Reviews in Mathematical Physics - Volume 13

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February 20, 2001 19:1 WSPC/148-RMP

S0129055X01000582

Reviews in Mathematical Physics, Vol. 13, No. 1 (2001) 1–28 c World Scientific Publishing Company

CONNES LOTT MODEL BUILDING ON THE TWO-SPHERE

J. A. MIGNACO∗ , C. SIGAUD and F. J. VANHECKE Instituto de F´ısica, UFRJ, Ilha do Fund˜ ao, Rio de Janeiro, Brasil E-mail : {mignaco,sigaud,vanhecke}@if.ufrj.br A. R. DA SILVA Instituto de Matem´ atica, UFRJ, Ilha do Fund˜ ao, Rio de Janeiro, Brasil E-mail : [email protected]

Received 26 May 1999 In this work we examine generalized Connes–Lott models, with C ⊕ C as finite algebra, over the two-sphere. The Hilbert space of the continuum spectral triple is taken as the space of sections of a twisted spinor bundle, allowing for nontrivial topological structure (magnetic monopoles). The finitely generated projective module over the full algebra is also taken as topologically non-trivial, which is possible over S 2 . We also construct a real spectral triple enlarging this Hilbert space to include “particle” and “anti-particle” fields. Keywords: Noncommutative geometry, Connes–Lott model, Real spectral triples. PACS numbers: Primary 20C15, 20C40

1. Introduction In previous work [15], we studied the Connes–Lott program with the complex algebra A1 = C(S 2 ; C) of continuous complex-valued functions on the sphere. The Hilbert space HI , on which this algebra was represented, consisted of one of the minimal left ideals I± of the algebra of sections of the Clifford bundle over S 2 with a standard scalar product. On this Hilbert space the Dirac operator was taken as DI = i(d − δ) restricted to each of the ideals I± . Using the Bott projector P = 12 (1 + n · σ), acting on the free module A1 ⊕ A1 , we constructed projective modules M over A1 . These modules are classified by the homotopy classes of the mappings n : S 2 → S 2 i.e. by π2 (S 2 ) = Z. In Dirac’s interpretation, each integer corresponds to a magnetic monopole at the center of the sphere with magnetic charge g quantised by eg/4π = (n/2)~. In the present paper we extend the study of topologically non trivial aspects in noncommutative geometry, to the product algebra A = A1 ⊗A2 , where A2 = C⊕C. ∗ Partially

supported by CNPq, Brasil. 1


2

S0129055X01000582

J. A. Mignaco et al.

It is clear that A ' C(S 2 × {a, b}; C), where {a, b} denotes a two-point space, as in the original Connes–Lott paper [5]. In Sec. 2 we construct the Hilbert space on which A1 is represented. This Hilbert space H(s) generalizes HI above, and is made of sections of what we call a Pensov spinor bundle of (integer or semi-integer) weight s, using a taxonomy introduced by Staruszkiewicz [19]. A generalized Dirac operator D(s) acting on these Pensov spinors is defined and, for s = ±1/2, we recover the K¨ ahler spinors of [15], while for s = 0 the usual Dirac spinors are obtained. These spinors may actually be identified with sections of twisted spinor bundles or, from a more physical viewpoint, as usual Dirac spinors interacting with a magnetic monopole of charge g given by eg/4π~ = s. The projective modules over A, following Connes–Lott, are constructed in Sec. 3 as M = P(A ⊕ A) where P = (Pa = 1, Pb = 12 (1 + n · σ)). In Connes’ work [2], the smooth manifold is four-dimensional so that, taking the four-sphere S 4 as an example, we get mappings n : S 4 → S 2 classified by π4 (S 2 ) = Z2 . However, the local unitary transformations acting as Pb → U † Pb U are also classified by homotopy classes π4 (U (2)) = π4 (SU (2)) = π4 (S 3 ) = Z2 . It follows thata all Bott projectors define modules isomorphic to the module obtained from Pb = 12 (1 + σ3 ), considered by Connes. In our case, considering the two-sphere, this does not happen since π2 (S 2 ) = Z and π2 (U (2)) = π2 (SU (2)) = π2 (S 3 ) = {1}. In Sec. 4 we sketch the construction of the full spectral triple and display the resulting Yang–Mills–Higgs action and covariant Dirac operator. For more details on this construction, we refer to [10, 22, 23].b The main new features in this action, as compared with Connes’ results, are the appearance of an additional monopole potential of strength eg/4π = (n/2)~, where n is the integer characterizing the homotopy class of Pb , and the fact that the Higgs doublet is not globally defined on S 2 but transforms as a Pensov field of weight ±n/2. Concerning the covariant N Dirac operator D∇ acting on the “particle sector” Hp = M A H, the novelty is that, whilst the “a-doublet” remains a doublet of Pensov spinors of weight s, the “b-singlet” metamorphoses in a Pensov spinor of weight s + n/2. If one insists on a comparison with the standard electroweak model on S 2 , this means that righthanded electrons see a different magnetic monopole than the left-handed and this is not really welcome. We also make some comments on the Euclidean chirality problem which consists in the identically vanishing of the matter Lagrangian when a pure chiral theory is aimed for. In Sec. 5 we introduce a real Dirac–Pensov spectral triple by doubling the Hilbert space as H1 = H(s) ⊕ H(−s) . It is seen that, with the same Hdis as before, it is not possible to define a real structure. However, a more general discrete Hilbert space H2 = CNaa ⊕ CNab ⊕ CNba ⊕ CNbb , as considered in [11, 18], allows for the construction of a real structure on Hnew = H1 ⊗ H2 . The covariant Dirac operator a We bA

are indebted to prof. Balachandran of Syracuse University for discussions on this point. lengthy but comprehensive calculation is available at hep-th/9904171.


S0129055X01000582

Connes–Lott Model Building on the Two-Sphere

3

N N on M A Hnew A M∗ can also be defined and it is furthermore seen that, with the use of such a non trivial projective module, the Abelian gauge fields are not slain, as they are when M = A [22]. Clearly this model building led us far from a toy electroweak model. The main purpose however is not to reproduce such a model on the two-sphere, but rather to examine some of the topologically nontrivial structures in model building with the simplest manifold allowing for such possibilities. 2. The Hilbert Space of Pensov Spinors on S 2 The standard atlas of the two-sphere S 2 = {(x, y, z) ∈ R3 |x2 + y 2 + z 2 = 1} consists of two charts, the boreal, HB = {(x, y, z) ∈ S 2 | − 1 < z ≤ +1}, and austral chart, HA = {(x, y, z) ∈ S 2 | − 1 ≤ z < +1}, with coordinates 1 2 + iξB =+ ζB = ξB

x + iy 1+z

in HB ,

1 2 + iξA =− ζA = ξA

x − iy 1−z

in HA .

In the overlap HB ∩ HA , they are related by ζA ζB = −1 and the usual spherical coordinates (θ, ϕ), given by ζB = −1/ζA = tan θ/2 exp iϕ, are nonsingular. In each chart, dual coordinate bases of the complexified tangent and cotangent spaces are 1 ∂ ∂ ∂ ∂ 1 ∂ ∂ ∗ = −i 2 , ∂ = ∗ = +i 2 , ∂= ∂ζ 2 ∂ξ 1 ∂ξ ∂ζ 2 ∂ξ 1 ∂ξ {dζ = dξ 1 + idξ 2 ,

dζ ∗ = dξ 1 − idξ 2 } .

In HB ∩ HA they are related by ! dζB ∗ dζB

=

ζB 2

0

0

∗2 ζB

!

dζA ∗ dζA

! .

Real and complex Zweibein fields are given by 2 i 2 2 ∗ i ∗ θ = dζ, θ = dζ , θ = dξ ; i = 1, 2 , q q q and

ei =

q ∂ ; i = 1, 2 2 ∂ξ i

,

q ∂ q ∂ , e∗ = e= , 2 ∂ζ 2 ∂ζ ∗

where q = 1 + |ζ|2 . The Euclidean metric in R3 induces the standard metric on the sphere g=

2 4 δij dξ i ⊗ dξ j = 2 (dζ ∗ ⊗ dζ + dζ ⊗ dζ ∗ ) q2 q

= δij θi ⊗ θj =

1 ∗ (θ ⊗ θ + θ ⊗ θ∗ ) . 2

(2.1)


4

S0129055X01000582


A rotation of the real Zweibein by an angle α ! ! ! θ˜1 cos α sin α θ1 ⇒ = θ2 − sin α cos α θ˜2 becomes diagonal for the complex Zweibein ! ! θ˜ exp(−iα) θ ⇒ = ∗ ∗ ˜ 0 θ θ

θ1 θ2

!

0 exp(iα)

!

θ θ∗

! .

(2.2)

This means that the complexified cotangent bundle splits, in an SO(2) invariant way, into the direct sum of two line bundles ^

(1,0)

(T ∗ S 2 )C =

^

(0,1)

(T ∗ S 2 ) ⊕

(T ∗ S 2 )

with one-dimensional local bases of sections given by {θ} and {θ∗ }. In the overlap HB ∩ HA , the Zweibein in HA and in HB are related by θA = (cAB )−1 θB ,

∗ ∗ θA = cAB θB ,

(2.3)

∗ ∗ = ζA /ζA = exp(2iϕ), ϕ being the with the transition function cAB = ζB /ζB azimuthal angle, well defined (modulo 2π) in HB ∩ HA . V(0,1) ∗ 2 V(1,0) ∗ 2 (T S ) and (T S ) are written as Σ(+1) = σ (+1) θ and Sections of (−1) (−1) ∗ = σ θ such that, in the overlap HB ∩ HA , σ (±1) |A = (cAB )±1 σ (±1) |B . Σ Following Staruszkiewicz, who refers to Pensov [19], we call such a field a Pensov scalar of weight (±1). An Hermitian structure on these line bundles is introduced by Z σ (±1)∗ τ (±1) ω , (Σ(±1) , T (±1) )±1 = S2

where ω = θ1 ∧ θ2 = (1/2i)θ∗ ∧ θ is the invariant volume element on S 2 . The question is now addressed to define Pensov scalars of weight s on S 2 . In general this would require a cocycle condition on transition functions in triple overlaps. However, since the sphere is covered by only two charts, it is enough that the overlap equation σ (s) |A = (cAB )s σ (s) |B be well defined. Now, (cAB )s = exp(2isϕ) is well defined when 2s takes integer values, the corresponding line bundle and also its space of sections will be denoted by P (s) . Actually we have reproduced in a way, rather pedestrian but more accessible to physicists, the well known fact that all line bundles over CP (1) = S 2 are tensor products of the tautological line bundle or of its dual, P (+1/2) and P (−1/2) in our notation. The integer 2s is identified with the integer representing an element of the ˇ 2 (S 2 , Z) = Z, classifying ˇ second Cech cohomology group of S 2 with integer values, H 2 the line bundles over the sphere S . The Hermitian structure on P (s) is given by Z (s) (s) σ (s)∗ τ (s) ω . (2.4) (Σ , T )s = S2


S0129055X01000582


5

On spheres with round metric g = δij θi ⊗ θj , the Levi–Civita connection is given by the one-form Γ = 12 Γ[ij] E[ij] , where Γ[ij] = ξ i θj − ξ j θi , which takes values in the Lie algebra so(N ) whose basis is denoted by {E[ij] }. The covariant derivative is then obtained as ∇LC θi = −Γk` ⊗ θ` with the vector representation (E[ij] )k` = δik δj` − δjk δi` . For the complexified Zweibein (2.1), with Γ = −Γ∗ = − 12 {ζ ∗ θ − ζθ∗ }, we may write ∇LC θ = −Γ ⊗ θ ,

∇LC θ∗ = −Γ∗ ⊗ θ∗ .

(2.5)

Let Θ(s) be a basis of the sections of P (s) , then it is easy to check that ∇LC Θ(s) = −sΓ ⊗ Θ(s) defines a connection generalising (2.5) above. This connection maps P (s) in (T ∗ (S 2 ))C ⊗ P (s) . Now the space of complex-valued one-forms (T ∗ (S 2 ))C is isomorphic to P (+1) ⊕ P (−1) , so that ∇LC is actually a mapping ∇LC : P (s) 7→ P (s+1) ⊕ P (s−1) : Σ(s) 7→ ∇LC Σ(s) = (dσ (s) − sΓσ (s) ) ⊗ Θ(s) . Projecting ∇LC Σ(s) on each term in the sum P (s+1) ⊕ P (s−1) we obtain ∇LC Σ(s) =

1 ˇ (s) (s+1) ˇ† (s) (s−1) (/δ σ Θ + /δ s σ Θ ), 2 s

where the “edth” operators of Newman and Penrose [16] are given by ∂q ∂σ (s) ∂ + s σ (s) , /δˇs σ (s) = q −s+1 (q s σ (s) ) = q ∂ζ ∂ζ ∂ζ ∂σ (s) ∂q ∂ † − s ∗ σ (s) . /δˇs σ (s) = q s+1 ∗ (q −s σ (s) ) = q ∂ζ ∂ζ ∗ ∂ζ

(2.6)

V(1,0) † ⊗ P (s) ' P (s+1) , /δˇs : P (s) → These edth operators are mappings /δˇs : P (s) → V(0,1) ⊗ P (s) ' P (s−1) and may be viewed as a covariant form of the Dolbeault differentials ∂ and ∂¯ on the Riemann sphere. With respect to the scalar product of Pensov scalars (2.4), they are formally anti-adjoint †

(σ (s+1) , /δˇs τ (s) )s+1 = (−/δˇs+1 σ (s+1) , τ (s) )s .

(2.7)

A Dirac spinor on S 2 can be defined as a section of the spinor bundle, which is V the Whitney sum P (−1/2) ⊕ ( (1,0) ⊗ P (−1/2) ), and the Dirac operator is given by † D(0) = −i(/δˇ(−1/2) ⊕ /δˇ(+1/2) ). V(1,0) ⊗ P (s−1/2) ) = Sections Ψ(s) of the twisted spinor bundles P (s−1/2) ⊕ ( (s−1/2) (s+1/2) ⊕P , baptized Pensov spinors, are acted on by the twisted Dirac P operator locally expressed as   ! ! † 0 /δˇs+1/2 σ (s−1/2) σ (s−1/2)   = −i . (2.8) D(s) σ (s+1/2) σ (s+1/2) 0 /δˇs−1/2


6

S0129055X01000582


With the complex representation of the real Clifford algebrac C`(2, 0) ! ! 0 1 0 i , γ2 ⇒ , γ1 ⇒ 1 0 −i 0 (s−1/2) acting on ψ(s) = σσ(s+1/2) , the Dirac operator can also be written as D(s) ψ(s) = −iγ k ∇LC k,(s) ψ(s) ,

(2.9)

(2.10)

where the covariant derivative of the spinor ψ(s) is given by ∇LC k,(s) ψ(s) =

q ∂ψ(s) 1 (s) [ij] + Σ[ij] Γk ψ(s) . 2 ∂ξk 2

(s) 0 reduces to 14 [γ1 , γ2 ] for s = 0. Here, Σ[12] = is 10 01 + −i/2 0 +i/2 The transformation law for s-Pensov spinorsd under a local Zweibein rotation (2.2) is given by ! ! ! σ (s−1/2) σ 0(s−1/2) σ (s−1/2) (s) 7→ = exp{αΣ[12] } , σ (s+1/2) σ 0(s+1/2) σ (s+1/2) where (s) exp{αΣ[12] }

= exp{isα}

exp(−iα/2)

0

0

exp(+iα/2)

! .

The Clifford action of γ3 = iω yields a grading on the Pensov spinors ! ! ! ψ (s) 1 0 ψ (s−1/2) = , (γ3 )2 = 1 , γ3 ψ (s+1) ψ (s+1/2) 0 −1 such that the Dirac operator (2.8) is odd D(s) γ3 + γ3 D(s) = 0 . According to (2.4), the scalar product of two Pensov spinors is defined as hΦ(s) |Ψ(s) i = (Σ(s−1/2) , T (s−1/2) )s−1/2 + (Σ(s+1/2) , T (s+1/2) )s+1/2 .

(2.11)

†

The adjointness (2.7) of −i/δˇs−1/2 and −i/δˇs+1/2 implies that the Dirac operator is formally self-adjoint with respect to this scalar product. After completion, P (s−1/2) ⊕ P (s+1/2) becomes a bona fide Hilbert space H(s) on which D(s) acts as a self-adjoint (unbounded) operator. It can be completely diagonalized with specp trum given by ± (j + 1/2)2 − s2 , where j = |s|, |s| + 1, . . . . The corresponding real Clifford algebra C`(p, q) is defined by γ k γ ` + γ ` γ k = 2ηk` , where the flat metric tensor ηk` is diagonal with p times +1 and q times −1. This entails some differences with other work using the Clifford algebra C`(0, n) for Riemannian manifolds instead of C`(n, 0) used here. d A Pensov spinor of weight s can be interpreted as a usual Dirac spinor on S 2 , interacting with a Dirac monopole of strength s. Indeed, in the expression of the covariant derivative, the term . s k 12 ∗ isγ k Γ12 k is the Clifford representative of the one-form (potential) µs = isθ Γk = 1+|ζ|2 (ζ dζ − ∗ ζdζ ), which, in {HB ; cos θ 6= +1}, takes the usual form µs |B = is(1 − cos θ)dφ. c The


S0129055X01000582


7

eigenspinors can be constructed in terms of the monopole harmonics of Wu and Yang [24]. In particular it follows that, for Dirac spinors i.e. s = 0, and only in this case, there are no zero eigenvalues. When s 6= 0, zero is an eigenvalue, 2|s| times 0 D− degenerate. The index of the Dirac operator D(s) = + (s) is then obtained as D(s)

0

+ − )) − dim(ker(D(s) )) = 2|s| . ind(D(s) ) = dim(ker(D(s)

Naturally, the index formula follows directly from Hodge’s theoreme identifying the kernel of the Dirac operator with the cohomology space over S 2 with values in the sheaf of sections of the line bundle P (s−1/2) and the index with its Euler number. 3. Projective Modules over A Through the Gel’fand-Na˘ımark construction, the topology of M = S 2 × {a, b} is encoded in the complex C ∗ -algebra of continuous complex-valued functions on M = S 2 × {a, b}. However, in order to get a fruitful use of a differential structure, we have to restrict this C ∗ -algebra to its dense subalgebra of smooth functions. This proviso made, let {f, g, . . .} denote elements of A = C(M ) and let the value of f at a pointf p = {x, α} ∈ M , be written as f (p) = fα (x). The modules of interest will be the free right A-module of rank two, identified with A2 , and P its projective submodules. The vectors of A2 are of the form X = i=1,2 Ei f i , P∞ where f i ∈ A and {Ei ; i = 1, 2} is a basis of A2 . Let Ω• (A) = k=0 Ω(k) (A) denote the universal differential envelope of A. Elements of Ω(k) (A) can be realized, see e.g. [7], as functions on the Cartesian product of (k + 1) copies of M , vanishing on neighbouring diagonals, i.e. F (p0 , p1 , . . . , pk ) = 0 if, for some i, pi = pi+1 . The product in Ω• (A) is obtained by concatenation, e.g. if F ∈ Ω(1) (A) and G ∈ Ω(2) (A) then their product F · G ∈ Ω(3) (A) is represented by (F · G)(p0 , p1 , p2 , p3 ) = F (p0 , p1 )G(p1 , p2 , p3 ) . The differential d acts on f ∈ Ω(0) (A) and on F ∈ Ω(1) (A) as follows: (df )(p0 , p1 ) = f (p1 ) − f (p0 ) , (dF )(p0 , p1 , p2 ) = F (p1 , p2 ) − F (p0 , p2 ) + F (p0 , p1 ) . The involution,g defined in A by (f † )(p) = (f (p))∗ , extends to Ω(k) (A) as (F † )(p1 , p2 , . . .) = (F (. . . , p2 , p1 ))∗ . e see

e.g. Berline et al. [1] Chap. 3.6. this section points of S 2 are denoted by x, y, . . ., while α, β, . . . will assume values in the twopoint space {a, b}. Points of M = S 2 × {a, b} are thus written as p = {x, α}, q = {y, β}, etc. and the value of a function F at (p, q, . . .) will also be expressed as Fα,β,... (x, y, . . .). g Note that d(f † ) = −(df )† , f ∈ A and, more generally, if F ∈ Ω(k) (A), then d(F † ) = (−1)k+1 (dF )† . f In


8

S0129055X01000582


A (universal) connection on A2 is given, in the basis {Ei ; i = 1, 2}, by an Ω(1) i (A)-valued 2 × 2 matrix ((ω))k . It acts on X = Ei f i as O (df i + ((ω))ik f k ) . (3.1) ∇free (X) = Ei A

Let X = Ei f and Y = Ei g be two vectors of A2 , then the standard Hermitian product with values in A is given by X X (f i )† δij g j = (f i )† g i . h(X, Y ) = i

i

i,j •

2

It extends as Ω (A)-valued on (A

N A

i •

Ω (A)) × (A2

N A

Ω• (A)) as

h(X ⊗ F, Y ⊗ G) = F † h(X, Y )G . The connection is Hermitian if d(h(X, Y )) = h(X, ∇free Y ) − h(∇free X, Y ). `†

¯

i

For the product above, this yields ((ω))j = δ ik ((ω))k δ`j ¯ or ¯

((ω))j,αβ (x, y) = δ ik (((ω))k,βα (y, x))∗ δ`j ¯ . i

`

(3.2)

The action of the connection (3.1) is represented by i

(∇free X)iαβ (x, y) = fβi (y) − fαi (x) + ((ω))k,αβ (x, y)fβk (y) . A finitely generated projective module M, submodule of A2 , can be defined by an idempotent and Hermitian endomorphism P of A2 , i.e. P2 = P and P† = P, where the adjoint E† of an endomorphism E is defined by h(X, E† Y ) = h(EX, Y ). The projector is given by a 2 × 2 matrix ((P ))ij with entries in A. It is represented by i

((P ))j,α (x). M is then the image of P. The hermiticity of the projector guarantees that h, restricted to M, defines a Hermitian product in M. In the Connes–Lott model [5], the projectors are of the form i

((P ))j,a (x) = δji

i

and ((P ))j,b (x) =

1 (1 + n(x) · σ)ij , 2

where σ are the Pauli matrices and n(x) is a real unit vector, mapping S 2 7→ S 2 so that the projectors are classified by π2 (S 2 ) = Z. Furthermore, since π2 (U (2)) = π2 (SU (2)) = π2 (S 3 ) = {1}, projectors, belonging to different homotopy classes, cannot be unitarily equivalent. target target and HA . In The target sphere S 2 also has two coordinate charts HB B these charts, the projector ((Pb )) can be written as ((Pb )) = |νB ihνB |, respectively ((PbA )) = |νA ihνA |, where νB , respectively νA , is the complex coordinate of n in target target , respectively HA . Here we have used the Dirac ket- and bra-notation: HB 1 1 1 ∗ , |νB i = p , hνB | = p 1 νB 2 2 νB 1 + |νB | 1 + |νB | 1 1 −νA ∗ , hνA | = p |νA i = p 1 . −νA 2 2 1 1 + |νA | 1 + |νA |


S0129055X01000582


9

a (x)i An element X of A2 is represented by the column matrix X ⇒ |f , where |fb (x)i fb1 (x) fa1 (x) |fa (x)i = f 2 (x) and |fb (x)i = f 2 (x) . It belongs to M if PX = X, which yields a

b

target it is expressed as no restriction on |fa (x)i but there is one on |fb (x)i. In HB B |fb i = |νB ifb , where

1 ∗ 2 (fb1 + νB fb ) . fbB = hνB |fb i = p 1 + |νB |2 target one has |fb i = |νA ifbA , where In the same way, in HA

1 ∗ 1 (−νA fb + fb2 ) . fbA = hνA |fb i = p 2 1 + |νA | As representatives of the homotopy class [n] ∈ π2 (S 2 ) ≡ Z, we choose a mapping n target target , respectively HA transforming HB , respectively HA of the range S 2 , into HB of the target S 2 . Such a choice ish n−1 n−1 2 2 ζB ζA νB (x) = ζ , ν (x) = ζA ; n ∈ Z , (3.3) B A ∗ ∗ ζB ζA where ζB , ζA are the complex coordinates of x ∈ S 2 . In the overlap HA ∩ HB , with transition function cAB given by (2.3), we have fbA (x) = (cAB (x))n/2 fbB (x) and this tells us that fb (x) is a Pensov scalar of weight n/2. This means that on the b-copy of the sphere S 2 we again have a twisted line bundle. Here it is characterized ˇ 2 (S 2 , Z) which classifies the line bundles over S 2 by their by π2 (S 2 ) isomorphic to H first Chern class. In the rest of this paper we shall omit the A and B labels except when relating quantities in HA with those in HB in the overlap HA ∩ HB . |fa i |ga i and Y ⇒ |νig with scalar Elements of M are thus represented by X ⇒ |νif b b product (hP (X, Y ))a (x) = (fa1 (x))∗ ga1 (x) + (fa2 (x))∗ ga2 (x) , (hP (X, Y ))b (x) = (fb (x))∗ gb (x) . Under an active gauge transformation X 7→ UX: |(UX)a (x)i = ((Ua (x)))|fa (x)i , |(UX)b (x)i = |ν(x)iub (x)|fb (x) ,

(3.4)

where ((Ua (x))) ∈ U (2) and ub (x) ∈ U (1). The induced connection in M is given by ∇X = P∇free X and reads: |(∇X)αβ (x, y)i = ((Pα (x)))(|fβ (y)i − |fα (x)i) + ((Aαβ (x, y)))|fβ (y)i . h Note

that this choice is different from the one in previous work [15].

(3.5)


10

S0129055X01000582


The 2 × 2 matrices ((Aαβ (x, y))) are given by ((Aaa (x, y))) = ((ωaa (x, y))) , ((Aab (x, y))) = |Φab (x, y)ihν(y)| , ((Aba (x, y))) = |ν(x)ihΦba (x, y)| , ((Abb (x, y))) = |ν(x)iωb (x, y)hν(y)| ,

(3.6)

where we have introduced the Ω(1) (A)-valued ket- and bra-vectors |Φab (x, y)i = ((ωab (x, y)))|ν(y)i , hΦba (x, y)| = hν(x)|((ωba (x, y))) ,

(3.7)

and the universal one-form ωb (x, y) = hν(x)|((ωbb (x, y)))|ν(y)i .

(3.8)

The hermiticity condition (3.2) yields; +

((ωaa (x, y))) = ((ωaa (y, x))) , (ωb (x, y))∗ = ωb (y, x) , |Φab (x, y)i+ = hΦba (y, x)| .

(3.9)

In HB ∩ HA , B −n/2 , |ΦA ab (x, y)i = |Φab (x, y)i(cAB (y)) n/2 hΦB hΦA ba (x, y)| = (cAB (x)) ba (x, y)| ,

ωbA (x, y) = (cAB (x))n/2 ωbB (x, y)(cAB (y))−n/2 .

(3.10)

The action of ∇ on X ∈ M is obtained using (3.6)–(3.8): |(∇X)aa (x, y)i = |fa (y)i − |fa (x)i + ((ωaa (x, y)))|fa (y)i , |(∇X)ab (x, y)i = |Hab (x, y)ifb (y) − |fa (x)i , |(∇X)ba (x, y)i = |ν(x)i[hHba (x, y)|fa (y)i − fb (x)] , |(∇X)bb (x, y)i = |ν(x)i[fb (y) − fb (x) + (ωb (x, y) + mb (x, y))fb (y)] ,

(3.11)

where |Hab (x, y)i = |Φab (x, y)i + |ν(y)i , hHba (x, y)| = hΦba (x, y)| + hν(x)| .

(3.12)

and the “monopole” connection mb (x, y) appears as mb (x, y) = hν(x)|ν(y)i − 1 .

(3.13)


S0129055X01000582


11

As seen from (3.10) and (3.12), the off-diagonal connections |Hab (x, y)i, hHba (x, y)| and also ωb (x, y) transform homogeneously from HB to HA but mb (x, y) transforms with the expected inhomogeneous term n/2 B mb (x, y)(cAB (y))−n/2 mA b (x, y) = (cAB (x))

+ (cAB (x))n/2 [(cAB (y))−n/2 − (cAB (x))−n/2 ] .

(3.14)

The curvature of the connection is defined by: |∇2 Xi = ((R))|Xi. It is a rightN module homomorphism M → M A Ω(2) (A) given in the basis {Ei } by the 2 × 2 matrix with values in Ω(2) (A) 2

((R)) = ((P ))(d((A)))((P )) + ((A)) + ((P ))(d((P )))(d((P )))((P )) , or, within the used realisation, by ((Rαβγ (x, y, z))) = ((Pα (x)))(((Aβγ (y, z))) − ((Aαγ (x, z))) + ((Aαβ (x, y))))((Pβ (z))) + ((Aαβ (x, y)))((Aβγ (y, z))) + ((Pα (x)))(((Pβ (y))) − ((Pα (x))))(((Pγ (z))) − ((Pβ (y))))((Pγ (z))) .

(3.15)

A connection ∇ compatible with the Hermitian structure in M implies in a selfadjoint curvature ¯

†

((R))ij = δ i` ((R))k` δkj ¯ .

(3.16)

The active gauge transformation (3.4) acts on the right on the space of connections as ∇ 7→ ∇U = U−1 ◦ ∇ ◦ U. The action of ∇U on X is given by a similar expression as in (3.5) with the matrices ((A)) replaced by ((AU )) = ((U))−1 ((A))((U)) + ((P ))((U−1 ))(d((U)))((P )) which is explicitly given by U (x, y))) = ((Ua (x)))−1 ((ωaa (x, y)))((Ua (y))) ((ωaa

+ (((Ua (x))) U (x, y)i = ((Ua (x))) |Hab

−1

−1

(((Ua (y))) − ((Ua (x)))) ,

|Hab (x, y)iub (y) ,

U (x, y)| = (ub (x))−1 hHba (x, y)|((Ua (y))) , hHba −1 mb (x, y)ub (y) mU b (x, y) = (ub (x))

ωbU (x, y) = (ub (x))−1 ωb (x, y)ub (y) + (ub (x))−1 (ub (y) − ub (x)) .

(3.17)

It is thus seen that |Hab (x, y)i, hHba (x, y)| and the monopole connection (3.13) mb (x, y) transform homogeneously under an active gauge transformation, while ((ωaa (x, y))) and ωb (x, y) have the expected inhomogeneous terms U−1 dU, u−1 du.


12

S0129055X01000582


4. The Spectral Triple {A, H, D, Γ} Let us briefly sketch the construction of the full spectral triple. It is obtained as the product of the Dirac–Pensov triple for the algebra A1 with a discrete spectral triple for the algebra A2 . Following again Connes’ prescription a` la lettre, we take the discrete Hilbert space as Hdis = CNa ⊕ CNb with chirality χdis given as +1 on the a-sector and −1 on the b-sector. The discrete Dirac operator, odd with respect to the grading defined by χdis , is given by ! 0 M+ , Ddis = M 0 where M is a Nb × Na matrix describing the phenomenology of the masses. The tensor product of A1 with A2 yields A = C(S 2 × {a, b}; C) and the Hilbert space is H = H(s) ⊗ Hdis . The total Dirac operator D = D(s) ⊗ 1a+b + γ3 ⊗ Ddis acts on H as ! ! ! D(s) ⊗ 1a γ3 ⊗ M + ψ(s),a ψ(s),a = . D ψ(s),b γ3 ⊗ M D(s) ⊗ 1b ψ(s),b It is odd with respect to the grading Γ = γ3 ⊗ χdis . Eliminating the so-called “junk” in the induced representation of the universal differential envelope Ω• (A), yields bounded operators Ω•D (A) in H. The standard use of the Dixmier trace and of Connes’ trace theorem allows then (see [5, 10, 13, 23]) to define a scalar product of operators in Ω•D (A), which is then used to construct the Yang–Mills–Higgs action. The universal connection in M, given by the matrices ((Aαβ (x, y))) of (3.6), is (1) represented by an operator in ΩD (A):   (1) (0) σab γ3 ⊗ M + −ic(σa ) ⊗ 1a , π(F ) =  (0) (1) σba γ3 ⊗ M −ic(σb ) ⊗ 1b where the σ (k) ’s are matrix-valued differential k-forms given by σa(1) (x) = ((αa (x))) ≡ (ek,y ((ωaa (x, y))))|y=x θxk , (1)

σb (x) = αb (x)((Pb (x))) ≡ (ek,y ωb (x, y))|y=x θxk ((Pb (x))) , (0)

σab (x) = ((|Φab (x, x)ihν(x)|)) , (0)

σba (x) = ((|ν(x)ihΦba (x, x)|)) ,

(4.1)

and c(σ (k) ) denotes the Clifford representation of the k-form σ (k) : c(σi1 ···ik θi1 ∧ · · · ∧ θik ) = σi1 ···ik γ i1 · · · γ ik . The monopole connection (3.13) also yields a differential one-form µb (x) = (ek,y mb (x, y))|y=x θxk = hν(x)|(d|ν(x)i) , =

1/2 (ν(x)∗ dν(x) − ν(x)dν(x)∗ ) . 1 + |ν(x)|2

(4.2)


S0129055X01000582


13

It is also convenient to introduce the Higgs field doublets |ηab (x)i = |Hab (x, x)i = |Φab (x, x)i + |ν(x)i , hηba (x)| = hHab (x, x)| = hΦba (x, x)| + hν(x)| .

(4.3)

From (3.17) it follows that, under an active gauge transformation, the differential forms (4.1) and (4.2) behave as −1

((αU a )) = ((Ua ))

((αa ))((Ua )) + ((Ua ))

−1

d((Ua )) ,

−1 αb (ub ) + (ub )−1 dub = αb + (ub )−1 dub , αU b = (ub ) −1 µb (ub ) = µb , µU b = (ub ) −1

U i = ((Ua )) |ηab

U |ηab iub , hηba | = (ub )−1 hηba |((Ua )) .

On the other hand, under a passive gauge transformation HB → HA , according to (3.10) and (3.14), they transform as B A B ((αA a )) = ((αa )), αb = αb , A B A B i = |ηab i(cAB )−n/2 , hηba | = (cAB )+n/2 hηba |, |ηab B +n/2 −1 d(cAB )−n/2 = µB dcAB . µA b = µb + (cAB ) b − (n/2)(cAB )

This means that the Higgs fields {|ηab i; hηba |} are actually Pensov scalars of weight {−n/2; +n/2} and that the monopole potential cannot be represented by a globally defined one-form on the sphere, but acquires the inhomogeneous term −(n/2)(cAB )−1 dcAB in HB ∩ HA . (1) The canonical projection on ΩD (A) yields a connection in M and its curvature is represented by πD (R) of the form ! πD (R)[aa] πD (R)[ab] (4.4) πD (R) = πD (R)[ba] πD (R)[bb] defined by (0)

+ πD (R)[aa] = −c(ρ(2) aaa ) ⊗ 1a + ρaba ⊗ [M M ]N T , (1)

πD (R)[ab] = −ic(ρab )γ3 ⊗ M + , (1)

πD (R)[ba] = −ic(ρba )γ3 ⊗ M , (2)

(0)

πD (R)[bb] = −c(ρbbb ) ⊗ 1b + ρbab ⊗ [M M + ]N T . (k)

Here, the differential forms ρ··· ’s are 2 × 2-matrix valued. The diagonal elements of πD (R) are given by ρ(2) aaa = ((Fa )) = d((αa )) + ((αa )) ∧ ((αa )) , (2)

ρbbb = Fb ((Pb )) = (dαb + dµb )((Pb )) ,

(4.5)


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S0129055X01000582

J. A. Mignaco et al. (0)

ρaba = |ηab ihηba | − ((Id)) , (0)

ρbab = (hηba |ηab i − 1)((Pb )) .

(4.6)

The off-diagonal elements are given in terms of the covariant differentials of the Higgs fields (4.3): |∇ηab i = d|ηab i + ((αa ))|ηab i − |ηab i(αb + µb ) , h∇ηba | = dhηba | − hηba |((αa )) + (αb + µb )hηba | .

(4.7)

They read (1)

ρab = |∇ηab ihν| ,

(1)

ρba = |νih∇ηba | .

Finally, the traceless matricesi are given by [M + M ]N T = M + M −

1 tr{M + M } , Na

[M M + ]N T = M M + −

1 tr{M M + } . Nb

The Yang–Mills–Higgs action reads Z Z λ + trmatrix {((Fa )) ∧ ?((Fa ))} + Nb (Fb )∗ ∧ ?Fb Na SY MH (∇D ) = 2π S2 S2 Z + 2 tr{M M +} h∇ηba | ∧ ?|∇ηab i S2 2

Z

+ tr{[M + M ]N T }

?((hηba |ηab i − 1)2 + 1) S2

2

Z

+ tr{[M M + ]N T }

?(hηba |ηab i − 1)2 .

(4.8)

S2

Next, we examine the particle sector constructing the tensor product over A of the right A-module M with the (left-module) Hilbert space H. It results in a Hilbert N space Hp = M A H, with scalar product induced by the scalar product in H and the Hermitian structure h in the module M. A generic element of Hp can be N written as kΨp ii = Ei A Ψi , where Ψi ∈ H obeys π(Pji )Ψj = Ψi . In the model considered here, H = H(s) ⊗ (CNa ⊕ CNb ) and the projective module is M = PA2 , with P defined by the homotopy class [n] in (3.3). A state kΨp ii describing particles, is thus represented by: 1 ψa (x) , each with Na (1) A pair of Pensov spinors of H(s) , given by |ψa (x)i = ψ 2 a (x) values of the generation index. i Note

that when Na = Nb = N and M is a scalar matrix, these traceless matrices [M + M ]NT and (0) [M M + ]NT vanish and there is no ρα term in πD (R). Physically this implies that, in order to have a Higgs mechanism, a nontrivial mass spectrum is necessary!


S0129055X01000582


15

(2) A single Pensov spinor ψb (x) of H(s+n/2) , with a Nb -valued generation index, ψb1 (x) such that |ψb (x)i = ψ 2 (x) = |ν(x)iψb (x) in HB . b

With kΨp ii as above, (D∇ kΨp ii) is represented by ! ! |ψa i (D∇ )aa (D∇ )ab (D∇ )ba

|νiψb

(D∇ )bb

,

(4.9)

(D∇ )aa = D(s) ⊗ 1a − ic((αa )) ⊗ 1a , (D∇ )ab = ((|ηab ihν|)) ⊗ γ3 M + , (D∇ )ba = ((|νihηba |)) ⊗ γ3 M , (D∇ )bb = ((Pb ))D(s) ((Pb )) ⊗ 1b − ic(αb )((Pb )) . A grading in Hp is defined by Γp : X

O

Ψ 7→ X

A

O

ΓΨ

(4.10)

A

and the covariant Dirac operator is odd with respect to this grading D∇ Γp + Γp D∇ = 0 .

(4.11)

Now, hν|D(s) |νiψb = D(s) ψb − ic(mb )ψb and with our choice (3.3) of the representative of the homotopy class [n] ∈ Z, we obtainj ((Pb ))D(s) |νiψb = |νiD(s+n/2) ψb .

(4.12)

Substituting (4.12) in (4.9) yields finally (D∇ kΨp ii)a = (D(s) − ic((αa )))|ψa i + |ηab iγ3 M + ψb , (D∇ kΨp ii)b = |νi[(D(s+n/2) − ic(αb ))ψb + γ3 M hηba |ψa i] .

(4.13)

The matter action functional is then constructed as SMat (kΨp ii, ∇D ) = (kΨp ii; D∇ kΨp ii) Z ?{hψa |(D(s) − ic((αa )))|ψa i + hψa |ηab iγ3 M + ψb = S2

+ (ψb )+ γ3 M hηba |ψa i + (ψb )+ (D(s+n/2) − ic(αb ))ψb } . (4.14) In this sector, the novelty is that, whilst the “a-doublet” remains a doublet of Pensov spinors of weight s, the “b-singlet” metamorphoses in a Pensov spinor of weight s + n/2. If one insists on a comparison with the standard electroweak model j Note that with a different choice in (3.3), a globally defined differential one-form would be added to D(s+n/2) and this can always be absorbed in αb .


16

S0129055X01000582


on S 2 , this means that right-handed electrons see a different magnetic monopole than the left-handed and this is not really welcome. If an Euclidean chiral theory is aimed for, then Γp kΨp ii = kΨp ii implies that the action (4.14) vanishes identically due to the oddness of the Dirac operator (4.11). A proposed way out, as in [23], is just to make an easy switch going to an indefinite Minkowski type metric changing the ψ + to a ψ¯ = ψ + γ 0 so that the presence of the γ 0 provides an extra factor minus one and the action does not vanish. Such a “usual incantation” [13] appears highly unaesthetic and rather unsatisfactory. It seems necessary to double the Hilbert space in order to deal with this issue. This can be achieved introducing a Hilbert space Hp¯ of “anti-particle” states. The need of doubling the fermion fields also arises in the usual Euclidean quantum field theory, where the fermion fields are operator valued in Fock space, in order to cure inconsistent hermiticity properties of the propagators [6, 17]. Alternative proposals were made by [14] and more recently by [21]. Related comments by [12] in a non-commutative geometric setting, should also be mentioned. Here, however, we choose to remain with the primary interpretation of kΨp ii as a state in the Hilbert space Hp represented by Euclidean wave functions. This means that in this work we endeavour an Euclidean one-particle (plus one would-be anti-particle) field theory, which, in a path integral formalism, may hopefully lead to a proper quantum theory. 5. Real Spectral Triples 5.1. The real Pensov Dirac spectral triple The complex conjugation K transforms a Pensov field of weight s, σ (s) , into a Pensov field σ (s)∗ of weight −s. Besides the Hilbert space of Pensov spinors ψ(+) of weight s, denoted here as H1(+) , we also introduce H1(−) , with spinors ψ(−) of weight −s. A real structure will be induced by a pair of anti-linear mappings J1(±) : H1(±) 7→ H1(∓) , which are required to preserve the real Clifford-algebra module structure of the spinor spaces ¯ 1(±) ψ(±) , J1(±) λψ(±) = λJ J1(±) (γ k ψ(±) ) = αγ k (J1(±) ψ(±) ) ,

(5.1)

where we allow for α to be a sign factor ±1. With J1(±) ψ(±) = a(±s) C1,α Kψ(±) ,

(5.2)

where a(±s) is an arbitrary complex number, we should have −1 = αγ k . C1,α (γ k )∗ C1,α

In the chiral representation (2.9), we may choose C1,α =

.

0 1 α 0


S0129055X01000582


17

The adjoint of J1(±) is given by t Kψ(∓) . (J1(±) )+ ψ(∓) = a(±s) C1,α

(5.3)

The unitarity requirement on J1(±) restricts a(±s) be a phase factor. On H1 = H1(+) ⊕ H1(−) , the antilinear isometry defined by ! 0 J1(−) (5.4) J1 = J1(+) 0 obeys J1

γk

0

0

γk

! =α

γk

0

0

γk

! J1 .

If we require J21 = 1 11 ,

with 1 = ±1 ,

(5.5)

the phases a(±s) are related by a(−s) = α1 a(+s) and J1(−) = 1 (J1(+) )+ . The antilinear mappings J1(±) intertwine with the Dirac operators D(±s) as: J1(±) D(±s) = −αD(∓s) J1(±) .

(5.6)

On H1(+) , the Dirac operator is chosen as D1(+) = D(s) , but on H1(−) we may choose D(−s) up to a sign. Let 01 be another arbitrary sign factor, then the choice D1(−) = −α01 D(−s) ,

(5.7)

yields a Dirac operator D1 = D1(+) ⊕ D1(−) intertwining with J1 as J1 D1 = 01 D1 J1 .

(5.8)

The representation of A1 = C(S 2 ; C) in H1 is obtained by taking two copies of the representation in H(s) : ! ! ! ψ(+) f (x) 0 ψ(+) (x) . (5.9) (x) = π1 (f ) ψ(−) ψ(−) (x) 0 f (x) In general, a real structure J1 induces a representation of the opposite algebra Ao1 by: π1o (f ) = J1 (π1 (f ))+ J+ 1 , so that the Hilbert space H1 becomes an A1 bimodule. Here A1 is Abelian, and with the representation π1 above, we have π1o (f ) = π1 (f ). Since [D1 , π1 (f )] = −iκ(+)0c(df ) −iκ 0c(df ) , the first-order conditionk (−)

[[D1 , π1 (f )], π1o (g)] = 0 ,

(5.10)

notational convenience, we have defined κ(+) = +1 and κ(−) = −α01 so that D1(±) = κ(±)D(±s) .

k For


18

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which is needed to define a connection in bimodules [8], is satisfied. Since J1(±) γ3 = −γ3 J1(±) , the chirality in H1 will be taken as χ1 = γ3 ⊕ γ3 .

(5.11)

With this choice: J1 χ1 = 001 χ1 J1 ,

with 001 = −1 .

(5.12)

The -sign table of Connes [3, 4] for a two-dimensional triple is satisfied choosing 1 = −1 and 01 = +1, but for the moment these choices are left open. The spectral triple T1 = {A1 , H1 , D1 , χ1 , J1 } is actually a 0-sphere real spectral triple as defined in [3]. For our pragmatic purposes, an S 0 -real spectral triple may be defined as a real spectral triple with an Hermitian involution σ 0 commuting with π(A1 ), D1 , χ1 and anticommuting with J1 . It is implemented in H1 = H1(+) ⊕ H1(−) by: σ 0 = 11(+) ⊕ (−11(−) ) .

(5.13)

The doubling of the Hilbert space is justified if we interpret the Pensov spinors of H(s) as usual (Euclidean!) Dirac spinors interacting with a magnetic monopole of strenght s. It seems then natural to consider the (Euclidean!) anti-particle fields as Dirac spinors “seeing” a monopole of strength −s i.e. as Pensov spinors of H(−s) . 5.2. The real discrete spectral triple Proceeding further, as in Sec. 4, we have to compose the above S 0 -real “Dirac– Pensov” spectral triple T1 with a real discrete spectral triple T2 = {A2 , H2 , D2 , χ2 , J2 } over the algebra A2 = C ⊕ C. The most general finite Hilbert space allowing a A2 -bimodule structurel is given by the direct sum M CNαβ , (5.14) H2 = α,β

where α and β vary over {a, b} and where Nαβ are integers. Its elements are of the form  aa  ξ  ab  ξ   (ξ) =   ξ ba  ,   ξ bb l For

a general discussion on real discrete spectral triples, we refer to ([11, 18])


S0129055X01000582


19

where each ξ αβ is a column vector with Nαβ rows. An element λ = (λa , λb ) of A1 acts on the left on H2 by   λa 1Naa 0 0 0    0 0 0  λa 1Nab  , (5.15) π2 (λ) =  0  0 λb 1Nba  0  0

0

0

λb 1Nbb

λa 1Naa

0

0

0

0

λb 1Nab

0

0

0

0

λa 1Nba

0

0

0

0

λb 1Nbb

and on the right by    π2o (λ) =   

   .  

(5.16)

Although A2 is an Abelian algebra and, as such, isomorphic to its opposite algebra, it is not a simple algebra so that, in general, π2 (λ) 6= π2o (λ). The discrete real so that C2 structure J2 = C2 K relates both actions through π o (λ) = J2 π(λ)+ J−1 2 intertwines the two representations π2o (λ) = C2 π2 (λ)C2−1 .

(5.17)

. This requires that Nab = Nba = N and, since we require J2 to be anti-unitary, the basis in H2 may be chosen such that   1Naa 0 0 0    0 0  0 1N . (5.18) C2 =   0 0 0  1N   0 0 0 1Nbb This implies that J22 = 2 12 ,

with

2 = +1 .

(5.19)

The chirality, χ2 , defining the orientation of the spectral triple is the image of a Hochschild 0-cycle, i.e. an element of A2 ⊗ Ao2 . This implies that χ2 is diagonal and χαβ = ±1 on each subspace CNαβ . Furthermore, demanding thatm J2 χ2 = 002 χ2 J2 ,

with

002 = +1 ,

(5.20)

we should require that 00 2 = −1, then Naa = Nbb = 0 and χab = −χba and the corresponding odd Dirac operator would not satisfy the first order condition.

m If


20

S0129055X01000582


requires χab = χba = χ0 so that the chirality in H2 reads   χaa 1Naa 0 0 0     0 0 0 χ0 1 N . χ2 =    0 0 0 χ0 1 N   0 0 0 χbb 1Nbb

(5.21)

There are three possibilities leading to a non trivial Hermitian Dirac operator, odd with respect to this chirality: (a) +χaa = +χ0 = −χbb = ±1 (b) −χaa = +χ0 = +χbb = ±1 (c) −χaa = +χ0 = −χbb = ±1 The corresponding Dirac operators have the form (a), (b)



0

 0 Da =  0  K (c)

0

0

0

0

0

0

A

B

K+





 A+  ; B+   0



0

 0 B Dc =   A0  0

0

 B Db =  A  K

B 0+

A0+

0

0

0

0

A

B

0

B+

A+

0

0

0

0

0

0

K+



 0  . 0   0



 A+  . B+   0

The first-order condition [[D2 , π2 (λ)], π2o (µ)] = 0, satisfied in Case (c), implies that in Cases (a) and (b) K must vanish. If we asssume J2 D2 = 02 D2 J2 ,

with 02 = +1 ,

(5.22)

then B = A∗ ;

B 0 = A0∗ .

(5.23)

It should be stressed that, in order to have a non trivial Dirac operator, necessarily N 6= 0. This confirms that the discrete Hilbert space Hdis used in Sec. 4 does not allow for a real structure in the above sense. At last, it can be shown [11, 18] that noncommutative Poincaré duality, in the discrete case, amounts to the non T degeneracy of the intersection matrix with elements αβ = χαβ Nαβ . This non degeneracy condition in Cases (a) and (b) reads Naa Nbb + N 2 6= 0 and is always satisfied. In Case (c) it is required that Naa Nbb − N 2 6= 0 and if all N ’s should be equal, this would not be satisfied. If we insist on equal N ’s, which is not strictly


S0129055X01000582


21

necessary, we are limited to the models (a) and (b) with representations given by (5.15), (5.16). In the sequel we consider only Case (a). The treatment of Case (b) is similar. The Dirac operator and the chirality assignments are thus     0 0 0 0 1N 0 0 0     0 0  0 0 A+  0 0  1N 0   . ; χ2 = χ  (5.24) D2 =  0 B+  0  0 1N  0 0  0  0 A B 0 0 0 0 −1N 5.3. The product The product of T1 with T2 yields the total triple T = {A, H, D, χ, J} with algebra A = A1 ⊗ A2 . Since T1 is S 0 -real, the product is also S 0 -real with Hermitian involution Σ0 = σ 0 ⊗ 12 . The total Hilbert space H = H1 ⊗ H2 is decomposed as H = H(+) ⊕ H(−) , where H(±) = H1(±) ⊗ H2 with elements represented by column matrices   ψ(±)aa (x)    ψ(±)ab (x)    (5.25) Ψ(±) (x) =  ,  ψ(±)ba (x)    ψ(±)bb (x) where each ψ(±)αβ (x) is a Pensov spinor of H1(±) with N “internal” indices, which we do not write down explicitly. The total Dirac operator is D = D1 ⊗ 12 + χ1 ⊗ D2 ,

(5.26)

and the chirality is given by χ = χ 1 ⊗ χ2 .

(5.27)

The continuum spectral triple T1 of Sec. 5.1 is of dimension two and its real structure J1 obeys J21 = 1 11 ;

J1 D1 = 01 D1 J1 ;

J1 χ1 = 001 χ1 J1 ,

where 001 = −1 was fixed but 1 and 01 were independent free ±1 factors. The discrete triple T2 of Sec. 5.2 is of zero dimension and its real structure J2 obeys J22 = 2 12 ; with 2 = 02 = 002 = +1.

J2 D2 = 02 D2 J2 ;

J2 χ2 = 002 χ2 J2 ,


22

S0129055X01000582


The real structure J of the product triple should obey J2 = 1;

JD = 0 DJ;

Jχ = 00 χJ .

(5.28)

If we require that Connes’ sign table be satisfied, i.e. • for T1 , n1 = 2 and 1 = −1, 01 = +1, 001 = −1, • for T2 , n2 = 0 and 2 = 02 = 002 = +1 for n2 = 0, • for the product T , n = 2 and = −1, 0 = +1, 00 = −1, it is seen that, with J = J1 ⊗ J2 , the sign table for the product is not obeyed. Indeed, such a J implies the consistency conditions = 1 2 ,

0 = 01 = 001 02 ,

00 = 001 002

(5.29)

and the second condition is not satisfied. If we keep the same Dirac operator (5.26), it is the definition of J that should be changedn to J = J1 ⊗ (J2 χ2 ) ,

(5.30)

and with this J the consistency conditions become = 1 2 002 ,

0 = 01 = −001 02 ,

00 = 001 002

(5.31)

and these are satisfied. In the rest of this section, we shall assume that these choices are made. Also, in order to simplify the forthcoming formulae, we take α = −1 so that C1 = −10 10 and a(+s) = a(−s) = +1 which imply that J1(−) = −J1(+) + = J1(+) and D1(±) = D(±s) . The change of J2 to J2 χ2 will not change the representation π2o since π2 is even with respect to χ2 . The S 0 -real structure implies that π and π o , D and χ are block diagonal in the decomposition of H and we obtain the representations: let f ∈ C(S 2 , C ⊕ C), then π(f ) = π(+) (f ) ⊕ π(−) (f );    π(±) (f (x)) =       o (f (x)) =  π(±)  

nA

o o π o (f ) = π(+) (f ) ⊕ π(−) (f ) ,

fa (x)1N

0

0

0

0

fa (x)1N

0

0

0

0

fb (x)1N

0

0

0

0

fb (x)1N

fa (x)1N

0

0

0

0

fb (x)1N

0

0

0

0

fa (x)1N

0

0

0

0

fb (x)1N

(5.32)    ,      ;  

(5.33)

general examination of the sign table in relation to the product of two real spectral triples is made in [20].


S0129055X01000582


the Dirac operator: D = D(+) ⊕ D(−) is given by  D1(±) 1N 0 0   0 0 D1(±) 1N D(±) =   0 0 D1(±) 1N  0

γ3 A

the real structure: J = J 0 (+) C2 χ2 yields  C1 1N   0 J(±) = χ0   0  0

0 J(−) , 0

0



 γ3 A+  ; γ3 B +   D1(±) 1N

γ3 B

the chirality: χ = χ(+) ⊕ χ(−) , is given by  0 0 γ3 1N   0 0 γ3 1N χ(±) = χ0   0 0 γ3 1N  0

0

23

(5.34)



0

  ;  

0 0

(5.35)

−γ3 1N

exchanges H(+) and H(−) and J(±) = C1 ⊗

0

0

0

0

C1 1N

0

C1 1N

0

0

0

0

−C1 1N

   K.  

(5.36)

5.4. The “Real” Yang Mills Higgs action o of (5.33), with the Dirac operator of The representations π(±) of (5.32) and π(±) • (5.34), induce representations of Ω (A). Let F ∈ Ω(1) (A), then, using the same techniques as in Sec. 4, we obtain   (1) −ic(σa )1N 0 0 0     (1) (0) 0 σab γ3 A+  0 −ic(σa )1N  . π(±) (F ) =    (1)   0 0 −ic(σb )1N 0   (0) (1) 0 −ic(σb )1N 0 σba γ3 A Introducing the 2N × 2N matrix M(2.a) = 00 A0 , we may also write   (1) (0) −ic(σa )12N σab γ3 M + . π(±) (F ) =  (5.37) (0) (1) σba γ3 M −ic(σb )12N

In particular, the connection is represented by such an expression with the same differential forms σ (k) as in (4.1). For the curvature, represented by a universal two-form G ∈ Ω(2) (A), the unwanted differential ideal J is removed using an or(2) thogonality condition as before. The resulting scalar product in ΩD (A) is the same


24

S0129055X01000582


for representatives in H(+) or H(−) and is used to construct the Yang–Mills–Higgs action which is essentially twice the action (4.8) obtained in Sec. 4: Z λ + (trmatrix {((Fa )) ∧ ?((Fa ))} + Fb∗ ∧ ?Fb ) + 2 tr{A+ A} 2N SY MH (∇D ) = π 2 S Z 1 + 2 + 2 (tr{A A}) h∇ηba | ∧ ?|∇ηab i + tr{(A A) } − × 2N S2 Z 2 ?(2(hηba |ηab i − 1) + 1) . (5.38) × S2

5.5. The “Real ” covariant Dirac operator Matter in this “real spectral triple” approach is represented by states of the covariN N ant Hilbert space HCov = M A H A M∗ . For the S 0 -real spectral triple, the Hilbert space is decomposed as H = H(+) ⊕ H(−) so that also HCov splits in a sum of “particle” and “antiparticle” Hilbert spaces: HCov = H(+p) ⊕ H(−p) , where each N N H(±p) = M A H(±) A M∗ has typical elements kΨ(±p ii. The projective module M = PA2 and its dual M∗ were examined in Sec. 3. In bases {Ei } and {E j } N N ∗ ∗ of the free modules A2 and A2 , we may represent a state of A2 A H(±) A A2 N N i i j as Ei A Ψ(±) j A E . It is a state kΨ(±p) ii of H(±p) if Ψ(±) j ∈ H(±) obeys o (Pj` )Ψ(±) k` . Ψ(±) ij = π(±) (Pki )π(±)

(5.39)

i i (x) = δki and Pk,b (x) = |ν(x)ii hν(x)|k (cf. Sec. 3) the vector Ψ(±) ij ∈ H(±) Since Pk,a is represented by the column vector  i  ((ψ(±)aa ))j     i |ψ i hν|   j (±)ab i  , (5.40) Ψ(±) j =    |νii hψ(±)ba |  j   |νii ψ(±)bb hν|j

where ((ψ(±)aa )) is a quadruplet and ψ(±)bb = hν|((ψ(±)bb ))|νi a singlet of Pensov spinor fields of spin weight ±s, while |ψ(±)ab i = ((ψ(±)ab ))|νi, respectively hψ(±)ba | = hν|((ψ(±)ba )), are doublets of Pensov spinors of weight (±s) − n/2, respectively (±s) + n/2. The covariant real structure JCov acts on kΨCov ii as  i`¯  k δ (C1 K((ψ(∓)aa ))` )δkj ¯    i`¯  δ (C Khψ | )hν|   1 ` j (∓)ba i . (5.41) (JCov kΨCov ii)(±) j = χ0     |νii (C1 K|ψ(∓)ab ik )δkj ¯    i −|νi (C1 Kψ(∓)bb )hν|j


S0129055X01000582


25

The covariant Dirac operator is also block diagonal: D∇ = D(+p) ⊕ D(−p) and is given by O O O O i k i ` j ψ(±) ` (Pj E ) = Ei π(±) (((A))k )ψ(±) kj Ej D(±p) (Ei Pk ) A

A

A

+ Ei

O A

+ 0 Ei

A

o π(±) (Pki )π(±) (Pj` )D(±) ψ(±) k`

O A

`

o π(±) (((A))j )ψ(±) i`

O

O

Ej

A

Ej ,

A

(5.42) where ((A)) is the 2 × 2 matrix of universal one-forms given in (3.6). It is represented by the matrix valued differential one- and zero-forms given in terms of ((αa )), αb , |Φab i and hΦba |, defined in (4.1) by: ((Aa )) = −iγ r ((αa,r )) , ((Ab )) = −iγ r αb,r |νihν| , ((Aab )) = |Φab ihν|γ3 , ((Aba )) = |νihΦba |γ3 . We obtain



  π(±) (((A))) =   

(5.43)

((Aa ))1N

0

0

0

((Aa ))1N

0

0

0

((Ab ))1N

0

((Aba ))A

0

o representative of ((A)) is computed as The π(±)  ((Aa ))1N 0 0   0 0 ((Ab ))1N o (((A))) =  0 π(±)  0 0 ((Aa ))1N 

0

0

−((Aab ))B



0

 ((Aab ))A+  .  0  ((Ab ))1N

0

(5.44)



   . (5.45) + −((Aba ))B  ((Ab ))1N 0

Substituting (5.44) and (5.45) in (5.42), we obtain ((D∇ kΨCov ii(±)aa )) = D1(±) ((ψ(±)aa )) − iγ r [((αa,r )), ((ψ(±)aa ))] , (−n/2)

|D∇ kΨCov ii(±)ab i = D1(±) |ψ(±)ab i + |ηab iγ3 A+ ψ(±)bb − iγ r (((αa,r ))|ψ(±)ab i − |ψ(±)ab iαb,r ) ; (+n/2)

hD∇ kΨCov ii(±)ba | = D1(±) hψ(±)ba | + hηba |γ3 B+ ψ(±)bb − iγ r (αb,r hψ(±)ba | − hψ(±)ba |((αa,r ))) ,


26

S0129055X01000582


D∇ kΨCov ii(±)bb = D1(±) ψ(±)bb + γ3 Ahηba |ψ(±)ab i + γ3 Bhψ(±)ba |ηab i . (−n/2)

(5.46)

(+n/2)

The Dirac operators, D1(±) = D(±s−n/2) and D1(±) = D(±s+n/2) , acting on Pensov spinor fields of spin weight ±s + n/2 or ±s − n/2, arise from (−n/2)

(D1(±) (|ψ(±)ab ihν|))|νi = D1(±) |ψ(±)ab i , (+n/2)

hν|(D1(±) (|νihψ(±)ba |)) = D1(±) hψ(±)ba | , where ((Pb )) = |νihν| is the representative, chosen in (3.3), of the homotopy class [n] ∈ π2 (S 2 ). The induced contribution of the “magnetic monopole” is hidden in this modification of the Dirac operator. The Higgs doublets of Pensov fields of weight ∓n/2, |ηab i and hηba | were defined in (4.3). A suitable action of the matter field would be SMat (kΨCov ii, ∇D ) = (kΨCov ii; D∇ kΨCov ii) .

(5.47)

But, if we aim for a theory admitting only chiral matter, i.e. if we restrict the Hilbert space to those vectors kΨCov ii(+) , obeying say ΓkΨCov ii(+) = +kΨCov ii(+) ,

(5.48)

then the above action vanishes identically. An action that does not vanish identically, would be SChiral (kΨCov ii, ∇D ) = (JCov kΨCov ii; D∇ kΨCov ii) .

(5.49)

It is easy to show that this action does not vanish identically if 00 = −1 ,

0 = +1 .

(5.50)

In two dimensions Connes’ sign table obeys the first but not the second condition.o It should however be stressed that Connes’ sign table, with its modulo eight periodicity, comes from representation theory of the real Clifford algebras and if we restrict our (generalized) spinors to Weyl spinors, we loose the Clifford algebra representation and the sign tables ceases to be mandatory. We could then return to Sec. 5.3 and, • with J = J1 ⊗ J2 , require conditions (5.29) to hold with 01 = −1 = 1 , or, • with J = J1 ⊗ (J2 χ2 ), require conditions (5.31) to hold with 01 = +1 = 1 . This recipe yields then a chiral action depending only on kΨCov ii(+) . We postpone the examination of this chirality problem, essential for a thorough understanding of the “neutrino paradigm”, to further work on C⊕ H on the four-dimensional sphere. o In

four dimensions it is the first condition that is not satisfied.


S0129055X01000582


27

6. Conclusions and Outlook The Connes–Lott model over the two-sphere, with C ⊕ C as discrete algebra, has been generalized such as to allow for a nontrivial topological structure. The basic Hilbert space H(s) was made of Pensov spinors which can be interpreted as usual spinors interacting with a Dirac monopole “inside” the sphere. Covariantisation of the Hilbert space H(s) ⊗ Hdis with a nontrivial projective module M induced a “spin” change in certain matter fields so that we obtained singlets and doublets of different spin content. The Higgs fields also acquired a nontrivial topology since they are no longer ordinary functions on the sphere, but rather Pensov scalars, i.e. sections of nontrivial line bundles over the sphere. A real spectral triple has also been constructed essentially through the doubling of the Pensov spinors so that the Hilbert space of the continuum spectral triple became H1 = H(s) ⊕ H(−s) . The discrete spectral triple had also to be extended in order that the first order condition (5.10) could be met. In contrast with the standard noncommutative geometry model of the standard model, in our model the continuum spectral triple has an S 0 -real structure while the discrete spectral triple has not. Some physical plausibility arguments for this were given in Sec. 5.1. It was also shown that the covariantisation of the real spectral triple with the nontrivial M allows the Abelian gauge fields to survive, while they are slain if covariantisation is done with a trivial module. Finally a possibility of solution to the problem of a non vanishing action of chiral matter has been indicated, paying the price of using a complex action. If we address the quantisation problem in a path integral formalism, let us first recall that we have Higgs fields which are Pensov scalars of a certain weight (s1 ) and matter fields which are Pensov spinors of some H(s2 ) . It is thus tempting to assume that the Higgs fields should be even Grassmann variables if s1 is integer valued and odd Grassmann variables if it is half-integer. In the same vein the matter fields should be odd or even Grassmann variables if s2 is integer or half integer valued. A thorough examination of this issue is however beyond the scope of this work. References [1] N. Berline, E. Getzler and M. Vergne, Heat Kernels and Dirac Operators, Springer, Berlin, 1996. [2] A. Connes, Noncommutative Geometry, Acad. Press, London, 1994. [3] A. Connes, J. Math. Phys. 36 (1995) 6194. [4] A. Connes, Comm. Math. Phys. 182 (1996) 155. [5] A. Connes and J. Lott, Nucl. Phys. (Proc. Suppl.) 18B (1990) 29. [6] J. Fr¨ ohlich and K. Osterwalder, Helv. Phys. Acta 47 (1974) 781. [7] R. Coquereaux, J. Geom. Phys. 6 (1989) 425. [8] M. Dubois-Violette and T. Masson, Lett. Math. Phys. 37 (1996) 467. [9] C. Jayewardena, Helv. Phys. Acta 61 (1988) 636. [10] D. Kastler and T. Sch¨ ucker, Rev. Math. Phys. 8 (1996) 205. [11] T. Krajewski, J. Geom. Phys. 28 (1998) 2. [12] F. Lizzi, G. Mangano, G. Miele and G. Sparano, hep-th/9610035.


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[13] C. P. Mart´ın, J. M. Gracia-Bond´ıa and J. V´ arilly, Phys. Rep. 294 (1998) 363. [14] M. R. Mehta, Phys. Rev. Lett. 65 (1990) 1983; Mod. Phys. Lett. A 6 (1991) 2811. [15] J. A. Mignaco, C. Sigaud, F. J. Vanhecke and A. R. Da Silva, Rev. Math. Phys. 9 (1997) 689. [16] E. Newman and R. Penrose, J. Math. Phys. 7 (1966) 863. [17] K. Osterwalder and R. Schrader, Helv. Phys. Acta 46 (1973) 277. [18] M. Paschke and A. Sitarz, MZ-TH/96-40, Mainz, 1996. [19] A. Staruszkiewicz, J. Math. Phys. 8 (1967) 2221; G. Pensov, Compt. Rend. Acad. Sci. URSS 54 (1946) 563. [20] F. J. Vanhecke, Lett. Math. Phys. 50 (1999) 157, and math-phys/9903029. [21] P. van Nieuwenhuizen and A. Waldron, Phys. Lett. B 389 (1990) 29; also hep-th/ 9610035 and A. Waldron hep-th/9702057. [22] J. C. V´ arilly, “An introduction to noncommutative geometry”, Monsaraz lectures, Lisbon, 1997. [23] J. C. V´ arilly and J. M. Gracia-Bond´ıa, J. Geom. Phys. 12 (1993) 223. [24] T. T. Wu and C. N. Yang, Phys. Rev. D12 (1975) 3845; Nucl. Phys. B107 (1976) 365.


00062


ENTROPY OF BOGOLIUBOV AUTOMORPHISMS OF CAR AND CCR ALGEBRAS WITH RESPECT TO QUASI-FREE STATES

S. V. NESHVEYEV Institute for Low Temperature Physics & Engineering Lenin Ave 47, Kharkov 310164, Ukraine E-mail : [email protected]

Received 21 July 1999 Revised 13 March 2000 We compute the dynamical entropy of Bogoliubov automorphisms of CAR and CCR algebras with respect to arbitrary gauge-invariant quasi-free states. This completes the research started by Størmer and Voiculescu, and continued in works of Narnhofer– Thirring and Park–Shin.

1. Introduction and Formulation of Main Result One of the most beautiful results in the theory of dynamical entropy is the formula for the entropy of Bogoliubov automorphisms of the CAR-algebra with respect to quasi-free states obtained by Størmer and Voiculescu [12] in 1990. They proved it under the assumption that the operator determining the quasi-free state has pure point spectrum. Since then several papers devoting to the computation of the entropy of Bogoliubov automorphisms have appeared. Narnhofer and Thirring [9] and Park and Shin [11] proved the formula for some operators with continuous spectrum. The latter paper contains also a similar result for the CCR-algebra. On the other hand, Bezuglyi and Golodets [1] proved an analogous formula for Bogoliubov actions of free abelian groups. While the cases considered in [9] and [11] required a non-trivial analysis, the proof of Størmer and Voiculescu is very elegant. It relies on an axiomatization of certain entropy functionals on the set of multiplicity functions. The main axiom there stems from the equality hω (α) = n1 hω (αn ). Thus their method can not be directly applied to groups without finite-index subgroups. Instead, we can “cut and move” multiplicity functions without changing the entropy (see Lemma 5.1 below). This observation together with the methods developed in [1] allowed to prove (under the same restrictions on quasi-free states) an analogue of Størmer–Voiculescu’s formula for Bogoliubov actions of arbitrary torsion-free abelian groups [7]. In this paper we will show that, in fact, the formula holds without any restrictions on the 29


30

00062

S. V. Neshveyev

operator determining the quasi-free state. We will prove also an analogous result for the CCR-algebra. We will consider only the case of single automorphism, since in view of the methods of [7] the case of arbitrary torsion-free abelian group gives nothing but more complicated notations. So the main result of the paper is as follows. Theorem 1.1. Let U be a unitary operator on a Hilbert space H, αU the corresponding Bogoliubov automorphism of the CAR or the CCR algebra over H, A a bounded (A ≤ 1 for CAR) positive operator commuting with U and determining a quasi-free state ωA . Let Ua = U |Ha be the absolutely continuous part of U, Z ⊕ Z ⊕ Z ⊕ Hz dλ(z) , Ua = zdλ(z) , A|Ha = Az dλ(z) Ha = T

T

T

a direct integral decomposition, where λ is the Lebesgue measure on the torus T (λ(T) = 1). Then Z Tr(η(Az ) + η(1 − Az ))dλ(z) , CAR : hωA (αU ) = T

Z CCR : hωA (αU ) =

T

Tr(η(Az ) − η(1 − Az ))dλ(z) .

Corollary 1.1. The necessary condition for the finiteness of the entropy is that Az has pure point spectrum for almost all z ∈ T. Corollary 1.2. If the spectrum of the unitary operator is singular, then the entropy is zero. For CAR, the latter corollary is already known from [12]. Finally, for systems considered in [9] and [11], Theorem 1.1 may be reformulated as Corollary 1.3. Let I be an open subset of R, ω a locally absolutely continuous function on I, ρ a bounded (ρ ≤ 1 for CAR) positive measurable function on I. Let U and A be the operators on L2 (I, dx) of multiplication by the functions eiω and ρ, respectively. Then Z 1 [η(ρ(x)) + η(1 − ρ(x))]|ω 0 (x)|dx , CAR : hωA (αU ) = 2π I Z 1 [η(ρ(x)) − η(1 − ρ(x))]|ω 0 (x)|dx . CCR : hωA (αU ) = 2π I The paper is organized as follows. Section 2 contains some preliminaries on entropy and algebras of canonical commutation and anti-commutation relations. In Sec. 3 we prove that the entropies don’t exceed the values of the integrals in Theorem 1.1. The opposite inequality is proved in Secs. 4 and 5. In Sec. 4 we obtain a lower bound for the entropy in the case where the unitary operator has


00062

Entropy of Bogoliubov Automorphisms of CAR and CCR

31

Lebesgue spectrum and the operator determining the quasi-free state is close to a scalar operator. In Sec. 5, first, using the observation mentioned above we extend the estimate of Sec. 4 to arbitrary unitaries, and then prove the required inequality. There are also two appendices to the paper. The results of [6] show that modular automorphisms can have the K-property (in the sense of Narnhofer and Thirring [8]). This observation combined with the results of the present paper allow to construct on the hyperfinite III1 -factor a simple example of non-conjugate K-systems with the same finite entropy. This is done in Appendix A. Appendix B contains an auxiliary result on decomposable operators. 2. Preliminaries Recall the definition of dynamical entropy [4]. Let (A, φ, α) be a C ∗ -dynamical system, where A is a C ∗ -algebra, φ a state on A, α a φ-preserving automorphism of A. By a channel in A we mean a unital completely positive mapping γ : B → A of a finite-dimensional C ∗ -algebra B. The mutual entropy of channels γi : Bi → A, i = 1, . . . , n, with respect to φ is given by Hφ (γ1 , . . . , γn ) = sup

X

η(φi1 ···in (1)) +

i1 ,...,in

n X X

(k)

S(φ ◦ γk , φik ◦ γk ) ,

k=1 ik

P (k) where η(t) = −t log t, S(·, ·) the relative entropy, φik = i1 ,...,in φi1 ···in , and the ik fixed P supremum is taken over all finite decompositions φ = φi1 ···in of φ in the sum of positive linear functionals. If A is a W ∗ -algebra and φ is a normal faithful state, φ (x)) for some then any positive linear functional ψ ≤ φ on A is of the form φ(·σ−i/2 x ∈ A, 0 ≤ x ≤ 1, where σtφ is the modular group corresponding to φ. Thus Hφ (γ1 , . . . , γn ) = sup

X

η(φ(xi1 ···in )) +

i1 ,...,in

n X X

(k)

S(φ(γk (·)), φ(γk (·)σ−i/2 (xik ))) ,

k=1 ik

where the supremum is taken over all finite partitions of unit. The entropy of the automorphism α with respect to a channel γ and the state φ is given by hφ (γ; α) = lim

n→∞

1 Hφ (γ, α ◦ γ, . . . , αn−1 ◦ γ) . n

The entropy hφ (α) of the system (A, φ, α) is the supremum of hφ (γ; α) over all channels γ in A. We refer the reader to [4, 8, 10, 12] for general properties of entropy. Lemma 2.1. Let (A, φ, α) be a C ∗ -dynamical system, {An }∞ n=1 a sequence of a sequence of completely positive unital α-invariant subalgebras of A, {Fn }∞ n=1 mappings Fn : A → An such that kFn (x)−xkφ → 0 as n → ∞, for any x ∈ A. Then hφ (α) ≤ lim inf hφ (α|An ) . n→∞


32

00062

S. V. Neshveyev

Proof. The result follows from the continuity of mutual entropy in k kφ -topology: see the proof of Lemma 3.3 in [12]. Though the possibility of An 6⊂ An+1 is important for applications to actions of more general groups (see the proof of Theorem 4.1 in [7]), we will use this lemma only when An ⊂ An+1 . Then the existence of Fn ’s is not necessary, as the following result shows. Lemma 2.2. Let A be a C ∗ -algebra, φ a state on A, {An }∞ n=1 an increasing seS quence of C ∗ -subalgebras such that n πφ (An ) is weakly dense in πφ (A). Then, for any channel γ : B → A and any ε > 0, there exist n ∈ N and a channel γ˜ : B → An such that kγ − γ˜kφ < ε. Proof. This follows from the identification of completely positive maps Matd (C) → A with positive elements in Matd (A) [3] and, in fact, is implicitly contained in [4]. We include a proof for the convenience of the reader. Without loss of generality we may suppose that B = Matd (C). The channels B → A are in one-to-one correspondence with positive elements Q ∈ Matd (A) such P ˜ that k Qkk = 1. By Kaplansky’s density theorem, there exists a net {Qi }i ⊂ S n Matd (πφ (An )) such that ˜ i −→ πφ (Q) strongly . Q

˜i ≤ 1 , 0≤Q ˜ i to an element Qi ∈ We can lift Q Q(i; δ)kl =

X

i

S n

Matd (An ), 0 ≤ Q ≤ 1. For δ > 0, set

−1/2 Q(i)jj + dδ

X −1/2 (Q(i)kl + δkl δ) Q(i)jj + dδ .

j

Let γi,δ : B →

S n

j

An be the corresponding channel, γi,δ (ekl ) = Q(i; δ)kl . Then lim lim kγ − γi,δ kφ = 0 .

δ→0

i

Now recall some facts concerning CAR and CCR algebras [2]. Let H be a Hilbert space. The CAR-algebra A(H) over H is a C ∗ -algebra generated by elements a(f ) and a∗ (f ), f ∈ H, such that the mapping f 7→ a∗ (f ) is linear, a(f )∗ = a∗ (f ) and a∗ (f )a(g) + a(g)a∗ (f ) = (f, g)1 ,

a(f )a(g) + a(g)a(f ) = 0 .

Each unitary operator U on H defines a Bogoliubov automorphism αU of A(H), αU (a(f )) = a(U f ). The fixed point algebra A(H)e = A(H)αU for U = −1 is called the even part of A(H). Each operator A on H, 0 ≤ A ≤ 1, defines a quasi-free state ωA on A(H), ωA (a∗ (f1 ) · · · a∗ (fn )a(gm ) · · · a(g1 )) = δnm det((Afi , gj ))i,j .


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33

If Ker A = Ker(1 − A) = 0, then ωA is a KMS-state with respect to the group given by σtωA (a(f )) = a(B it f ) ,

where B =

A . 1−A

(2.1)

If U and A commute, then ωA is αU -invariant. If H = K ⊕ L, then A(K) and A(L)e commute, and we have A(H)α1⊕−1 = A(K) ∨ A(L)e ∼ = A(K) ⊗ A(L)e . If K is an invariant subspace for A, then ωA |A(K)⊗A(L)e = ωA |A(K) ⊗ ωA |A(L)e . In particular, there exists an ωA -preserving conditional expectation (IdA(K) ⊗ ωA (·)|A(L)e ) ◦

1 + α1⊕−1 2

onto A(K). If dim K = n < ∞, then A(K) is a full matrix algebra of dimension 22n . In particular, for any f ∈ H, kf k = 1, the algebra A(Cf ) is isomorphic to Mat2 (C), and we define matrix units for it as e11 (f ) = a(f )a∗ (f ) ,

e22 (f ) = a∗ (f )a(f ) ,

e12 (f ) = a(f ) ,

e21 (f ) = a∗ (f ) . (2.2)

The restriction of a quasi-free state ωA to A(Cf ) is given by the matrix ! 1−λ 0 , where λ = (Af, f ) . (2.3) 0 λ The CCR-algebra U(H) over H is a C ∗ -algebra generated by unitaries W (f ), f ∈ H, such that W (f )W (g) = ei

Im(f,g) 2

W (f + g) .

A representation π of U(H) is called regular, if the mapping R 3 t 7→ π(W (tf )) is strongly continuous for each f ∈ H. For any such a representation, the generator Φπ (f ) of the group {π(W (tf ))}t is defined, π(W (tf )) = eitΦπ (f ) . Then annihilation and creation operators are defined as aπ (f ) =

Φπ (f ) + iΦπ (if ) √ , 2

a∗π (f ) =

Φπ (f ) − iΦπ (if ) √ . 2

These are closed unbounded operators affiliated with π(U(H))00 , aπ (f )∗ = a∗π (f ), a∗π (f ) depends on f linearly, and for any f , g ∈ H we have the commutation relations aπ (g)a∗π (f ) − a∗π (f )aπ (g) = (f, g)1 ,

aπ (g)aπ (f ) − aπ (f )aπ (g) = 0


34

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S. V. Neshveyev

on a dense subspace. In the sequel we will suppress π in the notations of annihilation and creation operators. Each unitary operator U on H defines a Bogoliubov automorphism αU of U(H), αU (W (f )) = W (U f ) . Each positive operator A on H defines a quasi-free state ωA on U(H), ωA (W (f )) = e− 4 kf k

2

1

− 12 (Af,f )

.

The cyclic vector ξωA in the GNS-representation belongs to the domain of any operator of the form a# (f1 ) · · · a# (fn ), where a# means either a∗ or a, and (a∗ (f )a(g)ξωA , ξωA ) = (Af, g) . If Ker A = 0, then ωA is separating (i.e. ξωA is separating for πωA (U(H))00 ), and σtωA (W (f )) = W (B it f ) ,

where B =

A , 1+A

so that # # # it # it ∆it ωA a (f1 ) · · · a (fn )ξωA = a (B f1 ) · · · a (B fn )ξωA .

(2.4)

If H = K ⊕ L, then U(H) ∼ = U(K) ⊗ U(L). If K is an invariant subspace for A, then ωA = ωA |U (K) ⊗ ωA |U (L) , so that there exists an ωA -preserving conditional expectation IdU (K) ⊗ ωA |U (L) onto U(K). If K is finite-dimensional, then every regular representation π of U(K) is quasiequivalent to the Fock representation, in particular, π(U(K))00 is a factor of type I∞ (if K 6= 0). Thus for any regular state ω on U(K) (so that the mapping t 7→ ω(W (tf )) is continuous) the von Neumann entropy of the continuation ω ¯ of the state ω to πω (U(K))00 is defined. We will denote it by S(ω) (in fact, the notion of ω) [10]). If entropy of state can be defined for all C ∗ -algebras, and then S(ω) = S(¯ for π(U(K))00 K = Cf , kf k = 1, we define a system of matrix units {eij (f )}∞ i,j=0 as follows: ekk (f ) is the spectral projection of a∗ (f )a(f ) corresponding to {k}, ek+n,k (f ) = =

k! (k + n)! k! (k + n)!

1/2

a∗ (f )n ekk (f )

1/2 ek+n,k+n (f )a∗ (f )n .

(2.5)

In particular, if ωA is a quasi-free state on U(H), for any f ∈ H, kf k = 1, we obtain a system of matrix units {eij (f )}i,j in πωA (U(H))00 , and ωA (eij (f )) = δij

λi , (1 + λ)i+1

where λ = (Af, f ) .

(2.6)


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35

(This is equivalent to the fact that if A is of trace class, then the quasi-free state ωA is given in the Fock representation by the density operator TrΓ(B) Γ(B) , where Γ is the operator of second quantization.) Note for the future use that ∞ X

η(λk (1 + λ)−k−1 ) = η(λ) − η(1 + λ) .

(2.7)

k=0

In the sequel we will write C(H) instead of A(H) and U(H) in the arguments that are identical for CAR and CCR. The following result is known, but we will give a proof for the reader’s convenience. Lemma 2.3. Let H be finite-dimensional, ωA a quasi-free state on C(H). Then (i) CAR S(ωA ) = Tr(η(A) + η(1 − A)), CCR S(ωA ) = Tr(η(A) − η(1 + A)); (ii) if H = H1 ⊕ H2 , then S(ωA ) ≤ S(ωA |C(H1 ) ) + S(ωA |C(H2 ) ). Proof. Let Pi be the projection onto Hi , Ai = Pi A|Hi . Set Mi = πωAi (U(Hi ))00 , M = πωA (U(H))00 . Since all regular representations of U(Hi ) are quasi-equivalent, we may consider Mi as a subalgebra of M . Since M1 is a type I factor, we have M = M1 ⊗ (M10 ∩ M ), whence M = M1 ⊗ M2 . Thus the assertion (ii) for CCR is the usual subadditivity of von Neumann entropy. Turning to CAR, let us first note that if M is a full matrix algebra, ω a state on M and α an automorphism of M , then S(ω) ≤ S(ω|M α ), and the equality holds ˜ be the density operator for ω iff ω is α-invariant. Indeed, let Q (respectively Q) α (respectively ω|M ). Since the canonical trace on M α is given by the restriction of ˜ = 1, hence the canonical trace Tr on M , we have Tr Q ˜ ≥ 0, S(ω|M α ) − S(ω) = Tr Q(log Q − log Q) ˜ i.e. Q ∈ M α . and the equality holds iff Q = Q, Applying this to CAR, we obtain S(ωA ) ≤ S(ωA |A(H1 )⊗A(H2 )e ) ≤ S(ωA |A(H1 ) ) + S(ωA |A(H2 )e ) = S(ωA |A(H1 ) ) + S(ωA |A(H2 ) ) . We see also that if Hi is an invariant subspace for A, then S(ωA ) = S(ωA |C(H1 ) ) + S(ωA |C(H2 ) ) . So, in proving (i) it is enough to consider one-dimensional spaces, for which the result follows immediately from (2.3), (2.6) and (2.7). Lemma 2.4. Let U be a unitary operator on H, {Pn }∞ n=1 a sequence of projections in B(H), Pn U = U Pn , Pn → 1 strongly, Hn = Pn H. Then, for the Bogoliubov automorphism αU and any αU -invariant quasi-free state ωA on C(H), we have hωA (αU ) ≤ lim inf hωA (αU |C(Hn ) ) . n→∞


36

00062

S. V. Neshveyev

Proof. Let C be an operator commuting with Pn for all n ∈ N. Let En be the ωC -preserving conditional expectation of C(H) onto C(Hn ) defined above. Then kEn (x)−xkωA → 0 for any x ∈ C(H). Indeed, for CAR we have even the convergence in norm, that follows from kEn k = 1 and ka(f )k = kf k: if x = a# (f1 ) · · · a# (fm ) then m X a# (Pn f1 ) · · · a# (Pn fk−1 ) x = a# (Pn f1 ) · · · a# (Pn fm ) + k=1

× a# (fk − Pn fk )a# (fk+1 ) · · · a# (fm ) , P whence kEn (x) − xk ≤ 2 m k=1 kPn f1 k · · · kPn fk−1 k · kfk − Pn fk k · kfk+1 k · · · kfm k. For CCR, the assertion follows from the equalities En (W (f )) = e− 4 k(1−Pn )f k

2

1

kW (f ) − W (g)k2ωA = 2 − 2 Re(e

− 12 (C(1−Pn )f,(1−Pn )f )

i Im(f,g) 2

W (Pn f ) ,

ωA (W (f − g))) .

Thus we can apply Lemma 2.1. 3. Upper Bound for the Entropy In this section we will prove that the entropies do not exceed the values of the integrals in Theorem 1.1. There exists a Hilbert space K and a unitary operator V on K such that Ua ⊕ V has countably multiple Lebesgue spectrum. Set ˜ =H ⊕K, H

˜ =U ⊕V , U

A˜ = A ⊕ 0 .

˜ → Then, due to the existence of an ωA˜ -preserving conditional expectation C(H) ˜ U ˜ , A) ˜ C(H), we have hωA (αU ) ≤ hωA˜ (αU˜ ). On the other hand, the passage to (H, does not change the value of the integral in Theorem 1.1. So, without loss of generality we may suppose that Ua has countably multiple Lebesgue spectrum. If the value of the integral is finite, then Az has pure point spectrum for almost all z ∈ T. Then we can represent Ha as the sum of a countable set of copies of L2 (T, dλ) in such a way that U and A act on the nth copy as multiplications by functions z and λn (z), respectively (see Appendix B). By Lemma 2.4, we may restrict ourselves to the sum of a finite number of copies of L2 (T). Thus we suppose Ha =

m0 M

L2 (T) ,

Ua =

m=1

M=1

and we have to prove that CAR : hωA (αU ) ≤

m0 Z X m=1

CCR : hωA (αU ) ≤

m0 M

T

A|Ha =

m0 M

λm (z) ,

m=1

(η(λm (z)) + η(1 − λm (z)))dλ(z) ,

m0 Z X m=1

z,

T

(η(λm (z)) − η(1 + λm (z)))dλ(z) .


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37

Let H0 be the m0 -dimensional subspace of H spanned by constant functions in L L each copy of L2 (T). Then Ha = n∈Z U n H0 . For n ∈ N, set Hn = nk=0 U k H0 . We state that 1 S(ωA |C(Hn−1 ) ) . n→∞ n

hωA (αU ) ≤ lim

(3.1)

For CAR, this is implicitly contained in the proof of Lemma 5.3 in [12]. So we will consider CCR only. For a finite set X, we denote by Mat(X) the C ∗ -algebra of linear operators on l2 (X). Let {exy }x,y∈X be the canonical system of matrix units for Mat(X). Following Voiculescu (see Lemmas 5.1 and 6.1 in [13]), for X ⊂ H, we introduce unital completely positive mappings iX : Mat(X) → U(H) , iX (exy ) =

jX : U(H) → Mat(X) ,

1 W (x)W (y)∗ , |X|

jX (a) = PX πτ (a)PX ,

where τ denotes the unique trace on U(H) (τ (W (f )) = 0 for f 6= 0), PX is the projection onto the subspace Lin{πτ (W (x))ξτ |x ∈ X} ⊂ Hτ identified with l2 (X), and |X| is the cardinality of X. Then (iX ◦ jX )(W (f )) =

|X ∩ (X − f )| W (f ) ∀ f ∈ H . |X|

Hence, for any subspace K of H, there exists a net {Xi }i of finite subsets of K such that k(iXi ◦ jXi )(a) − ak−→ 0 ∀ a ∈ U(K). i

Let Hs = H Ha be the subspace corresponding to the singular part of the spectrum of U . By Lemma 2.2, in computing the entropy we may consider only the S channels in m U(Hs ⊕ Hm ). If γ is a channel in U(Hs ⊕ Hm ) = U(Hs ) ⊗ U (Hm ), then it can be approximated in norm by a channel of the form (iX ⊗iZ )◦(jX ⊗jZ )◦γ, where X ⊂ Hs and Z ⊂ Hm . Hence, it suffices to consider only the channels iX ⊗iZ . So, let γ = iX ⊗ iZ : Mat(X) ⊗ Mat(Z) → U(Hs ) ⊗ U(Hm ) = U(Hs ⊕ Hm ). Set L = Lin X. Fix ε > 0. By Lemma 5.1 in [12], there exist n0 ∈ N and a sequence of k k projections {Qn }∞ n=n0 in B(Hs ) such that dim Qn ≤ εn and k(U − Qn U )|L k ≤ ε (n,k) for k = 0, . . . , n − 1. Define a channel iX : Mat(X) → U(Hs ), (n,k)

iX

(exy ) = =

1 W (Qn U k x)W (Qn U k y)∗ |X| 1 − i Im(Qn U k x,Qn U k y) e 2 W (Qn U k (x − y)) . |X|

On the other hand, we have (αkU ◦ iX )(exy ) =

1 1 − i Im(U k x,U k y) W (U k x)W (U k y)∗ = e 2 W (U k (x − y)) . |X| |X|


38

00062

S. V. Neshveyev (n,k)

We may conclude that there exists an upper bound for kαkU ◦iX −iX only on ε, kAk, |X| and kXk = max{kxk |x ∈ X}. Set (n,k)

γn,k = iX

kωA depending

⊗ (αkU ◦ iZ ) .

Then kαkU ◦ γ − γn,k kωA is bounded by a value depending only on ε, kAk, kXk, |X| and |Z|. By Proposition 4.3 in [4], ◦ γ) − HωA (γn,0 , γn,1 , . . . , γn,n−1 )| < nδ , |HωA (γ, αU ◦ γ, . . . , αn−1 U

(3.2)

where δ = δ(ε, kAk, kXk, |X|, |Z|)−→ 0. Since γn,k ’s are channels in U(Qn Hs ⊕ ε→0

Hm+n−1 ), we have HωA (γn,0 , γn,1 , . . . , γn,n−1 ) ≤ S(ωA |U (Qn Hs ⊕Hm+n−1 ) ) .

(3.3)

By Lemma 2.3, S(ωA |U (Qn Hs ⊕Hm+n−1 ) ) ≤ S(ωA |U (Qn Hs ) ) + S(ωA |U (Hm+n−1 ) )

(3.4)

and S(ωA |U (Qn Hs ) ) ≤ (η(kAk) − η(1 + kAk)) dim Qn Hs ≤ εn(η(kAk) − η(1 + kAk)) .

(3.5)

From (3.2)–(3.5) we conclude that 1 S(ωA |U (Hn−1 ) ) . n Because of the arbitrariness of ε, the proof of (3.1) is complete. Applying Lemma 2.3, we obtain hωA (γ; αU ) ≤ δ + ε(η(kAk) − η(1 + kAk)) + lim

n→∞

CAR : hωA (αU ) ≤ S(ωA |A(H0 ) ) =

m0 X

(η(λm ) + η(1 − λm )) ,

m=1

CCR : hωA (αU ) ≤ S(ωA |U (H0 ) ) =

m0 X

(η(λm − η(1 + λm ))) ,

m=1

R where λm = T λm (z)dλ(z). Applying these inequalities to the operator U n and using the equality hωA (αU ) = n1 hωA (αU n ), we may conclude that CAR : hωA (αU ) ≤

m0 X n 1 X (η(λmnk + η(1 − λmnk ))) , n m=1

(3.6)

m0 X n 1 X (η(λmnk − η(1 − λmnk ))) , n m=1

(3.7)

k=1

CCR : hωA (αU ) ≤

k=1

where λmnk = n

R

k n k−1 n

λm (e2πit )dt.

To complete the proof of our assertion, we apply the following lemma to (3.6) and (3.7).


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39

Lemma 3.1. Let g be a bounded measurable function, f a continuous function. Then ! Z Z nk n 1 1X f n g(t)dt = f (g(t))dt . lim k−1 n→∞ n 0 n k=1

Proof. Define a linear operator Fn on L1 (0, 1), Z nk k−1 k , h(t)dt on . (Fn h)(τ ) = n k−1 n n n Then Fn → id pointwise-norm. Indeed, since kFn k = 1, it suffices to prove the assertion for continuous functions, for which it is obvious. Thus Fn g → g in mean, hence in measure. By virtue of the of f , we conclude that f ◦ R 1 uniform continuity R1 Fn g → f ◦ g in measure, whence 0 f ◦ Fn g dt → 0 f ◦ g dt. 4. Lower Bound for the Entropy: Basic Estimate The aim of this section is to prove the following estimate. Proposition 4.1. For given ε > 0 and C > 0 (C < 1 for CAR), there exists δ > 0 such that if Spec A ⊂ (λ0 − δ, λ0 + δ) for some λ0 ∈ (0, C) and the spectrum of U n has Lebesgue component for some n ∈ N, then CAR : hωA (αU |A(H)e ) ≥ CCR : hωA (αU ) ≥

1 (η(λ0 ) + η(1 − λ0 ) − ε) ; n

1 (η(λ0 ) − η(1 + λ0 ) − ε) . n

First, we will prove that if f ∈ H is close to be an eigenvector for A, then the restriction of the state ωA to C(Cf ) ⊗ (C(Cf )0 ∩ C(H)) is close to the product of the restrictions. Lemma 4.1. Let {eij }i,j be a system of matrix units in a W ∗ -algebra M, e = P k ekk , ω a normal faithful state on M. Then, for any x ∈ M commuting with the matrix units, we have X 1/2 1/2 1/2 kλj σ−i/2 (ekj ) − λk ekj kω kxk# |ω(ekk x) − λk ω(x)| ≤ 2 λk k1 − ekω + ω , j ∗ ∗ 1/2 and λk = ω(ekk ). where kxk# ω = (ω(x x) + ω(xx ))

Proof. Let ξ = ξω and J = Jω be the cyclic vector and the modular involution corresponding to ω. We have 1/2

1/2

1/2

λj ω(ekk x) = λj ((λj Jejk − λk ekj )ξ, Jejk xξ) + λk (ekj Jx∗ ξ, (λj Jejk − λk ekj )ξ) + λk (xξ, Jejj ξ) , 1/2

1/2

1/2


40

00062

S. V. Neshveyev

whence |λj ω(ekk x) − λk (xξ, Jejj ξ)| ≤ (λj kxkω + λk kx∗ kω )kλj σ−i/2 (ekj ) − λk ekj kω 1/2

1/2

1/2

1/2

1/2

1/2

≤ 2kxk# ω kλj σ−i/2 (ekj ) − λk ekj kω .

(4.1)

Further, X 1/2 ω(ekk x) − λj ω(ekk x) = ω(1 − e)|ω(ekk x)| ≤ k1 − ekω λk kxkω ,

(4.2)

and X = λk |(xξ, J(1 − e)ξ)| ≤ λ1/2 k1 − ekω kxkω . λ (xξ, Je ξ) − λ ω(x) k jj k k

(4.3)

j

j

Summing up (4.1)–(4.3), we obtain the desired estimate. Recall that in Sec. 2 we introduced a system of matrix units {eij (f )}i,j in πωA (C(H))00 (f ∈ H, kf k = 1). In the sequel we will identify C(H) with its image in B(HωA ). Lemma 4.2. CAR: For given ε > 0, there exists δ > 0 such that if Spec A ⊂ (0, 1) and

1/2 1/2

λ A

f− f

1−A 1−λ 0, C > 0 and k, j ∈ Z+ , there exists δ > 0 such that if Spec A ⊂ (0, C) and

1/2 1/2

λ A

f− f

1+A 1+λ k, i.e. m = k + n for some n ∈ N. We have (see (2.5)) 1/2 (n − 1)! . ek+n,k+n = ckn a∗ (f )k+1 en−1,k+n , where ckn = (k + n)! Using (2.4), we obtain ∆1/2 ek+n,k+n ξ = ckn Jek+n,n−1 Ja∗ (B 1/2 f )k+1 ξ . Since k(a∗ (f1 )k+1 − a∗ (f2 )k+1 )ξk is bounded by a value which depends only on k, kAk, kfi k and kf1 − f2 k (this is most easily seen from the explicit description of the GNS-representation in terms of the Fock representation, see Example 5.2.18 in [2]), we conclude that σ−i/2 (ek+n,k+n )ξ is close to β

k+1 2

ckn Jek+n,n−1 Ja∗ (f )k+1 ξ

when δ is sufficiently small. But then ekk σ−i/2 (ek+n,k+n )ξ is close to β

k+1 2

ckn Jek+n,n−1 Jekk a∗ (f )k+1 ξ = 0 .

It remains to consider the case j 6= k. As above, we may suppose that j > k, j = k + n for some n ∈ N. We have 1/2 k! . ek+n,k = dkn a∗ (f )n ekk , where dkn = (k + n)!


42

00062

S. V. Neshveyev

As above, we conclude that σ−i/2 (ek+n,k )ξ is close to β 2 dkn Jekk Ja∗ (f )n ξ n

for sufficiently small δ, so ωA (ek,k+n σ−i/2 (ek+n,k )) is close to β 2 dkn (ek,k+n a∗ (f )n ξ, Jekk ξ) = β 2 (ekk ξ, Jekk ξ) = β 2 ωA (ekk σ−i/2 (ekk )) . n

n

n

λk , (1+λ)k+1 1/2

As we have proved, ωA (ekk σ−i/2 (ekk )) can be made close to λk = n 2

then β ωA (ekk σ−i/2 (ekk )) is close to

n λ ( 1+λ )2

·

λk (1+λ)k+1

= (λk λk+n )

but

.

Lemma 4.3. For given N ∈ N and ε > 0, there exists δ = δ(ε, N ) > 0 such that if A is an abelian W ∗ -algebra, ω a normal faithful state on A, B ⊂ A a W ∗ -subalgebra, P and {xi }N i=1 a family of projections in A such that i xi = 1 and |ω(xi y) − ω(xi )ω(y)| ≤ δkyk then

X

η(ω(xi yj )) ≥

i,j

X

∀ y ∈ B, i = 1, . . . , N ,

η(ω(xi )) +

i

X j

for any finite family of projections {yj }j in B with

η(ω(yj )) − ε

P j

yj = 1.

Proof. Cf. [6, Lemma 3.2]. The proof of Theorem 3.1 in [6] shows that Lemma 4.3 is also valid for nonabelian A (with xi ∈ B 0 ∩ A) and without the requirement that xi ’s and yj ’s are projections, but we will not use this fact. Proof of Proposition 4.1. Consider the case of CCR-algebra. There exists δ1 > 0 such that ε ∀ λ0 ∈ (0, C) ∀ λ ≥ 0 : |λ − λ0 | < δ1 . |η(λ) − η(1 + λ) − η(λ0 ) + η(1 + λ0 )| < 6 We can find N ∈ N such that ∞ X λk ε η < (1 + λ)k+1 6

∀ λ ∈ (0, C + δ1 ) .

k=N

Then using (2.7), we obtain N −1 X λk ε η > η(λ0 ) − η(1 + λ0 ) − (1 + λ)k+1 3 k=0

∀ λ0 ∈ (0, C) ∀ λ ≥ 0 : |λ − λ0 | < δ1 .

(4.4)

By assumptions of Proposition, there exists f ∈ H such that {U kn f }k∈Z is an PN −1 orthonormal system in H. Set pk = ekk (f ), k = 0, . . . , N −1, and pN = 1− k=0 pk .


00062


43

Let P be the algebra generated by pk , k = 0, . . . , N . Then hωA (αU ) ≥ lim

k→∞

≥ lim

k→∞

1 Hω (P, αU (P), . . . , αkn−1 (P)) U kn A 1 k(n−1) Hω (P, αnU (P), . . . , αU (P)) kn A

1 k→∞ kn i

≥ lim

N X

(k−1)n

η(ωA (pi0 αnU (pi1 ) · · · αU

(pik−1 )))

0 ,...,ik−1 =0

1X S(ωA |P , ωA (·σ−i/2 (pj ))|P ) . n j=0 N

+

(4.5)

We want to prove that if Spec A ⊂ (λ0 − δ, λ0 + δ) with sufficiently small δ, then the first term in (4.5) is close to n1 (η(λ0 ) − η(1 + λ0 )) to within nε , while the second term is close to zero. Start with the second term. We have N X

S(ωA |P , ωA (·σ−i/2 (pj ))|P ) =

j=0

N N X X

ωA (pk σ−i/2 (pj ))

j=0 k=0

× (log ωA (pk σ−i/2 (pj )) − log ωA (pk ))   N N X X η(ωA (pk )) − η(ωA (pk σ−i/2 (pj ))) . = j=0

k=0

By Lemma 4.2, ωA (pk σ−i/2 (pj )) can be made arbitrary close to δkj ωA (pk ) (more precisely, we can state that this is true for j ≤ N − 1, but since ωA (pk σ−i/2 (pN )) = ωA (pN σ−i/2 (pk )) and ωA (pN ) − ωA (pN σ−i/2 (pN )) =

N −1 X

ωA (pN σ−i/2 (pk )) ,

k=0

this holds for all k, j ≤ N ). Hence, there exists δ2 ∈ (0, δ1 ) such that if Spec A ⊂ (λ0 − δ2 , λ0 + δ2 ), λ0 ∈ (0, C), then N X

ε S(ωA |P , ωA (·σ−i/2 (pj ))|P ) > − . 3 j=0

Turning to the first term in (4.5), set ε ,N + 1 , ε1 = δ 3 where δ(·, ·) is from Lemma 4.3. Find N1 ∈ N such that N21 ε1 C + δ2 . < 1 + C + δ2 8N

(4.6)

(4.7)


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Then

NX 1 −1

ekk (f )

1 −

k=0

k η − (k − 1) , ≥k j+1 3 (1 + λ) 3 j=0 j=0 N X

ε ≥ ··· 3

(4.10)

where λ = (Af, f ) ∈ (λ0 − δ3 , λ0 + δ3 ). It follows from (4.4), (4.6) and (4.10) that we may take δ = δ3 . The proof for CAR is similar, and we omit the details. 5. Lower Bound for the Entropy: End of the Proof In this section we will complete the proof of the lower bound for the entropy. By virtue of the existence of an ωA -preserving conditional expectation C(H) → C(Ha ), we have hωA (αU ) ≥ hωA (αUa ). So we may suppose that U has absolutely continuous spectrum. First, we will extend Proposition 4.1 to arbitrary unitaries. The main step here is the following observation. Lemma 5.1. Let Un be a unitary operator on Hn , n ∈ N, and {zn }∞ n=1 ⊂ T. L∞ 0 00 Consider two unitary operators U and U on H = n=1 Hn , U0 =

∞ M n=1

Un ,

U 00 =

∞ M n=1

z n Un .


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45

Then hωA (αU 0 ) = hωA (αU 00 ) for any αU 0 - and αU 00 -invariant quasi-free state ωA on C(H). For CAR, the same holds for the restrictions of the automorphisms to the even part A(H)e of the algebra. Proof. For CAR, this was proved in [7, Lemma 2.4]. Though for CCR the result is valid by similar reasons, this case requires some additional arguments, since the local algebras are not finite-dimensional. L∞ Consider the unitary operator V = n=1 zn . We state that there exists a set {Ci }i of finite-dimensional C ∗ -subalgebras of C(H)00 (⊂ B(HωA )) such that αV (Ci ) = Ci and hωA (α) = sup hωA (Ci ; α) i

for any ωA -preserving automorphism α. Suppose the statement is proved. Then, since αU 00 = αV αU 0 = αU 0 αV , we have αkU 00 (Ci ) = αkU 0 (Ci ) ∀ k ∈ Z, and hence hωA (Ci ; αU 00 ) = hωA (Ci ; αU 0 ) ∀ i, whence hωA (αU 00 ) = hωA (αU 0 ). So it remains to prove the existence of Ci ’s. For each n ∈ N, choose an increasing S sequence {Hnk }∞ k=1 of finite-dimensional subspaces of Hn such that k Hnk is dense in Hn . Set Kn = H1n ⊕ · · · ⊕ Hnn . Then Kn is finite-dimensional, Kn ⊂ Kn+1 , ∪Kn is dense in H. Since V Kn = Kn , for CAR we may take Cn = A(Kn ) (respectively, for the even part we may take A(Kn )e ) [7, Lemma 2.4]. For CCR, we can not take U(Kn )’s, since they are infinitedimensional. However there exist finite-dimensional subalgebras of U(Kn )00 that are still invariant under αV . Namely, for any finite-dimensional subspace K of H and any n ∈ N, we define a finite-dimensional C ∗ -subalgebra Un (K) of U(H)00 as follows. Let NK be the number operator corresponding to K, i.e., NK = a∗ (f1 )a(f1 ) + · · · + a∗ (fm )a(fm ) , where f1 , . . . , fm is an orthonormal basis in K. This is a selfadjoint operator affiliated with U(K)00 , its spectrum is Z+ (see [2]). Let Pn (K) be the spectral projection of NK corresponding to [0, n − 1]. Set Un (K) = Pn (K)U(K)00 Pn (K) + C(1 − Pn (K)) . The algebra Un (K) is finite-dimensional, since in the Fock representation of U(K) Ln−1 k the projection Pn (K) is the projection onto the first n components k=0 S K of the symmetric Fock space over K, and any regular representation of U(K) is quasi-equivalent to the Fock representation. If V K = K, then αV (NK ) = NK and S αV (U(K)) = U(K), hence αV (Un (K)) = Un (K). Since n U(Kn )00 is weakly dense S in U(H)00 , and m Um (Kn ) is weakly dense in U(Kn )00 , by Lemma 2.2 we conclude that any channel in U(H)00 can be approximated in strong operator topology by a


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channel γ in Um (Kn ) for some m, n ∈ N. But then hωA (γ; α) ≤ hωA (Um (Kn ); α). Thus we may take Cmn = Um (Kn ). Lemma 5.2. Let X1 , X2 be measurable subsets of T, λ(X1 ), λ(X2 ) > 0. Then there exist a measurable subset Y of X1 , λ(Y ) > 0, and z ∈ T such that zY ⊂ X2 . Proof. See, for example, Lemma 3.5 in [7]. Now we can extend Proposition 4.1 to arbitrary unitaries (with absolutely continuous spectrum). Consider a direct integral decomposition Z ⊕ Z ⊕ Hz dλ(z) , U = zdλ(z) , H= T

T

and set X = {z ∈ T|Hz 6= 0}. Lemma 5.3. For given ε > 0 and C > 0 (C < 1 for CAR) there exists δ > 0 such that if Spec A ⊂ (λ0 − δ, λ0 + δ) for some λ0 ∈ (0, C), then CAR : hωA (αU |A(H)e ) ≥ λ(X)(η(λ0 ) + η(1 − λ0 ) − ε) ; CCR : hωA (αU ) ≥ λ(X)(η(λ0 ) − η(1 + λ0 ) − ε) . Proof. Consider the case of CAR-algebra. Choose δ > 0 as in the formulation of P 1 Proposition 4.1. Let {nk }∞ k=1 ⊂ N be a sequence such that λ(X) = k nk . The Zorn lemma and Lemma 5.2 ensure the existence of an at most countable set {X1m }m of disjoint measurable subsets of X and a set {z1m }m ⊂ T such that G 1 zkm Xkm mod 0 (5.1) = exp 2πi 0, nk m holds for k = 1. Proceeding by induction, we obtain a countable measurable partition {Xkm }k,m of X and a countable subset {zkm }k,m of T such that (5.1) holds for all k ∈ N. Let Hkm be the spectral subspace for U corresponding to the set Xkm . L Set Hk = m Hkm , and define a unitary operator Uk on Hk , M zkm U |Hkm . Uk = m

By Lemma 5.1 and Proposition 4.1, we have hωA (αU |A(Hk )e ) = hωA (αUk |A(Hk )e ) ≥

1 (η(λ0 ) + η(1 − λ0 ) − ε) . nk

For any k0 ∈ N, there exists an ωA -preserving conditional expectation A(H) → Nk0 k=1 A(Hk )e (see Remark 4.2 in [12]). By virtue of the superadditivity of the entropy [12, Lemma 3.4], we conclude that


00062


hωA (αU |A(H)e ) ≥

k0 X

47

hωA (αU |A(Hk )e )

k=1

≥

k0 X 1 nk

! (η(λ0 ) + η(1 − λ0 ) − ε) .

k=1

Letting k0 → ∞, we obtain the estimate we need. The proof for CCR is similar, and we omit it. We are already able to complete the proof in the case where Az has continuous spectrum on a set of positive measure, i.e. we now show Corollary 1.1. Proof of Corollary 1.1. We will consider only the case of CAR-algebra. Fix δ0 ∈ (0, 12 ) and take ε ∈ (0, η(δ0 )). Let δ be as in the formulation of Lemma 5.3 with C = 1 − δ0. For any Borel subset X of R, let 1X (A) be the spectral projection of A corresponding to X. Then Z ⊕ 1X (Az )dλ(z) . 1X (A) = T

Define a measurable function φX on T, ( 1, φX (z) = 0,

1X (Az ) 6= 0 , otherwise .

By Lemma 5.3, we conclude that if X is a Borel subset of (λ0 − δ, λ0 + δ) for some λ0 ∈ (δ0 , 1 − δ0 ), then Z hωA (αU |A(1X (A)H)e ) ≥ (η(λ0 ) + η(1 − λ0 ) − ε) φX (z)dλ(z) T

Z ≥ η(1 − δ0 ) ·

T

φX (z)dλ(z) ,

(5.2)

where we have used the inequality η(λ0 ) + η(1 − λ0 ) ≥ η(δ0 ) + η(1 − δ0 ). Let t0 = δ0 < t1 < · · · < tm = 1 − δ0 , tk − tk−1 < δ. Then by the same reasons as in the proof of Lemma 5.3, we obtain from (5.2) the inequality Z X m φ(tk−1 ,tk ] (z)dλ(z) . hωA (αU ) ≥ η(1 − δ0 ) · T k=1

Letting max(tk − tk−1 ) → 0, we conclude that if hωA (αU ) < ∞, then (δ0 , 1 − δ0 ) ∩ Spec Az is finite for almost all z ∈ T. Since δ0 is arbitrary, Az has pure point for almost all z provided the entropy is finite. It remains to consider the case where Az has pure point spectrum for almost all z. Then (see Appendix B) H=

N M n=1

L2 (Xn , dλ) ,


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where Xn is a measurable subset of T, N ≤ ℵ0 , U and A act on L2 (Xn ) as multiplications by functions z and λn (z), respectively. We must prove that N Z X (η(λn (z)) + η(1 − λn (z)))dλ(z) , CAR : hωA (αU ) ≥ n=1

CCR : hωA (αU ) ≥

Xn

N Z X n=1

(η(λn (z)) − η(1 + λn (z)))dλ(z) .

Xn

Again, consider only the case of CAR-algebra. Using the superadditivity as above, we see that it suffices to estimate hωA (αU |A(H)e ) supposing N = 1. As in the proof of Corollary 1.1, fixing δ0 > 0, ε > 0 and choosing t0 = δ0 < t1 < · · · < tm = 1 − δ0 , we obtain m Z X (η(tk ) + η(1 − tk ) − ε)dλ(z) hωA (αU |A(H)e ) ≥ k=1

{tk−1 0, is a trace class semigroup in HΛ . For example, these assumptions are valid when V (x) ≥ Kx2 + B, x ∈ R1 ,

for some K > 0, B ∈ R1 ,

(1.9)

and without loss of generality we will confine ourselves to this case. Further assumptions on the growth of V at infinity (like |V (x)| ≤ C(1 + |x|)R+1 with R ≥ 1, C > 0) and on the behaviour of its derivative V 0 , depending on the questions under consideration, will be specified later on. An important class of models which can be treated by our technique consists of the so called P (ϕ) (lattice) models. In that case the one particle potential has the concrete form V = λP where P is any polynomial of even degree P (x) = p2m x2m + · · · + p1 x + p0 ,

p2m > 0 ,

(1.10)

and λ is a coupling constant, λ > 0. 2. Euclidean Gibbs States of Quantum Lattice Systems: Temperature Loop Space Representation Let us move on to the problem of constructing quantum temperature (Gibbs) states for the lattice system (1.1). The Gibbs state of the system in the domain Λ for an


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inverse temperature 0 < β < ∞ is defined as a functional ωβ,Λ on the algebra AΛ of the form Tr(Ae−βHΛ ) (A ∈ AΛ ) . (2.1) Gβ,Λ (A) = Tr(e−βHΛ ) Naturally, one tries to define the Gibbs state of the infinite-dimensional system given by the heuristic Hamiltonian (1.2) as the functional Gβ on the algebra Aloc obtained from (2.1) by passing to the thermodynamic limit Gβ (A) = lim Gβ,Λn (A) Λn →Zd

(A ∈ Aloc ) .

(2.2)

But, as for unbounded spin systems in quantum statistical mechanics, the problem of a rigorous definition of equilibrium states themselves remains open. The main difficulty here is caused by the noncommutativity associated with the construction of a suitable operator of reflection in the algebra of observables (cf. [34]). S. Albeverio and R. Høegh-Krohn ([4], 1975) have initiated an approach to the construction and study of quantum Gibbs states using through Euclidean (rigorously defined) path space integrals (see also [26–28, 77, 101, 105, 106] for the further extension of the method and related work). This approach transforms the problem of constructing the states Gβ as functionals on the algebra of observables for the infinite system at the inverse temperature β into the problem of studying some Gibbs measures νβ on the space Ωβ of periodic trajectories (loops) on [0, β] with values in d RZ . More exactly, in full analogy to Euclidean field theory, the Gibbs state Gβ is uniquely recovered from the corresponding temperature Green functions, which in their turn can be defined as moments of a measure νβ . From the above reasoning the measures νβ will be called Euclidean Gibbs states for the system (1.1) in the temperature loop space representation. Namely, let Sβ be a circle of length β (sometimes it will be convenient to interpret Sβ as the interval [0, β] with identified ends). The spaces C(Sβ ) and L2 (Sβ ) consist of all continuous respectively square integrable (relative to Lebesgue measure) functions x : Sβ → R1 , equipped with the sup-norm k · kC(Sβ ) respectively 1/2

L2 -norm k · kL2 (Sβ ) := h·, ·iL2 . For the corresponding Borel σ-algebras we have B(C(Sβ ) = B(L2 (Sβ )) ∩ C(Sβ ) . As the configuration space we introduce the loop space (“loop lattice”) Ωβ := C(Sβ )Z = {X(·) = {xk (·)}k∈Zd |X : Sβ → RZ , xk (·) ∈ C(Sβ )} d

d

(2.3)

endowed with the product topology and with the Borel σ-algebra B(Ωβ ) (= σalgebra generated by the cylinder sets {X ∈ Ωβ | (xk )k∈Λ ∈ BΛ ∈ B(C(Sβ )Λ ), Λ ⊂ Zd with |Λ| < ∞}) . The Euclidean measure νβ corresponding to a Gibbs state Gβ of the system (1.1) has the heuristic representation 1 exp{−Eβ [X(·)]}k∈Zd × dxk (u) , (2.4) dνβ (X(·)) = Zβ k∈Zd u∈Sβ


00065

Glauber Dynamics for Quantum Lattice Systems

where Z

"

Eβ [X(·)] = Sβ

# X 1 X 1 X 2 x˙ k (u) + akj xk (u)xj (u) + V (xk (u)) du 2 2 d d d k∈Z

k,j∈Z

59

(2.5)

k∈Z

is the Euclidean action functional of the system and Zβ is a normalizing factor. In full analogy with classical statistical mechanics (cf. [46, 81, 143]), a rigorous meaning can be given to the measure νβ by the Dobrushin–Lanford–Ruelle formalism as an infinite volume Gibbs distribution with spin space C(Sβ ). Here we will explain only the general framework, for more detailed considerations we refer to our papers [15–17]. The Euclidean Gibbs measure νβ is described by a corresponding family of local specifications νβ,Λ , Λ ⊂ Zd , |Λ| < ∞. To construct a reference measure ×k∈Zd dσβ (xk (·)) one should first specify the following one-particle distribution on C(Sβ ) which heuristically looks like ( Z ) 1 1 2 x˙ (u) + V (x(u)) du ×u∈Sβ dx(u) . dσβ (x(·)) = exp − (2.6) Zβ Sβ 2 2 ˜ Since V (x) ≥ Kx2 + B with K > 0, we have the decomposition V (x) = K 2 x + V (x) with inf R1 V˜ (x) > −∞. Let ∆β be the Laplace–Beltrami operator on the circle Sβ and let γβ,K be the Gaussian measure on (L2 (Sβ ), B(L2 (Sβ )) with zero mean value and correlation operator (−∆β + K1l)−1 , i.e. Z −1 1 eihφ,xiL2 dγβ,K = e− 2 h(−∆β +K1l) φ,φiL2 , φ ∈ L2 (Sβ ) . L2 (Sβ ) ∗ (Cβ (Sβ )) = 1, Actually, the set C(Sβ ) of continuous loops has full measure, i.e. γβ,K and the measure γβ on the space C(Sβ ), B(C(Sβ )) can be viewed as the canonical realization of the well-known oscillator bridge process of length β (see e.g. [150]). We then define (by analogy with the Feynman–Kac formula) ( Z ) 1 ˜ exp − (2.7) V (x(u) du dγβ,K (x) dσβ (x) := Zβ Sβ

as a probability measure on (C(Sβ ), B(C(Sβ ))), and moreover, the measure σβ has all the moments of the form Eσβ [kxkM C(Sβ ) ] < ∞ ,

M ∈ N,

(2.8)

due to the corresponding property of the Gaussian measure γβ,K . One can also use an equivalent approach via the Kolmogorov theorem and define σβ in terms of the ∂2 integral kernels q(t; x, y) of the semigroup exp t( 12 ∂x 2 − V ), t ≥ 0, in L2 (R, dx). Namely, for any finite system of ordered points on the circle (uj )nj=0 ⊂ Sβ , uj ≤ uj+1 , un+1 = u0 , and for any Borel sets (Bj )nj=0 ⊂ B(R), we have:


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σβ ({X(·) ∈ Ωβ |X(u0 ) ∈ B0 , . . . , X(un ) ∈ Bn }) Z n Y 1 q(uj+1 − uj ; xj+1 , xj ) ×nj=0 dxj . = u0 ,...,un Zβ B0 ×···×Bn j=0 Note that the heuristic presentation (2.6) of the measure σβ originates from a standard lattice approximation used in constructive field theory (cf. [83, 150]). The local specifications νβ,Λ , |Λ| < ∞, are defined as stochastic kernels on (Ωβ , B(Ωβ )) in the following way: ∀ B ∈ B(Ωβ ), Y ∈ Ωβ Z 1 νβ,Λ (B|Y ) := exp{−Iβ,Λ (X|Y )}1lB (xΛ × yΛc ) ×k∈Λ dσβ (xk ) . Zβ,Λ (Y ) Ωβ,Λ (2.9) Here Ωβ,Λ := C(Sβ )Λ ,

xΛ := (xk )k∈Λ ∈ Ωβ,Λ ,

Zβ,Λ (ξ) is a normalization constant, and  Z  X 1  akj xk (u)xj (u) + IβΛ (X|Y ) := 2 Sβ  {k,j}⊂Λ

 X {k,j}⊂Zd k∈Λ,j∈Λc

 akj xk (u)yj (u)  du

(2.10)

is the harmonic pair interaction in the volume Λ under the external boundary condition yΛc := (yj )j∈Λc , Λc := Zd \ Λ. Obviously, each νβ,Λ (dX|Y ) is concentrated on configurations of the form X = (xΛ , yΛc ) and Iβ,Λ (X|Y ) is a continuous funcd tion with respect to X, Y ∈ Ωβ (or even X, Y ∈ L2 (Sβ )Z ); below we will refer to the latter as the regularity property of Iβ,Λ . Heuristically, we have in a way corresponding to (2.4), (2.5), νβ,Λ (dX(·)|Y (·)) =

1 exp{−Eβ,Λ[X(·)|Y (·)]} × dxk (u) Zβ,Λ k∈Zd

(2.11)

u∈Sβ

with the finite volume Euclidean action functional # Z " X X 1 2 Eβ,Λ [X(·)|Y (·)] = Iβ,Λ (X(·)|Y (·)) + x˙ (u) + V (xk (u)) du . Sβ 2 k∈Λ

(2.12)

k∈Λ

An essential point is that for the stochastic kernels (2.9) the consistency condition holds ([46, 47, 81, 143]): for all Λ ⊂ Λ0 , |Λ0 | < ∞, B ∈ B(Ωβ ), and Y ∈ Ωβ Z νβ,Λ0 (νβ,Λ (B|·)|Y ) := νβ,Λ0 (dX|Y )νβ,Λ (B|X) = νβ,Λ0 (B|Y ) . (2.13) Ωβ

Definition 2.1. A probability measure νβ on (Ωβ , B(Ωβ )) is called Euclidean Gibbs state in the temperature loop space (TLS) representation (corresponding to the lattice system (1.1) at inverse temperature β) if and only if it satisfies the


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61

Dobrushin–Lanford–Ruelle (DLR) equilibrium equations: for all Λ ⊂ Zd , |Λ| < ∞, and B ∈ B(Ωβ ) Z νβ (dX)νβ,Λ (B|X) = νβ (B) , νβ νβ,Λ (B) := Ωβ

i.e., νβ νβ,Λ (B) = νβ .

(2.14)

We denote by Gβ the set of all Gibbs measures for our system at a fixed inverse temperature β. But the definition, as it was given, is too extensive, and the usual way to avoid Gibbs states without physical relevance is to impose some growth conditions at infinity [83]. We will restrict our considerations to the subset Gβt of tempered Gibbs measures specified by the following assumption on their moments (hkxk kL2 (Sβ ) iνβ )k∈Zd ∈ S 0 (Zd ) ,

(2.15)

where, as usual, S 0 (Zd ) is the space of slowly increasing sequences over Zd (for its definition see also Sec. 4). Here and further on we write Z f (X) dνβ (X) (2.16) hf iνβ := Eνβ (f ) = Ωβ

for any νβ integrable f : Ωβ → R. Depending on each specific class of models, the study of the Euclidean measure νβ , e.g. the existence and uniqueness problem for it, can be performed by one or more of the following methods, which originated from classical statistical mechanics and allow a generalization to quantum lattice systems: (i) General criteria for existence of limiting Gibbs distributions (quantum analogs of the theorem of R. Dobrushin [26, 46, 47]); (ii) Superstability estimates (extension of results of D. Ruelle [147], and J. Lebowitz and E. Presutti [116] to the quantum case) [141]; (iii) Method of cluster expansions [4, 112, 140–142]; (iv) Method of correlation inequalities [10–12, 77, 107]; (v) Reflection positivity method for Gibbs states with periodic boundary conditions [27–30, 105, 106]. Note that all these methods look into the relationship between the Gibbs measure νβ and the thermodynamic limits of local Gibbs specifications νβ,Λ when Λ % Zd . Regarding the quantum lattice system with Hamiltionian (1.1), the simplest way to show that Gβt 6= ∅ is to observe that the interaction is superstable as soon as i 1h (2.17) K + a(0) − max |a(k)| > 0 , 2 k∈Zd which is the case under the assumptions (1.7), (1.9). Then, as was shown in [141], ⊂ Ωtβ (for instance, such that for a subclass of boundary conditions Y ∈ Ωt,0 β supk∈Zd kyk kL2 (Sβ ) < ∞), the family of local specifications νβ,Λ , Λ ⊂ Zd , |Λ| < ∞,


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has at least one accumulating point νβ from the subset of so-called Ruelle type “superstable” Gibbs measures Gβt,0 ⊂ Gβt . Moreover, from [15, 16] we get the following criterion on the uniqueness of νβ ∈ Gβt : Proposition 2.1. Suppose that the anharmonic self-interaction has the form V = V0 + W, where V0 ∈ C 2 (R1 ) is a convex function with polynomially bounded derivatives, that is, ∃ b, C > 0, R ∈ N : ∀ x ∈ R1 V000 (x) ≥ 2b ,

(l)

|V0 (x)| ≤ C(1 + |x|R+1 ), l = 0, 1, 2 ;

(2.18)

the potential W describes the presence of (possible) wells in the interaction V and is given by a bounded function W ∈ Cb (R1 ) , Then |Gβt | = 1 provided

δ(W ) := sup W − inf1 W < ∞ . R1

X k6=0

R

(2.19)

|a(k)|eβδ(W )

a(0) + 2b

< 1.

(2.20)

Trivially, (2.20) holds for any convex self-interaction V = V0 and dynamical matrix A = (a(k − j))k,j∈Zd satisfying the condition (1.7). In the present paper we suggest the method of stochastic quantization to be used for studying the limit measure (2.4). The efficiency of this method for our class of systems, as compared with the methods mentioned above, will be discussed later on. 3. The General Scheme of the Stochastic Quantization Method: Main Assumptions and Notations A complete realization of the stochastic quantization procedure (or “stochastic dynamics” method) applied to a lattice system with the Euclidean action Eβ [X], at the inverse temperature β, involves the following stages: 1. Existence. Constructing a random process Xt = {xk,t (u)}k∈Zd ,u∈Sβ , t ≥ 0, as the unique solution of the Langevin stochastic evolution equation 1 δEβ [X(·)] ∂ xk,t (u) = − (Xt (u)) + w˙ k,t (u) , ∂t 2 δxk (·)

k ∈ Zd .

(3.1)

Here t ≥ 0 is an “artificial” or “computer” time (the time in the sense of stochastic quantization), w˙ k,t (u) is a Gaussian white noise on Ωβ × [0, ∞) (heuristically, E w˙ k1 ,t1 (u1 )w˙ k2 ,t2 (u2 ) = δ(k1 − k2 )δ(t1 − t2 )δ(u1 − u2 ) δE [X(·)]

β for any k1 , k2 ∈ Zd , t1 , t2 ≥ 0, u1 , u2 ∈ Sβ ) and δx is a functional (variational) k (·) derivative of the energy Eβ [X(·)] associated with the configuration X(·) ∈ Ωβ .

2. Long-time asymptotic analysis of the stochastic dynamics. Establishing the ergodicity of the process Xt , t ≥ 0, (i.e. the existence and uniqueness of an


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invariant measure µβ,inv on a suitable state space X of trajectories and the weak convergence of the laws µ(Xt ) of the random variables Xt to µβ,inv as t → ∞) or describing its nonergodic behaviour (depending on the choice of an initial law µ0 ). A principal point of the whole scheme is the identity of the expectations given by the ergodic theorem: Z Z 1 t E[f (Xs )]ds = f (X)dµβ,inv (X) (f ∈ Cb (X )) . (3.2) lim t→∞ t 0 X 3. Identification of the quantum Gibbs distributions ν β (as limiting points of {νβ,Λ , |Λ| < ∞}, Λ % Zd ) with the invariant (reversible) distributions µβ,inv of the process Xt , t ≥ 0 (as limiting points of the laws {µ(Xt )}t≥0 , t → ∞). Generally speaking, under natural assumptions on the system (1.1) we have only a priori the inclusion Gβt ⊂ I for the sets of all tempered Gibbs and invariant distributions respectively. Hence, we can substitute the integration with respect to the unknown Gibbs measure (2.4) by the construction of a stochastic time dynamics governed by the Langevin equation. The applications of the method of stochastic dynamics in classical statistical mechanics goes back to the paper of R. J. Glauber ([78], 1963) on the time-dependent stochastics of the Ising model. In quantum physics it was introduced as a numerical tool, a so-called “stochastic quantization procedure”, by the physicists G. Parisi and Y. Wu ([144], 1981). From the physical point of view, the Langevin equation (3.1) means the following [130]: We insert the random forces d k w˙ k,t and the friction forces ∂x ∂t , k ∈ Z , into the system (1.1), they compensate each other and bring the system to the equilibrium over a long period of the “time” t ≥ 0. The mathematical background of the method can already be founded in the early papers of A. Kolmogorov and reflects the well-known relation in the finitedimensional case between the drift coefficient b of a diffusion process, supposed to be of the gradient type, and the density ρ of its reversible distribution ([57, 103]): dxt = b(xt )dt + dwt , 1 b(x) = − ∇ρ(x) , x ∈ Rd . 2 After calculating the variational derivative in (3.1) and taking into account (1.5) and (1.6), we arrive at an infinite system of stochastic parabolic differential equations  2   ∂ xk,t (u) = 1 ∂ xk,t (u) + fk (xk,t (u), Xt (u)) + w˙ k,t (u) ∂t 2 ∂u2 (3.3)   d k ∈ Z (t > 0, u ∈ Sβ ) . dµinv (x) = exp(−ρ(x))dx ,

The drift coefficients fk : R1 × RZ → R1 , 1 1 fk (x, {xj }j∈Zd ) = − V 0 (x) − 2 2 d

X j∈Bρ (k)

a(k − j)xj ,

(3.4)


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are supposed to be at least one time continuously differentiable in x ∈ R1 and d smooth in {xj }j∈Zd ∈ RZ , that is quite enough for our applications. We stress that our goal here is not to release (as much as possible) local regularity assumptions on the drifts fk (x, X), k ∈ Zd , but to study the qualitative behaviour of the system (3.3) when Λ % Zd , t → ∞. Moreover, due to the infinite number of components, d d the global drift term (fk )k∈Zd : RZ → RZ is generally a singular mapping in any d −p fixed weighted Hilbert space l2 (Zd ) ⊂ RZ (see Sec. 4). Throughout this paper we consider the following assumptions (A), (B) on the potential V ∈ C 2 (R) to be always satisfied: (A) (at most polynomial growth) ∃ C > 0, ∃ R ≥ 1 : ∀ x ∈ R1 1 |fk (x, 0)| = V 0 (x) ≤ C(1 + |x|)R ; 2

(3.5)

(B) (l.-dissipativity, i.e. dissipativity involving a linear term lx) ∃ l ∈ R1 : ∀ x, y ∈ R1 1 (x − y)(fk (x, 0) − fk (y, 0)) = − (x − y)(V 0 (x) − V 0 (y)) ≤ l(x − y)2 . (3.6) 2 The polynomial growth condition (3.5) is only of a technical nature: it will be crutially used in the proof of Lemmas 6.2 and 7.2 below. Note also that the assumptions (A), (B) imply the so-called one-sided linear growth condition xfk (x, 0) ≤ lx2 + C|x|. An elementary corollary of (3.5) and (3.6) are also the estimates: ∀ X = {xk }k∈Zd , Y = {yk }k∈Zd ∈ RZ , ! 12 X 1 R 2 |fk (x, X)| ≤ C(1 + |x| ) + kak xj 2 d

∀ x, y ∈ R1 ,

(3.7)

j∈Bρ (k)

and (xk − yk )(fk (xk , X) − fk (yk , Y )) ≤

ε 1 (xk − yk )2 l − a(0) + 2 4 X 1 + kak20 (xj − yj )2 4ε

(3.8)

0 0, a(j) being as in (1.5), and kak := kAe0 kl2 (Zd ) =

X j∈Bρ (k)

! 12

a (j − k) 2

,

kak0 :=

X

! 12 a (j − k) 2

.

0 0, ∃ b2 ∈ R1 : ∀ x, y ∈ R1 1 (x − y)(fk (x, 0) − fk (y, 0)) = − (x − y)(V 0 (x) − V 0 (y)) 2 ≤ −b1 (x − y)2 + b2 .

(3.9)


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65

And lastly, in Sec. 11 we will consider the so called strong dissipative stochastic systems for which the most restrictive condition holds (D) (st.-dissipativity) ∃ b > 0 : ∀ x, y ∈ R1 1 (x − y)(fk (x, 0) − fk (y, 0)) = − (x − y)(V 0 (x) − V 0 (y)) 2 ≤ −b(x − y)2 .

(3.10)

Besides this, as follows from the definition of the dynamical matrix A = (a(k − j))k,j∈Zd in Sec. 1, we always have the estimate (E) ∃ α > 0 ∀ X = {xk }k∈Zd ∈ l2 (Zd ), X 1 xk fk (0, X) = − 2 d k∈Z

X

a(k − j)xk xj

j∈Bρ (k)

1 α = − hAX, Xil2 (Zd ) ≤ − kXk2l2(Zd ) . (3.11) 2 2 We stress that the terminology concerning the different modifications (B)–(D) of the dissipativity property is by no mean of common use, we only introduce it in order to simplify some notations and descriptions in this paper. Also it should be noted that the polynomial lattice models (1.10) gives us an important class of interactions satisfying a priori the assumptions (A)–(C). Before going over to discuss in details the system (3.3), let us mention the main difficulties contained in it. The first one is the infinite number of equations. In case the continuous parameter u ∈ Sβ is absent, we obtain a lattice system of locally interacting diffusions xk,t , k ∈ Zd , of the following type ( dxk,t = fk (xk,t , Xt )dt + dwk,t (3.12) k ∈ Zd (t > 0) . The corresponding infinite-dimensional processes Xt ∈ RZ have been studied rather extensively, especially in connection with the problem of constructing a Glauber stochastic dynamics for classical Gibbs measures, see e.g. [10–13, 40, 42, 43, 47, 65, 66, 75, 91, 95, 117–119, 149]. The second problem, which arises even for a bounded region Λ ⊂ Zd , is due to the presence of the singular random forces w˙ k,t (u) (which are clearly outside the scope of the standard theory of partial differential equations). Note that the single equation d

1 ∂2 ∂ xt (u) = xt (u) + f (xt (u)) + w˙ t (u) (t > 0, u ∈ Sβ ) (3.13) ∂t 2 ∂u2 describes the motion of a closed Brownian string [69]. Finite systems of such SPDE’s, under different regularity conditions on the random forces w˙ t (u) and drifts fk , are of great interest in mathematical physics, especially in statistical hydrodynamics (e.g. stochastic Navier–Stokes, Burgers equations; see e.g. [1, 50, 62] and references therein), in the kinetic theory of phase transitions (e.g. reaction-diffusion,


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Ginzburg–Landau equations; see e.g. [61, 67, 69–72, 86–88, 89, 100, 102, 109, 122– 125, 133, 143]) and in stochastic quantization of Euclidean field theory (see e.g. [4, 21, 22, 32, 33, 49, 96–97, 98, 126–128, 144, 147, 148, 160]). Hence, our quantum lattice system (3.3) can be viewed as a composition of the two well-known models (3.12) and (3.13) of infinite-dimensional stochastic dynamical systems. In the present paper we elaborate some common methods for dealing with (3.3), (3.12) and (3.13). Our considerations are strong enough to cover the known optimal results for classical lattice systems (3.12) and Brownian strings (3.13). 4. Definition of a Generalized Solution for the Langevin Equations The first step in the study of the stochastic evolution equation (3.3) is the choice of a state space X for a solution Xt . To discuss the behaviour of the system when |k| → +∞ let us introduce the space of sequences on Zd , namely the scale of Hilbert spaces   # 12 "   X d (1 + δ|k|)2p x2k < ∞ , p ∈ Z1 , δ > 0 . l2δ,p (Zd ) := X ∈ RZ | kXklδ,p = 2   d k∈Z

(4.1) Obviously for any fixed p ∈ Z 1 the spaces l2δ,p (Zd ), δ > 0, are topologically isometric with l2p (Zd ) := l21,p (Zd ) through the equivalence of the norms p δ,p C1 (δ, p)k · kδ,p l2 ≤ k · kl2 ≤ C2 (δ, p)k · kl2

(C1 , C2 > 0) .

(4.2)

0

For every p0 > p + d/2 we have a Hilbert–Schmidt embedding l2p (Zd ) ⊂ l2p (Zd ), so that we can define the nuclear space of fastly decreasing sequences S(Zd ) =

lim

p=1,2,...

pr l2p (Zd )

and its dual one of slowly increasing sequences S 0 (Zd ) =

lim

p=1,2,...

ind l2−p (Zd ) .

The spaces S and S 0 give a Schwartz setting for l2 (Zd ) := l20 (Zd ). The only reason to use the continuous scale of spaces l2δ,p (Zd ), δ > 0, is our need to control the monotonicity properties of the dynamical matrix A = (a(k −j))k,j∈Zd provided A ≥ 0 in l2 (Zd ). Namely, due to the finite range of interaction (a(k − j) = 0, |k − j| > ρ) and the local proximity of the norms k · kl2 when δ → 0, we have the following assertion. Lemma 4.1. (see [53]). Suppose that hAX, Xil2 ≥ αkXk2l2 ,

X ∈ l2 (Zd )


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for some α ∈ R1 . Then ∀ ε > 0 ∀ p > d/2∃ δ0 > 0 : ∀ δ, 0 < δ ≤ δ0 hAX, Xil−p,δ ≥ (α − ε)kXk2l−p,δ , 2

2

X ∈ l2−p (Zd ) .

Remark 4.1. The choice of spaces l2−p,δ (Zd ), δ > 0, p > d/2, which suit the assertion of Lemma 4.1, is not unique. As in [52], we can introduce one more continuous scale of weighted Hilbert spaces   # 12 "   X d ˜l−δ (Zd ) := X ∈ RZ | kXk˜−δ = e−δ|k| x2k < ∞ , δ > 0. (4.3) 2 l 2   d k∈Z

The latter case can be treated in the same way as the former one, but the scale (4.1) is more precise for the lattice systems with polynomial interactions. Besides, the statement of Lemma 4.1 is still valid for the dynamical matrix A with infinite range of interaction (ρ = ∞), but fastly decreasing, i.e. such that {a(k)}k∈Zd ∈ S(Zd ). Remark 4.2. We will rely repeatedly on the following elementary estimate which holds for the weight sequence (1 + |k|)−2p , k ∈ Zd : X (1 + |j|)−2p ≤ |Bρ (k)|(1 + ρ)2p (1 + |k|)−2p (4.4) j∈Bρ

where |Bρ (k)| is the cardinality in Zd of the ball Bρ (k) := {j ∈ Zd | |j − k| ≤ ρ} with its centre at k ∈ Zd and radius ρ > 0. In order to study the coordinate processes xk,t , k ∈ Zd , we will use the classical Schwartz spaces (C ∞ (Sβ ) := D(Sβ ) ⊂ L2 (Sβ ) ⊂ D0 (Sβ ), h·, ·iL2 := h·, ·iL2 (Sβ ) ), the Lebesgue spaces (Lr (Sβ ), r ≥ 1) and the Sobolev spaces (Wrq (Sβ ), r ≥ 1, q ∈ R), understood in the usual sense with respect to the Lebesgue measure on Sβ . We will d seek the solution Xt = {xk,t }k∈Zd in the spaces of the functional sequences HZ p,δ p,δ or l2 (H) ∼ = l2 (Zd ) ⊗ H, where H is one of the functional spaces listed above. Actually, to formulate the existence and uniqueness theorem for Xt , it is enough to restrict our choice of a state space X to the following ones l2−p (Lβ,r ) := l2−p (Zd , Lr (Sβ )) , S 0 (Cβ ) := S 0 (Zd , C(Sβ )) ,

l2−p (Cβ ) := l2−p (Zd , C(Sβ )) , d Zd p> , r≥1 . Ωβ := C(Sβ ) 2

(4.5)

The spaces are equipped with the σ-algebra B(X ) of their Borel subsets. Let a probability space (Ω, F , P) with a filtration {Ft }t≥0 be given, as well as a family Wt = {wk,t }k∈Zd of independent D0 (Sβ )-valued standard {Ft }-Brownian motions indexed by the Schwartz space D(Sβ ). This means that for any k1 , k2 ∈ Zd , t1 , t2 ≥ 0, ϕ1 , ϕ2 ∈ D(Sβ ) E{hwk1 ,t1 , ϕ1 iL2 hwk2 ,t2 , ϕ2 iL2 } = δ(k1 − k2 ) min(t1 , t2 )hϕ1 , ϕ2 iL2 .

(4.6)


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Let X = { x k }k∈Zd be a F0 -measurable random variable in some l2−p0 (Cβ ), p0 > d/2, independent of the Brownian motions {wk,t }k∈Zd . Let µ0 be the distribution of X. ◦

Definition 4.1. By a generalized (g.-)solution (generalized, in the commonly accepted sense of PDE’s) of the Cauchy problem to the system (3.3) with initial data ◦ X ∈ l2−p0 (Cβ ) we mean a continuous process Xt ∈ l2−p (Cβ ), with some p ≥ p0 , defined on the probability space (Ω, F , P) and adapted to the filtration {Ft }t≥0 such that for every ϕ ∈ D(Sβ ), almost surely, #  Z t "  1 ◦  xk,s , ∆ϕ + hfk (xk,s , Xs ), ϕiL2 ds hxk,t , ϕiL2 = h x k + wk,t , ϕiL2 + 2 0 L2   k ∈ Zd (t ≥ 0) . (4.7) Since we consider a solution to (3.3) as a continuous process in l2−p (Cβ ), here we present sufficient conditions, analogous to Kolmogorov–Prokhorov’s criterion, for the continuity of the process Xt ∈ l2−p (Cβ ), t ≥ 0, and for the tightness of a family {µ}M of measures on C([0, ∞), l2−p (Cβ )). Lemma 4.2. Assume that for any T > 0 there exist positive constants α, γ, p∗ , 1 α > , p∗ < α(p − d/2) , γ and a positive sequence {ck }k∈Zd ∈ l2−p∗ (Zd ), such that for the random variables xk,t (u) ∈ R1 one has E|xk,t (u) − xk0 ,t0 (u0 )|α ≤ ck (|t − t0 |2+γ + |u − u0 |2+γ ) for all u, u0 ∈ Sβ , t, t0 ∈ [0, T ], k ∈ Zd . Then the process Xt = {xk,t (u)}k∈Zd ,u∈Sβ admits a continuous modification in l2−p (Cβ ). Lemma 4.3. Assume that for any T > 0 there exist positive constants α, γ, p∗ , 1 α > 1 + , p∗ < α(p − d/2) − d , γ and a positive sequence {ck }k∈Zd ∈ l2−p∗ (Zd ), such that for a family {µ}M of probability measures on C([0, ∞), l2−p (Cβ )) one has (a) supM Eµ |xk,0 (0)|α ≤ ck , (b) supM Eµ |xk,t (u) − xk,t0 (u0 )|α ≤ ck (|t − t0 |2+γ + |u − u0 |2+γ ) for all u, u0 ∈ Sβ , t, t0 ∈ [0, T ], k ∈ Zd . Then {µ}M is tight in the sense of weak convergence. The proof of Lemmas 4.2, 4.3 is contained in [156]. The methods are rather standard and based on arguments commonly used for the proof of multidimensional versions of Kolmogorov–Prokhorov’s criterion (see e.g. [84, 155]).


