Communications in Mathematical Physics - Volume 244

Commun. Math. Phys. 244, 1 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-0990-6 Published online: 2 December...

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Commun. Math. Phys. 244, 1 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-0990-6 Published online: 2 December 2003 c Springer-Verlag 2003

Communications in

Mathematical Physics

Editorial The Journal’s Mission and Standards of Presentation

The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards. The above is a broad calling, as it encompasses different subfields, which in practice have varying foundations, different subcommunities of practitioners, and which may be at different stages of progress towards mathematical maturity. As a primary instruction, we call upon the authors to keep both the above stated goals and the broad audience in mind, and we offer the following suggestions to this end. • State the results in a broadly accessible way. • Bear in mind that the case for publication in CMP is enhanced by a clear indication of the work’s relation to physics issues, or to mathematics which have sprung from such a relation. • Ensure that the status of the results presented in the work is unambiguously understood: whether they are established rigorously, i.e. proven within a mathematically precise framework, or according to other standards, for example granting the validity of some particular physical framework. • As possible, formulate clear definitions and state theorems, as facilitated by the formalism used. Conjectures, clearly stated as such, also have their place. Finally, it is important to put the work into context, stating for a broad audience both the background and the essential advances being made. While none of this is new, and the best papers in CMP have always exemplified these standards, the breadth of our readership as well as of our contributors makes it appropriate to reemphasize the importance of these requirements. They can be met in different ways; one possibility is to include an introductory section aimed at a broader readership than the body of the paper, where one may assume a somewhat higher level of familiarity with the subject. Both authors and referees are asked to consider these requirements. In some areas this may be viewed as an addition of another hurdle to an already demanding refereeing process, but we see this as an important part of the growth of mathematics and physics. We hope that the authors, referees, and readers will agree that the goals, which include a strong field and a vital mathematical physics community, are well worth this effort. The Editors

Commun. Math. Phys. 244, 3–27 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-0973-7

Communications in


Projectively and Conformally Invariant Star-Products C. Duval1 , A.M. El Gradechi2,3 , V. Ovsienko4 1

Université de la Méditerranée and CPT-CNRS, Luminy Case 907, 13288 Marseille, Cedex 9, France. E-mail: [email protected] 2 Faculté des Sciences, Université d’Artois, 62307 Lens, France 3 CPT-CNRS, Luminy Case 907, 13288 Marseille, Cedex 9, France. E-mail: [email protected] 4 Institut Girard Desargues, Université Claude Bernard Lyon 1, 69622 Villeubanne, Cedex, France. E-mail: [email protected] Received: 24 December 2002 / Accepted: 28 May 2003 Published online: 18 November 2003 – © Springer-Verlag 2003

Abstract: We consider the Poisson algebra S(M) of smooth functions on T ∗ M which are fiberwise polynomial. In the case where M is locally projectively (resp. conformally) flat, we seek the star-products on S(M) which are SL(n+1, R) (resp. SO(p +1, q +1))invariant. We prove the existence of such star-products using the projectively (resp. conformally) equivariant quantization, then prove their uniqueness, and study their main properties. We finally give an explicit formula for the canonical projectively invariant star-product.

1. Introduction The deformation quantization program initiated in the seventies [3] was aimed at defining an autonomous quantization method based on Gerstenhaber’s general theory of deformation of algebraic structures [28]. The original idea was to view quantum mechanics as a one-parameter deformation of classical mechanics, more precisely, a one-parameter deformation of the algebraic structures underlying classical mechanics. If P is a Poisson manifold, then C ∞ (P ) is naturally equipped with two algebraic structures, namely, the associative and commutative pointwise multiplication and the Lie algebra defined by the Poisson bracket. The deformed algebraic structure, describing the quantum mechanical counterpart of (C ∞ (P ), ·, {·, ·}) is (C ∞ (P )[[]], ), where the operation , called star-product, is an associative (but non-commutative) product on C ∞ (P )[[]] deforming the commutative multiplication in the direction of the Poisson bracket. More precisely: Definition 1.1. Let P be a Poisson manifold and C ∞ (P ) the space of smooth complex-valued functions on P . A star-product on P is an associative algebra structure on C ∞ (P )[[]], denoted , and given by a linear map from C ∞ (P ) ⊗ C ∞ (P ) to C ∞ (P )[[]], extended by linearity to C ∞ (P )[[]] ⊗ C ∞ (P )[[]], such that

4

C. Duval, A.M. El Gradechi, V. Ovsienko ∞

F G=F ·G+

i (i)r Br (F, G). {F, G} + 2

(1.1)

r=2

In the mathematical literature is a formal parameter, whereas in physical applications is Planck’s constant. There are usually three extra requirements for star-products : C1. The constant function 1 is the unit of (C ∞ (P )[[]], ), namely 1 F = F 1 = F ; C2. The star-product is symmetric, viz F G = G F ; C3. The bilinear maps Br are given by bidifferential operators. Note that Condition C2 is sometimes called the parity condition. The first reported star-product appeared in the work of Grœnewold [31]. It was derived from the Weyl-Wigner quantization on P = R2n . It is nowadays more commonly known as the Moyal star-product ; Moyal actually obtained the Lie algebra bracket associated with Grœnewold’s star-product [39]. This first star-product was later on rediscovered by Vey [44]. The general problem of existence of star-products was raised in [3]. Using cohomological techniques, De Wilde and Lecomte [16] proved the existence of star-products on any symplectic manifold. A geometric proof of the same result together with an algorithmic construction was obtained by Fedosov [24, 25] (see [45] for a survey of this construction and [41] for an alternative approach). More recently, Kontsevich proved an existence theorem for an arbitrary Poisson manifold, giving explicit formulæ for P = Rn [34].An operadic and a quantum field theoretic interpretations of Kontsevich’s result were later on given respectively by Tamarkin [43], and Cattaneo and Felder [11]. The problem of the uniqueness of star-products is usually studied modulo equivalence (see Sect. 2.2 for definitions and [32, 14] for recent developments). However, extra conditions can sometimes be imposed to single out a canonical star-product. For instance, Gutt [33] proved that the Moyal star-product is the unique (Sp(2n, R) R2n )-invariant and covariant star-product on R2n . The notion of a G-invariant star-product, where G is a Lie group of Poisson automorphisms of P , was introduced in [3] (see Sect. 2.1 for definitions). Existence of a G-invariant star-product on a symplectic manifold was proved by Lichnerowicz [38] for any compact Lie group G of symplectomorphisms. More recently, Fedosov [26] constructed a G-invariant star-product on a symplectic manifold endowed with a G-invariant symplectic connection. In this article, we deal with cotangent bundles P = T ∗ M equipped with their canonical symplectic structure, and restrict considerations to the Poisson algebra S(M) of smooth functions on T ∗ M polynomial on fibers. We furthermore assume M to be a smooth n-dimensional manifold endowed with either a projectively or a conformally flat structure, i.e., M admits a (locally defined) action of either SL(n + 1, R) or SO0 (p + 1, q + 1), the connected component of the pseudo-orthogonal group with n = p + q. The basic example of a projectively (resp. conformally) flat manifold is RPn (resp. (S p × S q )/Z2 ). Denote by G either the projective or the conformal group. We study, in the present article, G-invariant star-products on T ∗ M, where the G-action is the canonical lift of the natural action on the base. Our first result, Theorem 5.1, establishes the uniqueness of a G-invariant homogeneous star-product on S(M). Our second result, Theorem 5.7, proves the uniqueness of a G-invariant star-product modulo G-equivalence and reparametrization.

Projectively and Conformally Invariant Star-Products

5

Let us emphasize that we do not assume conditions C1, C2 and C3 a priori. It turns out that C1 and C2 are automatically satisfied while C3 doesn’t hold; in fact, the maps Br in (1.1) are pseudo-differential bilinear operators. Our G-invariant star-products cannot be obtained by Fedosov’s or Kontsevich’s constructions, as the latter lead to bidifferential star-products. The existence of G-invariant star-products on S(M) is based on the existence of a G-equivariant quantization map [37, 23] (see also [22]). The latter is the unique (up to normalization) isomorphism of G-modules, Qλ : S(M) → Dλ (M), where Dλ (M) is the space of differential operators acting on tensor densities of degree λ. Such a quantization map defines a G-invariant associative product on S(M) which turns out to be a star-product for λ = 21 as proved in [9, 23]. The existence and uniqueness results of the present article represent the deformation quantization counterparts of those obtained for G-equivariant quantization. In both situations invariance properties ensure uniqueness. The pseudo-differential nature of the G-invariant star-products has been revealed by Brylinski [9] and Astashkevich and Brylinski [2]. In the latter reference, invariant starproducts on minimal nilpotent coadjoint orbits of semi-simple Lie groups have been investigated. These results are closely related to ours since these orbits are punctured cotangent bundles T ∗ M \ M; nevertheless the Poisson algebras considered in [2] are smaller than S(M). Moreover, our approach provides explicit formulæ in the projective case, answering a question raised in [2]. The paper is organized as follows. In Sect. 2 we recall the notions of invariant and equivalent star-products, and we give a short account on equivariant quantization for cotangent bundles. In Sect. 3, we define projective and conformal geometries and determine the ring of projectively/conformally invariant linear operators on S(M). The existence of G-invariant star-product on T ∗ M, along with a few of their properties, are proved in Sect. 4. Sect. 5 contains our uniqueness theorems. In Sect. 6, we give an autonomous derivation of the canonical projectively invariant star-product on S(RPn ), based only on projective invariant theory. Explicit formulæ are then provided. We end this paper, with Sect. 7, where we gather our conclusion, a discussion and a few perspectives.

2. Invariant Star-Products and Equivariant Quantization In this section we introduce the general notions of invariance and covariance of starproducts with respect to a Hamiltonian action of a connected Lie group G.

2.1. Invariant, covariant and strongly invariant star-products. First of all, let us give the precise definition of an invariant star-product already mentioned in the Introduction. Definition 2.1. Given a Poisson action of a Lie group G on a Poisson manifold P , a star-product on C ∞ (P ) is called G-invariant if g ∗ (F G) = g ∗ F g ∗ G

(2.1)

for all F, G ∈ C ∞ (P )[[]] and g ∈ G. In the case where the G-action is Hamiltonian one has the following supplementary notions:

6

C. Duval, A.M. El Gradechi, V. Ovsienko

Definition 2.2. Consider a Hamiltonian G-action on a Poisson manifold P with associated equivariant moment map J : P → g∗ , where g∗ is the dual of the Lie algebra g of G. A star-product on P is called a) G-covariant if JX JY − JY JX = i {JX , JY },

(2.2)

b) strongly G-invariant if JX F − F JX = i {JX , F }

(2.3)

for all F ∈ C ∞ (P )[[]] and X, Y ∈ g, where JX is the Hamiltonian function on P corresponding to X. Remark 2.3. Note that a different terminology is sometimes attached to this last notion in the literature. What we call here strong G-invariance corresponds to the notion of preferred observables in [3, 18] and to Property IP2 in [1]. Beware that, in the latter reference, strong invariance means covariance and invariance. Let us now recall the following useful result. Proposition 2.4 ([1]). If a star-product is strongly G-invariant, then it is both G-invariant and G-covariant. Proof. Using the definition (2.3) of strong invariance, we write i{JX , F G} = JX F G − F G JX = JX F G − F J X G + F J X G − F G J X = i ({JX , F } G + F {JX , G}) which is nothing but the infinitesimal version of formula (2.1) expressing the invariance property. The G-invariance of the star-product then follows from the connectedness of G. As for covariance, it is an immediate consequence of strong invariance. Remark 2.5. The converse of Proposition 2.4 is proved in [1] under the additional assumption of a transitive G-action. 2.2. Equivalence, G-equivalence and reparametrization. In the traditional classification of star-products one introduces a notion of equivalence. Two star-products, and , are called equivalent if there exists a formal series = Id + i1 + (i)2 2 + · · · , where i :

C ∞ (P )

→

C ∞ (P )

(2.4)

are some linear operators, such that

(F ) (G) = (F G).

(2.5)

Usually, one also allows for formal changes of the parameter of deformation: µ : i → i +

∞

ak (i)k ,

(2.6)

k=2

where ak ∈ R, in order to comply with Property C2 from the Introduction. For G-invariant star-products it is natural to consider the notion of G-equivalence.


7

Definition 2.6 ([38]). Two equivalent G-invariant star-products are called G-equivalent if each map i in (2.4) is G-equivariant. The condition for two star-products to be G-equivalent is much stronger than the usual condition of equivalence (see [4] for recent developments). 2.3. Equivariant quantization and the associated invariant star-product. Equivariant quantization as developed in [37, 23, 21, 22] applies to cotangent bundles. From here on we restrict ourselves to P = T ∗ M endowed with its canonical symplectic form. Let S(M) ⊂ C ∞ (T ∗ M) be the space of (complex-valued) functions on T ∗ M polynomial on fibers, and D(M) be the space of linear differential operators acting on C ∞ (M). The space S(M) is the space of symbols of operators in D(M); it has a natural grading S(M) =

∞

Sk (M)

(2.7)

k=0

by the degree of homogeneity. Let Fλ (M) be the space of tensor densities of degree λ ∈ C on M, i.e., the space of sections of the complex line bundle |n T ∗ M|λ ⊗ C. In local coordinates such densities are of the form f (x 1 , . . . , x n ) |dx 1 ∧ · · · ∧ dx n |λ

(2.8)

with f ∈ C ∞ (M). Denote Dλ (M) the space of linear differential operators on Fλ (M); it has a natural filtration Dλ0 (M) ⊂ Dλ1 (M) ⊂ · · · ⊂ Dλk (M) ⊂ · · · such that S(M) = gr(Dλ (M)). Definition 2.7. A quantization map is an invertible linear map Qλ : S(M) → Dλ (M)[] which preserves the principal symbol in the following sense: for a homogeneous polynomial F ∈ Sk (M), the principal symbol of the differential operator Qλ (F ) is equal to (i)k F . There is a natural action of the group of diffeomorphisms, Diff(M), on Fλ (M), denoted by gλ : Fλ (M) → Fλ (M) for all g ∈ Diff(M). We will rather use the corresponding action of the Lie algebra of vector fields, Vect(M), which is given by LλX f = Xi

∂f ∂X i + λ f ∂x i ∂x i

(2.9)

for all X = Xi ∂/∂x i ∈ Vect(M), with the local identification Fλ (M) ∼ = C ∞ (M) made in (2.8). (We will use Einstein’s summation convention throughout this article.) Note that the expression (2.9) is, indeed, independent of the choice of a coordinate system. The canonical lift of the Diff(M)-action to T ∗ M is automatically Hamiltonian with moment map J given by JX = ξi X i ∈ S1 (M).

(2.10)

8


Definition 2.8. Consider a Lie group G ⊂ Diff(M). A quantization map Qλ is called G-equivariant if Qλ (g ∗ F ) = gλ−1 ◦ Qλ (F ) ◦ gλ

(2.11)

for all g ∈ G and F ∈ S(M). The above formula plays a central rôle in the forthcoming developments. We will need its infinitesimal guise LλX ◦ Qλ (F ) − Qλ (F ) ◦ LλX = Qλ (LX F )

(2.12)

for all X ∈ g, where LX stands for the canonical lift to T ∗ M of the fundamental vector field associated with X. From such a quantization map, we immediately obtain an associative product given by F λ G = Q−1 λ (Qλ (F ) ◦ Qλ (G)).

(2.13)

Note that this product is not necessarily of the form (1.1). However, Condition C1 is automatically satisfied. The following proposition is a direct consequence of the above definitions. Proposition 2.9. If Qλ is a G-equivariant quantization map on S(M), then the associative product on S(M) given by (2.13) is G-invariant. One wonders if there exists some extra condition sufficient to insure strong G-invariance of the G-invariant associative product (2.13). The next proposition introduces a natural geometric property of the quantization map that leads to the desired result. Proposition 2.10. If Qλ is a G-equivariant quantization map on S(M), which furthermore satisfies the following condition: Qλ (JX ) = i LλX

(2.14)

for all X ∈ g, then the associative product on S(M) given by (2.13) is strongly G-invariant. Proof. Let X ∈ g and F ∈ S(M), then, using successively (2.13), (2.14), and (2.12), we get JX λ F − F λ JX = (Qλ )−1 [Qλ (JX ), Qλ (F )] = (Qλ )−1 i LλX , Qλ (F ) = i LX F = i {JX , F }, where the last equality stems from the definition of the moment map. The proof that (2.3) holds is complete.


9

3. Projectively/Conformally Invariant Operators We gather here definitions and results that will be used throughout the paper. Those mainly concern projective/conformal differential geometry. We will consider the Lie groups G = SL(n + 1, R) and G = SO0 (p + 1, q + 1) together with their homogeneous spaces M = RPn and M = (S p × S q )/Z2 , respectively. From here on, G will stand for either of the two groups above and M for either of the corresponding homogeneous spaces. In the framework of Weyl’s invariant theory [46], we will introduce, for each geometry, G-invariant linear operators on T ∗ M which will serve as our main tools.

3.1. The projective and conformal symmetries. The real projective space of dimension n is an SL(n+1, R)-homogeneous space. More precisely, RPn = SL(n+1, R)/Aff(n, R), where Aff(n, R) = GL(n, R) Rn is an affine subgroup of SL(n + 1, R). Let x 1 , x 2 , . . . , x n be an affine coordinate system on RPn , the fundamental vector fields associated with the SL(n + 1, R)-action on RPn are then given by : ∂ , ∂x i

xi

∂ , ∂x j

xi xj

∂ , ∂x j

(3.1)

with i, j = 1, . . . , n. The vector fields (3.1) correspond to translations, linear transformations and inversions, respectively; they generate a flag of Lie algebras Rn ⊂ aff(n, R) ⊂ sl(n + 1, R). The sphere S n with its canonical metric is a conformally flat manifold. The same is true for (S p × S q )/Z2 in the case of signature p − q. Those are homogeneous spaces SO(p +1, q +1)/CE(p, q), where CE(p, q) = CO(p, q)Rn is the conformal Euclid∗ , and n = p + q. ean group, CO(p, q) = SO(p, q) R+ The fundamental vector fields associated with the SO0 (p + 1, q + 1)-action on (S p × S q )/Z2 in an “anallagmatic” coordinate system are given (see, e.g., [19]) by ∂ , ∂x i

xi

∂ ∂ − xj i , j ∂x ∂x

xi

∂ , ∂x i

xj x j

∂ ∂ − 2xi x j j , i ∂x ∂x

(3.2)

where i, j = 1, . . . , n, and where indices are raised and lowered using the standard metric g of (S p × S q )/Z2 . The vector fields (3.2) correspond to translations, rotations, homotheties and inversions, respectively; they generate a flag of Lie algebras Rn ⊂ e(p, q) ⊂ ce(p, q) ⊂ o(p + 1, q + 1). These two groups of transformations, G, define respectively the projective and the conformal geometries; their Lie algebras, g, spanned by the vector fields (3.1) and (3.2) are finite-dimensional maximal Lie subalgebras of Vect(M), see [37, 5]. We also introduce, for convenience, the notation H ⊂ G for the affine Lie subgroups H = Aff(n, R) in the projective case, and H = CE0 (p, q) in the conformal case. The corresponding Lie subalgebras will be denoted by h ⊂ g.

10


3.2. Affine and Euclidean invariant operators. Since the group Diff(M) of diffeomorphisms of M admits a canonical lift to T ∗ M, let us lift, accordingly, the action of G. The search for G-invariant linear operators on S(M) will be dealt with in two stages. We first consider the affine (resp. Euclidean) subgroup and determine the algebra of Aff(n, R)-invariant (resp. (SO0 (p, q) Rn )-invariant) operators; in the next section we will then enforce full G-invariance. A classical result from invariant theory shows that the commutant of Aff(n, R) in End(S(M)) is generated by the following two operators: E = ξi

∂ , ∂ξi

D=

∂ ∂ ∂x i ∂ξi

(3.3)

which span the Lie algebra aff(1, R). Indeed, an Aff(n, R)-invariant linear operator mapping Sk (M) into S (M) is proportional to Dk− (see, e.g., [46, 29]). The commutant of Aff(n, R) in End(S(M)) is, hence, given by series in E and D, convergent on S(M). It has been shown in [23] that the commutant of SO0 (p, q) Rn in End(S(M)) is generated by the operators R = ξ i ξi ,

E = ξi

∂ , ∂ξi

T=

∂ ∂ ∂ξ i ∂ξi

(3.4)

whose commutation relations are those of sl(2, R), together with G = ξi

∂ , ∂x i

D=

∂ ∂ , ∂x i ∂ξi

=

∂ ∂ , ∂x i ∂xi

(3.5)

which span the Heisenberg Lie algebra h1 . The operators (3.4) and (3.5) span the Lie algebra sl(2, R) h1 . 3.3. Projectively and conformally invariant operators. It is noteworthy that E commutes with the lift of any diffeomorphism of M. One may ask if, upon restriction to G ⊂ Diff(M), there exist other linear operators on T ∗ M that commute with G. The answer is negative in the projective case and positive in the conformal case. Proposition 3.1. The commutant of SL(n + 1, R) in End(S(M)) is generated by E. Proof. An affinely invariant linear operator is a series in E and D of the form A=

∞

Ps (E) Ds ,

(3.6)

s=0

where Ps is a series in one variable. Let Xi = x i x j ∂/∂x j be the i th generator of inversions in (3.1). Straightforward computation (see [37]) yields the commutation relation

∂ LXi , D = (2E + n + 1) ◦ . ∂ξi

One then checks that

LXi , A =

∞ s=0

sPs (E)(2E + n + s) Ds−1 ◦

∂ . ∂ξi

(3.7)

This expression vanishes if and only if Ps = 0 for all s ≥ 1. Hence A = P0 (E) is a necessary condition for A to commute with the SL(n + 1, R)-action.


11

The conformal counterpart of the above statement is as follows. Proposition 3.2. The commutant of SO0 (p +1, q +1) in End(S(M)) is the commutative associative algebra generated by E and the operator R0 = R ◦ T. Proof. A sketch of this proof was given in [23]; for the sake of completeness we give the details here. Let us consider an operator Z on the space of polynomials of degree k, namely S k (M) =

k

S (M),

=0

and commuting with the canonical lift of SO0 (p + 1, q + 1). It is, according to classical invariant theory [46, 29], a differential operator, polynomial in the generators (3.4) and (3.5). We therefore seek a differential operator Z on T ∗ M which commutes with the SO0 (p + 1, q + 1)-action. Its principal symbol σ (Z) is a function on T ∗ (T ∗ M), polynomial on fibers. More precisely, if (ζi , y i ) denote the conjugate variables to (x i , ξi ) respectively, then σ (Z) is polynomial in the variables ξi , ζi , y i . The function σ (Z) has to be annihilated by the canonical lifts to T ∗ (T ∗ M) of all generators (3.2) of the conformal Lie algebra o(p + 1, q + 1). Let us assume that σ (Z) is ce(p, q)-invariant and consider then invariance with respect to inversions whose i th generator is Xi = xj x j ∂/∂x i − 2xi x j ∂/∂x j . Its canonical lift to T ∗ (T ∗ M) is given by Xi = xj x j ∂ − 2xi x j ∂ L ∂x ∂x j i ∂ ∂ j ∂ +2xi ξj −y + ζj ∂y j ∂ζj ∂ξj ∂ ∂ ∂ ∂ ∂ ∂ j −2x ξi j − ξj i + yi j − yj + ζi j − ζj ∂ξ ∂ξ ∂y ∂ζ ∂ζ i ∂yi ∂ ∂ ∂ +2 ξi yj , (3.8) − y i ξj − ξj y j ∂ζj ∂ζj ∂ζ i Xi σ (Z) = 0. Now, invariance with and the invariance with respect to inversions reads L respect to ce(p, q) clearly implies that σ (Z) is annihilated by the first three terms in (3.8), so that ∂ ∂ ∂ ξi yj σ (Z) = 0 (3.9) − y i ξj − ξj y j ∂ζj ∂ζj ∂ζ i for all i = 1, . . . , n. Lemma 3.3. Equation (3.9) implies ∂σ (Z) =0 ∂ζi for all i = 1, . . . , n.

(3.10)

12


Proof. The determinant of the matrix Aij = y i ξj − ξ i yj + ξk y k δji intervening in (3.9) is det(A) = ξi ξ i yj y j (ξk y k )n−2 which is non-zero on the comple ment of a lower-dimensional smooth submanifold of T ∗ (T ∗ M). By e(p, q)-invariance, the operator Z is a polynomial in the differential operators (3.4) and (3.5), see Sect. 3.2. Furthermore, invariance with respect to the generator of homotheties X0 = x i ∂/∂x i shows that Z is in fact a polynomial in R0 = R ◦ T,

E,

G0 = G ◦ T,

0 = ◦ T.

D,

(3.11)

The principal symbols of the last three operators are σ (G0 ) = ξi ζ i yj y j ,

σ (D) = ζi y i ,

σ (0 ) = ζi ζ i yj y j .

These three polynomials are algebraically independent for n > 1. Condition (3.10) implies then that Z depends only on E and R0 . Note that if n = 1, we find R0 = E(E −1) in agreement with Proposition 3.1. We have thus shown that, for all k, any Z ∈ End(S k (M)) commuting with the SO0 (p + 1, q + 1)-action is polynomial in E and R0 . This completes the proof of Proposition 3.2. 4. Existence of Projectively and Conformally Invariant Star-Products Taking advantage of the results obtained in equivariant quantization (see [37, 23, 9]) and of Proposition 2.9, one defines a G-invariant star-product on T ∗ M . In this section we give a brief account on the projectively and conformally equivariant quantizations and discuss the main properties of the associated invariant star-products. 4.1. Construction of G-invariant star-products. It has been proved in [37, 23] that, for any λ ∈ C, there exists a unique G-equivariant quantization map Qλ : S(M) → Dλ (M)[] on T ∗ M. In a local coordinate system, one can locally identify S(M) and Dλ (M) through the normal ordering prescription: P i1 ...ik ξi1 · · · ξik → (i)k P i1 ...ik

∂ ∂ ··· i , i 1 ∂x ∂x k

(4.1)

where P i1 ...ik is a smooth function of (x 1 , . . . , x n ). The explicit formula of Qλ is only known in the projective case; it is given, in an adapted coordinate system, and using the identification (4.1), by the series [22] Qλ =

∞

Cr (E) (iD)r ,

(4.2)

1 (E + (n + 1)λ)r , r! (2E + n + r)r

(4.3)

r=0

where E and D are as in (3.3) and Cr (E) =


13

where (a)r := a(a + 1) · · · (a + r − 1) is the Pochhammer symbol. The expression (4.2) is well defined globally on T ∗ M since M is projectively flat. An important feature of the quantization map (4.2) is that it is homogeneous in the following sense. Let us assign a degree to the deformation parameter , more precisely, we put deg = 1. Then Qλ preserves the total degree on S(M)[]. In other words, one has Proposition 4.1. The quantization map Qλ commutes with the Euler operator: ∂ E = E + . ∂

(4.4)

Proof. This follows from the commutation relation [E, D] = −D and the expression (4.2). In the conformal case we have no explicit formula for the SO0 (p + 1, q + 1)-equivariant quantization map. However, one can guarantee [23] that Qλ is also homogeneous in this case. A G-invariant star-product on T ∗ M can be obtained from such a G-equivariant quantization map. Proposition 4.2 ([23, 9]). The associative product associated with Qλ through (2.13) is a star-product if and only if λ = 21 . The proof consists in checking that λ = 21 is the only value of λ for which the first-order term in of the associative product (2.13) coincides with the Poisson bracket. Note, however, that the uniqueness of Q 1 does not a priori insure the uniqueness of 2 a G-invariant star-product. 4.2. Main properties. For the constructed G-invariant star-products, Condition C1 from Sect. 1 is satisfied. We will show below that Condition C2 also holds. Definition 4.3. A star-product on the space S(M) will be called homogeneous, if all the bilinear operators Br in (1.1) are homogeneous of degree r, that is, if they preserve the grading (2.7) according to Br : Sk (M) ⊗ S (M) → Sk+−r (M),

(4.5)

or, equivalently, if E is a derivation of the star-product algebra. Proposition 4.4. The G-invariant star-product (2.13) obtained from the G-equivariant quantization map Q 1 is symmetric, homogeneous and strongly G-invariant. 2

Proof. The quantization map Q 1 is symmetric, namely, it satisfies 2

Q 1 (F )∗ = Q 1 (F ) 2

2

for all F ∈ S(M) [37, 23, 21], where Q 1 (F )∗ denotes the formal adjoint operator with 2

respect to the natural pairing on compactly supported 21 -densities. Using the definition (2.13) of the star-product, one now gets the symmetry condition C2.

14


Homogeneity of the quantization map Q 1 readily implies the homogeneity of the 2 corresponding star-product. The projectively and the conformally equivariant quantization maps Q 1 coincide up 2 to second-order terms, namely, in both cases one has Q 1 = Id + 2

i D + O( 2 ) 2

in any coordinate system (cf. [37, 23]). One easily verifies that Q 1 satisfies con2 dition (2.14). By Proposition 2.10, the associated G-invariant star-products are thus strongly G-invariant. Condition C3 fails to be satisfied (as proved in [9] and [2] for a subalgebra of S(M)). Each term Br is a pseudo-differential bilinear operator, while its restriction Br |Sk (M)⊗S (M) is a bidifferential operator, just like Q 1 |Sk (M) is a differential operator, see [37]. Hence 2 the constructed star-product is local, namely, for all F, G ∈ S(M), Supp(F G) ⊂ Supp(F ) ∩ Supp(G), see Lemma 5.3 below. 5. Uniqueness of G-Invariant Star-Product Our goal is to show that the star-products constructed in Sect. 4.1 with the help of the Gequivariant quantization map are the unique G-invariant star-products where, as above, G = SL(n + 1, R) and G = SO0 (p + 1, q + 1), respectively. We prove uniqueness in two different settings: 1. in the class of homogeneous G-invariant star-products, 2. in the class of all G-invariant star-products modulo formal reparametrizations and G-equivalence.

5.1. Homogeneous star-products. Homogeneity of a star-product (see Definition 4.3) is a very natural property from a physical standpoint. Indeed, if one considers as a physical constant whose dimension is that of an action (i.e., the dimension of Planck’s constant which is also the inverse dimension of the Poisson bracket on T ∗ M), then the physical dimension of the star-product F G of two observables is the same as that of their product F G, when is homogeneous. This is a direct consequence of the fact that Br has the same physical dimension as −r , which follows from associativity. In other words a homogeneous star-product is dimensionless. On the other hand, homogeneous star-products were thoroughly studied in the mathematical literature. For instance, De Wilde and Lecomte proved [15] that any two homogeneous star-products on a cotangent bundle are equivalent (in the sense of the definitions of Sect. 2.2). The G-invariant star-products constructed in Sect. 4.1 are also homogeneous (see Proposition 4.4). The first main result of this paper is Theorem 5.1. There exists a unique homogeneous G-invariant star-product on the space of symbols S(M).


15

Proof. In Sect. 4 we proved the existence of a homogeneous G-invariant star-product on S(M). We will now prove its uniqueness. Let and be two homogeneous G-invariant star-products. Let us assume that the first r − 1 terms of these star-products coincide, and use induction over r. The difference Br −Br is a G-invariant homogeneous Hochschild 2-cocycle. Indeed, associativity of the star-product implies that δBr depends only upon Bi with i < r, where the Hochschild coboundary of a 2-cochain B is given by δB(F, G, H ) = F B(G, H ) − B(F G, H ) + B(F, GH ) − B(F, G)H,

(5.1)

implying that δ(Br − Br ) = 0. Let C be a Hochschild 2-cocycle on S(M). Assume now that C is homogeneous as in (4.5) and G-invariant. As a bilinear map, C decomposes as a sum C1 + C0 , where C1 and C0 are, respectively, the skew-symmetric and the symmetric parts of C. We will need the following well-known result. Proposition 5.2. For any local Hochschild 2-cocycle C on S(M), the skew-symmetric part C1 is a bivector, and the symmetric part C0 is the coboundary of a local 1-cochain. This statement is an important result in deformation theory. It was first established in the differentiable case [44] and was later on generalized to local cocycles in [10] (let us mention that this result also holds for continuous cocycles [13, 40]). In order to apply Proposition 5.2, we will prove that each term Br of a G-invariant star-product is local, a result that generalizes Theorem 5.1 in [37]. Lemma 5.3. Any linear G-invariant operator B : Sk (M) ⊗ S (M) → Sm (M) with m ≤ k + is local. Proof. We must prove that Supp(B(F, G)) ⊂ Supp(F ) ∩ Supp(G) for all F ∈ Sk (M) and G ∈ S (M). Suppose that one of the arguments, F or G, vanishes in a neighbourhood of some x ∈ M; we will prove that B(F, G)(x) = 0. Let us now locally identify M with Rn and consider the subalgebra R Rn of g generated by the Euler vector field, E, and the translations. Using translation-invariance, we may, hence, assume x = 0. We will embed Sk (Rn ) ⊗ S (Rn ) into Sk+ (R2n ) and notice that F ⊗ G vanishes in a neighbourhood of x = 0 in R2n . It remains to show that if B : Sk+ (R2n ) → Sm (Rn ) is a linear map which commutes with the action of homotheties LE , then for all H ∈ Sk+ (R2n ) that vanishes in a neighbourhood of x = 0, we have B(H )(0) = 0, provided m ≤ k + . But the proof of the latter statement coincides with that of Theorem 5.1 in [37]. The building blocks of the operators Br are the H-invariant operators listed in (3.11). These operators never increase the degree of homogeneity in ξ = (ξi ), hence Lemma 5.3 applies. We are now able to use Proposition 5.2 and consider C1 and C0 separately. The assertion of Theorem 5.1 will follow from Lemmas 5.4 and 5.6 below. Lemma 5.4. There is no non-zero G-invariant bivector on T ∗ M with coefficients in S(M) homogeneous of degree r ≥ 2. Proof. There is clearly no non-zero such bivector W : Sk (M) ⊗ S (M) → Sk+−r (M), for r > 2. For r = 2, if it exists, it is necessarily of the form W = Wij ∂/∂ξi ∧ ∂/∂ξj with coefficients Wij of degree 0 in ξ . Since W is invariant with respect to the generators of translations, ∂Wij /∂x s = 0 for all s = 1, . . . , n. But, in this case, W cannot be invariant with respect to homotheties.

16


We thus have proved that there is no non-zero bivector invariant with respect to the (n + 1)-dimensional Lie algebra of translations and homotheties. This Lie algebra is a Lie subalgebra of both sl(n + 1, R) and o(p + 1, q + 1). Lemma 5.4 is proved. Remark 5.5. Note that, in the proofs of Lemmas 5.3 and 5.4, we only needed invariance with respect to a subalgebra of g. Lemma 5.6. There is no non-zero G-invariant Hochschild 2-coboundary C0 on the associative commutative algebra S(M) homogeneous of degree r ≥ 2. Proof. Suppose that such a C0 exists. Being a coboundary, it is of the form C0 = δA, where δA(F, G) = F A(G) − A(F G) + A(F )G

(5.2)

for some linear map A : Sk (M) → Sk−r (M), with r ≥ 2. Let us prove that A is G-invariant. Since C0 is G-invariant, then, for any X ∈ g, the linear map LX (A) = [LX , A] is a Hochschild 1-cocycle on S(M). Indeed one has δ ◦ LX = LX ◦ δ. Thus, LX (A) is a derivation on S(M). Therefore, this is a vector field on T ∗ M polynomial in ξ and, hence, LX (A) cannot decrease the degree by more than 1. Note, however, that LX (A) : Sk (M) → Sk−r (M) with r ≥ 2 since, again, LX (A) = LX ◦ A − A ◦ LX and LX preserves Sk (M) for any vector field X on M. It follows that LX (A) = 0 for all X ∈ g and thus A is G-invariant. The classification of G-invariant linear maps on S(M) is given by Proposition 3.1 and Proposition 3.2. Being homogeneous of degree zero in ξ , a non-zero G-invariant element A of End(S(M)) cannot decrease the degree. Lemma 5.6 is proved. Lemmas 5.4 and 5.6 imply that Br − Br = 0 for r ≥ 2. This completes the proof of Theorem 5.1. The unique homogeneous G-invariant star-product will be called G-canonical. According to Proposition 4.4, this G-canonical star-product is the one associated with the G-equivariant quantization map Q 1 from Sect. 4. The same proposition also states 2 that it is both symmetric and strongly G-invariant. 5.2. Uniqueness up to G-equivalence and reparametrization. The following theorem is the second main result of this paper. Theorem 5.7. The G-canonical star-product on the space of symbols S(M) is the unique G-invariant star-product modulo formal reparametrizations and G-equivalence. Proof. Let and be two G-invariant star-products. Let us assume that there exists a G-invariant formal series (2.4) and a reparametrization (2.6) intertwining the first r − 1 terms of these star-products, and use induction over r. Using this equivalence we can assume that and coincide up to the (r − 1)th order term. The difference Br − Br is then a G-invariant Hochschild 2-cocycle. As in Sect. 5.1 we consider the decomposition C = C1 + C0 , where C1 and C0 are, respectively, the skew-symmetric and the symmetric parts of C. By Proposition 5.2, C1 is a bivector and C0 is a coboundary.


17

We will need the following two lemmas. Lemma 5.8.

(i) In the projective case, the canonical Poisson bivector =

∂ ∂ ∧ i ∂ξi ∂x

(5.3)

on T ∗ M is the unique (up to an overall multiplicative constant) G-invariant bivector. (ii) In the conformal case with n = 2, the canonical Poisson bivector on T ∗ M is the unique (up to an overall multiplicative constant) G-invariant bivector. (iii) In the conformal case with n = 2, there are two G-invariant bivectors on T ∗ M, namely the canonical Poisson bivector and the Poisson bivector =

∂ ∂ 1 ij g ξi ξj σk ∧ , 2 ∂ξk ∂ξ

(5.4)

where g = gij dx i dx j represents a conformal class of (pseudo-)Riemannian metrics and σ = 21 σk dx k ∧ dx stands for the surface element of (M, g). Proof. Consider an arbitrary bivector field W on T ∗ M. In any local coordinate system it is of the form ∂ ∂ ∂ ∂ ∂ ∂ j ∧ j + Wij (ξ, x) ∧ , (5.5) W = W ij (ξ, x) i ∧ j + Wi (ξ, x) ∂x ∂x ∂ξi ∂x ∂ξi ∂ξj j

where the coefficients W ij (ξ, x), Wi (ξ, x) and Wij (ξ, x) are functions of x i , ξi which are polynomial in ξ . Choose an adapted coordinate system related to the projective or conformal structure on M respectively (see Sect. 3.1). Since W is G-invariant, it commutes with the action of the generators of translations, that is, with the vector fields Xi = ∂/∂x i , where i = 1, . . . , n. It follows that the coefficients of W are independent of x i . Furthermore, W is invariant with respect to the action of the homothety vector field X0 = x i ∂/∂x i . The canonical lift of X0 to T ∗ M is LX0 = x i ∂/∂x i −ξi ∂/∂ξi . One immediately obtains the following homogeneity conditions: 1. the coefficient W ij (ξ ) has to be homogeneous in ξ of degree −2, j 2. the coefficient Wi (ξ ) has to be homogeneous in ξ of degree 0, 3. the coefficient Wij (ξ ) has to be homogeneous in ξ of degree 2, j

so that W ij (ξ ) = 0, while Wi (ξ ) are constant, and Wij (ξ ) = Wijk ξk ξ are quadratic polynomials. A G-invariant bivector (5.5) is, therefore, a sum of two independent j G-invariant bivectors W0 = Wi ∂/∂ξi ∧ ∂/∂x j and W2 = Wijk ξk ξ ∂/∂ξi ∧ ∂/∂ξj . Considering now invariance with respect to the linear subgroup of G entails that W0 represents an invariant in (Rn )∗ ⊗ Rn and W2 an invariant in 2 (Rn )∗ ⊗ S 2 Rn with respect to the standard linear action of SL(n, R) in the projective case, and SO0 (p, q) in the conformal case. A classical result of invariant theory (see [46, 29]) yields W0 = c0 with c0 ∈ C and W2 = 0, except for n = 2, in the conformal case, where W2 = c2 with c2 ∈ C. Hence, we have proved that the bivectors (5.3), and (5.4) for n = 2 in the conformal case, are the only bivectors invariant with respect to the affine subgroup of G. To complete the proof, one checks that the bivectors (5.3) and (5.4) are invariant with respect to inversions, i.e., the quadratic vector fields in (3.1) and (3.2).

18


Lemma 5.9. Every G-invariant Hochschild 2-coboundary C0 on the associative commutative algebra S(M) is of the form C0 = δA, where A is a G-invariant linear map on S(M). Proof. The 2-coboundary C0 is local thanks to Lemma 5.3. This clearly implies that any 1-cochain A such that C0 = δA is local, cf. Proposition 5.2. Given a G-invariant Hochschild 2-coboundary C0 = δA, we will prove that there such that δ A = δA and LX (A) = 0 for all X ∈ g. Clearly, Gexists a linear map A invariance of C0 = δA implies LX (δA) = 0 for any X ∈ g. Thus, δ(LX (A)) = 0 which means that LX (A) is a vector field. A local operator A is a locally given, according to the Peetre theorem [42], by a differential operator; in an arbitrary coordinate system, A = A(0) + A(1) + A(2) + · · · + A(m) ,

(5.6)

where A(i) =

s1 ...si

At1 ···ti 1 (x, ξ )

i1 +i2 =i

2

∂ ∂ ∂ ∂ · · · si ··· . s 1 1 ∂x ∂x ∂ξt1 ∂ξti2

(5.7)

Choose a coordinate system adapted to either the projective or the conformal structure. Consider first the action of the affine Lie subalgebra, h ⊂ g, that is, h = aff(n, R) in the projective case and h = ce(p, q) in the conformal case, introduced in Sect. 3.1. For each component A(i) , except for A(1) , one has LX (A(i) ) = 0, where X ∈ h, since = A − A(1) ; this operator is of the form (5.7) and thus cannot be a vector field. Put A this operator satisfies LX (A) = 0 for all X ∈ h and, obviously, δ A = δA. In particular, invariance with respect to translations guarantees that the coefficients in (5.7) are independent of x. is of the form (3.6); for the In the projective case, an affinely invariant operator A is given by (3.7). This is a vector field if and only generators Xi of inversions, LXi (A) = P0 (E) and thus LXi (A) = 0, see Proposition 3.1. if Ps = 0 for all s ≥ 1. Hence A in a different form, namely In the conformal case, let us rewrite the expression of A = A (0) + A (1) + A (2) + · · · + A (t) , A where t ≤ m and (j ) = A s1 ...sj A

∂ ∂ · · · sj ; ∂x s1 ∂x

s1 ...sj is a differential operator in ξ with polynomial coefficients in ξ . Each each A (j ) is invariant with respect to translations and homogeneous in x of degree −j . term A (j ) is homogeneous in ξ of deInvariance with respect to homotheties implies that A gree −j , that is, (j ) ] = −j A (j ) . [E, A (j ) ) is homogeneous Let Xi be the i th generator of inversions. The operator LXi (A is a vector in ξ of degree −j , since LXi (E) = 0, cf. Proposition 3.2. Hence, LXi (A) field only if LXi (A(j ) ) = 0 for j ≥ 2 since it is polynomial in ξ . belongs to the ring generated by the operators Because of its h-invariance, A E,

R0 = R ◦ T,

D,

G0 = G ◦ T,

0 = ◦ T,


19

(1) is then where these operators have been defined in (3.4), (3.5) and (3.11). The term A necessarily of the form (1) = α D + β G0 A where α and β are polynomials in E and R0 . A direct computation yields ∂ ∂ ∂ LXi (A(1) ) = 2α ξi T − 2E − 2β R0 −n + 2ξi T . ∂ξi ∂ξi ∂ξi Every term in this expression, except for −2nα ∂/∂ξi , is a differential operator of order > 1 for any α and β. Thus, the right-hand side can be a non-zero vector field only if α is a non-zero constant. On the other hand, −2β R0 ∂/∂ξi is, at least, a thirdorder term unless β is zero. But, the remaining terms 2α ξi T and −4αE ∂/∂ξi are of (1) ) = 0. order 2 and linearly independent. One concludes that α = 0 and thus LXi (A (0) ) = 0. Finally, the term A(0) is obviously a polynomial in E and R0 and, hence, LXi (A We have thus proved that LXi (A) = 0 for all i = 1, . . . , n. Lemma 5.9 is proved. Let us resort to Lemmas 5.8 and 5.9 to complete the proof. The G-invariant Hochschild 2-cocycle C = Br − Br is a sum C = C1 + C0 . The symmetric part C0 is a Hochschild coboundary and, by Lemma 5.9 is of the form C0 = δA, where A is a G-invariant 1-cochain. This term can be removed by a G-equivalence map = Id + (i)r A. Under the hypotheses of parts (i) and (ii) of Lemma 5.8, the skew-symmetric part C1 is proportional to the canonical Poisson bivector, that is, to the first-order term B1 . It can be removed by a reparametrization i → i + c (i)r for some c ∈ R. Theorem 5.7 is proved for the first two options, (i) and (ii), of Lemma 5.8. In the conformal case and for n = 2 (part (iii) of Lemma 5.8), the skew-symmetric part C1 is a linear combination of the canonical Poisson bivector and of the bivector in (5.4). By a reparametrization map we can remove the canonical Poisson bivector but not the bivector . Let us, indeed, show that, if Br − Br = C1 = k, then necessarily k = 0. We associate to the star-products and the corresponding star-commutators [F, G] =

1 (F G − G F ) . i

(5.8)

Since the two star-products are associative, the corresponding star-commutators satisfy the Jacobi identity. Put J (F, G, H ) = [F, [G, H ] ] + (cyclic) and consider the difference J (F, G, H ) − J (F, G, H ). By assumption, this expression has to be identically zero. Since the two star-products coincide up to order r − 1 in i, this difference is trivially zero up to order r − 2. Straightforward computation shows that the (r − 1)th order term in the above difference is equal to 2k[ , ](F, G, H ), where [ , ] is the Schouten bracket of and . Jacobi identities for and -commutators then lead to k [ , ] = 0. Lemma 5.10. The two Poisson bivectors and are not compatible.

20


Proof. The Schouten bracket is

∂ ∂ ∂ ∂ [ , ] = 2 ∧ ∧ ξ1 1 + ξ2 2 ∂ξ1 ∂ξ2 ∂x ∂x G = 2 ∧ , R

where G and R are as in (3.5) and (3.4). This expression does not vanish.

Thus, the constant k in the above formula has to vanish. This completes the proof of Part (iii). Theorem 5.7 is proved. Lemmas 5.8–5.10 can be summarized as the following: Proposition 5.11. The second G-invariant Hochschild cohomology space is 2 R , in the conformal case for n = 2 HH2G(S(M); S(M)) = R, otherwise and the cup product in the first instance is non-zero. This result could have been derived from Kontsevich’s [34] or Fedosov’s [27] classification of equivalence classes of deformations. Remark 5.12. Theorem 5.7 does not guarantee uniqueness of a star-product but of a class of G-invariant star-products. Together with Propositions 3.1 and 3.2 this leads to an explicit description of all G-invariant star-products. Indeed, they are all obtained from the G-canonical homogeneous star-product by the equivalence (2.5) and reparametrization (2.6); the equivalence map is given in terms of the G-invariant operators E in the projective case and E and R0 in the conformal case. 5.3. Uniqueness up to G-equivalence and reparametrization, G-covariance and homogeneity. In this section we compare our uniqueness theorems with those obtained for the Moyal star-product in [33]. The Moyal star-product is the unique, up to reparametrization, (Sp(2n, R) R2n )-invariant star-product on R2n . It was also proved that it is uniquely selected within its reparametrization class by furthermore requiring its covariance. The (Sp(2n, R) R2n )-equivalence class of the Moyal star-product has a single element since the (Sp(2n, R) R2n )-commutant in End(C ∞ (R2n )) is trivial so that there are no non-zero invariant Hochschild 2-coboundaries. One may wonder if in our present setting G-covariance plays a similar role, namely, that of an extra condition that selects the canonical G-invariant star-product of Sect. 5.1 within its reparametrization and G-equivalence classes described in Sect. 5.2. The answer is negative; however we have Proposition 5.13. If two G-invariant and G-covariant star-products on S(M) are equivalent up to reparametrization, then they coincide. Proof. Let and be two G-invariant and G-covariant star-products on S(M) belonging to the same reparametrization class. Their G-covariance translates into (see (2.2)) : JX JY − JY JX = i {JX , JY } = JX JY − JY JX

(5.9)


21

for all X, Y ∈ g. On the other hand reparametrization equivalence means that there exist a formal power series (2.6) such that F G = (i)r Br (F, G) = (µ(i))r Br (F, G). r≥0

r≥0

Using this equation, one rewrites the right-hand side of (5.9) in terms of , with µ(i) as deformation parameter. Now, using the left hand side of (5.9) one gets µ(i) = i, from which the conclusion follows. An analog of the above statement, where the reparametrization equivalence is replaced by G-equivalence, does not hold. Indeed, one shows using an argument similar to the one in the above proof, that two G-invariant and G-covariant star-products on S(M) in the same G-equivalence class, do not necessarily coincide. So, covariance does not play the same role for G as it does for Sp(2n, R) R2n . However, a simple verification shows that, for the Moyal star-product, homogeneity has exactly the same effect as (Sp(2n, R) R2n )-covariance. Hence, the G-canonical and the Moyal star-products are uniquely determined by two simple conditions, namely, invariance and homogeneity. 6. Explicit Formula for the Projectively-Invariant Star-Product In this section we compute the explicit formula of the canonical homogeneous projectively-invariant star-product. This solves a problem raised in [2]. Projective invariance will be dealt with in two stages. We first consider invariance with respect to an affine subgroup Aff(n, R) of SL(n + 1, R) and determine the affineinvariant bilinear operators on S(RPn ). Those will be used to write down an Ansatz for the star-product we are looking for. We will then enforce full projective invariance by further demanding that inversions preserve the star-product. This will give rise to Eq. (6.11) and (6.12) below. Another system of equations will arise from the associativity requirement (see (6.14)). The unique solution of the complete system of equations will be given explicitly at the end of this section. 6.1. Autonomous derivation from the invariance principle. We need to classify the bilinear Aff(n, R)-invariant differential operators on S(Rn ). For that purpose, let us resort to the natural embedding S(Rn ) ⊗ S(Rn ) → S(R2n )

(6.1)

and denote by (x, ξ, y, η) the natural coordinate system on T ∗ Rn ×T ∗ Rn . The operators of divergence with respect to the first and the second arguments Dxξ (F, G) = D(F ) G,

Dyη (F, G) = F D(G),

(6.2)

where D is as in (3.3), and the operators of contraction Dxη (F, G) =

∂ ∂

F (ξ, x)G(η, y) ,

η=ξ,y=x ∂x i ∂ηi

(6.3)

Dyξ (F, G) =

∂ ∂

F (ξ, x)G(η, y)

i η=ξ,y=x ∂y ∂ξi

(6.4)

are obviously Aff(n, R)-invariant differential operators. Restricting ourselves to homogeneous components, we get the following:

22


Proposition 6.1. Every bilinear differential operator Sk (Rn ) ⊗ S (Rn ) → Sm (Rn )

(6.5)

invariant with respect to the action of Aff(n, R), is a homogeneous polynomial in Dξ x , Dξ y , Dηx and Dηy of degree k + − m. This enables us to write the most general Aff(n, R)-invariant bilinear operation S(Rn ) ⊗ S(Rn ) → S(Rn )[]. According to Theorem 5.7, we will express it as a termwise homogeneous formal series which, when restricted to Sk (Rn ) ⊗ S (Rn ), takes the form F G=

∞

(i)r Brk, (F, G),

(6.6)

r=0

where Brk, is a bidifferential operator, homogeneous of degree r in Dξ x , Dξ y , Dηx , Dηy , viz

γ k, α β δ Bα,β,γ D D D D F (ξ, x)G(η, y) Brk, (F, G)(ξ, x) =

,δ ξ y ηx ξ x ηy η=ξ,y=x

α+β+γ +δ=r

(6.7) k, with constant coefficients Bα,β,γ ,δ . Since we seek a star-product, we have to impose k, B0,0,0,0 =1

(6.8)

and k, k, = −B0,1,0,0 = B1,0,0,0

1 2

(6.9)

in order to get the multiplication and Poisson bracket as the first two terms as in Eq. (1.1). Expressions (6.6) and (6.7) constitute our Ansatz for an SL(n + 1, R)-invariant starproduct on T ∗ RPn . It now remains to impose to the operation (6.6) the following conditions: (i) invariance with respect to inversions, and (ii) associativity. 6.2. Projective invariance. Let Xi = x i x j ∂x j be the i th generator of inversions. Denote by LXi = x i x j

∂ ∂ ∂ ∂ ∂ ∂ − x j ξj − x i ξj + y i y j j − y j ηj − y i ηj ∂x j ∂ξi ∂ξj ∂y ∂ηi ∂ηj

its canonical lift to T ∗ (R2n ). Invariance with respect to inversions translates into the following equations:

γ k, α β δ Bα,β,γ =0 (6.10) ,δ LXi , Dξ y Dηx Dξ x Dηy

α+β+γ +δ=r

η=ξ,y=x


23

at each order r ∈ N. The latter yield the following system of equations: k, k, (α + 1)(α + δ − )Bα+1,β,γ ,δ + (β + 1)(β + δ − )Bα,β+1,γ ,δ k, k, = (γ + 1)(n + 2k − γ − 1)Bα,β,γ +1,δ + (α + 1)(β + 1)Bα+1,β+1,γ ,δ−1 (6.11)

and k, k, (β + 1)(β + γ − k)Bα,β+1,γ ,δ + (α + 1)(α + γ − k)Bα+1,β,γ ,δ k, k, = (δ + 1)(n + 2 − δ − 1)Bα,β,γ ,δ+1 + (α + 1)(β + 1)Bα+1,β+1,γ −1,δ . (6.12)

6.3. Associativity. If F ∈ Sk (Rn ), G ∈ S (Rn ), and H ∈ Sm (Rn ) the associativity condition takes the form r

k,+m−j

Br−j

(F, Bj,m (G, H )) =

j =0

r

k+−j,m

Br−j

(Bjk, (F, G), H )

(6.13)

j =0

for all r ∈ N. Equation (6.13) then reads r

k,+m−j

Bα,β,γ ,δ (Dξ y + Dξ z )α (Dηx + Dζ x )β

j =0 α+β+γ +δ=r−j γ ×Dξ x (Dηy + Dηz

× =

+ D ζ y + D ζ z )δ

α +β +γ +δ =j r

β

γ

α δ Bα,m ,β ,γ ,δ Dηz Dζ y Dηy Dζ z

k+−j,m

Bα,β,γ ,δ (Dξ z + Dηz )α (Dζ x + Dζ y )β

j =0 α+β+γ +δ=r−j

×(Dξ x + Dξ y + Dηx + Dηy )γ Dδζ z α β γ δ × Bαk, ,β ,γ ,δ Dξ y Dηx Dξ x Dηy .

(6.14)

α +β +γ +δ =j

6.4. Explicit solution of the system. We solve the system of equations (6.11), (6.12) and k, k, (6.14), by first determining the components Bα,β,0,0 , then Bα,β,γ ,0 and, finally, the full k, expression Bα,β,γ ,δ .

6.4.1. First stage. Identifying in the associativity equation (6.14) the coefficients of the r−j j k,+m k+,m k, monomials Dξ z Dζ x , one readily finds Br−j,j,0,0 = Br−j,j,0,0 . Thus, Bα,β,0,0 depends only on k + ; we write k, Bα,β,0,0 = Cα,β (k + ).

(6.15)

24

C. Duval, A.M. El Gradechi, V. Ovsienko r−j −1

Using again (6.14), we identify the coefficients of the monomials Dξ z r−j −1

and Dξ z

j

Dζ x Dηx

j

Dζ x Dξ y , respectively, to get the following system: k,+m k+,m k+,m k, (j + 1)Br−j −1,j +1,0,0 = Br−j −1,j,1,0 + Br−j −1,j,0,0 B0,1,0,0 , k,+m k+,m k+−1,m k, = Br−j (r − j )Br−j,j,0,0 −1,j,1,0 + Br−j −1,j,0,0 B1,0,0,0 .

Resorting to the invariance equation (6.11) for α = r − j − 1, β = j , and γ = δ = 0, we obtain the supplementary equation k, k, (r − j )(r − j − − 1)Br−j,j,0,0 + (j + 1)(j − )Br−j −1,j +1,0,0 k, −(n + 2k − 1)Br−j −1,j,1,0 = 0.

The previous three equations together with (6.9) and (6.15) imply 1 (r − j )(r − n − 2k)Cr−j,j (k) + (n + 2k − 2j − 1)Cr−j −1,j (k − 1) = 0. 2 The latter equation, supplemented with (6.8), yields then

k, Bα,β,0,0

(−1)β = (α + β)!

1 (n−1)+k+−β 1 (n−1)+k+−α 2

2

α

β

n+2k+2−α−β

.

(6.16)

α+β

6.4.2. Second stage. Here we only use the first invariance equation (6.11) with δ = 0. Long but straightforward calculations lead to k, Bα,β,γ ,0 =

γ 1 (α + 1)r (β + 1)s γ !(n + 2k − γ )γ r+s=γ r k, × (α − )r (β − )s Bα+r,β+s,0,0 ,

(6.17)

where the last term is as in (6.16). 6.4.3. Last stage. A reverse iterative computation on δ using the second invariance equation (6.12) finally leads to the sought for result k, Bα,β,γ ,δ

1 = δ!(n + 2 − δ)δ

r+s+t=δ

(−1)

s

δ r, s, t

×(α + 1)r (α + 1)s (β + 1)s (β + 1)t (α + γ − k)r (β + γ − k)t k, ×Bα+r+s,β+s+t,γ −s,0 ,

(6.18)

where the first line contains the trinomial coefficient and the last one is given by (6.17).


25

6.5. Symmetry condition. Proposition 6.2. The symmetry condition C2 translates for the Ansatz (6.6)–(6.7) into k, α+β+γ +δ ,k Bα,β,γ Bβ,α,δ,γ . ,δ = (−1)

(6.19)

Proof. If F ∈ Sk (Rn ), and G ∈ S (Rn ), we immediately get from Condition C2 that Brk, (F, G) = (−1)r Br,k (G, F ). Then, a change of dummy variables in (6.7) completes the proof.

It turns out that our star-product given by (6.6), (6.7) and (6.18) automatically satisfies the symmetry condition (6.19). Although this is not transparent from the expression (6.18), it is however a direct consequence of Proposition 4.4 and Theorem 5.1. 7. Conclusion, Discussion and Outlook In this work we have proved the existence and uniqueness of a canonical G-invariant star-product on T ∗ M for G = SL(n+1, R) (resp. G = SO0 (p +1, q +1) and M = RPn (resp. (S p × S q )/Z2 ). We have, moreover, given an explicit formula for the canonical projectively invariant star-product. For both geometries, the canonical star-product so obtained is symmetric, homogeneous, strongly G-invariant (hence G-covariant), but not differential. These properties, except for the last one, are shared with the Moyal star-product on R2n . Theorem 5.1 shows that the homogeneity condition supplementing G-invariance uniquely determines the canonical G-invariant star-product on S(M). Likewise, the Moyal star-product is also uniquely specified by (Sp(2n, R) R2n )-invariance and homogeneity. This allows us to draw a parallel between our canonical G-invariant starproduct and Moyal’s, namely, they are uniquely determined by the same two simple conditions : invariance and homogeneity. Of course, this parallel is far from complete, since, for instance, G and Sp(2n, R) R2n do not have the same geometric status; the action of the former on T ∗ M is lifted from that on M, which is not the case for the latter. Furthermore, it is clear that, for the projective and the conformal cases, there is no G-invariant (symplectic) connection on T ∗ M, since G does not act on the bundle of linear frames of T ∗ M. Hence, no Fedosov [26] canonical G-invariant star-product can be constructed. Besides, Fedosov’s construction would have led to a star-product given by bidifferential operators. The generalization of the existence and uniqueness theorems for projectively/conformally invariant star-products on T ∗ M in the case of a non-flat projective/conformal connection on M remains an open problem. In a recent work [6], Bordemann has taken a significant step in this new direction, by investigating the projectively equivariant quantization on a cotangent bundle of a manifold with a non flat projective structure (see also [21 and 7]). Note also that since the canonical star-products studied in this work may be considered as the projective/conformal analogs of the Moyal star-product, they may play a similar role as the latter in a construction a` la Fedosov of a star-product on a symplectic manifold with a Cartan projective/conformal symplectic connection. In the case n ≥ 2, let us mention that the explicit form of the conformally invariant star-product is, so far, out of reach. This was already the situation for the conformally equivariant quantization map [23] (see also [21]).

26


In the conformal case with n = 2, Theorem 5.7 holds for star-products of the form (1.1) with the standard Poisson bracket on T ∗ M as first-order term. However, one could easily construct, in this case, another G-invariant star-product with the Poisson bracket (5.4) as first-order term. It would be interesting to give a physical status to this second, somewhat “exotic”, star-product. In the case of dimension n = 1, our results are related to earlier work by Cohen, Manin and Zagier [12]. The projective and the conformal algebras are, in this case, isomorphic to sl(2, R). Moreover, the canonical projectively and conformally invariant star-products coincide by uniqueness and thus the explicit formulæ given in Sect. 6.4 correspond to the one obtained in [12] for λ = 21 . Acknowledgements. It is a pleasure to thank Ranee Brylinski, Simone Gutt, Pierre Lecomte and John Rawnsley for valuable help and encouragement. This work was done while the second author was visiting CPT as a délégué CNRS; he thanks CNRS for granting him a délégation and the Université d’Artois for consenting a one year leave of absence. The second and third authors both thank the CPT for hospitality.

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27

21. Duval, C., Ovsienko, V.: Conformally equivariant quantum Hamiltonians. Selecta Math. (N.S.) 7(3), 291–320 (2001) 22. Duval, C., Ovsienko, V.: Projectively equivariant quantization and symbol calculus: Noncommutative hypergeometric functions. Lett. Math. Phys. 57(1), 61–67 (2001) 23. Duval, C., Lecomte, P., Ovsienko, V.: Conformally equivariant quantization: Existence and uniqueness. Ann. Inst. Fourier. 49(6), 1999–2029 (1999) 24. Fedosov, B.V.: Formal quantization. In: Some topics of modern mathematics and their applications to problems of mathematical physics (in Russian), Moscow, 1985, pp. 129–136 25. Fedosov, B.V.: A simple geometrical construction of deformation quantization. J. Diff. Geom. 40(2), 213–238 (1994) 26. Fedosov, B.V.: Non-abelian reduction in deformation quantization. Lett. Math. Phys. 43(2), 137–154 (1998) 27. Fedosov, B.V.: Deformation quantization and index theory. Mathematical Topics. 9, Berlin: Akademie Verlag, 1996 28. Gerstenhaber, M.: On the deformation of rings and algebras. Ann. of Math. 79(2), 59–103 (1964) 29. Goodman, R., Wallach, N.: Representations and invariants of the classical groups. Encyclopedia of Mathematics and its Applications 68, Cambridge: Cambridge University Press, 1998 30. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics. Reading, MA: Addison-Wesley, 1992 31. Grœnewold, H.J.: On the principles of elementary quantum mechanics. Physica 12, 405–460 (1946) 32. Gutt, S.: Variations on deformation quantization. Math. Phys. Stud. 21, Dordrecht: Kluwer Acad. Publ., 2000, pp. 217–254 33. Gutt, S.: Contribution a` l’étude des espaces symplectiques homogènes. Acad. Roy. Belg. Cl. Sci. Mém. Collect. 8o (2) 44(6), (1983) 34. Kontsevich, M.: Deformation quantization of Poisson manifolds I. q-alg/9709040 35. Lecomte, P.B.A.: Classification projective des espaces d’opérateurs différentiels agissant sur les densités. C. R. Acad. Sci. Paris Sér. I Math. 328(4), 287–290 (1999) 36. Lecomte, P.B.A.: On the cohomology of Sl(m + 1, R) acting on differential operators and Sl(m + 1, R)-equivariant symbol. Indag. Math., N.S. 11(1) 95–114 (2000) 37. Lecomte, P.B.A., Ovsienko, V.: Projectively invariant symbol calculus. Lett. Math. Phys. 49(3), 173–196 (1999) 38. Lichnerowicz, A.: Déformations d’algèbres associées a` une variété symplectique (les ∗ν -produits). Ann. Inst. Fourier 32(1), 157–209 (1982) 39. Moyal, J.E.: Quantum mechanics as a statistical theory. Proc. Cambridge Philos. Soc. 45, 99–124 (1949) 40. Nadaud, F.: On continuous and differential Hochschild cohomology. Lett. Math. Phys. 47(1), 85–95 (1999) 41. Omori, H., Maeda,Y.,Yoshioka, A.: Weyl manifolds and deformation quantization. Adv. Math. 85(2), 224–255 (1991) 42. Peetre, J.: Une caractérisation abstraite des opérateurs différentiels, Math. Scand. 7, 211–218 (1959) and 8, 116–120 (1960) 43. Tamarkin, D.E.: Another proof of M. Kontsevich formality theorem. math.QA/9803025 44. Vey, J.: Déformation du crochet de Poisson sur une variété symplectique. Comment. Math. Helv. 50(4), 421–454 (1975) 45. Weinstein, A.: Deformation quantization. Séminaire Bourbaki, 1993/94, Astérisque 227, 389–409 (1995) 46. Weyl, H.: The Classical Groups. Princeton, NJ: Princeton University Press, 1946 Communicated by H. Spohn


Communications in


Random Matrix Ensembles Associated to Compact Symmetric Spaces Eduardo Duenez ˜ American Institute of Mathematics, The Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218, USA. E-mail: [email protected] Received: 10 October 2002 / Accepted: 23 June 2003 Published online: 25 November 2003 – © Springer-Verlag 2003

Abstract: We introduce random matrix ensembles that correspond to the infinite families of irreducible Riemannian symmetric spaces of type I. In particular, we recover the Circular Orthogonal and Symplectic Ensembles of Dyson, and find other families of (unitary, orthogonal and symplectic) ensembles of Jacobi type. We discuss the universal and weakly universal features of the global and local correlations of the levels in the bulk and at the “hard” edge of the spectrum (i. e., at the “central points” ±1 on the unit circle). Previously known results are extended, and we find new simple formulas for the Bessel Kernels that describe the local correlations at a hard edge. 1. Introduction Local correlations between eigenvalues of various ensembles of random unitary, orthogonal or symplectic matrices, in the limit when their size tends to infinity, are known to exhibit universal behavior in the bulk of the spectrum. Dyson’s “Threefold Way” [14] predicts that this behavior is to depend only on the symmetry type of the ensemble (unitary, orthogonal or symplectic). Unfortunately, for general ensembles this conjecture remains open, though in the unitary case (modeled after the Gaussian Unitary Ensemble) the universality of the local correlations has been proven for some classes of families [9, 7, 3, 2]. In the orthogonal and symplectic cases the extension of results known for Gaussian ensembles is technically more complicated but some more recent work deals with families of such ensembles [26]. Most of the focus has been on non-compact (Gaussian and the like) matrix ensembles. In the present article we study families of compact (circular) ensembles including, in particular, Dyson’s circular ensembles: the COE, CUE and CSE [11]. In Sect. 2 we fit Dyson’s ensembles into the framework of the theory of symmetric spaces. In Sect. 3 we proceed to associate a matrix ensemble to every family of irreducible compact symmetric spaces (all of these are known by the work of

This research has been supported in part by the FRG grant DMS–00–74028 from the NSF

30

E. Dueñez Table 1. Parameters of the probability measure of the eigenvalues for ensembles of type I Type AI (COE) A II (CSE) A III BD I D III CI C II

G/K

Parameters

U (R)/O(R)

β = 1 (not Jacobi)

U (2R)/U Sp(2R)

β = 4 (not Jacobi)

U (2R + L)/U (R + L) × U (R) O(2R + L)/O(R + L) × O(R) SO(4R)/U (2R) SO(4R + 2)/U (2R + 1) U Sp(2R)/U (R) U Sp(4R + 2L)/ U Sp(2R + 2L) × U Sp(2R)

β β β β β

= 2, (a, b) = (L, 0) = 1, (a, b) = ((L − 1)/2, −1/2) = 4, (a, b) = (0, 0) = 4, (a, b) = (2, 0) = 1, (a, b) = (0, 0)

β = 4, (a, b) = (2L + 1, 1)

Cartan [4, 5]). If G/K is a compact symmetric space, then K is the fixed-point set of an involution of G, and using this involution and a realization of G as a matrix group we embed G/K → G, thus realizing it as a matrix ensemble, cf., Theorem 1. By the Weyl integration formula, the measure on G is decomposed as a product of measures on K × A × K, where A is an abelian torus and it is the induced measure on A that yields the probability measure of the eigenvalues, cf., Theorem 2. As for specific examples, the most well-known of the compact symmetric spaces are the classical groups of orthogonal, unitary and symplectic matrices, for which questions about universality have known answers [17]. These are the so-called “Type II” spaces. Zirnbauer [30], on the other hand, has constructed the “infinitesimal” versions of the other (type I) ensembles, namely their tangent spaces at the identity element, which is enough to derive their eigenvalue measures. Our Theorem 1, however, describes the “global” ensembles associated to the infinite families of compact symmetric spaces of type I in a very explicit manner analogous to Dyson’s description of his circular ensembles. Besides Dyson’s COE and CSE, the other compact matrix ensembles of type I are of Jacobi type in the sense that their joint eigenvalue measure is given by dν(x1 , . . . , xR ) ∝

1≤j −1 (depending on the ensemble, see Table 1) and β = 1, 2, 4 (the “symmetry parameter”) in the orthogonal, unitary and symplectic cases, √ respectively. Here, the “free” eigenvalues come in R pairs (or quadruples) xj ± −1yj – excluding eigenvalues equal to +1 forced by the symmetry built into the ensemble. Also, R stands for the rank of the corresponding symmetric space, and our interest is in the semiclassical limit of the eigenvalue statistics as R → ∞ (here L ≥ 0 is a fixed integer: different values of this parameter yield different ensembles.) The name “Jacobi ensembles” comes from the intimate connection between the measure (1) and the classical Jacobi polynomials on the interval [−1, 1]. In Sect. 4 we state the main results on the level density (Theorem 3) and the universality of the local correlations (Theorem 4) in the bulk of the spectrum for general unitary, orthogonal and symplectic Jacobi ensembles (previous results of Nagao and Forrester [23] are insufficient for our purposes). We rely on work of Adler et al [1]. At the “hard edges” ±1 of the interval, Dyson’s universality breaks down and we obtain simple formulas for the Bessel kernel in terms of which the hard edge correlations are expressed. In a nutshell, for Jacobi ensembles:

Matrix Ensembles Associated to Symmetric Spaces

31

• Away from the “hard edge” x = ±1, the local correlations follow the universal law of the GOE (β = 1), GUE (β = 2) or GSE (β = 4). Namely, in terms of local parameters ξj around a fixed zo = cos αo ∈ (−1, 1) so that xj = cos(αo + (π/R)ξj ), these local correlations are given by (n) Lβ (zo ; ξ1 , . . . , ξn ) = DET(K¯ β (ξj , ξk ))n×n ,

(2)

where DET stands for either the usual (β = 2) or quaternion (β = 1, 4) determinant, and Kβ is the (scalar or quaternion) Sine kernel (cf., Eqs. (74)–(78).) • At the hard edge zo = +1, the local correlations depend on the parameter a of the Jacobi ensemble as well as on β. In terms of local parameters ξj > 0 with xj = cos((π/R)ξj ) the same expression (2) holds except that the kernel K¯ β is to (a) be replaced by a Bessel kernel Kˆ β (ξ, η) given by Eqs. (84)–(86). At the hard edge zo = −1 the result is equivalent upon replacing a by b. 2. Dyson’s Circular Ensembles as Symmetric Spaces For motivational purposes in this section we review the constructions of the circular ensembles of Dyson and their probability measures of the eigenvalues in a manner in which the theory of Riemannian symmetric spaces is brought into play. The Circular Unitary Ensemble (CUE) is the set S = S(N ) of all N × N unitary matrices H , endowed with the unique probability measure dµ(H ) that is invariant under left (also right) multiplication by any unitary matrix. This requirement makes the measure invariant under unitary changes of bases, hence the ensemble’s name. In the study of statistics of eigenvalues, the relevant probability measure is the one induced by dµ(H ) on the torus A = A(N ) ⊂ S(N ) consisting of unitary diagonal matrices A = {diag(λ1 = eiθ1 , . . . , λN = eiθN )},

(3)

where = (θ1 , . . . , θN ) ∈ [0, 2π)N , say. To be more precise, let us denote by K = K(N) the unitary group of N × N matrices (its underlying set is just S(N)). Then we have a surjective mapping K ×A S (k, a) → H = kak −1 ,

(4)

and correspondingly there exists a probability measure dν(a) on A such that, for any continuous function f ∈ C(S), f (H )dµ(H ) = f (kak −1 )dν(a)dHaar(k), (5) S

K

A

where we denote by dHaar(k) the unique translation-invariant probability measure on K (so here dHaar = dµ). This measure dν(a) can be pulled back to some measure on the space [0, 2π)N of angles which, abusing notation, we denote by dν( ). The measure dν() (or dν( )) is the so-called probability measure of the eigenvalues (for the CUE). We have [21], 2 dν( ) ∝ Van(ei ) d on [0, 2π )N . (6)

32

E. Dueñez

Here the symbol “∝” stands for proportionality up to a constant (depending only on N ), d = dθ1 . . . dθN is the usual translation-invariant measure on the space of angles , ei = (eiθ1 , . . . , eiθN ) and, for a vector x = (x1 , . . . , xN ), Van(x) is the Vandermonde determinant (xk − xj ). (7) Van(x) = det (xjk−1 ) = N×N

1≤j −1 and any real B: θ A (N + A + 1) θ θ B (A,B) sin cos PN (cos θ ) = N −A JA (N θ) + E, (95) 2 2 N! sin θ where N has the same meaning as in (93) and the error term E is given by θ 1/2 O(N −3/2 ) if c/N ≤ θ ≤ π − , E= θ A+2 O(N A ) if 0 < θ ≤ c/N,

(96)

where c, are arbitrary but fixed positive constants, and the constants implied by the O symbol depend on A, B, c, only. The restriction to A > −1, however, is too strong for some purposes, and we will need the following formula, also due to Szegö [27] (reproduced in [29]): θ tan(θ/2) θ −B θ −A (A,B) PN cos 1− (cos θ ) = sin 2 2 sin θ 2θ (97) × JA (N θ ) + R, with N as in (93). Here A, B are arbitrary reals. The error term R satisfies: 1 θ 2 −A O(N −3/2 ) if c/N ≤ θ ≤ π − , R= O(N A−2 ) if 0 < θ ≤ c/N,

(98)

where c, are fixed positive numbers, and the constants implied by the O symbol depend only on A, B, c, . It must be noted, however, that the error term R of (98) does not depend on θ on the range 0 < θ < c/N , which makes this formula less useful than (95) with the error term (96) for θ in this range. Recall Stirling’s asymptotic formula for the Gamma function: 1 1 log (x) = x − log x − x + log 2π + O(x −1 ), as x → ∞. (99) 2 2 The Bessel functions of the first kind are defined by the series ∞ z ν z2k Jν (z) = , z ∈ C\(−∞, 0], ν ∈ R; (100) (−1)k 2k 2 2 k!(ν + k + 1) k=0

they satisfy, among many others, the relations: ν Jν (z) = Jν−1 (z) − Jν (z), z ν Jν (z) = −Jν+1 (z) + Jν (z), z 1 Jν (z) = [Jν−1 (z) − Jν+1 (z)], 2 2ν Jν+1 (z) = Jν (z) − Jν−1 (z), z d ν [z Jν (z)] = zν Jν−1 (z), dz d −ν [z Jν (z)] = −z−ν Jν+1 (z). dz

(101) (102) (103) (104) (105) (106)


51

We also have Jν = 2

∞

∞

Jν+2k+1 ,

(107)

for ν > −1.

(108)

k=0

Jν (t)dt = 1

0

5.2. Asymptotics of the unitary Jacobi kernel. In this section we recall the proofs of some of the results of Nagao and Wadati [24], which will be needed later on in the analysis of the orthogonal and symplectic cases. (A,B) Using the Christoffel-Darboux summation formula [28], the scalar kernel KN2 can be written in the form (A,B)

KN 2

(x, y) =

2−A−B (N + 1)(N + A + B + 1) 2N + A + B (N + A)(N + B) (A,B) (A,B) (A,B) (A,B) ! PN (x)PN−1 (y) − PN−1 (x)PN (y) × w(x)w(y) , (109) x−y

for x = y, and (A,B)

KN 2

(x, x) =

2−A−B (N + 1)(N + A + B + 1) 2N + A + B (N + A)(N + B) (A,B) (A,B) (A,B) (A,B) ×w(x)[PN (x)PN−1 (x) − PN−1 (x)PN (x)]. (110)

We observe that the kernel KN2 given by (109) and (110) is well-defined for A, B > −c for any real constant c provided N is sufficiently large. First consider the global level density ρ(x) = lim N −1 K(x, x). N→∞

(111)

Using Darboux’s formula (93) together with the identity (92) in the expression (110) for the kernel, we find: (a,b)

KN2 (x, x) =

√

N

π 1 − x2

+ O(1),

(112)

where the implied constant depends only on for −1 + ≤ x ≤ 1 − . Equation (112) proves (69) (in the unitary case). A density function D = D(x1 , . . . , xn ) defines a measure D dx1 . . . dxn . Under a (monotonically increasing or decreasing) differentiable change of variables xj = X(uj ), this density is transformed into the density   n |X (uj )| D(X(u1 ), . . . , X(un )). (113) D(u1 , . . . , un ) =  j =1

If the density D is given as a determinant with a (scalar) kernel K(x, y), namely D = det(K(xj , xk ))n×n , then the change of variables reflects itself in the kernel in the following fashion:

52

E. Dueñez

Lemma 1. After the (monotonic) differentiable change of variables u → x = X(u), the correlation functions are given as the determinant (59) defined using the kernel K(u, v) =

!

|X (u)X (v)|K(X(u), X(v)).

(114)

√ This is clear since the" introduction of the factor |X (u)X (v)| results in multiplying n the determinant (59) by j =1 |X (uj )|. The localization at some −1 < zo = cos αo < 1 given by the change of variables (71) leads us to consider the limit # ! $−1 (a,b) (a,b) K¯ 2 (ξ, η) = lim N ρ(x)ρ(y) KN2 (x, y) N→∞ (115) (a,b) = lim (Nρ(zo ))−1 KN2 (x, y), N→∞

with x, y related to ξ, η by (71), which from Darboux’s formula (93) can be easily seen to be the Sine Kernel (74), independently of the value of zo (as long as −1 < zo < 1), for any real a, b, and the limit is attained uniformly on compacta. For the localization at zo = +1 (αo = 0) – localization at zo = −1 is analogous provided a and b are interchanged, we use the same change of variables (71) with ξ n > 0. To compute the limit (a,b) Kˆ 2 (ξ, η) = lim

N→∞

# ! $−1 (a,b) N ρ(x)ρ(y) KN2 (x, y)

= lim (Nρ(zo ))−1 KN2 (x, y), (a,b)

(116)

N→∞

we use Szegö’s formulas (95), (97), in conjunction with (109) and (110): (a) Kˆ 2 (ξ, η) =

√ ξη [πξ Ja (π ξ )Ja (π η) − Ja (π ξ )π ηJa (π η)]. ξ 2 − η2

(117)

Using the derivation formula (102) we rewrite this kernel in the form (84). For the case ξ = η we start with the expression (110) and use the derivation formula (92) to find: (a) (a) ρˆ2 (ξ ) = Kˆ 2 (ξ, ξ ) π (π ξ )Ja (π ξ ) − π ξ Ja (π ξ ) Ja+1 (π ξ )]. = [Ja (π ξ )Ja+1 (π ξ ) + πξ Ja+1 2 (118)

Applying the derivation formula (101) and the recurrence formula (104) this can be rewritten in the form (80).

5.3. Asymptotics of the orthogonal Jacobi kernel. We start with some general remarks. If a density P = P (x1 , . . . , xn ) is given as a quaternion determinant with a self-dual matrix kernel K(y, x) = K(x, y)D , namely P = qdet(Q(xj , xk ))n1 , then under a differentiable change of variables xj = X(uj ) the density is still given as a quaternion determinant.


53

Lemma 2. After a (monotonic) differentiable change of variables u → x = X(u), a density function P (x1 , . . . , xn ) = qdet(K(xj , xk )) defined in terms of some self-dual matrix kernel (δ = 0, 1) S(x, y) I (x, y) − δ(x − y) K(x, y) = D(x, y) S T (x, y) with

I (x, y) = −

(119)

(120)

y

S(x, z)dz,

(121)

x

D(x, y) = ∂x S(x, y), S (x, y) = S(y, x) T

(122) (123)

is transformed into the density P(u1 , . . . , un ) = qdet(K(uj , uk )), where

(124)

K(u, v) =

S(u, v) I(u, v) − δ(u − v) , D(u, v) S T (u, v)

S(u, v) = S(X(u), X(v))|X (v)| = ±S(X(u), X(v))X (v), v I(u, v) = − S(u, w)dw,

(125) (126) (127)

u

D(u, v) = ∂u S(u, v),

(128)

S (u, v) = S(v, u). T

(129)

For the proof, we need first: Lemma 3. Let H = H D = Jn H T JnT be a 2n × 2n self-dual complex matrix. Let kj , j = 1, 2, . . . , n be arbitrary complex constants. Set K = diag(k1 , . . . , kn ). Then the matrices H1 = diag(I, K)H diag(K, I )

H2 = diag(−I, K)H diag(−K, I )

(130)

(where I = In is the n × n identity matrix) are both self-dual, and qdet(H1 ) = det(K) qdet(H ) = qdet(H2 ).

(131)

The verification that H1 and H2 are self-dual is trivial. On the other hand, since (qdet X)2 = det X for any self-dual matrix X, we have that (qdet(H1 ))2 = (qdet(H2 ))2 = (det(K))2 det(H ) = (det(K))2 (qdet(H ))2 .

(132)

Hence Eq. (131), which is an equality between polynomials in the entries of the matrices involved, must hold up to a sign. Setting K = In we see that the first equality in (131)

54

E. Dueñez

holds, and setting K = −In , so H2 = −H , the validity of the second equality in (131) is equivalent to the easy fact that qdet(−H ) = (−1)n qdet H = det(−In ) qdet H . Proceeding to the proof of Lemma 2, we first observe that, after the change of variables u → x, the density P (x1 , . . . , xn ) transforms into the density P(u1 , . . . , un ) = P (X(u1 ), . . . , X(un ))

n

|X (uj )|.

(133)

j =1

We apply Lemma 3 with H = (K(X(uj ), X(uk )))n×n and kj = |X (uj )| to conclude that (124) holds with either of the two kernels (we write X(u, v) for (X(u), X(v))) S(X(u, v))|X (v)| ±(I − δ)(X(u, v)) K± (u, v) = . (134) ±D(X(u, v))|X (u)||X (v)| S T (X(u, v))|X (u)| The plus and minus signs correspond to applying the first and second of the equalities in (131), respectively. If x → u preserves orientation, then we observe that (X(u) − X(v)) = (u − v) and conclude by a simple application of the chain rule and a change of variables in the integral that the kernel K+ coincides with K from (125) for the choices (126)–(129). If x → u reverses orientation, we choose the minus signs, observe that (X(u)−X(v)) = −(u−v) and proceed exactly as before to see that K− coincides with (125) in this case. Lemma 2 explains the relations (78) between the entries of the limiting kernels K¯ β and also of Kˆ β (β = 1, 4). The relations certainly hold when R is finite after applying the change of variables (71) to the the matrix kernel KRβ so as to obtain another kernel KRβ . They can be shown to continue to hold in the limit either by noting that the sequence of scalar kernels {SRβ (ξ, η)}∞ R=0 is a normal sequence of analytic functions (i.e., it converges uniformly on compacta), or by direct verification that each of T } converges to the correct limit as R → ∞. In the sequences {SRβ }, {IRβ }, {KRβ }, {SRβ what follows we will only consider the limit of the quantity SRβ which alone determines the matrix kernel KRβ . Let A = 2a + 1, B = 2b + 1, where a, b are the parameters of the orthogonal Jacobi ensemble. Assume also that R is even. Observe that A, B > −1 if a, b > −1. The (a,b) summation formula of Adler et al [1] expresses the orthogonal kernel SR1 using the (A,B) unitary kernel KR−1,2 and another term. As we shall see, this other term is negligible in the localized limit (in the bulk of the spectrum), but it does contribute to the edge limit. (a,b) The summation formula for the quantity SR1 (x, y) of (60) is as follows [1]: 1 − x 2 (A,B) (a,b) K (x, y) + cR−2 ψR−1 (y)ψR−2 (x). (135) SR1 (x, y) = 1 − y 2 R−1,2 Here denotes the integral operator (cf., Eq. (61)) 1 (x − y)f (y)dy, (f )(x) =

(136)

−1

and we have set (A,B)

ψN (t) = ψN and

(A,B)

(t) = (1 − t)(A−1)/2 (1 + t)(B−1)/2 PN

(t)

(137)


cN = 2−A−B−1

55

(N + 2)(N + A + B + 2) . (N + A + 1)(N + B + 1)

(a,b)

(138)

(a,b)

The quantity SR1 determines the entries of the matrix kernel KR1 as per Eqs. (62)– (64). From Stirling’s formula (99), the asymptotic behavior of the coefficient cN is cN ∼ 2−A−B−1 N 2 ,

as N → ∞.

(139)

Lemma 4. For any real A, B: (A,B)

lim ψN

N→∞

(cos φ) = 0

(140)

for 0 < φ < π, uniformly on compacta. This follows immediately from Darboux’s formula (93). This lemma is, however, insufficient to understand the asymptotics of the function ψN as N → ∞ since it says nothing about the behavior of ψN near the edge. First we note: Lemma 5. For A > −1 and B arbitrary: lim N −1 ψN

(cos(φ/N )) = 2

A+B 2

(cos(φ/N )) sin(φ/N ) = 2

A+B 2

(A,B)

N→∞

(A,B)

lim ψN

N→∞

JA (φ) , φ

(141)

JA (φ).

(142)

The limits hold uniformly on compact subsets of (0, ∞). These follow from Szegö’s formula (95). Lemma 6. For A, B real with A > −1 and any 0 < θ < π we have:

θ

lim N

N→∞

0 θ/N

lim N

N→∞

0

(A,B)

(cos φ) sin φ dφ = 2

A+B 2

(A,B)

(cos φ) sin φ dφ = 2

A+B 2

ψN ψN

(143)

,

θ

(144)

JA . 0

These follow again from Szegö’s formula (95) and Eq. (108). When −1 < A < 0, the dependence on θ of the second of the error terms in (96) is critical to ensure that the contribution of this error term to the integral is negligible (in particular, this lemma cannot be proven using the alternate formula (98) unless A > 0.) Corollary 1. For −1 < A, B and 0 < θ < π: (A,B)

lim N(ψN

N→∞

(A,B)

lim N (ψN

N→∞

)(cos θ ) = 0,

)(cos(θ/N )) = 2

A+B 2

(145)

θ

1−

JA 0

=2

A+B 2

∞

JA . θ

(146)

56

E. Dueñez (A,B)

(B,A)

This follows from the previous lemma applied to both ψN and ψN . We also used (108) to obtain the last equality. We localize at some zo = cos αo ∈ (−1, 1) using the change of variable x → ξ of (71). The limit to consider is (a,b) (a,b) (a,b) S¯1 (ξ, η) = lim (Nρ(y))−1 SR1 (x, y) = lim (Nρ(zo ))−1 SR1 (x, y). R→∞

R→∞

(147)

By the lemmas above, the second term on the right-hand side of (135) is negligible 2

is 1 in the limit. Thus, the limit (147) is equal to the in the limit. Also, the factor 1−x 1−y 2 limiting unitary kernel, namely the Sine Kernel, whence the expression (78). As for the central point, let us now localize at z = +1. Using the summation formula (135), Lemma 5 and Corollary 1, we readily find: % & πξ ξ ˆ (2a+1) π a ˆ S1 (ξ, η) = (ξ, η) + J2a+1 (π η) 1 − J2a+1 (t)dt K η 2 2 0 ∞ ξ ˆ (2a+1) π = (ξ, η) + J2a+1 (π η) J2a+1 (t)dt. (148) K η 2 2 πξ As we remarked already, the conditions a > −1 and A > −1 are equivalent since A = 2a + 1. Thus we have derived a weak universality law for the local correlations at the central points ±1 for any a, b > −1. Lemma 7. Let κα (x, y) = xJα+1/2 (x)Jα−1/2 (y) − Jα−1/2 (x)yJα+1/2 (y). Then x y κα±1/2 (x, y) − κα∓1/2 (x, y) y x 2 x − y2 =∓ Jα−1/2∓1/2 (x)Jα−1/2±1/2 (y). √ xy

(149)

(This equation stands for two different equations, one with the top signs and another with the bottom signs.) We prove the equation with the choice of the top signs (the other case is analogous). Indeed, expanding the left-hand side we obtain: x 3/2 Jα+1 (x)y −1/2 Jα (y) − x 1/2 Jα (x)y 1/2 Jα+1 (y) −x 1/2 Jα (x)y 1/2 Jα−1 (y) + x −1/2 Jα−1 (x)y 3/2 Jα (y).

(150)

The central terms can be combined into −2αx 1/2 Jα (x)y −1/2 Jα (y) using the identity (104) and expanded using this same identity into −x 3/2 Jα−1 (x)y −1/2 Jα (y) − x 3/2 Jα+1 (x)y −1/2 Jα (y). Two terms cancel out, and the remaining two factor to give the right-hand side of (149). We now have, using Lemma 7, √ ξ ˆ (A) ξη K2 (ξ, η) = 2 κA+1/2 (π ξ, π η) η ξ − η2 √ η ξη = κA−1/2 (π ξ, π η) + π JA (π ξ )JA−1 (π η) ξ ξ 2 − η2


η ˆ (A−1=2a) (ξ, η) − π JA−1 (π ξ )JA (π η), K ξ 2

= and similarly

57

(151)

ξ ˆ (A) K (ξ, η) = η 2

From (107):

πξ

η ˆ (A+1) (ξ, η) + π JA+1 (π ξ )JA (π η). K ξ 2

JA ± 2JA∓1 (π ξ ) =

0

(152)

πξ

(153)

JA∓2 . 0

(a) The last two equations provide alternative forms of the kernel Sˆ1 , namely % & πξ η ˆ (2a) π (a) Sˆ1 (ξ, η) = K2 (ξ, η) + J2a+1 (π η) 1 − J2a−1 (t)dt , ξ 2 0 % & πξ π η ˆ (2a+2) (a) Sˆ1 (ξ, η) = (ξ, η) + J2a+1 (π η) 1 − J2a+3 (t)dt . K2 ξ 2 0 ' ∞ As before, the terms in brackets can be replaced by πξ .

(154) (155)

5.4. Asymptotics of the symplectic Jacobi kernel. Here we set A = a − 1, B = b − 1, where a, b are the parameters of the symplectic Jacobi ensemble. Note that here a, b > (A,B) as in (137), the −1 corresponds to A, B > −2. With cN as in (138) and ψN = ψN summation formula in this case reads 1 1 − x 2 (A,B) 1 (a,b) SR4 (x, y) = K (x, y) − c2R−1 ψ2R (y)δψ2R−1 (x), (156) 2 1 − y 2 2R,2 2 where the operator δ acts by

δf (x) =

1

(157)

f (t)dt. x

The formula (156) only holds verbatim when a > 0 (that is, A, B > −1), since the (A,B) integral defining δψN is divergent for A ≤ −1. However, we note that the skew orthogonal polynomials of the second kind are analytic functions of the parameters a, b > −1 (corresponding to A, B > −2), hence the kernel KN4 is an analytic function on a, b > −1. Thus, we must find a suitable analytic continuation of (156) valid (A,B) for A, B > −2. First we remark that, although the original kernel K2R,2 of unitary Jacobi ensembles is defined for A, B > −1, Eq. (109) is well-defined and analytic for A, B > −2 if R > 1 (which we will assume). We write 1 (A,B) (A,B) δψN (x) = (1 − t)(A−1)/2 (1 + t)(B−1)/2 PN (t)dt x

=

x

1

(A,B)

(1 − t)(A−1)/2 (1 + t)(B−1)/2 (PN (A,B)

+PN

1

(1) x

(A,B)

(t) − PN

(1 − t)(A−1)/2 (1 + t)(B−1)/2 dt.

(1))dt (158)

58

E. Dueñez

The first integral on the right-hand side is well-defined and analytic for A > −2. The $ # (A,B) (cf., Eq. (91)) vanishes for A = −1, which is sufficient to (1) = A+N term PN N extend the second integral on the right-hand side to a well-defined analytic function on the range A > −2. It is easy to rewrite that integral as an incomplete Beta function and use well-known results to achieve the extension, but one can also proceed elementarily as follows. Integrating the second integral by parts we obtain, for A > −1: 1 A+N (1 − t)(A−1)/2 (1 + t)(B−1)/2 dt N x 2 A+N = (1 − x)(A+1)/2 (1 + x)(B−1)/2 A+1 N 1 B −1 A+N + (1 − t)(A+1)/2 (1 + t)(B−3)/2 dt. (159) A+1 N x Observe that

1 1 A+N A+N = , A+1 N N N −1

(160)

and the latter is an analytic function of all A. Then both terms on the right-hand side of (159) are analytic functions of A > −2 for −1 < x ≤ 1, so this last equation provides the analytic extension of the integral (158) defining δψN (x), which is sensu stricti undefined for A ≤ −1, to an analytic function on A > −2. The rest of the reasoning is analogous to that in the orthogonal case. The only technical difficulty arises because the error term (98) in Szegö’s formula does not depend on θ in the range 0 < θ ≤ c/N , effectively making the reasoning of the previous section inapplicable when −2 < A ≤ −1. This is to be expected since the summation formula only makes sense after being analytically continued. In what follows we prove that the various limits of the kernel do in fact depend analytically on the parameter A, thus allowing the expressions obtained for A > −1 to be extended to A > −2. Using Szego’s formula (97) (valid for all A), there is no problem to obtain this variant of Lemma 6: Lemma 8. For any A, B, θ real and 0 < ψ < π we have: φ A+B (A,B) lim N ψN (cos ψ) sin ψ dψ = 2 2 N→∞

θ/N

Lemma 9. Using Eq. (159), the expression θ/N (A,B) N ψN (cos φ) sin φ dφ

∞

JA .

(161)

θ

(162)

0

can be analytically continued to a regular function on A > −2. As N → ∞, this function tends to a limit which is also analytic for A > −2 and coincides with (144) for A > −1. We change variables φ → φ/N. As before, we split the integral to rewrite (162) in the form & % θ φ A φ B φ (A,B) (A,B) 2(A+B)/2 sin cos PN cos (1) dφ − PN 2N 2N N 0


+2

(A+B)/2

(A,B) PN (1)

θ

φ sin 2N

0

59

A

φ cos 2N

B

(A,B)

PN

(1) dφ.

(163)

The first of these terms is analytic for A > −2, the second one has an analytic continuation given by (159). It is easy to see that this second term has the asymptotic behavior: θ φ A φ B (A,B) sin cos 2(A+B)/2 PN (1) dφ 2N 2N 0 A+1 B−A 1 A+N φ ∼2 2 (164) N N −1 N # $ as N → ∞, and from Stirling’s formula (99), the binomial coefficient A+N N−1 = (A+N +1) A+1 ), hence this second term is asymptotically negligible. As for (N)(A+2) = O(N the first term in (163), we first write φ (A,B) (A,B) PN cos (1) − PN N φ 1 ψ ψ (A,B) =− cos P sin dψ N 0 N N N N + A + B + 1 φ (A+1,B+1) ψ ψ =− cos PN −1 sin dψ, (165) 2N N N 0 where we have used the derivation formula (92). We can now use Szegö’s formula (95) to (A+1,B+1) since A + 1 > −1. The upshot is that the limit of (162) as N → ∞ estimate PN−1 can be written as the following integral, which is an analytic function of A > −2: θ φ (A+B)/2 φ A ψ −A JA+1 (ψ)dψ dφ. (166) −2 0

0

Using the Bessel function identity (106) we can simplify the above integral, for A > −1: θ (A+B)/2 JA (φ)dφ, (167) 2 0

which is in agreement with Lemma 6. We note that the expression (167) can be easily continued to an analytic function of A > −2 without the need to rewrite it as the double integral (166). Namely, using (107) we have, for A > −1, θ θ JA (φ)dφ = JA+1 (θ ) + JA+2 (φ)dφ. (168) 0

0

The expression on the right-hand side is analytic for A > −2 and provides the desired analytic continuation. The global level density is derived identically to the previous section. The limiting (a) kernel in the bulk of the spectrum is given by the sum of two terms: S¯2 (2ξ, 2η) and another term which is negligible in the limit. For the central point z = +1, the lemmas above yield the following expression for the limiting kernel: 2πξ ξ ˆ (A) π (a) Sˆ4 (ξ, η) = K2 (2ξ, 2η) − JA (2π η) JA (t)dt, (169) η 2 0

60

E. Dueñez

where the last integral is to be understood in the sense of Eq. (168) for A ≤ −1. Using Eqs. (151) and (152) together with (153) and Eq. (169) above, the kernel can be rewritten in either of the forms: 2πξ η ˆ (a) π (a) Sˆ4 (ξ, η) = K2 (2ξ, 2η) − Ja−1 (2πη) Ja+1 (t)dt, (170) ξ 2 0 2πξ η ˆ (a−2) π (a) (2ξ, 2η) − Ja−1 (2π η) Ja−3 (t)dt. (171) K2 Sˆ4 (ξ, η) = ξ 2 0 Acknowledgement. I wish to thank Prof. Peter Sarnak for his continued encouragement and guidance as my Ph. D. thesis advisor as well as Brian Conrey for making my stay at AIM possible.

References 1. Adler, M., Forrester, P.J., Nagao, T., van Moerbeke, P.: Classical skew orthogonal polynomials and random matrices. J. Statist. Phys. 99(1–2), 141–170 (2000) 2. Albeverio, S., Pastur, L., Shcherbina, M.: On asymptotic properties of certain orthogonal polynomials. Mat. Fiz. Anal. Geom. 4(3), 263–277 (1997) 3. 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) ´ La géométrie des groupes simples. Ann. Math. Pura Appl. 4, 209–256 (1927) 4. Cartan, E.: ´ Sur certaines formes riemannienes remarquables des géométries a groupe fondamental 5. Cartan, E.: ´ simple. Ann. Sci. Ecole Norm. Sup. 44, 354–467 (1927) 6. Chadha, S., Mahoux, G., Mehta, M.L.: A method of integration over matrix variables. II. J. Phys. A 14(3), 579–586 (1981) 7. Deift, P., Kriecherbauer, T., T-R McLaughlin, K., Venakides, S., Zhou, X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52(12), 1491– 1552 (1999) 8. Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou., X.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52(11), 1335–1425 (1999) 9. Percy, Deift, A., Alexander, Its, R., Xin Zhou.: A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics. Ann. Math. (2) 146(1), 149–235 (1997) 10. Dueñez, E.: Random Matrix Ensembles associated to Compact Symmetric Spaces. PhD thesis, Princeton: Princeton University, 2001 11. Dyson, F.J.: Statistical theory of the energy levels of complex systems. I. J. Math. Phys. 3, 140–156 (1962) 12. Dyson, F.J.: Statistical theory of the energy levels of complex systems. II. J. Math. Phys. 3, 157–165 (1962) 13. Dyson, F.J.: Statistical theory of the energy levels of complex systems. III. J. Math. Phys. 3, 166–175 (1962) 14. Dyson, F.J.: The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics. J. Math. Phys. 3, 1199–1215 (1962) 15. Gaudin, M.: Sur la loi de l’espacement limit des valeurs propres d’une matrice aléatoire. Nucl. Phys. 25, 447–458 (1961) 16. Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products. New York: Academic Press, 1965 17. Katz, N.M., Sarnak, P.: Random matrices, Frobenius eigenvalues, and monodromy. Providence, RI: American Mathematical Society, 1999 18. Mehta., M.L.: A method of integration over matrix variables. Commun. Math. Phys. 79(3), 327–340 (1981) 19. Mehta, M.L., Gaudin., M.: On the density of eigenvalues of a random matrix. Nucl. Phys. 18, 420–427 (1960) ´ 20. Mehta, M.L.: Matrix Theory. Selected Topics and Useful Results. Les Editions de Physique, Les Ulis cedex, France, first enlarged edition, 1989 21. Mehta, M.L.: Random Matrices. Boston, MA: Academic Press Inc., Second edition, 1991


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22. Mehta, M.L., Mahoux, G.: A method of integration over matrix variables. III. Indian J. Pure Appl. Math. 22(7), 531–546 (1991) 23. Taro Nagao, Peter, Forrester., J.: Asymptotic correlations at the spectrum edge of random matrices. Nucl. Phys. B 435(3), 401–420 (1995) 24. Nagao, T., Wadati., M.: Correlation functions of random matrix ensembles related to classical orthogonal polynomials. J. Phys. Soc. Japan 60(10), 3298–3322 (1991) 25. Nagao, T., Wadati., M.: Correlation functions of random matrix ensembles related to classical orthogonal polynomials. II. J. Phys. Soc. Japan 61(1), 78–88 (1992) 26. Stojanovic, A.: Une approche par les polynômes orthogonaux pour des classes de matrices aléatoires orthogonalement et symplectiquement invariantes:Application a` l’universalité de la statistique locale des valeur propres. Technical report, BiBoS Universität Bielefeld, 2000 27. Szegö, G.: Asymptotische Entwicklungen der Jacobischen Polynome. Schr. Königsb. Gelehr. Ges. Nat.wiss. Kl. 10, 35–112 (1933) 28. Szegö, G.: Orthogonal polynomials. Providence, RI: American Mathematical Society, 1975, American Mathematical Society, Fourth edition, Colloquium Publications, Vol. XXIII 29. Szegö, G.: Collected papers. Vol. 2. Boston, Mass: Birkhäuser, 1982, 1927–1943, Edited by Richard Askey. 30. Zirnbauer, M.R.: Riemannian symmetric superspaces and their origin in random-matrix theory. J. Math. Phys. 37(10), 4986–5018 (1996) Communicated by P. Sarnak


Communications in


Classification of Two-Dimensional Local Conformal Nets with c < 1 and 2-Cohomology Vanishing for Tensor Categories Yasuyuki Kawahigashi1, , Roberto Longo2, 1

Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153-8914, Japan. E-mail: [email protected] 2 Dipartimento di Matematica Università di Roma “Tor Vergata” Via della Ricerca Scientifica, 1, 00133 Roma, Italy. E-mail: [email protected] Received: 14 April 2003 / Accepted: 1 July 2003 Published online: 13 November 2003 – © Springer-Verlag 2003

Abstract: We classify two-dimensional local conformal nets with parity symmetry and central charge less than 1, up to isomorphism. The maximal ones are in a bijective correspondence with the pairs of A-D-E Dynkin diagrams with the difference of their Coxeter numbers equal to 1. In our previous classification of one-dimensional local conformal nets, Dynkin diagrams D2n+1 and E7 do not appear, but now they do appear in this classification of two-dimensional local conformal nets. Such nets are also characterized as two-dimensional local conformal nets with µ-index equal to 1 and central charge less than 1. Our main tool, in addition to our previous classification results for one-dimensional nets, is 2-cohomology vanishing for certain tensor categories related to the Virasoro tensor categories with central charge less than 1.

1. Introduction The subject of Conformal Quantum Field Theory is particularly interesting in two spacetime dimensions and has indeed been intensively studied in the last two decades with important motivations from Physics (see e.g. [11]) and Mathematics (see e.g. [14]). Basically the richness of structure is due to the fact that the conformal group (with respect to the Minkowskian metric) is infinite dimensional in 1 + 1 dimensions. Already at the early stage of investigation, it was realized that such infinite dimensional symmetry group puts rigid constrains on structure and the problem of classification of all models was posed and considered as a major aim. Indeed many important results in this direction were obtained, in particular the central charge c > 0, an intrinsic quantum label associated with each model, was shown to split in a discrete range c < 1 and a continuous one c ≥ 1, see [2, 17, 19] and refs in [19].

Supported in part by JSPS. Supported in part by GNAMPA and MIUR.

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Y. Kawahigashi, R. Longo

The main purpose of this paper is to achieve a complete classification of the twodimensional conformal models in the discrete series. In order to formulate such a statement in a precise manner, we need to explain our setting. The essential, intrinsic structure of a given model is described by a net A on the twodimensional Minkowski spacetime M [22]. With each double cone O (an open region which is the intersection of the past of one point and the future of a second point) one associates the von Neumann algebra A(O) generated by the observables localized in O (say smeared fields integrated with test functions with support in O). The net A : O → A(O) is then local and covariant with respect to the conformal group. One may restrict A to the two light rays x ± t = 0 and obtain two local conformal nets A± on R, hence on its one point compactification S 1 . So we have an irreducible two-dimensional subnet B(O) ≡ A+ (I+ ) ⊗ A− (I− ) ⊂ A(O) , where O = I+ × I− is the double cone associated with the intervals I± of the light rays. The structure of A, thus the classification of local conformal nets, splits in the following two points: • The classification of local conformal nets on S 1 . • The classification of irreducible local extension of chiral conformal nets. Here a chiral net is a net that splits in the tensor product of two one-dimensional nets on the light rays. Now the conformal group of M is Diff(S 1 ) × Diff(S 1 ) 1 thus, restricting the projective unitary covariance representation to the two copies of Diff(S 1 ), we get Virasoro nets Vir c± ⊂ A± with central charge c± . If there is a parity symmetry, then c+ = c− , so we may talk of the central charge c ≡ c± of A. If c < 1, it turns out that A is completely rational [32] and the subnet Vir c ⊗ Vir c ⊂ A has finite Jones index, where Vir c ⊗ Vir c (O) ≡ Vir c (I+ ) ⊗ Vir c (I− ). The classification of two-dimensional local conformal nets with central charge c < 1 and parity symmetry thus splits in the following two points: (a) The classification of Virasoro nets Vir c on S 1 with c < 1. (b) The classification of irreducible local extensions with finite Jones index of the twodimensional Virasoro net Vir c ⊗ Vir c . Point (a) has been completely achieved in our recent work [31]. The Virasoro nets on S 1 with central charge less than one are in bijective correspondence with the pairs of AD2n -E6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. Among other important aspects of this classification, we mention here the occurrence of nets that are not realized as coset models, in contrast to a long standing expectation. (See Remarks after Theorem 7 of [34] on this point. Also, Carpi and Xu recently made progress on classification for the case c = 1 in [10, 54], respectively.) The aim of this paper is to pursue point (b). We shall obtain a complete classification of the two-dimensional local conformal nets (with parity) with central charge in the discrete series. To this end we first classify the maximal nets in this class. Maximality here means that the net does not admit any irreducible local conformal net extension. Maximality will turn out to be also equivalent to the triviality of the superselection structure or to µ-index equal to one, that is Haag duality for a disconnected union of finitely many double cones. 1 More precisely Diff(S 1 ) × Diff(S 1 ) is the conformal group of the Minkowskian torus S 1 × S 1 , the conformal completion of M = R × R (light ray decomposition), and the covariance group is a central extension of Diff(S 1 ) × Diff(S 1 ), see Sect. 2.

Classification of 2D Local Conformal Nets

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It is clear at this point that our methods mainly concern Operator Algebras, in particular Subfactor Theory, see [50]. Indeed this was already the case in our previous one-dimensional classification [31]. The use of von Neumann algebras not only provides a clear formulation of the problem, but also suggests the path to follow in the analysis. Our strategy is the following. The dual canonical endomorphism of Vir c ⊗ Vir c ⊂ A decomposes as Zij ρi ⊗ ρ¯j (1) θ= ij

(i.e. the above is the restriction to Vir c ⊗ Vir c of the vacuum representation of A), where {ρi }i are representatives of unitary equivalence classes of irreducible DHR endomorphisms of the net Vir c . Since µA = 1, it turns out, by using the results in [32], that the matrix Z is a modular invariant for the tensor category of representations of the Virasoro net Vir c [41], and such modular invariants have been classified by Cappelli-Itzykson-Zuber [9]. We shall show that this map A → Z sets up a bijective correspondence between the set of isomorphism classes of two-dimensional maximal local conformal nets with parity and central charge less than one on one hand and the list of Cappelli-Itzykson-Zuber modular invariant, on the other hand. We first prove that the correspondence A → Z is surjective. Indeed, by our previous work [31], Z can be realized by α-induction as in [5] for extensions of the Virasoro nets. (See [38, 52, 3, 6, 7, 4] for more on α-induction.) Then Rehren’s results in [48] imply that θ defined as above (1) is the canonical endomorphism associated with a natural Qsystem, and we have a corresponding local extension A of Vir c ⊗ Vir c and this produces the matrix Z in the above correspondence. To show the injectivity of the correspondence note that, due to the work of Rehren [47], we have an inclusion Vir c (I+ ) ⊗ Vir c (I− ) ⊂ A+ (I+ ) ⊗ A− (I− ) ⊂ A(O), where A+ ⊗ A− is the maximal chiral subnet. By assumption, A+ and A− are isomorphic with central charge c < 1, thus they are in the discrete series classified in [31]. Moreover Z determines uniquely the isomorphism class of A± and an isomorphism π from a fusion rule of A+ onto that of A− so that the dual canonical endomorphism λ on A+ ⊗ A− decomposes as αi ⊗ α¯ π(i) , (2) λ= i

where {αi }i is a system of irreducible DHR endomorphisms of A+ = A− . If Z is a modular invariant of type I, the map π is trivial, so the dual canonical endomorphism has the same form of the Longo-Rehren endomorphism [38]. Thus the classification is reduced to classification of Q-systems in the sense of [36] having the canonical endomorphism of the form given by Eq. (1). This type of classification of Q-systems, up to unitary equivalences, was studied by Izumi-Kosaki [27] as a subfactor analogue of 2-cohomology of (finite) groups. In our setting, we now have a 2-cohomology group of a tensor category, while the 2-cohomology of Izumi-Kosaki does not have a group structure in general. The group operation comes from a natural composition of 2-cocycles. Then the crucial point in our analysis is the vanishing of this 2-cohomology

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for a certain tensor category as we will explain below, and this vanishing implies that the dual Q-system for the inclusion A+ ⊗ A− ⊂ A has a standard dual canonical endomorphism as in the Longo-Rehren Q-system [38], namely A+ ⊗ A− ⊂ A is the “quantum double” inclusion constructed in [38]. At this point, as we know the isomorphism class of A± by our previous classification [31], it follows that the isomorphism class of A is determined by Z. If the modular invariant is of type II, then π gives a non-trivial fusion rule automorphism, however π is actually associated with an automorphism of the tensor category acting non-trivially on irreducible objects [4]. We may then extend our arguments of 2-cohomology vanishing and deal also with this case. It turns out that the automorphism π is an automorphism of a braided tensor category. We thus arrive at the following classification: the maximal local two-dimensional conformal nets with c < 1 and parity symmetry are in a bijective correspondence with the pairs of the A-D-E Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1, namely Z is a modular invariant listed in Table 1 (end of Sect. 5). Note that Dynkin diagrams of type D2n+1 and E7 do appear in the list of present classification of two-dimensional conformal nets, but they were absent in the one-dimensional classification list [31]. Now, as we shall see, the two-dimensional local conformal net B in the discrete series is a finite-index subnet of a maximal local conformal net A. Moreover A and B have the same two-dimensional Virasoro subnet. Using this, we then obtain the classification of all local two-dimensional conformal nets with central charge less c < 1. The non-maximal ones correspond bijectively to the pairs (T , α), where T is a proper sub-tensor category of the representation tensor category of Vir c and α is an automorphism of T . There are at most two automorphisms, thus two possible nets for a given T . The complete list is given in Table 2 (end of Sect. 6). As we have mentioned, a crucial point in our analysis is to show the uniqueness up to equivalence of the Q-system associated with the canonical endomorphism of the form (2) in our cases. To this end we consider a cohomology associated with a representation tensor category that we have to show to vanish in our case. Note that our 2-cohomology groups are a generalization of the usual 2-cohomology groups of finite groups, so they certainly do not vanish in general. Before concluding this introduction we make explicit that our classification applies as well to the local conformal nets with central charge less than one on other twodimensional spacetimes. Indeed if N is two-dimensional spacetime that is conformally equivalent to M, namely conformally diffeomorphic to a subregion on the Einstein cylinder S 1 × R, we may then consider the local conformal nets on N that satisfy the double cone KMS property. These nets are in one-to-one correspondence with the local conformal nets on Minkowski spacetime M, see [21], and so one immediately reads off our classification in these different contexts. An important case where this applies is represented by the two-dimensional de Sitter spacetime. 2. Two-Dimensional Completely Rational Nets and Central Charge Let M be the two-dimensional Minkowski spacetime, namely R2 equipped with the metric dt 2 − dx 2 . We shall also use the light ray coordinates ξ± ≡ t ± x. We have the decomposition M = L+ × L− where L± = {ξ : ξ± = 0} are the two light ray lines. A double cone O is a non-empty open subset of M of the form O = I+ × I− with I± ⊂ L± bounded intervals; we denote by K the set of double cones.


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The Möbius group P SL(2, R) acts on R ∪ {∞} by linear fractional transformations, hence this action restricts to a local action on R (see e.g. [8]), in particular if F ⊂ R has compact closure there exists a connected neighborhood U of the identity in P SL(2, R) such that gF ⊂ R for all g ∈ U. It is convenient to regard this as a local action on R of the universal covering group P SL(2, R) of P SL(2, R). We then have a local (product) action of P SL(2, R)×P SL(2, R) on M = L+ ×L− . Clearly P SL(2, R)×P SL(2, R) acts by pointwise rescaling the metric dξ+ dξ− , i.e. by conformal transformations. A local Möbius covariant net A on M is a map A : O ∈ K → A(O), where the A(O)’s are von Neumann algebras on a fixed Hilbert space H, with the following properties: • Isotony. O1 ⊂ O2 ⇒ A(O1 ) ⊂ A(O2 ). • Locality. If O1 and O2 are spacelike separated then A(O1 ) and A(O2 ) commute elementwise (two points ξ1 and ξ2 are spacelike if (ξ1 − ξ2 )+ (ξ1 − ξ2 )− < 0). • Möbius covariance. There exists a unitary representation U of P SL(2, R)×P SL(2, R) on H such that, for every double cone O ∈ K, U (g)A(O)U (g)−1 = A(gO),

g ∈ U,

with U ⊂ P SL(2, R) × P SL(2, R) any connected neighborhood of the identity such that gO ⊂ M for all g ∈ U. • Vacuum vector. There exists a unit U -invariant vector , cyclic the O∈K A(O). • Positive energy. The one-parameter unitary subgroup of U corresponding to time translations has positive generator. The 2-torus S 1 × S 1 is a conformal completion of M = L+ × L− in the sense that M is conformally diffeomorphic to a dense open subregion of S 1 × S 1 and the local action of P SL(2, R) × P SL(2, R) on M extends to a global conformal action on S 1 × S 1 . But in general the net A does not extend to a Möbius covariant net on S 1 × S 1 ; this is related to the failure of timelike commutativity (note that a chiral net, i.e. the tensor product of two local nets on S 1 , would extend), indeed we have a covariant unitary representation of P SL(2, R) × P SL(2, R) and not of P SL(2, R) × P SL(2, R). Let however G be the quotient of P SL(2, R) × P SL(2, R) modulo the relation (r2π , r−2π ) = (id, id) (spatial 2π-rotation is the identity). Proposition 2.1. The representation U of P SL(2, R) × P SL(2, R) factors through a representation of G. The above proposition holds as a consequence of spacelike locality, it is a particular case of the conformal spin-statistics theorem and can be proved as in [20]. Because of the above Prop. 2.1, A does extend to a local G-covariant net on the Einstein cylinder E = R × S 1 , the cover of the 2-torus obtained by lifting the time coordinate from S 1 to R. Explicitly, M is conformally equivalent to a double cone OM of E. By parametrizing E with coordinates (t , θ), −∞ < t < ∞, −π ≤ θ < π , the transformation ξ± = tan( 21 (t ± θ))

(3)

is a diffeomorphism of the subregion OM = {(t , θ) : −π < t ± θ < π } ⊂ E with M, which is a conformal map when E is equipped with the metric ds 2 ≡ dt 2 − dθ 2 .

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G acts globally on E and the net A extends uniquely to a G-covariant net of E with U the unitary covariant action (see [8]). We shall denote by the same symbol A both the net on M and the extended net on E. If O1 ⊂ M (or O1 ⊂ E) we shall denote by A(O1 ) the von Neumann algebra generated by the A(O)’s as O varies in the double cones contained in O1 . If O ∈ K we shall denote by O the one-parameter subgroup of G defined as follows: O = g W g −1 if W is a wedge, W is the boost one-parameter group associated with W , and gO = W with g ∈ P SL(2, R) × P SL(2, R), see [23]. We collect in the next proposition a few basic properties of a local Möbius covariant net. The proof is either in the references or can be immediately obtained from those. All the statements also hold true (with obvious modifications) in any spacetime dimension. We shall use the lattice symbol ∨ to denote the von Neumann algebra generated. Proposition 2.2. Let A be a local Möbius covariant net on M as above. The following hold: (i) Double cone KMS property. If O ⊂ E is a double cone, then the unitary modular group associated with (A(O), ) has the geometrical meaning itO = U ( O (−2π t)) [8]. (ii) Haag duality on E; wedge duality on M. If O ⊂ E is a double cone then A(O ) = A(O) . Here O is the causal complement of O in E (note that O is still a double cone.) In particular A(W ) = A(W ) , where W is a wedge in M, say W = (−∞, a) × (−∞, b) and W its causal complement in M, thus W = (a, ∞) × (b, ∞) [8, 21]. (iii) Modular PCT symmetry. There is a anti-unitary involution on H such that A(O) = A(−O), U (g) = U (θ (g)) and θ = . Here O is any double one in E and θ is the automorphism of G associated with space and time reflection [8]. (iv) Additivity. Let O be a double cone and{Oi } a family of open sets such that i Oi contains the axis of O. Then A(O) ⊂ i A(Oi ) [15]. (v) Equivalence between and uniqueness of the vacuum. A is irreduc irreducibility A(O) = B(H)), iff A is irreducible on E, iff is the ible on M (that is O∈K unique U -invariant vector (up to a phase) [20]. (vi) Decomposition into irreducibles. A has a unique direct integral decomposition in terms of local irreducible Möbius covariant nets. If A is conformal (see below) then the fibers in the decomposition are also conformal [20]. By the above point (vi) we shall always assume our nets to be irreducible. Let Diff(R) denote the group of positively oriented diffeomorphisms of R that are smooth at infinity (with the identification R = S 1 {∞}, Diff(R) is the subgroup of Diff(S 1 ) of orientation preserving diffeomorphisms of S 1 that fix the point ∞). By identifying M with the double cone OM ⊂ E as above, we may identify elements of Diff(R) × Diff(R) with conformal diffeomorphisms of OM . Such diffeomorphisms uniquely extend (by periodicity) to global conformal diffeomorphisms of E. Namely the element (r2π , id) of G generates a subgroup of G (isomorphic to Z) for which OM is a fundamental domain in E. We may then extend an element of Diff(R) × Diff(R) from OM to all E by requiring commutativity with this Z-action; this is the unique conformal extension to E. Let Conf(E) denote the group of global, orientation preserving conformal diffeomorphisms of E. Conf(E) is generated by Diff(R) × Diff(R) and G (note that Diff(R) × Diff(R) intersects G in the “Poincaré-dilation” subgroup). Indeed if ϕ ∈ Conf(E), then


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ϕOM is a maximal double cone of E, namely the causal complement of a point. Thus there exists an element g ∈ G such that gOM = ϕOM . Then ψ ≡ g −1 ϕ maps OM onto OM and so ψ ∈ Diff(R)×Diff(R) and ϕ = g ·ψ. Note that, by the same argument, any element of Conf(E) is uniquely the product of an element of Diff(R) × Diff(R), a space rotation and time translation on E. A local conformal net A on M is a Möbius covariant net such that the unitary representation U of G extends to a projective unitary representation of Conf(E) (still denoted by U ) so that the extended net on E is covariant. In particular U (g)A(O)U (g)−1 = A(gO),

g∈U,

if U is a connected neighborhood of the identity of Conf(E), O ∈ K, and gO ⊂ M for all g ∈ U. We further assume that U (g)XU (g)−1 = X,

g ∈ Diff(R) × Diff(R) ,

(4)

if X ∈ A(O1 ), g ∈ Diff(R) × Diff(R) and g acts identically on O1 . We may check the conformal covariance on M by the local action of Diff(R) × Diff(R). Given a Möbius covariant net A on M and a bounded interval I ⊂ L+ we set A+ (I ) ≡ A(O) (5) O=I ×J

(intersection over all intervals J ⊂ L− ), and analogously define A− . By identifying L± with R we then get two Möbius covariant local nets A± on R, the chiral components of A, but for the cyclicity of ; we shall also denote A± by AR and AL . By the Reeh-Schlieder theorem the cyclic subspace H± ≡ A± (I ) is independent of the interval I ⊂ L± and A± restricts to a (cyclic) Möbius covariant local net on R on the Hilbert space H± . Since is separating for every A(O), O ∈ K, the map X ∈ A± (I ) → X H± is an isomorphism for any interval I , so we will often identify A± with its restriction to H± . Proposition 2.3. Let A be a Möbius covariant (resp. conformal) net on M. Setting A0 (O) ≡ A+ (I+ ) ∨ A− (I− ), O = I+ × I− , then A0 is a Möbius covariant (resp. conformal) subnet of A, there exists a consistent family of vacuum preserving conditional expectations εO : A(O) → A0 (O) and the natural isomorphism from the product A+ (I+ )·A− (I− ) to the algebraic tensor product A+ (I+ )A− (I− ) extends to a normal isomorphism between A+ (I+ ) ∨ A− (I− ) and A+ (I+ ) ⊗ A− (I− ). Proof. By the double cone KMS property in Prop. 2.2, for any given double cone O, A0 (O) is globally invariant under the modular group of A0 (O) w.r.t. the vacuum state. Hence, by the Takesaki theorem [50], there exists a vacuum preserving conditional expectation εO : A(O) → A0 (O). εO is given by εO (X)E = EXE, X ∈ A0 (O), where E is the orthogonal projection onto A0 (O). By the Reeh-Schlieder theorem E is independent of O, thus if O˜ is a double cone containing O we have εO˜ A(O) = εO , namely the εO ’s form a consistent family. Clearly A0 is a Möbius covariant subnet, as the unitary Möbius representation is generated by modular unitary one-parameter subgroups. In particular A0 is a factor. For similar reasons there exists a normal faithful expectation from A0 (O) to A+ (I+ ) and A− (I− ). Since A+ (I+ ) and A− (I− ) are commuting factors and A0 (O) is a factor, it follows that A0 (O) is naturally isomorphic to A+ (I+ ) ⊗ A− (I− ).

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Thus we may identify H+ ⊗ H− with H0 ≡ A0 (O) and A+ (I+ ) ⊗ A− (I− ) with A0 (O). Now suppose that A is conformal. The following corollary is immediate. Corollary 2.4. If A is conformal then A0 ≡ A+ ⊗ A− is also conformal, moreover A0 extends to a local Diff(S 1 ) × Diff(S 1 )-covariant net on the 2-torus, namely A± are local conformal nets on S 1 . Assuming A to be conformal we set Vir + (I ) ≡ U (g) : g ∈ Diff(I ) × {id} , I ⊂ L+ , Vir − (I ) ≡ U (g) : g ∈ {id} × Diff(I ) , I ⊂ L− ,

(7)

Vir(O) ≡ Vir + (I+ ) ∨ Vir − (I− ), I± ⊂ L± .

(8)

(6)

Proposition 2.5. Vir ± (I ) ⊂ A± (I ), I ⊂ L± , and Vir + (I+ ) ∨ Vir − (I− ) is naturally isomorphic to Vir + (I+ ) ⊗ Vir − (I− ), I± ⊂ L± . Vir ± is the Virasoro subnet of A± . Proof. Given an interval I+ ⊂ L+ and g ∈ Diff(I+ ) × {id} then, by property (4) and Haag duality on E (Prop. 2.2), U (g) belongs to A(O), where O = I+ × I− is a double cone, for all intervals I− ⊂ L− . Hence, by definition (5), U (g) ∈ A+ (I+ ). So Vir + (I+ ) ⊂ A+ (I+ ) and there is an analogous containment Vir − ⊂ A− . By Prop. 2.3 we then have a natural isomorphism Vir + (I+ )∨Vir − (I− ) Vir + (I+ )⊗ Vir − (I− ). The last statement is immediate because the restriction of U to H± implements the covariance unitary representation for A± . The central charge of Vir ± is denoted by c± and is called the central charge of A. In our case c+ = c− and we then refer to the common value c of c± as the central charge of A. In this paper we shall use only the a priori weaker form of conformal covariance given by the above proposition. Indeed we shall just need that A± are conformal nets on S 1 , with central charge less than one. Proposition 2.6. For every double cone O, Vir(M) ∩ A(O) = C (the coset net is trivial). If c± < 1 then Vir(O) ⊂ A(O) is an irreducible inclusion with finite Jones index. Proof. The proof is analogous to the one of [31, Prop. 3.5]. The second statement follows because Vir(O) is completely rational if c± < 1, see [31] and next section. Indeed, by a recent result by Köster [34], the local irreducibility Vir(O) ∩A(O) = C holds without assuming c± < 1, but we do not need this in our paper. Thus the left and right mover subalgebras A± are rich, and our problem is to classify the extensions of A0 , indeed of Vir. It is easy to see that A0 is the unique maximal chiral subnet of A, namely it coincides max max with the subnet Amax L ⊗ AR in Rehren’s work [47, 48]. That is to say AL (O) ⊗ 1 = max max max A(O) ∩ U ({id} × P SL(2, R)) and similarly for AR . Indeed AL ⊗ AR , being chiral, is clearly contained in A0 ; on the other hand A+ commutes with U id×P SL(2, R) max so A+ ⊂ Amax L and analogously A− ⊂ AR .


