correct action (4.26) for Yang-Mills theory on a bundle with Chern number q which contains the non-vanishing value (4.27) for the background magnetic field . Again, this latter change is equivalent to adding boundary terms to the action which do not contribute to the classical dynamics of the theory and hence are not relevant to our analysis based on instanton contributions. The only essential role of is, as we will see, to set the zero-point of theYang-Mills action in the instanton picture. Therefore, a shift in will at most result in multiplying the fixed module partition function (5.17) by overall constants dependent only on the topological numbers (p, q). 6. Yang-Mills Theory on a Noncommutative Torus: Rational Case Given the partition function (5.17) for Yang-Mills theory which is well-defined on a given module Ep,q of sections of some bundle, we can now use Morita equivalence to obtain an explicit formula for Yang-Mills theory on a torus with rational noncommutativity parameter θ from the commutative case. Morita equivalence in this case refers to the mapping between noncommutative tori which is generated by the infinite discrete group SO(2, 2, Z) ∼ = SL(2, Z)×SL(2, Z), where one of the SL(2, Z) factors coincides with the discrete automorphism group of the ordinary torus T2 . It provides a one-to-one

Instanton Expansion of Noncommutative Gauge Theory in Two Dimensions

371

correspondence between modules associated with different topological numbers and noncommutativity parameters. Here we will be interested in the transformations from modules corresponding to rational values of θ to modules with vanishing θ . In fact, there is an extended version of the correspondence known as gauge Morita equivalence [41, 48] which augments the mapping of tori with transformations of connections between modules, and leads to a rescaling of the area and coupling constant to give a symmetry of Yang-Mills theory as we have defined it in (2.26). The entire noncommutative quantum field theory is invariant under this extended equivalence [15] which coincides with the standard open string T-duality transformations [41, 51]. We will use this invariance property to construct the noncommutative gauge theory for rational values of the deformation parameter θ .

6.1. Gauge Morita equivalence. We begin by summarizing the basic transformation rules of Morita equivalence of noncommutative gauge theories [1, 3, 44]. In two dimensions, Morita equivalences of noncommutative tori are generated by the group elements

mn ∈ SL(2, Z) , r s

(6.1)

where we concentrate on the SL(2, Z) subgroup which acts only on the K¨ahler modulus of T2 . The full duality group acts on the K-theory ring K0 (Aθ ) ⊕ K1 (Aθ ) in a spinor representation of SO(2, 2, Z) and the topological numbers (p, q) of a module E = Ep,q transform as mn p p = . (6.2) r s q q The noncommutativity parameter θ transforms under a discrete linear fractional transformation θ =

mθ + n . rθ + s

(6.3)

From these rules it follows that under the gauge Morita equivalence parameterized by (6.1) the dimensions of modules are changed according to dim E =

dim E . |rθ + s|

(6.4)

The invariance of the noncommutative Yang-Mills action (2.27) then dictates the corresponding transformation rules for the area element of T2 , the Yang-Mills coupling constant, and the magnetic background as R = |rθ + s| R , g 2 = |rθ + s| g 2 , = (rθ + s)2 −

r(rθ + s) . 2π R 2

(6.5)

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6.2. The partition function for rational θ . Let us now consider the effect of such transformations on the module partition function defined in (5.17). Under the gauge Morita equivalence parameterized by (6.1), the ordinary Yang-Mills gauge theory (5.17) is mapped onto a noncommutative gauge theory with rational-valued noncommutativity parameter θ = n/s. The classical action of the theory is invariant, and constant curvature connections are mapped into one another [1, 48]. Thus Morita equivalence maps solutions of the equations of motion between the commutative and rational noncommutative cases. The localization of the partition function onto classical solutions is therefore not affected by the transformation. The topological numbers of the submodules comprising partitions which define solutions of the classical equations of motion also map into each other in the two cases. In particular, the total number |ν | of submodules in a partition is invariant under the Morita duality. The only component $of the partition function (5.17) we have left to examine is the pre-exponential factor a≥1 (2g 2 R 2 a 3 )−νa /2 /νa !. The symmetry factors νa ! associated with a partition are preserved, and so from the transformation rules (6.5) for θ = 0 it follows that this component is invariant only if the integer a 3 appearing here transforms according to the scaling a a = (6.6) |s| under the Morita equivalence. But (6.6) is exactly the rescaling (6.4) of the dimension of a projective module in this case. It follows that the indices a in the pre-exponential factors of (5.17) should be interpreted as the (integer) dimensions of submodules in the commutative gauge theory, and this fact provides a Morita covariant interpretation of these indices which leads immediately to the appropriate generalization of the formula (5.17) to rational-valued θ = 0. We are now in a position to write down an explicit expression for the partition function of quantum Yang-Mills theory on the module Ep,q corresponding to a rational, non-integer noncommutativity parameter θ . The only modifications required are the counting and dimensions of modules which, in contrast to the commutative case, are no longer integer-valued. We order the submodules in a given partition (p , q ) according to increasing dimension, 0 < dim Ep1 ,q1 ≤ dim Ep2 ,q2 ≤ dim Ep3 ,q3 ≤ . . . .

(6.7)

Let νa be the number of submodules in this sequence that have the a th least dimension, which we denote by dima . Then the integer |ν | = νa (6.8) a≥1

still gives the total number of submodules in a partition, and we may write the partition function for rational θ as −νa /2 2g 2 R 2 (dima )3 Zp,q (g 2 , θ ) = (−1)|ν | e −S(p ,q ;θ) . (6.9) νa ! (p , q )∈Pp,q (θ)

a≥1

This expression provides a direct evaluation ofYang-Mills theory on a torus with rational noncommutativity parameter θ , without recourse to Morita equivalence with the commutative theory. Note that in the case when all submodules in the partitions have integer dimension, the formula (6.9) reduces to that for the partition function of Yang-Mills theory on a commutative torus in (5.17).

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6.3. Relation between commutative and rational noncommutative gauge theories. The arguments which led to the expression (6.9) give an interesting way to see the well-known connections betweenYang-Mills theory on a noncommutative torus with rational-valued θ and Yang-Mills theory defined on a commutative torus. Consider the gauge theory defined on the Heisenberg module Ep,q over the noncommutative torus with deformation parameter θ = n/s, where n and s are relatively prime positive integers. It can be verified from the definition (4.4) that any projective module over such a torus has a Murray-von Neumann dimension of at least 1/s. Since the total dimension of the module Ep,q is p − nq/s in this case, a partition (p , q ) which obeys the constraints (4.24) and which consists of submodules of dimensions greater than or equal to 1/s has at most p − nq/s = ps − qn 1/s

(6.10)

components. Since Morita equivalence preserves the number of submodules in a partition, any gauge theory which is dual to this rational noncommutative one must admit partitions with ps − qn components. On the other hand, for U (N ) Yang-Mills theory defined on a commutative torus, we know that due to the constraints (4.24), the maximum number of submodules in a partition is N (corresponding to ν1 = N and νa = 0 ∀a > 1). We conclude that Yang-Mills theory on a Heisenberg module Ep,q over the noncommutative torus with θ = n/s is Morita equivalent to a U (N ) commutative gauge theory of rank N = ps − qn. This result agrees with how the rank of the noncommutative gauge theory appeared at the end of Sect. 4.1. Notice that Morita equivalence maps submodules of the U (ps − qn) commutative gauge theory, as defined in the previous section, onto submodules of the noncommutative gauge theory on the Heisenberg module Ep,q as defined in Sect. 4.1. The effect of this mapping on dimensions of projective modules is to divide by s. This includes the irreducible finite-dimensional representation of the Weyl-’t Hooft algebra generated by the Zˆ i in (2.1) as follows. The infinite-dimensional center of the algebra An/s is generated by the elements zi = (Zˆ i )s , i = 1, 2 which, in an irreducible unitary representation, can be taken to be complex numbers of unit modulus. The center can thereby be identified with the commutative algebra C ∞ (T2 ) of smooth functions on the ordinary torus T2 , i.e. An/s may be regarded as a twisted matrix bundle over C ∞ (T2 ) of topological charge n whose fibers are s × s complex matrix algebras Ms . In particular, there is a surjective algebra homomorphism π : An/s → Ms , sending the Zˆ i to the corresponding SU (s) shift and clock matrices, under which the entire center of An/s is mapped to C. In the language of s Heisenberg modules this representation corresponds to the finite-dimensional factor C s 2 2 of the separable Hilbert space Es = L (T ) ⊗ C , which allows for twisted boundary conditions on functions of the ordinary torus T2 leading to the appropriate Weyl-’t Hooft algebra in this case. The irreducible finite-dimensional representation of the algebra is thereby associated with a free module Es = Es,0 of vanishing Chern class. Therefore, the localization of the partition function of quantum Yang-Mills theory on a rational noncommutative torus is determined entirely by contributions from classical solutions associated with Heisenberg modules as we have described them above. By construction, this includes the Morita equivalent projective modules over the ordinary torus. 7. Yang-Mills Theory on a Noncommutative Torus: Irrational Case Finally, we come to the case of irrational θ . We claim that the formula (6.9) gives the Yang-Mills partition function as a sum over partitions (p , q ) consisting of pairs of

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integers satisfying the constraints (4.24). Before justifying this claim, let us describe the quantitative differences in the formula (6.9) between the rational and irrational cases. In fact, the analytical structure of the partition functions in the two cases is very different due to the drastic differences of the partitions in Pp,q (θ ) which contribute to the functional integral. Recall from the previous section that in the case of rational θ , all modules have dimension at least 1/s, and this fact was the crux of the existence of the mapping between the rational and commutative gauge theories. In contrast, when θ is irrational, submodules with arbitrarily small dimension can contribute to a partition which characterizes a critical point of the Yang-Mills action. As such, there is no a priori upper bound on the number of submodules in a partition of Pp,q (θ ). While for deformation parameter θ = n/s all partitions contain at most ps − qr submodules of Ep,q of dimension at least 1/s, in the irrational case there are no such global bounds on the elements of Pp,q (θ ). It is this fact that prevents Yang-Mills theory on a noncommutative torus with irrational-valued θ from being Morita equivalent to some commutative gauge theory of finite rank, and indeed in this case the algebra Aθ has a trivial center. As a consequence, in contrast to the rational case, the Yang-Mills partition function on an irrational noncommutative torus receives contributions from partitions containing arbitrarily many submodules. However, it is possible to show that any partition corresponding to a fixed finite action solution of the noncommutative Yang-Mills equations of motion contains only finitely many components. By using a Morita duality transformation (6.5) we can transform the action so that = 0. Consider a partition (p , q ) ∈ Pp,q (θ ) on which the Yang-Mills action has the value S(p , q ; θ) = ξ ∈ R+ . Since (4.26) is a sum of positive terms, this implies that qk2 ≤ ξ (pk − qk θ) for each k ≥ 1. But the constraints (4.24) imply 0 < pk − qk θ ≤ p − qθ

(7.1)

qk2 ≤ ξ (p − qθ ) .

(7.2)

for each k ≥ 1, and hence From (7.2) it follows that qk can range over only a finite number of integers, and hence from (7.1) the same is also true of pk , which establishes the result. In particular, we can pick out the minimum dimension submodule Ep1 ,q1 in a given partition (p , q ) and order the submodules according to increasing Murray-von Neumann dimension as in (6.7). The definition (6.8) still makes sense and hence so does the expression (6.9) for the partition function, provided that one now allows for partitions with arbitrarily large (but finite) numbers of submodules. Incidentally, this line of reasoning also shows that the set of values of the noncommutative Yang-Mills action on the critical point set Pp,q (θ ) is discrete, as is required of a Morse function [36]. Let us now indicate the reasons why (6.9) is the correct result for the partition function of Yang-Mills theory on a noncommutative torus with irrational θ . First of all, notice that the localization arguments of Sect. 3 which give the functional integral as a sum over critical points of the Yang-Mills action are independent of the particular value of θ . In a direct evaluation, the pre-exponential factors in (6.9) would be determined by performing the functional Grassmann integrations and taking the t → ∞ limit in the localization formula (3.36). In this formula, θ is a continuous parameter and we do not expect the calculation of contributions from Gaussian fluctuations to depend on the rationality of θ. Thus the pre-exponential factors in (6.9) yield the value of the fluctuation determinant at each critical point in the semi-classical expansion of the partition function. Moreover, as emphasized in Sect. 6.3, the contributing submodules to this expansion are always

Instanton Expansion of Noncommutative Gauge Theory in Two Dimensions

375

Heisenberg modules, which are the only projective modules in the irrational case. In this regard it is interesting to note the role of the alternating sign factors (−1)|ν | in (6.9). The global minimum of the action, which has |ν | = 1, is the only stable critical point of the theory. According to general stationary phase arguments [34], a classical solution with n unstable modes is always weighted with a phase − e π i n/2 in our normalization. Thus each submodule in a partition which defines a critical point corresponds to a local extremum of the noncommutative Yang-Mills action which is unstable in two directions. Going back to the topological sum (5.16), we see that, as is the usual case in U (p) gauge theory, each unit charge instanton configuration yields 2p − 2 negative modes. The instanton configurations will be studied in more detail in Sect. 9. Secondly, consider an approximation to the partition function for irrational θ by rational theories. Formally, this requires a limit θ = limm nm /sm with both nm → ∞ and sm → ∞ as m → ∞. As we have seen in the previous section, the minimum dimension of a submodule which is permitted over the noncommutative torus Anm /sm is 1/sm . Consequently, any rational approximation to the partition function would contain partitions of arbitrarily small dimension, as we expect to see for irrational values of the noncommutativity parameter θ. With these pieces of evidence at hand, we thereby propose that the partition function of noncommutative gauge theory on a Heisenberg module Ep,q over a two-dimensional torus is given for all values of the deformation parameter θ by the expression

Zp,q (g 2 , θ ) =

(−1)|ν |

(p , q )∈Pp,q (θ)

×

|ν | k=1

−νa /2 2g 2 R 2 (pa − qa θ)3 a≥1

νa !

2 & 1 qk q , (7.3) exp − 2 2 pk − qk θ − 2g R pk − q k θ p − qθ %

where the integer a labels the νa submodules of dimension dima = pa − qa θ . This formula exhibits the anticipated universality between the irrational and rational cases [37], a feature which we will see more of in the following. Note that the contributions from classical solutions containing submodules of very small Murray-von Neumann dimension are exponentially suppressed in (7.3). In what follows we will explore some applications of this formalism. 8. Smoothness in θ An important issue surrounding noncommutative field theories in general is the behaviour of the partition function and observables as functions of the noncommutativity parameter θ . For example, the poles at θ = 0 which arise from perturbative expansions are the earmarks of the UV/IR mixing phenomenon [55]. However, it is not yet clear in the continuum field theories whether this is an artifact of perturbation theory or if it persists at a nonperturbative level.9 A clearer understanding of the behaviour of the nonperturbative theory as a function of θ is therefore needed to fully address such issues. Related to this problem is the question of approximation of irrational noncommutative field theories by rational ones. If the quantum field theory is at least continuous in θ 9 In the lattice regularization of noncommutative field theories [15], UV/IR mixing persists at a fully nonperturbative level as a kinematical effect.

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then it can be successively approximated by rational theories. In particular, this would lead to a hierarchy of Morita dual descriptions in terms of quasi-local degrees of freedom [56] and also finite-dimensional matrix model approximations to the continuum noncommutative field theory [16]. It has also been suggested that smooth behaviour of physical quantities in θ could, by Morita equivalence, imply very stringent constraints on ordinary large N gauge theories on tori [57]. These issues have been further addressed recently in [58]. While physically one would not expect to be able to measure a distinction between rational-valued and irrational-valued observables, it has been observed that in certain examples and at high energies, generic non-BPS physical quantities exhibit discontinuous effects as functions of the deformation parameter, due to the multifractal nature of the renormalization group flows in these cases [59]. For instance, when θ is an irrational number, the cascade of Morita equivalent descriptions is unbounded as the energy of the system increases and no quasi-local description of the theory is possible beyond a certain energy level. To provide some different insight into these problems, in this section we will analyze the behaviour of quantum Yang-Mills theory on the noncommutative two-torus as a function of θ, using its representation (7.3) as a sum over partitions associated with the Heisenberg module Ep,q . As we have seen, each critical point of the Yang-Mills action is determined by a partition which is a list of pairs of integers |ν | (p , q ) = (pk , qk ) labelling submodules that obey the constraints (4.24) on their k=1 dimensions and Chern numbers. We will now develop a graphical technique for constructing solutions of these constraints which will serve as a useful method for obtaining solutions of the noncommutative Yang-Mills equations of motion. This method makes no distinction between rational or irrational θ and smoothly interpolates between the two cases. We will then use it to prove the smoothness of the Yang-Mills partition function (7.3) as a function of the noncommutativity parameter θ . This continuity result is in agreement with an analysis, based on continued fraction approximations, of the behaviour of classical averages on a fixed projective module [60].

8.1. Graphical determination of classical solutions. Consider the integral lattice K0 (Aθ ) of K-theory charges, which we will view as a subset of the plane R2 . Each point (pk , qk ) on this lattice corresponds to an isomorphism class Epk ,qk of projective modules over the noncommutative torus. Through each such point we draw a line in R2 of constant (positive) dimension according to the equation p − qθ = pk − qk θ , k = 1, 2, 3, . . . .

(8.1)

These lines all have slope θ −1 . For irrational values of θ , there is a unique solution, (p, q) = (pk , qk ), to (8.1) for each k, and hence there is only one point of the integer lattice on each line. Consequently there is an infinite number of parallel lines of constant dimension in any region of the K-theory lattice. On the other hand, if θ = n/s is a rational number, then there are infinitely many solutions (p, q) of Eq. (8.1) for each k and hence a large degeneracy of lattice points lying on each line. In this case there are only a finite number of lines of constant dimension in any region of the K-theory lattice. For a given Heisenberg module En,m , there are two important lines of constant dimension which will enable the enforcing of the constraints (4.24). These are the lines p−qθ = 0 and p − qθ = n − mθ . A partition which yields a critical point of the Yang-Mills

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377

Fig. 1. Graphical representation of the partition (1, 1) , (1, −1) , (2, 3) , (1, 0) ∈ P5,3 (θ ) which defines a solution of the noncommutative Yang-Mills equations of motion on the projective module E5,3 . The sequence of lines in R2 of constant dimension for the case θ = 1/2 is depicted. The dashed line goes through the sequence of points whose successive differences make up the elements of the partition

action on En,m is found by taking a sequence of points lying on lines of strictly increasing dimension, beginning at the origin (0, 0) and terminating at the point (n, m). Taking the difference of the coordinates of successive points gives the topological numbers (nk , mk ) of the submodules in the partition. The choice of a sequence of points which lie on lines of strictly increasing dimension guarantees that each submodule is of positive dimension. Fixing the initial and final points ensures that the constraints on the total dimension and Chern number are satisfied. An illustrative example of this procedure for the module E5,3 is depicted in Fig. 1. All finite sequences of points obeying these rules give all possible solutions of the constraints (4.24), and hence all critical points of the noncommutative Yang-Mills action corresponding to all solutions of the equations of motion. Note that the integer |ν |, counting the total number of submodules in a partition, may in this way be regarded as a topological invariant of the associated graphs. 8.2. Proof of θ -smoothness. Having determined all partitions graphically for fixed θ , we can now study how semi-classical quantities vary with a change of θ . From n−mθ = dim En,m , we see that θ is the inverse slope of lines of constant dimension in the (p, q) plane. Thus a change in θ amounts to a change in slope of the lines of constant dimension. For partitions a small change in θ leads to a small change in the dimensions of submodules in a partition but leaves the number |ν | of submodules and their topological numbers unchanged. The partition function (7.3) clearly varies smoothly under such variations of the noncommutativity parameter. This smooth behaviour terminates when a change in θ leads to a violation of the requirement that each submodule of a partition be of positive dimension. Such a condition can occur when a submodule of very small dimension to the right of the line p − qθ = 0 is pushed through to negative dimension by an infinitesimal variation of θ. For example, the partition depicted in Fig. 1 represents a valid solution of Yang-Mills theory on the module E5,3 for all θ < 1. As θ approaches unity, the dimension of the first submodule E1,1 vanishes. Thus at θ = 1, the constraints

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(4.24) defining partitions are violated and this partition is abruptly removed from the list P5,3 (θ ) of partitions which contribute to the partition function. Of course such an elimination occurs for any partition containing a submodule of vanishing dimension10 and it would appear in general that this leads to a discontinuity in the partition function as a function of θ. There are also various “degenerate” cases that appear to lead to discontinuities, such as those partitions for which the dimension of a component which doesn’t appear first in the list vanishes, or those where the dimensions of two submodules become equal when θ is varied. This leads to a reordering of the submodules and therefore a discontinuous change in the graphical representation of the previous subsection. However, these latter cases do not affect the partition sum in a discontinuous way, and thus only the former types of discontinuities appear to remain. In fact this is not the case and the partition function is smooth in θ. The reason is that the contribution to the partition function (7.3) from partitions with submodules of vanishing dimension are exponentially suppressed, since the Boltzmann weight associated with such topological numbers (n, m) is of order e −1/ dim En,m . Consequently, the partition function has already exponentially damped any contribution from a partition before it is discontinuously dropped due to the positive dimension constraint. It is easy to see that even though derivatives of the partition function with respect to θ will generate singular pre-exponential factors when submodule dimensions vanish, these singularities are all trumped by exponential suppression from the action. Thus all derivatives of the partition function with respect to θ are also finite and continuous. Note that, in the context of rational approximations to irrational values of the noncommutativity parameter, this analysis also shows that perturbations about any rational value of θ will miss exponentially small contributions to the partition function, which may be related to some of the peculiarities observed in the rational approximations of irrational noncommutative gauge theories [59]. The θ -smoothness proof can also be extended to physical (gauge invariant) observables which are at most polynomially singular in θ for modules of vanishing dimension. One such class of observables are the “topological” observables obtained by differentiating the partition function (3.27) with respect to the Yang-Mills coupling constant, n n ∂ n 1 2 ˆ , φˆ φ ln Z (g , θ) = p,q conn ∂g 2 8π 2 R 2 ( n ' n 1 2 = tr N φ(xr ) , (8.2) 8π 2 R 2 r=1

conn

where the brackets · · · conn denote connected correlation functions with respect to the functional integral (3.27), and x1 , . . . , xn are arbitrary points on T2 . For n = 1 this observable is proportional to the average energy of the system Tr Ep,q Fˆ 2ˆ on the A Heisenberg module Ep,q . 9. Instanton Moduli Spaces The expansion (7.3) of the partition function of gauge theory on a noncommutative torus has a natural interpretation as a sum over noncommutative instantons in two dimensions, 10 Recall that the components of a partition are partially ordered according to increasing submodule dimension.

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in the sense that we have defined them at the beginning of Sect. 4. They are classified topologically by the homotopy classes (p, q) of the space of Hermitian projectors Gr θ , and they are inherently nonperturbative since their action contribution to the path inte2 gral is of order e −1/g . However, the semi-classical expansion does not organize the contributions from classical solutions into gauge orbits. Different partitions (p , q ) may give contributions which should be identified as coming from the same instanton. In this section we will discuss the rearrangement of the series (7.3) into a sum over gauge inequivalent critical points and describe the structure of the moduli spaces of instantons that arise, comparing them with those of ordinary Yang-Mills theory in two dimensions. 9.1. Weak coupling limit. We will begin with the weak coupling limit g 2 → 0 of noncommutative gauge theory as it is the simplest case. Due to the invariance property (3.12), the moduli space Mp,q (θ ) of constant curvature connections on the Heisenberg module Ep,q modulo noncommutative gauge transformations has a natural symplectic structure inherited from the symplectic two-form (3.9) on Cp,q . As shown in Sect. 3.3, the partition function Zp,q (g 2 = 0, θ) formally computes the symplectic volume of Mp,q (θ ). Let us first describe this moduli space [37]. By using a Morita duality transformation (6.5) of the background magnetic flux if necessary, we can assume that f = 0 in (4.6) without loss of generality. We therefore need to classify the irreducible representations determined by the Heisenberg module (4.7). As discussed in Sect. 4.1, the Weyl-’t Hooft algebra (4.8) has N irreducible components, where N is the rank of the noncommutative gauge theory given by (4.13). On the other hand, in Sect. 6.3 we saw that each such irreducible representation has a pair of complex moduli (z1 , z2 ) generating the center of the Weyl-’t Hooft algebra. Thus the inequivalent irreducible representations of the ˜ 2 which matrix algebra (4.8) are labelled by a pair of complex numbers ζ = (z1 , z2 ) ∈ T live on a commutative torus dual to the original noncommutative torus. q ˜ 2 are the corresponding irreducible representations, then the If Wζ ⊂ C , ζ ∈ T Heisenberg module (4.7) decomposes into irreducible Aθ -modules according to (9.1) Ep,q = L2 (R) ⊗ Wζ1 ⊕ · · · ⊕ WζN . Gauge transformations which preserve the constant curvature condition (4.6) are finitedimensional unitary matrices in U (q). Central elements of the Weyl-’t Hooft algebra are represented by diagonal matrices with respect to the decomposition (9.1). There is a residual gauge symmetry which acts by permutation of the N summands in (9.1) as the permutation group N , and therefore the moduli space of constant curvature connections associated with the module Ep,q over the noncommutative torus is the symmetric orbifold [37] N ˜ 2 ≡ T˜ 2 (9.2) / N Mp,q (θ ) = SymN T of dimension 2N. This space is identical to the moduli space of flat bundles for commutative Yang-Mills theory on an elliptic Riemann surface with structure group U (N ) [35], 2 i.e. MN (θ = 0) = Hom π1 (T ) , U (N) / U (N ), since the maximal torus of U (N ) is U (1)N consisting of diagonal matrices and its discrete Weyl subgroup is precisely the symmetric group N . The standard symplectic geometry on (9.2) possesses conical N singularities on the coincidence locus, i.e. the “diagonal” subspace of T˜ 2 .

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Let us now consider this result in light of the instanton expansion. We take = 0 without loss of generality. The g 2 → 0 limit of the Boltzmann factor in the partition function (7.3) is non-vanishing only in the zero instanton sector qk = 0 ∀k ≥ 1. But the constraints (4.24) with all qk = 0 are just equivalent to those we encountered in Sect. 5 in the commutative limit θ = 0, with rank N = p. The same is true of the fluctuation determinant factors in (7.3), and hence the partition function at weak coupling is given by Zp,q (g = 0, θ) = 2

lim

g 2 →0

ν :

#

a

N (−1)νa 2 2 3 −νa /2 , 2g R a νa !

(9.3)

aνa =N a=1

where we have eliminated the (constant) exponential factor by a suitable renormalization of the quantum field theory [29]. The independence of (9.3) in the noncommutativity was observed in Sect. 3.3, where we saw that the auxiliary field φ in (3.27) was essentially a commutative field. In this way, the theory at g 2 = 0 eliminates all dependence on the parameters θ and R, and it is identical to topological Yang-Mills theory on the commutative two-torus [29]. This feature of the weakly coupled gauge theory is in agreement with the coincidence of the moduli space of the zero instanton sector in the commutative and noncommutative cases. It should be stressed though that, in contrast to the commutative case, by Morita equivalence the expressions for the moduli space (9.2) and partition function (9.3) over the noncommutative torus hold generically for all (not necessarily flat) constant curvature connections. In other words, in the noncommutative case the gauge quotiented level sets of the moment map µ on Cp,q are all equivalent. The partition function has non-analytic behaviour in theYang-Mills coupling constant as g 2 → 0, with a pole of order |ν | in g for each partition. We can relate the singularities arising in (9.3) in a very precise way to the orbifold singularities of the moduli space (9.2) which appear whenever the Heisenberg module Ep,q is reducible. The latter singular points come from the fixed point set of the action of the permutation group N on N T˜ 2 , which is straightforward to describe. As in Sect. 5, let C[ν ] = C[ν1 , . . . , νN ] be the conjugacy class of a given element σ ∈ N . An elementary cycle of length a N ˜2 leaves an N-tuple (ζ1 , . . . , ζN ) ∈ T invariant only if the a points on which it acts coincide. It follows that the fixed point locus of any permutation σ in the conjugacy class C[ν ] is given by C[ν ] N N νa ˜2 ˜2 T T = .

(9.4)

a=1

On each such fixed point set there is still the action of the stabilizer subgroup C(ν ) of N , which consists of all elements σ ∈ N that commute with σ and is given explicitly in terms of semi-direct products as C(ν ) =

N

νa Zνaa .

(9.5)

a=1

Here the symmetric group νa permutes the νa cycles of length a, while each cyclic group Za acts within one particular cycle of length a. Distinct singular points of the symmetric orbifold (9.2) then arise at the C(ν ) invariants of the fixed point loci (9.4).

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Only the subgroups νa of the centralizer (9.5) act non-trivially on (9.4). The singular point locus of the moduli space of constant curvature connections is thereby obtained as the disjoint union over the conjugacy classes C[ν ] of N of the strata C[ν ] N N 2 ˜ T / C(ν ) = Symνa T˜ 2 .

(9.6)

a=1

Given the result (9.6), the interpretation of the singularities in the formal orbifold volume (9.3) is now clear. It is a sum over the connected components, labelled by conjugacy classes in N , of the total singular locus of the orbifold symmetric product (9.2). Within each conjugacy class (partition) C[ν ], the contribution νa from a gauge equivalence ˜2 class of connections associated with a toroidal factor T in (9.6) is weighted by a singular fluctuation determinant (−1)νa (2g 2 R 2 a 3 )−νa /2 . Recall from Sect. 6.2 that the module dimension factor a 3 here ensures invariance under Morita duality. Gauge invariance dictates that the total contribution from the νa cycles of length a (submodules of rank a) be divided by the appropriate residual symmetry factor νa ! which is the order of the local orbifold group νa acting in (9.6). Thus the conical singularities of the zero instanton sector are not smoothed out by the noncommutativity, as one might have naively expected [61, 63], and the moduli spaces of flat connections are the same in both commutative and noncommutative cases. The corresponding partition functions (9.3) represent the contribution of the global minimum µ−1 (0) to the localization formula for the functional integral. We shall now analyze how these properties change as one moves away from the weak coupling limit of the noncommutative gauge theory. As we will see, the orbifold singularities for coincident instantons on the moduli space still persist. Geometrically, the noncommutative instantons of two dimensional gauge theory on a torus remain point-like and hence have no smoothing effect on the conical singularities ˜ 2 where two or more points come together. that occur on SymN T 9.2. Instanton partitions. To count instantons labelled by a generic partition (p , q ) consisting of non-zero Chern numbers qk , we need to arrange the expansion (7.3) into a sum over gauge inequivalent classical solutions. The essential problem which arises is the isomorphism Emp,mq ∼ = ⊕m Ep,q of projective modules. Partitions of either side of this isomorphism lead to gauge equivalent contributions to the partition function and, in particular, from (4.16) it follows that the minimizing connections on Emp,mq and Ep,q have the same constant curvature. Thus we need to refine the definition of partition given in Sect. 4.2 somewhat so as to combine submodules which yield the same constant curvature and hence prevent the over-counting of distinct noncommutative Yang-Mills stationary points [36]. This we do by writing any submodule dimension in the form pk − qk θ = Nk pk − qk θ , (9.7) where Nk = gcd(pk , qk ), and the integers pk and qk are relatively prime. The corresponding curvature (4.16) is independent of the noncommutative rank Nk , and so we should also restrict to submodules for which each K-theory charge (pk , qk ) is distinct. Therefore, we restrict the counting of critical points of the noncommutative Yang-Mills action to the sets of integers (N , p , q ) ≡ (Na , pa , qa ) which satisfy, in addition a≥1

to the constraints (4.24), the requirements that Na > 0, pa and qa are relatively prime,

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and the pairs of integers (pa , qa ) are all distinct. We shall refer to such a collection of integers as an “instanton partition”. The additional constraints imposed on an instanton partition guarantee that we do not count as distinct those partitions which contain some submodules that can themselves be decomposed into irreducible components. Let us look at the structure of the moduli space Mp,q (N , p , q ; θ) associated with an instanton partition (N , p , q ) [36]. We want to determine the space of gauge orbits c on submodule decompoof the associated critical point connections ∇ˆ cl = a≥1 ∇ˆ (a) sitions Ep,q = ENa pa ,Na qa . (9.8) a≥1

Since each constant curvature on ENa pa ,Na qa is distinct and any gauge transformation Uˆ ∈ G(Ep,q ) preserves the constant curvature conditions, every Uˆ is also a unitary operator on each instanton submodule ENa pa ,Na qa → ENa pa ,Na qa . It follows that the instanton moduli space is given by Mp,q (N , p , q ; θ ) = MNa pa ,Na qa (θ ) , (9.9) a≥1

where each MNa pa ,Na qa (θ ) is the moduli space of constant curvature connections on the Heisenberg module ENa pa ,Na qa . From (9.2) we thus find that (9.9) can be written in terms of a product of symmetric orbifolds as [36] Mp,q (N , p , q ; θ ) = SymNa T˜ 2 . (9.10) a≥1

This result generalizes the instanton moduli space (9.2) which corresponds to the global minimum of the noncommutative Yang-Mills action on Ep,q . 9.3. Examples. To get a feel for how the moduli spaces (9.10) classify the reorganization of the partition function (7.3) into a sum over distinct instanton contributions, let us consider two very simple examples [33]. For θ = 0 and p = 2 the partition function is easily written in the form e −q /4g R 1 Z2,q (g 2 , 0) = − ) + 2 2 4g R 16g 2 R 2 2

2

2

∞

e

−

1 2g 2 R 2

q12 +(q−q1 )2

.

(9.11)

q1 =−∞

When the Chern number q is odd, this is a sum over inequivalent instanton configurations, and the two terms in (9.11) are associated respectively with the smooth moduli spaces

M2,q

M2,q (1, 2, q; 0) = T˜ 2 , ˜2 . (1, 1, q1 ) , (1, 1, q2 ) ; 0 = T˜ 2 × T

(9.12) (9.13)

˜ 2 , representing the Heuristically, with the appropriate symmetry factors, each factor T single instanton moduli space, contributes a mode with fluctuation determinant

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) −1/ 16g 2 R 2 . On the other hand, when q = 2q is even there is a term in the infinite series in (9.11) which yields the same Boltzmann weight as the first term, and so these two terms should be combined to give * + 1 1 2 2 2 Z2,2q (g 2 , 0) = e −q /g R − ) + 2 2 4g R 16g 2 R 2 − 1 q 2 +(2q −q1 )2 1 1 2 R2 2g + 2 2 . (9.14) e 4g R q1 =q

Again the last term in (9.14) may be attributed to contributions from instantons in the smooth moduli space (9.13) with q = 2q and q1 = q2 . The two gauge equivalent instanton contributions to the first term are attributed with the singular moduli space in this case, M2,2q (2, 1, q ; 0) = Sym2 T˜ 2 .

(9.15)

˜ 2 with the disjoint The singular locus of the symmetric orbifold (9.15) is Sym2 T˜ 2 T sets corresponding to the identity and order two elements of the cyclic group Z2 , respectively. As in Sect. 9.1, the sum of contributions to the first term in (9.14) are readily seen to be those associated with the components of the total singular point locus of (9.15). For θ = 0 and p = 3 the partition function is given by e −q /6g R 1 + Z3,q (g 2 , 0) = − ) 2 R2 2 2 32g 54g R 2

2

2

1 − 2 6(2g R 2 )3/2

∞

∞

e

q1 =−∞ ∞

e

−

−

1 4g 2 R 2

1 2g 2 R 2

2q12 +(q−q1 )2

q12 +q22 +(q−q1 −q2 )2

.

(9.16)

q1 =−∞ q2 =−∞

For any q ∈ / 3 Z the expression (9.16) can be written as a sum over distinct instanton contributions as Z3,q (g 2 , 0) 2 2 2 ∞ 1 1 e −q /6g R − 21 2 2q12 +(q−q1 )2 + − e 4g R =− ) 32g 2 R 2 6(2g 2 R 2 )3/2 q =−∞ 54g 2 R 2 1 ∞ − 1 (2q1 −q)2 +(2q1 −q2 )2 +q 2 1 2 2 R2 2g − e 6(2g 2 R 2 )3/2 q =−∞ q2 =q1 1 ∞ 1 − 21 2 q12 +q22 +(q−q1 −q2 )2 2g R (9.17) − e 6(2g 2 R 2 )3/2 q =−∞ q1 =q mod 2

2

corresponding respectively to the instanton moduli spaces

M3,q

˜2 , M3,q (1, 3, q; 0) = T (1, 1, q1 ) , (2, 1, q2 ) ; 0 = T˜ 2 × Sym2 T˜ 2 ,

(9.18) (9.19)

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˜2 × T ˜2 , M3,q (1, 1, q1 ) , (1, 2, q2 ) ; 0 = T ˜2 × T ˜2 × T ˜2 . M3,q (1, 1, q1 ) , (1, 1, q2 ) , (1, 1, q3 ) ; 0 = T

(9.20) (9.21)

˜ 2 in the Note again how the fluctuation determinants in (9.17) weight each factor of T corresponding moduli space, and how the second term incorporates the sum over sin˜ 2 in (9.19). For q = 3q , the second term in gularities of the symmetric orbifold Sym2 T (9.17) yields a contribution to the global minimum for q1 = q , and we have * + 1 1 1 2 /2g 2 R 2 2 −3q Z3,3q (g , 0) = e −) + − 32g 2 R 2 6(2g 2 R 2 )3/2 54g 2 R 2 1 1 − 21 2 2q12 +(3q −q1 )2 4g R + − e 32g 2 R 2 6(2g 2 R 2 )3/2 q1 =q ∞ − 1 (2q1 −3q )2 +(2q1 −q2 )2 +q 2 1 2 2 2 − e 2g R 6(2g 2 R 2 )3/2 q =−∞ q2 =q1 1 ∞ 1 − 21 2 q12 +q22 +(3q −q1 −q2 )2 2g R . − e 6(2g 2 R 2 )3/2 q =−∞ q1 =3q mod 2

2

(9.22) The last three terms in (9.22) may again be attributed to contributions associated with the instanton moduli spaces (9.19)–(9.21), respectively, with q = 3q and q1 = q2 = q3 . The first term represents the gauge equivalent instanton contributions coming from replacing the smooth moduli space (9.18) by the singular one M3,3q (3, 1, q ; 0) = Sym3 T˜ 2 ,

(9.23)

with each fluctuation determinant associated with the singular points of the orbifold (9.23) corresponding to the three conjugacy classes of the symmetric group 3 . These two simple examples illustrate the general technique involved in reorganizing the sum (7.3) over critical points into distinct instanton contributions. They can be deduced, as above, from the singularity structures of the totality of instanton moduli spaces (9.10) corresponding to a Heisenberg module. The Boltzmann weight associated with an instanton partition (N , p , q ) is given by

e −S(N,p

, q ;θ)

=

e

−

1 2g 2 R 2

Na qa2 /(pa −qa θ)

,

(9.24)

a≥1

and about it there is a finite number of quantum fluctuations representing a finite, but nontrivial, perturbative expansion in g −1 . These fluctuations are determined by the singular locus of the corresponding symmetric orbifolds in (9.10). The combinatorial problem of summing over all such instanton partitions is in general quite involved, especially for irrational values of θ when there are infinitely many partitions. However, repeating analogous arguments to those around (7.1,7.2) shows that an instanton partition contains only finitely many components. Thus the perturbative expansion around each instanton contribution contains only finitely many terms, although in the irrational case the exponential prefactor is no longer a polynomial of set order. It is amusing that, within

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the class of noncommutative gauge theories, Morita equivalence allows such a moduli space classification of the instanton contributions even in the commutative case. Such a characterization is otherwise not possible because one only knows the structure of the moduli space of flat connections of commutative gauge theory on T2 . Notice also that for θ = 0 the instanton sums are no longer given by elementary theta-functions. Finally, let us note that the instantons which contribute to the semi-classical expansion of noncommutative gauge theory that we have developed are reminiscent of the solitons on noncommutative tori which arise as solutions of open string field theory describing unstable D-branes wrapping a two-dimensional torus in the background of a constant B-field [64, 65]. An extremum of the tachyon potential is described by a projector of the algebra Aθ , and leads to an effective gauge theory on the corresponding projective module determined by the tachyon. The remaining string field equations of motion are then solved by direct sum decompositions of the given Heisenberg module as we have described them in this paper. A special instance of this are the fluxon solutions which describe the finite energy instantons, carrying quantized magnetic flux, of gauge theory on the noncommutative plane [4–7]. In the present setting these are the classical solutions associated with partitions consisting of only the full module, giving the global minimum of the Yang-Mills action. For the module Ep,q , these solutions have gauge field strength (4.16) and partition function fluxon 2 ˜ (g , θ ) ZN,q

2

e −Nq /2g R (p −q θ) =) . 2g 2 R 2 N 3 (p − q θ)3 2

2

(9.25)

In the large area limit R → ∞ with the dimensionful noncommutativity parameter θ˜ = 2π R 2 θ finite, this is the contribution to the functional integral, along with the appropriate Gaussian fluctuation factor, from a fluxon of magnetic charge q = N q in gauge group rank N . The sum over all q ∈ Z in this limit determines the expansion of noncommutative gauge theory on R2 in terms of fluxons [10]. Acknowledgements. We thank C.-S. Chu, J. Gracia-Bond´ıa, G. Landi, F. Lizzi, B. Schroers and G. Semenoff for helpful discussions and correspondence. We also thank the referee for prompting us to correct several technical inconsistencies in the original version of this paper. This work was initiated during the PIMS/APCTP/PITP Frontiers of Mathematical Physics Workshop on “Particles, Fields and Strings” at Simon Fraser University, Vancouver, Canada, July 16–27 2001. The work of R.J.S. was supported in part by an Advanced Fellowship from the Particle Physics and Astronomy Research Council (U.K.). L.D.P. would like to thank the MCTP for their hospitality during his visit.

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40. Fairlie, D.B., Fletcher, P., Zachos, C.K.: Infinite Dimensional Algebras And A Trigonometric Basis For The Classical Lie Algebras. J. Math. Phys. 31, 1088 (1990) 41. Seiberg, N., Witten, E.: String theory and noncommutative geometry. JHEP 9909, 032 (1999) [arXiv:hep-th/9908142] 42. Blau, M., Thompson, G.: Topological Gauge Theories Of Antisymmetric Tensor Fields. Ann. Phys. 205, 130 (1991) 43. Pimsner, M., Voiculescu, D.: Exact sequences for K-groups and Ext-groups of certain crossed product C ∗ -algebras. J. Oper. Theor. 4, 93 (1980) 44. Rieffel, M.A.: The cancellation theorem for projective modules over irrational rotation C ∗ -algebras. Proc. London Math. Soc. (3) 47, 285 (1983) 45. Connes, A.: C ∗ -algebras and differential geometry. Compt. Rend. Acad. Sci. Paris Ser. A 290, 599 (1980) [arXiv:hep-th/0101093] 46. Rieffel, M.A.: Projective modules over higher-dimensional noncommutative tori. Can. J. Math. 40, 257 (1980) 47. Connes, A., Douglas, M.R., Schwarz, A.: Noncommutative geometry and matrix theory: Compactification on tori. JHEP 9802, 003 (1998) [arXiv:hep-th/9711162] 48. Schwarz, A.: Morita equivalence and duality. Nucl. Phys. B 534, 720 (1998) [arXiv:hep-th/9805034] 49. van Baal , P., van Geemen, B.: A Simple Construction Of Twist Eating Solutions. J. Math. Phys. 27, 455 (1986) 50. Lebedev, D.R., Polikarpov, M.I.: Extrema Of The Twisted Eguchi-Kawai Action And The Finite Heisenberg Group. Nucl. Phys. B 269, 285 (1986) 51. Pioline, B., Schwarz, A.: Morita equivalence and T-duality (or B versus θ ). JHEP 9908, 021 (1999) [arXiv:hep-th/9908019] 52. Seiberg, N.: A note on background independence in noncommutative gauge theories, matrix model and tachyon condensation. JHEP 0009, 003 (2000) [arXiv:hep-th/0008013] 53. Migdal, A.A.: Recursion Equations In Gauge Field Theories. Sov. Phys. JETP 42, 413 (1975) [Zh. Eksp. Teor. Fiz. 69, 810 (1975)] 54. Rusakov, B.E.: Loop Averages And Partition Functions In U(N) Gauge Theory On Two-Dimensional Manifolds. Mod. Phys. Lett. A 5, 693 (1990) 55. Minwalla, S., VanRaamsdonk, M., Seiberg, N.: Noncommutative perturbative dynamics. JHEP 0002, 020 (2000) [arXiv:hep-th/9912072] 56. Hashimoto, A., Itzhaki, N.: On the hierarchy between noncommutative and ordinary supersymmetric Yang-Mills. JHEP 9912, 007 (1999) [arXiv:hep-th/9911057] 57. Guralnik, Z., Troost, J.: Aspects of gauge theory on commutative and noncommutative tori. JHEP 0105, 022 (2001) [arXiv:hep-th/0103168] 58. Alvarez-Gaum´e, L., Barb´on, J.L.F.: Morita duality and large N limits. Nucl. Phys. B 623, 165 (2002) [arXiv:hep-th/0109176] 59. Elitzur, S., Rabinovici, E., Pioline, B.: On the short distance structure of irrational noncommutative gauge theories. JHEP 0010, 011 (2000) [arXiv:hep-th/0009009] 60. Landi, G., Lizzi, F., Szabo, R.J.: From large N matrices to the noncommutative torus. Commun. Math. Phys. 217, 181 (2001) [arXiv:hep-th/9912130] 61. Nekrasov, N.A., Schwarz, A.: Instantons on noncommutative R4 and (2, 0) superconformal sixdimensional theory. Commun. Math. Phys. 198, 689 (1998) [arXiv:hep-th/9802068] 62. Kapustin, A., Kuznetsov, A., Orlov, D.: Noncommutative instantons and twistor transform. Commun. Math. Phys. 221, 385 (2001) [arXiv:hep-th/0002193] 63. Gopakumar, R., Headrick, M., Spradlin, M.: On noncommutative multisolitons. Commun. Math. Phys. 233, 355 (2003) [arXiv:hep-th/0103256] 64. Martinec, E.J., Moore, G.W.: Noncommutative solitons on orbifolds. arXiv:hep-th/0101199 65. Krajewski, T., Schnabl, M.: Exact solitons on noncommutative tori. JHEP 0108, 002 (2001) [arXiv:hep-th/0104090] Communicated by M.R. Douglas

Commun. Math. Phys. 243, 389–412 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0969-3

Communications in

Mathematical Physics

A Generalized Hypergeometric Function II. Asymptotics and D4 Symmetry S.N.M. Ruijsenaars Centre for Mathematics and Computer Science, P.O.Box 94079, 1090 GB Amsterdam, The Netherlands Received: 28 November 2002 / Accepted: 22 May 2003 Published online: 11 November 2003 – © Springer-Verlag 2003

Abstract: In previous work we introduced and studied a function R(a+ , a− , c; v, v) ˆ that generalizes the hypergeometric function. In this paper we focus on a similarity-transformed function E(a+ , a− , γ ; v, v), ˆ with parameters γ ∈ C4 related to the couplings 4 c ∈ C by a shift depending on a+ , a− . We show that the E-function is invariant under all maps γ → w(γ ), with w in the Weyl group of type D4 . Choosing a+ , a− positive and γ , vˆ real, we obtain detailed information on the |Re v| → ∞ asymptotics of the E-function. In particular, we explicitly determine the leading asymptotics in terms of plane waves and the c-function that implements the similarity R → E. Contents 1. 2. 3. 4.

Introduction . . . . . . . . . . D4 -Invariance: First Steps . . . Asymptotics: The Key Results Proofs of Theorems 1.1 and 1.2

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389 395 399 406

1. Introduction This paper may be viewed as a continuation of our previous paper Ref. [1]. The “relativistic” hypergeometric function R(a+ , a− , c0 , c1 , c2 , c3 ; v, v) ˆ at issue here was first introduced in Ref. [2], as an eigenfunction of a relativistic Hamiltonian of Calogero-Moser type. This generalizes the fact that in suitable variables the 2 F1 -function is an eigenfunction of a nonrelativistic Hamiltonian of Calogero-Moser type. Likewise, the discretization of 2 F1 yielding the Jacobi polynomials generalizes to discretizations of R yielding the Askey-Wilson polynomials [3, 4]. Our lecture notes [2] and Ref. [1] (henceforth referred to as I) contain extensive background material and references to related work. More recent surveys pertinent to our R-function include Refs. [5, 6].

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We have occasion to invoke various results from I, some of which we begin by summarizing. This summary also serves to collect some definitions and notation from I, with an eye on making this paper and its sequel Ref. [7] somewhat more self-contained. We begin by recalling how R(a+ , a− , c; v, v) ˆ is defined via an integral involving our “hyperbolic gamma function” G(a+ , a− ; z) as a building block. First, we introduce quantities as ≡ min(a+ , a− ), al ≡ max(a+ , a− ), a ≡ (a+ + a− )/2, α ≡ 2π/a+ a− , (1.1) s1 ≡ c0 + c1 − a− /2, s2 ≡ c0 + c2 − a+ /2, s3 ≡ c0 + c3 ,

(1.2)

cˆ0 ≡ (c0 + c1 + c2 + c3 )/2.

(1.3)

Second, we take a+ , a− > 0 and c ∈ R4 unless explicitly stated otherwise. To ease the exposition, we also choose at first Re v and Re vˆ positive and s1 , s2 , s3 ∈ (−a, a). Then we can define the R-function by the contour integral 1 F (c0 ; v, z)K(c; z)F (cˆ0 ; v, ˆ z)dz, (1.4) R(c; v, v) ˆ = (a+ a− )1/2 C where F (b; y, z) ≡

G(z + y + ib − ia) G(z − y + ib − ia) , G(y + ib − ia) G(−y + ib − ia)

K(c; z) ≡

3 G(isj ) 1 . G(z + ia) G(z + isj )

(1.5)

(1.6)

j =1

(To unburden the notation, we often suppress the dependence on the parameters a+ , a− .) As concerns the G-function occurring in these formulas, we recall from I Appendix A that it can be written as G(a+ , a− ; z) = E(a+ , a− ; z)/E(a+ , a− ; −z),

(1.7)

where E(z) is an entire function (closely related to Barnes’ double gamma function) vanishing solely in the points + ≡ ia + zkl , zkl

k, l ∈ N,

(1.8)

where zkl ≡ ika+ + ila− .

(1.9)

Thus G has the same zeros as E and poles located only at − + ≡ −zkl , zkl

k, l ∈ N.

(1.10)

As a consequence of these pole/zero features, the function K has four upward pole sequences for z ∈ i[0, ∞), whereas F has two downward pole sequences starting at z = ±y − ib.

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Finally, the contour C is given by a horizontal line Im z = d, indented (if need be) so that it passes above the points −v − ic0 , −vˆ − i cˆ0 in the left half plane and the points v −ic0 , vˆ −i cˆ0 in the right half plane, and so that it passes below 0. Thus the four upward pole sequences of the integrand lie above C and the four downward ones lie below C. The integrand has exponential decay as |Re z| → ∞, uniformly for Im z in compact subsets of R, so that the choice of d is immaterial. The analyticity properties of the R-function are known in considerable detail. In particular, it extends to a meromorphic function in all of its eight arguments, provided a+ , a− stay in the right half plane (RHP). Moreover, the (eventual) pole locations are explicitly known. Specifically, introducing new parameters γ0 ≡ c0 − a+ /2 − a− /2, γ1 ≡ c1 − a− /2, γ2 ≡ c2 − a+ /2, γ3 ≡ c3 ,

(1.11)

and defining γˆ0 , . . . , γˆ3 by (1.16) below, the function ˆ Rren (a+ , a− , c; v, v)

3

E(a+ , a− ; δv + iγµ )E(a+ , a− ; δ vˆ + i γˆµ ),

(1.12)

G(a+ , a− ; isj )−1 · R(a+ , a− , c; v, v), ˆ

(1.13)

δ=+,− µ=0

where ˆ ≡ Rren (a+ , a− , c; v, v)

3 j =1

is analytic in RHP2 × C6 , cf. I Theorem 2.2. Hence Rren can only have poles for + + δv = −iγµ + zkl , δ vˆ = −i γˆµ + zkl , δ = +, −, µ = 0, 1, 2, 3, k, l ∈ N. (1.14)

The factors G(isj ) in K are convenient for normalization purposes. We are taking them out in the renormalized function Rren , however, since they give rise to poles (and zeros) that are independent of the variables v and v. ˆ These factors are also absent in the E-function, which is the main object of study in this paper. We now proceed to define this function. To this end we introduce the c-function c(a+ , a− , p; y) ≡

3 1 G(a+ , a− ; y − ipµ ), p ∈ C4 , G(a+ , a− ; 2y + ia)

(1.15)

µ=0

and dual parameters pˆ ≡ Jp, where J is the self-adjoint and orthogonal matrix   1 1 1 1 1  1 1 −1 −1  J ≡  . 2 1 −1 1 −1 1 −1 −1 1

(1.16)

(1.17)

Notice that this entails cˆ and γˆ are again related by (1.11). Introducing the function 2 2 χ (a+ , a− , p) ≡ exp(iα[p · p/4 − (a+ + a− + a+ a− )/8]), p ∈ C4 ,

(1.18)

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we now set E(a+ , a− , γ ; v, v) ˆ ≡ χ (a+ , a− , γ )Rren (a+ , a− , c; v, v)/c(a ˆ ˆ + , a− , γ ; v)c(a+ , a− , γˆ ; v).

(1.19)

The E-function just defined has various symmetry properties that are readily established from its definition. These include (cf. the paragraph in I containing Eq. (2.7)): ˆ = E(a+ , a− , γ ; v, v), ˆ λ > 0, (scale invariance), (1.20) E(λa+ , λa− , λγ ; λv, λv) ˆ = E(a+ , a− , γ ; v, v), ˆ E(a− , a+ , γ ; v, v)

(parameter symmetry),

ˆ v) = E(a+ , a− , γ ; v, v), ˆ E(a+ , a− , γˆ ; v,

(self − duality),

ˆ = −u(a+ , a− , γ ; v)E(a+ , a− , γ ; v, v), ˆ E(a+ , a− , γ ; −v, v)

(1.21) (1.22)

(reflection symmetry), (1.23)

where the u-function is defined by u(a+ , a− , p; y) ≡ −c(a+ , a− , p; y)/c(a+ , a− , p; −y).

(1.24)

Using the relation (cf. (1.2), (1.11)) sj = γ0 + γj + a,

j = 1, 2, 3,

(1.25)

it is also straightforward to check that E is invariant under any permutation of γ1 , γ2 , γ3 . In fact, however, E has a far stronger “hidden” γ -symmetry. Specifically, E is invariant not only under all permutations of γ0 , γ1 , γ2 , γ3 , but also under flipping the sign of any pair of γµ ’s. As is well known, the resulting invariance group, which we denote by W , is the Weyl group of the Lie algebra D4 . We state the symmetry just explained in the following theorem, which is a principal result of this paper. Theorem 1.1 (D4 symmetry). The E-function satisfies ˆ = E(a+ , a− , γ ; v, v), ˆ ∀w ∈ W. E(a+ , a− , w(γ ); v, v)

(1.26)

Morally speaking, the D4 invariance of E follows from its being a joint eigenfunction of four independent AOs (analytic difference operators) that are W-invariant. These AOs arise by similarity transforming the four Askey-Wilson type AOs from I with the above c-function, and their W-invariance is an obvious corollary of Lemmas 2.1 and 2.2. We would like to stress that the Askey-Wilson polynomials do not exhibit D4 -invariance for any choice of normalization. The three-term recurrence obtained by taking vˆ = i cˆ0 + ina− , n ∈ N,

(1.27)

in the pertinent R-function AO is not even S4 -invariant, cf. Eqs. (3.33)–(3.39) in I. (More precisely, it cannot be rendered S4 -invariant via any change of parameters such as (1.11).) For the similarity-transformed AO, however, this discretization of vˆ gives rise to a three-term recurrence of the form a0 c1 P1 + b0 P0 = 2 cos(2rv)P0 , an cn+1 Pn+1 + bn Pn + Pn−1 = 2 cos(2rv)Pn , n > 0, (1.28)

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instead of I (3.34), with an , bn , cn given by I (3.37)–(3.39). The S4 -invariance of the new recurrence coefficients (hence of the resulting polynomials with P0 ≡ 1) now follows from Lemmas 2.1 and 2.2. To be sure, S4 -invariance with this normalization is not a new result. Indeed, Askey and Wilson already pointed out S4 -invariance for their polynomials with a normalization that is connected to the above normalization by a manifestly S4 -invariant similarity, cf. Eqs. (1.24)–(1.27) in Ref. [3]. From our viewpoint, the reason that the D4 -symmetry of the E-function breaks down to S4 -symmetry of the corresponding Askey-Wilson polynomials is the non-invariance of the v-discretization ˆ (1.27) under any sign changes of γ0 , γ1 , γ2 and γ3 . (Recall that cˆ0 equals µ γµ /2 + a.) Even so, within the polynomial context the matrix J (1.17) is already important to understand self-duality properties, and this fact generalizes to the many-variable AskeyWilson (or Koornwinder [8]) polynomials, cf. Ref. [9]. This may be viewed as a hint for the existence of a D4 -symmetric interpolation (which for the case of one variable does exist, as transpires from our results). To appreciate this, recall that D4 admits outer automorphisms connecting the defining and spinor representations of SO(8) (“triality”). In this picture, the matrix J connects the weights of the defining representation and the even spinor representation, as is readily verified. It is therefore a natural question whether other interpolations of the Askey-Wilson polynomials can also be made D4 -symmetric by a suitable similarity. Specifically, we are thinking of the interpolations of Gr¨unbaum and Haine [10], and the special linear combination of the Ismail-Rahman functions [11] studied by Suslov [12, 13] and by Koelink/Stokman [14]. (See Ref. [5] for a comparison of the latter interpolations to our R-function.) The same question can be asked for the |q| = 1 Askey-Wilson function recently introduced by Stokman [15]. We proceed to comment on our proof of Theorem 1.1. As already indicated, the D4 invariance of the four AOs for which E is an eigenfunction “explains” why E is itself D4 -invariant. For a complete proof, though, we cannot extract enough information from the joint eigenfunction property. For one thing, very little is known in general about joint eigenspaces of commuting AOs. For another, even for the case at hand, where we know that E(γ ; v, v) ˆ and E(w(γ ); v, v), ˆ w ∈ W , satisfy the same four AEs (analytic difference equations), there is no general result yielding proportionality. (See Sect. 1 of Ref. [16] for an appraisal of the general situation.) As will become clear below, the main problem to prove Theorem 1.1 consists in showing that a Casorati determinant of E(γ ; v, v) ˆ and E(w(γ ); v, v) ˆ vanishes identically. The pertinent result of Sect. 2 is that a certain (a priori unknown) periodic multiplier in the Casorati determinant has no poles (Lemma 2.3). The known pole locations (1.14) of Rren are the key to obtain this result. However, we are not able to prove within the “algebraic” context of Sect. 2 that the multiplier actually vanishes. We can only show this in Sect. 4, after establishing (in Sect. 3) the asymptotic behavior of the E-function as Re v → ∞ in a suitable strip. The relevant result (Lemma 3.2) involves a restriction on the parameters, which is however solely a consequence of our proof strategy. Indeed, after using Lemma 3.2 to complete the proof of Theorem 1.1, the resulting W -invariance of E can be exploited to extend the domain of validity of our asymptotics results to arbitrary parameters. More precisely, defining the parameter set ≡ (0, ∞)2 × R4 , the restricted parameter set

(1.29)

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∗ ≡ \ {(a, a, 0, 0, 0, 0) | a > 0},

(1.30)

and the leading asymptotics function ˆ ≡ exp(iαv v) ˆ − u(a+ , a− , γˆ ; −v) ˆ exp(−iαv v), ˆ α = 2π/a+ a− , Eas (a+ , a− , γ ; v, v) (1.31) we obtain the following theorem. (We only detail the Re v → ∞ asymptotics of E; thanks to the reflection symmetry (1.23) and the u-asymptotics (3.6), the Re v → −∞ asymptotics is immediate from this.) Theorem 1.2 (Asymptotics). Letting (σ, a+ , a− , γ , δ, v) ˆ ∈ [1/2, 1) × × (0, ∞)2 , we have ˆ < C(σ, a+ , a− , γ , δ, v) ˆ exp(−σ αas v), |(E − Eas )(a+ , a− , γ ; v, v)|

(1.32)

for all v > δ. Here, C is a positive continuous function on [1/2, 1) × × (0, ∞)2 . Next, let (a+ , a− , γ , δ, v) ˆ ∈ ∗ × (0, ∞)2 . Then we have ˆ < C(a+ , a− , γ , δ, Im v, v) ˆ exp(−ρ(a+ , a− , γ )Re v), |(E − Eas )(a+ , a− , γ ; v, v)| (1.33) for all v ∈ C satisfying Re v > δ. Here, C is a positive continuous function on ∗ × (0, ∞) × R × (0, ∞) and ρ is a positive continuous function on ∗ . Finally, let a+ = a− = a, γ = 0 and (σ, a, δ, τ, v) ˆ ∈ [1/2, 1) × (0, ∞)2 × [0, 1) × (0, ∞). Then we have ˆ < C(σ, a, δ, τ, v) ˆ exp(−σ (1 − τ )αaRe v), |(E − Eas )(a, a, 0; v, v)|

(1.34)

for all v ∈ C satisfying Re v > δ, |Im v| ≤ τ a,

(1.35)

with C continuous on [1/2, 1) × (0, ∞)2 × [0, 1) × (0, ∞); moreover, |E(a, a, 0; v, v)| ˆ < C(a, δ, Im v, v), ˆ

(1.36)

for all v ∈ C satisfying Re v > δ, with C continuous on (0, ∞)2 × R × (0, ∞). Admittedly, our proofs of Theorems 1.1 and 1.2 are not exactly straightforward. Of course, we cannot exclude the existence of more direct proofs, avoiding the entanglement of W-invariance and asymptotics that characterizes our proof strategy. In any event, we have tried to render our reasoning more accessible (and possibly more amenable to shortcuts) by opting for an exposition that isolates readily understood key features before turning to detailed technical arguments. At the end of Sect. 4 we also indicate why a direct proof of Theorem 1.2 for arbitrary γ ∈ R4 and Im v ∈ R seems intractable. To conclude this introduction, we would like to mention that the result of Theorem 1.1 is by its nature “best possible”, whereas it is likely that the bounds in Theorem 1.2 are not optimal. (For example, there is evidence that (1.33) holds true with ρ = σ αas , σ ∈ [1/2, 1), for all (a+ , a− , γ ) ∈ , but we are only able to prove this for Im v = 0, cf. (1.32).) On the other hand, the results encoded in Theorem 1.2 are sufficiently strong to handle problems arising in the Hilbert space context of our next paper (Ref. [7]) in this series.

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2. D4 -Invariance: First Steps We begin by recalling from I that R(a+ , a− , c; v, v) ˆ is a joint eigenfunction of four Askey-Wilson type AOs, two acting on v and two on v, ˆ cf. I (3.1)–(3.5). It transpires from I (3.3) why the parameters cµ , µ = 0, 1, 2, 3, can be viewed as coupling constants: When they all vanish, the coefficients of the AOs are constant. On the other hand, this parametrization breaks symmetry properties that only become visible in terms of the shifted parameters γµ . Therefore, we switch to AOs with coefficients depending on γ , obtained from their counterparts in loc. cit. via the parameter change (1.11). To be specific, we first define the coefficient function

4 3µ=0 cosh(π[y − ipµ − ia− /2]/a+ ) C(a+ , a− , p; y) ≡ − , (2.1) sinh(2πy/a+ ) sinh(2π [y − ia− /2]/a+ ) which is manifestly invariant under permutations of the four parameters pµ . Now we define the AO A(a+ , a− , p; y) ≡ C(a+ , a− , p; y) exp(−ia− ∂/∂y) + (y → −y) + Vb (a+ , a− , p; y), (2.2) where Vb (a+ , a− , p; y)

  3 π  pµ + a −  . ≡ −C(a+ , a− , p; y) − C(a+ , a− , p; −y) − 2 cos a+

(2.3)

µ=0

Then the fourAOs I (3.2) amount to A(a+ , a− , γ ; v), A(a− , a+ , γ ; v), A(a+ , a− , γˆ ; v) ˆ and A(a− , a+ , γˆ ; v). ˆ Furthermore, their action on the meromorphic function R(a+ , a− , c(γ ); v, v) ˆ yields eigenvalues 2 cosh(2π v/a ˆ + ), 2 cosh(2π v/a ˆ − ), 2 cosh(2π v/a+ ) and 2 cosh(2π v/a− ), respectively. It is plain from the above definitions that the “external field” Vb (p; y) (2.3) is S4 -invariant in p. It is not obvious, but true that it is actually D4 -invariant. This stronger symmetry is manifest from a second formula for Vb , obtained in the next lemma. Lemma 2.1. We have Vb (a+ , a− , p; y) =

d− (p/a+ ) cosh(2πy/a+ ) + d+ (p/a+ ) cos(π a− /a+ ) , (2.4) sinh(2π[y − ia− /2]/a+ ) sinh(2π [y + ia− /2]/a+ )

with d± (p) ≡ C(p) ± S(p),

S(p) ≡ 4

3

(2.5)

sin(πpµ ),

(2.6)

cos(πpµ ).

(2.7)

µ=0

C(p) ≡ 4

3 µ=0

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Proof. Clearly, the functions on the right-hand sides of (2.3) and (2.4) both have period ia+ and limit 0 as |Re y| → ∞. Thus we need only show equality of residues at their poles in a period strip, choosing a+ /a− irrational to ensure the poles are simple. To this end, we note first that the poles due to the factors sinh(±2πy/a+ ) in (2.3) cancel by virtue of evenness. Likewise, by evenness it suffices to compare residues at y = ia− /2 and y = ia. Their equality amounts to two equations that are solved by (2.5)–(2.7).

Obviously, A(a+ , a− , p; y) is S4 -invariant, but not D4 -invariant. (Indeed, the coefficients of the shifts are not invariant under any sign flips, cf. (2.1).) We now define a similarity-transformed AO that is D4 -invariant. Specifically, we set A(a+ , a− , p; y) ≡ c(a+ , a− , p; y)−1 A(a+ , a− , p; y)c(a+ , a− , p; y).