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5. Stochastic Quantization of the Harmonic System: The Ornstein Uhlenbeck Process Here we will deal with the case V = 0 so that the lattice system is harmonic and ◦ the solution Xt , t ≥ 0, with a nonrandom initial datum X , is a Gaussian process. This is the starting point for constructing a solution to the full nonlinear problem. ∂2 Let the operator ∆β with the domain D(∆β ) be the closure of ∂u 2 in L2 (Sβ ) (i.e. ∆β be the Laplace–Beltrami operator on the circle Sβ ). We introduce L = 12 (−∆β ⊗ 1l + 1l ⊗ A) as a positive self-adjoint operator in l2 (Zd ) ⊗ L2 (Sβ ). Then the Langevin equation for the harmonic system (1.8) can be written in the operator form as a linear evolution equation dXt = −LXt dt + dWt .

(5.1)

It has a unique solution given by the Ornstein–Uhlenbeck process ◦

Xt = e−tL X +

Z

t

e−(t−s)L dWs

(5.2)

0

in the space S 0 (Zd )⊗L2 (Sβ ). The convergence of the stochastic quantization method for the harmonic systems has been studied in [108]. There it was shown, in particular, that under the assumption A ≥ m2 1l, m > 0, the process Xt is ergodic and its limit distribution µβ = limt→∞ µt (which is Gaussian with mean zero and correlation operator (2L)−1 ) coincides with the limit Gibbs measure νβ = limΛ→Zd νβ,Λ on Ωβ defined formally by the expression (2.4). Suppose further that the dynamical matrix A is not uniformly positive and there exist generalized eigenvectors ψ ∈ S 0 (Zd ), Aψ = 0. Then the distributions µt of the solution Xt , t ≥ 0, with ◦ ◦ the initial datum X ∈ S 0 (Zd ) ⊗ L2 (Sβ ) converge to µβ when L X 6= 0, or to ◦ ◦ µβ (· − X ) ⊥ µβ in the opposite case X (·) = ψ ⊗ 1l(·). On the other hand, one can prove that the infinite volume Gibbs measure γβ (·|Y ), which is constructed using the fixed boundary condition Y ∈ S 0 (Zd ) ⊗ L2 (Sβ ), coincides also with µβ when LY 6= 0, or with µβ (· − Y ) when Y = ψ ⊗ 1l. Hence, for quantum harmonic systems we have a one-to-one correspondence between the nonergodicity of the stochastic dynamics (5.1) and the phase transition phenomenon for the Gibbs states (2.4). ◦ Moreover, the role of the initial distributions X in the stochastic quantization procedure is similar to that played by boundary conditions Y when passing to the infinite volume Zd . We remark that the assumption A ≥ m2 1l is equivalent to the substitution of (− 21 ∆β + 12 m2 ) for (− 12 ∆β ) in Eq. (3.3). The case A = m2 1l will now be considered in more details. First we need more detailed information about the semigroup et∆β , t ≥ 0, and the fundamental solution qβ,m (t; u, v) = e−

m2 2

t

qβ (t; u, v) ,

qβ,m (0; u, v) = δ(u − v) ,

∂ of the heat equation ( ∂t − 12 (∆β − m2 ))x(t, u) = 0.


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As well known, the operator (− 12 ∆β ) has the discrete spectrum λn = n ∈ Z, and the complete orthonormal system of eigenfunctions 1 i2πnu ϕn (u) = √ e β , β

2π 2 n2 β2 ,

n ∈ Z,

in L2 (Sβ ). The operators et∆β , t ≥ 0, generate a contractive C0 -semigroup in the space C(Sβ ), Lp (Sβ ), W2p (Sβ ), p ≥ 1, and for f ∈ L2 (Sβ ) one has (et∆β f )(u) ∈ C ∞ ((0, +∞) × Sβ ). The kernel qβ ∈ C ∞ ((0, +∞) × Sβ × Sβ ) 1

of the operator e 2 t∆β , t > 0, can be represented by the uniformly convergent series X e−λn t ϕn (u)ϕn (v) qβ (t; u, v) = n∈Zd

=

1 X − 2πβ2 2n2 t i2πn(u−v) β e e β d n∈Z

=

1 ϑ3 β

u − v − 2πβ22 t ;e β

(5.3)

P 2 where ϑ3 (ω, r) = n∈Zd rn ei2πnωt , ω ∈ C 1 , is the classical Jacobi ϑ3 -function with parameter |r| < 1 (see e.g. [25]). Furthermore, applying the Poisson summation formula, we obtain the estimate 1 X − 2πβ2 2n2 t 1 X − n2 β 2 e = √ e 2t , (5.4) |qβ (t; u, v)| ≤ |qβ (t; 0, 0)| = β 2πt d d n∈Z n∈Z with the asymptotic expansions β2 1 + O(e− π2 t ) , t → 0 , |qβ (t; 0, 0)| = √ 2πt √ 2 32 ∂qβ β2 π β + O(e− π2 t ) , t → 0 . ∂t (t; 0, 0) = 4β π 2 t

On the other hand, we have qβ (t; u, v) =

X

(5.5)

q(t; u + nβ, v) ,

n∈Zd (u−v)2

1 where q(t; u, v) := √2πt e− 2t is the fundamental solution to the heat equation 1 on R . Taking into account (5.3)–(5.5), one can easily prove the following estimates on the functions qβ,m (t; u, v) and

∆qβ,m (s; v) := qβ,m (t − s; u, v)χ[0,t) (s) − qβ,m (t0 − s; u0 , v)χ[0,t0 ) (s)


00065


Z

(a)

t 0

kqβ,m (s; u, ·)k2L2 (Sβ ) ds ≤

1 β (

≤ Z

(b)

t

Z 0

X

t

e−(2λn +m

2

)s

71

ds

n∈Zd

C 0 (β, m) ,

m > 0,

C(β)(1 + T ) ,

m = 0;

kqβ,m (s; u, ·)kL2 (Sβ ) ds ≤ C 00 (β, m) ,

m > 0;

0

Z

(c)

t∨t0

0

k∆qβ,m (s, ·)k2L2 (Sβ ) ds ≤ C(β; α, γ)[|∆u|α + |∆t|γ + |∆t|] , 0 ≤ α < 1,

(d) Z

(

t∨t0

k∆qβ,m (s, ·)kL2 (Sβ ) ds ≤

0

0≤γ
0, m = 0,

for all 0 ≤ t0 < t < T < ∞ ,

u, u0 ∈ Sβ ;

∆t = t − t0 ,

∆u = u − u0 .

(5.6)

When A = m2 1l and V ≡ 0, the system (3.1) decouples into a sequence of non-interacting linear equations   dxk,t (u) = 1 (∆β − m2 )xk,t (u) + dwk,t (u) 2 (5.7)  k ∈ Zd (t > 0, u ∈ Sβ ) . ◦

The corresponding Cauchy problem with the initial datum X ∈ l2−p (Cβ ), p > d/2, has the◦ unique solution Gt = {gk,t }k∈Zd ∈ l2−p (Cβ ) explicitly given by the sum ◦ X t + G t = Gt of mutually independent random variables x k,t (u) = (et 2 (∆β −m ) x)(u) , Z t ◦ g k,t (u) = hqβ,m (t − s; u, ·), dwk,s (·)iL2 . ◦

1

2

◦

(5.8) (5.9)

0

We observe that the Ito stochastic integral in (5.9) R t is originally defined by the isometry in L2 (Ω, dP ) (through the finiteness of 0 kqβ,m (t − s; u,◦ ·)k2L2 ds due to (5.6a)) only for fixed t ≥ 0, u ∈ Sβ . The required properties for G t , t ≥ 0, to be a regular l2−p (Cβ ), l2−p (Lβ,r )-valued process are collected in the next lemma (for details see [156]). ◦

Lemma 5.1. The process G t , t ≥ 0, has a continuous modification in the spaces ◦ l2−p (Cβ ), l2−p (Lβ,r ), p > d/2, r ≥ 1. Its components g k,t (u), k ∈ Zd , make up the


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independent Gaussian random fields indexed by t ≥ 0, u ∈ Sβ with mean zero and covariance Z t∧t0 ◦ ◦ 0 0 E( g k,t (u), g k0 ,t0 (u )) = δ(k − k ) qβ,m (t + t0 − 2s; u, u0 ) ds 0 β β X e = m 2 2 2 2 n∈Zd 2π n + β 2 h i 2 2 2 2 2 2 −( 2πβ 2n + m2 )|t−t0 | −( 2πβ 2n + m2 )(t+t0 ) × e . −e i2πn(u−v)

(5.10)

Moreover, the following estimates hold : (i) ∀ r ∈ N, u, u0 ∈ Sβ , t, t0 ∈ [0, T ], 0 < T < ∞: E| g k,t (u) − g k,t0 (u0 )|2r ≤ C(r; α, γ)(|u − u0 |α + |t − t0 |γ + |t − t0 |)r ◦

◦

(5.11)

with 0 ≤ α < 1, 0 ≤ γ < 12 ; (ii) ∀ Q, R, r ≥ 1, p > d/2, 0 < T < ∞: " E

X

#Q −2p

(1 + |k|)

◦

sup 0≤t≤T

k∈Zd

E

k g k,t k2R L2rR

< ∞,

(5.12)

< ∞;

(5.13)

2Q

◦

sup k G t kl−p (Cβ ) 2

0≤t≤T

(iii) If m > 0, then under the same conditions " sup E t≥0

X

#Q −2p

(1 + |k|)

◦

k g k,t k2R L2rR

< ∞,

(5.14)

< ∞.

(5.15)

k∈Zd ◦

sup Ek G t k2Q l−p (C t≥0

2

β)

Proof. At first, we note the following fact (see e.g. the proof of Kolmogorov’s criterion in [84, Sec. 5, Chap. III and Sec. 4, Chap. VI] or in [159, Chap. 1]): If for a process ξ(τ ), τ ∈ [a, b], we have E|ξ(τ ) − ξ(τ 0 )|σ ≤ H|τ − τ 0 |1+σ ,

τ, τ 0 ∈ [a, b] ,

(5.16)

with some σ, δ > 0, then it possesses a continuous modification (again denoted by ξ) such that ∀ L > 0, H 0 P sup |ξ(τ ) − ξ(τ )| > L ≤ C(σ, δ) σ (b − a)1+δ . L τ,τ 0 ∈[a,b]


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Thus, obviously, for 0 < σ 0 < σ − 1, 0 ∞ X ((l + 1)L)σ 0 σ0 σ0 1+δ ≤ L + C(σ, δ)H(b − a) E sup |ξ(τ ) − ξ(τ )| (lL)σ τ,τ 0 ∈[a,b] l=1

0

0

≤ Lσ + C 0 (σ, σ 0 , δ)H(b − a)1+δ Lσ −σ , 1+δ

and, taking L := H 1/δ (b − a) σ we find that 0 0 0 σ0 E sup |ξ(τ ) − ξ(τ )| ≤ C 00 (σ, σ 0 , δ)H σ /σ (b − a)(1+δ)σ /σ .

(5.17)

τ,τ 0 ∈[a,b]

◦

Secondly, the Gaussian properties of the random variables g k,t and the equality (5.10) are direct consequences of applying the Ito stochastic calculus to the definition (5.9). Inequality (5.11) follows from the estimate (5.6) and the fact that the difference between two jointly Gaussian random variables is again a Gaussian variable and any higher moment of this difference is just a power of a second moment multiplied by a constant. ◦Relying on Lemma 4.2, the inequality (5.11) implies the continuity of the process G t in the spaces l2−p (Cβ ) ⊂ l2−p (Lβ,r ). Next, since we are dealing with a series of identically distributed random vari◦ older inequality, to show ables g k,t , for checking (5.12) it is enough, in view of the H¨ that "Z #( Qr ∧1) E

◦

sup k g k,t k2RQ L2rR ≤ C(β)

0≤t≤T

Q

sup | g k,t |2rR( r ∨1) du ◦

E

< ∞.

0≤t≤T

Sβ

(5.18) As a consequence of (5.11), (5.16) and (5.17) we have that 0 ◦ ◦ sup E sup | g k,t (u) − g k,t (u0 )|2Q ≤ C(Q0 ) and

E

(5.19)

u,u0 ∈Sβ

t≥0

◦

◦

sup | g k,t (u) − g

|t−t0 |≤1

k,t0

2Q0

|

≤ C 0 (Q0 )

(5.20)

for Q0 ≥ 1 uniformly in u ∈ Sβ . The latter inequality, together with g k,t (u) = 0 when t = 0, gives us (5.18) and therefore (5.12). Analogously, (5.19) implies that ◦

0

◦

Ek g k,t k2Q C(Sβ ) < ∞ ,

t ≥ 0,

Q0 ≥ 1 .

Moreover, using inequality (3.b) in [35] (which in turn is a sequel of the Garsia– Rodemich–Rumsey lemma [83]), one can deduce from (5.11) that    E  sup

u,u0 ∈Sβ t,t0 ∈[0,T ]

| g k,t (u) − g k,t0 (u0 )|2r   ≤ C(r, γ; T ) (|u − u0 | + |t − t0 |)γr  ◦

◦

(5.21)


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for r > 4 and 0 < γ < 1/2 − 2/r. This means, in particular, that for all Q ≥ 1, 2Q ◦ < ∞, (5.22) E sup k g k,t kC(Sβ ) 0≤t≤T

which provides (5.13). And finally, let now m > 0. In a similar manner, for checking (5.14) it is enough to show that 2rR( r ∨1) ( r ∧1) ] < ∞. sup Ek g k,t k2RQ L2rR ≤ C(β) sup sup [E| g k,t (u)| ◦

Q

◦

Q

(5.23)

t≥0 u∈Sβ

t≥0 ◦

Since g k,t (u) is Gaussian, (5.21) follows from the estimate (due to (5.6a)) Z t ◦ 2 E| g k,t (u)|2 = E kqβ,m (t − s; u, v)k2L2 ds ≤ C 0 (β, m) < ∞ ,

(5.24)

0

which holds uniformly in t ≥ 0 and u ∈ Sβ . To end with this, (5.19) and (5.24) gives us (5.15). ◦

Remark 5.1. Actually, the estimate (5.11) tells us that the process G t is locally H¨ older continuous in the following sense: ∀ ε > 0 ◦

G ∈ C 4 −ε ([0, +∞), l2−p (Cβ )) 1

and g k,t ∈ C 2 −ε (Sβ ) for any k ∈ Zd , 1

◦

t ≥ 0.

Corollary 5.1. Let m > 0, then Lm = 12 (−∆β + m2 1l)◦ is a positive self-adjoint ◦ operator in L2 (Sβ ) with TrL2 (Sβ ) L−1 m < ∞. The process G t = ( g k,t )k∈Zd , t ≥ 0, is −p Gaussian in the space l2 (Lβ,2 ), p > d/2, with covariance 0 0 1 ◦ ◦ 0 Eh g k,t , ϕiL2 h g k0 ,t0 , ϕ0 iL2 = δ(k − k 0 ) h(e−|t−t |Lm − e−(t+t )Lm )L−1 m ϕ, ϕ iL2 , 2

ϕ, ϕ0 ∈ L2 (Sβ ) .

(5.25)

Moreover, for any n ∈ N ◦

sup Ek G t k2n l−p (L t≥0

2

β,2 )

n n ≤ 2n · n!(TrL2 (Sβ ) L−1 m ) k1lkl−p (Zd ) ,

(5.26)

2

and for some α0 = α0 (p0 ) > 0 ◦ 2 αk G t k −p l (Lβ,2 ) 2 < C(β, m; p0 ) E e

(5.27)

uniformly in t ≥ 0, p > p0 > d/2 and 0 ≤ α ≤ α0 . Proof. The proof is based on standard Gaussian analysis in a Hilbert space, see e.g. [31]. ◦

◦

The free Gaussian process G t and the Ornstein–Uhlenbeck process Gt = G t + X t will play a fundamental role in our stochastic quantization scheme. So, the ◦


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solution of the linear equation (5.1) with an arbitrary dynamical matrix A, as well as the solution of (3.1) in the general case when anharmonic one-body interactions be treated as perturbations of Gt , t ≥ 0. Let V (xk ) are present, will conveniently ◦ us observe that the process G t , t ≥ 0, is not a martingale and its sample paths ◦ g k,t (u), k ∈ Zd , are polynomially bounded continuous functions on [0, +∞) × Sβ . 6. Equivalence with a System of Stochastic Integral Equations: Mild and Limit Solutions The commonly used approach for the study of the system of stochastic differential equations (3.3) consists of replacing it with the equivalent system of stochastic integral equations  Z tZ   xk,t (u) = gk,t (u) + qβ (t − s; u, v)fk (xk,s (v), Xs (v))dvds (6.1) Sβ 0   k ∈ Zd (t ≥ 0, u ∈ Sβ ) . Here for simplicity m = 0 and Gt = {gk,t }k∈Zd is a solution to the Cauchy problem ◦ for the linear system (5.7) with an initial datum X ∈ l2−p (Cβ ). The meaning of mild (m.-)solutions X ∈ C([0, +∞), l2−p (Cβ )), p > p0 , to (6.1) (solutions of SPDE’s (3.3) rewritten in their integral form (6.1) are usually called mild ones) is similar to that defined in Sec. 4 (by fixing the probability space (Ω, F , P ) with the filtration {Ft }t≥0 and pairing of both sides of (6.1) in distributional sense with test functions ϕ ∈ D(Sβ )). Lemma 6.1. If there exists a m.-solution Xt ∈ l2−p (Cβ ), t ≥ 0, in the sense of Definition 4.1, of the system of stochastic integral equations (6.1), then Xt is also a g.-solution to the system of stochastic differential equations (3.3) and vice versa. Proof. The proof of Lemma 6.1 follows quite standard lines for evolution equations and is related with that given in [41, 97]. First suppose that Xt is a solution to (6.1) and put it into the right-hand side of (4.7). To shorten notation, let Z tZ qβ (t − s; u, v)fk (xk,s (v), Xs (v))dvds , k ∈ Zd , yk,t (u) = 0 ◦

Sβ

◦

so that Xt = X t + G t + Yt . For fixed ϕ ∈ D(Sβ ) we need to calculate Z t 1 ◦ ◦ x k,x + g k,s + yk,s , ∆ϕ ds . 2 0 L2 ◦

1

◦

From the definition of x k,t (u) = (e 2 t∆β x k )(u) it follows that Z t 1 ◦ ◦ ◦ x k,s ∆ϕ ds = h x k,t − x k , ϕiL2 . 2 0 L2

(6.2)


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Changing the order of integration, we get that ) Z t Z t Z (Z t Z 1 1 0 yk,s , ∆ϕ ds = χ[s0 ,+∞) (s)qβ (s − s ; u, v) ∆ϕ(u)duds 2 2 Sβ Sβ 0 0 0 L2 × fk (xk,s0 (v), Xs (v))dvds0 Z t = hyk,t , ϕiL2 − hfk (xk,s , Xs ), ϕiL2 ds .

(6.3)

0

Quite similarly, by using the stochastic version of Fubini’s theorem (cf. [97]) we have that Z t 1 ◦ g k,s , ∆ϕ ds 2 0 L2 + Z t *Z Z t 1 0 χ[s0 ,+∞) (s) ∆ϕ(u)qβ (s − s ; u, ·)dsdu, dwk,s0 (·) = 2 Sβ 0 0 L2

◦

= h g k,t − wk,t , ϕiL2 .

(6.4)

Summation of Eqs. (6.2)–(6.4) completes the proof of the direct part of the lemma. Conversely, let Xt be a solution to (3.3). From (6.2) it follows # Z t " 1 ◦ ◦ xk,s − x k,s , ∆ϕ hxk,t − x k,t , ϕiL2 = + hfk (xk,s , Xs ), ϕiL2 ds 2 0 L2 + hwk,t , ϕiL2 .

(6.5) i2πnu

Let us take a total set of eigenfunctions ϕn (u) = √1β e β , n ∈ Z, of the operator 1 1 2 ∆β , i.e. 2 ∆β ϕn = λn ϕn . Substitution of ϕn into (6.5) gives us Z

◦

t

hxk,t − x k,t , ϕn iL2 = λn

◦

hxk,s − x k,s , ϕn iL2 ds Z

0 t

hfk (xk,s , Xs ), ϕiL2 ds + hwk,t , ϕiL2 .

+ 0

Applying Itˆ o’s formula to the process ξ(s) := e−λn (t−s) hxk,s − x k,s , ϕn iL2 , 0 ≤ s ≤ t, we find that Z t ◦ e−λn (t−s) hfk (xk,s , Xs ), ϕn iL2 ds hxk,t − x k,t , ϕn iL2 = ◦

0

Z

+

t

he−λn (t−s) ϕn , dwk,s iL2 = hyk,t + g k,t , ϕn iL2 .

0

Therefore, Xt is a solution to (6.1).

◦


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Denote by Φ the nonlinear mapping given by the right-hand side of the formula (6.1), i.e.  Z t  ◦ ◦ e(t−s)∆β fk (xk,s , Xs )ds (ΦX)k,t = et∆β x k + g k,t + (6.6) 0  k ∈ Zd (t ≥ 0) . The next assertion enables us to make a first step in studying the regularity properties of the solution Xt as a fixed point of Φ. Lemma 6.2. Φ defines a continuous bounded mapping from the spaces L∞ ([0, T ], l2−p0 (Lβ,r )) into the spaces C([0, T ], l2−p (Lβ,r )) and C([t0 , T ], l2−p (Cβ )) for 0 < t0 < ◦ T < ∞, r > 2R, p > p0 R + d/2. Under the given initial data X ∈ l2−p0 (Cβ ) one older continuous in the can take t0 = 0. Moreover, the process (ΦX)t is locally H¨ following sense: ∀ ε > 0 ΦX ∈ C 4 −ε ((0, +∞), l2−p (Cβ )) 1

and

(ΦX)k,t ∈ C 2 −ε (Sβ ), k ∈ Zd , t > 0 . 1

In the particular case where V 0 is a polynomial of degree R, we can take r = 2R. ◦

Proof. As we noted in Remark 5.1, the process G t is Hölder continuous in l2−p (Cβ ), ◦ ◦ p > d/2. Since X t depends linearly on the initial data X , it suffices to recall the trivial estimates on the C0 -semigroup et∆β , t ≥ 0, in the spaces Cβ and Lβ,r : ket∆β kLβ,2 →Cβ ≤ qβ (t; 0, 0) , ket∆β kLβ,r ≤ 1 ,

t ≥ t0 > 0 ,

ket∆β kCβ ≤ 1 ,

t ≥ 0.

So we need only to consider Yt which stands for the integral term in (6.6). By (3.7), (5.6), it is valid for any k ∈ Zd , u, u0 ∈ Sβ , 0 ≤ t0 < t ≤ T < ∞ and ε > 0, |yk,t (u) − yk,t0 (u0 )|2 ! Z Z Z tZ t 2 ≤ (∆qβ ) (s; v)dvds Sβ

0

0

! fk2 (xk,s (v), Xs (v))dvds

Sβ

1 2 −ε

+ |u − u0 |1−ε ) ≤ C(ε, T )(|t − t0 | Z T X 1 + kxk,s k2R × Lβ,2R + 0

! kxj,s k2Lβ,2 ds .

(6.7)

0≤|j−k|≤ρ

Hence we get that Yt ∈ C(Sβ )Z and moreover, relying on (4.2), that Yt ∈ l2−p (Cβ ) with the estimate d

kYt − Yt0 k2l−p (C 2

β)

1 ≤ C(ε, T, p0 , p, R)|t − t0 | 2 −ε 1 + sup kXt k2R −p l 0 (L 0≤t≤T

2

β,2R )

.

(6.8)

From (6.7), noting that Y0 = 0, we obtain the inclusion Y ∈ C([0, +∞), l2−p (Cβ )) and the boundedness of the mapping Φ.


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To verify the continuity of the mapping Φ let us choose any sequence X (M) , M ∈ N, converging to X in the norm of L∞ ([0, T ], l2−p0 (Lβ,r )). Just as before, we have (M) 2 kl−p (C ) β 2

sup kYt − Yt

0≤t≤T

≤ C(T )

X

−2p

Z

T

(1 + |k|)

(M)

(M)

kfk (xk,t , Xt ) − fk (xk,t , Xt

0

k∈Zd

)k2Lβ,2 dt . (6.9)

Due to the continuity of fk (x, X) on R × RZ , almost surely all integrands in (6.9) tend to zero when M → ∞. Besides, since r < 2R, there exists δ > 0 such that Z TZ X (M) (M) −2p (1 + |k|) |fk (xk,t (v), Xt (v))|2+δ dvds . sup d

M

Sβ

0

k∈Zd

≤ C(T, p0 , p, δ) sup sup

M∈N 0≤t≤T

n (M) 1 + kXt krl−p0 (L 2

o β ,r)

< ∞.

Therefore the uniform integrability enables us to pass to the limit in (6.9). When the nonlinear term V 0 in the drifts fk , k ∈ Zd , is a polynomial of degree R, we can directly apply to (6.9) the Lebesgue dominated convergence theorem because of the implication: kx(M) − xkL2R (Sβ ) −−−→ 0 =⇒ kV 0 (x(M) ) − V 0 (x)kL2 (Sβ ) −−−→ 0 , M→∞

M→∞

which concludes the proof of the lemma. In the broader sense, by a mild solution to the system (6.1) we will mean any process X ∈ C([0, +∞), l2−p (Lβ,r )) which is a fixed point of the mapping Φ. Moreover, relying on the ideas of [51, 52] and [86], one can further extend the notion of a solution to the nonlinear equation (3.3). Along with the generalized and mild solutions, we will also consider a limit (or constructible, a concept due to [86]) solution defined as follows. Definition 6.1. A process X ∈ C([0, +∞), l2−p (Lβ,r )) is called a limit (l.-) solution ◦ of the Cauchy problem (3.3) with the initial data X ∈ l2−p (Lβ,r ) if there exists a sequence of generalized (=mild) solutions {X (M) }M∈N ⊂ C([0, +∞), l2−p (Cβ )) such (M) that kXt − Xt kl−p (Lβ,r ) → 0, M → ∞, uniformly on any finite interval [0, T ], 2 T > 0. Obviously, the latter definition is connected with the appropriate choice of a ◦ sequence of initial data { X (M) }M∈N ⊂ l2−p0 (Cβ ), p0 ≤ p, so that ◦

◦

k X (M) − X kl−p (Lβ,r ) → 0 , 2

M → ∞.

The extended notion of l.-solution enables us to avoid the problem of how to pass to the limit in the nonlinear terms fk and check up the identity of both sides in (M) (3.3) and (6.1) for Xt = limM→∞ Xt . However, this notion of solution is strong


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enough for the study of the long-time behaviour and invariant distributions for the Langevin dynamics. 7. Finite Volume Approximation for the System of SPDE’s One possible way for constructing a solution to the system (3.3) in the whole infinite volume Zd is based on a finite volume approximation to be discussed in this section. Moreover, this is also a helpful step for the study of the relation between Euclidean Gibbs states and invariant distributions of the Langevin dynamics. Let ΛN , N ∈ N, be a sequence of finite subsets of Zd such that ΛN % Zd , N → ∞. Let {x1 , x2 , . . . xN }, N ∈ N, be the points of ΛN (with respect to an ◦ arbitrary ordering). For any fixed initial data X ∈ l2−p0 (Cβ ) and any ΛN let us consider the following system of stochastic differential equations derived from (3.3) / ΛN : by a cutoff procedure with respect to xk , k ∈  2 1 ∂ (N ) ∂ (N ) (N ) (N )   xk,t (u) = x (u) + fk (xk,t (u), Xt (u)) + w˙ k,t (u) , k ∈ ΛN , ∂t 2 ∂u2 k,t (7.1)   (N ) ◦ / ΛN (t > 0, u ∈ Sβ ) . xk,t (u) = x k (u) , k ∈ A generalized solution of (7.1) (N )

Xt

(N )

(N )

(N )

(N )

◦

◦

= {xk,t }k∈Zd = {x1,t , x2,t , . . . , xN,t , x N +1 , x N +2 , . . .}

is defined in the same way as that of (3.3) (recall that Xt is given by Definition 4.1). (N ) The procedure we will follow in Sec. 7, 8 is to show that the sequence {Xt }N ∈N converges when N → ∞ and its limit Xt is a generalized solution to the infinite volume system (3.3). The first question to discuss here is the solvability of the system (7.1) or, what is the same, due to Lemma 6.1, the solvability of the finite system of integral equations Z tZ  (N ) (N )   xk,t (u) = gk,t (u) + qβ (t − s; u, v)fk (xk,s (v), Xs(N ) (v))dvds , k ∈ ΛN ,  

0

(N ) xk,t (u)

◦

= x k (u) ,

Sβ

k∈ / ΛN

(t > 0, u ∈ Sβ ) . (7.2)

Suppose fk , k ∈ Zd , are globally Lipschitz continuous in (xk , {xj }j∈ΛN ), then the (N ) usual application of the Picard iteration method provides a unique solution Xt , t ≥ 0, for each N ∈ N. However, this is not sufficient to perform the stochastic quantization for our model. To weaken the restriction on fk , k ∈ Zd , one can use the general theory of stochastic semilinear equations with dissipative drifts (see e.g. [51–53, 114, 119, 122, 127, 129, 162]), as well as modify the treatment of Funaki’s string model (see [69, 97]) to cover the periodic case u ∈ Sβ . Both these ways lead to the following existence and uniqueness result (see also [156]). Lemma 7.1. Under the assumption (A), (B) on the drift terms fk , k ∈ Zd , for any ◦ X ∈ l2−p0 (Cβ ) and ΛN ⊂ Zd , |ΛN | < ∞, the finite volume systems (7.1), (7.2) have


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a unique generalized (=mild) solution Xt ∞ with some q ≥ 1, then

◦

∈ l2−p (Cβ ), p ≥ p0 . If Ek X |q−p0 l2

(Cβ )

0, u ∈ Sβ , and one can obtain (see [67, (N ) Chap. 1]) that yk,t (u) are continuously differentiable in t > 0 and twice so in u ∈ Sβ with the pointwise equality 1 ∂ 2 (N ) ∂ (N ) (N ) (N ) yk,t (u) = y (u) + fk (xk,t (u), Xt (u)) . ∂t 2 ∂u2 k,t (N )

(7.3)

(N )

Moreover, again due to Lemma 6.2, the mappings fk (xk,t , Xt ) : (0, T ) → C(Sβ ), older continuous, which gives us continuous Fréchet k ∈ ΛN , are also locally H¨ (N ) differentiability in t > 0 of yk,t ∈ C 2 (Sβ ) in all spaces L2r (Sβ ), r ≥ 1, and C(Sβ ), with the equality (7.3) now understood in the operator sense. Theorem 7.1. Let the assumptions (A), (B) be satisfied with the constant R ≥ 1 in ◦ the polynomial growth estimate (3.5) and let us take the initial data X ∈ l2−p0 (Cβ ). Then the sequence of finite volume approximations {X (N )}N ∈N converges almost surely, as N → ∞, in each of the spaces C([0, T ], l2−p (Lβ,2r )), where p > p0 R + d/2, r ≥ 1, 0 < T < ∞. Proof. We claim that (N ) (M) sup kXt − Xt kl−p (Lβ,2r ) = 0 = 1 . P lim N,M→∞

(N,M)

2

0≤t≤T (N )

(7.4)

(M)

Set Zt = Zt := Xt − Xt . Without loss of generality we may assume / ΛM . In the case where k ∈ ΛN , we have ΛM ⊃ ΛN . Obviously zk,t = 0 for k ∈ Z tZ (N ) (M) qβ (t − s; u, v) zk,t (u) = yk,t (u) − yk,t (u) = 0

(N )

Sβ

(M)

× [fk (xk,s (v), Xs(N ) (v)) − fk (xk,s (v), Xs(M) (v))]dvds ,

(7.5)


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and in order to estimate kzk,t kL2r let us integrate by parts using (3.8) and (7.3), Z d |zk,t (u)|2r du dt Sβ 2 ∂ |zk,t (u)|2r−2 zk,t (u) du ∂u Sβ

Z = −r(2r − 1) Z

(N )

+ 2r Sβ

(N )

(zk,t (u))2r−1 [fk (xk,t (u), Xt

(M)

(M)

(u)) − fk (xk,t (u), Xt

(u))]du

Z

|zk,t (u)|2r−2

≤ 2r Sβ

 1 1 1 |zk,t (u)|2 + kak20 ×  l − a(0) + 2 4 4



X

|zj,t (u)|2  du .

0 0 : k∈Zd γk Qkj ≤ Kγj , j ∈ Zd , P (c) k∈Zd γk sup0≤t≤T ζk (t) < ∞, Rt P (d) ζk (t) ≤ k∈Zd Qkj 0 ζj (s)ds + Bk , t ∈ [0, T ], k ∈ Zd . Then the following inequality holds: X X γk sup ζk (t) ≤ eT K γj Bj . k∈Zd

0≤t≤T

(7.11)

j∈Zd

Since the condition (c) is not always verified a priori, we present here a modification of Lemma 7.2 which turns out to be more useful in the applications we have in mind. Lemma 7.3. Let {γk > 0}k∈Zd , {Bk ≥ 0}k∈Zd be sequences on Zd and {Qkj }k,j∈Zd be a finite-diagonal matrix on Zd × Zd of the Toeplitz form, i.e. Qkj = Q(k − j) ≥ 0 ,

k, j ∈ Zd ,

∃ ρ > 0 : Q(k − j) = 0 ,

|k − j| > ρ .

Let {ζk (t)}k∈Zd be a sequence of non-negative measurable functions defined on a finite interval [0, T ]. Suppose that P (c0 ) k∈Zd γk0 sup0≤t≤T ζk (t) < ∞ with an auxiliary sequence {γk0 }k∈Zd on Zd such that 0 < γk0 ≤ γk , k ∈ Zd ,

and

0 γk−j < ∞, j ∈ Zd . 0 |k|→∞ γk

lim

If the conditions (a), (b), (d) of Lemma 7.2 are also satisfied, then the inequality (7.11) holds. Proof. According to our assumptions, the matrix Q generates a bounded operator in the Banach space ( ) X d Zd 0 γk |xk | < ∞ . l1,γ 0 (Z ) := X = {xk }k∈Zd ⊂ R , kXk1,γ 0 = k∈Zd

Let us denote by ϕ(t) = etQ B the unique solution in l1,γ 0 (Zd ) of the system Z t X γk Qkj ϕj (s)ds + Bk , t ∈ [0, T ] , k ∈ Zd . ϕk (t) = k∈Zd

0


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d d ˆ = sup Thus, taking ζ(t) 0≤s≤t ζ(s) ∈ l1,γ 0 (Z ), δ(t) = ϕ(t) − ζ(t) ∈ l1,γ 0 (Z ) and keeping in mind that Qkj ≥ 0, we obtain by a standard estimate Z t Z tZ s (Qδ)k (s)ds ≥ (Q2 δ)k (u)duds δk (t) ≥ 0

0

Z

0

0

(t − u)2 3 (Q δ)k (u)du 1!

(t − u)n n+1 (Q δ)k (u)du , (n − 1)!

t 0

t

t 0

Z

≥ ··· ≥ Z

Z

(t − u)(Q2 δ)k (u)du ≥

≥

Since

0

t

k ∈ Zd .

tn (t − u)n n+1 (Q kQkn+1 δ)k (u)du ≤ l1 ,γ 0 kδkl1 ,γ 0 → 0 , (n − 1)! (n − 1)!

n → ∞,

we get that 0 ≤ sup0≤s≤t ϕs (t) ≤ ϕk (t) = (etQ B)k , k ∈ Zd . But, due to the assumption (b), the matrix Q defines also a bounded operator in the space l1,γ (Zd ) ⊆ l1,γ 0 (Zd ) with the norm kQkl1 ,γ ≤ K. This yields finally that

sup ϕk (t)

≤ eT K kBkl1 ,γ .

0≤t≤T

k∈Zd l1 ,γ

Lemma 7.2, as applied to (7.6)–(7.8) with γk = (1 + |k|)−2p , k ∈ Zd , and " 2p # X 1 + |j| < ∞, Qkj K = sup 1 + |k| j∈Zd d k∈Z

gives us (N )

sup kXt

0≤t≤T

≤

X

(M) 2 kl−p (L ) β,2r 2

− Xt

−2p

(1 + |k|)

sup 0≤s≤t

k∈Zd

≤ 2eT K

X

(N,M) ζk (t)

(1 + |j|)−2p

X j∈ΛM \ΛN

◦

0≤t≤T

j∈ΛM \ΛN

≤ 4eT K

(M)

sup kxj,t k2L2r + k x j k2L2r

(1 + |j|)−2p sup sup kxj,t k2L2r . (N )

N ∈N 0≤t≤T

(7.12)

This implies that {X (N ) }N ∈N is a Cauchy sequence in C([0, T ], l2−p(Lβ,2r )) as soon as on the last line of (7.12) we have the remainder of a converging series, a fact that we shall now prove. Lemma 7.4. For all p > p0 R + d/2, r ≥ 1, 0 < T < ∞, we have ) ( X (N ) 2 −2p (1 + |j|) sup sup kxj,t kL2r < ∞ = 1 . P j∈ΛM \ΛN

N ∈N 0≤t≤T

(7.13)


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Proof. In view of Fatou’s lemma, it suffices to show that with probability one ( ) X (N ) 2 −2p (7.14) sup (1 + |k|) sup sup kxk,t kL2r < ∞ . M∈N

(N )

N ≤M 0≤t≤T

k∈Zd

(N )

(N )

(N )

= {yk,t }k∈Zd ), then for k ∈ ΛN , Z tZ (N ) (N ) (N ) yk,t (u) = xk,t (u) − gk,t (u) = qβ (t − s; u, v)fk (xk,s (v), Xs(N ) (v))dvds .

Set Yt

:= Xt

− Gt (with Yt

Sβ

0

After an integration by parts, we have the following chain of estimates Z d (N ) |y (u)|2r du dt Sβ k,t Z (N ) (N ) (N ) 2r−1 1 (N ) ∆β yk,t (u) + fk (xk,s (u), Xs (u)) du (yk,t (u)) = 2r 2 Sβ !1− 1r Z Z 1 (N ) (N ) 2r 2r ≤ 2r l − a(0) + 1 |yk,t (u)| du + r |yk,t (u)| du 2 Sβ Sβ ×

  Z 

!1r |fk (gk,t (u), Gt (u))|2r du

Sβ

so that (N ) kyk,t k2L2r

Z

t

≤ (2(l + 1) + |a(0)|) 0

Z + 0

1 + kak20 2

(N ) kyk,s k2L2r ds

!r1  (N ) |yj,t (u)|2r du ,  Sβ

Z

X 0 ρ ,

(7.18)


00065


and

Bk = T sup 0≤t≤T

kfk (gk,t , Gt )k2L2r

85

+2

sup 0≤t≤T

kgk,t k2L2r

+

◦

k x k k2L2r

.

Applying Lemma 7.2 with ζk = ηk , γk = (1 + |k|)−2p , k ∈ Zd , and   2p X X 1 + |j|   < ∞, K= Qkj 1 + |k| d d (M)

j∈Z

k∈Z

we obtain that X

(1 + |k|)−2p ηk

(M)

X

(t) ≤ eT K

k∈Zd

(1 + |j|)−2p Bj ,

j∈Zd

or X

(1 + |k|)−2p sup

(N )

sup kxk,t k2L2r

N ≤M 0≤t≤T

k∈Zd

≤2

X

(1 + |k|)−2p

sup kgk,t k2L2r + eT K Bk .

(7.19)

0≤t≤T

k∈Zd

In order to conclude the proof of (7.14), we need to check that the series on the right-hand side of (7.19) is finite, i.e. with probability one X (1 + |k|)−2p sup kfk (gk,t , Gt )k2L2r + sup kgk,t k2L2r < ∞ . 0≤t≤T

k∈Zd

0≤t≤T

But due to (3.7) and (4.2), the latter is apparent from the following propositions taken from Sec. 5: X ◦ ◦ (1 + |k|)−2p sup k x k,t k2R X ∈ l2−p0 (Cβ ) , p > p0 R + d/2 , L2rR < ∞ , 0≤t≤T

k∈Zd

X

(1 + |k|)−2p sup k g k,t k2R L2rR < ∞ , ◦

p > d/2 .

(7.20)

0≤t≤T

k∈Zd

This proves Lemma 7.4 and thus also Theorem 7.1. ◦

Remark 7.1. Under the above specified conditions, for a random initial data X ∈ ◦ < ∞ with some q ≥ 1, we have the convergence l2−p0 (Cβ ) such that Ek X k2Rq −p0 " lim

N,M→∞

E

l2

(Cβ )

X

# (N ) kXt

−

(M) Xt k2q −p l 0 (L

0≤t≤T

for all p > p0 Rq + d/2, r ≥ 1 and 0 < T < ∞.

2

β,2r )

= 0,

(7.21)


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The proof of this fact is in perfect analogy with the proof of the assertion (7.4). Using (7.12) and the Lebesgue dominated theorem, the desired convergence in (7.21) will follow from the estimate #q " X (N ) 2 −2p (1 + |k|) sup sup kxk,t kL2r < ∞ . (7.22) sup E M∈N

N ≤M 0≤t≤T

k∈Zd

Because of Lemma 5.1, the validity of (7.22) is equivalent to #q " X (N ) 2 −2p (1 + |k|) sup sup kyk,t kL2r < ∞ . E N ≤M 0≤t≤T

k∈Zd

(7.23)

But, from (7.19), we see that the estimate (7.23) is implied by the following one ( ) X ◦ ◦ 2Rq 2Rq (7.24) (1 + |k|)−2p Ek x k kL2rR + E sup k g k,t kL2rR < ∞ , 0≤t≤T

k∈Zd

which holds true according to our assumptions and Lemma 5.1. ◦

Remark 7.2. As it seen from (7.20) and (7.24), in the case of initial data X = 0 ◦ ◦ ∗ < ∞, p∗ ≥ 0, the sequence of finite volume approxior X ∈ l2p (Cβ ), Ek X k2Rq p∗ l2 (Cβ )

mations {X (N ) }N ∈N converges, as N → ∞, in the sense of (7.4) and (7.21) in each of the spaces C([0, T ], l2−p(Lβ,2r )), r ≥ 1, p > d/2. 8. Existence and Uniqueness of the Solution We are now ready to establish the central results about solvability of the infinite system of SPDE’s under consideration. Lemma 8.1. Let the assumption (A), (B) be satisfied and take the initial data ◦ X ∈ l2−p0 (Cβ ). Then there exists a unique solution Xt , t ≥ 0, to the Cauchy problem (3.3) that is a continuous process in every l2−p (Cβ ), p > p(p0 , R). ◦

Proof. Existence. Let X ∈ l2−p0 (Cβ ) and let ΛN % Zd , N → ∞, be a sequence of (N ) finite volume solutions Xt , N ∈ N, as constructed in Sec. 7. This means, relying on the notation (6.2), that   (X (N ) )k = (ΦX (N ) )k , k ∈ ΛN , (8.1)  (X (N ) ) = x◦ , k∈ / ΛN . k k According to Theorem 7.1, the sequence X (N ) has a limit, as N → ∞, in every C([0, T ], l2−p (Lβ,2r )), p > p0 R + d/2, r ≥ 1, 0 < T < ∞, let us call it X. From Lemma 6.2, the sequence ΦX (N ) converges, as N → ∞, to ΦX in the spaces C([0, T ], l2−p∗ (Cβ )), p∗ > pR + d/2, 0 < T < ∞. Passing to the limit on both sides of the equality (8.1), we find that (ΦX)k = (X)k for all k ∈ Zd , i.e. X is a


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fixed point of the mapping Φ. Hence by using Lemma 6.2 again, we get the desired inclusion X ∈ C([0, +∞], l2−p∗ (Cβ )), p∗ > p0 R2 + (R + 1)d/2. Uniqueness. Let X, X 0 ∈ C([0, +∞], l2−p (Cβ )) be two solutions of (3.3) starting ◦ from X ∈ l2−p0 (Cβ ). It is necessary to show that P (Xt = Xt0 , t ≥ 0) = 1 ,

(8.2)

but this is a straightforward corollary of the next Theorem 8.1. Remark 8.1. Through Lemma 8.1, we are not sure that the generalized solution ◦ will not go beyond the space l2−p0 (Cβ ) containing the initial value X . Actually, we have proved the solvability of the Cauchy problem (3.3) in the state space ◦ S 0 (Cβ ) = limp=1,2,... ind l2−p (Cβ ) only. In particular, for initial data X = 0 or ◦ ∗ X ∈ l2p (Cβ ), p∗ ≥ 0, we have constructed the g.-solution X ∈ C([0, ∞), l2−p0 (Cβ )), p > (R+ 1)d/2. These results are preliminary and will be improved in Theorems 8.2 and 8.3. The next proposition gives the continuous dependence of solutions on initial data. Theorem 8.1. If Xt , Xt0 are generalized solutions of the system (3.3) with the ◦ ◦ initial data X , X 0 respectively, then we have almost surely ◦

◦

(a)

kXt − Xt0 kl−p (Lβ,2r ) ≤ eKt k X − X 0 kl−p (Lβ,2r ) ,

(b)

kXt − Xt0 kl−p (Cβ ) ≤ eKt k X − X 0 kl−p (Cβ ) ,

2

2

◦

2

with the constant K = K(p) =

 1 kak0 2

◦

2

"

X

# 12 (1 + |k|)2p

t ≥ 0,

t ≥ 0,

(8.3)

 − a(0) + l ,

(8.4)

0 0, k ∈ Zd , 1 ∂2 ∂ zk,t (u) = zk,t (u) + fk (xk,t (u), Xt (u)) − fk (x0k,t (u), Xt0 (u)) . ∂t 2 ∂u2

(8.6)


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In view of (3.6) and (8.6), an integration by parts yields Z Z d |zk,t (u)|2r du ≤ 2r (zk,t (u))2r−1 dt Sβ Sβ × [fk (xk,s (v), Xs (v)) − fk (x0k,s (v), Xs0 (v))]du Z |zk,t (u)|2r−2 ≤ 2r Sβ

 ε 1 1  |zk,t (u)|2 + kak20 × l− a(0)+ 2 4 4ε

X

 |zj,t (u)|2du

0 0). Then by the H¨ older inequality, we get for kzk,t kL2r 6= 0 X ε 1 d kzk,t k2L2r ≤ 2l − a(0) + kzk,t k2L2r + kak20 kzj,t k2L2r . dt 2 2ε 0 0 ∀ p > d/2 ∃ δ0 > 0 : ∀ 0 < δ < δ0 ◦

◦

kXt − Xt0 klδ,−p (Lβ,2 ) ≤ e(l−α/2+ε)t k X − X 0 klδ,−p (Lβ,2 ) . 2

(8.12)

2

The estimate (8.12) is in exact agreement with [53]. Theorem 8.2. Let the assumptions (A), (B) be satisfied and take the initial data ◦ X ∈ l2−p0 (Cβ ) with p0 > (R + 1)d/2. Then there exists a unique generalized solution X ∈ C([0, +∞), l2−p0 (Cβ )) to the Cauchy problem (3.3). ◦

∗

Proof. Consider an arbitrary sequence of initial data { X 0 }M∈N ⊂ l2p (Cβ ), p∗ ≥ 0, ◦ approximating X in the norm of l2−p0 (Cβ ). Let {X (M) }M∈N be a corresponding


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sequence of g.-solutions: in the notation (6.2) this means that for all M ∈ N, k ∈ Zd (X (M) )k = (ΦX (M) )k .

(8.13)

As was mentioned in Remark 8.1, we have the inclusions X (M) ∈ C([0, +∞), l2−p (Cβ )), p > (R + 1)d/2. But from the estimate (8.3) it follows that the sequence {X (M) }M∈N is a fundamental sequence in the spaces C([0, T ), l2−p0 (Cβ )) ,

p0 > (R + 1)d/2 ,

0 < T < ∞.

Let us denote its limit point by X (which does not depend on the choice of an ◦ ◦ approximating sequence of initial data X (M) → X , M → ∞). Passing to the limit in (8.13) on the basis of Lemma 6.2, we conclude that for every fixed k ∈ Zd (X)k = (ΦX)k , i.e. X ∈

C([0, +∞), l2−p0 (Cβ ))

is the unique g.-solution to (3.3).

The following existence and uniqueness theorem for limit solutions holds. Theorem 8.3. Let the assumptions (A), (B) be satisfied and take the initial data ◦ X ∈ l2−p0 (Lβ,2r ) with r ≥ 1, p0 > d/2. Then there exists a unique limit solution X ∈ C([0, +∞), l2−p0 (Lβ,2r )). If r > R (for V being a polynomial we can take r ≥ R) and p > p(p0 , R) = p0 R + d/2, then, moreover, X ∈ C 4 −ε ((0, +∞), l2−p (Cβ )), xk,t ∈ C 2 −ε (Sβ ), ε > 0, t > 0, k ∈ Zd , 1

1

and X is the unique mild solution in the sense that it satisfies the system of integral equations (6.1). ◦

∗

Proof. Consider an arbitrary sequence of initial data { X (M) } ⊂ l2p (Cβ ), p∗ ≥ 0, ◦ approximating X in the norm of l2−p0 (Lβ,2r )). Making use of Remark 7.2, the corresponding g.-solutions X (M) belong to C([0, +∞), l2−p0 (Lβ,2r )) ∩ C((0, +∞), l2−p (Cβ )), p > p0 R + d/2. From the estimate (8.3), the sequence {X (M) }M∈N is fundamental in the spaces C([0, T ), l2−p0 (Lβ,2r )), 0 < T < ∞. According to Definition 6.1, its limit X is a unique l.-solution to the Cauchy problem (3.3). On the other hand, by Lemma 6.2, the convergence X (M) = ΦX (M) → ΦX, M → ∞, takes place in the spaces C([0, +∞), l2−p (Lβ,2r )) and C((0, +∞), l2−p (Cβ )) provided r > R, p > p0 R + d/2. Hence, under the last assumptions, X = ΦX is a unique 1 1 m.-solution to (6.1), such that X ∈ C 4 −ε ([0, +∞), l2−p (Cβ )) and xk,t ∈ C 2 −ε (Sβ ) for any ε > 0, t > 0, k ∈ Zd . ◦

Remark 8.5. Let X ∈ l2−p (Cβ ), p > (R + 1)d/2, and let X be the corresponding g.-solution constructed in Theorem 8.2. Taking again an approximating sequence of ◦ ∗ initial data { X (M) }M∈N ⊂ l2p (Cβ ), p∗ ≥ 0, and combining Lemma 6.2, Remark 7.2 and the estimate (8.8), it is readily seen that X can also be obtained as the thermodynamical limit in the space C([0, +∞), l2−p (Cβ )) of any sequence of finite volume


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solutions X (N ) defined by (7.2) when ΛN % Zd , N → ∞. The analogous finite volume approximation takes place in C([0, +∞), l2−p (Lβ,2r )), r ≥ 1, p > d/2, for the l.-solution constructed in Theorem 8.3. In addition, applying Remark 7.2 to the ◦ < ∞, we get that solutions {X (M) }M∈N such that Ek X (M) k2Rq p∗ l2 (Cβ )

E

sup kXt − 0≤t≤T

(N ) Xt k2q l−p (Lβ,2r )

→ 0,

N → ∞,

(8.14)

2

◦

◦

in the case of all random initial conditions X such that Ek X k2q l−p (L 2

β,2r )

< ∞.

Corollary 8.1. Under the assumption of Theorem 8.3, the following implications are valid for the l.-solutions, when q ≥ 1, r ≥ 1, p > d/2 and 0 < T < ∞: ! X ◦ q q (8.15) kXt kl−p (L ) < ∞ . Ek X kl−p (L ) < ∞ ⇒ E β,2r

2

0≤t≤T

2

β,2r

Proof. Because of Theorem 8.1, it suffices to check (8.15) for the solution Xt0 , ◦ t ≥ 0, starting at t = 0 from X 0 = 0. As before this property is an immediate sequel of Remark 7.2. Corollary 8.2. Under the assumptions of Theorem 8.2 the following implications are valid for the g.-solutions when q ≥ 1, p > (R + 1)d/2 and 0 < T < ∞: ◦ (8.16) Ek X kql−p (C ) < ∞ =⇒ E sup kXt kql−p (C ) < ∞ . 2

β

0≤t≤T

2

β

Proof. Due to Theorem 8.1, it is again enough to prove (8.16) only for the solution ◦ Xt0 , t ≥ 0, starting at t = 0 from X 0 = 0. The required property results immediately from Corollary 8.1, Lemmas 6.2 and 5.1(iii). Example 8.1. Let us apply the previous considerations to the particular case of a finite system of locally interacting diffusions (3.12), namely  # Z t" X  1  0  a(k − j)xj,s + V (xk,s ) ds + wk,t ,  xk,t = xk,0 − 2 0 (8.17) j∈Bρ (k)     k ∈ Zd (t ≥ 0) . The corresponding analogue of Theorem 8.3 gives us the existence and uniqueness of the limit solution X ∈ C([0, +∞), l2−p (Zd )), p > d/2. But, since V 0 ∈ C(R1 ) and the dynamical matrix A = (a(k − j))k,j∈Zd is bounded in all l2−p (Zd ), we are able to pass to the limit on the both sides of (8.17) and thereby find that X is the strong solution in the usual sense. Because in this case the continuous parameter u ∈ Sβ is absent, we do not have to deal with the Ornstein–Uhlenbeck process and with the notion of mild solution, thus the assumption (A) in Sec. 3 on the growth of V 0 can


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be omitted. The only sufficient condition for the unique solvability of the system (8.17) is l.-dissipativity, i.e. (B) : ∃ l ∈ R1 : ∀ x, y ∈ R1 1 − (x − y)(V 0 (x) − V 0 (y)) ≤ l(x − y)2 . 2 The same sufficient condition follows from the general criteria for existence and uniqueness of strong solutions proved in [119]. Example 8.2. For the model (3.13) of a single Brownian string on Sβ with a polynomial drift V 0 , Theorems 8.2 and 8.3 give us existence and uniqueness respectively of the generalized (=mild) solution X ∈ C([0, +∞), C(Sβ )) and the limit solution ◦ X ∈ C([0, +∞), L2r (Sβ )), r ≥ 1. If in the latter case X ∈ L2r (Sβ ) with r ≥ R, then X is also the unique mild solution. Applying the general theory of dissipative stochastic evolution equations in Banach spaces, exactly the same results have been obtained accordingly in [51] for the case of state space C(Sβ ), and respectively in [52] for L2r (Sβ ). However, the direct application in [51, 52] of the general dissipativity methods to the classical and quantum lattice systems (3.12) and (3.3) does not produce optimal results (the results of [51, 52] are in this regard also weaker than those particulary of [119] and [18]). 9. Solution as a Markov Process: Relation between Invariant, Reversible and Gibbs Distributions In analogy with ordinary stochastic differential equations, the solution of the Langevin equation is also a Markov process. Here we restrict ourselves to mentioning some fundamental definitions and facts from the theory of Markov processes, in particular, about their long-time behaviour, and show how they are applicable in our case. Lemma 9.1. Let the assumptions of the existence and uniqueness Theorem 8.3 be fulfilled. Then the l.-solution X ∈ C([0, +∞), l2−p (Lβ,2r )), r ≥ 1, p > d/2, forms a time-homogeneous Markov process with respect to (Ω, F , P, {Ft }t≥0 ): namely, for any 0 ≤ t0 < t < ∞ and Borel set ∆ ∈ B(l2−p (Lβ,2r )) almost surely we have P {Xt ∈ ∆|Ft0 } = P {Xt ∈ ∆|Xt0 } = µt−t0 (Xt0 , ∆) .

(9.1)

The transition probabilities µt−t0 (X, ∆) are defined by µt−t0 (X, ∆) = P (Xt (t0 , X) ∈ ∆) ,

t ≥ 0,

where Xt (t0 , X) ∈ l2−p (Lβ,2r ) is the l.-solution to the system of equations  Z Z tZ    xk,t (u) = qβ (t − t0 ; u, v)xk,t0 (v)dv + qβ (t − s; u, v)dwk,s (v)dv    Sβ t0 Sβ    Z tZ (9.2) qβ (t − s; u, v)fk (xk,s (v), Xs (v))dvds , +    t0 Sβ       k ∈ Zd (t ≥ t , u ∈ S ) 0

β


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with the nonrandom initial data X = Xt0 ∈ l2−p (Lβ,2r ) at t = t0 . Proof. The proof is standard (see e.g. [51, 85, 151]) and based on two main propositions, precisely, the unique solvability of the systems (6.1) and (9.2) (hence a solution Xt of the system (6.1) coincides a.s., when t ≥ t0 , with a solution Xt (t0 , Xt0 ) of the system (9.2) starting from Xt0 ) and the measurability of solutions in their dependence on the initial data. As follows from the general theory of Markov processes, the transition probabilities µt (X, ∆) satisfy the Kolmogorov–Chapman equality in the phase space X = l2−p (Lβ,2r ) Z µt0 (X, dY )µt−t0 (Y, ∆) µt (X, ∆) = l−p 2 (Lβ,2r )

(0 ≤ t0 < t < ∞, X ∈ l2−p (Lβ,2r ), ∆ ∈ B(l2−p (Lβ,2r ))) .

(9.3)

With the solution Xt there is associated a transition semigroup Tt f , t ≥ 0, in the space of all bounded measurable functions f on l2−p (Lβ,2r ) and an adjoint semigroup µTt , t ≥ 0, in the space P := P(l2−p (Lβ,2r )) of all probability Borel measures µ on l2−p (Lβ,2r ), which act according to the formulas Z Z (9.4) (Tt f )(X) = f (Y )µt (X; dY ) , (µTt )(∆) = µ(dX)µt (X; ∆) . As seen from Theorems 8.1 and 8.3, the semigroup Tt f , t ≥ 0, is Feller in the sense that it preserves the space Cb (l2−p (Lβ,2r )) of all bounded continuous functions f on l2−p (Lβ,2r ). The semigroup Tt f , t ≥ 0, is sometimes (see e.g. [91, 163]) called the stochastic dynamics (corresponding to the quantum lattice system (1.1), (2.4)) to distinguish it from the Langevin (Glauber) dynamics Xt , t ≥ 0, given by the solutions of Eq. (3.3). (However, as we already mentioned, sometimes by stochastic dynamics one just understands the stochastic quantization dynamics given by the solutions of (3.3)). Definition 9.1. A probability Borel measure µinv on l2−p (Lβ,2r ) is said to be an invariant distribution for the Markov process Xt if µinv Tt = µinv , t ≥ 0. ◦

In other words, the corresponding solution Xt ( µ) with the initial distribution ◦ ◦ µ := µ( X ) is a stationary process if and only if µ = µinv . This means ◦

◦

Ef (Xt (µinv )) = Ef ( X ) or hTt f iµinv = hf iµinv for any t ≥ 0 and f ∈ Cb (l2−p (Lβ,2r )). Definition 9.2. A Markov process Xt , t ≥ 0, is called ergodic if (a) it has exactly one invariant distribution µinv , (b) the weak convergence limt→∞ µTt = µinv takes place for all µ ∈ P.

(9.5)


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We will denote by J (X ) the set of all invariant distributions of our Markov process Xt , t ≥ 0, considered with a fixed state space X . Also P(X ) will be the set of all probability Borel measures on X . With obvious modifications, the corresponding definitions can be given for the g.-solution Xt , t ≥ 0, of the system (3.3) in the state space X = l2−p (Cβ ), p > (R + 1)d/2. Remark 9.1. The formula ˜ , µ(∆) = µ ˜ (∆) (∆ ∈ B(l2−p0 (Cβ ) ,

˜ ∩ l−p0 (Cβ ) , ∆=∆ 2 ˜ ∈ B(l−p (Lβ,2r ))) ∆ 2

establishes the one-to-one correspondence between invariant measures µinv on ˜inv on l2−p (Lβ,2r ), p ≥ p0 , such that µ ˜inv (l2−p0 (Cβ )) = 1. Moreover, l2−p0 (Cβ ) and µ from the unique solvability of Eq. (3.3) in both state spaces X1 ⊂ X2 (= l2−p0 (Lβ,2r ), r ≥ 1, p0 > d/2, or l2−p (Cβ ), p > (R + 1)d/2) it follows that J (X1 ) ⊂ J (X2 )

and J (X1 ) = J (X2 ) ∩ P(X1 ) .

(9.6)

Note that there is no problem with the measurability of the solution Xt , t ≥ 0, in different state spaces l2−p0 (Cβ ) ⊂ l2−p0 (Lβ,2r ) since for any two separable Banach spaces with a continuous embedding X1 ⊂ X2 we have B(X1 ) = B(X2 ) ∩ X1 (see cf. [36, 158]). In Sec. 5 we have discussed the ergodic properties of solutions to the linear problem based on their explicit representation (5.2). In the case of a noncompact state space and nonlinear interactions, things are much more complicated and it is not always trivial even to determine whether a nonlinear problem has any invariant distribution, to say nothing about the ergodicity. Section 10 below will be devoted to the study of invariant measures for the system of SPDE’s (3.3). In particular, we will use the notion of ultimate boundedness which goes back for the finite dimensional case to the papers [131, 161], and respectively for the infinite dimensional case to [94]. Definition 9.3. Let Xt , t ≥ 0, be a Markov process with values in a Banach space X . We say that the nth moment of the process Xt is (a) ultimately bounded if ◦

lim EkXt ( X )knX ≤ K < ∞

t→∞ ◦

for all initial data X ∈ X , (b) exponentially ultimately bounded if ◦

◦

EkXt ( X )knX ≤ K1 + K2 e−Lt k X knX , with some K1 , K2 , L > 0.

◦

∀X ∈X ,

(9.7)


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Proposition 9.1 (see [94, 131]). Let the nth moment of Xt , t ≥ 0, be ultimately bounded for some n > 0. Then for any invariant measure µ of the Markov process Xt : Z kXknX µ(dX) ≤ K < ∞ (9.8) E H

where K is the constant given in Definition 9.3(a). The proof of (9.6) follows from Fatou’s lemma and the ergodic theorem for invariant distributions. An analogous proposition can be formulated also for exponential moments. More detailed information about ergodic properties of the system (3.3) can be obtained for the special class of so-called strongly (st.-) dissipative stochastic evolution systems (see Sec. 11). The following general criterium of ergodicity is appropriate in this case. Proposition 9.2 (see [152]). Let Xt , t ≥ 0, be a Markov process with values in a complete measurable metric space (X , ρ), and let Tt , t ≥ 0, be the corresponding transition semigroup. Then Xt is ergodic if the two conditions hold: (i) Tt is contractive in the space of Lipschitz continuous functions on X , i.e. [Tt f ]Lip ≤ e−Lt [f ]Lip ,

t ≥ 0,

(9.9)

(Y )| supX6=Y |f (X)−f ; ρ(X,Y )

where L > 0 and [f ]Lip := (ii) there exists some X0 ∈ X such that Z ρ(X, X0 )δX0 Tt (dX) < ∞ , X

t ≥ 0,

(9.10)

(with δX0 being the δ-measure concentrated at point X0 ). It should be mentioned that the cited criterium has been applied in [146] to the study of infinite-dimensional diffusion processes associated with classical Gibbs measures. Using Proposition 9.2, in Sec. 11 we will offer sufficient conditions for the ergodicity of solutions of the Langevin equation for quantum lattice systems. ◦ Next, we characterize the subclass R ⊆ J of initial distributions µ, for which ◦ the corresponding process Xt ( µ), t ≥ 0, is not only stationary but also reversible. ◦

Definition 9.4. Let QT ( µ) be the probability law on the trajectory space ◦ C([0, T ], l2−p (Lβ,2r )), 0 ≤ T < ∞, of the l.-solution Xt ( µ), t ≥ 0, starting at t = 0 ◦ ◦ ◦ from X , µ( X ) = µ. A probability Borel measure µrev on l2−p (Lβ,2r ) is said to be a reversible distribution of the Markov process Xt , t ≥ 0, if the time-reversed process ˆ t := XT −t (µrev ), 0 ≤ t ≤ T , has the same law QT (µrev ). X Equivalently, using the Markov property (2.3), µrev is reversible iff ◦

◦

E[f (Xt (µrev ))g( X )] = E[f ( X )g(Xt (µrev ))] or hTt f, giL2 (dµrev ) = hf, Tt giL2 (dµrev )

(9.11)


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97

for any t ≥ 0 and f , g ∈ Cb (l2−p (Lβ,2r )). An analogous definition can be also given in the state spaces X = l2−p (Cβ ), p > (R + 1)d/2. We will denote by R(X ) the set of all reversible distributions of the Markov process Xt , t ≥ 0, treated in a fixed state space X . The following basic proposition relating Gibbs and reversible distributions is true: Proposition 9.3. (see [113]). Let the assumptions (A), (B) be satisfied. Then for any µ ∈ P(l2−p (Cβ )), with some p > (R + 1)d/2, we have the characterization µ ∈ R(l2−p (Cβ )) ⇐⇒ µ ∈ Gβ .

(9.12)

Corollary 9.1. Since any µ ∈ Gβt is supported on some l2−p∗ (Cβ ), p∗ > 0, (see the corresponding Definition (2.15)), then the inclusion holds Gβt ⊂

[

R(l2−p (Cβ )) ⊂

p∈N

[

J (l2−p (Cβ )) .

(9.13)

p∈N

We note that for classical lattice systems (3.12) the identity (9.12) between the classes of Gibbs and reversible distributions G = R was studied in [40, 54, 65, 66, 75, 91, 146, 149]. For the finite-particle case (3.13), but with a continuous parameter u ∈ R1 which corresponds to β = ∞, the equivalence between Gibbs property and reversibility has been established in [70, 96]. The next basic proposition gives the description of pure Gibbs states νβ (= extreme points of the convex set Gβt ) through the L2 -ergodicity of the corresponding stochastic dynamics Tt , t ≥ 0. Namely, the following generalization of the early result of Holley and Stroock [91] on the Ising model holds true for the quantum lattice system (3.3): Proposition 9.4 (see [13, Theorem 4.6]). Let the assumptions (A), (B) be satisfied. Then the characterization holds νβ ∈ P(l2−p (R1 )) ∩ Gβt , p > d/2 ,

is extreme

Z ⇔

(Tt f − hf iνβ )2 dνβ −→ 0, t → ∞ ,

for all f ∈ L2 (dνβ ) .

(9.14)

Another open problem is to find general conditions implying the identity between the Gibbs states of quantum lattice models and invariant distributions of the Langevin dynamics. A priori we have the inclusion Gβt ⊂ J and for classical lattice systems some sufficient conditions on Gβt = J were derived in [65, 66]. The situation is completely studied, both in classical and quantum cases, only for the strongly dissipative stochastic dynamics where the identity Gβt = R = J holds trivially due to the uniqueness of νβ = µrev = µinv (see Sec. 11).


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10. Energy Inequalities for SPDE’s and a priori Estimates on Invariant and Gibbs Measures The main goal of this section is to obtain the information about the existence, moment estimates, support of the Gibbs distributions via the qualitative analysis of the Langevin equation. The problem of a priori estimates on moments of tempered Gibbs states for classical lattice systems was firstly discussed in [39]. From now we impose one more condition on the drift terms fk (x, X), k ∈ Zd , namely b.-dissipativity, (C) ∃ b1 > 0, b2 ∈ R1 such that for ∀ x, y ∈ R1 1 (x − y)(fk (x, 0) − fk (y, 0)) = − (x − y)(V 0 (x) − V 0 (y)) ≤ −b1 (x − y)2 + b2 . (10.1) 2 A typical example when (C) holds with any constant b1 = b−ε, ε > 0, is the case (already treated in Proposition 2.1) of V = V0 + W ∈ C 2 (R1 ) being a sum of a strictly convex function V0 , such that 1/2 inf q∈R1 V000 (q) ≥ b > 0, and a bounded function W ∈ Cb1 (R1 ) with δ(W 0 ) := supq∈R1 W 0 (x) − inf q∈R1 W 0 (x) < ∞. In particular, this includes the class of all polynomial self-interactions with V = P given by (1.9). Moreover, when deg P ≥ 4, then it is obviously easy to construct a decomposition P = V0 + W with any arbitrary large b > 0. Below we will conveniently consider the Ornstein–Uhlenbeck process Gm t = ◦ m X t + Gtm , t ≥ 0, together with a small parameter m2 > 0 to be inserted into the definitions (5.8) and (5.9). The following energy inequalities take place for the the solution Xt from the Ornstein–Uhlenbeck process deviation Ytm = Xt − Gm t of ◦ m Gm t provided X0 = G0 = X . Lemma 10.1. Let the assumptions (A)–(C) and (E) be satisfied and let α + 2b1 − m2 > 0 . ◦

◦

Then for a given initial data X ∈ l2−p0 (Cβ ), Ek X k2R −p0 l2

the corresponding g.-solution Xt , t ≥ 0, in the space following estimates hold : sup EkYtm k2l−p (L

(a)

2

t≥0

(b)

Z ∃ K1 , K2 > 0 : E

t2

kYtm k2l−p (W 1 (S 2

t1

β,2 )

2

(0 < t1 < t2 < ∞) .

< ∞, p0 > d/2, and for

(Cβ ) l2−p (Cβ ),

p > p0 R + d/2, the

< ∞,

β ))

dt ≤ K1 + K2 (t2 − t1 ) (10.2)

Proof. Based on Theorem 8.2, the corresponding g.-solution Xt , t ≥ 0, is a continuous process in l2−p0 (Cβ ). Keeping in mind that Z tZ m m yk,t (u) = xk,t − gk,t (u) = qβm (t − s; u, v)fk (xk,s (v), Xs (v))dvds , 0

Sβ


00065


with the identity in the space C(Sβ ) ∂ m 1 2 m y = ∆β + m yk,t + fk (xk,t , Xt ) , ∂t k,t 2

k ∈ Zd ,

99

t > 0,

let us integrate by parts in perfect analogy to the proof of Theorems 7.1 and 8.1 when r = 1, Z 1 d m 2 m 2 m ky k = 2 yk,t (u) (∆β + m )yk,t (u) + fk (xk,t (u), Xt (u)) du dt k,t L2 2 Sβ Z ≤− Sβ

2 Z ∂ m yk,t (u) du + m2 ∂u

m |yk,t (u)|2 du

Sβ

Z

m yk,t (u)fk (xk,t (u), Xt (u))du

+2 Sβ

∂ m 2 2 m 2 m m

y ≤ −

∂u k,t − (2b1 − m − ε)kyk,t kL2 − h(AY )k,t , yk,t iL2 L2 1 m 2 + kfk (gk,t , Gm t )kL2 + 2b2 , ε

t > 0.

(10.3)

Since m > 0, thus by Lemma 5.1 # " X ◦ −2p m m 2 sup E < ∞. (1 + |k|) kfk (gk,t , Gt )kL2 (Sβ ) ≤ const 1 + Ek X k2R −p0 l (C ) t≥0

2

k∈Zd

β

(10.4) Setting 0

m 2 Mt ηˆk (t) = Ekyk,t kL2 e ,

M0 = α/2 + b1 − m2 /2 − ε > 0 ,

∂ m and dropping the terms with ∂u yk,t (u) in the first stage, we pass by Lemma 7.2 from (10.3) to the following estimate on the continuous vector-function ηˆ(t) = (ˆ ηk (t))k∈Zd , t ≥ 0, in the spaces l2−p,δ (Zd ), p > p0 R + d/2, with enough small 0 < δ < δ(ε) and ε > 0: 0

kˆ η (t)k2l−p,δ (Zd ) ≤ C(δ, ε, b1 , b2 ) 2

e2M t − 1 , 2M0

t ≥ 0.

Hence (10.2a) also holds, i.e. ◦ sup EkYtm kl−p,δ (Lβ,2 ) ≤ const 1 + Ek X k2R < ∞. −p0 l (C ) t≥0

2

2

(10.5)

β

Recall that the norms k · kl−p (Lβ,2 ) and k · kl−p,δ (Lβ,2 ) are equivalent in the sense 2 2 of (4.2).


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We return again to the estimate (10.3). Relying on (10.5) and summing over all k ∈ Zd , we obtain that for any 0 < t1 < t2 < ∞

Z t2

∂ m 2 m 2

E Yt dt EkYt2 kl−p,δ (L ) +

−p,δ β,2 2 ∂u t1 (Lβ,2 ) l 2

k2l−p,δ (L ≤ EkYtm 1 2

β,2 )

− 2M0

Z

t2

t1

EkYtm k2l−p,δ (L 2

β,2 )

dt

◦ + C(t2 − t1 ) 1 + Ek X k2R , −p0 l (C )

(10.6)

β

2

which is to say that Z k2l−p,δ (L EkYtm 2 2

β,2

+ )

t2

t1

∂ m 2

Y E dt ≤ K1 + K2 (t2 − t1 ) ,

∂u t −p,δ (Lβ,2 ) l

(10.7)

2

◦

◦

uniformly in all X ∈ l2−p0 (Cβ ) such that Ek X k2R −p0 l2

(Cβ )

< N with fixed N < ∞. ◦

◦

The proof of the lemma is complete. Note that for X = 0 or nonrandom X ∈ p∗ l2 (Cβ ), p∗ ≥ 0, one can take any p > (R + 1)d/2 in (10.2). Theorem 10.1. Let Xt , Xt0 be generalized solutions of the system (3.3) with initial ◦ ◦ data X , X 0 respectively. Then under the assumptions (A)–(C) and (E) it holds almost surely (a)

kXt − Xt0 k2l−p,δ (L 2

(b)

β,2 )

◦ ◦ b2 β 2 k1k2l−p,δ (Zd ) , ≤ e−2Mt k X − X 0 k2l−p,δ (L ) + β,2 2 2 M

2 Z t2

∂

(Xt − X 0 ) dt t

∂u t1 l−p,δ (Lβ,2 )

t ≥ 0,

2

◦

◦

≤ k X − X 0 k2l−p,δ (L 2

+

β,2 )

b2 β 2 k1k2l−p,δ (Zd ) +2b2 (t2 −t1 ) , 2 M

0 < t1 < t2 < ∞ , (10.8)

with the constant M = α/2 + b1 − ε/2 > 0 and for all p ∈ N, 0 < δ < δ0 (ε, p), ε > 0, such that the right-hand side of the estimate (10.8) is finite. Proof. The proof is in full analogy with the proof of Theorem 8.1 and Lemma 10.1. Obviously, for zk,t = xk,t − x0k,t we have kzk,t2 k2L2 ≤ kzk,t1 k2L2 #

2 Z t2 "

∂ 2

− +

∂u zk,t + 2b1 kzk,t kL2 − h(AZ)k,t , zk,t iL2 + 2b2 dt . t1

L2

(10.9)


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101

Since by Theorem 8.1 Z ∈ L∞ ([0, T ], l2−p,δ (Lβ,2 )) ,

0 < T < ∞,

summing over k ∈ Zd in (10.9) and applying the scalar Gronwall inequality, we obtain the required estimate (10.8a): b2 k1kl−p,δ (Lβ,2 ) , t ≥ 0 . 2 2 2 M Combining (10.9) and (10.10), we get accordingly the estimate (10.8b)

Z t2

∂ 2 ◦ b2 β 2 2

Zt Z k1k2l−p,δ (Zd ) k dt ≤ k + −p,δ

∂u −p,δ l2 (Lβ,2 ) 2 M t1 (Lβ,2 ) l kZt k2l−p,δ (L

◦

β,2 )

≤ e−2Mt k Z k2l−p,δ (L

β,2 )

+

(10.10)

2

+ 2b2 (t2 − t1 ) ,

0 < t 1 < t2 < ∞ .

(10.11)

Corollary 10.1. The assertions of Lemma 10.1 can be extended to all g.-solutions ◦ X ∈ C([0, +∞), l2−p (Cβ )), p > (R + 1)d/2, starting from random initial data X ∈ ◦ l2−p (Cβ ) such that Ek X k2l−p (C ) < ∞. β

2

Proof. The proof is based on comparison, (i.e. “independent coupling”) using Theorem 10.1, of the given solution Xt , t ≥ 0, with the solution Xt0 , t ≥ 0, starting ◦ from X 0 = 0. More precisely, in the norm of l2−p (Lβ,2 ) ◦

EkYt k2 ≤ 4E(kYt0 k2 + kXt − Xt0 k2 + k X t k2 ) and

Z

t2

E t1

Z

∂ 2

Yt dt ≤ 4E

∂u

!

2

∂ 0 2 ∂

∂ ◦ 2 0

Y + (Xt − X ) + X t dt , t

∂u t

∂u

∂u

t2

t1

which together with the estimates ◦

◦

k x k,t kL2 (Sβ ) ≤ k x k kL2 (Sβ ) , Z

t2

t1

◦

◦

h∆β x k,t , x k,t iL2 (Sβ ) dt ≤

1 −2t1 ∆β ◦ ◦ ◦ h(e − e−2t2 ∆β ) x k , x k iL2 (Sβ ) ≤ k xk2L2 (Sβ ) 2

give us respectively (10.2) and (10.3). Corollary 10.2. The assertions (10.2a) and (10.8a) can be extended to all l.solutions X ∈ C([0, +∞), l2−p (Lβ,2 )), p > d/2, starting from random initial data ◦ ◦ X ∈ l2−p (Lβ,2 ) such that Ek X k2l−p (L ) < ∞. 2

β,2

Proof. The proof follows immediately from the definition of l.-solution. Theorem 10.2. The Markov process Xt ∈ l2−p (Lβ,2 ), p > d/2, possesses the exponential ultimate boundeness of moments in the sense of Definition 9.4 provided


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α/2 + b1 > 0. Namely, for any nonrandom initial data X ∈ l2−p (Lβ,2 ) and n ∈ N, we have ◦

EkXt ( X )k2n l−p (L 2

◦

β,2 )

≤ K1 (n) + K2 (n)e−2Mnt k X k2n l−p (L 2

(10.12)

β,2 )

with the same M > 0 as in (10.8a). Proof. It suffices again, in view of (10.8a), to check that sup EkXt0 k2n l−p (L 2

t≥0

0 and small enough m2 > 0, ε > 0 and 0 < δ < δ(ε). Using the elementary inequality a2n−2 b2 ≤ σ 2 a2n +

1 σ 2n−2

b2n ,

a, b, σ > 0 ,

we derive from (10.14) that EkYtm k2n ≤ EkYtm k2n − 2n(M0 − σ 2 ) 2 1 Z

t2

+ const

◦

Z

t2

EkYtm k2n dt

t1 ◦

m 2n (1 + Ek{fk ( g m k,t , G t )}k∈Zd k ) dt ,

0 ≤ t 1 ≤ t2 < ∞ .

t1

(10.15) Gronwall’s lemma, as applied to (10.15) with a sufficiently small σ > 0, gives us < ∞, which proves (10.13). that supt≥0 EkYtm k2n l−p,δ (L ) 2

β,2

Corollary 10.3. As follows from Proposition 9.1, the estimate (10.12) ensures the uniform boundedness of polynomial moments with respect to all invariant distributions µ ∈ J (l2−p (Lβ,2 )): Z sup kXk2n dµ(X) ≤ K1 (n) < ∞ , n ∈ N . (10.16a) l−p (L ) µ∈J

l−p 2 (Lβ,2 )

2

β,2


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Since in the system (3.3) the drift terms fk (x, X), k ∈ Zd , are translation invariant, we have the same estimates (10.8)–(10.15) in all l2−p spaces constructed by the shifted sequences of weights {(1 + |k − k0 |)−2p }k∈Zd , k0 ∈ Zd . This yields the following uniform bound on the moments of every coordinate xk , k ∈ Zd : Z kxk k2n n ∈ N. (10.16b) sup sup Lβ,2 ≤ K1 (n) < ∞ , µ∈J k∈Zd

l−p 2 (Lβ,2 )

The latter implies that µ(l2−p0 (Lβ,2 )) = 1 and hence due to (9.6) J (l2−p (Lβ,2 ) = J (l2−p0 (Lβ,2 )) for any fixed p0 > d/2. Remark 10.1. Suppose that the drifts fκ (x, X), κ ∈ Zd , have no more than linear growth, i.e. (3.5) is valid with R = 1. Then, under the assumptions of Theorem 10.2, there exists κ0 = κ0 (p0 ) > 0 such that Z κkXk2−p l (Lβ,2 ) 2 e dµ(X) ≤ K < ∞ (10.17a) sup l−p 2 (Lβ,2 )

µ∈J

for any p ≥ p0 > d/2 and 0 ≤ κ ≤ κ0 . The proof of (10.17a) is quite similar to the proof of Theorem 10.2. In view of Proposition 9.1, it suffices to verify the ultimate boundedness of exponential ◦ moments of the Markov process Xt ( X ), i.e. that ◦

lim Ee

κkXt ( X )k2−p l2

(Lβ,2 )

t→∞

≤K

(10.18)

◦

uniformly for any nonrandom initial data X ∈ l2−p (Lβ,2 ). But due to (10.8a), one ◦ m starting from X 0 = 0. should check (10.18) only for the solution Xt0 = Gm t + Yt Because of Corollary 5.1, we really need to prove that lim Ee

κkYtm k2−p,δ l2

0. From (10.14), using the elementary inequality 2

2

2

κeκa (−a2 + b2 ) ≤ −eκa + eκb

a, b, κ > 0 ,

we come to the estimate in the norm of l2−p,δ (Lβ,2 ): Z t m 2 m m 2 κ e 2M k{fk (gk,s ,Gs )}k∈Zd k e2Ms ds , eκkYt k ≤ e−2Mt 1 + const

(10.20)

t ≥ 0.

0

The latter, together with (5.20), gives us (10.19) for small enough 0 < κ < κ0 . As soon as (10.17a) is proved, considering all possible sequences of weights {(1 + |k − k0 |)−2p }k∈Zd , k0 ∈ Zd , we also derive the inequality Z κkx k2 e k Lβ,2 dµ(X) ≤ K < ∞ . (10.17b) sup sup µ∈J k∈Zd

l−p 2 (Lβ,2 )

Remark 10.2. For the system of interacting diffusions (3.12) the assumptions (B), (C) and (E) would suffice to prove the ultimate boundedness of polynomial


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and exponential moments of solutions. In fact we do not need any information on the growth of V 0 , since Itô’s lemma (see e.g. [44, 51]) is applicable to the strong solution Xt , t ≥ 0, in the spaces l2−p,δ (Zd ), p > d/2, δ > 0. In fact, we have in the notation of Example 8.2, ◦ Z t κkXt k2−p,δ κ| X k2−p,δ κkXs k2−p,δ l2 l2 l2 ≤ Ee + κE e Ee 0

× (−hAXs , Xs il−p,δ − hXs , V 0 (Xs )il−p,δ 2

+ 2κkXsk2l−p,δ 2

2

⊂

+ kO : l2 −→

2 l−p,δ k2H.−Sh. )ds

Z

◦

≤ Ee

κk X k2−p,δ l

2

t

+ κ(α + 2b1 − ε − 2κ)E ⊂

+ κ(b2 + kO : l2 −→

l2−p,δ k2H.−Sh. )E

Z

kXs k2l−p,δ e

κkXs k2−p,δ l

2

ds

2

0

t κkX k2 s −p,δ l

e

2

ds .

(10.21)

0

Using again the inequality (10.20), we conclude that for any 0 < M < α/2 + b1 and small enough 0 < δ < δ(ε), 0 < κ < κ0 we have Ee

κkXt k2−p,δ l

2

◦

≤e

κk X k2−p,δ l

2

◦

e−2Mt + const

(10.22)

l2−p (Zd ),

uniformly for all nonrandom initial data X ∈ p > d/2. We remind however that an additional localization procedure would be required to complete the arguments since we do not know in advance that all expectations in (10.21) are finite. Remark 10.3. Analogous estimates are available in the spaces l2−p,δ (Lβ,2r ) and l2−p,δ (Cβ ) with r ≥ 1, p > d/2, δ > 0. Namely, suppose that (A)–(C) and (E) hold and  " #1/2  X 1  > 0. L(p, δ) = b1 + (1 + δ|k|)2p a(0) − kak0 (10.23) 2 0 (R + 1)d/2 and L(p0 , δ) > 0 we have also that kXt − Xt0 k2l−p0 (C 2

◦

β)

◦

≤ e−2L(p0 ,δ)t k X − X 0 k2l−p0 ,δ (C ) + 2

β

b2 β 2 k1k2l−p,δ (Zd ) . 2 L(p0 , δ)

(10.25)


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105

The statement of Theorem 10.2 about the exponential ultimate boundedness of ◦ + Y , t ≥ 0, be the moments can also be modified in a similar way. Let Xt0 = G m t t ◦ g.-solution starting from X 0 = 0. Now, repeating the arguments (10.13)–(10.15), we get for small enough σ > 0 the following estimate in the norm of l2−p,δ (Lβ,2r ): Z t2 2n m 2n 2 k ≤ EkY k − 2n(L(p, δ) − σ ) EkYtm k2n dt EkYtm t1 2 1 t1

Z

t2

+ C(n, σ)

◦

◦

m 2n (1 + Ek{fk ( g m k,t , G t )}k∈Zd k ) dt ,

0 ≤ t 1 ≤ t2 < ∞ .

t1

(10.26) Since m > 0, then by (5.15) ◦

◦

m 2n sup Ek{fk ( g m k,t , G t )}k∈Zd kl−p (C 2

t≥0

β)

< ∞.

(10.27)

Fatou’s lemma, as applied to (10.16) and (10.27), gives us immediately that sup EkXt0 k2n l−p (C 2

t≥0

β)

≤ sup lim EkXt0 k2n l−p (L t≥0 p∈N

2

β,2r )

< ∞.

(10.28)

Combining (10.24), (10.25) and (10.28), we get that for any nonrandom initial ◦ data X ∈ l2−p0 (Lβ,2r ), 1 ≤ r ≤ ∞, and n ∈ N, ◦

EkXt ( X )k2n −p δ l 0 (L 2

◦

≤ K1 (n, p0 , δ) + K2 (n)e−2L(p0 ,δ)nt k X k2n −p ,δ l 0 (L

β,2r )

2

β,2r )

.

(10.29)

In full analogy with (10.16a, b), from (10.29) the uniform estimates on the moments of all invariant measures µ ∈ J (l2−p (Lβ,2µ )), 1 ≤ r ≤ ∞, follow Z sup kXk2n dµ(x) ≤ K1 (n, p0 ) < ∞ , n ∈ N , (10.30a) −p l 0 (L ) µ∈J

−p0

l2

2

(Lβ,2r )

β,2r

Z sup sup µ∈J k∈Zd

l−p 2 (Lβ,2r )

kxk k2n (Lβ,2r ) ≤ K1 (n, p0 ) < ∞ ,

n ∈ N.

(10.30b)

Obviously, as compared with the case r = 1, the sufficient condition (10.23) is too rough. But here we will not go into detail on how to improve it. As was mentioned in the beginning of this section, the condition (10.23) holds for all polynomial models (1.9) with deg P ≥ 4. For fixed p > d/2, the condition (10.23), with small enough 0 < δ ≤ δ(p), is a consequence of the following assumption: (F)

1 b1 + (a(0) − kak0 (|Bρ (0) − 1)) > 0 . 2

(10.31)

For convenience we sum up all the previous work on the a priori estimates for the invariant (and hence, Gibbs) distributions in the next theorem. Theorem 10.3. Let the assumptions (A)–(C) and (E) be satisfied and let b1 + α/2 > 0. Then all invariant measures µ of the Markov process Xt , t ≥ 0, (treated


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in the scale of spaces l2−p (Lβ,2r ), l2−p (Cβ ), r ≥ 1, p > d/2) are supported on T −p0 (Lβ,2 ) and have finite polynomial moments: p0 >d/2 l2 D E 2n < ∞, n ∈ N. (10.32) sup sup hkxk k2n Lβ,2 iµ ≤ sup kXkl−p0 (L ) µ∈J k∈Zd

2

µ∈J

β,2

µ

If the additional assumption (F) holds, then all the invariant measures are supported on l2−p0 (Lβ,2r ) ∩ l2−p (Cβ ) with any fixed r ≥ 1, p0 > d/2, p > (R + 1)d/2 and have finite moments D E 2n kXk i ≤ sup < ∞, sup sup hkxk k2n −p µ Lβ,2r l 0 (L ) µ∈J k∈Zd

µ∈J

2

µ

β,2r

D E 2n < ∞, sup sup hkxk k2n Cβ iµ ≤ sup kXkl−p0 (C )

µ∈J k∈Zd

µ∈J

2

β

µ

n ∈ N.

(10.33)

S Corollary 10.4. Since Gβt ⊂ p∈N J (l2−p (Cβ )) (according to Corollary 9.1), then under the assumptions of Theorem 10.3 the following a priori estimates hold true for all tempered Gibbs distributions νβ ∈ Gβt , sup sup hkxk k2n Lβ,2r iνβ < ∞ ,

t k∈Zd νβ ∈Gβ

sup sup hkxk k2n Cβ iνβ < ∞ ,


n ∈ N.

(10.34)

Remark 10.4. The statements (10.2b), (10.8b) and (10.46) tell us that under the assumption (F) one has sup sup hkxk k2C α (Sβ ) iνβ < ∞ ,

(10.35)


where C α (Sβ ) is the space of Hölder continuous functions with exponent 0 ≤ α < 1/2. Remark 10.5. All the estimates in Sec. 8, 10 are based in reality on some general dissipativity properties of our system in the Banach spaces l2−p (Lβ,2r ), l2−p (Cβ ) (a sufficient condition for this, uniformly in r ≥ 1, is given by (F)). An abstract approach to dissipative stochastic evolution equations was developed in the recent papers [53, 143]. But in our case we need more detailed information about the solution, and for this reason we have given an independent approach for our models. The behaviour of solutions of some SPDE’s in the scale of H¨ older spaces was also studied in [38, 71, 143]. In the end of this section we present the existence theorems for invariant respectively Gibbs distributions. Theorem 10.4. Let the assumptions (A)–(C) and (E) be satisfied and let α/2 + 0 b1 > 0. Then J (l2−p (Cβ )) 6= ∅ for all p0 > (R + 2)d/2. Proof. The proof uses quite standard techniques (cf. [37, Proposition 5.1]) and is based on a time average procedure. For simplicity let us take the initial data ◦ X = 0 and let X ∈ C([0, ∞), l2−p (Cβ )), p > (R + 1)d/2, be the corresponding


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g.-solution defined on the probability space (Ω, F, P ). We denote by µt := µ(Xt ) Rt 0 the distribution of Xt on l2−p (Cβ ), p0 > p + d/2, and by µ ¯t := 1t 0 µs ds its Cesáro mean. It suffices to prove that the family {¯ µt }t≥0 is tight in the sense of weak convergence. As well known, by the Feller property of Tt , t ≥ 0, any limit point 0 µt }t≥0 will be an invariant measure µinv ∈ J (l2−p (Cβ )). But µ ¯ t can be µ∞ for {¯ viewed as the distribution of the random variable X(s, ω) := Xs (ω)◦ realized on m the probability space (Ω × [0, t], F × B([0, t]), P × ds/t). Since Xs = G m s + Ys , it can readily be◦ observed (cf. [36]) that the tightness of laws of the random variables ◦ G m (s, ω) := G m Y m (s, ω) := Ysm (ω) will provide the tightness of laws of s (ω) and ◦ m their sums X(s, ω) = G (s, ω) + Y m (s, ω). Due to Lemma 10.1, we have sup E(P × ds ) kY m (s, ω)k2l−p (W 1 (S t

t≥0

2

2

β ))

< ∞.

(10.36)

As usual, denote by C α (Sβ ) the Banach space of Hölder continuous functions on Sβ with the norm kf kC α(Sβ ) := sup |f (u)| + u∈Sβ

sup

u,u0 ∈Sβ u6=u0

|f (u) − f (u0 )| . |u − u0 |α

(10.37)

Relying on the compactness of the embedding 0

⊂

l2−p (W21 (Sβ )) −→ l2−p (C α (Sβ )) ,

p0 > p + d/2 ,

0 ≤ α < 1/2 ,

and the general Prokhorov criteria on the tightness of a family of distributions in a separable metric space (see e.g. [47, 59]), we conclude from (10.36) that 0 {µ(Y m (s, ω))}t≥0 is tight in l2−p (C α◦ ). As for the Gaussian process G m t , according to Lemma 5.1 and estimates (5.6a, c), we have that for all r ∈ N and u, u0 ∈ Sβ ◦

2r E(P ) | g m ≤ C(m, r) , k,t (u)|

m > 0,

E(P ) | g k,t (u) − g k,t (u0 )|2r ≤ C(α, r)|u − u0 |αr , ◦m

◦m

(10.38a) 0 ≤ α < 1,

(10.38b)

b) with the conditions◦ of uniformly in k ∈ Zd and t ≥ 0. Comparing (10.38a, ◦ m )}t≥0 of the process G m Lemma 4.3, we obtain the tightness of the laws {µ( G t , ◦ t −p m t ≥ 0, in any space l2 (Cβ ), p > d/2. Therefore {µ( G t (s, ω))}t≥0 also should be 0 tight in l2−p (Cβ ). All that gives the tightness of {¯ µt }t≥0 in l2−p (Cβ ), p0 > (R+2)d/2, 0 and the relation µ∞ := w. limn→∞ µ ¯tn ∈ J (l2−p (Cβ )) 6= ∅. Remark 10.6. Under the assumptions of Theorem 10.4, for nonrandom initial data X ∈ l2−p0 (Cβ ), p0 > d/2, and for the corresponding g.-solution X ∈ C([0, ∞), µt (X)}t≥0 is tight in the l2−p (Cβ )), p > p0 R + d/2, the family of distributions {¯ 0 space l2−p (Cβ ), p0 > p0 R + d. Furthermore, examining the estimates (10.38) more 0 closely, one can obtain the tightness of {¯ µt (X)}t≥0 in the spaces l2−p (C α (Sβ )), p0 > p0 R + d, 0 ≤ α < 1/2.


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Combining Theorems 10.3 and 10.4, we get that J (l2−p0 (Lβ,2 )) 6= 0 ∀ p0 > d/2 and, if in addition (F) holds, then J (l2−p (Cβ )) 6= ∅ ∀ p > (R + 1)d/2. Theorem 10.5. Let the assumptions (A)–(C) and (E) be satisfied and let α/2 + b1 > 0. Then Gβt 6= ∅, and moreover ∃ νβ ∈ Gβt such that νβ ∈ R(l2−p (Cβ )) and D E kXk2l−p (C α (S )) < ∞ , 0 ≤ α < 1/2 , p > d/2 . (10.39) 2

β

νβ

Proof. As usual, we should look at the thermodynamic limit of local specifications νβ,ΛN , ΛN % Zd , which are given by the definition (2.9). Utilizing the semiboundedness of the self-interaction V and using Girsanov’s transform, one can prove in a standard way (cf. [69, 70, 97]) the following description of the finite volume Gibbs distributions as the invariant distributions of the corresponding finite volume stochastic dynamics. Proposition 10.1. Under the above assumptions and for every fixed boundary condition Y ∈ Ωβ ∈ C(Sβ ) (and moreover, Y ∈ L2 (Sβ )), the finite volume Gibbs distribution νβ,Λ (dX|Y ), |Λ| < ∞, is the unique invariant (and moreover reversible) distribution of the cutoff system (see Sec. 7)  ∂ (Λ) 1 ∂ 2 (Λ)  (Λ) (Λ)  x (u) + fk (xk,t (u), Xt (u)) + ω˙ k,t (u) , k ∈ Λ ,  xk,t (u) = ∂t 2 ∂u2 k,t (10.40)    x(Λ) (u) = y (u) , k ∈ / Λ (t > 0, u ∈ S ) , k

k,t

β

which is ergodic when considered in the state space (C(Sβ ))Λ (respectively ◦ ◦ (L2 (Sβ ))Λ ) and with initial data x Λ := ( x k )k∈Λ . ◦

Now take for simplicity X = Y = 0. The tightness of {νβ,ΛN (dX|0)}N ∈N in l2−p (Cβ ), p0 > p + d/2, will be evident from the uniform integrability condition in l2−p (C α (Sβ )), 0 < α < 1/2, p > d/2: D E < ∞. (10.41) sup kXk2l−p (C α (S )) 0

N ∈N

(N )

2

β

νβ,ΛN

Let Xt , t ≥ 0, be the g.-solutions (given by Lemma 7.1) to the corresponding finite-volume systems (10.40) and let P (N ) be the projectors on the finitedimensional subspaces in l2 (Zd ) spanned by the vectors {ek }k∈ΛN . Then applying for every N ∈ N the generalization of Fatou’s lemma for weakly random R converging (N ) (N ) 1 t variables [47] to the tight family of distributions {¯ µt := t 0 µ(Xs )ds}t≥0 (see the notation and proof of Theorem 10.4), for the validity of (10.41) it suffices to show that Z 1 t EkXs(N ) k2l−p (C α ) ds < ∞ . (10.42) sup sup 2 N ∈N t≥0 t 0


00065


Define

 (N ) ◦ (N )  yk,t = xk,t − g m k,t , 

109

k ∈ ΛN ,

(N )

k∈ / ΛN ,

yk,t = 0 ,

and repeat the scheme of the proof of Lemma 10.1 in order to estimate kYt k2l−p (C α ) . Namely, like (10.3), for k ∈ ΛN

∂ (N ) 2 d (N ) 2 (N ) 2 2

ky k ≤ − yk,t

− (2b1 − m − ε)kyk,t kL2 dt k,t L2 ∂u L2

2

(N )

− h(AP (N ) Y (N ) )k,t , yk,t iL2 ◦ 1 ◦ + kfk ( g k,t , P (N ) G t )k2L2 + 2b2 , t > 0 . (10.43) ε Since α/2 + b1 − ε > 0, we can find a ◦small enough m > 0 such that by Lemma 5.1 ◦ m G G for the Ornstein–Uhlenbeck process t := t , t ≥ 0, we have ◦

◦

sup sup Ekfk ( g k,t , P (N ) G t )k2l−p (L

N ∈N t≥0

2

β,2 )

< ∞.

(10.44)

Following the reasoning of (10.4)–(10.7) and observing that hP (N ) Y, AP (N ) Y il−p,δ (Zd ) ≥ (α − ε)kP (N ) Y k2l−p,δ (Zd ) , 2

we get that 1 sup sup N ∈N t≥0 t

Z 0

2

t

EkYs(N ) k2l−p (W 1 ) ds < ∞ . 2

(10.45)

2

It remains to show that for all k ∈ Zd ◦

sup Ek g k,t k2C α (Sβ ) < ∞ .

(10.46)

t≥0

But as (5.11) tells us, for any r ∈ N, 0 ≤ σ < 1 E| g k,t (u) − g k,t (u0 )|2r ≤ C(r, σ)|u − u0 |σr , u, u0 ∈ Sβ . ◦

◦

Then by Inequality (3.b) from [35]: r ◦ ◦ | g k,t (u) − g k,t (u0 )| sup E sup d. The integrability property (10.39) follows from the Fatou’s lemma for weakly converging random variables. Note that every accumulation point of {νβ,ΛN (dX|Y )}N ∈N is Gibbs, since the locality condition for the family of local specifications νβ,Λ holds [81]. More precisely,


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due to the regularity property of the interaction Iβ,Λ (see Sec. 2), for every f ∈ Cb (l2−p (Cβ )) the local expectation hf iνβ,Λ (dX|Y ) is a continuous function of the boundary conditions Y ∈ Ωβ and, by Lebesgue’s theorem, one can pass to the limit ΛN % Zd in the consistency equation (2.13). And finally, (9.12) implies the inclusion νβ ∈ R(l2−p (Cβ )). ◦

Remark 10.7. For nonrandom initial data X ∈ l2−p0 (Cβ ), p0 > d/2, and for the corresponding g.-solution X ∈ C([0, +∞), l2−p (Cβ )), p > p0 R + d/2, the family of ◦ 0 distributions {¯ µt ( X )}t≥0 is tight in l2−p (Cβ ), p0 > p0 R+d. Furthermore, examining ◦ the estimates (10.33) more closely, one can obtain the tightness of {¯ µt ( X )}t≥0 in 0 the spaces l2−p (C α (Sβ )), p0 > p0 R + d, 0 ≤ α < 1/2. 11. Ergodic Properties of Strongly Dissipative Systems Our prime interest here concerns the systems (3.3) and (6.1) under the assumption of strongly dissipativity. This means, recalling the definition (3.10), that (D) ∃ b > 0 such that for ∀ x, y ∈ R1 1 (x − y)(fk (x, 0) − fk (y, 0)) = − (x − y)(V 0 (x) − V 0 (y)) ≤ −b(x − y)2 . (11.1) 2 Again, as in Sec. 10, it is convenient to extract a mass term m2 > 0 from the potential V and the dynamical matrix A = (a(k − j))k,j∈Zd , A ≥ α1l, and thereby substitute in the equations (− 12 ∆β + m2 1l) for − 21 ∆β . With the assumption (D) the system (3.3) possesses the exponential loss of memory of the initial data. According to Theorem 10.1 and Remark 8.4, we have with probability one ◦

◦

kXt − Xt0 kl−p,δ (Lβ,2 ) ≤ e−Mt k X − X kl−p,δ (Lβ,2 ) , 2

2

t ≥ 0,

(11.2) ◦

for any two l.-solutions Xt , Xt0 ∈ l2−p (Lβ,2 ), p > d/2, starting from X respectively ◦ X 0 . The relation (11.2) holds true with the constant M = α + b − ε > 0 and in all spaces l2−p,δ (Lβ,2 ) where 0 < δ < δ0 (p, ε). Moreover, as follows from Sec. 8, under the stronger assumption   X 12 1 (1 + δ|k|)2p  > 0 , L(p, δ) := b + a(0) − kak0 (11.3) 2 0 (R + 1) d2 , ◦

◦

kXt − Xt0 kl−p,δ (Cβ ) ≤ e−L(p,δ)t k X − X 0 kl−p,δ (Cβ ) , 2

2

t ≥ 0,

(11.4)

1 L := b + (a(0) − kak0 (|Bρ(0)| − 1)) > 0 , 2 and for l.-solutions Xt , Xt0 ∈ l2−p (Lβ,2r ), p0 > d/2, ◦

◦

kXt − Xt0 kl−p,δ (Lβ,2r ) ≤ e−L(p,δ)t k X − X 0 kl−p,δ (Lβ,2r ) , 2

2

t ≥ 0,

(11.5)


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uniformly on r ∈ N. Again one can only suppose that L := b +

1 (a(0) − kak0 (|Bρ(0)| − 1)) > 0 , 2

(11.6)

then (11.4), (11.5) hold obviously with 0 < δ < δ0 (p, ε). Note that all the estimates are valid (with the same constants M, L) also for any finite volume solutions XtN , ◦ ◦ 0 ◦ ◦ Xt N with initial values X and X 0 such that x k = x 0k , k ∈ Zd \ΛN , i.e. for example, ◦

0

◦

kXtN − Xt N kl−p,δ (Lβ,2 ) ≤ e−Mt k X − X 0 kl−p,δ (Lβ,2 ) , 2

t ≥ 0.

2

(11.7)

The foregoing properties imply immediately the ergodicity of the process Xt , t ≥ 0, as we shall see in the next Theorem 11.1. Let the assumptions (A)–(E) be satisfied. If b + α/2 > m2 > 0, then the Markov process Xt , t ≥ 0, solving the system of stochastic integral equations (3.3), is ergodic in every l2−p (Lβ,2 ), p > d/2. If, moreover, 1 b + (a(0) − kak0 (|Bρ(0)| − 1)) > m2 > 0 , 2 then Xt , t ≥ 0, is ergodic in every l2−p,δ (Lβ,2 ), p > d/2, r ≥ 1. Proof. First we consider X = l2−p,δ (Lβ,2 ) with 0 < δ < δ0 (p, ε) and ε = m2 > 0. The verification of the conditions (i), (ii) of Proposition 9.1 reduces to the estimates (i)

◦

◦

k(Tt f )( X ) − (Tt f )( X 0 )kl−p,δ (Lβ,2 ) 2

≤ [f ]Lip EkXt − Xt0 kl−p,δ (Lβ,2 ) 2

≤e (ii)

−Mt

◦

◦

[f ]Lip k X − X 0 kl−p,δ (Lβ,2 ) , 2

EkXt kl−p (Lβ,2 ) < ∞ , 2

t ≥ 0,

t ≥ 0,

(11.8) (11.9) ◦

◦

to be valid for any two l.-solutions Xt , Xt0 with nonrandom initial values X , X 0 respectively. But just the same estimates are already given by (11.2) and (8.15). For X = l2−p (Lβ,2r ) the proof relies on (11.5) and (8.15). Remark 11.1. Let us discuss in more detail the ergodic criteria for Markov processes stated in Proposition 9.1. In fact, this criteria reduces the problem of the existence and uniqueness of an invariant measure to the application of a fixed point theorem in the space µ of probability measures on X equipped with a Wasserstein metric. Let (X , ρ) be a complete separable metric space, P be the space of all probability measures on (X , B(X )) and Z ρ(X, X0 )dµ(X) < ∞ with some X0 ∈ X . P1 = µ ∈ P, X


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The Wasserstein distance R is defined for two measures µ1 , µ2 ∈ P by R(µ1 , µ2 ) := inf Eρ(ϕ1 , ϕ2 ) ,

(11.10)

ϕ1 ,ϕ2

where the infimum is taken over all pairs of X -valued random variables ϕ1 , ϕ2 with distributions µ1 and µ2 respectively. Proposition 11.1 (see [46]). Let µ0 ∈ P and U (µ0 ) := {µ ∈ P|R(µ0 , µ) < ∞}. Then (U (µ0 ), R) is a complete metric space. If µn → µ, n → ∞, in the Wasserstein metric, then µn → µ, n → ∞, in the topology of weak convergence. An equivalent definition of the Wasserstein metric R is given by the Kantorovich– Rubinshtein dual relation. Namely, for any µ1 , µ2 ∈ P1 Z Z (11.11) R(µ1 , µ2 ) = sup f dµ1 − f dµ2 , f ∈Lip1 (X )

where

X

( Lip1 (X ) :=

f : X → R1 , [f ]Lip

X

|f (X) − f (Y )| := sup ρ(X, Y ) X6=Y

) ≤ 1.

(11.12)

Moreover, (P1 , R) is a complete metric space and for µ, µ1 , µ2 , . . . ∈ P1 we have (cf. [59, 145]), Z w ρ(X, X0 )(µn − µ)(dX) → 0, n → ∞ . R(µn , µ) → 0 ⇔ µn −→ µ , X

Actually, in the definition (11.10) one can take infinum over a smaller class of functions than Lip1 (X ). For example, when X = H is a separable Hilbert space, ∞ (H) with bounded it suffices to consider only smooth cylinder functions f ∈ Ccyl 0 first order derivative kf kH ≤ 1. Since we need more comprehensive information about the invariant measure, let us explain how to obtain it using the definition (11.10). Given X = l2−p,δ (Lβ,2 ) and X0 ∈ X , denote by µt = δX0 Tt = µt (X0 ; dX) the transition probabilities of the Markov process Xt . According to (9.8), (11.8), µt (X0 ; dX) ∈ P1 ⊂ U (δX0 ) , and by (9.7), (11.2) R(µt , µt+∆t ) =

Z Z sup f (X)dµt+∆t (X) − f (Y )dµt (Y )

[f ]Lip ≤1

≤ e−Mt

Z

X

X ⊗X

X

kX − Y kX dµ∆t dµ0

= e−Mt EkX(∆t; X0 ) − X0 kl−p,δ (Lβ,2 ) .

(11.13)

2

Hence the stabilization of translation probabilities takes place, i.e. ∃ lim µt = lim µt (X0 ; dX) = µ∞ (X0 ) ∈ P1 ⊂ U (δX0 ) . t→∞

t→∞

(11.14)


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The rate of convergence is exponential: R(µt , µ∞ ) ≤ e−Mt C(X0 ) ,

t ≥ 0,

(11.15)

with the constant C(X0 ) = kX0 kX + sup EkX(s; X0 )kX < ∞ ,

(11.16)

s≥0

which is finite because of (10.12). Due to the Kolmogorov–Chapman equation, the limit measure µ∞ = µ∞ (X0 ) is invariant: ∀ f ∈ Cb (X ) Z Z Z f (X)(µ∞ Tt )(dX) = µ∞ (dY ) f (X)µt (Y ; dX) X

X

X

Z

= lim

s→∞

X

Z

µs (X0 ; dY )

Z = lim

s→∞

X

f (X)µt (Y ; dX)

X

Z

f (X)µs+t (X0 ; dY ) =

X

f (X)µ∞ (dX) .

The weak convergence µt (X00 ; dX) −→ µinv (dX) := µ∞ (dX) , w

t → ∞,

(11.17)

holds for all X00 ∈ X as well. Let Xt , Xt0 be corresponding solutions with initial ◦ ◦ values X = X0 , X 0 = X00 respectively, then Z Z 0 f (X)µt (X0 ; dX) − f (X)µt (X0 ; dX) X

Z ≤

X

|f (Xt ) − f (Xt0 )|dP → 0 ,

t → ∞,

Ω

where we have used the inclusion f ∈ Cb (X ), the estimate (11.2) and the Lebesgue dominated convergence theorem. In a similar way, w

µTt −→ µ∞ ,

t → ∞,

(11.18)

for all Borel measures on l2−p,δ (Lβ,2 ). Actually, due to (9.4) and (11.16), one has: ∀ f ∈ Cb (X ) Z f (X)(µTt )(dX) X

Z

Z f (X)

= X

−−−→ t→∞

X

Z

Z µ(dY )µt (Y ; dX) = Z

µ(dY ) X

X

Z µ(dY )

X

f (X)µ∞ (dX) =

Z X

X

f (X)µt (Y ; dX)

f (X)µ∞ (dX) .

The relation (11.17) implies also the uniqueness of µinv = µ∞ .


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The same arguments are valid in the phase space X = l2−p,δ (Lβ,2r ), r ≥ 1. Hence, recalling the definition (11.10) of the Wasserstein metric, we obtain the following statement which is exactly equivalent to (11.14). Corollary 11.1. Under the assumption of Theorem 11.1 there exists a unique invariant measure µinv for the transition semigroup Tt , t ≥ 0. Moreover, the rate of convergence to equilibrium is exponential : ∀ f ∈ Lip1 (X ) Z f (Y )dµinv (Y )| ≤ e−Mt C(X)[f ]Lip . (11.19) |(Tt f )(X) − X

Remark 11.2. The estimate (11.14), in terms of the Wasserstein metric, has been obtained in [18] for the case of quantum lattice systems; for classical lattice systems one can find its analogues in [146, 152]. The exponential convergence in the Wasserstein metric for continuous models in Rd was studied in [70]. A more recent develoment [53] is concerned with a general result about the convergence (11.16) of transition semigroups corresponding to strictly dissipative stochastic systems. Corollary 11.2. According to the Birkhoff ergodic theorem for Markov processes (cf. [151, Sec. 1, 2]), the identity of time and phase expectations takes place. For any f ∈ L1 (l2−p (Lβ,2 ), dµinv ) and for µinv -almost all X ∈ l2−p (Lβ,2 ) a nonrandom limit exists Z Z 1 t lim (Ts f )(X)ds = f (X)dµinv (X) . (11.20) t→+∞ t 0 X Recall that here by the definition (9.4) Z Z f (Y )µt (X; dY ) = f (Xt )dP (Tt f )(X) = l−p 2 (Lβ,2 )

Ω ◦

and Xt , t ≥ 0, is the l.-solution of (3.39) with the initial data X = X. It was shown above that for f ∈ Cb (l2−p (Lβ,2 )) we have in (11.19) even a pointwise identity on l2−p (Lβ,2 ). Corollary 11.3. Due to Remark 9.1 and under the assumption of Theorem 11.1, we get also the uniqueness of the invariant measure for Xt , t ≥ 0, considered as a ◦ Markov process in the spaces l2−p (Cβ ), p > (R + 2)d/2. Let X = 0 and µt (dX) := µt (0; dX). The process Xt , t ≥ 0, is ergodic in l2−p (Lβ,2 ) with the unique invariant ¯t , where µ ¯t := measure µinv := w. − limt→∞ µt . Moreover µinv := w. − limt→∞ µ R −p 1 t µ ds. By Theorem 10.3 the family {¯ µ } is tight on l (C ), which means t t≥0 β 2 t 0 s that µinv (l2−p (Cβ )) = 1. We end this section by showing that for the st.-dissipative system (3.3) the two limit procedures, namely in t → ∞ and ΛN → Zd , commute. Suppose that the assumptions of Theorem 11.1 are fulfilled. Let us fix a boundary condition Y = {yk }k∈Zd ∈ l2−p (Lβ,2 ) and a finite volume ΛN ⊂ Zd . We ◦ ◦ consider initial values X (N ) ∈ l2−p (Lβ,2 ) such that x k = yk , k ∈ Zd \ΛN . Let


00065

Glauber Dynamics for Quantum Lattice Systems (N )

(N )

(N )

115

(N )

Xt = {x1,t , x2,t , . . . , xN,t , yN +1 , yN +2 , . . .}, t ≥ 0, be the corresponding so◦ lutions of the system of integral equations (7.2) starting from X (N ) , which are continuous processes on l2−p (Lβ,2 ) due to Lemma 7.1. Then, by virue of the esti(N ) mate (11.6), {xk,t }k∈ΛN is an ergodic Markov process in the space (L2 (Sβ ))ΛN ,

conand hence the distributions µt on l2−p (Lβ,2 ) of the random variables Xt (N ) −p verge weakly as t → ∞ to some limit measure µ∞ (dX|Y ) on l2 (Lβ,2 ). Actually, (N ) µ∞ ({X|xk = yk , k ∈ Zd \ΛN }|Y ) = 1. ◦ Let X = l2−p,δ (Lβ,2 ) and for simplicity we may next take X (N ) = Y . By analogy with the proof of (11.14), we derive from (11.6) that Z Z (N ) (N ) (N ) (N ) f (Y )dµt (Y ) R(µt , µt+∆t ) = sup f (X)dµt+∆t (X) − (N )

[f ]Lip ≤1

≤ e−Mt

Z

(N )

X

X

(N )

X ⊗X

(N )

kX − Y kX dµ∆t dµ0

= e−Mt EkX∆t − Y kl−p,δ (Lβ,2 ) (N )

(11.21)

2

and (N )

R(µt

(N ) (dX), µ∞ (dX|Y )) ≤

e−Mt (N ) EkX1 − Y kl−p,δ (Lβ,2 ) . 2 1 − e−M

(11.22)

An important point is that, as a consequence of the estimate (8.14), for 0 < T < ∞ (N ) 2 kl−p (L ) β,2 2

sup EkXt − Xt

0≤t≤T

→ 0,

N → ∞.

(11.23)

Therefore we have, uniformly in N ∈ N, (N )

R(µt

(N ) (dX), µ∞ (dX|Y )) ≤ Ce−Mt ,

t ≥ 0,

with a constant C > 0 depending only on the boundary condition Y ∈ Moreover, from (11.22) we have the estimate Z (N ) (N ) R(µt , µt ) = sup (f (Xt ) − f (Xt ))dP [f ]Lip ≤1

(11.24) l2−p (Lβ,2 ).

Ω

(N )

≤ EkXt − Xt

kl−p,δ (Lβ,2 ) → 0 , 2

N → ∞,

(11.25)

uniformly in t ∈ [0, T ], for the solutions Xt , Xt starting from Y ∈ l2−p (Lβ,2 ). To complete the stochastic quantization procedure for the st.-dissipative systems, we state the following: (N )

Theorem 11.2. Let the assumption (A)–(E) be satisfied and b/2 + α > m2 > 0. Fix the boundary condition Y ∈ l2−p (Lβ,2 ) with some p > d/2. Then the limit(N ) (N ) ing distributions µ∞ (dX|Y ) for the finite volume solutions Xt converge, when d ΛN → Z , N → ∞, in the Wasserstein metric, and hence weakly on l2−p (Lβ,2 ), to the unique invariant distribution µinv = µ∞ for the infinite volume solution Xt .


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Proof. The statement follows immediately from the inequality (N )

(N ) R(µ∞ (dX), µ∞ (dX/Y )) ≤ R(µ∞ , µt ) + R(µt , µt (N )

+ R(µt

)

(N ) (dX), µ∞ (dX/Y ))

(11.26)

and the estimates (11.14), (11.23) and (11.24). Remark 11.3. All the previous results hold also in the space l2−p (Lβ,2r ), p > d/2, r ≥ 1, as soon as (11.6) holds. Due to Corollary 9.1, the uniqueness of the invariant distributions µinv ∈ J (l2−p (Cβ )) yields the uniqueness of the tempered Gibbs states νβ ∈ Gβt . Here we would also like to present a direct proof of the uniqueness of Gibbs states in the case of convex interactions using the fundamental Dobrushin criterion and estimates on the Dobrushin coefficients via the corresponding stochastic dynamics. But we note that Theorem 11.3 itself is only a trivial case of Proposition 2.1 which in turn is the main result of our paper [15]. Theorem 11.3. Let the assertions (A)–(E) be satisfied and let X |a(j)| < a(0) + 2b . j6=0

Then

|Gβt |

= 1.

Proof. We look at the one-dimensional Gibbs distributions νk (dxk |Y ) in the volumes Λk = {k}, k ∈ Zd , with the extended spin space X := L2 (Sβ ) and boundary d conditions Y ∈ L2 (Sβ )Z , which are given by νk (∆|Y ) = νβ,{k} ({X ∈ Ωβ , xk ∈ ∆ ∩ C(Sβ )}|Y ) ,

∆ ∈ B(L2 (Sβ )) .

(11.27)

For every k, j ∈ Zd , k 6= j, we are interested in the dependence of the measure νk (dxk |Y ) on the values yj of the boundary condition Y at site j. This can be described by the Dobrushin matrix (Ckj )k,j∈Zd , where Ckj :=

sup Y,Y 0 ∈L2 (Sβ )Z ∀ i6=j:yi =yi0

d

R(νk (·, Y ), νk (·, Y 0 )) kyj − yj0 kL2 (Sβ )

(11.28)

(the Wasserstein distance R is also taken w.r.t. spin space L2 (Sβ )). Due to the Dobrushin uniqueness criterion (cf. [47, 58, 63, 115]), it is sufficient to prove that X Ckj < 1 . (11.29) sup k∈Zd j∈Zd ,j6=k

According to Proposition 10.1, for every fixed boundary condition Y ∈ L2 (Sβ )Z , {k} the stochastic dynamics xt := xk,t (Y ) ∈ L2 (Sβ ), t ≥ 0, (which is given by (10.36) d


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117

and corresponds to the one-point Gibbs distribution νk (·|Y )) is ergodic. Moreover, from (11.25), R(νk (·, Y ), νk (·, Y 0 )) ≤ lim (Ekxt − x0t k2 )1/2 . t→∞

(11.30)

But in full analogy with (10.9), d kxt − x0t k2L2 ≤ (a(0) + 2b)kxt − x0t k2L2 + |a(k − j)| kxt − x0t kL2 kyj − yj0 kL2 , dt and hence by the Gronwall inequality

lim (Ekxt − x0t k2 )1/2 ≤

t→∞

|a(k − j)| kyj − yj0 kL2 . a(0) + 2b

(11.31)

From (11.30), (11.31) we conclude that X j6=k

Ckj < 1 provided

X

|a(j)| < a(0) + 2b .

j6=0

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CHARGED SECTORS, SPIN AND STATISTICS IN QUANTUM FIELD THEORY ON CURVED SPACETIMES∗

D. GUIDO∗ Dipartimento di Matematica, Universit` a della Basilicata I-85100 Potenza, Italy E-mail : [email protected] R. LONGO∗ AND J. E. ROBERTS∗ Dipartimento di Matematica, Universit` a di Roma “Tor Vergata” I-00133 Roma, Italy E-mail : [email protected] E-mail : [email protected] R. VERCH Institut f¨ ur Theoretische Physik, Universit¨ at G¨ ottingen D-37073 G¨ ottingen, Germany E-mail : [email protected]

Received 24 June 1999

Dedicated to Sergio Doplicher on the occasion of his sixtieth birthday The first part of this paper extends the Doplicher–Haag–Roberts theory of superselection sectors to quantum field theory on arbitrary globally hyperbolic spacetimes. The statistics of a superselection sector may be defined as in flat spacetime and each charge has a conjugate charge when the spacetime possesses non-compact Cauchy surfaces. In this case, the field net and the gauge group can be constructed as in Minkowski spacetime. The second part of this paper derives spin-statistics theorems on spacetimes with appropriate symmetries. Two situations are considered: First, if the spacetime has a bifurcate Killing horizon, as is the case in the presence of black holes, then restricting the observables to the Killing horizon together with “modular covariance” for the Killing flow yields a conformally covariant quantum field theory on the circle and a conformal spin-statistics theorem for charged sectors localizable on the Killing horizon. Secondly, if the spacetime has a rotation and PT symmetry like the Schwarzschild–Kruskal black holes, “geometric modular action” of the rotational symmetry leads to a spin-statistics theorem for charged covariant sectors where the spin is defined via the SU (2)-covering of the spatial rotation group SO(3).

∗ Supported

by GNAFA and MURST. 125


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Contents 1. Introduction 1.1. Algebraic quantum field theory on curved spacetime: general setting 1.2. Superselection sectors 1.3. Covariant sectors and univalence (spin) 1.4. Tomita–Takesaki theory and symmetry 1.5. Modular inclusion and conformal theories on the circle 1.6. Description of contents 2. Some Spacetime Geometry 2.1. Generalities 2.2. Appendix to Chapter 2 3. Superselection Structure in Curved Spacetimes 3.1. Introduction 3.2. The selection criterion 3.3. Localized endomorphisms 3.4. The left inverse and charge transfer 3.5. Sectors of a fixed-point net 3.6. Appendix to Chapter 3 4. The Conformal Spin and Statistics Relation for Spacetimes with Bifurcate Killing Horizon 4.1. Spacetimes with bKh 4.2. Conformal spin-statistics relation 4.3. Appendix to Chapter 4 5. The Spin and Statistics Relation for Spacetimes with Rotation Symmetry 5.1. Geometric assumptions 5.1.1. Assumptions (a), (b), (c) and (c0 ) in some spacetimes 5.1.2. Spherically symmetric black holes 5.2. Quantum field theories on spacetimes with rotation symmetry 5.2.1. Spin and Statistics under property (c) 5.2.2. Spin and Statistics under property (c0 ) 5.3. Appendix. Equivalence between local and global intertwiners in Minkowski spacetime Acknowledgments References

126 127 129 131 131 132 133 135 135 139 139 139 141 143 148 150 152 163 163 168 175 177 177 179 179 182 183 189 191 195 195

1. Introduction General Relativity is a theory of gravitation with a geometric interpretation. A solution to the Einstein–Hilbert equations describes a curved spacetime manifold, whose curvature is related to the distribution of matter. Quantum Field Theory on the other hand arose as a theory for describing finitely many elementary particles and the underlying mathematical structure is that of a net of noncommutative von Neumann algebras of local observables. There have been many attempts to fuse the two theories to obtain a theory of Quantum Gravity but, as is well known, the basic problems remain unsolved and their solution would seem to be still a long way off. There is however one theory describing the effects of gravitation on quantum systems and this is Quantum Field Theory on a Curved Spacetime, where the


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gravitational field is treated as a background field so that the backreaction of the quantum system is ignored. Of course, this approximation cannot be expected to remain valid down to distances comparable to the Planck length. Progress in the field was initially hampered not only by the difficulties of handling interactions, well known from Minkowski space, but also through using a mathematical formalism which was not general enough. Nor did it help that no really interesting physical effects were found. This last point changed dramatically with the advent of Black Hole Thermodynamics and more particularly with the well known Hawking effect whereby a quantum effect causes a black hole to radiate thermally [36, 63]. More recently, the field has evolved rapidly on the mathematical side, too, primarily thanks to adopting methods and concepts from algebraic quantum field theory as e.g. in the work of [28, 35, 41, 47, 63]. But there have been other important developments, too. In particular, the discovery by Radzikowski that the Hadamard condition is equivalent to a wavefront set condition [13, 51] is worth mentioning. This has led to ambitious rigorous work on perturbative quantum field theory in curved spacetime by Brunetti and Fredenhagen [12]. Very recent work in algebraic quantum field theory [19, 52] contributes to clarifying the structure of quantum field theories on anti-de Sitter spacetime and its conformal boundary, an issue which has nowadays attracted great attention. The DHR analysis of superselection sectors in Minkowski spacetime is a good illustration of the effectiveness of algebraic quantum field theory in treating structural and conceptual problems. The aim of this paper is to lay the foundations of superselection theory in quantum field theory on curved spacetimes and to derive some first results. We find it advantageous to proceed by recalling, for the benefit of the nonexpert reader, the basic ideas and features of algebraic quantum field theory relevant to the two main themes of this paper: the general theory of superselection sectors and the connection between Tomita–Takesaki modular theory of von Neumann algebras and spacetime symmetries, particularly in the context of covariant superselection sectors. Our presentation will be simplified, with full details appearing in the main body of the paper. Readers familiar with superselection theory and the relations between modular theory and symmetry in algebraic quantum field theory may wish to turn directly to the outline of the contents in Sec. 1.6 where relations to other papers are indicated.

1.1. Algebraic quantum field theory on curved spacetimes: general setting In formulating algebraic quantum field theory on a curved spacetime one assumes the underlying spacetime to be described by a smooth manifold M (of any dimension ≥ 2) together with a Lorentzian metric g. The quantum system in question is supposed to be described by an inclusion preserving map K 3 O 7→ A(O) assigning


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to each member O in a collection K of subregions of M a C ∗ -algebra A(O). Usually, K is chosen to be a base forthe topology of M (we will specify K later on). The motivating idea is that A(O) contains the observables which can be measured at times and locations within the spacetime region O and that the way these algebras relate to each other for different regions O essentially fixes the physical content of the theory [35]. The collection K of subregions of M need not be directed under inclusion, but we shall nevertheless refer to K 3 O 7→ A(O) as a net of local algebras. If K is directed, then one can form the “quasilocal algebra”, i.e. the smallest C ∗ -algebra containing all the local algebras A(O). It is the norm closure of the union of the local S algebras, O A(O). In the generic case where K is not directed, this possibility is denied to us. But one can still expect Hilbert space representations of the inclusionpreserving map K 3 O 7→ A(O). More precisely, we say that a representation of K 3 O 7→ A(O) is a consistent family {πO }O∈K of representations of the local algebras A(O) by bounded operators on a common Hilbert space Hπ , i.e. πO1 A(O) = πO whenever O1 ⊃ O. For the known examples of quantum field theories on globally hyperbolic spacetimes and (conformal) quantum field theories on S 1 , such representations exist in abundance. (There are indications to the contrary for non-globally-hyperbolic spacetimes [39, 40]. The present paper is restricted to quantum field theory on globally hyperbolic spacetimes and the above notion of representation suffices.) Every representation {πO }O∈K yields states on the local algebras A(O) since each normal state ω on B(Hπ ) restricts to a state ωO (A) := ω(πO (A)) ,

A ∈ A(O)

of the local algebra. Not every consistent family of local states corresponds to a physical state of the system; nor can all representations of the observable net be considered as physical so one needs criteria to select physical representations. In practice, one begins with some collection of physical representations and uses them to construct others. In what follows, we compile a brief list of criteria to be fulfilled by such an initial collection P of physical representations of the net K 3 O 7→ A(O) of local observables on a curved spacetime (M, g). (1) πO , O ∈ K is faithful for each {πO }O∈K ∈ P. Otherwise the description of the system by the net of local algebras K ∈ O 7→ A(O) would contain redundancies. (2) Locality: The algebras πO (A(O)) and πO0 (A(O0 )) commute elementwise if the regions O and O0 cannot be connected by a causal curve. (3) Irreducibility and Duality: P consists of irreducible representations, i.e. representations {πO }O∈K fulfillinga )0 ( [ πO (A(O)) = C 1 . O∈K a A0

= {B ∈ B(H) : BA = AB ∀ A ∈ A} denotes the commutant of A ⊂ B(H).


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These representations are required to fulfill essential duality, i.e. the net, \ πO1 (A(O1 ))0 , K 3 O 7→ Adπ (O) := O1

is local, where the intersection is taken over all O1 ∈ K causally disjoint from O. This property is stronger than locality but not as strong as Haag duality which demands that πO (A(O))00 = Adπ (O) for all O ∈ K. This latter property means that the von Neumann algebras πO (A(O))00 cannot be enlarged by adding elements of B(Hπ ) without violating the locality condition. 0 }O∈K are two members of P, (4) Local Equivalence: Whenever {πO }O∈K and {πO 0 π there is for each O ∈ K a unitary UO : H → Hπ such that 0 (A)UO , UO πO (A) = πO

A ∈ A(O) .

(5) Covariance: For each {πO }O∈K ∈ P there is an (anti-)unitaryb representation G 3 γ 7→ Uπ (γ) of a (subgroup of) the spacetime isometry group G on Hπ so that Uπ (γ)πO (A(O))Uπ (γ)∗ = πγO (A(γO)) ,

γ ∈ G, O ∈ K .

Obviously, if the underlying spacetime (M, g) has a trivial isometry group, this condition is void. If (M, g) is Minkowski spacetime, there is typically a distinguished vacuum representation πvac in P which is irreducible and covariant and possesses a cyclic vacuum vector Ωvac ∈ Hπvac invariant under the action of Uπvac . Moreover, a vacuum representation fulfills the spectrum condition, i.e. the time-translations in any Lorentz frame have positive generator. In more general spacetimes, one can usually not select a distinguished vacuum representation by similar requirements since, in the absence of a sufficiently large isometry group, there is no analogue of the 5vacuum vector nor of the spectrum condition. However, one expects that a collection of physical representations P can still be selected in quantum field theory on curved spacetimes, even if there is no single preferred representation. For a Klein–Gordon field on any four-dimensional globally hyperbolic spacetime the representations induced by pure quasifree Hadamard states have been shown to form a collection P satisfying the conditions listed above [60]. 1.2. Superselection sectors We assume now that a curved spacetime (M, g), a net K 3 O 7→ A(O) of local algebras on this spacetime background and a collection P of physical representations fulfilling the conditions stated above have been given. To simplify notation, we denote a representation {πO }O∈K of the net of local algebras simply by π. b That

is, Uπ (γ) is anti-unitary if γ reverses the time-orientation, otherwise it is unitary.


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Picking an irreducible physical representation π 0 ∈ P as reference, another irreducible representation π (not necessarily belonging to P) is said to satisfy the selection criterion for localizable charges if, given O ∈ K, there is a unitary VO 0 between the representation Hilbert spaces Hπ and Hπ such that 0 (A)VO , VO πO1 (A) = πO 1

A ∈ A(O1 ) ,

for all regions O1 ∈ K causally disjoint from O. Irreducible representations which fulfill this selection criterion and are globally unitarily equivalent are said to carry the same charge, or to define the same superselection sector. The selection criterion thus selects representations π differing from the reference representation by some “charges” which can be localized in any spacetime region O (and are then not detectable in spacetime regions situated acausally to O). This form of localizability does not apply to all kinds of charges, e.g. electric charge is not localizable in this way (cf. [35] and references therein for further discussion). Yet for certain general types of charges, like flavours in strong interactions, this description is appropriate and hence a useful starting point. The notion of localized charge and superselection sector now apparently depends on the chosen reference representation π 0 (typically the vacuum representation in the case of flat spacetime), but as physical representations are required to be locally equivalent, the charge structure, being given by the structure of the space of intertwining operators of representations fulfilling the selection criterion, is expected 0 to be independent of that choice. Here, a bounded operator T : Hπ → Hπ is called an intertwiner for the representations π and π 0 of K 3 O 7→ A(O) if 0 (A)T , T πO (A) = πO

A ∈ A(O) ,

O ∈ K.

A crucial point is that the space of intertwiners admits a product having the formal properties of a tensor product. The statistics of the charges in the theory reflects the behaviour of this product under interchange of factors. Under certain general conditions, e.g. if the Cauchy surfaces of the spacetime are not compact, each charge has a conjugate charge and then the statistics of each charge can be characterized by a number, its statistics parameter. This number can be split into its phase and modulus being, respectively, the statistics phase and the inverse of the statistical dimension. (The latter is defined to be ∞ if the statistics parameter equals 0 and one says that the superselection sector has infinite statistics. We shall only consider superselection sectors having finite statistics.) If the statistics phase takes the values ±1, then the (para-)Bose/Fermi alternative holds in that there is a conventional description in terms of Bose and Fermi fields commuting or anticommuting when localized in causally disjoint regions. This is the generic situation in physical spacetime dimension. In lower spacetime dimension, braid group statistics may occur and the statistics phase may take values different from ±1. In previous papers [24, 25] it was shown that, in Minkowski spacetime, one can construct a field net together with a unitary action of a compact (global) gauge


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group containing the observable net A as fixed points so that the superselection sectors correspond naturally to the equivalence classes of irreducible representations of the gauge group. A similar result will turn out to hold in curved spacetime as well. As so little input is used (essentially only the physically motivated selection criterion and local commutativity of the observables) this result clearly demonstrates how effective the operator algebraic approach to quantum field theory can be. 1.3. Covariant sectors and univalence (spin) Our notion of spin on curved spacetime involves a group G of isometries although there ought to be a more general notion not involving symmetries. For this reason, we assume covariance of our reference representation π 0 . A superselection sector described by a representation π is covariant if there eπ (Γ) of the universal covering e 3 Γ 7→ U exists an (anti-)unitary representation G π group of G on H with eπ (Γ)◦πO , πγO ◦ αγ = Ad U

e, Γ∈G

O ∈ K,

where Γ 7→ γ denotes the covering projection. We may now consider continuous curves [0, 2π] 3 t 7→ Γ(t) whose projection [0, 2π] 3 t 7→ γ(t) is a cycle, i.e. a closed curve possessing no closed sub-curves. eπ (Γ(2π)) = sπ · 1 eπ (Γ(2π)) may be different from 1, but as π is irreducible, U U where sπ is a complex number of modulus 1. When the cycle γ([0, 2π]) has the geometric interpretation of a “spatial rotation by 2π”, then it is appropriate to refer to the phase factor sπ as the “spin”, or more precisely, the univalence of the charge represented by π.c Then, the spin-statistics connection is said to hold if, for all covariant superselection sectors of the theory, the univalence equals the statistics phase. 1.4. Tomita Takesaki theory and symmetry Let us next summarize some basic points of the modular theory for von Neumann algebras by Tomita and Takesaki [59]. Given a von Neumann algebra N on a Hilbert space H together with a cyclic and separating unit vector Ω ∈ H, the antilinear operator S : N Ω → N Ω defined by S(AΩ) := A∗ Ω admits a minimal closed extension with polar decomposition S¯ = J∆1/2 where J is anti-unitary. J is referred to as modular conjugation and {∆it }t∈R as modular unitary group associated with the pair N , Ω; one refers to Ad J as the antilinear modular morphism associated with N , Ω and usually denoted it by j. These modular objects satisfy JN J = N 0 and ∆it N ∆−it = N , t ∈ R. Moreover, a state ω on a C ∗ -algebra A is a KMSstate (thermal equilibrium state) at inverse temperature β with respect to a oneparametric group {αt }t∈R of automorphisms of A if and only if c We

do not wish to discuss how sπ depends on the different possible “rotations”. It suffices to say that in the relevant cases the above procedure assigns an invariant sπ to any covariant superselection sector.


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πω ◦ αt = Ad ∆−iβt/2π ◦ πω where πω is the GNS-representation of ω and {∆it }t∈R is the modular group of πω (A)00 , Ωω , Ωω being the GNS-vector. Thus the modular group may, in certain situations, have a physical (dynamical) significance. Furthermore, Bisognano and Wichmann showed [4] that, in Wightman’s setting of quantum fields in Minkowski spacetime, the modular objects associated with pairs A(W ), Ω, where A(W ) is the von Neumann algebra of observables in a certain “wedge-region”d W and Ω the vacuum vector, induce spacetime transformations. That is, if JW , {∆it W }t∈R denote the corresponding modular objects, then there are elements jW , ΛW,t in the Poincaré group so that Ad JW A(O) = αjW (A(O)) = A(jW (O)) ,

(1.1)

Ad ∆it W A(O) = αΛW,t (A(O)) = A(ΛW,t (O)) ,

(1.2)

for all open subregions O of Minkowski spacetime, all t ∈ R and all wedge-regions W. Further investigations (e.g. [5, 8, 15, 17, 31]) relate spacetime symmetries and modular objects and indicate that vacuum states in Minkowski spacetime can possibly be characterized through the geometric meaning of the modular objects associated with A(W ), Ω for a certain class of wedge-regions W . This idea has been pursued in non-flat spacetimes with a sufficiently rich group of isometries and a suitable class of wedge-regions, such as de Sitter spacetime and, to some extent, Schwarzschild–Kruskal spacetime, too [9, 10, 57]. There are indications that physical states of quantum field theory on arbitrary spacetime manifolds can be distinguished by the “geometrical action” of the corresponding modular objects for a certain class of regions, understood in sufficient generality. The reader is referred to [18] and references therein for considerable further discussion. In Minkowski spacetime, the geometric action of the modular objects associated with wedge-algebras A(W ) and the vacuum vector Ω has important consequences for the relation between spin and statistics. It can be derived either from geometric modular action [43], i.e. the geometric action of the modular conjugations as in (1.1), or from modular covariance [32], meaning the geometric action of the modular group as in (1.2). Similarly, for conformal quantum field theories on the circle S 1 where modular objects and conformal symmetry are intimately related, there is a spin-statistics relation, as will be briefly summarized in the next section. 1.5. Modular inclusion and conformal theories on the circle In this section we summarize the connection between conformally covariant theories on the circle S 1 and halfsided modular inclusions established by Wiesbrock [65, 66, 67, 68]. wedge region is any Poincar´ e transform of the set {(x0 , . . . , xn ) : 0 < x1 , 0 ≤ |x0 | < x1 } in Minkowski spacetime.

dA


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We briefly recall what is meant by a conformally covariant theory on the circle S 1 (see e.g. [31, 33] for further details). This is a net (or precosheaf) I 7→ M(I) taking proper open subintervals I of S 1 to von Neumann algebras M(I) on a Hilbert ¯ ⊂ M(I)0 . Moreover, there exists a space HM so that locality holds, i.e. M(S 1 \I) unitary strongly continuous positive energy representation U of P SL(2, R) acting covariantly, U (g)M(I)U (g)∗ = M(gI), and preserving a unit vector ΩM , cyclic for the von Neumann algebra generated by the M(I)’s. (In other words, the theory is given in a reference “vacuum representation”.) The theory may be equivalently described as a net of von Neumann algebras indexed by intervals on the real line, identified as the circle with one point removed. Using the Cayley transform, conformal transformations on the circle correspond to fractional linear transformations on the line. Modular transformations have a geometric meaning and Haag duality holds for any conformal theory on the circle, ¯ = M(I)0 [14]. Haag duality on the line, M(R \ I) ¯ = M(I)0 , namely M(S 1 \ I) holds precisely when the net I 7→ M(I) is strongly additive [34], i.e. if M(I) = M(I1 ) ∨ M(I2 ) whenever the union of I1 and I2 yields I up to at most a single point. We recall that a ±hsm inclusion (N ⊂ M, Ω) is given by a pair N ⊂ M of von Neumann algebras on some Hilbert space together with a unit vector Ω, cyclic and separating for both N and M, such that ∆it N ∆−it ⊂ N for all ∓t ≥ 0, where ∆it , t ∈ R, is the modular group of M, Ω. A ±hsm inclusion (N ⊂ M, Ω) is called standard if Ω is cyclic for N 0 ∩ M, too (hsm abbreviates “half sided modular”). An interesting result of Wiesbrock ([65, 66] see also [34]) asserts that there is a one-to-one correspondence between strongly additive conformally covariant theories on S 1 and standard ±hsm inclusions. The rotations of S 1 form a subgroup of the covering group of P SL(2, R). Let π eπ the be a Hilbert space representation of a covariant superselection sector and U associated unitary representation of the covering group of P SL(2, R). Assuming eπ has positive energy, the generator of rotations in the unitary representation that U e Uπ has a lowest eigenvalue Lπ . Then the conformal spin of the superselection sector, or rather, its univalence, is defined by sπ = e2πiLπ . For superselection sectors with positive energy in a conformally covariant theory on S 1 , the univalence equals the statistics phase, which may be any complex number of modulus 1 [33].

1.6. Description of contents We now describe the contents of the subsequent chapters. In Chapter 2 we summarize several notions of spacetime geometry needed here. Lemma 2.2, of relevance to superselection theory, asserts that the set of pairs of causally separated points in a globally hyperbolic spacetime is connected. Chapter 3 contains the general framework for superselection theory in curved spacetimes, patterned conceptually on the DHR analysis in Minkowski spacetime


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([23], cf. also [35, 54] and references given there). It will be formulated for nets K 3 O 7→ A(O) of operator algebras in a reference representation with general index sets K possessing a partial ordering and a causal disjointness relation. Thus quantum fields on arbitrary globally hyperbolic spacetimes in any dimensions, with compact or non-compact Cauchy surfaces, as well as quantum field theory on the circle, can be treated on an equal footing. The existence of statistics is established in this generality. If the index set K is directed, all the other basic results known for superselection theory on Minkowski spacetime, classification of statistics, existence of charge conjugation and construction of field algebra and gauge group (cf. [25]) can again be shown to hold. Chapter 4 begins with a summary of the geometry of spacetimes with a bifurcate Killing horizon following Kay and Wald [41]. We introduce a family of wedge-regions Ra , a > 0 which are copies of the canonical right wedge shifted by a in the affine geodesic parameter on the horizon (a similar construction can be carried out for the left wedge). We suppose that we are given a net of von Neumann algebras O 7→ A(O) in the representation of a state which is, in restriction to the subnet of observables which are localized on the horizon, a KMS-state at Hawking temperature for the Killing flow. Thus on the horizon we have modular covariance and are consequently in Wiesbrock’s situation of half-sided modular inclusion [65]. Using Haag duality and additivity of the net, it follows that the maximal subnet of observables localized on the horizon is a conformally covariant family of von Neumann algebras. Restricting the original net of von Neumann algebras to the Killing horizon thus yields a conformal quantum field theory on S 1 . A conformal spin is therefore assigned to a superselection sector of the original theory, localizable on the horizon, and the conformal spin-statistics connection [33] holds. This approach has, however, the drawback of applying only to horizon-localizable charges, and this may be quite restrictive. In Chapter 5 we introduce a class of spacetimes with a special rotation symmetry and certain adapted wedge-regions. Essentially we assume that there is a group of symmetries, to be viewed as rotations, generated by pairs of time-reversing wedgereflections mapping wedge-regions onto each other. In the Schwarzschild–Kruskal spacetime, for example, these wedge-regions can be envisaged as the causal completions of “halves” of the canonical Cauchy-surfaces chosen so that rotating by π about a suitable axis maps each such half onto its causal complement. These wedge-regions differ from the usual canonical “right” and “left” wedges (R and L in Chap. 4) and lie in a sense transverse to the latter. Then we consider a net of von Neumann algebras O 7→ A(O) over such a spacetime in a representation where the full isometry group acts covariantly. Moreover we suppose that there is an isometry-invariant state and that the modular conjugations associated with the vacuum vector and the von Neumann algebras A(W ) for the said class of wedges W induce the geometric action of the wedge-reflections. This form of geometric modular action will allow us to define the rotational spin of a covariant superselection


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sector and to derive the spin and statistics connection using a variant of arguments presented in [46]. 2. Some Spacetime Geometry 2.1. Generalities In the present section we summarize some notions about causal structure of Lorentzian manifolds, thereby establishing our notation. Standard references for this section include [3, 37, 50, 62]. We begin by recalling that a curved spacetime (M, g) is a 1 + s-dimensional (s ∈ N), Lorentzian manifold. In other words, it is a 1 + s-dimensional orientable, Hausdorff, second countable C ∞ -manifold equipped with a smooth Lorentzian metric g having signature (+, −, . . . , −). A continuous, (piecewise) smooth curve γ : I → M , defined on a connected subset I of R and having tangent γ, ˙ is called a timelike curve whenever g(γ, ˙ γ) ˙ > 0, a causal curve if g(γ, ˙ γ) ˙ ≥ 0, and a lightlike curve if g(γ, ˙ γ) ˙ = 0 while γ˙ 6= 0, for all parameter values t. A spacetime (M, g) is called time-orientable if there exists a global timelike (non-vanishing) vector field ξ on M . Such a vector field induces a time-orientation: a causal curve γ is called future-directed or past-directed according as g(ξ, γ) ˙ > 0 or g(ξ, γ) ˙ < 0. We shall henceforth tacitly assume our spacetimes to be time-orientable with a given time-orientation. A future-directed causal curve γ : I → M is said to have a future (past)endpoint if γ(t) converges to some point in M as the parameter t approaches sup I (inf I). Correspondingly one defines the past (future)-endpoints of past-directed causal curves. A future (past)-directed causal curve is said to start at a point p ∈ M provided that p is the past (future)-endpoint of γ. Moreover, one calls a future (past)-directed causal curve future (past)-inextendible if it possesses no future (past)-endpoint. For any subset O of M one defines the sets J ± (O) as consisting of all points in M lying on future(+)/past(–)-directed causal curves that start at some point in O. Then J ± (O) are called the causal future(+)/causal past(–) of O. The set J(O) := J + (O) ∪ J − (O) is then referred to as the causal set of O. The subsets D± (O) of M are, for given O ⊂ M , defined as the collection of all those points p ∈ M such that every past(+)/future(–)-inextendible causal curve starting at p meets O. One calls D± (O) the future(+)/past(–) -domain of dependence of O, and D(O) := D+ (O) ∪ D− (O) the domain of dependence of O. One says that two points p and q in M are causally disjoint, in symbols p ⊥ q, if there are open neighbourhoods U of p and V of q such that there is no causal curve connecting U and V (i.e. U ∩ J(V ) = ∅ = V ∩ J(U )). Correspondingly one calls two subsets P and Q of M causally disjoint if p ⊥ q holds for all pairs p ∈ P and q ∈ Q; this will be abbreviated as P ⊥ Q.