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2.1. Complete rationality. Let A be a local conformal net on the two-dimensional Minkowski spacetime M. We shall say that A is completely rational if the following three conditions hold: a) Haag duality on M. For any double cone O we have A(O) = A(O ) . Here O is the causal complement of O in M. b) Split property. If O1 , O2 ∈ K and the closure of O¯ 1 of O1 is contained in O2 , the natural map A(O1 ) · A(O2 ) → A(O1 ) A(O2 ) extends to a normal isomorphism A(O1 ) ∨ A(O2 ) → A(O1 ) ⊗ A(O2 ) . c) Finite µ-index. Let E = O1 ∪ O2 ⊂ M be the union of two double cones O1 , O2 such that O¯ 1 and O¯ 2 are spacelike separated. Then the Jones index [A(E ) : A(E)] is finite. This index is denoted by µA , the µ-index of A. The notion of complete rationality has been introduced and studied in [32] for a local net C on R. If C is conformal, the definition of complete rationality strictly parallels the above one in the two-dimensional case. In general, the above (one-dimensional version) of the above three conditions must be supplemented by the following two conditions: d) Strong additivity. If I1 , I2 ⊂ R are open intervals and I is the interior of I1 ∪ I2 , then C(I ) = C(I1 ) ∨ C(I2 ). e) Modular PCT symmetry. There is a vector , cyclic and separating for all the C(I )’s, such that if a ∈ R the modular conjugation J of (C(a, ∞), ) satisfies J C(I )J = C(I + 2a), for all intervals I . If C is conformal, then d) and e) follows from a), b), c). In any case all conditions a) to e) have the strong consequences on the structure of A [32]. In particular µC = d(ρi )2 , i

where the ρi form a system of irreducible sectors of C. Returning to the two-dimensional local conformal net A, consider the time-zero net C(I ) ≡ A(O), where I is an interval of the t = 0 line in M and O = I is the double cone with basis I . Note that C is local but not conformal (positivity of energy does not hold). However C inherits all properties from a) to e) from A. Thus we may define A to be completely rational by requiring C to be completely rational. In this way all results in [32] immediately apply to the two-dimensional context. 3. Modular Invariance and µ-Index of a Net Rehren raised a question in [49, p. 351, lines 8–13] about modular invariants arising from a decomposition of a two-dimensional net and its µ-index. Müger has then solved the problem affirmatively in [41]. We recall some notions and results necessary for our work here. In [47–49], Rehren studied 2-dimensional local conformal quantum field theory B(O) which irreducibly extends a given pair of chiral theories A = AL ⊗AR . That is, the mathematical structure studied there is an irreducible inclusion of nets, AL (I ) ⊗ AR (J ) ⊂ B(O), where I, J are light ray intervals and O is a double cone I × J . Note that here

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AL and AR can be distinct. For such an extension, we decompose the dual canonical endomorphism θ on AL ⊗ AR as Zij αiL ⊗ αjR , θ= ij

where {αiL }i and {αjR }j are systems of irreducible DHR endomorphisms of AL and AR , respectively. The matrix Z = (Zij ) is called a coupling matrix. The two nets AL and AR define S- and T -matrices, SL , TL , SR , TR , respectively, as in [46]. We are interested in the case where the S-matrices are invertible. (By the results in [32], this invertibility, which is called non-degeneracy of the braiding, holds if the nets are completely rational in the sense of [32].) Then Rehren considered when the following two intertwining relations hold. TL Z = ZTR ,

SL Z = ZSR .

(9)

Note that if AL = AR and the non-degeneracy of the braiding holds, this condition implies the usual modular invariance of Z. (We always have Z00 = 1 and Zij ∈ {0, 1, 2, . . . }.) He considered natural situations where the above equalities (9) hold, but also pointed out that it is not necessarily valid in general by showing a very easy counter-example to the intertwining property (9). He then continues as follows. “A possible criterium to exclude models like the counter examples, and hopefully to enforce the intertwining property, could be that the local 2D theory B does not possess nontrivial superselection sectors, but I have no proof that this condition indeed has the desired consequences.” Müger [41] has proved that this triviality of the superselection structures is indeed sufficient (and necessary) for the intertwining property (9), when the nets AL and AR are completely rational. Theorem 3.1 (Muger ¨ [41]). Under the above conditions, the following are equivalent. 1. The net B has only the trivial superselection sector. 2. The µ-index µB is 1. 3. The matrix Z has the intertwining property (9), TL Z = ZTR ,

SL Z = ZSR .

In the case where we can naturally identify AL and AR , the above theorem gives a relation between the classification problem of the modular invariants and the classification problem of the local extension of AL ⊗ AR with µ-index equal to 1. 4. Longo-Rehren Subfactors and 2-Cohomology of a Tensor Category Let M be a type III factor. We say that a finite subset ⊂ End(M) is a system of endomorphisms of M if the following conditions hold, as in [5, Def. 2.1]. 1. 2. 3. 4.

Each λ ∈ is irreducible and has finite statistical dimension. The endomorphisms in are mutually inequivalent. We have idM ∈ . ¯ is the conjugate For any λ ∈ , we have an endomorphism λ¯ ∈ such that [λ] sector of [λ].


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5. The set is closed under composition and subsequent irreducible decomposition, i.e., ν with [λ][µ] =

ν [ν] for any λ, µ ∈ , we have non-negative integers Nλ,µ N ν∈ λ,µ as sectors. Two typical examples of systems of endomorphisms are as follows. First, if we have a subfactor N ⊂ M with finite index, then consider representatives of unitary equivalence classes of irreducible endomorphisms appearing in irreducible decompositions of powers γ n of the canonical endomorphism γ for the subfactor. If the set of representatives is finite, that is, if the subfactor is of finite depth, then we obtain a finite system of endomorphisms. Second, if we have a local conformal net A on the circle, we consider representatives of unitary equivalence classes of irreducible DHR endomorphisms of this net. If the set of representatives is finite, that is, if the net is rational, then we obtain a finite system of endomorphisms of M = A(I ), where I is some fixed interval of the circle. Recall the definition of a Q-system in [36]. Let θ be an endomorphism of a type III factor. A triple (θ, V , W ) is called a Q-system if we have the following properties: V W V ∗V W ∗W V ∗W

∈ Hom(id, θ ), ∈ Hom(θ, θ 2 ), = 1, = 1, = θ (V ∗ )W ∈ R+ ,

W 2 = θ (W )W, θ (W ∗ )W = W W ∗ .

(10) (11) (12) (13) (14) (15) (16)

Actually, it has been proved in [39] that Condition (16) is redundant. (It has been also proved in [27] that Condition (15) is redundant if (16) is assumed.) In this case, θ is a canonical endomorphism of a certain subfactor of the original factor. For a finite system as above, Longo and Rehren constructed a subfactor M ⊗ M opp ⊂ R in [38, Prop. 4.10] such that the dual canonical endomorphism has a decomposition θ = λ∈ λ ⊗ λopp , by explicitly writing down a Q-system (θ, V , W ). We, however, could have an inequivalent Q-system for the same dual canonical endomorphism θ. (We say that two Q-systems (θ, V1 , W1 ) and (θ, V2 , W2 ) are equivalent if we have a unitary u ∈ Hom(θ, θ ) satisfying V2 = uV1 ,

W2 = uθ (u)W1 u∗ .

This equivalence of Q-systems is equivalent to inner conjugacy of the corresponding subfactors [27].) We study this problem of uniqueness of the Q-systems below. Classification of Q-systems for a given dual canonical endomorphism was studied as a subfactor analogue of 2-cohomology of a group in [27]. We show that for a Longo-Rehren Q-system, we naturally have a 2-cohomology group of a tensor category, while 2-cohomology in [27] is not a group in general. Suppose we have a family (Cλµ )λ,µ∈ with Cλµ ∈ Hom(λµ, λµ). An intertwiner ν ∈ End(Hom(ν, λµ)) for ν ∈ by composition Cλµ naturally defines an operator Cλµ from the left. For λ, µ, ν, π ∈ , we have a decomposition Hom(σ, λµ) ⊗ Hom(π, σ ν). Hom(π, λµν) = σ ∈

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We have

σ Cλµ ⊗ Cσπν ∈ End(Hom(π, λµν))

σ ∈

according to this decomposition. We similarly have π τ Cλτ ⊗ Cµν ∈ End(Hom(π, λµν)) τ ∈

based on the last expression of the decompositions Hom(π, λµν) ∼ Hom(π, λτ ) ⊗ λ(Hom(τ, µν)) = τ ∈

∼ =

Hom(π, λτ ) ⊗ Hom(τ, µν).

τ ∈

We now consider the following conditions. Definition 4.1. We say that a family (Cλµ )λ,µ∈ is a unitary 2-cocycle of , if the following conditions hold: 1. For λ, µ ∈ , each Cλµ is a unitary operator in Hom(λµ, λµ). 2. For λ ∈ , we have Cλid = 1 and Cidλ = 1. 3. For λ, µ, ν, π ∈ , we have σ π τ Cλµ ⊗ Cσπν = Cλτ ⊗ Cµν σ ∈

τ ∈

as an identity in End(Hom(π, λµν)) with respect to the above decompositions of Hom(π, λµν). We always assume unitarity for Cλµ in this paper, so we simply say a 2-cocycle for a π unitary 2-cocycle. For a 2-cocycle (Cλµ )λ,µ∈ , we define Cλµν ∈ End(Hom(π, λµν)) by σ Cλµ ⊗ Cσπν . σ ∈

Similarly, we can define µ µ ···µ

Cλ11λ22···λn m ∈ End(Hom(µ1 µ2 · · · µm , λ1 λ2 · · · λn )). (Note that well-definedness follows from Condition 3 in Definition 4.1.) In this notaλµ tion, we have Cλµ ∈ End(Hom(λµ, λµ)) and this endomorphism is given as the left multiplication of Cλµ ∈ Hom(λµ, λµ) on Hom(λµ, λµ), where the product strucλµ ture on Hom(λµ, λµ) is given by composition. In this way, we can identify Cλµ ∈ End(Hom(λµ, λµ)) and Cλµ ∈ Hom(λµ, λµ). We next consider a strict C ∗ -tensor category T , with conjugates, subobjects, and direct sums, whose objects are given as finite direct sums of endomorphisms in . We then study an automorphism of T such that (λ) and λ are unitarily equivalent for all objects λ in T . For all λ ∈ , we choose a unitary uλ with (λ) = Ad(uλ ) · λ. By adjusting with (Ad(uλ ))λ∈ , we may and do assume that (λ) = λ. Then such an automorphism gives a family of automorphisms µ µ ···µ

λ11λ22···λnm ∈ Aut(Hom(µ1 µ2 · · · µm , λ1 λ2 · · · λn )),


75

for λ1 , λ2 , · · · , λn , µ1 , µ2 , · · · , µm ∈ , with the compatibility condition k νλ11νλ22···ν ···λn =

µ µ ···µ

µ1 ,µ2 ,··· ,µm ∈

k λ11λ22···λnm ⊗ νµ11νµ22···ν ···µm

on the decomposition

Hom(ν1 ν2 · · · νk , λ1 λ2 · · · λn ) =

Hom(µ1 µ2 · · · µm , λ1 λ2 · · · λn )

µ1 ,µ2 ,··· ,µm ∈

⊗Hom(ν1 ν2 · · · νk , µ1 µ2 · · · µm ). µ µ ···µ

It is clear that a family (Cλ11λ22···λn m ) arising from a 2-cocycle (Cλµ ) is an automorphism of a tensor category in this sense. Conversely, suppose that we have an automorphism of a tensor category acting on objects trivially as above. Then using the isomorphism Hom(λµ, λµ) ∼ =

Hom(ν, λµ) ⊗ Hom(λµ, ν),

ν∈

the family (νλµ ) gives a unitary intertwiner in Hom(λµ, λµ). We denote this intertwiner by Cλµ and then it is clear that the family (Cλµ ) gives a 2-cocycle in the above sense. Thus in this correspondence, we can identify a 2-cocycle on and an automorphism of the tensor category arising from that fixes each object in the category. We now have the following definition. ) Definition 4.2. (1) We say that 2-cocycles (Cλµ )λµ and (Cλµ λµ are equivalent if we have a family (ωλ )λ of scalars of modulus 1 such that ν Cλµ = ων /(ωλ ωµ )C λµ ∈ End(Hom(ν, λµ)). ν

If a 2-cocycle (Cλµ )λµ is equivalent to (1)λµ , then we say that it is trivial. ν ’s are scalar operators (2) We say that a 2-cocycle (Cλµ )λµ is scalar-valued if all Cλµ on Hom(λµ, ν). (3) We say that an automorphism of the tensor category as above is trivial if we have a family (ωλ )λ of scalars of modulus 1 satisfying µ µ ···µ

λ11λ22···λnm = ωµ1 · · · ωµm /(ωλ1 · · · ωλn ). Note that if a 2-cocycle is trivial, then it is scalar-valued, in particular. We now recall the definition of the Longo-Rehren subfactor [38, Prop. 4.10] as follows. (See [40, 43, 44] for related or more general definitions.) Let = {λk | k = 0, 1, . . . , n} be a finite system of endomorphisms of a type III factor

M where λ0 = id. We choose a system {Vk | k = 0, 1, . . . , n} of isometries with nk=0 Vk Vk∗ = 1 in the factor M ⊗ M opp , where M opp is the opposite algebra of M and we denote the anti-linear isomorphism from M onto M opp by j . Then we set ρ(x) =

n k=0

Vk ((λk ⊗ λk )(x))Vk∗ , opp

76


for x ∈ M ⊗ M opp , where λopp = j · λ · j −1 . We set V = V0 ∈ Hom(id, ρ) and define W ∈ Hom(ρ, ρ 2 ) as follows: n dk dl opp Vk (λk ⊗ λk )(Vl )Tklm Vm∗ , W = wdm k,l,m=0

where dk is the statistical dimension of λk , w is the global index of the system, w =

n 2 k=0 dk , and m

Tklm

=

Nkl

(Tklm )i ⊗ j ((Tklm )i ).

i=1

is the structure constant, dim Hom(λm , λk λl ), and {(Tklm )i | i = 1, 2, . . . , Nklm } is a fixed orthonormal basis of Hom(λm , λk λl ). Note that the operator Tklm does not depend on the choice of the orthonormal basis. Proposition 4.10 in [38] says that the triple (ρ, V , W ) is a Q-system. Thus we have a subfactor M ⊗ M opp ⊂ R with index w corresponding to the dual canonical endomorphism ρ. We call this a Longo-Rehren subfactor arising from the system . Furthermore, if is a subsystem of all the irreducible DHR endomorphisms of a local conformal net A, then any Q-system having this dual canonical endomorphism gives an extension B ⊃ A ⊗ Aopp . This 2-dimensional net B is local if and only if ε(ρ, ρ)W = W by [38, Prop. 4.10], where ε is the braiding. In general, if the system

has a braiding ε, and this condition ε(ρ, ρ)W = W holds, we say that the Q-system (ρ, V , W ) satisfies locality. We now would like to characterize a general Q-system having the same dual canonical endomorphism ρ. First, we have the following simple lemma. Here Nklm

Lemma 4.3. Let F, F be finite dimensional complex Hilbert spaces and j an anti-linear isomorphism from F onto F . For any vector ξ ∈ F ⊗ F , we define a linear map A : F → F by ξ = k Aξk ⊗ j (ξk ), where {ξk } is an orthonormal basis of F . Then this linear map A is independent of the choice of the orthonormal basis {ξk }. Proof. This is straightforward by the anti-isomorphism property of j .

The next theorem gives our characterization of Q-systems. Theorem 4.4. Let , ρ, V , W be as above. If another triple (ρ, V , W ) with W ∈ Hom(ρ, ρ 2 ) is a Q-system, we have a 2-cocycle (Cλµ )λ,µ∈ such that n dk dl opp W = Vk (λk ⊗ λk )(Vl )(Cλk λl ⊗ 1)Tklm Vm∗ . (17) wdm k,l,m=0

Conversely, if we have a 2-cocycle (Cλµ )λ,µ∈ , then the triple (ρ, V , W ) with W defined as in (17) is a Q-system. The Q-system (ρ, V , W ) is equivalent to the above canonical Q-system (ρ, V , W ) if and only if the corresponding 2-cocycle (Cλµ )λ,µ∈ is trivial, if and only if the corresponding automorphism of the tensor category arising from is trivial. Moreover, suppose that the system has a braiding ε± . Then the Q-system (ρ, V , W ) satisfies locality if and only if the corresponding 2-cocycle (Cλµ )λ,µ∈ satisfies the following symmetric condition: − + Cµλ ελµ , Cλµ = εµλ

(18)


77

for all λ, µ ∈ . If this symmetric condition holds, the corresponding automorphism of the tensor category arising from is an automorphism of a braided tensor category. Proof. If (ρ, V , W ) with W ∈ Hom(ρ, ρ 2 ) is a Q-system, then we have a system of intertwiners (Cλµ )λ,µ∈ such that identity (17) holds and the intertwiners (Cλµ ) are uniquely determined by Lemma 4.3. Expanding both sides of identity (15), we obtain the following identity: n dk dl dm opp opp opp Vk (λk ⊗ λk )(Vl )(λk λl ⊗ λk λl )(Vm ) w 2 dp k,l,m,p,q=0

× (λk ⊗ λk )((Cλl λm ⊗ 1)Tlm )(Cλk λq ⊗ 1)Tkq Vp∗ n dk dl dm opp opp opp = Vk (λk ⊗ λk )(Vl )(λk λl ⊗ λk λl )(Vm ) w 2 dp opp

q

p

k,l,m,p,r=0

× (Cλk λl ⊗ 1)Tklr (Cλr λm ⊗ 1)Trm Vp∗ . p

(19)

We decompose Hom(λp , λk λl λm ) ∼ = ∼ =

n q=0 n

Hom(λp , λk λq ) ⊗ Hom(λq , λl λm ) Hom(λr , λk λl ) ⊗ Hom(λp , λr λm ),

r=0

as above, and apply Lemma 4.3 to the above identity (19) to obtain Condition 3 in Definition 4.1. Similarly, Condition 2 in Definition 4.1 follows from identity (14). We next prove unitarity of Cλµ ∈ Hom(λµ, λµ). First note that the operators ¯ ¯ Cλidλ¯ , Cλid ¯ λ are scalar multiples of the identity because Hom(id, λλ), Hom(id, λλ) are both 1-dimensional. Since the triple (ρ, V , W ) also satisfies identity (16), we expand both sides of identity (16) and use Lemma 4.3 as in the above arguments. Then we obtain the following: The intertwiner space Hom(λµ, νσ ) for λ, µ, ν, σ ∈ can be decomposed in two ways as follows: Hom(λ, ντ ) ⊗ Hom(τ µ, σ ) (20) Hom(λµ, νσ ) ∼ = τ ∈

∼ =

Hom(λµ, π ) ⊗ Hom(π, νσ ).

(21)

π∈

On one hand, Lemma 4.3 applied to the left-hand side of identity (16) produces a map in End(Hom(λµ, νσ )) which maps Ti ⊗ Sj∗ ∈ Hom(λ, ντ ) ⊗ Hom(τ µ, σ ), identified with ν(Sj∗ )Ti ∈ Hom(λµ, νσ ), to ν(Sj∗ Cτ∗µ )Cντ Ti ∈ Hom(λµ, νσ ), where Ti and Sj are isometries in Hom(λ, ντ ) and Hom(σ, τ µ), respectively. On the other hand, Lemma 4.3 applied to the right-hand side of identity (16) produces a map in End(Hom(λµ, νσ )) which maps Ti ∗ ⊗ Sj ∈ Hom(λµ, π ) ⊗ Hom(π, νσ ), identified with Sj Ti ∗ ∈ ∗ ∈ Hom(λµ, νσ ), where T and S are isometries in Hom(λµ, νσ ), to Cνσ Sj Ti ∗ Cλµ i j Hom(π, λµ) and Hom(π, νσ ), respectively. These two maps are equal in

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End(Hom(λµ, νσ )). In the above decomposition (21), we set λ = σ = id and µ = ν, then we have τ = µ¯ and π = µ in the summations. With Frobenius reciprocity as in [25] and the above identity of two maps in End(Hom(λµ, νσ )), we obtain the identity id = 1. Cµidµ¯ Cµµ ¯

(22)

We next apply identity (13) to (17) and obtain the following equality:

ν dλ dµ Kλµ = wdν ,

(23)

λ,µ∈

ν = Tr((C ν )∗ C ν ) and Tr is the non-normalized trace on where we have set Kλµ λµ λµ Hom(λµ, λµ). Setting ν = id in (23), we obtain

dλ2 |Cλidλ¯ |2 = w,

λ∈

which, together with (22), implies |Cλidλ¯ | = 1 for all λ ∈ . In the above decomposition (21), we now set λ = id, then we have τ = ν¯ and π = µ in the summations. With Frobenius reciprocity as in [25] and the above identity of two maps in End(Hom(λµ, νσ )), we obtain the identity Cνidν¯ ν((Cνσ¯ µ )∗ T˜ )Rν ν¯ =

dµ µ C T, dν dσ νσ

(24)

for all T ∈ Hom(µ, νσ ), where T˜ ∈ Hom(¯ν µ, σ ) is the Frobenius dual of T and Rν ν¯ ∈ Hom(id, ν ν¯ ) is the canonical isometry. This identity (24), Condition 3 in Definition 4.1, already proved, and identity (22) imply the following identity: µ ˜∗ Cνσ¯ µ T , Cνσ¯ µ S = (Cνidν¯ )∗ Rν∗¯ν ν¯ (Cνσ S )Cνσ¯ µ T ∗ = (Cνidν¯ )∗ Cνid ¯νS T = T , S,

where we have T , S ∈ Hom(σ, ν¯ µ) and the inner product is given by T , S = S ∗ T ∈ C. This is the desired unitarity of Cν¯ µ . The converse also holds in the same way and the remaining parts are straightforward. It is easy to see that we can multiply 2-cocycles and the multiplication on the equivalence classes of 2-cocycles is well-defined. In this way, we obtain a group and this is called the 2-cohomology group of (or of the corresponding tensor category). It is also easy to see that the multiplication gives the composition of the corresponding automorphisms of the tensor category. The part of the above theorem on a bijective correspondence between Q-systems (ρ, V , W ) with locality and automorphisms of the braided tensor category has been also announced by Müger in [41].


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Example 4.5. If all the endomorphisms in are automorphisms, then the fusion rules determine a finite group G. It is easy to see that the Longo-Rehren Q-system gives a crossed product by an outer action of G and the above 2-cohomology group for is isomorphic to the usual 2-cohomology group of G. Furthermore, if the system has a braiding, then the group G is abelian. In this case, the symmetric condition of a cocycle means cg,h = ch,g for the corresponding usual 2-cocycle c of the finite abelian group G. It is well-known that such a 2-cocycle is trivial. (See [1, Lemma 3.4.2], for example.) When all the 2-cocycles for are trivial, we say that we have a 2-cohomology vanishing for . Thus, 2-cohomology vanishing implies uniqueness of the Longo-Rehren subfactor in the following sense. Corollary 4.6. Let be as above. If we have a 2-cohomology vanishing for and ρ = λ∈ λ ⊗ λopp is a dual canonical endomorphism for a subfactor M ⊗ M opp ⊂ P , then this subfactor is inner conjugate to the Longo-Rehren subfactor M ⊗ M opp ⊂ R. 5. 2-Cohomology Vanishing and Classification In this section, we first study a general theory of 2-cohomology for a C ∗ -tensor category and then apply it to the tensor categories related to the Virasoro algebra. We consider a strict C ∗ -tensor category T (with conjugates, subobjects, and direct sums) in the sense of [13, 39] and we assume that we have only finitely many equivalence classes of irreducible objects in T and that each object has a decomposition into a finite direct sum of irreducible objects. Such a tensor category is often called rational. We may and do assume that our tensor categories are realized as those of endomorphisms of a type III factor. Choose a system of endomorphisms of a type III factor M corresponding to the C ∗ -tensor category T . Suppose we have a 2-cocycle (Cλµ )λ,µ∈ . We introduce some basic notions. Suppose that we have σ ∈ such that for any λ ∈ , there exists k ≥ 0 such that λ ≺ σ k . Then we say that σ is a generator of . In the following, we consider only the case σ = σ¯ . In this case, we say that has a self-conjugate generator σ . Suppose σ ∈ is a self-conjugate generator of . We further assume that for all λ, µ ∈ , we have dim Hom(λσ, µ) ∈ {0, 1}. In this case, we say that multiplications by σ have no multiplicities. Take λ1 , λ2 , λ3 , λ4 ∈ and assume dim Hom(λ1 σ, λ2 ) = dim Hom(σ λ1 , λ3 ) = dim Hom(λ3 σ, λ4 ) = dim Hom(σ λ2 , λ4 ) = 1. Choose isometric intertwiners T1 ∈ Hom(λ2 , λ1 σ ), T3 ∈ Hom(λ3 , σ λ1 ),

T2 ∈ Hom(λ4 , σ λ2 ), T4 ∈ Hom(λ4 , λ3 σ ).

Then the composition T4∗ T3∗ σ (T1 )T2 is in Hom(λ4 , λ4 ) = C. This value is the connection as in [42, 14, Chapter 9]. We denote this complex number by W (λ1 , λ2 , λ3 , λ4 ). (Note that this value depends on T1 , T2 , T3 , T4 though they do not appear in the notation.) If all these complex numbers are non-zero, then we say that the connections of with respect to the generator σ are non-zero. This condition is independent of the choices of

80


isometric intertwiners Tj ’s, because we now assume that multiplications by σ have no multiplicities. Suppose we have a map g : → Z/2Z. For an endomorphism σ that is a direct sum of elements λj ’s with g(λj ) = k ∈ Z/2Z, we also set g(σ ) = k. If we have g(λµ) = g(λ) + g(µ), then we say that has a Z/2Z-grading. An endomorphism λ ∈ is called even [resp. odd] when g(λ) = 0 [resp. g(λ) = 1]. Theorem 5.1. Suppose we have a finite system of endomorphisms with a self-conjugate generator σ ∈ satisfying all the following conditions: 1. Multiplications by σ have no multiplicities. 2. One of the following holds: (a) We have σ ≺ σ 2 . (b) The system has a Z/2Z-grading and the generator σ is odd. 3. The connections of with respect to the generator σ are non-zero. 4. For any λ, ν1 , ν2 ∈ with ν1 ≺ σ n , ν2 ≺ σ n , λ ≺ σ ν1 , and λ ≺ σ ν2 , we have µ ∈ with µ ≺ σ n−1 , ν1 ≺ σ µ, and ν2 ≺ σ µ. Then any 2-cocycle (Cλµ )λµ of is trivial. Before presenting a proof, we make a comment on Condition 4. Consider the Bratteli diagram for the higher relative commutants of a subfactor σ (M) ⊂ M. We number the steps of the Bratteli diagrams as 0, 1, 2, . . . . Then Condition 4 says the following. (Recall that σ is self-conjugate.) Suppose we have vertices corresponding to ν1 and ν2 at the nth step of the Bratteli diagrams, and they are connected to the vertex λ in the n + 1st step. Then there exists a vertex µ in the n − 1st step that is connected to ν1 and ν2 . Note that if ν1 and ν2 already appear in the n − 2nd step, then this condition trivially holds by taking µ = λ. Thus, if the subfactor σ (M) ⊂ M is of finite depth, then checking finitely many cases is sufficient for verifying Condition 4, and this can be done by drawing the principal graph of the subfactor σ (M) ⊂ M. Proof. Using Conditions 1, 3 and 4, we first prove that the unitary operator Cσλσ ···σ ∈ End(Hom(λ, σ σ · · · σ )) is scalar for any λ ∈ . Let the number of σ ’s in Cσλσ ···σ be k and we prove the above property Cσλσ ···σ ∈ C by induction on k. Note that the intertwiner space Hom(λ, σ σ · · · σ ) is decomposed as Hom(λ1 , σ σ ) ⊗ Hom(λ2 , λ1 σ ) ⊗ · · · ⊗ Hom(λ, λk−2 σ ), and each of the spaces Hom(λ1 , σ σ ) ⊗ Hom(λ2 , λ1 σ ) ⊗ · · · ⊗ Hom(λ, λk−2 σ ) is one-dimensional by Condition 1. Each such one-dimensional subspace gives a nonzero eigenvector of the unitary operator Cσλσ ···σ with eigenvalue Cσλ1σ Cλλ12σ · · · Cλλk−2 σ , and what we have to prove is these eigenvalues are all identical. Note that the decomposition of Hom(λ, σ σ · · · σ ) as above is depicted graphically in Fig. 1. Another picture Fig. 2 gives another decomposition into a direct sum of one-dimensional eigenspaces.


81

λ

q

λk−2

q

q @

@ @

··· λ3 λ2 λ1

q

σ

q @

@

@ @

q

q @

@

@

@ q @ @ @ @ @ @q @q @q σ

σ

@ @

@

@ @q

···

σ

σ

Fig. 1. Decomposition into a direct sum of one-dimensional eigenspaces λ

q

q @ λk−2 @q ···

q

σ

q @ λ3 @q λ @ 2 @q λ1 @q @ @q q q

q

···

σ

σ

σ

σ

Fig. 2. Decomposition into a direct sum of one-dimensional eigenspaces

Roughly speaking, what we prove is that if a unitary matrix has several “different” decompositions into direct sums of one-dimensional eigenspaces, then the unitary matrix needs to be a scalar multiple of the identity matrix. First, let k = 2. By Condition 1, the space Hom(λ, σ σ ) is one-dimensional for any λ ∈ , so we obviously have Cσλσ ∈ C. Suppose now we have Cσλσ ···σ ∈ C for any λ ∈ if the number of σ ’s is less than or equal to k. We will prove Cσλσ ···σ ∈ C for any λ ∈ when the number of σ ’s is k + 1. µ First note that we have Cσλσ ···σ Cλσ ∈ C by the induction hypothesis and Condition 1. What we have to prove is that this scalar is independent of λ when µ is fixed. That is, suppose we have λ, λ , µ ∈ , λ ≺ σ k , λ ≺ σ k , µ ≺ λσ , µ ≺ λ σ . We will prove µ

µ

Cσλσ ···σ Cλσ = Cσλσ ···σ Cλ σ ∈ C.

82


By Condition 4, there exists ν ∈ such that ν ≺ σ k−1 , λ ≺ σ ν, and λ ≺ σ ν. Then there exists τ ∈ such that τ ≺ νσ and µ ≺ σ τ . Note that we have µ

µ

Cσλσ ···σ Cλσ = Cσλν Cσν σ ···σ Cλσ ∈ C, µ

where the number of σ ’s in Cσν σ ···σ is k − 1. The scalar Cσλν Cλσ is the eigenvalue µ of the operator Cσ νσ corresponding to the eigenvector given by the one-dimensional τ C µ is the eigenintertwiner space Hom(λ, σ ν)⊗Hom(µ, λσ ). Similarly, the scalar Cνσ στ µ value of the same operator Cσ νσ corresponding to the eigenvector given by the onedimensional intertwiner space Hom(τ, νσ ) ⊗ Hom(µ, σ τ ). Condition 3 implies that these two eigenvectors are not orthogonal, thus the two eigenvalues are equal, because µ the operator Cσ νσ has an orthonormal basis of eigenvectors and thus it is normal. In this way, we obtain the identities

µ

µ

τ Cσλν Cλσ = Cνσ Cσµτ = Cσλν Cλ σ ,

which implies µ

µ

µ

µ

Cσλσ ···σ Cλσ = Cσλν Cσν σ ···σ Cλσ = Cσλν Cσν σ ···σ Cλ σ = Cσλσ ···σ Cλ σ ∈ C,

as desired, where the numbers of σ ’s in Cσλσ ···σ , Cσν σ ···σ , and Cσλσ ···σ are k, k − 1, and k, respectively. We next prove the triviality of the cocycle C by using Condition 2. First we assume we have 2 (a) of the assumptions in the theorem, that is, σ ≺ σ 2 . σ . Set ωid = 1. Since id ≺ σ 2 , the condition Cσσ σ σ ∈ C implies that Cσσ σ Cσσ σ = Cσidσ Cidσ σ σ −1 By unitarity of C in Theorem 4.4, we have |Cσ σ | = 1, we thus set ωσ = (Cσ σ ) ∈ C. 2 (Recall that we have already proved Cσσ σ is a scalar.) Then this implies Cid σ σ = ωid /ωσ . For λ ∈ not equivalent to id, σ , we choose a minimum positive integer k with λ ≺ σ k . We set ωλ = ωσk Cσλ···σ ∈ C, where the number of σ ’s in Cσλ···σ is k. For any m > k, we can represent the scalar Cσλ···σ , where σ appears for m times, as Cσσ σ · · · Cσσ σ Cσλ···σ , where the number of Cσσ σ ’s is m − k and the number of σ ’s in Cσλ···σ is k. This implies Cσλ···σ = ωλ /ωσm , where the number of σ ’s in Cσλ···σ is m. Now choose arbitrary λ, µ, ν ∈

with λ ≺ σ l , µ ≺ σ m . We can represent Cσν ···σ ∈ C with σ appearing for l + m times, µ ν Cλ as the product Cλµ σ ···σ Cσ ···σ , where σ ’s appear for l and m times, respectively, and then we obtain ων ν ωλ ωµ Cλµ = l+m , ωσl ωσm ωσ ν ω ω = ω . Unitarity in Theorem 4.4 gives ω ω = 0; we thus have which gives Cλµ λ µ ν λ µ ν Cλµ = ων /(ωλ ωµ ). We next deal with the case 2 (b), that is, we now assume that the system has a Z/2Z-grading and the generator σ is odd. We first set ωid = 1. Since id ≺ σ 2 , we next set ωσ to be a square root of (Cσidσ )−1 . (Note that |Cσidσ | = 1 by unitarity in Theorem 4.4.) It does not matter which square root we choose. For λ ∈ not equivalent to id, σ , we choose a minimum positive integer k with λ ≺ σ k in the same way as above in the case of 2 (a). We again set ωλ = ωσk Cσλ···σ ∈ C, where the number of σ ’s in Cσλ···σ is k. For any m > k, we can represent the scalar Cσλ···σ , where σ appears for m times, as Cσidσ · · · Cσidσ Cσλ···σ , where the number of Cσidσ ’s is (m − k)/2 and the number of σ ’s in Cσλ···σ is k, because m − k is now even, due to the Z/2Z-grading. Then we obtain Cσλ···σ =

1 ωσm−k

ωλ ωλ = m, ωσk ωσ


83

where the number of σ ’s in Cσλ···σ is m. Then the same argument as in the above case of 2 (a) proves the triviality of the cocycle Cλµ . Remark 5.2. The 2-cohomology does not vanish in general, as is well-known in the finite group case. For example, if the system arises from an outer action of a finite group G = Z/2Z × Z/2Z, it is known that we have a non-trivial unitary 2-cocycle for this group G. So as in Example 4.5, the 2-cohomology for the corresponding tensor category does not vanish. In [31, Theorem 2.4, Theorem 4.1], we have classified local extensions of the conformal nets SU (2)k and Vir c with k = 1, 2, 3, . . . and c = 1−6/m(m+1), m = 2, 3, 4, . . . . (Here the symbol Vir c denotes the Virasoro net with central charge c.) We use the symbols SU (2)k and Vir c also for the corresponding C ∗ -tensor categories. We also say that the corresponding C ∗ -tensor categories of these local extensions of the nets SU (2)k and Vir c are extensions of the tensor categories SU (2)k and Vir c . Furthermore, the tensor category SU (2)k has a natural Z/2Z-grading and the even objects make a sub-tensor category. We call it the even part of SU (2)k . We then have the following theorem. Theorem 5.3. Any finite system of endomorphisms corresponding to one of the following tensor categories has a self-conjugate generator σ satisfying all the Conditions in Theorem 5.1, and thus, we have 2-cohomology vanishing for these tensor categories: 1. The SU (2)k -tensor categories and their extensions. 2. The sub-tensor categories of those in Case 1. 3. The Virasoro tensor categories Vir c with c < 1 and their extensions. 4. The sub-tensor categories of those in Case 3. Proof. We deal with the following cases separately. Here for the extensions of SU (2)k tensor categories and the Virasoro tensor categories Vir c , we use the labels by (pairs of) Dynkin diagrams as in [31, Theorem 2.4, Theorem 4.1], which arise from the labels of modular invariants by Cappelli-Itzykson-Zuber [9]. (These also correspond to the type I modular invariants listed in Table 1 in this paper.) Note that the braiding does not matter now, so we ignore the braiding structure here. 1. The SU (2)k -tensor categories and their extensions. (a) Tensor categories An . (b) Tensor categories D2n . (c) Tensor category E6 . (d) Tensor category E8 . 2. The (non-trivial) sub-tensor categories of those in Case 1. (a) The group Z/2Z. (b) The even parts of the SU (2)k -tensor categories. 3. The Virasoro tensor categories Vir c with c < 1 and their extensions. (a) Tensor categories (An−1 , An ). (b) Tensor categories (A4n , D2n+2 ). (c) Tensor categories (D2n+2 , A4n+2 ). (d) Tensor category (A10 , E6 ). (e) Tensor category (E6 , A12 ). (f) Tensor category (A28 , E8 ). (g) Tensor category (E8 , A30 ). 4. The (non-trivial) sub-tensor categories of those in Case 3. (a) The sub-tensor categories of those in Case 3(a).

84


(b) (c) (d) (e) (f) (g)

The sub-tensor categories of those in Case 3(b). The sub-tensor categories of those in Case 3(c). The sub-tensor categories of those in Case 3(d). The sub-tensor categories of those in Case 3(e). The sub-tensor categories of those in Case 3(f). The sub-tensor categories of those in Case 3(g).

Case 1(a). We label the irreducible objects of the tensor category Ak+1 with 0, 1, 2, . . . , k, as usual. Let σ be the standard generator 1. Condition 1 of Theorem 5.1 clearly holds. Since the fusion rule of the tensor category SU (2)k has a Z/2Z-grading and this generator 1 is odd, Condition 2(b) also holds. Now the connection values with respect to this σ are the usual connection values of the paragroup Ak+1 as in [42, 30, 14, Sect. 11.5], and they are non-zero and Condition 3 holds. The multiplication rule by the generator σ is described with the usual Bratteli diagram for the principal graph Ak+1 as in [28, 14, Chapter 9], so we see that Condition 4 holds. Case 1(b). The irreducible objects of the tensor category are labeled with the even vertieven for this tensor category.) ces of the Dynkin diagram D2n . (So we also use the name D2n If 2n = 4, then this tensor category is given by the group Z/3Z, and we can verify the conclusion directly, so we assume that n > 2. We label σ as in Fig. 3. Then we can easily verify Conditions 1, 2(a) and 4. We next verify Condition 3. We label four irreducible objects as in Fig. 4. (If n = 3, we set λ1 = id.) Note that the connection with respect to the generator σ has a principal graph as in Fig. 5. (See [24], for example, for the fusion rules of a subfactor with principal graph D2n .) We first claim that if the vertices λ3 and λ4 are not involved, then the connection values with respect to the generator σ are non-zero. As in [3, II, Sect. 3], we may assume that (1) the irreducible objects of the tensor category are realized as {α0 , α2 , . . . , α2n−4 , α2n−2 , (2)

α2n−2 }, arising from α-induction applied to the system SU (2)4n−4 having the irreducible objects {0, 1, 2, . . . , 4n − 4}. (Note that it does not matter whether we use α + or α − , so we have dropped the ± symbol.) We denote, by W (i, j, k, l), the connection value with respect to the generator σ = α2 given by the square in Fig. 6. (Note that the value W (i, j, k, l) depends on the choices of intertwiners, but the absolute value |W (i, j, k, l)| is independent of such choices, since the intertwiner spaces are now all one-dimensional.) For example, assume n > 4 and consider the connection value W (α4 , α6 , α4 , α6 ). By σ

id

r @

r @

@

@ @r

r @

r

r

@

···

@ @r

@r

Fig. 3. The principal graph for the subfactor D2n λ1

id

r @

@

@ @r

···

λ2

r @

λ3

@

r @

@ @r

r

@

@ @r

Fig. 4. The principal graph for the subfactor D2n

λ4

r

Classification of 2D Local Conformal Nets id

σ

id

σ

r @

85

r @

1

2

3

4

1

2

3

4

r r rH r r @ @HH @ @ ··· @ @ H @ @ @ H @ @ @ @ @r @r@ @r H Hr @ @r @r r Fig. 5. The principal graph for the subfactor σ (M) ⊂ M i q

q k

qj q l

even Fig. 6. A connection value for D2n

[3, II, Sect. 3], all four intertwiners involved in this connection come from the intertwiners for SU (2)4n−4 , and thus the connection value is given by the connection W (4, 6, 4, 6) for SU (2)even 4n−4 with respect to the generator 2. This value is given as a single term of 6j -symbols of SU (2)4n−4 and it is non-zero by [29]. The general case is dealt with in the same method. Thus, we consider the remaining case where all four vertices of the connection value are one of λ1 , λ2 , λ3 , λ4 . Below, we denote the vertices λ1 , λ2 , λ3 , λ4 simply by 1, 2, 3, 4. Denote the statistical dimensions of 1, 2, 3, 4 by d1 , d2 , d3 , d4 respectively. Their explicit values are as follows: d1 = d2 =

2n−5 4n−2 π , π sin 4n−2 2n−3 sin 4n−2 π , π sin 4n−2

sin

d3 = d 4 =

1 π . 2 sin 4n−2

(25)

For a fixed pair (i, l), we denote the unitary matrix (W (i, j, k, l))j,k by Wil . Using the bi-unitarity Axioms 1 and 4 in [14, Chap. 10], originally due to [42], we compute several matrices Wil below. Recall that the renormalization Axiom 4 in [14, Chap. 10] now implies |W (i, j, k, l)| =

dj dk |W (j, i, l, k)|. d i dl

If i = 1 and l = 3, 4, then the entries in Wil are again given as single terms of the 6j -symbols of SU (2)4n−4 and thus, they are non-zero. The unitary matrices W13 and W14 have size 1 × 1, so the entries are obviously non-zero. The unitary matrix W21 has a size 2 × 2, and all the entries in Wil are again given as single terms of the 6j -symbols of SU (2)4n−4 and thus, they are non-zero. The unitary matrix W22 has a size 4 × 4. The entry W (2, 1, 1, 2) is non-zero because we have already seen that W11 has no zero entries and we have the renormalization axiom. Similarly, the entries W (2, 2, 1, 2), W (2, 1, 2, 2), W (2, 3, 1, 2), W (2, 1, 3, 2), W (2, 4, 1, 2), and W (2, 1, 4, 2) are non-zero.

86


The entry W (2, 2, 2, 2) is also given as a single term of the 6j -symbols of SU (2)4n−4 and thus, it is non-zero. We assume W (2, 3, 2, 2) = 0 and will derive a contradiction. Using the renormalization axiom twice, we obtain W (2, 2, 3, 2) = 0. Another use of the renormalization axiom gives W (3, 2, 2, 2) = 0. Since the 2 × 2 matrix W32 is unitary, this implies |W (3, 4, 2, 2)| = 1. The renormalization axiom then gives |W (2, 2, 3, 4)| = 1. Since the 2 × 2 matrix W24 is unitary, this gives W (2, 3, 3, 4) = W (2, 2, 2, 4) = 0. These two equalities then give W (3, 4, 2, 3) = 0 and W (2, 2, 4, 2) = 0 with the renormalization axiom, respectively. Thus we have verified the (2, 4)-entry of the 4×4 unitary matrix W22 is zero. Similarly, its (4, 2)-entry is also zero. The identity W (3, 4, 2, 3) = 0 and unitarity of the 2 × 2 matrix W33 give |W (3, 2, 2, 3)| = 1. The renormalization axiom then produces |W (2, 3, 3, 2)| = d3 /d2 . The 1×1 matrix W43 is unitary, thus the renormalization axiom gives |W (2, 4, 3, 2)| = d3 /d2 . Similarly, we obtain |W (2, 3, 4, 2)| = d3 /d2 . The √ 1 × 1 matrix W13 is unitary, thus the renormalization axiom gives |W (2, 1, 3, 2)| = d1 d3 /d2 . Now we use the orthogonality of the second and third row vectors of the 4×4 unitary matrix W22 . We have so far obtained that the (2, 3), (2, 4), (3, 2)-entries are zero and the (3, 1)-entry is non-zero. We thus know that the (2, 1)-entry is zero, but this is a contradiction because we have already seen above that the (2, 1)-entry W (2, 1, 2, 2) is non-zero. We have thus proved W (2, 3, 2, 2) = 0. By a similar method, we can prove that W (2, 4, 2, 2), W (2, 2, 3, 2) and W (2, 2, 4, 2) are all non-zero. We next assume W (2, 3, 3, 2) = 0. For the same reason as above, we obtain √ d1 d3 |W (2, 3, 1, 2)| = |W (2, 4, 1, 2)| = |W (2, 1, 3, 2)| = |W (2, 1, 4, 2)| = , (26) d2 d3 |W (2, 3, 4, 2)| = |W (2, 4, 3, 2)| = . (27) d2 Since W (2, 3, 3, 2) = 0, the renormalization axiom implies W (3, 2, 2, 3) = 0. Since the 2 × 2-matrix W33 is unitary, √ we obtain |W (3, 2, 4, 3)| = 1. The renormalization axiom gives |W (2,√3, 3, 4)| = d3 /d2 . Unitarity of the 2 × 2-matrix W24 then gives |W (2, 2, 2, 4)| = d3 /d2 , which then gives |W (2, 2, 4, 2)| = |W (2, 4, 2, 2)| = d3 /d2 with the renormalization axiom. The identities (25), together with a simple computation of trigonometric functions, give d1 d3 + 2d32 = d22 .

(28)

Since the third row vector, the fourth row vector, and the third column vector of the unitary matrix W22 have a norm 1, this identity (28), together with (26), (27) gives |W (2, 2, 3, 2)| = |W (2, 3, 2, 2)| = d3 /d2 and W (2, 4, 4, 2) = 0. Thus the matrix A = (Aj k )j k = (|W (2, k, j, 2)|)j k is given as follows, where α, β, γ are non-negative real numbers, √ √   d1 d3 d1 d3 β  α d2 d2     d3  d 3  β  γ  d2 d2  √ . (29)  d1 d3 d3 d3    0   d2   √ d2 d2  d1 d3 d3 d3  0 d2 d2 d2


87

Orthogonality of the first and third row vectors of W22 implies √ √ d1 d3 d3 d1 d3 d3 α +β . d2 d2 d22

(30)

Since the first row vector of W22 has a norm 1, we also have α2 + β 2 = 1 −

2d1 d3 . d22

(31)

The Cauchy-Schwarz inequality with (30), (31), we obtain √ d1 d 3 2d1 d3 d1 + d 3 1 − , d2 d22 which, together with (28), implies d1 d3 d1 + d3 2d32 − d1 d3 . This implies d12 2d3 , which gives sin2

2n − 5 1 π 4n − 2 2

(32)

by (25). This inequality (32) fails, if we have (2n − 5)/(4n − 2) > 1/4, that is, n > 9/2. Since we now assume n ≥ 3, this has produced a contradiction and we have shown W (2, 3, 3, 2) = 0, unless n = 3, 4. We deal with the remaining two cases n = 3, 4 by direct computations of the connection as follows. If n = 3, we have the Dynkin diagram D6 . A subfactor with principal with D6 is realized as the asymptotic inclusion [42, p. 137], [14, Def. 12.23], [26, Sect. 2], of a subfactor with principal graph A4 as in [43, Sect. III.1], [14, p. 663], [26, Theorem 4.1]. Thus the tensor category D6even is realized as a self-tensor product of the tensor category of Aeven and that our current generator σ is realized as a tensor product of the standard 4 generators in two copies of Aeven 4 . As in Case 2 (b) below, the connection values are non-zero for Aeven , thus our current connection values are also non-zero as products of 4 two non-zero values. We finally deal with the case n = 4. We label the even vertices of the principal graph D8 as in Fig. 7. We continue the computations of |W (i, j, k, l)|’s using the matrix (29), where the non-negative real numbers α, β, γ have been defined. The renormalization axiom gives 0

1

r @

2

@

r @

@ @r

@

@ @r

3

r @ @

r

@ @r

Fig. 7. The principal graph for the subfactor D8

4

r

88


√ |W (1, 1, 2, 1)| = d2 /d1 |W (1, 1, 1, 2)| and unitarity of the 2 × 2-matrix W12 gives |W (1, 2, 2, 2)| = |W (1, 1, 1, 2)|. So we have d2 |W (1, 1, 2, 1)| = |W (1, 2, 1, 1)| = |W (1, 2, 2, 2)| = |W (2, 1, 2, 2)| = β, (33) d1 again by the renormalization. We also have |W (1, 2, 1, 1)| = β.

(34)

Unitarity of the 1×1-matrix W02 gives |W (0, 1, 1, 2)| = 1 and thus, the renormalization axiom gives √ d2 |W (1, 0, 2, 1)| = |W (1, 2, 0, 1)| = , (35) d1 since d0 = 1. Similarly, unitarity of the 1 × 1-matrix W01 gives 1 |W (1, 0, 1, 1)| = |W (1, 1, 0, 1)| = √ , d1

(36)

and unitarity of the 1 × 1-matrix W00 gives 1 . d1

(37)

d2 d2 |W (2, 1, 1, 2)| = α. d1 d1

(38)

|W (1, 0, 0, 1)| = We also have |W (1, 2, 2, 1)| =

Thus the 3 × 3-matrix B = (Bj k )j k = (|W (1, k, j, 1)|)j k is given as follows, where δ is a non-negative real number, by (33), (34), (35), (36), (37), (38),  √  1 1 d2  d √  d1 d1   1  1  √ (39) δ β   d . √ 1   d2 d2  β α d1 d1 The first row vector of the matrix (29) has a norm 1, thus we have α2 + β 2 = 1 −

2d1 d3 . d22

(40)

The third row vector of the matrix (39) has a norm 1, thus we have d22 2 d2 2 + β + α = 1. d12 d12

(41)


89

Eqs. (40) and (41) give the following value for β 2 : β2 =

d22 − 2d1 d3 − d12 + d2 d22 − d12

.

(42)

Note that the denominator is not zero. Let t be the index of the subfactor with principal graph D8 . (That is, t = 4 cos2 π/14.) Then the Perron-Frobenius theory gives the following identities: d1 = t − 1, d2 = t 2 − 3t + 1, t 3 − 5t 2 + 6t − 1 d3 = . 2 Then these imply d22 − 2d1 d3 − d12 + d2 = 0 in (42); we thus obtain β = 0, which has been already excluded above. We have thus reached a contradiction and shown W (2, 3, 3, 2) = 0. Similarly, we can prove W (2, 4, 4, 2) = 0. The unitary matrix W34 has a size 1 × 1, so the renormalization axiom implies W (2, 4, 3, 2) = 0. Similarly, we have W (2, 3, 4, 2) = 0. We have thus proved that all the entries of W22 are non-zero. The unitary matrix W23 has a size 2 × 2. If this matrix has a zero entry, we have either W (2, 2, 2, 3) = W (2, 4, 4, 3) = 0 or W (2, 2, 4, 3) = W (2, 4, 2, 3) = 0. The former case, together with the renormalization axiom, implies W (2, 2, 3, 2) = 0, which is already excluded in the above study of W22 . The latter case gives |W (2, 4, √4, 3)| = 1, which, together with the renormalization axiom, implies |W (4, 2, 3, 4)| = d2 /d4 > 1 by (25). This is against the unitarity axiom and thus cannot happen. The 2 × 2 unitary matrix W24 is dealt with in a similar way to the case W23 . The unitary matrices W31 and W34 also have size 1 × 1, so the entries are again non-zero. The matrices W32 and W33 have size 2 × 2. The entries of W32 have the same absolute values as the entries of W23 , so the above arguments for W23 show that they are non-zero. We next consider W33 . If this 2 × 2 unitary matrix contains a zero entry, then we have either W (3, 2, 2, 3) = W (3, 4, 4, 3) = 0 or W (3, 2, 4, 3) = W (3, 4, 2, 3) = 0. The former case, together with the renormalization axiom, implies W (2, 3, 3, 2) = 0, which is already excluded in the above study of W22 . The latter case, together with the renormalization axiom, implies W (2, 3, 3, 4) = 0, which is already excluded in the above study of W24 . The four matrices W4l can be dealt with in the same way as above for W3l . Thus we are done for Case 1(b). Case 1(c). Only fusion rules and 6j -symbols matter, and the braiding does not matter, for the Conditions in Theorem 5.1, so our tensor category can be identified with SU (2)2 and this is a special case of Case 1 above. Case 1(d). In a similar way to the above case, this tensor category can be identified with the even part of the tensor category SU (2)3 , so this is a special case of Case 2(b) below. Case 2(a). This is trivial. Case 2(b). We label the irreducible objects of the tensor category SU (2)k with index 0, 1, 2, . . . , k, as above. (We also use the name Aeven k+1 for this tensor category.) Let σ be the generator 2 this time. Conditions 1 and 2(a) of Theorem 5.1 clearly hold.

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Since all 6j -symbols for SU (2)k have non-zero values as in [29], Condition 3 holds, in particular. The multiplication rule by the generator σ is described with the even steps of the usual Bratteli diagram for the principal graph Ak+1 as in [28, 14, Chap. 9], so we see that Condition 4 holds. Case 3(a). This is the Virasoro tensor category with central charge c = 1 − 6/n(n + 1). We recall the description of the irreducible objects in the tensor category given by [53, Theorem 4.6] applied to SU (2)n−1 ⊂ SU (2)n−2 ⊗SU (2)1 , as follows. (Also see [31, Sect. 3] for our notations.) We now have a net of subfactors Vir c ⊗ SU (2)n−1 ⊂ SU (2)n−2 ⊗ SU (2)1 with finite index and apply the α-induction to this inclusion. The irreducible representations of the net Vir c are labeled as {σj,k | j = 0, 1, . . . , n − 2,

k = 0, 1, . . . , n − 1,

j + k ∈ 2Z}.

Xu’s result [53, Theorem 4.6] then shows the following. First, the systems {σj,k } and {ασj,k ⊗id } have the isomorphic fusion rules and 6j -symbols. Furthermore, the latter system is isomorphic to the system {(λ j ⊗ id)(αid×λk ) | j = 0, 1, . . . , n − 2,

k = 0, 1, . . . , n − 1,

j + k ∈ 2Z},

where {λk | k = 0, 1, . . . , n − 1} and {λ j | j = 0, 1, . . . , n − 2} are the system of irreducible DHR endomorphisms of the nets SU (2)n−1 and SU (2)n−2 , respectively. This system has further isomorphic fusion rules and 6j -symbols to the system {λ j ⊗ λk | j = 0, 1, . . . , n − 2,

k = 0, 1, . . . , n − 1,

j + k ∈ 2Z},

(43)

of irreducible DHR endomorphisms of the net SU (2)n−2 ⊗ SU (2)n−1 . (Note that we have a restriction j + k ∈ 2Z, so this system is a subsystem of all the irreducible DHR endomorphisms of the net SU (2)n−2 ⊗ SU (2)n−1 .) As in [31, Sect. 3], we can identify the system of these σj,k ’s with the system of characters of the minimal models [11, Subsect. 7.3.4] whose fusion rules are given in [11, Subsect. 7.3.3]. We take the DHR endomorphism σ1,1 as σ in Theorem 5.1 and then, from these fusion rules, we easily see that Condition 1 holds. It is also easy to see that we have a natural Z/2Z-grading such that σ is an odd generator, so Condition 2(b) holds. By considering the connection of the system (43), we know that the connection value with respect to the generator σ is a product of the two connection values of the systems SU (2)n−2 and SU (2)n−1 with respect to the standard generators. Since these two connection values for SU (2)n−2 and SU (2)n−1 are the usual connection values for the paragroups labeled with the Dynkin diagrams An−1 and An , and they are non-zero by [42, 30, 14, Section 11.5], we conclude that Condition 3 holds. From the fusion rule described as above, we verify that Condition 4 also holds. (Recall the comment on Condition 4 after the statement of Theorem 5.1 and draw the principal graph for a subfactor given by σ1,1 .) Case 3(b). The tensor category is produced with α-induction and a simple current extension of index 2 as in [3, II, Sect. 3]. The fusion rules and 6j -symbols are given by even a direct product of the two systems Aeven 4n and D2n+2 . We can use the direct product of the σ in Fig. 3 and the σ in Case 2(b) as the current σ for Theorem 5.1. Then Conditions 1, 2(a), and 4 easily follow and the connection values are non-zero as products of non-zero values in Cases 1(b) and 2. Case 3(c). This case is proved in a similar way as the above proof of case 3(b).


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Case 3(d). The tensor category is produced with α-induction as in [31, Sect. 4.2]. The irreducible objects of the tensor category are labeled with pairs (j, k) with j = 0, 1, . . . , 9 and k = 0, 1, 2 with j + k ∈ 2Z. The fusion rules of the objects {(j, 0) | j = 0, 1, . . . , 9} obey the A10 fusion rule and those of {(0, 0), (0, 1), (0, 2)} obey the A3 fusion rule. Let σ be the object (1, 1). Then as in Case 1, we can verify Conditions 1, 2(b), 3 and 4. Case 3(e). This case is proved in a similar way to the above proof of case 3(d). Case 3(f). The tensor category is again produced with α-induction as in [31, Sect. 4.2]. The fusion rules and 6j -symbols are given as the direct product of the two syseven tems Aeven 28 and A4 . The irreducible objects of the former system are labeled with 0, 2, . . . , 26 as usual, and the latter system is given as {id, τ } with τ 2 = id ⊕ τ . Then we can choose (14, τ ) as σ and verify Conditions 1, 2(a), 3 and 4, using the same arguments as in Cases 2 and 3(b). Case 3(g). This case is proved in a similar way to the above proof of case 3(f). Case 4(a). Now, the only non-trivial sub-tensor categories are Z/2Z, SU (2)n−2 , even SU (2)even n−2 , SU (2)n−1 , SU (2)n−1 and the even parts with respect to the Z/2Z-grading described in the above proof of Case 3(a). The conclusion trivially holds for the first case. The next four cases have been already dealt with in Cases 1(a) and 2(b). In the last case, we can identify the tensor category with the direct product of two tensor even categories SU (2)even n−2 and SU (2)n−1 . We use the same labeling of the irreducible DHR sectors as in the proof of Case 3(a) and then we can use the generator σ2,2 as σ in Theorem 5.1. even Case 4(b). The only non-trivial sub-tensor categories we have are now Aeven 4n and D2n+2 . Thus, we have the conclusion by Cases 2(b) and 1(b), respectively. Case 4(c). This case is proved in a similar way to the above proof of Case 4(b). Case 4(d). The only non-trivial sub-tensor categories we have are now Z/2Z, Aeven 10 , their direct product, and A3 . We can deal with the group Z/2Z trivially. The cases Aeven 10 and A3 are particular cases of Cases 2(b) and 1(a), respectively. For the case of the direct product of Aeven 10 and Z/2Z, we can choose σ = (2, 2) in the notation of the proof for Case 3(d). Case 4(e). This case is proved in a similar way to the above proof of Case 4(d). Case 4(f). The only non-trivial sub-tensor categories we have are now Aeven and Aeven 4 28 . Both are special cases of Case 2(b). Case 4(g). This case is proved in a similar way to the above proof of Case 4(f). Remark 5.4. We have the following application of the above theorem. Consider the tensor category corresponding to the WZW-model SU (2)28 . Regard the irreducible objects as irreducible endomorphisms of a type III factor M and label them as id = λ0 , λ1 , λ2 , . . . , λ28 as usual. Then the endomorphism γ = λ0 ⊕ λ10 ⊕ λ18 ⊕ λ28 is a dual canonical endomorphism and uniqueness of the Q-system (γ , V , W ) up to unitary equivalence was shown in [33, Sect. 6] based on a result in vertex operator algebras. (This uniqueness was used in our previous work [31].) Izumi has also given another proof of this uniqueness with a more direct method. We remark that our above theorem also gives a different proof of this uniqueness as follows. We may assume that M is injective. Suppose that we have two endomorphisms ρ1 , ρ2 of M such that ρ1 ρ¯1 = ρ2 ρ¯2 = γ . As in [6, Prop. A.3], we can prove that the two subfactors ρ1 (M) ⊂ M and ρ2 (M) ⊂ M have the isomorphic higher relative commutants, and then we conclude by [45, Cor. 6.4] that the two subfactors are isomorphic via θ ∈ Aut(M). We then may and do assume ρ2 = θ · ρ1 and now we have θ · γ · θ −1 = γ . Since γ = λ0 ⊕ λ10 ⊕ λ18 ⊕ λ28 and powers of γ produce all of λ0 , λ2 , λ4 , . . . , λ28 ,

92


we know that [θ · λ2j · θ −1 ] = [λ2j ] for j = 0, 1, 2, . . . , 14, where the square brackets denote the unitary equivalence classes. Then we have a map θ : Hom(λ, µ) t → θ (t) ∈ Hom(θ · λ · θ −1 , θ · µ · θ −1 ) giving an automorphism of the tensor category generated by powers of γ . By Case 2 of Theorem 5.3, this automorphism θ is trivial in the sense of Definition 4.2. The automorphism θ sends the Q-system (γ , V1 , W1 ) for ρ1 to the one (γ , V2 , W2 ) for ρ2 , and now the triviality of θ implies that these two systems are unitarily equivalent. Using the above Theorem 5.3, we obtain the following classification result of 2dimensional completely rational nets. The meaning of the condition that the µ-index is 1 will be further studied in the next section. Consider a 2-dimensional local completely rational conformal net B with central charge c = 1 − 6/m(m + 1) < 1 and µ-index µB = 1. By [47], we have inclusions max AL ⊗ AR ⊂ Amax L ⊗ AR ⊂ B, max where AL , AR , Amax L , AR are one-dimensional local conformal nets. By assumption, max have the same central charge c. Rehren’s result [47, Cor. 3.5] and and A Amax L R our results [32, Prop. 24] together imply that the fusion rules of the systems of entire max irreducible DHR endomorphisms of the two nets Amax L , AR are isomorphic, and our max previous result [31, Theorem 5.1] implies that the two nets Amax L , AR are isomorphic as max max nets. Since both AL , AR contain Vir c as subnets, we obtain an irreducible inclusion Vir c ⊗ Vir c ⊂ B. A decomposition of a vacuum sector of B restricted on Vir c ⊗ Vir c produces a decomposition matrix (Zλµ )λµ , where λ, µ are representatives of unitary equivalence classes of irreducible DHR endomorphisms of the net Vir c . Since µB = 1, by Theorem 3.1, due to Müger [41], we know that this matrix Z is a modular invariant of the Virasoro tensor category Vir c and such modular invariants have been classified by Cappelli-Itzykson-Zuber [9] as in Table 1. We claim that this correspondence from B to Z is bijective.

Theorem 5.5. The above correspondence from B to Z gives a bijection from the set of isomorphism classes of such two-dimensional nets to the set of modular invariants Z in Table 1. Table 1. Modular invariants for the Virasoro tensor category Vir c m n 4n 4n + 1 4n + 2 4n + 3 11 12 17 18 29 30

Labels for modular invariants in [9] (An−1 , An ) (D2n+1 , A4n ) (A4n , D2n+2 ) (D2n+2 , A4n+2 ) (A4n+2 , D2n+3 ) (A10 , E6 ) (E6 , A12 ) (A16 , E7 ) (E7 , A18 ) (A28 , E8 ) (E8 , A30 )

Type I II I I II I I II II I I


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Proof. We first prove that this correspondence is surjective. Take a modular invariant Z in Table 1. By [31, Subsecs. 4.1, 4.2, 4.3], we conclude that this modular invariant can be realized with α-induction as in [5, Cor. 5.8] for extensions of the Virasoro nets. Then Rehren’s results in [48, Theorem 1.4, Prop. 1.5] imply that we have a corresponding Q-system and a local extension B ⊃ Vir c ⊗ Vir c and that this B produces the matrix Z in the above correspondence. We next show injectivity of the map. Suppose that we have inclusion max AL ⊗ AR ⊂ Amax L ⊗ AR ⊂ B,

where AL , AR are isomorphic to Vir c and that this decomposition gives a matrix Z. We have to prove that the net B is uniquely determined up to isomorphism. Recall that the max nets Amax L and AR are among those classified by [31, Theorem 5.1]. As we have seen max above, AL and Amax R are isomorphic as nets and we can naturally identify them. This max isomorphism class and an isomorphism π from a fusion rule of Amax L onto that of AR are uniquely determined by Z by [31, Theorem 5.1]. (Also see [4].) max If the modular invariant is of type I, then we can naturally identify Amax L and AR max ⊂ B has and the map π is trivial. Then the Q-system for the inclusion Amax ⊗ A L R a standard dual canonical endomorphisms as in the Longo-Rehren Q-system and the above results Corollary 4.6, Theorems 5.1, 5.3 imply that this Q-system is equivalent to the Longo-Rehren Q-system. If the modular invariant is of type II, then we have a non-trivial fusion rule automorphism π. We then know by [4, Lemma 5.3] that this fusion rule automorphism π actually gives an automorphism of the tensor category acting non-trivially on irreducible objects. The same arguments as in the proof of Theorem 4.4 show that 2-cohomology vanishing implies uniqueness of the Q-system. Again, the above results Corollary 4.6, Theorems 5.1, 5.3 give the 2-cohomology vanishing, thus we have the desired uniqueness of the max Q-system for the inclusion Amax L ⊗ AR ⊂ B. Remark 5.6. In the case when the modular invariant Z above is of type II, the automorphism π of the tensor category above is actually an automorphism of a braided tensor category, as seen from the above proof. In the above classification, we have shown 2-cohomology vanishing without assuming locality. In the context of classification of two-dimensional nets, this means that max any (relatively local irreducible) extension B of Amax L ⊗ AR with µ-index being 1 is automatically local. 6. The µ-Index, Maximality of Extensions, and Classification of Non-Maximal Nets In Theorem 5.5, we have classified 2-dimensional completely rational local conformal nets and central charge less than 1 under the assumption that the µ-index is 1. In this section, we clarify the meaning of this condition on the µ-index. As we have seen above, this condition is equivalent to triviality of the superselection structure of the net. We further show that this condition is equivalent to maximality of extensions of the 2-dimensional net, when we have a parity symmetry for the net B. Here the net B is said to have a parity symmetry if we have a vacuum-fixing unitary involution P such that P B(O)P = B(pO), where p maps x + t → x − t in the two-dimensional Minkowski space. In this case, P clearly implements an isomorphism of AL and AR and thus, an max isomorphism of Amax L and AR .

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Suppose we have a local extension C of the two-dimensional completely rational local conformal net B and the inclusion B ⊂ C is strict. Then we have µB > µC ≥ 1 by [32, Prop. 24]. That is, if the net B is not maximal with respect to local extensions, then we have µB > 1. This argument does not require a parity symmetry condition. Conversely, suppose we have µB > 1. Then the results in [41] show that the dual max ⊂ B is of the form canonical endomorphism for the inclusion Amax ⊗A λ⊗π(λ), L R max are local extensions of Vir and λ runs through a proper where both Amax and A c L R subsystem of the system of the irreducible DHR endomorphisms of the net Amax and L π is an isomorphism from such system onto another subsystem of irreducible DHR max and Amax are in the classification list of endomorphisms of the net Amax R . Both AL R [31, Theorem 5.1], and now they are isomorphic. Recall that at least one of the two subsystems is a proper subsystem, since µB > 1, and the parity symmetry condition now implies that both subsystems are proper. First suppose that the map π is trivial. Then the Q-system for the inclusion Amax L ⊗ Amax ⊂ B is the usual Longo-Rehren Q-system arising from the subsystem by CorolR lary 4.6, Theorem 5.1 and Case 4 of Theorem 5.3. Then, Izumi’s Galois correspondence [26, Theorem 2.5] shows that we have a further extension C ⊃ B such that the Q-system max ⊂ C is the Longo-Rehren Q-system using the entire system of the for Amax L ⊗ AR irreducible DHR endomorphisms of Amax L and the index [C : B] is strictly larger than 1. We know that the extension C arising from the Longo-Rehren Q-system is local. That is, the net B is not maximal with respect to local extensions. Next suppose that the map π is non-trivial. By checking the representation categories of the local extensions of the Virasoro nets classified in [31, Theorem 5.1], we know that only such non-trivial isomorphisms arise from interchanging of 2j and 4n − 2 − 2j of the system SU (2)4n−2 , where j = 0, 1, . . . , 2n − 1, or the well-known non-trivial even . In both cases, the map π can be extended to an automorphism of the system D10 automorphism of the entire system of irreducible DHR endomorphism of Amax L and we can obtain a proper extension C ⊃ B in a similar way to the above case. Thus, again, the net B is not maximal with respect to local extensions. We summarize these proper sub-tensor categories of the extensions of the Virasoro tensor categories Vir c (c < 1) with trivial or non-trivial automorphisms as in Table 2. Each entry “nontrivial” means that we have a unique nontrivial automorphism for the sub-tensor category. For example, the sub-tensor category SU (2)even of (A7 , A8 ) appears in the case n = 8 of the 4th entry 6 having a trivial automorphism and the case n = 2 of the 5th entry having a nontrivial automorphism. We thus have exactly two non-maximal local conformal nets for this sub-tensor category. Thus we have proved that the net B with parity symmetry has µB = 1 if and only if it is maximal with respect to local extensions. In such a case, we say that B is a maximal net. These results, together with Theorem 5.5, imply the following main theorem of this paper immediately. Theorem 6.1. The above correspondence from B to Z in Theorem 5.5 gives a bijection from the set of isomorphism classes of such maximal two-dimensional nets with parity symmetry and central charge less than 1 to the set of modular invariants Z in Table 1. Furthermore, the above discussions on the possible proper sub-tensor categories of the extensions of the Virasoro tensor categories Vir c (c < 1) with trivial or non-trivial automorphisms imply that non-maximal two-dimensional local conformal nets with parity symmetry and central charge less than 1 are classified according to Table 2, since we have 2-cohomology vanishing for all these tensor categories by Theorem 5.3.


95

Table 2. Proper sub-tensor categories of extensions of the Virasoro tensor categories Virc with automorphisms m n n n n 4n n 4n − 1 n 4n 4n − 1 4n + 1 4n + 1 4n + 1 4n + 2 4n + 2 4n + 2 11 11 11 11 11 12 12 12 12 12 17 18 29 29 29 30 30 30

Tensor category (An−1 , An ) (An−1 , An ) (An−1 , An ) (An−1 , An ) (A4n−1 , A4n ) (An−1 , An ) (A4n−2 , A4n−1 ) (An−1 , An ) (A4n−1 , A4n ) (A4n−2 , A4n−1 ) (A4n , D2n+2 ) (A4n , D2n+2 ) (A4n , D2n+2 ) (D2n+2 , A4n+2 ) (D2n+2 , A4n+2 ) (D2n+2 , A4n+2 ) (A10 , E6 ) (A10 , E6 ) (A10 , E6 ) (A10 , E6 ) (A10 , E6 ) (E6 , A12 ) (E6 , A12 ) (E6 , A12 ) (E6 , A12 ) (E6 , A12 ) (A16 , D10 ) (D10 , A18 ) (A28 , E8 ) (A28 , E8 ) (A28 , E8 ) (E8 , A30 ) (E8 , A30 ) (E8 , A30 )

Sub-tensor category {id} Z/2Z SU (2)n−2 SU (2)even n−2 SU (2)even 4n−2 SU (2)n−1 SU (2)even 4n−2 even SU (2)even n−2 × SU (2)n−1 even SU (2)4n−2 × SU (2)even 4n−1 even SU (2)even 4n−3 × SU (2)4n−2 {id} SU (2)even 4n−1 even D2n+2 {id} SU (2)even 4n+1 even D2n+2 {id} Z/2Z SU (2)2 SU (2)even 9 Z/2Z × SU (2)even 9 {id} Z/2Z SU (2)2 SU (2)even 11 Z/2Z × SU (2)even 11 even D10 even D10 {id} SU (2)even 3 SU (2)even 27 {id} SU (2)even 3 SU (2)even 29

Automorphism trivial trivial trivial trivial nontrivial trivial nontrivial trivial nontrivial nontrivial trivial trivial trivial trivial trivial trivial trivial trivial trivial trivial trivial trivial trivial trivial trivial trivial nontrivial nontrivial trivial trivial trivial trivial trivial trivial

Theorem 6.2. The non-maximal two-dimensional local conformal nets with parity symmetry and central charge less than 1 are classified bijectively, up to isomorphism, according to the entries in Table 2. Acknowledgements. A part of this work was done during a visit of the first-named author to Università di Roma “Tor Vergata”. Another part was done while both authors stayed at the Mathematisches Forschungsinstitut Oberwolfach for a miniworkshop “Index theorems and modularity in operator algebras”. We thank M. Izumi for useful discussions. We gratefully acknowledge the support of GNAMPA-INDAM and MIUR (Italy), Grants-in-Aid for Scientific Research, JSPS (Japan) and the Mathematisches Forschungsinstitut Oberwolfach.

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Communications in


Nonlinear Stability of Boundary Layers of the Boltzmann Equation, I. The case M∞ < −1 Seiji Ukai1 , Tong Yang2 , Shih-Hsien Yu2 1 2

Department of Applied Mathematics, Yokohama National University, Yokohama, Japan Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, P.R. China

Received: 4 March 2003 / Accepted: 3 July 2003 Published online: 11 November 2003 – © Springer-Verlag 2003

Abstract: This is a continuation of the paper [15] on nonlinear boundary layers of the Boltzmann equation where the existence is established and shown to be strongly dependent on the Mach number M∞ of the Maxwellian state at far field. In this paper, when M∞ < −1, we will show that the linearized operator has the exponential decay in time property and therefore a bootstrapping argument yields nonlinear stability of the boundary layers. 1. Introduction and Main Result The nonlinear Milné problem can be stated as follows. Consider the 3-dimensional halfspace D = {(x, y, z) ∈ R3 |x > 0}, in which the mass density F of gas particles is assumed constant on each plane parallel to the boundary ∂D = {x = 0} although the particle motion is 3-dimensional. That is, F is assumed to be a function of position x (but not of y, z) and particle velocity ξ = (ξ1 , ξ2 , ξ3 ) ∈ R3 . Here, ξ1 stands for the velocity component along the x-axis. Then, F is governed by the stationary Boltzmann equation  x > 0, ξ ∈ R3 ,  ξ1 Fx = Q(F, F ), (1.1) F| = Fb (ξ ), ξ1 > 0, (ξ2 , ξ3 ) ∈ R2 ,  x=0 F → M∞ (ξ ) (x → ∞), ξ ∈ R3 , where M∞ (ξ ) = M[ρ∞ , u∞ , T∞ ](ξ ) =

ρ∞ |ξ − u∞ |2 , exp − (4πT∞ )3/2 2T∞

(1.2)

is a Maxwellian with constants ρ∞ > 0, u∞ = (u∞,1 , u∞,2 , u∞,3 ) ∈ R3 , and T∞ > 0 which are the macroscopic components in the particle distribution F . By a shift of the

100

S. Ukai, T. Yang, S.-H. Yu

variable ξ in the direction orthogonal to the x-axis, we can assume without loss of generality that u∞,2 = u∞,3 = 0, and then, the sound speed and Mach number of this equilibrium state are given by c∞ =

5 T∞ , 3

M∞ =

u∞,1 , c∞

(1.3)

respectively, see [4]. Here, Q, the collision operator, is a bilinear integral operator 1 Q(F, G) = F (ξ )G(ξ∗ ) + F (ξ∗ )G(ξ ) − F (ξ )G(ξ∗ ) − F (ξ∗ )G(ξ ) 2 R3 ×S 2 (1.4) ×q(ξ − ξ∗ , ω) dξ∗ dω, with ξ = ξ − [(ξ − ξ∗ ) · ω] ω,

ξ∗ = ξ∗ + [(ξ − ξ∗ ) · ω] ω,

(1.5)

where “·” is the inner product of R3 . We restrict ourselves to the hard sphere gas for which the collision kernel q is given by q(ζ, ω) = σ0 |ζ · ω|, where σ0 is the surface area of the hard sphere. The existence of stationary solutions, called boundary layer solutions, to the problem (1.1) is studied recently in [15]. The result there shows that the existence of boundary layer solutions depends on the Mach number M∞ at x = ∞. When M∞ = 0, ±1, a solvability condition is given implicitly so that the co-dimensions of the manifold for boundary data Fb (ξ ) is obtained. In the simplest case, i.e., M∞ < −1, there is no extra solvability condition because all the information at infinity goes into the layer, which means that as long as the boundary data Fb is close to the Maxwellian at x = ∞ under some suitable norm, the boundary layer solution always exists. As the first step, to study the stability of the boundary layer solutions obtained in [15], we will study the case when M∞ < −1. The main reason why this case is easiest is that the linearized problem has exponential decay phenomena. And this decay estimate is easier to be handled in the bootstrapping argument for nonlinear stability. For the other case, the decay rate should be algebraic as for the Cauchy problem so that it is more difficult and will be pursued by authors in the future. For the boundary layer problem, there are a lot of results on the linear existence, stability and the numerical computation, cf. [1, 2, 5–8, 12–14]. Since we will discuss the stability problem in this paper, we will not present their works in details. The main result in this paper can be stated as follows. Let F¯ = F¯ (x, ξ ) be the stationary solution to the problem (1.1). Consider the initial boundary value problem,  F + ξ 1 Fx    t F |t=0 F |x=0    F

= Q(F, F ), = F0 (x, ξ ), = Fb (ξ ), → M∞ (ξ ) (x → ∞),

t > 0, x > 0, ξ ∈ R3 , x > 0, ξ ∈ R3 , t > 0, ξ1 > 0, (ξ2 , ξ3 ) ∈ R2 , t > 0, ξ ∈ R3 .

(1.6)

Nonlinear Stability of Boundary Layers of the Boltzmann Equation

101

Theorem 1.1. When M∞ < −1, under the assumption that |Fb (ξ ) − M∞ (ξ )| ≤ 0 Wβ (ξ ),

3 ξ ∈ R+ ,

β > 5/2,

with the weight function Wβ (ξ ) defined in (2.1) and 0 being a sufficiently small positive constant, there exists a boundary layer solution F¯ (x, ξ ) to (1.1) proved in [15]. For (1.6), when [[F0 (x, ξ ) − F¯ (x, ξ )]] < 1 with β > 5/2, where 1 > 0 is a sufficiently small constant and the norm [[·]] is defined in (2.28), there exists a unique solution F (t, x, ξ ) to the problem (1.6) which decays exponentially in time to the stationary solution F¯ (x, ξ ). In other words, the boundary layer solution in this case is nonlinearly stable. Remark 1.2. We prove the global existence in the setting of the contraction mapping principle associated to the reduced problem (2.7) related to the quantity F − F¯ , in the space endowed with the norm (2.30). Hence, the asymptotic stability is a straightforward consequence of it. As for the existence, the method in [11] may work for (1.1). The proof of our theorem is given in the following section. We will first consider two semigroups associated with two linearized problems of (1.6) and show that they both have exponential decay property. Then by applying the bootstrapping argument and the smallness of the strength of the boundary layer, we will have the nonlinear stability result stated in Theorem 1.1. In the following, c is used to denote a generic positive constant. 2. Stability Analysis The stability problem to (1.6) can be discussed in two steps. The first step is to consider the corresponding linearized problem by the energy method for L2x,ξ and then the bootstrapping argument for L∞ x,ξ . The exponential decay in time estimate obtained in the first step can be used in the second step for nonlinear stability by using Grad’s estimate on the nonlinear Boltzmann collision term to obtain an a priori estimate on the solution for the application of the fixed point theorem. In the following, we will use the following weighted function: 1/2 Wβ (ξ ) = (1 + |ξ |)−β M[1, u∞ , T∞ ](ξ ) , (2.1) with β ∈ R. First, we shall look for the solution of (1.6) in the form F (t, x, ξ ) = M∞ (ξ ) + W0 (ξ )f (t, x, ξ ),

(2.2)

where W0 is the weight of (2.1) with β = 0. Then, the problem (1.6) reduces to  f + ξ1 fx − Lf = (f ), t > 0, x > 0, ξ ∈ R3 ,    t f |t=0 = f0 (x, ξ ), x > 0, ξ ∈ R3 , (2.3) f |x=0 = a0 (ξ ), t > 0, ξ1 > 0, (ξ2 , ξ3 ) ∈ R2 ,    f → 0 (x → ∞), t > 0, ξ ∈ R3 , where

a0 = W0−1 Fb − M∞ ,

102


and Lf = W0−1 Q(M∞ , W0 f ) + Q(W0 f, M∞ ) ,

(f ) = (f, f ),

(2.4)

with

(f, g) = W0−1 Q(W0 f, W0 g). The operator L is linear while the remainder is quadratic, both acting only on the variable ξ . The following properties (and nothing else) from them will be used in the p 3 ∞ sequel. Set Lξ = Lp (Rξ3 ) and L∞ ξ,β = L (Rξ , Wβ (ξ )dξ ). Proposition 2.1. For the hard sphere model, the following holds with some positive constants ν0 , ν1 , k0 , k1 , k2 depending only on ρ∞ , u∞ , T∞ . (i) L has the decomposition L = −ν(ξ ) × +K, where ν(ξ ) is a positive function satisfying ν0 ≤ ν(ξ ) ≤ ν0−1 (1 + |ξ |),

ξ ∈ R3 ,

whereas K is an integral operator Kh =

R3

K(ξ, ξ )h(ξ )dξ

with the kernel enjoying the estimate 2

|K(ξ, ξ )| ≤ k0 (|ξ − ξ | + |ξ − ξ |−1 )e−k1 |ξ −ξ | . (ii) L is non-positive self-adjoint on L2ξ , with the estimate (Lh, h)L2 ≤ −ν1 ||(1 + |ξ |)1/2 P ⊥ h||2L2 , ξ

(2.5)

ξ

where P ⊥ = I − P , P being the orthogonal projection onto the null space N of L. (iii) K has the regularizing property that it is bounded as an operator ∞ K : L∞ ξ,β → Lξ,β+1

and

K : L2ξ → L∞ ξ

for all β ≥ 0. (iv) The bilinear operator (f, g) enjoys the estimate ||ν −1 (f, g)||L∞ ≤ k3 ||f ||L∞ ||g||L∞ ξ,β ξ,β ξ,β for all β.


103

Proof. For ρ∞ = 1, u∞ = 0, and T∞ = 1, that is, for the case of the standard Maxwellian M 0 (ξ ) = M[1, 0, 1](ξ ), all the statements in the above are found in, e.g. [4], pp. 197-198, except for (2.5) which is stated in [6]. Let ν 0 (ξ ) and K 0 (ξ, ξ ) be ones corresponding to the standard Maxwellian M 0 . Their explicit formulas go back to [10, 3] (see also [4], pp. 196–197). Since M[ρ∞ , u∞ , T∞ ](ξ ) = αM 0 (γ (ξ − u∞ )), 3/2

1/2

for α = ρ∞ /T∞ and γ = 1/T∞ , it follows from (2.4) that ν(ξ ) = c0 ν 0 (γ (ξ − u∞ )),

K(ξ, ξ ) = c0 K 0 (γ (ξ − u∞ )), γ (ξ − u∞ )),

with c0 = α/γ = ρ∞ /T∞ , whence the proposition follows for the general Maxwellian. This proposition is also valid for Grad’s cut-off hard potential [9] with due modification, particularly with (|ξ | + 1)δ (δ ∈ [0, 1]) in place of (|ξ | + 1) in (2.5). Since the model we consider is the hard sphere (δ = 1), we can let f = e−σ x g in (2.3) and control by (2.5) (and by P ) the term σ ξ1 appearing in the deduced problem  g + ξ1 gx − σ ξ1 g − Lg = e−σ x (g), t > 0, x > 0, ξ ∈ R3 ,    t x > 0, ξ ∈ R3 , g|t=0 = g0 (x, ξ ), (2.6) g|x=0 = a0 (ξ ), t > 0, ξ1 > 0, (ξ2 , ξ3 ) ∈ R2 ,    3 g → 0 (x → ∞), t > 0, ξ ∈ R . Now, denote the stationary boundary layer solution to (2.6) by g¯ and let the initial g0 be a small perturbation of g. ¯ Then the stability problem we consider can be formulated as follows:  g˜ t + ξ1 g˜ x − σ ξ1 g˜ − Lg˜ = e−σ x {L¯ g˜ + (g)}, ˜ t > 0, x > 0, ξ ∈ R3 ,   

g˜ t=0 = g˜ 0 (x, ξ ), x > 0, ξ ∈ R3 ,

g˜ x=0 = 0, t > 0, ξ1 > 0, (ξ2 , ξ3 ) ∈ R2 ,    g˜ → 0 (x → ∞), t > 0, ξ ∈ R3 , (2.7) ¯ g). ˜ where g˜ = g − g, ¯ g˜0 = g0 − g¯ and L¯ g˜ = 2 (g, Let S(t) be the solution operator (semi-group) of the linear problem   ht + ξ1 hx − σ ξ1 h − Lh = 0, t > 0, x > 0, ξ ∈ R3 , h = 0 (ξ1 > 0), h → 0(x → ∞), t > 0, ξ ∈ R3 , 

x=0 h t=0 = h0 (x, ξ ), x > 0, ξ ∈ R3 .

(2.8)

Then we have h = S(t)h0 . For the case M∞ < −1, the L2 decay estimate for (2.8) is easy to establish. Recall that in this case, the operator A = P ξ1 P introduced in our previous paper [15] is negative definite on N , whereas L is also negative definite on N ⊥ with the estimate (2.5). Here, P and N are as in Proposition 2.1. Now for a small σ > 0, a straightforward energy estimate gives 1 d ||h(t)||2 + < |ξ1 |h0 , h0 >− +ν2 ||(1 + |ξ |)1/2 h(t)||2 ≤ 0, 2 dt

104


with a constant ν2 > 0 (say ν2 = ν1 /2), where ||·|| = ||·||L2 , < ·.· >− = (·, ·)L2 (ξ1 >0) , x,ξ

and h0 = h|x=0 . This implies that d ν2 t e ||h(t)||2 + eν2 t 2 < |ξ1 |h0 (t), h0 (t) >− +ν2 ||(1 + |ξ |)1/2 h(t)||2 ≤ 0. dt Then it follows that t eν2 t ||h(t)||2 + eν2 t 2 < |ξ1 |h0 (t), h0 (t) >− +ν2 ||(1 + |ξ |)1/2 h(t)||2 dt ≤ ||h0 ||2 , 0

(2.9) and ||S(t)h0 || ≤ e−κt ||h0 ||,

κ=

ν2 . 2

(2.10)

As for the existence analysis, we want to prove the following estimate which is sufficient for the application of the fixed point theorem to get the global existence of the solution to the nonlinear problem (2.5), ||S(t)h0 ||β ≤ ce−κt ||h0 ||β + ||h0 ||L2 , (2.11)

x,ξ

Wβ (ξ )dxdξ = L∞ for β ≥ 0, where || · ||β is the norm of the space β . In order to prove (2.11), we first consider another simpler linear solution operator. Let ν(ξ ) be as in Proposition 2.1(i) and let S0 (t) be the solution operator (semi-group) of   ht + ξ1 hx − σ ξ1 h + ν(ξ )h = 0, t > 0, x > 0, ξ ∈ R3 , h = 0 (ξ1 > 0), h → 0(x → ∞), t > 0, ξ ∈ R3 , (2.12) 

x=0 h t=0 = h0 (x, ξ ), x > 0, ξ ∈ R3 . L∞ x,ξ

The solution to the above linear initial boundary value problem has the following explicit expression: h = S0 (t)h0 = e−(ν(ξ )−σ ξ1 )t χ (x − ξ1 t)h0 (x − ξ1 t, ξ ),

(2.13)

where χ (y) is the usual characteristic function for y > 0. Based on this expression and with the lower bound ν(ξ ) ≥ ν0 > 0, a simple calculation yields the following estimate on S0 : ||S0 (t)h0 ||X ≤ ce−(2κ−ε)t ||h0 ||X ,

(2.14)

with κ chosen to be min(ν0 , ν2 )/2, for some small constant > 0. Here the space X 2 can be either L∞ β or Lx,ξ . From (2.8) and (2.12), we have t  S(t)h0 = S0 (t)h0 + 0 S0 (t − s)KS(s)h0 ds     = m−1  j =0 Ij (t) + Jm (t)   (t)h I0 (t) = S 0 0t (2.15)  I (t) = S (t − s)KIj −1 (s)ds = (S0 K) ∗ Ij −1 j 0  0    J (t) = (S0 K) ∗ (S0 K) ∗ · · · ∗ (S0 K) ∗ h,   m m


105

with h = S(t)h0 . Here and hereafter, “∗” stands for the convolution in t. By using the estimate (2.14) and the regularizing property of the compact operator K in Proposition 2.1(iii), we have for β ≥ j ≥ 0, ||Ij (t)||β ≤ cj e(−2κ+ε)t ||h0 ||β−j .

(2.16)

The estimate on Jm is more complicated and can be stated in the following bootstrapping lemma. Lemma 2.2. For β ≥ 0, we have ||Jβ+3 (t)||β ≤ ce−κt ||h0 ||L2 . x,ξ

Proof. First, again by the regularizing property of K in Proposition 2.1(iii), we have C t ||Jβ+3 (t)||β ≤ (t − τ )β e−(2κ−)(t−τ ) ||J2 ||L∞ (L2 ) (τ )dτ, (2.17) x ξ β! 0 where J2 (t) = (S0 K) ∗ (S0 K) ∗ h = S0 ∗ J¯,

(2.18)

with J¯ = KS0 K ∗ h =

t

t

KS0 (t − s)Kh(s)ds =

0

J¯0 (t − s, s)ds.

(2.19)

0

We now estimate J¯0 (t, s) as follows. Here, we need to use some integral property of the compact operator K. By definition, we have J¯0 (t, s) = KS0 (t)Kh(s) K(ξ, ξ )K(ξ , ξ )e−(ν(ξ )−σ ξ1 )t χ (y)h(s, y, ξ )dξ dξ , =

(2.20)

R3 ×R3

where y = x − ξ1 t. Hence, |J¯0 (t, s)| ≤ e−(ν0 −ε)t

R×R3

where K0 (ξ, ξ1 , ξ ) ≡

K0 (ξ, ξ1 , ξ )χ (y)|h(s, y, ξ )|dξ1 dξ , R2

(2.21)

|K(ξ, ξ )||K(ξ , ξ )|dξ2 dξ3 ,

with ξ = (ξ1 , ξ2 , ξ3 ). Notice that the estimate of the kernel K(ξ, ξ ) stated in Proposition 2.1(i) gives |K(ξ, ξ )| dξ = |K(ξ , ξ )| dξ ≤ C0 , 3 3 R R |K(ξ, ξ )| dξ2 dξ3 ≤ C1 , R2

106


where C0 and C1 are some positive constants depending only on the parameters ρ∞ , u∞ , T∞ . Thus, we have K0 (ξ, ξ1 , ξ )dξ1 dξ = |K(ξ, ξ )| |K(ξ , ξ )| dξ dξ ≤ C02 , R×R3 R3 ×R3 K0 (ξ, ξ1 , ξ )dξ ≤ C0 |K(ξ , ξ ) dξ2 dξ2 ≤ C0 C1 . R3

R2

By (2.21) and the Schwartz inequality,

J¯0 (t, s) 2 ≤ e−2(2κ−)t K0 (ξ, ξ1 , ξ ) dξ1 dξ R2 ×R3

2 K0 (ξ, ξ1 , ξ )χ (y) h(s, y, ξ ) dξ1 dξ × R2 ×R3

2 2 −2(2κ−)t K0 (ξ, ξ1 , ξ )χ (y) h(s, y, ξ ) dξ1 dξ . (2.22) ≤ C0 e R2 ×R3

Therefore, we have x

J¯0 (t, s) 2 dξ x>0 R3

2 2 −2(2κ−)t ≤ C 0 C0 C1 e χ (y) h(s, y, ξ ) dξ1 dξ R×R3

2

c −2(2k−ε)t ∞ dy dξ h(s, y, ξ ) = e 3 t R 0 c ≤ e−2(2k−ε)t e−2ks h0 2L2 . (2.23) x,ξ t

J¯0 (t, s) 2L∞ (L2 ) = sup ξ

Here, we have used the L2 decay estimate (2.10). Hence (2.19) and (2.23) give t

J¯(t) L∞ (L2 ) ≤

J¯0 (t − s, s) L∞ (L2 ) ds x x ξ ξ 0 t e−(2κ−)(t−s) ≤c e−κs h0 L2 ds √ x,ξ t −s 0 t −(κ−)(t−s) e ≤ c e−kt (2.24) ds h0 ≤ ce−κt h0 . √ t −s 0 This and (2.14), (2.18) give

J2 (t) L∞ (L2 ) = S0 ∗ J¯ ≤ x

ξ

t 0

≤c ≤ ce

e−(2κ−ε)(t−s) J¯(s) L∞ (L2 ) ds x

t

ξ

e−(2κ−ε)(t−s) e−κs ds h0 L2

0 −κt

x,ξ

h0 L2 . x,ξ

(2.25)

Plugging this into (2.17) yields

t c ||Jβ+3 (t)||β ≤ e (t − τ )β e−(κ−)(t−τ ) dτ ||h0 ||L2 x,ξ β! 0 −κt ≤ ce ||h0 ||L2 . −κt

x,ξ

And this completes the proof of the lemma.

(2.26)


107

This lemma and (2.16) complete the proof of the L∞ β decay estimate (2.11). In order to estimate the nonlinear term (g) ˜ and the coupling term L¯ g˜ in (2.7) by Proposition 2.1(iv), we also need the following lemma. Lemma 2.3. When β ≥ 0, for the two semigroups S0 and S, we have ||S0 ∗ ν(ξ )h||β (t) ≤ ce−κt sup {eκτ ||h||β (τ )}, ||S ∗ ν(ξ )h||β (t) ≤ ce

0≤τ ≤t −κ/2t

{ sup (eκ/2τ ||h||β (τ )) + sup (eκ/2τ ||νh||L2 (τ ))}, 0≤τ ≤t

0≤τ ≤t

x,ξ

both for every function h(t, x, ξ ) with the relevant norm bounded. Proof. First, by the special property of the semigroup S0 and the linear growth rate of ν(ξ ), we have t ||S0 ∗ νh||β ≤ sup (1 + |ξ |β )e−(ν(ξ )−σ ξ1 )(t−s) χ (x − ξ1 s)ν(ξ )|h(s, x − ξ1 s, ξ )|ds x,ξ

0

≤ e−κt sup {eκτ ||h||β (τ )} sup 0≤τ ≤t

ξ

t

e−(ν(ξ )−κ−σ ξ1 )(t−s) ν(ξ )ds

0

≤ ce−κt sup {eκτ ||h||β (τ )}. 0≤τ ≤t

To give the estimate for S, we use the relation between S and S0 , S = S0 + S0 ∗ KS. First, write (2.11) as ||S(t)h0 ||β ≤ ce−κt [[h0 ]]β ,

(2.27)

[[·]]β = || · ||β + || · ||L2 .

(2.28)

with x,ξ

We assume β ≥ 1 but the proof is similar for other β. By the regularizing property of the operator K again, we have t ||S0 ∗ KS ∗ νh||β ≤ e−κ(t−s) ||KS ∗ νh||β (s)ds 0 t ≤c e−κ(t−s) ||S ∗ νh||β−1 (s)ds 0 t s ≤c e−κ(t−s) e−κ(s−τ ) [[νh]]β−1 (τ )dτ ds (by (2.27)) 0 0 t κ/2τ ≤ c sup {e [[νh]]β−1 (τ )} e−κ(t−s) e−κ/2s sds 0≤τ ≤t

0

≤ ce−κ/2t sup {eκ/2τ [[νh]]β−1 (τ )}. 0≤τ ≤t

108


Combining this with the estimate for S0 , we have −κ/2t κ/2τ κ/2τ sup e ||h||β (τ ) + sup e [[νh]]β−1 (τ ) . ||S ∗ ν(ξ )h||β (t) ≤ ce 0≤τ ≤t

0≤τ ≤t

Recalling the linear growth of ν(ξ ) and the definition (2.28) completes the proof of the lemma. By using the estimates in the above lemmas and (2.27), we can now construct a global solution to the nonlinear problem (2.7). The definition of the semigroup implies that ˜ (2.29) g˜ = S(t)g˜ 0 + S ∗ {e−σ x (L¯ g˜ + (g))}. Write the right-hand side by [g]. ˜ We have ˜ ||[g]|| ˜ β ≤ ||S(t)g˜ 0 ||β + ||S ∗ {νν −1 e−σ x (L¯ g˜ + (g))}|| β ≤ ce−κ/2t [[g˜ 0 ]]β + sup eκ/2τ ||e−σ x νν −1 (L˜ g¯ + (g))|| ˜ β (τ ) τ ≥0

+ sup e τ ≥0

κ/2τ

||e

−σ x

νν

−1

(L˜ g¯ + (g))|| ˜ L2

x,ξ

(τ )

≤ ce−κ/2t {[[g˜ 0 ]]β + ||g|| ¯ β |||g||| ˜ + |||g||| ˜ 2 }, where |||h||| = sup{eκ/2t ||h||β (t)}.

(2.30)

t≥0

In the above we have used the estimate in Proposition 2.1(iv) and the relation ∞ ||e−σ x νh||2L2 ≤ e−2σ x dx ν 2 (ξ )(1 + |ξ |)−2β dξ ||h||2β x,ξ R3 0 5 . = c||h||2β , β > 2 Consequently, we have ¯ β |||g||| ˜ + |||g||| ˜ 2 ), |||[g]||| ˜ β ≤ c([[g˜ 0 ]]β + ||g|| and similarly, ˜ + |||g˜ + h||||| ˜ ˜ ˜ β ≤ c(||g|| ¯ β |||g˜ − h||| g˜ − h|||), |||[g] ˜ − [h]||| with the same constant c. ¯ which follows from the The smallness assumption on [[g¯ 0 ]]]β and that on |||g||| smallness assumption on the boundary data a0 in (2.6) now assure that the nonlinear map is a contraction map in a small ball of the Banach space defined with the norm (2.30) and therefore a unique fixed point exists. This implies, taking into account the choice of the norm (2.30), that (2.7) has a unique global in time solution converging exponentially to 0 as t → ∞ in the norm (2.28). Thus Theorem 1.1 follows. Acknowledgement. The research of the first author was supported by Grant-in Aid for Scientific Research (C) 136470207, Japan Society for the Promotion of Science (JSPS). The research of the second author was supported by the Competitive Earmarked Research Grant of Hong Kong CityU 1092/02P# 9040737. The research of the third author was supported by the Competitive Earmarked Research Grant of Hong Kong # 9040645.


109

References 1. Aoki, K., Nishino, K., Sone, Y., Sugimoto, H.: Numerical analysis of steady flows of a gas condensing on or evaporating from its plane condensed phase on the basis of kinetic theory: Effect of gas motion along the condensed phase. Phys. Fluids A 3, 2260–2275 (1991) 2. Bardos, C., Caflish, R.E., Nicolaenko, B.: The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas. Comm. Pure Appl. Math. 49, 323–352 (1986) ´ 3. Carleman, T.: Sur La Théorie de l’Equation Intégrodifférentielle de Boltzmann. Acta Mathematica 60, 91–142 (1932) 4. Cercignani, C., Illner, R., Purvelenti, M.: The Mathematical Theory of Dilute Gases. Berlin: SpringerVerlag, 1994 5. Cercignani, C.: Half-space problem in the kinetic theory of gases. In: E. Kröner, K. Kirchgässner, (eds.), Trends in Applications of Pure Mathematics to Mechanics, Berlin: Springer-Verlag, 1986, pp. 35–50 6. Coron, F., Golse, F., Sulem, C.: A classification of well-posed kinetic layer problems. Commun. Pure Appl. Math. 41, 409–435 (1988) 7. Golse, F., Perthame, B., Sulem, C.: On a boundary layer problem for the nonlinear Boltzmann equation. Arch. Rat. Mech. Anal. 103(1), 81–96 (1988) 8. Golse. F., Poupaud, F.: Stationary solutions of the linearized Boltzmann equation in a half-space. Math. Methods Appl. Sci. 11, 483–502 (1989) 9. Grad, H.: Asymptotic Theory of the Boltzmann Equation. In: Rarefied Gas Dynamics, J.A. Laurmann, (ed.), Vol 1, 26, New York: Academic Press, 1963, pp. 26–59 10. Hilbert, D.: Grundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen. (German) New York, N.Y.: Chelsea Publishing Company, 1953, pp. xxvi+282 11. Lions, P.-L.: Conditions at infinity for Boltzmann’s equation. Commun. Partial Diff. Eqs. 19, 335– 367 (1994) 12. Sone, Y.: Kinetic Theory of Evaporation and Condensation-Linear and Nonlinear Problems. J. Phys. Soc. Japan 45(1), (1978) 13. Sone, Y.: Kinetic Theory and Fluid Dynamics. Berlin: Birkhäuser, 2002 14. Ukai, S.: On the half-space problem for the discrete velocity model of the Boltzmann equation. In: Advances in Nonlinear Partial Differential Equations and Stochastic, Kawashima, T. Yangisawa, (eds.), Series on Advances in Mathematics for Applied Sciences, Vol. 48, Singapore–New York: World Scientific, 1998, pp. 160–174 15. Ukai, S., Yang, T., Yu, S.-H.: Nonlinear Boundary Layers of the Boltzmann Equation: I, Existence. Commun. Math. Phys. 236, 373–393 (2003) Communicated by H.-T. Yau


Communications in


Young Wall Realization of Crystal Graphs for Uq (Cn(1) ) Jin Hong1 , Seok-Jin Kang2, , Hyeonmi Lee2, 1 2

National Security Research Institute, 161 Gajeong-Dong, Yuseong-Gu, Daejeon 305-350, Korea. E-mail: [email protected] Korea Institute for Advanced Study, 207-43 Cheongryangri-dong, Dongdaemun-Gu, Seoul 130-722, Korea. E-mail: [email protected]; [email protected]

Received: 7 August 2002 / Accepted: 8 July 2003 Published online: 13 November 2003 – © Springer-Verlag 2003

Abstract: We give a realization of crystal graphs for basic representations of the (1) quantum affine algebra Uq (Cn ) using combinatorics of Young walls. The notion of splitting blocks plays a crucial role in the construction of crystal graphs. 1. Introduction In [10] and [13], Kashiwara and Lusztig independently developed the crystal basis theory (or canonical basis theory) for integrable modules over quantum groups associated with symmetrizable Kac-Moody algebras. A crystal basis can be viewed as a basis at q = 0 and has a structure of colored oriented graph, called the crystal graph, defined by Kashiwara operators. The crystal graphs have many nice combinatorial features reflecting the internal structure of integrable modules over quantum groups. For example, one can compute the characters of integrable representations by finding an explicit combinatorial description of crystal graphs. Moreover, the crystal graphs have extremely simple behavior with respect to taking the tensor product. Thus, the crystal basis theory provides us with a very powerful combinatorial method of studying the structure of integrable modules over quantum groups. Let Uq (g) be a quantum group associated with a symmetrizable Kac-Moody algebra g and let V (λ) denote the irreducible highest weight Uq (g)-module with a dominant integral highest weight λ. One of the most interesting problems in crystal basis theory is to find an explicit realization of the crystal graph B(λ) of V (λ), called the irreducible highest weight crystal with highest weight λ. When g is a classical Lie algebra, the crystal graph B(λ) can be realized as the set of semistandard Young tableaux of a given shape satisfying certain additional conditions [11]. This work was supported by KOSEF Grant # 98-0701-01-5-L and theYoung Scientist Award, Korean Academy of Science and Technology. This work was supported by BK21 Project, Mathematical Sciences Division, Seoul National University.

112

J. Hong, S.-J. Kang, H. Lee (1)

For the quantum affine algebras of type An−1 , Misra and Miwa constructed the crystal graphs B(λ) for basic representations using colored Young diagrams [14]. Their idea (1) was extended to construct crystal graphs for irreducible highest weight Uq (An−1 )-modules of arbitrary higher level [4]. The crystal graphs constructed in [4] and [14] can be parameterized by certain paths which arise naturally in the theory of solvable lattice modules. Motivated by this observation, Kang, Kashiwara, Misra, Miwa, Nakashima, and Nakayashiki developed the theory of perfect crystals for quantum affine algebras and gave a realization of crystal graphs B(λ) over classical quantum affine algebras of arbitrary higher level in terms of paths [7–9]. In [5], Kang introduced the notion of Young walls as a new combinatorial scheme for realizing the crystal graphs for quantum affine algebras. The Young walls consist of colored blocks with various shapes that are built on a given ground state wall, and they can be viewed as generalizations of colored Young diagrams. For the classical quantum (1) (2) (1) (2) (2) affine algebras of type An (n ≥ 1), A2n−1 (n ≥ 3), Dn (n ≥ 4), A2n (n ≥ 1), Dn+1 (1)

(n ≥ 2), and Bn (n ≥ 3), the crystal graphs B(λ) of the basic representations were realized as the affine crystals consisting of reduced proper Young walls. However, for (1) the quantum affine algebras of type Cn (n ≥ 2), the problem of Young wall realization of crystal graphs was left open. The purpose of this paper is to fill in this missing part: we develop the combinatorics (1) of Young walls for the quantum affine algebras Uq (Cn ) (n ≥ 2) and give a realization of crystal graphs B(λ) for the basic representations as the sets of reduced proper Young walls. This case is more difficult to deal with than the other classical quantum affine algebras, because the level-1 perfect crystals for this case are intrinsically of level-2. The notion of splitting blocks was introduced to overcome this difficulty. We believe this notion will play a crucial role in the construction of higher level irreducible highest weight crystals for all classical quantum affine algebras. 2. Quantum Group Uq (Cn(1) ) and its Level-1 Perfect Crystal We refer the readers to the references cited in the introduction, or to the books on quantum groups [2, 3] for the basic concepts on quantum groups and crystal bases. Familiarity with at least the following concepts will be assumed: quantum group, crystal basis, irreducible highest weight crystal, (abstract) crystal, perfect crystal, ground state path, λ-path, signature (of a path), path space. A clear understanding of the Young wall theory for any one of the affine types developed prior to this work will be immensely helpful in reading this paper, although not a logical prerequisite. Let us fix basic notations here. • • • • • • • • •

(1)

(1)

Uq = Uq (Cn ) : quantum group of type Cn (n ≥ 2). I = {0, 1, . . . , n} : index set. (1) A = (aij )i,j ∈I : generalized Cartan matrix of type Cn . ∨ P = (⊕i∈I Zhi ) ⊕ Zd : dual weight lattice. h = C ⊗Z P ∨ : Cartan subalgebra. αi , δ, i : simple root, null root, fundamental weight. P = (⊕i∈I Zi ) ⊕ Zδ : weight lattice. (1) ei , Ki±1 , fi , q d : generators of Uq (Cn ). (1) Uq = Uq (Cn ) : subalgebra of Uq generated by ei , Ki±1 , fi (i ∈ I ).

(1)

Young Walls for Uq (Cn )

• • • • • •

113

P = ⊕i∈I Zi : classical weight lattice. wt, wt : (affine) weight, classical weight. B(i ) : irreducible highest weight crystal of highest weight i . (1) B (1) : level-1 perfect crystal of type Cn . e˜i , f˜i : Kashiwara operators. P(i ) : the set of i -paths (with crystal structure).

We cite two theorems that are crucial for our work. The first is the path realization of irreducible highest weight crystals. Theorem 2.1. The path space is isomorphic to the irreducible highest weight crystal : B(i ) ∼ = P(i ). (1)

Below is a perfect crystal of type Cn , introduced in [6]. We use a special case of this result. (1)

Theorem 2.2. A level-1 perfect crystal of type Cn is given as follows: B (1) = (x1 , . . . , xn |x¯n , . . . , x¯1 ) xi , x¯i ∈ Z≥0 , ni=1 (xi + x¯i ) = 0 or 2 . For b = (x1 , . . . , xn |x¯n , . . . , x¯1 ), the action of the Kashiwara operator f˜i on B (1) is given as follows. For i = 0,   if x1 ≥ x¯1 , (x1 + 2, x2 , . . . , x¯2 , x¯1 ) ˜ f0 b = (x1 + 1, x2 , . . . , x¯2 , x¯1 − 1) if x1 = x¯1 − 1,  (x , x , . . . , x¯ , x¯ − 2) if x1 ≤ x¯1 − 2. 1 2 2 1 For i = 1, . . . , n − 1, (x1 , . . . , xi − 1, xi+1 + 1, . . . , x¯1 ) if xi+1 ≥ x¯i+1 , f˜i b = (x1 , . . . , x¯i+1 − 1, x¯i + 1, . . . , x¯1 ) if xi+1 < x¯i+1 . For i = n, fñ b = (x1 , . . . , xn − 1|x¯n + 1, . . . , x¯1 ). The action of the Kashiwara operator e˜i on B (1) is given as follows. For i = 0,   if x1 ≥ x¯1 + 2, (x1 − 2, x2 , . . . , x¯2 , x¯1 ) e˜0 b = (x1 − 1, x2 , . . . , x¯2 , x¯1 + 1) if x1 = x¯1 + 1,  (x , x , . . . , x¯ , x¯ + 2) if x1 ≤ x¯1 . 1 2 2 1 For i = 1, . . . , n − 1, (x1 , . . . , xi + 1, xi+1 − 1, . . . , x¯1 ) if xi+1 > x¯i+1 , e˜i b = (x1 , . . . , x¯i+1 + 1, x¯i − 1, . . . , x¯1 ) if xi+1 ≤ x¯i+1 . For i = n, eñ b = (x1 , . . . , xn + 1|x¯n − 1, . . . , x¯1 ).

114

J. Hong, S.-J. Kang, H. Lee

The remaining maps describing the crystal structure on B (1) are given below : 1

ϕ0 (b) = 1 − (xi + x¯i ) + (x¯1 − x1 )+ , 2 n

i=1

ϕi (b) = xi + (x¯i+1 − xi+1 )+ (i = 1, . . . , n), ϕn (b) = xn , 1

ε0 (b) = 1 − (xi + x¯i ) + (x1 − x¯1 )+ , 2 n

i=1

εi (b) = x¯i + (xi+1 − x¯i+1 )+ (i = 1, . . . , n), εn (b) = x¯n , wt(b) =

n

(ϕi (b) − εi (b))i .

i=0

Here, (x)+ = max(0, x). (1)

The perfect crystal for Uq (C2 ) used in [1] is different from the one given in this theorem. (1)

Example 2.3. The following is a drawing of the level-1 perfect crystal for Uq (C2 ) in the form of the above theorem. Readers familiar with the crystal basis theory will notice the Uq (C2 )-crystal B(21 ) ⊂ B(1 ) ⊗ B(1 ) in the drawing. This is what we meant (1) by the level-2 nature of the perfect crystal for Uq (Cn ) in the introduction. (1, 0|0, 1) 1

1

(1, 0|1, 0) 2

(0, 1|0, 1)

0

0

(1, 1|0, 0) 1

2

(0, 1|1, 0) 1

(2, 0|0, 0)

2

(0, 0|1, 1) 2

(0, 2|0, 0)

1 (0, 0|2, 0)

0

1 (0, 0|0, 2)

0 (0, 0|0, 0)

3. New Realization of the Level-1 Perfect Crystal In this section, we construct the set of slices and obtain a new realization for the level-1 perfect crystal B (1) . 3.1. Slices. A slice is what will later become a column in our Young walls. The basic ingredient of our discussion will be the following colored blocks. 0

: half-unit height, unit width, unit depth.

i

: unit height, unit width, unit depth (i = 1, . . . , n).

To simplify drawings, we shall use just the frontal view when representing a set of blocks stacked in a wall of unit thickness.

(1)


1

4 2 0 2

1 1

115 0 2 0 0 0 3

1 2 2 3 2

3 1 0

←→

1

1

4 2 0 2

1 1

0 2 0 0 0 3

1 2 2 3 2

3 1 0

1

A set of finitely many blocks, stacked in one column, following the pattern

     

n ···

covering blocks

1 0 0 1 ···

      

1 0 covering block → 0

            

supporting blocks

(1)

is called a level- 21 slice of type Cn . For those with previous Young wall experience, we stress that the bottom of the column must be a 0-block as given in the above pattern. We see that, in this repeating pattern, an i-block appears twice in each cycle for i = 0, . . . , n − 1. To distinguish the two places, we have given names to these positions or blocks. A covering block is one that is closer to the n-block that sits below it than to the position for n-block above it. If it is the other way around, it is a supporting block. Notice that, by convention, each n-block is both a supporting block and a covering block. Any consecutive sequence of blocks in a level- 21 slice that contains one n-block and two i-blocks for each i = 0, . . . , n − 1 is called a δ. If we may place an i-block on top of some level- 21 slice and still obtain a level- 21 slice, we shall call that place an i-slot. The notions of covering i-slot or supporting i-slot is self-explanatory. Remark 3.1. We warn the reader that, even though δ = α0 + 2α1 + · · · + 2αn−1 + αn (1) for Uq (Cn ), we are using two 0-blocks for our definition of δ. This is because we shall always be using 0-blocks in pairs. For example, in applying f˜0 action, two 0-blocks will be added. We may add a δ to a level- 21 slice or remove a δ from a big enough level- 21 slice c (1) and write this as c ± δ. For example, when dealing with Uq (C2 ), we have 1 0 0 1 2 1 0 0

+δ =

1 0 0

.

Definition 3.2. An ordered pair C = (c1 , c2 ) of level- 21 slices is a level-1 slice of type (1) Cn , if c1 ⊂ c2 ⊂ c1 + δ and if it contains an even number of 0-blocks. Each ci is called the i th layer of C. The set of all level-1 slices is denoted by S (1) .

116


We shall often just say slice, when dealing with level-1 slices. Mentally, we picture a level-1 slice as two columns with the first layer placed in front of the second layer, rather than as an ordered pair. We explain how to draw a slice with the following example.

( c1 =

1 0 0

,

c2 =

1

1

1

2

2

2

1 0 0

1 0 0

←→

)

C =

=

1 0 0

We now explain the notion of splitting an i-block in a level-1 slice. Suppose that the top part of some level-1 slice C takes one of the following two shapes:

···

···

n

0 0 ···

i−1

···

i

i

···

···

i−1

C=

0 0

or

C=

n

for some 0 < i < n. To split an i-block in such a level-1 slice, means to break off the top half of the i-block and to place it on top of the (i − 1)-block, so that it looks like

···

n

0 0 ···

i/2 i−1

i/2

···

···

···

i/2 i−1

···

i/2

C =

0 0

or

C =

n

.

The “i/2” written in the cut off i-blocks are supposed to convey the idea that this is a half of the i-block. We will never split a 0-block, but splitting an n-block may be done similarly. n

n/2 n−1

C=

0 0

···

···

n−1

−→

C =

0 0

.

(1)

0 0

0 0

C=

n

···

n/2 n−1

···

n−1

···

117

···


−→

C =

n/2

.

Simply put, splitting an i-block (i = 0) is breaking off the top half of a covering i-block and placing it in a supporting i-slot. Remark 3.3. The result obtained after splitting an i-block in a slice will not be considered a level-1 slice. As it will become clearer when we deal with Young walls in the following sections, splitting is supposed to be a temporary act, used to see things from a different point of view. Remark 3.4. If it is possible to split an i-block in some slice, splitting a block of color different from i in the same column is not possible. So it makes sense to split a column if possible. Any non-split bock is a whole block. Hence a whole 0-block is of half-unit height and whole blocks of any other color is of unit height. We now explain how to apply some action, which we denote by f˜i (i = 0, . . . , n), on the set S (1) ∪ {0}. For i = 1, . . . , n, we go through the following steps, until we see a matching case, either to add one i-block to the slice, or to take the result as zero. (1) The f˜i action on zero is zero. (2) If i =

n and splitting an (i + 1)-block is possible, we take the result to be zero. (3) If neither of the slots at the top of the two level- 21 slices are for i-blocks, the result is zero. (4) If just one of the two slots is for an i-block, place an i-block in the slot. (5) If we have two i-slots at the top, and either i = n or they are of the same kind (supporting or covering), do as follows: • If they are at different heights, place an i-block in the first layer, i.e., the lower slot. • Otherwise, place an i-block in the second layer, i.e., the back slot. (6) If we’ve come this far, we must have two i-slots of different kinds. Place an i-block in the covering slot. To apply f˜0 , we follow the next steps. (1) (2) (3) (4) (5)

The f˜0 action on zero is zero. If it is possible to split a 1-block, the result is zero. If there are no 0-slots available, the result is zero. If the top of both layers are 0-slots, place a 0-block in each of the two slots. If only one of the slots is a 0-slot, place two 0-blocks in that layer.

We also define e˜i action on S (1) ∪ {0}. For i = 0, the action of e˜i on a slice removes one i-block or sends it to zero following the next set of rules. (1) The e˜i action on zero is zero. (2) If i =

n and splitting an (i + 1)-block is possible, we take the result to be zero.

118


(3) If neither of the blocks at the top of the two level- 21 slices are i-blocks, the result is zero. (4) If just one of the two top blocks is an i-block, remove the i-block. (5) If the top of both level- 21 slices are i-blocks, and if either i = n or they are of the same kind, do as follows: • If they are at different heights, remove the i-block in the second layer, i.e., the higher block. • Otherwise, remove the i-block in the first layer, i.e., the closer block. (6) We must now have two i-blocks of different types at the top of the two level- 21 slices. Remove the supporting i-block. To apply e˜0 , we use the following steps. (1) (2) (3) (4) (5)

The e˜0 action on zero is zero. If it is possible to split a 1-block, the result is zero. If neither of the two top blocks are 0-blocks, the result is zero. If the top of both layers are 0-blocks, remove a 0-block from each of the two layers. If only one of the top blocks is a 0-block, remove two 0-blocks in that layer.

3.2. The perfect crystal. In this subsection, we give a new realization for the level-1 (1) perfect crystal of type Cn by moding out the repetitive part from the set of slices. Definition 3.5. We may add a δ to a slice C = (c1 , c2 ), by changing this into C + δ = (c2 , c1 + δ). If, for the same slice C, we have c2 ⊃ δ, we may also remove a δ from C, by changing C into C − δ = (c2 − δ, c1 ). Definition 3.6. Two slices C and C are related, denoted by C ∼ C , if one of the two slices may be obtained from the other by adding finitely many δ. Define C (1) = S (1) / ∼ . Proposition 3.7. The actions f˜i and e˜i , previously defined on the set of level-1 slices, (1) gives the set C (1) a Uq (Cn )-crystal structure. Proof. Consider the map C → C +δ, where C denotes a slice. We may easily check that each of the steps used in defining f˜i and e˜i actions on the set of level-1 slices commutes with this map. So the Kashiwara operators f˜i and e˜i are well-defined on C (1) . We may now define various other maps as usual: εi (C) = max{n | e˜in C ∈ C (1) }, ϕi (C) = max{n | f˜in C ∈ C (1) },

wt(C) = ϕi (C) − εi (C) i . i

(1)


119

Checking that these maps satisfy all of the following relations defining a crystal structure is straightforward : ϕi (C) = εi (C) + hi , wt(C) wt(e˜i C) = wt(C) + αi wt(f˜i C) = wt(C) − αi

for all i ∈ I ,

if e˜i C ∈ C (1) , if f˜i C ∈ C (1) ,

εi (e˜i C) = εi (C) − 1, ϕi (e˜i C) = ϕi (C) + 1 if e˜i C ∈ C (1) , εi (f˜i C) = εi (C) + 1, ϕi (f˜i C) = ϕi (C) − 1 if f˜i C ∈ C (1) , f˜i C = C if and only if C = e˜i C for C, C ∈ C (1) , i ∈ I , if ϕi (C) = −∞ for C ∈ C (1) , then e˜i C = f˜i C = 0. Hence we have a Uq (Cn )-crystal structure on C (1) . (1)

Recall the finite Uq (Cn )-crystal B (1) given in Sect. 2. We shall define a map from to C (1) . The following preliminary mapping is first needed: (1)

B (1)

(0, . . . , 0|0, . . . , 0)

−→

0

(1, 0, . . . , 0|0, . . . , 0)

−→

0 0

···

i−1

(0, . . . , 0, 1, 0, . . . , 0|0, . . . , 0)

−→

0 0

(1 at the i th place from left)

···

n

(0, . . . , 0|1, 0, . . . , 0)

−→

0 0

···

i

···

n

(0, . . . , 0|0, . . . , 0, 1, 0, . . . , 0)

−→

0 0

(1 at the i th place from the right)

Now, to map an element of B (1) to C (1) , we first write the element as a sum of two elements, and map it to C (1) , using the above preliminary mapping. The following few

120


examples in the case of C2 should make this clearer. (0, 0|0, 0) = (0, 0|0, 0) + (0, 0|0, 0) −→

,

0 2

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

−→

1 0 0

,

−→

1 0 0

.

Of course, the right-hand side should be taken as the equivalence class in S (1) / ∼, represented by the drawing. It is easy to see that this correspondence does not depend on which of the two summands we decide to map to the first layer. For example, we could have done 0 0 1 2

(1, 0|1, 0) = (0, 0|1, 0) + (1, 0|0, 0)

−→

1 0 0

for the second example above. This might look different at first sight, but you can check that the drawing on the right-hand side belongs to the same equivalence class as the one given above. Theorem 3.8. The map defined above is an isomorphism of Uq (Cn )-crystals, (1)

B (1) ∼ = C (1) . Proof. There is a natural map from S (1) to B (1) . It may be moded out by the equivalence relation on S (1) to obtain the inverse of the above defined map. Hence the map is bijective. It is a lengthy but straightforward case-by-case comparison to verify that the Kashiwara operators of Theorem 2.2 and those of this section are compatible under this bijection. Remark 3.9. If we change the definition of a slice slightly to be a column which extends infinitely downward, we may state that the set of slices S (1) is a realization for the affinization of B (1) . We close this section with an example that could be of help in understanding the proof of Theorem 3.8. (1)

Example 3.10. The following is a drawing of the level-1 perfect crystal for Uq (C2 ) in the form of C (1) . Readers may want to compare this with Example 2.3.

(1)


121 1 2 1 0 0

1

1

2 1 0 0

1 2 1

2

0

0

1 0 0 1

2

2 1 1

1 2

2

2

1

1 0 0

0 0

1 1 2

2

0

0 0 1

A close study of this example will convince the reader that choosing to use two half-height blocks (and not a single unit-height block) for the f˜0 action was a natural decision. 4. Young Walls In this section, we define the set of reduced proper Young walls. We also define a crystal structure on the set of proper Young walls. 4.1. Level-1 Young walls. We line up the level- 21 slices defined earlier and consider blocks stacked in the following pattern.

···

···

···

n

n

n

n

···

···

1 0 0 1

···

1 0 0 1

···

1 0 0 1

···

1 0 0 1

1 0 0

1 0 0

1 0 0

1 0 0

(1)

Definition 4.1. A level- 21 weak Young wall of type Cn is a set of blocks, or halves of blocks, that satisfies the following conditions: • It is stacked in the pattern given above. • Except for the rightmost column, there is no free space to the right of any block. Remark 4.2. In previous works ([1, 5]), level-1 Young walls were defined to be built on ground state walls. Implicitly, it was also assumed that the building process was done in finite steps. Neither of these conditions are imposed on a level- 21 weak Young wall. For example, the wall

··· ··· ···

0 0

0 0

0 0

1 0 0

1 0 0

2

2

1 0 0

1 0 0

122


is a level- 21 weak Young wall of type Cn , but not a level-1 Young wall in the sense given in previous works [1, 5]. We also do not allow the empty wall to be considered a level- 21 weak Young wall. For a level- 21 weak Young wall Y , we define Y + δ to be the level- 21 weak Young wall obtained by adding a δ to each and every column of Y . Here, a δ is a connected sequence of blocks that contain one n-block and two i-blocks for each i = 0, 1, . . . , n − 1. (1)

Definition 4.3. An ordered pair Y = (Y1 , Y2 ) is a level-1 weak Young wall of type Cn , if it satisfies the following conditions. • Each Yi is a level- 21 weak Young wall. • In each column, any halves of blocks for each color add up to form a whole block. That is, any split blocks come in matching pairs. The level- 21 weak Young wall Yi is called the i th layer of the Young wall Y. Definition 4.4. A level-1 weak Young wall Y = (Y1 , Y2 ) is a level-1 Young wall, if it satisfies the following conditions. • It contains only whole blocks. • Y1 ⊂ Y2 ⊂ Y1 + δ. • Each column contains an even number of 0-blocks. In short, a level-1 Young wall is a level-1 weak Young wall obtained by concatenating level-1 slices. 4.2. Reduced proper Young walls. The i th column of a level-1 Young wall Y = (Y1 , Y2 ) is denoted by Y(i). We choose to number them so that the rightmost column is named Y(0). Note that each column of a level-1 Young wall is a level-1 slice. So we shall utilize the previous notation for drawing level-1 slices when drawing level-1 Young walls. That is, we shall color the first layer gray. The i th column of the j th layer Yj is denoted by Yj (i). We could view the same Yj (i) also as the j th layer of the i th column of Y. Normally, splitting some block in a column of a level-1 Young wall would not give us a Young wall, nor even a weak Young wall. Definition 4.5. Let us be given a level-1 Young wall Y = (Y1 , Y2 ). The Young wall Y is proper if it satisfies the following conditions: • When we split every possible column of Y (see Remark 3.4), the end result Y = (Y1 , Y2 ) is a level-1 weak Young wall. • For each of the two level- 21 weak Young wall Yj in the end result, none of the columns of integer height have the same height. The set of all level-1 proper Young walls is denoted by F. Since a column of a Young wall is a slice, we can add a δ to or remove a δ from a column. To add a δ to a column Y(i) = (Y1 (i), Y2 (i)) means to change this into Y(i) + δ = (Y2 (i), Y1 (i) + δ).

(1)


123

Here, the “+δ” on the right hand side should be understood in the level- 21 sense. Similarly, if Y2 (i) ⊃ δ, we may remove a δ from the same column by changing it into Y(i) − δ = (Y2 (i) − δ, Y1 (i)). Definition 4.6. A column in a level-1 proper Young wall is said to contain a removable δ, if the Young wall is still proper after removing a δ from that column. A proper Young wall is reduced, if none of its columns contain a removable δ. The set of all level-1 reduced proper Young walls is denoted by Y. 4.3. The crystal structure. Recall that each column of a Young wall is a slice, so that we have the actions f˜i and e˜i defined on them (Sect. 3.1). Definition 4.7. (1) A column in a level-1 proper Young wall is k times i-admissible, if k is the maximal number of times we may act f˜i to the column while remaining a proper Young wall. (2) A column in a level-1 proper Young wall is k times i-removable, if k is the maximal number of times we may act e˜i to the column while remaining a proper Young wall. Remark 4.8. Recall that we add two 0-blocks to a column when applying f˜0 . So, being k times 0-admissible will imply that we can place 2k number of 0-blocks and still obtain a proper Young wall. Remark 4.9. Even for i = 0, the number of slots in a column of a Young wall, in which a single i-block may be placed while remaining a proper Young wall, does not necessarily equal the number of times a column is i-admissible. 1

···

2

2

1 0 0

1 0 0

···

(1)

In the above Young wall of type C2 , the left column is only once 1-admissible. But we may place a 1-block in either of the two i-slots in the left column and still obtain a proper Young wall. Remark 4.10. The property of a Young wall column being i-admissible depends on the column which sits to the right of the column in consideration. Likewise, being i-removable depends on the left column. The action of the Kashiwara operators f˜i and e˜i on a level-1 proper Young wall is defined as follows: (1) For each column of the Young wall, write under them x-many 1 followed by y-many 0, if the column is x times i-removable and y times i-admissible. (2) From the (half-)infinite list of 0 and 1, successively cancel out each (0, 1) pair to obtain a finite sequence of 1 followed by some 0 (reading from left to right). (3) Let f˜i act on the column corresponding to the left-most 0 remaining (as an operator acting on slices). Set it to zero if no 0 remains. (4) Let e˜i act on the column corresponding to the right-most 1 remaining (as an operator acting on slices). Set it to zero if no 1 remains.

124


The 0 and 1 placed under the Young wall in the above process are called i-signature of the respective columns or of the Young wall. It is clear from the definition that the result obtained after the action of f˜i or e˜i is still a proper Young wall. We may now define various other maps as before. For a proper Young wall Y , we define εi (Y ) = max{n | e˜in Y is nonzero}, ϕi (Y ) = max{n | f˜in Y is nonzero},

ϕi (Y ) − εi (Y ) i . wt(Y ) =

(4.1) (4.2) (4.3)

i

Checking that these maps satisfy all of the relations defining a crystal structure is (1) straightforward. Hence we have a Uq (Cn )-crystal structure on the set of proper Young walls. Proposition 4.11. The set F of all level-1 proper Young walls forms an (abstract) (1) Uq (Cn )-crystal.