(2.8)

Then we have the following explicit formula for A. Lemma 2.2. The AO (2.8) can be written A(a+ , a− , p; y) = exp(−ia− ∂/∂y) + Va (a+ , a− , p; y) exp(ia− ∂/∂y) +Vb (a+ , a− , p; y),

(2.9)

with Va (a+ , a− , p; y)

16 3µ=0 cosh(π[y + ipµ + ia− /2]/a+ ) cosh(π [y − ipµ + ia− /2]/a+ ) . (2.10) ≡ sinh(2πy/a+ ) sinh(2π[y + ia− /2]/a+ )2 sinh(2π [y + ia− ]/a+ ) Proof. We recall the G-function satisfies the AEs G(a+ , a− ; z + iaδ /2) = 2 cosh(π z/a−δ ), G(a+ , a− ; z − iaδ /2)

δ = +, −.

(2.11)

Using the δ = − AE and the definition (1.15) of the c-function, we obtain C(a+ , a− , p; y) = c(a+ , a− , p; y)/c(a+ , a− , p; y − ia− ).

(2.12)

Thus the coefficient of the shift y → y − ia− in (2.8) equals 1, as asserted in (2.9). Moreover, for the coefficient of the y-shift over ia− we obtain Va (y) = C(−y)C(y + ia− ), and hence (2.10) follows from (2.1).

(2.13)

It is immediate from Lemmas 2.1 and 2.2 that A(a+ , a− , p; y) is invariant under taking p → w(p) for all w ∈ W . Therefore, E(a+ , a− , w(γ ); v, v), ˆ w ∈ W , is a joint eigenfunction of the four AOs A(a+ , a− , γ ; v), A(a− , a+ , γ ; v), A(a+ , a− , γˆ , v), ˆ A(a− , a+ , γˆ ; v), ˆ denoted briefly as A+ , A− , Aˆ + , Aˆ − , resp. For later purposes we note that due to (2.12)–(2.13) we have Va (a+ , a− , p; y) = u(a+ , a− , p; y + ia− )/u(a+ , a− , p; y),

(2.14)

with u defined by (1.24). We also observe that (2.10) entails invariance of Va under arbitrary sign flips of pµ . (This is not true for Vb , since S(p) (2.6) changes sign under an odd number of flips.)

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Next, we point out the crucial relation J W J = W,

(2.15)

which is readily verified. More specifically, it is useful to note that a permutation of γ1 , γ2 , γ3 yields the same permutation of γˆ1 , γˆ2 , γˆ3 , whereas the transpositions γ0 ↔ γj transform as follows: γ → (γ1 , γ0 , γ2 , γ3 ) ⇒ γˆ → (γˆ0 , γˆ1 , −γˆ3 , −γˆ2 ),

(2.16)

γ → (γ2 , γ1 , γ0 , γ3 ) ⇒ γˆ → (γˆ0 , −γˆ3 , γˆ2 , −γˆ1 ),

(2.17)

γ → (γ3 , γ1 , γ2 , γ0 ) ⇒ γˆ → (γˆ0 , −γˆ2 , −γˆ1 , γˆ3 ).

(2.18)

As a consequence, the dual c-function c(γˆ ; v) ˆ occurring in the definition (1.19) of the E-function is invariant under permutations leaving γ0 fixed, but not under arbitrary permutations, in contrast to the c-function c(γ ; v). Before turning to the W-invariance of E announced above, it is expedient to define a fundamental domain DW for the W-action on R4 . This domain may be viewed as the closure of a Weyl chamber, and it is particularly suited for our later purposes. Specifically, we set DW ≡ {γ ∈ R4 | γ0 ≤ γ1 ≤ γ2 ≤ 0, γ3 ∈ [γ2 , −γ2 ]}.

(2.19)

It is readily verified that this entails J DW = DW .

(2.20)

For the remainder of this section we fix vˆ ∈ (0, ∞) and parameters satisfying a+ , a− , γ0 , γ1 , γ2 , γ3 linearly independent over Q.

(2.21)

γ (j ) ≡ wj (γ ), wj ∈ W, j = 1, 2, w1 = w2 ,

(2.22)

Ej (v) ≡ E(a+ , a− , γ (j ) ; v, v), ˆ j = 1, 2.

(2.23)

Next, we set

and introduce

We are now going to study the Casorati determinant D(v) ≡ E1 (v + ia− /2)E2 (v − ia− /2) − (i → −i),

(2.24)

pertinent to the eigenvalue AEs A+ Ej (v) = 2 cosh(2π v/a ˆ + )Ej (v), j = 1, 2.

(2.25)

We need some well-known features of Casorati determinants, detailed for example in Sect. 1 of Ref. [16]. Our goal is to show that D(v) vanishes identically. We are only able to prove this by obtaining a contradiction from the assumption that this is not the case. Furthermore, we can only arrive at this contradiction in Sect. 4, after determining the |Re v| → ∞ asymptotics of E in Sect. 3.

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In this section, however, we derive a key feature of D(v), assuming from now on D(v) is not identically 0. For a start, this assumption together with (2.25) entails that D(v) satisfies the first order AE, D(v + ia− /2) 1 = . D(v − ia− /2) Va (a+ , a− , γ ; v)

(2.26)

(Cf. Eqs. (1.1)–(1.6) in Ref. [16].) By (2.14), it now follows that the function m(v) ≡ −D(v)u(a+ , a− , γ ; v + ia− /2)

(2.27)

is a meromorphic ia− -periodic function. Since we explicitly know the (eventual) poles of D(v) and the poles of the G-function, we are able to show that m(v) cannot have poles. This is the gist of the next lemma and its proof, with which we conclude this section. Lemma 2.3. Assume D(v) does not vanish identically. Then m(v) is an entire ia− -periodic function that does not vanish identically. Proof. We have already seen that m(v) is an ia− -periodic meromorphic function. To show that m(v) is pole-free, we first note that from (1.24), (1.15) and the reflection equation G(−z) = 1/G(z) (cf. (1.7)), we have −u(a+ , a− , γ ; v + ia− /2) = G (v)/G(2v + ia− + ia)G(2v + ia− − ia), (2.28) with G (v) ≡

3

G(v + ia− /2 + iδγµ ).

(2.29)

δ=+,− µ=0

We now introduce E˜j (v) ≡ Ej (v)/G(2v + ia), j = 1, 2,

(2.30)

˜ D(v) ≡ E˜1 (v + ia− /2)E˜2 (v − ia− /2) − (i → −i),

(2.31)

and rewrite m(v) as ˜ m(v) = G (v)D(v)G(2v − ia− + ia)/G(2v + ia− − ia) ˜ sinh(2πv/a− )/ sinh(2π v/a+ ), = G (v)D(v)

(2.32)

where we used the G-AEs (2.11). ˜ Next, we study the poles of G (v) and D(v). Clearly, the poles of G (v) are located at v = iδγµ − ia+ /2 − ika+ − ila− , δ = +, −, µ = 0, 1, 2, 3, k ∈ N, l ∈ N∗ , (2.33) ˜ cf. (1.7)–(1.10). Turning to D(v), we deduce using (2.23), (1.19), (1.15) and (1.7) that we have E˜j (v) = χ (γ )c(J γ (j ) ; v) ˆ −1 Rren (c(γ (j ) ); v, v) ˆ

3 (j ) E(−v + iγµ ) (j )

µ=0

E(v − iγµ )

.

(2.34)

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Now the function v → Rren (c(γ (j ) ); v, v) ˆ

3

E(−v + iγµ(j ) )E(v + iγµ(j ) )

(2.35)

µ=0

is entire, cf. the paragraph containing (1.11). Therefore, E˜j (v) can only have poles at the zero locations of the function 3

3

E(v − iδγµ(j ) ) =

δ=+,− µ=0

E(v − iδγµ ).

(2.36)

δ=+,− µ=0

˜ These are given by iδγµ + ia + zkl , cf. (1.8), so we finally conclude that D(v) can only have poles at v = iδγµ + ia+ /2 + ika+ + ila− , δ = +, −, µ = 0, 1, 2, 3, k, l ∈ N.

(2.37)

The upshot is that eventual poles of m(v) sinh(2π v/a+ ) must be located at the points (2.33) and (2.37). Let us now assume that m(v) has a pole at v = v0 , so as to derive a contradiction. To begin with, by ia− -periodicity our assumption entails that m(v) has poles at all points v = v0 + ij a− ,

j ∈ Z.

(2.38)

Only one of these poles can be matched by a pole of the factor 1/ sinh(2π v/a+ ) in (2.32), since we have a+ /a− ∈ / Q. Save for at most one pole, therefore, all poles (2.38) must be located at (2.33) or (2.37). Furthermore, since (2.37) consists of upward pole sequences and (2.33) of downward ones, the poles (2.38) must be located at (2.37) for j → ∞ and at (2.33) for j → −∞. From this we see that iv0 can be written in two ways as a Q-linear combination of a+ , a− , γ , with distinct coefficients of a+ . In view of our requirement (2.21), this yields the desired contradiction.

3. Asymptotics: The Key Results In this paper we focus on the asymptotic behavior of E(a+ , a− , γ ; v, v) ˆ for |Re v| → ∞ with the parameters a+ , a− positive. In this case the |Re z| → ∞ asymptotics of G(a+ , a− ; z) obtained in I Theorem A.1 simplifies considerably. Indeed, in I (A.23)– (A.24) we have φ± = φmax = φmin = 0, and the asymptotics domain I (A.32) reduces to A ≡ {z ∈ C | Re z > al }, al = max(a+ , a− ).

(3.1)

Setting 2 2 + a− )/48 + f (a+ , a− ; z)]), z ∈ ±A, (3.2) G(a+ , a− ; z) = exp(∓iα[z2 /4 + (a+

it now follows from I Theorem A.1 that for σ ∈ [1/2, 1) we have |f (a+ , a− ; z)| < C(σ, a+ , a− , Im z) exp(−σ αas |Re z|), as = min(a+ , a− ), z ∈ ±A, (3.3)

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where C is continuous on [1/2, 1)×(0, ∞)2 ×R. (Here and below, we find it convenient to formulate decay bounds that are uniform on compact subsets of a given set S in terms of positive continuous functions on S, generically denoted by C.) Next, we recall that for a+ , a− positive, G(a+ , a− ; z) has no poles and zeros for |Re z| > 0. It readily follows that there exist functions C± (a+ , a− , δ, Im z) that are continuous on (0, ∞)3 × R and such that 0 < C− < |G(a+ , a− ; z)| exp(−αIm z|Re z|/2) < C+ ,

(3.4)

for all z ∈ C satisfying |Re z| > δ. In the sequel, we will frequently invoke these estimates. The asymptotic behaviors of the c-function (1.15) and u-function (1.24) as |Re y| → ∞ easily follow from (3.1)–(3.3). Specifically, we obtain

∓1 c(p; ±y) exp αy < Cc exp(−σ αas Re y), (3.5) p /2 + a χ (p) − 1 µ µ

|u(p; ±y)χ (p)∓2 − 1| < Cu exp(−σ αas Re y),

(3.6)

for all y ∈ C satisfying Re y > al ; the functions Cs = Cs (σ, a+ , a− , p, Im y) with s = c, u are continuous on [1/2, 1) × × R, with defined by (1.29). At this point it is expedient to add two more estimates whose verification is routine. Specifically, we see from (2.4) and (2.10) that there exist functions Cs (a+ , a− , p, δ) with s = a, b that are continuous on × (0, ∞) and such that |Va (a+ , a− , p; ±y) − 1| < Ca exp(−αa− Re y),

(3.7)

|Vb (a+ , a− , p; ±y)| < Cb exp(−αa− Re y),

(3.8)

for all y ∈ C satisfying Re y > δ. For the remainder of this section we choose vˆ ∈ [r− , r+ ], 0 < r− < r+ ,

(3.9)

Re v > r+ + al .

(3.10)

This ensures that the two downward pole sequences due to F (cˆ0 ; v, ˆ z) are at a distance at least r− from the imaginary axis and the ones due to F (c0 ; v, z) at a distance larger than vˆ + al . Moreover, the integrand Iren (γ ; v, v, ˆ z) ≡ (a+ a− )−1/2

F (γ0 + a; v, z)F (γˆ0 + a; v, ˆ z) ,

3 G(z + ia) j =1 G(z + ia + itj )

tj ≡ γ0 + γj = γˆ0 + γˆj , j = 1, 2, 3,

(3.11)

(3.12)

of the function Rren (cf. (1.13) and (1.2)–(1.6), (1.11)) has simple poles in the z-plane at ±vˆ − i cˆ0 . We continue to study the residues of Iren at these points. To do so, we recall that the residue of G(z) at z = −ia is given by Res G(z)|z=−ia = i(a+ a− )1/2 /2π,

(3.13)

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401

cf. I (A.20). The residue at z = vˆ − i cˆ0 is therefore given by i F (c0 ; v, vˆ − i cˆ0 )G(2vˆ − ia) i = c(γˆ ; −v)F ˆ (c0 ; v, vˆ − i cˆ0 ),

3 2π 2π µ=0 G(vˆ + i γˆµ )

(z = vˆ − i cˆ0 ). (3.14)

Likewise, the residue at −vˆ − i cˆ0 yields i c(γˆ ; v)F ˆ (c0 ; v, −vˆ − i cˆ0 ), 2π

(z = −vˆ − i cˆ0 ).

(3.15)

Let us now obtain the Re v → ∞ asymptotics of these residues multiplied by the factor −2π iχ (γ )/c(γ ; v)c(γˆ ; v), ˆ

(3.16)

in anticipation of their contribution to the E-function asymptotics (recall (1.19)). For (3.14) we obtain R+ (γ ; v, v) ˆ = −u(γˆ ; −v)χ ˆ (γ )F (c0 ; v, vˆ − i cˆ0 )/c(γ ; v),

(3.17)

for which we have by (3.5) and (3.1)–(3.3), ˆ + u(γˆ ; −v) ˆ exp(−iαv v)| ˆ < C(σ, a+ , a− , γ , Im v, v) ˆ exp(−σ αas Re v), |R+ (γ ; v, v) (3.18) for all v ∈ C satisfying Re v > vˆ + al ; the function C is continuous on [1/2, 1) × × (0, ∞)2 . Likewise, for the residue (3.15) we get R− (γ ; v, v) ˆ = χ (γ )F (c0 ; v, −vˆ − i cˆ0 )/c(γ ; v),

(3.19)

and |R− (γ ; v, v) ˆ − exp(iαv v)| ˆ < C(σ, a+ , a− , γ , Im v, v) ˆ exp(−σ αas Re v).

(3.20)

Clearly, (3.18) and (3.20) reveal the plane wave terms featuring in Eas (1.31). To exploit this, however, we must bound the contour integral shifted across the poles at z = ±vˆ − i cˆ0 . We now turn to this task. We begin by defining the shifted contour. Save for three eventual indentations, it is the horizontal line Im z = −cˆ0 − η,

η ∈ (0, as ).

(3.21)

The η-restriction ensures that all poles of the downward sequences starting at ±v−i ˆ cˆ0 are below the contour, except those at ±v−i ˆ cˆ0 . (Later on, we impose further case-dependent restrictions on η.) The eventual indentations are defined as follows. Setting m ≡ min(0, a − s1 , a − s2 , a − s3 ),

(3.22)

we indent the contour downwards at the imaginary axis, so that its distance to i[m, ∞) is bounded below by d ≡ min(r− /2, as /2).

(3.23)

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S.N.M. Ruijsenaars

(Hence no indentation is needed for m ≥ −cˆ0 + as /2, for instance.) Likewise, if need be we indent the contour upwards at Re z = ±Re v, so that it stays at a distance as /2 from ±Re v + i(−∞, −c0 + as ]. (We will let Im v vary over [−as , as ], so this ensures that the pole sequences starting at ±v − ic0 stay below the contour.) We have depicted the situation in Fig. 1 for a choice of parameters where the three indentations are needed, choosing them rectangular for convenience. The poles of Iren (3.11) lie at ±vˆ − i cˆ0 and on the half lines symbolized by the vertical arrows. We have also chosen Im v = −as . Denoting the shifted contour just defined by Cs , we deduce from the above that we have (E − R+ − R− )(γ ; v, v) ˆ = χ (γ )[c(γ ; v)c(γˆ ; v)] ˆ −1 Iren (γ ; v, v, ˆ z)dz. (3.24) Cs

We may and will put the four z-independent G-functions in Iren in front of the integral. This yields a v-independent prefactor χ (γ )[c(γˆ ; v)G( ˆ vˆ + i γˆ0 )G(−vˆ + i γˆ0 )]−1

(3.25)

that is bounded as (a+ , a− , γ ) varies over -compacts and vˆ varies over [r− , r+ ]. Thus we may and shall omit it in estimating E − R+ − R− . By (3.5) and (3.4), the v-dependent prefactor P (γ ; v) ≡ [c(γ ; v)G(v + iγ0 )G(−v + iγ0 )]−1

(3.26)

obeys |P (a+ , a− , γ ; v)| < CP (a+ , a− , γ , δ, Im v) exp(α(γˆ0 − γ0 + a)Re v),

(3.27)

for all v ∈ C satisfying Re v > δ, with CP continuous on × (0, ∞) × R. Therefore, we are left with estimating the integral IL (γ ; v, v, ˆ z)dz, (3.28) L(γ ; v, v) ˆ = Cs

where IL (γ ; v, v, ˆ z) ≡

G(z + v + iγ0 )G(z − v + iγ0 )G(z + vˆ + i γˆ0 )G(z − vˆ + i γˆ0 ) .

G(z + ia) 3j =1 G(z + ia + itj ) (3.29)

To this end we begin by observing that (3.4) entails |IL | < Ct (a+ , a− , γ , Im v, v, ˆ Im z) exp(∓α(2aRe z−Im vRe v)), ±Re z > Re v+as /2, (3.30) with Ct continuous on × R × [r− , r+ ] × R. Therefore the integrals over the tails of Cs are majorized by C exp(−α(2a ∓ Im v)Re v).

(3.31)

Here and from now on, we use the symbol C to denote a positive function of the parameters a+ , a− , γ and η that can be chosen independent of vˆ ∈ [r− , r+ ], Im v ∈ [−as , as ] and Re v ∈ [r+ + al , ∞), and that is continuous for (a+ , a− , γ ) in the parameter set at issue.

Generalized Hypergeometric Function II

403

In connection with this convention, we add three remarks. First, though the parameter set on which (3.31) holds true is all of (1.29), it will be necessary to restrict the parameters in various ways later on. Second, we repeat that we are going to fix the shift parameter η in (3.21) in a case-dependent way. (Of course, the total contour integral is η-independent, but since we can only estimate it piecemeal, the pieces do have dependence on η.) Third, at this stage there seems to be no reason to restrict Im v to [−as , as ], but the need for this restriction will become apparent shortly. Next, we combine these estimates with the bound (3.27) on the prefactor to obtain upper bounds C exp(α(γˆ0 − γ0 − a ± Im v)Re v)

(3.32)

for the tail contributions to E − R+ − R− . Clearly, for these contributions to converge to 0 as Re v → ∞, it suffices that we have |Im v| < γ0 − γˆ0 + a.

(3.33)

At this point we should stress that we do not know whether (3.33) is necessary for convergence to 0. This restriction is however essential for our analysis to yield the desired convergence. (See also our remarks at the end of Sect. 4 in this connection.) Indeed, for later purposes we need to let Im v vary at least over [−as , as ]. Thus we see that (3.33) can only hold on all of the latter interval when the parameters satisfy γ0 − γˆ0 + a − as > 0.

(3.34)

Obviously, this condition is not valid for arbitrary parameters. In Sect. 4, we will see that for a+ = a− it does hold true on a suitable fundamental domain for the W -action. But when a+ equals a− , (3.34) is plainly false for γ = 0. It should be stressed that these arguments apply irrespective of the choice of shift parameter η. (Recall that (3.30) holds true for all Im z ∈ R.) Therefore, we proceed in two stages. For the remainder of this section, we choose the parameters in a subset r of defined by the restrictions as ∈ (0, al /8],

(3.35)

γ ≤ as /2,

(3.36)

where · denotes the euclidean norm on R4 . This entails |γµ |, |γˆµ | ≤ as /2 (recall J is orthogonal), so that we have γ0 − γˆ0 + a − as > al /4.

(3.37)

Moreover, the requirement (3.36) is clearly W -invariant. Accordingly, we can use the asymptotics results obtained in this section to complete the proof of Theorem 1.1 in the next one. Once we have shown W -invariance of E, we can proceed to the second stage, where we choose γ in a fundamental domain for which γ0 − γˆ0 ≥ 0 with equality only for γ = 0. Due to W -invariance, we can then deduce the asymptotic behavior for arbitrary γ , obtaining Theorem 1.2. After this sketch of our proof strategy, we turn to estimating the contour integral L(γ ; v, v) ˆ (3.28). The pertinent result is the following lemma.

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Lemma 3.1. Assume the parameters a+ , a− and γ are restricted by (3.35) and (3.36), and the variables vˆ and v by (3.9), (3.10) and Im v ∈ [−as , as ].

(3.38)

|L(γ ; v, v)| ˆ < C exp(−α(γˆ0 − γ0 + a + η)Re v),

(3.39)

Then we have

with C defined below (3.31). Proof. We have already seen that the integrals over the right and left tail are bounded above by (3.31). Since we have |Im v| ≤ as , we obtain an upper bound C exp(−αal Re v)

(3.40)

for both tail integrals. Next, we examine the integral over the right indentation. On this piece of the contour, the G-function G(z − v + iγ0 ) in (3.29) is bounded independently of Re v, and so we deduce from (3.4), |IL | < C exp(α[−4aRe z + (Im z + γ0 )(Re v − Re z) + Im v(Re v + Re z)]/2). (3.41) Since |Re z−Re v| ≤ as /2 on the indentation, and since Im v ≤ as , this can be majorized by |IL | < C exp(−αal Re v).

(3.42)

As the length of the indentation is bounded, its contribution is bounded by (3.40). Proceeding analogously for the left indentation, we see that its contribution is once more bounded by (3.40). Next, we study the contribution of the middle indentation. We need only consider the two v-dependent G-functions in (3.29), since the remaining ones stay bounded. Thus we get from (3.4) |IL | < C exp(α[(Im z + γ0 )Re v + Im vRe z]).

(3.43)

Now Im v and Re z stay bounded and we have Im z ≤ −γˆ0 − a − η on the middle indentation. Hence we obtain |IL | < C exp(−α(γˆ0 − γ0 + a + η)Re v),

(3.44)

and since the indentation length is bounded, its contribution to L is majorized by C exp(−α(γˆ0 − γ0 + a + η)Re v).

(3.45)

We proceed with the horizontal piece of the contour where Re z varies from d (3.23) ˆ we can proceed just as for the midto Re v − as /2, cf. Fig. 1. On the part from d to v, dle indentation, from which we deduce its contribution is bounded by (3.45). On the remainder, however, we must take all G-functions into account. Doing so, we obtain from (3.4), |IL | < C exp(α[(Im z + γ0 )Re v + (Im v − Im z − γ0 − 2a)Re z]), vˆ < Re z < Re v − as /2.

(3.46)

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Fig. 1. The shifted contour Cs in the z-plane

Consider now the coefficient A of Re z. Since Im z = −cˆ0 − η and Im v ≤ as , we have A ≡ Im v − Im z − γ0 − 2a ≤ B ≡ −al /2 + as /2 + γˆ0 − γ0 + η.

(3.47)

Using (3.35), (3.36) and η < as , we obtain B < −al /8.

(3.48)

Hence the contribution of the horizontal part Re z ∈ [v, ˆ Re v − as /2] is again majorized by (3.45). Finally, repeating this analysis for the horizontal part Re z ∈ [−Re v + as /2, −d], we obtain once more an upper bound (3.45) for its contribution. The upshot is that the contribution to L for |Re z| ≤ Re v − as /2 is bounded above by (3.45), and the remaining contribution by (3.40). The difference of the Re v-coefficients equals −αB, so by (3.48) it is positive. Therefore, the estimate (3.39) follows.

From (3.39) we obtain an upper bound on the shifted contour integral L(γ ; v, v) ˆ for Re v → ∞, so we can deduce information on the asymptotics of E from our previous analysis. In order to obtain the same type of decay bound from the two residue contributions and from the shifted contour integral, we choose η = σ as .

(3.49)

Now we are prepared for our next lemma, which concludes this section. Lemma 3.2. Letting (σ, a+ , a− , γ , v) ˆ ∈ [1/2, 1) × r × [r− , r+ ], we have |(E − Eas )(γ ; v, v)| ˆ < C(σ, a+ , a− , γ , v) ˆ exp(−σ αas Re v),

(3.50)

for all v ∈ C satisfying Re v > r+ + al , |Im v| ≤ as , with C continuous on [1/2, 1) × r × [r− , r+ ]. Proof. In view of (3.18) and (3.20), it suffices to show |(E − R+ − R− )(γ ; v, v)| ˆ < C(σ, a+ , a− , γ , v) ˆ exp(−σ αas Re v).

(3.51)

Recalling (3.24)–(3.28), we need only multiply (3.27) and (3.39) to see that (3.51) follows from our choice (3.49).

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4. Proofs of Theorems 1.1 and 1.2 With Lemmas 2.3 and 3.2 at our disposal, it is not difficult to prove the W -invariance of the E-function, as will now be detailed. Proof of Theorem 1.1. We recall that for vˆ > 0 and parameters restricted by (2.21), we have already obtained information on the Casorati determinant D(v) (2.24), cf. (2.27) and Lemma 2.3. In addition to (2.21), we at first restrict the parameters by requiring (3.35)–(3.36). For notational convenience, we also choose a− = as .

(4.1)

(On account of the parameter symmetry (1.21), this choice is not a restriction.) Now (3.36) is a W -invariant restriction, so that Lemma 3.2 applies to Ej (v), j = 1, 2, cf. (2.22)–(2.23). In particular, it follows from Lemma 3.2 that Ej (v) remains bounded as Re v → ∞ for vˆ ∈ [r− , r+ ] and |Im v| ≤ as . Next, we recall from (3.6) that u(γ ; v) remains bounded as |Re v| → ∞, uniformly for Im v in R-compacts. Therefore, both factors on the rhs of (2.27) remain bounded as Re v → ∞, uniformly for |Im v| ≤ a− /2. In view of the reflection symmetry (1.23), this is also the case for Re v → −∞. It follows that the entire ia− -periodic function m(v) remains bounded as |Re v| → ∞, uniformly for vˆ ∈ [r− , r+ ] and |Im v| ≤ a− /2. From Liouville’s theorem we then infer that m(v) is constant. We continue to invoke Lemma 3.2 once more, this time to show the constant equals 0. First, we note that we may write (cf. (1.15) and (1.24))

3 µ=0 G(y − ipµ )G(y + ipµ ) u(p; y) = − . (4.2) G(2y + ia)G(2y − ia) From this we obtain u(w(p); y) = u(p; y), ∀w ∈ W.

(4.3)

Recalling (2.15), we deduce that the u-functions in the asymptotics (3.50) of Ej (v) satisfy u(J γ (j ) ; −v) ˆ = u(J γ ; −v), ˆ j = 1, 2.

(4.4)

As a consequence, the finite part of the asymptotics of Ej (v) (as Re v → ∞ with vˆ ∈ [r− , r+ ] and |Im v| ≤ a− ) is the same for j = 1 and j = 2. But then we have lim

Re v→∞

D(v) = 0, |Im v| ≤ a− /2,

(4.5)

and so the constant vanishes, as announced. The upshot is that the function m(v) vanishes identically. Hence the assumption of Lemma 2.3 is false, i.e., D(v) vanishes identically. This entails that the quotient Q(a+ , a− , γ , w1 , w2 , v; ˆ v) ≡ E(a+ , a− , w1 (γ ); v, v)/E(a ˆ ˆ (4.6) + , a− , w2 (γ ); v, v), satisfies Q(a+ , a− , γ , w1 , w2 , v; ˆ v + ia− /2) = Q(a+ , a− , γ , w1 , w2 , v; ˆ v − ia− /2), cf. (2.24).

(4.7)

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We have now proved ia− -periodicity of Q(v) for vˆ ∈ [r− , r+ ] and parameters a+ , a− , γ restricted by (2.21), (3.35), (3.36) and (4.1). But for Re v, Re vˆ > 0 (say), the functions E1 and E2 are real-analytic in a+ , a− , γ for a+ , a− > 0 and γ ∈ R4 , and for fixed parameters they are meromorphic in v and vˆ (as follows from I). Therefore, (4.7) holds true for parameters a+ , a− > 0, γ ∈ R4 , and variables v, vˆ ∈ C. Now E1 and E2 are also invariant under a+ ↔ a− , cf. (1.21). Hence the same is true for Q. But then we have Q(a+ , a− ; v + ia+ ) = Q(a− , a+ ; v + ia+ ) = Q(a− , a+ ; v) = Q(a+ , a− ; v), (4.8) so Q(v) has both period ia− and period ia+ . Choosing a+ , a− > 0 with a+ /a− ∈ / Q, it follows that Q(v) is constant. By denseness and real-analyticity in a+ , a− , we see that Q(v) is constant for all a+ , a− > 0. A priori, the constant could depend on a+ , a− , γ , w1 , w2 and v, ˆ however. To show that this is not so, we reconsider (4.6), with vˆ and the parameters restricted so that Lemma 3.2 applies. Then E1 (v) and E2 (v) have equal asymptotics as v → ∞, so that Q(v) equals 1. Invoking once again analyticity, we deduce Q(v) = 1 for arbitrary parameters and variables. Hence Theorem 1.1 follows.

Now that we have proved W -invariance of E, we need only determine the Re v → ∞ asymptotics of E for γ in a fundamental domain to handle γ ∈ R4 . (As already became clear from the analysis leading to (3.34), our proof strategy cannot be directly applied to arbitrary γ .) Specifically, we choose γ ∈ diag(−1, 1, 1, −1)DW ,

(4.9)

with DW defined by (2.19). Hence we have 0 ≤ |γ3 | ≤ −γ2 ≤ −γ1 ≤ γ0 .

(4.10)

γˆ0 ≤ γ0 /2,

(4.11)

γ0 − γˆ0 + a − as ≥ γ0 /2 + (al − as )/2.

(4.12)

This entails

so that

If we now reconsider the contribution of the tail integrals for this γ -choice (cf. the paragraph containing (3.34)), we are led to a dichotomy. To be specific, for γ0 > 0 or al > as we see that it converges exponentially to 0 as Re v → ∞, with a rate that can be chosen uniformly for |Im v| ≤ as . But when we have both γ0 = 0 (which entails γ = 0) and al = as = a, then we only get exponential convergence to 0 for |Im v| ≤ a − , > 0, with a rate that goes to 0 as → 0. On the other hand, the tail contribution does remain bounded as Re v → ∞, uniformly for |Im v| ≤ a, cf. (3.31)–(3.32). After these introductory observations, we are prepared to complete the proof of Theorem 1.2. Proof of Theorem 1.2. Since Eas (a+ , a− , γ ; v, v) ˆ (1.31) is manifestly W -invariant and E(a+ , a− , γ ; v, v) ˆ is also W -invariant (as proved in Theorem 1.1), we may and will restrict γ by (4.9). Moreover, both functions are uniformly bounded on any compact subset of the set × {Re v > 0} × (0, ∞), since they are real-analytic in a+ , a− , γ , vˆ

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and analytic in v on the latter set. Therefore, we need only prove the bounds in the theorem for vˆ varying over an arbitrary interval [r− , r+ ] with 0 < r− < r+ , and for δ equal to r+ + al . Hence we can follow the reasoning in Sect. 3. Specifically, recalling the analysis leading to (3.18) and (3.20), we deduce that to obtain (1.32) , it suffices to show ˆ < C(σ, a+ , a− , γ , v) ˆ exp(−σ αas v), |(E − R+ − R− )(a+ , a− , γ ; v, v)|

(4.13)

for all v > r+ + al , with C continuous on [1/2, 1) × × [r− , r+ ]. Adapting the arguments below (3.24), we first obtain (3.31) and (3.32) with Im v = 0. Due to our γ -choice (4.9), we have γˆ0 − γ0 ≤ 0, so the tail contributions to E − R+ − R− are majorized by C exp(−αav).

(4.14)

On the right indentation we may invoke (3.41) with Im v = 0. Arguing as before, we see that its contribution to E − R+ − R− is bounded by C exp(α(γˆ0 − γ0 − a)v).

(4.15)

Repeating the reasoning for the left indentation, we obtain again an upper bound (4.15) for its contribution. Since γˆ0 − γ0 ≤ 0, the bound (4.15) is majorized by (4.14). Turning to the middle indentation, we are once more led to (3.45), so its contribution to E − R+ − R− is bounded by C exp(−αηv).

(4.16)

(Recall we need to multiply by (3.27).) More generally, we obtain this bound on the part of the contour where we have −vˆ < Re z < v. ˆ Next, we invoke (3.46) with Im v = 0. The coefficient of Re z equals cˆ0 + η − γ0 − 2a = γˆ0 − γ0 + η − a ≤ η − a < 0,

(4.17)

since η < as . Choosing η = σ as from now on, so that Im z + γ0 = −γˆ0 + γ0 − a − σ as

(4.18)

on this part of the contour, we deduce that its contribution to E − R+ − R− is bounded by C exp(−σ αas v).

(4.19)

Likewise, the part Re z ∈ [−v + as /2, −d] leads to (4.19). Putting the pieces together, we obtain (4.13) and hence (1.32). We proceed with the special case γ = 0, a+ = a− = as = al = a. Thus we have c0 = cˆ0 = a and m = 0 (cf. (3.22)), while (3.27) reduces to |P (a, a, 0; v)| < CP (a, a, 0, δ, Im v) exp(αaRe v).

(4.20)

As we have already pointed out, in this case our estimates on the contribution of the tail integrals to the rhs of (3.24) yield an upper bound C exp(α(−a + |Im v|)Re v),

(4.21)

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cf. (3.32). For |Im v| ≤ τ a with τ ∈ [0, 1), this is bounded by a multiple of exp((τ − 1)αaRe v), but for |Im v| ≤ a we can only deduce boundedness. Proceeding as in the proof of Lemma 3.1, we reach the same conclusion for the right and left indentations. Since m = 0 in this special case, no middle indentation occurs. For Re z ∈ [−v, ˆ v] ˆ we obtain as before from (3.4), |IL | < C exp(−α(a + η)Re v).

(4.22)

Combining this with (4.20), we see that the contribution of this line segment to the rhs of (3.24) is majorized by C exp(−αηRe v).

(4.23)

Turning to the bound (3.46), we see it reduces to |IL | < C exp(α[(−a − η)Re v + (Im v − a + η)Re z]), vˆ < Re z < Re v − a/2. (4.24) Taking Im v = a, we infer from (4.20) that the contribution of this line segment cannot be bounded by (4.23). (Recall C is by definition independent of Im v ∈ [−a, a].) On the other hand, fixing η ∈ (0, a), we get for Im v ≤ a: Re v−a/2−ia−iη Re v | dzIL (γ ; v, v, ˆ z)| < C exp(α(−a − η)Re v) dxeαηx v−ia−iη ˆ

0

< C exp(−αaRe v).

(4.25)

Combined with (4.20), this yields boundedness for |Im v| ≤ a. Fixing τ ∈ [0, 1) and σ ∈ [1/2, 1) and taking Im v ≤ τ a, η = σ (1 − τ )a, we have Im v − a + η ≤ (σ − 1)(1 − τ )a < 0. Then the contribution of the pertinent integral to (3.24) is bounded above by C(σ, a, τ, v) ˆ exp(−αηRe v), η = σ (1 − τ )a,

(4.26)

where C is continuous on [1/2, 1) × (0, ∞) × [0, 1) × [r− , r+ ]. Repeating these arguments for the line segment −Re v + a/2 < Re z < −v, ˆ we obtain the same conclusion. The upshot is that the rhs of (3.24) is bounded uniformly for |Im v| ≤ a and vˆ ∈ [r− , r+ ], whereas for |Im v| ≤ τ a with τ ∈ [0, 1), this can be improved to (4.26). Recalling our previous results (3.18) and (3.20), we see that we have now proved the decay assertion (1.34), whereas (1.36) has only been shown to hold for |Im v| ≤ a. To extend (1.36) to Im v ∈ R, we invoke the AE E(v − ia, v) ˆ + Va (v)E(v + ia, v) ˆ + Vb (v)E(v, v) ˆ = 2 cosh(2π v/a)E(v, ˆ v), ˆ (4.27) and the bounds (3.7)–(3.8) on Va and Vb . Specifically, taking v → v + ia in (4.27), we obtain (1.36) for Im v ∈ [−2a, −a]. Clearly, we can now proceed in a recursive, strip-by-strip fashion to deduce (1.36) for Im v ∈ (−∞, −a]. Multiplying (4.27) by Va (v)−1 , we can take v → v − ia to obtain (1.36) for Im v ∈ [a, 2a], and then recursively for Im v ∈ [a, ∞). Thus we have now proved Theorem 1.2 for the special case γ = 0, a+ = a− . It remains to prove (1.33) for vˆ ∈ [r− , r+ ], Re v > r+ + al and γ satisfying (4.9), with a+ = a− in case γ0 = 0. Setting r ≡ γ0 − γˆ0 + a − as ,

(4.28)

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we have r > 0 (cf. (4.12)), and the tail contributions to E − R+ − R− are bounded above by C exp(−αrRe v)),

(4.29)

cf. (3.32). Using the bound (3.41) for the right indentation and its counterpart for the left one, we deduce that their contribution is also majorized by (4.29). Turning to the middle indentation, we conclude as before that its contribution to L is bounded above by (3.45). Combining this with (3.27), we see that its contribution to E − R+ − R− is majorized by C exp(−αηRe v).

(4.30)

Again, we obtain the same result for the line segments d ≤ |Re z| ≤ v. ˆ Next, consider the bound (3.46). In the present case, the coefficient A of Re z satisfies A ≤ as + γˆ0 + η − γ0 − a = η − r.

(4.31)

Thus we can ensure that it is negative by choosing η in (0, min(as , r)). Doing so, the contribution of this piece of Cs to E − R+ − R− is again bounded above by (4.30). For −Re v + as /2 < Re z < −v, ˆ we reach the same conclusion. In summary, we have |(E − R+ − R− )(γ ; v, v)| ˆ < C exp(−αηRe v), η ∈ (0, min(as , r)),

(4.32)

with r given by (4.28). Recalling (3.18) and (3.20), we deduce |(E − Eas )(γ ; v, v)| ˆ < C exp(−ρRe v), ρ ≡ α min(σ as , r).

(4.33)

As a consequence, (1.33) follows with Im v ∈ [−as , as ]; for γ restricted by (4.9), we can choose for instance ρ(a+ , a− , γ ) = α min(as /2, γ0 − γˆ0 + a − as ),

(4.34)

and this choice can be extended to arbitrary (a+ , a− , γ ) ∈ ∗ by requiring that ρ be W -invariant. Finally, to extend (1.33) to Im v ∈ R, we exploit the eigenvalue AE for E that involves v-shifts over ±ias . To avoid clumsy formulas, we assume from now on as = a− . (The case as = a+ reduces to obvious notation changes. Alternatively, we can invoke (1.21).) Accordingly, we start from the AE, F (v − ia− ) + Va (a+ , a− , γ ; v)F (v + ia− ) + Vb (a+ , a− , γ ; v)F (v) = 2 cosh(αa− v)F ˆ (v),

(4.35)

ˆ Just as in the special case already handled, this can be obeyed by E(a+ , a− , γ ; v, v). done recursively, so we only detail the first step. Specifically, taking v → v + ia− in the AE (4.35), we see that we may write (E −Eas )(v, v) ˆ = − exp(iαv v)+u( ˆ γˆ ; −v) ˆ exp(−iαv v)−V ˆ ˆ a (v+ia− )E(v+2ia− , v) +[exp(αa− v) ˆ + exp(−αa− v) ˆ − Vb (v + ia− )]E(v + ia− , v). ˆ (4.36) Letting Im v ∈ [−2a− , −a− ], we are entitled to use (1.33) for the E-functions on the rhs. Doing so, and using also (3.7)–(3.8), we see that the eight terms staying away from 0 as Re v → ∞ cancel pairwise. Noting ρ < αa− , the remaining terms manifestly have decay O(exp(−ρRe v)). Therefore, (1.33) holds for Im v ∈ [−2a− , −a− ].

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With Theorem 1.2 proved, we can shed more light on the asymptotics of the shifted contour integral L(γ ; v, v) ˆ (3.28), both for γ that do not satisfy (3.34) and for |Im v| > as . Indeed, from (3.18), (3.20) and Theorem 1.2 we deduce ˆ = O(exp(−ρRe v)), Re v → ∞, (E − R+ − R− )(γ ; v, v)

(4.37)

where we have ρ > 0 for arbitrary (fixed) γ ∈ R4 \ {0} and uniformly for Im v in arbitrary R-compacts. Choosing in particular Im v ∈ [−as , as ], we may invoke (3.24), and all of the arguments leading to (3.33) apply as well. The point is now that (3.4) also yields lower bounds on the prefactors (3.25), (3.26) of the same type as the upper bounds. (That is, (3.25) is bounded away from 0, and we may also take |P | → |P |−1 in (3.27).) From this we see that for arbitrary a+ , a− and γ = 0 we have L(γ ; v, v) ˆ = O(exp([α(γ0 − γˆ0 − a) − ρ]Re v)), ρ > 0, Re v → ∞,

(4.38)

uniformly for |Im v| ≤ as ; provided we enlarge the right and left indentations of Cs (so that the pole sequences starting at ±v − ic0 stay below Cs ), we obtain (4.38) even for Im v in an arbitrary R-compact. We now explain why this estimate is remarkable. To this end we point out that when we use (3.2)–(3.3) to bound IL (3.29) for Re z > Re v + al , it becomes clear that we have not only |IL | ∈ exp(−α(2aRe z − Im vRe v))(C− , C+ ), 0 < C− < C+ ,

Re z > Re v + al , (4.39)

but also that we are estimating away only one diverging phase (as Re v → ∞), viz., the factor exp(−iα(Re v)2 /2). Since this factor is z-independent, it is quite plausible (though it does not rigorously follow) that the modulus of the integral over Re z > Re v + al is bounded below by K exp(α(Im v − 2a)Re v), with K > 0. Assuming this is indeed the case, we see from (4.38) that whenever we have Im v−a ≥ γ0 − γˆ0 , the quotient of the integral over the contour tail Re z > Re v +al and the integral L over the whole contour Cs diverges exponentially as Re v → ∞. This comparison reveals why our piecemeal reasoning cannot be directly applied to arbitrary Im v and γ . In this connection we should also draw the reader’s attention to the phase factor exp(iα(Re z)2 /2) that is estimated away in (3.46). Since it is divergent on the pertinent interval, it can (and apparently does) supply the cancellations that must occur for Im v − a ≥ γ0 − γˆ0 . In view of this state of affairs, it may be regarded as a felicitous circumstance that there does exist an (Im v)- and γ -window that is not only accessible via piecewise estimates, but large enough to open up the asymptotics of E for arbitrary Im v and γ . Acknowledgements. We would like to thank the referee for some valuable suggestions.

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3. Askey, R., Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Am. Math. Soc. 319, (1985) 4. Gasper, G., Rahman, M.: Basic hypergeometric series. In: Encyclopedia of Mathematics and its Applications. 35, Cambridge: Cambridge Univ. Press 1990 5. Ruijsenaars, S. N. M.: Special functions defined by analytic difference equations. In: Proceedings of the Tempe NATO Advanced Study Institute “Special Functions 2000”, NATO Science Series Vol. 30, Bustoz, J., Ismail, M., Suslov, S., eds., Dordrecht: Kluwer, 2001, pp. 281–333 6. Ruijsenaars, S. N. M.: Sine-Gordon solitons vs. relativistic Calogero-Moser particles. In: Proceedings of the Kiev NATO Advanced Study Institute “Integrable structures of exactly solvable two-dimensional models of quantum field theory”, NATO Science Series Vol. 35, Pakuliak, S., von Gehlen, G., eds., Dordrecht: Kluwer, 2001, pp. 273–292 7. Ruijsenaars, S. N. M.: A generalized hypergeometric function III. Associated Hilbert space transform. To appear in Commun. Math. Phys. - DOI 10.1007/s00220-003-0970-x 8. Koornwinder, T. H.: Askey-Wilson polynomials for root systems of type BC. Contemp. Math. 138, 189–204 (1992) 9. van Diejen, J. F.: Self-dual Koornwinder-Macdonald polynomials. Invent. Math. 126, 319–339 (1996) 10. Gr¨unbaum, F. A., Haine, L.: Some functions that generalize the Askey-Wilson polynomials. Commun. Math. Phys. 184, 173–202 (1997) 11. Ismail, M. E. H., Rahman, M.: The associated Askey-Wilson polynomials. Trans. Am. Math. Soc. 328, 201–237 (1991) 12. Suslov, S. K.: Some orthogonal very-well-poised 8 φ7 -functions. J. Phys. A: Math. Gen. 30, 5877– 5885 (1997) 13. Suslov, S. K.: Some orthogonal very-well-poised 8 φ7 -functions that generalize Askey-Wilson polynomials. The Ramanujan Journal 5, 183–218 (2001) 14. Koelink, E., Stokman, J. V.: The Askey-Wilson function transform. Int. Math. Res. Notes No. 22, 1203–1227 (2001) 15. Stokman, J. V.: Askey-Wilson functions and quantum groups. Preprint, math.QA/0301330 16. Ruijsenaars, S. N. M.: Relativistic Lam´e functions revisited. J. Phys. A: Math. Gen. 34, 10595–10612 (2001) Communicated by L. Takhtajan

Commun. Math. Phys. 243, 413–448 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0970-x

Communications in

Mathematical Physics

A Generalized Hypergeometric Function III. Associated Hilbert Space Transform S.N.M. Ruijsenaars Centre for Mathematics and Computer Science, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands Received: 28 November 2002 / Accepted: 22 May 2003 Published online: 11 November 2003 – © Springer-Verlag 2003

Abstract: For generic parameters (a+ , a− , c) ∈ (0, ∞)2 × R4 , we associate a Hilbert space transform to the “relativistic” hypergeometric function R(a+ , a− , c; v, v) ˆ studied in previous papers. Restricting the couplings c to a certain polytope, we show that the (renormalized) R-function kernel gives rise to an isometry from the even subspace of L2 (R, w( ˆ v)d ˆ v) ˆ to the even subspace of L2 (R, w(v)dv), where w( ˆ v) ˆ and w(v) are positive and even weight functions. We prove that the orthogonal complement of the range of this isometry is spanned by N ∈ N pairwise orthogonal functions. The latter are in essence Askey-Wilson polynomials, arising from the R-function by choosing vˆ = iκn , with κ0 , . . . , κN−1 distinct negative numbers. The two commuting analytic difference operators acting on the variable v for which R is a joint eigenfunction, give rise to two commuting self-adjoint Hamiltonians on the even subspace of L2 (R, w(v)dv). We explicitly determine the relation of the time-dependent scattering theory for these dynamics to their joint spectral transform. Contents 1. Introduction . . . . . . . . . . . . . . . 2. Associating Hilbert Space Operators to E 3. Scattering Theory and Isometry of F . . 4. Bound States and Spectral Resolutions . Appendix A. Proofs of Lemmas 2.2 and 2.3 Appendix B. Proofs of Lemmas 4.1 and 4.2

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413 421 425 429 440 444

1. Introduction As is well known, the linear ordinary second-order differential equation − (x) + V (x)(x) = E(x),

(1.1)

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S.N.M. Ruijsenaars

can be solved explicitly in terms of the hypergeometric function 2 F1 (a, b, c; w) for certain coefficients V (x). The interpretation and further study of the explicit solutions depend on the context, however. For the 2 F1 -solutions at issue, one can distinguish three related, yet distinct contexts. Within the first context, namely elementary quantum mechanics, one views (1.1) as a time-independent Schr¨odinger equation (with Planck’s constant = 1 and particle mass m = 1/2), and one is interested in real x, real-valued V (x), and real E for which (x) is square-integrable (“bound states”) or has plane wave asymptotics as x → ±∞ (“scattering states”). Whenever these can be found explicitly (as is the case for the V (x) associated with 2 F1 ), the problem is viewed as “solved”. This viewpoint dates back to Dirac’s monograph Ref. [1]; for special V (x), solutions to (1.1) in this spirit can be found for instance in Ref. [2]. In the second context, namely the mathematically rigorous theory of unbounded self-adjoint operators on Hilbert space (which dates back to von Neumann’s monograph Ref. [3]), the existence of suitable solutions to (1.1), “explicit” or not, is viewed as a stepping stone to construct a realization of the spectral theorem for the Schr¨odinger operator on the lhs of (1.1) in terms of the solutions for real E (eigenfunction transform). In this way, the results of the first context, where matters of self-adjointness, orthogonality and completeness are treated heuristically, can be corroborated by solid functional analysis. The corresponding theory of eigenfunction expansions is the Weyl-Titchmarsh-Kodaira (WTK) theory, a textbook account of which can be found in Ref. [4], for example. Both in Fl¨ugge’s [2] and in Titchmarsh’s [4] monograph, one can find worked examples of V (x) for which the 2 F1 -function is relevant. These include not only hyperbolic potentials of the form V (x) = λ1 / sinh2 x + λ2 / cosh2 x on the line, but also their trigonometric counterparts on the circle, V (x) = λ1 / sin2 x + λ2 / cos2 x. In the latter case, the Hilbert space viewpoint entails that one may restrict attention to the Jacobi polynomials, which arise via a suitable discretization of 2 F1 . The third context is also rigorous, but much more focused and of a quite different flavor, compared to the second one. Specifically, we have in mind the context of “rank-1 harmonic analysis”. The oldest and most well-known instance of this context is that of Legendre polynomials vs. the rotation group. However, widening attention to rank-1 symmetric spaces and then interpolating root multiplicities, one can tie in all 2 F1 -related eigenfunction transforms with this context. (See for instance Koornwinder’s account Ref. [5].) The quantum dynamics that lead to 2 F1 -eigenfunctions may be viewed as the centerof-mass two-particle Hamiltonians of suitable Calogero-Moser-Sutherland N -particle systems. (The connection of these integrable quantum systems with harmonic analysis was first pointed out by Olshanetsky and Perelomov, cf. their survey Ref. [6].) These systems admit a relativistic generalization, and the generalized hypergeometric function R(a+ , a− , c; v, v) ˆ under consideration in this series of papers was introduced in this setting [7]. From a mathematical point of view, the relativistic generalization leads to time-independent Schr¨odinger equations that are analytic difference equations (AEs), instead of the differential equations arising for the nonrelativistic systems. Using physical variables, the type of AE relevant for the R-function is of the form C(x)[(x − i/mc) − (x)] + C(−x)[(x + i/mc) − (x)] = (E − E0 )(x), (1.2) with Planck’s constant, m the particle mass, and c the speed of light. For the meromorphic coefficients C(x) at issue, this equation can be similarity-transformed to an

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415

equation of the form ψ(z − i) + Va (z)ψ(z + i) + Vb (z)ψ(z) = Eψ(z),

(1.3)

where z is a dimensionless (scaled) variable. For general meromorphic coefficients Va , Vb , very little is known about (1.3), compared to (1.1) with V (x) meromorphic. (For example, it is not known whether the meromorphic solutions form a two-dimensional vector space over the field of meromorphic functions of period i.) However, for the quite special trigonometric coefficients leading to the Askey-Wilson polynomials [8, 9], various questions from the viewpoints of Hilbert space theory and harmonic analysis (on rank-1 quantum groups and symmetric spaces) have been answered. (See e.g. Refs. [10–16].) There are also recent Hilbert space results involving certain interpolations of the Askey-Wilson polynomials [17–19]. Our “relativistic” interpolation R(a+ , a− , c; v, v) ˆ is associated with hyperbolic coefficients in (1.3), and is quite different from the ones just mentioned (which involve the Ismail-Rahman 8 φ7 -functions [20]). Returning to the above sketch of the three contexts for 2 F1 -type transforms, we can characterize the current situation by stating that to date no suitable harmonic analysis context for our R-tranform has been found, whereas in Refs. [21, 22] (henceforth denoted by I and II) we obtain information that does not go beyond the first, theoretical physics context (although our results are rigorous). In the present paper, we use our previous work to handle the key issues within the second context. To be sure, this is a somewhat procrustean appraisal, inasmuch as there exists no body of heuristic knowledge regarding what “should be the case” for the type of operators and eigenfunctions at hand. Moreover, the WTK-theory cannot be readily applied either. Indeed, from our previous explicit results on reflectionless analytic difference operators (see Ref. [23] and references given there), it transpires that a general functional-analytic approach to analytic difference operators is quite elusive. Staying with the much smaller class of analytic difference operators (AOs) associated with the R-function, we have already studied the functional-analytic aspects of a tiny subclass of reflectionless AOs in Ref. [24]. In the latter setting, we explicitly exhibited breakdown of self-adjointness and isometry for large couplings, phenomena that have no analogs for Schr¨odinger type Hamiltonians and their eigenfunction transforms. It is likely that these anomalies are present for the general-parameter R-function transform as well, but in this paper we only study a region in the parameter space where quite satisfactory Hilbert space results can be obtained. Before sketching our results, we collect some conventions and definitions that are used throughout this paper. Almost all of the latter can be found scattered over our previous papers I and II, but we summarize them here for ease of reference. On the other hand, we refer the reader to I and II for the definitions of the R-function and the associated AOs, since these are too substantial to repeat. First, in this paper we work with parameters (a+ , a− , p) in the set ≡ (0, ∞)2 × R4 .

(1.4)

We often use the quantities a ≡ (a+ + a− )/2, α ≡ 2π/a+ a− , as ≡ min(a+ , a− ), al ≡ max(a+ , a− ), (1.5)

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and dual parameters



pˆ ≡ Jp,

 1 1 1 1 1  1 1 −1 −1  J ≡  . 2 1 −1 1 −1 1 −1 −1 1

(1.6)

As a rule, we do not employ the couplings c ∈ R4 , but parameters γ related to c by a shift, viz., c(γ ) ≡ (γ0 + a, γ1 + a− /2, γ2 + a+ /2, γ3 ).

(1.7)

Note that the shift belongs to the fixed point subspace of J , so that cˆ is related to γˆ by the same formula. We often suppress the dependence on a+ , a− , and sometimes the dependence on γ as well. We mostly work with the renormalized R-function and the E-function (as compared to the R-function). We now recall their definitions, given the definition of the R-function (for which we refer to I and II). The former function is given by (cf. I (2.32) and II (1.13)) Rren (a+ , a− , c(γ ); v, v) ˆ ≡ ρ(a+ , a− , γ )R(a+ , a− , c(γ ); v, v), ˆ ρ(a+ , a− , γ ) ≡ 1/

3

G(a+ , a− ; iγ0 + iγj + ia),

(1.8) (1.9)

j =1

where G(a+ , a− ; z) is the hyperbolic gamma function (cf. I Appendix A). Secondly, E is defined by E(a+ , a− , γ ; v, v) ˆ ≡ χ (a+ , a− , γ )Rren (a+ , a− , c(γ ); v, v) ˆ /c(a+ , a− , γ ; v)c(a+ , a− , γˆ ; v). ˆ

(1.10)

Here, χ is the function 2 2 + a− + a+ a− )/8]), χ (a+ , a− , p) ≡ exp(iα[p · p/4 − (a+

(1.11)

ˆ since which is clearly a phase factor for (a+ , a− , p) ∈ . (We also have χ (p) = χ (p), J is orthogonal.) Finally, the c-function is given by c(a+ , a− , p; y) ≡

3 1 G(a+ , a− ; y − ipµ ). G(a+ , a− ; 2y + ia)

(1.12)

µ=0

With these key quantities at hand, we begin by pointing out that there are three distinct Hilbert space formulations, each of which has its pros and cons. The first one consists in viewing Rren (c(γ ); v, v), ˆ v, vˆ ∈ R, as the kernel of a map between Hilbert spaces Hw ≡ L2s (R, w(γ ; v)dv), Hwˆ ≡ L2s (R, w(γˆ ; v)d ˆ v). ˆ

(1.13)

Here the function w(p; y) is a positive even weight function (given by (1.15)), and the suffix s stands for the even subspace. (We recall that Rren is even in v and v, ˆ cf. I (2.10).) ˆ γˆ ; v) ˆ 1/2 , v, vˆ ∈ The second one consists in viewing w(γ ; v)1/2 Rren (c(γ ); v, v)w( (0, ∞), with square roots chosen positive, as the kernel of a map between Hilbert spaces H ≡ L2 ((0, ∞), dv),

Hˆ ≡ L2 ((0, ∞), d v). ˆ

ˆ In the third we employ E(γ ; v, v) ˆ as the kernel of a map between H and H.

(1.14)

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417

Obviously, the first two pictures are unitarily equivalent. The third one is unitarily equivalent to the previous ones, too. The point is that the w-function in (1.13) is defined by w(p; y) ≡ 1/c(p; y)c(p; −y),

(1.15)

and that we have c(a+ , a− , p; −y) = c(a+ , a− , p; y), (a+ , a− , p, y) ∈ × R.

(1.16)

(Here and below, the overbar denotes complex conjugation or closure, depending on context.) The connection between the second and third picture can therefore be made via the square root of the u-function u(p; y) ≡ −c(p; y)/c(p; −y).

(1.17)

Since this is a phase for parameters in and real y, the square roots of u(γ ; v) and ˆ resp. Hence the announced u(γˆ ; v) ˆ yield unitary multiplication operators on H and H, unitary equivalence follows. The advantages of the E-kernel include its D4 -symmetry and plane wave asymptotics established in II. These features can be exploited in a quite direct fashion within the third picture, and so we have opted for the latter in this paper. In particular, for this third setting we need the AOs, A+ ≡ A(a+ , a− , γ ; v), A− ≡ A(a− , a+ , γ ; v).

(1.18)

Here, A(a+ , a− , p; y) is of the form A(a+ , a− , p; y) = exp(−ia− ∂y ) + Va (a+ , a− , p; y) exp(ia− ∂y ) +Vb (a+ , a− , p; y),

(1.19)

with coefficients Va and Vb given by II (2.10) and II (2.3)–(2.7), resp.; moreover, exp(α∂) denotes translation over α ∈ C∗ . For our present purposes it suffices to mention that Vb (y) is real-valued for real y and that Va (y) admits the representation Va (a+ , a− , p; y) = u(a+ , a− , p; y + ia− )/u(a+ , a− , p; y),

(1.20)

cf. II (2.14); moreover, we recall the eigenvalue AEs, ˆ = Eδ (v)E(v, ˆ v), ˆ Aδ E(v, v)

δ = +, −,

(1.21)

where we have introduced Eδ (y) ≡ 2 cosh(2πy/aδ ), δ = +, −.

(1.22)

We proceed by sketching how we associate Hilbert space operators to the E-function and to the AOs A± . First, we define the function space C ≡ C0∞ ((0, ∞)), ˆ For φ ∈ C and v > 0 the integral which is obviously dense in H and H. ∞ Iφ (v) ≡ d vE(γ ˆ ; v, v)φ( ˆ v), ˆ 0

(1.23)

(1.24)

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S.N.M. Ruijsenaars

is absolutely convergent. The function Iφ (v), v ∈ (0, ∞), is real-analytic and has a meromorphic extension whose poles can only occur at pole locations of E, all of which are on the imaginary axis (cf. (2.13)–(2.15) below). Denoting the meromorphic extension by the same symbol, the action of the AOs A± on Iφ (v) now yields meromorphic functions Jφ,± (v). For v > 0 the latter are given by the integrals ∞ Jφ,± (v) = d vE(γ ˆ ; v, v)E ˆ ± (v)φ( ˆ v). ˆ (1.25) 0

Taking these assertions for granted (they are substantiated in Sect. 2), it is immediate that for all φ ∈ C such that Iφ (v) = 0, one also has Jφ,± (v) = 0. (Observe that this is not at all obvious from a direct comparison of (1.24) and (1.25).) Now the only obstruction to the function v → Iφ (v) being in H is that E(γ ; v, v) ˆ might have a pole at v = 0. Indeed, we have enough information on the v → ∞ asymptotics of E for vˆ ∈ supp(φ) to deduce Iφ (v) ∈ H when E(γ ; v, v) ˆ has a finite limit for v ↓ 0. More is true: absence of a pole at v = 0 entails that for all r− , r+ satisfying 0 < r− < r+ , the map C → H, φ(v) ˆ → Iφ (v),

(1.26)

ˆ with a bound ˆ of H, gives rise to a bounded operator on the subspace L2 ([r− , r+ ], d v) depending only on r− and r+ , cf. Lemma 2.2. The regularity property is in particular satisfied for all (a+ , a− , γ ) in the polytope P ≡ {(a+ , a− , p) ∈ | |pµ | < a, µ = 0, 1, 2, 3}.

(1.27)

All of our results hold true for a set Pe that is slightly larger than P , but to ease the exposition in this introduction, we assume (a+ , a− , γ ) ∈ P from now on. Defining Fφ ≡ (a+ a− )−1/2 Iφ (·), φ ∈ C,

(1.28)

we have Fφ ∈ H. We now associate operators on the space P ≡ FC

(1.29)

to the AOs A+ , A− (denoted by the same symbols) by setting Aδ Fφ ≡ FMδ φ,

δ = +, −,

φ ∈ C,

(1.30)

where Mδ denotes the operator of multiplication by Eδ (v). ˆ (This is well defined, since Fφ = 0 implies FMδ φ = 0, as we have already pointed out.) Now whenever F is isometric, this definition leads to operators that are not only symmetric, but also essentially self-adjoint on P. (Indeed, C is a core for the self-adjoint multiplication operators Mδ .) The problem is, however, that a direct isometry proof appears quite intractable. We therefore proceed differently. First, we prove that the operator As with the smallest step size as is symmetric on P. This involves considerable work, whereas essential self-adjointness of As on P is a rather direct consequence of Nelson’s analytic vector theorem [25].

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419

The results just summarized are obtained in Sect. 2. The proofs of Lemmas 2.2 and 2.3, which deal with the operators F and As , resp., are rather long and technical. Therefore, we have relegated them to Appendix A. In the auxiliary Lemma 2.1, we obtain a result that is of interest in itself, namely real-valuedness of the R-function for real parameters and variables. In Sect. 3 we use this input and time-dependent scattering theory [26] to show that F is an isometry. Specifically, we compare the unitary one-parameter group exp(−itAs )− on P to a “free” evolution exp(−itA0,s ) on H, which we proceed to define. To begin with, observe that the operator R F0 : Hˆ → H, φ(v) ˆ → l.i.m.R→∞ (a+ a− )−1/2 d vE ˆ 0 (v, v)φ( ˆ v), ˆ (1.31) 0

ˆ ≡ exp(iαv v) ˆ − exp(−iαv v), ˆ E0 (v, v)

(1.32)

is unitary (i.e., an isometry onto H); indeed, F0 is basically the sine transform. Denoting now the self-adjoint operator of multiplication by 2 cosh(2π v/a ˆ l ) on Hˆ by Ms , A0,s is the self-adjoint operator on H defined by A0,s F0 ψ ≡ F0 Ms ψ,

ψ ∈ Ds ,

(1.33)

where Ds denotes the domain of Ms . It should be noted that for φ ∈ C, the functions ψ(v) ≡ (F0 φ)(v), ψM (v) ≡ (F0 Ms φ)(v), v ∈ (0, ∞),

(1.34)

are restrictions of entire functions that are related by ψM (v) = ψ(v − ias ) + ψ(v + ias ).

(1.35)

Thus the free Hilbert space operator A0,s corresponds to the free AO, exp(−ias ∂v ) + exp(ias ∂v ).

(1.36)

The point is now that we can invoke the v → ∞ asymptotics of E obtained in II to show that the wave operators exist. In the process, we conclude that FF0∗ equals the t → ∞ wave operator, entailing isometry. The corresponding S-operator equals F0 u(γˆ ; ·)F0∗ . These results are detailed in Corollary 3.2 and Theorem 3.3, with Lemma 3.1 containing a key technical result. Provided the dual parameters (a+ , a− , γˆ ) belong to P as well (which is e.g. the case for |γ | < a), it is not hard to see that F is in fact unitary. For such parameters the dual AOs also give rise to self-adjoint Hilbert space operators, as we show at the end of Sect. 3. Section 4 is concerned with the far richer situation where F ∗ is not isometric and bound states appear on the stage. This happens whenever max(|γˆ0 |, |γˆ1 |, |γˆ2 |, |γˆ3 |) > a.

(1.37)

The D4 -invariance of the E-function and the lack of D4 -invariance of the c- and w-functions play an important role in this setting. The crux is that only for γ satisfying γˆ0 < −a the weight function w(γ ; v) decreases exponentially for v → ∞, cf. (4.8)–(4.9) below. In this connection we mention that for the special case γb ≡ (b − a, −a− /2, −a+ /2, 0),

(1.38)

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S.N.M. Ruijsenaars

we studied the w-function in considerable detail in Sect. VA of Ref. [27]. (To establish the equality asserted here, the duplication formula for the hyperbolic gamma function should be used, cf. Eqs. (3.24)–(3.25) in loc. cit.) Though the results we obtained there can be easily generalized to the arbitrary-γ w-function occurring here, the key difference with the special “A1 ” case (1.38) is that (1.37) can be satisfied. (To have (a+ , a− , γb ) ∈ P , we need b ∈ (0, 2a). This implies (a+ , a− , γˆb ) ∈ P , precluding (1.37).) We exploit the D4 -invariance of the E-kernel by choosing γ in a fundamental domain (Weyl chamber) for the action of the D4 Weyl group W in which (1.37) amounts to γˆ0 < −a. This choice enables us to tie in the bound states with the R-function in a quite simple way. To be sure, the W -invariance of E entails the formula ˆ ˆ Rren (c(γ (1) ); v, v) c(γ (1) ; v)c(J γ (1) ; v) = , γ (j ) ≡ wj (γ ), wj ∈ W, j = 1, 2, (2) (2) (2) Rren (c(γ ); v, v) ˆ c(γ ; v)c(J γ ; v) ˆ (1.39) which can be used to transform results involving Rren (and R) to other Weyl chambers. But with our choice we need only consider v-values ˆ for which R(v, v) ˆ reduces to a polynomial in cosh(αas v). More specifically, for γˆ0 < −a the exponential decay rate of the positive even weight function w(γ ; v) on R determines the number N of orthogonal polynomials in cosh(αas v) of degrees n = 0, . . . , N − 1 in L2 (R, w(γ ; v)dv), and we show that these polynomials can be obtained (up to normalization) by choosing vˆ = i cˆ0 + inas in R(v, v). ˆ Furthermore, our bound state results in Sect. 4 entail that these polynomials are in essence Askey-Wilson polynomials, in the sense that their recurrence coefficients (cf. (4.39)–(4.46) below) can be obtained by analytic continuation of the Askey-Wilson coefficients. Returning to our Hilbert space formulation in terms of H (1.14) and the E-kernel, the ˆ span all of H. Our method to arrive at this completeness property bound states and F(H) deserves special mention. Indeed, it involves an unexpected and striking application of the Christoffel-Darboux formula, which is well known in orthogonal polynomial theory [28]. Moreover, it reveals in a quite explicit way that the symmetry violation of the dual AOs is linked to the isometry violation of F ∗ . The proofs of Lemmas 4.1 and 4.2 are relegated to Appendix B. Here we handle the analytic aspects of orthogonality and completeness along lines similar to the proof of Lemma 2.3 in Appendix A. As a final item of independent interest, we mention that we obtain the integral of the weight function (1.15) as a spin-off of our completeness proof, cf. Theorem 4.3. To be specific, it reads 0

∞

3

µ=0 G(v

+ iγµ )G(−v + iγµ )

G(2v − ia)G(−2v − ia)

dv = (a+ a− )

1/2

0≤µ<ν≤3 G(iγµ + iγν + ia) .