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In the present paper we will primarily be interested in globally hyperbolic spacetimes. A spacetime (M, g) is globally hyperbolic if it can be smoothly foliated in acausal Cauchy surfaces. Here, an acausal Cauchy surface C is a smooth hypersurface in M such that each causal curve in (M, g) without endpoints meets C exactly once. This implies that C is indeed acausal, i.e. p ⊥ q holds for all distinct p, q ∈ C. By a (smooth) foliation of (M, g) in acausal Cauchy surfaces we mean a diffeomorphism F : R × Ξ → M where Ξ is an s-dimensional smooth manifold such that F ({t} × Ξ) is, for each t ∈ R, an acausal Cauchy surface in (M, g), and the curves R 3 t 7→ F (t, q), q ∈ Ξ, are timelike and endpointless. Thus, the foliation-parameter t plays the role of a “time-parameter”. One may give a broader definition of Cauchy surfaces which are not necessarily acausal, by defining a Cauchy surface as a C 0 hypersurface C such that C ∩ int J ± (C) = ∅ and D(C) = M . With this definition, a Cauchy surface is allowed to have lightlike parts. Such a broader definition of Cauchy surfaces is often useful. However, it is a remarkable fact that the existence of a single, not necessarily acausal Cauchy surface in (M, g) already implies that (M, g) is globally hyperbolic in the above sense [21, 30, 62]. Whilst the question of whether physical spacetime models are necessarily globally hyperbolic has been discussed in the literature (see [20, 62, 64] and references given there), it is certainly the case that a great number of the prominent spacetime models are globally hyperbolic, like Schwarzschild–Kruskal, deSitter, the Robertson–Walker models, and many others, including of course Minkowski spacetime. One may therefore regard the class of globally hyperbolic spacetimes as being sufficiently general and comprising many examples of physical interest. Note that global hyperbolicity in no way presupposes the presence of spacetime symmetries. At this point we recall some properties of causal sets; for their proof and further discussion, we refer to the indicated references. Whenever N ⊂ M and (M, g) is globally hyperbolic, then: N compact implies J ± (N ) closed, N compact implies that J(N ) ∩ C is compact for each Cauchy surface C, N compact implies D(N ) compact. Furthermore, J + (N+ ) ∩ J − (N− ) is empty or compact for all compact N+ , N− ⊂ M . Moreover, in (time-orientable) spacetimes (M, g), a time-orientation preserving isometry τ of (M, g) satisfies τ (J ± (O)) = J ± (τ (O)) ,

(2.1)

for O ⊂ M . It is moreover worth mentioning that for any two subsets P and Q of a globally hyperbolic spacetime (M, g) we have P ⊥ Q if and only if P ⊂ Q⊥ , where the causal complement Q⊥ of Q ⊂ M is defined by Q⊥ := M \J(Q), see e.g. [42, Proposition 8.1]. We need to consider special regions of a globally hyperbolic spacetime (M, g) namely those causally closed regions generated by an open subset of a Cauchy surface. More particularly we are interested in regular diamonds defined as follows. A set of the form O = int D(G) is a regular diamond provided O⊥ is non-void and


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(i) G is an open subset of an acausal Cauchy-surface C, and G is compact and contractible to a point in G, (ii) ∂G, the boundary of G, is a (possibly multiply connected) locally flat embedded, two-sided topological submanifold of C which is an embedded C ∞ -submanifold near to points in each of its connected components. We refer to [11, 61] for the precise definition of locally flat embeddings and twosidedness. Intuitively, these two conditions are substitutes for the existence of an oriented normal vector field over ∂G. These regularity properties serve to prove the following assertion: Lemma 2.1. Let O be a regular diamond and p ∈ O⊥ . Then there exists another regular diamond O1 with O ∪ {p} ⊂ O1 . A rough sketch of the proof will be given in Sec. 2.2, the Appendix to this chapter. The reader is referred to [61] for a detailed proof. A double cone in Minkowski space is, of course, a regular diamond. Double cones may be generalized easily to curved spacetime. They are sets of the form int(J − ({v + }) ∩ J + ({v − })) with v + ∈ int J + ({v − }). However, double cones need not have the property analogous to Lemma 2.1, think e.g. of a spacelike strip in Minkowski spacetime. Nor is it clear that a double cone is a regular diamond. For this reason, it is not clear, even for simple free fields, whether duality is satisfied for such regions. We expect the requirement of essential duality (cf. Sec. 1.1) to be realistic for regular diamonds, in particular, as their bases are assumed contractible. Furthermore, duality for regular diamonds has already been established for the Klein–Gordon field [27, 60] and can presumably be verified for other free fields. For these reasons, we have chosen to use the collection K of regular diamonds rather than the collection of double cones whose causal complement has non-empty interior as an index set in a globally hyperbolic spacetime. Given a spacetime (M, g), we introduce the set XM,g := {(x, y) ∈ M × M : x ⊥ y}

(2.2)

of pairs of causally disjoint points in M . According to the definition of causal disjointness, this set is an open subset of M × M . The subsequent assertion about XM,g will prove to be important in discussing the statistics of superselection sectors in the next chapter. It may be known to experts, but as we have not found it in the literature, we put it on record here. Lemma 2.2. Let (M, g) be a globally hyperbolic spacetime. Then XM,g is pathwise connected except when the Cauchy surfaces in (M, g) are noncompact and 1– dimensional in which case there are precisely two path–components corresponding to x being causally to the left or to the right of y.


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Proof. Let F : R × Ξ → M be a foliation in acausal Cauchy surfaces and write C := F ({0} × Ξ). We first show that it suffices to restrict one’s attention to the Cauchy surface C. More precisely, we show that Y := {(x, y) ∈ C × C : x ⊥ y} is a strong deformation retract of XM,g . In fact, using F to parametrize M and defining h : XM,g × I → XM,g by h(t, ξ; t0 , ξ 0 ; s) := ((1 − s)(t + s(t0 − t)), ξ; (1 − s)t0 , ξ 0 ) we have a homotopy of the identity on XM,g onto the projection, (t, ξ; t0 , ξ 0 ) 7→ (0, ξ; 0, ξ 0 ), onto C leaving C fixed. The only non-trivial point is to show that the image of h lies in XM,g and this is where the causal structure enters. However, two remarks suffice: first, causal disjointness reduces to disjointness on an acausal Cauchy surface and hence is preserved if we pass from one acausal Cauchy surface to another by changing the value of t. Secondly, if we take causally disjoint points xi = F (ti , ξi ), i = 1, 2 with distinct values of t then the curve γ : [inf{t1 , t2 }, sup{t1 , t2 }] 3 t 7→ F (t, ξ1 ) is timelike and connects x1 with that Cauchy surface of the foliation containing x2 . Its range must lie in {x2 }⊥ or there would be a causal curve coming arbitrarily close to connecting x1 and x2 , contrary to assumption. We now know that the inclusion of Y in XM,g induces an isomorphism in homotopy and, in particular, an isomorphism of path-components. Now unless C is one dimensional and non-compact, the complement of a point of C is path-connected and Y is then also path-connected. If C is one dimensional and non-compact it is isomorphic to R so that Y has two path-components. When XM,g has two components, we use the foliation F : R × R → M into acausal Cauchy surfaces to distinguish the “right” component from the “left” component as that containing pairs (x, y), where the spatial component of y is greater than that of x. In fact, this distinction depends only on the nowhere vanishing spacelike vector field ξ induced by the foliation. Given such a field ξ, a spacelike curve I 3 t 7→ γ(t) is called right-directed if g(ξ, γ) ˙ > 0 and left-directed if g(ξ, γ) ˙ < 0 for one and hence all values of t. (A different choice of ξ would at most lead to interchanging “right-directed” and “left-directed” since in two spacetime dimensions the set of spacelike vectors at each point has two components.) The orientation of spacelike curves defined in this way can now be used to specify the two connected components of XM,g in the case of a non-compact Cauchy surface. The right component is that containing (γ(0), γ(1)) for the endpoints γ(0) and γ(1) of some and hence any right-directed spacelike curve γ. This follows from the previous description in terms of the foliation since the spatial component is strictly increasing along such a curve.


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2.2. Appendix to Chapter 2 Proof of Lemma 2.1 (Sketch). Let O = int D(G) be a regular diamond, G ⊂ C where C is an acausal Cauchy-surface, and p ∈ O⊥ . Choose a C ∞ -foliation F : R × Σ → M of M into smooth, acausal Cauchy surfaces. Then for each y ∈ Σ, the curves t 7→ F (t, y) are inextendible, futuredirected timelike curves. Therefore, given any acausal Cauchy surface C0 , each of these curves intersects C0 exactly once, at the parameter value t = τC0 (y). The function τC0 : Σ → R is a smooth function and one has C0 = {F (τC0 (y), y) : y ∈ Σ}. Furthermore, the map ΦC,C0 : C → C0 induced by F (τC (y), y) 7→ F (τC0 (y), y) is a diffeomorphism. Using the results of [11], one can show that there is an open neighbourhood U ¯ of G in C possessing the same properties (i) and (ii) as G, i.e. U is the base of a regular diamond. It is also not difficult to show (cf. [61]) that there exists an acausal Cauchy surface C0 containing p and with the additional property that ¯ ∩ C0 ⊂ ΦC,C0 (U ) =: U0 . J(G) The latter property means there are acausal Cauchy surfaces C0 passing through ¯ This entails that O0 := int D(U0 ) contains O. p and coming arbitrarily close to G. Since ΦC,C0 is a diffeomorphism, U0 satisfies (i) and (ii) with respect to the Cauchy surface C0 . It remains to show that U0 ∪ {p} is contained in a subset U1 of C0 satisfying (i) and (ii) with respect to the Cauchy surface C0 . This is done by connecting a point in a smooth part of ∂U0 by a smooth curve λ to p and by attaching to U0 a suitable smooth deformation of a tubular normal neighbourhood of λ. This yields the required set U1 ; properties (i) and (ii) follow by construction as does O ∪ {p} ⊂ int D(U1 ) =: O1 .

3. Superselection Structure in Curved Spacetimes 3.1. Introduction In this section, we adapt the basic notions and results of the theory of superselection sectors to curved spacetime, limiting ourselves to globally hyperbolic spacetimes. As we shall see, the basic theory goes through smoothly in the case of globally hyperbolic spacetimes with a noncompact Cauchy surface and much of it in the case of a compact Cauchy surface. The geometry of spacetime fortunately enters the long analysis only in establishing a few specific points. We can therefore limit ourselves to clarifying these points and otherwise just quoting the consequences. We let K denote the set of regular diamonds in M , ordered under inclusion. If M is globally hyperbolic with a non-compact Cauchy surface, K may not be directed although it will be in cases of interest. However, when M is globally hyperbolic with a compact Cauchy surface, K will never be directed and we shall meet problems


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akin to those on the circle. The more complicated structures involved have been relegated, as far as possible, to the appendix to this chapter. The set of double cones in M whose causal complement has non-empty interior is even less likely to be directed. Both sets have in common that they form a base for the topology of M and we will consider our nets of observables as being defined over K with the general philosophy that they can be extended to other regions, if necessary. In fact, we will consider a wider class of regions in subsequent chapters. Now, the geometry of spacetime enters the analysis only through the partially ordered set K and its relation of causal disjointness, introduced below. In view of further applications and despite the degree of abstraction involved, we have emphasised the relevant properties of K. The selection criterion for localized charges in Minkowski space uses the vacuum representation as a reference. Although there is no such preferred representation in curved spacetime, one expects there to be a preferred collection of representations satisfying the conditions listed in Sec. 1.1. In the case of the Klein–Gordon field on a four dimensional globally hyperbolic spacetime, we may take the representations induced by the pure quasifree Hadamard states [60]. We shall choose one of these representations as our reference representation and, whilst our sectors will depend on this choice, the superselection structure will not since this depends only on the net of von Neumann algebras. By (4) of Sec. 1.1, any two preferred representations generate the same net of von Neumann algebras. We will denote our reference representation by π 0 and its Hilbert space by H0 . Once the reference representation has been fixed, it is just the causal structure of Minkowski space that plays a role in the superselection criterion for localized charges. For this reason, the analysis of the superselection structure adapts well to curved spacetime. The causal structure enters in the form of the relation ⊥ of causal disjointness, defined in Chap. 2, and here to be considered as a relation on the ordered set K, satisfying (a) O1 ⊥ O2 ⇒ O2 ⊥ O1 . (b) O1 ⊂ O2 and O2 ⊥ O3 ⇒ O1 ⊥ O3 . (c) Given O1 ∈ K, there exists an O2 ∈ K such that O1 ⊥ O2 . We write O⊥ := {O1 ∈ K : O1 ⊥ O}. As explained above, the geometry of spacetime enters through the partially ordered set K together with the relation ⊥ of causal disjointness. Hence we have to pass from geometric or topological properties of (M, g) to properties of (K, ⊥). We will need to know whether certain partially ordered sets are connected, a notion defined in the appendix. But the basic idea is to move from one element O1 of K to a nearby element O2 , where nearby means that there is a third element O3 containing O1 and O2 . A finite series of such moves constitutes a path. K is connected if any two elements can be connected by a path. By virtue of Lemma 3A.1, we know that K is connected and, see Lemma 2.2, that O⊥ is connected except when M is two dimensional with a non–compact Cauchy surface.


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Lemma 2.2, itself, asserts that the set XM,g of pairs of spacelike separated points is pathwise connected again unless M is two dimensional with a noncompact Cauchy surface. Since pairs of elements of K form a base for the topology in the product space, we can again conclude by Lemma 3A.1 that the graph G ⊥ of the relation ⊥ is connected, G ⊥ = {O1 , O2 : O1 ⊥ O2 } . In the exceptional case, XM,g has two pathwise connected components. Indeed the causal complement of a point is no longer connected but decomposes into a “left” causal complement and a “right” causal complement. These are the basic geometric considerations determining the statistics. The remaining condition used in Sec. 3.3, the surjectivity of the projection from Gc⊥ , a connected component of G ⊥ , to K has no geometric relevance seeing that it is automatically satisfied in the context of globally hyperbolic spacetimes. Thus, as will follow from the results of Sec. 3.3, in a globally hyperbolic spacetime of dimension greater than 2, we get a net of symmetric tensor W ∗ -categories, (Tt , εc ), whereas in a 2-dimensional spacetime we shall in general get a braided tensor W ∗ category with two different braidings ε` and εr corresponding to the left and right causal complements of a double cone. Obviously, ε` = εr∗ , where ε∗ is defined by ε∗ (ρ, σ) = ε(σ, ρ)∗ . The next basic step is to establish the properties of charge conjugation. The basic tool here is a left inverse. The physical idea behind constructing left inverses is that of transferring charge to spacelike infinity and a geometric property is obviously involved. Expressed as a property of our partially ordered set K we need to assume the existence of a net On of elements of K such that given O ∈ K there exists an n0 with On ⊥ O for n ≥ n0 . We will say that such a net On tends spacelike to infinity. Such a net obviously exists whenever K is directed but it continues to exist for an arbitrary globally hyperbolic spacetime with a noncompact Cauchy surface. The question of whether one can find a suitable substitute for globally hyperbolic spacetimes with compact Cauchy surfaces is still open, a defect mitigated by the circumstance that a left inverse exists as a consequence of the equality of local and global intertwiners, postulated in Chap. 5. In this way, we establish in Sec. 3.4 the classification of statistics and the existence of charge conjugation for finite statistics for the case of a globally hyperbolic spacetime of dimension greater than two. 3.2. The selection criterion Our discussion of superselection theory in this and in subsequent sections is in terms of a partially ordered set K together with a binary relation ⊥. The necessary properties will be introduced as needed and there will be no specific reference to spacetime. We have adopted this procedure for clarity and with future applications


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in mind. Thus the best choice of K in a curved spacetime is not altogether clear. We have already, for example, thought fit to use regular diamonds in place of double cones. On the other hand, we might like to go beyond strictly localized charges and work with spacelike cones or to replace causal disjointness by its Euclidean counterpart, disjointness, as when working on the circle. In fact, we shall need to use results on superselection structure on the circle in Chap. 4 and, although these results have been developed previously [29, 33], the formalism presented here includes this case and allows a uniform approach to all such problems. We shall also simplify the exposition by making use of the freedom to modify the binary relation on K. Thus this degree of abstraction is now called for even if we have not been able to derive all results in an adequate generality.e Two nets A and B of ∗ -subalgebras of B(H0 ) over K are said to be relatively local if A(O1 ) ⊂ B(O2 )0 , whenever O1 ⊥ O2 . This relation fulfills the analogues of (a), (b) and (c) above. Furthermore, there is a maximal net, the dual net Ad , which is relatively local to A. It is given by \ Ad (O) = {A(O1 )0 : O1 ⊥ O} . Since Add is the largest net local relative to Ad , A ⊂ Add . However A ⊂ B implies B d ⊂ Ad , so that Ad = Addd . A net A is said to be local if A ⊂ Ad and then Add ⊂ Ad = Addd so that Add is local, too. We now compute the double dual: \ \ _ ˆ . Ad (O1 )0 = A(O) Add (O) = O1 ⊥O

ˆ O⊥O1 O⊥O 1

Definition 3.1. A representation π of the net A is said to satisfy the selection criterion if π O⊥ ' π 0 O⊥ ,

O ∈ K.

When K is directed this means that for each O there is a unitary VO such that VO π(A) = AVO ,

A ∈ A(O1 ) ,

O1 ∈ O⊥ ,

where, to simplify notation in the sequel, we have omitted the symbol π 0 for the reference representation. We write T ∈ (π, π 0 ) to mean that T intertwines the representations π and π 0 and let Rep⊥ A denote the W ∗ -category whose objects are the representations of A satisfying the selection criterion and whose arrows are the intertwiners between these representations. As far as superselection theory goes, the following result allows one to replace the original net by its bidual. e Baumg¨ artel

and Wollenberg [2] treat nets over partially ordered sets with a relation of causal disjointness. In their applications to superselection structure they assume among other properties that the partially ordered set is directed. When the partially ordered set is not directed, their notion of representation depends on a choice of enveloping quasilocal algebra.


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The Extension Theorem. If each O⊥ is connected, every object π of Rep⊥ A admits a unique extension to an object of Rep⊥ Add . Furthermore there is a canonical isomorphism of the corresponding W ∗ -categories. This result is proved as Theorem 3A.4 of the Appendix. How to proceed when O is not connected is exemplified by the well known case of a two-dimensional Minkowski space and we will not attempt a general analysis here. The theory of superselection structure rests on two assumptions. The first is a property derived by Borchers in Minkowski space as a consequence of additivity, locality and the spectrum condition. Here it involves the dual net, Ad . ⊥

Definition 3.2. A net Ad satisfies Property B if given O, O1 and O2 in K such that O ⊥ O2 , and O, O2 ⊂ O1 and a projection E 6= 0 in Ad (O), there is an isometry W ∈ Ad (O1 ) with W W ∗ = E. Lemma 3.3. If Ad satisfies Property B, the set of representations satisfying the selection criterion is closed under direct sums and (non-trivial ) subrepresentations. In other words, the W ∗ -category Rep⊥ A has direct sums and (non-zero) subobjects. The proof of this lemma will be omitted as it in no way differs from its Minkowski counterpart [23]. The characteristic assumption of superselection theory is a duality assumption. Definition 3.4. A net A is said to satisfy duality if A = Ad and essential duality if Add = Ad . To simplify notation, we shall suppose here that our net satisfies duality but, as a consequence of the Extension Theorem, the results remain valid under the weaker assumption of essential duality, whenever each O⊥ is connected. In the Appendix, we have adopted the cohomological approach to superselection structure as this provides the most natural expression of the selection criterion. In the main text, we shall pursue the alternative strategy of working in terms of localized endomorphisms rather than 1-cocycles. 3.3. Localized endomorphisms When K is directed, the analysis of superselection structure rests on the following simple construction: let π be a representation satisfying the selection criterion, pick a unitary VO as above and set ρ(A) := VO π(A)VO∗ ,

A ∈ A.

Obviously ρ is a representation of A on H0 unitarily equivalent to π but, in fact, ρ(A) ⊂ A. To see this, pick O1 , O2 ∈ K, O1 ⊃ O, O1 ⊥ O2 and B ∈ A(O2 ) then, writing V for VO , ρ(A)B = V π(A)V ∗ B = V π(AB)V ∗ = V π(BA)V ∗ = BV π(A)V ∗ = Bρ(A) ,


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hence ρ(A) ∈ Ad (O1 ) = A(O1 ), as required. Furthermore, ρ is localized in O, i.e. ρ(AB) = ρ(A)B ,

B ∈ A(O1 ) ,

A ∈ A , O1 ⊥ O

and we refer to ρ as a localized endomorphism. Now if ρ and ρ0 are localized endomorphisms, an intertwiner R for the corresponding representations is automatically in A. For suppose ρ and ρ0 are localized in O and A ∈ A(O1 ), O1 ⊥ O, then RA = Rρ(A) = ρ0 (A)R = AR so that R ∈ Ad (O) = A(O). We can thus write R ∈ (ρ, ρ0 ) without specifying whether we treat ρ as a representation or as an endomorphism and, when studying superselection sectors, Rep⊥ A may be replaced by the full subcategory Tt of End A. End A is a tensor C ∗ -category and we use the tensor product notation. Thus if S ∈ (σ, σ 0 ), we write R⊗S to denote the intertwiner Rρ(S) ∈ (ρσ, ρ0 σ 0 ). We characterize Tt by characterizing the corresponding set ∆t of endomorphisms. The representation corresponding to ρ ∈ ∆t satisfies the selection criterion precisely when, given O ∈ K, there is an equivalent endomorphism σ localized in O. We then call ρ transportable since, transporting ρ by a suitable unitary U ∈ A, it can be localized in any given O ∈ K. ∆t is thus the set of transportable localized endomorphisms and ∆t (O) shall denote the subset of endomorphisms localized in O. Lemma 3.5. If ρ, ρ0 ∈ ∆t then ρρ0 ∈ ∆t . Proof. As the product of endomorphisms localized in O is again localized in O, it suffices to observe that if U ∈ (ρ, σ) and U 0 ∈ (ρ0 , σ 0 ) are unitary then U ⊗ U 0 ∈ (ρρ0 , σσ 0 ) is unitary. Thus the unitary equivalence class of ρρ0 depends only on the unitary equivalence classes of ρ and ρ0 and, regarding charge as the quality distinguishing one sector from another, this defines a composition of charges. When K is not directed, this simple scheme must be modified. The basic complication is that localized endomorphisms are now not defined on the whole net A. Instead, an endomorphism ρ localized in O is just defined on the net O1 7→ A(O1 ) with O ⊂ O1 and has the property that ρ(A(O1 )) ⊂ A(O1 ). As explained in detail in the Appendix, we have a net O 7→ Tt (O) of tensor W ∗ -categories, the objects of Tt (O) are the transportable endomorphisms localized in O. It is also shown in the Appendix how a representation π satisfying the selection criterion gives rise to objects of Tt (a), a ∈ Σ0 and how an interwiner T ∈ (π, π 0 ) between two such representations leads to arrows ta , a ∈ Σ0 , between the corresponding objects of Tt (a). We can no longer study superselection sectors replacing Rep A⊥ by Tt (O), more precisely, we have a faithful ∗ -functor from Rep A⊥ to Tt (O) but cannot assert that it is an equivalence of W ∗ -categories. Thus, when K is


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not directed, Tt (O) may not give a description of superselection sectors. Nevertheless, as we shall see, an analysis of localized endomorphisms still provides useful information. The basic step in this analysis is to investigate the relation between causal disjointness and commutation of localized endomorphisms and their intertwiners. It is natural to say that an intertwiner T ∈ Tt (O) is localized in O, but we need a finer notion because we may have T ∈ (ρ1 , ρ0 ) where ρi ∈ ∆t (Oi ) with Oi ⊂ O. In this case, we refer to O1 as being an initial support and O0 as being a final support of T . As explained in the Appendix, we consider the set Σ1 of 1-simplices in K as a partially ordered set and let Σ⊥ 1 denote the subset of 1-simplices b with ∂1 b ⊥ ∂0 b with the induced order. ⊥ Lemma 3.6. Let Σ⊥ 1,c be a connected component of Σ1 , and suppose that given 0 O0 ∈ K, there is a b ∈ Σ⊥ 1,c with ∂0 b = O0 . Let Ti ∈ (ρi , ρi ) be arrows in some Tt (O) then

T0 ⊗ T1 = T1 ⊗ T0 , if there are b, b0 ∈ Σ⊥ 1,c so that ∂0 b and ∂1 b are initial supports of T0 and T1 and 0 0 ∂0 b and ∂1 b are final supports of T0 and T1 . Proof. We first show that T0 ρ0 (T1 ) = T1 ρ1 (T0 ). This relation is trivial if T0 and ˆ T1 are causally disjoint in the sense that there is a ˆb ∈ Σ⊥ 1 such that ∂0 b contains ˆ an initial and final support of T0 and ∂1 b an initial and final support of T1 . The idea of the proof is to reduce to this trivial case. Replace T0 and T1 by T2 = T0 ◦ U0 and T3 = T1 ◦ U1 , where U0 ∈ (ρ2 , ρ0 ) and U1 ∈ (ρ3 , ρ1 ) are unitary. Then T 2 ⊗ T 3 = T 0 ⊗ T 1 ◦ U0 ⊗ U1 ,

T 3 ⊗ T 2 = T 1 ⊗ T 0 ◦ U1 ⊗ U0 ,

ˆ for O ˆ sufficiently large. Thus if U0 and U1 to be understood as valid in some Tt (O) are causally disjoint, the validity or not of our relation is unaffected by the passage from T0 , T1 to T2 , T3 . But b and b0 lie in a connected component Σ⊥ 1,c by hypothesis, so after a finite number of steps we can arrange that the initial and final supports of both intertwiners coincide. This is again the trivial case so T0 ρ0 (T1 ) = T1 ρ1 (T0 ), as required. It only remains to show that ρ 0 ρ1 − ρ1 ρ0 = 0 . The above computations show that the kernel of the left hand side does not change if we shift to ρ2 and ρ3 . However, by hypothesis, given O ⊃ b0 , we can find ˆb ∈ Σ⊥ 1,c with ∂0ˆb = O and we can take ρ3 ∈ ∆t (∂1ˆb), when ρ0 ρ3 (A) = ρ0 (A) = ρ3 ρ0 (A) ,

A ∈ A(O) ,

completing the proof. After this one crucial lemma, the standard results on the existence of a braiding follow without further geometric input. Of course the braiding will, in general, continue to depend on the choice of connected component.


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⊥ Theorem 3.7. Let Σ⊥ 1,c be a connected component of Σ1 . If the projection mapping b 7→ ∂0 b from Σ⊥ 1,c to K is surjective then there is a unique intertwiner-valued function (ρ0 , ρ1 ) 7→ εc (ρ0 , ρ1 ) ∈ (ρ0 ρ1 , ρ1 ρh0 ) such that (a) εc (ρ00 , ρ01 ) ◦ T1 ⊗ T2 = T2 ⊗ T1 ◦ εc (ρ0 , ρ1 ) , Ti ∈ (ρi , ρ0i ) , i = 0, 1, (b) εc (ρ0 , ρ1 ) = 1ρ0 ρ1 , if there is a b ∈ Σ⊥ 1,c such that ρi ∈ ∆t (∂i b) i = 0, 1.

Proof. The uniqueness claim tells us how to go about defining εc : given ρ1 , ρ2 pick b ∈ Σ⊥ 1,c and unitaries Ui ∈ (ρi , τi ) where τi ∈ ∆t (∂i b) and we have no option but to set εc (ρ1 , ρ2 ) = U2∗ ⊗ U1∗ ◦ U1 ⊗ U2 . By Lemma 3.6, such a choice, however made, automatically satisfies (b). We have εc (ρ01 , ρ02 ) = U20 ∗ ⊗ U10 ∗ ◦ U10 ⊗ U20 , where Ui0 ∈ (ρ0i , τi0 ) and with the supports of τ10 and τ20 chosen appropriately. Set Si = Ui0 ◦Ti ◦Ui∗ then, by Lemma 3.6, S1 ⊗S2 = S2 ⊗S1 and rearranging this identity gives (a) and completes the proof of the theorem. Corollary 3.8. Under the hypothesis of Theorem 3.7 (a) εc (ρ1 ρ2 , ρ3 ) = εc (ρ1 , ρ3 ) ⊗ 1ρ2 ◦ 1ρ1 ⊗ εc (ρ2 , ρ3 ), (b) εc (ρ1 , ρ2 ρ3 ) = 1ρ2 ⊗ εc (ρ1 , ρ3 ) ◦ εc (ρ1 , ρ2 ) ⊗ 1ρ3 , ⊥ ¯ ¯ ¯ ¯ If b ∈ Σ⊥ 1,c implies b ∈ Σ1,c , where |b| = |b|, ∂0 b = ∂1 b and ∂1 b = ∂0 b, then c c (c) ε (ρ2 , ρ1 ) ◦ ε (ρ1 , ρ2 ) = 1ρ1 ,ρ2 . Proof. These equalities follow easily from the formula εc (ρ1 , ρ2 ) = U2∗ ⊗ U1∗ ◦ U1 ⊗ U2 used to define εc in the proof of Theorem 3.7. As a consequence of (a) and (b) or by direct computation, we also have εc (ρ, ι) = εc (ι, ρ) = 1ρ . In virtue of (a) and (b), if K is directed, the pair (Tt , εc ) is a braided tensor W ∗ category and when (c) holds, too, we get a symmetric tensor W ∗ -category. In the general case we get a net O 7→ (Tt (O), εc ) of braided or symmetric tensor W ∗ categories, where the terminology implies that the inclusion Tt (O1 ) ⊂ Tt (O2 ) for O1 ⊂ O2 is not only a tensor ∗ -functor but also preserves the braiding. In view of the above results, it is obviously important to be able to compute the connected components of Σ⊥ 1 . We first localize and try to compute the connected components of ⊥ Σ⊥ 1 (O) := {b ∈ Σ1 : |b| ⊂ O}

before trying to compute those of Σ⊥ 1 . Needleess to say, neither step can be carried through at this level of generality but we shall carry them through when K is the set of regular diamonds in a globally hyperbolic spacetime.


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Note that Σ⊥ 1 (O) is closely related to the local graph of the relation ⊥, G ⊥ (O) := {O1 , O0 : O1 , O0 ⊂ O, O1 ⊥ O0 } . There is an obvious order-preserving injection i : G ⊥ (O) → Σ⊥ 1 (O). We simply consider Oi as ∂i b and O as |b|. Conversely, we have an order-preserving surjection ⊥ ⊥ s : Σ⊥ 1 (O) → G (O) mapping b to (∂1 b, ∂0 b). b lies in the same component of Σ1 (O) 0 0 as i ◦ s(b). Hence if s(b) and s(b ) lie in the same component, so do b and b , thus we ⊥ have computed the components of Σ⊥ 1 (O) in terms of those of G (O). Now if O is a regular diamond in a globally hyperbolic spacetime, then O itself with the induced metric is a globally hyperbolic spacetime with a non-compact Cauchy surface and the connected components have been computed in Lemma 2.2. For passing from the local to the global computation, the strategy is to look for ⊥ coherent choices of components for the Σ⊥ 1 (O), i.e. we want a component Σ1,c (O) for each O such that ⊥ ⊥ Σ⊥ 1,c (O1 ) = Σ1,c (O2 ) ∩ Σ1 (O1 ) ,

O1 ⊂ O2 .

⊥ Lemma 3.9. Given a coherent choice of components O 7→ Σ⊥ 1,c (O), then Σ1,c := ⊥ ⊥ {b ∈ Σ1 : b ∈ Σ1,c (|b|)} is a component of Σ1 .

Proof. K being connected, the result will follow from Lemma 3A.3 once we show that ⊥ ⊥ Σ⊥ 1,c (O) = Σ1,c ∩ Σ1 (O) .

But if b ∈ Σ⊥ 1,c (O), |b| ⊂ O and since we have a coherent choice of components, (|b|) giving an inclusion. The reverse inclusion is trivial, completing the b ∈ Σ⊥ 1,c proof. Now when K denotes the set of regular diamonds in a globally hyperbolic space⊥ time with dimension ≥ 2, then Σ⊥ 1 (O) has a single component so that Σ1 is connected by Lemma 3.9. It remains to consider the case of a globally hyperbolic spacetime of dimension two. We know that each Σ⊥ 1 (O) now has two components and that one passes from one component to the other by reversing the orientation of the 1-simplices. We need a way of specifying a coherent choice of components. If the Cauchy surfaces are non-compact, then G ⊥ also has two components and one passes from one component to the other by interchanging the two double cones. Hence mapping b to ∂1 b × ∂0 b must map the two components of Σ⊥ 1 (O) into dif⊥ ⊥ ferent components of G . Denoting the two components of G by G`⊥ and Gr⊥ , the inverse images under the above map give us a coherent choice of components. Lemma 3.9 then shows us that Σ⊥ 1 has precisely two components and that one passes from one component to the other by reversing the orientation of 1-simplices. On the other hand, in a globally hyperbolic spacetime (M, g) of dimension two with compact Cauchy surfaces, we know from the discussion in Sec. 3.1 that G ⊥ is


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connected. However, Σ⊥ 1 continues to have two components and we need a different procedure for making a coherent choice of local components. To this end, we pick a nowhere vanishing timelike vector field and restricting this to a regular diamond O, we have, by the discussion following Lemma 2.2, a coherent way of distinguishing the left and right components of the set of spacelike points in the regular diamond and hence left and right components of G ⊥ (O) and Σ⊥ 1 (O). Thus by Lemma 3.9, ⊥ Σ1 has two connected components and one passes from one component to the other by reversing the orientation of 1-simplices. 3.4. The left inverse and charge transfer The classification of statistics makes essential use of left inverses. When K is directed, we may proceed as follows. Definition 3.10. A positive linear mapping φ on B(H0 ) is called a left inverse of a representation π of A on H0 if φ(Aπ(B)) = φ(A)B ,

A ∈ B(H0 ), B ∈ A ,

and φ(1) = 1 .

There are some simple facts to be noted: first, a positive mapping is automatically self-adjoint, φ(A∗ ) = φ(A)∗ so that we have φ(π(A)B) = Aφ(B), A, B ∈ A. Secondly, if π(B) = B, then φ(B) = B. Thus φ inherits any localization properties of π. In particular, if π is localized in O φ(A) = A

for A ∈ A(O2 ) ,

O2 ⊥ O

and, by duality, if O ⊂ O1 then φ(A(O1 )) ⊂ A(O1 ). Consequently φ maps A into A. Furthermore one may show that φ(A∗ A) ≥ φ(A)∗ φ(A) and kφk ≤ 1. The complications involved when K is not directed are treated in the Appendix where the relations with the left inverse of a localized endomorphism and the left inverse of a cocycle are also discussed. Once we have left inverses, we may proceed to the classification of statistics. We suppose we have permutation statistics. The basic result, stated abstractly, is as follows. Theorem 3.11. Let ρ be an object in a symmetric tensor C ∗ -category (T , ε) and φ a left inverse of ρ with φρ,ρ (ε(ρ, ρ)) = λ1ρ for some scalar λ. Then λ ∈ {0} ∪ {±d−1 : d ∈ N} and depends only on the equivalence class of ρ. The Young tableaux associated with the representations of Pn on (ρn , ρn ), n ≥ 1 are all Young tableaux : (a) whose columns have length ≤ d, if λ = d−1 (para-Bose statistics of order d ); (b) whose rows have length ≤ d if λ = −d−1 (para-Fermi statistics of order d ); (c) without restriction, if λ = 0 (infinite statistics). Note that when ρ is irreducible, φρ,ρ (ε(ρ, ρ)) is automatically a scalar, called the statistics parameter of ρ. d is referred to as the statistics dimension and the


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sign is the statistics phase, κρ and corresponds to the Bose–Fermi alternative. In general, we say that ρ has infinite statistics if there is a left inverse φ with φρ,ρ (ε(ρ, ρ)) = 0. Otherwise ρ is said to have finite statistics. Assuming our category T has subobjects, ρ has finite statistics if and only if ρ is a finite direct sum of irreducible objects with finite statistics. In the cases where we can have braid statistics there is, of course, no correspondingly complete classification, not even if we invoke the special setting of a two dimensional Minkowski space. However, many partial results are known in that case and the proofs presumably generalize without essential modification. As explained in Sec. 3.1, to deduce the existence of a left inverse, we assume that K has an asymptotically causally disjoint net On . Thus, given O ∈ K there is an n0 with On ⊥ O for n ≥ n0 . Under such a hypothesis, every representation π satisfying the selection criterion can be obtained as a limit of unitary transformations. Physically, this would be interpreted as creating charge by transferring it from spacelike infinity. We pick unitary intertwiners Un ∈ (πn , π) where πn is localized in On . The corresponding unitary transformation σUn , σUn (A) := Un AUn∗ , may be interpreted as an operation which transfers charge from On to O. Now if A ∈ A(O0 ) and n is sufficiently large so that O0 ⊥ On then σUn (A) = π(A) so that, as far as A is concerned, we have created a charge in O. In the limit as n → ∞ this holds for all A ∈ A and we have Lemma 3.12. limk→∞ kUk AUk∗ − π(A)k = 0, A ∈ A. The physical idea is now to create the conjugate charge in O by transferring charge to spacelike infinity. More prosaically, we would like to get a left inverse by replacing Uk by Uk∗ and taking a limit. This will indeed be the case although the limiting procedure is more delicate and we cannot work in the strong topology (i.e. pointwise norm topology) for linear mappings on A. We consider the space M of bounded linear mappings on B(H0 ) equipped with the pointwise σ–topology, i.e. a net φn from M converges to φ if φn (A) converges to φ(A) in the σ–topology for each A ∈ A. The important fact for our purposes is that the unit ball M1 of M is compact in this topology, M1 = {φ ∈ M : kφk ≤ 1} . Lemma 3.13. The net σUn∗ possesses at least one limit point in M. Every limit point of this net is a left inverse of π. The set of all left inverses of ρ is a non empty compact convex subset of M. We omit the proof as it is identical with that already given for Minkowski space [23]. The existence of an asymptotically causally disjoint net On is also used in the analysis of left inverses but there are no new geometric properties involved. Another important aspect of superselection structure which does not involve spacetime symmetries is the existence of a complete field net with gauge symmetry


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describing the superselection sectors [25]. This clearly involves no further input of a geometric nature as it is based on Corollary 6.2 of [24] which refers to a single C ∗ -algebra rather than a net of von Neumann algebras. We leave to the reader the task of formulating a precise result so as to avoid having to introduce the relevant definitions from [25].

3.5. Sectors of a fixed-point net Although we have now succeeded in adapting the main results of superselection theory to globally hyperbolic spacetimes with non-compact Cauchy surface, there is another important aspect to be discussed. As we have seen the Selection Criterion has a natural mathematical extension to curved spacetime. In Minkowski space, however, it is further justified by there being a simple mechanism producing examples of such sectors. Under rather general conditions, it suffices to begin with a field net F in its vacuum representation and a group of unitaries, a gauge group, compact in the strong operator topology, and inducing automorphisms of the field net. Then defining an observable net A as the fixed-point net: A(O) := F (O)G , the resulting representation decomposes as a direct sum of irreducible representations satisfying the selection criterion. The equivalence classes of these representations ˆ of equivalence classes of irreducible, contiare in 1–1 correspondence with the set G nuous, unitary representations of G and the irreducible representation correspondˆ has multiplicity d(ξ), the dimension of ξ. The question is whether ing to ξ ∈ G these results continue to hold in curved spacetime. The original result in [23] does not, as it stands, apply to curved spacetime as it involves translations and the cluster property. However the variant given in [25] involves only structural elements and geometric properties compatible with curved spacetime and therefore can be stated here as a result on superselection sectors in curved spacetime. In fact, the following result is valid for a directed set K with a binary relation ⊥ such that given O ∈ K, there exists O1 , O2 ∈ K with O, O1 ⊂ O2 and O ⊥ O1 . This condition is related to our use of the Borchers Property. Theorem 3.14. Let F be a field net over K acting irreducibly on a Hilbert space H equipped with a strongly compact group G of unitaries inducing automorphisms of the net F . We define the observable net A to be the fixed-point net : A(O) := F (O)G ,

O ∈ K.

We assume that the subspace H0 of G-invariant vectors is separable and that A is represented irreducibly on H0 , satisfying duality there and having the Borchers Property. Furthermore, H0 is supposed to be cyclic for each F (O) and F (O1 ) and A(O2 ) to commute whenever O1 ⊥ O2 . Then A0 = G00 and letting π denote the defining representation of A on H


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π=

X

d(ξ)πξ ,

151

ˆ, ξ∈G

ξ

where the πξ are inequivalent irreducible representations satisfying the selection ˆ denotes the set of equivalence classes of continuous irreducible criterion and G unitary representations of G. Despite this positive result, we must examine the assumptions carefully to see whether they remain reasonable in the context of curved spacetime. To test the assumptions we turn to the examples of scalar free fields defined using quasifree Hadamard states [60]. It is known that duality holds for the Klein–Gordon field on a globally hyperbolic spacetime for regular diamonds and that the associated von Neumann algebra is the hyperfinite type III1 factor and hence satisfies the Borchers property. However, at least in the context of Theorem 3.13, this must be regarded as a field net rather than an observable net. Furthermore, we actually use ˜ of causal disjointness to pass from cocycles to duality for the modified relation ⊥ localized endomorphisms in the next section. This strengthened form of duality is equivalent to the original form whenever the nets are inner regular, as is the case for ˆ the Klein–Gordon field. An even stronger form of duality, ⊥-duality, is used in the discussion of left inverses in the next section. However, our basic result on regular diamonds, Lemma 2.1, shows that it is in fact equivalent to ⊥-duality for additive nets. As is well known, a geometric property is involved in passing from duality for the fields to duality for the observables. We give here a variant on the proof of Theorem 4.3 of [53], not a priori requiring each irreducible representation of the gauge group to be realized on Hilbert spaces in F . In view of the Z2 -graded structure of a field net, it is appropriate to define its dual net by \ F d (O) = {F t (O1 )0 : O1 ⊥ O} . Here F t , the twisted field net, can be defined as the transform of F under the unitary transformation 2−1/2 (1 + iV ), where V is the gauge transformation changing the sign of Fermi fields, see e.g. [23]. Theorem 3.15. Let F be a field net over K on a Hilbert space H satisfying twisted duality under a compact group of unitaries G inducing automorphisms of the net F . Let H0 , the subspace of G-invariant vectors, be cyclic for each F (O). Then the fixed–point net A satifies duality for each O ∈ K provided O⊥ is connected. Proof. Let E denote the projection onto H0 then the conditional expectation m of F onto A may either be defined by integrating over the action of G or by m(F )E = EF E , Now (AE )d (O) =

\ O1 ⊥O

(AE (O1 )0 ) =

F ∈F. \ O1 ⊥O

(EF t (O1 )E H0 )0 .


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Since E is cyclic and separating for each F t (O) and A(O) = m(F (O)), (EF t (O1 )E H0 )0 = (EF t (O1 )0 E) H0 . Now using the fact that E is separating for each F t (O1 )0 and that O⊥ is pathconnected, we obtain \ F t (O1 )0 E H0 = A(O) , (AE )d (O) = E O1 ⊥O

since F satisfies twisted duality. What is still missing is a result allowing one to pass from the Borchers Property for the field net to the corresponding property of the observable net. 3.6. Appendix to Chapter 3 In this Appendix, we begin by introducing various notions we shall need in connection with the partially ordered set K. We recall [54] that a 0-simplex a of the partially ordered set P is just an element of P and a 1-simplex b consists of two 0-simplices denoted ∂0 b and ∂1 b contained in a third element |b| of P called the support of b. More generally, an n-simplex is an order-preserving map into P from the set of subsimplices of the standard n-simplex, ordered under inclusion. Σn (P) or just Σn will denote the partially ordered set of n-simplices of P with the pointwise ordering. A partially ordered set P is connected if given a, a0 ∈ Σ0 (P), there is a path from a to a0 in P, i.e. if there exist b0 , b1 , . . . , bn ∈ Σ1 (P) with ∂0 b0 = a, ∂1 bn = a0 and ∂0 bi = ∂1 bi−1 , i = 1, 2, . . . , n. Obviously, if P is not connected, it is a disjoint union of its connected components. We will be taking for P not only subsets of K with the induced order but also of K × K with the product ordering. These notions are related to topological notions in the following way. Lemma 3A.1. Let P be a base for the topology of a space M and ordered under inclusion and suppose the elements of P are open, (non-empty) and path-connected. Then an open subset X of M is path–connected if and only if PX := {O ∈ P : O ⊂ X} is connected. Proof. Any two points of X are contained in elements of PX so if this is connected and each of its elements are path-connected the two points can be joined by a path in X. Conversely, given O0 , O1 ∈ PX , there is a path in X beginning in O1 and ending in O0 , if X is pathwise connected. Since P is a base for the topology, it is easy to construct a path in PX joining O1 and O0 . A subset S of P of the form PX has the property that O ∈ S and O1 ⊂ O implies O1 ∈ S. Such subsets are referred to as sieves. If P is a base for the topology of M then a sieve S is a base for the topology of the open subset XS := ∪{O : O ∈


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S}. The connected components of a partially ordered set are sieves, the union or intersection of sieves is again a sieve. Corollary 3A.2. Under the hypotheses of Lemma 3 A.1, the connected components of P are of the form PX , where X runs over the path–connected components of M. We let Open(M ) denote the set of open subsets of M ordered under inclusion and Sieve(K) the set of sieves of K, then defining for a open set X of M, µ(X) to be the set of O ∈ K contained in X, µ is an injective order-preserving map from Open(M ) to Sieve(K). If we define ν(S) := XS , then ν is order-preserving and a left inverse for µ. The following result will prove useful in calculating the connected components of a partially ordered set. Lemma 3A.3. Let i 7→ Pi be an order-preserving map from a partially ordered set I to the set of sieves of a partially ordered set P ordered under inclusion. Suppose S that P = i∈I Pi . Let C ⊂ P and set Ci := C ∩ Pi then C is a union of components of P if and only if Ci is a union of components of Pi for each i ∈ I. If I is connected and Ci is either empty or a component of Pi , i ∈ I, then C is a component of P. Proof. If C is a union of components and b ∈ Σ1 (Pi ) with ∂1 b ∈ Ci then b ∈ Σ1 (P) so ∂0 b ∈ C ∩ Pi = Ci and Ci is a union of components. Conversely, if each Ci is a union of components and b ∈ Σ1 (P) with ∂1 b ∈ C, then |b| ∈ Pi for some i. But Pi is a sieve so b ∈ Σ1 (Pi ) and ∂1 b ∈ C ∩ Pi . Since Ci is a union of components, ∂0 b ∈ Ci ⊂ C so C is a union of components. Now C is a component, if any given pair a ∈ Ci and a0 ∈ Ci0 can be joined by a path in C. But I being connected, we may as well suppose i and i0 have an upper bound j ∈ I. If Cj is a component, a and a0 can even be joined by a path in Cj , completing the proof of the lemma. Now an automorphism g of a partially ordered set P such that given O ∈ P there is a b ∈ Σ1 (P) with ∂1 b ⊂ O and ∂0 b ⊂ gO obviously leaves each connected component of P globally invariant. If G is a connected topological group acting continuously on a topological space M and P is a base for the topology of M , then it is easy to see that given O ∈ K there is a O1 ∈ K and a neighbourhood N of the unit in G such that N O1 ⊂ O. It follows that G leaves any path-component of P globally invariant. Of course, this may also be deduced from Corollary 3A.2. After these generalities on partially ordered sets, we turn to the theory of superselection sectors and need a partially ordered set K equipped with a binary relation ⊥ satisfying (a), (b) and (c) of Sec. 3.1. Note that (b) just says that O⊥ is a sieve ˜ and ⊥ ˆ defined by supplementing of K. There are two derived binary relations ⊥ O1 ⊥ O2 by requiring that there exists an O3 ∈ K such that O1 ⊥ O3 , O2 ⊥ O3 or such that O1 , O2 ⊂ O3 ,


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respectively. These relations automatically satisfy (a) and (b) but (c) remains to be checked and will not prove to be a problem in our applications to curved spacetime. ˜ or ⊥ ˆ is idempotent and if K is directed, all The operation of passing from ⊥ to ⊥ three relations coincide. Furthermore, by Proposition 5.6, the corresponding notions of duality coincide for additive nets when K is the set of regular diamonds in a globally hyperbolic spacetime. If O1 ⊥ O2 and O3 has non-trivial causal complement in O2 , i.e. if there exists an ˜ 3 . Now a regular diamond is a O4 with O3 ⊥ O4 , O3 , O4 ⊂ O2 then trivially O1 ⊥O union of a sequence of smaller regular diamonds with non-trivial causal complement in the original regular diamond. Thus when K is the set of regular diamonds, the ˜ is, in this sense, a boundary effect. difference between the relations ⊥ and ⊥ ˜ ˆ The difference between ⊥ and ⊥ merely reflects the potential difficulty of finding suitably large regular diamonds. If we replace the set K of regular diamonds by ˜ of sieves in K with non-trivial causal complement, defining the causal the set K T ˜ = ⊥. ˆ In fact, complement S ⊥ of a sieve S to be the sieve S ⊥ := O∈S O⊥ , then ⊥ ⊥ ⊥ ⊥ ˆ ˜ if S1 ⊥S2 , then (S1 ∪ S2 ) = S1 ∩ S2 6= ∅ so that S1 ⊥S2 . If K is a base of open sets of a topological space M and the relation ⊥ on K is induced by a relation ⊥ on Open(M ) satisfying (a) and (b) of Sec. 3.1 and which S is local in the sense that if X ∈ Open(M ) and X ⊂ i Oi , then Oi ⊥ O for all i implies X ⊥ O. This condition is obviously satisfied by the relation of causal disjointness on a globally hyperbolic spacetime. It implies that µ(X ⊥ ) = µ(X)⊥ . We also have ν(S)⊥ = ν(S ⊥ ) for any sieve S in K. Lemma 3A.4. When restricted to causally closed open sets and sieves, the maps µ and ν are inverses of one another. Proof. If S is a sieve and X := ν(S), then µ(X)⊥ = S ⊥ . If S is causally closed, so is X since µ is injective. On the other hand, if X is causally closed and we set S := µ(X), then S ⊥⊥ = µ(X ⊥⊥ ) = µ(X) and S is causally closed. It remains to show that S = µν(S) if S is causally closed. But, in this case, S ⊂ µν(S) ⊂ µν(S)⊥⊥ = S ⊥⊥ = S , completing the proof. By a representation π of a net of von Neumann algebras A over K we mean normal representations πO of A(O) on a Hilbert space Hπ such that πO1 is πO2 restricted to A(O1 ), whenever O1 ⊂ O2 in K. If G is a group of automorphisms of K and (A, α) is a covariant net then a covariant representation is a pair (π, U ) consisting of a representation π of A and a unitary representation of G on Hπ such that U (g)πO (A) = πgO (αg (A))U (g), A ∈ A(O), g ∈ G. We now provide a cohomological interpretation of superselection sectors leading to a proof of the Extension Theorem of Sec. 3.1. To enter into the spirit of the cohomological interpretation, we regard O⊥ , O ∈ K as being a covering of K, the


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causal covering. The selection criterion selects those representations that are trivial on the causal cover and these representations allow a cohomological description in analogy with locally trivial bundles. For each a ∈ Σ0 we pick a unitary Va such that Va πO (A) = AVa ,

A ∈ A(O) ,

O⊥a

and set z(b) := V∂0 b V∂∗1 b , b ∈ Σ1 . Obviously if O ∈ |b|⊥ , z(b) ∈ A(O)0 thus z(b) ∈ Ad (|b|). Furthermore, z(∂0 c)z(∂2 c) = z(∂1 c) ,

c ∈ Σ2

so that z is a unitary 1-cocycle with values in the dual net Ad . We consider such 1-cocycles as objects of a category Z 1 (Ad ), where an arrow t in this category from z to z 0 is a ta ∈ Ad (a), a ∈ Σ0 , such that t∂0 b z(b) = z 0 (b)t∂1 b ,

b ∈ Σ1 .

∗

This makes Z (A ) into a W -category. Note that kta k is independent of a. If we were to make a different choice Va0 of unitaries Va , then setting z 0 (b) := 0 V∂0 b V∂0∗1 b and wa := Va0 Va∗ , we see that wa ∈ Ad (a) and w∂0 b z(b) = z 0 (b)w∂1 b . Thus w ∈ (z, z 0 ) is a unitary and the 1-cocycle attached to π is defined up to unitary equivalence in Z 1 (Ad ). More generally, if T ∈ (π, π 0 ) and π and π 0 are trivial on the causal cover and z and z 0 are associated cocycles defined by unitaries Va and Va0 , as above, set 1

d

ta := Va0 T Va∗ , a ∈ Σ0 . Then ta ∈ Ad (a) and t∂0 b z(b) = V∂00 b T V∂∗0 b V∂0 b V∂∗1 b = V∂00 b T V∂∗1 b = V∂00 b V∂0∗1 b V∂01 b T V∂∗0 b = z 0 (b)t∂1 b , so that t ∈ (z, z 0 ). Conversely, if t ∈ (z, z 0 ) then T := Va0∗ ta Va is independent of a so that 0 (A)T , T πO (A) = πO

A ∈ A(O) ,

O∈K

and we clearly have a close relation between Z 1 (Ad ) and the W ∗ -category Rep⊥ A of representations of A trivial on the causal cover. However, any cocycle z arising from such a representation has two special properties that may not be shared by a general 1-cocycle. First, z is trivial on B(H0 ), i.e. there are unitaries Va , a ∈ Σ0 , on H0 such that z(b) = V∂0 b V∂∗1 b , b ∈ Σ1 . If K is directed then Σ∗ (K) admits a contracting homotopy [54]. In this case every 1-cocycle of Ad is trivial in B(H0 ). In general, if we consider the graph with vertices Σ0 and arrows Σ1 then the category generated by this graph has as arrows the paths in K. Thus every 1-cocycle extends to a functor from this category. When z is trivial on B(H0 ) then z(p) for a path p depends only on the endpoints ∂0 p and ∂1 p of the path. Conversely, if z(p) just depends on the endpoints of p and K is connected, then z is trivial on B(H0 ). To see this we pick a base point a0 ∈ Σ0 , then


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given a ∈ Σ0 a path pa with ∂0 pa = a and ∂1 pa = a0 and finally define y(a) = z(pa ). z(p)y(∂1 p) = y(∂0 p), so we have trivialized z in B(H0 ). Secondly, for any path p, z(p)Az(p)∗ = A whenever A ∈ A(O) and ∂0 p, ∂1 p ∈ O⊥ . The full subcategory of Z 1 (Ad ) whose objects satisfy these two conditions will be denoted by Zt1 (Ad ). The following simple result shows that the second condition is automatically satisfied in an important special case. Lemma 3A.5. If O⊥ is connected, then any object z of Z 1 (Ad ), trivial on B(H0 ) satisfies z(p)Az(p)∗ = A ,

∂0 p , ∂1 p ∈ O⊥ , A ∈ A(O) .

Proof. Since O⊥ is connected, it suffices to prove the result when the path p is a 1-simplex b with |b| ∈ O⊥ . But then, z(b) ∈ Ad (|b|) ⊂ A(O)0 . Having discussed these two conditions, we can give our cohomological characterization of the selection criterion. Theorem 3A.6. The W ∗ -categories Rep⊥ A and Zt1 (Ad ) are equivalent. Proof. We pick unitaries Vaπ , a ∈ Σ0 , as above, for each object π of Rep⊥ A. Given an arrow T ∈ (π, π 0 ) in that category, we define for b ∈ Σ1 , a ∈ Σ0 F (π)(b) = V∂π0 b V∂π∗ ; 1b

0

F (T )a := Vaπ T Vaπ∗ .

Then F is a faithful ∗ -functor and our computations above show that it is full. Hence, it remains to show that each object z of Zt1 (Ad ), is equivalent to an object in the image of F . We show this by constructing a representation π z . We pick unitaries Va , a ∈ Σ0 , on H0 such that z(b) = V∂0 b V∂∗1 b , b ∈ Σ1 , and define z (A) = Va∗ AVa , πO

a ∈ O⊥ ,

A ∈ A(O) .

This is well defined since K is connected and for any path p with ∂0 p, ∂1 p ∈ O⊥ we have z(p) ∈ A(O)0 . Furthermore, the definition respects the net structure since z z (A) = πO (A) , πO 1 2

A ∈ A(O1 ) ,

O1 ⊂ O2 .

Hence we get a representation of the net A, trivial on the covering by construction, and V∂0 b V∂∗1 b = z(b) is an associated 1-cocycle. This completes the proof. We now consider the problem of extending representations of a net A, trivial on the causal cover, to representations of the bidual net Add , again trivial on the causal cover. Theorem 3A.7. If each O⊥ is connected, every object π of Rep⊥ A admits a unique extension to an object of Rep⊥ Add . Furthermore there is a canonical isomorphism of W ∗ -categories Rep⊥ A and Rep⊥ Add .


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Proof. Let Va , a ∈ Σ0 be unitaries realizing the equivalence of π and π 0 on a⊥ . Then z(b) := V∂0 b V∂∗1 b , b ∈ Σ1 is an associated object of Zt1 (Ad ). Since each O⊥ is connected, z is at the same time an object of Zt1 (Addd ) by Lemma 3.A.4. If we define π Õ (A) := Va∗ AVa ,

A ∈ Add (O) ,

a ∈ O⊥ ,

this gives a well defined element of Rep⊥ Add just as in the proof of Theorem 3A.6. Furthermore, π ˜ obviously extends π by the choice of the Va . If we make another ˜ remains unchanged and is consechoice Va0 of the Va then Va0 Va∗ ∈ Ad (a) so that π quently the unique extension of π to an object of Rep⊥ Add . Passing to the extensions does not change the intertwiners by Theorem 3A.6. For the further development of superselection theory, we must assume duality A = Ad , although essential duality would do whenever each O⊥ is connected. We ˜ shall even need to assume ⊥–duality, but this coincides with duality in curved spacetime whose status is commented on in Sec. 4.2. The next goal is to show that sectors have a tensor structure. More precisely, we shall show that Z 1 (A) has a canonical structure of a tensor W ∗ -category arising by adjoining endomorphisms. If A is a net of von Neumann algebras, then there is an associated net O 7→ End A(O) of tensor W ∗ -categories. End A(O) has as objects the normal endomorphisms of the net O1 7→ A(O1 ), i.e. normal endomorphisms ρO1 of A(O1 ) compatible with the net structure. An arrow T ∈ (ρ, σ) in End A(O) is a T ∈ A(O) such that T ρ(A) = σ(A)T ,

A ∈ A(O1 ) , O ⊂ O1 .

The tensor structure is defined on the lines of Sec. 3.3 and the net structure is given by the obvious restriction mappings. The construction of appropriate endomorphisms is just a variant on that already used to pass from a 1-cocycle z ∈ Zt1 (Ad ) to a representation π z . Given a ∈ Σ0 , and A ∈ A(O), a ⊂ O pick a path p with ∂0 p = a and ∂1 p ∈ O⊥ and set y(a)(A) := z(p)Az(p)∗ . y(a)(A) is independent of the choice of p since z ∈ Zt1 (Ad ). Given X ∈ A(O1 ) with O1 ⊥ O, O2 with O2 ⊥ O and O2 ⊥ O1 and choosing ∂1 p = O2 , we see that ˜ y(a)(A) and X commute so that y(a)(A) ∈ A(O) by ⊥-duality. Thus y(a) is an object of End A(a). But y(a) is not only localized in a in the sense of net automorphisms but also in the sense of superselection theory in that y(a)(A) = A whenever A ∈ A(O1 ) where O1 ∈ a⊥ and O1 , a ⊂ O, since the endpoints of p lie in O1⊥ . We write ∆(a) to denote the objects of End A(a) satisfying this second localization conditon and denote by T (a) the corresponding full tensor C*-subcategory of End A(a). Lemma 3A.8 Let p be a path with ∂1 p, ∂0 p ⊂ O . Then z(p)y(∂1 p)(A) = y(∂0 p)(A)z(p),

A ∈ A(O).


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Proof. Given A ∈ A(O) and a path p with ∂1 p, ∂0 p ⊂ O, pick paths p0 , p00 with ∂0 p0 = ∂1 p, ∂0 p00 = ∂0 p and ∂1 p0 , ∂1 p00 ∈ O⊥ , then z(p)y(∂1 p)(A) = z(p)z(p0 )Az(p0 )∗ = z(p00 )Az(p00 )∗ z(p) = y(∂0 p)(A)z(p) , as required. Furthermore if t ∈ (z, zˆ), A ∈ A(O) and p is a path with ∂0 p = a ⊂ O and ∂1 p ⊂ O⊥ then z (p)∗ ta = yˆ(a)(A)ta . ta y(a)(A) = ta z(p)Xz(p)∗ = zˆ(p)t∂1 p Az(p)∗ = zˆ(p)Aˆ In other words ta ∈ (y(a), yˆ(a)) . These results admit the following interpretation.

˜ Theorem 3A.9 Let A be a net over (K, ⊥) satisfying ⊥-duality. If z is a 1-cocycle of A trivial in B(H0 ) then (y, z) is a 1-cocycle in the net T of tensor W ∗ -categories and the map z 7→ (y, z) together with the identity map on arrows is an isomorphism of Zt1 (A) and Zt1 (T ). Now, T being a net of tensor W ∗ -categories, Z 1 (T ) is itself a tensor W ∗ – category. Given 1-cocycles (y1 , z1 ) and (y2 , z2 ), their tensor product is the 1-cocycle (y, z), where y(a) = y1 (a)y2 (a),

z(b) = z1 (b)y1 (∂1 b)(z2 (b)).

If both (y1 , z1 ) and (y2 , z2 ) are trivial in B(H0 ) then so is their tensor product. The tensor product on arrows is defined as follows: if ti maps from (yi , zi ) to (yi0 , zi0 ) for i = 1, 2, then the tensor product t1 ⊗ t2 is given by (t1 ⊗ t2 )a = t1,a y1 (a)(t2,a ). This completes our goal of describing superselection structure in terms of a tensor W ∗ -category. Note that we could have used the subnet Tt in place of T defined by requiring an object ρ of T (O) to be transportable, i.e. there exists a map a 7→ ρa , where ρa is an object of T (a) and ρa = ρ when a = O and a map Σ1 3 b 7→ u(b), where u(b) is an arrow from ρ∂1 b to ρ∂0 b in T (|b|). In fact the tensor W ∗ -categories Zt1 (T ) and Zt1 (Tt ) are canonically isomorphic. In Sec. 3.3, we showed how to get a net (Tt , εc ) of braided tensor W ∗ -categories and it is a simple general fact that this leads to a braided tensor W ∗ -category, (Zt1 (Tt ), εc ). We need only set εc (z, z 0 )a := ε(y(a), y 0 (a)). Since this expression obviously acts correctly on the arrows evaluated in a and the laws for a braiding hold for each a, the only point that has to be checked is that εc (z, z 0 ) is an arrow from z × z 0 to z 0 × z. However, if b ∈ Σ1 , z(b) ∈ (ρ∂1 b , ρ∂0 b ) in Tt (|b|) and similarly for z 0 (b). Thus z 0 (b) × z(b) ◦ ε(ρ∂1 b , ρ0∂1 b ) = ε(ρ∂0 b , ρ0∂0 b ) ◦ z(b) × z 0 (b), as required.


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Thus the cohomological approach leads to a braided tensor W ∗ -category describing superselection structure and in the context of globally hyperbolic spacetimes this is even a symmetric tensor W ∗ -category for spacetime dimensions ≥ 2. It should be noted that except when K is directed, we have not given a direct description of this structure in terms of transportable localized endomorphisms. In particular, it is not clear that every transportable localized endomorphism arises from a 1-cocycle. Furthermore, if ρ and σ are in ∆t (O) and T is a bounded operator on the ambient Hilbert space, such that (Zt1 (Tt ),εc )

T ρ(A) = σ(A)T,

A ∈ A(O1 ), O ⊂ O1 ,

then T commutes with A(O2 ) for O2 ⊥ O provided there is a O1 with O, O2 ⊂ O1 . ˆ to be This means, we would need duality with respect to the modified relation ⊥ able to conclude that T ∈ A(O) and hence that T is an arrow from ρ to σ in Tt (O). Conversely, if π and π 0 are representations satisfying the selection criterion and restricting to endomorphisms ρ and ρ0 in ∆t (O) then it is not clear that an arrow T ∈ (ρ, ρ0 ) in Tt (O) will at the same time intertwine π and π 0 . These points should be bourne in mind, when, in the main body of the text, we avoid the cohomological description and put the emphasis on transportable localized endomorphisms. To proceed with the analysis of statistics, we need to use left inverses and we examine, at this point, the notions involved and the relations between them. If π is a representation of A on H0 then we define a left inverse φ of π to be given by unital positive linear mappings φO on B(H0 ) compatible with the net inclusions and satisfying φO (AπO (B)) = φO (A)B,

A, B ∈ A(O).

Note that if πO (B) = B then φO (B) = B. If π is localized in O in the sense that πO1 (A) = A ,

O ⊥ O1 ,

A ∈ A(O1 ) ,

ˆ 2 and O ⊂ O2 , then φ is localized in O in the same sense. Furthermore, if O1 ⊥O then φO2 (A)B = BφO2 (A) for A ∈ A(O2 ) and B ∈ A(O1 ). In fact, picking O3 with O1 , O2 ⊂ O3 we have φO2 (A)B = φO3 (A)B = φO3 (AπO3 (B)) = φO3 (AπO1 (B)) = φO3 (AB). Since A and B commute, we interchange them and reverse the steps to conclude that φO2 (A) and B commute. This proves the following result. Lemma 3A.10 If φ is a left inverse for a representation π localized in O then φ ˆ φO A(O1 ) ⊂ A(O1 ) for is localized in O and if duality holds for the relation ⊥, 1 O ⊂ O1 . The restriction of π to the net O1 7→ A(O1 ), O1 ⊃ O is a localized endomorphism ρ and an object of the tensor W ∗ -category End A(O). The above notion of


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left inverse adapts easily to localized endomorphisms. If ρ is localized in O, a left inverse of ρ is a family O1 ⊃ O 7→ φO1 of unital positive linear mappings on the A(O1 ), compatible with the net inclusions and satisfying φO1 (AρO1 (B)) = φO1 (A)B,

A, B ∈ A(O1 ).

Obviously, a left inverse for ρ considered as a representation yields a left inverse for the endomorphism ρ on restriction. If ρ¯ is a conjugate for ρ then we get a left inverse φ for ρ by setting φO1 (A) := V ∗ ρ¯O1 (A)V,

A ∈ A(O1 ), O1 ⊃ O,

where V ∈ (id, ρ¯ρ) is an isometry. The restriction of π to the net O1 7→ A(O1 ), O1 ⊃ O is a localized endomorphism ρ and an object of the tensor W ∗ -category End A(O). We now show that a left inverse φ for ρ induces a left inverse of ρ in the categorical sense [48]. In other words, we need a set φσ,τ : (ρσ, ρτ ) → (σ, τ ), of linear mappings where σ, τ are objects of the category. These have to be natural in σ and τ, i.e. given S ∈ (σ, σ 0 ) and T ∈ (τ, τ 0 ) we have φσ0 ,τ 0 (1ρ ⊗ T ◦ X ◦ 1ρ ⊗ S ∗ ) = T ◦ φσ,τ (X) ◦ S ∗ , X ∈ (ρσ, ρτ ), and furthermore to satisfy φσν,τ ν (X ⊗ 1π ) = φσ,τ (X) ⊗ 1ν , X ∈ (ρσ, ρτ ) for each object ν. We will require that φ is positive in the sense that φσ,σ is positive for each σ and normalized in the sense that φι,ι (1ρ ) = 1ι . We say that φ is faithful if φσ,σ is faithful for each object σ. Now, given T ∈ (ρσ, ρτ ), we recall that T ∈ A(O). Hence we set φσ,τ (T ) = φO (T ) and since φO (T ) ∈ A(O) by Lemma 3A.10, we conclude without difficulty that we get a left inverse for ρ in this way. On the other hand, if we are dealing with a representation satisfying the selection criterion then we know that, by passing to an associated 1-cocycle, we get a field a 7→ y(a) of localized endomorphisms under the weaker assumption that duality ˜ In this case, we would actually like a left inverse for the holds for the relation ⊥. 1-cocycle considered as an object of the tensor W ∗ -category Zt1 (A). To this end, we pick, for each of the associated endomorphisms y(a) a left inverse φa and ask whether a 7→ φa (ta ) is an arrow from z 0 to z 00 , whenever a 7→ ta is an arrow from z × z 0 to z × z 00 . Thus ta ∈ A(a) and (z × z 00 )(b)t∂1 b = t∂0 b (z × z 0 )(b).


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It follows that z 00 (b)φ∂1 b (t∂1 b ) = φ∂1 b (y(∂1 b)(z 00 (b))t∂1 b ) = φ∂1 b (z(b)∗ t∂0 b z(b))z 0 (b) and we deduce the following lemma. Lemma 3A.11 If z ∈ Zt1 (A) and a 7→ y(a) is the associated field of endomorphisms. Then a field a 7→ φa of left inverses of the y(a) defines a left inverse for z by the formula φz0 ,z00 (t)a := φa (ta ) provided φ∂0 b = φ∂1 b Ad z(b)∗ for b ∈ Σ1 . There is no a priori reason to suppose that every left inverse for a 1-cocycle arises from such a field of left inverses. In particular a map t ∈ (z, z 0 ) 7→ ta ∈ (y(a), y 0 (a)) might not be surjective. We can also not just begin with a left inverse φa for y(a) since it is not clear that we get a field of left inverses using the cocycle. However, if we assume, as in Sec. 3.4, that K has an asymptotically causally disjoint net On , then we can construct left inverses for 1-cocycles. If z is an object of Zt1 (A), we denote by z(a, n), the evaluation of z on a path p with ∂0 b = a and ∂1 b = On . This is independent of the chosen path. We now define φa (X) to be a Banach–limit over n of z(a, n)∗ Xz(a, n). Then φa is a positive linear map satisfying a (A)), φa (X)A = φa (XπO

A ∈ A(O).

Furthermore, from the cocycle identity we have φ∂0 b = φ∂1 b Ad(z(b)). Since each φa defines a left inverse for y(a), we have constructed a left inverse for z by Lemma 3A.11. One sometimes wishes to consider nets defined over a wider class of regions than say just the set of regular diamonds. Thus in Sections 4 and 5, we are interested in defining the von Neumann algebras of wedge regions. Furthermore, another reason for wanting von Neumann algebras associated with large rather than small regions is that we can only compose endomorphisms if we find a joint localization region for the endomorphisms involved. We consider here the task of extending the domain of definition of the net in the context of the present formalism where K is a partially ordered set commenting on the relation with regions of spacetime afterwards. Thus instead of a region, we use the notion of a sieve S, see above, and consider the ˜ of sieves S of K such that neither S nor S ⊥ are the empty set, ordered set K under inclusion. To each such sieve S, we associate the von Neumann algebra A(S) generated by the A(O) with O ∈ S in the defining representation. We now show that a representation π of A satisfying the selection criterion has a natural extension to a representation of the net S 7→ A(S). We pick for each a ∈ Σ0 a unitary Va such that πO (A) = Va∗ AVa ,

A ∈ A(O), O ∈ a⊥ ,


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and then define πS (A) := Va∗ AVa ,

A ∈ A(S), a ∈ S ⊥ .

Note that this expression is well defined being independent of the choice of a ∈ S ⊥ since if a0 ∈ S ⊥ then Va0 Va∗ ∈ ∩O∈S A(O)0 = A(S)0 . In the same way, we see that πS is independent of the choice of a 7→ Va . Note, too that we get a representation of the extended net in that if S1 ⊂ S2 then πS1 is the restriction of πS2 to A(S1 ). Obviously, an intertwiner T ∈ (π, π 0 ) over K remains an ˜ so that effectively Rep⊥ A remains unchanged when we extend intertwiner over K the net. That part of the formalism related to the concept of localized endomorphism is however sensitive to extending the net. Although localized endomorphisms do not play the same fundamental role as 1-cocycles, we have found it convenient to use them in developing the theory. The problems involved in using them are two: they are not defined on the whole net and the natural map (z, z 0 ) 7→ (y(a), y 0 (a)) may not be surjective. Extending the net improves matters in that localized endomorphisms are then defined on more operators and hence have fewer intertwiners. Since ˆ localized endomorphisms require subsets satisfying ⊥-duality, we benefit from the ˜ ˆ ˜ equality ⊥ = ⊥ on K. Supposing we have as usual a field a 7→ y(a), a ∈ Σ0 , of localized endomorphisms ˜ 1 and a ⊥ O1 , y(a)(A(O)) ⊂ derived from a 1-cocycle, then we know that if O⊥O ˜ holds A(O1 )0 . Hence y(a)(A(S)) ⊂ ∩O1 ∈S ⊥˜ A(O1 )0 . We conclude that if ⊥-duality for S in the defining representation in the sense that ˜

A(S) = A(S ⊥ )0 , then y(a) acts as an endomorphism of A(S), y(a)(A(S)) ⊂ A(S). Now if S sat˜ ˜ isfies ⊥-duality then so does S ⊥ . Furthermore, A(S) = A(S ⊥ )0 ⊃ A(S ⊥⊥ ). Thus A(S) = A(S ⊥⊥ ). Hence, we may as well restrict attention to causally closed sieves and choose as our index set the set L of non-trivial causally closed sieves S for ˜ which ⊥-duality holds either for S or for S ⊥ . This choice has the disadvantage of depending on the theory under consideration but it allows a smooth treatment ˜ of endomorphisms. In particular, if ⊥-duality holds for S and a ∈ S ⊥ then the endomorphism y(a) associated with a 1-cocycle satisfies y(a)(A(S ⊥ )) ⊂ A(S ⊥ ), ˜ ˜ because, as we have seen above, duality holds for S ⊥ and A(S ⊥ ) = A(S ⊥ ). ˜ We shall be assuming ⊥-duality for the elements of K. Thus K ⊂ L and {O⊥ : ˜ Let us call two localized O ∈ K} ⊂ L. Thus L is both coinitial and cofinal in K. ˜ and endomorphisms comparable if they are both localized in a common sieve in K hence in some element of L. In this case, it makes sense to talk about intertwining operators between the two localized endomorphisms. If ρi is localized in Si , i = 1, 2, then ρ1 and ρ2 are comparable, if and only if S1 ∩ S2 6= ∅.


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We turn now to the notion of left inverse. If we consider π as a representation of the extended net S 7→ A(S), then there is an obvious modification of the notion of left inverse as we just need to replace O everywhere by S. Suppose π is localized ˆ in S and φ is a left inverse for π, then given S1 ⊃ S and O ∈ S1⊥ , we remark that there is a sieve S2 with O ∈ S2 and S1 ⊂ S2 . Given A ∈ A(S1 ) and B ∈ A(O) we have φS1 (A)B = φS2 (A)B = φS2 (AπS2 (B)) = φS2 (AπO (B)) = φS2 (AB). Since A and B commute, we interchange them and reverse the steps to conclude ˜ =⊥ ˆ on K, ˜ this proves the following that φS1 (A) and B commute. Recalling that ⊥ result. Lemma 3A.12 Let φ be a left inverse for a representation π of the extended net ˜ holds for S1 , φS1 A(S1 ) ⊂ S 7→ A(S) localized in S. Then, if S ⊂ S1 and ⊥-duality A(S1 ). 4. The Conformal Spin and Statistics Relation for Spacetimes With Bifurcate Killing Horizon In the present chapter, we shall specialize our considerations to the class of spacetimes with a bifurcate Killing horizon (bKh), whose definition we now summarize, following Kay and Wald [41]. The interested reader is strongly recommended to consult this reference for further details not spelled out here. The main purpose here is to show that, from the original theory, we can construct a family of local algebras localized on the horizon, which possesses a conformal symmetry. Therefore horizon-localized superselection sectors have a conformal spin and we prove that this coincides with their statistics phase. 4.1. Spacetimes with bKh A spacetime with a bKh is a triple (M, g, τt ) where (M, g) is a four-dimensional, globally hyperbolic spacetime, although spacetimes with a bKh generalize to other spacetime dimensions. (τt )t∈R is a non-trivial one-parameter group of isometries of (M, g), assumed to be C ∞ , and hence the flow of a Killing vector field ξ on M for the metric g. We often refer to (τt )t∈R as the Killing flow (of the spacetime with bKh). We shall assume that (M, g) is orientable and that the set Σ ⊂ M of fixed points of (τt )t∈R is a two-dimensional smooth, acausal, orientable, connected submanifold of M . It is worth noting that Σ, when compact, automatically lies in some Cauchy surface, see [41] for a proof. From this data we can construct the bKh, h, as follows: at each point p ∈ Σ we choose a pair of linearly independent, lightlike, future-oriented vectors χA (p), χB (p) ∈ Tp M , normal to Σ. They are unique up to scalars and they may be chosen so that Σ 3 p 7→ χA (p) and Σ 3 p 7→ χB (p) are smooth vector fields along


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Σ since M and Σ are orientable. Now let γAp and γBp be the maximal geodesics with tangents χA (p) and χB (p) at p ∈ Σ, respectively. Since (τt ) leaves each p ∈ Σ fixed, it maps each of the curves γAp and γBp into itself. Moreover, γAp and γAp0 do not intersect for p 6= p0 , and the same holds with B in place of A. Now one defines sets hA and hB to be the lightlike hypersurfaces in M formed by the γAp and γBp , respectively, as p ranges over Σ. Then h := hA ∪ hB is the bKh, and one distinguishes the following subsets: + hR A := (hA \Σ) ∩ J (Σ) ,

− hL A := (hA \Σ) ∩ J (Σ) ,

− hR B := (hB \Σ) ∩ J (Σ) ,

+ hL B := (hB \Σ) ∩ J (Σ) .

The Killing vector field ξ is conventionally assumed to be future oriented on hR A. The bKh divides the spacetime M locally into four disjoint parts, F := J + (Σ), P := R + R R − L L + L L J − (Σ), R := (J − (hR A )\hA ) ∩ (J (hB )\hB ) and L := (J (hB )\hB ) ∩ (J (hA )\hA ), the future, past, right and left parts of the spacetime respectively. To give a rather simple illustration, consider (M, g) as Minkowski spacetime (of dimension 4).a Then choose an inertial coordinate system and define Σ as the twodimensional hyperplane {(x0 , x1 , x2 , x3 ) ∈ R4 : x0 = x1 = 0}. There is a smooth, one-parameter group τt = Λt , t ∈ R, of pure Lorentz transformations leaving Σ fixed; they are defined by Λt (x0 , x1 , x2 , x3 ) := (cosh(t)x0 + sinh(t)x1 , sinh(t)x0 + cosh(t)x1 , x2 , x3 ) .

(4.1)

Then h = hA ∪ hB is a bKh, where hA = {(u, u, x2 , x3 ) : u ∈ R, (x2 , x3 ) ∈ R2 } and hB = {(v, −v, x2 , x3 ) : v ∈ R, (x2 , x3 ) ∈ R2 }. Here, the regions R and L correspond to the usual “right wedge” and “left wedge” regions in Minkowski spacetime. Other important examples of spacetimes with a bKh include e.g. de Sitter and Schwarzschild–Kruskal spacetimes, as well as the Schwarzschild–de Sitter spacetimes and (certain regions of the) Kerr–Newman spacetimes. (The latter have at least two bKhs with different surfaces gravities, see below. This leads [41] to conclude that there are no regular, Killing-flow invariant states of the free scalar field on such spacetimes.) Let us now look at how the Killing flow acts on the bKh in greater detail. Each of the geodesic generators γAp of the hA -part of the bKh is defined on some interval of γAp , with affine parameter U , such Ip . We may choose an affine parametrization d γAp U=0 = χA (p) for all p ∈ Σ. This parametrizes that γAp (U = 0) = p and dU all the geodesics and, since the vector field χA (p) depends smoothly on p ∈ Σ, by assumption, the affine parametrization of the curves γAp depends smoothly on p ∈ Σ. Since γAp is left invariant under the Killing flow, Ip must be invariant under a (non-trivial) smooth representation of the additive group R (with 0 as the only fixed point), and thus Ip = R. A similar result holds for the domains of the geodesic generators γBp of hB . Therefore, each point q ∈ hA is uniquely determined by the pair (U, p), where q = γAp (U ). Hence we have a diffeomorphism ψA : hA → R × Σ


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assigning to q ∈ hA the pair (U, p) ∈ R × Σ with q = γAp (U ). f As explained below, certain choices of χA and χB turn out to be particularly useful for our purposes and lead to the following relation (cf. [41], see also [58]): τt ◦ ψA −1 (U, p) = ψA −1 (eκt U, p) ,

t, U ∈ R, p ∈ Σ ,

(4.2)

where the number κ > 0, called the surface gravity, is an invariant of the bKh under consideration. (For the Schwarzschild-Kruskal spacetime of a black hole with mass mbh > 0, κ is proportional to mbh . The reader is referred to [41],[62] for more information about the notion of surface gravity.) Constructing a diffeomorphism ψB : hB → R × Σ, similarly, where ψB (q) = (V, q) iff q = γBp (V ), the affine geodesic parameter being now denoted by V , one can show that τt ◦ ψB −1 (V, p) = ψB −1 (e−κt V, p) ,

t, V ∈ R, p ∈ Σ ,

(4.3)

with the same κ > 0 as in the previous equation. There are a few other geometric actions on hA and hB , induced by identifying these parts of the bKh with R × Σ via the maps ψA and ψB . First, there are the affine translations à ◦ ψA −1 (U, p) := ψA −1 (U + a, p) , à ◦ ψB

−1

(V, p) := ψB

−1

(V + a, p) ,

(4.4) a, U, V ∈ R, p ∈ Σ .

(4.5)

In contrast to the dilations on hA and hB , induced by restricting the Killing flow to the bKh, the translations will not, in general, extend to isometries of the full spacetime. Another action is the (affine) reflection,g ι ◦ ψA −1 (U, p) := ψA −1 (−U, p) , ι ◦ ψB

−1

(V, p) := ψB

−1

(−V, p) ,

(4.6) U, V ∈ R, p ∈ Σ .

(4.7)

Again, ι need not extend to an isometry of the full spacetime of the bKh. However, Kay and Wald [41] have shown that, if the spacetime with bKh is analytic, there is a neighbourhood N of h and an orientation and chronology-reversing isometry j of N (“horizon reflection”) commuting with the action of (τt ) which reflects the affine parameter of geodesics passing orthogonally through Σ. In the next step, we shall specify some families of regions analogous in some respects to the “shifted wedges” in Minkowski spacetime. With their help, we can then formulate a version of geometric modular action for quantum field theories on spacetimes with a bKh in the operator-algebraic framework. To begin with, we note (cf. [41]) that the parts F , P , R and L of a spacetime with bKh (see above) that ψA depends on the choice of the vector field Σ 3 p 7→ χA (p) along Σ. It may be rescaled at each point: χ Ã (p) = φ(p)χA (p), with φ : Σ → R a smooth, strictly positive function, would serve just as well when constructing hA . A similar remark applies to the hB -horizon. g The definitions of ` and ι involve ψ a A (or ψB ) so these quantities, cf. the previous footnote, depend on the scaling freedom when choosing ψA (or ψB ). f Notice


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satisfy F ∩P = Σ,

F ∩R = ∅,

F ∩L = ∅,

P ∩L = ∅.

P ∩R = ∅,

(4.8)

Thus, as we have already seen from the example above, R and L may be viewed as playing the role of the right and left wedge regions in Minkowski spacetime. If ˜ := F ∪ P ∪ L ∪ R, then M ˜ , L and R, with the appropriate restrictions of g M as Lorentzian metric, are globally hyperbolic spacetimes. It may, however, happen ˜ 6= M , see [41] for examples. As we shall later assume that M = M ˜ , this that M possibility need not concern us. One can see from (4.2) and (4.3) that the regions ˜ is also F , P , R and L are invariant under the Killing flow (τt ). This implies that M invariant under (τt ). For open intervals (a, b) with a < b and a, b ∈ R ∪ {±∞}, we now define hA (a, b) := {ψA −1 (U, p) : a < U < b, p ∈ Σ} ;

(4.9)

with an analogous definition of hB (a, b). Notice that with this notation, hR A = hA (0, ∞) ,

hL A = hA (−∞, 0) .

(4.10)

The “shifted right wedge” can then be defined as Ra := R \ cl J − (hA (−∞, a))

(4.11)

for a > 0, where cl means “closure”. Lemma 4.1. τt (Ra ) = Reκt ·a

for all

t ∈ R, a ≥ 0 .

(4.12)

Proof. Since (τt ) is a group of isometries leaving R invariant, τt (Ra ) = τt R \cl J − (hA (−∞, a)) = τt (R) \τt (cl J − (hA (−∞, a))) = R \ cl J − (hA (−∞, eκt · a)) = Reκt ·a .

(4.13)

Similarly, setting L−a := L \ cl J + (hA (−∞, −a))

(4.14)

for a > 0 (!), we find as before that τt (L−a ) = L−e−κt ·a ,

t ∈ R, a > 0 .

(4.15)

In this section, a non-void open O ⊂ M is called a diamond if it is of the form O = int D(G) where G is an open subset of a Cauchy surface C (not necessarily


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acausal) such that ∂G is continuous and O⊥ non-void; moreover O or O⊥ is required to be connected. Below we study nets of von Neumann algebras indexed by the diamond regions in a given spacetime with bKh. Hence we would like the regions Ra and L−a to be diamonds. Our task is thus to verify this if χA and χB are chosen suitably. By assumption, there is an acausal Cauchy surface C passing through Σ. Let C1 be another acausal Cauchy surface lying strictly in the future of C, i.e. C1 ⊂ int J + (C) = J + (C)\C. Then we suppose that χA has been chosen such that each point q ∈ hA ∩ C1 has affine parameter U = 1, which means that q = ψ −1 (1, p) for some p ∈ Σ. Clearly such a choice is always possible (it amounts to a suitable choice of the smooth rescaling function φ : Σ → R). Under the Killing flow τt we get a family Ceκt := τt (C1 ), t ∈ R, of acausal Cauchy surfaces (not necessarily forming a foliation) having the property that each q ∈ hA ∩ Ceκt is represented as −1 κt (e , p) with suitable p ∈ Σ. Obviously, a similar construction can be carried q = ψA out with a Cauchy surface C−1 lying strictly in the past of C and leads to family of acausal Cauchy surfaces C−eκt = τt (C−1 ). (Moreover, similar constructions can be made for χB , hB .) As we first chose C1 and then adjusted χA to give all points of C1 ∩ hA affine parameter U = 1 it is not obvious that we can choose C−1 to give all points of C−1 ∩ hA affine parameter U = −1. It would suffice if there were a global isometry of M acting as a horizon-reflection symmetry j since then one may simply choose C−1 = j(C1 ). The existence of such an isometry will be required later, but not for the next lemma, where an arbitrary pair of Cauchy surfaces C1 and C−1 with the indicated properties is assumed given, and the corresponding vector fields (−) χA and χA assumed chosen so that each point on C1 ∩ hA has affine parameter U = 1 with respect to χA and each point on C−1 ∩ hA affine parameter U = −1 (−) with respect to χA . ˜ , then R⊥ = L, L⊥ = R and R, L and Ra , L−a , a > 0, are Lemma 4.2. If M = M diamonds. Proof. By assumption, we have M = F ∪ P ∪ R ∪ L, and F ∪ P = J(Σ). Since Σ is part of a Cauchy surface, it follows that Σ⊥ = int D(C\Σ). Hence R ∪ L = int D(C\Σ). Now define CR := C∩R, CL := C∩L. Then CR ∩CL = ∅ since L∩R = ∅ (see [41]), and CL ∪CR = C\Σ. Therefore we obtain int D(C\Σ) = int D(CR ∪CL ) = int D(CL ) ∪ int D(CR ) where the last equality is a consequence of the fact that CL and CR are disjoint open subsets of a Cauchy surface. The boundary of CL and CR is in both cases the smooth manifold Σ. Hence L = int D(CL ) and R = int D(CR ) are diamonds, and since CL and CR are disjoint and their union yields C up to the common boundary Σ of CL and CR , this entails R⊥ = L and L⊥ = R. Now we define the following sets: Σa := Ca ∩ hA , CaR := Ca ∩ R, CaL := Ca ∩ L, CaF = Ca ∩ F . One can see that Ca ∩ P = ∅, for there would otherwise be causal curves joining pairs of points on Ca and this is excluded. It follows that Ca = CaL ∪ CaR ∪ CaF is the union of three disjoint parts, and int D(CaR ) = (CaF ∪ CaL )⊥ . The common boundary of CaR and CaL ∪ CaF is the smooth man-


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ifold Σa , implying that int D(CaR ) is a diamond. Moreover, it is obvious that hA (a, ∞) ⊂ J + (Σa ), hA (−∞, a) ⊂ J − (Σa ), and by standard arguments it follows that J + (Σa ) = cl J + (hA (a, ∞)) and J − (Σa ) = cl J − (hA (−∞, a)). Let us check that Ra = int D(CaR ). First we notice that int D(CaR ) ⊂ R is fairly obvious (R is causally closed, i.e. R⊥ ⊥ = R, and CaR is an acausal hypersurface in R), and so is int D(CaR ) = (CaF ∪ CaL )⊥ ⊂ Σ⊥ a = M \J(Σa ), implying int D(CaR ) ⊂ Ra . To show the reverse inclusion it is sufficient to prove that Ra ∩ cl J(CaF ∪ CaL ) = ∅. We have cl J(CaF ∪ CaL ) = cl J(CaF ) ∪ cl J(CaL ) and CaL ⊂ L and R = L⊥ imply that R ∩ cl J(CaL ) = ∅. Now consider an arbitrary past-directed causal curve γ starting at some point on CaF . For γ to meet Ra , it must intersect hA . However, any intersection of γ with hA must be contained in hA (−∞, a] since γ is past-directed and we have seen that hA (a, ∞) ⊂ J + (Σa ) ⊂ J + (CaF ). Thus, since only the part of γ lying in the causal past of its intersection with hA can enter R, γ never meets Ra = R\cl J − (hA (−∞, a)), showing that cl J(CaF )∩Ra = ∅. Therefore Ra = int D(CaR ) is a diamond. An analogous argument works for L−a . 4.2. Conformal spin-statistics relation Our aim in this subsection will be to show that the net O 7→ A(O) on a spacetime with bKh induces a net of von Neumann algebras (a, b) 7→ C(a, b), indexed by the open intervals (a, b) of the real line and allowing an extension to a conformally covariant theory on the circle S 1 . Moreover, we shall see that this net is to be viewed as containing precisely the observables localized arbitrarily close to the hA -horizon. (A similar construction works for the hB -horizon). The variant of Wiesbrock’s results on modular inclusion [65] which is needed to show this may be familiar to experts, but for the reader’s convenience we present the arguments in an appendix to this chapter (Sec. 4.3). Earlier results [33, 34] on the spin-statistics connection for conformally covariant theories on S 1 then apply, yielding a conformal spin-statistics theorem for the subnets of the initial theory consisting of observables concentrated on the parts hA and hB of the horizon. We begin with a spacetime with a bKh, (M, g, τt , Σ, h), where we assume ˜ (cf. Sec. 4.1). Furthermore, we assume given a net henceforth that M = M K 3 O 7→ A(O) assigning to each member O in the collection K of regions in M a von Neumann algebra A(O) on a Hilbert space HA . For convenience, we shall work not with K, the collection of regular diamonds ordered under inclusion, but extend the domain of our observable net A in the canonical way to include a larger collection L of open subsets of our spacetime. As discussed in the appendix to Sec. 3, this choice does not change the superselection structure in that each representation satisfying the selection criterion based on K extends uniquely to a representation satisfying the selection criterion based on L, the intertwining operators thereby remaining unchanged. Again as discussed in the Appendix to Sec. 3, the formalism changes only in so far as the localized endomorphisms are now defined on larger al-


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gebras and this proves to be an advantage. We choose L to be the set of non-empty causally closed subsets S of M with non-empty causal complements such that for ˜ the given net A ⊥-duality holds either for S or for S ⊥ . By virtue of Lemma 3A.4, this is the same as the partially ordered set L defined in the Appendix to Sec. 3 in terms of sieves. We recall, too, that if A is additive, or even inner regular, as a net ˜ over K, then ⊥-duality coincides with ⊥-duality. Indeed, even though we assume duality for all diamonds, it is actually used only for two kinds of regions, the translated wedges La and Ra , and some tubular neighborhoods of the horizon intervals hA (a, b) or hB (a, b), which are in turn tubular neighborhoods in hA or hB of a suitable translation of Σ. We observe that the obstructions to duality are usually homological in nature, and that is why duality is generally assumed to hold for regular diamonds. On the other hand the surface Σ, even though not necessarily homologically trivial, is often relatively trivial, meaning that k-cycles in Σ which are trivial in M are trivial in Σ too. In the following we shall consider the subnet of O 7→ A(O) generated by the observables located arbitrarily close to the (half) horizon hA . Let us adopt the setting of Lemma 4.2 and start with a given acausal Cauchy surface C containing Σ and choose an acausal Cauchy surface C1 lying strictly in the future of C and the vector field χA so that each point on C1 ∩ hA has affine parameter U = 1. Then we define for 0 < a < b < ∞, \ R (a, b) := {A(O) : O ⊃ hA (a, b)} (4.16) BA O

where the intersection is taken over diamonds O. Likewise, one may also assume that another acausal Cauchy surface C−1 , lying strictly in the past of C, has been (−) selected and that another (possibly identical) copy χA of χA has been chosen to give each point of C−1 ∩ hA an affine parameter U = −1. Correspondingly, we set for −∞ < −b < −a < 0 ; \ L (−b, −a) := {A(O) : O ⊃ hA (−b, −a)}. (4.17) BA O

Finally, with these assumptions, one may also define \ BA (a0 , b0 ) := {A(O) : O ⊃ hA (a0 , b0 )},

(4.18)

O

for −∞ < a0 < b0 < ∞. Substituting B for A in the above, algebras BB (a, b), BB (a0 , b0 ) can be defined and all results formulated in the sequel for the algebras BA hold with obvious modifications for the algebras BB too. R/L

Lemma 4.3. Suppose that the net O 7→ A(O) satisfies the following assumptions: W (I) Irreducibility: O∈K A(O) = B(H). W S (II) Additivity: O ⊂ i∈I Oi , Oi , O ∈ K ⇒ A(O) ⊂ i∈I A(Oi ). (implying locality). (III) Haag duality: A(O⊥ ) = A(O)0 , O ∈ K


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Then R (a, b) = A(Ra ) ∩ A(Rb )0 BA L (−b, −a) BA BA (−a0 , b)

(4.19) 0

= A(L−a ) ∩ A(L−b ) 0

0

= A(L−a0 ) ∩ A(Rb )

(4.20) (4.21)

0

for all 0 < a < b < ∞, −a < 0. Proof. We shall only give the proof of the first equality, since the remaining cases are completely analogous, requiring some largely obvious notational changes. We recall that Ca = τln a/κ (C1 ) for any a > 0, and also the notation Σa = Ca ∩ hA , CaR = Ca ∩ R, CaF = Ca ∩ F and CaL = Ca ∩ L used in the proof e a := (Ra )⊥ = int D(CaL ∪ CaF ), of Lemma 4.2. Then we define the subsets L + − Fa := J (Σa ) and Pa := J (Σa ), and analogous sets with a replaced by b. Next, we define C ∨ := CaL ∪CaF ∪hA (a, b)∪CbR , and aim at demonstrating that this set is a Cauchy surface. It is fairly obvious that C ∨ is achronal, i.e. C ∨ ∩ int J ± (C ∨ ) = ∅. e a ∪ Rb ∪ Fa ∪ Pb where the sets forming It is also not difficult to check that M = L the union are pairwise disjoint except for the intersection Fa ∩ Pb = hA (a, b). Now e a or Rb , it must let γ be an arbitrary endpointless causal curve in M . If γ enters L ∨ intersect CaL ∪ CaF or CbR , hence C . Suppose that γ enters Fa . Since Fa is paste a or Pb , as γ would otherwise have compact, γ must intersect one of the regions Rb , L a past-endpoint. On the other hand, a causal curve without endpoint intersecting Fa can only meet Pb if it intersects hA (a, b), too. Hence, if γ enters Fa , it must also intersect C ∨ . Using the same argument with obvious modifications for the case that γ enters Pb , one arrives at the same conclusion. This shows that every causal curve without endpoints in M intersects C ∨ , implying M = D(C ∨ ), and therefore C ∨ is a Cauchy surface. Now we note that int D(U ) ⊃ hA (a, b) for each open neighbourhood U of hA (a, b) in C ∨ since J(hA (a, b)) = Pb ∪ Fa has empty intersection with cl(C ∨ \U ). Thus hA (a, b) is an intersection of diamonds. Moreover, whenever O ⊃ hA (a, b) is any diamond, it is obvious that we can find some open subset U of C ∨ with piecewise smooth boundary hA (a, b) ⊂ U ⊂ O ∩ C ∨ , implying hA (a, b) ⊂ int D(U ) ⊂ O. Hence, to establish the lemma, it suffices to consider diamonds of the form O = int D(U ). Obviously, the causal complement O⊥ of each such O may be written as ⊥ ⊥ ⊥ ⊥ ea = ∪ OL where OR = O⊥ ∩ Ra = int D(CaR \U) and OL = O⊥ ∩ L O⊥ = OR ⊥ ⊥ int D((CaL ∪ CaF )\U) are both diamonds. Notice that the union of OL and OR e a and Rb , respectively. Consequently we have over all O = int D(U ) yield L _ _ \ A(O) = ( A(O)0 )0 = ( A(O⊥ ) )0 O

O

=(

_

O ⊥ A(OL

∪

⊥ 0 OR ))

O

= (

_

⊥ ⊥ 0 A(OL ) ∨ A(OR ))

O

e a ) ∨ A(Rb ) )0 = ( A(Ra )0 ∨ A(Rb ) )0 = ( A(L = A(Ra ) ∩ A(Rb )0 ,


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where the second equality follows from Haag duality, the third has been justified above, the fourth and fifth equalities use additivity and the last but one again follows from Haag duality. The formulation of the subsequent result necessitates introducing further assumptions and related notation. W R R (a, ∞) := We shall write BA b>a BA (a, b), and define the other horizonalgebras associated with unbounded intervals in a similar manner by additivity. R R (0, ∞)Ω, (Ω) := BA Let Ω ∈ H be a unit vector vector, then we denote by HA L L HA (Ω) := BA (−∞, 0)Ω and HA (Ω) := BA (−∞, ∞)Ω the Hilbert subspaces generated by applying the various algebras of observables concentrated on the hA -horizon on that vector. We say that (B R (0, ∞), Ω) is a standard pair if Ω is separating for R (Ω). The B R (0, ∞). It is by definition cyclic with respect to the Hilbert subspace HA R modular objects (with respect to HA (Ω)) of such a standard pair will be denoted by JR,Ω , ∆R,Ω . The like objects for L in place of R are defined similarly. In the following, we shall focus attention on the next two assumptions: (IV) Geometric modular group on the horizon: There is a unit vector Ω ∈ H so that R (0, ∞), Ω) is a standard pair, and (i) (BA −it R R −2πt/κ a, ∞) , ∆it R,Ω BA (a, ∞) ∆R,Ω = BA (e

(4.22)

L (0, ∞), Ω) is a standard pair, and (ii) (BA −it L L 2πt/κ a) , ∆it L,Ω BA (−∞, −a) ∆L,Ω = BA (−∞, −e

(4.23)

for all a > 0, t ∈ R, where κ > 0 is the surface gravity of the bKh. (V) Geometric modular conjugation on the horizon: For the Ω as in (IV), we have R L (Ω) = HA (Ω) and moreover HA (Ω) = HA R L (a, ∞)JR,Ω = BA (−∞, −a) , JR,Ω BA

a ≥ 0.

(4.24)

Let us now assume that the net O 7→ A(O) satisfies assumptions (I–IV). Thus R R L (1, ∞) ⊂ BA (0, ∞), Ω) is a +hsm inclusion and (BA (−∞, −1) ⊂ we see that (BA L BA (−∞, 0), Ω) is a -hsm inclusion. Then the results of [65, 1] yield two continuous unitary groups U R/L (a), a ∈ R, having positive/negative spectrum and satisfying the following relations for a > 0: R it R 2πt a) , ∆−it R U (a)∆R = U (e

−it L L 2πt ∆it a) , L U (a)∆L = U (e

JR U R (a)JR = U R (−a) ,

JL U L (a)JL = U L (−a) ,

R R (0, ∞)U R (−a) = BA (a, ∞) , U R (a)BA L L (−∞, 0)U L (−a) = BA (−∞, −a) , U L (a)BA

where we have dropped the index Ω on the modular objects to simplify notation. R and Without further assumptions, U R and U L are unrelated and so are the nets BA


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L BA . However, if we suppose that (V) holds, too, then it follows from the way these unitaries are constructed (cf. [65]), that JR U R (a)JR = U L (a), a ∈ R. Therefore we obtain the following:

Corollary 4.4. Under assumptions (I–IV) the nets of horizon-algebras indexed by the intervals of the half real lines, R (a, b) , (a, b) 7→ BA

0 < a < b < ∞,

L (−b, −a) , (−b, −a) 7→ BA

−∞ < −b < −a < 0 ,

extend to local conformal nets I 7→ MR (I) and I 7→ ML (I) of von Neumann R (a, b)Ω and HL = B L (−b, −a)Ω, algebras on S 1 on the Hilbert spaces H0R = BA 0 0 respectively (where the 0 < a < b < ∞ are arbitrary). If (V) is assumed, too, then the net (a0 , b0 ) 7→ BA (a0 , b0 ) on the full real line extends to a local conformal net I 7→ M(I) on HA (Ω). Proof. The first part is a variant of Wiesbrocks’s result [65, 66], cf. also [34]. We supply the relevant argument as Proposition 4A.2 in Sec. 4.3. If assumption (V) is added so that JR intertwines U R and U L , the adjoint action of U R (a) on the net BA is geometrically correct, i.e. U R (a)BA (a0 , b0 )U R (−a) = BA (a0 + a, b0 + a), a ∈ R, a0 < b0 . Thus the net BA together with its dilation and translation symmetries coincides with both C R and C L (derived from the nets R L and BA as in Proposition 4A.2) and their respective translation and dilation BA symmetries. Thus the corresponding extensions to conformally covariant theories coincide. Condition (IV) may be viewed as a weak form of the Hawking–Unruh effect: an observer moving with the Killing flow of the bKh registers a thermal ensemble in the “vacuum” state (see [57, 63]). The term “vacuum” here means a state invariant under the space-time isometries and fulfilling additional stability conditions, in fact (IV) and (V) may be viewed as a weak form of such conditions, namely applying to the subsystem of observables concentrated on the horizon. As the group of affine translations along the geodesic generators of the horizon has positive generator derived from the modular inclusion of horizon-algebras, Ω can be justly interpreted as a vacuum vector for the horizon-algebras (cf. the principle of geometric modular action [17] or modular covariance [15]). Clearly, if Ω induces a KMS-state for the Killing flow at the Hawking temperature on A(R), then (IV, i) follows by Lemma 4.3. Likewise, if Ω induces a KMS-state for the Killing flow at negative Hawking temperature on A(L), then (IV, ii) follows by Lemma 4.3. The motivation for Condition (V) is that, a horizon (or wedge) reflection symmetry should be implemented in a “vacuum” representation by the modular conjugations JR , in analogy with the Bisognano–Wichmann result for quantum fields in Minkowski space 4, 5, 6, 57. Our condition is actually a bit weaker in that JR need not implement a point-transformation of the underlying spacetime manifold.


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However, Condition (V) implicitly imposes a relation between the horizon segments hA (−∞, −a) and hA (a, ∞). We finally comment on whether these assumptions are realistic. For the free scalar field, conditions (I, II) hold generally in representations induced by quasifree Hadamard states (for O ∈ K based on relatively compact subsets of Cauchy surfaces, and, in more special cases, even when the base is unbounded), see [60]. The Hartle–Hawking state, i.e. the candidate for the “vacuum” state of the free scalar field on the Schwarzschild–Kruskal spacetime, should also satisfy all the assumptions [38, 41] ((III) has not been checked in the generality formulated here, but a version of (III) sufficient to imply the spin-statistics theorem in the sequel does hold). As is known from the Bisognano–Wichmann result [4], the assumptions are fulfilled for local von Neumann algebras generated by (finite-component) Wightman fields in Minkowski spacetime ((III) then holds for wedge-regions and this suffices to establish the spin-statistics relations [32, 43]). Results of Borchers [5, 6] yield (III–V) generally for algebraic quantum field theories in two spacetime dimensions. With additional conditions these generalize to higher dimensions [7, 8, 67]. Now we can formulate the conformal spin and statistics theorem. Our aim is to define the spin of a sector as the conformal spin on the horizon. To this end we need to restrict to considering sectors that are horizon localizable, namely having a representative which acts trivially on the algebras BA (a, b)0 for some a, b ∈ R (or the same for the B horizon). However this is not sufficient in general because the sector on the horizon may not be covariant. As shown in [34] covariance of localized endomorphisms with finite statistics is automatic when the net is strongly additive, which is always the case for the dual net. Unfortunately extending a sector on a conformal net to a sector on the dual net may produce soliton sectors. Therefore we shall only consider those sectors which are not only horizon localizable, but also dual localizable, namely which give rise to a localized sector on the dual net of the horizon conformal net. Clearly if we have a dual localizable sector on the net O 7→ A(O) satisfying assumptions (I–V) with non-zero statistical parameter λ, we obtain a covariant sector on the dual net on the horizon with the same statistical parameter, since this is determined by the intertwiners. The following theorem is now a simple consequence of the conformal spin and statistics theorem in [33]. Theorem 4.5. Let O 7→ A(O) be a theory on a spacetime with bKh satisfying assumptions (I–V) and ρ a dual localizable sector with finite statistics. Then ρ gives rise to a covariant sector on the dual net on the horizon, therefore a conformal spin sρ is defined, and the conformal spin and statistics relation holds, namely sρ = κρ . To conclude Sec. 4.2, we give some remarks. The idea of passing from a quantum field theory initially formulated over a fourdimensional spacetime to observables concentrated on a lightlike hypersurface (i.e. pieces of a bKh) is not a new one and once was popular in quantum field theory under the keyword “infinite momentum frame”. [44, 56] are just two references in this direction. This matter is studied for the first time in the operator algebraic


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framework in [26]. One motivation is that symmetries may be enhanced by restricting to a subtheory concentrated on a lightlike hypersurface, a particularly attractive possibility for quantum field theory in curved spacetimes where symmetries of the underlying four-dimensional spacetime are rather limited. As proved in this section, for bKh spacetimes restricting to the horizon does indeed give conformally covariant nets. Sewell [57] was the first to observe that this allows one to formulate a Bisognano– Wichmann theorem relating to the Hawking effect for quantum fields on blackhole spacetimes, in the setting of a Wightman field theory (see e.g. [63] for further discussion). In this context, two papers rigorously established related results for free field models [38, 22]. Kay and Wald [41] realized that such results may be generalized to spacetimes with a bKh and obtained strong theorems for free fields in this setting. An operator-algebraic version of aspects of Sewell’s work appears in [58] where the nets BA (a, b) are used. We ought to mention that in general it is not very clear how “big” the algebras BA (a, b) (or BB (a, b)) are in the original algebras A(O). If Ω is cyclic for BA (a, b) then it is reasonable to expect that sectors are horizon localizable (on the A-horizon). Moreover in this case the conformal net on the horizon is strongly additive by definition, therefore it coincides with its dual net (cf. [34]), and then horizon-localizability and dual localizability are equivalent. It is known that Ω is cyclic for BA (a, b) when free fields on the n-dimensional Minkowski space are considered, n 6= 2. We give here a simple argument based on [16]. By a “free field” on Minkowski space we here mean a local net A of von Neumann algebras indexed by regions of Minkowski space which can be constructed by second quantization from a net K of real vector spaces in a complex Hilbert space H, plus the usual assumptions of Poincaré covariance, positive energy, and in particular the Bisognano–Wichmann property and irreducibility: ∩W A(W ) = CI. Working in the first quantization space H from now on, we first observe that irreducibility means ∩W K(W ) = {0} and, by the Bisognano–Wichmann property, this is equivalent to there being no fixed vectors for the action of the Poincaré group on H. Then, by a theorem of Mackey (cf. e.g. [69], Proposition 2.3.5), the absence of invariant vectors for the whole Poincaré group is equivalent, when n 6= 2, to the absence of invariant vectors for any given translation, hence the spectrum of the generator of any light-like translation is strictly positive, i.e. zero is not an eigenvalue. Now, given two wedges W1 , W2 , the cyclicity of the vacuum vector Ω for A(W1 )∩ A(W2 ) is equivalent to (K(W1 ) ∩ K(W2 )) + i(K(W1 ) ∩ K(W2 )) being dense in H, this being in turn equivalent to having {v ∈ dom(sW1 ) ∩ dom(sW2 ) : sW1 v = sW2 v} dense in H, where sWj denotes the “first quantized” Tomita operator defined by sWj (χ + iφ) := χ − iφ, χ, φ ∈ K(Wj ), [16]. When W1 = {(t, x) : x1 > |x0 |} and W2


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is a translation of the causal complement of W1 , W2 = {(t, x) : x1 − c < −|t − c|}, c > 0, the situation met when considering the vector space associated with the interval (0, c) on the A-horizon, this is in turn equivalent, again by the Bisognano– Wichmann property, to the density of the space 1/2

1/2

1/2

1/2

{v ∈ dom(δ1 T (c)δ1 ) : T (c)δ1 T (c)δ1 v = v},

(4.25)

where a → T (a) denotes the representation of the light-like translations along the Ahorizon and δ1 denotes the “first quantized” modular operator for the space K(W1 ). This property clearly depends only on the restriction of the representation of the Poincaré group to the subgroup P1 generated by boosts and light-like translations with strictly positive generator (relative to the wedge W1 ). As the logarithm of the generator of translations and the generator of the boosts give rise to (and are determined by) a representation of the CCR in one dimension, the strictly positive energy representations of P1 have a simple structure: they are always a multiple of the unique irreducible representation. Therefore the density of the space in Eq. (4.25) holds either always or never, and hence can be checked in the irreducible case. But this is the case of the current algebra on the circle, where cyclicity holds by conformal covariance. Of course, the vector Ω is not expected to be cyclic in general for the algebra generated by the BA (a, b), and it might even happen that BA (a, b) contains only multiples of the identity. Field nets giving rise to non-trivial superselection sectors of the observable net localizable on the horizon can easily be constructed just by requiring the vacuum to be cyclic for the horizon field algebras. However it is not clear, in general, how strong the requirement of dual localizability is. 4.3. Appendix to Chapter 4 For the benefit of the non-expert reader, we present in detail in this Appendix the arguments leading from the results in [34, 65, 66] to Corollary 4.5. To begin with, we state a result about modular inclusions needed in the following. Lemma 4A.1. Let (N ⊂ M, Ω) be a pair of von Neumann algebras with a unit vector Ω cyclic and separating for M and such that ∆it N ∆−it ⊂ N for all −t ≥ 0 (or t ≥ 0), where ∆it , t ∈ R, is the modular group of M, Ω. Then M = ∨t∈R ∆it N ∆−it if and only if Ω is cyclic for ∆it N ∆−it for some (hence for any) t ∈ R. Proof. If Ω is cyclic for ∆it N ∆−it for a given t, then it is cyclic for ∨t∈R ∆it N ∆−it , too. However this von Neumann algebra is invariant under the modular group of M, and hence coincides with M by Takesaki’s theorem. Conversely, let ξ be orthogonal to ∆it N ∆−it Ω. Then for any x ∈ ∆it N ∆−it we have xΩ ∈ dom(∆1/2 ), hence the function z 7→ (∆iz xΩ, ξ) is analytic on the strip −i/2 < =z < 0 and continuous on the boundary. But as we have a +hsm inclusion, it vanishes for negative real z and hence everywhere. Thus ξ is orthogonal to ∨t∈R ∆it N ∆−it Ω = MΩ, completing the proof.


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Proposition 4A.2. Let (N ⊂ M, Ω) be a a pair of von Neumann algebras with a unit vector Ω which is cyclic and separating for M and such that ∨t∈R ∆it N ∆−it = M and ∆it N ∆−it ⊂ N for all −t ≥ 0, where ∆it , t ∈ R, is the modular group of M, Ω. Then, setting H0 = (N ∩ (∆−i N ∆i )0 )Ω C(a, b) = (∆−i

log a 2π

N ∆i

log a 2π

(4.26)

) ∩ (∆−i

log b 2π

N ∆i

log b 2π

)0 H0 ,

0 < a < b , (4.27)

the family (a, b) 7→ C(a, b) extends to a local conformal net of von Neumann algebras acting on the Hilbert space H0 . Proof. Set Na = ∆−i

log a 2π

N ∆i

log a 2π

,

a > 0.

By the previous lemma Ω is cyclic for Na , a > 0, therefore we may apply a result of Wiesbrock and Araki–Zsido ([1, 65]) to the +hsm inclusion (N ⊂ M, Ω) and get a one parameter group of unitaries U (a) on H with positive generator satisfying ∆−it U (a)∆it = U (e2πt a) JU (a)J = U (−a) . Hence we have Na = U (a)MU (a)∗ ,

a ≥ 0,

and this equation is used to define Na for negative a. We now set C(a, b) = Na ∩ Nb0 H0 , C(−∞, b) = ∨aa C(a, b) ,

−∞ < a < b < +∞ −∞ < b < +∞ −∞ < a < +∞

and the definition of C(a, b) clearly agrees with (4.26) when 0 < a < b < ∞. Furthermore, H0 = N ∩ Ne02π Ω = C(1, e2π )Ω. Moreover, the operators J, ∆ restricted to H0 give the modular conjugation and operator of (C(0, ∞), Ω). Similarly, using the results of [65, 1] anew, the restriction of U (a) to H0 (again denoted by U (a)) coincides with the unitary group derived from the +hsm inclusion (C(1, ∞) ⊂ C(0, ∞), Ω). Now a standard Reeh–Schlieder argument, based on the positivity of the generator of U (a), shows that C(−∞, b)Ω is independent of b, while the “modular” Reeh–Schlieder argument in Lemma 4A.1 shows that C(a, b)Ω is independent of a ∈ (−∞, b). Thus the inclusion (C(1, ∞) ⊂ C(0, ∞), Ω) is standard. We have proved that H0 = C(a, b)Ω for any −∞ ≤ a < b ≤ +∞, and that C gives a translation-dilation covariant net of von Neumann algebras on H0 . Then we get a conformally covariant net by a result of Wiesbrock ([66], see also [34]).


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5. The Spin and Statistics Relation for Spacetimes with Rotation Symmetry In this section, we present a proof of the spin and statistics relation for superselection sectors on a globally hyperbolic spacetime with some rotational symmetry. The main assumption here is the existence of a suitable family of regions, called wedges, each being equipped with a reflection mapping it to its causal complement and of a net of von Neumann algebras with a common cyclic vector whose modular conjugations implement the said reflections, in the spirit of [18] and [46]. Moreover we assume the existence of rotational spacetime symmetries, rotating a wedge to its causal complement and belonging to the commutator of the spacetime symmetry group. As we shall see, our geometric assumptions are satisfied in many interesting spacetimes and form the geometric basis for the rotational spin and statistics theorem, explained in more detail below. 5.1. Geometric assumptions A spacetime with rotation and reflection symmetry is a quadruple (M, W, G+ , j), where M is a globally hyperbolic spacetime, W is a family of open subregions called wedges, G+ is a Lie group of proper (i.e. orientation preserving) transformations of M and j is a map from W to the antichronous (i.e. time reversing) reflections in G+ ; we write it as W 7→ jW . We denote the orthochronous subgroup of G+ by G↑+ and the identity component of G+ by G0 . The universal covering of G0 is denoted e The e The Z2 action implemented by any jW on G0 lifts to an action on G. by G. quadruple has to satisfy the following properties: (a) j leaves W globally invariant and verifies jW (W ) = W ⊥ and jgW = gjW g −1 , W ∈ W, g ∈ G+ . (b) There is a W ∈ W and an element h in the Lie algebra of G0 such that (1) (2) (3) (4)

exp(2πh) is the identity in G0 , jW exp(th)jW = exp(−th), exp(πh)W = W ⊥ , T 0≤t≤π/2 exp(th)W is non-empty.

(c) h belongs to the commutator of the Lie algebra of G0 . f=W f and j f W = f are called orthogonal if jW W Remark 5.1. Two wedges W , W W π W . It is easy to see that W and exp( 2 h)W are orthogonal. Indeed, making use of assumptions (b 2) and (b 3), we get π π π h W = exp − h jW W = exp − h W ⊥ jW exp 2 2 2 π π h W. = exp − h exp(πh)W = exp 2 2 The second equation is proved analogously.


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We shall also consider spaces where property (c) is replaced by the following property: f , orthogonal to W , such that j f commutes with (c0 ) There exists a wedge W W exp(th). Remark 5.2. (i) Assumption (a) has to be seen as a part of the definition of a wedge. The first part of property (a) says that any wedge is G+ -equivalent to its causal complement, hence a wedge is in some sense “a half” of M or, more precisely, is the causal completion of “a half” of a Cauchy surface. The second part means that jW commutes with the stabilizer of W and, when G+ acts transitively on W, says that j is determined by its value on one wedge. (ii) Properties (b) describe the rotation symmetry. In view of property (b 1) we call the group elements exp(th) rotations. Property (c) ensures that all characters e are trivial on the cycle {exp(th), t ∈ [0, 2π]}, since the latter belongs to of G e where all characters are trivial. As we shall the commutator subgroup of G see, this makes the spin well defined. (iii) The element jW , seen as an automorphism of the Lie algebra of G0 , has eigenvalues 1 and −1 and by (a) the eigenspace corresponding to 1 consists of generators of transformations preserving W . Therefore (b 2) essentially says that not all rotations preserve W . More precisely, W may be rotated to its spacelike complement by (b 3). (iv) Property (b 4) mainly expresses the fact that 2π is the minimal period of the one-parameter group exp(th). (v) Property (b) is stated for one wedge W , but then holds for any wedge in the family W0 := {gW : g ∈ G+ }. We are of course interested in the case where the cycle {exp(th), t ∈ [0, 2π]} is not homotopy trivial and hence gives rise to a non-trivial notion of spin. However this is not needed for the proof of the spin and statistics theorem nor do we require that the exp(2πh) generate the homotopy group of G0 . f are mutually orthogonal. It (vi) Property (c0 ) implies that W , eπ/2h W and W ↑ also implies that rW := jW jW f is an involution in G+ and that exp(2th) = [exp(th), rW ], where the square brackets here denote the multiplicative commutator. As a consequence, the rotations exp(th) belong to the commutator subgroup of G↑+ . In this sense (c0 ) is a weak form of (c). G↑+ and G0 do not always coincide. Of course rW ∈ G↑+ , but we do not require that rW belongs to G0 . (vii) Property (b) fixes the the generator h up to a sign, indeed (b 1) fixes the generator up to an integer, (b 3) implies this integer to be odd, and (b 5) requires this integer to be 1 or −1. When the spacetime is two-dimensional, i.e. when the Cauchy surface is 1-dimensional, the orientation fixes a direction on any spacelike curve (from left to right). In this case we choose the sign in such a way that the element h generates a rotation in the prescribed direction.


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5.1.1. Assumptions (a), (b), (c) and (c0 ) in some spacetimes In the case of the n-dimensional Minkowski spacetime M n , a wedge is any G+ transform of the region W = {|x0 | < x1 } if n > 2, and of the region {x > 0} if n = 1. Taking jW to be the reflection w.r.t. the edge of the wedge, the map j turns out to be uniquely defined by property (a). When G+ is the proper Poincaré group and n ≥ 3, property (b) holds with W as above and h as the generator of rotations in the (x0 , x1 )-plane. Indeed the proper orthochronous Poincaré group is perfect, hence property (c) is obviously satisfied. f = {|x0 | < x2 }. If n ≥ 4 then (c0 ) is satisfied too, with W When G+ is the proper conformal group, properties (b) and (c) are satisfied for any n ≥ 1, h being the generator of a suitable group of (conformal) rotations. Property (c0 ) is satisfied when n ≥ 3, W being as before, h being the generator of f a double cone with spherical basis centred on rotations in the (x0 , x1 )-plane and W the origin. Since the n-dimensional de Sitter spacetime Dn may be defined as the hyperboloid x20 + 1 = |x|2 in M n+1 , the wedges can be defined as the intersection of this hyperboloid with the wedges in M n+1 whose edge contains the origin. Then properties (b), (c) or (c0 ) hold for Dn if and only if properties (b), (c) or (c0 ) hold for M n+1 (with Poincaré symmetry), respectively. Note that the Cauchy surface of Dn is compact and the same is true for M n with conformal symmetry, since in this case the quantum field theories actually live on (a covering of) the Dirac–Weyl compactification of M n (cf. [14]). Whenever the spin makes sense in the above examples, i.e. whenever (c) or (c0 ) holds, the group G↑+ has no non-trivial finite dimensional representations, a much stronger requirement than (c) or (c0 ). In this case the spin and statisitics relation may be proved as in [46]. Moreover, in these examples, modular covariance makes sense, i.e. there is a natural definition of the geometric action of ∆it , furthermore, the Bisognano– Wichmann property has been proved for Wightman fields ([4, 10]), wedges separate spacelike points and every double cone is an intersection of wedges. Therefore geometric modular conjugation follows from modular covariance (as in [32], cf. [18]) and modular covariant free fields may be constructed canonically as in [16] by second quantizing (anti-)unitary representations of G+ . We now describe a class of spacetimes where these additional features do not hold, namely where the group admits one-dimensional representations and the wedges do not separate points. Nevertheless, these cases are still covered by the spin and statistics theorem we are going to present below. 5.1.2. Spherically symmetric black holes We call spherically symmetric black holes those spacetimes (K, gK ) whose structure is very similar to the Schwarzschild–Kruskal spacetime, i.e. they are isometric to X × S n , X being the set of points (x0 , x1 ) ∈ R2 with x20 − x21 < µ2 , µ ∈ R ∪ {∞},


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with the metrich ds2K = a(x20 − x21 )(dx20 − dx21 ) − b(x20 − x21 )d2 σ , where d2 σ is the usual Riemannian metric on the sphere S n and a and b are smooth, strictly positive functions. Then the hypersurface x0 = 0 is a Cauchy surface and (K, gK ) is globally hyperbolic. The structure of such spacetimes is in some respects similar to that of Minkowski spacetime. For instance, if points in X × S n are represented as (x0 , x1 , σ), then one may define a one-parametric group of isometries Λt , t ∈ R, by replacing the pair (x2 , x3 ) ∈ R2 by σ ∈ S n in (4.1) and then define Σ and hA and hB , correspondingly. Hence (K, gK ) has the structure of a spacetime with bKh, where the Killing flow is τt = Λt , t ∈ R. Moreover, there is a horizon reflection j(x0 , x1 , σ) = (−x0 , −x1 , σ) which is a PT symmetry, i.e. an orientation and chronology-reversing isometry. Let us investigate further the isometries of such spacetimes. To simplify the matter a bit, we assume that (K, gK ) does not admit translations in the X-part of K = X × S n as symmetries. (This is not really a restriction; our findings can be modified by taking the semidirect product of the translational symmetry group TX with the non-translational symmetry group G in the presence of such symmetries. For our treatment of the connection between rotational spin and statistics, translational symmetries are irrelevant.) Since (K, gK ) is orientable and time-orientable, we consider the groups G+ and G↑+ of proper (i.e. orientation preserving) and proper orthochronous (i.e. time-orientation preserving) isometries, respectively. In the following, we describe the proper orthochronous subgroup G↑+ . The form of the metric tensor gK and the assumed triviality of TK imply that all isometries leave Σ globally fixed and that an elementof G↑+ acting trivially on Σ has to be an element of the Killing flow. Conversely, orientation preserving isometries of Σ, i.e. elements of SO(n + 1), naturally give rise to symmetries in ˆ 0 , x1 , σ) = (x0 , x1 , Rσ), R ∈ SO(n + 1), gives an isometry of G↑+ . Indeed, R(x (K, gK ). To extend orientation reversing isometries of Σ to orientation preserving isometries of K, we obviously need a different procedure. To this end we note that each orientation reversing isometry of Σ ≡ S n can be written as a product of a rotation in SO(n + 1) and an equatorial reflection rQ , where Q denotes the S n−1 equator of fixed points of such a reflection. More precisely, rQ reflects points on S n about Q along the great circles orthogonal to the equator Q. In other words, rQ acts as a reflection of the normal geodesic spray of Q in S n . Note that such equatorial reflections generate the action of O(n + 1) on S n . In fact, if an equator Q1 is inclined at angle φ to an equator Q2 , then rQ1 rQ2 is a rotation by 2φ about the axis defined by the intersection of Q1 and Q2 . Now choosing a normalized, timelike, future-oriented, rotation-invariant normal vector field ξ0 along Σ there is a unique normalized, spacelike, outward-oriented, rotation invariant normal vector field ξ1 along Σ such that ξ0 + ξ1 is parallel to the h It

is customary to write coordinate indices as upper indices, but our deviating from this convention is unlikely to cause confusion.


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vector field χA . It is therefore equivalent to choosing an orthonormal frame on the X-component of K = X × S n . Moreover the Killing flow acts transitively on the set of such possible choices. An equatorial reflection rQ extends to an orientation and chronology-preserving isometry rˆQ,0 ∈ G↑+ by setting rˆQ,0 := (x0 , −x1 , rQ σ) and we define rˆQ,t := ˆQ,t ∈ G↑+ is an involution. Λt rˆQ,0 Λ−1 t , where Λt , t ∈ R, is the Killing flow. Each r On the other hand, by the above observation, each involution in G↑+ restricting to some rQ on Σ must be of the form rˆQ,t for some t ∈ R. Clearly, rˆQ,t determines a unique normalized, spacelike, outward-oriented, rotation invariant normal vector field ξ1 along Σ which is anti-invariant under rˆQ,t . Thus G↑+ is generated by the Killing flow, the (extensions of the) orientation preserving isometries of Σ and the reflections rˆQ,t so that G↑+ ≡ (R × SO(n + 1)) ×σ Z2 , where σ denotes the conjugation by rˆQ,0 , for some given equator Q. Consequently G0 ≡ (R × SO(n + 1)) and G+ , being generated by G↑+ and the horizon reflection j, is isomorphic to (R × SO(n + 1)) ×σ Z2 × Z2 . The following lemma obviously holds. Lemma 5.3. On a spherically symmetric black hole, the commutator subalgebra of the Lie algebra of the identity component G0 of the group of proper isometries is isomorphic to so(n+1). The commutator subgroup of G0 is isomorphic to SO(n+1). We now show that the reflection symmetries rˆQ,t are naturally associated with wedge-like subregions of K. Indeed, given a normalized, spacelike, outward-oriented, rotation-invariant normal vector field ξ1 along Σ, its (two-sided, maximally extended) geodesic spray gives a geodesic-foliated Cauchy surface containing Σ, and it is easy to see that all such Cauchy surfaces arise in this way. Therefore, given a reflection rˆQ,t and an open hemisphere E in Σ ≡ S n with ∂E = Q, we may consider the open causal completion W (E, t) of the part of the Cauchy surface generated by the spacelike vectors determined by rˆQ,t and based on E. Put difˆ0 Λ−1 ˆt := Λt E ˆ0 := {(0, x1 , σ) : x1 ∈ R, σ ∈ E} and E ferently, defining E t , ˆt ) where D(Eˆt ) is the domain of dependence of t ∈ R, then W (E, t) = int D(E ˆt . We also mention that the edge of the wedge W (E, t) is the spacelike cylinE der generated by the geodesic spray of the vectors of ξ1 based on ∂E, i.e. the set Λt {(0, x1 , σ) : x1 ∈ R, σ ∈ ∂E}. Hence each W (E, t) is a diamond. The set of such wedge-regions will be denoted by W0 . The following proposition immediately follows. Proposition 5.4. (i) W (E, t)⊥ = rˆ∂E,t W (E, t) = W (E 0 , t), where E 0 denotes the interior of the complement of E. ˆ (E, t) = W (RE, t), for any R ∈ SO(n + 1). (ii) RW (iii) Λs W (E, t) = W (E, s + t). (iv) The group G↑+ acts transitively on the family W of wedges W (E, s). (v) The group G↑+ is generated by the reflections rˆQ,t .


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Now we show that these spacetimes fit in the scheme proposed at the beginning of this section. Let us define W as W0 ∪ {R} ∪ {L}, jR = jL as the horizon reflection j and jW (E,t) = jR rˆ∂E,t . Proposition 5.5. If n ≥ 2 then properties (a), (b), (c) and (c0 ) hold. If n = 1 then properties (a), (b) and (c0 ) hold. Proof. Proposition 5.4 immediately gives (a). Then let W = W (E, t) and choose h ∈ so(n + 1) as an eigenvector with eigenvalue −1 of jW (E,t) , normalized in such a way that exp(ϑh) is a rotation through an angle ϑ. Then property (b) is obviously f = R we get property (c0 ). When n ≥ 2, (c) follows by satisfied and choosing W Lemma 5.3. 5.2. Quantum field theories on spacetimes with rotation symmetry Now we consider a net O 7→ A(O) of von Neumann algebras indexed by elements O ∈ K ∪ W where K is the set of regular diamonds and W is a set of wedges with the properties discussed in the previous section; this net describes the observables of a local quantum theory on M . We require irreducibility, additivity and Haag duality as in assumptions (I–III) of Sec. 4.2 and, moreover, (VI) Reeh–Schlieder property: There exists a unit vector Ω (vacuum) cyclic for the von Neumann algebras associated with all wedge regions. (VII) Geometric modular conjugation: JW A(O)JW = A(jW O) , where O is any regular diamond and JW denotes the modular conjugation associated with the algebra A(W ) and the vector Ω, cf. Sec. 4.2. (VIII) Covariance: There exists a unitary representation U of the group G↑+ such that U (g)Ω = Ω for any g ∈ G↑+ , U (g)A(O)U (g)∗ = A(gO) for any g ∈ G↑+ and any regular diamond O and JW U (g)JW = U (jW gjW ) for any wedge W. Let us note that, under the previous hypotheses, the representation U extends to an (anti)-unitary representation of G+ with a geometric action on the net verifying U (jW ) = JW . Proposition 5.6. Under the above assumptions, the net satisfies duality for the ˆ namely relation ⊥, \ A(O1 )0 A(O) = ˆ O1 ⊥O

ˆ if O1 ⊥ O and ∃O2 ∈ K : O1 , O ⊂ O2 . where (cf. Appendix to Sec. 3) O1 ⊥O


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˜x ∈ K Proof. Let O1 ⊥ O. By Lemma 2.1ainer, for any x ∈ O1 there exist Ox , O ˆ ˜ such that O ⊥ Ox and O, Ox ⊂ Ox , in particular O⊥Ox . Then \ \ \ A(Ox )0 = A(O1 )0 = A(O) A(O) ⊆ O1 ⊥O x∈O1

O1 ⊥O

where the first equality follows by additivity and the second by duality. (IX) equivalence of local and global intertwiners: Given a representation π satisfying the selection criterion and localized in a wedge W , let ρW denote the associated endomorphism of A(W ), then (π, π) = (ρW , ρW ) . Remark 5.7. (i) This assumption implies factoriality for the algebras associated with wedge regions, that irreducibility of representations coincides with irreducibility on a wedge and that the equality (π, π 0 ) = (ρW , ρ0W ) holds for pair of representations (see [33]). (ii) Assumption (IX) has been shown to follow from dilation invariance [55], and it is conjectured that it already follows from the existence of a non-trivial scaling limit. We give an explicit proof of its validity for Minkowski space of any dimension in the Appendix to this section. (iii) If we assume G↑+ to be continuously represented by automorphisms αg , G+ to be generated by {jW , W ∈ W} and Ad JW1 JW2 = αg , with g = jW1 jW2 , we get covariance (VIII). Moreover we obtain algebraic covariance for any sector with ↑ uger [49], finite statistics, namely ρ ' αg ρα−1 g , g ∈ G+ . By an argument of M¨ this implies that any sector is covariant w.r.t. a continuous representation of a central extension of G↑+ . When the wedges separate spacelike points, i.e. regular diamonds are intersections of wedges, geometric modular conjugation (VII) also follows (cf. [18]). 5.2.1. Spin and Statistics under property (c) Theorem 5.8. Let π be a representation satisfying the selection criterion and localized in O ⊂ W. Suppose the associated endomorphism ρW of the von Neumann algebra of the wedge W has finite index. Let j be the antilinear morphism implemented by the modular conjugation of (A(W ), Ω). Then j · π · j is a conjugate of π and π has finite statistics. Remark 5.9. To inclusions of von Neumann algebras one can assign an invariant, a positive number called the index (cf. [45] and refs. cited there). The index of the endomorphism ρW is that assigned to the inclusion ρW (A(W )) ⊂ A(W ). For discussion of the relation between the statistical dimension of a superselection sector in quantum field theory in Minkowski spacetime and the index of its associated localized endomorphisms, the reader is again referred to [45].


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Proof. Pick a representation π 0 equivalent to π and localized in W ⊥ . Then arguing as in [31], we see that jπj and π 0 yield conjugate endomorphisms of the von Neumann algebra of the wedge W ⊥ . The next step is to deduce from Assumption IX that jπj and π are conjugate representations. This circumstance is obscured by the fact that the product even of localized representations is defined only up to equivalence. For this reason, we use cocycles from Zt1 (A) instead of representations, recalling Theorem 3A.5. We have a faithful tensor ∗ -functor Zt1 (A) → T (a) taking a cocycle z into the associated endomorphism y(a) in a and an arrow t into ta . If a ⊂ W , then there is a tensor ∗ -functor from T (a) into the category of endomorphisms of the von Neumann algebra of the wedge W , mapping an object ρ onto its restriction to the algebra of the wedge ρW and acting as the identity on arrows. Assumption IX means that the composition of these functors is even full. Thus if y(a)W and y¯(a)W are the images of z and z¯ and are conjugates, z and z¯ are conjugates. If z is a cocycle associated with π 0 and z¯ is a cocycle associated with jπj, then the endomorphisms of A(W ⊥ ) obtained by restriction are conjugates and so are the equivalent endomorphisms y(a)W and y¯(a)W . Hence z has a left inverse and finite statistics. By assumption, JW implements a spacetime reflection consisting of a time reversing (since JW is anti-unitary) and a space reversing transformation since, preserving the overall orientation, it has to reverse the orientation of any globally invariant Cauchy surface. Therefore the previous theorem is indeed a PCT theorem. In the following we choose a rotationally symmetric spacetime (M, W, G+ , j) satisfying properties (a), (b) and (c), a local net O 7→ A(O) verifying the above e assumptions and an irreducible, G-covariant, superselection sector with finite statistics. If π is a representation obeying the selection criterion with finite statistics, as above, let ρ be a localized endomorphism defined using an associated cocycle. The standard left inverse for the cocycle gives us a left inverse φ for ρ, cf. Lemma 3A.10. When the statistics operator ε(ρ, ρ) is uniquely defined, namely when the spacetime dimension is greater than or equal to 3, φρ,ρ (ε(ρ, ρ)) is an intertwiner between π and itself. Therefore, when π is irreducible, it is a complex number, cf. Sec. 3.4. When the dimension of a Cauchy surface is one, there are two choices for the statistics and correspondingly two choices for the statistics parameter. In this case, we choose the statistics operator ε associated with the connected component of G ⊥ where the 1-simplices have the chosen orientation (cf. Remark 5.2(vii)). Let us note that, by Assumption IX, a left inverse exists even when a Cauchy surface is compact. The preceding theorem shows that the statistics phase is well defined. In fact, the same is true for the spin, as the following proposition shows. Proposition 5.10. Let π be a representative of the given sector and (π, Uπ ) a covariant representation. Then:


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(i) The quantity s := Uπ (exp(2πh)) is a complex number of modulus one depending only on the equivalence class of π and not on the representation Uπ . It is called the spin of the sector. (ii) Given Uπ , let ν := Ad V · π be an equivalent representation, then (ν, Uν ) is a covariant representation, where Uν := V Uπ V ∗

(5.1)

does not depend on the intertwiner V. Proof. (i) Since π is irreducible, Uπ is fixed up to a one-dimensional representation. By Assumption (c), one-dimensional representations are trivial on exp(th), hence Uπ (exp(2πh)) does not depend on the chosen representation. Since exp(2πh) is ˜ is a central element, the identity element in G0 , the corresponding element in G so Uπ (exp(2πh) is a scalar by irreducibility. Equation 5.1 shows that s does not depend on the representative π. (ii) is obvious. A priori s depends on the Lie algebra element h. However, this possibility is ruled out a posteriori by the spin and statistics relation. In the following, we fix the assignment π 7→ Uπ for any representative π, as described in the above proposition. Now we may state the main theorem of this section. The proof will require some lemmas. Theorem 5.11. Let us consider a local net O 7→ A(O) on a rotationally symmetric spacetime (M, W, G+ , j), satisfying the above assumptions (I–III), (VI–IX), and ˜ an irreducible G-covariant superselection sector with finite statistics on such a net. Then the spin of the sector agrees with its statistics phase. Let π be a representative of a sector with finite statistics, let O be contained in a wedge W and let ρ be an object of End A(O) associated with π and set ρ¯ := j · ρ · j , where j is the modular antilinear morphism associated with A(W ) and Ω. ρ¯ is an object of End A(jW O). Let V denote the Araki–Connes–Haagerup standard implementation (cf. e.g. [33]) of the restriction of ρ to A(W ). ˜ be a wedge orthogonal to W. Let Lemma 5.12. (cf. Lemma 3.1 of [33]) Let W ˜ ), then (id, ρ ˜ ρ¯ ˜ ) is one¯W ¯ to A(W ρW ˜ and ρ ˜ denote the restrictions of ρ and ρ W W i ˜ ˜ ¯W dimensional and V ∈ (id, ρW ˜ρ ˜ ) ∩ A(O), where O is any element of L containing ˜⊂W ˜. O and jW O with O Proof. We remark that the existence of conjugates for finite statistics depends on Assumption IX and was discussed in the proof of Theorem 5.8 . Since we are ˜ is in L if ⊥-duality ˜ ˜ or for O ˜ ⊥ , cf. the discussion at the beginning that O holds either for O of Sec. 4.2.

i Recall


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dealing with a sector, Assumption IX implies that (id, ρW ¯W ˜ρ ˜ ) is one-dimensional ˜ In fact, let z yield ρ in O, i.e. y(a) = ρ for a = O, then the and contained in A(O). cocycle z¯, defined by z¯(b) = j(z(jW b)) ,

b ∈ Σ1 ,

˜ ¯ and |ˆb| = O. yields j · ρ · j in a ¯ = jW O. Let ˆb ∈ Σ1 be defined by ∂0ˆb = a, ∂1ˆb = a A simple computation shows that y(a)(¯ z (ˆb))V A = y(a)¯ y (a)(A)y(a)(¯ z (ˆb))V ,

˜ ). a ∈ A(W

˜ Hence Thus by Assumption IX, y(a)(¯ z (ˆb))V ∈ A(O). But z¯(ˆb) ∈ A(|ˆb|) = A(O). ˜ ¯W V ∈ A(O) as claimed. Obviously, an isometry V in (id, ρW ˜ ρ ˜ ) will implement ˜ ρW . Now a simple computation shows that j(V ) ∈ (id, ρ¯jW W ˜ ρjW W ˜ ). But jW W = ˜ since W and W ˜ are orthogonal. Hence, we may suppose that V = j(V ) and W differs at most by a sign from the standard implementation of the restriction of ρ to A(W ). Let π be a representative of a sector with finite statistics and let z be an associated cocycle. Let O be a diamond contained in the intersection of two wedges W1 and W2 and ρ the object of End A(O) associated with z. Write ji for the modular antilinear morphism associated with A(Wi ) and ρ¯i for ji · ρ · ji , i = 1, 2. Lemma 5.13. Let ρ, ρ¯i and Wi , i = 1, 2, be as above and suppose there exists a ˜ with W2 = gW1 . The following identity between representations of the net g ∈G O1 7→ A(O1 ), O1 ⊃ O, holds: π ρ¯1 = Ad Uπ (j1 gj1 g −1 )π ρ¯2 Ad U (j1 gj1 g −1 )∗ , ˜ and where g 7→ j1 gj1 denotes by abuse of notation the action of j1 lifted to G j1 := jW1 . Proof. We have J2 = U (g)J1 U (g)∗ , hence J1 J2 = U (j1 gj1 g −1 ) and j1 j2 = Ad U (j1 gj1 g −1 ), therefore ρ2 Ad U (j1 gj1 g −1 )∗ . ρ¯1 = Ad U (j1 gj1 g −1 )¯ Thus by covariance ρ2 Ad U (j1 gj1 g −1 )∗ ρ¯ ρ1 = ρ Ad U (j1 gj1 g −1 )¯ = Ad Uπ (j1 gj1 g −1 )ρ¯ ρ2 Ad U (j1 gj1 g −1 )∗ . Lemma 5.14. Let ρ, W1 and W2 and g be as in the previous lemma. Then there is a (unique) complex number c(ρ, W1 , g) of modulus one such that Uπ (j1 gj1 g −1 )V2 U (j1 gj1 g −1 )∗ = c(ρ, W1 , g)V1 .