5. Irreducible Highest Weight Crystals In this section, we show that the set of all reduced proper Young walls built on some ground state wall is isomorphic to the irreducible highest weight crystal of appropriate highest weight. 5.1. Ground state walls. Let us denote by P, the set of all half-infinite tensor product of elements from the perfect crystal B (1) . In particular, we have P(k ) ⊂ P. By using the composition of maps ∼

S (1) −→ C (1) −→ B (1) on each column of a proper Young wall, we may define a map : F −→ P.

(5.1)

Now, fix any ground state path bk ∈ P and consider its inverse image −1 (bk ). For (1) example, when dealing with Uq (C2 ), all of the following level-1 proper Young walls are sent to b0 under the map .

(1)


0

0

0

125

0

0 1

0 1

0 1

2

2

2

0

1 0 0

0

1 0 0

1 0 0

0 1

0 1

0 1

0 1

0 1

0 1

2

2

2

2

2

2

0

0

0

1 0 0

1 0 0

0

0

1 0 0

1 0 0

1 0 0

0

0

0

0

0

0

0

0

0

0

1 0 0

0 1

0 1

0 1

0 1

0 1

0 1

0 1

0 1

0 1

0 1

0 1

0 1

0 1

0 1

0 1

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

1 0 0

1 0 0

1 0 0

1 0 0

1 0 0

1 0 0

1 0 0

1 0 0

1 0 0

1 0 0

1 0 0

1 0 0

1 0 0

1 0 0

1 0 0

0

k

k

k

k

k

···

···

···

···

n

n

n

n

n

,

···

0

···

0

···

0

···

0

···

Y0 =

···

We can see that three of these are reduced. Let us denote by Yk ∈ F the unique element of −1 (bk ) that is contained in all other elements of −1 (bk ). That is, we take it to be the smallest element of −1 (bk ). It is clear that each Yk is a reduced proper Young wall. Explicitly, we have

···

···

···

···

Yk =

···

k−1 k−1 k−1 k−1 k−1 0 0

0 0

0 0

0 0

0 0

n

n

n

n

n

(for k = 1, . . . , n − 1),

···

···

···

···

Yn =

···

n−1 n−1 n−1 n−1 n−1 0 0

0 0

0 0

0 0

0 0

.

Definition 5.1. The level-1 reduced proper Young wall Yk is called the ground state wall of weight k . Any level-1 proper Young wall that contains only finitely many more blocks than Yk is said to have been built on Yk . 5.2. Irreducible highest weight crystal. Definition 5.2. The set of all level-1 proper Young walls built on Yk is denoted by F(k ). We will denote by Y(k ) the set of all level-1 reduced proper Young walls built on Yk .

126


Proposition 5.3. The set F(k ) of level-1 proper Young walls built on Yk forms a (1) Uq (Cn )-crystal. Proof. It is clear that it forms a Uq (Cn )-subcrystal of F. So it suffices to give an affine weight to each Young wall which is compatible with other maps. Recall the previous definition of classical weight of a Young wall given by (4.3). We may define the affine weight of a Young wall Y ∈ F(k ) by (1)

1 wt(Y) = wt(Y) − (number of 0-blocks in Y \ Yk ) δ. 2

(5.2)

In particular, the affine weight of Yk is k . It is straightforward to verify that F(k ) (1) forms a Uq (Cn )-crystal with this definition of affine weights. Notice that the image of Yk under given by (5.1) is the ground state path bk of P (k ). Hence it makes sense to restrict the domain and range of as : Y(k ) −→ P(k ).

(5.3)

We claim that this is a bijection. To check the surjectivity, we may explicitly construct a reduced proper Young wall which maps to the path in question, and in the process, we will notice that the condition reduced forces it to be chosen uniquely. Here is our main theorem, which shows that this bijection is a crystal isomorphism. (1)

Theorem 5.4. The set Y(k ) is a Uq (Cn )-subcrystal of F(k ), and it is isomorphic to the irreducible highest weight crystal B(k ). The rest of this section is devoted mostly to proving this theorem. Let us denote by the inverse of the map (5.3), but with the range enlarged as : P(k ) −→ F(k ). To prove the theorem, it suffices to show that this is a strict crystal morphism. Then the image, which we have already seen to be Y(k ), would be a subcrystal of F(k ). The second statement follows from Theorem 2.1. We shall focus our efforts on showing that this map commutes with the Kashiwara operator f˜i . Other parts of the proof are similar or easy. Let us first review the action of the Kashiwara operator f˜i on a path element p = · · · ⊗ p(j ) ⊗ · · · ⊗ p(1) ⊗ p(0) ∈ P(k ). (1) Under each p(j ), write εi (p(j ))-many 1 followed by ϕi (p(j ))-many 0. (2) From the (half-)infinite list of 0 and 1, successively cancel out each (0, 1) pair to obtain a finite sequence of 1 followed by some 0 (reading from left to right). (3) Let f˜i act on the p(j ) corresponding to the left-most 0 remaining. Set it to zero if no 0 remains. This is quite similar to the action of f˜i onYoung walls defined in Sect. 4.3. Now, recalling the definition of and Theorem 3.8, we find that, to prove Theorem 5.4, it suffices to prove the following two lemmas. Lemma 5.5. A Kashiwara operator acts on the j th tensor component of a path p if and only if it acts on the j th column of the Young wall (p). We also have f˜i (p) = 0 (or e˜i (p) = 0) for some path p if and only if f˜i ( (p)) = 0 (respectively, e˜i ( (p)) = 0).

(1)


127

Lemma 5.6. The set Y(k ) ∪ {0} is closed under the action of Kashiwara operators. Before giving the proofs for these lemmas, we shall illustrate the signature cancellation comparison between path and Young wall with two examples. Example 5.7. Consider the following part of a path: · · · ⊗ (0, . . . , 0|0, . . . , 0, 2) ⊗ (2, 0, . . . , 0|0, . . . , 0) ⊗ · · · . Suppose we are dealing with i = 0 case. The signature that should be under the left element (0, . . . , 0, 2) is 00 and that for the right element (2, 0, . . . , 0) is 11. After cancelling out the (0, 1) pairs, we are left with nothing. Now consider columns of the reduced proper Young wall which corresponds to this path under the map .

···

1

0 0 1

or

···

···

1 . . . n . . .

0 0 1 . . . n . . .

0 0 1

0 0 1

···

The two drawings are δ-shifts of each other and it does not matter which of the two drawings we use. When dealing with the case i = 0, under the left column we would write 0 and under the right column we would write 1. After (0, 1)-pair cancellation, we are again left with nothing. The signatures under path description and the Young wall description agree after cancelling out (0, 1) pairs. We give one more example which is a bit more complicated. Example 5.8. Consider the following part of a path and columns of the reduced proper Young wall which corresponds to this path under the map . · · · ⊗ (0, . . . , 0|0, . . . , 0) ⊗ (0, . . . , 0|0, . . . , 0) ⊗ · · · ···

0 1

0 1

···

As before, the reader is free to use a δ-shift of this Young wall. When dealing with f˜0 , the signatures to be written under them are given in the following table: path left ε ϕ 1 0

right ε ϕ 1 0

Young wall left right ε ϕ ε ϕ ? · · ?

The first question mark in the above Young wall table signifies that the number of 1 that should be written there depends on the column that sits to left of the left column. Likewise, the second question mark is to signify that the number of 0 to be written depends on the column that comes to its right. The two dots imply that no 0 and 1, respectively, should be written there.

128


So in this case, we do not know the complete signature to be written under the Young wall columns. Hence, a straightforward comparison of signatures after cancellations of (0, 1) pairs is not possible. But still, we can verify that what is left of the left-ϕ and right-ε signatures, after the (0, 1)-pair cancellation, is the same for the path and Young wall. In this example, they both amount to nothing. Proof of Lemma 5.5. It suffices to check that, for all possible left-right pairs of perfect crystal elements and their correspondingYoung wall elements, what remains after (0, 1)cancellation of left-ϕ and right-ε signatures agree. (The right-most column may be dealt with in a similar way.) Let us deal with the i = 0 case first. The following notation will be used to denote various columns of Young walls. k

k

column :

0 0 1

0 0 1

0 0 1

0 0 1

0 0 1

notation :

0

00

10

11

1

Here, the top k-blocks can be anything that comes between the supporting 1-block and the covering 2-block (inclusive), but may not be the covering 1-block. Columns that are related (Definition 3.6) are denoted by the same notation. Notice that we have taken the signature of the corresponding perfect crystal element for the notation of each Young wall columns. The following table lists all left-right pairs for which the signatures to be written under the path description and Young wall description are not trivially the same. The signatures in the table body are what should be written as the left-ϕ and right-ε signatures under the two Young wall columns. right

10

11

1

0

·

1

· / 01

00

0

01

0

10

·

1

·

left

The case when the left column is 0 and the right column is 1 breaks up into two cases. Depending on the two top k-blocks for the left and right columns, which may be distinct, the left-ϕ and the right-ε signatures to be placed under the two columns could be either nothing or 01. In the latter case the signature agrees trivially with that of the path description. We can easily see that the signatures agree with that of the corresponding path description in all the above cases after (0, 1)-pair cancellations. For all other possible left-right pairs not covered in this table, the left-ϕ and right-ε signatures to be placed under the Young wall columns agree exactly with the corresponding path signatures, that is, even before the (0, 1)-pair cancellations. For 0 < i < n, the following notation will be used.

(1)


129

i

i

i

(k = i, i−1)

i+1

(k = i, i+1)

i+1

(k = i, i−1)

i+1

(k = i, i+1)

i

n

i

n

i

n

i

i+1

i

i−1

i

i+1

i

i

i−1

k

i+1

i

i+1

i+1

k

0

00

00

00

i

i−1

i

i−1

i i

i−1

k

0

10

10

10

k

1

11

11

11

1

Again, we have used notation that reflect the signature of the corresponding perfect crystal element. Underlines and overlines show whether the i-blocks and i-slots of the slice are in supporting or covering positions. The following table gives the Young wall signatures for all the nontrivial pairs. right

10

10

1

11

11

0

·

· / 01

· / 01

1

1 / 011

00

0

0

0

·

01

00

0

0

0

0 / 001

01

00

0

0

· / 01

left

10

11

1

01

01

0 / 001

0

01

·

0

·

1 / 011

1

· / 01

10

·

·

10

·

·

·

·

·

10

·

1

1

· / 01

1

1

1

· / 01

1

1

·

The blank slots are where the signatures agree trivially. As before, many of the cases break up into subcases, depending on the top k-blocks for 0, 0, 1, and 1. For the remaining i = n case, the following notation will be used. n n−1

n n

n

n

n−1

n−1

n−1

k

0

n−1 k

00

10

11

1

Here, the top k-blocks for 0 and 1 may be taken to be anything except for the n-block and the supporting (n − 1)-block. It could also be two 0-blocks. The following table lists all nontrivial pairs. right

10

11

1

0

·

1

· / 01

00

0

01

0

10

·

1

·

left

This completes the proof of Lemma 5.5.

130


Proof of Lemma 5.6. Suppose that Y(k ) ∪ {0} is not closed under f˜i . That is, there exists some Y ∈ Y(k ) for which f˜i Y ∈ Y(k ) ∪ {0}. We assume that f˜i has acted on the j th column of Y and set p = −1 (Y ). Then, by Lemma 5.5 the action of f˜i on p would also have been on the j th tensor component of p. Since f˜i Y ∈ F(k ), we may remove δ’s finitely many times from f˜i Y to obtain a reduce properYoung wall Y . The number of δ’s removed is nonzero since f˜i Y ∈ Y(k ). Note that f˜i (p) = −1 (Y ). Let us apply e˜i to both f˜i (p) and Y . We have p = e˜i (f˜i (p)) and the action of e˜i on f˜i (p) would have been on the j th tensor component. By Lemma 5.5, the action of e˜i of Y will also be on the j th column. Recall that we started out with a reduced proper Young wall Y , added a block to the th j column to obtain f˜i Y , removed finitely many δ’s to obtain Y , and, finally, removed a block from the j th column of Y to obtain e˜i Y . Hence the proper Young wall e˜i Y may be obtained from the reduced proper Young wall Y by removing finitely many δ’s. We may now remove finitely many δ’s from e˜i Y to obtain a reduced proper Young wall Y which also corresponds to p = −1 (Y ) under the map . This contradicts the fact that only one element of Y(k ) corresponds to a given p under the map . This completes the proof of this lemma. Since the proofs of Lemmas 5.5 and 5.6 are complete, Theorem 5.4 has been proved. We close this paper with an example. (1)

Example 5.9. Following is the top part of the crystal graph Y(0 ) for Uq (C2 ).

2 0

0

0 0

1

1 0 0

2 1 0 0

1

1

2 1 0 0

1 2 1 0 0 0 2 1 0 0 0

1

1 0 0 0 0

0

2 0

1 2 1 0 0 0 0

1

2 0

0

1 2 1 0 0 0

2

2 1 0 0 0 2 1 0 0 0 0

1

1 2 1 0 0 0 0

···

1 2 1 1 0 0 0 0

···

1 2 1 0 0 0

···

0

2 1

2 1 0 0 0 0 2 1 1 0 0 0 0

···

···

References (1)

1. Hong, J., Kang, S.-J.: Crystal graphs for basic representations of the quantum affine algebra Uq (C2 ). In: Representations and quantizations (Shanghai, 1998), Beijing: China High. Educ. Press, 2000, pp. 213–227 2. Hong, J., Kang, S.-J.: Introduction to Quantum Groups and Crystal Bases. Graduate Studies in Mathematics 42, Providence, RI: Am. Math. Soc., 2002 3. Jantzen, J.C.: Lectures on Quantum Groups. Graduate Studies in Mathematics 6, Providence, RI: Am. Math. Soc., 1996

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sl(n)) at 4. Jimbo, M., Misra, K.C., Miwa, T., Okado, M.: Combinatorics of representations of Uq ( q = 0. Commun. Math. Phys. 136(3), 543–566 (1991) 5. Kang, S.-J.: Crystal bases for quantum affine algebras and combinatorics of Young walls. Proc. Lond. Math. Soc. (3) 86, 29–69 (2003) 6. Kang, S.-J., Kashiwara, M., Misra, K.C.: Crystal bases of Verma modules for quantum affine Lie algebras. Compositio Math. 92(3), 299–325 (1994) 7. Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Affine crystals and vertex models. In: Infinite analysis, Part A, B (Kyoto, 1991), River Edge NJ: World Sci. Publishing, 1992, pp. 449–484 8. Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Perfect crystals of quantum affine Lie algebras. Duke Math. J. 68(3), 499–607 (1992) 9. Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Vertex models and crystals. C. R. Acad. Sci. Paris Sér. I Math. 315(4), 375–380 (1992) 10. Kashiwara, M.: On crystal bases of the q-analogue of universal enveloping algebras. Duke Math. J. 63(2), 465–516 (1991) 11. Kashiwara, M., Nakashima, T.: Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra 165(2), 295–345 (1994) 12. Kuniba, A., Misra, K.C., Okado, M., Takagi, T., Uchiyama, J.: Crystals for Demazure modules of classical affine Lie algebras. J. Algebra 208(1), 185–215 (1998) 13. Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3(2), 447–498 (1990) 14. Misra, K., Miwa, T.: Crystal base for the basic representation of Uq ( sl(n)). Commun. Math. Phys. 134(1), 79–88 (1990)

Communicated by L. Takhtajan


Communications in


A New Class of Obstructions to the Smoothness of Null Infinity Juan Antonio Valiente Kroon Max Planck Institut für Gravitationphysik, Albert Einstein Institut, Am Mühlenberg 1, 14476 Golm, Germany. E-mail: [email protected] Received: 12 November 2002 / Accepted: 14 July 2003 Published online: 5 November 2003 – © Springer-Verlag 2003

Abstract: Expansions of the gravitational field arising from the development of asymptotically Euclidean, time symmetric, conformally flat initial data are calculated in a neighbourhood of spatial and null infinities up to order 6. To this end a certain representation of spatial infinity as a cylinder is used. This setup is based on the properties of conformal geodesics. It is found that these expansions suggest that null infinity has to be non-smooth unless the Newman-Penrose constants of the spacetime, and some other higher order quantities of the spacetime vanish. As a consequence of these results it is conjectured that similar conditions occur if one were to take the expansions to even higher orders. Furthermore, the smoothness conditions obtained suggest that if time symmetric initial data which are conformally flat in a neighbourhood of spatial infinity yield a smooth null infinity, then the initial data must in fact be Schwarzschildean around spatial infinity.

1. Introduction Penrose introduced the seminal idea that the gravitational field of isolated systems can be conveniently described by means of the notion of asymptotic simplicity [21] 1 . Central to the concept of asymptotic simplicity is the idea – expectation – that the conformal boundary of the spacetime – null infinity, I – should possess a smooth differentiable structure. This approach to the description of isolated bodies in General Relativity is usually known as Penrose’s proposal – see e.g. [17, 18]. Static and stationary spacetimes have been shown to be (weakly) asymptotically simple, with a smooth null infinity [6]. However, the main purpose behind introducing the concept of asymptotic simplicity is to provide a suitable framework for the discussion of radiation. In spite of its elegance and Current address: Institut f¨ ur Theoretische Physik der Universität Wien, Boltzmanngasse 5, 1180 Wien, Austria. 1 It will be assumed that the reader is familiar with the ideas of the so-called conformal framework to describe the properties of isolated bodies and the concept of asymptotic flatness. For a recent review, the reader is remitted to [18].

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aesthetical appeal, Penrose’s proposal is of little use if one is not able to prove that there exists a big family of non-trivial (in the radiative sense) asymptotically simple solutions to the Einstein Field Equations. A programme aimed to investigate the existence of such solutions, and provide a conclusive “answer” to Penrose’s proposal has been started by Friedrich – see e.g. [10–16]. His strategy is based on the use of the so-called Conformal Field Equations, which allow us to work and prove existence statements directly in the conformally rescaled, “unphysical” spacetime. Along these lines, Friedrich has been able to prove a semiglobal existence result that ensures that hyperboloidal initial data close to Minkowski data yields an asymptotically simple development which includes the point i + (future timelike infinity) [14]. Recently, Chru´sciel & Delay [2] – retaking an old idea by Cutler & Wald [5] – using a refined version of some initial data set constructed by Corvino [4] have been able to prove the existence of a big class of non-trivial (radiative) asymptotically simple spacetimes. Corvino’s initial data are constructed so that they are Schwarzschildean in a neighbourhood of spatial infinity, i 0 . This means that the radiation content in the spacetime arising from the development of the data is somehow special. This can be seen directly from the fact that the Newman-Penrose constants of the spacetime are zero [8, 26]. Chru´sciel & Delay’s result is no doubt very important. However, it is not as general as one would like. It has been suspected for a long time now that the region of spacetime where null infinity and spatial infinity meet is somehow problematic – see e.g. [22]. From the analysis of the hyperboloidal initial value problem it turns out that the smoothness of null infinity is preserved by the evolution if smooth data sufficiently close to Minkowski data are prescribed. The latter indicates that somehow the “decision” of having a smooth structure at null infinity is made in an arbitrarily small neighbourhood of spatial infinity. In some sense, Corvino’s data avoids all the intricacies and complications of this region of spacetime by setting the asymptotic end in the simplest way which is consistent with the presence of a non-vanishing ADM mass on the initial hypersurface. In connection with this, Friedrich [16] has performed a detailed first analysis of the behaviour of the gravitational field arising from asymptotically Euclidean, time symmetric initial data in the region where null infinity “touches” spatial infinity. By means of a novel representation of spatial infinity in which the point i 0 of the standard conformal picture is blown up to a cylinder I – the cylinder at spatial infinity – a certain regularity condition on the CottonBach tensor and symmetrised higher order derivatives of it has been obtained. The hope was that this regularity condition would ensure the smoothness of null infinity, at least in the region close to i 0 and I .A subsequent analysis by Friedrich & Kánnár [19] of the first orders of some expansions that can be obtained by evaluating the Conformal Field Equations at I , lead to conjecture that Friedrich’s regularity condition is the only condition one has to impose on time symmetric, initial data possessing an analytic compactification in order to obtain a development with smooth null infinity [18]. More precisely, Conjecture (Friedrich, 2002). There exists an integer k∗ > 0 such that for given k ≥ k∗ the time evolution of an asymptotically Euclidean, time symmetric, conformally smooth initial data set admits a conformal extension to null infinity of class C k near spacelike infinity, if the Cotton-Bach spinor 2 satisfies the condition, D(as bs · · · Da1 b1 babcd) (i) = 0, 2

s = 0, 1, . . . , s∗

The Cotton-Bach tensor Bij k is related to the Cotton-Bach spinor via: Bij k → babce df + babdf ce .

This correspondence is carried out by the Infeld-van der Waerden symbols.

New Class of Obstructions to the Smoothness of Null Infinity

135

for a certain integer s∗ = s∗ (k). If the extension is of class C ∞ then the condition should hold to all orders. The Cotton-Bach spinor alluded to in the previous conjecture can be regarded as the 3-dimensional analogue of the Weyl spinor in the sense that it characterises locally the conformal flatness of a manifold – i.e. it vanishes in a conformally flat region of a 3-dimensional manifold. The condition provided in Friedrich’s conjecture can be shown to be conformally invariant and thus it is an asymptotic condition on the initial data. More remarkably, it it can be shown to be satisfied by asymptotic Euclidean static data without restricting the multipole structure [15]. The objective of this paper is to provide a further insight into the conjecture above. It will turn out that the conjecture, as it stands, is false. In order to see why this is the case, the expansions of Kánnár & Friedrich will be carried to an even higher order. This requires the implementation of the Conformal Field Equations on a computer algebra system (Maple V). It should be emphasized that despite the use of the computer to perform the expansions, the results here presented are exact up to the order carried out. In order to simplify our discussion, the analysis will be restricted to developments of time symmetric initial data which are conformally flat near infinity. This class of data satisfies the regularity condition of the conjecture in the simplest possible way while still providing a big enough family of data. The time symmetry requirement stems from the fact that Friedrich’s analysis has only yet been carried for this class of data. A similar analysis of initial data with non-vanishing second fundamental form lies still in the future. However, some first steps have already been carried out [7]. The conformal flatness of the data ensures that the initial data satisfies the regularity condition trivially. Again, the construction of non-conformally flat data satisfying Friedrich’s regularity condition is a non-trivial endeavour whose undertaking will be left for future studies. In the light of the results here presented, it turns out that the restriction to the class of conformally flat data is not a drawback. Furthermore, it is not hard to guess how the results could generalise in the case of general time symmetric data. The principal result of our investigation is the following, Theorem (Main theorem). Necessary conditions for the development of initial data which are time symmetric, conformally flat in a neighbourhood Ba (i) of (spatial) infinity to be smooth at the intersection of null infinity and spatial infinity are that the Newman(5) Penrose constants Gk , k = 0, . . . , 4 and the higher order Newman-Penrose constants, (6) Gk , k = 0, . . . , 6 vanish. A more precise formulation of the theorem, including the definition of the Newman-Penrose constants and the higher order Newman-Penrose constants in terms of the initial data will be given in the main text. It is just mentioned that the Newman-Penrose constants are a set of absolutely conserved quantities along null infinity. They are expressed in terms of integrals on cuts of null infinity and their value is independent of the cut considered. For asymptotically simple spacetimes containing the point i + in their conformal completion – e.g. those spacetimes arising from Friedrich’s existence result of hyperboloidal data [14] – , the constants can be shown to correspond to the value of the rescaled Weyl tensor at i + . It is noted that the conditions appearing in the main theorem are fulfilled by the Schwarzschild initial data. The theorem has been obtained from the analysis, up to order p = 6, of expansions constructed from the solutions of the transport equations induced by the conformal field equations upon evaluation on the cylinder at spatial infinity. From the evidence provided by the equations it is not unreasonable to conjecture that similar

136

J.A.Valiente Kroon

conditions arise if one were to obtain expansions to even higher orders. In [16] it was shown that the whole setup of the cylinder at spatial infinity is completely regular for Schwarzschildean data. Thus, the hypothetic new conditions must be satisfied by the Schwarzschild initial data. It is on these grounds that the following conjecture is put forward: Conjecture (New conjecture). If an initial data set which is time symmetric and conformally flat in a neighbourhood Ba (i) of the point i yields a development with a smooth null infinity, then the initial data are in fact Schwarzschildean in Ba (i). Again, a more technical version of the conjecture is given in the main text. This conjecture can be understood as some kind of rigidity result – i.e. the only asymptotically simple spacetimes which arise from data which is conformally flat near infinity are those of the Chru´sciel-Delay type. For example, the Brill-Lindquist data – which is conformally flat – do not satisfy the conditions stated in the main theorem, and thus it is not asymptotically simple – see the main text for the details. However, one could always use the Corvino-Chru´sciel-Delay techniques to produce a modified Brill-Lindquist data whose development possesses a smooth null infinity. The article is structured as follows: in Sect. 2 a brief summary of the description of spacetime in the neighbourhood of spatial and null infinities in terms of the cylinder at spatial infinity is given. This digest has the intention of fixing the notation and conventions to be used in the calculations described in the present article. Particular attention is paid to the spatial 2-spinor formalism and to the expansions of functions on S 3 in terms of unitary representations of SU (2, C). The reader is, in any case, remitted to [16] for a more extensive discussion. In Sect. 3, the conformal field equations written in the conformal geodesic gauge are discussed. The initial data for the latter in the case of an asymptotically Euclidean, time symmetric, conformally flat initial hypersurface are described. The transport equations implied by the conformal field equations on the cylinder at spatial infinity are also introduced. Section 4 contains the new results to be presented in the article. Here a description of the solution of the transport equations is done. Due to the large size of the expressions involved, the description will be focused on what we believe are the most relevant features of the solutions. However, it should be emphasized that everything has been explicitly calculated. The main conclusions extracted from the calculations is presented as our main theorem. In order to understand the meaning of the conditions presented in the main theorem, the Schwarzschild solution is discussed in this context, and a conjecture is formulated. 2. Spacetime in a Neighbourhood of Spatial Infinity Let (M, gµν ) be a vacuum spacetime arising as the development of asymptotically Euclidean, time symmetric, conformally analytic initial data ( S, hαβ ). Later, we will restrict the class of initial data sets under discussion to those conformally flat around infinity. Assume for simplicity that S possesses only one asymptotic end. Let i be the infinity corresponding to that end. The point i is obtained by conformally compactifying the initial hypersurface S, with an analytic conformal factor which can be obtained from solving the time symmetric constraint equations. The compact 3 dimensional manifold obtained in this way will be denoted by S, and the conformally rescaled 3-metric by hαβ . Assume that the 3-metric hαβ is analytic in an open ball Ba (i) of S with radius a centered on i. Let ρ denote the geodesic distance along geodesics starting at i. The radius a of the ball Ba (i) is chosen such that Ba (i) is geodesically convex.


137

∩M be the domain of influence of the ball Ba (i) ∩ S. Intuitively, one Now, let N expects N to cover a region of spacetime “close to null and spatial infinities”. In reference [16] it has been shown that once the time symmetric constraint equations have been solved, a certain gauge based on the properties of conformal geodesics can be introduced. Let τ be the parameter along these curves. This gauge has the property of to producing a conformal factor which can be in turn used to rescale the region N obtain a “finite representation” N , of spacetime in a neighbourhood of spatial and null infinities. The relevant conformal factor is given by, κ2 = κ −1 1 − τ 2 2 , (1) ω where ω is given by, 2 , ω= √ |Da D a |

(2)

and κ is a smooth function depending on ρ and the “angular coordinates”– see below – such that κ = κ ρ with κ (i) = 1. The function κ contains the remaining piece of conformal freedom in our setting. Throughout this work, space 2-spinors will be systematically used. In order to avoid problems with vanishing frame vectors on surfaces diffeomorphic to spheres, our discussion will be carried out not on N but on a subbundle Ca,κ of the frame bundle over N . The subbundle Ca,κ can be shown to be a 5-dimensional submanifold of R × R × SU (2, C) with structure group U (1). More precisely, we define Ca,κ to be given by, ω ω Ca,κ = (τ, ρ, t) ∈ R × R × SU (2, C)| 0 ≤ ρ < a, − ≤ τ ≤ . (3) κ κ The projection π of Ca,κ into N corresponds to the Hopf map SU (2, C) → SU (2, C)/U (1) ≈ S 2 . Scalar fields and tensorial fields on N are lifted to Ca,κ . Their “angular” dependence will be then given in terms of functions of t ∈ SU (2, C). The manifold Ca,κ has the following important submanifolds, I = {(τ, ρ, t) ∈ Ca,κ | ρ = 0, |τ | < 1}, I ± = {(τ, ρ, t) ∈ Ca,κ | ρ = 0, τ = ±1}, ω I ± = (τ, ρ, t) ∈ Ca,κ | ρ > 0, τ = ± . κ

(4a) (4b) (4c)

2.1. Space 2-spinors. Consider the antisymmetric spinors ab , ab , a, b = 0, 1. These satisfy 01 = 1, 01 = 1. Let τ aa denote the tangent vector to the conformal geodesics parametrised by τ . We set,

τ aa = 0 a 0 a + 1 a 1 a .

(5)

Differential operators X+ , X− and X on SU (2, C) can be introduced by considering a basis of the Lie algebra of SU(2,C) and then looking at (complex) linear combinations of the real left invariant vectors fields generated by the basis of the Lie algebra on SU (2, C). In particular, the vector field X is chosen so that it generates U (1). The field X± , X satisfy the following commutation relations: [X, X+ ] = 2X+ ,

[X, X− ] = −2X− ,

[X+ , X− ] = −X.

(6)

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J.A.Valiente Kroon

With the help of X± one can construct the following (frame) spinor fields, + − 0 1 caa = caa ∂τ + caa ∂ρ + caa X+ + caa X− ,

(7)

on Ca,κ .3 √ The use of a space spinor formalism based on the vector field 2∂τ = τ aa caa allows to perform our whole discussion in terms of quantities without primed indices. Accordingly, we write 1 caa = √ τaa ∂τ − τ ba cab 2

(8)

b

0 ∂ + c1 ∂ + c+ X + c− X . The connection is represented with cab = τ(a cb)b = cab τ ab ρ ab + ab − by coefficients abcd which can be decomposed in the form,

1 1 abcd = √ ξabcd − χ(ab)cd − ab fcd , 2 2

(9)

where the fields entering in the decomposition possess the following symmetries: χabcd = χab(cd) , ξabcd = ξ(ab)(cd) , fab = f(ab) . The curvature will be described by the rescaled conformal Weyl spinor φabcd = φ(abcd) , and by the spinor field abcd = ab(cd) which encodes information relative to the Ricci part of the curvature. For the purpose of writing g the field equations it will be customary to consider it decomposed in terms of g cd and (ab)cd . For latter use it is noted that an arbitrary four indices spinor Xabcd can be writi ten in terms of the “elementary spinors” abcd with i = 0, . . . , 4, ac xbd + bd xac , ac ybd + bd yac , ac zbd + bd zac , and habcd where, xab =

√ 0 1 2(a b) ,

1 yab = − √ a 1 b 1 , 2

1 zab = √ a 0 b 0 , 2

(10)

and, (e

f

g

i abcd = (a b c d)

h)i

,

habcd = −a(c d)b .

(11)

The notation (abcd)i means that the indices are to be symmetrised and then i of them set to 1. We write, 0 1 2 3 4 Xabcd = X0 abcd + X1 abcd + X2 abcd + X3 abcd + X4 abcd +Xx (ac xbd + bd xac ) + Xy (ac ybd + bd yac ) +Xz (ac zbd + bd zac ) + Xh habcd .

(12)

3 A knowledge of the spinorial fields is equivalent to a knowledge of the metric. The tensorial expres caa , where sions for the frame can be recovered using the Infeld-van der Waerden symbols via eµˆ = σµaa ˆ the indices with caret are frame indices. The metric tensor is then recovered from the completeness relation for the frame.


139

2.2. Expansions of functions on Ca,κ . In order to obtain our expansions of the gravitational field around spatial infinity, we will require to decompose the functions arising into their diverse spherical (harmonic) sectors. It is recalled that any function real analytic complex value function f on SU (2, C) can be expanded in terms of some functions √ m + 1 Tm kj forming a complete set in L2 (µ, SU (2, C)) where µ is the standard Haar measure in SU (2, C). One has, f (t) =

m m ∞

fm,k,j Tm kj .

m=0 j =0 k=0

The functions Tm kj can be shown to be related with the standard spin-weighted spherical harmonics. The operators X± and X introduced in the previous section can be seen to yield, upon application to the Tm kj functions the following, X+ Tm kj = βm,j Tm kj −1 , X− Tm kj = −βm,j +1 Tm kj +1 ,

with βm,j

= j (m − j + 1).

(13a) (13b)

A function f on SU (2, C) is said to have spin weight s if Xf = 2sf . This definition can be readily extended to functions on Ca,κ . As it will be seen later, all the quantities we will work with will have a well defined spin weight. Let f be an analytic function f on Ca,κ with an integer spin weight s. Now, consider a spinorial symmetric analytic function on Ca,κ , λa1 ···a2r with essential components λj = λ(a1 ,··· a2r )j , 0 ≤ j ≤ 2r, of spin weight s = r − j . Then, the components of the function will possess expansions of the form, λj =

∞

λj,p ρ p ,

(14)

p=0

where the coefficients λj,p can in turn be decomposed in terms of the functions Tm kj , as λj,p =

q(p)

2q

λj,p;2q,k T2q

k q−r+j ,

(15)

q=|r−j | k=0

where 0 ≤ |r − j | ≤ q(p) ≤ ∞. An expansion of the latter type will be referred to as an expansion of type q(p). The conformal field equations are nonlinear. Thus, when expanding them, one finds products of T -functions. These products can in turn be reexpressed as a linear combination of T ’s. More precisely, one has: Lemma 1. Multiplying T functions. The following holds: T2n1 k1l1 × T2n2 k2l2 =

n 1 +n2

(−1)n+n1 +n2 C(n1 , n1 − l1 ; n2 , n2 − l2 ; n, n1 + n2 − l1 − l2 )

n=q0

×C(n1 , n1 − k1 ; n2 , n2 − k2 ; n, n1 + n2 − k1 − k2 ) ×T2n n+k1 +k2 −n1 −n2n+l1 +l2 −n1 −n2 ,

(16)

140

J.A.Valiente Kroon

where q0 = max{|n1 −n2 |, n1 +n2 −k1 −k2 , n1 +n2 −l1 −l2 }, and C(l1 , m1 ; l2 , m2 ; l, m) denotes the standard Clebsch-Gordan coefficients of SU (2, C).4 3. The Conformal Evolution Equations Using the conformal geodesic gauge and the 2-spinor decomposition, it can be shown that the extended conformal field equations given in [16] imply the following evolution g µ equations for the unknowns v = (cab , ξabcd , fab , χ(ab)cd , (ab)cd , g cd ), µ = 0, 1, ±, ef

0 0 ∂τ cab = −χ(ab) cef − fab , α = ∂τ cab

ef α −χ(ab) cef ,

(17a) (17b)

1 ef ∂τ ξabcd = −χ(ab) ξef cd + √ (ac χ(bd)ef + bd χ(ac)ef )f ef 2 √ 1 f f e − 2χ(ab)(c fd)e − (ac f bd + bd f ac ) − i µabcd , (17c) 2 1 ef f ∂τ fab = −χ(ab) fef + √ f ab , (17d) 2 ef

∂τ χ(ab)cd = −χ(ab) χef cd − (cd)ab + ηabcd , √ ef ∂τ (ab)cd = −χ(ab) (ab)ef − ∂τ ηabcd + i 2d e(a µb)cde , √ g ef g ∂τ g ab = −χ(ab) g ef + 2d ef ηabef ,

(17e) (17f) (17g)

where 1 (φabcd + τa a τb b τc c τd d φ a b c d ), 2 i = − (φabcd − τa a τb b τc c τd d φ a b c d ), 2

ηabcd = µabcd

(18)

denoting respectively the electric and magnetic part of φabcd with respect to the field ∂τ . The quantities , ∂τ and dab , given by formulae (1) and (23) are known directly from the initial data. Thus, Eqs. (17a)–(17g) are essentially ordinary differential equations for the components of the vector v. The most important part of the propagation equations corresponds to the evolution equations for the spinor φabcd derived from the Bianchi identities, the Bianchi propagation equations: √ 0 0 α α )∂τ φ0 + 2c00 ∂τ φ1 − 2c01 ∂α φ0 + 2c00 ∂α φ 1 ( 2 − 2c01 = (20011 − 81010 )φ0 + (40001 + 81000 )φ1 − 60000 φ2 , 4

Some other used notations in the physics literature are: C(l1 , m1 ; l2 , m2 ; l, m) ≡< l1 , l2 ; m1 , m2 |l, m >, ≡ C(l1 , l2 , l|m1 , m2 , m).

.

(19a)


141

√

0 0 α α 2∂τ φ1 − c11 ∂τ φ0 + c00 ∂τ φ2 − c11 ∂α φ0 + c00 ∂ α φ2 = −(41110 + f11 )φ0 + (20011 + 41100 − 2f01 )φ1 + 3f00 φ2 − 20000 φ3 ,(19b) √ 0 0 α α 2∂τ φ2 − c11 ∂τ φ1 + c00 ∂τ φ3 − c11 ∂α φ1 + c00 ∂α φ 3 = −1111 φ0 − 2(1101 + f11 )φ1 + 3(0011 + 1100 )φ2 −2(0001 − f00 )φ3 − 0000 φ4 , (19c) √ 0 0 α α 2∂τ φ3 − c11 ∂τ φ2 + c00 ∂τ φ4 − c11 ∂α φ2 + c00 ∂α φ4 = −21111 φ1 − 3f11 φ2 + (21100 + 40011 + 2f01 )φ3 − (40001 − f00 )φ4 ,(19d) √ 0 0 α α ( 2 + 2c01 )∂τ φ4 − 2c11 ∂τ φ3 + 2c01 ∂α φ4 − 2c11 ∂α φ 3 = −61111 φ2 + (41110 + 80111 )φ3 + (21100 − 80101 )φ4 . (19e)

To the latter we add a set of three equations, also implied by the Bianchi identities which we refer as to the Bianchi constraint equations, 0 0 0 α α α c11 ∂τ φ0 − 2c01 ∂τ φ1 + c00 ∂τ φ2 + c11 ∂α φ0 − 2c01 ∂α φ1 + c00 ∂α φ 2 = −(2(01)11 − 41110 )φ0 + (20011 − 4(01)01 − 41100 )φ1 +6(01)00 φ2 − 20000 φ3 , 0 ∂τ φ 1 c11

0 − 2c01 ∂τ φ 2

0 + c00 ∂τ φ 3

α + c11 ∂ α φ1

α − 2c01 ∂ α φ2

= 1111 φ0 − (4(01)11 − 21101 )φ1 + 3(0011 − 1100 )φ2 −(20001 − 4(01)00 )φ3 − 0000 φ4 , 0 ∂τ φ 2 c11

0 − 2c01 ∂ τ φ3

0 + c00 ∂τ φ 4

α + c11 ∂α φ 2

α − 2c01 ∂ α φ3

= 21111 φ1 − 6(01)11 φ2 + (40011 + 4(01)01 −21100 )φ3 − (40001 − 2(01)00 )φ4 .

(20a)

α + c00 ∂α φ 3

(20b)

α + c00 ∂α φ 4

(20c)

3.1. The initial data. As pointed out in the introduction, only asymptotically Euclidean, time symmetric, analytically conformally flat initial data will be considered in our discussion. A number of simplifications arise under these assumptions. In particular, around i, the conformal factor of the initial hypersurface can be written as, =

ρ2 , (1 + ρW )2

(21)

where W (i) = m/2, m the ADM mass of the initial hypersurface. The function W satisfies the Yamabe equation, which under our assumptions reduces to the Laplace equation. Therefore W is harmonic, and thus can be written as ∞

W =

m 1 k + wp,2p,k T2p p ρ p , 2 p! 2p

(22)

p=1 k=0

where the coefficients wp,2p,k , p = 1, 2, . . . , k = 0, . . . , 2p are complex numbers satisfying the regularity condition w p,2p,k = (−1)i+k wp,2p,2p−k , so that W is a real valued function. As mentioned in Sect. 2, a crucial property of our setup based on the properties of conformal geodesics is that it renders a conformal factor – see formula (1) – for the region of spacetime under discussion. Furthermore, solving the conformal geodesic

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J.A.Valiente Kroon

equations also yields a 1-form dab , which appears in the propagation equations (17f) and (17g). Under our assumptions of time symmetry and conformal flatness it is given by xab − ρ 2 Dab W dab = 2ρ . (23) (1 + ρW )3 Once the function κ of Sect. 2 has been chosen, the initial data for the conformal propagation equations (17a)–(17g) is given by abcd = −

κ2 D(ab Dcd) ,

(24a)

κ3 D(ab Dcd) , 2 0 1 cab = 0, cab = κxab , κ κ c+ = zab , c− = yab , ρ ρ

√ κ 1 ξabcd = 2 (ac xbd + bd xac ) − (ac Dbd κ + bd Dac κ) , 2ρ 2κ χ(ab)cd = 0, fab = Dab κ, φabcd =

(24b) (24c) (24d) (24e) (24f)

where Dab , the spinorial covariant derivative of the initial hypersurface S is given by Dab µcd = xab ∂ρ µcd +

1 1 zab X+ µcd + yab X− µcd − γab ec µed − γab ed µec , ρ ρ

(25)

where the flat connection coefficients γabcd are given by γabcd =

1 (ac xbd + bd xac ), 2ρ

(26)

for a given differentiable spinorial function µab . 3.2. The transport equations. Equations (17a)–(17g) can be concisely written in the form ∂τ v = Kv + Q(v, v) + Lφ,

(27)

where K and Q are respectively a linear and a quadratic function with constant coefficients, whereas L is a linear function depending on the coordinates via , ∂τ and dab . For the Bianchi propagation equations (19a)–(19e) one can write √ µ 2E∂τ φ + Aab cab ∂µ φ = B(abcd )φ, (28) µ

where now, E denotes the (5 × 5) unit matrix, Aab cab are (5 × 5) matrices, and B(abcd ) is a linear (5 × 5)-matrix valued function of the connection coefficients abcd . In the sequel, given an unknown u we will write, u(0) = u|I . The objects , ∂τ and dab from which L is constructed, vanish on I . Thus, L(0) = L|I = 0, and consequently Eqs. (27) and (28) decouple from each other. The system of equations for the v unknowns, Eq. (27), turns out to be an interior system upon evaluation on the cylinder


143

at spatial infinity. Its initial data can be read from the restriction of the initial data (24a)(24f) to I . It can be seen that this restriction, irrespectively of the choice of the function κ coincides with the initial data of Minkowski spacetime. With this information in hand, the system for v (0) can be readily solved yielding, (0)

abcd = 0, 0 (0) ) (cab

(0)

(0)

χ(ab)cd = 0,

= −τ xab ,

fab = 0,

1 (0) (cab )

= 0,

(0)

ξabcd = 0,

− (0) (cab )

= yab ,

(29a) + (0) (cab )

= zab . (29b)

1 in the system (28) satisfies, From this solution it follows that the matrix Aab cab 1 Aab cab |I = 0.

(30)

This particular result will be crucial in our later discussion. As a consequence of it, the system (28) implies another interior system on I , as no ρ-derivatives will arise upon evaluation on I . It can be solved giving, (0)

2 . φabcd = −6mabcd

(31)

Because of the fact that the whole system of conformal field equations reduces to an interior system on I – something that does not happen with normal characteristics – we call it a total characteristic. The idea of interior systems previously discussed can be generalised by applying p times ∂ρ to Eqs. (27) and (28) and then evaluating on I . In this way one obtains a hierarchy of interior systems for the unknowns u(p) = ∂ρ u|I . These quantities can be used to construct formal expansions of the form, u=

1 u(p) ρ p p!

(32)

p=0

for the field quantities. The resulting equations, which will be referred generically to as transport equations, are of the form, ∂τ v (p) = Kv (p) + Q(v (0) , v (p) ) + Q(v (p) , v (0) ) p−1

+ Q(v (j ) , v (p−j ) ) + L(j ) φ (p−j ) + L(p) φ (0) ,

(33)

j =1

which correspond to the propagation equations of the v unknowns, Eqs. (17a)-(17g). From the Bianchi propagation equations (19a)-(19e) one gets, √

0 (0) C (0) ∂τ φ (p) + Aab (cab 2E + Aab (cab ) ) ∂C φ (p) p

p (j ) µ (0) B(abcd )φ (p−j ) − Aab (cab )(j ) ∂µ φ (p−j ) , (34) = B(abcd )φ (p) + j j =1

where C = ±. The systems (33) and (34) can be regarded as systems for the unknowns v (p) and φ (p) if the lower order quantities v (j ) and φ (j ) , 0 ≤ k ≤ p − 1 are known. The two systems are decoupled from each other, and accordingly one would firstly solve the system (33) and then feed its solution into the system (34) which now could in turn be solved.

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J.A.Valiente Kroon

I+

I+ x0 i0

ρ

S

I−

I−

Fig. 1. Spacetime close to spatial and null infinities: to the left the standard representation of spatial infinity as a point i 0 ; to the right the representation where spatial infinity is envisaged as a cylinder

Because of (29a) and (29b), the matrix accompanying the ∂τ derivative in the system (34) is given by, √

√ 0 (0) 2E + Aab (cab (35) = 2 diag(1 + τ, 1, 1, 1, 1 − τ ). ) As a consequence of this, the symbol of the system looses rank at τ = ±1 – the system degenerates there. The points τ = ±1 correspond precisely to the sets I ± – cf. (4b)– the sets where “null infinity touches spatial infinity”. It is exactly this particular feature of the field equations that forces us to undertake a complicated and detailed analysis of the system (28). From an heuristic point of view the degeneracy at the sets I ± can be understood as a consequence of the change of behaviour of the conformal boundary of the spacetime with regard to the conformal field equations: the cylinder at spatial infinity I is a total characteristic, while the I ± are “only” normal characteristics – i.e. only proper subsets of the field equations reduce to interior systems on either I + or I − . Now, standard theory of symmetric hyperbolic systems guarantees, for a given order p, the existence of solutions to the joint system (33)–(34) for any subset of I . However, for the sets I ± the degeneracy implied by Eq. (35) the usual energy estimates provide no information precisely at the points one is interested the most – a similar phenomenon occurs with the original system (27)–(28). Thus, one needs to devise non-standard methods in order to address existence issues – see for example the discussions in [9, 25]. A first analysis of the transport equations carried out in [16] reveals that the effect of the degeneracy described in the previous paragraph is to produce solutions of the transport equations containing terms of the form, (1 − τ )m (1 + τ )n ln(1 ± τ ), with the effect that the solutions to the transport equations do not extend smoothly to the sets I ± . A necessary condition to preclude the appearance of such logarithmic terms is given in the following theorem. Theorem 1 (Friedrich, 1997). The solutions u(p) of the transport equations extend smoothly to the sets I ± only if the condition D(as bs · · · Da1 b1 babcd) (i) = 0,

s = 0, 1, . . .

(36)

is satisfied at all orders s. It is not satisfied for some s, the solution develops logarithmic singularities at I ± .


145

This theorem is the basis to the conjecture by H. Friedrich mentioned in the introduction. The analysis leading to the latter theorem is essentially an analysis of the homogeneous parts of the transport equations (33)-(34). A careful look at the complete equations reveals that even if the regularity condition (36) is satisfied there are still other potential sources of logarithmic terms. In Sect. 4 we shall discuss these further obstructions to the smoothness of null infinity. So far, the function κ introduced in Eq. (1) has been required to be of the form κ = ρκ , with κ analytic and such that κ (i) = 1, but otherwise it has remained unspecified 5 . Two choices consistent with the requirement κ = ρκ will be considered here. The first, κ = ρ is the simplest non-trivial one. For this choice I + in a neighbourhood of I + is concave, while I − in a neighbourhood of I − would be convex. The choice κ = ρ has the virtue of rendering the simplest possible analytic expressions, both for the initial data (24a)-(24f) and the solution of the transport equations (33)-(34). Unfortunately, it is hard to attach to it some geometrical significance other that its simplicity. The other choice to be considered is κ = ω. This choice is fine as under our assumptions ω = ρ + O(ρ). With this choice,

= ω−1 1 − τ 2 , (37) so that I ± near I ± are described by the hypersurfaces τ = ±1, ρ > 0 respectively: null infinity will be composed of two parallel planes, formally similar to the case of Minkowski – see [25] 6 . Consequently, the system of conformal field equations (27)(28) degenerate not only on I ± but also on I ± . Thus, the choice κ = ω has more geometrical and analytic relevance. As a drawback it renders more complicated analytic expressions. For future use, we note the following result on the expansion types of the diverse unknowns appearing in the transport equations (33) and (34). Its proof comes from inspection [16]. 1 (p) (p) (p) Lemma 2. The functions cab − ρxab , v , φ p = 1, 2, . . . , are of expansion type p − 2, p − 1, and p respectively.

4. Solving the Transport Equations Given Time Symmetric, Conformally Flat Initial Data On simplicity and aesthetical grounds it is natural to wonder whether the regularity condition, Eq. (36), is the only requirement one has to impose on the initial data in order to obtain solutions to the transport equations which are smooth – see for example the discussion in [18]. Before trying to prove some statement along these lines, it is of clear interest to calculate some further orders in the expansions. The rationale behind it being firstly to verify whether the conjecture still holds, and secondly to try to find some patterns in the solutions that one could exploit in an eventual abstract proof. In order to simplify the calculations, we have chosen to restrict our attention to those time 5 It is noted that the simple choice κ = 1 would lead to the standard representation of spatial infinity as a point – see the figure. The requirement of κ being of the form κ = ρκ ensures that spatial infinity is blown up to a cylinder. See Fig. 1. 6 This similarity is in some aspects deceiving, as generically when m = 0 the generators of null infinity, although confined to the planes τ = ±, are bent and may rotate a spin frame that is parallelly transported along it.

146

J.A.Valiente Kroon I+

I+

I+ I

τ

I0

S

ρ

I− I−

I−

Fig. 2. Effect of the choice of the function κ on the representation of null infinity near spatial infinity. To the left with the choice κ = ρ; to the right the choice κ = ω so that null infinity corresponds to the hypersurfaces τ = ±1

symmetric initial data sets which are conformally flat near infinity. These satisfy the regularity condition (36) in a trivial way. Thus, they represent the simplest (non-trivial) class of initial data sets one can look at. Their simplicity is somehow deceiving, and should not be regarded as a drawback on the kind of insight that can be gained through them. A great deal about the solutions of the Einstein field equations has been learned from the analysis of this class of initial data – see e.g. [1, 20, 26]. The already “large” transport equation systems (33)–(34) do not give an appropriate dimension of the computational difficulties one has to face if one is to take the expansions carried in [19] to even higher orders. However, the calculations one has to carry out are suitable to a treatment using a computer algebra system. In order to analyse with ease the solutions of the transport equations some scripts in the computer algebra system Maple V have been written. With the aim to work with a system of ordinary differential equations, the Maple V scripts take the transport equations (33) and express them in terms of the functions T2n kl according to Lemma 2. This involves expanding products of the form T2n1 k1l1 × T2n2 k2l2 using formula (16). The resulting ordinary equations are then solved exactly using the ordinary differential equation solver of Maple V. Full details of the computer algebra implementation will be presented elesewhere. Because of the largeness of the expressions contained in the solutions, we have opted to provide a description of the qualitative features of the expansions rather than a complete list of all the terms calculated. In particular, attention will be focused on the solutions to the transport equations arising from the Bianchi identities, the functions (p) φj . It should be emphasized that this does not mean that the solutions to the v unknown transport equations are not important. They are also crucial: a tiny mistake in the calculation of their solutions would destroy the whole structure of the solutions. However, as the discussion in the previous section has pointed out, the logarithmic terms that destroy the smoothness of null infinity appear firstly in the components of the Weyl spinor. In order to perform our expansions a number of assumptions have been made. We list them here for the purposes of a quick reference. Assumptions. It will be assumed that: (i) the initial data set is asymptotically Euclidean and time symmetric. (ii) In a neighbourhood Ba (i) of i the initial data is assumed to be conformally flat. The function W appearing in the conformal factor of the initial hypersurface – see Eq. (21) – is a solution of the Laplace equation admitting in Ba (i) a decomposition


147

of the form, 1 1 m+ Wi ρ i + O(ρ 9 ), 2 i! 8

W =

(38)

i=1

where Wi =

2i

wi,2i,k T2i ki ,

(39)

k=0

with the coefficients wi,2i,k , i = 1, . . . , 7, k = 0, . . . , 2i complex numbers satisfying w i,2i,k = (−1)i+k wi,2i,2i−k so that W is a real valued function. This is in consistency with the decomposition given in Eq. (22). (iii) Likewise, the components of the vector unknowns v (p) and φ (p) admit on I expank sions in terms of Tj l functions consistent with Lemma 2. (iv) The two following choices of the function κ – see Eq. (1) – will be considered: κ1 = ρ, κ2 = ω. The result of the calculations under these assumptions are now described. 4.1. The orders p = 0, . . . , 4. Firstly, calculations for the orders p = 1, p = 2 and p = 3 were undertaken. The results are in complete agreement with those given by Friedrich & Kánnár when reduced to the case of conformally flat initial data. The solutions at order p = 0 can be schematically written as, (0)

φ0 = 0,

(40a)

(0) φ1 (0) φ2 (0) φ3 (0) φ4

= 0,

(40b)

= −m,

(40c)

= 0,

(40d)

= 0,

(40e)

independently of the choice of κ. At order p = 1 one has, (1)

φ0 = 0,

(41a)

(1) φ1

(41b)

(1)

φ2

= −3 X+ W1 (1 − τ )2 , 1 4 = m2 τ − 3τ 2 + 6 W1 (τ 2 − 1), 2

(1)

(41c)

φ3 = 3 X− W1 (1 + τ )2 ,

(41d)

(1) φ4

(41e)

= 0,

148

J.A.Valiente Kroon (1)

(1)

when κ = ρ. The expressions for φ0 , φ1 , (1) similar. That of φ2 is given by, 1 4 (1) φ2 = m2 τ − 3τ 2 − 2

(1)

(1)

φ3 and φ4 for the choice κ = ω are 3 2

+ 6 W1 (τ 2 − 1).

(41f)

At order p = 2 one has, (2)

φ0 = f1 (τ )X+ X+ W2 ,

(42a)

(2) φ1 (2) φ2 (2) φ3 (2) φ4

= f2 (τ )mX+ W1 + f3 (τ )X+ W2 ,

(42b)

= f4 (τ )m + f5 (τ )mW1 + f6 (τ )W2 ,

(42c)

= −f2 (−τ )mX− W1 − f3 (−τ )X− W2 ,

(42d)

= f1 (−τ )X− X− W2 ,

(42e)

3

where fi (τ ), i = 1, . . . 6 are polynomials on τ . Their explicit form is not relevant for our purposes. The polynomials are slightly different for each of the choices of κ, but are of the same order. Similarly, the components of the Weyl tensor at order p = 3 are of the form, (3)

(43a)

(3) φ1 (3) φ2

= g4 (τ )X+ W3 + g5 (τ )mX+ W2 + g6 (τ )W1 X+ W1 + g7 (τ )m X+ W1 , (43b)

φ0 = g1 (τ )X+ X+ W3 + g2 (τ )mX+ X+ W2 + g3 (τ )(X+ W1 )2 , 2

= g8 (τ )W3 + g9 (τ )mW2 + g10 (τ )(W1 )2 +g11 (τ )2 m2 W1 + g12 (τ )m4 + g13 (τ )b,

(3) φ3

(43c)

= −g4 (−τ )X− W3 − g5 (−τ )mX− W2 −g6 (−τ )W1 X− W1 − g7 (−τ )m2 X− W1 ,

(3) φ4

(43d)

= g1 (−τ )X− X− W3 + g2 (−τ )mX− X− W2 + g3 (−τ )(X− W1 ) . 2

(43e)

Again, the functions gi (τ ), i = 1, . . . , 13 are polynomials, while b = 2w1,2,0 w1,2,2 − 2 w1,2,1 . The first fully new result corresponds to the order p = 4. Here, again, the solutions are still fully regular and polynomial: (4)

φ0 = h1 (τ )X+ W1 X+ W2 + h2 (τ )W1 X+ X+ W2 + h3 (τ )m(X+ W1 )2 +h4 (τ )mX+ X+ W3 + h5 (τ )X+ X+ W4 , (4) φ1

= h6 (τ )m X+ W1 + h7 (τ )W2 X+ W1 + h8 (τ )W1 X+ W2 +h9 (τ )mW1 X+ W1 + h10 (τ )mX+ W3 + h11 (τ )X+ W4 ,

(4) φ2

= h12 (τ )m + h13 (τ )b + h14 (τ )m W1 + h15 (τ )W1 W2

(44a)

3

5

(44b)

3

+h16 (τ )m(W1 )2 + h17 (τ )mW3 + h18 (τ )W4 ,

(44c)

(4) φ3

= −h6 (−τ )m X− W1 − h7 (−τ )W2 X− W1 − h8 (−τ )W1 X− W2 −h9 (−τ )mW1 X− W1 − h10 (−τ )mX− W3 − h11 (−τ )X− W4 ,

(4) φ4

= h1 (−τ )X− W1 X− W2 + h2 (−τ )W1 X− X− W2 + h3 (−τ )m(X− W1 ) +h4 (−τ )mX− X− W3 + h5 (−τ )X− X− W4 . (44e)

3

(44d) 2

Again, the functions hi (τ ), i = 1, . . . , 16 are polynomials depending on the choice of κ.


149

4.2. The first obstructions to smoothness: Orders p = 5, 6.. The calculation of the solutions to the Bianchi transport equations up to order p = 4 have shown that all of them are polynomial, and thus smooth at I ± . Consequently, the v unknowns also happen to be polynomial. The first modifications to this behaviour occur at the rather high order p = 5. Feeding the solution of the transport equations up to order p = 4 into the v transport equations (33) with p = 5 and solving one finds again that the components of v (5) are again polynomial. However, the solution of the Bianchi transport equations are of the form, (5) φ0 = C0 m2 G(5) (1 − τ )7 ln(1 − τ )

+(1 + τ )3 (351 − 150τ + 48τ 2 − 10τ 3 + τ 4 ) ln(1 + τ ) + k0 (τ ), (45a) (5) φ1 = C1 m2 G(5) (1 − τ )6 (2τ + 5) ln(1 − τ )

−(2τ 3 − 15τ 2 + 48τ − 75)(1 + τ )4 ln(1 + τ ) + k1 (τ ), (45b) (5) φ2 = C2 m2 G(5) (1 − τ )5 (τ 2 + 5τ + 8) ln(1 − τ )

+(1 + τ )5 (τ 2 − 5τ + 8) ln(1 + τ ) + k2 (τ ), (45c) (5) φ3 = C1 m2 G(5) (1 − τ )4 (2τ 3 + 15τ 2 + 48τ + 75) ln(1 − τ )

−(1 + τ )6 (2τ − 5) ln(1 + τ ) + k3 (τ ), (45d) (5) φ4 = C0 m2 G(5) (1 − τ )3 (351 + 150τ + 48τ 2 + 10τ 3 + τ 4 ) ln(1 − τ )

(45e) +(1 + τ )7 ln(1 + τ ) + k4 (τ ), where C0 , C1 , C3 are non-relevant non-zero numerical factors, ki (τ ) i = 0, . . . , 4 are polynomials depending on m, W1 , W2 , W3 , W4 , their X± derivatives and products of them. Most remarkably, G(5) =

4

(5)

Gk T4 k2 ,

(46)

k=0

where,

√ 2 (5) , G0 = mw2,4,0 − 2 6w1,2,0 √ (5) G1 = mw2,4,1 − 4 3w1,2,0 w1,2,1 , (5) G2 (5) G3 (5) G4 (5)

=

2 mw2,4,2 − 4w1,2,1

√

− 4w1,2,0 w1,2,2 ,

= mw2,4,3 − 4 3w1,2,1 w1,2,2 , √ 2 = mw2,4,4 − 2 6w1,2,2 .

(47a) (47b) (47c) (47d) (47e)

Thus, the coefficients Gk are (besides an irrelevant numerical factor) the NewmanPenrose constants of the development of the time symmetric conformally flat initial data – [19]. Plugging the solutions (45a)–(45e) into the transport equations for p = 6 one finds that the solutions of the sectors of the form T4 kj develop terms containing ln2 (1 ± τ ) and

150

J.A.Valiente Kroon (p)

ln(1 ± τ ). Furthermore, the sectors T6 kj +1 in φj contain logarithms. More precisely, the solution will be of the form, (6) φj = G(5) l1j (τ ) ln2 (1 − τ ) + l2j (τ ) ln(1 − τ ) + l3j (τ ) ln2 (1 + τ )

+l4j (τ ) ln(1 + τ ) + G(6) l5j (τ ) ln(1 − τ ) + l6j (τ ) ln(1 + τ ) + l7j (τ ), (48) where j = 0, . . . , 7, and lkj (τ ) are polynomials. The coefficients G(6) are new obstructions to the smoothness of the solutions. These will be discussed a bit later. It is not hard to imagine that from this point onward, terms containing ln(1 ± τ ) and higher order powers of them will spread all around the solutions to the transport equations. Instead of analysing this phenomenon, we will rather focus on the smooth solutions. Setting the Newman-Penrose constants to zero, the solution at order p = 6 is of the form, (6) φ0 = D0 m3 G(6) (2 + τ )(1 − τ )8 ln(1 − τ ) + (−254 + 233τ − 128τ 2

(49a) +46τ 3 − 10τ 4 + τ 5 )(1 + τ )4 ln(1 + τ ) + l0 (τ ), (6) φ1 = D1 m3 G(6) (23 + 20τ + 5τ 2 )(1 − τ )7 ln(1 − τ )

−(233 − 256τ + 138τ 2 − 40τ 3 + 5τ 4 )(1 + τ )5 ln(1 + τ ) + l1 (τ ), (49b) (6) φ2 = D2 m3 G(6) (64 + 69τ + 30τ 2 + 5τ 3 )(1 − τ )6 ln(1 − τ )

+(−64 + 69τ ) − 30τ 2 − 5τ 3 )(1 + τ )6 ln(1 + τ ) + l2 (τ ), (49c) (6) φ3 = D1 m3 G(6) (233 + 256τ + 138τ 2 + 40τ 3 + 5τ 4 )(1 − τ )5 ln(1 − τ )

(49d) +(23 − 20τ + 5τ 2 )(1 + τ )7 ln(1 + τ ) + l3 (τ ), (6) φ4 = D0 m3 G(6) (254 + 233τ + 128τ 2 + 46τ 3 + 10τ 4 + τ 5 )(1 − τ )4 ln(1 − τ )

+(−2 + τ )(1 + τ )8 ln(1 + τ ) + l4 (τ ), (49e) with l0 (τ ), . . . , l4 (τ ) polynomials, and G(6) =

6

(6)

Gk T6 k3 ,

(50)

k=0 (6)

where the coefficients Gk , k = 0, . . . , 6 are given in terms of initial data quantities by, √ (6) 3 , (51a) G0 = m2 w3,6,0 − 12 10w1,2,0 √ (6) 2 G1 = m2 w3,6,1 − 12 30w1,2,0 w1,2,1 , (51b) √ √ 2 (6) 2 2 (51c) G2 = m w3,6,2 − 24 6w1,2,0 w1,2,1 − 12 6w1,2,0 w1,2,2 , (6)

3 , G3 = m2 w3,6,3 − 72w1,2,0 w1,2,1 w1,2,2 − 24w1,2,1 √ √ (6) 2 2 2 , G4 = m w3,6,4 − 24 6w1,2,1 w1,2,2 − 12 6w1,2,0 w1,2,2

(51d) (51e)


√ (6) 2 G5 = m2 w3,6,5 − 12 30w1,2,1 w1,2,2 , √ (6) 3 . G6 = m2 w3,6,6 − 12 10w1,2,2

151

(51f) (51g)

In analogy to the order p = 5 we will refer to these coefficients as to the order 6 Newman-Penrose “constants”. One is naturally bound to ask whether the coefficients G(6) are actually associated to some conserved quantities at null infinity in the same way that the coefficients G(5) are. This consideration is beyond the scope of the present article, and will be analysed in detail in future work. It is pointed out, as a plausibility argument, that in the analysis of polyhomogeneous Bondi expansions carried out in for example [3, 23, 24], the first logarithmic terms appearing in the expansions were associated with a conserved quantity on null infinity. As it can be seen from our discussion, if G(5) = 0, then the first logarithmic terms appearing in our expansions are precisely those associated to G(6) . Notwithstanding, the expansions described here are based in the conformal geodesics gauge, while those in [3, 23, 24] use the so-called Bondi gauge. Thus, one would have to look in detail into the possible appearance of logarithmic terms in the transformation connecting the two gauges. Theorem (Main theorem, precise formulation). Necessary conditions for the development of initial data which is time symmetric and conformally flat in a neighbourhood Ba (i) of (spatial) infinity to be smooth on the set I ∪ I + ∪ I − are that the Newman(5) Penrose constants, Gk , k = 0, . . . , 5, should vanish. Furthermore, the “higher order” (6) Newman-Penrose constants, Gk , k = 0, . . . , 6, should also vanish. If only the coeffi(5) cients Gk vanish then the rescaled Weyl spinor is at most C 2 on a neighbourhood of either I + or I − . If both G(5) and G(6) vanish then generically the Weyl tensor will be at most of class C 3 of the latter neighbourhoods. From the last result one can extract directly the following (important) consequence: Corollary 1. The regularity condition (36) is not a sufficient condition for the smoothness at I ∪ I + ∪ I − of the development of asymptotically Euclidean, time symmetric initial data sets. This corollary is, thus, a negative answer to the conjecture raised in [18]. It is nevertheless surprising that the obstructions to the smoothness of null infinity arise at such a high order in the expansions. In order to acquire a deeper understanding of why this is the case would require in turn an abstract understanding of the algebraic structure of the transport equations (33) and (34). It is not unreasonable to reckon that some group theoretical properties of the whole setup play a major role here. It is worth mentioning that the logarithmic singularities contained in the solutions of the transport equations are associated with the conformal structure, and not artifacts of a choice of gauge – see the discussion regarding this in [16]. Furthermore, because of the hyperbolic nature of the propagation equations, it is very likely that the logarithmic terms in the solutions of the transport equations will propagate along the generators of null infinity. The details of this propagation, however, still have to be worked out in full detail. The theorem also suggests the following unexpected conjecture: Conjecture. The developments of the (non-degenerate) Brill-Lindquist and Misner initial data sets possess non-smooth null infinities.

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By non-degenerate Brill-Lindquist and Misner data it is understood that neither the individual masses of the individual holes nor the separation parametre are zero, in which case data is strictly Schwarzschildean – and thus its development has a smooth null infinity. In [8, 26] the Newman-Penrose constants of the Brill-Lindquist and Misner data sets [1, 20] have been calculated using the formula found by Friedrich & Kánnár [19]. Due to the axial symmetry, there is only one non-vanishing Newman-Penrose constant. Because of the non-vanishing Newman-Penrose constants, their asymptotic expansions will contain logarithmic terms as indicated in Sect. 4. In order to prove the conjecture one would then require an existence statement for the development. This is a very hard task and may involve considerations regarding Cosmic Censorship. That the development of these two initial data sets possesses a non-smooth null infinity may have implications in the description of their late time behaviour in terms of linear perturbations on a Schwarzschild background. It is however noted that the Corvino-Chru´sciel-Delay techniques provide the possibility of constructing modified Brill-Lindquist and Misner data which are exactly Schwarzschild data outside a compact set. These data sets would, in principle, admit a smooth null infinity, but again the details of this cannot be filled with the techniques currently available. Some remarks regarding the theorem come also into place: Remark 1. The calculations for the order p = 7 are already beyond the capabilities of Maple V – the expressions involved are too large for the simplification routines of the computer algebra system, even for the Origin computer available at the Albert Einstein Institute. Nevertheless, the calculations of axially symmetric situations are still possible. These have been carried out for the orders p = 7 and p = 8 inclusive. Assuming that both G(5) and G(6) vanish, the solutions of the Bianchi transport equations have again the expected form:

(7) (52) φj = Ej m3 G(7) m1j (τ ) ln(1 − τ ) + m2j (τ ) + m3j (τ ), where now due to the axial symmetry there is only one order 7 Newman-Penrose constant,

4 T8 44 . (53) G(7) = m3 w4,8,4 − 192w1,2,1 If G(7) vanishes in turn,

(8) φj = Fj m4 G(8) n1j (τ ) ln(1 − τ ) + n2j (τ ) + n3j (τ ),

(54)

5 T10 55 . G(8) = m4 w5,10,5 − 1920w1,2,1

(55)

with,

From this evidence it is not too hard to guess the following general formula for the obstructions of the smoothness of null infinity in the axially symmetric situation,

p−3 (56) G(p) = mp−4 wp−3,2p−6,p−3 − 2p−4 (p − 4)!(w1,2,1 )p−3 T2p−6 p−3 . The proof of such a formula is nevertheless beyond our current understanding of the transport equations. The significance of the latter expression will be discussed shortly.


153

Remark 2. In the non-axially symmetric case, it is conjectured that obstructions to the smoothness of null infinity (Generalised Newman-Penrose constants) are given in terms of the initial data by the following expressions, (p) s0 s1 s2 Gk = mp−4 wp−3,2p−6,k − cp,k;s0 ,s1 ,s2 w1,2,0 w1,2,1 w1,2,2 , (57) s0 +s1 +s2 =k

with k = 0, 1, . . . , p, where the coefficients cp,k;s0 ,s1 ,s2 are some numerical constants. 4.3. Obstructions to smoothness and the Schwarzschild initial data. As mentioned before, the Newman-Penrose constants of the Schwarzschild spacetime are all zero. Thus, in order to gain some insight into the significance of the expressions (56) and (57) it is convenient to see what occurs in the case of the Schwarzschild initial data. It is not hard to calculate an expression for initial data for the Schwarzschild spacetime on the slice of time symmetry. On this slice the initial 3-metric is conformally flat. Thus, the required (harmonic) conformal factor can be calculated directly from the Green’s function for the three dimensional Laplace equation in spherical coordinates. The Green’s function is given by,

G(r, θ, φ; r , θ , φ ) =

n ∞ n=0

4π r n Yn,m (θ, φ)Yn,m (θ , φ ) n+1 , 2n + 1 r m=−n

(58)

for r > r . The latter expression can be lifted into the frame bundle Ca,κ and written in terms of k the functions Tj l so that, ∞

W =

m n r T2n 2n−kn (t ) T2n kn ρ n , 2 2n

(59)

n=0 k=0

where (t , r ) denote the coordinates of the singularity of the Green function on the frame bundle. Thus, the T2n 2n−kn (t ) are fixed complex numbers. We write, 1 n (60) mr n!T2n 2n−kn (t ), 2 for n = 0, 1, . . . and k = 0, . . . , 2n. The latter coefficients are not independent but related to each other via recurrence relations, (n + 1)(2n + 1)w1,2,1 × wn,2n,k = (k + 1)(2n − k + 1)wn+1,2(n+1),k+1 +r 2 n(n + 1) k(2n − k)wn−1,2(n−1),k−1 . (61) wn,2n,k =

These can be readily obtained from similar recurrence relations holding for the spherical harmonics. The Schwarzschild solution has an obvious axial symmetry. If one considers an orientation of the coordinate system in a way that makes this symmetry of the initial data explicit – the singularity of the Green’s function is set along the z axis – one ends up with a much simplified expression, ∞ n−1 2 W = (w1,2,1 )n T2n nn ρ n , (62) m n=0

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so that the only non-vanishing wn,2n,k coefficients are given by, wn,2n,n =

2 m

n−1 (w1,2,1 )n n! .

(63)

One obtains precisely this expression if one requires that the hierarchy of axial NewmanPenrose constants, formula (56), vanishes at every order p. It is however important to point out that this is not a proof but again a plausibility argument as formula (56) has only been verified up to order p = 8. A much more involved calculation shows that the expressions one obtains for the coefficients w2,4,0 , . . . , w2,4,4 and w3,6,0 , . . . , w3,6,6 by setting G(5) and G(6) to zero are precisely those one has for the Schwarzschild initial data. This involves the use of the recurrence relation (61). Now, the function W is a solution to W = 0, and thus analytic. This in turn implies that the function W one deduces from requiring the coefficients (56) to vanish at all orders p is exactly Schwarzschildean. This evidence leads to the following, Conjecture (Precise formulation). For every k > 0 there exists a p = p(k) such that the time evolution of an asymptotically Euclidean, time symmetric, conformally smooth initial data set which is conformally flat near i admits a conformal extension to null infinity of class C k near spacelike infinity, if and only if the initial data set is Schwarzschildean to order p(k) near i. The latter is the successor of the conjecture, in the context of time symmetric conformally flat, of the conjecture by Friedrich [18] to which reference was made in the introduction. It is, however, worth mentioning that if proved true, the conjecture here stated would constitute a rigidity result associated with the notion of asymptotic simplicity.

5. Conclusions and Extensions The results presented in the main theorem together with the considerations leading to the conjecture put forward in Sect. 4 seem to suggest that no gravitational radiation should be present around spatial infinity if one is to have a smooth null infinity. In other words, the notion of smooth null infinity seems to be incompatible with the presence of radiation around i 0 . Whether this behaviour of spacetime in the neighbourhood of spatial infinity has some implications on either the demeanour of the sources of the gravitational field in the infinite past or on the nature of incoming radiation traveling from past null infinity is a natural – but hard to answer – question. A natural extension of the calculations here described would be to consider what happens with more general (i.e. not conformally flat) time symmetric initial data. If the time symmetric initial data is not conformally flat, then the Newman-Penrose constants are given in terms of the initial data by, √ 2 (5) = mw2,4,0 − 2 6w1,2,0 G − 0 (5) G 1 (5) G 2

1 (2) √ R0 , 254 6 √ 1 (2) = mw2,4,1 − 4 3w1,2,0 w1,2,1 − √ R1 , 254 6 1 (2) 2 = mw2,4,2 − 4w1,2,1 − 4w1,2,0 w1,2,2 − √ R2 , 254 6


√ (5) = mw2,4,3 − 4 3w1,2,1 w1,2,2 − G 3 √ 2 (5) = mw2,4,4 − 2 6w1,2,2 G − 4

155

1 (2) √ R3 , 254 6

1 (2) √ R4 , 254 6

(2)

where Rk are coefficients appearing in the expansion of the Ricci scalar r of the initial 3-metric, around spatial infinity. In the view of the present results one would expect to (5) appearing as obstructions to the smoothness of null infinity. have the quantities G k (6) of the higher order NewSimilarly, one would expect to obtain generalisations G k (6) man-Penrose constants Gk by adding suitable expressions containing the Ricci scalar. Ultimately, one would like to prove that there is an infinite hierarchy of such quantities as obstructions to the smoothness of I. The relation of these quantities with static initial data should then be analysed, at least for the first orders. It could well be the case that a time symmetric initial data set will yield a development with smooth null infinity only if it is static in a neighbourhood of spatial infinity. In relation to this, it is noted that the asymptotically Euclidean static initial data satisfy the regularity condition (36) – see [15]. With regard to a proof to the conjecture presented in this article, it is noted that it would require a much deeper understanding of the properties of the transport equations at spatial infinity than the one currently available. In particular one would like to be able to discuss its algebraic properties in a much more abstract way, and without having to resort to “explicit expressions” as it was done here. As mentioned before, group theoretical properties of the setup should play a crucial role here. Acknowledgements. I thank H. Friedrich who suggested the problem and helped with discussions and clarifications all along the long winding road. I also thank S. Dain, M. Mars and R. Vera from whom I have benefited through several discussions and comments on early versions of the manuscript. The late stages of this research have been funded by a Lise Meitner grant (M-690-N09) of the FWF, Austria. The computer algebra calculations in Maple V have been carried out at the facilities of the Max Planck Institut für Gravitationsphysik, Albert Einstein Institut in Golm, Germany.

References 1. Brill, D.R., Lindquist, R.W.: Interaction energy in geometrostatics. Phys. Rev. 131, 471 (1963) 2. Chru´sciel, P.T., Delay, E.: Existence of non-rivial, vacuum, asymptotically simple spacetimes. Class. Quantum Grav. 19, L71 (2002) 3. Chru´sciel, P.T., MacCallum, M.A.H., Singleton, D.B.: Gravitational waves in general relativity XIV. Bondi expansions and the “polyhomogeneity” of I. Phil. Trans. Roy. Soc. Lond. A 350, 113 (1995) 4. Corvino, J.: Scalar curvature deformations and a gluing construction for the Einstein constraint equations. Commun. Math. Phys. 214, 137 (2000) 5. Cutler, C. Wald, R.M.: Existence of radiating Einstein-Maxwell solutions which are C∞ on all of I + and I − . Class. Quantum Grav. 6, 453 (1989) 6. Dain, S.: Initial data for stationary spacetimes near spacelike infinity. Class. Quantum Grav. 18, 4329 (2001) 7. Dain, S., Friedrich, H.: Asymptotically flat initial data with prescribed regularity at infinity. Commun. Math. Phys. 222, 569 (2001) 8. Dain, S., Valiente Kroon, J.A.: Conserved quantities in a black hole collision. Class. Quantum Grav. 19, 811 (2002) 9. Friedrich, H.: Spin-2 fields on Minkowski space near space-like and null infinity. Class. Quantum Grav. 20, 101 (2003) 10. Friedrich, H.: On the existence of analytic null asymptotically flat solutions of Einstein’s vacuum field equations. Proc. Roy. Soc. Lond. A 381, 361 (1981)

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11. Friedrich, H.: On the regular and the asymptotic characteristic initial value problem for Einstein’s vacuum field equations. Proc. Roy. Soc. Lond. A 375, 169 (1981) 12. Friedrich, H.: Cauchy problems for the conformal vacuum field equations in General Relativity. Commun. Math. Phys. 91, 445 (1983) 13. Friedrich, H.: On purely radiative space-times. Commun. Math. Phys. 103, 35 (1986) 14. Friedrich, H.: On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure. Commun. Math. Phys. 107, 587 (1986) 15. Friedrich, H.: On static and radiative space-times. Commun. Math. Phys. 119, 51 (1988) 16. Friedrich, H.: Gravitational fields near space-like and null infinity. J. Geom. Phys. 24, 83 (1998) 17. Friedrich, H.: Einstein’s equation and conformal structure. In: The Geometric Universe. Science, Geometry and the work of Roger Penrose, S.A. Huggett, L.J. Mason, K.P. Tod, S.T Tsou, N.M.J. Woodhouse, (eds.), Oxford: Oxford University Press, 1999, p. 81 18. Friedrich, H.: Conformal Einstein evolution. In: The conformal structure of spacetime: Geometry, Analysis, Numerics, J. Frauendiener and H. Friedrich, (eds.), Lecture Notes in Physics, Berlin-Heidelberg-New York: Springer, 2002, p. 1 19. Friedrich, H., Kánnár, J.: Bondi-type systems near space-like infinity and the calculation of the NP-constants. J. Math. Phys. 41, 2195 (2000) 20. Misner, C.W.: The method of images in geometrodynamics. Ann. Phys. 24, 102 (1963) 21. Penrose, R.: Asymptotic properties of fields and space-times. Phys. Rev. Lett. 10, 66 (1963) 22. Penrose, R.: Zero rest-mass fields including gravitation: Asymptotic behaviour. Proc. Roy. Soc. Lond. A 284, 159 (1965) 23. Valiente Kroon, J.A.: Conserved Quantities for polyhomogeneous spacetimes. Class. Quantum Grav. 15, 2479 (1998) 24. Valiente Kroon, J.A.: Logarithmic Newman-Penrose Constants for arbitrary polyhomogeneous spacetimes. Class. Quantum Grav. 16, 1653 (1999) 25. Valiente Kroon, J.A.: Polyhomogeneous expansions close to null and spatial infinity. In: The Conformal Structure of Spacetimes: Geometry, Numerics, Analysis J. Frauendiner and H. Friedrich, (eds.), Lecture Notes in Physics, Berlin-Heidelberg-New York: Springer, 2002, p. 135 26. Valiente Kroon, J.A.: Early radiative properties of the developments of time symmetric conformally flat initial data. Class. Quantum Grav. 20, L53 (2003) Communicated by H. Nicolai

Commun. Math. Phys. 244, 157–185 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-0989-z

Communications in


“Extrinsic” and “Intrinsic” Data in Quantum Measurements: Asymptotic Convex Decomposition of Positive Operator Valued Measures Andreas Winter Department of Computer Science, University of Bristol, Merchant Venturers Building, Woodland Road, Bristol BS8 1UB, UK. E-mail: [email protected] Received: 12 September 2001 / Accepted: 21 July 2003 Published online: 25 November 2003 – © Springer-Verlag 2003

Abstract: We study the problem of separating the data produced by a given quantum measurement (on states from a memoryless source which is unknown except for its average state), described by a positive operator valued measure (POVM), into a “meaningful” (intrinsic) and a “not meaningful” (extrinsic) part. We are able to give an asymptotically tight separation of this form, with the “intrinsic” data quantified by the Holevo mutual information of a certain state ensemble associated to the POVM and the source, in a model that can be viewed as the asymptotic version of the convex decomposition of POVMs into extremal ones. This result is applied to a similar separation therorem for quantum instruments and quantum operations, in their Kraus form. Finally we comment on links to related subjects: we stress the difference between data and information (in particular by pointing out that information typically is strictly less than data), derive the Holevo bound from our main result, and look at its classical case: we show that this includes the solution to the problem of extrinsic/intrinsic data separation with a known source, then compare with the well–known notion of sufficient statistics. The result on decomposition of quantum operations is used to exhibit a new aspect of the concept of entropy exchange of an open dynamics. An appendix collects several estimates for mixed state fidelity and trace norm distance, that seem to be new, in particular a construction of canonical purification of mixed states that turns out to be valuable to analyze their fidelity.

I. The Problem Consider a quantum system, represented by a Hilbert space H (which we assume to be of dimension d < ∞ in the sequel), and a measurement on this system, described by a positive operator valued measure (POVM) a = (a1 , . . . , am ), aj ∈ B(H) such that aj ≥ 0 and j aj = 11.

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Following [22] and [29] we shall be concerned with the question “How much information is obtained by a?”, beginning with a clarification what this question should mean at all. Imagine that a family of states (represented by density operators) ρi on H is given, let us say witha priori probabilities pi , such that the density operator of this source of states is ρ = i pi ρi , then the “information” in question could mean the information in j about i, and one way to quantify it would be given by Shannon’s mutual information [25] I (i ∧ j ). Note that this is in general less than the amount of raw data, which is operationally quantified by the entropy of the distribution of the j : H (λ), with λj = Tr (ρaj ), due to Shannon’s source coding Theorem [25]. This choice however is rather arbitrary: asking about the identity of the state from a list. Why not allow a different list, or ask for some property of the state. Also, mutual information is a measure of correct identification; but what if we need only “almost correct” identification, as in quantum statistical detection theory [11]? It seems hence that specifying the information in measurement results, or even only the amount, in an operationally satisfying way, is problematic, and one reason might be the complementarity of quantum mechanics: qualitatively, accessing some observable property optimally entails rather poor performance for others. Nevertheless, it is quite obvious intuitively that in almost any POVM there is “quantum noise”, i.e. redundancy put into the j by the very quantum mechanical probability rule, most simply due to nonorthogonality of the operators aj , for example in an overcomplete system (see e.g. [15]). Our approach will thus be from the opposite end: instead of attempting the impossible, defining what “useful” means in any circumstances, we adopt a very simple criterion of uselessness: statistical independence from the measured states, because independent randomness can be generated from outside without accessing the quantum system. On the other hand we do not permit a distortion of the measurement itself, so that we are forced to consider a simulation of the original measurement by means of, first, a random choice ν of a measurement a(ν) from a list and, second, computation of a result from the outcome of this measurement and the random choice, such that the statistical distribution of these results is indistinguishable from the ones of the original measurements, on any prepared state. Because we can absorb the computation of the results into the labelling of the a(ν) , this means that we aim at finding such POVMs, whose indices are labelled by the same j as a and probabilities xν , such that a=

xν a(ν) , i.e. ∀j aj =

ν

(ν)

xν a j .

(1)

ν

(The operators must be the same because otherwise there would be states that induce distinguishable outcome distributions. Below we will introduce an element of approximation into this scheme). Why should we want to do such a decomposition, interesting though the structure (ν) exhibited (convex set of POVMs) might be mathematically? Observe that each a has (ν) (ν) of j conditional its distribution of outcomes, with the probabilities λj = Tr ρaj on ν. Shannon’s source coding Theorem [25] quantifies the amount of data in such a source as the (Shannon) entropy (ν) (ν) H λ(ν) = −λj log λj , j

“Extrinsic” and “Intrinsic” Data in Quantum Measurements

159

by compression (we note that in this paper all logs and exps are to basis 2). Hence, on average, one needs H (j |ν) := xν H λ(ν) ν

bits to faithfully compress the data (j ), given ν as side–information. This motivates the study of the function (ν) δ(ρ, a) := min H (j |ν) : a = , xν a

(2)

ν

which is the minimum data rate (in Shannon’s sense) for exact reconstruction of the data. Example 1. Look at a qubit system, C2 , with basis {|0, |1}: there let us consider the five “Chrysler” states (in analogy to the “Mercedes” trine states)

πt πt |et = cos |0 + sin |1, for t = 0, . . . , 4. 5 5 The collection a = ( 25 |et et |)t=0,... ,4 is a POVM, and we can determine its decompositions into extremal ones: these latter are given by putting weights on the |et et |, and it is straightforward that for an extremal POVM at most 3 can be nonzero (as the “Chrysler” states form a pentagon on the Bloch sphere equator). In fact, every extremal must be of the form α|et et |, β|et+2 et+2 |, β|et+3 et+3 | , t = 0, . . . , 4, indices understood modulo 5. From here one can determine the weights to be

2π 2 2π −2 1 α = 1 − cot sin ≈ 0.5528, β = ≈ 0.8944. 5 2 5 For simplicity now look at the maximally mixed state ρ = 21 11, for which it is unimportant which decomposition into these extremal POVMs is chosen, as all contributions ν will give the same Shannon entropy:

α β β δ(ρ, a) = H (j |ν) = H , , 2 2 2

β β = H 1 − β, , = H (1 − β, β) + β ≈ 1.5447. 2 2 In contrast, the main Theorem 2 below will achieve a rate of H (ρ) = 1, asymptotically. The computation of δ(ρ, a) is an interesting problem in its own right (in particular the question if anything can be gained on δ by considering multiple copies, i.e. the additivity problem), however we take a different approach, bearing in mind that the operational content of Shannon’s theorem involves block coding – i.e., a large number l of independent copies of the simple system described above, and an arbitrarily small yet nonzero error probability. Thus we are really decomposing the POVM a⊗l = aj1 ⊗ · · · ⊗ ajl j l ∈{1,... ,m}l ,

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where we have introduced the notation j l = j1 . . . jl for a string of symbols, used henceforth. And the error introduced through block compression entails that instead of Eq. (1) we will only have xν A(ν) , (3) a⊗l ≈ A = ν

where the ≈ sign is made precise to mean “average approximation of outcome statistics”: assuming an ensemble {σk , qk } with k qk σk = ρ ⊗n , there is the joint distribution of input k and output j l when applying a⊗l , γ (k, j l ) = qk Tr (σk aj l ),

(4)

(k, j l ) = qk Tr (σk Aj l ).

(5)

and likewise for A:

Then we require that, independent of the particular ensemble, 1 1

γ −

1 = |γ (k, j l ) − (k, j l )| ≤ . 2 2 l

(CP)

k,j

(It is not difficult to see that Eq. (1) raised to the l th tensor power, together with Shannon compression of the outcomes of a(ν1 ) ⊗· · ·⊗a(νl ) for the probable ν1 . . . νl yields exactly that.) Indeed we can, using the abbreviation ω = ρ ⊗l , rewrite Eq. (4) as √ √ γ (k, j l ) = Tr ω−1/2 qk σk ω−1/2 ωaj l ω , observing that the Sk = ω−1/2 qk σk ω−1/2 form a POVM on H⊗l (this fact was observed before, and used in [17] to classify all ensembles with a given average state). Similarly √ √

(k, j l ) = Tr Sk ωAj l ω , and we can rewrite and estimate the left-hand side of (CP) as follows: 1 √ √ 1 Tr Sk ω(a l − A l ) ω

γ −

1 = j j 2 2 l j

k

1 √ √ ω(a l − A l ) ω , ≤ j j 1 2 l j

with the trace norm · 1 . So (CP) is in fact implied by 1 √ ⊗l √ ⊗l ρ (Aj l − aj l ) ρ ≤ . 1 2 l

(CM)

j

Notice that the condition can be phrased in a particularly nice way introducing the quantum operations ϕ∗⊗l : σ −→ Tr (σ aj l )|j l j l |, (6) jl

∗ : σ −→

jl

Tr (σ Aj l )|j l j l |.

(7)


SOURCE

k

a

l

161

jl

DATA RECORD

ml

K

ml

Fig. 1. The source represents a number of possible states encountered by the POVM, but there is no way of knowing which is present (apart from the a priori distribution). The data produced by the measurement is then stored in a record. The rates of these processes are represented by the sizes of the different boxes and width of the data flow arrows: originally the rates of the source and of the measurement outcomes are both large

Namely, for a purification π of ρ, (CM) is easily seen to be equivalent to 1 (8) (id ⊗ ∗ )(π ⊗l ) − id ⊗ φ∗⊗l (π ⊗l ) ≤ . 1 2 The organization of the paper is as follows: In Sect. II we will present our main Theorem 2 and its proof, which is much more satisfying than results in previous work [22, 29], that can now be regarded as precursors: they are shown to easily follow from Theorem 2 in Sect. III. Sect. IV is concerned with the asymptotic optimality of our main theorem, a strong converse result, Theorem 8. After this, in Sect. V we apply our result to a kind of asymptotic normal form of completely positive trace preserving maps (operations as well as instruments), and present an extensive discussion in Sect. VI: we restate our observation from [29] that one ought to distinguish obtained data from information, give a new, conceptually simple proof of the Holevo bound, remark on the classical case of the main theorem (which includes the problem of separating extrinsic and intrinsic data under a known source ensemble), comment on the related concept of sufficient statistics, and discuss the bearing of our results on the concept of entropy exchange of an open dynamics of a system. We close with a challenging open problem. An appendix features several not widely known facts about the mixed state fidelity, in particular introducing canonical purifications of mixed states, a second appendix collects properties of typical sequences and typical subspaces, used in the main text. II. Separating Extrinsic and Intrinsic Data We want to represent (up to a small deviation as specified by the (CM) condition) a⊗l as a convex combination of POVMs A(ν) , with positive weights xν , ν = 1, . . . , N, each being defined on the set [m]l and having a small number M of sequences on which it is (ν) supported (i.e. where Aj l = 0): this is an even stronger requirement than the entropy condition we had considered in the introduction. Performing A amounts to choosing a ν (with probability xν ), and performing A(ν) , which itself can generate at most M different outcomes: the ν–part of the produced data is obviously independent of the incoming signal, while the measurement outcome (conditional on the ν chosen) contains the useful information. Our central result is: Theorem 2. For the state ρ and the POVM a define a canonical ensemble {ρˆj , λj }, with λj = Tr (ρaj ),

ρˆj =

1√ √ ρaj ρ. λj

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SOURCE

k

jl

A

DATA RECORD

M

ml

K ν RANDOMNESS

N Fig. 2. A nice way of picturing the content of Theorem 2 is in the form of an elaborate bottleneck between source and outcomes: it is supplied from outside with the extrinsic data ν, and conditional on this and the incoming k produces the intrisic data j l . Only the intrinsic data are correlated to the signal k, while the extrinsic data (though evidently an indispensable part of the whole data) is independent of it. To put it pointedly: while it is difficult and possibly ambiguous to speak of “useful data”, one can clearly identify data of no import in all respects: the unrelated randomness ν. This is put into focus by Theorem 2, and our concept of usefulness is just the remainder after extracting as much uselessness as possible

There exist POVMs A(ν) on [m]l , ν = 1, . . . , N, each supported on a set of cardinality at most M, where √ √ M = exp lI (λ; ρ) ˆ + O( l) , N = exp l H (λ) − I (λ; ρ) ˆ + O( l) , such that for A =

1 N

ν

A(ν) condition (CM) is satisfied.

The characteristic constant in the exponent is I (λ; ρ) ˆ = H (ρ) −

λj H (ρˆj ),

j

the entropy defect of the ensemble (Lebedev and Levitin [20]), or the quantum mutual information between a sender producing the letter j with probability λj and a receiver getting the letter state ρˆj (see [16, 24]). It is the difference between the von Neumann entropy H (ρ) = −Tr ρ log ρ of the ensemble and its conditional entropy H (ρ|λ) ˆ = j λj H (ρˆj ). Observe that not only ρ can be recovered from this ensemble (as its average), but also the POVM a: aj = ρ −1/2 λj ρˆj ρ −1/2 . This construction is known as the “square root measurement” [14], or “pretty good measurement” [10]. We shall give the proof of Theorem 2 in a minute, after a few preparations. A central part of the argument is the following auxiliary result from [2] that we state separately: Lemma 3 (Ahlswede, Winter [2], Thm. A.19). Let X1 , . . . , XM be independent identically distributed (i.i.d.) random variables with values in the algebra L(K) of linear operators on K, which are bounded between 0 and 11. Assume that the average


163

EXµ = σ ≥ s 11. Then for every 0 < η < 1/2,  

M 1  η2 s Pr Xµ ∈ [(1 ± η)σ ] ≤ 2 dim K exp −M , M  2 ln 2 µ=1

where [(1 ± η)σ ] = [(1 − η)σ ; (1 + η)σ ] is an interval in the operator order: [A; B] = {X ∈ B(K) : A ≤ X ≤ B}. We shall use the concepts of typical and conditionally typical subspaces in the form of [28], which we collect in Appendix B. l let Proof of Theorem 2. Define the following operators: for j l ∈ Tλ,δ l l l l ξj l = lρ,δ lρ,δ ˆ (j )ρˆj l ρ,δ ˆ (j )ρ,δ .

We choose δ = m 2d , so that

l ) ≥ 1 − , S := λ⊗l (Tλ,δ

(9)

≥ 1 − ,

(10)

Tr ξj l

which is true by Chebyshev’s inequality and Eqs. (B.2) and (B.3), specifying later. Notice that in this way Tr ω ≥ 1 − 2 for ω = λj l ξj l . l j l ∈Tλ,δ

By Eq. (B.8) we have lρ,δ ωlρ,δ ≥ αlρ,δ , √ with α = exp(−lH (ρ) − O( l)). Define now to be the projector onto the subspace spanned by the eigenvectors of ω with eigenvalue ≥ α. By construction we find Tr ≥ 1 − 3 for = S −1 ω . (ν) l , ν = 1, . . . , N, Now let ξj l = ξj l and define i.i.d. random variables Jµ ∈ Tλ,δ µ = 1, . . . , M by λj l =: Lj l . Pr{Jµ(ν) = j l } = S l . That is, we consider N independent sets of M independent choices each, from Tλ,δ Observe that = EξJ (ν) , the expected value of the random operators ξJ (ν) . µ µ We shall show that with high probability the following conditions hold: M 1 ξJ (ν) ∈ [(1 ± )], µ M

(Iν )

N,M 1 δJ (ν) ∈ [(1 ± )L ]. µ NM

(II)

µ=1

for all ν, and

ν,µ=1

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This is most easily seen with the help of Lemma 3: according to it

3α , Pr{¬Iν } ≤ 2Tr exp −M 2β ln 2

with

2γ , −N M 2 ln 2

Pr{¬II} ≤

l 2|Tλ,δ | exp

√ l } ≥ exp −lH (λ) − Kmδ l , γ = min{λj l : j l ∈ Tλ,δ

compare Eq. (B.8). Choosing M and N according to the theorem’s statement will force the sum of these probabilities to be less than 1, i.e. with positive probability all the events (Iν ) and (II) happen. (ν) Let us assume we fix now values for the Jµ such that all equations (Iν ) and (II) are satisfied. Then we may define operators   S 1  (ν) ξJ (ν) ω−1/2 ω−1/2  Aj l = µ 1+ M (ν) µ:Jµ =j l

=

(ν)

S |{µ : Jµ = j l }| −1/2 ω ξj l ω−1/2 . 1+ M

We check that for each ν these form a sub–POVM (i.e., a collection of positive operators with sum upper bounded by 11): using (Iν ) and the definitions of and ω we find   M (ν) √ √ S 1 ω Aj l  ω = ξJ (ν) ≤ S = ω ≤ ω ≤ ω. µ 1 + M l µ=1

j

Finally, we check that condition (CM) holds: it is sufficient to do this for the sub–POVM constructed, because then we can distribute the remaining operator weight to fill up to 11 arbitrarily. We calculate directly from the definitions: 1 √ √

ω(aj l − Aj l ) ω 1 2 jl (µ) 1 S|{νµ : Jµ = j l }| = ξj l λj l ρˆj l − 2 (1 + )NM l

1 j 1 1 1 1 ≤ (1 − S) + Lj l ρˆj l − ξj l 1 + L − δJ (ν) µ 2 2 2 N M l νµ l j ∈Tλ,δ

1 ≤ + Lj l ρˆj l − ξj l 1 + 2 2 2 l j l ∈Tλ,δ

1 1 ≤+

ρˆ l − ξj l 1 + ξj l − ξj l 1 . Lj l 2 j 2 l l j ∈Tλ,δ

1

(11)


165

By the definition of ξj l , using Eq. (10) and Lemma 4 below, we can bound the first of √ the two terms in brackets by + 2. It remains to estimate the second: consider = Lj l ξj l , l j l ∈Tλ,δ

and recall that ξj l = ξj l , hence = . By construction we have l j l ∈Tλ,δ

Lj l Tr ξj l ≥ 1 − 3,

thus, using Lemma 4 with each of the ξj l and employing concavity of the square root function, we end up with l j l ∈Tλ,δ

√ 1 Lj l ξj l − ξj l 1 ≤ 6, 2

which allows us to estimate (11) by 2 +

√ √ 2 + 6.

Here is the lemma that we needed in the proof: it says that a POVM element that is likely to respond to a state acts “gently” on it in the sense of little disturbance. Lemma 4 (Lemma V.9 of [28]). For a state ρ and and an operator 0 ≤ X ≤ 11, if √ √ √ Tr (ρX) ≥ 1 − λ, then ρ − Xρ X ≤ 8λ. 1 The same is true if ρ is only a subnormalized density operator. III. Previous Approaches The question addressed in the present paper of quantifying the “amount of information obtained by a quantum measurement” has been posed before, in the works [22] and [29], with mathematical modellings different from ours, though there is an evolution leading from the first to the present: In [22] the POVM a was assumed to maximize a certain Bayesian gain (there called “fidelity”) pi Tr (ρi aj )Fij , F (a) = ij

to achieve the optimal (i.e. maximal) value Fopt . On blocks of length l the gain (or fidelity) function was extended by defining Fi l j l = 1l lk=1 Fik jk . This definition has the easily checked property that the gain on blocks of length l, F (a⊗l ) = pi l Tr (ρi l aj l )Fi l j l , (12) il j l

equals the single letter expression F (a). Note that in this way the maximum Bayesian gain is still Fopt (which can be seen from Eq. (13) below). Then the following theorem was shown:

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Theorem 5 (Massar, Popescu [22]). For > 0 and l large enough there exists a POVM A with fidelity F (A) ≥ Fopt − and M ≤ exp l(H (ρ) + ) ,

many outcomes among the j l .

This result was interpreted as saying that about any property of the ensemble states, as encoded in the Bayesian gain matrix Fij , one can learn at most one bit per qubit. In [29] this was extended and clarified as follows: observe that for any POVM A = (Ajµl )µ=1,... ,M one has F (A) =

pi l

µ

il

=

1 l

k=1

l k=1

i

1 Fik jµk l l

Tr (ρi l Ajµl )

pi Tr (ρi (A|k)j )Fij ,

where (with [l] = {1, . . . , l})    (A|k)j = Tr =k  ρ ⊗[l]\k ⊗ 11k Ajµl  = ρ −1 Tr =k ρ ⊗l 

=

µ: jµk =j



(13)

j





√ √ ρ −1 Tr =k  ρ ⊗l  Ajµl  ρ ⊗l  ρ −1 .

µ: jµk =j

 Ajµl  (14)

µ: jµk =j

For each k, the collection ((A|k)j )j =1,... ,m obviously is a POVM on H. We may assume (as we shall do in the sequel) that the |Fij | are bounded by 1: then the fidelity condition of Theorem 5, reading |F (A) − F (a)| ≤ , is implied by ∀k

pi Tr ρi (A|k)j − pi Tr (ρi aj ) ≤ .

(C0)

(C1)

ij

This is itself implied by ∀k∀i

|Tr (ρi (A|k)j ) − Tr (ρi aj )| ≤ ,

(C2)

j

which in turn follows from ∀k

(A|k)j − aj ≤ ,

(C3)

j

with the operator norm · . It was then proved Theorem 6 (Winter, Massar [29]). Given > 0, there exists a POVM A = (Ajµl )µ=1,... ,M with √ M ≤ exp lI (λ; ρ) ˆ +C l (where C is a constant depending only on , d and m), and such that (C3) is satisfied.


k

SOURCE

A

167

µ

M

DATA REC.

M

K Fig. 3. In [22] and [29] the original POVM is replaced by an “equivalent” one (as made precise in Theorems 5 and 6) with many fewer outcomes. So, POVM and data record need much less rate of processing and storage, respectively. Of course, compared to Theorem 2 we lose many potential measurement results in constructing the new POVM

This theorem is in an asymptotic sense the best possible (such an optimality was missing in [22]): Theorem 7 (Winter, Massar [29]). Let 0 < ≤ (λ0 /2)2 , with λ0 = minj λj . Then for any POVM A = (Ajµl )µ=1,... ,M such that (C3) holds, one has 2 3 M ≥ exp l I (λ; ρ) ˆ + 2 log 2 . λ0 λ0 d Here we want to show that the Theorems 5 and 6 may be obtained as corollaries of Theorem 2. (ν) Proof of Theorem 5. Choose xν and A according to Theorem 2, such that condition (CM) is satisfied for A = ν xν A(ν) , with some > 0 (which implies that also (CP) is satisfied with the same ). Then, assuming without loss of generality that |Fij | ≤ 1, we get immediately out of Eq. (12) that

|F (A) − F (a⊗l )| ≤ . Since we assume that a maximizes F we conclude, using linearity of F in the POVM: Fopt − = F (a⊗l ) − ≤ F (A) = xν F (A(ν) ). ν

This finally means that for at least one ν, F (A(ν) ) ≥ Fopt − ,

√ ˆ + O( l) ≤ which is what we wanted to prove: recall that A(ν) has M ≤ exp lI (λ; ρ) √ exp lH (ρ) + O( l) many outcomes. Note that the latter estimate is met with equality if and only if a is maximally refined (i.e., consists of rank–1 operators only), so regardless of a, H (ρ) is the rate of intrinsic data of any probing of the ensemble. Note further that our derivation does not depend on the particular structure of the block–fidelity: obviously we can as well conclude for any ensemble {σk , qk } with average ω and any fidelity matrix Fkj l that qk |Tr (σk Aj l ) − Tr (σk aj l )|Fkj l ≤ F , F (A) − F (a⊗l ) ≤ kj l

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with F := maxkj l |Fkj l |. If now F ≤ O F (a⊗l ) for l → ∞ then we get (for sufficiently large l) F (A) ≥ (1 − )F (a⊗l ). Of course, as explained in Eqs. (C0)–(C3), Theorem 5 is really a corollary of Theorem 6. So, we continue to prove the latter: (ν) , ν = 1, . . . , N like in Proof of Theorem 6. Assume that a collection of POVMs A Theorem 2 is chosen, with probabilities xν , such that A = ν xν A(ν) satisfies (CM). Define i.i.d. random variables T1 , . . . , TQ , each with Pr{Tq = ν} = xν . We want to study the random POVMs A(Tq ) , and especially their mean

1 (Tq ) A= A . Q Q

q=1

Observe that EA = EA(Tq ) = A . Recall the definition of marginal POVMs. Obviously, by linearity of this definition, we have Q 1 (Tq ) (A|k) = (A |k) Q q=1

and

E(A|k) = E(A(Tq ) |k) = (A |k).

From condition (CM) and the monotonicity of the trace norm under partial trace we get now, for every k, 1 √ √ ρ (A |k)j − aj ρ ≤ . 1 2

(15)

j

√ √ Denoting the smallest nonzero eigenvalue of √ √ any of the ρaj ρ by u, and√choosing √ small enough, this assures that ρ(A |k)j ρ restricted to the support of ρaj ρ is lower bounded by u/2. Then we can apply Lemma 3 and obtain

√ 2u √ √ √ Pr ρ(A|k)j ρ ∈ (1 ± ) ρ(A |k)j ρ on supp ρˆj ≤ 2d exp −Q . 4 ln 2 Thus we can estimate the sum of these probabilities over all k = 1, . . . , l and j = 1, . . . , m to less than 1 if Q≥1+

4 ln 2 log(2dlm). 2u

This implies that there exist actual values of the Tq such that for all k, 1 √ √ √ √ ρ(A|k)j ρ|supp ρˆj − ρaj ρ ≤ 2, 1 2 j

(16)


169

√ √ where we observed that the ρ(A |k)j ρ all have trace at most 1, and have used √ √ Eq. (15). Hence we get (with kj = Tr (ρ(A|k)j ) and kj ! Pkj = ρ(A|k)j ρ) kj Tr ! Pkj |supp ρˆj ≥ 1 − 2, j

and using Lemma 4, this gives 1 √ √ √ √ √ ρ(A|k)j ρ|supp ρˆj − ρ(A|k)j ρ ≤ 2 . 1 2

(17)

j

Now (16) and (17) yield 1 √ √ √ ρ (A|k)j − aj ρ ≤ 2 + 2 . 1 2 j

Denoting the minimal eigenvalue of ρ by r (which we assumed to be positive) this readily implies √ 1 (A|k)j − aj ≤ 2 + 2 , 2 rd j

and we are done, since A has only MQ many possible outcomes.

IV. Strong Converse In this section we prove the asymptotic optimality of the separation of the measurement from Theorem 2. To be precise, it is Theorem 8. Whenever there are POVMs A(ν) on [m]l , ν = 1, . . . , N, each supported on at most M elements, and probability weights xν > 0, such that A = ν xν A(ν) satisfies condition (CM), for some < 1, then √ √ M ≥ exp lI (λ; ρ) ˆ − O( l) , MN ≥ exp lH (λ) − O( l) , where the constants depend only on . Proof. Let us begin with the second inequality: by construction the set R ⊂ [m]l of possible outcomes of A has cardinality at most MN . Denoting by the distribution of outcomes according to A, i.e. j l = Tr ρ ⊗l Aj l , from (CM) we get immediately 1 ⊗l

λ − 1 ≤ , 2 which in turn implies

λ⊗l (R) ≥ (R) − = 1 − .

(18)

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A. Winter

l , By a well known trick [30] the lower bound now follows: we consider R = R ∩ Tλ,δ with δ = 1− 2m , whence we have, using Chebyshev’s inequality

λ⊗l (R ) ≥

1− . 2

Using the fact (compare Eq. (B.8)) √ λj l ≤ exp −lH (λ) + Kmδ l ,

l ∀j l ∈ Tλ,δ

we conclude MN ≥ |R| ≥ |R | ≥

√ 1− exp lH (λ) − Kmδ l . 2

Now for the first inequality: introduce the ensembles {! Pj l , j l }j l with (ν)

(ν) (ν) j l ! Pj l =

(ν)

√ (ν) √ ωAj l ω,

all of which have average ω. Then we define the (subnormalized) density operators (ν) l (ν) l l " Pj l = lρ,δ ˆ (j )Pj l ρ,δ ˆ (j ),

l l l ρ˜j l = lρ,δ ˆ (j )ρj l ρ,δ ˆ (j ), l , with δ = for j l ∈ Tλ,δ

4mn 1− .

Then by Chebyshev inequality and Eq. (B.2)

1 1− λj l ρ˜j l ≤ , ω − 2 2 l l j ∈Tλ,δ 1

while from (CM) we get (ν) (ν) 1 1+ " λj l ρ˜j l − xν j l Pj l ≤ =: . 2 l l 2 l ν j ∈Tλ,δ j l ∈Tλ,δ

(19)

1

These immediately imply ν

xν

l j l ∈Tλ,δ

j l Tr " Pj l ≥ 1 − , (ν)

(ν)

so there exists at least one ν such that (ν) (ν) j l Tr " Pj l ≥ 1 − . l j l ∈Tλ,δ

Now consider the (subnormalized) density operators # # (ν) (ν) (ν) ! j l = ! Pj l lρ,δ Pj l , ˆ

(20)


171

which evidently satisfy θ :=

l j l ∈Tλ,δ

(ν)

(ν)

j l j l ≤

(ν) (ν) Pj l = ω. j l !

jl

Denoting with the projection onto the support of θ and inserting Tr j l = Tr " Pj l , we arrive at Tr (ω) ≥ 1 − , (ν)

(ν)

from where we conclude

√ rank = Tr ≥ exp lH (ρ) − O( l) .

This follows by a standard reasoning (which we take from [28]): for F = lρ,δ lρ,δ , choosing δ large enough, we get 1 − Tr lρ,δ ωlρ,δ = Tr (ωF ) ≥ . 2 By Eq. (B.8) the inequality follows. √ (ν) On the other hand each of the j l has rank at most exp lH (ρ|P ˆ ) + O( l) , and we deduce our claim. We may relax a bit the condition of the theorem regarding the parameter M: if we allow the different POVMs A(ν) to have different numbers Mν of possible outcomes, then we can prove the slightly stronger estimate √ xν Mν ≥ exp lI (λ; ρ) ˆ − O( l) M := ν

(while the second inequality obviously holds for ν Mν ). To see this go back to Eq. (20) and observe that by a Markov inequality argument √ √ (ν) (ν) j l Tr " Pj l ≥ 1 − ≥ 1 − , Pr x ν : whence the claim directly follows. Remark 9. While in the above proof we assumed the property (CM) for < 1, we conjecture that (CP) for all sources with average ω, with < 1, is sufficient to arrive at its conclusion. Let us inspect this possibility along the lines of the proof: crucial were the estimates (18) and (19), the former being an immediate consequence of (CP), so we would have to show this only for the latter. However, this demonstration has escaped us so far. Finally, a comment on why this converse is strong: optimality of Theorem 2 is proved already by our observation in the previous section that it implies Theorem 6, and the lower bound of Theorem 7. However, closer inspection of this lower bound reveals that it coincides with the upper bound only in the limit → 0. For positive it leaves room for a tradeoff between compression and error (not untypical for the type of error concept we had used). This is known in information theory as a weak converse [30]. The strong converse in contrast shows optimality of the upper bound in the asymptotic limit l → ∞, with any bounded away from 1.

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V. Asymptotic Decomposition of Instruments and Operations An interesting generalization of our main theorem arises from the point of view that POVMs are just a special case of general open dynamics: the most general form of evolution is a completely positive, trace preserving linear map ϕ∗ from states on H to states on K. Such a map can (non–uniquely) be represented in the Kraus form ϕ∗ : π −→

m j =1

Vj πVj∗ ,

(21)

where Vj : H → K are C–linear and j Vj∗ Vj = 11. The representation can be made unique by considering it as a partial measurement, and including the outcome j : extend the output system to K ⊗ J , and modify the map ϕ∗ to " ϕ∗ : π −→

m j =1

Vj πVj∗ ⊗ |j j |.

(Technically this will amount to a change of the Kraus operators, too, but we will not need the details here.) This is the notion of an instrument (Davies and Lewis [7]). One can see that it is representable in Kraus form, too, so we will in the sequel always look at a particular Kraus representation. In analogy to the question about POVMs of this work we would like to approximate (ν) ϕ∗⊗l by the average of some ∗ , ν = 1, . . . , N, each of which should have a Kraus representation with a small number of contributing operators. As is well known this number is the dimension of the ancillary system (environment) sufficient to emulate the effect of the operation by a unitary interaction and subsequent partial trace. Its logarithm is an upper bound on the “information leakage” from the system to the environment. Note that (apart from looking at approximation) we are considering here the problem of convex decomposition of completely positive maps, like we did before for POVMs. Of course, every completely positive map has a decomposition into such extremal ones, with possibly fewer terms in the Kraus representation. For this one can employ a theorem of Choi [6], saying that ϕ∗ from Eq. (21) is extremal if and only if the family of operators Vj∗ Vk is linearly independent (in particular, then m ≤ d). We show now how to solve this problem as a consequence of Theorem 2, with an additional reasoning mainly directed to quantum state fidelities: Formally, we are looking for a family of maps ⊗l ⊗l (ν) ∗ : B(H ) −→ B(K ),

σ −→

M

Wµ(ν) σ Wµ(ν)∗

(22)

µ=1

(ν) and probabilities xν such that for ∗ = ν xν ∗ and any ensemble {σk , qk } with average ω = ρ ⊗l the following condition holds: k

1 qk ϕ∗⊗l (σk ) − ∗ (σk ) ≤ . 1 2

(CO)


173

In fact, there is an appealing way to state them all together, and strengthen the content at the same time: for a purification π of ρ on an extended system H ⊗ H we ask for 1 (CO*) (ϕ∗ ⊗ id)⊗l (π ⊗l ) − (∗ ⊗ id⊗l )(π ⊗l ) ≤ . 1 2 Indeed, this implies (CO): just observe that by choosing a POVM (Tk ) on H⊗l one can “induce” any ensemble {σk , qk } on H⊗l for ω, in the following sense: qk σk = Tr H⊗l π ⊗l (11 ⊗ Tk ) . How to do this is explained in detail in [17] (or see Appendix A below). Note that this generalizes the implication of (CP) from (CM), discussed earlier, when we view the POVMs as the quantum operations Eqs. (6) and (7). Conversely, assuming (CO) for all ensembles for ω does unfortunately not imply (CO*) with a comparable error parameter. (Examples are not hard to construct for which (CO) holds with a small while the bound in (CO*) is close to 1.) With ϕ∗ there is associated the POVM a = (aj = Vj∗ Vj : i = j, . . . , m), and with this goes the ensemble {ρˆj , λj }, as before. Theorem 10. With the above notation and > 0 there exist quantum operations in the form of Eq. (22), with √ √ ˆ + O( l) , M ≤ exp lI (λ; ρ) ˆ + O( l) , N ≤ exp l H (λ) − I (λ; ρ) (ν) and such that ∗ = N1 ν ∗ satisfies (CO*). These bounds are asymptotically best possible if ϕ∗ is an instrument. Proof. Let A(ν) and xν be the POVMs and probabilities constructed in Theorem 2 from a⊗l and ω = ρ ⊗l , and let A = ν xν A(ν) . We use the notation from the proof of this theorem and from Sect. IV: √ (ν) √ √ √ (ν) (ν) (ν) ρaj ρ = λj ρˆj , ωAj l ω = j l ! x ν j l . Pj l , j l = ν

(ν) the ! Pj l

Note that by the proof of Theorem 2 either are 0 or equal to ! Pj l := Introduce the unitaries Uj by the polar decomposition √ √ √ Vj ρ = Uj ρVj∗ Vj ρ = Uj λj ρˆj ,

S 1+ ξj l .

(23)

(ν)

and let Uj l = Uj1 ⊗ · · · ⊗ Ujl . Now define Wj l by letting (ν) √ Wj l ω

#

(ν) (ν) = Uj l j l ! Pj l ,

(24)

and observe that for fixed ν only M of them are nonzero, and that for fixed j l these (ν) are all multiples of each other. Hence these operators define a quantum operation ∗ (ν) according to the theorem, and ∗ = ν xν ∗ .

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√ With these definitions we check that (CO*) is satisfied: using π ⊗l = ( ω ⊗ √ 11⊗l )|I I |( ω ⊗ 11⊗l ) (see Lemma 14 in Appendix A) and Eqs. (23) and (24) we calculate ⊗l (ϕ∗ ⊗ id⊗l )(π ⊗l ) − (∗ ⊗ id⊗l )(π ⊗l ) 1 (ν) (ν)∗ ≤ xν Wj l ⊗ 11⊗l π ⊗l Wj l ⊗ 11⊗l Vj l ⊗ 11⊗l π ⊗l Vj∗l ⊗ 11⊗l − ν

jl

λ l = ρˆj l ⊗ 11⊗l |I I | ρˆj l ⊗ 11⊗l j jl

! −j l Pj l ⊗ 11⊗l |I I | ! Pj l ⊗ 11⊗l 1 ⊗l ⊗l ≤ λ − 1 + λj l ρˆj l ⊗ 11 |I I | ρˆj l ⊗ 11⊗l jl

⊗l ⊗l ! ! − |I I | Pj l ⊗ 11 Pj l ⊗ 11 .

1

(25)

1

The last line here is estimated as follows: the first term is bounded by 2 (see the proof of Theorem 8), and for the other we use Lemma 14 in Appendix A: observe that for each j l the two terms inside the trace norm are the canonical purifications of ρˆj l and ! Pj l , respectively. Thus we get

⊗l ⊗l ! ! ρˆ l ⊗ 11⊗l |I I | ρˆ l ⊗ 11⊗l − |I I | P P ⊗ 1 1 ⊗ 1 1 l l j j j j 1 √ 4 ! ≤ 2 2 ρˆj l − Pj l 1 , and using concavity of the root function and the estimate of Eq. (11) we can upper bound the last line of Eq. (25) by O( 1/8 ). If ϕ∗ is an instrument any approximate convex decomposition of ϕ∗⊗l implies a similar decomposition for the POVM a⊗l . Hence Theorem 8 gives the optimality of the bounds for M and N . Interestingly, the bounds of Theorem 10 depend on the Kraus representation (21) of the map ϕ∗ : all other such representations are related by unitary transforms, i.e. VJ σ VJ∗ ϕ∗ (σ ) = J

if and only if VJ =

UJj Vj ,

j

with a unitary matrix (UJj )Jj of complex numbers. (This is essentially a consequence of the uniqueness up to unitaries of the Stinespring dilation [26] of ϕ, which implies the Kraus representation. This fact is also discussed in detail in [23]). This motivates the introduction of (ρ; ϕ∗ ) :=

min

Kraus repr. of ϕ∗

I (λ; ρ), ˆ

(26)


175

i.e. the minimum rate of the parameter M in decompositions of ϕ∗ according to Theorem 10. Note that, according to [23], the minimum of H (λ) over all Kraus representations is exactly Se , the entropy exchange of the map ϕ∗ (with respect to ρ). For a discussion see Subsect. VIE below. VI. Discussion We have introduced a separation into extrinsic and intrinsic data of a quantum measurement. It was shown to have definite minimal rates for either of these, and that it encompasses all previously known results on “meaningful” data in quantum measurements. A particular advantage of Theorem 2 before Theorems 5 and 6 is that it not even requires a new POVM (which might be experimentally difficult to realize). Instead, it can be understood as a mere re–interpretation of the data delivered by a⊗l : in fact, by (ν) our construction in the proof of Theorem 2 for all ν and j l either Aj l is 0 or very close to a multiple of aj l , in the sense of (CM). Hence the random variable N, defined as a function of j l : (ν) Tr ωA xν jl (ν) Pr N = ν|j l = , (27) Tr ωAj l = xν λj l Tr (ωaj l ) (up to a scaling factor, close to 1 for typical j l ), is almost independent from the source ensemble {σk , qk } in (CP). More precisely, 1 (ν) qk xν Tr (σk Aj l ) − qk Tr (σk aj l ) Pr{N = ν|j l } ≤ , 2 l kνj

and in fact, we even have 1 √ √ √ (ν) √ xν ωAj l ω − Pr{N = ν|j l } ωaj l ω ≤ . 1 2 l νj

This means that one can reproduce the statistics of the whole diagram in Fig. 2 from the outcomes of a⊗l , by inventing the ν distributed according to Eq. (27). This gives a new view on the extrinsic/intrinsic separation: rather than replacing the original POVM by a fancy construction, one can from the original data j l compute the extrinsic data ν, and conditional on that the intrinsic part. Then one can sucessfully pretend that this separation was delivered by the mixture of the POVMs A(ν) .

A. Data vs. information. One (as it turns out, rather careless) interpretation of our result could be that the “useful” information produced by the POVM a amounts to I (λ; ρ). ˆ This in itself is not yet precise, so let‘s fix “information” to mean “communicable informa tion” in the sense of Shannon [25]: for any source {σi , µi } with average i µi σi = ρ the source and measurement outcome are random variables X and Y with a joint distribution Pr{X = i, Y = j } = µi Tr (σi aj ),

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and the mutual information of these is I (X ∧ Y ) = H (X) + H (Y ) − H (XY ). We repeat here the discussion of [29] regarding the relation between this quantity and I (λ; ρ): ˆ Observe first that the joint distribution of X and Y can be rewritten as √ √ Pr{X = i, Y = j } = Tr ρ −1/2 µi σi ρ −1/2 ρaj ρ = λj Tr (ρˆj Si ), where the Si = ρ −1/2 µi σi ρ −1/2 form a POVM (compare [17] where this correspondence between POVMs and ensembles was used to classify ensembles with given density matrix). But here the Holevo bound [13] applies, with the ensemble {ρˆj , λj }, and thus we have proved: Theorem 11. Let {σi , µi } be any ensemble whose average state i µi σi equals ρ. Define random variables X, Y with joint distribution Pr{X = i, Y = j } = µi Tr (σi aj ) (this is the probability for σi to occur and that j is observed on this state). Then I (X ∧ Y ) ≤ I (λ; ρ). ˆ Note that in general maximization over the ensemble {σi , µi } (yielding the accessible information Jρ (a) = Iacc (λ; ρ), ˆ because in the above proof it corresponds to an information maximization over the POVM Si ) does not achieve the upper bound: see [13], where it is shown that it does if and only if all the ρˆj commute. Furthermore, by a result from [12], Jρ ⊗l (a⊗l ) = lJρ (a), hence the gap remains even asymptotically! For further discussion of this point we refer the reader to [29], Sect. VII C. We record here only the consequence that one ought to distinguish between data (collected by measurement) and information (about a property of the states): the latter is never larger than the former, and typically in quantum situations it is strictly less. However, this seems nothing to worry about: after all, this is an observation quite familiar from our experience, though it is worth stressing that in the present context it is a purely quantum phenomenon. Peter Shor has remarked the notable fact that in the presence of entanglement, however, this distinction disappears: the entanglement–assisted capacity [3] for the quantum–classical channel that is represented by our POVM, i.e. ϕ∗ from Eq. (6), with the average of the sent symbols required to be ρ (this means that in the formula for the entanglement–assisted capacity one has to put a purification of ρ) coincides with our I (λ; ρ)! ˆ In fact, our result can be understood as a weak version of the conjectured “Quantum Reverse Shannon Theorem” [3, 4], for quantum–classical channels. To end this part of the discussion note that the bound of Theorem 2 in the case of a maximally refined measurement is simply the von Neumann entropy H (ρ) of the source, and this regardless of the nature of the POVM and of the source. In this sense, there is “democracy among measurements”, at least the maximally refined ones.


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It is thus appealing to view our result as a dual to the creation of a density operator by mixing pure states: it is well known that in any representation ρ = i pi σi , with pure states σi , H (p) ≥ H (ρ), with equality iff the σi are mutually orthogonal eigenstates of ρ: hence, H (ρ) is the minimum entropy needed to generate ρ. In the present work we identify H (ρ) as the maximum entropy of measurement data correlated to ρ. B. Holevo bound. Here we show how to turn around the previous argument to actually prove the Holevo information bound. The statement is as follows: Theorem 12 (Holevo [13]). Let {ρˆj , λj }j =1,... ,m be an ensemble of states with average ρ = j λj ρˆj , and (Si )i=1,... ,n a POVM. Define the joint distribution of random variables Y , X to be Pr{Y = j, X = i} = λj Tr (ρˆj Si ). Then we have the inequality I (Y ∧ X) ≤ I (λ; ρ) ˆ = H (ρ) −

(28)

λj H (ρˆj ).

j

Proof. To begin with, observe that Eq. (28) may be rewritten as Pr{Y = j, X = i} = pi Tr (σi aj ), √ √ with aj = ρ −1/2 λj ρˆj ρ −1/2 and the ensemble {σi , pi }, where pi σi = ρSi ρ. Now consider i.i.d. realizations X1 , Y1 , . . . , Xl , Yl of the pair X, Y . We shall apply Theorem 2 to a⊗l and ρ ⊗l , with parameter 0 < < 1. Hence, for A = ν xν A(ν) and the ensemble {σi l , pi l } the condition (CP) holds. Let us define random variables , ϒ by (ν)

Pr{ϒ = j l , = i l , N = ν} = xν pi l Tr (σi l Aj l ), so that

Pr{ϒ = j l , = i l } = pi l Tr (σi l Aj l ).

Then we may calculate (with f () := (log m + 2 log n)) lI (Y ∧ X) = I (X l ∧ Y l ) ≤ I ( ∧ ϒ) + lf () + 4 ≤ I ( ∧ Nϒ) + lf () + 4 = I ( ∧ N) + I ( ∧ ϒ|N) + lf () + 4 ≤ 0 + log M + lf () + 4 √ ≤ lI (λ; ρ) ˆ + O( l) + lf () + 4. Only classical entropy relations have been used: line 2 is by Lemma 13 stated below, line 3 is by data processing, as υ is a function of ν and µ, line 4 is a standard identity, and line 5 by independence of ν and ξ and the standard inequality I (ξ ∧ µ|ν) ≤ H (µ). Now divide by l and let l → ∞: I (Y ∧ X) ≤ I (λ; ρ) ˆ + (log m + 2 log n). As > 0 was arbitary, the theorem follows.

Lemma 13 (Fano [8]). Let P and Q be probability distributions on a set with finite cardinality a, such that 21 P − Q 1 ≤ λ. Then |H (P ) − H (Q)| ≤ λ log a + 2H (λ, 1 − λ).

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The reader may want to compare this proof to our earlier one in [29]: despite similarities they are conceptually completely different! In fact, there we introduced the Holevo mutual information as a certain fidelity measure (which may seem slightly artificial) and applied Theorem 6, while here we directly exploit the “bottleneck” nature of our main result (compare again Fig. 2), thus providing a much more natural approach. C. Fixed source ensemble and classical case. Our approach has concentrated on universal properties of the POVM, leaving the source as free as possible. What happens if we fix the source {ρi , pi }? Note that the whole situation is fully classical now, as we only have to regard the correlation between source issues X = i and measurement results Y = j. Thus it is modelled by the classical case of the initial problem: the source is {|ii|, pi }, and the POVM b consists of operators bj = Tr (ρi aj )|ii|. i

This model has the same joint statistics of i and j as the above described one (most generally, bj can be any operator with eigenbasis {|i}). Now observe the following: as long as the POVMs A(ν) are diagonal in the basis l {|i }, too (this is the classicality condition for the POVMs), the validity of (CP) for all ensembles with average P = pi |ii| i

is implied by its validity for the ensemble {|ii|, pi }. This is because source states ρi and i |ii|ρi |ii| produce the same statistics, so only sources consisting of mixtures of the |ii| have to be considered. The condition (CP) for them clearly is implied by its validity for {|ii|, pi }. At this point Theorems 2 and 8 can be applied: because the induced ensemble for source state P and POVM b is {σˆ j , λj }, with λj =

pi Tr (ρi aj ) = Tr (ρaj ),

i

σˆ j =

1 pi Tr (ρi aj )|ii|, λj i

we obtain I (X ∧ Y ), that is the Shannon mutual information between the source and the measurement, as the rate of intrinsic data. More precisely, we can perform a data separation by postprocessing, according to the prescription of the beginning of this section, Eq. (27), into extrinsic ν, almost independent of i l , and intrinsic j l depending on i l and ν. However, this is not exactly what we set out to do initially: Theorem 2 allows us to decompose the bj l into convex combinations of operators (ν) (ν) βj l |i l |i l i l |, Bj l = il

but it is not clear that these can be obtained from POVMs A(ν) , in the sense that (ν) (ν) ∀ν∀j l ∀i l βj l |i l = Tr ρi l Aj l .