G(i 3µ=0 γµ + 3ia)

(1.40) Here, G(z) = Ghyp (a+ , a− ; z) is the hyperbolic gamma function from Ref. [27] and the parameters are restricted by γµ ∈ (−a, a), µ = 0, 1, 2, 3,

3 µ=0

γµ < −2a.

(1.41)

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421

This explicit formula may be viewed as a hyperbolic counterpart of the “trigonometric” Askey-Wilson weight function integral [8, 9]. Indeed, provided the latter is expressed in terms of the trigonometric gamma function from Ref. [27], it takes basically the same form as (1.40). To detail this, we reparametrize Eqs. (6.1.1)–(6.1.2) in Ref. [9] by setting q → e−2a , a → e−α0 −a , b → e−α1 −a , c → e−α3 −a , d → e−α3 −a .

(1.42)

Then the Askey-Wilson integral can be written π 3 µ=0 Gt (θ + iαµ )Gt (−θ + iαµ ) 0

dθ Gt (2θ − ia)Gt (−2θ − ia) 0≤µ<ν≤3 Gt (iαµ + iαν + ia) = 2πGt (ia) .

Gt (i 3µ=0 αµ + 3ia)

(1.43)

Here we have Gt (θ ) ≡ Gtrig (1/2, 2a; θ ), a ∈ (0, ∞] ∪ {iπ + b | b ∈ (0, ∞)},

(1.44)

with Gtrig (r, a; z) the trigonometric gamma function from loc. cit.; moreover, in this case one can choose αµ + a ∈ (0, ∞] ∪ {iπ + β | β ∈ (0, ∞)}, µ = 0, 1, 2, 3.

(1.45)

(To verify that (1.43) amounts to Eq. (6.1.1) in Ref. [9], the duplication formula for the trigonometric gamma function is helpful, cf. Eq. (3.148) in Ref. [27].) 2. Associating Hilbert Space Operators to E and the A∆Os A± We begin this section with an auxiliary lemma. The main observation of the lemma is interesting in its own right: we show that for all parameters a+ , a− > 0, c ∈ R4 and variables v, vˆ ∈ R, the function R(a+ , a− , c; v, v) ˆ takes real values. (In the “nonrelativistic” limit, this reduces to the real-valuedness of 2 F1 (a, b, c; w) for b = a and c, w real, cf. I.) We phrase the lemma in terms of conjugate meromorphic functions. To be specific, when M is a meromorphic function on Cn , the conjugate M ∗ is defined by M ∗ (Z) = M(Z), Z ∈ Cn .

(2.1)

Lemma 2.1. Letting (a+ , a− , γ ) ∈ , the functions R(c(γ ); v, v) ˆ and Rren (c(γ ); v, v) ˆ on C2 are self-conjugate, whereas E ∗ (γ ; v, v) ˆ = χ (γ )−2 u(γ ; v)u(γˆ ; v)E(γ ˆ ; v, v). ˆ

(2.2)

Proof. Since a+ , a− > 0, we can invoke the integral representation

∞ sin(2tz) z dt G(a+ , a− ; z) = exp i − , |Im z| < a. t 2 sinh(a+ t) sinh(a− t) a+ a− t 0 (2.3) This yields the conjugate G-function G∗ (a+ , a− ; z) = G(a+ , a− ; −z).

(2.4)

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From (1.12) we now obtain c∗ (p; y) = c(p; −y).

(2.5)

Recalling (1.10) and (1.17), it is clear that (2.2) amounts to Rren (v, v) ˆ being self-conjugate. This in turn is equivalent to self-conjugacy of R(v, v), ˆ since the renormalizing G-factors in (1.8)–(1.9) are real for (a+ , a− , γ ) ∈ . It is therefore enough to show R = R ∗ . By analytic continuation, this property will follow if we can prove real-valuedness of R(v, v) ˆ for parameters satisfying I (1.28)– (1.29) and positive variables v, v. ˆ Then Fig. 1 in I applies and we may replace the integration contour C for the R-function by the shifted contour C˜ ≡ {z ∈ C | Im z = −η, η ∈ (0, min(c0 , cˆ0 ))}.

(2.6)

Next, we consider the building blocks of the integrand I (v, v, ˆ z) given by I (1.33). From (2.4) we have F ∗ (y, z) = F (y, −z), K ∗ (z) = K(−z),

(2.7)

I ∗ (v, v, ˆ z) = I (v, v, ˆ −z).

(2.8)

so that

With the above parameter restrictions and positive variables v, v, ˆ we now deduce from ˜ (2.8) and the C-integral representation ∞ −1/2 R(v, v) ˆ = (a+ a− ) dxI (v, v, ˆ x − iη), (2.9) −∞

that we have R(v, v) ˆ = (a+ a− )−1/2

∞ −∞

dxI (v, v, ˆ −x − iη) = R(v, v). ˆ

Hence R is indeed real-valued for v, vˆ > 0.

(2.10)

Following our outline in the Introduction, we now embark on studying the integral Iφ (v) (1.24). To this end we first need to discuss the analyticity properties of the E-function in more detail. Using the c-function definition (1.12) and the relation G(z) = E(z)/E(−z) of the hyperbolic gamma function G(z) to the entire function E(z) (cf. I (A.40)), we can rewrite (1.10) as 3 E(v − iδγµ )E(vˆ − iδ γˆµ ), E(v, v) ˆ = χ G(2v + ia)G(2vˆ + ia)H (v, v) ˆ δ=+,− µ=0

(2.11) with ˆ H (v, v) ˆ ≡ Rren (v, v)

3 δ=+,− µ=0

E(δv + iγµ )E(δ vˆ + i γˆµ ).

(2.12)

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423

The point of doing so is that H (v, v) ˆ is entire (holomorphic) in v and vˆ and real-analytic in the parameters, cf. I Theorem 2.2. Furthermore, the zeros of E(z) and their multiplicity are known, cf. I Appendix A. Therefore, the representation (2.11) enables us to deduce the location of eventual poles of E and their (maximal) multiplicity. Specifically, we recall E(z) has zeros only for z = ia + zkl , zkl ≡ ika+ + ila− , k, l ∈ N,

(2.13)

and the multiplicity of a zero ia + zk0 l0 is given by the number of distinct pairs k, l ∈ N for which zkl equals zk0 l0 . (The zero at z = ia is therefore always simple.) As a consequence, for fixed parameters in and vˆ ∈ (0, ∞), the function E(v, v) ˆ is meromorphic in v with poles that can only occur at the zero locations v = iδγµ + ia + zkl , δ = +, −, k, l ∈ N,

(2.14)

of the E-function product in the denominator of (2.11), and at the poles v = −ia − zkl /2, k, l ∈ N,

(2.15)

of G(2v + ia). In particular, for parameters that do not belong to the hyperplanes δγµ + a + ka+ + la− = 0, δ = +, −, µ = 0, 1, 2, 3, k, l ∈ N,

(2.16)

we deduce regularity of E(v, v) ˆ at v = 0. We will show shortly that regularity at v = 0 implies Iφ (v) (1.24) is square-integrable on (0, ∞). But first we isolate some properties of Iφ (v) for arbitrary parameters. We start by pointing out that the integral Iφ (v) yields a function that is analytic for v not equal to (2.14) and (2.15). Moreover, for the latter v-values eventual poles have an order that is smaller than or equal to the pole order of the function 3

G(2v + ia)/

E(v − iδγµ ).

(2.17)

δ=+,− µ=0

(These assertions are plain from the representation (2.11) and holomorphy of H (v, v).) ˆ Next, consider the action of the AOs A± (1.18) on the meromorphic function Iφ (v). Again using (2.11), we have for Re v > 0 (say), ∞ (A+ Iφ )(v) = d v[E(v ˆ − ia− , v) ˆ + Va (v)E(v + ia− , v) ˆ + Vb (v)E(v, v)]φ( ˆ v) ˆ 0 ∞ d vE(v, ˆ v)E ˆ + (v)φ( ˆ v), ˆ (2.18) = 0

and likewise

(A− Iφ )(v) =

∞ 0

d vE(v, ˆ v)E ˆ − (v)φ( ˆ v), ˆ

(2.19)

ˆ given by (1.22). These formulas will be invoked when we associate Hilbert with Eδ (v) space operators to the AOs Aδ . However, Iφ (v) need not be regular at v = 0 for arbitrary parameters in , which is why we need the regularity assumption in the next lemma.

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On the other hand, for later purposes it is crucial that the regularity property holds true for parameters varying over a subset r of that is larger than P (1.27). The set r is defined by allowing three γµ ’s to vary over (−a, a), whereas the remaining one is allowed to vary over (−2a, 2a). Note that W r = r ,

(2.20)

where W is the Weyl group of D4 (cf. II), and that r is open, connected and simplyconnected, but not convex. Lemma 2.2. Assume E(v, v) ˆ has no pole at v = 0. Then Iφ (v) belongs to H. Assuming supp(φ) ⊂ [r− , r+ ] with 0 < r− < r+ , we moreover have Iφ (·) < C(r− , r+ )φ,

(2.21)

where C is a positive continuous function on {(r− , r+ ) ∈ (0, ∞)2 | r− < r+ }. Finally, suppose (a+ , a− , γ ) ≡ ∈ r . Then E(; v, v) ˆ has no pole at v = 0 and Iφ (; ·) ∈ H is strongly continuous on r . The proof of this lemma can be found in Appendix A. With the regularity assumption of the lemma in effect from now on, we can define an operator F : C → H, φ → (a+ a− )−1/2 Iφ (·).

(2.22)

Although we know from the lemma that this operator is well defined, we do not know much else at this stage. In particular, we do not know whether it is bounded, and the only thing we know about its range is that it consists of real-analytic functions on [0, ∞) that admit a meromorphic extension with certain pole properties. Even so, this suffices to associate Hilbert space operators to the AOs A± , as we have detailed already in the Introduction, cf. (1.29)–(1.30). We emphasize that the former are naturally associated to the latter, in as much as the Hilbert space action (1.30) on the vector Fφ ∈ H is the restriction of the AO-action on the meromorphic function (Fφ)(v), cf. (2.18)–(2.19). Two other key properties of the Hilbert space operators A± are also quite easily established from the above. First, they leave P invariant. (Indeed, the operators M± leave C invariant.) Second, we have estimates Anδ Fφ ≤ Ccδn φ, n ∈ N, δ = +, −,

(2.23)

where C and c± depend only on supp(φ). (This readily follows from (2.21).) By virtue of these features, the vectors in P are analytic vectors for A± . By Nelson’s analytic vector theorem [25], symmetry of Aδ on P therefore yields essential self-adjointness on P. Accordingly, we should try and answer the question: Are the operators A± symmetric on P? Our next lemma gives a partial answer to this question. Later on, we will answer it in more detail, but we can only do so after combining the lemma with results involving scattering theory, obtained in the next section. Lemma 2.3. Assume (a+ , a− , γ ) belongs to P (1.27). Then the operators A± are well defined on P (1.29), and the operator As with step size as is symmetric.

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425

We prove this lemma in Appendix A. In the next section we will show that both operators A± are symmetric. As already explained above, the estimates (2.23) combined with the lemma imply essential self-adjointness of As on P. Denoting the self-adjoint closure by the same symbol, we obtain a unitary one-parameter group exp(−itAs ) on the closure P. In general, P = Ran(F)− is a proper subspace of H. Whenever this is the case, we extend As provisionally to a self-adjoint operator on H by choosing it equal to an arbitrary bounded self-adjoint operator on the orthogonal complement P ⊥ . We cannot avoid this provisional extension, since at this point we cannot yet prove that P ⊥ is actually spanned by finitely many pairwise orthogonal functions that are restrictions to v ∈ (0, ∞) of joint eigenfunctions of the AOs A± with real eigenvalues. The results we are now going to obtain via scattering theory are independent of the above extension. (Our final choice will be detailed in Sect. 4.) 3. Scattering Theory and Isometry of F Thanks to the above analysis, we now have a unitary one-parameter group exp(−itAs ) on H available for parameters in P . From its definition, it is immediate that it fulfills the intertwining relation exp(−itAs )Fφ = F exp(−itMs )φ, φ ∈ C.

(3.1)

At this point, however, we still do not know whether F is bounded on H. A principal result of this section is that F is actually isometric. This involves a comparison of the interacting evolution exp(−itAs ) to the free evolution exp(−itA0,s ) defined in the Introduction, cf. (1.31)–(1.33). The following lemma is the key to isometry. We will use it for parameters in P , but its proof only involves the assumption that E(v, v), ˆ vˆ > 0, is regular at v = 0. Lemma 3.1. Assume the parameters = (a+ , a− , γ ) ∈ are such that E(; v, v), ˆ vˆ > 0, has no pole at v = 0. Then we have lim (F − F0 ) exp(−itMδ )φ = 0, δ = +, −, φ ∈ C,

(3.2)

lim (F − F0 m) exp(−itMδ )φ = 0, δ = +, −, φ ∈ C,

(3.3)

t→∞

t→−∞

where m denotes the operator of multiplication by u(a+ , a− , γˆ ; −v). ˆ Proof. Letting φ ∈ C with supp(φ) ⊂ [r− , r+ ], 0 < r− < r+ , we have (F − F0 ) exp(−itMδ )φ2 2 ∞ r+ 1 = dv dp(E − E0 )(v, p) exp(−itEδ (p))φ(p) . a+ a− 0 r−

(3.4)

Now when we change variables p → y = Eδ (p), we see that the p-integral yields a function b(t, v) that converges to 0 for t → ∞ by the Riemann-Lebesgue lemma. Recalling (A.1)–(A.3) and using the regularity assumption, we also deduce that we have |b(t, v)| < C on R × [0, ∞). By dominated convergence, it follows that the v-integral over an interval (0, R) converges to 0 as t → ∞.

426

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In the integral over [R, ∞) with R > 1 (say), we telescope E − E0 with Eas and estimate in the obvious way to obtain an upper bound 2(I1 + I2 )/a+ a− , with ∞ r+ I1 ≡ dv dp|D(v, p)φ(p)|2 , (3.5) r−

R

∞

I2 ≡ 1

dv

r+

r−

2 dp[u(γˆ ; −p) − 1] exp(−iαvp − itEδ (p))φ(p) .

(3.6)

Using the uniform bound (A.3), we see that I1 can be made arbitrarily small by choosing R large enough. We now write exp(−iαvp − itEδ (p)) = i[αv + 4πtaδ−1 sinh(2πp/aδ )]−1 ∂p exp(−iαvp − itEδ (p)), (3.7) and integrate by parts in I2 . Then we readily verify that I2 can be made arbitrarily small by choosing t large enough. As a result, (3.4) can be made as small as we please by choosing t large enough. Therefore, (3.2) follows. The proof of (3.3) proceeds along the same lines, so it need not be spelled out. We will come back to our assumptions in this lemma shortly, but first we obtain a crucial corollary. Corollary 3.2. Let (a+ , a− , γ ) ∈ P . Then the operator family exp(itAs ) exp(−itA0,s ) F0 , t ∈ R, has strong limits U± for t → ±∞ given by U+ = F,

U− = Fu(γˆ ; ·).

(3.8)

The operator F is an isometry and the S-operator S ≡ U+∗ U−

(3.9)

equals multiplication by u(γˆ ; v). ˆ Proof. Using (3.1) and (1.33), we obtain for all φ ∈ C Fφ − exp(itAs ) exp(−itA0,s )F0 φ = (F − F0 ) exp(−itMs )φ.

(3.10)

By (3.2), this has limit 0 for t → ∞. Likewise, we have Fm∗ φ − exp(itAs ) exp(−itA0,s )F0 φ = (Fφ − F0 m) exp(−itMs )m∗ φ, (3.11) with limit 0 for t → −∞ due to (3.3). Since the family exp(itAs ) exp(−itA0,s )F0 consists of unitaries, the first assertion and isometry of F result. Isometry of F implies F ∗ F = 1, so the S-operator (3.9) equals u(γˆ ; ·). Now that we know F is isometric, we can conclude that both operators A± are essentially self-adjoint on P. Defining a one-parameter unitary group exp(−itAl ) just as we did for As (cf. the end of Sect. 2), we see from Lemma 3.1 that in Corollary 3.2 we may replace As and A0,s by Al and A0,l , resp. More is true: Assuming ∞ M(y) ∈ CR ([0, ∞)), M (y) > 0, y ∈ [0, ∞),

(3.12)

Generalized Hypergeometric Function III

427

we obtain self-adjoint operators A and A0 on P and H, resp., by setting AFψ ≡ FM(·)ψ, A0 F0 ψ ≡ F0 M(·)ψ, ψ ∈ DM ,

(3.13)

with DM the natural domain of M(·). Replacing Mδ by M(·) in Lemma 3.1, it still holds with obvious changes. Thus Corollary 3.2 follows with As → A, A0,s → A0 . (This is a manifestation of the invariance principle for the wave operators [26].) Another important point should be made next. At this stage we do not yet know whether the two properties of isometry of F and symmetry of A± on P can ever hold for parameters not in P . However, given the regularity assumption of Lemma 3.1, we now know that these properties are equivalent. Since we have already shown that the regularity assumption is met for parameters in r , the question arises whether F is isometric for parameters in r \ P . From results obtained later on it will transpire that this is not true in general. To continue, however, we show that isometry of F (hence symmetry of A± on P) still holds true on a set Pe that is slightly larger than P . Specifically, Pe is defined by allowing one γµ to vary over [−a, a], whereas the three remaining ones vary over (−a, a). Thus Pe is a subset of r and satisfies W Pe = Pe ,

(3.14)

just as r , cf. (2.20). This extension of P arises in particular for what we have called the even, attractive sine-Gordon channel in Ref. [29]. Theorem 3.3. The operators F(), = (a+ , a− , γ ) ∈ Pe ,

(3.15)

form a W -invariant family of isometries that is strongly continuous on Pe . Proof. We have already shown isometry of F for ∈ P . To prove isometry for parameters in Pe \ P , we need only consider γ ≡ (−a, γ1 , γ2 , γ3 ) with |γj | < a, j = 1, 2, 3. (Indeed, W -invariance of F is clear from W -invariance of E.) Letting ∈ (0, a), we have γ () ≡ (−a +, γ1 , γ2 , γ3 ) ∈ P and γ () → γ for → 0. From the last assertion of Lemma 2.2 we now deduce F(γ )φ = s · lim F(γ ())φ, ∀φ ∈ C. ↓0

(3.16)

ˆ The strong continuity Hence F(γ ) is isometric on C, so it extends to an isometry on H. of the family on Pe now follows from isometry and strong continuity of F()φ, φ ∈ C, on Pe . Thus far, we have not discussed the adjoint F ∗ of F. A moment’s thought suffices to see that it is defined on C ⊂ H whenever E(v, v) ˆ is not only regular at v = 0 for vˆ > 0 (which we required to define F, cf. (2.22)), but also regular at vˆ = 0 for v > 0. Indeed, in that case we clearly have ∞ ˆ φ(v) → (a+ a− )−1/2 F ∗ : C → H, dvE(γ ; v, v)φ(v). ˆ (3.17) 0

We now analyze this dual regularity condition. First, let us note that (2.2) yields E(γ ; v, v) ˆ = χ (γ )−2 u(γ ; v)u(γˆ ; v)E(γ ˆ ; v, v), ˆ v, vˆ > 0.

(3.18)

428

S.N.M. Ruijsenaars

Next, we recall the self-duality relation E(γ ; v, v) ˆ = E(γˆ ; v, ˆ v),

(3.19)

cf. II (1.22). For (a+ , a− , γ ) ∈ r it need not be true that we have (a+ , a− , γˆ ) ∈ r , so in this case we cannot conclude from (3.19) that the dual regularity condition is satisfied. On the other hand, it is not difficult to check that we have J Pe ⊂ r .

(3.20)

Thus (3.18) and (3.19) imply that on Pe the dual regularity requirement is met. This is in accordance with Theorem 3.3: For (a+ , a− , γ ) ∈ Pe the operator F is isometric, so that its adjoint F ∗ is bounded. More generally, we summarize some salient features of F ∗ in the following theorem. Theorem 3.4. The operators F ∗ (), = (a+ , a− , γ ) ∈ Pe ,

(3.21)

form a W -invariant family of partial isometries that is strongly continuous on Pe ; moreover, their action on C is given by (3.17), and they are isometric when (a+ , a− , γˆ ) ∈ Pe . Proof. Theorem 3.3 implies that the family consists of W -invariant partial isometries and that it is weakly continuous on Pe . From (3.20) we obtain regularity at vˆ = 0 and hence (3.17). Combining (3.17)–(3.20) with the last assertion of Lemma 2.2, we see that ˆ → F ∗ ()φ, is strongly continuous for all φ ∈ C. Since F ∗ () has the map Pe → H, norm 1, strong continuity of the operator family follows. Finally, for (a+ , a− , γˆ ) ∈ Pe we deduce isometry of F ∗ from (3.17)–(3.19) and isometry of F on Pe . Consider next the dual AOs, Aˆ + ≡ A(a+ , a− , γˆ ; v), ˆ Aˆ − ≡ A(a− , a+ , γˆ ; v), ˆ

(3.22)

cf. II Sect. 2. Their action on E(γ ; v, v) ˆ yields eigenvalues E± (v). But when they act on the meromorphic function (F ∗ φ)(v), φ ∈ C, then (3.17)–(3.18) reveal this does not yield the functions (F ∗ Mδ φ)(v). The point is that the u-function in (3.18) spoils the eigenfunction property. This circumstance is one of the drawbacks of the use of the E-function for Hilbert space purposes. (It does not occur for the first two pictures described in the Introduction.) But it can be remedied by switching to AOs, A˜ ± ≡ u(γˆ ; v) ˆ Aˆ ± u(γˆ ; v) ˆ −1 .

(3.23)

Indeed, we obtain in the same way as for A± the relations (A˜ ± F ∗ φ)(v) = (F ∗ M± φ)(v).

(3.24)

In our Hilbert space formulation, therefore, it is more natural to work with A˜ ± . For a better appreciation of (3.23), we should add that (1.18) and (1.20) entail u(y)A(a+ , a− , p; y)u(y)−1 = V˜a (a+ , a− , p; y) exp(−ia− ∂y ) + exp(ia− ∂y ) + Vb (a+ , a− , p; y),

(3.25)

Generalized Hypergeometric Function III

429

with V˜a (a+ , a− , p; y) ≡ u(a+ , a− , p; y)/u(a+ , a− , p; y − ia− ).

(3.26)

Let us now fix (a+ , a− , γ ) ∈ Pe . Then we can use (3.24) to obtain Hilbert space operators A˜ ± : F ∗ C ⊂ Hˆ → F ∗ C,

(3.27)

but the reasoning in the proof of Lemma 2.3 cannot be “dualized” whenever (a+ , a− , γˆ ) ∈ / Pe . The difficulty is that in that case we encounter poles inside the pertinent closed contour. This state of affairs will play a pivotal role in the next section. Here we add that when we do have (a+ , a− , γˆ ) ∈ Pe , we have already shown isometry of F ∗ , cf. Theorem 3.4. Therefore, F is in fact unitary in that case, and the self-adjoint closures of the operators A± on FC and A˜ ± on F ∗ C are unitarily equivalent to multiplication by E± (v) ˆ and E± (v), resp. Since J is orthogonal, parameters in Pe give rise to dual parameters in Pe for |γ | < a. But this is not a necessary condition. Indeed, recalling the equivalences Jp = p ⇔ (Jp)0 = p0 ⇔ p0 = p1 + p1 + p3 ,

(3.28)

we see that F is unitary on the fixed point subset Pe,sd ≡ {(a+ , a− , p) ∈ Pe | Jp = p}

(3.29)

of Pe , and on Pe,sd the norm of γ can be arbitrarily close to 2a. (Take γ = (b, b, b, −b), b = a − , to check this.) 4. Bound States and Spectral Resolutions Letting (a+ , a− , γ ) ∈ Pe , we have now established that F is an isometry. As a consequence, we have operator identities F ∗ F = 1, FF ∗ = 1 − Qbs ,

(4.1)

where Qbs : H → H is the orthogonal projection on Ran(F)⊥ = P ⊥ . In particular, defining Pˆe ≡ {(a+ , a− , p) ∈ | (a+ , a− , Jp) ∈ Pe },

(4.2)

we have seen at the end of Sect. 3 that (a+ , a− , γ ) ∈ Pe ∩ Pˆe ⇒ Qbs = 0.

(4.3)

The set Pe ∩ Pˆe for which F is unitary is a proper subset of Pe , however. In this section we obtain a quite detailed picture of P ⊥ for all (a+ , a− , γ ) ∈ Pe . Indeed, as it turns out, the space P ⊥ can be explicitly understood via a discretization of the R-function that yields (an analytic continuation of) the Askey-Wilson polynomials, cf. I Sect. 3. The simplest polynomial is of course the constant one, and we begin by elucidating its role in the present Hilbert space context.

430

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To this end we recall how it arises from the R-function: We have (for generic parameters) R(c; v, i cˆ0 ) = 1,

(4.4)

cf. I (3.22)–(3.26). We also recall that the joint eigenfunction property of the constant function is plain from the structure of the AOs A± (c; v) given by I (3.2), just as their eigenvalues 2 cos(αa∓ cˆ0 ). (Indeed, the operators Tzv −1 annihilate the constant function for any value of the shift parameter z.) It is therefore obvious from the similarity transformation connecting A± (c; v) and the AOs A± that the function 1/c(γ ; v) obeys A± [c(γ ; v)]−1 = 2 cos(αa∓ [ γµ /2 + a])[c(γ ; v)]−1 . (4.5) µ

Setting γ w ≡ w(γ ), w ∈ W,

(4.6)

the W -invariance of A± (proved in II) entails γµw /2 + a])[c(γ w ; v)]−1 . A± [c(γ w ; v)]−1 = 2 cos(αa∓ [

(4.7)

µ

For w ∈ S4 , this gives no new information, since c(γ ; v) is S4 -invariant, cf. (1.12). But for generic γ ∈ R4 neither c(γ ; v) nor the eigenvalues are invariant under an even number of sign flips of γ0 , . . . , γ3 . Therefore we obtain (generically) eight distinct joint eigenfunctions with eight distinct joint eigenvalues! We proceed to discuss the question whether these functions belong to H, choosing (a+ , a− , γ ) ∈ Pe from now on. This choice entails that the joint eigenfunctions 1/c(γ w ; v) are regular at v = 0, so we need only inspect their v → ∞ behavior. The latter can be inferred from II (3.5): It is given by c(γ w ; v)−1 = exp[αv( γµw /2 + a)]χ (γ )−1 µ

× (1 + O(exp[−σ αas v])),

v → ∞,

(4.8)

with σ < 1 and the bound uniform on -compacts. Hence we have c(γ w ; ·)−1 ∈ H ⇔ (J γ w )0 < −a, (a+ , a− , γ ) ∈ Pe .

(4.9)

This result yields a first reason why we now switch to a suitable fundamental domain for the W -action, namely the set DW given by II (2.19). More specifically, until further notice we consider (a+ , a− , γ ) in the parameter set Pe≤ ≡ {(a+ , a− , p) ∈ Pe | p ∈ DW }.

(4.10)

−a < γ0 ≤ γ1 ≤ γ2 ≤ |γ3 | ≤ 0,

(4.11)

−a = γ0 < γ1 ≤ γ2 ≤ |γ3 | ≤ 0.

(4.12)

Thus we have either

or

Generalized Hypergeometric Function III

431

In both cases, it easily follows from (1.6) that the dual parameters (a+ , a− , γˆ ) belong to ≤ r ≡ {(a+ , a− , p) ∈ r | p ∈ DW }.

(4.13)

Pˆe≤ ≡ {(a+ , a− , p) | (a+ , a− , Jp) ∈ Pe≤ },

(4.14)

The set

is however a proper subset of ≤ r . (Fixing a+ , a− , the volumes of the two resulting sets in R4 differ by a factor 2, since J is orthogonal.) It is also important to observe that we have a decomposition ≤ ≤ ≤ r = Pe ∪ Pbs ,

(4.15)

where ≤ Pbs ≡ {(a+ , a− , p) ∈ r | p0 ∈ (−2a, −a), −a < p1 ≤ p2 ≤ |p3 | ≤ 0}.

(4.16)

Returning to (4.9) with the above assumption (a+ , a− , γ ) ∈ Pe≤ in force, we see that there are two cases to consider. In the first case we have (a+ , a− , γˆ ) ∈ Pe≤ . Then 1/c(γ ; v) is not square-integrable, and neither are its W -transforms. In the second case ≤ , so that γˆ0 ∈ (−2a, −a). This entails that the function we have (a+ , a− , γˆ ) ∈ Pbs 1/c(γ ; v) is in H, whereas none of the (at most) seven other functions obtained via sign flips of γ is in H, cf. (4.9). The key point is now that in the second case the joint eigenfunction 1/c(γ ; v) is orthogonal to P, as we will prove shortly. More generally, we will show that P ⊥ is spanned by functions in H of the form ψn (; v) = Pn (; cosh(αas v))/c(; v), n ∈ N, = (a+ , a− , γ ),

(4.17)

where Pn (u) denotes the degree-n polynomials from I Theorem 3.2. Taking this for granted, it is immediate from the c-function asymptotics (4.8) which of these functions satisfy the H-restriction: The number N of functions ψ0 , . . . , ψN−1 in H equals the largest integer such that cˆ0 + (N − 1)as < 0,

cˆ0 ∈ (−as /2 − al /2, 0).

(4.18)

It should be noted that this entails the restrictions al > (2N − 3)as , N > 1, cˆ0 ∈ as [−N, −N + 1).

(4.19) (4.20)

After these introductory observations we turn to a complete analysis for parameters satisfying ∈ Pe≤ , ∈ / Pˆe≤ .

(4.21)

(For ∈ Pe≤ ∩ Pˆe≤ the operator F is unitary, and little more can be said than we already did in Sect. 3.) A crucial role in this analysis is played by the relations 3

γµ ∈ (−4a, −2a), γµ + γν ∈ (−2a, 0), 0 ≤ µ < ν ≤ 3,

µ=0

which readily follow from (4.21). (Recall (4.10)–(4.12) and (4.14) to see this.)

(4.22)

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Our first task is to make the relation to the results in I Sect. 3 more precise. Indeed, our present choice of parameters differs from the one made there. To begin with, consider the renormalizing factor (1.9) in (1.8). From (4.22) and (2.3) we deduce ρ(γ ) ∈ (0, ∞),

(4.23)

so this factor does not give rise to trouble. Next, we compare the v-values ˆ vˆn ≡ i cˆ0 + inas , n = −1, 0, . . . , N,

(4.24)

to the (eventual) pole locations ±i vˆ = −γˆµ − a + kas + mal , k, m ∈ N∗ ,

(4.25)

ˆ (4.24) are releof Rren , cf. I (2.35). (As will become clear shortly, only the v-values vant in the present context.) The numbers on the rhs of (4.25) are positive, whereas vˆ−1 , . . . , vˆN−1 belong to the lower half plane due to (4.18). Hence we first study the equations γˆ0 + nas = γˆµ − kas − mal , µ = 0, 1, 2, 3, k, m ∈ N∗ ,

(4.26)

for n = −1, . . . , N − 1. Obviously, for µ = 0 they have no solutions. For µ = p > 0 we recall γˆ0 − γˆp = γq + γr > −2a, {p, q, r} = {1, 2, 3},

(4.27)

to infer that they have no solutions for n ≥ 0, whereas for n = −1 there is a unique solution, provided γˆ0 − γˆp = −al ,

p = 1, 2, 3.

(4.28)

Thus R(v, v) ˆ may have a pole for vˆ = vˆ−1 , whose multiplicity equals at most the number of p satisfying (4.28). But v = vˆ0 is not among the possible pole locations, so from I (3.26) we obtain R(v, vˆ0 ) = 1.

(4.29)

The number −i vˆN belongs to [0, as ), so it cannot equal the rhs of (4.25) for µ = 0, 1, 2. But for µ = 3 we get a unique solution for γˆ0 + γˆ3 + N as = 0.

(4.30)

Hence R(v, v) ˆ may have a simple pole at vˆ = vˆN , provided (a+ , a− , γ ) satisfies (4.30). In order to arrive at a polynomial recurrence, we now use the eigenvalue AE for the dual AO As (ˆc; v) ˆ with step size as , cf. I (3.28)–(3.29). Here it is more convenient to start from II (2.1)–(2.3), however. Thus we consider ˆ vˆ + ias ) − R(v, v)] ˆ + C(al , as , γˆ ; v) ˆ C(al , as , γˆ ; −v)[R(v, × [R(v, vˆ − ias ) − R(v, v)] ˆ + 2 cos(αas c0 )R(v, v) ˆ = 2uR(v, v), ˆ (4.31) where u ≡ cosh(αas v),

(4.32)

Generalized Hypergeometric Function III

433

and where the parameters (a+ , a− , c(γ )) of the R-function are suppressed. (Note that R is invariant under a− ↔ a+ when it is parametrized by γ instead of c.) Now from II (2.1) we see that vˆ = −vˆ0 is neither a pole nor a zero of C(al , as , γˆ ; v). ˆ But vˆ = vˆ0 yields a simple zero whenever cˆ0 = −al /2, −al /2 + as /2, −al + as /2,

(4.33)

γˆ0 − γˆp = −al , p = 1, 2, 3.

(4.34)

and

This suffices to deduce by continuity lim C(al , as , γˆ ; v)[R(v, ˆ vˆ − ias ) − R(v, v)] ˆ = 0.

v→ ˆ vˆ0

(4.35)

(Indeed, the remaining terms in (4.31) are regular at vˆ = vˆ0 for all parameters at issue.) As an interesting consequence of (4.35) we mention in passing that when (4.34) holds, we must have R(v, vˆ−1 ) = 1, cˆ0 ∈ {−al /2, −al /2 + as /2, −al + as /2}.

(4.36)

(Indeed, in this case we have C(al , as , γˆ ; vˆ0 ) = 0.) This can also be deduced from the contour shift argument yielding (4.29), cf. the paragraph containing I (3.26). Next, II (2.1) entails that the v-values ˆ ±vˆ1 , . . . , ±vˆN−2 and vˆN−1 are not poles of C(al , as , γˆ ; v). ˆ Thus we may now define an ≡ C(al , as , γˆ ; −vˆn ), n = 0, 1, . . . , N − 2, cn ≡ C(al , as , γˆ ; vˆn ), n = 1, . . . , N − 1, which yields explicitly 4 3µ=0 sin(π[γˆ0 + γˆµ + (n + 1)as ]/al ) an = , sin(π[2cˆ0 + 2nas ]/al ) sin(π[2cˆ0 + (2n + 1)as ]/al )

cn =

4

3

µ=0 sin(π[γˆ0

− γˆµ + nas ]/al )

sin(π[2cˆ0 + 2nas ]/al ) sin(π[2cˆ0 + (2n − 1)as ]/al )

,

(4.37) (4.38)

n = 0, . . . , N − 2, (4.39)

n = 1, . . . , N − 1. (4.40)

From this we obtain using (4.19), (4.20) and (4.22), an ∈ (−∞, 0), cn ∈ (−∞, 0),

n = 0, . . . , N − 2, n = 1, . . . , N − 1.

(4.41) (4.42)

Putting the pieces together, we see that we have R(a+ , a− , c(γ ); v, i γˆ0 + ia + inas ) = Pn (a+ , a− , γ ; cosh(αas v)), n = 0, . . . , N − 1,

(4.43)

where Pn (u) is a polynomial of degree n with real coefficients, uniquely determined by P0 (u) ≡ 1 and the recurrence a0 P1 (u) + b0 = 2u,

(4.44)

434

S.N.M. Ruijsenaars

an Pn+1 (u) + bn Pn (u) + cn Pn−1 (u) = 2uPn (u), n = 1, . . . , N − 2,

(4.45)

with b0 ≡ 2 cos(αas c0 ) − a0 , bn = 2 cos(αas c0 ) − an − cn ,

n = 1, . . . , N − 2. (4.46)

The corresponding functions ψn (; v) (4.17) satisfy Al ψn (v) = El ψn (v), As ψn (v) = Es,n ψn (v),

n = 0, . . . , N − 1, n = 0, . . . , N − 1,

(4.47) (4.48)

where El ≡ 2 cos(2π cˆ0 /as ), Es,n ≡ 2 cos(2π [cˆ0 + nas ]/al ), n = 0, . . . , N − 1,

(4.49) (4.50)

and A± are the AOs (1.18). From (4.18) we also infer −2 ≤ Es,0 < · · · < Es,N−1 < 2.

(4.51)

As already noted, it is obvious from the structure of the meromorphic functions ψn (v), n = 0, . . . , N − 1, that their restrictions to (0, ∞) belong to H. We are now prepared for the following lemma. Lemma 4.1. With (4.21) in effect and N the largest integer satisfying (4.18), the functions ψ0 , . . . , ψN−1 are pairwise orthogonal; moreover, they satisfy Qbs ψm = ψm , m = 0, . . . , N − 1,

(4.52)

where Qbs is defined by (4.1). We have relegated the proof of this lemma to Appendix B. The idea of the proof is quite simple: We use distinctness of the As -eigenvalues Es,n and Es (v), ˆ vˆ > 0, and symmetry of As reinterpreted as a Hilbert space operator. In this connection we recall the provisional extension to P ⊥ of the operators A± we made in Sects. 2 and 3: In view of this lemma we may and will choose the extension on the N -dimensional subspace of P ⊥ spanned by ψ0 (v), . . . , ψN−1 (v) in accordance with (4.47) and (4.48). Next, we are going to show that the latter subspace coincides with P ⊥ ; to use quantum-mechanical parlance, this entails that the joint eigenfunctions E(v, v), ˆ vˆ > 0, and ψ0 (v), . . . , ψN−1 (v) form a complete orthogonal set. Compared to our previous proofs in this paper, the completeness proof for which we now prepare the ground may seem counterintuitive (if not opaque) on first acquaintance. We have not found any direct proof, but as a bonus of our indirect reasoning we obtain additional information that is of considerable interest in itself. The starting point of our arguments is the following observation. At this stage we already know that our assumption (4.21) entails that F ∗ is not isometric. Specifically, from Lemma 4.1 we see that Ran(F)⊥ is at least N -dimensional. But this implies that the Hilbert space operators we have associated to the dual AOs A˜ ± (cf. the end of Sect. 3) cannot be symmetric. (Indeed, if one of them would be symmetric, we could use a dualized scattering theory argument to prove that F ∗ is an isometry, a contradiction.)

Generalized Hypergeometric Function III

435

Now when we inspect the dual version of our symmetry proof (i.e., the proof of Lemma 2.3 in Appendix A), we see that the only way this proof can break down is that a nonzero residue sum arises for the relevant closed contour. For the AO A˜ l this is all we know (that is, we do not know the residues explicitly), but for A˜ s we can actually find the residues in closed form. The crux is now that the resulting explicit symmetry breakdown identity can be exploited not only to prove completeness, but also to obtain the norms of the above functions ψ0 (v), . . . , ψN−1 (v)! The pertinent residue sum involves apart from R(v, vˆN−1 ) = PN−1 (u) the function R(v, vˆN ), whose features we now elucidate. For the parameters at issue, it can have two anomalies. First, for the special case cˆ0 + (N − 1)as = −as /2,

(4.53)

it reduces to PN−1 (u). One way to appreciate this is to observe that its eigenvalue Es,N (given by (4.50)) reduces to Es,N−1 when (4.53) holds. To check the asserted equality, however, the behavior of the recurrence relation for vˆ = vˆN should be inspected: The point is that when we define aN−1 by (4.37) (yielding (4.39) with n = N − 1), we arrive at a pole for parameters satisfying (4.53). The second anomaly can also be seen from (4.39) with n = N − 1: For the special case (4.30), the coefficient aN−1 vanishes, so that the highest degree coefficient of PN (u) develops a pole. Thus the eventual simple pole we already found in the paragraph containing (4.30) is indeed present. For parameters that do not satisfy (4.53) and (4.30) we have aN−1 ∈ R∗ .

(4.54)

R(v, vˆN ) = PN (cosh(αas v)),

(4.55)

Hence we obtain

where PN (u) is the degree-N polynomial obtained from (4.45) with n = N − 1 and bN −1 defined by (4.46), and so we may and will define ψN (; v) by (4.17) for n = N , too. Next, we study E(γ ; v, v) ˆ (1.10) for vˆ = vˆ0 , . . . , vˆN−1 . At these v-values ˆ the factor Rren (v, v) ˆ is regular, as we have already established. But the factor 1/c(γˆ ; v) ˆ gives rise to simple poles, so that the bound states ψ0 (v), . . . , ψN−1 (v) arise as the residues of E(v, v) ˆ at these poles. To be more precise, we have Res E(γ ; v, v)| ˆ v= ˆ vˆn = χ (γ )ρ(γ )ψn (γ ; v)rn (γ ), n = 0, . . . , N − 1,

(4.56)

rn (a+ , a− , γ ) ≡ Res (1/c(a+ , a− , γˆ ; v))| ˆ v= ˆ vˆn , n = 0, . . . , N − 1.

(4.57)

with

We are now prepared for the following lemma, which is proved in Appendix B. Lemma 4.2. Suppose = (a+ , a− , γ ) satisfies (4.21) and, moreover, cˆ0 + (N − 1/2)as = 0, cˆ0 + N as = 0, γˆ0 + γˆ3 + N as = 0.

(4.58) (4.59) (4.60)

436

S.N.M. Ruijsenaars

Then we have for all φ1 , φ2 ∈ C, (A˜ s F ∗ φ1 , F ∗ φ2 ) − (F ∗ φ1 , A˜ s F ∗ φ2 ) = N

∞

∞

dv1 φ1 (v1 ) 0

dv2 φ2 (v2 )s(v1 , v2 ), 0

(4.61) with N ≡ 2iπρ(γ )2 rN−1 (γ )/a+ a− c(γˆ ; −vˆN ), s(v1 , v2 ) ≡ u(γ ; v2 )[ψN (v1 )ψN−1 (v2 ) − (v1 ↔ v2 )].

(4.62) (4.63)

As announced, it still seems unclear why the result of this lemma has a bearing on our Hilbert space completeness problem. (Note in particular that ψN (v) is not squareintegrable over (0, ∞).) We proceed to clarify the connection. First, we need renormalized polynomials P˜n (u) ≡ (−)n kn Pn (u),

n = 0, . . . , N − 1,

(4.64)

where k0 ≡ 1,

kn ≡

n

(aj −1 /cj )1/2 , n = 1, . . . , N − 1.

(4.65)

j =1

Due to (4.41) and (4.42), the radicands are positive, and we choose the positive square root. Likewise, we introduce a˜ n ≡ (an cn+1 )1/2 ∈ (0, ∞), n = 0, . . . , N − 2.

(4.66)

Then we easily verify that for n = 0, . . . , N − 2, we obtain a self-adjoint recurrence a˜ n P˜n+1 (u) + bn P˜n (u) + a˜ n−1 P˜n−1 (u) = 2uP˜n (u),

(4.67)

with bn given by (4.46) and a˜ −1 P˜−1 (u) ≡ 0.

(4.68)

Now with the assumptions of the lemma in effect, (4.45) is still valid for n = N − 1, with aN−1 , cN−1 and bN−1 given by (4.39), (4.40) and (4.46), resp. Defining a˜ N−1 ≡ (−)N−1 kN−1 aN−1 , P˜N (u) ≡ PN (u),

(4.69)

it readily follows that (4.67) holds for n = N − 1, too. We now observe that (4.67) entails an identity 2(t − u)

N−1

P˜n (t)P˜n (u) = a˜ N−1 [P˜N (t)P˜N−1 (u) − P˜N (u)P˜N−1 (t)],

(4.70)

n=0

whose proof is elementary. This identity (known as the Christoffel-Darboux formula) yields the link to our completeness problem. Setting ψ˜ n (v) ≡ P˜n (u)/c(γ ; v), n = 0, . . . , N,

(4.71)

Generalized Hypergeometric Function III

437

it amounts to ψN (v1 )ψN−1 (v2 ) − (v1 ↔ v2 ) =

N−1 Es (v1 ) − Es (v2 ) ψ˜ n (v1 )ψ˜ n (v2 ). 2 kN−1 aN−1 n=0

(4.72)

Thus we can rewrite (4.61) as (F ∗ Ms φ1 , F ∗ φ2 ) − (F ∗ φ1 , F ∗ Ms φ2 ) N−1 = N˜ [(φ1 , ψ˜ n )(ψ˜ n , Ms φ2 ) − (Ms φ1 , ψ˜ n )(ψ˜ n , φ2 )],

(4.73)

n=0

with 2 N˜ ≡ N /kN−1 aN−1 .

(4.74)

Now Ms maps C onto C. Setting φj ≡ M−1 s χj , j = 1, 2, we deduce that we have the following relation between bounded operators on H: −1 ∗ −1 ˜ ˜ ˜ −1 FF ∗ M−1 s − Ms FF = N (Ms Q − QMs ),

˜ ≡ Q

N−1

ψ˜ n ⊗ ψ˜ n .

(4.75)

(4.76)

n=0

Recalling (4.1), we see that this amounts to ˜ ˜ [M−1 s , Qbs − N Q] = 0.

(4.77)

Since M−1 s is a bounded multiplication operator with simple spectrum [0, 1/2], it follows that there exists a bounded function µ(v) such that ˜ = µ(·). Qbs − N˜ Q

(4.78)

Acting on ψ˜ n , this yields by Lemma 4.1 (1 − N˜ (ψ˜ n , ψ˜ n ))ψ˜ n = µ(·)ψ˜ n , n = 0, . . . , N − 1,

(4.79)

so µ(v) is constant. Acting on a nonzero vector ψ ∈ Ran(F), it follows that this constant equals 0. As a consequence, we have proved ˜ Qbs = N˜ Q,

(4.80)

(ψ˜ n , ψ˜ n ) = 1/N˜ , n = 0, . . . , N − 1,

(4.81)

and

under the assumptions of Lemma 4.2. More generally, we have the following theorem.

438

S.N.M. Ruijsenaars

Theorem 4.3. With (4.21) in effect and N the largest integer satisfying (4.18), the functions ψ0 , . . . , ψN−1 satisfy (ψm , ψn ) = νn−1 δmn ,

m, n = 0, . . . , N − 1,

(4.82)

where ν0 ≡ (a+ a− )−1/2 G(i

3

γµ + 3ia)/

G(iγµ + iγν + ia),

(4.83)

0≤µ<ν≤3

µ=0

νn ≡ ν0 kn2 , n = 1, . . . , N − 1,

(4.84)

and kn is defined by (4.65), (4.39) and (4.40). Moreover, the orthogonal projection Qbs defined by (4.1) can be written Qbs =

N−1

νn ψ n ⊗ ψ n .

(4.85)

n=0

Proof. We first prove the theorem for parameters satisfying the restrictions (4.58)–(4.60). Then Lemma 4.2 applies and we may invoke the above consequences. Specifically, combining (4.80), (4.76), (4.71), (4.64) and (4.17), we obtain Qbs = N˜

N−1

kn2 ψn ⊗ ψn .

(4.86)

n=0

Since we have already shown pairwise orthogonality in Lemma 4.1, it remains to verify the equality ν0 = N˜ .

(4.87)

To this end we first combine (4.74) and (4.65) to get N˜ = N

N−1 j =1

cj /

N−1

aj .

(4.88)

j =0

Next, we use (4.37), (4.38) and the representation ˆ = c(γˆ ; v)/c( ˆ γˆ ; vˆ − ias ), C(al , as , γˆ ; v)

(4.89)

cf. II (2.12). Since 1/c(γˆ ; v) ˆ has a simple pole for vˆ = vˆ0 , . . . , vˆN−1 , this yields N−1

cj = r0 /rN−1 ,

(4.90)

aj = c(γˆ ; −vˆ0 )/c(γˆ ; −vˆN ).

(4.91)

j =1

cf. (4.57), whereas N−1 j =0

Generalized Hypergeometric Function III

439

Combining (4.88), (4.90) and (4.91) with (4.62), we obtain N˜ = 2iπρ 2 r0 /a+ a− c(γˆ ; −vˆ0 ).

(4.92)

To calculate r0 , we recall (1.12) and the relation Res (1/G(z))|z=ia = (a+ a− )1/2 /2iπ,

(4.93)

cf. I (A.20). From this we get r0 =

(a+ a− )1/2 G(2i cˆ0 + ia) . 2iπ 3j =1 G(i cˆ0 − i γˆj )

(4.94)

Using (1.12) once more, we also calculate c(γˆ ; −vˆ0 ) =

3

G(−i cˆ0 − i γˆj ) = ρ(γ ),

(4.95)

j =1

cf. (1.9). Substituting (4.94) and (4.95) in (4.92), we obtain (4.87), so we have now proved the theorem for parameters obeying (4.58)–(4.60). To lift the latter restrictions, we use a continuity argument. First, we recall that ψ0 , . . . , ψN−1 ∈ H were defined without these restrictions, and shown to be pairwise orthogonal in Lemma 4.1. It follows from their definition that these vectors depend continuously (in the strong topology) on (a+ , a− , γ ) ∈ Pe≤ , provided cˆ0 varies over the interval as [−N, −N + 1). Hence the first assertion of the theorem now follows by continuity. As just shown, the rhs of (4.85) is strongly continuous on the pertinent parameter set. Since F and F ∗ are strongly continuous on Pe by Theorems 3.3 and 3.4, the projection Qbs is strongly continuous on Pe as well, and (4.85) results. We have now completed our results for parameters in Pe≤ (4.10). Consider next parameters (a+ , a− , γ ) ∈ Pe . Acting with a suitable w ∈ W , we obtain (a+ , a− , γ w ) ∈ Pe≤ . Since the eigenfunction transform F is W -invariant, so are F ∗ and the bound state projection Qbs . In particular, the dimension of Qbs (H) is determined by the relation between the number (J γ w )0 = − max(|γˆ0 |, |γˆ1 |, |γˆ2 |, |γˆ3 |),

(4.96)

and as , al (recall (4.18)), and Qbs can be expressed in terms of the functions ψn (γ w ; v) in the same way as for (a+ , a− , γ ) ∈ Pe≤ . Next, the W -invariance of the AOs A± entails that their action on the latter functions is again given by (4.47)–(4.50). Thus we may and will choose the extension of the Hilbert space operators A± on P to P ⊥ in accordance with these formulas. We summarize their spectral properties in the following theorem. Theorem 4.4. Letting (a+ , a− , γ ) ∈ Pe , the Hilbert space operators A± are essentially self-adjoint on the direct sum of P and Hbs , where Hbs ≡ Span (ψ0 , . . . , ψN−1 ) = P ⊥ .

(4.97)

They have pure point spectrum on Hbs given by (4.47)–(4.51), and simple absolutely continuous spectrum [2, ∞) on P, and they commute. Proof. It suffices to recall that the isometry F from Hˆ onto P intertwines the action of ˆ A± on P and the action of multiplication by 2 cosh(αa∓ v) ˆ on C ⊂ H.

440

S.N.M. Ruijsenaars

Appendix A. Proofs of Lemmas 2.2 and 2.3 In this appendix we prove Lemmas 2.2 and 2.3 and add a comment on the latter proof. Proof of Lemma 2.2. The key to the proof is the uniform v → ∞ asymptotics established in II Theorem 1.2. Specifically, setting Eas (v, v) ˆ ≡ exp(iαv v) ˆ − u(γˆ ; −v) ˆ exp(−iαv v), ˆ α = 2π/a+ a− ,

(A.1)

D(v, v) ˆ ≡ (E − Eas )(v, v), ˆ

(A.2)

|D(v, v)| ˆ < C(, v) ˆ exp(−αas v/2), = (a+ , a− , γ ),

(A.3)

we obtain from II (1.32),

for all v > a (say), where C is continuous on × (0, ∞). Let us now fix ∈ such that E(; v, v) ˆ is regular at v = 0. Then we infer from (A.1)–(A.3) that we have |D(v, v)| ˆ < C(v) ˆ exp(−αas v/2), v ≥ 0,

(A.4)

where C is continuous on (0, ∞). Next, writing E = Eas + D, we note that the kernel Eas yields a bounded operator Hˆ → H. (Indeed, u(γˆ ; −v) ˆ is a phase for real vˆ and the plane waves give rise to Fourier transformations.) To prove Iφ (v) ∈ H and the bound (2.21), it is therefore sufficient to show that the integral 2 ∞ r+ dv dpD(v, p)φ(p) , (A.5) Dφ = 0

r−

is bounded above by Cφ2 , with C continuous in r− , r+ . For this purpose we invoke the Schwarz inequality. It yields ∞ r+ Dφ ≤ φ2 dv dp|D(v, p)|2 , 0

(A.6)

r−

so from the bound (A.4) we get Dφ ≤ φ (αas ) 2

−1

r+

dpC(p)2 .

(A.7)

r−

Hence (2.21) follows. We proceed to prove regularity at v = 0 for ∈ r . In view of (2.20), we need only handle the case γ0 ∈ (−2a, −a]. Moreover, recalling (2.14), it suffices to consider γ0 of the form γ0 = −a − nas ∈ (−2a, −a], n ∈ N, as = min(a+ , a− ).

(A.8)

The crux is that for this choice of γ0 and |γj | < a, j = 1, 2, 3, the E-function product in (2.11) has a simple zero at v = 0, cf. the paragraph containing (2.13). Since H (v, v) ˆ is holomorphic and the factor G(2v + ia) has a simple zero for v = 0, too, we deduce that E(v, v) ˆ has no pole at v = 0, as asserted.

Generalized Hypergeometric Function III

441

It remains to prove strong continuity of the map r → H, → Iφ (; ·). To this end we fix 0 ∈ r and choose δ0 > 0 such that the ball | − 0 | ≤ δ0 belongs to r (with | · | denoting the euclidean norm on R6 ). Fixing R > 0, we assert that the integral r+ 2 R dv dp[E(; v, p) − E(0 ; v, p)]φ(p) , | − 0 | ≤ δ, δ ∈ (0, δ0 ], 0

r−

(A.9) can be made arbitrarily small by a suitable δ-choice. To prove this assertion, suppose first that 0 is such that the E-function product in the denominator of (2.11) has no zero for v = 0. In that case no zero occurs for | − 0 | ≤ δ ≤ δ1 , with δ1 ≤ δ0 small enough, so that E(; v, p) is jointly continuous on the set | − 0 | ≤ δ1 , v ∈ [0, R], p ∈ [r− , r+ ]. This entails uniform continuity on the latter set, so that the assertion easily follows in this (generic) case. Consider next the special case that 0 gives rise to a zero at v = 0 of the E-function product. Since 0 ∈ r , this zero is simple. The simple zero of G(2v + ia) at v = 0 then ensures that E(; v, p) remains bounded near v = 0, uniformly for | − 0 | ≤ δ and p ∈ [r− , r+ ]. To be more specific, the function H (; v, v) ˆ in (2.11) is not only entire in v and v, ˆ but also real-analytic for ∈ , cf. I Theorem 2.2. Discontinuous behavior for (, v) → (0 , 0) is therefore caused solely by the quotient of the principal parts of G(2v + ia) and the pertinent E-function near v = 0. The coefficients of these principal parts do not vanish and are continuous for | − 0 | ≤ δ with δ small enough, so the relevant discontinuity is of the form v/(v − if ()), with f () real-valued and continuous for | − 0 | ≤ δ and f (0 ) = 0. A dominated convergence argument now shows that the discontinuity is innocuous, and so the assertion follows again. Thus it remains to control the v-integral over (R, ∞), uniformly for | − 0 | ≤ δ. To do so, we telescope the integrand using Eas , and estimate in the obvious way to obtain the upper bound 2 2 ∞ r+ ∞ r+ 3 dv dpD(; v, p)φ(p) + 3 dv dpD(0 ; v, p)φ(p) R

r−

∞

+3 0

dv

R

r+

r−

r−

2 dp[Eas (; v, p) − Eas (0 ; v, p)]φ(p) .

(A.10)

Thanks to the uniform bound (A.3), we can now render the first two terms as small as we please by choosing R large enough, uniformly for | − 0 | ≤ δ0 . Once R is fixed, we can handle the v-integral over [0, R] by choosing δ ≤ δ0 sufficiently small, as already detailed. We are therefore reduced to proving strong continuity of r+ φas (; v) ≡ dpEas (; v, p)φ(p), (A.11) r−

as → 0 . Recalling r (A.1), this is not hard. Indeed, it is routine to check that the maps (0, ∞) → H, α → r−+ dp exp(±iαvp)φ(p) are strongly continuous. Telescoping in the obvious ˆ → u(γˆ ; −v)φ( way, one can then use strong continuity of the map → H, ˆ v) ˆ and boundedness of Fourier transformation to complete the proof.

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Proof of Lemma 2.3. The parameter set at issue belongs to r , so that (1.30) yields well-defined operators Aδ : P → H. To prove symmetry of As , we should show that D(φ1 , φ2 ) ≡ (Aδ Fφ1 , Fφ2 ) − (Fφ1 , Aδ Fφ1 ), φ1 , φ2 ∈ C,

(A.12)

vanishes when Aδ equals As . We detail the case as = a− (so that δ = +), the case as = a+ then following from invariance of E under a+ ↔ a− , cf. II (1.21). By (1.30), we have R ∞ 1 D(φ1 , φ2 ) = lim dv dp[E(v, p)φ1 (p)]− a+ a− R→∞ 0 0 ∞ × dqE(v, q)φ2 (q)[E+ (p) − E+ (q)]. (A.13) 0

Using Fubini’s theorem and the conjugacy relation (2.2), this can be rewritten as ∞ ∞ a+ a− D(φ1 , φ2 ) = lim dpφ1 (p) dqφ2 (q)IR (p, q), (A.14) R→∞ 0

where

IR (p, q) ≡

0

R

0

dvχ −2 u(γ ; v)u(γˆ ; p)E(v, p)E(v, q)[E+ (p) − E+ (q)].

(A.15)

By (1.10), the integrand in (A.15) equals 1 1 · Rren (v, p)Rren (v, q)[E+ (p) − E+ (q)], (A.16) c(γˆ ; −p)c(γˆ ; q) c(γ ; v)c(γ ; −v) ˆ is even in v, cf. I (2.10).) which shows that the integrand is even in v. (Recall Rren (v, v) R R −1 Thus we may replace 0 by 2 −R in (A.15). Doing so, we use the A+ -eigenvalue relation to obtain R 2 2χ IR (p, q) = u(γˆ ; p) dvu(v)([E(v − ia− , p)E(v, q) −R

+Va (v)E(v + ia− , p)E(v, q)] − [p ↔ q]). Using (1.20), we can rewrite 2χ 2 IR as R u(γˆ ; p) dv[J (v − ia− /2, p, q) − J (v + ia− /2, p, q)], −R

(A.17)

(A.18)

with J (v, p, q) ≡ u(v + ia− /2)E(v − ia− /2, p)E(v + ia− /2, q) − (p ↔ q).

(A.19)

We now study J (v, p, q) for |Im v| ≤ a− /2. First, we recall u(v) = −

3

G(v − iδγµ )/G(2v + ia)G(2v − ia).

(A.20)

δ=+,− µ=0

Using also the representation (2.11), the relation G(z) = E(z)/E(−z) and the G-AE, G(2v − ia− + ia)/G(2v + ia− − ia) = sinh(2πv/a− )/ sinh(2π v/a+ ),

(A.21)

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443

we see that the v-dependence of the first term on the rhs of (A.19) is given by the factor sinh(2πv/a− )H (v − ia− /2, p)H (v + ia− /2, q) . sinh(2π v/a+ ) δ,µ E(−v − ia− /2 − iδγµ )E(v − ia− /2 − iδγµ )

(A.22)

The point is now that this factor is pole-free in the strip |Im v| ≤ a− /2 for (a+ , a− , γ ) ∈ P. To prove this, we first observe that the poles at ±v = ia + zkl + ia− /2 + iδγµ ,

k, l ∈ N, δ = +, −, µ = 0, 1, 2, 3,

(A.23)

due to the E-product are outside the critical strip. For a+ = a− the sinh-quotient equals 1, so that it yields no poles either. For a− < a+ the poles due to sinh(2π v/a+ ) are outside the strip, so we have now shown that (A.22) is pole-free for parameters in P (1.27). As a consequence, we may invoke Cauchy’s theorem to obtain a− /2 χ 2 IR (p, q) = −iu(γˆ ; p) dyJ (R + iy, p, q), (A.24) −a− /2

where we also used evenness of the integrand (recall (A.16)). Thus we have from (A.14), ∞ ia+ a− χ 2 D(φ1 , φ2 ) = lim dpφ1 (p)u(γˆ ; p) R→∞ 0 a− /2 ∞ dqφ2 (q) dyJ (R + iy, p, q), (A.25) × 0

−a− /2

for all parameters in P . In order to handle the R → ∞ limit of the boundary term, we first recall (cf. II (3.6)) u(γ ; v) = χ (γ )2 + O(exp(−αas v/2)), Re v → ∞,

(A.26)

where the bound is uniform for Im v varying over an arbitrary compact subset of R. Fixing next (a+ , a− , γ ) ∈ P with (a+ , a− , γ ) = (a, a, 0), we use the estimate II (1.33) to infer ˆ + O(exp(−ρv)), ρ > 0, Re v → ∞, E(v, v) ˆ = Eas (v, v)

(A.27)

where the bound is uniform for Im v in R-compacts and vˆ in (0, ∞)-compacts. (For parameters (a, a, 0) we do not know whether this holds true.) Combining these estimates with (A.1) and (A.19), we see that in (A.25) we may replace E by Eas in J (R + iy, p, q). Doing so, we conclude by a simple application of Fubini’s theorem, the Riemann-Lebesgue lemma and dominated convergence that the resulting integral vanishes for R → ∞. Hence the lemma follows for parameters in P that are not of the form (a, a, 0). To handle the latter parameters, we invoke the strong continuity of Iφ proved in Lemma 2.2. Specifically, starting from (a, a, , 0, 0, 0) with ∈ (0, a) (say), we have already shown that D (φ1 , φ2 ) = (F Mφ1 , F φ2 ) − (F φ1 , F Mφ2 )

(A.28)

vanishes, the notation being clear from context. By virtue of the strong continuity property, the limit ↓ 0 now yields the desired symmetry for parameters (a, a, 0).

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Our proof does not apply to the operator Al with step size al (for a+ = a− , of course), since in that case there are poles within the strip |Im v| ≤ al /2 that arises in the course of the proof. Even so, Al is also symmetric on P, as we demonstrate in another way in Sect. 3. Therefore, the residue sum arising for Al via the above reasoning must actually vanish. (It seems intractable to see this directly.) Appendix B. Proofs of Lemmas 4.1 and 4.2 In this appendix we prove Lemmas 4.1 and 4.2. Proof of Lemma 4.1. We prove the lemma for the case as = a− , so that As = A+ is given by II (2.9)–(2.10). (The case as = a+ can be handled via notation changes.) Taking N > 1, we first show ψm ⊥ ψn for 0 ≤ m < n ≤ N − 1. Setting Imn ≡ (E+,m − E+,n )(ψm , ψn ),

(B.1)

we need only prove Imn = 0 (since E+,m < E+,n by (4.51)). Now the polynomials are even and real-valued for v real, so from (1.16) we obtain ∞ Pm (cosh(αas v))Pn (cosh(αas v)) 2Imn = (E+,m − E+,n ) . (B.2) dv c(−v)c(v) −∞ Using (4.48) and (1.17)–(1.19), this implies ∞ −2Imn = dvu(v)([ψm (v − ia− ) + Va (v)ψm (v + ia− )]ψn (v) − (m ↔ n)). −∞

(B.3)

By (1.20), this is of the form ∞ −2Imn = dv[Jmn (v − ia− /2) − Jmn (v + ia− /2)]. −∞

(B.4)

Here we have Jmn (v) ≡ u(v + ia− /2)ψm (v − ia− /2)ψn (v + ia− /2) − (m ↔ n),

(B.5)

which we rewrite as Jmn (v) = [P s (v − ia− /2)])Pm (cosh[αas (v + ia− /2)]) − (m ↔ n)] n (cosh[αa c(δv − ia− /2). (B.6) δ=+,−

Since Jmn (v) decays exponentially as |Re v| → ∞ (uniformly for |Im v| ≤ a− /2), it is now clear that we need only show that the function 1/c(v−ia− /2)c(−v−ia− /2) has no poles for |Im v| ≤ a− /2. Recalling (1.12) and the reflection equation G(−z) = 1/G(z) (cf. (2.3)), we obtain 3 1 G(2v − ia− + ia) 1 = . c(v − ia− /2)c(−v − ia− /2) G(2v + ia− − ia) G(δv − ia− /2 − iγµ ) δ=+,− µ=0

(B.7)

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Using (A.21), this can be written

1 sinh(2πv/a− ) E(δv + ia− /2 + iγµ ) = , sinh(2πv/a+ ) E(δv − ia− /2 − iγµ ) δ c(δv − ia− /2)

(B.8)

δ,µ

where a− < a+ , since N > 1. Now the poles due to the E-product in the denominator are outside the strip |Im v| ≤ a− /2 unless γ0 = −a. But in that case the simple poles at v = ±ia− /2 are matched by zeros of sinh(2π v/a− ). Any poles due to 1/ sinh(2π v/a+ ) are outside the critical strip as well, since a− < a+ . The upshot is that (B.8) has no poles for |Im v| ≤ a− /2, so that Imn = 0, as claimed. To prove (4.52), we need only show (ψm , Fφ) = 0 for all φ ∈ C. Now we have ∞ Pm (cosh(αas v)) ∞ dv dqE(v, q)φ(q). (B.9) (ψm , Fφ) = (a+ a− )−1/2 c(−v) 0 0 Since E(v, q) remains bounded and ψm (−v) decays exponentially for Re v → ∞, we may interchange the integrals to obtain ∞ dqφ(q)Im (q), (B.10) (ψm , Fφ) = (a+ a− )−1/2 0

with Im (q) ≡ −

1 2

∞

−∞

dvu(v)ψm (v)E(v, q), q > 0.