(5.2)


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Proof. By Lemma 5.12, V1 ∈ (id, ρW ¯W ˜1ρ ˜ 1 ). Furthermore, by the previous lemma, Uπ (j1 gj1 g −1 )V2 U (j1 gj1 g −1 )∗ belongs to the same one-dimensional space of intertwiners. Lemma 5.15. Let ρ and σ be two endomorphisms associated with a given sector as above, the first localized in W1 ∩ W2 , W2 = gW1 , the second in hW1 ∩ hW2 , ˜ Then c(ρ, W1 , g) = c(σ, hW1 , hgh−1 ). g, h ∈ G. Proof. We first observe that if σ = Ad W ∗ ρ for some unitary W ∈ A(W1 ∩ W2 ), then Viρ = W ∗ Ji W ∗ Ji Viσ and this implies that c(σ, W1 , g) = c(ρ, W1 , g). Then −1 ), where αh = Ad U (h), because we note that c(ρ, W1 , g) = c(α−1 h ραh , hW1 , hgh U (h) establishes an isomorphism between the original structure and the structure transformed by h. Since α−1 h ραh and σ are associated with the same sector and both localized in hW1 ∩ hW2 and hgh−1 hW1 = hW2 , the result now follows. The previous lemma shows that for the given sector there is a well defined function c(W, g) satisfying c(W, g) = c(hW, hgh−1 ) whenever W ∩ gW 6= ∅. Lemma 5.16. Let W ∈ W. Then the function g 7→ c(W, g) is a local group ˜ such that W ∩ gW ∩ ghW 6= ∅, we have character, namely for any g, h ∈ G c(W, g)c(W, h) = c(W, gh) . Proof. Choose associated endomorphisms localized in W ∩ gW ∩ ghW and denote the involutions associated with W and gW by j1 and j2 , respectively. Then from the definition of c for the pairs (W, g) and (gW, hgW ) and the equality (j1 gj1 g −1 )(j2 hj2 h−1 ) = j1 gj1 g −1 (gj1 g −1 )h(gj1 g −1 )h−1 = j1 hgj1 (hg)−1 one obtains the relation c(W, g)c(gW, h) = c(W, gh) , which means that the function c is a local groupoid character. Then, making use of Lemma 5.15 we get c(W, g)c(W, h) = c(W, ghg −1 ) = c(W, (ghg −1 )g) = c(W, gh) . In Proposition 5.10, we only used properties (b 1), (b 2). The rest of the argument makes essential use of further properties, more precisely, (b 2) and (b 3) are used in the following proposition, whilst (b 3) and (b 4), or rather, the orthogonality of Remark 5.1, are used to conclude the proof of Theorem 5.11. Proposition 5.17. Under the given assumptions, we have c(W, exp π2 h) = 1.


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Proof. Since g 7→ c(W, g) is a local representation, it is locally trivial on the come hence, by assumption (c), there exists ε > 0 such that c(W, exp th) = 1 mutator of G, for |t| ≤ ε. Because of assumption (b 4) the result follows by applying Lemma 5.16 sufficiently often. Lemma 5.18. Let ρ be an endomorphism associated with the sector and localized in O ⊂ W1 ∩ W2 , where W1 and W2 are orthogonal wedges, W2 := exp( π2 h)W1 (cf. Remark 5.1). Let the standard implementations of its restriction to the algebras A(W1 ), A(W2 ) be denoted by V1 , V2 as before. Then the statistics parameter λρ can be written as λρ = V1∗ V2∗ V1 V2 . Proof. As in [33], Lemma 3.5, we first show λρ = ρ(V1∗ )V1 ; indeed if ρ0 is localized in W1 ∩ W2⊥ and u is a unitary in (ρ, ρ0 ) in End A(W1 ), then u ∈ A(W1 ). Since (W1 ∪ W2 )⊥ 6= ∅, u∗ A = u∗ ρ0 (A) = ρ(A)u∗ , for A ∈ A(W2 ). But V1 ∈ A(W2 ) by Lemma 5.12. Thus ρ(V1∗ )V1 = u∗ V1∗ uV1 . Now ρ¯1 := j1 ·ρ·j1 is localized in W1⊥ ∩W2 and, again since (W1 ∪ W2 )⊥ 6= ∅, ρ, ρ0 and ρ¯1 are comparable and ρˆ1 (u) = u. Thus V1∗ uV1 = φ(u), where φ is the left inverse of ρ. Hence ρ(V1∗ )V1 = u∗ φ(u) = φ(ε(ρ, ρ)) = λρ . Now V2 ∈ A(W1 ) and implements ρ on A(W2 ). ρ¯2 := j2 · ρ · j2 is localized in W1 ∩ W2⊥ and since (W1 ∪ W2 )⊥ 6= ∅, ρ¯2 (V1 ) = V1 so we have V1∗ V2∗ V1 V2 = V1∗ φ(V1 ) = φ(ρ(V1∗ )V1 ) = φ(λρ ) = λρ .

(5.3)

Before proceeding to the proof of Theorem 5.11, we prove a result about orthogonal wedges. Lemma 5.19. Given two orthogonal wedges W1 , W2 with reflections j1 and j2 , there is a region O with non-empty causal complement which is invariant under j1 and j2 . Proof. Take O1 and O2 orthogonal to each other and contained in W1 ∩ W2 , and set O = O1 ∪ j1 O1 ∪ j2 O1 ∪ j1 j2 O1 . Clearly O is causally disjoint from O2 and invariant under j1 and j2 . Proof of Theorem 5.11. We follow the reasoning of [33]. Consider the two orthogonal wedges W1 , and W2 = exp( π2 h)W1 as in the preceding lemma and choose a representative endomorphism localized in a regular diamond O ∈ W1 ∩ W2 and ˜ containing O, j1 O, j2 O and j1 j2 O. Then ρ¯ ρ1 j2 ρ¯ ρ1 j2 = chosen such that there is an O ˜ V1 J2 V1 J2 and V2 J1 V2 J1 intertwine from ρ2 j1 and are objects of End A(O). ρ¯ ρ2 j1 ρ¯ ˜ Thus βρ := (V1 J2 V1 J2 )∗ V2 J1 V2 J1 is a the identity to this object in End A(O). scalar and we first show that it belongs to (0, 1], as in Lemma 3.4 in [33], observing that βρ = V1∗ U (exp πh)V1∗ V2 U (exp πh)V2 .

(5.4)


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Then, by (5.2) with g = exp π2 h and its adjoint and using the equation c(W, exp π2 h) = 1, proved in Proposition 5.17, we get V2∗ V1 = sρ U (exp πh)V1∗ V2 U (exp πh) .

(5.5)

Inserting this equation into the expression for the statistics parameter given by Lemma 5.18 and comparing with (5.4) we obtain λρ = V1∗ V2∗ V1 V2 = sρ V1∗ U (exp πh)V1∗ V2 U (exp πh)V2 = sρ βρ and the result follows easily. We conclude this subsection showing that the Spin and Statistics relation makes sense and is true for reducible covariant representations, too. Clearly the result follows from the irreducible case once we can show that the irreducible subrepresentations are still covariant. Proposition 5.20. Let π be a representation satisfying the selection criterion and e Then there exists a covariant with finite statistics and covariant under the group G. representation (π, Uπ ), where Uπ acts trivially on π(A)0 . Uπ is unique up to a onedimensional representation and any other choice of Uπ may be written as a product of Uπ and a representation Uπ0 contained in π(A)0 . In particular, each irreducible e component of π is G-covariant. Proof. Since π has finite statistics, π(A)0 and hence the centre of π(A) are finite eπ is trivial on eπ ) is a covariant representation, U dimensional. Therefore if (π, U eπ implements automorphisms of π(A), it implements an the centre. Then, since U e by automorphisms of π(A)0 , preserving any factorial component. Thus action of G this action is implemented by a unitary representation U 0 in π(A)0 . Then g ∈ e→U er (g)U 0 (g)∗ is a representation of G e acting trivially on π(A)0 . Clearly such a G representation decomposes into representations of the irreducible components of π, e so these are G-covariant. Remark 5.21. The given proof of the spin and statistics relation does not rely on the continuity of the representations U or Uπ . Even Proposition 5.20 does not require continuity, because it relies on the fact that a connected Lie group acts trivially on a finite set and this is true without assuming continuity. 5.2.2. Spin and Statistics under property (c’) Now we give a proof of the Spin and Statisitics Theorem for rotationally symmetric spacetimes satisfying (c0 ) rather than (c), such as the 3-dimensional Schwarzschild– Kruskal spacetime, for example. Recall that in this case there is an involution rW := jW jW0 ∈ G↑+ anticommuting with h (cf. Remark 5.2(vi)). Let us denote the subgroup of G↑+ generated by G0 and rW by G1 . If rW does not belong to G0 , G1 is isomorphic to G0 ×σ Z2 , where σ = Ad rW . In the same


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e1 ≡ G e ×σ Z2 , where, by an abuse of notation, σ way we can consider the group G e lifted to G is still denoted by σ. We shall also denote the corresponding element in e 1 to G1 . e1 by rW . Clearly the covering map extends to an epimorphism from G G e e 1 -covariant, We want to show that any G-covariant sector with finite statistics is G too. Proposition 5.22. Let π be an irreducible representation of A obeying the selece Then it is tion criterion, with finite statistics, and covariant under the group G. e covariant under G1 . / G0 . Since r ≡ rW is the Proof. Of course we may restrict to the case rW ∈ product of jW with jW0 , such reflections are implemented by the corresponding modular involutions J, J0 , and jρj is equivalent to j0 ρj0 , both being conjugate endomorphisms of ρ, there exists a unitary Ur intertwining ρ and α(r)ρα(r). Since r2 = 1, Ur2 implements the trivial action on A, hence, ρ being irreducibile, Ur2 is a constant and we may choose Ur selfadjoint. Then Ur Uρ (rgr)Ur is another e realizing the covariance of ρ. Since ρ is irreducible, we get representation of G e Applying this relation Ur Uρ (g)Ur = χ(g)Uρ (rgr), where χ(g) is a character of G. twice, we get Uρ (g) = χ(g)χ(rgr)Uρ (g), namely χ(g)χ(rgr) = 1. Now observe that, since χ is a Lie group representation, it is the exponential of a Lie algebra morphism e is simply connected, κ/2 exponentiates κ from the Lie algebra of G0 to R. Since G √ to a character, which we denote by χ, whose square gives χ, and we get √ √ Ur χ(g −1 )Uρ (g)Ur = χ(rg −1 r)Uρ (rgr) , √ e1 . namely χ(g −1 )Uρ (g) and Ur yield the required representation of G Theorem 5.23. Let (M, G+ , j, W) be a rotationally symmetric spacetime satisfying properties (a), (b) and (c0 ), (A, U, Ω) a covariant net verifying the mentioned e axioms (I–III), (VI–IX), and let ρ be a G-covariant sector with finite statistics. Then the spin and statistics relation holds. Proof. By property (c0 ), sρ does not depend on Uρ , as observed in Remark 5.2 (vi). Concerning the relation between spin and statistics, we may define a funce1 , as in the proof of Theorem 5.11, which is indeed a local tion c(W, g), g ∈ G e1 verify W ∩ gW ∩ ghW 6= ∅, we have group representation namely, if g, h ∈ G c(W, g)c(W, h) = c(W, gh). Setting r˜ := exp( π2 h)rW exp(− π2 h), we get π π h rW exp − h W = exp(πh)rW W = W r˜W = exp 2 2 and r˜ exp(th)˜ r = exp(−th). Hence, for sufficiently small t, c(W, exp(2th)) = c(W, r˜)c(W, r˜ exp(th))c(W, exp(th)) = c(W, r˜ exp(th)˜ r )c(W, exp(th)) = c(W, exp(−th))c(W, exp(th)) = 1 .


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The proof now continues as in Theorem 5.11. As before, the Spin and Statistics relation for reducible representations follows ˜ 1 -covariant, and this as soon as we prove that the irreducible representations are G is a consequence of Proposition 5.20 and Proposition 5.22. Remark 5.24. Generally speaking, asking for an irreducible endomorphism to be covariant corresponds to asking for a projective representation of the group G0 , namely a representation of a central extension of G0 by some subgroup of T1 implementing the action of G0 on ρ(A). This means that there may be an extension at the Lie algebra level, not only a covering. However, we are not aware of any physical example where non-trivial Lie algebra central extensions exist (for the Poincaré group on the two-dimensional Minkowski space, such non-trivial extensions exist, but are incompatible with the positive energy requirement). As a consequence, we have only treated the case of the universal covering. 5.3. Appendix. Equivalence between local and global intertwiners in Minkowski spacetime In this appendix we prove that Assumption IX concerning the equivalence of local and global intertwiners holds for sectors localized in a wedge region of a Minkowski space of arbitrary dimension. The argument is a straightforward adaptation of that given in [33] for a conformal net on S 1 . In the following, A is a net of von Neumann algebras on the (d + 1)-dimensional Minkowski spacetime. We assume Poincaré covariance with positive energy and uniqueness of the vacuum, additivity and Haag duality A(O) = A(O0 )0 if O is either a double cone or a wedge region. If ρ, σ are endomorphisms of A localized in the wedge region W , we consider their intertwiner space (ρW , σW ) := {T ∈ A(W ) : σ(A)T = T ρ(A), ∀ A ∈ A(W )}. By duality we always have (ρ, σ) ⊂ (ρW , σW ). Theorem A5.1. Let W be a wedge region and ρ, σ be endomorphisms with finite dimension localized in a double cone O ⊂ W. Then (ρW , σW ) = (ρ, σ) . Namely, if T ∈ (ρW , σW ) then T intertwines the representations ρ and σ of A. In the following ρ denotes an endomorphism with finite dimension of the quasilocal observable C ∗ -algebra A localized in a double cone O contained in the wedge W . We may assume that W = {x ∈ Rd+1 : −x1 > |x0 |}. We shall denote by R2 the 2-dimensional (x0 − x1 )-plane and by P the corresponding 2-dimensional Poincaré group, namely the semidirect product of the


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2-dimensional translations {T (x)}x∈R2 and boosts {Λ(s)}s∈R associated to W : each g ∈ P can be written uniquely as a product g = T (x)Λ(s). The endomorphism ρ restricts to an endomorphism of the C ∗ -algebra associated with W + x and then extends to the von Neumann algebra A(W + x), for x1 > 0, hence giving rise to an endomorphism of the C ∗ -algebra A∞ , the norm closure of S x∈R2 A(W + x). We will still use ρ to denote this endomorphism. Since P is simply connected, there is a unitary representation Uρ of P expressing the covariance of ρ with respect to P βg (A) = Uρ (g)AUρ (g)∗ = zρ (g)U (g)AU (g)∗ zρ (g)∗ ,

A ∈ A∞ , g ∈ P .

(5.6)

As the cocycle zρ is a local operator by Haag duality (this is the essential point about the 2-dimensional (x0 −x1 )-net inherited from the higher dimensional original net) β is an action of P by automorphisms of A∞ . We consider now the semigroup P0 , the semidirect product of the boosts Λ(s) with the positive translations, where we say that T (x) is positive if x ∈ R2 with x1 > |x0 |. P0 is an amenable semigroup and we need an invariant mean m constructed as follows: first we average (with an invariant mean) over positive translations and R then over boosts. Observe that f → P0 f (g)dm(g) gives an invariant mean on all P vanishing on f if, for any given s ∈ R, the map x ∈ R2 → f (T (x)Λ(s)) vanishes on a right wedge. Then we associate to m the completely positive map Φ of A∞ to B(H) given by Z zρ (g)∗ Azρ (g)dm(g) , A ∈ A∞ . Φ(A) := P0

Lemma A5.2. Φ is a left inverse of ρ on A∞ . Moreover Φ is locally normal, i.e. has normal restriction to A(W + x). x ∈ R2 , and P-invariant, namely Φ = α−1 g Φβg ,

g ∈ P.

We have set αg ≡ Ad U (g). Proof. Let A belong to A(W + x), x ∈ R2 . By (5.6) Z αg (ρ(αg−1 (A)))dm(g) = A Φ(ρ(A)) = P0

because of the above property of m since the integrand is constantly equal to A on the set g ∈ P0 : g −1 W ∩ O = ∅. Then the localization of ρ and Haag duality imply that the range of Φ is contained in A∞ . Setting E = ρ · Φ gives a conditional expectation of A∞ onto the range of ρ that restricts to a conditional expectation Ex of A(W + x) onto ρ(A(W + x)) if W +x ⊃ O. Since ρW +x is assumed to have finite index, Ex is automatically normal [45]. Therefore Φ A(W + x) = ρ−1 W +x Ex is normal for x = (0, x1 ) with x1 > 0, hence for any x.


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Concerning the P-invariance of Φ we have, making use of the cocycle condition, Z −1 −1 ∗ zρ (h) βg (A)zρ (h)dm(h) αg Φβg (A) = αg = α−1 g Z = P0

Z

P0

P0

zρ (h)∗ zρ (g)αg (A)zρ (g)∗ zρ (h)dm(h)

zρ (hg −1 )∗ Azρ (hg −1 )dm(h) = Φ(A) .

Corollary A5.3. ϕ = ωΦ is a locally normal β-invariant state on A∞ , where ω = ( · Ω, Ω). Proof. We have ϕβg = ωΦβg = ωαg Φ = ωΦ = ϕ and ϕ is locally normal because both ω and Φ are locally normal. Let {πϕ , ξϕ , Hϕ } be the GNS triple associated with the above state ϕ and V be the unitary representation of P on Hϕ given by Vg Aξϕ = βg (A)ξϕ for A ∈ A∞ . Notice that V is strongly continuous because ϕ is locally normal. Lemma A5.4. If ρ is irreducible then Z βg (x)dm(g) , x ∈ A∞ . ϕ(x) = P0

Proof. If A ∈ A(W +x) and B ∈ A∞ is localized in a double cone, the commutator function R2 3 x 7→ [βT (x)Λ(s) (A), ρ(B)] = βT (x)Λ(s) ([A, ρ(α−1 T (x)Λ(s) (B)]) vanishes on R R ρ(B)dm(g)] = [β (A), ρ(B)]dm(g) = 0. a right wedge, hence [ P0 βg (A), P0 g R Since ρ is locally normal, P0 βg (A)dm(g) commutes with every ρ(A(W + x)), thus with ρ(A∞ ); but ρ being irreducible, it is therefore a scalar equal to its vacuum expectation value: Z Z Z βg (A)dm(g) = ω(βg (A))dm(g) = ω(zg∗ Azg )dm(g) = ωΦ(A) = ϕ(A) , P0

P0

P0

as ω is normal and α-invariant. Corollary A5.5. If ρ is irreducible, the two-parameter unitary translation group V (T (x)) satisfies the spectrum condition. Proof. One may repeat the proof of Corollary 2.7 of [33] for each of the oneparameter light-like unitary translation groups. Corollary A5.6. If ρ is irreducible, ϕ is faithful on

S

ρ(A(W + x)).

Proof. A∞ is a simple C ∗ -algebra since it is the inductive limit of type III factors (that are simple C ∗ -algebras). Therefore πϕ is one-to-one and the statement will follow if we show that ξϕ if cyclic for Bx ≡ ρ(A(W + x))0 , x1 > 0. To this end we


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may use a classical Reeh–Schlieder argument. If ψ ∈ H is orthogonal to Bx ξϕ , and x−y ∈ W , then for all A ∈ By we have (Aξϕ , V (T (x))ψ) = 0 for x in a neighborhood of 0, thus for all x ∈ R2 by the spectrum condition shown by Corollary A5.5. Hence, S setting αx ≡ αT (x) and βx ≡ βT (x) , ψ is orthogonal to ( x βx (By ))ξϕ , thus ψ = 0 S because x βx (By ) is irreducible since !0 \ \ [ βx (By ) = βx (ρ(A(W + y)) = ρ(αx (A(W + y))) x

x

=ρ

\ x

! αx (A(W + y))

x

=

\

A(W + x) = C

x

by the local normality of ρ. Proposition A5.7. (ρW +x , ρW +x ) does not depend on the wedge W + x ⊃ O. Proof. We begin with the case where ρ is irreducible and assume for convenience that O ⊂ W . Notice then that (ρW , ρW ) is finite-dimensional and, by covariance, globally βg -invariant with g in the subgroup of boosts because these transformations preserve W . Therefore (ρW , ρW )ξϕ is a finite-dimensional subspace of Hϕ globally invariant for V (Λ(s)), s ∈ R. By Proposition B.3 of [33] we thus have V (T (x))Aξϕ = Aξϕ for every element A ∈ (ρW , ρW ), thus βT (x) (A) = A because ξϕ is separating. It follows that if A ∈ (ρW , ρW ) and B ∈ A(W ) [A, ρ(αg (B))] = βg ([βg−1 (A), ρ(B)]) = βg ([A, ρ(B)]) = 0 namely A ∈ (ρW , ρW ) ⇒ A ∈ (ρ, ρ) = C . Since the converse implication is obvious by wedge duality we have the equality of the two intertwiner spaces. Now if ρ is any endomorphism with finite index, (ρ, ρ) is finite-dimensional because (ρ, ρ) ⊂ (ρW , ρW ) and ρ decomposes into a direct sum of irreducible endomorphisms of A∞ which are covariant, therefore the preceding analysis shows that (ρW , ρW ) = (ρ, ρ) in this case, too. Since (ρ, ρ) is translation invariant, we get (ρW +x , ρW +x ) = (ρ, ρ) whenever O ⊂ W + x and, since x was arbitrary, the result follows. Proof of Theorem A5.1. The case σ = ρ follows immediately by Proposif tion A5.6: if T ∈ (ρW , ρW ) then T also belongs to (ρW f , ρW f ) for any wedge W ⊃ W hence by additivity T is a self-intertwiner of ρ on the whole algebra A. To handle the general case, consider a direct sum endomorphism η := ρ ⊕ σ localized in W , then dim(ηW , ηW ) = dim(ρW , ρW ) + dim(σW , σW ) + 2 dim(ρW , σW )


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while dim(η, η) = dim(ρ, ρ) + dim(σ, σ) + 2 dim(ρ, σ) therefore dim(ρW , σW ) = dim(ρ, σ) and since we always have (ρ, σ) ⊂ (ρW , σW ) these two intertwiner spaces coincide. Acknowledgments R.V. has been in part supported through the Operator Algebras Network funded by the EU under contract CHRX-CT94-0566. R.V. also wishes to thank all the members of the operator algebra group at the Dipartimento di Matematica, Universit` a di Roma “Tor Vergata”, for their kind hospitality in 1996. Three of the authors (D.G., J.R., R.V.) would like to thank the Erwin Schr¨ odinger Institute, Vienna, as well as the Organizers of the Workshop on Quantum Field Theory in September 1997, D. Buchholz, H. Narnhofer and J. Yngvason, for the opportunity of participating in the workshop. The excellent working conditions provided a basis for discussions relevant to the present paper. We would also like to thank K.-H. Rehren for pointing out a gap in an earlier version of Sec. 4.3. References [1] H. Araki and L. Zsido, “Extension of the structure theorem of Borchers and its application to half-sided modular inclusions” manuscript, preliminary version (1995), to appear. [2] H. Baumg¨ artel and M. Wollenberg, Causal Nets of Operator Algebras, Akademie Verlag, Berlin, 1992. [3] J. K. Beem and P. E. Ehrlich, Global Lorentzian Geometry, Marcel Dekker, New York, 1981. [4] J. J. Bisognano and E. H. Wichmann, “On the duality condition for quantum fields”, J. Math. Phys. 17 (1976) 303. [5] H.-J. Borchers, “The CPT-theorem in two-dimensional theories of local observables”, Commun. Math. Phys. 143 (1992) 315. [6] H.-J. Borchers, “On modular inclusion and spectrum condition”, Lett. Math. Phys. 27 (1993) 311. [7] H.-J. Borchers, “When does Lorentz invariance imply wedge duality?”, Lett. Math. Phys. 35 (1995) 39. [8] H.-J. Borchers, “Half-sided modular inclusions and the construction of the Poincar´ e Group”, Commun. Math. Phys. 179 (1996) 703. [9] H.-J. Borchers and D. Buchholz, “Global properties of vacuum states in de Sitter space”, Ann. Inst. H. Poincar´ e 70 (1999) 23. [10] J. Bros, H. Epstein and U. Moschella, “Analyticity properties and thermal effects for general quantum field theory on de Sitter space-time”, Commun. Math. Phys. 196 (1998) 535. [11] M. Brown, “Locally flat imbeddings of topological manifolds”, Annals of Math. 75 (1962) 331. [12] R. Brunetti and K. Fredenhagen, “Interacting quantum fields in curved space: Renormalizability of ϕ4 ”, in Proceedings of the Conference “Operator Algebras and Quantum Field Theory” held in Rome, July 1996, eds. S. Doplicher, R. Longo, J. Roberts


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DIFFUSION PROCESSES ON PATH SPACES WITH INTERACTIONS

YUU HARIYA Graduate School of Mathematics, Nagoya University Chikusa-ku, Nagoya, 464-8602, Japan E-mail : [email protected] HIROFUMI OSADA Graduate School of Mathematics, Nagoya University Chikusa-ku, Nagoya, 464-8602, Japan E-mail : osadamath.nagoya-u.ac.jp

Received 1 October 1999 Revised 12 April 2000 We construct dynamics on path spaces C(R; R) and C([−r, r]; R) whose equilibrium states are Gibbs measures with free potential ϕ and interaction potential ψ. We do this by using the Dirichlet form theory under very mild conditions on the regularity of potentials. We take the carr´ e du champ similar to the one of the Ornstein–Uhlenbeck process on C([0, ∞); R). Our dynamics are non-Gaussian because we take Gibbs measures as reference measures. Typical examples of free potentials are double-well potentials and interaction potentials are convex functions. In this case the associated infinite-volume Gibbs measures are singular to any Gaussian measures on C(R; R). Keywords: Path valued diffusion, Dirichlet forms, Gibbs

1. Introduction In [13] Gibbs measures with two types of potentials had been constructed on the infinite-volume path space C(R; R). Such measures are defined through the so-called DLR equation with respect to a (formal) Hamiltonian H given by ZZ Z 1 ψ(x − y, z(x) − z(y))dxdy , z ∈ C(R; R) . (1.1) H(z) = ϕ(z(x))dx + 2 Here the functions ϕ : R → R ∪ {∞} and ψ : R2 → R ∪ {∞} describe the free potential and the interaction potential, respectively. Once such Gibbs measures µ are established on C(R; R), we are led to dynamical issues, namely, the construction of suitable dynamics with µ as their invariant probability measures, and the investigation of their properties such as spectral gap, log-Sobolev inequality and the ergodicity of dynamics. In the present paper we construct the dynamics as a first step of dynamical issues. 199


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In the finite-volume path space Wr = C([−r, r]; R) the dynamics {Xt }0≤t 1, δ ≥ 1) .

Then there exists a (ϕ, ψ)-Gibbs measure µ which is translation invariant (see [13]). Note that this Gibbs measure is a continuous version of Ising ferro magnet with an unbounded spin. By the presence of the term a4 and |a|δ (if δ 6= 2), we see µ is not a Gaussian measure. In addition, since we consider infinite-volume measures, µ is singular to any Gaussian measures on W . Example 2.2 (hard core potential) Another example of interest is ( α(a4 − a2 ) (a > 0) , ϕ(a) = ∞ (a ≤ 0) , ψ(x, a) = β(1 + |x|)−γ |a|δ

(β ≥ 0, γ > 1, δ ≥ 1) .

The existence of finite-volume (ϕ, ψ)-Gibbs measures is trivial, and the first author has succeeded in proving the existence of Gibbs measures for these potentials on the infinite-volume path space W . Since state spaces of these diffusions are restricted on C(R; R+ ) in infinite volume and restricted on C([−r, r]; R+ ) in finite volume, one may regard these as reflecting barrier Brownian motions. The existence of the associated diffusion on finite-volume path spaces C([−r, r]; R+ ) follows from Theorem 2.1 and the general theory of Dirichlet forms (see [3], [11]). Note that those two examples satisfy the assumptions (A.1) and (A.2) introduced in the next subsection. 2.2. Construction of bilinear forms and main results We endow W with the compact uniform topology. Let B(W ) be the corresponding Borel σ-field. Let W 0 be the topological dual of W . Remark 2.1. Each l ∈ W 0 is of the form Z z(x)dν(x) , l(z) = R

z∈W

(2.2)

for some function ν of bounded variation, that is constant outside of some finite interval [−r0 , r0 ] of R, r0 ∈ N. This follows from Riesz–Markov theorem (see, e.g., [9, Chap. XI, Sec. 2]).


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203

Recall the definition of Gibbs measure µ. The conditional distributions of µ have densities with respect to pinned Brownian motions on each finite-volume path space Wr . With this in mind, we introduce the reproducing kernel Hilbert space (RKHS) Hr of pinned Brownian motions on [−r, r] (or, its extention on the infinite-volume path space W ) given by Hr : = h ∈ W ; h is absolutely continuous, h(x) = 0 if |x| ≥ r , Z R

2 ˙ |h(x)| dx < +∞

with the inner product

(2.3)

Z hh1 , h2 iHr :=

R

h˙ 1 (x)h˙ 2 (x)dx .

When we deal with the finite-volume path space Wr , Hr , as a subset of Wr , means the ordinary RKHS of pinned Brownian motions on [−r, r]. We will use the same symbol for the notational simplicity. We note that the triplicate (Wr0 , Hr , Wr,0 ) forms an abstract Wiener space. Here Wr0 = {w ∈ Wr ; w(±r) = 0} and Wr,0 is the distribution of the pinned Brownian motion conditioned zero at x = ±r. Now we proceed to the construction of bilinear forms on L2 (W ; µ). Define the set D of continuous functions on W by D := {u(z) = f (l(z))(z ∈ W ); f ∈ Cb∞ (Rn ) , l = (l1 , . . . , ln ) ,

li ∈ W 0 , 1 ≤ i ≤ n, n ∈ N} ,

where Cb∞ (Rn ) denotes the set of all infinitely differentiable functions on Rn with bounded derivative of all order. Let L2 (W → Hr ; µ) denote the set of Hr -valued random variables on (W, µ) which are square-integrable in the sense of Hr -norm. For u = f (l) ∈ D, we define the gradient operator ∇r : D → L2 (W → Hr ; µ) as follows: X ∂i f (l(z))li . (2.4) ∇r u(z) := i

Here each li is regarded as an element of Hr by the canonical identification of Hr and its dual Hr0 : W 0 ⊂ Hr0 ∼ = Hr . Notice that ∇r u(z) is the unique element of Hr satisfying h∇r u(z), hiHr = where

∂u(z) ∂h

∂u(z) ∂h

for any h ∈ Hr ,

is the Fréchet derivative in the direction of h:

u(z + sh) − u(z) ∂u(z) = lim . s→0 ∂h s Consider the associated carré du champ 1 Dr [u, v] := h∇r u, ∇r viHr . 2

(2.5)

(2.6)


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Note that Dr [·, ·] is well-defined on D, because (2.5) is independent of the represenb be the tation used for u ∈ D. Since we consider bilinear forms over L2 (W ; µ), let D corresponding set of equivalence classes, where two functions are identified if they b we are equal µ-almost everywhere. In order that Dr [·, ·] be also well-defined on D, assume: (A.0’) For u, v ∈ D such that u = v µ-a.e., ∇r u = ∇r v µ-a.e. for each r ∈ N . Note that (A.0’) is implied by (A.0) introduced below. We set Z b. Dr [u, v](z)dµ(z) , u, v ∈ D Er (u, v) := W

b in the following. For simplicity D means D Next we construct a bilinear form on L2 (Wr ; µr,ξ ) similarly to (Er , D). Let Dr be the set of continuous functions on Wr given by Dr := {u(w) = f (l(w))(w ∈ Wr ); f ∈ Cb∞ (Rn ) , l = (l1 , . . . , ln ) , li ∈ Wr0 ,

1 ≤ i ≤ n, n ∈ N} .

Here Wr0 denotes the dual of Wr with respect to the topology induced by the supbr be the corresponding set of equivalence classes in L2 (Wr ; µr,ξ ). norm on Wr . Let D In an obvious manner we regard ∇r as the mapping ∇r : Dr → L2 (Wr → Hr ; µr,ξ ). We also regard Dr [·, ·], defined through (2.6), as the carré du champ over L2 (Wr ; br , we assume: µr,ξ ). In order that Dr be well-defined on D (A.0) For µ∗r -a.e. ξ ∈ Wr∗ , ∇r u = ∇r v µr,ξ -a.e. if u, v ∈ Dr satisfy u = v µr,ξ -a.e. We define

Z Dr [u, v](w)dµr,ξ (w) ,

Er,ξ (u, v) :=

br . u, v ∈ D

Wr

br for simplicity. In the following Dr means D In our first main theorem we prove the closability of (Er,ξ , Dr ), that is, we will give a sufficient condition for (2.7): (Er,ξ , Dr ) is closable on L2 (Wr ; µr,ξ ) for µ∗r -a.e.ξ ∈ Wr∗ .

(2.7)

We assume: (A.1) ϕ is bounded from below and {ϕ = ∞} is connected. Moreover, ϕ is bounded on An for each n ∈ N. Here An := {a ∈ R; ϕ(a) < ∞, |a| < n}. (A.2) ψ is represented as ψ(x, a) = ψ1 (x)ψ2 (a) with even functions ψ1 and ψ2 satisfying (P.1) and (P.2), respectively: (P.1) ψ1 satisfies, for some C1 > 0 and γ > 1, 0 ≤ ψ1 (x) ≤ C1 (1 + |x|)−γ

for all x ∈ R .


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(P.2) ψ2 is convex and there exist C2 , p > 0 such that |ψ2 (a)| ≤ C2 ep|a|

for all a ∈ R .

(A.3) µ is a (ϕ, ψ)-Gibbs measure that satisfies Z −γ p|z(x)| dx < ∞ = 1 . µ z ∈ W ; (1 + |x|) e R

Remark 2.2. (i) We do not assume any smoothness of potentials; the measurability is enough as for the regularity of potentials. (ii) Hr,ξ may take the value ∞ but is bounded from below by (A.1) and (A.2). Theorem 2.1. Assume (A.1)–(A.3). Then (2.7) holds. If (Er,ξ , D(Er,ξ )) is quasi-regular and local, then by applying the Dirichlet form theory developed in [3] and [11], we see there exists a diffusion associated with (Er,ξ , D(Er,ξ )). Indeed, we obtain: Corollary 2.1. Let (Er,ξ , D(Er,ξ )) denote the closure of (Er,ξ , Dr ) on L2 (Wr ; µr,ξ ) obtained by Theorem 2.1. Then (Er,ξ , D(Er,ξ ), L2 (Wr ; µr,ξ )) is a quasi-regular Dirichlet space and the associated reversible Wr -valued diffusion X = {Xt }t≥0 exists. Remark 2.3. (i) Since 1 ∈ D(Er,ξ ) and, by [14, Proposition 3.1] (see also [15, Sec. 4(a)]), we see (Er,ξ , D(Er,ξ )) is capacity tight. Other axioms for the quasiregularity are trivial. The local property follows from the argument in [11, Example V.1.12]. (ii) For reader’s intuition we will give an SDE representation for X in Sec. 3. Next we turn to results for infinite-volume dynamics. We begin with: Proposition 2.1. (i) Suppose that (2.7) holds. Then (Er , D) is closable on L2 (W ; µ). (ii) Let (Er , D(Er )) be the closure of (Er , D) on L2 (W ; µ). Then there exists diffusion associated with (Er , D(Er )) on L2 (W ; µ). (iii) The Dirichlet form (Er , D(Er )) is local and quasi-regular on L2 (W ; µ). Note that Dr is increasing in the sense that Dr [u, u] ≤ Dr+1 [u, u] for all u ∈ D(Er+1 ) and r ∈ N. So we define Z D[u, u]dµ for u ∈ D(E) . D[u, u] := lim Dr [u, u] , E(u, u) := r→∞

Here D(E) := {u ∈

W

T

r∈N D(Er ); supr∈N Er (u, u) < ∞}. We assume:

(A.4) For µ-a.e. z ∈ W 1 r→∞ 2r

Z

r

z(x)dx exists .

w(z) := lim

−r


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Remark 2.4. This condition is not a strict one. If µ is translation invariant in R x with |z(0)|µ(dz) < ∞, or more generally, absolutely continuous with respect to such a measure, then (A.4) is satisfied. This follows from the individual ergodic theorem immediately. By (A.4) we deduce (E, D(E)) is densely defined (see Lemma 7.3). Theorem 2.2. (i) Assume (2.7) and (A.4). Then (E, D(E)) is a densely-defined closed form on L2 (W ; µ). (ii) Let {Gα }α>0 be the resolvent of (E, D(E)). Let {Gr,α }α>0 , r ∈ N, be the resolvents of (Er , D(Er )), r ∈ N. Then Gr,α converges to Gα strongly in L2 (W ; µ) as r → ∞ for all α > 0. Remark 2.5. (i) So far we have not yet succeeded in proving the quasi-regularity of (E, D(E)). However, Theorem 2.2 shows we have constructed some infinite-volume dynamics in the category of Markovian semi-group. (ii) For the subset D0 of D given by (7.1) in Sec. 7, we have D0 ⊂ D(E) (see Lemma 7.1(ii)). Hence by Theorem 2.2(i), we see (E, D0 ) is closable on L2 (W ; µ). We now consider the quotient space W ∼ of W under the equivalence relation ∼ such that z1 ∼ z2 if and only if z1 = z2 + const.. We can regard (E, D0 ) as the closable bilinear form on W ∼ and prove the quasi-regularity of its closure. We thus obtain the associated diffusion on the infinite-volume quotient path space W ∼ . We will return to this somewhere else. Iwata [7] (in the case ψ ≡ 0) and Funaki [4–6] constructed W -valued dynamics with µ as their invariant probability measures by means of stochastic partial differential equations (SPDEs). These dynamics are essentially different from ours: indeed, if we represent them by Dirichlet forms, Iwata and Funaki’s dynamics correspond to the carré du champ given by Dcy [·, ·] :=

1 hD·, D·iL2 (R) , 2

where D denotes some differential operator acting on L2 (R). Notice that D and the inner product are taken over L2 (R) (not over H). We will give their SPDEs in Sec. 3 for comparison with ours. We refer to [8] for the case of two parameter dimension. To give an outline: In Sec. 3 we comment on the SDE representation corresponding to (Er,ξ , D(Er,ξ )). We also remark on the choices of carré du champs. In Sec. 4 we give a preliminary lemma. In Sec. 5 we prove Theorem 2.1. Keys to the proof are (i) disintegration and (ii) the monotone convergence theorem for closable forms. Using these two tools, we shall show the closability of (Er,ξ , Dr ) is reduced to that of some bilinear form on R. In Sec. 6 we prove proposition (2.1)(i) plays an important role also there. In Sec. 7 we prove Theorem 2.2. For simplicity we will write Dr [u] for Dr [u, u] afterwards.


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3. SDE Representation In this section we give remarks on the SDE representation for (Er,ξ , D(Er,ξ )). We also remark on another choice of carré du champs over L2 (Wr , µrξ ) and L2 (W ; µ); corresponding SPDEs are also introduced. We emphasize that underlying S(P)DEs are formal ones and we do not solve those here. Remark 3.1. (i) Let mr,ξ be the mean function of the pinned Brownian motion under Wr,ξ : mr,ξ (x) = {ξ(r)(r + x) + ξ(−r)(r − x)}/2r ,

x ∈ [−r, r] ,

and let ρr be the covariance function: ρr (x, y) = r + x ∧ y − (r + x)(r + y)/2r ,

x, y ∈ [−r, r] .

Here x ∧ y := min{x, y}. Note that ρr (x, ·) ∈ Hr for each fixed x by the definition of RKHS. Let Wr0 := {w ∈ Wr ; w(±r) = 0} and let B = {Bt }t≥0 be a Wr0 valued Brownian motion. The diffusion process X = {Xt ∈ Wr }t≥0 associated with (Er,ξ , D(Er,ξ )) on L2 (Wr ; µr,ξ ) satisfies the following (formal) SDE: 1 dXt (x) = − {Xt (x) − mr,ξ (x) + (∇r Hr,ξ (Xt ))(x)}dt 2 x ∈ [−r, r] ,

+ dBt (x) ,

(3.1)

where the term ∇r Hr,ξ is the functional derivative of Hr,ξ . This can be written as follows: for w ∈ Wr Z (∇r Hr,ξ (w))(x) = ρr (x, y)ϕ0 (w(y))dy |y|≤r

ZZ

+ |y1 |,|y2 |≤r

ρr (x, y1 )

∂ ψ(y1 − y2 , w(y1 ) − w(y2 ))dy1 dy2 ∂a

ZZ +

|y1 |≤r 0, of (E , D ) 2 strongly in L (W ; µ) as n → ∞.

Proof. See, for example, [11, proposition I.3.7] for (i) and [16, Theorem 3.1] for (ii). 5. Proof of Theorem 2.1 In this section we prove Theorem 2.1. For each n ∈ N, let Γn be an open subset of Wr given by Γn := {w; w(x) ∈ int An for all x ∈ [−r, r]} .


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Here An is the set defined in (A.1) and int An is its interior. Since {An }n∈N is increasing by definition, {Γn }n∈N is also increasing. Let Γ0 := {w ∈ Wr ; w(x) ∈ int{ϕ < ∞} for all x ∈ [−r, r]}. Then we have [ Γn = Γ0 . (5.1) n∈N

Lemma 5.1. Assume (A.1)–(A.3). Then, for µ∗r -a.e. ξ ∈ Wr∗ , there exists a constant Cr,ξ (n) such that Hr,ξ (w) ≤ Cr,ξ (n)

for all w ∈ Γn .

Proof. By (A.2) and (2.1), Hr,ξ (w) is written as Hr,ξ = I1 + 12 I2 + I3 , where Z ϕ(w(x))dx , I1 (w) = |x|≤r

ZZ I2 (w) = ZZ I3 (w) =

ψ1 (x − y)ψ2 (w(x) − w(y))dxdy ,

|x|,|y|≤r

|x|≤r 0 such that the infrared regularity condition, λ/ων ∈ L2 (Rd ), holds for ν > 0, which raise the


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bottom of the angular frequency ω(k) of bosons (see [7]): ων (k) := ω(k) + ν ,

ν > 0,

ων ) , Hb (ν) := dΓ(ˆ

(1.12) (1.13)

√ µ σ3 ⊗ I + I ⊗ Hb (ν) + 2ασ1 ⊗ φS (λ) , (1.14) 2 where ν means the lower bound of the angular frequency which we can observe precisely by an equipment. Of course, we shall remove “ν” later by taking the limit ν ↓ 0 such as making the precision better. For simplicity, we put HSB (ν) :=

HSB (0) := HSB .

(1.15)

For a linear operator T on a Hilbert space, we denote its domain by D(T ). It is well-known that HSB (ν) is self-adjoint on D(HSB (ν)) = D(I ⊗ Hb (ν)) ,

(1.16)

and bounded from below for all α ∈ R

(1.17)

and every ν ≥ 0 by [6, Proposition 1.1(i)] since σ1 is bounded now. For a self-adjoint operator T bounded from below, we denote by E0 (T ) the infimum of the spectrum σ(T ) of T : E0 (T ) = inf σ(T ) . In this paper, when T is a Hamiltonian, we call E0 (T ) the ground state energy of T even if T has no ground state. For HSB (ν) (ν ≥ 0) we set ESB (ν) := E0 (HSB (ν)) . It is well known that for ν > 0

2

2

µ µ −2α2 kλ/ων k20 2 λ 2 λ − − α √ ≤ ESB (ν) ≤ − e − α √ 2 ων 0 2 ων 0

(1.18)

by easy estimate and the variational principle ([3, Theorem 2.4]) and [12, p. 161]). In this paper, we give new upper and lower bounds for ESB (0) in the massless case. The idea to obtain the bounds is very simple as follows: we know the model of a quantum harmonic oscillator (QHO) with the RWA-interaction is derived from the linear coupling model of the QHO coupled with a Bose field by neglecting the terms which were not observed by an equipment (see [24, Remark on RWA]) . Such the approximation is called the rotating wave approximation (RWA). The Wigner–Weisskopf (WW) model has the same form as the model of QHO with the RWA-interaction, so we can regard WW model as the model derived from the


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Remarks on the Ground State Energy of the Spin-Boson Model

225

spin-boson (SB) model through RWA as well as QHO with the RWA-interaction. But there is a difference between WW model and QHO with the RWA-interaction. Namely, WW and SB models describe models of the spin coupled with a Bose field. So, although QHO with the RWA-interaction cannot restore the neglected terms, WW can do it in the sense of (2.8) and (3.17) below. Thus, we prove Theorem 3.1 (upper bound) in terms of WW model by regarding WW as an approximation of SB model, and using the difference as mentioned above and the variational principle. Moreover, this idea with the simple means which Lieb and Yamazaki already used in [36] for the polaron model brings Theorem 3.2 (lower bound). 2. Spectral Properties of Wigner Weisskopf Model To obtain the bounds for the ground state energy of SB model, we use the properties of WW model [7, 11, 29, 38, 53]. So, in this section, we describe fundamental properties of the Wigner–Weisskopf model. We define a matrix c by ! 0 0 c := . (2.1) 1 0 And let Hb (0) := Hb , ω0 (k) := ω(k) ,

(2.2) k ∈ Rd .

(2.3)

Then, for every µ0 ∈ R and ν ≥ 0, we define a Hamiltonian Hα (µ0 ; ν) of the Wigner–Weisskopf model by Hα (µ0 ; ν) := µ0 c∗ c ⊗ I + I ⊗ Hb (ν) + α(c∗ ⊗ a(λ) + c ⊗ a(λ)∗ ) ! Hb (ν) + µ0 αa(λ) . = Hb (ν) αa(λ)∗

(2.4)

We call Hα (µ0 ; ν) the Wigner–Weisskopf Hamiltonian. We may put for ν = 0 Hα (µ0 ) := Hα (µ0 ; 0) .

(2.5)

Remark 2.1. The Wigner–Weisskopf model is one of several examples √ of the ∗ generalized √ spin-boson model. We know it if we put B1 ≡ (c + c)/ 2, B2 ≡ i(c∗ − c)/ 2; λ1 ≡ λ and λ2 ≡ iλ. It is easy to prove that Hα (µ0 ; ν) is self-adjoint on D(Hα (µ0 ; ν)) = D(I ⊗ Hb (ν)) ,

(2.6)

and bounded from below

(2.7)


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for every ν ≥ 0 by [6, Proposition 1.1(i)] since each Bj is bounded, and ˜ α (µ0 ; ν)U1 = Hα (µ0 ; ν) U1∗ H

for every ν ≥ 0 ,

(2.8)

where the unitary operator U1 is given by U1 := σ1 ⊗ I =

0

I

I

0

! ,

(2.9)

˜ α (µ0 ; ν) is given by and H ˜ α (µ0 ; ν) := µ0 cc∗ ⊗ I + I ⊗ Hb (ν) + α(c∗ ⊗ a(λ)∗ + c ⊗ a(λ)) H ! αa(λ)∗ Hb (ν) . = αa(λ) Hb (ν) + µ0

(2.10)

Remark 2.2. For µ0 < 0, the above Wigner–Weisskopf Hamiltonian Hα (µ0 ; ν) was treated in [7, Theorem 6.15]. On the other hand, for µ0 ≥ 0, Hα (µ0 ; ν) was treated in [29, §6] with ν > 0, and [7, Theorm 6.14] with ν ≥ 0. As we did in [7, §6.2] we introduce a function Dµα0 ,ν for µ0 ∈ R and ν ≥ 0 by Z Dµα0 ,ν (z) := −z + ε0 − α2

dk Rd

|λ(k)|2 , ων (k) − z

(2.11)

defined for all z ∈ C such that |λ(k)|2 /|z − ων (k)| is Lebesgue integrable on Rd . Remark 2.3. It is well-known that the Wigner–Weisskopf model is the simplified Lee model [32, 34, 52] & [50, §5.2], and the solution of Dεα0 ,ε1 ,ν (z) = 0 gives the renormalized mass for the Lee model. As we mentioned in [7, §6.2], Dµα0 ,ν (z) is defined in the cut plane Cν := C \ [ν, ∞) ,

ν≥0

(2.12)

and analytic there. It is easy to see that Dµα0 ,ν (x) is monotone decreasing in x < ν. Hence, the limit α dα ν (µ0 ) := lim Dµ0 ,ν (x) x↑ν

Z = −ν + µ0 − α2 lim t↓0

exists. Actually, for a.e. k ∈ Rd ,

dk Rd

|λ(k)|2 ων (k) − ν + t

(2.13)


00072


0
0 and lim = t↓0 ων (k) − ν + t ων (k) − ν + t ω(k) ω(k)

227

(a.e.) ,

√ and we assumed λ/ ω ∈ L2 (Rd ) in (A.1), moreover set ων (k) := ω(k) + ν(ν > 0, k ∈ Rd ). So, by the Lebesgue dominated convergence theorem, we have Z 2 dα ν (µ0 ) = −ν + µ0 − α

dk Rd

|λ(k)|2 . ω(k)

(2.14)

We may put for ν = 0 Dµα0 (z) := Dµα0 ,0 (z) ,

(2.15)

dα (µ0 ) := dα 0 (µ0 ) .

(2.16)

The Wigner–Weisskopf model has a conservation law for a kind of the particle number in the following sense: We define NP :=

1 + σ3 ⊗ I + I ⊗ Nb , 2

(2.17)

which was already introduced in [29, §6], where Nb is the boson number operator, Nb := dΓ(1) =

X

`P (`) .

(2.18)

`=0

Here (2.18) is the spectral resolution of Nb , and P (`) is the orthogonal projection onto the `-particle space in Fb for each ` ∈ {0} ∪ N. The spectral resolution of NP is given as X `P` , (2.19) NP = `=0

where

 1 − σ3   ⊗ P (0) P` = 1 +2 σ 1 − σ3 3   ⊗ P (`−1) + ⊗ P (`) 2 2

if ` = 0 , (2.20) if ` ∈ N .

Hα (µ0 ; ν) is reduced by P` F for every α ∈ R and each ` ∈ {0} ∪ N, i.e., P` Hα (µ0 ; ν) ⊂ Hα (µ0 ; ν)P` , which means that D(P` Hα (µ0 ; ν)) ⊂ D(Hα (µ0 ; ν)P` ) , P` Hα (µ0 ; ν)Ψ = Hα (µ0 ; ν)P` Ψ

for Ψ ∈ D(P` Hα (µ0 ; ν))

(2.21)


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(see [31, p. 278] ). So, for every α ∈ R, Hα (µ0 ; ν) is decomposed to the direct sum of H`,α (µ0 ; ν)’s as Hα (µ0 ; ν) =

∞ M

H`,α (µ0 ; ν) ,

(2.22)

`=0

where H`,α (µ0 ; ν) is self-adjoint on the closed subspace F` defined by F` := P` F

(2.23)

for each ` ∈ {0} ∪ N and F=

∞ M

F` .

(2.24)

`=0

The proof of the above statement is that, for instance, we have only to extend [31, Problem 3.29] to its infinite version by repeating [31, Problem 3.29] with the closedness of Hα (µ0 ; ν). We call F` the `-sector. We define a vector Ω0 ∈ F0 by ! ! 0 0 0 . (2.25) ⊗ Ω0 = Ω := 1 Ω0 Then, we have kΩ0 k = 1 .

(2.26)

For every f ∈ D(ˆ ων ) (ν ≥ 0), we define a vector Ω1 (f ) ∈ F1 by ! ! ! 1 0 Ω0 1 ∗ . ⊗ Ω0 + ⊗ a(f ) Ω0 = Ω (f ) := a(f )∗ Ω0 0 1

(2.27)

Then, we have kΩ1 (f )k = (1 + kf k20 )1/2 .

(2.28)

When a zero Eµα0 ,ν of Dµα0 ,ν (z) exists in (−∞, ν), we define a function by gµα0 ,ν (k) := −α

λ(k) ∈ D(ˆ ων ) , ων (k) − Eµα0 ,ν

k ∈ Rd .

(2.29)

Especially, we may put for ν = 0 Eµα0 := Eµα0 ,0 ,

(2.30)

gµα0 := gµα0 ,0 .

(2.31)

For a self-adjoint operator T , we denote the set of all essential spectra of T by σess (T ), and pure point spectra by σpp (T ).


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By the definition (2.4) of the Hamiltonian Hα (µ0 ; ν), the free Hamiltonian of the Wigner–Weisskopf model is H0 (µ0 ; ν) for every µ0 ∈ R and ν ≥ 0. Then, it is clear that σpp (H0 (µ0 ; ν)) = {0, µ0 } ,

(2.32)

σess (H0 (µ0 ; ν)) = [min{0, µ0 }, ∞) ,

(2.33)

0 and µ0 are simple ,

(2.34)

the unique eigenvector of 0 is Ω0 ∈ F0 ,

(2.35)

and the unique eigenvector of µ0 is Ω1 (0) ∈ F1 .

(2.36)

2.1. Review The following theorem follows from [7, Proposition 6.13, Theorems 6.14 and 6.15] with a slight revision. Theorem 2.1. (a) Let ν, dα ν (µ0 ) ≥ 0. Then, 0 ∈ σpp (Hα (µ0 ; ν)) ,

(2.37)

σess (Hα (µ0 ; ν)) = [ν, ∞) .

(2.38)

In particular, 0 is the ground state energy of Hα (µ0 ; ν) with its unique ground state Ω0 ∈ F 0 . √ 2 2 (b) Let dα ν (µ0 ) < 0 < ν and α kλ/ ων k0 ≤ µ0 . Then, {0, Eµα0 ,ν } ⊂ σpp (Hα (µ0 ; ν)) ,

(2.39)

σess (Hα (µ0 ; ν)) = [ν, ∞) ,

(2.40)

with 0 ≤ Eµα0 ,ν < ν. In particular, 0 is the ground state energy of Hα (µ0 ; ν). Moreover, √ if α2 kλ/ ων k20 < µ0 , then 0 < Eµα0 ,ν ; 0 is simple , and Ω0 ∈ F0 is the unique ground state of Hα (µ0 ; ν) ,

(2.41)

√ if α2 kλ/ ων k20 = µ0 , then 0 = Eµα0 ,ν ; Ω0 ∈ F0 and Ω1 (gµα0 ,ν ) ∈ F1

(c) Let

are the degenerate ground states of Hα (µ0 ; ν) . √ < 0 < ν and µ0 < α2 kλ/ ων k20 . Suppose that !

2

kλk20 2 λ , 2ν − µ0 > α √ − M (α, µ0 , ων ) + ων 0 M (α, µ0 , ων )

(2.42)

dα ν (µ0 )

where

Z M (α, µ0 , ων ) :=

dk Rd

|λ(k)|2 . √ ων (k) − µ0 + α2 kλ/ ων k20

(2.43)

(2.44)


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Then,

Eµα0 ,ν

{Eµα0 ,ν , 0} ⊂ σpp (Hα (µ0 ; ν)) ,

(2.45)

σess (Hα (µ0 ; ν)) = [Eµα0 ,ν + ν, ∞) ,

(2.46)

Eµα0 ,ν

< 0. In particular, is the ground state energy of Hα (µ0 ; ν) with with its ground state Ω1 (gµα0 ,ν ) ∈ F1 . Proof. Parts (a) and (b) follows from [7, Proposition 6.13, Theorem 6.14] . For (2.41), we did not state the simpleness of 0 in [7, Theorem 6.14(ii)] . But, if we √ use α2 kλ/ ων k20 < µ0 instead of [7, (6.60)], then we get α2 a(λ)∗ a(λ) ≤ µ0 Hb (ν). So, we obtain the simpleness in the same way as the proof of that in [7, Proposition 6.13(i)] by using the argument in [29] which derives [29, (6.14)] from [29, (6.3)] In order to prove part (c), we have only to modify [7, Theorem 6.15] . Here we note the following. By applying [7, Proposition 6.13(ii)] to our assumption, we have E0 (Hα (µ0 ; ν)) ≤ Eµα0 ,ν < µ0 , which is essential in [7, (6.72)] in the proof of [7, Theorem 6.15] , not µ0 < 0. Remark 2.4. We are also interested in the case for the large absolute value of the coupling constant (i.e. |α| 1). Fix µ0 and make |α| so large. Then, we have α dα ν (µ0 ) < 0. Thus, we have to investigate the case for dν (µ0 ) < 0 to investigate the case for large |α|. See Theorem 2.2 below. Remark 2.5. In [ν, ∞) for ν ≥ 0, we can make a different eigenvalue from both of Eµα0 ,ν and 0 by adding some conditions to ω(k) and λ(k) as we mentioned in [7, Remark 6.4] . Namely, as an effect of the scalar Bose field, a new eigenvalue appears in (ν, ∞). Remark 2.6. It is easy to check that

λ 2

√ − M (α, µ0 , ων ) > 0 .

ων 0

(2.47)

Let µ0 ≥ 0. Then, if ν = 0, then (2.43) does not hold by (2.47). Let µ0 < 0. Then, by the definition (2.44), we get Z |λ(k)|2 kλk20 dk = M (α, µ0 , ων ) < −µ0 −µ0 Rd since µ0 < 0 now, which implies that the left hand side of (2.43) > −µ0 since µ0 < 0 < M (α, µ0 , ων ). Thus, (2.43) is meaningful for the case of massive bosons only. In Theorem 2.1(c), we cannot show the ground state energy of Hα (µ0 ) for the massless bosons under the condition dα (µ0 ) < 0 alone as we remarked in Remark 2.6, but if we add the condition (S), then we can determine the pure point


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spectra of Hα (µ0 ) completely for the massless bosons by using Skibsted’s result [45, Theorem 3.1] . So, we need the following condition: T (S) λ(1) ∈ M0 M−2 , where λ(1) (k) :=

(d − 1)λ(k) ∂ λ(k) + , ∂|k| 2|k|

k ∈ Rd

(2.48)

considered as a distribution on C0∞ (Rd \ {0}). Remark 2.7. Assuming (S) practically amounts to assuming the infrared regularity condition, namely not the infrared singularity condition (1.8). Proposition 2.1. Assume (A.1), (S) and (1.8). Let ω(k) = |k| and dα (µ0 ) < 0. Then, σpp (Hα (µ0 )) = {Eµα0 , 0} ,

(2.49)

σess (Hα (µ0 )) = [Eµα0 , ∞)

(2.50)

1 . 4kλ(1) k20

(2.51)

for all α ∈ R with α2
0. Moreover, by and (2.52), we get Eµα0 < 0. The existence of a ground state follows from [18, Theorem 1] which is an improvement of [6, Theorem 1.3]. On the other hand, set β := α2 kλ(1) k20 . Then, we get 2(1 − β)−1 < 3 by (2.51). Thus, by [45, Theorem 3.7] which is an improvement of [29], we know that the total number of all eigenvectors of Hα (µ0 ; ν) is less that 3, namely 2. Therefore, we obtain (2.49), parts (b) and (c). (2.50) follows from [4, Theorem 3.6].


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2.2. Existence of a non-perturbative ground state We use the following condition in the main theorem below: (A.2) The functions, ω(k), λ(k), are continuous, and lim ω(k) = ∞ .

(2.53)

|k|→∞

Moreover, there exist constants γω > 0 and Cω > 0 such that |ω(k) − ω(k 0 )| ≤ Cω |k − k 0 |γω (1 + ω(k) + ω(k 0 )) ,

k, k 0 ∈ Rd .

(2.54)

Theorem 2.2. Let ν ≥ 0. Assume (A.1). Then, (a) there exists αWW (ν) > 0 such that {Eµα0 ,ν , 0} ⊂ σpp (Hα (µ0 ; ν))

(2.55)

with E0 (Hα (µ0 ; ν)) < min{Eµα0 ,ν , 0} ,

(2.56)

σess (Hα (µ0 ; ν)) = [E0 (Hα (µ0 ; ν)) + ν, ∞)

(2.57)

for every α ∈ R with |α| > αWW (ν). (b) Let ν > 0 (massive bosons). Assume (A.2) in addition. Then, there exists a ground state ΨWW ∈ F of Hα (µ0 ; ν), namely Hα (µ0 ; ν)ΨWW = E0 (Hα (µ0 ; ν))ΨWW , such that {E0 (Hα (µ0 ; ν)), Eµα0 ,ν , 0} ⊂ σpp (Hα (µ0 ; ν)) , / F0 ∪ F1 ΨWW ∈

with (2.56)

(2.58) (2.59)

for every α ∈ R with |α| > αWW (ν). (c) Let ν = 0 (massless bosons). Assume (A.2), ∇ω ∈ L∞ (Rd ) and (1.8) in addition. Then, there exists a ground state ΨWW ∈ F of Hα (µ0 ; 0) such that (2.56), (2.58) and (2.59) hold for every α ∈ R with |α| > αWW (0). Remark 2.8. When the case of massive bosons (ν > 0), we can apply the regular perturbation theory to WW model for sufficiently small absolute value of the coupling constant α, and then Theorem 3.1 says that we get either Eµα0 ,ν or 0 as the ground state energy. Theorem 2.2 means that, for sufficiently large absolute value of the coupling constant, a non-perturbative ground state appears as an influence of the scalar Bose field, and its ground state energy is so low that we cannot conjecture it by the regular perturbation theory for sufficiently small absolute value of the coupling constant. We will clarify the other non-perturbative properties of ΨWW in [8]. For other models, the similar phenomenon were investigated by Hiroshima and Spohn [27].


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The following Figures 1–4 show the spectra which we had found, so not all: (I) For |α| < αWW (ν): (I-a) Let dα ν (µ0 ) ≥ 0. Then Point Spectra

Essential Spectrum

PA PPP AA PPPP AU ν PPP x

h

µ0 moves

0

?

Ground State Energy

PPq

?

Excited State Energy

?

(I-b) Let dα ν (µ0 ) < 0. √ (I-b-1) If µ0 > α2 kλ/ ων k20 , then Point Spectra

P A PPP PP A PP A PP ν µ0 movesPP U A q P

x

x

Essential Spectrum

h

0 Eµα0 Excited State Energy Ground State Energy

?


√ (I-b-2) If µ0 = α2 kλ/ ων k20 , then Point Spectrum

P PPP PP P

x

ν

Essential Spectrum PP

PP

h

µ0 moves

0 = Eµα0 Degenerate Ground State Energy

P q P

?


√ (I-b-3) If µ0 < α2 kλ/ ων k20 , and all other hypothese in Theorem 2.1(c) hold, then Point Spectra

P A PPP PP A PP A PP PP U Eµα0 + ν µ0 moves A q P

x

x

Eµα0 0 Excited State Energy Ground State Energy

Essential Spectrum

h

?


Appearance or disappearance of depends on the condition for λ by an effect of the scalar Bose field as non-purterbative eigenvalue. Fig. 1.

Spectra we had found for WW model (I) for ν > 0.


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Masao hirokawa

(II) For |α| > αWW (ν): If all hypothesse in Theorem 2.2(b) hold, then Point Spectra

?

PP AA PPPP PP A PP PP U A q µ0 moves P g.s.e.+ν

x

Ground State Energy

x

Essential Spectrum

h

Eµα0 0 Excited State Energies

?


Appearance or disappearance of depends on the condition for λ, and F appears by an effect of the scalar Bose field. Both of F and are non-perturbative eigenvalues. Fig. 2.

Spectra we had found for WW model (II) for ν > 0

For |α| < αWW (0): (I-a) If dα (µ0 ) ≥ 0, then Point Spectra

P AAPPPPP PPPP AAU Pq x h PP

Essential Spectrum

?

µ0 moves

0 Ground State Energy

?


?

(I-b) If all hypotheses in Proposition 2.1 hold, then Point Spectra

Essential Spectrum

A A

x

A AU

x

h

?

µ0 moves

Eµα0 0 Excited State Energy Ground State Energy

Appearance or disappearance of depends on the condition for λ by an effect of the scalar Bose field as non-perturbative eigenvalue. Fig. 3.

Spectra we had found for WW model (I) for ν = 0.

(II) For |α| > αWW (0): If all hypotheses in Theorem 2.2 (c) hold, then Point Spectra

?

P A PPP PP A PP A PP PP U A q µ0 moves P

x

Eµα0

Ground State Energy

x

Essential Spectrum

h

?

0

Excited State Energies


Appearance or disappearance of depends on the condition for λ, and F appears by an effect of the scalar Bose field. Both of F and are non-perturbative eigenvalues. Fig. 4.

Spectra we had found for WW model (II) for ν = 0.


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235

2.3. Proof of Theorem 2.2 Here, we set the following condition |λ(k)|2 is not Lebesgue integrable for all x ∈ (ν, ∞), and |ων (k) − x| we prove the following lemma: (D)ν The function

Lemma 2.1. Let ν ≥ 0, and Ψ ∈ F be an eigenvector of Hα (µ0 ; ν). (a) If Ψ ∈ F0 , then Ψ = cΩ0 for some c ∈ C \ {0}. 1 α (b) If Ψ ∈ F1 is a ground state and dα ν (µ0 ) 6= 0, then Ψ = cΩ (gµ0 ,ν ) for some α c ∈ C \ {0}, and dν (µ0 ) < 0. (c) If Ψ ∈ F1 with eigenvalue below ν, then Ψ = cΩ1 (gµα0 ,ν ) for some c ∈ C \ {0}, and dα ν (µ0 ) < 0. (d) Fix ν ≥ 0 arbitrarily, and assume dα ν (µ0 ) 6= 0 and (D)ν . If Ψ ∈ F1 , then Ψ = cΩ1 (gµα0 ,ν ) for some c ∈ C \ {0}, and dα ν (µ0 ) < 0. Proof. It is easy to see that there exists just one eigenvector Ω0 in F0 but the constant-times Ω0 . So, we have part (a). In the case of parts (b), (c) and (d), we can write as Ω0 ∈ F1 , g ∈ D(ˆ Ψ =: ων ) , (2.60) a(g)∗ Ω0 and Hα (µ0 ; ν)Ψ = EΨ .

(2.61)

If Ψ is a ground state and dα ν (µ0 ) 6= 0, then E ≤ 0 since 0 ∈ σpp (Hα (µ0 ; ν)) by [7, Proposition 6.13] , and E 6= 0 since dα ν (µ0 ) 6= 0. So especially E < ν. By (2.61), we get (µ0 − E + α(λ, g)0 )Ω0 = 0 ,

(2.62)

a({αλ + (ων − E)g})∗ Ω0 = 0 .

(2.63)

Considering (Ω0 , L.H.S. of (2.62))Fb = 0, we get µ0 − E + α(λ, g)0 = 0 .

(2.64)

Considering (a(f )∗ Ω0 , L.H.S. of (2.63))Fb = 0 for every f ∈ L2 (Rd ), we get (f, αλ + (ων − E)g)0 = 0 , so that αλ + (ων − E)g = 0 in L2 (Rd ), which implies that g(k) = −α

λ(k) ων (k) − E

a.e. k ∈ Rd .

(2.65)


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Inserting (2.65) into (2.64), we have

Z

µ0 − E − α2

dk Rd

|λ(k)|2 = 0. ων (k) − E

α So, E is a zero of below ν. Suppose that dα ν (µ0 ) ≥ 0. Then Dµ0 ,ν (z) has α no zero in Cν by [7, Lemma 6.5(i)] , which is a contradiction. Thus, dν (µ0 ) < 0. So, by [7, Lemma 6.5(ii)] , Dµα0 ,ν (z) (z ∈ C \ [ν, ∞)) has a unique simple zero. Thus, E = Eµα0 ,ν , and g = cgµα0 ,ν for some c ∈ C \ {0}. Thus, we have part (b), and the proof of part (b) includes that of part (c). Under assumptions in part (d), Dµα0 ,ν (z) does not have a zero in [ν, ∞). Because (D)ν implies that Dµα0 ,ν (z) does not have a zero in (ν, ∞) by the similar reason in [7, Remark 6.4] . And dα ν (µ0 ) 6= 0 implies that E < ν in the same way above. Therefore, part (d) can be proved in the same way as part (b).

Dµα0 ,ν (z)

To show the existence of a non-perturbative ground state for sufficiently large |α|, we prove the following lemma. Lemma 2.2. Fix µ0 arbitrarily. Let ν ≥ 0. Then, even under infrared singularity condition (1.7), we have

2

2

α2 λ µ0 2 λ

− (2.66) min{0, µ0 } − α √ ≤ E0 (Hα (µ0 ; ν)) ≤ √ . ων 0 2 4 ων 0 Thus, for any parameter α = αt (t > 0) and sequence α = αn (n ∈ N) with |α| → ∞ as t ↑ ∞ and n ↑ ∞, lim E0 (Hα (µ0 ; ν)) = −∞ ,

(2.67)

|α|→∞

2

λ 2 λ 1 E0 (Hα (µ0 ; ν))

− √ ≤ lim sup ≤ − √ . ων α2 4 ων 0

|α|→∞

(2.68)

0

Remark 2.9. For (2.68), we cannot apply [6, Proposition 1.4] √ to E0 (Hα (µ0 ;∗ν)) ∗ ≡ (c + c)/ 2 and B2 ≡ i(c − because [6, (A.6)] does not hold now, namely B 1 √ c)/ 2 are not commutative. Proof of Lemma 2.2. We can define a self-adjoint operator Htest (ν) acting in F for ν ≥ 0 by µ0 α I ⊗ I + I ⊗ Hb (ν) + √ σ1 ⊗ φS (λ) Htest (ν) := 2 2   µ0 α √ φS (λ) Hb (ν) +   2 2 =  α (2.69) µ0  √ φS (λ) Hb (ν) + 2 2


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237

with D(Htest (ν)) = D(I ⊗ Hb (ν)). Then we get Htest (ν) =

1 ˜ α (µ0 ; ν)} {Hα (µ0 ; ν) + H 2

on D(I ⊗ Hb (ν)), which implies that, for all Ψ ∈ D(I ⊗ Hb (ν)) = D(Htest (ν)) = ˜ α (µ0 ; ν)) with kΨkF = 1, D(Hα (µ0 ; ν)) = D(H (Ψ, Htest (ν)Ψ)F = ≥

1 ˜ α (µ0 ; ν)Ψ)F } {(Ψ, Hα (µ0 ; ν)Ψ)F + (Ψ, H 2 1 ˜ α (µ0 ; ν))} . {E0 (Hα (µ0 ; ν)) + E0 (H 2

˜ α (µ0 ; ν)) by (2.8), we have Since E0 (Hα (µ0 ; ν)) = E0 (H E0 (Htest (ν)) ≥ E0 (Hα (µ0 ; ν))

for every ν ≥ 0 .

(2.70)

By [6, Proposition 1.4] , we get √ α2 µ0 − kλ/ ων k20 ≤ E0 (Htest (ν)) . 2 4

(2.71)

α Let ΨV H be the ground state of the van Hove model Hb (ν) + √ φS (λ) for ν > 0. 2 ! ΨV H √ µ0 α2 − kλ/ ων k20 . is the eigenvector of Htest (ν) with its eigenvalue Then 2 4 ΨV H So, by (2.71), we have E0 (Htest (ν)) =

√ α2 µ0 − kλ/ ων k20 2 4

for ν > 0. At last, by [7, Proposition 3.2(iii)] , we have E0 (Htest (ν)) =

√ α2 µ0 − kλ/ ων k20 2 4

for ν ≥ 0 .

(2.72)

By (2.70) and (2.72), we obtain the right inequality of (2.66). We can obtain the left inequality of (2.66) by [7, (2.21)] . The equalities (2.67) and (2.68) follows from (2.66) directly. Proof of Theorem 2.2. We have only to consider large coupling constants α’s α such that dα ν (µ0 ) < 0 by Remark 2.4 (so, Eµ0 ,ν < 0 by [7, Proposition 6.13(ii)]) . Then, (2.55) follows from [7, Proposition 6.13(ii)] . Fix ν ≥ 0 arbitrarily. We use the reductive absurdity. Suppose that there is no such an αWW (ν) that (2.56) holds for every α ∈ R with |α| > αWW (ν). Then, there exists a sequence {αn }n∈N such that E0 (Hαn (µ0 ; ν)) = Eµα0n,ν (< 0) and |αn | → ∞ as n → ∞. Since Eµα0n,ν is the zero of Dµα0n,ν (z), we get Z |λ(k)|2 αn 2 Eµ0 ,ν = µ0 − αn dk . ων (k) − Eµα0n,ν Rd


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Thus, we get Eµα0n,ν µ0 = 2 − α2n αn

Z dk Rd

|λ(k)|2 0. ων (k) − Eµα0n,ν

(2.73)

By (2.67) in Lemma 2.2 and Lebesgue’s dominated convergence theorem, we have lim inf (RHS of (2.73)) = lim (RHS of (2.73)) = 0 . n→∞

n→∞

(2.74)

By (2.68) in Lemma 2.2, we have √ √ 1 −kλ/ ων k20 ≤ lim sup(LHS of (2.73)) ≤ − kλ/ ων k20 , 4 n→∞ which contradicts to (2.74). Thus, we obtain (2.56). (2.57) follows from [5, Theorem 3.3]. Let ν > 0 now. Then, [6, (A.1)] is satisfied because the operator A in [6] is c∗ c now, and we assumed [6, (A.2), (A.4) and (A.5)] in our (A.1) √ and ∗ ≡ (c + c)/ 2 and (A.2). The condition [6, (A.3)] is satisfied for all α because B 1 √ ∗ B2 ≡ i(c − c)/ 2 are bounded operators, namely all aj ’s in [6] are zero now. Thus, by [6, Theorem 1.2] , Hα (µ0 ; ν) has a ground state ΨWW for ν > 0, which implies (2.58). The existence of a ground state ΨWW for the case of massless bosons (i.e. ν = 0) is due to [18, Theorem 1] which is an improvement of [6, Theorem 1.3] . (2.59) follows from Lemma 2.1. 3. Ground State Energy of Spin-Boson Model 3.1. New upper and lower bounds for the ground state energy In this subsection, we give new upper and lower bounds for the ground state energy in the case of massless bosons. Theorem 3.1 (upper bound without infrared cutoff ). Assume (A.1). For the Hamiltonian HSB of the spin-boson model without infrared cutoff (i.e. even under the infrared singularity condition (1.7)), an upper bound is given as follows: ESB (0) ≤ −

µ 2α 1.

(4.12)

By Corollary 3.1 and (4.11), we have the existence of δ < cµ,α in (4.5). Conditions (3.3) in (A.3) and (3.6) in (A.4) imply that Z Z |λ(k)|2 |λ(k)|2 2 dk < 2(1 − δ)α dk 2(1 − δ)α2 µ ω(k) + 2 ω(k) Rd Rd Z |λ(k)|2 by (3.2) < 2α2 dk ω(k) Rd Z |λ(k)|2 < µ − 2α2 by (3.6) in (A.4) dk ω(k) Rd Z |λ(k)|2 = µ 1 − α2 dk ω(k) µ2 Rd Z |λ(k)|2 2 dk ≤ µ 1−α ω(k)(ω(k) + µ2 ) Rd by the second of (3.27). So, we get Z 2 2 µ>α dk|λ(k)|

2(1 − δ) µ µ + ω(k)(ω(k) + 2 ) ω(k) + µ2 Rd Z Z |λ(k)|2 |λ(k)|2 − 2δα2 dk dk , = 2α2 ω(k) ω(k) + µ2 Rd Rd

which implies that −

µ − δα2 2

Z dk Rd

|λ(k)|2 < −α2 ω(k) + µ2

Z dk Rd

|λ(k)|2 . ω(k)

(4.13)

Inequality (4.6) follows from (3.4) and (4.13). The limit (4.7) follows from (4.1), (4.2) and (4.6). We obtain (4.8) by (3.4) in Corollary 3.1 and (3.8) in Theorem 3.2.


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Acknowledgments H. Spohn proposed the problem of expressing ESB (0) independently of the existence of its ground state to me when I held discussions on [25] with him, though I had assumed the existence in [25]. So, this is the beginning of the problem I dealt with in this paper. I wish to thank him for giving me the beginning of the problem. I argued the problem about ESB (0) given by the limit (4.2) of the explicit expression at for ESB (ν) in [25] with V. Bach and A. Elgart when I visited Technische Universit¨ Berlin during September 8–10, ’98. Then the above problem for (4.1)–(4.4) on the survival of µ arose. I wish to thank them for arrangements of my visiting Technische Universit¨ at Berlin and the hospitality. I am indebted to A. Arai for useful discussions which proofs in this paper were based on. I thank H. Spohn and F. Hiroshima for their hospitality at Technische Universit¨ at M¨ unchen during April 15–22, ’99, and discussing Spohn’s unpublished results. I wish to express H. Spohn, R. A. Minlos, H. Ezawa, K. Watanabe, K. Yasue, M. Jibu, F. Hiroshima, and the referee for valuable advice. I wish to thank J. Derezi´ nski for discussing several aspects about the generalized spin-boson model at the summer school “Schr¨ odinger Operators and Related Topics,” Sh¯ onan Village Center, July 5–9, ’99, and also C. Gérard for telling me how to get his recent result which broke through a wall in Theorem 2.2(c). My research is supported by the Grant-In-Aid No. 11740109 for Encouragement of Young Scientists from Japan Society for the Promotion of Science.

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COMPLETIONS OF 2-TORSION KN-ALGEBRAS OF GENUS 1

L. GUERRINI∗ Department of Mathematics, Universit` a di Roma “Tor Vergata” Via della Ricerca Scientifica, I-00133, Rome, Italy E-mail : [email protected]

Received 14 January 1999 Revised 20 October 1999 Krichever–Novikov algebras KN of genus 1 with markings which are two 2-torsion points are related to a family Wf of deformations of the Witt algebra W, where f varies in the space of even polynomials with vanishing constant terms. An isomorphism between the formal (resp. analytic) completion of these KN-algebras with those of the Witt algebra is proved. Central extensions of these algebras are also defined and their formal completion is proved to be isomorphic to that of the Virasoro algebra Vir. For paola, my parents and my brothers.

1. Introduction In 1987, Krichever and Novikov [7] started the study of Lie algebras of meromorphic vector fields on compact Riemann surfaces of arbitrary genus, which are holomorphic outside two distinguished points P− , P+ (called markings). These algebras are nowadays called Krichever–Novikov algebras (or KN-algebras) and they can be interpreted as a generalization of the Witt algebra W. In fact in the case of a compact Riemann surface of genus 0 the surface is the Riemann sphere and choosing the two markings to be 0 and ∞ we obtain the Witt algebra. The generalization of these algebras to the case of an arbitrary (but finite) number of markings was then independently started by Dick [3] and Schlichenmaier [8, 9]. We are interested in the Lie algebra of meromorphic vector fields on compact Riemann surfaces of genus 1 with markings P− = −z0 and P+ = +z0 . A compact Riemann surface of genus 1 is a torus T and it is represented as a quotient C/L, L being a lattice in C, L = Z(2ω1 ) ⊕ Z(2ω3 ) with ω1 , ω3 ∈ C and Im(ω3 /ω1 ) > 0. In the following we shall write z for both a point in C and its image in C/L. Since the vector field d/dz has no zeros or poles on T , the map f (z) 7→ f (z)(d/dz) is a vector space isomorphism from the field of meromorphic functions on T , holomorphic on T \{P−, P+ }, to our KN-algebra. Meromorphic functions on the torus are nothing else but periodic meromorphic functions, that is elliptic functions. As an application ∗ Work

supported by the istituto nazionale di alta matematica “f. severi”. 253


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of the Riemann–Roch theorem [10], a basis for the elliptic functions with poles at −z0 , +z0 is 1 1 σ m− 2 (z − z0 )σ(z + 2mz0 ) m∈Z+ Vm (z) = 1 2 σ m+ 2 (z − z0 ) (σ denoting the Weierstrass sigma function). Note Vm (z) has a zero of order m − 12 at P+ and a pole of order m + 12 at P− . Moreover the double periodicity requires a simple zero at −2mz0 . A basis for the KN-algebra is therefore d 1 m∈Z+ . em (z) = Vm (z) dz 2 Note that the choice of the point z0 is not free. In fact if z0 is a n-torsion point of T , that is nz0 = 0 (n ∈ Z) , then there is a cancellation of poles and zeros in the expression of the Vm ’s and so they do not build a basis. In [2], it was shown that the following choice of 2-torsion points, P− = 0 ,

P+ =

1 , 2

as markings with lattice L = h1, τ i, Im(τ ) > 0, − 21 ≤ Re τ < 12 , makes the structure of the KN-algebra more transparent and a connection to the Witt algebra can be investigated. In fact this algebra turns out to be embedded into a two parametric family of Lie algebras Lp,q (p, q ∈ C). Moreover this family has a one-dimensional li = z i+1 , i ∈ Z and bracket central extension. The family Lp,q has bases b  lm ]p,q = (m − n)(b lm+n + pb lm+n+2 + qb lm+n+4 ) , [b ln , b    lβ ]p,q = [b lα , b lβ ] , [b lα , b    b b lα+n + (n − α + 1)pb lα+n+2 + (n − α + 2)qb lα+n+4 , [lα , ln ]p,q = (n − α)b with α, β denoting odd indices and n, m even indices. The KN-algebras are obtained for the special values p = 3e1 , where e 1 = ℘τ

1 , 2

q = (e1 − e2 )(e1 − e3 )

e2 = ℘τ

1 (1 + τ ) , 2

e 3 = ℘τ

1 τ 2

.

For p = q = 0 the family Lp,q reduces to the Witt algebra W and so it is a deformation of it. But in the case of the KN-algebra this can never happen for any τ , because it is well known that (e1 − e2 )(e1 − e3 ) is always different from zero.


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In [4–6] the deformations Lp,q were generalized to a large family Wf of deformations of W, where the parameter f varies in an infinite dimensional vector space. Moreover the existence of an isomorphism c, cf ' W W where b refers to suitable completions of W, was proved. The aim is now to show similar results for the algebras KN . Moreover since the Witt algebra is known to have a central extension, the Virasoro algebra, we would like to extend our results to the case in which we work with the completions of the centrally extended algebras. In Sec. 2, we consider KN-algebras with markings P− = 0, P+ = ω1 . An expression for a basis is given and the Lie algebra structure is described. Note that one of the two markings can always be chosen to be 0 by translation. Any other choice of a 2-torsion point as P+ between ω2 = ω1 + ω3 and ω3 would give a similar situation. In Sec. 3, we recall results about the family Wf and prove the existence of an isomorphism between formal and smooth completions of our KN-algebras with those of the Witt algebra. It is interesting to compare these results with the remark in [7] that the KNalgebras may be densely imbedded inside the smooth Witt algebra. In Sec. 4, we consider central extensions Lep,q of the two parametric family of Lie algebras Lp,q and show the existence of an isomorphism between the formal e d ' Vd ir. completions of Lep,q and Vir. In particular we have KN 2. The Lie Algebra KN of Meromorphic Vector Fields on a Torus Let T be a torus with lattice L = Z(2ω1 ) ⊕ Z(2ω3 )

ω3 ω1 , ω3 ∈ C, Im >0 , ω1

and let’s consider the following two markings P− = 0 and P+ = ω1 . Let KN be the Lie algebra of meromorphic vector fields on T = C/L, which are holomorphic on T \{P−, P+ }. In order to describe all functions on the torus T let’s consider the so-called Weierstrass ℘-function X 1 1 1 − 2 . ℘(z) = 2 + z (z − ζ)2 ζ ζ∈L\{0}

Note that this series is convergent to a holomorphic function for all z ∈ / L. It is even, doubly periodic and has a pole of second order at the lattice points, and no other. Moreover, if we differentiate ℘(z) term by term we obtain X 2 , ℘0 (z) = − (z − ζ)3 ζ∈L


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that is a doubly periodic, odd function with a pole of order 3 at the lattice points. The functions ℘ and ℘0 are of course linearly independent but algebraically dependent since (℘0 (z))2 = 4(℘(z) − e1 )(℘(z) − e2 )(℘(z) − e3 )

(2.1)

with e1 = ℘(ω1 ) ,

e2 = ℘(ω2 ) ,

e3 = ℘(ω3 ) (ω2 = ω1 + ω3 ) .

Note that the ei ’s are all distinct and satisfy the identity e1 + e2 + e3 = 0 . Let {Lj }j∈Z denote the following infinite set of functions   L2k = (℘(z) − e1 )k  L2k+1 = 1 ℘0 (z) (℘(z) − e1 )k−1 2 with k ∈ Z. Proposition 2.1. The functions {Lj }j∈Z are a basis for the space of meromorphic functions on T, which are holomorphic outside {P− , P+ }. Proof. Up to reindexing and rewriting, this is exactly the same basis introduced in [2] (with ω1 = 12 and ω3 = τ2 ). A basis for the Lie algebra KN of meromorphic vector fields on T = C/L, holomorphic on T \{P−, P+ }, is therefore given by Ei (z) = Li (z)

d dz

(i ∈ Z) .

For convenience the index shift bi = Ei+1 E will be used. Moreover note that the ℘-Weierstrass function is actually a function of z, ω1 and ω3 . Therefore we should write bi (z|ω1 , ω3 ) bi (z) = E E and p = p(ω1 , ω3 ) ,

q = q(ω1 , ω3 )

in order to remind their dependence on the lattice. Corollary 2.1. The Lie algebra structure of the Krichever–Novikov algebra KN is  b b b b b   [En , Em ] = (m − n)(Em+n + pEm+n+2 + q Em+n+4 ) ,  bβ ] = (β − α)E bα+β , bα , E [E    b b bα+n + p(n − α − 1)E bα+n+2 + q(n − α − 2)E bα+n+4 , [Eα , En ] = (n − α)E


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where α, β denote odd indices as well as m, n even indices and p = 3e1 ,

q = (e1 − e2 )(e1 − e3 ) .

Proof. It is straightforward using the definition of [·, ·] and the identity (2.1). 3. Formal and Analytic Completions of the Lie Algebra KN Let W be the Lie algebra of vector fields on the unit circle S 1 with finite Fourier series expansion and bracket d dg d dh d −h . = g g ,h dz dz dz dz dz This algebra is called the Witt algebra and through the assignment g(z) → g(z)d/dz it can be identified with the Lie algebra of Laurent polynomials X cj z j (cj ∈ C) g= |j|≤N

with bracket [g, h] = gh0 − g 0 h . In [4–6] a family Wf of deformations of the Lie algebra W were defined, the parameter f varying in the space E of all even polynomials with vanishing constant terms. Its vector space structure is the same as W and bracket  (g, h odd) , [g, h]f = (1 + f )[g, h]    (g, h even) , [g, h]f = [g, h]    1 0 [g, h]f = (1 + f )[g, h] + 2 f gh (g even, h odd) , with g, h ∈ W and 0 denoting differentiation with respect to z. For f = 0, Wf reduces to the Witt algebra W and for f = pz 2 + qz 4 to the family Lp,q in which the Krichever–Novikov algebras KN are embedded. b c c o completion of W (resp. Wf ), where f ∈ E = n Let W (resp. Wf ) be the adic 2 4 b b b f = f1 z + f2 z + · · · , fj ∈ C , that is replace the underlying vector space of Laurent polynomials by formal Laurent series X cj z j . j≥−N

Let (ak )k>1 and (bk )k>1 be scalars uniquely determined from the power series identities (in the variable z)  ∞ Y   (1 + ak z k ) = 1 , (1 + z)    k=1

    

∞ Y (1 + bk z k )2 = 1. 1 + ak z k

k=1


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Note that, for all k > 1, explicit formulae for the ak ’s and bk ’s can be given [5]  X k d (−1)k k   a + = (−1) d add ,  k   k k  1≤d 1, let Tk : W maps defined by ( ak if g is odd , k Tk g = ck,g f g , ck,g = bk if g is even . These maps Tk are the basic tools in the construction of an isomorphism Sf cf and W. c In fact Sf can be built as between the Lie algebras W Sf = :

∞ Y

(1 + Tk ) :

k=1

where :

: means the product is taken as

(1 + T1 )(1 + T2 )(1 + T3 ) · · · . Q c The requirement heuristically is that : N k=1 (1 + Tk ) : reduces Wf to a Lie algebra N c whose bracket is congruent to the bracket in W c module terms of degree > N +1 W f in f and f 0 . The result is the following [5]. Theorem 3.1. The map Sf = lim (1 + T1 )(1 + T2 ) · · · (1 + TN ) N →∞

c cf and W. (limit taken in the adic topology) gives an isomorphism between W c∞ ) be the smooth completion of W (resp. W cf ), that is we c∞ (resp. W Let W f replace the underlying vector space by ( ) X ∞ 1 n −M gn z , |gn | = O(|n| ) for all M > 0 . C (S ) = g = n∈Z


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Let now the parameter f ∈ Eb∞ that are even. The smooth analogue of Theorem 3.1 is the following [5]. Theorem 3.2. If sup|f | < 1 and kf k1 < 1, k · k1 denoting the l1 -norm, then the map Sf = :

∞ Y

(1 + Tk ) :

k=1

c∞ . c∞ and W is an isomorphism between W f In particular, if f is a Laurent polynomial, X ui z 2i , f= i

then it is sufficient to have the condition kf k1 < 1, X |ui | < 1 . i

Let now apply these results to the KN algebras introduced in Sec. 2. Theorem 3.3. Let f = pz 2 + qz 4 . Then the KN-algebra KN is isomorphic to the algebra Lp,q , where p = 3e1 , q = (e1 − e2 )(e1 − e3 ). Moreover d 'W c. KN Proof. Using the map bi → b li E

(i ∈ Z) ,

we have an isomorphism between KN and Lp,q . The statement now follows from Theorem 3.1 taking f = pz 2 + qz 4 .

For the smooth case, let recall the following useful expressions [1] for the functions e1 , e2 , e3 : )  2 ( ∞ X π η h2k 1    e1 = +2 + ,   ω1 ω1 4 (1 − h2k )2   k=1    2 X ∞  π η h2k−1 +2 , e2 = −  ω1 ω1 (1 + h2k−1 )2  k=1    2 X  ∞   π η h2k−1   ,  e3 = − ω − 2 ω (1 − h2k−1 )2 1 1 k=1


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where h = eπiτ , τ =

ω3 , Im τ > 0, and η is a constant. ω1

Theorem 3.4. Let ω3 = iλω1 with λ ∈ R>0 . If |ω1 | is sufficiently large, for the corresponding KN-algebra KN we have the isomorphism of smooth completions d KN

∞

c∞ . 'W

Proof. As |ω1 | → ∞, |e1 |, |e2 | and |e3 | become sufficiently small and in particular we have |p(ω1 , ω3 )| < 12 and |q(ω1 , ω3 )| < 12 . The statement now follows from Theorem 3.3. Remark 3.1. These results are interesting because they essentially say that the KN-algebra and the Witt algebra can be identified at the completion level. We refer c∞ . again to the remark in [7] about the dense imbedding of KN-algebras in W 4. Central Extensions of KN and its Formal Completions In [2] a one-dimensional central extension Lep,q of the two parametric family Lp,q was defined. This is a short exact sequence 0 → Cλ → Lep,q → Lp,q with bracket lj ]ep,q = [b li , b lj ]p,q ⊕ χ eij [b li , b

(i, j ∈ Z) ,

with 



X X − χ eij =   clik ckjl k 2, − |Dπ |2−p := W Res(|Dπ |2−p ) is a positive quadratic form on each Aσ , with unique minimum πσ . The representation πσ is given by A acting as multiplication operators on the Hilbert space L2 (X, Sσ ) and Dπσ as the Dirac operator of the lift of the Levi– Civita connection to the spin bundle R Sσ .√ p R R gd x where R is the scalar curvature For p > 2 − |Dπσ |2−p = − (p−2)c(p) 12 X and c(p) =

2[p/2] . (4π)p/2 Γ(p/2 + 1)

(65)

Remark. Since, as is well known, every spin manifold gives rise to such data, the above theorem demonstrates a one-to-one correspondence (up to unitary equivalence and spin structure preserving isometries) between spin structures on spin manifolds and real commutative geometries. 4.2. Proof of (1) and (2) Without loss of generality, we will make the simplifying assumption that π is faithful on A. This allows us to identify A with π(A) ⊂ B(H), and we will simply write A. ¯ is a C ∗ -subalgebra of B(H). Then As π is a ∗ -homomorphism, the norm closure, A, ¯ is a compact, Hausdorff the Gelfand–Naimark theorem tells us that X = Spec(A) space. Since A is dense in its norm closure, each state on A (defined with respect to the C ∗ norm of A) extends to a state on the closure, by continuity. Recall that we are assuming that A is imbued with a norm such that the C ∗ -condition is satisfied for elements of A; thus here we mean continuity in the norm. Hence ¯ The connectivity of such a space is equivalent to the nonSpec(A) = Spec(A). existence of nontrivial projections in A¯ ∼ = C(X). So let p ∈ A¯ be such that p2 = p. Then [D, p] = [D, p2 ] = p[D, p] + [D, p]p = 2p[D, p] .

(66)

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So (1 − 2p)[D, p] = 0 implying that [D, p] = 0. By the irreducibility of π, we must have p = 1 or p = 0. Hence A contains no non-trivial projections and X is connected. Note that the irreducibility also implies that [D, a] 6= 0 unless a is a scalar. Also, as there are no projections, any self-adjoint element of A has only continuous spectrum. The reader will easily show that Eq. (64) does define a metric on X, [20]. The topology defined by this metric is finer than the weak∗ topology, so functions continuous for the weak∗ topology are automatically continuous for the metric. Furthermore, elements of A are also Lipschitz, since for any a ∈ A, k[D, a]k ≤ 1, we have |a(x) − a(y)| ≤ d(x, y). Thus for any a ∈ A we have |a(x) − a(y)| ≤ k[D, a]kd(x, y). Later we will show that the metric and weak∗ topologies actually ¯ is finitely generated. This agree. This will follow from the fact that A, and so A, ∼ ¯ also implies the separability of C(X) = A, which is equivalent to the metrizability of X. This will complete the proof of (1) and (2), but it will have to wait until we have learned some more about A. The last point of (2) is that the metric is invariant under unitary transformations. That is if U : H → H is unitary [U DU ∗ , U aU ∗ ] = U [D, a]U ∗ ⇒ k[U DU ∗ , U aU ∗ ]k = k[D, a]k .

(67)

So while D will be changed by a unitary change of representation D := Dπ → U DU ∗ := DUπU ∗ = Dπ + U [Dπ , U ∗ ] ,

(68)

commutators with D change simply. For this reason, when we only need the unitary equivalence class of π, we drop the π, and write Ω∗D (A) for π(Ω∗ (A)), where the D is there to remind us that this is the representation of Ω∗ (A) induced by D and the first order condition. 4.3. Proof of (3) and remainder of (1) and (2) Before beginning the proof of (3), which will also complete the proof of (1) and (2), let us outline our approach, as this is the longest, and most important, portion of the proof. 4.3.1. Generalities This section deals with the various bundles involved, their Hermitian structures and their relationships. We also analyse the structure of Ω∗D (A) and Λ∗D (A), particularly in relation to Hochschild homology. 4.3.2. X is a p-dimensional topological manifold We show that the elements of A involved in the Hochschild cycle X ai0 [D, ai1 ] · · · [D, aip ] π(c) = Γ = i

(69)

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provided by the axioms generate A. This is done in two steps. Results from 4.3.1 show that Γ is antisymmetric in the [D, aij ], and this is used to show that Ω1D (A) is finitely generated by the [D, aij ] appearing in Γ. The second step then uses the long exact sequence in Hochschild homology to show that A is finitely generated by the aij and 1 ∈ A. In the process we will also show that X is a topological manifold. 4.3.3. X is a smooth manifold We show here that A is C ∞ (X), and in particular that X is a smooth manifold. After proving that the weak∗ and metric topologies on X agree, we show that A is closed under the holomorphic functional calculus, so that the K-theories of A and C(X) agree. At this point we will have completed the proof of (1) and (2). 4.3.4. X is a spinc manifold The form of the operators D, |D| and D2 is investigated. The main result is that D2 is a generalised Laplacian while D is a generalised Dirac operator, in the sense of [27]. This allows us to show that Ω∗D (A) is (at least locally) the Clifford algebra of the complexified cotangent bundle. This is sufficient to show that the metric given by Eq. (64) agrees with the geodesic distance on X. As the representation of Ω∗D (A) is irreducible, we will have completed the proof that X is a spinc manifold. 4.3.5. X is spin It is at this point that we utilise the real structure. Furthermore, we reformulate Connes’ result to allow a representation of the Clifford algebra of an indefinite metric. This will necessarily involve a change in the underlying topology, which we do not investigate here. 4.3.6. Generalities The axiom of finiteness and absolute continuity tells us that \ Dom Dm H∞ = m≥1

is a finite projective A module. This tells us that H∞ ∼ = eAN , as an A module, for 2 some N and some e = e ∈ MN (A). Furthermore, from what we know about A and Spec(A), H∞ is also isomorphic to a bundle of sections of a vector bundle over X, say H∞ ∼ = Γ(X, S). These sections will be of some degree of regularity which is at least continuous as A ⊂ C(X). This bundle is also imbued with an Hermitian structure (· , ·)E : H∞ × H∞ → A such that (aψ, bη)S = a∗ (ψ, η)S b etc., R which provides us with an interpretation of the Hilbert space as H = L2 (X, S, −(· , ·)|D|−p ). We will return to the important consequences of the Hermitian structure and the measure theoretic niceties of the above interpretation later.

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As mentioned earlier, A is a Frechet space for the locally convex topology coming from the family of seminorms kak ,

kδ(a)k ,

∀a ∈ A,

kδ 2 (a)k, . . .

δ(a) = [|D|, a] .

(70)

Note that the first semi-norm in this family is the C ∗ -norm of A, and that δ n (a) makes sense for all a ∈ A by hypothesis. As the first semi-norm is in fact a norm, this topology is Hausdorff. In fact our hypotheses allow us to extend these seminorms to all of Ω∗D (A), and it too will be complete for this topology. Now let us turn to the differential structure. The first things to note are that DH∞ ⊂ H∞ , AH∞ ⊂ H∞ and k[D, a]k < ∞ ∀ a ∈ A. The associative algebra Ω∗D (A) is generated by A and [D, A], so Ω∗D (A)H∞ ⊂ H∞ . In other words Ω∗D (A) ⊂ End(Γ(X, S)) ∼ = End(eAN ) ∼ = {B ∈ MN (A) : Be = eB} .

(71)

The most important conclusion of these observations is that Ω∗D (A) and so Ω1D (A) are finite projective over A, and so are both (sections of) vector bundles over X, the former being (the sections of) a bundle of algebras as well. To see that Γ(X, S) is an irreducible module (of sections) for the algebra (of sections) Ω∗D (A), we employ Poincaré Duality. Suppose that the representation of Ω∗D (A) is reducible. Then by the finite projectiveness of both H∞ and Ω∗D (A), it decomposes as a finite sum of irreducible representations. To begin, let us assume that all the irreducible components are equivalent. Then Ω∗D (A) breaks up as a direct sum of equivalent blocks and there are projections pi , i = 1, . . . , n say, such that D=

n X

pi Dpj ,

i,j=1

n X

pi = 1 .

(72)

i=1

Elements of A evidently commute with these projections, and by the block diagonality of Ω∗D (A), [D, a] = Thus

n X

pi [D, a]pi

∀a ∈ A.

(73)

i=1

P i6=j

pi [D, a]pj = 0 and writing D=

n X

pi Dpi + B

(74)

i=1

with B = B ∗ , we have [B, a] = 0 for all a ∈ A. Hence B is an A-linear operator, and considering its action on any generating set for H∞ shows that it is bounded. The point of this is that regarding (H, D, Γ) as a K-cycle, the operator D, and so the cycle, is operator homotopic to D0 =

n X i=1

pi Dpi .

(75)

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As all the irreducible components of the representation are equivalent, the projec˜ = pi Dpi for any of the pi we have tions pi are all equivalent. Writing D ˜ ⊗ Cn , Γ ⊗ Idn ] [H, D, Γ] = [H, D0 , Γ] = [H ⊗ Cn , D ˜ Γ] ∈ K∗ (X) . = n[H, D, Hence the map

\

µ : K ∗ (X) → K∗ (X)

(76)

(77)

˜ Γ]. In given by the cap product with [H, D, Γ] sends [1] ∈ K ∗ (X) to n[H, D, ˜ particular, (n − 1)[H, D, Γ] is not in the image of µ, contradicting Poincaré Duality. A similar argument applies for a finite number of inequivalent irreducible representations appearing in the above decomposition. So what is Ω∗D (A)? The central idea for studying this algebra is the first order condition. When we construct this representation of Ω∗ (A) from π and A using D, the first order condition forces us to identify the left and right actions of A on Ω∗D (A), at least in the commutative case. Assuming as we are that the representation is faithful on A, we see that the ideal ker π is generated by the first order condition, ker π = hωa − aωia∈A,ω∈Ω∗ (A) = hfirst order conditioni .

(78)

So for ω = δf of degree 1 and a ∈ A, aδf − δf a ∈ ker π and (δf )(δa) + (δa)(δf ) ∈ δ ker π .

(79)

Equation (78) ensures that π ◦ b = 0, as Image(b) = ker π, so that we have a well-defined faithful representation of Hochschild homology π : HH∗ (A) → Ω∗D (A) .

(80)

If we write d = [D, ·], we see that the existence of junk is due to the fact that π ◦ δ 6= d◦π, and that this may be traced directly to the first order condition. Let us continue to write Ω∗D (A) := π(Ω∗ (A)) and also write Λ∗D (A) := π(Ω∗ (A))/π(δ ker π), and note that the second algebra is skew-commutative, and a graded differential algebra for the differential d = [D, ·]. Note that this notation differs somewhat from the usual,[6]. Note that Ω1D (A) and Λ1D (A) are the same finite projective A module, and we denote them both by Γ(X, E) for some bundle E → X, where as before we do not specify the regularity of the sections, only that they are at least continuous for the weak∗ topology and Lipschitz for the metric topology. The next point to examine is δ ker π = Image(δ ◦ b). This is easily seen to be generated by ker π = Image(b) and graded commutators (δa)ω − (−1)|ω| ωδa, for ω ∈ Ω∗ (A) and a ∈ A. Thus the image of π◦δ◦b in Ω∗D (A) is junk, and this is graded commutators. As elements of the form appearing in Eq. (79) generate π(δ ker π), it is useful to think of Eq. (79) as a kind of “pre-Clifford” relation. In particular, controlling the representation of elements of δ ker π will give rise to a representation

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of the Clifford algebra as well as the components of the metric tensor. More on that later. To help our analysis, define σ : Ω∗ (A) → Ω∗ (A)

by

σ(a) = a

for all a ∈ A

and σ(ωδa) = (−1)|ω| (δa)ω ,

for |ω| ≥ 0 .

Then, [2], we have b ◦ δ + δ ◦ b = 1 − σ, on Ω∗ (A). As π ◦ b = 0, and Image(π ◦ δ ◦ b) = Junk, we have Image(π ◦ (1 − σ)) = Junk. Thus while π(Ω∗ (A)) = Ω∗D (A) ∼ = Ω∗ (A)/hImage(b)i ,

(81)

passing to the junk-free situation gives b ∗ (A) . Ω∗ (A)/hImage(b) , Image(1 − σ)i ∼ = Λ∗D (A) ∼ =Ω

(82)

It is easy to see that b(1 − σ) = (1 − σ)b, so that ker b is preserved by 1 − σ. In fact, 1 − σ sends Hochschild cycles to Hochschild boundaries. For if bc = 0 for some element c ∈ Cn (A), then (1 − σ)c = (bδ + δb)c = bδc

(83)

which is a boundary. So ker b is mapped into Image(b) under 1 − σ and so when we quotient by Image(1 − σ) we do not lose any Hochschild cycles. So, π descends to a faithful representation of Hochschild homology with values in Λ∗D (A). In general, the Hochschild homology groups of a commutative and unital b ∗ (A) as a direct summand, [2], but we have shown that in fact algebra contain Ω c∗ (A) . HH∗ (A) ∼ = Λ∗D (A) ∼ =Ω

(84)

This is certainly a necessary condition for the algebra A to be smooth, but more important for us at this point is that all Hochschild cycles are antisymmetric in elements of Ω1D (A). In particular, π(c) = Γ 6= 0 in ΛpD (A) and is totally antisymmetric. 4.3.7. X is a p-dimensional topological manifold We claim that the elements aij , i = 1, . . . , n j = 1, . . . , p appearing in the Hochschild cycle Γ, along with 1 ∈ A, generate A as an algebra over C. Without loss of generality we take aij to be self-adjoint for i, j ≥ 1. Furthermore, we may also assume that k[D, aij ]k = 1. To show that the aij generate, we first show that the [D, aij ] generate Ω1D (A). Let Γ be the (totally antisymmetric) representative of the Hochschild p-cycle provided by the axioms. We write da := [D, a] for brevity, and similarly we write d for the action of [D, ·] on forms.

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Recall that Γ = π(c), and note that for any a ∈ A π(1 − σ)(cδa) = Γda − (−1)p daΓ = (1 − (−1)p (−1)p−1 )Γda = 2Γda .

(85)

Thus Γda is junk and so contains a symmetric factor, and da ∧ Γ = 0 for all a ∈ A. In order to show that the daij generate Ω1D (A) as an A bimodule, we need to show that da ∧ Γ = 0 implies that da is a linear combination of the daij . To do this, first write Γ=

n X

ai0 dai1 · · · daip =

i=1

n X

Γi .

(86)

i=1

Now suppose that n = 1, so that Γ = a0 da1 · · · dap . Then if (da ∧ Γ)(x) = (a0 da ∧ da1 · · · dap )(x) = 0

(87)

for all x ∈ X, elementary exterior algebra tells us that da(x) is a linear combination of da1 (x), . . . , dap (x) in each fibre. So if n > 1, the only thing we need to worry about is cancellation in the sum X (88) da ∧ Γi . Without loss of generality, we can assume that at each x ∈ X there is no cancellation in the sum X (89) Γi (x) . So for all I, J ⊂ {1, . . . , n} with I ∩ J = ∅, X X Γi (x) 6= − Γj (x) . i∈I

(90)

j∈J

If there were such terms we could simply remove them anyway, and we know in doing so we do not remove all the terms Γi as Γ(x) 6= 0 for all x ∈ X. Now suppose that for some x ∈ X and some I, J ⊂ {1, . . . , n} with I ∩ J = ∅ we have   ! X X da ∧ Γi (x) = −  da ∧ Γj  (x) . (91) i∈I

j∈J

If da(x) is a linear combination of any of the terms appearing in these Γi ’s, we are done. So supposing that da is linearly independent of the terms appearing in P i I∪J Γ , we have X X Γi (x) = − Γj (x) (92) i∈I

j∈J

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contradicting our assumption on Γ. Thus we may assume that no terms cancel, which shows that (da ∧ Γi )(x) = 0

(93)

for each i = 1, . . . , n. Hence if Γi (x) 6= 0, da(x) is linearly dependent on dai1 (x), . . . , daip (x). This also shows that if Γi , Γj are both nonzero at x ∈ X, then they are linearly dependent at x. These considerations show that the daij generate Ω1D (A) as an A bimodule. As a consequence, the daij also generate Λ∗D (A) as a graded differential algebra. From what we have already shown, this algebra is precisely Λ∗A Ω1D (A) = Λ∗A Γ(E) = Γ(Λ∗ E) .

(94)

Now the daij generate and any p + 1 form in them is zero from the above argument, while we know that ΛpD (A) 6= {0} because Γ ∈ ΛpD (A). Also, for all x ∈ X, we know that Γ(x) 6= 0, so each fibre Λp Ex is nontrivial. Lastly, using the antisymmetry and non-vanishing of Γ, it is easy to see that for all x ∈ X there is an i such that the daij (x), j = 1, . . . , p, are linearly independent in Ex . For if, say, dai1 (x)

=

p X

cj daij (x)

(95)

j=2

then inserting this expression into the formula for Γ and using the antisymmetry shows that dai1 dai2 · · · daip (x) = 0 .

(96)

If this happened for all i at some x ∈ X we would have a contradiction of the non-vanishing of Γ(x). Hence we can always find such an i. Putting all these facts together, and recalling that X is connected, we see that E has rank p as a vector bundle, and moreover, for all x ∈ X there is an index i such that the daij (x) form a basis of Ex . Later we will see that E is essentially the (complexified) cotangent bundle. We now have the pieces necessary to show that A is in fact finitely generated by the aij . Suppose that the functions aij do not separate the points of X. Define an equivalence relation on X by x ∼ y ⇔ aij (x) = aij (y) ∀ i, j .

(97)

Then by adding constants to the aij if necessary, there is an equivalence class B such that aij (B) = {0} ∀ i, j .

(98)

So the aij generate an ideal haij i whose norm closure is C0 (X \ B). The fact that Λ∗D (A) is complete in the topology determined by the family of seminorms provided by δ, and is a locally convex Hausdorff space for this topology, shows that the

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topological Hochschild homology is Hausdorff, and allows us to use the long exact sequence in topological Hochschild homology. We have the exact sequence I

P

0 → haij i → A → A/haij i → 0

(99)

as well as a norm closed version 0 → C0 (X \ B) → C(X) → C(B) → 0.

(100)

The former sequence, being a sequence of locally convex algebras, induces a long exact sequence in topological Hochschild homology. The bottom end of this looks like · · · → Λ2 (A/haij i) → Λ1D (haij i) → Λ1D (A) → Λ1D (A/haij i) I

P

→ haij i → A → A/haij i → 0 . From what we have shown, every element of ΛnD (A) is of the form X ˆ 2⊗ ˆ · · · ⊗a ˆ n) ˆ 1 ⊗a ω= (a⊗a

(101)

(102)

where a ∈ A and ak ∈ haij i for each 1 ≤ k ≤ n. The map induced on homology by P : A → A/haij i is easy to compute: X X ˆ (a1 )⊗ ˆ · · · ⊗P ˆ (an )) ˆ · · · ⊗a ˆ n) = ˆ 1⊗ (P (a)⊗P P∗ (a⊗a =

X ˆ 0···⊗ ˆ 0) = 0 . (P (a)⊗

(103)

So ΛnD (A/haij i) = 0 for all n ≥ 1. The case n = 1 says that δP (a) = 1 ⊗ P (a) − P (a) ⊗ 1 = 0 ⇒ P (a) ∈ C · 1 .

(104)

Hence C(B) = C and B = {pt}. As an immediate corollary we see that all the equivalence classes of ∼ are singletons, so A is generated in its Frechet topology by the elements aij . Now take the natural open cover of X given by the open sets U i = {x ∈ X : [D, ai1 ], . . . , [D, aip ] 6= 0} .

(105)

From what we have already shown, over this open set we obtain a local trivialisation E|U i ∼ = U i × Cp .

(106)

|aij (x) − aij (y)| ≤ k[D, aij ]|F kd(x, y)

(107)

As

where F is any closed set containing x and y, we see that the aij are constant off U i . By altering these functions by adding scalars, we see that we can take their value off U i to be zero. Thus haij ij ⊆ C0 (U i ). Noting that the daij provide a generating set for Ω1D (AU i ) over AU i (the closure of the functions in A vanishing off U i for the Frechet topology), the previous argument shows that the aij generate AU i in the

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Frechet topology and C0 (U i ) in norm. The inessential detail that AU i is not unital may be repaired by taking the one point compactification of U i or simply noting that the above argument runs as before, but now the only scalars are zero, whence the equivalence class B is empty. We are now free to take as coordinate charts (U i , ai ) where ai = (ai1 , . . . , aip ) : i U → Rp . As both the aij and the akj generate the functions on U i ∩ U k , we i : Rp → Rp with may deduce the existence of continuous transition functions fjk compact support such that i (ak1 , . . . , akp ) on the set U i ∩ U k . aij = fjk

(108)

As these functions are necessarily continuous, we have shown that X is a topological manifold, and moreover the map a = (a1 , . . . , an ) : X → Rnp is a continuous embedding. 4.3.8. X is a smooth manifold We can now show that X is a smooth manifold. On the intersection U i ∩ U k , the functions can be taken to be generated by either ai1 , . . . , aip or ak1 , . . . , akp . Thus we may write the transition functions as i = aij = fjk

∞ X

pN (akj )

(109)

N =0

where the pN are homogenous polynomials of total degree N in the akj . As the aij generate A in its Frechet topology, we may assume that this sum is convergent for all T the seminorms kδ n (·)k. Also, Ω∗D (A) ⊂ n≥1 Dom δ n and [D, ·] : Ω∗D (A) → Ω∗D (A), showing that the sequence ∞ X

[D, pN ]

(110)

N =0

converges. Since D is a closed operator, the derivation [D, ·] can be seen to be closed as well. Thus, over the open set U i ∩ U k , we see that the above sequence converges to [D, aij ], so [D, aij ] =

p X ∞ X ∂pN [D, akl ] , k ∂a l l=1 N =0

(111)

where we have also used the first order condition. Consequently, the functions ∞ X ∂pN ∈ A ⊂ C(X) ∂akl N =0

(112)

are necessarily continuous. This allows us to identify i ∂fjk

∂akl

=

∞ X ∂pN . ∂akl N =0

(113)

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i Applying the above argument repeatedly to the functions ∂ajkk , shows that fjk l is a C ∞ function. Hence X is a smooth manifold for the metric topology and A ⊆ C ∞ (X). In particular, the functions aij are smooth. Conversely, let f ∈ C ∞ (X). Over any open set V ⊂ U i we may write

f=

∞ X

pN (a1 , .., ap )

(114)

n=0

where we have temporarily written aj := aij . As f is smooth, all the sequences ∞ X X ∂ |α| f ∂ |α| pN = (115) α α α ∂ 1 a1 · · · ∂ p ap ∂ 1 a1 · · · ∂ αp ap N =0 |α|=n |α|=n P αp n 1 converge, where α ∈ N is a multi-index. Let pN = |α|=N Cα aα 1 · · · ap and let PM sM = N =0 pN be the partial sum. Then

X

[|D|, sM ] =

nj p X M X X X

n −k

CN an1 1 · · · aj j

[|D|, aj ]ak−1 · · · anp p j

N =0 |α|=N j=1 k=1

=

nj p X M X X X

α −1

j 1 Cα aα 1 · · · aj

p · · · aα p [|D|, aj ]

n=0 |α|=N j=1 k=1

+

nj p X M X X X

α −k

j 1 Cα aα 1 · · · aj

p [[|D|, aj ], ak−1 · · · aα p ] j

n=0 |α|=N j=1 k=1

= G1M +

p X ∂sM j=1

∂aj

[|D|, aj ] .

(116)

To show that f ∈ Dom δ, we must show that G1M can be bounded independent of M , the other term being convergent by the smoothness of f and the boundedness of [|D|, aj ] for each j. We have the following bound kG1M k

≤

nj p X M X X X

α −k

j 1 kCα aα 1 · · · aj

p [[|D|, aj ], ak−1 · · · aα p ]k j

N =0 |α|=N j=1 k=1

≤

nj p X M X X X

2|CN |ka1 kn1 · · · kaj knj −1 · · · kap knp k[|D|, aj ]k

N =0 |α|=N j=1 k=1

=2

p M X X ∂ pÑ N =0 j=1

∂aj

(ka1 k, . . . , kap k)k[|D|, aj ]k ,

where pÑ (x1 , . . . , xp ) =

X |α|=N

αp 1 |Cα |xα 1 · · · xp .

(117)

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The absolute convergence of the sequence of real numbers ∞ X ∂pN (ka1 k, . . . , kap k) ∂aj

(118)

N =0

now shows that kG1M k can be bounded independtly of M . Thus the sequence [|D|, sM ] converges, and as [|D|, ·] is a closed derivation, it converges to [|D|, f ]. Hence k[|D|, f ]k < ∞, and f ∈ Dom δ. Applying δ twice gives X

[|D|, [|D|, sM ]] = G2M +

|α|=2

+

∂ 2 sM [|D|, aj ]αj [|D|, ak ]αk α j ∂ k ak

∂ αj a

X ∂sM δ 2 (aj ) . ∂aj

(119)

|α|=1

The second two terms can be bounded independently of M by the smoothness of f . The term G2M is a sum of commutators and double commutators which can be bounded independently of M in exactly the same manner as G1M . This shows that k[|D|, [|D|, f ]]k < ∞

(120)

and f ∈ Dom δ 2 . Continuing this line of argument shows that f ∈ Dom δ n for all n, and so f ∈ A. Consequently, A = C ∞ (X), and the seminorms kδ n (·)k determine the C ∞ topology on A. To show that the weak∗ and metric topologies agree, it is sufficient to show that convergence in the weak∗ topology implies convergence in the metric topology, as the metric topology is automatically finer. ∗ ¯ So let {∂hik }∞ k=1 be a weak convergent sequence of pure states of A (or A). Thus there is a pure state φ such that for all f ∈ A, |φk (f ) − φ(f )| → 0 .

(121)

As A is commutative, we know that every pure state is a ∗-homomorphism, and P writing the generating set of A as a1 , . . . , anp we have for f = pN , φk (f ) =

∞ X

pN (φk (ai ))

(122)

N =0

and this makes sense since the sum is convergent in norm. The next aspect to address is the norm of [D, f ]. Recalling that k[D, ai ]k = 1, we have

X

∗ np

∂f ∂f 2 ∗

[D, ai ] [D, aj ] k[D, f ]k = ∂a ∂a j

i,j=1 i

np ∗ X ∂f ∂f (x) (x) ≤ sup ∂aj x∈X i,j=1 ∂ai

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np ∗ X ∂f ∂f = sup (a(x)) (a(x)) ∂xj a(x)∈a(X) i,j=1 ∂xi = kdf k2 , regarding f : Rnp → C ,

(123)

where a : X → Rnp is our (smooth) embedding and xi are coordinates on Rnp . Thus kdf k ≤ 1 ⇒ k[D, f ]k ≤ 1. Any function f : Rnp → C satisfying kdf k ≤ 1 is automatically Lipschitz (as a function on Rnp ). So |φk (f ) − φ(f )| = |f (φk (ai )) − f (φ(ai ))| ≤ |φk (ai ) − φ(ai )| → 0 as k → ∞ .

(124)

Hence sup{|φk (f ) − φ(f )| : k[D, f ]k ≤ 1} = sup{|f (φk (ai )) − f (φ(ai ))| : kdf k ≤ 1} ≤ {|φk (ai ) − φ(ai )|} → 0

(125)

so φk → φ in the metric. So the two topologies agree. As a last note on these issues, it is important to point out that A is stable under the holomorphic functional calculus. If f : X → C is in A, then we may (locally) regard it as a smooth function f : Rp → C of ai1 , . . . , aip for some i. So let g : C → C be holomorphic. Then g ◦ f ◦ ai

(126)

is patently a smooth function on X. Thus the K-theory and K-homology of A and A¯ coincide, [6]. 4.3.9. X is a spinc manifold We have been given an Hermitian structure on H∞ , (· , ·)S , and as Ω1D (A) is finite projective, we are free to choose one for it also. Regarding Ω∗D (A) as a subalgebra of End(H∞ ), any non-degenerate Hermitian form we choose is unitarily equivalent to ([D, a], [D, b])Ω1 := p1 Tr([D, a]∗ [D, b]), where p is the fibre dimension of Ω1D (A). We have shown this is a non-degenerate positive definite quadratic form. Over each U i , we have a local trivialisation (recalling that we have set Ω1D (A) = Γ(X, E)) E|U i ∼ = U i × Cp .

(127)

As X is a smooth manifold, we can also define the cotangent bundle, and as the ai are local coordinates on each U i , we have ∗ X|U i ∼ TC = U i × Cp .

(128)

It is now easy to see that these bundles are locally isomorphic. Globally they may ∗ X to be not be isomorphic, though. The reason is that while we may choose TC ∗ 1 T X ⊗ C globally, we do not know that this is true for ΩD (A). Nonetheless, up

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to a possible U (1) twisting, they are globally isomorphic. It is easy to show using i our change of coordinate functions fjk that up to this possible phase factor the two bundles have the same transition functions. For the next step of the proof we require only local information, so this will not affect us. Later we will use the real structure to show that Ω∗D (A) is actually untwisted. From the above comments, we may easily deduce that Λ∗D (A)|U i ∼ = Γ(Λ∗C (T ∗ X))|U i .

(129)

The action of d = [D, ·] on this bundle may be locally determined, since we know that Λ∗D (A) is a skew-symmetric graded differential algebra for d. First d2 = 0, and d satisfies a graded Liebnitz rule on Λ∗D (A). Furthermore, from the above local isomorphisms, given f ∈ A, df |U i =

p p X X ∂f ∂f i i [D, a ] = da . j i ∂aj ∂aij j j=1 j=1

(130)

By the uniqueness of the exterior derivative, characterised by these three properties, [D, ·] is the exterior derivative on forms. We shall continue to write d or [D, ·] as convenient. Let us choose a connection compatible with the form (· , ·)S ∇ : H∞ → Λ1D (A) ⊗ H∞

(131)

∇(aξ) = [D, a] ⊗ ξ + a∇ξ .

(132)

Note that from the above discussion, this notion of connection agrees with our usual idea of covariant derivative. Denote by c the obvious map c : End(H∞ ) ⊗ H∞ → H∞

(133)

and consider the composite map c ◦ ∇ : H∞ → H∞ . We have (c ◦ ∇)(aξ) = [D, a]ξ + c(a∇ξ) ,

∀ a ∈ A, ξ ∈ H∞

= [D, a]ξ + ac(∇ξ)

(134)

whereas D(aξ) = [D, a]ξ + aDξ

∀ a ∈ A, ξ ∈ H∞ .

(135)

Hence, on H∞ , (c ◦ ∇ − D)(aξ) = a(c ◦ ∇ − D)ξ

(136)

so that c ◦ ∇ − D is A-linear, or better, in the commutatant of A. Thus if c ◦ ∇ − D is bounded, it is in the weak closure of Ω∗D (A). However, as (c ◦ ∇ − D)H∞ ⊆ H∞ , it must in fact be in Ω∗D (A). The point of these observations is that if c ◦ ∇ − D is bounded, then as ∇ is a first order differential operator (in particular having terms of integral order only) so is D (as elements of Ω∗D (A) act as endomorphisms of H∞ , and so are order zero operators). So let us show that c ◦ ∇ − D is bounded. We know

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H∞ ∼ = eAN for some N and e ∈ MN (A). As both D and ∇ have commutators with e in Ω∗D (A) (because D, c ◦ ∇ : H∞ → H∞ ) there is no loss of generality in setting e to 1 for our immediate purposes. So, simply consider the canonical generating set of H∞ over A given by ξj = (0, . . . , 1, . . . , 0), j = 1, . . . , N . Then, there are bji , cji ∈ A such that X j X j bi ξj , Dξi = ci ξj . (137) c ◦ ∇ξi = j

j

As c ◦ ∇ − D is A-linear, this shows that c ◦ ∇ − D is bounded. Hence D is a first order differential operator. As the difference c ◦ ∇ − D is in Ω∗D (A), c ◦ ∇ − D = A, for some element of Ω∗D (A). However, as c ◦ ∇ = D + A is a connection (ignoring c), A ∈ Ω1D (A). Thus over U i , we may write the matrix form of D as k Dm

p X

=

j=1

αkjm

∂ k + βm ∂aj

(138)

k , αkjm are bounded for each k, m. Similarly we write the square of D as where βm

(D2 )nm =

X

Anjkm

j,k

X ∂2 ∂ n n + Bkm + Cm ∂aj ∂ak ∂ak

(139)

k

with all the terms A, B, C bounded, so that (as a pseudodifferential operator) |D|nm =

X k

where E, F are bounded and X

n Ekm

∂ n + Fm ∂ak

(140)

n m Ekm Ejp = Ankjp

(141)

m

et cetera. We will now show that the boundedness of [|D|, [D, a]], required by the axioms, tells us that the first order part of |D| has a coefficient of the form f IdN , for some f ∈ A. With the above notation, X ∂[D, a]m p n n Ekm [|D|, [D, a]]p = ∂ak k,m

+

X k,m

n n m (Ekm [D, a]m p − [D, a]m Ekp )

∂ + [F, [D, a]]np . ∂ak

(142)

For this to be bounded, it is necessary and sufficient that [Ek , [D, aj ]] = 0, for all j, k = 1, . . . , p. As [|D|, [D, aj ][D, ak ]] = [D, aj ][|D|, [D, ak ]] + [|D|, [D, aj ]][D, ak ] ,

(143)

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and the commutant of Ω∗D (A) restricted to U i is the weak closure of A restricted to U i , the matrix Ek must be scalar over A for each k (not A00 since |D|H∞ ⊆ H∞ ). n n = f k δm , for some fk ∈ A. Since Thus Ekm Ankjp = fk fj δpn

(144)

the leading order terms of D2 also have scalar coefficients. Using the first order condition we see that [D, aj ][D, ak ] + [D, ak ][D, aj ] = [[D2 , aj ], ak ] = [[D2 , ak ], aj ] ,

(145)

i := ([D, aij ], [D, aik ])Ω1 , we have and denoting by gjk

1 i Tr([D, aj ][D, ak ] + [D, ak ][D, aj ]) = −2 Re(gjk ), p

(146)

since [D, aj ]∗ = −[D, aj ]. Now (145) is junk (since it is a graded commutator), and we are interested in the exact form of the right hand side. This is easily computed in terms of our established notation, and is given by Ajk + Akj = 2fk fj IdN .

(147)

Taken together, we have shown that [D, aj ][D, ak ] + [D, ak ][D, aj ] = [[D2 , ak ], aj ] = Akj + Ajk = 2fk fj IdN i = −2 Re(gjk )IdN .

(148)

This proves that i )), by the universality (1) The [D, aij ] locally generate Cliff(Ω1D (ai1 , . . . , aip ), Re(gjk of the Clifford relations. Also, from the form of the Hermitian structure on i ) is a nondegenerate quadratic form. Ω1D (A), Re(gjk i ). (2) The operator D2 is a generalised Laplacian, as fk fj = −Re(gjk |D|

(3) From (2), we have the principal symbols σ2D (x, ξ) = kξk2 Id, σ1 (x, ξ) = kξkId, for (x, ξ) ∈ T ∗ X|U i , the total space of the cotangent bundle over U i . This tells us that |D|, D2 and D are elliptic differential operators, at least when restricted to the sets U i . With a very little more work one can also see that σ1D (x, ξ) = ξ·, Clifford multiplication by ξ. (4) As Ω∗D (A)|U i ∼ = Cliff(T ∗ X)|U i , and H∞ is an irreducible module for Ω∗D (A), we see that S is the (unique) fundamental spinor bundle for X; see [21,appendix]. (5) D = c ◦ ∇ + A, where ∇ is a compatible connection on the spinor bundle, and A is a self-adjoint element of Ω1D (A). (Using the above results one can now show that c ◦ ∇ is essentially self-adjoint, whence A must be self-adjoint.) 2

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(6) It is possible to check that the connection on Λ∗D (A) ⊗ Γ(X, S) given by the graded commutator [∇, ·] is compatible with (· , ·)Ω1D . Hence ∇ is the lift of a compatible connection on the cotangent bundle. The existence of an irreducible representation of Cliff(T ∗ X ⊗ L) for some line bundle L shows that X is a spinc manifold. Before completing the proof that X is in fact spin, we briefly examine the metric. It is now some time since Connes proved that his “sup” definition of the metric coincided with the geodesic distance for the canonical triple on a spin manifold, [22]. We will reproduce the proof here for completeness. All one needs to know in order to show that these metrics agree is that for a ∈ A the operator [D, a] = P i j (∂a/∂aj )[D, aj ] is (locally, so over U for each i) Clifford multiplication by the gradient. Then Connes’ proof holds with no modification: 1/2 X ∗ ∂a ∂a [D, aj ] [D, ak ] k[D, a]k = sup ∂aj ∂ak x∈X j,k

1/2 ∗ X i ∂a ∂a = sup gjk ∂aj ∂ak x∈X j,k = kakLip := sup x6=y

|a(x) − a(y)| . dγ (x, y)

(149)

In the last line we have defined the Lipschitz norm, with dγ (· , ·) the geodesic distance on X. The constraint k[D, a]k ≤ 1 forces |a(x)−a(y)| ≤ dγ (x, y). To reverse the inequality, we fix x and observe that dγ (x, ·) : X → R satisfies k[D, dγ (x, ·)]k ≤ 1. Then sup{|a(x) − a(y)| : k[D, a]k ≤ 1} = d(x, y) ≥ |dγ (x, y) − dγ (x, x)| = dγ (x, y) .

(150)

Thus the two metrics d(· , ·) and dγ (· , ·) agree. 4.3.10. X is spin In discussing the reality condition, we will need to recall that Cliff r,s module multiplication is, [21], (1) R-linear for r − s ≡ 0, 6, 7 mod 8 (2) C-linear for r − s ≡ 1, 5 mod 8 (3) H-linear for r − s ≡ 2, 3, 4 mod 8. To show that X is spin, we need to show that there exists an irreducible representation of Ω∗D (AR ), where AR = {a ∈ A : a = a∗ }. This is a real algebra with trivial involution. We will employ the properties of the real structure to do this,

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also extending the treatment to cover representations of Cliff r,s , with r + s = p. This requires some background on Real Clifford algebras, [21, 23]. Let Cliff(Rr,s ) be the Real Clifford algebra on Rr ⊕ Rs with positive definite quadratic form and involution generated by c : (x1 , . . . , xr , y1 , . . . , ys ) → (x1 , . . . , xr , −y1 , . . . , −ys ) for (x, y) ∈ Rr ⊕ Rs . The map c has a unique antilinear extension to the complexification Cliff(Rr,s ) = Cliff(Rr,s ) ⊗ C given by c ⊗ cc, where cc is complex conjugation. Note that all the algebras Cliff r,s with r + s the same will become isomorphic when complexified, however this is not the case for the algebras Cliff(Rr,s ) with the involution. If we forget the involution, or if it is trivial, then Cliff(Rr,s ) ∼ = Cliff r+s and Cliff(Rr,s ) ∼ = Cliff r+s . A Real module for Cliff(Rr,s ) is a complex representation space for Cliff r,s , W , along with an antilinear map (also called c) c : W → W such that c(φw) = c(φ)c(w)

∀ φ ∈ Cliff(Rr,s ) ,

∀w ∈ W .

(151)

It can be shown, [21], that the Grothendieck group of Real representations of Cliff(Rr,s ) is isomorphic to the Grothendieck group of real representations of Cliff r,s , and as every Real representation of Cliff(Rr,s ) automatically extends to Cliff(Rr,s ), the latter is the appropriate complexification of the algebras Cliff r,s . It also shows that KR-theory is the correct cohomological tool. Pursuing the KR theme a little longer, we note that (1, 1)-periodicity in this theory corresponds to the (1, 1)-periodicity in the Clifford algebras Cliff r,s ∼ = Cliff r−s,0 ⊗ Cliff 1,1 ⊗ · · · ⊗ Cliff 1,1

(152)

where there are s copies of Cliff 1,1 on the right hand side. As Cliff 1,1 ∼ = M2 (R) is a real algebra (as well as Real), this shows why the R, C, H-linearity of the module multiplication depends only on r − s mod 8. We take Cliff 1,1 to be generated by 12 and v = (v1 , v2 ) ∈ R2 by setting ! v1 v2 v= −v1 −v2 and the multiplication is just matrix multiplication ! 0 1 − (v1 w1 − v2 w2 )12 v · w = (v2 w1 − v1 w2 ) 1 0 ! 0 −1 = v∧w − (v, w)1,1 12 . −1 0

(153)

We take Cliff(R1,1 ) to be generated by (v1 , iv2 ) and we see that the involution is then given by complex conjugation. The multiplication is matrix multiplication with ! 0 −i . (154) v · w = −(v, w)2 12 + v ∧ w −i 0

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Thus we may always regard the involution on Cliff(Rr,s ) ∼ = Cliff r−s,0 ⊗ Cliff(R1,1 ) ⊗ · · · ⊗ Cliff(R1,1 ) ⊗ C

(155)

as 1 ⊗ cc ⊗ cc · · ·⊗ cc ⊗ cc. This is enough of generalities for the moment. In our case we have a complex representation space Γ(X, S), and an involution J such that J(φξ) = (JφJ ∗ )(Jξ) ,

∀ φ ∈ Ω∗D (A) ,

∀ ξ ∈ Γ(X, S) .

(156)

So we actually have a representation of Cliff(Rr,s ) with the involution on the algebra realised by J ·J ∗ . It is clear that J ·J ∗ has square 1, and so is an involution, and we set s = number of eigenvalues equal to −1. Then from the preceeding discussion it is clear that J · J ∗ = 1Cliff p−2s,0 ⊗ cc ⊗ · · · ⊗ cc ⊗ cc

(157)

with s copies of cc acting on s copies of Cliff(R1,1 ) and with the behaviour of J|Cliff p−2s,0 determined by p − 2s mod 8 according to table (51). It is clear that J · J ∗ reduces to 1 on the positive definite part of the algebra, as it is an involution with all eigenvalues 1 there. This implies that J · J ∗ preserves elements of the form φ⊗ 1 ⊗ · · ·⊗1 ⊗ 1C where φ ∈ Cliff p−2s,0 . However, we still need to fix the behaviour of J, and this is what is determined by p − 2s mod 8. So we claim that we have a representation of Cliff p−s,s (T ∗ X, (J · J ∗ , ·)Ω1 ) provided the behaviour of J is determined by p − 2s mod 8 and table (51). Two points: First, this reduces to Connes’ formulation for s = 0; second, the metric (J · J ∗ , ·)Ω1 has signature (p − s, s) and making this adjustment corresponds to swapping between the multiplication on Cliff(R1,1 ) and Cliff 1,1 . Similarly we replace (· , ·)S with (J· , ·)S . In all the above we have assumed that 2s ≤ p. If this is not the case, we may start with the negative definite Clifford algebra, Cliff 0,2s−p , and then tensor on copies of Cliff 1,1 . Note that it is sufficient to prove the reduction for 0 < p ≤ 8 and s = 0. This is because the extension to s 6= 0 involves tensoring on copies of Cliff(R1,1 ) for which the involution is determined, whilst raising the dimension simply involves tensoring on a copy of Cliff 8 = M16 (R), and this will not affect the following argument. These simplifications reduce us to the case J · J ∗ = 1 ⊗ cc on Ω∗D (A). To complete the proof, we proceed by cases. The first case is p = 6, 7, 8. As J 2 = 1 and JD = DJ, J = cc. We set ΓR (X, S) to be the fixed point set of J. Then restricting to the action of Ω∗D (AR ) on ΓR (X, S), J is trivial. Hence we may regard the representation π as arising as the complexification of this real representation. As φ = Jφ = φJ = JφJ ∗ on ΓR (X, S), the action can only be R-linear. From the fact that [D, J] = 0, we easily deduce that ∇J = 0, so that J is globally parallel. Thus there is no global twisting involved in obtaining Ω∗D (A) from Cliff(T ∗ X). Hence X is spin.

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In dimensions 2, 3, 4, not only does J commute with Ω∗D (AR ), but i does also (we are looking at the action on Γ(X, S), not ΓR (X, S)). So set e=J,

f = i,

g = Ji ,

(158)

note that e2 = f 2 = g 2 = −1, and observe that the following commutation relations hold: ef = −f e = g ,

f g = −gf = e ,

ge = −eg = f .

(159)

Thus regarding e, f, g and Ω∗D (AR ) as elements of homR (Γ(X, S), Γ(X, S)), we see that Γ(X, S) has the structure of a quaternion vector bundle on X, and the action of Cliff(T ∗ X) is quaternion linear. As in the last case, ∇J = 0, so that the Clifford bundle is untwisted and so X is spin. The last case is p = 1, 5. For p = 1, the fibres of Ω∗D (AR ) are isomorphic to C, and we naturally have that the Clifford multiplication is C-linear. For p = 5, the fibres are M4 (C), and as J 2 = −1, we have a commuting subalgebra spanR {1, J} ∼ = C. Note that the reason for the anticommutation of J and D is that D maps real functions to imaginary functions, for p = 1, and so has a factor of i. Analogous statements hold for p = 5. In particular, removing the complex coefficients, so passing from D to ∇, we see that ∇J = 0, and so X is spin. Note that in the even dimensional cases when π(c)J = Jπ(c), π(c) ∈ Ω∗D (AR ). When they anticommute, π(c) is i times a real form. This corresponds to the behaviour of the complex volume form of a spin manifold on the spinor bundle. Compare the above discussion with [21]. It is interesting to consider whether we can recover the indefinite distance from (J · J ∗ , ·)Ω1 . We will not address the issue here, but simply point out that in the topology determined by (J · J ∗ , ·), our previously compact space is no longer necessarily compact, and so can not agree with the weak∗ topology. It is worth noting that if J · J ∗ has one or more negative eigenvalues and ∇ is compatible with the Hermitian form (J· , ·)S , then D = c ◦ ∇ is hyperbolic rather than elliptic. So many remaining points of the proof, relying on the ellipticity of D, will not go through for the pseudo-Riemannian case. We will however point out the occasional interesting detail for this case. So for all dimensions we have shown that X is a spin manifold with A the smooth functions on X acting as multiplication operators on an irreducible spinor bundle. Thus (3) is proved completely. 4.4. Completion of the proof 4.4.1. Generalities and proof of (4) To prove (4), note that if we make a unitary change of representation, the metric, the integration defined via the Dixmier trace, and the absolutely continuous spectrum of the aij (i.e. X), are all unchanged. The only object in sight that varies in any

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important way with unitary change of representation is the operator D. The change of representation induces an affine change on D: D → U DU ∗ = D + U [D, U ∗ ] .

(160)

This in itself shows that the connected components of the fibre over [π] → dπ (· , ·) are affine. To show that there are a finite number of components, it suffices to note that a representation in any component satisfies the axioms, (recall that a spin structure for one metric canonically determines one for any other metric, [21]), and so gives rise to an action of the Clifford bundle, and so to a spin structure. As there are only a finite number of these, we have proved (4). The only items remaining to be proved are, for p > 2, R (1) − |D|2−p is a positive definite quadratic form on each Aσ with unique minimum πσ (2) This minimum is achieved 6 , the Dirac operator on Sσ R R for D = D (3) − |6D|2−p = − (p−2)c(p) Rdv. 12 X These last few items will all be proved by direct computation once we have narrowed down the nature of D a bit more. As an extra bonus, we will also be able to determine the measure once we have this extra information. Recall the condition for compatibility of a connection ∇S on S with the Hermitian structure (· , ·)S as [D, (ξ, η)S ] = (ξ, ∇S η)S − (∇S ξ, η)S ,

∀ ξ, η ∈ Γ(X, S) . Λ∗D (AR )

(161) Λ∗+1 D (AR )

→ is a Given such a connection, the graded commutator [∇ , ·] : ∗ connection compatible with the metric on ΛD (AR ). If instead we have a connection compatible with (J· , ·)S , then [∇S , ·] is compatible with (J · J ∗ , ·)Ω1D . Note that we are really considering differential forms with values in Γ(X, S), so actually have a connection [∇S , ·] : Λ∗D (A) ⊗ Γ(X, S) → Λ∗+1 D (A) ⊗ Γ(X, S). Beware of confusing S the notation here, for [∇ , ·] uses the graded commutator, while [D, ·](a[D, b]) = [D, a][D, b]. The torsion of the connection [∇S , ·] on T ∗ X is defined to be T ([∇S , ·]) = d − ◦ [∇S , ·], where d = [D, ·] and is just antisymmetrisation. Then from what has been proved thus far, we have S

D = c ◦ ∇S + T ,

[D, ·] = c ◦ [∇S , ·] + c ◦ T ([∇S , ·]) ,

(162)

on Γ(X, S) and Ω∗D (AR )⊗Γ(X, S) respectively. Here c is the composition of Clifford multiplication with the derivation in question. On the bundle Λ∗D (A) ⊗ Γ(X, S) we have already seen that [D, ·] is the exterior derivative. The T in the expression for D is the lift of the torsion term to the spinor bundle. Any two compatible connections on S differ by a 1-form, A say, and by virtue of the first order condition, adding A to ∇S does not affect [∇S , ·], and so in particular ∇S would still be the lift of a compatible connection on the cotangent bundle. As U [D, U ∗ ] is self-adjoint, for any representation π, the operator Dπ is the Dirac

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operator of a compatible connection on the spinor bundle. Note that as D is selfadjoint, the Clifford action of any such 1-form A must be self-adjoint on the spinor bundle. It is important to note that for every unitary element of the algebra, u say, gives rise to a unitary transformation U = uJuJ ∗ . If we start with D, and conjugate by U , we obtain D + u[D, u∗ ] + 0 Ju[D, u∗ ]J ∗ . If the metric is positive definite, then JAJ ∗ = −A∗ for all A ∈ Ω1D (A). Thus all of these gauge terms (or internal fluctuations, [1]) vanish in the positive definite, commutative case. This corresponds to the Clifford algebra being built on the untwisted cotangent bundle, so that we do not have any U (1) gauge terms. Moreover it is clear that the most general form of D in the real case is D + A + 0 JAJ ∗ for A a self-adjoint 1-form. The above discussion shows these vanish in the positive definite commutative case. In the indefinite case we find non-trivial gauge terms associated with timelike directions. To see this, note that every element of Ω1D (A) is of the form A + iB, where each of A and B are real, so anti-self-adjoint. Possible gauge terms are of the form iB, as they must be self-adjoint. If we assume that B is timelike (i.e. JBJ ∗ = −B), and set (u[D, u∗ ])t to be the timelike part of u[D, u∗ ], then U (D + iB)U ∗ = D + iB + JiBJ ∗ + u[D, u∗ ] + Ju[D, u∗ ]J ∗ = D + iB − iJBJ ∗ + u[D, u∗ ] + Ju[D, u∗ ]J ∗ = D + 2iB + 2(u[D, u∗ ])t .