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(ν) For this to hold the vectors βj l |i l l (for all j l ) must belong to the cone spanned by the i vectors (ψ|ρi l |ψ)i l . It is conceiveable that under this condition the obtainable intrinsic data rate increases. We have to leave this interesting question for the moment. For classical sources and measurements we thus obtain that intrinsic data equals mutual information. On the other hand, we can come back to their being distinct in truly quantum situations: we pointed out in Subsect. VIA that the maximum of I (X ∧ Y ) over all sources with average ρ gives the accessible information Iacc (λ; ρ) ˆ of the ensemble {ρˆj , λj }, which in general is less than I (λ; ρ). ˆ The difference can be accounted for by considering that the sources in this maximization are of the special i.i.d. type (on l–blocks), while (CM) implies (CP) even for sources of entangled states, as long as their average is ω = ρ ⊗l . This should be viewed especially in the light of the conjecture implied in Subsect. VIF. D. Sufficient statistics. The reader familiar with classical statistical theories may have been reminded by our above discussion of the concept of sufficient statistics, at least when the quantum source and the observation are essentially classical, i.e. when all the ρi and aj commute: the former are then just probability distributions and the latter form a statistical decision rule, with distribution of j conditional on i denoted q(j |i). As there is also a distribution pi on the i we have here a statistical model in the sense of estimation theory (we refer the reader to [19] for detailed explanations). We will consider the values of i and j as random variables: then a sufficient statistics is a random variable k which is a function of j (whose distribution conditional on i we denote q(k|i)), ˜ such that the distribution of j conditional on k is independent of i: Pr{j |k} = Pr{j |k, i}

∀i.

Let us denote these conditional probabilities by r(j |k). This implies that we can simulate the distribution of j conditional on i from k: r(j |k)q(k|i). ˜ q(j |i) = k

In words, to each entry k of the new data record there exists a distribution on the j of the original data record such that the latter’s distribution is recovered as a convolution; in terms of stochastic maps q is factorized into q˜ and r: q˜

r

i −→ k −→ j. On the other hand, our Theorem 2 provides something appearing to be dual to this (apart from holding only approximately and in an asymptotic setting; these things are easily introduced in sufficient statistics, too): a random variable ν with distribution x, independent of i and j , and conditional on it a stochastic map aν (j |i) such that xν aν (j |i). q(j |i) = ν

In a diagram:

$

"ν Q

ν ↓ %&

' Rν

i −→ µ −→ j.

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Like k in the case of sufficient statistics, the pair µν is a function of j , but unlike there, where q˜ and r were stochastic maps with independent sources of randomness (when stochastic maps are viewed as set function valued random variables, this is expressed by " and R draw their randomness from the same the independence of q˜ and r), the maps Q source ν. In summary, there is no direct isomorphism between our concept of data reduction and sufficient statistics (which, too, can be used to reduce the entropy of data sets): the " and R are independent. latter appears as a special case where the maps Q E. Entropy exchange. We want to discuss an application of Theorem 10 to the entropy exchange of quantum operations, introduced by Schumacher [23] (and previously by Lindblad [21]): for a quantum operation ϕ∗ in the form (21) it is defined as Se (ρ; ϕ∗ ) = H (W ), with Wj k = Tr (Vj ρVk∗ ). It can be shown to be independent of the Kraus representation, by identifying it with the entropy increase in an initially pure environment of the system by a Stinespring dilation of ϕ∗ , see [23]. In the latter work a number of interesting relations between Se and other entropic quantities are shown. In particular, returning to the notation of Sect. V, it is shown that there is a (in this sense, minimal) Kraus representation of ϕ∗ such that H (λ) = Se (ρ; ϕ). Because of I (λ; ρ) ˆ ≤ H (λ) (this is simply data processing inequality [1]), we conclude (ρ; ϕ∗ ) ≤ Se (ρ; ϕ∗ ). By the derivation this quantity may be dubbed genuinely quantum entropy exchange of a channel, as it is that part of the noise that cannot be accounted for classically. From a different point of view, in fact also the maximum of I (λ; ρ) ˆ over all Kraus representations of ϕ∗ (compare Eq. (26)) is interesting: in a cryptographic setting, where ϕ∗ connects users A and B, and is controlled by an eavesdropper E, it is the amount of data collected by E about A’s messages in the worst case. A deeper investigation of these concepts is relegated to another occasion. F. An open problem. An interesting and challenging question is about the amount of data collected by a under the hypothesis of an arbitrarily varying source (AVS), instead of the i.i.d. model considered here: An AVS is a collection of source ensembles {ρis , pis } (with average state ρs ), labelled by s ∈ S, which we make into a discrete memoryless source by considering the ensembles (labelled by s l ∈ S l ) {ρi l s l , pi l s l }i l . The idea is that at each position k = 1, . . . , l the source may be arbitrarily in one of the internal states s ∈ S. We have no – not even statistical information – about s, so our data separation must work for all s l ∈ S l : formally the condition on A = ν xν A(ν) is 1 ∀s l ω(s l )(aj l − Aj l ) ω(s l ) ≤ , (AVCM), 1 2 l j

where ω(s l ) = ρs1 ⊗ · · · ⊗ ρsl is the average state of the source when in internal state sl .


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A natural candidate for the minimum data rate of the A(ν) seems to be ( ) max I (λ; ρ) ˆ : ρ ∈ conv{ρs : s ∈ S} , √ √ with λj ρˆj = ρaj ρ, and conv denoting the closed convex hull. If this is true, then in particular the quantity (a) = max I (λ; ρ) ˆ ρ

is the amount of data collected by a, regardless of any source ensemble. Acknowledgements. I am indebted to Serge Massar for his introducing me to the problem addressed in this paper and for interesting discussions, and to Hiroshi Nagaoka for pointing out to me the possible relation between the present approach and sufficient statistics. Thanks to Peter Shor who supplied the insight that the difference between data and information disappears in the presence of entanglement. I thank Masanao Ozawa for pointing out to me that Theorem 10, initially only formulated for operations, is in fact valid for instruments. Part of this work was done during my stay at the ERATO project “Quantum Computation and Information”, Tokyo (August/September 2001). I thank the members of the project for their hospitality, and especially Keiji Matsumoto for discussions on the content of the appendix, on which I also enjoyed conversation with Richard Jozsa and Masahide Sasaki. Last but not least, special thanks are due to Marco P. Carota for constant encouragement during the course of this work.

Appendix A. Canonical Purifications In this appendix we collect a few facts about mixed state fidelity and a certain kind of purification of mixed states, which we call canonical, that seem not to be widely known. These are used in the main text, but seem to be of interestin their own right. √ For the state ω on H1 consider a purification |ψ = i ri |i ⊗ |i on a bipartite system H1 ⊗ H2 , that we already have put in Schmidt polar form. Then on both systems there exist (R–linear) complex conjugation maps with respect to the basis {|i}: |φ = αi |i −→ αi |i =: |φ. i

Then, with |I =

i

i

|i ⊗ |i, it can be checked that

√ √ √ √ |ψψ| = ( ω ⊗ 11)|I I |( ω ⊗ 11) = (11 ⊗ ω)|I I |(11 ⊗ ω), see also the following Lemma 14. Then √ √ 11 ⊗ Sk |ψψ| 11 ⊗ Sk = 11 ⊗ Sk ω |I I | 11 ⊗ Sk ω √ √ = qk (11 ⊗ Uk ) (11 ⊗ τk )|I I |(11 ⊗ τk ) (11 ⊗ Uk∗ ) = qk (11 ⊗ Uk )|tk tk |(11 ⊗ Uk∗ ), √ √ the third qk τk = ωSk ω on H2 , and the polar decomposition √ √ line introducing √ Sk ω = Uk qk τk , the fourth the canonical purification |tk on H1 ⊗ H2 of τk (with respect to |I I |), see Lemma 14 below. By this lemma we can infer Tr H2 |ψψ|(11 ⊗ Sk ) = qk Tr H2 |tk tk | = qk τk ,

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with the complex conjugated operator τk , which is defined as τk = |φi φi |, if τk = |φi φi |. i

i

Note that this is uniquely defined, regardless of the convex decomposition chosen, and in particular independent of the phases of the |φi . The ensemble {τk , qk } has average ω = ω, and conversely, the above formulas show how to induce any ensemble {σk , qk } for ω on H1 : let Sk = ω−1/2 qk σk ω−1/2 (this was noted before in [17] in the context of classifying ensembles with a given density operator). Lemma 14 (“Pretty good purifications”). Consider orthonormal bases of spaces H1 and H2 , both denoted {|i}, and introduce |I = i |i ⊗ |i. As before, we denote the complex conjugation with respect to this basis by . Then for a state ρ = i αi |ψi ψi | (in diagonalized form), √ √ √ |rr| = ρ ⊗ 11 |I I | ρ ⊗ 11 , with |r = αi |ψi ⊗ |ψi , i

is a purification of ρ. We call it the canonical purification with respect to |I . (Note that this definition makes sense as it is independent of phases in the |ψi ). If |ss| is the canonical purification of another state σ then for the fidelity between these: √ √ 2 F (|r, |s) = |r|s|2 = Tr ρ σ . (A.1) Furthermore

√ √ ρ σ ≥ 1 − ρ − σ 1 , (A.2) 1 |rr| − |ss| ≤ 4 4 ρ − σ 1 . (A.3) 1 2 Proof. The formula for the canonical purification is a straightforward calculation. With its help, it is also straightforward to check the fidelity identity, Eq. (A.1). Now for the last two estimates: begin with √ √ √ √ √ √ √ √ 1 − Tr ρ σ = Tr ρ ρ − σ ≤ ρ( ρ − σ ) 1 √ √ √ ≤ ρ 2 ρ − σ 2 ≤ |ρ − σ | = ρ − σ 1 , Tr

2

invoking two nontrivial inequalities: in the third line we use Cor. IV.2.6 of [5] (which is a kind of Hölder or Cauchy–Schwarz inequality), in the fourth line Thm. X.1.3 from the same book. Finally, use the well known identity 1 |rr| − |ss| = 1 − F (|r, |s) 1 2 to obtain √ 1 |rr| − |ss| = 1 − |r|s|2 ≤ 2 1 − |r|s| 1 2 √ √ √ √ = 2 1 − Tr ρ σ ≤ 2 4 ρ − σ 1 , which we wanted to show.


183

√ √ √ √ Remark 15. Observe Tr ρ σ ≤ ρ σ 1 , the square of this latter quantity being known as the (mixed state) fidelity √ By theorems of Uhlmann [27] and Jozsa [18] the √[18]. mixed state fidelity F (ρ, σ ) = ρ σ 21 equals the maximum over the pure state fidelities of all possible purifications of ρ and σ . Because of well known relations between mixed state fidelity and trace norm distance (see [9]), more precisely 1−

F (ρ, σ ) ≤

1

ρ − σ 1 ≤ 1 − F (ρ, σ ), 2

(A.4)

the lemma tells us that at least for (mixed state) fidelity close to 1 the canonical purifications are not too far off the optimum with respect to (pure state) fidelity. Appendix B. Typical Sequences and Subspaces For a probability distribution P on the finite set X define set of typical sequences (with δ > 0) √ TPl ,δ = x l : ∀x |N (x|x l ) − lPx | ≤ δ l Px (1 − Px ) , where N(x|x l ) counts the number of occurrences of x in the word x l = x1 . . . xn . For a state ρ fix eigenstates e1 , . . . , ed (with eigenvalues R1 , . . . , Rd ) and define for δ > 0 the typical projector as et1 ⊗ · · · ⊗ etl . lρ,δ = l t l ∈TR,δ

For a collection of states ρˆj , j = 1, . . . , m, and j l ∈ [m]l define the conditional typical projector as * Ij l lρ,δ ρˆ ,δ , ˆ (j ) = j

j I

where Ij = {k : jk = j } and ρˆj ,δ is meant to denote the typical projector of the state j ρˆj on the subsystem composed of the tensor factors Ij in the tensor product of l factors. From [28] we cite the following properties of these projectors: Tr (ρ ⊗l lρ,δ ) ≥ 1 − l Tr (ρˆj l lρ,δ ˆ (j )) ≥ 1 −

d , δ2

(B.1)

md , δ2

(B.2)

m2 d , δ2 √ ≤ exp lH (ρ) + Kdδ l , Tr (ρˆj l lρ,δ ) ≥ 1 −

Tr lρ,δ

Tr lρ,δ

d ≥ 1− 2 δ

√ exp lH (ρ) − Kdδ l ,

(B.3) (B.4) (B.5)

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√ l Tr lρ,δ ˆ j l ) + Kmdδ l , ˆ (j ) ≤ exp lH (ρ|P

√ md l l Tr ρ,δ exp lH ( ρ|P ˆ (j ) ≥ 1 − l ) + Kmdδ l , j ˆ δ2

(B.6) (B.7)

for an absolute constant K > 0, and the empirical distribution Pj l of letters j in the word j l : 1 Pj l (j ) = N (j |j l ). l Finally, with √ √ α = exp −lH (ρ) − Kdδ l , α = exp −lH (ρ) + Kdδ l , √ β = exp −lH (ρ|P ˆ j l ) + Kmdδ l ,

√ β = exp −lH (ρ|P ˆ j l ) − Kmdδ l ,

we have α lρ,δ ≥ lρ,δ ρ ⊗l lρ,δ ≥ αlρ,δ ,

(B.8)

l l l l l l l β lρ,δ ˆ (j ) ≤ ρ,δ ˆ (j )ρˆj l ρ,δ ˆ (j ) ≤ βρ,δ ˆ (j ).

(B.9)

References 1. Ahlswede, R., Löber, P.: Quantum Data Processing. IEEE Trans. Inf. Theory 47(1), 474–478 (2001) 2. Ahlswede, R., Winter, A.: Strong converse for identification via quantum channels. IEEE Trans. Inf. Theory 48(3), 569–579 (2002) 3. Bennett, C.H., Shor, P.W., Smolin, J.A., Thapliyal, A.V.: Entanglement–assisted capacity of a quantum channel and the reverse Shannon theorem. IEEE Trans. Inf. Theory 48(10), 2637–2655 (2002) 4. Bennett, C.H., Devetak, I., Harrow, A., Shor, P.W., Winter, A.: The Quantum Reverse Shannon Theorem. In preparation 5. Bhatia, R.: Matrix Analysis. Graduate Texts in Mathematics 169, Berlin-New York: Springer Verlag, 1997 6. Choi, M.-D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10, 285–290 (1975) 7. Davies, E.B., Lewis, J.T.: An operational approach to quantum probability. Commun. Math. Phys. 17, 239–260 (1970) 8. Fano, R.M.: Class Notes for Transmission of Information. Course 6.574, MIT, Cambridge MA, 1952. See also R. M. Fano, Transmission of Information, New York: Wiley and Sons, 1961 9. Fuchs, C.A., van de Graaf, J.: Cryptographic Distinguishability Measures for Quantum–Mechanical States. IEEE Trans. Inf. Theory 45(4), 1216–1227 (1999) 10. Hausladen, P., Wootters, W.K.: A ‘pretty good’ measurement for distinguishing quantum states. J. Modern Opt. 41(12), 2385–2390 (1994) 11. Helstrom, C.W.: Quantum Detection and Estimation Theory. New York: Academic Press, 1976 12. Holevo, A.S.: Information–theoretical aspects of quantum measurement. Probl. Inf. Transm. 9(2), 110–118 (1973) 13. Holevo, A.S.: Bounds for the quantity of information transmitted by a quantum channel. Probl. Inf. Transm. 9(3), 177–183 (1973) 14. Holevo, A.S.: Asymptotically optimal hypotheses testing in quantum statistics. Theor. Probability Appl. 23(2), 411–415 (1979) 15. Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. Amsterdam: North Holland, 1982 16. Holevo, A.S.: The Capacity of the Quantum Channel with General Signal States. IEEE Trans. Inf. Theory 44(1), 269–273 (1998)


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17. Hughston, L.P., Jozsa, R., Wootters, W.K.: A complete classification of quantum ensembles having a given density matrix. Phys. Lett. A 183(1), 14–18 (1993) 18. Jozsa, R.: Fidelity for mixed quantum states. J. Mod. Optics 41, 2315–2323 (1994) 19. Lehmann, E.L., Casella, G.: Theory of Point Estimation. 2nd edition, Springer Texts in Statistics, Berlin-New York: Springer, 1998 20. Lebedev, D.S., Levitin, L.B.: The maximum amount of information transmissible by an electromagnetic field. Dokl. Akad. Nauk (SSSR) 149(6), 1299–1302 (1963) (Russian). [English translation: Soviet Physics Dokl. 8, 377–379 (1963)] 21. Lindblad, G.: Quantum entropy and quantum measurements. In: C. Bendjaballah, O. Hirota, S. Reynaud, (eds.), Quantum Aspects of Optical Communications, Lecture Notes in Physics, Vol. 378, Berlin: Springer Verlag, 1991, pp. 71–80 22. Massar, S., Popescu, S.: Amount of information obtained by a quantum measurement. Phys. Rev. A 61, 062303 (2000) 23. Schumacher, B.: Sending entanglement through noisy quantum channels. Phys. Rev. A 54(4), 2614– 2628 (1996) 24. Schumacher, B., Westmoreland, M.D.: Sending classical information via noisy quantum channels. Phys. Rev. A 56(1), 131–138 (1997) 25. Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948) 26. Stinespring, W.F.: Positive functions on C∗ –algebras. Proc. Am. Math. Soc. 6, 211–216 (1955) 27. Uhlmann, A.: The ‘transition probability’ in the state space of a ∗–algebra. Rep. Math. Phys. 9, 273–279 (1976) 28. Winter, A.: Coding Theorem and Strong Converse for Quantum Channels. IEEE Trans. Inf. Theory 45(7), 2481–2485 (1999) 29. Winter, A., Massar, S.: Compression of quantum–measurement operations. Phys. Rev. A. 64 012311 (2001) 30. Wolfowitz, J.: Coding Theorems of Information Theory. 2nd edition, Berlin: Springer Verlag, 1964 Communicated by H. Araki


Communications in


A Statistical Approach to the Asymptotic Behavior of a Class of Generalized Nonlinear Schrödinger Equations Richard S. Ellis1, , Richard Jordan2, , Peter Otto1, , Bruce Turkington1, 1

Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA. E-mail: [email protected]; [email protected]; [email protected] 2 Dynamics Technology, Inc., 1555 Wilson Boulevard, Suite 320, Arlington, VA 22209, USA. E-mail: [email protected] Received: 4 April 2003 / Accepted: 23 July 2003 Published online: 13 November 2003 – © Springer-Verlag 2003

Abstract: A statistical relaxation phenomenon is studied for a general class of dispersive wave equations of nonlinear Schrödinger-type which govern non-integrable, non-singular dynamics. In a bounded domain the solutions of these equations have been shown numerically to tend in the long-time limit toward a Gibbsian statistical equilibrium state consisting of a ground-state solitary wave on the large scales and Gaussian fluctuations on the small scales. The main result of the paper is a large deviation principle that expresses this concentration phenomenon precisely in the relevant continuum limit. The large deviation principle pertains to a process governed by a Gibbs ensemble that is canonical in energy and microcanonical in particle number. Some supporting MonteCarlo simulations of these ensembles are also included to show the dependence of the concentration phenomenon on the properties of the dispersive wave equation, especially the high frequency growth of the dispersion relation. The large deviation principle for the process governed by the Gibbs ensemble is based on a large deviation principle for Gaussian processes, for which two independent proofs are given.

1. Introduction Many dynamical models of physical systems governed by nonlinear partial differential equations exhibit a typical long-time behavior in which coherent structures organize on the large spatial scales while turbulent fluctuations dominate the small scales [19]. Perhaps the most familiar setting for this behavior is two-dimensional or quasi-geostrophic This research was supported in part by grants from the Department of Energy (DE-FG02-99ER25376) and from the National Science Foundation (NSF-DMS-0202309). This research was partially supported by a Mathematical Sciences Postdoctoral Research Fellowship from the National Science Foundation. This research was supported in part by grants from the Department of Energy (DE-FG02-99ER25376) and from the National Science Foundation (NSF-DMS-0207064).

188

R.S. Ellis, R. Jordan, P. Otto, B. Turkington

fluid turbulence, where the coherent structures are large-scale steady motions, such as shear flows or vortices, and the turbulent background is a vorticity field that fluctuates on the small scales [12, 32]. The zonal jets and embedded vortical spots in the active weather layer of Jupiter are especially persistent and conspicuous examples of this phenomenon [13, 29]. These coherent structures have been shown to be realizable as the equilibrium states in a statistical model of the geophysical fluid dynamical system [8, 33]. Another physical system exhibiting this behavior is two-dimensional magnetohydrodynamics. Long-time numerical simulations of MHD turbulence show that the systems end in states in which the magnetic and velocity fields fluctuate on small scales around a steady mean state on the large scales [4]. Equilibrium statistical models have also been able to capture these coherent structures [20, 24]. This statistical relaxation phenomenon is shared by certain dispersive wave systems, for which the generic coherent structures are solitary waves that interact with a disorganized background of wave radiation [9, 10, 15, 34]. For instance, non-integrable, focusing, nonlinear Schrödinger equations in a bounded domain organize after a long evolution into a single solitary wave coupled with small-scale fluctuations [22, 23]. This self-organization behavior has been shown to be consistent with relaxation to a statistical equilibrium state, both qualitatively and quantitatively [25]. Motivated by this fundamental phenomenon exhibited by many complex systems, we devote the present paper to a detailed analysis of the equilibrium statistical behavior of a particular class of dispersive wave systems. Specifically, we consider a class of generalized nonlinear Schrödinger (GNLS) equations on a bounded domain D in Rd with appropriate boundary conditions. These systems govern the dynamics of a complex field ψ(x, t), x ∈ Rd , t ∈ R, via the equation iψt + Lψ + f (|ψ|2 )ψ = 0.

(1.1)

L denotes an unbounded linear operator on the complex Hilbert space L2c (ρ) of squareintegrable functions on D with respect to a measure ρ; ·, · denotes the inner product on L2c (ρ). It is assumed that L is symmetric and that the spectrum of −L consists of positive eigenvalues λk satisfying ∞ 1/λ k < ∞. In addition, the corresponding k=1 eigenfunctions ek are assumed to be real functions that form an orthonormal basis of L2c (ρ). We choose to focus our analysis on these systems for two reasons. First, they are widely considered to be prototypes for dynamical systems that exhibit organization of coherent structures within turbulence, and accordingly there is a rich literature on their phenomonology [9, 10, 15, 34]. Second, they are simple enough to be amenable to a complete and rigorous analysis by the methods of equilibrium statistical mechanics. In one space dimension (d = 1) the basic example of this class is L = ∂ 2 /∂x 2 on D = [0, ] with Lebesgue measure ρ on [0, ] and with homogeneous Dirichlet boundary conditions, where < ∞. In this case, we refer to (1.1) as the basic NLS equation. Our analysis also applies to the operator L = ∂ 2 /∂x 2 + U (x), where U (x) is a suitable potential, and to other boundary conditions such as -periodic conditions. For the basic NLS equation and for this wider class of NLS equations in one dimension, the eigenvalues λk grow like k 2 as k → ∞. In (1.1) we restrict our attention to smooth nonlinearities f that satisfy f (0) = 0,

sup

|f (a)| < ∞ ;

(1.2)

a∈[0,∞)

e.g., f (|ψ|2 ) = b|ψ|2 /(1 + |ψ|2 ) with scale factor b. Nonlinearities with these properties arise in physical applications as large-amplitude corrections to the cubic NLS

Statistical Approach to Asymptotic Behavior of NLS Equations

189

equation, and they are referred to as bounded or saturated nonlinearities [31]. Our analysis applies to both the focusing GNLS, for which f (a) > 0 , and the defocusing GNLS, for which f (a) < 0. Our main interest, however, is on the focusing case since the formation of coherent solitary waves is the dominant mechanism in that case. The restriction to bounded nonlinearities excludes blow-up of solutions and the collapse of waves due to self-focusing. We impose this restriction because our goal is to analyze statistical equilibrium ensembles of regular solutions that model the long-time average behavior of the system. Accordingly, we choose GNLS equations for which solutions exist and are regular for all time. An interesting generalization included in our analysis is to pseudo-differential operators L whose eigenvalues λk grow like k α as k → ∞ with α > 1. The GNLS equation (1.1) then resembles the equation introduced by Majda, McLaughlin and Tabak (MMT) in their study of weak turbulence closure theories [28]. From the standpoint of equilibrium statistical mechanics, we are interested in how the phenomenon of concentration into a coherent structure depends on α. In contrast to the MMT equations, we restrict our analysis to bounded nonlinearities in ψ itself; the MMT equations pertain to homogeneous, cubic nonlinearities in Mψ, where M is another pseudo-differential operator with eigenvalues that grow like k −σ . While it would be possible to study a broader class of such equations, the class of GNLS equations (1.1) is sufficiently broad to exhibit the typical behavior of the statistical equilibrium states and to show how this behavior depends upon the linear frequencies of the dispersive wave system. The object of our analysis is the statistical equilibrium description of the complex dynamical system (1.1) via classical Gibbsian statistics. Our choice of distribution on phase space is a mixed Gibbs ensemble that is canonical with respect to the energy invariant and microcanonical with respect to the particle number invariant. For the GNLS equation (1.1), the Hamiltonian, or energy functional, is 1 . 1 H (ψ) = − Lψ, ψ − F (|ψ|2 ) dρ, (1.3) 2 2 D a where F is related to the nonlinearity f by F (a) = 0 f (s) ds. The particle number, or wave action, is half the L2c (ρ)-norm squared: . 1 Q(ψ) = |ψ|2 dρ. 2 D The resulting statistical description rests on these two exact invariants of the GNLS equation (1.1), together with the conservation of phase volume under the Hamiltonian dynamics. In order to keep our development concise, we intentionally suppress the momentum invariant by breaking the x-translation invariance of the system. To this end, we consider (1.1) in a bounded domain D with homogeneous boundary conditions ψ = 0 imposed on ∂D. Alternatively, we could consider an operator L with a potential under periodic boundary conditions to obtain similar results. While the statistical equilibrium NLS equation has been the focus of several analyses, including [1, 7, 27, 30, 35], our approach and our results differ fundamentally from those investigations. In particular, previous investigators have constructed Gibbs distributions that are Wiener-type measures having infinite mean energy. Our interest, on the other hand, centers on modeling the ensemble-average behavior of regular solutions to (1.1) from initial conditions having given, finite, mean energy H (ψ 0 ) = E and given particle number Q(ψ 0 ) = N . Our motivation derives from numerical studies of

190


the underlying GNLS equation which show that from generic initial conditions, such as a field of waves emerging from a modulational instability, the dynamics approximately realize a Gibbs ensemble after a sufficient time [22, 23]. A spectral analysis of these numerical solutions identifies an approximate dimension n = n(T ) of the phase space that supports the Gibbs distribution after a long, but finite, time T . Moreover, a continuum limit is achieved as T → ∞, in the sense that n(T ) goes to infinity at a definite rate with T . This observed behavior of regular solutions to (1.1) strongly suggests that the relevant continuum limit for a statistical equilibrium theory is the one obtained from the Gibbs states of the spectrally-truncated dynamics on n eigenmodes with fixed mean energy E and fixed particle number N as n → ∞. Accordingly, this limit is the focus of our analysis. While we establish rigorous results about these statistical equilibrium states, we do not address the theoretical problem of proving ergodicity of this dynamics. Rather we accept the ergodic hypothesis on the basis of convincing numerical evidence [22, 23]. Our results pertain to a continuum limit n → ∞ of a sequence of Gibbs distributions on n-dimensional phase spaces corresponding to spectrally-truncated, Hamiltonian dynamics having a finite number of degrees of freedom n. As has been noted elsewhere [25, 27], these Gibbs ensembles are necessarily microcanonical in Q, since a Gibbs canonical ensemble with respect to both H and Q can be divergent for a focusing nonlinearity. We therefore use a mixed ensemble in which the microcanonical condition Q = N is imposed on the canonical distribution in H with an inverse temperature β that is rescaled by n so that the mean energy, H = E, remains finite as n → ∞. The study of the limiting behavior of these mixed ensembles is perfectly suited to analysis by large deviation techniques. Our main result is a large deviation principle demonstrating that the ground-state solitary waves are the most probable macroscopic states in the relevant continuum limit, in the sense that the mixed ensembles concentrate on these ground states in the L2c (ρ)-norm. This large deviation principle may be considered as a mathematically rigorous statement of the explanation of the observed formation of large-scale coherent structures within small-scale wave turbulence given in [25], where an asymptotically exact mean-field theory was developed and compared with direct numerical simulations. The outline of the paper is as follows. In Sect.2, we construct the statistical equilibrium model based on a spectral truncation of the GNLS dynamics and motivate the mixed ensemble based upon the invariants H and Q. In Sect.3 we state the main theorem, a large deviation principle for a sequence of finite-dimensional fields with respect to the mixed ensemble introduced in Sect.2. The main theorem is proved in Sect.4 using a basic large deviation theorem for Gaussian processes, for which two independent proofs are given. Both of these proofs require that the linear frequencies λk satisfy ∞ k=1 1/λk < ∞ (see Cond. 2.1). Finally, in Sect.5 we display the results of some Monte-Carlo simulations of the mixed ensemble in one space dimension. Besides demonstrating the concentration phenomenon numerically when ∞ k=1 1/λk < ∞, these simulations exhibit the change in behavior when this growth condition does not hold.

2. Statistical Equilibrium Description of GNLS Dynamics The GNLS equation (1.1) is considered on a bounded domain D in Rd . The nonlinearity is bounded, in the sense that f satisfies the conditions (1.2). The operator L defining the linear part of the GNLS equation is assumed to satisfy the following condition:


191

Condition 2.1. L is a symmetric operator on L2c (ρ). The spectrum of −L consists of ∞ positive eigenvalues λk satisfying k=1 1/λk < ∞. The corresponding eigenfunctions ek are real functions that form an orthonormal basis of L2c (ρ). A number of important examples underlie the general theory. Example 2.2. (a) The basic example is L = ∂ 2 /∂x 2 on D = [0, ] with homogeneous Dirichlet boundary conditions and ρ Lebesgue measure on [0, ], where < ∞. In √ this case, for each k ∈ N λk = (kπ/)2 and ek = 2/ sin(kπ x/). (b) Let p be a C 2 function on D = [0, ] satisfying inf x∈[0,] p(x) > 0, q a negative continuous function on [0, ], and ρ Lebesgue measure on [0, ]. For ξ ∈ L2c (ρ) we define dξ . d Lξ = p(x) − q(x)ξ(x) dx dx with homogeneous Dirichlet boundary conditions. By standard Sturm-Louiville theory, L satisfies Condition 2.1. As in the basic example given in part (a), the eigenvalues λk of L grow like k 2 as k → ∞ [3, Thm. 10.9]. d (c) Let D be any bounded on D, {λk }∞ k=1 any positive ∞domain in R , ρ any measure ∞ sequence satisfying k=1 1/λk < ∞, and {ek }k=1 any real orthonormal basis of L2c (ρ). We denote by ·, · the inner product on L2c (ρ). The operator L defined for any ξ ∈ L2c (ρ) by ∞ . Lξ = − λk ξ, ek ek k=1

satisfies Condition 2.1. Such operators L include a class of pseudodifferential operators that arise in weak turbulence theory [28], for which the boundary conditions are periodic, the Fourier basis functions ek are trigonometric, and eigenvalues λk are powers k α . Condition 2.1 limits the power to α > 1. We proceed with the definition of the probabilistic model, which pertains to a sequence of finite-dimensional approximations to the GNLS equation (1.1). For these approximations we choose a spectral truncation of the GNLS-dynamics [7, 25, 35]. The same ideas can be applied to other discrete approximations such as finite-difference [26]. . With respect to the basis {ek } of L2c (ρ), let Wn = span {e1 , . . . , en } be the n-dimensional subspace consisting of functions . (x) = ψk ek (x), n

ψ

(n)

(x) = u

(n)

(x) + iv

(n)

(2.1)

k=1

with arbitrary complex coefficients ψk = uk + ivk . For each fixed n, the field ψ (n) takes values in L2c (ρ) and corresponds to an n-dimensional microstate for the model; that is, . . a point ψ = (ψ1 , . . . , ψn ) in the phase space n = Cn or equivalently n = R2n . The microscopic dynamics for this model is governed by (n) iψt + Lψ (n) + P (n) f (|ψ (n) |2 )ψ (n) = 0, (2.2) where P (n) denotes the orthogonal projection that maps L2c (ρ) onto Wn . This spectral truncation of the GNLS equation (1.1) is equivalent to a system of ordinary differential

192


equations for the real Fourier coefficients uk and vk , k = 1, . . . , n, having a canonical Hamiltonian form; namely, duk ∂Hn f ((u(n) )2 + (v (n) )2 )v (n) ek dρ = , = λ k vk − dt ∂vk D dvk ∂Hn f ((u(n) )2 + (v (n) )2 )u(n) ek dρ = − = −λk uk + dt ∂uk D with Hamiltonian Hn (ψ) = Hn (u1 , v1 , . . . , un , vn ) n 1 . 1 2 = λk |ψk | − F (|ψ (n) |2 ) dρ 2 2 D k=1 1 1 (n) (n) F (|ψ (n) |2 ) dρ = − Lψ , ψ − 2 2 D ≡ Dn (ψ) + n (ψ). The functions Dn and n are defined by this display, and . 1

(ψ) = − F (|ψ|2 ) dρ; 2 D

(2.3)

(2.4)

hence for ψ ∈ Wn , n (ψ) = (ψ (n) ). Clearly, Hn (ψ) equals H (ψ (n) ), the restriction to Wn of the functional H defined in (1.3). The spectrally truncated particle number, . Qn (ψ) = Q(ψ (n) ), is also an invariant of the microscopic dynamics (2.2) and is given by . 1 Qn (ψ) = Qn (u1 , v1 , . . . , un , vn ) = 2

D

1 |ψk |2 . 2 n

|ψ (n) |2 dρ =

(2.5)

k=1

We define the statistical equilibrium model by a Gibbs ensemble on the 2n-dimensional phase space n , in which Hn is treated canonically and Qn is treated microcanonically. We refer to this ensemble as the mixed ensemble, and we denote it by PβN (dψ), where N ∈ [0, ∞) is a given particle number and β > 0 is a given inverse temperature. Formally, the mixed ensemble is the probability distribution . N Pn,β (dψ) =

1 exp(−βHn (ψ)) δ(Qn (ψ) − N ) Vn (dψ), Zn (β, N )

(2.6)

where Zn (β, N ) is the normalizing constant . Zn (β, N ) = exp(−βHn (ψ)) ds(ψ). {Qn =N}

. Here Vn (dψ) = nk=1 duk dvk is the phase volume on n , and ds(ψ) is the hypersurface area on the sphere Qn (ψ) = N , which is the support of the distribution (2.6). This choice of ensemble can be motivated intuitively from the known dynamical behavior of numerical solutions to (1.1). Long-time simulations of the dynamics show that, while the energy H is sensitive to the fluctuations that develop on the small scales,


193

the particle number Q depends on the coherent structure on the large scale. This phenomenon is related to the phenomenological description of weak turbulence in which there is a flux of energy to small scales and of particle number to large scales. Physical reasoning then suggests that the appropriate ensemble be canonical in H , since energy is in contact with a bath of turbulent small-scale waves, and that it be microcanonical in Q, since the particle number is contained in the coherent large-scale waves which are isolated from the turbulent bath. Let us define this mixed ensemble precisely as a conditional probability measure. We return to the decomposition Hn (ψ) = Dn (ψ) + n (ψ) given in (2.3). An easy calculation given in part (a) of Proposition 4.3 shows that for any bounded nonlinearity f , σ > 0 can be chosen sufficiently large so that (ξ ) + σ Q(ξ ) ≥ 0 for all ξ ∈ L2c (ρ). It follows that for all n ∈ N and ψ ∈ Wn , Since Dn (ψ) =

n (ψ) + σ Qn (ψ) = (ψ) + σ Q(ψ) ≥ 0. 1 − 2 Lψ (n) , ψ (n) ≥ 0, we have for all ψ ∈ Wn

Hn (ψ) + σ Qn (ψ) = Dn (ψ) + n (ψ) + σ Qn (ψ) ≥ 0. It is worth noting that such a σ also exists for a wider class of nonlinearities f ; e.g., unbounded, but subcritical nonlinearities. We then construct the following σ -regularized canonical measure: 1 . Pn,β (dψ) = exp(−β[Hn (ψ) + σ Qn (ψ)]) Vn (dψ), (2.7) Zn (β) which exists and is normalizable. The normalizing constant Zn (β) is given by . Zn (β) = exp(−β[Hn (ψ) + σ Qn (ψ)]) Vn (dψ). n

By contrast, when σ = 0, it is known that Zn (β) diverges for certain focusing nonlinearities since Hn goes to −∞ in some directions of the phase space n [25, 27]. Ideally, we would like to define the mixed ensemble to be . N Pn,β (dψ) = Pn,β ( dψ | Qn (ψ) = N ); (2.8) namely, a regular conditional distribution given the microcanonical constraint Qn = N . In this formulation, the mixed ensemble (2.8) is independent of the choice of σ and coincides with the formal expression (2.6). In order to avoid technicalities involving regular conditional distributions, we will consider, in place of (2.8), the conditional measure . N,ε Pn,β (dψ) = Pn,β ( dψ | Qn (ψ) ∈ [N − ε, N + ε] ), (2.9) where ε is a positive parameter defining the thickened shell [N − ε, N + ε]. For suitable values of N, all sufficiently large n, and all ε > 0, Pn,β {Qn (ψ) ∈ [N − ε, N + ε]} > 0, N,ε (dψ) is well defined [see (4.11)]. The main theand so the conditional probability Pn,β orem in this paper, stated in Theorem 3.1, is the large deviation principle on L2c (ρ) for N,ε ψ (n) with respect to Pn,nβ in the continuum limit n → ∞ followed by ε → 0; N is kept fixed while β has been replaced by the mean-field scaling nβ. With this scaling the ensemble mean energy Hn tends to a finite limit E. In contrast to the definition (2.8), N,ε the conditional measures Pn,nβ are no longer independent of σ because of the presence of ε in the definition (2.9). However, the rate function in Theorem 3.1 is independent of σ .

194


3. Statement of Main Theorem: LDP for Mixed Ensemble Earlier investigations in [22, 25] give theoretical and numerical evidence that, for the basic NLS equation in which L = ∂ 2 /∂x 2 and ρ is Lebesgue measure on [0, ], the random field ψ (n) defined in (2.1) concentrates on the set of ground states eiθ ϕ(x) of the NLS equation in the continuum limit n → ∞ with fixed E and N . In [25] a meanfield approximation is developed to explain this phenomenon, and long-time numerical simulations of the freely-evolving dynamics support the theory [22]. Alternatively, this phenomenon of concentration on the set of ground states can be demonstrated by implementing Monte-Carlo simulations of the mixed Gibbs ensemble in the continuum limit; this approach is used in Sect.5 of the present paper. Our main goal in the present paper is to formulate and prove a large deviation principle that holds for ψ (n) with respect to the N,ε mixed ensemble Pn,nβ defined in (2.9) and that is valid for general operators L satisfying Condition 2.1. This large deviation principle constitutes a precise and rigorous statement of the concentration phenomenon that occurs in the continuum limit. We start with two definitions. Let X be a Hilbert space, J a function mapping X into [0, ∞], {µn , n ∈ N} a sequence of probability measures on X , {µεn , n ∈ N, ε > 0} a family of probability measures on X , and an a positive sequence tending to ∞. J is called a rate function if for each M < ∞ the set {ξ ∈ X : J (ξ ) ≤ M} is compact. For A a subset of X we write J (A) for inf{J (ξ ) : ξ ∈ A}. The sequence µn is said to satisfy a large deviation principle (LDP) on X with the scaling constants an and the rate function J if for each closed subset F of X lim sup n→∞

1 log µn {F } ≤ −J (F ) an

and for each open subset G of X lim inf n→∞

1 log µn {G} ≥ −J (G). an

Similarly, as n → ∞ and ε → 0, the collection µεn is said to satisfy an LDP on X with the scaling constants an and the rate function J if for each closed subset F of X lim sup lim sup n→∞

ε→0

1 log µεn {F } ≤ −J (F ) an

and for each open subset G of X lim inf lim inf ε→0

n→∞

1 log µεn {G} ≥ −J (G). an

The main result in this paper is the LDP stated in Theorem 3.1. We first indicate the 2 form of the rate function. For ξ = ∞ k=1 ξ, ek ek ∈ Lc (ρ), the Hamiltonian introduced in (1.3) can be written as H (ξ ) = D(ξ ) + (ξ ), where (ξ ) is defined in (2.4), and ∞

. 1 λk |ξ, ek |2 . D(ξ ) = 2 k=1

In terms of the square root of the positive, symmetric operator −L,

1 √ √ −Lξ 2 if ξ ∈ dom( −L) √ D(ξ ) = 2 ∞ if ξ ∈ L2c (ρ) \ dom( −L) ,


195

where √

∞

. −Lξ = λk ξ, ek ek . k=1

For N ∈ [0, ∞) we also introduce . ¯ )= E(N inf{H (ξ ) : ξ ∈ L2c (ρ), Q(ξ ) = N }.

(3.1)

¯ We call E¯ the coherent energy function since E(N) is the energy of the coherent structure with particle number N . The function E¯ is lower semicontinuous and bounded below; indeed, by (4.12) E¯ differs by a constant from the function E˜ defined in (4.10), which by part (a) of Proposition 4.5 is nonnegative and lower semicontinuous. For N ∈ [0, ∞) and ξ ∈ L2c (ρ), the rate function in Theorem 3.1 is defined to be

¯ ) if Q(ξ ) = N . H (ξ ) − E(N N (3.2) J (ξ ) = ∞ otherwise. Theorem 3.1 will be proved in the next section. Theorem 3.1. The L2c (ρ)–valued process ψ (n) is defined in (2.1) and the mixed ensemN,ε ble Pn,nβ in (2.9). We fix β > 0, take N ∈ [0, ∞), and assume Condition 2.1. Then as N,ε -distributions of ψ (n) satisfy the LDP on L2c (ρ) with the n → ∞ and ε → 0, the Pn,nβ scaling constants nβ and the rate function J N defined in (3.2). Heuristically, the LDP means that the elements ξ¯ ∈ L2c (ρ) that minimize H subject to the constraint Q = N are the overwhelmingly most probable states with respect to the mixed ensemble in the continuum limit n → ∞ followed by ε → 0. This set of constrained minimizers is the set of equilibrium macrostates or ground states; we denote it by E N . For an equilibrium macrostate ξ¯ we have J N (ξ¯ ) = 0, while for any ξ ∈ L2c (ρ) that is not an equilibrium macrostate we have J N (ξ ) > 0. We now consider, for any r > 0, the complement of an r-neighborhood of the equilibrium set E N and define . j (r) = inf{J N (ξ ) : dist(ξ, E N ) ≥ r > 0},

the distance being taken in the L2c (ρ)-norm. Then j (r) > 0. From the large deviation upper bound in Theorem 3.1, we infer that N,ε Pn,nβ {dist(ψ (n) , E N ) ≥ r > 0 } ≤ e−nβj (r)/2 → 0 as n → ∞ , ε → 0.

Thus any set of ξ ∈ L2c (ρ) that lies a positive distance from the equilibrium set E N has an exponentially small probability of being observed for sufficiently large n and sufficiently small ε > 0. This property of E N justifies calling it the set of equilibrium macrostates. An LDP can be viewed as an exponential-order refinement of the law of large numbers [11, 16]. From this viewpoint, we might expect the random field ψ (n) (x) to satisfy an LDP in the continuum limit because it is the sum of component fields ψk ek (x) that are asymptotically independent. In essence, this insight is the basis for the mean-field approximation used in [25], which relies on the smallness of the fluctuations of ψ (n) (x) in the L2c (ρ)-norm. As we will see in the next section, the proof of the LDP for ψ (n) depends crucially on the continuity of the functionals Q and with respect to the L2c (ρ)topology [Prop. 4.3]. These properties that are needed to prove the LDP are intimately related to the properties used to derive the mean-field theory.

196


4. Proof of Theorem 3.1 Given β > 0 we introduce the following Gaussian measures on the phase space n = R2n : . Gn,β (dψ) =

1 exp(−βDn (ψ)) Vn (dψ), Cn,β

(4.1)

where Cn,β is the normalizing constant . Cn,β = exp(−βDn (ψ)) Vn (dψ). R2n

The proof of Theorem 3.1 is based on the LDP of ψ (n) with respect to the measures Gn,nβ , where β in (4.1) has been replaced by nβ. The motivation for introducing these measures is that the canonical measures Pn,nβ can be expressed in terms of Gn,nβ [see (4.8)], and hence the LDP for Pn,nβ follows directly from that for Gn,nβ [Thm. 4.1]. In N,ε -distributions of ψ (n) stated in Theorem 3.1 is derived from turn, the LDP for the Pn,nβ the LDP for the Pn,nβ -distributions of ψ (n) . The LDP for the Gn,nβ -distributions of ψ (n) is stated in the next theorem and is proved using a corollary of Baldi’s Theorem stated in [11, Cor. 4.5.27]. After giving this proof, we sketch a second proof using an LDP for Gaussian processes proved by Bolthausen [5]. The next theorem states the LDP on the complex Hilbert space of square-integrable functions L2c (ρ); any ξ ∈ L2c (ρ) can be written as ξ 1 +iξ 2 , where ξ 1 and ξ 2 are elements of the corresponding real Hilbert space L2 (ρ). Since both proofs of Theorem 4.1 are based on results formulated for real spaces, we will prove an equivalent LDP replacing L2c (ρ) by the topologically equivalent Hilbert space L2 (ρ) × L2 (ρ); this equivalence is defined by the correspondence ξ = ξ 1 + iξ 2 ∈ L2c (ρ) ↔ (ξ 1 , ξ 2 ) ∈ L2 (ρ) × L2 (ρ). Theorem 4.1. The L2c (ρ)-valued process ψ (n) is defined in (2.1) and the Gaussian measures Gn,β in (4.1). We fix β > 0 and assume Condition 2.1. Then as n → ∞, the Gn,nβ -distributions of ψ (n) satisfy the LDP on L2c (ρ) with the scaling constants nβ and the rate function 1 1 2 . 1 I (ξ ) = I (ξ 1 + iξ 2 ) = λk |ξk1 + iξk2 |2 = λk (ξk ) + (ξk2 ) , 2 2 ∞

∞

k=1

k=1

(4.2)

. where ξk1 + iξk2 = ξ 1 , ek + iξ 2 , ek . Alternatively, the rate function I (ξ ) equals

1 √ √ −Lξ 2 if ξ ∈ dom( −L) √ D(ξ ) = 2 ∞ if ξ ∈ L2c (ρ) \ dom( −L).

(4.3)

Proof. The equality of the quantities defined in (4.2) and in (4.3) is immediate. The function ψ (n) defined in (2.1) can be written as ψ (n) =

n k=1

uk ek (x) + i

n k=1

vk ek (x) ≡ ψ (n),1 + iψ (n),2 .


197

. . Setting ψk1 = uk and ψk2 = vk , we have ψ (n),1 = nk=1 ψk1 ek and ψ (n),2 = nk=1 ψk2 ek . Because L2c (ρ) and L2 (ρ) × L2 (ρ) are topologically equivalent, proving an LDP for the Gn,nβ -distributions of ψ (n) on L2c (ρ) is equivalent to proving an LDP for the Gn,nβ -distributions of (ψ (n),1 , ψ (n),2 ) on L2 (ρ) × L2 (ρ). The inner product on L2 (ρ) × L2 (ρ) is ∞

. (ξ 1 , ξ 2 ), (θ 1 , θ 2 ) = ξ 1 , θ 1 + ξ 2 , θ 2 = ξkα θkα , 2

α=1 k=1

. . where for α = 1, 2 ξkα = ξ α , ek and θkα = θ α , ek . We begin the proof by computing, for ϕ ∈ L2 (ρ) × L2 (ρ),

1 . c(ϕ) = lim exp nβ ϕ, ψ (n) Gn,nβ (dψ) log n→∞ nβ n n R ×R 2 n 1 1 1 α α α 2 α = lim dψk exp nβ ϕk ψk − λk (ψk ) log n→∞ nβ Cn,nβ 2 α=1 k=1 R n 1 nβ (ϕk1 )2 + (ϕk2 )2 = lim log exp n→∞ nβ 2 λk k=1

=

∞ 1 (ϕ 1 )2 + (ϕ 2 )2 k

2

k=1

k

λk

(4.4)

.

By Condition 2.1 λk > 0 and λk → ∞; hence 0 ≤ c(ϕ) ≤ const · ϕ2 < ∞. Because of the relatively simple form of c, it is elementary to check that c is Gateaux differentiable and is weakly continuous on L2 (ρ) × L2 (ρ). Because c is a sum of quadratic terms, it is also straightforward to calculate its Legendre-Fenchel transform. For ξ = (ξ 1 , ξ 2 ) ∈ L2 (ρ) × L2 (ρ), this function is given by . I (ξ ) = = =

sup

{ϕ, ξ − c(ϕ)} ∞ 2 )2 (ϕ 1 (ϕk1 )2 1 k + − sup ϕk2 ξk2 − 2 λk 2 λk ϕ 2 ∈R

ϕ∈L2 (ρ)×L2 (ρ) ∞ sup ϕk1 ξk1 1 ϕ k=1 k ∈R ∞ 1 1 2 λk (ξk ) 2 k=1

k=2

k

+ (ξk2 )2 .

(4.5)

The function I (ξ ) calculated in the preceding display coincides with the function I (ξ ) defined in (4.2). By Corollary 4.5.27 in [11], we will be able to conclude that the Gn,nβ -distributions of (ψ (n),1 , ψ (n),2 ) on L2 (ρ) × L2 (ρ)—and thus the Gn,nβ distributions of ψ (n) on L2c (ρ)—satisfy the LDP with rate function I (ξ ) after we show that the Gn,nβ -distributions of (ψ (n),1 , ψ (n),2 ) are exponentially tight; i.e., for any K < ∞ there exists a compact set A such that lim sup n→∞

1 log Gn,nβ {(ψ (n),1 , ψ (n),2 ) ∈ Ac } < −K. nβ

198


In order to prove the exponential tightness, we define for M < ∞ the level sets . AM = {ξ ∈ L2 (ρ) × L2 (ρ) : I (ξ ) ≤ M}. We first prove that the sets AM are compact by showing that any sequence ξ (n) = (ξ (n),1 , ξ (n),2 ) in AM has a subsequence converging to an element of AM . Since ξ (n) ∈ (n),α 2 ) ≤ M; thus for each k AM , we have for all k ∈ N, n ∈ N, and α = 1, 2 21 λk (ξk (n),α (n ),α has a convergent subsequence. For α = 1, 2 let ξ1 1 be the convergent and α ξk (nk+1 ),α (n),α subsequence of ξ1 , and for each k and α let ξk+1 be the convergent subsequence (nk ),α

of ξk

(nk ),α

. For each k and α there exists ξkα such that limnk →∞ ξk

= ξkα . A diagonal-

(n),α ˆ

ξk = ξkα for each ization argument yields a subsequence nˆ ∈ N such that limn→∞ ˆ . ∞ 1 2 k and α. The quantity ξ = k=1 (ξk , ξk )ek is an element of AM . Indeed, by Fatou’s lemma 2 ∞ 2 ∞ 1 1 (n),α ˆ λk (ξkα )2 ≤ lim inf λk (ξk )2 ≤ M. 2 2 n→∞ ˆ α=1 k=1

α=1 k=1

. ∞ (n),1 ˆ (n),2 ˆ ˆ = , ξk )ek . In order to complete the proof For each nˆ we define ξ (n) k=1 (ξk (n) ˆ 2 2 (n) ˆ that AM is compact, we show that ξ → ξ in L (ρ) × L (ρ). Since ξ and ξ are in AM , the finiteness of ∞ 1/λ assumed in Condition 2.1 implies that uniformly over k k=1 nˆ ∞ 2

(n),α ˆ

(ξk

− ξkα )2 ≤ 2

α=1 k=1

∞ 2

(n),α ˆ 2

(ξk

) + (ξkα )2

α=1 k=1

≤4

∞ M k=1

λk

< ∞.

Hence by the dominated convergence theorem lim ξ

n→∞ ˆ

(n) ˆ

− ξ = lim 2

n→∞ ˆ

2 ∞

(n),α ˆ

(ξk

− ξkα )2 = 0.

α=1 k=1

This concludes the proof that AM is compact in L2 (ρ) × L2 (ρ). We complete the proof of Theorem 4.1 by showing that the Gn,nβ -distributions of (ψ (n),1 , ψ (n),2 ) are exponentially tight. For any M < ∞ Chebyshev’s inequality yields Gn,nβ {(ψ (n),1 , ψ (n),2 ) ∈ AcM } n 1 1 2 2 2 = Gn,nβ λk (ψk ) + ψk ) > M 2 k=1 n nβ 1 2 1 nβ 2 2 exp λk (ψk ) + ψk ) Gn,nβ (dψ) ≤ exp − M 2 Cn,nβ Rn ×Rn 4 k=1


199

nβ α 2 dψkα exp − (ψ ) λ 2 n k k nβ 4 R = exp − M nβ 2 k=1 α=1 exp − λk (ψkα )2 dψkα 2 R nβ = exp − M 2n . 2 It follows that lim sup n→∞

1 log 2 M log Gn,nβ {(ψ (n),1 , ψ (n),2 ) ∈ AcM } ≤ − + . nβ 2 β

Since AM is compact and M can be taken arbitrarily large, the proof is complete.

Before proving Theorem 3.1 we sketch a second proof of Theorem 4.1 using an LDP for Gaussian processes proved by Bolthausen [5]. Let (, F, ) be a probability space on which is defined a doubly indexed sequence of independent, N (0, 1) Gaussian random variables gkα indexed by k ∈ N and α = 1, 2. In terms of the eigenvalues and eigenfunctions of L introduced in Condition 2.1, we define for n ∈ N, ω ∈ , and x ∈ D the independent mean-0 Gaussian processes ek (x) . 1 gk (ω) √ y (n),1 = y (n),1 (ω, x) = λk k=1 n

and

ek (x) . 2 y (n),2 = y (n),2 (ω, x) = gk (ω) √ . λk k=1 n

(4.6)

These processes take values in L2 (ρ), and y (n),1 + iy (n),2 take values in L2c (ρ). The basic NLS equation is defined by L = ∂ 2 /∂x 2 on [0, ]. Inserting into (4.6) the eigenvalues and eigenfunctions given in Example 2.2(a), we have for α = 1, 2 and x ∈ [0, ], y (n),α (ω, x) =

√ n 2 α sin(kπ x/) . gk (ω) π k k=1

It is well known that with probability 1, as n → ∞ these processes converge in L2 (dx) to independent Brownian bridges on [0, ]. The processes y (n),α are closely related to the processes used by Wiener in his construction of Brownian motion [21, pp. 21-22]. The probability–1 convergence of y (n),α in the general case of (4.6) is the basis of the second proof of Theorem 4.1. We will prove the convergence in a moment. √ We need the following lemma relating the distributions of (y (n),1 + iy (n),2 )/ β and ψ (n) . The routine proof is omitted. Lemma 4.2. Fix β > 0. Then as measures on L2c (ρ), the -distributions of (y (n),1 + √ iy (n),2 )/ β and the Gn,β -distributions of ψ (n) are equal. In particular, replacing β by √ nβ, we see that the -distributions of (y (n),1 +iy (n),2 )/ nβ and the Gn,nβ -distributions of ψ (n) are equal.

200


This lemma allows √ us to prove Theorem 4.1 by showing that the -distributions of (y (n),1 + iy (n),2 )/ nβ on L2c (ρ)—equivalently, the -distributions of (y (n),1 , y (n),2 )/ √ nβ on L2 (ρ) × L2 (ρ)—satisfy the LDP with the scaling constants nβ and the rate function I (ξ ) defined in (4.2). We first prove that as n → ∞, with -probability 1 the Gaussian processes y (n),1 and y (n),2 defined in (4.6) converge in L2 (ρ) to the independent Gaussian processes ∞

∞

. 1 ek . 2 ek gk √ and y 2 = gk √ . y1 = λk λk k=1 k=1 On the product space × D, we define for n ∈ N and α = 1, 2 ∞

. α |ek | . α |ek | = |gk | √ and y α = |gk | √ . λk λk k=1 k=1 n

y

(n),α

Since the gkα are independent, N (0, 1) Gaussian random variables and the ek are orthonormal in L2 (ρ), (y α )2 d × dρ = lim (y (n),α )2 d × dρ ×D

= =

n→∞ ×D ∞ k=1 D ∞ k=1

(gkα )2

(ek )2 d dρ λk

1 . λk

Since this sum is finite by Condition 2.1 and |y α | ≤ |y α |, we have y α ∈ L2 (ρ) -a.e. and thus y α ∈ L2 (ρ) -a.e. The bound |y α − y (n),α |2 ≤ (2y α )2 ∈ L1 (ρ) -a.e. allows us to apply the dominated convergence theorem, which yields the desired limit: |y α − y (n),α |2 dρ → 0 −a.e. D

This completes the proof that with -probability 1 y (n),1 and y (n),2 converge in L2 (ρ) to y 1 and y 2 . The probability–1 convergence just proved implies the weak convergence {(y (n),1 , y (n),2 )/ β ∈ dξ 1 × dξ 2 } ⇒ {(y 1 , y 2 )/ β ∈ dξ 1 × dξ 2 } on L2 (ρ) × L2 (ρ). (4.7) By Theorem 2 in [5] and the discussion √ following that theorem, we conclude that the -distributions of (y (n),1 , y (n),2 )/ nβ on L2 (ρ) × L2 (ρ)—and thus the Gn,nβ distributions of ψ (n) on L2c (ρ)—satisfy the LDP with scaling constants nβ and rate function 1 . 1 hβ (ξ ) = · sup {ϕ, ξ − log Mβ (ϕ)}, β β ϕ∈L2 (ρ)×L2 (ρ) where . Mβ (ϕ) =

expϕ, (y 1 , y 2 )/ β (dϕ).


201

In order to calculate Mβ (ϕ), we use the bound sup

n∈N

exp(t(y (n),1 , y (n),2 )) d < ∞ for all t > 0,

which follows from the weak convergence (4.7) [5, p. 427]. Applying Lemma 4.2 and calculating as in (4.4), we find that for ϕ = (ϕ 1 , ϕ 2 ) ∈ L2 (ρ) × L2 (ρ) Mβ (ϕ) = lim

n→∞

expϕ, (y (n),1 , y (n),2 )/ β d

= lim expϕ, ψ (n) Gn,β (dψ) n→∞ R2n ∞ 1 (ϕk1 )2 + (ϕk2 )2 = exp . 2β λk k=1

Via a calculation as in (4.5), we conclude that for ξ = (ξ 1 , ξ 2 ) ∈ L2 (ρ) × L2 (ρ) ∞

1 1 λk (ξk1 )2 + (ξk2 )2 ). h(ξ ) = β 2 k=1

Since this equals the function I (ξ ) defined in (4.2), our sketch of the second proof of Theorem 4.1 is complete. We now turn to the proof of Theorem 3.1, which states the LDP for the distributions N,ε of ψ (n) with respect to the conditional measures Pn,nβ defined in (2.9). Before proving this theorem, we must establish several properties of the functionals Q and F appearing N,ε in Pn,nβ . We recall that for ξ ∈ L2c (ρ) . 1 Q(ξ ) = 2

. 1 |ξ |2 dρ and (ξ ) = − 2 D

F (|ξ |2 ) dρ D

. a and that for a ≥ 0 F (a) = 0 f (s) ds. Proposition 4.3. The following properties are valid. (a) For any σ > f ∞ we have (ξ ) + σ Q(ξ ) ≥ 0 for all ξ ∈ L2c (ρ). (b) Both Q and are continuous functionals on L2c (ρ). Proof. (a) For any ξ ∈ L2c (ρ),

(ξ ) + σ Q(ξ ) = −

1 2

F (|ξ |2 ) dρ + D

σ 2

|ξ |2 dρ ≥ − D

The last expression is positive provided σ > f ∞ .

f ∞ σ ξ 2 + ξ 2 . 2 2

202


(b) The continuity of Q is obvious. To prove the continuity of , we note that for 0 ≤ a ≤ b < ∞ , |F (b) − F (a)| ≤ f ∞ |b − a|. Hence for ξ and ζ in L2c (ρ), | (ξ ) − (ζ )| f ∞ ≤ |ξ |2 − |ζ |2 dρ + |ζ |2 − |ξ |2 dρ 2 {|ξ |≥|ζ |} {ξ | f ∞ so that (ξ ) + σ Q(ξ ) ≥ 0 for any ξ ∈ L2c (ρ) [Prop. 4.3(a)]. We are now ready to prove Theorem 3.1, which states N,ε the LDP for the distributions of ψ (n) with respect to the conditional measures Pn,nβ . We first express the measures Pn,nβ in terms of Gn,nβ : 1 exp(−nβ[Hn (ψ) + σ Qn (ψ)]) Vn (dψ) Zn (nβ) 1 = exp(−nβ[ (ψ (n) ) + σ Q(ψ (n) )]) Gn,nβ (dψ), ˆ Zn (nβ)

. Pn,nβ (dψ) =

(4.8)

where Zˆ n (nβ) denotes the normalizing constant . ˆ Zn (nβ) = exp(−nβ[ (ψ (n) ) + σ Q(ψ (n) )]) Gn,nβ (dψ). n

In order to prove the LDP for the measures Pn,nβ , we need a definition. Let J be a rate function on L2c (ρ). A sequence of measure µn on L2c (ρ) is said to satisfy the Laplace principle on L2c (ρ) with the scaling constants nβ and the rate function J if for all bounded continuous functions h, 1 log lim exp(−nβh) dµn = − inf {h(ξ ) + J (ξ )}. n→∞ nβ ξ ∈L2c (ρ) L2c (ρ) As proved in Theorems 1.2.3 and 1.2.5 in [14], µn satisfies the Laplace principle on L2c (ρ) with the rate function J if and only if µn satisfies the LDP on L2c (ρ) with the rate function J . The measures Pn,nβ defined in (2.7) have the form of a canonical ensemble with interaction function +σ Q. We prove the LDP for the Pn,nβ -distributions of ψ (n) by proving the Laplace principle for these distributions. By Theorem 4.1 the Gn,nβ -distributions of ψ (n) satisfy the LDP on L2c (ρ) with rate function D(ξ ). Since (ξ ) + σ Q(ξ ) ≥ 0 for any ξ ∈ L2c (ρ) [Prop. 4.3 (a)], for any bounded continuous function h , + σ Q + h


203

is bounded below. Hence the Laplace principle for the Pn,nβ -distributions of ψ (n) is a consequence of Theorem 1.3.4 in [14]. A straightforward calculation (see, e.g., the proof of Thm. 3.1 in [6]) allows us to express the rate function in terms of the Hamiltonian . H (ξ ) = D(ξ ) + (ξ ): J (ξ ) = D(ξ ) + (ξ ) + σ Q(ξ ) − inf{D(ξ ) + (ξ ) + σ Q(ξ ) : ξ ∈ L2c (ρ)} = H (ξ ) + σ Q(ξ ) − inf{H (ξ ) + σ Q(ξ ) : ξ ∈ L2c (ρ)}.

(4.9)

We have proved the following result. Theorem 4.4. As n → ∞, the sequence Pn,nβ (ψ (n) ∈ dξ ) satisfies the LDP on L2c (ρ) with the scaling constants nβ and the rate function J defined in (4.9). N,ε We now address the question of when the conditional measure Pn,nβ is well defined. To this end, we introduce . ˜ )= E(N inf{J (ξ ) : ξ ∈ L2c (ρ), Q(ξ ) = N }, (4.10)

which is nonnegative and finite for all N ∈ [0, ∞). Because Q is a continuous functional on L2c (ρ), the preceding theorem and the contraction principle imply that E˜ is a rate function—and thus is lower semicontinuous—and also yield the LDP stated in part (a) of the next proposition [11, Thm. 4.2.1]. If one applies the large deviation lower bound to the ˜ ˜ ), open set (N −ε, N +ε) ⊂ [N −ε, N +ε] and uses the bound E((N −ε, N +ε)) ≤ E(N then one obtains part (b). Part (b) implies that for any N ∈ [0, ∞), all sufficiently large n, and all ε > 0 we have Pn,nβ {Q(ψ (n) ) ∈ [N − ε, N + ε]} > 0.

(4.11)

N,ε is well defined. Hence for these values of N , n, and ε, the conditional measure Pn,nβ

Proposition 4.5. (a) As n → ∞, the sequence Pn,nβ {Q(ψ (n) ) ∈ dx} satisfies the LDP ˜ In particular, E˜ is on R with the scaling constants nβ and the rate function E. nonnegative and lower semicontinuous on R. (b) For N ∈ [0, ∞) and any ε > 0, lim inf n→∞

1 ˜ ) > −∞. log Pn,nβ {Q(ψ (n) ) ∈ [N − ε, N + ε]} ≥ −E(N nβ

˜ Substituting into the definition of E˜ the formula We indicate other expressions for E. (4.9) for J , we obtain . ˜ )= E(N inf{J (ξ ) : ξ ∈ L2c (ρ), Q(ξ ) = N } = inf{H (ξ ) + σ Q(ξ ) : ξ ∈ L2c (ρ), Q(ξ ) = N } − inf{H (ξ ) + σ Q(ξ ) : ξ ∈ L2c (ρ)} = inf{H (ξ ) : ξ ∈ L2 (ρ), Q(ξ ) = N } + σ N − inf{H (ξ ) + σ Q(ξ ) : ξ ∈ L2c (ρ)}. . ¯ )= Recalling the function E(N inf{H (ξ ) : ξ ∈ L2c (ρ), Q(ξ ) = N } introduced in (3.1), we see that ˜ ) = E(N ¯ ) + σ N − inf{H (ξ ) + σ Q(ξ ) : ξ ∈ L2c (ρ)}. E(N

(4.12)

204


Since E˜ is nonnegative and lower semicontinuous, it follows that E¯ is bounded below and lower semicontinuous. N,ε We now complete the proof of Theorem 3.1, which states the LDP for the Pn,nβ (n) distributions of ψ . This is carried out by proving that as n → ∞ and ε → 0 the N,ε sequence Pn,nβ (ψ (n) ∈ dξ ) satisfies the LDP on L2c (ρ) with the scaling constants nβ and the rate function J N defined in (3.2). The function Q defining the conditioning in this measure is continuous. If it were also bounded, then the desired LDP would be a consequence of Theorem 3.2 in our paper [17]. However, a quick examination of the proof of that theorem reveals that only the continuity of Q is required, not its boundedness (specifically, in the application of the contraction principle in the proof of Prop. 3.1 in [17]). For N ∈ [0, ∞), Theorem 3.2 in [17], with its proof modified as just described, N,ε shows that as n → ∞ and ε → 0 the sequence Pn,nβ (ψ (n) ∈ dξ ) satisfies the LDP on 2 Lc (ρ) with the scaling constants nβ and the rate function

˜ . J (ξ ) − E(N) if Q(ξ ) = N JÑ (ξ ) = ∞ otherwise. Substituting the formula (4.9) for J into the definition of JÑ and using (4.12) to relate ¯ we see that JÑ equals the function J N defined in (3.2). This yields the desired E˜ and E, N,ε LDP for Pn,nβ (ψ (n) ∈ dξ ). The proof of Theorem 3.1 is concluded. 5. Monte-Carlo Simulations Here we summarize some numerical computations that display the variety of behaviors N,ε that are exhibited by the mixed Gibbs ensembles Pn,nβ (dψ (n) ) in the continuum limit as n → ∞. In particular, we implement a Monte-Carlo sampling procedure to probe the dependence of the concentration phenomenon on Condition 2.1. To this end we consider operators √ L on L2c [0, π ] with Dirichlet boundary conditions and with eigenfunctions ek (x) = 2 sin(kx), and we choose the corresponding eigenvalues to be λk = −k α for 0 < α < +∞. Our main concern is to distinguish the case when α > 1, for which Condition 2.1 holds and hence our LDP applies, from the case when α ≤ 1, for which a concentration behavior may or may not occur. Our Monte-Carlo procedure is a modification of the standard Metropolis algorithm [2] appropriate to the mixed ensemble, which is canonical with respect to the energy Hn and microcanonical with respect to the particle number Qn . The microcanonical constraint is enforced exactly at each step of the Markov chain that defines the Metropolis algorithm; at each step two components of the random state ψ ∈ n are updated in a manner that preserves the spherical constraint, n1 |ψk |2 = 2N . To improve the sampling properties of the algorithm, a form of simulated annealing is used. That is, the sampling procedure is implemented in two stages: first at a small β (high temperature) and then at the prescribed β (given temperature). To exhibit the concentration behavior predicted by the LDP for ψ (n) , we sample N,ε the mixed Gibbs ensemble Pn,nβ (with ε → 0+) for the three values n = 16, 64, 256 with β fixed. In all our computations the underlying GNLS equation has the saturated nonlinearity f (|ψ|2 ) =

b |ψ|2 (1 + |ψ|2 )

(5.1)


n = 16

(a) α = 2.0 β = 0.1 b = 10

n = 64

n = 256

1.5

1.5

1.5

1

1

1

0.5

0.5

0.5

0

0

1

2

3

0

0

1

n = 16

(b) α = 0.5 β = 1.0 b = 1.0

205

2

3

0

1.5

1

1

1

0.5

0.5

0.5

1

2

3

0

0

1

2

2

3

n = 256

1.5

0

1

n = 64

1.5

0

0

3

0

0

1

2

3

Fig. 1a,b. Samples from mixed Gibbs ensemble for α = 2.0 and α = 0.5

with scale factor b. First we set α = 2.0, β = 0.1, and b = 10. These parameters are chosen to yield a ground state of approximately unit amplitude and unit width. Figure 1a displays the three plots for this sequence, each plot composed of five representative samples of |ψ| drawn from the Monte-Carlo sampled ensemble. The expected behavior under Condition 2.1 is clearly demonstated; namely, the fluctuations visible for n = 16 decrease for n = 64 and almost disappear for n = 256, for which all five samples remain close to the ground state. Next we set α = 0.5, β = 1.0, and b = 1.0. This choice of α furnishes an example of the behavior of the mixed Gibbs ensemble in the continuum limit when Condition 2.1 is violated. As in Fig. 1a, the parameters β and b are fixed so that the ground state has approximately unit amplitude and unit width. The sequence of three plots displayed in Fig.1b shows greater fluctuations than the corresponding plots in Fig.1a. As n increases, both the spatial scale and the magnitude of the fluctuations decreases, so that for n = 256 each of the five samples is a near-ground state on the large scale having small fluctuations on the small scales. Figure 1b appears to show a concentration around the ground state even in the case when α = 0.5, but possibly with a slower rate of convergence than for α > 1, and possibly in a weaker norm than the L2c (ρ)-norm. We do not know whether Condition 2.1 is necessary as well as sufficient for an LDP for the process ψ (n) in the L2c (ρ)-topology. On the one hand, the displayed computations for α < 1 and other computed results for α < 1 not included here suggest that a weaker condition may be sufficient for such an LDP. On the other hand, the proof of Theorem 4.1, the basic LDP for Gaussian processes on which our main Theorem 3.1 is based, is not valid for α < 1. This suggests

206


that the concentration property breaks down when α < 1. A more exhaustive numerical investigation might help to resolve this question, but such a computationally intensive study will not be pursued here. A second set of numerical computations is shown in Fig. 2. Here we investigate the dependence of the concentration phenomenon and the associated ground states on the strength of the nonlinearity in the GNLS equation. Specifically, we vary the parameter b > 0 in the saturated nonlinearity given in (5.1). Since the GNLS equation is focusing, an increase of b is expected to localize and intensify the ground state solitary wave. In Fig. 2a this effect is displayed for b = 10, 20, 100, with α = 2.0, β = 10 and n = 64; in Fig. 2b it is displayed for b = 1, 2, 10, with α = 0.5, β = 100 and n = 64. As in Fig. 1, these plots exhibit five samples of |ψ| for each ensemble. Since β is relatively large in each of these cases, each sample has the shape of a ground state. For large b, whether or not the summability of 1/λk in Condition 2.1 holds, we observe a ground state that is a highly localized solitary wave; an extreme case of this is given in Fig. 2b for b = 10. From the point of view of the present paper, the most noteworthy effect displayed in Fig. 2 is the presence of approximate translates of the exact ground state among the Monte Carlo samples. Indeed, for high β (low temperature) the energy of all the displayed samples is close to the ground state energy, and the samples themselves are often close to being translates of the exact ground state. This effect is straightforwardly explained by the fact that, for relatively localized states, the energy H is only slightly different among all translations of a ground state that are not too close to the boundary. According to our main LDP in Theorem 3.1, the rate function is simply the energy difference between a candidate state and the ground state(s) [see (3.2)]. Thus, while we have adopted Dirichlet

b = 10

b = 20

1

b = 100

1.5

3 2.5

0.8 (a) α = 2.0 β = 10 n = 64

1

2

0.6 1.5 0.4 0.5

1

0.2 0

0.5 0

1

2

3

0

0

1

b=1

2

3

0

0

1

b=2

2

3

b = 10

1

15 2

0.8 (b) α = 0.5 β = 100 n = 64

1.5

10

0.6 1

0.4

5 0.5

0.2 0

0

1

2

3

0

0

1

2

3

0

0

1

Fig. 2a,b. Samples from mixed Gibbs ensemble for increasing values of b

2

3


207

boundary conditions to reduce the translational invariance of the GNLS equation and thereby to simplify the presentation of our results, we have found that in the strongly focusing regime an approximate translational invariance persists. Of course, this effect diminishes as the Monte Carlo simulations are carried out for increasing n. In summary, the concentration phenomenon that is precisely expressed in our main LDP is definitely borne out by numerical sampling of the mixed ensembles over a wide range of parameters, even though this phenomenon is somewhat complicated by the near translation-invariance of localized ground states. In fact, the range of parameters for which the LDP holds may be wider than the range covered by our main theorem. Acknowledgement. The authors thank Adam Eisner, who provided the Monte-Carlo method and code used in Sect. 5 to carry out the simulations of the mixed ensemble.

References 1. Bidegaray, B.: Invariant measures for some partial differential equations. Physica D 82, 340–364 (1995) 2. Binder, K., Heermann, D.W.: Monte Carlo Simulation in Statistical Physics. Fourth edition. Springer Series in Solid-State Sciences, Vol. 80, Berlin: Springer-Verlag, 2002 3. Birkhoff, G., Rota, G.-C.: Ordinary Differential Equations. Second edition. Waltham: Blaisdell Publishing Co., 1969 4. Biskamp, D.: Nonlinear Magnetohydrodynamics. Cambridge Monographs in Plasma Physics. Cambridge: Cambridge Univ. Press,1993 5. Bolthausen, E.: On the probability of large deviations in Banach spaces. Ann. Probab. 12, 427–435 (1984) 6. Boucher, C., Ellis, R.S., Turkington, B.: Derivation of maximum entropy principles in twodimensional turbulence via large deviations. J. Stat. Phys 98, 1235–1278 (2000) 7. Bourgain, J.: Periodic nonlinear Schrödinger equation and invariant measures. Commun. Math. Phys. 166, 1–26 (1994) 8. Bouchet, F., Sommeria, J.: Emergence of intense jets and Jupiter’s Great Red Spot as maximumentropy structures. J. Fluid Mech. 464, 165–207 (2002) 9. Cai, D., Majda, A.J., McLaughlin, D.W., Tabak, E.G.: Spectral bifurcations in dispersive wave turbulence. Proc. Nat. Acad. Sci. 96, 14216–14221 (1999) 10. Cai, D., McLaughlin, D.W.: Chaotic and turbulent behavior of unstable 1D nonlinear dispersive waves. J. Math. Phys. 41, 4125–4153 (2000) 11. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Second edition. New York: Spring-Verlag, 1998 12. DiBattista, M.T., Majda, A.J., Grote, M.J.: Meta-stability of equilibrium statistical structures for prototype geophysical flows with damping and driving. Physica D 151, 271–304 (2001) 13. Dowling, T.E.: Dynamics of Jovian atmospheres. Ann. Rev. Fluid Mech. 27, 293–334 (1995) 14. Dupuis, P., Ellis, R.S.: A Weak Convergence Approach to the Theory of Large Deviations. New York: John Wiley & Sons, 1997 15. Dyachenko, S., Zakharov, V.E., Pushkarev, A.N., Shvets, V.F., Yan’kov, V.V.: Soliton turbulence in nonintegrable wave systems. Soviet Phys. JETP 69, 1144–1147 (1989) 16. Ellis, R.S.: Entropy, Large Deviations and Statistical Mechanics. New York: Springer-Verlag, 1985 17. Ellis, R.S., Haven, K., Turkington, B.: Large deviation principles and complete equivalence and nonequivalence results for pure and mixed ensembles. J. Stat. Phys. 101, 999–1064 (2000) 18. Gikhman, I.I., Skorohod, A.V.: The Theory of Stochastic Processes I. Trans. by S. Kotz, Berlin: Springer-Verlag, 1974 19. Hasegawa, A.: Self-organization processes in continuous media. Adv. Phys. 34, 1–42 (1985) 20. Isichenko, M.B., Gruzinov, A.V.: Isotopological relaxation, coherent structures, and Gaussian turbulence in two-dimensional magnetohydrodynamics. Phys. Plasmas 1, 1802–1816 (1994) 21. Itô, K., McKean, H.P.: Diffusion Processes and Their Sample Paths. New York/Berlin: Academic Press/Springer Verlag, 1965 22. Jordan, R., Josserand, C.: Self-organization in nonlinear wave turbulence. Phys. Rev. E 61, 1527– 1539 (2000) 23. Jordan, R., Josserand, C.: Statistical equilibrium states for the nonlinear Schrödinger equation. Math. Comp. Simulation 55, 433–447 (2001)

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24. Jordan, R., Turkington, B.: Ideal magnetofluid turbulence in two dimensions J. Stat. Phys. 87, 661– 695 (1997) 25. Jordan, R., Turkington, B., Zirbel, C.L.: A mean-field statistical theory for the nonlinear Schrödinger equation. Physica D 137, 353–378 (2000) 26. Kevrekidis, P.G., Rasmussen, K.O., Bishop, A.R.: The discrete nonlinear Schrödinger equation: A survey of recent results. Int. J. Mod. Phys. B. 15, 2833–2900 (2001) 27. Lebowitz, J.L., Rose, H.A., Speer, E.R.: Statistical mechanics of a nonlinear Schrödinger equation. J. Stat. Phys., 50, 657–687 (1988) 28. Majda, A.J., McLaughlin, D.W., Tabak, E.G.: A one-dimensional model for dispersive wave turbulence. J. Nonlinear Sci. 7, 9–44 (1997) 29. Marcus, P.S.: Jupiter’s Great Red Spot and other vortices. Annual Rev. Astronomy and Astrophys. 31, 523–573 (1993) 30. McKean, H.P.: Statistical mechanics of nonlinear wave equations IV. Cubic Schrödinger. Commun. Math. Phys.168, 479–491 (1995) 31. Rasmussen, J.J., Rypdal, K.: Blow-up in nonlinear Schroedinger equations–I: A general review. Physica Scripta 33, 481–504 (1986) 32. Segre, E., Kida, S.: Late states of incompressible 2d decaying vorticity fields. Fluid Dyn. Res. 23, 89–112 (1998) 33. Turkington, B., Majda, A.J., Haven, K., DiBattista, M.: Statistical equilibrium predictions of jets and spots on Jupiter. Proc. Nat. Acad. Sci. USA 98, 12346–12350 (2001) 34. Zakharov, V.E., Pushkarev, A.N., Shvets, V.F.,Yan’kov,V.V.: Soliton turbulence. JETP Lett. 48, 83–86 (1988) 35. Zhidkov, P.E.: On an invariant measure for a nonlinear Schrödinger equation. Soviet Math. Dokl. 43, 431–434 (1991) Communicated by P. Constantin


Communications in


PCT Theorem for the Operator Product Expansion in Curved Spacetime Stefan Hollands Enrico Fermi Institute, Department of Physics, University of Chicago, 5640 Ellis Ave., Chicago, IL 60637, USA. E-mail: [email protected] Received: 17 December 2002 / Accepted: 20 June 2003 Published online: 27 November 2003 – © Springer-Verlag 2003

Abstract: We consider the operator product expansion for quantum field theories on general analytic 4-dimensional curved spacetimes within an axiomatic framework. We prove under certain general, model-independent assumptions that such an expansion necessarily has to be invariant under a simultaneous reversal of parity, time, and charge (PCT) in the following sense: The coefficients in the expansion of a product of fields on a curved spacetime with a given choice of time and space orientation are equal (modulo complex conjugation) to the coefficients for the product of the corresponding charge conjugate fields on the spacetime with the opposite time and space orientation. We propose that this result should be viewed as a replacement of the usual PCT theorem in Minkowski spacetime, at least in as far as the algebraic structure of the quantum fields at short distances is concerned. 1. Introduction The operator product expansion [1, 2] states that the product of any finite number of field operators localized at nearby points can be approximated by a sum of products of c-number coefficient functions of the coordinates of the points relative to a reference point, times fields that are localized at the reference point. Furthermore, of these coefficient functions, only finitely many are singular as the spacetime points approach the reference point. In mathematical symbols, if the fields in the theory are denoted by the generic symbol φ (i) (with (i) a label that distinguishes the various kinds of fields), then the operator expansion is an expansion of the form c(i) (y1 , y2 , . . . , yn )φ (i) (x). (1) φ (1) (y1 )φ (2) (y2 ) · · · φ (n) (yn ) ∼ (i)

The notation “∼” means that the difference between the expectation value in any “reasonable” state of the left side and the expectation value of a suitable finite partial sum on

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right side goes to zero as the points y1 , . . . , yn approach the reference point1 , x. Moreover, the rate at which this difference goes to zero can be made arbitrary by including sufficiently many terms in the partial sum. In practice, the operator product expansion is useful to find approximate expressions at short distances (high momenta) for the expectation value of a product of n fields when the corresponding expectation values of the singly localized fields φ (i) on the right side of Eq. (1) are known, for example experimentally, and where the coefficients c(i) can be calculated. Such techniques have been used successfully e.g. to gain insights into the internal structure of hadrons. In Minkowski spacetime, model-independent derivations of the operator product expansion from first principles within an axiomatic framework, including a precise specification of the nature of states for which it holds, have been given in various contexts [3–7]. The derivation that is in our opinion most general and physically best motivated seems to be that of [7], (which is based in turn on earlier work by [8, 9]) and our analysis builds partly on the results and ideas of this work. More formal proofs of the validity of operator product expansion order by order in perturbation theory within quantum field theoretic models derived from a renormalizable Lagrangian had in fact been established much earlier [2]. In general curved spacetimes, a derivation of the operator product expansion from first principles is not available at present2 . In this paper, we will not investigate this important issue, but consider instead the simpler question which properties of the operator product expansion can be derived in curved spacetime if one assumes that such an expansion exists in a suitable sense and that it has certain general model-independent properties. Specifically, we are going to derive the following result about the invariance of the operator product expansion under parity, time, and charge (PCT) in a general 4-dimensional analytic curved spacetime: If the operator product expansion has the properties3 (L) that the distributional coefficients c(i) depend only locally on the metric (in a generally covariant manner), (M) that they satisfy a suitable “microlocal” spectrum condition [10], (A) that they vary analytically under analytic variations of the spacetime metric, then the operator product expansion will automatically have an invariance under reversal of the space and time orientations of the spacetime, and charge conjugation of the fields (if the theory contains any charged fields). Since the coefficients c(i) in the operator product expansion can be viewed in some sense as “structure constants” of the algebra of quantum fields, one can interpret this result as showing that the algebraic structure of the quantum fields in curved spacetime is invariant under PCT at short distances, at least under the above general assumptions. The status of our assumptions is the following: We will argue that property (M) is satisfied for the operator product expansion in Minkowski spacetime as constructed in [7], and we will show elsewhere [11] that properties (L), (M), and (A) are satisfied for perturbative constructions of the operator product expansion in an arbitrary curved spacetime. We would now like to qualify our statement about the PCT invariance of the operator product expansion and distinguish it from corresponding statements about (global) PCT invariance. For quantum field theories in Minkowski spacetime satisfying the Wight1 One can of course take advantage of translation invariance in Minkowksi spacetime to set the reference point x to the origin in Minkowkski space, which is done in usual formulations of the operator product expansion. We are avoiding this since it has no invariant meaning on a curved spacetime. 2 Heuristically, one expects that if an operator product expansion holds for a theory in Minkowski spacetime, then it also holds for the corresponding theory in curved spacetime, since, essentially by the “Einstein equivalence principle,” the short distance behavior of a quantum field theory in curved spacetime should be the “same” as that of the corresponding field theory in Minkowski spacetime. 3 We emphasize in particular that we do not assume that our model is derived from a Lagrangian.

PCT Theorem for the Operator Product Expansion in Curved Spacetime

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man-axioms or the axioms of algebraic quantum field theory, one can show [12–14] that PCT is always implemented by an anti-unitary operator on the vacuum Hilbert-space of the theory, in the sense that φ(x)−1 = i F (−1)M φ(−x)∗ , where M is the number of unprimed spinor indices of the field, and where F is zero if the field is bosonic, and one if it is fermionic. If the theory has an operator product expansion like Eq. (1), then one easily finds that this expansion has a similar invariance under PCT by acting with on both sides of this expansion. Hence, in Minkowski spacetime, the PCT invariance of the operator product expansion is a simple and direct consequence of the (global) PCT invariance of the theory. On the other hand, it is clear that for a quantum field theory defined only on a single, fixed curved spacetime, one cannot in general even formulate the notion of PCT symmetry in the same manner as in Minkowski spacetime, since a generic spacetime does not possess any isometries analogous to parity (x → −x) and time reversal (t → −t) in Minkowski spacetime. Nevertheless, a notion of (global) PCT symmetry can naturally be formulated in a theory that is consistently given on all oriented and time oriented globally hyperbolic spacetimes in the sense of a generally covariant quantum field theory as recently introduced in [15, 16]: It is natural to say that a generally covariant quantum field theory is globally PCT invariant if the algebras of observables4 corresponding to any given spacetime equipped with the opposite orientations are isomorphic, and if any quantum field on the spacetime with the original orientation is mapped to the charge conjugate field on the spacetime with the opposite orientation under this isomorphism. In Minkowski spacetime, the map t → −t, x → −x is an isometry reversing the space and time orientations, so the notion of PCT invariance of a theory as just stated for general curved spacetimes reduces to the usual notion of PCT invariance Minkowski spacetime (with the isomorphism given by A → A∗ ). It is not known at present whether and under what circumstances PCT invariance as formulated above holds in general curved spacetimes. Consequently, a corresponding PCT invariance of the operator product expansion in curved spacetime does not automatically follow in the same straightforward manner as in Minkowski spacetime. As we show in this paper, PCT invariance of the operator product expansion can nevertheless be proven under the above general and model-independent assumptions (M), (L), and (A). As we have already mentioned, this result may be viewed as an “infinitesimal” version of the PCT-theorem in curved spacetime, in the sense that it proves PCT-invariance of the algebraic relations between the quantum fields at short distances. Our strategy for proving this result is the following. Using that the coefficients in the operator product expansion depend locally and covariantly on the spacetime metric and the spacetime orientations in a covariant manner, and using that this dependence is analytic, we show, using the ideas of [17], that each coefficient can be expanded into a sum of terms, each of which is a product of a curvature tensor at the reference point, x, times a Lorentz-invariant Minkowski space distribution in the Riemannian normal coordinates of the points yi relative to x. The PCT invariance of the coefficients in the operator product expansion is then seen to follow if these Minkowski space distributions are invariant (up to permutation of the arguments and a combinatorical factor) under a reflection of the Riemannian normal coordinates of the n points yi about the origin. In order to show that this invariance indeed holds, we use the microlocal spectrum condition to show that our Minkowski space distributions arise as the boundary value of certain analytic functions. The desired invariance is then shown using the transformation properties of these analytic functions under complex Lorentz transformations on a 4 The algebra of observables associated with a given spacetime is the abstract *-algebra generated by quantum fields smeared with testfunctions of compact support in the spacetime.

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suitable complex domain by methods that are similar to the proof of the PCT-theorem in Minkowski spacetime [12]. We remark that, while our proof makes essential use of the fact that the spacetime is real analytic, we expect that our result can be generalized to spacetimes that are only smooth by approximating such spacetimes with a sequence of real analytic spacetimes and by making suitable additional assumptions about the “continuity” of the operator product expansion (of the kind introduced in [17]) under such approximations. The organization of this paper is as follows. In Sect. 2, we recall the notion of a generally covariant quantum field theory in curved spacetime. In Sect. 3, we give a precise formulation of the operator product expansion in curved spacetime and in Sect. 4 we state our technical assumptions concerning the properties of this expansion. Section 5 contains the main result of this paper (Theorem 5.1). Our conventions and notations related to the spacetime geometry are as follows: We view a spacetime as a triple M = (M, gab , o), where M is a 4-dimensional manifold, gab is a metric tensor of signature (+ − −−), and o denotes space and time orientations, represented by a tuple (T , abcd ), where T is a time function on M and abcd is a nowhere vanishing volume form. Throughout, we assume the manifold structure of M and the spacetime metric gab to be real analytic. We denote (abstract) tensor indices by lower case letters of the Roman alphabet and the components of a tensor in a coordinate chart by letters of the Greek alphabet.

2. Mathematical Formulation of Quantum Field Theory on Curved Spacetimes The usual formulations of quantum field theory on Minkowski spacetime rely heavily on the existence of a preferred vaccum state and the special properties of that state. The existence of such a state is, in turn, tied up with the special symmetries of Minkowski spacetime, and indeed, there is no preferred state, nor even any preferred Hilbert space construction that can be singled out for special consideration on a generic curved spacetime. Moreover, apart from a very limited class of spacetimes such as static ones, most Lorentzian spacetimes cannot be viewed as a real section of a complex spacetime that also possesses a real, Euclidean section, so a formulation of quantum field theories on generic Lorentzian spacetimes via Euclidean methods such as the Euclidean path-integral is not possible5 in general. Fortunately, there is a simple, and fully satisfactory way to formulate quantum field theory in curved spacetime which bypasses all of these problems, namely the so-called “(generally covariant) algebraic approach to quantum field theory”[15, 16] (for a review of the algebraic approach to quantum field theory in Minkowski spacetime, see [18]). In this framework, a (generally covariant) quantum field theory is viewed as an assignment that associates with every oriented and time-oriented spacetime M ≡ (M, gab , o) an abstract *-algebra6 A(M) with unit whose elements are the observables7 of the theory. 5 By this we do not mean that it is not worthwhile to study the Euclidean path integral in curved space, or other related quantities, such as e.g. “effective actions”. What we mean is that the physical interpretation of such quantities and their properties is very unclear unless the Euclidean spacetime under consideration has a real, Lorentzian section. This is a very severe restriction that excludes essentially all spacetimes that are not static. 6 In [15], these algebras were assumed to be C∗ -algebras. This is too restrictive for the purposes of the present paper since we also want A(M) to contain unbounded elements. 7 We use the term “observable” somewhat sloppily since we will allow in the algebras also quantities that anti-commute (rather than commute) at spacelike separations.


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The features of locality and general covariance of a quantum field theory are reflected in the following consistency properties of this assignment: Consider a situation in which we are given an isometric embedding χ : N → M of a spacetime N into a spacetime M which preserves the causal structure and the orientations, meaning that if o = (T , abcd ) is the orientation of M, then (χ ∗ T , χ ∗ abcd ) coincides with the orientation of N . Then we postulate that there exists an injective *-homomorphism αχ : A(N ) → A(M).

(2)

Furthermore, if χ1 and χ2 are isometric embeddings with the above properties such that the composition χ1 ◦ χ2 can be defined (and consequently defines again an isometric embedding with these properties), then we postulate that αχ1 ◦χ2 = αχ1 ◦ αχ2 .

(3)

The existence of the algebraic isomorphism αχ in Eq. (2) with the property (3) formalizes the idea that observables associated with a spacetime N that is isometric to a globally hyperbolic subregion of a larger spacetime M can be viewed via αχ as observables in the larger spacetime satisfying the same algebraic relations. This can be interpreted as saying that the algebraic relations between the observables depend locally and covariantly on the metric. If M is Minkowski spacetime with a given choice of orientations, then the orientation and causality preserving (global) isometries of M are given precisely by the translations x → x + a, where a ∈ R4 , together with the proper orthochronous Lorentz ↑ transformations x → x, where ∈ L+ . Thus, in the special case of Minkowski spacetime, our axioms say that the Poincaré group acts on the algebra of observables by a group of *-automorphisms α{,a} . Requirements Eq. (2) and Eq. (3) may therefore be viewed as a replacement of the notion of Lorentz-covariance of a quantum field theory in Minkowski spacetime by the notion of general covariance. In order to formulate the notion of local commutativity respectively local anticommutativity in this algebraic framework, we need to assume that algebraic elements A ∈ A(M) can be uniquely decomposed into a “bosonic” and a “fermionic” part. This is formalized by requiring that there exists a *-automorphism γM for every oriented spacetime M with the property (γM )2 = 1 and γM = γN ◦ αχ whenever χ is an orientation and causality preserving isometric embedding from N into M. We can then uniquely decompose A = A+ + A− , where γM (A± ) = ±A± , and we call A+ the bosonic and A− the fermionic part of A. Given now two isometric embeddings χi : Ni → M, i = 1, 2, such that the image of N1 under χ1 in M is spacelike related to the image of N2 under χ2 , then our requirement of local (anti-) commutativity is [αχ1 (A1 ), αχ2 (A2 )]γ = 0 for all A1 ∈ A(N1 ) and A2 ∈ A(N2 ), where AB + BA [A, B]γ = AB − BA

A, B fermionic, A or B bosonic,

(4)

(5)

is the graded commutator. The algebras of observables, A(M), were referred to as “abstract”, because it has not been assumed that its elements are represented as linear operators on some particular Hilbert space. This is of great conceptual advantage, because there exist in general

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many inequivalent representations of which no particular one can be singled out for special consideration. The quantum states are simply all linear functionals ω : A(M) → C, A → A ω from the algebra associated with that spacetime with values in the complex numbers, which are positive in the sense that A∗ A ω ≥ 0 for all A ∈ A(M), and which are normalized in the sense that ? ω = 1, where ? is the identity element. By formulating the theory in terms of abstract algebras, we have therefore avoided predjudicing ourselves towards the particular class of states that can be represented as vectors or density matrices in some particular representation. States of particular interest may be singled out for example in spacetimes which happen to have symmetries or suitable asymptotic regions, or in models with additional internal symmetries, but we emphasize that the question whether such choices are possible is not in any way related to the algebraic structure of A(M), and hence does not affect the formulation of the quantum field theory. A local covariant (scalar) field, φ, is an assignment which associates with every spacetime M a linear map φM : D(M) → A(M),

f → φM (f ),

(6)

from the space D(M) of all smooth compactly supported functions on M to A(M). The locality and covariance property of the field is encoded in the requirement that αχ (φN (f )) = φM (χ∗ f ),

(7)

whenever χ : N → M is an orientation and causality preserving isometric embedding of a spacetime N into a spacetime M, and where χ∗ f denotes the testfunction on M corresponding to the testfunction f on N via the map χ . The above transformation law (7) expresses (a) that the field φ(x) is constructed entirely out of the metric in an arbitrary small neighborhood of the point x, and (b) that it is constructed out of the metric in a generally covariant way. In the case when M is Minkowski spacetime and χ = {, a} is an element of the Poincaré group, Eq. (7) specializes to α{,a} (φ(x)) = φ(x + a), which is the familiar special relativistic transformation law for a scalar field. The above definition of local covariant quantum field of scalar type can be generalized in a relatively straightforward manner to fields of arbitrary spinor type. The main new issue is that the definition of spinors curved spacetime requires the existence and specification of a spin structure. We will consequently assume that the spacetimes under consideration can be equipped with a spin structure (for matters related to spinors in curved spacetime, see Appendix B). Since we want the quantum fields of spinor type to be elements in A(M) after smearing with a suitable testfunction, we will now view these algebras as depending not only on the spacetime metric and orientations, but also on the particular choice of spin structure, if several inequivalent spin structures are possible. The above locality and covariance property (2) and the local (anti-) commutativity property (4) of the assignment M → A(M) are then formulated in terms of embedding maps χ : N → M which not only preserve the metric structure and orientation, but in addition also lift to a homomorphism between the spin structures on N respectively M. Moreover, local and covariant quantum fields of spinor type are defined as assignments φM : D(M; F(M)) → A(M),

f → φM (f ),

(8)

where F(M) is the appropriate tensor product of vector bundles V(M), V (M), V ∗ (M) and V ∗ (M) corresponding respectively to the unprimed, primed, upper primed, and upper unprimed spinor indices of the field. The local and covariant transformation property of these fields is formulated as in the scalar case, Eq. (7), in terms of embedding


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maps χ : N → M. The new feature is that these maps are now also required to preserve orientation and time orientation and to lift to a corresponding map between the respective spinor structures. Moreover, the symbol χ∗ f in Eq. (7) now denotes the compactly supported section in the vector bundle F(M) that corresponds to the compactly supported section f in the bundle F(N ) via the lift of the map χ to the spin structures over N and M. For the quantum field theories that we consider in this paper, we assume that there are countably many local covariant fields, which we shall denote by the generic symbol φ (i) , where i ∈ N is a label that distinguishes the various fields. Note that, since the grading maps satisfy γM = γN ◦ αχ for every orientation and causality preserving isometric embedding, we can consistently decompose any local and covariant field into its bosonic and fermionic parts for all spacetimes. Thus, without loss of generality, we can assume that a local covariant field is either bosonic or fermionic. We emphasize however that we do not assume that all half odd-integer spin fields are fermionic and that all integer spin fields are bosonic. Such a relation between spin and statistics has been proven recently by Verch [19], but the technical assumptions made in [19] are not identical with the technical assumptions we will be making here. The proof of our main result on the other hand does not rely on the spin-statistics relation, so we will avoid assuming the spin-statistics relation in this paper. We emphasize that the axiomatic framework we have set up so far says nothing a priori about the relation between the field observables and corresponding algebras associated with globally isometric spacetimes carrying different orientations. However, it is natural to conjecture that the framework (or, more likely, some extended version or variant thereof) implies a symmetry of the theory if the orientations of the spacetime are changed in a way corresponding to the reversal of time and parity in Minkowski spacetime. Namely, let M = (M, gab , o) be a given globally hyperbolic spacetime with orientations o = (T , abcd ), and let M = (M, gab , −o) be the same spacetime with orientations −o = (−T , abcd ), i.e., with the same orientation and the opposite time orientation. In this situation, a (as yet, hypothetical) PCT-theorem in curved spacetime can be formulated as follows: There exists, for each M, an anti-linear *-isomorphism, P CT : A(M) → A(M), θM

(9)

such that P CT ◦ θ P CT = id, θM

M

(10)

which is consistently given for all M in the following sense. If χ : N → M is a causality and orientation preserving embedding (lifting to a homomorphism of the spin structures over N and M compatible with o) and if χ : N → M is the corresponding embedding between the spacetimes with orientation −o, then P CT P CT ◦ αχ = αχ¯ ◦ θN . θM

(11)

P CT is The above consistency condition formally expresses the demand that the map θM P CT locally and covariantly constructed out of the metric. It implies in particular that θM maps local, covariant fields over M to local covariant fields over M. By analogy to P CT can be chosen in such Minkowski spacetime, we conjecture that, more precisely, θM a way that P CT (φM (f )) = i F (−1)M φM (f )∗ , θM

(12)

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where F is 0 or 1 if the field is bosonic resp. fermionic, and where M is the number of unprimed spinor indices of the field. It is maybe not immediately obvious how the above formulation of the PCT-property in the generally covariant framework is related to the usual PCT-theorem in Minkowski space, and so we briefly explain this point. In Minkowski spacetime consider the map χ : x → −x. In the framework of Wightman field theory, one proves [13, 12] the existence of an anti-unitary map on the Hilbert space on which the fields are represented as operators, such that φM (f )−1 = i F (−1)M φM (χ∗ f )∗ for all local fields on Minkowski spacetime M = (R4 , ηab , o) with a given set of orientations o. The map χ is a global orientation preserving isometry8 between Minkowski spacetime with a given set of orientations o, and Minkowski spacetime M = (R4 , ηab , −o) with the orientations −o. Hence, by the general covariance principle Eq. (7) applied to the special case of Minkowski spacetime, we get a corresponding *-isomorphism αχ : A(M) → A(M) satisfying αχ (φM (f )) = φM (χ∗ f ). Thus, the composition P CT θM ≡ αχ ◦ Ad

(13)

(where Ad (A) = A−1 ) defines a PCT-map with the desired properties in the special case of Minkowski spacetime. Note that, in Minkowski spacetime, it is not actually necessary to consider spacetimes with opposite orientations and one can simply view Ad : A(M) → A(M) itself as the PCT-map (as is of course done in all discussions of the PCT theorem in Minkowski space). However, it is clearly not possible to generalize this formulation of PCT to general curved spacetimes without symmetries analogous to x → −x. On the other hand, P CT does not rely on the existence of any the above formulation of PCT via the maps θM such symmetries. As we argue in remark (3) following our main Theorem 5.1 below, the P CT in the sense of results of this paper in some sense prove the existence of the maps θM an asymptotic expansion of the algebra structure near a spacetime point (for real analytic P CT with the above spacetimes). Thus, our results support the conjecture that the maps θM properties indeed exist. 3. Formulation of the Operator Product Expansion in Curved Spacetime In the last section we have reviewed the formulation of quantum field theory in curved spacetime as an assignement of spacetimes with *-algebras of observables, and we have introduced local, covariant quantum fields as suitable assignments of spacetimes with elements in the algebra of observables associated with the spacetime. We now wish to study quantum field theories in curved spacetime that possess in addition an operator product expansion. Let (M) be the space of all complex linear functionals on A(M), (M) = {σ : A(M) → C | σ (c1 A1 + c2 A2 ) = c1 σ (A1 ) + c2 σ (A2 )}.

(14)

We say that such a functional is real, if σ (A∗ ) = σ (A) for all A. Quantum states ω : A → A ω are normalized and positive elements of (M). The proof [7] of the operator product expansion in Minkowski spacetime suggests that one should view the coefficients c(i) appearing in the operator product expansion (1) as being the n-point functions of certain “standard” linear functionals σ (i) on A(M), 8

Note that χ is an orientation reversing isometry from M to M.


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where (i) is a label that distinguishes the various local and covariant fields in the theory. We will adapt this viewpoint in our formulation of the operator product expansion in curved spacetime. Our (as yet, still formal) definition of a local, covariant quantum field theory possessing an operator product expansion is then as follows: Definition 3.1. We say that a local covariant quantum field theory (with only scalar fields) possesses an operator product expansion, if for any space and time oriented spacetime M and any point x ∈ M there exist linear functionals (i)

σM,x ∈ (M)

(15)

such that (i)

σM,x ◦ γM = (−1)

F (i)

(i)

σM,x ,

F

(i)

=

0 if φ (i) is bosonic, 1 if φ (i) is fermionic,

(16)

and

n

(jk )

−

φM (yk )

k=1

ω

N

(i)

σM,x

i=1

n

(jk )

φM (yk )

(i)

φM (x)

k=1

ω

→ 0,

(17)

as (y1 , . . . , yn ) → (x, . . . , x) and as N → ∞, for all suitable states ω on A(M), and any collection of fields φ (j1 ) , . . . , φ (jn ) . Remarks. (1) The coefficients c(i) in our previous expression for the operator product expansion (1) correspond to the standard functionals σ (i) in the above formulation via (i)

(i)

(j )

(j )

cM,x (y1 , . . . , yn ) = σM,x (φM1 (y1 ) · · · φMn (yn )),

(18)

where we have now put a subscript “M, x” on the coefficients c(i) in order to indicate the dependence on the spacetime and the reference point, x. Condition Eq. (16) expresses the demand that each term in the operator product expansion has the same fermion number modulo 2. (2) The above definition can be generalized in a straightforward way to theories that contain not only scalar fields but fields of arbitrary spinor type, in which case all quantities depend in addition on a choice of spin structure over M which is compatible with the space and time orientations (see Appendix B for details). Since local covariant fields of spinor type take testfunctions as entries that are sections in a vector bundle F(M) (i) corresponding to the spinor type of the field, it is natural in this case to view σM,x as linear functionals on A(M) taking values not in C but instead in the complex vector space Fx (M), where we mean the fibre of this vector bundle over x. To make the above definition mathematically precise, we still need to specify (a) the precise nature of the states ω that are allowed in Eq. (17), as well as the nature (i) of the functionals σM,x . (b) the precise sense in which the expression (17) tends to 0. We now turn to these tasks.

218

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Given a spacetime M, a collection φ (j1 ) , . . . , φ (jn ) of local covariant fields and a functional σ ∈ (M), we consider the multi-linear functional ×n D(M) → C,

(j )

(j )

(f1 , . . . , fn ) → σ (φM1 (f1 ) · · · φMn (fn ))

(19)

on the n fold cartesian product of the space of testfunctions on M, where for simplicity we assume that all the fields are scalar. The regularity properties of a functional σ may be specified by specifying regularity properties for the linear functionals (19) for an arbitrary set of local covariant fields. Firstly, we will ask that the linear functionals (19) are distributions on ×n M, i.e., that they are continuous with respect to the Laurent-Schwarz topology on the spaces of testfunctions9 ×n D(M). Among these, we now further restrict our attention to those functionals σ for which the distributions (19) have a particular singularity structure specified by the following “microlocal spectrum condition” [10]: (j )

(j )

WFA (σ (φM1 (y1 ) · · · φMn (yn ))) ⊂ M ,

(20)

where WFA is the “analytic wave front set” [20] of a distribution10 , and where M ⊂ T ∗ (×n M)\{0} is defined in terms of the geometry as follows: Let G(p) be a “decorated embedded graph” in M. By this we mean an embedded graph in M whose vertices are points x1 , . . . , xn in M and whose edges, e, are piecewise smooth curves11 γ in M connecting the vertices. Each such edge e is equipped with a future pointing timelike or null coparallel covectorfield (pe )a , meaning that γ˙ a ∇a (pe )b = 0,

g ab (pe )a (pe )b ≥ 0,

(pe )a ∇a T > 0,

(21)

where T is the time function that defines the time orientation of M. If e is an edge in G(p) connecting the points xi and xj with i < j , then we denote s(e) = i its source and t (e) = j its target. With this notation, we define

M = (x1 , k1 ; . . . ; xn , kn ) ∈ T ∗ (×n M)\{0} | ∃ decorated graph G(p) with vertices x1 , . . . , xn such that ki = pe − pe ∀i . (22) e:s(e)=i

e:t (e)=i

We will denote by

A (M) = σ ∈ (M) | WFA σ

n

(jk )

φM (yk )

⊂ M

(23)

k=1

the space of all linear functionals such that Eq. (20) holds for an arbitrary set of local covariant fields. Our operator product expansion will be required to hold only for states ω ∈ A (M). The analytic wave front set of φ (i) (x) ω is then empty, meaning that this expression is not just a distribution, but in fact an analytic function in x. This 9 Strictly speaking, we should demand that our functionals are continuously defined on the space D(×n M) rather than continuous multilinear functionals on ×n D(M). However, by the “Schwartz Nuclear Theorem”, these requirements are actually equivalent.

10 Our convention for the Fourier transform in Rm is fˆ(k) = (2π)−m/2 e+ikx f (x) d m x, which is opposite to the convention used in [20]. It follows from this that our definition of the analytic wave front set is minus the definition given in [20]. 11 We note that a more restrictive notion of a microlocal spectrum condition would be obtained if we would replace “piecewise smooth curve” by “causal curve” or “null-geodesic”.


219

implies in particular that the products of distributions implicit in our operator product expansion (17) are automatically well-defined. (i) We furthermore require that, for all x ∈ M, the standard functionals σM,x are such that Eq. (20) is satisfied in some neighborhood of x; in other words, we require: (M) For every x ∈ M there exists an isometric embedding χ : N → M preserving the orientations such that x is in the image of N under χ , and such that the linear functional on A(N ) defined by (i)

A(N ) A → σM,x (αχ (A)) ∈ C

(24)

is an element of A (N ). We have thus accomplished (a). The above microlocal spectrum condition (or rather, an analogous “C ∞ ”-version thereof) was first proposed by [10], as a replacement for the usual spectrum condition on vacuum states in Minkowski spacetime. It was shown in [10] that it is satisfied in any Wightman quantum field theory in Minkowski spacetime, as well as for so-called quasifree “Hadamard states” [21] in linear quantum field theories in curved spacetimes12 . The above analytic version of this condition is natural in analytic spacetimes and was first proposed in [16]. It is discussed in [23] in connection with long-range correlations in quantum field theories on curved spacetimes, and it was used in [24] to prove the PCT and spin-statistics theorem for certain non-local field theories on Minkowski spacetime. Our motivation for imposing the microlocal spectrum condition (20) on the operator product expansion comes from the following facts. It was shown in [7] that the operator product expansion in Minkowski spacetime will hold typically only for states ω that are well-behaved at high energies (for example energy-bounded), and that the standard (i) functionals σM,x can be chosen energy-bounded. On the other hand, one can show that if M is Minkowski spacetime, then every functional with bounded energy satisfies the microlocal spectrum condition. More specifically, assume that the algebra of observables corresponding to Minkowski space admits a faithful representation on a Hilbert space on which the group of automorphisms αa associated with the translations byµa four vector a µ is implemented by a strongly continuous group of unitaries, αa (A) = eia Pµ A e−ia Pµ , with self-adjoint generator P satisfying the spectrum condition, specP ⊂ V¯ + , where V¯ + is the closure of the future lightcone in Minkowksi spacetime, and where A has been identified with the linear operator on the Hilbert space representing it. We say that a functional σ ∈ (M) in Minkowski space has finite energy below p 0 (relative to some Lorentz frame) if σ (A) = σ (Ep0 AEp0 ) ∀A ∈ A(M),

(25)

where Ep0 denotes the projector on the spectral subspace of the Hamiltonian P 0 corresponding to energies less than p0 , and where we have assumed that σ can be identified with a functional on the image of A(M) under the represenation. Then such a σ satisfies the microlocal spectrum condition (20) in Minkowski spacetime (a formal proof of this statement, which follows closely a similar argument invented in [10], is given in Appendix A). Furthermore, it seems to be the case that the coefficients in the operator product 12 In fact, it can be shown that the Hadamard states as defined in [21] are precisely the states whose twopoint function satisfies a suitably strengthened version (see footnote 11 on p. 218) of the C ∞ -microlocal spectrum condition [22] (for a discussion of the analytic case, see [23]).

220

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expansion for free fields in analytic curved spacetimes satisfy our analytic microlocal spectrum condition [11], and we expect this also to be true for perturbatively defined self-interacting quantum field theories in curved spacetimes. We next turn to our second task (b) to explain the precise sense in which the expression (17) converges to zero. An investigation of the operator product expansion for free fields shows that one can certainly not expect that expression (17) tends to zero in the sense of a convergent sequence of functions, or rather, distributions. Rather, one can only expect that this expression has an arbitrarily low scaling degree as (y1 , . . . , yn ) → (x, . . . , x) for large N in any given state ω ∈ A (M). The model-independent derivation [7] of the operator product expansion in Minkowski spacetime from first principles leads to the same conclusion13 . We will consequently formulate the convergence of the operator product expansion by demanding that the scaling degree of the expression (17) becomes arbitrarily small when N → ∞. Let u be a distribution on an open, convex neighborhood X of Rn . If λ is a positive number less than 1 and f is a smooth compactly supported function on X, we define another such function fλ by setting fλ (y) = λ−n f (x+λ(y−x)). The scaling degree [25], δ, of u at the point x is defined as δ = inf{γ ∈ R+ | lim λγ u(fλ ) = 0 λ→0

∀f ∈ D(X)}.

(26)

The scaling degree of a distribution thus characterizes the strength of its singularity at x. It is a completly local concept in that it depends only on the behaviour of u near x and can be generalized in an invariant manner to distributions on a manifold X by localizing u in a chart near a point x in the manifold. The precise sense in which we assume the operator product expansion to converge is then the following: We ask that for every δ < 0, we can find an N such that the scaling degree of the distribution defined by the left side of expression (17) at (y1 , . . . , yn ) = (x, . . . , x) is less than δ. This accomplishes (b). We note however that the PCT-invariance of the operator product expansion that we are going to state and prove in Sect. 5 will follow independently of any assumptions made about the convergence of this expansion at small distances – in other words, property (b) will not be used at all in the proof given in Sect. 5 (see also the remark following Theorem 5.1).

4. Technical Assumptions About the OPE In the last section we have given a mathematically precise formulation of the operator product expansion in a curved spacetime. In order to be able to prove our main result that the operator product expansion (17) has a PCT-invariance, we will now make the following further assumptions about the nature of this expansion: (i)

(L) The standard functionals σM,x have a local and covariant dependence on the spacetime metric and orientations. (i) (A) The standard functionals σM,x have a suitable analytic variation under analytic variations of the spacetime metric. 13 We remark however that the convergence properties of the operator product expansion established in [7] are stronger than the convergence properties postulated here in that they hold uniformly for all states with energy below some arbitrary p0 .


221

Our motivation for imposing (L) and (A) comes from the fact that these properties are satisfied in free field theories in curved spacetime and are also expected to hold in perturbatively defined interacting quantum field theories in curved spacetime [11]. As we explain below, condition (L) is equivalent to the local and covariant dependence on the metric of the coefficients c(i) in the operator product expansion (1). Since these coefficients can be viewed, in some sense, as structure constants for the algebraic relations between the quantum fields at short distances, we may view (L) as a strengthened version of the general covariance property of the quantum field theory under consideration. We now discuss the precise form (L) and (A) in turn. In order to formulate our condition that the standard functionals depend locally and x covariantly on the metric, it is useful to first define an equivalence relation ∼ between linear functionals in (M) relative to a point x ∈ M by declaring two such functionals ϕ1 and ϕ2 to be equivalent if they coincide when restricted to some neighborhood of the point x, where the restriction of a linear functional on A(M) to a globally hyperbolic neighborhood O ⊂ M is defined in the obvious way by viewing A(O) as a subalgebra of A(M) via the *-isomorphism Eq. (2) corresponding to the embedding O ⊂ M. The assignment of pairs (M, x) consisting of oriented, time oriented spacetimes M and a (i) point x ∈ M, to the functionals σM,x is then said to be local and covariant if x

(i)

(i)

σM,χ(x) ◦ αχ ∼ σN ,x

(27)

for any orientation and causality preserving isometric embedding χ : N → M, where we have assumed for simplicity that all fields in the theory are scalar. In the case when the theory contains spinor fields as well, we consider causality and orientation preserving isometric embeddings χ that in addition lift to a corresponding map between the spin-structures over N and M respectively. If F(N ) is the vector bundle over N corresponding to the spinor type of the field φ (i) , then, as explained in the remark following (i) Def. 3.1, the functional σN ,x should be viewed as taking values not in C, but in the finite (i)

dimensional vector space Fx (N ), and likewise for the functional σM,χ(x) . The analog of Eq. (27) for spinor fields is then (i)

x

(i)

σM,χ(x) ◦ αχ ∼ χ∗ σN ,x ,

(28)

where χ∗ : Fx (N ) → Fχ(x) (M) is the linear map induced by χ . The above locality and covariance conditions (27) and (28) imply that the n-point functions (18) of our standard functionals are distributions which are locally and covariantly constructed out of the metric and the orientations near the reference point, x. Namely if χ : N → M is an orientation and causality preserving isometric embedding, then it follows immediately from the transformation law of the fields (7) and the functionals (27) that χ ∗ cM,χ(x) (y1 , . . . , yn ) = cN ,x (y1 , . . . , yn ) (i)

(i)

(29)

in the sense of distributions for all yj in some neighborhood of the point x, where χ ∗ denotes the pull-back of a distribution, defined by analogy with the pull back of a smooth density. The reader may wonder why we are not demanding equality in Eq. (27) rather x than only equivalence under ∼, or alternatively, why we do not impose that relation (29) holds for all yj in N , rather than some neighborhood of the point x. The reason for this is that we typically expect the coefficients (18) to contain expressions like the geodesic distance, sM (y1 , y2 ), between two points in M near x. Now the geodesic distance

222

S. Hollands

between two points is not a quantity that is locally constructed out of the metric, since the geodesic distance between two points in a spacetime N (even if it can be defined unambiguously) can be made shorter by embedding N into a suitably chosen larger spacetime M. Therefore, it is not in general true that χ ∗ sM = sN for the geodesic distance. On the other hand, it is true that χ ∗ sM = sN when both sides are restricted to a suitably small neighborhood O of x. Our locality and covariance condition as stated above requires only that there is some region, O, such that both sides of (27) are equal upon restriction to O, but we have not imposed any requirements upon the size of O, which could vary arbitrarily so far as the embedding varies. For technical reasons, we must also impose the additional condition that O can be chosen uniformly in the following sense as the embedding χ varies. We ask that for every spacetime M and point x, there exists an open neighborhood X of the identity in the space Diff A x (M) of analytic diffeomorphisms on M leaving x fixed (i) (i) such that σM,x ◦ αχ = σχ ∗ M,x for all χ ∈ X , when restriced to some fixed O. We view this additional requirement as part of our definition of locality and covariance of the functionals in the operator product expansion. We next want to formulate condition (A) that the local, covariant functionals in the operator product expansion have an analytic dependence under analytic variations of the spacetime metric. For this, we consider 1-parameter families of real analytic metrics (s) gab on M which vary analytically with respect to a real parameter s ∈ I = (a, b) in the sense that (s)

gab − (ds)a (ds)b

(30)

is a real analytic metric on the real analytic 5-dimensional manifold I × M. Since the standard functionals in the operator product expansion have already been assumed to be locally and covariantly constructed out of the metric, we therefore obtain from the family (s) of metrics gab a corresponding family of functionals (labelled by the parameter s) associated with this family of metrics. Our analyticity requirement (A) is then, in essence, that all n-point functions of these functionals have a suitable analytic dependence on the parameter s. A complication arises from the fact that these n-point functions are not analytic functions but rather only distributions, so we must first consider the question what we actually mean by the statement that a family of distributions depends analytically on a parameter. Following [17], we make the following definition. Definition 4.1. We say that a family of distributions u(s) on an analytic manifold X depends analytically on the parameter s ∈ I = (a, b) with respect to a family of conic sets K (s) ⊂ T ∗ (X) \ {0} if (a) the dependence on s of the family of distributions u(s) on X is such that can be viewed as a distribution u˜ on X˜ = I × X and if (b) it holds that ˜ ∈ T ∗ (X) ˜ \ {0} | f (s) (x) = x, ˜ ⊂ {(x, ˜ k) ˜ WFA (u)

˜ ∈ K (s) }, (x, t f (s) (x)k)

(31)

˜ f (s) is the differential where f (s) : X → X˜ maps any point x ∈ X to x˜ = (s, x) ∈ X, t (s) ˜ of this map viewed as a linear map T (X) → T (X), and f denotes the transpose of ˜ → T ∗ (X). this linear map, acting between T ∗ (X) A detailed discussion and motivation of this definition is given in [17, App. A]; here ˜ is any distribution satisfying (31), we only note the following facts. Firstly, if u˜ ∈ D (X) then by the results of [20, Thm. 8.5.1], the pull-back of this distribution and all of its


223

s-derivatives by the map f (s) exists as a distribution on X for any s ∈ I and defines an analytic family u(s) of distributions in the sense of the above definition, with each member satisfying WFA (u(s) ) ⊂ K (s) . In the special case when the cones K (s) are empty for all s, we consequently have that WFA (u(s) ) = ∅, so each u(s) is an analytic function on X. The set (31) is then empty as well and the family is consequently jointly analytic in s and x. Thus, when K (s) is empty, our definition of the analytic dependence on a parameter coincides with the natural notion for analytic functions. With this definition in mind, we now state the precise form of condition (A). Let (s) (M, gab ) be a family of analytic spacetimes whose metrics vary analytically with s, and suppose that there is a corresponding analytic family of time functions T (s) and (s) volume forms abcd for all s, which thus define a family of space and time oriented (i) spacetimes M(s). We say that σM,x depend analytically on the metric if there is a neighborhood N of x such that the restriction of (i)

(j )

(j )

(f1 , . . . , fn ) → σM(s),x (φM1 (s) (f1 ) · · · φMn(s) (fn ))

(32)

to ×n D(N ) is a family of distributions that depends analytically on s with respect to the family of conic sets M(s) defined in Eq. (22) for every set of local, covariant fields φ (j1 ) , . . . , φ (jn ) . 5. PCT-Invariance of the Operator Product Expansion We are now going to formulate our main result about the PCT invariance of the operator product expansion in curved spacetime. Let M be a globally hyperbolic spacetime with metric gab and space-time orientation o = (T , abcd ), which admits a spin-structure. Let M be the spacetime whose manifold structure and metric coincides with that of our original spacetime, but whose space and time orientation is given by −o = (−T , abcd ), i.e., are reversed relative to those of the original spacetime. Since the definition of spinors involves a choice of orientation, the notion of spinors on M and M will not coincide. Therefore, in order to formulate a relation between the operator product expansions on M and M involving spinors, one needs to identify spinors on M with spinors on M. As we show in Appendix B, it is always possible to choose the spinor structures on M and M in such a way that a natural identification is possible, namely, we get a map I˜ : V(M) → V(M)

(33)

between the corresponding associated vector bundles of which the spinors are elements. In the following, we shall therefore always assume that the spin structures over M and M have been chosen so that such an identification is possible. The same remarks apply to the bundles V ∗ (M), V (M), V ∗ (M), as well as their tensor products. Theorem 5.1. Suppose that a local, covariant quantum field theory possesses an operator product expansion in the sense of Def. 3.1, and suppose that the standard functionals (i) σM,x in this operator product expansion satisfy (L), (M), and (A). Then the dependence of these standard functionals on the space and time orientations is expressed by the relation (i) (i) (j ) (j ) (i) (j ) (i) (jn ) φ (yn ) · · · φ 1 (y1 ) , i F (−1)M σM,x φM1 (y1 ) · · · φMn (yn ) = i F (−1)M σ M,x

M

M

(34)

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S. Hollands

for any finite number of local covariant fields, and any oriented and time oriented spacetime admitting a spin structure, and all yj in some open neighborhood of x. Here, F = F (j1 ) + · · · + F (jn ) , M = M (j1 ) + · · · + M (jn ) ,

(35)

with M (i) the nunber of unprimed spinor indices of the field φ (i) , 0 if φ (i) is bosonic, (i) F = 1 if φ (i) is fermionic,

(36)

and it is understood that the map I˜ is used to identify the spinor indices corresponding to the space-time orientation +o on the left side with the spinor indices corresponding to the space-time orientation −o on the right side of the above equation. Remarks. (1) The proof given below shows that relation (34) holds true for any family (i) of functionals σM,x with the properties (L), (M), (A), and Eq. (16). The fact that these functionals define an operator product expansion with property (17) does not play any role in our proof. We also re-emphasize that it is neither assumed nor used anywhere in the proof that the spin-statistics relation holds for the fields, i.e., it is not assumed that half odd-integer spin fields are fermionic and that integer spin fields are bosonic. (2) It follows from condition (16) on the functionals σ (i) , that the number F is even respectively odd if and only if φ (i) is bosonic respectively fermionic. (3) The theorem can be reformulated as an invariance condition of the theory under PCT as follows: If φM is a local covariant field on M with N primed and M unprimed spinor C on M, by indices, define a corresponding charge-conjugate field, φM C (f ) = i F (−1)M φM (f )∗ , φM

(37)

C is now a local, covariant field that has M primed and N unprimed spinor so that φM (i) indices. Consider the distributions cM,x defined in terms of the standard functionals (i)

σM,x by Eq. (18). These distributions are the coefficients appearing in the operator (j )

product expansion of the product of the fields φM , j = 1, . . . , n, on a spacetime M with a given space and time orientation, (i) (1) (n) (i) φM (y1 ) . . . φM (yn ) ∼ cM,x (y1 , . . . , yn ) φM (x), (38) (i)

where “∼” in the above relation is understood in the precise sense of Def. 3.1. It follows from Eq. (34) that the distributional coefficients in the operator product expansion of (j )C the charge conjugate fields φ (yj ) on the spacetime M with the opposite space and M time orientation relative to that of M are given by the complex conjugated coefficients (i)

cM,x for the spacetime M, i.e., φ

(1)C

M

(y1 ) . . . φ

(n)C

M

(yn ) ∼

(i)

(i)

cM,x (y1 , . . . , yn ) φ

(i)C (x). M

(39)


225

Relations (38) and (39) say that the operator product expansions on M and M are equivalent. If these relations were moreover honest equations rather than only asymp(i) totic relations as (y1 , . . . , yn ) → (x, . . . , x), then the cM,x could be viewed as structure constants of the algebra of fields A(M), and relations (38) and (39) could be viewed P CT : φ (f ) → φ C (f ) sending any smeared local covariant as saying that the map θM M M

field on M to its charge conjugate on M defines an (anti-linear) isomorphism between the field algebra A(M) and the field algebra A(M) of the spacetime associated with the opposite space and time orientation. In this sense, our theorem may be viewed as an analog of the PCT theorem in Minkowski spacetime as described in Sect. 2. (4) Throughout this paper, we are restricting attention to theories in spacetime dimension d = 4. Let us therefore briefly comment on what happens to our results and constructions in other dimensions. In even spacetime dimensions, our main Theorem 5.1 as well as the method of proof are basically unchanged. Some minor differences arise only from the fact that the notion of a spinor field is now based on the covering homomorphism ↑ Spin0 (d − 1, 1) → L+ of the d-dimensional proper orthochronous Lorentz group by the spin group. In odd spacetime dimensions d, the reversal of parity and time (x 0 , x 1 , . . . x d−1 ) → (−x 0 , −x 1 , . . . , −x d−1 ) in Minkowski spacetime corresponds to changing the orientations as (T , ab...c ) → (−T , −ab...c ), where we note the difference to the case of even d. However, it is well known that otherwise reasonable theories in Minkowski spacetime may fail to possess an invariance under PCT in this sense14 , and it is therefore not expected that there is an analog of our main result 5.1 in odd dimensions with regard to the change of orientations (T , ab...c ) → (−T , −ab...c ). On the other hand, there is an analog of the PCT theorem in Minkowski spacetime associated with the transformation (x 0 , x 1 , x 2 , . . . x d−1 ) → (−x 0 , −x 1 , x 2 , . . . , x d−1 ) corresponding to the change of orientations (T , ab...c ) → (−T , ab...c ). It can be shown that our main result 5.1 as well as the method of proof generalizes also to odd dimensions with regard to the change of orientations (T , ab...c ) → (−T , ab...c ). Some differences concerning the prefactors in our invariance property (34) arise only from the fact that the identification of the spin-bundles over M respectively M (see Appendix B for the case d = 4) have to be done in a different way than in even dimensions. Proof of Theorem 5.1, scalar bosonic case. For simplicity, we will first treat the special case when all fields in the theory are scalar and bosonic. Then Eq. (34) reduces to (1) (n) (n) (1) σM,x φM (y1 ) · · · φM (yn ) = σM,x φ (yn ) · · · φ (y1 ) , M M

(40)

for all yj in some neighborhood N of x, where we have set (jk ) = (k) without loss of generality, and where we have dropped the superscript (i) on σM,x to simplify the notation. Thus, when only scalar fields are present in the theory, our theorem will be proven if we can prove Eq. (40). We will prove Eq. (40) using a particular family of metrics which interpolates analytically between the metric gab and the Minkowski metric. The construction of this family is as follows. In a convex normal neighborhood around the point x, we introduce 14 The failure of the usual proof of the PCT theorem in odd dimensions can be traced back to the fact that the transformation (x 0 , x 1 , . . . x d−1 ) → (−x 0 , −x 1 , . . . , −x d−1 ) is not contained in the complexified proper, orthochronous Lorentz group in odd dimensions.