(B.11)

(Here we used evenness of the integrand and (1.17).) Hence it suffices to show Im (q) vanishes. To this end we note that we have E+ (q) − E+,m > 0, cf. (1.22) and (4.51). Thus we need only prove (E+ (q) − E+,m )Im (q) vanishes. Using the AEs satisfied by ψm (v) and E(v, q), we obtain ∞ dvu(v)([ψm (v − ia− ) + Va (v)ψm (v + ia− )]E(v, q) 2[E+ (q) − E+,m ]Im (q) = −∞

−ψm (v)[E(v − ia− , q) + Va (v)E(v + ia− , q)]). Proceeding as before, the rhs can be written ∞ dv[Jm (v − ia− /2, q) − Jm (v + ia− /2, q)], −∞

(B.12)

(B.13)

with Jm (v, q) ≡ u(v + ia− /2)[ψm (v − ia− /2)E(v + ia− /2, q) −ψm (v + ia− /2)E(v − ia− /2, q)].

(B.14)

Using (2.11), (A.20) and (A.21), we see that the v-dependence of the first term on the rhs is given by sinh(2π v/a− ) µ E(−v+ia− /2+iγµ ) · Pm (cosh[αas (v−ia− /2)])H (v+ia− /2, q) . sinh(2π v/a+ ) δ,µ E(−v − ia− /2 − iδγµ ) · µ E(v − ia− /2 − iγµ ) (B.15)

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There are now two cases to consider. First, let a− < a+ . Then we conclude as before that (B.15) has no poles for |Im v| ≤ a− /2. The second term has a similar v-dependence and yields no poles either. Thus the integral Im (q) vanishes. In the second case a− = a+ , we have m = 0 in (B.15), and for γ0 = −a we can no longer infer that (B.15) has no pole for v = ia− /2. But by continuity of (ψ0 , Fφ) for a− → a+ we obtain once again (ψ0 , Fφ) = 0. Proof of Lemma 4.2. We proceed along the same lines as in the proof of Lemma 2.3, cf. Appendix A. Thus we handle the case as = a− and use (3.24) and (3.18) to rewrite ˜ 1 , φ2 ) ≡ (A˜ s F ∗ φ1 , F ∗ φ2 ) − (F ∗ φ1 , A˜ s F ∗ φ2 ) D(φ as 1 lim a+ a− R→∞

∞

dxφ1 (x) 0

0

∞

dyφ2 (y)I˜R (x, y),

(B.16)

(B.17)

where I˜R (x, y) ≡

0

R

d vχ ˆ −2 u(γˆ ; v)u(γ ˆ ; y)E(x, v)E(y, ˆ v)[E ˆ + (x) − E+ (y)].

(B.18)

We can now use the Aˆ + -eigenvalue AE in the same way as in our previous proof to obtain R 2χ 2 I˜R (x, y) = u(γ ; y) d v[ ˆ J˜(vˆ − ia− /2, x, y) − J˜(vˆ + ia− /2, x, y)], (B.19) −R

where J˜(v, ˆ x, y) ≡ u(γˆ ; vˆ + ia− /2)E(x, vˆ − ia− /2)E(y, vˆ + ia− /2) − (x ↔ y). (B.20) The v-dependence ˆ of the first term on the rhs is given by (A.22) with γµ , v, p, q → γˆµ , v, ˆ x, y. In this case, two of the poles among ±vˆ = ia + zkl + ia− /2 + iδ γˆµ , k, l ∈ N, δ = +, −, µ = 0, 1, 2, 3,

(B.21)

belong to the critical strip |Im v| ˆ ≤ a− /2, viz., ±vˆ = i cˆ0 + i(N − 1/2)a− .

(B.22)

(Recall (4.20) to see this.) Due to our assumptions (4.58) and (4.59), these poles are simple and not on the boundary of the strip. Furthermore, writing the first term as −

χ2 Rren (x, vˆ − ia− /2)Rren (y, vˆ + ia− /2) , c(γ ; x)c(γ ; y) c(γˆ ; −vˆ − ia− /2)c(γˆ ; vˆ − ia− /2)

(B.23)

we see that due to our assumption (4.60) we can express the residues at (B.22) in terms of PN−1 (u) and PN (u). (Note that the pole in the highest degree coefficient of PN (u) arising for γˆ0 + γˆ3 + N as = 0 is matched by a zero arising from 1/c(γˆ ; −vˆ − ia− /2). Thus it does not contradict our pole analysis yielding (B.22), where the restriction (4.60) did not show up.)

Generalized Hypergeometric Function III

447

Following again the arguments in the proof of Lemma 2.3, we now see that the analog of (A.24) reads a− /2 2˜ dt J˜(R + it, x, y) + iπ u(γ ; y)r (x, y), (B.24) χ IR (x, y) = −iu(γ ; y) −a− /2

where r denotes the sum of the residues at the poles (B.22). The limit of the boundary term for R → ∞ can now be treated in the same way as before, which shows that it vanishes. We are, therefore, left with ∞ ∞ iπ −2 ˜ χ dxφ1 (x) dyφ2 (y)u(γ ; y)r (x, y), (B.25) D(φ1 , φ2 ) = a+ a− 0 0 and it remains to determine the residues. The residue of (B.23) at vˆ = i cˆ0 +i(N −1/2)a− reads −χ 2 ρ 2 rN−1 ψN−1 (x)ψN (y)/c(γˆ ; −vˆN ),

(B.26)

with rN−1 given by (4.57). Likewise, the residue at vˆ = −i cˆ0 − i(N − 1/2)a− is given by χ 2 ρ 2 rN−1 ψN (x)ψN−1 (y)/c(γˆ ; −vˆN ).

(B.27)

Finally, the residue sum for the second term on the rhs of (B.20) is clearly equal to that of the first term. Hence (4.61) follows. Acknowledgements. This paper was completed during our November 2002 stay at the Max Planck Institute in Munich (Heisenberg Institute). We would like to thank the Institute for its financial support and E. Seiler for his invitation. The referee is thanked for numerous useful suggestions and comments, and for pointing out a recent preprint by Stokman [30], in which the hyperbolic Askey-Wilson integral (1.40) is obtained in a quite different way.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Dirac, P.A.M.: Principles of quantum mechanics. Oxford: Oxford University Press, 1930 Fl¨ugge, S.: Practical quantum mechanics. New York: Springer, 1974 von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Berlin: Springer, 1932 Titchmarsh, E.C.: Eigenfunction expansions associated with second-order differential equations, Part I. Oxford: Oxford University Press, 1962 Koornwinder, T.H.: Jacobi functions and analysis on noncompact semisimple Lie groups. In: Special functions: Group theoretical aspects and applications, Mathematics and its applications, Askey, R.A., Koornwinder, T.H., Schempp, W., (eds.), Dordrecht: Reidel, 1984, pp. 1–85 Olshanetsky, M.A., Perelomov, A.M.: Quantum integrable systems related to Lie algebras. Phys. Reps. 94, 313–404 (1983) Ruijsenaars, S.N.M.: Systems of Calogero-Moser type. In: Proceedings of the 1994 Banff summer school Particles and fields, CRM Ser. in Math. Phys., Semenoff, G., Vinet, L., (eds.), New York: Springer, 1999, pp. 251–352 Askey, R., Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Am. Math. Soc. 319 (1985) Gasper, G., Rahman, M.: Basic hypergeometric series. In: Encyclopedia of Mathematics and its Applications. 35, Cambridge: Cambridge Univ. Press, 1990 Noumi, M., Mimachi, K.: Askey-Wilson polynomials as spherical functions on SUq (2). In: Quantum groups, Lect. Notes in Math. Vol. 1510, New York: Springer, 1992, pp. 98–103 Floreanini, R., Vinet, L.: Quantum algebras and q-special functions. Ann. Phys. (NY) 221, 53–70 (1993)

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12. Koornwinder, T.H.: Askey-Wilson polynomials as zonal spherical functions on the SU (2) quantum group. SIAM J. Math. Anal. 24, 795–813 (1993) 13. Koelink, H.T.: Askey-Wilson polynomials and the quantum SU (2) group: Survey and applications. Acta Appl. Math. 44, 295–352 (1996) 14. Dijkhuizen, M.S., Noumi, M.: A family of quantum projective spaces and related q-hypergeometric orthogonal polynomials. Trans. Am. Math. Soc. 350, 3269–3296 (1998) 15. Rosengren, H.: A new quantum algebraic interpretation of the Askey-Wilson polynomials. Contemp. Math. 254, 371–394 (2000) 16. Noumi, M., Stokman, J.V.: Askey-Wilson polynomials: An affine Hecke algebraic approach. To appear in Proceedings of the 2000 SIAG Laredo summer school Orthogonal polynomials and special functions, Marcellan, F., van Assche, W., Alvarez-Nodarse, R., (eds), Nova, Science 17. Suslov, S.K.: Some orthogonal very-well-poised 8 φ7 -functions. J. Phys. A: Math. Gen. 30, 5877– 5885 (1997) 18. Suslov, S.K.: Some orthogonal very-well-poised 8 φ7 -functions that generalize Askey-Wilson polynomials. The Ramanujan J. 5, 183–218 (2001) 19. Koelink, E., Stokman, J.V.: The Askey-Wilson function transform. Int. Math. Res. Notes (22), 1203– 1227 (2001) 20. Ismail, M.E.H., Rahman, M.: The associated Askey-Wilson polynomials. Trans. Am. Math. Soc. 328, 201–237 (1991) 21. Ruijsenaars, S.N.M.:A generalized hypergeometric function satisfying four analytic difference equations of Askey-Wilson type. Commun. Math. Phys. 206, 639–690 (1999) 22. Ruijsenaars, S.N.M.: A generalized hypergeometric function II. Asymptotics and D4 symmetry. To appear in Commun. Math. Phys. - DOI 10.1007/s00220-003-0969-3 23. Ruijsenaars, S.N.M.: A new class of reflectionless AOs and its relation to nonlocal solitons. Regular and Chaotic Dynamics 7, 351–391 (2002) 24. Ruijsenaars, S.N.M.: Hilbert space theory for reflectionless relativistic potentials. Publ. RIMS Kyoto Univ. 36, 707–753 (2000) 25. Reed, M., Simon, B.: Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. New York: Academic Press, 1975 26. Reed, M., Simon, B.: Methods of modern mathematical physics. III. Scattering theory. New York: Academic Press, 1979 27. Ruijsenaars, S.N.M.: First order analytic difference equations and integrable quantum systems. J. Math. Phys. 38, 1069–1146 (1997) 28. Szeg¨o, G.: Orthogonal polynomials. AMS Colloquium Publications Vol. 23, Providence, RI: Am. Math. Soc., 1939 29. Ruijsenaars, S.N.M.: Sine-Gordon solitons vs. relativistic Calogero-Moser particles. In: Proceedings of the Kiev NATO Advanced Study Institute “Integrable structures of exactly solvable two-dimensional models of quantum field theory”, NATO Science Series Vol. 35, Pakuliak, S., von Gehlen, G., (eds.), Dordrecht: Kluwer, 2001, pp. 273–292 30. Stokman, J.V.: Hyperbolic beta integrals. Preprint, math.QA/0303178 Communicated by L. Takhtajan

Commun. Math. Phys. 243, 449–459 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0971-9

Communications in

Mathematical Physics

Quantum Even Spheres Σ2n q from Poisson Double Suspension F. Bonechi1,2 , N. Ciccoli3 , M. Tarlini1,2 1 2 3

INFN Sezione di Firenze, Firenze, Italy. E-mail: [email protected] Dipartimento di Fisica Universit`a di Firenze, Firenze, Italy. E-mail: [email protected] Dipartimento di Matematica, Universit`a di Perugia, Perugia, Italy. E-mail: [email protected]

Received: 9 December 2002 / Accepted: 30 May 2003 Published online: 13 November 2003 – © Springer-Verlag 2003

Abstract: We define even dimensional quantum spheres q2n that generalize to higher dimension the standard quantum two-sphere of Podle´s and the four-sphere q4 obtained in the quantization of the Hopf bundle. The construction relies on an iterated Poisson double suspension of the standard Podle´s two-sphere. The Poisson spheres that we get have the same kind of symplectic foliation consisting of a degenerate point and a symplectic R2n and, after quantization, have the same C ∗ –algebraic completion. We investigate their K-homology and K-theory by introducing Fredholm modules and projectors.

1. Introduction In the seminal paper [13] by Podle´s on SUq (2)-covariant quantum two-spheres a family 2 of deformations Sq,d was introduced depending on d ∈ R and q < 1. He defined three 2 : the case d = 0, the so-called standard sphere, distinct quantum topological spaces Sq,d d > 0, the non standard sphere, and d = −q 2n , n = 1, 2 . . . , the exceptional case. The geometry of these quantum spaces can be nicely interpreted by looking at the underlying Poisson geometry and considering the sphere as a patching of symplectic leaves. There exists, in fact, a 1–parameter family of SU (2)–covariant Poisson bivec2 can be seen as a quantization of such structures tors on the sphere S2 such that Sq,d ([15]). In the case d = 0 the symplectic foliation is made of a point and a symplectic R2 ; after quantization, the symplectic plane is quantized to K, the algebra of compact 2 ) is operators, and the degenerate point survives as a character. The C ∗ -algebra C(Sq,0 ˜ and satisfies the following then isomorphic to the minimal unitization of compacts K, 2 ) → C → 0. In the case d > 0 the symplectic exact sequence 0 → K → C(Sq,0 foliation consists of an S1 -family of degenerate points, and two symplectic disks. After 2 ) satisfies 0 → K ⊕ K → C(S2 ) → C(S1 ) → 0. The excepquantization C(Sq,d q,d

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F. Bonechi, N. Ciccoli, M. Tarlini

2 tional spheres correspond to the symplectic case and after quantization C(Sq,−q 2n ) is isomorphic to the finite dimensional algebra Mn (C) of matrices. In higher dimension, there exist the so-called euclidean spheres Sqn introduced in [8] as quantum homogeneous spaces of SOq (n + 1); let us notice that in odd dimension they coincide with the so-called Vaksman-Soibelman spheres Sq2n+1 ([18]) introduced as quantum quotients Uq (n)/Uq (n − 1), as observed in [10]. This family of spheres represents a generalization of the Podle´s d = 1 case. In fact they satisfy the following exact sequences (see [10, 11, 18]):

0 → K ⊕ K → C(Sq2n ) → C(Sq2n−1 ) → 0, 0 → C(S1 ) ⊗ K → C(Sq2n+1 ) → C(Sq2n−1 ) → 0 . This behaviour reflects exactly the Poisson level where each S2n contains an equator S2n−1 as a Poisson submanifold and the two remaining hemispheres are open symplectic leaves of dimension 2n. In particular all these spheres contain an S1 -family of degenerate points. As there are no symplectic forms on higher dimensional spheres, exceptional spheres are possible only in dimension 2. In this paper we introduce the family of even spheres q2n which generalize the d = 0 Podle´s sphere. They all share with it the same kind of symplectic foliation; their spaces of leaves consist of two points, one of which is open. The idea of the construction relies on two classical subjects: double suspension and Poisson geometry. The suspension idea is certainly not new and, in fact, appears in many of the papers devoted to the construction of particular deformations of the four-sphere ([5, 3, 6, 16]). When one considers double suspension, however, more interesting possibilities appear: on one hand one could define a purely classical double suspension already at an algebraic level by adding a pair of central selfadjoint generators and modding out suitable relations. A different kind of double suspension was considered, at the C ∗ –algebra level, in [11]. There the authors consider the non-reduced double suspension of a C ∗ –algebra A as the middle term S 2 A of a short exact sequence 0 → A ⊗ K → S 2 A → C(S1 ) → 0 for a suitable fixed Busby invariant. Let us remark that such a non reduced double suspension always has a naturally defined S1 –family of characters. All euclidean spheres were reconstructed in this way starting either from a two–point space or from S1 . In this paper we will consider the reduced double suspension or reduced topological product S 2 X = S2 × X/S2 ∨ X, where, given p ∈ S2 and q ∈ X, S2 ∨ X = (S2 × q) ∪ (p × X). It is quite natural to look at the interaction between the double suspension and Poisson geometry. A word has to be said about the fact that while suspension is an essentially topological construction, Poisson bivectors are of differential nature, so that, in principle, on the double suspension of a given manifold there’s no reason to have a manifold structure, let alone a Poisson bracket. Still whenever the manifold structure is there one can ask whether such a Poisson structure arises. More precisely the double suspension of a manifold M can be seen as a topological quotient S 2 M of S2 × M. If we are given a Poisson bivector on the 2–sphere and a Poisson bivector on M we can ask whether the quotient map S2 × M → S 2 M coinduces a Poisson bracket on the quotient. If this is the case we then look for its quantization.

Quantum Even Spheres q2n from Poisson Double Suspension

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From this point of view the classical double suspension quantizes a double suspension with respect to the trivial Poisson structure on S2 while Hong–Szyma´nski construction corresponds to the standard symplectic structure on R2 attached along an S1 which will survive as a family of 0–leaves on the suspension. In this paper the double suspension is built by assuming the Podle´s d = 0 Poisson structure on the two-sphere and considering S2n as the double suspension of S2(n−1) . We will show that a Poisson double suspension of spheres exists for each n and that it quantizes both at an algebraic level and at the C ∗ –algebra level. The spheres q2n that we obtain have all the same symplectic foliation of the two-sphere, and their quantizations are topologically equivalent, i.e. they are the minimal unitization of compacts and satisfy 0 → K → C(q2n ) → C → 0 . We then have a quite extreme case of quantum degeneracy: these quantum spaces, whatever is the classical dimension, are all topologically equivalent to a zero dimensional compact quantum space. This is an extreme manifestation of a well known fact that quantum spaces associated to quantum groups have lower dimension than the classical one. Moreover this reminds us of canonical quantization in which the Weyl quantization of C0 (R2n ) is K, for each n. The opposite behavior is represented by the so called θ -deformation, whose behavior is almost classical (see [5, 4]). In the case of the four-sphere q4 , the algebra that we get is that obtained in [1, 2], in the context of a quantum group analogue of the Hopf principal bundle S7 → S4 . It is unclear whether all even spheres q2n can be obtained as coinvariant subalgebras in quantum groups. In Sect. 2 we introduce the Poisson double suspension that iteratively defines the Poisson even spheres and we study their symplectic foliation. In Sect. 3 we introduce the quantization Pol(q2n ) at the level of polynomial functions, we classify the irreducible representations in bounded operators and then show that the universal C ∗ -algebra C(q2n ) is the minimal unitization of compacts. We introduce Fredholm modules for each of these spheres. In Sect. 4, we give the non trivial generator of K0 (C(q2n )) and compute its coupling with the character of the previously introduced Fredholm modules. 2. The Standard Poisson Structure Let us define a point on S2 × . . . × S2 by giving to it coordinates ((α1 , τ1 ), . . . (αn , τn )), where |αi |2 = τi (1 − τi ). Let M be the matrix whose entries are Mii = 0, Mij = 1 and Mj i = 1/2 if i < j . Let M τk ik t = τi . (1) a i = αi k

i

Since 2M |ai |2 = τi (1 − τi ) τk ik i

i

k

= τ1 (1 − τ1 )τ22 . . . τn2 + τ2 (1 − τ2 )τ1 τ32 . . . τn2 + . . . + τn (1 − τn )τ1 . . . τn−1 = t (1 − t) , then relation (1) defines a projection into S2n . Let us call : S2 × . . . × S2 → S2n such a projection. One can verify that this map is equivalent to the iterated reduced double

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suspension of a two-sphere, with preferred point the North Pole α = τ = 0. In fact the map from the cartesian product S2 × X to the reduced topological product S 2 X is the unique continuous map which is a homeomorphism everywhere but on the counter image of a point over which its fiber is S2 ∨X. Starting from the two-sphere and iterating this procedure one defines a map from S2 × . . . × S2 to S2n that is a homeomorphism everywhere but on the North Pole, where its fiber is the topological join of n copies of 2 . . × S2 . This map is the projection . S × . n−1

Let us equip S2 with the standard Poisson structure, i.e. the limit structure of the Podle´s standard two sphere ([13, 15]), and S2 × . . . × S2 with the product Poisson structure. The brackets among polynomial functions are defined by giving {αi , τj } = −2δij αi τi ,

{αi , αj∗ } = 2δij (τi2 − αi αi∗ ) .

(2)

We prove the following result. Proposition 1. The map is a Poisson map. The coinduced brackets on S2n read: {ak , a } = ak a (k < ) , {ak , a∗ } = −3ak a∗ (k = ) , {ai , t } = −2ai t , {ak , ak∗ } = 2t 2 + 2 a a∗ − 2ak ak∗ .

(3)

Proof. The result is obtained by explicitly computing the brackets on S2 × . . . × S2 . The only relation that deserves some attention is the last one. By direct computation we obtain 2M {ak , ak∗ } = 2τk2 τi ki − 2ak ak∗ . i

Let us show by induction on k that 2M τk2 τi ki = t 2 + a a∗ . i

It is clearly true for k = 1. Let it be true for k. We then have 2M 2 2 2 τi k+1i = τ1 . . . τk τk+1 . . . τn2 = τ1 . . . (τk2 + αk αk∗ )τk+1 . . . τn2 τk+1 i

= t2 +

a a∗ + ak ak∗ = t 2 +

a a∗ .

(S2n , {, })

is usually called coinduced from the bracket on The Poisson manifold S2 × . . . × S2 (see [17]). Its symplectic foliation is described in the following proposition. Proposition 2. There are two distinct symplectic leaves in (S2n , {, }): i) a zero dimensional leaf given by the north pole PN =(ai = 0, t = 0); ii) R2n = S2n \ PN . The Poisson brackets on R2n read:

{zk , z } = zk z (k < ) , {zk , zk∗ } = 2 1 + z z∗ ,

{zk , z∗ } = zk z∗ (k = ) .

≤k

(4)

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Proof. It is clear that the north pole PN defined by ai = t = 0 is a degenerate point. We are going to show that R2n = S2n \ PN is symplectic. Relations (4) are obtained by direct computation of the brackets among the complex coordinates zi = ai /t. (n) Let us define the 2n × 2n antisymmetric matrix S (n) as Sij = {wi , wj }, where for w2k−1 = zk and w2k = zk∗ , for k = 1, . . . , n. It is clear that Sij = Sij i, j = 1 . . . , 2(n − 1). To compute the determinant of this matrix let us introduce a set of 2n fermionic variables ηi , i.e. ηi ηj + ηj ηi = 0. The pfaffian of S (n) can be expressed as (n) 1

Pf (S (n) ) = dη2n . . . dη1 e 2 ij Sij ηi ηj

2(n−1) (n) 1 2(n−1) (n) Sij ηi ηj = dη2n . . . dη1 e 2 ij e i=1 ηi Ji eS2n−1,2n η2n−1 η2n , (n)

(n)

(n−1)

(n)

where Ji = Si,2n−1 η2n−1 + Si,2n η2n . By using standard rules for fermionic integration (see for instance [14]) and taking into account that Ji Jj = 0 we get (n) Pf (S (n) ) = Pf (S (n−1) ) dη2n dη2n−1 (1 + S2n−1,2n η2n−1 η2n ) = Pf (S (n−1) )S2n−1,2n = Pf (S (n−1) ){zn , zn∗ } = 2Pf (S (n−1) ) 1 + |z |2 . (n)

≤n

Since Pf (S (1) ) = 2(1+|z|2 ) we conclude that Pf (S (n) ) = 0 and that R2n is symplectic.

The Poisson structure on the other chart R2n = S2n \{t = 1} is symplectic everywhere but the origin. In [19] Zakrzewski introduced a family of SU (n)-covariant Poisson structures on R2n with this foliation. It is an interesting problem to understand the relations between them. 3. Quantization of the Standard Poisson Structure 2 ) of the Podle´s standard sphere is generated by {α, α ∗ , τ }, where τ The algebra Pol(Sq,0 is real, with the following relations:

ατ = q 2 τ α ,

q 2 α ∗ α = τ (1 − τ ),

αα ∗ = q 2 α ∗ α + (1 − q 2 )τ 2 .

2 ) with bounded Let 0 < q < 1. There are two irreducible representations of Pol(Sq,0 2 ) → operators, the first is one dimensional (α) = (τ ) = 0, the second σ : Pol(Sq,0 2 B( (N)) is defined by

σ (α)|n = q n−1 (1 − q 2n )1/2 |n − 1 , σ (τ )|n = q 2n |n .

(5)

2 )⊗n → B(2 (N)⊗n ) as the nth –tensor product of the repreLet us define σ ⊗n : Pol(Sq,0 2 ). sentation σ , we denote by {αi , αi∗ , τi } the generators of the i th – Pol(Sq,0

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Definition 3. Let us define Pol(q2n ) as the algebra generated by {ai , ai∗ , t}ni=1 with relations ai t = q 2 t ai ,

ai aj = q −1 aj ai ,

ai ai∗ = q 2 ai∗ ai + q 2 (1 − q 2 )

ai aj∗ = q 3 aj∗ ai (i < j ) ,

a∗ a + (1 − q 2 )t 2 ,

n

q 2 ai∗ ai = t − t 2 .

i=1

Proposition 4. The mapping σn : Pol(q2n ) → B(2 (N)⊗n ) given by σn (ai ) = σ ⊗n αi τkMik ,

σn (t) = σ ⊗n

k

τi ,

(6)

i

is a representation of Pol(q2n ). Proof. From the relation σ ⊗n (αk τkMik ) = q Mik σ ⊗n (τkMik αk ), where Mij is the matrix with Mii = 0, Mij = 1 and Mj i = 1/2 for i < j , it is straightforward to verify the first line of relations. In order to prove the relations in the second line we need the following equality (we will omit the application of σ ⊗n ): τi2

τk2Mik = t 2 + q 2

k

a∗ a .

For i = 1 it is true, we will verify that it is true for i + 1 assuming it true for i: 2 τi+1

k

2Mi+1k

τk

2 = τ1 · · · τi τi+1 · · · τn2 2 · · · τn2 = τ1 · · · τi−1 (q 2 αi∗ αi + τi2 )τi+1 2M = q 2 ai∗ ai + τi2 τk ik = q 2 ai∗ ai + t 2 + q 2 a∗ a

= t2 + q2

k

a∗ a .

To verify the modulus relation we simply need the same computation of the classical case presented at the beginning of Sect 2.

1 Remark 5. The semiclassical limit, defined by {f, g} = limq→1 1−q [f, g], of the relations of Proposition 4 coincides with the Poisson structure defined by the map in Proposition 1. 2 . In the n = 2 case we get the For n = 1 we obviously get the Podle´s sphere Sq,0 same quantum four sphere introduced in [1] by setting t = R , a1 = a and a2 = b∗ . It is also easy to verify that (t) = (ak ) = 0 defines the only character of Pol(q2n ) for each n.

Proposition 6. If ϕ : Pol(q2n ) → B(H) is an irreducible representation on some Hilbert space, then ϕ = or ϕ = σn .

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Proof. In order to prove the existence of an eigenvector of ϕ(t) we will adapt the proof of Theorem 4.5 in [9]. By using relations we see that ϕ(t −t 2 ) > 0 so that Sp(ϕ(t)) ⊂ [0, 1]. If Sp(ϕ(t)) = {0}, then ϕ(t) = 0 and ϕ = ; if Sp(ϕ(t)) = {1}, then ϕ(t) = 1 and this contradicts relations; if Sp(ϕ(t)) = {0, 1}, then λ = 0 would be an eigenvalue and Ker(ϕ(t)) would be an invariant subspace. So in order to have ϕ irreducible and ϕ = we must have Sp(ϕ(t)) \ {0, 1} = ∅. Let λ ∈ Sp(ϕ(t)) \ {0, 1} and let {ξs } be a set of approximate unit eigenvectors, i.e. unit vectors such that lims→∞ ||ϕ(t)ξs − λξs || = 0. By writing t − t 2 = λ(1 − λ) + (t − λ)(1 − λ − t) we get ||ϕ(t − t 2 )ξs || ≥ |λ(1 − λ)| − ||ϕ(t − λ)ϕ(1 − λ − t)ξs || ≥ |λ(1 − λ)| − ||ϕ(t − λ)ξs || ||ϕ(1 − λ − t)|| ≥ C |λ(1 − λ)| , for s bigger than some so and for some C > 0. Moreover we have n n ∗ ≤ ϕ(a a )ξ ϕ(ak∗ ) ||ϕ(ak )ξs || ≤ C n||ϕ(ak(s) )ξs || , k k s k=1

k=1

where and k(s) are such that ||ϕ(ak∗ )|| ≤ C and ||ϕ(ak )ξs || ≤ ||ϕ(ak(s) )ξs || for all k. We conclude that for each s > so there exists 1 ≤ k(s) ≤ n such that ||ϕ(ak(s) )ξs || ≥ C |λ(1 − λ)|. We can define νs = ϕ(ak(s) )ξs /||ϕ(ak(s) )ξs || and verify that they are approximating unit eigenvectors for q −2 λ; in fact C

q −2 ||ϕ(ak(s) (t − λ))ξs || ||ϕ(ak(s) )ξs || q −2 λ(1 − λ) ||ϕ(ak(s) )|| ||ϕ(t − λ)ξs || . ≤ C

||(ϕ(t) − q −2 λ)νs || =

We then showed that if λ ∈ Sp(ϕ(t)) \ {0, 1}, then q −2 λ ∈ Sp(ϕ(t)). In order to keep Sp(ϕ(t)) bounded it is necessary that for each λ there exists k such that q −2k λ = 1, i.e. Sp(ϕ(t)) \ {0} = {q 2k , k ∈ N}. Since each q 2k is isolated, we conclude that it is an eigenvalue. Let ψ be the eigenvector corresponding to λ = 1: since it is the biggest eigenvalue we get that ϕ(ai )ψ = 0. By direct computation one recovers: ϕ(ai )

→ j

ϕ(aj∗ )mj ψ = q 3 j i mj q 2(mi −1) (1 − q 2mi ) → → ∗m ∗m ∗ (m −1) ϕ aj j a i i aj j ψ . j
j >i

(7)

Let us define, with m = (m1 , . . . , mn ), →

2

−1/2 ai∗ mi ψ , C m = q −( i mi ) + i mi (mi +1)/2 (q 2 ; q 2 )mi , ψ m = Cm ϕ i

where (α; q)s =

s

j =1 (1 − q

i j −1 α)

||ϕ(ai∗ )ψ m ||2 = q 2

j ≤i

and (α; q)0 = 1. Formula (7) implies mj 4

q

j >i

mj

(1 − q 2(mi +1) )||ψ m ||2 ,

from which we conclude that ψ m = 0 for each m. The space generated by {ψ m } is invariant and it coincides with H. Finally it can be verified that the mapping T : 2 (N)⊗n → H defined by |m1 , · · · , mn → ψ m intertwines the representations σn and ϕ. Since one can verify that the ψ m ’s are orthonormal, we conclude that T is unitary.

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The universal C ∗ -algebra generated by Pol(q2n ) is then the norm closure of σn (Pol(q2n )). Since σn (ai ) and σn (t) are trace-class operators, then σn (Pol(q2n )\C) ⊂ K. By using Proposition 15.16 of [7], which states that a norm-closed ∗-subalgebra A of K, such that the representation A → K is irreducible, coincides with K, we prove the following result. ˜ the minimal Corollary 7. The C ∗ –algebra generated by Pol(q2n ) is isomorphic to K, unitization of compacts. Remark 8. The C ∗ –algebra of quantum even spheres is independent of the classical dimension. This is not as strange as it may appear; the C ∗ –algebra level often reflects the topology of the space of leaves on the underlying Poisson bracket which is the same in all cases.

Remark 9. As already hinted in the introduction, starting from any C ∗ –algebra A with at least one character εx and from a quantum two-sphere B with a character ε0 one could define a topological double suspension:

Sq2 A := {f ∈ A ⊗ B (εx ⊗ id)(f ) = (id ⊗ ε0 )(f ) ∈ C} .

(8)

It is then a trivial remark that such construction applied to the standard Podle´s sphere is 2(n+1) ). Less trivial is the fact that such stable; this means that Sq2 (C(q2n )) K˜ C(q algebras quantize a whole family of polynomial quantum even spheres. By comparing characters it can be seen that our double suspension is different from the (unreduced) double suspension introduced in [11].

Thanks to Corollary (7) we can conclude that K 0 (C(q2n )) = Z2 for each n, see [12]; each polynomial sphere Pol(q2n ) will provide different representatives of the same class in K-homology. Let us describe them explicitly, along the same lines of [12]. The first one [ ] is the pullback by : C(q2n ) → C of the generator of K 0 (C). By analogy with the classical case we say that its character computes the rank of the vector bundle. Let us describe the second and more generator. interesting Let the Hilbert space be σn 0 01 2 ⊗2 2 ⊗2 H = (N) ⊕ (N) and π = , F = . We have that (H, π, F ) 0 10 is a 1–summable Fredholm module whose character is the zero cyclic cocycle tr σn = tr (σn − ). Let us denote with [tr σn ] its class; we say that tr σn computes the charge. In particular, from (6) we get tr σn (1) = 0 ,

tr σn (t) =

1 . (1 − q 2 )n

(9)

Since tr σn is the restriction to Pol(q2n ) of the trace defined on trace class operators, ˜ = K 0 (C) ⊕ K 0 (K) we have that K 0 (C) is generated by [ ] and K 0 (K) by in K 0 (K) [tr σn ].

Quantum Even Spheres q2n from Poisson Double Suspension

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4. Algebraic Projectors and the Chern–Connes Pairing Let us now come to the construction of non-trivial quantum vector bundles on spheres. Since C(q2n ) = K˜ we have that K0 (C(q2n )) = Z2 . In this section we will introduce two algebraic generators, i.e. two projectors with entries in Pol(q2n ) whose classes generate K0 . The first one is the trivial one [1]; in order to get the non-trivial generator let us go back to the classical case. The non-trivial generator of K-theory for the classical even sphere can be explicitly 2n written in the following way. Let G2n 2k ∈ M2k (S ) be defined iteratively by 2n ∗ G2k ak+1 , G2n G2n = 0 =1−t . 2(k+1) ak+1 1 − G2n 2k It is easy to verify that G2n ≡ G2n 2n is an idempotent for n ≥ 0 defining the vector bundle E2n of rank 2n−1 and charge −1. This construction has the following geometrical interpretation. Let i : S2n → S2(n+1) defined by i(t, a1 , . . . , an ) = (t, a1 , . . . , an , 0) be an embedding of S2n in S2(n+1) and let i ∗ (E2(n+1) ) be the pullback vector bundle on S2n . , where E is the conjugated vector bundle on It is clear that i ∗ (E2(n+1) ) = E2n ⊕ E2n 2n 2n S of charge 1. In the quantum case not every step of this procedure can be obviously extended. In 2(n+1) , i.e. there are no algebra projections from fact there is no embedding of q2n in q 2(n+1)

) to Pol(q2n ). It is possible to adapt the procedure in order to produce a Pol(q rank 2n−1 idempotent for q2n ; but it is convenient to lift the construction to Rq2n+1 , a deformation of the odd linear space where the even sphere lives. Definition 10. Let us define Pol(Rq2n+1 ) as the algebra generated by {xi , xi∗ , y = y ∗ }ni=1 with relations xi xj = q −1 xj xi , xi xj∗ = q 3 xj∗ xi , xi xi∗ = q 2 xi∗ xi + q 2 (1 − q 2 ) x∗ x + (1 − q 2 )y 2 .

xi y = q 2 y xi ,

i<j ,

It is clear that Pol(q2n ) = Pol(Rq2n+1 )/In , where In is the ideal generated by

n ∗ 2 i xi xi − y + y and ai = p(xi ), t = p(y) if p is the projection map. Let 2n+1 φn : Pol(Rq ) → Pol(Rq2n+1 ) be the algebra automorphism defined by φn (xi ) = q 2 xi and φn (y) = q 2 y. The existence of this automorphism, that doesn’t pass to the quotient, is actually the reason for this lifting. 2n ∈ M (R2n+1 ) using the following recursive For each 0 ≤ k ≤ n let us define e2k 2k q formula: 2n 2n x ∗ C2k e2k 2n k+1 , e02n = 1 − y , = (10) e2(k+1) 2n x 2n C2k k+1 1 − φn (e2k ) q2

2n is the diagonal complex matrix given by where C2k 2n C2(k−1) 0 2n C2k = ∈ M2k (C) , C02n = q . 2n 0 qC2(k−1)

We need the following lemma.

(11)

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Lemma 11. For each such that k < ≤ n we have that 2n 2n ∗ 2n ∗ 2n M2n 2k, ≡ e2k C2k x − C2k x φ(e2k ) = 0 .

(12)

Proof. We prove the result by induction on k. For k = 0 it is equivalent to yx∗ = q 2 x∗ y. Let us suppose (12) true for k and let us show it for k + 1. By direct computation we 2n 2n 2n 2n −1 get [M2n 2(k+1), ]11 = M2k, , [M2(k+1), ]22 = −q φn (M2k, ) and [M2(k+1), ]12 = 2n )2 (x ∗ x ∗ −qx ∗ x ∗ ). Using the inductive hypothesis and relations in Pol(R2n+1 ) q(C2k q k+1 k+1 we conclude that M2n = 0.

2(k+1), We now prove the main result of this section. Proposition 12. For each k ≤ n we have k 2n 2 2n 2n 2 (e2k ) − e2k = q2 xi∗ xi − y(1 − y) q −2 (C2k ) .

(13)

i=1

Proof. We show the result by induction on k. For k = 0 it is easy to see that it is true. Let us suppose it true for k. By direct computation, using the inductive hypothesis and Eq. (12) we get

2n 2n )2 − e2(k+1) (e2(k+1)

2n )2 (e2(k+1)

2n − e2(k+1)

11

22

k+1 2n 2 = q2 xi∗ xi − y(1 − y) q −2 (C2k ) ,

=

i=1 2n 2 ∗ (C2k ) xk+1 xk+1

k 2n 2 ∗ = (C2k ) xk+1 xk+1 + q 4 xi∗ xi − y(1 − q 2 y)

2n 2 2 = (C2k ) q

2n 2n (e2(k+1) )2 − (e2(k+1) )

2n 2 2n + φn (e2k ) − φn (e2k )

i=1 k+1

xi∗ xi − y(1 − y)

i=1 12

2n 2n ∗ 2n ∗ 2n = e2k C2k xk+1 − C2k xk+1 φ(e2k )=0.

2n we finally get the result Recalling the iterative definition of C2(k+1) k+1 2n 2n 2n )2 − e2(k+1) = q2 xi∗ xi − y(1 − y) q −2 (C2k+1 )2 . (e2(k+1)

i=1 2n ) ∈ M n (Pol( 2n )). Definition 13. Let G2n = p(e2n 2 q

It is clear from (13) that, for each n > 0, G2n is a projector; let us denote with [G2n ] its class in K-theory (both algebraic and topological). By using the recursive definition 2n we can compute the matrix trace of G . In fact the equation of e2k 2n 2n 2n 2n − φn (e2k ) , Tr e02n = 1 − y , Tr e2(k+1) = 2k + Tr e2k 2n ) = 2k−1 − (1 − q 2 )k y (for k ≥ 1), so that we have is solved by Tr(e2k

Tr(G2n ) = 2n−1 − (1 − q 2 )n t .

(14)

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459

By recalling the definition of the Fredholm modules [ ] and [tr σn ], we compute their Chern–Connes pairing with G2n : [ ], [G2n ] = 2n−1 ,

[tr σn ], [G2n ] = −1 .

Corollary 14. The projector G2n defines a non trivial class both in K0 (Pol(q2n )) and K0 (C(q2n )). By using the nondegeneracy of the pairing with K-homology it is now clear that [1] ˜ = K0 (C) ⊕ K0 (K). and 2n−1 [1] − [G2n ] generate K0 (K) Acknowledgements. F.B. wants to thank L. Dabrowski, G. Landi and E. Hawkins for useful discussions on the subject.

References 1. Bonechi, F., Ciccoli, N., Tarlini, M.: Non-commutative instantons on the 4–sphere from quantum groups. Commun. Math. Phys. 226, 419–432 (2002) 2. Bonechi, F., Ciccoli, N., Tarlini, M.: Quantum 4–sphere: The infinitesimal approach. Banach Center Publ., in press 3. Brzezinski, T., Gonera, C.: Non-commutative 4–spheres based on all Podle´s 2–spheres and beyond. Lett. Math. Phys. 54, 315–321 (2000) 4. Connes, A., Dubois-Violette, M.: Noncommutative finite-dimensional manifolds, I. Spherical manifolds and related examples. Commun. Math. Phys. 230, 539–579 (2002) 5. Connes, A., Landi, G.: Noncommutative manifolds, the instanton algebra and isospectral deformations. Commun. Math. Phys. 221, 141–159 (2001) 6. Dabrowski, L., Landi, G., Masuda, T.: Instantons on the quantum sphere S4q . Commun. Math. Phys. 221, 161–168 (2001) 7. Doran, R.S., Fell, J.M.G.: Representations of *–Algebras, Locally Compact Groups, and Banach *–Algebraic Bundles: Vol I. New York: Academic Press, 1988 8. Faddeev, L., Reshetikhin, N., Takhtajan, L.: Quantization of Lie Groups and Lie Algebras. Lengingrad Math. J. 1, 193 (1990). [Alg. Anal. 1, 178 (1990)] 9. Hajac, P.M., Matthes, R., Szymanski, W.: Quantum Real Projective Space, Disc and Sphere. Algebr. Represent. Th. 6, 169–192 (2003) 10. Hawkins, E., Landi, G.: Fredholm modules for quantum Euclidean spheres. math.KT 0201039 11. Hong, J.H., Szyma´nski, W.: Quantum spheres and projective spaces as graph algebras. Commun. Math. Phys. 232, 157–188 (2002) 12. Masuda, T., Nakagami, Y., Watanabe, J.: Noncommutative differential geometry on the quantum two sphere of Podle´s I: An algebraic viewpoint. K–theory 5, 151–175 (1991) 13. Podle´s, P.: Quantum Spheres. Lett. Math. Phys. 14, 193–202 (1987) 14. Ramond, P.: Field Theory: A modern primer. Reading, MA: Addison-Wesley, 1990 15. Sheu, A.J.L., (with an appendix by Lu, J.H., Weinstein, A.): Quantization of the Poisson SU(2) and its Poisson homogeneous space – the 2-sphere. Commun. Math. Phys. 135, 217–232 (1991) 16. Sitarz, A.: More non commutative 4–spheres. Lett. Math. Phys. 55, 127–131 (2001) 17. Vaisman I.: Lectures on the geometry of Poisson manifolds. Progress in Math., 118, Basel Boston: Birkh¨auser Verlag, 1994 18. Vaksman, L.L., Soibelman, Ya.: Algebra of functions on the quantum group SU (N + 1) and odd dimensional quantum spheres. Leningrad Math. J. 2, 1023–1042 (1991) 19. Zakrzewski, S.: Poisson structures on R2n having only two symplectic leaves: The origin and the rest. In: J. Grabowski, P. Urba´nski, (eds.), Poisson Geometry. Banach Center Publ. 51, Warszaw, 2000 Communicated by L. Takhtajan

Commun. Math. Phys. 243, 461–470 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0982-6

Communications in

Mathematical Physics

On Smooth Cauchy Hypersurfaces and Geroch’s Splitting Theorem Antonio N. Bernal, Miguel S´anchez Dpto. de Geometr´ıa y Topolog´ıa, Facultad de Ciencias, Fuentenueva s/n, 18071 Granada, Spain Received: 21 April 2003 / Accepted: 30 June 2003 Published online: 11 November 2003 – © Springer-Verlag 2003

To Professor J.K. Beem, in the year of his retirement Abstract: Given a globally hyperbolic spacetime M, we show the existence of a smooth spacelike Cauchy hypersurface S and, thus, a global diffeomorphism between M and R × S. 1. Introduction In a classical article published in 1970, Geroch [10] proved the equivalence, for a spacetime, between global hyperbolicity and the existence of a Cauchy hypersurface S. As he stated clearly, the results were obtained at a topological level. In fact, he proved the existence of a continuous time function t : M → R such that each level t =constant is a (topological) Cauchy hypersurface and, then, M is homeomorphic to R × S. The improvement of these topological results in smooth differentiable ones is important not only as a typical mathematical challenge but also from the theoretical viewpoint: Cauchy hypersurfaces are the natural subsets where initial conditions to differential equations (as Einstein’s equation) are posed. Moreover, any achronal hypersurface N can be seen as a graph on a Cauchy hypersurface; this allows to study several properties which involve the differentiable structure of N (as mean curvature), provided that the splitting is smooth; see for example, [7]. In general, continuous maps between smooth manifolds can be approximated by smooth maps. Thus, it is reasonable to expect that, with some effort, one could fill the details to strengthen the topological results. Some authors have claimed that function t (and, thus, the Cauchy hypersurface) can be smoothed and, therefore, M is diffeomorphic to R × S; for example, see the end of the proof of Proposition 6.6.8 in the very influential book by Hawking and Ellis [12] (or the more recent book [22, p. 209]). In fact, this has been assumed in very different contexts where smoothness seems unavoidable (see, for example, [8, 13, Chap 8 or 21]). The second-named author has been partially supported by a MCyT-FEDER Grant BFM2001-2871C04-01.

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Nevertheless, as far as we know, general smoothing procedures have resisted the attempts of formalisation. Budic and Sachs [4] proved C 1 -smoothing of time functions for deterministic globally hyperbolic spacetimes. Then, Seifert [19] claimed the existence of a general procedure for smoothing time functions. But his proof is complicated and seems unclear; no cleaner version of this result has been published since then. Later on, Dieckmann [6, 5] clarifies some points in Geroch’s proof, but cites Seifert at the crucial point for smoothing (see [6, proof of Theorem]). In general, in spite of these two references, most specialists in pure Lorentzian Geometry do not affirm that a smooth Cauchy hypersurface must exist, even when the context could suggest it (say, expressions such as “consider a globally hyperbolic spacetime with a smooth Cauchy hypersurface S” are commonly used, when necessary). For example, among our references clearly posterior to Seifert’s article, see [1, 2, 7, 9, 14, 16]. Summing up, Sachs and Wu [18, p. 1155] claimed in 1977: ...This is one of the folk theorems of the subject. It is not difficult to prove that every Cauchy surface is in fact a Lipschitzian hypersurface in M [17]. However, to our knowledge, an elegant proof that his Lipschitzian submanifold can be smoothed out to such an N above is still missing. This “folk question” is regarded as open in the first edition of the classical book by Beem and Ehrlich [1, p. 31], and remains open in the second edition of 1996, with Easley [2, p. 65]. As far as we know, no formalisation of the quoted result has been obtained since then. On the other hand, it is worth mentioning that some properties of volume functions (which appear in Geroch’s proof) have been studied in [5] (see also [2, pp. 65–69]), and some smooth splitting results in different contexts have been obtained, see for example, [14, 2, Chap 14]. The aim of the present article is to give a simple, self-contained and detailed proof of the following result: Theorem 1. Any globally hyperbolic spacetime admits a smooth spacelike Cauchy hypersurface S0 and, then, it is diffeomorphic to R × S0 . This paper is organised as follows. Among the preliminaries in Sect. 2, we state what can be asserted from Geroch’s splitting theorem, Lemma 2, Proposition 5. In Sect. 3 some technical properties of Cauchy hypersurfaces are proven. We do not try to give general properties here; plainly, we give direct proofs to the results needed later, for the sake of clarity and completeness. We prove, essentially (compare with [3, 9]): any closed spacelike hypersurface N which lies in one side of a Cauchy hypersurface is achronal (Proposition 7) and, thus, if it lies between two Cauchy hypersurfaces, then it is a Cauchy hypersurface too (Corollary 11). The necessity of the hypotheses is discussed in Remarks 8, 10. In Sect. 4 we prove Theorem 1, according to the following two steps. Fix two Cauchy hypersurfaces, one of them, S1 , in the past of the other one, S2 . Roughly: (1) For each point p ∈ S2 a smooth function hp with compact support can be constructed such that ∇hp is timelike (or 0) in the past of S2 (Lemma 12). This function is constructed from the square of time-separation, which is either null or a quadratic polynomial in normal coordinates and, thus, essentially smooth (see formula (7)). (2) By using technical properties related to paracompactness (Lemma 13), a locally finite set of these functions can be summed in such a way that the sum, h, restricted to J + (S1 ) ∩ J − (S2 ), admits regular level hypersurfaces which are spacelike and do not touch each Si (Proposition 14).

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2. Preliminaries. Geroch’s Result We will follow usual conventions in Lorentzian Geometry as in [2, 10, 16–18]. In particular, a spacetime M will be a connected n−manifold n ≥ 2 endowed with a timeoriented Lorentzian metric g. Differentiability C k , k ∈ N ∪ {∞}, (N = {1, 2, . . . }), will be assumed for both the manifold and the metric, and the term “smooth” will mean “C k -differentiable”. Sometimes an auxiliary complete Riemannian metric gR (with associated norm · R ) will be needed. Recall that the existence of a Riemannian metric on any paracompact manifold (as those admitting a Lorentzian metric, see the discussion above Lemma 13) is well-known; it can be chosen complete by using Whitney’s embedding theorem (moreover, any Riemannian metric admits a conformal metric which is complete [15]). M is globally hyperbolic if it is strongly causal and J + (p) ∩ J − (q) is compact for any p, q ∈ M. In this case, it is not hard to prove that, for any two compact subsets K1 , K2 the set K = J + (K1 ) ∩ J − (K2 )

(1)

is compact too. A hypersurface H in M is a embedded topological (n − 1)-submanifold without boundary. H can be regarded as a subset of M and, then, H will be closed if it is a closed subset of M. A spacelike hypersurface is a embedded C r -hypersurface (r ∈ {1, . . . k}) such that its tangent space at each point is spacelike. A Cauchy hypersurface in M is a subset S that is met exactly once by every inextendible timelike curve in M. Then, S will be a closed achronal connected (topological) hypersurface and it is intersected by any inextendible causal curve [16, Lemma 14.29] (the intersection may be a closed geodesic segment instead a single point, if the curve is lightlike there). The following result should be well-known (see for example, [10, p. 444, Property 7], [16, Prop. 14.31]), even though we include its proof for the sake of completeness and further referencing. Lemma 2. Let M be a (C k -)spacetime which admits a C r -Cauchy hypersurface S, r ∈ {0, 1, . . . k}. Then M is C r -diffeomorphic to R × S and all the C r -Cauchy hypersurfaces are C r diffeomorphic. Proof. It is well-known that any spacetime admits a smooth timelike vector field T . Moreover, T can be assumed to be complete (otherwise, choose any auxiliary complete Riemannian metric gR and take the complete vector field T / T R ). Let φ be the flow of T and consider the map: :R×S →M

(s, x) → φs (x).

As S is a Cauchy hypersurface, then is bijective and, by construction, is a C r diffeomorphism (in the case r = 0, is a homeomorphism -use the classical Brouwer theorem on the invariance of the domain). Let S be any other C r -Cauchy hypersurface. Putting −1 (z) = (s(z), ρ(z)), it is clear that the map

S

→ S, z → ρ(z) is a

C r -diffeomorphism.

(2)

Remark 3. We emphasize that, in this result, each hypersurface at constant s is not necessarily a Cauchy hypersurface. Geroch proved in [10] (see Sect. 5, Theorem 11, plus footnote 26):

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Proposition 4. Assume that the spacetime M is globally hyperbolic. Then there exists a continuous and onto map t : M → R satisfying: (1) Sa := t −1 (a) is a Cauchy hypersurface, for all a ∈ R. (2) t is strictly increasing on any causal curve. In the proof of this result, t is obtained by a famous argument: essentially, choose a measure on M with finite total volume and put t (z) = ln vol(J − (z))/vol(J + (z)) (3) (see also Theorem 3.26 in [2] as well as pp. 65–72, for a discussion about the admissible measures a` la Dieckmann [5]). Then, as a consequence one has: Proposition 5. Let M be a globally hyperbolic spacetime and S one of its Cauchy hypersurfaces. Then there exist a homeomorphism : M → R × S,

z → (t (z), ρ(z)),

(4)

which satisfies: (a) Each level hypersurface St = {z ∈ M : t (z) = t} is a Cauchy hypersurface. (b) Let γx : R → M be the curve in M characterized by: (γx (t)) = (t, x),

∀t ∈ R.

Then the continuous curve γx is timelike in the following sense: t < t ⇒ γx (t) << γx (t ). Sketch of proof. Define t (z) as in (3). By Proposition 4, S := t −1 (0) is a Cauchy hypersurface, and one only has to choose ρ from −1 as in the proof of Lemma 2.

Notice that, if function t were smooth, then the property (b) would imply directly that ∇t is everywhere causal. Even more, if γ were an integral curve of ∇t and γ (t0 ) were lightlike, γ could not be lightlike on some open interval containing t0 (as t is taken from Proposition 4, its item (2) holds). Notice that Lemma 2 and Proposition 5 give two types of topological splittings , for M, being the curve s → (s, x) timelike and smooth for , and the hypersurface t =constant a Cauchy hypersurface for . In what follows, the properties of the Cauchy hypersurfaces of a globally hyperbolic spacetime M will be studied. Then, we will assume that a topological splitting either as in Lemma 2 or as in Proposition 5 is fixed, and we will drop , writting simply M = R × S. 3. Some Properties of Cauchy Hypersurfaces First, recall the following technical result. Lemma 6. Let M be a spacetime and N a closed connected spacelike hypersurface. (1) A closed curve that meets N exactly once and then transversally is not freely homotopic to a closed curve that does not meet N. (2) If N separates M (i.e., M\N is not connected) then N is achronal.

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Assertion (1) comes from intersection theory; it is a particular case of, for example, [11, Sect. 2.4, Theorem in p. 78]. The proof of (2) can be seen in [16, Lemma 14.45(2)]. Proposition 7. Let M be a spacetime which admits a Cauchy hypersurface S. Then any closed connected spacelike hypersurface N which does not intersect S is achronal. Proof. By Lemma 6(2) it is enough to show that N separates M. Otherwise, reasoning by contradiction, recall that there exists a closed curve γ : [−1, 1] → M which intersects N exactly once and, then, transversally. In fact, it is enough to take γ on some closed interval [−, ], 0 < < 1 transversal to N . Then, use that M\N is open and connected (thus, connected by arcs) to join γ (−) and γ ().

Consider the functions and ρ : M → S in the proof of Lemma 2. In order to obtain a contradiction with Lemma 6(1), it is enough to prove that γ and ρ ◦ γ are freely homotopic. As the homeomorphism is constructed from the flow φ of vector field T , there exists a continuous function µ such that ρ ◦ γ (t) = φµ(t) (γ (t)) for all t ∈ [−1, 1]. Then H (s, t) = φsµ(t) (γ (t)) is the required free homotopy.

∀(s, t) ∈ [0, 1] × [−1, 1]

Remark 8. Neither the hypothesis “closed” nor the hypothesis on the intersection (N ∩ S = ∅) can be dropped, as one can check easily. In fact, consider a cylinder R × S1 (S1 = R/2π Z), endowed with the Lorentzian metric −dt 2 + dθ 2 , which admits the Cauchy hypersurface S = {−1}× S1 . For positive slope c < 1, the curve t (θ) = c θ corresponds with a helix H , which is a non-achronal complete spacelike closed hypersurface and crosses S. If the constant c is replaced by a function c(θ ) with 0 < c(θ ˙ )θ + c(θ ) < 1 and limθ→±∞ c(θ )θ , then the corresponding helix H does not cross S (in fact, it lies strictly between two Cauchy hypersurfaces). But H fails to be closed and is neither achronal. Recall that, as N is achronal, then it can be seen as a graph on an open subset of the Cauchy hypersurface. In order to assume that this graph is defined on all the hypersurface, the following result gives a sufficient condition. Proposition 9. Let M be a globally hyperbolic spacetime and S1 , S2 be two disjoint Cauchy hypersurfaces, with S1 ⊂ J − (S2 ), that is, U = I + (S1 ) ∩ I − (S2 ) = ∅. Consider the topological splitting M = R × S1 in Lemma 2. Then, any closed connected spacelike hypersurface S ⊂ U = I + (S1 ) ∩ I − (S2 ) is a graph on all S1 , i.e.: there exists a continuous function λ : S1 → (0, ∞) such that S = {(λ(x), x) : x ∈ S1 }. Proof. First, let us see that the Cauchy hypersurface S2 is a graph too. Consider the canonical projections ρ : R × S 1 → S1 ,

πR : R × S1 → R,

where we have written ρ ≡ πS1 consistently with (2). By the construction of the topological splitting in Lemma 2 from the (timelike) integral curves of T , it is clear that ρ|S2 : S2 → S1 is continuous and injective. Interchanging the roles of S1 and S2 , ρ|S2 is a homeomorphism and S2 is a graph: S2 = {(ρ|S2 )−1 (x) : x ∈ S1 } = {(λ2 (x), x) : x ∈ S1 }, where λ2 (x) = πR ◦ (ρ|S2 )−1 (x) for all x ∈ S1 .

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For the hypersurface S, as it is achronal (Proposition 7), the restriction ρ|S : S → S1 is continuous and injective. Thus, from the theorem of the invariance of the domain, ρ|S yields a homeomorphism between S and an open subset U1 := ρ(S) ⊆ S1 . Therefore, S is a graph on U1 : S = {(ρ|S )−1 (x) : x ∈ U1 } = {(λ(x), x) : x ∈ U1 }, for some continuous function λ on U1 . We only must prove that U1 is closed in S1 . Previously, recall that from the construction of the topological splitting R × S1 and the inclusion S ⊂ I + (S1 ) ∩ I − (S2 ): x << (λ(x), x) << (λ2 (x), x),

∀x ∈ U1 .

(5)

Now, let {xn }n be a sequence in U1 ⊂ S1 which converges to a point x0 in the closure of U1 . In order to prove that x0 ∈ U1 , let K1 = {xn : n ∈ N} ∪ {x0 } ⊂ S1 , K2 = {(λ2 (xn ), xn ) : n ∈ N} ∪ {(λ2 (x0 ), x0 )} ⊂ S2 , and define K = J + (K1 ) ∩ J − (K2 ), which is compact (see (1)). From (5), K contains the sequence {(λ(xn ), xn ) : n ∈ N} ⊂ S.

(6)

As S is closed, this sequence is contained in the compact subset KS = K ∩ S. Thus, it converges, up to a subsequence, to a point (λ 0 , x0 ) of S. But x0 = π(λ 0 , x0 ) = π(lim(λ(xn ), xn )) = lim xn = x0 . n

That is, x0 =

x0

∈ π(S) = U1 , as required.

n

Remark 10. Even though, by Proposition 7, only one of the inclusions S ⊂ I + (S1 ) or S ⊂ I − (S2 ) is enough to ensure the achronality of S, both inclusions are needed for Proposition 9. The reason relies on the central role of inequality (5), and it is not difficult to obtain a counterexample if one of them is removed. In fact, consider on R2 the warped metric g = −dt 2 + f (t)2 dx 2 with f > 0 and ∂t future-directed. Each hypersurface t =constant is a Cauchy hypersurface, because the spacetime is conformal to (I × R, g ∗ = −ds 2 + dx 2 ), where I is some interval of R and ds = dt/f . Now, the graph S of a smooth curve x → t (x) is a spacelike hypersurface if and only if |dt/dx| < f (t (x)). But f can grow fast enough in such a way that the inextendible domain of t (x) is a finite interval, and S will be closed (eventually complete, with the induced metric) but not a graph on all S1 . As a concrete example, consider f (t) = cosh t for all t (the spacetime is then isometric to the universal covering of 2-dimensional de Sitter spacetime), choose S1 as the 2 hypersurface √ t ≡ −1 and put t (x) = tg (x/4), which satisfies the required2 inequality |dt/dx| = t(1+t)(x)/2 < f (t (x)). Recall that the hypersurface S = {(tg (x/4), x) : x ∈ (−2π, 2π )}, which lies in I + (S1 ), is not only closed but also complete (the g-length of the graph (t (x), x) restricted to both x ∈ (−2π, 0) and x ∈ (0, 2π ) is infinity). Finally, notice that this example can be easily modified to make S1 compact (take the quotient cylinder generated from the isometry (t, x) → (t, x + 4π )). Thus, as a straightforward consequence we obtain the following result (an alternative proof can be found in [9, Cor. 2]):

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Corollary 11. Let S1 and S2 be two disjoint Cauchy hypersurfaces of M with S1 ⊂ I − (S2 ). Then: Any closed connected spacelike hypersurface S contained in U = I + (S1 ) ∩ I − (S2 ) is a Cauchy hypersurface. Proof. From Lemma 7, S is achronal, and we only must check that each inextendible timelike curve γ crosses it. As S is a graph on S1 , it separates R × S1 in two disjoint open subsets: W − = {(t, x) : t < λ(x)}, which includes S1 , and W + = {(t, x) : t > λ(x)}, which includes S2 . As γ crosses the Cauchy surfaces S1 and S2 , it will cross S too.

4. Smooth Cauchy Hypersurfaces In what follows we will use convex open subsets of M. Recall that an open set is called convex if it is a (starshaped) normal neighborhood of all its points; every point p ∈ M has a convex neighborhood Cp (Cp can be also chosen simple in the terminology of [17, 5], i.e. convex with compact closure included in a bigger convex set); some properties of this set in relation to causality can be seen in [16, 14.2]. When a convex set C is regarded as a spacetime, the time–separation or Lorentzian distance on C has especially good properties. In particular, it is not only continuous but also smooth whenever it does not vanish (see, for example, [16, Lemmas 5.9 and 14.2(1)]). Lemma 12. Let M be a globally hyperbolic spacetime with a topological splitting R ×S as in Proposition 5. Let t1 < t2 and denote the corresponding Cauchy hypersurfaces St1 ≡ S1 , St2 ≡ S2 . Fix p ∈ S2 , and a convex neighborhood of p, Cp ⊂ I + (S1 ). Then there exists a smooth function hp : M → [0, ∞) which satisfies: (i) hp (p) = 1. (ii) The support of hp (i.e., the closure of h−1 p (0, ∞)) is compact and included in + Cp ∩ I (S1 ). (iii) If q ∈ J − (S2 ) and hp (q) = 0 then ∇hp (q) is timelike and past-pointing. Proof. Choose p ∈ I − (p) ∩ I + (S1 ) such that J + (p ) ∩ J − (S2 ) ⊂ Cp and define hp on I − (S2 ) as the C k function: τ (p , p)−2k · τ (p , q)2k if k < ∞, hp (q) = (7) −2 −2 if k = ∞ eτ (p ,p) · e−τ (p ,q) where τ is the time-separation on Cp regarded as a spacetime (hp is regarded as 0 on I − (S2 )\Cp ). Now, construct any C k extension of hp out of I − (S2 ) such that the support of hp is included in Cp and hp ≥ 0.1 Obviously, conditions (i) and (ii) are fulfilled and, in order to check (iii), take into account that the gradient of −τ (p , ·)2 /2 at q is the velocity of the unique (timelike, future-pointing) geodesic σ : [0, 1] → Cp from p to q.

1 This extension can be easily carried out because the support of h on J − (S ) lies in the compact p 2 subset K = J + (p ) ∩ J − (S2 ) ⊂ Cp . Fix an open neighbourhood U of K included in Cp , consider the covering (U, Cp \K) of Cp , and take the subordinate partition of the unity {µ, µ } with support(µ) ⊂ U . Then, hp can be chosen on all Cp as the expression in (7) multiplied by µ.

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Recall that the existence of a Lorentzian metric on M implies its paracompactness, i.e., for any open covering W of M there exists a locally finite refinement [20, Vol II, Addendum 1]. We will need a technical property related to paracompactness. Essentially, take any complete distance dR on M compatible with its topology, and assume that there exist a dR -bound of the diameter of the elements of the covering W. Then there exist a locally finite subcovering W of W. Lemma 13. Let dR be the distance on M associated to any auxiliary complete Riemannian metric gR . Let S be a closed subset of M and W = {Wα , α ∈ I} a collection of open subsets of M which cover S. Assume that each Wα is included in an open subset Cα and the dR -diameter of each Cα is smaller than 1. Then there exist a subcollection W = {Wj : j ∈ N} ⊂ W which covers S and is locally finite (i.e., for each p ∈ ∪j Wj there exists a neighborhood V such that V ∩ Wj = ∅ for all j but a finite set of indexes). Moreover, the collection {Cj : j ∈ N} (where each Wj ∈ W is included in the corresponding Cj ) is locally finite too. Proof. Fix p ∈ M and consider the open and closed balls, resp., Bp (r), B¯ p (r) of center p and radius r > 0 for the distance dR . As each closed ball is compact, the following subsets are compact too: Km = B¯ p (m)\Bp (m − 1), (M ⊂ ∪m Km ), Sm = Km ∩ S, (S ⊂ ∪m Sm ), for all m ∈ N. From the compactness of Sm , a finite subset {W1m , . . . Wkm m } ⊂ W covers Sm . Then, take: W = {Wkm : m ∈ N, k = 1, . . . km }. Clearly, W covers S, and W (as well as the corresponding collection of the Cj ’s) is locally finite because, if |m − m | ≥ 3, then Wkm ∩ Wk m = ∅ (use the definition of the s, S ’s and the inequality diam W < 1, for all α). Km

m α Proposition 14. Let M be a globally hyperbolic spacetime with topological splitting R × S and fix t1 < t2 , Si , as in Lemma 12. Then there exists a smooth function h : M → [0, ∞) which satisfies: 1. h(t, x) = 0 if t ≤ t1 . 2. h(t, x) > 1/2 if t = t2 . 3. The gradient of h is timelike and past-pointing on the open subset V = h−1 ((0, 1/2))∩ I − (S2 ). As a consequence, Ssh = h−1 (s) ∩ J − (S2 ) is a closed smooth spacelike hypersurface which lies in I + (S1 ) ∩ I − (S2 ), for each s ∈ (0, 1/2). Proof. Fix a complete auxiliary distance dR as in Lemma 13, take, for any p ∈ S2 , a convex neighborhood Cp with diameter smaller than 1 and the corresponding function hp as in Lemma 12. Let Wp = h−1 p ((1/2, ∞)) (Wp ⊂ Cp ). Obviously, W = {Wp , p ∈ S2 }

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covers the closed hypersurface S2 and lies in the hypothesis of Lemma 13. Now, if W = {Wi ; i ∈ N} is the locally finite subcovering given by this lemma, and for each Wi (≡ Wpi ), hi is the corresponding function with support Ci (Wi ⊂ Ci ), we put: h= hi , i

which is well-defined and smooth because of the local finiteness of the supports of the hi ’s. Clearly, h satisfies Items 1, 3 because each hi satisfies them, and Item 2 is satisfied because each hi (p) > 1/2 for any p ∈ Wi , and S2 ⊂ ∪i Wi . To prove the last assertion, notice that, by Item 3, any s ∈ (0, 1/2) is a regular value of the restriction of h to I − (S2 ); thus, Ssh is a spacelike hypersurface, and it is also closed in I − (S2 ). Nevertheless, the closure of I − (S2 ) is S2 and, by Item 2, no frontier point of I − (S2 ) can be a frontier point of Ssh . Thus, Ssh is closed in M, as required.