(163)

Thus we can find non-trivial gauge terms in timelike directions. Since we are unequivocably in the manifold setting now, and as we shall require the symbol calculus to compute the Wodzicki residue, we shall now change notation. In traditional fashion, let us write γ µ γ ν + γ ν γ µ = −2g µν 1S

(164)

γ a γ b + γ b γ a = −2δ ab 1S

(165)

for the curved (coordinate) and flat (orthonormal) gamma matrices respectively. Let σ k , k = 1, . . . , [p/2], be a local orthonormal basis of Γ(X, S), and a ∈ π(A). Then the most general form that Dπ can take is X 1 X γ µ (∂µ a)σ k + aγ µ ωµab γ a γ b σ k Dπ (aσ k ) = 2 µ µ,a 2, and has unique minimum T = 0. The value of W Res(D2−p ) at the minimum is just the other term involving the scalar curvature. Hence, in the even dimensional case, we have completed the proof of Theorem 4.1. 4.4.3. The odd-dimensional case For the odd-dimensional case (p = 2m + 1) we begin with the observation that |D|−p+2 = D−2m |D|. As we already know a lot about D−2 , the difficult part here will be the absolute value term. So consider the following σ−p (|D|2−p ) = σ1 (|D|)σ−2m−2 (|D|−2m ) + σ0 (|D|)σ−2m−1 (|D|−2m ) X ∂ξµ σ1 (|D|)∂xµ σ−2m−1 (|D|−2m ) + σ−1 (|D|)σ−2m (|D|−2m ) − i µ

−i

X

∂ξµ σ0 (|D|)∂xµ σ−2m (|D|−2m )

µ

−

1X 2 ∂ σ1 (|D|)∂x2µ xν σ−2m (|D|−2m ) . 2 µ,ν ξµ ξν

(209)

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This tells us that the only terms to compute are σ1 (|D|), σ0 (|D|) and σ−1 (|D|), the other terms having been computed earlier. It is a simple matter to convince oneself that σ(|D|) has terms of integral order only by employing X ∂ξµ σ(|D|)∂xµ σ(|D|) + etc. . (210) σ(D2 ) = σ(|D|2 ) = σ(|D|)σ(|D|) − i µ

Clearly σ1 (|D|) = kξk, which we knew anyway from the multiplicativity of principal symbols. Also iaµ ξµ + b = 2kξk(σ0 (|D|) + σ−1 (|D|) + σ−2 (|D|)) X ∂ξµ σ1 (|D|)∂xµ σ0 (|D|) + σ0 (|D|)2 + 2σ−1 (|D|)σ0 (|D|) − i µ

−i

X

∂ξµ σ0 (|D|)∂xµ σ1 (|D|) − i

µ

−i

X

X

∂ξµ σ1 (|D|)∂xµ σ−1 (|D|)

µ

∂ξµ σ0 (|D|)∂xµ σ0 (|D|) − i

µ

X

∂ξµ σ−1 (|D|)∂xµ σ1 (|D|)

µ

−

1X 2 1X 2 ∂ξµ ξν σ1 (|D|)∂x2µ xν σ1 (|D|) − ∂ σ1 (|D|)∂x2µ xν σ0 (|D|) 2 µ,ν 2 µ,ν ξµ ξν

−

1X 2 ∂ σ0 (|D|)∂x2µ xν σ1 (|D|) + order2 or less . 2 µ,ν ξµ ξν

Looking at the terms of order 1, we have iaµ ξµ = 2kξkσ0 (|D|) − i

X

∂ξµ kξk∂xµ kξk ,

(211)

(212)

µ

or, in Riemann normal coordinates, σ0 (|D|) = RN

1 iaµ ξµ . 2kξk

(213)

The terms of order 0 are more difficult, and we find that b = RN 2kξkσ−1 (|D|) −

1 aµ ξµ aν ξν 4kξk2

− iξ µ ∂xµ σ0 (|D|) − i∂ξµ σ0 (|D|)∂xµ σ1 (|D|) 1 (214) − ∂ξ2µ ξν σ1 (|D|)∂x2µ xν σ1 (|D|) . 2 Remembering that the derivative of an expression in Riemann normal form is not the Riemann normal form of the derivative, we eventually find that 1 b = RN, mod kξk 2σ−1 (|D|) − aµ ξµ aν ξν 4 1 1 ρσ + aν,µ ξν ξ µ + δ µν Rµν ξρ ξσ . 2 12

(215)

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In the above, as well as expressing the result in Riemann normal coordinates and ρσ ξρ ξσ , since we know that mod kξk, we have omitted a term proportional to ξ µ ξ ν Rµν this will vanish when averaged over the cosphere bundle. This gives us an expression for σ−1 (|D|) in terms of aµ and b. Indeed, with the same omissions as above we have 1 1 σ−1 (|D|) = RN, mod kξk b + aµ ξµ aν ξν 2 2 1 1 ρσ − aν,µ ξν ξ µ − δ µν Rµν ξρ ξσ . 8 24 Completing the tedious task of calculation and substitution yields σ−p (|D|2−p ) = RN, mod kξk − −

(216)

p(p − 2) ν (p − 2) b+ a,µ ξν ξ µ 2 4

p(p − 2) µ p(p − 2) µν ρσ a ξµ aν ξν + δ Rµν ξρ ξσ . 8 24

(217)

We note that the factor p(p − 2) arises from 4m2 − 1 = (2m + 1)(2m − 1) = p(p − 2). Using the experience gained from the even case, we have no trouble integrating this over the cosphere bundle, giving Z (p − 2) √ c(p) R gdp x W Res(|D|2−p ) = − 12 X Z 3 √ tabc tabc gdp x . (218) + (p − 2)c(p) X 2 Again, this expression clearly has a unique minimum (for p > 1 and odd) given by the Dirac operator of the Levi–Civita connection. From the results of [24] and the above calculations, if we twist the Dirac operator by some bundle W , the symbol will involve the “twisting curvature” of some connection on W . This does not influence the Wodzicki residue, and so the above result will still hold, except that the minimum is no longer unique. If we have no real structure J, and so are dealing with a spinc manifold, we have the same value at the minimum, though it is now reached on the linear subspace of self-adjoint U (1) gauge terms. This completes the proof of the theorem. 5. The Abstract Setting In presenting axioms for noncommutative geometry, Connes has given sufficient conditions for a commutative spectral triple to give rise to a classical geometry, but has not given a simple abstract condition to determine whether an algebra has at least one geometry. In our setup we did not try to remedy this situation, but merely to flesh out some of Connes’ ideas enough to give the proof of the above theorem. In light of this proof, we offer a possible characterisation of the algebras that stand a chance of fulfilling the axioms. The main points are that

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(1) ∃c ∈ Zn (A, A ⊗ Aop ) such that π(c) = Γ, (2) π(Ω∗ (A)) ∼ = γ(Cliff(T ∗ X)) whilst π(Ω∗ (A))/π(δ(ker π)) is the exterior algebra ∗ of T X. This second point may be seen as a consequence of the first order condition and the imposition of smoothness on both A and π(Ω∗ (A)). The other interesting feature of the (real) representations of Ω∗ (A), is that if the algebra is noncommutative we have self-adjoint real forms, and so gauge terms. With this in mind we should regard Ω∗ (A), or at least its representations obeying the first order condition, as a generalised Clifford algebra which includes information about the internal (gauge) structure as well. Since this algebra is built on the cotangent space, the following definition is natural. Definition 5.1. A pregeometry is a dense subalgebra A of a C ∗ -algebra A such that Ω1 (A) is finite projective over A. This is in part motivated by definitions of smoothness in algebraic geometry, and provides us with our various analytical constraints. Let us explore this. The hypothesis of finite projectiveness tells us that there exist Hermitian structures on Ω1 (A). Let us choose one, (· , ·)Ω1 . We can then extend it to Ω∗ (A) by requiring homogenous terms of different degree to be orthogonal and (δ(a)δ(b), δ(c)δ(d))Ω∗ = (δ(a), δ(c))Ω1 (δ(b), δ(d))Ω1 ,

(219)

and so on. Then we can define a norm on Ω∗ (A) by the following equality: kδakΩ∗ = k(δa, δa)kA .

(220)

As (δa, δa)Ω1 = (δa, δa)∗Ω1 and kδakΩ∗ = k(δa)∗ kΩ∗ we have kδa(δa)∗ kΩ∗ = kδak2Ω∗ .

(221)

So Ω∗ (A) is a normed ∗ -algebra satisfying the C ∗ -condition, and so we may take the closure to obtain a C ∗ -algebra. What are the representations of Ω∗ (A)? Let π : Ω∗ (A) → End(E)

(222)

be a ∗ -morphism, and E a finite projective module over A. Thus π|A realises E as E∼ = AN e for some N and some idempotent e ∈ MN (A). As E is finite projective, we have nondegenerate Hermitian forms and connections. Let (· , ·)E be such a form, and ∇π be a compatible connection. Thus ∇π : E → π(Ω1 (A)) ⊗ E ∇π (aξ) = π(δa) ⊗ ξ + a∇π ξ (· , ·)E : E ⊗ E → π(A) (∇π ξ, ζ)E − (ξ, ∇π ζ)E = π(δ(ξ, ζ)) .

(223)

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If we denote by c the obvious map c : End(E) ⊗ E → E ,

(224)

Dπ = c ◦ ∇π : E → E .

(225)

Dπ (aξ) = π(δa)ξ + aDπ ξ

(226)

Dπ (aξ) = [Dπ , a]ξ + aDπ ξ

(227)

then we may define

Comparing

and

we see that π(δa) = [Dπ , π(a)]. Note that Dπ depends only on π and the choice of Hermitian structure on Ω1 (A). This is because all Hermitian metrics on E are equivalent to X (228) (ξ, ζ) = ξi ζi∗ . This in turn tells us that the definition of compatibility with (· , ·)E reduces to compatibility with the above standard structure. The dependence on the structure on Ω1 (A) arises from the symmetric part of the multiplication rule on Ω∗ (A) being determined by (· , ·)Ω1 . If we are thinking of (· , ·)Ω as “g” in the differential geometry context, then it is clear that Dπ should depend on it if it is to play the role of Dirac operator. Thus it is appropriate to define a representation of Ω∗ (A) as follows. Definition 5.2. Let A ⊂ A be a pregeometry. Then a representation of Ω∗ (A) is a ∗ -morphism π : Ω∗ (A) → End(E), where E is finite projective over A and such that the first order condition holds. In the absence of an operator D, we interpret the first order condition as saying that π(Ω0 (A)) lies in the centre of π(Ω∗ (A)), at least in the commutative case. In general, we simply take it to mean that the action of π(Aop ) commutes with the action of π(Ω∗ (A)). Next, it is worthwhile pointing out that representations of Ω∗ (A) are a good place to make contact with Connes description of cyclic cohomology via cycles, [6], though this will have to await another occasion. In this definition we encode the first order condition by demanding that π(Ω∗ (A)) is a symmetric π(A) module in the commutative case. In the noncommutative case that we discuss below, we will require that the image of Aop commutes with the image of Ω∗ (A). Let us consider the problem of encoding Connes’ axioms in this setting. The first thing we require is an extension of these results to Ω∗ (A) ⊗ Aop . Since a left module for A is a right module for Aop , we shall have no problem in extending these definitions if we demand that [π(a), π(bop )] = 0 for all a, b ∈ A. Since Ω∗ (A) is a C ∗ -algebra, any representation of it on Hilbert space lies in the bounded operators. This deals with the first two items of Definition 3.1. The real structure will clearly remain as an independent assumption. What remains?

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We do not know that A is “C ∞ (X)” in the commutative case yet. Examining the foregoing proof, we see that we first needed to know that the elements involved in the Hochschild cycle c generated π(A), which came from π(c) = Γ. Then we needed to show that the condition π(a), [Dπ , π(a)] ∈

∞ \

Dom(δ m )

(229)

implied that π(a) was a C ∞ function and that π(Ω∗ (A)) was the smooth sections of the Clifford bundle. Recall that δ(x) = [|Dπ |, x]. So having a representation, we obtain Dπ , and we can construct |D|π if Dπ is self-adjoint. This will follow from a short computation using the fact that ∇π is compatible. Then we say that π is a smooth representation if π(Ω∗ (A)) ⊂

∞ \

Dom(δ m ) .

(230)

This requires only the finite projectiveness of Ω∗ (A) to state, though this is not necessarily sufficient for it to hold. As E ∼ = eAN , this also ensures that Dπ : E → E is well-defined. Further, in the commutative case we see immediately that Dπ is an operator of order 1. Thus any pseudodifferential parametrix for |Dπ | is an operator of order −1. We can then use Connes’ trace theorem that |Dπ |−p ∈ L(1,∞) . R to state The imposition of Poincaré duality then says that − |Dπ |−p 6= 0. It is not clear how this works in the general case. So, a pregeometry is a choice of “C 1 ” functions on a space. Given a first order representation π of the universal differential algebra of A provides an operator Dπ of order 1. We use this to impose a further restriction (smoothness) on the representation π and algebra A. Definition 5.3. Let π : Ω∗ (A) → End(E) be a smooth representation of the pregeometry A ⊂ A. Then we say that (A, Dπ , c) is a (p, ∞)-summable spectral triple if (1) c ∈ Zp (A, A ⊗ Aop ) is a Hochschild cycle with π(c) = Γ (2) Poincaré duality is satisfied R (3) E is a pre-Hilbert space with respect to −(· , ·)E |D|−p π . Definition 5.4. A real (p, ∞)-summable spectral triple is a (p, ∞)-summable spectral triple with a real structure. It is clear that in the commutative case this reformulation loses no information. This approach may be helpful in relation to the work of [28]. By employing extra operators and imposing supersymmetry relations between them, the authors show that all classical forms of differential geometry (K¨ ahler, hyperk¨ ahler, Riemannian. . .) can also be put into the spectral format. Examining their results

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show that the converse(s) may also be proved in a similar way to this paper, provided the correct axioms are provided. The elaboration of these axioms may well be aided by the above formulation, but this will have to await another occasion. Acknowledgments I would like to take this opportunity to thank Alan Carey for his support and assistance whilst writing this paper, as well as Steven Lord and David Adams for helpful discussions. I am also indebted to A. Connes and J. C. Varilly for pointing out serious errors and omissions in the original version of this paper, as well as providing guidance in dealing with those problems. References [1] A. Connes, “Gravity coupled with matter and the foundations of non-commutative geometry”, Comm. Math. Phys. 182 (1996) 155–176. [2] J. L. Loday, Cyclic Homology, Springer-Verlag, 1992. [3] G. Landi, “An introduction to noncommutative spaces and their geometry”, (preprint), hep-th/9701078. [4] N. E. Wegge-Olsen, K-Theory and C ∗ -algebras, Oxford University Press, 1993. [5] R. G. Swan, “Vector bundles and projective modules”, Trans. Am. Math. Soc. 105 (1962) 264–277. [6] A. Connes, Noncommutative Geometry, Academic Press, 1994. [7] P. Baum and R. Douglas, “Index theory, bordism and K-homology”, Invent. Math. 75 (1984) 143–178. [8] P. Baum and R. Douglas, “Toeplitz operators and poincare duality”, Proc. Toeplitz Memorial Conf., Tel Aviv, Birkhauser, Basel, pp. 137–166 in 1982. [9] P. Baum, R. Douglas and M. Taylor, “Cycles and relative cycles in analytic Khomology”, J. Diff. Geom. 30 (1989) 761–804. [10] P. Baum and R. Douglas, “K-homology and index theory”, Proc. Symp. Pure Math. 38 (1982) 117–173. [11] A. Connes, “Noncommutative geometry and reality”, J. Math. Phys. 36(11) (1995) 6194–6231. [12] M. Wodzicki, “Local invariants of spectral asymmetry”, Invent. Math. 75 (1984) 143–178. [13] D. Voiculescu, “Some results on norm-ideal perturbations of Hilbert space operators, 1 and 2”, J. Operator Theory 2 (1979) 3–37; 5 (1981) 77–100. [14] A. Connes, “The action functional in noncommutative geometry”, Comm. Math. Phys. 117 (1988) 673–683. [15] A. Connes and H. Moscovici, “The local index formula in noncommutative geometry”, Geometric and Functional Analysis 5 (1995) 174–243. [16] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1966, 1976. [17] A. Connes, “Geometry from the spectral point of view”, Lett. Math. Phys. 34 (1995) 203–238. [18] F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967. [19] M. Atiyah, K-Theory, W. A. Benjamin Inc, 1967. [20] M. Rieffel, “Metrics on states from actions of compact groups”, (preprint) archives math.OA/9807084 v2.

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[21] H. Lawson and M. Michelsohn, Spin Geometry, Princeton University Press, 1989. [22] A. Connes, “Compact metric spaces, Fredholm modules and hyperfiniteness”, Ergodic Theory and Dynamical Systems 9 (1989) 207–220. [23] M. Atiyah, “K-theory and reality”, Quart. J. Math. Oxford 17 (1966) 367–386. [24] W. Kalau and M. Walze, “Gravity, non-commutative geometry and the Wodzicki residue”, J. Geom. Phys. 16 (1995) 327–344. [25] D. Kastler, “The Dirac operator and gravitation”, Comm. Math. Phys. 166 (1995) 633–643. [26] P. Gilkey, Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem, Publish or Perish, 1984. [27] N. Berline, E. Getzler and M. Vergne, Heat Kernels and Dirac Operators, 2nd ed., Springer-Verlag, 1996. [28] J. Fr¨ olich, O. Grandjean and A. Recknagel, “Supersymmetric quantum theory and differential geometry”, Comm. Math. Phys. 193 (1998) 527–594.

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NORM RESOLVENT CONVERGENCE TO MAGNETIC ¨ SCHRODINGER OPERATORS WITH POINT INTERACTIONS

HIDEO TAMURA Department of Mathematics, Okayama University Okayama 700-8530, Japan E-mail : tamuramath.okayama-u.ac.jp

Received 21 April 1999 The Schr¨ odinger operator with δ-like magnetic field at the origin in two dimensions is not essentially self-adjoint. It has the deficiency indices (2, 2) and each self-adjoint extension is realized as a differential operator with some boundary conditions at the origin. We here consider Schr¨ odinger operators with magnetic fields of small support and study the norm resolvent convergence to Schr¨ odinger operator with δ-like magnetic field. We are concerned with the boundary conditions realized in the limit when the support shrinks. The results obtained heavily depend on the total flux of magnetic field and on the resonance space at zero energy, and the proof is based on the analysis at low energy for resolvents of Schr¨ odinger operators with magnetic potentials slowly falling off at infinity.

1. Introduction In the present work we study the norm resolvent convergence to Schr¨ odinger operators with δ-like magnetic fields in two dimensions. We work in the two-dimensional space R2 with generic point x = (x1 , x2 ), and write H(A, V ) = (−i∇ − A)2 + V =

2 X

(−i∂j − aj )2 + V ,

∂j = ∂/∂xj ,

j=1

for the Schr¨ odinger operator with magnetic potential A(x) : R2 → R2 and electric potential V (x) : R2 → R. The magnetic field b(x) is defined as b = ∇ × A = ∂1 a2 − ∂2 a1 for A(x) = (a1 (x), a2 (x)), and the quantity Z α = (2π)−1 b(x) dx is called the total flux of field b, where the integration with no domain attached is taken over the whole space. This abbreviation is used throughout. 465

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Let b, V ∈ C0∞ (R2 ) be given smooth functions with compact support and let A(x), ∇ × A = b, be the magnetic potential associated with field b(x). We now consider the Schr¨ odinger operator Hε = H(µ(ε)Aε , λ(ε)Vε ) = (−i∇ − µ(ε)Aε )2 + λ(ε)Vε

(1.1)

for 0 < ε 1 small enough, where Aε (x) = ε−1 A(x/ε) ,

Vε (x) = ε−2 V (x/ε)

and the two real parameters µ(ε) and λ(ε) behave like µ(ε) = 1 + µε2σ (1 + o(1)) ,

λ(ε) = 1 + λε2σ (1 + o(1)) ,

ε → 0,

(1.2)

for some σ > 0. The magnetic potential Aε has the field bε (x) = ε−2 b(x/ε), which has small support but preserves the flux Z Z −1 −1 bε (x) dx = (2π) b(x) dx = α (2π) independent of ε. The operator Hε formally defined above admits a unique selfadjoint realization in the space L2 = L2 (R2 ). We denote by the same notation Hε this realization with domain D(Hε ) = H 2 (R2 ) (Sobolev space of order two). The aim here is to study the convergence as ε → 0 of resolvent (Hε + i)−1 in operator norm. We assume that α 6∈ Z is not an integer. For brevity, we assume throughout the entire discussion that 0 < α < 1. The magnetic potential A(x) is not uniquely determined. If, for example, we define Φ(x) = (a1α (x), a2α (x)) = (−∂2 ϕ(x), ∂1 ϕ(x)) with ϕ(x) = (2π)−1

(1.3)

Z log|x − y|b(y) dy ,

then ∇ × Φ = ∆ϕ = b, and hence Φ becomes the potential associated with field b(x). As is easily seen, Φ(x) behaves like Φ(x) = Aα (x) + O(|x|−2 ) ,

|x| → ∞ ,

(1.4)

where Aα (x) = α(−x2 /|x|2 , x1 /|x|2 ) .

(1.5)

It should be noted that magnetic potentials never decay faster than O(|x|−1 ) at infinity, even if b(x) is assumed to be of compact support. If two magnetic potentials ˜ A(x) and A(x) have the same field, then A˜ = A + ∇g for some real function g(x). Hence H(µ(ε)A˜ε , λ(ε)Vε ) = exp(iµ(ε)gε )H(µ(ε)Aε , λ(ε)Vε ) exp(−iµ(ε)gε )

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467

with gε (x) = g(x/ε), and both the Hamiltonians become unitarily equivalent to each other. Thus we fix one of such magnetic potentials. In Sec. 2, we specify the more precise form of potential A(x), which has the property A(x) = Aα (x) for |x| 1 large enough. The present problem is motivated by the recent works [1, 7]. Let A(x) be as above. When ε → 0, Aε (x) → Aα (x) for x 6= 0 and bε (x) = ∇ × Aε = ε−2 b(x/ε) → 2παδ(x) in D0 (distributional sense). Thus Hε formally converges to the Hamiltonian Hα = H(Aα , 0) = (−i∇ − Aα )2

(1.6)

on D(Hα ) = C0∞ (R2 \{0}). The limit Hamiltonian has a δ-like magnetic field at the origin and it is not essentially self-adjoint. According to the results obtained by [1, 7], Hα has the deficiency indices (2, 2), and it has a family of self-adjoint extensions {H U } parameterized by 2 × 2 unitary mapping U from one deficiency subspace to the other one. Each self-adjoint operator H U is realized as a differential operator with some boundary conditions at the origin. Let (r, θ) be the polar coordinate system over R2 . If u belongs to the domain D(H U ), then u behaves like u(x) = (u−0 r−α + u+0 rα + o(rα )) + (u−1 r−(1−α) + u+1 r1−α + o(r1−α ))eiθ + o(r) ,

r → 0,

with some coefficients u±l , l = 0, 1, and there exist 2 × 2 matrices B± for which the boundary condition is described as the relation ! ! u−0 u+0 + B+ =0 B− u−1 u+1 between these coefficients. For example, the operator H AB with domain n o lim |u(x)| = 0 D(H AB ) = u ∈ L2 : Hα u ∈ L2 , |x|→0

= {u ∈ L2 : Hα u ∈ L2 ,

u−0 = u−1 = 0}

(1.7)

is known as the Aharonov–Bohm Hamiltonian, where Hα u = (−i∇ − Aα )2 u is understood in D0 , and the scattering and spectral problems for such a Hamiltonian have been studied by [2, 10]. We are interested in the boundary conditions realized at the origin in the limit ε → 0. Let Jε : L2 → L2 ,

(Jε f )(x) = ε−1 f (x/ε)

be the unitary operator, so that (Jε∗ f )(x) = ε f (εx) and (Hε + i)−1 = ε2 Jε (H(ε) + iε2 )−1 Jε∗ ,

(1.8)

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where H(ε) = ε2 Jε∗ Hε Jε = H(µ(ε)A, λ(ε)V ) → H(A, V ) ,

ε → 0.

Thus the results strongly depend on the spectral properties at low energy of the operator H(A, V ), and, in particular, on the structure of the resonance space E1 at zero energy. Roughly speaking, E1 is defined as E1 = {u ∈ L2loc (R2 ) : u(x) is bounded ,

H(A, V )u = 0}/E0

with zero eigenspace E0 = {u ∈ L2 : H(A, V )u = 0}. If α is not an integer, then we can prove that dim E1 ≤ 2. The main theorem is formulated as Theorem 6.1 in Sec. 6. The precise formulation requires some additional assumption (see Assumption (A)). We here mention the obtained results somewhat loosely. We further assume that max(α, 1 − α) < σ < 1 for σ > 0 in (1.2) (see Remark 6.2). If dim E1 = 0, then Hε is shown to be convergent to the Aharonov–Bohm Hamiltonian H AB with boundary conditions u−0 = u−1 = 0 in norm resolvent sense. If, in particular, V (x) ≥ 0, then it follows that dim E1 = 0 and the convergence above is obtained. If, on the other hand, dim E1 = 2, then E1 is spanned by a pair of two independent resonance functions (ρ0 , ρ1 ) taking the form ρl (x) = r−ν eilθ + gl ,

ν = |l − α| ,

with some gl ∈ L2 , and Hε is shown to converge to H U with boundary conditions u+0 = u+1 = 0. The case dim E1 = 1 is rather complicated. The resonance function ρ(x) spanning E1 is represented as a linear combination ρ(x) = c0 r−α + c1 r−(1−α) eiθ + g ,

g ∈ L2 ,

for some nonzero vector (c0 , c1 ) 6= 0, and Hε is shown to converge to different self-adjoint extensions H U according to the flux α and to the ratio c = c1 /c0 . The problem of norm resolvent convergence has been already studied in the case without magnetic potentials in a lot of works. An extensive list of related literatures can be found in the book [5]. Among them, the work [4] has dealt with the Schr¨ odinger operator ˜ ε)−1 )V (x/ε) , −∆ + ε−2 λ((log

˜ λ(s) = λ1 s + λ2 s2 + o(s2 ) ,

s → 0,

in two dimensions, where V (x) falls off rapidly at infinity. We follow in principle the same idea as developed in the case without magnetic fields (see [5] and the references quoted there). However the obtained results are quite different, and several technical improvements and new devices are required for magnetic Schrödinger operators. As is seen from (1.4), A(x) does not fall off rapidly at infinity, even if the field b(x) is of compact support. This makes it difficult to treat H(A, V ) as a short-range perturbation to the free Hamiltonian −∆. In addition, the structure of resonance space E1 heavily depends on the flux α of field b(x). Thus the argument undergoes a lot of changes at many stages of the proof.

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The Hamiltonian H U obtained from Hα by self-adjoint extension may be regarded as solvable models in quantum mechanics. We have the explicit information on spectral quantities such as negative eigenvalues, resonances and scattering amplitudes for H U ([1, 7]). Hence it is important to establish the convergence of resolvents in operator norm. As a consequence, we can know these quantities approximately for Hε with 0 < ε 1 small enough. The detailed matter will be discussed elsewhere ([11]). 2. Magnetic Schr¨ odinger Operators We specify the precise form of magnetic potential A(x) associated with given field b ∈ C0∞ (R2 ). For brevity, we assume throughout the whole exposition that b(x) and V (x) have support supp b ,

supp V ⊂ {x ∈ R2 : |x| < 1}

(2.1)

in the unit ball. The aim of this section is to construct A(x) with property |x| ≥ 2 .

A(x) = Aα (x) = α(−x2 /|x|2 , x1 /|x|2 ) ,

(2.2)

Let Φ(x) = (a1α (x), a2α (x)) be defined by (1.3). Then it behaves like ∂xβ Φ(x) = ∂xβ Aα (x) + O(|x|−2−|β| ) ,

|x| → ∞ ,

(2.3)

and, in particular, we have x1 a1α (x) + x2 a2α (x) = O(|x|−1 ). This enables us to define aα (x) as Z ∞ (x1 a1α (sx) + x2 a2α (sx)) ds , aα (x) = − 1

which is smooth in R \ {0} and obeys ∂xβ aα (x) = O(|x|−1−|β| ) at infinity. 2

Lemma 2.1. Let aα (x) be as above. Then Φ(x) is represented as Φ(x) = Aα (x) + ∇aα (x) + E(x) , where E(x) = (e1 (x), e2 (x)) is given by Z ∞ sx2 b(sx) ds , e1 (x) = 1

x 6= 0 ,

Z

∞

e2 (x) = −

sx1 b(sx) ds 1

and it has support in the unit ball. Proof. We set bjk (x) = ∂j akα (x) − ∂k ajα (x) for Φ = (a1α , a2α ), so that b(x) = b12 (x) = −b21 (x). Then a simple calculation yields Z ∞ (ajα (sx) + s(d/ds)ajα (sx) + sxk bjk (sx)) ds ∂j aα (x) = − 1

with k 6= j, and hence we obtain

Z

∞

∂j aα (x) = ajα (x) − 1

sxk bjk (sx) ds − lim Rajα (Rx) R→∞

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by partial integration. By (2.3), RΦ(Rx) → Aα (x) as R → ∞. This completes the proof. We now introduce a basic cut-off function. Let χ ∈ C ∞ ([0, ∞)) be a smooth nonnegative function such that χ(s) = 1

for 0 ≤ s ≤ 1 ,

χ(s) = 0

for s ≥ 2 .

(2.4)

We set χ0 (x) = χ0 (|x|) = χ0 (r) = χ(r) ,

χ∞ (x) = χ∞ (r) = 1 − χ0 (x) ,

and define the magnetic potential A(x) in question by A(x) = χ∞ (x)Aα (x) + B(x) = Bα (x) + B(x) ,

(2.5)

where B = aα ∇χ0 + χ0 Φ and Bα (x) = χ∞ (x)Aα (x) = αχ∞ (r)(−x2 /|x|2 , x1 /|x|2 ) .

(2.6)

By definition, it is obvious that A(x) has the required property (2.2) and it follows from Lemma 2.1 that Φ(x) admits the decomposition Φ = (χ∞ + χ0 )Φ = A(x) + ∇(χ∞ aα ) , because χ∞ (x)E(x) = 0 by construction. This implies that A(x) still has the field b(x). We now fix A(x) = (a1 (x), a2 (x)) as in (2.5) throughout the remaining sections. 3. Self-Adjoint Extensions and Boundary Conditions We write R(ζ; H) = (H − ζ)−1 , Im ζ 6= 0, for the resolvent of self-adjoint operator H and f ⊗ g = (·, g)L2 f for the integral operator with kernel f (x)¯ g (y), where (, )L2 or (,) denotes the scalar product in L2 = L2 (R2 ). As previously stated, Hε is formally convergent to the Hamiltonian Hα = H(Aα , 0) = (−i∇ − Aα )2 ,

D(Hα ) = C0∞ (R2 \ {0}) ,

which has a δ-like magnetic field at the origin. In this section, we make a brief review on Krein’s theory about the self-adjoint extension of Hα (see [3, 6, 9] for details). We also indicate that such a self-adjoint extension is realized as a differential operator with some boundary conditions at the origin. ¯ α of Hα is symmetric but is not self-adjoint. In fact, it has the 3.1. The closure H deficiency indices (2, 2) ([1, 7]). To see this, we consider the equation (Hα∗ ∓ i)u = 0 for u ∈ L2 . Since Hα∗ admits the polar coordinate decomposition X ⊕(−∂r2 + (ν 2 − 1/4)r−2 ) , ν = |l − α| , Hα∗ ' l∈Z

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the kernel Σ± = Ker(Hα∗ ∓ i) is spanned by two independent solutions ξ+l (x) = Nl Hν (τ+ r)eilθ ∈ Σ+ ,

ξ−l (x) = Nl eiνπ/2 Hν (τ− r)eilθ ∈ Σ−

(3.1)

(1)

with l = 0, 1, where Hν (z) = Hν (z) is the Hankel function of first kind, and √ √ τ+ = i = eiπ/4 , τ− = −i = ei3π/4 , Im τ± > 0 . The constant Nl = 2−1/2 (cos(νπ/2))1/2

(3.2)

is chosen so that ξ±l ∈ L2 is normalized as Z 2 |ξ±l (x)|2 dx = 1 , kξ±l kL2 = R2

while the phase factor eiνπ/2 is chosen so that ξ+l (x) − ξ−l (x) → 0 ,

r = |x| → 0 .

(3.3)

Thus we have dim Σ± = 2, and hence Hα has the deficiency indices (2, 2). We now consider the self-ajoint extension of Hα . According to Krein’s theory, we can construct a family of self-adjoint extensions {H U } parameterized by 2 × 2 unitary matrix U = (Ulm )0≤l,m≤1 . We use the same notation U to denote the unitary operator X Ulm (ξ−l ⊗ ξ+m ) : Σ+ → Σ− . U= 0≤l,m≤1

Then the self-adjoint extension H U of Hα acts as ¯ α ψ0 + i(Id − U )ψ+ HU u = H on the domain D(H U ) = {u ∈ L2 : u = ψ0 + (Id + U )ψ+ ,

¯α) , ψ0 ∈ D(H

ψ+ ∈ Σ+ } ,

where Id is the identity operator. Lemma 3.1. Let J be the isometry operator defined by ¯ α + i)−1 : Σ⊥ → Σ⊥ , ¯ α − i)(H J = −(H + − U ¯ where Σ⊥ ± = Ran(Hα ± i) is the orthogonal complement of Σ± . Then R(−i; H ) is represented as

R(−i; H U ) = (1/2i)(Id + (U ⊕ J)) . Proof. Let f ∈ L2 and set u = (1/2i)(Id + (U ⊕ J))f . We decompose f into f = h + g with h ∈ Σ+ and g ∈ Σ⊥ + . Then u = (1/2i)((Id + J)g + (Id + U )h) .

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¯ ¯ ¯ α ). Since g ∈ Σ⊥ g for some g˜ ∈ D(H + = Ran(Hα + i), we can write g = (Hα + i)˜ ¯ α ), and hence u ∈ D(H U ). We calculate Thus (Id + J)g = 2i˜ g ∈ D(H ¯ α + i)(Id + J)g + (H U + i)(Id + U )h) . (H U + i)u = (1/2i)((H Then we obtain (H U + i)u = g + h = f , so that u = R(−i; H U )f . This proves the lemma. By the above lemma, the family of self-adjoint extensions {H U } satisfies the resolvent identity X 0 0 (Ulm − Ulm )(ξ−l ⊗ ξ+m ) . R(−i; H U ) − R(−i; H U ) = (1/2i) 0≤l,m≤1

We now denote by the special notation H U = H AB ,

U = −Id = −(δlm )0≤l,m≤1 ,

(3.4)

the self-adjoint extension H U with U = −Id. This extension will be later shown to coincide just with the Aharonov–Bohm Hamiltonian with domain (1.7). We choose U 0 = −Id in the resolvent identity above to obtain that X (Ulm + δlm )(ξ−l ⊗ ξ+m ) . R(−i; H U ) = R(−i; H AB ) + (1/2i) 0≤l,m≤1 U

If, conversely, R(−i; H ) is represented through X Mlm (ξ−l ⊗ ξ+m ) R(−i; H U ) = R(−i; H AB ) +

(3.5)

0≤l,m≤1

with some 2 × 2 matrix M = {Mlm }0≤l,m≤1 , then U is determined by U = 2iM − Id .

(3.6)

3.2. We briefly discuss the boundary conditions which the extension H U fulfills at the origin. If we take account of the behavior as |z| → 0 of Hankel function Hν (z) with ν = |l − α|, then u ∈ D(H U ) behaves like u(x) = (u−0 r−α + u+0 rα + o(rα )) + (u−1 r−(1−α) + u+1 r1−α + o(r1−α ))eiθ + o(r) ,

r → 0,

(3.7)

for some u±l , l = 0, 1. According to [7], there exist 2 × 2 matrices B± such that rank (B− , B+ ) = 2, (B− , B+ ) being regarded as a matrix of size 2 × 4, and that ! ! u−0 u+0 + B+ = 0. (3.8) B− u−1 u+1 This relation determines the boundary conditions at the origin. Since Hν (z) = Jν (z) + iNν (z) =

i (e−iνπ Jν (z) − J−ν (z)) sin νπ

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by formula, it follows from (3.1) that ξ+l (x) = with τ+ =

iNl (e−iνπ Jν (τ+ r) − J−ν (τ+ r))eilθ sin νπ

√ i = eiπ/4 again, and hence ξ+l (x) behaves like ξ+l (x) = (a+l r−ν + b+l rν + o(rν ))eilθ ,

r → 0,

where a+l =

−iNl (τ+ /2)−ν , sin νπ Γ(1 − ν)

b+l =

iNl (τ+ /2)ν −iνπ e . sin νπ Γ(1 + ν)

Similarly we have ξ−l (x) = (a−l r−ν + b−l rν + o(rν ))eilθ ,

r → 0,

where a−l = with τ− =

−iNl (τ− /2)−ν iνπ/2 e , sin νπ Γ(1 − ν)

b−l =

iNl (τ− /2)ν −iνπ/2 e sin νπ Γ(1 + ν)

√ −i = ei3π/4 . The four coefficients above satisfy the relations b+l = e−iνπ b−l .

a+l = a−l ,

(3.9)

¯α) Recall that u ∈ D(H U ) takes the form u = ψ0 + (Id + U )ψ+ for some ψ0 ∈ D(H and ψ+ ∈ Σ+ . If we write X X p+m ξ+m → U ψ+ = p−m ξ−m , U : ψ+ = m=0,1

m=0,1

then we have u−l = a+l p+l + a−l p−l ,

u+l = b+l p+l + b−l p−l

by (3.7), and hence u−0 u−1

! = Ua

p+0

!

u+0

,

p+1

!

u+1

p+0

= Ub

p+1

! ,

where

Ua =

Ub =

a+0 0 b+0 0

0

!

a+1 ! 0 b+1

+

+

!

a−0

0

0

a−1 ! 0

b−0 0

b−1

U,

U.

Thus (3.8) is obtained by eliminating (p+0 , p+1 ) from the two relations above.

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There does not seem to be a canonical way to calculate B± from given unitary matrix U . We here consider several special cases for later references. We now write the unitary matrix U as ! a −¯b iη (3.10) , |a|2 + |b|2 = 1 , a, b ∈ C . U = U (η, a, b) = e b a ¯ If, for example, U = U (0, −1, 0) = −Id, then (3.9) yields that Ua = 0 and Ub is invertible, so that u−0 = u−1 = 0. Hence we have D(H U ) = {u ∈ L2 : Hα u ∈ L2 ,

u−0 = u−1 = 0}

for U = −Id. Thus the Aharonov–Bohm Hamiltonian H AB with domain (1.7) is obtained as the self-adjoint extension H U with U = −Id. Next we consider the case ! −iαπ −e 0 U = U (π/2, ei(1/2−α)π , 0) = . 0 eiαπ Then it follows again from (3.9) that Ub = 0 and Ua is invertible, and hence D(H U ) = {u ∈ L2 : Hα u ∈ L2 ,

u+0 = u+1 = 0}

for U as above. By repeated use of similar argument, we have D(H U ) = {u ∈ L2 : Hα u ∈ L2 ,

u+0 = u−1 = 0}

for U = U ((1 − α/2)π, e

−iαπ/2

−e−iαπ

, 0) =

0

!

−1

0

,

and D(H U ) = {u ∈ L2 : Hα u ∈ L2 ,

u−0 = u+1 = 0}

for U = U ((1 + α)π/2, e

i(1−α)π/2

, 0) =

−1 0

0 eiαπ

! .

Finally we study the case U = U (3π/4, a, b) for α = 1/2, where 1 a= √ 2

1 − |c|2 1−i , 1 + |c|2

2ic 1 , b= √ 2 1 + |c|2

(3.11)

c = c1 /c0 ,

for nonzero vector (c0 , c1 ) 6= 0. If c0 = 0, then c is understood as c = ∞. Let ! 1 c¯ 1 Pc = 1 + |c|2 c |c|2

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be the orthogonal projection associated with t (c0 , c1 ), and let Pc⊥ be defined by ! c |c|2 −¯ 1 ⊥ . Pc = Id − Pc = 1 + |c|2 −c 1 Then a direct computation yields

√ U = U (3π/4, a, b) = (ei3π/4 / 2)(Id + i(Pc⊥ − Pc )) (1 + i) ⊥ (1 − i) i3π/4 √ Pc + √ Pc = −Pc⊥ + i Pc . =e 2 2

If α = 1/2, then ν = |l − α| = 1/2 for l = 0, 1, and a+0 = a+1 = a−0 = a−1 ,

b+0 = b+1 ,

b−0 = b−1 = ib+0

in (3.9). Hence Ua = a+0 (Id + U ) = a+0 (1 + i)Pc ,

Ub = b+0 (Id + i U ) = b+0 (1 − i)Pc⊥

with nonzero constants a+0 and b+0 . Thus we have D(H U ) = {u ∈ L2 : Hα u ∈ L2 ,

Pc⊥t (u−0 , u−1 ) = 0 ,

Pct (u+0 , u+1 ) = 0}

for U as in (3.11). 3.3. The relations (3.5) and (3.6) yield the strategy to study the problem on the norm resolvent convergence. We end this section by explaining it briefly. We again set H(ε) = ε2 Jε∗ Hε Jε = H(µ(ε)A, λ(ε)V ) → H(A, V ) ,

ε → 0,

(3.12)

where Jε : L2 → L2 is the unitary operator defined by (1.8). Then we have Hε = ε−2 Jε H(ε)Jε∗ by definition, so that R(−i; Hε ) = ε2 Jε R(−iε2 ; H(ε))Jε∗ . We further define H0ε = H(µ(ε)Aε , 0) ,

H0 (ε) = ε2 Jε∗ Hε0 Jε = H(µ(ε)A, 0) .

(3.13)

Then Hε = H0ε + λ(ε)Vε ,

H(ε) = H0 (ε) + λ(ε)V

and we obtain V 1/2 R(−iε2 ; H(ε)) = Z(ε)V 1/2 R(−iε2 ; H0 (ε)) by use of the resolvent identity, where Z(ε) = (Id + λ(ε)V 1/2 R(−iε2 ; H0 (ε))|V |1/2 )−1 and V is decomposed into the product V = V 1/2 × |V |1/2 = (Vˆ |V |1/2 )|V |1/2 ,

Vˆ = sgn V = V /|V | .

(3.14)

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If we further take account of V 1/2 R(−iε2 ; H0 (ε))Jε∗ = ε−2 (R(i; H0ε )Jε V 1/2 )∗ , then we arrive at the basic representation R(−i; Hε ) = R(−i; H0ε ) − λ(ε)ε−2 (R(−i; H0ε )Jε |V |1/2 )Z(ε)(R(i; H0ε )Jε V 1/2 )∗ . In fact, this is derived through the computation R(−i; Hε ) = ε2 Jε R(−iε2 ; H(ε))Jε∗ = R(−i; H0ε ) − λ(ε)R(−i; H0ε )Jε V R(−iε2 ; H(ε))Jε∗ = R(−i; H0ε ) − λ(ε)R(−i; H0ε )Jε |V |1/2 Z(ε)V 1/2 R(−iε2 ; H0 (ε))Jε∗ . We take the limit ε → 0 in the relation above. To do this, we have to analyze the behavior as ε → 0 of the following three operators: R(−i; H0ε ) ,

R(±i; H0ε )Jε |V |1/2 ,

Z(ε)

appearing on the right side. We can show that R(−i; H0ε ) → R(−i; H AB ) and that X Mlm (ξ−l ⊗ ξ+m ) R(−i; Hε ) → R(−i; H AB ) + 0≤l,m≤1

for some 2 × 2 matrix M = {Mlm }0≤l,m≤1 . This determines M and then U is obtained from M through (3.6). This is a strategy. The remaining sections are devoted to analyzing the three operators above. 4. Resolvent Analysis at Low Energy Let H0 (ε) = H(µ(ε)A, 0) be as in (3.13). As ε → 0, we have H0 (ε) → Lα := H(A, 0) = (−i∇ − A)2 .

(4.1)

To analyze the three operators in the previous section, we have to study the behavior at low energy of resolvent R(k 2 ; Lα ), |k| 1, with Im k > 0. The standard way for analyzing the resolvent at low energy is based on the relation R(k 2 ; Lα ) = (Id + R(k 2 ; H0 )(Lα − H0 ))−1 R(k 2 ; H0 ) obtained from the resolvent identity, where H0 = −∆ is the free Hamiltonian. However, as already stated, the perturbation Lα − H0 is not necessarily of shortrange class even for the field b(x) compactly supported, and this does not work for the pair (H0 , Lα ). On the other hand, the difference Lα − H AB between Lα and the Aharonov–Bohm Hamiltonian H AB becomes a perturbation of short-range class by property (2.2), but the domain D(H AB ) does not coincide with that of Lα , D(Lα ) = H 2 (R2 ). It should be noted that even the form domain is different from each other. This makes it difficult to use the above relation also for the pair (H AB , Lα ). We introduce another auxiliary operator Kα = H(Bα , 0) = (−i∇ − Bα )2

(4.2)

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with magnetic potential Bα (x) defined by (2.6). By definition, Kα has the same domain as Lα , and Lα − Kα is a perturbation of short-range class. In addition, Kα admits the partial wave expansion in angular momentum. This enables us to expand R(k 2 ; Kα ) asymptotically in k, |k| 1, and to analyse the behavior at low energy of resolvent R(k 2 ; Lα ) through the above relation applied to the pair (Kα , Lα ). 4.1. We work in the space L2 (R+ ) with R+ = (0, ∞) in the first half of the section. Let (,) or (, )L2 (R+ ) denote the scalar product in L2 (R+ ). We define the operator Ul by Z 2π r1/2 f (rθ)e−ilθ dθ : L2loc (R2 ) → L2loc (Ra+ ) (4.3) (Ul f )(r) = (2π)−1/2 0

for each l ∈ Z, so that the formal adjoint operator Ul∗ is calculated as (Ul∗ )g(x) = (2π)−1/2 r−1/2 g(r)eilθ : L2loc (R+ ) → L2loc (R2 ) , where L2loc(R+ ) stands for the set of locally square integrable functions over [0, ∞). If f, g ∈ L2 (R2 ), then it follows that X (Ul f, Ul g)L2 (R+ ) . (f, g)L2 (R2 ) = l∈Z

We further define the unitary operator jε : L2 (R+ ) → L2 (R+ ) ,

(jε f )(r) = ε−1/2 f (r/ε) .

(4.4)

Let Kα be as above and let Π be defined by Π = ∂r − 1/2r. As stated above, Kα has the partial wave decomposition X X Ul∗ Kl Ul ' ⊕Kl Kα = l∈Z

l∈Z

and hence we have R(k 2 ; Kα ) =

X

Ul∗ R(k 2 ; Kl )Ul ,

l∈Z

where Kl = −∂r2 + ((l − αχ∞ (r))2 − 1/4)r−2 = Π∗ Π + (l − αχ∞ (r))2 r−2 acts on L2 (R+ ), χ∞ being as in (2.6), with domain n o D(Kl ) = u ∈ L2 (R+ ) : Kl u ∈ L2 (R+ ) , lim r−1/2 u(r) < 0 . r→0

We now fix l0 1 large enough. To analyze the behavior at low energy of resolvent R(k 2 ; Kl ) acting on L2 (R+ ), we further introduce another auxiliary operator Λl = Π∗ Π + ql

(4.5)

for each l ∈ Z, where ql (r) = l2 r−2 (0 < r < 1) ,

ql (r) = ν 2 r−2 (r > 1) ,

ν = |l − α| ,

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for |l| ≤ l0 , and ql (r) = ν 2 r−2 for |l| > l0 . The operator Λl is self-adjoint with the same domain as Kl . This is true even for |l| > l0 1 large enough. 4.2. We write G(·, ·; T ) for the integral kernel of operator T . We study the Green kernel G(r, p; R(k 2 ; Λl )) with Im k > 0, when |l| ≤ l0 . Consider the equation (Λl − k 2 )u = 0. Let fl = fl (r; k) be a solution regular at the origin and let gl = gl (r; k) be a solution falling off rapidly at infinity. Such solutions are given by ( r1/2 Jl (kr) 0 1 , ( d1l (k)r1/2 Jl (kr) + d2l (k)r1/2 Nl (kr) 0 < r ≤ 1 gl (r; k) = r > 1, r1/2 Hν (kr) Nl (z) being the Neumann function, and hence we have G(r, p; R(k 2 ; Λl )) = −(1/Dl (k))fl (r ∧ p; k)gl (r ∨ p; k)

(4.6)

with r ∧ p = min(r, p) and r ∨ p = max(r, p), where the four coefficients above are determined so as to satisfy the connecting conditions at r = 1 and Dl (k) = W (fl , gl )(k) = fl (r; k)gl0 (r; k) − fl0 (r; k)gl (r; k) is the Wronskian of two independent solutions fl and gl , Dl (k) being independent of r > 0. Since W (Jl , Nl )(z) = (2/π)z −1 ,

W (Jν , J−ν )(z) = −2(sin νπ/π)z −1

by formula, Dl (k) is calculated as Dl (k) = W (fl , gl )(k) = d2l (k)kW (Jl , Nl )(k) = (2/π)d2l (k)

(4.7)

and the four coefficients are determined as follows : 0 (k)Jl (k)) , c1l (k) = (π/2 sin νπ)k(J−ν (k)Jl0 (k) − J−ν

c2l (k) = (π/2 sin νπ)k(Jl (k)Jν0 (k) − Jl0 (k)Jν (k)) , d1l (k) = (π/2)k(Hν (k)Nl0 (k) − Hν0 (k)Nl (k)) , d2l (k) = (π/2)k(Jl (k)Hν0 (k) − Jl0 (k)Hν (k)) . The next lemma is verified by making use of the asymptotic formula as z → 0 of the Bessel functions Jp (z). We skip the proof. It is easy but a little tedious. Lemma 4.1. Dl (k) has the following asymptotic properties as k → 0: (1) 1/Dl (k) = O(|k|ν−l ). If l = 0, 1, then 1/Dl (k) = −i(Γ(1 − ν)/(l + ν)) sin νπ(k/2)ν−l (1 + O(|k|2ν )). (2) c2l (k)/Dl (k) = O(|k|2ν ). (3) c1l (k)/Dl (k) = −iπ/2 + O(|k|2ν ).

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We still assume that |l| ≤ l0 . The equation Λl u = 0 has the following pair (ul , vl ) of linearly independent solutions normalized as W (ul , vl ) = −1 : If l = 0, then  r ≤ 1,  ν −1/2 r1/2 u0 (r) = 1  ν −1/2 (rν+1/2 + r−ν+1/2 ) r > 1 , 2 ( ν −1/2 (r1/2 − νr1/2 log r) r ≤ 1 , v0 (r) = r > 1, ν −1/2 r−ν+1/2 with ν = |l − α| = α for l = 0, and if |l| ≤ l0 with l 6= 0, then   (|l| + ν)−1/2 r|l|+1/2 ul (r) = 1  (|l| + ν)−1/2 ((1 + |l|/ν)rν+1/2 + (1 − |l|/ν)r−ν+1/2 ) 2   1 (l + ν)−1/2 ((1 − ν/|l|)r|l|+1/2 + (1 + ν/|l|)r−|l|+1/2 ) vl (r) = 2  (|l| + ν)−1/2 r−ν+1/2

r ≤ 1, r > 1, r ≤ 1, r > 1.

The lemma below is also proved by use of the asymptotic formula of Bessel functions. We omit the proof here. Lemma 4.2. Let fl (r; k), |l| ≤ l0 , be as above. Then fl (r; k) = (l + ν)1/2 (k/2)l (ul (r) + O(|k|2 )) ,

|k| → 0 ,

locally uniformly in r ≥ 0. Next we consider the case |l| > l0 . The resolvent R(k 2 ; Λl ) with Im k > 0 has the Green kernel G(r, p; R(k 2 ; Λl )) = (iπ/2)r1/2 p1/2 Jν (k(r ∧ p))Hν (k(r ∨ p)) and the equation Λl u = 0 has the two independent solutions ul (r) = (2ν)−1/2 rν+1/2 ,

vl (r) = (2ν)−1/2 r−ν+1/2

with normalization W (ul , vl ) = −1. Let χR (r) = χ(r/R) for the basic cut-off function χ ∈ C0∞ ([0, ∞)) as in (2.4), and let L2com (I0 ) = {f ∈ L2 (R+ ) : supp f ⊂ I0 } for the interval I0 = [0, 4). We denote by B(L2com (I0 ) → L2loc(R+ )) the class of all operators T such that χR T : L2com (I0 ) → L2 (R+ ) is bounded for any R > 0, when restricted to the subspace L2com (I0 ). We say that T (k) ∈ B(L2com (I0 ) → L2loc (R+ )) is of class Op(k γ ) and op(k γ ), if T (k) obeys the bound kχR T (k)k = O(|k|γ ) and kχR T (k)k = o(|k|γ ) as |k| → 0, respectively. We sometimes use the same notation T (k) ∈ B(X → Y) for a bounded operator T (k) from Hilbert space X to Y.

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We now study the asymptotic behavior as k → 0 of R(k 2 ; Λl ) for each l ∈ Z. Let R(0; Λl ) : L2com (I0 ) → L2loc (R+ ) be defined by Z ∞ G(r, p; R(0; Λl ))f (p) dp (4.8) (R(0; Λl )f )(r) = 0

with G(r, p; R(0; Λl )) = ul (r ∧ p)vl (r ∨ p) . Then we obtain the following proposition on the asymptotic behavior of R(k 2 ; Λl ). The proof is again based on the asymptotic formula of Bessel functions, although it requires a little tedious computation. We skip the proof. For details, see Proposition 4.1 of [11]. Proposition 4.1. Let the notation be as above and, in particular, let ν denote ν = |l − α| for l ∈ Z. Then R(k 2 ; Λl ) with Im k > 0 has the following asymptotic properties in B(L2com (I0 ) → L2loc (R+ )). (1) If l = 0 or 1, then R(k 2 ; Λl ) = R(0; Λl ) + γl (k)(ul ⊗ ul ) + Op(|k|2 ) , where γl (k) = (2ν/(l − ν))βl (k)(1 + βl (k))−1

(4.9)

with βl (k) = −((l − ν)Γ(1 − ν)/(l + ν)Γ(1 + ν))e−iνπ (k/2)2ν . (2) If l 6= 0, 1, then R(k 2 ; Λl ) = R(0; Λl ) + Op(|k|2 ) . (3) If, in particular, |l| > l0 1, then r−2 R(k 2 ; Λl ) = r−2 R(0; Λl ) + |l|−1 Op(|k|2 ) uniformly in l. Remark 4.1. We can prove that G(r, p; R(k 2 ; Λl )) = G(r, p; R(0; Λl )) + γl (k)ul (r)ul (p) + O(|k|2 ) ,

l = 0, 1 ,

locally uniformly in (r, p) ∈ [0, ∞) × [0, ∞). For later references, we note that γl (k) behaves like γl (k) = γl k 2ν (1 + O(|k|2ν )) ,

|k| → 0 ,

where γl = −(2νΓ(1 − ν)/(l + ν)Γ(1 + ν))2−2ν e−iνπ ,

l = 0, 1 .

4.3. The resolvent R(k 2 ; Kα ) is represented as X X Ul∗ R(k 2 ; Kl )Ul ' ⊕R(k 2 ; Kl ) . R(k 2 ; Kα ) = l∈Z

l∈Z

(4.10)

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We study the behavior at low energy of resolvent R(k 2 ; Kl ) with Im k > 0. To do this, we begin by proving the following simple but useful lemma on the uniqueness of solution to Kα u = 0. Lemma 4.3. Let u ∈ L2loc(R2 ) be a solution to equation Kα u = 0. If u satisfies Z |u(x)|2 dx < ∞ , lim sup R−2 R→∞

R l0 1, then r−2 R(0; Λl ) : L2com (I0 ) → L2loc (R+ ) is defined and it satisfies the bound kχR r−2 R(0; Λl )k = O(|l|−2 ) ,

|l| → ∞ .

Proof. By (4.8), R(0; Λl ) has the integral kernel ul (r ∧ p)vl (r ∨ p). If |l| > l0 , then u = R(0; Λl )f is a L2 solution to Kl u = f for f ∈ L2com (I0 ). By the density argument, we may assume that f is a smooth function vanishing near r = 0. Then u ∈ C ∞ ((0, ∞)) behaves like u(r) = O(rν+1/2 ) as r → 0, and hence r−2 u ∈ L2 (R+ ). We take the scalar product between r−2 u and equation Kl u = f . Then a simple computation using integration by parts shows that |l|2 kr−2 uk ≤ ckf k for c > 0 independent of l, where k · k denotes the L2 norm in L2 (R+ ). This proves the lemma. Lemma 4.6. The operator Id + Wl R(0; Λl ) : L2com (I0 ) → L2com (I0 ) has an inverse for all l ∈ Z and Tl = (Id + Wl R(0; Λl ))−1 : L2com (I0 ) → L2com (I0 )

(4.11)

is bounded uniformly in l. Proof. If |l| > l0 , then the lemma follows from Lemma 4.5 at once. We consider the case |l| ≤ l0 . Since Wl R(0; Λl ) : L2com (I0 ) → L2com (I0 ) is a compact operator, it suffices to show that w + Wl R(0; Λl )w = 0

for w ∈ L2com (I0 ) ⇒ w = 0 .

To see this, we set v = R(0; Λl )w. Then v satisfies Λl v = w, so that Kl v = Λl v + Wl v = 0. As is easily seen, v behaves like v(r) ∼ r−ν+1/2 at infinity and it satisfies the assumptions in Lemma 4.4. Thus we can conclude that w = 0. As previously stated, Kl and Λl have the same domain. Hence we have R(k 2 ; Kl ) = R(k 2 ; Λl )(Id + Wl R(k 2 ; Λl ))−1

(4.12)

by the resolvent identity, and there exists a limit R(0; Kl ) = lim R(k 2 ; Kl ) = R(0; Λl )Tl : L2com (I0 ) → L2loc (R+ ) k→0

(4.13)

by Proposition 4.1 and Lemma 4.6. As is easily seen, Tl is expressed as Tl = Id − Wl R(0; Kl ). If, in particular, |l| ≤ l0 , then Tl∗ = Id − R(0; Kl )Wl : L2loc (R+ ) → L2loc (R+ ) is well defined.

(4.14)

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Lemma 4.7. The resolvent R(k 2 ; Kl ) with Im k > 0 has the following asymptotic expansion in B(L2com (I0 ) → L2loc(R+ )). (1) If l = 0 or 1, then there exists γ˜l (k) such that R(k 2 ; Kl ) = R(0; Kl ) + γ˜l (k)(Tl∗ ul ⊗ Tl∗ ul ) + Op(|k|2 ) , where γ˜l (k) behaves like γ˜l (k) = γl (k)(1 + O(|k|2ν )). (2) If l 6= 0, 1, then R(k 2 ; Kl ) = R(0; Kl ) + Op(|k|2 ) uniformly in |l| 1. Remark 4.3. The consatant γ˜l (k) is explicitly given by γ˜l (k) = γl (k)(1 + γl (k)(Tl Wl ul , ul )L2 (R+ ) )−1 but the discussion below does not require this representation. Proof. (1) The proof uses (4.12). Set El (k) = γl (k)(ul ⊗ ul ) for l = 0, 1. Then it follows from Proposition 4.1 and Lemma 4.6 that R(k 2 ; Kl ) has the expansion R(k 2 ; Kl ) = R(0; Kl ) + Rl (k) + Op(|k|2 ) in B(L2com (I0 ) → L2loc (R+ )), where Rl (k) = −R(0; Λl )Tl + (R(0; Λl ) + El (k))(Id + Tl Wl El (k))−1 Tl . We calculate Rl (k) as Rl (k) = (Id − R(0; Λl )Tl Wl )El (k)(Id + Tl Wl El (k))−1 Tl = (Id − R(0; Kl )Wl )El (k)(Id + Tl Wl El (k))−1 Tl . Hence Rl (k) takes the form Rl (k) = Tl∗ El (k)(Id + Tl Wl El (k))−1 Tl = γ˜l (k)(Tl∗ ul ⊗ Tl∗ ul ) for some γ˜l (k) having the property in the lemma. Thus (1) is proved. (2) Recall that Wl takes the form Wl (r) = wl (r)r−2 for |l| 1, where wl satisfies wl = O(|l|). By Lemma 4.6, Tl : L2com(I0 ) → L2com (I0 ) is bounded uniformly in l, and by Proposition 4.1(3), Wl (R(k 2 ; Λl ) − R(0; Λl )) : L2com(I0 ) → L2com (I0 ) obeys the bound O(|k|2 ) uniformly in l. Thus (2) also follows again from (4.12). We turn back to the space L2 = L2 (R2 ). We use similar notation L2com (Σ0 ) and ˜l (x) be defined by B(L2com (Σ0 ) → L2loc (R2 )) for Σ0 = {x ∈ R2 : |x| < 4}. Let u u˜l (x) = (Ul∗ Tl∗ ul )(x)

(4.15)

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for l = 0, 1, where Ul : L2loc (R2 ) → L2loc (R+ ) is defined by (4.3). Then the following lemma can be obtained as an immediate consequence of Lemma 4.7. In particular, the second statement follows immediately from elliptic estimate. Lemma 4.8. (1) The resolvent R(k 2 ; Kα ) with Im k > 0 has the asymptotic expansion X γ˜l (k)(˜ ul ⊗ u ˜l ) + Op(|k|2 ) R(k 2 ; Kα ) = R(0; Kα ) + l=0,1

in B(L2com (Σ0 ) → L2loc (R2 )). (2) A similar expansion holds true for ∇R(k 2 ; Kα ) with natural modifications. 4.4. We now proceed to the resolvent R(k 2 ; Lα ) in question. We set W = Lα − Kα = H(A, 0) − H(Bα , 0) ,

W∗ = W .

By (2.2), this is a first order differential operator with smooth coefficients supported in {x ∈ R2 : |x| < 2}. We repeat the same argument as used to prove Lemma 4.6 (see also Remark 4.2) to obtain that T = (Id + W R(0; Kα ))−1 : L2com (Σ0 ) → L2com (Σ0 )

(4.16)

is well defined and that the limit R(0; Lα ) = lim R(k 2 ; Lα ) = R(0; Kα )T : L2com (Σ0 ) → L2loc (R2 ) k→0

(4.17)

exists. We can also show that T = Id − W R(0; Lα )

(4.18)

and hence T ∗ = Id − R(0; Lα )W : L2loc (R2 ) → L2loc (R2 ) is also well defined. We further define ˜l (x) = (T ∗ Ul∗ Tl∗ ul )(x) , ωl (x) = T ∗ u

l = 0, 1 ,

(4.19)

for u˜l defined by (4.15). Proposition 4.2. Let the notation be as above. Then there exists γαl (k) such that R(k 2 ; Lα ) with Im k > 0 has the asymptotic expansion X γαl (k)(ωl ⊗ ωl ) + Op(|k|2 ) R(k 2 ; Lα ) = R(0; Lα ) + l=0,1

in

B(L2com (Σ0 )

→

L2loc (R2 )),

where γαl (k) behaves like

γαl (k) = γl (k)(1 + o(1)) = γl k 2ν (1 + o(1)) , with γl defined by (4.10).

|k| → 0 ,

(4.20)

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Proof. The proof is done in almost the same way as in the proof of Lemma 4.7. We start with the relation R(k 2 ; Lα ) = R(k 2 ; Kα )(Id + W R(k 2 ; Kα ))−1 obtained from the resolvent identity. Set X γ˜l (k)(˜ ul ⊗ u ˜l ) . E(k) = l=0,1

Then it follows from Lemma 4.8 that R(k 2 ; Lα ) = R(0; Lα ) + R(k) + Op(|k|2 ) in B(L2com (Σ0 ) → L2loc (R+ )), where R(k) is calculated as   ∞ X R(k) = T ∗ E(k)(Id + T W E(k))−1 T = T ∗ E(k)  (−1)j (T W E(k))j  T . j=0

Hence R(k) takes the form R(k) =

X

γlm (k)(ωl ⊗ ωm )

0≤l,m≤1

for some 2×2 matrix (γlm (k))0≤l,m≤1 , where the diagonal component γll (k) satisfies γll (k) = γl (k)(1 + O(|k|2ν )) and the off-diagonal one obeys the bound γlm (k) = O(|k|2 ). This completes the proof. 5. Resonance Space In this section we define the resonance space at zero energy of operator H(A, V ) = H(A, 0) + V = Lα + V and we mention its important properties as a series of lemmas. These properties are required to formulate the main theorem in Sec. 6. Lemma 5.1. Let f ∈ L2com(Σ0 ) with Σ0 = {x ∈ R2 : |x| < 4}. Then R(0; Lα )f ∈ L2 ⇔ (f, ωl ) = (f, ωl )L2 = 0 ,

l = 0, 1 ,

for ωl defined by (4.19). Proof. Recall the definition (4.8) of the integral operator R(0; Λl ). If g ∈ L2com (I0 ) with I0 = [0, 4) again, then it follows that R(0; Λl )g ∈ L2 (R+ ) for l 6= 0, 1, and R(0; Λl )g ∈ L2 (R+ ) ⇔ (g, ul )L2 (R+ ) = 0

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for l = 0, 1. We can easily see that R(0; Λl ) : L2com (I0 ) → L2 (R+ ) is bounded uniformly in |l| 1. By (4.13), R(0; Kl ) = R(0; Λl )Tl and hence ! X ∗ Ul R(0; Kl )Ul f ∈ L2 ⇔ (f, Ul∗ Tl∗ ul ) = 0 R(0; Kα )f = l∈Z

for l = 0, 1. If we further use the relation (4.17), then we have R(0; Lα )f ∈ L2 ⇔ (f, T ∗ Ul∗ Tl∗ ul ) = (f, ωl ) = 0 for l = 0, 1. This proves the lemma. We follow the idea due to [8] to define the resonance space in question. Let M = {u ∈ L2 : (Id + V 1/2 R(0; Lα )|V |1/2 )u = 0} , where V 1/2 = Vˆ |V |1/2 with Vˆ = V /|V | again. Since V 1/2 R(0; Lα )|V |1/2 : L2 → L2 is compact, dim M < ∞. We further define another subspace of the weighted L2 space L2−1 (R2 ) = L2 (R2 ; hxi−2 dx). As is seen from (4.17), R(0; Lα )V : L2−1 (R2 ) → L2−1 (R2 ) is well defined. Thus we can define E = {u ∈ L2−1 (R2 ) : (Id + R(0; Lα )V )u = 0} . Lemma 5.2. If v = R(0; Lα )|V |1/2 u for u ∈ L2 , then R(0; Lα )Lα v = v. Proof. By assumption, Lα v = |V |1/2 u ∈ L2 , and hence Lα v has compact support. We set w = R(0; Lα )Lα v − v. Then w belongs to L2−1 (R2 ) and solves the equation Lα w = Lα (R(0; Lα )Lα v − v) = 0 . However such a solution identically vanishes by Lemma 4.3 (see Remark 4.2). This completes the proof. Lemma 5.3. Let M and E be as above. Then R(0; Lα )|V |1/2 : M → E is injective and surjective. Proof. Let u ∈ M. We assert that v = R(0; Lα )|V |1/2 u ∈ E. Since Lα v = |V |1/2 u, we have Lα v + V v = |V |1/2 (u + V 1/2 v) = |V |1/2 (Id + V 1/2 R(0; Lα )|V |1/2 )u = 0 . By Lemma 5.2, v = R(0; Lα )Lα v, and hence v + R(0; Lα )V v = R(0; Lα )(Lα v + V v) = 0 . This proves that v ∈ E. If R(0; Lα )|V |1/2 u = 0, then it is obvious that u = 0, and it follows that R(0; Lα )|V |1/2 : M → E is injective. It is also easily seen that R(0; Lα )|V |1/2 is surjective. Let v ∈ E. We set u = −V 1/2 v. Then u ∈ L2 and it satisfies (Id + V 1/2 R(0; Lα )|V |1/2 )u = −V 1/2 (Id + R(0; Lα )V )v = 0 .

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Hence u ∈ M, and v ∈ E is represented as v = −R(0; Lα )V v = R(0; Lα )|V |1/2 u . This completes the proof. We further define the following two subspaces: M0 = {u ∈ M : (u, |V |1/2 ωl ) = 0 for l = 0, 1} ,

E0 = E ∩ L 2 .

Obviously u ∈ E0 means that u is a bound state associated with zero eigenvalue of H(A, V ). Lemma 5.4. Let M0 and E0 be as above. Then R(0; Lα )|V |1/2 : M0 → E0 is injective and surjective. Proof. We set v = R(0; Lα )|V |1/2 u for u ∈ M0 . By Lemma 5.3, v ∈ E and it solves H(A, V )v = Lα v + V v = 0. Since (|V |1/2 u, ωl ) = 0 for l = 0, 1, we further have v ∈ L2 by Lemma 5.1. Thus the mapping R(0; Lα )|V |1/2 : M0 → E0 is well defined, and it is easily seen that it is injective. To prove that it is surjective, we again set u = −V 1/2 v for v ∈ E0 . Then v = R(0; Lα )|V |1/2 u ∈ L2 . Hence it follows again from Lemma 5.1 that (u, |V |1/2 ωl ) = 0 for l = 0, 1. This implies that u ∈ M0 , and R(0; Lα )|V |1/2 is shown to be surjective. We now denote by P : L2 → M, P ∗ = P , the orthogonal projection onto M. Let M1 ⊂ M be the subspace spanned by linear combinations of two elements ψl = P |V |1/2 ωl ∈ L2 with l = 0, 1, although ψ0 and ψ1 are not necessarily linearly independent. The space M can be decomposed into the orthogonal sum M = M0 ⊕ M1 ,

M0 ⊥ M1 ,

and Lemmas 5.3 and 5.4 enable us to decompose E into the direct sum (not necessarily orthogonal) E = E0 ⊕ E1 , where E1 = {u ∈ E : u = R(0; Lα )|V |1/2 v ,

v ∈ M1 } .

(5.1)

It follows from Lemma 5.4 that dim E1 = dim M1 ≤ 2. We call u ∈ E1 a resonance state at energy zero of H(A, V ). Lemma 5.5. The geometric null space M coincides with the algebraic null space. Proof. To prove the lemma, it suffices to show that: u = (Id + V 1/2 R(0; Lα )|V |1/2 )u1 ∈ M

for u1 ∈ L2 ⇒ u = 0 .

To see this, we calculate (u, Vˆ u) = (u1 , Vˆ (Id + V 1/2 R(0; Lα )|V |1/2 )u) = 0 . Since u ∈ M satisfies Vˆ u = −|V |1/2 R(0; Lα )|V |1/2 u, we have (|V |1/2 u, R(0; Lα)|V |1/2 u) = (Lα v, v) = 0

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for v = R(0; Lα )|V |1/2 u ∈ L2−1 (R2 ). We repeat the the same argument as in the proof of Lemma 4.3 to conclude that v = 0. This implies that u = 0 and the proof is complete. As a consequence of Lemma 5.5, there exists a projection (not necessarily orthogonal) Q : L2 → M with Q2 = Q and a bounded operator Y3 : L2 → L2 such that QY3 = Y3 Q = 0 and (Id + V 1/2 R(0; L)|V |1/2 )Y3 = Y3 (Id + V 1/2 R(0; L)|V |1/2 ) = Id − Q .

(5.2)

We write P0 : L2 → M0 ,

P1 : L2 → M1 ,

P = P0 + P1 ,

(5.3)

for the orthogonal projections on M0 and M1 , and we define the projections Γ1 , Γ2 and Γ3 (not necessarily orthogonal) as Γ1 = P1 Q ,

Γ2 = P0 Q ,

Γ3 = Id − P Q = Id − Q ,

(5.4)

where P : L2 → M again denotes the orthogonal projection. This family of projections has the properties Γ1 + Γ2 + Γ3 = Id ,

Γi Γj = δij Γj ,

(5.5)

where δij is the Konecker notation. By definition, we see that Γ2 : L → M0 and Γ1 : L2 → M1 are also the projections on M0 and M1 , although these are not necessarily orthogonal. We consider functions spanning the resonance space E1 . We divide the case into dim E1 = 2 and dim E1 = 1. Assume that dim E1 = 2. Then dim M1 = 2. We again set 2

ψl = P |V |1/2 ωl = P1 |V |1/2 ωl ∈ M1 ,

l = 0, 1 ,

(5.6)

for ωl defined by (4.19). Then ψ0 and ψ1 are linearly independent and span M1 . We further introduce ηl ∈ M1 with property (ηl , |V |1/2 ωm ) = (ηl , ψm ) = δlm ,

0 ≤ l, m ≤ 1 ,

(5.7)

and we define ρl = R(0; Lα )|V |1/2 ηl = R(0; Kα )T |V |1/2 ηl ∈ E1 ,

l = 0, 1 ,

(5.8)

for ηl ∈ M1 as above. By Lemmas 5.3 and 5.4, ρ0 and ρ1 are linearly independent and span the resonance space E1 . We now recall that T : L2com(Σ0 ) → L2com (Σ0 ) is well defined (see (4.16)) and that Tm : L2com (I0 ) → L2com(I0 ), I0 = [0, 4), is bounded uniformly in m ∈ Z (see Lemma 4.6). Thus (Um T |V |1/2 ηl )(r) has support in I0 , and we have (Tm Um T |V |1/2 ηl )(r) = 0 for r > 4. Hence ρl (x) takes the form ! X ∗ Um R(0; Km )Um T |V |1/2 ηl ρl = m∈Z

=

X m=0,1

! ∗ Um R(0; Λm )Tm Um

T |V |1/2 ηl + g

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for some g ∈ L2 . If r > 4, then we calculate (R(0; Λm )Tm Um T |V |1/2 ηl )(r) = (Tm Um T |V |1/2 ηl , um )L2 (R+ ) vm (r) = (ηl , |V |1/2 ωm )vm (r) = (ηl , ψm )vm (r) = δlm vm (r) for m = 0, 1, where (ul , vl ) is the pair of linearly independent solutions to Λl u = 0 (see Sec. 4.2). This yields that ρl (x) = (Ul∗ vl )(x) + g = (2π)−1/2 r−1/2 vl (r)eilθ + g = (2π(l + ν))−1/2 r−ν eilθ + g

(5.9)

with another g ∈ L . Next we consider the case dim E1 = dim M1 = 1. Let ψl be defined by (5.6). Assume that ψ0 6= 0. Then ψ0 spans M1 , and ψ1 is represented through the relation 2

ψ1 = βψ0

(5.10)

with some constant β ∈ C. We normalize ψ0 as η˜0 = ψ0 /(ψ0 , ψ0 ), so that (ψ0 , η˜0 ) = 1 and hence (ψ1 , η˜0 ) = β. We further define ρ = R(0; Lα )|V |1/2 η˜0 ∈ E1 for η˜0 ∈ M1 as above. Then ρ spans the resonance space E1 . We repeat the same calculation as in the case dim E1 = 2 to obtain that ρ takes the form ¯ ∗ v1 )(x) + g ρ(x) = (U0∗ v0 )(x) + β(U 1 ¯ − α))−1/2 r−(1−α) eiθ + g = (2πα)−1/2 r−α + β(2π(2 with g ∈ L2 . If ψ0 = 0 (and hence ψ1 6= 0), then E1 is spanned by ρ(x) behaving like ρ(x) = (U1∗ v1 )(x) + g = (2π(2 − α))−1/2 r−(1−α) eiθ + g ,

g ∈ L2 .

In any case, E1 is spanned by function ρ(x) of the form ρ(x) = c0 r−α + c1 r−(1−α) eiθ + g ,

g ∈ L2 ,

with some nonzero vector (c0 , c1 ) 6= 0. We note that c0 = 0 ⇔ ψ0 = 0 .

(5.11)

If, in particular, α = 1/2, then

√ ¯ 3)r−1/2 eiθ ρ(x) ∼ π −1/2 r−1/2 + π −1/2 (β/

and hence it follows that

for c0 6= 0.

√ ¯ 3 c = c1 /c0 = β/

(5.12)

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6. Main Theorem In this section we formulate the main theorem. We first enumerate the assumptions. We have assumed that b, V ∈ C0∞ (R2 ) are real smooth functions with compact support and that the total flux α of field b(x) satisfies a0 < α < 1. In addition to these assumptions, we make the following assumption : Assumption (A). Pb Y (µ, λ)Pb : E0 → E0 is invertible, where Pb : L2 → E0 is the orthogonal projection on the zero eigenspace E0 of H(A, V ), and Y (µ, λ) is defined by Y (µ, λ) = µ

2 X

(aj (−i∂j − aj ) + (−i∂j − aj )aj ) − λV

(6.1)

j=1

for the real numbers µ and λ in (1.2), where aj (x) is the component of the magnetic potential A = (a1 , a2 ). Theorem 6.1. Let the notations and assumptions be as above and let E1 be the resonance space at zero energy of H(A, V ). Assume in addition that max(α, 1 − α) = αmax < σ < 1

(6.2)

for σ > 0 as in (1.2). Then Hε → H U ,

ε → 0,

in norm resolvent sense. The limit Hamiltonian H U obtained from Hα by selfadjoint extension takes different forms according as dim E1 = 0, 1 or 2. (1) Assume that dim E1 = 0. Then H U = H AB and hence D(H U ) = {u ∈ L2 : Hα u ∈ L2 , u−0 = u−1 = 0} . (2) Assume that dim E1 = 2. Then D(H U ) = {u ∈ L2 : Hα u ∈ L2 , u+0 = u+1 = 0} . (3) Assume that dim E1 = 1, and denote by ρ(x) = c0 r−α + c1 r−(1−α) eiθ + g, g ∈ L2 , the resonance function spanning E1 for nonzero vector (c0 , c1 ) 6= 0. (3a) Let 0 < α < 1/2. If c0 6= 0, then D(H U ) = {u ∈ L2 : Hα u ∈ L2 , u+0 = u−1 = 0} , and if c0 = 0, then D(H U ) = {u ∈ L2 : Hα u ∈ L2 , u−0 = u+1 = 0} .

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(3b) Let 1/2 < α < 1. If c1 6= 0, then D(H U ) = {u ∈ L2 : Hα u ∈ L2 , u−0 = u+1 = 0} and if c1 = 0, then D(H U ) = {u ∈ L2 : Hα u ∈ L2 , u+0 = u−1 = 0} . (3c) Let α = 1/2. Then D(H U ) = {u ∈ L2 : Hα u ∈ L2 , Pc⊥t (u−0 , u−1 ) = 0, Pct (u+0 , u+1 ) = 0} , a where Pc is the orthogonal projection associated with t (c0 , c1 ), and Pc⊥ = Id − Pc . Remark 6.1. Theorem 6.1 is invariant under the gauge transformation. Let g ∈ C ∞ (R2 ) be a real function falling off at infinity and set ˜ ε = H(µ(ε)A˜ε , λ(ε)Vε ) , H

A˜ = A + ∇g .

˜ ε is convergent to H U in norm resolvent sense. For example, this applies to Then H the magnetic potential Φ(x) defined by (1.3). We will sketch out a proof in Sec. 9 after completing the proof of Theorem 6.1. Remark 6.2. The assumption (6.2) in Theorem 6.1 is not essential. The results extend to other cases. If, for example, dim E1 = 0, the theorem remains true for 0 < σ < 1, and if dim E1 > 0, the results are further divided into the following four cases : σ = αmax ,

αmin < σ < αmax ,

σ = αmin ,

0 < σ < αmin ,

where αmin = min(α, 1 − α). The limit Hamiltonian H U may have a negative eigenvalue only in the cases σ = αmax and σ = αmin . We here skip the detailed matter. We end the section by making several comments on Assumption (A). (1) The assumption is important. For example, consider the case µ(ε) = λ(ε) = 1, so that µ = λ = 0, and we have Hε = ε−2 Jε H(A, V )Jε∗ . Assume that H(A, V ) has a bound state at zero energy. Then (A) is violated, and Hε also has a bound state at zero energy for all ε > 0. On the other hand, H U does not have a bound state at zero energy for any unitary matrix U . Thus the norm resolvent convergence is in general not expected without any assumption. (2) The assumption is fulfilled when λ 6= 0 and |µ/λ| 1. Assume that µ = 0. Then Y (µ, λ) = −λV . We shall prove that Pb V Pb : E0 → E0 is invertible. To see this, it suffices to show that if (V ϕ, ϕ) = 0 for ϕ ∈ E0 , then ϕ = 0. Since (Lα ϕ, ϕ) = −(V ϕ, ϕ) = 0, we can conclude by the same argument as in the proof of Lemma 4.3 that ϕ = 0. Thus (A) is fulfilled for λ 6= 0 and hence it is satisfied when |µ/λ| 1.

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(3) The assumption is invariant under gauge transformation A → A + ∇g. To prove this, it is enough to show that (Y (A)ϕ, ψ) = (Y (A + ∇g)eig ϕ, eig ψ)

(6.3)

for ϕ, ψ ∈ E0 , where Y (A) is defined by Y (A) =

2 X

(aj (−i∂j − aj ) + (−i∂j − aj )aj ) = −(d/ds)H(sA, 0)|s=1

(6.4)

j=1

for magnetic potential A = (a1 , a2 ). If ϕ ∈ E0 , then H(A + ∇g, V )eig ϕ = 0 and hence (igeig ϕ, H(A + ∇g, 0)eig ψ) + (H(A + ∇g, 0)eig ϕ, igeig ψ) = 0 . Thus we get (6.3) by differentiating the both side of the relation (H(sA, 0)ϕ, ψ) = (H(s(A + ∇g), 0)eisg ϕ, eisg ψ) . 7. Preliminary Step The present section is devoted to the preliminary step toward the proof of the main theorem. We keep the same notation as in the previous sections, and we set χ1 (x) = χ1 (r) = χ(r/2) for the basic cut-off function χ ∈ C0∞ ([0, ∞)) with property (2.4). This function has support in Σ0 = {x : |x| < 4} and χ1 = 1 on {x : |x| ≤ 2}. The aim here is to prove the following two propositions. Proposition 7.1. Let τ+ = eiπ/4 and τ− = ei3π/4 again, and let Lε be defined by Lε = ε−2 Jε Lα Jε∗ = ε−2 Jε H(A, 0)Jε∗ = H(Aε , 0) with Aε (x) = ε−1 A(x/ε). Then there exists παl (k) such that X παl (τ+ ε)(ξ+l ⊗ ωl )χ1 + Op(ε2 ) R(i; Lε )Jε χ1 = ε l=0,1

R(−i; Lε )Jε χ1 = ε

X

e−iνπ/2 παl (τ− ε)(ξ−l ⊗ ωl )χ1 + Op(ε2 )

l=0,1

in B(L → L ), where ξ±l ∈ L2 is defined by (3.1), and παl (k) behaves like 2

2

παl (k) = πl k ν (1 + o(1)) ,

|k| → 0 ,

(7.1)

with πl = iπ −1/2 (Γ(1 − ν)/(l + ν)1/2 )(sin νπ/(cos(νπ/2))1/2 )2−ν .

(7.2)

Proposition 7.2. Let Lε be as above and let H AB be the Aharonov–Bohm Hamiltonian defined by (3.4). Then Lε is convergent to H AB R(±i; Lε ) → R(±i; H AB ) , in norm resolvent sense.

ε → 0,

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7.1. We begin by proving Proposition 7.1. This is verified through a series of lemmas. Let Λl and Kl be as in Sec. 4. We set Λlε = ε−2 jε Λl jε∗ ,

Klε = ε−2 jε Kl jε∗ .

Then R(±i; Λlε ) = ε2 jε R(±iε2 ; Λl )jε∗ ,

R(±i; Klε ) = ε2 jε R(±iε2 ; Kl )jε∗ ,

(7.3)

where jε : L2 (R+ ) → L2 (R+ ) is the unitary operator defined by (4.4). We recall the notation : Nl and Dl (k) are defined by (3.2) and (4.7), respectively, and (ul , vl ) denotes the pair of independent solutions to equation Λl u = 0 (see Sec. 4.2). Lemma 7.1. (1) Assume that l = 0 or 1. Then R(±i; Λlε )jε χ1 = (2π)1/2 Nl επl (τ± ε)(e±l ⊗ ul )χ1 + Op(ε2 ) in B(L2 (R+ ) → L2 (R+ )), where e±l (r) = r1/2 Hν (τ± r) and πl (k) = −(2π)−1/2 ((l + ν)1/2 /Nl )((k/2)l /Dl (k)) = O(|k|ν ) obeys the same asymptotic formula as in (7.1). (2) If l 6= 0, 1, then kR(±i; Λlε )jε χ1 k = O(ε2 ) uniformly in l ∈ Z as a bounded operator acting on L2 (R+ ). Proof. For brevity, we prove the lemma for the + case only. (1) We first note that 1/Dl (k) behaves as in Lemma 4.1(1), and hence πl (k) obeys (7.1). By (7.3), R(i; Λlε )jε has the kernel G(r, p; R(i; Λlε )jε ) = ε3/2 G(r/ε, p; R(iε2 ; Λl )) . Hence we have

Z

cε

Z

0

∞

|G(r, p; R(i; Λlε )jε )χ1 (p)| dp 2

dr = O(ε4 )

0

by Proposition 4.1 (see Remark 4.1). On the other hand, it follows from (4.6) that G(r, p; R(i; Λlε )jε )χ1 (p) = −(ε3/2 /Dl (k))gl ((r/ε) ∨ p; k)fl ((r/ε) ∧ p; k)χ1 (p) with k = τ+ ε. We take c 1 so large that c > p ≥ 0 for p ∈ supp χ1 . If r > cε, then G(r, p; R(i; Λlε )jε )χ1 (p) = −(ε/Dl (k))r1/2 Hν (τ+ r)fl (p; k)χ1 (p) with k = τ+ ε again, and also the integral obeys the bound Z cε r|Hν (τ+ r)|2 dr = O(ε2−2ν ) , ν = |l − α| , 0

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for l = 0, 1. Thus Lemma 4.2 yields the desired relation. (2) We calculate kR(i; Λlε )jε χ1 k2 = kχ1 jε∗ R(−i; Λlε )R(i; Λlε )jε χ1 k = O(ε2 )kχ1 (R(iε2 ; Λl ) − R(−iε2 ; Λl ))χ1 k . Hence (2) follows again from Proposition 4.1. Lemma 7.2. (1) Assume that l = 0 or 1. Let e±l (r) be as in Lemma 7.1. Then there exists π ˜l (k) such that πl (τ± ε)(e±l ⊗ Tl∗ ul )χ1 + Op(ε2 ) R(±i; Klε )jε χ1 = (2π)1/2 Nl ε˜ ˜l (k) again obeys (7.1) as |k| → 0. in B(L2 (R+ ) → L2 (R+ )), where π (2) If l 6= 0, 1, then kR(±i; Klε )jε χ1 k = O(ε2 ) uniformly in l ∈ Z. Proof. (1) Recall that Wl = Kl − Λl has support in the interval [0, 2), so that Wl = χ1 Wl . If we use (7.3), then we obtain R(i; Klε ) = R(i; Λlε ) − ε2 jε R(iε2 ; Λl )χ1 Wl R(iε2 ; Kl )jε∗ by the resolvent identity, and hence R(i; Klε )jε χ1 = R(i; Λlε )jε χ1 (Id − Wl R(iε2 ; Kl )χ1 ) . Since Id−R(0; Kl )Wl = Tl∗ by (4.14), (1) follows from Lemmas 4.7 and 7.1(2). This is proved in exactly the same way as Lemma 7.1(2). We have only to use Lemma 4.7 in place of Proposition 4.1. Lemma 7.3. Let Kε be defined by Kε = ε−2 Jε Kα Jε∗ = ε−2 Jε H(Bα , 0)Jε∗ = H(Bε , 0) '

X

⊕Klε

l∈Z

with Bε (x) = ε−1 Bα (x/ε). Then X π ˜l (τ+ ε)(ξ+l ⊗ u˜l )χ1 + Op(ε2 ) R(i; Kε )Jε χ1 = ε l=0,1

R(−i; Kε )Jε χ1 = ε

X

e−iνπ/2 π ˜l (τ− ε)(ξ−l ⊗ u˜l )χ1 + Op(ε2 )

l=0,1

˜l = Ul∗ Tl∗ ul is again defined by (4.15). in B(L2 → L2 ), where u Proof. We prove the lemma for the + case only. If we use the relation Ul Jε = jε Ul , then X Ul∗ (R(i; Klε )jε χ1 )Ul , R(i; Kε )Jε χ1 = l∈Z

and we have (2π)1/2 Nl (Ul∗ e+l )(x) = Nl Hν (τ+ r)eilθ = ξ+l (x) by Definition 3.1. Thus the lemma follows from Lemma 7.2 at once.

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Proof of Proposition 7.1. The proof is almost the same as that of Lemma 7.2. Recall that the coefficients of W = Lα − Kα are all supported in {x : |x| ≤ 2}. Hence χ1 W = W χ1 = W . Since R(i; Kε ) = ε2 Jε R(iε2 ; Kα )Jε∗ ,

R(i; Lε ) = ε2 Jε R(iε2 ; Lα )Jε∗ ,

we can write R(i; Lε ) = R(i; Kε ) − ε2 Jε R(iε2 ; K)W R(iε2 ; Lα )Jε∗ by use of the resolvent identity, and hence R(i; Lε )Jε χ1 = R(i; Kε )Jε χ1 (Id − W R(iε2 ; Lα )χ1 ) . ˜l . Thus the proposition By (4.18), T = Id − W R(0; Lα ), and by (4.19), ωl = T ∗ u follows from Proposition 4.2 and Lemma 7.3. 7.2. The second proposition is also verified through a series of lemmas. The Aharonov–Bohm Hamiltonian H AB admits the partial wave decomposition H AB ' P AB , where l∈Z ⊕Hl HlAB = −∂r2 + (ν 2 − 1/4)r−2 = Π∗ Π + ν 2 r−2 is self-adjoint with domain n D(HlAB ) = u ∈ L2 (R+ ) : (Π∗ Π + ν 2 r−2 )u ∈ L2 (R+ ) , The Green kernel of

R(±i; HlAB )

(7.4)

o lim r−1/2 u(r) = 0 .

r→0

is given by

G(r, p; R(±i; HlAB )) = (iπ/2)r1/2 p1/2 Jν (τ± (r ∧ p))Hν (τ± (r ∨ p)) .

(7.5)

If, in particular, |l| > l0 1, then HlAB = Λl = ε−2 jε Λl jε∗ = Λlε

(7.6)

and these three operators have the same domain. We start with the following lemma. Lemma 7.4. Assume that |l| ≤ l0 . Let Ωε = {(r, p) : 0 < r < cε

or 0 < p < cε}

for some c > 0. Then: RR (1) R RΩε |G(r, p; R(±i; HlAB ))|2 drdp = O(ε2+2ν ) + O(ε4 ) 2 2+2ν ) + O(ε4 ). (2) Ωε |G(r, p; R(±i; Λlε ))| drdp = O(ε Proof. We prove the lemma for the + case only. (1) As is well known, the Hankel function Hν (z) behaves like Hν (z) = O(|z|−ν ) as |z| → 0, and it falls off exponentially at infinity, provided that Im z > 0. If we make use of this property, the desired bound can be easily obtained. In fact, the integral over {(r, p) ∈ Ωε : r > p} is evaluated as Z 1 Z cε p2ν+1 r−2ν+1 dr dp + O(ε2+2ν ) = O(ε4 ) + O(ε2+2ν ) . O(1) 0

p

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(2) By (7.3), we obtain Z Z Z Z |G(r, p; R(i; Λlε ))|2 drdp = ε4 |G(r, p; R(iε2 ; Λl ))|2 drdp , Ωε

(7.7)

Ω

where Ω = {(r, p) : 0 < r < c or 0 < p < c}. We evaluate the integral on the right side of (7.7). We may assume that c > 1. The Green kernel G(r, p; R(iε2 ; Λl )) has the representation (4.6). By Proposition 4.1 (see Remark 4.1 also), we have Z cZ c |G(r, p; R(iε2 ; Λl ))|2 drdp = O(1) . 0

0

On the other hand, we use Lemmas 4.1 and 4.2 to obtain that Z ∞ Z c Z ∞ 2 2 |G(r, p; R(iε ; Λl ))| dr dp = O(ε2ν ) r|Hν (τ+ εr)|2 dr . 0

c

Since

c

Z

∞

r|Hν (τ+ εr)|2 dr = O(ε−2 ) + O(ε−2ν )

c

by a change of variables, (2) follows at once. Lemma 7.5. kR(i; Λlε ) − R(i; HlAB )k = O(ε2ν ) + O(ε2 ) uniformly in l ∈ Z. Proof. By (7.6), it suffices to prove the lemma for |l| ≤ l0 . We write G(r, p) for the Green kernel G(r, p; R(i; Λlε )). If r > p > cε with c > 1, then it follows from (7.3) and (4.6) that G(r, p) is represented as G(r, p) = −(1/Dl (k))r1/2 p1/2 Hν (τ+ r)(c1l (k)Jν (τ− p) + c2l (k)J−ν (τ+ p)) with k = τ+ ε. Since J−ν (z) = e−iνπ Jν (z) + i sin νπHν (z) by formula, we have by Lemma 4.1 that G(r, p) = (1 + O(ε2ν ))G(r, p; R(i; HlAB )) + O(ε2ν )r1/2 p1/2 Hν (τ+ p)Hν (τ+ r) for (r, p) as above. A similar relation remains true when p > r > cε. Hence Lemma 7.4 completes the proof. Lemma 7.6. kR(i; Klε ) − R(i; HlAB )k = O(ε2ν ) + O(ε2 ) uniformly in l ∈ Z, and hence kR(i; Kε ) − R(i; H AB )k = o(1) ,

ε → 0.

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We prepare three lemmas to prove the lemma above. Lemma 7.7. Assume that |l| ≤ l0 . Then the inverse Z1l (ε) = (Id + χ1 R(iε2 ; Λl )Wl )−1 : L2 (R+ ) → L2 (R+) is bounded uniformly in ε. ˜1 (r) = χ1 (r)/r. Then Lemma 7.8. Assume that |l| > l0 1. Let χ ˜1 k = O(|l|−2 ) , kχ ˜1 R(iε2 ; Λl )χ

|l| → ∞ ,

uniformly in ε. Lemma 7.9. Let χ ˜1 be as above. Then ˜1 k = |l|−1/2 O(ε2 ) kR(±i; Λl)jε χ uniformly in |l| ≥ l0 . These three lemmas are proved after completing the proof of Lemma 7.6. Proof of Lemma 7.6. We set Rlε = R(i; Klε )−R(i; Λlε ). By Lemma 7.5, it suffices to show that kRlε k = O(ε2ν ) + O(ε2 )

(7.8)

uniformly in l ∈ Z. We first consider the case |l| ≤ l0 . As in the proof of Lemma 7.2, we obtain R(i; Klε ) = R(i; Λlε ) − ε2 jε R(iε2 ; Λl )Wl χ1 R(iε2 ; Kl )jε∗ . The operator χ1 R(iε2 ; Kl ) on the right side is represented χ1 R(iε2 ; Kl ) = Z1l (ε)χ1 R(iε2 ; Λl ) by use of the resolvent identity. Z1l (ε) being as in Lemma 7.7. Hence we have Rlε = −ε−2 (R(i; Λlε )jε χ1 )Wl Z1l (ε)(R(−i; Λlε )jε χ1 )∗ . Since kR(±i; Λlε )jε χ1 k = O(ε1+ν ) + O(ε2 ) by Lemma 7.1, Lemma 7.7 implies (7.8) with |l| ≤ l0 . Next we prove (7.8) for the case |l| > l0 . Recall that Wl (r) takes the form Wl = χ1 (r)wl (r)r−2 , where wl (r) has support in [0, 2) and obeys the bound wl (r) = O(|l|) as |l| → ∞. By Lemma 7.8, there exists the inverse ˜1 R(iε2 ; Λl )(wl /r))−1 : L2 (R+ ) → L2 (R+ ) Z2l (ε) = (Id + χ bounded uniformly in ε and l, |l| 1, and hence ˜1 R(iε2 ; Λl ) . χ ˜1 R(iε2 ; Kl ) = Z2l (ε)χ

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Thus we repeat the same argument as in the case |l| ≤ l0 to obtain that ˜1 )wl Z2l (ε)(R(−i; Λlε )jε χ ˜1 )∗ . Rlε = −ε−2 (R(i; Λlε )jε χ Hence it follows form Lemma 7.9 that kRlε k = O(ε2 ) uniformly in |l| ≥ l0 . Thus the proof of Lemma 7.6 is now complete. We prove the three lemmas above which remain unproved. Proof of Lemma 7.7. As in the proof of Lemma 4.6, we can show that (Id + χ1 R(0; Λl )Wl )−1 : L2 (R+ ) → L2 (R+ ) exists. Hence the lemma follows from Proposition 4.1 (1). Proof of Lemma 7.8. As in the proof of Lemma 4.5, the proof is done by use of integration by parts. We skip the details. Proof of Lemma 7.9. We repeat the same argument as in the proof of Lemma 7.1 (2) to obtain ˜1 k2 = O(ε2 )kχ ˜1 (R(iε2 ; Λl ) − R(−iε2 ; Λl ))χ ˜1 k . kR(i; Λl )jε χ Hence the lemma follows from Proposition 4.1(3) by interpolation. We further require one lemma to prove Proposition 7.2 in question.

Lemma 7.10. kR(i; Kε )Jε χ1 ∇k = O(ε) . Proof. We repeat the same argument as used in the proof of Lemma 7.1(2) to obtain that kR(i; Kε )Jε χ1 ∇k2 = O(ε2 )k∇χ1 (R(iε2 ; Kα ) − R(−iε2 ; Kα ))χ1 ∇k . By Lemma 4.8, we have kχ1 R(±iε2 ; Kα )χ1 k = O(1), and hence k∇χ1 R(±iε2 ; Kα )χ1 ∇k = O(1) by elliptic estimate. This proves the lemma. Proof of Proposition 7.2. The proof is almost the same as that of Lemma 7.6. We set Rε = R(i; Lε ) − R(i; Kε ). Then Rε = −ε−2 (R(i; Kε )Jε χ1 )W (Id + χ1 R(iε2 ; Kα )W )−1 (R(−i; Kε )Jε χ1 )∗ . We can show as in the proof of Lemma 7.7 that the inverse (Id + χ1 R(iε2 ; Kα )W )−1 : L2 → L2 is bounded uniformly in ε. By Lemma 7.10, kR(i; Kε )Jε χ1 W k = O(ε), and by Lemma 7.3, k(R(−i; Kε )Jε χ1 )∗ k = kR(−i; Kε )Jε χ1 k = o(ε) . Thus kRε k = o(1), and hence Lemma 7.6 completes the proof.

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8. Proof of Main Theorem In the present and next sections we prove the main theorem (Theorem 6.1). As stated in Sec. 3, the proof is based on the relation R(−i; Hε ) = R(−i; H0ε ) − λ(ε)ε−2 (R(−i; H0ε )Jε |V |1/2 )Z(ε)(R(i; H0ε )Jε V 1/2 )∗ and the strategy consists of analyzing the behavior as ε → 0 of the three operators R(−i; H0ε ), R(±i; H0ε )Jε |V |1/2 and Z(ε) appearing on the right side of the above relation. 8.1. We first study the resolvent R(−i; H0ε ) of operator H0ε = H(µ(ε)Aε , 0). The aim is to prove the following proposition. Proposition 8.1. R(i; H0ε ) → R(i; H AB ) ,

ε → 0,

in operator norm .

The proof requires the two lemmas below. Lemma 8.1. Let X denote the multiplication by hxi = (1 + |x|2 )1/2 and let s ∼ 1 be in a small interval around one. Then X −1 R(iε2 ; H(sA, 0))X −1 : L2 → L2 is bounded uniformly in ε and s. Proof. We prove the lemma for Lα = H(A, 0) with s = 1 only. Recall that Kα = P H(Bα , 0) admits the partial wave expansion Kα ' l∈Z ⊕ Kl , where Kl = Π∗ Π + (l − αχ∞ (r))2 r−2 . By Lemma 4.7, R(iε2 ; Kl ) : L2com (I0 ) → L2loc(R+ ) is bounded uniformly in ε and l. Since α 6∈ Z by assumption, we have (l − αχ∞ (r))2 r−2 > cr−2 ,

r 1,

for some c > 0 independent of l. We take the scalar product between the equation (Kl + iε2 )u = f , f ∈ L2com (I0 ), and the solution u ∈ L2 (R+ ). Then a simple calculation using partial integration shows that (1 + r)−1 R(iε2 ; Kl ) : L2com(I0 ) → L2 (R+ ) is uniformly bounded, and also it follows by duality that R(iε2 ; Kl )(1 + r)−1 : L2 (R+ ) → L2loc (R+ ) is uniformly bounded. We repeat the argument using partial integration to obtain that (1 + r)−1 R(iε2 ; Kl )(1 + r)−1 : L2 (R+ ) → L2 (R+ ) is bounded uniformly in ε and l, and hence it follows that X −1 R(iε2 ; Kα )X −1 : L2 → L2 is bounded uniformly in ε. By Proposition 4.2, R(iε2 ; Lα ) : L2com(Σ0 ) → L2loc (R2 )

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is bounded uniformly in ε, and W = Lα − Kα is a first order differential operator with smooth coefficients supported in {x : |x| < 2}. Thus the lemma is verified by use of the resolvent identity. Lemma 8.2. Let Xε denote the multiplication by hx/εi and let s ∼ 1 be again in a small interval around one. Then ε−1 Xε−1 R(±i; H(sAε , 0)) : L2 → L2 is bounded uniformly in ε and s. Proof. We again prove the lemma for Lε = H(Aε , 0) with s = 1 only. We calculate ε−2 kXε−1 R(i; Lε )k2 = ε−2 kJε∗ Xε−1 R(i; Lε )k2 = ε−2 kX −1 Jε∗ R(i; Lε )k2 = ε−2 kX −1 Jε∗ R(i; Lε )R(−i; Lε )Jε X −1 k = O(ε−2 )kX −1 Jε∗ (R(i; Lε ) − R(−i; Lε ))Jε X −1 k . Since Jε∗ R(i; Lε )Jε = ε2 R(iε2 ; Lα ), the lemma follows from Lemma 8.1. Proof of Proposition 8.1. By the resolvent identity, R(i; H0ε ) − R(i; Lε ) = −R(i; Lε )(H(µ(ε)Aε , 0) − H(Aε , 0))R(i; H0ε ) . The magnetic potential Aε (x) obeys |Aε (x)| = O(ε−1 )hx/εi−1 . If we write H0ε = Π1 (ε)2 + Π2 (ε)2 , then it follows that Πj (ε)R(±i; H0ε ) : L2 → L2 is uniformly bounded. Hence we have R(i; H0ε ) − R(i; Lε ) → 0 ,

ε → 0,

by Lemma 8.2. Thus the proposition follows from Proposition 7.2 at once. 8.2. We proceed to the second operator R(±i; H0ε )Jε |V |1/2 . We shall prove the following proposition. Proposition 8.2. There exists ζ±l (k) such that   X ζ+l (τ+ ε)(ξ+l ⊗ ωl )χ1  + op(ε1+2σ ) + Op(ε2 ) R(i; H0ε )Jε χ1 = ε  l=0,1

 R(−i; H0ε )Jε χ1 = ε 

X

 e−iνπ/2 ζ−l (τ− ε)(ξ−l ⊗ ωl )χ1  + op(ε1+2σ ) + Op(ε2 )

l=0,1

in B(L2 → L2 ), where ζ±l (k) behaves like ζ±l (k) = πl k ν (1 + o(1)) , with πl defined by (7.2).

|k| → 0 ,

(8.1)

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Before proving the proposition, we look at the function ωl = T ∗ u ˜l defined by (4.19). By (4.18) and (4.14), ωl is further written as ˜l − R(0; Lα )W u ˜l ωl = u = Ul∗ ul − Ul∗ R(0; Kl )Wl ul − R(0; Lα )W u ˜l . We see that ωl solves the equation Lα ωl = 0. In fact, we calculate ul = Kα Ul∗ Tl∗ ul Lα ωl = (Lα − W )˜ = Ul Kl Tl∗ ul = Ul (Kl − Wl )ul = Ul Λl ul = 0 . Since Ul∗ ul behaves like (Ul∗ ul )(x) = (2π)−1/2 ((l + ν)1/2 /(2ν))rν eilθ + o(1) = hl (x) + o(1) at infinity and since hl fulfills Lα hl = Kα hl = 0 for |x| > 2, it follows from Lemma 4.3 (see Remark 4.2 also) that ωl (x) is represented as ωl = (1 − χ2 )hl + R(0; Lα )[Lα , χ2 ]hl , where χ2 (x) = χ(r/4) = χ(|x|/4) and [Lα , χ2 ] again denotes the commutator [Lα , χ2 ] = Lα χ2 − χ2 Lα . Note that (1 − χ2 )χ1 = 0, so that χ1 ωl = χ1 R(0; Lα )[Lα , χ2 ]hl

(8.2)

with hl = (2π)−1/2 ((l + ν)1/2 /2ν)rν eilθ . We further prepare two lemmas to prove the above proposition. Lemma 8.3. If s ∼ 1, then both the operators X −1 R(0; H(sA, 0))X −1 , ∇R(0; H(sA, 0))X −1 : L2 → L2 are bounded. Proof. By Lemma 8.1, X −1 R(iε2 ; Lα )X −1 : L2 → L2 is bounded uniformly in ε. We take the limit ε → 0 to obtain the boundedness of X −1 R(0; Lα )X −1 . We shall show that ∇R(0; Lα )X −1 : L2 → L2 is bounded. If we write Lα = H(A, 0) = Π1 (A)2 + Π2 (A)2 , then Πj (A)R(0; Lα )X −1 : L2 → L2 is bounded. Since A(x) = O(|x|−1 ) at infinity, the boundedness of ∇R(0; Lα )X −1 is obtained from that of X −1 R(0; Lα )X −1 , and the proof is complete. Lemma 8.4. If s ∼ 1, then the mapping s → X −1 R(0; H(sA, 0))X −1 is differentiable as a function with values in B(L2 → L2 ), and X −1 ((d/ds)R(0; H(sA, 0))|s=1 )X −1 = X −1 R(0; Lα )Y (A)R(0; Lα )X −1 , where Y (A) is again defined by (6.4).

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Proof. The relation in the lemma can be easily derived by a formal computation. However Lemma 8.3 enables us to justify this computation. We can show in a similar way that R(0; H(sA, 0)) is C 2 -smooth and that R(0; H(µ(ε)A, 0)) = R(0; Lα ) + µε2σ R(0; Lα )Y (A)R(0; Lα ) + op(ε2σ )

(8.3)

in B(L2com (Σ0 ) → L2loc (R2 )). We now prove Proposition 8.2 in question. Proof of Proposition 8.2. The magnetic potential µ(ε)Aε (x) has the flux µ(ε)α 6∈ Z for ε small enough. We apply Proposition 7.1 to R(i; H0ε ) with H0ε = H(µ(ε) Aε , 0). We write ωlε (x) for the ωl -function corresponding to H0 (ε) = H(µ(ε)A, 0). This is defined by (8.2) with Lα = H(A, 0) and ν = |l − α| replaced by H0 (ε) and |l − µ(ε)α|, respectively. By Lemma 8.4, we have kχ1 (ωlε − ωl )kL2 = O(ε2σ ) .

(8.4)

Similarly we denote by ξ±lε ∈ L2 the ξ±l -function associated with H(µ(ε)Aα , 0), which solves (H(µ(ε)Aα , 0)∗ ∓ i)ξ±lε = 0 . This function is defined by (3.1) with ν replaced by |l − µ(ε)α| and it obeys the bound kξ±lε − ξ±l kL2 = O(ε2σ ). Hence it follows from (8.4) that (ξ±lε ⊗ ωlε )χ1 = (ξ±l ⊗ ωl )χ1 + Op(ε2σ ) in B(L2 → L2 ). Thus Proposition 7.1 implies that   X ζ+l (τ+ ε)(ξ+l ⊗ ωl )χ1  + op(ε1+2σ ) + Op(ε2 ) R(i; H0ε )Jε χ1 = ε  l=0,1

for some ζ+l (k) having the property in the proposition. A similar argument applies to R(−i; H0ε )Jε χ1 and the proposition is verified. 8.3. The remaining subsections are devoted to analyzing the behavior as ε → 0 of the inverse Z(ε) = A(ε)−1 ,

A(ε) = Id + λ(ε)V 1/2 R(−iε2 ; H0 (ε))|V |1/2

(8.5)

with H0 (ε) = H(µ(ε)A, 0). To do this, we often use the following lemma due to [8] (Lemma 3.12) in the future discussion. Lemma 8.5. Let Xj , Yj , j = 0, 1, be vector spaces and let A : X1 → Y1 . Assume that B : X0 → X1 is surjective and C : Y1 → Y0 is injective. Define A = CAB : X0 → Y0 . If A−1 exists, then A−1 = BA−1 C. We consider the operator A(ε) defined above. We apply Proposition 4.2 to R(k 2 ; H0 (ε)) with k = τ− ε. If we make use of (8.3) and (8.4), then the resolvent R(−iε2 ; H0 (ε)) admits the expansion

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R(−iε2 ; H0 (ε)) = R(0; Lα ) +

X

503

δl (k)(ωl ⊗ ωl )

l=0,1

+ µR(0; Lα )Y (A)R(0; Lα )ε2σ + op(ε2σ ) in B(L2com (Σ0 ) → L2loc (R2 )), and hence A(ε) obeys   X δl (k)(ωl ⊗ ωl ) |V |1/2 A(ε) = (Id + V 1/2 R(0; Lα )|V |1/2 ) + V 1/2  l=0,1

+ V 1/2 R(0; Lα )(µY (A)R(0; Lα ) + λ)|V |1/2 ε2σ + op(ε2σ )

(8.6)

in B(L2com (Σ0 ) → L2 ), where δl (k) behaves like δl (k) = γl k 2ν (1 + o(1)) ,

|k| → 0 ,

(8.7)

with the constant γl defined by (4.10). Let {Γj }3j=1 be the family of projections (not necessarily orthogonal) defined by (5.4). We shall mention several properties of these projections as a series of lemmas, which are required for studying the inverse Z(ε) in question. Lemma 8.6. (1)

(Id + V 1/2 R(0; Lα )|V |1/2 )Γ1 = 0 ,

(2)

Γ∗1 Vˆ (Id + V 1/2 R(0; Lα )|V |1/2 ) = 0 ,

(Id + V 1/2 R(0; Lα )|V |1/2 )Γ2 = 0 . Γ∗2 Vˆ (Id + V 1/2 R(0; Lα )|V |1/2 ) = 0 .

Proof. (1) is obvious by definition. Since |V |1/2 = Vˆ × V 1/2 , we have Γ∗1 Vˆ (Id + V 1/2 R(0; Lα )|V |1/2 ) = Γ∗1 (Id + |V |1/2 R(0; Lα )V 1/2 )Vˆ = 0 by adjoint, and hence (2) is obtained. Lemma 8.7. Let El be defined by El = ωl ⊗ ωl for l = 0, 1. Then Γ∗2 |V |1/2 El |V |1/2 Γ2 = 0 and Γ∗1 |V |1/2 El |V |1/2 Γ2 = 0 ,

Γ∗2 |V |1/2 El |V |1/2 Γ1 = 0 ,

Γ3 |V |1/2 El |V |1/2 Γ2 = 0 ,

Γ∗2 |V |1/2 El |V |1/2 Γ3 = 0 .

Proof. Recall that Γ2 is the projection from L2 onto M0 = {u ∈ M : (u, |V |1/2 ωl ) = 0

for l = 0, 1} .

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Hence we have El |V |1/2 Γ2 = 0 and Γ∗2 |V |1/2 El = 0. Thus all the relations in the lemma follow at once. Lemma 8.8. There exists a bounded inverse Y3 = (Γ3 (Id + V 1/2 R(0; Lα )|V |1/2 )Γ3 )−1 : Ran Γ3 → Ran Γ3 . Proof. The lemma is nothing but (5.2). Lemma 8.9. There exists an inverse Y2 = (Γ∗2 |V |1/2 R(0; Lα )(µY (A)R(0; Lα ) + λ)|V |1/2 Γ2 )−1 : Ran Γ∗2 → Ran Γ2 . Proof. It follows from Lemma 8.6 that |V |1/2 Γ2 = −V R(0; Lα )|V |1/2 Γ2 , and hence R(0; Lα )(µY (A)R(0; Lα ) + λ)|V |1/2 Γ2 = R(0; Lα )Y (µ, λ)R(0; Lα )|V |1/2 Γ2 , where Y (µ, λ) is defined by (6.1). By assumption (A), Pb Y (µ, λ)Pb : E0 → E0 is invertible, and by Lemma 5.4, R(0; Lα )|V |1/2 : M0 = Ran Γ2 → E0 is injective and surjective. This proves the lemma. The analysis on the inversion of Γ∗1 |V |1/2 El |V |1/2 Γ1 = Γ∗1 |V |1/2 (ωl ⊗ ωl )|V |1/2 Γ1 : Ran Γ1 → Ran Γ∗1 is divided into the three cases according as dim E1 = 0, 1 or 2. 8.4. We first deal with the case dim E1 = 2. We again take ηl ∈ M1 as in (5.7) and we define the projection Ql as Ql = ηl ⊗ ηl∗ : L2 → L2 , where

ηl∗

l = 0, 1 ,

∈ L denotes the basis dual to ηl with the following properties: 2

(ηm , ηl∗ ) = (ηm , ηl∗ )L2 = δlm ,

ηl∗ ⊥ Ran Γ2 ⊕ Ran Γ3 ,

Ran Γ2 = M0 .

The projection Ql is easily shown to satisfy the relations Q 0 + Q 1 = Γ1 ,

Qi Qj = δij Qj ,

0 ≤ i, j ≤ 1 .

The following two lemmas can be verified by use of property (5.7). We omit the proof here. Lemma 8.10. Let El be again defined by El = ωl ⊗ ωl . Then El |V |1/2 Qm = 0 ,

Q∗m |V |1/2 El = 0

for l 6= m, 0 ≤ l, m ≤ 1. Lemma 8.11. The operator Q∗l |V |1/2 El |V |1/2 Ql is represented as Q∗l |V |1/2 El |V |1/2 Ql = ηl∗ ⊗ ηl∗ : Ran Ql → Ran Q∗l and it has the inverse

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Yl = (Q∗l |V |1/2 El |V |1/2 Ql )−1 = ηl ⊗ ηl : Ran Q∗l → Ran Ql for l = 0, 1. We now set Q2 = Γ2 and Q3 = Γ3 . Then the family {Qj }3j=0 of projections preserves the same properties as in (5.5). We apply Lemma 8.5 to the operator A(ε) : X1 → Y1 ,

X1 = Y1 = L2 ,

defined by (8.5) under the situation B = (k −α Q0 , k −(1−α) Q1 , ε−σ Q2 , Q3 ) : X0 → X1 C = t (k −α Q∗0 Vˆ , k −(1−α) Q∗1 Vˆ , ε−σ Q∗2 Vˆ , Q3 ) : Y1 → Y0 with k = τ− ε, where X0 =

3 X j=0

 ⊕ Ran Qj ,

Y0 = 

2 X

 ⊕ Ran Q∗j  ⊕ Ran Q3 .

j=0

It is obvious that B is surjective. We show that C is injective. Assume that Cu = 0 for u ∈ L2 . Then Q3 u = 0 and Q∗ Vˆ u = (Q∗0 + Q∗1 + Q∗2 )Vˆ u = 0 . Hence u is in M and Qu = u. This implies that (|V |1/2 u, R(0; Lα)|V |1/2 u) = (Vˆ u, V 1/2 R(0; Lα )|V |1/2 u) = −(Vˆ u, Qu) = 0 . If we set v = R(0; Lα )|V |1/2 u, then v ∈ L2−1 (R2 ) and (Lα v, v) = 0. We can obtain v = 0, using the same argument as in the proof of Lemma 4.3, and hence it follows that u = 0. Thus C is shown to be injective. Recall that A(ε) admits the expansion (8.6) in B(X1 → Y1 ) and that δl (k) with k = τ− ε obeys the asymptotic formula (8.7) with γl defined by (4.10). Since σ > αmax = max(α, 1 − α) by assumption (6.2), it follows from Lemmas 8.6–8.11 that the inverse A(ε)−1 of the matrix representation A(ε) = CA(ε)B : X0 → Y0 exists for ε, 0 < ε 1, small enough and takes the form   Y0 /γ0 0 0 0    0 0 0  Y1 /γ1 −1   + op(ε0 ) A(ε) =   0 0 0 Y 2   0 0 0 Y3 in B(Y0 → X0 ). Thus Z(ε) = BA(ε)−1 C is expanded as −2(1−α)

−2α /γ0 )ε−2α Q0 Y0 Q∗0 Vˆ + (τ− Z(ε) = (τ−

/γ1 )ε−2(1−α) Q1 Y1 Q∗1 Vˆ

+ ε−2σ Q2 Y2 Q∗2 Vˆ + Q3 Y3 Q3 + B(op(ε0 ))C ,

(8.8)

provided that dim E1 = 2. 8.5. Next we consider the case dim E1 = 1. We recall the notation. Let ψl ∈ M1 be defined by (5.6). Then at least one of ψ0 and ψ1 never vanishes. If we assume that

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ψ0 6= 0, then ψ1 = βψ0 for some β ∈ C (see (5.10)). We take η˜0 = ψ0 /(ψ0 , ψ0 ), so that (ψ0 , η˜0 ) = 1 and (ψ1 , η˜0 ) = β. As in the previous subsection, we now define the projection ˜ 0 = η˜0 ⊗ η˜0∗ : L2 → M1 , Q where η˜0∗ ∈ L2 is the basis dual to η˜0 such that (˜ η0 , η˜0∗ ) = 1 ,

η˜0∗ ⊥ Ran Γ2 ⊕ Ran Γ3 .

If ψ0 = 0 and hence ψ1 6= 0, then we take η˜1 and η˜1∗ in a similar way, and we define ˜ 1 = η˜1 ⊗ η˜1∗ : L2 → M1 . the projection Q Lemma 8.12. Let Y˜l be defined by Y˜l = η˜l ⊗ η˜l for l = 0, 1, and let X δl (k)(ωl ⊗ ωl ) , k = τ− ε , E(k) = l=0,1

be as in (8.6). Then one has: (1) If ψ0 6= 0, then ˜ ∗0 → Ran Q ˜0 , ˜ 0 )−1 = δ(k)−1 Y˜0 : Ran Q ˜ ∗0 |V |1/2 E(k)|V |1/2 Q (Q where δ(k) = δ0 (k) + |β|2 δ1 (k) for β as above. (2) If ψ0 = 0, then ˜ 1 )−1 = δ1 (k)−1 Y˜1 : Ran Q ˜ ∗ → Ran Q ˜1 . ˜ ∗ |V |1/2 E(k)|V |1/2 Q (Q 1 1 Proof. We calculate ˜ 0 = |(ψl , η˜0 )|2 (˜ ˜ ∗ |V |1/2 (ωl ⊗ ωl )|V |1/2 Q η0∗ ⊗ η˜0∗ ) . Q 0 This proves (1), and the same argument applies to (2) also. We discuss only the case α = 1/2 in some detail. Assume that ψ0 6= 0. If α = 1/2, then ν = |l − α| = 1/2 for l = 0, 1, and δ(k) in Lemma 8.12 behaves like δ(k) = γk(1 + o(1)) ,

|k| → 0 ,

by (8.7), where γ = −2(1 + |β|2 /3)e−iπ/2 .

(8.9)

We again set Q2 = Γ2 and Q3 = Γ3 , and apply Lemma 8.5 to A(ε) : X1 → Y1 under the situation ˜ 0 , ε−σ Q2 , Q3 ) : X0 → X1 , B = (k −1/2 Q ˜ ∗ Vˆ , ε−σ Q∗ Vˆ , Q3 ) : Y1 → Y0 C = t (k −1/2 Q 0 2 with k = τ− ε again, where X1 = Y1 = L2 and ˜ 0 ⊕ Ran Q2 ⊕ Ran Q3 , X0 = Ran Q

˜ ∗0 ⊕ Ran Q∗2 ⊕ Ran Q3 . Y0 = Ran Q

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Then the matrix representation A(ε) = CA(ε)B has the inverse of the form   Y˜0 /γ 0 0    + op(ε0 ) A−1 (ε) =  0 0 Y 2   0 0 Y3 in B(Y0 → X0 ), and hence Z(ε) has the expansion −1 ˜ 0 Y˜0 Q ˜ ∗ Vˆ + ε−2σ Q2 Y2 Q∗ Vˆ /γ)ε−1 Q Z(ε) = (τ− 0 2

+ Q3 Y3 Q3 + B(op(ε0 ))C

(8.10)

in the case α = 1/2, provided that ψ0 6= 0. If ψ0 = 0, we repeat the same argument to obtain that −1 ˜ 1 Y˜1 Q ˜ ∗1 Vˆ + ε−2σ Q2 Y2 Q∗2 Vˆ /γ1 )ε−1 Q Z(ε) = (τ−

+ Q3 Y3 Q3 + B(op(ε0 ))C .

(8.11)

A similar argument applies to the case α 6= 1/2. We briefly discuss the case 0 < α < 1/2. Assume that ψ0 6= 0. Since α satisfies α < 1 − α, δ(k) behaves like δ(k) = γ0 k −2α (1 + o(1)). Hence we obtain −2α ˜ 0 Y˜0 Q ˜ ∗ Vˆ + ε−2σ Q2 Y2 Q∗ Vˆ /γ0 )ε−2α Q Z(ε) = (τ− 0 2

+ Q3 Y3 Q3 + B(op(ε0 ))C ,

(8.12)

˜ 0 , ε−σ Q2 , Q3 ) and C = t (k −α Q ˜ ∗ Vˆ , ε−σ Q∗ Vˆ , Q3 ). If ψ0 = 0, then where B = (k −α Q 0 2 we have −2(1−α)

Z(ε) = (τ−

˜ 1 Y˜1 Q ˜ ∗1 Vˆ + ε−2σ Q2 Y2 Q∗2 Vˆ /γ1 )ε−2(1−α) Q

+ Q3 Y3 Q3 + B(op(ε0 ))C , ˜ 1 , ε−σ Q2 , Q3 ) and C = t (k −(1−α) Q ˜ ∗1 Vˆ , ε−σ Q∗2 Vˆ , Q3 ). where B = (k −(1−α) Q The case dim E1 = 0 is the most simple to deal with. We have Z(ε) = ε−2σ Q2 Y2 Q∗2 Vˆ + Q3 Y3 Q3 + B(op(ε0 ))C , where B = (ε−σ Q2 , Q3 ) and C = t (ε−σ Q∗2 Vˆ , Q3 ). 9. Completion of Proof In this section we complete the proof of Theorem 6.1. We analyse the second operator M (ε) = ε−2 R(−i; H0ε )Jε |V |1/2 Z(ε)(R(i; H0ε )Jε V 1/2 )∗ appearing on the right side of the basic relation.

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(1) We first consider the case dim E1 = 2. We follow the notation in Sec. 8.4. By Proposition 8.2, we have   X e−iνπ/2 ζ−l (τ− ε)(ξ−l ⊗ ωl )|V |1/2  R(−i; H0ε )Jε |V |1/2 = ε  l=0,1

+ op(ε1+2σ ) + Op(ε2 ) and



(R(i; H0ε )Jε V 1/2 )∗ = ε 

X

 ζ+l (τ+ ε)V 1/2 (ωl ⊗ ξ+l ) + op(ε1+2σ ) + Op(ε2 )

l=0,1

in B(L2 → L2 ), where ζ±l (k) obeys (8.1) as |k| → 0. We further have the relations Q∗l |V |1/2 (ωm ⊗ ξ+m ) = δlm (ηl∗ ⊗ ξ+m ) (ξ−m ⊗ ωm )|V |1/2 Ql = δlm (ξ−m ⊗ ηl∗ ) by (5.7), and Q∗2 |V |1/2 (ωl ⊗ ξ+l ) = 0 ,

(ξ−l ⊗ ωl )|V |1/2 Q2 = 0

by Lemma 8.7. Hence we use the expansion (8.8) for Z(ε) to obtain that X Mlm (ξ−l ⊗ ξ+m ) + op(ε0 ) M (ε) = − 0≤l,m≤1

in B(L → L ), where Mlm = 0 for l 6= m and 2

2

τ+ /τ− )ν e−iνπ/2 = −(|πl |2 /γl )e−i3νπ/2 . Mll = −(|πl |2 /γl )(¯ Thus it follows from Proposition 8.1 that R(−i; Hε ) = R(−i; H AB ) +

X

Mll (ξ−l ⊗ ξ+l ) + op(ε0 )

l=0,1

in B(L2 → L2 ). If we use the formula Γ(1 − ν)Γ(1 + ν) = νπ/ sin νπ, then we can calculate Γ(1 − ν)Γ(1 + ν) sin2 νπ |πl |2 =− eiνπ = −eiνπ sin(νπ/2) γl 2νπ cos(νπ/2) by (4.10) and (7.2), so that Mll = e−iνπ/2 sin(νπ/2). This implies that 2iMll − 1 = 2i(cos(νπ/2) − i sin(νπ/2)) sin(νπ/2) − 1 = −e−iνπ . Hence the unitary matrix U = 2iM − Id is determined as ! 0 −e−iαπ = U (π/2, ei(1/2−α)π , 0) , U= 0 eiαπ

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where the notation U (η, a, b) is used with the meaning ascribed in (3.10). This yields the desired result for the case dim E1 = 2 (see Sec. 3.2). The case dim E1 = 0 is much easier to deal with. We can prove that R(−i; Hε ) = R(−i; H AB ) + op(ε0 ) and hence the desired result follows at once. (2) Next we consider the case dim E1 = 1. We use the notation in Sec. 8.5. We again discuss only the case α = 1/2 in some details. Let ρ(x) ∼ c0 r−1/2 + c1 r−1/2 eiθ be the resonance function spanning √ E1 . Assume that c0 6= 0 and hence ψ0 6= 0 (see √ (5.11)). Then β¯ = 3(c1 /c0 ) = 3c by (5.12), and Z(ε) has the expansion (8.10). −1 ˜ 0 Y˜0 Q ˜ ∗0 Vˆ in (8.10), /γ)ε−1 Q The main contribution comes from the first term (τ− 2 −iπ/2 by (8.9). We repeat the same argument as in the case where γ = −2(1 + |c| )e dim E1 = 2 to obtain that X Mlm (ξ−l ⊗ ξ+m ) + op(ε0 ) R(−i; Hε ) = R(−i; H AB ) + 0≤l,m≤1

in B(L2 → L2 ), where ¯m /γ)(¯ τ+ /τ− )1/2 e−iπ/4 (|V |1/2 ωl , η˜0 )(˜ η0 , |V |1/2 ωm ) Mlm = −(πl π ¯m /γ)e−i3π/4 (|V |1/2 ωl , η˜0 )(˜ η0 , |V |1/2 ωm ) . = −(πl π We calculate Mlm . Note that (|V |1/2 ω0 , η˜0 ) = (ψ0 , η˜0 ) = 1 and √ c. (|V |1/2 ω1 , η˜0 ) = (ψ1 , η˜0 ) = (βψ0 , η˜0 ) = β = 3¯ The constant πl is again given by (7.2) with ν = 1/2. Since Γ(1/2) =

√ π, we have

¯m /γ = −i2−3/2 (l + 1/2)−1/2 (m + 1/2)−1/2 (1 + |c|2 )−1 . πl π Thus we have the desired relation M = (Mlm )0≤l,m≤1 = (1/2i)(U (3π/4, a, b) + Id) after a simple calculation, where a and b are the constants in (3.11). The same relation is also obtained in the case c0 = 0, if we use the expansion (8.11) for Z(ε), and a similar argument applies to the case α 6= 1/2. If, for example, 0 < α < 1/2 and c0 6= 0, then we use (8.12) to obtain that Mlm = 0 for l 6= m, and τ+ /τ− )α e−iαπ/2 = e−iαπ/2 sin(απ/2) , M00 = −(|π0 |2 /δ0 )(¯ and hence U = 2iM − Id =

−e−iαπ

0

0

−1

M11 = 0 ,

! = U ((1 − α/2)π, e−iαπ/2 , 0) .

We skip the details for the other cases. Thus the proof of the main theorem is now completed. ˜ ε is also convergent to H U in norm resolvent We end the paper by showing that H sense, where ˜ ε = H(µ(ε)A˜ε , λ(ε)Vε ) = exp(iµ(ε)gε )Hε exp(−iµ(ε)gε ) , H

A˜ = A + a∇g ,

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is as in Remark 6.1, and gε (x) is defined by gε (x) = g(x/ε) with g ∈ C ∞ (R2 ) falling off at infinity. We give only a sketch for a proof. We first note that gε (x) → 0 uniformly in |x| ≥ c > 0 as ε → 0, c > 0 being fixed arbitrarily. We assert that kexp(igε )χR(−i; H AB ) − χR(−i; H AB )k → 0 ,

ε → 0,

(9.1)

where χ denotes the multiplication by χ(r), χ ∈ C0∞ ([0, ∞)) being the basic cut-off function. To prove this, we write X Ul∗ R(−i; HlAB )Ul R(−i; H AB ) = l∈Z

by use of partial wave decomposition. It is easy to see that kχR(−i; HlAB )k = o(1) ,

|l| → ∞ ,

as an operator acting on L2 (R+ ), and we have kexp(igε )Ul∗ χR(−i; HlAB )Ul − Ul∗ χR(−i; HlAB )Ul k → 0 ,

ε → 0,

for l fixed. In fact, the second convergence holds true in the Hilbert–Schmidt norm by the dominated convergence theorem. These facts imply (9.1) and hence it follows that kexp(iµ(ε)gε )R(−i; H AB ) exp(−iµ(ε)gε ) − R(−i; H AB )k → 0 ,

ε → 0.

(9.2)

By the dominated convergence theorem again, we have kexp(iµ(ε)gε )(ξl ⊗ ξ−l ) exp(−iµ(ε)gε ) − ξl ⊗ ξ−l k → 0 ,

ε → 0,

˜ ε. for l = 0, 1. This, together with (9.2), yields the norm resolvent convergence of H References [1] R. Adami and A. Teta, “On the Aharonov–Bohm Hamiltonian”, Lett. Math. Phys. 43 (1998) 43–53. [2] Y. Aharonov and D. Bohm, “Significance of electromagnetic potential in the quantum theory”, Phys. Rev. 115 (1959) 485–491. [3] N. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Vol. 2, Pitman, 1981. [4] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn and H. Holden, “Point interaction in two dimensions : Basic properties, approximation and applications to solid state physics”, J. reine angew. Math. 380 (1987) 87–107. [5] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, Springer-Verlag, 1988. [6] L. Dabrowski and H. Grosse, “On nonlocal point interactions in one, two, and three dimensions”, J. Math. Phys. 26 (1985) 2777–2780. [7] L. Dabrowski and P. Stovicek, “Aharonov–Bohm effect with δ-type interaction”, J. Math. Phys. 39 (1998) 47–62. [8] A. Jensen and T. Kato, “Spectral properties of Schr¨ odinger operators and time-decay of the wave functions”, Duke Math. J. 46 (1979) 583–611. [9] M. Reed and B. Simon, Methods of Modern Mathematical Physics, II. Fourier Analysis, Self-Adjointness, Academic Press, 1975.

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[10] S. N. M. Ruijsenaars, “The Ahanorov–Bohm effect and scattering theory”, Ann. of Phys. 146 (1983) 1–34. [11] H. Tamura, “Magnetic scattering at low energy in two dimensions”, Nagoya Math. J. 155 (1999) 95–151.

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S0129055X01000740


STABILITY OF GROUND STATES IN SECTORS AND ITS APPLICATION TO THE WIGNER WEISSKOPF MODEL

ASAO ARAI Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan E-mail : [email protected] MASAO HIROKAWA Department of Mathematics, Faculty of Science, Okayama University Okayama 700-8530, Japan E-mail : [email protected]

Received 2 March 2000 We consider two kinds of stability (under a perturbation) of the ground state of a self-adjoint operator: the one is concerned with the sector to which the ground state belongs and the other is about the uniqueness of the ground state. As an application to the Wigner–Weisskopf model which describes one mode fermion coupled to a quantum scalar field, we prove in the massive case the following: (a) For a value of the coupling constant, the Wigner–Weisskopf model has degenerate ground states; (b) for a value of the coupling constant, the Wigner–Weisskopf model has a first excited state with energy level below the bottom of the essential spectrum. These phenomena are nonperturbative. Keywords: Fock space, Wigne–Weisskopf model, ground state, ground state energy, stability, conservation law, first excited state. Mathematics Subject Classifications 2000: 81Q10, 47B25, 47N50

1. Introduction Let H be a Hilbert space and H0 a self-adjoint operator on H, bounded from below. Let I be an open interval of R containing the origin 0 and {H(α)}α∈I be a family of self-adjoint operators acting in H with H(α) bounded from below for every α ∈ I such that H(0) = H0 .

(1.1)

For a linear operator T on a Hilbert space, we denote its domain (respectively spectrum, point spectrum) by D(T ) (respectively σ(T ), σp (T )). If T is self-adjoint and bounded from below, then E0 (T ) := inf σ(T ) > −∞

(1.2)

is called the ground-state energy of T . We say that T has a ground state if ker(T − E0 (T )) 6= {0}; a non-zero vector in ker(T − E0 (T )) is called a ground state of T . 513

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The ground state of T is said to be unique (respectively degenerate) if dim ker(T − E0 (T )) = 1 (respectively ≥ 2). In this paper we are concerned with stabilities of ground states of H(α) in the parameter α ∈ I. In particular we are interested in the following two kinds of stability: (S.1) (Stability in sectors) Suppose that H has an orthogonal decomposition H = H0 ⊕ H1

(1.3)

with Hj (j = 0, 1) being a closed subspace of H such that, for all α ∈ I, H(α) is reduced by each Hj . In the context of quantum field theory, where H describes the Hilbert space of state vectors for the model under consideration, each Hilbert space Hj is called a sector. Suppose that H0 has a ground state in H0 . Then a natural question is: To which sector do the ground states of H(α) belong? (S.2) Uniqueness of ground states of H(α). As for (S.2), there are already fundamental results available (e.g. [6, Chapter VII], [9, Sec. XII.2]). We apply these results in a more restricted situation to obtain a stronger result. On the other hand, to our best knowledge, the problem (S.1) seems not to have been considered, at least, on an abstract level. In Sec. 2 we prove abstract results on problem (S.1) and degeneracy of ground states. These results are applied to a special class of self-adjoint operators in Sec. 3. In the last section we consider the Wigner–Weisskopf model (WW model) which describes one mode fermion coupled to a quantum scalar field [10]. We apply the results of Sec. 3 to this model in the massive case to establish the following properties: (a) For a value of the coupling constant, the WW model has degenerate ground states; (b) for a value of the coupling constant, the WW model has a first excited state with energy level below the bottom of the essential spectrum. We want to emphasize that these phenomena are nonperturbative and may be effects due to a strong coupling of the one mode fermion and the quantum scalar field. 2. Stability of Ground States in Sectors: Abstract Results 2.1. Main results We denote the resolvent of H(α) (α ∈ R) by Rz (α) := (H(α) − z)−1 ,

z ∈ ρ(H(α)) ,

(2.1)

where ρ(A) denotes the resolvent set of a closed operator A. We set E0 (α) := E0 (H(α)) ,

α∈I.

(2.2)

Our basic assumptions are as follows: (A.1) For all z ∈ C \ R, Rz : α → Rz (α) is continuous on I in operator norm.

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(A.2) For each α ∈ I, there exists a constanat Cα > 0 such that, for all sufficiently small |κ|, E0 (α + κ) ≥ Cα .

(2.3)

(A.3) For all α ∈ I, E0 (α) is an isolated eigenvalue of H(α) (hence H(α) has a ground state). A solution to the stability problem (S.1) is given in the following theorem: Theorem 2.1. Assume (A.1)–(A.3) and that H has the orthogonal decomposition (1.3) such that, for all α ∈ I, H(α) is reduced by H0 . Suppose that, for all α ∈ I, the ground state of H(α) is unique and that the ground state of H0 is in H0 . Then, for all α ∈ I, the ground state of H(α) is in H0 . This theorem can be used to show a degeneracy of ground states: Corollary 2.1. Assume (A.1)–(A.3) and that H has the orthogonal decomposition (1.3) such that, for all α ∈ I, H(α) is reduced by H0 . Suppose that the ground state of H0 is unique and in H0 . Moreover, suppose that there exists an α0 ∈ I such that H(α0 ) has a ground state which is not in H0 . Then, for some α0 ∈ I \ {0}, the ground state of H(α0 ) is degenerate. Proof. If the conclusion does not hold, then the ground state H(α) is unique for all α ∈ I. Hence, by Theorem 2.1, the ground state of H(α) is in H0 for all α ∈ I. But this contradicts the assumption that H(α0 ) has a ground state which is not in H0 . To prove Theorem 2.1, we establish two lemmas. Lemma 2.1. Assume (A.1) and (A.2). Then the ground state energy E0 (α) is continuous in α ∈ I. Proof. Fix α ∈ I arbitrarily. By (A.2), there exists a constant γα ∈ R such that, for all sufficiently small |κ|, γα ∈ ρ(H(α + κ)) and γα < E0 (α + κ). Assumption (A.1) implies that kRγα (α + κ) − Rγα (α)k → 0 (κ → 0). Hence 1 1 = lim kRγα (α + κ)k = kRγα (α)k = , κ→0 E0 (α + κ) − γα κ→0 E0 (α) − γα lim

which implies that limκ→0 E0 (α + κ) = E0 (α). Thus the desired result follows. Lemma 2.2. Assume (A.1)–(A.3). Suppose that, for all α ∈ I, the ground state H(α) is unique. Let Ψ0 (α) be a normalized ground state of H(α). Then, for all α ∈ I, lim (Ψ0 (α + κ), Ψ0 (α))Ψ0 (α + κ) = Ψ0 (α) .

κ→0

(2.4)

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Proof. For each α ∈ I, we denote by Pα (·) the spectral measure of H(α). Fix α ∈ I arbitrarily. By (A.3), there exists a constant a, b ∈ R ∩ ρ(H(α)) such that a < E0 (α) < b and (a, b) ∩ σ(H(α)) = {E0 (α)}. By (A.1) and a general fact [7, Theorem VIII.23(b)], kPα+κ ((a, b)) − Pα ((a, b))k → 0(κ → 0) .

(2.5)

Hence, by [9, p. 14, Lemma], dim Ran Pα+κ ((a, b)) = dim Ran Pα ((a, b)) = 1 for all sufficiently small |κ|. By Lemma 2.1, E0 (α + κ) ∈ (a, b) for all |κ| < δ with some constant δ > 0. Hence, for all |κ| < δ, Pα+κ ((a, b)) is the orthogonal projection onto ker(H(α + κ) − E0 (α + κ)), which implies that Pα+κ ((a, b))Ψ0 (α) = (Ψ0 (α + κ), Ψ0 (α))Ψ0 (α + κ). On the other hand, (2.5) implies that Pα+κ ((a, b))Ψ0 (α) → Pα ((a, b))Ψ0 (α) = Ψ0 (α)(κ → 0). Thus (2.4) follows. Proof of Theorem 2.1. Let Ψ0 (α) be a normalized ground state of H(α). By the uniqueness of the ground state of H(α), either Ψ0 (α) ∈ H0 or Ψ0 (α) ∈ H1 . By the present assumption, Ψ0 (0) ∈ H0 . Suppose that there existed a sequence {αn }∞ n=1 such that αn → 0 (n → ∞) and Ψ0 (αn ) ∈ H1 . Hence (Ψ0 (αn ), Ψ0 (0)) = 0 for all n ≥ 1. Then, by applying Lemma 2.2 to the case α = 0, we have Ψ0 (0) = 0. But this is a contradiction. Thus there exists a constant δ > 0 such that, for all |α| < δ, we have α ∈ I and Ψ0 (α) ∈ H0 . Let α− := inf{α ∈ I|Ψ0 (α) ∈ H0 } ,

α+ := sup{α ∈ I|Ψ0 (α) ∈ H0 } .

Then, by the above fact, α− < 0 < α+ . We first consider the case I = (c, d) with −∞ < c < 0 < d < ∞. We show that α− = c, α+ = d. Suppose that α+ < d. Then there exists a sequence {αn }∞ n=1 such that αn → α+ (n → ∞) and Ψ0 (αn ) ∈ H0 . Suppose that Ψ0 (α+ ) ∈ H1 . Then (Ψ0 (αn ), Ψ0 (α+ )) = 0. Applying Lemma 2.2 to the case α = α+ , we have Ψ0 (α+ ) = 0. But this is a contradiction. Hence Ψ0 (α+ ) ∈ H0 . Then, in the same way as above, we can show that there exists a constant α0 ∈ (α+ , d) such that Ψ0 (α0 ) ∈ H0 . Hence, by the definition of α+ , α0 ≤ α+ . But this is a contradiction. Thus α+ = d. Similarly we can show that α− = c. The same method works in the other cases of I. The proof of Theorem 2.1 shows in an obvious way that Theorem 2.1 can be generalized to the case of other eigenvectors of H(α): Theorem 2.2. Assume (A.1) and that H has the orthogonal decomposition (1.3) such that, for all α ∈ I, H(α) is reduced by H0 . Suppose that, for each α ∈ I, H(α) has an isolated eigenvalue E(α) such that dim ker(H(α) − E(α)) = 1, E(·) is continuous on I and ker(H0 −E(0)) ⊂ H0 . Then, for all α ∈ I, ker(H(α)−E(α)) ⊂ H0 .

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2.2. Uniqueness of ground states We first prove a general fact on the stability of uniqueness of eigenvectors of H(α). Proposition 2.1. Assume (A.1). Suppose that, for each α ∈ I, there exist constants E(α) ∈ R, δα > 0 and Kα > 0 such that [E(α) − δα , E(α) + δα ] ∩ σ(H(α)) = {E(α)}

(2.6)

and, for all |κ| < Kα , [E(α) − δα , E(α) + δα ] ∩ σ(H(α + κ)) = {E(α + κ)} ,

(2.7)

so that E(α) is an eigenvalue of H(α). Suppose that dim ker(H0 − E(0)) = 1. Then, for all α ∈ I, dim ker(H(α) − E(α)) = 1. Proof. Let a0 := E(0) − δ0 , b := E(0) + δ0 . As in the proof of Lemma 2.2, we see that, for all |α| < δ with some δ > 0 sufficiently small, dim Ran Pα ((a0 , b0 )) = dim Ran P0 ((a0 , b0 )) = 1. By (2.7), Ran Pα ((a0 , b0 )) = ker(H(α) − E(α)), |α| < δ. Hence dim ker(H(α) − E(α)) = 1, |α| < δ. Let a− := inf{α ∈ I| dim ker(H(α) − E(α)) = 1} a+ := sup{α ∈ I| dim ker(H(α) − E(α)) = 1} . By the above fact, we have a− < 0 < a+ . Consider the case I = (c, d) with −∞ < c < 0 < d < ∞. We show that a− = c, a+ = d. Suppose that a+ < d. Then there exists a sequence {αn }∞ n=1 such that αn → a+ (n → ∞) and dim ker(H(αn ) − E(αn )) = 1. Suppose that dim ker(H(a+ ) − E(a+ )) ≥ 2. We have for all n ≥ n0 with some n0 ≥ 1 dim Ran Pαn ((E(a+ ) − δa+ , E(a+ ) + δa+ )) = dim Ran Pa+ ((E(a+ ) − δa+ , E(a+ ) + δa+ )) . Hence, for all n ≥ n0 , dim Ran Pαn ((E(a+ ) − δa+ , E(a+ ) + δa+ )) ≥ 2. By (2.7), Ran Pαn ((E(a+ ) − δa+ , E(a+ ) + δa+ )) = ker(H(αn ) − E(αn )) ,

n ≥ n0 ,

which implies dim Ran Pαn ((E(a+ ) − δa+ , E(a+ ) + δa+ )) = 1. But this is a contradiction. Thus a+ = d. Similarly we can show that a− = c. The same method works in the other cases of I. We consider a sufficient condition for (2.6) and (2.7) to hold in the case E(α) = E0 (α). Let E1 (α) := inf{σ(H(α)) \ {E0 (α)}} .

(2.8)

Proposition 2.2. Assume (A.1) and (A.2). Suppose that, for every α ∈ I, there exists a constant Lα > 0 such that α ± Lα ∈ I , inf 0≤|κ|≤Lα

{E1 (α + κ) − E0 (α + κ)} > E0 (α) −

(2.9) inf 0≤|κ|≤Lα

E0 (α + κ) .

(2.10)

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Then H(α) satisfies (2.6) and (2.7). Proof. Fix α ∈ I arbitrarily. By (2.10), there is a real constant Mα such that inf 0≤|κ|≤Lα

{E1 (α + κ) − E0 (α + κ)} > Mα > E0 (α) −

inf 0≤|κ|≤Lα

E0 (α + κ) .

(2.11)

Hence, for every κ with 0 ≤ |κ| ≤ Lα , we have Mα < E1 (α + κ) − E0 (α + κ) .

(2.12)

In particular, putting κ = 0, we have E0 (α) + Mα < E1 (α) .

(2.13)

By the second inequality in (2.11), there exists a constant δα such that 0 < δα < M α +

inf

0≤|κ|≤Lα

E0 (α + κ) − E0 (α) .

(2.14)

By (2.12) and (2.14), we have E0 (α) + δα < Mα +

inf

0≤|κ0 |≤Lα

E0 (α + κ0 )

≤ (E1 (α + κ) − E0 (α + κ)) − E0 (α + κ) = E1 (α + κ) for 0 ≤ |κ| ≤ Lα , which, together with Lemma 2.1 and (2.9), implies (2.6) and (2.7). Propositions 2.1 and 2.2 immediately yield the following theorem. Theorem 2.3. Let the assumption of Proposition 2.2 be satisfied. Suppose that the ground state of H0 is unique. Then, for all α ∈ I, the ground state of H(α) is unique. A sufficient condition for (2.9) and (2.10) to hold is given in the following proposition. Proposition 2.3. Assume (A.1) and (A.2). Suppose that E0 (α) < E1 (α) for all α ∈ I, and E1 (α) is continuous in α ∈ I. Then (2.9) and (2.10) hold. Proof. Fix α ∈ I arbitrarily. Let ε be such that 0 0 such that if 0 ≤ |κ| ≤ K1,α , then α ± K1,α ∈ I and |E1 (α) − E1 (α + κ)| < ε .

(2.17)

Lα := min{K0,α , K1,α } .

(2.18)

Let

Then α ± Lα ∈ I, i.e. (2.9) holds. By Lemma 2.1, there exists a constant κ0 with 0 ≤ |κ0 | ≤ Lα such that inf

0≤|κ|≤Lα

Hence we have E0 (α) −

inf

0≤|κ|≤Lα

E0 (α + κ) = E0 (α + κ0 ) .

E0 (α + κ) = |E0 (α) − E0 (α + κ0 )| < ε .