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S. Hollands

Riemannian normal coordinates for our metric gab , denoted y α = (y 0 , y 1 , y 2 , y 3 ), so that the point x has the coordinates y α = 0, and so that the coordinate components gµν of the metric satisfy gµν (0) = ηµν at this point. On this convex normal neighborhood of (s) x, we define a family of metrics gab , s ∈ I = (−1 − c, 1 + c), c > 0 via its coordinate components by sn ∂ n gµν (0) (s) α (y ) = ηµν + y α1 · · · y αn α . (41) gµν n! α ...α ∂y 1 · · · ∂y αn n≥1

1

n

(0)

It is obvious from this expression that (in our convex normal neighborhood) gab is the (1) (s) flat, Minkowskian metric, that gab is equal to the original metric, and that gab has an analytic dependence on the parameter s ∈ I . Since the statement of the theorem is com(s) pletely local, we may pass from M to a neighborhood of x on which all metrics gab are defined globally and which are globally hyperbolic with respect to these metrics for all s ∈ I . Furthermore, we will from now on view M as a neighborhood of the origin in R4 by identifying points y ∈ M with their Riemannian normal coordinates, viewed as points in R4 . If (T , abcd ) is the time function respectively volume form defining the time and (1) space orientation, +o, of the spacetime (M, gab ), then it is clear that ∇a T will remain (s) timelike with respect to the metrics gab in a neighborhood of the point x for all s ∈ I . By shrinking M further if necessary, we can therefore assume without loss of general(s) ity that T defines a corresponding time orientation on the spacetimes (M, gab ) for all s ∈ I . Similar remarks apply to the space orientation, as well as the reversed orientations −o = (−T , abcd ). We have therefore defined 1-parameter families of oriented and time oriented spacetimes (s)

M(s) = (M, gab , +o),

(s)

M(s) = (M, gab , −o).

(42)

By our assumption (A), we know that σM,x are local and covariant functionals that vary analytically under analytic variations of the metric. This means by definition that we can pick a neighborhood N of the point x such that the restriction to ×n N of the (j ) family of n-point functions σM(s),x ( φM(s) (yj )) is a family of distributions which varies analytically with s with respect to the cones M(s) , and we may thus differentiate this family with respect to the parameter s and set s = 0 afterwards. An analogous statement holds of course for the family of spacetimes M(s) with the opposite orientations. If Eq. (40) holds for all spacetimes, then this gives (1) (n) dk dk (n) (1) σ (y ) · · · φ (y ) = σM(0),x φ (yn ) · · · φ (y1 ) , φ 1 n M (0),x M(0) M (0) k k M (0) M (0) ds ds (43) (0)

for all (y1 , . . . , yn ) ∈ ×n N , where M(0) is Minkowski spacetime (M, gab ) with space and time orientation +o, and where M(0) is Minkowski spacetime with the space and time orientation −o. This equation is therefore a necessary condition for our theorem to be true. Our first major step in the proof is to show that it is also sufficient. It follows immediately from the microlocal spectrum condition together with the transformation rules of the analytic wave front set under diffeomorphisms that (n) (1) WFA σM,x φ (yn ) · · · φ (y1 ) ⊂ π ∗ M , (44) M

M


227

where π is the permutation π=

1 2 ... n , n n − 1 ... 1

(45)

and where the set M is defined as in (22), but with the orientations reversed relative to the original orientations, o = (T , abcd ). We claim that π ∗ M = M .

(46)

there exists a decorated graph G(p) Let (y1 , k1 ; . . . ; yn , kn ) ∈ M , which means that with (pe )a ∇a T > 0 such that ki = e:i=s(e) pe − e:i=t (e) pe for all i. (Remember that if e is an edge joining yj and yk with j < k, then j = s(e) and k = t (e).) We need to show ¯ that (yn , kn ; . . . ; y1 , k1 ) ∈ M . Consider the graph G(p) whose edges and vertices are identical to the edges and vertices of G(p), but which are decorated with the covectors −pe , each of which is future pointing with respect to the time function −T on M. Note that the notion of source s¯ (e) and target t¯(e) relative to the ordering (yn , . . . , y1 ) of the vertices is opposite to the above notion of source and target relative to the ordering ¯ (y1 , . . . , yn ), so that if e is an edge in G(p) joining yj and yk with j < k, then j = t¯(e) and k = s¯ (e) relative to the ordering (yn , . . . , y1 ). Itis a trivial consequence of these definitions that ki can be written alternatively as ki = e:i=¯s (e) (−pe )− e:i=t¯(e) (−pe ) for all i, which displays (yn , kn ; . . . ; y1 , k1 ) as the element in M associated with the ¯ graph G(p). Let u be the distribution on ×n M given by the difference between the left and right side of Eq. (40). Equations (44) and (46) show that u is the difference of two distributions whose analytic wave front set is contained in M . Hence, by the rules for calculating the analytic wave front set of sums of distributions, we have WFA (u) ⊂ M . By a similar argument, if u(s) is the family of distributions defined as the difference of the left minus right side of Eq. (40) with M and M replaced by M(s) and M(s), then u(s) is an analytic family of distributions on ×n N relative to the conic sets M(s) . Furthermore, since we assume that Eq. (43) holds, we have that d k (s) u (y1 , . . . , yn )|s=0 = 0 ds k

∀k, (y1 , . . . , yn ) ∈ ×n N .

(47)

We need to show that it follows that u(s) (y1 , . . . , yn ) = 0 for all s ∈ I and (y1 , . . . , yn ) ∈ ×n N . Since we have identified M with an open neighborhood of the origin in R4 via Riemannian normal coordinates (with the point x corresponding to the origin under this identification), we may take the neighborhood N to be the ball Br of radius r around the origin in R4 with respect to the Euclidean norm y2 =

3

|y µ |2 ,

(48)

µ=0

defined by our choice of coordinates y µ . It is known [10, Lem. 4.2] that each component ( M )(y1 ,... ,yn ) , of M in the cotangent space T(y∗ 1 ,... ,yn ) (×n M) \ {0}, is a proper15 , 15

A cone is said to be proper if it does not contain any straight line.

228

S. Hollands

closed convex cone, which we identify with a proper, closed convex cone in R4n \ {0}. Moreover, it can be seen that r > 0 can be chosen so small that ( M(s) )(y1 ,... ,yn ) ⊂ C, (49) (y1 ,... ,yn )∈×n Br ,s∈I

where C is a proper, closed, convex cone in R4n . Our claim that u(s) (y1 , . . . , yn ) = 0 now follows from the following lemma, which we shall prove in Appendix C. Lemma 5.1. Let u(s) ∈ D (X), X an open subset of Rm , be a family of distributions that depends analytically on s ∈ I = (a, b) with respect to conic sets K (s) ⊂ X × (C \ {0}), where C is a closed, proper convex cone in Rm . Suppose that d k (s) u |s=s0 = 0 ds k

(50)

for all k and some s0 ∈ I . Then u(s) = 0 for all s ∈ I . Thus, we have obtained the important intermediate result that our theorem will be proven if Eq. (43) can be shown. It follows immediately from the definition of our analytic family of metrics that the map χ : y → −y satisfies χ ∗ gab = gab . (s)

(−s)

(51)

Moreover, it is clear that χ reverses the space and time orientation in the sense that χ ∗ o = (χ ∗ T , χ ∗ abcd ) defines the same space and time orientation as −o = (−T , abcd ). This shows that χ is an orientation preserving isometry between the space and time orientated spacetimes M(s) and M(−s) given in Eq. (42). By the locality and covariance property (7) of the fields, we therefore have that χ ∗ φM(−s) (y) = φM(−s) (−y) = αχ (φM(s) (y)),

(52)

and by the locality and covariance of the standard functionals, we have that x

σM(−s),x ◦ αχ ∼ σM(s),x . Putting this together, we therefore know that (1) (1) (n) σM(s),x φM(s) (y1 ) · · · φM(s) (yn ) = σM(−s),x φ

M(−s)

(53)

(n) (−yn ) , M(−s)

(−y1 ) · · · φ

(54) for all (y1 , . . . , yn ) ∈ ×n Br , all s ∈ I , for some r > 0. If we now differentiate both sides of this equation k times with respect to s at s = 0, and substitute the result into Eq. (43), we get the important intermediate result that the theorem will be proven if we can show (1) dk (n) σ (y1 ) · · · φM(0) (yn ) φ ds k M(0),x M(0) (n) dk (1) = (−1)k k σM(0),x φM(0) (−yn ) · · · φM(0) (−y1 ) , ds

(55)


229

for all k, (y1 , . . . , yn ) ∈ ×n Br and some r > 0, where M(0) is Minkowski spacetime (0) (M, gab ) with the orientation +o. Thus, by the preceding steps, we have managed to transform the original problem of proving identity (40) between distributions associated (1) with spacetimes (M, gab ) with space and time orientations +o respectively −o, to the problem of proving relations (55) for a set of distributions on Minkowski spacetime (0) (M, gab ) with a single orientation, +o. The remainder of this proof therefore consists in showing that these relations are indeed true. For this, we need to analyze the s-derivatives of the distributions (32) for our particular family of spacetime metrics (41) and orientation +o. Such an analysis was carried out in a similar context in [17, Thm. 4.1] in order to derive a “scaling expansion” for certain distributions that arise in the context of perturbative interacting quantum field theories in curved spacetimes. The properties of the distributions considered in [17] which enter the analysis are (a) that they are locally and covariantly constructed from the metric near a reference point, x, (b) that they depend analytically on the metric in analytic spacetimes, (c) that they depend smoothly on the metric in smooth spacetimes, and (d) that they have a certain scaling behavior under rescalings of the metric by a constant conformal factor. Using only these properties, it was shown that the k th derivative with respect to s of the family of these distributions corresponding to the spacetime metrics defined in (41) can be written as a linear combination of curvature terms of the appropriate “dimension”, times Lorentz-invariant Minkowski space distributions (they also satisfy other properties, but these are not relevant in the present context). Inspection of the proof of this statement given in [17] shows that in analytic spacetimes, it only relies on (a) and (b) above, but not on (c) and (d). Furthermore, one easily sees that the arguments given in [17] will still be valid when properties (a) and (b) are replaced by the essentially identical properties (L) and (A) assumed for our distributions (18). (In fact, the precise form of our conditions (L) and (A) has been chosen precisely so that the arguments of [17] are still valid.) We therefore conclude that the expression on the left side of Eq. (55) can be decomposed into a sum of curvature terms of the appropriate dimension, times Lorentz invariant distributions in Minkowski spacetime. However, since assumptions (L) and (A) are weaker than the requirements (a) and (b) used in [17, Thm. 4.1] in that they hold only for an arbitrary small neighborhood of the reference point, x, one gets only the weaker result that the Minkowski space distributions are in fact only defined in some neighborhood of the origin (which we take to be a ball), and that they are invariant only under those Lorentz transformations that are sufficiently close to the identity. More precisely, there exists an r > 0 such that (1) dk (n) σM(0),x φM(0) (y1 ) · · · φM(0) (yn ) ds k ) = C (J )µ1 ...µj (x) Wµ(J1 ...µ (y1 , . . . , yn ) j

(56)

j µ1 ...µj

for all (y1 , . . . , yn ) ∈ ×n Br , where W (J ) are tensor-valued distributions on ×n Br and where (J ) = (12 . . . n) is a shorthand for the indices labelling the fields. The expressions C (J ) are the coordinate components in Riemannian normal coordinates of curvature tensors that are polynomials C (J ) m1 ...mj (gab (x), Rabcd (x), . . . , ∇(e1 · · · ∇ek−2 ) Rabcd (x))

(57)

of the metric, the Riemann tensor and its covariant derivatives at x. Each monomial in C (J ) contains precisely k derivatives of the metric, implying that

230

S. Hollands

j = k mod 2.

(58)

The W (J ) have the further property: (i) There exists an r, δ > 0 such that W (J ) (y1 , . . . , yn ) = D()W (J ) (y1 , . . . , yn )

(59)

↑

for all ∈ L+ with − 1 < δ and all (y1 , . . . , yn ) ∈ ×n Br−δ , where the norm of a linear transformation is defined using the Euclidean norm (48), and where D() is the tensor representation ν ...ν

ν

D()µ11 ...µj j = νµ11 · · · µjj .

(60)

Furthermore, since the restriction of σM(s),x to a sufficiently small neighborhood of x satisfies the microlocal spectrum condition for the cone M(s) , and since this family has an analytic dependence on s with respect to these cones, we have, by the same arguments as in the remark at the end of Sect. 4 of [17], that (ii) There exists an r > 0 such that the restriction of W (J ) to ×n Br has analytic wave front set WFA (W (J ) ) ⊂ M(0) .

(61) (0)

Here, M(0) is the cone (22) defined with respect to the Minkowskian metric gab and orientation +o. Using our Riemannian normal coordinates (y 0 , y 1 , y 2 , y 3 ) to identify M with a subset of the ball Br , and assuming that ∇a y 0 is future pointing with respect to +o, it can be written as

M(0) = (y1 , k1 ; . . . ; yn , kn ) ∈ T ∗ (×n Br ) \ {0} ∃pij ∈ V¯+ , n ≥ j > i ≥ 1: pij − pj i for all i , (62) ki = j :j >i

j :j

where V¯ + is the closure of the forward light cone V + in Minkowski space, V ± = {k ∈ R4 | ηµν k µ k ν > 0,

±k 0 > 0}.

(63)

Besides the above properties (i) and (ii) for the W (J ) in the expansion (56), we will now derive one more property, (iii), from the fact that the coefficients in our operator product expansion are not just arbitrary local covariant distributions with a specific analytic dependence on the metric, but arise in fact from a set of linear functionals on the algebras of observables A(M). To exploit this fact, we consider the multilinear maps on ×n D(M) defined by (j ) (1) (j +1) (n) (64) (f1 , . . . , fn ) → σM,x φM (f1 ) · · · φM (fj ), φM (fj +1 ) · · · φM (fn ) . Then by Eq. (4), the right side of the above equation will vanish if the supports of fj (1) and fj +1 are spacelike related16 in M with respect to the metric gab . Consider now the 16 We say that two closed sets K and K are spacelike related if there exist open neighborhoods, O 1 2 1 and O2 such that x1 and x2 are spacelike for every pair of points (x1 , x2 ) ∈ O1 × O2 .


231

(s)

analytic family of metrics gab constructed above in Eq. (41) and a set of testfunctions on M such that the supports of fj and fj +1 are spacelike related with respect to the flat (0) Minkowskian metric gab . Then the supports of fj and fj +1 will continue to be spacelike (s) related also with respect to the metrics gab for sufficiently small s. Consequently, (1) (j ) (j +1) (n) σM(s),x φM(s) (f1 ) · · · φM(s) (fj ), φM(s) (fj +1 ) · · · φM(s) (fn ) = 0 (65) will hold provided that s is sufficiently small and provided fj and fj +1 are spacelike (0) related with respect to the flat Minkowskian metric gab . Differentiating this relation k times at s = 0, we therefore find that (f1 , . . . , fn ) →

(k) (1) dk (k+1) σM(s),x φM(s) (f1 ) · · · φM(s) (fk ), φM(s) (fk+1 ) k ds (n) · · · φM(s) (fn ) |s=0

(66)

is a distribution on ×n D(M) that vanishes whenever the supports of fj and fj +1 are (0) spacelike separated with respect to the Minkowskian metric gab . Since the distributions W (J ) are related to the above distributions Eq. (66) via the scaling expansion (56), we have found the following property of the distributions W (J ) : (iii) There exists an r > 0 such that if yj < r and yi and yi+1 are spacelike to each other with respect to the Minkowski metric ηµν , then there holds W (J ) (y1 , . . . , yi , yi+1 , . . . , yn ) = W (πi,i+1 J ) (y1 , . . . , yi+1 , yi , . . . , yn ), (67) where (πi,i+1 J ) stands for (1 . . . (i + 1)i . . . n). We have argued so far that the theorem will be proven if we can show Eq. (55). If we now substitute the expansion (56) into Eq. (55), and use that j = k modulo 2, we see Eq. (55) will follow if we can show that there is a r > 0 such that W (J ) (y1 , . . . , yn ) = (−1)j W (πJ ) (−yn , . . . , −y1 )

(68)

for all yj ∈ Br in the sense of distributions, where π is the permutation (45). Since we already know that the distributions W (J ) satisfy properties (i), (ii) and (iii) above, the proof of the theorem will therefore be complete once we have established the following proposition: (J )

Proposition 5.1. Suppose that Wµ1 ...µj are tensor-valued distributions on ×n Br for which there holds (i), (ii) and (iii). Then there is some r > 0 such that W (J ) satisfies Eq. (68) within ×n Br . The remainder of this section is devoted to the proof of this proposition. A key ingredient in our proof is the following theorem about distributions whose analytic wave front set is contained in the dual of an open, convex cone [20, Thm. 8.4.15]: Theorem 5.2. Let u be a distribution on X ⊂ Rm with WFA (u) ⊂ X × K D , where17 K D = {k ∈ Rm | k · x ≤ 0 ∀x ∈ K}

(69)

17 Our definition of the dual cone differs trivially from that employed in [20, Thm. 8.4.15] since our convention for the Fourier transform is opposite to the convention employed in [20].

232

S. Hollands

is the dual of an open convex cone K ⊂ Rm , with k · x the standard inner product in Rm . If X0 ⊂ X is an open subset with compact closure X¯ 0 ⊂ X, then one can find a γ > 0 and a function U analytic in {x + iy ∈ Cm | x ∈ X0 , y ∈ K, y < γ } such that u is the boundary value of U , u(f ) = lim U (x + iy)f (x) d m x (70) y∈K,y→0

which we write as u(x) = B. V. U (x + iy).

(71)

y∈K,y→0

Consider the linear transformation f : (ξ1 , . . . , ξn ) → (y1 , . . . , yn ) on R4n defined by f : yi = ξi + ξi+1 + · · · + ξn

for all i,

so that ξn = yn and ξi = yi − yi+1 for all i = n. For (ξ1 , . . . , ξn ) ∈ sufficiently small, we define a distribution w(J ) by w (J ) (ξ1 , . . . , ξn ) = W (J ) (f (ξ1 , . . . , ξn )),

(72) ×n B

r,

r > 0 (73)

which expresses W (J ) in terms of relative coordinates about the “center of mass” point ξn = yn . By the rules for calculating the analytic wave front set of the pull back of a distribution under an analytic map, we have WFA (w (J ) ) = f ∗ WFA (W (J ) )

= (ξ1 , 1 ; . . . ; ξn , n ) ∈ T ∗ (×n Br ) \ {0} ∃(y1 , k1 ; . . . ; yn , kn ) ∈ WFA (W (J ) ) : i = k1 + · · · + ki , yi = ξi + ξi+1 + · · · + ξn . (74) WFA (W (J ) ) if and only if there exists a By (ii), we know that (y1 , k1 ; . . . ; yn , kn ) is in set of covectors pij ∈ V¯ + , i < j such that ki = j :j i pij for all i. Thus, if (ξ1 , 1 ; . . . ; ξn , n ) is in WFA (w (J ) ), then we must have   i i  i = (75) kj = pj l − plj  j =1

j =1

=

l:l>j

l:l<j

(p1j + p2j + · · · + pij ),

(76)

j :j >i

where the equality in the second line can be proved by induction in i. Thus, i ∈ V¯+ for all i, and in particular n = 0. We have thus shown that WFA (w (J ) ) ⊂ (×n Br ) × (V¯ + × · · · × V¯ + × {0}).

(77)

We now use this information about the analytic wave front set of w (J ) to show that w (J ) is the boundary value of some analytic function. The closed forward lightcone is the dual of the open past lightcone, V¯ + = (V − )D , therefore V¯ + × · · · × V¯ + × {0} = (V − × · · · × V − × R4 )D ⊂ R4n ,

(78)


233

so Eq. (77) tells us in conjunction with the above theorem that w (J ) is the boundary value w(J ) (ξ1 , . . . , ξn ) =

B. V.

ηj →0,ηj ∈V − ∀j =n

w (J ) (ξ1 + iη1 , . . . , ξn + iηn )

(79)

of a function w(J ) (ζ1 , . . . , ζn ) that is holomorphic in the domain Tn = {(ζ1 , . . . , ζn ) ∈ C4n | ζj < r for all j , Im ζj ∈ V − for all j = n}

(80)

for some r > 0. Thus, we have shown that property (ii) of the distributions W (J ) implies that w(J ) is the boundary value of a function that is holomorphic on Tn . We next want to show that the analytic functions w(J ) (ζ1 , . . . , ζn ) are Lorentz invari↑ ant. For this let ∈ L+ with − 1 < δ, and consider the function (ζ1 , . . . , ζn ) → w(J ) (ζ1 , . . . , ζn ) − D()w (J ) (ζ1 , . . . , ζn )

(81)

for (ζ1 , . . . , ζn ) ∈ Tn such that ζj < r − δ for all j . The boundary value of this function as Im ζj → 0 vanishes by (i). Therefore, by the “edge-of-the-wedge theorem” (see e.g. [12, Thm. 2.17]), this function itself has to vanish, w (J ) (ζ1 , . . . , ζn ) = D()w (J ) (ζ1 , . . . , ζn )

(82)

↑

for all ∈ L+ such that − 1 < δ and such that (ζ1 , . . . , ζn ) and (ζ1 , . . . , ζn ) are in Tn . We finally would like to use property (iii) to infer a corresponding property for w (J ) . Let (ξ1 , . . . , ξn ) ∈ ×n Br , (y1 , . . . , yn ) = f (ξ1 , . . . , ξn ) such that all the difference vectors yi − yj are spacelike related with respect to ηµν . Then since w (J ) = f ∗ W (J ) , and since W (J ) satisfies (iii), we conclude that there exists an r > 0 such that n (J ) (πJ ) w (ξ1 , . . . , ξn−1 , ξn ) = w −ξn−1 , . . . , −ξ1 , (83) ξi , i=1

in the sense of distributions. We have thus altogether shown that properties (i), (ii) and (iii) for W (J ) imply that (J w ) = f ∗ W (J ) , with f given by (72), is the boundary value of an analyic function w(J ) (ζ1 , . . . , ζn ) on the domain Tn , satisfying Eqs. (82) and (83). When expressed in terms of w (J ) , the assertion (68) of the proposition reads n (J ) j (πJ ) w (ξ1 , . . . , ξn−1 , ξn ) = (−1) w ξn−1 , . . . , ξ1 , − (84) ξi , i=1

in the sense of distributions on ×n Br for some r > 0. We have therefore reached the important intermediate conclusion that the proposition will be shown if we can show that Eq. (84) holds for any w(J ) which is the boundary value of an analytic function on the domain Tn (see Eq. (79)) satisfying Eqs. (82) and (83). One notes that Eqs. (79), (82) and (83) satisfied by the functions w (J ) resemble properties of the Wightman functions in Minkowski spacetime [12] (when expressed in relative coordinates). Moreover, relation Eq. (84) which we want to prove resembles a property of the Wightman functions expressing the PCT invariance of the corresponding

234

S. Hollands

Wightman field theory. Our proof of Eq. (84) will follow closely the proof of the PCT theorem for Wightman field theories in Minkowski spacetime, see especially [12]. There are, however, two important differences between our functions w (J ) and the Wightman functions (expressed in relative coordinates) in Minkowski spacetime. Firstly, our functions W (J ) are by contrast with the Wightman functions not translation invariant, so the relations (83) reflecting the local commutativity are not identical to the corresponding relations for the Wightman functions. Secondly, our distributions w (J ) as well as their analytic extensions are defined only locally and the Lorentz invariance Eq. (82) holds a priori only for Lorentz transformations in a neighborhood of the identity transformation. On the other hand, one finds that global translation invariance and invariance under the full Lorentz group play an important role once one looks at the details of the proof (see e.g. [12]) of the PCT theorem in Minkowski space. For these reasons, the arguments given e.g. in [12] cannot be taken over wholesale, but must be carefully adapted. First, we notice that the transformation law Eq. (82) of the w(J ) not only holds for proper Lorentz transformations such that − 1 < δ and such that (ζ1 , . . . , ζn ) and (ζ1 , . . . , ζn ) are in Tn , but moreover for any rotation of the form 

 1 0 0 0 0 cos ϕ sin ϕ 0 R(ϕ) =  . 0 − sin ϕ cos ϕ 0 0 0 0 1

(85)

This can easily be proven by noting that such a rotation leaves the region Tn invariant and that it can be written as a product of N rotations R(ϕ/N ), each of which satisfy R(ϕ/N )−1 < δ. The invariance then follows by applying the transformation rule (82) to each such small rotation in turn and using the group character of the transformation rule. We will now show that the transformation law can be further generalized to more general transformations by invoking the analyticity of the functions w (J ) . Consider the abelian group of complex Lorentz transformations 

sinh(α + iβ) 0 0 1  (α + iβ) =  0 0 cosh(α + iβ) 0

 0 cosh(α + iβ) 0 0  , 1 0 0 sinh(α + iβ)

α, β ∈ R,

(86)

which corresponds to a real, proper orthochronous Lorentz transformation if β = 0. The action of such a transformation on a complex vector, ζ → ζˆ = (α + iβ)ζ , can be written as ζˆ + = eα+iβ ζ + ,

ζˆ − = e−α−iβ ζ − ,

ζˆ 1 = ζ 1 ,

ζˆ 2 = ζ 2 ,

(87)

√ where we have introduced the notation ζ ± = (ζ 0 ± ζ 3 )/ 2 for every complex four vector. We now have (compare [12, Thm. 2.11]): Lemma 5.2. The functions w(J ) (ζ1 , . . . , ζn ) possess a unique, single-valued analytic continuation to the extended tube domain Tn = {((α + iβ)ζ1 , . . . , (α + iβ)ζn ) | (ζ1 , . . . , ζn ) ∈ Tn , α, β ∈ R} ⊂ C4n , (88)


235

which transforms as w (J ) ((α + iβ)ζ1 , . . . , (α + iβ)ζn ) = D((α + iβ))w (J ) (ζ1 , . . . , ζn )

(89)

for all (ζ1 , . . . , ζn ) ∈ Tn and all α, β. Proof. We already know by Eq. (82) that Eq. (89) holds if β = 0, if (ζ1 , . . . , ζn ) as well as ((α)ζ1 , . . . , (α)ζn ) are in Tn and if α is in a sufficiently small real neighborhood of 0. Since (α) is real analytic in α, both sides of Eq. (82) define analytic functions of α in this real neighborhood of 0. Therefore Eq. (89) holds for all α + iβ in a sufficiently small complex neighborhood of 0 such that ((α + iβ)ζ1 , . . . , (α + iβ)ζn ) ∈ Tn . If (ζ1 , . . . , ζn ) is in Tn but ((α + iβ)ζ1 , . . . , (α + iβ)ζn ) is not in Tn , then the right side of Eq. (89) is initially not defined and we try to define it by the left side in this case. It may happen that a point (ξ1 , . . . , ξn ) ∈ Tn can be reached in different ways from elements in Tn , i.e., that it can be written as ((α1 + iβ1 )ζ1 , . . . , (α1 + iβ2 )ζn ) or ((α2 + iβ2 )ρ1 , . . . , (α2 + iβ2 )ρn ), where (ζ1 , . . . , ζn ) and (ρ1 , . . . , ρn ) are in Tn . Unless these different ways of writing (ξ1 , . . . , ξn ) ∈ Tn give rise to the same definition of w(J ) (ξ1 , . . . , ξn ), our proposed extension of w (J ) will not be single valued. Thus, the nontrivial task is to show that Eq. (89) holds when (ζ1 , . . . , ζn ) and ((α + iβ)ζj , . . . , (α + iβ)ζn ) are in Tn , where α = α1 − α2 and β = β1 − β2 . We already know that Eq. (89) holds when α + iβ is sufficiently close to 0. Therefore, by the well known method of analytic continuation by overlapping neighborhoods combined with the group character of the transformation law Eq. (89), this equation will follow if we can show that there exists a continuous curve t → γ (t) = α(t) + iβ(t) such that γ (0) = 0, γ (1) = α + iβ and such that ((γ (t))ζ1 , . . . , (γ (t))ζn ) ∈ Tn for all 0 ≤ t ≤ 1. Thus, our construction of the analytic extension of w(J ) to the extended domain Tn will be complete if we can construct such a curve γ . Without loss of generality we can assume that 0 ≤ β ≤ π . Our proposal for the curve γ is then 2tα for 0 ≤ t ≤ 21 , γ (t) = (90) α + i(2t − 1)β for 21 ≤ t ≤ 1. We need to show that ζj (t) = (γ (t))ζj satisfies ζj (t) < r,

Im ζj (t) ∈ V −

for all j = 1, . . . , n and t ∈ [0, 1].

(91)

In order to prove the first relation, we note that for any complex four√vector ζ , we have that ζ 2 = |ζ + |2 + |ζ − |2 + |ζ 1 |2 + |ζ 2 |2 , where ζ ± = (ζ 0 ± ζ 3 )/ 2. For t ∈ [0, 21 ], this gives ζj (t)2 = | e2tα ζj+ |2 + | e−2tα ζj− |2 + |ζj1 |2 + |ζj2 |2 ≤ 21 ζj (0)2 + 21 ζj ( 21 )2 ,

(92)

by the convexity of the exponential function. For t ∈ [ 21 , 1], this gives ζj (t)2 = | eα+i(2t−1)β ζj+ |2 + | e−α−i(2t−1)β ζj− |2 + |ζj1 |2 + |ζj2 |2 = ζj (1)2 < r 2 .

(93)

It follows straightfowardly from these relations that ζj (t) < r for t ∈ [0, 1] and all j . In order to prove the second relation, we note that, since (2tα) are real restricted

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Lorentz transformations, we have that Im ζj (t) = (2tα) Im ζj ∈ V − for t ∈ [0, 21 ]. To show that Im ζj (t) ∈ V − also for all t ∈ [ 21 , 1], it is sufficient show that Im ζj (t)µ nµ > 0 for any real, future pointing timelike or null vector n, for all j . We have µ

µ

(94)

sin β Im ζj (t)µ nµ = sin βτ Im ζj (1)µ nµ + sin β(1 − τ ) Im ζj ( 21 )µ nµ ,

τ = 2t − 1. (95)

Im ζj (t)µ nµ = sin β(2t − 1) Re ζj nµ + cos β(2t − 1) Im ζj nµ , which implies that

The case β = 0 is trivial, and the case β = π cannot occur, since otherwise we would have Im ζj (1)µ nµ < 0 for some future pointing timelike or null vector n, which cannot be since Im ζj (1) ∈ V − by assumption. We can therefore assume that 0 < β < π. The above equation then displays Im ζj (t)µ nµ as a positive linear combination of two positive numbers for 0 ≤ τ ≤ 1. This proves that Im ζj (t) ∈ V − for 21 ≤ t ≤ 1 and hence altogether that (ζ1 (t), . . . , ζn (t)) ∈ Tn for all 0 ≤ t ≤ 1. Consider now a point (ζ1 , . . . , ζn ) ∈ Tn . Then we know that w (J ) (ζ1 , . . . , ζn ) transforms according to the transformation law Eq. (82) for any rotation R(ϕ) as in Eq. (85), and any complex Lorentz transformations (α + iβ). Applying this transformation rule in particular to the product (iπ)R(π) = −1, we find that w(J ) (−ζ1 , . . . , −ζn ) = (−1)j w (J ) (ζ1 , . . . , ζn )

(96)

for all (ζ1 , . . . , ζn ) ∈ Tn , since D(−1) = (−1)j . Moreover, since both sides of this equation are analytic functions, we find that this equation holds in fact for all (ζ1 , . . . , ζn ) in the extended domain Tn . We will now use this equation to get a relation between the w (J ) for real arguments. We note however that we cannot straightforwardly take the boundary value of both sides of the above equation as Im ζj → 0 in order to get such a relation since Im ζj has the opposite sign on both sides of the above equation and Im ζj = 0 can therefore not be approached from within V − on both sides. To circumvent this problem, one considers special real points in Tn defined as follows (compare [12, Thm. 2.12]): Let n be the spacelike vector in R4 given by (0, 0, 0, 1), and consider the open, proper, convex and spacelike cone K in R4 defined by the equation ξ µ nµ > ξ . Lemma 5.3. The extended domain Tn includes the set Jn = {(ζ1 , . . . , ζn ) ∈ R4n | ζj < r, ζj ∈ K},

(97)

and Jn is an open, real domain in R4n . Proof. The last statement is obvious. We must show that if (ξ1 , . . . , ξn ) ∈ Jn , then there is α, β such that ((α + iβ)ξ1 , . . . , (α + iβ)ξ1 ) is in Tn . One calculates that  3 ξj 0  Im (iβ)ξj = − sin β  (98)  0  , (iβ)ξj = ξj . 0 ξj


237

µ

By definition, ξj ∈ K means that ξj nµ > ξj and ξj < r, where n = (0, 0, 0, 1). The first condition implies that ξj3 > (|ξj0 |2 + · · · + |ξj3 |2 )1/2 ≥ |ξj0 |, showing that Im (iβ)ξj ∈ V − for all j and any 0 < β < π, and the second condition shows that (iβ)ξj < r for all j . This proves the lemma. One now notes that if (ζ1 , . . . , ζn ) ∈ Jn , and (y1 , . . . , yn ) = f (ζ1 , . . . , ζn ), then it follows that the difference vectors yj − yk are all spacelike (and non-zero) since yj − yk = ζi ∈ K, (99) j ≤i 0,

abcd (e0 )a (e1 )b (e2 )c (e3 )d > 0.

(117)

)a is another such frame, then there is a unique proper orthochronous LoClearly, if (eµ )a = ν (e )a . The frame bundle, F (M), is defined rentz transformation such that (eµ µ ν as the union of the spaces Fx (M) as x runs over all points in M. It has the stucture of a principal fibre bundle over M, whose structure group is the proper orthochronous ↑ Lorentz group L+ which acts upon elements in each fibre Fx (M) by transforming the orthonormal frames. Spinors over M can be defined if and only if there exists a principal fibre bundle S(M), called “spin-bundle”, with structure group SL2 (C) and base manifold M that covers the frame bundle in the sense that there is an onto map f : S(M) → F (M) such that the group action of SL2 (C) on the spin bundle corresponds to the group action of the Lorentz group on the frame bundle via the covering ↑ homomorphism SL2 (C) → L+ . A spin bundle need not exist in a general curved spacetime, and if it exists, it need not be unique. The situation is however rather simple in the case when the spacetime M is simply connected, π1 (M) = 0. In that case, a (necessarily unique) spin bundle will exist if and only if π1 (F (M)) = Z2 , and the spin bundle is in fact simply given by the the universal covering space of the frame bundle F (M). (Remember that the universal covering space X˜ of a topological space X is the space of equivalence classes of continuous paths γ : [0, 1] → X with γ (0) = x0 , where two paths are equivalent if they can be composed to a closed path in X that is homotopic to the trivial path given by γ (λ) = x0 for all λ ∈ [0, 1].) Let χ : N → M be an isometric, orientation and time orientation preserving embedding. If (eµ )a is an oriented and timeoriented orthonormal frame on N , then clearly χ∗ (eµ )a will be such a frame over M and this defines an embedding χ∗ : F (N ) → F (M). This embedding lifts to a corresponding map between the covering spaces by defining its action on a path γ : [0, 1] → F (N ) to be the path γ : [0, 1] → F (M) given by γ (λ) = χ∗ γ (λ) for all λ ∈ [0, 1], because it is easily seen that the equivalence class of γ only depends on the equivalence class of γ . Thus, if both N and M are simply connected and π1 (F (M)) = Z2 , then also π1 (F (N )) = Z2 , and we get a natural embedding map

χ∗ : S(N ) → S(M)

(118)

which is compatible with the action of the group SL2 (C) on these spaces. (This can be ↑ seen by noting that, since SL2 (C) is the universal cover of L+ , every L ∈ SL2 (C) can ↑ be identified with an equivalence of continuous paths γL : [0, 1] → L+ starting at 1 ↑ and ending at , the element in L+ covered by L.) If the spacetime M is not simply connected, i.e. π1 (M) = G = 0, then one can show that a spin-bundle S(M) covering the frame bundle will exist if and only if the fundamental group of the frame bundle is isomorphic to a direct product ψ : π1 (F (M)) ∼ = Z2 × G

(119)


241

in the sense that every element of the form (g1 , e2 ) (with e2 the identity element in G) corresponds via ψ to a path in F (M) that is homotopic to a path lying within a single fiber Fx (M), and that each path in F (M) corresponding to an element of the form (e1 , g2 ) (with e1 the identity in Z2 ) projects down to a path in M that is homotopic to a path representing g2 ∈ π1 (M). In this case, one can define a spin bundle S(M) as the space of equivalence classes of continuous paths in F (M), where two such paths are now regarded as equivalent if their composition can be continuously deformed to a path that corresponds to the group element of the form (e1 , g2 ) under the isomorphism ψ. Since ψ is not necessarily unique18 , there may now exist several inequivalent constructions of S(M), each corresponding to a particular choice for the isomorphism ψ. An isometric embedding χ : N → M between spacetimes admitting a spin structure, can therefore be lifted to a map χ∗ as in Eq. (118) for one and only one choice of spin-structure over N. Spinors in the spacetime M are constructed as elements in the vector bundles that are associated with the principal fibre bundle S(M). These are defined as follows. On the cartesian product S(M) × C2 we define an equivalence relation ∼ by declaring two elements (s, v) and (s , v ) to be equivalent if there is an element L ∈ SL2 (C) such that s = L−1 s and v = D(L)v, where L−1 s denotes the action of a group element L on an element s in the principal fibre bundle, and where D is the fundamental representation of SL2 (C) on C2 . The space of equivalence classes V(M) = (S(M) × C2 )/ ∼

(120)

is then seen to be a vector bundle over M with each fibre isomorphic to C2 . Classical spinors fields over M (with an upper unprimed spinor index) are by definition sections in this vector bundle. This construction can be varied by replacing the space C2 by ∗ ¯ 2∗ of antilinear the dual space C2 of complex linear functionals on C2 , or the space C 2 2 2∗ ¯ dual to C ¯ , and by replacing the representation D functionals on C , or the space C by the appropriate representations of SL2 (C) on these spaces. We shall denote the corresponding vector bundles over M by V ∗ (M), V ∗ (M) and V (M), respectively. They correspond to spinors with a lower unprimed, lower primed and upper primed index. Suppose that we are given an isometric, orientation and causality preserving embedding, χ : N → M between two oriented spacetimes N and M and suppose that each spacetime has a spin structure such that χ lifts to a map χ∗ as in Eq. (118) between the corresponding spin bundles. In this situation, we automatically get a map χ∗ : V(N ) → V(M),

[(s, v)]∼ → [(χ∗ (s), v)]∼

(121)

between the corresponding associated spin bundles (and likewise the bundles V (M), V ∗ (M) and V (M) as well as their tensor products). We finally explain the dependence of the above construction of spinors on the choice of space and time orientation of the spacetime. Let F (M) be the bundle of frames that are orthogonal with respect to the metric gab and that are oriented and time oriented with respect to a time and space orientation o = (T , abcd ) on M, and let F (M) be the bundle of orthonormal frames that are oriented with respect to the opposite time 18 It is clear that the different possible choices for ψ are in 1-to-1 correspondence with the non-unity automorphisms of the group Z2 × G. These in turn are easily seen to be in 1-to-1 correspondence with the normal subgroups H of G such that G/H = Z2 .

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and space orientation, −o = (−T , abcd ). Then these bundles are naturally isomorphic under the map I : F (M) → F (M),

(eµ )a → (−eµ )a ,

(122)

since a tetrad (eµ )a is positively oriented with respect to o if and only if the tetrad (−eµ )a is positively oriented with respect to −o. As explained above, a construction of a spinor bundle S(M) covering the frame bundle F (M) is equivalent to a choice of isomorphism ψ : π1 (F (M)) → Z2 × G, and a construction of a spinor bundle S(M) in the spacetime M with the opposite orientations is likewise equivalent to a choice of isomorphism ψ : π1 (F (M)) → Z2 × G. It is possible to see that the map I will lift to a corresponding bundle isomorphism I˜ between the spinor bundles S(M) and S(M) if and only if ψ and ψ are compatible in the sense that ψ ◦ I ◦ ψ −1 is the identity homomorphism in π1 (M) × Z2 . By changing ψ or ψ if necessary, we can therefore always assume that there is indeed a natural map I˜ identifying the spin bundles S(M) and S(M). From the above constructions it is then clear that this map will induce a corresponding map I˜ : V(M) → V(M)

(123)

between the corresponding associated vector bundles of which the spinors are elements, and similar statements hold for the bundles V (M), V ∗ (M), V ∗ (M), as well as their tensor products. This provides us with a natural identification of spinors over the spacetimes M and M. C. Proof of Lemma 5.1 For the convenience of the reader, we repeat the statement of the lemma: Lemma C.1. Let u(s) ∈ D (X), X an open subset of Rm , be a family of distributions that depends analytically on s ∈ I = (a, b) with respect to conic sets K (s) ⊂ X × (C \ {0}), where C is a closed, proper convex cone in Rm . Suppose that d k (s) u |s=s0 = 0 ds k

(124)

for all k and some s0 ∈ I . Then u(s) = 0 for all s ∈ I . Proof. Let us set X˜ = I × X, x˜ = (s, x), and view the family of distributions u(s) as ˜ Since the set C is proper, closed and convex, we may defining a distribution u˜ ∈ D (X). without loss of generality assume that C = {k ∈ Rm | k · n ≥ δnk}, where “dot” is the standard inner product for vectors in Rm , where · is the corresponding norm, where 0 < δ < 1 and where n is some nonzero vector in Rm . Let n˜ = (0, n) ∈ Rm+1 and consider the quantity c=

inf

˜ (x, ˜ k)∈WF ˜ X˜ 0 A (u)

k˜ · n˜ , ˜ kn ˜

(125)

˜ We claim that c > 0. Since the analytic where X˜ 0 is any closed compact subset of X. ˜ X˜ 0 . If wave front is closed, the infimum is achieved for some (x˜0 , k˜0 ) ∈ WFA (u) c ≤ 0, we have consequently 0 ≥ k˜0 · n˜ = k0 · n,


k0 = t f

(s)

(x0 )k˜0 ,

243

x˜0 = f (s) (x0 ),

(126)

where f (s) : X → X˜ is the embedding map. It is therefore not possible that k0 ∈ C, unless k0 = 0. This is however in contradiction with the assumption of the lemma, since the analyticity of u(s) with respect to s implies that when (x˜0 , k˜0 ) ∈ WFA (u), ˜ then necessarily k0 ∈ C \ {0}. We must therefore have that c > 0, and consequently that ˜ WFA (u) ˜ X˜ 0 ⊂ X˜ 0 × C,

˜ C˜ = {k˜ ∈ Rm+1 | k˜ · n˜ ≥ cn ˜ k},

(127)

and we may assume without loss of generality that c < 1. The cone C˜ is the dual of the ˜ x. ˜ By Thm. 5.2, we open cone consisting of all x˜ ∈ Rm+1 , such that x˜ · n˜ > (1 − c)n can therefore conclude that there is a function U˜ that is analytic in the complex domain consisting of all x˜ + i y˜ ∈ Cm+1 for which y˜ · n˜ > (1 − c)n ˜ y, ˜ and x˜ ∈ X˜ 0 so that u˜ is the boundary value of U˜ as y˜ → 0, u(s, ˜ x) =

B. V. √

(t,y)→0,n·y>(1−c)n

y2 +t 2

U˜ (s + it, x + iy),

(128)

where we are now writing x˜ = (s, x), y˜ = (t, y), and where we have used the definition n˜ = (0, n). We may set t = 0 on the right side of this equation when y · n > (1 − c)ny, and take k derivatives with respect to s of both sides of the equation. Setting s = s0 and using the assumption of the lemma, this gives 0=

dk ˜ U (s0 , x + iy) y→0,n·y>(1−c)ny ds k B. V.

∀k.

(129)

We already know that the function x + iy → d k /ds k U˜ (s0 , x + iy) is analytic when n · y > (1 − c)ny, and we have now found that its distributional boundary values as y → 0 vanish. We therefore conclude, by the “edge-of-the-wedge theorem” (see e.g. [12, Thm. 2.17]), that this function itself has to vanish. Therefore U˜ (s, x + iy) =

∞ (s − s0 )k d k ˜ U (s0 , x + iy) = 0 k! ds k

(130)

k=0

for sufficiently small |s − s0 |, and hence for all s in I . Thus, by Eq. (128), u(s, ˜ x) = u(s) (x) = 0 in the sense of distributions for all s ∈ I . References 1. Wilson, K.: Non-Lagrangian models of current algebras. Phys. Rev. 179, 1499 (1969); Anomalous dimensions and the breakdown of scale invariance in perturbation theory. Phys. Rev. D2(8), 1478–1493 (1970) 2. Zimmermann, W.: Normal products and the short distance expansion in the perturbation theory of renormalizable interactions. Ann. Phys. 77, 570 (1973) [Lect. Notes Phys. 558, 278 (2000)] 3. Wilson, K., Zimmermann, W.: Operator product expansions and composite field operators in the general framework of quantum field theory. Commun. Math. Phys. 24, 87–106 (1972) 4. Fredenhagen, K., Jörss, M.: Conformal Haag-Kastler nets, pointlike localized fields and the existence of operator product expansions. Commun. Math. Phys. 176, 541–554 (1996) 5. Schlieder, S., Seiler, E.: Remarks concerning the connection between properties of the 4-point function and the Wilson-Zimmermann expansion. Commun. Math. Phys 31, 137–159 (1973) 6. Luscher, M.: Operator product expansions on the vacuum on conformal quantum field theory in two space-time dimensions. Commun. Math. Phys. 50, 23 (1976)

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7. Bostelmann, H.: Lokale Algebren und Operatorprodukte am Punkt. Doktorarbeit, Universität Göttingen, 2000, available at http://www.lqp.uni-goettingen.de/papers/00/12/00121700.html 8. Fredenhagen, K., Hertel, J.: Local algebras of observables and pointlike localized fields. Commun. Math. Phys. 80, 555–561 (1986) 9. Buchholz, D., Wichmann, E.H.: Causal independence and the energy level density of states in local quantum field theory. Commun. Math. Phys. 106, 321 (1986) 10. Brunetti, R., Fredenhagen, K., Köhler, M.: The microlocal spectrum condition and Wick polynomials on curved spacetimes. Commun. Math. Phys. 180, 633–652 (1996) [arXiv:math-ph/9903028] 11. Hollands, S., Wald, R.M.: Work in progress 12. Streater, R.F., Wightman, A.S.: PCT, Spin and Statistics and All That. New York: Benjamin, 1964 13. Pauli, W.: Exclusion Principle, Lorentz Group and Reflection of Space-Time and Charge. In: Niels Bohr and the Devolopment of Physics, W. Pauli (ed.), New York: Pergamon Press, 1955, p. 30; R. Jost, Eine Bemerkung zum CPT-Theorem. Helv. Phys. Acta 30, 409 (1957) 14. Borchers, H.J., Yngvason, J.: On the PCT-theorem in the theory of local observables. [arXiv:mathph/0012020] 15. Brunetti, R., Fredenhagen K., Verch, R.: The generally covariant locality principle: A new paradigm for local quantum physics. Commun. Math. Phys. 237, 31–68 (2003) 16. Hollands, S., Wald, R.M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289 (2001) [arXiv:gr-qc/0103074] 17. Hollands, S., Wald, R.M.: Existence of local covariant time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309 (2002) [arXiv:gr-qc/0111108] 18. Haag, R.: Local Quantum Physics. Berlin: Springer-Verlag, 1992 19. Verch, R.: A spin-statistics theorem for quantum fields on curved spacetime manifolds in a generally covariant framework. Commun. Math. Phys. 223, 261 (2001) [arXiv:math-ph/0102035] 20. Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Berlin: Springer-Verlag, 1983 21. Kay, B.S., Wald, R.M.: Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate killing horizon. Phys. Rep. 207, 49–136 (1995) 22. Radzikowski, M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179, 529 (1996) 23. Strohmaier, A., Verch, R., Wollenberg, M.: Microlocal analysis of quantum fields on curved spacetimes: Analytic wavefront sets and Reeh-Schlieder theorems. J. Math. Phys. 43, 5514–5530 (2002) 24. Soloview, M.A.: PCT, spin-statistics, and analytic wave front set. Theor. Math. Phys. 121, 1377 (1999) 25. Steinmann, O.: Perturbation expansions in axiomatic field theory. Lect. Notes Phys. 11, Berlin: Springer Verlag, 1971 26. Wald, R.M.: General Relativity. Chicago: Unversity of Chicago Press, 1984 Communicated by H. Nicolai


Communications in


A Strong Regularity Result for Parabolic Equations Qi S. Zhang Department of Mathematics, University of California, Riverside, CA 92521, USA. E-mail: [email protected] Received: 7 January 2003 / Accepted: 9 July 2003 Published online: 18 November 2003 – © Springer-Verlag 2003

Abstract: We consider a parabolic equation with a drift term u+b∇u−ut = 0. Under the condition divb = 0, we prove that solutions possess dramatically better regularity than those provided by standard theory. For example, we prove continuity of solutions when not even boundedness is expected. 1. Introduction We aim to study the parabolic equation u(x, t) + b(x, t)∇u(x, t) − ut (x, t) = 0, (x, t) ∈ Rn × (0, ∞),

(1.1)

where is the standard Laplacian, b is a vector valued function and n ≥ 2. Standard existence and regularity theory for this kind of equations has existed for several decades. For instance when |b| ∈ Lp (Rn ), p > n, the fundamental solution of (1.1) has a local in time Gaussian lower and upper bound ([A]). Hence bounded solutions are Hölder continuous. In this paper we study the regularity problem of (1.1) for much more singular functions b. Several factors provide strong motivations for studying these kind of problems. The first is to investigate a possible gain of regularity in the presence of the singular drift term b. This line of research has been followed in the papers [St, KS, O, CrZ, Se]. Under the condition |b| ∈ Ln (Rn ), Stampacchia [St] proved that bounded solutions of u + b∇u = 0 are Hölder continuous. In the paper [CrZ], Cranston and Zhao proved that solutions to this equation are continuous when b is in a suitable Kato class, |b(y)| i.e. limr→0 supx |x−y|≤r |x−y| n−1 dy = 0. In the paper [KS] Kovalenko and Semenov

proved the Hölder continuity of solutions when |b|2 is independent of time and is sufficiently small in the form sense. See the next paragraph for a statement of their condition. This result was recently generalized in [Se] to equations with leading term in divergence form. In [O], Osada proved, among other things, that the fundamental solution of (1.1)

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has global Gaussian upper and lower bound when b is the derivative of bounded functions (in distribution sense) and divb = 0. More recently in the paper [LZ], Hölder continuity of solutions to (1.1) was established when |b|2 is form bounded and div b = 0. See the next page for a description. We should mention that many authors have also studied the regularity property of the related heat equation u + V u − ut = 0. Here V is a singular potential. We refer the reader to the papers by Aizenman and Simon [AS], Simon [Si] and references therein. It is worth remarking that the current situation exhibits fundamentally new phenomena comparing with that case. Another motivation comes from the study of nonlinear equations involving gradient structures. These include the Navier Stokes equations, which can be regarded as systems of parabolic equations with very singular first order terms. Our result provides a different proof of the well known fact that weak solutions to the two dimensional Navier-Stokes equations are smooth (Corollary 2). For the three dimensional Navier-Stokes equations, it is interesting to note that the singularity of the velocity field is covered by our theorem (see the discussion at the end of the introduction). In this paper we actually go much beyond the above kinds of singularities. A special case of our result states that, under the assumption that divb = 0, weak solutions of p (1.1) are bounded as long as b ∈ Lloc (Rn ) with p > n/2, n ≥ 4. For a time dependent vector field b ∈ L2loc (Rn × (0, ∞)), it suffices to assume the general form bounded condition: for a fixed m ∈ (1, 2], and φ ∈ C0∞ (Rn × (0, ∞)), Rn

|b(x, t)|m φ 2 dxdt ≤ k

Rn

|∇x φ(x, t)|2 dxdt,

where k is independent of φ. When m = 2 and k is sufficiently small, we are in a situation covered by the paper [KS] and [LZ]. The most interesting case is when m is close to 1. It is widely assumed that solutions of (1.1) can be regular only if the above inequality holds for m ≥ 2. However Theorem 1.1 below proves that weak solutions to (1.1) are bounded as long as m > 1. In fact they are Lipschitz in the spatial direction. Hence b can be almost twice as singular as allowed by standard theory, provided that divb = 0. The above class of the drift term b includes and much exceeds the generalized Kato class that has been studied in several interesting papers [ChZ, CrZ, G]. These functions in general are not the derivative of bounded functions considered in [O] either (see Remark 1.1 below). Here is an example. Let b = b(x1 , x2 , x3 ) be a vector field in R3 . If b has a local singularity of the form |x|c1+ with a small > 0, then b is in this class. In contrast c all previous results at best allow singularities in the form of |x| . In addition to the unexpected regularity result, we also prove that the fundamental solution of (1.1) satisfies a global Gaussian like upper bound. An interesting feature of the bound is that it is not just a perturbation. This means that the global bound holds when b satisfies (1.2) with no additional smallness condition. Using the approximation result in Sect. 2, the fundamental solution here means the minimum of pointwise limits of the fundamental solution of (1.1) with b replaced by a sequence of smooth and divergence free vector fields. An interesting fact is that the bound is no longer Gaussian, reflecting the contribution given by the singularity of the drift term b. In the Kato class case, local in time Gaussian bounds for the heat kernel with singular drift terms were obtained in [Z], which was extended in [LS] recently.

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247

In this paper we use the following definition of weak solutions. Definition 1.1. Let D ⊆ Rn be a domain and T ∈ (0, ∞]. A function u such that u, |∇u| ∈ L2loc (D ×[0, T ]) is a weak solution to (1.1) if: for any φ ∈ C0∞ (D ×(−T , T )), there holds T T (u∂t φ − ∇u∇φ)dxdt + b∇u φ dxdt = − u0 (x)φ(x, 0)dx. 0

0

D

D

D

The assumption that ∇u is square integrable can be weakened. However we are not seeking full generality here. Now we are ready to state the main theorem of the paper. Theorem 1.1. Suppose b satisfies: (1) b ∈ L2loc (Rn × [0, ∞)) and divb(·, t) = 0; (2) for a fixed m ∈ (1, 2] and any φ ∈ C ∞ (Rn × (0, ∞)) with compact support in the spatial direction, m 2 |b| φ dxdt ≤ k |∇φ|2 dxdt, (1.2) Rn

Rn

where k is independent of φ. Then the following statements hold. (i) Weak solutions of (1.1) are locally bounded. (ii) Weak solutions of (1.1) are Hölder continuous when b = b(x) and m = 2. (iii) Let G be the fundamental solution of (1.1). There exist positive constants c1 and c2 such that, for any x, y ∈ Rn and t > s > 0, 

 c (t−s) |x−y|m |x−y|2  ,  [(t−s)|B(x,√1t−s)|]m/(2(m−1)) exp −c2 (t−s)m−1 + exp −c2 t−s

G(x, t; y, s) ≤ m  c√1 |x−y|2  exp −c2 |x−y| , t − s ≥ 1.  m−1 + exp −c2 t−s |B(x, t−s)|

t − s ≤ 1,

(t−s)

Remark 1.1. Note that the above upper bound reduces to the standard Gaussian upper bound when m = 2. This case was recently investigated in [LZ]. Part (ii), follows from [LZ], is here for completeness. If b ∈ Lp (Rn ) with p > n/2, then it is well known that (1.2) is satisfied. See [Si]. Hence we have Corollary 1. Let u be a weak solution of the elliptic equation u + b∇u = 0. Suppose p b ∈ Lloc (Rn ), p > n/2, n ≥ 4, and divb = 0. Then u is a bounded function. Note that without the assumption of divb = 0, it is known ([St]) that u is Hölder p continuous when b ∈ Lloc with p = n. Remark 1.2. Due to its importance and potential applications, we single out part of the result of Theorem 1.1 in the three dimensional case as a corollary. Corollary 2. Let D ⊆ Rn , n = 2, 3. Assume |b| ∈ L∞ ([0, T ], L2 (D)) and divb = 0. Suppose u is a weak solution of (1.1) in D × [0, T ]. Then u is locally bounded. In particular weak solutions to the two dimensional Navier-Stokes equation is smooth when t > 0.

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Q.S. Zhang

Proof. It is enough to prove that the above condition on b alone implies that condition (1.2) is satisfied for some m > 1. Here is a proof when n = 3. The case when n = 2 is dealt with similarly. Let us take m = 4/3 and p = 2/m = 3/2. Then, by Hölder’s inequality,

T

0

|b|4/3 φ 2 dxdt ≤ D

T

=

|b| dx 2

φ dx

T

|b|2 (x, t)dx

≤ C sup

dt

D

1/3 φ 6 dx

dt

D

2/3

T

|b|2 (x, t)dx 0

D

(p−1)/p

dt

0

D

φ 2p/(p−1) dx

1/3

D 2/3

≤ sup

t∈[0,T ]

D 6

D

t∈[0,T ]

1/p |b|mp dx

0 2/3

0

T

|∇φ|2 dxdt. D

The last step is by Sobolev imbedding. Now let u = (u1 (x, t), u2 (x, t)) be a weak solution to the 2 − d Navier-Stokes equation u − u∇u − ∇P − ut = 0,

div u = 0.

Then the curl of u, denoted by w, is a scalar satisfying w + u∇w − wt = 0. By definition, u ∈ L∞ ((0, ∞), L2 (R2 )). So Theorem 1.1 shows that w is bounded when t > 0. Hence u is smooth too when t > 0.

Discussion. Here we would like to speculate on some possible links between the regularity problem of the 3-d Navier-Stokes equation and Corollary 2. Let u be a Leray-Hopf solution to the 3-d Navier-Stokes equation u − u∇u − ∇P − ut = 0,

div u = 0,

|u(·, 0)| ∈ L2 (R3 ).

Then it is well known that u(·, t)L2 (Rn ) is non-increasing and hence uniformly bounded. Therefore assuming only the pressure term P is sufficiently regular locally, then Corollary 2 implies that u is bounded and hence smooth. It seems that all previous regularity results either make some global restrictions on P or on the initial value u0 . We mention the recent interesting result of Seregin and Sverak [SS]. The authors proved that u is smooth provided that u0 ∈ W 1,2 and P is bounded from below. See also [BG]. It would be interesting to see how far the method in this paper may go for the system case. The rest of the paper is organized as follows. In Sect. 2 we show some approximation results of solutions of (1.1) under some singular drift term. Theorems 1.1 will be proven in Sect. 3.

Regularity

249

2. Preliminaries Since the drift term b in (1.1) can be much more singular than those allowed by the standard theory, the existence and uniqueness of weak solutions of (1.1) can not be taken for granted. In order to proceed first we need to prove some approximation results. The next proposition shows that Eq. (1.1) possesses weak solutions even when b satisfies the assumption of Theorem 1.1. Proposition 2.1. Let b be given as in Theorem 1.1 and bk be a sequence of smooth divergence free vector fields. Suppose bk → b in L2 (D × [0, T ]) norm and let uk be the unique solution to   uk + bk ∇uk − ∂t uk = 0, in D × [0, T ] (2.1) uk (x, t) = 0, (x, t) ∈ ∂D × [0, T ],  u (x, 0) = u (x), u ∈ L2 (Rn ). k 0 0 Then there exists a subsequence of {uk }, still denoted by {uk }, which converges weakly in L2 (D × [0, T ]) to a solution of (1.1). Proof. Since divbk = 0, multiplying Eq. (2.1) by uk and integrating, one easily obtains

T

0

D

|∇uk |2 dxdt +

D

u2k (x, T )dx =

Hence there exists a function u such that u, |∇u| ∈ of {uk }, still denoted by {uk }, such that uk → u,

D

× [0, T ]) and a subsequence

L2 (D × [0, T ]);

weakly in

∇uk → ∇u,

L2 (D

u20 (x)dx.

L2 (D × [0, T ]).

weakly in

We will prove that u is a solution to (1.1). Clearly uk satisfies, for any φ ∈ C0∞ (D × [0, T ),

T

(uk ∂t φ − ∇uk ∇φ)dxdt + D 0 =− u0 (x)φ(x, 0)dx.

T

0

D

bk ∇uk φ dxdt (2.2)

D

By the weak convergence of uk and ∇uk , we have 0

T

D

(uk ∂t φ − ∇uk ∇φ)dxdt →

Next, notice that T 0

D

T

= 0

D

0

bk ∇uk φdxdt −

T

T

D

(u∂t φ − ∇u∇φ)dxdt,

k → ∞. (2.3)

b∇u φdxdt

0

D

(bk − b)∇uk φdxdt +

T 0

D

b(∇uk − ∇u) φdxdt.

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Q.S. Zhang

By the strong convergence of bk and the weak convergence of ∇uk , we see that T T uk bk ∇φdxdt − b∇u φdxdt → 0, k → ∞. (2.4) 0

0

D

By (2.2) and (2.4) we obtain T (u∂t φ − ∇φ)dxdt + 0

D

i.e. u is a solution to (1.1).

T

D

b∇u φdxdt = −

0

D

u0 (x)φ(x, 0)dx, D

Proposition 2.2. Suppose b ∈ C ∞ (Rn × [0, ∞)) ∩ L∞ and divb = 0. Let G be the fundamental solution of (1.1). Then, for any x ∈ Rn and t > s > 0, G(x, t; y, s)dy = 1, G(x, t; y, s)dx = 1. Rn

Rn

Proof. Since b is smooth and bounded, G is smooth and has local Gaussian upper bound. Hence we have d G(x, t; y, s)dy = [−y G(x, t; y, s) + b(y, s)∇y G(x, t; y, s)]dy = 0. ds Rn Rn

The other equality is proved similarly.

Proposition 2.3. Let Q ≡ D × [0, T ] with D ⊆ Rn being a smooth domain and T > 0. Suppose that b ∈ C ∞ (Q) ∩ L∞ (Q) and f ∈ L1 (Q). Suppose u is a weak solution to   u + b∇u − ut = f, in Q (2.5) u(x, t) = 0, (x, t) ∈ ∂D × [0, T ]  u(x, 0) = 0. Here the boundary condition is in the sense that u ∈ L2 ([0, T ], W01,2 (D)). Then t u(x, t) = − G(x, t; y, s)f (y, s)dyds. 0

D

Here G is the Green’s function of (1.1) with initial Dirichlet boundary condition in Q. Proof. This result is trivial when f is bounded and smooth. When f is just L1 , it is known too. Here we present a proof for completeness. Let ψ be a smooth function in Q. Since b is bounded and smooth, standard theory shows that the following backward problem has a unique smooth solution:   η − b∇η + ηt = ψ, in Q (2.6) η(x, t) = 0, (x, t) ∈ ∂D × [0, T ]  u(x, T ) = 0. Moreover

η(y, s) = −

T

G(x, t; y, s)ψ(x, t)dxdt.

s

D

(2.7)

Regularity

251

Since u is a weak solution to (2.5), we have, by definition [−∇u∇η + b∇uη + uηt ]dxdt = f ηdxdt. Q

Q

Using integration by parts we have u[η − b∇η + ηt ]dxdt = f ηdxdt. Q

Q

By this, (2.6) and (2.7), we deduce T uψdxdt = − f (y, s) 0

Q

D

T

G(x, t; y, s)ψ(x, t)dxdt. D

s

That is

T

uψdxdt = − Q

t G(x, t; y, s)f (y, s)dyds ψ(x, t)dxdt.

0

D

0

D

The proposition follows since ψ is arbitrary.

Proposition 2.4. Suppose u is a weak solution of Eq. (1.1) in the cube Q = D × [0, T ], where b satisfies the condition in Theorem 1.1. Here D is a domain in Rn . Then u is the L1loc limit of functions {uk }. Here {uk } is a weak solution of (1.1) in which b is replaced by smooth, divergence free bk such that bk → b strongly in L2 (Q), k → ∞. Proof. First we select a sequence of smooth, bounded, divergence free bk such that bk → b strongly in L2 (Q), k → ∞. Let D ⊂ D be a smooth sub-domain of D. Then the following problem has a weak solution uk :   uk + bk ∇uk − (uk )t = 0, in Q = D × (0, T ) (2.8) uk (x, t) = u(x, t), (x, t) ∈ ∂D × [0, T ]  u (x, 0) = u(x, 0). k Clearly uk − u is a weak solution to the following:   (uk − u) + bk ∇(uk − u) − (uk − u)t = (b − bk )∇u, (uk − u)(x, t) = 0, (x, t) ∈ ∂D × [0, T ]  (u − u)(x, 0) = 0. k

in

Q = D × (0, T )

(2.9) Here the boundary condition is in the sense that uk − u ∈ L2 ([0, T ], W01,2 (D )). By our assumptions on b, bk and ∇u, we know that (b − bk )∇u ∈ L1 (Q ). Since bk is bounded and smooth, Proposition 2.3 shows that t (uk − u)(x, t) = − Gk (x, t; y, s)(b − bk )∇u(y, s)dyds. 0

D

252

Q.S. Zhang

Here Gk is the Green’s function of u + bk ∇u − ut = 0 in Q with Dirichlet initial boundary value condition. By Proposition 2.2, or the local version of it, we have Gk (x, t; y, s)dx ≤ 1. D

Hence D

Hence

|uk − u|(x, t)dx ≤

t 0

D

|b − bk ||∇u(y, s)|dyds.

D

|uk − u|(x, t)dx ≤ b − bk L2 (Q ) ∇uL2 (Q ) → 0.

This proves the proposition.

3. Proof of Theorem 1.1 Using the approximation result of Sect. 2, we may and do assume that the vector field b is bounded and smooth. The beginning of the proof generally follows the classical strategy of using test functions to establish L2 − L∞ bounds and weighted estimates for solutions of (1.1). However it is well known that this method does not provide a sharp global upper bound in the presence of lower order terms and the vector field b can not be as singular as we are assuming. For instance there is usually an extraneous ect term when t is large. Nevertheless, by using the special structure of the drift term and exploiting a special role of the divergence of b, we show that this classical method can be refined to derive sharp global bounds. In order to overcome the singularity of the drift term b, we need to construct a refined test function. This is the key step in proving the bounds. We divide the proof into five steps. For the sake of clarity we draw a flow chart for the proof: Step 1: Energy estimates using refined test function ⇒ Step 2: L∞ bound for weak solutions ((i) of Theorem 1.1) ⇒ Step 3: Weighted estimates ⇒ Step 4: Gaussian like upper bound ((iii) of Theorem 1.1); Step 5: Proof of (ii) Step 1. Caccioppoli inequality (energy estimates). Let u be a solution of (1.1) in the parabolic cube Qσ r = B(x, σ r) × [t − (σ r)2 , t]. Here x ∈ Rn , σ > 1, r > 0 and t > 0. By direct computation, for any rational number p ≥ 1, which can be written as the quotient of two integers with the denominator being odd, one has up + b∇up − ∂t up = p(p − 1)|∇u|2 up−2 .

(3.1)

Here the condition on p is to ensure that up makes sense when u changes sign. One can also just work on positive solutions now and prove the boundedness of all solutions later. See Step 6 at the end of the section.

Regularity

253

Choose ψ = φ(y)η(s) to be a refined cut-off function satisfying supp φ ⊂ B(x, σ r); φ(y) = 1, y ∈ B(x, r);

|∇φ| C ≤ , 0 ≤ φ ≤ 1; δ φ ((σ − 1)r)

here δ ∈ (0, 1). By scaling it is easy to show that such a function exists, supp η ⊂ (t − (σ r)2 , t); η(s) = 1, s ∈ [t − r 2 , t]; |η | ≤ 2/((σ − 1)r)2 ;

0 ≤ η ≤ 1.

Denoting w = up and using wψ 2 as a test function on (3.1), one obtains (w + b∇w − ∂s w)wψ 2 dyds = p(p − 1) |∇u|2 w 2 u−2 ≥ 0. Qσ r

Qσ r

Using integration by parts, one deduces ∇(wψ 2 )∇wdyds ≤ b∇w(wψ 2 )dyds − Qσ r

Qσ r

Qσ r

(∂s w)wψ 2 dyds.

(3.2)

By direct calculation, 2 ∇(wψ )∇wdyds = ∇[(wψ)ψ]∇wdyds Qσ r Q σr = [ ∇(wψ)( ∇(wψ) − (∇ψ)w) + wψ∇ψ∇w]dyds Q σr = |∇(wψ)|2 − |∇ψ|2 w 2 dyds. Qσ r

Substituting this to (3.2), we obtain |∇(wψ)|2 dyds ≤ b∇w(wψ 2 )dyds − (∂s w)wψ 2 dyds Qσ r Qσ r Qσ r + |∇ψ|2 w 2 dyds.

(3.3)

Qσ r

Next notice that 1 (∂s w)wψ 2 dyds = (∂s w 2 )ψ 2 dyds 2 Qσ r Q σr 1 =− w 2 φ 2 η∂s ηdyds + w 2 (y, t)φ 2 (y)dy. 2 Qσ r B(x,σ r) Combining this with (3.3), we see that 1 2 |∇(wψ)| dyds + w 2 (y, t)φ 2 (y)dy 2 B(x,σ r) Qσ r ≤ (|∇ψ|2 + η∂s η) w 2 dyds + b(∇w)(wψ 2 )dyds ≡ T1 + T2 . Qσ r

Qσ r

(3.4)

254

Q.S. Zhang

The first term on the right-hand side of (3.4) is already in good shape. So let us estimate the second term as follows: b(∇w)(wψ 2 )dyds T2 = Qσ r 1 1 = bψ 2 ∇w 2 dyds = − div(bψ 2 )w 2 dyds 2 Qσ r 2 Qσ r 1 1 =− divb(ψw)2 dyds − b∇(ψ 2 )w 2 dyds 2 Qσ r 2 Q σr 1 =− divb(ψw)2 dyds − b(∇ψ)ψw 2 dyds 2 Qσ r Qσ r 2 =− b(∇ψ)ψw dyds. Qσ r

Here we just used the assumption that divb = 0. The next paragraph contains the key argument of the paper. Notice that for δ ∈ (0, 1), a ∈ (0, 2) and m ∈ (1, 2], T2 ≤ | b(∇ψ)ψw 2 dyds| Q σr ∇ψ =| bψ 1+δ |w|2−a δ |w|a dyds| ψ Qσ r 1/m ≤ |b|m ψ (1+δ)m |w|(2−a)m dyds Qσ r (m−1)/m |∇ψ| × . ( δ )m/(m−1) |w|am/(m−1) dyds Qσ r ψ Take a, δ so that (2 − a)m = 2, Then 2 am/(m − 1) = a 2−a

(1 + δ)m = 2.

2 − 1 = 2, 2−a

δ = (2/m) − 1 < 1.

These and the assumption on the cut-off function ψ show that 1/m (m−1)/m c 2 T2 ≤ |b|m (ψw)2 dyds w dyds . m/(m−1) Qσ r Qσ r [(σ − 1)r] This implies for any > 0, T2 ≤ m |b|m (ψw)2 dyds + C Qσ r

−m/(m−1) [(σ − 1)r]m/(m−1)

By our assumptions on b, |b|m (ψw)2 dyds ≤ k Qσ r

Qσ r

w 2 dyds. Qσ r

|∇(ψw)|2 dyds.

(3.5)

Regularity

255

Substituting the above to (3.5), we can find k1 < 1/2 and k2 > 0 such that b(∇w)(wψ 2 )dyds| |T2 | = | Q σ r 1 2 ≤ k1 |∇(ψw)| dyds + k2 w 2 dyds. ((σ − 1)r)m/(m−1) Qσ r Qσ r

(3.6)

Combining (3.4) with (3.6), we reach 2 |∇(wψ)| dyds + w 2 (y, t)φ 2 (y)dy Qσ r B(x,σ r) C ≤ w 2 dyds, r ≤ 1, (3.7) ((σ − 1)r)m/(m−1) Qσ r C 2 2 2 |∇(wψ)| dyds + w (y, t)φ (y)dy ≤ w 2 dyds, r ≥ 1. 2 ((σ − 1)r) Qσ r B(x,σ r) Qσ r (3.7 ) Step 2. L2 − L∞ bounds. It is known that (3.7) implies the following L2 − L∞ estimate via Moser’s iteration. 1 u2 dyds, r ≤ 1. (3.8) sup u2 ≤ C |Qr |m/(2(m−1)) Q2r Qr Here m > 1. Also, (3.7 ) shows sup u2 ≤ C Qr

1 |Qr |

Indeed, by Hölder’s inequality,

2(1+(2/n)) (φw) ≤ Rn

Rn

(3.8 )

r ≥ 1.

u2 dyds, Q2r

(φw)

2n/(n−2)

(n−2)/n

2/n Rn

Using the Sobolev inequality, one obtains 2/n

(φw)2(1+(2/n)) ≤ C (φw)2 Rn

Rn

(φw)

2

.

|∇(φw)|2 .

The last inequality, together with (3.7) implies, for some C1 > 0, θ

2pθ −m/(m−1) 2p u ≤ C C1 (rτ ) u , Qσ r

Qσ r

where θ = 1 + (2/n). When the dimension n is odd or u ≥ 0, we set τi = 2−i−1 , σ0 = 1, σi = σi−1 − τi = 1 − 1i τj , p = θ i . The above then yields, for some C2 > 0,

u Qσi+1 r (x,t)

2θ i+1

≤C

C2i+1 r −m/(m−1)

θ

u Qσi r (x,t)

2θ i

.

256

Q.S. Zhang

After iterations the above implies

θ −i−1 u

2θ i+1

≤ C θ

−j −1

Qσi+1 r (x,t)

−(j +1)θ −j −1

C2

(r −m/(m−1) )θ

−j

u2 . Qr

Letting i → ∞ and observing that j∞=0 θ −j = (n + 2)/2, we obtain sup u2 ≤ Qr/2

C r m(n+2)/(2(m−1))

u2 . Qr

This proves (3.8) either for odd n or for all n and nonnegative u. Similarly one proves (3.8 ). In case n is even and u changes sign, we just regard u as a solution of Eq. (1.1) in Rn+1 × (0, T ). Then the L∞ bound of u follows from the above. Step 3. Weighted estimate. Let G be the heat kernel of (1.1). For a fixed λ ∈ R and a fixed bounded function ψ such that |∇ψ| ≤ 1, we write fs (y) = eλψ(y)

G(y, s; z, 0)e−λψ(z) f (z)dz.

Here and later the integral takes place in Rn if no integral region is specified. Direct computation shows that ∂s ||fs ||22

=2

(∂s fs (y))fs (y)dy λψ(y) =2 e fs (y) ∂s G(y, s; z, 0)e−λψ(z) f (z)dzdy

λψ(y) −λψ(z) =2 e G(y, s; z, 0)e fs (y)y f (z)dz dy

λψ(y) −λψ(z) G(y, s; z, 0)e +2 e fs (y)b(y)∇y f (z)dz dy

≡ J1 + J2 .

(3.9)

Following standard computation, we see that

J1 ≤ −2

|∇fs (y)| dy + 2cλ 2

2

Next we estimate J2 . For simplicity we write u(y, s) = e−λψ(y) fs (y),

fs (y)2 dy.

(3.10)

Regularity

257

which is a solution to u + b∇u − ∂t u = 0 in Rn × (0, ∞). Then J2 = 2 eλψ(y) fs (y)b(y)∇y u(y, s)dy = −2 div eλψ(y) fs (y)b(y) u(y, s)dy λψ(y) = −2 ∇ e fs (y) b(y)u(y, s)dy − 2 eλψ(y) fs (y)u(y, s)divb(y)dy = −2λ eλψ(y) fs (y)u(y, s)∇ψ(y)b(y)dy λψ(y) −2 e u(y, s)∇fs (y)b(y)dy − 2 fs (y)2 divb(y)dy = −2λ fs (y)2 ∇ψ(y)b(y)dy − 2 fs (y)∇fs (y)b(y)dy − 2 fs (y)2 divb(y)dy 2 = −2λ fs (y) ∇ψ(y)b(y)dy − fs (y)2 divb(y)dy. In the last step we have used integration by parts. Hence J2 = −2λ fs (y)2 ∇ψ(y)b(y)dy.

(3.11)

Using an argument similar to that in the middle of Step 2, we see that |bfs2−a ∇ψfsa |dy J2 ≤ 2λ Qσ r 1/m (m−1)/m m (2−a)m m/(m−1) am/(m−1) ≤ |b| fs dy 2λ (|∇ψ|) fs dy . Here, as before a ∈ (0, 2) and (2 − a)m = 2, am/(m − 1) = 2. It follows that, for any > 0, m 2 −1/(m−1) m/(m−1) fs (y)2 dy. λ J2 ≤ |b| fs (y) dy + c Combining this with the estimate for J1 , we have 2 2 m/(m−1) 2 2 + λ ) fs (y) dy + |b|m fs (y)2 dy. ∂s ||fs ||2 ≤ −2 |∇fs (y)| dy + c1 (λ Here c1 may depend on . Writing F (s) ≡ ||fs ||22 ,

H (s) ≡ −2

|∇fs (y)|2 dy +

|b|m fs (y)2 dy,

the above differential inequality can be written as ∂s F (s) ≤ c1 (λm/(m−1) + λ2 )F (s) + H (s).

258

Q.S. Zhang

Hence

s

F (s) ≤ ecz(λ)s F (0) + eCz(λ)s

e−z(λ)τ H (τ )dτ,

0

where z(λ) ≡ λm/(m−1) + λ2 . That is

s |∇(fτ (y)e−z(λ)τ/2 )|2 dydτ F (s) ≤ ecz(λ)s F (0) + eCz(λ)s − 2 0 s m −z(λ)τ/2 2 + |b| (fs (y)e ) dydτ . 0

Taking sufficiently small and using the condition on b we conclude that m/(m−1) +λ2 )s

||fs ||22 ≤ ecz(λ)s ||f ||22 = ec(λ

||f ||22 .

(3.12)

Step 4. Gaussian-like upper bound. For simplicity we only prove the bound for G(x, t; y, 0). We just prove the inequality t ≤ 1. When t ≥ 1, the situatioin is simpler and the proof is omitted. Now consider the function u(y, s) = e−λψ(y) fs (y) which is a solution to u + b∇u − ∂t u = 0 in Rn × (0, ∞). Here ψ is a function such that |∇ψ| ≤ 1 and √ whose precise values are to be chosen later. Applying (3.8) with Q√t/2 (x, t) = B(x, t/2) × (3t/4, t), we obtain t 1 u(x, t)2 ≤ C √ u2 . |Q t/2 (x, t)|m/(2(m−1)) 3t/4 B(x,√t/2) From (3.12), it follows that t 1 u2 |Q√t/2 (x, t)|m/(2(m−1)) 3t/4 B(x,√t/2) t 1 =C √ e2λ[ψ(x)−ψ(z)] fs2 |Q t/2 (x, t)|m/(2(m−1)) 3t/4 B(x,√t/2) √ t m/(m−1) +λ2 )t ≤ Ce2λ t ||f ||22 . √ m/(2(m−1)) ec(λ [t|B(x, t)|] √ Taking the supremum over all f ∈ L2 (B(y, t)) with ||f ||2 = 1, we find that G(x, t; z, 0)2 dz e2λ[ψ(x)−ψ(y)] √ e2λψ(x) u(x, t)2 ≤ Ce2λψ(x)

≤ Ce

B(y, t/2) √ 4λ t+c(λm/(m−1) +λ2 )t

[t|B(x,

√

t . t)|]m/(2(m−1))

Note that the second entries of the heat kernel G satisfies the equation u − ∇(bu) + ∂s u = 0.

(3.13)

Regularity

259

Hence it satisfies u − b∇u + ∂s u = 0. Therefore we can use (3.8) backward on the second entries of the heat kernel to conclude, from (3.13), that t/4 1 2 G(x, t; y, 0) ≤ C G(x, t; z, s)2 dzds √ |Q√t/2 (y, t)|m/(2(m−1)) 0 B(y, t/2) √ t2 m/(m−1) +λ2 )t−2λ[ψ(x)−ψ(y)] ≤C . √ m/(m−1) e4λ t+c(λ [t|B(x, t)|] √ This shows, since λ t ≤ c1 + c2 λ2 t, G(x, t; y, 0)2 ≤ C

t2 m/(m−1) +λ2 )t−2λ[ψ(x)−ψ(y)] . √ m/(m−1) ec(λ [t|B(x, t)|]

Now we select ψ so that ψ(x) − ψ(y) = |x − y|. Then it follows G(x, t; y, 0)2 ≤ C

t2 m/(m−1) +λ2 )t−2λ|x−y| ≡ C(t)eQ(λ) . (3.14) √ m/(m−1) ec(λ [t|B(x, t)|]

Here for simplicity, we write Q(λ) ≡ c(λm/(m−1) + λ2 )t − 2λ|x − y|. Now we choose λ to be a positive number satisfying λ1/(m−1) + λ = a|x − y|/t,

(3.15)

where a > 0 will be chosen in a moment. Then Q(λ) = cλ(λ1/(m−1) + λ)t − 2λ|x − y| = (ca − 2)λ|x − y|. Taking a = 1/c, we see that Q(λ) = −λ|x − y|.

(3.16)

Next we consider two separate cases. Case 1. |x − y|/t ≥ 1. Then from (3.15), there exists c0 > 0 such that λ ≥ c0 . Hence λ ≤ c1 λ1/(m−1) because m ≤ 2. By (3.15), λ ≥ c2 (|x − y|/t)m−1 . This shows, via (3.16), Q(λ) ≤ −c3

|x − y|m . t m−1

(3.17)

Case 2. When |x−y|/t ≤ 1. In this case (3.15) implies that λ ≤ c0 and hence λ1/(m−1) ≤ c1 λ. Therefore, by (3.15), λ ≥ c2 |x − y|/t. Hence Q(λ) ≤ −c3 |x − y|2 /t.

(3.18)

Substituting (3.17) and (3.18) to (3.14), we obtain

c1 t |x − y|m |x − y|2 G(x, t; y, 0) ≤ . +exp −c2 √ m/(2(m−1)) exp −c2 m−1 t t [t|B(x, t)|] This proves the upper bound for G.

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Step 5. Proof of (ii). Since the proof is identical to that in [LZ], we omit the details. Final Remark. Using a Nash type estimate, it is easy to prove that G(x, t; y, s) ≤ c/(t − s)n/2 . It would be interesting to combine this bound with the bound in (iii) to get a sharper bound. The same could be said about the lower bound. Acknowledgement. We thank Professor Vitali Liskevich and Victor Shapiro for helpful conversations.

References [A]

Aronson, D.G.: Non-negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa 22, 607–694 (1968) [AS] Aizenman, M., Simon, B.: Brownian motion and Harnack inequality for Schrdinger operators. Comm. Pure Appl. Math. 35(2), 209–273 (1982) [BG] Berselli, Luigi, C., Galdi, Giovanni, P.: Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations. Proc. Am. Math. Soc. 130(12), 3585–3595 (2002) [ChZ] Chen, Z.Q., Zhao, Z.: Diffusion processes and second order elliptic operators with singular coefficients for lower order terms. Math. Ann. 302(2), 323–357 (1995) [CrZ] Cranston, M., Zhao, Z.: Conditional transformation of drift formula and potential theory for 1 2 + b()∇. Commun. Math. Phys. 112(4), 613–625 (1987) [G] Gerhard, W.D.: The probabilistic solution of the Dirichlet problem for 21 + a, ∇ + b with singular coefficients. J. Theoret. Probab 5(3), 503–520 (1992) d ) generated [KS] Kovalenko, V.F., Semenov, Yu.A.: Co -semigroups in the spaces Lp (R d ) and C(R by + b.∇. (Russian) Teor. Veroyatnost. i Primenen. 35(3), 449–458 (1990); Translation in Theory Probab. Appl. 35(3), 443–453 (1990) [LS] Liskevich, V., Semenov, Y.: Estimates for fundamental solutions of second-order parabolic equations. J. Lond. Math. Soc. (2) 62(2), 521–543 (2000) [LZ] Liskevich, V., Zhang, Q.S.: Extra regularity for parabolic equations with drift terms. Manuscripta Math., to appear [MS] Milman, P.D., Semenov, Y.: Disingularizing weights and the heat kernel bounds. Preprint, 1998 [O] Osada, H.: Diffusion processes with generators of generalized divergence form. J. Math. Kyoto Univ 27(4), 597–619 (1987) [Se] Semenov, Y.A: Hölder continuity of bounded solutions of parabolic equations. Preprint, 1999 [Si] Simon, B.: Schrödinger semigroups. Bull. AMS 7, 447–526 (1982) [St] Stampacchia, G.: Le problème de Dirichlet pour les e´ quations elliptiques du second ordre a` coefficients discontinus. (French) Ann. Inst. Fourier (Grenoble) 15(1), 189–258 (1965) ˇ [SS] Seregin, G., Svera’k, V.: Navier-Stokes equations with lower bounds on the pressure. Arch. Ration. Mech. Anal. 163(1), 65–86 (2002) [Z] Zhang, Q.S.: Gaussian bounds for the fundamental solutions of ∇(A∇u) + B∇u − ut = 0. Manuscripta Math. 93(3), 381–390 (1997) Communicated by B. Simon


Communications in


On the Representation Theory of Virasoro Nets Sebastiano Carpi Dipartimento di Scienze, Università “G. d’Annunzio” di Chieti-Pescara, Viale Pindaro 42, 65127 Pescara, Italy. E-mail: [email protected] Received: 1 July 2003 / Accepted: 23 July 2003 Published online: 25 November 2003 – © Springer-Verlag 2003

Abstract: We discuss various aspects of the representation theory of the local nets of von Neumann algebras on the circle associated with positive energy representations of the Virasoro algebra (Virasoro nets). In particular we classify the local extensions of the c = 1 Virasoro net for which the restriction of the vacuum representation to the Virasoro subnet is a direct sum of irreducible subrepresentations with finite statistical dimension (local extensions of compact type). Moreover we prove that if the central charge c is in a certain subset of (1, ∞), including [2, ∞), and h ≥ (c − 1)/24, the irreducible representation with lowest weight h of the corresponding Virasoro net has infinite statistical dimension. As a consequence we show that if the central charge c is in the above set and satisfies c ≤ 25 then the corresponding Virasoro net has no proper local extensions of compact type. 1. Introduction The idea that the formulation of relativistic quantum physics in terms of local nets of von Neumann algebras (see e.g. [27]) provides a natural framework for the classification of two-dimensional conformal field theories was already present in the late eighties in the work of Buchholz, Mack and Todorov [3]. As an illustration of this idea these authors classified the local conformal nets over S 1 (compactified light ray) whose common “germ” is the U(1) chiral current algebra, namely the local nets extending the one generated by a U(1) current. In the same paper they suggested the more general (and ambitious) classification program of conformal field theories S 1 whose “germ” is the Virasoro algebra Vir. In other words, they proposed to classify the local extensions of the Virasoro nets, i.e. the local nets of von Neumann algebras on S 1 which are generated by the positive energy unitary irreducible representations with lowest weight 0 (vacuum representations) of Vir or equivalently (see [45]) by a chiral energy-momentum tensor

Supported in part by the Italian MIUR and GNAMPA-INDAM.

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T (z), cf. [5]. Since the equivalence class of a Virasoro net is completely determined by the value of the central charge c in the corresponding representation of Vir, one has to classify the local extensions of a family of nets labelled by a positive real number c and this is a clearly well defined problem which in fact turns out to be equivalent to the one of classifying diffeomorphism covariant nets on the circle. In a recent remarkable paper [33] Kawahigashi and Longo have been able to solve the above problem for all the Virasoro nets with c < 1 and subsequently they used this result to classify all local conformal nets on the two-dimensional Minkowski space-time, having parity symmetry and central charge less than 1 [34]. The extension of their results in the c ≥ 1 region appears to be a very important and difficult challenge. In the transition from c < 1 to c ≥ 1 two drastic differences are immediately evident. The first is that the Virasoro nets with c ≥ 1 are all known to be non-rational. The second is that they are all expected to have irreducible sectors with infinite statistical dimension [49], a fact that has been proved by the present author in the case c = 1 [8]. Rationality and the absence of irreducible sectors with infinite statistical dimension play a fundamental role in the classification of c < 1 conformal nets and hence, some of the main ideas in [33] do not apply for the remaining values of the central charge. The purpose of this paper is to give some new insight in the understanding of the representation theory of c ≥ 1 Virasoro nets and their local extensions with the above problems in mind, especially concerning the role of infinite statistical dimension. Our main results are the classification of local extensions of compact type (see Definition 3.2) for the Virasoro net with c = 1 (Theorem 3.5)1 and the proof that if the central charge c is in a certain subset of (1, ∞) containing [2, ∞), then the irreducible positive energy representations with lowest weight h ≥ (c − 1)/24 of the corresponding Virasoro net have infinite statistical dimension (Theorem 5.1 ), a fact that is expected to hold for every c > 1 and h > 0 [49]. As a consequence of the latter result we show that if c ∈ [2, 25] then the corresponding Virasoro net has no proper local extensions of compact type (Theorem 5.7). As in the c = 1 case [8], we use oscillator (Fock) representations of Vir to obtain the result on infinite statistical dimension but the argument we have found for c > 1 is more intricate and relies in part on recent results of S. Köster [38]. Besides these main results we also provide the proof of some relevant properties of the Virasoro nets which seem to appear at most implicitly in the literature, like the fact that every irreducible positive energy representation of a Virasoro net on a separable Hilbert space comes from a representation of the Virasoro algebra (Prop. 2.1) or the fact that every local extension of a Virasoro net is diffeomorphism covariant (Prop. 3.7).

2. Preliminaries 2.1. Conformal nets on S 1 and diffeomorphism covariance. Let I be the set of nonempty, nondense, open intervals of the unit circle S 1 = {z ∈ C : |z| = 1}. A conformal net on S 1 is a family A = {A(I ) : I ∈ I} of von Neumann algebras, acting on an infinite-dimensional separable complex Hilbert space HA , satisfying the following properties: (i) Isotony. A(I1 ) ⊂ A(I2 ), if I1 ⊂ I2 , I1 , I2 ∈ I. 1

An equivalent result as been independently obtained by F. Xu [55]

(1)

On the Representation Theory of Virasoro Nets

263

(ii) Locality. A(I1 ) ⊂ A(I2 ) , if I1 ∩ I2 = ∅, I1 , I2 ∈ I.

(2)

(iii) Möbius covariance. There exists a strongly continuous unitary representation U of PSL(2, R) in H such that U (g)A(I )U (g)−1 = A(gI ), I ∈ I, g ∈ PSL(2, R),

(3)

where PSL(2, R) acts on S 1 by Möbius transformations (cf. Appendix A). (iv) Positivity of the energy. The representation U has positive energy, namely the conformal Hamiltonian L0 , which generates the restriction of U to the one-parameter subgroup of rotations r(ϑ), has nonnegative spectrum. (v) Existence and uniqueness of the vacuum. There exists a unique (up to a phase) U -invariant unit vector ∈ HA . (vi) Cyclicity of the vacuum. is cyclic for the algebra A(S 1 ) := I ∈I A(I ). Some consequences of the axioms are [22, 25, 19]: (vii) Reeh-Schlieder property. For every I ∈ I, is cyclic and separating for A(I ). (viii) Bisognano-Wichmann property. If I is the modular operator associated to A(I ) and then itI = U (I ((2π )t),

(4)

where I is the one parameter subgroup of PSL(2, R) of special conformal transformations preserving I . (ix) Haag duality. For every I ∈ I, A(I ) = A(I c ),

(5)

where I c denotes the interior of S 1 \I . (x) Factoriality. The algebras A(I ) are type III1 factors. (xi) Irreducibility. A(S 1 ) = B(HA ), where B(HA ) denotes the algebra of all bounded linear operators on HA . (xii) Additivity. If S ⊂ I is a covering of the interval I then A(I ) ⊂ A(J ). (6) J ∈S

Furthermore, it follows easily from the strong continuity of the representation U that conformal nets are continuous from outside, namely A(I ) = A(J ). (7) J ⊃I

A conformal net A is said to be split if given two intervals I1 , I2 ∈ I such that the closure I1 of I1 is contained in I2 , there exists a type I factor N (I1 , I2 ) such that A(I1 ) ⊂ N (I1 , I2 ) ⊂ A(I2 ). If Tr(t L0 ) < ∞ for every t ∈ (0, 1) then A is split [13, Theorem 3.2].

(8)

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A is said to be strongly additive if for every I, I1 , I2 ∈ I with I1 , I2 obtained by removing a point from I we have A(I1 ) ∨ A(I2 ) = A(I ).

(9)

It is often convenient to identify S 1 /{−1} with the real line R. With this identification, the family of nonempty open bounded intervals of R corresponds to the family I0 = {I ∈ I : −1 ∈ / I }. The restriction A0 of a conformal net A to I0 can be considered as a net on R. Moreover, since I0 is directed under inclusion, one can define the quasi-local C ∗ -algebra (still denoted A0 ) ( I ∈I0 A(I ))−||·|| ) as C ∗ -inductive limit of the local von Neumann algebras A(I ), I ∈ I0 . We now briefly discuss diffeomorphism covariance. Let Diff + (S 1 ) the group of orientation preserving diffeomorphisms of the circle. It is an infinite dimensional Lie group modelled on the real topological vector space Vect(S 1 ) of smooth real vector fields on S 1 with the usual C ∞ topology [46, Sect.6]. Its Lie algebra coincides with Vect(S 1 ) with the bracket given by the negative of the usual brackets of vector fields. Hence if g(z), f (z), z = eiϑ , are real valued functions in C ∞ (S 1 ) then [g(eiϑ )

d d d d d , f (eiϑ ) ] = ( g(eiϑ ))f (eiϑ ) − ( f (eiϑ ))g(eiϑ ) . dϑ dϑ dϑ dϑ dϑ

(10)

d ∈ Vect(S 1 ) with the In this paper we shall often identify the vector field g(eiϑ ) dϑ corresponding real function g(z) ∈ C ∞ (S 1 ). Following [33] for every I ∈ I we shall denote by Diff(I ) the subgroup of Diff + (S 1 ) whose elements are the diffeomorphisms of the circle which act as the identity on I c . Note that Diff(I ) does not coincide with the group of diffeomorphisms of the open interval I , as the notation might erroneously suggest. By a strongly continuous projective unitary representation V of Diff + (S 1 ) on a Hilbert space we shall always mean a strongly continuous homomorphism of Diff + (S 1 ) into the quotient U(H)/T of the unitary group of H by the circle subgroup T. The restriction of the representation V to the Möbius subgroup of Diff + (S 1 ) always lifts to a unique strongly continuous unitary representation U of the universal covering group R) of PSL(2, R). We shall say that V extends U and that V is a positive energy PSL(2, R), namely if the correrepresentation if U is a positive energy representation of PSL(2, sponding conformal Hamiltonian L0 , which generates the restriction of U to the lifting r˜ (ϑ) of the one-parameter subgroup r(ϑ) of rotations, has nonnegative spectrum. Note that although for γ ∈ Diff + (S 1 ), V (γ ) is defined only up to a phase as an operator on H, expressions like V (γ )T V (γ )∗ for T ∈ B(H) or V (γ ) ∈ M for a (complex) subspace M ⊂ B(H) are unambiguous and will be used in the following. We shall say that a conformal net on S 1 is diffeomorphism covariant if there is a strongly continuous projective unitary representation V of Diff + (S 1 ) on HA extending U ◦ q ( where U is the original unitary representation of PSL(2, R) making A Möbius R) → PSL(2, R) denotes the covering map) and such that, for covariant and q : PSL(2, every I ∈ I,

V (γ )A(I )V (γ )∗ = A(γ I ), γ ∈ Diff + (S 1 ), V (γ )AV (γ )∗ = A, γ ∈ Diff(I ), A ∈ A(I c ).

(11) (12)


265

2.2. Representations of conformal nets. A representation of a conformal net A is a family π = {πI : I ∈ I}, where πI is a (unital) representation of A(I ) on a fixed Hilbert space Hπ , such that πJ |A(I ) = πI if I ⊂ J, I, J ∈ I.

(13)

Irreducibility, direct sums and unitary equivalence of representations of a conformal net can be defined in a natural way, see [22, 25]. If Hπ is separable then, since the local von Neumann algebras A(I ), I ∈ I, are factors, π is automatically locally normal, namely πI is normal for each I ∈ I, see [51]. Hence, πI (A(I )) is a type III1 factor. The unitary equivalence class representation π on a separable Hilbert space is called a sector and denoted [π ]. If π is irreducible then we say that [π ] is an irreducible sector (also called superselection sector). The defining representation π0 of a conformal net A on the Hilbert space HA is called the vacuum representation. The corresponding sector is called the vacuum sector and HA is said to be the vacuum Hilbert space of A. A representation π is said to be covariant if there is a strongly continuous unitary R) on Hπ such that representation Uπ of PSL(2, R), I ∈ I. AdUπ (g) ◦ πI = πq(g)I ◦ AdU (q(g)), g ∈ PSL(2,

(14)

If Uπ can be chosen to be a positive energy representation, then π is said to be covariant with positive energy. In this case one can always choose Uπ to be with positive energy and inner, namely such that R)) ⊂ π(A) := Uπ (PSL(2,

πI (A(I )),

(15)

I ∈I

and this choice is unique, see [36] and (the proof of) [1, Lemma 5.14]. Given a covariant representation π of A on a separable Hilbert space Hπ one has the (isomorphic) type III subfactors πI (A(I )) ⊂ πI c (A(I c )) , I ∈ I [22]. Hence the corresponding (minimal) index [πI c (A(I c )) : πI (A(I ))] [29, 39, 41] is independent of I ∈ I and the statistical dimension d(π) of π is given by 1

d(π) = [πI c (A(I c )) : πI (A(I ))] 2 .

(16)

A representation ρ of a conformal net A on its vacuum Hilbert space HA is said to be localized in an interval I0 ∈ I if ρI0c is the identity representation. As a consequence of Haag duality if a representation ρ is localized in I0 and I ∈ I contains I0 then ρI is an endomorphism of A(I ) whose index is the square of the statistical dimension of the representation ρ. Moreover, for every interval I0 ∈ I and every representation π of A on a separable Hilbert space one can find a representation ρ, localized in I0 and unitarily equivalent to π, see [22, 25]. The restriction to A0 of a representation ρ localized in some I0 ∈ I0 is called a DHR endomorphism and in fact yields an endomorphism of the quasi-local C ∗ -algebra A0 , see [43, Sect. 3]. The set of DHR endomorphisms is a semigroup (under composition) and it has a natural (DHR) unitary braiding, see [20, 22]. As usual we shall denote (ρ, σ ) the unitary braiding operator associated to the DHR endomorphisms ρ and σ .

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2.3. Subsystems. A conformal subsystem (or subnet) of a conformal net A is a family B = {B(I ) : I ∈ I} of nontrivial von Neumann algebras acting on HA such that: B(I ) ⊂ A(I ) I ∈ I; U (g)B(I )U (g)−1 = B(gI ) I ∈ I, g ∈ PSL(2, R); B(I1 ) ⊂ B(I2 ) if I1 ⊂ I1 , I1 , I2 ∈ I.

(17) (18) (19)

We shall use the notation B ⊂ A for conformal subsystems. Note that B is not in general a conformal net in the precise sense of the definition since it does not satisfy property (vi) (cyclicity of the vacuum) unless B = A. However one gets a conformal restricting the algebras B(I ), I ∈ I, and the representation U to the closure HB net B ) is an isomorphism for every of B(S 1 ). Since the map b ∈ B(I ) → b|HB ∈ B(I I ∈ I, because of the Reeh-Schlieder property, as usual, we shall often use the symbol specifying, if necessary, when B acts on HA or on HB . B instead of B, Let π be the defining representation of the conformal net B ⊂ A on the Hilbert space HA (i.e. the restriction to B of the vacuum representation of A). Because of the separability of HA , for every I0 ∈ I we can find a representation θ on the vacuum Hilbert space HB of B, which is unitarily equivalent to π and is localized in I0 . Then if I0 ⊂ I ∈ I, θI is a dual canonical endomorphism for the subfactor B(I ) ⊂ A(I ), namely there is a canonical endomorphism (in the sense of [41]) for the latter whose restriction to B(I ) coincides with θI , see [44, Prop. 3.4] and [43, Sect. 3.3]. 2.4. Virasoro nets and their representations. Let Vir denote the Virasoro algebra, i.e. the complex Lie algebra spanned by Ln , n ∈ Z and a central element κ with relations [Ln , Lm ] = (n − m)Ln+m + δn+m,0

n3 − n κ. 12

(20)

We shall denote L(c, h) the unique positive energy irreducible unitary representation of Vir with lowest weight h and central charge c (see e.g. [14, 30]). The conformal Hamiltonian L0 is diagonalizable on the corresponding Vir-module (still denoted L(c, h)) with spectrum contained in h + N0 2 and containing the lowest weight h. Moreover, the central element κ acts as multiplication by the real number c. Positivity of the energy implies h ≥ 0 and unitarity (or hermiticity) means that there is a positive definite sesquilinear form (·, ·) on L(c, h) such that (ξ, Ln ψ) = (L−n ξ, ψ),

(21)

for ξ, ψ ∈ L(c, h), n ∈ Z. The above conditions give restrictions on the values of the pair (c, h). In fact either c ≥ 1 and h ≥ 0 or we have a pair (c(m), hp,q (m)), m ∈ N, where 6 (m + 2)(m + 3)

(22)

((m + 3)p − (m + 2)q)2 − 1 , 4(m + 2)(m + 3)

(23)

c(m) = 1 − and hp,q (m) = 2

In this paper N0 (resp. N) denotes the set of nonnegative (resp. positive) integers.


267

p = 1, ..., m + 1, q = 1, ..., p, (discrete series representations). For later convenience we shall denote D ⊂ [ 21 , 1) the set of discrete values of the central charge in Eq. (22). Accordingly the set of allowed values of the central charge is D ∪ [1, ∞). Now let H(c, h) be the Hilbert space completion of the module L(c, h). Then the Virasoro algebra acts on H(c, h) by unbounded operators on the common invariant domain L(c, h) ⊂ H(c, h) which can in fact be identified with the subspace Hf in (c, h) of finite energy vectors, i.e. the linear span of the eigenvectors of the conformal Hamiltonian. The (chiral) energy-momentum tensor T(c,h) (z), z = eiϑ ∈ S 1 associated to L(c, h), is defined by the formal power series T(c,h) (z) = Ln z−n−2 . (24) n∈Z

For a function on S 1 , ϑ → f (eiϑ ) with finite Fourier series (trigonometric polynomial), the operator T(c,h) (f ) = L n fn , (25) n∈Z

where

2π

fn = 0

dϑ −inϑ f (eiϑ ), e 2π

(26)

belongs to Vir and hence is well defined on Hf in (c, h) and leave it invariant. Also the following (formal) notation is used

T(c,h) (f ) =

S1

zdz T(c,h) (z)f (z) = 2πi

2π 0

dϑ T(c,h) (eiϑ )ei2ϑ f (eiϑ ). 2π

(27)

The Virasoro net A(Vir,c) can be defined as in [5] as the net generated by the energymomentum tensor Tc (z) := T(c,0) (z) in the representation of lowest weight 0 on H(c, 0) =: HA(Vir,c) . First of all one can show that the map f → Tc (f ) extends (uniquely) to an operator valued distribution (Wightman field) on the invariant domain C ∞ (L0 ), the subspace of smooth vectors for L0 . Moreover the linear energy-bounds established in [5] (also cf. [24]) imply that for every smooth real valued function f , Tc (f ) is essentially self-adjoint (on any core for L0 ) and that eiTc (f1 ) commutes with eiTc (f2 ) when the real smooth functions f1 and f2 have disjoint supports (in fact these properties also hold in the representations with h > 0). It follows that the net of von Neumann algebras defined by A(Vir,c) (I ) = {eiTc (f ) : f ∈ C ∞ (S 1 ), real, supp f ⊂ I } , I ∈ I.

(28)

is local, and in fact one can verify all the other axioms of a conformal net. In particular the representation U of PSL(2, R) is obtained by integrating the self-adjoint part of the (complex) Lie subalgebra of Vir spanned by L−1 , L0 , L1 and the vacuum vector is the (normalized) lowest weight vector in L(c, 0). An alternative construction is obtained by integrating the representations L(c, 0) of Vir to the corresponding projective unitary representations of Diff + (S 1 ). In fact as shown by Goodman and Wallach [24] (cf. also [53]), for each allowed pair (c, h) there

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is a unique strongly continuous projective unitary representation V(c,h) of Diff + (S 1 ) on H(c, h) satisfying V(c,h) (exp(f )) = p(eiT(c,h) (f ) )

(29)

for every real smooth function (vector field) f on S 1 . Here exp(f ) ∈ Diff + (S 1 ) denotes the exponential of the vector field f , namely t → exp(tf ) is the unique one-parameter group of diffeomorphisms generated by f , and p : U(H(c, h)) → U(H(c, h))/T denotes the quotient map. Then the net A(Vir,c) can be defined by A(Vir,c) (I ) = {V(c,0) (γ ) : γ ∈ Diff(I )} ,

(30)

I ∈ I. The two definitions are equivalent because the group generated by the exponentials of smooth vector fields with support in I ∈ I is dense in Diff(I ), see [40, Sect. V.2]. From the second one the diffeomorphism covariance of the Virasoro nets is explicit. As a consequence of the finiteness of the (vacuum) Virasoro characters χ (t) := Tr(t L0 ) for every t ∈ (0, 1) the Virasoro nets are split for every allowed value of the central charge. For c ≤ 1 the Virasoro nets are strongly additive [33, 55] while for c > 1 they are not [5]. We now discuss some properties of the representation theory of the Virasoro nets that we shall need in the following. Let H(c, h) be the Hilbert space completion of Vir module L(c, h) as at the beginning of this subsection and let T(c,h) (z) be the corresponding energy-momentum tensor. A representation of A(Vir,c) on H(c, h) will be denoted πhc if for every I ∈ I and every real smooth real function f on S 1 with support in I , the following holds: πhc I (eiTc (f ) ) = eiT(c,h) (f ) .

(31)

It is immediate to verify that if a representation satisfying Eq. (31) exists, then it is unique. More complicated is to demonstrate the existence of such representations. Of course the vacuum representation π0c exists for every allowed value of c i.e. for each c ∈ D ∪ [1, ∞). If c < 1 and h is a corresponding allowed value of the lowest weight, then the representation πhc exists as a consequence of the Goddard, Kent, Olive coset construction [23] and the local equivalence of positive energy representations of the loop groups LSU(2) at fixed level [22, 54], cf. [40, V.3.3.2] and [33, Sect. 3]. If c ≥ 1 the existence of πhc has been proved by D. Buchholz and H. Schulz-Mirbach for every h ≥ (c − 1)/24. Finally if c ∈ (D + 1) ∪ [2, ∞), then c − 1 is an allowed value of the central charge. Then using the embedding A(Vir,c) ⊂ A(Vir,c−1) ⊗ A(Vir,1) , one can easily construct, for every h ≥ 0, the representation πhc as a subrepresentation of the restriction to A(Vir,c) of π0c−1 ⊗ πh1 .3 As far as we know the existence of the representation πhc for the remaining allowed pairs (c, h) is still an important open problem. Proposition 2.1. If π is an irreducible covariant representation with positive energy of the Virasoro net A(Vir,c) on a separable Hilbert space Hπ then it is unitarily equivalent to πhc for some h ≥ 0. 3

I learned this argument in an unpublished manuscript of D. Buchholz [6].


269

Proof. Let V(c,0) be the unique projective unitary representation of Diff + (S 1 ) on HA(Vir,c) such that Eq. (29) holds with h = 0. From [37, Sect. 2] (cf. also [33, Lemma 3.1]) we know that there is a strongly continuous positive energy projective unitary representation Vπ of Diff + (S 1 ) on Hπ such that p(πI (V(c,0) (γ ))) = Vπ (γ ) for each I ∈ I and γ ∈ Diff(I ). Then it follows from the irreducibility and local normality of π that Vπ is f in irreducible. As a consequence of Theorem A.1 in the Appendix , there is on Hπ a positive energy representation Rπ of theVirasoro algebra with central charge c ∈ D∪[1, ∞), which is unitarily equivalent to L(c , h) for some h ≥ 0. Let T π (z) =

Lπn z−n−2

n∈Z

be the corresponding energy-momentum tensor. Then, for every real smooth vector field f on S 1 , we have Vπ (exp(f )) = p(eiT

π (f )

).

It follows that if I ∈ I and the support of f is contained in I , πI (eiTc (f ) ) = eiαI (f ) eiT

π (f )

,

where αI (f ) is a real constant. Now, it is fairly easy to check that there is a (necessarily unique ) distribution α such that α(f ) = αI (f ) for every I ∈ I and every real function f with support is contained in I and that Möbius covariance implies that α = 0. Hence we have the equality πI (eiTc (f ) ) = eiT

π (f )

,

which implies c = c. The conclusion then follows because the representation Rπ of Vir is unitarily equivalent to L(c, h) for some h ≥ 0. We conclude this subsection with the following proposition. Proposition 2.2. Let π be a positive energy covariant representation of the Virasoro net A(Vir,c) on a separable Hilbert space and let Uπ be the corresponding unique inner R). Assume that Uπ (˜r (2π )) ∈ C1. Then the following unitary representation of PSL(2, hold: (a) The representation π is a direct sum of irreducible covariant positive energy representations. (b) There exists a unique strongly continuous projective unitary representation Vπ of Diff + (S 1 ) on Hπ satisfying p(πI (eiTc (f ) )) = Vπ (exp(f )),

(32)

for every I ∈ I and every real smooth function f with support contained in I . Moreover, this representation satisfies Vπ (γ ) ∈ πI (A(Vir,c) (I )) ∀I ∈ I, ∀γ ∈ Diff(I ),

(33)

R). Vπ (q(g)) = p(Uπ (g)) ∀g ∈ PSL(2,

(34)

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S. Carpi

Proof. The net A(Vir,c) as the split property and hence, as a consequence of [35, Prop. 56], π has a direct integral decomposition ⊕ π= πλ dµ(λ), X

where, for almost every λ, πλ is an irreducible representation of A(Vir,c) on a separable R) we also have Hilbert space H(λ). Since Uπ (g) ∈ π(A(Vir,c) ) for each g ∈ PSL(2, the decomposition Uπ (g) =

⊕

Uλ (g)dµ(λ). X

If hπ is the lowest eigenvalue of Lπ0 we have by assumption Uπ (˜r (2π )) = e2πihπ . Hence Uλ is, for almost every λ, a positive energy representation satisfying Lλ0 ≥ hπ and Uλ (˜r (2π )) = e2πihπ . It follows that, for almost every λ, πλ is an irreducible covariant representation of A(Vir,c) with positive energy which, because of Prop. 2.1, is unitarily equivalent to πhcπ +nλ for some nλ ∈ N0 . Now let Xn = {λ ∈ X : πλ πhcπ +n }. Then, it follows from [35, Lemma 60], that {X n : n ∈ N0 } is a family of pairwise disjoint measurable subsets of X such that µ(X\ n∈N0 Xn ) = 0. Hence ⊕ π πλ dµ(λ), n∈N0 Xn

⊕ and since Xn πλ dµ(λ) is unitarily equivalent to a (possibly zero) multiple of πhcπ +n , (a) follows. Now it follows from (a) and Prop.2.1 that on the dense subspace C ∞ (Lπ0 ) of smooth vectors for Lπ0 there is a projective representation η of the Lie algebra of smooth real vector fields on S 1 by essentially skew-adjoint operators satisfying eη(f ) = πI (eiTc (f ) ) if I ∈ I and suppg ⊂ I . Moreover η satisfies the assumptions in [53, Theorem 5.2.1] (cf. the proof of [53, Theorem 6.1.1] and the discussion in [38, Appendix]). Hence it can be integrated to a unique strongly continuous projective unitary representation of the covπ ering group of Diff + (S 1 ) which, since by assumption e2πiL0 = e2πihπ , factors through Diff + (S 1 ) giving a representation Vπ satisfying Eqs. (32) and (34). The remaining claim in (b) then follows easily. 3. Local Extensions Definition 3.1. We define a local extension of a conformal net A to be a conformal net (B, U, HB ) together with a conformal subsystem C ⊂ B such that the corresponding conformal net C on HC is isomorphic to A and such that U (PSL(2, R)) ⊂ C(S 1 ).

(35)

In agreement with the notation for conformal subsystems, since A and Care isomorphic, we shall often identify A and C and accordingly we shall write A ⊂ B instead of C ⊂ B. Condition (35) implies that C ⊂ B is a full subsystem, namely that C(S 1 ) ∩ B(I ) = C1 I ∈ I.

(36)


271

It prevents trivial extensions of the type A ⊂ A ⊗ C, cf. [3]. For finite index subsystems condition (35) is automatically satisfied and we don’t know any example of a full conformal subsystem violating it. Note that in the literature the term “local extension” is often used in a weaker sense (see e.g. [44]). A class of examples of local extensions is obtained by considering fixed point subsystems under compact group actions. More precisely given a conformal net B and a strongly compact group G of (vacuum preserving) internal symmetries of B one can define the fixed point subsystem A ≡ B G . This kind of construction is paradigmatic in the algebraic approach to quantum field theory, see [15, 18]. One has A(S 1 ) = G (cf. [18, Theorem 3.6]) and since U and G commute (see [22, Lemma 2.22]), condition (35) is satisfied. Hence B is a local extension of A in the sense of Definition 3.1. If π is the identical representation of A on HB one has π= d(ξ )πξ , (37) ˆ ξ ∈G

ˆ is the set of equivalence classes of irreducible unitary representations of G, where G the πξ are mutually inequivalent irreducible covariant representations of A (with trivial univalence) appearing with multiplicity d(ξ ) equal to the dimension of the representations of G of class ξ and satisfying d(πξ ) = d(ξ ), see [18, 28, 47] and [41, I Sect. 7]. Moreover, the vacuum representation of A is associated to the trivial one dimensional representation of G and the corresponding Hilbert space HA coincides with the subspace of G-invariant vectors of HB . We denote by ≡ B the semigroup of the DHR endomorphisms of A0 which are ˆ Then the (DHR) unitarily equivalent to a finite direct sum of representations πξ , ξ ∈ G. braiding on gives in fact a permutation symmetry, is a dual of G in the sense of Doplicher-Roberts duality theory [16, 17] and one can recover the local extension B by the Doplicher-Roberts reconstruction theorem [18], see [47, Prop. 3.8] for the necessary adaptations to conformal nets on S 1 . More generally, let A be a conformal net on S 1 and let be a semigroup of DHR endomorphisms of A0 , all covariant with finite dimension. Assume that the DHR braiding on is in fact a permutation symmetry (para-Bose statistics for the endomorphisms in ) and that is specially directed in the sense of [16, Sect. 5]. Then DoplicherRoberts construction provides a local extension B of A and a strongly compact group G of vacuum preserving internal symmetries of B such that A coincides with the fixed point net B G . Note that by [17, Theorem 3.4] (see also [18, Lemma 3.7]) if has direct sums, subobjects and conjugates then it is specially directed. In the following we shall use the notation B = A for the net obtained through the above Doplicher-Roberts cross product construction. The decomposition in Eq. (37) suggests the following generalization of the local extensions with compact group action discussed above, cf. [44, Sect. 5]. Definition 3.2. A local extension B of a conformal net A is of compact type if the corresponding representation π of A on HB satisfies π= ni πi , (38) where the πi are (necessarily covariant with positive energy) mutually inequivalent irreducible subrepresentations of π having finite statistical dimension and appearing with multiplicity ni .

272

S. Carpi

Although we did not assume in Definition 3.2 any bound on the multiplicities ni , these turn out to be finite as a consequence of the following proposition, cf. [33, Prop.2.3] and [10] for related results. Proposition 3.3. Let B be a local extension of compact type of a conformal net A on S 1 and let π be the corresponding representation of A on HB . Then the following hold: (a) On HB we have A(I ) ∨ A(I c ) = A(S 1 ),

I ∈ I.

(39)

(b) The local extension B is irreducible, namely A(I ) ∩ B(I ) = C1,

I ∈ I.

(40)

(c) Every irreducible representation of A is contained in π with finite (possibly zero) multiplicity. Proof. Let θ be a representation of A localized in I ∈ I and equivalent to π . Then for J ⊃ I , θJ is a dual canonical endomorphism for the subfactor A(J ) ⊂ B(J ). By assumption π is a direct sum of covariant representations with finite statistical dimension. Hence we can find isometries V i ∈ A(I ), i ∈ N, with orthogonal ranges, sat isfying Ei := Vi Vi ∗ ∈ θ (A(S 1 )) , i∈N Ei = 1 and such that the representations σ i defined by σ i J (·) = Vi ∗ θJ (·)Vi , J ∈ I are irreducible, covariant, localized in I and with finite statistical dimension. If T ∈ θI (A(I )) ∩ A(I ) then Vi ∗ T Vj σ j I (·) = σ i I (·)Vi ∗ T Vj for i, j ∈ N and hence by the equivalence of local and global intertwiners for localized representations with finite dimension [25, Theorem 2.3] we have Vi ∗ T Vj σ j J (·) = σ i J (·)Vi ∗ T Vj for every J ∈ I. It follows that Ei T Ej ∈ θ(A(S 1 )) and hence T ∈ θ (A(S 1 )) . Since T ∈ θI (A(I )) ∩ A(I ) was arbitrary and θ(A(S 1 )) ⊂ θI (A(I )) ∩A(I ) by Haag duality, we conclude that θ (A(S 1 )) = θI (A(I ))∨θI c (A(I c )). Hence π(A(S 1 )) = πI (A(I )) ∨ πI c (A(I c )) which proves (a). Now, recalling that U (PSL(2, R)) ⊂ A(S 1 ), by definition of local extensions, we find C1 = A(S 1 ) ∩ B(I ) and hence (b) follows from (a) and locality. Finally (c) follows from [28, p. 39]. Since the defining extensions of the fixed point nets under compact groups of internal symmetries and the finite index extensions are of compact type we can conclude that (b) of Prop. 3.3 generalizes the irreducibility results for conformal subsystems in [7, Prop. 2.1] and [13, Cor. 2.7] (the latter in the local case). Remark 3.4. If B is a local extension of compact type of a conformal net A on S 1 then it follows from Proposition 3.3 (and its proof) that A(I ) ⊂ B(I ), I ∈ I is an irreducible discrete inclusion of infinite factors in the sense of [28, Def. 3.7]. We now consider the Virasoro net A(Vir,1) with c = 1. By [49, Prop. 4] A(Vir,1) is the fixed point net under the action of SO(3) on the conformal net ASU(2)1 associated to the level one vacuum representation of the loop group LSU(2). The corresponding representation π of A(Vir,1) on HASU(2)1 satisfies (2j + 1)πj12 , (41) π= j ∈N0

where πj12 is the representation of A(Vir,1) with lowest weight j 2 . As a consequence d(πj12 ) = 2j + 1 for each j ∈ N [49, Cor. 6].


273

We can consider the permutation symmetric semigroup of DHR endomorphisms of A(Vir,1) which are localized in some I ∈ I0 and equivalent to a finite direct sum of representations of the type πj12 , j ∈ N0 . Then, as discussed above, ASU(2)1 can be identified with the Doplicher-Roberts cross product A(Vir,1) . Now let B be a local extension of compact type of A(Vir,1) and let π be the corresponding representation of A(Vir,1) on HB . By Prop. 2.1 and [8, Theorem 4.4] every irreducible subrepresentation of π is equivalent to a DHR endomorphism in (note that only subrepresentations with integer lowest conformal energy can appear) and hence π σ i, (42) i∈N

where σi ∈ , for each i ∈ N. The local extensions of A(Vir,1) with the above property have been independently classified by the author (cf. the announcement in [32]) and by Feng Xu [55, Sect. 4.2.2]. The resulting possibilities are described in the following theorem (we outline our original proof below). Theorem 3.5. A local extension B of A(Vir,1) is of compact type if and only if B is isomorphic to a fixed point net AH SU(2)1 for some closed subgroup H of SO(3). Proof. The “if part” is a straightforward consequence of the fact that ASU(2)1 is an extension of compact type of A(Vir,1) , cf. Eq. (41). Now let B be an extension of compact type of A(Vir,1) and let π be the corresponding representation of A(Vir,1) on HB . Given a representation θ of A(Vir,1) localized in I ∈ I0 and unitarily equivalent to π (so that if J ⊃ I , θJ is a dual canonical endomorphism for the inclusion A(Vir,1) (J ) ⊂ B(J )) we deduce from Eq. (42) that θ is equivalent to a (possibly infinite) direct sum of DHR endomorphisms in the permutation symmetric semigroup defined after Eq. (41). It follows that the monodromy operator M (ρ, θ ) := (ρ, θ ) (θ, ρ) is trivial (i.e. equal to 1) for every ρ ∈ . We now use the extension of DHR endomorphisms as defined in [44, Prop. 3.9] (cf. also [50, Sect. 3.4.7]) and which is called α-induction in [2]. For every ρ ∈ , the triviality of the monodromy operator M (ρ, θ ) implies that its extension αρ (we use the notation in [2]) to B0 is still localized in an interval in I0 , see [44, Prop. 3.9]. Now the crucial point is that the functorial properties α-induction (called homomorphism properties in [2]) imply that α := {αρ : ρ ∈ } is still a specially directed permutation symmetric semigroup of Möbius covariant (bosonic) DHR endomorphisms of B0 . These functorial properties have been established in [11, 12] for inclusions of local nets on the four dimensional Minkowski space-time and in [2] for finite index nets of the subfactor on the real line. Due to the triviality of the monodromy (which is automatic in four space-time dimensions) one can use the arguments in [11, 12] (see also [9, Sect. 2] for an overview) to get the desired structure on α . Hence, as recalled at the beginning of this section, we can use the Doplicher-Roberts cross product construction to define a local extension B α of the conformal net B. The next point is that the proof of [12, Theorem 3.5] can be adapted to our situation to show that there is a natural inclusion (up to isomorphism) ASU(2)1 = A(Vir,1) ⊂ B α (compatible with A(Vir,1) ⊂ B ) and in fact it turns out that B α is a local extension of ASU(2)1 . But the latter conformal net has no proper local extensions (see e.g. [3, 33]) and hence we conclude that B α = ASU(2)1 . Accordingly B = AH SU(2)1 for some closed subgroup H of SO(3) as claimed.

274

S. Carpi

The above proof relies on specific properties of the representation category net A(Vir,1) , namely on the fact that the subcategory of representations with finite statistical dimension (and trivial univalence) is permutation symmetric, a fact that appears to be rather exceptional for conformal nets on S 1 . However in the case of local nets on the four dimensional space-time similar ideas have been used by the author and R. Conti to study local extensions in a fairly general context [10]. As matter of fact the above mentioned investigation in [10] inspired our proof of Theorem 3.5. Coming back to conformal nets on S 1 we remark that there are well known local extensions of the Virasoro net with c = 1 which are not conformal subsystems of ASU(2)1 (see e.g. [3] ) and hence are not of compact type as a consequence of Theorem 3.5. However F. Xu has made further progress and classified the local extensions B of the c = 1 Virasoro net such that the corresponding representation of A(Vir,1) on HB contains a subrepresentation equivalent to some πj12 , j ∈ N [55, Theorem 4.6]. The above condition is called the “spectrum condition” in [55] where it is conjectured that all nontrivial extensions of the c = 1 Virasoro net have to satisfy it. This motivates the following definition: Definition 3.6. A local extension B of a conformal net A is maximally non-compact if the corresponding representation π of A on HB satisfies the following condition: the only subrepresentation of finite statistical dimension of π is the vacuum subrepresentation. From the previous discussion we can conclude that a local extension of the c = 1 Virasoro net satisfies Xu’s spectrum condition if and only if it is not maximally noncompact. No examples of maximally non-compact extensions of this net seem to be known. We shall however exhibit in Sect. 5 various examples of maximally non-compact extensions for the Virasoro nets with c > 1. We conclude this section with the following proposition. Proposition 3.7. Let B be a local extension of the Virasoro net A(Vir,c) . Then the following hold: (a) A(Vir,c) (I ) ∩ B(I ) = C1, for every I ∈ I; (b) The net B is diffeomorphism covariant. Proof. Let π be the representation of A(Vir,c) on HB associated with the local extension B. If V is the corresponding strongly continuous projective unitary representation of Diff + (S 1 ) on HB given by (b) of Prop. 2.2, then V (γ ) ∈ A(Vir,c) (I ) if γ ∈ Diff(I ), for each I ∈ I. Moreover, for every g ∈ PSL(2, R) we have V (g) = p(U (g)), where U is the representation, makes B Möbius covariant. Hence, it follows from [38, Theorem 12] that A(Vir,c) (I ) ∩ B(I ) = U (PSL(2, R)) ∩ B(I ) = C1, which proves (a). Now let I be a given interval in I and let γ ∈ Diff + (S 1 ) be such that γ I = I . Since γ preserves the orientation it must keep fixed the boundary points of I. An elementary argument (which we omit here) then shows that for every J ∈ I containing the closure of I we can find a diffeomorphism γ J ∈ Diff(J ) with γ J |I = γ |I , i.e. γ −1 γ J ∈ Diff(I c ). Since V (γ −1 γ J ) ∈ A(Vir,c) (I c ) ⊂ B(I c ), we find V (γ J )B(I )V (γ J )∗ = V (γ )V (γ −1 γ J )B(I )V (γ −1 γ J )∗ V (γ )∗ = V (γ )B(I )V (γ )∗ , and hence V (γ )B(I )V (γ )∗ ⊂ B(J ), for every J ∈ I containing the closure of I . Thus, being conformal nets continuous from outside, we conclude that V (γ )B(I )V (γ )∗ ⊂


275

B(I ). If γ is arbitrary we can always find a g ∈ PSL(2, R) such that gI = γ I . It follows that V (γ )B(I )V (γ )∗ = U (g)V (g −1 γ )B(I )V (g −1 γ )∗ U (g)∗ ⊂ B(gI ) = B(γ I ), and hence, for every I ∈ I, γ ∈ Diff + (S 1 ), we have V (γ )B(I )V (γ )∗ = B(γ I ), and also (b) is proved.

Remark 3.8. If B is a diffeomorphism covariant net on S 1 and V is the corresponding projective unitary representation of Diff + (S 1 ), one can define a covariant subsystem C of B by C(I ) = {V (γ ) : γ ∈ Diff(I )}

I ∈ I.