Proof of Theorem 1. To obtain the smooth hypersurface, apply Corollary 11 to any hypersurface Ssh yielded by Proposition 14. For the diffeomorphism, use Lemma 2.

Acknowledgements. The authors acknowledge Prof. P. Ehrlich’s clarifying comments on Geroch’s theorem and subsequent references.

References 1. Beem, J.K., Ehrlich, P.E.: Global Lorentzian geometry. Monographs and Textbooks in Pure and Applied Math. 67, New York: Marcel Dekker, Inc., 1981 2. Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian geometry. Monographs Textbooks Pure Appl. Math. 202, New York: Dekker Inc., 1996 3. Budic,R., Isenberg, J., Lindblom, L., Lee, Yasskin, P.B.: On determination of Cauchy surfaces from intrinsic properties. Commun. Math. Phys. 61(1), 87–95 (1978) 4. Budic, R., Sachs, R.K.: Scalar time functions: Differentiability. In: Differential geometry and relativity, Math. Phys. Appl. Math., Vol. 3, Dordrecht: Reidel, 1976, pp. 215–224 5. Dieckmann, J.: Volume functions in general relativity. Gen. Relativity Gravitation 20(9), 859–867 (1988) 6. Dieckmann, J.: Cauchy surfaces in a globally hyperbolic space-time. J. Math. Phys. 29(3), 578–579 (1988) 7. Ecker, K., Huisken, G.: Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes. Commun. Math. Phys. 135(3), 595–613 (1991) 8. Furlani, E.P.: Quantization of massive vector fields in curved space-time. J. Math. Phys. 40(6), 2611–2626 (1999) 9. Galloway, G.J.: Some results on Cauchy surface criteria in Lorentzian geometry. Illinois J. Math. 29(1), 1–10 (1985) 10. Geroch, R.: Domain of dependence. J. Math. Phys. 11, 437–449 (1970) 11. Guillemin, V., Pollack, A.: Differential Topology. Englewood Cliffs, NJ: Prentice-Hall Inc., 1974 12. Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. Cambridge Monographs on Mathematical Physics, No. 1., London-New York: Cambridge University Press, 1973 13. Masiello, A.: Variational methods in Lorentzian geometry. Pitman Res. Notes Math. Ser. 309, Harlow: Longman Sci. Tech., 1994 14. Newman, R.P.A.C.: The global structure of simple space-times. Commun. Math. Phys. 123(1), 17–52 (1989) 15. K. Nomizu, H. Ozeki, The existence of complete Riemannian metrics Proc. Am. Math. Soc. 12, 889–891 (1961) 16. O’Neill, B.: Semi-Riemannian geometry with applications to Relativity. New York: Academic Press Inc., 1983 17. Penrose, R.: Techniques of Differential Topology in Relativity. CBSM-NSF Regional Conference Series in Applied Mathematics, Philadelphia: SIAM, 1972

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18. Sachs, R.K., Wu, H.: General relativity and cosmology. Bull. Am. Math. Soc. 83(6), 1101–1164 (1977) 19. Seifert, H.-J.: Smoothing and extending cosmic time functions. Gen. Relativity Gravitation 8(10), 815–831 (1977) 20. Spivak, M.: A comprehensive introduction to differential geometry Vol. II. Second edition. Wilmington, DE: Publish or Perish, Inc., 1979 21. Uhlenbeck, K.: A Morse theory for geodesics on a Lorentz manifold. Topology 14, 69–90 (1975) 22. Wald, R.M.: General Relativity. Chicago, IL: Univ. Chicago Press, 1984 Communicated by G.W. Gibbons

Commun. Math. Phys. 243, 471–483 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0983-5

Communications in

Mathematical Physics

On the Interaction of Nearly Parallel Vortex Filaments Carlos E. Kenig1 , Gustavo Ponce2 , Luis Vega3 1 2 3

Department of Mathematics, University of Chicago, Chicago, IL 60637, USA Department of Mathematics, University of California, Santa Barbara, CA 93106, USA Departamento de Matematicas, Universidad del Pais Vasco, Apartado 644, 48080 Bilbao, Spain

Received: 10 June 2003 / Accepted: 7 July 2003 Published online: 11 November 2003 – © Springer-Verlag 2003

Abstract: The system for the interaction of nearly parallel vortex filaments deduced by Klein, Majda and Damodaran is considered. Local existence for the associated initial value problem with data which are small perturbations of perfect parallel filaments is proved. Assumptions on the unperturbed configurations and on the initial parameters which guarantee the existence of global regular solutions (no finite time collapse) are deduced.

1. Introduction Consider the initial value problem (IVP)  j − k  ∂t j = J αj j ∂σ2 j + J 2k , |j − k |2 k=j   j (σ, 0) = 0,j (σ ),

j = 1, .., N,

(1)

where j (σ, t) = (xj , yj )(σ, t) with σ, t ∈ R. The system (1) was introduced by Klein, Majda, and Damodaran [K-M-D] as a simplified model describing the time evolution of N -vortex filaments nearly parallel to the z-axis. Here j (σ, t) represents the position of the j th filament, σ parametrizes the z-axis, t is the time, j is the circulation of the j th vortex filament, αj denotes a vortex structure parameter appearing in the asymptotic analysis used to deduce the equations, see [C-T, K-M], and

0 −1 J = . 1 0

(2)

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C.E. Kenig, G. Ponce, L. Vega

The system Xj (t) − Xk (t) d 2 k , Xj (t) = J dt |Xj (t) − Xk (t)|2

j = 1, .., N,

(3)

k=j

where Xj (t) = (xj , yj )(t), describes the motion of N -point vortices in the flow plane as well as the evolution of N line filaments perpendicular to the plane flow. On the other hand, the equation representing the motion of a self-interaction vortex filament Z = Z(σ, t) was derived in the works of Da Rios, Hama, and Arms and Hama, see [R, A-H], and references therein, ∂t Z = ctˆ × ∂σ2 Z,

(4)

where tˆ is the unit tangent vector and c is a constant depending on the circulation and on the reference time chosen. By linearizing this equation about a filament parallel to the z-axis and evaluating it in the perturbed filament Zj (σ, t) one gets the equation (see [K-M-D, M-B]) ∂t Zj = J αj j ∂σ2 Zj .

(5)

Hence, the system (1) reflects both the self-interaction of each filament and the interaction between them. We introduce the complex notation (σ, t) = x(σ, t) + iy(σ, t),

X(t) = x(t) + iy(t),

(6)

so multiplication by the matrix J becomes multiplication by i. An interesting question arising from system (1) is to find conditions for the existence of finite time collapse solutions of (1), i.e. t0 , σ0 ∈ R and j, k ∈ {1, .., N}, j = k such that j (σ0 , t0 ) = k (σ0 , t0 ), as well as for the existence of global in time smooth solutions. Another relevant question is related to the existence and the stability of special solutions of the system (1). In [K-M-D], Klein, Majda and Damodaran studied the system for a pair of nearly parallel filaments, 1 − 2 , |1 − 2 |2 1 − 2 , ∂t 2 = i 2 ∂σ2 1 − 2i |1 − 2 |2

∂t 1 = i ∂σ2 1 + 2i2

(7)

i.e. the system (1) with complex notation, N = 2, identical vortex core parameters αj = 1, j = 1, 2, 1 = 1 and 2 ∈ [−1, 1] − {0}. In this setting, after a possible time rescaling, 2 measures the circulation ratio of the two filaments. They considered the stability of the linearized system associated to (7) evaluated in a pair of perfect parallel filaments and showed that for positive circulation ratio 2 > 0 the linearized system is stable, and for negative circulation ratio 2 < 0 it is unstable. They also presented numerical evidences which suggest that solutions of (7) with 2 < 0 which are perturbations of straight-line point vortex pairs collapse in finite time. For positive circulation ratio 2 > 0 their numerical results predict the global existence of solutions to (7) which are perturbations of straight-line point vortex pairs. Also concerning the system (1) in [L-M] Lions and Majda gave a statistical equilibrium mechanics formulation for this problem of nearly parallel filaments.

On the Interaction of Nearly Parallel Vortex Filaments

473

Before we state our results concerning the IVP (1) it is convenient to recall some known facts concerning the system (3), see [A, N]. Solutions Xj (t) = (xj , yj )(t) = x(t) + iy(t), j = 1, .., N of (3) preserve the following quantities: H =2

j k ln |Xj (t) − Xk (t)|,

I=

N

j ((xj (t))2 + (yj (t))2 ),

(8)

j =1

j
and x¯0 =

N

j xj (t),

j =1

y¯0 =

N

j yj (t).

(9)

j =1

The point (x0 , y0 ) =

1 (x¯0 , y¯0 ), ∗

when ∗ =

N

j = 0,

(10)

j =1

is called the center of vorticity. In the case N = 2 the distance between the vortices d = |(X1 − X2 )(0)| is a constant of their motion and the pair of point vortices rotate rigidly about the center of vorticity with angular frequency ω = ∗ /(2πd 2 ). When ∗ = 0 they translate with speed ((12 + 22 )/2)1/2 /2πd. In the case N = 3 (assuming 1 ≥ 2 > 0) there is only one fixed equilibrium, a stationary solution (X1 , X2 , X3 )(t) = (X1 , X2 , X3 )(0), ∀ t ∈ R, and is given by the following values of the initial parameters: X2 (0) − X1 (0) = 2 (X1 (0) − X3 (0))/ 3 and 1 2 + 2 3 + 3 1 = 0. Also there are two possible relative equilibrium (the shape of the initial configuration may rotate and translate but does not change its size or form): (i) (X1 (0), X2 (0), X3 (0)) forms an equilateral triangle with side d > 0. In this case, if ∗ = 0 it rotates about the center of vorticity with angular frequency ω = ∗ /2π d 2 . When ∗ = 0 it translates with speed ((12 + 22 + 32 )/2)1/2 /2πd. (ii) the points (X1 (0), X2 (0), X3 (0)) are collinear and f1 (l12 , l23 , l31 ) = A = (r(r − l12 )(r − l23 )(r − l31 ))1/2 , r = (l12 + l23 + l31 )/2, A = A(t) being the area of the triangle (X1 (t), X2 (t), X3 (t)) so A(0) = 0, and lj k = lj k (t) = |(Xj − Xk )(t)| satisfies d f1 (l12 , l23 , l31 ) = 0. dt In the setting (i) the equilateral triangle is unstable if 3 < 0 and 1−1 + 2−1 + 3−1 > 0. When N = 3 the collapse occurs if the following conditions hold: the initial configuration is not an equilibrium, h = 1/ 1 + 1/ 2 + 1/ 3 = 0, ∗ I − (x¯0 )2 − (y¯0 )2 = 0 (see (8)-(10)), and the triangle (X1 , X2 , X3 )(0) is positive oriented.

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For the case N ≥ 4 vortices having identical strength and placed at each vertex of a regular N -polygon form a relative equilibrium. It rotates about the center of vorticity with angular velocity ω = (N − 1)/4πd 2 , with d denoting the side of the polygon. This configuration is (linearly) unstable for N > 7. Also one gets an equilibrium of N + 1 vortices by setting N of them with identical strength at each vertex of a regular N -polygon and the N + 1 vortex of arbitrary strength at the center of vorticity. A system of N ≥ 2 vortices having the same strength sign cannot have a fixed equilibrium (stationary solution). For further results and references see [N, A-N-S-T-V]. To state our results we need to introduce the perturbed system. Since the filaments are nearly parallel we define the complex valued functions uj = uj (σ, t) as j (σ, t) = Xj (t) − uj (σ, t),

j = 1, .., N.

(11)

Using that j − k Xj − Xk − |j − k |2 |Xj − Xk |2 j − k − (Xj − Xk ) 1 1 = + (X − X ) − j k |j − k |2 |Xj − Xk |2 |j − k |2 uj − uk = |Xj − Xk − (uj − uk )|2 2 Re (Xj − Xk )(u¯ j − u¯ k ) − |uj − uk |2 , (12) +(Xj − Xk ) |Xj − Xk − (uj − uk )|2 |Xj − Xk |2 we obtain the following IVP for the uj ’s:  uj − u k   ∂t uj = iαj j ∂σ2 uj + 2i k   |Xj − Xk − (uj − uk )|2   k=j  2 Re (Xj − Xk )(u¯ j − u¯ k ) − |uj − uk |2 , +2i k (Xj − Xk )   |Xj − Xk − (uj − uk )|2 |Xj − Xk |2   k=j    uj (σ, 0) = u0,j (σ ), j = 1, .., N.

(13)

Theorem 1. Assume N = 3 and αj , j > 0,

α1 1 = α2 2 = α3 3 = k0 .

(14)

If (X1 (0), X2 (0), X3 (0)) forms an equilateral triangle with side d > 0, then there exists δ = δ(d; (j )3j =1 ; k0 ) > 0 so that for any (u0,1 , u0,2 , u0,3 ) ∈ (H 1 (R))3 with µ=

3

u0,j 1,2 < δ,

(15)

j =1

· 1,2 denoting the H 1 -norm, the IVP (1) has a unique global solution (1 , 2 , 3 )(σ, t) which can be written as j (σ, t) = Xj (t) − uj (σ, t),

j = 1, 2, 3,

(16)

On the Interaction of Nearly Parallel Vortex Filaments

475

where (X1 , X2 , X3 )(t) is the unique solution of the system (3) with data (X1 (0), X2 (0), X3 (0)) and (u1 , u2 , u3 )(σ, t) is the unique solution of the IVP (13), with uj ∈ C(R : H 1 (R)),

j = 1, 2, 3.

(17)

Moreover, k0 j 2

|∂σ j (σ, t)| dσ + 2

j k

j
|(j − k )(σ, t)|2 −1 d2

2 dσ ≤ λ, (18)

where the constant λ > 0 depends on d and µ with λ(µ) ↓ 0 as µ ↓ 0, and d/4 ≤ j (·, t) − k (·, t)∞ ≤ 4d,

j, k = 1, 2, 3.

(19)

In addition, if (u0,1 , u0,2 , u0,3 ) ∈ (H m (R))3 , m ∈ Z+ , m > 1, then uj ∈ C(R : H m (R)),

j = 1, 2, 3.

(20)

Theorem 2. If N = 2 under the corresponding modified versions of (14)-(15) the conclusions of Theorem 1 hold for any initial configuration (X1 (0), X2 (0)) with d = |X1 (0) − X2 (0)| > 0. Remarks. (a) The estimate (19) tells us that at any plane z = σ0 the configuration (1 , 2 , 3 )(σ0 , t) remains close to an equilateral triangle for all time t ∈ R. This is consistent with the results for the IVP (3) described before. The factor 4 in (19) can be replaced by any ρ > 1 by taking µ in (15) sufficiently small. In fact, we will show that ρ = ρ(µ) ↓ 1 as µ ↓ 0. (b) We observe that under the hypothesis α1 1 = α2 2 = α3 3 = k0 in (14) the system in (1) is Galilean invariant. Thus, if (1 , 2 , 3 )(σ, t) is a solution of (1) given in Theorems 1-2, corresponding to data (0,1 , 0,2 , 0,3 )(σ ), then ˜ 1, ˜ 2, ˜ 3 )(σ, t) = e−ik0 ν t eiνσ (1 , 2 , 3 )(σ − 2k0 νt, t) ( 2

(21)

solves the IVP (1) with data ˜ 0,2 , ˜ 0,3 )(σ ) = eiνσ (0,1 , 0,2 , 0,3 )(σ ). ˜ 0,1 , (

(22)

Notice that solutions in (21) are not necessarily small perturbations of filaments perpendicular to the xy-plane. If in addition, we assume 1 = 2 = 3 , fix the origin at the center of vorticity and use the Galilean invariance of the solution of (3) and (1), Xj (t) = d e2π(j −1)i/3 e2π iωt ,

ω = ∗ /2π d 2 ,

j = 1, 2, 3,

(23)

we get the one parameter family of solutions of the system in (1), j,ν (σ, t) = d eiνσ e2π(j −1)i/3 ei(2πω−k0 ν

2 )t

,

j = 1, 2, 3.

(24)

,

(25)

Thus, taking ∗

ν=ν =

2πω k0

1/2

=

∗ k0 d 2

1/2

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C.E. Kenig, G. Ponce, L. Vega

one obtains a stationary solution of the system in (1) in the form of triple helix j,ν ∗ (σ, t) = d eiν

∗σ

e2π(j −1)i/3 ,

j = 1, 2, 3.

(26)

By rescaling a solution j (σ, t), j = 1, .., N, of the system in (1) we get a family of solutions η,j (σ, t) = η−1 j (ησ, η2 t), j = 1, . . . , N, η ∈ R. In particular, applying this rescaling to (26) we get a one parameter family of stationary solutions of (1). Under the appropriate modification of the values of the constants involved, formula (26) provides a stationary solution of the system in (1) for any N ≥ 2. Instead of (23) one starts with N vortices of equal strength placed at each vertex of a regular N-polygon. A similar remark applies to the system of N + 1 vortices N of them of equal strength set at the corners of a regular N -polygon with the N + 1 vortex of arbitrary strength located at the center of vorticity. We recall that the IVP (3) with j > 0, j = 1, .., N , N ≥ 2 does not have any stationary solution. Theorems 1-2 imply that this stationary solution is stable under small perturbation in the case N = 2, 3. (c) It will follow from our proof that the case of periodic perturbations, i.e. (13) with periodic small data (u0,1 , u0,2 , u0,3 ), is simpler since as we will see the conservation laws in Sect. 3 can be used directly. (d) In the case N = 2, under the additional assumption 1 = 2 , (1) can be rewritten in terms of (φ1 , φ2 )(σ, t) = (1 + 2 , 1 − 2 )(σ, t) as  ∂t φ1 = i∂σ2 φ1 , 4iφ2 (27) ∂t φ2 = i∂σ2 φ2 + . 2 |φ2 | In this case where the problem reduces to one scalar nonlinear equation, and the results in Theorem 2 can be obtained by applying the arguments in [Z] (Theorem III.3.1). To prove Theorems 1-2 we first obtain an appropriate local existence theory for the IVP (1) (Theorem 3) which allows us to point out the a priori estimates needed to extend these local solutions to global ones. This will be done in Sect. 2. In Sect. 3 we shall prove Theorem 1 (the proof of Theorem 2 is similar so it will be omitted) by establishing the a priori estimates described in the previous section. 2. Local Existence Theory for the IVP (1) To obtain solutions of (1) it suffices to consider the IVP’s (13) and the corresponding one for the system in (3). We begin by considering an unperturbed configuration given by the point vortices (Xj (0))N j =1 with inf |Xj (0) − Xk (0)| ≥ d > 0.

k=j

(28)

By the local existence theory and the continuous dependence of the solution upon the initial parameters for ODE systems one has that there exists T0 = T0 (N ; (Xj (0))N j =1 ,

On the Interaction of Nearly Parallel Vortex Filaments

477

N N (j )N j =1 ) > 0 such that (3) with initial data (Xj (0))j =1 has unique solution (Xj (t))j =1 in the time interval [0, T0 ] with

inf

inf |Xj (t) − Xk (t)| ≥ d/2.

t∈[0,T0 ] k=j

(29)

Theorem 3. Under the hypothesis (29) and assuming that αj , j = 0, j = 1, . . . , N, N there exists δ = δ(d; N ; (j )N j =1 ; (αj )j =1 ) > 0 such that for any 1 u0,j ∈ H (R), j = 1, . . . , N, with N

u0,j 1,2 ≤ δ,

(30)

j =1 N N there exists T ∈ (0, T0 ] with T = T (δ; d; N ; (j )N j =1 ; (αj )j =1 ; (u0,j 1,2 )j =1 ) > 0 such that the IVP (13) has a unique local solution

uj ∈ C([0, T ] : H 1 (R)), j = 1, .., N,

(31)

with inf Xj (t) − Xk (t) − (uj (·, t) − uk (·, t))∞ ≥ d/4.

inf

t∈[0,T ] k=j

(32)

Moreover, the map data-solution u0 = (u0,1 , .., u0,N ) → u(·, t) = (u1 , .., uN )(·, t) from a neighborhood of the origin in (H 1 (R))N into C([0, T ] : (H 1 (R))N ) is locally Lipschitz. Proof. We define βj = αj j . For v ∈ C([0, T0 ] : (H 1 (R))N ) define the operator (v)j (t) = e

iβj t∂σ2

t u0,j + 2i

k=j

0

+(Xj − Xk )(

2

eiβj (t−t )∂σ

k (

vj − vk |Xj − Xk − (vj − vk )|2

2 Re (Xj − Xk )(v¯j − v¯k ) − |vj − vk |2 ))(t )dt . (33) |Xj − Xk |2 |Xj − Xk − (vj − vk )|2

Thus, 2

∂σ (v)j (t) = eiβj t∂σ ∂σ u0,j t 2 +2i eiβj (t−t )∂σ k { 0

k=j

∂σ (vj − vk ) |Xj − Xk − (vj − vk )|2

(2Re (Xj − Xk )(∂σ (v¯j − v¯k )) + ∂σ |vj − vk |2 )(vj − vk ) − |Xj − Xk − (vj − vk )|4 (2Re (Xj − Xk )(∂σ (v¯j − v¯k )) + ∂σ |vj − vk |2 )(Xj − Xk ) + · |Xj − Xk |2 |Xj − Xk − (vj − vk )|2 (Xj − Xk )(2Re (Xj − Xk )(v¯j − v¯k ) − |vj − vk |2 )· −( |Xj − Xk |2 |Xj − Xk − (vj − vk )|4 × (2Re (Xj − Xk )(∂σ (v¯j − v¯k )) + ∂σ |vj − vk |2 )}(t )dt . (34)

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C.E. Kenig, G. Ponce, L. Vega

For δ > 0 sufficiently small (depending on d) we can assume that if v ∈ C([0, T ] : (H 1 (R))N ) with sup

N

t∈[0,T0 ] j =1

vj (·, t)1,2 ≤ 2δ,

(35)

then inf

inf Xj (t) − Xk (t) − (vj (·, t) − vk (·, t))∞ ≥ d/4.

t∈[0,T0 ] k=j

(36)

Introducing the notations |||vj |||T ,p = sup vj (·, t)p ,

p ∈ [1, ∞], T ∈ (0, T0 ],

(37)

t∈[0,T ]

and || =

N

|j |,

(38)

j =1

from (33) it follows that 10N || T |||vj − vk |||T ,2 d2 k=j 100N|| T + |||vj − vk |||T ,∞ |||vj − vk |||T ,2 , d3

|||(v)j |||T ,2 ≤ u0,j 2 +

(39)

k=j

and from (34) |||∂σ (v)j |||T ,2 ≤ ∂σ u0,j 2 +

10N || T |||∂σ (vj − vk )|||T ,2 d2 k=j

3 400N||(1 + d) T + ( |||vj − vk |||lT ,∞ ) |||∂σ (vj − vk )|||T ,2 . d4

(40)

l=1

Using that 1/2

1/2

|||vj − vk |||T ,∞ ≤ |||vj − vk |||T ,2 |||∂σ (vj − vk )|||T ,2 , |||vj − vk |||T ,2 ≤ |||vj |||T ,2 + |||vk |||T ,2 , |||∂σ (vj − vk )|||T ,2 ≤ |||∂σ vj |||T ,2 + |||∂σ vk |||T ,2 ,

(41)

one sees that for T > 0 sufficiently small, N j =1

(|||(v)j |||T ,2 + |||∂σ (v)j |||T ,2 ) ≤ 2

N j =1

u0,j 1,2 ≤ 2δ.

(42)

On the Interaction of Nearly Parallel Vortex Filaments

479

Similarly we can show that for T > 0 sufficiently small, N

(|||((v − w))j |||T ,2 + |||∂σ ((v − w))j |||T ,2 )

j =1

1 (|||(v − w)j |||T ,2 + |||∂σ (v − w)j |||T ,2 ), 4 N

≤

(43)

j =1

which proves that the operator is a contraction in the ball of radius 2δ (see (30)) centered at the origin of the space C([0, T ] : (H 1 (R))N ). This gives us the desired local solution u(·, t) = (u1 (·, t), .., uN (·, t)). From the above proof it follows that this local existence result can be extended to any interval [0, T ∗ ] if the following a priori estimates were available: inf

inf

j =k t∈[0,T ∗ ]

(Xj − Xk )(t) − (uj − uk )(·, t)∞ ≥ d/4,

(44)

and for some M > 0 (independent of T ∗ ) sup (uj − uk )(·, t)∞ ≤ M.

t∈[0,T ∗ ]

(45)

Under appropriate assumptions on the coefficients j ’s, αj ’s and the unperturbed configuration (Xj (0))N j =1 we shall establish the a priori estimates (44)-(45) in the time interval [0, ∞) which allows to extend the local solution obtained above to any time interval (global solution). 3. Global Existence Theory for the IVP (1.1) Formally solutions of the system (1) satisfy the following conservation laws (see [K-M-D, M-B]) (using the summation convention) 1 H = − αj j2 |∂σ j (σ, t)|2 dσ + j k ln |(j − k )(σ, t)|2 dσ, (46) 2 j
A = j

|j (σ, t)|2 dσ,

(47)

and in addition if βj = αj j = k0 ,

j = 1, .., N,

then one has the conservation law 1 j k |(j − k )(σ, t)|2 dσ. I= 2

(48)

(49)

j
The conservation law H(t) = H(0) is a consequence of the Hamiltonian structure of the system (1), A(t) = A(0) is due to the rotation invariant property of the set of solutions of (1), and I(t) = I(0) is the average distance functional.

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C.E. Kenig, G. Ponce, L. Vega

In our setting, for nearly parallel vortex filaments, the conservation laws (46), (47) and (49) are not finite, since all the integrands tend to a constant different than zero at infinity. We shall modify them by defining |(j − k )(σ, t)|2 1 dσ, j k ln H∗ = − k0 j |∂σ j (σ, t)|2 dσ + 2 d2 j
(50) and 1 I = j k 2 ∗

j
|(j − k )(σ, t)|2 − 1 dσ, d2

(51)

which are also preserved by the flow. To expect the finiteness of the quantities H∗ and I ∗ we need |Xj (0) − Xk (0)| = d > 0,

j, k = 1, .., N.

(52)

Therefore we consider only two cases : (i) N = 2 with any initial configuration, (ii) N = 3 with the initial configuration (X1 (0), X2 (0), X3 (0)) given by an equilateral triangle with side equals d. We recall that if the initial configuration of a three point vortices system is an equilateral triangle then it does not change shape or size as it moves with the fluid. In other words, the solution of (3) with initial values as in (52) with N = 3 satisfies |Xj (t) − Xk (t)| = d,

∀t ∈ R,

j, k = 1, 2, 3, j = k.

(53)

We shall study the case (ii). To obtain a positive conserved quantity we consider k0 j ∗ ∗ −H + 2I = j k |∂σ j (σ, t)|2 dσ + 2 j
(55)

satisfies f (1) = f (1) = 0 and f (x) > 0 so it follows that for x ∈ [1/4, 4] , f (x) ≥

(x − 1)2 . 2

(56)

Hence, from (54) one has (formally) that, if d/2 ≤ |j − k | ≤ 2d, for all σ , and a fixed t, − H∗ + 2I ∗ = λ 2 |(j − k )|2 k0 j 2 j k − 1 dσ ≥ 0. |∂σ j | dσ + ≥ 2 d2 j
(57)

On the Interaction of Nearly Parallel Vortex Filaments

481

We shall use that λ > 0 depends on d and on the size of the initial data µ, see (15). Moreover, it will be shown that λ(µ) ↓ 0 as µ ↓ 0. From (57) we have that Gj,k (σ, t) =

|(j − k )(σ, t)|2 − 1 ∈ L2 (R : dσ ), d2

j, k = 1, 2, 3, j = k. (58)

Since ∂σ Gj,k (σ, t) =

¯ k )∂σ (j − k ) + (j − k )∂σ ( ¯j − ¯ k) ¯j − ( , d2

(59)

combining Gagliardo-Nirenberg and Young’s inequalities it follows that (j − k )(t)2∞ 2 d |(j − k )(t)|2 +1 ≤ − 1 2 d ∞ 1/2 1/2 ¯ ¯ ( |(j − k )(t)|2 j − k )(∂σ j − ∂σ k )(t) − 1 ≤ 1+c 2 2 d d 2 2 1/2 λ1/2 (j − k )(t)∞ 1/2 ≤ 1 + c 1/2 (∂σ j − ∂σ k )(t)2 d d 1/2 1/2 λ (j − k )(t)∞ ≤ 1 + c 1/2 d d 1/2 1 (j − k )(t)2∞ λ 4/3 ≤1+ + 4c . 2 d2 d 1/2

(60)

Therefore, (j − k )(t)2∞ ≤ 2d 2 (1 + 4c(λ/d 1/2 )4/3 ),

(61)

if d/2 ≤ |j − k | ≤ 2d, for all σ ∈ R at that t. A similar argument shows that, if d/2 ≤ |j − k | ≤ 2d, at t, for all σ ∈ R, 1/2 1/2 |(j − k )(t)|2 |(j − k )(t)|2 (∂σ j − ∂σ k )(t)2 − 1 ≤ c − 1 d2 d2 d 1/2 ∞ 2 cλ ≤ 1/2 . (62) d Thus as long as the local solution exists in the time interval [0, T ] and d/2 ≤ |j − k | ≤ 2d, for all (σ, t) ∈ R × [0, T ], we have that

|(j − k )(σ, t)|2

≤ cλ . sup sup

− 1 (63)

d 1/2 2 d t∈[0,T ] σ ∈R Therefore, for λ small enough for all (σ, t) ∈ R × [0, T ] it follows that cλ 1/2 d/2 < d 1 − 1/2 ≤ |j (σ, t) − k (σ, t)| d = |Xj (t) − Xk (t) − (uj − uk )(σ, t)| ≤ d(1 + cλ/d 1/2 )1/2 < 2d.

(64)

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C.E. Kenig, G. Ponce, L. Vega

This combined with the continuity in (σ, t) of |(j − k )(σ, t)|2 and the smallness of λ gives (64) for any T > 0. In particular, from (53) one has that |||uj − uk |||T ,∞ = sup (uj − uk )(t)∞ ≤ 3d.

(65)

t∈[0,T ]

As we remark before a priori estimates (65)-(66) guarantee that the local solution can be extended to any time interval (global solution). To justify the finiteness and smallness of the quantity λ defined in (57) we observe that for x ∈ [1/4, 4], f (x) = − ln(x) + x − 1 ≤ 100(x − 1)2 .

(66)

Taking x = |j − k |2 /d 2 and using (52), (53), and (11) it follows that

2 2 |j − k |2 |Xj − Xk − (uj − uk )|2 (x − 1) = −1 = −1 d2 d2 2 −2Re(X¯ j − X¯ k )(uj − uk ) + |uj − uk |2 = , d2 2

(67)

which is an L1 -function whenever uj ∈ H 1 (R), j = 1, 2, 3. Moreover, by (67)-(68) one has that ∗ ∗ ∗ ∗ 0 ≤ λ = −H (t) + 2I (t) = −H (0) + 2I (0) k0 j ≤ |∂σ u0,j (σ )|2 dσ 2 2 −2Re(X¯ j − X¯ k )(0)(u0,j − u0,k ) + |u0,j − u0,k |2 j k dσ +100 d2 j
This implies that λ(µ) ↓ 0 as µ = above.

3

j =1 u0,j 1,2

↓ 0 which was used in our proof

Acknowledgement. The authors would like to thank P. K. Newton for useful conversations regarding this work. C. E. Kenig and G. Ponce were supported by NSF grants. L. Vega was supported by an MCEyT grant.

References [A] Aref, H.: Motion of three vortices. Phys. Fluids 22, 393–400 (1979) [A-N-S-T-V] Aref, H., Newton, P.K., Stremler, M.A., Tokieda, T., Vainchtein, D.L.: Vortex Crystals. Preprint [A-H] Arms, R.J., Hama, F.R.: Localized-induction concept on a curved vortex and motion of an elliptic vortex ring. Phys. Fluids 8, 553–559 (1965)

On the Interaction of Nearly Parallel Vortex Filaments [C-T] [K-M] [K-M-D] [L-M] [M-B] [N] [R] [Z]

483

Callegari, A.J., Ting, L.: Motion of a curve filament with decaying vortical core and axial velocity. SIAM J. Appl. Math. 35, 148–175 (1978) Klein, R., Majda, A.: Self-stretching of a perturbed vortex filament I. The asymptotic equations for deviations from a straight line. Physica D 49, 323–352 (1991) Klein, R., Majda, A., Damodaran, K.: Simplified equations for the interaction of nearly parallel vortex filaments. J. Fluid Mech. 288, 201–248 (1995) Lions, P.L., Majda, A.: Equilibrium statistical theory for nearly parallel vortex filaments. Comm. Pure Appl. Math. 53, 76–142 (2000) Majda, A., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge: Cambridge University Press, 2001 Newton, K.: The N-Vortex Problem. New York: Springer, 2001 Ricca, R.L.: The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics. Fluid Dynamics Research 18, 245–268 (1996) Zhidkov, P. E.: Korteweg-de Vries and Nonlinear Schr¨odinger Equations: Qualitative Theory. Lectures Notes 1756, New York: Springer, 2001

Communicated by P. Constantin

Commun. Math. Phys. 243, 485–540 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0980-8

Communications in

Mathematical Physics

Derivation of the Euler Equations from Quantum Dynamics∗ Bruno Nachtergaele1,∗∗ , Horng-Tzer Yau2,∗∗∗ 1 2

Department of Mathematics, University of California at Davis, Davis, CA 95616, USA. E-mail: [email protected] Courant Institute, New York University, 251 Mercer Street, New York, NY 10012, USA. E-mail: [email protected]

Received: 17 December 2002 / Accepted: 10 July 2003 Published online: 11 November 2003 – © B. Nachtergaele and H.-T. Yau 2003

Abstract: We derive the Euler equations from quantum dynamics for a class of fermionic many-body systems. We make two types of assumptions. The first type are physical assumptions on the solution of the Euler equations for the given initial data. The second type are a number of reasonable conjectures on the statistical mechanics and dynamics of the Fermion Hamiltonian.

Contents 1. Main Notations . . . . . . . . . . . . . . . . . . . . . 1.1 A convention . . . . . . . . . . . . . . . . . . . 2. Introduction . . . . . . . . . . . . . . . . . . . . . . . 2.1 Schr¨odinger and Euler dynamics . . . . . . . . . 2.2 Local equilibrium . . . . . . . . . . . . . . . . . 2.3 Assumptions and the main theorem . . . . . . . . 2.4 Outline of the proof . . . . . . . . . . . . . . . . 3. Relative Entropy Identity and High Momentum Cutoff 3.1 Entropy identity . . . . . . . . . . . . . . . . . . 3.2 High-momentum cutoff . . . . . . . . . . . . . . 4. Construction of Local Commuting Observables . . . . 4.1 Construction of an isometric embedding . . . . . 4.2 Commuting local conserved quantities . . . . . . 4.3 Local average of currents . . . . . . . . . . . . .

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486 487 487 489 492 493 496 497 497 499 501 501 503 504

∗ c 2003 B. Nachtergaele and H.-T. Yau. This article may be reproduced in its entirety for noncommercial purposes. ∗∗ Research partially supported by NSF # DMS-0070774. ∗∗∗ Research partially supported by NSF # DMS-0072098, the Veblen Fund from the Institute for Advanced Study and a Fellowship from the MacArthur Foundation.

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5. Bounds on the Currents . . . . . . . . . . . . . . . 6. Local Ergodicity . . . . . . . . . . . . . . . . . . . 6.1 General properties of limiting states . . . . . 6.2 Extension to unbounded conserved quantities 6.3 Basic estimate . . . . . . . . . . . . . . . . . 6.4 Proof of main ergodic lemma . . . . . . . . . 7. Relative Entropy Estimate . . . . . . . . . . . . . 7.1 Reduction to large deviation . . . . . . . . . 7.2 Conclusion of the relative entropy estimate theorem . . . . . . . . . . . . . . . . . . . . 8. Thermodynamics and Large Deviation . . . . . . . 8.1 Local Gibbs state with independent subcubes 8.2 Large deviation for commuting variables . . . 8.3 Proof of Lemma 7.2 . . . . . . . . . . . . . . 9. Calculation of the Currents . . . . . . . . . . . . . 10. The Virial Theorem . . . . . . . . . . . . . . . . . 11. Appendix. The Entropy Inequality . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . and proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . of the . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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505 507 510 512 514 516 516 518 519 520 521 523 527 528 532 539

1. Main Notations µ ..... j ...... w .....

W

.....

w

.....

∇

.....

u

......

u0x

.....

ux

.....

u4x A

..... .....

Typical index labeling the five conserved quantities. Typical index referring to time, j = 0, and space, j = 1, 2, 3, components. µ w = (wµ ) = (wj,x ),µ = 0, . . . 4, j = 1, 2, 3, are the components of the current densities. µ = 0 is the particle current, µ = 1, 2, 3, indexes the three components of the momentum current, µ = 4 is the energy current. j = 1, 2, 3 refers to the three spatial directions of the current. All quantities denoted by W∗ , are understood to be given by l 3 w∗ , for any subsript ∗ . Underlined vectors have an additional component, referring to time, or, in the case of the currents, the conserved quantities. ∇ = (∂t , ∇) is the four-component gradient, including the derivative with respect to time. u = (u0 , · · · , u4 ) = w0 := w0 are the five conserved quantities: particle number, three components of the momentum, and energy. u0x = nx is the quantum observable for the particle number density at microscopic space point x. ux = (u1x , u2x , u3x ), the quantum observables for the three components of the momentum density at x. u4x = hx is the quantum observables for the energy density at x. µ A = (Aj ) are the classical currents appearing on the RHS of the Euler equations (µ = 0, · · · , 4, j = 1, 2, 3). They are defined by A0j = q j , Aij = δij P + qi qj /q0 , A4j = q j (q4 + P )/q0 .

Derivation of the Euler Equations from Quantum Dynamics

A

.....

q

......

q

......

v ...... e˜ . . . . . . P (e, ρ) ε

......

I

......

I

......

+ ... ux, . . . . λε (t, x)

487

µ

A = (Aj ) are the classical currents augmented with the conserved quantities (j = 0, vanishing current in the time direction): A0 = q = A0 . q = (q 0 , · · · , q 4 ): q 0 = ρ is the classical particle density, q 1 , q 2 , q 3 are the three components of the classical momentum density, and q 4 = e is the classical energy density. These quantities are a function of macroscopic space and time. This notation is also used for the expectation value of these quantities in a quantum Gibbs state. q = (q 1 , q 2 , q 3 ) are the three components of the classical momentum density. v = q/ρ is the classical, macroscopic mean velocity per particle. e˜ = e/ρ is the classical, macroscopic mean energy per particle. The thermodynamic pressure as a function of the energy and particle densities. Appears in the Euler equations and is defined as the quantum statistical pressure for the Fermion system under consideration. ε is the scaling parameter relating the macroscopic coordinates X, T , with the microscopic coordinates x, t: X = εx, T = εt. The hydrodynamic limit is the limit ε → 0. The embedding from a collection of independent subcubes of periodic boundary condition to the cube ε−1 . The embedding from a subcube of periodic boundary condition to the cube ε−1 . local conservative quantities in a cube of size centered at x. cube of width centered at the origin. λε (t, x) = λ(εt, εx).

1.1. A convention. In the course of our arguments, we will encounter a large (but finite!) number of error terms. Therefore, we introduce a common notation that conveys all relevant information about these error terms. For k ≥ 1, y any list of symbols, let y (x) denote a real-valued function of x ∈ Rk , with the property that lim lim · · · lim y (x) = 0, xk xk−1

x1

where the limits are determined by the names x1 , . . . , xk of the variables. E.g., any term denoted by κ,X (ε, ) has the property lim lim κ,X (ε, ) = 0.

→∞ ε→0

(1.1)

The limits are to be taken in the specified order and the quantities denoted by y = (y1 , . . . , yl ) are kept fixed in the limits. The limit for ε is ε → 0, as this is the limit we are considering and similarly the limit for is unambiguously → ∞. The actual value of any y (x) may vary from occurrence to occurrence. For quantities used in error bounds about which no claim of convergence to zero is made, we will usually use the notation Cy , where y lists the relevant parameters the constant may depend on. 2. Introduction The fundamental laws of non-relativistic microscopic physics are Newton’s and Schr¨odinger equations in the classical and the quantum case respectively. These equa-

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tions are impossible to solve for large systems and macroscopic dynamics is therefore modeled by phenomenological equations such as the Euler or the Navier-Stokes equations. Although they were derived centuries ago from continuum considerations, they are in principle consequences of the microscopic physical laws and should be viewed as secondary equations. It was first observed by C. Morrey [12] in the fifties that the Euler equations become “exact” in the Euler limit, provided that the solutions to the Newton’s equation are “locally” in equilibrium. Morrey’s original work was far from rigorous and the meaning of “local equilibrium” was not clear. It is nevertheless a very original idea and it contributed significantly to the later development of the hydrodynamical limits of interacting particle systems, see [20] for a review. Instead of considering general classical dynamical systems with two body interactions, a different approach is to prove as much as possible for some simplified models. Outstanding examples are the works by Boldrighini, Dobrushin, and Suhov [1], and Sinai [19] in the case of one space dimension, and the more recent work by Eyink and Spohn [5] who study a d-dimensional classical system of non-interacting particles. In terms of a rigorous proof of Morrey’s idea, however, significant progress has only been made rather recently [16]. This long delay is mostly due to a serious lack of tools for analyzing many-body dynamics, in the classical case and even more so in the quantum case. In this paper, we derive the Euler equations from microscopic quantum dynamics, extending the relative entropy method of [22, 16] to the quantum cases. Our main result was announced in [14]. As we want to consider the genuine quantum dynamics for a system with short-range pair interactions, we cannot take a semiclassical limit. Although one-particle quantum dynamics converges to Newtonian dynamics in the semiclassical limit, this limit does not commute with the scaling limit needed for the Euler equation. This is most clearly seen in the pressure function, for which quantum corrections survive at the macroscopic scale. In fact, one of the conclusions of our work is that under rather general conditions, the pressure function is the only place where the quantum nature of the underlying system, in particular the particle statistics, survive in the Euler limit. The Euler equations have traditionally been derived from the Boltzmann equation both in the classical case and in the quantum case, see Kadanoff and Baym [9] for the quantum case. Since the Boltzmann equation is valid only in very low density regions, these derivations are not satisfactory, especially in the quantum case where the relationship between the quantum dynamics and the Boltzmann equation is not entirely clear. There were, however, two approaches based directly on quantum dynamics. The first was due to Born and Green [2], who used an early version of what was later called the BBGKY hierarchy, together with moment methods and some truncation assumptions. A bit later, Irving and Zwanzig [8] used the Wigner equation, moment methods and truncations to accomplish a similar result. These two approaches rely essentially on the moment method with the Boltzmann equation replaced by the Schr¨odinger equation. Unlike in the Boltzmann case, where one can do asymptotic analysis to justify this approach, it seems unlikely that this can be done for the Schr¨odinger dynamics. One of the benefits of our approach is that we develop a general strategy applicable to all situations where a number of reasonable assumptions are satisfied. We believe that our general assumptions, which are discussed in detail in Sect. 2.3, hold for a large class of physical models. We regard proving the properties that we assume as an important, although rather challenging, research project in quantum statistical mechanics. The main novelty of our work lies in the fact that, for the first time, the relative entropy method is applied to a quantum mechanical system. This requires solving a number of technical problems which, not surprisingly, all stem from the fact that the

Derivation of the Euler Equations from Quantum Dynamics

489

local observables corresponding to the globally conserved quantities of the dynamics, are represented by non-commuting operators. This is mainly discussed in Sect. 6. Although our goal is a derivation of the Euler equations, the relative entropy method indeed constructs an approximate solution to the underlying many body dynamics based on solution of Euler equations and the concept of local Gibbs states. It thus establishes the key role played by the Euler equations: they are not just a set of conservation laws but, with the correct choice for the pressure function, they actually dictate the leading approximation to the many body classical or quantum dynamics. Thus, the Euler equations may also be used to obtain information about the solutions of the many-body Schr¨odinger equation. 2.1. Schr¨odinger and Euler dynamics. We begin by considering N particles on R3 , evolving according to the Schr¨odinger equation i∂t ψt (x1 , · · · , xN ) = H ψt (x1 , · · · , xN ), where the Hamiltonian is given by H =

N −j j =1

2

+

W (xi − xj ) .

(2.1)

1≤i<j ≤N

Here, W is a two-body short-ranged stable isotropic pair interaction and ψt (x1 , · · · , xN ) is the wave function of particles at time t. We only consider Fermions (such as electrons, but for simplicity we ignore spin) and thus the state space HN is the subspace of antisymmetric functions in L2 (R3N ), i.e., ψ(xσ1 , · · · , xσN ) = (−1)σ ψ(x1 , · · · , xN ), for any permutation σ of {1, · · · , N }. It is more suitable not to fix the total number of particles and to use the second quantization terminology. In fact, it would be extremely cumbersome to work through all arguments without the second quantization formalism. The state space of the particles, called the Fermion Fock space, is the direct sum of HN : N H := ⊕∞ N=0 H . Define the annihilation and creation operators ax and ax+ by √ (ax )N (x1 , · · · , xN ) = N + 1 N+1 (x, x1 , · · · , xN ), (2.2) N 1 (ax+ )N (x1 , · · · , xN ) = √ (−1)j −1 δ(x − xj ) N−1 (x1 , · · · , xj , · · · , xN ) , N j =1

(2.3) where, as usual, means “omit”. ax and ax+ are to be interpreted as operator-valued distributions [3]. The annihilation operator ax is simply the adjoint of ax+ with respect to the standard inner product of the Fock space with Lebesgue measure dx. These operators satisfy the canonical anticommutation relations [ax , ay+ ]+ := ax ay+ + ax ay+ = δ(x − y), [ax+ , ay+ ]+ = [ax , ay ]+ = 0 ,

(2.4)

where δ is the delta distribution. The derivatives of these distributions with respect to the parameter x are denoted by ∇ax and ∇ax+ . With this notation, we can express the Hamiltonian as H = H0 + V , where the kinetic energy is given by 1 H0 = ∇ax+ ∇ax dx, ¯ 2

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and the potential energy V =

1 2

dxdyW (x − y)ax+ ay+ ay ax .

It is more convenient to put the Schr¨odinger equation into the operator form, which is sometimes called the Schr¨odinger-Liouville equation. Denote the density matrix of the state at time t by γt . Only normal states, which can be represented by density matrices, will be considered in the time evolution. Then the Schr¨odinger equation is equivalent to i∂t γt = δH γt

with δH γt := [H, γt ].

(2.5)

The conserved quantities of the dynamics are the number of particles, the three components of the momentum and the energy. The local densities of these quantities are denoted by u = (uµ ), µ = 0, · · · , 4, and are given by the following expressions: u0x = nx = ax+ ax , i j j ux = px = [∇j ax+ ax − ax+ ∇j ax ], j = 1, 2, 3, 2 1 1 u4x = hx = ∇ax+ ∇ax + dyW (x − y)ax+ ay+ ay ax . 2 2

(2.6)

We also introduce the notation u = (u1 , u2 , u3 ). This convention will be followed for the rest of the paper. We use boldface for the vector of the conservative quantities and use the frac for the vector consisting only the components 1, 2, 3. Let denote a cube of width centered at the origin. The subscript may be omitted if it plays no active role. We shall adopt the convention that unbounded observables on will be defined with periodic boundary conditions. E.g., the number of particles in , the total momentum, and the total energy of the particles in , respectively, are defined by N = dx nx , j j P = dx px , j = 1, 2, 3, H = dx hx .

In other words, we shall always view as a three-dimensional torus. We slightly generalize the definition of the grand canonical Gibbs states to include a parameter for the total momentum of the system: the Lagrange multiplier α. We will work under the assumption that the temperature and chemical potential are in the one-phase region of the phase diagram of the system under consideration such that the thermodynamic limit is unique. The finite volume Gibbs states are then given by the following formula: ωβ,α,µ (X) =

Tr Xe−β(H0, +V −α·P −µN ) . Tr e−β(H0, +V −αP −µN )

(2.7)

Derivation of the Euler Equations from Quantum Dynamics

491

The infinite volume Gibbs states ωβ,α,µ are the limiting points of the finite volume ones. It is convenient to denote the parameters (β, α, µ) by λ = (λµ ), µ = 0, · · · , 4 with λ0 = βµ, λj = βα j , λ4 = β. Define (notice the sign convention) λ·u=

3

λµ uµ − λ4 u4

(2.8)

µ=0

and

λ, u = ||−1

dxλ(x) · u(x).

These notations allow us to give a compact formula for the unique, translation invariant Gibbs state (defined with constant λ), as well as for the states describing local equilibrium (defined with x-dependent λ): ωλ = e|| λ,u /Z (λ),

(2.9)

where Z (λ) is the partition function Z (λ) = Tr e|| λ,u .

(2.10)

The pressure as a function of the constant vector λ, is defined by ψ(λ) = lim ||−1 log Z (λ). →∞

Denote the expectation value of the conservative quantities in an infinite-volume equilibrium state, ωλ introduced following (2.7), by q = (q 0 , · · · , q 4 ). Then we have ∂ψ = ωλ (uµ ). ∂λµ

(2.11)

Explicitly, 1 ω (N ), || λ 1 q = ωλ (px ) = lim ωλ (P ), 3 →R || 1 e = ωλ (hx ) = lim ωλ (H ). 3 || →R

ρ = ωλ (nx ) = lim

→R3

Notice that q and e are momentum and energy per volume. Again, we will work under the assumption that these parameters stay in the onephase region, the limiting Gibbs state is unique and these definitions are unambiguous. Although momentum is preferable as a quantum observable, we also introduce the velocity in order to be able to compare with the classical case. The velocity field v(x) has to be defined as a mean velocity of the particles in a neighborhood of x. Therefore we have v(x) = q(x)/ρ(x). We also introduce the energy per particle defined by e˜ = e/ρ. The usual Euler equations are written in terms of ρ, v, and e. ˜ In order to derive the Euler equations, we need to perform a rescaling. So we shall put all particles in a torus ε−1 of size ε −1 and use (X, T ) = (εx, εt) to denote the

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macroscopic coordinates. For all equations in this paper periodic boundary conditions are implicitly understood. The Euler equations for the five conserved quantities, which arise in the limit → 0, are given by ∂ ∂ρ (ρvj ) = 0, + ∂T ∂Xj 3

j =1

3 ∂ ∂ ( ρvk ) ∂ P (e, ρ) = 0, ρvj vk + + ∂T ∂Xj ∂Xk

(2.12)

j =1

3 ∂ ( ρ e˜ ) ∂ ρ ev ˜ j + vj P (e, ρ) = 0. + ∂T ∂Xj j =1

These equations are in form identical to the classical ones but all physical quantities are computed quantum mechanically. In particular, P (e, ρ) is the thermodynamic pressure computed from quantum statistical mechanics for the microscopic system. It is a function of X and T only through its dependence on e and ρ. If no velocity dependent forces act between the molecules of the fluid under consideration (we consider only a pair potential), the pressure is independent of the velocity. The conservative quantities q = (q 0 , · · · , q 4 ), are related to density, momenta and energy as follows: q0 = ρ ,

q i = ρ vi ,

q 4 = e = ρ e. ˜

(2.13)

In other words q 1 , q 2 , q 3 , and q 4 are momenta and energy per volume instead of per particle as in the usual Euler equation (2.12). We rewrite the Euler equations in the following form: 3 ∂q µ X µ + ∇i Ai (q) = 0 , ∂T

µ = 0, 1, 2, 3, 4 .

(2.14)

i=1

The matrix A is determined by comparison with the Euler equations: A0j = q j , Aij = δij P + qi qj /q0 , A4j

(2.15)

= q (q4 + P )/q0 . j

2.2. Local equilibrium. To proceed we need a microscopic description of local equilibrium and a microscopic prescription to compute the pressure from quantum statistical mechanics. Suppose we are given macroscopic functions q(X). We wish to find a local Gibbs state with the conserved quantities given by q(X). The local Gibbs states are states locally in equilibrium. In other words, in a microscopic neighborhood of any point x ∈ T 3 the state is given by a Gibbs state. More precisely, we wish to find a local Gibbs state with the expected values of the energy, momentum, and particle number per unit volume at X given by q(X). To achieve this, we only have to adjust the parameter

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λ at every point X. More precisely, we choose λ(X) such that Eq. (2.11) holds at every point, i.e., ∂ψ(λ(X)) = q µ (X). ∂λµ (X) If we denote the solution to the Euler equation by q(X, T ), then we can choose in a similar way a local Gibbs state with given conserved quantities at the time T . Define the local Gibbs state 1 ωtε = (2.16) exp ε−3 λ(εt, ε·), uε−1 , cε (t) µ

where cε (t) is the normalization constant. Clearly, we have that ωtε (ux ) = q µ (εx, εt) to leading order in ε. Our construction of local equilibrium states is consistent with the abstract framework discussed in [21]. Later we will need the following relation for the normalization constant cε (t): d log cε (t) = εTr ωtε ε−3 ∂t λ(εt, ε·), uε−1 . dt

(2.17)

Since the inner product almost exclusively is taken on ε−1 , we shall drop this subscription or replace it by , ε−1 for the rest of this paper. In summary, the goal is to show that, in the limit ε → 0, the following diagram commutes: q(X, 0)   local equilibrium γ0

Euler −−−−→

Schr¨odinger −−−−−−−−→

q(X, T )

limit ε ↓ 0 of expectation of locally averaged observables γε−1 T

As smooth solutions of the Euler equations are guaranteed to exist only up to a finite time [10], say T0 , we will formulate our assumptions on the dynamics of the microscopic system for a finite time interval as well, say t ∈ [0, T0 /ε]. Note the cutoff assumptions below would hold automatically for lattice models.

2.3. Assumptions and the main theorem. Our main result is stated in Theorem 2.1 below. First, we state the assumptions of the theorem with some brief comments. There are three kinds of assumptions. The first category of assumptions could be called physical assumptions on the solution of the Euler equations that we would like to obtain as a scaling limit of the underlying dynamics, and on the pair interaction potential of this system. I. One-phase regime. We assume that the pair potential, W , is C 1 radial and supported in a ball of radius R. Furthermore, we assume that W is stable in the sense that W (x) = W0 (x) + W1 (x), where W0 ≥ 0 , W0 (0) > 0 and W1 is positive definite. (2.18)

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Here, the positive-definiteness of W1 refers to the sesquilinear form, not the function itself. I.e., W1 (xi − xj )zi zj , for all, n ≥ 1, xi , xj ∈ R3 , zi , zj ∈ C. 1≤i,j ≤n

Such potentials automatically satisfy the usual super-stability property [18]. In particular, they are stable, i.e., there is a constant B ≥ 0 such that, for all N ≥ 2, x1 , . . . , xN ∈ R3 , W (xi − xj ) ≥ −BN. 1≤1<j ≤N

Of the Fermion system with potential W we assume that there is an open region D ⊂ R2 , which we will call the one-phase region, such that the system has a unique limiting Gibbs state and a regular pressure function for all values of particle density and energy density (ρ, e) ∈ D. The solution of the Euler equations we consider, q(X, T ), will be assumed to C 1 in X for T ∈ [0, T0 ], and have local particle and energy density in the one-phase region for all times T ∈ [0, T0 ]. I.e., (ρ(X, T ), e(X, T )) ∈ D, for all X ∈ 1 and T ∈ [0, T0 ]. The next category of assumptions is on the local equilibrium states for the Fermion system that we construct and on their time-evolution under the Schr¨odinger equation. II. Cutoff assumptions. Suppose that γt is the solution to the Schr¨odinger equation (2.5) with a local equilibrium state as initial condition, constructed with the parameters derived from a solution of the Euler equations (with the appropriate pressure function) for times t in a finite interval [0, T0 ], that does not leave the one-phase region. We make the following two assumptions: 1. High-momentum cutoff assumption: Let Np (t) = Tr γt ap+ ap , where ap# is the Fourier transform of ax# . Then there is a constant c > 0 such that for all t ≤ T0 /ε, 2 (2.19) εd dpecp Np (t) ≤ CT0 , where CT0 , is constant only depending on T0 . 2. Non-implosion assumption: There is a constant CT0 (not necessarily the same as CT0 in the previous paragraph) such that for all t ≤ T0 /ε,

2 d ≤ CT0 , dxnx ny dy (2.20) Tr γt ε

|x−y|≤2R

where R is the range of the interaction W . Finally, we have an assumption on the set of the time-invariant ergodic states of the Fermion system. To state this assumption we need the notion of relative entropy, of a normal state γ with respect to another normal state ω. Let γ and ω denote the density matrices of these states. The relative entropy, S(γ | ω), is defined by Tr {γ (log γ − log ω)} if ker ω ⊂ ker γ S(γ |ω) = . +∞ otherwise

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For a pair of translation invariant locally normal states, one can show existence of the relative entropy density [15], defined by the limit s(γ |ω) = lim ε 3 S(γε−1 | ωε−1 ) , ε↓0

where γε−1 and ωε−1 denote the density matrices of the normal states obtained by restricting γ and ω to the observables localized in ε−1 = ε−1 1 . The existence of the limit can be proved under more general conditions on the finite volumes, but this is unimportant for us. III. Ergodicity assumption (“Boltzmann Hypothesis”). All translation invariant, ergodic with respect to space translations, stationary (i.e., time invariant) states to the Schr¨odinger equation with the Hamiltonian H are Gibbs states with the same Hamiltonian provided they satisfy the following assumptions: 1) the density and energy is in one phase region. 2) The relative entropy density with respect to some Gibbs state is finite. We expect that the cutoff assumptions hold for the solutions γt of the Schr¨odinger equation that we employ, but for now there is no complete proof that it holds for Gibbs states other than the free Fermi gas. For Gibbs states in the high temperature region we expect these assumptions can be proved by using some type of cluster expansion methods. A partial result in this direction has been obtained recently, in the case of Bosons, by Gallavotti, Lebowitz, and Mastropietro in [6]. For the rest of this paper, we shall assume this cutoff assumption for the solution to the Schrodinger equations as well as the Gibbs states in the one phase regions considered in this paper. We wish to point out that in the treatment of the classical case in [16] the cut-off assumption 2 was not needed. There is however no proof for the cut-off assumption 1 even in the classical case. (In [16], the usual quadratic kinetic energy was replaced by one with bounded derivatives with respect to momentum. So the cut-off assumption 1 is not needed too.) The cutoff assumptions are technical in nature. For Fermion models on a lattice instead of in the continuum, no cut-off assumptions are required. The Boltzmann hypothesis on the other hand is a fundamental problem in statistical physics. A version of it was proved to hold for a classical ideal gas by Eyink and Spohn in [5]. Gurevich and Suhov [7] proved that a stationary Gibbs state to a classical dynamics with a Hamiltonian H has to be a Gibbs state with the same Hamiltonian. Under the assumption that the stationary measures velocity distribution has no correlation (a weaker assumption than in [7]), the Boltzmann hypothesis was proved for classical gas with two-body interaction [16]. Our main result is the following theorem. We also expect it to hold for Bosons with a super-stable interaction. Theorem 2.1. Suppose that q(X, T ) is a smooth solution to the Euler equation in one phase region up to time T ≤ T0 . Let ωtε be the local Gibbs state with conserved quantities given by q(X, T ). Suppose that the cutoff assumptions and the ergodicity assumption hold. Let γt be the solution to the Schr¨odinger equation (2.5) and γ0 = ω0ε . (Note that γt depends on ε.) Then we have lim

sup

ε→0 0≤t≤ε−1 T

0

s(γt |ωtε ) = 0.

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In other words, ωtε is a solution to the Schr¨odinger equation (2.5) in entropy sense. In particular, for any smooth function f on , we have, for all 0 ≤ T ≤ T0 , 3 lim ε dxf (εx) γε−1 T (ux ) − q(T , εx) = 0. ε→0

ε−1

The proof of this theorem will be given in Sect. 7.2. Illustrated with a diagram, the main theorem says ωqε 0  

Euler equation −−−−−−−−−−→

ωqε T  limε↓0 s(γ −1 |ωε )=0 ε T qT

Schr¨odinger equation γ0 −−−−−−−−−−−−−−−→ γε−1 T Notice that we have proved more than just convergence to the Euler equation. We have shown that the local-equilibrium Gibbs state constructed from the evolution of the Euler equations solves the many-body Schr¨odinger equation, approximately in entropy sense.

2.4. Outline of the proof. The basic structure of our proof follows the relative entropy approach of [16, 22]. The aim is to derive a differential inequality for the relative entropy between the solution to the Schr¨odinger equation and a time-dependent local Gibbs state constructed to reproduce the solution of the Euler equations. The time derivative of the relative entropy can be expressed as an expectation of the local currents with respect to the solution to the Schr¨odinger equation. Since we do not know the solution well enough, this expectation can not be computed. Step 1. Replace the local microscopic currents by macroscopic currents. The basic idea in the hydrodynamical limit is first to show that the local space time average of the solution is time invariant. From the Boltzmann hypothesis, ergodic time invariant states are Gibbs. For Gibbs states, we can replace the local microscopic currents by macroscopic currents. This is the first step. In the quantum setting, there are several crucial issues we need to address. 1a: Construct a commuting version of the local conserved quantities. Recall macroscopic currents are functions of the local conserved quantities, i.e., density, momentum and energy. For the microscopic quantum system, the local conservative quantities are operators which commute only up to boundary terms. In order to express the macroscopic currents as functions of the local conserved quantities, we need either to prove that the non-commutativity does not affect the macroscopic currents or we need to construct some commuting version of the local conserved quantities. As the first approach seems very difficult to carry out, we follow the second one and construct a commuting version of local conservative quantities in Sect. 4. 1b: Restriction to the one phase region. Since the Boltzmann hypothesis holds only in the one phase region, we have to exclude the region outside the one phase region. To perform this restriction to the one-phase region we would normally multiply the observables by some cutoff function. In our case however, the cutoff function does not commute with the local currents. This seemingly trivial multiplication by a cutoff function illustrates the kind of technical problems we have to address in this work. Our approach to this is presented in Sect. 6.

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1c: Virial Theorem. Even assuming the local ergodic states are Gibbs in the one phase region, we still have to compute the macroscopic currents from the microscopic currents. This requires a virial Theorem, which we provide in Sect. 10. Step 2. Estimate all errors by local conservative quantities. As will become clear, the errors associated with the cutoff of the one phase region are difficult to control directly. We shall bound them by the local conserved quantities. This will be carried out in Sect. 5 and 6. Step 3. Derive a differential inequality of the entropy with error term given by a large deviation formula. After Step 2, we have an expression of the derivative of the entropy in terms of local (commuting) conservative quantities. Since these quantities commute, by an entropy inequality, we can bound it by a large deviation expression. Notice that it is crucial that we control everything by commuting objects. There is no large deviation theory for non-commuting observables. After this step, the standard relative entropy method provides the rest of the argument. Technically speaking, the main difficulty to study a quantum mechanical system, in comparison with a classical one, can be traced back to the non-commutativity of the algebra of observables. E.g., suppose A and B are two self-adjoint operators representing observables of the system. A simple inequality, such as |A + B| ≤ |A| + |B|, which is used numerous times in estimates for classical systems, is false, and so is |AB| ≤ |A| |B|. Therefore, there are essentially no absolute values taken in our proof and we estimate all quantities by commuting versions of the locally conserved quantities. Of course, these inequalities hold with the absolute value replaced by the norm. However, we will frequently deal with error terms that are expectations of unbounded observables, such as, e.g., the high-momentum contributions to the energy. Clearly, norm estimates are useless in this situation. 3. Relative Entropy Identity and High Momentum Cutoff 3.1. Entropy identity. The first step in the derivation of a diffential inequality for the relative entropy is to compute the derivative. Suppose γt is a solution to the Schr¨odinger equation. Recall that one has d Tr A(t)B(t) = Tr A (t)B(t) + A(t)B (t), dt

d Tr eA(t) = Tr eA(t) A (t) (3.1) dt

and d S(γt ) = 0. dt

(3.2)

Thus we have for any time-dependent density matrix ρt the identity d S(γt |ρt ) = Tr γt {−iδH − ∂t } log ρt . dt

(3.3)

This identity replaces the relative entropy inequality in [16, 22]. Thus, by (2.16), d s(γt |ωtε ) = Tr γt {−iδH − ∂t } λε (t, ·), uε−1 − ε 3 log cε (t) , (3.4) dt where , ε−1 = , ε−1 .

498

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Let w denote the current tensor with components wk,x defined by i (3.5) [∇k ax+ ax − ax+ ∇k ax ], 2 1 = ∇j ax+ ∇k ax + ∇k ax+ ∇j ax 2 (x − y)j (x − y)k + + 1 − (3.6) dy W (x − y) ax ay ay ax , k = 1, 2, 3, 2 |x − y| i = − ∇k ax+ ax − ax+ ∇k ax 4 i + (3.7) dy W (x − y) ∇k ax+ ay+ ay ax − ax+ ay+ ay ∇k ax 4 (x − y)k (x − y)j + i − dy W (x − y) ax ∇j ay+ ay ax − ax+ ay+ ∇j ay ax , 4 |x − y| (3.8)

0 wk,x = pxk = j

wk,x

4 wk,x

where we have used the rotation invariance of the potential to write dW (r) |r=|x| . dr We have the following proposition. See Sect. 9 for the derivation of these expressions for the current. W (x) =

Proposition 3.1. Let λ (ε) be defined by the equation iδH λε (t, ·), uε−1 = ε

3

∇j λε (t, ·), wj (t)ε−1 + λ (ε).