(2.19)

Since E1 (α) − E0 (α) is continuous in α ∈ I, there exists a constant κ1 with 0 ≤ |κ1 | ≤ Lα such that inf 0≤|κ|≤Lα

{E1 (α + κ) − E0 (α + κ)} = E1 (α + κ1 ) − E0 (α + κ1 ) .

Hence we have by (2.15), (2.16), (2.17) and (2.19) inf

0≤|κ|≤Lα

{E1 (α + κ) − E0 (α + κ)}

= E1 (α + κ1 ) − E0 (α + κ1 ) = (E1 (α + κ1 ) − E1 (α)) + (E0 (α) − E0 (α + κ1 )) + (E1 (α) − E0 (α)) ≥ 2ε + (E1 (α) − E0 (α)) >ε > E0 (α) −

inf

0≤|κ|≤Lα

E0 (α + κ) .

Thus (2.10) follows. Theorem 2.3 and Proposition 2.3 imply the following theorem: Theorem 2.4. Assume (A.1), (A.2) and that E0 (α) < E1 (α) for all α ∈ I and E1 (α) is continuous in α ∈ I. Suppose that the ground state of H0 is unique. Then, for all α ∈ I, the ground state of H(α) is unique.

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3. A Special Class of Self-Adjoint Operators Let HI be a symmetric operator on H satisfying the following condition: (B.1) D(H0 ) ⊂ D(HI ) and there exist constants a, b > 0 such that, for all ψ ∈ D(H0 ), kHI ψk ≤ akH0 ψk + bkψk .

(3.1)

We define T (α) := H0 + αHI

(3.2)

with α ∈ R a coupling constant. Let Ia be an open interval from −1/a to 1/a: 1 1 . (3.3) Ia := − , a a By the Kato–Rellich theorem (e.g. [8, Theorem X.12]), for all α ∈ Ia , T (α) is self-adjoint with D(T (α)) = D(H0 ) and bounded from below with b|α| , |α|(a|E0 | + b) , (3.4) E0 (T (α)) ≥ E0 − max 1 − a|α| where E0 := E0 (H0 ) .

(3.5)

We assume the following: (B.2) For all α ∈ Ia , E0 (T (α)) is an isolated eigenvalue of T (α). Theorem 3.1. Assume (B.1), (B.2) and that H has the orthogonal decomposition (1.3) such that, for all α ∈ Ia , T (α) is reduced by H0 . Suppose that, for all α ∈ Ia , the ground state T (α) is unique and that the ground state of H0 is in H0 . Then, for all α ∈ Ia , the ground state of T (α) is in H0 . Corollary 3.1. Assume (B.1), (B.2) and that H has the orthogonal decomposition (1.3) such that, for all α ∈ Ia , T (α) is reduced by H0 . Suppose that the ground state of H0 is unique and in H0 . Moreover, suppose that there exists an α0 ∈ Ia such that T (α0 ) has a ground state which is not in H0 . Then, for some α0 ∈ Ia \ {0}, the ground state of T (α0 ) is degenerate. We prove these results by applying Theorem 2.1 and Corollary 2.1. To do this we need a lemma. Let Qz (α) := (T (α) − z)−1 ,

z ∈ ρ(T (α)) .

(3.6)

Lemma 3.1. Assume (B.1). Then, for all z ∈ C \ R, the operator-valued function: α → Qz (α) is continuous on Ia in operator norm topology.

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Proof. Fix α ∈ Ia and z ∈ C \ R arbitrarily. Since D(T (α)) = D(T (α + κ)) = D(H0 ) for every κ ∈ R with α + κ ∈ I, we have Qz (α + κ) − Qz (α) − κQz (α + κ)HI Qz (α) .

(3.7)

For Ψ ∈ D(H0 ), we have by the triangle inequality and (3.1) kH0 Ψk ≤ kT (α)Ψk + |α|kHI Ψk ≤ kT (α)Ψk + a|α|kH0 Ψk + b|α|kΨk . Hence kH0 Ψk ≤

b|α| 1 kT (α)Ψk + kΨk , 1 − a|α| 1 − a|α|

where |α| satisfies that 0 < |α| < 1/a. Putting this into (3.1), we obtain ab|α| a kT (α)Ψk + + b kΨk , kHI Ψk ≤ 1 − |α|a 1 − a|α|

(3.8)

which implies that HI Qz (α) is bounded. Since kQz (α + κ)k ≤ 1/|=z|, we obtain kQz (α + κ) − Qz (α)k ≤

|κ| kHI Qz (α)k → 0 |=z|

as κ → 0. Hence the desired result follows. Proof of Theorem 3.1. By the present assumption, (3.4) and Lemma 3.1, the assumption of Theorem 2.1 with H(α) = T (α) and I = Ia is satisfied. Thus the assertion follows. Remark 3.1. Assume (B.1) and fix α ∈ Ia arbitrarily. Then T (α+κ) is an analytic family of type (A) near κ = 0. This follows from (3.8) and a general fact [9, p. 16, Lemma]. Remark 3.2. In the case where HI is infinitesimally small with respect to H0 , Theorem 3.1, Corollary 3.1 and Lemma 3.1 hold with Ia = R. We can obtain results on uniqueness of ground states of T (α) by applying the results in Sec. 2.2 to the operator T (α). But we omit writing down them. 4. Application to the WW Model In this section we apply the main results of Sec. 3 to the WW model. We first recall the definition of the WW model. We take a Hilbert space of bosons to be Fb := Fb (L2 (Rd )) :=

∞ M [⊗ns L2 (Rd )] n=0

(4.1)

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(d ∈ N) the symmetric Fock space over L2 (Rd ) (⊗ns K denotes the n-fold symmetric tensor product of a Hilbert space K, ⊗0s K := C). In this paper, we set both of ~ (the Planck constant divided by 2π) and c (the speed of light) one, i.e. ~ = c = 1. Let ω : Rd → [0, ∞) be Borel measurable such that 0 < ω(k) < ∞ for almost everywhere (a.e.) k ∈ Rd with respect to the d-dimensional Lebesgue measure and Hb := dΓ(ω) , the second quantization of the multiplication operator on L2 (Rd ) by the function ω [8, Sec. X.7]. Let λ be a function on Rd . We assume the following (W.1) and (W.2): (W.1) The function λ is continuous on Rd , not identically zero with λ, λ/ω ∈ L2 (Rd ). (W.2) The function ω(k) is continuous with lim ω(k) = ∞ ,

(4.2)

|k|→∞

and there exist constants γω > 0 and Cω > 0 such that |ω(k) − ω(k 0 )| ≤ Cω |k − k 0 |γω (1 + ω(k) + ω(k 0 )) , We define a matrix c by c :=

0

0

1

0

k, k 0 ∈ Rd .

(4.3)

! .

(4.4)

The Hamiltonian HWW (α) of the WW model is defined by HWW (α) := H0 + αHI

(4.5)

H = C2 ⊗ F b

(4.6)

acting in

with H0 := µ0 c∗ c ⊗ I + I ⊗ Hb ,

(4.7)

HI := c∗ ⊗ a(λ) + c ⊗ a(λ)∗ ,

(4.8)

where µ0 , α ∈ R \ {0} are constant parameters and a(·) (respectively I) denotes the annihilation operator on Fb (respectively identity operator). It is easy to prove the following fact: Lemma 4.1. (i) The operator HI is infinitesimally small with respect to H0 . (ii) For all α ∈ R, HWW (α) is self-adjoint with D(HWW (α)) = D(H0 ) and bounded from below.

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The WW model has a conservation law for a kind of the particle number in the sense described below. Let σ3 be the third of the Pauli matrices: ! 1 0 (4.9) σ3 := 0 −1 and define NP :=

1 + σ3 ⊗ I + I ⊗ Nb , 2

(4.10)

where Nb := dΓ(I) is the boson number operator. The operator NP was introduced in [5, Sec. 6]. Let P (`) be the orthogonal projection onto the `-particle space of Fb (` ≥ 0). Then we have X `P (`) . (4.11) Nb = `=0

The spectral resolution of NP is given by X `P` , NP =

(4.12)

`=0

where

 1 − σ3   ⊗ P (0)  2 P` :=  1 − σ3 1 + σ3   ⊗ P (`−1) + ⊗ P (`) 2 2

if ` = 0 , (4.13) if ` ∈ N .

It is easy to see that, for every α ∈ R and each ` ∈ {0} ∪ N, P` HWW (α) ⊂ HWW (α)P` .

(4.14)

Hence HWW (α) is reduced by P` H. Let H0 := (P0 + P1 )H

(4.15)

H1 := H0⊥ (the orthogonal complement of H0 ) .

(4.16)

H = H0 ⊕ H1 .

(4.17)

and

Then

The following lemma easily follows: Lemma 4.2. (i) For each α ∈ R, HWW (α) is reduced by Hj , j = 1, 2. (ii) H0 has a unique ground state in H0 .

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Let E0WW (α) := E0 (HWW (α))

(4.18)

µ := ess. inf ω(k) ≥ 0 .

(4.19)

and k∈Rd

We say that the WW model is massive (respectively massless) if µ > 0 (respectively µ = 0). Proposition 4.1 ([1, Remark 3.1], [3, Proposition 6.10(i)]). σess (HWW (α)) = [E0WW (α) + µ, ∞) , where σess (·) denotes essential spectrum. We define Z |λ(k)|2 , dk Dµα (z) := −z + µ0 − α2 ω(k) − z Rd The limit

Z Cµ := lim t↓0

dk Rd

z ∈ Cµ := C \ [µ, ∞) .

|λ(k)|2 ω(k) − µ + t

(4.20)

(4.21)

exists or is infinity. In the former case, Cµ > 0 by (W.1). It is easy to see that Dµα (x) is monotone decreasing in x < µ. Hence the limit α dα µ := lim Dµ (x) x↑µ

(4.22)

exists or is −∞ and 2 dα µ = −µ + µ0 − α Cµ .

Let

µ −µ  0 Cµ β0 :=  0

if 0 < Cµ < ∞ ,

(4.23)

if Cµ = ∞

and 2 Aµ := {α ∈ R|−∞ ≤ dα µ < 0} = {α ∈ R|α > β0 } .

For all α ∈ Aµ , there exists a unique zero EWW (α) of Dµα (z): Z |λ(k)|2 2 . dk EWW (α) = µ0 − α ω(k) − EWW (α) Rd

(4.24)

(4.25)

Proposition 4.2 ([4, Theorem 2.3(b), (c)]). Let α ∈ Aµ . Assume either (i) µ > 0 or (ii) µ = 0 with ∇ω ∈ L∞ (Rd ). Then there exists a constant αWW ∈ Aµ ∩ (0, ∞) such that, for all |α| > αWW , {E0WW (α), EWW (α), 0} ⊂ σp (HWW (α))

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with E0WW (α) < min{EWW (α), 0} and / H0 . Ψ0 (α) ∈ Let E1WW (α) := inf{σ(HWW (α)) \ {E0WW (α)}}

(4.26)

and ε0 := min{0, µ0 } ,

ε1 := max{0, µ0 } .

(4.27)

Note that, if E1WW (α) is an eigenvalue of HWW (α), then each eigenvector corresponding to it physically describes one of the first excited states of the WW model. Theorem 4.1. Let µ > 0. Then: (i) There exists a constant α0 ∈ Aµ such that HWW (α0 ) has degenerate ground states. (ii) There exists a constant α1 ∈ Aµ such that E1WW (α1 ) is an eigenvalue of HWW (α1 ) and E1WW (α1 ) < E0WW (α1 ) + µ = inf σess (HWW (α1 )) .

(4.28)

Moreover, if 0 < µ < |µ0 |, then E1WW (α1 ) < ε1 .

(4.29)

Proof. (i) Since µ > 0, it follows from [2, Theorem 1.2] that, for all α ∈ R, HWW (α) has a ground state and E0WW (α) is an isolated eigenvalue of HWW (α). These facts together with Lemmas 4.1, 4.2, Proposition 4.2 imply that the assumption of Corollary 3.1 with T (α) = HWW (α) is satisfied. Hence there exists a constant α0 6= 0 such that the ground state of HWW (α0 ) is degenerate. If α0 6∈ Aµ so that 0 dα µ ≥ 0, then, by [3, Theorem 6.14(i)], HWW (α0 ) has a unique ground state. But this is a contradiction. (ii) By Proposition 4.1, we have for all α ∈ R E0WW (α) < E1WW (α) ≤ E0WW (α) + µ . Suppose that, for all α ∈ R \ {0}, E1WW (α) = inf σess (HWW (α)) = E0WW (α) + µ . By an application of Lemma 2.2, E0WW (α) is continuous in α ∈ R. Hence so is E1WW (α). Then, by an application of Theorem 2.4, for all α ∈ R, the ground state of HWW (α) is unique. But this contradicts part (i). Hence there exists a constant α1 6= 0 such that (4.28) holds and E1WW (α1 ) is an eigenvalue of HWW (α1 ). We show

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α that α1 ∈ Aµ . If µ0 < 0, then dα µ < 0 for all α ∈ R, which implies Aµ = R (µ0 < 0). α1 Hence α1 ∈ Aµ . Let µ0 > 0. Suppose that dµ ≥ 0. Then, by [3, Theorem 6.14(i)] 1 we have E1WW (α1 ) = E0WW (α1 ) + µ, which contradicts (4.28). Hence dα µ < 0. Therefore α1 ∈ Aµ . Finally we prove (4.29). Let µ < |µ0 |. Since 0 ∈ σp (HWW (α)) for all α ∈ R by [3, Proposition 6.13], we have

E1WW (α1 ) < E0WW (α1 ) + µ ≤ 0 + µ = µ . We first consider the case 0 < µ0 . In this case, ε0 = 0, ε1 = µ0 . Hence E1WW (α1 ) < ε1 . We next consider the case µ0 < 0. In this case, ε0 = µ0 and ε1 = 0. Since α1 ∈ Aµ 1 (i.e. dα µ < 0), we have by [3, Proposition 6.13(ii)] 0, EWW (α1 ) ∈ σp (HWW (α)) with EWW (α1 ) < 0. Since µ0 < 0, we have Z Dµα1 (µ0 ) − α21

dk Rd

|λ(k)|2 < 0. ω(k) − µ0

This implies that EWW (α1 ) < µ0 , since Dµα1 (x) is monotone decreasing in x < µ and Dµα1 (EWW (α1 )) = 0. Hence we have E1WW (α1 ) < E0WW (α1 ) + µ ≤ EWW (α1 ) + µ < µ0 + µ < 0 = ε1 . Thus (4.29) follows. Remark 4.1. One may think of why the reduced part H1 (α) of HWW (α) to the second sector H1 is not studied directly. A reason for that is just a technical one: If |α| is large, then it seems to be difficult to prove directly the existence of a ground state of H1 (α). Another reason is a theoretical one: One may expect that the ground states of H(α) move among various sectors as |α| becomes larger. To see this behavior, one may need study the Hamiltonian HWW (α) “globally” (cf. [4]). Remark 4.2. A theoretical meaning of the abstract theory in Sec. 2 is that it gives a class of families of self-adjoint operators to which a regular perturbation theory can be applied to ensure the stability of ground states in sector. In particular, a key condition for the uniqueness of ground states is the local uniformness of the spectral gap as formulated in (2.7). The WW model is an example such that (i) if the modulus of the coupling constant is sufficiently small, then the ground state is unique and stable in sector, so that it is in the class mentioned above; but (ii) if the modulus of the coupling constant is large enough, then the stability of ground states in sector breaks down and degenerate ground states may appear. Remark 4.3. Let µ > 0. Then it follows from the analytic perturbation theory that, for all sufficiently small |α|, the ground state of HWW (α) is unique. Hence

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Theorem 4.1(i) is a nonperturbative effect which may be due to a strong coupling of the one mode fermion and the quantum scalar field. Remark 4.4. Generally speaking, in a quantum field model, it is difficult to prove nonperturbatively the existence of an eigenvalue corresponding to the first excited states of the model. There are many papers stating the possibility of the existence of the first excited states in a nonperturbative way, but, to authors’ best knowledge, there is few papers proving nonperturbatively the real existence of those. In this sense, Theorem 4.1(ii) may have a meaning. Note that, if 0 < µ < |µ0 | = ε1 − ε0 , then ε1 is an embedded eigenvalue of H0 . Hence, in this case, we cannot apply the analytic perturbation theory even in the case where |α| is small. But, in this case too, Theorem 4.1(ii) holds, showing that, in the WW model, the embedded eigenvalue does not necessarily disappear under the perturbation αHI . In the case 0 < µ < |µ0 |, Theorem 4.1(ii) also is a nonperturbative effect. Remark 4.5. The phenomena described in Theorem 4.1 do not occur in the region of the coupling constant treated by H¨ ubner and Spohn [5, Sec. 6] and ourselves in [3, Theorem 6.14(i)]. Remark 4.6. We may expect that, in the massless case too (i.e. µ = 0), Theorem 4.1(i) holds. Acknowledgments One (M. H.) of the authors would like to thank H. Spohn, F. Hiroshima, R. A. Minlos, H. Ezawa and K. Watanabe for their valuable advices. Research of M. H. is supported by the Grant-In-Aid No. 11740109 for Encouragement of Young Scientists from Japan Society for the Promotion of Science (JSPS). A. A. is supported by the Grant-in-Aid No. 11440036 for Scientific Research from the Ministry of Education, Science, Sports and Culture. References [1] A. Arai, “Essential spectrum of a self-adjoint operator on an abstract Hilbert space of Fock type and applications to quantum field Hamiltonians”, J. Math. Anal. Appl. 246 (2000) 189–216. [2] A. Arai and M. Hirokawa, “On the existence and uniqueness of ground states of a generalized spin-boson model”, J. Funct. Anal. 151 (1997) 455–503. [3] A. Arai and M. Hirokawa, “Ground states of a general class of quantum field Hamiltonians”, Rev. Math. Phys. 12 (2000) 1085–1135. [4] M. Hirokawa, “Remarks on the ground state energy of the spin-boson model. An application of the Wigner–Weisskopf model” to appear in Rev. Math. Phys. [5] M. H¨ ubner and H. Spohn, “Spectral properties of the spin-boson Hamiltonian”, Ann. Inst. Henri. Poincaré 62 (1995) 289–323. [6] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin Heidelberg, New York, 1980.

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[7] M. Reed and B. Simon, Methods of Modern Mathematical Physics Vol. I, Academic Press, New York, 1975. [8] M. Reed and B. Simon, Methods of Modern Mathematical Physics Vol. II, Academic Press, New York, 1975. [9] M. Reed and B. Simon, Methods of Modern Mathematical Physics Vol. IV, Academic Press, New York, 1978. [10] V. F. Weisskopf and E. P. Wigner, “Berechnung der nat¨ urlichen Linienbreite auf Grund der Diracschen Lichttheorie”, Z. Phys. 63 (1930) 54–73.

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ON THE INTEGRABILITY OF A CLASS OF ` EQUATIONS MONGE AMPERE

J. C. BRUNELLI Universidade Federal de Santa Catarina, Departamento de F´ısica — CFM Campus Universit´ ario — Trindade, C. P. 476, CEP 88040–900 Florian´ opolis, SC — BRAZIL E-mail : [email protected] ¨ M. GURSES and K. ZHELTUKHIN Department of Mathematics, Bilkent University, 06533, Ankara, Turkey E-mail : [email protected] E-mail : [email protected]

Received 25 March 2000 Revised 5 June 2000 We give the Lax representations for the elliptic, hyperbolic and homogeneous second order Monge–Amp` ere equations. The connection between these equations and the equations of hydrodynamical type give us a scalar dispersionless Lax representation. A matrix dispersive Lax representation follows from the correspondence between sigma models, a two parameter equation for minimal surfaces and Monge–Amp` ere equations. Local as well nonlocal conserved densities are obtained.

1. Introduction The nonlinear partial differential equation in 1 + 1 dimensions 2 = −k Utt Uxx − Utx

(1)

is the second order Monge–Ampère equation. Here we will be interested in the case where k is a constant. For k = 1 we have the hyperbolic Monge–Ampère equation which is equivalent [1] to the Born–Infeld equation [2]. The choice k = −1 yields the elliptic Monge–Ampère equation that is related [3, 4] to the equation for minimal surfaces [5]. Finally, k = 0 corresponds to the homogeneous Monge–Ampère equation that can be shown to be related to the Bateman equation [6]. The Born–Infeld, minimal surfaces and Bateman equations can be treated simultaneously as (k 2 + φ2x )φtt − 2φx φt φxt + (k 2 α + φ2t )φxx = 0

(2)

where α ≡ k2 − k − 1

(3)

and we should keep in mind the trivial identities αk = −k and αk = −k . 2

529

2

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J. C. Brunelli, M. G¨ urses & K. Zheltukhin

The Born–Infeld equation was introduced in 1934 as a nonlinear generalization of Maxwell’s electrodynamics. It is the simplest wave equation in 1 + 1 dimensions, that preserves Lorentz invariance and is nonlinear. This equation is integrable [7, 8] and has a multi-Hamiltonian structure [9]. The Bateman equation was introduced in 1929 and is related with hydrodynamics. This equation has a very interesting behavior [10]. If φ(x, t) is a solution of (2), for k = 0, so is any function of it (covariance of (2)). Also, (2) can be derived from an infinite class of inequivalent Lagrangian densities and is form invariant under arbitrary linear transformations of the (x, t) coordinates. The equation for minimal surfaces gives the surface z = φ(x, t) in the three-dimensional space that spans a given contour and has the minimum area. This is the Plateau’s problem and has interest both in physics and mathematics. In this paper we will obtain Lax representations for (1) and (2) since both systems are related. A scalar dispersionless Lax representation as well a matrix dispersive Lax representation will be given. As far as the authors can say this is the first example of a system where both Lax pairs are present. In fact our results suggest that many other systems, which have both an infinite number of local and nonlocal charges, are likely to have such characteristic. This paper is organized as follows. In Sec. 2 we review the Bianchi transformation which relates (1) and (2). This is the Proposition 2.1 that unifies the results obtained in [1, 3, 4]. With this transformation we can easily translate results from the system (1) to system (2) and vice-versa. The existence of this Bianchi transformation is due to the fact that both (1) and (2) can be rewritten in a hydrodynamic type equation (polytropic gas). In Sec. 3, using results from [8, 11, 12], we obtain the dispersionless Lax representation of (1) (Proposition 3.1) and write the two sets of local conserved charges densities for the Monge–Ampère equation. In Sec. 4 we generalize the results of [5, 13, 14] concerning the matrix Lax representation for minimal surfaces through its correspondence with the sigma model. We obtain a matrix Lax representation for a two parameter equation for minimal surfaces which includes (2) for particular choices of the parameter (Proposition 4.4). From this Lax representation we give the nonlocal conserved charges densities of the system. In Sec. 5 we write explicitly the Lax representations, obtained in the previous sections, for the Monge–Ampère system (1) using the Bianchi transformation (Proposition 5.1). Finally we present our conclusions in Sec. 6. 2. Bianchi Transformation In order to see the connection between (1) and (2) (see Eq. (16)) we have to express these equations in the form of equations of hydrodynamic type [15]. From now on the reader should keep in mind that k is limited to the special values −1, 0, 1 and α to the corresponding values given by (3). Following [9, 16] we first introduce the potentials a and b, defined as a = Ux b = Ut .

(4)

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On the Integrability of a Class of Monge–Amp` ere Equations

531

Then, Eq. (1) can be expressed as a first order system k(at − bx ) = 0 , bt =

1 2 (b − k) , ax x

(5)

which is the natural starting point for a Hamiltonian treatment of Monge–Ampère equations (1). Now, introducing bx , ax v = ax , u=−

(6)

the Monge–Ampère equation can be written in the following hydrodynamic type equation form ut + uux + kv −3 vx = 0 ,

(7)

k(vt + (uv)x ) = 0 . Equation (2) follows from the Lagrangian q L = k 2 + φ2x + αφ2t .

(8)

We stress that the Bateman equation can be obtained from a large class of inequivalent Lagrangian. However, we will use this one and the limit k → 0 will give us results for the Bateman equation. Since (8) has no φ dependence (2) can be written as a conservation law given by ∂L ∂L + ∂t = 0. (9) ∂x ∂φx ∂φt This result allows us to rewrite (2) as a set of coupled first order nonlinear equations. Following [9, 16] let us express (2) as the integrability condition of a first-order system given by ψx = − ψt =

∂L αφt = −p , 2 ∂φt k + φ2x + αφ2t

∂L φx = p . ∂φx k 2 + φ2x + αφ2t

(10)

Introducing the variables r = φx , s = ψx ,

(11)

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we get from (10)

r φt = −αs r ψt = r

k 2 + r2 , 1 − αs2

(12)

1 − αs2 . k 2 + r2

So, the one-forms

r dφ = rdx − αs r

k 2 + r2 dt , 1 − αs2

1 − αs2 dt dψ = sdx + r k 2 + r2 are exact and its closure give us the equations s αrs k 2 + r2 rx − α sx , rt = − p (1 − αs2 )3 (k 2 + r2 )(1 − αs2 ) s st = k 2

(13)

(14) αrs 1 − αs sx . rx − p 2 (k 2 + r2 )3 (k + r2 )(1 − αs2 ) 2

Now the amazing fact is that Eq. (14) is also related with Eq. (7) by a special transformation. For the case k = 1 this transformation is known as the Verosky transformation [9]. The following k generalized Verosky transformation αrs , u = p (k 2 + r2 )(1 − αs2 ) (15) p 2 2 2 kv = −k (k + r )(1 − αs ) links (14) with (7). This result can be easily checked if we derive (15) with respect to time, use (14) and observe that from (15) it follows that k(uv) = −αk(rs) , k 2 v 2 − k 4 + αk 2 u2 v 2 = k 2 r2 − αk 4 s2 . From the diagram Eq. (1) ⇒ U

Eq. (4)

−→ a, b

Eq. (6)

−→

 ⇒ U = U (u, v)        ⇓      Eq. (7) u, v

⇑ Eq. (2) ⇒ φ

Eq. (11)

−→ r, s

Eq. (15)

−→

we are led to the proposition [1, 3, 4]:

u, v

     (   u = u(φ)     ⇒  v = v(φ)

⇒ U = U (φ) ,

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533

Proposition 2.1. The Monge–Ampère Eqs. (1) and (2) are related by the following Bianchi transformation k − φ2t , Utt = p k 2 + φ2x + αφ2t −φx φt , Utx = p 2 k + φ2x + αφ2t

(16)

−(k 2 + φ2x ) . Uxx = p k 2 + φ2x + αφ2t 3. Dispersionless Lax Representation: Local Conserved Charges Equation (7) for k = 1 corresponds to the equations of isentropic, polytropic gas dynamics with the adiabatic index γ = −1 [9]. This system is known as a Chaplygin gas [17]. For k = 0, (7) is the Riemann equation [11] and in this case the transformation (15) give us u = − φφxt . In [12] the polytropic gas dynamics [18] equations ut + uux + v γ−2 vx = 0 ,

γ≥2

(17)

vt + (uv)x = 0

were derived from the following dispersionless nonstandard Lax representation L = pγ−1 + u +

v γ−1 −(γ−1) p , (γ − 1)2

γ≥2 (18)

γ (γ − 1) ∂L = {(L γ−1 )≥1 , L} . ∂t γ

Here {A, B} =

∂B ∂A ∂A ∂B − ∂x ∂p ∂x ∂p

γ

and (L γ−1 )≥1 stands for the purely nonnegative (without p0 terms) part of the 1 polynomial in p. In (18) L γ−1 was expanded around p = ∞. A Lax description for the Chaplygin gas like equations vx ut + uux + β+2 = 0 , β ≥ 1 v (19) vt + (uv)x = 0 was obtained in [8] in connection with the Born–Infeld equation and it is given by L = p−(β+1) + u +

v −(β+1) β+1 p , (β + 1)2

β≥1

(20)

with β (β + 1) ∂L = {(L β+1 )≤1 , L} ∂t β

(21)

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J. C. Brunelli, M. G¨ urses & K. Zheltukhin 1

where L β+1 is expanded around p = 0. In view of these results we have the proposition: Proposition 3.1. For β = 1, the Lax operator L = p−2 + u +

k 2 p 4v 2

(22)

where 1 (L1/2 )≤1 = p−1 + up 2 reproduces (7). In terms of the variables a and b the Lax representation (22) assumes the form L = p−2 −

bx k + 2 p2 ax 4ax

(23)

∂L = 2{(L1/2 )≤1 , L} ∂t and yields the Monge–Ampère equation as expressed in (5). This proposition is the first main result of our paper. This is a dispersionless Lax representation, a dispersive one will be obtained in Sec. 5 (see Proposition 5.1). Conserved charges for the Chaplygin gas like Eq. (19) can be easily obtained from (20) through [8, 12] β+2

Hn = Tr Ln+ β+1 ,

n = 0, 1, 2, 3, . . . .

(24)

1 β+1

around p = 0. An This conserved charges were obtained by expanding L alternate expansion around p = ∞ is possible and it gives us a second set of conserved charges through 1 ˜ n = Tr Ln− β+1 , H

n = 0, 1, 2, 3, . . . .

(25)

Both set of densities for (24) and (25) can be expressed in closed form [8]. They are (n+1)(β+1)+1 (β+1)

Hn = (n + 1)!Cn+1

[ n+1 2 ]

X

−

m=0

×

`=0

−1 `(β + 1) + 1

!

v −m(β+1) un−2m+1 , m!(n − 2m + 1)! (−β − 1)m

˜ n = n!(−β − 1) H ×

m Y

2 β+1

n(β+1)−1 (β+1)

Cn

[2] X

n

m Y

m=0

`=0

v −m(β+1)+1 un−2m . m!(n − 2m)! (−β − 1)m

−1 `(β + 1) − 1

!

(26)

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The first densities Hn for the Monge–Ampère are H0 = − H1 =

5 1 (3b2 + k) , 8 a2x x

H2 = − H3 =

3 bx , 2 ax

35 bx 2 (b + k) , 16 a3x x

(27)

63 1 (5b4 + 10b2x k + k 2 ) , 128 a4x x .. .

˜ n are and the first densities H ˜ 0 = −2k 2 ax , H ˜ 1 = −k 2 bx , H ˜ 2 = − 3 k 2 1 (b2x + k) , H 4 ax

(28)

˜ 3 = 5 k 2 bx (b3x + 3k) , H 8 a2x .. . 4. Minimal Surfaces and Sigma Models In this section we will generalize some results of [5, 13, 14] where a matrix Lax representation for the minimal surface equation (Eq. (2) with k = −1) was obtained. Let g be a 2 × 2 matrix function with components k1 + a2 ab k2 + b2 , g12 = g21 = and g22 = , ω ω ω where k1 and k2 are arbitrary constants, not vanishing simultaneously and g11 =

εω 2 = k1 k2 + k1 b2 + k2 a2 ,

where

ε = ±1 .

Thus, det g = ε. Note that ε is not fully independent of k1 and k2 . ω 2 > 0 when we are dealing with real fields. In the case of complex fields ε is independent of k1 and k2 . The sigma model equation can be written as ∂α (g αβ g −1 ∂β g) = 0 , αβ

where g are the components of g tation of (29) is εαβ ∂β ψ =

λ2

−1

(29)

. As shown in [5] and [13] the Lax represen-

1 [λg αβ − εεαβ ](g −1 ∂β g)ψ , +ε

(30)

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where εαβ is Levi–Civita tensor with ε12 = 1, λ is the spectral parameter, det g = ε and ε = ±1. Now, let us see how a Lax representation for (2) can be obtained from (30). First, let M3 be a three-dimensional manifold with metric ds2 |M3 = k1 dt2 + k2 dx2 + dz 2

(k1 6= 0, k2 6= 0)

and z = φ(t, x) define a graph of a regular surface S in M3 . The induced metric on S is given by ds2 |S = (k1 + φ2t )dt2 + (k2 + φ2x )dx2 + 2φx φt dxdt . If a = φt and b = φx , then g is a metric tensor on S. Surface S is called minimal if its mean curvature H vanishes. Minimality condition leads to the equation g αβ ∂α ∂β φ = 0 , or (k1 + φ2t )φxx − 2φx φt φxt + (k2 + φ2x )φtt = 0 .

(31)

There is a parametrization of the minimal surfaces where the minimality condition reduces to the Laplace equation in two dimensions. Let X : S → M3 define a parametrization of S in M3 . This parametrization is called isothermal [19, 20], if hXu Xu i = εhXv Xv i ,

(32)

hXu Xv i = 0 (ε = ±1) .

(33)

Proposition 4.1. S is a minimal surface if and only if Xuu + εXvv = 0, where X is an isothermal parametrization. A connection between the above two different parametrizations may be obtained from the following two propositions: Proposition 4.2. Let z = φ(t, x) define a regular surface S. Parametrization X : S → M3 is isothermal if and only if the following equations are satisfied (k1 + φ2t )tu = −ωxv − φt φx xu , (k2 + φ2t )tv = −ωxu − φt φx xv . Proof. Equation (33) can be written as xu (k2 xv + φ2x xv + φt φx tv ) + tu (k1 tv + φt φx xv + φ2t tv ) = 0 and it is equivalent to the system ( tu = λ−1 [(k2 + φ2x )xv + φt φx tv ] , xu = λ−1 [(k1 + φ2t )tv + φt φx xv ] .

(34)

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537

Inserting expressions for tu and xu into (32), it can be found that ελ2 = (k1 k2 + k1 φ2x + k2 φ2t ) . Hence, λ = ω and

(

tu = ω −1 [(k2 + φ2x )xv + φt φx tv ] , xu = ω −1 [(k1 + φ2t )tv + φt φx xv ] .

That is equivalent to (34). Propositions 4.1 and 4.2 imply next proposition: Proposition 4.3. Let x and t be harmonic functions of u and v. Let a differentiable function φ(t, x) be defined by (34). Then the function φ(t, x) is a harmonic function of u and v if and only if it satisfies the minimality condition (31). Let us consider Eq. (31), where k1 and k2 are arbitrary constants. We have four distinct cases: (i) k1 k2 > 0. (a) k1 > 0, k2 > 0. This is equivalent to the equation of minimal surface in R3 or elliptic Monge–Ampère equation (k1 = k2 = −k = 1). (b) k1 > 0, k2 < 0. This is equivalent to the equation of minimal surface in M3 (three-dimensional Minkowski space with metric (1, 1, −1)). (ii) k1 k2 < 0. This is equivalent to the Born–Infeld equation (which is the equation of a minimal surface in a three-dimensional Minkowski space with metric (−1, 1, 1)) or hyperbolic Monge–Ampère equation (−k1 = k2 = k = 1). We have the following cases which do not arise from the embedding problem in M3 : (iii) k1 k2 = 0, but not simultaneously vanishing. This is a new type of equation. (iv) k1 = k2 = 0. This is Bateman equation or homogeneous Monge–Ampère equation (k1 = k2 = k = 0). The above propositions relate different parametrizations of the minimal surfaces. Proof of Proposition 4.1 can be found in reference [19] and proofs of Propositions 4.2 and 4.3, although lengthy, are straightforward. The next proposition is very important since it provides the Lax pair for systems that include Eq. (2): Proposition 4.4. Let φ be a differential function of t, x and let a = φt , b = φx . Then Eq. (31) solves the sigma model equation (29), if k1 , k2 not vanish simultaneously. If k1 = k2 = 0 Eq. (31) solves the sigma model equation (29) for another matrix g, namely, φt φx , g12 = g21 = b1 and g22 = a2 , g11 = a1 φx φt

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where a1 , a2 , b1 are constants. The Lax pair of (31) is then given by (30). Proof. The second half of the proposition is relatively easier to prove, hence we shall give a detailed proof of the first part. Let ρ = w2 , then the metric gαβ and its inverse g αβ can be written in a nice form 1 gαβ = √ (ηαβ + φ,α φ,β ) ρ det η α β √ αβ αβ φ φ g = ρ η − ρ

(35)

where ηαβ are the components of the diagonal matrix η with η11 = k1 , η22 = k2 and η αβ are the components of the inverse matrix η −1 . Indices are raised and lowered by η αβ and ηαβ respectively. For instance φα = η αβ φ,β . Here the Einstein summation convention is assumed. In this index notation ρ = det(η)(1 + η αβ φ,α φ,β ). The above form of the metric and its inverse are valid also when η is not a zero matrix but det(η) = 0. This means that either k1 = 0, k2 6= 0 or k1 6= 0, k2 = 0. In each case the components of the metric and its inverse are all finite. The analysis given below will not depend on whether det η is zero or not. To this end we let det η = 1 for simplicity. The minimality condition g αβ φ,αβ = 0 reduces to φα α =

φα ρα . 2ρ

(36)

Minimality condition (36) implies also ∂µ g µν = 0 . Hence the sigma model equation (29) to be proved takes the form hµν ∂ν [g αγ ∂µ gγβ ] = 0 where hαβ =

(37)

√ ρgαβ . It is straightforward to show that

αγ ∂µ gβγ = − (g −1 ∂µ g)α β =g

1 ρ,µ α 1 ρ,µ α 1 α δβ − φ φ,β + φ,β φα µ + φ φµβ . 2 ρ 2 ρ ρ

(38)

To proceed we shall present some identities. The following identity is valid only in two dimensions φ,αµ φ,βγ − φαβ φµγ = λ0 (ηαµ ηβγ − ηαβ ηγµ ) 2 where λ0 = − 21 [φαβ φαβ − (φα α ) ]. Using this identity and the minimality condition (36) we have the following

ρµ φ,βγ − ρ,β φ,µγ = 2λ0 (φ,µ ηβγ − φ,β ηγµ ) , ρµ φµβ = φα α ρ,β − 2λ0 φ,β , hαβ ∂α

1 2λ0 ∂β ρ + 2 (1 + ρ) = 0 . ρ ρ

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Utilizing these identities we get hαβ φµαβ = − hαβ φµα φνβ = − hαβ ∂α (φνβ φµ ) = −

2λ0 µ φ , ρ λ0 ν λ0 ν δ − φ φµ , ρ µ ρ λ0 ν 3λ0 ν δ − φ φµ , ρ µ ρ

ρ,α hαβ ∂β (φν φµ ) = −4λ0 φν φµ . Now applying ∂ν to (38) then multiplying by hµν and using the above identities (by virtue of the minimality condition (36)) it is easy to show (37) and hence ∂ν (g µν g −1 ∂µ g) = 0 . In the next section we will use the last proposition to obtain the Lax representations for the Monge–Ampère equations (1). In doing so we will return to our original parameter k instead of working with the parameters k1 and k2 . It is just a matter of scale transformation either in formula for ds2 |M3 or in Eq. (31) (redefining x and t) to give k1 = ±1 and k2 = ±1. Also, we will set ε = 1 in the next section. 5. Matrix Lax Representation: Nonlocal Conserved Charges Now we can write the Lax pairs for (1). First, let us give the Lax pairs for (2) more explicitly. Equation (30) can be rewritten in the form 1 ∂ψ = 2 [λ(g 11 A + g 12 B) − B]ψ , ∂x λ +1

(39)

1 ∂ψ =− 2 [λ(g 21 A + g 22 B) + A]ψ , ∂t λ +1 where A = g −1 ∂t g ,

B = g −1 ∂x g .

(40)

From (40) it follows the identity ∂A ∂B − − [A, B] = 0 . ∂x ∂t The integrability of (39) yields the equations

(41)

det g = 1 ,

(42)

(g 11 A + g 12 B)t + (g 21 A + g 22 B)x = 0 .

(43)

From the Proposition 4.4 we have for k 6= 0 1

g= p −k(1 + φ2x ) + φ2t

−k + φ2t

φt φx

φt φx

1 + φ2x

! ,

(44)

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√ and for k = 0 (setting a1 = a2 = 2 and b1 = 1)  √ φ t 2 1   φx g= √ φx  . 2 1 φt

(45)

With this choice (41) and (42) are trivial identities and (43) is identical to Eq. (2), i.e. to the minimal surface equation for k = −1, Born–Infeld equation for k = 1 and Bateman equation for k = 0. The Bianchi transformation (16) for k 6= 0 assumes the form √ −k + φ2t −kUtt = p , −k(1 + φ2x ) + φ2t √ φx φt −kUtx = p , −k(1 + φ2x ) + φ2t

(46)

√ 1 + φ2x −kUxx = p −k(1 + φ2x ) + φ2t and (44) in terms of U can be written as √ g = −k

Utt

Utx

Utx

Uxx

! .

(47)

In this way (39) with (47) give us the matrix Lax representation for the hyperbolic Monge–Ampère equation (k = 1) and elliptic Monge–Ampère equation (k = −1). Let us observe that (1) for k 6= 0 follows from (42) while Eqs. (41) and (43) are trivial identities. We can also express (47) in terms of variables a and b defined in (4) by ! √ bt bx (48) g = −k b x ax and (5) follows easily since at = bx is a trivial identity and det g = −k(bt ax −b2x ) = 1. The Bianchi transformation (16) for k = 0 yields Utt φt = , φx Utx

φx Uxx = φt Utx

(49)

and (45) in terms of U can be written as  √ Utt 2 1   Utx g= (50) √ Uxx  2 1 Utx and det g = 1 reproduces (1) for k = 0. In terms of the variables a and b we have  √ bt 1   2 bx (51) g= √ ax  2 1 bx

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which give us (5) for k = 0. So, we have the following proposition: Proposition 5.1. The Lax pair (39) with (48) or (51) yields the Monge–Ampère equations as expressed in (5) for k 6= 0 and k = 0, respectively. This proposition is the second main result of our paper. This is a matrix dispersive Lax representation. In Sec. 3, using the dispersionless Lax representation for the Monge–Ampère equations (1), we were able to derive two sets of infinite number of local conserved charges. Now, using (39) it will possible to find infinitely nonlocal conserved ones. Let us denote M = −(g 11 A + g 12 B) and N = g 21 A + g 22 B, then the Lax pair (39) can be written as (λ2 + 1)ψx = −λM ψ − g −1 gx ψ , (λ2 + 1)ψt = −λN ψ − g −1 gt ψ , or (gψ)x = −λgM ψ − λ2 gψx , (gψ)t = −λgN ψ − λ2 gψt .

(52)

Let us assume that the function ψ is analytical in the parameter λ and can be expanded as ψ = ψ0 + λψ1 + λ2 ψ2 + · · · .

(53)

Then, (52) imply ψ0 = g −1 , (gψ1 )x = −gM g −1 , (gψ1 )t = −gN g −1 , (gψ2 )x = gx g −1 + gM g −1∂x−1 (gM g −1 ) ,

(54)

(gψ2 )t = gt g −1 + gN g −1 ∂t−1 (gN g −1 ) , .. . and we have now infinitely many conserved laws in the form (Xn )x = (Tn )t where the densities are X1 = N , T1 = M , X2 = g −1 gt + (∂t−1 N )N , T2 = g .. .

−1

gx +

(∂x−1 M )M

(55) ,

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Now we can use (49) and (51) to express the densities Tn in terms of variables a and b. 6. Conclusion In this paper we have obtained the Lax representation of the Monge–Ampère equations (1). In Sec. 2 the Bianchi transformation relating Eqs. (1) and (2) was given (Proposition 2.1). This transformation allowed us to translate results obtained for one equation to the other. In Sec. 3 the dispersionless Lax pair for (1) as well the local conserved densities were given (Proposition 3.1). In Sec. 4 the correspondence between sigma models and a two parameter equation for minimal surfaces was given and the matrix Lax pair for Eq. (2) was obtained (Proposition 4.4). A Lax representation for the system (1) as well the nonlocal conserved densities were given in Sec. 5 (Proposition 5.1). The algebra of the local and nonlocal charges that follows from (27), (28) and (55) as well the multi-Hamiltonian formulation of the Monge–Ampère equations (1) will be the subject of a future publication. Some results on this line for the second order homogeneous Monge–Ampère equation were already obtained in [21, 22]. As we have pointed, the homogeneous Monge–Amp` ere equation has an infinite number of inequivalent Lagrangians and somehow this should be reflected in its Lax representation. This also deserves further clarifications. Acknowledgments This work was partially supported by the Scientific and Technical Research Council ¨ ITAK), ˙ ¨ of Turkey (TUB Turkish Academy of Sciences (TUBA) and CNPq, Brazil. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

O. I. Mokhov and Y. Nutku, Lett. Math. Phys. 32 (1994) 121. M. Born and L. Infeld, Proc. R. Soc. London A144 (1934) 425. K. J¨ orgens, Math. Annal. 127 (1954) 130. E. Heinz, Nach. Akad. Wissensch. in G¨ ottingen Mathem.-Phys. Klasse IIa (1952) 51. M. G¨ urses, Lett. Math. Phys. 44 (1998) 1. H. Bateman, Proc. R. Soc. A125 (1929) 598. B. M. Barbishov and N. A. Chernikov, Sov. Phys. JETP 24 (1966) 93. J. C. Brunelli and A. Das, Phys. Lett. B426 (1998) 57. M. Arik, F. Neyzi, Y. Nutku, P. J. Olver and J. Verosky, J. Math. Phys. 30 (1988) 1338. D. B. Fairlie, J. Govaerts and A. Morozov, Nucl. Phys. B373 (1992) 214. J. C. Brunelli, Rev. Math. Phys. 8 (1996) 1041. J. C. Brunelli and A. Das, Phys. Lett. A235 (1997) 597. M. G¨ urses and A. Karasu, Int. J. Mod. Phys. A6 (1991). M. G¨ urses, Lett. Math. Phys. 26 (1992). B. Dubrovin and S. Novikov, Russ. Math. Surv. 44 (1989) 35. Y. Nutku, J. Math. Phys. 26 (1985) 1237.

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[17] K. P. Stanyukovich, Unsteady Motion of Continuous Media, Pergamon, New York, 1960, p. 137. [18] P. J. Olver and Y. Nutku, J. Math. Phys. 29 (1988) 1610. [19] M. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, New Jersey, 1976. [20] U. Dierken, S. Hildebrandt, A. K¨ unster and O. Wohlrab, “Minimal Surfaces I”, Grundlehren der Mathematishen Wissenschaften, No. 295, Springer-Verlag, BerlinHeidelberg, 1992. ¨ Sarioˇ [21] Y. Nutku and O. glu, Phys. Lett. A173 (1993) 270. [22] Y. Nutku, J. Phys. A29 (1996) 3257.

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GENERALIZED r-MATRIX STRUCTURE AND ALGEBRO-GEOMETRIC SOLUTION FOR INTEGRABLE SYSTEM

ZHIJUN QIAO Institute of Mathematics, Fudan University, Shanghai 200433, P. R. China Fachbereich 17, Mathematik Informatik, Universit¨ at-GH Kassel, Heinrich-Plett-Str. 40, D-34109 Kassel, Germany and Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA E-mail : [email protected], [email protected]

Received 12 February 2000 Revised 9 May 2000 The purpose of this paper is to construct a generalized r-matrix structure of finite dimensional systems and an approach to obtain the algebro-geometric solutions of integrable nonlinear evolution equations (NLEEs). Our starting point is a generalized Lax matrix instead of the usual Lax pair. The generalized r-matrix structure and Hamiltonian functions are presented on the basis of fundamental Poisson bracket. It can be clearly seen that various nonlinear constrained (c-) and restricted (r-) systems, such as the c-AKNS, c-MKdV, c-Toda, r-Toda, c-Levi, etc, are derived from the reductions of this structure. All these nonlinear systems have r-matrices, and are completely integrable in Liouville’s sense. Furthermore, our generalized structure is developed to become an approach to obtain the algebro-geometric solutions of integrable NLEEs. Finally, the two typical examples are considered to illustrate this approach: the infinite or periodic Toda lattice equation and the AKNS equation with the condition of decay at infinity or periodic boundary. Keywords: Lax matrix, r-matrix structure, integrable system, algebro-geometric solution. Mathematics Subject Classification 2000: 35Q53, 58F07, 35Q35

1. Introduction Completely integrable systems are widespreadly applied in fields theory, fluid mechanics, nonlinear optics and other fields of nonlinear sciences. The new development of integrability theory can be roughly divided into three stages. The first one was the direct use of Lax equations for some systems such as the Calogero–Moser system [33] and the Euler rigid equation [32], which were allowed for integration. The second one was the so-called “algebraization”, i.e. the tools 545

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of Lie algebras, Kac–Moody algebras were used to sysmetically construct a large class of soliton equations and integrable systems [3, 43], and simultaneously present the Lax representations of soliton equations and Hamiltonian structures. The third one is being developed and witnessed through the use of nonlinearization method [8] to generate finite dimensional integrable systems. These systems can be the Bargmann system [12, 35], the C. Neumann system [12, 35], the higher-order constrained flows or symmetric constrained flows [4, 5], and the stationary flows of soliton equations [51]. Indeed, with the help of this method, many new completely integrable systems were successively found [12, 35]. In this way, each integrable system is generated through making nonlinearized procedure for a concrete spectral problem or Lax pair, and has its own characteristic property. Then a natural question arises whether or not there is a unified structure such that it can contain those concrete integrable systems? Recently, the study of r-matrix for nonlinear integrable systems brings a great hope to dealing with this problem. Semenov–Tian–Shansky ever gave the definition of r-matrix [45], and used the r-matrix to construct Lie algebra and new Poisson bracket [44] in a given Lie algebra and corresponding coadjoint orbit. The main idea of Semenov–Tian–Shansky and Reyman was how to obtain the new Poisson bracket from a given r-matrix and an element of Lie algebra. Here our thought is how to present r-matrix structure from a given Lax matrix and the standard Poisson bracket: ?

L, {· , ·} =⇒ r-matrix . In the present paper, we give a sure answer for the above question. We propose an approach to generate finite dimensional integrable systems by beginning with the so-called generalized Lax matrix instead of the usual Lax pair. Another main result of this paper is to deal with the algebro-geometric solutions of integrable nonlinear evolution equations (NLEEs). It is well-known that the ideal aim for nonlinear equations is to obtain their explicit solutions. According to the nonlinearization method, solutions of integrable NLEEs can have the parametric representations [10] or involutive representations [11], and also have numeric representations in the discrete case [42]. However, these representations of solutions are not given in an explicit form. Thus, an open question is how to obtain their explicit forms. In the paper we would like to give solutions of integrable NLEEs in the form of algebro-geometric Θ-functions. The algebro-geometric solutions for some soliton equations with the periodic boundary value problems were known since the works of Lax [28], Dubrovin, Mateev and Novikov [20]. Similar results for the periodic Toda case were obtained slightly later by Date and Tanaka [14]. Afterwards, the relations between commutative rings and ordinary linear periodic differential operators and between algebraic curves and nonlinear periodic difference equations were discussed by Krichever [27]. The technique they used is the Bloch eigenfunctions, the spectral theory of linear periodic operators, and some analysis tools on Riemann surfaces.

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547

In the first example of this paper, we give the algebro-geometric solution of the periodic or infinite Toda lattice equation. Our method is a constrained approach, connecting finite dimensional integrable systems with integrable NLEEs, instead of the usual spectral techniques and Bloch eigenfunctions which are often available to the periodic boundary problems. The results with the periodic boundary conditions are included in ours. In the second example, we consider the well-known AKNS equation. The Ablowitz–Kaup–Newell–Segur (AKNS) equations are a very important hierarchy [1] of NLEEs in soliton theory. It can turn out that the KdV, MKdV, NLS, sine-Gordon, sinh-Gordon equations etc. All these equations can be solvable by the inverse scattering transform (IST) [24], and usually have N -soliton solutions [2]. But the algebro-geometric solutions of the AKNS equations seem not to be obtained. We shall deal with this problem by using our constrained procedure. The considered AKNS equation is under the case of decay at infinity or periodic boundary condition. The whole paper is organized as follows. We first introduce a generalized Lax matrix in the next section, then construct a generalized r-matrix structure and a generalized set of involutive Hamiltonian functions in Sec. 3. All those Hamiltonian systems have Lax matrices, r-matrices, and are therefore completely integrable in Liouville’s sense. In Sec. 4 it can be clearly seen that various nonlinear constrained (c-) and restricted (r-) integrable flows, such as the c-AKNS, c-MKdV, c-Toda, r-Toda, c-WKI, c-Levi, etc, can be derived from the reductions of this structure. Moreover, the following interesting facts are given in Secs. 5, 6, 7, respectively: — Several pairs of different integrable systems share the same r-matrices with the good property of being non-dynamical (i.e. constant). In particular, a discrete and a continuous dynamical system possess the common Lax matrix, r-matrix, and even completely same involutive set. Additionally, on a symplectic submanifold integrability of the restricted Hamiltonian flow (for continuous case) and symplectic map (for discrete case) are described by introducing the DiracPoisson bracket. They also have the same r-matrix but being dynamical. — A pair of constrained integrable systems, produced by two gauge equivalent spectral problems, possesses different r-matrices being non-dynamical. — New integrable systems are generated through choosing new r-matrices from our structure, and the associated spectral problems are also new. In the last section, as a development of the generalized structure, through considering the relation lifting finite dimensional system to infinite dimensional system and using the algebro-geometric tools we present an approach for obtaining the algebro-geometric solution of integrable NLEEs. To illustrate the procedure we take the periodic or infinite Toda lattice equation and the AKNS equation with the condition of decay at infinity or periodic boundary as the examples. Before displaying our main results, let us first give some necessary notation: dp ∧ dq stands for the standard symplectic structure in Euclidean space R2N = {(p, q)|p = (p1 , . . . , pN ), q = (q1 , . . . , qN )}; h· , ·i is the standard inner product in

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RN ; in the symplectic space (R2N , dp ∧ dq) the Poisson bracket of two Hamiltonian functions F, G is defined by [6] N X ∂F ∂G ∂F ∂G ∂F ∂G ∂F ∂G , , − − ; (1.1) = {F, G} = ∂qi ∂pi ∂pi ∂qi ∂q ∂p ∂p ∂q i=1 I and ⊗ stand for the 2 × 2 unit matrix and the tensor product of matrix, respectively; λ1 , . . . , λN are N arbitrarily given distinct constants; λ, µ are the two different spectral parameters; Λ = diag(λ1 , . . . , λN ), I0 = hq, qi, J0 = hp, qi, K0 = hp, pi, I1 = hΛp, pihΛq, qi, J1 = hΛp, qi, a0 , a1 = const.. Denote all infinitely times differentiable functions on real field R by C ∞ (R). 2. A Generalized Lax Matrix Consider the following matrix (called Lax matrix) L(λ) =

A(λ)

B(λ)

C(λ)

−A(λ)

! (2.1)

where A(λ) = a−2 (I1 , J1 )λ−2 + a−1 (J0 )λ−1 + a0 + a1 λ +

B(λ) = b−1 (I0 , J0 )λ−1 + b0 (J0 ) −

N X j=1

C(λ) = c−1 (J0 , K0 )λ−1 + c0 (J0 ) +

qj2 , λ − λj

N X j=1

p2j , λ − λj

N X pj qj , λ − λj j=1

(2.2)

(2.3)

(2.4)

with some undetermined functions a−2 , a−1 , b−1 , c−1 , b0 , c0 ∈ C ∞ (R). Now, in order to produce finite dimensional integrable systems directly from the Lax matrix (2.1), we need an inevitable assumption. Assumption (A): {A(λ), A(µ)}, {A(λ), B(µ)}, {A(λ), C(µ)}, {B(λ), B(µ)}, {B(λ), C(µ)}, and {C(λ), C(µ)} are expressed as some linear combinations of A(λ), A(µ), B(λ), B(µ), C(λ), C(µ) with the cofficients in C ∞ (R). Then we have the following lemma. Lemma 2.1. Under Assumption (A), L(λ) only contains the following cases: (1) If a−2 6= const., a0 = b0 = c0 = a1 = 0, a−1 = −J0 , b−1 = I0 , and c−1 = −K0 , then a−2 satisfies the relation I1 = (J1 + a−2 )2 + f (a−2 ); if a−2 = const. 6= 0 and a0 = b0 = c0 = a1 = 0, then a−1 = −J0 , b−1 = I0 , c−1 = −K0 , or a−1 = const., b−1 = I0 + f1 (J0 ), c−1 = −K0 + g1 (J0 ), where f1 , g1 satisfy the relation f1 g1 = −J02 − 2a−1 J0 + const..

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(2) (3) (4) (5)

a−2 = a−1 = b−1 = c−1 = b0 = c0 = a1 = 0, and a0 = const. a−2 = b−1 = a0 = b0 = c0 = a1 = 0, c−1 = −K0 , and a−1 satisfies a0−1 6= 0. a−2 = a−1 = b−1 = c−1 = b0 = a1 = 0, a0 = const., and c0 6= 0. a−2 = c−1 = b0 = a1 = 0, a−1 , a0 = const., b−1 = I0 + g(J0 ), and c0 satisfies (c0 g)0 = −2a0 . (6) a−2 = c−1 = a0 = b0 = c0 = a1 = 0, a−1 = J0 + const., and b−1 = I0 . (7) If a−2 = a0 = b0 = c0 = a1 = 0, then there are the following five subcases: (i) a−1 = const., c−1 = −K0 + f2 (J0 ), and b−1 = I0 + g2 (J0 ); (ii) a−1 = −J0 , b−1 = I0 , and c−1 = K0 ; (iii) a−1 = −J0 + const., and b−1 = b−1 (J0 ), c−1 = c−1 (J0 ) satisfy (b−1 c−1 )0 = 2a−1 ; (iv) a−1 = −J0 + const., b−1 = I0 , and c−1 = c−1 (J0 ); (v) a−1 = −J0 + const., c−1 = −K0 , and b−1 = b−1 (J0 ). (8) a−2 = a−1 = b−1 = c−1 = 0, a0 , a1 = const., b0 6= 0, c0 6= 0, and b0 , c0 satisfy the relation (b0 c0 )0 = −2a1 . (9) a−2 = a−1 = b−1 = c−1 = 0, c0 , a1 , a0 = const., and b0 6= 0. (10) a−2 = b−1 = c0 = a1 = 0, a−1 , a0 = const., c−1 = −K0 + h(J0 ), and b0 satisfies the relation (b0 h)0 = −2a0 . The above all functions f, g, h, fi , gi (i = 1, 2) are in C ∞ (R), and “0 ” means d dJ0 . Proof. Through some calculations we have λ µ ∂a−2 hΛp, pi B(λ) − B(µ) {A(λ), A(µ)} = 2 ∂I1 µ2 λ2 λ µ ∂a−2 hΛq, qi C(λ) − 2 C(µ) +2 ∂I1 µ2 λ 1 ∂a−2 1 − 2 (hΛp, pi(b−1 − I0 ) − hΛq, qi(c−1 + K0 )) +2 2 ∂I1 λ µ λ ∂a−2 µ − (hΛp, pib0 + hΛq, qic0 ) , +2 ∂I1 λ2 µ2 1 db0 ∂b−1 1 B(λ) − B(µ) + 2 (B(λ) − B(µ)) ∂J0 µ λ dJ0 1 ∂b−1 db0 ∂b−1 db0 1 − I0 + b0 − b−1 , +2 λ µ ∂I0 dJ0 ∂J0 dJ0 1 dc0 1 ∂c−1 (−C(λ) + C(µ)) − C(λ) + C(µ) + 2 {C(λ), C(µ)} = 2 ∂J0 µ λ dJ0 1 ∂c−1 dc0 ∂c−1 dc0 1 K 0 + c0 − c−1 , +2 − + λ µ ∂K0 dJ0 ∂J0 dJ0

{B(λ), B(µ)} = 2

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{A(λ), B(µ)} =

2 da−1 2 (−B(µ) + B(λ)) − B(µ) λ−µ λ dJ0 2µ ∂a−2 ∂a−2 2 ∂b−1 B(λ) − 2 B(µ) + 2 hΛq, qiA(µ) + µ ∂I0 λ ∂J1 ∂I1 ∂b−1 ∂a−2 ∂a−2 ∂b−1 ∂a−2 J1 + + 2a−2 λ−2 µ−1 − 2hΛq, qi 2 ∂I0 ∂I1 ∂I0 ∂J1 ∂I1 ∂b−1 da−1 ∂b−1 da−1 +2 − I0 − b−1 + b−1 + b−1 λ−1 µ−1 ∂I0 dJ0 ∂I0 dJ0 ∂a−2 ∂a−2 − 2 2hΛq, qi (J0 + a−1 ) + (I0 − b−1 ) λ−2 ∂I1 ∂J1 ∂a−2 ∂a−2 − 2 −b0 + 2a0 hΛq, qi λ−2 µ ∂J1 ∂I1 + 2b0

{A(λ), C(µ)} =

2 da−1 2 (C(µ) − C(λ)) + C(µ) λ−µ λ dJ0 2µ ∂a−2 ∂a−2 2 ∂c−1 C(λ) + 2 C(µ) + 2 hΛp, piA(µ) + µ ∂K0 λ ∂J1 ∂I1 ∂c−1 ∂a−2 ∂a−2 ∂c−1 ∂a−2 J1 + − 2a−2 λ−2 µ−1 + 2hΛp, pi 2 ∂K0 ∂I1 ∂K0 ∂J1 ∂I1 ∂c−1 da−1 ∂c−1 da−1 +2 K0 − c−1 − c−1 − c−1 λ−1 µ−1 ∂K0 dJ0 ∂K0 dJ0 ∂a−2 ∂a−2 −2 −2hΛp, pi (J0 + a−1 ) + (K0 + c−1 ) λ−2 ∂I1 ∂J1 ∂a−2 ∂a−2 − 2 c0 + 2a0 hΛp, pi λ−2 µ ∂J1 ∂I1 − 2c0

{B(λ), C(µ)} =

∂a−2 −2 2 da−1 −1 ∂b−1 −1 λ − 2b0 µ − 4a1 hΛq, qi λ µ , dJ0 ∂I0 ∂I1

∂a−2 −2 2 da−1 −1 ∂c−1 −1 λ − 2c0 µ − 4a1 hΛp, pi λ µ , dJ0 ∂K0 ∂I1

4 ∂b−1 4 (−A(µ) + A(λ)) + A(µ) λ−µ λ ∂I0 1 ∂c−1 1 ∂b−1 dc0 db0 4 ∂c−1 A(λ)+2 + + B(λ)+2 C(µ) − µ ∂K0 µ ∂J0 dJ0 λ ∂J0 dJ0 ∂c−1 ∂b−1 −2 −1 + 1 a−2 µ λ + 4 + 1 a−2 λ−2 µ−1 +4 − ∂I0 ∂K0

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+2

551

∂c−1 ∂b−1 ∂b−1 ∂c−1 ∂b−1 ∂c−1 I0 + 2 J0 + K0 ∂J0 ∂I0 ∂I0 ∂K0 ∂J0 ∂K0

∂b−1 ∂c−1 ∂b−1 ∂c−1 − c−1 − b−1 − 2 a−1 + 2 a−1 + 2a−1 λ−1 µ−1 ∂J0 ∂J0 ∂I0 ∂K0 ∂b−1 ∂b−1 dc0 dc0 ∂b−1 +2 − c0 + I0 − b−1 − 2 a0 λ−1 ∂J0 ∂I0 dJ0 dJ0 ∂I0 ∂c−1 ∂c−1 db0 db0 ∂c−1 +2 − b0 + K0 − c−1 + 2 a0 µ−1 ∂J0 ∂K0 dJ0 dJ0 ∂K0 db0 dc0 ∂b−1 ∂c−1 −2 c0 + b0 + 2a1 + 4 a1 µ−1 λ − 4 a1 λ−1 µ . dJ0 dJ0 ∂K0 ∂I0 According to Assumption (A), the terms that do not contain A(λ), A(µ), B(λ), B(µ), C(λ), C(µ) in the above six equalities, are zero. After discussing these terms, we can obtain every result in Lemma 2.1. 3. Generalized r-Matrix Structure and Integrable Hamiltonian Systems Let L1 (λ) = L(λ) ⊗ I, L2 (µ) = I ⊗ L(µ). In the following, we search for a general 4 × 4 r-matrix structure r12 (λ, µ) such that the fundamental Poisson bracket [21]: (3.1) {L(λ) ⊗, L(µ)} = [r12 (λ, µ), L1 (λ)] − [r21 (µ, λ), L2 (µ)] P 3 holds, where r21 (λ, µ) = P r12 (λ, µ)P , P = 12 i=0 σi ⊗ σi , and σi0 s are the standard Pauli matrices. For the given Lax matrix (2.1) and the Poisson bracket (1.1), we have the following theorem. Theorem 3.1. Under Assumption (A), r12 (λ, µ) =

2 P +S µ−λ

(3.2)

is an r-matrix structure satisfying (3.1), where   2λ ∂a 2 da−1 ∂a−2 2 ∂b−1 2λ −2 + hΛq, qi 0   µ2 ∂J1 µ dJ0 µ ∂J0 µ2 ∂I1    2 ∂c−1 2λ ∂a−2  dc0   0 − 2 hΛq, qi 2  dJ0 µ ∂K0 µ ∂I1  .  S=  2 ∂b−1 db0   − 2λ hΛp, pi ∂a−2 − 0 −2   µ2 ∂I1 µ ∂I0 dJ0    2λ ∂a−2 2 ∂c−1 2 da−1  2λ ∂a−2 0 hΛp, pi − + µ2 ∂I1 µ ∂J0 µ2 ∂J1 µ dJ0 Proof. Under Assumption (A), we have

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λ µ B(λ) − 2 B(µ) µ2 λ λ µ ∂a−2 hΛq, qi C(λ) − 2 C(µ) , +2 ∂I1 µ2 λ 1 db0 ∂b−1 1 B(λ) − B(µ) + 2 (B(λ) − B(µ)) , {B(λ), B(µ)} = 2 ∂J0 µ λ dJ0 ∂a−2 {A(λ), A(µ)} = 2 hΛp, pi ∂I1

{C(λ), C(µ)} = 2

∂c−1 ∂J0

1 dc0 1 (−C(λ) + C(µ)) , − C(λ) + C(µ) + 2 µ λ dJ0

{A(λ), B(µ)} =

2 da−1 2 ∂b−1 2 (−B(µ) + B(λ)) − B(µ) + B(λ) λ−µ λ dJ0 µ ∂I0 ∂a−2 2µ ∂a−2 B(µ) − 2 hΛq, qiA(µ) , − 2 λ ∂J1 ∂I1

{A(λ), C(µ)} =

2 da−1 2 ∂c−1 2 (C(µ) − C(λ)) + C(µ) + C(λ) λ−µ λ dJ0 µ ∂K0 ∂a−2 2µ ∂a−2 C(µ) + 2 hΛp, piA(µ) , + 2 λ ∂J1 ∂I1

{B(λ), C(µ)} =

4 ∂b−1 4 ∂c−1 4 (−A(µ) + A(λ)) + A(µ) − A(λ) λ−µ λ ∂I0 µ ∂K0 1 ∂b−1 dc0 db0 1 ∂c−1 + + B(λ) + 2 C(µ) , +2 µ ∂J0 dJ0 λ ∂J0 dJ0

which complete the proof. In general, Eq. (3.2) is a dynamical r-matrix structure, i.e. dependent on canonical variables pi , qi [7]. Now, we turn to consider the determinant of L(λ) −det L(λ) = =

1 Tr L2 (λ) = A2 (λ) + B(λ)C(λ) 2 2 X i=−4

Hi λi +

N X j=1

Ej , λ − λj

(3.3)

where H−4 = a2−2 ,

(3.4)

H−3 = 2a−2 a−1 ,

(3.5)

H−2 = a2−1 + 2a−2 a0 + b−1 c−1 − 2a−2 hΛ−1 p, qi ,

(3.6)

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H−1 = 2a−2 a1 + 2a−1 a0 + b−1 c0 + b0 c−1 − 2a−2 hΛ−2 p, qi − 2a−1 hΛ−1 p, qi − b−1 hΛ−1 p, pi + c−1 hΛ−1 q, qi ,

(3.7)

H0 = a20 + 2a−1 a1 + b0 c0 + 2a1 hp, qi ,

(3.8)

H1 = 2a0 a1 ,

(3.9)

H2 = a21 ,

(3.10)

−1 Ej = (2a−2 λ−2 j + 2a−1 λj + 2a0 + 2a1 λj )pj qj −1 2 2 + (b−1 λ−1 j + b0 )pj − (c−1 λj + c0 )qj − Γj ,

Γj =

N X

(pj qk − pk qj )2 , j = 1, 2, . . . , N . λj − λk

k=1,k6=j

(3.11) (3.12)

Let Eq. (3.3) be multiplied by a fixed multiplier λk (k ∈ Z), then it leads to 2 k−1 N X X X λkj Ej 1 k λ · Tr L2 (λ) = Hl λl+k + Fi λk−1−i + 2 λ − λj i=0 j=1 l=−4 −1 X

=

Hl−k λl +

l=k−4

+

k+2 X l=k

k−1 X

(Hl−k + Fk−1−l )λl

l=0 N X λkj Ej Hl−k λ + , λ − λj j=1 l

(3.13)

where Fm =

N X

λm j Ej , m = 0, 1, 2, . . . ,

(3.14)

j=1

which read Fm = 2a−2 hΛm−2 p, qi + 2a−1 hΛm−1 p, qi + 2a0 hΛm p, qi + 2a1 hΛm+1 p, qi + b−1 hλm−1 p, pi + b0 hΛm p, pi − c−1 hΛm−1 q, qi − c0 hΛm q, qi X (hΛi p, pihΛj q, qi − hΛi p, qihΛj p, qi) . −

(3.15)

i+j=m−1

Because there is an r-matrix structure satisfying Eq. (3.1), one can obtain r12 (λ, µ), L1 (λ)] − [¯ r21 (µ, λ), L2 (µ)] , {L2 (λ) ⊗, L2 (µ)} = [¯

(3.16)

where r¯ij (λ, µ) =

1 1 X X

k l L1−k (λ)L1−l 1 2 (µ) · rij (λ, µ) · L1 (λ)L2 (µ) ,

k=0 l=0

i = 12, j = 21 .

(3.17)

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Thus, 4{Tr L2 (λ), Tr L2 (µ)} = Tr{L2 (λ) ⊗, L2 (µ)} = 0 .

(3.18)

So, by Eq. (3.13) we immediately obtain Theorem 3.2. Under Assumption (A), the following equalities {Ei , Ej } = 0, {Hl , Ej } = 0, {Fm , Ej } = 0 ,

(3.19)

i, j = 1, 2, . . . , N, l = −4, . . . , 2, m = 0, 1, 2, . . . , hold . Hence, the Hamiltonian systems (Hl ) and (Fm ) ∂Hl , ∂p ∂Fm , = ∂p

(Hl ) : qx = (Fm ) : qtm

∂Hl , ∂q ∂Fm , =− ∂q

px = − ptm

l = −4, . . . , 2 ,

(3.20)

m = 0, 1, 2, . . . ,

(3.21)

are completely integrable in Liouville’s sense. Corollary 3.1. All composition functions f (Hl , Fm ), f ∈ C ∞ (R) are completely integrable Hamiltonians in Liouville’s sense. 4. Reductions For the various cases of Lemma 2.1, we give the corresponding reductions of r-matrix structure r12 (λ, µ) in this section. The following numbers of title coincide with the ones in Lemma 2.1, i.e. the corresponding conditions are coincidental. Before giving our reductions, we’d like to re-stress the two “terminologies” used usually in the theory of integrable systems in order to avoid some confusions: one is “constrained system”, which means the finite dimensional Hamiltonian system or symplectic map in R2N under the Bargmann-type constraint; the other “restricted system”, which means the finite dimensional Hamiltonian system or symplectic map on some symplectic submanifold in R2N under the Neumann-type constraint. In the future we shall follow this principle. (1) r12 (λ, µ) = 

1

 0 S= 0  0

0

0

0

0

0

0

0

0

0

∂a−2 λ 2λ ∂a−2 λ P +2 · 2S + 2 · Q, µ(µ − λ) ∂J1 µ ∂I1 µ2 

 0 , 0  1



0

  0 Q=  −hΛp, pi  0

0

hΛq, qi

0

0

0

0

hΛp, pi

0

(4.1)

0



 −hΛq, qi  .  0  0

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Particularly, with f (a−2 ) = −1, Eq. (4.1) exactly reads as the r-matrix of the constrained WKI (c-WKI) system. With a−2 = const. 6= 0, Eq. (3.2) reads as the 2λ P of ellipsoid geodesic flow [26], or reads r-matrix r12 (λ, µ) = µ(µ−λ)   0 0 0 f10   0 0 −1 0  2 2  , P + S, S = r12 (λ, µ) =  µ−λ µ 0 −1 0 0   0 0 −g10 0 which is a new r-matrix structure. For simplicity, below write “0 ” =

d dJ0 .

(2) r12 (λ, µ) =

2 P. µ−λ

(4.2)

This is nothing but the r-matrix of the well-known constrained AKNS (c-AKNS ) system [8]. (3)  2 2 P + S, r12 (λ, µ) = µ−λ µ

a0−1

  0 S=  0  0

0

0

0

−1

0

0

0

0



0

 0   , a0−1 6= 0 . 0  

(4.3)

a0−1

In particular, with a−1 = −J0 , Eq. (4.3) reads as the r-matrix of the constrained LZ (c-LZ ) system [12]. (4)  r12

2 P + c00 S , = µ−λ

0

 1 S= 0 

0

0

0

0

0

0

0



 0 . 0  0

(4.4)

0 0 −1 √ With c0 = −2 J0 , Eq. (4.4) reads as the r-matrix of the constrained Hu (c-H ) system [12]. (5)  0

r12 (λ, µ) =

2 P +S, µ−λ

  0  c0  S=  0  0

1 0 g µ 0 2 − µ 0

 0 0 0 c00

0

    . 1 0 − g  µ  0

0

(4.5)

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With b−1 = I0 , c0 = const., Eq. (4.5) reads as the r-matrix of the constrained Qiao (c-Q) system [38]. (6)  2 2 P + S, r12 (λ, µ) = µ−λ µ

1

 0 S= 0  0

0

0

0

0

1

0

0

0

0



 0 . 0  1

(4.6)

This is a new r-matrix . (7)(i)  2 2 P + S, r12 (λ, µ) = µ−λ µ

g20

0

0

 0  f2 S= 0 

0

−1

−1

0

0

0

0

0



 0 . 0  0

(4.7)

This is also a new r-matrix . (ii) r12 (λ, µ) =

2λ P. µ(µ − λ)

(4.8)

This is the r-matrix of the constrained Heisenberg spin chain (c-HSC ) system [39]. (iii)  2 2 P − S, r12 (λ, µ) = µ−λ µ

1

 0 S= 0  0

0

0

0

0

0

0

0

c0−1

0



 0  . b0−1   1

(4.9)

With b0−1 = −1, c0−1 = 1, Eq. (4.9) becomes the r-matrix of the constrained Levi (c-L) system [35]. (iv)  2 2 P − S, r12 (λ, µ) = µ−λ µ

1

 0 S= 0  0

0

0

0

0

1

0

0

c0−1

0



 0 . 0  1

(4.10)

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This is a new r-matrix . (v)  r12 (λ, µ) =

2 2 P − S, µ−λ µ

1

 0 S= 0  0

0

0

0

1

0

0

0

0



0

 0  . b0−1   1

(4.11)

This is also a new r-matrix . (8)  2 P +S, r12 (λ, µ) = µ−λ

0

 0  2c0 S=  0 

2b00

0

0

0

0

0

0



 0 . 0  0

(4.12)

0 0 0 √ With b0 = c0 = J0 , a1 = − 21 , Eq. (4.12) reads as the r-matrix of the constrained Tu (c-T ) system [12]. (9)  r12 (λ, µ) =

2 P +S, µ−λ

0

 0 S = b00  0  0

1

0

0

0

0

0

0

0

0



 0 . −1   0

(4.13)

With b0 = −J0 , Eq. (4.13) reads as the common r-matrix of the constrained Toda (c-Toda) system (a discrete system) and the constrained CKdV (c-CKdV ) system (a continuous system), which will be seen in Sec. 5.1. (10) 

r12 (λ, µ) =

2 P +S, µ−λ

0

 1 0  h µ S=  0    0

b00

0

0

−

0 0

2 µ 0

1 − h0 µ

0



  0   . −b00     0

(4.14)

With h = const., b0 = 0, Eq. (4.14) reads as the r-matrix of the constrained MKdV (c-MKdV ) system [37]. Proof. For simplicity, we only present the proof in Cases 1 and 10, other cases are similar.

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Case 1. With a−2 6= const., the matrix S becomes  2λ 2 ∂a−2 2λ ∂a−2 0 hΛq, qi 2  µ2 ∂J1 − µ µ ∂I1   2  0 0 −  µ S =  2λ ∂a 2 −2  − hΛp, pi 0 −  µ2 ∂I1 µ   ∂a−2 2λ hΛp, pi 0 0 µ2 ∂I1

 0

     .   0   2λ ∂a−2 2  − µ2 ∂J1 µ

2λ ∂a−2 − 2 hΛq, qi µ ∂I1

Substituting S into Eq. (3.2) and sorting it, we can obtain Eq. (4.1), where ), for any f (a−2 ) ∈ C ∞ (R). a−2 satisfies the relation I1 = (J1 + a−2 )2 + f (a−2p 1 + hΛp, pihΛq, qi − hΛp, qi. Particularly, choosing f (a−2 ) = −1 yields a−2 = Thus Eq. (4.1) reads r12 (λ, µ) =

2λ 2λ λ 1 Q, P − 2S + 2 p µ(µ − λ) µ µ 1 + hΛp, pihΛq, qi

while the corresponding Lax matrix L(λ) becomes ! N X hq, qiλ−1 l11 1 + L(λ) = −1 λ − λj −hp, piλ −l11 j=1

(r − W KI)

pj qj

−qj2

p2j

−pj qj

! ,

where p l11 = ( 1 + hΛp, pihΛq, qi − hΛp, qi)λ−2 − hp, qiλ−1 . Set an auxiliary matrix M1 as follows  −λ  M1 = M1 (λ) =   hΛp, pi λ −p 1 + hΛp, pihΛq, qi

 hΛq, qi p λ 1 + hΛp, pihΛq, qi  .  λ

Then the Lax equation Lx = [M1 , L] p is equivalent to the following finite dimensional Hamilton system ( H−4 ): p  ∂ H−4 hΛq, qi   ,   qx = −Λq + p1 + hΛp, pihΛq, qi Λp = p ∂p p (4.15) ( H−4 ) :  ∂ H hΛp, pi −4   , Λq = −  px = Λp − p ∂q 1 + hΛp, pihΛq, qi with p

H−4 = a−2 = −hΛp, qi +

p 1 + hΛp, pihΛq, qi ,

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which is obviously integrable by Theorem 3.2. Let hΛp, pi hΛq, qi , v = −p . u= p 1 + hΛp, pihΛq, qi 1 + hΛp, pihΛq, qi p Then ( H−4 ) is nothing but the WKI spectral problem [49] ! −λ λu y yx = λv λ

559

(4.16)

with the above two constraints (4.16), λ = λj , and y = (qj , pj )T , j = 1, . . . , N . That means that (r − W KI) is the r-matrix of the integrable constrained WKI (c-WKI) system (4.15). Other subcases in Case 1 can be similarly proven. The proof of Case 10 can be found in Ref. [37]. In this case, the corresponding constrained system is reduced to the well-known MKdV spectral problem [48]. Remark 4.1. From the above formulae (4.1)–(4.14), the r-matrices of Cases 2, 6, and 7(ii) are non-dynamical. But in fact, for other cases we can also obtain non-dynamical r-matrices if choosing some special functions, for instance, in Eq. (4.3) setting a−1 such that a0−1 = const. leads to a non-dynamical one. Of course, we can also get dynamical r-matrices, for instance, in Eq. (4.4) choosing √ c0 = −2 J0 yields a dynamical one. Remark 4.2. Equations (4.1)–(4.14) cover most r-matrices of 2 × 2 constrained systems. But among them there are also some new r-matrices and finite dimensional integrable systems like Cases 6, 7(i), 7(iv), and 7(v) (also see Sec. 7). Their Lax matrices are altogether unified in Eq. (2.1). So, quite a large number of finite dimensional integrable systems are classified or reduced from the viewpoint of Lax matrix and r-matrix structure. 5. Different Systems Sharing the Same r-Matrices In the above r-matrices, we find some pairs of different integrable systems sharing the common r-matrices. Now, we present these results as follows. 5.1. The constrained Toda and CKdV flows Let us consider the following 2 × 2 traceless Lax matrix [36] (corresponding to Case 9 in Sec. 4)   1 − λ hp, qi   LT C = LT C (λ, p, q) =  2  + L0 1 λ −1 2 ! AT C (λ) BT C (λ) , (5.1) ≡ CT C (λ) −AT C (λ)

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where L0 = L0 (λ, p, q) =

N X j=1

1 λ − λj

−pj qj

p2j

−qj2

pj qj

! .

(5.2)

The determinant of Eq. (5.1) leads to EjT C P 1 1 λ Tr(LT C )2 (λ) = − λ3 + hp, qiλ + 2HC + N , j=1 2 2 λ − λj

(5.3)

EjT C = λj pj qj − p2j − hp, qiqj2 − Γj , j = 1, 2, . . . , N ,

(5.4)

where Γj is defined by Eq. (3.12) and the Hamiltonian function HC is 1 1 1 (5.5) HC = − hp, pi + hΛq, pi − hq, qihp, qi . 2 2 2 Viewing the variables q and p as the functions of continuous variables x, we have the following Hamiltonian canonical equation (HC ):  1 1 ∂HC   = − Λp + hq, qip + hp, qiq ,  px = − ∂q 2 2 (5.6) 1 1 ∂HC    qx = = −p + Λq − hq, qiq , ∂p 2 2 which is nothing but the coupled KdV (CKdV) spectral problem [29]   1 1 v − λ+ u   2 2 (5.7) ψx =  ψ 1 1 λ− u −1 2 2 with the two constraints (Bargmann-type) u = hq, qi ,

v = hp, qi ,

(5.8)

T

λ = λj and ψ = (pj , qj ) . So, (HC ) coincides with the constrained CKdV (c-CKdV) flow. Let us consider endowing with an auxiliary 2 × 2 matrix MT as follows   0 g   2 2 (5.9) MT =  1 λ − hq, qi  , g = hΛq, qi − hp, qi − hq, qi . − g g Then, we have the following theorem. Theorem 5.1. The discrete Lax equation (LT C )0 MT = MT LT C , (LT C )0 = LT C (λ, p0 , q 0 ) is equivalent to a finite dimensional symplectic map HT : R (p0 , q 0 ), which is called the constrained Toda (c-Toda) flow :  0   p = gq ,  q 0 = Λq − p − hq, qiq .  g

(5.10) 2N

→ R

2N

, (p, q) 7→

(5.11)

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Proof. Through direct calculations, one can readily show (5.10) ⇔ (5.11) and (HT )∗ (dp ∧ dq) = dp ∧ dq. When we understand the above two matrices (LT C )0 and MT in the following C , MT → MT n (i.e. q → qn , p → pn , here n is the discrete sense: (LT C )0 → LTn+1 variable), and set ( 1 un = ±(hΛqn , qn i − hpn , qn i − hqn , qn i2 ) 2 , (5.12) vn = hqn , qn i , the constrained Toda flow (5.11) is none other than the well-known Toda spectral problem Lψn ≡ (E −1 un + vn + un E)ψn = λψn ,

Efn = fn+1 ,

E −1 fn = fn−1

(5.13)

with the above constraint (5.12), λ = λj and ψn = qn,j . Theorem 5.1 shows that the constrained Toda flow (HT ) has the discrete Lax representation (5.10). Equation (5.12) is a kind of discrete Bargmann constraint [42] of the Toda spectral problem (5.13). The Hamiltonian systems (HT ) and (HC ) share the common Lax matrix (5.1). Thus, they have the following same r-matrix:   0 1 0 0   0 0 0 0 2  (5.14) P − S, S =  r12 (λ, µ) =  0 0 0 −1  , µ−λ   0 0 0 0 which is proven to satisfy the classical Yang–Baxter equation (YBE) [rij , rik ] + [rij , rjk ] + [rkj , rik ] = 0, i, j, k = 1, 2, 3 .

(5.15)

5.2. The restricted Toda and CKdV flows Let us now consider the case on a symplectic manifold. We restrict the Toda and CKdV flows on the following symplectic submanifold M in R2N 1 (5.16) M = (q, p) ∈ R2N |F ≡ hq, qi − 1 = 0, G ≡ hq, pi − = 0 . 2 Let us first introduce the Dirac bracket 1 {f, g}D = {f, g} + ({f, F }{G, g} − {f, G}{F, g}) , 2 which is easily proven to be a Poisson bracket on M . According to the idea of Ref. [36], the following Lax matrix   1 ! 0 − AR (λ) BR (λ)   2 TC TC LR = LR (λ, p, q) =   + L0 ≡ 1 CR (λ) −AR (λ) 0 2

(5.17)

(5.18)

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yields −λ2 det LTRC =

N TC X λ2j ER,j 1 2 1 C λ Tr (LTRC )2 = λ2 + hp, qiλ + 2HR + , 2 4 λ − λj j=1

(5.19)

where C = HR

1 1 1 hΛp, qi − hq, qihp, pi + hp, qi2 , 2 2 2

TC TC = ER,j (p, q) = pj qj − Γj , i = 1, . . . , N , ER,j

(5.20) (5.21)

and L0 is defined by Eq. (5.2). An important observation is: if we consider the Hamiltonian canonical equation produced by Eq. (5.20) in R2N , then this equation is exactly the well-known constrained AKNS flow, which will be discussed in the next subsection. Now, we first consider the Hamiltonian canonical equation restricted on M : C C C ) : qx = {q, HR }D , px = {p, HR }D , (HR

(5.22)

which reads as the following finite dimensional system:  1 1  px = − Λp + (hΛq, qi − 1)p + hp, piq ,    2 2   1 1 (5.23) qx = −p + Λq − (hΛq, qi − 1)q ,  2 2      hq, qi = 1, hq, pi = 1 . 2 This is actually the CKdV spectral problem (5.7) with the two constraints (Neumann-type) [13] u = hΛq, qi − 1,

v = hp, pi ,

(5.24)

and λ = λj , ψ = (pj , qj ), j = 1, 2, . . . , N . So, the finite dimensional system (5.23) coincides with the restricted CKdV (r-CKdV ) flow . Let us return to the Lax matrix (5.18). After endowing with an auxiliary matrix MT,R as follows   0 a   (5.25) MT,R =  1 λ−b , − a a a2 = hΛq − p, Λq − pi + hΛq, qi − hΛq, qi2 , b = hΛq, qi − 1 , we have the following theorem. Theorem 5.2. The discrete Lax equation (LTRC )0 MT,R = MT,R LTRC ,

(LTRC )0 = LTRC (λ, p0 , q 0 )

(5.26)

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is equivalent to a discrete Neumann type of finite dimensional symplectic map HT : (p, q)T → (p0 , q 0 )T  0 p = aq ,    0 q = a−1 (Λq − p − bq) , (5.27)   1  hq, qi = 1, hq, pi = , 2 which is called the restricted Toda (r-Toda) flow . Remark 5.1. If we understand the above two matrices (LTRC )0 and MT,R in the following sense: (LTRC )0 → (LTRC )n+1 , MT,R → (MT,R )n (i.e. q → qn , p → pn , a → an , b → bn , here n is the discrete variable), then the restricted Toda flow (5.27) on the symplectic submanifold M = {(q, p) ∈ R2N |hq, qi = 1, hq, pi = 12 } is nothing but the discrete Neumann system studied by Ragnisco [41]. Let LTR1C = LTRC (λ, p, q) ⊗ I and LTR2C = I ⊗ LTRC (µ, p, q). Then, under the Dirac bracket (5.17) we obtain the following theorem. Theorem 5.3. The Lax matrix LTRC (λ, p, q) defined by Eq. (5.18) satisfies the following fundamental Dirac–Poisson bracket {LTRC (λ) ⊗, LTRC (µ)}D = [r12 (λ, µ), LTR1C (λ)] − [r21 (µ, λ), LTR2C (µ)]

(5.28)

with a dynamical r-matrix r12 (λ, µ) =

2 P − S12 (λ, µ), µ−λ

r21 (µ, λ) = P r12 (µ, λ)P ,

where S12 = (E11 − E22 ) ⊗ E12 + E11 ⊗

+ E12 ⊗

0

−BR (µ)

CR (µ)

0

CR (µ)

0

0

0

! + E22 ⊗

(5.29)

!

0

2AR (µ)

0

CR (µ)

! (5.30)

P and P = 12 (I + 3j=1 σj ⊗ σj ) is the permutation matrix, σj (j = 1, 2, 3) are the Pauli matrices, and Eij stands for the 2 × 2 matrix with the i-th line and jth column element 1 and other elements 0. This theorem ensures that Eq. (5.21) satisfies TC TC , ER,j }D = 0, i, j = 1, . . . , N . {ER,i

(5.31)

TC 0 0 TC (p , q ) = ER,i (p, q) as well as For the r-Toda flow (5.27), we have ER,i Pn 1 TC E = hp, qi = from the discrete Lax equation (5.26). Thus, in the set i=1 R,i 2 TC N TC TC TC }j=1 , only ER,1 , ER,2 , . . . , ER,N are independent on M . {ER,j −1

Theorem 5.4. The restricted Toda flow HT is completely integrable, and its T C N −1 }i=1 . independent and invariant (N − 1)-involutive system is {ER,i

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For the restricted CKdV flow on M, we have 1X TC λj ER,j 2 j=1 N

C = HR

(5.32)

which implies C TC , ER,j }D = 0 , {HR

j = 1, 2, . . . , N .

(5.33)

Thus, the following theorem holds. C ) is completely integrable, and its indepenTheorem 5.5. The r-CKdV flow (HR T C N −1 }k=1 . dent (N − 1)-involutive system is also {ER,k

Remark 5.2. As shown in this and last subsection, the r-Toda (i.e. Neumanntype) and the r-CKdV flows, and the c-Toda (i.e. Bargmann-type) and the c-CKdV flows respectively share the completely same Lax matrix, r-matrix and involutive conserved integrals. Thus, we say that the finite dimensional integrable CKdV flow both restricted and constrained is the interpolating Hamiltonian flow of invariant of the corresponding Toda integrable symplectic map. 5.3. The constrained AKNS and Dirac (D) flows From now on we assume: L0 = L0 (λ, p, q) =

N X j=1

1 λ − λj

!

pj qj

−qj2

p2j

−pj qj

.

(5.34)

Let us again consider Eq. (5.18), and rewrite it as the following version: ! 1 0 AKN S AKN S =L (λ, p, q) = + L0 , (5.35) L 0 −1 while we introduce L

D

D

= L (λ, p, q) =

0

1

−1

0

! + L0 .

(5.36)

Then we have 1 2 λ Tr(LAKN S )2 (λ) = λ2 + 2λhp, qi + hp, qi2 2 + 2HAKN S +

N X λ2j EjAKN S j=1

λ − λj

,

(5.37)

1 2 λ Tr(LD )2 (λ) = λ2 + λ(hq, qi + hp, pi) 2 N X λ2j EjD 1 , − (hp, pi + hq, qi)2 + 2HD + 4 λ − λj j=1

(5.38)

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where 1 HAKN S = hΛp, qi − hq, qihp, pi , 2 1 1 HD = (hΛq, qi + hΛp, pi) + (hp, qi2 − hq, qihp, pi) 2 2 1 (hp, pi + hq, qi)2 , 8 = 2pj qj − Γj , j = 1, . . . , N ,

(5.40)

+

EjAKN S

(5.39)

(5.41)

EjD = p2j + qj2 − Γj , j = 1, . . . , N . Thus, HAKN S and HD generate the following two Hamiltonian systems  ∂HAKN S   = −hq, qip + Λq ,  qx = ∂p (HAKN S ) :  ∂HAKN S   px = − = hp, piq − Λp ; ∂q  1 ∂HD    qx = ∂p = hp, qiq + 2 (hp, pi − hq, qi)p + Λp , (HD ) :  1 ∂HD   px = − = −hp, qip − (hp, pi − hq, qi)q − Λq . ∂q 2

(5.42)

(5.43)

(5.44)

It can be easily seen that (HAKN S ) and (HD ) are changed to the well-known Zakharov–Shabat–AKNS spectral problem [52] ! λ u y (5.45) yx = v −λ and the Dirac spectral problem [30] yx =

−v

λ−u

−λ − u

v

! y

(5.46)

with the constraints u = −hq, qi, v = hp, pi, λ = λj , y = (qj , pj )T , and the constraints u = − 21 (hp, pi − hq, qi), v = −hp, qi, λ = λj , y = (qj , pj )T , respectively. Therefore (HAKN S ) and (HD ) coincide with the constrained AKNS (c-AKNS) system and the constrained Dirac (c-D) system, respectively. Let LJ1 (λ) = LJ (λ) ⊗ I, LJ2 (µ) = I ⊗ LJ (µ) (J = AKN S, D). Then we have the following theorem. Theorem 5.6. The Lax matrices LJ (λ) (J = AKN S, D) defined by Eq. (5.35) and Eq. (5.36) satisfy the fundamental Poisson bracket {LJ (λ) ⊗, LJ (µ)} = [r12 (λ, µ), LJ1 (λ)] − [r21 (µ, λ), LJ2 (µ)] .

(5.47)

Here the r-matrices r12 (λ, µ), r21 (µ, λ) are exactly given by the following standard r-matrix 2 P , r21 (µ, λ) = P r12 (µ, λ)P , (5.48) r12 (λ, µ) = µ−λ

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

1

 0 P = 0  0

0

0

0

1

1

0

0

0

0



 0 1 = 0  2 1

I+

3 X

! σi ⊗ σi

.

(5.49)

i=1

So, the c-AKNS and c-D flows share the same standard r-matrix (5.48), which is obviously non-dynamical. However, the two constrained flows, produced by Eqs. (5.45)’s and (5.46)’s extensive spectral problems (6.1) and (6.2) (they are gauge equivalent), have different r-matrices (see Sec. 6). Remark 5.3. In fact, the r-matrix r12 (λ, µ) in the case of the c-AKNS and c-D flows can be also chosen as ! a b 2 ˜ S˜ = P + I ⊗ S, (5.50) r12 (λ, µ) = µ−λ c d where the elements a, b, c, d can be arbitrary C ∞ -functions a(λ, µ, p, q), b(λ, µ, p, q), c(λ, µ, p, q), d(λ, µ, p, q) with respect to the spectral parametres λ, µ and the dynamical variables p, q. This shows that for a given Lax matrix, the associated r-matrix is not uniquely defined (there are even infinitely many r-matrices possible). Here we give the simplest case: a = b = c = d = 0, i.e. the standard r-matrix (5.48). 5.4. The constrained Harry–Dym (HD) and Heisenberg spin chain (HSC ) flows The constrained Harry–Dym system describes the geodesic flow on an ellipsoid and shares the same r-matrix with the constrained Heisenberg spin chain (HSC). To prove this, we consider the following Lax matrices: ! −hp, qiλ−1 λ−2 + hq, qiλ−1 HD HD = L (λ, p, q) = (5.51) + L0 , L −hp, piλ−1 hp, qiλ−1 ! −hp, qiλ−1 hq, qiλ−1 HSC HSC =L (λ, p, q) = (5.52) + L0 . L −hp, piλ−1 hp, qiλ−1 Here LHSC is included in the generalized Lax matrix (2.1), but LHD is not. We need two associated auxiliary matrices   0 1   (5.53) MHD =  hΛp, pi , λ 0 − 2 hΛ q, qi ! −iλhΛp, qi iλhΛq, qi , i2 = −1 . (5.54) MHSC = −iλhΛp, pi iλhΛp, qi

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Theorem 5.7. The Lax representations = [MHD , LHD ] , LHD x

(5.55)

= [MHSC , LHSC ] LHSC x

(5.56)

respectively give the following fininte dimensional Hamiltonian flows:  ∂HHD   q = p = ,  x   ∂p T QN −1  ∂HHD hΛp, pi (HHD ) : Λq = − , px = − 2   hΛ q, qi ∂q T QN −1     hΛq, qi = 1 ;  ∂HHSC   ,  qx = ihΛq, qiΛp − ihΛp, qiΛq = ∂p (HHSC ) :  ∂HHSC   px = ihΛp, qiΛp − ihΛp, piΛq = − , ∂q

(5.57)

(5.58)

with the Hamiltonian functions HHD =

hΛp, pi 1 hp, pi − (hΛq, qi − 1) , 2 2hΛ2 q, qi

(5.59)

1 1 ihΛp, pihΛq, qi − ihΛp, qi2 . 2 2

(5.60)

HHSC =

In Eq. (5.57) T QN −1 is a tangent bundle in R2N : T QN −1 = {(p, q) ∈ R2N |F ≡ hΛq, qi − 1 = 0, G ≡ hΛp, qi = 0} .

(5.61)

Obviously, (5.57) is equivalent to qxx +

hΛqx , qx i Λq = 0, hΛq, qi = 1 , hΛ2 q, qi

(5.62)

which is nothing but the equation of the geodesic flow [26] on the surface hΛq, qi = 1 in the space RN and also coincides with the constrained HD (c-HD) flow [9]. In addition, (5.58) becomes the Heisenberg spin chain spectral problem [47] ! −iλw −iλu y , i2 = −1 , (5.63) yx = −iλv iλw with the constraints u = −hΛq, qi, v = hΛp, pi, w = −hΛp, qi, λ = λj , y = (qj , pj )T . Thus, Eq. (5.58) reads as the constrained Heisenberg spin chain (c-HSC) flow [39]. Their Lax matrices (5.51) and (5.52) share all elements except one, namely ! 0 λ−2 . 0 0

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This element does not affect the calculations concerning the fundamental Poisson bracket, one can readily deduce that the c-HD flow and the c-HSC flow possess the same non-dynamical r-matrix r12 (λ, µ) =

2λ P, r21 (µ, λ) = P r12 (µ, λ)P . µ(µ − λ)

(5.64)

Remark 5.4. The r-matrix (5.64) of the c-HD and c-HSC flows can be also chosen as 2λ P + I ⊗ S˜ . (5.65) r12 (λ, µ) = µ(µ − λ) Evidently, (5.64) is the simplest case: S˜ = 0 of (5.65). 5.5. The constrained G and Q flows In this subsection, we introduce the following Lax matrices:   1 −1 −1 + hp, qi λ hq, qiλ  2   + L0 , LG = LG (λ, p, q) =    1 −1 + hp, qi λ 0 − 2 LQ = LQ (λ, p, q) = If we set

−λ−1

hq, qiλ−1

0

λ−1

! + L0 .

 1 (hp, pi − hq, qi) − 1   α MG =  , 1 1 (hp, pi − hq, qi + 1)λ λ α α   1 1 hΛq, qi   λ + 2β 2 hΛq, qihp, pi β  MQ =    1 1 −λ − 2 hΛq, qihp, pi − hp, piλ β 2β 

with α=

p

1 − λ α

(hp, pi − hΛq, qi)2 − 4hΛq, pi,

(5.66)

β = 1 − hp, qi ,

(5.67)

(5.68)

(5.69)

(5.70)

then, by a lengthy and straightforward calculation we obtain the following theorem. Theorem 5.8. The following Lax representations G LG x = [MG , L ]

(5.71)

Q LQ x = [MQ , L ] ,

(5.72)

and

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where the first one is restricted to the surface M1 = {(p, q) ∈ R2N |hp, qi = 0, hΛq, qihp, pi + hΛq, pi = 0} in the space R2N , respectively produce the finitedimensional systems:  1   qx = (−Λq + (hp, pi − hΛq, qi)p) − p , α   p = 1 (Λp + (hp, pi − hΛq, qi)Λq) + Λq , x α

(5.73)

 1 1    qx = Λq + hΛq, qip + 2 hp, pihΛq, qiq , β 2β 1 1    px = −Λp − hp, piΛq − 2 hp, pihΛq, qip . β 2β

(5.74)

and

Equations (5.73) and (5.74) turn out to be the spectral problem studied by Geng (simply called G-spectral problem) [22] yx =

−λu

v−1

λ(v + 1)

λu

! y

(5.75)

with the constraint condition u=

1 1 , =p α (hp, pi − hΛq, qi)2 − 4hΛq, pi

v=

hp, pi − hΛq, qi hp, pi − hΛq, qi , = p α (hp, pi − hΛq, qi)2 − 4hΛq, pi

λ = λj , y = (qj , pj )T , and the spectral problem proposed by Qiao (simply called Q-spectral problem) [38] 



1 λ − uv  2 yx =  λv

u 1 −λ + uv 2

 y

with the constraint condition u=

hΛq, qi hΛq, qi = , β 1 − hq, pi

v=

hp, pi −hp, pi =− , β 1 − hq, pi

λ = λj , y = (qj , pj )T , respectively.

(5.76)

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So, Eqs. (5.73) and (5.74) are the constrained Geng (c-G) flow and the constrained Qiao (c-Q) flow, and they have the same non-dynamical r-matrix:   0 0 0 0   0 0 0 0 2 2   = σ− ⊗ σ + . P − S, S =  (5.77) r12 (λ, µ) =  µ−λ µ 0 1 0 0 0 0 0 0 Here, the r-matrix r12 (λ, µ) can be also chosen as r12 (λ, µ) =

2 2 P − S + I ⊗ S˜ . µ−λ µ

(5.78)

Equation (5.77) is the simplest case: S˜ = 0 of (5.78). We have already seen that the r-matrix r12 (λ, µ) satisfying the fundamental Poisson bracket is not unique (in fact, infinitely many) and is usually composed of two parts, the first one being their main term, and the second one being the ˜ Usually, to prove the integrability we choose their main term common term I ⊗ S. as the simplest r-matrix. 6. An Equivalent Pair with Different r-Matrices This section reveals the following interesting fact: a pair of constrained systems, produced by two gauge equivalent spectral problems, possesses different r-matrices. In 1992, Geng introduced the following spectral problem [23] ! iλ − iβuv u (6.1) , i2 = −1 φx = M φ, M = v −iλ + iβuv where u and v are two scalar potentials, λ is a spectral parameter and β is a constant, and discussed its evolution equations and Hamiltonian structure. Equation (6.1) is apparently an extension of the AKNS spectral problem (5.45). Two years later the author considered an extension of the Dirac spectral problem (5.46) [40] ! −is λ + r + β(s2 − r2 ) ¯ ψ, M ¯ = , (6.2) ψx = M is −λ + r − β(s2 − r2 ) where r, s are two potentials, and obtained a finite dimensional involutive system being not equivalent to that one in Ref. [23]. But, the spectral problems (6.1) and (6.2) are gauge equivalent via the following transformation [50] ! 1 1 ψ = Gφ, G = , (6.3) i −i v = i(r − s), u = −i(r + s). In Ref. [50], Wadati and Sogo discussed the gauge transformations of some spectral problems like Eq. (5.63).

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Now, we discuss their r-matrices. Let us consider the following two Lax matrices: ! 1 + 2iβhp, qi 0 − iL0 , (6.4) LGX = LGX (λ, p, q) = 0 −1 − 2iβhp, qi LQZ = LQZ (λ, p, q)   =

0 1 − + β(hp, pi + hq, qi) 2

 1 − β(hp, pi + hq, qi)  2  + L0 . 0

(6.5)

Then calculating their determinants leads to the following Hamiltonian systems  hp, pihq, qi ∂HGX hq, qi    qx = ∂p = Λq + iβ (1 + 2iβhp, qi)2 q − 1 + 2iβ p , (6.6)  hp, pihq, qi ∂HGX hp, pi   px = − = −Λp − iβ q , p + ∂q (1 + 2iβhp, qi)2 1 + 2iβ and  ∂HQZ   qx =   ∂p      2hp, qiq + (hp, pi − hq, qi)p 4hp, qi2 + (hp, pi − hq, qi)2   , p−  = Λp − β (1 − 2β(hp, pi + hq, qi))2 1 − 2β(hq, qi + hp, pi) (6.7) ∂HQZ   = − p  x   ∂q     2hp, qip − (hp, pi − hq, qi)q 4hp, qi2 + (hp, pi − hq, qi)2   , q+  = −Λq + β (1 − 2β(hp, pi + hq, qi))2 1 − 2β(hq, qi + hp, pi) with the Hamiltonian functions HGX = ihΛq, pi −

hp, pihq, qi 2(1 + 2iβhp, qi)

(6.8)

and HQZ =

1 4hp, qi2 + (hp, pi − hq, qi)2 1 hΛp, pi + hΛq, qi − . 2 2 4 − 8β(hp, pi + hq, qi)

(6.9)

Obviously, Eqs. (6.6) and (6.7) become Eqs. (6.1) and (6.2) with the constrants u=−

hp, pi hq, qi , v= , 1 + 2iβhp, qi 1 + 2iβhp, qi

(6.10)

λ = λj , φ = (qj , pj )T , j = 1, . . . , N ; and the constraints s=

−2ihp, qi , 1 − 2β(hq, qi + hp, pi)

r=

−hp, pi + hq, qi , 1 − 2β(hq, qi + hp, pi)

(6.11)

λ = λj , ψ = (qj , pj )T , j = 1, . . . , N , respectively. Thus, the finite dimensional Hamiltonian systems (6.6) and (6.7) are respectively the constrained flows of the

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spectral problems (6.1) and (6.2). Since they have Lax matrices (6.4) and (6.5), then the r-matrices of Eqs. (6.6) and (6.7) are respectively:   1 0 0 0   0 0 0 0 2  , P + 4iβS, S =  r12 (λ, µ) = (6.12)  µ−λ 0 0 0 0 0 0 0 1 and



0

  0 2 P + 2βS, S =  r12 (λ, µ) =  0 µ−λ  −1

0

0

0

1

1

0

0

0

−1



 0 , 0  0

(6.13)

which are apparently different. 7. New Integrable Systems In this section, three new integrable systems are generated as the representatives from our generalized r-matrix structure. (1) The first system is given by Case 6 in Sec. and involutive systems are respectively  1   0 2 2 P + S, S =  r12 (λ, µ) = 0 µ−λ µ  0

4. The corresponding r-matrix 0

0

0

0

1

0

0

0

0



 0 , 0  1

(7.1)

and −1 2 Ej1 = 2(hp, qi + c)λ−1 j pj qj + hq, qiλj pj − Γj , j = 1, . . . , N ,

(7.2)

1 ) defined by where c ∈ R. Thus, the finite dimesional Hamiltonian systems (Fm PN 1 m 1 Fm = j=1 λj Ej , m = 0, . . ., i.e. 1 = 2(hp, qi + c)hλm−1 p, qi + hq, qihλm−1 p, pi Fm j j X (hΛi q, qihΛj p, pi − hΛi q, pihΛj p, qi) −

(7.3)

i+j=m−1

are completely integrable. Particularly, with m = 2 the Hamiltonian system (F21 ):  ∂F21   qx = = 2cΛq − 2hΛq, qip + 4hΛp, qiq + 4hp, qiΛq ,  ∂p (7.4) 1   p = − ∂F2 = −2cΛp + 2hp, piΛq − 4hΛp, qip − 4hp, qiΛp ,  x ∂q

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is a new integrable system, which becomes the following spectral problem ! (2c + 4v)λ + 4u −2w φ φx = 2sλ −(2c + 4v)λ − 4u

573

(7.5)

with the constraint conditions u = hΛp, qi, v = hp, qi, w = hΛq, qi, s = hp, pi, and λ = λj , φ = (qj , pj )T , j = 1, . . . , N . Apparently, the spectral problem (7.5) is new. (2) The second system is produced by Case 7(i) in Sec r-matrix and involutive systems are respectively  0 g20 0  0  f2 0 −1 2 2 P + S, S =  r12 (λ, µ) =  µ−λ µ −1 0 0 0 0 0

4. The corresponding 0



 0 , 0  0

(7.6)

and −1 2 Ej2 = 2cλ−1 j pj qj + (hq, qi + g2 )λj pj 2 + (hp, pi − f2 )λ−1 j qj − Γj , j = 1, . . . , N,

(7.7)

where c ∈ R, f2 = f2 (hp, qi), g2 = g2 (hp, qi) ∈ C ∞ (R). Hence, the Hamiltonian PN system (F22 ) defined by F22 = j=1 λ2j Ej2 , i.e. F22 = 2chΛp, qi + 2hΛp, qihp, qi + g2 hΛp, pi − f2 hΛq, qi

(7.8)

is completely integrable. Meanwhile the Hamiltonian system (F22 ): qx =

∂F22 = 2cΛq + 2hΛq, piq + 2hp, qiΛq ∂p + 2g2Λp + hΛp, pig20 q − hΛq, qif20 q ,

px = −

(7.9)

∂F22 = −2cΛp − 2hΛq, pip − 2hp, qiΛp ∂q

− hΛp, pig20 p + 2f2 Λq − hΛq, qif20 p ,

(7.10)

can be also related to a new 2 × 2 spectral problem with some constraint conditions. (3) The third system is derived by Case 7(iv) in Sec. r-matrix and involutive systems are respectively  1 0 0  0 0 0 2 2 P − S, S =  r12 (λ, µ) =  µ−λ µ 0 0 1 0

0

c0−1

4. The corresponding 0



 0 , 0  1

(7.11)

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and −1 2 −1 2 Ej3 = 2(−hp, qi + c)λ−1 j pj qj + hq, qiλj pj − c−1 λj qj − Γj

(7.12)

where c ∈ R and c−1 = c−1 (hp, qi) ∈ C ∞ (R). The following Hamiltonian system (F23 ):  ∂F23   = 2cΛq − hΛq, qi(c0−1 q + 2p) ,  qx = ∂p (7.13)  ∂F 3   px = − 2 = −2cΛp + 2(c−1 + hp, pi)Λq + c−1 hΛq, qip , ∂q is one of their products, where F23 = 2chΛp, qi − hΛq, qi(c−1 + hp, pi) .

(7.14)

In general, with any c−1 Eq. (7.13) can’t be changed to a 2×2 spectral problem with some constraints. But with two special c−1 : c−1 = 0 and c−1 = hp, qi, Eq. (7.13) can respectively become the spectral problem [31] ! 2cλ −2v φ (7.15) φx = 2uλ −2cλ with the constraint conditions u = hp, pi, v = hΛq, qi, and the spectral problem ! 2cλ − v −2v φ (7.16) φx = 2uλ −2cλ + v with the constraint conditions u = hp, q + pi, v = hΛq, qi. Here in Eqs. (7.15) and (7.16) λ = λj , φ = (qj , pj )T , j = 1, . . . , N are set. Equation (7.16) is a new spectral problem. We can consider further new integrable systems generated by Theorem 3.1. The above procedure actually gives an approach how to connect an r-matrix of finite dimensional system with a spectral problem, which is closely associated with integrable NLEEs. 8. Algebro-Geometric Solutions In this section, we connect the integrable NLEEs with the finite dimesional integrable flows, and solve them with a form of algebro-geometric solutions. Here, we take two examples: one being the periodic or infinite Toda lattice equation, the other the AKNS equation with the condition of decay at infinity or periodic boundary. 8.1. Toda lattice equation The Toda hierarchy associated with Eq. (5.13) is derived as follows: ! un = JGnj , j = 0, 1, 2, . . . vn t j

(8.1)

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n −1 T where {Gnj = J −1 KGnj−1 }∞ j=0 is the Lenard sequence, G−1 = (αun , β) ∈ Ker J, ∞ for all α = α(tj ), β = β(tj ) ∈ C (R), the two symmetric operators K, J are   1 un (E − E −1 )un un (E − 1)vn , K =2 vn (1 − E −1 )un 2(u2n E − E −1 u2n )

J=

0

un (E − 1)

(1 − E −1 )un

0

! .

(8.2)

In particular, with j = 0, β = 1, Eq. (8.1) reads as the Toda lattice u˙ n = un (vn+1 − vn ) ,

v˙ n = 2(u2n − u2n−1 ),

(8.3)

which can be changed to x ¨n = 2(e2(xn+1 −xn ) − e2(xn −xn−1 ) )

(8.4)

via the following transformation un = exn+1 −xn ,

vn = x˙ n .

(8.5)

It is easy to prove the following theorem. Theorem 8.1. ∞ ˆ (1) ˆ (2) T ˆ (1) ˆ (2) ˆ n = (G (1) Let G n , Gn ) , ∀ Gn , Gn ∈ C (R). Then the operator equation

ˆ n ) − L∗ (J G ˆ n )L ˆ n ), L] = L∗ (K G [V (G possesses the operator solution ˆ (1) ) − un G ˆ (1) ) + un G ˆ (2) E (8.6) ˆ (2) E −1 + 1 ((E −1 un G ˆ n ) = −(E −1 un )G V (G n n n n 2 where [· , ·] is the usual commutator; the operator L is defined by Eq. (5.13); L∗ (ξ) = E −1 ξ1 + ξ2 + ξ1 E , ∀ ξ = (ξ1 , ξ2 )T , ξ1 , ξ2 ∈ C ∞ (R). ˆ n = Gn , j = −1, 0, 1, . . . , then the Toda hierarchy (2) Let us choose the special G j (8.1) has the following Lax representation of operator form Ltj = [W (Gnj ), L] , j = 0, 1, 2, . . . , Pj where the operator W (Gnj ) = k=0 V (Gnk−1 )Lj−k .

(8.7)

Particularly, the standard Toda Equation (8.4) possesses the Lax representation of operator form Lt = [W (Gn0 ), L], where the operator W (Gn0 ) = exn+1 −xn E − exn −xn−1 E −1 , and un , vn in L are substituted by Eq. (8.5). We have shown that the c-Toda flow and the c-CKdV flow share a common nondynamical r-matrix, and in particular, this ensures the integrability of their flows. A calculation of determinant yields their common N -involutive systems Eα = λα pα qα − p2α − hp, qiqα2 −

N X β6=α,β=1

(qα pβ − pα qβ )2 , λα − λβ

α = 1, . . . , N,

(8.8)

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which are independent and invariant (i.e. Eα (λ, p, q) = Eα (λ, p0 , q 0 )). Apparently, PN the functions Fs = α=1 λsα Eα , s = 0, 1, 2, . . . , are given by Fs = hΛs+1 p, qi − hΛs p, pi − hp, qihΛs q, qi X (hΛj p, pihΛk q, qi − hΛj p, qihΛk q, pi) −

(8.9)

j+k=s−1

and {Fm , Fl } = 0, ∀ m, l ∈ Z + implies the Hamiltonian systems (Fs ) are completely integrable. Let (p0 (ts ), q0 (ts ))T be a solution of the initial problem ! ! ! ! p0 p p −∂Fs /∂q ∂ = , . (8.10) = ∂ts q q0 ∂Fs /∂p q t =0 s

Set pn (ts ) qn (ts )

! = HTn

p0 (ts )

! (8.11)

q0 (ts )

where HT is defined by Eq. (5.11). Now, we rewrite Eq. (5.12) as a map f : R2N → R2 defined by f : (pn , qn )T 7→ (un , vn )T .

(8.12)

Then, we have the following theorem. Theorem 8.2. (un (ts ), vn (ts ))T = f (pn (ts ), qn (ts )) satisfies the Toda hierarchy ! un d (8.13) = JGns , s = 0, 1, . . . . dts vn Particularly, with s = 0 the following calculable method ! ! ! ! un (t) p0 F0 p0 (t) H n pn (t) f → → → q0 q0 (t) qn (t) vn (t)

(8.14)

produces a solution of the Toda lattice Equation (8.3). Thus, the standard Toda Equation (8.4) has the following formal solution Z (8.15) xn (t) = hqn (t), qn (t)idt . We shall concretely give the expression hqn (t), qn (t)i. Let us rewrite the element CT C (λ) of Eq. (5.1) as CT C (λ) ≡ −

Q(λ) , K(λ)

K(λ) =

N Y

(λ − λα ) ,

(8.16)

α=1

and choose N distinct real zero points µ1 , . . . , µN of Q(λ). Then, we have Q(λ) =

N Y j=1

(λ − µj ) ,

hq, qi =

N X α=1

λα −

N X j=1

µj .

(8.17)

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577

Let πj = AT C (µj ) ,

(8.18)

then it is easy to prove the following proposition. Proposition 8.1. {µi , µj } = {πi , πj } = 0 ,

{πj , µi } = δij ,

i, j = 1, 2, . . . , N ,

i.e. πj , µj are conjugated, and thus they are the seperated variables [46]. PN Write det LT C (λ) = −A2T C (λ) − BT C (λ)CT C (λ) = − 14 λ2 − α=1 P (λ) , − K(λ)

(8.19)

Eα λ−λα

=

where Eα is defined by Eq. (8.8), and P (λ) is an N + 2 order polyno-

mial of λ whose first term’s coefficient is we choose the generating function W =

N X

Wj (µj , {Eα }N α=1 )

j=1

=

1 4,

then πj2 =

N Z X j=1

µj (n)

µj (0)

P (µj ) K(µj ) ,

j = 1, . . . , N. Now,

s P (λ) dλ K(λ)

(8.20)

where µj (0) is an arbitrarily given constant. Let us view Eα (α = 1, . . . , N ) as actional variables, then angle-coordinates Qα are chosen as ∂W , α = 1, . . . , N, Qα = ∂Eα i.e. QN N Z µk (n) X k6=α,k=1 (λ − λk ) p dλ , α = 1, . . . , N . (8.21) ω ˜α , ω ˜α = Qα = 2 K(λ)P (λ) k=1 µk (0) Hence, on the symplectic manifold (R2N , dEα ∧ dQα ) the Hamiltonian function PN F0 = α=1 Eα produces a linearized flow   Q˙ α = ∂F0 , ∂Eα (8.22)  ˙ Eα = 0 . Thus

  Qα (n) = Q0α + t + cα n ,

cα =

 E (n) = E (n − 1) , α α

R µk (n+1) k=1 µk (n)

PN

ω ˜α ,

(8.23)

0 where cα are dependent on actional variables {Eα }N α=1 , and independent of t; Qα is an arbitrarily fixed constant. Choose a basic system of closed paths αi , βi , i = 1, . . . , N of Riemann surface ¯ µ2 = P (λ)K(λ) with N handles. ω Γ: ˜ j (j = 1, . . . , N ) are exactly N linearly ¯ independent holomorphic differentials of the first kind on this Riemann surface Γ. PN ˜ l , i.e. ωj satisfy ω ˜ j are normalized as ωj = l=1 rj,l ω I I ωj = δij , ωj = Bij , αi

βi

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where B = (Bij )N ×N is symmetric and the imaginary part Im B of B is a positive definite matrix. R µk (n) P By Riemann Theorem [25] we know: µk (n) satisfies N k=1 µk (0) ωj = φj , φj = 4 PN 0 φj (n, t) = l=1 rj,l (Ql + t + cl n), j = 1, . . . , N , if and only if µk (n) are the ˜ ) = Θ(A(P ) − φ − K) which zero points of the Riemann-Theta function RΘ(P RP P has exactly N zero points, where A(P ) = ( P0 ω1 , . . . , P0 ωN )T , φ = φ(n, t) = (φ1 (n, t), . . . , φN (n, t))T , K ∈ CN is the Riemann constant vector, P0 is an ¯ arbitrarily given point on Riemann surface Γ. Because of [18] I 1 ˜ ) = C1 (Γ) ¯ λd ln Θ(P (8.24) 2πi γ ¯ has nothing to do with φ; γ is the boundary of simple where the constant C1 (Γ) ¯ along closed connected domain obtained through cutting the Riemann surface Γ paths αi , βi . Thus, we have a key equality N X

¯ − Resλ=∞1 λd ln Θ(P ˜ ) − Resλ=∞2 λd ln Θ(P ˜ ) µk (n) = C1 (Γ)

(8.25)

k=1

p p where ∞1 := (0, P (z −1 )K(z −1 )|z=0 ), ∞2 := (0, − P (z −1 )K(z −1 )|z=0 ). Through a lengthy careful calculation and combining Eq. (8.17), we obtain N X d Θ(φ(n, t) + K + η1 ) ¯ λα − C1 (Γ) + ln (8.26) hqn (t), qn (t)i = dt Θ(φ(n, t) + K + η2 ) α=1 RP where the jth component of ηi (i = 1, 2) is ηi,j = ∞0i ωj . By the Riemann surface R ∞1 R ∞1 PN P ˜ l which implies properties, we can also have N l=1 rj,l cl = ∞2 ωj = l=1 rj,l ∞2 ω Q R ∞1 R ∞1 N i6=l,i=1 (λ−λi ) √ ˜ l = P0 dλ. So, the standard Toda Equation (8.4) has the cl = ∞2 ω P (λ)K(λ)

following explicit solution, called algebro-geometric solution xn (t) = ln

Θ(U n + V t + Z) + Cn + Rt + const . Θ(U (n + 1) + V t + Z)

(8.27)

ˆ C, ˆ V = R ˆ J, ˆ Z = RQ ˆ 0 + K + η1 with Cˆ = (c1 , . . . , cN )T , Jˆ = where U = R PN T 0 0 0 T ¯ while matrix R ˆ = λα − C1 (Γ), (1, . . . , 1) , Q = (Q1 , . . . , QN ) , R = PNα=1 H ˜ l = δij , and C is certain (rj,l )N ×N is determined by the relation l=1 rj,l αi ω constant which can be determined by the algebro-geometric properties on the ¯ [16]. The symmetric matrix B = (Bij )N ×N in Θ function is Riemann surface Γ H PN ˜ l = Bij . determined by l=1 rj,l βi ω Hence, the algebro-geometric solution of Toda lattice Equation (8.3) is  Θ2 (U (n + 1) + V t + Z)  x −x C   un (t) = e n+1 n = e · Θ(U (n + 2) + V t + Z)Θ(U n + V t + Z) , (8.28)  Θ(U n + V t + Z)   vn (t) = x˙ n = R + d ln . dt Θ(U (n + 1) + V t + Z)

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Obviously, the algebro-geometric solution un (t) and vn (t) given by Eq. (8.28) are quasi-periodic functions, and they are periodic iff U = M N , where M is a N dimensional integer column vector. It is easy to see that Eq. (8.28) is the finite-band solution of Toda lattice Equation (8.3) if λ1 , . . . , λN are chosen as the eigenvalues of Toda spectral problem (5.13). 8.2. AKNS equation In Sec. 5.3 we have shown that the constrained AKNS flow shares a common r-matrix with the constrained Dirac flow, therefore they are integrable. Now, we derive the algebro-geometric solution for the second order AKNS Equation (8.31). It is well-known that the AKNS hierarchy is given by ! u = JGj , j = 0, 1, 2, . . . , (8.29) v t j

−1

KGj−1 }∞ j=0

is the Lenard sequence, with G−1 = (0, 0)T and where {Gj = J ∂ T , ∂∂ −1 = ∂ −1 ∂ = 1). G0 = (v, u) , the two symmetric operators K, J are (∂ = ∂x ! ! 0 −1 ∂ − 2u∂ −1 v 2u∂ −1 u , J =2 . (8.30) K= −2v∂ −1 v 1 0 ∂ − 2v∂ −1 u A representative equation (j = 2) of Eq. (8.29) is 1 1 ut = − uxx + u2 v, vt = vxx − v 2 u, t = t2 . 2 2

(8.31)

The independent N -involutive system of the constrained AKNS flow is expressed by Eq. (5.41). Similarly, we consider the following Hamiltonian functions FsAKN S =

N X

λsj EjAKN S

j=1

= 2hΛs p, qi −

X

(hΛj p, pihΛk q, qi − hΛj p, qihΛk q, pi) .

(8.32)

j+k=s−1

Let (p(x, ts ), q(x, ts ))T be the involutive solution of the consistent Hamiltonian canonical equations (HAKN S ), (FsAKN S ). Then, we have the following theorem. Theorem 8.3. u = −hq(x, tj ), q(x, tj )i, v = hp(x, tj ), p(x, tj )i, j = 0, 1, 2, . . . , satisfy the higher-order AKNS Equation (8.29). Particularly, Eq. (8.31) is solved with the following solution: u = −hq(x, t2 ), q(x, t2 )i, v = hp(x, t2 ), p(x, t2 )i ,

(8.33)

where (p(x, t2 ), q(x, t2 ))T is the involutive solution of the consistent Hamiltonian systems (HAKN S ), (F2AKN S ).

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In the following procedure we shall express Eq. (8.33) in a form of algebrogeometric solution. To do so, let us rewrite Eq. (5.35) as follows: ! AAKN S (λ) BAKN S (λ) AKN S (8.34) = L CAKN S (λ) −AAKN S (λ) where AAKN S (λ) = 1 +

N X j=1

N X

BAKN S (λ) = −

j=1 N X

CAKN S (λ) =

j=1

1 pj qj , λ − λj

(8.35)

1 q2 , λ − λj j

(8.36)

1 p2 . λ − λj j

(8.37)

Note that BAKN S (λ), CAKN S (λ) can be changed to the following fractional form: BAKN S (λ) ≡ −

hq, qiQB (λ) , K(λ)

CAKN S (λ) ≡

hp, piQC (λ) , K(λ)

(8.38)

where hq, qiQB (λ) =

N X

qj2

j=1

hp, piQC (λ) =

N X j=1

K(λ) =

N Y

N Y

(λ − λk ) ,

k=1,k6=j

p2j

N Y

(λ − λk ) ,

k=1,k6=j

(λ − λj ) .

j=1 B Respectively choosing N − 1 (N > 1) distinct real zero points µB 1 , . . . , µN −1 and C µC 1 , . . . , µN −1 of QB (λ) and QC (λ) leads to N −1 X

µB k = A1 −

k=1

A1 −

N −1 X

hΛq, qi , hq, qi

!2 −

µB k

k=1

A1 −

N −1 X k=1

N −1 X k=1

µC k = A1 −

hΛp, pi , hp, pi

N −1 X

2 2 (µB k ) = 2A2 − A1 + 2

hΛ2 q, qi , hq, qi

(8.40)

2 2 (µC k ) = 2A2 − A1 + 2

hΛ2 p, pi , hp, pi

(8.41)

k=1

!2 µC k

−

N −1 X k=1

(8.39)

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PN PN where A1 = j=1 λj , A2 = k,j=1,j 0, denote with Cn = (xn , xn+1 ], n ∈ Z. Suppose that the potential V is real and satisfies, hV : (i) (ii)

V ∈ C2 (Cn ) if n > N and V ∈ C0 (Cn ) for |n| < N . 0 00 , sup V∞,n < +∞ , sup V∞,n , sup V∞,n

n≥N

n≥N

n≥N

0 = supx∈Cn |Vn0 |, here Vn is the restriction of V on Cn , V∞,n = supx∈Cn |Vn |, V∞,n 00 00 c and V∞,n = supx∈Cn |Vn |. Denoting with A , the complement of the set A ⊂ R, then we have

Theorem 1.1. There exists F ⊂ (−∞, 0) with a full Lebesgue measure (|F c | = 0) such that for all F ∈ F, there exists a full Lebesgue measure S ⊂ R, such that H(F ) is purely absolutely continuous on S. Moreover σac (H(F )) = R. These results cover in particular the situation where the potential V consists in some bounded sequence of square well potentials. It is also related to the one obtained in [14], where by using some probabilistic method, it is shown that for the Gaussian white noise, the spectrum is singular for all F < 0 with a transition. In the case of regular stark operators, the stability of the a-c spectrum is usually obtained, by applying the Mourre theory ([16, 7, 15]) or the spectral deformation method as in [4]. An alternative proof, using the subordinacy criteria of Gilbert and Pearson is given in [5]. In our case, we use a method introduced in [17] which allows some estimates to be derived on the resolvent for a general class of Schr¨ odinger operators. This method which is described in Sec. 2 only involves existence and smoothness assumptions on the solutions of the associated differential equation. In Sec. 3, we then prove Theorem 1.1. Our proof can be easily adapted to cover the case of slowly growing potentials V , but in order to avoid some technicalities, here we have supposed V bounded. We discuss these different extensions of Theorem 1.1 in Sec. 4. 2. Resolvent Estimates and Stability of the Absolutely Continuous Spectrum The goal of this section is to get some general estimates on the resolvent of the operators

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Absolutely Continuous Spectrum for Singular Stark Hamiltonians

H(F ) = −

d2 + V + Fx, dx2

F 0, I = (x0 , +∞) and denote with χ a smooth R ε x (1−χ)φ2 dt and for function of I, define for ε > 0 the weight functions K = e x0 −α α > 0, hxiα = (1 + |x|) . We have the following estimate, Theorem 2.1. Let E ∈ S and α > 1/2. There exits a finite constant C(E, F ) such that for all u, v ∈ L2 (R) and kφuk, kφvk < ∞ we have sup |(u, hxiα Kε (H(F ) − E − iε)−1 Kε hxiα v)| ≤ C(E, F )kφuk kφvk .

(2.3)

1>ε>0

Remark 2.1. (i) In the free case i.e. V = 0 then S = R, since for all energy E, |ϕ1 |, |ϕ2 | = O((F x)−1/4 ) and |ϕ01 |, |ϕ02 | = O((F x)1/4 ) near +∞ and |ϕ1 |, |ϕ2 | = O((F x)−1/4 e 3F (F x−E) 2

3/2

) and |ϕ01 |, |ϕ02 | = O((F x)1/4 e 3F (F x−E) 2

3/2

)

near −∞. Moreover, by using the Liouville–Green approximation of solutions for second order differential equations [18], we see that this behavior holds for all E ∈ R if V ∈ Cb2 so that S = R in this case too. We will show in the next section that this property is also true for some bounded discontinuous potentials V . (ii) By our assumptions h1, h2 and the estimate (2.3) we get for all E ∈ S and α > 1/2, sup kgα Kε (H(F ) − E − iε)−1 Kε gα k ≤ C(E, F ) ,

(2.4)

1>ε>0

for some function gα satisfying gα (x) = O(hxiα ) near +∞ and gα (x) = O(hxiα /φ) near −∞. Proof of Theorem 2.1. Choose an real energy E ∈ S and consider for ε > 0 the following family operators, H(ε, F ) = H(F ) − 2ε(φ2 )0 + ε2 φ4

(2.5)

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on L2 (R), the function φ is defined as above. This family coincides with H(F ) in the limit ε = 0, we will precise below the sense of this limit. Due to h1 and h2 and the Faris–Lavine theorem [20], {H(ε, F ), ε > 0} is a family of essentially selfadjoint operators on C∞ 0 (R). The main property of such a −i(ϕ01 +iϕ02 ) (φ2 )0 1 family is an consequence of a well known fact [21]. Let f = ϕ1 +iϕ = 2 − i φ2 , φ 2 then f satisfies the Riccati equation if 0 − f 2 = V − E a.e. on R. From this, we get the following decomposition valid in operator sense on C∞ 0 (R), H(ε, F ) − E − 2iε = (p − f − iεφ2 )(p + f + iεφ2 ) ,

(2.6)

d . i dx

As we will show below the identity (2.6) where p denotes the momentum, p = involves only invertible operators. By a straightforward calculus we get (i)

A+ (ε) = (p + f + iεφ2 ) = φa+ (ε)φ−1 , a+ (ε) = p + φ−2 + iεφ2

(ii)

A− (ε) = (p − f − iεφ2 ) = φ−1 a− (ε)φ, a− (ε) = p − φ−2 − iεφ2 ,

(2.7)

on C∞ 0 (R). Some standard arguments (see e.g. [19, 20]) show that the families of first order differential operators, {A+ (ε), A− (ε), ε > 0} and {a+ (ε), a− (ε), ε > 0} are families of closable operators on L2 (R), we denote by the same symbol their closed extension. From (2.7), for ε > 0 we compute explicitly an inverse for a+ (ε) and a− (ε) on C∞ 0 (R), Z x −θ(x) (ε)u)(x) = e e+θ(t) u(t)dt , (i) (a−1 + θ(x) (ii) (a−1 − (ε)u)(x) = e

where θ(x) = ε we have

Rx x0

Z

−∞

−∞

e−θ(t) u(t)dt ,

(2.8)

x

φ2 dt − i

Rx x0

φ−2 (t)dt. Since Im{a+ (ε)} = −Im{a− (ε)} = εφ2 ,

−1 2 Im{φa−1 + (ε)φ} = ε|φa+ (ε)φ|

(2.9)

C∞ 0 (R).

By the Cauchy–Schwartz inequality, this implies that in the form sense on −1 −1 2 −1 . φa+ (ε)φ is a bounded operator on C∞ 0 (R), then on L (R) and kφa+ (ε)φk ≤ ε −1 The same holds for the operator φa− (ε)φ. Now a direct calculation show that in the form sense on C∞ 0 (R), for ε > 0, −1 (2.10) (H(ε, F ) − E − 2iε)−1 = φa−1 + (ε)φk − kφa− (ε)φ , R R x x where k(x) = ie−2θ(x) −∞ e2θ(t) φ−2 (t)dt = −1/2(1 − 2εe−2θ(x) −∞ φ2 e2θ(t) dt) and verifies supx |k(x)| < 1. Since (2.10) involves only bounded operators, this identity is valid in bounded operators sense on L2 (R). By the same arguments we also have,

φ2 A+ (ε)(H(ε, F ) − E − 2iε)−1 = φa−1 − (ε)φ

(2.11)

in bounded operators sense on L2 (R). The formulas (2.10) and (2.11) are consequences of the factorization (2.6) and will give an uniform estimate on the resolvent

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of the approximate operator H(ε, F ) and then on the resolvent of H(F ). In particular, it is proven in [17], Lemma 2.1. Let E ∈ S and α > 1/2. Then for all u, v ∈ L2 (R) we have −1 kuk kvk . sup |(u, hxiα a−1 ± (ε)hxiα v)| ≤ α

(2.12)

ε>0

Moreover, then for all u, v ∈ L2 (R) and kφuk, kφvk < ∞, (2.10) and (2.12) imply sup |(uhxiα (H(ε, F ) − E − 2iε)−1 hxiα v)| ≤ 2α−1 kφuk kφvk .

(2.13)

ε>0

From (2.13), we now derive an estimate on the resolvent of H(F ) by using the perturbation theory between the operators, Kε (H(ε, F ) − E − 2iε)−1 Kε and Kε (H(F ) − E − 2iε)−1 Kε . We have in the form sense on C∞ 0 (R), for all ε > 0, Kε (H(ε, F ) − E − 2iε) − (H(F ) − E − 2iε)Kε = εWε − 2iεKε (χ − 1)φ2 A+ (ε)

(2.14)

where, Wε = Kε (−χ0 φ2 −2χ(φ2 )0 +2i(1−χ)+εχ2 φ4 ) which under our assumption is a bounded function, notice that this fact is due to the decay property of the weight Kε . Hence by (2.11) and (2.14), for all ε > 0, we have the geometric resolvent equation (H(F ) − E − 2iε)−1 Kε = Kε (H(ε, F ) − E − 2iε)−1 + (H(F ) − E − 2iε)−1 εWε (H(ε, F ) − E − 2iε)−1 + (H(F ) − E − 2iε)−1 2iε(χ − 1)Kε φa−1 − (ε)φ ,

(2.15)

which is valid in the bounded operator sense on L2 (R) and by the Cauchy Schwartz inequality, this implies that there exists a strictly positive constant c, such that for all u, v ∈ L2 (R), |(u, Kε (H(F ) − E − 2iε)−1 Kε v)| ≤ c(|(K2ε u, (H(F, ε) − E − 2iε)−1 v)| + |(u, Kε (H(F ) − E − 2iε)−1 Kε u)|1/2 (|(v, (H(F, ε) − E − 2iε)−1 v)|1/2 1/2 + |(v, Kε φa−1 )) . − (ε)φKε v)|

(2.16)

Taking u = v = hxiα w, kφwk < ∞, then the quadratic estimate (2.16) together with (2.12) and (2.13) yield to sup |(whxiα Kε (H(F ) − E − 2iε)−1 Kε hxiα w)| ≤ ckφwk2

(2.17)

1>ε>0

for an another strictly positive constant c. Then the theorem follows from (2.16) and (2.17) in the same way as above. We can prove now the main result of this section, Theorem 2.2. Suppose that S is a borel set with |S| > 0. Then the operator H(F ) is purely absolutely continuous on S. Let Sˆ = σac (H(F )) ∩ S, then |Sˆc | = 0.

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Proof. Theorem 2.2 follows from the Theorem 2.1 by a standard way ([2, 3, 21, 17]) once we have proved that for E ∈ S there exists some vectors u ∈ L2 (R) for which lim Im(u, (H(F ) − E − 2iε)−1 u) 6= 0 ,

(2.18)

ε→0

i.e. H(F ) has spectrum in a neighbourhood of E. To prove this fact, we do not use some results of e.g. [20], here we proceed as follow. Firstly we show that this is true for H(ε, F ) by using a second representation of the resolvent, valid in the bounded operators sense on L2 (R), let ε > 0, we have, (H(ε, F ) − E − 2iε)−1 −1 −1 2 −1 = 1/2(φa−1 − (ε)φ − φa+ (ε)φ − 2iεφa+ (ε)φ a− (ε)φ) .

(2.19)

This last formula which is more convenient here than (2.10), is derived from the factorization (2.6). In particular suppose that there exists a vector u ∈ L2 (R) with compact support such that lim Im(u, φa−1 − (ε)φu) = 0 and

lim Im(u, φa−1 + (ε)φu) 6= 0 .

ε→0

ε→0

(2.20)

Then (2.19) and (2.20) together with the following estimate: −1 2 −1 −1 Im(u, φa−1 ε|(u, φa−1 + (ε)φ a− (ε)φu)| ≤ η Im(u, φa+ (ε)φu) + η − (ε)φu)

(2.21)

for all η > 0, proves (2.18) for H(ε, F ). Now (2.20) is implied by the conditions Z +∞ R i x φ−2 (t)dt (i) e x0 φ · u(x)dx = 0 and −∞

Z

+∞

e

(ii)

−i

Rx x0

φ−2 (t)dt

−∞

φ · u(x)dx 6= 0 ,

(2.22)

since in particular, Im(u, φa−1 − (ε)φu) Z

+∞

2 2ε

φ e

=ε

Rx x0

φ2 (t)dt

−∞

Z

x

e

i

Rt x0

φ−2 (s)ds−ε

−∞

Rt x0

φ2 (s)ds

2 ! φ · udt dx

and if u has a compact support and verifies (2.22(i)), then the Lebesgue theorem gives, limε→0 Im(u, φa−1 − (ε)φu) = 0. Clearly (2.22(ii)) implies lim Im(u, φa−1 + (ε)φu) 6= 0.

ε→0

˜ of {x ∈ Rx > x0 } and denote with χ e the characteristic Choose a compact subset, K R t −2 ˜ function of K. Then since x0 φ (t)dt is a strictly positive function for x > x0 , i

Rx

φ−2 (t)dt

−i

Rx

φ−2 (t)dt

x0 χe ˜ x0 φ and χe ˜ φ are two independent L2 -vectors. From this last remark, it is easy to show that there exists a vector u ˜, supp(˜ u) ⊂ supp(χ) ˜ satisfying (2.22) and then (2.20). By using the same arguments we also have,

u, φKε a−1 u) = 0 and lim Im(˜ − (ε)Kε φ˜

ε→0

lim Im(˜ u, φKε a−1 u) 6= 0 , + (ε)Kε φ˜

ε→0

(2.23)

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since Kε is uniformly bounded with respect to ε. From the above analysis we now prove (2.18) for H(F ), by the geometric resolvent equation: Kε (H(ε, F ) − E − 2iε)−1 Kε = (H(F ) − E − 2iε)−1 K2ε − (H(F ) − E − 2iε)−1 εWε (H(ε, F ) − E − 2iε)−1 Kε + (H(F ) − E − 2iε)−1 2iε(χ − 1)φa−1 − (ε)φKε ,

(2.24)

which is valid in the bounded operator sense on L2 (R) for all ε > 0, so by the Cauchy Schwartz inequality, |Im(u, Kε (H(ε, F ) − E − 2iε)−1 Kε u)| ≤ const.(|Im(u, (H(F ) − E − 2iε)−1 K2ε u)| + |Im(u, (H(F ) − E − 2iε)−1 u)| + |Im(u, Kε φa−1 − (ε)φKε u)|)

(2.25)

for a const. > 0 and all u ∈ L2 (R). Taking now u˜ with compact support satisfying (2.20) and (2.23), then the quadratic estimate (2.25) gives (2.18) for H(F ). Notice that we also use for any u ∈ L2 (R), (see Remark 2.1(ii)) lim |Im(u, (H(F ) − E − 2iε)−1 u)|

ε→0

= lim |Im(u, (H(F ) − E − 2iε)−1 K2ε u)| . ε→0

(2.26)

3. Application to Some Bounded and Singular Perturbations In this section, we prove Theorem 1.1, so consider the Stark operator on L2 (R), d2 + V + Fx, F < 0, (3.1) dx2 where the potential V is defined as in Sec. 1. We denote the jump of the potential − + − V at point xn by δn i.e. V (x+ n ) − V (xn ) = δn , xn = xn + i0, xn = xn − i0. Under our assumptions, {δn ; n ∈ N } is bounded sequence of real numbers. H(F ) = −

Proof of Theorem 1.1. We will show that there exists a set F ⊂ (−∞, 0) of full measure such that for all F ∈ F , the set S defined in Sec. 2 is of full measure. This is done by applying the Theorem 2.1 of the last section, so we have to know the asymptotic behavior of the solutions of the Shr¨ odinger equation, H(F )ϕ = Eϕ ,

E ∈ R, F < 0 .

(3.2)

Notice that here we will use some elements of the standard theory of o.d.e. which can be easily extended to such a differential equation. Let F < 0 and E ∈ R, we

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first study the solutions of (3.2) on the region Ω+ = {x, E − V − F x > 0} which due to our assumption on V , contains a neighbourhood of +∞. Let xn0 ∈ Ω+ for some integer n0 > 0 large enough, then for n ≥ n0 , the standard Liouville–Green approximation [18] gives a fundamental set of C2 -solutions of (3.2) on each cell Cn , ψn+ (x) = fn−1/4 (x)eiγn (x) (1 + εn (x)) ; ψn− (x) = fn−1/4 (x)e−iγn (x) (1 + ε¯n (x))

Rx

(3.3)

1/2

where γn (x) = xn fn (t)dt, fn = E − Vn − F x and the error function εn is defined as in [18], it satisfies |εn (x)|

and |fn1/2 ε0n (x)| Z

≤ eνn (x) − 1 ;

|Vn00 |

x

νn (x) =

3/2

+

4fn

xn

5|Vn0 + F | 5/2

! dt .

(3.4)

16fn

Denote with γn = γn (xn+1 ) and εn = εn (xn+1 ). Notice that for all n ≥ n0 the wronskian of (ψn+ , ψn− ), W (ψn+ , ψn− ) = 2i and since xn ≥ cn1−µ for some constant c > 0, then for n0 large enough, |εn (x)|

and |fn1/2 ε0n (x)| = O(nµ/2−3/2 ) .

(3.5)

From this, we want to show that there exists solutions of (3.2), ϕ1 , ϕ2 ∈ AC (R) such that (ϕ1 (xn0 ), ϕ01 (xn0 )) = (1, 0) and (ϕ2 (xn0 ), ϕ02 (xn0 )) = (0, 1) which satisfy h1 for some F ∈ (−∞, 0) and E ∈ R. We prove this fact for ϕ1 , by the same arguments this will follow for the solution ϕ2 . For all n ≥ n0 and x ∈ Cn , by (3.3) such a solution is expressed as, 1

ϕ1 (x) = αn ψn+ (x) + βn ψn− (x) ,

(3.6)

and has to satisfy the continuity conditions at x = xn , + − − + − − αn ψn+ (x+ n ) + βn ψn (xn ) = αn−1 ψn−1 (xn ) + βn−1 ψn−1 (xn ) , 0

0

0

0

+ − − + − − αn ψn+ (x+ n ) + βn ψn (xn ) = αn−1 ψn−1 (xn ) + βn−1 ψn−1 (xn ) .

(3.7)

This determines αn , βn in terms of αn−1 , βn−1 and then by induction ϕ1 . The initial ¯n and then writing αn−1 = |αn−1 |eiθn−1 , we conditions and (3.6) imply that βn = α get from (3.7) for n ≥ n0 and n0 large enough, αn = |αn−1 |((1 + εn )(1/2(Xn + Xn−1 ) + 0(nµ/2−3/2 ))ei(θn−1 +γn−1 ) + (1 + ε¯n )(1/2(Xn − Xn−1 ) + 0(nµ/2−3/2 ))e−i(θn−1 +γn−1 ) ) where Xn =

− 1/4 (fn (x+ n )/fn−1 (xn ))

(3.8)

and then

¯n e−2i(θn−1 +γn−1 ) ) |αn |2 = |αn−1 |2 (An + Bn e2i(θn−1 +γn−1 ) + B

(3.9)

with An = |1 + εn |2 (1/2(Xn2 + Xn−2 ) + 0(nµ/2−3/2 )) = 1 + 0(nµ/2−3/2 ) , Bn = (1 + εn )2 (1/4(Xn2 − Xn−2 ) + 0(nµ/2−3/2 )) = 0(δn nµ−1 ) .