(43)

Arguing as in the proof of Prop. 2.1 it can be shown that the conformal net C on S 1 is isomorphic to A(Vir,c) for some c ∈ D ∪ [1, ∞). It follows that the correspondence between diffeomorphism covariant nets on S 1 and local extensions of the Virasoro nets is one-to-one, cf. [33]. 4. On the Oscillator Representations of the Virasoro Nets with c > 1 Let (AU(1) , U, HU(1) ) be the conformal net generated by the U(1) chiral current algebra, see [3, 5]. The Hilbert space HU(1) and the net AU(1) can be identified with the Fock space eH1 , where H1 is acted on by the irreducible representation of PSL(2, R) of lowest weight 1, and with the corresponding second quantization net respectively [26]. f in We denote HU(1) the dense subspace of finite energy vectors, i.e. the algebraic direct f in

sum of the L0 eigenspaces. Then HU(1) carries the unique irreducible lowest weight representation of the oscillator (Heisenberg) algebra [Jn , Jm ] = nδn+m,0

m, n ∈ Z, J0 = q1,

(44) (45)

with lowest weight q = 0, see [3 and 30, Sect. 2.2]. The corresponding lowest weight f in vector is the vacuum vector and for ξ, ψ ∈ HU(1) , n ∈ Z we have (ξ, Jn ψ) = (J−n ξ, ψ), q

q

(46)

(hermiticity). Note that defining Jn := Jn , J0 = q1 we obtain a unitary representation of the oscillator algebra with arbitrary lowest weight q ∈ R. The U(1) current J (z), z = eiϑ ∈ S 1 is defined as an operator valued distribution by Jn z−n−1 (47) J (z) = n∈Z

and the common invariant domain for the smeared field operators

dz J (z)u(z) u ∈ C ∞ (S 1 ) J (u) = S 1 2πi

276

S. Carpi

can be chosen to be the subspace C ∞ (L0 ) of smooth vectors for L0 . For a real function u ∈ C ∞ (S 1 ), J (u) is essentially self-adjoint and the unitary operators W (u) := eiJ (u) with u ∈ C ∞ (S 1 ) real, supp u ⊂ I generate AU(1) (I ) for every I ∈ I. Moreover the Weyl relations hold: W (u)W (v) = W (u + v)e−

dz S 1 4π i u (z)v(z)

(48)

,

d u(z) = −ie−iϑ for real smooth functions u, v, where u (z) denotes the derivative dz d iϑ dϑ u(e ). As shown in [3] (see also [4]) for every q ∈ R there is a covariant irreducible representation of AU(1) (BMT-automorphism) γq on HU(1) such that

γq I (W (u)) = eiq

dz −1 S 1 2π i z u(z)

W (u) = eiJ

q (u)

,

for I ∈ I, u ∈ C ∞ (S 1 ) with support in I . Here the field J q (z) is defined by q Jn z−n−1 = J (z) + qz−1 . J q (z) =

(49)

(50)

n∈Z

γq1 and γq2 are inequivalent if q1 = q2 . Moreover, if ϕ is a real smooth function such that −iϕ (z) = z−1 q for z ∈ I then γq I (·) = AdW (−ϕ)(·),

(51)

and hence γq is locally implementable by Weyl unitaries. In fact Eq. (51) can be used to define the representation γq . Note that γ0 is the vacuum representation of AU(1) and that for every I ∈ I, we have γq I (AU(1) (I )) = AU(1) (I ).

(52)

We now come to the oscillator representations of the Virasoro algebra. For λ, q ∈ R, n ∈ Z the operators (λ,q)

Ln

= δn,0

1 q q λ2 q + : J−j Jj +n : +iλnJn , 2 2

(53)

j ∈Z

where the colons denote normal ordering, define a positive energy unitary representaf in tion R(λ, q) of the Virasoro algebra on HU(1) with central charge c = 1 + 12λ2 , see (0,0)

e.g. [30, Sect. 3.4]. Since L0 coincides with L0 (by the Sugawara formula) we have (λ,q) 2 2 = L0 +(λ +q )/2 and hence is a lowest energy vector for these representations L0 with energy (λ2 + q 2 )/2. We associate to the above representations the energy-momentum tensors T (λ,q) (z) defined by (λ,q) T (λ,q) (z) = Ln z−n−2 . (54) n∈Z

Then the following holds (see [21, Remark 4.2]) 1 1 λ2 d q 2 (λ,q) T (z) = : J (z) : −iλ + J q (z) + 2 , 2 z dz 2z

(55)


277

and hence, recalling that J q (z) = J (z) + qz−1 , T (λ,q) (z) =

1 q 1 d λ2 + q 2 . : J (z)2 : + J (z) − iλ + J (z) + 2 z z dz 2z2

For f ∈ C ∞ (S 1 ) the smeared field operator

dz (λ,q) T (λ,q) (f ) = (z)f (z) T S 1 2πi

(56)

(57)

is well defined on the domain C ∞ (L0 ) and leaves it globally invariant. Moreover we see from Eq. (56) that the field T (λ,q) (z) is local with respect to J (z) in the sense that if f, u ∈ C ∞ (S 1 ) have disjoint supports, the operators T (λ,q) (f ) and J (u) commute on C ∞ (L0 ) and that T (λ,q) (f ) is hermitian if f is a real function. Finally, it follows from [5, Sect. 2] (cf. also [24]) that T (λ,q) (f ) is essentially self-adjoint for each real valued (λ,q) (f ) commutes with W (u) if the support smooth function f and that in this case eiT of the real function u is disjoint from the one of f . We now define an isotonous net B (λ,q) on HU(1) by : f ∈ C ∞ (S 1 ), real, supp f ⊂ I } ,

(λ,q) (f )

B (λ,q) (I ) = {eiT

(58)

for I ∈ I. As a consequence of the above discussion and of Haag duality for AU(1) we obtain the following proposition: Proposition 4.1. For every I ∈ I we have B (λ,q) (I ) ⊂ AU(1) (I ).

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The net B (λ,q) so defined it is not in general a conformal subsystem of AU(1) . And in fact it can be shown that B (λ,q) transforms covariantly with respect to the representation U making AU(1) Möbius covariant only for (λ, q) = (0, 0), B (0,0) being the (c = 1) (λ,q) = L0 + (λ2 + q 2 )/2 implies Virasoro subnet of AU(1) . However, the equality L0 2 rotation covariance for every (λ, q) ∈ R , namely U (r(ϑ))B (λ,q) (I )U (r(−ϑ)) = B (λ,q) (r(ϑ)I )

ϑ ∈ R, I ∈ I.

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We shall need the following two lemmata: Lemma 4.2. For every pair (λ, q) ∈ R2 and every I ∈ I the following holds: B (λ,q) (I ) = γq I (B (λ,0) (I )). C ∞ (S 1 )

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−iϕ (z)

z−1 q

Proof. Let ϕ, f ∈ be real functions such that = f in suppf ⊂ I . For ξ, ψ ∈ HU(1) a straightforward calculation shows that

for z ∈ I and

(ξ, W (−ϕ)T (λ,0) (f )ψ) = (T (λ,q) (f )ξ, W (−ϕ)ψ). Since Hf in is a common core for T (λ,0) (f ) and T (λ,q) (f ) it follows that W (−ϕ)eiT

(λ,0) (f )

W (ϕ) = eiT

(λ,q) (f )

,

and hence, recalling Eq. (51), γq I (eiT

(λ,0) (f )

) = eiT

(λ,q) (f )

,

cf. [5, p. 123] and [4, p. 361]. The conclusion then follows from the definition of B (λ,q) (I ) given in Eq. (58).

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Lemma 4.3. The representation R(λ, q) defined after Eq. (53) is irreducible for every λ = 0 and q ∈ R. Proof. The character χ(λ,q) (t), t ∈ (0, 1) of the representation R(λ, q) is given by (λ,q)

χ(λ,q) (t) = Tr(t L0

)=t

λ2 +q 2 2

p(t),

∞

where p(t) = n=1 (1−t n )−1 = Tr(t L0 ) and hence the conclusion follows since, by [30, Eq. (3.15) and Prop. 8.2], it coincides with the character of the irreducible representation L(c, h) of Vir with central charge c = 1 + 12λ2 and lowest weight h = (λ2 + q 2 )/2. Corollary 4.4. Let A(Vir,c) be the Virasoro net with central charge c = 1 + 12λ2 , λ = 0 and let πhc be the (irreducible) representation of A(Vir,c) with lowest weight h = (λ2 + q 2 )/2 as defined in Subsect. 2.4. Then there is a representation π(λ,q) of A(Vir,c) on HU(1) , unitarily equivalent to πhc , such that for every I ∈ I the following holds: π(λ,q) I (A(Vir,c) (I )) = B(λ,q) (I ).

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We shall need the following proposition in the next section. Proposition 4.5. Let A(Vir,c) be a Virasoro net with c > 1. Then, if h ≥ (c − 1)/24 we have d(πhc ) = d(c), where d(c) ∈ [1, ∞] does not depend on h and satisfies d(c) > 1. Proof. The assumption on the range of c and h implies that we can find λ = 0 and q ∈ R such that c = 1 + 12λ2 and h = (λ2 + q 2 )/2. Then it follows from Corollary 4.4 that d(πhc ) = d(π(λ,q) ) and we have to show that the latter does not depend on q. By Eq. (16) and Corollary 4.4 we find d(π(λ,q) )2 = [B(λ,q) (I c ) : B(λ,q) (I )],

I ∈ I.

From Proposition 4.1 and Haag duality for AU(1) it follows that B(λ,q) (I ) ⊂ AU(1) (I ) = AU(1) (I c ) ⊂ B(λ,q) (I c ) , and hence, using the multiplicativity of the minimal index [42] (cf. the proof of [8, Prop. 3.1]), that d(π(λ,q) )2 = [AU(1) (I ) : B(λ,q) (I )] · [AU(1) (I c ) : B(λ,q) (I c )]. Now, using Lemma 4.2 and Eq. (52) we find, for an arbitrary J ∈ I, [AU(1) (J ) : B(λ,q) (J )] = [γq J (AU(1) (J )) : γq J (B(λ,0) (J ))] = [AU(1) (J ) : B(λ,0) (J )], and hence d(π(λ,q) )2 = [AU(1) (I ) : B(λ,0) (I )] · [AU(1) (I c ) : B(λ,0) (I c )] does not depend on q. Finally if d(c) = 1 then, for every I ∈ I, π(λ,q) (A(Vir,c) (I )) = AU(1) (I ), which is impossible since AU(1) is strongly additive (see [5, 26]) while A(Vir,c) it is not.


279

5. Sectors with Infinite Dimension and Maximally Non-Compact Local Extensions Let D ⊂ [ 21 , 1) be the set of allowed values of the central charge in the discrete series representations of Vir as defined in Subsect. 2.4 and let c ∈ (D + 1) ∪ [2, ∞). Then c − 1 is an allowed value of the central charge and the tensor product net A(Vir,c−1) ⊗ ASU(2)1 is a local extension of A(Vir,c) . The representation of A(Vir,1) on HASU(2)1 contains the irreducible lowest weight representation πj12 , j ∈ N0 with multiplicity 2j + 1 (see Eq. (41)) and hence the multiplicity m(c, j ) of πjc2 in the representation of A(Vir,c) on HA(Vir,c−1) ⊗ HASU(2)1 satisfies m(c, j ) ≥ 2j + 1 for every j ∈ N0 . We are now ready to prove the following theorem, cf. [8, Theorem 4.4] and the guess in [49, Sect. 2]. Theorem 5.1. If c ∈ (D + 1) ∪ [2, ∞) and h ≥ (c − 1)/24 then d(πhc ) = ∞. Proof. Let π be the representation of A(Vir,c) in HA(Vir,c−1) ⊗ HASU(2)1 as described above. Then, as explained in Sect. 2, π is unitarily equivalent to a representation θ on HA(Vir,c) localized in an interval I0 ∈ I and for every I ∈ I with I0 ⊂ I θI is a dual canonical endomorphism for the inclusion A(Vir,c) (I ) ⊂ A(Vir,c−1) (I ) ⊗ ASU(2)1 (I ), which is irreducible because of Prop. 3.7. Now let ρjc2 be a representation of A(Vir,c) on HA(Vir,c) , unitarily equivalent to πjc2 and localized in I0 and let I ∈ I be an interval containing I0 . As shown just before the statement of this theorem the multiplicity m(c, j ) of the representation ρjc2 in θ satisfies m(c, j ) ≥ 2j + 1. Hence (by Haag duality) the endomorphism ρjc2 is contained in θI with multiplicity n(c, j ) ≥ 2j + 1 for each I √ j ∈ N0 . Now, it follows from Prop. 4.5 that d(ρjc2 ) = d(c), for each j ≥ (c − 1)/24, where d(c) does not depend on j . Let us assume that d(c) < ∞. Then √ by [25, Cor. 2.10] ρjc2 is an irreducible endomorphism of A(Vir,c) (I ) for every j ≥ (c − 1)/24 and by I √ [28, p. 39] we conclude that 2j + 1 ≤ n(c, j ) ≤ d(c)2 for every j ≥ (c − 1)/24, in contradiction with the assumption d(c) < ∞. Hence d(c) = ∞ and the conclusion follows from Prop. 4.5. Corollary 5.2. If c ∈ (D + 1) ∪ [2, ∞) and B is a local extension of compact type of A(Vir,c) then the index [B : A(Vir,c) ] is finite. Proof. Let π be the representation of A(Vir,c) on HB defined by the local extension B. Only representations with integer lowest weight can appear in the decomposition of π . But there are only a finite number of positive integers m satisfying m < (c − 1)/24 and hence, by Theorem 5.1, only a finite number of irreducible DHR sectors can appear in the decomposition of π. Now, recalling that the inclusion A(Vir,c) (I ) ⊂ B(I ), I ∈ I, is irreducible, the conclusion follows from (the proof of) [33, Prop. 2.3]. Now let G be a simply connected compact Lie group with simple Lie algebra Lie(G) and let k be a positive integer. We denote by AGk the conformal net associated to the vacuum representation of the corresponding Loop group (or affine Lie algebra) at level k (see [22, 49, 52, 54]). As it is well known, the Sugawara formula (see e.g. [14, Sect. 15.2] and [30, Sect.10.1]), implies that the net AGk is a local extension of the Virasoro net A(Vir,c) with central charge c ≡ c(Gk ) =

dim(G)k , k + h∨

(63)

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where h∨ is the dual Coxeter number of Lie(G), cf. [22, Sect. III.7] and [49, Sect.1]. The central charge c(Gk ) is bounded by r ≤ c(Gk ) ≤ dim(G),

(64)

where r is the rank of Lie(G) and the lower bound is saturated only for simply laced Lie algebras at level k = 1. Note that c(Gk ) < 2 implies that r = 1 and thus that G = SU(2). In the latter case we have c(SU(2)k ) = 3k/(k + 2). If k ≥ 4 we have c(SU(2)k ) ≥ 2. The remaining possibilities are c(SU(2)1 ) = 1, c(SU(2)2 ) = 1 + 1/2 and c(SU(2)3 ) = 1 + 4/5. We summarize the above discussion in the following lemma. Lemma 5.3. If Gk = SU(2)1 then c(Gk ) ∈ (D + 1) ∪ [2, ∞). Recall that there is a strongly continuous representation of G in the (unitary) group of internal symmetries of AGk leaving the vacuum invariant. This representation is not in general faithful and its kernel coincides with the (finite) center Z(G) of G. It is known that the fixed point net AG Gk satisfies A(Vir,c) ⊂ AG Gk ⊂ AGk ,

c = c(Gk ),

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see [49]. In particular, being G/Z(G) infinite, the index [AGk : A(Vir,c) ] is infinite. Corollary 5.4. If Gk = SU(2)1 then the local extension AGk of A(Vir,c) , c = c(Gk ), is not of compact type. Proof. Due to Lemma 5.3 we can apply Corollary 5.2 and the conclusion follows from [AGk : A(Vir,c) ] = ∞.

The following consequence of Corollary 5.4 has been pointed out by K.-H. Rehren in [49] with a different argument based on the comparison of characters. It can also be proved using [55, Theorem 2.4] and the fact that A(Vir,c) is not strongly additive when c > 1. Corollary 5.5. If Gk = SU(2)1 , then the inclusion A(Vir,c) ⊂ AG Gk is proper. The next result shows that maximally non-compact local extensions naturally appear for the Virasoro nets with c > 1. Proposition 5.6. If Gk = SU(2)1 and c = c(Gk ) ≤ 25 then AGk is a maximally non-compact local extension of A(Vir,c) . Proof. The representation π of A(Vir,c) in HAGk can only have irreducible subrepresentations with a nonnegative integer lowest weight. Since by assumption (c − 1)/24 ≤ 1, it follows from Theorem 5.1 that the only subrepresentation π with finite dimension is the vacuum representation. Hence the extension is maximally non-compact. For SU(N ) h∨ = N and hence c(SU(N )k ) = k(N 2 − 1)/(N + k) and we see that Prop. 5.6 gives an infinite series of maximally non-compact extensions of the c > 1 Virasoro nets. Examples are: SU(2)k , k > 1; SU(3)k , SU(4)k , SU(5)k , k arbitrary; SU(N )1 , 2 < N ≤ 26. Actually the same proof of Prop. 5.6, together with Prop. 2.1, gives the following stronger result. Theorem 5.7. If c ∈ (1 + D) ∪ [2, 25] then every local extension of the Virasoro net A(Vir,c) is maximally non-compact. In particular A(Vir,c) has no local extensions of compact type.


281

A. Appendix In this appendix we give a differentiability result for the representations of Diff + (S 1 ) which is used in the proof of Prop. 2.1. This result has been essentially obtained by T. Loke [40] (cf. also [54] for analogous results for loop groups) and here we consider the necessary modifications we need in this paper. We shall closely follow the discussion in [40, Chap. I]. An element of the group Mob of Möbius transformations of S 1 is given by a map z → αz+β , where α, β are complex numbers satisfying |α|2 − |β|2 = 1. Mob is a Lie βz+α subgroup of Diff + (S 1 ) isomorphic to PSL(2, R). The corresponding Lie subalgebra of Vect(S 1 ) is spanned by the vector fields x := − sin ϑ

d d d , y := − cos ϑ , h := , dϑ dϑ dϑ

(66)

whose brackets are given by [h, x] = −y, [h, y] = x, [x, y] = h.

(67)

More generally, for each n ∈ N, the vector fields 1 1 1 d d d , yn := − cos nϑ , hn := , xn := − sin nϑ n dϑ n dϑ n dϑ

(68)

span isomorphic Lie subalgebras of Vect(S 1 ) each associated to a Lie subgroup Mobn of Diff + (S 1 ). Clearly Mob1 = Mob and it is not hard to see that, for each n > 1, Mobn is isomorphic to an n-fold covering of PSL(2, R) Mob and that the corresponding covering map transforms the one-parameter group exp(thn ) into the one-parameter subgroup r(t) of rotations of PSL(2, R). Now let V be a strongly continuous projective unitary representation of Diff + (S 1 ) on a separable Hilbert space. For every n ∈ N, the restriction of V to Mobn lifts to a R). Note that exp(2π hn )n = 1 strongly continuous unitary representation Un of PSL(2, n n and hence Un (˜r (2π)) = Un (exp(2πhn )) = χn 1 for a suitable complex number χn of modulus one. In particular U (˜r (2π)) has finite spectrum for each n ∈ N. Now let n1 Xn , n1 Yn and ni (L0 +cn ), cn ∈ R, c1 = 0, be the skew-adjoint generators of the one-parameter groups of unitaries Un (exp(txn )), Un (exp(tyn )), and Un (exp(thn )), respectively. On the dense subspace Dn ⊂ H of C ∞ vectors for the representation Un the above operators define a representation of the Lie algebra (67) and hence we have on Dn , [iL0 , Xn ] = −nYn , [iL0 , Yn ] = nXn , [Xn , Yn ] = in(L0 + cn ), n ∈ N.

(69)

If V is a positive energy representation, since the unitary operator ei2πL0 acts as multiplication by a complex number, the spectrum of L0 is pure point and every eigenvalue is of the form h + n, where h ≥ 0 is the lowest eigenvalue of L0 and n is a nonnegative integer. Now let Hf in be the linear span of the eigenvectors of L0 . Loke has shown in [40, Sect. I.1] that if a positive energy representation V is such that the eigenspaces of L0 are all finite-dimensional, then Dn . (70) Hf in ⊂ n∈N

282

S. Carpi

Moreover he proved that the operators L0 , Ln := iYn − Xn and L−n := iYn + Xn , n ∈ N define a unitary representation of Vir on Hf in and that the corresponding energymomentum tensor Ln z−n−2 (71) T (z) = n∈Z

extends to an operator valued distribution on the subspace of smooth L0 vectors such that T (f ) is essentially self-adjoint on Hf in for each f ∈ Vect(S 1 ) and satisfies p(eiT (f ) ) = V (exp(f )).

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The finite dimensionality of the L0 eigenspace is used in [40] to infer that for each n ∈ N, R) the representation Un is a direct sum of positive energy representations of PSL(2, and that Dn ⊃ Hf in . However these facts hold for every positive energy representation V , as a consequence of the proposition below (applied to each representation Un ) and hence the results of Loke described above hold (without any essential modification in the proofs) also if the finite dimensionality of the eigenspaces of L0 is not assumed. Moreover the representation of Vir on Hf in so obtained can be seen to be irreducible (and hence unitarily equivalent to some L(c, h), cf. [30, Remark 3.5]) if and only if the corresponding projective representation V of Diff + (S 1 ) is irreducible, cf. Lemma 2.2. in [40, Sect. I.2]. R) Proposition A.1. Let U be a strongly continuous unitary representation of PSL(2, on a separable Hilbert space H and let L0 be the self-adjoint generator of the restriction of U to the lifting r˜ (t) of the one-parameter rotation subgroup of PSL(2, R). Assume that the spectrum of L0 is bounded from below and that the one of U (˜r (2π )) is finite. Then the following hold: (a) U is a positive energy representation (i.e. L0 has a nonnegative spectrum) and it is completely reducible to a direct sum of irreducible subrepresentations. (b) Every eigenvector of L0 is a smooth vector for the representation U . Proof. If U is assumed to be irreducible then the positivity of the energy follows from the bound on the spectrum of L0 as a consequence of the classification of the irreducible R) [48] (cf. [40, Sect. I.1.3]) and hence the positive energy representations of PSL(2, condition for U follows in general by direct integral decomposition. Then (a) follows e.g. from [36, Lemma 8]. As a consequence there is an increasing sequence 0 = n1 < n2 ... of nonnegative integers (which is possibly finite) and a decomposition Hk H= k

such that the restriction of U to Hk is a (possibly infinite) multiple of an irreducible representation with lowest weight h + nk . Hence if ψ is an eigenvector of L0 corresponding to the eigenvalue λ we can write (ψk , ψ)ψk , ψ= nk ≤λ−h

where ψk ∈ Hk is a normalized eigenvector of L0 . Since every eigenvector of the generator of rotations in an irreducible representation R) is smooth (see e.g. [48, Sect. I.1]), each ψk is smooth and hence ψ is of PSL(2, smooth vector for the representation U so that also (b) is proved.


283

We can summarize the discussion in this appendix in the following theorem, cf. [40, Sect. I.2.4]. Theorem A.1. Let V be a strongly continuous positive energy projective unitary irreducible representation of Diff + (S 1 ) on a (necessarily separable) Hilbert space H. Then V is unitarily equivalent to the unique projective unitary representation V(c,h) which integrates the Vir-module L(c, h) for some c > 0, h ≥ 0. In particular the corresponding generator of rotations L0 has finite-dimensional eigenspaces. Acknowledgements. The author would like to thank R. Conti, S. Köster and R. Longo for discussions, explanations and comments. Theorem 3.5 has been announced at the Miniworkshop “Conformal Field Theory. An Introduction” held in Rome in March 2003. The author thanks the organizers D. Guido and (again) R. Longo for the invitation.

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Communications in


Extremal Projectors of q-Boson Algebras Toshiki Nakashima Department of Mathematics, Sophia University, Tokyo 102-8554, Japan. E-mail: [email protected] Received: 9 July 2002 / Accepted: 26 July 2003 Published online: 13 November 2003 – © Springer-Verlag 2003

Abstract: We define the extremal projector of the q-boson Kashiwara algebra Bq (g) and study their basic properties. Applying their properties to the representation theory of the category O(Bq (g)), whose objects are “upper bounded” Bq (g)-modules, we obtain its semi-simplicity and the classification of simple modules. 1. Introduction In [3], we studied the so-called q-boson Kashiwara algebra, in particular, a kind of q-vertex operators and their 2 point functions. We found therein an interesting object . But at that time we did not reveal its whole properties, as the “Extremal Projector”. Tolstoy,V.N., et al., introduced the notion of “Extremal Projectors” for Lie (super)algebras and quantum (super) algebras, and made extensive study of their properties and applied them to representation theory, (see [2, 7, 8]) and references therein). In the present paper, we shall re-define the extremal projector for the q-boson algebras, clarify its properties and apply it to the representation theory of q-boson algebras. To be more precise, let {ei , fi , q h | i ∈ I, h ∈ P ∗ } be the generators of the q-boson algebra Bq (g). The extremal projector is an element in B q (g)(some completion of Bq (g)) which satisfies the following; ei = fi = 0, 2 = , ak bk = 1, k

for some ak ∈ Bq+ (g) and bk ∈ Bq− (g) (see Theorem 5.2). Let O(B) be the category of “upper bounded” Bq (g)-modules (see Sect. 3). By using the above properties of , we

shall show that the category O(B) is semi-simple and classify its simple modules. In [1], Kashiwara gave the projector P for the q-boson algebra of sl2 -case in order to define the crystal base of Uq− (g). He used it to show the semi-simplicity of O(Bq (sl2 )). So our is a generalization of his projector P to arbitrary Kac-Moody algebras.

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The organization of this article is as follows: In Sect.2, we review the definitions of quantum algebras and q-boson Kashiwara algebras and their properties. In Sect.3, we introduce the category of modules of the q-boson algebras O(B), which we treat in the sequel. In Sect.4, we review the so-called Drinfeld Killing form and using it we define some element C in the tensor product of q-boson algebras, which plays a significant role of studying extremal projectors. In Sect.5, we define extremal projectors for the q-boson algebras and involve their important properties. In the last section, we apply it to show the semi-simplicity of the category O(B) and classify the simple modules in O(B). In [3] we gave the proof of its semi-simplicity, but there was a quite big gap. Thus, the last section is devoted to closing that gap. We can find an elementary proof of the semi-simplicity of the category O(B) in e.g.[9]. 2. Quantum Algebras and q-Boson Kashiwara Algebras We shall define the algebras playing a significant role in this paper. First, let g be a symmetrizable Kac-Moody algebra over Q with a Cartan subalgebra t, {αi ∈ t∗ }i∈I the set of simple roots and {hi ∈ t}i∈I the set of coroots, where I is a finite index set. We define an inner product on t∗ such that (αi , αi ) ∈ Z≥0 and hi , λ = 2(αi , λ)/(αi , αi ) for λ ∈ t∗ . Set Q = ⊕i Zαi , Q+ = ⊕i Z≥0 αi and Q− = −Q+ . We call Q a root lattice. Let P be a lattice of t∗ i.e. a free Z-submodule of t∗ such that t∗ ∼ = Q ⊗Z P , and P ∗ = {h ∈ t|h, P ⊂ Z}. Now, we introduce the symbols {ei , ei , fi , fi (i ∈ I ), q h (h ∈ P ∗ )}. These symbols satisfy the following relations:

q 0 = 1, and q h q h = q h+h , q h ei q −h = q h,αi ei , q h ei q −h = q h,αi ei ,

(2.1) (2.2) (2.3)

q h fi q −h = q −h,αi fi , q h fi q −h = q −h,αi fi ,

(2.4) (2.5)

[ei , fj ] = δi,j (ti − ti−1 )/(qi − qi−1 ),

h ,α ei fj = qi i j fj ei + δi,j , h ,α fi ej = qi i j ej fi + δi,j , 1−hi ,αj (1−hi ,αj −k) (k) (−1)k Xi Xj Xi k=0 for Xi = ei , ei , fi , fi ,

(2.6) (2.7) (2.8)

= 0, (i = j ),

(2.9)

where q is transcendental over Q and we set qi = q (αi ,αi )/2 , ti = qihi , [n]i = (qin − (n) qi−n )/(qi − qi−1 ), [n]i ! = nk=1 [k]i and Xi = Xin /[n]i !. Now, we define the algebras Bq (g), B q (g) and Uq (g). The algebra Bq (g) (resp. B q (g)) is an associative algebra generated by the symbols {ei , fi }i∈I (resp. {ei , fi }i∈I ) and q h (h ∈ P ∗ ) with the defining relations (2.1), (2.3), (2.4), (2.7) and (2.9) (resp. (2.1), (2.2), (2.5), (2.8) and (2.9)) over Q(q). The algebra Uq (g) is the usual quantum algebra generated by the symbols {ei , fi }i∈I and q h (h ∈ P ∗ ) with the defining relations (2.1),(2.2),(2.4), (2.6) and (2.9) over Q(q). We shall call algebras Bq (g) and B q (g) the q-boson Kashiwara algebras ([1]). Furthermore, we define their subalgebras

Extremal Projectors of q-Boson Algebras

287

T = q h |h ∈ P ∗ = Bq (g) ∩ B q (g) ∩ Uq (g), ∨

Bq∨ (g) (resp. B q (g)) = ei , fi (resp. ei , fi )|i ∈ I ⊂ Bq (g) (resp. B q (g)), +

Uq+ (g) (resp. Uq− (g)) = ei (resp. fi )|i ∈ I =: B q (g) (resp. Bq− (g)), Uq≥ (g) (resp. Uq≤ (g)) = ei (resp. fi ), q h |i ∈ I, h ∈ P ∗ , −

∨

Bq+ (g) (resp. B q (g)) = ei (resp. fi )|i ∈ I ⊂ Bq∨ (g) (resp. B q (g)), ≤

Bq≥ (g) (resp. B q (g)) = ei (resp. fi ), q h |i ∈ I, h ∈ P ∗ ⊂ Bq (g) (resp. B q (g)). We shall use the abbreviated notations U , B, B, B ∨ ,· · · for Uq (g), Bq (g), B q (g), Bq∨ (g),· · · if there is no confusion. For β = mi αi ∈ Q+ we set |β| = mi and ± = {u ∈ U ± |q h uq −h = q ±h,β u (h ∈ P ∗ )}, U±β − and call |β| a height of β and Uβ+ (resp. U−β ) a weight space of U + (resp. U − ) with a −

weight β (resp. −β). We also define Bβ+ and B −β by the similar manner.

Proposition 2.1 ([3]). (i) We have the following algebra homomorphisms : : U −→ U ⊗ U , (r) : B −→ B ⊗ U , (l) : B −→ U ⊗ B and (b) : U −→ B ⊗ B given by (q h ) = (r) (q h ) = (l) (q h ) = (b) (q h ) = q h ⊗ q h ,

(2.10)

(ei ) = ei ⊗ 1 + ti ⊗ ei , (fi ) = fi ⊗ ti−1 + 1 ⊗ fi , (r) (ei ) = (qi − qi−1 ) · 1 ⊗ ti−1 ei + ei ⊗ ti−1 , (r) (fi ) = fi ⊗ ti−1 + 1 ⊗ fi , (l) (ei ) = ei ⊗ 1 + ti ⊗ ei , (l) (fi ) = (qi − qi−1 )ti fi ti ei (b) (ei ) = ti ⊗ + ei ⊗ 1, qi − qi−1 t −1 f (b) (fi ) = 1 ⊗ fi + i i−1 ⊗ ti−1 , qi − q i

(2.11) (2.12) ⊗ 1 + ti ⊗ fi , (2.13)

(2.14)

and extending these to the whole algebras by the rule: (xy) = (x)(y) and (i) (xy) = (i) (x)(i) (y) (i = r, l, b). (ii) We have the following anti-isomorphisms S : U −→ U and ϕ : B −→ B given by S(fi ) = −fi ti , S(q h ) = q −h , S(ei ) = −ti−1 ei , 1 ϕ(ei ) = − ei , ϕ(fi ) = −(qi − qi−1 )fi , ϕ(q h ) = q −h , qi − qi−1 and extending these to the whole algebras by the rule: S(xy) = S(y)S(x) and ϕ(xy) = ϕ(y)ϕ(x). Here S is called a anti-pode of U . We also denote ϕ|U ≥ = ϕ|B ≥ by ϕ.

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We obtain the following triangular decomposition of the q-boson Kashiwara algebra: Proposition 2.2. The multiplication map defines an isomorphism of vector spaces: ∼

Bq− (g) ⊗ T ⊗ Bq+ (g) −→ Bq (g), u1 ⊗ u 2 ⊗ u 3 → u1 u2 u3 . Proof. By [1, (3.1.2)], we have

ei fj

n (m)

 min(n,m)  2nm+(n+m)i−i(i+1)/2 min(n,m) (m−i) n−i   fi qi ei , if i = j, i = i=0  nmh  i ,αj (m) n q fj ei , otherwise. i

By this formula and the standard argument, we can show the proposition.

We define weight completions of L(1) ⊗ · · · ⊗ L(m) , where L(i) = B or U (see [6]).

(1) ⊗

(m) = lim L(1) ⊗ · · · ⊗ L(m) /(L(1) ⊗ · · · ⊗ L(m) )L+,l ,

···⊗

L L ←− l

+ + where L+,l = ⊕|β1 |+···+|βm |≥l L(1) β1 ⊗ · · · ⊗ L(m) βm . (Note that U ∼ = U − ⊗ T ⊗ U + and − + (r) B∼ = B ⊗ T ⊗ B . ) The linear maps , , S, ϕ, multiplication, etc. are naturally extend for such completions.

3. Category O(B) Let O(B) be the category of left B-modules such that (i) Any object M has a weight space decomposition M = ⊕λ∈P Mλ , where Mλ = {u ∈ M | q h u = q h,λ for any h ∈ P ∗ }. (ii) For any element u ∈ M there exists l > 0 such that ei1 ei2 · · · eil u = 0 for any i1 , i2 , · · · , il ∈ I . The similar category O(B ∨ ) for Bq (g)∨ is introduced in [1], which is defined with the above condition (ii). In [1], Kashiwara mentions that the category O(B ∨ ) is semi-simple though he does not give an exact proof. Here we give a proof of the semi-simplicity of O(B) in Sect 6. Here for λ ∈ P we define the B-module H (λ) by H (λ) := B/Iλ , where the left ideal Iλ is defined as Iλ := Bei + B(q h − q h,λ ). i

h∈P ∗

In Sect. 6, we shall also show that {H (λ)|λ ∈ P } is a set of representatives of isomorphism classes of simple modules.


289

4. Bilinear Forms and Elements C Proposition 4.1 ([4–6]).

(i) There exists the unique bilinear form ,

: U ≥ × U ≤ −→ Q(q),

satisfying the following; x, y1 y2 = (x), y1 ⊗ y2 , x1 x2 , y = x2 ⊗ x1 , (y),

q h , q h = q −(h|h ) , T , fi = ei , T = 0,

(x ∈ U ≥ , y1 , y2 ∈ U ≤ ), (x1 , x2 ∈ U ≥ , y ∈ U ≤ ),

(h, h ∈ P ∗ ),

ei , fj = δij /(qi−1 − qi ),

where ( | ) is an invariant bilinear form on t. (ii) The bilinear form , enjoys the following properties:

xq h , yq h = q −(h|h ) x, y, for x ∈ U ≥ , y ∈ U ≤ , h, h ∈ P ∗ . (4.1) − + For any β ∈ Q+ , , |U + ×U − is non-degenerate and Uγ , U−δ = 0, β −β if γ = δ. (4.2) We call this bilinear form the Drinfeld-Killing form of U . β For β = i mi αi ∈ Q+ (mi ≥ 0), set kβ := i timi , and let {xr }r be a basis of −β − Uβ+ and {yr }r be the dual basis of U−β with respect to the Drinfeld-Killing form. We + − denote the canonical element in Uβ ⊗ U−β with respect to the Drinfeld-Killing form by Cβ := xrβ ⊗ yr−β . r

We set C :=

U − = U + ⊗

B − . (1 ⊗ kβ−1 )(1 ⊗ S −1 )(Cβ ) ∈ U + ⊗

(4.3)

β∈Q+

The element C satisfies the following relations: Proposition 4.2.

(i) For any i ∈ I , we have (ti−1 ⊗ ei )C = C(ti−1 ⊗ ei + (qi − qi−1 )ti−1 ei ⊗ 1),

(fi ⊗ ti−1

+ 1 ⊗ fi )(ϕ ⊗ 1(C)) = (ϕ

⊗ 1(C))(fi ⊗ ti−1 ).

(4.4) (4.5)

Bq (g) and (4.5) is the equation in Here note that (4.4) is the equation in Uq (g)⊗

Bq (g). Bq (g)⊗ (ii) The element C is invertible and the inverse is given as q −(β,β) (kβ ⊗ kβ−1 )(S −1 ⊗ S −1 )(Cβ ). (4.6) C −1 = β∈Q+

Proof. The proof of (4.5) has been given in [3, 6.2]. Thus, let us show (4.4). For that purpose, we need the following lemma:

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β −β − Lemma 4.3. For β ∈ Q+ , let Cβ = r xr ⊗yr be the canonical element in Uβ+ ⊗U−β as above and set Cβ := (1 ⊗ S −1 )(Cβ ). Then for any β ∈ Q+ and i ∈ I , we have −1 )(Cβ+α )] = (1 ⊗ kβ−1 )(Cβ )(ti−1 ei ⊗ (qi − qi−1 ) · 1) [ti−1 ⊗ ei , (1 ⊗ kβ+α i i ∈ Uq (g) ⊗ Bq (g), (4.7)

where we use the identification Bq− (g) = Uq− (g). − Proof. Applying ·, z ⊗ 1 on both sides of (4.7), where z ∈ U−β−α , we obtain i

(·, z ⊗ 1)(L.H.S.of(4.7)) =

−1 ti−1 xrβ+αi , z ⊗ ei kβ+α S −1 (yr−β−αi ) i r

−1 −xrβ+αi ti−1 , z ⊗ kβ+α S −1 (yr−β−αi )ei i −1 −1 = q −(αi ,β+αi ) ei kβ+α S −1 (z) − kβ+α S −1 (z)ei i i −1 = kβ+α (ei S −1 (z) − S −1 (z)ei ), i

(·, z ⊗ 1)(R.H.S.of(4.7)) =

xrβ ti−1 ei , z ⊗ (qi − qi−1 )kβ−1 S −1 (yr−β ).

(4.8)

r − − we can define v ∈ U−β uniquely by For z ∈ U−β−α i

(z) = 1 ⊗ z + fi ⊗ vti−1 + · · · . By the property of the Drinfeld Killing form, we have xrβ ti−1 ei , z = ei ⊗ xrβ ti−1 , (z)

= ei ⊗ xrβ ti−1 , 1 ⊗ z + fi ⊗ vti−1 + · · · = ei , fi xrβ ti−1 , vti−1

=

qi−2 xrβ , v. −1 qi − q i

Thus, R.H.S. of (4.8) = −qi−2 kβ−1 S −1 (v).

(4.9)

Here in order to complete the proof of Lemma 4.3, let us show: ei S −1 (z) − S −1 (z)ei = −qi−2 ti S −1 (v).

(4.10)

Without loss of generality, we may assume that z is in the form z = fi1 fi2 · · · fik ∈ − U−β−α (β + αi = αi1 + · · · + αik ). For β = j mj αj , we shall show this by induction i on mi for fixed i ∈ I . If mi = 0, z is in the form z = z fi z , where z and z are monomials of fj ’s not including fi . By S −1 (fj ) = −tj fj and ei (tj fj ) = (tj fj )ei (i = j ) we have ei S −1 (z ) = S −1 (z )ei ,

ei S −1 (z ) = S −1 (z )ei .

(4.11)


291

Hence, we obtain ei S −1 (z) = S −1 (z )(−ei ti fi )S −1 (z ) = S −1 (z )(−ti fi ei − qi−2 ti )S −1 (z ) = S −1 (z )(−ti fi )S −1 (z )ei − qi−2 S −1 (z )ti S −1 (z ) = S −1 (z )S −1 (fi )S −1 (z )ei − q (β

−α ,α ) i i

ti S −1 (z z ),

where β = wt (z ). Therefore, for mi = 0, we have L.H.S. of (4.10) = −q (β

−α ,α ) i i

In the case mi = 0 we can easily obtain v = q (β R.H.S. of (4.10) = −q (β

−α ,α ) i i

,α ) i

ti S −1 (z z ).

z z and then

ti S −1 (z z ) = L.H.S. of (4.10).

Thus, the case mi = 0 has been shown. Suppose that mi > 0. We divide z = z z such that mi < mi and mi < mi , where mi ( resp. mi ) is the number of fi including in z (resp. z ). Writing (z ) = 1 ⊗ z + fi ⊗ v ti−1 + · · · ,

(z ) = 1 ⊗ z + fi ⊗ v ti−1 + · · · ,

and calculating (z z ) directly, we obtain v = z v + q (β

,α ) i

v z .

(4.12)

By the hypothesis of the induction, ei S −1 (z) = ei S −1 (z )S −1 (z ) = (S −1 (z )ei − qi−2 ti S −1 (v ))S −1 (z ) = S −1 (z )ei S −1 (z ) − qi−2 ti S −1 (z v )

= S −1 (z )(S −1 (z )ei − qi−2 ti S −1 (v )) − qi−2 ti S −1 (z v ) = S −1 (z z )ei − qi−2 ti (S −1 (z v ) + q (β

,α ) i

S −1 (v z ))

= S −1 (z)ei − qi−2 ti S −1 (v).

Note that in the last equality, we use (4.12). Now, we have completed the proof of Lemma 4.3. − Proof of Proposition 4.2. If β ∈ Q+ does not include αi , since ei and S −1 (z) (z ∈ U−β ) commute with each other by (4.11), we have

(ti−1 ⊗ ei )(1 ⊗ kβ−1 )(Cβ ) = (1 ⊗ kβ−1 )(Cβ )(ti−1 ⊗ ei ).

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Thus, we have (ti−1 ⊗ ei )C − C(ti−1 ⊗ ei ) = (ti−1 ⊗ ei )(1 ⊗ kγ−1 )(Cγ ) − (1 ⊗ kγ−1 )(Cγ )(ti−1 ⊗ ei ) γ ∈Q+

=

−1 −1 (ti−1 ⊗ ei )(1 ⊗ kβ+α )(Cβ+α ) − (1 ⊗ kβ+α )(Cβ+α )(ti−1 ⊗ ei ) i i i i

β∈Q+

=

−1 [ti−1 ⊗ ei , (1 ⊗ kβ+α )(Cβ+α )] i i

β∈Q+

=

(1 ⊗ kβ−1 )(Cβ )((qi − qi−1 )ti−1 ei ⊗ 1)

(by Lemma 4.3)

β∈Q+

= C((qi − qi−1 )ti−1 ei ⊗ 1). Then we obtain (4.4). (β,β) Next, let us show (ii). Set C := q (1 ⊗ kβ )(S ⊗ 1)(Cβ ). By [6, Sect.4], we −1 −1 (β,β) have C := q (kβ ⊗ kβ )(Cβ ). Here note that = (S −1 ⊗ S −1 )(C) = =

q (β,β) (1 ⊗ S −1 ){(1 ⊗ kβ )(Cβ )}

q (β,β) {(1 ⊗ S −1 )(Cβ )}(1 ⊗ kβ−1 ) (1 ⊗ kβ−1 )(1 ⊗ S −1 )(Cβ )

= C. Thus, we obtain C −1 = (S −1 ⊗ S −1 )(C−1 ) = q (β,β) {(S −1 ⊗ S −1 )(Cβ )}(kβ ⊗ kβ−1 ) = q −(β,β) (kβ ⊗ kβ−1 )(S −1 ⊗ S −1 )(Cβ ), and complete the proof of Proposition 4.2.

Uq− (g) = Remark. By the explicit form of C −1 in (4.6), we find that C −1 ∈ Uq+ (g)⊗

Bq− (g). Uq+ (g)⊗ 5. Extremal Projectors Let C be as in Sect.4. We define the extremal projector of Bq (g) by := m ◦ σ ◦ (ϕ ⊗ 1)(C) =

kβ−1 S −1 (yr−β )ϕ(xrβ ),

(5.1)

β∈Q+ , r

where m : a ⊗ b → ab is the multiplication and σ : a ⊗ b → b ⊗ a is the permutation. Here note that is a well-defined element in B q (g).


293

Example 5.1 ([1, 3]). In sl2 -case, the following is the explicit form of . 1 n = q 2 n(n−1) (−1)n f (n) e . n≥0

Theorem 5.2. The extremal projector enjoys the following properties: (i) ei = 0, fi = 0 (∀i ∈ I ). 2 (ii) = . (iii) There exists ak ∈ Bq− (g)(= Uq− (g)), bk ∈ Bq+ (g) such that

ak bk = 1.

k

q∨ (g). (iv) is a well-defined element in B Proof. It is easy to see (iv) by the explicit forms of the anti-pode S, the anti-isomorphism ϕ and in (5.1). The statement (ii) is an immediate consequence of (i). So let us show (i) and (iii). The formula fi = 0 has been shown+in [3]. Thus, we −shall show ei = 0. Here let us write C = k ck ⊗ dk , where ck ∈ Uq (g) and dk ∈ Bq (g). Thus, we have =

dk ϕ(ck ).

k

Equation (4.4) can be written as follows: ti−1 ck ⊗ ei dk = ck ti−1 ⊗ dk ei + (qi − qi−1 )ck ti−1 ei ⊗ dk . k

(5.2)

k

Applying m ◦ σ ◦ (ϕ ⊗ 1) on both sides of (5.2), we get ei dk ϕ(ck )ti = dk ei ti ϕ(ck ) − dk ei ti ϕ(ck ) = 0, k

k

k

and then ei ti = 0, which implies the desired result since ti is invertible. Next, let us see (iii). By the remark in the last section, we can write

Bq− (g). C −1 = bk ⊗ ak ∈ Uq+ (g)⊗ k

Then, 1⊗1=

bk cj ⊗ ak dj .

j,k

Applying m ◦ σ ◦ (ϕ ⊗ 1) on both sides of (5.3), we obtain ak dj ϕ(cj )ϕ(bk ) = ak ϕ(bk ). 1= j,k

Here setting bk := ϕ(bk ), we get (iii).

k

(5.3)

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6. Representation Theory of O(B) As an application of the extremal projector , we shall show the following theorem; Theorem 6.1. (i) The category O(B) is a semi-simple category. (ii) The module H (λ) is a simple object of O(B) and for any simple object M in O(B) there exists some λ ∈ P such that M ∼ = H (λ). Furthermore, H (λ) is a rank one free Bq− (g)-module. In order to show this theorem, we need to prepare several things. For an object M in O(B), set K(M) := {v ∈ M | ei v = 0 for any i ∈ I }. Lemma 6.2. For an object M in O(B), we have · M = K(M).

(6.1)

ei

= 0 for any i ∈ I . Thus, it is trivial to see that Proof. By Theorem 5.2(i), we have · M ⊂ K(M). Owing to the explicit form of , we find that B 1− ∈ q (g)ei . i

Therefore, for any v ∈ K(M) we get (1 − )v = 0, which implies that · M ⊃ K(M). Lemma 6.3. For an object M in O(B), we have M = Bq− (g) · (K(M)). (6.2) Proof. By Theorem 5.2(iii), we have 1 = k ak bk (ak ∈ Bq− (g), bk ∈ Bq+ (g)). For any u ∈ M, u= ak (bk u). k

By Lemma 6.2, we have bk u ∈ K(M). Then we obtain the desired result. Proposition 6.4. For an object M in O(B), we have Im(fi )). M = K(M) ⊕ (

(6.3)

i

Proof. By (6.2), we get

M = K(M) + (

Im(fi )).

i

Thus, it is sufficient to show

Im(fi )) = {0}. K(M) ∩ ( i

(6.4)

Let u be a vector in K(M)∩( i Im(fi )). Since u ∈ i Im(fi ), there exist {ui ∈ M}i∈I such that u = i∈I fi ui . By the argument in the proof of Lemma 6.2, we have u = u for u ∈ K(M). It follows from Theorem 5.2(i) that (fi )ui = 0, u = u = i∈I

which implies (6.4).


295

Lemma 6.5. If u, v ∈ M (M is an object in O(B)) satisfies v = u, then there exists P ∈ Bq (g) such that v = P u. β

Proof. By the definition of the category of O(B), there exists l > 0 such that ϕ(xr )u = 0 for any r and β with |β| > l. Thus, by the explicit form of in (5.1), we can write kβ−1 S −1 (yr−β )ϕ(xrβ ))u, v = u = ( |β|≤l, r

which implies our desired result.

Proof of Theorem 6.1. Let L ⊂ M be objects in the category O(B). We shall show that there exists a submodule N ⊂ M such that M = L ⊕ N . Since K(M) (resp. K(L)) is invariant by the action of any q h , we have the weight space decomposition: K(M)λ (resp. K(L) = K(L)λ ). K(M) = λ∈P

λ∈P

There exist subspaces Nλ ⊂ K(M)λ such that K(M)λ = K(L)λ ⊕ Nλ , which is a decomposition of a vector space. Here set N := ⊕λ Nλ . We have K(M) = K(L) ⊕ N. Let us show M = L ⊕ Bq (g) · N.

(6.5)

Since M = Bq (g) · (K(M)) = Bq (g)(K(L) ⊕ N ), we get M = L + Bq (g) · N . Let us show L ∩ Bq (g) · N = {0}.

(6.6)

For v ∈ L ∩ Bq (g) · N we have by Theorem 5.2 (iii), v= ak (bk v). k

It follows from v ∈ L that bk v ∈ K(L), and from v ∈ Bq (g) · N that bk v ∈ (Bq (g) · N) = N . These imply bk v ∈ K(L) ∩ N = {0}. Hence we get v = 0 and then (6.5). Next, let us show (ii). As an immediate consequence of Proposition 2.2 we can see that H (λ) is a rank one free Bq− (g)-module. Let πλ : Bq (g) → H (λ) be the canonical projection and set uλ := πλ (1). Here we have H (λ) = Bq− (g) · uλ = Q(q)uλ + Im(fi ). i

It follows from this, Proposition 6.4 and Q(q)uλ ⊂ K(H (λ)) that H (λ) = Q(q)uλ ⊕ i Im(fi ) and then · H (λ) = K(H (λ)) = Q(q)uλ .

(6.7)

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In order to show the irreducibility of H (λ), it is sufficient to see that for arbitrary u(= 0), v ∈ H (λ), there exists P ∈ Bq (g) such that v = P u. Set v = Quλ (Q ∈ Bq− (g)). By Theorem 5.2 (iii), we have u= ak (bk u) = 0. k

Then, for some k we have bk u = 0, which implies that cbk u = uλ for some non-zero scalar c. Therefore, by Lemma 6.5, there exists some R ∈ Bq (g) such that uλ = Ru and then we have v = Quλ = QRu. Thus, H (λ) is a simple module in O(B). Suppose that L is a simple module in O(B). First, let us show dim(K(L)) = 1.

(6.8)

For x, y(= 0) ∈ K(L), there exists P ∈ Bq (g) such that y = P x. Since x ∈ K(L), we can take P ∈ Bq− (g). Because y ∈ K(L) and K(L) ∩ i Im(fi ) = {0}, we find that P must be a scalar, say c. Thus, we have y = cx, which derives (6.8). Let u0 be a basis vector in K(L). The space K(L) is invariant by the action of any q h and then, u0 ∈ Lλ for some λ ∈ P . Therefore, since H (λ) is a rank one free Bq− (g)-module, the map φλ : H (λ) −→ L P uλ → P u0 ,

(P ∈ Bq− (g)),

is a well-defined non-trivial homomorphism of Bq (g)-modules. Thus, by Schur’s lemma, we obtain H (λ) ∼ = L. Acknowledgement. The author would like to thank Y. Koga for valuable discussions and A.N. Kirillov for introducing the papers [2, 7] to him.

References 1. Kashiwara, M.: On crystal bases of the q-analogue of universal enveloping algebras. Duke Math. J. 63, 465–516 (1991) 2. Khoroshkin, S.M., Tolstoy, V.N.: Exremal projector and universal R-matrix for quantized contragradient Lie (super) algebras. In: Quantum Groups and related topics, Gielerak et al, (eds.), 1992, pp. 23–32 3. Nakashima, T.: Quantum R-matrix and Intertwiners for the Kashiwara algebras. Commum. Math. Phys. 164, 239–258 (1994) 4. Rosso, M.: Analogues de la forme de Killing et du théorème d’Harish-Chandra pour les groupes ´ Norm. Sup. 23, 445–467 (1990) quantiques. Ann. scient.Ec, 5. Rosso, M.: Certaines formes bilinéaires sur les groupes quantiques et une conjecture de Schechtman et Varchenko. C.R. Acad. Sci. Paris Ser. 1 Math. 314(1), 5–8 (1992) 6. Tanisaki, T.: Killing forms, Harish-Chandra isomorphisms, and universal R-matrices for quantum algebras. Int. J. Mod. Phys. A7(Suppl. 1B), 941–961 (1992) 7. Tolstoy, V.N.: Extremal projectors for Quantized Kac-Moody superalgebras and some of their applications. In: Quantum Groups, (Clausthal, 1989), Lecture Notes in Physics 370, Berlin: Springer, 1990, pp. 118–125 8. Tolstoy, V.N.: Projection operator method for quantum groups. In: Special Functions 2000: Current perspective and future directions, J.Bustoz, et al., (eds.), NATO Science Series II, 30, Amsterdam: Kluwer Acad. Publishers, 2001, pp. 457–488, arXiv:math.QA/0104045 9. Tan, Y.: The q-analogue of bosons and Hall algebras. Comm. Algebra 30(9), 4335–4347 (2002) Communicated by Y. Kawahigashi


Communications in


Cantor Spectrum for the Almost Mathieu Operator Joaquim Puig Dept. de Matemàtica Aplicada i Anàlisi, Univ. de Barcelona, Gran Via 585, 08007 Barcelona, Spain. E-mail: [email protected] Received: 24 March 2003 / Accepted: 29 July 2003 Published online: 18 November 2003 – © Springer-Verlag 2003

Abstract: In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, Hb,φ x n = xn+1 + xn−1 + b cos (2π nω + φ) xn on l 2 (Z) and its associated eigenvalue equation to deduce that for b = 0, ±2 and ω Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called “Ten Martini Problem” for these values of b and ω. Moreover, we prove that for |b| = 0 small or large enough all spectral gaps predicted by the Gap Labelling theorem are open.

1. Introduction. Main Results In this paper we study the nature of the spectrum of the Almost Mathieu operator (Hb,φ x)n = xn+1 + xn−1 + b cos(2πωn + φ)xn ,

n∈Z

(1)

on l 2 (Z), where b is a real parameter, ω is an irrational number and φ ∈ T = R/(2π Z). Since for each b this is a bounded self-adjoint operator, the spectrum is a compact subset of the real line which does not depend on φ because of the assumption on ω. This spectrum will be denoted by σb . For b = 0, this set is the interval [−2, 2]. The understanding of the spectrum of (1) is related to the dynamical properties of the difference equation xn+1 + xn−1 + b cos(2πωn + φ)xn = axn ,

n∈Z

(2)

for a ∈ R, which is sometimes called the Harper equation. In what follows we will assume that the frequency ω is Diophantine:

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Definition 1. We say that a real number ω is Diophantine whenever there exist positive constants c and r > 1 such that the estimate c |sin 2πnω| > |n|r holds for all n = 0. The nature of the spectrum of this operator has been studied intensively in the last twenty years (for a review, see Last [27]) and an open problem has been to know whether the spectrum is a Cantor set or not, which is usually referred to as the “Ten Martini Problem”. In this paper we derive two results on this problem. The first one is non-perturbative: Corollary 1. If ω is Diophantine, then the spectrum of the Almost Mathieu operator is a Cantor set if b = 0, ±2. Here, we prefer to call this result a corollary, rather than a theorem, because the proof requires just a combination of reducibility, point spectrum and duality developed quite recently for the Almost Mathieu operator and the related eigenvalue equation. The argument is in fact reminiscent of Ince’s original argument for the classical Mathieu differential equation (see [19]). In the critical case |b| = 2, Y. Last proved in [26] that the spectrum of the Almost Mathieu operator is a subset of the real line with zero Lebesgue measure and that it is a Cantor set for the values of ω which have an unbounded continued fraction expansion, which is a set of full measure. This last result has been obtained recently for the remaining Diophantine frequencies by Avila & Krikorian [2]. The Cantor structure of the spectrum of the Almost Mathieu operator can be better understood if we make use of the concept of rotation number, which can be defined as follows. Let (xn )n∈Z be a non-trivial solution of (2), for some fixed a, b, φ. Let S(N ) be the number of changes of sign of such a solution for 1 ≤ n ≤ N , adding one if x(N) = 0. Then the limit S(N) N→∞ 2N lim

exists, it does not depend on the chosen solution x, nor on φ and it is denoted by rot(a, b). A more complete presentation of this object can be found in Sect. 2. Here we only mention some properties which relate it to the spectrum of Hb,φ : Proposition 1 ([3, 12, 18, 23]). The rotation number has the following properties: (i) The rotation number, rot(a, b), is a continuous function of (a, b) ∈ R2 . (ii) For a fixed b, the spectrum of (1), σb , is the set of a0 ∈ R, such that a → rot(a, b) is not locally constant at a0 . (iii) (Gap labelling) If I is an open, non-void interval in the resolvent set of (1), ρb = R − σb , then there is an integer k ∈ Z such that 2rot(a, b) − kω ∈ Z for all a ∈ I. That is, rot(a, b) =

1 {kω} , 2

where {·} denotes the fractional part of a real number.

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299

3 2.5 2 1.5 1 0.5 0 0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 1. Numerical computation √ of the ten biggest spectral gaps for the Almost Mathieu operator with different values of b and ω = ( 5 − 1)/2. They correspond to the first |k| such that {kω}/2 belongs to [1/4, 1/2]. The coupling parameter b is in the vertical direction whereas the spectral one, a, is in the horizontal one. Note that for b = 0, all gaps except the upper one are collapsed

From this theorem we conclude that the resolvent set is the disjoint union of countably (or finitely) many open intervals called spectral gaps, possibly void, and which can be uniquely labelled by an integer k called the resonance. If the closure of a spectral gap degenerates to a point we will say that it is a collapsed gap and otherwise that it is a non-collapsed gap. See Fig. 1 for a numerical computation of the biggest gaps in the spectrum of the Almost Mathieu operator for several values b. In particular if, for a fixed b, all the spectral gaps are open and the frequency ω is irrational, then the spectrum σb is a Cantor set. The question of the non-collapsing of all spectral gaps is sometimes called the Strong (or Dry) Ten Martini Problem. However, if non-collapsed gaps are dense in the spectrum, then this is still a Cantor set, although some (perhaps an infinite number) of collapsed gaps may also coexist. Now we can formulate the second corollary in this paper: Corollary 2. Assume that ω ∈ R is Diophantine. Then, there is a constant C = C(ω) > 0 such that if 0 < |b| < C or 4/C < |b| < ∞ all the spectral gaps of the spectrum of the Almost Mathieu operator are open. Before ending this introduction we give a short account of the existing results (to our knowledge) on the Cantor spectrum of the Almost Mathieu operator for |b| = 0, 2. The Cantor spectrum for the Almost Mathieu operator was first conjectured by Azbel [4] and Kac, in 1981, conjectured that all the spectral gaps are open. The problem of the Cantor structure of the spectrum was called the “Ten Martini Problem” by Simon [32] (and remained as Problem 4 in [33]). Sinai [34], proved that for Diophantine ω’s and sufficiently large (or small |b|), depending on ω, the spectrum σb is a Cantor set. Choi, Elliott & Yui [8] proved that the spectrum σb is a Cantor set for all b = 0 when ω is a

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Liouville number obeying the condition ω − p < D −q , q for a certain constant D > 1 and infinitely many rationals p/q. In particular, this means that for a Gδ -dense subset of pairs (b, ω) the spectrum is a Cantor set, which is the Bellissard-Simon result [5]. For results on Cantor spectrum for continuous quasi-periodic and almost periodic Schrödinger operators see Moser [28], Johnson [22], Eliasson [17] and Puig & Simó [31]. Nevertheless, collapsed gaps appear naturally in quasi-periodic Schrödinger operators, as it was shown by Broer, Puig & Simó [6] and there are examples which do not display Cantor spectrum, see De Concini & Johnson [11]. Finally, let us mention that, if we consider the case of rational ω, all spectral gaps, apart from the middle one, are open if b = 0. This result was proved by van Mouche [35] and Choi, Elliott & Yui [8]. Let us now outline the contents of the present paper. In Sect. 2 we introduce some of the tools needed to prove our two main results. These include the different definitions of the rotation number, the concept of reducibility of linear quasi-periodic skew-products and the duality for the Almost Mathieu operator. In Sect. 3 we apply the reducibility results by Eliasson to prove Corollary 2. Finally, in Sect. 4, the proof of of Corollary 1 is given, which is based on a result of non-perturbative localization by Jitomirskaya. 2. Prerequisites: Rotation Number, Reducibility, Duality and Lack of Coexistence Rotation number. The rotation number for quasi-periodic Schrödinger equations is a very useful object with deep connections to the spectral properties of Schrödinger operators. It is also related to the dynamical properties of the solutions of the associated eigenvalue equation. This allows several equivalent definitions, which we shall now try to present. The rotation number was introduced for continuous time quasi-periodic Schrödinger equations by Johnson & Moser [23]. The discrete version was introduced by Herman [18] (which is also defined for quasi-periodic skew-product flows on SL(2, R) × T) and Delyon & Souillard [12] (which is the definition given in the introduction). We will now review these definitions, their connection and some important properties. Herman’s definition is dynamical. Here we follow the presentation by Krikorian [25]. Write Eq. (2) as a quasi-periodic skew-product flow on R2 × T, un+1 = A(θn )un

θn+1 = θn + 2π ω,

(3)

setting un = (xn , xn−1 )T and A(θ ) =

a − b cos θ −1 , 1 0

(4)

which belongs to SL(2, R) the group of bidimensional matrices with determinant one. The quasi-periodic flow can also be defined on SL(2, R) × T considering the flow given by Xn+1 = A(θn )Xn ,

θn+1 = θn + 2π ω,

(5)


301

with X0 ∈ SL(2, R). This can be seen as the iteration of the following quasi-periodic cocycle on SL(2, R) × T: SL(2, R) × T −→ SL(2, R) × T (X, θ ) → (A(θ )X, θ + 2π ω) ,

(6)

which we denote by (A, ω). We will now give Herman’s definition of the rotation number of a quasi-periodic cocyle like (6) with A : T → SL(2, R) homotopic to the identity. For a general A : T → SL(2, R), this last property is not always true, since SL(2, R) is not simply connected. Indeed, its first homotopy group is isomorphic to Z, with generator the rotation R1 : T → SL(2, R) given by cos θ − sin θ R1 (θ ) = sin θ cos θ for all θ ∈ T. In our case, the Almost Mathieu cocyle (4) is homotopic to the identity. Let S1 be the set of unit vectors of R2 and let us denote by p : R → S1 the projection given by the exponential p(t) = eit , identifying R2 with C. Because of the linear character of the cocyle, the continuous map 1 F : S1 × T −→ S ×T A(θ )v (v, θ ) → , θ + 2π ω A(θ )v

(7)

is also homotopic to the identity. Therefore, it admits a continuous lift F˜ : R×T → R×T of the form: F˜ (t, θ ) = (t + f (θ, t), θ + 2π ω) such that f (t + 2π, θ + 2πω) = f (t, θ ) and p (t + f (t, θ)) =

A(θ )p(t) A(θ )p(t)

for all t ∈ R and θ ∈ T. The map f is independent of the choice of F˜ up to the addition of a constant 2π k, with k ∈ Z. Since the map θ → θ + 2π ω is uniquely ergodic on T for all (t, θ ) ∈ R × T, the limit N−1 1 ñ f F (t, θ ) N→∞ 2πN

lim

n=0

exists, it is independent of (t, θ ) and the convergence is uniform in (t, θ ), see Herman [18] and Johnson & Moser [23]. This object is called the fibered rotation number, which will be denoted as ρf (a, b), and it is defined modulus Z. For instance, if A0 ∈ SL(2, R) is a constant matrix, then the fibered rotation number of the cocycle (A0 , ω), for any irrational ω, is the absolute value of the argument of the eigenvalues divided by 2π . Using a suspension argument (see Johnson [24]) it can be seen that, for the Almost Mathieu cocycle (like for any quasi-periodic Schrödinger cocycle), the fibered rotation number coincides with the Sturmian definition given in the introduction. Note that this

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last rotation number, rot(a, b), belongs to the interval [0, 1/2], whereas the fibered rotation number, f (a, b), is an element of R/Z. They can be both linked by means of the integrated density of states, see Avron & Simon [3]. Let kL (a, b, φ) be (L − 1)−1 times the number of eigenvalues less than or equal to a for the restriction of Hb,φ to the set {1, . . . , L − 1}, for some φ ∈ T, with zero boundary conditions at both ends 0 and L. Then, as L → ∞, the kL (a, b, φ) converge to a continuous function k(a, b), which is the integrated density of states. The basic relations are 2rot(a, b) = k(a, b)

and

2f (a, b) = k(a, b) + l,

for a suitable integer l ∈ Z. In particular, 1 rot(a, b) = ρf (a, b) (mod Z). 2 In what follows, the arithmetic nature of the rotation number will be of importance. We will say that the rotation number is rational or resonant with respect to ω if there exists a constant k ∈ Z such that rot(a, b) = {kω}/2 or equivalently, f (a, b) = kω/2 modulus 21 Z. Also, we say that it is Diophantine with respect to ω whenever the bound rot(a, b) − {kω} = min ρf (a, b) − kω − l ≥ K , 2 2 2 |k|τ l∈Z holds for all k ∈ Z − {0} and suitable fixed positive constants K and τ . Reducibility. A main tool in the study of quasi-periodic skew-product flows is its reducibility to constant coefficients. Reducibility is a concept defined for the continuous and discrete case (for an introduction see the reviews by Eliasson [14, 15] and, for more references, the survey [30] by the author). A quasi-periodic skew-product flow like (3), or a quasi-periodic cocycle like (6), with A : T → SL(2, R), is said to be reducible to constant coefficients if there is a continuous map Z : T → SL(2, R) and a constant matrix B ∈ SL(2, R), called the Floquet matrix, such that the conjugation A(θ )Z(θ ) = Z(θ + 2πω)B

(8)

is satisfied for all θ ∈ T. When ω is rational, in which case the flow is periodic, any skew-product flow is reducible to constant coefficients. Even in this periodic case, it is not always possible to reduce with the same frequency ω, but with ω/2. If there is a reduction to constant coefficients like (8), then a fundamental matrix of solutions of (3), Xn+1 (φ) = A(2πnω + φ)Xn (φ),

n ∈ Z,

with X0 : T → SL(2, R) continuous, has the following Floquet representation: Xn (φ) = Z(2πnω + φ)B n Z(φ)−1 X0 (φ)

(9)

for all n ∈ Z and φ ∈ T. This gives a complete description of the qualitative behaviour of the flow (3). The rotation number of a quasi-periodic cocycle is not invariant through a conjugation like (8). There are however the following easy relations:


303

Proposition 2. Let ω be an irrational number and (A1 , ω) and (A2 , ω) be two quasiperiodic cocycles on SL(2, R) × T homotopic to the identity, being ρ1 and ρ2 the corresponding fibered rotation numbers. Assume that there exists a continuous map Z : T → SL(2, R) such that A1 (θ )Z(θ ) = Z(θ + 2πω)A2 (θ ) for all θ ∈ T. Then, if k ∈ Z is the degree of Z, ρ1 = ρ2 + kα modulus Z. This proposition shows that, for any fixed irrational frequency ω, the class of quasiperiodic cocycles with rational rotation number (resp. with Diophantine rotation number) is invariant under conjugation, although the rotation number itself may change. Also, that whenever a quasi-periodic skew-product flow in SL(2, R) × T is reducible to a Floquet matrix with trace ±2, the rotation number must be rational. Duality and lack of coexistence. To end this section, let us present a specific feature of the Almost Mathieu operator or, rather, of the associated eigenvalue equation which is in the basis of our arguments. It is part of what is known as Aubry duality or simply duality: Theorem 1 (Avron & Simon [3]). For every irrational ω, the rotation number of (2) satisfies the relation rot(a, b) = rot(2a/b, 4/b)

(10)

for all b = 0 and a ∈ R. According to Proposition 1 this means that the spectrum σ4/b , for b = 0 is just a dilatation of the spectrum σb . In particular, σb is a Cantor set (resp. none of the spectral gaps of σb is collapsed) if and only if σ4/b is a Cantor set (resp. none of the spectral gaps of σ4/b is collapsed). In the proof of our two main results we will use the following argument, which is analogous to Ince’s argument for the classical Mathieu periodic differential equation (see [19] §7.41). In principle, the eigenvalue equation of a general quasi-periodic Schrödinger operator may have two linearly independent quasi-periodic solutions with frequency ω (or ω/2). One may call this phenomenon coexistence of quasi-periodic solutions, in analogy with the classical Floquet theory for second-order periodic differential equations. A trivial example of this occurs in the Almost Mathieu case for b = 0 and suitable values of a. Let us now show that in the Almost Mathieu case this does not happen if b = 0, i.e. two quasi-periodic solutions with frequency ω of the eigenvalue equation cannot coexist. Let (xn )n∈Z satisfy the equation xn+1 + xn−1 + b cos(2πωn + φ)xn = axn ,

n∈Z

(11)

for some a, b = 0 and φ. If it is quasi-periodic with frequency ω, there exists a continuous function ψ : T → R such that xn = ψ(2πωn + φ) for all n ∈ Z. The Fourier coefficients of ψ, (ψm )m∈Z satisfy the following equation: b 2 cos(2π ωm)ψm + (ψm+1 + ψm−1 ) = aψm , 2

m ∈ Z,

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which is equivalent to ψm+1 + ψm−1 +

4 2a cos(2π ωm)ψm = ψm , b b

m ∈ Z.

(12)

Since ψ is at least continuous, then (ψm )m∈Z belongs to l 2 (Z). Now the reason for the absence of coexisting quasi-periodic solutions is clear. Indeed, if (yn )n∈Z is another linearly independent quasi-periodic solution of (11) with frequency ω, say yn = χ (2π ω+φ), for some continuous χ , then the sequence of the Fourier coefficients of χ , (χm )m∈Z , would be a solution of (12) belonging to l 2 (Z). The sequences (ψm )m∈Z and (χm )m∈Z would be two linearly independent solutions of (12) which belong both to l 2 (Z). This is a contradiction, because for bounded potentials, like the cosine, we are always in the limit-point case (see [7, 9] for the continuous case). In our discrete case, this is even simpler, since any solution in l 2 (Z) of the eigenvalue equation must tend to zero at ±∞. Hence, the existence of two linearly independent solutions belonging both to l 2 (Z) would be in contradiction with the preservation of the Wronskian. Therefore, two quasi-periodic solutions with frequency ω cannot coexist if b = 0. A similar argument shows that quasi-periodic solutions of the form (−1)n ψ(2πωn + φ),

(13)

for a continuous ψ : T → R cannot coexist. Finally, note that the coexistence of two quasi-periodic solutions with frequency ω of Eq. (11) is equivalent to the reducibility of the corresponding two-dimensional skewproduct flow (3), with the identity as Floquet matrix. Similarly the coexistence of two quasi-periodic solutions of the type (13) is equivalent to the reducibility of the flow with minus the identity as Floquet matrix. 3. The Strong Ten Martini Problem for Small (and Large) |b| In this section we will show that for 0 < |b| < C, where C > 0 is a suitable constant, and for |b| > 4/C all spectral gaps are open. The theorem from which we will derive Corollary 2 is due to Eliasson and it was originally stated for the continuous case, based on a KAM scheme. It can be adapted to the discrete case to obtain the following: Theorem 2 ([16, 17]). Assume that ω is Diophantine with constants c and r. Then there is a constant C(c, r) such that, if |b| < C(c, r) and rot(a, b) is either rational or Diophantine, then the quasi-periodic skew-product flow

xn+1 xn

=

a − b cos θn −1 1 0

xn xn−1

,

θn+1 = θn + 2π ω

(14)

on R2 × T is reducible to constant coefficients, with Floquet matrix B, by means of a quasi-periodic (with frequency ω/2) and analytic transformation. Moreover, if a is at an endpoint of a spectral gap of σb , then the trace of B is ±2, being B = ±I if, and only if, the gap collapses. Finally, if B = ±I then the transformation Z can be chosen to have frequency ω.


305

For other reducibility results in the context of quasi-periodic Schrödinger operators see Dinaburg & Sinai [13] and Moser & Pöschel [29] for the continuous case and Krikorian [25] and Avila & Krikorian [2] for the discrete case. Taking into account the arguments from the previous section, Corollary 2 is immediate. Indeed, let |b| < C, where C is the constant given by the theorem for a fixed Diophantine frequency ω. Then the skew-product flow (14) is reducible to constant coefficients and the Floquet matrix has trace ±2 if a is an endpoint of a spectral gap. Moreover the gap is collapsed if, and only if, the Floquet matrix B is ±I . Since we have seen in the previous section that (14) for b = 0 cannot be reducible to these Floquet matrices, Corollary 2 follows. 4. Non-Perturbative Localization and Cantor Spectrum for b = 0 In this section we will see how Corollary 1 is a consequence of the following theorem on non-perturbative localization, due to Jitomirskaya: Theorem 3 ([20]). Let ω be Diophantine. Define the set of resonant phases as the set of those φ ∈ T such that the relation 1 |sin (φ + πnω)| < exp −|n| 2r (15) holds for infinitely many values of n, r being the constant in the definition of a Diophantine number. Then, if φ ∈ and |b| > 2 the operator Hb,φ has only pure point spectrum with exponentially decaying eigenfunctions. Moreover, any of these eigenfunctions (ψn )n∈Z satisfies that 2 log ψn2 + ψn+1 |b| β(b) = − lim = log . (16) |n|→∞ 2|n| 2 Now we prove Corollary 1. Let |b| > 2. Then, according to Theorem 3, the operators Hb,0 and Hb,π have only pure point spectrum with exponentially decaying eigenfunctions. The eigenvalue equation associated to these operators has the following properties: Lemma 1. Let (xn )n∈Z be a solution of the difference equation xn+1 + xn−1 + b cos(2πnω + φ)xn = axn ,

n ∈ Z,

for some constants a, b and φ ∈ T. Then, if φ = 0, π , (x−n )n∈Z is also a solution of this equation. Let us consider the operator Hb,0 . According to Theorem 3, there exists a sequence of eigenvalues (a k (b))k∈Z with eigenvectors (ψ k (b))k∈Z , exponentially localized and which form a complete orthonormal basis of l 2 (Z). Moreover the set of eigenvalues (a k (b))k∈Z must be dense in the spectrum σb . Again, we do not write the dependence on b for simplicity in what follows. None of these eigenvalues can be repeated, since we are in the limit point case. Writing each of the ψ k as ψ k = (ψnk )n∈Z , we define ψ˜ k (θ ) =

k∈Z

ψnk eikθ ,

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for θ ∈ T. All these functions belong to Cβa (T, R), the set of real analytic functions of T with analytic extension to | θ | < β and they are even functions of θ , because of Lemma 1 (here we have applied again that we are in the limit point case). Passing to the dual equation, we obtain that, for each k ∈ Z, the sequence (ψ˜ k (2π ωn))n∈Z is a quasi-periodic solution of xn+1 + xn−1 +

4 2a cos θn xn = xn , b b

θn+1 = θn + 2π ω

n ∈ Z,

(17)

provided a is now replaced by a k . We are now going to see that 2a k /b is at an endpoint of a spectral gap and that this is collapsed. To do so we will use reducibility as in the proof of Theorem 2. For a direct proof that 2a k /b is at an endpoint of a gap (it has rational rotation number), see again Herman [18]. The fact that (ψ˜ k (2π ωn))n∈Z is a quasi-periodic solution of (17) means that, for all θ ∈ Td , the following equation is satisfied: k k 2a ψ˜ (2π ω + θ) ψ˜ (4π ω + θ ) − b4 cos θ −1 b = . 1 0 ψ˜ k (2π ω + θ ) ψ˜ k (θ ) The following lemma shows that, if this is the case, then the quasi-periodic skew-product flow k 2a 4 xn xn+1 b − b cos θn −1 = , θn+1 = θn + 2π ω (18) xn xn−1 1 0 is reducible to constant coefficients. Lemma 2. Let A : T → SL(2, R) be a real analytic map, with analytic extension to | θ| < δ for some δ > 0. Assume that there is a nonzero real analytic map v : T → R2 , with analytic extension to | θ| < δ such that v(θ + 2πω) = A(θ )v(θ ) holds for all θ ∈ T. Then, the quasi-periodic skew-product flow given by un+1 = A(θn )un ,

θn+1 = θn + 2π ω,

(19)

with (un , θn ) ∈ R2 × T for all n ∈ Z is reducible to constant coefficients by means of a quasi-periodic transformation which is analytic in | θ| < δ and has frequency ω. Moreover the Floquet matrix can be chosen to be of the form 1 c B= (20) 0 1 for some c ∈ R. Proof. Since v = (v1 , v2 )T does not vanish, d = v12 + v22 is always different from zero and the transformation v1 (θ ) −v2 (θ )/d(θ ) Z(θ ) = , v2 (θ ) v1 (θ )/d(θ )


307

is an analytic map Z : T → SL(2, R). The transformation Z defines a conjugation of A with B 1 , being A(θ )Z(θ ) = Z(2πω + θ )B 1 (θ ), which means that B 1 is

B 1 (θ ) =

1 (θ ) 1 b12 , 0 1

1 : T → R. The transformed skew-product flow, defined by the for some analytic b12 1 matrix B is reducible to constant coefficients because it is in triangular form, the fre1 is analytic. Indeed, if y quency ω is Diophantine and b12 12 : T → R is an analytic solution of the small divisors equation 1 1 y12 (2πω + θ ) − y12 (θ ) = b12 (θ ) − [b12 ],

θ ∈ T,

1 ] is the average of b1 (see [1]), then the transformation where [b12 12 1 y12 Y (θ ) = 0 1

conjugates B 1 with its averaged part:

B = [B ] = 1

which is in the form of (20).

1 ] 1 [b12 0 1

Thus, applying this lemma, the flow (18) is reducible to constant coefficients with Floquet matrix B, of the form (20). That is, there exists a real analytic map Z : T → SL(2, R) such that A(θ )Z(θ ) = Z(θ + 2π ω)B

(21)

for all θ ∈ T. Moreover, since the trace of B is 2, the rotation number of (17) is rational, so that we are at the endpoint of a gap, which we want to show is non-collapsed. By the arguments of Sect. 2, we rule out the possibility of B being the identity. Indeed, this would imply the coexistence of two quasi-periodic analytic solutions with frequency ω, which does happen in the Almost Mathieu case. Therefore B = I and, thus, c = 0 in the definition above. If B = I , it is a well-known fact of Floquet theory that 2a k /b lies at the endpoint of a non-collapsed gap (see, for example, the monograph [36] for classical Floquet theory or [6] for the continuous and quasi-periodic Schrödinger case). For the sake of self-completeness we sketch the argument. We will see that there exists a α0 > 0 such that if 0 < |α| < α0 and α is either positive or negative (depending on the sign of c) then 2a k /b + α lies in the resolvent set of σ4/b . To do so, we will show that, for these values of α, the skew-product flow k 2a 4 xn xn+1 + α − cos θ −1 n b b = , θn+1 = θn + 2π ω (22) xn xn−1 1 0

308

J. Puig

has an exponential dichotomy (see Coppel [10]) which implies that 2a k /b + α ∈ σ4/b (see Johnson [21]). The reduction given by Z transforms this system into 2 2 1 + α z11 z12 − cz11 c + α −cz11 z12 + z12 yn+1 = yn , 2 −αz11 1 − αz11 z12 θn+1 = θn + 2πω, (23) where yn ∈ R2 are the new variables. The zij are the elements of the matrix Z and we have used the relations given by (21) and the special form of A and B. In the same calculation, we also see that (z11 (2πnω))n∈Z is a quasi-periodic solution of Eq. (17) and that it is not identically zero. Using averaging theory (see, for example, Arnol d [1]), system (23) can be transformed into 2 ] 2 ] 1 + α [z11 z12 ] − c[z11 c + α −c[z11 z12 ] + [z12 yn+1 = + M yn 2 ] −α[z11 1 − α[z11 z12 ] (24) θn+1 = θn + 2π ω by means of a conjugation in SL(2, R), with M analytic in both θ and α (in some narrower domains) and of order α 2 . The time-independent part of the above system is hyperbolic if cα < 0. Therefore, if |α| = 0 is small enough the time-dependent system (24) has an exponential dichotomy for cα < 0. Hence 2a k /b + α does not belong to σ4/b . Since this works for all a k , (which are dense in the spectrum), σ4/b is a Cantor set. By duality the result is also true for σb . This ends the proof of Corollary 1. Remark 1. The same can be done for the operator Hb,π instead of Hb,0 . In this case the Floquet matrix has trace −2. The corresponding point eigenvalues correspond to ends of non-collapsed gaps and are dense in the spectrum. Acknowledgements. The author is indebted to Hakan Eliasson, Raphael Krikorian and Carles Simó for stimulating discussions and comments on this problem. He would like to thank the anonymous referee for ´ his useful suggestions. He also wants to thank the Centre de Mathématiques at the Ecole Polytechnique for hospitality. Help from the Catalan grant 2000FI71UBPG and grants DGICYT BFM2000-805 (Spain) and CIRIT 2000 SGR-27, 2001 SGR-70 (Catalonia) is also acknowledged.

References 1. Arnol d, V.I.: Geometrical methods in the theory of ordinary differential equations. Vol. 250 of Grundlehren der Mathematischen Wissenschaften. New York: Springer-Verlag, 1983 2. Avila, A., Krikorian, R.: Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Preprint, 2003 3. Avron, J., Simon B.: Almost periodic Schrödinger operators II. The integrated density of states. Duke Math. J. 50, 369–391 (1983) 4. Azbel, M.Ya.: Energy spectrum of a conduction electron in a magnetic field. Soviet Phys. JETP. 19, 634–645 (1964) 5. Bellissard, J., Simon, B.: Cantor spectrum for the almost Mathieu equation. J. Funct. Anal. 48(3), 408–419 (1982) 6. Broer, H.W., Puig, J., Simó, C.: Resonance tongues and instability pockets in the quasi-periodic Hill-Schrödinger equation. Commun. Math. Phys. 241 (2-3), 467–503 (2003) 7. Carmona, R., Lacroix, J.: Spectral theory of random Schrödinger operators. The Probability and its Applications. Basel-Boston: Birkhäuser, 1990 8. Choi, M.D., Elliott, G.A., Yui, N.: Gauss polynomials and the rotation algebra. Invent. Math. 99(2), 225–246 (1990) 9. Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. New York-Toronto-London: McGraw-Hill Book Company, Inc., 1955


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10. Coppel, W.A.: Dichotomies in stability theory. Lecture Notes in Mathematics, Vol. 629, Berlin: Springer-Verlag, 1978 11. DeConcini, C., Johnson R.A.: The algebraic-geometric AKNS potentials. Ergodic Theory Dynam. Syst. 7(1), 1–24 (1987) 12. Delyon, F., Souillard, B.: The rotation number for finite difference operators and its properties. Commun. Math. Phys. 89(3), 415–426 (1983) 13. Dinaburg, E.I., Sinai,Y.G.: The one-dimensional Schrödinger equation with quasi-periodic potential. Funkt. Anal. i. Priloz. 9, 8–21 (1975) 14. Eliasson, L.H.: One-dimensional quasi-periodic Schrödinger operators – dynamical systems and spectral theory. In: European Congress of Mathematics, Vol. I (Budapest, 1996), Basel: Birkhäuser, 1998, pp. 178–190 15. Eliasson, L.H.: Reducibility and point spectrum for linear quasi-periodic skew-products. In: Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), number Extra Vol. II, (electronic), 1998, pp. 779–787 16. Eliasson, L.H.: On the discrete one-dimensional quasi-periodic Schrödinger equation and other smooth quasi-periodic skew products. In: Hamiltonian systems with three or more degrees of freedom (S’Agaró, 1995), Volume 533 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Dordrecht: Kluwer Acad. Publ., 1999, pp. 55–61 17. Eliasson, L.H.: Floquet solutions for the one-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146, 447–482 (1992) 18. Herman, M.R.: Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2. Comment. Math. Helv. 58(3), (1983) 19. Ince, E.L.: Ordinary Differential Equations. New York: Dover Publications, 1944 20. Jitomirskaya, S.Y.: Metal-insulator transition for the almost Mathieu operator. Ann. Math. (2) 150(3), 1159–1175 (1999) 21. Johnson, R.: The recurrent Hill’s equation. J. Diff. Eq. 46, 165–193 (1982) 22. Johnson, R.: Cantor spectrum for the quasi-periodic Schrödinger equation. J. Diff. Eq. 91, 88–110 (1991) 23. Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 84, 403–438, (1982) 24. Johnson, R.A.: A review of recent work on almost periodic differential and difference operators. Acta Appl. Math. 1(3), 241–261 (1983) 25. Krikorian, R.: Reducibility, differentiable rigidity and Lyapunov exponents for quasi-periodic cocycles on T × SL(2, R). Preprint 26. Last, Y.: Zero measure spectrum for the almost Mathieu operator. Commun. Math. Phys. 164(2), 421–432 (1994) 27. Last, Y.: Almost everything about the almost Mathieu operator. I. In: XIth International Congress of Mathematical Physics (Paris, 1994), Cambridge MA: Internat. Press, 1995, pp. 366–372 28. Moser, J.: An example of schrödinger equation with almost periodic potential and nowhere dense spectrum. Comment. Math. Helv. 56, 198–224 (1981) 29. Moser, J., Pöschel, J.: An extension of a result by Dinaburg and Sinai on quasi-periodic potentials. Comment. Math. Helv. 59, 39–85 (1984) 30. Puig, J.: Reducibility of linear differential equations with quasi-periodic coefficients: A survey. Barcelona: Preprint University of Barcelona, 2002, Available at http://www.maia. ub.es/∼puig/preprints/qpred.ps 31. Puig, J, Simó, C.: Analytic families of reducible linear quasi-periodic equations. In progress 2003 32. Simon, B.: Almost periodic Schrödinger operators: A review. Adv. Appl. Math. 3(4), 463–490 (1982) 33. Simon, B.: Schrödinger operators in the twenty-first century. In: Mathematical physics 2000, London: Imp. Coll. Press, 2000, pp. 283–288 34. Sinai, Ya.G.: Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential. J. Statist. Phys. 46(5-6), 861–909 (1987) 35. van Mouche, P.: The coexistence problem for the discrete Mathieu operator. Commun. Math. Phys. 122(1), 23–33 (1989) 36. Yakubovich, V.A., Starzhinskii V.M.: Linear differential equations with periodic coefficients. 1, 2. New York-Toronto, Ont: Halsted Press [John Wiley & Sons], 1975 Communicated by B. Simon


Communications in


Towards a Quantum Analog of Weak KAM Theory Lawrence C. Evans Department of Mathematics, University of California, Berkeley, CA 94720, USA Received: 3 February 2003 / Accepted: 31 July 2003 Published online: 18 November 2003 – © Springer-Verlag 2003

Abstract: We discuss a quantum analogue of Mather’s minimization principle for Lagrangian dynamics, and provide some formal calculations suggesting the corresponding Euler–Lagrange equation. We then rigorously construct from the dual eigenfunctions of a certain non-selfadjoint operator a candidate ψ for a minimizer, and recover aspects of “weak KAM” theory in the limit as h → 0. Regarding our state ψ as a quasimode, we furthermore derive some error estimates, although it remains an open problem to improve these bounds. 1. Introduction This paper proposes an extension of Mather’s variational principle [M1, M2, M-F] and Fathi’s weak KAM theory[F1, F2, F3] to quantum states. We interpret “weak KAM” theory to mean the application of nonlinear PDE methods, mostly for first–order equations, towards understanding the structure of action minimizing measures solving Mather’s problem. As explained in the introduction to [E-G], a goal is interpreting these measures as providing a sort of “integrable structure”, governed by an associated “effective Hamiltonian” H¯ , in the midst of otherwise possibly very chaotic dynamics. The relevant PDE are a nonlinear eikonal equation and an associated continuity (or transport) equation. In this work we attempt to extend this viewpoint and some of the techniques to a quantum setting, in the semiclassical limit. We do so by suggesting an analogue of Mather’s action minimization problem for the Lagrangian L(v, x) = 21 |v|2 − W (x), where the potential W is periodic, and formally computing the first and second variations. Thus motivated, we next build a candidate state ψ for a minimizer and discuss at length its properties. As in the nonquantum case, we come fairly naturally upon an eikonal PDE (with some extra terms) and an exact continuity equation. We next send h → 0 and show how the usual structure of weak KAM theory appears in this limit. Supported in part by NSF Grant DMS-0070480 and by the Miller Institute for Basic Research in Science, UC Berkeley

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More interesting is understanding if our ψ is a good quasimode, that is, a decent approximate solution of an appropriate eigenvalue problem. This turns out to be so, although our error bounds are too weak to allow for any deductions about the spectrum. Much of the interest in the following calculations centers upon our minimizing, subject to certain side conditions, the expected value of the Lagrangian, namely the expression h2 |Dψ|2 − W |ψ|2 dx, n 2 T and not the expected value of the Hamiltonian, h2 |Dψ|2 + W |ψ|2 dx. Tn 2 It will turn out that owing to the constraints a minimizer of the former is approximately a critical point of the later. But the key question, unresolved here, is determining when the error terms are of order, say, o(h) in L2 . The calculations below represent improvements upon some ideas developed earlier in [E1]. An interesting recent paper of Anantharaman [A] presents a somewhat similar approach within a probabilistic framework, and Holcman and Kupka’s forthcoming paper [H-K] is related. Likewise, Gomes [G] found some related constructions for his stochastic analogue of Aubry-Mather theory. We later discuss also some formal connections with the “stochastic mechanics” approach to quantum mechanics of Nelson [N] and also with homogenization theory for divergence–structure second–order elliptic PDE. Action minimizing measures. Hereafter Tn denotes the flat torus in Rn , the unit cube with opposite faces identified. We are given a smooth and periodic potential function W : Tn → R and a vector V ∈ Rn . The Lagrangian is 1 2 |v| − W (x) 2 and the corresponding Hamiltonian is L(v, x) :=

(v ∈ Rn , x ∈ Tn ),

1 2 (p ∈ Rn , x ∈ Tn ). |p| + W (x) 2 Mather’s minimization problem is to find a Radon measure µ on the velocity–position configuration space Rn × Tn to minimize the generalized action 1 2 A[µ] := (1.1) |v| − W dµ, n n 2 R T H (p, x) :=

subject to the requirements that µ ≥ 0, µ(Rn × Tn ) = 1, v · Dφ dµ = 0 for all φ ∈ C 1 (Tn ), Rn Tn

and

(1.2) (1.3)

Rn Tn

v dµ = V .

The identity (1.3) is a weak form of flow invariance.

(1.4)

Towards a Quantum Analog of Weak KAM Theory

313

Quantum action minimizing states. We propose as a quantum version of Mather’s problem to find ψ minimizing the action h2 A[ψ] := (1.5) |Dψ|2 − W |ψ|2 dx, Tn 2 subject to the constraints that |ψ|2 dx = 1, n T ¯ ¯ · Dφ dx = 0 (ψDψ − ψD ψ) Tn

and

(1.6) for all φ ∈ C 1 (Tn ),

h ¯ ψDψ − ψD ψ¯ dx = V . 2i Tn

(1.7)

(1.8)

Here h denotes a positive constant. We always suppose that ψ has the Bloch wave form ψ =e

iP ·x h

ψˆ

for some P ∈ Rn and a periodic function ψˆ : Tn → C. If ψ is smooth, condition (1.7) reads ¯ ¯ = 0. div(ψDψ − ψD ψ)

(1.7 )

¯ This is the analogue of the flow invariance. The vector field j := ψDψ −ψD ψ¯ represents the flux. Remark. While it is presumably possible to introduce some sort of quantization for more general Lagrangians than L(v, x) = 21 |v|2 − W (x), most of the subsequent analysis would fail: we will from Sect. 3 onward rely upon some Cole-Hopf type transformations that depend upon the precise structure of this Lagrangian. 2. First and Second Variations, Local Minimizers In this section we provide some formal calculations concerning the first and second variations of our problem (1.5)−(1.8). These heuristic deductions motivate the constructions and computations in Sects. 3–8. Let us take the complex-valued state in polar form ψ = aeiu/ h ,

(2.1)

u=P ·x+v

(2.2)

where the phase u has the structure

for some Tn -periodic function v. Thus ψ has the requisite Bloch wave form. The action is then h2 a2 |Da|2 + |Du|2 − W a 2 dx, (2.3) A[ψ] = 2 Tn 2

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and the constraints (1.6)–(1.8) become a 2 dx = 1, Tn

div(a 2 Du) = 0, a 2 Du dx = V . Tn

(2.4) (2.5) (2.6)

2.1. First variation. Let {(u(τ ), a(τ ))}−1≤τ ≤1 be a smooth one-parameter family satisfying (2.4)–(2.6), with (u(0), a(0)) = (u, a). We suppose also that for each τ ∈ (−1, 1), we can write u(τ ) = P (τ ) · x + v(τ ), where P (τ ) ∈

Rn

and v(τ ) is Tn –periodic. Define h2 a 2 (τ ) j (τ ) := |Da(τ )|2 + |Du(τ )|2 − W a 2 (τ ) dx, 2 Tn 2

and hereafter write = Theorem 2.1. We have

(2.7)

d dτ . j (0)

= 0 for all variations if and only if h2 |Du|2 − a = a +W −E 2 2

(2.8)

for some real number E. We interpret (2.8) as the Euler–Lagrange equation for our minimization problem, and call ψ = aeiu/ h a critical point if this PDE is satisfied. Proof. 1. We first compute j = h2 Da · Da + aa |Du|2 + a 2 Du · Du − 2W aa dx. Tn

Next, differentiate (2.5), (2.6): div(2aa Du + a 2 Du ) = 0, 2aa Du + a 2 Du dx = 0, Tn

(2.9) (2.10)

and set τ = 0. Recall also that Du = P + Dv. Multiply (2.9) by v, integrate by parts, then take the inner product of (2.10) with P . Add the resulting expressions to find 2aa |Du|2 + a 2 Du · Du dx = 0. Tn

Hence j (0) =

h2 Da · Da − aa |Du|2 − 2W aa dx 2 |Du|2 h =2 + W a dx. a − a − 2 2 Tn Tn


315

Then j (0) = 0 for all such a provided h2 |Du|2 − a − + W a = −Ea, 2 2 for some real constant E. This is so since the variation a must satisfy the identity a a dx = 0, (2.11) Tn

which we obtain upon differentiating (2.4).

Remark. The foregoing deduction depends upon the implict assumption that we can construct a wide enough class of variations to permit our concluding (2.8) from the integral identities involving a . We will return to this point in Sect. 8.

2.2. Second variation. We next differentiate j twice with respect to τ : Theorem 2.2. If ψ = aeiu/ h is a critical point, then 2 2 2 2 2 2 |Du| j (0) = h |Da | + a |Du | − 2(a ) + W − E dx. 2 Tn Proof. 1. We have j =

Tn

(2.12)

h2 |Da |2 + h2 Da · Da + (a )2 |Du|2

+aa |Du|2 + 4aa Du · Du + a 2 |Du |2 +a 2 Du · Du − 2W (a )2 − 2W aa dx.

(2.13)

Differentiating (2.9), (2.10) again, we find div(2(a )2 Du + 2aa Du + 4aa Du + a 2 Du ) = 0 and

Tn

2(a )2 Du + 2aa Du + 4aa Du + a 2 Du dx = 0.

Set τ = 0. Multiply (2.14) by v, integrate, multiply (2.15) by P , and add: 2(a )2 |Du|2 + 2aa |Du|2 + 4aa Du · Du + a 2 Du · Du dx = 0. Tn

2. We employ this equality in (2.13): 2 |Du|2 h j (0) = +W + h2 |Da |2 2a − a − a 2 2 Tn

(2.14)

(2.15)

(2.16)

−(a )2 |Du|2 + a 2 |Du |2 − 2W (a )2 dx 2 2 2 2 |Du| = + W + a 2 |Du |2 dx 2a (−Ea) + h |Da | − 2(a ) 2 Tn |Du|2 = + W − E dx. h2 |Da |2 + a 2 |Du |2 − 2(a )2 2 Tn

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We have used here the identity Tn

a a + (a )2 dx = 0,

derived by twice differentiating (2.4).

2.3. Local minimizers. We continue to write ψ = aeiu/ h , and now assume as well that a>0

in Tn .

(2.17)

Observe that this follows from (2.8) and the strong maximum principle, provided a and u are smooth enough. Theorem 2.3. If ψ = aeiu/ h is a critical point and (2.17) holds, then

j (0) =

2 a a |Du | + a D dx > 0, n a 2

2

2

T

(2.18)

provided a = 0. In this case we call ψ = aeiu/ h a local minimizer. Proof. The Euler–Lagrange equation (2.8) asserts that −

h2 a = a 2

|Du|2 +W −E . 2

Hence 2 h a dx a 2 |Du |2 + h2 |Da |2 − 2(a )2 − 2 a Tn |Da|2 Da · Da = dx a 2 |Du |2 + h2 |Da |2 + h2 (a )2 2 − 2h2 a a a Tn 2 a 2 2 2 2 = a |Du | + h a D dx. n a

j (0) =

T

The last term is strictly positive, unless a ≡ λa for some constant λ = 0. But this is impossible, since a a dx = 0. Tn

3. Some Useful Identities Motivated by the foregoing calculations, our aim now is constructing an explicit state ψ, which will turn out to be a critical point, and indeed a local minimizer, of A[·], subject to (1.6) − (1.8). We start with two linear problems.


317

3.1. Dual eigenfunctions. Consider the dual eigenvalue problems: 2 − h2 w + hP · Dw − W w = E 0 w in Tn w is Tn -periodic

(3.1)

and

2

− h2 w ∗ − hP · Dw ∗ − W w ∗ = E 0 w ∗ w ∗ is Tn -periodic,

in Tn

(3.2)

where E 0 = E 0 (P ) ∈ R is the principal eigenvalue. Note carefully the minus signs in front of the potential W . We may assume the real eigenfunctions w, w ∗ to be positive in Tn and normalized so that ww ∗ dx = 1. (3.3) Tn

Furthermore, we can take w, w∗ and E 0 to be smooth in both the variables x and P . We employ a form of the Cole–Hopf transformation, to define v := −h log w (3.4) v ∗ := h log w∗ . Then

w = e−v/ h ∗ w ∗ = ev / h ,

and a calculation shows that − h2 v + 21 |P + Dv|2 + W = H¯ h (P ) v is Tn -periodic

(3.5)

in Tn

(3.6)

and

h ∗ 2 v ∗ v is

+ 21 |P + Dv ∗ |2 + W = H¯ h (P ) Tn -periodic,

in Tn

(3.7)

for |P |2 H¯ h (P ) := − E 0 (P ). 2 Standard PDE estimates applied to (3.6) and (3.7) provide the bounds |Dv|, |Dv ∗ | ≤ C, for a constant C depending only upon P and the potential W .

(3.8)

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3.2. Continuity and eikonal equations. Define σ := ww ∗

(3.9)

and u := P · x +

v + v∗ . 2

(3.10)

Note that although w, w∗ , v, v ∗ , u and σ depend on h, we will for notational simplicity mostly not write these functions with a subscript h. The importance of the product (3.9) of the eigenfunctions is noted also in Anantharaman [A]. According to (3.3), n σ > 0 in T , σ dx = 1. Tn

Theorem 3.1. (i) We have div(σ Du) = 0 in Tn .

(3.11)

(ii) Furthermore, h 1 1 |Du|2 + W − H¯ h (P ) = (v − v ∗ ) − |Dv − Dv ∗ |2 2 4 8

in Tn .

(3.12)

We call (3.11) the continuity (or transport) equation, and regard (3.12) as an eikonal equation with an error term on the right-hand side. Proof. 1. We compute h div(w∗ Dw − wDw ∗ ) = h(w∗ w − ww ∗ ) 2 2 2 h ∗ h ∗ = w w − w w h 2 2 2 = (w ∗ (−E 0 w − W w + hP · Dw) h −w(−E 0 w ∗ − W w ∗ − hP · Dw ∗ )) = 2(w ∗ P · Dw + wP · Dw ∗ ) = 2P · Dσ. But ∗ Dv ∗ Dv 1 w −w w = − σ (Dv + Dv ∗ ), w∗ Dw − wDw ∗ = w∗ − h h h and therefore 1 P · Dσ + div σ D(v + v∗ ) = 0. 2 This is (3.11).


319

2. Recalling the formula 1 1 |a − b|2 + |a + b|2 = |a|2 + |b|2 , 2 2 we compute 2 1 1 1 ∗ P + D(v + v ) = |(P + Dv) + (P + Dv ∗ )|2 2 2 8 1 1 1 = |P + Dv|2 + |P + Dv ∗ |2 − |Dv − Dv ∗ |2 . 4 4 8 Hence 1 |P + Dv|2 + W − H¯ h (P ) 2 1 1 1 + |P + Dv ∗ |2 + W − H¯ h (P ) − |D(v − v ∗ )|2 2 2 8 1 1 h ∗ 1 h v + − v − |D(v − v ∗ )|2 , = 2 2 2 2 8

1 1 |Du|2 + W − H¯ h (P ) = 2 2

owing to (3.6), (3.7).

Remark. We also have the identities h − σ − div ((P + Dv)σ ) = 0, 2 h − σ + div (P + Dv ∗ )σ = 0. 2

(3.13) (3.14)

For a quick derivation, observe first that h 1 σ = div D(v ∗ − v)σ . 2 2 Add and substract this from (3.11).

3.3. Integral identities involving Du and D 2 u. To simplify notation, we will hereafter write dσ := σ dx. Theorem 3.2. These formulas hold: 1 1 |Du|2 + W dσ = H¯ h (P ) + |Dv − Dv ∗ |2 dσ, n n 2 8 T T 1 2 2 2 2 ∗ 2 |D u| + |D v − D v | dσ = − W dσ. 4 Tn Tn

(3.15)

(3.16) (3.17)

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L.C. Evans

Proof. 1. In view of (3.12), 1 h |Du|2 + W − H¯ h (P ) dσ = (v − v ∗ )σ dx 4 Tn Tn 2 1 − |Dv − Dv ∗ |2 σ dx. 8 Tn v ∗ −v

But σ = ww∗ = e h , and therefore 1 h D(v ∗ − v) 2 ¯ |Du| + W − Hh (P ) dσ = − σ dx D(v − v ∗ ) · 4 Tn h Tn 2 1 − |Dv − Dv ∗ |2 σ dx 8 Tn 1 = |Dv − Dv ∗ |2 dσ. 8 Tn 2. Now differentiate the identity (3.12) twice with respect to xk : Du · Duxk xk + Duxk · Duxk + Wxk xk h 1 = (v − v ∗ )xk xk − D(v − v ∗ ) · D(v − v ∗ )xk xk 4 4 1 − D(v − v ∗ )xk · D(v − v ∗ )xk . 4 Multiply by σ and integrate: Du · Duxk xk + Duxk · Duxk + Wxk xk dσ Tn h 1 ∗ = (v − v )xk xk dσ − D(v − v ∗ ) · D(v − v ∗ )xk xk dσ 4 Tn 4 Tn 1 − D(v − v ∗ )xk · D(v − v ∗ )xk dσ. 4 Tn According to Theorem 3.1, the first term on the left vanishes. We integrate by parts in the first term on the right, and thereby derive an expression that cancels the second term on the right. Next sum on k, to derive (3.17). ¯h 4. First and Second Derivatives of H As explained in [E-G], the behavior of various expressions as functions of P is important: Theorem 4.1. We have D H¯ h (P ) = D 2 H¯ h (P ) =

Tn

Du dσ,

2 2 DxP u ⊗ DxP u dσ 1 2 + D 2 (v − v ∗ ) ⊗ DxP (v − v ∗ ) dσ. 4 Tn xP

(4.1)

Tn

In particular, H¯ h is a convex function of P .

(4.2)


321

Our notation means that the (l, m)th component of thefirst term on the right-hand side of (4.2) is Tn uxi Pl uxi Pm dσ and of the second term is 41 Tn (v −v ∗ )xi Pl (v −v ∗ )xi Pm dσ. Proof. 1. We differentiate (3.1) and (3.2) with respect to Pk : h2 wPk + hP · DwPk − W wPk − E 0 wPk = EP0 k w − hwxk , 2 h2 − wP∗ k − hP · DwP∗ k − W wP∗ k − E 0 wP∗ k = EP0 k w ∗ + hwx∗k . 2

−

(4.3) (4.4)

Multiply (4.3) by w∗ and integrate over Tn . Since w ∗ solves (3.2), we deduce 0 ∗ ∗ Dww dx = − Dv ww dx = − Dv dσ. DE (P ) = h Tn

Tn

Tn

Similarly, we multiply (4.4) by w and integrate: Dw ∗ w dx = − DE 0 (P ) = −h Tn

Then D H¯ h (P ) = P − DE 0 (P ) = P +

Tn

Dv ∗ dσ.

1 Dv + Dv ∗ dσ = Du dσ. 2 Tn Tn

2. Next, differentiate the identity (3.12) with respect to Pk and Pl : h H¯ h,Pk Pl = Du · DuPk Pl + DuPk · DuPl − (v − v ∗ )Pk Pl 4 1 1 + D(v − v ∗ ) · D(v − v ∗ )Pk Pl + D(v − v ∗ )Pk · D(v − v ∗ )Pl . 4 4 Multiply by σ and integrate, to discover h ¯ Du · DuPk Pl + DuPk · DuPl dσ − (v − v ∗ )Pk Pl dσ Hh,Pk Pl = 4 Tn Tn 1 + D(v − v ∗ ) · D(v − v ∗ )Pk Pl dσ 4 Tn 1 + D(v − v ∗ )Pk · D(v − v ∗ )Pl dσ. 4 Tn In view of Theorem 3.1 and the periodicity of uPk Pl , the first term on the right vanishes. v ∗ −v

Since σ = ww∗ = e h , we can integrate by parts in the third term on the right, obtaining an expression that cancels the fourth term. Formula (4.2) results. As an application of Theorem 4.1, we modify some ideas from [E-G] and [E2] to discuss the effects of a nonresonance condition on the asymptotics as h → 0. We will explain in the next section that as h → 0 the functions H¯ h converge uniformly on compact sets to the convex function H¯ , the effective Hamiltonian in the sense of Lions–Papanicolaou–Varadhan [L-P-V]. Let us suppose that H¯ is differentiable at P and that V = D H¯ (P ) satisfies V · m = 0

for each m ∈ Zn , m = 0.

(4.5)

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L.C. Evans

Theorem 4.2. Suppose also that D 2 H¯ h (P ) is bounded as h → 0. Then lim (DP u) dσ = (X) dX h→0 Tn

Tn

(4.6)

for each continuous, Tn -periodic function . We discuss in [E-G] that this statement is consistent with the classical assertion that the Hamiltonian dynamics in the X, P variables, where X := DP u, correspond to the trivial motion X˙ = V , P˙ = 0. Proof. Fix any m ∈ Zn , and observe that the function e2πim·DP u = e2πim·x eπim·DP (v+v

∗)

is Tn -periodic. Consequently

2πim·DP u 0= Du · D e e2πim·DP u mk uxj uxj Pk dσ dσ = 2πi Tn Tn = 2π i e2πim·DP u mk H¯ h,Pk dσ n T e2πim·DP u mk (uxj uxj Pk − H¯ h,Pk ) dσ + 2π i

(4.7)

Tn

= : 2π i(A + B). We claim now that B = O(h) as h → 0. To confirm this, notice first that our differentiating (3.12) gives the identity uxj uxj Pk − H¯ h,Pk =

h 1 (v − v ∗ )Pk − (v − v ∗ )xj (v − v ∗ )xj Pk ; 4 4

and therefore h (uxj uxj Pk − H¯ h,Pk )σ = (v − v ∗ )xj Pk σ x . j 4 Hence

B=

Tn

e2πim·DP u mk (uxj uxj Pk − H¯ h,Pk )σ dx

h e2πim·DP u mk (v − v ∗ )xj Pk σ x dx j 4 Tn hπ i =− e2πim·DP u ml uxj Pl mk (v − v ∗ )xj Pk dσ. 2 Tn =

So

|B| ≤ Ch

Tn

2 2 |DxP u|2 + |DxP (v − v ∗ )|2 dσ = O(h)

according to (4.2), since D 2 H¯ h (P ) is bounded. This proves (4.8).

(4.8)


But then (4.7) implies m · D H¯ h (P )

Tn

e

323

2πim·DP u

dσ = |B| ≤ O(h).

We will see in Sect. 5 that H¯ h → H¯ , locally uniformly. Since therefore D H¯ h (P ) → D H¯ (P ) = V and since m · V = 0, we deduce that e2πim·DP u dσ = 0. lim h→0 Tn

This limit holds for all m = 0, and hence lim (DP u) dσ = h→0 Tn

for each Tn -periodic function .

Tn

(X) dX

5. An Identity Involving Exact Solutions of the Eikonal Equation Assume next that vˆ is a Lipschitz continuous almost everywhere solution of the cell problem 1 ˆ 2 + W = H¯ (P ) in Tn 2 |P + D v| (5.1) n vˆ is T -periodic. The term on the right-hand side of (5.1) is the effective Hamiltonian in the sense of Lions– Papanicolaou–Varadhan [L-P-V], a central assertion of which is that for a given vector P problem (5.1) is solvable in the sense of viscosity solutions. (Our H¯ corresponds to Mather’s function α, and is equivalent also to Mañe´ ’s constant.) Write uˆ := P · x + v; ˆ

(5.2)

so that 1 |D u| ˆ 2 + W = H¯ (P ) 2

almost everywhere in Tn .

(5.3)

Theorem 5.1. This formula holds: 2 1 1 D(v + v ∗ ) − D vˆ dσ + 1 |Dv − Dv ∗ |2 dσ = H¯ (P ) − H¯ h (P ). (5.4) 2 Tn 2 8 Tn Proof. We employ the identity 1 1 1 |a − b|2 + a · (b − a) = |b|2 − |a|2 2 2 2 with 1 a = P + D(v + v ∗ ) = Du, b = P + D vˆ = D u, ˆ 2

324

L.C. Evans

to discover

2 1 1 1 D(v + v ∗ ) − D vˆ dσ + Du · D(vˆ − (v + v ∗ )) dσ n 2 Tn 2 2 T 1 1 2 = |D u| ˆ + W dσ − |Du|2 + W dσ. n n 2 2 T T

Owing to Theorem 3.1 and the periodicity of v, v ∗ and v, ˆ the second term on the left vanishes. Consequently (5.3) and (3.16) imply 2 1 1 D(v + v ∗ ) − D vˆ dσ = H¯ (P ) − H¯ h (P ) − 1 |Dv − Dv ∗ |2 dσ. 2 Tn 2 8 Tn As an application, we have the estimate: Theorem 5.2. (i) For each P ∈ Rn , H¯ h (P ) ≤ H¯ (P ) ≤ H¯ h (P ) + O(h) as h → 0.

(5.5)

2 1 D(v + v ∗ ) − D vˆ dσ + |Dv − Dv ∗ |2 dσ = O(h). Tn 2 Tn

(5.6)

(ii) Hence

Proof. 1. According to (5.4), H¯ h (P ) ≤ H¯ (P ). In addition, we have the minimax formula 1 2 ¯ H (P ) = inf max |P + Dv| + W (x) . v∈C 1 (Tn ) x∈Tn 2

(5.7)

(See for instance the appendix of [E2] for a quick proof due to A. Fathi.) Furthermore, standard PDE estimates deduce from (3.6), (3.7) the one-sided second derivative bounds vξ ξ ≤ C

and vξ∗ξ ≥ −C,

(5.8)

for some constant C and any unit vector ξ . Then formula (3.12) implies 2 1 1 ∗ ¯ ) P + D(v + v + W ≤ Hh (P ) + Ch. 2 2 Consequently, we can deduce from (5.7) that

2 1 1 ∗ ¯ H (P ) ≤ max P + D(v + v ) + W (x) ≤ H¯ h (P ) + Ch. 2 x∈Tn 2 2. Statement (ii) follows from (5.4), (5.5).

6. Quantum Lagrangian Calculations In this section we draw some connections between the minimization problems, both classical and quantum, discussed in Sect. 1–2, and the explicitly constructed state ψ studied in Sect. 3–5.


325 v ∗ −v

6.1. Quantum action. As before, we have a = σ 1/2 = e 2h and we continue to write iu ψ = ae h . Recall as well that the action of ψ is h2 |Dψ|2 − W |ψ|2 dx. A[ψ] := Tn 2 We next demonstrate that ψ satisfies the Euler–Lagrange equation (2.8). Theorem 6.1. We have h2 a 1 |Du|2 + W − H¯ h (P ) = − in Tn . 2 2 a In particular, ψ is a critical point of the action A[·], subject to (1.6)–(1.8).

(6.1)

According to Theorem 2.3, ψ is a local minimizer as well. I conjecture that ψ is in fact a global minimizer, but am unable to prove this. Proof. Since a = e

v ∗ −v 2h

, we compute h2 v−v∗ v∗ −v

a − = − e 2h e 2h 2 a 2 ∗ v −v h2 v−v∗ 1 1 ∗ ∗ 2 2h =− e (v − v) + 2 |D(v − v)| e 2h 2 2h 4h h 1 = (v − v ∗ ) − |D(v − v ∗ )|2 4 8 1 = |Du|2 + W − H¯ h (P ), 2 the last equality being (3.12). We now compute the action of ψ. To do so, we first introduce L¯ h , the Legendre transform of H¯ h , and also write ¯ Vh := D Hh (P ) = Du dσ. (6.2) h2

Tn

Theorem 6.2. The quantum action of ψ is A[ψ] = L¯ h (Vh ). Proof. Let us employ (3.11) and (3.13), to deduce h2 A[ψ] = |Dψ|2 − W |ψ|2 dx n 2 T h2 a2 = |Da|2 + |Du|2 − W a 2 dx 2 Tn 2 2 h 1 2 2 2 2 = |Da| dx + |Du| + W a 2 dx |Du| a dx − Tn 2 Tn Tn 2 1 1 ∗ 2 ∗ = P + D(v + v ) · Du dσ |Dv − Dv | dσ + 8 Tn 2 Tn 1 − |Du|2 + W dσ n 2 T = P · Vh − H¯ h (P ) = L¯ h (Vh ).

(6.3)

326

L.C. Evans

6.2. Convergence as h→0. We now determine the behavior of σ and u, defined by (3.9), (3.10), as h → 0. For the remainder of this section we for clarity add subscripts “h” to display the dependence on this parameter. Thus σh = wh wh∗ ,

uh = P · x +

vh + vh∗ , etc. 2

Define a measure µh on velocity-position configuration space by requiring (v, x) dµh := (Duh (x), x) dσh Rn Tn

Tn

(6.4)

for each continuous : Rn × Tn → R. In view of estimate (3.8) the measures {µh }h>0 have uniformly bounded support, and we can consequently obtain a sequence hj → 0 and a probability measure µ such that µhj µ

weakly as measures.

We may also suppose that Vhj → V

in Rn .

(6.5)

Theorem 6.3. The measure µ solves Mather’s minimization problem (1.1)–(1.4). Proof. 1. First we check that µ satisfies the constraints (1.2)–(1.4), the first of which is clear since µ is a probability measure. If furthermore φ ∈ C 1 (Tn ), then v · Dφ dµ = lim Duhj · Dφ dσhj = 0, j →∞ Tn

Rn Tn

since div(σh Duh ) = 0. This is (1.3); and (1.4) similarly holds since v dµ = lim Duhj dσhj = lim Vhj = V . j →∞ Tn

Rn Tn

j →∞

2. Recall from (5.5) that H¯ h → H¯ , uniformly on compact sets. Therefore Theorem 6.2 and (6.5) imply ¯ ) = lim L¯ hj (Vhj ) L(V j →∞

= lim

j →∞ Tn

h2j 2

|Dahj |2 +

ah2j

|Duhj |2 − W ah2j dx 1 − Dvh∗j |2 dσhj + lim |Duhj |2 − W dσhj (6.6) j →∞ Tn 2 2

1 |Dvhj j →∞ 8 Tn 1 2 = lim |v| − W dµhj j →∞ Rn Tn 2 1 2 |v| − W dµ = A[µ]. = n n 2 R T = lim

Here we recalled (5.6) to ensure that the first term on the third line goes to 0.


327

3. Suppose now ν is any other measure satisfying (1.2) − (1.4). Let u = η ∗ u, ˆ where η is a standard mollifier and as before uˆ solves the eikonal equation (5.3) for any ¯ ). Then P ∈ ∂ L(V 1 |Du |2 + W ≤ H¯ (P ) + C 2 for some constant C, everywhere on Tn . Therefore 1 2 1 2 1 |v| − W dν ≥ |v| + |Du |2 dν − H¯ (P ) − C A[ν] = n n n n 2 2 2 R T R T ≥ v · Du dν − H¯ (P ) − C n Tn R = v · Dv dν + P · V − H¯ (P ) − C Rn Tn

¯ ) − C. = L(V ¯ ) is less than or equal to the action of any measure This holds for each > 0, and so L(V ν satisfying (1.2) − (1.4). But then (6.6) guarantees that the measure µ is a minimizer. Anantharaman [A] provides interesting and additional information about the limit minimizing measure. Limits of the eikonal equations. Upon passing if necessary to a further subsequence, we deduce from (3.6), (3.7) that vhj → v, vh∗j → v ∗

uniformly on Tn ,

where v, v ∗ are viscosity solutions of the respective PDE 1 |P + Dv|2 + W = H¯ (P ) 2

in Tn

(6.7)

and 1 − |P + Dv ∗ |2 − W = −H¯ (P ) 2

in Tn .

(6.8)

In particular the Lipschitz continuous functions v, v ∗ solve (6.7), (6.8) almost everywhere. Therefore uhj → u := P · x +

v + v∗ uniformly on Tn , 2

where 1 |Du|2 + W ≤ H¯ (P ) 2

almost everywhere.

Finally, write σ := projx µ

(6.9)

328

L.C. Evans

for the projection µ onto Tn . Then σhj σ

weakly as measures.

Applying the regularity theory from [E-G], we deduce that Dv, Dv ∗ , and therefore Du, exist for each point in spt σ , and 1 |Du|2 + W = H¯ (P ) 2

on spt σ.

(6.10)

7. Quantum Hamiltonian Calculations 7.1. Quasimodes. The observations in the last section show that our construction in Sect. 3 is a sort of semiclassical “quantization” of weak KAM theory. We next show that ψ built above is an approximate solution of the stationary Schrödinger equation −

h2 ψ + W ψ = Eψ 2

in Tn . v ∗ −v

Notice the plus sign in front of the potential W . As usual, a = σ 1/2 = e 2h and iu ψ = ae h . Then h2 ih div(a 2 Du) h2 a 1 − ψ + W ψ − Eψ = ψ − |Du|2 + W − E ψ − ψ 2 2 2 a2 2 a = : A + B + C. (7.1) In view of Theorem 3.1, B ≡ 0. Now take E = H¯ h (P ). According to Theorem 6.1, A ≡ C; that is, the formal O(1)–term identically equals the formal O(h2 )–term in the expansion (7.1). Therefore h2 1 2 ¯ − ψ + W ψ − Eψ = 2 |Du| + W − Hh (P ) ψ (7.2) 2 2 for E = H¯ h (P ). It is sometimes useful to rewrite this as −

h2 ψ = (|Du|2 + W − H¯ h (P ))ψ. 2

Theorem 7.1. If E = H¯ h (P ), −

h2 ψ + W ψ − Eψ = O(h), 2

the right hand side estimated in L2 (Tn ).

(7.3)


329

Proof. Define the remainder term 1 2 ¯ R := 2 |Du| + W − Hh (P ) ψ. 2

(7.4)

Then 2 1 h 1 2 ∗ ∗ 2 |R| dx = dσ (v − v ) − |D(v − v )| 4 Tn 8 Tn 4 h2 h = ((v − v ∗ ))2 − (v − v ∗ )|D(v − v ∗ )|2 16 Tn 16 1 ∗ 4 + |D(v − v )| dσ. 64 Observe now that

Since

1 16

>

h − (v − v ∗ )|D(v − v ∗ )|2 dσ 16 Tn v ∗ −v h =− (v − v ∗ )|D(v − v ∗ )|2 e h dx 16 Tn 1 = |D(v − v ∗ )|2 D(v − v ∗ ) · D(v ∗ − v) dσ 16 Tn h + D(v − v ∗ ) · D 2 (v − v ∗ )D(v − v ∗ ) dσ 8 Tn 1 =− |D(v − v ∗ )|4 dσ 16 Tn h + D(v − v ∗ ) · D 2 (v − v ∗ )D(v − v ∗ ) dσ. 8 Tn

1 64 ,

Tn

we derive for some constant C the estimate: 2 ∗ 4 2 |R| dx + |D(v − v )| dσ ≤ Ch |D 2 (v − v ∗ )|2 dσ. Tn

Tn

(7.5)

We deduce finally from (7.5) and (3.17) that R is of order at most O(h) in L2 (Tn ).

Remark. The O(h)–error term is not especially good, and indeed M. Zworski has outlined for me some other constructions building quasimodes with similar error estimates in quite general circumstances. Estimate (7.5) does show that if E = H¯ h (P ) and if |D 2 (v − v ∗ )|2 dσ = o(1), (7.6) Tn

we would then have the better error bound −

h2 ψ + W ψ − Eψ = o(h) 2

as h → 0

(7.7)

in L2 (Tn ). We may hope that assertion (7.6) is true in some generality, although it can fail, as the following shows:

330

L.C. Evans

Example. Assume that P = 0 and that the potential W attains its maximum at a unique point x0 ∈ Tn , where W (x0 ) < 0. In this situation we can readily check that H¯ (0) = W (x0 ). Since P = 0, we can take w ≡ w∗ ; whence v ≡ −v ∗ and so u ≡ 0. Then according to (3.16) and (5.15), lim W dσ = W (x0 ) = max W. (7.8) Tn

h→0 Tn

So the weak limit of the measures σ as h → 0 is the unit mass at x0 . Consequently, the identity (3.17) implies lim |D 2 (v − v ∗ )|2 dσ = −4W (x0 ) > 0. (7.9) h→0 Tn

This example is from the forthcoming paper of Y. Yu [Y], who provides a very complete analysis of our problem in one dimension. In particular, if n = 1 and H¯ (P ) > min H¯ , the error term is indeed o(h) in L2 as h → 0. 7.2. Comparison with stochastic mechanics. There are formal connections with the Guerra–Morato and Nelson variational principle in stochastic quantum mechanics, as set forth in Nelson [N], Guerra–Morato [G-M], Yasue [Ya], Carlen [C], etc. (I thank A. Majda for some of these references.) I attempt here to explain the link by recasting their form of the action into our setting and notation, disregarding the probabilistic interpretations. In effect, then, the action of Guerra–Morato becomes 1 ∗ ˜ A[ψ] := (7.10) Dv · Dv − W a 2 dx Tn 2 v ∗ −v

for ψ = aeiu/ h , a = e 2h , u = 21 (v + v ∗ ), P = 0. (The Lagrangian density 1 ∗ 2 2 Dv · Dv − W a here should be compared with formula (93) in Guerra–Morato [G-M]. See also (14.31) in Nelson [N].) We rewrite this, observing that −

a2 a2 h2 |Da|2 + |Du|2 = Dv ∗ · Dv. 2 2 2

Consequently ˜ A[ψ] =

Tn

−

h2 a2 |Da|2 + |Du|2 − W a 2 dx. 2 2

(7.11)

This action differs from ours due to the sign change in the first term. ˜ Theorem 7.2. Let ψ = aeiu/ h be a smooth critical point of the action A[·], subject to the constants (1.6)–(1.8). Then for some real constant E: −

h2 ψ + W ψ = Eψ 2

in Tn ,

and so ψ is an exact solution of this stationary Schrödinger equation.

(7.12)


331

Proof. We deduce, as in the proof of Theorem 2.1, that h2 |Du|2 − a + a + W − E = 0, 2 2 with a sign change as compared with (2.8). Thus if we write out (7.1), we have − and B ≡ 0, A + C ≡ 0.

h2 ψ + W ψ − Eψ = A + B + C, 2

8. Connections with Linear Elliptic Homogenization This section works out some relationships between H¯ h and homogenization theory for divergence–structure, second order elliptic PDE: see for instance Bensoussan–Lions– Papanicolaou [B-L-P]. Our conclusions are very similar to those of Capdeboscq [Cp], and the calculations in Pedersen [P] are related as well. Let A = ((aij )) be symmetric, positive definite, and Tn -periodic. Suppose U is a bounded, open subset of Rn , with a smooth boundary. We consider this boundary value problem for an elliptic PDE with rapidly varying coefficients: − aij xε uεxi x = f in U j

uε = 0

on ∂U .

Then uε u weakly in H01 (U ), u solving the limit problem −a¯ ij uxi xj = f in U u = 0 on ∂U . The effective diffusion coefficient matrix A¯ = ((a¯ ij )) is determined as follows [B-L-P]. For j = 1, . . . , n, let χ j solve the corrector problem

j − akl χxk = (aj l )xl in Tn xl (8.1) χ j is Tn -periodic. Let us then for i, j = 1, . . . , n define j a¯ ij := aij − akl χxi k χxl dx. Tn

(8.2)

Theorem 8.1. Define Vh = D H¯ h (P ). Then ¯ = Vh , AP

(8.3)

where A¯ = ((a¯ ij )) is the effective diffusion coefficient matrix corresponding to A := a 2 I = ((a 2 δij )). Notice in (8.4) that a 2 = σ = ww∗ depends on both h and P .

(8.4)

332

L.C. Evans

Proof. For the special case of the diagonal matrix A given by (8.4), the corrector PDE (8.1) reads j −(a 2 χxk )xk = (a 2 )xj in Tn (8.5) χ j is Tn -periodic. Now u = x · P + 21 (v + v ∗ ) solves div(σ Du) = 0, and therefore

1 −div a 2 D(v + v ∗ ) = div(a2 P) = D(a2 ) · P. 2

Hence v + v∗ = Pi χ i . 2

(8.6)

Consequently, for j = 1, . . . , n, uxj dσ = (χ i + xi )xj Pi dσ Vh,j = Tn Tn j = Pj − χ i Pi σxj dx = Pj + χ i Pi (a 2 χxk )xk dx n n T T 2 i j ¯ )j . = Pj − Pi a χxk χxk dx = (AP Tn

Remark (Constructing a family of variations). We return finally to a point left open in Sect. 2. Recall that we introduced a one parameter family of variations {(u(τ ), a(τ ))}−1≤τ ≤1 satisfying the constraints (2.4)–(2.6), with (u(0), a(0)) = (u, a). We assumed also that u(τ ) = P (τ ) · x + v(τ ),

(8.7)

for P (τ ) ∈ Rn and v(τ ) is Tn -periodic. da was arbitrary, To finish up the proof of Theorem 2.1 we needed to know that a := dτ subject only to the integral identity (2.11). We show next we can indeed do so, provided a>0

in Tn .

(8.8)

To confirm this, take a smooth function a(·) of τ satisfying (2.4) and a(0) = a. For a given P ∈ Rn , we invoke the Fredholm alternative to solve −div(a 2 (τ )Dv) = div(a 2 (τ )P ) for a periodic function v = v(τ ). Then u(τ ) = P ·x +v(τ ) solves div(a(τ )2 Du(τ )) = 0, and the issue is whether for small τ we can select P = P (τ ) so that a 2 (τ )Du(τ ) dx = V . (8.9) Tn


333

We next introduce the function = { 1 , . . . , n } defined by (P ) := a 2 Du dx = P + a 2 Dv dx. Tn

Tn

Now since − div(a 2 Dv) = div(a 2 P ), we have −(a 2 vxk Pl )xk = (a 2 )xl ; and so vPl = χ l in the notation above. Therefore, as in the proof of Theorem 8.1, we can calculate k 2 Pl = δkl + a vxk Pl dx = δkl − (a 2 )xk vPl dx Tn Tn = δkl + (a 2 vxi Pk )xi vPl dx = δkl − a 2 vxi Pk vxi Pl dx = a¯ kl .