(3.9)

j =1

Then λ (ε) is an error term which satisfies the condition lim Tr γ λ (ε) = 0.

ε→0

These expressions of the microscopic currents seemingly bear no relationship to the macroscopic currents in the Euler equations, even when one assumes that γt is locally Gibbs. This difficulty already appears in the classical case. But by reasonably straightforward computation and application of a quantum version of the virial theorem 10.1 one can show that indeed these currents correspond to the standard Euler equations given in (2.12). µ µ Define ∇0 = ∂t and w0,x = ux . We have {−iδH − ∂t } ε 3 λε (t, ·), uε−1 = −ε

3

∇j λε (t, ·) , wj (t)ε−1 := εG(λε , a + , a),

j =0

(3.10) where λε (t, x) = λ(t, x). Introduce the notations ∇ = (∇0 , ∇), w = (w0 , w) and A•B =

3 3 j =0 µ=0

µ µ Aj B j

−

3 j =0

A4j Bj4 .

(3.11)

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Then we can rewrite the last expression as {−iδH − ∂t } ε 3 λε (t, ·), uε−1 = − ∇λε (t, ·) • w(t)ε−1 .

(3.12) µ

If we wish to emphasize the dependence on the operator, we shall write wj,x = µ wj,x (a + , a). From now on, we shall drop the subscript ε −1 in , ε−1 . 3.2. High-momentum cutoff. Most of the estimates we need are obtained using bounded versions of the creation and annihilation operators, i.e., suitable so-called smeared operators. Physically, this corresponds to introducing a high-momentum cutoff. The precise form of the cutoff will be important for us, as we will have strict requirements on the behavior of the error terms for the proof to go through. Let φˆ M be a smooth function such that 2 2 1. |φˆ M (p) − 1| ≤ e−M for |p| ≤ M and |φˆ M (p)| ≤ e−M for |p| ≥ 2M. 2 2. The support of φM is bounded in a ball of radius eM . To construct such a function, let g be a smooth function supported in |x| ≤ 2 such that g(p) ˆ ≤ C[1 + p 2 ]−3 . Define gλ (x) = g(x/λ)λ−3/2 . Let hM be a smooth function such that hˆ M (p) = 1 for |p| ≤ M and hˆ M (p) = 0, for |p| ≥ 2M . Let φM = (gλ ∗ gλ )hM .

2

Notice that φM = 1. Let λ = eM . Then we can check easily Properties 1 and 2. 2 Although φM is supported in a ball of radius eM , its mass is concentrated in a ball of radius M −1 . More precisely, there is a constant c such that hM (x)dx ≤ e−crM . |x|≥r

Define + ax,M =

φM (x − y)ay+ = aφ+x,M ,

ax,M =

φ(x − y)ay = aφx,M ,

where φx,M = φM (x − y). In our setting φx,M = φx,M . Define + + = a∇φ , ∇ax,M x,M

∇ax,M = a∇φx,M .

± ± and ax,M are bounded operators localized in a ball of radius eM . Notice that ∇ax,M We now perform the preliminary truncation. Denote the cutoff version of the current by 2

+ , aM ). wj,x,M = wj,x (aM

(3.13)

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Notice that wj,x,M is bounded. The difference between the kinetic energy term in the energy current with and without cut-off can be calculated using + + ∇ax+ ax − ∇ax,M ax,M = ∇bx,M ax + ∇ax+ bx,M ,

where + + bx,M = ax+ − ax,M .

Lemma 3.2. For any state γ that satisfies the cutoff assumptions and for G defined by (3.10), we have 2 + d ε Tr γ dx ∇bx,M ax + ∇ax+ bx,M ≤ e−cM . Proof. By using the Fourier transform, we have + 2 + ˆ dx∇bx,M ax = dp(1 − φ(p))p a p ap . Let Np (t) = Tr γ ap+ ap . Then Tr γ dp(1 − φˆ M (p))p 2 ap+ ap = dp(1 − φˆ M (p))p 2 Np . The lemma is thus a simple consequence of the Chebeshev inequality and the definition of φM . From the Schwarz inequality d + dx∇ax dyW (x − y)ay+ ay ax Tr γ ε 1/2

2 1/2 + + Tr γ ax dyW (x − y)ny ax ≤ ε Tr γ dx∇ax ∇ax ≤ C. d

Thus for any state γ satisfy the cutoff assumptions, we have 2 + + Tr γ εd dxdyW (x − y) ∇ax+ ay+ ay ax − ∇ax,M ay,M ay,M ax,M ≤ e−cM .

(3.14)

Lemma 3.3. For any state γ satisfy the cutoff assumptions, we have + Tr γ εd [G(λ, a + , a) − G(λ, aM , aM )] ≤ e−cM . 2

Thus we have d d + 1 , aM ) − ε 3 log cε (t) + EM s(γt |ωtε ) = εTr γt G(λε , aM dt dt

(3.15)

(3.16)

1 ≤ Ce−cM . The precise form of the last estimate is crucial as we shall see later with EM on. We recall the crucial relative entropy inequality [15]: for all self-adjoint observables h, and for any δ > 0, : 2

γ (h) ≤ δ −1 log Tr eδh+log ω + δ −1 S(γ |ω). A proof will be given in the appendix.

(3.17)

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4. Construction of Local Commuting Observables The conserved quantities commute as global observables on with periodic boundary conditions. In fact, this is essential for the classical equations of motion to make sense. For any bounded quasi-local observable X on the Fock space define ˆ X(q) = Tr ωλ X,

(4.1)

where the chemical potential λ is the dual of q in the sense of (2.11). We define Xˆ only for arguments in the one phase region. Local averages of the densities of the conserved quantities, however, do not commute due to boundary effects. Therefore, we cannot extend the functions Xˆ to functions of the operator-valued local densities of the conserved quantities, which is what we would like to do. To circumvent this difficulty, in this section we construct commuting versions of the local conserved quantities. We have for any smooth function J , Tr γt J (εt, ε · · · ), X = Tr γt

Av J (εt, εx)

x∈ε−1

Av

|z−x|≤/2

τz X + (ε),

where Av(·), stands for the average of its argument over the domain indicated in the · subscript, and limε→0 (ε) = 0 for any fixed. Denote u := u := τx u dx the local conservative quantities and we would like to replace the microscopic current Av τz X by a certain function of the local |z−x|≤/2

conservative quantities τz u . Unfortunately the components of u do not commute and functions of u are not well-defined. In fact, even the definition of u is ambiguous since we did not specify the boundary condition. Intuitively, the components of u actually commute up to boundary terms, and the ambiguity should be negligible in the limit → 0. Since it is rather difficult to control these boundary terms in a simple way, we construct in the following a commuting version of the local conservative quantities. 4.1. Construction of an isometric embedding. Let f be a smooth function with √ f (s) = 1/ 2 if s ≤ 0 = 0 if s ≥ 1 and

f (1) = f (1) = f (1) = 0 . For any given η, 0 < η < 1/2, let g(t) = 1 − f

2

1 2

g(t) = f

−t η

t− η

1/2 ,

1 2

0 ≤ t ≤ 1/2,

g(t) = g(−t),

,

t ≥ 1/2, t ∈R.

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Then g is smooth, supported in |t| ≤ 1/2 + η and g 2 (t + j ) = 1. j ∈Z

Let χ(x) = g(x 1 )g(x 2 )g(x 3 ). Then

χ 2 (t + j ) = 1.

(4.2)

j ∈Z3

Let α ± := τα ± be a cube of size ± 4η centered at α. Let χα (x) = χ ((x − α)/) be a smooth function supported in τα +2η ⊂ α + and χα (x) = 1 in τα −2η ⊃ α − . We collect these relations in the following: α − = τα −4η ⊂ τα −2η ⊂ {x : χα (x) = 1} ⊂ {x : χα (x) = 0} ⊂ τα +2η ⊂ τα +4η = α + .

(4.3)

There is a wide range of choices for η. The main restrictions we need are η → ∞,

η→0.

We shall choose, for simplicity of notation η = −1/2 for the rest of this paper. Recall the configuration space S() is the space S() = {x := (x1 , · · · , xn ) : n ∈ {0} ∪ N, xj ∈ for all j }. Denote by () the space of antisymmetric functions from the configuration space S() to the complex numbers. With the standard L2 inner product, () is a Hilbert space. Define Iα from (α + ) to () by

χα (xj ) ψ(x1 , · · · , xn ). (Iα ψ)(x1 , · · · , xn ) = j

Usually we shall take = ε−1 and omit the labels

and α whenever they are obvious or unimportant. It is crucial that j χαj (xj ) is symmetric w.r.t. permutations of x so that Iα ψ is antisymmetric as a function of x. Define I ∗ to be the adjoint of I , i.e., we have (I ∗ f, g) = (f, Ig). Let X be an observable X on α defined by dx1 · · · dxk dy1 · · · dyk f (x1 , . . . , xk ; y1 , . . . , yk )ax+1 ,α · · · ax+k ,α ayk ,α · · · ay1 ,α , X= α+

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where f is a distribution (kernel) with support in (α + )×2k . Here we have labelled the operators by α to emphasize the cube α. We can check the identity: I ∗ XI = dx1 · · · dxk dy1 · · · dyk χα (x1 ) · · · χα (xk )χα (y1 ) · · · χα (yk ) ×f (x1 , . . . , xk ; y1 , . . . , yk )ax+1 · · · ax+k ayk · · · ay1

(4.4)

as an operator on the torus ε−1 . From this definition the pull-backs of all observables we need, i.e., I ∗ XI for a conserved quantity or current given by X, can easily be computed. By using the appropriate distribution kernels f , observables involving derivatives are included. E.g.,

∗ + ∗ + I dxf (x)∇ax ax I = I − dyδ (y − x) dxf (x)ay ax I α+ α+ = − dyδ (y − x) dxf (x)χα (x)χα (y)ay+ ax = dxf (x) χα (x)2 ∇ax+ ax + χα (x)∇χα (x)ax+ ax . (4.5) For the kinetic energy we have

I∗ dxf (x)∇j ax+ ∇j ax I = dxf (x) χα (x)2 ∇j ax+ ∇j ax + (∇j χα (x))2 ax+ ax α+ +χα (x)∇j χα (x)(∇j ax+ ax + ax+ ∇j ax ) . (4.6) If we take f = 1α − (x), we have f (x)χα (x) = 0. Together with χα (x) = 1 in τα −2η ⊃ α − , we have

∗ + (4.7) I dx∇ax ax I = dx ∇ax+ ax , α− α−

I∗ dx∇j ax+ ∇j ax I = dx∇j ax+ ∇j ax . α−

α−

4.2. Commuting local conserved quantities. Let Hα + , Pα + be the total energy and momentum operators on α + with periodic boundary condition. Then Hα + and Pα + commute with each other and also with the number operator Nα + . Denote uα + = −3 (Pα + , Nα + , Hα + ). Since the image of I is in the domain of Hα + and Pα + , the operator I ∗ uα + I is welldefined. Since the components of uα + commute, the function ˆ α + )I I ∗ X(u is now well-defined. We shall use the notation + ux, := uα + ,

+ −3 −3 n+ x, := Nα + , hx, := Hα +

when α is centered at x. When x = 0, we shall omit the subscript x.

(4.8)

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4.3. Local average of currents. Let −3 wα ± = dxwx , α±

Wα ± =

α±

dxwx

be the average over the cube α ± of the currents wx , where we have divided the integration by 3 which is approximately the volume to the cube α ± . By definition wx is an operator on the torus ε−1 . Since ax can be viewed as an operator on α + with periodic + boundary condition for x ∈ α , wα ± can be understood as an operator on the cube α + as well. We shall use the same symbol in both contexts. + Recall the cutoff version of the current wx,M = wx (aM , aM ) (3.13). Thus we can define the cutoff version of the current wM,α ± = −3 dxwM,x . (4.9) α±

Here wM,α ± can be viewed as an operator either on the torus ε−1 or α + with periodic boundary condition. 2 Recall that ax,M are bounded operators localized in a ball of radius eM centered at

x. Thus the support of ax,M is contained in τα −2√ for x ∈ α − = τα −4√ , as long √ 2 as > eM , which we shall assume from now on. Since χα (x) = 1 for x ∈ τα −2√ , following the proof of (4.7) we have the identity

I ∗ wM,α − I = wM,α − ;

(4.10)

here wM,α − is understood as an operator on the torus ε−1 on the right side and as an operator on α + on the left side. Define the notation − wM,x, = −3 dy wM,y , − x,

− √ where − x, = τx −4 = τx .

From (4.10), the boundedness of aM,x and simple counting of the number of terms, we have the following lemma. Lemma 4.1. For any state γ , and any smooth function J , we have − Tr γ J (ε·), wM = Tr γ Av Jε (x) I ∗ wM,x, I + M (ε, ), x

(4.11)

where the error term M (ε, ) vanishes in the sense given by (1.1) and Av = ε3 dx. x

ε−1

Applying this lemma to a smooth function J (εt, εx) and average over t, we have − Av Tr γt J (ε·, εt), wM = Av γt Jε (t, ·) , I ∗ wM,·, I + M (ε, ).

t≤T /ε

t≤T /ε

(4.12)

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5. Bounds on the Currents The aim of this section is to show that the currents with momentum cutoff, i.e., the quantities W M,α ± , can be bounded by a multiple of the Hamiltonian plus particle number. This is the content of the following lemma. Lemma 5.1. The following operator inequalities hold: W M,α + ≤ CM Hα + + Nα + .

(5.1)

Note that the dependence on M in the right hand side is linear. Similarly inequality holds if W M,α + on the left side is replaced by W M,α − . As it is essential for the proof of this lemma, we first recall a standard stability result based on the superstability conditions we have assumed on the potential W . Stated in words, the result says that a large class of two-body quantities can be bounded in terms of the two-body interaction and the particle number. We state this result as a lemma for functions but it obviously extends to the corresponding second quantized observables. Lemma 5.2. Suppose U is a positive bounded function with compact support on R3 and W is a superstable potential stated in the sense of (2.18) . Then there is a δ > 0 such that δ

N

U (xα − xβ ) ≤

α=β

N

W (xα − xβ ) + N .

α=β

The proof of Lemma 5.2 is contained in Ruelle’s book [18] or see [16]. Proof of Lemma 5.1. First, we treat the one-particle (i.e., quadratic in the ax# ’s) terms and show that they can be bounded by the kinetic energy term and chemical potential term of the Hamiltonian. We will use the following inequalities several times without further reference: for any pair of bounded operators A and B, and c > 0, one has A∗ B + B ∗ A ≤ cA∗ A + c−1 B ∗ B and A + B ≤ |A| + |B|. From the last inequality it follows that for a bounded family of self-adjoint operators Ax , and a real valued L1 -function f , one has f (x)Ax dx ≤ |f (x)||Ax |dx. Note that one cannot replace the LHS of these inequalities by their absolute values unless all terms commute. We start with bounding the momentum components of WM : i + + 0 Wk,M,α + = dx ∇k ax,M ax,M − ax,M ∇k ax,M 2 + α + + ≤ dx ∇k ax,M ∇k ax,M + ax,M ax,M . (5.2) α+

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We would like to obtain bounds by multiples the Hamiltonian and the number operator without momentum cut-off M. For this we use the following inequalities: + ax,M ≤ C|φM | ∗ ax+ ax , ax,M + ∇ax,M ∇ax,M + ∇ax,M ∇ax,M + ax,M ax,M

(5.3) (5.4)

≤

CM|∇φM | ∗ ax+ ax , |φM | ∗ ∇ax+ ∇ax ,

≤

CM|∇φM | ∗ ∇ax+ ∇ax .

(5.6)

≤

(5.5)

In the above expressions the convolutions are with respect to the variable x in the RHS. The four inequalities are proved in almost identical fashion. E.g., the first inequality is obtained as follows: + ax,M = dzdwφM (x − z)az+ φM (x − w)aw ax,M 1 + dz dw|φM (x − z)||φM (x − w)| [az+ az + aw ≤ aw ] 2 ≤ |φM | ∗ ax+ ax , where we have used that |φM | = 1. For (5.4) and (5.6) one also has to use ∇φM 1 ≤ CM, for a suitable constant C, but otherwise the proofs are the same. Now, we can finish 0 the bound of Wk,Mα , by using (5.3) and (5.5) in (5.2). We obtain 0 + + ≤ dx |φ |(x − y) ∇ a ∇ a + a a Wk,M,α + M k y k y y y α+

≤ C(H0,α + Nα ). By stability of the potential the kinetic energy term in this bound can be replaced by the full Hamiltonian, up to an adjustment to the constant C. This completes the proof of the 0 . lemma for Wk,Mα For the other one-particle terms of WM,α one proceeds in the same way. E.g., for the last inequality one starts from + + + + ax,M − ax,M ∇k ax,M ≤ M∇k ax,M ∇k ax,M + M −1 ax,M ax,M . −i ∇k ax,M The rest of the argument is the same. In summary, the results are i + + dx ∇k ax,M ax,M − ax,M ∇k ax,M ≤ C(Hα + + Nα + ) 2 α+ + dx ∇j ax,M ∇k ax,M ≤ C(Hα + + Nα + ) α+ i + + − dx ∇k ax,M ax,M − ax,M ∇k ax,M ≤ CM(Hα + + Nα + ). 4 α+ The two-particle terms (quartic in the ax# ’s) appearing in (3.6) and (3.8), can follow the same procedure. E.g., to bound the middle term of (3.8), we start from i + + + + ∇k ax,M ay,M ay,M ax,M − ax,M ay,M ay,M ∇k ax,M 4 + + + + ≤ Max,M ay,M ay,M ax,M + M −1 ∇ax,M ay,M ay,M ∇ax,M . (5.7)

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The first term can further be bounded in a way similar to (5.3): + + + ax,M ay,M ay,M ax,M ≤ dz|φM (z − y)|ax,M ay+ ay ax,M + = duφM (u − y)ay+ ax,M ax,M ay ≤ du dv|φm |(u − y)|φm |(v − x)ax+ ay+ ay ax . The same quantity appears in (3.6). In both cases, after integration, we get something of the form: M dx dy[ du dv|φM |(u − x)G(u − v)|φM |(v − y)]ax+ ay+ ay ax , α+

α+

where G is a non-negative function of compact support. To estimate this term, we use Lemma 5.2 to obtain a bound of the last expression of the form CM times the potential energy. The second term in the RHS of (5.7) gives rise to M −1 dx dyM[ du dv|∇φM |(u − x)G(u − v)|φM |(v − y)]ax+ ay+ ay ax α+

α+

and something of the same form for the third term in (3.8), which, with another application of Lemma 5.2, can also be bounded by the CM times the potential energy. This completes the proof of the lemma. 6. Local Ergodicity Recall that by assumption the solution up to time t ≤ T0 /ε of the Euler equations has density and energy taking values in a compact set strictly contained in the one phase region of the phase diagram of the fermion systems. Let σ κ be a smooth function supported in the one phase region such that σ κ = 1 on this compact set. Furthermore, we require that as κ → 0, σ κ becomes the characteristic function of a compact neighborhood of this set contained in the one phase region. Since the phase transition region depends only on the density and energy, σ κ needs to depend only on the density and energy. We will take σ κ (e, n) of the form σ1κ (e)σ2κ (ρ), where σ1κ and σ2κ are some smoothed versions of the characteristic functions on a set of sufficiently high e and sufficiently low ρ, respectively. The aim of this section is to prove the following theorem. Recall Xˆ is defined in (4.1). Theorem 6.1. For all smooth functions J , and X any one of the components of wM we have − Av γt Av J (εt, εx) I ∗ Xx, I

t≤T /ε

x

≤ Av Tr γt Av J (εt, εx) t≤T /ε

x

+ + − + × I ∗ σ˜ κ Xˆ σ˜ κ )(ux, )I +I ∗ (1 − σ κ (ux, ))Xx, (1 − σ κ (ux, ))I +J,κ,X (ε, , a), (6.1) √ where σ˜ κ = σ κ (2 − σ κ ).

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The function σ˜ κ behaves essentially the same way as σ κ , i.e., it is a smooth version of a characteristic function supported in the one-phase region. As a first step towards the proof of Theorem 6.1, we partition ε−1 into cubes of size aε−1 , where a is a sufficiently small positive constant. For any z ∈ ε−1 , let Q = z,aε−1 denote the cube of size aε −1 centered at z. For any bounded quasi-local observable Z, define the average of Z in the cube Q by ZQ = Av τy Z. y∈Q

We also divide the time interval [0, ε−1 T ], into disjoint intervals of size 2aε −1 and label the centers by t1 , · · · tn , n = T /(2a) (the nth interval is [tn − a, tn + a] ∩ [0, ε −1 T ]). Since J is a smooth function, −1

Jε (t, ·), Z = n

n j =1

Av J (εtj , εz) z

Av

|t−tj |≤a/ε

Tr γt Zz,aε−1

+ Z,J (a, ε), (6.2)

where lima→0 limε→0 Z,J (a, ε) = 0 and the average is over z ∈ aε−1 Z3 ∩ ε−1 . For Q, a, j fixed, define a family of states labelled by ε consisting of the states defined by γεQ,j (Z) = Q,j

Then {γε

Av

|t−tj |≤a/ε

Tr γt ZQ .

| ε > 0} is w∗ -precompact and, hence, has at least one limit point.

Lemma 6.2. Let ω be the Gibbs state on ε−1 defined in (2.9) with = ε−1 and the chemical potential λ := λ¯ is chosen to be λ¯ = Avx λ(0, εx), where λ(0, ·) are the parameters for the initial condition defined in (2.16). Then for any t ≥ 0 the relative entropy ¯ ≤C s(γt | ω) for some constant C depending only on the initial value λ(0, ·). Proof. Recall the initial state is ω0ε = Then we have ¯ s(ω0ε |ω)

=

dxω0ε

1 exp ε −3 λε (0, ·) , u . cε (0)

¯

λε (0, ·) , u − λ , u + ε 3 log cε (0) − ε 3 log Zε−1 .

(6.3)

(6.4)

¯ ≤ C. From a simple direct Since each term on the right side is bounded, we have s(ω0ε |ω) calculation, we know that s(γtε |ω) ¯ is a constant of motion. This proves the lemma.

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Lemma 6.3. Fix the parameter a and let η be any limit point of {γε | ε > 0}. Then η is a translation invariant, time invariant state of the dynamics. Furthermore, the specific relative entropy of η with respect to the translation invariant state ωλ , satisfies the bound s(η|ωλ ) ≤ Cλ a −3 . Proof. The invariance under space and time translations is an immediate consequence of the scaling by ε−1 . Since the proof for the quantum case is parallel to that of the classical case, we refer the reader to [16] for a proof of the classical case. To show that the specific relative entropy with respect to ωλ is finite, we start form Lemma 6.2 stating that the relative entropy ε 3 S(γtε | ωλε ) ≤ C for a suitable constant C. The operations of averaging over translations in a cube Q and over times in an interval [ti −a/ε, ti +a/ε], are completely positive, therefore, by the monotonicity (or convexity) of the relative entropy (see, e.g., [15]), we have ε3 S(

Av γtε ◦ τy | ωλε ) ≤ C.

Av

|t−tj ≤a/ε y∈Q

The relative entropy is also monotone with respect to restriction to the algebra of observables of a subvolume. Therefore we have ε d ε ε S( Av Av γt ◦ τy Q ωλ Q ) ≤ C. |t−tj ≤a/ε y∈Q Q,j

Now, η is a limiting point of {γε

γεQ,j =

| ε > 0}, where Av

Av γtε ◦ τy

|t−tj |≤a/ε y∈Q

Q

.

Therefore, by the lower semicontinuity of the specific relative entropy, we can conclude 1 Q,j ε S(γ | ω s(η | ωλ ) = lim ) ε λ Q ε→0 (2aε −1 )3 ≤ (2a)−3 lim sup ε 3 S(γεQ,j | ωλε ) ≤ C(2a)−3 . ε

Q,j

Consider any limiting point η of {γε | ε > 0}. Since η is translation invariant, we can decompose it into ergodic components (with respect to space translations) and there is a probability measure µ supported on ergodic states ω such that η = ω µ(dω). The key property of η is the following lemma. Lemma 6.4. Let η be as above, and X ∈ A0 . Then there is M (, κ) such that

∗ − η(I X I ) − η I ∗ σ˜ κ Xˆ σ˜ κ )(u+ )I + I ∗ (1 − σ κ (u+ ))X − (1 − σ κ (u+ ))I ≤ M (, κ), √ where σ˜ κ = σ κ (2 − σ κ ).

(6.5)

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Proof of Theorem 6.1 assuming Lemma 6.4. Let

κ + − ∗ − ∗ κ ∗ κ + κ + ˆ Z = I X I − I σ˜ X σ˜ )(u )I + I (1 − σ (u ))X (1 − σ (u ))I . It is crucial that Z is a bounded and local observable. Theorem 6.1 now follows immediately from (6.2) and Lemma 6.4. The rest of this section is devoted to prove Lemma 6.4. We shall drop the labels ± on X and u, etc. for the rest of this section. 6.1. General properties of limiting states. We now prove a number of results for the ergodic components of the limit points η. At this point η depends on a macroscopic space point z, and a macroscopic time tj , and in principle also on the subsequence, but we will eventually see that η is in fact independent of the subsequence. Lemma 6.5. Let γn , γ be normal states on a von Neuman algebra A, and γn → γ weakly. Suppose that A is a non-negative self-adjoint operator affiliated with A, such that γn (A) is bounded by a constant M, uniformly in n. Then, limn→∞ γn (A) exists and satisfies γ (A) ≤ lim γn (A). n→∞

Proof. Let A = λdEλ be the spectral resolution of A. As A is affiliated with A, the k projections Pk = 0 dEλ belong to A, and γn (A) = supk γ (Ak ), where Ak = APk . The supremum is finite by the assumptions. Therefore, lim γn (A) = lim sup γn (Ak ) ≥ sup lim γn (Ak ) = γ (A). n

n

k

k

n

Recall that h and n (we omit the superscripts +) are the average of the local conservative quantities H and N (4.8). Let e and ρ be determined by e = lim ω(I ∗ h I ), →∞

and ρ = lim ω(I ∗ n I ). →∞

(6.6)

Where necessary, we will indicate the dependence on ω by e(ω), and ρ(ω). In the following lemma we prove the existence and finiteness of these limits when the parameter a is fixed. Lemma 6.6. For µ−almost all states ω, the limits e and ρ of (6.6) are finite. For the state η = ω µ(dω), we have the following bounds: lim sup η(I ∗ h I ) ≤ Ca −3 e0 , →∞

Proof. We have

(6.7)

→∞ Q,j

Av Av γε

all boxes Q

lim sup η(I ∗ n I ) ≤ Ca −3 ρ0 .

(h ) = e0 , where e0 is the initial total energy. This is a

direct consequence of the fact that the energy is conserved by the dynamics. Therefore, for each box Q, we have Q,j

Av γε Q

(h ) ≤ Ca −3 e0 .

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By Lemma 6.5 with A = I ∗ h I , it follows that η(I ∗ h I ) ≤ lim Av γε

Q,j

ε→0 Q

(h ) ≤

Q

Q,j

Av γε Q

(h ) ≤ Ca −3 e0 ,

which implies the bound for the energy (6.7). The proof for the particle density is the same. As η = ω µ(dω), it then also follows that e(ω) and ρ(ω) are finite for µ− almost all ω. observable in the local algebra A0 ⊂ AR3 . E.g., A = Let A be a bounded dx dyf (x, y)ax+ ay , where f (x, y) = 0 unless x, y ∈ 0 . For concreteness, we assume that 0 contains the origin. We will also use the notation 1 Av(A) = dx τx (A). || Lemma 6.7. Suppose lim η = 0. For every translation invariant ergodic state ω on AR3 , any bounded local observable A and any continuous function f , we have the limit lim ω(I∗ f (Av A)I ) = f (ω(A)).

(6.8)

l

Proof. The proof rests on the following property of I : for A ∈ A0 , we have I ∗ τx (A)I = τx (A),

if τx (0 ) ⊂ − .

(6.9)

− Denote by int = {x ∈ | τx (0 ) ⊂ }. Note that

| \ int diam 0 | =: δ ≤ 2η + l | |

(6.10)

and that lim δ = 0. First, consider the function f (x) = x. Then, using the property (6.9), we have 1 1 ∗ ∗ ω(I Av(A)I ) = ω( τx (A)) + ω(I τx (A) I ). | | int | | \int Without loss of generality we may assume ω(A) = 0. Using the definition of δ (6.10), and the isometry property of I , we find ω(I ∗ Av(A)I ) ≤ (1 − δ ) 1 dx ω(τ (A)) + δ A. x int |int |

The two terms in the RHS tend to zero, the first due to the ergodicity of ω, the second because δ → 0. Next, we prove by induction the result for f (x) = x n , for all n ≥ 1. Suppose we have the result for f (x) = x n−1 , i.e., lim ω(I ∗ (Av A)n−1 I ) = 0.

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Then, by the same arguments as above, we have the estimate ∗ n n n lim ω(I (Av(A)) I ) ≤ (1 − δ ) ω((Av (A)) ) + C(1 − (1 − δ )n ) int and the result follows by the ergodicity of ω. For arbitrary continuous functions f , (6.8) can now be obtained by approximating f by polynomials, uniformly on [−A, A]. This proves the lemma. Lemma 6.7 can trivially be extended as follows: Corollary 6.8. For any bounded local observables X, Y, A ∈ A0 , and continuous functions f and g, we have that

∗ ∗ lim ω(I Xf (Av A)Y g(Av A)I ) − f (ω(A))g(ω(A))ω(I XY I ) = 0.

6.2. Extension to unbounded conserved quantities. Lemmas 6.7 and Corollary 6.8 are general properties of ergodic states applied to bounded observables. We now show how the one-phase region cut-off functions, which depend on unbounded but conserved quantities, can be included. This is a difficult step and we will have to use the special forms of the conserved quantities. The key technical estimate is contained in Lemma 6.10. We remark that a naive application of Schwarz’ inequality to prove Lemma 6.9 would produce expressions with six or more creation or annihilation operators about which we have no control. Q,j

Lemma 6.9. Let η be any limiting point of {γε | ε > 0}, let X be one of the components of w, and let X the averaged version of X. Then the following limits vanish: (6.11) lim η I ∗ B X [σ1κ (h )σ2κ (n ) − σ1κ (e)σ2κ (ρ)]I = 0, →∞ ∗ (6.12) lim η I B X [σ2κ (n ) − σ2κ (ρ)]I = 0, →∞

for B = 1 or B = σ1κ (h )σ2κ (n ).

Here e = e(η) = e(ω) dµ(ω), and similarly for ρ. In particular, we have ! " lim η I ∗ σ κ (h , n )X σ κ (h , n ) − σ κ (e, ρ)X σ κ (e, ρ) I = 0, →∞

(6.13)

and the same result holds if σ κ (h , n )X σ κ (h , n ) is replaced by σ κ (h , n )X or by X σ κ (h , n ). Proof. We start with the case B = 1. Recall, σ κ (e, ρ) = σ1 (e)σ2 ρ). There exist bounded functions σ˜ 1κ and σ˜ 2κ such that κ σi (x) − σiκ (y) = (x − y)σ˜ iκ (x, y), for i = 1, 2. Using these functions we can write σ1κ (h )σ2κ (n ) − σ1κ (e)σ2κ (ρ) = (σ1κ (h ) − σ1κ (e))σ2κ (n ) + σ1κ (e)(σ2κ (n ) − σ2κ (ρ)) = σ2κ (n )σ˜ 1κ (h , e)(h − e) + σ1κ (e)σ˜ 2κ (n , ρ)(n − ρ).

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Therefore, for a suitable bounded function f , for any ergodic state ω, we can write ω(I ∗ σ κ (h , n )X σ κ (h , n )) = ω(I ∗ Xσ2κ (N )f (h , e)(h − e)I ) " ! = ω I Xσ2κ (n )f (h )[hB − eB ]I ! " +ω I ∗ Xσ2κ (n )f (h )(h − hB )I ,

(6.14)

where hB =

1 HB | |

with HB = H Ind(H ≤ B| |), so that HB ≤ B| |, and HB ↑ H as B → ∞. Introduce eB (ω) = lim

→∞

1 ω(I ∗ HB I ) | |

and use Schwarz’ inequality to obtain ! " ω I ∗ Xσ2κ (n )f (h ) hB − e I ! " ∗ κ ≤ ω I ∗ Xσ2κ (n )f (h ) hB − eB I + (eB (ω) − e(ω)) ω(I σ2 (n )f (h )I ) 2 " " ! ! I ≤ δω I ∗ X (σ2κ (n ))2 f (n )2 X ∗ I + δ −1 ω I ∗ hB − eB +(eB (ω) − e(ω)) ω(I ∗ σ2κ (n )f (h )I ). Now, we integrate over ω with respect to the measure µ, and take absolute values. The first term is uniformly bounded in . As hB is bounded, the integrand of the second term vanishes for each ω, in the limit → ∞, by Lemma 6.7. As the integrand is bounded uniformly in ω, the integral vanishes as well. For the third term we use the argument of (6.6) to show that it vanishes in the limit B → ∞: Q,j −3 lim sup η(hB − h ) lim ε d |γε,t (H B − H )|. ≤ lim sup lim Av γε (hB − h ) ≤ Ca B

B

ε

Q

ε

The RHS is independent of t and ε, and vanishes as B → ∞. For the second term of (6.14), we first apply Schwarz’ inequaltity: ! " ω I ∗ Xσ2κ (n )f (h ) h − hB I ! " ! " κ −1 ∗ B ≤ δω I ∗ Xσ2κ (n )f (h ) hB − h )σ (n )XI + δ ω I − h f (h h I . 2 As before, the last term vanishes in the limit B → ∞. Since (h − hB ) ≤ h , the first term is bounded by ! " ω I ∗ Xσ2κ (n )f (h )h f (h )σ2κ (n )XI . As f is bounded, we have that f (h )h f (h ) ≤ Ch . Therefore, after integration over ω, and with the use of Lemma 6.10, we obtain the bound δ(Cη(I ∗ X(hl + n )I ) + C),

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which can be shown to be bounded in terms of the corresponding expectation in γt,ε , as before. In conclusion, as B and δ are arbitrary, we have proved (6.11) for B = 1. It is straight forward to adapt the argument to prove also (6.12) and the case B = σ1κ (h )σ2κ (n ).

6.3. Basic estimate. In the previous proof the following lemma was used. It provides a bound on the Hamiltonian sandwiched by bounded operators. Lemma 6.10. For µ-almost all translation invariant ergodic states ω, and X the averaged version of one of the components of wM (which are all self-adjoint), we have ω I ∗ X σ2κ (n )h σ2κ (n )X I ≤ Cω I ∗ [h + n ]I + C, (6.15) where the constant is independent of ε, but may depend on a, M. Proof. Since X is particle number preserving, X commutes with n . Therefore, we can rewrite the quantity we need to estimate as ω I ∗ X σ2κ (n )h σ2κ (n )X I ) = ω I ∗ σ2κ (n )X h X σ2κ (n )I . h is the sum of two terms, a kinetic energy and a potential energy term, which we will treat separately. First, we consider the kinetic energy term: ∇ax+ ∇ax , defined with periodic boundary conditions. We start from the identity X ∇ax+ ∇ax X = ∇ax+ X X ∇ax + ∇ax+ X [∇ax , X ] + [X , ∇ax+ ]∇ax X . (6.16) + Note that X is a linear combination of linear and quadratic terms in au,M av,M (see + + + (2.6,3.5-3.8)). Therefore, commutators of the form [ax,M ay,M , ax ], [ax,M ay,M , ∇ax+ ], etc., are bounded operators. More precisely, there is a constant CM . such that

[X , ax+ ] ≤ CM −3 ,

and [X, ∇ax+ ] ≤ CM −3 .

(6.17)

These bounds will be used repeatedly in the following estimates. E.g., applied to the first term of (6.16), they yield ∇ax+ XX∇ax ≤ CM ∇ax+ ∇ax . To bound the second and third term we first apply Schwarz’ inequality: ! " ω I ∗ σ2κ (n )[X , ∇ax+ ]∇ax X σ2κ (n )I ! " ≤ δω I ∗ σ2κ (n )[X , ∇ax+ ][X , ∇ax+ ]∗ σ2κ (n )I ! " +δ −1 ω I ∗ σ2κ (n )X ∇ax+ ∇ax X σ2κ (n )I . The first term of the RHS is bounded and the last term can be re-absorbed into the quantity we started out to estimate. Thus, for the kinetic energy term and any of the X , we have an estimate of the form " ! " ! ω I ∗ σ2κ (n )X h0, X σ2κ (n )I ≤ Cω I ∗ σ2κ (n )h0, σ2κ (n )I + C.

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Similarly, for the potential energy we start from the identity X ax+ ay+ ay ax X = ax+ ay+ X X ay ax + ax+ ay+ X [ay ax , X ] + [X , ax+ ay+ ]ay ax X and the bound ax+ ay+ X X ay ax ≤ Cax+ ay+ ay ax . For the commutator terms we have ax+ ay+ X [ay ax , X ] = ax+ ay+ Xay [ax , X ] − ax+ ay+ X[ay , X ]ax , which can be estimated using Schwarz’ inequality: ! " 2 Re ω I ∗ σ2κ (n )ax+ ay+ X ay [ax , X ]σ2κ (n )I ! " ≤ ω I ∗ σ2κ (n )ax+ ay+ X2 ay ax σ2κ (n )I " ! +ω I ∗ σ2κ (n )[ax , X ]ay+ ay [ax , X ]σ2κ (n )I . We use ax+ ay+ X2 ay ax ≤ CM ax+ ay+ ay ax for the first term. For the second term we use the identity [ax , X]∗ ay+ ay [ax , X] = ay+ [ax , X]∗ [ax , X]ay + ay+ [ax , X]∗ [ay , [ax , X]] +[[ax , X]∗ , ay+ ]ay [ax , X]. The first term of the RHS is bounded by CM ay+ ay . The other two terms can be bounded by CM ay+ ay + CM by repeating the same procedure once more (first apply Schwarz’ inequality, then use (6.17)). We conclude that ! " dx dyW (x − y)ω I ∗ σ2κ (n )X ax+ ay+ ay ax X σ2κ (n )I ! " ≤C dx dy|W |(x − y) ω I ∗ σ2κ (n )ax+ ay+ ay ax σ2κ (n )I ! " ∗ κ + ω I σ2 (n )ay+ ay σ2κ (n )I + C . Now from the super-stability estimate, we have + + dx dy|W |(x − y)ax ay ay ax ≤ C dx C

Thus,

dyW (x − y)ax+ ay+ ay ax + N .

! " dx dyW (x − y) dyω I ∗ σ2κ (n )X ax+ ay+ ay ax X σ2κ (n )I ! " C−3 dx dyW (x − y) dyω I ∗ σ2κ (n )ax+ ay+ ay ax σ2κ (n )I ! " +C dx dyW (x − y) dyω I ∗ σ2κ (n )n σ2κ (n )I .

−3

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The last term is bounded. Combining these estimates, we have ! " ! " ω I ∗ σ2κ (n )X h X σ2κ (n )I ≤ Cω I ∗ σ2κ (n )h σ2κ (n )I + C. Since h ≤ h + Cn , h + Cn ≥ 0 and [h , n ] = 0, we have ! " ! " ω I ∗ σ2κ (n )h σ2κ (n )I ≤ ω I ∗ [h + Cn ]1/2 σ2κ (n )σ2κ (n )[h + Cn ]1/2 I ! " ≤ Cω I ∗ [h + n ]I + C. ! " We can prove that ω I ∗ [h + n ]I is bounded by using Lemma 6.5.

6.4. Proof of main ergodic lemma. We can now prove Lemma 6.4. Proof. Recall the decomposition of η into its spatially ergodic components: η = µ(dω)ω. Since X is bounded, by Lemma 6.9 there is κ,X () such that # # ω(I ∗ (1 − σ κ )X (1 − σ κ )I ) + ω(I ∗ σ κ (2 − σ κ )Xˆ σ κ (2 − σ κ )I ) ˆ ) + κ,X (), = (1 − σ κ (ω))2 ω(I ∗ X I ) + σ κ (ω)(2 − σ κ (ω))ω(I ∗ XI where σ κ (ω) = σ κ (lim ω(h ), lim ω(n )). Therefore, # # η(I ∗ X I ) − η(I ∗ (1 − σ κ )X (1 − σ κ )) − η(I ∗ σ κ (2 − σ κ )Xˆ σ κ (2 − σ κ )I ) = µ(dω)[1 − (1 − σ κ (ω))2 − σ κ (ω)(2 − σ κ (ω))]ω(I ∗ X I ) − µ(dω)σ κ (ω)(2 − σ κ (ω))ω{I ∗ (Xˆ − X )I } + κ, (X). As 1 − (1 − x)2 − x(2 − x) = 0, the first term vanishes identically. The middle term vanishes by the hypothesis that the only ergodic states of finite specific relative entropy in the one-phase region are the Gibbs states. The support of the function σ κ (ω)(2 − σ κ )(ω)) is such that only these Gibbs states contribute to the integral. The integrand vanishes ˆ by the definition of Xˆ (4.1), since we have ω(X) = X(lim ω(u )). This concludes Lemma 6.4. 7. Relative Entropy Estimate We now summarize the estimates on the relative entropy we have so far. For any 0 ≤ T ≤ T0 , we write s(γt | ωtε )t=ε−1 T = ε−1 T

Av

0≤t≤ε−1 T

d s(γt | ωtε ). dt

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We compute the rate of change of entropy by (3.4), (3.10) and (3.16) to have + 1 Av , aM ) − ε 2 ∂t log cε (t) + EM , s(γt | ωtε )t=ε−1 T = T Tr γt G(λε , aM 0≤t≤ε−1 T

1 ≤ Ce−cM (3.16). where G is defined in (3.10) and EM Recall the meaning of the various length scales and cut-off parameters: ε is the ratio of the macroscopic to microscopic length scale, M is the high-momentum cut-off, is the length scale in the isometry I employed to define commuting local versions of the conserved quantities, a is a length scale for averaging needed to make use of local ergodicity, κ is the length scale used to smooth the characteristic function of the one-phase region, and δ is a small parameter used in applications of the entropy inequality. Recall the convention 2

A•B =

3 3

µ

µ

Aj B j −

j =0 µ=0

3

A4j Bj4 .

j =0

+ We now apply Theorem 6.1 to estimate Tr γt G(λε , aM , aM ) by + Av T r γt G(λε , aM , aM ) ≤ T1 + T2 ,

t≤T /ε

where T1 and T2 are defined as follows: T1 = − T2 = −

Av

0≤t≤ε−1 T

Av

0≤t≤ε−1 T

! " ˆ M,x σ˜ κ I , γt Av ∇λ(εt, εx) • I ∗ σ˜ κ w x ! " γt Av ∇λ(εt, εx) • I ∗ (1 − σ κ )(ux, )wM,x (1 − σ κ )(ux, ) I . x

ˆ which we state as the following lemma. We need to compute w Lemma 7.1. We have the following identities: µ

µ

wˆ j = Aj ,

j = 0, . . . , 3, µ = 0, · · · , 4,

µ

where the functions Aj are given in (2.15). These relations follow directly from the definition of the Gibbs states, the expressions µ µ for Aj in (2.15), the calculation of the currents wj in Sect. 9 and the virial theorem proved in Sect. 10. By construction of the ωtε , the time derivative of log cε (t) can be expressed as d 2 ε log cε (t) = ε3 dx(∂t λ · q)(εt, εx), dt where q is the solution of the Euler equations that we are considering. Recall the following identity about the Euler equations: 3 j =1

Aj (q(X) ) · ∇j λ(q(X) ) dX = 0.

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Recall also A0 (q) = q. We can rewrite d 2 3 3 dx(∂t λ · q)(εt, εx) = ε dx(∇λ · A)(εt, εx). ε log cε (t) = ε dt Together with Lemma 7.1, we have T1 −

d 2 ε log cε (t) dt

=−

Av

0≤t≤ε−1 T

ˆ ˆ M,x σ˜ κ (ux, )I − w(q(εt, γt Av ∇λ(εt, εx) • I ∗ σ˜ κ w εx)) . x

Denote ∇λ∞ = ∇λ∞ + ∂t λ∞ and introduce the functions 1 ˆ M σ κ )(u) − w(q) ˆ M (λ, u) = ∇λ• (σ κ w , 2 (λ, u) = ∇λ∞ (1 − σ κ )(u) h + n (1 − σ κ )(u) , where h, n are the energy and density components to u and q is the dual variable of λ defined in (2.11). We can bound wM in T2 by the cutoff Lemma 5.1. Thus we have T1 + T 2 −

! " d 2 1 ε log cε (t) ≤ Av γt Av I ∗ {−M + M 2 }(λ(εt, εx), ux, )I . x dt 0≤t≤ε−1 T

Therefore, we have s(γt | ωtε )t=ε−1 T ≤ T

Av

0≤t≤ε−1 T

" ! 1 1 γt Av I ∗ {−M + M 2 }(λ(εt, εx), ux, )I + EM , x

(7.1) 1 ≤ Ce−cM (3.16). where EM 2

7.1. Reduction to large deviation. Recall the standard thermodynamics pressure is defined by ψ(λ) = lim −3 log Z,λ . →∞

Define the entropy s(q ) = sup[λq − ψ(λ)] λ

and the rate function (notice we also use I for the embedding into the standard torus ε−1 ) I (q , λ) = s(q ) + ψ(λ) − λ · q .

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The rate function has the following property: I (q , λ ) ≥ 0,

I (q, λ ) = 0,

where q = ∂ψ(λ)/∂λ. Furthermore, if the Gibbs state with chemical potential λ is in the one phase region, we have Hess I (q, λ ) ≥ c1l for some c > 0. The main large deviation estimate we shall use is given in the following lemma. This lemma will be proved in Sect. 8.3. Lemma 7.2. Suppose λ is a bounded smooth function so the Gibbs state with chemical potential λ(x) is in the one phase region for all x. For any bound smooth function G that satisfies the condition |G(λ, q)| ≤ C(e + ρ),

(7.2)

where e is the energy and ρ is the density. Then there is a δ0 > 0 depending only on C and a convex functional I˜ such that for all 0 < δ ≤ δ0 , I˜(q , λ) = I (q , λ) in a small neighborhood of q = ∂ψ(λ)/∂λ and " ! + lim lim γ Av I ∗ G(λ(εx), ux, )I →∞ ε→0

≤

x

dX sup G(λ(X)), q (X) ) − δ −1 I˜(q (X), λ(X) ) + δ −1 lim s(γ | ωλε ). q (X)

ε→0

Here the sup is over all functions q (X). 7.2. Conclusion of the relative entropy estimate and proof of the main theorem. We now apply Lemma 7.2 to estimate (3.17). Since we need the bound (7.2), 1 + M 2 }. Thus we have for any δ ≤ δ , we set G = M −1 {−M 0 " ! + 1 − Av γt Av I ∗ {−M + M 2 }(λ(εt, εx), ux, )I ≤ R6 + δ −1 M s(γt | ωtε ), 0≤t≤ε−1 T

where R6 =

x

dx sup q

1 {−M + M 2 }(λ(x)), q (x) ) − δ −1 M I˜(q (x), λ(x) ) ,

(7.3)

where I˜ is related to the rate function defined in Lemma 7.2. We now estimate the 1 on M. dependence of M Lemma 7.3. There is a constant c > 0 such that 1 M (λ, q ) = 1 (λ, q ) + e−cM , 2

where

ˆ κ )(q ) − w(q) ˆ 1 (λ, q ) = ∇λ• (σ κ wσ .

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This lemma can be proved following the idea of the proofs of Lemmas 3.2 and 3.3. It is part of our assumptions that the Gibbs states satisfy the cutoff assumptions. Now, we can conclude the relative entropy estimate and the proof of Theorem 2.1. Proof of Theorem 2.1. Recall q = ∂ψ(λ)/∂λ. Clearly, 1 (q(εt, εx), λ(εt, εx)) = 0. The first derivative ∂ 1 (λ(εt, εx), q(εt, εx) ) =0 ∂q(εt, εx) is equivalent to the Euler equation as checked in [16]. Recall 2 (λ(X) , q (X) ) is nonzero only when q (X) is away from q(X). Thus we have for |q | ≤ C, {− 1 + M 2 }(λ(X) , q (X) ) ≤ CM(q (X) − q(X))2 . Furthermore, from the definition of j we have for all q , {− 1 + M 2 }(λ(X), q (X) ) ≤ CM |q |(X) + 1 . Since I˜(q (X) , λ(X) ) ≥ 0 and I˜(q (X), λ(X) ) = 0 only when q (X) = q(X), for δ small enough we have 2 sup {− 1 + M 2 }(q (X) , λ(X) ) − δ −1 M I˜(q (X) , λ(X) ) ≤ e−cM . q

We thus have s(γt |ωtε ) ≤ δ −1 Mε

t 0

s(γt |ωtε ) + Ce−cM + M (, κ) + (, a) + CM e−cl 2

3

dt .

By integrating this inequality (i.e., using Gronwall’s inequality), and using the fact that t ≤ T0 , we arrive at the bound −1 2 3 s(γt |ωtε ) ≤ δ −1 MT0 eδ MT0 Ce−cM + M (, κ) + (, a) + CM e−cl . Taking the limits limκ→0 lima→0 lim→∞ limε→0 , we get the inequality lim s(γt |ωtε ) ≤ Cδ −1 MT0 eδ

→0

−1 MT −cM 2 0

.

We can now let M → ∞ and conclude the proof of Theorem 2.1. We emphasize that we need the error term stemming from the high-momentum cutoff to be smaller than e−CM for any C > 0 in order to have our results hold for t ≤ CT0 for arbitrary T0 . This is guaranteed by the Maxwellian bound in the cutoff assumption II.1, expressed by (2.19). 8. Thermodynamics and Large Deviation We now prepare the way for the proof of Lemma 7.2. Our approach to large deviations for quantum Gibbs states and local Gibbs states is quite different from the explicit analysis in [11] for the ideal gases. We first introduce the following local Gibbs state with independent subcubes.

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8.1. Local Gibbs state with independent subcubes. Divide the torus = ε−1 into unions of non-overlapping cubes of size . To fix the grid, we assume that the origin is the center of one small cube. Denote a typical cube by α. Recall the configuration space S(α + ) and define the configuration space S((+) ) = ⊗α∈Z3 ∩ S(α + ). An element in this configuration space can be denoted by x = (· · · , (αj , xj ), · · · ),

xj ∈ αj+ .

The (Fock) function space ((+) ) is the L2 space of antisymmetric functions on S((+) ). Notice that S((+) ) = S(+ ) and ((+) ) = (+ ). Recall = ε−1 is a torus. Define I from () to ((+) ) (cf. [4]) by (I ψ)(x ) =

χαj (xj ) ψ(x) .

j

The crucial fact is that I is an isometry. Lemma 8.1. I is an isometric embedding, i.e., φ = Iφ. Proof. Recall from the construction of χ the relation (4.2) implies that χα2 (x) = 1.

(8.1)

α∈Z3

We can prove the isometry by the following identity: For any two wave functions f and g, we have 2 (If, Ig) = dxj |χαj (xj )| f¯(x)g(x) = (f, g). α

αj

j

For isometric embeddings, we have the following useful bound. Lemma 8.2. Suppose I : H1 → H2 is an isometric embedding. Then ∗ Tr H1 eI AI ≤ Tr H2 eA .

Proof. We can assume that H1 is just a subspace of H2 and I is the natural embedding. Let φj be the orthonormal eigenvectors of A in H1 . Then the claim follows from the following Peierls’ inequality: Suppose φj are orthonormal. Then

e(φj ,Aφj ) ≤ Tr eA .

j

The following lemma shows that I ∗ XI = I ∗ XI for a suitable class of observables.

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Proposition 8.3. Suppose X is the observable X= dx1 · · · dxk dy1 · · · dyk f (x1 , . . . , xk ; y1 , . . . , yk )ax+1 ,α · · · ax+k ,α ayk ,α · · · ay1 ,α . α+

Then we have I ∗ XI = I ∗ XI. Proof. The following identity is a direct consequence of the definition of I: For n ≥ k, ayk ,α · · · ay1 ,α Iψ((z1 , α1 ), . . . , (zn , αn )) = ayk ,α · · · ay1 ,α χα1 (z1 ) · · · χαn (zn )ψ(z1 , . . . , zn ) = δα1 ,α · · · δαk ,α χα (y1 ) · · · χα (yk )χαk+1 (zk+1 ) · · · χαn (zn ) ×ψ(y1 , . . . , yk , zk+1 , . . . , zn ) . It follows that, for any φ, ψ ∈ L2 (×n ), n ≥ k, (φ, I ∗ XIψ) = dx1 · · · dxk dy1 · · · dyk f (x1 , . . . , xk ; y1 , . . . , yk ) α+

=

×(axk ,α · · · ax1 ,α Iφ, ayk ,α · · · ay1 ,α Iψ) dzk+1 · · · dzn dx1 · · · dxk dy1 · · · dyk δα1 ,α · · · δαk ,α

α1 ,... ,αn

×χαk+1 (zk+1 )2 · · · χαn (zn )2 χα (x1 ) · · · χα (xk )χα (y1 ) · · · χα (yk ) ×f (x1 , . . . , xk ; y1 , . . . , yk )φ(x1 , . . . , xk , zk+1 , . . . , zn ) × ψ(y1 , . . . , yk , zk+1 , . . . , zn ). The sum over α1 , . . . , αn can be carried out using the Kronecker deltas and (8.1). As n ≥ k, φ, and ψ are arbitrary, we have I ∗ XI = I ∗ XI by the formula of I in (4.4). We now construct a “special local Gibbs state”. Recall that uα + is defined in (4.8) and its component commutes. For a smooth λ, let ω˜ λε, be the state 1 ω˜ λε, = Tr exp ε−3 Av λ(εα) · I ∗ uα + I , α c˜ε, (λ) where the average of α is over α ∈ Z3 ∩ and c˜ε, (λ) is the partition function defined by c˜ε, (λ) = Tr exp ε−3 Av λ(εα) · I ∗ uα + I . α

(+) (ε−1 ).

Here the trace is over Assume for the moment we can drop the I and the small cubes are independent. Then c˜ε, (λ) can be computed easily. The following lemma asserts that this is essentially correct. Recall the partition function defined in (2.10). Lemma 8.4.

" ! lim lim ε 3 log cε (λ) − Av −3 log Z (λ(εα)) α →∞ ε→0 " ! 3 = lim lim ε log c˜ε, (λ) − Av −3 log Z (λ(εα)) = 0. →∞ ε→0

α

(8.2)

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Proof. Upper bound: We first state an upperbound to ε 3 log c˜ε, . Notice that it is an inequality with no limits or other constants, ε3 log c˜ε, (λ) ≤ Av −3 log Z (λ(εα)). α

(8.3)

From Lemma 8.3, we have log c˜ε, (λ) = log Tr exp ε−3 I ∗ Av λ(εα) · uα + I . α

If we can neglect I ∗ and I, then (8.3) follows from the fact that different cubes are considered independent. Lemma 8.2 shows that we can remove I ∗ and I to have an upper bound. This concludes the proof of (8.3). Lower bound: Since λ is fixed, we shall drop it in the subscript. Consider the entropy 0 ≤ s ωε | ω˜ ε, = R2 + ε 3 log c˜ε, − ε 3 log cε , where R2 = ωε ε3

dxλ(εx) · ux − Av λ(εα) · I ∗ uα + I . α

From Lemma 4.1, we have lim lim R2 = 0.

→∞ ε→0

We have thus proved that lim lim ε3 log c˜ε, − ε 3 log cε ≥ 0.

→∞ ε→0

(8.4)

To conclude Lemma 8.4, we now obtain a lower bound on cε . This is the standard procedure on the thermodynamics and we shall √ give only a sketch. We first √ divide the cube of size ε−1 into cubes of size (1 − ) with corridors of size 2 . Now we impose the Dirichlet boundary conditions on the boundary to obtain au upper bound on the kinetic energy. The partition function is bounded below by restricting the configurations so that there is no particle on the corridors. Now there are no interactions between different cubes and we obtain a lower bound of cε in terms of √ average over Dirichlet boundary conditioned partition functions in cubes of size (1 − ). Since we can take η → 0 after → ∞ and partition functions independent of boundary conditions, we have thus proved that lim lim ε 3 log cε ≥ Av −3 log Z (λ(εα)).

→∞ ε→0

This concludes the lemma.

α

8.2. Large deviation for commuting variables. Recall that uα + is defined in (4.8) and its components commute. We shall take x = 0 and denote uα + by u . The following lemma

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is a standard application of large deviation theory (or thermodynamics) to commuting variables. Lemma 8.5. Suppose λ is a fixed constant so the Gibbs state with chemical potential λ is in the one phase region. For any bound smooth function, G satisfies that |G(λ, q)| ≤ C(e + ρ), where e is the energy and n is the density. Let ωλ, be the finite volume Gibbs state defined by ωλ, (X) =

1 Tr exp λ, u X Z (λ)

with periodic boundary condition. Since the components of u are commuting, we have " ! 1 Tr exp λ, u + δ3 G(λ, u ) = ωλ, exp δ3 G(λ, u ) . Z (λ) Then there is a δ0 > 0 depending only on C and a convex functional I˜ such that for all 0 ≤ δ ≤ δ0 , I˜(q , λ) = I (q , λ) in a small neighborhood of q = ∂ψ(λ)/∂λ and ! " | |−1 log ωλ, exp δ3 G(λ, u ) ≤ sup δG(λ, q ) − I˜(q , λ ) . q

Here the sup is over all constants q . We first sketch the idea of the proof for Lemma 8.5: The rate function I can be understood in the following way: The probability to find the u with a given value q is given by exp[−| |s(q )] with the entropy given by s(q ) = sup[λ · q − ψ(λ)]. We now write Tr

1 Z (λ)

λ

exp λ, u as dq exp | | λ · q − ψ(λ) − s(q ) .

This gives the last variational formula. Proof of Lemma 8.5. We shall drop the constant parameter λ in G in this proof. Since the components of u commute, we can define the joint distribution ν (du) of u w.r.t. the state ωλ, . Thus ! " ! " ωλ, exp δ3 G(λ, u ) = dµ (u) exp δ3 G(λ, u ) . We now approximate the integral by the summation so that dµ (u) exp δ3 G(λ, u) ≤ Pλ, [|u − εm| ≤ ε] exp δ3 G(εm) , m∈Zd

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where Gε (y) = sup G(x), |x−y|≤ε

and Pλ, denotes the probability of the event described in its argument, with respect to the state ωλ, . We can bound the summation by ε −5 dx Pλ, [|u − x| ≤ ε] exp δ3 Gε (x) . We have Pλ, |u − x| ≤ ε ≤ Pλ, ξ · u ≥ ξ · x − |ξ |ε for all ξ . Notice that from the Chebeshev inequality we have 3 3 3 Pλ, ξ · u ≥ ξ · x − |ξ |ε ≤ e− ξ ·x+ |ξ |ε dµ (u) e ξ ·u . Let ψ (λ) be the pressure defined by ψ (λ) = −3 log Tr so that

3 ξ ·u

dµ (u) e

3 λ·u

e

= exp 3 ψ (ξ + λ) − ψ (λ) .

Thus

Pλ, ξ · u ≥ ξ · x − |ξ |ε ≤ exp − 3 ξ · x − ψ (ξ + λ) − ε|ξ | + ψ (λ)

for all λ4 + ξ4 > 0. In particular, we have Pλ, |u − x| ≤ ε ≤ exp − 3

sup

ξ · x − ψ (ξ + λ) − ε|ξ | + ψ (λ) .

−η−1 ≤ξj +λj ≤η−1 , j =0···3 η≤ξ4 +λ4 ≤η−1

The existence of thermodynamics states that lim ψ (λ) − ψ(λ)| = 0 →∞

uniformly in a compact interval away from λ4 = 0. Fix a small constant η > 0. Define ξ · x − ψ(ξ ) . sup s˜η (x) = −η−1 ≤ξj ≤η−1 , j =0···3 η≤ξ4 ≤η−1

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We have

sup −η−1 ≤ξj +λj ≤η−1 , j =0···3 η≤ξ4 +λ4 ≤η−1

ξ · x − ψ (ξ + λ) − ε|ξ |

≤ s˜η (x) − λ · x − C(η, λ)ε + C , where lim C = 0.

→∞

Define I˜η (x) = s˜η (x) − λ · x + ψ(λ). Thus we have Pλ, |u − x| ≤ ε ≤ exp − 3 I˜η (x) − C − C(η, λ)ε . We now have the estimate −3 log dµ (u) exp δ3 G(λ, u) ≤ −3 log ε−5 dx Pλ, |u − x| ≤ ε exp δ3 Gε (x) −3 ≤ log dx exp −3 I˜η (x) − δGε (x) + C + C(η, λ)ε + −3 | log ε|. The error vanishes in the limit limε→0 lim→∞ . The integration can be calculated using the Laplace method to give lim

→∞

−3

log

dx exp −3 I˜η (x) − δGε (x) ≤ sup δGε (λ, q ) − I˜η (q , λ ) . q

Clearly, we have lim sup δGε (λ, q ) − I˜η (q , λ ) = sup δG(λ, q ) − I˜η (q , λ ) .

ε→0 q

q

We now collect some properties for s˜η and I˜η . Notice that, by definition, s˜η and I˜η are still convex. Furthermore, we can check that if η is small then I˜η (x) = I (x) in a small neighborhood of q. This proves the lemma.

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8.3. Proof of Lemma 7.2. Recall α indices disjoint subcubes of width . By the entropy inequality (3.17), we have " ! − Av γε Av I ∗ G(λ(εα), uα + )I ≤ R + δ −1 M s(γε | ω˜ tε, ), 0≤t≤ε−1 T

α

where, for any δ > 0, R = εd δ −1 log Tr

1 exp ε−3 Av I ∗ λ(εα) · uα + − δG(λ(εα), uα + ) I . α c˜ε, (λ) (8.5)

We first control the last term s(γε | ω˜ tε, ) by writing it as s(γε | ω˜ tε, ) = s(γε | ωtε ) + ε 3 log c˜ε (λ) − ε 3 log c˜ε, (λ) + R4 , where R4 = γε ε3

dxλ(εx) · ux − Av λ(εα) · I ∗ uα + I . α

From the Lemma 4.1, we have lim lim R4 = 0.

→∞ ε→0

We now estimate R. Using the argument in the proof of Lemma 8.3, we can drop the operator I in (8.5) to have an upper bound. Since the cubes indexed by α are independent, we have ! " R ≤ δ −1 Av Qα − ε 3 log c˜ε, (λ) − Av −3 log Z (λ(εα)) , α

α

where −3

Qα =

log Tr Z (λ(εα))

−1

3

exp

λ(εα) · uα + − δG(λ(εα), uα + )

.

! " The last term ε 3 log c˜ε, (λ) − Av −3 log Z (λ(εα)) vanishes by Lemma 8.4. Notice α that the components of uα + commute. Thus the trace is over a functional of commuting operators and we are essentially the same as in the classical theory. Thus we can apply Lemma 8.5 to estimate Qα . Summarizing, we have " ! lim lim γ Av I ∗ G(λ(εα), uα + )I α →∞ ε→0 ≤ dX sup G(λ(X)), q (X) ) − δ −1 I˜(q (X), λ(X) ) + δ −1 lim s(γ | ωλε ) + R4 . q (X)

ε→0

Notice that the right side of the inequality is independent of the location of the grid. If we average the grid over the cube of size , we can replace the left side of the inequality from averaging over α to averaging over all points x on the torus. This proves Lemma 7.2. We now state a corollary to Lemma 7.2.

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Corollary 8.6. Suppose λ is a bounded smooth function so the Gibbs state with chemical potential λ(X) is in the one phase region for all X. Suppose γε is a sequence of states such that the specific entropy s(γε | ωλε ) satisfies lim s(γε | ωλε ) = 0.

ε→0

For any bound smooth function J on the unit torus, we have lim γε ε 3 dx J (εx) · u(εx) = dX J (X) · q(X). ε→0

Proof. Since J is bounded smooth, from Lemma 4.1 we have 3 + dx J (εx) · u(εx) − Av J (εx) · ux, = 0. lim lim γ ε x

→∞ ε→0

We now apply Lemma 7.2 to have + − q(εx) lim lim γε Av J (εx) · ux, x

→∞ ε→0

≤

dX sup J (X) · q (X) − q(X) − δ −1 I˜(q (X), λ(X)) + δ −1 lim s(γε |ωλ ). q (X)

ε→0

Since I˜ ≥ 0 and I˜(q (X), λ(X)) = 0 only when q (X) = q(X), the sup is bounded by Cδ. To see this, consider the model problem sup x − δ −1 x 2 ≤ δ. x

Recall the assumption limε→0 s(γε |ωλ ) = 0. Since we can choose δ arbitrarily small, we prove the corollary. 9. Calculation of the Currents µ

The current density operators wk,x are implicitly defined by (3.9), i.e., for any test functions J = (J µ ), µ = 0, . . . , 4, they should satisfy iδH (

dx J, ux ) −

3

terms containing second and higher deriva ∇k J, wk,x ) = tives of the J µ integrated with densities of bounded expectation. k=1

If we apply the same sign convention for dot products with w as the convention adopted µ in (2.8) for u, this means we are looking for the definition of wk,x , such that, for any test function J , the following formal identity holds: i

dxJ (x)[H, uµ x] =

3

µ

dx∇k J (x)wk,x + integrals with higher derivatives of J ,

k=1

(9.1) where H is the formal Hamiltonian H =

dxhx , with hx as defined in (2.6).

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In order to compute the commutators we use the canonical anticommutation relations (2.4) and integration by parts. The commutation relations involving derivatives such as ∇k ax , etc., are most easily derived by taking derivatives of the appropriate commutation relations without derivatives. E.g., the identity au+ av , ax+ ay = δ(x − v)au+ ay − δ(y − u)ax+ av follows directly from (2.4) and, by taking derivatives with respect to u, also leads to ∇k au+ av , ax+ ay = δ(x − v)∇k au+ ay + δk (y − u)ax+ av , where δk is the derivative of the delta distribution with respect to the k th component. It is straightforward to derive all other necessary relations in the same way. E.g., [∇k ay+ ∇k ay , ax+ ax ] = −δk (x − y)∇k ay+ ax + δk (x − y)∇k ay ax+ .