(3.10)

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Formula (3.9) gives us an inductive procedure to determine all of the αn ’s for large n. In particular this implies, |αn | = |αn−1 |(1 + 0(nµ−1 )) ,

(3.11)

which clearly is not sufficient here but useful for the sequel. To show αn are uniformly bounded for large n, we note that by (3.8) together with (3.11), cos(θn ) = (1 + 0(nµ−1 )) cos(θn−1 + γn−1 ) , sin(θn ) = (1 + 0(nµ−1 )) sin(θn−1 + γn−1 ) .

(3.12)

Setting now θn = θn−1 + γn−1 + en , we get from (3.12), |sin(en /2) sin(θn + γn + en /2)| , |sin(en /2) cos(θn + γn + en /2)| = 0(nµ−1 ) .

(3.13)

For n large enough, this leads to |sin(en /2)| = 0(nµ−1 ), so θn = θn−1 + γn−1 + en

mod 2π

(3.14)

where the error term en satisfies, en = 0(nµ−1 ). P Pn Let Γn = n−1 n0 γk and En = n0 ek , from (3.9), (3.10) and (3.14), we have for n0 large enough and n > n0 , n Y Ck δk 2 2 µ/2−3/2 1 + 1−µ cos(Γk + Ek ) + O(k |αn | = |αn0 | ) (3.15) k k>n0

where Ck , k ∈ N is some bounded sequence of real numbers, the phase Γk = O(k 3/2(1−µ) ) and the error Ek = O(k µ ); 0 < µ < 1/4 (Ek = O(log k) if µ = 0). This shows that the boundedness of the αn ’s is implied by the boundedness of the sequence, S(n, E, F ) =

n X Ck δk cos(Γk + Ek ) , k 1−µ 0

(3.16)

k>n0

which is given by the Lemma 2.1 below, for almost every (E, F ) ∈ R × (−∞, 0). Hence for such a (E, F ), by using (3.3), (3.6) and the boundedness of αn for large n, ϕ1 , ϕ2 is a fundamental set of solution of (3.2) satisfying h1. We want to prove that for E ∈ R, (ϕ1 , ϕ2 ) satisfy h2 . To do this, we will show that there exists at least a solution of ϕ1 , ϕ2 which grows exponentially in some ˆ 0 the Liouville–Green approximation neighbourhood of −∞. Let n ˆ 0 < −N , for n < n gives a fundamental set of C2 -solutions of (3.2) on Cn , ψˆn+ (x) = fn−1/4 (x)eγn (x) (1 + ε+ n (x)) ; ψˆn− (x) = fn−1/4 (x)e−γn (x) (1 + ε− n (x))

(3.17)

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where now γn (x) = satisfies

R xn+1 x

1/2

fn (t)dt, fn (x) = Vn + F x − E and the error function ±

|ε± n (x)|

0 νn (x) and |fn1/2 (ε± −1; n ) (x)| ≤ e ! Z xn+1 5|Vn0 + F | |Vn00 | + νn (x) = dt ; + 3/2 5/2 x 4fn 16fn ! Z x 00 0 5|V | + F | |V n n dt . + νn− (x) = 3/2 5/2 xn 4fn 16fn

(3.18)

− As above let γn = γn (xn+1 ), ε+ n = εn (xn ); εn = εn (xn+1 ), on the other hand the n0 | is chosen large enough, wronskian, W (ψˆn+ , ψˆn− ) = 2 and if |ˆ

|ε± n (x)| ;

0 µ/2−3/2 |fn1/2 (ε± ) and γn = O(|n|1/2−3µ/2 ) . n ) (x)| = O(|n|

(3.19)

ˆ = Then for all n < n ˆ 0 and x ∈ Cn , all solutions of (3.2) are of the form, ϕ(x) + − ˆ ˆ αn ψn (x) + βn ψn (x) and the continuity conditions at xn implies αn−1 =

(Xn + Xn−1 ) (Xn−1 − Xn ) (1 + O(|n|−ν ))αn eγn + (1 + O(|n|−ν ))βn e−γn , 2 2

βn−1 =

(Xn−1 − Xn ) (Xn + Xn−1 ) (1 + O(|n|−ν ))αn eγn + (1 + O(|n|−ν ))βn e−γn , 2 2 (3.20)

where Xn are defined as above and ν = 3/2 − µ/2. Let ! n ˆ0 X 0 and βn = βn0 exp γn αn = αn exp k=n

n ˆ0 X

! γn

,

k=n

0 by (3.20), (α0n−1 , βn−1 ) = T (α0n , βn0 ) where T is a 2 × 2 matrix satisfying |T | = ˆ 0 }, {βn0 , n < n ˆ 0 } are 1 + O(|n|2(µ−1) ), then since by assumption µ < 1/4, {α0n , n < n two uniformly bounded sequences. Moreover (3.20) gives for n < n ˆ0, 0 0 |2 = |α0n−1 |2 (1 + O(|n|−ν )) − |βn−1 |2 (1 + O(|n|−ν ))e−2γn . |α0n−1 |2 − |βn−1

(3.21)

ˆ Suppose that |αnˆ 0 | > |βnˆ 0 | via the choice of the initial conditions of an solution ϕ, ˆ 0 , |αn | > |βn | and |α0n−1 |2 > |α0n |2 ((1 + then (3.21) implies for large |ˆ n0 | and n < n ˆ 0 , |α0n | ≥ const. > 0, hence O(|n|−ν )). This implies in particular that for all n < n for x ∈ Cn such a solution satisfies, 2

|ϕ| ˆ ∼ |αn |/(F x)1/4 e 3F (F x−E)

3/2

and |ϕˆ0 | ∼ |αn |(F x)1/4 e 3F (F x−E) 2

3/2

.

(3.22)

By using some standard arguments of the o.d.e. theory, one solution (at least) of the fundamental system ϕ1 , ϕ2 has the same behavior as the one given in (3.22), then it is easy to verify that (ϕ1 , ϕ2 ) satisfy h2 for all E ∈ R and F < 0. Hence by applying some usuals arguments (see e.g. [2]), the theorem is proven. Lemma 3.1. For almost every (E, F ) ∈ R × (−∞, 0), there exits a finite constant C(E, F ) and |S(n, E, F )| ≤ C(E, F ).

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Proof. Let n0 > 0 large enough and for n ≥ n0 , γn0 = Consider the sum, n n X X Ck δk i(Γk +Ek ) e = S(n, E, F ) = k 1−µ

where Γ0k =

Pk

k>n0

j>n0

R xn+1 xn

ei(Γk +Ek −Γk ) k 1−µ 0

k>n0

597

(E−Vn ) (−F t)1/2 + (−F dt. t)1/2

! 0

Ck δk eiΓk ,

(3.23)

γj0 . We have

|(Γk + Ek − Γ0k ) − (Γk−1 + Ek−1 − Γ0k−1 )| = |(γk − γk0 ) + ek | = O(k µ−1 ) for k large enough. The Abel’s transformation gives 0 n X sn ei(Γk−1 +Ek−1 −Γk−1 ) ˜ S(n, E, F ) = 1−µ + sk , Ck n k 2−2µ 0

(3.24)

k>n0

Pk 0 where sk = l>n0 Cl δl eiΓl and C˜k , k ∈ N is an another bounded sequence of real 0 numbers. To study the sum S, we use some arguments of the probability theory [22]. Hence let I = (I1 , I2 ], I1 6= I2 an interval of (0, +∞), we denote by ERI the expectaI tion value with respect to the parameter a = (−F )1/2 , EI = |I|−1 I12 · da. Since, for any p ≥ n0 , the phase Γ0p = O(p3/2(1−µ) and 3/2(1 − µ) > 1, a straightforward computation shows that, EI (|sp |2 ) ≤ const. p

(3.25)

for some const. > 0. n Then let S1 (n, E, F ) = ns1−µ , we have from (3.25), EI (|S1 |2 ) ≤ const. n−1+2µ and by using some arguments of [22] (Chp. I.4), this implies S1 (nξ , E, F ) → 0 as n → +∞ a.e. a ∈ I, for any 1/(2µ) > ξ > 1/(1 − 2µ). Moreover, there exists a positive integer m such that mξ ≤ n ≤ (m + 1)ξ , n > n0 , then since S1 (n, E, F ) ≤ 1 S1 (mξ , E, F ) + n2µ− ξ , S1 (n, E, F ) → 0 as n → +∞ a.e. a ∈ I and then a.e. a ∈ (0, +∞). i(Γk−1 +Ek−1 −Γ0 ) P k−1 sk , by using the Cauchy– Let now, S2 (n, E, F ) = n 0 C˜k e 2−2µ

k>n0

k

Schwartz inequality we get for all η > 0 and small, |S2 (n, E, F )|2 ≤ const.

n X k>n00

|sk |2 3−4µ−2η k

.

P for some const. > 0. Considering now the sum k Uk (F ) with Uk (F ) = the estimate (3.25) gives,   n n X X Uk  = EI (Uk ) ≤ const. EI  k≥1

k≥1

(3.26) |sk |2 k3−4µ−2η

,

(3.27)

P for some uniform const. > 0, this proves that j EI (Uk ) < ∞ and then EI P P ( j Uk ) < ∞. It follows that k Uk (F ) < +∞ a.e. on I so S2 (n, E, F ) is uniformly bounded a.e. on I and then a.e. on (0, +∞).

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Hence there exists a subset A of (0, +∞) with a full measure such that if a ∈ A, S(n, E, F ) = S1 (n, E, F )+S2 (n, E, F ) is uniformly bounded, by usual arguments the lemma follows. 4. Concluding Remarks In this section we discuss some extensions of Theorem 1.1. Suppose that the potential V satisfies hV (i) and for large n, V∞,n = 0(nξ ), 0 < ξ < 1/4 − µ and X

0 V∞,n

n3/2−µ/2

< ∞,

X

00 V∞,n

n3/2−µ/2

< ∞.

(4.1)

Then by using some slight modifications of the method used in Sec. 2, we can see that the solutions of the differential equation satisfy h1 and h2 for all real energies and a.e. F ∈ (−∞, 0), so that Theorem 1.1 holds in this case. Notice also that, if we consider the case where δn = 0; n ∈ Z i.e. V is bounded and differentiable except at points {xn }, then the conclusions of Theorem 1.1 also holds for any strictly monotone sequence {xn } satisfying |xn − xn−1 | ≥ c|n|−µ with µ ≤ 1 for large n. Moreover, in this case the operator H(F ) is purely absolutely continuous on R for all F < 0. References [1] P. Ao, “Absence of localization in energy space of a Bloch electron driven by a constant electric force”, Phys. Rev. B41 (1989) 3998–4001. [2] L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators, Springer Verlag Berlin, 1992. [3] H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon, Schr¨ odinger Operators, Springer Verlag, 1987. [4] J. Asch and P. Briet, “Lower bounds on the width of Stark–Wannier types resonances”, Comm. Math. Phys. 179 (1996) 725–735. [5] A. Kiselev, “Absolutely continuous spectrum of perturbed Stark operator”, Trans. Amer. Math. Soc. 352(1) (2000) 243–256. [6] S. N. Naboko and A. B. Pushnitskii, “Point spectrum on a continuous spectrum for weakly perturbed Stark type operators”, Fonc. Anal. Appl. 29(4) (1995) 248–257. [7] J. Sahbani, “Propagation theorems for some classes of pseudo-differential operators”, J. Math. Anal. Appl. 211 (1997) 481–497. [8] F. Bentosela and P. Briet, “Stark resonances for random potential of Anderson type”, Ann. Inst. Henri Poincaré 71(5) (1999) 497–538. [9] V. Grecchi, M. Maioli and A. Sacchetti, “Lifetimes of the Wannier–Stark resonances and perturbation theory”, Comm. Math. Phys. 185 (1997) 359–378. [10] F. Delyon, B. Souillard and B. Simon, “From pure point to continuous spectrum in disordered systems”, Ann. Inst. Henri Poincaré 42(3) (1985) 283–309. [11] M. Maioli and A. Sacchetti, “Absence of the absolutely continuous spectrum for Stark–Bloch operators with strongly singular periodic potentials”, J. Phys. A: Math. Gene. 28 (1995) 1101–1106. [12] J. E. Avron, P. Exner and Y. Last, “Periodic Schr¨ odinger operators with large gaps and Wannier–Stark ladders”, Jour. Math. Phys. 36(9) (1995) 4561–4570.

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[13] J. Asch, P. Exner and P. Duclos, “Stability of driven systems with growing gaps, quantum rings and Wannier ladders”, Jour. Stat. Phys. 92 (1998) 1053–1069. [14] N. Minami, “Random Schr¨ odinger operators with a constant electric field”, Ann. Inst. Henri Poincaré 56(3) (1992) 307–344. [15] F. Bentosela, R. Carmona, P. Duclos, B. Simon, B. Souillard and R. Weder, “Schr¨ odinger operators with an electric field and random or deterministic potential”, Comm. Math. Phys. 88 (1983) 333–339. [16] E. Mourre, “Absence of singular spectrum for certain selfadjoint operators”, Comm. Math. Phys. 68 (1981) 391–408. [17] P. Briet and E. Mourre, “Some resolvent estimatesfor Sturm Liouville operators”, J. Math. Anal. Appl. 201 (1996) 867–879. [18] F. W. Olver, Asymptotics and Special Functions, Academic Press, 1974. [19] T. Kato, Perturbation Theory for Linear Operators, Springer, 1980. [20] M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, New-York, 1975. [21] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, New York, 1978. [22] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, A. Wiley Interscience Publication, 1978.

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CORRECTIONS TO “INTERACTING QUANTUM FIELDS”

GLENN ERIC JOHNSON Litton-TASC, 4801 Stonecroft Blvd. Chantilly, VA 20151-3822 E-mail : [email protected]

Received 31 October 2000

The author reports that the two incorrect assertions in [1] are resolved and a nontrivial quantum field theory with the reported scattering amplitudes has been completed [3]. This answers in the affirmative the speculation in [2]. A constraint of nonnegative Euclidean times (τ > 0) in [1] achieves an inner product on quantum field model states but also precludes reaching those boundary values of the Wightman functions for which scattering was established. Removal of the constraint and a strengthened demonstration of positivity is completed in [3]. Nonnegativity of an inner product on quantum field states is established in the physical spacetime domain. This inner product is used to construct the Hilbert space of quantum field states. Construction of self-adjoint field operators acting in the constructed Hilbert space remains unattainable since the algebra of sampling functions for which the Wightman functions provide a nonnegative form is necessarily not a *-algebra. Also, although minimally modified from the conventional form, subtraction of a divergent two-point function requires that the inner product be in a more general form than the conventional multiplicatively positive generalized function. These results do not preclude an algebra of quantum field operators realized in alternative forms and yielding the constructed vacuum expectation values (VEV) when evaluated for the model nonnegative sesquilinear form. The establishment of positivity uses augmentation functions added to the truncated Wightman functions of the QFT model derived from convoluted generalized white noise [4,1]. The augmentation functions “complete the square” in the statement of physical domain positivity. The augmentation functions violate the spectral condition and do not analytically extend back to complex time differences. However, those components of the resulting positive Wightman functions that violate the spectral condition are made arbitrarily small as the scale for a physically trivial component of the model diverges. The scattering and production predictions of the model are invariant to this scale. Alternatively, validity of the Wightman axioms can be considered asymptotic, becoming valid for states in the asymptotic (scattering) regions. 601

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This model provides an explicit set of Wightman functions for which scattering and particle production is established together with the locality, spectrum, Poincaré invariance, cluster decomposition, and positivity conditions. The description of the Wightman functions is in terms of elementary generalized functions, and the model is formulated for any number of spacetime dimensions. The approach suggests technical modifications for QFT descriptions (innovative selections for the class of functions for which the Wightman functions provide an inner product) by explicitly describing a nontrivial realization for one set of modified requirements. The selected functions are denoted sampling functions and are a class of generalized functions with no intersection with conventional test functions. References [1] G. Johnson, “Interacting quantum fields”, Rev. Math. Phys. 11 (1999) 881-928. [2] G. Johnson, “Comments on ‘Interacting quantum fields’ ”, Rev. Math. Phys. 12 (2000) 687-689. [3] G. Johnson, “Interacting quantum fields, II”, unpublished, 2000. [4] S. Albeverio, H. Gottschalk and J.-L. Wu, “Convoluted generalized white noise, Schwinger functions and their analytic continuation to Wightman functions”, Rev. Math. Phys. 8 (1996) 763-817.

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THE STRUCTURE OF SECTORS ASSOCIATED WITH LONGO REHREN INCLUSIONS II. EXAMPLES

MASAKI IZUMI Department of Mathematics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan E-mail : [email protected]

Received 10 July 2000 Dedicated to Sergio Doplicher on the occasion of his sixtieth birthday As an application of the general theory established in the first part, we determine the structure of Longo–Rehren inclusions for several systems of sectors arising from endomorphisms of the Cuntz algebras. The E6 subfactor and the Haagerup subfactor are included among these examples, and the dual principal graphs and the S and T -matrices for their Longo–Rehren inclusions are obtained. We also construct several new subfactors using endomorphisms of the Cuntz algebras, and determine their tube algebra structure.

1. Introduction This is the second part of our analysis initiated in [14] about Longo–Rehren inclusions, or equivalently asymptotic inclusions, which are considered to be a subfactor version of quantum double construction. For the history and significance of the subject, we refer to the introduction of the first part. (See also [3, 5–10, 17, 21, 22, 26, 28, 31] for related topics.) In the previous paper, we have established a general description of Longo–Rehren inclusions in terms of the notion of half braidings. This second part is devoted to applying our general machinery to concrete examples. Such an attempt has been already started also by J. B¨ ockenhauer–D. E. Evans–Y. Kawahigashi [7], where Longo–Rehren subfactors arising from α-induction are discussed. In [12], we constructed several endomorphisms of the Cuntz algebras with finite indices by giving explicit formulae of the images of the canonical generators. These formulae are essentially the same as the table of 6j symbols in good cases; in fact the Turaev–Viro–Ocneanu type three manifold invariant of the E6 subfactor is calculated by K. Suzuki and M. Wakui [29] based on them. In this paper, we explicitly calculate the tube algebras Tube ∆ for these systems ∆ of endomorphisms, which provide a lot of information of Longo–Rehren inclusions for these systems, such as the dual principal graphs and the associated S and T -matrices. More precisely, the general theory of the first part tells that the algebra structure of the Tube ∆ gives the dual principal graph, and that explicit description of a system 603

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of matrix units of Tube ∆ gives S and T -matrices. For example, we give the dual principal graphs and the S and T -matrices for the Longo–Rehren inclusions of the E6 subfactors and the Haagerup subfactors. Note that the principal graphs of the Longo–Rehren inclusion for the E6 case are also obtained in [7] among other interesting examples. This paper is organized as follows. In Sec. 2, we review basic facts obtained in the first part. In Secs. 3 and 4, we give a system of matrix units of Tube ∆ for two examples obtained in [12]; we treat Tambara–Yamagami categories associated with (1) finite abelian groups [30] and even sectors for the D5 subfactors. These examples are relatively easy to deal with, and are instructive for readers to get the basic idea of our method. In Sec. 5, we analyze a system of equations on finite abelian groups G corresponding to some class of endomorphisms of the Cuntz algebras. A part of these equations has been obtained in [12] just to construct subalgebras with finite Watatani indices. We give necessary and sufficient conditions that these endomorphisms obey certain fusion rules, and extend the equations. There are plenty of solutions for the equations, and they give rise to new subfactors and new 6j symbols. Since the simplest case G = Z/2Z corresponds to even sectors of the E6 subfactors, we may regard them as generalized E6 subfactors. In Sec. 6, we investigate the structure of Tube ∆ for these systems of endomorphisms, and consequently obtain the S and T -matrices for them. In Sec. 7, we deduce another system of equations on finite abelian groups G of odd order, which corresponds to even sectors of the Haagerup subfactor in the simplest case G = Z/3Z. Using this and Longo’s Q-system technique [20], we give another proof of existence of Haagerup subfactor shown in [1, 11]. There exists a solution for the equations other than the Haagerup subfactor case, which may be regarded as a generalized Haagerup subfactor. Section 8 is devoted to the calculation of Tube ∆ for the Haagerup subfactor.

2. Preliminaries Throughout this paper, we use the same definitions and notation as in the first part unless we specifically state. In this section, we collect the most frequently used ones among them. Let M be an infinite factor and ∆ = {ρξ }ξ∈∆0 be a finite system of endomorphisms of M satisfying the conditions in [14, Sec. 2]. The tube algebra Tube ∆ of ∆, which is a finite dimensional C∗ -algebra, is defined as a linear space by, Tube ∆ :=

M

(ρξ · ρζ , ρζ · ρη ) .

ξ,η,ζ∈∆

When X ∈ (ρξ · ρζ , ρζ · ρη ) is regarded as an element of Tube ∆, it is denoted by (ξ ζ|X|ζ η). The product and the ∗-operation are defined by

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605

Nν

(ξ ζ|X|ζ η)(ξ 0 ζ 0 |Y |ζ 0 η 0 ) := δη,ξ0

ζ,ζ 0 X X

ν≺ζζ 0

(ξ ν|T (νζ,ζ 0 )∗i ρζ (Y )Xρξ (T (νζ,ζ 0 )i )|ν η 0 ) ,

i=1

(2.1) ¯ ¯(ρξ (R ¯ ∗ )X ∗ )Rζ |ζ¯ ξ) , (ξ ζ|X|ζ η)∗ := d(ζ)(η ζ|ρ ζ ζ

(2.2)

where {T (νζ,ζ 0 )i }i is an orthonormal basis of (ρν , ρζ ρζ 0 ). We denote by Aξ,η the L linear subspace ζ∈∆0 (ρξ ρζ , ρζ ρη ) ⊂ Tube ∆. Then, we have Aξ,η Aξ0 ,η0 ⊂ δξ0 ,η Aξ,η0 ,

A∗ξ,η = Aη,ξ .

(2.3)

Thus, Aξ := Aξ,ξ is a ∗-subalgebra of Tube ∆. We define a faithful linear functional ϕ∆ on Tube ∆ by (2.4) ϕ∆ ((ξ ζ|X|ζ η)) := d(ξ)2 δξ,η δζ,e X , L which is a trace on ξ∈∆ Aξ . The following easy lemma is useful in order to construct systems of matrix units of Tube ∆ from those of Aξ : Lemma 2.1. Let 0 ∈ ∆0 with ρ0 = id. (1) A0 is isomorphic to the fusion algebra of ∆. (2) Let 1ξ := (ξ 0|1|0 ξ). Then, 1ξ is the unit of Aξ . ¯ ξ R∗ |ξ¯ ξ). Then, tξ is a central unitary in Aξ satisfying (3) Let tξ := d(ξ)(ξ ξ|R ξ t∗ξ = (ξ ξ|1|ξ ξ). The T -matrix of ∆ is given by X tξ = T . ξ∈∆0

(4) Every minimal projection in Aξ is minimal in Tube ∆ as well. Proof. (1) and (2) are straightforward. (3) is essentially obtained in [14, Sec. 3]. (2.3) shows Aξ = 1ξ (Tube ∆)1ξ , which proves (4). Let A ⊃ B be the Longo–Rehren subfactor of ∆ and ι the inclusion map of B into A. For a finite direct sum σ of endomorphisms in ∆, a system of unitary operators {Eσ (ξ)}ξ∈∆0 is called a half braiding of σ with respect to ∆ if it satisfies the following: (1) Eσ (ξ) ∈ (σρξ , ρξ σ). (2) For every X ∈ (ρζ , ρξ ρη ), the following holds: XEσ (ζ) = ρξ (Eσ (η))Eσ (ξ)σ(X) . Two half braidings {Eσ (ξ)}ξ∈∆0 and {Eσ0 (ξ)}ξ∈∆0 are equivalent if there exists a unitary u ∈ (σ, σ) such that Eσ (ξ) = ρξ (u)Eσ0 (ξ)u∗

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holds for all ξ ∈ ∆0 . In general, σ may have several inequivalent half braidings. Therefore, we use the notation Eσα = {Eσα (ξ)}ξ∈∆0 where α is a parameter to distinguish different half braidings. σ ˜ α denotes the corresponding endomorphism of B defined in [14, Sec. 4]. Let D(∆) be the set of endomorphisms ρ ∈ End(B)0 such that [ι][ρ] is a finite direct sum of sectors in {[ρξ ⊗ idop ][ι]}ξ∈∆0 . We call D(∆) the quantum double of ∆. Lemma 4.5 and Theorem 4.6 of [14] show that D(∆) is nothing but the set of endomorphisms of the form σ ˜ α . The B − B sectors associated with A ⊃ B, those contained in some power of the restriction of the canonical endomorphism to B, belong to D(∆). When σ ˜ α is irreducible, we can construct a system of matrix units of a simple nξ ⊂ (ρξ , σ) component of Tube ∆ as follows: We fix an orthonormal basis {Wσ (ξ)i }i=1 and set Eσα (ξ)(η,i),(ζ,j) = ρξ (Wσ (ζ)∗j )Eσα (ξ)Wσ (η)i ∈ (ρη · ρξ , ρξ · ρζ ) .

(2.5)

Then, thanks to [14, Theorem 4.10], we can construct a system of matrix units σ α ) by {e(˜ σ α )(η,i),(ζ,j) } and the corresponding minimal central projection z(˜ X d(σ) d(ξ)(η ξ|Eσα (ξ)(η,i),(ζ,j) |ξ ζ) , e(˜ σ α )(η,i),(ζ,j) := p λ d(η)d(ζ) ξ z(˜ σα ) =

X

e(˜ σ α )(η,i),(η,i) .

(2.6)

(2.7)

η,i

As stated in the remark at the end of [14, Sec. 4], the number σα ) dim((ρξ ⊗ idop )ι, ι˜ is given by the rank of z(˜ σ α )Aξ , which is the number of the edges between two σ α ]. vertices of the dual principal graph corresponding to [(ρξ ⊗ idop )ι] and [˜ ∆ ∆ Let S and T be the two matrices defined in [14, Sec. 3]. (To avoid possible ˜α − µ ˜β confusion, we use the notation S ∆ and T ∆ instead of S and T ). Then, σ matrix element of S ∆ is given by d(σ) X d(ξ)φξ (Eµβ (η)∗(ξ,i),(ξ,i) Eσα (ξ)∗(η,j),(η,j) ) . (2.8) Sσ˜∆α ,˜µβ = λ ξ,i

˜ α . Then, ωσ˜ α is given by Let ωσ˜ α be the eigenvalue of T ∆ corresponding to σ ωσ˜ α = d(ξ)φξ (Eσα (ξ)(ξ,i),(ξ,i) ) .

(2.9)

In [12], we showed that all the examples of endomorphisms of the Cuntz algebras On we are going to deal with in this paper can extend to the weak closure M := On00 , which are type III factors, in the GNS representations of some KMS states. Moreover, the extended endomorphisms have the same system of intertwiners as the original ones. Therefore in what follows, though we introduce endomorphisms of the Cuntz algebras, we use the same symbols for their extension to the weak closure.

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Before ending this section, we recall the following easy but useful lemma: Lemma 2.2 ([12, Lemma 2.6]). Let ρ be a unital endomorphism of the Cuntz algebra On with the canonical generators {S1 , S2 , . . . , Sn }. We fix 1 ≤ i ≤ n and set Tj := Si∗ ρ(Sj )Si . If {T1 , T2 , . . . , Tn } satisfy the Cuntz algebra relation, then Sk∗ ρ(x)Si = 0 for k 6= i and all x ∈ On . In consequence, σ(·) := Si∗ ρ(·)Si is a unital endomorphism and Si ∈ (σ, ρ). 3. Tambara Yamagami Categories In this section, we calculate the tube algebras for Tambara–Yamagami categories classified in [30]. Let G be a finite abelian group and h·, ·i : G × G → T be a non-degenerate symmetric pairing of G and itself: hg, hi = hh, gi is a character for each variable and hg, hi = 1 for all h ∈ G implies g = 0. We consider the Cuntz algebra On with the canonical generators {Sg }g∈G , where n is the cardinality of G. In [12, Example 3.7], we constructed the following example of a system of endomorphisms αg , ρ, g ∈ G: αg (Sh ) = Sg+h ,

h ∈ G,

ρ(Sh ) = U (h)

1 X √ Sk n

U (h) =

X

! U (h)∗ ,

h ∈ G,

k∈G

hh, kiSk Sk∗ ,

h ∈ G.

k∈G

α is an action of G on On and {U (g)}g∈G is a unitary representation of G in On . √ ρ has a statistical dimension n. These satisfy, αg · ρ = ρ ,

g ∈ G,

ρ · αg = Ad(U (g)) · ρ , g ∈ G , X Sg αg (x)Sg∗ , x ∈ On , ρ2 (x) =

(3.1) (3.2) (3.3)

g∈G

αg (U (h)) = hg, hiU (h) , g, h ∈ G , X Sh−g Sh∗ , g ∈ G . ρ(U (g)) =

(3.4) (3.5)

h∈G

We set ρg := αg for g ∈ G and ρ+ := ρ, ρ− := θ·ρ, where θ is a gauge automorphism defined by θ(Sg ) = −Sg , g ∈ G. Let ∆+ = {ρg }g∈G ∪ {ρ+ } , ∆− = {ρg }g∈G ∪ {ρ− } . Then, ∆+ and ∆− obey the following fusion rules:

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[ρg ][ρh ] = [ρg+h ] ,

g, h ∈ G ,

[ρg ][ρ± ] = [ρ± ][ρg ] = [ρ± ] , M [ρg ] . [ρ± ][ρ± ] =

g ∈ G,

g∈G

In fact, Tambara and Yamagami classified categories having the above fusion rules under a mild condition [30]. We call them Tambara–Yamagami categories. ∆+ and ∆− with all the possible pairings h·, ·i exhaust all Tambara–Yamagami categories. Now we determine the structure of Tube ∆± . Basic intertwiner spaces are as follows: (ρg+h , ρg ρh ) = C , (ρ± , ρ± ρg ) = CU (g) ,

(ρ± , ρg ρ± ) = C , (ρg , ρ± ρ± ) = CSg .

Using these, we can obtain the linear space structure of Tube ∆± . Let g 6= h ∈ G. Then, Ag,h = C(g ± |U (h)| ± h) , M C(g k|1|k g) ⊕ C(g ± |U (g)| ± g) Ag,g =

(3.6) (3.7)

k∈G

A± =

M

C(± g|U (g)∗ |g ±) ⊕

g∈G

M

C(± ± |Sg Sg∗ | ± ±) ,

(3.8)

g∈G

Ag,± = A±,g = {0} .

(3.9)

Let σ be a finite direct sum of endomorphisms in ∆± . Then, the defining conditions of half braidings for σ are Eσ (h + k) = ρh (Eσ (k))Eσ (h) ,

(3.10)

Eσ (±) = ρh (Eσ (±))Eσ (h) ,

(3.11)

U (h)Eσ (±) = ρ± (Eσ (h))Eσ (±)σ(U (h)) ,

(3.12)

Sh Eσ (h) = ρ± (Eσ (±))Eσ (±)σ(Sh ) .

(3.13)

It is straightforward to show the following technical lemma: Lemma 3.1. For a function f : G → C, we define the Fourier transform fˆ : G → C by 1 X hg, hif (h) , g ∈ G . fˆ(g) = √ n h∈G

Then,

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ˆ (1) fˆ(g) = f (−g), g ∈ G. (2) Let a : G → T be a function satisfying a(g)a(h) = hg, hia(g + h) ,

g, h ∈ G .

(3.14)

Then, |ˆ a(0)| = 1 and a ˆ(g) = a ˆ(0)a(g). (3) If moreover a(g) = a(−g) holds for all g ∈ G, we have a(g)2 = hg, gi. We fix a : G → T satisfying the conditions in (2) and (3) above. Note that such a always exists. For example, let √G be a cyclic group G = Z/nZ and assume that the 2π −1gh by pairing is given by hg, hi = e n . If n is odd, there is only one such a given √ √ π −1 2 g − n g − π n−1 g2 .√If n is even, there are two of such given by a+ (g) = e a(g) = (−1) e g − π n−1 g2 and a− (g) = (−1) e . First, we determine the half braidings of ρg and ρ± . Lemma 3.2. Let g, h ∈ G. (1) We define Egi , i = 0, 1 by Egi (h) := hg, hi ∈ (ρg ρh , ρh ρg ) ,

h ∈ G,

Egi (±) := (−1)i a(g)U (g) ∈ (ρg ρ± , ρ± ρg ) . Then, Egi , i = 0, 1 are half braiding for ρg , g ∈ G with respect to ∆± , and they exhaust all the half braidings for ρg . (g,i) (2) We fix one of the square roots of ±ˆ a(g) and denote it by ωg . We define E± , i = 0, 1 by (h) := hg, hia(h)U (h)∗ ∈ (ρ± ρh , ρh ρ± ) , X (g,i) hg, kia(k)Sk Sk∗ ∈ (ρ2± , ρ2± ) . E± (±) := ±(−1)i ωg (g,i)

E±

k∈G (g,i) E± ,

g ∈ G, i = 0, 1 are half braidings for ρ± with respect to ∆± , and Then, they exhaust all the half braidings for ρ± . Proof. (1) Thanks to (3.7), if Eg is a half braiding for ρg , we have Eg (h) ∈ C and there exists a scalar c(g) ∈ T satisfying Eg (±) = c(g)U (g). On the other hand, when such unitaries Eg (h) and Eg (±) are given, the defining conditions for Eg to be a half braiding are Eg (h + k) = ρh (Eg (k))Eg (h) , Eg (±) = ρh (Eg (±))Eg (h) , U (h)Eg (±) = ρ± (Eg (h))Eg (±)ρg (U (h)) , Sh Eg (±) = ρ± (Eg (±))Eg (±)ρg (Sh ) , where h, k ∈ G. Thanks to (3.4), these are equivalent to Eg (h) = hg, hi and c(g)2 = hg, gi. Using (3) of Lemma 3.1 and (3.5), we get the result.

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(2) Thanks to (3.8), if E± is a half braiding, we have E± (h) ∈ CU (g)∗ and X CSg Sg∗ . E± (±) ∈ g∈G

Thus there exist scalars ξ(g), η(g), ω ∈ T, g ∈ G satisfying η(0) = 1 and E± (g) = ξ(g)U (g)∗ , X ω η(g)Sg Sg∗ . E± (±) = ±¯ g∈G

On the other hand, when such unitaries E± (g), E± (±) are given, the defining conditions for E± to be a half braiding are E± (h + k) = ρh (E± (k))E± (h) ,

(3.15)

E± (±) = ρh (E± (±))E± (h) ,

(3.16)

U (h)E± (±) = ρ± (E± (h))E± (±)ρ± (U (h)) ,

(3.17)

Sh E± (h) = ρ± (E± (±))E± (±)ρ± (Sh ) ,

(3.18)

where h, k ∈ G. (3.15) and (3.16) are equivalent to the following two conditions respectively: ξ(h + k) = hh, kiξ(h)ξ(k) , η(k) = hk, −hiξ(h)η(k − h) ,

h, k ∈ G , k, h ∈ G .

Therefore, (3.15) and (3.16) together are equivalent to the condition that η satisfies (3.14) and ξ(h) = η(−h), h ∈ G. Note that if this is the case, there exists g ∈ G satisfying η(h) = hg, hia(h) and ξ(h) = hg, hia(h) for all h ∈ G. When (3.15) and (3.16) hold, the right-hand side of (3.18) in h = 0 case is ω ¯2 X η(h)η(k)ρ(Sk Sk∗ )Sh ±ρ(E± (±))E± (±)ρ(S0 ) = ± √ n h,k∈G

ω ¯2 X = ±√ η(h)η(k)U (k)ρ(S0 S0∗ )U (k)∗ Sh n h,k∈G

=±

ω ¯2 X η(h + k)U (k)ρ(S0 ) n h,k∈G

√ η (0)S0 S0∗ ρ(S0 ) = ±¯ ω 2 nˆ = ±¯ ω 2 ηˆ(0)S0 . Therefore, this coincides with the left-hand side if and only if a(−g) = ±ˆ a(g) . ω 2 = ±ˆ

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Now we show that if (3.15), (3.16), and h = 0 case of (3.18) hold, so do (3.17) and (3.18), which finishes the proof. Using h = 0 case of (2.18) and (3.16), we get Sh = ρh (S0 ) = ρh (ρ± (E± (±))E± (±)ρ± (S0 )) = ρ± (E± (±))ρh (E± (±))ρ± (S0 ) = ρ± (E± (±))E± (±)E± (h)∗ ρ± (S0 ) = ρ± (E± (±))E± (±)ρ± (Sh )E± (h)∗ , which shows (3.18). Using the definition of U (h) and (3.18), we get ρ± (U (h)) = E± (±)∗ ρ± (E± (±)∗ )U (h)ρ± (E± (±))E± (±) . Iterating this into the right-hand side of (3.17) and using (3.16), we get ρ± (E± (h))E± (±)ρ± (U (h)) = ρ± (E± (h)E± (±)∗ )U (h)ρ± (E± (±))E± (±) = ρ± (ρh (E± (±)∗ ))U (h)ρ± (E± (±))E± (±) = U (h)E± (±) . (g,i)

Remark. (1) The half braiding E± was essentially obtained in [13]. (2) If E is a braiding for ∆± , there exist ig , jg ∈ {0, 1} for g ∈ G and s, t ∈ G, k, l ∈ {0, 1} such that i

E(g, ·) = Egg , (s,k)

E(±, ·) = E±

,

j

E(·, g)∗ = Egg , E(·, ±)∗ = E±

(t,l)

,

which are equivalent to hg, hi = hg, hi , (−1)ig = hs, gi ,

g, h ∈ G ,

(−1)jg = ht, gi g ∈ G ,

ωs ωt = (−1)k+l hs + t, gia(g)2 ,

g ∈ G.

The first condition implies that G is a direct sum of Z/2Z, and there always exist ig and jg satisfying the second condition. In this case, since G 3 g 7→ hg, gi is a homomorphism, there exist a unique element g0 satisfying hg, gi = hg, g0 i for all g ∈ G. It is easy to show that there exists k, l such that the third condition holds if and only if a ˆ(0)2 = a(g0 ) and s + t = g0 . In fact, for G = Z/2Z and G = Z/2Z × Z/2Z, this holds for any pairing h·, ·i. If we further assume that the braiding E is non-degenerate, D(∆± ) is just a direct product of ∆± and its opposite. Theorem 3.5 below shows that this is the case only if G = Z/2Z. Indeed, this case corresponds to the category of sectors for the Ising model [2]. Using (2.6) and (2.7), we get the following mutually orthogonal minimal central projections of Tube ∆± :

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" # X √ 1 z(˜ ρig ) = (g h|Egi (h)|h g) + n(g ± |Egi (±)| ± g) 2n h∈G

(−1)i 1 X hg, hi(g h|1|h g) + √ a(g)(g ± |U (g)| ± g) , = 2n 2 n

(3.19)

h∈G

z(f ρ±

(g,i)

" # √ 1 X (g,i) (g,i) )= (± h|E± (h)|h ±) + n(± ± |E± (±)| ± ±) . (3.20) 2n h∈G

ρ± Since dim A± = 2n and z(f

(g,i)

) ∈ A± , we get

Corollary 3.3. Under the above notation, we have M Cz(f ρ± (g,i) ) . A± = g∈G,i=0,1

Thanks to (3.9), Tube ∆± is a direct sum of two C∗ -subalgebras A1 and A± where M Ag,h . A1 := g,h∈G

Thus, to determine the structure of Tube ∆± , we need to investigate the structure of A1 . Lemma 3.4. We set V g (h) := (g h|1|h g) , X(g, h) := (g ± |U (h)| ± h) ,

g, h ∈ G , g, h ∈ G ,

which form a basis of A1 . Then, we have (1) For fixed g ∈ G, the map G 3 h 7→ V g (h) gives a unitary representation of G in Ag . Let E g (h) :=

1 X hh, kiV g (k) . n k∈G

Then, {E g (h)}h∈G are mutually orthogonal non-zero projections. (2) X(g, h)∗ = hg, hiX(h, g) holds and X(g, h)X(k, l) = δg,l δh,k nhg, hiE g (h) . Proof. (1) The first part follows from a direct application of the definitions of the tube algebra operations to this case.

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¯ ± = ±Se . Using (2.2), we have (2) Note that we can choose as R± = Se , R √ X(g, h)∗ = ± n(h ± |ρ± (ρg (Se∗ )U (h)∗ )Se | ± g) √ = n(h ± |U (g)ρ(Se∗ )U (g)∗ ρ(U (h)∗ )Se | ± g) √ = n(h ± |U (g)ρ(Se∗ )U (g)∗ Sh | ± g) √ = nhg, hi(h ± |U (g)ρ(Se∗ )∗ Sh | ± g) = hg, hi(h ± |U (g)| ± g) = hg, hiX(h, g) . Applying (2.1), we get X(g, h)X(k, l) = δh,k (g ± |U (h)| ± h)(h ± |U (l)| ± l) X (g p|Sp∗ ρ± (U (l))U (h)ρg (Sp )|p l) = δh,k p∈G

= δh,k

X

∗ (g p|Sp+l U (h)Sg+p |p l)

p∈G

= δh,k

X

∗ hp + l, hi(g p|Sp+l Sg+p |p l)

p∈G

= δg,l δh,k hg, hi

X

hh, pi(g p|1|p g)

p∈G

= δg,l δh,k nhg, hiE g (h) . The above lemma shows that if g 6= h ∈ G, ) ( 1 hg, hi g h E (h), √ X(g, h), √ X(h, g), E (g) n n forms a system of 2 by 2 matrix units. We denote by A{g,h} the linear span of these, which is a C∗ -subalgebra of A1 . If {g, h} 6= {k, l}, (2) of Lemma 3.4 shows that A{g,h} and A{k,l} are mutually orthogonal. Moreover, (3.19) implies z(˜ ρig ) =

E g (g) +

(−1)i √ a(g)X(g, g) n

2

,

which shows that A{g,h} is orthogonal to Cz(˜ ρik ) for all k ∈ G. Therefore A{g,h} is a simple component of Tube ∆± . We fix an arbitrary linear order of G. Then, we have M M A{g,h} ⊕ Cz(˜ ρig ) . A1 = g≺h,g6=h

g∈G,i=0,1

For g 6= h ∈ G, we set σg,h := ρg ⊕ ρh . The above shows that σg,h has only one equivalence class of half braidings that gives rise to an irreducible sector in D(∆± ).

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There exists a half braiding Eσg,h corresponding to the above matrix units through (2.6). Summing up these, we conclude Theorem 3.5. Under the above notation, we have (1)

M

Tube ∆± =

M

Cz(˜ ρig ) ⊕

g∈G,i=0,1

Cz(f ρ±

(g,i)

M

)⊕

g∈G,i=0,1

A{g,h} .

g≺h,g6=h

There exists a half braiding Eσg,h for σg,h such that the corresponding system of matrix units of A{g,h} is 1 X g hh, ki(g k|1|k g) , e(σg g,h )g,g = E (h) = n k∈G

1 X hg, ki(h k|1|k h) , n

h e(σg g,h )h,h = E (g) =

k∈G

e(σg g,h )g,h

1 1 = √ X(g, h) = √ (g ± |U (h)| ± h) , n n

hg, hi hg, hi X(h, g) = √ (h ± |U (g)| ± g) . e(σg g,h )h,g = √ n n (2) We have Eσg,h (k)g,g = hh, ki ,

Eσg,h (k)h,h = hg, ki ,

Eσg,h (±)g,h = U (h) ,

Eσg,h (±)h,g = hg, hiU (g) .

The other components of Eσg,h are zero. Using (2.8) and (2.9), we obtain Theorem 3.6. The matrix elements of the S and T -matrices of the Longo–Rehren inclusions of Tambara–Yamagami categories are give as follows: 2

∆ Sρf±i ,f j g ρh

hg, hi , = 2n

∆

∆

= Sρf±i ,ρf (h,j) = Sρf±(h,j) ,f ρ i ±

g

±

∆

g

∆

± = Sρf±i ,σg = Sσg ,f ρ i g

h,k

∆

Sρf±(g,i) ,ρf (h,j) = ±

±

h,k

g

(−1)i hg, hi √ , 2 n

hg, h + ki , n

(−1)i+j ωg ωh X hk − (g + h), ki , 2n k

∆

∆

± = 0, Sρf±(g,i) ,σg = Sσg ,ρf (g,i) ±

h,k

h,k

±

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± Sσg = ,σ^ 0 0 g,h

g ,h

615

1 [hg, h0 ihh, g 0 i + hg, g 0 ihh, h0 i] , n

ωρfg i = hg, gi , a(g))1/2 , ωρf± (g,i) = (−1)i ωg = (−1)i (±ˆ = hg, hi . ωσg g,h ∗ Proof. Note that φρg = ρ−1 g and φρ± is given by S0 ρ± (·)S0 , which is a trace with equal weights on (ρ2± , ρ2± ). Thus, we get

2

∆ Sρf±i ,f j g ρh

∆

Sρf±i ,ρf (h,j) ±

g

1 j ∗ i ∗ hg, hi E (g) Eg (h) = , = 2n h 2n √ (−1)i hg, hi n i ∆± ∗ (h,j) ∗ √ E , = Sρf (h,j) ,f (±) E (g) = i = ± g ρg ± 2n 2 n

∆

∆

± = Sρf±i ,σg = Sσg ,f ρ i g

h,k

h,k

∆

Sρf±(g,i) ,ρf (h,j) = ±

±

=

g

2 i ∗ hg, h + ki Eg (h) Eσh,k (g)∗h,h = , 2n n

n (h,j) (g,i) φρ (E (±)∗ E± (±)∗ ) 2n ± ± (−1)i+j ωg ωh X hg + h, kia(k)2 2n k∈G

=

(−1)i+j ωg ωh X hk − (g + h), ki , 2n k

∆ Sρf±(g,i) ,σg ± h,k

=

∆

± = Sσg ,σ^ 0 0 g,h

g ,h

=

∆± Sσg (g,i) f ± h,k ,ρ

√ 2 n (g,i) φρ± (E± (h)∗ Eσh,k (±)∗h,h ) = 0 , = 2n

2 [Eσ (g)∗0 0 Eσ (g 0 )∗g,g + Eσg0 ,h0 (g)∗h0 ,h0 Eσg,h (h0 )∗g,g ] 2n g0 ,h0 g ,g g,h 1 [hg, h0 ihh, g 0 i + hg, g 0 ihh, h0 i] , n

ωρfg i = Egi (g) = hg, gi , ωρf± (g,i) =

√ ±(−1)i ωg X (g,i) √ nφρ± (E± (±)) = hg, hia(h) = (−1)i ωg , n h

= Eσg,h (g)g,g = hg, hi . ωσg g,h

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√ When G = Z/2Z,√ a(1) = −1 gives a solution of (3.14). In this case, we may √ √ π −1 π −1 choose ω0 = ω1 = e 8 for ∆+ , and ω0 = ω1 = −1e 8 for ∆− . Thus, S ∆± for (0,0) (0,1) (1,0) (1,1) , ρf , ρf , ρf , σg the basis corresponding to ρ˜00 , ρ˜10 , ρ˜01 , ρ˜11 , ρf ± ± ± ± 0,1 is as follows:   √ √ √ √ 1 1 1 1 2 2 2 2 2 √ √ √ √    1 2 − 2 − 2 − 2 2 1 1 1 −   √ √ √ √   2 2 − 2 − 2 −2  1 1 1  1   √ √ √ √  1 2 2 −2  1 1 1 − 2 − 2  √ √ √ √  1 .  2 − 2 2 − 2 0 0 2 −2 0   √ √ √ 4 √  2 − 2 0 0 −2 2 0  2 − 2  √ √ √ √  2 − 2 − 2 2 2 −2 0 0 0  √ √ √ √    2 − 2 − 2 2 −2 2 0 0 0   2 2 −2 −2 0 0 0 0 0 Remark. Let ν be a A − B sector defined by (ρ± ⊗ idop ) · ι, where ι is the inclusion map of B into A. Then, we have   M op  op [ρg ⊗ ρop [ν ν¯] = [ρ± ⊗ idop ]  −g ] ⊕ [ρ± ⊗ ρ± ] [ρ± ⊗ id ] g∈G

=

M

op [ρg ⊗ ρop h ] ⊕ n[ρ± ⊗ ρ± ] ,

g,h∈G

[¯ ν ν] =

M

[¯ι][ρg ⊗ idop ][ι] =

g∈G

=

M

[¯ι][ι][˜ ρ0g ]

g∈G



M

[˜ ρ10 ] ⊕ ρ00 ] ⊕ [˜

g∈G

=

M

M

  ρ0g ] [σg 0,h ] [˜

h∈G\{0}

M

[˜ ρig ] ⊕

g∈G,i=0,1

2[σg g,h ] .

g≺h,g6=h

This means that A ⊃ (ρ± ⊗ idop )(B) is a depth 2 inclusion and the corresponding Kac algebra B and the dual Kac algebra Bˆ are n2 times

z }| { B∼ = C ⊕ · · · C ⊕M (n, C) , 2n times

n(n−1) 2

times

}| { z }| { z Bˆ ∼ = C ⊕ · · · C ⊕ M (2, C) ⊕ · · · M (2, C) . Since D(∆± ) has a braiding, the representation category of B has a braiding as well. In particular, G = Z/2Z case shows that Kac–Paljutkin’s 8-dimensional Kac algebra

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[18] has a braiding, which can be also shown by considering G = Z/2Z × Z/2Z case directly. (See the remark after Lemma 3.2 and [30]). Note that this type of a Kac algebra always comes from a semi-direct product group with a cocycle [16]. (1)

4. D5

Subfactors

In [12, Example 3.2], we introduced a period two automorphism α and an endomorphism ρ of the Cuntz algebra O3 = C ∗ {S1 , S2 , S3 }. Let w be a complex number satisfying w3 = 1, and U ∈ O3 be a unitary defined by U := S1 S1∗ + S2 S2∗ − S3 S3∗ . Then, α and ρ are defined by α(S1 ) = S2 , ρ(S1 ) =

α(S2 ) = S1 ,

α(S3 ) = −S3 ,

S3 S3 S1 + S2 + √ , 2 2

ρ(S2 ) = U ρ(S1 )U , ¯ ρ(S3 ) = w

S1 − S2 ∗ √ S3 + wS3 (S1 S1∗ − S2 S2∗ ) . 2

ρ has a statistical dimension 2. α and ρ satisfy the following: α2 = id ,

(4.1)

α ·ρ = ρ,

(4.2)

ρ · α = Ad(U ) · ρ ,

(4.3)

ρ2 (x) = S1 xS1∗ + S2 α(x)S2∗ + S3 ρ(x)S3∗ ,

(4.4)

α(U ) = U ,

(4.5)

ρ(U ) = S1 S2∗ + S2 S1∗ − S3 U S3∗ .

(4.6)

We set ρ0 := id, ρ1 := α, ρ2 := ρ, and ∆ := {ρ0 , ρ1 , ρ2 }. Note that ∆ for 2π w = 1, e± 3 are isomorphic to even sectors of the subfactors with principal graph (1) D5 [15]. In particular when w = 1, ∆ is equivalent to the representation category of the symmetric group S3 . We have the following fusion rules: [ρ1 ]2 = [ρ0 ] , [ρ1 ][ρ2 ] = [ρ2 ][ρ1 ] = [ρ2 ] , [ρ2 ]2 = [ρ0 ] ⊕ [ρ1 ] ⊕ [ρ2 ] . Basic intertwiner spaces are as follows: (ρ0 , ρ21 ) = C , (ρ0 , ρ22 ) = CS1 ,

(ρ2 , ρ1 ρ2 ) = C , (ρ1 , ρ22 ) = CS2 ,

(ρ2 , ρ2 ρ1 ) = CU , (ρ2 , ρ22 ) = CS3 .

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Linear space structure of Tube ∆ is given as follows: Ai,i = C(i 0|1|0 i) ⊕ C(i 1|1|1 i) ⊕ C(i 2|U i |2 i) ,

i = 0, 1 ,

(4.7)

A0,1 = C(0 2|U |2 1) ,

(4.8)

A1,0 = C(1 2|1|2 0) ,

(4.9)

Ai,2 = C(i 2|S3 |2 2) ,

i = 0, 1 ,

A2,i = C(2 2|U i S3∗ |2 i) ,

(4.10)

i = 0, 1

(4.11)

A2,2 = C(2 0|1|0 2) ⊕ C(2 1|U |1 2) ⊕ C(2 2|S1 S1∗ |2 2) ⊕ C(2 2|S2 S2∗ |2 2) ⊕ C(2 2|S3 S3∗ |2 2) .

(4.12)

Let σ be a finite direct sum of sectors in ∆. Then, the defining conditions of the half braiding for σ are 1 = α(Eσ (1))Eσ (1) ,

(4.13)

Eσ (2) = α(Eσ (2))Eσ (1) ,

(4.14)

U Eσ (2) = ρ(Eσ (1))Eσ (2)σ(U ) ,

(4.15)

S1 = ρ(Eσ (2))Eσ (2)σ(S1 ) ,

(4.16)

S2 Eσ (1) = ρ(Eσ (2))Eσ (2)σ(S2 ) ,

(4.17)

S3 Eσ (2) = ρ(Eσ (2))Eσ (2)σ(S3 ) .

(4.18)

Using these, we can show the following lemma in a similar way as in the proof of Lemma 3.2. Lemma 4.1. (1) There is only one half braiding for ρi , i = 0, 1 given by Eρ0 (0) = Eρ0 (1) = Eρ0 (2) = 1 , Eρ1 (0) = Eρ1 (1) = 1 , (2) There are exactly three half braidings Eρj2 (0) = 1 ,

Eρj2 ,

Eρ1 (2) = −U . j = 0, 1, 2 for ρ2 given by

Eρj2 (1) = −U ,

Eρj2 (2) = ωj (S1 S1∗ − S2 S2∗ ) + ωj S3 S3∗ , 2πj

√

−1

1+3j

±2π

√ −1

for w = 1, and ωj = w− 3 for w = e 3 . where ωj is e 3 (3) The minimal central projections corresponding to the above half braiding are (0 0|1|0 0) + (0 1|1|1 0) + 2(0 2|1|2 0) , 6 (1 0|1|0 1) + (1 1|1|1 1) − 2(1 2|U |2 1) , z(˜ ρ1 ) = 6

z(˜ ρ0 ) =

z(˜ ρj2 ) =

(2 0|1|0 2) − (2 1|U |1 2) + 2(2 2|Eρj2 (2)|2 2) . 3

(4.19) (4.20) (4.21)

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To determine the half braidings of reducible sectors, we deal with the tube algebra Tube ∆ directly. Lemma 4.2. The following hold : ρ0 ), p1 , p2 ∈ (1) A0 and A1 are isomorphic to C3 with the minimal projections z(˜ ρ1 ), q1 , q2 ∈ A1 where A1 and z(˜ p1 =

(0 0|1|0 0) + (0 1|1|1 0) − (0 2|1|2 0) , 3

p2 =

(0 0|1|0 0) − (0 1|1|1 0) , 2

q1 =

(1 0|1|0 1) + (1 1|1|1 1) + (1 2|U |2 1) , 3

q2 =

(1 0|1|0 1) − (1 1|1|1 1) . 2

(2) (1 2|1|2 0)∗ = (0 2|U |2 1) , (0 2|U |2 1)(1 2|1|2 0) = 3p1 , (1 2|1|2 0)(0 2|U |2 1) = 3q1 , (3) (0 2|S3 |2 2)∗ = w(2 2|S3∗ |2 0) , ¯ 2, (0 2|S3 |2 2)(2 2|S3∗|2 0) = 2wp (2 2|S3∗ |2 0)(0 2|S3|2 2) =

w ¯ w ¯ (2 0|1|0 2) + (2 1|U |1 2) 2 2 ¯ 2|S2 S2 |2 2) . + w(2 ¯ 2|S1 S1 |2 2) + w(2

(4) (1 2|S3 |2 2)∗ = −w(2 2|U S3∗ |2 1) , ¯ 2, (1 2|S3 |2 2)(2 2|U S3∗|2 1) = −2wp w ¯ w ¯ (2 2|U S3∗ |2 1)(1 2|S3 |2 2) = − (2 0|1|0 2) − (2 1|U |1 2) 2 2 ¯ 2|S2 S2∗ |2 2) . + w(2 ¯ 2|S1 S1∗ |2 2) + w(2 ρi2 ), i = 0, 1, 2, E0 , and E1 (5) A2 is isomorphic to C5 with minimal projections z(˜ where (2 0|1|0 2) + (2 1|U |1 2) (2 2|S1 S1∗ |2 2) + (2 2|S2 S2∗ |2 2) + , E0 := 4 2 E1 :=

(2 0|1|0 2) + (2 1|U |1 2) (2 2|S1 S1∗ |2 2) + (2 2|S2 S2∗ |2 2) − . 4 2

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Proof. (1) Since A0 and A1 are C∗ -algebras with dim A0 = dim A1 = 3, they are isomorphic to C3 . Calculating the products of the natural linear basis, we can obtain all the characters of these algebras. The minimal projections are obtained from them. It is straightforward to show (2), (3), and (4). Since p2 and q2 are minimal projections in A0 and A1 , so are in Tube ∆ thanks to Lemma 2.1, (4). (3) and (4) above imply that E0 is equivalent to p2 , and that E1 is equivalent to q2 , which shows that E0 and E1 are minimal projections of Tube ∆. (4.12) and Lemma 4.1 show that A2 is a 5 dimensional C∗ -algebra with mutually orthogonal three central projections. Therefore A2 ∼ = C5 . We set Aρ0 ⊕ρ1 := span{p1 , (0 2|U |2 1), (1 2|1|2 0), q1} , Aρ0 ⊕ρ2 := span{p2 , (0 2|S3 |2 2), (2 2|S3∗|2 0), E0 } , Aρ1 ⊕ρ2 := span{q2 , (1 2|S3 |2 2), (2 2|S3∗ |2 1), E1 } . The above lemma shows that these are ∗-subalgebras of Tube ∆ all isomorphic to the 2 by 2 matrix algebra. Note that these are mutually orthogonal to each other ρ1 ), z(˜ ρi2 ), i = 0, 1, 2. Moreover, these span Tube ∆. and orthogonal to z(˜ ρ0 ), z(˜ Therefore we get Theorem 4.3. Under the above notation, the following hold : (1) ρ1 ) ⊕ Cz(˜ ρ02 ) ⊕ Cz(˜ ρ12 ) ⊕ Cz(˜ ρ22 ) Tube ∆ = Cz(˜ ρ0 ) ⊕ Cz(˜ ⊕ Aρ0 ⊕ρ1 ⊕ Aρ0 ⊕ρ2 ⊕ Aρ1 ⊕ρ2 . (2) There are half braidings Eρ0 ⊕ρ1 , Eρ0 ⊕ρ2 , and Eρ1 ⊕ρ2 for ρ0 ⊕ ρ1 , ρ0 ⊕ ρ2 , and ρ1 ⊕ ρ2 respectively, such that the corresponding systems of matrix units of Aρ0 ⊕ρ1 , Aρ0 ⊕ρ2 , and Aρ1 ⊕ρ2 are as follows: e(ρ^ 0 ⊕ ρ1 )0,0 =

(0 0|1|0 0) + (0 1|1|1 0) − (0 2|1|2 0) , 3

e(ρ^ 0 ⊕ ρ1 )0,1 =

(0 2|U |2 1) √ , 3

e(ρ^ 0 ⊕ ρ1 )1,1 =

(1 0|1|0 1) + (1 1|1|1 1) + (1 2|U |2 1) , 3

e(ρ^ 0 ⊕ ρ2 )0,0 =

(0 0|1|0 0) − (0 1|1|1 0) , 2

e(ρ^ 0 ⊕ ρ2 )0,2 =

(0 2|S3 |2 2) √ , 2

e(ρ^ 0 ⊕ ρ1 )1,0 =

e(ρ^ 0 ⊕ ρ2 )2,0 =

(1 2|1|2 1) √ , 3

w(2 2|S3∗ |2 0) √ , 2

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e(ρ^ 0 ⊕ ρ2 )2,2 =

(2 0|1|0 2) + (2 1|U |1 2) (2 2|S1 S1∗ |2 2) + (2 2|S2 S2∗ |2 2) + , 4 2

e(ρ^ 1 ⊕ ρ2 )1,1 =

(1 0|1|0 1) − (1 1|1|1 1) , 2

e(ρ^ 1 ⊕ ρ2 )1,2 =

(1 2|S3 |2 2) √ , 2

e(ρ^ 1 ⊕ ρ2 )2,2 =

(2 0|1|0 2) + (2 1|U |1 2) (2 2|S1 S1∗ |2 2) + (2 2|S2 S2∗ |2 2) − . 4 2

e(ρ^ 1 ⊕ ρ2 )2,1 =

−w(2 2|U S3∗ |2 1) √ , 2

(3) We have Eρ0 ⊕ρ1 (0)0,0 = Eρ0 ⊕ρ1 (0)1,1 = 1 ,

Eρ0 ⊕ρ1 (0)0,1 = Eρ0 ⊕ρ1 (0)1,0 = 0 ,

Eρ0 ⊕ρ1 (1)0,0 = Eρ0 ⊕ρ1 (1)1,1 = 1 ,

Eρ0 ⊕ρ1 (1)0,1 = Eρ0 ⊕ρ1 (1)1,0 = 0 ,

√ 1 3U , Eρ0 ⊕ρ1 (2)0,0 = − , Eρ0 ⊕ρ1 (2)0,1 = 2 2 √ 1 3 , Eρ0 ⊕ρ1 (2)1,1 = . Eρ0 ⊕ρ1 (2)1,0 = 2 2 Eρ0 ⊕ρ2 (0)0,0 = Eρ0 ⊕ρ2 (0)2,2 = 1 , Eρ0 ⊕ρ2 (1)0,0 = −1 ,

Eρ0 ⊕ρ2 (0)0,2 = Eρ0 ⊕ρ2 (0)2,0 = 0 , Eρ0 ⊕ρ2 (1)2,2 = U ,

Eρ0 ⊕ρ2 (1)0,2 = Eρ0 ⊕ρ2 (1)2,0 = 0 Eρ0 ⊕ρ2 (2)0,0 = 0 , Eρ0 ⊕ρ2 (2)2,0 = wS3∗ ,

Eρ0 ⊕ρ2 (2)0,2 = S3 ,

Eρ0 ⊕ρ2 (2)2,2 = S1 S1∗ + S2 S2∗ ,

Eρ1 ⊕ρ2 (0)1,1 = Eρ1 ⊕ρ2 (0)2,2 = 1 , Eρ1 ⊕ρ2 (1)1,1 = −1 ,

Eρ1 ⊕ρ2 (0)1,2 = Eρ1 ⊕ρ2 (0)2,1 = 0 , Eρ1 ⊕ρ2 (1)2,2 = U ,

Eρ1 ⊕ρ2 (1)1,2 = Eρ1 ⊕ρ2 (1)2,1 = 0 , Eρ1 ⊕ρ2 (2)1,1 = 0 , Eρ1 ⊕ρ2 (2)2,1 = −wU S3∗ ,

Eρ1 ⊕ρ2 (2)1,2 = S3 ,

Eρ1 ⊕ρ2 (2)2,2 = −S1 S1∗ − S2 S2∗ .

From this theorem, we can obtain the dual principal graph of the Longo–Rehren (1) inclusions for the D5 subfactors as in Fig. 1. Using (2.8), (2.9), and the half braidings obtained above, we can calculate the S and T -matrices. Theorem 4.4. The matrix elements of the S and T -matrices of the Longo–Rehren inclusions of ∆ with respect to the basis corresponding to ρ˜0 , ρ˜1 , ρ˜02 , ρ˜12 , ρ˜22 , ρ^ 0 ⊕ ρ1 , ρ^ 0 ⊕ ρ2 , ρ^ 1 ⊕ ρ2 are given as follows:

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-

/

0

2

4

6

8

:

@

A

B

C

D

+

Y

E

G

H

J

K

L

N

O

P

Q

R

?

\Z

T

U

V

W

X

M

[ ^

[

S

] _

^

]

] \ Z Y

_ [

^ [

^

]

\ Z

Y

] ^

[

_ [

^

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Y

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^

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Y

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Y

[ Z

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\

_ [

^

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_ [

^ ]

\ [

Z Y

^

]

Z Y

_ [

^ ]

\ [

^

]

#

(

*

Fig. 1.

(1) When w = 1,

S∆



1

3

(2) When w = e±  1  1   2    2 1 ∆ S =  6  2   2   3  3

3

−1

!

$

%

&

)

*

1

2

2

2

2

3

1

2

2

2

2

−3

2

4

−2

−2

−2

0

2

−2

−2

4

−2

0

2

−2

4

−2

−2

0

2

−2

−2

−2

4

0

−3

0

0

0

0

3

−3

0

0

0

0

−3

T ∆ = Diag(1, 1, 1, e √

The dual principal graph of L–R inclusion.

 1  2   1 2 =  6 2  2   3 

2π

2π

√ −1 3

, e−

2π

√

−1

3

3



 −3   0   0  , 0  0   −3   3

, 1, 1, −1)

, 1 1 2 2 2 2

2

2

2 4π 4 cos 9 8π 4 cos 9 2π 4 cos 9 −2

2

2 8π 4 cos 9 2π 4 cos 9 4π 4 cos 9 −2

2 2π 4 cos 9 4π 4 cos 9 8π 4 cos 9 −2

2

3

2

−3

−2

0

−2

0

−2

0

4

0

−3

0

0

0

0

3

−3

0

0

0

0

−3

T ∆ = Diag(1, 1, w− 3 , w− 3 , w− 3 , 1, 1, −1) , 1

where we choose w 3 = e± 1

2π

√ 9

−1

.

4

7

3



 −3    0    0  .   0   0   −3   3

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Remark. When w = 1, D(∆) is isomorphic to the quantum double of the symmetric group S3 because ∆ is isomorphic to the representation category of S3 . We conjecture that when w 6= 1, D(∆) is isomorphic to the quantum double with an appropriate 3-cocycle (cf. [15]). 5. E 6 Subfactor and Generalization In this section, we analyze endomorphisms introduced in [12, Example 3.5]. Let G be a finite abelian group of order n. We fix a non-degenerate symmetric pairing h·, ·i : G × G → T and define the Fourier transform fˆ of a function f : G → C as in Lemma 3.1. We consider two functions a : G → T, b : G → C and c ∈ T satisfying the following conditions: a(0) = 1 ,

a(g) = a(−g) ,

a(g + h)hg, hi = a(g)a(h) ,

g, h ∈ G ,

a(g)b(−g) = b(g) , √ c n X + b(g) = 0 , d

(5.1) (5.2) (5.3)

g∈G

X

b(g + h)b(g) = δh,0 −

g∈G

1 , d

(5.4)

√ where d = (n + n2 + 4n)/2 satisfying d2 = nd + n. Using the Fourier transform, (5.3) and (5.4) are equivalent to the following two respectively: ˆb(0) = − c , d

(5.5)

δg,0 1 . |ˆb(g)|2 = − n d

(5.6)

Let O2n be the Cuntz algebra with the canonical generators {Sg , Th }g,h∈G . We set 1 X hg, hiTh . Tˆg := √ n h∈G

Then, {Sg , Tˆh }g,h∈G is also a set of canonical generators. We consider an action α of G on O2n and a unitary representation U of G in O2n defined by αg (Sh ) = Sg+h ,

g, h ∈ G

αg (Th ) = hg, hiTh , (αg (Tˆh ) = Th−g ) , g, h ∈ G . X X hg, hiSh Sh∗ + Th−g Th∗ . U (g) = h∈G

h∈G

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Thanks to (5.3) and (5.4), we can define an endomorphism ρ by " # 1 X 1 X hg, hiSh + √ a(h)Th−g T−h U (g)∗ , ρ(Sg ) = d d h∈H

h∈G

c X a(g)c X hk, gihh, kiSh Tk∗ + √ hh, gihh, kiTh Sk Sk∗ ρ(Tg ) = √ n nd h,k∈G h,k∈G X + a(h)b(g + h)hk, giTh+k T−h Tk∗ k,h∈G

X c X ∗ Tˆ−(g+k) Sk Sk∗ = √ Sh Tˆg−h + a(g)c d h∈G k∈G X + a(h)b(g + h)hk, giTh+k T−h Tk∗ . k,h∈G

It is easy to show αg · ρ = ρ ,

ρ · αg = Ad(U (g)) · ρ ,

g ∈ G.

(5.7)

Lemma 5.1. We have (1) Sg ∈ (αg , ρ2 ). (2) ρ(U (g)) =

X

Sh−g Sh∗ + a(g)

h∈G

=

X

Sh−g Sh∗ +

h∈G

X

X

∗ hg, hiTh U (g)Th−g

h∈H ∗ a(h)a(h − g)Th U (g)Th−g .

h∈H

(3) ρ2 (U (g)) =

X

Sh αh (U (g))Sh∗ +

h∈G

X

Th ρ(U (g))Th∗ .

h∈G

(Note that there is a typographic error in [13, (3.5.10)] and the above formula in (2) is correct.) Proof. (1) (5.7) implies the general statement from that for g = 0 case. First we claim, ρ(U (g))Sh = Sh−g ,

g, h ∈ G .

Indeed, we have ρ(U (g))S0 =

X

hg, hiρ(Sh Sh∗ )S0 +

h∈G

X h∈G

ρ(Tl−g Tl∗ )S0 .

(5.8)

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The first term is equal to X X hg, hiU (h)ρ(S0 S0∗ )U (h)∗ S0 = hg, hiU (h)ρ(S0 S0∗ )S0 h∈G

h∈G



=

=

1 ∗ nS−g S−g + d

X

 hg, hiTk−h Tk∗  ρ(S0 )

h,k∈G

1 X n S−g + √ a(k)hg, hiTk−h T−k . 2 d d d h,k∈G

The second term is equal to   X c X c ¯ 1 c¯ X  √ S−g + √ √ ρ(Tl−g )Tˆl = √ a(h)b(l − g + h)hk, giTh+k T−h  n d l∈G d l∈G d h,k∈G =

X c¯ n S−g + √ a(h)b(l)hk, giTh+k T−h d nd h,k,l∈G

=

n 1 X S−g − √ a(h)hk, giTh+k T−h . d d d h,k∈G

Therefore, we have ρ(U (g))S0 = S−g . Applying αh to the both sides, we get (5.8). Thanks to Lemma 2.2, in order to prove the statement it suffices to show ∗ 2 S0 ρ (x)S0 = x for the generators. If this is the case for x = S0 , we would get S0∗ ρ2 (Sg )S0 = S0∗ ρ2 (αg (S0 ))S0 = S0∗ ρ(U (g)ρ(S0 )U (g)∗ )S0 = Sg∗ ρ2 (S0 )Sg = αg (S0∗ ρ2 (S0 )S0 ) = αg (S0 ) = Sg 0. Thus, we consider only the cases x = S0 and x = Tg , which can be shown by direct computation using S0∗ ρ(Sg ) =

1 U (g)∗ , d

c S0∗ ρ(Tg ) = √ Tˆg∗ , d

ρ(Tg )S0 = a(g)cTˆ−g S0 .

(2) We define a projection P ∈ O2n by X Sg Sg∗ . P = g∈G

Then (5.8) shows that ρ(U (g)) commutes with P and satisfies X Sh−g Sh∗ . ρ(U (g))P = h

We claim Th∗ ρ(U (g))Th−g = a(g)hg, hiU (g) ,

h ∈ G.

(5.9)

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Indeed, we have Th∗ ρ(U (g))Th−g = T0∗ U (h)ρ(U (g))U (h)∗ T−g = T0∗ ρ(αh (U (g)))T−g = hg, hiT0∗ ρ(U (g))T−g . Thus, it suffices to show the claim for h = 0, which can be shown by direct computation. Note that the right-hand side of (5.9) is unitary. Thus, 1 = Th∗ ρ(U (g))ρ(U (g)∗ )Th = Th∗ ρ(U (g))P ρ(U (g)∗ )Th +

X

Th∗ ρ(U (g))Tk Tk∗ ρ(U (g)∗ )Th

k∈G ∗ = Th∗ ρ(U (g)∗ )Th−g Th−g ρ(U (g))Th ,

which shows Th∗ ρ(U (g))Tk = 0 for k 6= h − g. Therefore we get X Th Th∗ ρ(U (g))Tk Tk∗ ρ(U (g)) = ρ(U (g))P + h,k

=

X

Sh−g Sh∗ + a(g)

h∈G

X

∗ hg, hiTh U (g)Th−g .

h∈H 2

(3) Thanks to (1), we know that ρ (U (g)) commutes with P and X Sh αh (U (g))Sh∗ . ρ2 (U (g))P = h∈G

We claim Tˆh∗ ρ2 (U (g))Tˆh = ρ(U (g)) ,

g, h ∈ G .

(5.10)

Thanks to Tˆh = α−h (Tˆ0 ), the general statement follows from that for h = 0, which can be shown by direct computation using (2). In the same way as in the proof of (2), we get Tˆh∗ ρ2 (U (g))Tˆk = 0 for h 6= k, and so X X Tˆh ρ(U (g))Tˆh∗ . Sh αh (U (g))Sh∗ + ρ2 (U (g)) = h∈G

Since span{Th }h∈G

h∈G

= span{Tˆh }h∈G , we get the statement.

As we saw in [12], (1) of Lemma 5.1 shows that the image of ρ has Watatani index d2 . Next we obtain a necessary and sufficient condition that {Tg }g∈G are intertwiners between ρ and ρ2 . Lemma 5.2. The following are equivalent: (1) Tˆ0∗ ρ2 (T0 )Tˆ0 = ρ(T0 ). (2) ρ2 (x) =

X h∈G

Sh αh (x)Sh∗ +

X h∈G

Th ρ(x)Th∗ ,

x ∈ O2n .

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(3) ρ(T0∗ ρ(T0 ))Tˆ0 = ρ(T0∗ )Tˆ0 ρ(T0 ). Proof. (1) ⇒ (2) First we claim that the following always hold: X X Sh αh (Sg )Sh∗ + Th ρ(Sg )Th∗ . ρ2 (Sg ) = h∈G

h∈G

Thanks to (5.7) and (2) of Lemma 5.1, it suffices to show the claim for g = 0, which can be shown by direct computation. If Tˆ0∗ ρ2 (T0 )Tˆ0 = ρ(T0 ) holds, we have Tˆ0∗ ρ2 (Tg )Tˆ0 = Tˆ0∗ ρ2 (U (g)∗ T0 )Tˆ0 = ρ(U (g)∗ )Tˆ0∗ ρ2 (T0 )Tˆ0 = ρ(U (g)∗ T0 ) = ρ(Tg ) , where we use (3) of Lemma 5.1. Thus, Lemma 2.2 implies Tˆ0 ∈ (ρ, ρ2 ). Using Tˆg = α−g (Tˆ0 ), we get Tˆg ∈ (ρ, ρ2 ) as well. Therefore (2) holds. (2) ⇒ (3) This is obvious. (3) ⇒ (1) First we claim the following always holds: ρ(Sg∗ ρ(T0 ))Tˆ0 = ρ(Sg∗ )Tˆ0 ρ(T0 ) . Indeed, g = 0 case can be shown by direct computation, and in general we have ρ(S ∗ ρ(T0 ))Tˆ0 = ρ(αg (S ∗ ρ(T0 )))Tˆ0 = U (g)ρ(S ∗ ρ(T0 ))U (g)∗ Tˆ0 g

0

0

= U (g)ρ(S0∗ ρ(T0 ))Tˆ0 = U (g)ρ(S0∗ )Tˆ0 ρ(T0 ) = ρ(αg (S0∗ ))U (g)Tˆ0 ρ(T0 ) = ρ(Sg∗ ))Tˆ0 ρ(T0 ) . Next, we show the following equality assuming that it is the case for g = 0: ρ(Tg∗ ρ(T0 ))Tˆ0 = ρ(Tg∗ )Tˆ0 ρ(T0 ) . Indeed, we have ρ(Tg∗ ρ(T0 ))Tˆ0 = ρ(T0∗ U (g)ρ(T0 ))Tˆ0 = ρ(T0∗ ρ(αg (T0 ))U (g))Tˆ0 = a(g)ρ(T0∗ ρ(T0 ))Tˆg U (g) = a(g)α−g (ρ(T0∗ ρ(T0 ))Tˆ0 )U (g) = a(g)α−g (ρ(T0∗ )Tˆ0 ρ(T0 ))U (g) = a(g)ρ(T0∗ )Tˆg ρ(T0 )U (g) = ρ(T0∗ U (g))Tˆ0 U (g)∗ ρ(T0 )U (g) = ρ(Tg∗ )Tˆ0 ρ(α−g (T0 )) = ρ(Tg∗ )Tˆ0 ρ(T0 ) . Therefore, we get ρ2 (T0 )Tˆ0 =

X

ρ(Sg Sg∗ )ρ2 (T0 )Tˆ0 +

g∈G

=

X

X

ρ(Tg Tg∗ )ρ2 (T0 )Tˆ0

g∈G

ρ(Sg Sg∗ )Tˆ0 ρ(T0 ) +

g∈G

= Tˆ0 ρ(T0 ) ,

X g∈G

ρ(Tg Tg∗ )Tˆ0 ρ(T0 )

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which implies (1). Theorem 5.3. The conditions in the previous lemma are equivalent to the following: ˆ(0) = 1 c3 a ˆb(g) = cb(g) , X g∈G

(5.11)

g ∈ G,

c b(g + h)b(g + k)b(g) = hh, kib(h)b(k) − √ , d n

(5.12) h, k ∈ G .

(5.13)

Proof. We show that (3) of Lemma 5.2 is equivalent to (5.11)–(5.13). The left hand-side of (3) is X c¯ X ρ(Sg Sg∗ )Tˆ0 + b(g)ρ(Tg Tg∗ )Tˆ0 ρ(T0∗ ρ(T0 ))Tˆ0 = √ n g∈G

g∈G

X c¯ X = √ U (g)ρ(S0 S0∗ )U (g)∗ Tˆ0 + b(g)ρ(Tg Tg∗ )Tˆ0 n g∈G

g∈G

X √ ∗ = c¯ n(S0 S0∗ + Tˆ0 Tˆ0∗ )ρ(S0 S0∗ )Tˆ0 + c a(g)b(g)ρ(Tg )S−g S−g +

X

g∈G

a(k)b(g)b(g − k)ρ(Tg )Tˆg Tk∗

g,k∈G

c¯ X c¯ X = √ a(k)S0 Tk∗ + √ a(h)a(k)Tˆ0 T−h Tk∗ nd d d k∈G h,k∈G X X c ∗ + b(g)Tˆ0 S−g S−g +√ a(k)b(g)b(g − k)S0 Tk∗ d g∈G g,k∈G X + a(h)a(k)b(g + h)b(g)b(g − k)Tˆ0 T−h Tk∗ g,h,k∈G

  2 X X c X c ¯ = √ a(k)  + b(g)b(g − k) S0 Tk∗ + b(−g)Tˆ0 Sg Sg∗ d d k∈G g∈G g∈G   X X c¯ + a(h)a(k) √ + b(g)b(g − h)b(g − h − k) Tˆ0 T−h Tk∗ . nd g∈G

h,k∈G

The right-hand side is X

c2 X b(h)Tˆ0 Th∗ ρ(T0 ) = √ S0 Tk∗ nd h∈G k∈G X X ∗ ˆb(−k)Tˆ0 Sk S + + c¯ b(h + k)b(−h)Tˆ0 T−h Tk∗ . k

ρ(T0∗ )Tˆ0 ρ(T0 ) = cS0 S0∗ ρ(T0 ) +

k∈G

h,k∈G

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Thus, (3) of Lemma 5.2 is equivalent to (5.12) and X ca(k) c¯2 , b(g)b(g − k) = √ − n d g∈G

X g∈G

c¯ b(g)b(g − h)b(g − h − k) = a(h)a(k)b(−h)b(h + k) − √ . nd

Thanks to (5.1) and (5.2), the second equation is equivalent to (5.13). Using (5.12), (5.2), and (5.6), we can compute the left-hand side of the first one as follows: X X X ˆb(l)ˆb(−l)hk, li = c¯2 b(g)b(g − k) = b(l)b(−l)hk, li g∈G

l∈G

= c¯2

X

l∈G

a(l)|b(l)|2 hk, li = c¯2

l∈G

=

X

a(l)

l∈G

1 δl,0 − n d

hk, li

c2 a ˆ(k) c¯2 c2 a ˆ(0)a(k) c¯2 √ √ − = − . n d n d

Thus, the first one is equivalent to (5.11) under the presence of (5.12). Remark. h = 0 case of (5.13) follows from the other conditions (5.1)–(5.4), (5.11), (5.12). Therefore, when we have a solution of (5.1)–(5.4), (5.11), (5.12) and we need to check whether (5.13) holds, it suffices to consider the cases of h 6= 0, k 6= 0; we need to check n(n − 1)/2 equations. The above theorem shows that every solution of the equations (5.1)–(5.4) and (5.11)–(5.13) gives rise to a category obeying the following fusion rules: [αg ][αh ] = [αg+h ] ,

g, h ∈ G ,

[αg ][ρ] = [ρ][αg ] = [ρ] , g ∈ G , M [αg ] ⊕ n[ρ] . [ρ][ρ] = g∈G

As we saw in [12], G = {0} case corresponds to even sectors of the A4 subfactor, and G = Z/2Z case corresponds to those of the E6 subfactors. Example 5.1 ([12, Example 3.4]). Let G = Z/2Z = {0, 1}. Then, there is only one non-degenerate symmetric pairing hk, li = (−1)kl . There exist exactly two solutions of (5.1)–(5.4) and (5.11)–(5.13): one given below and its complex conjugate. √ a(0) = 1 , a(1) = −1 , π

√

−1

e− 4 1 , b(0) = − , b(1) = √ d 2 √ √ √ √ 7π −1 1 − 3 + (1 + 3) −1 √ , c = e 12 = 2 2 √ d = 1 + 3.

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The complex conjugate of this solution is employed in [29] to compute the Turaev– Viro–Ocneanu type three manifold invariant for E6 subfactors, which detects orientations of several manifolds. In Appendix, we present several solutions of (5.1)–(5.4) and (5.11)–(5.13) for groups G = Z/3Z, Z/4Z, Z/2Z×Z/2Z, Z/5Z. There also exist finite abelian groups without solutions (See Proposition A.1). Problem 5.1. For each prime p, construct a solution of (5.1)–(5.4) and (5.11)– (5.13) for G = Z/pZ. In the rest of this section and next section, we assume that ρ comes from a, b, and c satisfying (5.1)–(5.4) and (5.11)–(5.13). Let M be the weak closure of O2n in the GNS representation of the KMS state considered in [12]. We still use the same symbol ρ for its extension to M . As we saw in [12, Sec. 6], the von Neumann algebra generated by ρ(M ) and {U (g)}g∈G can be identified with the crossed product of L := ρ(M ) o G. We computed in [12] the sector of the canonical endomorphism for the inclusion M ⊃ L, which is [id] ⊕ [ρ] . Consequently, we have n+2+

√

n2 + 4n

. 2 In G = Z/2Z case, the principal graph of M ⊃ L is E6 . The principal graph of M ⊃ L for a group G of order n is as in Fig. 2. [M : L] = d + 1 =

-

0

0

0

0

0

0

.

0

0

0

0

0

0

.

.

.

.

1

1

1

1

1

1

.

/

1

/

1

/

1

1

1

1

,

Fig. 2.

.

The principal graph of M ⊃ ρ(M ) o G for #G = n case.

6. Tube ∆ for E 6 Subfactor and Generalization Let ρ and αg be as in the previous section. We set ∆ = {αg }g∈G ∪ {ρ} .

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The purpose of this section is to determine the structure of Tube ∆ and obtain their S and T -matrices. Although the above notation of ∆ is slightly out of our convention (we used to employ the notation ∆ = {ρξ }ξ∈∆0 ), we believe that there is no possibility of confusion. We sometimes use notation g instead of αg for simplicity. We set λ to be the global index of ∆: p n(n + 4) + n n(n + 4) 2 . λ = n + d = n(d + 2) = 2 We fix an arbitrary linear order ≺ of G as in Sec. 3. Necessary intertwiner spaces for our purpose are g, h ∈ G ,

(αg+h , αg αh ) = C1 , (ρ, αg ρ) = C1 ,

(ρ, ραg ) = CU (g) ,

g ∈ G,

(αg , ρ2 ) = CSg , g ∈ G , M CTg . (ρ, ρ2 ) = g∈G

Thus, linear space structure of the tube algebra is given as follows: let g, h ∈ G, g 6= h, then M C(g k|1|k g) ⊕ C(g ρ|U (g)|ρ g) , (6.1) Ag,g = k∈G

Ag,h = C(g ρ|U (h)|ρ h) , M C(g ρ|Tk |ρ ρ) , Ag,ρ =

(6.2) (6.3)

k∈G

Aρ,g =

M

C(ρ ρ|U (g)Tk∗ |ρ g) ,

(6.4)

k∈G

Aρ,ρ =

M

C(ρ k|U (k)∗ |k ρ) ⊕

k∈G

⊕

M

C(ρ ρ|Sk Sk∗ |ρ ρ)

k∈G

M

C(ρ ρ|Tk Tl∗ |ρ ρ) .

(6.5)

k,l∈G

Let σ be a finite direct sum of sectors in ∆. Then, the defining conditions of the half braiding for σ are g, h ∈ G

(6.6)

g ∈ G,

(6.7)

U (g)Eσ (ρ) = ρ(Eσ (g))Eσ (ρ)σ(U (g)) ,

g ∈ G,

(6.8)

Sg Eσ (g) = ρ(Eσ (ρ))Eσ (ρ)σ(Sg ) ,

g ∈ G,

(6.9)

Tg Eσ (ρ) = ρ(Eσ (ρ))Eσ (ρ)σ(Tg ) ,

g ∈ G.

(6.10)

Eσ (g + h) = αg (Eσ (h))Eσ (g) , Eσ (ρ) = αg (Eσ (ρ))Eσ (g) ,

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First we determine the half-braidings of αg . Lemma 6.1. Let g ∈ G. Then, there exists only one half braiding of g with respect to ∆ given by Eg (h) = hg, hi ,

h ∈ G,

Eg (ρ) = a(g)U (g) . The corresponding minimal and central projection of Tube ∆ is z(˜ g) =

d 1 X hg, hi(g h|1|h g) + a(g)(g ρ|U (g)|ρ g) . λ λ

(6.11)

h∈G

Proof. If Eg is a half braiding, we have Eg (h) ∈ (αg+h , αg+h ) = C1 and Eg (ρ) ∈ (ρ, ραg ) = CU (g). It is easy to show that the above is the only solution for (6.6)–(6.10). We set xg := a(g)(g ρ|U (g)|ρ g) , xg,h := (g ρ|U (h)|ρ h) , yg := (g ρ|T−g |ρ ρ) , yg (h) := (g ρ|T−h |ρ ρ) ,

g ∈ G, g 6= h ∈ G , g ∈ G, g 6= h ∈ G .

It is straightforward to show the following: Lemma 6.2. Let g ∈ G. (1) The map G ∈ h 7→ (g h|1|h g) gives a unitary representation of G in Ag . Thus, we can define projections pg (h) ∈ Ag , h ∈ G by pg (h) =

1 X hh, ki(g k|1|k g) . n k∈G

(2) xg is a self-adjoint element satisfying pg (h)xg = xg pg (h) = δg,h xg . x2g = npg (g) + nxg . (3) Let qg = =

d2 pg (g) − dxg λ d d+1 X hg, ki(g k|1|kg) − a(g)(g ρ|U (g)|ρ g) . λ λ k∈G

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Then, qg is a minimal projection. Ag has the following direct sum decomposition as an algebra: M Cpg (h) ⊕ Cqg ⊕ Cz(˜ g) . Ag = h∈G\{g}

(4) For g 6= h ∈ G, we have x∗g,h = hg, hixh,g , xg,h xh,g = hg, hinpg (h) . (5) We have c¯ X a(k)hg + k, hi(ρ ρ|U (g)Tk∗ |ρ g) , (g ρ|Th |ρ ρ)∗ = √ n k∈G

λ xg = qg , yg yg∗ = n pg (g) − d d+1 yg (h)yg (h)∗ = npg (h) . (6) The coefficient of (ρ h|U (h)∗ |h ρ) in yg∗ yg is 2

hh, gi a(h) . d Thanks to Lemma 6.2, we can construct systems of matrix units for some simple components of Tube ∆. We introduce reducible endomorphisms πg := αg ⊕ ρ ,

g ∈ G,

σg,h := αg ⊕ αh ⊕ ρ , and set e(f πg )g,g := qg ,

d e(f πg )g,ρ := √ yg , nλ

d e(f πg )ρ,g := √ yg∗ , nλ

e(f πg )ρ,ρ :=

d+1 ∗ y yg , λ g

πg )g,g + e(f πg )ρ,ρ , z(f πg ) := e(f πg )s,t }s,t=g,ρ , Aπfg := span{e(f e(σg g,h )g,g := pg (h) ,

e(σg g,h )h,h := ph (g) ,

1 e(σg g,h )g,h := √ xg,h , n 1 e(σg g,h )g,ρ := √ xg (h) , n

hg, hi xh,g , e(σg g,h )h,g := √ n 1 ∗ e(σg g,h )ρ,g := √ xg (h) , n

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1 e(σg g,h )h,ρ := √ xh (g) , n e(σg g,h )ρ,ρ :=

1 ∗ e(σg g,h )ρ,h := √ xh (g) , n

1 xg (h)∗ xg (h) , n

g g g z(σg g,h ) := e(σ g,h )g,g + e(σ g,h )h,h + e(σ g,h )ρ,ρ , := span{e(σg Aσg g,h )s,t }s,t=g,h,ρ . g,h Corollary 6.3. Under the above notation, we have (1) Aπfg , (g ∈ G) and Aσg g,h , (g ≺ h ∈ G) are mutually orthogonal simple components of Tube ∆ with systems of matrix units {e(f πg )s,t }s,t=g,ρ ,

{e(σg g,h )s,t }s,t=g,h,ρ .

(2) Let Eπg and Eσg,h be the corresponding half braidings of πg and σg,h . Then, Eπg (k)g,g = hg, ki , Eπg (ρ)g,g = −

k ∈ G,

a(g) U (g) , d+1

Eπg (h)ρ,ρ = hg, hi2 a(h)U (h)∗ , Eσg,h (k)g,g = hh, ki ,

g, h ∈ G ,

Eσg,h (k)h,h = hg, ki ,

k ∈ G,

Eσg,h (ρ)g,g = Eσg,h (ρ)h,h = 0 . Note that {e(f πg )ρ,ρ }g∈G ∪ {e(σg g,h )ρ,ρ }g≺h∈G are central projections in Aρ . Let A1 ⊂ Aρ be the subalgebra of the elements orthogonal to these projections. Then, we have n(n + 3) n(n − 1) = , dim A1 = dim Aρ − n − 2 2 M M M Cz(f αg ) ⊕ Aπfg ⊕ Aσg ⊕ A1 . Tube ∆ = g,h g∈G

g∈G

g≺h,g6=h

Theorem 6.4. There exist exactly n(n + 3)/2 half braidings Eρj , j ∈ J for ρ with respect to ∆. We have the following decomposition: M M M M Cz(f αg ) ⊕ Cz(˜ ρj ) ⊕ Aπfg ⊕ Aσg . Tube ∆ = g,h g∈G

j∈J

g∈G

g≺h,g6=h

Proof. It suffices to show that A1 is abelian, or more strongly, Aρ is abelian. Let Vg := a(g)(ρ g|U (g)∗ |g ρ) ,

g ∈ G,

Xg := (ρ ρ|Sg Sg∗ |ρ ρ) ,

g ∈ G,

Yg,h := (ρ ρ|Tg Th∗ |ρ ρ) ,

g, h ∈ G .

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Then, direct computation shows that the map G 3 g 7→ Vg is a unitary representation of G in Aρ and g, h ∈ G ,

Vg Xh = Xh Vg = a(g)hg, hiXg+h , Vg Yh,k = Yh,k Vg = a(g)hg, h − kiYh,k−g ,

g, h, k ∈ G .

Thus, to prove commutativity of Aρ , it suffices to show it for {X0 , Yg,0 }g∈G . Indeed, direct computation shows a(g) a(g) X X + b(k − g)Yk,−g , g d2 d

X0 Yg,0 = Yg,0 X0 =

k∈G

c X c2 X hg − k, h − kia(k)b(k)Vk + a(k)hg + h, kiXk Yg,0 Yh,0 = √ nd n k∈G

k∈G

X

+ a(g + h)

hg + h, l − kia(l − k)b(l)b(k − g)b(k − h)Yk,l ,

k,l∈G

which shows the statement. The above theorem shows that the dual principal graph of the Longo–Rehren inclusion for the E6 subfactor is as in Fig. 3, which is also obtained in [7]. Now, we deduce the equations determining the half braidings for ρ. Let Eρ = {Eρ (g), Eρ (ρ)}g∈G , be a half braiding for ρ. Since Eρ (g) ∈ (ραg , ρ), Eρ (ρ) ∈ (ρ2 , ρ2 ), there exists µ(g) ∈ T such that Eρ (g) = µ(g)U (g)∗ , g ∈ G , X X CSg Sg∗ + CTg Th∗ . Eρ (ρ) ∈ g∈G

g,h∈G

Lemma 6.5. The defining conditions (6.6)–(6.10) of half braidings for ρ are equivalent to the following: q

q

b c

r

t

u

v

w

x

z

q

j {

r

t

u

.

v

w

x

z

b {

h

r

t

.

1

u

v

w

x

z

{

.

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

a

.

. .

.

.

.

.

. .

h

i

f

b

c

Fig. 3.

e

c

c

g

i

i

i

i

h

j

j

k

j

l

j

n

j

o e

h

b

h

The dual principal graph of L–R inclusion for G = Z/2Z case.

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µ(g)µ(h) = hg, hiµ(g + h) , αg (Eρ (ρ)∗ )Eρ (ρ) = µ(g)U (g)∗ ,

g, h ∈ G ,

g ∈ G,

U (g)ρ(Eρ (ρ))Eρ (ρ) = ρ(Eρ (ρ))Eρ (ρ)ρ(U (g)) ,

(6.12) (6.13)

g ∈ G,

(6.14)

S0 = ρ(Eρ (ρ))Eρ (ρ)ρ(S0 ) ,

(6.15)

T0 Eρ (ρ) = ρ(Eρ (ρ))Eρ (ρ)ρ(T0 ) .

(6.16)

Proof. It is obvious that (6.6) and (6.12) are equivalent and that (6.7) and (6.13) are identical. If Eρ satisfies (6.9) and (6.10), we get (6.14) using the definition of U (g). Thus, (6.12)–(6.16) are necessary conditions for Eρ to be a half braiding. On the other hand, assume that Eρ satisfies (6.12)–(6.16). To prove that Eρ is a half braiding, we need to show that (6.8), (6.9), and (6.10) hold. Indeed using (6.13) and (6.14), we have ρ(Eρ (g))Eρ (ρ)ρ(U (g)) = ρ(αg (Eρ (ρ)∗ )Eρ (ρ))Eρ (ρ)ρ(U (g)) = ρ(αg (Eρ (ρ)∗ ))U (g)ρ(Eρ (ρ))Eρ (ρ) = U (g)ρ(Eρ (ρ)∗ )ρ(Eρ (ρ))Eρ (ρ) = U (g)Eρ (ρ) , which shows (6.8). (6.13) and (6.15) imply Sg Eρ (g) = Sg αg (Eρ (ρ)∗ )Eρ (ρ) = αg (S0 Eρ (ρ)∗ )Eρ (ρ) = αg (ρ(Eρ (ρ))Eρ (ρ)ρ(S0 )Eρ (ρ)∗ )Eρ (ρ) = ρ(Eρ (ρ))αg (Eρ (ρ))ρ(S0 )αg (Eρ (ρ)∗ )Eρ (ρ) = ρ(Eρ (ρ))Eρ (ρ)Eρ (g)∗ ρ(S0 )Eρ (g) = ρ(Eρ (ρ))Eρ (ρ)ρ(Sg ) . (6.13), (6.16), and (6.8) imply Tg Eρ (ρ) = U (g)∗ T0 Eρ (ρ) = U (g)∗ ρ(Eρ (ρ))Eρ (ρ)ρ(T0 ) = ρ(α−g (Eρ (ρ)))U (g)∗ Eρ (ρ)ρ(T0 ) = ρ(Eρ (ρ)Eρ (−g)∗ ))U (−g)Eρ (ρ)ρ(T0 ) = ρ(Eρ (ρ))Eρ (ρ)ρ(U (−g))ρ(T0 ) = ρ(Eρ (ρ))Eρ (ρ)ρ(Tg ) . Therefore, Eρ is a half braiding. (6.12) shows that there exists a unique element τ ∈ G satisfying µ(g) = hτ, gia(g), g ∈ G. We have µ ˆ(0) = a ˆ(0)a(τ ).

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Theorem 6.6. Let µ : G → T be a function satisfying (6.12), and τ ∈ G be as above. (1) We set 1 X µ(−g)U (g) . Uρ := √ n g∈G

Then, Uρ ∈ (ρ2 , ρ2 ) is a unitary satisfying αg (Uρ∗ )Uρ = µ(g)U (g)∗ . (2) Let Vρ ∈ (ρ2 , ρ2 ) be a unitary normalized as Vρ S0 = S0 . Then, Eρ (ρ) = ω ¯µ ˆ(0)Vρ Uρ with ω ∈ T satisfies (6.13)–(6.14) if and only if the following hold : There exists a function ξ : G → T satisfying X X Sg Sg∗ + ξ(g)Tg Tg∗ , (6.17) Vρ = g∈G

g∈G

√ ˆ = ω 2 a(τ )ˆ a(0) − ξ(0) c¯

X

n , d

b(g + k)ξ(k) = ω 2 a(τ )ˆ a(0)ξ(g + τ ) −

k∈G

√ ˆ = ω 2 a(τ )ˆ ˆ − n δg,0 , nµ(g)b(g)ξ(g) a(0)ξ(g) d a(0) , ξ(g)ξ(τ − g)a(g)a(τ − g) = cωˆ X

(6.18) √ n , d

g ∈ G,

g ∈ G,

g ∈ G,

(6.19)

(6.190 ) (6.20)

ξ(k)b(k − g)b(k − h)

k∈G

= b(g + h − τ )ξ(g)ξ(h)c2 a(g − h) −

c2 , d

g, h ∈ G .

(6.21)

((6.19) is equivalent to (6.190 ) via Fourier transformation.) (3) The eigenvalue of the T -matrix corresponding to the half braiding Eρ (ρ) is ω. Proof. (1) is easy. ¯µ ˆ(0)Vρ Uρ , (6.13) is equivalent to αg (Vρ ) = Vρ , g ∈ G, and so (2). For Eρ (ρ) = ω Vρ is as in (6.17). We show that (6.14) automatically holds for such Eρ (ρ). Indeed, we have ρ(Eρ (ρ)∗ )U (g)ρ(Eρ (ρ))Eρ (ρ) = ρ(Uρ∗ Vρ∗ )ρ(αg (Vρ Uρ ))U (g)Eρ (ρ) = ρ(Uρ∗ αg (Uρ ))U (g)Eρ (ρ) = µ(g)ρ(U (g))U (g)Eρ (ρ) .

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Thus, (6.14) is equivalent to ρ(U (g))U (g)Vρ = µ(g)Vρ Uρ ρ(U (g))Uρ∗ =

µ(g) X µ(−k)µ(−h)Vρ U (h)ρ(U (g))U (k)∗ n h,k∈G

=

µ(g) X hh, giµ(−k)µ(−h)Vρ ρ(U (g))U (h − k) n h,k∈G

=

µ(g) X hh + k, giµ(−k)µ(−(h + k))Vρ ρ(U (g))U (h) n h,k∈G

1 X hk, h − giVρ ρ(U (g))U (h) = n h,k∈G

= Vρ ρ(U (g))U (g) , which can be easily shown. Next we show that (6.15) is equivalent to (6.18) and (6.19). We have X X U (g)ρ(S0 S0∗ )U (g)∗ S0 + ξ(g)ρ(Tg Tg∗ )S0 ρ(Vρ∗ )S0 = g∈G

g∈G

X

c¯ X = U (g)ρ(S0 S0∗ )S0 + √ ξ(g)ρ(Tg )Tˆg d g∈G g∈G =

=

√ ˆ n nξ(0) (S0 S0∗ + Tˆ0 Tˆ0∗ )ρ(S0 ) + S0 d d X c¯ +√ ξ(g)a(l)b(g + l)Tk+l T−l nd g,k,l∈G n + d2

√ ˆ ! nξ(0) S0 d

  √ X 1 X n + c¯ +√ a(l)  ξ(g)b(g + l) Tˆ0 T−l . d d l∈G g∈G On the other hand,

  X X 1 X 1 1 µ(k)ρ(Uρ )Vρ U (−k)  Sg + √ a(l)Tl T−l  ρ(Uρ )Vρ Uρ ρ(S0 ) = √ n d d l∈G g∈G k∈G 1 X µ(k)hg, kiρ(Uρ )Vρ Sg = √ nd g,k∈G

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1 X +√ a(l)µ(k)ρ(Uρ )Vρ Tl+k T−l nd k,l∈G =

1X 1 X µ ˆ(−g)ρ(Uρ )Sg + √ a(l)ξ(h)µ(h − l)ρ(Uρ )Th T−l d nd h,l∈G g∈G

=

µ ˆ(0) X µ(−g)ρ(Uρ )Sg d g∈G

1 X +√ a(l)ξ(h)µ(h − l)ρ(Uρ )Th T−l . nd h,l∈G The first term is equal to µ ˆ(0) X µ ˆ(0) X √ µ(−g)µ(k)ρ(U (−k))Sg = √ µ(−g)µ(k)Sg+k d n d n g,k∈G

g,k∈G

µ ˆ(0) X = √ µ(−g)hg, kiSg d n g,k∈G

√ nˆ µ(0) S0 . = d The second term is X 1 √ a(l)ξ(h)µ(−k)µ(h − l)ρ(U (k))Th T−l n d h,k,l∈G =

X 1 √ a(l − k)ξ(h − k)a(k)µ(−k)µ(h − l)hk, hiTh T−l n d h,k,l∈G

=

X 1 √ a(l)hl, kiξ(h − k)µ(−k)µ(h − l)hk, hiTh T−l n d h,k,l∈G

1 X = √ a(l)ξ(k)µ(k)µ(−l)hk, liTˆ0 T−l . nd k,l∈G Thus, (6.15) is equivalent to (6.18) and √ X ˆ2 (0) X nµ(l) ω ¯ 2µ √ + c¯µ(l) ξ(k)µ(k)hk, li = b(g − l)ξ(g) , n d

l ∈ G.

g∈G

k∈G

By Fourier transformation transformation, this is equivalent to √ c¯ X nˆ µ(k) 2 2 +√ ˆ (0)ξ(k)µ(k) = b(g − l)µ(l)ξ(g)hk, li ω ¯ µ d n g,l∈G

√ c¯ X ˆ nˆ µ(0)µ(k) + b(h)hh, g − liµ(l)ξ(g)hk, li = d n g,h,l∈G

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√ 1 X nˆ µ(0)µ(k) +√ hh, giˆ µ(h + k)b(h)ξ(g) d n g,h∈G   √ X 1 n +√ hh, g + kiµ(h)b(h)ξ(g) =µ ˆ(0)µ(k)  d n =

g,h∈G



 √ X 1 n +√ hh, g + k − τ ia(h)b(h)ξ(g) =µ ˆ(0)µ(k)  d n g,h∈G



 √ X 1 n +√ hh, g + k − τ ib(−h)ξ(g) =µ ˆ(0)µ(k)  d n g,h∈G



 √ X n ˆb(g + k − τ )ξ(g) + =µ ˆ(0)µ(k)  d g∈G



 √ X n +c b(g + k − τ )ξ(g) , =µ ˆ(0)µ(k)  d g∈G

where we use (5.2) and (5.12) several times. Thus, (6.15) is equivalent to (6.18) and (6.19). Finally, we show that (6.16) is equivalent to (6.19) and (6.20) under the presence of the equations so far obtained. Since U (g) commutes with ρ(T0 ) for every g ∈ G, so does Uρ . Thus, (6.16) is equivalent to ¯µ ˆ (0)Vρ ρ(Tρ )Vρ∗ . ρ(Uρ∗ Vρ∗ )T0 = ω We start with the left-hand side: X X ρ(Sg Sg∗ )T0 + ξ(g)ρ(Tg Tg∗ )T0 ρ(Vρ∗ )T0 = g∈G

=

X

g∈G

U (−k)ρ(S0 S0∗ )T−k

k∈G

+

X g∈G

"

X a(g)c X ξ(g)ρ(Tg ) √ Sl Sl∗ + a(k)b(g − k)hg, kiTk Tk∗ n l∈G

k∈G

1 X 1 X a(k)hh, kiSh Tk∗ + a(h)a(k)Th+k T−h Tk∗ = √ d d d h,k∈G h,k∈G X c +√ ξ(g)a(k)b(g − k)hh, kiSh Tk∗ nd g,h,k∈G

#

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+

1 n

X

hh, gihh, kiξ(g)Th Sk Sk∗

g,h,k∈G

X

+

641

a(h)b(g + h)ξ(g)a(k)b(g − k)Th+k T−h Tk∗

g,h,k∈G

ω ¯ 2µ 1 X ˆ (0) X ∗ ˆ = √ ξ(τ − k)a(k)hh, kiSh Tk∗ + √ hh, kiξ(h)T h Sk Sk n nd h,k∈G h,k∈G   X X 1 + a(h)a(k)  + b(g + h)ξ(g)b(g − k) Th+k T−h Tk∗ , d g∈G

h,k∈G

where we use (6.19) in the last equality. Therefore, ρ(Uρ∗ Vρ∗ )T0 is ρ(Uρ∗ Vρ∗ )T0 =

ω ¯ 2µ ˆ(0) X √ µ(l)ξ(τ − k)a(k)hh, kiρ(U (l))Sh Tk∗ n d h,k,l∈G +

1 X ˆ µ(l)hh, kiξ(h)ρ(U (l))Th Sk Sk∗ n h,k,l∈G

  X µ(l)a(h)a(k) 1 X  + √ b(g + h)ξ(g)b(g − k) n d

+

g∈G

h,k,l∈G

× ρ(U (l))Th+k T−h Tk∗ =

ω ¯ 2µ ˆ(0) X √ µ(l)ξ(τ − k)a(k)hh + l, kiSh Tk∗ n d h,k,l∈G +

1 X ˆ − l)Th Sk S ∗ µl)hh, k − lia(l)ξ(h k n h,k,l∈G

  X µ(l)a(h)a(k)hl, ki 1 X  + √ b(g + h − l)ξ(g)b(g − k) n d

+

g∈G

h,k,l∈G

× Th+k T−h Tk∗ =

ω ¯ 2µ ˆ(0)2 X √ ξ(τ − k)hk, k − τ ihh, kiSh Tk∗ nd h,k∈G X 1 X +√ hh, τ − hihh, kiξ(τ − h)Th Sk Sk∗ + a(h)a(k) n h,k∈G

h,k∈G

 X b(g + h − l)µ(l)ξ(g)b(g − k)hl, ki µ ˆ (0)µ(k)  Th+k T−h Tk∗ . √ + × d n 

g,l∈G

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The summation in the coefficient of Th+k T−h Tk∗ is 1 X √ b(−l)a(l)µ(g + h)hl, τ − (g + h + k)iξ(g)b(g − k)hg + h, ki n g,l∈G

1 X µ(g + h)hl, τ − (g + h + k)iξ(g)b(g − k)b(l)hg + h, ki = √ n =

X

g,l∈G

µ(g + h)ξ(g)b(g − k)ˆb(τ − (g + h + k))hg + h, ki

g∈G

X

= c¯

b(τ − (g + h + k))µ(g + h)ξ(g)b(g − k)hg + h, ki

g∈G

X

= c¯

µ(g + h)ξ(g)a(g + h + k − τ )b(g − k)b(g + h + k − τ )hg + h, ki

g∈G

= a(k − τ )c

X

ξ(g)b(g − k)b(g + h + k − τ ) ,

g∈G

where we use (5.2) and (5.12) several times. Thus, the coefficient of Th+k T−h Tk∗ in ρ(Uρ∗ Vρ∗ )T0 is   X a ˆ (0) + c¯ ξ(g)b(g − k)b(g + h + k − τ ) . a(h)a(k)2 a(τ )hτ, ki  d g∈G

On the other hand, the right-hand side is c¯ ωµ ˆ (0) X ξ(k)hh, kiSh Tk∗ ω ¯µ ˆ(0)Vρ ρ(T0 )Vρ∗ = √ nd h,k∈G c¯ω ¯µ ˆ(0) X √ ξ(h)hh, kiTh Sk Sk∗ n h,k∈G X +ω ¯µ ˆ(0) ξ(h + k)ξ(k)b(−h)Th+k T−h Tk∗ . +

k,h∈G

Therefore, (6.16) is equivalent to (6.19) and (6.20). (3) Thanks to (2.9), it suffices to show dφρ (Eρ (ρ)) = ω for the left inverse φρ . Note that the restriction of φρ to (ρ2 , ρ2 ) is a trace given by 1 X ∗ 1X ∗ Sg xSg + Tg xTg . φρ (x) = 2 d d g∈G

g∈G

Thus, we get

  X X ω ¯µ ˆ(0) X 1 µ(−h)  Sg∗ Vρ U (h)Sg + Tg∗ Vρ U (h)Tg  dφρ (Eρ (ρ)) = √ n d h∈G



g∈G

g∈G

 X X X 1 ω ¯µ ˆ(0) µ(−h)  hg, hi + ξ(g)Tg∗ U (h)Tg  = √ n d h∈G

g∈G

g∈G

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  X ω ¯µ ˆ (0) X n = √ µ(−h)  δh,0 + δh,0 ξ(g) n d h∈G g∈G   ω ¯µ ˆ (0)  n X + ξ(g) = √ d n g∈G

= ω, which finishes the proof. Remark. Theorem 6.4 assures that there exist exactly n(n + 3)/2 solutions for (6.18)–(6.21). Thus, once we happen to know that there exist exactly n(n + 3)/2 solutions for (6.18)–(6.20) in a concrete example, we don’t need to check (6.21). In fact, it could be cumbersome to check (6.21) if the order of the group G is not very small. We denote by Eρj , j ∈ J the half braidings for ρ, and by µj , τj , ωj , ξj , Uj , and Vj the quantities appearing in the above theorem for Eρ = Eρj . ¯ For Eρj , we introduce the dual half braiding Eρj as in [14, Theorem 4.6, (iv)] defined by Eρj (h) = dS0∗ ρ(Eρj (h)∗ αh (S0 )) , ¯

h ∈ G,

Eρj (ρ) = dS0∗ ρ(Eρj (ρ)∗ ρ(S0 )) . ¯

¯

Note that ρ˜j is the conjugate endomorphism of ρ˜j . Using the fact αg ρ = ρ for g ∈ G, we can also introduce g · j ∈ J through Eρg·j (h) = Eg (h)αg (Eρj (h)) ,

h ∈ G,

Eρg·j (ρ) = Eg (ρ)αg (Eρj (ρ)) . Proposition 6.7. Under the above notation, the following hold : (1) τ¯j = −τj , ω¯j = ωj , ξ¯j (h) = ξj (τj + h), h ∈ G. (2) τg·j = τj − 2g, ωg·j = hg, g − τj iωj , ξg·j (h) = ξj (g + h), h ∈ G. Proof. (1) By definition, we have Eρj (h) = dµj (h)S0∗ ρ(U (h)Sh ) = µj (h)hh, hiU (h)∗ ¯

= µj (−h)U (h)∗ , which shows τ¯j = −τj . ω¯j = ωj holds in general (or can be shown by direct ˆ¯j (0). Thus, computation.) Since a(g) = a(−g), g ∈ G, we have µ ˆj (0) = µ 2

ˆj (0) dTl∗ S0∗ ρ(Uj∗ Vj∗ ρ(S0 ))U¯j∗ Tl ξ¯j (l) = ωj2 µ 2

= ωj2 µ ˆj (0) dTl∗ S0∗ ρ(Uj∗ Vj∗ )U¯j∗ Tl ρ(S0 )

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ωj2 µ ˆj (0) d X = µj (p)µj (q)Tl∗ S0∗ ρ(U (p)Vj∗ )U (−q)Tl ρ(S0 ) n p,q∈G

2

ˆj (0) d X ωj2 µ µj (p)µj (q − l)Tl∗Sp∗ ρ(Vj∗ )Tq ρ(S0 ) = n p,q∈G

" X ˆj (0) d X ωj2 µ ∗ ∗ µj (p)µj (q − l)Tl Sp U (g)ρ(S0 S0∗ )U (g)∗ = n 2

p,q∈G

+

X

g∈G

# ξj (g)ρ(Tg Tg∗ ) Tq ρ(S0 ) .

g∈G 2

ˆj (0) d/n) is The first term (up to constant ωj2 µ X µj (p)µj (q − l)Tl∗ Sp∗ U (g)ρ(S0 S0∗ )U (g)∗ Tq ρ(S0 ) g,p,q∈G

=

X

µj (p)µj (q − l)hg, piTl∗ Sp∗ ρ(S0 S0∗ )Tg+q ρ(S0 )

g,p,q∈G

=

1 d2

X

µj (p)µj (q − l)hg, piδl,g+q

g,p,q∈G

√ X √ n nˆ n µj (0) = 2 µ ˆj (−g)µj (−g) = . d d2 g∈G

2

ˆj (0) d/n) is The second term (up to constant ωj2 µ X µj (p)µj (q − l)ξj (g)Tl∗ Sp∗ ρ(Tg Tg∗ )Tq ρ(S0 ) g,p,q∈G

c X ∗ µj (p)µj (q − l)ξj (g)Tl∗ Tˆg−p ρ(Tg∗ )Tq ρ(S0 ) =√ d g,p,q∈G X c µj (p)µj (q − l)ξj (g)b(g + l)hp, q − li =√ nd g,p,q∈G =

ncˆ µj (0) X ξj (g)b(g + l) . d g∈G

Thanks to (6.19), this is equal to √ ˆj (0)2 ξ(τj + l) n nˆ n¯ ωj2 µ µj (0) − . d d2 Thus we get the result.

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(2) By definition, we have Eρg·j (h) = hg, hiαg (µj (h)U (h)∗ ) = µj (h)hg, hi2 U (h)∗ . ˆg·j (0) = µ ˆj (0)µj (2g). Direct computation shows This implies τg·j = τj − 2g and µ U (g)αg (Uj ) = µj (g)µj (2g)Ug·j . Thus, we get ˆj (0)a(g)U (g)Vj αg (Uj ) Eρg·j (ρ) = ωj µ µj (0)a(g)U (g)Vj U (g)∗ Ug·j = ωj µj (2g)µj (g)ˆ = ωj µj (g)a(g)ˆ µg·j (0)U (g)Vj U (g)∗ Ug·j . This implies Vg·j = U (g)Vj U (g)∗ and ωg·j = a(g)2 hg, τj iωj . Therefore we get the result. Remark. When the order of G is odd, every solution of (6.18)–(6.21) can be obtained from that with τ = 0 through (2). Theorem 6.8. The S and T -matrices for ∆ are given as follows: 2

Sα∆ f f g ,α h

hg, hi , = λ

g, h ∈ G , 2

= Sπ∆ = Sα∆ f f f f g ,π g h h ,α

(d + 1)hg, hi , λ

(d + 2)hg, h + ki , λ

= Sσ∆ = Sα∆ g f f g g ,σ g h,k h,k ,α ∆ Sα∆ = ρj = Sρ f f ej ,α g ,e g

g, h ∈ G ,

dhτj , gi , λ

g, h, k ∈ G ,

g ∈ G,

2

∆ = Sπf f g ,π h

∆ = Sσ∆ Sπf g πg = g g ,σ h,k h,k ,f

hg, hi , λ

g, h ∈ G ,

(d+2)hg, h + ki , λ

∆ ∆ Sπf ρj = Sρ πg = − ej ,f g ,e

Sσ∆ = ^ g 0 0 g,h ,σ g ,h

g, h, k ∈ G ,

dhτj , gi , λ

g ∈ G,

(d + 2) [hg, h0 ihh, g 0 i + hg, g 0 ihh, h0 i] , λ

g, h, g 0 , k 0 ∈ G ,

∆ = 0, Sσ∆ ρj = Sρ g ej ,σg g,h ,e g,h

Sρe∆j ,e ρl =

ωj ωl X hτj + τl , gihg, gi λ g∈G

+

ˆj (0)ˆ µl (0) X dωj ωl µ ξj (g)ξl (h)hτj −τl , h−gihh−g, h−gi , nλ g,h∈G

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ωαfg = ωπfg = hg, gi , = hg, hi , ωσg g,h

g ∈ G,

g, h ∈ G ,

ωρej = ωj .

Proof. Using (2.8), Lemma 6.1, Corollary 6.3, and Theorem 6.6 we get

= Sα∆ f f g ,α h

2 1 1 Eh (g)∗ Eg (h)∗ = hg, hi , λ λ

Sα∆ f f g ,π h

Sπ∆ f f g h ,α

2

=

d+1 (d + 1)hg, hi Eg (h)∗ Eπh (g)∗h,h = , = λ λ

= Sσ∆ = Sα∆ g f f g g ,σ g h,k h,k ,α ∆ = Sα∆ ρj = Sρ f f ej ,α g ,e g

d+2 (d + 2)hg, h + ki Eg (h)∗ Eσh,k (g)∗h,h = , λ λ

dhτj , gi d da(g)µ(g) Eg (ρ)∗ Eρj (g)∗ = = , λ λ λ

d+1 [Eπh (g)∗h,h Eπg (h)∗g,g + dφρ (Eπh (g)∗ρ,ρ Eπg (ρ)∗g,g )] λ " 2# 2 2 dhg, hi hg, hi d+1 = , hg, hi − = λ d+1 λ

∆ = Sπf f g ,π h

∆ = Sσ∆ Sπf g πg = g g ,σ h,k h,k ,f

=

(d + 2)hg, h + ki , λ

∆ ∆ = Sπf ρj = Sρ f ej ,α g ,e g

Sσ∆ = ^ g 0 0 g,h ,σ g ,h

d+2 [Eπg (h)∗g,g Eσh,k (g)∗h,h + dφρ (Eπg (h)∗ρ,ρ Eσh,k (ρ)∗h,h )] λ

d+1 dhτj , gi dφρ (Eρj (g)∗ Eπg (ρ)∗g,g ) = − , λ λ

(d + 2) [Eσg0 ,h0 (g)∗g0 ,g0 Eσg,h (g 0 )∗g,g λ + Eσg0 ,h0 (g)∗h0 ,h0 Eσg,h (h0 )∗g,g + dφρ (Eσg0 ,h0 (g)∗ρ,ρ Eσg,h (ρ)∗g,g )]

=

(d + 2) [hg, h0 ihh, g 0 i + hg, g 0 ihh, h0 i] , λ

∆ = Sσ∆ ρj = Sρ g ej ,σg g,h ,e g,h

d+2 dφρ (Eρj (g)∗ Eσg,h (ρ)∗g,g ) = 0 . λ

ˆ j (−g)Sg . Thus, Note that we have Uj∗ Sg = µ

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d2 d2 ωj ωl µ ˆj (0)ˆ µl (0) φρ (Eρl (ρ)∗ Eρj (ρ)∗ ) = φρ (Ul∗ Vl∗ Uj∗ Vj∗ ) λ λ   X ˆj (0)ˆ µl (0)  X ∗ ∗ ∗ ∗ ∗ ωj ωl µ Sg Ul Vl Uj Vj Sg + d Tg∗ Ul∗ Vl∗ Uj∗ Vj∗ Tg  = λ

Sρe∆j ,e ρl =

g∈G

=

g∈G

ˆj (0)ˆ µl (0) X ωj ωl µ µ ˆl (−g)ˆ µj (−g) λ g∈G

+

ˆj (0)ˆ µl (0) dωj ωl µ nλ

X

ξj (g)µj (h)µl (k)Tg∗ U (k)Vl∗ U (h)Tg

g,h,k∈G

ωj ωl X = µl (−g)µj (−g) λ g∈G

+

ˆj (0)ˆ µl (0) dωj ωl µ nλ

X

∗ ξj (g)µj (h)µl (k)Tg+k Vl∗ Tg−h

g,h,k∈G

ωj ωl X = hτj + τl , gihg, gi λ g∈G

+

ˆj (0)ˆ µl (0) X dωj ωl µ ξj (g)ξl (g − h)µj (h)µl (−h) nλ g,h∈G

ωj ωl X hτj + τl , gihg, gi = λ g∈G

+

ˆj (0)ˆ µl (0) X dωj ωl µ ξj (g)ξl (h)hτj − τl , h − gihh − g, h − gi . nλ g,h∈G

The eigenvalues of the T -matrix can be obtained easily from (2.9), Lemma 6.1, Corollary 6.3, and Theorem 6.6. Using [29, Lemma 4.4], we get the following: Corollary 6.9. Let HΣg be the Hilbert space of the closed genus g surface Σg in the TQFT of ∆. Then, dim HΣg =

n2g−1 (n − 1) ng (n + 3)(n + 4)g−1 + 2 2 " √ √ 2g−2 √ √ 2g−2 # n+4+ n n+4− n 2g−1 g−1 (n + 4) + . +n 2 2

Example 6.1. Let G = Z/2Z. The solutions of (6.18)–(6.21) for Example 5.1 are given as follows:

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τ

ω

Eρ1

0

e

Eρ2

0

e

Eρ3

1

e

Eρ4

1

e

Eρ5

1

π

µ ˆ(0)

√ −1 3

e

π

√ −1 4

π

√ −1 4

4π

√ −1 3

e

5π

√ −1 5

e−

π

√ −1 4

5π

√ −1 5

e−

π

√ −1 4

√ − −1

e−

π

√ −1 4

ξ(0)

ξ(1)

√ √ 1 − −1 7π −1 √ e 12 2 √ √ 1 + −1 7π −1 √ e 12 2 p √ √ √ 1 − 3 + −2 3 5π −1 e 6 2 p √ √ √ 1 − 3 − −2 3 5π −1 e 6 2

√ √ 1 + −1 7π −1 √ e 12 2 √ √ 1 − −1 7π −1 √ e 12 2 p √ √ √ 1 − 3 − −2 3 5π −1 e 6 2 p √ √ √ 1 − 3 + −2 3 5π −1 e 6 2

e

√ −5π −1 6

e

√ −5π −1 6

From Theorem 6.7, we get   d ˆ (0)ˆ µl (0)(ξj (0) − ξj (1))(ξl (0) − ξl (1)) , τj + τl = 0 j ωj ωl  2 µ ∆ Sρej ,e ρl = λ  2 + dµ ˆj (0)ˆ µl (0)(ξj (0) + ξj (1))(ξl (0) + ξl (1)) , τj + τl = 1. 2 e α Therefore, the S and T -matrices with respect to the basis corresponding to id, f1 , j π1 , σg ρ e , j = 1, 2, · · · , 5, are as follows: π f0 , f 0,1 ! A B 1 , S∆ = λ tB C 

1

1

d+1

d+1

1

d+1

d+1

d+1

1

1

d+1

1

1

−(d + 2)

d+2

−(d + 2)

 1   A= d + 1  d + 1 d+2



d

 d   B=  −d   −d 0 

d

 d   C=  −d   −d 2d

d

d

d

d

−d

−d

−d

−d

−d

−d

d

d

0

0

0

−(d + 2)    d+2  ,  −(d + 2) 

d

0 

−d    −d  ,  d  0

d

−d

−d

d

d

d

d

√ −(d + 2) −1 √ (d + 2) −1

√ (d + 2) −1 √ −(d + 2) −1

−2d

0

0

d



d+2

2d



−2d    0  ,  0  0

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√ √ √ √ √ π −1 4π −1 5π −1 5π −1 T ∆ = Diag 1, −1, 1, −1, 1, e 3 , e 3 , e 6 , e 6 , − −1 . (Complex conjugate of) these matrices are also obtained in [29]. 7. Haagerup Subfactor and Generalization In [11], U. Haagerup√proved that there exists no finite depth subfactor of index between 4 and√(5 + 13)/2, and he claimed that there exists a unique subfactor of index (5 + 13)/2, which we call the Haagerup subfactor. In [1], Asaeda and Haagerup gave a proof of the existence of this √ subfactor, and they also constructed another finite depth subfactor of index (5 + 17)/2. The fusion rules of one of the even systems of the sectors for the Haagerup subfactor is as follows: [α]3 = [id] ,

[α][ρ] = [ρ][α−1 ] ,

[ρ]2 = [id] ⊕ [ρ] ⊕ [αρ] ⊕ [α2 ρ] . In this section, we construct a system of endomorphisms of the Cuntz algebra O4 obeying the above fusion rules, and give a new proof of the existence of the Haagerup subfactor using Longo’s Q-system [20]. We start with the following lemma, which can be shown by direct computation. Lemma 7.1. Let M be an infinite factor and ρ an irreducible endomorphism of statistical dimension d > 1. We assume that S0 ∈ (id, ρ2 ) and T0 ∈ (ρ, ρ2 ) are isometries satisfying S0∗ ρ(S0 ) =

1 , d

T0∗ ρ(T0 ) = −

1 T0∗ ρ(S0 ) = √ T0 , d

1 S0∗ ρ(T0 ) = √ T0∗ , d

d 1 + S0 S0∗ + T0 T0∗ . d−1 d−1

We take isometries V1 , V2 ∈ M satisfying the O2 relation V1 V1∗ + V2 V2∗ = 1, and set γ1 (x) := V1 xV1∗ + V2 ρ(x)V2∗ ,

x∈M,

1 1 V1 + √ V2 ρ(V1 )V2∗ + W1 := √ d+1 d+1 r d−1 V2 ρ(V2 )T0 V2∗ . + d+1

r

d V2 ρ(V2 )S0 V1∗ d+1

Then, (γ1 , V1 , W1 ) is a Q-system : V1 ∈ (id, γ1 ) and W1 ∈ (γ1 , γ12 ) are isometries satisfying 1 , V1∗ W1 = γ1 (V1 )∗ W1 = √ d+1 W1∗ γ1 (W1 ) = W1 W1∗ ,

γ1 (W1 )W1 = W12 .

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In consequence, there exists a subfactor M ⊃ L ⊃ γ1 (M ) whose canonical endomorphism is γ1 . Let n be an odd number and G be a finite abelian group of order n. Let On+1 be the Cuntz algebra with the canonical generators {S0 , Tg }g∈G . We consider a function A : G × G → C satisfying the following conditions: g, h ∈ G ,

A(g, h) = A(h, g) ,

(7.1)

A(g, h) = A(−h, g − h) = A(h − g, −g) , A(g, 0) = δg,0 − X

1 , d−1

A(g + k, h)A(k, h) = δg,0 −

k∈G

X

g, h ∈ G ,

g ∈ G, δh,0 , d

(7.2) (7.3)

g, h ∈ G ,

(7.4)

g, h, k, l ∈ G ,

(7.5)

A(m, g + h)A(m + k, g)A(m + l, h)

m∈G

= A(g + l, k)A(h + k, l) − where d = (n +

δg,0 δh,0 , d

√ n2 + 4)/2 satisfying d2 = 1 + nd. (7.3) implies X g∈G

1 A(g, 0) = − . d

(7.6)

We definite an action α of G on On+1 and an endomorphism ρ ∈ End(On+1 ) by αg (S0 ) = S0 , ρ(S0 ) =

αg (Th ) = Th+2g ,

g, h ∈ G ,

1 1 X S0 + √ Tg Tg , d d g∈G

1 ∗ + T−g S0 S0∗ ρ(Tg ) = √ S0 T−g d X + A(g + h, g + k)Th Tg+h+k Tk∗ ,

g ∈ G.

h,k∈G

Thanks to (7.4) and (7.6), ρ is well-defined. Theorem 7.2. Let A be a function satisfying (7.1)–(7.4), and α and ρ be as above. Then, the following hold : (1) αg · ρ = ρ · α−g , g ∈ G, and S0 ∈ (id, ρ2 ). (2) If A further satisfies (7.5), then Tg ∈ (αg · ρ, ρ2 ).

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Proof. (1). It is easy to show the first relation. In a similar way as in the proof of Theorem 5.3, we can show that S0 ∈ (id, ρ2 ) is equivalent to S0∗ ρ2 (S0 )S0 = S0 , ρ(S0∗ ρ(Tg ))S0 = ρ(S0∗ )S0 Tg ,

ρ(Th∗ ρ(Tg ))S0 = ρ(Th∗ )S0 Tg ,

g, h ∈ G ,

which follows from (7.1), (7.4), and (7.6). (2). Thanks to (1), it suffices to show the statement for g = 0, which is again equivalent to T0∗ ρ2 (S0 )T0 = ρ(S0 ) , ρ(S0∗ ρ(Tg ))T0 = ρ(S0∗ )T0 ρ(Tg ) ,

g ∈ G,

ρ(Th∗ ρ(Tg ))T0 = ρ(Th∗ )T0 ρ(Tg ) ,

g, h ∈ G .

Indeed, using (7.1) and (7.4), we get T0∗ ρ2 (S0 )T0 =

1 ∗ 1 X ∗ T0 ρ(S0 )T0 + √ T0 ρ(Tg Tg )T0 d d g∈G

" # X 1 X T02 ∗ ∗ = √ +√ A(g, g + k)Tg+k Tk δg,0 S0 S0 + d d d g∈G k∈G " ×

δg,0 S0 X √ + A(g + h, g)Th Tg+h d h∈G

#

S0 1 X T2 +√ A(g, h)A(h, g)Th Th = ρ(S0 ) . = √0 + d d d d g,h∈G The second equation easily follows from (7.1) and (7.2). The left-hand side of the third one is ! X ∗ ∗ ∗ A(g + h, g + k)Tg+h+k Tk T0 ρ(Th ρ(Tg ))T0 = ρ δg+h,0 S0 S0 + k∈G

=

δg+h,0 ρ(S0 )T0∗ √ d +

X k∈G

A(g + h, g + k)ρ(Tg+h+k ) "

" δk,0 S0 S0∗

+

X l∈G

# A(k, k +

# S0 T0∗ X ∗ √ + Tl Tl T0 + A(g + h, g)T−(g+h) S0 S0∗ d l∈G " X δg+h+k+l,0 S0 √ + A(g + h, g + k)A(k, k + l) d k,l∈G

δg+h,0 = d

∗ l)Tl Tk+l

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+

X

# ∗ A(g + h + k + p, g + h + k + l)Tp Tg+h+k+l+p Tk+l

p∈G

=

1 X δg+h,0 S0 T0∗ ∗ √ +√ A(g + h, g + k)A(k, −(g + h))S0 T−(g+h) d d d k∈G + A(g + h, g)T−(g+h) S0 S0∗ +

δg+h,0 X Tl Tl T0∗ d l∈G

+

X

A(g + h, g + q)A(g + h + p + q, g + h + k)

k,p,q∈G

× A(q, k)Tp Tg+h+k+p Tk∗ =

1 X δg+h,0 S0 T0∗ ∗ √ +√ A(g +h, g +k)A(g +h, g +k+h)S0 T−(g+h) d d d k∈G + A(g + h, g)T−(g+h) S0 S0∗ +

δg+h,0 X Tl Tl T0∗ d l∈G

+

X

A(q, g + h + k)A(q − h − p, g + h)A(q − g − h − p, k)

k,p,q∈G

× Tp Tg+h+k+p Tk∗ . On the other hand, the right-hand side is X ∗ A(h, h + l)Tl Th+l ρ(Tg ) + δh,0 S0 S0∗ ρ(Tg ) ρ(Th∗ )T0 ρ(Tg ) = l∈G

=

X

A(h, l)Tl−h Tl∗ ρ(Tg ) +

l∈G

=

∗ δh,0 S0 T−g √ d

∗ δh,0 S0 T−g √ + A(h, −g)T−(g+h) S0 S0∗ d X + A(h, h + p)A(g + h + p, g + k)Tp Tg+h+p+k Tk∗ . p,k∈G

Thus, the third equation holds. Note that we have T0∗ ρ(T0 ) = S0 S0∗ +

X k∈G

A(0, k)Tk Tk∗ = S0 S0∗ + T0 T0∗ −

1 X Tk Tk∗ , d−1 k∈G

which shows that ρ satisfies the condition of Lemma 7.1. Therefore, to prove the existence of the Haagerup subfactor, it suffices to solve the Eqs. (7.1)–(7.5) for

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G = Z/3Z. Before giving a solution, we look up (7.5) more carefully in order to obtain solutions for more general groups. Lemma 7.3. Let A be a function satisfying (7.1)–(7.4). Then, (1) In either of the following special cases, (7.5) is equivalent to (7.5)0 below : (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)

g = 0, h = 0, g + h = 0, k = 0, l = 0, g + l = 0, h + k = 0, g + l = k, h + k = l.

A(p + r, q)A(q, r) − A(−r, p)A(p, q − r) =

δp,0 − δq,0 , d−1

p, q, r ∈ G .

(7.5)0

(2) If g 6= 0, h 6= 0, g 6= h, (7.5)0 implies

√ d . |A(g, h)| = d−1

(3) Assume that G is a cyclic group Z/nZ and A satisfies (7.5)0 . Then, there exists j(g) ∈ T, g = 1, 2, · · · , n − 1 satisfying j(k)j(n − k) = j(n − 1) and √  j(h)j(g)j(h − g) d    0 −∞ .

x>1

(1.2)

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We shall also assume that n−1 R(X) ⊂ [0, ∞) σ −∆X + 4n

769

(1.3)

where ∆X is the Laplace–Beltrami operator on X, R(X) is the scalar curvature of X and n = dim X. Although we assumed the inclusion relation, our main interest is the case that the equality holds in (1.3). As examples, the Euclidean space and the hyperbolic space are included in this class. Passing to the Lorentzian metric ds2X − dt2 on X × R and using conformal equivalence, our class also covers the Schwarzschild metric. We consider a classical field of a scalar particle obeying the following Klein– Gordon equation (∂t2 + L0 + L1 (t))u = 0 ,

(1.4)

n−1 R(X) + m2 , L1 (t) = ai (t)∂i + a0 (t) , (1.5) 4n where m ≥ 0 is the mass of the particle. The coefficients ai (t) of the perturbation term L1 (t) depend also on the space variables, which are omitted for simplicity’s sake. L1 (t) is assumed to be formally self-adjoint. We rewrite (1.4) into a first order system of equations L0 = −∆X +

i∂t ψ = H(t)ψ = (H0 + V (t))ψ .

(1.6)

Under a suitable decay assumption on ai (t), the classical scattering operator S associated with (1.6) is introduced. The operator A defined by S = 1 − iA is called the classical scattering amplitude. The next step is to solve the following equation AF = A + iAP− AF

(1.7)

with A as an input, where P− is the projection onto the negative spectral subspace for H0 . This is a well-known procedure in external QFT and has alredy been studied by the above cited authors [2, 18, 13, 12], and logically we have only to accept (1.7) and to solve it. However Eq. (1.7) is closely related with Feynman’s propagator in QED, which also has a close connection to the inverse scattering theory of Faddeev. Therefore we shall elucidate the derivation of (1.7) in detail. Our main purpose is to implement the classical scattering operator S on the Fock space. It is well-known that the implementability of S is equivalent to that P+ SP− , P− SP+ are Hilbert–Schmidt, where P+ is the projection onto the positive spectral subspace for H0 . The Hilbert–Schmidt property of P± SP∓ is proved by introducing a parametrics of −∆X in the form of pseudo-differential operators. Our main theorem is Theorem 7.3. For the massless case (m = 0), one cannot in general expect the same result as above. The main difficulty is that one needs detailed spectral properties of L0 at the bottom of the spectrum. In this paper we shall only deal with the case of asymptotically flat and hyperblic spaces (see Theorems 8.5 and 8.7).

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Let Ω be the vacuum of the Fock space. Then the implemented S-matrix S satisfies (Ω, SΩ) = [det(1 − (B+− )∗ B+− ]1/2 ,

B+− = P+ AF P− .

(1.8)

An interesting consequence of (1.8) is the following equivalence 1 − |(Ω, SΩ)|2 > 0 ⇐⇒ P+ AP− 6= 0 ⇐⇒ P− AP+ 6= 0 .

(1.9)

The left-hand side is an assertion in quantum field theory and states that the probability of creation of particles from the vacuum is positive. The assertins P+ AP− 6= 0, P− AP+ 6= 0 are those in classical field theory and state that particles with negative (positive) energy in the past turn into a positive (negative) energy state due to the scattering by L1 (t). This is the so-called Klein’s paradox, a sort of prelude of Dirac’s theory of positron (see [4]). We shall discuss (1.9) again in Sec. 6. There are many interesting subjects related to external QFT. They are seen in [6, 19] and [20]. We shall assume the integrability in t on ai (t). This rapid decay assumption excludes many physically interesting examples. This is one of the limitations of external QFT. The plan of this paper is as follows. In Sec. 2, we state the assumptions on the manifold X. The classical S-matrix is constructed in Sec. 3. The derivation of Eq. (1.7) is explained in Sec. 4. The solvability of (1.7) is well-known. However we reproduce it in Sec. 5 in order to prove the equivalence (1.9). In Sec. 6, we construct the Fock space and state the main theorem (Theorem 6.1). The Hilbert–Schmidt property of P± SP∓ is proved in Secs. 6 and 7. Section 8 is devoted to study the massless case. The notation used in this paper is standard. We use Einstein’s summation conP vention: i ai bi = ai bi . For two Banach spaces X and Y , B(X; Y ) denotes the set of all bounded operators from X to Y , and B(X) = B(X; X). For a set M and a measure dσ on M , L2 (M ; dσ) is the space of L2 -functions on M with respect to dσ. 2. Invariant Wave Equations on Riemannnian Manifolds 2.1. Klein Gordon equation Let X be an n-dimensional Riemannian manifold equipped with Riemannian metric gij (x)dxi dxj . We call the following equation the Klein–Gordon equation on X: (∂t2 − ∆X + V0 (x))u = 0 , V0 (x) =

n−1 R(X) + m2 , 4n

(2.1) (2.2)

where ∆X is the Laplace–Belrami operator on X, R(X) is the scalar curvature of X and m is a non-negative constant. This equations is a natural object by the following reason of conformal invariance.

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2.2. Conformal invariance Let Y be an N -dimensional Lorentzian manifold endowed with Lorentzian metric PN −1 α 2 N 2 gαβ dy α dy β , which is reduced to α=1 (dy ) − (dy ) at each point of Y . Let √ 1 αβ α g = | det(gαβ )| and = √g ∂α ( gg ∂β ), ∂α = ∂/∂y , and let R(Y ) be the scalar curvature of Y . Suppose we are given another Lorentzian metric gαβ dy α dy β on Y satisfying (gαβ ) = e2ϕ (gαβ )

(2.3)

¯ be the D’Alembertian associated with for some function ϕ on Y . Then letting this Lorentzian mteric and R(Y ) be the associated scalar curvature, we have N −2 N −2 −(N +2)ϕ/2 ¯ R(Y ) = e R(Y ) e(N −2)ϕ/2 . (2.4) − − 4(N − 1) 4(N − 1) This well-known formula shows that N −2 R(Y ) u = 0 − 4(N − 1)

(2.5)

is a conformally invariant wave equation on Y . Now we multiply the above Riemannian manifold X by the time axis, and regard the resulting Y = X × R as the Lorentzian manifold with the Lorentzian metric gij (x)dxi dxj −(dt)2 . Note that R(Y ) = R(X) in this case. By the term of Fulling [6], a Lorentzian metric of the form gij (x)dxi dxj − e2ϕ(x) (dt)2 is called static. Suppose that two metrics of X (gij ) and (gij ) are conformal: (gij ) = e2ϕ (gij ). The above argument shows that when m = 0, Eq. (2.1) is the canonical wave equation for the static metric because of the conformal change gij (x)dxi dxj − e2ϕ(x) (dt)2 = e2ϕ(x) (gij (x)dxi dxj − (dt)2 ) . 2.3. Model spaces We have in mind the space having the following properties as a model for X. (M-1) X ' (0, ∞) × M . Namely, X is diffeomorphic to the product of (0, ∞) and a compact Riemannian manifold M of dimension n − 1. (M-2) The metric of X is ds2 = (dx)2 + e2λ(x) gM , where x ∈ (0, ∞) and gM is the metric on M, and λ(m) (x) ∈ L∞ ((1, ∞)) , inf λ(x) > −∞ .

x>1

(M-3) σ(−∆X +

n−1 4n R(X))

⊂ [0, ∞).

∀m ≥ 1,

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Let us note that these properties imply that ∂2 ∂ + e−2λ ∆M , + (n − 1)λ0 (2.6) ∂x2 ∂x being the Laplace–Beltrami operator on M , the volume element of X is ∆X =

∆M

e(n−1)λ dxdM ,

(2.7)

dM being the volume element of M , and that ∂2 (n − 1)2 0 2 n − 1 00 (n−1)λ/2 −(n−1)λ/2 (λ ) + λ + e−2λ ∆M . (2.8) ∆X e = − e ∂x2 4 2 To compute the scalar curvature, let us recall the following well-knwon lemma (see e.g. [1]). Lemma 2.1. Let X be an n-dimensional Riemannian manifold equipped with two metrics gij dxi dxj and gij dxi dxj . Suppose they are conformal : (gij ) = e2ϕ (gij ) for some ϕ. Let R and R be the associated scalar curvatures. Then e2ϕ R = R − 2(n − 1)ϕii , n−2 ϕi ϕi , ϕi = ∂i ϕ , ϕi = g ij ϕj , 2 where ∂i = ∂/∂xi , and ∇i is the covariant differentiation. ϕii = ∇i ϕi +

Using this lemma and the following conformal relation (dx)2 + e2λ gM = e2λ (e−2λ (dx)2 + gM ) , one can show R(X) = e−2λ R(M ) − (n − 1)(2λ00 + n(λ0 )2 ) ,

(2.9)

where R(M ) is the scalar curvature of M . In fact, letting R0 be the scalar curvature associated with the metric e−2λ (dx)2 + gM , we have by Lemma 2.1 e2λ R(X) = R0 − 2(n − 1)λii . Obviously R0 = R(M ). Choosing x as the 1st coordinate, we have n−2 n−2 λ1 λ1 = (λ1 )0 + Γ111 λ1 + λ1 λ1 . λii = ∇1 λ1 + 2 2 Since λ1 = λ0 e2λ and Γ111 = −λ0 , we get (2.9). Recall that if M is the space of constant curvature R(M ) = (n − 1)(n − 2)k , where k is the sectional curvature of M . In particulr if M = S R(S

n−1

) = (n − 1)(n − 2) .

Let us denote the eigenvalues of −∆M by 0 = µ0 < µ1 ≤ µ2 ≤ · · · ≤ µk ≤ · · · .

(2.10) n−1

, (2.11)

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Then if R(M ) is constant, we have the following unitary equivalence by virtue of (2.8) and (2.9) n − 1 00 n − 1 −2λ n−1 d2 −2λ R(X) ' ⊕∞ λ e + + R(M ) + µ e −∆ + − , k k=0 4n dx2 2n 4n where each operator in the parenthesis is defined in L2 ((0, ∞); dx). Example 2.2. X = Rn = Euclidean space. In this case, letting x ∈ (0, ∞) be the radial variable and M = S n−1 , we have ds2 = (dx)2 + x2 gM ,

R(X) = 0 .