Tn

Tn

In other words, D(P ) = A¯ and the latter matrix is nonsingular. Hence the Implicit Function Theorem ensures for small τ that we can find P = P (τ ) satisfying (8.9). References [A]

Anantharaman, N.: Gibbs measures on path space and viscous approximation to action-minimizing measures. Trans. AMS, 2003 [B-L-P] Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. Amsterdam: North-Holland, 1978 [Cp] Capdeboscq, Y.: Homogenization of a neutronic critical diffusion problem with drift. Proc. Royal Soc. Edinburgh 132, 567–594 (2002) [C] Carlen, E.: Conservative diffusions. Commun. Math. Phys. 94, 293–315 (1984) [E1] Evans, L.C.: Effective Hamiltonians and quantum states. Seminaire Equations aux Dérivées Partielles, Ecole Polytechnique 2000–2001 [E2] Evans, L.C.: Some new PDE methods for weak KAM theory. Calculus of Variations and PDE 17, 159–177 (2003) [E-G] Evans, L.C., Gomes, D.: Effective Hamiltonians and averaging for Hamiltonian Dynamics I. Arch. Rat. Mech. Anal. 157, 1–33 (2001) [F1] Fathi, A.: Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. C. R. Acad. Sci. Paris Sr. I Math. 324, 1043–1046 (1997) [F2] Fathi, A.: Solutions KAM faibles conjuguées et barrières de Peierls. C. R. Acad. Sci. Paris Sr. I Math. 325, 649–652 (1997) [F3] Fathi, A.: Weak KAM theory in Lagrangian Dynamics, Preliminary Version. Lecture notes 2001 [G] Gomes, D.: A stochastic analogue of Aubry-Mather theory. Nonlinearity 15, 581–603 (2002) [G-M] Guerra, F., Morato, L.: Quantization of dynamical systems and stochastic control theory. Phys. Rev. D 27, 1774–1786 (1983) [H-K] Holcman, D., Kupka, I.: Singular perturbations and first order PDE on manifolds. Preprint, 2002 [L-P-V] Lions, P.-L., Papanicolaou, G., Varadhan, S. R. S.: Homogenization of Hamilton–Jacobi equations. Unpublished, circa 1988 [M1] Mather, J.: Minimal measures. Comment. Math Helv. 64, 375–394 (1989) [M2] Mather, J.: Action minimizing invariant measures for positive definite Lagrangian systems. Math. Zeits. 207, 169–207 (1991) [M-F] Mather, J., Forni, G.: Action minimizing orbits in Hamiltonian systems. In: Transition to Chaos in Classical and Quantum Mechanics, Lecture Notes in Math 1589, S. Graffi, ed., BerlinHeidelberg-New York: Springer, 1994

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[N] [P]

Nelson, E.: Quantum Fluctuations. Princeton, NJ: Princeton University Press, 1985 Pedersen, F.B.: Simple derivation of the effective mass equation using a multiple-scale technique. Euro. J. Phys. 18, 43–45 (1997) Yasue, K.: Stochastic calculus of variations. J. Funct. Anal. 41, 327–340 (1981) Yu, Y.: Error estimates for a quantization of a Mather set. To appear

[Ya] [Y]

Communicated by P. Constantin


Communications in


A Positive Mass Theorem for Spaces with Asymptotic SUSY Compactification Xianzhe Dai Department of Mathematics, University of California, Santa Barbara, CA 93106, USA. E-mail: [email protected] Received: 16 June 2003 / Accepted: 8 August 2003 Published online: 25 November 2003 – © Springer-Verlag 2003

Abstract: We prove a positive mass theorem for spaces which asymptotically approach a flat Euclidean space times a Calabi-Yau manifold (or any special honolomy manifold except the quaternionic Kähler). This is motivated by the very recent work of HertogHorowitz-Maeda [HHM]. In general relativity, isolated gravitational systems are modelled by asymptotically flat spacetimes. The spatial slices of such spacetime are then asymptotically flat Riemannian manifolds. That is, Riemannian manifolds (M n , g) such that M = M0 ∪ M∞ with M0 compact and M∞ Rn − BR (0) for some R > 0 so that in the induced Euclidean coordinates the metric satisfies the asymptotic conditions gij = δij + O(r −τ ), ∂k gij = O(r −τ −1 ), ∂k ∂l gij = O(r −τ −2 ).

(0.1)

Here τ > 0 is the asymptotic order and r is the Euclidean distance to a base point. The total mass (the ADM mass) of the gravitational system can then be defined via a flux integral [ADM, LP] 1 m(g) = lim (∂i gij − ∂j gii ) ∗ dxj . (0.2) R→∞ 4ωn SR Here ωn denotes the volume of the n − 1 sphere and SR the Euclidean sphere with radius R centered at the base point. If τ > n−2 2 and n ≥ 2, then m(g) is independent of the asymptotic coordinates xi , and thus is an invariant of the metric. The positive mass theorem [SY1, SY2, SY3, Wi1] says that this total mass is nonnegative provided one has nonnegative local energy density. Theorem 0.1 (Schoen-Yau, Witten). Suppose (M n , g) is an asymptotically flat spin manifold of dimension n ≥ 3 and of order τ > n−2 2 . If the scalar curvature R ≥ 0, then m(g) ≥ 0 and m(g) = 0 if and only if M = Rn .

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Remark. The scalar curvature R is the local energy density. According to string theory [CHSW], our universe is really ten dimensional, modelled by M 3,1 × X, where X is a Calabi-Yau 3-fold. This is the so called Calabi-Yau compactification, which motivates the spaces we now consider. We consider the complete Riemannian manifolds (M n , g) such that M = M0 ∪ M∞ with M0 compact and M∞ (Rk − BR (0)) × X for some R > 0 and X a compact simply connected Calabi-Yau manifold (or with any other special honolomy except Sp(m) · Sp(1)) so that the metric on M∞ satisfies ◦

◦

◦

◦ ◦

g =g +h, g = gRk + gX , h = O(r −τ ), ∇ h = O(r −τ −1 ), ∇ ∇ h = O(r −τ −2 ). (0.3) ◦

◦

Here ∇ is the Levi-Civita connection of g , τ > 0 is the asymptotical order. We will call M a space with asymptotic SUSY compactification. The mass for such a space is then defined by ◦ ◦ 1 m(g) = lim (∇ ea0 gj a − ∇ e0 gaa ) ∗ dxj dvol(X). (0.4) j R→∞ 4ωk vol(X) SR ×X ◦

Here {ea0 } = { ∂x∂ i , fα } is an orthornormal basis of g , the ∗ operator is the one on the Euclidean factor, the index i, j run over the Euclidean factor and the index α runs over X while the index a runs over the full index of the manifold. In fact, this reduces to 1 m(g) = lim (∂i gij − ∂j gaa ) ∗ dxj dvol(X). R→∞ 4ωk vol(X) SR ×X Remark. If τ >

k−2 2

and k ≥ 2, then m(g) is independent of the asymptotic coordinates.

Our main result is Theorem 0.2. Let (M, g) be a complete spin manifold as above and the asymptotic order τ > k−2 2 and k ≥ 3. If M has nonnegative scalar curvature, then m(g) ≥ 0 and m(g) = 0 if and only if M = Rk × X. Remark. The result extends without change to the case with more than one end. Remark. Just like in the usual case, the restriction k ≥ 3 has to do with getting the correct spin structure at the ends. See Sect. 5 for additional comments regarding the spin structures of the ends. Our motivation comes from a very recent work of Hertog-Horowitz-Maeda [HHM] on the Calabi-Yau compactifications. Using the existence result of Stolz [S1, S2] on metrics of positive scalar curvature, they constructed classical configurations which have regions of (arbitrarily large) negative energy density as seen from the four dimensional perspective. This should be contrasted with the positivity (nonnegativity) of the total mass, as guaranteed by Theorem 0.2. According to [HHM], physical consequences of the negative energy density include possible violation of Cosmic Censorship and new thermal instability. The Lorentzian version of Theorem 0.2 will be discussed in a separate paper.

Positive Mass Theorem for Spaces with Asymptotic SUSY Compactification

337

1. Manifolds with Special Holonomy For a complete Riemannian manifold (M n , g), the holonomy group Hol(g) (with respect to a base point) is the subgroup of O(n) generated by parallel translations along all loops at the base point. For simply connected irreducible nonsymmetric spaces, Berger has given a complete classification of possible holonomy groups, namely, SO(n) which is the generic situation, U (m) (if n = 2m) which is Kähler, SU (m) for Calabi-Yau, Sp(m) · Sp(1) (if n = 4m) which is called quaternionic Kähler, Sp(m) which is called hyper-Kähler, Spin(7) (if n = 8), and G2 (if n = 7). Except for the generic and Kähler cases, the rest are called special holonomy. If a Riemannian manifold (M, g) is spin, then one can consider spinors φ on M which are sections of the spinor bundle S. The Levi-Civita connection ∇ of g lifts to a connection of the spinor bundle, which will still be denoted by the same notation. In fact, any metric connections lift in the same way. The Dirac operator Dφ = ei · ∇ei φ, where ei is a local orthonormal basis of M and ei · is the Clifford multiplication. A spinor φ is parallel if ∇φ = 0. Implicitly, all these depend on the underlying spin structure, which is in one-to-one correspondence with elements of H 1 (M, Z2 ) [LM]. Thus, for simply connected manifolds, one has a unique spin structure. It seems that the issue of spin structure in this context is a subtle one, deserving further study. (See also Sect. 5.) All manifolds with special holonomy, with the exception of the quaternionic Kähler ones, carry nonzero parallel spinor. In fact, one has the following theorem of McKenzie Wang [Wa]. Theorem 1.1. Let (M, g) be a complete, simply connected, irreducible Riemannian spin manifold and N be the dimension of parallel spinors. Then N > 0 if and only if the holonomy group is one of SU (m), Sp(m), Spin(7), G2 . Remark. Wang [Wa] actually characterizes each special holonomy by the number of parallel spinors. Remark. Manifolds with parallel spinors are called supersymmetric (SUSY) in physics literature. 2. Proof of Theorem 0.2 Our proof is an extension of Witten’s spinor proof [Wi1]. Here we follow the idea of Anderson and Dahl [AnD] and use the following alternative formula for the Lichnerowicz formula. Lemma 2.1. Given a spinor φ on a Riemannian spin manifold, define a 1-form α via α(X) = (∇X + X · D)φ, φ . Then div α =

R 2 |φ| + |∇φ|2 − |Dφ|2 . 4

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Proof. Choose an orthonormal basis ea such that ∇ea = 0 at the given point. Then (Einstein summation enforced) div α = (∇ea α)(ea ) = ea (α(ea )) = (∇ea + ea · D)φ, ∇ea φ + ∇ea (∇ea + ea · D)φ, φ

= |∇φ|2 − |Dφ|2 + (δab + ea · eb ·)∇ea ∇eb φ, φ . The last term is just

1 1 R [ea ·, eb ·]∇ea ∇eb φ, φ = [ea ·, eb ·]R(ea , eb )φ, φ = |φ|2 2 4 4

by the usual calculation as in the Lichnerowicz formula [LM].

Therefore, for any compact domain ⊂ M,

R 2 2 2 |φ| + |∇φ| − |Dφ| dvol(g) 4 = (∇ea + ea · D)φ, φ int(ea ) dvol(g), ∂ = (∇ν + ν · D)φ, φ dvol(g|∂ ),

(2.5)

∂

where ea is an orthonormal basis of g and ν is the unit outer normal of ∂. Also, here int(ea ) is the interior multiplication by ea . In particular, for a harmonic spinor φ, i.e., Dφ = 0, the left-hand side of (2.5) will be nonnegative provided R ≥ 0. On the other hand, if the harmonic spinor φ can be chosen so that it is asymptotic to a parallel spinor at infinity and we choose the domain so that ∂ = SR × X, then we will show that the right-hand side of (2.5) converges to the mass (up to a positive normalizing constant). Thus, for the first part of our theorem, we are left with two tasks. First, we need to show the existence of harmonic spinors which are asymptotic to a parallel spinor. Second, we need to show that the limit of the boundary term converges to the mass. The existence of the harmonic spinor is dealt with in Sect. 4 (Lemma 4.1) after the necessary analysis in the next section and the computation of the limit of the boundary term is also left to Sect. 4 (Lemma 4.2). We now continue with the proof of the rigidity. If m(g) = 0, then it follows that φ is a (nonzero) parallel spinor on M. This implies that M is Ricci flat, as 1 ea · R(ea , X)φ = − Ric(X) φ. 2 Thus, we are in a position to use the splitting theorem of Cheeger-Gromoll [CG]. To find lines in M, we start with sequences of pairs of points pi , qi in M∞ (Rk −BR (0))×X. When R is sufficiently large, one can choose pi , qi so that their distance is comparable to their Euclidean distance. It follows that one can construct a line in M this way. Similarly, we can construct k lines in M that are almost perpendicular to each other. It follows that M = Rk × X.


339

3. Fibered Boundary Calculus We will use the fibered boundary calculus of Melrose-Mazzeo [MM] (and further developed by Boris Vaillant in his thesis [V] and in [HHMa]) to solve for the harmonic spinor with the correct asymptotic behavior. The change of variable r = x1 makes metric into what is called fibered boundary metric, which is defined in the more general setting as follows. Consider a complete noncompact Riemannian manifold (M, g). Assume that M has π a compactification M¯ such that ∂ M¯ comes with a fibration structure F → ∂ M¯ −→ B. ¯ the metric g has the form Moreover, in a neighborhood of the boundary ∂ M, g=

dx 2 π ∗ (gB ) + + gF , 4 x x2

(3.6)

where x is a defining function of the boundary, i.e., x = 0 on ∂ M¯ and dx = 0 on the boundary. Also, gB is a metric on the base B, gF is a family of fiberwise metrics. Thus, in the setting of spaces with asymptotic SUSY compactification, one has a trivial fibration S k−1 × X and x = 1r . ¯ and ∂M, ∂ M¯ interchangeably. For a manifold with We will use the notation M, M, boundary, the Lie algebra of b-vector fields consists of vector fields tangent to the boundary Vb (M) = {V | V is tangent to the boundary ∂M}. The Lie algebra of vector fields associated with the fibered boundary metric is Vf b = {V ∈ Vb (M) | V is tangent to the fibers F at ∂M, V x = O(x 2 )}.

(3.7)

If y is a local coordinate of B and z is a local coordinate of F , then Vf b is spanned by x 2 ∂x , x∂y , ∂z . The fibered boundary vector fields Vf b generate the ring of fibered boundary differential operators. The Dirac operator D associated to the fibered boundary metric is such a fibered boundary differential operator of first order. Define the L2 and Sobolev spaces as follows: L2 (M, S) = L2 (M, S; dvol(g)) = L2 (M, S,

dxdydz ), x 2+l

if dim B = l, Lp,2 (M, S) = { φ ∈ L2 (M, S) | ∇V1 · · · ∇Vj φ ∈ L2 (M, S), ∀j ≤ p, Vi ∈ Vb }. For γ ∈ R, the space of conormal sections of order γ is defined to be Aγ (M, S) = { φ ∈ C ∞ (M, S) | ∇V1 · · · ∇Vj φ| ≤ Cx γ , ∀j, Vi ∈ Vb }, while the space of polyhomogeneous sections is A∗phg (M, S) = { φ ∈ A∗ (M, S) | φ ∼

Nj

Reγj →∞ k=0

ψj k x γj (log x)k , ψj k ∈ C ∞ (∂M, S) }.

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Here the expansion is the usual asymptotic expansion, uniform with all the derivatives. We usually specify all possible pairs (γj , Nj ) that can appear in the expansion and the collection of (γj , Nj ) is called the index set. Assume that ker DF has constant dimension so it forms a vector bundle on the base B. Let 0 be the orthogonal projection onto ker DF and ⊥ = I − 0 . The following is a summary of the results developed in [MM, V, HHMa]. Theorem 3.1. Suppose that a is not an indicial root of 0 x −1 D 0 . Then D : x a L1,2 (M, S) → x a+1 0 L2 (M, S) ⊕ x a ⊥ L2 (M, S) is Fredholm. If Dφ = 0 for φ ∈ x a L2 (M, S), then φ is polyhomogeneous with exponents in its expansion determined by the indicial roots of 0 x −1 D 0 and truncated at a. If Dξ = ψ for ψ ∈ Aa (M, S) and ξ ∈ x c−1 0 L1,2 (M, S) ⊕ x c ⊥ L1,2 (M, S) and c < a, then ξ ∈ 0 AIphg (M, S) + Aa (M, S). For the precise definition of the indicial root, and in particular, the indicial root of 0 x −1 D 0 , we refer the reader to [MM, HHMa]. For our purpose, we only note that it is a discrete set. ◦

Remark. Strictly speaking, only g is a fibered boundary metric in the pure sense but it is easy to see that the result generalize to the metric g. In any case, the metric perturbation produces only a lower order term (cf. Sect. 4). Lemma 3.2. If R ≥ 0 and a >

k−2 2

is not an indicial root, then

D : x a L1,2 (M, S) → x a+1 0 L2 (M, S) ⊕ x a ⊥ L2 (M, S) is an isomorphism. Proof. We first see that it is injective. If Dφ = 0 for φ ∈ x a L2 (M, S), then by Theorem 3.1, φ ∈ Aaphg (M, S). Now, from (2.5), |∇φ|2 +

R 2 ∇ν φ, φ dvol(∂). |φ| dvol = 4 ∂

By taking so that ∂ = Sr × X and r → ∞ we see that the right hand side goes to zero since φ ∈ Aaphg (M, S) and a > k−2 2 . It follows then by the assumption R ≥ 0 that φ is parallel and hence zero. Now, if ω is in the cokernel of D, then, by the Fredholm property, ω ∈ x a+1 0 L2 (M, S) ⊕ x a ⊥ L2 (M, S) and ω is a weak solution of Dirac equation: ω, Dξ = 0, ∀ξ ∈ x a L1,2 (M, S). It follows by the regularity part of Theorem 3.1, ω ∈ Aaphg (M, S). Therefore the same argument as above shows ω = 0.


341

4. Computation of the Mass ◦

◦

◦

◦ ◦

Recall that g =g +h with g = gRk + gX and h = O(r −τ ), ∇ h = O(r −τ −1 ), ∇ ∇ ◦ h = O(r −τ −2 ). Let ea0 be the orthonormal basis of g which consists of ∂x∂ i followed by an orthonormal basis fα of gX . Orthonormalizing ea0 with respect to g gives rise an orthonormal basis ea of g. Moreover, 1 ea = ea0 − hab eb0 + O(r −2τ ). 2

(4.8)

This gives rise to a gauge transformation ◦

A : SO(g ) ea0 → ea ∈ SO(g) which identifies the corresponding spin groups and spinor bundles. ◦

To compare ∇ and ∇ , in particular their lifts to the spinor bundles, one introduces a ◦

new connection ∇ 0 = A◦ ∇ ◦A−1 . This connection is compatible with the metric g but has a torsion ◦

◦

0 Y − ∇Y0 X − [X, Y ] = −(∇ X A)A−1 Y + (∇ Y A)A−1 X. T (X, Y ) = ∇X

(4.9)

The difference of ∇ and ∇ 0 is then expressible in terms of the torsion 0 2 ∇X Y − ∇X Y, Z = T (X, Y ), Z − T (X, Z), Y − T (Y, Z), X ,

(4.10)

where we use the metric g for the inner product , . Since ∇ and ∇ 0 are both g-compatible, their induced connections on the spinor bundle differ by ∇ea − ∇e0a = −

1 ◦ (ωbc (ea )− ωbc (ea ))eb ec , 4

(4.11)

b,c

where eb , ec act on the spinors by the Clifford multiplication and the connection 1-forms ◦

◦

ωbc (ea ) = ∇ea eb , ec , ωbc (ea ) = ∇ ea eb , ec . From (4.10) and (4.9) we obtain ∇ea − ∇e0a =

◦ 1 ◦ (∇ eb gac − ∇ ec gab )eb ec + O(r −2τ −1 ) 8

(4.12)

b=c

for the difference of the two connections acting on spinors. Lemma 4.1. There exists a harmonic spinor on (M, g) which is asymptotic to a parallel spinor at infinity.

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Proof. Our manifold M = M0 ∪ M∞ with M0 compact and M∞ (Rk − BR (0)) × X. Since k ≥ 3 and X is simply connected, the end M∞ is also simply connected, and therefore has a unique spin structure coming from the product of the restriction of the spin structure on Rk and the spin structure on X. Now pick a unit norm parallel spinor ψ0 of (Rk , gRk ) and a unit norm parallel spinor ψ1 of (X, gX ). Then φ0 = A(ψ0 ⊗ ψ1 ) defines a spinor of M∞ . We extend φ0 smoothly inside. Then ∇ 0 φ0 = 0 outside the compact set. Thus, it follows from (4.12) that ∇φ0 = O(r −τ −1 ).

(4.13)

We now construct our harmonic spinor by setting φ = φ0 + ξ and solve Dξ = −Dφ0 ∈ O(r −τ −1 ). By using Lemma 3.2, adjusting τ slightly if necessary so that it is not one of the indicial roots, we have a solution ξ ∈ O(r −τ ). Lemma 4.2. For the harmonic spinor φ constructed above, we have lim (∇ea + ea · D)φ, φ int(ea ) dvol(g) = ωk vol(X)m(g). R→∞ SR ×X

Proof. By (2.5),

SR ×X

= Re Now,

(∇ea + ea · D)φ, φ int(ea ) dvol(g)

SR ×X

(∇ea + ea · D)φ, φ int(ea ) dvol(g).

1 (∇ea + ea · D)φ, φ = [ea ·, eb ·] ∇eb φ, φ 2 1 1 = [ea ·, eb ·] ∇eb φ0 , φ0 + [ea ·, eb ·] ∇eb φ0 , ξ 2 2 1 1 + [ea ·, eb ·] ∇eb ξ, φ0 + [ea ·, eb ·] ∇eb ξ, ξ . 2 2

(4.14)

The second term and the last term are O(r −2τ −1 ) and therefore contribute nothing in the limit. For the third term, one notices that if β is the n − 2 form, β = [ea ·, eb ·] φ, ψ int(ea ) int(eb ) dvol(g) (Einsterin summation here and below), then d β = −2 [ea ·, eb ·] ∇eb φ, ψ int(eb ) dvol(g) + [ea ·, eb ·] φ, ∇eb ψ int(eb ) dvol(g) = −4 [ea ·, eb ·] ∇eb φ, ψ int(eb ) dvol(g) − φ, [ea ·, eb ·] ∇eb ψ int(eb ) dvol(g) which yields [ea ·, eb ·] ∇eb φ, ψ int(eb ) dvol(g) = ∂

∂

(4.15)

φ, [ea ·, eb ·] ∇eb ψ int(eb ) dvol(g).


343

It follows then that the third term is similarly dealt with as the second. Thus the only contribution is coming from the first term, for which we note that 1 1 0 [ea ·, eb ·] ∇eb φ0 , φ0 = [ea ·, eb ·](∇eb − ∇eb )φ0 , φ0 2 2 ◦ 1 ◦ = (∇ ec gbd − ∇ ed gbc ) [ea ·, eb ·] ec · ed · φ0 , φ0

16 c=d −2τ −1

+O(r

)

by (4.12). Now ◦ 1 ◦ (∇ ec gbd − ∇ ed gbc ) [ea ·, eb ·] ec · ed · φ0 , φ0

16 c=d

=

◦ 1 ◦ (∇ ec gbd − ∇ ed gbc ) ea · eb · ec · ed · φ0 , φ0

8 c=d

+

◦ 1 ◦ (∇ ec gad − ∇ ed gac ) ec ·, ed · φ0 , φ0

8 c=d

1 ◦ 1 ◦ = ∇ ec gbd ea · eb · ec · ed · φ0 , φ0 + ∇ ed gbb ea · ed · φ0 , φ0

8 8 c=d

c=d

◦ 1 ◦ + (∇ ec gbd − ∇ ed gbc ) ec · ed · φ0 , φ0

8 c=d

1 ◦ 1 ◦ = ∇ ec gbb ea · ec · φ0 , φ0 + ∇ eb gbd ea · ed · φ0 , φ0

8 4 c=d

c=d

◦ 1◦ 1 ◦ + (∇ ec gbd − ∇ ed gbc ) ec · ed · φ0 , φ0 . ∇ ed gbb ea · ed · φ0 , φ0 + 8 8 c=d

c=d

For the last equality, we use ec · ed · = 21 [ec ·, ed ·] for c = d, and [ec ·, ed ·] skew-hermitian to see that its real part is zero. Finally, one uses ea · ed · = 21 [ea ·, ed ·] − δad and the skew-hermitian property of the commutators to obtain

◦ 1 1 ◦ Re [ea ·, eb ·] ∇eb φ0 , φ0 = (∇ eb gab − ∇ ea gbb )|φ0 |2 + O(r −2τ −1 ). 2 4 This yields

lim

R→∞ SR ×X

(∇ea + ea · D)φ, φ int(ea ) dvol(g)

= lim

R→∞ SR ×X

◦ 1 ◦ (∇ eb gab − ∇ ea gbb )|φ0 |2 int(ea ) dvol(g). 4

To see that this reduces to the definition of the mass, we first note that one can replace ea by ea0 in the integrand on the right-hand side, producing only an error of O(r −2τ −1 ), then replace dvol(g) by dxdvolX with a similar error term.

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5. Negative Energy Solutions in Kaluza-Klein Theory It was observed by Witten that positive energy theorems do not extend immediately to Kaluza-Klein theory [Wi2]. He observed that there are two zero energy solutions on a space asymptotic to M4 × S 1 which should lead to perturbatively negative energy solutions. The explicit negative energy solutions were constructed later in [BP, BH]. The following example is from [BH]. The analytically continued Reissner-Nordström metric ds 2 = (1 − where r ≥ r+ = m +

2m q 2 2m q 2 −1 2 − 2 )dθ 2 + (1 − − 2 ) dr + r 2 d2 , r r r r 2πr 2

+ m2 + q 2 , θ ∈ R/r+ −m Z and d2 is the standard metric on the

2-sphere. This is a scalar flat metric on R2 × S 2 and asymptotic to R3 × S 1 at infinity. The mass can be computed via (0.4), which is m(g) =

1 r+ − m m . 2 2 2πr+

(5.16)

2πr 2

+ For fixed asymptotic geometry, i.e., fixed circle size r+ −m = l, this can be made arbitrarily negative if one takes m < 0 sufficiently large, while q = 0 is chosen appropriately (which will necessarily be large as well). The reason here is that the end R3 × S 1 , and in particular, S 1 has the wrong spin structure! Recall that S 1 has two spin structures which correspond to the trivial double cover of S 1 and the nontrivial double cover of S 1 . Here, since S 1 bounds the disk inside, it has the spin structure corresponding to the nontrivial double cover. It therefore has no parallel spinor.

Acknowledgement. This work is motivated and inspired by the work of Gary Horowitz and his collaborators [HHM]. The author is indebted to Gary for sharing his ideas and for interesting discussions. The author would also like to thank Is Singer for bringing them together and for useful discussion. Thanks are also due to Xiao Zhang for useful comments.

References [AnD] [ADM] [AsHa] [AsHo] [Ba1] [Ba2] [BP] [BH] [CG]

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Communications in


One-Dimensional Behavior of Dilute, Trapped Bose Gases Elliott H. Lieb1, , Robert Seiringer1,2, , Jakob Yngvason2 1 2

Department of Physics, Jadwin Hall, Princeton University, P. O. Box 708, Princeton, NJ 08544, USA Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, 1090 Vienna, Austria

Received: 12 May 2003 / Accepted: 14 August 2003 Published online: 25 November 2003 – © E.H. Lieb, R. Seiringer, J. Yngvason 2003

Abstract: Recent experimental and theoretical work has shown that there are conditions in which a trapped, low-density Bose gas behaves like the one-dimensional delta-function Bose gas solved years ago by Lieb and Liniger. This is an intrinsically quantum-mechanical phenomenon because it is not necessary to have a trap width that is the size of an atom – as might have been supposed – but it suffices merely to have a trap width such that the energy gap for motion in the transverse direction is large compared to the energy associated with the motion along the trap. Up to now the theoretical arguments have been based on variational - perturbative ideas or numerical investigations. In contrast, this paper gives a rigorous proof of the one-dimensional behavior as far as the ground state energy and particle density are concerned. There are four parameters involved: the particle number, N , transverse and longitudinal dimensions of the trap, r and L, and the scattering length a of the interaction potential. Our main result is that if r/L → 0 and N → ∞ the ground state energy and density can be obtained by minimizing a one-dimensional density functional involving the Lieb-Liniger energy density with coupling constant ∼ a/r 2 . This density functional simplifies in various limiting cases and we identify five asymptotic parameter regions altogether. Three of these, corresponding to the weak coupling regime, can also be obtained as limits of a three-dimensional Gross-Pitaevskii theory. We also show that Bose-Einstein condensation in the ground state persists in a part of this regime. In the strong coupling regime the longitudinal motion of the particles is strongly correlated. The Gross-Pitaevskii description is not valid in this regime and new mathematical methods come into play.

c

2003 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. Work partially supported by U.S. National Science Foundation grant PHY 01-39984. Erwin Schr¨ odinger Fellow, supported by the Austrian Science Fund.

348

E.H. Lieb, R. Seiringer, J. Yngvason

1. Introduction The technique of trapping and cooling atoms, that led to the first realization of BoseEinstein condensation (BEC) in dilute alkali gases in 1995 [5, 18], has recently opened the possibility for experimental studies, in highly elongated traps, of Bose gases that are effectively one-dimensional. Some of the remarkable properties of ultracold one-dimensional Bose systems with delta function interactions, analyzed long ago [22, 23], may thus become accessible to experimental scrutiny in the not too distant future. Among these are pseudo-fermionic behavior [12], the absence of BEC in a dilute limit [21, 36, 14, 34, 10], and an excitation spectrum different from that predicted by Bogoliubov’s theory [23, 17, 20]. The paper [33] by Olshanii triggered a number of theoretical investigations on the transitions from 3D to an effective 1D behavior with its peculiar properties, see, e.g., [6–8, 11, 13, 19, 32, 35, 39]; systems showing the first evidence of such a transition have recently been prepared experimentally [4, 15, 16, 38]. Until now the theoretical work on the dimensional cross-over in elongated traps has either been based on variational calculations, starting from a three-dimensional deltapotential [7, 13, 33], or on numerical Monte Carlo studies [1, 3] with more realistic, genuine 3D potentials but particle numbers limited to the order of 100. This work is important and has led to valuable insights, in particular about different parameter regions [8, 35, 32], but a more thorough theoretical understanding is clearly desirable since this is not a simple problem. In fact, it is evident that for a potential with a hard core the true 3D wave function does not approximately factorize in the longitudinal and transverse variables and the effective one-dimensional potential can not be obtained by simply integrating out the transverse variables of the 3D potential. In this sense the problem is more complicated than in a somewhat analogous situation of atoms in extremely strong magnetic fields [2, 30], where the Coulomb interaction behaves like an effective onedimensional delta potential when the magnetic field shrinks the cyclotron radius of the electrons to zero. In that case the delta potential can be obtained formally by integrating out the variables transverse to the field in a suitable scaled Coulomb potential. With a hard core, on the other hand, where the energy is essentially kinetic, this method will not work since it would immediately introduce impenetrable barriers in 1D. The onedimensional effective interaction emerges only if the kinetic part of the Hamiltonian and the potential are considered together. In the present paper we start with an arbitrary, repulsive 3D pair potential of finite range and prove rigorously that in a well defined limit the ground state energy and particle density of the system are described exactly by a one-dimensional model with delta-function interaction. This is a highly quantum-mechanical phenomenon with no classical counterpart, since a 1D description is possible even though the transverse trap dimension is much larger than the range of the atomic forces. It suffices that the energy gap associated with the transverse confinement is much larger than the internal energy per particle. While the three-dimensional density remains low (in the sense that distance between particles is large compared to the three-dimensional scattering length) the one-dimensional density can either be high or low. We remark that, in contrast to three-dimensional gases, high density in one dimension corresponds to weak interactions and vice versa [22]. In this paper we shall always be concerned with large particle number, N , which is appropriate for the consideration of actual experiments. In order to make precise statements we shall typically take the limit N → ∞ but the reader can confidently apply these limiting statement to finite numbers like N = 100.

One-Dimensional Behavior of Dilute, Trapped Bose Gases

349

Besides N , the parameters of the problem are the scattering length, a, of the two-body interaction potential, and two lengths, r and L, describing the transverse and the longitudinal extension of the trap potential, respectively. To keep the introductory discussion simple let us first think of the case that the particles are confined in a box with dimensions r and L. The three-dimensional particle density is then ρ 3D = N/(r 2 L) and the one-dimensional density ρ 1D = N/L. The case of quadratic or more general trapping potentials will be considered later. We begin by describing the division of the space of parameters into two basic regions. This decomposition will eventually be refined into five regions, but for the moment let us concentrate on the basic dichotomy. In earlier work [27, 26] we have proved that the three-dimensional Gross-Pitaevskii formula for the energy (including its limiting “Thomas-Fermi” case) is correct to leading order in situations in which a is small and N is large. This energy has two parts: The energy necessary to confine the particles in the trap, which is roughly ( 2 /2m)N (r −2 + L−2 ), plus the internal energy of interaction, which is ( 2 /2m)N 4π aρ 3D . The trouble is that while this formula is correct for a fixed confining potential in the limit N → ∞ with a 3 ρ 3D → 0, it does not hold uniformly if r/L gets small as N gets large. In other words, new physics can come into play as r/L → 0 and it turns out that this depends on the ratio of a/r 2 to ρ 1D = N/L . As we shall show, the two basic regimes to consider in highly elongated traps, i.e., when r L, are • The one-dimensional limit of the three-dimensional Gross-Pitaevskii/“Thomas-Fermi” regime • The “true” one-dimensional regime. The former is characterized by aL/r 2 N → 0, while in the latter regime aL/r 2 N is of the order one or even tends to infinity (which is referred to as the Girardeau-Tonks1 region). These two regimes correspond to high one-dimensional density (weak interaction) and low one-dimensional density (strong interaction), respectively. The significance of the combination aL/r 2 N can be understood by noting that it is the ratio of the 3D energy per particle, ∼ aρ 3D ∼ N a/r 2 L, to the 1D energy ∼ (ρ 1D )2 = (N/L)2 . Physically, the main difference between the two regimes is that for strong interactions the motion of the particles in the longitudinal direction is highly correlated, while in the weak interaction regime it is not. Mathematically, this distinction also shows up in our proofs. In both regimes the internal energy of the gas is small compared to the energy of confinement which is of order N/r 2 . However, this in itself does not imply a specifically one-dimensional behavior. (If a is sufficiently small it is satisfied in a trap of any shape.) One-dimensional behavior, when it occurs, manifests itself by the fact that the transverse motion of the atoms is uncorrelated while the longitudinal motion is correlated (very roughly speaking) in the same way as pearls on a necklace. Thus, the true criterion for 1D behavior is that aL/r 2 N is of the order unity and not merely the condition that the energy of confinement dominates the internal energy. The starting point for our investigations is the Hamiltonian for N spinless Bosons in a confining 3D trap potential and with a short range, repulsive pair interaction. We find 1 We call this the Girardeau-Tonks region only because many authors refer in the present context to Tonks [41]. In our opinion this should really be called the Girardeau region because it was he who first understood how to compute the spectrum of a 1D quantum-mechanical hard core gas and who understood that the Fermi-Dirac wave functions played a role [12]. Tonks was interested in the positive temperature partition functions of a hard core classical gas – a very different and much simpler question.

350


it convenient to write the Hamiltonian in the following way (in appropriate units): HN,L,r,a =

−j + Vr⊥ (xj⊥ ) + VL (zj ) +

N j =1

va (|xi − xj |)

(1.1)

1≤i<j ≤N

with x = (x, y, z) = (x⊥ , z), Vr⊥ (x⊥ ) =

1 ⊥ ⊥ V (x /r), r2

VL (z) =

1 V (z/L), L2

va (|x|) =

1 v(|x|/a) , a2

(1.2)

where r, L, a are variable scaling parameters while V ⊥ , V and v are fixed. The interaction potential v is supposed to be nonnegative, of finite range and have scattering length 1; the scaled potential va then has scattering length a. The external trap potentials V and V ⊥ confine the motion in the longitudinal (z) and the transversal (x⊥ ) directions, respectively, and are assumed to be locally bounded and tend to ∞ as |z| and |x⊥ | tend to ∞. To simplify the discussion we find it also convenient to assume throughout that V is homogeneous of some order s > 0, namely V (z) = |z|s , but weaker assumptions, e.g. asymptotic homogeneity [28], would in fact suffice. The case of a simple box with hard walls is realized by taking s = ∞, while the usual harmonic approximation is s = 2. Moreover, to avoid unnecessary technicalities we shall assume that V ⊥ is polynomially bounded at infinity, but our results certainly also hold for faster growing potentials, or even finite domains with Dirichlet boundary conditions. Units are chosen so that = 1 and 2m = 1. It is understood that the lengths associated with the ground states of −d 2 /dz2 + V (z) and −⊥ + V ⊥ (x⊥ ) are both of the order 1 so that L and r measure, respectively, the longitudinal and the transverse extensions of the trap. We denote the ground state energy of (1.1) by E QM (N, L, r, a) and the ground state particle QM density by ρN,L,r,a (x). In parallel with the three-dimensional Hamiltonian we consider the Hamiltonian for n Bosons in one dimension with delta interaction and coupling constant g ≥ 0, i.e., 1D = Hn,g

n

−∂j2 + g

j =1

δ(zi − zj ) ,

(1.3)

1≤i<j ≤n

where ∂j = ∂/∂zj . We consider this Hamiltonian for the zj in an interval of length in the thermodynamic limit, → ∞, n → ∞ with ρ = n/ fixed. The ground state energy per particle in this limit is independent of boundary conditions and can, according to [22], be written as e01D (ρ) = ρ 2 e(g/ρ) ,

(1.4)

with a function e(t) determined by a certain integral equation. Its asymptotic form is e(t) ≈ 21 t for t 1 and e(t) → π 2 /3 for t → ∞. Thus e01D (ρ) ≈ 21 gρ for g/ρ 1

(1.5)

and e01D (ρ) ≈

π2 2 ρ for g/ρ 1 . 3

(1.6)


351

Taking ρe01D (ρ) as a local energy density for an inhomogeneous one-dimensional system we can form the energy functional √ E[ρ] = (1.7) |∇ ρ(z)|2 + VL (z)ρ(z) + ρ(z)3 e(g/ρ(z)) dz R

with ground state energy defined by minimizing E[ρ] over all normalized densities ρ, i.e., E(N, L, g) = inf E[ρ] : ρ ≥ 0 , ρ(z)dz = N . (1.8) R

By standard methods (cf.,e.g., [27]) one can show that there is a unique minimizer, i.e., a density ρN,L,g (z) with ρN,L,g (z)dz = N and E[ρN,L,g ] = E(N, L, g). Here it is important to note that t → t 3 e(1/t) is convex. We define the mean 1D density of this minimizer to be 2 1 ρ¯ = ρN,L,g (z) dz . (1.9) N R In a rigid box, i.e., for s = ∞, ρ¯ is simply N/L, but in more general traps it depends also on g besides N and L. The order of magnitude of ρ¯ in the various parameter regions will be described in the next section. Our main result relates the 3D ground state energy of (1.1), E QM (N, L, r, a), to the 1D density functional energy E(N, L, g) for a suitable g in the large N limit provided r/L and a/r are sufficiently small. To state this precisely, let e⊥ and b(x⊥ ) denote the ground state energy and the normalized, nonnegative ground state wave function of −⊥ + V ⊥ (x⊥ ), respectively. The corresponding quantities for −⊥ + Vr⊥ (x⊥ ) are e⊥ /r 2 and br (x⊥ ) = (1/r)b(x⊥ /r). In the case that the trap is a cylinder with hard walls b is a Bessel function; for a quadratic V ⊥ it is a Gaussian. In any case, b is a bounded function and, in particular, b ∈ L4 (R2 ). Hence we can define g by 8πa g= 2 |b(x⊥ )|4 d 2 x⊥ = 8πa |br (x⊥ )|4 d 2 x⊥ . (1.10) 2 2 r R R Our main theorem is: Theorem 1.1 (From 3D to 1D). Let N → ∞ and simultaneously r/L → 0 and a/r → 0 in such a way that r 2 ρ¯ · min{ρ, ¯ g} → 0. Then lim

E QM (N, L, r, a) − N e⊥ /r 2 =1. E(N, L, g)

(1.11)

¯ g} → 0 is the same Note that because of (1.5) and (1.6) the condition r 2 ρ¯ · min{ρ, as e01D (ρ) ¯ 1/r 2 ,

(1.12)

i.e., the average energy per particle associated with the longitudinal motion should be much smaller than the energy gap between the ground and first excited state of the confining Hamiltonian in the transverse directions. (The precise meaning of is that the ratio of the left side to the right side tends to zero in the limit considered.) Note also that while

352


the one-dimensional density can be either high or low (compared to g), the gas is always dilute in a three-dimensional sense in the limit considered, i.e., a 3 ρ 3D ∼ a 2 g ρ¯ 1. The two regimes mentioned previously correspond to specific restrictions on the size of the ratio g/ρ¯ as N → ∞, namely g/ρ¯ 1 for the limit of the 3D Gross-Pitaevskii regime (weak interaction/high 1D density), and g/ρ¯ 1 for the “true” one-dimensional regime (strong interaction/low 1D density). We shall now describe briefly the finer division of these regimes into five regions altogether. Three of them (Regions 1–3) belong to the weak interaction regime and two (Regions 4–5) to the strong interaction regime. In each of these regions the general functional (1.7) can be replaced by a different, simpler functional, and the energy E(N, L, g) in Theorem 1.1 by the ground state energy of that functional. The five regions are • Region 1, the Ideal Gas case: g/ρ¯ N −2 , with ρ¯ ∼ N/L, corresponding to a non-interacting gas in an external potential. • Region 2, the 1D GP case: g/ρ¯ ∼ N −2 , with ρ¯ ∼ N/L, described by a 1D GrossPitaevskii energy functional with energy density 21 gρ 2 . • Region 3, the 1D TF case: N −2 g/ρ¯ 1, with ρ¯ ∼ (N/L)(NgL)−1/(s+1) , where s is the degree of homogeneity of the longitudinal confining potential V . This region is described by a Thomas-Fermi type functional with energy density 21 gρ 2 , without a gradient term. • Region 4, the LL case: g/ρ¯ ∼ 1 , with ρ¯ ∼ (N/L)N −2/(s+2) , described by an energy functional with the Lieb-Liniger energy (1.4), without a gradient term. • Region 5, the GT case: g/ρ¯ 1, with ρ¯ ∼ (N/L)N −2/(s+2) , described by a functional with energy density ∼ ρ 3 , corresponding to the Girardeau-Tonks limit of the LL energy density. We note that the condition g/ρ¯ ∼ 1 means that Region 4 requires the gas cloud to have aspect ratio r/L¯ of the order N −1 (a/r) or smaller, where L¯ ≡ N/ρ¯ ∼ LN 2/(s+2) is the length of the cloud. Experimentally, such small aspect ratios are quite a challenge and the situations described in [4, 15, 16, 38] are still rather far from this regime. It may not be completely out of reach, however. The condition a/r → 0 is automatically fulfilled in Regions 1–4, provided (1.12) holds, since (a/r)2 = r 2 (g ρ)(g/ ¯ ρ) ¯ = r 2 ρ¯ 2 (g/ρ) ¯ 2 1 by (1.12) if g/ρ¯ is bounded. Moreover, as discusssed in the next section, in Regions 1–2 the condition r/L → 0 implies (1.12) and hence a/r → 0 , and in Region 4, a/r → 0 implies (1.12). The hypotheses of Theorem 1.1 are thus not entirely independent. In the next Sect. 2 we define the various energy functionals more precisely and also QM discuss the 1D behavior of the density ρN,L,r,a (x), separately for each region. Moreover, we prove, in Subsect. 2.7, that Regions 1–3 can be reached as limiting cases of a 3D Gross-Pitaevskii theory. In this sense, the behavior in these regions contains remnants of the 3D theory, which also shows up in the fact that BEC prevails in Regions 1 and 2, as discussed in Sect. 5. Heuristically, these traces of 3D can be understood from the fact ¯ gives that in Regions 1–3 the 1D formula for the energy per particle g ρ¯ ∼ aN/(r 2 L), the same result as the three-dimensional formula [31], i.e., scattering length times threedimensional density. This is no longer so in Regions 4 and 5 and different mathematical methods are required. Despite significant differences the proof of Theorem 1.1 has some basic strategies in common with the (considerably simpler) proof of the Gross-Pitaevskii limit Theorem in [27] (see also [31] and [26]). The upper bound for the energy in Regions 1–3, given


353

in Subsect. 4.1, is the simplest estimate and can be obtained by a method analogous to that of [27]. For the lower bound, and also for the upper bound in Regions 4–5, one considers first finite numbers of particles in boxes with Neumann or Dirichlet boundary conditions, and subsequently puts these boxes together to treat inhomogeneous external potentials and the infinite particle number limit. For the lower bound of the energy in the boxes Dyson’s Lemma [9, 31, 26], which converts a “hard” potential into a “soft” potential at the expense of sacrificing kinetic energy, is an essential tool. The main differences compared to [27] are on the one hand due to the fact that in Regions 1–3 the lower bounds in [27] are not valid because they are not uniform in the shape of the trap, and on the other hand due to the correlations in the Lieb-Liniger wave function for the longitudinal motion in Regions 4–5. For the latter reason even the proof of the upper bound in Regions 4–5 is considerably more involved than for Regions 1–3. Section 3 contains the main technical estimates for the boxes. We consider here the 3D Hamiltonian for a finite number of particles on a finite interval in the longitudinal direction with Neumann or Dirichlet boundary conditions and a confining potential Vr⊥ in the transverse directions. We estimate its energy from above and below in terms of the energy of a 1D Hamiltonian with delta interactions. In Sect. 4 we apply these results to prove Theorem 1.1. In the last Sect. 5 we consider the question of Bose-Einstein condensation and prove that it holds in Regions 1 and 2. Part of the results of this work were announced in [29]. 2. The Five Parameter Regions We shall now discuss the simplifications of Theorem 1.1 in the five different parameter regions. Besides the ground state energy we also consider the convergence of the QM quantum mechanical density ρN,L,r,a (x) averaged over the transverse variables, i.e., of QM QM ρˆN,L,r,a (z) := ρN,L,r,a (x⊥ , z)d 2 x⊥ . (2.1) We state the results in the form of five theorems, one for each region. These theorems will be proved in Sect. 4. Together they imply Theorem 1.1 because the energy functionals involved are all limiting cases of the general functional (1.8). The methods for proving the latter statement are fairly standard (cf., e.g., [27, 28] for similar computations). Since a proof of all five limit theorems for the functional (1.8) would be largely repetitious, we shall limit ourselves to giving a proof of two of them in Subsect. 2.6, as an example. 2.1. The Ideal Gas Region. This region corresponds to the trivial case where the interaction is so weak that it effectively vanishes in the large N limit and everything collapses to the ground state of −d 2 /dz2 + VL (z). By scaling the ground state

energy and wave −2 e and L−1/2 ρ (z/L), where e function of this latter operator can be written as L

and ρ (z) are the corresponding quantities for −d 2 /dz2 + V (z). Theorem 2.1 (Ideal gas limit). Suppose r/L → 0 and NgL ∼ N aL/r 2 → 0 as N → ∞. Then (N/L2 )−1 E QM (N, L, a, r) − N e⊥ /r 2 → e (2.2) and

354


(N/L)−1 ρˆN,L,r,a (Lz) → ρ (z) QM

(2.3)

weakly in L1 (R). 2.2. The 1D Gross-Pitaevskii Region. This region is described by the 1D GP density functional √ GP EL,g |∇ ρ(z)|2 + VL (z)ρ(z) + 21 gρ(z)2 dz (2.4) [ρ] = R

corresponding to the high density approximation (1.5) of the interaction energy in (1.7). Its ground state energy GP GP E (N, L, g) = inf EL,g [ρ] : ρ ≥ 0 , ρ(z)dz = N (2.5) R

has the scaling property E GP (N, L, g) = (N/L2 )E GP (1, 1, NgL)

(2.6)

GP (z) satisfies and likewise, the minimizer ρN,L,g GP GP ρN,L,g (Lz) = (N/L)ρ1,1,NgL (z) .

(2.7)

Theorem 2.2 (1D GP limit). Suppose r/L → 0 and NgL ∼ N aL/r 2 is fixed as N → ∞. Then (N/L2 )−1 E QM (N, L, a, r) − N e⊥ /r 2 → E GP (1, 1, NgL) (2.8) and GP (N/L)−1 ρˆN,L,r,a (Lz) → ρ1,1,NgL (z) QM

(2.9)

weakly in L1 (R). Remark. If r/L → 0 and NgL stays bounded, as in Regions 1 and 2, condition (1.12) is automatically satisfied, because g ρr ¯ 2 ∼ aN/L ∼ (r/L)2 (NgL). Likewise, a/r → 0, −1 because a/r ∼ (r/L)N (NgL). 2.3. The 1D “Thomas-Fermi” Region. This region is a limiting case of the previous one in the sense that NgL ∼ N aL/r 2 → ∞, but a/r is sufficiently small so that g/ρ¯ ∼ (aL/N r 2 )(N aL/r 2 )1/(s+1) → 0, i.e., the high density approximation in (1.5) is still valid. Here s is the degree of homogeneity of V and the explanation of the factor (NaL/r 2 )1/(s+1) ∼ (NgL)1/(s+1) is as follows: The linear extension L¯ of the minimizGP ¯ ∼ g(N/L), ¯ which ing density ρN,L,g is for large values of NgL determined by VL (L) 1/(s+1) −2 ¯ gives L ∼ (NgL) L. In addition condition (1.12) requires g ρ¯ r , which means that (N a/L)(N aL/r 2 )−1/(s+1) → 0. If NgL ∼ N aL/r 2 → ∞ the gradient term in the functional (2.5) becomes negligible compared to the other terms. In fact, by a simple scaling,


355

N (NgL)s/(s+1) L2

2 × (NgL)−(s+2)/(s+1) |∇ ρ(z)| ˜ +V (z)ρ(z) ˜ + 21 ρ(z) ˜ 2 dz, (2.10)

GP EL,g [ρ] =

R

where the scaled density ρ˜ is determined by ρ(z) = (N/L¯ TF )ρ(z/ ˜ L¯ TF ) ,

with L¯ TF := (NgL)1/(s+1) L .

(2.11)

This leads to the functional TF EL,g [ρ]

=

R

VL (z)ρ(z) + 21 gρ(z)2 dz

(2.12)

whose ground state energy E

TF

(N, L, g) = inf

TF EL,g [ρ]

: ρ ≥ 0,

R

ρ(z)dz = N

(2.13)

has the scaling property E TF (N, L, g) = (N/L2 )(NgL)s/(s+1) E TF (1, 1, 1) .

(2.14)

TF The minimizer ρN,L,g satisfies TF TF ρN,L,g (L¯ TF z) = (N/L¯ TF )ρ1,1,1 (z)

(2.15)

and can be computed explicitly: TF ρ1,1,1 (z) = [µTF − V (z)]+ ,

(2.16)

TF (z)dz = 1. where [t]+ = max{t, 0} and µTF is determined by the normalization ρ1,1,1 Because of a formal similarity with the Thomas-Fermi energy functional for fermions, which also has no gradient terms, the functional (2.12) is in the literature usually referred to as a “Thomas-Fermi” functional, but the physics is, of course, quite different. The limit theorem in this region is Theorem 2.3 (1D “TF limit”). Suppose, as N → ∞, r/L → 0, NgL ∼ N aL/r 2 → ∞, but g/ρ¯ ∼ (a L¯ TF /N r 2 ) → 0 and g ρr ¯ 2 ∼ N a/L¯ TF → 0 with L¯ TF given in (2.11). Then (N/L2 )−1 (NgL)−s/(s+1) E QM (N, L, a, r) − N e⊥ /r 2 → E TF (1, 1, 1) (2.17) and TF (N/L¯ TF )−1 ρˆN,L,r,a (L¯ TF z) → ρ1,1,1 (z) QM

weakly in L1 (R).

(2.18)

356


2.4. The Lieb-Liniger Region. This region corresponds to the case g/ρ¯ ∼ 1, so that neither the high density (1.5) nor the low density approximation (1.6) is valid and the full LL ¯ ∼ energy (1.4) has to be used. The extension L¯ of the system is now determined by VL (L) 2 2/(s+2) 1/(s+1) ¯ ¯ ¯ (N/L) which leads to LLL = LN , in contrast to LTF = L(NgL) in the TF case. Condition (1.12) means in this region that N r/L¯ LL ∼ N s/(s+2) r/L → 0. Since Nr/L¯ LL ∼ (ρ/g)(a/r), ¯ this condition is automatically fulfilled if g/ρ¯ is bounded away from zero and a/r → 0. Conversely, if g/ρ¯ is bounded, then (1.12) implies a/r → 0. The energy functional is LL VL (z)ρ(z) + ρ(z)3 e(g/ρ(z)) dz (2.19) EL,g [ρ] = R

with corresponding energy

LL [ρ] : ρ ≥ 0 , ρ(z)dz = N . E LL (N, L, g) = inf EL,g R

(2.20)

Introducing the density parameter γ := N/L¯ LL = (N/L)N −2/(s+2)

(2.21)

we can write the scaling relation of the functional as E LL (N, L, g) = N γ 2 E LL (1, 1, g/γ ) .

(2.22)

The minimizer, ρ LL (z), satisfies LL LL (L¯ LL z) = γρ1,1,g/γ (z) . ρN,L,g

(2.23)

Theorem 2.4 (LL limit). Suppose r/L → 0 and a/r → 0, with g/γ > 0 fixed as N → ∞. Then (2.24) (N γ 2 )−1 E QM (N, L, a, r) − N e⊥ /r 2 → E LL (1, 1, g/γ ) and LL (z) γ −1 ρˆN,L,r,a (L¯ LL z) → ρ1,1,g/γ QM

(2.25)

weakly in L1 (R). 2.5. The Girardeau-Tonks Region. This region corresponds to impenetrable particles, i.e, the limiting case g/ρ¯ → ∞ and hence the formula (1.6) for the energy density. As in Region 4, the mean density is here ρ¯ ∼ γ = (N/L)N −2/(s+2) . The energy functional is GT VL (z)ρ(z) + (π 2 /3)ρ(z)3 dz (2.26) EL [ρ] = R

with energy

ρ(z)dz = N , E GT (N, L) = inf ELGT [ρ] : ρ ≥ 0 , R

which can be written

(2.27)


357

E GT (N, L) = N γ 2 E GT (1, 1) .

(2.28)

GT (z) has the form The minimizer ρ1,1 GT ρ1,1 (z) = π −1 [µGT − V (z)]+ , 1/2

(2.29)

with µGT determined by the normalization. Note that its shape is different from that of (2.16), which makes it possible to distinguish experimentally the TF regime from the GT regime. The scaling relation for the minimizer is GT ¯ GT ρN,L (LLL z) = γρ1,1 (z) .

(2.30)

Theorem 2.5 (GT limit). Suppose r/L → 0 and a/r → 0, with g/γ → ∞ as N → ∞. Then (2.31) (N γ 2 )−1 E QM (N, L, a, r) − N e⊥ /r 2 → E GT (1, 1) and GT (z) γ −1 ρˆN,L,r,a (L¯ LL z) → ρ1,1 QM

(2.32)

weakly in L1 (R).

2.6. Limiting cases of the general energy functional. As already stated, the proof of Theorem 1.1 in Sect. 4 consists in comparing the ground state energy of the manybody Hamiltonian (1.1) with the ground state energies of the functionals defined in Subsects. 2.1–2.5 in the various parameter domains. To link these special cases to the functional (1.7) it then remains to show that the ground state energy of (1.7) coincides with that of the functionals in Subsects. 2.1–2.5 in the appropriate asymptotic limits. The proof of this follows the same pattern as the derivation of the 3D TF limit from 3D GP theory in [27, 28] and we shall here only give explicit proofs for the functionals in Theorems 2.3 and 2.4 as examples. The limit theorems for the density are derived from the energy convergence by variation of the external potential in a standard way (cf., e.g. [28]). Proposition 2.1. If N → ∞, NgL → ∞, but g L¯ TF /N = gL(NgL)1/(s+1) /N → 0, then

(N/L2 )(NgL)s/(s+1)

−1

E(N, L, g) → E TF (1, 1, 1)

(2.33)

and

N/L¯ TF

weakly in L1 (R).

−1

TF ρN,L,g L¯ TF z → ρ1,1,1 (z)

(2.34)

358


Proof. With ρ˜ the scaled density given by (2.11) the energy functional (1.7) can be written

2 s/(s+1) 2 ˜ (NgL)−(s+2)/(s+1) |∂ ρ(z)| E[ρ] = (N/L )(NgL) R + V (z)ρ(z) ˜ + ρ(z) ˜ 3 N (g L¯ TF )−1 e(g L¯ TF /N ρ(z)) ˜ dz . (2.35) Now te(1/t) ≤

1 2

for all t [22], so

−1 (N/L2 )(NgL)s/(s+1) E[ρ]

2 ≤ ˜ + V (z)ρ(z) ˜ + 21 ρ(z) ˜ 2 dz . (NgL)−(s+2)/(s+1) |∂ ρ(z)| R

(2.36)

Let jε (z) = (2ε)−1 exp(−|z|/ε) . TF ∗ j . Then, since |∂j | = ε −1 j and ρ(z)dz ˜ = 1, Define ρ˜ = ρ1,1,1 ε ε ε

2 |∂ ρ(z)| ˜ dz ≤ 1/(4ε 2 ) < ∞ , and (2.36) implies −1 TF TF E(N, L, g) ≤ E1,1 [ρ1,1,1 ∗ jε ] lim sup (N/L2 )(NgL)s/(s+1)

(2.37)

(2.38)

(2.39)

N→∞

TF 2 TF ∗ jε )2 ≤ (ρ1,1,1 ) because jε = 1, and in the limit considered. Moreover, (ρ1,1,1 TF s TF ρ1,1,1 ∗ jε (z)|z| dz → ρ1,1,1 (z)|z|s dz (2.40) TF = [µTF − |z|s ] is continuous and of compact support.) for ε → 0. (Note that ρ1,1,1 + Hence −1 lim sup (N/L2 )(NgL)s/(s+1) E(N, L, g) ≤ E TF (1, 1, 1) . (2.41) N→∞

On the other hand, dropping the positive gradient term in (2.35) gives −1 2 s/(s+1) V (z)ρ(z) ˜ + ρ(z) ˜ 3 Me(1/M ρ(z)) E[ρ] ≥ ˜ dz , (N/L )(NgL) R

(2.42)

with M = N/(g L¯ TF ). Note that M → ∞ in the limit considered here. The functional on the right side of (2.42) has as minimizer −1 (M) (µ − V (z))]+ , ρ (M) (z) = [fM

(2.43)

−1 where fM is the inverse of the function fM (t) = d/dt[Mt 3 e(1/tM)] and µ(M) is cho (M) sen so that ρ = 1. Note also that t −1 fM (t) → 1 as M → ∞, uniformly on [δ, ∞[


359

TF , for every δ > 0. From this it follows easily that ρ (M) converges uniformly to ρ1,1,1 given by (2.16), as M → ∞. With ρ = ρN,L,g , the minimizer of E, we thus obtain from (2.42), −1 lim inf (N/L2 )(NgL)s/(s+1) E(N, L, g) ≥ E TF (1, 1, 1) . (2.44) N→∞

To prove the corresponding result (2.34) for the density we pick a C ∞ function Y of compact support together with an ε > 0 and replace VL (z) by ε (NgL)s/(s+1) Y (L−1 (NgL)−1/(s+1) z) L2

1 = 2 (NgL)s/(s+1) V (z ) + εY (z ) L

VL (z) +

(2.45)

with z = z/L¯ TF = L−1 (NgL)−1/(s+1) z. While V (z ) + εY (z ) is not strictly homogeneous of order s, it is asymptotically homogeneous in the sense of Def. 1.1 in [28] and as in the proof of Lemma 2.3 in [28] this is sufficient for (2.33), now with the modified external potential (2.45). Since both (1.8) and the TF energy are concave in ε, the derivative with respect to ε can be exchanged with the limits N → ∞, (NgL) → ∞, giving (2.34) in the sense of distributions. Since the densities have norm 1, the convergence holds also weakly in L1 (R). Proposition 2.2. If N → ∞ with g/γ fixed, where γ = N/L¯ LL = (N/L)N −2/(s+2) , then (N γ 2 )−1 E(N, L, g) → E LL (1, 1, g/γ )

(2.46)

LL (z) γ −1 ρN,L,g (L¯ LL z) → ρ1,1,g/γ

(2.47)

and

weakly in L1 (R). Proof. With L¯ LL = LN 2/(s+2) we define the scaled density ρ˜ by ˜ L¯ LL ) . ρ(z) = (N/L¯ LL )ρ(z/ The energy functional (1.7) can then be written

2 N −2 |∂ ρ(z)| ˜ + V (z)ρ(z) ˜ + ρ(z) ˜ 3 e g/(γ ρ(z)) ˜ dz . E[ρ] = Nγ 2 R

(2.48)

(2.49)

The lower bound (N γ 2 )−1 E(N, L, g) ≥ E LL (1, 1, g/γ )

(2.50)

follows simply by dropping the positive gradient term from the right side of (2.49). LL For the upper bound take jε as in (2.37) and define ρ˜ = jε ∗ ρ1,1,g/γ to obtain (N γ 2 )−1 E[ρ] ≤ and hence

N −2 LL + E1,1,g/γ [ρ] ˜ 4ε 2

(2.51)

360

E.H. Lieb, R. Seiringer, J. Yngvason LL LL lim sup(N γ 2 )−1 E(N, L, g) ≤ E1,1,g/γ [jε ∗ ρ1,1,g/γ ]

(2.52)

N→∞

for all ε > 0. The convergence LL LL LL LL lim E1,1,g/γ [jε ∗ ρ1,1,g/γ ] = E1,1,g/γ [ρ1,1,g/γ ] = E LL (1, 1, g/γ )

ε→0

(2.53)

LL follows by continuity of |z|s and t 2 e(t) and uniform convergence of jε ∗ ρ1,1,g/γ to LL ρ1,1,g/γ . The convergence of the densities follows as in the previous proposition by perturbing the external potential, this time replacing VL (z) by

VL (z) + εγ 2 Y (L−1 N −2/(s+2) z) = γ 2 V (z ) + εY (z ) (2.54)

with z = z/L¯ LL = L−1 N −2/(s+2) z.

2.7. One-dimensional GP as limit of three-dimensional GP. We shall now demonstrate that the ground state energy in Regions 1–3 can be obtained as a limit of the three-dimensional Gross-Pitaevskii energy. The latter is defined by the energy functional GP E3D [ ] = |∇ (x)|2 + Vr⊥ (x⊥ ) + VL (z) | (x)|2 + 4π a| (x)|4 d 3 x . R3

(2.55)

We denote its ground state energy, i.e, the infimum over all with GP (N, L, r, a). It satisfies the scaling relation E3D

| |2 = N , by

GP GP E3D (N, L, r, a) = (N/L2 )E3D (1, 1, r/L, N a/L) .

(2.56)

GP and E GP for N = 1 and Because of (2.56) and (2.6) it is sufficient to compare E3D L = 1.

Theorem 2.6. Let g be given by (1.10). In the limit r → 0 and a → 0, GP (1, 1, r, a) − e⊥ /r 2 E3D →1, E GP (1, 1, g)

(2.57)

uniformly in g as long as r 2 E GP (1, 1, g) → 0. Proof. We denote the minimizer of the one-dimensional GP functional (2.4) with N = 1, L = 1 and g fixed by φ(z)2 . Taking br (x⊥ )φ(z) as the trial function for the 3D functional (2.55) and using the definition (1.10) of g we obtain without further ado the upper bound GP E3D (1, 1, r, a) ≤ e⊥ /r 2 + E GP (1, 1, g)

(2.58)

for all r and a. For a lower bound we consider the one-particle Hamiltonian Hr,a = −⊥ + Vr⊥ (x⊥ ) − ∂z2 + V (z) + 8π abr (x⊥ )2 φ(z)2 .

(2.59)


361

Taking the 3D Gross-Pitaevskii minimizer (x) for N = 1, L = 1, as the trial function we get GP inf spec Hr,a ≤ E3D (1, 1, r, a) − 4πa 4 + 8π a br2 φ 2 2 GP ≤ E3D (1, 1, r, a) + 4πa br4 φ 4 g GP = E3D (1, 1, r, a) + (2.60) φ4 . 2 On the other hand, inf spec Hr,a can be bounded below by Temple’s inequality [40], which says that for any Hamiltonian H with lowest eigenvalues E0 < E1 and expectation value H < E1 in some state, E0 ≥ H −

(H − H )2 . E1 − H

(2.61)

We apply this to H = Hr,a and the state defined by br φ. Here g e⊥ H = 2 + E GP (1, 1, g) + r 2

φ4 ,

(2.62)

and since E1 ≥ e˜⊥ /r 2 with e˜⊥ > e⊥ , (2.62) is smaller than E1 for r 2 E GP (1, 1, g) small enough. Moreover, (H − H )(br φ) = (8πabr2 − g)φ 3 br

(2.63)

and thus

(8πa)2 br6 − g 2 ≤ φ 6 (8πa)2 br4 br 2∞ 2 ≤ const. gφ∞ g φ 4 ≤ const. E GP (1, 1, g)2 ,

(H − H )2 =

φ6

(2.64)

where we used [28, Lemma 2.1] to bound gφ2∞ by the GP energy. Combining (2.60), (2.61) and (2.62) we thus get GP E3D (1, 1, r, a) − e⊥ /r 2 ≥ E GP (1, 1, g) 1 − const. r 2 E GP (1, 1, g) , and the proof is complete.

(2.65)

Remark. In combination with Theorem 2.2 this result demonstrates a fortiori that the three-dimensional GP limit theorem in [27] holds even if r/L → 0, provided N aL/r 2 stays bounded. A more direct proof of this fact, closer to the lines of [27], is certainly possible, but it requires redoing all estimates keeping track of the dependence on r/L.

362


3. Finite n Bounds Before we give the proof of our main Theorem 1.1 in Sect. 4, we will explain briefly the strategy, and give some auxiliary results in this section. In particular, we will derive upper and lower bounds on the ground state energy of (1.1) with the external potential VL (z) replaced by a box with Dirichlet and Neumann boundary conditions, respectively. To obtain bounds on the full Hamiltonian (1.1), space will be divided in the z-direction into small boxes of side length , and the bounds of this section will be used in every box. The reason for this is twofold: first, this allows to consider an essentially homogeneous system, without the additional difficulty of the external potential VL (z), and secondly, by varying the particle number in each box can be controlled. This is necessary, since the bounds we obtain in every box will not be uniform in the particle number. Since the particle number in the boxes will be small (compared to N ), we denote it by n. In this section, we study the Hamiltonian H =

n

−j + Vr⊥ (xj⊥ ) +

j =1

va (|xi − xj |)

(3.1)

1≤i<j ≤n QM

QM

on L2 ((R2 × [0, ])n ). Let ED (n, , r, a) and EN (n, , r, a) denote its ground state energy with Dirichlet and Neumann boundary conditions, respectively. The following theorem gives upper and lower bounds on the ground state energy of (3.1) in terms of the ground state energy of the 1D Hamiltonian (1.3). Its proof will be given in Subsects. 3.1 and 3.2. A crucial step in the proof of the lower bound will be the use of the “Dyson Lemma” (see [9, 31]), which converts the “hard” interaction potential va into a “soft” one. With this new “soft” potential perturbation theory can be done, and rigorous bounds are obtained with the use of Temple’s inequality (2.61). Theorem 3.1 (Finite n bounds). Let ED1D (n, , g) and EN1D (n, , g) denote the ground state energy of (1.3) on L2 ([0, ]n ), with Dirichlet and Neumann boundary conditions, respectively, and let g be given by (1.10). Then there is a finite number C > 0 such that a 1/8 nr a 1/8 ne⊥ QM 1+ . (3.2) EN (n, , r, a) − 2 ≥ EN1D (n, , g) 1 − Cn r r r Moreover, QM ED (n, , r, a) −

a 1/3 ne⊥ na 2 1D , 1+ 2 ≤ ED (n, , g) 1 + C r2 r r

(3.3)

provided the term in square brackets is less than 1. Let us comment briefly on the error terms in (3.2) and (3.3). As already mentioned, in the proof of Theorem 1.1 we will divide space in the z-direction into small boxes of ¯ where side length . The number of particles in each box will be roughly n ∼ N /L, L¯ ≡ N/ρ¯ is the extension of the system in the z-direction. The n-dependence of the error term in (3.2) restricts us essentially to have a finite particle number n, i.e., that ¯ ¯ , n ∼ N/L¯ ∼ 1, or ∼ L/N. But for (3.2) to be useful we need r, i.e., r L/N or, in other words, r should be of the order of the mean particle spacing, or smaller. ¯ , r is much bigger than the mean particle spacing, and we have to use For r L/N a different strategy, similar to the one used in the 3D problem [31, 27]. This will be


363

necessary for the lower bound in Regions 1–3. The result is stated in Theorem 3.2. For its proof it will be necessary to use the box method also in x⊥ -direction, similar to the 3D case considered in [27]. However, one cannot use directly the results from there, one has to be careful to retain uniformity in r/L. Likewise, (3.3) will not be useful for an upper bound in all the Regions 1–5. The ¯ reason is the last term in (3.3), where we want g ∼ g L/N 1, which is only fulfilled in Regions 1–4. For Region 5, we use a different upper bound, given in the following Theorem. The proof of Theorem 3.2 will be given in Subsects. 3.3 and 3.4. Theorem 3.2 (Additional energy bounds). With the same notations as in Thm. 3.1,   ⊥ 2g ne n na 1/8 QM −1/14   EN (n, , r, a) − 2 ≥ + 1−C n r 2 r2  √ 1/4 4/39 na r na  + 1+ √ + . (3.4) r n Moreover, denoting the range of va by R0 , QM

ED (n, , r, a) −

ne⊥ π 2 n3 (1 + 1/n) (1 + 1/2n) ≤ , r2 3 2 (1 − (n − 1)R0 /)2

(3.5)

provided (n − 1)R0 < . ¯ )1/3 Remark. By definition, R0 ∼ a. Equation (3.4) will be used with ∼ (r 2 L/N 2 1/3 ¯ ¯ and n ∼ N /L. √ In this case we have, in Regions 1–3, 1, √ ng ∼ a(N/r L) ¯ 1/3 1 and r/( n) ∼ 1. na/ ∼ a ρ¯ 1, na/r ∼ a(N/r 2 L) The following four subsections contain the proofs of Theorems 3.1 and 3.2. Throughout, C denotes a constant independent of the parameters, although the value of different C’s may be different.

3.1. Upper bound for Theorem 3.1. In this subsection we are going to prove (3.3). We use the variational principle. Let ψ denote the ground state of (1.3) with Dirichlet boundary conditions, normalized by ψ|ψ = 1, and let G and F be given by G(x1 , . . . , xn ) = ψ(z1 , . . . , zn )

n

br (xj⊥ )

(3.6)

j =1

and F (x1 , . . . , xn ) =

f (|xi − xj |) .

(3.7)

i<j

Here f is a monotone increasing function, with 0 ≤ f ≤ 1 and f (t) = 1 for t ≥ R for some R ≥ R0 . For t ≤ R we shall choose f (t) = f0 (t)/f0 (R), where f0 is the solution to the zero-energy scattering equation for va [31, 27]. Note that f0 (R) = 1 − a/R for

364


R ≥ R0 , and f0 (t) ≤ t −1 min{1, a/t}. We use as a trial wave function for (3.1) the function (3.8)

(x1 , . . . , xn ) = G(x1 , . . . , xn )F (x1 , . . . , xn ) . (2) (2) Let ρψ denote the two-particle density of ψ, normalized by ρψ (z, z )dzdz = 1. Since F is 1 whenever no pair of particles is closer together than a distance R, we can estimate the norm of by   | ≥ G2 1 − θ (R − |xi − xj |) i<j

n(n − 1) (2) =1− ρψ (z, z )br (x⊥ )2 br (y⊥ )2 θ(R − |x − y|)dzdz d 2 x⊥ d 2 y⊥ 2 n(n − 1) (2) ≥1− ρψ (z, z )br (x⊥ )2 br (y⊥ )2 2 × θ (R − |x⊥ − y⊥ |)dzdz d 2 x⊥ d 2 y⊥ n(n − 1) =1− br (x⊥ )2 br (y⊥ )2 θ (R − |x⊥ − y⊥ |)d 2 x⊥ d 2 y⊥ 2 n(n − 1) πR 2 ≥1− b44 , (3.9) 2 r2 where we used Young’s inequality [24] in the last step. Using (3.10) | − j | = − F 2 Gj G + G2 |∇j F |2 and the Schrödinger equation Hn,g ψ = ED1D ψ, we can write the expectation value of (3.1) as n δ(zi − zj )| |H | = ED1D + 2 e⊥ | − g | r i<j   n + G2  (3.11) |∇j F |2 + va (|xi − xj |)|F |2  . j =1

Now, since 0 ≤ f ≤ 1 and n

|∇j F |2 ≤ 2

j =1

f

≥ 0 by assumption, F 2 ≤ f (|xi − xj |)2 , and

f (|xi − xj |)2 + 4

i<j

i<j

f (|xk − xi |)f (|xk − xj |) .

(3.12)

k 0 we can find κ > 0 so that for any τ and t ∈ [0, 1], D(τ, µˆ N 0,1 ) < implies dF (µˆ N , ν τ ) < κ. tXN +(1−t)XN t 1

0

Therefore, for any t1 , · · · , tn ∈ [0, 1], for any (X0N , X1N ) ∈ {d(µˆ N ˆN 0 , µ0 ) < δ; d(µ 1 , µ1 ) < N δ; D(µˆ 0,1 , τ ) < }, The Chebyshev inequality yields P( max dF (µˆ N , µτtk ) > η|X1N ) ≤ nC(κ + XN 1≤k≤n

tk

1 ) N

√ with µτt = µXt the distribution of Xt = tX1 + (1 − t)X0 + t (1 − t)S when the law of (X0 , X1 ) is τ . Hence for any η, when κ (i.e. ) is small enough and N large enough, , µτtk ) < η|X1N ) > P( max dF (µˆ N XN 1≤k≤n

tk

1 . 2

Hence P(d(µˆ N ˆN 1 , µ1 ) < δ; D(µ 0,1 , τ ) < ) ≤ 2P(d(µˆ N ˆN ˆN , µτtk ) < η). 1 , µ1 ) < δ; D(µ 0,1 , τ ) < , max dF (µ XN 1≤k≤n

tk

We arrive at, for small enough and any τ ∈ M0,1 , 1 log P(d(µˆ N ˆN 1 , µ1 ) < δ; D(µ 0,1 , τ ) < ) 2 N→∞ N 1 τ ≤ lim sup 2 log P( max dF (µˆ N tk , µtk ) < δ). 1≤k≤n N→∞ N

lim sup

Using the large deviation upper bound for the law of (µˆ N t , t ∈ [0, 1]) from [16], we deduce lim sup N→∞

1 β min log P(d(µˆ N inf 1 , µ1 ) < δ) ≤ − 2 N 2 1≤p≤M max1≤k≤n dF (νt

τp k ,µtk )≤δ

S(ν).

First Order Asymptotics of Matrix Integrals

565

We can now let go to zero, and then with δ going to zero, and then n going to infinity, to obtain, since S is a good rate function, that lim sup lim sup δ→0

N→∞

1 β log P(d(µˆ N 1 , µ1 ) < δ) ≤ − 2 N 2

inf

τ :τ ◦X0−1 =µ0 τ ◦X1−1 =µ1

S(µτ ).

Since it was also proved in [16] that lim inf lim inf δ→0

we conclude that

N→∞

1 β log P(d(µˆ N inf S(ν), 1 , µ1 ) < δ) ≥ − 2 0 N 2 νν0 =µ =µ 1

inf S(ν) =

ν0 =µ0 ν1 =µ1

inf

τ :τ ◦X0−1 =µ0 τ ◦X1−1 =µ1

1

S(µτ ).

Hence, if FBB(µ0 , µ1 ) is the set of laws of free Brownian bridges between µ0 and µ1 , i.e FBB(µ0 , µ1 ) = {µτ , τ ◦ X0−1 = µ0 , τ ◦ X1−1 = µ1 }, we have seen that inf{S(ν), ν0 = µ0 , ν1 = µ1 } = inf{S(ν), ν ∈ FBB(µ0 , µ1 )}. To finish the proof of Theorem 2.7, we need to show that FBB(µ0 , µ1 ) is a closed subset of C([0, 1], P(R)) so that indeed the infimum is reached in FBB(µ0 , µ1 ). Observe here that µτ does depend only partially on τ since it only depends on {νtτ , t ∈ [0, 1]}. Noting that νtτ (x p ) =

p

t r τ (Pr,p (X1 − X0 , X0 ))

r=0

with Pr,p (X, Y ) the sum over all the monomial functions with total degree p and degree r in X, we see that µτ only depends on the restriction of τ to polynomial functions P ∈ S = {Pr,p , 0 ≤ r ≤ p < ∞}. Of course, ,C 2p + Y 2p ) ≤ 2C 2p , ∀p ∈ N} MS 0,1 = {τ |S , τ ∈ M0,1 , τ (X

is closed for the dual topology generated by the polynomial functions of S. Here C denotes a common uniform bound on X0 and X1 , and we have ,C FBB(µ0 , µ1 ) = {µτ |S , τ ∈ M0,1 } = {µκ , κ ∈ MS 0,1 }. ,C κ We denote, for κ ∈ MS 0,1 and t ∈ [0, 1], νt ∈ P(R) the distribution of tX1 + (1 − t)X0 when the joint distribution of (X0 , X1 ) restricted to S is κ. Then, µκt = νtκ + σt (1−t) . We now show that FBB(µ0 , µ1 ) is a closed set of C([0, 1], P(R)), which insures, since S is a good rate function on C([0, 1], P(R)), that the infimum is achieved on FBB(µ0 , µ1 ). Indeed, if µn is a sequence of FBB(µ0 , µ1 ) given by {νtκn + σt (1−t) , t ∈ [0, 1]}, the weak convergence of µn implies the weak convergence of κ n . Indeed, for any p ∈ N, any t ∈ [0, 1], µnt (x p ) = νtκn (x p ) + Pt (µnt (x l ), l ≤ p − 1)

566

A. Guionnet

with a polynomial function Pt . Hence, by induction, the convergence of (µnt (x p ))p∈N (recall that µn is supported by [−C − 2, C + 2] for any n so that weak

convergence is equivalent to moment convergence) results in the convergence of νtκn (x p )) p∈N , and again, since (νtκn )n∈N is supported by [−C, C], with the weak convergence of νtκn towards some probability measure νt . Since this convergence holds for any t ∈ [0, 1], we can expend the moments in powers of the time variable to conclude that κn converges ,C towards κ ∈ MS 0,1 . Again by free convolution calculus, this convergence results in the convergence of µn towards µκ ∈ FBB(µ0 , µ1 ). Hence, FBB(µ0 , µ1 ) is closed. 4.2. Proof of Lemma 2.4. In [16] (see (2.13) and Lemma 2.10) O. Zeitouni and I proved that for any path ν ∈ C 1 ([0, 1], P(R)), there exists a path ν , such that lim sup S 0,1 (ν , ) = Sµ0 (ν). ,↓0

This path was constructed as follows. Let P be the Cauchy law with parameter and set µ = P ∗ µ as the convoluted path with the Cauchy law. Moreover, if 0 = t1 < t2 < . . . < tn = 1 with ti = (i − 1), we set, for t ∈ [tk , tk+1 [, νt , = νt k +

(t − tk ) [νtk+1 − νt k ].

Let us therefore consider S 0,1 (ν , ). Because we took the convolution with respect to the Cauchy law, the Hilbert transform H νt , is well defined, and actually a continuously differentiable function with respect to the time variable and an analytic function with respect to the space variable. Henceforth, in the supremum defining S 0,1 (ν , ), we can actually make the change of function f (t, x) → f (t, x) − log |x − y|dνt , (y). Observing that, with νi = νi ∗ P for i ∈ {0, 1},

1

∂t 0

1

log |x − y|−1 dνt , (y) dνt , (x)dt = (ν1 ) − (ν0 ) , 2

we find that 1 1 1

(H νt , )2 dνt , dt (ν1 ) − (ν0 ) + 2 2 0 + sup f1 dµ 1 − f0 dν0

S 0,1 (ν , ) = −

f ∈Cb2,1 ([0,1]×R)

1 0,1 − − < f, f >ν , 2 0 1 1 1

(H νt , )2 dµ , ≥ − (ν1 ) − (ν0 ) + t dt. 2 2 0

1

∂t ft dνt , dt

Noticing that 0

1

[] 1

(H νt , )2 dµ , t dt

=

k=0

(H νt k )2 dνt k


567

converges since t → H νt and t → νt are continuous for any ν ∈ C([0, 1], P(R)), we arrive at 1 1 1

(H νt )2 dνt dt. (4.1) lim inf S 0,1 (ν , ) ≥ − (ν1 ) − (ν0 ) + ↓0 2 2 0 Remark that for t ∈ {0, 1}, 1 (νt ) = log |x−y|−1 dP ∗νt (x)dP ∗νt (y) = log((x−y)2 + 2 )−1 dνt (x)dνt (y). 2 Hence, the monotone convergence theorem asserts that lim (νt ) = (νt ). ↓0

In particular, if (ν0 ) is finite, (4.1) implies that (ν1 ) is also bounded below, and therefore bounded since S 0,1 (ν) < ∞ implies that ν1 (x 2 ) < ∞, and consequently that (ν1 ) is bounded above. Now, recall that for any ρ ∈ L3 , Tricomi [27] p. 169 asserts that π2 1 ρ(x)2 = (Hρ)2 (x) − H (ρ(Hρ))(x) , 2 2 so that π2 (Hρ)2 (x)ρ(x)dx = (ρ(x))3 dx. 3 Since, for any > 0, νt is absolutely continuous with respect to Lebesgue measure with density ρt ∈ L3 (dx) for almost all t ∈ [0, 1], (4.1) implies that 1 ρt (x)3 dxdt ≤ C 0

with a finite constant C independent of . Consequently, for any Lipschitz function f , by Holder’s inequality, ! ! ! ! ! ! ! ! ! ft (x)dνt (x)dt ! ≤ sup |ft |L + ! ft (x)ρ (x)dxdt ! t ! ! ! ! t∈[0,1]

≤ sup |ft |L + C t∈[0,1]

so that letting go to zero, we obtain ! ! ! ! ! ft (x)dνt (x)dt ! ≤ C ! !

1

1

3 2

|ft (x)| dxdt

23 ,

0

3 2

|ft (x)| dxdt

23 ,

0

3

an inequality which extends readily to L 2 (dxdt) by density. As a consequence, dνt (x)dt dxdt, dνt (x)dt = ρt (x)dxdt and ρt (x) converges towards ρt almost surely. We conclude by Fatou’s lemma that +∞ > lim inf lim inf S 0,1 (ν , ) ↓0

↓0

π2 1 1 lim inf (ρt (x))3 dxdt. ≥ − ((ν1 ) − (ν0 )) + ↓0 2 6 0 1 2 π 1 (ρt (x))3 dxdt. = − ((ν1 ) − (ν0 )) + 2 6 0

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Acknowledgements. I am very much indebted to C. Villani and O. Zeitouni whose careful reading of preliminary versions of the manuscript, wise remarks and encouragements were crucial in this research. I am also very grateful to D. Serre and Y. Brenier for stimulating discussions.

References 1. Ben Arous, G., Guionnet, A.: Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy. Prob. Th. Rel. Fields 108, 517–542 (1997) 2. Bercovici, H., Voiculescu, D.: Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42, 733–773 (1993) 3. Biane, P.: On the Free convolution with a Semi-circular distribution. Indiana Univ. Math. J. 46, 705–718 (1997) 4. Biane, P., Capitaine, M., Guionnet, A.: Large deviation bounds for matrix Brownian motion. Invent. Math. 152, 433–459 (2003) 5. Brenier, Y.: Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Comm. Pure. Appl. Math. 52, 411–452 (1999) 6. Brézis, H.: Functional analysis. Paris: Masson, 1983 7. Cabanal-Duvillard, T., Guionnet, A.: Large deviations upper bounds and non commutative entropies for some matrices ensembles. Ann. Probab. 29, 1205–1261 (2001) 8. Cabanal-Duvillard, T., Guionnet, A.: Discussions around non-commutative entropies. Adv. Math. 174, 167–226 (2003) 9. Chadha, S., Madhoux, G., Mehta, M.L.: A method of integration over matrix variables II. J. Phys. A. 14, 579–586 (1981) 10. Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R.: New results on the equilibrium measure for logarithmic potentials in the presence of an external field. J. Approx. Theory 95, 388–475 (1998) 11. Dembo, A.,Zeitouni, O.: Large deviations techniques and applications. Second edition, BerlinHeidelberg-New York: Springer, 1998 12. Ercolani, N.M., McLaughlin, K.D.T-R.: Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques, and applications to graphical enumeration. To appear in Int. Math. Res. Notes, 2003 13. Eynard, B.: Eigenvalue distribution of large random matrices, from one matrix to several coupled matrices. Nucl. Phys. B. 506, 633–664 (1997) 14. Eynard, B.: Random matrices. http://www-spht.cea.fr/lectures-notes.shtml 15. Guionnet, A.: Large deviation upper bounds and central limit theorems for band matrices. Ann. Inst. H. Poincaré Probab. Statist. 38, 341–384 (2002) 16. Guionnet, A., Zeitouni, O.: Large deviations asymptotics for spherical integrals. J. Funct. Anal. 188, 461–515 (2002) 17. Guionnet, A. Zeitouni, O.: Addendum to: Large deviations asymptotics for spherical integrals. To appear in J. Funct. Anal. (2004) 18. Harer, J., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math. 85, 457–485 (1986) 19. Loeper, G.: The inverse problem for the Euler-poisson system in cosmology. Preprint, 2003 20. Mahoux, G., Mehta, M.: A method of integration over matrix variables III. Indian J. Pure Appl. Math. 22, 531–546 (1991) 21. Matytsin, A.: On the large N-limit of the Itzykson-Zuber integral. Nucl. Phys. B411, 805–820 (1994) 22. Matytsin, A., Zaugg, P.: Kosterlitz-Thouless phase transitions on discretized random surfaces. Nucl. Phys. B497, 699–724 (1997) 23. Mehta, M.L.: Random matrices. 2nd ed., New York-London: Academic Press, 1991 24. Mehta, M.L.: A method of integration over matrix variables. Comm. Math. Phys. 79, 327–340 (1981) 25. Serre, D.: Sur le principe variationnel des e´ quations de la mécanique des fluides parfaits. Math. Model. Num. Anal. 27, 739–758 (1993) 26. Szarek, S., Voiculescu, D.: Volumes of restricted Minkowsky Sums and the Free analogue of the Entropy Power Inequality. Commun. Math. Phys. 178, 563–570 (1996) 27. Tricomi, F.G.: Integral equations. New York: Interscience, 1957 28. Voiculescu, D.: Limit laws for random matrices and free products. Invent. Math. 104, 201–220 (1991) 29. Voiculescu, D.: The analogues of Entropy and Fisher’s Information Measure in Free Probability Theory, V : Noncommutative Hilbert Transforms. Invent. Math. 132, 189–227 (1998) 30. Voiculescu, D.: Lectures on free probability theory. In: Sptinger Lecture Notes Mathematics 1738, Berlin-Heidelberg-New York: Springer-Verlag, 2000, pp. 283–349 31. Wigner, E.: On the distribution of the roots of certain symmetric matrices. Ann. Math. 67, 325–327 (1958)


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32. Zinn-Justin, P.: Universality of correlation functions of hermitian random matrices in an external field. Commun. Math. Phys. 194, 631–650 (1998) 33. Zinn-Justin, P.: The dilute Potts model on random surfaces. J. Stat. Phys. 98, 245–264 (2000) 34. Zvonkin, A.: Matrix integrals and Map enumeration: an accessible introduction. Math. Comput. Mod. 26, 281–304 (1997) Communicated by M. Aizenman


Communications in


Integrability Versus Separability for the Multi-Centre Metrics Galliano Valent1,2 1

Laboratoire de Physique Théorique et des Hautes Energies, Unité associée au CNRS UMR 7589, 2 Place Jussieu, 75251 Paris Cedex 05, France 2 CNRS Luminy Case 907, Centre de Physique Théorique, 13288 Marseille Cedex 9, France Received: 28 November 2002 / Accepted: 12 September 2003 Published online: 17 December 2003 – © Springer-Verlag 2003

Abstract: The multi-centre metrics are a family of euclidean solutions of the empty space Einstein equations with self-dual curvature. For this full class, we determine which metrics do exhibit an extra conserved quantity quadratic in the momenta, induced by a Killing-Stäckel tensor. Our systematic approach brings to light a subclass of metrics which correspond to new classically integrable dynamical systems. Within this subclass we analyze on the one hand the separation of coordinates in the Hamilton-Jacobi equation and on the other hand the construction of some new Killing-Yano tensors. 1. Introduction The discovery of the generalized Runge-Lenz vector for the Taub-NUT metric [8] has been playing an essential role in the analysis of its classical and quantum dynamics. As shown in [5] this triplet of conserved quantities gives quite elegantly the quantum bound states as well as the scattering states. The Killing-Stäckel tensors, which are the roots of the generalized Runge-Lenz vector of Taub-NUT, have been derived in [10] using purely geometric tools. As a result the classical integrability of the Taub-NUT metric was established. The classical integrability of the Eguchi-Hanson metric was obtained in [15] where the Hamilton-Jacobi equation was separated. This result was further generalized in [10] to cover the twocentre metric. Despite these successes, a systematic analysis of the full family of the multi-centre metrics was still lacking. It is the aim of this article to fill this gap. In Sect. 2 we have gathered a summary of known properties of the multi-centre metrics, their geodesic flow and some basic concepts about Killing-Stäckel tensors. In Sect. 3 we obtain the most general structure of the conserved quantity associated to a Killing-Stäckel tensor: it is a bilinear form in the momenta. Taking this quadratic structure as a starting point, we obtain the system of equations which ensure that such a kind of quantity is preserved by the geodesic flow. This system is analyzed and simplified. Its most important consequence is that the existence of an extra conserved quantity

572

G. Valent

is related to the existence of an extra spatial Killing (besides the tri-holomorphic one), which may be either holomorphic or tri-holomorphic. In Sect. 3 we first consider the case of an extra spatial Killing which is holomorphic. We find that the extra conserved quantity does exist for the following families, with (minimal) isometry U (1) × U (1): 1. The most general two-centre metric, with the potential V = v0 +

m1 m2 + . |r + c| |r − c|

Our approach explains quite simply why there are three extra conserved quantities for Taub-NUT and only one for Eguchi-Hanson, and their very different nature. 2. A first dipolar breaking of Taub-NUT, with potential V = v0 +

m F · r + 3 . r r

3. A second dipolar breaking of Taub-NUT with potential V = v0 +

m + E · r. r

In the Taub-NUT limit E → 0 there appears a triplet of extra conserved quantities: the generalized Runge-Lenz vector of [8]. The classical integrability of these three dynamical systems follows from our analysis. In Sect. 4 we consider the case of an extra spatial Killing which is tri-holomorphic, with (minimal) isometry group still U (1) × U (1). We find four different families of metrics, which share with the previous ones their classical integrability and, using appropriate coordinates, with potentials: 1. In the first case

aξ ξ 2 − c2 + bη c2 − η2 V = v0 + . ξ 2 − η2

2. In the second case V = v0 + m

cos(2φ) . r2

3. In the third case V =

aξ + bη . ξ 2 + η2

4. And in the fourth case V = v0 + mx. As an application we work out in Sects. 5 and 6 the separation of variables for the Hamilton-Jacobi equation which gives also a check of the results obtained in the former sections. Eventually we present in Sect. 7 some new Killing-Yano tensors, and some conclusions in Sect. 8.

Multi-Centre Conservation Law

573

2. The Multi-Centre Metrics 2.1. Background material. These euclidean metrics on M4 have at least one Killing = ∂t and have the local form vector K g=

1 (dt + )2 + V γ , V

V = V (x),

= i (x) dx i ,

(1)

where the x i are the coordinates on γ . They are solutions of the empty space Einstein equations provided that : 1. The three dimensional metric γ is flat. Using cartesian coordinates x i we can write γ = d x · d x.

(2)

dV = ∗ d.

(3)

2. Some monopole equation holds γ

Notice that the integrability condition for the monopole equation is V = 0, hence these metrics display an exact linearization of the empty space Einstein equations. They have been derived in many ways [14, 7, 11, 12]. In this last reference the geometric meaning of the cartesian coordinates xi was obtained: they are nothing but the momentum maps of the complex structures under the circle action of ∂t . Let us summarize some background knowledge on the multi-centre metrics for further use. Taking for canonical vierbein 1 E0 = √ (dt + ), V

Ea :

Ei =

√ V dxi

(4)

and defining as usual the spin connection ωab and the curvature Rab by dEa + ωab ∧ Eb = 0,

Rab = dωab + ωas ∧ ωsb ,

one can check that these metrics have a self-dual spin connection: (−)

ωi

≡ ω0i −

1 ij k ωj k = 0, 2

⇒

(−)

Ri

= 0,

which implies the self-duality of their curvature. It follows that they are hyperkähler and hence Ricci-flat. The complex structures are given by the triplet of 2-forms (−)

i

= E0 ∧ Ei −

1 1 ij k Ej ∧ Ek = (dt + ) ∧ dxi − V ij k dxj ∧ dxk , 2 2

(5)

which are closed, in view of the hyperkähler property of these metrics. Let us note that the self-duality of the complex structures and of the spin connection are opposite and that the Killing vector ∂t is tri-holomorphic. = ∂t , which reads It is useful to define the Killing 1-form, dual of the vector K K=

dt + , V

and plays some role in characterizing the multi-centre metrics.

(6)

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G. Valent

Among these characterizations let us mention: 1. For the multi-centre metrics the differential dK has a self-duality opposite to that of the connection. A proof using spinors may be found in [17] and without spinors in [6]. 2. The multi-centre metrics possess at least one tri-holomorphic Killing. For a proof see [10]. 2.2. Geodesic flow. The geodesic flow is the Hamiltonian flow of the metric considered as a function on the cotangent bundle of M4 . Using the coordinates (t, xi ) we will write a cotangent vector as i dxi + 0 dt. The symplectic form is then ω = dxi ∧ di + dt ∧ d0 , and we take for hamiltonian 1 1 H = g µν µ ν == 2 2

1 2 2 (i − 0 i ) + V 0 . V

(7)

(8)

For geodesics orthogonal to the U (1) fibers and affinely parametrized by λ the equations for the flow allow on the one hand to express the velocities dt ∂H 2 i i ˙t ≡ = 0 − , = V + dλ ∂0 V V (9) ∂H dxi 1 = x˙i ≡ = pi , pi = i − 0 i , dλ ∂i V and on the other hand to get the dynamical evolution equations ∂H = 0, ∂t ∂H ˙i =− ⇒ ∂xi ˙0 =−

(t˙ + i x˙i ) q ≡ 0 = , (a) V (10) H q p˙ i = − q 2 ∂i V + (∂i s − ∂s i ) ps . (b) V V

Relation (10a) expresses the conservation of the charge q, a consequence of the U (1) isometry of the metric. For the multi-centre metrics, use of relation (3) brings the equations of motion (10b) to the nice form ˙p = H − q 2 ∇ V + q p ∧ ∇V . (11) V V The conservation of the energy 1 pi2 V 1 2 H = + q V = (x˙i2 + q 2 ) = gµν x˙ µ x˙ ν 2 V 2 2

(12)

is obvious since it expresses the constancy of the length of the tangent vector x˙ µ along a geodesic.


575

2.3. Killing-Stäckel tensors and their conserved quantities. A Killing-Stäckel (KS) tensor is a symmetric tensor Sµν which satisfies ∇(µ Sνρ) = 0.

(13)

Let us observe that if K and L are two (possibly different) Killing vectors their symmetrized tensor product K(µ Lν) is a KS tensor. So we will define irreducible KS tensors as the ones which cannot be written as linear combinations, with constant coefficients, of symmetrized tensor products of Killing vectors. For a given KS tensor Sµν the quadratic form of the velocities: S = Sµν x˙ µ x˙ ν

(14)

is preserved by the geodesic flow. In all what follows we will look for KS tensors, under the assumptions A1: The KS tensor is preserved by Lie dragging along the tri-holomorphic Killing vector: L Sµν = 0, K˜

K˜ = ∂t .

(15)

A2: We will consider generic values of H and q = 0. Furthermore, instead of focusing ourselves on the KS tensor Sµν , whose usefulness is just to produce the conserved quantity S, let us rather examine more closely the structure of the conserved quantity induced by such a KS tensor. From relation (14) we obtain the following ansatz for the conserved quantity we are looking for: S = Aij (xk ) pi pj + 2q Bi (xk ) pi + C(xk ),

(16)

where the various unknown functions, as a consequence of A1, are independent of the coordinate on the U (1) fiber. It is interesting to notice that the knowledge of S is equivalent to the knowledge of the K-S tensor: using (9) one can express S in terms of the velocities and, going backwards, compute the K-S tensor components from relation (14). Imposing the conservation of S under the geodesic flow gives: Proposition 1. Under Assumptions A1 and A2 the quantity S, given by (16), is conserved iff the following equations are satisfied 1 a)

q · L V = 0,

b) c) d)

∂(k Aij ) = 0, q(∂(i Bj ) − As(i j )su ∂u V ) = 0, ∂i C + 2(H − q 2 V ) Ais ∂s V − 2 q 2 ist Bs ∂t V = 0.

B

(17)

We are now in a position to explain why we assumed, in A2, that q should not vanish. Indeed for q = 0 the relations (17a) and (17c) are trivially true and we are left with ∂(k Aij ) = 0,

∂i C + 2H Ais ∂s V = 0,

while the conserved quantity S reduces to S = Aij (x) pi pj + C(x). 1

Assumption A2 implies that H − q 2 V does not vanish identically.

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It is interesting to notice that, formally, S is preserved by the hamiltonian flow induced by the classical hamiltonian [16] H=

p 2 − H V, 2

where now H appears as some constant parameter. However the assumption that q = 0 leads to a reduced system which has only three degrees of freedom and as such may exhibit integrability. Since we are interested in genuine four dimensional integrability we have to exclude such a possibility. Let us proceed to the discussion of the system (17). Relation (17a) shows that there are two possible situations: 1. Either the potential V has one (or more) spatial symmetries, with Killing K, and then B has to be conformal to this Killing vector, 2. Or the potential has no spatial symmetry, and in this case B = 0. Let us show that this last possibility does not give any new conserved quantity. Indeed relation (17c) can be written [A, R] = 0,

(R)ij = isj ∂s V .

(18)

Since V has no Killing the matrix R is a generic matrix in the Lie algebra so(3). By the Schur lemma it follows that A has to be proportional to the identity matrix and this does trivialize the corresponding conserved quantity S. So the unique possibility left is the first one. Let us notice that K lifts up to an isometry of the 4 dimensional metric. We have obtained: Proposition 2. The number of extra conserved quantities, having the structure (16), of a multi-centre metric is at most equal to the number of extra spatial Killing vectors it = ∂t . does possess, besides the tri-holomorphic Killing K Using this result we can discuss the triaxial generalization of the Eguchi-Hanson metric, with a tri-holomorphic su(2), discovered in [2]. Its potential and cartesian coordinates were given in [9] and the potential has no spatial Killing. From the previous proposition it follows that this metric will not exhibit a conserved quantity of the form (16) for generic values of H and q = 0. 2.4. Transformations of the system. As observed above, the vector B has to be conformal to the Killing K. So we define the conformal factor F such that Bi = − F Ki .

(19)

The conserved quantity (16) becomes S = Aij (x) pi pj − 2 q F Ki pi + C(x),

(20)

and Eq. (17c) transforms into K(i ∂j ) F + As(i j )su ∂u V = 0.

(21)

Taking its trace we see that L F = 0, showing that V and F must have the same K

Killing.


577

Contracting (21) with ∂j V gives Lemma 1. Equation (21) has for a consequence: (dV · dF )K + (A[dV ] ∧ dV ) = 0,

A[dV ] = Ais ∂s V dx i .

(22)

We can proceed to: Proposition 3. The relation (21) is equivalent (except possibly at the points where the norm of the Killing K vanishes) to the relations: A[K] = a(x) K, a) (23) b) |K|2 dF − A[(K ∧ dV )] + (A[K] ∧ dV ) = 0. Proof. Contracting relation (21) with Kj gives relation b), while contracting with Ki Kj we have stu Ks A[K]t ∂u V = 0

⇒

A[K]i = a(x)Ki + b(x)∂i V ,

(24)

which is not relation a). To complete the argument we first contract relation (21) with iab Ka ; after some algebra we get Kj iab ∂i F Ka + 2A[K]j ∂b V + A[dV ]b Kj − Ks A[dV ]s δj b − Ass Kj ∂b V = 0, (25) which, upon contraction with A[K]b , gives eventually (A[K]s ∂s V ) A[K]i = {− stu Ks A[K]t ∂u F + Ass A[K]t ∂t V −A[K]s A[dV ]s } Ki .

(26)

Let us now suppose that A[K]s ∂s V = 0. The previous relation shows that in (24) we must have b(x) = 0, hence A[K]s ∂s V = 0 which is a contradiction. Let us prove that the converse is true. From (23b) we get |K|2 K(i ∂j ) F + (K(j Ai)s Kt tsu + A[K]s K(j i)su )∂u V = 0.

(27)

Use of the identity Ais Kt Kj tsu ∂u V = (|K|2 Ais j su − A[K]i Kt j tu )∂u V

(28)

and of relation (23a) leaves us with (21), up to division by |K|2 . Notice that |K|2 vanishes at the fixed points under the Killing action, i. e. in subsets of zero measure in R3 . We can give, using (23a) and the identity −A[(K ∧ dV )] = (A[K] ∧ dV ) − Ass (K ∧ dV ) + (K ∧ A[dV ]),

(29)

a simpler form to the relation (23b): Lemma 2. The relation (23b) is equivalent to |K|2 dF + (2a − Tr A) (K ∧ dV ) + (K ∧ A[dV ]) = 0. For further use let us prove:

(30)

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G. Valent

Lemma 3. To the spatial Killing K, leaving the potential V invariant, there corresponds a quantity Q invariant under the geodesic flow given by Q = Ki pi + qG,

i(K)F = − dG.

with

(31)

Proof. We start from L V = 0. Since K is a Killing we have L( dV ) = d(L V ) = 0, K

K

K

and (3) implies that L d = 0. The closedness of d implies d(i(K)d) = 0, and K

since our analysis is purely local in R3 , we can define η dG = −i(K) d,

⇒

(K ∧ dV ) = dG.

(32)

Then we multiply (10b) by pi and get successively Ki p˙ i = (Ki˙pi ) − K˙ i pi = (Ki˙pi ) = which concludes the proof.

q ˙ Ki (∂i s − ∂s i )ps = −q x˙s ∂s G = −q G, V

Let us point out that if we use the coordinate φ adapted to the Killing K˜ = ∂φ , we can write the connection = G dφ, where G does not depend on φ. 2.5. Integrability equations. We will derive now the integrability conditions for Eqs. (17c) and (17d). The first one was written using forms in (30) while the second one is dC + 2(H − q 2 V )A[dV ] + 2q 2 F (K ∧ dV ) = 0.

(33)

It can now be proved : Proposition 4. The integrability condition for (33) is d A[dV ] = 0

⇒

A[dV ] = dU

and

L U = 0. K

(34)

Proof. The integrability condition is obtained by differentiating (33). We get 2(H − q 2 V ) d A[dV ] + 2q 2 A[dV ] ∧ dV + 2q 2 dF ∧ (K ∧ dV ) +2q 2 F d (K ∧ dV ) = 0.

(35)

The last term in this equation vanishes in view of (32). Furthermore we have the identity specific to three dimensional spaces dF ∧ (K ∧ dV ) = −(K · dF ) dV + (dV · dF ) K = (dV · dF ) K because K is a symmetry of F. Relation (35) simplifies to 2(H − q 2 V ) d A[dV ] + 2q 2 [(dV · dF ) K + (A[dV ] ∧ dV )] = 0, and lemma 1 implies the closedness of A[dV ]. Since our analysis is purely local, the existence of U is a consequence of Poincaré’s lemma. The relations L U = i(K) dU = i(K) A[dV ] = (A[K] · dV ) = a(K · dV ) = a L V = 0 K

show the invariance of U under the Killing K.

K


579

Let us now turn to Eq. (30). We will prove: Proposition 5. The integrability condition for (30) is (2a − Tr A)dV + dU = |K|2 dτ,

L dτ = 0, K

(36)

for some one form τ. Proof. Let us define the 1-form Y = (2a − Tr A)dV + dU.

(37)

It allows to write (30) and its integrability condition as K ∧Y K ∧Y , δ = 0, dF = − |K|2 |K|2

(38)

or switching to components Ki δ

Y |K|2

+

Y s ∂ s K i − K s ∂ s Yi = 0. |K|2

(39)

Let us examine the last terms. Since a and Tr A are invariant under the Killing K, we obtain Ys ∂s Ki − Ks ∂s Yi = −(2a − Tr A)∂i (Ks ∂s V ) − ∂i (Ks ∂s U ),

(40)

and both terms vanish because V and U are invariant under K. We are left with the vanishing of the divergence of Y /|K|2 from which we conclude (local analysis!) that it must have the structure dτ for some 1-form τ . From its definition it follows that dτ is invariant under K. Using this result we can simplify (30) to dF + (K ∧ dτ ) = dF − i(K)dτ = 0.

(41)

Collecting all these results we have: Proposition 6. Under Assumptions A1 and A2, the quantity S = Aij (x) pi pj − 2 q F Ki pi + C(x) is preserved by the geodesic flow of the multi-centre metrics provided that the integrability constraints V = 0,

A[dV ] = dU,

(2a − Tr A) dV + dU = |K|2 dτ

(42)

and the following relations hold: a)

L V = 0,

b) c) d)

A[K] = a K, ∂(k Aij ) = 0, dF = i(K) dτ, d(C + 2H U ) + 2q 2 (−V dU + F dG) = 0,

K

(43) (K ∧ dV ) = dG.

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G. Valent

2.6. Classification of the spatial Killing vectors. An important point, in view of classification, is whether the extra spatial Killing is tri-holomorphic or not. This can be checked thanks to: Lemma 4. The spatial Killing vector Ki ∂i is tri-holomorphic iff ist ∂[s Kt] = 0. Otherwise it is holomorphic. Proof. From [3] we know that, for an hyperkähler geometry, a Killing may be either holomorphic or tri-holomorphic. As shown in [10] such a vector will be tri-holomorphic iff the differential of the dual 1-form K = Ki dxi has the self-duality opposite to that of the complex structures. A computation shows that this is equivalent to the vanishing of 1 1 dK (−) = − ij k ∂[j Kk] E0 ∧ Ei − ist Es ∧ Et , 2 2 from which the lemma follows.

Since we are working in a flat three dimensional flat space, there are essentially two different cases to consider: 1. The Killing K generates a spatial rotation, which we can take, without loss of generality, around the z axis. In this case we have Ki pi = Lz , and this Killing vector is holomorphic with respect to the complex structure J3 , defined in Sect. 2. 2. The Killing K generates a spatial translation, which we can take, without loss of generality, along the z axis. In this case we have the Ki pi = pz , and this Killing vector is tri-holomorphic. We will discuss successively these two possibilities, under the simplifying additional assumption: A3: the K-S tensor Sµν is also preserved by Lie dragging along the extra spatial Killing vector K L Sµν = 0. K

3. One Extra Holomorphic Spatial Killing Vector Equation (43b) states that Aij is a Killing tensor in flat space. As shown in [13] such a Killing tensor is totally reducible to symmetrized tensor products of Killing vectors


581

and involves 20 free parameters. It is most conveniently written in terms of A(p, p) ≡ Aij p i p j . One has:   α L2x + β L2y + γ L2z + 2µ Ly Lz + 2ν Lz Lx + 2λ Lx Ly A(p, p) = +a1 px Ly + a2 px Lz + b1 py Lx + b2 py Lz + c1 pz Lx  +c2 pz Ly + d1 px Lx + d2 py Ly + aij pi pj .

(44)

The constraint (A 3) for the rotational Killing, requires L Aij = 0, which allows to K

bring (44) to the form z + a33 pz2 + a11 p 2 + δ pz Lz . A(p, p) = α(L2x + L2y ) + γ L2z + b (p ∧ L)

(45)

We note that the parameter γ corresponds to a reducible piece which is just the square of Lz . We will take γ = α for convenience. The parameter a11 is easily seen, upon integration of the remaining equations in (17), to give rise, in the conserved quantity S, to the full piece a11 (p 2 − 2H V + q 2 V 2 ),

(46)

which vanishes thanks to the energy conservation (12). So we can take a11 = 0. The second relation in (43b) implies the vanishing of δ. Hence, with slight changes in the notation, we end up with 2 + c2 pz2 + b (p ∧ L) z. A(p, p) = a L

(47)

Let us note that the parameters a and b are real while the parameter c may be either real or pure imaginary. To take advantage of the rotational symmetry around the z axis we use the coordinates x=

√ ρ cos φ,

y=

√ ρ sin φ,

z,

and write the connection = G dφ. By Lemma 3, this symmetry gives for conserved quantity Jz = Lz + q G = x y − y x .

(48)

From the system (43) one can check that the functions F and U are to be determined from F,ρ = (az + b/2)V,z − a/2 V,z (49) F,z = 2(az2 + bz − c2 )V,ρ − (az + b/2)V,z and

U,ρ = z(az + b)V,ρ − 21 (az + b/2)V,z . U,z = −2ρ(az + b/2)V,ρ + (aρ + c2 )V,z

(50)

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G. Valent

3.1. The two-centre metric. This case corresponds to the choice a = 1 and c = 0. Since a = 1, we can get rid of the constant b by a translation of the variable z. So, without loss of generality, we can take b = 0 and use the new variables r± = x 2 + y 2 + (z ± c)2 . We get the relations ∂r+ F = −c ∂r+ V ,

∂r− F = +c ∂r− V ,

which imply V = f (r+ ) + g(r− ),

F = −c(f (r+ ) − g(r− )).

Imposing to the potential V the Laplace equation we have m1 m2 m1 m2 = −c, + , F = −c − V = v0 + r+ r− r+ r−

(51)

i.e. we recover the most general 2-centre metric. Let us recall that only the double TaubNUT metric, given by real m1 = m2 , is complete. If in addition we take the limit v0 → 0, we are led to the Eguchi-Hanson [4] metric. One has then to check the integrability constraint (34) and to determine the functions U and C,2 U = −cz,

C = −2(H − q 2 V )U − q 2 r 2 2 ,

r 2 = x 2 + y 2 + z2 . (52)

Let us observe that the conserved quantity which we obtain may be real even if c is pure imaginary. In this case m1 = m may be complex, but if we take m2 = m the functions V and c are real, as well as S. One obtains quite different metrics (as first observed in the particular case of Eguchi-Hanson metric): real c corresponding to type II metric and c pure imaginary corresponding to type I metric, in the terminology of [4]. The final form of the conserved quantity for the two-centre metric is therefore  2 + c2 pz2 + 2 qc Lz + 2cz (H − q 2 V ) − q 2 r 2 2 ,  SI = L z+c z−c m1 m2 m1 m2 (53) + , = − , G = m1 + m2 .  V = v0 + r+ r− r+ r− r+ r− The relation of our results with the separability of the Hamilton-Jacobi equation for the two-centre metric, obtained in [10], will be discussed in the next section. From the very definition of the coordinates r± it is clear that the previous analysis is only valid for c = 0. The special case c = 0 (it is a singular limit), giving a first dipolar breaking of the Taub-NUT metric, will be examined now. 3.2. First dipolar breaking of Taub-NUT. This case corresponds to the choice a = 1 and c = 0. Since a = 1, we can again get rid of the parameter b. Then relation (49) for F implies V = w0 (r) + w1 (r) z, F,r = −rw1 (r), r = x 2 + y 2 + z2 . (54) 2

We discard constant terms in the function C.


583

Imposing the Laplace equation we obtain V = v0 +

m z + Ez + F 3 , r r

F =−

E 2 F r + . 2 r

(55)

The integrability relations for U require that E = 0 and we have z U =F , r

C = −2F

(3z2 − r 2 ) z z (H − q 2 V ) − 2mq 2 F 2 − q 2 F 2 . r r r4

The final form of the conserved quantity is therefore  2 2  2 − 2 q F Lz − 2F z (H − q 2 v0 ) + q 2 F 2 (x + y ) ,  SI I = L r r r4 x2 + y2 z  V = v + m + F z , G=m −F . 0 r r3 r r3

(56)

(57)

Let us now consider the case a = 0, which leads to a second dipolar breaking of Taub-NUT. 3.3. Second dipolar breaking of Taub-NUT. This case corresponds to the choice a = 0 and b = 1. The relation (49) shows that by a translation of z we can take, without loss of generality, c = 0. From the integrability of F we deduce V = f (x 2 + y 2 + (z − c)2 ) + g(z). Hence by a translation of z we can set c to 0. We are left with V = f (r) + g(z),

F =

1 (f (r) − g(z)). 2

(58)

F =

1 m − Ez . 2 r

(59)

Imposing the Laplace equation yields V = v0 +

m + Ez, r

Then the integrability conditions for U and C are satisfied and we obtain U=

mz E 2 − (x + y 2 ), 2r 4

C = −2U (H − q 2 v0 ) − 2q 2 mE

(x 2 + y 2 ) . r

(60)

The final form of the conserved quantity is therefore   

z−q SI I I = (p ∧ L) V = v0 +

m r

+ E z,

(x 2 + y 2 ) − E z Lz − 2U (H − q 2 v0 ) − 2q 2 mE , (61) r r G = m rz + E2 (x 2 + y 2 ).

m

For E = 0 we are back to the Taub-NUT metric. In this case the spatial isometries are lifted up from u(1) to su(2). As a result we have now three possible Killings to start with (1)

K i pi = L x

(2)

Ki pi = Ly

(3)

Ki pi = Lz ,

(62)

584

G. Valent

and we expect that the conserved quantity found above should be part of a triplet. The two missing conserved quantities can be constructed following the same route which led to SI I I using the new available spatial Killings given by (62). We recover −q mL + m(q 2 v0 − H ) r , S = p ∧ L r r

SI I I (E = 0) ≡ Sz .

(63)

Lemma 3 lifts up Jz , given by (48), to a triplet of conserved quantities + q m r, J = L r

(64)

r S = p ∧ J + m(q 2 v0 − H ) , r

(65)

which allows to write

on which we recognize the generalized Runge-Lenz vector discovered by Gibbons and Manton [8]. We have therefore obtained, for the three hamiltonians HI , HI I (F = 0) and HI I I , corresponding respectively to the extra conserved quantities SI , SI I and SI I I , (the proof of their irreducibility with respect to the Killing vectors is easy) a set of four independent conserved quantities: H,

q = 0 ,

Jz ,

S,

which can be checked to be in involution with respect to the Poisson bracket. Hence we conclude: Proposition 7. The three hamiltonians HI , HI I (F = 0) and HI I I , defined above are integrable in the Liouville sense. 4. One Extra Tri-Holomorphic Spatial Killing Vector This time we have for Killing Ki pi = pz . Imposing (A 3) for the translational invariance and the constraint A[K] ∝ K restricts A(p, p) to have the form A(p, p) = a L2z − 2b px Lz + 2c py Lz +

2

aij pi pj .

(66)

i,j =1

We have omitted a term proportional to pz2 since it is reducible. The functions F and U, which depend only on the coordinates x and y, using the system (43), are seen to be determined by F,x = A12 V,x − A11 V,y U,x = A11 V,x + A12 V,y (67) F,y = A22 V,x − A12 V,y U,y = A12 V,x + A22 V,y with A11 = ay 2 + 2by + a11 , A22 = ax 2 + 2cx + a22 , A12 = −axy − bx − cy + a12 . In order to organize the subsequent discussion, let us observe:

(68)


585

1. For a = 0, we may take a = 1. The spatial translations in the xy-plane allow to take b = c = 0, and a rotation a12 = 0 as well. Hence we are left with A(p, p) = L2z + (a11 − a22 ) px2 + a22 (px2 + py2 ). Adding the reducible term a22 pz2 we recover the piece a22 p 2 which can be discarded, as already explained in Sect. 4. So we will take for our first case A1 (p, p) = L2z − c2 px2 ,

c ∈ R ∪ iR,

c = 0.

(69)

2. Our second case, which is the singular limit c → 0 of the first case, corresponds to A2 (p, p) = L2z .

(70)

3. For a = 0, a first translation allows to take a12 = 0, while the second one allows the choice a11 = a22 and the corresponding term a11 (px2 + py2 ) is disposed of as in the first case. Eventually a rotation will bring b to zero and c to 1. Our third case will be A3 (p, p) = py Lz .

(71)

4. For a = b = c = 0, we can discard px2 + py2 and we are left with our fourth case A4 (p, p) = α py2 + β px py .

(72)

We will state the results obtained for these four cases without going through the detailed computations, which are greatly simplified by the use of the complex coordinate w = x + iy. In all four cases the metric will have the form g=

1 (dt + )2 + V (dz2 + dwdw), V

= G dz.

(73)

4.1. First case. Writing the conserved quantity as S1 = L2z − c2 2x − 2c2 F 0 z + c2 (2v0 U + D)20 − 2c2 U H,

c = 0,

(74)

where z = pz + G 0 and • V + iG = v0 + 2m √ •

w w2

+ c2

w+w , U + iF = −m √ w 2 + c2

,

v0 ∈ R, D = −2|m|2

m ∈ C, (w 2 + w 2 + |w|2 + c2 ) . (75) √ | w 2 + c2 |2

In the particular case where v0 = 0, c ∈ R, m ∈ R (resp. v0 = 0, c ∈ R, m ∈ iR) the metric reduces to the Bianchi VII0 (resp. Bianchi VI0 ) multi-centre metric. The integrability of their geodesic flow was first proved in [1].

586

G. Valent

4.2. Second case. Writing the conserved quantity as S2 = L2z − 2F 0 z + 2v0 U 20 − 2U H,

(76)

we have: • •

m V + iG = v0 + 2 , w w U + iF = m . w

v0 ∈ R,

m ∈ C, (77)

4.3. Third case. Writing the conserved quantity as S3 = y Lz − 2F 0 z + (2v0 U + D)20 − 2U H,

(78)

we have: • •

m V + iG = v0 + 2 √ , w m w−w U + iF = − √ , 2 w

v0 ∈ R,

m ∈ C,

w+w D = |m|2 √ 2 . | w|

(79)

4.4. Fourth case. In this case we take for the driving term A4 (p, p) = α py2 + β px py . Using the freedom of rotations in the xy-plane, at the level of the metric, we can take V = v0 + mx,

G = my.

(80)

S4 = α S4 + β S4 ,

(2)

(81)

S4 = 2y + (z − my 0 )2 , (2) S4 = x y − V 0 (z − my 0 ) − my H.

(82)

This time there are two conserved quantities (1)

given by

(1)

We added reducible terms of the form 2z and z 0 to get a simpler final form. The metric exhibits one further tri-holomorphic Killing vector and a corresponding conserved quantity ∂y − mz ∂t

⇒

y − mz 0 .

Let us close the algebra of the conserved quantities under Poisson bracket. For the Killing vectors we recover a Bianchi II Lie algebra {0 , z } = 0,

{z , y − mz 0 } = m 0 ,

{y − mz 0 , 0 } = 0.

The K-S tensors are invariant under the Killing vectors action, and it may be interesting to note that the Schouten bracket of the two K-S tensors is vanishing. This hamiltonian is therefore super-integrable.


587

To conclude this Sect. let us notice that, among the four potentials considered, only the second one and the fourth one are uniform functions in the three dimensional flat space. As was the case when the extra spatial Killing was holomorphic, we have obtained for the four hamiltonians considered in this section, a set of (at least) four conserved quantities H,

q = 0 ,

z ,

S,

and in all the four cases S is irreducible with respect to the Killing vectors. One can check that these four independent conserved quantities are in involution with respect to the Poisson bracket, hence we have: Proposition 8. The four hamiltonians determined in this Sect. are integrable in the Liouville sense. As is well known the existence of K-S tensors is related to the separability of the Hamilton-Jacobi (H-J) equation, or equivalently to the separability of the Schrödinger equation. In the next sections we will analyze the separability of the H-J equation according to the nature of the extra Killing vector. 5. H-J Separability: Extra Holomorphic Killing We write the metric g=

1 (dt + G dφ)2 + V (γ1 dξ12 + γ2 dξ22 + γ3 dφ 2 ), V

(83)

= ∂t and L = ∂φ , where which makes apparent the two commuting Killing vectors K only the first one is tri-holomorphic. The hamiltonian is 2φ 22 G2 + γ3 V 2 2 G 1 21 H = . (84) + 0 − 0 φ + + 2γ3 V γ3 V 2γ3 V 2V γ1 γ2 Since the γi ’s depend only on ξ1 and ξ2 , it follows that 0 and φ are conserved. 5.1. The two-centre case. The H-J equation separability was first used in [10] to get the corresponding K-S tensor. This reference is muddied by so many misprints that we will present its results anew. Separability relies here on the use of spheroidal coordinates ξ1 = ζ, ξ2 = λ, defined by y = c (ζ 2 − 1)(1 − λ2 ) sin φ, z = c ζ λ. x = c (ζ 2 − 1)(1 − λ2 ) cos φ, This implies γ1 = c2

ζ 2 − λ2 , ζ2 − 1

γ2 = c2

ζ 2 − λ2 , 1 − λ2

γ3 = c2 (ζ 2 − 1)(1 − λ2 ).

588

G. Valent

The potential and connection are V = v0 +

σ ζ − δλ , c(ζ 2 − λ2 )

G=

σ λ(ζ 2 − 1) + δζ (1 − λ2 ) , ζ 2 − λ2

with σ = m1 + m2 and δ = m1 − m2 . The hamiltonian is 2 (ζ − 1) 2ζ + (1 − λ2 ) 2λ (φ − G 0 )2 1 V H = 2 + 2 + 20 . 2c V (ζ 2 − λ2 ) (ζ − 1)(1 − λ2 ) 2

(85)

(86)

The separation constants 3 are 2φ ζ −2δ 2 0 φ − 2c(v0 cζ 2 + σ ζ )H Cζ = (ζ 2 − 1) 2ζ + 2 ζ − 1 ζ − 1 2 δ 2 2 2 + + v c ζ + 2v cσ ζ 20 , 0 0 ζ2 − 1

(87)

and 2φ λ −2σ 0 φ + 2c(v0 cλ2 + δλ)H Cλ = (1 − λ2 ) 2λ + 2 2 1 − λ 1 − λ σ2 2 2 2 + − v0 c λ − 2v0 cδλ 20 . 1 − λ2

(88)

The knowledge of these separation constants is of paramount importance since it reduces the integration of the H-J equation to quadratures. Indeed writing S = t 0 + φ φ + A(ζ ) + B(λ), dA dB in (87) and λ by in (88) to get the relevant dζ dλ separated differential equations. In practice the final integrations may be quite tough. Some algebra allows us to relate the conserved quantity obtained in Sect. 3 to the separation constants, with the final simple result

one has just to replace ζ by

SI = Cλ − (σ 2 + δ 2 )20 .

(89)

In [10] it was conjectured that in the Taub-NUT limit c → 0 this separation constant could be related to some component of the generalized Runge-Lenz vector. We can check that this is not true since, using relation (53), we get 2 − δ 2 20 . lim SI = L

c→0 3

In all that follows each couple of separation constants add up to zero.

(90)


589

5.2. First dipolar breaking. The H-J equation does separate in spherical coordinates ξ1 = r, ξ2 = θ, for which we have γ1 = 1,

γ2 = r 2 ,

γ3 = r 2 sin2 θ,

and V = v0 +

m cos θ +F 2 , r r

G = m cos θ − F

sin2 θ . r

The separation constants in the H-J equation are F F2 2 2 2 2 Cr = r r + 2 0 φ + v0 r + 2v0 mr + 2 20 − 2(v0 r 2 + mr)H, r r

(91)

(92)

and Cθ = 2θ +

2φ sin2 θ

−2m

cos θ 0 φ + sin2 θ

m2 F cos θ 20 − 2F cos θ H. + 2v 0 sin2 θ (93)

The relation with the K-S tensor of Sect. 3 is SI I = Cθ − m2 20 . 5.3. Second dipolar breaking. The H-J equation does separate in parabolic coordinates ξ1 = ξ, ξ2 = η, for which we have γ1 =

(ξ + η) , 4ξ

γ2 =

(ξ + η) , 4η

γ3 = ξ η,

and V = v0 +

2m E + (ξ − η), ξ +η 2

G=m

ξ −η E + ξ η. ξ +η 2

(94)

The separation constants in the H-J equation are 2φ E m E Cξ = 4ξ 2ξ + +2 − ξ 0 φ − 2 m + v 0 ξ + ξ 2 H ξ ξ 2 2 2 2 m E + + 2v0 m + (v02 + 3mE)ξ + v0 Eξ 2 + ξ 3 20 , ξ 4

(95)

and 2φ E m E Cη = 4η 2η + −2 + η 0 φ − 2 m + v 0 η − η 2 H η η 2 2 2 2 m E + + 2v0 m + (v02 − 3mE)η − v0 Eη2 + η3 20 . η 4

(96)

The relation with the K-S tensor of Sect. 3 is SI I I = − 21 Cξ . Having settled the case of an extra holomorphic Killing vector let us now consider the case of an extra tri-holomorphic Killing vector.

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G. Valent

6. H-J Separability: Extra Tri-Holomorphic Killing We write the metric in the form g=

1 (dt + G dz)2 + V dz2 + γ1 dξ12 + γ2 dξ22 , V

(97)

where the coordinates ξ1 and ξ2 will be appropriate coordinates in the xy-plane which = ∂t and L = ∂z , both will ensure separability. The two commuting Killing vectors K tri-holomorphic, are apparent. The hamiltonian is 2z 22 V 2 + G2 2 G 1 21 H = . (98) + 0 − 0 z + + 2V V 2V 2V γ1 γ2 It follows that 0 and z are conserved.

6.1. First case. We restrict ourselves to the case of real c and use elliptic coordinates ξ1 = ξ and ξ2 = η in the xy-plane defined by x=

1 c

(ξ 2 − c2 )(c2 − η2 ),

y=

1 ξ η. c

For convenience, we will define

ξ = ξ ξ 2 − c2 ,

η = η c2 − η2 .