(9.2)

There are essentially three cases to consider: i) µ = 0, ii) µ = 1, 2, 3, and iii) µ = 4. 0 : As n commutes with the potential part of the Hamili) µ = 0: calculation of wk,x x tonian we only have to consider the kinetic energy term, which can be computed using (9.2). After integrating by parts, we get 1 1 + + i dxdy J (x)[ ∇ay ∇ay , ax ax ] = dx ∇J (x) i[∇k ax+ ax − ax+ ∇ax ]. 2 2 0 as given By comparing this result and (9.1) we find agreement with the definition of wk,x in (3.5). Note that, in this case, no higher order derivatives of J appear. j ii) µ = j = 1, 2, 3: calculation of wk,x . Now, both the kinetic energy term and potential energy term yield non-trivial contributions. First, we compute the kinetic energy term, 1 1 j dxdyJ (x)[∇j ax+ ax , ∇ay+ ∇ay ] + h.c., i dxdyJ (x)[ ∇ay+ ∇ay , px ] = 2 4

where here and in the following h.c. stands for the adjoint of the preceding term(s). By using the commutation relations and integration by parts we find the following expression for this quantity: i

=− =

1 ∇k ay+ ∇k ay ] 2 3

j

dxdyJ (x)[px ,

k=1

3 dxJ (x) ∇j ax+ ax + ax ∇j ax+ − ∇k J (x)ax+ ax + h.c.

3 k=1

+2

k=1

dxJ (x) ∇k ∇j ax+ ∇k ax + ∇k ax+ ∇k ∇j ax

3 k=1

dx∇k J (x)∇j ax+ ∇k ax −

dx∇j J (x)ax+ ax + h.c.

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After further integration by parts and reorganization the result can be written as i =

1 j ∇k ay+ ∇k ay , px ] 2 3

dxdyJ (x)[ 3 k=1

k=1

dx∇k J (x)

1 ∇j ax+ ∇k ax + ∇k ax+ ∇j ax 2

1 + ∇k ∇j J (x) ∇k ax+ ax + ax+ ∇k ax . 4 The first term of the RHS in this expression determines the first term (3.6). Note that j this time higher derivative terms appear that are not included in the definition of wk,x , but they contribute to the error terms. To calculate the contribution from the potential energy term in the Hamiltonian, we start from the identity ∇j au+ au , ax+ ay+ ay ax = δ(x − u)∇j au+ ay+ ay ax + δ(y − u)ax+ ∇j au+ ay ax +∇j δ(y − u)ax+ ay+ au ax + ∇j δ(x − u)ax+ ay+ ay au , which leads to 1 dxdyW (x − y) duJ (u) [∇j au+ au − au+ ∇j au ], ax+ ay+ ay ax 2 = dxdyW (x − y) J (x)[∇j ax+ ay+ ay ax + h.c.] + J (y)[ax+ ∇j ay+ ay ax + h.c.] +[∇j J (x) + ∇j J (y)]ax+ ay+ ay ax = − dxdy J (x)∇j,x W (x − y) + J (y)∇j,y W (x − y) ax+ ay+ ay ax . (9.3) Due to the spherical symmetry of the potential we have ∇j W (x − y) = W (x − y)

(x − y)j . |x − y|

(9.4)

Using this identity we can write (9.3) in the form (x − y)j + + − (J (x) − J (y))W (x − y) a a ay ax . |x − y| x y As the range of W is finite by assumption, we can Taylor expand J (x) − J (y) to rewrite this quantity in the following form: −

3 k=1

dxdy∇k J (x)W (x − y)

(x − y)k (x − y)j + + a x a y ay ax |x − y|

+higher order derivatives of J .

(9.5)

Recall that, by definition, only the coefficients of the first order derivatives of J are included in the w tensor. Therefore, combining (9.2) and (9.5) and also including the j appropriate factors 1/2 and - signs, we find the expression for wk,x claimed in (3.6).

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4 . The calculation of the energy current proceeds in iii) µ = 4: calculation of wk,x the same way as the previous cases, but there are more terms and terms with higher derivatives. The contribution from the kinetic energy in the Hamiltonian to the kinetic energy current is, up to a trivial constant, given by

i

dudxJ (u)[∇au+ ∇au , ∇ax+ ∇ax ] 3

=i

dudxJ (u)δk,l (u − x) ∇k au+ ∇l ax − ∇l ax+ ∇k au

k,l=1

=i

3

dx∇k J (x) ∇k ax+ ax − ax+ ∇k ax ,

k=1

where δk,l is shorthand for ∇k ∇l δ. This yields the first term of the energy current. The potential energy term in the Hamiltonian does not contribute to the potential energy portion of the energy current due to the fact that the following commutators vanish: au+ av+ av au , ax+ ay+ ay ax = 0. To calculate its contribution to the kinetic energy current we start from

J (u) ∇au+ ∇au , ax+ ay+ ay ax

=

3

duJ (u) δk (u − x)∇k au+ ay+ ay ax + δk (u − y)ax+ ∇k au+ ay ax

k=1

−δk (u − y)ax+ ay+ ∇k au ax − δk (u − x))ax+ ay+ ay ∇k au

= −∇x (J (x)∇ax+ )ay+ ay ax − ax+ ∇y (J (y)∇ay+ )ay ax +ax+ ay+ ∇y (J (y)∇ay )ax + ax+ ay+ ay ∇x (J (x)∇ax ). By multiplying this expression by W (x−y), and integrating over x and y, and integrating by parts, we find

=

dxdyduW (x − y)J (u) ∇au+ ∇au , ax+ ay+ ay ax 3

J (x)∇k,x W (x − y) ∇k ax+ ay+ ay ax − ax+ ay+ ay ∇k ax

k=1

+

3 k=1

J (y)∇k,y W (x − y) ax+ ∇k ay+ ay ax − ax+ ay+ ∇k ay ax .

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The contribution of the kinetic energy to the potential energy current is obtained in a similar way. The result is dxdyduW (x − y)J (x) ax+ ay+ ay ax , ∇au+ ∇au =−

3

∇k,x (J (x)W (x − y)) ∇k ax+ ay+ ay ax − ax+ ay+ ay ∇k ax

k=1

−

3

J (x)∇k,y W (x − y) ax+ ∇k ay+ ay ax − ax+ ay+ ∇k ay ax .

k=1

The last four terms become the last two terms of the energy current: i dxJ (x)[hx , H ] i 4 3

=

− +

∇k J (x) ∇k ax+ ax − ax+ ∇k ax

k=1 3

i 4 i 4

k=1 3

(∇k J )(x)W (x − y) ∇k ax+ ay+ ay ax − ax+ ay+ ay ∇k ax (J (y) − J (x))∇k,y W (x − y) ax+ ∇k ay+ ay ax − ax+ ay+ ∇k ay ax .

k=1

By the same argument as for (9.5), the last term can be rewritten in the form 3 (x − y) ⊗ (x − y) +i ∇k J (x) W (x − y) |x − y| k=1 × ax+ ∇ay+ ay ax − ax+ ay+ ∇ay ax + O(J ). 10. The Virial Theorem The purpose of this section is to relate the expectation values of the RHS of the dynamical equations (3.9) in a Gibbs state with specified values of the densities of the conserved quantities (local equilibrium), to these quantities themselves in order to obtain a closed set of equations. To achieve this we will make use of canonical transformations relating Gibbs states with respect to reference frames with different velocities. This will allow us to use reflection symmetry of Gibbs states at zero total momentum. A second element we will need is the Virial Theorem to relate the so-called virial to the thermodynamic pressure. We start with the latter. For the convenience of the reader we first recall the main definitions. Consider a system of particles in a finite volume ⊂ Rd , interacting via a pair potential W . The pressure at inverse temperature β and chemical potential µ, P (β, µ), is defined by P (β, µ) = lim

→Rd

1 log Tr e−β(H0, +V −µN ) , β||

(10.1)

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where

533

1 dx ∇ax+ ∇ax , 2 1 V = dxdy W (x − y)ax+ ay+ ay ax , 2 N = dx ax+ ax .

H0, =

The trace is taken over the Fermion Fock space with one-particle space L2 (). For our purposes, we can simply consider to be a cube of side L, and define the operators with periodic boundary conditions. We will write V (W ) when we wish to indicate the pair potential function explicitly. By our general assumptions, the limit (10.1) exists and we will restrict ourselves to the one-phase region of the phase diagram. In particular we assume that the pressure is continuously differentiable. Gibbs states at non-vanishing total momentum are defined by introducing an additional Lagrange multiplier for the momentum as follows: 1 Tr Xe−β(H0, +V −α·P −µN ) , →Rd Z(λ)

ωλ (X) = lim

(10.2)

where λ = (β, α, µ), α = (α1 , α2 , α3 ), are constants, and P is the total momentum operator in the volume defined by i dx ∇ax+ ax − ax+ ∇ax , P = 2 and Z(λ) = Tr e−β(H0, +V −αP −µN ) is the partition function. The kinetic energy density is defined by ekin (β, α, µ) = lim

→Rd

1 ωβ,π,µ (H0, ). ||

The limits → R3 exist and are independent of the boundary conditions under general stability assumptions [18]. We will use the abreviations ωβ,µ = ωβ,0,µ and ekin (β, µ) = ekin (β, 0, µ). The virial of the potential W in the volume is denoted by V (W ) and is defined by 1 dxdy ∇W (x − y) · (x − y)ax+ ay+ ay ax , (10.3) V (W ) = 2 and the density of the local density virial is given by 1 dy ∇W (x − y) · (x − y)ax+ ay+ ay ax . νx = 2 R3 As we have assumed that W has compact support, νx is well-defined.

(10.4)

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Due to Galileo invariance, the Gibbs states for different values of α are related by a canonical transformation, which is why in the statistical mechanics of global equilibrium situations the total momentum is usually assumed to vanish. The canonical transformations relating ωβ,µ and the states ωβ,α,µ , are defined as follows. Let s ∈ Rd , and consider the unitary Us on L2 (Rd , dx) defined by (Us ψ(x)) = eis·x ψ(x). The second quantization of Us implements an automorphism γs on the Fermion algebra given by γs (a(f )) = a(Us f ),

γs (a + (f )) = a + (Us f ).

One can easily verify that the action of γs on the operator-valued distributions ax , ∇ax , and their adjoints, is given by: γs (ax ) = e−is·x ax , γs (∇ax ) = e−is·x ∇ax − ise−is·x ax ,

γs (ax+ ) = eis·x ax+ , γs (∇ax+ ) = eis·x ∇ax+ + iseis·x ax+ .

Clearly, γs−1 = γ−s . With these relations it is easy to check that γs (∇ax+ ∇ax ) = ∇ax+ ∇ax + |s|2 ax+ ax + is · (ax+ ∇ax − ∇x+ ax ). Hence, the kinetic energy transforms as follows: 1 γs (H,0 ) = H,0 + |s|2 N − s · P . 2 In the same way we see that

(10.5)

γs (N ) = N , γs (P ) = P − sN , γs (V ) = V , γs (V (W )) = V (W )).

(10.6) (10.7) (10.8) (10.9)

It follows that 1

γs (e−β(H −µN ) ) = e−β(H −s·P −(µ− 2 |s|

2N )

.

By putting s = α, replacing µ by µ − 21 |α|2 , we obtain 1

e−β(H −α·P −µN ) = γα (e−β(H −(µ+ 2 |α|

2 )N )

).

As the trace is invariant under γα , this implies 1

Z(λ) = Tr e−β(H −α·P −µN ) = Tr e−β(H −(µ+ 2 |α|

2 )N )

˜ = Z(λ),

(10.10)

where, for λ = (β, α, µ), we define λ˜ = (β, 0, µ + 21 |α|2 ). Using this relation between partition functions and the invariance of the trace under canonical transformations, we immediately get 1 Tr γα (X)e−β(H −α·P −µN ) Z(λ) 1 1 2 Tr Xe−β(H −(µ+ 2 |α| )N ) = ωλ˜ (X). = ˜ Z(λ)

ωλ (γα (X)) =

(10.11)

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In combination with (10.6) this implies ωλ (N ) = ωλ˜ (N ), ˜ and we will simply write ρ. Also and hence ρ(λ) = ρ(λ), ωλ˜ (P ) = αωλ (N ). If we apply this to X = H,0 and combine this with (10.5), to relate the kinetic energy densities of ωλ and ωλ˜ , we find 1 1 ekin (β, α, µ) = ekin (β, µ + |α|2 ) + |α|2 ρ, 2 2

(10.12)

where we have also used ωλ˜ (P ) = 0. The relation (10.10) between partition functions immediately implies the following property of the pressure: 1 P (β, α, µ) = P (β, 0, µ + |α|2 ), 2

(10.13)

One interpretation of this relation is that the chemical potentials at different values of α, when regarded as a function of the particle density ρ, satisfy 1 µα (ρ) = µ0 (ρ) − |α|2 . 2 We can now prove the virial theorem in the form we need. Theorem 10.1 (Virial Theorem). For a three-dimensional translation invariant system with a continuously differentiable pressure function, one has

1 1 2 2 ekin (β, α, µ) − |α| ρ − lim ωβ,α,µ (V (W )) = dP (β, α, µ). d 2 || →R The quantity between square brackets can be considered as the gauge invariant kinetic energy. Proof. Suppose that the theorem holds for α = 0. We can then use a canonical transformation to obtain the result for arbitrary α as a consequence of (10.11): ωλ (V(W )) = ωλ (γα−1 (V(W ))) = ωλ˜ (V(W )) 1 1 = 2ekin (β, µ + α 2 ) − dP (β, µ + |α|2 ). 2 2 By using (10.12) and (10.13), this is equivalent to the statement of the theorem. We now prove the theorem for α = 0. As the pressure is independent of the boundary conditions, we can use periodic boundary conditions to compute it, i.e., we choose to be a d-dimensional torus. For t > 0, let t be the torus rescaled by t. Then |t| = t d ||, and Ut : L2 (t, dx) → L2 (, dx) : (Ut ψ)(x) = t d/2 ψ(tx) is unitary. The Laplacians on and t are related as follows: Ut t Ut∗ = t −2 .

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This relation carries over to the kinetic energy in second quantization: Ut H0,t Ut∗ = t −2 H0, , where we have used the same notation for the corresponding unitary on the Fock space F(L2 ()) with one-particle space L2 (). Similarly, one easily finds that the scaling behavior of the potential energy terms in the Hamiltonians is as follows: Ut Vt (W )Ut∗ = V ((W (t·)). and for the particle number we have Ut Nt Ut∗ = N . By using these unitary equivalences we obtain 1 log Tr F (L2 (t)) e−β(H0,t +Vt (W )−µNt ) →Rd |t| 1 −2 log Tr F (L2 ()) e−β(t H0, +V (W (t·))−µN ) . = lim d d →R t ||

P (β, µ) = lim

This shows that the last expression is independent of t. Setting its derivative in t = 1 equal to zero yields the following equation 2ekin (β, µ) − lim

→Rd

1 ωβ,µ (V (W )) − dP (β, µ) = 0. ||

In order to close the dynamical equations, we need to express the expectation values of j the currents wk , given in (3.5-3.8), in the states ωλ in terms of the expectations of the conserved quantities uj of (2.6). j

Proposition 10.2. The expectations of the local currents wk in a Gibbs state ωλ are given by 0 ) = ωλ (ukx ), ωλ (wk,x j

ωλ (wk,x ) = αj αk ωλ (u0x ) + δk,j P (λ), 4 ωλ (wk,x ) = αk ωλ (u4x ) + αk P (λ). j

where P (λ) is the pressure defined in (10.1) and ux are the local densities of the five conserved quantities defined in (2.6). With the definitions of (2.15) and (4.1), this is ˆ Explicitly: q 0 = ρ, and equivalent to A = w. 0 ) = αk ρ = q k , ωλ (wk,x j

ωλ (wk,x ) = αj αk ρ + δj,k P = q j q k /q 0 + δj,k P , 4 ωλ (wk,x ) = αk (q 4 + P ) = q k (q 4 + P )/q 0 .

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Proof. The first equation, j = 0, follows directly from (3.5), (10.7), and (10.11). The j expressions for wk,x , j = 1, 2, 3, contain the virial of the potential W , which we can relate to the thermodynamic pressure by using the virial theorem, Theorem 10.1: (x − y)j (x − y)k + + 1 j a x a y ay ax . wk,x = ∇j ax+ ∇k ax − W (x − y) 2 |x − y| For j = k, the expectation of the second term vanishes as it changes sign under rotation over π about the j th axis, which is a symmetry of the potential and the Gibbs states. Due to the rotation invariance of the potential, we also have W (x)x/|x| = ∇W (x). Therefore, the expectation of the second term in a Gibbs state ωλ is given by 1 − δj,k ωλ (νx ). 3 j

4 , we will transform To treat the first term of wk,x , as well as the first two terms of wk,x these terms to a frame where the Gibbs state has zero total moment, so that we can more easily use invarance under reflections in space. E.g., from (10.11) we get

ωλ (∇j ax+ ∇k ax ) = ωλ˜ (γα (∇j ax+ ∇k ax ))

= ωλ˜ (∇j ax+ ∇k ax ) + αj αk ωλ˜ (ax+ ax ) + iωλ˜ (αj ax+ ∇k ax − αk ∇j ax+ ax ).

As the total momentum has zero expectation in ωλ˜ , the last term vanishes for all j, k = 1, 2, 3. By reflection symmetry and the definition of the kinetic energy we have ωλ˜ (∇j ax+ ∇k ax ) =

2 1 ekin (β, µ + |α|2 ). 3 2

By combining the above relations we obtain

1 1 j ωλ (wk,x ) = αj αk ωλ (u0x ) + δj,k 2ekin (β, µ + |α|2 ) − ωλ (νx ) 3 2 1 = αj αk ωλ (u0x ) + P (β, α, µ), β where, for the last equality, we have used the virial theorem and (10.13). 4 ), we need to consider the following expecTo compute the energy current, ωλ (wk,x tations: iωλ (∇k ax+ ay+ ay ax − ax+ ay+ ay ∇k ax ), iωλ (ax+ ∇j ay+ ay ax − ax+ ay+ ∇j ay ax ), iωλ (∇k ax+ ax − ax+ ∇k ax ). Again, we use (10.11) to relate these expectations to expectations in ωλ˜ . The first expectation becomes: iωλ˜ (∇k ax+ ay+ ay ax − ax+ ay+ ay ∇k ax ) + 2αk ωλ˜ (ax+ ay+ ay ax ).

(10.14)

The first term of this expression vanishes by symmetry. In the same way we find iωλ (ax+ ∇j ay+ ay ax − ax+ ay+ ∇j ay ax ) = 2αj ωλ˜ (ax+ ay+ ay ax ).

(10.15)

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We treat the third expression with similar arguments: iωλ (∇k ax+ ax − ax+ ∇k ax ) = iωλ˜ (γα ((∇k ax+ ax − ax+ ∇k ax )) = iωλ˜ (∇k ax+ ax + 2i∇k ax+ (α · ∇)ax − |α|2 ∇k ax+ ax −iαk ax+ ax + 2αk ax+ (α · ∇)ax + iαk |α|2 ax+ ax ) + complex conjugate = −αk ωλ˜ (2|α|2 ax+ ax + 4∇k ax+ ∇k ax − 2ax+ ax ). Then, by using integration by parts and reflection symmetry we get the following expression: iωλ (∇k ax+ ax − ax+ ∇k ax ) = −4αk

1 2 5 ˜ . |α| ωλ˜ (u0x ) + ekin (λ) 2 3

(10.16)

Recall the expression for the energy current: i ∇k ax+ ax − ax+ ∇k ax 4 i + dyW (x − y) ∇k ax+ ay+ ay ax − ax+ ay+ ay ∇k ax 4 (x − y)k (x − y)j + i − W (x − y) ax ∇j ay+ ay ax − ax+ ay+ ∇j ay ax . 4 |x − y|

4 wk,x (t) = −

Using (10.16), we see that the expectation of the first term in ωλ equals αk

5 1 2 0 ˜ ekin (λ) + |α| ωλ (ux ) . 3 2

For the middle term we use (10.15) and find 1 αk ωλ ( 2

dyW (x − y)ax+ ay+ ay ax ).

Similarly, for the last term we get

1 − αk ω λ ( 2

(x − y)k (x − y)j 1 dy W (x − y) ax+ ay+ ay ax ) = − αk ωλ (νx ) |x − y| 3

2 ˜ = αk P (λ) − ekin (λ) , 3

where we have used the definition of νx (10.4) and the virial theorem (Theorem 10.1). By combining the three terms and applying the relation (10.12) one obtains the 4 ) given in the statement of this proposition. expression for ωλ (wk,x

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539

11. Appendix. The Entropy Inequality Our arguments rely in a crucial way on the following entropy inequality (3.17): For any pair of density matrices γ and ω, and for all self-adjoint h, and any δ > 0, one has γ (h) ≤ δ −1 log Tr eδh+log ω + δ −1 S(γ |ω).

(11.1)

The inequality holds in the more general context of normal faithful states on a von Neumann algebra [17]. Here we give a proof for density matrices that emphasizes the connection with the variational principle of statistical mechanics. Proof. Let h be self-adjoint, and β > 0. The variational principle of statistical mechanics [18] states that 1 − Tr e−βH = inf Tr γ H − β −1 S(γ ) , γ β where the infimum is taken over density matrices γ , and S(γ ) := −Tr γ log γ , is the von Neumann entropy of γ . For any non-singular density matrix ω, define H = −(β −1 (h + log ω), and take β = δ, use S(γ |ω) = Tr γ (log γ − log ω) = −Tr γ log(ω) − S(γ ), and rearrange the resulting inequality to obtain (11.1).

Equality in (11.1) holds if and only if γ =

eh+log ω . Tr eh+log ω

The inequality (11.1) can also be turned around: S(γ |ω) ≤ γ (h) − log Tr eh+log ω ,

(11.2)

and one can then take the sup over h to obtain a characterization of the relative entropy (as was done [17]): S(γ |ω) = sup γ (h) − log Tr eh+log ω (11.3) h

with equality iff ω = e−h /Tr e−h , i.e., iff h = log Dω + constant × 1l. In contrast to the classical case, if log ω and h do not commute, we generally have log Tr eh+log ω = log Tr ωeh . However, due to the Golden-Thompson inequality, i.e., for any pair of self-adjoint A and B, T reA+B ≤ T reA eB , we still have T rγ h − log Tr ωeh ≤ S(γ |ω). Whenever ω and γ do not commute, the equality will be strict for all h.

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References 1. Boldrighini, C., Dobrushin, R. L., Suhov, Yu. M.: One-dimensional hard rod caricatures of hydrodynamics. J. Stat. Phys. 31, 577–616 (1983) 2. Born, M., FRS, Green, H. S.: A general kinetic theory of liquids. IV. Quantum mechanics of fluids Proc. Roy. Soc. A 191 (1947) 168. Reprinted in M. Born, FRS, H.S Green,: A general kinetic theory of liquids. Cambridge: Cambridge University Press, 1949 3. Bogolubov, N. N., Logunov, A. A., Todorov, I. T.: Introduction to Axiomatic Quantum Fields Theory. Reading, Massachusetts: W.A. Benjamin, Inc., 1975 4. Conlon, J. G., Lieb, E. H.,Yau, H. T.: The Coulomb gas at low temperature and low density. Commun. Math. Phys. 125, 153–218 (1989) 5. Eyink, G. L., Spohn, H.: Space-time invariant states of the ideal gas with finite number, energy, and entropy density. In: On Dobrushin’s Way. From Probability Theory to Statistical Physics. Amer. Math. Soc. Transl. Ser. 2, 198, Providence, RI: Am. Math. Soc., 2000, pp. 71–89 6. Gallavotti, G., Lebowitz, J.L., Mastropietro, V.: Large deviations in rarefied quantum gases. condmat/0107295 7. Gurevich, B. M., Suhov, Y.M.: Stationary solutions of the Bogolyubov hierarchy equations in classical statistical mechanics IV. Commun. Math. Phys. 84, 333–376 (1984) 8. Irving, J.H., Zwanzig, R.W.: The statistical mechanics theory of transport processes. V. Quantum hydrodynamics. J. Chem. Phys. 19, 1173–1180 (1951) 9. Kadanoff, L.P., Baym, G.: Quantum Statistical Mechanics. New York: W.A. Benjamin, 1962 10. Klainerman, S., Majda, A.: Compressible and incompressible fluids. Comm. Pure Appl. Math 35, 629–651 (1982) 11. Lebowitz, J. L., Lenci, M., Spohn, H.: Large deviations for ideal quantum systems. In: Probabilistic Techniques in Equilibrium and Nonequilibrium Statistical Physics. J. Math. Phys. 41, 1224–1243 (2000) 12. Morrey, C. B.: On the derivation of the equations of hydrodynamics from statistical mechanics. Commun. Pure Appl. Math. 8, 279–290 (1955) 13. Nachtergaele, B., Verbeure, A.: Groups of canonical transformations and the virial-Noether theorem. J. Geom. Phys. 3, 315–325 (1986) 14. Nachtergaele B.,Yau, H.-T.: Derivation of the Euler equations from many-body quantum mechanics. In: Li, Tatsien (ed.), Proceedings of the International Congress of Mathematicians, Vol. III, Beijing: Higher Education Press, 2002, pp. 467–476 15. Ohya, M., Petz, D.: Quantum Entropy and its Use. Berlin-Heidelberg-New York: Springer Verlag, 1993 16. Olla, S., Varadhan, S.R.S., Yau, H.-T.: Hydrodynamical limit for a Hamiltonian system with weak noise. Commun. Math. Phys. 155, 523–560 (1993) 17. Petz, D.: A variational expression for the relative entropy. Commun. Math. Phys. 114, 345–349 (1988) 18. Ruelle, D.: Statistical Mechanics. Reading, Massachusetts: W.A. Benjamin, 1969 19. Sinai, Ya. G.: Dynamics of local equilibrium Gibbs distributions and Euler equations. The onedimensional case. Selecta Math. Sov. 7, 279–289 (1988) 20. Spohn, H.: Large Scale Dynamics of Interacting Particles. New York: Springer-Verlag, 1991 21. Sen, R. N., Sewell, G. L.: Fibre bundles in quantum physics. J. Math. Phys. 43, 1323–1339 (2002), archived as mp arc 01-439 22. Yau, H.-T.: Relative entropy and the hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 22, 63–80 (1991) Communicated by J.L. Lebowitz

Commun. Math. Phys. 243, 541–555 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0984-4

Communications in

Mathematical Physics

Higgs Fields, Bundle Gerbes and String Structures Michael K. Murray, Daniel Stevenson Department of Pure Mathematics, University of Adelaide, Adelaide, SA 5005, Australia. E-mail: [email protected]; [email protected] Received: 11 July 2001 / Accepted: 22 September 2003 Published online: 11 November 2003 – © Springer-Verlag 2003

Abstract: We use bundle gerbes and their connections and curvings to obtain an explicit formula for a de Rham representative of the string class of a loop group bundle. This is related to earlier work on calorons. 1. Introduction In this paper we bring together calorons (monopoles for the loop group), bundle gerbes and string structures to produce a formula for the string class of a principal bundle with structure group the loop group. If K is a compact Lie group and L(K) is the group of smooth maps γ from [0, 2π] to K such that γ (0) = γ (2π ), it is well known [18] that there is a central extension → L(K) → 0, 0 → U (1) → L(K) is the Kac-Moody group. If P → M is a principal bundle with strucwhere L(K) ture group L(K) it has a characteristic class, the string class, in H 3 (M, Z) which is bundle. The string class was first introduced by the obstruction to lifting P to a L(K) Killingback in [12] in the case that K = Spin(n) and later considered in [13] and [3], where it was shown that there is a universal string class in H 3 (BL Spin(n), Z), where BL Spin(n) is the classifying space of the loop group of the spin group. In [16] Murray introduced the lifting bundle gerbe. This is a bundle gerbe whose ˆ Dixmier-Douady class is the obstruction to a principal G bundle lifting to a principal G bundle when ˆ →G→0 0 → C× → G is a central extension. In the case that G = L(K) the Dixmier-Douady class of the lifting bundle gerbe is the string class. We use the methods of [16] to calculate an explicit

Both authors acknowledge the support of the Australian Research Council.

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formula for the Dixmier-Douady class of the lifting bundle gerbe when G is the loop group and hence provide an explicit differential three form representative for the de Rham image of the string class in real cohomology. This three form is defined in terms of a connection and Higgs field for the loop group bundle. In [8] it was shown that there is a bijective correspondence between principal bundles over a manifold M with structure group L(K) and K bundles over S 1 × M. This was used to set up a correspondence between periodic instantons, or calorons, and loop group valued monopoles. In particular a connection for the K bundle corresponded to a connection and Higgs field for the L(K) bundle. We apply this correspondence to show that the string class of an L(K) bundle on M is the integral over the circle of the Pontrjagin class of the corresponding K bundle over S 1 × M. Finally we relate these results to earlier work [12, 4] on string structures. Recall that if Q is a K bundle over a manifold X we can take loops everywhere and form a loop group bundle P = L(Q) over M = L(X). In that case it was known from work of Killingback [12] that the Pontrjagin class of the bundle Q in H 4 (X, Z) transgressed to define the string class in H 3 (L(X), Z). The transgression consists of pulling back by the evaluation map ev : S 1 × L(X) → X and pushing down to L(X) by integrating over the circle. If we apply the correspondence of [8] to the principal L(K) bundle L(Q)(L(X), L(K)) it produces a K bundle over S 1 × L(X) and we show that this is just the pull-back of Q by the evaluation map. This recovers the result of Killingback [12]. As the present work was being completed a preprint was received from Kiyonori Gomi [9] which also defines connections and curvings on the lifting bundle gerbe using the notion of reduced splittings and building on results in [1]. We discuss the relationship between reduced splittings and Higgs fields in 5.3. Calorons originally arose in consideration of finite-temperature instantons [10] or periodic instantons, that is instantons on S 1 × R3 . The idea of relating them to loop group monopoles on R3 was due to Garland [8] and is a form of “fake” dimensional reduction. The physical significance of this reduction is not known to the authors. 2. Some Preliminaries 2.1. C× bundles. Let P → X be a C× bundle over a manifold X. We shall denote the fibre of P over x ∈ X by Px . Recall [1] that if P is a C× bundle over a manifold X we can define the dual bundle P ∗ as the same space P but with the action p ∗ g = (pg −1 )∗ and, that if Q is another such bundle, we can define the product bundle P ⊗ Q by (P ⊗ Q)x = (Px × Qx )/C× , where C× acts by (p, q)w = (pw, qw −1 ). We denote an element of P ⊗Q by p⊗q with the understanding that (pw)⊗q = p⊗(qw) = (p⊗q)w for w ∈ C× . It is straightforward to check that P ⊗ P ∗ is canonically trivialised by the section x → p ⊗ p∗ , where p is any point in Px . If P and Q are C× bundles on X with connections µP and µQ then P ⊗ Q has an induced connection we denote by µP ⊗ µQ . The curvature of this connection is RP + RQ , where RP and RQ are the curvatures of µP and µQ respectively. The bundle P ∗ has an induced connection whose curvature is −RP . 2.2. Simplicial spaces. Recall [7] that a simplicial manifold X is a collection of manifolds X0 , X1 , X2 , X3 , . . . with maps di : Xp → Xp−1 for i = 0, . . . , p, and sj : Xp →

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Xp+1 for j = 0, . . . , p, satisfying the, so-called, simplicial identities: di dj = dj −1 di , i < j, si sj = sj +1 si , i ≤ j,   sj −1 di , i < j di sj = id, i = j, i = j + 1  s d , i > j + 1. j i−1

(1) (2) (3)

Let p (M) denote the space of all differentiable p forms on a manifold M. Define a homomorphism δ : n (Xp ) → n (Xp+1 ) by δ=

p

(−1)i di∗ .

i=0

It is straightforward to check that δ 2 = 0 and it clearly commutes with the exterior derivative d. Hence we have a complex δ

δ

δ

δ

δ

n (X0 ) → n (X1 ) → n (X2 ) → · · · → n (Xp ) → · · · .

(4)

We remark that from a simplicial space we can define a topological space called its realisation1 . The double complex p (X q ) with the differentials d and δ has total cohomology the real cohomology of the realisation. If P → Xp is a C× bundle then we can define a C× bundle over Xp+1 denoted δ(P ) by δ(P ) = d1−1 (P ) ⊗ d2−1 (P )∗ ⊗ d3−1 (P ) ⊗ . . . . If s is a section of P then it defines δ(s) a section of δ(P ) and if µ is a connection on P with curvature R it defines a connection δ(µ) on δ(P ) with curvature δ(R). If we consider δ(δ(P )) it is a product of factors and because of the simplicial identities (1) every factor occurs with its dual so δ(δ(P )) is canonically trivial. If s is a section of P then under this identification δδ(s) = 1 and moreover if µ is a connection on P then δδ(µ) is the flat connection on δδ(P ) with respect to δ(δ(s)). If X is a simplicial space then a simplicial line bundle [2] is a C× bundle P over X1 with a section s of δ(P ) over X2 with the property that δ(s) is the canonical section of δ 2 (P ). In much of the subsequent discussion we will be interested in submersions π : Y → M. Recall that as a consequence of being a submersion π admits local sections, that is for every x ∈ M there is an open set U containing x and a local section s : U → Y . Given a submersion π : Y → M we denote by Y [2] = Y ×π Y the fibre product of Y with itself over π, that is the subset of pairs (y, y ) in Y × Y such that π(y) = π(y ). More generally we denote the pth fold fibre product by Y [p] . It is straightforward to show that, as a consequence of π being a submersion, Y [p] ⊂ Y p is a submanifold. For p + 1 = 1, 2, . . . we have p projection maps πi : Y [p+1] → Y [p] for i = 1, 2, . . . , p + 1 given by omitting the i th factor, so πi (y1 , y2 , . . . , yp+1 ) = (y1 , . . . , yi−1 , yi+1 , . . . , yp+1 ).

1

It makes no difference for the discussion in this paper if it is the fat or geometric realisation.

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The spaces X0 = Y, X1 = Y [2] , . . . define a simplicial manifold. For this simplicial manifold we can augment the complex (4) by adding at the beginning the space n (M) and the map which pulls back forms from M to Y to obtain a complex π∗

δ

δ

δ

δ

n (M) → n (Y ) → n (Y [2] ) → · · · → n (Y [p] ) → · · · .

(5)

It is a fundamental result of [16] that the complex (5), has no cohomology. So if η ∈ n (Y [p] ) satisfies δ(η) = 0 then we can solve the equation η = δ(ρ) for some ρ ∈ n (Y [p−1] ) (we define Y [0] = M). Note that a solution ρ is not unique, any two solutions ρ and ρ will differ by δ(ζ ) for some ζ ∈ n (Y [p−1] ). 3. Central Extensions We give here details of a method of constructing central extensions presented in [17]. Let G be a Lie group. Recall that from G we can construct a simplicial manifold N G = {N Gp } with NGp = G p with face operators di : G p+1 → G p defined by   i = 0, (g2 , . . . , gp+1 ), di (g1 , . . . , gp+1 ) = (g1 , . . . , gi−1 gi , gi+1 , . . . , gp+1 ), 1 ≤ i ≤ p − 1,  (g , . . . , g ), i = p. 1 p Consider a central extension

π C× → Gˆ → G.

Following Brylinski and McLaughlin [2] we think of this as a C× bundle Gˆ → G with a product M : Gˆ × Gˆ → Gˆ covering the product m = d1 : G × G → G. Because this is a central extension we must have that M(pz, qw) = M(p, q)zw for ˆ given by any p, q ∈ Gˆ and z, w ∈ C× . This means we have a section s of δ(G) s(g, h) = p ⊗ M(p, q) ⊗ q for any p ∈ Gˆg and q ∈ Gˆh . This is well-defined as pw ⊗ M(pw, qz) ⊗ qz = pw ⊗ M(p, q)(wz)−1 ⊗ qz = p ⊗ M(p, q) ⊗ q. Conversely any such section gives rise to an M. Of course we need an associative product and it can be shown that M being associative is equivalent to δ(s) = 1. To actually make Gˆ into a group we need more than multiplication, we need an identity eˆ ∈ Gˆ and an inverse map. It is straightforward to check that if e ∈ G is the identity then, because M : Gˆe × Gˆe → Gˆe , there is a unique eˆ ∈ Gˆe such that M(e, ˆ e) ˆ = e. ˆ It is also straightforward to deduce the existence of a unique inverse. Hence we have the result from [2] that a central extension of G is a C× bundle P → G together with a section s of δ(P ) → G × G such that δ(s) = 1. In [2] this is phrased in terms of simplicial line bundles. For our purposes we need to phrase this result in terms of differential forms. We call a connection for Gˆ → G, thought of as a C× bundle, a connection for the central extension. An isomorphism of central extensions with connection is an isomorphism of ˆ Denote by bundles with connection which is a group isomorphism on the total space G. C(G) the set of all isomorphism classes of central extensions of G with connection.

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ˆ be a connection on the bundle Gˆ → G and, as before, denote by Let µ ∈ 1 (G) ˆ → G × G. Let α = s ∗ (δ(µ)). We then have that δ(µ) the connection one-form for δ(G) ∗ ∗ 2 δ(α) = (δ(s) )(δδ(µ)) = (1) (δ (µ)) = 0 as δ 2 (µ) is the flat connection on δ 2 (P ). Also dα = s ∗ (dδ(µ)) = δ(R). In more detail α and R satisfy: d0∗ α

d0∗ R − d1∗ R + d2∗ R = dα, − d1∗ α + d2∗ α − d3∗ α = 0.

(6) (7)

Let (G) denote the set of all pairs (α, R), where R is a closed, 2π i integral, two form on G and α is a one-form on G × G with δ(R) = dα and δ(α) = 0. We have constructed a map C(G) → (G). In the next section we construct an inverse to this map by showing how to define a central extension from a pair (α, R). For now notice that isomorphic central extensions with connection clearly give rise to the same (α, R) and that if we vary the connection, which is only possible by adding on the pull-back of a one-form η from G, then we change (α, R) to (α + δ(η), R + dη). 3.1. Constructing the central extension. Recall that given R we can find a principal C× bundle P → G with connection µ and curvature R which is unique up to isomorphism. It is a standard result in the theory of bundles that if P → X is a bundle with connection µ which is flat and π1 (X) = 0 then P has a section s : X → P such that s ∗ (µ) = 0. Such a section is not unique, of course it can be multiplied by a (constant) element of C× . Consider now our pair (R, α) and the bundle P . As δ(R) = dα we have that the connection δ(w) − π ∗ (α) on δ(P ) → G × G is flat and hence we can find a section s such that s ∗ (δ(w)) = α. The section s defines a multiplication by s(p, q) = p ⊗ M(p, q)∗ ⊗ q. Consider now δ(s), this satisfies δ(s)∗ (δ(δ(w))) = δ(s ∗ (δ(w))) = δ(α) = 0. On the other hand the canonical section 1 of δ(δ(P )) also satisfies this so they differ by a constant element of the group. This means that there is a w ∈ C× such that for any p, q and r we must have M(M(p, q), r) = wM(p, M(q, r)). Choose p ∈ Gˆe , where e is the identity in G. Then M(p, p) ∈ Gˆe and hence M(p, p) = pz for some z ∈ C× . Now let p = q = r and it is clear that we must have w = 1. So from (α, R) we have constructed P and a section s of δ(P ) with δ(s) = 1. However s is not unique but this is not a problem. If we change s to s = sz for some constant z ∈ C× then we have changed M to M = Mz. As C× is central, multiplying by z is an isomorphism of central extensions with connection. So the ambiguity in s does not change the isomorphism class of the central extension with connection. Hence we have constructed a map

(G) → C(G) as required. That it is the inverse of the earlier map follows from the definition of α as s ∗ (δ(µ)) and the fact that the connection on P is chosen so its curvature is R.

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3.2. An explicit construction. First we show how the pair (α, R) can be used to recover the original central extension with connection. We will then show this gives a construction of a central extension from any pair (α, R) ∈ (G). Finally we have to show that this gives an inverse to the map defined in the preceding section. Let P G denote the space of all paths in G which begin at the identity and define p : P G → G to be the map which evaluates the endpoint of the path. We can use this map to pullback the central extension Gˆ → G to a central extension of P G by C× defined by ˆ = {(f, g) ˆ | f (1) = π(g)}. ˆ p−1 (G) Because P G is contractible this must be trivial and indeed we can map (f, z) ∈ P G ×C× to (f, fˆ(1)z), where fˆ is the (unique) lift of f to a horizontal path in Gˆ starting at the identity. The two projection maps define a commuting diagram: P G × C× → Gˆ ↓ ↓ PG →G ˆ induces a product on P G × C× which must take the form The product on p−1 (G) (f, z)(g, w) = (f g, c(f, g)zw) for some C× valued cocycle on P G. Using methods similar to those in [15] it can be shown that the cocycle is given by c(f, g) = exp α . (f,g)

ˆ As in [15] we can now identify the kernel of the homomorphism P G × C× → G. This is all pairs (h, z) such that the holonomy of the connection around the loop h is equal to z−1 . As we are assuming that G is simply connected we can extend any loop h to a map h˜ from a disk D into G and define H (h, R) = exp R . ˜ h(D)

Notice that the 2π i integrality of R implies that H (h, R) is well-defined. The kernel is therefore the subgroup of all pairs (h, H (h, R)−1 ). We can now see how to define a central extension given the pair (α, R). First we define c(f, g) by c(f, g) = exp( (f,g) α) and it can be checked, as in [15], that this co-cycle makes P G × C× into a group. Then define H (h, R) as above and consider the subset of all pairs (h, H (h, R)−1 ). It can be shown that this is a normal subgroup and the quotient defines the central extension. We define a connection 1-form µ on the principal C× bundle Gˆ using the same technique as in [16]. We use the map P G × C× → Gˆ to pullback the connection one-form µ on Gˆ to a connection one-form µ. ˜ A straightforward calculation shows that this is given by z−1 dz + µ, ˆ

Higgs Fields, Bundle Gerbes and String Structures

where µˆ =

[0,2π]

547

ι(0, d ) ev∗ R dt

and ev : PG × [0, 2π ] → G is the evaluation map ev(f, t) = f (t). In the case that we start with a pair (α, R) it can be shown that this connection ˆ descends to a connection on G. We now have a procedure for constructing from a given central extension a pair (α, R) and from a pair (α, R) a central extension. It is clear from the construction that if we start with a central extension, construct (α, R) and then construct a central extension we get back to where we started from. Consider now what happens if we start with an (α, R), ˜ It is straightforward to construct the central extension and then construct a pair (α, ˜ R). show that we have

and

π ∗ (α) ˜ = c−1 dc + δ(µ), ˆ ∗ ˜ ˆ π (R) = d µ.

Using the definition of c and µˆ we can show that α and R satisfy the same equations and hence deduce that α˜ = α and R˜ = R as π ∗ is injective. 3.3. The non-simply connected case. Although we do not need it below we note here how one can deal with the case that G is not simply connected. This needs the theory of differential characters [5]. Let Z1 (M) be the group of all closed smooth one-cycles. We add to our pair (α, R) a homorphism h : Z1 (M) → U (1) satisfying h(∂(σ )) = exp R σ

for every two-cycle σ . The pair (h, R) is a differential character and for the U (1) bundle Gˆ → G the map h is the holonomy. In addition to the requirement that δ(R) = dα and δ(α) = 0 we require that for any closed one-cycle γ in G × G we have δ(h)(γ ) = exp α . γ

With these conditions central extensions are determined by triples (α, R, h). If we change the connection on Gˆ → G by adding on the pull-back of one-form η then α and R change as before and h changes by multiplying by the integral of η over any element of Z1 (M). 4. Loop Groups There are a number of variants of the loop group that we wish to consider. To define these let K be a compact group and consider first L(K) = {γ : [0, 2π] → K | γ (0) = γ (2π )}; this has a subgroup of based loops L0 (K) = {γ : [0, 2π] → K | γ (0) = γ (2π ) = 1}.

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We assume here that the maps γ are smooth on [0, 2π ]. Now map [0, 2π ] to the circle S 1 by θ → exp(iθ ). We therefore have identified (K) = C ∞ (S 1 , K), the space of smooth maps from the circle to K with a subgroup of L(K) and we let 0 (K) = (K) ∩ L0 (K). These groups are all Frechet Lie groups and their Lie algebras are the analogous spaces of maps of [0, 2π] to k and denoted by L(k), L0 (k), (k) and 0 (k). In the case where G = L(K) there is a well known expression for the curvature R of – see [18]. We can also write down a 1-form α on a left invariant connection on L(K) L(K) × L(K) such that δ(R) = dα and δ(α) = 0. These are: i R= , ∂θ dθ, (8) 4π S 1 i α= d ∗ , d0∗ Z dθ. (9) 2π S 1 2 Here denotes the Maurer-Cartan form on the Lie group K, that is (k)(kX) = X, Z is the function on L(K) defined by Z(k) = (∂θ k)k −1 and , is an invariant inner product normalised so that the longest root has length squared equal to 2. Note that R is left invariant and that α is left invariant in the first factor of G × G. For later use we record here some identities relating and Z. At a point g ∈ LK we have ∂θ = ad(g −1 )(dZ),

(10)

∂θ (ad(g −1 (X))) = ad(g −1 )([X, Z]) + ad(g −1 )∂θ X.

(11)

and if X is in L(k) then

5. Lifting Bundle Gerbes and the String Class 5.1. The string class. Consider a central extension C× → Gˆ → G π

and let P be a principal bundle over a manifold M with structure group G. There is a characteristic class in H 3 (M, Z) which is the obstruction to lifting P to a Gˆ bundle. This is easily described in Cech cohomology. Let {Uα }α∈I be a good cover of M with respect to which P has transition functions gαβ . As the double intersections are contractible we can lift the transition functions to gˆ αβ with values in Gˆ such that π gˆ αβ = gαβ . Because −1 g gβγ gαγ αβ = 1 it follows that −1 dαβγ = gˆ βγ gˆ αγ gˆ αβ

defines a class in H 2 (M, C× ) which vanishes if and only if the bundle lifts to Gˆ [16]. A standard calculation with the short exact sequence Z → C → C× identifies this class with a class in H 3 (M, Z). In the case that G is the loop group of Spin(n) and Gˆ the central extension of the loop group, this class is called the string class [12].

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5.2. Lifting bundle gerbes. We first briefly review the notion of a bundle gerbe from [16] cast in the language of simplicial line bundles. Let π : Y → M be a submersion. Recall that it defines a simplicial space Y [p] . A bundle gerbe over a manifold M is a pair (P , Y ), where P is a simplicial line bundle over the simplicial space defined by Y . This means that P is a C× bundle over Y [2] . A bundle gerbe P has a characteristic class DD(P ) in H 3 (M; Z) associated to it – the Dixmier-Douady class. We refer to [16] for the definition and properties of DD(P ). We can realise the image of the Dixmier-Douady class in real cohomology in a manner analogous to the chern class. A bundle gerbe (P , Y, M) can be equipped with a bundle gerbe connection. This is a connection ∇ on P which is compatible with the bundle gerbe product in the sense that δ(∇) is the flat connection for the trivialisation s of δ(P ). It is not hard to see that bundle gerbe connections always exist, indeed if ∇ is any connection then δ(s ∗ (∇)) = 0 and hence from (5) s ∗ (∇) = δ(ρ) for some ρ on Y [2] and ∇ − π ∗ (ρ) is a bundle gerbe connection. If F∇ denotes the curvature of the bundle gerbe connection ∇, then it is easy to see that we have δ(F∇ ) = 0. Hence we can solve the equation F∇ = δ(f ) for some f ∈ 2 (Y ). A choice of f is called a curving for the bundle gerbe connection ∇. Since F∇ is closed, we get δ(df ) = 0 and so we have df = 2π iπ ∗ ω for some necessarily closed 3-form ω on M. The 3-form ω is called the three curvature of the bundle gerbe connection ∇ and curving f . In [16] it is shown that ω is an integral 3-form and represents the image of the Dixmier-Douady class DD(P ) of ω in H 3 (M, R). Suppose that G is a Lie group forming part of a central extension C× → Gˆ → G. Recall from [16] that we can associate to a principal G bundle P (M, G) a bundle gerbe (P˜ , P , M) – the so called lifting bundle gerbe. P˜ → P [2] is the pullback P˜ = τ −1 Gˆ of Gˆ → G by the natural map τ : P [2] → G defined by p2 = p1 τ (p1 , p2 ) for p1 and p2 points of P lying in the same fibre. τ satisfies the property τ (p1 , p2 )τ (p2 , p3 ) = τ (p1 , p3 ) for points p1 , p2 and p3 all lying in the same fibre. P˜ inherits a bundle gerbe ˆ The Dixmier-Douady class of the bundle gerbe P˜ meaproduct from the product in G. ˆ It sures the obstruction to lifting the structure group of the principal G bundle to G. follows that when G is a loop group the Dixmier-Douady class is (the image in real cohomology of) the string class of the bundle P . From the principal bundle we can construct a simplicial space as above, and if τ : P [2] → G is defined by p2 = p1 τ (p1 , p2 ) then we can define τ : P [k+1] → G k by τ (p1 , . . . , pk+1 ) = (τ (p1 , p2 ), . . . , τ (pk , pk+1 ). It can be checked that this is a simplicial map, that is it commutes with the face and degeneracy maps. It follows that pullback of differential forms by τ ∗ commutes with ˆ Then the natural connection µ˜ = τ ∗ µ on P˜ is δ. Suppose that µ is a connection on G. not a bundle gerbe connection on P˜ . Indeed from the equation δ(µ) = π ∗ (α) we see that δ(τ ∗ (µ)) = τ ∗ (α). However the form β = τ ∗ (α) satisfies δ(β) = τ ∗ (δ(α)) = 0 as

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δ(α) = 0. So we can solve the equation δ() = β for some 1-form on P [2] . Then the connection µ− ˜ is a bundle gerbe connection on P˜ . Its curvature is given by τ ∗ (R)−d, where R = dµ is the curvature of µ. The curving is therefore a two-form f on P satisfying δ(f ) = τ ∗ (R) − d for some 2-form f on P . From f the Dixmier-Douady class ω is obtained as df = π ∗ (ω). To proceed further we need to concentrate on a specific example so we will let G = L(K). So we have a principal L(K) bundle P (M, L(K)) on M for K a compact Lie group, we can form the lifting bundle gerbe (P˜ , P , M) associated to the central extension → L(K), C× → L(K) is the Kac-Moody group. We can form a bundle gerbe connection ∇ on P˜ where L(K)

in the manner described above using the natural connection on L(K) and the 1-form α on L(K) × L(K). We will show that it is possible to write down an expression for the three curvature ω of the bundle gerbe connection ∇. Suppose we have chosen a connection 1-form A on the principal G bundle P → M. Then this is a one-form on P with values in g. It is straightforward to show that π1∗ (A) = ad(τ −1 )π2∗ (A) + τ ∗ ().

(12)

Let τij (p1 , . . . , pk ) = τ (pi , pj ). Then using β = (τ12 × τ23 )∗ α, the definition of α from (9) and the identity (12) we obtain i −1 ∗ π ∗ A − ad(τ12 )−1 π23 A, ∂θ (τ23 )τ23

dθ, β= 2π S 1 13 where π12 (p1 , p2 , p3 ) = p3 , etc. Define a one-form on P [2] by =

i 2π

S1

π2∗ A, τ ∗ (Z) dθ.

(13)

It can be shown that δ() = β. To solve the equation τ ∗ R − d = δ(f ) for some choice of curving f we first need an explicit expression for τ ∗ R − d. Using (9), the standard fact that d = −(1/2)[d, d] and the identities (10) and (11) we obtain i ∗ τ R − d = π ∗ (A), ∂θ π1∗ (A) − π2∗ (A), ∂θ π2∗ (A) 4π S 1 1 − [π2∗ (A), π2∗ (A)], τ ∗ (Z) − 2 π2∗ (dA), τ ∗ (Z) dθ. Recalling that F , the curvature of A satisfies F = dA + 1/2[A, A] we have that τ ∗ R − d is equal to i i A, ∂θ A − π ∗ (F ), τ ∗ (Z) dθ, δ 4π S 1 2π S 1 2 where δ = π1∗ − π2∗ . We want to solve τ ∗ R − d = δ(f ) so we now need to write the two-form i π ∗ (F ), τ ∗ (Z) dθ 2π S 1 2

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as δ of a two-form on P . To this end, choose a map : P → C ∞ ([0, 2π ], k) satisfying (pg) = ad(g −1 )(p) + g −1

∂g . ∂θ

(14)

We call any such function a twisted Higgs field for P . As a convex combination of twisted Higgs fields is a twisted Higgs field and they clearly exist when P is trivial, it is straightforward to use a partition of unity on M to construct a twisted Higgs field for P . Notice that a twisted Higgs field will not generally live in L(k). However if we start with a bundle P (M, (K)) then the twisted Higgs field can be required to take values in (k). Choose then a twisted Higgs field for P . It satisfies ad(τ )π1∗ () = π2∗ () + τ ∗ (Z) and we have π2∗ (F ), τ ∗ (Z) = π1∗ ( F, ) − π2∗ ( F, ). Finally we can write τ ∗ R − d = δ(f ), where f is the two-form on P defined by f =

i 2π

S1

1 A, ∂θ A − F, dθ, 2

and using the Bianchi identity for F we obtain df = −

i 2π

S1

F, ∇ dθ,

where ∇ = d + [A, ] − ∂θ A. It is straightforward to show that ∇ is equivariant for the adjoint action of L(K) and hence descends to a section of the adjoint bundle of P just as F descends to a two-form on M with values in the adjoint bundle of P . Finally we have Theorem 5.1. Let P → M be a principal L(K) bundle and let A be a connection for P with curvature F and be a twisted Higgs field for P . Then the string class of P is represented in de Rham cohomology by the three-form 1 − 2 4π

S1

F, ∇ dθ,

(15)

where ∇ = d + [A, ] − ∂θ A.

(16)

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5.3. Reduced splittings. In [9] the concept of reduced splitting is used to calculate a connection and curving for the lifting bundle gerbe. In this section we explain how this fits in with our constructions in the case of the loop group. First we have Definition 5.2 ([9]). Define the group cocycle Z : L(K) × L(k) → iR by (0, Z(g, X)) = ad(gˆ −1 )((X, 0)) − (ad(g −1 )(X), 0), as L(K) ⊕ iR and gˆ is a lift of g to the central where here we assume a splitting of L(K) extension. The group cocycle Z encapsulates the information of the central extension in a similar manner to our α. Indeed in the case of the loop group with the extension defined by the (R, α) in (9) then we have i −1 Z(g , X) = −α(1, g)(X, 0) = X, ∂θ (g)g −1 dθ. 2π S 1 Definition 5.3 ([9]). A reduced splitting for a loop group principal bundle P (M, L(K)) is a map : P × L(k) → R which is linear in the second factor and satisfies (p, X) = (pg, ad(g −1 )(X)) + Z(g −1 , X). A straightforward calculation shows that if is a Higgs field then i (p, X) = , X dθ 2π S 1 is a reduced splitting.

5.4. The path fibration. We will illustrate the above discussion for the case of the L0 (K) bundle PK(K, L0 (K)), where PK → K is the path fibration of the compact, simple and simply connected group K – see [4]. In this case there is a canonical closed 3-form on K representing the image of the generator of H 3 (K; Z) = Z inside H 3 (K; R) given by 1

,

],

, [ ω3 = 48π 2

is the right invariant Maurer-Cartan form on K see for example [1]. We will where show that for the path fibration the string class (15) is ω3 . First we need to define a connection. The tangent space at p ∈ PK is the set of all vector fields along the path p and such a vector field is vertical if it vanishes at 2π . A right invariant splitting is given by Hp = {θ → (θ/2π)Rp(θ) X |∈ k}. Clearly this satisfies Rg∗ Hp = Hpg and the corresponding connection one-form is A=−

θ

). ad(p −1 )π ∗ ( 2π

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A calculation shows that the curvature of A is 2 θ θ

), π ∗ (

)]. ad(p−1 )[π ∗ ( − F = 8π 2 4π A suitable Higgs field is given by (p) = p −1

∂p , ∂θ

and another calculation shows that ∇ =

1

). ad(p −1 )π ∗ ( 2π

Putting these into the formula for the string class (15) gives the required result. 6. Calorons and the String Class In [8] as part of the study of calorons a correspondence was introduced between K bundles P˜ on a manifold M × S 1 and (K) bundles P on M. This correspondence also related a connection A˜ on P˜ to a connection A on P and a section of the twisted adjoint bundle. We will describe this correspondence and show that the integral over the circle of the Pontrjagin class of A˜ is a representative for the string class (15). If P → M is an (K) bundle we define P˜ = (P × K × S 1 )/ (K), where (K) acts on P × K × S 1 by g(p, k, θ ) = (pg −1 , g(θ )k, θ ). Letting [p, k, θ] denote the equivalence class of (p, k, θ) we have a right K action on P˜ given by [p, k, θ ]h = [p, kh, θ ] and a projection map π : P˜ → M ×S 1 defined by π([p, k, θ]) = (π(p), θ ). The fibres of π are the orbits of K and P˜ is a principal K bundle. Starting instead with a principal K bundle P˜ → M × S 1 we define a bundle P → M whose fibre at m is all the sections of P˜ restricted to {m} × S 1 . This is clearly acted on by (K). Given a connection 1-form A on P (M, (K)) we can define a connection 1-form A˜ on P˜ by pushing forward the 1-form on P × K × S 1 which is also denoted by A˜ and is given by A˜ = ad(k −1 )(A) + + ad(k −1 )dθ .

(17)

˜ A] ˜ for the connection A˜ we obtain: If we compute the curvature R˜ = d A˜ + 1/2[A, R˜ = ad(k −1 )(F + ∇()dθ ),

(18)

where ∇() is defined as in (16) and F denotes the curvature of the connection A on the principal (K) bundle P . The Pontrjagin form of the connection A on the bundle over S 1 × M is −

1 ˜ ˜ 1 R, R = − 2 ( F, F + 2 F, ∇ ) dθ, 2 8π 8π

(19)

and integrating over the circle gives the string class of the bundle P (M, (K)). We have hence proved

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Theorem 6.1. Let P → M be an (K) bundle and P˜ → M × S 1 the corresponding K bundle. Then the string class of P is obtained from the Pontrjagin class of P˜ by integrating over the circle. This result enables an easy proof of a result of Killingback also proved in [4, 9]. We start with a K bundle Q → X and let P and M be the loop spaces of Q and X respectively. Then P is an (K) bundle over M and Killingback shows that the string class of P is obtained from the Pontrjagin class of Q by pulling back by the evaluation map ev : M × S 1 → X

(20)

and integrating over the circle. If P is the space of loops in Q there is a natural map P × K × S 1 → Q given by (p, k, θ) → p(θ )k. This is constant on the orbits of the (K) action and hence defines a map P˜ → Q which is K equivariant and covers the evaluation map (20). This shows that P˜ is the pull-back of Q by the evaluation map. Hence the Pontrjagin class of P˜ is the pull-back by the evaluation map of the Pontrjagin class of Q. Thus the string class of P is the integral over the circle of the pull-back by the evaluation map of the Pontrjagin class of Q. 7. Conclusion If we try and apply the caloron construction to an L(K) bundle with a connection we obtain a K bundle on S 1 × M which is only smooth on (0, 2π ). There should be a theory of K bundles on [0, 2π] × M with connection which patch together over {0} × M and {2π} × M in such a way as to recover the result of [8] relating L(K) bundles on M to such bundles. Notice that this approach could be used to show that any L(K) bundle over M has a three class which we could define by transgressing the Pontrjagin class of the induced K bundle over S 1 × M. However it would not be clear that this three class was the string For this we needed the theory of bundle gerbes class, the obstruction to lifting to L(K). and the connections and curvings which provide a bridge between the Cech description of the string class and a de Rham realisation of it. References 1. Brylinski, J.-L.: Loop spaces, characteristic classes and geometric quantization. Progr. Math. 107, Boston, MA: Birkh¨auser, 1993 2. Brylinski, J.-L., McLaughlin, D.A.: The geometry of degree-four characteristic classes and of line bundles on loop spaces. I. Duke Math. J. 75(3), 603–638 (1994) 3. Carey, A.L., Crowley, D., Murray, M.K.: Principal bundles and the Dixmier-Douady class. Commun. Math. Phys. 193, 171–196 (1998) 4. Carey, A.L., Murray, M.K.: String structures and the path fibration of a group. Comm. Math. Phys. 141(3), 441–452 (1991) 5. Cheeger, J., Simons, J.: Characteristic forms and geometric invariants. In: Geometry and topology. Lecture Notes in Mathematics 1167, Berlin-Heidelberg-New York: Springer, 1985 6. Coquereaux, R., Pilch, K.: String Structures on Loop Bundles. Commun. Math. Phys. 120, 353–378 (1989) 7. Dupont, J.L.: Curvature and characteristic classes. Lecture Notes in Mathematics, Vol. 640. Berlin: Springer, 1978 8. Garland, H.K., Murray, M.K.: Kac-Moody monopoles and periodic instantons. Commun. Math. Phys. 120(2), 335–351 (1988)

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9. Gomi, K.:Connections and curvings on lifting bundle gerbes. J. Lond. Math. Soc. 67(2), 510–526 (2003) 10. Gross, D.J., Pisarski, R.D., Yaffe, L.G.: QCD and temperature. Rev. Mod. Phys. 53, 43–80 (1981) 11. Koboyashi, S., Nomizu, K.: Foundations of Differential Geometry. New York: Interscience, John Wiley and Sons, 1963 12. Killingback, T.: World-sheet anomalies and loop geometry. Nucl. Phys. B288, 578–588 (1987) 13. McLaughlin, D.A.: Orientation and String Structures on Loop Space. Pac. J. Math. 155, 1–31 (1992) 14. Mickelsson, J.: Kac-Moody groups, topology of the Dirac determinant bundle and fermionization. Commun. Math. Phys. 110, 173–183 (1987) 15. Murray, M.K.: Another construction of the central extension of the loop group. Comm. Math. Phys. 116, 73–80 (1988) 16. Murray, M.K.: Bundle gerbes. J. London Math. Soc. 54(2), 403–416 (1996) 17. Murray, M.K., Stevenson, D.: Yet another construction of the central extension of the loop group. In: Proceedings of the National Research Symposium on Geometric Analysis and Applications at the Centre for Mathematics and its Applications, The Australian National University, 2000 18. Pressley, A., Segal, G.: Loop groups. Oxford: Clarendon Press, 1986 Communicated by R.H. Dijkgraaf

Commun. Math. Phys. 243, 557–582 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0985-3

Communications in

Mathematical Physics

Mirror Symmetry on Kummer Type K3 Surfaces Werner Nahm1 , Katrin Wendland2 1 2

Physikalisches Institut, Universit¨at Bonn, Nußallee 12, 53115 Bonn, Germany. E-mail: [email protected] Dept. of Physics and Astronomy, UNC at Chapel Hill, 141 Phillips Hall, CB #3255, Chapel Hill, NC 27599, USA. E-mail: [email protected]

Received: 5 September 2001 / Accepted: 2 October 2003 Published online: 11 November 2003 – © Springer-Verlag 2003

Abstract: We investigate both geometric and conformal field theoretic aspects of mirror symmetry on N = (4, 4) superconformal field theories with central charge c = 6. Our approach enables us to determine the action of mirror symmetry on (non-stable) singular fibers in elliptic fibrations of ZN orbifold limits of K3. The resulting map gives an automorphism of order 4, 8, or 12, respectively, on the smooth universal covering space of the moduli space. We explicitly derive the geometric counterparts of the twist fields in our orbifold conformal field theories. The classical McKay correspondence allows for a natural interpretation of our results. 1. Introduction We investigate the version of mirror symmetry [Di2, L-V-W, G-P] which was found by Vafa and Witten for orbifolds of toroidal theories [V-W], and which was generalized to the celebrated Strominger/Yau/Zaslow conjecture [S-Y-Z]. Since the conceptual issues of mirror symmetry for N = (4, 4) superconformal field theories on first sight are different and more controversial than for strict N = (2, 2) theories, we first discuss the latter, as a preparation. An N = (2, 2) superconformal field theory is a fermionic conformal field theory (CFT) together with a marking, i.e. a map from the standard super–Virasoro algebra into the operator product expansion (OPE) of this theory. Due to the marking the theory has well defined left and right handed U (1) charges Ql , Qr . Markings which differ by Ql , Qr gauge transformations are identified. Results on N = (2, 2) deformation theory [Di2] show that for given central charge c a moduli space of N = (2, 2) superconformal field theories can be defined. Its irreducible components at generic points are Riemannian manifolds under the Zamolodchikov metric [Za] and have at most orbifold singularities. Note that the completion of the moduli space may contain points of extremal transitions which do not have CFT descriptions and will not be of relevance for our discussion. We shall restrict considerations to a connected part of the moduli space. By the above

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[Th2]. We also assume it has a unique smooth, simply connected covering space M that all theories in our moduli space include the spectral flow operators in their Hilbert spaces. Let us now consider strict N = (2, 2) theories, i.e. those for which the marking has no continuous deformations. Then with respect to the left and right U (1) action the canonically splits into two subbundles. The cover of the moduli tangent bundle T M = M 1 × M 2 [Di2, D-G2]. space has a corresponding canonical product realization M 0 Let ⊂ be the subgroup of elements which admit a factorization γ = γ1 γ2 such i (other automorphisms may exist that are related to the that γi ∈ only acts on M induces a factorization effect of monodromy [G-H-L, K-M-P]). The factorization of M 0 = M1 × M2 . The corresponding two subbundles of T M are distinM := M/ guished by the marking. The standard mirror automorphism of the super–Virasoro OPE which inverts the sign of one of the U (1) generators interchanges these subbundles. One expects that near some boundary component of the moduli space any of our theories has a geometric interpretation as supersymmetric sigma model on a space X with Ricci flat K¨ahler metric of large radius. Though there may be several boundary components of this kind which yield manifolds that cannot be deformed into each other as algebraic manifolds, but are birationally equivalent manifolds [A-G-M2, Ko3, D-L, Ba2], = M(X). we choose a unique X for ease of exposition, and we write M = M(X), M One possible deformation of X is given by the scale transformation of the metric. It turns out to belong to the tangent space of one of the factors, say M1 . Then M2 becomes the space of complex structures on X, and close to the boundary M1 corresponds to variations of the (instanton corrected) complexified K¨ahler structure. Under mirror symmetry the roles of the two factors are interchanged, such that M1 becomes the moduli space of complex structures on some other space X . This induces a duality between geometrically different Calabi Yau manifolds, an observation that has had a striking impact on both mathematics and physics (see [C-O-G-P, Mo1], and [C-K] for a more complete list of references). Let (X) be the space of sigma model Lagrangians on X (possibly with a marking of the homology of X), and (X)b ⊂ (X) a connected and simply connected boundary region whose points define a conformal field theory by some quantization scheme. Let σb (X) : (X)b → M(X) be the corresponding continuous map. Locally, σb (X) is a homeomorphism. By deformation, a mirror symmetry (X)b ∼ = (X )b induces ∼ an isomorphism M(X) = M(X ). Since isomorphic CFTs yield the same point in M(X), such an isomorphism cannot depend on the choice of the boundary region. When X = X the induced automorphism of M(X) corresponds to an automorphism of the CFT which changes the sign of one of the two U (1) currents and exchanges the corresponding supercharges. This automorphism changes the marking and therefore acts non-trivially on M(X). When a base point has been chosen, the local isomorphism σb (X) lifts to an inclusion σb (X) : (X)b → M(X) and analogously for X . When X = X and (X)b ∩ (X )b is non-empty, we choose a base point in the intersection. Then σb (X ) ◦ σb (X)−1 lifts to b : M(X) ). Sometimes there are canonical choices a mirror isomorphism γms → M(X for the base point, otherwise one obtains equivalent choices which correspond to mulb by an element of some subgroup of . tiplication of γms Given a Calabi-Yau manifold X one can construct the corresponding family of sigma models, the moduli space M(X), and the mirror Calabi-Yau manifold X . Since X, X determine (families of) classical geometric objects, it should be possible to transform one into the other by purely classical methods. Such constructions for mirror pairs have

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been proposed in the context of toric geometry [Ba1] as well as T–duality [V-W, S-Y-Z, K-S]. The basic case is given by theories with central charge c = 3, which correspond = H × H, where we shall to sigma models on a two-dimensional torus. Here one has M 1 ∼ 2 . use coordinates ρ, τ for the two copies of the upper complex half-plane H ∼ =M =M In our conventions on automorphisms, the fundamental group is given by the standard Orientation change, given by ρ → −ρ, SL(2, Z) × SL(2, Z) action on M. ¯ τ → −τ¯ , and space parity change, given by ρ → −ρ, ¯ τ → τ , are not considered as automorphisms, since they change the marking. For purely imaginary ρ, τ the theory is the (fermionic) product of circle models with squared radii r12 = ρ/τ , r22 = −ρτ . Hence our theory has a nonlinear sigma model description given by two Abelian U (1) currents j1 , j2 , and their N = 2 superpartners ψ1 , ψ2 , together with analogous right-handed fields  1 ,  2 , ψ 1 , ψ 2 , all compactified on a real torus. The ψi , ψ i are Majorana fermions. The eigenvalues of the four currents ji ,  i lie in a four-dimensional vector space with natural O(2)lef t × O(2)right and O(2, 2) actions. Due to the compactification they form a lattice of rank 4. The U (1) current of the left-handed N = 2 superconformal algebra is J = i :ψ2 ψ1:. Hence mirror symmetry can be induced by the OPE preserving map which leaves right-handed fields unchanged and transforms left-handed ones by (ψ1 , ψ2 ) → (−ψ1 , ψ2 ), (j1 , j2 ) → (−j1 , j2 ). For the fermionic sigma model on the first circle this is the T–duality map, i.e. r1 → (r1 )−1 . The existence of this OPE preserving map implies that M1 is isomorphic to M2 , as stated above. Summa mirror symmetry is rizing, close to the “boundary point” τ = i∞, ρ = i∞ of M given by the exchange of ρ and τ . This is fiberwise T-duality in an S1 fibration (with section) on the underlying torus and motivates the construction [V-W, S-Y-Z]. The cusp τ = i∞, ρ = i∞ corresponds to a limit where the base volume r2 becomes infinite, whereas the relative size of the fiber is arbitrarily small, e.g. for constant r1 . Since base space and fiber are flat, semiclassical considerations are applicable for arbitrary r1 , such that mirror symmetry yields a relation between two classical spaces. Conjecturally the idea carries over to suitable torus fibrations over more complicated base spaces of infinite volume. Mathematically, the expected map is given by a Fourier–Mukai type functor [Ko2, Mo2]. The construction of mirror symmetry by fiberwise T-duality also makes sense when X is a hyperk¨ahler manifold and the corresponding sigma model has a superconformal symmetry which is extended beyond N = (2, 2), though the relationship with CFT is somewhat different. The moduli space M no longer splits canonically, since there is no canonical N = (2, 2) subalgebra of the extended super Virasoro algebra. Moreover, there are no quantum corrections to the K¨ahler structure, such that each point of corresponds to a classical sigma model, with well defined Ricci flat metric M and B-field. In this situation classical geometries corresponding to different boundary components should be diffeomorphic, since for compact hyperk¨ahler manifolds this is implied by birational equivalence [Hu]. According to the previous arguments, mirror symmetry must yield an element γms of , which depends on the geometric interpretation. The picture developed so far is conjectural, but in some cases it can be verified, since the moduli space is entirely known. This is true for toroidal theories and for those theories with c = 6 whose Hilbert spaces include the spectral flow operators [Na4, A-M, N-W]. Every such theory admits geometric interpretations in terms of nonlinear sigma models either on tori or on K3, depending on the CFT. Thus any mirror symmetry relates two different geometric interpretations within the same moduli space M.