The first case corresponds to γ1 =

ξ 2 − η2 , ξ 2 − c2

γ2 =

ξ 2 − η2 , c2 − η2

V = v0 +

a ξ + b η , ξ 2 − η2

G=

−b ξ + a η . ξ 2 − η2

(99)

The separation constants in the H-J equation are Cξ = (ξ 2 − c2 )2ξ + v02 ξ 2 + 2v0 a ξ + (a 2 + b2 )(ξ 2 − c2 /2) 20 +2 b ξ 0 z + ξ 2 2z − 2(v0 ξ 2 + a ξ ) H,

(100)

and Cη = (c2 − η2 )2η + −v02 η2 + 2v0 b η + (a 2 + b2 )(η2 − c2 /2) 20 −2 a η 0 z − η2 2z + 2(v0 η2 − b η)H.

(101)

The relation with the K-S tensor obtained in Sect. 4 is S1 = −Cξ + c2 (2z + v02 20 − 2v0 H ).

(102)


591

6.2. Second case. We use polar coordinates ξ1 = r, ξ2 = φ in the xy-plane. The second case corresponds to γ1 = 1,

γ2 = r 2 ,

V = v0 + m

cos(2φ) , r2

G = −m

sin(2φ) . r2

(103)

The separation constants in the H-J equation are  2   Cr = r 2 (2 + 2 ) + v 2 r 2 + m 2 − 2v0 r 2 H, r z 0 0 r2   Cφ = 2 + 2m sin(2φ) 0 z + 2m cos(2φ) v0 2 − H . φ 0

(104)

The relation with the K-S tensor obtained in Sect. 4 is S2 = Cφ .

6.3. Third case. We use squared parabolic coordinates ξ1 = ξ, ξ2 = η in the xy-plane. The third case corresponds to γ1 = γ2 = ξ 2 + η2 ,

V =

aξ +bη , ξ 2 + η2

G=

bξ −aη . ξ 2 + η2

(105)

The separation constants in the H-J equation are

Cξ = 2ξ + (ξ z − b 0 )2 + 21 (a 2 − b2 )20 − 2aξ H, Cη = 2η + (η z + aη 0 )2 − 21 (a 2 − b2 )20 − 2bη H.

(106)

The relation with the K-S tensor obtained in Sect. 4 is S3 = − 21 Cξ . 6.4. Fourth case. We use cartesian coordinates ξ1 = x, ξ2 = y in the xy-plane. The fourth case corresponds to γ1 = γ2 = 1,

V = v0 + mx,

G = my.

(107)

The separation constants in the H-J equation are

Cx = 2x + V 2 20 − 2V H, Cy = 2y + (z − my 0 )2 .

(108) (1)

The relation with the K-S tensors obtained in Sect. 4 is merely S4 = Cy . As a conclusion of these last two sections let us observe that the separable coordinates, known for the various potentials V , lift up, without any modification, to separable coordinates for the four dimensional system. Let us turn now to the Killing-Yano tensors.

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G. Valent

7. Killing-Yano Tensors An antisymmetric tensor Yµν is a Killing-Yano (K-Y) tensor iff ∇(µ Yν)ρ = 0.

(109)

A complex structure is therefore a K-Y tensor. The usefulness of such a concept is related to the fact that the symmetrized tensor product of two K-Y tensors does give a K-S tensor, as can be checked by an easy computation. Clearly the triplet of complex structures shared by the multi-centre metrics is not very useful since it gives only trivial K-S tensors so we need extra K-Y tensors. It is the aim of this section to give new examples of these extra K-Y tensors which will give some explicit K-S tensors which do not satisfy Assumption A3. We have been able to obtain K-Y tensors for 1. The special case of the second dipolar breaking, corresponding to V = v0 + E z. 2. The fourth case with an extra tri-holomorphic Killing vector, with potential V = v0 + m x. Let us consider successively these two cases. 7.1. Special second dipolar breaking. For m = 0 the metric simplifies to g=

1 (2dt − Ey dx + Ex dy)2 + V (dx 2 + dy 2 + dz2 ), 4V

V = v0 + Ez. (110)

We have four Killing vectors ∂t ,

x ∂y − y ∂x ,

∂x +

Ey ∂t , 2

∂y −

Ex ∂t , 2

(111)

and the induced conserved quantities have simple Poisson brackets: 0 is central and for the remaining ones {Jz , px } = py ,

{Jz , py } = −px

{px , py } = E 0 ,

(112)

with Jz = x y − y x ,

px = x +

Ey 0 , 2

py = y −

Ex 0 . 2

Using the canonical vierbein one gets for the K-Y two-form Y = −E 2 E0 ∧ (x E1 + y E2 ) + E 2 (x E2 ∧ E3 + y E3 ∧ E1 ) +2EV E1 ∧ E2 .

(113)

From it and the complex structures we can construct four K-S tensors Y 2,

(−)

Si = Y i

(−)

+ i

Y,

i = 1, 2, 3.

We will quote the corresponding conserved quantities instead of the K-S tensors, for the ease of comparison with our earlier results:


593

Y2 → −4V (2x + 2y ) E2 +E 2 (x 2 + y 2 )V 20 + 4E φ (x x + y y ) − 2E 2 (x 2 + y 2 )H, S1 → 4EV 0 py − 4E φ px + 4E 2 x H, S2 → −4EV 0 px − 4E φ py + 4E 2 y H, S3 → 4E(px2 + py2 ). (114) Let us observe that S3 is reducible and that S1 and S2 do not satisfy Assumption A3, so we are left with Y 2 . Some algebra shows how it is related to the conserved quantity obtained in Sect. 3: SI I I (m = 0) = −

Y2 v0 − 2 S 3 − v 0 0 Jz , 3 4E 4E

(115)

so that, up to reducible terms, the two conserved quantities are one and the same. This case is quite similar to the Kerr metric (albeit much simpler) for which the Carter K-S tensor is in fact the square of some K-Y tensor.

7.2. The fourth case. Using the canonical vierbein one gets for the K-Y two-form (−)

Y = −my 2

(−)

− mz 3

+ 2V E2 ∧ E3 .

(116)

Defining pz = z − G 0 , we can write the induced conserved quantities: Y2 → −V 2y − V pz2 + my x y − myV 0 pz 4 m2 2 +mz x pz + mzV 0 y − (y + z2 )H, 2 S1 → 2y + pz2 , pz = z − G 0 , S2 → −x y + V 0 pz + my H, 4 S3 → −x pz − V 0 y + mz H. 4

(117)

We see that S1 and S2 were alredy obtained in Sect. 4. The other two are missing since (2) they don’t satisfy our Assumption A3. Notice also that the conserved quantity S4 cannot be obtained in that way. So this example is of some interest since it shows that there do exist K-S tensors which do not satisfy Assumption A3. However, since the corresponding conserved quantities do not commute with z , they are of no use to prove integrability. 8. Conclusion We have settled the problem of finding all the multi-centre metrics which do exhibit some extra conserved quantity, having the structure (16), under the assumptions A1 to A3. Since it is induced by a KS tensor, this conserved quantity is quadratic with respect to the momenta, and preserved by the geodesic flow. As we have observed, the existence of this extra conserved quantity is essential to obtain integrability.

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However one should keep in mind that our analysis does not cover all the integrable multi-centre metrics, since integrability could emerge from the existence of more complicated conserved quantities. In fact the concept of the Killing-Stäckel tensor can be generalized to symmetric (n, 0) tensors with n ≥ 3 such that ∇(λ Sµ1 ···µn ) = 0. It follows that the geodesic flow preserves the quantity Sµ1 ···µn x˙ µ1 · · · x˙ µn . The corresponding invariants will be cubic, quartic, etc... with respect to the momenta. Little is known about the existence of such conserved quantities, which could produce possibly new integrable multi-centre metrics. Let us conclude by putting some emphasis on the purely local nature of our analysis: it makes no difference between complete and non-complete metrics. For instance in Sect. 3 we have seen that the most general two-centre metric is integrable, however it is complete only for real m1 = m2 , i. e. for the double Taub-NUT metric. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Aliev, A.N., Hortacsu, M., Kalayci, J., Nutku, Y.: Class. Quant. Grav. 16, 189–210 (1999) Belinskii, V.A., Gibbons, G.W., Page, D.N., Pope, C.N.: Phys. Lett. B 76, 433–435 (1978) Boyer, C.P., Finley, J.D.: J. Math. Phys. 23, 1126–1130 (1982) Eguchi, T., Hanson, A.J.: Phys. Lett. B 74, 249–251 (1978) Feher, L.G., Horváthy, P.A.: Phys. Lett. B 183, 182–186 (1987) Gegenberg, J.D., Das, A.: Gen. Rel. Grav. 16, 817–829 (1984) Gibbons, G., Hawking, S.: Phys. Lett. B 78, 430–432 (1978) Gibbons, G.W., Manton, N.S.: Nucl. Phys. B 274, 183–224 (1986) Gibbons, G.W., Olivier, D., Ruback, P.J., Valent, G.: Nucl. Phys. B 296, 679–696 (1988) Gibbons, G.W., Ruback, P.J.: Commun. Math. Phys. 115, 267–300 (1988) Hitchin, N.: Math. Proc. Camb. Phil. Soc. 85, 465–476 (1979) Hitchin, N.: Monopoles, minimal surfaces and algebraic curves. In: NATO Advanced Study Institute n◦ 105, Montreal, Canada: Presses Université de Montreal, 1987 Katzin, H., Levine, J.: Tensor 16, 97 (1965) Kloster, S., Som, M., Das, A.: J. Math. Phys. 15, 1096–1102 (1974) Mignemi, S.: J. Math. Phys. 32, 3047–3054 (1991) Perelomov, A.M.: Integrable systems of classical mechanics and Lie algebras. Basel-Boston-Berlin: Birkhäuser Verlag, 1990 Tod, K.P., Ward, R.S.: Proc. Roy. Soc. London A 368, 411–427 (1979)

Communicated by H. Nicolai


Communications in


Ising Models with Four Spin Interaction at Criticality Vieri Mastropietro Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy Received: 30 January 2003 / Accepted: 12 June 2003 Published online: 12 December 2003 – © Springer-Verlag 2003

Abstract: We consider two bidimensional Ising models coupled by an interaction quartic in the spins, like in the spin representation of the Eight vertex or the Ashkin-Teller model. By Renormalization Group methods we write a convergent perturbative expansion for the specific heat and for the energy-energy correlation up to the critical temperature. A form of nonuniversality is proved, in the sense that the critical behaviour is described in terms of critical indices which are non trivial functions of the coupling. The logarithmic singularity of the specific heat of the Ising model is removed or changed in a power law (with a non universal critical index) depending on the sign of the interaction. 1. Main Results 1.1. Much of our understanding about phase transitions and critical behaviour of classical spin systems on a 2D lattice is based on some remarkable exact solutions. Onsager [O] solved the Ising model, in which the spins take two values and only nearest-neighbor two spin interactions are considered. Lieb [Li] and Baxter [B] solved respectively the Six vertex and Eight vertex models; in their original formulation such models are vertex models (to each site of a bidimensional lattice is associated a vertex with four arrows) but via a suitable identification of the parameters they can be written as two Ising models coupled by a four spin interaction [W]. The critical exponents describing the behaviour of the system close to the critical point can be exactly computed; it is remarkable that the critical indices in the Ising or in the vertex models are different. The exact solutions provide indeed a lot of detailed information about such integrable models; however even very small and apparently harmless modifications of them completely destroy their integrability. On the other hand one can hope that many relevant properties of the integrable models are quite “robust” under perturbations. It is believed that a universality property holds for the Ising model, in the sense that by adding to it, for instance, a next to nearest neighbor or a four spin interaction the critical indices remain unchanged. A universality property is believed to hold also for the Eight vertex

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V. Mastropietro

model; Kadanoff [K] by “operator algebra and scaling theory” found evidence that the Eight vertex model is in the same class of universality of the Ashkin-Teller model [AT], which is not integrable. Other evidence for such a conclusion was found in [PB] (using second order renormalization group arguments) and [LP, N] (by a heuristic mapping of both models into the massive Luttinger model describing interacting fermions in the continuum). The natural method to relate non-integrable models to integrable ones is given by the Renormalization Group (RG); this was known for a long time but the main open problem in this context was to implement such a method in a rigorous way. While RG methods were generally applied to the spin variables, it was realized in recent times that it can be convenient to do this in the fermionic representation of spin models. The fermionic representation of the Ising model was done in [SML, H, Ka, MW, S, ID] and it was shown that the correlations can be written as Grassmann integrals formally describing noninteracting fermions on a lattice in d = 1 + 1. In the same way Ising models with quartic interactions can be written as Grassmann integrals formally describing interacting nonrelativistic fermions. The rigorous analysis of Grassmann integrals for non-relativistic fermions via RG methods is quite well developed, starting from [G] and [BG1] (see also [BG] or [GM] for extensive reviews) and one can apply such methods to classical 2D spin systems (such methods were already applied to a closely related problem, the XYZ Heisenberg spin chain [BM]; the relation between the Eight vertex and XYZ model is well known, [Su, Ba]). Fermionic RG methods for classical spin models have been applied first in [PS] to the Ising model with a small next to nearest neighbor or four spin interaction. A form of universality was established in the sense that the interaction does not change certain critical indices; the fermionic interaction is, in this case, irrelevant in the RG sense and the fixed point of the RG transformation is the free one. The aim of the present paper is to study two Ising models coupled by an interaction quartic in the spins, such that both the Eight vertex and the Ashkin-Teller models are included: the system is, in general, non-integrable. The specific heat and the energyenergy correlation are written as Grassmann integrals and studied by RG methods. In such cases the fermionic interaction is marginal and the RG transformation has a line of fixed points. The critical behaviour is different with respect to the case of the Ising model, and it is described in terms of critical indices which are analytic non-trivial functions of λ. In agreement with [K] we find that the behaviour of the system is quite independent from the details of the quartic interaction. In our analysis no use is made of the Six or Eight vertex model exact solutions; we use instead some properties which can be deduced from the solution [ML] of the (massless) Luttinger model following a strategy first outlined (in purely fermionic models) in [BG1]. Our analysis establishes as a mathematically rigorous statement the statement in [K, LP, N, PB] that the Gaussian boson model, the massive Luttinger model, the Eight vertex and the Ashkin-Teller models are in the same class of universality. 1.2. We consider two Ising models coupled by a four spin interaction bilinear in the energy densities of the two sublattices. Given ∈ Z 2 a square lattice with side M (1) and periodic boundary condition, we call x = (x, x0 ) a site of . If σx = ±1 and (2) σx = ±1, we write the following Hamiltonian H (σ (1) , σ (2) ) = HI (σ (1) ) + HI (σ (2) ) + V (σ (1) , σ (2) ) M ≡ H,x (σ (1) , σ (2) ), x,x0 =1

(1.1)

Ising Models with Four Spin Interaction at Criticality

597

Fig. 1. The spins involved in the interaction of the models in Eq. (1.1). The heavy dots and lines or the light dots and lines mark the Ising lattices and the nearest neighbors Ising couplings. The ellipses symbolize the Ashkin–Teller four spin interactions and the circles the Eight vertex four spin interactions couplings

where, if α = 1, 2 M

HI (σ (α) ) = −J

(α)

x,x0 =1

(α)

(α) (α) [σx,x σ + σx,x σ ] 0 x+1,x0 0 x,x0 +1

(1.2)

is the Ising model hamiltonian and V (σ (1) , σ (2) ) is the interaction between the Ising systems V (σ (1) , σ (2) ) = −λ

M

(1)

(2)

(1)

(2)

(1) (1) σ σ (2) σ + σx,x σ σ (2) σ ] a [σx,x 0 x+1,x0 x,x0 x+1,x0 0 x,x0 +1 x,x0 x,x0 +1

x,x0 =1 (2) (1) (1) +b[σx,x σ σ (2) σ 0 x+1,x0 x,x0 x,x0 +1

(2) (2) (1) (1) + σx,x σ σ σ ] . 0 x,x0 +1 x−1,x0 +1 x,x0 +1 (1.3)

If b = 0 the Hamiltonian (1.1) coincides with the Hamiltonian of the spin representation [F] of the Ashkin-Teller model [AT]; if a = 0 it coincides with the spin representation [W] of the Eight vertex model. For a given observable O(x) localized near x we define the correlation < O(x)O(y) > =

where Z =

(1) (2) σx ,σx =±1 x∈

1 Z

e−H (σ

O(x)O(y)e−H (σ

(1) ,σ (2) )

,

(1.4)

(1) (2) σx ,σx =±1 x∈M

(1) ,σ (2) )

is the the partition function. The truncated

correlation of the observable O(x) is < O(x)O(y) >,T =< O(x)O(y) > − < O(x) > < O(y) > ,

(1.5)

598

V. Mastropietro

and the energy-energy truncated correlation function is given by (1.5) with O(x) = H,x (σ (1) , σ (2) ); the specific heat Cvλ is 1 < H,x (σ (1) , σ (2) )H,y (σ (1) , σ (2) ) >,T . ||→∞ ||

Cvλ = lim

(1.6)

x,y∈

If λ = 0 the model reduces to two independent Ising models and close to the critical temperature (equal for both) it is Jc (1.7) − 1| + C2 , J √ where C1 , C2 are positive constants and tanh Jc = 2 − 1, see [MW] Eq. (3.58). The truncated correlation of the observable O(x) = HI,x (σ (α) ) for λ = 0 has the property | < O(x)O(y) >T | ≤ Ce−A|t−tc ||x−y| with A, C suitable constants. We expect that the interaction changes the value of the critical temperature (i.e. of Jc ) by quantities O(λ). However it is convenient to keep the critical singularity at a λ-independent value; we shall show that this can be done by choosing properly the molecular energy parameter J as a function of λ. Therefore we consider the model (1.1) with Jr replacing J , and we shall choose √ Jr = J + O(λ) so that the critical coupling is precisely in correspondence of tanh−1 ( 2 − 1). Denoting by N an arbitrary positive integer, fixing a + b = 0 and with the notations √ def t ≡ tanh J , tanh Jr = tanh J + ν(λ) and tc ≡ 2 − 1, we shall rigorously derive the following result. Cv0 −C1 log |

Theorem. Assume a = 0 or b = 0. There are C, CN , C1 , C2 , τ, Z˜ 1 , positive λ–independent constants, such that for λ small enough one can uniquely define ν (λ), analytic in λ, so that the model in Eq. (1.1), (1.3) and with coupling Jr = J + ν (λ) is critical at t = tc . This means that, for |t − tc | > 0, lim H,x (σ (1) , σ (2) )H,y (σ (1) , σ (2) ),T = a (x, y) + b (x, y),

||→∞

(1.8)

and the bounds CN 1 , |x − y|2+2η1 1 + ( |x − y|)N CN 1 |b (x, y)| ≤ |x − y|2+τ 1 + ( |x − y|)N |a (x, y)| ≤

(1.9)

hold, with “correlation length” −1 and “critical indices” η1 , η2 given by = |t − tc |1+η2 , η1 (λ) = −a1 (a + b)λ + O(λ2 ) η2 (λ) = −a2 (a + b)λ + O(λ2 )

(1.10)

with a1 > 0, a2 > 0 constants and η1 , η2 analytic in λ. Furthermore if 1 ≤ |x| ≤ −1 the correlation is asymptotic to a in the sense that b is neglegible because a (x, y) =

1 1 + A(x − y) , Z˜ 2 (x − y)2+2η1 1

1 |A(x)| ≤ C |λ| + ( |x|) 2 .

(1.11)


599

Finally the specific heat Cvλ (1.6) verifies C1

1 1 [1 − 2η1 ] ≤ Cvλ ≤ C2 [1 − 2η1 ], 2η1 2η1

(1.12)

where C1 , C2 are positive constants. 1.3. The above result says that the interaction changes the value of the critical temperature and it qualitatively modifies the critical behaviour of the specific heat and of the energy-energy correlations. As t gets closer and closer to the critical temperature the logarithmic singularity of the specific heat in the Ising model is changed by the four spin interaction into a power law singularity with non-universal critical indices if λ (a + b) > 0; if λ (a + b) < 0 the specific heat is instead continuous, but higher derivatives of the free energy are singular, as one can check from the proof of the Theorem. Moreover one can distinguish two different regimes in the asymptotic behaviour of the energy- energy correlation function, discriminated by an intrinsic correlation length ξ of order |t − tc |−1−η2 with η2 = O(λ). If 1 |x − y| ξ , the bound for the correlation function is power-like while if ξ |x − y|, there is a faster than any power decay with rate of order ξ −1 . The splitting (1.8) and (1.9) might suggest that the fast decay is modulated by a power |x − y|−2−2η1 but it does not prove that because the first of (1.9) is an inequality rather than an asymptotic expression. We do not study the free energy directly at t = tc , therefore in order to show that t = tc is a critical point we must study some thermodynamic property like the specific heat by evaluating it at t = tc and M = ∞ and then verify that it has a singular behavior as t → tc . Moreover (1.11) holds uniformly for all |t − tc | > 0, hence we can draw the remarkable consequence that assuming continuity for t → tc , at fixed |x − y|, of the correlations in (1.8) we obtain at t = tc a power law behaviour with critical index η1 . We cannot exclude a discontinuity at t = tc of the correlation in (1.8), not even at fixed x − y, because, as it is the case in various models which can be studied up to the critical point, the case t precisely equal to tc cannot be discussed at the moment with our techniques in spite of the uniformity of our bounds as t → tc . In the case of the Eight vertex model our results are in agreement with the exact solution in [B] (see also [W]). For definiteness we have chosen V (σ (1) , σ (2) ) of the form (1.3) but the proof of the Theorem does not depend on the details of the interaction but only on a few general properties; one needs essentially that the interaction is short ranged and it is invariant under the same symmetry transformations which leave invariant the “free” hamiltonian HI (σ (1) ) + HI (σ (1) ). We will describe briefly how the proof of the theorem can be generalized in Appendix O. 1.4. The paper is organized in the following way. We begin to study the analyticity properties of the partition function. The starting point is the well known representation, due to [H, Ka, MW, S], of the Ising model partition function in terms of Grassmann integrals with a formal action which is quadratic. Also the partition function of the model (1.1) can be written in terms of Grassmann integrals, with a formal action which is however non-quadratic. By a suitable linear transformations, see §2, the Grassmann integrals can be written in a form which strongly resembles the partition function of a system of two interacting Dirac fermions on a lattice in d = 1 + 1; one fermion (called massive) has an O(1) mass, while the other (light fermion) has a mass O(t − tc ) i.e. vanishing at criticality. In §3 we “integrate out” the massive fermions, thus obtaining an effective theory in terms of the light fermions only. The integration of the light fields is much more

600

V. Mastropietro

difficult, as their mass is almost zero, and we perform a multiscale analysis based on Renormalization Group ideas, see §4; the result of such analysis is an integration procedure (or a resummmation prescription) for the partition functions which is written as a series in a number of functions which are called running coupling constants carrying a scale label h = 0, 1, . . . : for each scale there are only a few such running couplings. Contrary to the naive expansion in powers of λ (which cannot be convergent at t = tc ), such expansion is well defined arbitrarily close to the critical temperature if the running coupling constants are small enough. The running coupling constants verify a recursive relation expressing the running couplings on a given scale h as a function of the ones on the previous scales h < h: the latter function is usually called the Beta function and it is defined as long as its arguments are small enough. In §5 we show that the running coupling constants are indeed small, if ν is chosen properly and λ is small enough. In order to prove this one has to use two key results. The first is the exploitation of a number of symmetry cancellations to prove that a number of running coupling constants are exactly vanishing; such symmetries, which are manifest in the original spin variables, become quite involved in the fermionic representation. The second one is the decomposition of the beta function in the sum of many terms, in which only one of them is really crucial, while the others would have a small effect in the absence of the first one. One recognizes that such a crucial contribution to the Beta function of our model coincides with the Beta function of the Luttinger model: the latter Beta function was proved to be zero, as a consequence of its exact solution [ML], in [BGPS, GS, BM1] (see [BeM1] for a simplified proof). This means that the apparently largest contribution to the Beta function is essentially zero, if ν (λ) is properly chosen. Note also that, despite the vanishing of the Luttinger model the Beta function is believed to be a consequence of suitable Ward identities, to convert such an argument on a rigorous proof seems at the moment quite difficult, see [BeM1], hence the only rigorous proof of such a key result is the one in [BGPS, GS, BM1]. Finally in §5 we define an expansion for the correlation functions and the specific heat; it is similar to the one for the partition function, with the main difference that one has to introduce new terms in the action associated with the external fields introduced to express via functional integrals the correlation functions. The proof establishes rigorously a relationship between spin models with quartic interactions like the model (1.1) and the massive Luttinger model: in agreement with what was conjectured in [LP, N, PB]. Our results extend a previous paper [M1] (where Eq. (1.12) must replace Eq. (1.16) of [M1] which was incorrect). The analysis of ref. −a [M1] was restricted to the case |t − tc | ≥ e λ2 , where a is a suitable constant. The paper is self contained aside from a few technical lemmata proved in full detail in [BM]. A very important open problem is to obtain by such fermionic RG methods the asymptotic behaviour of the spin-spin correlation function; its fermionic representation is much more involved than the one for the specific heat or for energy-energy correlations which are the only correlations considered here. One can study also the cases in which the parameters J of the two Ising model hamiltonian are different so that there are two critical temperatures; new fermionic effective marginal interactions appear in such a case and universality will be probably found. Another possible extension is the analysis of four coupled Ising models; in this last case interacting spinning d = 1 fermions appear in the fermionic description, which are known to have a behaviour quite different from the spinless one (like in the d = 1 Hubbard model).


601

2. Fermionic Representation (α)

2.1. The partition function ZI of the Ising model with Hamiltonian HI (σ (α) ) in (1.3) can be written as a Grassmann integral; this is a classical result, mainly due to [Ka, H, MW, S] and rederived recently in §3 of [PS] to which we refer for a detailed proof. It is (α) (cosh J )B 2S (α) (α) S (α) ZI = (−1)S dHx(α) dH x dVx(α) dV x (−1)δγ e J ;γ , (2.1) 2 ε,ε =±

x∈M

def

def

where α = 1, 2 denotes the lattice, γ = (ε, ε ) and δγ is δ+,+ = 1, δ+,− = δ−,+ = def

δ−,− = 2, M = , B is the total number of bonds and S is the total number of sites, (α) (α) (α) (α) (α) [H x,x0 Hx+1,x0 + V x,x0 Vx,x0 +1 ] SJ,γ = tanh J +

x∈M (α) (α) [H x,x0 Hx,x 0

x∈M (α) (α) +Hx,x V 0 x,x0 (α)

(α)

(α)

(α)

(α)

(α)

(α)

(α) (α) + V x,x0 Vx,x + V x,x0 H x,x0 + Vx,x H x,x0 0 0

(α) + Vx,x H (α) ], 0 x,x0

(2.2)

(α)

where Hx , H x , Vx , V x are Grassmann variables verifying different boundary conditions depending on the label γ = (ε, ε ) which is not affixed explicitly, to simplify the notations, i.e. H x,x0 +M = εH x,x0 H x+M,x0 = ε H x,x0 (α) (α) (α) (α) Hx,x0 +M = εHx,x Hx+M,x0 = ε Hx,x 0 0 (α)

(α)

(α)

(α)

and identical definitions are set for the variables V (α) , V the set of k’s such that

(α)

ε, ε = ±,

(2.3)

. We call Dγ , for γ = ε, ε

2πn1 2π n0 (ε − 1)π (ε − 1)π + k0 = + (2.4) M 2M M 2M and −[M/2] ≤ n0 ≤ [(M − 1)/2], −[M/2] ≤ n1 ≤ [(M − 1)/2], n0 , n1 ∈ Z. We can write if k = (k0 , k), 1 1 (α) (α) (α) Hx(α) = 2 Hk e−ikx H x = 2 H k e−ikx , (2.5) M M k=

k∈Dε,ε

k∈Dε,ε

(α)

(α)

and similar expressions hold for Vx , V x .

(α) (α) (α) (α) The integration x dHx dH x or x dVx dV x is defined as a linear functional on the Grassmann algebra in the standard way: we recall it in Appendix A below. It will be convenient to use auxiliary models in which J is allowed to depend on α and on the bonds: i.e. we can imagine replacing the coupling J of each bond b joining the (α) nearest neighbors x, y on the lattice α by Jb = Jx,y . If J is not constant but it depends (α) (α) on the bonds, one expresses the partition function ZI (Jx,x ) by a formula similar to Eq. (2.1) in which SJ,γ , with γ = (ε, ε ), becomes (α) (α) (α) (α) (α) (α) (α) tanh Jx,x0 ;x+1,x0 H x,x0 Hx+1,x0 + tanh Jx,x0 ;x,x0 +1 V x,x0 Vx,x0 +1 SJ (α) ,γ = (α)

x

602

V. Mastropietro

+

(α) (α) (α) (α) (α) (α) (α) (α) [H x,x0 Hx,x + V x,x0 Vx,x + V x,x0 H x,x0 + Vx,x H x,x0 0 0 0

x (α) (α) +Hx,x V 0 x,x0

(α) + Vx,x H (α) ], 0 x,x0

and the factor (cosh J )B is replaced by

(2.6) (α)

b

cosh Jb .

2.2. The partition function of the model (1.1) with Jr replacing J is (1) (2) (1) (2) Z2I = e−HI (σ ) e−HI (σ ) e−V (σ ,σ ) .

(2.7)

(1) (2) σx =±1 σx =±1 x∈M x=M

Setting λ a = tanh(λ a), λb = tanh(λ b) we see that Z2I becomes 2I with (cosh λa cosh λb)2S times Z (1) (2) 2I = Z e−HI (σ ) e−HI (σ ) def

def

σ (1) =±1 σ (2) =±1 x∈M x∈M

·

x∈M

·

x∈M

·

x∈M

·

(2) (1) (1) [1 + λ aσx,x σ σ (2) σ ] 0 x+1,x0 x,x0 x+1,x0 (1) [1 + λ aσx,x σ σ (2) σ ] 0 x,x0 +1 x,x0 x,x0 +1 (1)

(2)

(2) (1) (1) x,x [1 + λbσ σ σ (2) σ ] 0 x+1,x0 x,x0 x,x0 +1 (1) x,x [1 + λbσ σ σ σ ], 0 x,x0 +1 x−1,x0 +1 x,x0 +1 (1)

(2)

(2)

(2.8)

x∈M

where HI (σ (α) ) are defined as in (1.3) with Jr replacing J . Note that

σx(α) σx e−HI (σ (α)

(α) σx =±1 x∈

(α) )

=

∂ (α) ∂Jx,x

(α)

(α)

ZI ({J }x,x )|{J (α) }={J } , x,x

(2.9)

r

where x, x are nearest neighbors on the lattice α, and from (2.6) (and remembering that (α) (α) (α) a = 0 or b = 0) this derivative gives an extra factor tanh Jr + sech2 Jr H x,x0 Hx+1,x0 in 2I , hence Z2I , as a Grassmann integral over the variables (2.1). We can therefore write Z H, V , H , V . The algebra is straightforward and we reproduce it in Appendix B, and 2I as a sum of sixteen partition functions labeled by the result is that we can express Z γ1 , γ2 = (ε(1) , ε (1) ), (ε(2) , ε (2) ) (corresponding to choosing each ε and ε as ±) γ1 ,γ2 , 2I = (cosh λa cosh λb)2S (−1)δγ1 +δγ2 Z (2.10) Z 2I γ1 ,γ2

each of which is given by a functional integral 2B 2S γ1 ,γ2 = (cosh Jr ) 2 Z 2I 4

2 α=1

(α)

(α) (α) SJ,γ α

dHx(α) dH x dVx(α) dV x

e

e−V ,

x∈M

(2.11)


603 (α)

(α)

(α)

where V is an expression containing linear or bilinear terms in H x Hx+1,x0 or V x (α)

Vx,x0 +1 , see (7.4). It is convenient to rewrite (2.11) in a form closer to an expression more familiar in the theory of fermionic ground states: our aim in fact is to reduce our critical point problem to a rather standard problem on the ground state of Fermi systems. −,−,−,− def − , i.e. the We shall consider for simplicity the partition function Z = Z 2I 2I partition function in which all Grassmann variables verify antiperiodic boundary conditions (see (2.3)). The other fifteen partition functions in (2.10) admit similar expressions. γ ,γ Furthermore it will appear that for |t − tc | > 0 the logarithm of Z2I1 2 divided by its expression for λ = 0 is insensitive to boundary conditions up to corrections which are exponentially small in the size M of the system in the thermodynamic limit in which M → ∞ (see Appendix G) so that it will turn out that it is sufficient to study just one of −,−,−,− is chosen here (arbitrarily). It is convenient the sixteen partition functions and Z 2I to perform the following change of variables [ID], α = 1, 2: H x + iHx(α) = ei 4 ψx(α) − ei 4 χx(α) H x − iHx(α) = e−i 4 ψ x − e−i 4 χ (α) x (α) (α) (α) V x + iVx(α) = ψx(α) + χx(α) V x − iVx(α) = ψ x + χ (α) (2.12) x (α)

π

π

(α)

π

(α)

π

which replaces the H, V , H , V variables with “Majorana variables” ψ (α) , χ α) . Subsequently we replace the Majorana variables with Dirac variables by setting 1 ∓ = √ (ψx(1) ± iψx(2) ), ψ1,x 2 1 ∓ χ1,x = √ (χx(1) ± iχx(2) ), 2

1 (1) (2) ∓ ψ−1,x = √ (ψ x ± iψ x ) , 2 1 ∓ (2) χ−1,x = √ (χ (1) x ± iχ x ) . 2

The final expression, see Appendix C for the algebra, is −,−,−,− = N P (dψ)P (dχ )eQ(χ,ψ)−V (χ,ψ) , − def = Z Z 2I 2I

(2.13)

(2.14)

(2.15)

where N is a suitable constant and, if φ denotes either ψ or χ , t (+)T + − dφk,ω dφk,ω exp −ξ A (k)ξ P (dφ) = Nφ−1 φ k , k 2M 2 k∈D−,− ω=±1 k∈D−,− i sin k + sin k0 −imφ (k) − − Aφ (k) = , ξ T k = (φk,1 , φk,−1 ) imφ (k) i sin k − sin k0 + + , φk,−1 ) (2.16) ξ +,T k = (φk,1 with Nφ a normalization constant, mφ defined, differently for φ = ψ (choose +) and for φ = χ (choose −), by √ t t (2.17) mφ (k) = (t − (± 2 − 1)) + (cos k0 + cos k − 2). 2 2 √ Remark. Note that we are interested in t close to tc = 2 − 1 hence, for t → tc , mχ is bounded away from 0 and therefore m−1 χ (0) defines a length scale which stays finite in this limit while mψ (0) → 0 and the corresponding length scale diverges. Note also that +,+,+,+ at t = tc is meaningless, as in that case N = 0 (as Nψ = 0); hence (2.15) for Z 2I the assumption |t − tc | > 0.

604

V. Mastropietro

Finally Q(χ , ψ) and V(χ , ψ) are obtained respectively from (7.10) and (7.5) in Appendix C through the change of variables (2.12), (2.13) and (2.14). The final expressions for them are rather intricate and we just extract from them a few properties which will be important in the following. Introducing the discrete derivatives of φ = ψ, χ as def

def

∂1 φx = φx+1,x0 − φx ,

∂0 φx = φx,x0 +1 − φx .

(2.18)

It turns out, see Appendix D, that Q and V are given by a sum of terms of the forms b;σ ,σ a;σ ,σ b;σ1 ,σ2 1 2 1 2 Ax;φ,ω or A (2.19) ,ω , ,ω Ax ;φ ,ω ;φ ,ω , ;φ ;φ x;φ,ω 1 2 1 2 x

1

x

2

where x = x or x = (x − 1, x0 + 1) with φ, φ , φ , φ ∈ {ψ, χ }, σ = ± and 1) If ω1 = ω2 then for a suitable numerical coefficient aσ1 ,σ2 ,ω,c,n it is, for n = 1, 2 and c = a, b,

σ1 σ2 1 ,σ2 Ac;σ x;φ,ω;φ ,ω = aσ1 ,σ2 ,ω,c,n φω,x ∂xn φω,x

with

(2.20)

1a) If n = 1 ∂xn = ∂x0 and aσ1 ,σ2 ,ω,c,1 is imaginary; 1b) If n = 2 ∂xn = ∂x and aσ1 ,σ2 ,ω,c,2 is real. 2) If ω1 = −ω2 then for suitable real numerical coefficients bσ1 ,σ2 ,ω,c,m , cσ1 ,σ2 ,ω,c,m it is

σ2 σ1 1 ,σ2 2a) Ac,σ x;φ,ω;φ ,−ω = ibσ1 ,σ2 ,ω,c,m ∂xm φω,x ∂xm φ−ω,x , ∂xm = ∂x0 if m = 1, ∂xm = ∂x if m = 2, σ σ1 1 ,σ2 2 2b) Ac,σ x;φ,ω;φ ,−ω = icσ1 ,σ2 ,ω,c,l φω,xl φ−ω,xl ,

(2.21)

with l = 1, 2, 3 and xl = x, xl = (x + 1, x0 ), xl = (x, x0 + 1) for l = 1, 2, 3 respectively. 2.3. The value of P (dφ)Q(φ), where Q(φ) is any monomial on the φ = −ψ, χ+variables, is given by the anticommutative Wick rule with propagator P (dφ) φx,ω φy,ω = (φ)

gω,ω (x − y) given by (φ)

gω,ω (x − y) =

2 −ik(x−y) −1 e [Aφ (k)]ω,ω . tM 2

(2.22)

k

If we set Qφ (k) = det Aφ (k) = − sin2 k0 − sin2 k − [mφ (k)]2 , then 1 − sin k0 + i sin k imφ (k) (k) = . A−1 φ −imφ (k) sin k0 + i sin k Qφ (k)

(2.23)

The following bounds hold for the propagators, for any N > 1 and for a suitable constant CN (φ) |gω,ω (x − y)| ≤ (φ)

1 CN , 1 + |d(x − y)| 1 + |mφ (0)d(x − y)|N

|gω,−ω (x − y)| ≤

|mφ (0)| log[1 + (|mφ (0)||d(x − y)|)−1 ]CN , 1 + |mφ (0)d(x − y)|N

(2.24) (2.25)


605

where d is a distance between x, y which takes into account the antiperiodicity of the boundary conditions that we are considering, namely π(x − y) M π(x0 − y0 ) M sin , sin . (2.26) d(x − y) = π M π M Note that the following parity properties hold: (φ) (φ) (x) = −gω,ω (−x), gω,ω

(φ)

(φ)

gω,−ω (x) = gω,−ω (−x) .

(2.27)

Remark. After the change of variables (2.12), (2.13) and (2.14) we have achieved writing − as (2.15), which can be naturally seen as the partition function of a system of two Z 2I kinds of bidimensional Dirac fermions on a lattice. The remark following (2.17) says that the χ -fields mass is O(1) while the ψ-fields mass is vanishing when t = tc ; hence the χ -fields will be called massive fields and the ψ-fields will be called light fields. In contrast with this interpretation note, however, that the interaction V has a quite nonstandard form; it is not invariant under global gauge transformations and is not given by products of density operators, unlike in the usual fermionic models. 3. Integration of Massive Fermions 3.1. Considering (2.15) we proceed to perform the Grassmann integration over the massive χ fields and to reduce the double integration over ψ, χ to an integration of a (more (1) involved) new exponential e−V (ψ) over the light fields ψ alone, − = N P (dψ) P (dχ )eQ(χ,ψ) e−V (ψ,χ) = P (dψ)eM 2 N (1) −V (1) (ψ) , (3.1) Z 2I where N (1) is a constant such that the effective potential V (1) (ψ) vanishes at ψ = 0 and P is suitably defined. Indeed we prove the following result. 3.2. Lemma 1. Assume a = 0 or b = 0. There exists ε and C such that, for |λ|, |ν| ≤ ε, 2n W2n,σ ,α,ω (x1 , .., x2n )∂ α1 ψxσ11,ω1 ...∂ α2n ψxσ2n V (1) = ,ω2n , n≥1 α,ω,σ x1 ,..,x2n

2n,σ ,α,ω (k1 , ...kn−1 )| ≤ M 2 C n ε n/2 , |W

n ≥ 2.

(3.2)

4 J + O(ε 2 ) The addends in (3.2) with n = 2 can be written, for l1 = 2(λ a + λb)sech r real, as + + − − l1 ψ1,x ψ−1,x ψ−1,x ψ1,x + W4,σ ,α,ω (x1 , .., x4 ) x1 ,..,x4 α1 +..α4 ≥1,σ

x

× ∂ α1 ψxσ11,ω1 ∂ α2 ψxσ22,ω2 ∂ α3 ψxσ33,ω3 ∂ α4 ψxσ44,ω4 .

(3.3)

The addend with n = 1 can be written, for ν1 = ν + O(ε), a1 , a2 = ν/2 + O(ε), as − + + − [−iων1 ψx,ω ψx,−ω + ψx,ω (iωa1 ∂0 − a2 ∂1 )ψx,ω ] ω x + W2,σ ,α,ω (x1 , x2 )∂ α1 ψxσ11,ω1 ∂ α2 ψxσ22,ω2 (3.4) x1 ,x2 {ω} α1 +α2 ≥2,σ1 ,σ2

606

V. Mastropietro

2,σ ,α,ω (k1 )| ≤ M 2 C|ε|. Finally making use of a general notawith ν1 , a1 , a2 real and |W tion for later reference, the Grassmann integration P (dψ) is PZ1 ,m1 ,C1 (dψ), where PZ1 ,m1 ,C1 (dψ) = N −1





k∈D−,− (k)−1 >0 C 1

ω=±1

1 (k)   tZ1 C (1) + − + −  dψk,ω dψk,ω exp  − ψ T (k)ψ k,ω k,ω ω,ω ,  M2 k∈D −,− (k)−1 >0 C 1

1 def T (1) (k) = C0 + µ0,0 (k) 1 (i sin k + sin k0 ) + µ1,1 (k)Z −1 Z −im1 − iµ1,2 (k)Z1−1 1 × 1 (i sin k − sin k0 ) + µ2,2 (k)Z −1 (3.5) Z im1 + iµ1,2 (k)Z1−1 1 √ √ 1 (k) ≡ 1, m1 = C0 (t − tc ), Z1 = 1, Z 1 = t [(2t + 2 2t) + with C0 = (t + 1 + 2)2 , C 2 √ (2 2 + 3 + t 2 )], µi,j (k) analytic functions in k of size O(k2 ) with µi,i (k), i = 1, 2, odd and µ1,2 (k) even and real; moreover C0 + µ0,0 ≥ 1 and det T (1) (k) is bounded above and below by two constants times −2t (1 − t 2 )(cos k0 + cos k1 − 2) + m21 . The proof of the above proposition is a repetition of standard arguments, see for instance [BGPS] or [BM]: the key is the Gram-Hadamard inequality applied along the lines of Lesniewski, [Le]. For completeness the details are in Appendix E and F. The result says that the integration of the massive fermions has the “only” effect over the remaining (non trivial) ψ–integration of modifying the propagator of the light ψ fields by a few trivial factors of O(1) (analytically dependent on λ for λ small). The only difficulty and novelty is that a detailed analysis of the bilinear and quartic terms in V (1) is necessary. In fact we have to show that the quadratic part can be writσ ψ −σ , or ψ σ ψ σ ten as in (3.4), saying that there are no terms of the form ψx,ω x,ω x,ω x,−ω or σ σ ψx,ω ∂ψx,−ω ; despite the fact that such terms are absent in V, they could be generated by the integration of the χ variables. This is not the case, as a consequence of symmetry properties verified by the model (1.1), as it will be shown in Appendix F. 4. Renormalization Group for Light Fermions − (2.15); after the integration 4.1. Multiscale analysis. We continue the analysis of Z 2I over the χ –fields we have to compute the Grassmann integral over the ψ–fields given by the r.h.s. of (3.1). The problem is quite different from the one treated in Sect. 3 because the ψ–field has propagator, (3.5), with “mass” O(t − tc ) which can be arbitrarily close to 0, and we need estimates that are uniform in this quantity. Therefore we shall proceed via a multiscale analysis following the techniques developed to study the ground state of one–dimensional Fermi systems in [BG], [BGPS] and [BM]. We introduce a scaling parameter γ > 1 which will be used to define a geometrically growing sequence of length scales 1, γ , γ 2 , γ 3 , . . . , i.e. of geometrically decreasing momentum scales γ h , h = 0, −1, −2, . . . . Let χ (k) ∈ C ∞ be a non-negative function such that 1 if |k| < 1/γ , χ(k) = χ (−k) = , where |k| = sin k02 + sin k 2 , (4.1) 0 if |k| > 1 ,


607

Fig. 2. The function χ (γ −h x), χ(γ −(h−1) x), f (γ −h x) def

and for h ≤ 0 integer define fh (k) = χ (γ −h k) − χ (γ −h+1 k) so that, for h < 0, it is χ(k) = 0h=h +1 fh (k) + χ (γ −h k) . Note that, if h ≤ 0, fh (k) = 0 for |k| < γ h−2 or |k| > γ h , and fh (k) = 1, if |k| = γ h−1 . Furthermore with our boundary conditions ε = ε = −, see (2.4), the def

√

momenta k = (k0 , k) are such that |k| > kM = πM2 . Therefore if we define the “minimum” momentum scale larger than kM (i.e. hM = min{h : γ h > kM }) it will be for all such k: 1=

1

fh (k)

f1 = 1 − χ (k) ,

(4.2)

h=hM

which can be visualized as in Fig. 2. Note that the fact that hM is finite plays essentially no role in the subsequent analysis; note also that we are making a multiscale decomposition around k = k0 = 0 as it is the only pole of the propagator corresponding to PZ1 ,m1 ,C1 (dψ). The purpose is to perform the integration over the light fermion fields in a iterative way. The iteration steps will be labeled by scale values h = 1, 0, −1, . . . , hM . The number of iterations will be −hM + 2 and after each iteration we shall be left with a “simpler” Grassmann integration to perform: it will be an integration with respect to a field ψ (≤h) , h = 0, −1, . . . , hM of (≤h) 2 (h) √ PZh ,mh ,Ch (dψ (≤h) ) e−V ( Zh ψ )−M Eh , V (h) (0) = 0 , (4.3) where the quantities PZh ,mh ,Ch (dψ), Zh , mh , Ch (k), V (≤h) (ψ), Eh have to be defined 2 2I , i.e. the value recursively and the result of the last iteration will be e−M E−1+hM ≡ Z of the partition function. 1 (k) The PZh ,mh ,Ch (dψ) integration is defined by (3.5) in which we replace Z1 , m1 , C by other quantities Zh , mh , Ch (k) with Ch (k)−1 =

h

fj (k),

(4.4)

j =hM

1 fixed to the value in (3.5) and Zh , mh recursively defined as discussed below; keeping Z moreover V

(h)

(ψ

(≤h)

)=

∞ 2n n=1

x1 ,...,x2n , σ ,ω,α

i=1

(h)

iW ∂ αi ψx(≤h)σ 2n,σ ,α,ω (x1 , ..., x2n ) . i ,ωi

(4.5)

608

V. Mastropietro

4.2. The localization operator. The effective potential V (h) will be rather involved: to define it recursively it will be convenient to identify in it a part that can be called “irrelevant” and the rest. Here the word irrelevant does not mean “negligible”: it identifies a part of V (h) which can be expressed as a (convergent) power series in terms of a number of parameters vh , h > h, which we call running coupling constants. The latter are also defined recursively and they can be isolated from the effective potential V (h) by acting on it with a “localization operator” L which extracts from the sum of monomials in the fields in (3.1) the terms of degree 2n = 2, 4 in the fields and from each of them it extracts the “local part”: for h ≤ 0 it acts on the kernels W by simplifying them as follows: 1) If 2n = 4, then we define ¯ ¯ ¯ (h) (h) LW 4,σ ,α,ω (k1 , k2 , k3 ) = W4,σ ,α,ω (k++ , k++ , k++ ) ,

(4.6)

π π where k¯ ++ = ( M , M ) is the smallest momentum allowed by the boundary conditions that we are using (see (2.4)). π π 2) If 2n = 2 and k¯ ηη = η M , then , η M 1 (h) M (h) ¯ LW2,σ ,α,ω (k) = W2,σ ,α,ω (kηη ) 1 + aM (η sin k + η sin k0 ) , (4.7) 4 π η,η =±1

π where aM M π sin M = 1. h 3) In all other cases LW 2n,σ ,α,ω (k1 , . . . , k2n−1 ) = 0.

Remark. Note that in the limit M → ∞ (4.7) becomes simply LW 2,σ ,α,ω (k) = [W2,σ ,α,ω (0) + sin k0 ∂k0 W2,σ ,α,ω (0) + sin k∂k W2,σ ,α,ω (0)], (h)

(h)

(h)

a(h)

(4.8)

hence LW 2,σ ,α,ω (k) has to be understood as a discrete version of the Taylor expansion up to order 1. Since aM = 1+O(M −2 ) this property would be true also if aM = 1; how (h) (k) = ever the choice (4.7) shares with (4.8) another important property, that is L2 W 2,σ ,ω (h) (k), see [BM]. LW (h)

2,σ ,ω

4.3. Relevant, marginal and irrelevant operators. By (4.6),(4.7) and the symmetry relations in Appendix F we can write LV (h) as: (≤h)

LV (h) (ψ (≤h) ) = (sh + γ h nh )Fm(≤h) + lh Fλ

(≤h)

+ z h Fζ

+ ah Fα(≤h) ,

(4.9)

where sh , nh , lh , zh , ah are real and, if |λ|, |ν| ≤ ε, s1 = O(m1 λ), z1 , a1 = O(λ), sech4 Jr + O(λ2 ), γ n1 = ν + O(λ); moreover l1 = 2(λ a + λb) 1 (≤h)− , (≤h)+ ψ iωψ k,ω k,−ω M2 k∈DM ω=±1 1 (≤h)+ ψ (≤h)+ (≤h)− (≤h)− ψ = 8 k1 ,+1 k2 ,−1 ψk3 ,−1 ψk4 ,+1 δ(k1 − k2 + k3 − k4 ) , M k1 ,...,k4 ∈DM 1 (≤h)− , (≤h)+ ψ = 2 i sin k ψ k,ω k,ω M

Fm(≤h) = (≤h)

Fλ

Fα(≤h)

k∈DM ω=±1

Ising Models with Four Spin Interaction at Criticality (≤h)

Fζ

=

1 (≤h)+ ψ (≤h)− , ω sin k0 ψ k,ω k,ω M2

609

(4.10)

k∈DM ω=±1

where δ(k) = 0 if k = 0 and δ(0) = 1. Applying the operations L to the kernels of the effective potential generates the sum in (4.9), i.e. a linear combination of the Grassmann monomials in (4.10) which, in the renormalization group language are called “relevant” operators (the first) and “marginal” operators (the three others); while applying the operations 1−L generates a sum of (infinitely many, in the limit M → ∞) monomials called irrelevant operators. Note that one can repeat the analysis in Appendix F to conclude that many terms, which could be a priori present in (4.9) are indeed absent. Hence the constants nh , sh , lh , (≤h)− or (≤h)+ ψ zh , ah are real and many possible marginal interactions (like k sin k ψ k,ω k,−ω (≤h)+ (≤h)− k ψk,ω ψk,ω ) are excluded. This remark is crucial in order to analyze the flow of the running coupling constant, see the next section: it shows that the number of relevant or marginal operators is far smaller than a priori one might expect, due to the symmetries in the hamiltonian. (≤h) Note also that we have written the coefficient of Fσ as sh + γ h nh according to a rule which will be specified in (4.17), (4.18) below and for the reasons explained in the subsequent remark. 4.4. Renormalization. We have set all definitions needed to define the recursive procedure leading to the definition of the running couplings and of the effective potentials. Suppose that Zk , mk ,√Ck , V (k) in (4.3) have√been defined for k = 1, 0, √. . . h + 1. Then we can write V (h) ( √Zh ψ (≤h) ) as LV (h) ( Zh ψ (≤h) ) + (1 − L)V (h) ( Zh ψ (≤h) ) and we split from LV (h) ( Zh ψ (≤h) ) in (4.9) the three terms quadratic in ψ (≤h) given (≤h) (≤h) (≤h) + Zh zh (Fζ + Fα ). by Zh sh Fσ Since such terms are quadratic we can imagine to include them in the “the free integration” PZh ,mh ,Ch (dψ (≤h) ) by simply replacing the integration symbol PZh ,mh ,Ch (dψ (≤h) ) by a new Grassmann integration symbol PZˆ h−1 ,mh−1 ,Ch (dψ (≤h) ) obtained from PZh ,mh ,Ch (dψ (≤h) ) via the substitutions of Zh , mh (k) with −1 (C0 + µ0,0 (k))zh ], Zˆ h−1 (k) = Zh [1 + t −1 Ch−1 Z 1 Zh mh−1 (k) = [mh (k) + Ch−1 (k)t −1 (C0 + µ0,0 (k))sh ] ; Zˆ h−1 (k)

(4.11)

√ (h) = V (h) − Zh sh Fσ(≤h) − and correspondingly by replacing V (h) ( Zh ψ (≤h) ) by V (≤h) (≤h) + Fα ). This means that the subtracted terms are imagined included in Zh zh (Fζ PZˆ h−1 ,mh−1 ,Ch as an algebraic check confirms. If exp(−M 2 th ) is a suitable constant factor fixing normalization of the two integrations we get (h) √ (≤h) PZh ,mh ,Ch (dψ (≤h) )e−V ( Zh ψ ) (≤h) (h) √ −M 2 th PZˆ h−1 ,mh−1 ,Ch (dψ (≤h) ) e−V ( Zh ψ ) , =e (4.12) and we try to express the r.h.s. as a double integral by writing ψ (≤h) = ψ (≤h−1) + ψ (h) .

610

V. Mastropietro

We shall call mh (0) ≡ mh and Zˆ h (0) ≡ Zh . The r.h.s of (4.12) can be written, as an algebraic check will confirm, as (≤h) (h) √ −M 2 th (≤h−1) PZh−1 ,mh−1 ,Ch−1 (dψ ) PZh−1 ,mh−1 ,f −1 (dψ (h) ) e−V ( Zh ψ ) , e h

(4.13) where we have set −1 C0 ) , Zh−1 = Zh (1 + zh t −1 Z 1

fh (k) = Zh−1

C −1 (k) Ch−1 (k) − h−1 Zh−1 Zˆ h−1 (k)

. (4.14)

Note that fh (k) has the same support of fh (k). The single scale propagator is

(h)

gω,ω (x − y) (h)− (h)+ ψy,ω = , PZh−1 ,mh−1 ,f−1 (dψ (h) ) ψx,ω h Zh−1 1 −ik(x−y) def (h) e fh (k)[Th−1 (k)]ω,ω , gω,ω (x − y) = tM 2

(4.15)

k

and Th (k) is defined by performing in (3.5) the replacement indicated in (4.4). −1 C0 zh | ≤ 1 , |C0 sh | ≤ |mh /2| and supk≥h | Zk | ≤ ec0 |λ| , the large distance If |Z 1 2 Zk−1 (h)

behavior of gω,ω (x − y) and of its (discrete) derivatives can be established in detail and one finds that it is characterized by a single lengh scale, namely γ −h . The analysis leads to naively expected results that will be exploited in the following and it is performed in Appendix H. (h) (k¯ ηη ) = sh + γ h nh We can now specify according to which rule the splitting W 2,σ ,ω in (4.9) will be done. We write (h)

(h)

(h)

(h)

(h)

gω,−ω (x − y) = gω,−ω (x − y) + gω,−ω (x − y), 1 −ik(x−y) imh (k) def (h) gω,−ω (x − y) = fh (k) 2 2 e , (4.16) 2 sin k 2 + m2 (k) sin k0 + Z tM 2 Z 1 1 h k (h)

gω,−ω and it does not vanish for mh = 0. We write and gω,−ω (x − y) is gω,−ω − a(h) + W b(h) (h) = W W 2,σ ,ω 2,σ ,ω 2,σ ,ω

(4.17)

a(h) given by definition by a sum of terms containing at least a propagator with W 2,σ ,ω (k)

gω,−ω (x − y), k > h and we set  1 sh = δω,−ω  4

η,η =±1

 a(h) (k¯ ηη ) W 2,σ ,ω

 1 γ h nh = δω,−ω  4

η,η =±1

 b(h) (k¯ ηη ) . W 2,σ ,ω (4.18)

a(h)

Such definitions imply that W2,σ ,ω is vanishing at t = tc for all h.


611

Remark. In a theory of fermions if there is no mass term in the action then no mass terms are generated by the Renormalization Group iterations, by local Gauge invariance. In our spin model this is not true, as the interaction is not Gauge invariant; hence even if t = tc (or m1 = 0) a mass term in the Renormalization Group iterations can be generated. Hence we collect all the relevant terms which are vanishing if t = tc , in sh , which we include in the fermionic integration; the “mass” has a non trivial flow producing at the end the critical index of the correlation length. The remaining terms are left in the effective interaction; they constitute the running coupling constant νh whose flow is controlled by the counterterm ν. √ √ (h) (h) , see (4.4), by a factor Zh / Zh−1 so We now rescale the kernels W2n,σ ,α,ω in V √ (h) ( Zh ψ (≤h) ) can be rewritten as that the effective potential V (h) ( Zh ψ (≤h) ) = V (h) ( Zh−1 ψ (≤h) ) ; V

(4.19)

and as a consequence, see (4.9), (h) ( Zh−1 ψ ≤h ) = γ h νh Zh−1 Fσ(≤h) + δh Zh−1 Fα(≤h) + λh (Zh−1 )2 F LV λ Zh 2 def Zh def Zh def nh , δh = (ah − zh ) , λh = ( ) lh . νh = Zh−1 Zh−1 Zh−1

(≤h)

,

(4.20)

def

We will call vh = (λh , δh , νh ) the running coupling constants and Zh , mh the renormalization constants. h by If we now define V (h−1) , E (h−1) (√Z (≤h−1) )−M 2 E (≤h) (h) √ h h−1 ψ e−V = PZh−1 ,mh−1 ,f−1 (dψ (h) ) e−V ( Zh−1 ψ ) , h

(4.21) √ h such that V (h−1) (0) = 0, we see that V (h−1) ( Zh−1 ψ (≤h−1) ) is of the form (4.4) with E h : this is checked by decomposing ψ (≤h) = ψ (≤h−1) +ψ (h) and and Eh−1 = Eh +th + E by means of the relation (which is, essentially, a definition of truncated expectations), h M 2E

+V

(h−1)

( Zh−1 ψ

(≤h−1)

∞ 1 (h) ( Zh−1 ψ (≤h) )) , (−1)n+1 EhT ,n (V )= n! n=1

(4.22) −1 gω,ω , see where EhT ,n denotes the truncated expectation of order n with propagator Zh−1 (4.15). (h) The above procedure allows us to write the kernels W2n,σ ,ω,α and E˜ h by a convergent expansion in the running coupling constants and the renormalization constants at higher scales; more exactly we will prove in Appendix I the following proposition. (h)

4.5. Lemma 2. Suppose that εh < ε, then if ε is small enough and if for some constant c1 , ! m ! ! Z ! 2 ! h ! ! h ! c1 εh | ≤ εh , max ≤ e |v sup , sup (4.23) ! ! ! ! ≤ ec1 εh , h m Z h >h h −1 h −1 h >h h >h

612

V. Mastropietro

then for a suitable M–independent constant c0 the kernels in (4.4) satisfy −hDk (Pv0 ) |W (c0 εh )max(1,n−1) 2n,σ ,ω,α (k1 , ..., k2n−1 )| ≤ γ (h)

where Dk (Pv0 ) = −2 + n + k and k =

2n

i=1 αi .

(4.24)

Moreover

h+1 | ≤ γ 2h c0 εh . (|nh | + |zh | + |ah | + |lh |) ≤ c0 εh , |sh | ≤ |mh | c0 εh , |E

(4.25)

5. The Flow of the Running Coupling Constants (h)

5.1. By the result in Sect. (4.5) it follows that the kernels W2n,σ ,ω,α in (4.4) are bounded as soon as the condition (4.23) on the running coupling constants vh and the renormalization constants Zh , mh , h > h are verified. Such quantities verify a set of recursive equations called Beta function relations of the form νh−1 = γ νh + βνh (ah , νh ; ...; a1 , ν1 ), mh−1 h = 1 + βm (ah , νh ; ...; a1 , ν1 ), mh

ah−1 = ah + βah (ah , νh ; ...; a1 , ν1 ), Zh−1 = 1 + βzh (ah , νh ; ...; a1 , ν1 ), (5.1) Zh

where ah = (λh , δh ). By explicit calculation of the lower order non-zero terms one finds, for h ≤ 0, βzh (ah , νh ; ...; a1 , ν1 ) = b1 λ2h + O(εh3 ) , h (ah , νh ; ...; a1 , ν1 ) = a2 λh + O(εh2 ) , βm

b1 > 0 , a2 > 0 .

(5.2)

It is possible to prove the following proposition, see Appendix L. 5.2. Lemma 3. There are positive constants ci 1, . . . , 6, such that for M large, |t −tc | > 0 and small enough and λ small enough one can uniquely define ν(λ) such that there exists an integer h∗ ≤ 0 such that for h∗ − 1 ≤ h ≤ 0, 1

∗

|δh | ≤ c1 |λ| , |νh | ≤ c6 |λ| (γ − 2 (h−h ) + γ κh ), |λh − λ| ≤ c1 |λ|3/2 , m 2 2 h γ −λc2 h < < γ −λc3 h , γ −c4 λ h < Zh < γ −c5 λ h . (5.3) m1 ∗ −2

The scale h∗ is such that γ k−1 ≥ 4|mk | for 1 ≥ k ≥ h∗ while γ h logγ |m1 | 1 − λc2

≤ h∗
0. The proof of (5.7) is done by comparing the expression for the correlation functions obtained by the exact solution in [ML] with their expression as series in terms of running coupling constants, see [BGPS, GS, BM1] and [BeM1]. Hence by (5.6), (5.7) and some properties of the functions βih (ah , νh ; ...; a1 , ν1 ) in (5.1) the above proposition on the flow follows, see Appendix L. (L)

5.5. The propagator corresponding to the integration of all the scales between h∗ and hM , (≤h∗ ) g ω,ω (x − y) ∗ (≤h∗ )− (≤h∗ )+ ≡ PZh∗ −1 ,mh∗ −1 ,Ch∗ (dψ (≤h ) )ψx,ω ψy,ω , (5.9) Zh∗ −1 obeys the same bound as the propagator of the integration of a single scale greater than h∗ , see (7.67) in Appendix H ; this property can used to perform the integration of all the scales ≤ h∗ in a single step. We define √ (≤h∗ ) ) 2 ∗ (h∗ ) e−M E≤h∗ = PZh∗ −1 ,mh∗ −1 ,Ch∗ (dψ (≤h ) ) e−V ( Zh∗ −1 ψ , (5.10) and in Appendix I it is proved that:

614

V. Mastropietro

5.6. Suppose that εh∗ < ε, then if ε is small enough and (4.23) holds with h = h∗ − 1, then ∗

≤h∗ | ≤ γ 2h c0 εh . |E

(5.11)

Then by the statements in §4.5, §5.2, §5.6 for λ small enough and ν suitably chosen we get a convergent expansion for the free energy −

1 1 ˆ− = h + th ] + E ≤h∗ + t≤h∗ . log Z [E 2I M2 ∗

(5.12)

h=h +1

h , E ≤h∗ are written by a convergent tree expansion, see Appendix I. The quantities E Note that the fact that hM is finite, which is due to the fact that we are considering the addend with γ (1) , γ (2) = −, −, −, − in (2.10), plays essentially no role in the analysis. ∗

Remark. γ h is a momentum scale and, roughly speaking, for momenta bigger than ∗ γ h the theory is “essentially” a massless theory (up to O(mh γ −h ) terms), while for ∗ ∗ momenta smaller than γ h it is a “massive” theory with mass O(γ h ) which can be integrated without multiscale decomposition. 6. Correlation Functions and the Specific Heat 6.1. In the preceding sections we have found a convergent expansion for the free energy; the latter is not interesting per se until we show that the free energy as a function of t − tc has some singularity at t = tc . In order to show that t = tc is a critical point we can study some correlation functions or some thermodynamic property like the specific heat by evaluating them at t = tc and M = ∞ and then verify that they have a singular behavior as t → tc . We shall study, for this purpose, the energy-energy correlation function (1.8) and the specific heat. We start by considering the following expression: (x, y) =< [HI,x (σ (1) ) + HI,x (σ (2) )][HI,y (σ (1) ) + HI,y (σ (2) )] >,T ,

(6.1)

where HI (σ (α) ) = x HI,x (σ (1) ), and HI (σ (α) ) is the Ising model hamiltonian defined in the first of (1.3). By using (2.9) we get, for x = y (x, y) =

(−1)

δγ1 +δγ2

γ1 ,γ2

γ ,γ

Z2I1 2 γ1 ,γ2 , (x − y), Z2I

(6.2)

where γ1 ,γ2 , (x − y) =

"# 2 α=1

− −

∂ (α)

∂Jx,x0 ;x,x0 +1

! ! V!

∂ (α)

∂Jr;y,y0 ;y+1,y0

{Jr }

! ! V!

! ! V!

∂

α sech2 Jr ∂t Sx,ε (a) ,ε (α) −

(α)

∂Jx,x0 ;x+1,x0 {Jr } $ # 2 α ; sech2 Jr ∂t Sy,ε (a) ,ε (α)

{Jr }

α=1

−

∂ (α)

∂Jy,y0 ;y,y0 +1

! ! V!

$%T {Jr }

.

(6.3)


615

If A1 , . . . , An are functions of the field, we are using the symbol < A1 ; . . . ; An >T to

(α) denote the truncated expectation w.r.t. the fermionic integration γ11,γ2 [ 2α=1 P (α) (α) ε ,ε Z2I

(dH (α) , dV (α) )]e−V of ni=1 Ai . Let us consider first the following expression, which gives the dominant large distance contribution %T 2 " 2 α α ˜ γ1 ,γ2 , (x − y) = ∂t Sx,ε(a) ,ε (α) ; ∂t Sy,ε(a) ,ε (α) . (6.4) α=1

α=1

Performing the change of variable (2.12), (2.13) we see that the r.h.s. of (6.4) can be ∂ ˜ γ1 ,γ2 , (x − y) = ∂ written as ∂φ(x) ∂φ(y) Sγ1 ,γ2 (φ)|φ=0 where, with the notation of (3.1), eSγ1 ,γ2 (φ) =

P (dψ)

P (dχ )eQ(χ,ψ)−V (ψ,χ) e

x

2 +∂t S 2 2 x,ε 1 ,ε 1 x,ε ,ε

φ(x) ∂t S 1

.

(6.5)

This is a new expression similar but not identical to the ones studied to obtain analyticity of the free energy for t = tc . We can study (6.5) in a similar way, by adapting the free energy analysis for the integration of Sγ1 ,γ2 (φ). def ˜ −,−,−,−, (x − y) def ˜ (x − y). One can Consider S−,−,−,− (φ) = S(φ) and = proceed as in §3 in order to integrate the massive χ fields and finding, for |λ| ≤ ε and with the notations of (3.1), (1) S (φ) M2N =e P (dψ)e−V (ψ)+B(φ,ψ) , (6.6) e where N is a normalization constant and B(ψ, φ) =

∞ ∞

···

m=1 n=1 σ ,α,ω x1

xm

···

y1

·

y2n

m &

· Bm,2n,σ ,α,ω (x1 , . . . , xm ; y1 , . . . , y2n )

i=1

where for n ≥ 2,

2n '&

φ(xi )

' ∂ αi ψyσii,ωi ,(6.7)

i=1

n

|Bm,2n,σ ,α,ω (x1 , . . . , xm ; y1 , . . . , y2n )| ≤ C n ε 2 ,

(6.8)

y1 ,...,y2n

and for n = 1 (1) − + iωZ1 φ(x)ψx,ω ψx,−ω + x

B1,2,σ ,α,ω (x; y1 , y2 )φ(x)

y1 ,y2 x {σ,ω} α1 +α2 ≥1

ψ), × ∂ α1 ψyσ11,ω1 ∂ α2 ψyσ22,ω2 + B(φ, (1)

where Z1 is an O(1) constant,

y1 ,y2

(6.9)

ψ) con|B1,2,σ ,α,ω (x; y1 , y2 )| ≤ C and B(φ, (1)

tains the terms with m ≥ 2. All kernels Bm,2n,σ ,α,ω and Z1 are analytic in λ, as follows by proceeding as in Appendix E. The symmetry considerations in Appendix F apply here as well and imply that the + − only possible local terms with n = m = 1 are of the form φ(x)ψx,1 ψx,−1 .

616

V. Mastropietro

6.2. We shall evaluate the integral, over the light fermions, in the r.h.s. of (6.6) in a way which is very close to that used for the integration of the r.h.s. of (3.1). We introduce the scale decomposition described in §4 and we perform iteratively the integration of the single scale fields, starting from the field of scale 1. After integrating the fields ψ (1) , ...ψ (h+1) , 0 ≥ h ≥ h∗ , we find 2 (h+1) (φ) (h) √ (≤h) (h) √ (≤h) eS (φ) = e−M Eh +S PZh ,mh ,Ch (dψ ≤h )e−V ( Zh ψ )+B ( Zh ψ ,φ) , (6.10) where PZh ,mh ,Ch (dψ (≤h) ) and V h are given by (4.4), respectively, while S (h+1) (φ), which denotes the sum over all terms dependent on φ but independent of the ψ field, and B (h) (ψ (≤h) , φ), which denotes the sum over all terms containing at least one φ field and two ψ fields, can be represented in the form S (h+1) (φ) =

∞ m=1 x1

B (h) (ψ (≤h) , φ) =

···

m &

(h+1) Sm (x1 , . . . , xm )

xm

' φ(xi ) ,

(6.11)

i=1

∞ ∞

···

···

·

xm y1 y2n m=1 n=1 α,ω x1 (h) · Bm,2n,σ ,α,ω (x1 , . . . , xm ; y1 , . . . , y2n )· m 2n & ' '& i . · φ(xi ) ∂ αi ψy(≤h)σ i ,ωi i=1 i=1

(6.12)

Since the field φ is equivalent, from the point of view of dimensional considerations, to two ψ fields, the only terms in the r.h.s. of (6.12) which are not irrelevant are those with m = 1 and n = 1, which are marginal. Repeating the symmetry considerations in Appendix F we can conclude that the only local terms with n = m = 1 and α1 = α2 = 0 (≤h)+ (≤h)− have the form φ(x)ψx,ω ψx,−ω . Hence we extend the definition of the localization operator L, so that its action on B (h) (ψ (≤h) , φ) is defined by its action on the kernels (h) Bm,2n,σ ,α,ω (x1 , . . . , xm ; y1 , . . . , y2n ): 1) if m = 1, n = 1, α1 = α2 = 0, then (h) LB1,2,σ ,α,ω (x1 ; y1 , y2 )

= δ(y1 − x1 )δ(y2 − x1 )

(h)

dz1 dz2 B1,2,σ ,α,ω (x1 ; z1 , z2 ), (6.13)

2) in all the other cases, (h)

LBm,2n,σ ,α,ω (x1 , ...xm ; y1 , ..., y2n ) = 0 .

(6.14)

Hence, by the symmetry reasons discussed in Appendix F, (1)

LB(h) (ψ (≤h) , φ) =

Zh (≤h) F , Zh 1

(6.15)


617

(1)

where Zh is a real number and (≤h) (≤h)+ (≤h)− (≤h)+ (≤h)− = φ(x)i[ψx,1 ψx,−1 − ψx,−1 ψx,1 ] . F1

(6.16)

x

In the expansion for the energy-energy correlation function there is then a renormal(1) ization constant more, namely Zh . With the notation of §4 we can write the integral in the r.h.s. of (6.10) 2 (≤h) (h) √ (≤h) (h) √ e−M th PZh−1 ,mh−1 ,Ch (dψ (≤h) )e−V ( Zh ψ )+B ( Zh ψ ,φ) −M 2 th PZh−1 ,mh−1 ,Ch−1 (dψ (≤h−1) ) · =e (≤h) (≤h) (h) √ (h) √ · PZh−1 ,mh−1 ,f−1 (dψ (h) )e−V ( Zh−1 ψ )+B ( Zh−1 ψ ,φ) , (6.17) h

√

√ (h) ( Zh−1 ψ (≤h) ) is defined as in (4.19) and B (h) ( Zh−1 ψ (≤h) , φ) = where√ V √ B (h) ( Zh ψ (≤h) , φ); moreover B (h−1) ( Zh−1 ψ (≤h−1) , φ) and S (h) (φ) are then defined through the relation analogue of (4.21), that is √

√

2

(h)

)+B ( Zh−1 ψ ,φ)−M Eh +S (φ) e−V ( Zh−1 ψ √ (h) (≤h) (≤h) (h) √ = PZh−1 ,mh−1 ,f−1 (dψ (h) )e−V ( Zh−1 ψ )+B ( Zh−1 ψ ,φ) . (6.18) (h−1)

(≤h−1)

(h−1)

(≤h−1)

h

As in §5.5, the fields of scale between h∗ and hM are integrated in a single step without any multiscale decomposition. Hence we define, in analogy to (5.10), ∗ ∗ h∗ def S (h ) (φ)−M 2 E = PZh∗ −1 ,mh∗ −1 ,Ch∗ (dψ (≤h ) ) e √ √ (≤h∗ ) )+B (h∗ ) (h∗ ) ( Zh∗ −1 ψ (≤h∗ ) ,φ) . (6.19) × e−V ( Zh∗ −1 ψ It follows that ∗

S(φ) = −M 2 EM + S (h ) (φ) = −M 2 EM +

1

S (h) (φ) ;

(6.20)

h=h∗

hence, if S2 (x, y) = (h)

∂ (h) ∂ ∂φ(x) ∂φ(y) S (φ)|φ=0 , ∗

˜ (x, y) = S (h ) (x, y) = 2

1 h=h∗

(h) S2 (x, y) .

(6.21)

˜ (x, y) = α (x, y) + β (x, y), where 6.3. It is shown in Appendix M that α (x, y) =

1

(Z (1) )2 (h ) h∨h (h) [gω,ω (x − y)g−ω,−ω (y − x)− Zh−1 Zh −1

h,h =h∗ ω=±1

(h) g+1,−1 (x

(h ) − y)g−1,+1 (y

− x)] +

1 h=h∗

(

(1)

Zh Zh

)2 (h),a

G

(x, y),

(6.22)

618

V. Mastropietro (h∗ )

(≤h∗ )

where h ∨ h = max{h, h } and gω1 ,ω2 (x) has to be understood as gω1 ,ω2 (x); moreover for all N > 0 there exists a constant CN such that (h),α

|∂xm1 ∂xm00 G

(x, y)| ≤ γ (2+m0 +m1 )h |λ1 |

CN h 1 + (γ |d(x

− y)|)N

.

(6.23)

β

For (x, y) the following bound holds: ( (1) )2 1 Zh CN m1 m0 β |∂x ∂x0 (x, y)| ≤ γ (2+m0 +m1 +τ )h , h Zh 1 + (γ |d(x − y)|)N ∗

(6.24)

h=h

where 0 < τ < 1 is a constant. A similar bound, by dimensional reasons, holds for ˜ (x, y) − −,−,−,−, (x, y). It is shown in Appendix M that sech4 (1)

Zh−1 (1) Zh

(1)

= 1 + zh = 1 + a1 λh + O(µ2h ) ,

so that there exist two constants c1 , c2 such that γ −λ1 c1 h
, where Ai , Bj are vectors in a Hilbert space with scalar product < ·, · >, then | det M| ≤ ||Ai || · ||Bi ||, (7.27) i

where || · || is the norm induced by the scalar product. Let H = Rs ⊗ H0 , where H0 is the Hilbert space of complex four dimensional vectors F (k) = (F1 (k), . . . , F4 (k)), Fi (k) being a function on the set D−,− , with scalar product < F, G >=

4 1 ∗ Fi (k)Gi (k), M2

(7.28)

k

i=1

and one checks that (χ)

GTij,i j = ti,i gω− ,ω+ (xij − yi j ) =< ui ⊗ Ax(f − ),ω(f − ) , ui ⊗ Bx(f + ),ω(f + ) > , l

ij

l

ij

i j

i j

(7.29) where ui ∈ Rs , i = 1, . . . , s, are the vectors such that ti,i = ui · ui , and (with Q(k) defined in (2.23)) 1 (− sin k0 + i sin k, 0, −imχ (k), 0), if ω = +1, ik x · Ax,ω (k) = e (0, imχ (k), 0, mχ (k)), if ω = −1, −Qχ (k) (7.30) 1 (1, 1, 0, 0), if ω = +1, ik y · Bx,ω = e (0, 0, 1, (sin k0 + i sin k)/mχ (k)), if ω = −1. −Qχ (k) Hence from (7.27) we immediately find |GTij,i j | ≤ C1n

(7.31)

where C1 is an O(1) constant. Finally we get xv0

∞ 1 |KPv (xv0 )| ≤ 0 n! v ,..,v n=1

1

n xv1 ,...,xvn

C1n

n (χ) [ |gω− ,ω+ (xl − yl )|] |vi (xvi )|, T

l∈T

i=1

(7.32)

where we have used that dP +T (t) = 1. The number of addends in T is bounded by n!C2n . Finally T and the i xvi form a tree connecting all points, so that using that the propagator is massive and that the interactions are short ranged xv ,...xvn 1

n (χ) n n 2 n is the number of T [ l∈T |gω− ,ω+ (xl − yl )|] i=1 |vi (xvi )| ≤ C3 |ε| M , where couplings O(ε). v0 | ≥ 4. Note that if to vi are associated only terms Let us consider the case |P from V(ψ, χ ), then n = n. Let us consider now the case in which there are end-points v0 | end-points associated to Q(ψ, χ ), which have O(1) coupling; there are at most |P


625

associated with Q(ψ, χ ). In fact in Q(ψ, χ ) there are only terms of the form ψχ, so at most the number of them is equal to the number of ψ fields. If we call nλ ≤ n the v0 |/2 − 1 ≥ |P v0 |/4; hence number of vertices quartic in the fields it is clear that nλ ≥ |P

|KPv (xv0 )| ≤ M 2 0

xv0

∞

n+|Pv0 | C |ε|n

(7.33)

v |/4 n=|P 0

v0 | ≥ 4. and the second of (3.2) holds for |P Consider now the case |Pv0 | = 2; in this case there are terms O(1), obtained when all the vi are associated with elements of Q(ψ, χ ). It is convenient to include all such terms in the gaussian integration, as they cannot be considered as perturbations (they are not O(ε)). Hence we define N P (dψ) = P (dψ) P (dχ )eQ(ψ,χ) (7.34) and, if < X >0 =

− + − + P (dψ)X, it holds < ψx,1 ψy,1 >0 =< ψx ψy >0 , < ψx,−1 ψy,−1 >0 (1)

(1)

− + =< ψ x ψ y >0 and < ψx,1 ψy,−1 >0 =< ψx ψ y >0 . By using the explicit expres(1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

sions for < ψx ψy >0 , < ψ x ψ y >0 , < ψx ψ y >0 in [MPW], (3.5) follows. In order to obtain (3.3) we single out the local part of the terms quartic in the fields; the fact that l1 = 2λ(a + b)sech4 J + O(ε 2 ) can be checked by an explicit computation of all the contributions with coupling O(λ) to W2 , noting that they can only be obtained contracting a term quartic in the χ fields with one of the addends of (7.10); each of such terms carries a derivative in the coordinate space, hence the Fourier transform of such terms is vanishing at zero momentum. 7.6. Appendix F: Symmetry cancellations in the effective potential. + − • There are no local terms in the r.h.s. of (3.2) of the form ψx,1 ψx,1 ; in fact by (2.13) − + ψx,1 = iψx ψx , but the system is invariant under the transformation ψx,1 (1)

(2)

(1)

(1)

ψ (1) , ψ , χ (1) , χ (1) → −ψ (1) , −ψ , −χ (1) , −χ (1) (2) (2) ψ (2) , ψ , χ (2) , χ (2) → ψ (2) , ψ , χ (2) , χ (2) ,

(7.35)

hence such terms cannot be present as they violate such symmetry. − − + + • There are no terms in the r.h.s. of (3.2) of the form ψ1,x ψ−1,y or ψ1,x ψ−1,y ; in fact, − − ψ−1,y = ψ1,x

1 (1) (1) (2) (2) (1) [ψ ψ y − ψx(2) ψ y + iψx(1) ψ y + iψx(2) ψ y ] 2 x

(7.36)

and the last two terms violate the symmetry (7.35). Moreover the first two terms are forbidden a) in the case b = 0 by the invariance under the symmetry (1) (1) (2) (2) (2) (2) (1) (1) ψx,x , χx,x , ψx,x , χx,x → ψx,x , χx,x , ψx,x , χx,x ; 0 0 0 0 0 0 0 0

(7.37)

b) in the case a = 0 by the invariance under the symmetry (1)

(1)

(1) (1) (2) (2) (2) (2) , χx,x , ψx,x , χx,x → ψx,x , χx,x , ψx+1,x0 −1 , χx+1,x0 −1 . ψx,x 0 0 0 0 0 0

(7.38)

626

V. Mastropietro

(1) (1) In fact consider in V (1) (3.1) the terms of the form x,y [ψx ψ¯ y w (1) (x, y) + (2) (2) ψx ψ¯ y w (2) (x, y)]; w (1) (x, y) is obtained by the truncated expectation EχT of a certain number of (V + Q)|ψ=0 , of a term

∂ (1) (V ∂ψx

+ Q)|ψ=0 and of a term

∂ (1) (V ∂ ψ¯ y

+ Q)|ψ=0 .

If we perform in the truncated expectation the change of variable (7.37) or (7.38)we get that (V + Q)|ψ=0 is invariant while ∂(1) (V + Q)ψ=0 is changed in ∂(2) (V + Q)ψ=0 and

∂ (1) (V ∂ ψ¯ y

∂ψx

+ Q)ψ=0 is changed in

∂ (2) (V ∂ ψ¯ y

∂ψx

+ Q)ψ=0 ; this shows that w(1) (x, y) =

w(2) (x, y). + + A similar argument can be repeated for ψ1,x ψ−1,y . + ψ + ; in fact, • There are no terms in the r.h.s. of (3.2) of the form ψω,x ψω,y or ψω,x ω,y − − ψ1,x ψ1,x =

' 1 & (1) (1) ψx ψy − ψx(2) ψy(2) + iψx(1) ψy(2) + iψx(2) ψy(1) 2

(7.39)

and we can proceed as in the previous case. • The model is invariant under complex conjugation and the exchange (α)

(α)

ψx(α) , ψ x → ψ x , ψx(α)

(α) (α) χx(α) , χ (α) x → χ x , χx ; (α)

(7.40)

(α)

this follows from the fact that, from (2.12), H , H (α) , V , V (α) , written in terms (α) of ψ , ψ (α) , χ (α) , χ (α) , are invariant under such transformation. Hence the coefficient of the lo+ + cal part of the quartic (non-vanishing) terms is real; in fact ψ1,x ψ1,x ψ−1,x ψ−1,x ≡ (1)

(1)

(2)

(2)

(0, 0, 0) must be equal, by the above invariance, to ψx ψ x ψx ψ x times w (1) (1) (2) (2) (0, 0, 0) = w ∗ (0, 0, 0). Finally the combiw ∗ (0, 0, 0)ψ x ψx ψ x ψx , hence w (1) (2) (2) (1) + − + − ψx,−1 + ψx,−1 ψx,1 is equal to i[ψx ψ x − ψx ψ x ] so it nation of local terms ψx,1 + − cannot be present as it violates the symmetry (7.35). On the other hand ψx,1 ψx,−1 − + − ψx,1 is equal to [ψx ψ x + ψx ψ x ]; hence the coefficient of the local part ψx,−1 (1)

(1)

(2)

(2)

(1)

(1)

(2)

(2)

is imaginary and odd in ω; in fact w (0)[ψx ψ x + ψx ψ x ] must be equal to (1) (1) (2) (2) w ∗ (0)[ψ x ψx +ψ x ψ x ], by the invariance under complex conjugation and (7.40), hence w (0) = − w∗ (0). • We consider now the addends with n = 1 in the r.h.s. of (3.2), + Wω1 ,ω2 (x, y)ψx,ω ψ− . (7.41) 1 y,ω2 x,y

We can represent Wω1 ,ω2 (x, y) as the sum over Feynman graphs g in the usual way (see for instance [GM]); the external lines are associated to the ψ fields, and to χ the internal lines are associated the propagators gω,ω (x − y); moreover the vertices 1 ,ε2 associated to the interaction are linear or bilinear in Ac,ε x;φ,ω1 ;φ ,ω2 . We show that

x

sin

πx πx0 Wω,−ω (x, 0) = Wω,−ω (x, 0) = 0 . sin M M x

ω,−ω (x, 0) and we call We can consider a single Feynman diagram value W g

(7.42)


627

1 ,ε2 1) nωa is the number in g of terms Aεx;φ,ω with ω1 = ω2 = ω; n1a + n−1 a = na . 1 ;φ ,ω2 ε1 ,ε2 2) nb is the number of Ax;φ,ω1 ;φ ,ω2 with ω1 = −ω2 = ω.

(χ)

3) nω+ is the number of diagonal propagators gω,ω . (χ) 4) n− is the number of non diagonal propagators gω,−ω .

g πx x Wω,−ω (x, 0) sin M (χ) and we use (2.27), then in each Feynman graph each propagator gω,ω (x) is replaced (χ) (χ) by (−1)δω,ω gω,ω (x). Moreover the propagators ∂gω,ω (x) are replaced by (−1)δω,ω +1 (χ) ∂x0 fx,x0 = fx,x0 − fx,x0 −1 and ∂x fx,x0 = fx,x0 − fx−1,x0 ; finally ∂gω,ω (x), where (χ) δω,ω (χ) gω,ω (x + a) is replaced by (−1) gω,ω (x − a), if a is a constant vector.

If we make the transformation xi → −xi in all the sums in

On the other hand we could equivalently write the interaction (1.3) as

V (σ (1) , σ (2) ) = −

M

(1)

(2)

(1)

(2)

(1) (1) λa[σx−1,x0 σx,x σ σ (2) + σx,x0 −1 σx,x σ σ (2) ] 0 x−1,x0 x,x0 0 x,x0 −1 x,x0

x,x0 =1 (1) (1) (2) σ σ (2) +λb[σx−1,x0 σx,x 0 x,x0 −1 x,x0 (1) (2) (1) (2) +σx,x0 −1 σx,x0 σx,x0 −1 σx+1,x0 −1 ]}

;

(7.43)

Equation (7.43) can be found from (1.3) making the change of variables x → −x, (α) (α) and then making the transformation σx → σ−x . Starting from this expression and repeating the computations in §2, §3 we get an expression similar to (2.11), where V is (α) (α) (α) (α) now an expression linear or bilinear in H x−1,x0 Hx,x0 or V x,x0 −1 Vx,x0 . From (2.12) it holds (α) (α) 1(α) 2(α) 3(α) V x,x0 −1 Vx,x =Q +Q +Q x x x , 0

(7.44)

where 1 (α) (α) (α) (α) (α) (α) (α) (α) 1(α) Q = [ψx,x0 −1 ψx,x − ψ x,x0 −1 ψ x,x0 + ψ x,x0 −1 ψx,x − ψx,x0 −1 ψ x,x0 ], x 0 0 4i 1 (α) (α) (α) (α) (α) (α) (α) 2(α) Q = [χx,x0 −1 χx,x − χ x,x0 −1 χ (α) x x,x0 + χ x,x0 −1 χx,x0 − χx,x0 −1 χ x,x0 ], 0 4i (7.45) 1 (α) (α) (α) (α) (α) (α) (α) 3(α) Q = [ψx,x0 −1 χx,x − ψ x,x0 −1 χ (α) x x,x0 − ψx,x0 −1 χ x,x0 + ψ x,x0 −1 χx,x0 0 4i (α) (α) (α) (α) (α) (α) (α) (α) +χx,x0 −1 ψx,x − χ x,x0 −1 ψ x,x0 − χx,x0 −1 ψ x,x0 + χ x,x0 −1 ψx,x ]. 0 0 (α)

(α)

A similar expression hold for H x−1,x0 Hx,x0 . Note that, looking for instance to the first of (7.45), we get (α) (α) ψx,x0 −1 ψx,x − ψ x,x0 −1 ψ x,x0 = ψx,x ∂ ψ (α) − ψ x,x0 ∂x0 ψ x,x0 0 0 x0 x,x0 (α)

(α)

(α)

(α)

(α)

(7.46)

and (α) (α) (α) (α) (α) (α) (α) (α) ψ x,x0 −1 ψx,x − ψx,x0 −1 ψ x,x0 = − ∂x0 ψ x,x0 + ψ x,x0 ψx,x ∂x0 ψx,x 0 0 0 (α)

(α)

+ψ x,x0 −1 ψx,x0 −1 .

(7.47)

628

V. Mastropietro

σ1 ,σ2 σ1 ,σ2 x Ax;φ,ω1 ;φ ,ω2 or x Ax;φ,ω1 ;φ ,ω2 σ ,σ 1 2 1, x0 − 1) and A x;φ,ω1 ;φ ,ω2 is identical

One verifies that V is a sum of the form

σ 1 ,σ 2 where x = x or x = (x + A x ;φ ,ω1 ;φ ,ω2 1 ,σ2 to Aσx;φ,ω up to the substitutions ∂ → ∂, x + 1 → x − 1 and x0 + 1 → x0 − 1. 1 ;φ ,ω2 Hence it holds

sin

x

πx πx sin Wω,−ω (x, 0) = (−1)na +n+ +1 Wω,−ω (x, 0) . M M x

(7.48)

It holds 2na + 2nb = 2(n+ + n− ) + 2 so that

sin

x

πx πx Wω,−ω (x, 0) = (−1)2na +nb −n− Wω,−ω (x, 0) sin M M x πx = (−1)nb −n− sin Wω,−ω (x, 0). M x

(7.49)

The number of fields with ω = 1 is 2n1a + nb and the number of external fields with ω = 1 is then 2n1a + nb − 2n1+ − n− = 2(n1a − n1+ ) + nb − n− which implies that nb − n− must be an odd numberif the number of external fields = 1 is 1. Hence with ω πx πx πx x sin M Wω,−ω (x, 0) = (−1) x sin M Wω,−ω (x, 0) so that x sin M Wω,−ω (x, 0) = 0. We consider now Wω,ω (x; 0); we have already proved that x Wω,ω (x; 0) = 0. We want to show that sin πx0 x

M

Wω,ω (x; 0) = iωα;

sin π x x

M

Wω,ω (x; 0) = β

with α, β real. From (2.23) we see that gω,−ω (x) is even in the exchange x → −x and imaginary. Moreover we can write gω,ω (k) =

sin2 k

−i sin k ω sin k0 1 2 + = gω,ω (k) + gω,ω (k) 2 2 2 + sin k0 + mχ sin k + sin2 k0 + m2χ (7.50)

1 (x) real, odd in the exchange x → −x and even in x → −x ; g 2 (x) is with gω,ω 0 0 ω,ω imaginary, even in the exchange x → −x and odd in x0 → −x0 . Remember that (see 1 ,σ2 §2.4) the coefficient of Aσx;φ,ω is a) imaginary if ω1 = ω2 and α = 1, ∂xα = ∂x0 ; 1 ;φ ,ω2 b) real if ω1 = ω2 and α = 2, ∂xα = ∂x ; c)imaginary if ω1 = −ω2 . Given a Feynman diagram g contributing to i x sin πx M Wω,ω , by parity it must be present a total odd num1 (x) propagators and ∂ derivatives from the interaction. Moreover by parity the ber of gω,ω x 2 (x) and ∂ from the interaction must be even. Finally as the external lines number of gω,ω x0 have the same ω index, the sum of the number of non diagonal propagators gω,−ω (x) plus 1 ,σ2 the number of Aσx;φ,ω with ω1 = −ω2 must be even. Hence x sinMπx Wω,ω (x; 0) 1 ;φ ,ω2 sin πx0 is real and ω-independent. In the same way one sees that x M Wω,ω (x; 0) = iωα.


629

7.7. Appendix G: Independence from boundary conditions. We show that, if |t −tc | > 0, ε(1),ε(2),ε(3),ε(4) = Z ε(1),ε(2),ε(3),ε(4) (Z ε(1),ε(2),ε(3),ε(4) )−1 , Z 2I 2I 0,2I

(7.51)

ε(1),ε(2),ε(3),ε(4)

where Z0,2I is given by (7.2) with λ = 0, is exponentially insensitive to boundary conditions. In particular we show that for |t − tc | > 0, λ small enough ! ! ! γ (1),γ (2) !! Z ! 2I (7.52) !log −,−,−,− ! ≤ |λ|M 2 e−c1 |t−tc |M , ! ! Z 2I

where c1 > 0 is a suitable constant. The above equation implies in particular that the partition function is non-vanishing; in fact, from (2.10) Z2I is (cosh λa cosh λb)2S times ε(1),ε(2),ε(3),ε(4) Z ε(1),ε(2),ε(3),ε(4) (−1)δγ1 +δγ2 Z 2I 0,2I ε

=

−,−,−,− Z0,2I Z 2I

−,−,−,− +Z 2I

(−1)δγ1 +δγ2

ε(1),ε(2),ε(3),ε(4)

2I

ε

× Z0,2I

ε(1),ε(2),ε(3),ε(4) Z 2I −1 −,−,−,− Z

(7.53)

,

ε(1),ε(2) where Z0,2I = ZI2 and ZI = ε(1),ε(2) (−1)δε(1),ε(2) ZI is the Ising model partition function. ε(1),ε(2) We recall that in §4 of [MW] it was proved that the limit M → ∞ of |ZI | if |t − tc | > 0 is exponentially independent from boundary conditions; moreover if ε(1),ε(2) have a positive limit, while if t − tc < 0 for any choice of ε1 , ε2 the functions ZI t − tc > 0 the limit of ZI+,+ is negative, and for the other choices the limit is a positive number. −,−,−,− Z0,2I ||λ|M 2 Hence by (7.52) the second addend in (7.53) is bounded by C|Z 2I e−c1 |t−tc |M so (7.53) is non-vanishing. In order to prove (7.52) we can write, see (7.12) and (7.20), 2 (1),ε(2),ε (2) ε(1),ε (α) (α) (α) (α) log Z˜ = P (dψ )P (dχ ) eQ−V , (7.54) 2I

(α)

α=1

(α)

εα ,εα

(α)

(α)

εα ,εα

ε(1),ε (1),ε(2),ε (2)

can be written as in (5.12). and proceeding as in §3 we see that log Z2I The terms E˜ h are the sum of addends of the form x1 ,..,xn Wε (x1 , .., xn ), with xi varying M M M in [− M 2 , 2 ] × [− 2 , 2 ] and the W are truncated expectations for which a formula like (7.26) holds. Note that W (x1 , .., xn ) is periodic with period M in any of its coordinates, for any ε; this follows from the fact that there is an even number of ψ, χ fields associated to any xi , and from the form of V. Moreover W (x1 , .., xn ) is translation invariant, so that we can fix one variable to (0, 0), for instance x1 ; hence it holds Wε (x1 , .., xn ) = Wε (0, x2 .., xn ) . (7.55) x1 ,..,xn

x1 ,..,xn

∗

∗∗

We can write x1 ,..,xn W as x1 ,..,xn W + x1 ,..,xn W , where ∗x1 ,..,xn is over xi varying ∗∗ M M M −c1 |t−tc |M) ), as in W there is surely in [− M x1 ,..,xn W is O(e 4 , 4 ] × [− 4 , 4 ]. Then

630

V. Mastropietro

a chain of propagators exponentially decaying connecting ∗ the point (0, 0) with a point M M M outside [− M , ] × [− , ]. On the other hand in x1 ,..,xn W we can use the Poisson 4 4 4 4 summation formula, stating that M−1 1 n2π απ f f (nM)(−1)αn , + = M M M n=0

(7.56)

n∈Z

(i)

where f is a 2π-periodic function and α = (0, 1). From (7.56) we find, if g,ε,ε (x, x0 ), i = ψ, χ is the propagator corresponding to Pε,ε (dψ) or Pε,ε (dχ ) (7.12), (i) (−1)nδε (−1)nδε g (i) (x − y + nM, x0 − y0 + n0 M) g,ε,ε (x − y, x0 − y0 ) = n,n0 ∈Z (i)

(i)

≡ g (x − y, x0 − y0 ) + δgε,ε (x − y, x0 − y0 ),

(7.57)

(i)

where g (i) (x, x0 ) = limM→∞ g,ε,ε (x, x0 ) and δε = 1 if ε = − and δε = 0 if ε = +. Note that the only dependence on boundary conditions in the r.h.s. of (7.57) is (i) M in δgε,ε (x − y, x0 − y0 ) and it holds, if |x − y| ≤ M 2 , |x0 − y0 | ≤ 2 , |δg (i) (x − y, x0 − y0 )| ≤ e−c2 |mi |M , (7.58) ∗ with a proper constant c2 . Hence all the terms in x1 ,..,xn W with at least a δg (i) (x − y, x0 − y0 ) are exponentially bounded, and the part with only g (i) (x − y, x0 − y0 ) is independent from boundary conditions. By (7.56) it holds that also the terms th are exponentially insensitive to boundary conditions. −1 C0 zh | ≤ 7.8. Appendix H:Asymptotic properties of the propagators on scale h. If |Z 1 Z 1 |λ| k 2 , |C0 sh | ≤ |mh /2| and supk≥h | Zk−1 | ≤ e , for λ, t − tc small enough, given the positive integers N, n0 , n1 and if n = n0 + n1 , it holds (h) |∂xn00 ∂xn1 gω,ω (x − y)| ≤ CN,n

(h)

|∂xn00 ∂xn1 gω,−ω (x − y)| ≤ CN,n |

γ h(1+n) , 1 + (γ h |d(x − y)|)N

mh γ h(1+n) | , γ h 1 + (γ h |d(x − y)|)N

(7.59)

(7.60)

where ∂x denotes the discrete derivative. This follows immediately from the compact support properties of f˜h (k) and the fact that (h)

dM (x − y)n1 dM (x0 − y0 )n0 gω,ω (x − y) 1 −ik(x−y) n1 n0 −1 −1 = e−iπ(xM n1 +x0 M n0 ) (−i)n0 +n1 2 e ∂k ∂k0 M k & ' × f˜h (k)[Th−1 (k)]ω,ω , where Th is the quadratic form associated to PZh−1 ,mh−1 ,Ch (dψ).

(7.61)


631

It will be useful to write (h)

(h) (h) (h) (x − y) = gL;ω,ω (x − y) + gω,ω (x − y) + gω,ω (x − y) gω,ω

(7.62)

with (h)

gL;ω,ω (x − y) =

1 −ik(x−y) fh (k) e , 1 ωk0 + i Z 1 k M2 −Z

(7.63)

k

which is of course obeying the bound (7.59). The decomposition (7.62) is related to the following identity: 1 1 1 [Th−1 (k )]ω,ω = + − −ωk −ω sin k0 + i sin k −ωk0 + ik 0 + ik −ω sin k0 + i sin k 1 + − . (7.64) −ω sin k0 + i sin k sin2 k02 + sin2 k + [mh−1 (k)]2 From (7.64) one shows that γ (2+n)h , 1 + (γ h |d(x − y))|N γ h(1+n) mh (h) |∂xn00 ∂xn1 gω,ω (x − y)| ≤ CN,n | h |2 . γ 1 + (γ h |d(x − y)|)N (h) gω,ω (x − y)| ≤ CN,n |∂xn00 ∂xn1

(7.65) (7.66)

(h)

Analogously the decomposition (4.16) is such that gω,−ω (x − y) verifies (7.60) and (h) gω,−ω (x − y), verifying (7.65). Finally note that, with the definition (5.9), it holds, given the positive integers N, n0 , n1 and putting n = n0 + n1 , that there exists a constant CN,n such that (≤h∗ ) |∂xn00 ∂xn1 gω,ω (x; y)|

∗

γ h (1+n) ≤ CN,n . ∗ 1 + (γ h |d(x − y)|)N

(7.67)

7.9. Appendix I: The integration of the ψ fields. It is possible to write V (h) in terms of Gallavotti-Nicolo trees We need some definitions and notations. 1) Let us consider the family of all trees which can be constructed by joining a point r, the root, with an ordered set of n ≥ 1 points, the endpoints of the unlabeled tree, so that r is not a branching point. n will be called the order of the unlabeled tree and the branching points will be called the non trivial vertices. The unlabeled trees are partially ordered from the root to the endpoints in the natural way; we shall use the symbol < to denote the partial order. Two unlabeled trees are identified if they can be superposed by a suitable continuous deformation, so that the endpoints with the same index coincide. Then the number of unlabeled trees with n end-points is bounded by 4n . We shall consider also the labeled trees (to be called simply trees in the following); they are defined by associating some labels with the unlabeled trees, as explained in the following items.

632

V. Mastropietro

Fig. 3. A tree with its scale labels

2) We associate a label h ≤ 0 with the root and we denote Th,n the corresponding set of labeled trees with n endpoints. Moreover, we introduce a family of vertical lines, labeled by an integer taking values in [h, 2], and we represent any tree τ ∈ Th,n so that, if v is an endpoint or a non trivial vertex, it is contained in a vertical line with index hv > h, to be called the scale of v, while the root is on the line with index h. There is the constraint that, if v is an endpoint, hv > h + 1; if there is only one end-point its scale must be equal to h + 2, for h ≤ 0. The tree will intersect in general the vertical lines in set of points different from the root, the endpoints and the non trivial vertices; these points will be called trivial vertices. The set of the vertices of τ will be the union of the endpoints, the trivial vertices and the non trivial vertices. Note that, if v1 and v2 are two vertices and v1 < v2 , then hv1 < hv2 . Moreover, there is only one vertex immediately following the root, which will be denoted v0 and can not be an endpoint; its scale is h + 1. 3) With each endpoint v of scale hv = +2 we associate one of the contributions to V (1) given by (3.2); with each endpoint v of scale hv ≤ 1 one of the terms in LV (hv −1) defined in (4.19). Moreover, we impose the constraint that, if v is an endpoint and hv ≤ 1, hv = hv + 1, if v is the non trivial vertex immediately preceding v. 4) If v is not an endpoint, the cluster Lv with frequency hv is the set of endpoints following the vertex v; if v is an endpoint, it is itself a (trivial) cluster. The tree provides an organization of endpoints into a hierarchy of clusters. 5) We introduce a field label f to distinguish the field variables appearing in the terms associated with the endpoints as in item 3); the set of field labels associated with the endpoint v will be called Iv . Analogously, if v is not an endpoint, we shall call Iv the set of field labels associated with the endpoints following the vertex v; x(f ), σ (f ) and ω(f ) will denote the space-time point, the σ index and the ω index, respectively, of the field variable with label f . 6) We associate with any vertex v of the tree a subset Pv of Iv , the external fields of v. These subsets must satisfy various constraints. First of all, if v is not an endpoint and v1 , . . . , vsv are the sv vertices immediately following it, then Pv ⊂ ∪i Pvi ; if v is an endpoint, Pv = Iv . We shall denote Qvi the intersection of Pv and Pvi ; this definition implies that Pv = ∪i Qvi . The subsets Pvi \Qvi , whose union will be made, by definition, of the internal fields of v, have to be non empty, if sv > 1, that is if v is a non-trivial vertex.


633

Given τ ∈ Tj,n , there are many possible choices of the subsets Pv , v ∈ τ , compatible with the previous constraints; let us call P one of these choices. Given P, we consider the family GP of all connected Feynman graphs, such that, for any v ∈ τ , the internal fields of v are paired by propagators of scale hv , so that the following condition is satisfied: for any v ∈ τ , the subgraph built by the propagators associated with all vertices v ≥ v is connected. The sets Pv have, in this picture, the role of the external legs of the subgraph associated with v. The graphs belonging to GP will be called compatible with P and we shall denote Pτ the family of all choices of P such that GP is not empty. As explained for instance in §3.2 of [BM] we can write, if h ≤ 0, V

(h)

Zh ψ

(≤h)

h+1 + M 2E

=

∞

Zh

|Pv0 |

n=1 τ ∈Th,n P∈Pτ

(≤h) (Pv0 )K (h+1) (xv0 ) , ψ τ,P

xv0

(7.68) where (≤h) (Pv ) = ψ

(≤h)

ψx(f ),ω(f )

(7.69)

f ∈Pv (j +1)

and Kτ,P (xv0 ) is a suitable function, which is obtained by summing the values of all the Feynman graphs compatible with P, see item 6) above, and applying iteratively in the vertices of the tree, different from the endpoints and v0 , the R-operation, starting from the vertices with higher scale. In order to control, uniformly in M, the various terms in (7.68) one has to exploit the Gram-Hadamard inequality (see Appendix E) and to take into account the R operation acting on the vertices of the tree, as explained in full detail in [BM], §3. The result of this analysis, which applies essentially unchanged in the present case, is the following bound (see (3.105) of [BM]), if k = i αi , (h+1) |Kτ,P (xv0 )| ≤ C n M 2 εhn γ −hDk (Pv0 ) · xv0

·

v not

e.p.

, |Pv | 2 |Pv | 1 sv |Pv |−|Pv | Zhv −[−2+ +z(P )] v 2 , γ C i=1 i sv ! Zhv −1 (7.70)

with −2 +

|Pv | 2

+ z(Pv ) > 0 and

  1 z(Pv ) = 2  0

if |Pv | = 4, if |Pv | = 2, otherwise.

(7.71)

The above bound admits a simple dimensional interpretation. If we erase the R operation from all the vertices of the tree, then z(Pv ) = 0 and (7.70) allow us to associate a |Pv | factor γ 2− 2 with any trivial or non-trivial vertex of the tree. This would allow us to control the sums over the scale labels and Pτ , provided that |Pv | were larger than 4 in all vertices, which is however not true. The effect of the R operation is to improve the |Pv | bound with the factor γ −z(Pv ) , so that there is a factor γ −[−2+ 2 +z(Pv )] smaller than 1 associated with all the vertices.

634

V. Mastropietro

In order to perform the sums note that the number of unlabeled trees is ≤ 4n ; fix an unlabeled tree, the number of terms in the sum over the various labels of the tree is bounded by C n , except the sums over the scale labels. In order to bound the sums over the scale labels and P we first use the inequality   |Pv | γ −[−2+ 2 +z(Pv )] ≤ γ −2α(hv −hv )  γ −2α|Pv |  , (7.72) v v not e.p. v not e.p. where v are the non-trivial vertices, and v is the non trivial vertex immediately preceding −2α(h −h ) v v v or the root. The factors γ in the r.h.s. of (7.72) allow us to bound the sums 1 ). over the scale labels by C n ; α is a suitable constant (one finds α = 40 Finally the sum over P can be bounded by using the following combinatorial inequalv ity, trivial for γ large enough. Let {pv , v ∈ τ } be a set of integers such that pv ≤ si=1 pvi for all v ∈ τ which are not endpoints; then pv γ − 40 ≤ C n . (7.73) p v not e.p. v It follows that

P |Pv0 |=2m

v not

γ−

|Pv | 40

≤

e.p.

v not

e.p.

pv

γ − 40 ≤ C n .

(7.74)

pv

7.10. Appendix L: The flow of running coupling constants. Choice of the counterterm ν. Let us call µh = supk≥h max{|λk |, |δk |}. Let us consider the first of Eqs. (5.1) for fixed values of ah , Zh−1 and mh−1 (k), h˜ ≤ h ≤ 1, if h˜ is a negative integer, satisfying the conditions µh ≤ ε 1 ≤ ε 0 , γ −c0 µh ≤

a0 γ h−1 ≥ 4|mh |,

mh−1 ≤ γ +c0 µh , mh

γ −c0 µh ≤ 2

Zh−1 2 ≤ γ +c0 µh Zh

(7.75)

(7.76)

for some constant c0 . We prove that, if ε¯ 0 is small enough, there exist some constants ε¯ 1 , κ, γ , c1 , B, and ¯ a family of intervals I (h) , h˜ ≤ h¯ ≤ 0, such that ε¯ 1 ≤ ε¯ 0 , 0 < κ < 1, 1 < γ < γ , ¯ ¯ ¯ ¯ ¯ I (h) ⊂ I (h+1) , |I (h) | ≤ c1 ε¯ 1 (γ )h and, if ν = ν1 ∈ I (h) , 1

¯

|νh | ≤ B ε¯ 1 [γ − 2 (h−h) + γ κh ] ≤ ε¯ 0 ,

h¯ ≤ h ≤ 1.

(7.77)

In order to show this, note that if |νh | ≤ ε 0 for h¯ ≤ h ≤ 1 and ε 0 is small enough, the r.h.s. of the first of (5.1) is well defined for h = h¯ and we can write = γ νh¯ + bh¯ + rh¯ , νh−1 ¯

(7.78)

¯

ν γ h−1 λ and r collects all terms of second or higher order in ε . In the where bh¯ = ch−1 0 h¯ h¯ ¯

tree expansion of βνh , there is no contribution from the trees with n ≥ 2 endpoints, which


635

are only of type ν or δ, because of the support properties of the single scale propagators; hence by (7.75) |rh¯ | ≤ c2 µh¯ ε0 . Let us now fix a positive constant c, consider the intervals bh bh − cε 1 , − + cε1 . J (h) = − γ −1 γ −1

(7.79)

By using (7.78) one can show by an inductive argument (see for instance §4.3 of [BM]) ¯ that there exists a decreasing family of intervals I (h) , h˜ ≤ h¯ ≤ 0, such that, if ν = ν1 ∈ ¯ ( h) I , then the sequence νh is well defined for h ≥ h¯ and satisfies the bound |νh | ≤ ε 0 . In order to prove the bound (7.77) we note that, if we iterate the first of (5.1), we can ¯ write, if h¯ ≤ h ≤ 0 and ν1 ∈ I (h) , νh = γ −h+1 ν1 +

1

γ k−2 βνk (νk , . . . , ν1 )

,

(7.80)

k=h+1

where now the functions βνk are thought of as functions of νk , . . . , ν1 only. If we put h = h¯ in (7.80), we get the following identity: ν1 = −

1

¯

γ k−2 βνk (νk , . . . , ν1 ) + γ h−1 νh¯ .

(7.81)

¯ k=h+1

Equations (7.80) and (7.81) are equivalent to νh = −γ −h

h

¯

γ k−1 βνk (νk , . . . , ν1 ) + γ −(h−h) νh¯ ,

h¯ < h ≤ 1 .

(7.82)

¯ k=h+1

By construction, see §4.4, βkν is given by the sum over trees with at least an end-point νk , k ≥ h or at least a propagator gω,−ω , see (4.16), or at least with an end-point at scale 2 to which is associated one of the terms in RV (1) . Hence, we can write βνh

= µh

1

ν h,k νk β γ −2κ(k−h) + γ κh µh Rhν ,

(7.83)

k=h

ν | ≤ C and κ is a constant. The second term in (7.83) comes from the where |Rhν |, |β h,k trees with at least a propagator gω,−ω or with an end-point at scale 2, and the first term from the trees with at least a νk end-point. The factor γ −2κ(k−h) in the r.h.s. of (7.83) follows from the simple remark that the bound over all the trees contributing to νh , which have at least one endpoint of fixed scale k > h, can be improved by a factor γ −η (k−h) , with η positive but small enough. It is sufficient to use (7.72), which allows to extract such a factor from the r.h.s. before performing the sum over the scale indices, and to choose η = 2κ, which is possible if κ is small enough. Let us now observe that the sequence νh , h¯ < h ≤ 1, satisfying (7.77) can be obtained (n) as the limit as n → ∞ of the sequence {νh }, h¯ < h ≤ 1, n ≥ 0, parameterized by ¯ νh¯ ∈ J (h+1) and defined recursively in the following way:

636

V. Mastropietro (0)

νh = 0 ,

h

νh = −γ −h (n)

(n−1)

γ k−1 βkν (νk

(n−1)

, . . . , ν1

¯

) + γ −(h−h) νh¯ ,

n≥1.

(7.84)

¯ k=h+1 (n)

In fact, by induction one verifies that, if ε1 is small enough, |νh | ≤ Cε 1 ≤ ε 0 , (n) (n−1) | ≤ (Cε 1 )n . In fact for so that (7.84) is meaningful, and maxh∗ 1 it follows by the fact that βkν (νk (n−2) (n−2) ν , . . . , ν1 ) can be written as a sum of terms in which there is at least one βk (νk (n−1) (n−2) − νh , h ≥ k, in place of the correspondendpoint of type ν, with a difference νh (n) ing running coupling, and one endpoint of type λ. Then νh converges as n → ∞, for h¯ < h ≤ 1, to a limit νh , satisfying (7.77) and the bound |νh | ≤ ε0 , if ε 1 is small enough. Hence, if ε 1 is small enough, by (7.83), |βkν | ≤ Cε 1

1

|νm |γ −2κ(m−k) + γ κk

.

(7.85)

+ γ −(h−h) } .

(7.86)

m=k

Hence (n) |νh |

≤ cε 1 {γ

−h

h k=h+1

γ

k

1

(n−1) −2κ(m−k) |νm |γ

+γ

κk

m=k

Let us now suppose that, for some constant cn−1 , 1

(n−1) | ≤ cn−1 ε 1 (γ κm + γ − 2 (m−h) ) ≤ ε 0 , |νm

(7.87)

(0)

which is true for n = 1, since νm = 0, if ε 1 is small enough. One then checks that the (n) same bound is verified by νm , if cn−1 is substituted with cn = c(1 + c4 cn−1 ε 1 ), where (n) c4 is a suitable constant. Hence, we can prove the bound (7.77) for νh = limn→∞ νh , for ε1 small enough. Proof of Lemma 3. We shall proceed by induction. The second part of (5.1) and the above analysis imply that, if λ is small enough, there exists an interval I (0) , whose size is of order λ, such that, if ν ∈ I (0) , then the bound (7.77) is satisfied, together with |λ0 − λ| ≤ C|λ|2 . Let us now suppose that the solution of (5.1) is well defined for h¯ ≤ h ≤ 0 and satisfies the conditions (7.75),(7.77), for any ν belonging to an interval ¯ I (h) . Suppose also that there exists a constant c0 , such that µh¯ ≤ c0 |λ| .

(7.88)

We want to prove that all these conditions are verified also if h¯ is substituted with h¯ − 1, if λ is small enough. The induction will be stopped as soon as the second condition in ¯ (7.75) is violated for some ν ∈ I (h) . We shall put ν equal to one of these values, so ∗ defining h as equal to h¯ + 1.


637

By using (5.5) we have ah−1 = ah¯ + βhα,L ¯ ¯ (ah¯ , . . . , ah¯ ) +

1 ¯ k=h+1

α α Dh,k ¯ + rh¯ (ah¯ , νh¯ ; . . . ; a1 , ν1 ; u) ,

(7.89)

where α = βhα,L (ah , . . . , ah , ak , ak+1 , . . . , a1 ) − βhα,L (ah , . . . , ah , ah , ak+1 , . . . , a1 ) . Dh,k (7.90) α admits a tree expansion similar to that of the On the other hand, one checks that Dh,k

functions βhα,L (ah , . . . , a1 ), with the property that all trees giving a non zero contribution must have an endpoint of scale h, associated with a difference λk − λh or δk − δh . Hence, if κ is the same constant in (7.83) and h ≤ 0, α | ≤ C|λh |γ −κ(k−h) |ak − ah | . |Dh,k

(7.91)

Let us now suppose that h¯ ≤ h ≤ 0 and that there exists a constant c0 , such that ¯

1

|ak−1 − ak | ≤ c0 |λ|3/2 [γ − 2 (k−h) + γ ϑk ] ,

h < k ≤ 0,

(7.92)

where ϑ = min{κ/2, η }. Equation(7.92) is certainly verified for k = 0, thanks to the second part of (5.1); we want to show that it is verified also if h is substituted with h − 1, if λ1 is small enough. By using (7.89), (5.6), (5.7) and (7.92), we get

¯

1

|ah−1 − ah | ≤ Cλh γ η h + C|λh |2 [γ − 2 (h−h) + γ ϑh ] 1 k 1 ∗ +Cc0 |λh |5/2 γ −κ(k−h) [γ − 2 (h −h ) + γ ϑh ] , 2

(7.93)

h =h+1

k=h+1

which immediately implies (7.92) with h → h − 1 and (7.88) with h¯ → h¯ − 1. The bound (7.93) implies also the first of (5.3). Finally the second of (5.3) follows from (5.2). Independence of ν from t − tc . We have shown that by choosing ν ∈ Ih∗ then (5.3) ∗ holds; such ν are parametrized by νh∗ ∈ J (h +1) . Assuming (7.75) and h¯ = hM , one can proceed as above to show that there exists a sequence νh , hM < h ≤ 1 such that (so that ν hM = 0) νh = −γ −h

h

γ k−1 βkν (νk , . . . , ν1 ) .

(7.94)

k=hM +1

If νh , h∗ < h ≤ 1 verify (7.82) with νh∗ = 0 it holds that ∗

|νh − νh | ≤ Cε 1 γ κh

h∗ ≤ h ≤ 1 ;

(7.95)

638

V. Mastropietro

this implies that one can choose ν = ν1 for any h∗ . Equation (7.95)) is proved by induction assuming that it holds for any k ≥ h + 1 and subtracting (7.82) with h = h∗ and νh∗ +1 = 0 from (7.94), finding νh − νh = −γ −h −γ −h

h

γ k−1 [βkν (νk , . . . , ν1 ) − βkν (νk , . . . , ν1 )]

k=h∗ +1 h∗

γ k−1 βkν (νk , . . . , ν1 ) .

(7.96)

k=hM +1

By using (7.83) and the inductive hypothesis, (7.95) follows. √ 7.11. Appendix M: Physical observables. The functionals B (h) ( Zh ψ (≤h) , φ) and S (h) (φ) defined in (6.11),(6.12) can be written in terms of a tree expansion similar to the one introduced in Appendix I. m of trees, which are We introduce, for each n ≥ 0 and each m ≥ 1, a family Th,n defined as in Appendix I, with some differences. m , the tree has n + m (instead of n) endpoints. Moreover, among 1) First of all, if τ ∈ Th,n the n + m endpoints, there are n endpoints, which we call normal endpoints, which are associated with a contribution to the effective potential on scale hv − 1. The m remaining endpoints, which we call special endpoints, are associated with a local term of the form (6.15); we shall say that they are of type Z (1) . 2) We associate with each vertex v a new integer lv ∈ [0, m], which denotes the number of special endpoints following v, i.e. contained in Lv .

In order to study the expansion of the correlation function (x, 0) ≡ (x), which follows from (6.21), we have to consider the trees with two special endpoints, whose space-points we shall denote x and y = 0; moreover, we shall denote by hx and hy the scales of the two special endpoints and by hx,y the scale of the smallest cluster containing both special endpoints. ˜ (x, y) = α (x, y) + β (x, y) is such that α (x, y) is given The decomposition 2 with endpoints v to which are associated only by the sum over trees belonging to Th,n

terms in LV (hv −1) or LB (hv −1) , and (x, y) is the sum over the remaining trees. The first two addends in (6.22) are the contribution from the trees with n = 0, while β

(1)

(

Zh Zh

(h),α

)2 G

(x) is given by the sum of trees with n ≥ 1, (h),α

G

(x) =

∞

h−1

n=1 hr =h∗ −1

(h,hr ),α

G

(x, τ, P) ,

(7.97)

τ ∈Th2 ,n,l P∈Pτ ,r r Pv0 =∅ hx,y =h

and, as proved in full detail in §5 of [BM], the following bound holds, see (5.60) of [BM], (h,hr ),α

|G

(x, τ, P)| ≤ (Cεh )n CN (2n + 1)N

γ 2h · 1 + [γ h d(x)]N


639

 ) (1) Zhy Zh 1 sv   · C i=1 |Pvi |−|Pv | (1) (1) sv ! Zhx −1 Zh Zhy −1 Zh v not e.p. |Pv |/2

−[−2+ |P2v | +lv +z(Pv ,lv )] · Zhv /Zhv −1 γ . , (

(1)

Zhx Zh

(7.98)

where z(Pv , lv ) = 1 if Pv = 4, lv = 0; z(Pv , lv ) = 2 if Pv = 2, lv = 0; z(Pv , lv ) = 1 if Pv = 2, lv = 1; z(Pv , lv ) = 0 in all other cases. We can now perform as in Appendix I the various sums in the r.h.s. of (7.97). There are some differences in the sum over the scale labels, but they can be easily treated. First of (1) (1) (1) (1) all, one has to take care of the factors (Zhx Zh )/(Zhx −1 Zh ) and (Zhy Zh )/(Zhy −1 Zh ),

with the only effect of adding to the final bound a factor γ C|λ|(hv −hv ) for each non-trivial vertex v containing one of the special endpoints and strictly following the vertex vx,y ; this has a negligible effect, thanks to a bound analogous to (7.72), valid in this case. The other difference is in the fact that, instead of fixing the scale of the root, we have now to fix the scale of vx,y . However, this has no effect, since we bound the sum over the scales with the sum over the differences hv − hv . (h)α The previous considerations are sufficient to get the bound (6.23) for G (x). An (h)β expression similar to (7.97) holds also for G (x); the extra factor γ τ h in the bound (6.24) (with respect to (6.23)) is due to the fact that the bound over all the trees which have at least one endpoint v of fixed scale hv = 2 can be improved by a factor γ τ h . It is sufficient to use (7.72), which allows to extract such a factor from the r.h.s. before performing the sum over the scale indices. (1) Note also that from (6.15), (6.17) we get (6.25) , where zh is given by (1) zh

=

∞

(1)

zh (τ, P) ,

(7.99)

n=1 τ ∈T 1 ,P∈Pτ ,Pv =(f1 ,f2 ) h,n 0

with |zh (τ, P)| ≤ C n εhn γ −h[D0 (Pv0 )+lv0 ] (1)

C

sv

i=1 |Pvi |−|Pv |

v not e.p. |Pv |/2 |Pv | 1 · γ −[−2+ 2 +lv +z(Pv ,lv )] . Zhv /Zhv −1 sv !

(7.100)

˜ (x, y) − −,−,−,−, (x, y) is given by a sum of terms in Finally note that sech4 Jr which three or four external φ fields are present. Essentially by power counting one gets a bound similar to (7.98) in which γ 2h is replaced by γ 3h or γ 4h depending if there are three or four external φ fields.

7.12. Appendix N: Perturbations of a single Ising model. If we consider the hamiltonian (1.1) with interaction given by (6.33) all the analysis in §2, §3 is still valid; the only place in which we have used the explicit form of V is in Appendix F, but the symmetry cancellations exploited there hold also in the case of V given by (6.33). The integration (≤h) = 0, of the light fermions is done exactly as in §4 but now in (4.9) and (4.20) Fλ (h) i.e. the term and quartic in the field is missing in LV ; the reason is that

640

V. Mastropietro

(≤h)+

ψx,1

(≤h)−

ψx,1

ψx,−1 ψx,−1 = ψ¯ x(≤h)(1) ψx(≤h)(1) ψ¯ x(≤h)(2) ψx(≤h)(2) , (≤h)+

(≤h)−

(7.101)

but such a term cannot be present as the (1) and (2) systems are independent. As a h , β h , β h are all O(ε γ κh ), if κ is a constant, for the same conconsequence, in (5.1) βm h z δ siderations used in Appendix L: there is no contribution from trees with only end-points of type ν or δ, because of the support properties of the single scale propagators. Hence h , β h , β h are given by a sum of trees with at least an end-point of scale h = 2 and by βm v z δ (7.72) the bound for them can be improved by a factor γ κh . Then, choosing ν properly, δh = O(λ), mh = m0 (1 + O(λ)), Zh = 1 + O(λ). For the same reasons the analysis in (1) §6 still holds but Zh = 1 + O(λ) and at the end (1.8)–(1.12) hold with η1 = η2 = 0. 7.13. Appendix O: Extensions of the main Theorem. It should be clear from the above analysis that the correlation function or the specific heat behaviour in (1.10) or (1.12) does not depend on the details of the interaction (1.3) but on a few general properties. In fact assume that V verifies the following properties. (1)

(2)

(2)

(1)

1) V is symmetric under the exchange {σx }x∈ , {σx }x∈ → {σx }x∈ , {σx }x∈ . This is true for the Ashkin-Teller Hamiltonian which is invariant under the operation (1) (2) (2) (1) σx,x0 , σx,x0 → σx,x0 , σx,x0 , and for the Eight vertex model which is invariant under (1) (2) (2) (1) σx,x0 , σx,x0 → σx,x0 , σx+1,x0 −1 for any x ∈ . 2) V is given by the sum of monomials in the spin variables each one of the form λv(x1 , .., xn )

n i=1

(α )

σx(αi i ) σx i

(7.102)

i

with αi = 1, 2 , xi , xi nearest neighbor, v(x1 , .., x2 ) short ranged and λ small. The above two properties ensure that the effective potential can be written in the form (3.1), with V given by a sum over short range monomials in the Grassmann variables ψ, χ . Moreover the analysis in Appendix F can be repeated, as the symmetries which were true in the Ashkin Teller or in the Eight vertex model are true also here, and the marginal or relevant terms in the Renormalization group analysis are the same as in the Eight vertex or Ashkin Teller models. Note that the interaction in the AshkinTeller or the Eight-vertex model verify an extra symmetry, namely a symmetry in the exchange x, x0 → x0 , x; such extra symmetry is however not used in our analysis. Finally: 3) V is such that in V (3.1) there is a non vanishing local term of the form + − + − [cλ + O(λ2 )]ψ1,x ψ1,x ψ−1,x ψ−1,x

(7.103)

with c = 0 a constant. If such conditions are verified, then a statement identical to the main Theorem follows. Acknowledgements. This paper was partly written in the stimulating atmosphere of the Institute for Advanced Studies, in Princeton. I am indebted to Prof. Spencer for his invitation and for many clarifying discussions about his work [PS]. I thank G.Benfatto, G.Gallavotti and A. Giuliani for many important remarks and suggestions.


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[Su]

Sutherland, S.B.: Two-Dimensional Hydrogen Bonded Crystals. J. Math. Phys. 11, 3183–3186 (1970) Spencer, T.: A mathematical approach to universality in two dimensions. Physica A 279, 250– 259 (2000) Schultz, T., Mattis, D., Lieb, E.: Two-dimensional Ising model as a soluble problem of many Fermions. Rev. Mod. Phys. 36, 856 (1964) Wu, F.W.: The Ising model with four spin interaction. Phys. Rev. B 4, 2312–2314 (1971)

[Spe] [SML] [W]

Communicated by G. Gallavotti

Communications in Mathematical Physics - Volume 244

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Communications in Mathematical Physics - Volume 244

Communications in Mathematical Physics - Volume 221

Communications in Mathematical Physics - Volume 220

Communications in Mathematical Physics - Volume 235

Communications in Mathematical Physics - Volume 223

Communications In Mathematical Physics - Volume 283

Communications In Mathematical Physics - Volume 270

Communications in Mathematical Physics - Volume 208

Communications in Mathematical Physics - Volume 186

Communications In Mathematical Physics - Volume 294

Communications in Mathematical Physics - Volume 217

Communications In Mathematical Physics - Volume 274

Communications in Mathematical Physics - Volume 239

Communications in Mathematical Physics - Volume 306

Communications in Mathematical Physics - Volume 264

Communications in Mathematical Physics - Volume 227

Communications in Mathematical Physics - Volume 184

Communications in Mathematical Physics - Volume 261

Communications in Mathematical Physics - Volume 225

Communications In Mathematical Physics - Volume 263

Communications in Mathematical Physics - Volume 211

Communications In Mathematical Physics - Volume 293

Communications in Mathematical Physics - Volume 246

Communications In Mathematical Physics - Volume 298

Communications in Mathematical Physics - Volume 234

Communications In Mathematical Physics - Volume 288

Communications in Mathematical Physics - Volume 304

Communications In Mathematical Physics - Volume 292

Communications in Mathematical Physics - Volume 233

Communications in Mathematical Physics - Volume 253

Communications in Mathematical Physics - Volume 222

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