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In this note, we explore a version of mirror symmetry on Kummer type K3 surfaces that was proven in a much more general context in [V-W] and actually led to the Strominger/Yau/Zaslow conjecture, see [S-Y-Z, G-W1, Mo2, vEn]. Namely, for a T 2 fibered K3 surface p : X −→ P1 with elliptic fibers and a section, mirror symmetry is induced by T–duality on each regular fiber of p. Among various maps known as mirror symmetry, this is the only one with general applicability. We show that it generalizes to the singular fibers and determine the induced map. It turns out to be of finite order 4, 8, or 12 in the different cases we discuss. Note that by construction, our mirror map depends on the respective geometric interpretations on orbifold limits of K3. In other words, we of the moduli space. It would be interesting to are forced to work on the cover M Our understand the precise effect of quotienting out by the automorphism group of M. approach enables us to read off the exact identification of twist fields in the relevant orbifold conformal field theory with geometric data on the corresponding Kummer type K3 surface. The role of “geometric” versus “quantum” symmetries is thereby clarified. The correct identification is also of major importance for the discussion of orbifold cohomology and resolves the objection of [F-G] to Ruan’s conjecture [Ru1] on the orbifold cohomology for hyperk¨ahler surfaces1 . For those Kummer type K3 surfaces discussed in this note, our results in fact prove part of Ruan’s conjecture. To make the paper more accessible to mathematicians, we do not use the language of branes for the geometric data, but a translation is not hard. This work is organized as follows: In Sect. 2 we discuss our mirror map on four-tori. In Sect. 3 we show how this map induces a mirror map on Kummer K3 surfaces X. In particular, we determine the induced map on the (non-stable) singular fibers of our elliptic fibration p : X −→ P1 . In Sect. 4 we give its generalization to other Kummer type surfaces, i.e. other non-stable singular fibers. Section 5 deals with the CFT side of the picture: As explained above, the mirror map is an automorphism on a given superconformal field theory. Its action on the bosonic part of the Hilbert space of ZN orbifold conformal field theories on K3, N ∈ {2, 3, 4, 6}, is determined independently from the results of Sects. 3 and 4. In Sect. 6 we use the previous results to read off an explicit formula that maps twist fields to cohomology classes on K3. This is interpreted in terms of the classical McKay correspondence. We close with a summary and discussion in Sect. 7.

2. The Mirror Map for Tori As pointed out in the Introduction, for an N = (2, 2) superconformal field theory on a two–dimensional orthogonal real torus with radii r, r and vanishing B–field, at large r , mirror symmetry is just the T–duality map r → r −1 for one radius, whereas r remains unchanged. This map is naturally continued to arbitrary values of r, r [V-W]. Now consider a toroidal theory on the Cartesian product T of two two-dimensional orthogonal tori with radii r1 , r3 , and r2 , r4 , respectively. Since the U (1) currents of the N = 2 superconformal algebras in the lower dimensional theories add up to give the U (1) current of the full theory, mirror symmetry is induced by r1 → (r1 )−1 , r2 → (r2 )−1 [V-W]. After a suitable choice of complex structure this is fiberwise T–duality on a special Lagrangian fibration (with section) of our four–torus. Hence we are discussing the version of mirror symmetry that was generalized in [S-Y-Z], see also [Mo2, vEn]. 1 The fact that our transformation resolves this objection was explained to us by Yongbin Ruan [Ru2] and goes back to an earlier observation by Edward Witten.

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561

Alternatively, the fibration can be understood in terms of a Gromov-Hausdorff collapse [K-S]. We wish to determine the corresponding map on the cover of the moduli space. Recall [E-T] that in the present case of N = (2, 2) superconformal field theories with central charge c = 6 we actually have extended, i.e. N = (4, 4) supersymmetry. By [Na4, Se, C, A-M, N-W], a theory in the corresponding moduli space is specified by the relative position of an even self-dual lattice L and a positive definite2 four-plane x in R4,4+δ ∼ = H even (Y, R) with δ = 0 or 16, depending on whether the theory is associated to a torus or a K3 surface Y . In terms of parameters (g, B) of nonlinear sigma models on Y , g an Einstein metric on Y and B a B-field, and for vanishing B–field, x is the positive eigenspace of the Hodge star operator ∗ in H even (Y, R), and L = H even (Y, Z). The symmetry group of x is SO(4) × O(4 + δ), such that = O + (4, 4 + δ; R)/ (SO(4) × O(4 + δ)) , M

(1)

as in which is indeed simply connected [Wo]. We remark that for δ = 16, the space M (1) is a partial completion of the smooth universal covering space of the actual moduli contains points which do not correspond space of N = (4, 4) SCFTs on K3. Namely, M with at least complex codito well-defined SCFTs [Wi1]. They form subvarieties of M mension one [A-G-M1]. These ill-behaved theories, however, will not be of relevance for the discussion below. For the torus we have δ = 0, and the denominator in (1) contains SO(4)lef t × SO(4)right , the elements of which act as rotations on the left- and right-handed charges Ql , Qr , analogously to the case of the two-dimensional torus discussed in the Introduction. The fundamental group of the moduli space is given by = Aut (L(Y0 )) ∼ = O + (4, 4 + δ; Z), where L(Y0 ) describes the base point. To determine the element of that acts as mirror symmetry, it is crucial to gain a detailed understanding of the map that associates a point in moduli space to given nonlinear sigma model data. In d dimensions, it is customary to specify a toroidal theory by a lattice with generators λ1 , . . . , λd ∈ Rd , i.e. T = Rd / spanZ {λ1 , . . . , λd }, and a B-field B ∈ Skew(d). Here Rd carries the standard metric. We choose a reference torus T0 given by the lattice Zd with standard orthonormal generators e1 , . . . , ed , and ∈ Gl + (d) such that λi = ei , i ∈ {1, . . . , d}. ˇ d which is idenThe group Gl + (d) has a natural representation on the dual space R tified with Rd by the standard metric. The corresponding image of is M := ( T )−1 . The vectors µi := Mei , i ∈ {1, . . . , d}, form a dual basis with respect to λ1 , . . . . . . , λd . ˇ d ). The image of under this repSimilarly, we have a natural representation on n (R n d resentation will be denoted (M). Note that (M) acts by multiplication with V −1 , where V = det( ) is the volume of the torus. In the standard description, the charge lattice of the theory is given by pairs √1 (Ql + 2 ˇ d ⊕ Rd ∼ Qr , Ql − Qr ) ∈ Rd,d , where it is natural to take Rd,d = R = H 1 (T0 , R) ⊕ H1 (T0 , R) with bilinear form (α, β) · (α , β ) = αβ + α β. 2

(2)

On cohomology, we generally use the scalar product (4) that is induced by the intersection form on the respective surface.

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The charge lattice is even and integral and is obtained as the image of the standard lattice Zd,d ∼ = H 1 (T0 , Z) ⊕ H1 (T0 , Z) under M 0 ?B (3) v( , B) = ∈ O + (d, d). 0 0 ? ˇ d . The correHere B appears as a skew symmetric linear transformation from Rd to R 2 d ˇ ) will be denoted b, such that b is a vector with components sponding element of (R bij = Bij with respect to the basis ei ∧ ej . We will also use its dual bˇ with components bˇij = k,l ij kl Bkl /2. As mentioned below (1), the rotations of the left and right charge lattices form a subgroup SO(d)lef t × SO(d)right ⊂ O + (d, d). Rotations which act on Ql and Qr in the same way are generated by 0 , ∈ so(d). 0 Rotations for which the respective actions on Ql , Qr are inverse to each other are generated by 0 , ∈ so(d). 0 To describe torus orbifolds we have to work with the lattice H even (T0 , Z) instead of H 1 (T0 , Z) ⊕ H1 (T0 , Z). The vector space H even (T0 , R) carries a bilinear form ·, · that we obtain from the intersection form upon Poincar´e duality, i.e. even (T0 , R) : a, b = a ∧ b. (4) ∀ a, b ∈ H T0

Accordingly, we have to use a half spinor representation s of O + (d, d). We now specialize to the case d = 4, where H 1 (T0 , R) ⊕ H1 (T0 , R) ∼ = H even (T0 , R). Hence the representation s can be obtained from v by triality [Di1, N-W, K-O-P, O-P]. In other words,  −1  2  V 0 0 1 bˇ − B 2 (5) s( , B) = V 1/2  0 2 (M) 0   0 ? −b  . 0 0 1 00 1 Here B2 = B, B as given in (4), and by bˇ we denote the dual of b as introduced above. The matrix s( , B) acts on L(T0 ) := H even (T0 , Z) ∼ = Z4,4 and is given with respect to the basis e1 ∧ e2 ∧ e3 ∧ e4 , ei ∧ ej , 1. Note that in [N-W] we have used a normalization of the scalar product on H 2 (T0 , Z) which differs by a factor of V from the above. For later use we note that analogously to (5) one determines   1 00 s(C) =  −cˇ ? 0  (6) 2 c 1 − C 2 as the triality conjugate of

Mirror Symmetry on Kummer Type K3 Surfaces

v(C) =

?0 −C ?

563

∈ O + (4, 4),

where c is the row vector with components cij = Cij with respect to ei ∧ ej . The sigma model on T0 with B = 0 is described by the lattice L(T0 ) and the positive definite four-plane x0 ⊂ H even (T0 , R) which is left invariant by the Hodge star operator ∗. The latter is given by

1 + e 1 ∧ e2 ∧ e3 ∧ e4 , e1 ∧ e3 + e4 ∧ e2 , x0 = spanR . (7) e1 ∧ e2 + e3 ∧ e4 , e1 ∧ e4 + e2 ∧ e3 For arbitrary sigma model parameters ( , B), x0 as in (7) remains the +1 eigen space of ∗, but we have H even (T , Z) ∼ = s( , B)L(T0 ) =: L. A point in the cover M of the moduli space is described by the relative position of L with respect to x0 , i.e. the pair x0 , s( , B). Since only the relative position counts, we can use Rx0 and Rs( , B) with arbitrary R ∈ O + (4, 4). To avoid confusion one should note that, as mentioned above, in [N-W] we have used Rs( , B) = s(V −1/4 , B). For the description of mirror symmetry on orthogonal tori, however, it is more convenient to choose R = ?. Let us now determine the lattice automorphism that acts as mirror symmetry. It suffices to consider B = 0 and a torus T with defining matrix = diag(r1 , . . . , r4 ), ri > 0. The generators of its even cohomology group H even (T , Z) will be denoted υ = µ1 ∧ · · · ∧ µ4 , µi ∧ µj , υ 0 = 1. By the above we need to find a map that leaves H even (T , Z) and the four–plane (7) invariant and induces r1 → (r1 )−1 , r2 → (r2 )−1 on the torus parameters. To this end, substituting ei = ri µi into (7) one in particular finds ± υ 0 ←→ ± µ1 ∧ µ2 , ± µ1 ∧ µ3 ←→ ± µ2 ∧ µ3 , γMS (T0 ) : ± υ ←→ ± µ3 ∧ µ4 , ± µ4 ∧ µ2 ←→ ± µ1 ∧ µ4 for the base point of the moduli space given by T0 and B = 0 [Na3]. To fix the signs, recall that T-duality in the x1 , x2 fiber of our T 2 fibration of T , as automorphism of the Grassmannian of four-planes in H 1 (T0 , R) ⊕ H1 (T0 , R), acts by conjugation with the element σ = (diag(−1, −1, 1, 1), ?) ∈ SO(4)lef t × SO(4)right ⊂ O + (4, 4). Correspondingly, its action on H even (T0 , R) is given by the spinor representation s(σ ) of this group element. Since σ is a rotation by π , the square of s(σ ) is −?. We will argue that (up to an irrelevant overall sign)  υ 0 −→ µ1 ∧ µ2 −→ −υ 0 ,    υ −→ µ3 ∧ µ4 −→ −υ, γMS (T0 ) : (8)  µ ∧ µ 3 −→ µ2 ∧ µ3 −→ −µ1 ∧ µ3 ,   1 µ4 ∧ µ2 −→ µ1 ∧ µ4 −→ −µ4 ∧ µ2 , where the lower two lines of (8) are induced by µ1 → µ2 → −µ1 . This can be explained as follows. Above, we have described mirror symmetry by the sign change of two left-handed current components (j1 , j2 ) → (−j1 , −j2 ), whereas the right-handed components are unchanged. Since only the relative rotation by π between the two chiralities is important, one may as well consider the maps (j1 , j2 ) → (j2 , −j1 ) for the left-handed components and (¯1 , ¯2 ) → (−¯2 , ¯1 ) for the right-handed ones. By the above, this rotation is described by 0 12 0 −1 vms = exp(π /2), where = . ∈ o+ (4, 4), 12 = 1 0 12 0

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We use (5) and (6) to translate this into the half spinor representation and find   T 0 −ωˇ 12 0 12 /2), where 12 =  ωˇ 12 0 ω12  , sms = exp(π T 0 −ω12 0 ˇ B. Since this gives the and ω12 , ωˇ 12 are obtained from 12 as explained above for b, b, transformation with respect to the basis υ, µi ∧ µj , υ 0 , the mirror symmetry transformation given by sms can be written as  υ 0 −→ µ1 ∧ µ2 , µ1 ∧ µ2 −→ −υ 0 ,    υ −→ µ3 ∧ µ4 , µ3 ∧ µ4 −→ −υ, γMS (T0 ) : (9)  µ ∧ µ µ2 ∧ µ3 −→ µ2 ∧ µ3 , 3 −→ µ1 ∧ µ3 ,   1 µ4 ∧ µ2 −→ µ4 ∧ µ2 , µ1 ∧ µ4 −→ µ1 ∧ µ4 . To obtain the mirror map γMS (T0 ) corresponding to the original transformation (j1 , j2 ) → (−j1 , −j2 ), (¯1 , ¯2 ) → (¯1 , ¯2 ) one composes γMS (T0 ) with the classical symmetry (j1 , j2 ) → (j2 , −j1 ), (¯1 , ¯2 ) → (¯2 , −¯1 ) and finds (8). When we discuss orbifolds with respect to threefold rotations in fiber and base, the transformation (8) is not always applicable. For such rotations, the forms υ, υ 0 , µ1 ∧ µ2 , and µ3 ∧ µ4 are invariant, but not necessarily the others. In particular, the four plane (7) is not invariant. The transformation γMS (T0 ) is well behaved in all cases, however. When we work with orbifolds of the corresponding fibered tori we always keep µ1 , µ2 as generators of periods in the fiber and µ3 , µ4 as generators of periods in the base, such that γMS (T0 ) lifts to a symmetry of the conformal field theory which commutes with the symmetries used for orbifolding. We stress that γMS (T0 ), γMS (T0 ) are lattice automorphisms of order 4, and γMS (T0 )◦ γMS (T0 ) = −?. It has also been observed before that γMS (T0 ) can be understood as [B-B-R-P, Di1, B-S, A-C-R+ 1, A-C-R+ 2], in full agreement hyperk¨ahler rotation in M with the above. With respect to the complex structure I given by (e1 − ie3 ) ∧ (e2 + ie4 ), the second line in (7) corresponds to the complex structure and the first to its orthogonal complement. γMS (T0 ) exchanges the two, in accord with the general notion of mirror symmetry discussed in the Introduction. Moreover, the T 2 fibration with fiber coordinates x1 , x2 is elliptic with respect to the complex structure J given by (e1 + ie2 ) ∧ (e3 + ie4 ) (with a holomorphic section), and therefore it is a special Lagrangian with respect to I (cf. [H-L]). Hence (8) also describes a version of mirror symmetry in the sense of [S-Y-Z, Mo2], obtained from r1 → (r1 )−1 , r2 → (r2 )−1 , as in [V-W]. 3. The Mirror Map for Kummer Surfaces Recall the classical Kummer construction of K3: Given a four-torus T , we have a Z2 symmetry induced by multiplication with −1 on R4 . By minimally resolving the 16 singularities of the corresponding Z2 orbifold of T and assigning volume zero to all exceptional divisors in the blow up we obtain an orbifold limit of K3, a Kummer surface X. In particular, there is a rational map π : T −→ X of degree 2 which is defined outside the fixed points. The T 2 fibration of T used in Sect. 2 induces a T 2 fibration p : X −→ P1 which is elliptic with respect to π∗ J and therefore special Lagrangian with respect to π∗ I. Note that the holomorphic section is not the π∗ image of the section

Mirror Symmetry on Kummer Type K3 Surfaces

565

in our fibration of T . We rather have to make sure that the fibration can be written in the Weierstraß form, such that each singular fiber can be labeled by its Kodaira type [Ko1]. The Poincar´e duals of the generic fiber and generic section are given in (11) below. Apart from the behavior at the four singular fibers, mirror symmetry as discussed above is induced by mirror symmetry on the torus. This was in fact proven more generally in [V-W] and generalized to the Strominger/Yau/Zaslow conjecture in [S-Y-Z]. It again suffices to specialize to the standard torus T0 , consider the corresponding Kummer surface X0 , and determine the automorphism of the lattice L(X0 ) that acts as mirror symmetry at this base point. To this end, let us recall the description of L(X0 ) as found in [Ni, N-W]. The orbifolding map π induces an injective map π∗ on cohomology such that3 π∗ H even (T0 , Z) ∼ = H even (T0 , Z)(2). We√embed H even (T0 , Z)(2) even in H (X0 , Z) by rescaling L(T0 ) = H even (T0 , Z) with 2. With this convention we have H even (X0 , R) = π∗ H even (T0 , R) ⊥ spanR {Ei |i ∈ I }, where I labels the 16 fixed points of the Z2 orbifolding, and the Ei project to the Poincar´e duals of the exceptional divisors in the blow up of these fixed points (see below). Since each divisor is a rational curve of self-intersection number −2 on X0 , the Ei generate a lattice Z16 (−2) ⊂ H even (X0 , R). On I one finds an affine F24 geometry4 [Ni], which we use to label the fixed points. The four–plane x0 ⊂ H even (T0 , R) given in (7) remains unchanged. By we denote the so-called Kummer lattice := spanZ Ei , i ∈ I ; 21 Ei , H ⊂ I a hyperplane . i∈H

∼ Its projection = onto H 2 (X0 , Z) is the minimal primitive sublattice which i , i ∈ I, of exceptional divisors. contains all Poincar´e duals E Let Pj,k := spanF2 (fj , fk ) ⊂ F24 with fj ∈ F24 be the j th standard basis vector and Qj,k := Pl,m such that {j, k, l, m} = {1, 2, 3, 4}. Then5 [Ni] M := √1 µj ∧ µk − 21 Ei+l , l ∈ I and 2

i∈Qj,k

generate a lattice isomorphic to H 2 (X0 , Z). The lattice L(T0 )(2) = H even (T0 , Z)(2) and M belong to L(X0 ), but the Ei do not, since π∗ L(T0 ) ⊥ cannot be embedded as a sublattice into L(X0 ). Instead, L(X0 ) = ∪ 0 }, where spanZ {M √ := M ∪ i := Ei + √1 υ, i ∈ I M υ 0 := √1 υ 0 + 41 Ei + 2υ; E 2

and

2

i∈I

: π, m ∈ Z 0 := π ∈ | ∀ m ∈ M

(10)

i is the two-form contribution to Ei , and vice [N-W]. It is important to note that E i onto (π∗ H even (T0 , R))⊥ . versa Ei is the orthogonal projection of the lattice vector E The observation that this gives the unique consistent embedding of π∗ L(T0 ) into L(X0 ) 3 Given a lattice , by (n) we denote the same Z module as with quadratic form scaled by a factor of n. 4 As usual, F , p prime, denotes the unique finite field with p elements. p 5 We remark that in [N-W] we missed to exchange P j,k with Qj,k , which amounts to translation from homology to cohomology by Poincar´e duality.

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implies that the B-field in a Z2 orbifold CFT on K3 has value 1/2 in direction of each exceptional divisor of the blow up [As, N-W]. This observation generalizes to all orbifold , CFTs on K3 [We]. The lattice of two-form contributions to vectors in is denoted in the following. With the above description of H even (X0 , Z) one checks that the Poincar´e duals of generic fiber and generic holomorphic section in our elliptic fibration p : X −→ P1 are given by √ 2µ3 ∧ µ4 , √1 µ1 ∧ µ2 − 21 E(0,0,0,0) + E(0,0,1,0) + E(0,0,0,1) + E(0,0,1,1) ,(11) 2

respectively. We will now determine the induced mirror map γMS (X0 ) ∈ = Aut (L(X0 )) ∼ = + O (4, 20; Z). The geometric description of mirror symmetry implies that the action on L(T0 )(2) is induced by the action of γMS (T0 ) on L(T0 ). To extend it to all of L(X0 ) we have to find images of Ei , i ∈ I, in ⊗ Q, such that the induced linear map is an automorphism of . We claim that with arbitrary K0 , M0 ∈ F2 and t0 := (0, 0, K0 , M0 ) ∈ I the following map will do: ∀ (I, J, K, M) ∈ F24 : γMS (X0 )(E(I,J,K,M) ) :=

1 (−1)iI +j J E(i,j,K,M)+t0 . 2

(12)

i,j ∈F2

First, it is easy to see that this map preserves scalar products and acts as involution on 0 . Since 0 is generated by Ei ± Ej , i, j ∈ I, and 21 i∈H Ei , H ⊂ I a hyperplane, is mapped into one finds that (12) maps 0 into itself. Next, we check that M ⊂ M H even (X0 , Z). Namely, there are πa,b ∈ 0 such that for all I, J, K, L ∈ F2 , Ei+(I,J,K,M) γMS (X0 ) √1 µ1 ∧ µ3 − 21 2

= =

√1 µ2 2

∧ µ3 −

√1 µ2 2

∧ µ3 −

1 2

1 2

i∈Q1,3

(−1)iI E(i,0,K,m)+t0

i,m∈F2

Ei+(0,0,K,0)+t0 + I π1,3 ,

i∈Q2,3

γMS (X0 )

√1 µ4 2

∧ µ2 −

=

√1 µ1 2

∧ µ4 −

1 2

=

√1 µ1 2

∧ µ4 −

1 2

1 2

Ei+(I,J,K,M)

i∈Q2,4

(−1)j J E(0,j,k,M)+t0

j,k∈F2

Ei+(0,0,0,M)+t0 + J π2,4 ,

i∈Q1,4

and π(I,J,K,M) ∈ 0 with γMS (X0 ) √1 µ3 ∧ µ4 − 2

1 2

Ei+(I,J,K,M)

i∈Q3,4

(0,0,K,M)+t0 , = − √1 υ − E(0,0,K,M)+t0 = −E (10)

2

Mirror Symmetry on Kummer Type K3 Surfaces

γMS (X0 ) = − √1

2

567

∧ µ2 − 21 Ei+(I,J,K,M) i∈Q1,2 υ 0 − 41 E(0,0,k,m)+t0 + (−1)I E(1,0,k,m)+t0 √1 µ1 2

k,m∈F2

+(−1)J E(0,1,k,m)+t0 + (−1)I +J E(1,1,k,m)+t0 = − √1 υ 0 − 41 Ei + π(I,J,K,M) .

2

i∈I

Since γMS (X0 )|0 is an involution, and γMS (X0 ) ◦ γMS (X0 )|⊥ = −?, this suf0 fices to prove consistency. From the above it also follows that up to automorphisms of (π∗ H even (T0 , Z))⊥ ∩ H even (X0 , Z), (12) gives the only consistent maps γMS (X0 ). We will be more precise about this point at the end of Sects. 4 and 6. Let us consider the actual geometric action of γMS (X0 ). From the above we can eas ∪ 0 }. Hence we have in particular ily write out the map on H even (X0 , Z) = spanZ {M found an explicit continuation of mirror symmetry as induced by fiberwise T–duality to the four singular fibers of p : X −→ P1 over x3 ∈ {0, r3 /2}, x4 ∈ {0, r4 /2}, i.e. with labels (K, M) ∈ F22 . Each of these singular fibers is of type I0∗ in Kodaira’s classification [Ko1, Th.6.2] with components dual to (i,j,K,M) , i, j ∈ F2 , and CK,M := √1 µ3 ∧ µ4 − 1 i+(0,0,K,M) . E E 2 2

i∈Q3,4

4 type Dynkin diagram The latter form CK,M corresponds to the center node of the D describing I0∗ . For simplicity, let us set t0 = (0, 0, 0, 0). Since for suitable π(I,J,K,M) ∈ √ 4 0 and with the generator υ = 2υ of H (X0 , Z) (cf. [N-W]), (I,J,K,M) ) (10) γMS (X0 )(E = γMS (X0 ) √1 υ + E(I,J,K,M) 2 1 = √ µ3 ∧ µ4 + 21 E(0,0,K,M) + (−1)I E(1,0,K,M) 2 +(−1)J E(0,1,K,M) + (−1)I +J E(1,1,K,M) = CK,M + π(I,J,K,M) − δI,0 δJ,0 υ, √ (0,0,K,M) − 2µ3 ∧ µ4 , γMS (X0 )(CK,M ) = −E

we see that up to signs and possible corrections in 0 and π∗ H even (T0 , Z), γMS (X0 ) 4 with each of its four legs. exchanges the center node of I0∗ ∼ =D Note also that for t0 = (0, 0, 0, 0) the Poincar´e duals (11) of generic fiber and generic section of our elliptic fibration are simply mapped onto − υ , − υ0 + υ under γMS (X0 ), 0 4 0 where υ, υ are the generators of H (X0 , Z), H (X0 , Z), respectively (see (10) and [N-W]). Next, let us investigate the monodromy [m], m ∈ SL(2, Z), of the regular two-tori in our fibration p : X −→ P1 around a singular fiber, where [m] denotes the conjugacy class of m in SL(2, Z). Since γMS (X0 ) acts by S ∈ SL(2, Z) on the modular parameter of the fiber, 0 −1 S= , 1 0

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the monodromy should transform as

m

−→

mT

−1

=

SmS −1 ,

(13)

i.e. [m] should remain invariant under mirror symmetry. In the present case, all singular fibers are of type I0∗ , hence by [Ko1, Th.9.1] their monodromy is −1 0 m= , (14) 0 −1 which is even invariant under (13). This is in accord with our construction, since our geometric interpretation of a K3 theory obtained as a Z2 orbifold on a Kummer surface with B = 0 for the underlying toroidal theory is indeed mapped into another such geometric interpretation. The mirror K3 is hence expected to be a singular Kummer surface again, with singular fibers of type I0∗ , and with monodromy (14) around each of them. 4. The Mirror Map for Kummer Type Surfaces In this section, we give the action of mirror symmetry on Kummer type K3 surfaces obtained as orbifold limits T /ZN , N ∈ {3, 4, 6} of K3. Since the proofs are analogous to the one for N = 2 that has been discussed at length in the previous section, we restrict ourselves to a presentation of the results. For explicit proofs, one needs the description of L(X0 ) in terms of π∗ L(T0 )ZN ∼ = L(T0 )ZN (N ) and the exceptional divisors obtained by minimally resolving all singularities, as given in [N-W, We]. Recall the ZN orbifold construction of K3, N ∈ {3, 4, 6}. Let ζN denote a generator of ZN , where ZN is realized as a group of N th roots of unity in C, and z1 , z2 complex coordinates on T , such that T = T2 × T 2 with elliptic curves T2 , T 2 . If both curves are ZN symmetric (note that the metric in general need not be diagonal with respect to z1 , z2 ), there is an algebraic ZN action on T given by ZN ζNl :

ζNl .(z1 , z2 ) = (ζNl z1 , ζN−l z2 ).

By minimally resolving the singularities of T /ZN and assigning volume zero to all components of the exceptional divisors we obtain a ZN orbifold limit X of K3. The fixed point set of ZN on T will be denoted I in general, and n(t) is the order of the stabilizer group of t ∈ I . For N = 3 we have I ∼ = F32 , and since the Z4 , Z6 orbifold limits can be obtained from the Z2 , Z3 orbifold limits by modding out an algebraic Z2 action, for N = 4, 6 we use I = F24 / ∼ and I = F24 ∪ F32 / ∼, respectively, where ∼ denotes the necessary identifications. To assure compatibility of the ZN action with our fibration p : T −→ T 2 , we choose z1 = x1 + ix2 , z2 = x3 + ix4 and obtain an induced fibration p : X −→ P1 as in the Z2 case. We can also restrict considerations to () () appropriate standard tori T0 , T0 for N = 4, N ∈ {3, 6}, respectively, X0 = T0 /ZN . Here, T0 = R4 / spanZ {e1 , . . . , e4 } with an orthonormal basis e1 , . . . , e4 as introduced in Sect. 2, and √ µ1 = e1 , µ2 = 21 e1 + 23 e2 , 4 √ T0 = R / spanZ {λ1 , . . . , λ4 } with µ3 = e3 , µ4 = 21 e3 + 23 e4

Mirror Symmetry on Kummer Type K3 Surfaces

569

for the dual basis µ1 , . . . , µ4 . The Z3 action then is given by ζ3 :

µ1 − → µ 2 − µ1 , µ3 − → −µ4 ,

µ2 − → −µ1 , µ4 − → µ3 − µ4 .

The exceptional divisor in the minimal resolution of a Zn(t) type fixed point t ∈ I has n(t) − 1 irreducible components with intersection matrix the negative of the Cartan matrix of the Lie group An(t)−1 . The projections to (π∗ L(T0 )ZN )⊥ of the Poincar´e duals (l) of these (−2) curves are denoted Et , l ∈ {1, . . . , n(t) − 1}, such that

(l)

(m)

Et , Et

  −2 for l = m, 1 for |l − m| = 1, =  0 otherwise.

We define Et :=

n(t)−1

(l) lEt ,

(0) Et

l=1

:= −

n(t)−1

(l)

Et ,

l=1 (l)

as well as a Zn(t) action on Et := spanZ {Et , l ∈ {1, . . . , n(t) − 1}} generated by (l) ϑ(Et )

:=

(l+1)

Et

if l < n(t) − 1,

(0) Et

if l = n(t) − 1,

(0)

such that (ϑ − 1)Et = n(t)Et .

(15)

(l)

As in Sect. 3 it suffices to determine the image of each Et under the extension of the mirror map of the underlying toroidal theory. For the Z4 orbifold, the latter is γMS (T0 ) as in (8), and for the Z3 and Z6 orbifolds we have to use γMS (T0 ) as defined in (9). We list the singular fibers of p : X −→ P1 in terms of Kodaira’s classification [Ko1, Th.6.2], all of which are non-stable. Recall also the labels of the corresponding extended Coxeter Dynkin diagrams: Z3 : I V ∗ + I V ∗ + I V ∗ , Z4 : I0∗ + I I I ∗ + I I I ∗ , 6 , I0∗ ∼ 4 , I I I ∗ ∼ 7 , I I ∗ ∼ 8 . IV ∗ ∼ =E =D =E =E

Z6 : I0∗ + I I ∗ + I V ∗ ;

()

By construction, our map γMS (X0 ) acts fiberwise (cf. (12)), hence it suffices to specify the map on each type of singular fibers. Accordingly, fixed points are only labeled by the fiber coordinates x1 , x2 in the following. Fibers of type I0∗ have been discussed in Sect. 3. Type I V ∗ occurs in both Z3 and Z6 orbifolds and contains three A2 type exceptional divisors with components Poincar´e (l) dual to Et , l ∈ {1, 2}, t ∈ F3 . We find t ∈ F3 , l ∈ {1, 2} :

γMS (X0 )(Et ) = − 13 (l)

ϑ l+kt (Ek ),

k∈F3

where we have chosen an origin 0 ∈ F3 and the standard scalar product on F3 .

(16)

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W. Nahm, K. Wendland

For I I I ∗ , which occurs in the Z4 orbifold, we have two A3 type exceptional divi(l) sors giving Ei , l ∈ {1, 2, 3}, i ∈ {(0, 0), (1, 1)}, and one A1 type exceptional divisor corresponding to E(1,0) . Then (l) ϑ l+2ik (E(k,k) ), i ∈ F2 , l ∈ {1, 2, 3} : γMS (X0 )(E(i,i) ) = − 21 (−1)l+i (E(1,0) ) − 41 γMS (X0 )(E(1,0) ) =

− 41 (ϑ

+ϑ ) 3

k∈F2

k

(17)

ϑ (E(k,k) ).

k∈F2

Finally, for I I ∗ we have one A5 , A2 , A1 type exceptional divisor each, corresponding (l) (l) to E0 , l ∈ {1, . . . , 5}, E1 , l ∈ {1, 2}, E(1,0) . Here, l ∈ {1, . . . , 5} : l ∈ {1, 2} :

γMS (X0 )(E0 ) = − 21 (−1)l (E(1,0) ) − 13 ϑ l (E1 ) − 16 ϑ l (E0 ), (l)

γMS (X0 )(E1 ) = 13 ϑ l (E1 ) − 16 (ϑ 0 + ϑ 3 )ϑ l (E0 ), (l)

γMS (X0 )(E(1,0) ) = − 21 E(1,0) − 16 (ϑ + ϑ 3 + ϑ 5 )(E0 ).

(18)

To prove the above, one has to check that scalar products are preserved and that the () generator υ 0 of H 0 (X0 , Z) (see (22)) is mapped onto a lattice vector as well as √ t(l) = Et(l) + 1 υ = Nυ (19) ∀ t ∈ I, l ∈ {1, . . . , n(t) − 1} : E n(t) υ , √ (l) () () (recall Et ∈ L(X0 ), in general, and that υ = N υ generates H 4 (X0 , Z); see [N-W, We] and compare to (10)). This is a straightforward calculation using [We]. For later convenience, (12), (16)-(18) can be summarized in the following formula: t ∈ I a fixed point of type Zn(t) : () (L) γMS (X0 )(Et )

=

− N1

(20)

ϑ

l+kt

(Ek ),

0≤l
where for n(t) = 2 we set Et := Et , and in the sum over k ∈ fiber each fixed point on () the torus is counted separately. As noted in Sect. 2, γMS (T0 ) is a lattice automorphism () of order 4, in fact a hyperk¨ahler rotation. For our extension (20) of γMS (T0 ) to the Kummer type lattice N generalizing we find γMS (X0 ) ◦ γMS (X0 )|N = −ϑ −1 ι, ()

()

where for t ∈ I, l ∈ {1, . . . , n(t) − 1} : ()

(21) ι:

(l) Et

−→

(l) E−t .

Hence γMS (X0 ) has order 4, 12, 8, 12 for N = 2, 3, 4, 6, respectively. For all fibers discussed above, the geometric action is analogous to that on I0∗ : Up () N , the generalization of to signs and possible corrections in π∗ H even (T0 , Z)ZN and () , γMS (X ) exchanges the center node of the extended Dynkin diagram that describes 0 the fiber with each component of its long legs. () () Of course, γMS (X0 )|N is not uniquely determined by the two conditions γMS (X0 ) () () () () ∈ Aut (L(X0 )) and γMS (X0 )|⊥ = γMS (T0 ). In fact, assume that γMS (X0 )g is N

Mirror Symmetry on Kummer Type K3 Surfaces

571

()

another consistent extension of γMS (T0 ). Then by definition g|⊥ = ?, and g acts as a N

()

lattice automorphism both on L(X0 ) and N . It is clear that g can incorporate arbitrary shifts of the fixed points in the direction of the base in our fibration p : X −→ P1 . In case N = 2 this corresponds to the freedom of choice of t0 ∈ I in (12). One checks that among the lattice automorphisms that permute fixed points, g must respect our fibration and therefore can only incorporate shifts on I or the map ι : t → −t, t ∈ I . The latter is nontrivial only in the Z3 case I ∼ = F32 , where it corresponds to the standard Z2 action on the underlying torus, i.e. the algebraic automorphism that is modded out from T 0 /Z3 to obtain T0 /Z6 . Having said this and using the form of (20), in the following (l) we may assume that g acts as a lattice automorphism on each Et = spanZ {Et , l ∈ {1, . . . , n(t) − 1}} separately. Since g is orthogonal, it must permute the roots in Et . Recall from [We, Th. 3.3] that υ0 =

√1 υ N

−

1 N BN

+ υ,

(l)

where BN ∈ N : ∀ t ∈ I, l ∈ {1, . . . , n(t) − 1} : BN , Et

(22) = 1.

t ) ∈ L(X ) with (19) together with the fact that g preserves scalar Then g(E 0 products implies g|Et = ϑ kt for some kt ∈ {1, . . . , n(t)}. Moreover, g must obey (l)

()

g( υ0) − υ0 =

1 N (1 − g)BN

∈ N ,

which restricts the possible combinations of kt that define g. In fact, with the results of [We] about the form of N , all in all we find that γMS (X0 ) is given by (20) up to the above mentioned permutations in I and the action of some g b ∈ Aut (N ) : g|bEt = ϑ kt , g b N1 BN = N1 BN + b, b ∈ N Et . (23) t∈I

For N ∈ {2, 3} these degrees of freedom are parametrized by Aff (I, FN ), since we have a natural identification gb

1:1

←→

k b ∈ Aff (I, FN ) :

g|bEt = ϑ k

b (t)

.

We will give an interpretation of this result at the end of Sect. 6. Note that by the above () () () () for any a ∈ Z, γMS (X0 )ϑ a = ϑ a γMS (X0 ) are extensions of γMS (T0 ) to L(X0 ) as well. Hence (21) shows that we can define mirror maps of order 4 on the Z3 and Z6 orbifold limits of K3. () We can also easily check the action of γMS (X0 ) on the monodromy around the singular fibers. Again from [Ko1, Th.9.1] we read off the monodromy matrices and -apart from I0∗ - find invariance under (13) for I I I ∗ type fibers only. It follows that the geometric interpretation as a Z4 orbifold limit obtained from a toroidal theory on a rectangular torus with vanishing B-field is mapped to another such geometric interpretation under mirror symmetry, as expected. For the Z3 and Z6 cases, on the other hand, the analogous statement is not true. Though the conjugacy class of the monodromy remains unchanged under γMS (X0 ) by definition, its representative changes under (13). This might not be surprising because of the modification (9) by a classical symmetry of order 4 that we had to perform on the hyperk¨ahler rotation γMS (T0 ) to gain our lattice automorphism γMS (T0 ). It also seems to be due to differences in the complex structure. Namely, for

572

W. Nahm, K. Wendland

rectangular tori one has a purely imaginary complex structure on the fiber, but for Z3 or Z6 orbifolds this is impossible. The images under mirror symmetry will not have purely imaginary K¨ahler parameters, in other words will have a non-vanishing B-field. It might be interesting to explore this point, which suggests a link between monodromy and complexified K¨ahler structure within classical geometry. In particular, it suggests that there are interesting subtleties resulting from a transition from the universal covering space of our moduli space to M. M 5. The Mirror Map for ZN Orbifold Conformal Field Theories ()

By our discussion in the Introduction, γMS (X0 ) is to be interpreted as an automorphism of the moduli space of N = (4, 4) superconformal on the smooth universal cover M and m field theories on K3 that identifies equivalent theories. Let m be a point in M its image under this automorphism. Given a path between m and m one can deform is smooth and simply connected, the deformation the CFT along this path. Since M (see e.g. [Ku]) is defined up to an element of the holonomy group of the moduli space. The quantum numbers of the fields, in particular the conformal dimensions change continuously under deformations, but in general the deformation is defined up to linear transformations among fields of identical quantum numbers. When there are no degeneracies, this just amounts to an arbitrary wave function renormalisation. When the conformal dimension of a field changes along a path, the perturbation integrals for the change of its n-point functions are logarithmically divergent and need a regularisation, which introduces the arbitrariness just mentioned. For chiral fields or their spectral flow partners, like the twist fields, such a logarithmic divergence does not occur. Their Since holonomy is trivial, too, as long as we deform within an orbifold component of M. our version of mirror symmetry is induced by T-duality on a c = 3 toroidal subtheory of the underlying c = 6 toroidal superconformal field theory, we can determine the action () of γMS (X0 ) independently from the geometric results of Sects. 3 and 4. This is the aim of the present section, where it suffices to restrict to the bosonic subsector of each of our theories. We remark that our technique is similar to that used in [B-E-R]. Orbifold CFTs have a generic W-algebra given by the invariant part of the current generated algebra on the underlying torus. The Hilbert space of such a theory decomposes into the ZN invariant W-algebra representations of the underlying toroidal theory and the so–called twisted sector. The latter consists of W-algebra representations with infinite quantum dimensions, the ground states of which are given by the twist fields. In the following, these twist fields are denoted Ttl , l ∈ {1, . . . , n(t) − 1}, where we have chosen Tt of order n(t) for each t ∈ I . We normalize them such that −l l Ttl , Ttl = lim |x| Ttl (x)Tt−l (24) (0) = δt,t δl,l 2 − ζn(t) − ζn(t) . x→0

Here ·, · denotes the standard scalar product on the Hilbert space which on (1, 1) fields induces the Zamolodchikov metric. In Sect. 6, we will see that the normalization (24) is the natural one. Twist fields and all the fields of the torus theory can be included together in n-point functions, if one admits ramifications of the world sheet at the twist field positions [H-V, D-F-M-S]. Denote by V (p, z) the vertex operator of momentum p = (pl , pr ) and con¯ = (p 2 /2, pr2 /2), where in the notations of Sect. 2 we have formal dimensions (h, h) l 1 p = (pl , pr ) = √ (Ql , Qr ), i.e. √1 (Ql + Qr , Ql − Qr ) ∈ v( , B)Z4,4 . The OPE of 2

2

Mirror Symmetry on Kummer Type K3 Surfaces

573

V (p, z) with the twist fields Ttl (x), t ∈ I , has a contribution given by the twist fields themselves. We write it in the form ¯ ¯ −h V (p, z)Ttl (x) = (z − x)−h (¯z − x) Wtt (p)Ttl (x) + other terms . (25) t ∈I, n(t)l =ln(t )

We will read off the commutation relations of the matrices W (p) from the four-point function

l V (p, z)V (p , z )Ttl (x)Ttl (y) = e(z, z , x, y)O(z, z , x, y)Tt−l (x)Tt (y), ¯

¯

e(z, z , x, y) := (z − x)−h (¯z − x) ¯ −h (z − x)−h (¯z − x) ¯ −h ¯

¯

× (z − y)−h (¯z − y) ¯ −h (z − y)−h (¯z − y) ¯ −h . The function O can be written as the product of two functions which are analytic resp. anti-analytic in their corresponding domains. Let us assume that Ttl , Ttl both correspond to Zn twists θ, in particular n = n(t)/ gcd(l, n(t)) = n(t )/ gcd(l , n(t )),

such that by moving z in a loop around x in Ttl (x) or around y in Ttl (y) in the opposite sense, V (p, z) becomes V (θp, z). Hence the V (p, z) are well defined on the nfold cover of the Riemann sphere, which is another Riemann sphere with coordinate ξ = ((z − x)/(z − y))1/n , and analogously η = ((z − x)/(z − y))1/n . For fixed x, y and the appropriate domains for z, z , the function O(z, z , x, y) is analytic on the cover, with poles given by the known OPE of the vertex operators. This yields n ! k k O(z, z , x, y) = o(x − y) (ξ − ζnk η)pl θ pl (ξ¯ − ζn−k η) ¯ pr θ pr , k=1

where ζn := exp(2πi/n) and o(x − y) is easy to calculate but irrelevant for our purpose. Taking the limits z → x and z → x in different orders one finds n !

W (p )W (p) =

kpθ k p

ζn

φ (p,p )

W (p)W (p ) = ζn n

W (p)W (p ),

k=1

where φn (p, p ) :=

n

kpθ k p and qq = ql ql − qr qr = Ql Qr + Qr Ql (2)

(26)

k=1

(see [Le, N-S-V]). Hence the W (p) form a Weyl algebra which is represented on the vector space spanned by the twist fields. For a given geometric interpretation, where we assume B = 0 on the underlying toroidal theory, the momentum state vertex operators are characterized by pl = pr . They therefore form an Abelian subalgebra of the Weyl algebra (26). With respect to this subalgebra, the representation decomposes into one-dimensional subrepresentations with ground states Ttl , each of which corresponds to a Zn(t)/ gcd(l,n(t)) twist on a Zn(t) type fixed point.

574

W. Nahm, K. Wendland

To see this explicitly note that for a geometric interpretation with B = 0 we have p = (pl , pr ) = with

√1 (µ + λ, µ − λ) 2

=: p(µ, λ)

(µ, λ) ∈ spanZ {µ1 , . . . , µ4 } ⊕ spanZ {λ1 , . . . , λ4 }.

Now (26) takes the form φn (p, p ) ≡

n k µθ k λ − µ θ k λ mod nZ.

(27)

k=1

In fact, the fixed point set I can be interpreted as part of the subgroup of elements of order n with n | N in the Jacobian torus H1 (T0 , R)/H1 (T0 , Z), where H1 (T0 , Z) ∼ = spanZ {λ1 , . . . , λ4 }, and we can use I → H1 (T0 , Q)/H1 (T0 , Z),

where nλ ∈ spanZ {λ1 , . . . , λ4 } for [λ] ∈ I.

Then (25) reads, after suitable normalizations of the vertex operators, ¯

¯ −h V (p(µ, λ), z)Ttl (x) = (z − x)−h (¯z − x)

lµ(nt)

ln(t )/n(t)

ζn(t) Tt

(x) + other terms ,

n(t)−1 " # " # 1 t := t + (1 − θ )−1 λ = t − n(t) kθ k λ ,

(28)

k=1

θ n = ?,

n = gcd(l, n(t)),

lφ (p,p )

W (p )W (p)Ttl = ζn(t)n

which indeed yields

W (p)W (p )Ttl .

()

Now recall that γMS (T0 ) as discussed in Sect. 2 is given by T-duality on the x1 , x2 directions of the torus T , which agrees with the Fourier-Mukai transform on the corresponding two-torus [Na1, Na2, Sc, B-vB], see also [Th1]. The standard Fourier transform on Rd extends to all measurable Abelian groups. On ZN it takes the form (N) : CN −→ CN , F

N jk N (N) (fk )N √1 F fj ζN = k=1 N

j =1

k=1

.

(N) to ZN ⊂ U (1) ⊂ C∗ on In the present context, however, it is natural to restrict F each component. Let tk ∈ ZN denote a generator in the k th component, then one has F (N) (tkl ) =

√1 N

N

j kl

tjl ζN ,

j =1

which for simplicity we keep on calling discrete Fourier transform in the following. Under T-duality, momentum states and winding states are interchanged, and the latter are characterized by pl = −pr . The ground states of the one-dimensional subrepresentations of the subalgebra of winding states are therefore obtained from the generators

Mirror Symmetry on Kummer Type K3 Surfaces

575

{Ttl , t ∈ I, l ∈ {1, . . . , n(t) − 1}} of the twisted sector by performing a ZN type discrete Fourier transform. By the above, the resulting map on twist fields has the general form

l ∈ {1, . . . , n(t) − 1} : F (Ttl ) =

ln(j )/n(t) ltj ζn(t) ,

l an(j ),n(t) Tj

(29)

j ∈ fiber: n(t)|ln(j )

with ∀ λ ∈ C :

F (λTtl ) = λF (Ttl )

l and the coefficients an(j ),n(t) to be determined. Here, we use the labeling of fixed points that was introduced in Sect. 4 and which is restricted to the fiber coordinates x1 , x2 . As in (20), each fixed point on the torus contributes separately to the sum j ∈ fiber. When N factorizes, the orbifolding can be done stepwise. Therefore, kl −1/2 l an,kn an,n , = k

l 1/2 l akn,n an,n . = k

and

1 to be determined. Since F must conserve the normalization, countThis leaves only an,n ing fixed points gives 1 −2 | = 2 − ζn − ζn−1 . |an,n

Up to a finite ambiguity, the phase can be determined from the order of F . Instead of a cumbersome direct determination, we fix it by consistency requirements with the results of Sects. 3,4 (see Sect. 6). This yields: l an,n

=

− √1 nn

n−1

mζnlm =

$

l −1 n n (1 − ζn ) .

m=1

Note in particular that n(t)−l F Ttl = F Tt = F (Ttl ). Explicitly we have in the Z2 case: t ∈ F22 :

F (Tt ) =

1 2

(−1)tj Tj ,

j ∈F22

in the Z3 case: t ∈ F3 :

iω F (Tt ) = − √

3

j ∈F3

ωtj Tj , ω = − 21 + i

√

3 2 ,

in the Z4 case: t ∈ F2 :

T(0,0) + (−1)t T(1,1) , √ 2 2 2 F (T(t,t) ) = 21 2(−1)t T(0,1) + T(0,0) + T(1,1) , 2 2 F (T(0,1) ) = √1 T(0,0) , − T(1,1) F (T(t,t) ) =

1+i 2

2

576

W. Nahm, K. Wendland

in the Z6 case: √

F (T0 ) = ω T0 , ω = 21 + i 23 , √ √ iω T02 + 2T1 , ω = − 21 + i 23 , F (T02 ) = − √ 3 √ 1 3 F (T0 ) = 2 T03 + 3T(1,0) , √ iω T1 − 2T02 , F (T1 ) = √ 3 √ F (T(1,0) ) = − 21 T(1,0) − 3T03 .

6. Mirror Symmetry, Discrete Fourier Transform, and the McKay Correspondence ()

In Sects. 4 and 5 we have independently derived the action of mirror symmetry γMS (X0 ) both in terms of geometric data (20) and CFT data (29). Now we shall relate the two approaches and give an interpretation in terms of the classical McKay correspondence. In general, we assume that the data of the ZN orbifold CFT as discussed in Sect. 5 possess a geometric interpretation on our ZN orbifold limit of K3 as discussed in Sect. 4. In view of our results on mirror symmetry, we may expect a linear map that intertwines between the pictures. More precisely, on the CFT side the twist fields Ttl generate defor(l) mations of the theory, which act linearly on the cycles Poincar´e dual to the Et . Thus they should naturally correspond to linear combinations of these cocycles. We denote the induced linear map from cocycles to twist fields by C. This map should act isometrically with respect to the standard metrics (24) and (4) up to a global sign, and it should obey ∀ t ∈ I, l ∈ {1, . . . , n(t) − 1} :

(l)

()

(l)

F C(Et ) = C γMS (X0 )(Et ).

(30)

On the CFT level, the sectors of a ZN orbifold theory carry distinct representations of the dual of ZN , with the untwisted sector corresponding to the trivial representation. In particular, one obtains nontrivial representations on all twist fields. More precisely, we have an action of the dual of Zn(t) on the twist fields Ttl at a given fixed point t ∈ I . In Sect. 5 we have already labeled the twist fields such that the action is generated by l multiplication with ζn(t) on Ttl . By abuse of notation we denote the generator of this Zn(t) action by ϑ. It gives the “quantum symmetry” of the orbifold conformal field theory that replaces the “geometric symmetry” on the underlying toroidal theory. It is well known that by modding out the orbifold CFT by the total “quantum” ZN symmetry we can reproduce the original toroidal theory. Geometrically, each fixed point t ∈ I is left invariant by a subgroup Zn(t) of ZN . This (l) induces an action of the dual of Zn(t) on Et = spanZ {Et , l ∈ {0, . . . , n(t)−1}}. Indeed, its generator ϑ has been introduced in (15) above. The map C can therefore be obtained by decomposing the integral Zn(t) action on Et into one-dimensional representations. We claim that this yields t ∈ I, l ∈ {1, . . . , n(t) − 1} :

(l)

C(Et ) :=

√1 n(t)

n(t)−1 k=1

lk ζn(t) Ttk .

(31)

Mirror Symmetry on Kummer Type K3 Surfaces

577

Together with (20), (29) one first checks (30). Moreover, the reality condition ∀ t ∈ I, l ∈ {1, . . . , n(t) − 1} :

(l)

(l)

C(Et ) = C(Et ),

n(t)−l

translates into Ttl = Tt , as expected. Finally, up to a prefactor, the Hermitian form on Et is given by the intersection product (4) on the cocycles, thus by the Cartan matrix of the extended Dynkin diagram. Since the latter can be written as 2 − ϑ − ϑ −1 , it commutes with ϑ and becomes diagonal in the basis Ttl of eigenvectors of ϑ. With C a unitary matrix with respect to this basis, −l l − ζn(t) of the the squared norms of the Ttl yield the corresponding eigenvalues 2 − ζn(t) Cartan matrix, in agreement with (24). All in all, (31) is checked to give an isometry, up to a global sign, on the entire twisted sector. Now (31) shows that (28) in fact induces an action of spanZ {µ1 , . . . , µ4 } ⊕ spanZ {λ1 , . . . , λ4 } on N . The action of spanZ {µ1 , . . . , µ4 } is given by ∀ µ ∈ spanZ {µ1 , . . . , µ4 }, t ∈ I, (l)

W (µ, 0)Et

l ∈ {1, . . . , n(t)−1} : (µ(nt)+l) (15)

(l)

= ϑ µ(nt) Et .

:= Et

(32)

The action of spanZ {λ1 , . . . , λ4 } is more complicated, at least in this basis. It translates directly into a natural action on the basis t ∈ I, l ∈ {1, . . . , n(t) − 1} :

C −1 (Ttk ) = √ (31)

n(t)−1 1 (l) ζ lk Et , n(t) l=0 n(t)

along with the corresponding Weyl algebra representation given in Sect. 5. In order to work with geometric objects, recall from our discussion in Sect. 4 that rather than the N onto H 2 (X () , Z). By υ0, υ we denote lattice N we have to consider its projection 0 () () 0 4 the generators of H (X0 , Z), H (X0 , Z), respectively, and recall from (19) that ∀ t ∈ I, l ∈ {1, . . . , n(t) − 1} :

t − Et = E (l)

(l)

1 υ. n(t)

Hence the Zn(t) symmetry (15) on the irreducible components of the exceptional divisor over a given fixed point t ∈ I actually is  t(l+1) if l < n − 1, E  (l) n−1 t (j ) ϑ E = t if l = n − 1, υ− E   j =1

i.e. ϑ( υ) = υ . By the above, the “quantum” symmetry ϑ has a straightforward geometric meaning. t(l) , l ∈ υ 0 and consider the induced action on the basis dual to {E We also set ϑ( υ0) = n−1 (j ) t(l) )∗ } ⊂ spanQ {E t(l) } the dual basis with respect t }: With {(E {1, . . . , n(t)−1}, υ− E j =1

t(l) } let t(l) } of Et = spanZ {E to the fundamental system {E 0

if l = 0, υ (l) (l+n(t)) (l) εt := = εt . ∀ l ∈ Z : εt t(l) )∗ if 0 < l < n(t) , υ 0 + (E

(33)

578

W. Nahm, K. Wendland

By (32), {ε (0) , . . . , ε(n(t)−1) } carries the representation analogous to (32), which obeys µ ∈ spanZ {µ1 , . . . , µ4 }, (l)

t ∈ I, l ∈ {1, . . . , n(t) − 1} : (µ(nt)+l)

W (µ, 0)εt = εt

n = n(t),

,

(34)

W (µ, λ)W (µ , λ ) = ϑ φn (p,p ) W (µ , λ )W (µ, λ), p = p(µ, λ), p = p(µ , λ ) with φn (p, p ) as in (27). This is readily interpreted in terms of ZN equivariant topological K-theory on the underlying torus T . Namely, consider a ZN equivariant flat line bundle L on T , then by [B-K-R] there is a corresponding line bundle π∗ L on our K3 surface X = T /ZN . More precisely, L is determined by the representation χt of ZN in each fiber over a fixed point t ∈ I . Then c1 (π∗ L) = c1 (χt ), t∈I

where c1 (χt ) is given by the classical McKay correspondence [Mc1, Mc2, G-S-V, Kn]: (l) t(l) )∗ , c1 (χt ) = (E

(l)

if χt

(l)

: C −→ C,

l χt (z) = ζn(t) z.

(l)

Hence εt , t ∈ I, l ∈ {0, . . . , n(t) − 1} as in (33) are the H 0 (X, Z) ⊕ H 2 (X, Q) components of the Mukai vectors $ " # E υ = [rkE] υ 0 + c1 (E) + (c2 − 21 c12 )(E)[X] + rk ch(E) A(X) 48 p1 (X) for all possible contributions E to π∗ L over t ∈ I . The coefficient of υ does not enter into the relation between line bundles and representations. () We therefore find that the action (34) of spanZ {λ1 , . . . , λ4 } ∼ = H1 (T0 , Z) agrees () () with a non-standard action of the subgroup I ⊂ H1 (T0 , R)/H1 (T0 , Z) of the Jacobian torus on ZN equivariant flat line bundles on T . It looks natural in the dual of the basis C −1 (Ttk ). Moreover, µ ∈ spanZ {µ1 , . . . , µ4 } acts by tensoring a line bundle on T with the bundle Lµ associated to χ , where z ∈ (Lµ )t , t ∈ I :

µ(n(t)t)

χ (z) = ζN

z.

Summarizing, (31) gives an isometry between the C-vector space spanned by the () twist fields of an orbifold CFT and a subspace of H ∗ (X0 , C), which allows us to identify the natural action of the Weyl algebra (26) on twist fields with an action of the same Weyl algebra on the ZN equivariant flat line bundles on T . It is interesting to note that the action of the Jacobian torus is rather non-trivial. It has been known before, of course, that such identifications should be possible [Wi2, Ga, D-G1, B-G-K, dB-D-H+ ]. Recall from (23) that our formula (20) was only determined up to certain permutations of the fixed points and an % action of g b that is% characterized by inducing an integral B–field shift by some b ∈ N / Et . Since N / Et , and for N ∈ {2, 3} equivalently t∈I

t∈I

Aff (I, FN ) parametrizes ZN equivariant flat line bundles on T , this freedom of choice translates nicely into possible choices of the origin in that parameter space. We have seen, though, that the choice of g b may influence the order of the resulting γMS (X0 ). We

Mirror Symmetry on Kummer Type K3 Surfaces

579

find that it then corresponds to a twisting of our mirror map, given by some equivariant flat line bundle on T . Finally, let us briefly explain the connection of our results to Chen and Ruan’s orbifold cohomology [C-R] on [T /G] as discussed in [Ru1, F-G]. In the present case of cyclic G = ZN it is isomorphic to n(t)−1

∗ HCR ([T /ZN ], C) ∼ = H ∗ (T , C)ZN ⊕

CT (t)ζ l , t∈I

l=1

n(t)

where the generators T (t)ζ l of the twisted sector corresponding to the fixed point t ∈ I n(t) have scalar product T (t)ζ l , T (t )ζ l := δt,t δl,−l . (35) n(t)

n(t )

In fact, if instead of (35) one uses the associated sesquilinear form6 and slightly different normalizations to identify T (t)ζ l with Ttl , it agrees with our metric in the n(t) twisted sector of the orbifold CFT. More importantly, this means that (31) can be used to prove that for all orbifold limits discussed in this note ∗ HCR ([T0 /ZN ], C) ∼ = H ∗ (X0 , C) ()

()

as metric spaces, confirming part of Ruan’s conjecture [Ru1, Conj.6.3] for these cases. 7. Conclusions In this note, we have analyzed a version of mirror symmetry on elliptically fibered K3 surfaces that is induced by fiberwise T-duality on nonsingular fibers. It is straightforward to determine this map for four-tori, which enables us to give the explicit action on ZN orbifold limits of K3, N ∈ {2, 3, 4, 6}, by the techniques of [N-W, We]. In the present case of N = (4, 4) superconformal field theories with central charge c = 6 the mirror map can be realized as automorphism on the lattice of integral cohomology on the underlying complex surface (torus or K3). While the order of mirror symmetry on the torus is 4, it can take values 4, 8, or 12 on our orbifold limits of K3. On the CFT side, the mirror map is induced by a ZN type discrete Fourier transform in the fiber acting on the twist fields of the orbifold conformal field theory. Since we are able to derive the mirror map in both the geometric and conformal field theoretic description independently, we can deduce the exact geometric counterparts of orbifold CFT twist fields. Moreover, the correspondence between the twist fields and ZN equivariant flat line bundles can be deduced directly. The natural “quantum” ZN symmetry in the twisted sector of the orbifold CFT, which can be modded out to retain the original toroidal theory, gains geometric meaning. In fact, by the classical McKay correspondence it can be traced back to properties of singularities already investigated in [Mu, Hi]. Our version of mirror symmetry agrees with the one proven more generally in [V-W], which was generalized to the celebrated Strominger/Yau/Zaslow conjecture [S-Y-Z]. We have avoided M-theory language, though, and restricted considerations to the underlying 6 As remarked in Footnote 1, this was explained to us by Yongbin Ruan [Ru2], and goes back to earlier observations by Edward Witten.

580

W. Nahm, K. Wendland

geometry. Together with the rich structure of the K3 moduli space this enables us to carry out the construction away from large volume or large complex structure limits in the moduli space and even without touching the issue of its compactification. Note that all our T 2 fibrations p : X −→ P1 have non-stable singular fibers. A large complex structure limit in the sense of [G-W2] has therefore not even been defined for those cases we are interested in. Our results on the explicit prescription for the identification of conformal field theoretic and geometric data might be of interest in their own right. We have pointed out how they resolve the objection in [F-G] to Ruan’s conjecture [Ru1, Conj.6.3] (see Footnotes 1 and 6; [Ru2]). Note also that we are working within the full CFT, without having to perform a topological twist or introduce boundary states to probe the geometry of our orbifold limits of K3. Acknowledgement. It is a pleasure to thank Paul Aspinwall, David R. Morrison, M. Ronen Plesser, Miles Reid, Daniel Roggenkamp, Yongbin Ruan, Eric Sharpe, and Bernardo Uribe for helpful discussions. W.N. thanks the LPTENS in Paris for an invitation, which led to the start of part of this work. K.W. was supported by U.S. DOE grant DE-FG05-85 ER 40219, TASK A.

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