Strictly speaking, X \ {0} has the structure of the model space. Example 2.3. Space of constant negative curvature −1/a2 . We employ the ball model X = {z ∈ Rn ; |z| < a} with metric 2 2a2 2 ((dr)2 + r2 gM ) , M = S n−1 , ds = a2 − r 2 where r = |z|. By the change of variable x = a log a+r a−r , we have X ' (0, ∞) × M and a2 ds2 = (dx)2 + (ex/a − e−x/a )2 gM . 4 Therefore (n − 1)(n − 3) 4 d2 n−1 ∞ R(X) ' ⊕k=0 − 2 + 2 x/a + µk , −∆ + 4n dx 4 a (e − e−x/a )2 n−1 . It is {µk }∞ k=0 being the eigenvalues of the Laplace–Beltrami operator on S 2 2 known that −∆X is absolutely continuous and σ(−∆X ) = [(n−1) /(4a ), ∞). Since (n−1)/(4n)R(X) = −(n−1)2 /(4a2 ), we have σ(−∆X +(n−1)/(4n)R(X)) = [0, ∞).

Example 2.4. Schwarzschild metric. This is the following metric on a 4dimensional space-time: a a −1 (dt)2 , r > a , (dr)2 + r2 gM − 1 − 1− r r where a > 0 and M = S 2 . By the change of variable x = −r − a log(r − a), this metric becomes r3 a {(dx)2 + f (x)gM − (dt)2 } , f (x) = . 1− r r−a Therefore, we consider the Riemannian manifold X = (−∞, ∞) × S 2 with metric ds2 = (dx)2 + f (x)gM , Letting λ =

1 2

M = S2 .

3

r log r−a , we have

3a 1 dλ = 2− , dx 2r r

d2 λ 3a2 4a 1 =− 4 + 3 − 2. 2 dx r r r

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Therefore we have R(X) = −

3a2 , 2r4

which leads us to the untary equivalence 1 a(r − a) r − a d2 + + µ − , −∆X + R(X) ' ⊕∞ k k=0 6 dx2 r4 r3 2 {µk }∞ k=0 being the eigenvalues of the Laplace–Beltrami operator on S . This shows that 1 σ −∆X + R(X) = [0, ∞) . 6

This space has two infinities. The first one is at x = −∞ (r = ∞), around which f (x) ∼ x2 and R(X) ∼ 0. This is the usual infinity and the space is asymptotically flat there. The second one is at x = ∞ (r = a), around which f (x) ∼ a3 ex/a , R(X) ∼ − 2a32 . This infinity is usually called the event horizon. When a = 1, X is asymptotic to the hyperbolic space near this infinity. 2.4. Assumptions on the manifold Let us now state the assumptions on our manifold. Let X be an n-dimensional Riemannian manifold. Suppose X is a union of finite number of open sets X = X0 ∪ X1 ∪ · · · ∪ Xk having the following properties. (A-1) X is connected and Xi ∩ Xj = ∅ if i, j ≥ 1, i 6= j. (A-2) X0 is compact. (A-3) Each Xi (1 ≤ i ≤ k) has the same structure as the model space. Namely (A-3-1) Xi is diffeomorphic to (ri , ∞) × Mi , where Mi is a compact Riemannian manifold of dimension n − 1. (A-3-2) On Xi , the Riemannian metric has the following form ds2 |Xi = (dx)2 + e2λi (x) gMi ,

x ∈ (ri , ∞) ,

where gMi is the Riemannian metric on Mi and (m)

λi

(x) ∈ L∞ ((ri , ∞)) ,

∀m ≥ 1.

inf λi (x) > −∞ .

x>ri

Let R(X) be the scalar curvature of X. We also assume that (A-4) σ(−∆X +

n−1 4n R(X))

⊂ [0, ∞).

Let us remark that by the above assumptions X is geodesically complete, hence the Laplace–Beltrami operator ∆X is essentially self-adjoint on C0∞ (X).

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3. Classical S-Matrix 3.1. Assumptions on the perturbations Let X be a Riemannian manifold satisfying the assumptions (A − 1) ∼ (A − 4) in Sec. 2, and let us consider the wave equation ∂t2 u + (L0 + L1 (t))u = 0 ,

(3.1)

n−1 R(X) + m2 . (3.2) 4n Until Sec. 8, we shall assume that m > 0. L1 (t) is a formally self-adjoint 1st order differential operator: L0 = −∆X + V0 (x) ,

V0 (x) =

L1 (t) = ai (t)∂i + b(t) ,

(3.3)

where a(t) = (a1 (t), . . . , an (t)) and b(t) are smooth vector and scalar fields on X depending smoothly on t ∈ R. For the sake of simplicity, we are omitting the space variables. To state the assumptions on a(t) and b(t), we need some preparations. For a tensor field A = (Aj1 j2 ... k1 k2 ... ) on X, we put Ai1 i2 ...

j1 j2 ...

k1 k2 ...

= ∇i1 ∇i2 · · · Aj1 j2 ...

k1 k2 ... ,

(3.4)

where (∇1 , . . . , ∇n ) is the covariant differentiation. At each point of X, let |A|m =

m X

|Ai1 ···ip

j1 j2 ...

k1 k2 ... A

i1 ···ip

j1 j2 ...

|

k1 k2 ... 1/2

(3.5)

p=0

denote the semi-norm of the derivatives up to order m of A. |A|m is a scalar function on X. We fix a partition of unity {χi }ki=0 on X such that χ0 ∈ C0∞ (X) ,

χ0 = 1

supp χi ⊂ Xi ,

on X0 ,

1 ≤ i ≤ k.

(3.6) (3.7)

On compact portions of X, one can state the assumptions in an invariant way. We assume: (EM-1) For any p, m ≥ 0 sup(|∂tp χ0 a(t)|m + |∂tp χ0 b(t)|m ) ∈ L1 (R) ∩ L∞ (R)

(3.8)

X

as a function of t. For 1 ≤ i ≤ k, χi a(t) and χi b(t) are tensor fields supported in the end Xi . The assumptions for them depend on the space structure of Xi ' (ri , ∞) × Mi . We fix a coordinate system on Mi . On each coordinate patch, we take local coordinates (θ1 , . . . , θn−1 ) and put Dj = e−λi (x)

∂ , ∂θj

Dn =

∂ . ∂x

(3.9)

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We assume that on this coordinate patch, L1 (t) takes the following form χi L1 (t) =

n X

aj(i) (t)Dj + b(i) (t) .

(3.10)

j=1

For a multi-index α = (α1 , . . . , αn ), let Dα = D1α1 · · · Dnαn . We assume (EM-2) For any α and p ≥ 0 sup(|∂tp Dα aj(i) (t)| + |∂tp Dα b(i) (t)|) ∈ L1 (R) ∩ L∞ (R)

(3.11)

Xi

for 1 ≤ i ≤ k, 1 ≤ j ≤ n. The following assumption is needed in Sec. 7. (EM-3) There exists an integer m > n/4 such that

p j

sup |Dα ∂t a(i) (t)|e(2m−1/2)λi (x) 2 Mi

L ((ri ,∞);dx)

Mi

L2 ((ri ,∞);dx)

p

sup |Dα ∂t b(i) (t)|e(2m−1/2)λi (x)

∈ L1 (R; dt) ,

(3.12)

∈ L1 (R; dt) .

(3.13)

3.2. 1st order systems Let us transform (3.1) into the 1st order system. Let ω=

p

L0 ,

1 ψ= 2

(ω)1/2

i(ω)−1/2

(ω)1/2

−i(ω)−1/2

!

u ut

! .

(3.14)

Then we have i∂t ψ = H(t)ψ

on H = (L2 (X))2 ,

H(t) = H0 + V (t) , ! ω 0 , H0 = 0 −ω V (t) =

v(t)

v(t)

−v(t)

−v(t)

(3.15) (3.16) (3.17)

! ,

v(t) = (2ω)−1/2 L1 (t)(2ω)−1/2 .

(3.18)

H(t) is not self-adjoint in H, although so is L0 + L1 (t) in L2 (X). However letting ! ! 1 0 0 0 , P− = , (3.19) P+ = 0 0 0 1 ! 1 0 , (3.20) J = P+ − P− = 0 −1

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we have H(t)∗ J = JH(t) .

(3.21)

Note that P+ (P− ) is the projection onto the positive (negative) spectral subspace for H0 . Let U0 (t, s) and U (t, s) be the evolution operators for H0 and H(t), respectively. The formula (3.21) implies that U (t, s)∗ J = JU (s, t) .

(3.22)

It follows from (3.8) and (3.11) that kV (t)kB(H) , the operator norm of V (t) on H, is integrable in t ∈ R. This implies that sup kU (t, s)kB(H) < ∞ .

(3.23)

t,s

It also shows the existence of wave operators W± (s) = lim U (s, t)U0 (t, s) .

(3.24)

W± (s)−1 = lim U0 (s, t)U (t, s) .

(3.25)

t→±∞

Moreover we have t→±∞

Let the scattering operator S be defined by S = W+ (0)−1 W− (0) .

(3.26)

Since W+ (s)∗ = JW+ (s)−1 J by (3.22), we have S ∗ JS = SJS ∗ = J .

(3.27)

We also call S the classical S-matrix. 4. Feynman’s Scattering Amplitude The classical scattreing amplitude A is defined from the classical S-matrix as follows: S = 1 − iA .

(4.1)

Our next aim is to solve the following equation AF = A + iAP− AF ,

(4.2)

with A as an input, where P− is defined by (3.19). From the logical point of view, we have only to accept this equation and to solve it. However, the derivation of this equation is worth explaining. Since it is of general character, we do it in an abstract setting.

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4.1. Feynman’s propagator Let H(t) be a linear operator depending on t defined in a Hilbert space H. Suppose that H(t) generates an evolution operator such that sup kU (t, s)kB(H) < ∞ . t,s

By a propagator of H(t), we simply mean a right inverse of i∂t − H(t). Among the many propagators, standard ones are the retarded and the advanced propagators defined by Z t U (t, s)f (s)ds , (4.3) Gret f = −i −∞

Z

∞

Gadv f = i

U (t, s)f (s)ds ,

(4.4)

t

acting for f ∈ L1 (R; H). They are the unique solution of the equation (i∂t − H(t))Gret f = f ,

Gret f → 0

(t → −∞) ,

(i∂t − H(t))Gadv f = f ,

Gadv f → 0 (t → ∞) . (0)

(0)

For a time-independent self-adjoint operator H0 , let Gret and Gadv be the associated retarded and advanced propagators. Suppose there exist two self-adjoint operators P± such that P+ + P− = 1, [P± , H0 ] = 0. The Feynman’s free propagator (0) GF is defined to be (0)

(0)

(0)

GF = P+ Gret + P− Gadv .

(4.5)

Usually, P+ (P− ) is the projection onto the positive (negative) spectral subspace for H0 . For H(t) = H0 + V (t), The Feynman’s perturbed propagator GF is a solution to the equation (0)

(0)

(0)

(0)

GF = GF + GF V GF = GF + GF V GF ,

(4.6)

where V = V (t). 4.2. Feynman’s scattering amplitude Suppose that kV (t)kB(H) ∈ L1 ((−∞, ∞); dt). Then the associated scattering amplitude A defined by (4.1) has the following expression. Lemma 4.1. Z Z ∞ eitH0 V (t)e−itH0 dt + A= −∞

∞

−∞

eitH0 V (t)Gret (V (·)e−i·H0 )dt .

(4.7)

Proof. Since kU (t, 0)W− (0) − U0 (t, 0)k → 0 (t → −∞) and kU (t, 0) − U0 (t, 0) W+ (0)−1 k → 0 (t → ∞), we see that for any ϕ ∈ H, u(t) = U (t, 0)W− (0)ϕ satisfies ku(t) − e−itH0 ϕk → 0 ku(t) − e−itH0 Sϕk → 0

(t → −∞) ,

(4.8)

(t → ∞) .

(4.9)

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Let u(t) = e−itH0 ϕ + v(t). Then v(t) satisfies (i∂t − H(t))v(t) = V (t)e−itH0 ϕ ,

v(t) → 0

(t → −∞) .

Therefore u(t) is written as u(t) = e−itH0 ϕ + Gret (V (·)e−i·H0 ϕ) .

(4.10)

Since v(t) also satisfies (i∂t − H0 )v(t) = V (t)u(t) , v(t) is rewritten as

Z v(t) = −i

t

e−i(t−s)H0 V (s)u(s)ds .

(4.11)

−∞

Therefore by virtue of (4.9)

Z

Sϕ = ϕ − i

∞

eitH0 V (t)u(t)dt .

−∞

This together with (4.10) proves the lemma. Now Feynman’s scattering amplitude AF is defined as Z ∞ Z ∞ eitH0 V (t)e−itH0 dt + eitH0 V (t)GF (V (·)e−i·H0 )dt . AF = −∞

(4.12)

−∞

Namely, we replace Gret by GF in (4.7). (0)

(0)

Lemma 4.2. Let T = GF − Gret . Then GF = Gret + (1 + Gret V )T (1 + V GF ) . Proof. We compute (0)

(0)

(1 + Gret V )(GF − Gret )(1 + V GF ) (0)

(0)

(0)

(0)

= (1 + Gret V )(GF + GF V GF ) − (Gret + Gret V Gret )(1 + V GF ) = (1 + Gret V )GF − Gret (1 + V GF ) = GF − Gret , (0)

(0)

where we have used (4.6) and the equation Gret = Gret + Gret V Gret . Theorem 4.3. AF = A + iAP− AF . Proof. We define operators ∈ B(H; L∞ (R; H)) Φ0 (t) = e−itH0 , Φret (t) = Φ0 (t) + Gret V Φ0 , ΦF (t) = Φ0 (t) + GF V Φ0 ,

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where Φ0 = Φ0 (t). Then we have by Lemma 4.2 ΦF (t) − Φret (t) = (GF − Gret )V Φ0 = (1 + Gret V )T (1 + V GF )V Φ0 = (1 + Gret V )T V ΦF . Noting that

Z (0)

(0)

T f = (GF − Gret )f = iP− we have

Z T V ΦF = iP− Φ0

∞

−∞

∞

e−i(t−s)H0 f (s)ds ,

−∞

eitH0 V (t)ΦF (t)dt = iP− Φ0 AF .

This shows ΦF (t) − Φret (t) = i(1 + Gret V )Φ0 P− AF . The theorem then follows by multiplying this equation by eitH0 V (t) and integrate in t. Remark 4.4. There is a close connection between the above arguments and Faddeev’s theory of multi-dimensional inverse scattering for Schrödinger operators [5]. He introduced a new Green’s function of the Laplacian, which is outgoing in some half space and incoming in another half space. The Faddeev scattering amplitude is defined in the same way as the usual one with the usual Green operator replaced by Faddeev’s Green operator. One then gets the equation similar to the one in Theorem 4.3. See also [9]. 5. Solvability of the Equation AF = A + iAP− AF Let us discuss the solvability of Eq. (4.2). Lemma 5.1. For , 0 = ±, let A0 = P AP0 . Then 1 − iA−− and 1 − iA++ are bijections. Proof. From the equation S ∗ JS = SJS ∗ = J, we have A∗ JA = i(JA − A∗ J) ,

AJA∗ = i(AJ − JA∗ ) .

(5.1)

They imply that (A++ )∗ A++ − (A−+ )∗ A−+ = iA++ − i(A++ )∗ ,

(5.2)

(A+− )∗ A+− − (A−− )∗ A−− = −iA−− + i(A−− )∗ ,

(5.3)

A++ (A++ )∗ − A+− (A+− )∗ = iA++ − i(A++ )∗ ,

(5.4)

A−+ (A−+ )∗ − A−− (A−− )∗ = −iA−− + i(A−− )∗ .

(5.5)

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Using these equations, we have (1 − iA++ )∗ (1 − iA++ ) = 1 + (A−+ )∗ A−+ ≥ 1 ,

(5.6)

(1 − iA++ )(1 − iA++ )∗ = 1 + A+− (A+− )∗ ≥ 1 ,

(5.7)

(1 − iA−− )∗ (1 − iA−− ) = 1 + (A+− )∗ A+− ≥ 1 ,

(5.8)

(1 − iA−− )(1 − iA−− )∗ = 1 + A−+ (A−+ )∗ ≥ 1 .

(5.9)

Since the terms in the right-hand side of (5.6) ∼ (5.9) are invertible self-adjoint operators, the lemma immediately follows. Lemma 5.2. 1 − iAP− and 1 − iAP+ are bijections. Proof. By decomposing the equation (1 − iAP− )Q = 1, we have Q++ − iA+− Q−+ = P+ , Q+− − iA+− Q−− = 0 , Q−+ − iA−− Q−+ = 0 , Q−− − iA−− Q−− = P− . Since 1 − iA−− is a bijection, we obtain Q, the right inverse of 1 − iAP− . By decomposing the equation R(1 − iAP− ) = 1, we have R++ = P+ , R+− − iR++ A+− − iR+− A−− = 0 , R−+ = 0 , R−− − iR−+ A+− − iR−− A−− = P− . Since 1 − iA−− is bijective, we get R, the left inverse of 1 − iAP− . The proof for 1 − iAP+ is similar. Since 1 − iAP− is bijective, we can solve Eq. (4.2) to obtain AF . Lemma 5.3. 1 − iP+ A and 1 + iP− AF are bijections. Proof. Let B0 = P AF P0 . By decomposing the equation (1 + iP− AF )Q = 1, we have Q++ = P+ , Q+− = 0 , Q−+ + iB−+ Q++ + iB−− Q−+ = 0 , Q−− + iB−+ Q+− + iB−− Q−− = P− .

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1 + iB−− is a bijection, because of the relation 1 + iB−− = 1 + i(1 − iA−− )−1 A−− = (1 − iA−− )−1 , which can be obtained by multiplying P− to the Eq. (4.2). Therefore 1 + iP− AF has a right inverse. In the same way, 1 + iP− AF is shown to have a left inverse. The proof for 1 − iP+ A is similar. We shall now discuss the properties of AF . Theorem 5.4. (1) AF = A + iAP− AF = A + iAF P− A. (2) P+ AP− = 0 ⇐⇒ P+ AF P− = 0. (3) P− AP+ = 0 ⇐⇒ P− AF P+ = 0. (4) P+ AP− = 0 ⇐⇒ P− AP+ = 0. Proof. From (4.2), we have A = AF (1 + iP− AF )−1 . The second equation in (1) easily follows from this. The assertions (2), (3) are easily proved by multiplying (1) by P± . Let us prove (4). Suppose P+ AP− = 0. Then by multiplying (5.1) by P± , we have P+ A∗ P− (1 − iAP− ) = 0. This proves P+ A∗ P− = 0, hence P− AP+ = 0. Conversely, if P− AP+ = 0, we get P− A∗ P+ (1 − iAP+ ) = 0. Therefore P− A∗ P+ = P+ AP− = 0. Theorem 5.5. Let SF = 1 − iJAF . Then SF∗ SF = SF SF∗ = 1. Proof. Using (5.1), we have (1 + iP− A∗ )J(1 − iA) = J(1 − iP+ A) . Since A∗ J = JA(1 − iA)−1 , we have using the above equation A∗F = A∗ (1 + iP− A∗ )−1 = JA(1 − iA)−1 J(1 + iP− A∗ )−1 = JA(1 − iP+ A)−1 J = JAF (1 + iP− AF )−1 (1 − iP+ A)−1 J = JAF (1 − iJAF )−1 J . This implies that A∗F AF = i(JAF − A∗F J), hence SF∗ SF = 1. Similarly we can prove SF SF∗ = 1. 6. Quantum Field Theory in External Forces We turn to the quantum field theory associated with the classical scattering system discussed in Sec. 3. See e.g. [15, 19] for details. Let H = (L2 (X))2 , H± = P± H and F = F (H+ ) ⊗ F(H− ), where F (H± ) = ⊕∞ n=0 H± ⊗s · · · ⊗s H± ,

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H± ⊗s · · · ⊗s H± being the n-fold symmetric tensor product of H± . Let a, b be annihilation operators on F (H+ ), F (H− ), respectively. They satisfy the canonical commutation relations [a(f ), a(g)] = [a(f )∗ , a(g)∗ ] = 0 , [a(f ), a(g)∗ ] = (f, g) , for f , g ∈ H+ , and the same formulas for b with f , g ∈ H− . The field operator is defined to be Ψ(f ) = a(P+ f ) + b(P− f )∗ ,

f ∈ H.

An operator U ∈ B(H) is said to be unitarily implementable on F , if there exists a unitary operator U on F such that Ψ(U ∗ f ) = U∗ Ψ(f )U ,

f ∈ H.

The most important theorem in the quantum field theory in external forces is the following Theorem 6.1. Let S be the classical S-matrix. Suppose that P+ SP− , P− SP+ are Hilbert–Schmidt operators. Then S is unitarily implementable on F . Let S be the implemented S-matrix on F . Then (Ω, SΩ) = [det(1 − (B+− )∗ B+− )]1/2 , B+− = P+ AF P− , Ω being the vacuum of F . For the proof, see e.g. [15]. Remark 6.2. 1 − |(Ω, SΩ)|2 is the probability of creation of particles from the vacuum and we have the following equivalences: 1 − |(Ω, SΩ)|2 > 0 ⇐⇒ P+ AP− 6= 0 ⇐⇒ P− AP+ 6= 0 . In fact, Theorem 6.1 implies that 1 − |(Ω, SΩ)|2 > 0 ⇐⇒ P+ AF P− 6= 0 . By Theorem 5.4, we have P+ AF P− 6= 0 ⇐⇒ P− AF P+ 6= 0 ⇐⇒ P+ AP− 6= 0 ⇐⇒ P− AP+ 6= 0 . These equivalences mean that the occurrence of creation of particles from the vacuum in quantum field theory, namely 1 − |(Ω, SΩ)|2 > 0, is equivalent to the occurrence of the transition from negative (positive) energy state in the remote past to a positive (negative) energy state in the remote future in classical field theory, namely P± AP∓ 6= 0. The latter is the physical content of the so-called Klein’s paradox pointed out just after the publication of Dirac’s paper on his equation

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of electron (1928). As is clear from the above equivalence, this Klein’s paradox is a counter part (in the classical theory) of the existence of anti-particle (in the quantum field theory). See also [4]. Behind quantum mechanics, there is classical mechanics. The semi-classical analysis studies the effect of classical mechanics in quantum mechanics. Just in the same way, there is classical field theory behind quantum field theory. The above equivalence is an example of emergence of the former in the latter. Therefore the remaining task is to prove the Hilbert–Schmidt property of P+ SP− , P− SP+ . We do it in an abstract formulation. It is convenient to introduce the notation for multiple commutators. For two operators P and Q in a Hilbert space H, let ad0 (P, Q) = P , n ≥ 1.

adn (P, Q) = [adn−1 (P, Q), Q] ,

Let us recall the following well-known commutator expansion (see e.g. [7]). Let f (x) ∈ C ∞ (R) be such that |f (k) (x)| ≤ Ck (1 + |x|)s−k ,

∀k ≥ 0.

Then if s < 0, we have for any self-adjoint operator A Z 1 ∂z F (z)(z − A)−1 dz ∧ dz , f (A) = 2πi C

(6.1)

where F (z) ∈ C ∞ (C) satisfies F (x) = f (x) ,

x ∈ R,

|∂z F (z)| ≤ Ck (1 + |z|)s−1−k |Im z|k ,

∀k ≥ 0.

(6.2)

We also have for any N ≥ 2 N −1 X

(−1)n−1 adn (P, A)f (n) (A) + RN (P, A) , n! n=1 Z 1 ∂z F (z)(A − z)−1 adN (P, A)(A − z)−N dz ∧ dz . RN (P, A) = 2πi C [P, f (A)] =

(6.3)

(6.4)

If there exists an integer k ≥ 0 such that s + k < N and adN (P, A)(A + i)−k is bounded, so is RN (P, A) and we have kRN (P, A)k ≤ CkadN (P, A)(A + i)−k k . This follows from (6.2), (6.4) and the following inequality k(A + i)k (A − z)−N k ≤ sup |(λ + i)k (λ − z)−N | ≤ C(1 + |z|)k |Im z|−N . λ∈R

Let us also remark that if, in addition, there exists an integer n ≥ 0 such that (A + i)−n adN (P, A)(A + i)−k is Hilbert–Schmidt, so is (A + i)−n RN (P, A) and k(A + i)−n RN (P, A)kHS ≤ Ck(A + i)−n adN (P, A)(A + i)−k kHS .

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Lemma 6.3. If A is positive definite and P, [P, A] are bounded, we have for any integer m ≥ 1, P A−m = A−m (P + Bm ) , where kBm k ≤ Cm k[P, A]k. Proof. Suppose A ≥ > 0. We take f (x) ∈ C ∞ (R) such that f (x) = x−m if x > and f (x) = 0 if x < 0. Then by (6.1) Z 1 ∂z F (z)(z − A)−1 [P, A](z − A)−1 dz ∧ dz . [P, f (A)] = 2πi C The lemma then follows from this by using (6.2) and the following inequality kAm (z − A)−1 [P, A](z − A)−1 k ≤ C(1 + |z|)m |Im z|−2 k[P, A]k . Let H0 be self-adjoint, and P± be the projections onto positive, negative spectral subspaces for H0 . Let H(t) = H0 + V (t), and Gret be the retarded propagator for H(t). We assume that (P.1) There exists a self-adjoint operator ω and a constant m > 0 such that H0 P± = ±ωP± ,

ω ≥ m.

(P.2) There exists n0 > 0 such that for any t ∈ R and i, j = 0, . . . , n0 , adi (∂tj V (t), ω) ∈ B(H), and ω −n0 adi (∂tj V (t), ω) is Hilbert–Schmidt. We put r(t) =

n0 X

(kadi (∂tj V (t), ω)k + kω −n0 adi (∂tj V (t), ω)kHS )

i,j=0

and assume that r(t) ∈ L1 (R) ∩ L∞ (R) . Let A be defined by Z Z ∞ itH0 −itH0 e V (t)e dt + A= −∞

∞

−∞

eitH0 V (t)Gret (V (·)e−i·H0 )dt .

(6.5)

Theorem 6.4. Under the assumptions (P.1), (P.2), P+ AP− and P− AP+ are Hilbert–Schmidt operators. To show the Hilbert–Schmidt property, we shall estimate multiple commutators of ω and V (t). This idea of using multiple commuator has already been employed by Palmer [11] and Ruijsenaars [16]. To explain the idea of the proof, let us consider the first term of the right-hand side of (6.5). Since eitH0 P± = e±itω P± , we have Z ∞ Z ∞ P+ eitH0 V (t)e−itH0 P− dt = P+ eitω V (t)eitω P− dt . (6.6) −∞

−∞

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Let us put M V (t) = −(2iω)−1 ([V (t), iω] + ∂t V (t)) .

(6.7)

Then since d itω (e V (t)eitω ) = 2iωeitω V (t)eitω + eitω [V (t), iω]eitω + eitω (∂t V (t))eitω , dt we have Z ∞ Z ∞ eitω V (t)eitω dt = eitω M n V (t)eitω dt . −∞

−∞

By the assumption (P.2), kM n0 V (t)kHS ∈ L1 (R), which proves that (6.6) is Hilbert–Schmidt. To estimate the second term of the right-hand side of (6.5), we need a little more notation. For 0 ≤ p ≤ n0 , Vp is the set of the finite sums of the following terms (2iω)−α(1) A1 (t)(2iω)−α(2) A2 (t) · · · (2iω)−α(n) An (t) ,

(6.8)

where n ≤ n0 + 1, α(j) is a non negative integer such that α(1) + α(2) + · · · + α(n) = p ,

(6.9)

and Aj (t) is one of the following operators adk (∂tr V (t), iω) ,

k, r ≤ α(j) .

(6.10)

Lemma 6.5. B(t) ∈ Vp is written as e , B(t) = ω −p B(t) e kB(t)k ≤ C(r1 (t))n0 ,

r1 (t) =

p X

kadi (∂tj V (t), ω)k .

i,j=0

If p = n0 , we also have with r(t) in (P.2) n0 e . kω −n0 B(t)k HS ≤ C(r(t)) The above constants C depend only on n0 and m. Proof. We have only to consider the case that B(t) has the form (6.8). Commuting An−1 (t) and ω −α(n) , we have by using Lemma 6.3 An−1 (t)ω −α(n) = ω −α(n) a(t) , X kadk (∂tr V (t), ω)k . ka(t)k ≤ C k,r≤α(n−1)+α(n)

We next commute An−2 (t) and ω −α(n−1)−α(n) to obtain An−2 (t)ω −α(n−1)−α(n) = ω −α(n−1)−α(n)−k b(t) , X kadk (∂tr V (t), ω)k . kb(t)k ≤ C k,r≤α(n−2)+α(n−1)+α(n)

Repeating this procedure and using (6.9), we obtain the lemma.

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Let us consider the following integral Z ∞ Z ∞ ··· W (tn )θ(tn − tn−1 )Wn−1 (tn−1 ) · · · θ(t2 − t1 )W1 (t1 )dt1 · · · dtn , Sn = −∞

−∞

(6.11) θ(t) = 1 (t > 0) ,

θ(t) = 0 (t < 0) .

Lemma 6.6. In the expression of Sn in (6.11) we let Wp (t) = eitH0 Bp (t)e−itH0 , where Bp (t) ∈ Vβ(p) , β(1) + · · · + β(n) = n0 . Then there exists a constant C, which depends only on n0 and m, such that n Z ∞ Cn R(t)dt , kSn kHS ≤ n! −∞ where R(t) = r(t)n0 , with r(t) in (P.2). Proof. The idea of the proof is the same as Lemma 6.5. Noting that Bp (t) = fp (t), where B fp (t) has the properties in Lemma 6.5, we commute Bp−1 (t) ω −β(p) B −β(p) . Repeating this procdure, we see that the integrand of Sn consists of a and ω finite sum of the terms of the form ω −n0 bn (tn )θ(tn − tn−1 )bn−1 (tn−1 ) · · · θ(t2 − t1 )b1 (t1 ) , kω −n0 bn (t)kHS ≤ CR(t) ,

kbi (t)k ≤ CR(t) ,

1 ≤ i ≤ n− 1,

where the constnt C depends only on n0 and m. Therefore Z ∞ Z ∞ ··· R(tn )θ(tn − tn−1 )R(tn−1 ) · · · θ(t2 − t1 )R(t1 )dt1 · · · dtn kSn kHS ≤ C n −∞

=

Cn n!

Z

−∞

∞

n

R(t)dt

.

−∞

Now we estimate the 2nd term of the right-hand side of (6.5). We put Z ∞ Z ∞ ··· W (tn+1 )θ(tn+1 − tn )W (tn ) · · · θ(t2 − t1 )W (t1 )dt1 · · · dtn+1 , In = −∞

−∞

(6.12) W (t) = eitH0 V (t)−itH0 .

(6.13)

By the perturbation expansion (0)

(0)

(0)

(0)

(0)

(0)

Gret = Gret + Gret V Gret + Gret V Gret V Gret + · · · ,

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we have

Z

∞

−∞

eitH0 V (t)Gret (V (·)e−i·H0 )dt =

∞ X

(−i)n In .

(6.14)

n=1

Inserting P+ + P− = 1 between θ(tk+1 − tk ) and W (tk ), P+ In P− becomes a sum of the terms Z ∞ Z ∞ ··· P+ W (tn+1 )θ(tn+1 − tn )P W (tn ) · · · θ(t2 − t1 )P0 W (t1 )P− dt1 · · · dtn+1 , −∞

−∞

(6.15) where , 0 = ±. Each integrand contains at least one of the following 3 factors: (a) P+ θ(tk+1 − tk )W (tk )θ(tk − tk−1 )P− , (b) P+ W (tn+1 )θ(tn+1 − tn )P− , (c) P+ θ(t2 − t1 )W (t1 )P− . Let us consider the case (a). (The other cases are dealt with similarly). Noting the formulas P+ W (t)P− = P+ eitω V (t)eitω P− , eitω V (t)eitω = (2iω)−1 we have Z ∞ −∞

d itω (e V (t)eitω ) + eitω (M V (t))eitω , dt

θ(tk+1 − tk )P+ W (tk )P− θ(tk − tk−1 )dtk = (2iω)−1 P+ W (tk+1 )P− − (2iω)−1 P+ W (tk−1 )P− Z ∞ + θ(tk+1 − tk )P+ eitk H0 (M V (tk ))e−itk H0 P− θ(tk − tk−1 )dtk . −∞

By doing the same manipulation in the last term, we have Z ∞ θ(tk+1 − tk )P+ W (tk )P− θ(tk − tk−1 )dtk −∞

= (2iω)−1 P+ eitk+1 H0

nX 0 −1

M j (V (tk+1 ))e−itk+1 H0 P−

j=0

− (2iω)−1 P+ eitk−1 H0

nX 0 −1

M j (V (tk−1 ))e−itk−1 H0 P−

j=0

Z

∞

+ −∞

θ(tk+1 − tk )P+ eitk H0 (M n0 V (tk ))e−itk H0 P− θ(tk − tk−1 )dtk .

We insert this equality into (6.15). Then the 3rd term of the right-hand side gives rise to a Hilbert–Schmidt operator by virtue of Lemma 6.5. The 1st and 2nd terms

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give rise to n-fold integrals similar to (6.15). Therefore, one can repeat the same arguments as above. Finally we obtain j Z ∞ n X Cj R(t)dt . kP+ In P− kHS ≤ j! −∞ j=max{n−n0 ,1}

This proves the left-hand side of (6.5), sandwitched by P± , is Hilbert–Schmidt. 7. Hilbert Schmidt Property on Riemannian Manifolds In this section we shall prove the property (P.2) for our manifold X. This can be proved by the standard calculus of pseudo-differential operators when X is compact. Therefore our main concern is the analysis on each end Xi . Let us start with the model space case. 7.1. Model space case Let X be the model space introduced in Sec. 2. Let 0 = µ0 < µ1 ≤ µ2 ≤ · · · and φ0 , φ1 , φ2 , · · · be the eigenvalues and the associated complete orthonormal system of eigenvectors of −∆M . Let us consider ∂2 + q(x) − e−2λ(x) ∆M , ∂x2

(7.1)

n − 1 00 (n − 1)2 0 (λ (x))2 + λ (x) 4 2

(7.2)

L=− q(x) =

in L2 ((0, ∞) × M ; dxdM ). We take ζ > 0 large enough so that q(x) + ζ > 1 and let pk (x, ξ; ζ) = (|ξ|2 + q(x) + e−2λ(x) µk + ζ)−m .

(7.3)

A direct computation shows that |∂xα ∂ξβ pk (x, ξ; ζ)| ≤ Cαβ (|ξ|2 + e−2λ(x) µk + ζ)−m−β/2 ,

(7.4)

˜ ∈ C ∞ (R) such that χ(x) = 0 where Cαβ is independent of k. We take χ(x), χ(x) (x < 2), χ(x) = 1 (x > 3), χ(x) ˜ = 0 (x < 1), χ(x) ˜ = 1 (x > 2) and define Pk (ζ) by ZZ ei(x−y)ξ χ(x)p ˜ (7.5) Pk (ζ)f = (2π)−1 k (x, ξ; ζ)χ(y)f (y)dydξ . R2

By (7.4), Pk (ζ) is a bounded operator on L2 ((0, ∞); dx) with norm uniformly bounded in k if m ≥ 0. For u ∈ L2 (X; dxdM ), we define P (ζ)u =

∞ X

(Pk (ζ)uk ) ⊗ φk ,

(7.6)

k=0

uk (x) = hu(x, ·), φk (·)i , where h , i is the inner product of L2 (M ).

(7.7)

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Lemma 7.1. Let m >n/4. Let a(x) be such that a(x)e(2m−1/2)λ(x) ∈L2 ((0, ∞); dx). Then a(x)P (ζ) is Hilbert–Schmidt on L2 (X; dxdM ) and ka(x)P (ζ)kHS ≤ Cka(x)e(2m−1/2)λ(x) kL2 ((0,∞);dx) . 2 Proof. Let {ψj (x)}∞ j=1 be a complete orthnormal system of L ((0, ∞); dx). Then since

P (ζ)(ψj ⊗ φk ) = (Pk (ζ)ψj ) ⊗ φk , we have X

ka(x)P (ζ)ψj ⊗ φk k2L2 (X;dxdM) =

XX

j,k

k

P

ka(x)Pk (ζ)ψj k2L2 ((0,∞);dx) .

j

2 j ka(x)Pk (ζ)ψj kL2 ((0,∞);dx)

is the square of the Hilbert–Schmidt norm of Since a(x)Pk (ζ), it is dominated from above by Z ∞Z ∞ |a(x)|2 (|ξ|2 + e−2λ(x) µk + ζ)−2m dxdξ −∞

0

Z

∞

Z

∞

= 0

−∞

|a(x)|2 e4mλ(x) [e2λ(x) (ζ + |ξ|2 ) + µk ]−2m dxdξ .

We take α > (n − 1)/2, β > 1/2 such that α + β = 2m. Then [e2λ(x) (ζ + |ξ|2 ) + µk ]−2m ≤ C(1 + µk )−α e−2βλ(x) (ζ + |ξ|2 )−β . We then have X ka(x)P (ζ)ψj ⊗ φk k2L2 (X;dxdM) j,k

≤C

X k

−α

Z

(1 + µk )

0

P

∞

Z

∞

−∞

|a(x)|2 e(4m−2β)λ(x) (ζ + |ξ|2 )−β dxdξ .

It is well-known that k (1 + µk )−α < ∞ if α > (n − 1)/2, which is proved by constructing a parametrix of (−∆M +ζ)−α by using the standard pseudo-differential calculus. This proves the lemma. 7.2. Hilbert Schmidt property on X We turn to the general case stated in Sec. 2.3. We take a patition on unity {χi }ki=0 satisfying (3.6) and (3.7). Lemma 7.2. Let m > n/4. For a function A on X, we put Ai (x) = supMi |Aχi |. Suppose A ∈ L∞ (X), Ai (x)e(2m−1/2)λi (x) ∈ L2 ((ri , ∞); dx) for 1 ≤ i ≤ k. Then A(L0 − z)−m is Hilbert–Schmidt for z 6∈ [0, ∞), and ! k X kAi (x)e(2m−1/2)λi (x) kL2 ((ri ,∞);dx) . kA(L0 − z)−m kHS ≤ C kAkL∞ (X) + i=1

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Proof. Since the scalar curvature R(X) is a bounded function, we have only to prove the lemma for A(−∆X + ζ)−m for large ζ. We take smooth functions χ ei (0 ≤ i ≤ k) such that χ e0 ∈ C0∞ (X) , supp χ ei ⊂ Xi ,

χ e0 = 1 on supp χ0 ,

χ ei = 1

1 ≤ i ≤ k.

on supp χi ,

e replaced by χi , χ ei . We put For 1 ≤ i ≤ k, we define P (i) (ζ) as in Sec. 7.1 with χ, χ P (ζ) = χ e0 (−∆X + ζ)−m χ0 +

k X

P (i) (ζ) .

i=1

Then as can be checked easily (−∆X + ζ)m P (ζ) = 1 + O(1/ζ) . Therefore for large ζ (−∆X + ζ)−m = P (ζ)(1 + O(1/ζ))−1 . As is well-known, χ e0 (−∆X + ζ)−m χ0 is Hilbert–Schmidt. This together with Lemma 7.1 proves Lemma 7.2. 7.3. Proof of the assumption (P.2) We shall assume (EM-1), (EM-2) and (EM-3) stated in Sec. 3.1. To consider the commutator adi (∂tj V (t), ω), we have only to look at adi (ω −1/2 ∂tj L1 (t)ω −1/2 , ω) . We first deal with ω −1/2 ∂tj L1 (t)ω −1/2 . Let f (x) ∈ C ∞ (R) be such that f (x) = 0 (x < −1), f (x) = x−1/4 (x ≥ m2 ). Then ω −1/2 = f (L0 ). Applying (6.3) to [∂tj L1 (t), f (L0 )], we have −1/2

ω −1/2 ∂tj L1 (t)ω −1/2 = ∂tj L0 (t)L0

+

N −1 X

−1/4

Cn adn (∂tj L1 (t), L0 )f (n) (L0 )L0

n=1 −1/4

+ RN (∂tj L1 (t), L0 )L0

,

(7.8)

Cn being a constant. It is easy to see that adn (∂tj L1 (t), L0 ) is a differential operator of order n + 1. Since f (n) (x) = O(x−1/4−n ) as x → ∞, we have −1/4 ∈ B(L2 (X)). (L0 + 1)n+1/2 f (n) (L0 )L0 j Now look at adn (∂t L1 (t), L0 ) on the end Xi . Then it is a finite sum of the differential monomial Dα (|α| ≤ n + 1), D being defined by (3.9), with coefficients satisfying (3.11) and (7.8). As is easily seen Dα (L0 + 1)−n−1/2 is bounded if −1/4 is bounded and that, |α| ≤ n + 1. This proves that adn (∂tj L1 (t), L0 )f (n) (L0 )L0 −m if we mulitiply L0 (m > n/4) from the left, it is Hilbert–Schmidt by virtue of Lemma 7.2. Moreover their norms are integrable in t ∈ R. The portion of adn (∂tj L1 (t), L0 ) on the compact part of X is easier to handle. In fact, the usual

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calculus of pseudo-differential operators applies here. By taking N large enough, −1/4 we can prove the same property for the remainder term RN (∂tj L1 (t), L0 )L0 . ∞ 1/2 (x ≥ m2 ). We Let g(x) ∈ C (R) be such that g(x) = 0 (x < −1), g(x) = x take the commutator of (7.9) and ω = g(L0 ). Then −1/4

[adn (∂tj L1 (t), L0 ), g(L0 )]f (n) (L0 )L0 =

N −1 X

−1/4

Ck adk+n (∂tj L1 (t), L0 )g (k) (L0 )f (n) (L0 )L0

k=1 −1/4

+ RN (adn (∂tj L1 (t), L0 ), L0 )f (n) (L0 )L0

.

−1/4

∈ B(L2 (X)). Since adk+n (∂tj L1 (t), L0 ) Note that (L0 +1)k+n g (k) (L0 )f (n) (L0 )L0 is a differential operator of order k + n + 1, the same arguments as above apply to this term. By repeating these procedures, one can prove (P.2). Theorem 7.3. Assume (EM-1), (EM-2) and (EM-3). Then P+ SP− , P− SP+ are Hilbert–Schmidt operators. Hence the assertions in Theorem 6.1 hold. 8. Massless Case In this section we consider the massless case (m = 0). In this case, the same results as in the previous section cannot be expected to hold in general, since the behavior of the resolvent (L0 − z)−1 near z = 0 plays an important role. We pick up two examples to elucidate how to overcome the difficulty: the asymptotically flat space and the hyperbolic space. 8.1. Asymptotically flat case On Rn let us consider a 2nd order formally self-adjoint elliptic operator n n X X ∂2 ∂ aij (x) + ai (x) + a0 (x) . L0 = − ∂x ∂x ∂x i j i i,j=1 i=1

(8.1)

We assume that the coefficients are smooth and satisfy for a constant ρ > 0 |∂xα (aij (x) − δij )| + |∂xα ak (x)| ≤ Cα hxi−ρ

(8.2)

for 1 ≤ i, j ≤ n, 0 ≤ k ≤ n and all α, where hxi = (1 + |x|2 )1/2 . Note that in this section, x denotes the variable in Rn . We also assume that σ(L0 ) = [0, ∞) .

(8.3)

Let R(z) be the resolvent of L0 : R(z) = (L0 − z)−1 ,

z 6∈ [0, ∞) .

(8.4)

Then, as is well-known, if λ > 0 is not an eigenvalue of L0 , there exists a limit R(λ ± i0) = lim↓0 R(λ ± i) in B(L2,s ; L2,−s ), s > 1/2, where Z 2,s 2 hxi2s |f (x)|2 dx < ∞ . f ∈ L ⇐⇒ kf k = Rn

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Main barriers in studying the low energy asymptotics of R(λ ± i0) as λ → 0 are the zero-eigenvalue and the zero-resonance. The latter roughly means the existence of non-trivial solutions of L0 ψ = 0 decaying slowly so that ψ 6∈ L2 (Rn ). Theorem 8.1. Suppose 0 is neither the eigenvalue nor the resonance of L0 . Then as λ → 0 R(λ ± i0) = O(1)

B(L2,s ; L2,−s )

in

provided (i) n ≥ 4, ρ > 4, s > n/2, (ii) n = 3, ρ > 5, s > 5/2, (iii) n = 2, ρ > 6, s > 3. This theorem follows from Theorems 4.3 and 8.1 of [10]. Corollary 8.2. Under the conditions in Theorem 8.1, there exists a δ > 0 such that Z δ λ−1/2 khxi−s R(λ ± i0)hxi−s kdλ < ∞ , (8.5) 0

where k · · · k denotes the norm of B(L2 (Rn )). We take χ0 (λ) ∈ C ∞ (R) such that χ0 (λ) = 1 (λ < δ/2), χ0 (λ) = 0 (λ > δ) and let χ1 (λ) = 1 − χ0 (λ). let f0 (λ) = χ0 (λ)λ−1/4 ,

f1 (λ) = χ1 (λ)λ−1/4 .

(8.6)

Lemma 8.3. Under the conditions in Theorem 8.1, f0 (L0 )hxi−s ∈ B(L2 (Rn )). Proof. Let K = f0 (L0 )hxi−s . Then −1/2

K ∗ K = hxi−s L0

χ0 (L0 )2 hxi−s .

The lemma then follows from Corollary 8.3 and Z δ 1 −1/2 2 λ−1/2 χ0 (λ)2 (R(λ + i0) − R(λ − i0))dλ . L0 χ0 (L0 ) = 2πi 0 We next consider the perturbation term L1 (t): L1 (t) =

n X i=1

bi (x, t)

∂ + b0 (x, t) . ∂xi

We assume that L1 (t) is formally self-adjoint and that for i = 0, 1, . . . , n, Z ∞ sup |hxi2s ∂xα ∂tp bi (x, t)|dt < ∞ , Z

(8.7)

(8.8)

−∞ x∈Rn

∞

−∞

khxi2s ∂xα ∂tp bi (x, t)kL2 (Rn ) dt < ∞ ,

(8.9)

for any α, p ≥ 0 and s > s(n), where s(n) = n/2 if n ≥ 4, s(3) = 5/2, s(2) = 3.

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Lemma 8.4. Under the assumptions in Theorem 8.1, (8.8) and (8.9), we have for any p ≥ 0 Z ∞ (kf1 (L0 )∂tp L1 (t)f0 (L0 )kHS + kf0 (L0 )∂tp L1 (t)f0 (L0 )kHS )dt < ∞ . −∞

Proof. Let χ2 (λ) ∈ C ∞ (R) be such that χ2 (λ) = 1 (λ < δ), χ2 (λ) = 0 (λ > 2δ). Then by using (6.3), we have hxi−s χ2 (L0 ) = Bhxi−s ,

(L0 + 1)m B ∈ B(L2 (Rn )) ,

∀m ≥ 0.

We then have bi (x, t)∂i f0 (L0 ) = bi (x, t)∂i hxis Bhxi−s f0 (L0 ) . By Lemma 8.3 and the assumption (8.9), one can easily prove the lemma by the standard pseudo-differential calculus. The above lemma shows that V (t) in (3.16) is a bounded operator in our case. Therefore one can define the classical S-matrix and the scattering amplitude A as in Sec. 3. Theorem 8.5. Suppose 0 is neither the eigenvalue nor the resonance of L0 . Assume ρ > 4 for n ≥ 4, ρ > 5 for n = 3 and ρ > 6 for n = 2. Suppose (8.8) and (8.9) are satisfied. Then P+ AP− and P− AP+ are Hilbert–Schmidt operators. Hence the same concluson as in Theorem 6.1 holds. Proof. Let f0 and f1 be as in (8.6) and put 1 fi (L0 )L1 (t)fj (L0 ) . 2 P Accordingly we split V (t) as V (t) = 0≤i,j≤1 Vij (t). As in Sec. 6, let us consider (6.6) to give the idea of the proof. By Lemma 8.4, V10 (t), V01 (t) and V00 (t) are Hilbert–Schmidt operators with norm integrable in t ∈ R. Take χ3 (λ) ∈ C ∞ (R) such that χ3 (λ) = 1 (λ > δ/2), χ3 (λ) = 0 (λ < δ/4), and put Vij (t) =

Mδ V (t) = −(2iω)−1 χ3 (L0 )([V (t), iωχ3 (L0 )] + ∂t V (t)) . Then we have Z

∞

−∞

Z eitω V11 (t)eitω dt =

∞

−∞

eitω Mδn0 V11 (t)eitω dt ,

and we can show that (6.6) is Hilbert–Schmidt. The second term of the right-hand side of (6.5) is treated similarly.

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8.2. Space of constant negative curvature Let X = (0, ∞) × M , M = S n−1 , be as in Example 2 of Sec. 2. Let us recall that L0 = −∆X −

(n − 1)2 , 4a2

(8.10)

a x/a (e − e−x/a ) . 2 In this section, x denotes the variable in (0, ∞). eλ(x) =

(8.11)

Lemma 8.6. Let s > 1/2, δ > 0. Then there exists a constant C > 0 such that if f (λ) = 0 for λ > δ Z δ √ |f (λ)| λdλ , khxi−s e−(n−1)λ(x)/2 f (L0 )e−(n−1)λ(x)/2 hxi−s k ≤ C 0

where k · k denotes the operator norm on L (X; dxdM ). 2

Let us admit this lemma for the moment. We take local coordinates (θ1 , . . . , θn−1 ) on M and put Dj = e−λ(x)

∂ , ∂θj

Dn =

∂ . ∂x

On the coordinate patch, the perturbation term L1 (t) is assumed to have the form L1 (t) =

n X

bj (x, θ, t)Dj + b0 (x, θ, t) .

(8.12)

j=1

We shall assume that for j = 0, 1, . . . , n sup |∂tp Dα bj (x, θ, t)| ∈ L1 (R) ∩ L∞ (R) ,

(8.13)

x,θ

holds for any α and p ≥ 0, and that there exists an integer m > n/4 such that for some s > 1/2

p ∈ L1 (R) (8.14)

sup |∂t Dα bj (x, θ, t)|e(2m+n−3/2)λ(x) hxis 2 θ

L ((0,∞);dx)

holds for any α, p ≥ 0 and 0 ≤ j ≤ n. Let f0 (λ) be as in (8.6). Then by Lemma 8.6 and the same arguments as in the proof of Lemma 8.3, we have f0 (L0 )e−(n−1)λ(x)/2 hxi−s ∈ B(L2 (X)) .

(8.15)

Using this fact, one can argue as in Sec.8.1 to show the following theorem. Theorem 8.7. Assume (8.13) and (8.14). Then P+ AP− , P− AP+ are Hilbert– Schmidt operators. Hence the same conclusion as in Theorem 6.1 holds. In fact, in view of the proof of Theorem 8.5, we have only to check that kf1 (L0 )∂tp L1 (t)f0 (L0 )kHS + kf0 (L0 )∂tp L1 (t)f0 (L0 )kHS ∈ L1 (R) . This is proved by using Lemma 7.1, (8.14) and (8.15).

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Now let us return to the proof of Lemma 8.6. For the sake of simplicity, we consider the case of a = 1. Let us consider the equation (L0 − z)u = 0 in the upperhalf space Hn = {(x0 , xn ); x0 ∈ Rn−1 , xn > 0}. Then by passing to the Fourier transformation with respect to x0 , we have the following equation −(xn ∂n )2 u + (n − 1)xn ∂n u + (xn |ξ 0 |2 − z)u = 0 ,

ξ 0 ∈ Rn−1 ,

∂n = ∂/∂xn . (8.16)

Letting u =

(n−1)/2 v, xn

we have

(x2n ∂n2 + xn ∂n − (x2n |ξ 0 |2 − z))v = 0 .

(8.17)

Ki√z (|ξ 0 |xn ),

Ii√z (|ξ 0 |xn ),

where We choose two linearly independent solutions Kν (x), Iν (x) are the modified Bessel functions. Using the fact that 1 Kν (x)Iν0 (x) − Iν (x)Kν0 (x) = , x we see that (L0 − z)−1 has the following integral kernel Z 0 0 0 0 0 −(n−1)/2 (n−1)/2 e 0 , xn , yn ; z)dξ 0 , (xn yn ) ei(x −y )·ξ R(ξ R(x , xn ; y , yn ; z) = (2π) ( e xn , yn ; z) = R(ξ,

Rn−1

Ki√z (|ξ 0 |xn )Ii√z (|ξ 0 |yn ) ,

xn > yn ,

Ii√z (|ξ 0 |xn )Ki√z (|ξ 0 |yn ) ,

yn > xn .

Using the identity Kν (x) =

π (I−ν (x) − Iν (x)) , 2 sin(νπ)

we have R(x0 , xn , y 0 , yn ; z) − R(x0 , xn , y 0 , yn ; z) =

√ 2 (2π)−(n−1)/2 (xn yn )(n−1)/2 sinh( zπ) iπ Z 0 0 0 ei(x −y )·ξ Ki√z (|ξ 0 |xn )Ki√z (|ξ 0 |yn )dξ 0 . × Rn−1

Stone’s formula implies that the spectral decomposition E(λ) of L0 has the following integral kernel Z √λ 0 0 −(n−1)/2 −2 (n−1)/2 π (xn yn ) dk2k sinh(πk) E(x , xn , y , yn ; λ) = (2π) 0

Z

0

0

0

dξ 0 ei(x −y )·ξ Kik (|ξ 0 |xn )Kik (|ξ 0 |yn ) .

× Rn−1

Therefore f (L0 ) has the following integral kernel 0

0

−(n−1)/2 −2

f (L0 ; x , xn , y , yn ) = (2π) Z ×

Rn−1

π

Z

∞

(n−1)/2

(xn yn )

√ dλf (λ) sinh(π λ)

0 0

0

0

dξ 0 ei(x −y )·ξ Ki√λ (|ξ 0 |xn )Ki√λ (|ξ 0 |yn ) .

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We assume that f (λ) = 0 for λ > δ. Since Kν (x) has the following estimate |Ki√λ (x)| ≤ Ce−x ,

x > 0,

where C is independent of 0 < λ < δ, we have Z δ Z √ |f (λ)| λdλ (xn yn )(n−1)/2 |f (L0 ; x0 , xn , y 0 , yn )| ≤ C 0

Z

√ |f (λ)| λdλ

δ

√ |f (λ)| λdλ .

0

≤C

√ xn yn n−1 xn + yn

δ

=C Z

0

e−|ξ |(xn +yn ) dξ 0

Rn−1

0

This proves that f (L0 ) has a bounded integral kernel on L2 (Hn ; x−n n dx1 · · · dxn ). This is also true on L2 (X; e(n−1)λ(x) dxdM ). This means that on L2 (X; dxdM ), it has an integral kernel e(n−1)λ(x)/2 K(x, θ, y, ω)e(n−1)λ(y)/2 , with the property sup |K(x, θ, y, ω)| < ∞ . Lemma 8.6 now follows from this fact. References [1] T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Springer, Berlin Heidelberg, New York, 1982. [2] J. Bellissard, “Quantized fields in interaction with external fields I: Exact solutions and perturbative expansions”, Commun. Math. Phys. 41 (1975) 235–266. [3] J. Bellissard, “Quantized fields in external fields II: Existence theorems”, Commun. Math. Phys. 46 (1976) 53–74. [4] P. J. M. Bongaarts and S. N. M. Ruijsenaars, “The Klein paradox as a many particle problem”, Ann. Phys. 101 (1976) 289–318. [5] L. D. Faddeev, “Inverse problem of quantum scattering theory”, Sov. Phys. Dokl. 5 (1976) 334–396. [6] S. A. Fulling, Aspects of Quantum Field Theory in Curved Space-Time, London Mathematical Society Student Texts 17, Cambridge University Press, 1989. [7] C. Gérard, H. Isozaki and E. Skibsted, “Commutator algebra and resolvent estimates”, pp. 69–82 in Spectral and Scattering Theory and Applications, Advanced Studies in Pure Mathematics 23, ed. K. Yajima, 1994. [8] S. W. Hawking, “Particle creation by black holes”, Commun. Math. Phys. 43 (1975) 199–220. [9] H. Isozaki, “Multi-dimensional inverse scattering theory for Schr¨ odinger operators”, Rev. in Math. Phys. 8 (1996) 591–622. [10] M. Murata, “Asymptotic expansions in time for solutions of Schr¨ odinger type equations”, J. Funct. Anal. 49 (1982) 10–56. [11] J. Palmer, “Scattering automorphism of the Dirac field”, J. Math. Anal. Appl. 64 (1978) 189–215.

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[12] J. Palmer, “Symplectic groups and the Klein–Gordon field”, J. Func. Anal. 27 (1978) 308–336. [13] S. N. M. Ruijsenaars, “Charged particles in external fields I: Classical theory”, J. Math. Phys. 18 (1977) 720–737. [14] S. N. M. Ruijsenaars, “Charged particles in external fields II: The quantized Dirac and Klein–Gordon theories”, Commun. Math. Phys. 52 (1977) 267–294. [15] S. N. M. Ruijsenaars, “On Bogolieubov transformation for systems of relativistic charged particles”, J. Math. Phys. 18 (1977) 517–526. [16] S. N. M. Ruijsenaars, “Gauge invariance and implementability of the S-operator for spin-0 and spin-1/2 particles in time-dependent external fields”, J. Funct. Anal. 33 (1979) 47–57. [17] G. Scharf, Finite Quantum Electrodynamics, Texts and Monographs in Physics, Springer-Verlag, Berlin Heidelberg, 1989. [18] R. Seiler, “Quantum theory of particles with spin zero and one half in external fields”, Commun. Math. Phys. 25 (1972) 127–151. [19] B. Thaller, The Dirac Equation, Springer-Verlag, Berlin Heidelberg, 1992. [20] G. Velo and A. S. Wightman (eds.), Invariant Wave Equations, Lecture Notes in Physis 73, Springer, 1977. [21] R. M. Wald, “On particle creation by black holes”, Commun. Math. Phys. 45 (1975) 9–34. [22] R. M. Wald, “Existence of the S-matrix in quantum field theory in curved spacetime”, Ann. Phys. 118 (1979) 490–510.

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QUANTIZATION OF KINEMATICS ON CONFIGURATION MANIFOLDS

H.-D. DOEBNER Arnold Sommerfeld Institut f¨ ur Mathematische Physik, Technische Universit¨ at Clausthal, Leibnizstr. 10, D-38678 Clausthal, Germany ˇTOV ˇ ÍCEK ˇ P. S and J. TOLAR Doppler Institute, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Bˇ rehov´ a 7, CZ-115 19 Prague 1, Czech Republic

Received 6 July 1999 This review paper is devoted to topological global aspects of quantal description. The treatment concentrates on quantizations of kinematical observables — generalized positions and momenta. A broad class of quantum kinematics is rigorously constructed for systems, the configuration space of which is either a homogeneous space of a Lie group or a connected smooth finite-dimensional manifold without boundary. The class also includes systems in an external gauge field for an Abelian or a compact gauge group. Conditions for equivalence and irreducibility of generalized quantum kinematics are investigated with the aim of classification of possible quantizations. Complete classification theorems are given in two special cases. It is attempted to motivate the global approach based on a generalization of imprimitivity systems called quantum Borel kinematics. These are classified by means of global invariants — quantum numbers of topological origin. Selected examples are presented which demonstrate the richness of applications of Borel quantization. The review aims to provide an introductory survey of the subject and to be sufficiently selfcontained as well, so that it can serve as a standard reference concerning Borel quantization for systems admitting localization on differentiable manifolds.

Contents 1. Introduction 2. Mackey’s System of Imprimitivity 2.1. The formalism of quantum mechanics 2.2. Symmetry and quantum mechanics 2.3. Localization, system of imprimitivity 2.4. Quantization on homogeneous spaces 2.5. Infinitesimal action on a G-space 2.6. Examples 3. Quantum Borel Kinematics: Localization 799

2 6 6 7 9 10 13 14 18

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ˇˇ H.-D. Doebner, P. S tov´ıˇ cek & J. Tolar

3.1 Generalized system of imprimitivity 3.2 Quantum Borel kinematics 3.3 Quasi-invariant measures, projection-valued measures 3.4 An example 4. Quantum Borel Kinematics: External Gauge Fields 4.1 External magnetic field 4.2 Construction of a class of generalized systems of imprimitivity 4.3 Construction in the associated vector bundle 4.4 The Case G = U (r) 4.5 Unitary equivalence of generalized systems of imprimitivity 4.6 Irreducibility of generalized systems of imprimitivity 4.7 Quantum Borel kinematics with vanishing external field 5. Quantum Borel Kinematics: Classfication 5.1 Classfication of generalized systems of imprimitivity via cocycles 5.2 Differentiable quantum Borel kinematics 5.3 Canonical representation of differentiable QBKr 5.4 Classification of differentiable QBKr ’s 5.5 Classification of elementary differentiable quantum Borel kinematics 5.6 Classification of quantum Borel kinematics of type 0 5.7 Elementary quantum Borel kinematics with vanishing external field 5.8 Examples Acknowledgments References

18 19 21 23 23 23 24 25 28 29 32 34 35 35 36 37 38 38 40 41 43 44 45

1. Introduction The successful development of quantum theory in this century shows convincingly that it provides perhaps the most universal language for the description of physical phenomena. In quantum theory, as in any other physical theory, two fundamental aspects can be distinguished: the mathematical formalism and the physical interpretation. At the basis of the most common mathematical formalism of quantum mechanics lies the notion of a complex separable Hilbert space H of, in general, infinite dimension. Normed vectors in H correspond to pure states of a quantum system, whereas quantal observables are represented by self-adjoint operators in H. However, only the rules of a physical interpretation enable one to use quantum theory for the description of physical systems. The principal general rule is Born’s statistical interpretation of the wave function. For each physical system, or at least for a certain class of them, it is further necessary to specify which operators in H are associated with physical observables measured by certain measuring devices. This means in particular that at least the operators of kinematical observables (position and momentum), and the dynamical evolution law of the system are to be specified.

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Quantization of Kinematics on Configuration Manifolds

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An important tool for the derivation of quantum models are quantization methods. The primary aim of quantization of a given classical system is to associate self-adjoint operators with classical observables. As a rule, two main methods are used. The first one is based on Bohr’s correspondence principle: the physical meaning of quantum operators is found by looking at their classical counterparts. In this way non-relativistic quantum mechanics was formulated by quantization of classical Hamiltonian mechanics, quantum theory of electromagnetic field by quantization of the Maxwell theory, etc. [42]. The correspondence principle can, of course, be the leading rule for quantization, if the observables already existed in a classical form. What should be done in the case of quantum observables without a classical analogue like the spin? Here the second method is often applicable, which uses invariance principles connected with the symmetries of the system. By Noether’s theorem the operators corresponding to conserved quantities can be found as generators of some projective representation of the symmetry group in H. As a far reaching application of this approach let us mention the relativistic quantum theory of elementary particles based on the irreducible unitary representations of the Poincar´ e group. Both methods were used from the very first days of quantum theory, always taking into account specific physical properties of the systems considered. The first method usually appears in non-relativistic quantum mechanics as canonical quantization [42], for systems with the Euclidean configuration space Rn . The position coordinates qj and the canonically conjugate momenta pk are quantized into selfadjoint position and momentum operators Qj , Pk (in a separable Hilbert space H), satisfying canonical commutation relations. This was originally discovered and mathematically formulated independently by W. Heisenberg and E. Schr¨ odinger in 1925–26. The uniqueness of the mathematical formulation up to unitary equivalence was then guaranteed by the Stone–von Neumann Theorem. Quantum mechanics on Rn became very soon a successful theory which has been able to correctly describe experimental findings in vast areas of quantum physics. However, in some cases it was necessary to look for a formulation of quantum mechanics when the configuration space of a system was not Euclidean [52]. For instance, in connection with the studies of rotational spectra of molecules and of deformed nuclei, quantum rotators were introduced as fundamental quantum models with configuration spaces S 1 (the circle), S 2 (the 2-sphere) and SO(3) (the rotation group). The textbook treatment of spinning top models (quantum mechanics of angular momentum) presents a successful application of the approach via invariance principles. There were also attempts to enforce canonical quantization in cases where global Cartesian coordinates do not exist on the configuration manifold M . A formal quantization of generalized coordinates qj and conjugate momenta pk was suggested [52] on a manifold M with the Riemann structure (metric tensor gjk with determinant g > 0): ∂ i~ ∂ Qj = qj , Pk = −i~ − (ln g) . (1.1) ∂qk 4 ∂qk

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Note that the additional term in Pk ’s makes them symmetric operators in H = √ L2 (M, dµ) with respect to the Riemann measure dµ = gdn q on M . The main difficulty encountered here is that operators (1.1) are not globally defined since, in general, qj are only local coordinates. It is therefore desirable to invent quantization methods which employ global geometric objects. On several occasions the formalism of quantum mechanics in connection with non-trivial topology of the configuration manifold lead to new non-classical effects. A deep and in its time not completely understood and recognized accomplishment in this direction was Dirac’s famous investigation [6] of a quantum charged particle (charge e) in the external magnetic field of a point-like magnetic monopole (magnetic charge g). If the singular Dirac monopole is placed at the origin of a Cartesian coordinate system in R3 , one deals in fact with quantum mechanics on a topologically non-trivial effective configuration manifold R3 \{(0, 0, 0)} (the three-dimensional Euclidean space with the origin excluded). Here the formalism of quantum mechanics in connection with non-trivial topology of the configuration manifold leads to an unexpected topological quantum effect originating from a peculiar behaviour of the phase of a wave function: Dirac discovered that a quantal description exists only under the condition that the dimensionless quantity eg/2π~ is an integer. Another phenomenon of this kind was noticed in 1959 by Y. Aharonov and D. Bohm [1]. The origin of the Aharonov–Bohm effect can be traced to a shift of the phase of wave function due to an external magnetic flux imposed on a charged particle. Here the effective Aharonov–Bohm configuration space is R3 \R, the threedimensional Euclidean space with a straight line excluded. In both mentioned cases the topologies of the configuration spaces differ from the trivial topology of the Euclidean space and play decisive rˆ ole in quantum theory. These remarks about the early history of quantum mechanics clearly point to the need for a systematic development of global quantization methods. For systems with sufficiently symmetric configuration or phase spaces, two modern approaches in the theory of group representations can be applied: (1) Mackey’s quantization on homogeneous configuration manifolds M = G/H [10, 23, 50]. Essentially, it is equivalent to the construction of systems of imprimitivity for a (locally compact, separable) group G, based on M = G/H. (2) The method of coadjoint orbits which play the role of homogeneous phase spaces [20, 47]. In the case of configuration or phase manifolds without geometric symmetries, two programs of global quantization were suggested: (3) Borel quantization on configuration manifolds [4, 11] which extends the notion of Schr¨ odinger systems [27]. (4) Geometric quantization on symplectic phase manifolds [20, 47].

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These methods have been elaborated to differing degrees of sophistication and have, in general, different classes of classical systems as their domains of applicability. Borel quantization is built on configuration spaces and reflects the topology of M . For physical applications it is important that it yields both important classification theorems and explicit relations for quantization of kinematical observables. Like canonical quantization, it is a two step procedure. In a first step the kinematics, i.e. position and momentum observables on M , is quantized. The time dependence is introduced in a second step with a quantum analogue of a second order Riemannian dynamics on M [4]. In its most general form it leads to Doebner–Goldin non-linear Schr¨ odinger equations [8]. Concerning other quantization methods respecting global properties of configuration or phase spaces we should especially mention: (5) The Feynman path integral method (it was used, e.g. in [28] for M = SO(3) and in [21] for configuration spaces of identical particles). (6) Quantization by deformation of classical mechanics [5]. (7) Dirac quantization of systems with constraints in phase space [41]. This review article is devoted to the mathematical exposition of quantum Borel kinematics. This method yields quantizations of kinematics for systems admitting localization on connected smooth finite-dimensional configuration manifolds without boundary. We restrict our consideration exclusively to paracompact manifolds which (by Whitney’s embedding theorem) can be regarded as submanifolds of Rn . In Sec. 2, the Hilbert space formalism of quantum mechanics, Wigner’s Theorem on symmetry transformations, and the notion of Mackey’s system of imprimitivity are briefly surveyed. The notion of quantum Borel kinematics is introduced in Sec. 3. In Sec. 4, a family of quantum Borel kinematics is constructed. This geometrical construction of quantum kinematics (Sec. 4.2) is based on the notion of a generalized system of imprimitivity for the family of one-parameter groups of diffeomorphisms (Sec. 3.1). It represents a generalization of quantum Borel kinematics of Ref. [4], especially in admitting an external gauge field with an arbitrary Abelian or compact structure group G. Thus the construction involves associated Cr -bundles with finite-dimensional fibres Cr . Section 4 is also devoted to questions of unitary equivalence and irreducibility (Secs. 4.5 and 4.6) of this class of quantum kinematics. Important special case of the vanishing external field is treated in Sec. 4.7. The classification of quantum Borel kinematics cannot be considered to be complete. In Sec. 5 theorems are stated which fully characterize them as well as two cases of complete classifications — elementary quantum Borel kinematics (Sec. 5.5) and quantum Borel r-kinematics of type 0 (Sec. 5.6). In these cases it is shown that the first and the second singular homology groups of the configuration manifold M are involved and provide the necessary topological tools for classification of quantizations.

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We have payed particular attention to a selection of proper examples which complement each chapter and demonstrate the richness of possible applications. From these examples, we mention a new derivation of the Dirac quantization condition from rotational symmetry [30] (Example 2.5), a topological description of the Aharonov–Bohm effect (Example 5.2), classification of elementary quantum Borel kinematics on arbitrary two-dimensional compact orientable manifolds (Example 5.3) as well as in the real projective space — topologically non-trivial part of the configuration space of the system of two identical particles [9] (Example 5.4). 2. Mackey’s System of Imprimitivity 2.1. The formalism of quantum mechanics In quantum mechanics, a separable Hilbert space H is associated with a quantum system we are going to describe. States of the system are represented by von Neumann’s statistical (density) operators — bounded self-adjoint positive operators in H with unit trace. The set of states W introduced in this way is convex; its extremal points are called pure states. The pure states are just the projectors on one-dimensional subspaces of H. We assumea with [53] that, to a measurement on the system, taking values in a set X endowed with a σ-algebra B(X) of measurable subsets, there corresponds a projection-valued measure F on X. To any measurable set S ∈ B(X), a projector S∞ P∞ F (S) is related such that F (X) = 1l and F ( i=1 Si ) = i=1 F (Si ), provided Si ∩ Sj = ∅ for i 6= j. If the system is in a state U ∈ W, then the formula pU (S) = Tr(U F (S)) gives the probability that the result of the measurement belongs to the set S ∈ B(X). The map pU : B(X) → R is evidently a probability measure on X. From the above considerations it is clear that in the quantal formalism crucial role is played by an orthocomplemented lattice of projectors onto subspaces of the Hilbert space H [53]. This lattice will be denoted by L; partial ordering of L is defined as follows: F1 ≤ F2 if and only if F1 F2 = F1 ; the complement: F ⊥ = 1l − F . Clearly, F1 ≤ F2 if and only if the corresponding subspaces are in inclusion; F ⊥ projects on the orthogonal complement. Important properties of the lattice of projectors are: W (ı) For any countable set F1 , F2 , . . . of elements from L there exist n Fn and V b n Fn in L; a We are going to use such spectral measures for position measurements in configuration space and assume tacitly that they are ideal and the state after the measurement can be described by a projection E(S)ψ of the original state ψ. There are several options for a description of a non-ideal localization — e.g. with W the use of positive operator-valued measures [40], Chap. 3. b The element A = n Fn is defined by the following properties: 1) A ≥ Fn for all n; 2) if B is any element of L such that Fn ≤ B for all n, then A ≤ B. In an analogous fashion, the element V C = n Fn is defined by: 1) C ≤ Fn for all n; 2) if D is any element of L such that Fn ≥ D for all n, then C ≥ D.

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(ıı) For F1 , F2 ∈ L and F1 ≤ F2 there exists an element P ∈ L such that P ≤ F1 and P ∨ F1 = F2 . This element is equal to P = F1⊥ F2 = (1l − F1 )F2 = F2 − F1 and is unique with this property. An orthocomplemented lattice satisfying (ı), (ıı) is called a logic. 2.2. Symmetry and quantum mechanics Let the configuration manifold — which we shall always denote by M — be a G-space of a symmetry group G. This means that an action of G on M is given, i.e. to each element g of G there corresponds a transformation g of M onto itself such that: (1) e.u = u, (2) g1 (g2 · u) = (g1 g2 ) · u, where g1 , g2 ∈ G, u ∈ M . Some important assertions, in particular Mackey’s Imprimitivity Theorem, can be stated provided the group G is locally compact and separable (i.e. with countable basis of the topology). In the following we shall restrict our considerations to the case when G is a finite-dimensional connected Lie group. It is just these groups that very often appear in physical applications. Let the manifold M be also connected and smooth, and the mapping (g, u) 7→ g · u be infinitely differentiable (C ∞ ). Now we would like to associate, to each symmetry transformation g of the configuration space M , a symmetry transformation of the quantum mechanical description. To be specific, we introduce two notions. Definition 2.1. An automorphism of a logic L is a one-to-one mapping α : L → L which satisfies (i) α(1l) = 1l, (ii) α(F ⊥ ) = α(F )⊥ , W∞ W∞ (iii) α( n=1 Fn ) = n=1 α(Fn ). Definition 2.2. A convex automorphism of the set of states W is a one-to-one mapping β : W → W with the following property: given positive real numbers P P P c1 , c2 , . . . such that n cn = 1, then β( n cn Un ) = n cn β(Un ). Now we can state the conditions on symmetry transformations of the quantum mechanical description in the following form: (a) There exists a homomorphism α : g 7→ α(g) from the group G into the group of automorphisms of the logic L, α(g) : L → L : F 7→ F g . (b) There exists a homomorphism β : g 7→ β(g) from the group G into the group of convex automorphisms of the set of states W, β(g) : W → W : U 7→ U g . (c) The probability does not change under the symmetry transformations, i.e. Tr(U g F g ) = Tr(U F ) ,

U ∈ W, F ∈ L .

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(d) Given a projection-valued measure F on a set X (endowed with a σ-algebra of measurable subsets) which corresponds to measurements on the system with values in X, there exists a homomorphism γ : g 7→ γ(g) from the group G into the group of measurable and one-to-one mappings from X onto itself. Every γ(g) induces an automorphism of the σ-algebra, B(X) → B(X) : S 7→ S g . We demand F (S g ) = F (S)g .

(2.1)

Automorphisms of the logic and convex automorphisms of the set of states are described by Wigner’s theorem: Theorem 2.1 ([53, Chap. VII.3]). Let H be a separable infinite-dimensional Hilbert space. Then: (1) All automorphisms of the logic L are of the form α(F ) = T F T −1 ,

F ∈ L,

where T is a fixed unitary or antiunitary operator in H. Two such operators induce the same automorphism of the logic if and only if they differ by a phase factor. (2) All convex automorphisms of the set of states W are of the form β(U ) = T U T −1 ,

U ∈W,

where T is a fixed unitary or antiunitary operator in H. Two such operators induce the same convex automorphism of the set of states if and only if they differ by a phase factor. Theorem 2.1 and conditions (a), (b) imply that to each action g ∈ G a pair of operators T (g), T 0 (g) is associated, both being unitary or antiunitary. To fulfil condition (c), operators T (g), T 0 (g) may differ by a phase factor at most. Hence these operators can be identified, T (g) = T 0 (g) ,

g ∈ G.

Since we consider only connected Lie groups, all operators T (g) will be unitary.c Let us denote by U (H) the group of unitary operators in H with strong topology. The centre Z of this group consists of operators z · 1l, z ∈ T 1 , where T 1 denotes the compact Lie group of complex numbers of unit modulus. The quotient group P (H) = U (H)/Z c This follows, on the one hand, from the fact that the composition of two antiunitary operators is unitary, and, on the other hand, from the fact that in some neighbourhood N of the unit element e there exists a ∈ N to each b ∈ N such that b = a2 ; it is well known that G is generated by the elements of N .

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is called the projective group of the Hilbert space H. The conditions (a)–(c) can be summarized in one requirement (abc) There exists a homomorphism h : G → P (H) . Moreover, we shall demand h to be measurable. In this case h is even continuous [54, Chap. VIII.5]. The condition (abc) can be reformulated with the use of the notion of a projective representation. Let π : U (H) → P (H) be the canonical homomorphism. For each given homomorphism h : G → P (H) there exists a measurable mapping V : G → U (H), V (e) = 1l, h = π ◦ V . The mapping V is called the projective representation of G. Two projective representations V , V 0 are called equivalent if there exists a measurable mapping z : G → T 1 such that V 0 (g) = z(g)V (g). The homomorphism h obviously determines the projective representation uniquely up to this equivalence. Given a projective representation V , there exists a measurable mapping m : G × G → T1 such that V (a)V (b) = m(a, b)V (ab) ,

a, b ∈ G .

The factor m(a, b) is called a multiplier of G; by definition it fulfils m(ab, c)m(a, b) = m(a, bc)m(b, c) ,

m(a, e) = m(e, a) = 1 .

Two multipliers m, m0 are equivalent if there exists a measurable mapping z : G → T 1 such that m0 (a, b) = z(ab)−1 z(a)z(b)m(a, b) . By definition a multiplier is exact (or trivial), if it is equivalent to 1. The set of all multipliers with pointwise multiplication forms an Abelian group; trivial multipliers form its invariant subgroup. The corresponding quotient group is referred to as the multiplier group for G; we shall denote it by M(G).d 2.3. Localization, systems of imprimitivity The discussion of condition (d) of Sec. 2.2 was postponed to this section, since its analysis requires the description of a concrete measurement on the system. For localizable systems the position measurements play a distinguished rôle. Results of position measurements are points of the configuration space M , i.e. X = M . So it is natural to consider the σ-algebra B(M ) of Borel subsets of M as the σ-algebra of measurable sets.e d For details see [54], Chap. X. e σ-algebra B(M ) is generated by

open subsets of manifold M .

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The starting point of Mackey’s quantization and of quantum Borel kinematics is the notion of localization of a quantum system on a configuration manifold M . It is mathematically modeled by a projection-valued measure E : S 7→ E(S) mapping Borel subsets S of M (S ∈ B(M )) into projection operators E(S) on a separable Hilbert space H subject to the usual axioms of localization. For convenience, these axioms are given below: E(S1 ∩ S2 ) = E(S1 ) · E(S2 ) , E(S1 ∪ S2 ) = E(S1 ) + E(S2 ) − E(S1 ∩ S2 ) , ! ∞ ∞ [ X E Si = E(Si ) for mutually disjoint Si ∈ B(M ) , i=1

i=1

E(M ) = 1l . For a given subset S ∈ B(M ), the projection E(S) corresponds to a measurement which determines whether the system is localized in S; its eigenvalues 1 (0) correspond to situations when the system is found completely inside (outside) S, respectively. According to (d), each action g ∈ G induces a Borel transformation of M onto itself. As already mentioned, the other three conditions (abc) imply the existence of a projective representation V of G. Hence (2.1) can be written in the form E(g · S) = V (g)E(S)V (g)−1 ,

(S g ≡ g · S) .

(2.2)

Definition 2.3. A pair (V, E) where V is a (projective) representation of a group G and E is a projection-valued measure on a G-space M , is called a (projective) system of imprimitivity for the group G, if (2.2) holds for all g ∈ G, S ∈ B(M ). Two projective systems of imprimitivity are equivalent if the corresponding projective representations are equivalent and if the projection-valued measures are equal. 2.4. Quantization on homogeneous spaces Stronger results can be obtained if the symmetry group G of M is sufficiently rich. More precisely, we shall turn our attention to homogeneous spaces. By definition, M is a homogeneous G-space if G acts transitively on M , i.e. to each pair of points u, u0 ∈ M there exists a transformation g ∈ G such that g · u = u0 . Let us fix a point u0 ∈ M . The isotropy subgroup of u0 in G will be denoted by H. It is well known that H is a closed Lie subgroup of the Lie group G. The space G/H of left cosets gH, g ∈ G, endowed with factor topology, can be given a differentiable (C ∞ ) structure, thus becoming a smooth manifold, and the mapping π : g 7→ g·u0 induces a diffeomorphism of G/H onto M ([45], Chap. II.3, II.4). Having identified G/H with M , the group G acts on M in the natural way, a: gH 7→ agH. The quadruple (G, π, M ; H) can be viewed as a principal fibre bundle. Let us note here that the requirement on G to be connected is not very restrictive provided M is connected:

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Proposition 2.1 ([45, Chap. II.4]). Let G be a finite dimensional Lie group acting transitively on a connected smooth manifold M and let G0 be the connected component of unity in G. Then G0 acts transitively on M, too. As shown by Mackey [23, 50], the transitive systems of imprimitivity (i.e. based on homogeneous spaces G/H) can be completely classified. The notions of irreducibility, unitary equivalence, direct sum decomposition, etc., can be taken over for the systems of imprimitivity in exact analogy with these notions for (projective) unitary representations [50, Chap. 1.2]. In order to investigate questions of irreducibility, direct sum decomposition, etc., a commuting ring C(V, E) is considered, which consists of all bounded operators in H commuting with E(S), V (g) for all S ∈ B(M ), g ∈ G. We have for instance the property that a system of imprimitivity (V, E) is irreducible if and only if the ring C(V, E) consists of multiples of the unit operator 1l only (Schur’s Lemma). Now, following G. W. Mackey, we are going to describe the canonical construction of transitive systems of imprimitivity. Let G be a locally compact group satisfying the second axiom of countability, H its closed subgroup. On the coset space G/H there exists a quasi-invariant measure defined on the σ-algebra of Borel subsets. A measure µ on G/H is called quasi-invariant with respect to the action of G, if for all g ∈ G the measures µ and µ ◦ g : S 7→ µ(g · S) are mutually absolutely continuous. Moreover, all quasi-invariant σ-finite measures on G/H are mutually absolutely continuous [54, Chap. VIII.4]. We fix a measure µ from this class. Further, let m be a multiplier of G and let L be a projective unitary representation of H with multiplier m restricted to H in a separable Hilbert space HL . Then we construct the Hilbert space H as the space of vector-valued functions ψ : G → HL satisfying (a) a 7→ hψ(a)q, f i is a Borel function on G for all f ∈ HL ; (b) ψ(ah) = m(a, h)L−1 h ψ(a), h ∈ H; (c) kψk < ∞, where k · k is the norm induced by the inner product Z (ψ, ψ 0 ) = hψ(a), ψ 0 (a)idµ(u) ; G/H

the integral is well-defined since, because of (b), the inner product hψ(a), ψ 0 (a)i in HL remains constant on the left cosets u = aH. Henceforth we shall identify two functions on G/H which are equal µ-almost everywhere. Then the projection-valued measure S 7→ E L (S) on G/H is canonically defined by [E L (S)ψ](a) = χ ˜S (a)ψ(a) ,

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where

( χ ˜S (a) =

1,

for aH ∈ S ,

0,

for aH ∈ / S.

The projective m-representation of G, g 7→ V L (g), is given by s dµ L [V (g)ψ](a) = (g −1 u)m(a−1 , g)ψ(g −1 a) , d(µ ◦ g) where dµ/d(µ ◦ g) is the Radon–Nikod´ ym derivative. The pair (V L , E L ) is called a canonical system of imprimitivity and its equivalence class does not depend on the choice of a quasi-invariant measure µ. Theorem 2.2 (The Imprimitivity Theorem [23]). Let G be a locally compact group satisfying the second axiom of countability, H its closed subgroup and m a multiplier of G. Let a pair (V, E) be a projective system of imprimitivity for G based on G/H with multiplier m. Then there exists an m-representation L of H such that (V, E) is equivalent to the canonical system of imprimitivity (V L , E L ). For any two m-representations L, L0 of the subgroup H the corresponding canonical systems of imprimitivity are equivalent if and only if L, L0 are equivalent. The commuting rings C(V L , E L ) and C(L) are isomorphic. The Imprimitivity Theorem shows how to obtain all systems of imprimitivity up to unitary equivalence, provided the multiplier group M(G) is known. More facts about the multiplier group can be given in the case when G is a connected and simply connected Lie group. Then every multiplier is equivalent to a multiplier of class C ∞ . These multipliers can be expressed in the form exp(ip) where p is called an infinitesimal multiplier. Let us introduce a coboundary operator δ on real skew-symmetric multilinear forms on the Lie algebra G via X ˆk , . . . , X ˆ j , . . . , Xn ) , δp(X1 , . . . , Xn ) = (−1)k+j−1 p([Xk , Xj ], X1 , . . . , X k<j

where p is any (n − 1)-form. Then the Abelian group of infinitesimal multipliers of G is isomorphic to the second cohomology group H 2 (G, R). Example 2.1. (i) H 2 (Rs , R) is isomorphic to the additive group of real skewsymmetric 2-forms on Rs ; given a 2-form p, then m : (x, y) 7→ exp[ip(x, y)] is a multiplier; M(R1 ) = {1}. (ii) M(T s ) = {1}, T s = Rs /Zs . (iii) If G is a connected and simply connected semi-simple Lie group, then M(G) = {1}. We note that any connected Lie group G can be replaced by its (connected and ˜ The action of g˜ ∈ G ˜ on M is simply connected) universal covering Lie group G. ˜ given by g˜ : u 7→ π(˜ g ) · u where π : G 7→ G is the covering homomorphism. As

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Example 2.4 (see Sec. 2.5) will show, the transition to the covering group can lead to richer results with reasonable physical interpretation. Remark 2.1. Detailed descriptions of general foundations of quantum mechanics can be found in [53, Chap. VI and VII]; of systems of imprimitivity in [54, Chap. IX], [23] and [10]; of multipliers in [54, Chap. X]. 2.5. Infinitesimal action on a G-space Let G be a connected Lie group and M a (not necessarily homogeneous) G-space. A one-parameter subgroup of G is a one-dimensional Lie subgroup including its parametrization, a : R → G : t 7→ a(t). There is a one-to-one correspondence between elements A of a Lie algebra G and one-parameter subgroups {a(t)} which can be expressed by a0 (0) = A ([47], Chap. I.6.4). This correspondence can be used to define the mapping exp : G → G : A 7→ a(1); then one has a(t) = exp(tA), and exp is a local diffeomorphism at the unit element of G. To each A ∈ G there corresponds a one-parameter subgroup {a(t) = exp(tA)} and a flow on M , (t, u) 7→ a(t) · u; the corresponding vector field on M will be denoted by DA . If F u : G → M is the mapping g 7→ g·u depending on u ∈ M , then obviously DA (u) = (dF u )e ·A. In the terminology of [38], an infinitesimal action is the mapping G → X (M ) : A 7→ DA , where X (M ) denotes the infinite-dimensional Lie algebra of smooth vector fields on M . Let N be the subgroup of ineffectively acting elements from G. If N = {e}, then G is said to act effectively on M . N is closed and normal, the factor group G/N is a Lie group acting effectively on M . Manifold M can be considered as a G/N -space if the action is given by G/N 3 gN : u 7→ g · u. In this way the ineffectively acting elements can be eliminated.f Theorem 2.3 ([38, Chap. III.3.7]). The infinitesimal action A 7→ DA is linear. For all A, B ∈ G one has [DA , DB ] = −D[A,B] . Hence the image of this mapping is a finite-dimensional Lie subalgebra in X (M ). The kernel is N , the Lie algebra of the group N of ineffective elements from G. The proof of the first part is based on a straightforward calculation [38, Chap. III.3.7]. For the last assertion we observe that, if DA = 0 and φA is the corresponding flow, then a(t) · u = φA (t, u) = u so {a(t)} ⊂ N . Now let us consider the opposite situation. Suppose we are given a finitedimensional Lie subalgebra G in X (M ) such that all vector fields from G are complete. Let G˜ be the Lie algebra with the same vector space as G but with a Lie ˜ is the connected and simply connected Lie group with bracket [·, ·]∼ = −[·, ·]. If G f However,

for N discrete it does not seem reasonable to eliminate N in this way; see Example 2.3.

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˜ is unique up to isomorphism), then according to [38], Chap. III.4.7, Lie algebra G˜ (G Theorem 6, to each u ∈ M there exists an open neighbourhood Bu and a uniquely ˜ on Bu (i.e. an action defined only for elements from some defined local action of G neighbourhood Ue of the unity) such that the associated infinitesimal action is identical with the mapping G˜ 3 X 7→ X|Bu . The neighbourhood Ue can be chosen small ˜ φX — the corresponding enough for exp to be a diffeomorphism on it. For X ∈ G, flow, v ∈ Bu , t ∈ R sufficiently small, we have exp(t · X) · v = φX (t, v). Since all vector fields X ∈ G˜ are complete, Ue can be chosen independently of u ∈ M . In this way ˜ into the group of diffeomorphisms of M . we obtain a local homomorphism from G ˜ is connected and simply connected, the domain of the local homomorphism Since G ˜ ([39], Chap. II and VII). can be unambiguously extended to the whole G 2.6. Examples Example 2.2. M = R1 , G = R1 — the group of translations. Both the multiplier group M(R1 ) and the isotropy subgroup are trivial. So in this case exactly one irreducible system of imprimitivity exists (up to unitary equivalence). Let H be a Hilbert space and (Q, P ) be a pair of self-adjoint operators in H satisfying the commutation relation QP − P Q = i~1l. Then if E : S 7→ E(S) is the spectral projection-valued measure of Q, and V (t) = exp(−itP/~), then the commutation relation is equivalent to the identity V (t)E(S)V (−t) = E(t + S) . So the Imprimitivity Theorem implies the Stone–von Neumann theorem (cf. [50], Chap. 2.5). Example 2.3. M = R2 , G = R2 — the group of translations. To each element A = (A1 , A2 ) ∈ G = R2 there corresponds a vector field DA = (A1 ∂/∂x1 ) + (A2 ∂/∂x2 ) ∈ X (R2 ). The isotropy subgroup is trivial. M(R2 ) is isomorphic to the group R under addition: if B ∈ R, then eB 2 2 mB : R × R → U (1) : (x, y) 7→ exp −i (x1 y2 − x2 y1 ) 2~ is a multiplier (e is an arbitrary fixed non-zero constant). The inequivalent irreducible systems of imprimitivity (V B , E B ) are labelled by B ∈ R: the Hilbert space is HB = L2 (R2 , dx1 dx2 ) and we find E B (S) : ψ 7→ χS · ψ and i B B V (exp(tA))ψ = exp − tP (DA ) ψ , t ∈ R , ~ where P B (X) is a self-adjoint operator, P B (X)ψ = (−i~X − eα(X))ψ ,

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α = (B/2)(x1 dx2 − x2 dx1 ), i.e. dα = Bdx1 ∧ dx2 , and ψ ∈ C0∞ (R2 ). The real number B can be given physical meaning: a particle with electric charge e moves on the plane R2 in an external magnetic field which is perpendicular to the plane and has constant value B (the sign reflects the orientation). Example 2.4. M = S 1 , G = U(1) — the group of rotations of the circle S 1 . Both M(U(1)) and the isotropy subgroup are trivial; hence there exists exactly one irreducible system of imprimitivity (up to unitary equivalence). Now let us replace ˜ = R. Then M(R) = {1} and the isotropy G = U(1) by its universal covering G subgroup H = 2πZ. The irreducible unitary representations of H are labelled by elements z ∈ R/2πZ: for Φ ∈ R let i LΦ : 2πk 7→ exp keΦ , k ∈ Z ; ~ 0

if (e/~)(Φ − Φ0 ) ∈ 2πZ, then LΦ = LΦ . We shall describe the system of imprimitivity for given Φ ∈ R. The Hilbert space HΦ consists of (equivalence classes of) functions ψ : R → C such that i ψ(x + 2πk) = exp − keΦ ψ(x) , (k ∈ Z) ~ almost everywhere; the inner product is defined by Z a+2π (ψ, ψ 0 ) = ψ(x)ψ 0 (x) dx ,

a ∈ R.

a

We have

i Φ V (t) = exp − tP , ~ Φ

where t ∈ R and P Φ = −i~d/dx is self-adjoint. The mapping

eΦx W : HΦ → H0 : ψ(x) 7→ exp i · ψ(x) 2π~

is unitary, H0 can be identified with L2 (S 1 , dϕ). We find W P Φ W −1 = −i~

d eΦ − , dϕ 2π

ϕ ∈ [0, 2π) .

A possible physical interpretation is connected with the Aharonov–Bohm effect [1]: a particle in R3 with electric charge e is moving on the circle x21 + x22 = 1, x3 = 0, and external magnetic flux Φ is concentrated along the x3 -axis passing through the centre of the circle. If (e/2π~)(Φ − Φ0 ) is an integer, then the two quantum kinematics with fluxes Φ 6= Φ0 lead to the same observable results (e.g. the same interference pattern).

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Example 2.5. M = S 2 , the symmetry group G = SO(3) is replaced by the quantum mechanical symmetry group SU(2) [10] acting on S 2 in the usual way ! ! x3 x1 − ix2 x1 − ix2 x3 7→ T T∗ , x1 + ix2 −x3 x1 + ix2 −x3 P where T ∈ SU(2) and points (x1 , x2 , x3 ) ∈ S 2 ⊂ R3 ( i x2i = 1) are identified with P subgroup the matrices i xi σi ; σ1 , σ2 , σ3 are the Pauli spin matrices. The isotropy τ 0 1 H of the north pole (0, 0, 1) consists of the diagonal matrices 0 τ¯ , τ ∈ T ' U(1), ¯ = 1, hence H ≡ U(1). If T is parametrized by α, β ∈ C, α ¯ α + ββ ! α −β¯ T = T (α, β) = , β α ¯ then the projection π : SU(2) → S 2 ' SU(2)/U(1) is given by ¯ . π(T ) = T σ3 T ∗ = (2 0, b ∈ R . 0 1 1 1 The isotropy subgroup H of the origin O consists of all matrices with b = 0. The irreducible unitary representations of H are of the form Lc : a 7→ aic , c ∈ R. The Lie algebra G consists of all matrices ! A1 A2 A= , A1 , A2 ∈ R . 0 0 gA

more detailed discussion of this example was given in [30]. The first treatments of the magnetic monopole using a connection in a fibre bundle appeared in [19, 36, 37].

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An element A ∈ G induces a vector field DA = (A1 x + A2 )d/dx ∈ X (R1 ). Let us investigate M(G): G is connected and simply connected, and a general real skew-symmetric 2-form p on G is of the form p : (A, B) 7→ K(A1 B2 − A2 B1 ) for some constant K ∈ R; but it is easily verified that p(A, B) = q([A, B]) where q : A 7→ KA2 is a 1-form on G, so M(G) is trivial (see Sec. 2.3). Thus the irreducible systems of imprimitivity are labelled by c ∈ R. Explicitly, Hc = L2 (R, dx), E c (S) acts via multiplication by indicator function χS (x), and x−b c −1/2 ic ln a e ψ . [V (a, b)ψ](x) = a a The self-adjoint generalized momentum operators P c (DA ), A ∈ G, defined by V c (exp(tA)) = exp(−itP c (DA )/~) , are of the form

1 P (X) = −i~ X + divX − ~c divX , 2 c

X = DA .

3. Quantum Borel Kinematics: Localization 3.1. Generalized system of imprimitivity Generally, for a given smooth manifold M there is, a priori, no geometric symmetry group. As indicated in [4, 11, 12, 27], the investigation of vector fields on M is a meaningful starting point. We denote by X (M ) the Lie algebra of smooth vector fields on M , by X0 (M ) its subalgebra of compactly supported vector fields, by Xc (M ) the family of all complete vector fields, X0 (M ) ⊂ Xc (M ). The flow φX of a complete vector field X represents a one-parameter group of diffeomorphisms {φX t }t∈R of M , also called a dynamical system on M . And, vice versa, every dynamical system is a flow of some (uniquely determined) complete vector field d [Xψ](u) = (ψ ◦ φX )(u) . t dt t=0 The family of dynamical systems on M will be denoted by D(M ). The following theorem summarizes some well-known facts from differential geometry [48, 51]. Theorem 3.1. Let f : M → M 0 be a diffeomorphism. Then f 0 : X (M ) → X (M 0 ), where (f 0 · X)f (u) = dfu (Xu ), is a Lie algebra isomorphism; the restriction f 0 : X0 (M ) → X0 (M 0 ) is also a Lie algebra isomorphism; f 0 : Xc (M ) → Xc (M 0 ) is a bijection. The mapping f D : D(M ) → D(M 0 ) : {φt }t∈R 7→ {f ◦ φt ◦ f −1 }t∈R is bijective and f D (φX ) = φf

0

·X

.

For every φX ∈ D(M ) the manifold M becomes a G-space for the group G = R. Attempting to generalize Mackey’s quantization (Secs. 2.3 and 2.4) we require that

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there exist: a Hilbert space H, a projection-valued measure E on M and unitary representations V X in H of the flows φX ∈ D(M ) such that V X (t)E(S)V X (−t) = E(φX t · S) ,

(3.1)

where the objects H, E do not depend on the choice of φX ∈ D(M ). Equation (3.1) is just a generalization of (2.2). Geometric shifts of Borel sets S ∈ B(M ) by flows φX t along complete vector fields X are represented in H by unitary operators V (t) such that (3.1) holds. Generalized momentum operators can then be introduced via Stone’s Theorem as (essentially self-adjoint) infinitesimal generators P (X) of the one-parameter groups of unitary operators — shifts in H of the localized quantum system, i X V (t) = exp − P (X)t , t ∈ R . ~ 3.2. Quantum Borel kinematics The quantization of “classical” Borel kinematics (B(M ), Xc (M )) thus requires [11] the imprimitivity condition (3.1) for the unitary representation of the flow of each complete vector field individually. Then we can state Definition 3.1. Quantum Borel kinematics is a pair (V, E), where E is a projection-valued measure on M in a separable Hilbert space H, and V associates with each φX ∈ D(M ) a homomorphism V X : R → U (H) such that the following conditions are satisfied: (1) Equation (3.1) holds for all t ∈ R, X ∈ Xc (M ), S ∈ B(M ); (2) The mapping P : X 7→ P (X) from the Lie algebra X0 (M ) into the space of essentially self-adjoint operators with common invariant dense domain in H is a Lie algebra homomorphism: P (X + aY ) = P (X) + aP (Y ) , [P (X), P (Y )] = −i~ P ([X, Y ] ;

(3.2) (3.3)

(3) Locality condition. If two flows φXi ∈ D(M ), i = 1, 2, after restriction on the set (−a, a) × S, a > 0, S ∈ B(M ), coincide, then the mappings R × H → H : (t, ψ) 7→ V Xi (t)ψ coincide on the domain (−a, a) × HS , where HS is the subspace of H projected out by E(S). If in (2) only linearity (3.2) is required, we shall call (V, E) a generalized system of imprimitivity for D(M ).h It will describe quantum Borel kinematics with external gauge field. h The

family D(M ) acts transitively on M (we suppose M to be connected, without boundary): for any two points u, v ∈ M there exists X ∈ X0 (M ), φX ∈ D(M ) such that φX (1, u) = v.

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It follows from condition (3) that, if V X is known for all X ∈ X0 (M ), then it is determined for all X ∈ Xc (M ); (3) further implies that P (X) are differential operators. Condition (2) may sometimes be too restrictive since (3.3) excludes a non-vanishing external gauge field on M ; in this connection see [12, 13, 33] and also Example 2.3, Example 2.5 and Sec. 4.7. The projection-valued measure E induces in a natural way a quantization Q of classical (smooth) real functions f : M → R on configuration space (e.g. coordinate functions, potentials, etc.). Not necessarily bounded, self-adjoint quantum position operators Q(f ) are uniquely determined by their spectral decompositions Z ∞ λdEλf , Q(f ) = −∞

where the spectral function Eλf is given by the spectral measure E f (∆) = E(f −1 (∆)) on subsets ∆ = (−∞, λ) of R. Equation (3.1) is then replaced by V X (t)Q(f )V X (−t) = Q(f ◦ φX −t ) ,

(3.4)

where f ∈ C ∞ (M, R), and implies a generalization of the Heisenberg commutation relations in terms of coordinate-independent objects [Q(f ), P (X)] = i~Q(X.f ) on D ⊂ H .

(3.5)

It is assumed that operators P (X), Q(f ) have a common invariant dense domain D in H. If an obvious relation [Q(f ), Q(g)] = 0 on D

(3.6)

for all f, g ∈ C ∞ (M, R) is still added, then (3.6), (3.5) and (3.3) define a Schr¨ odinger system in the sense of [27]. We can say that Borel quantization on a smooth configuration manifold M associates the generalized position Q(f ) and momentum operators P (X) with smooth functions f ∈ C ∞ (M, R) and smooth vector fields X ∈ X (M ), respectively. These quantum kinematical observables on M are globally defined, hence Borel quantization incorporates the global structure of M . Remark 3.1. The natural infinite-dimensional Lie algebra structure (3.6), (3.5) and (3.3) of quantum Borel kinematics should be compared with the non-relativistic local current algebra for a Schrödinger second quantized field over M = R3 studied in [18]: [ρ(f1 ), ρ(f2 )] = 0 , [ρ(f ), J(X)] = i~ρ(Xf ) , [J(X), J(Y )] = −i~J([X, Y ]) , where f, fi ∈ C ∞ (R3 , R), X, Y ∈ X (R3 ). There is an apparent algebraic correspondence of the local density operator ρ(f ) with Q(f ) and the local current operator

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J(X) with P (X). In both cases the representations of the algebraic structures yield possible quantum kinematics. However, in contrast to quantum Borel kinematics, where Q(f ) is a multiplication operator, local current algebra is more general, since also representations where ρ(f1 f2 ) is not equal to ρ(f1 )ρ(f2 ) are admitted. 3.3. Quasi-invariant measures, projection-valued measures The question of existence and uniqueness of a measure which is quasi-invariant with respect to all diffeomorphisms φ1 : u 7→ φ(1, u) for which φ ∈ D(M ), is answered by Theorem 3.2. The family of quasi-invariant measures on B(M ) is non-empty and, moreover, all measures in this family are mutually equivalent and form a unique invariant measure class.i After completion, those subsets in M which have measure zero are exactly measure zero sets in the sense of Lebesgue. Proof. The fact that the family of sets of zero measure in the sense of Lebesgue is invariant under diffeomorphisms is well known [44]. The existence part of the theorem can be seen as follows. Having embedded M in Rm (Whitney’s Theorem), we can consider a tubular neighborhood M ε of M in the normal bundle ([44], Chap. 2.3). Denoting by π : M ε → M the associated submersion, we can define µ(S) = λm (π−1 (S)) for S ∈ B(M ), where λm denotes the Lebesgue measure in Rm ; then µ is quasi-invariant. The assertion about uniqueness of the invariant measure class for M = Rn follows from the fact that the family of diffeomorphisms φ1 includes all translations and the assertion for the group of translations is known ([54], Chap. II.3). In general, M can be covered by a countable family of open sets, each of which is diffeomorphic to Rn and so the assertion is true again. Measure µ will be called differentiable if the mapping R → R : t 7→ µ(φt · S) is smooth for all S ∈ B(M ), φ ∈ D(M ). For instance, measure µ used in the proof of Theorem 3.2 is differentiable. Every manifold is locally orientable; having fixed an orientation on an open set U ⊂ M (dim M = n), then to every differentiable measure µ exactly one n-form ω exists on U such that Z Z f ·ω = f (u)dµ(u) for all f ∈ C0∞ (U ) ; U

U

ωu (X1 , . . . , Xn ) > 0 for every positively oriented basis in Tu U . For each X ∈ X (U ) we define a function divµ X on U by divµ X · ω = k! d(iX ω) , where the (n − 1)-form iX ω is defined by (iX ω)(X1 , . . . , Xn−1 ) = ω(X, X1 , . . . , Xn−1 ) . i The

invariant measure class is called the Lebesgue measure class.

(3.7)

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Equivalently, if φ = φX ∈ D(M ), then d dµ (φ−t · u)|t=0 = −divµ X . dt d(µ ◦ φt )

(3.8)

In local coordinates, if ω = ρ(u)du1 ∧ · · · ∧ dun , ρ > 0, then divµ X = X · ln ρ +

X ∂Xj j

∂uj

.

The structure of projection-valued measures on M is well known. According to [54], Chap. IX.4, we have a canonical representation of a localized quantum system: Let E be a projection-valued measure on M in a separable Hilbert space H. Then there exist two sequences {Kr }, {νr }, r = ∞, 1, 2, . . . , the first one consisting of Hilbert spaces, the other of measures on M such that dim Kr = r and νr , νs are mutually singularj for r 6= s. The projection-valued measure E is unitarily equivalent to the measure E 0 which acts via multiplication by indicator functions L of subsets in the Hilbert space H0 = r Hr , where Hr are the Hilbert spaces of vector-valued functions from M to Kr , Hr = L2 (M, Kr , νr ). The measures νr are determined uniquely up to equivalence. If only one νr is non-zero, the projectionvalued measure E is called homogeneous. Due to the transitivity of actions of the family D(M ) the following theorem holds (for details see [54], Chap. IX.5, IX.6): Theorem 3.3. If (V, E) is a generalized system of imprimitivity for D(M ), then E is homogeneous. The unique non-zero measure νr belongs to the Lebesgue measure class on M. Thus the canonical representation of a localized quantum system on M involves a smooth measure, i.e. a measure induced by the Lebesgue measure of the coordinate charts. An r-homogeneous localized quantum system of degree r > 1 can be interpreted as a quantum system with internal degrees of freedom; a 1-homogeneous localized quantum system will be called elementary as its E’s are related to elementary spectral measures. We shall need also Theorem 3.4 ([54, Chap. IX.2]). Let µ be a Borel measure on M, K a Hilbert space, H = L2 (M, K, µ), E a projection-valued measure on B(M ) acting in H via multiplication by indicator functions. Then any bounded operator B in H commuting with E(S) for all S ∈ B(M ) (B ∈ C(E)) is of the form B : ψ(u) 7→ b(u) · ψ(u) , where b is a Borel mapping from M into the space of bounded operators in K such that supu∈M |b(u)| < ∞. Function b is determined by B uniquely on M modulo a set of µ-measure zero. j i.e.

there exist Sr , Ss ∈ B(M ), Sr ∪ Ss = M , Sr ∩ Ss = ∅, νr (Ss ) = νs (Sr ) = 0.

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3.4. An example Quantum Borel kinematics are rather diverse (even for trivial configuration space Rn ), as the following example [33] shows. Let M = Rn with a fixed basis. To every P vector field X = k Xk ∂/∂xk we relate a matrix-valued function A(X) : Rn → Rn,n ,

[A(X)u ]i,j =

∂Xi (u) . ∂xj

It is straightforward to verify [A(X), A(Y )] − X · A(Y ) + Y · A(X) = −A([X, Y ]) .

(3.9)

Let L be a skew-Hermitean representation of the Lie algebra gl+ (n, R) in a Hilbert space HL . We define operators Q(f ), P (X) in H = L2 (Rn , HL , dx1 · · · dxn ) by 1 Q(f )ψ = f · ψ , P (X)ψ = −i~ X + div X − L(A(X)) . 2 Then using (3.9) and the identity X ·L(A(Y )) = L(X ·A(Y )), the pair (P, Q) can be shown to be a quantum Borel kinematic. Choosing the representation L in HL = C to be given by L(A) = −ic tr A , where c is a real constant, we obtain L(A(X)) = −ic div X . This is just an example of the divergence term which we shall encounter in Sec. 4.7. Remark 3.2. Consider the surjective mapping p : gl(n, R) → sl(n, R) : A 7→ A −

1 (tr A)1ln , n

where 1ln is the unit n × n-matrix. This mapping permits to associate with every representation L0 of sl(n, R) a representation L = L0 ◦ p of gl(n, R). Then our mapping P is the infinitesimal form of a representation of the group of diffeomorphisms of Rn induced from SL(n, R) (see [18] for n = 3). For general results concerning the “divergence-like” terms in the framework of systems of imprimitivity for the group of diffeomorphisms, see [31]. 4. Quantum Borel Kinematics: External Gauge Fields 4.1. External magnetic field In order to motivate our construction of quantum Borel kinematics with external field via generalized systems of imprimitivity for D(M ), let us consider quantum kinematics on R3 , for a charged particle in an external magnetic field B. Since

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div B = 0, the Poincaré lemma implies that there exists a vector potential A such that B = rot A. The Hilbert space is H = L2 (R3 , d3 x), and the Hamiltonian 2 e ~2 X ∂ − i Aj . H=− 2m j ∂xj ~ If another potential A0 , rot A0 = B, is chosen, then according to the Poincaré theorem there exists a real function λ such that A0 = A + grad λ. The Hamiltonian H is transformed as H 7→ H 0 = W HW −1 , where W is the unitary mapping ie W : H → H 0 : ψ 7→ exp λ ψ. ~ This common quantum mechanical scheme can be reformulated in geometric language [36]: H is the space of measurable sections in the trivial Hermitian complex line bundle R3 × C1 associated with the principal bundle R3 × T 1 ; X i(e/~)α , α = Aj dxj , is a localized connection 1-form; i(e/~)β ,

β = dα = B1 dx2 ∧ dx3 + cycl .

is the curvature 2-form;

x 7→ exp

ie λ(x) ~

is a transition function in the principal bundle relating two different trivializations. In general, if a magnetic field is given by a closed 2-form β on manifold M , a vector potential 1-form α such that β = dα need not exist. However, following [36], one can always define vector potentials αk locally, i.e. on open sets Uk (diffeomorphic to Rn ) such that {Uk } is an open covering of M . We require that vector potentials αk , αj be related on the intersection Uk ∩ Uj by a gauge transformation ~ ie ie iαj = iαk + d exp λ exp − λ = i(αk + dλ) , e ~ ~ where exp( ie ~ λ(x)) is a transition function. Then dαj = dαk . In this way we shall construct a principal bundle with typical fibre T 1 and connection {iαj }. 4.2. Construction of a class of generalized systems of imprimitivity We choose a measure µ from the Lebesgue measure class on M and four objects (P, G, Γ, L), where: P ≡ (P, π, M, G) (or shortly P (M, G)) is a principal bundle over M ; its typical fibre G is an Abelian or compact Lie group; Γ is a connection in P , and L is a unitary representation of G in a finite-dimensional Hilbert space K with inner product h·, ·i. We construct a separable Hilbert space H like in Sec. 2.3, consisting of vector-valued functions ψ : P 7→ K, such that

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(1) x 7→ hψ(x), f i is a Borel function on P for all f ∈ K, (2) ψ(xg) = L−1 g ψ(x), g ∈ G (ψ is an equivariant function), (3) kψk < ∞, where k · k is the norm induced by the inner product Z 0 (ψ, ψ ) = hψ(x), ψ 0 (x)idµ(u) . M

The integral is well defined as the integrand remains constant on the fibres. Two functions ψ, ψ 0 are identified if they coincide almost everywhere. The projectionvalued measure E on M is defined via multiplication by indicator functions: ( 1 , π(x) ∈ S , E(S)ψ = χ ˜S ψ , χ ˜S (x) = (4.1) 0 , π(x) ∈ / S. For φ ∈ D(M ) we define the unitary representation of the additive group R s dµ [V φ (t)ψ](x) = (4.2) (φ−t u)ψ(φ˜−t x) , u = π(x) . d(µ ◦ φt ) Here φ˜ denotes the horizontal lift of the flow φ on M . It is easily verified that the constructed pair (V, E) is a generalized system of imprimitivity for D(M ) in the sense of Definition 3.1. The linearity of mapping P required by this definition will be investigated in Sec. 4.3 (see (4.7)). The equivalence class of (V, E) does not depend on measure µ; if µ0 is another p 0 measure from the Lebesgue class, then H → H : ψ 7→ dµ/dµ0 ψ is the desired unitary mapping. For simplicity we shall suppose µ to be differentiable. We assumed G to be Abelian or compact in order to deal only with finite-dimensional unitary representations L of G. For dim K = ∇ we identify K with Cr endowed with the standard inner product. We say that the pair (V, E) is a generalized system of imprimitivity specified by the quadruple (P, G, Γ, L). We note that, for two diffeomorphic manifolds, there is a one-to-one correspondence between the equivalence classes of generalized systems of imprimitivity constructed in this way. 4.3. Construction in the associated vector bundle The associated vector bundle (F, π ¯ , M ; Cr ) will be constructed in the standard way (see e.g. [48, 51]). We introduce equivalence relation (x; ξ) ∼ (xg; L−1 g ξ) ,

g ∈ G,

on P × Cr and put F = P × Cr / ∼; the projection is π ¯ : F → M : [x; ξ] 7→ π(x). Each fibre becomes an r-dimensional Hilbert space: λ[x; ξ] + [x; ξ 0 ] = [x; λξ + ξ 0 ] , h[x; ξ], [x; ξ 0 ]i = hξ, ξ 0 i ,

π(x) = u .

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The inner product is well-define d since L is unitary. For the same reason we can relate a unitary mapping x ˆ to each x ∈ P : x ˆ : Cr → Fπ(x) : ξ 7→ [x; ξ] and we have x cg = x ˆ ◦ Lg . ¯ be the Hilbert space of measurable sections in the associated vector Let H bundle, having finite norm induced by the inner product Z (σ, τ ) = hσ(u), τ (u)idµ(u) . (4.3) M

¯ : ψ 7→ σ in a natural way: We can define a unitary mapping T : H → H σ(u) = [x; ψ(x)] ,

π(x) = u .

−1 The definition of T is correct since (x; ψ(x)) ∼ (xg; L−1 g ψ(x)) and Lg ψ(x) = ψ(xg). −1 −1 The inverse mapping T : σ 7→ ψ is given by ψ(x) = x ˆ ◦ σ(π(x)). Having performed the unitary transform T we replace (V, E) by a generalized system of ¯ in H. ¯ We shall compute explicit expressions. imprimitivity (V¯ , E) ¯ Clearly, E acts via multiplication by indicator functions

¯ E(S)σ = χS · σ .

(4.4)

In order to express V¯ we must first describe induced connection in the associated vector bundle F . Having Hermitian structure on the fibres, we consider only Hermitian connections on F , i.e. connections for which all linear isomorphisms C(ut ) : Fu0 → Fut (shortly Ct , see below), which belong to curves ut in M , are unitary. As is well known, there is a one-to-one correspondence between Hermitian connections and Hermitian covariant derivatives in F . A covariant derivative ∇ acting on smooth sections, ∇X : Sec F → Sec F , X ∈ X (M ), is Hermitian, if it satisfies (in addition to four conditions [48] defining the covariant derivative) the identity Xhσ, τ i = h∇X σ, τ i + hσ, ∇X τ i ; Sec F denotes the linear space of smooth sections in F . A connection Γ in P induces a Hermitian connection in F ; given a piecewise ˆ−1 smooth curve ut in M and its lift xt in P, then Ct : Fu0 → Fut , Ct = xˆt ◦ x 0 is the desired unitary mapping. Its definition does not depend on the starting point x0 , since (xc x0 g)−1 = x ˆt ◦ x ˆ−1 t g) ◦ (d 0 , and xt g is another horizontal lift of the curve ut . Now the Hermitian covariant derivative is defined by the limit 1 (∇X σ)(u0 ) = lim [Ct−1 ◦ σ(ut ) − σ(u0 )] , t→0 t

u˙ 0 = X .

We are now in the position to give explicit formula for V¯ φ (t): "s # dµ V¯ φ (t)σ(u) = C(φ−t u)−1 σ (φ−t u) . d(µ ◦ φt )

(4.5)

(4.6)

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Let Sec0 F denote the linear space of smooth and compactly supported sections in ¯ As before, a self-adjoint operator P¯ (X) is F ; the subspace Sec0 F is dense in H. defined by X i V¯ φ (t) = exp − P¯ (X)t ; ~ using (4.6), (4.5) and (3.6) we find 1 P¯ (X)σ = −i~ ∇X + divµ X σ , 2

σ ∈ Sec0 F .

(4.7)

Note that expression (4.7) implies that P¯ ∼ P is linear in X ∈ X0 (M ). Using unitary representation L we can associate a principal bundle P˜ (M, U(r)) to the principal bundle P (M, G). The construction is similar to that for F . On P × U(r) we introduce the equivalence relation (x; a) ∼ (xg; L−1 g a), g ∈ G, and put ˜ ˜ P = P × U(r)/ ∼, π ˜ : [x; a] 7→ π(x). Lie group U(r) acts on P via [x; a] · b = [x; ab], b ∈ U(r). The mapping f : P → P˜ : x 7→ [x; e] is a bundle homomorphism — we have f (xg) = f (x)Lg . With the help of this homomorphism we can transform the ˜ in P˜ , as described in the following theorem. connection Γ in P into a connection Γ ˜ be a prinTheorem 4.1 ([51, Chap. II.5]; [48]). Let f : P (M, G) → P˜ (M, G) cipal bundle homomorphism and Γ a connection in P. Then there exists a unique ˜ in P˜ such that the tangent mapping f∗ maps every horizontal subspace connection Γ ˜ of connection Γ onto a horizontal subspace of Γ. ˜ specified by (P˜ , U(r), Γ, ˜ id), with The generalized system of imprimitivity (V˜ , E) id: U(r) → U(r) being the identity mapping (fundamental representation), is equivalent to (V, E). We describe the corresponding unitary mapping. If ψ ∈ H, there ˜ (x)) = ψ(x). Indeed, if f (x) = f (x0 ), then ˜ such that ψ(f is a unique vector ψ˜ ∈ H 0 x = xg and Lg = e, hence ψ(x) = ψ(x0 ) for each ψ ∈ H. The mapping defined in ˜ since π this way is unitary and transforms E in E ˜ (f (x)) = π(x); V is transformed in V˜ since f preserves the connection. We can again associate a vector bundle F˜ to the principal bundle P˜ using the fundamental representation id. Both F˜ and F have the same base space and the same typical fibre. In fact, F and F˜ can be identified by the mapping W : F → F˜ : [x; ξ] 7→ [f (x); ξ] , where the homomorphism f : P → P˜ was described above. W is well-defined since −1 [f (xg); L−1 g ξ] = [f (x)Lg ; Lg ξ] = [f (x); ξ] .

W is surjective because (˜ x; ξ) ∼ (˜ xa−1 ; aξ) = (f (x); aξ) for each x ˜ = [x; a] ∈ P˜ . W is injective because, for each u ∈ M , the induced mapping Wu : Fu → F˜u is unitary. ˜ in F˜ which corresponds to There is again a Hermitian covariant derivative ∇ ˜ ˜ the Hermitian connection Γ in P . We can briefly say that the following diagram commutes:

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HH HH

f

P (M, G), Γ (L)

Hj

- P˜(M, U(r)), Γ˜ (id)

F (M, C ), ∇ r

For the corresponding generalized systems of imprimitivity we find the following commuting diagram:

HH HH

(V, E)

Hj ¯

- (V˜ , E)˜

¯ (V , E)

All these generalized systems of imprimitivity are mutually unitarily equivalent.

4.4. The Case G = U(r) With the help of the identical representation id of U(r) in Cr we can associate a vector bundle F (M, Cr ) to every principal bundle P (M, U(r)). To each x ∈ P there corresponds the unitary mapping xˆ which we henceforth denote by the same letter x, namely x : Cr → Fπ(x) : ξ 7→ [x; ξ]. Conversely, let w : Cr → Fπ(x) be a unitary mapping. Then w necessarily has the form ξ 7→ [x; ξ w ], where ξ 7→ ξ w is a unitary mapping Cr → Cr . Hence there exists a ∈ U(r) such that wξ = [x; aξ] = [xa; ξ] = xaξ. Every unitary mapping Cr → Fu which is represented by unitary mapping xa : Cr → Fu is equal to the composition x ◦ a since [xa; ξ] = [x; aξ]. Further, two points x, x0 ∈ P coincide as unitary mappings if and only if x = x0 . So we can return from the vector bundle F (M, Cr ) back to the principal bundle P (M, U(r)). The fibre Pu over u ∈ M consists of unitary mappings Cr → Fu and the structure group G = U(r) acts on P by composition a : x 7→ x ◦ a. Definition 4.1. Two principal bundles P (M, G), P 0 (M, G) over the same base space and with the same structure group are said to be isomorphic if there exists a diffeomorphism f : P → P 0 fulfilling (1) π 0 (f (x)) = π(x), (2) f (x ◦ a) = f (x)a for all a ∈ G. Two Hermitian vector bundles F (M, Cr ), F 0 (M, Cr ) are said to be isomorphic if there exists a diffeomorphism W : F → F 0 such that the restrictions Wu : Fu → Fu0 are unitary mappings for all u ∈ M . Lemma 4.1. Two principal bundles both with structure group U(r) are isomorphic if and only if the corresponding associated vector bundles (with typical fibres Cr ) are isomorphic.

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Proof. If f : P → P 0 is an isomorphism, then W : F → F 0 : [x; ξ] 7→ [f (x); ξ] is an isomorphism of the associated vector bundles. Conversely, let x ∈ P be a unitary mapping Cr → Fu ; then Wu ◦ x is unitary mapping Cr → Fu0 and there exists a unique x0 ∈ Pu0 such that x0 = Wu ◦ x. The mapping P → P 0 : x 7→ Wπ(x) ◦ x is the desired isomorphism. As already shown, a connection in a principal bundle can be carried over to the associated vector bundle. If G = U(r), L = id, this procedure can be inverted. Let us take a Hermitian connection in the vector bundle F (M, Cr ) associated with principal bundle P (M, U(r)). The connection relates a family of unitary mappings Ct : Fu0 → Fut to every piecewise smooth curve ut on M . If x0 ∈ P , π(x0 ) = u0 , then xt = Ct ◦ x0 will be the lift in P of the curve ut with the initial point x0 . We have (Ct ◦x0 )a = Ct ◦(x0 a). This lifting prescription determines a unique connection Γ in the principal bundle P . This correspondence between the connections in P and the Hermitian connections in F is one-to-one. Definition 4.2. Connections Γ, Γ0 in principal bundles P (M, G), P 0 (M, G), respectively, are said to be isomorphic, if there exists an isomorphism F : P → P 0 which maps connection Γ in connection Γ0 . Hermitian covariant derivatives ∇, ∇0 in Hermitian vector bundles F (M, Cr ), F 0 (M, Cr ), respectively, are said to be isomorphic if there exists an isomorphism W : F → F 0 such that ∇0 = W ∇W −1 ; more precisely, ∇0X σ(u) = Wu (∇X Wu−1 σ(u)) . Lemma 4.2. Connections Γ, Γ0 in principal bundles P (M, U(r)), P 0 (M, U(r)), respectively, are isomorphic if and only if the corresponding Hermitian covariant derivatives in the associated vector bundles (with typical fibres Cr ) are isomorphic. Proof. According to Lemma 4.1, to every isomorphism f : P → P 0 there exists an isomorphism W : F → F 0 fulfilling f (x) = Wπ(x) ◦ x, and conversely. If f transforms Γ in Γ0 and if xt is the lift in P of a curve ut with starting point x0 , then x0t = f (xt ) is the lift in P 0 of the same curve with starting point f (x0 ). We have : if Ct = xt ◦ x−1 : Fu0 → Fut , then Ct0 = x0t ◦ x,−1 = Wut Ct Wu−1 . Using 0 0 0 0 −1 the last relation we obtain ∇ = W ∇W . Conversely, the relation ∇0 = W ∇W −1 implies Ct0 = Wut Ct Wu−1 and so x0t = Wπ(xt ) ◦ xt . this means that the isomorphism 0 f : x 7→ Wπ(x) ◦ x preserves the connection. 4.5. Unitary equivalence of generalized systems of imprimitivity Theorem 4.2. Two generalized systems of imprimitivity (Vj , Ej ) specified by quadruples (Pj , Gj , ΓJ , Lj ), j = 1, 2, are unitarily equivalent, if and only if the corresponding covariant derivatives in the associated vector bundles are isomorphic.

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Proof. First we have the necessary condition that the Hilbert spaces of representations L1 , L2 should have the same dimension, say r. According to the results of ¯j ), Sec. 4.3 we can equivalently take the generalized systems of imprimitivity (V¯j , E j = 1, 2, constructed in the associated vector bundles and investigate their unitary equivalence. ¯1 → H ¯ 2 relating the two generalized Thus consider a unitary mapping W : H systems of imprimitivity. Let U (F1 , F2 ) denote the fibre bundle over M with each fibre over u ∈ M consisting of unitary mappings F1u → F2u . From the equality ¯1 (S)W −1 it follows that W is induced by a measurable section u 7→ E¯2 (S) = W E W (u) in the bundle U (F1 , F2 ). From the equality V¯2φ (t) = W V¯1φ (t)W −1 and from relation (4.6) one deduces that C2 (φt u) = W (φt u)C1 (φt u)W −1 (u)

(4.8)

holds for all φ ∈ D(M ), all t ∈ R and for almost all (depending on t) u ∈ M . Here Cj (φt u), j = 1, 2, are unitary mappings corresponding to the curve ut = φt u. We shall show that the measurable section u 7→ W (u), after proper redefinition on a set of measure zero, is smooth. It is sufficient to verify this assertion locally, i.e. to investigate the case M = Rn . Let us consider all constant vector fields on Rn , the corresponding flows and t = 1. Thus we have for each v ∈ Rn and almost all u ∈ Rn the equality W (u + v) = C2 (u + v)W (u)C1−1 (u + v) ,

(4.9)

where Cj (w) : Fju → Fjw , j = 1, 2. It follows from Fubini’s Theorem that, for almost all u, the equality (4.9) holds true for almost all v. We need one such u. Then we can say that for almost all w ∈ Rb we have W (w) = C2 (w)W (u)C1−1 (w) . Here the right-hand side depends differentiably on w and this proves our assertion that the section W can be defined in smooth manner. From the identity (4.8) ∇2X = W ∇1X W −1

for all X ∈ X (M )

(4.10)

now follows immediately. The converse part of the proof is easier. If covariant derivatives ∇1 , ∇2 are isomorphic, there exists a smooth section W in U (F1 , F2 ) which defines a unitary ¯1 → H ¯ 2 : σ(u) 7→ W (u)σ(u). This unitary mapping carries the generalized maping H ¯1 ) over to (V¯2 , E ¯2 ). Then it suffices to notice that system of imprimitivity (V¯1 , E (4.10) implies (4.8). Corollary 4.1. Two generalized systems of imprimitivity specified by quadruples (Pj , U(r), Γj , id), j = 1, 2, are equivalent if and only if the connections Γ1 , Γ2 are isomorphic. Notation 4.1. Let us consider a principal bundle P (M, G) with connection Γ, and fix a point x0 ∈ P, π(x0 ) = u0 . Then, for every piecewise smooth closed curve τ on

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M with base point u0 there exists a unique element g ∈ G such that x0 is a starting point and x0 g the end point of the lift of curve τ . We denote this element by c(τ ). Theorem 4.3. Let (Vj , Ej ) be generalized systems of imprimitivity specified by quadruples (Pj , Gj , Γj , Lj ), j = 1, 2, and let points x1 ∈ P1 , x2 ∈ P2 , u0 ∈ M satisfy π1 (x1 ) = π2 (x2 ) = u0 . The generalized systems of imprimitivity (V1 , E1 ), (V2 , E2 ) are equivalent if and only if there exists a unitary mapping W : HL1 → HL2 such that for all piecewise smooth closed curves τ on M with the base point u0 , L2 (c2 (τ )) = W L1 (c1 (τ ))W −1

(4.11)

is valid. Proof. We start with two remarks. (1) Taking x0 = xa instead of x, π(x) = π(x0 ) = u0 , then c0 (τ ) = a−1 c(τ )a. (2) According to Sec. 4.3, to a principal bundle P (M, G) with connection Γ we can ˜ there also exists a associate a principal bundle P (M, U(r)) with connection Γ; ˜ homomorphism f0 : P → P preserving the connection and fulfilling f0 (xg) = f0 (x)Lg . If x ˜ = f0 (x), π˜ (˜ x) = π(x) = u0 , then c˜(τ ) = L(c(τ )). The systems of imprimitivity for these two principal bundles are equivalent. It follows from these two remarks that the choice of a point x ∈ P, π(x) = u0 , plays no role and, moreover, we can restrict our considerations to the case G1 = G2 = U(r), L1 = L2 = id. We are thus considering principal bundles Pj (M, U(r)) with connections Γj , j = 1, 2. It suffices to show that condition (4.11) is valid if and only if connections Γ1 , Γ2 are isomorphic. If f : P1 → P2 is an isomorphism preserving the connection and if we choose x2 = f (x), then c1 (τ ) = c2 (τ ) holds for every closed curve τ with base point u0 . Conversely, let c2 (τ ) = bc1 (τ )b−1 for some b ∈ U (r). After having substituted x1 b for x1 we can suppose c1 (τ ) = c2 (τ ). We define a partial mapping f on some subset of P1 into P2 : if u(t), 0 ≤ t ≤ 1, is a piecewise smooth curve in M with starting point u0 and if x1 (t), x2 (t) are the lifts of this curve in P1 , P2 with starting points x1 , x2 , respectively, we put f (x1 (1)) = x2 (1). Let us investigate the case when two curves u(t), v(t), 0 ≤ t ≤ 1, u(0) = v(0) = u0 , have the same end point u(1) = v(1). Let xj (t), yj (t), j = 1, 2 be the lifts of curves u(t), v(t) with starting points x1 , x2 , respectively. There exists a unique a ∈ G such that y1 (1) = x1 (1)a. Let τ (t), 0 ≤ t ≤ 1, be a closed curve with base point u0 , coinciding with u(t) for 0 ≤ t ≤ 1 and with v(2 − t) for 1 ≤ t ≤ 2. Then we find c1 (τ ) = a−1 = c2 (τ ) and hence y2 (1) = x2 (1)a. Thus function f is well-defined. The domain of f consists of all points in P1 which can be connected with x1 by a horizontal curve. But this domain can be extended to the whole P1 by the relation f (xa) = f (x)a. In this way f becomes an isomorphism of principal bundles P1 → P2 and, by construction, preserves the connection.

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4.6. Irreducibility of generalized systems of imprimitivity Let us consider principal bundle P (M, G) with connection Γ. We associate a subgroup Φx of G to each point x ∈ P ; Φx consists of all g ∈ G such that the points x, xg lie on a common horizontal curve in P . The group Φx is called the holonomy group of the connection Γ with the base point x. It has following properties: (1) If x0 ∈ P can be connected with x by a horizontal curve, then Φx0 = Φx ; (2) Φxb = b−1 Φx b. So all holonomy groups are conjugate subgroups in G and we need not specify the base point. The restricted holonomy group Φ0 is the subgroup of Φ corresponding to horizontal lifts of those closed curves which are homotopic to 0. The groups Φ, Φ0 — being subgroups of G — are topological groups. Theorem 4.4 ([51, Chap. II.3], [48, 43]). The restricted holonomy group Φ0 is a connected Lie group, and it coincides with the arcwise connected component of unity in Φ. Moreover, the quotient Φ/Φ0 is finite or countable. The holonomy group Φ itself need not be a Lie group. But it can be equipped with a new topology which induces the original topology on Φ0 and the quotient group Φ/Φ0 is discrete. In this topology one can verify that the inclusion ı : Φ ,→ G : g 7→ g is a homomorphism of Lie groups. Definition 4.3. We say that a structure group G of a principal bundle P (M, G) is reducible to a Lie group G0 , if there exists a principal bundle P 0 (M, G0 ) and homomorphism f : P 0 → P such that f (x0 g 0 ) = f (x0 )f0 (g 0 ), π(f (x0 )) = π 0 (x0 ) with f0 : G0 → G being an injective homomorphism of Lie groups. Moreover, if bundles P, P 0 are endowed with connections Γ, Γ0 , respectively, and the homomorphism f preserves connection, we say that connection Γ is reducible to connection Γ0 . Theorem 4.5. Let a generalized system of imprimitivity (V, E) be specified by a quadruple (P, G, Γ, L). If connection Γ is reducible to a connection Γ0 on a principal bundle P 0 (M, G0 ) and L0 = L ◦ f0 denotes the representation of the Lie group G0 , then the generalized system of imprimitivity specified by the quadruple (P 0 , G0 , Γ0 , L0 ) is equivalent to (V, E). Proof. The assertion can be proved by a method completely analogous to that used at the end of Sec. 4.3. The desired unitary mapping can be constructed in terms of the injective homomorphism f : P 0 → P ; ψ 0 (x0 ) = ψ(f (x0 )). Theorem 4.6 ([51, Chap. II.6], [48]). Let P (M, G) be a principal bundle, Φ the holonomy group of connection Γ in P. Then the structure group G is reducible to a connection in the reduced principal bundle P 0 (M, Φ), the holonomy group of which is identical with Φ.

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Theorem 4.7. Let (V, E) be a generalized system of imprimitivity specified by a quadruple (P, G, Γ, L), Φ be the holonomy group of connection Γ, L0 be the restriction of representation L to the subgroup Φ. Then the commuting algebras C(V, E), C(L0 ) are isomorphic. Proof. In view of Theorems 4.5 and 4.6 it suffices to consider the case G = Φ. Let A˜ be a bounded operator H → H. If A˜ commutes with all E(S), S ∈ B(M ), then, according to Theorem 3.4, it is of the form ψ(x) 7→ A(x)ψ(x), where x 7→ A(x) is some measurable mapping from P into the space of operators in Cr such that ˜ A(xg) = L−1 g A(x)Lg is true on almost all fibres. Moreover, if A commutes with φ φ φ −1 ˜ all V (t), then A(x) = V (t)A(x)V (t) = A(φt · x) holds almost everywhere. We shall show that function x 7→ A(x) can be considered smooth after a redefinition on a set of measure zero. To show this local property, we can consider M = Rn , P (M, G) = Rn × G. Then for vectors v ∈ S n−1 ⊂ Rn the mapping y 7→ A(y) remains constant (almost everywhere) on horizontal lifts of straight lines in Rn with directions v and passing through u for almost all u. More precisely, if φvt (u) = u + tv, then for all (t, v) ∈ R × S n−1 and for almost all u ∈ Rn (depending on t and v) the equality A(u; g) = A(φ˜vt (u; g))

(4.12)

is valid for each g ∈ G. Using the Fubini Theorem we find that for almost all u ∈ Rn , (4.12) holds for all g ∈ G and almost all (t, v) ∈ R × S n−1 . Let us fix u with this property; we can assume u = 0. Then we construct an auxiliary section σ in P . For w ∈ Rn we lift the curve u(t) = tw, 0 ≤ t ≤ 1, in the given connection, choosing (0; e) for the starting point, and we put σ(w) to be equal to the final point of the lifted curve. Let A = A(0; e). Then for almost all w ∈ Rn and all g ∈ G, A(σ(w)g) = L−1 g ALg holds. This proves our assertion that x 7→ A(x) can be considered smooth. Namely, the mapping (w, g) 7→ σ(w)g is an automorphism of the principal bundle Rn × G. So let us suppose that function x 7→ A(x) is smooth and again use the equation A(x) = A(φ˜t · x), now being valid for all t, x. We find that the linear mapping dAx : Tx P → Cr,r , if restricted to the horizontal subspace, is zero. Hence A(x) is constant on horizontal curves in P . Since the holonomy group coincides with the structure group, arbitrary two points in P can be connected by a horizontal curve. Thus A(x) = A for all x ∈ P and we have A = A(x) = A(xg) = 0 L−1 g ALg . In this way we have associated a unique operator A ∈ C(L ) to each A˜ ∈ C(V, E). Conversely, one can relate a unique A˜ ∈ C(V, E) to each A ∈ C(L0 ) by means of ˜ the relation Aψ(x) = A · ψ(x). The one-to-one correspondence A˜ ↔ A is the desired isomorphism. Corollary 4.2. (V, E) is irreducible if and only if L0 is irreducible.

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4.7. Quantum Borel kinematics with vanishing external field Let G denote the Lie algebra of the group G, u(r) the Lie algebra of the group U(r) (consisting of skew-Hermitian (r × r)-matrices), and Ω the curvature 2-form of connection Γ. Ω is a 2-form on P taking values in G such that Ra∗ Ω = ad(a−1 )Ω holds for all a ∈ G, where Ra : P → P : x 7→ xa. By composition with the representation L0 of G we obtain a 2-form L0 ◦ Ω taking values in u(r). Under the homomorphism of principal bundles f : P → P˜ , P˜ = P˜ (M, U(r)) (see Sec. 4.3) ˜ of the connection Γ. ˜ L0 ◦ Ω is mapped into the curvature form Ω For a vector bundle F associated to P let End F denote a vector bundle over M , with fibres (over u ∈ M ) consisting of linear endomorphisms of fibres (from Fu into Fu ). 2-forms w on P taking values in u(r) and satisfying Ra∗ w = ad(a−1 )w, a ∈ U(r), are in one-to-one correspondence with 2-forms K on M taking skew-adjoint values in the space of sections Sec(End F ); the correspondence is expressed by the relation Ku (X, Y ) = x ◦ wx (X ∗ , Y ∗ ) ◦ x−1 , where π ˜ (x) = u and X ∗ , Y ∗ are horizontal lifts of X, Y with respect to the ˜ In this way to Ω ˜ a 2-form R is related and connection Γ. R(X, Y ) = [∇X , ∇Y ] − ∇[X,Y ]

(4.13)

holds [51, Chap. III.5]. R is the curvature of the covariant derivative ∇. A simple calculation using (4.7) then leads to [P (X), P (Y )] = −i~P ([X, Y ]) + (−i~)2 R(X, Y ) , since P ≈ P˜ . If R = 0 (this is true if and only if L0 ◦ Ω = 0), we say that the external field vanishes on the manifold. We see that the field vanishes if and only if [P (X), P (Y )] = −i~P ([X, Y ]) , see (3.3). We recall the mapping c : τ 7→ c(τ ) introduced in Sec. 4.5. If the field vanishes on the manifold, then the value L(c(τ )) ∈ U (r) depends only on the homotopy class of the curve τ . In this manner we obtain a representation Lc of the fundamental group π1 (M ) of manifold M . Let (M c , π c , M ; π1 (M )) be the universal covering of M . Since dim M c = dim M , there exists exactly one flat connection Γc on the principal bundle (M c , π c , M ; π1 (M )). Moreover, according to Theorem 4.3 (Eq. (4.11)) the generalized system of imprimitivity (V c , E c ) specified by the quadruple (M c , π c , Γc , Lc ) is equivalent to (V, E). Conversely, since the connection Γc is flat, the field on M will vanish for every generalized system of imprimitivity (V c , E c ), no matter which representation Lc of π1 (M ) is chosen. So we arrive at a canonical form for generalized systems of imprimitivity (or quantum Borel kinematics) with vanishing field (with flat connection). This form was already studied in detail (see [2, 3] and references therein). The results of Secs. 4.5 and 4.6 imply in this case:

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Theorem 4.8. Two generalized systems of imprimitivity (or two quantum Borel kinematics) with vanishing field (Vjc , Ejc ), j = 1, 2, are equivalent if and only if the representations Lc1 , Lc2 are equivalent. The commuting algebras C(V c , E c ), C(Lc ) are isomorphic. 5. Quantum Borel Kinematics: Classification 5.1. Classification of generalized systems of imprimitivity via cocycles The generalized systems of imprimitivity described by (4.1), (4.2) are not of the most general form. Starting from a characterization of Mackey’s systems of imprimitivity in terms of cocycles ([54], Theorem 9.11), the following theorem was proved in [26]: Theorem 5.1. Any r-homogeneous generalized system of imprimitivity on M is unitarily equivalent to a canonical one (V, E), with H = L2 (M, Cr , µ) for some smooth measure µ on M, (E(S)ψ)(u) = χS (u)ψ(u) for all ψ ∈ H and S ∈ B(M ), and s X

[V φ (t)ψ](u) = ξ X (t, φX −t (u))

dµ X (φX −t (u))ψ(φ−t (u)) d(µ ◦ φX t )

(5.1)

for all ψ ∈ H and all X ∈ Xc (M ). Equivalence classes of r-homogeneous generalized systems of imprimitivity are in one-to-one correspondence with equivalence classes of cocycles [ξ X ]. Here ξ X is a cocycle of R relative to the Lebesgue measure class on M with values in U(r), i.e. a Borel measurable map ξ X : R × M 7→ U(r) with ξ X (0, u) = 1 , X ξ X (s + t, u) = ξ X (s, φX t (u))ξ (t, u)

for almost all u ∈ M and almost all s, t ∈ R. Two cocycles ξjX , j = 1, 2, are called equivalent (cohomologous), if there is a Borel function ζ : R 7→ U(r), such that for all X ∈ Xc (M ) and t ∈ R, u ∈ M X −1 ξ2X (t, u) = ζ(φX . t (u))ξ1 (t, u)ζ(u)

Unfortunately, the classification given in Theorem 5.1 is not easy to handle, since the calculation of cocycles is rather tedious. To be more specific, one has to impose further conditions on the operators under consideration.

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5.2. Differentiable quantum Borel kinematics To gain further insight into the structure of the shift operators (5.1), one can perform a formal calculation. In particular, assume for the moment that the cocycles of the representation are smooth maps from R × M into U(r). Then, by formal differentiation of (5.1) with respect to t at t = 0, an expression for the generalized momentum operator P (X) is obtained, 1 P (X)ψ = −i~ Xψ + divµ (X) · ψ − i~ρ(X) · ψ , (5.2) 2 where ψ ∈ C0∞ (M, Cr ) and ρ(X)(u) :=

d X ξ (t, φX −t (u))|t=0 . dt

The first two terms on the right hand side of (5.2) are linear in the vector field X ∈ X (M ). Though the set of complete vector fields Xc (M ) is not a linear space — the sum of two complete vector fields may not be complete — it contains the “large” linear subset X0 (M ) of vector fields with compact support for which one can demand linearity (or demand “partial” linearity at least for all complete linear combinations of complete vector fields, cf. Theorem 3.1). Thus as a first additional assumption on P (X) we require (3.2), i.e. ρ(X) to be linear in that case. Using the formal expression (5.2), the commutator of P (X) and Q(f ) is obviously obtained again in the form (3.5). Finally, for the commutator [P (X), P (Y )] we obtain [P (X), P (Y )] = −i~P ([X, Y ]) − ~2 R(X, Y ) ,

(5.3)

R(X, Y ) = [ρ(X), ρ(Y )] + Xρ(Y ) − Y ρ(X) − ρ([X, Y ]) .

(5.4)

where

If ρ were a localized connection 1-form, (5.4) would represent the local definition of a curvature 2-form R of a Cr -bundle over M . The Jacobi identity for generalized momenta would then give us precisely the Bianchi identity DR = 0, where D is the covariant differential defined by the connection. In order to arrive at this point we had to assume differentiability of the shift operators (5.1) and of the functions in the domain of momentum operators. Now there are different ways of defining differentiable structures and thus differentiability on the set M ×Cr ; for a discussion of this point we refer to [3, 14]. On the other hand, we have already interpreted R as a curvature 2-form that is in general related to a connection on a Cr -bundle over M . This line of reasoning leads us to the following definition [14, 26]: Definition 5.1. Let M be a differentiable manifold, (V, E) an r-homogeneous generalized system of imprimitivity on M for r = 1, 2, . . . , and R a differential 2form on M with values in Hermitian operators on Cr . Let Q(f ) and P (X) denote the corresponding generalized position and momentum operators, respectively. Then

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(1) The quadruple (H, Q, P, R) is called an (R-compatible) quantum Borel rkinematics (QBKr ), if P is (partially) linear, satisfies (5.3), (5.4), and the common invariant domain D of Q(f )’s and P (X)’s for f ∈ C ∞ (M ) and X ∈ Xc (M ) is dense in H. It is local, if P is local, and elementary, if r = 1. (2) A quantum Borel kinematics is called differentiable, if it is equivalent to (H, Q, P, R) of standard form, constructed from the following ingredients: (1) Lebesgue measure µ on M ; (2) Hermitian vector bundle F (M ; Cr ) over M with fibres diffeomorphic to Cr equipped with Hermitian inner product h·, ·i; (3) A 2-form R with skew-adjoint values in the endomorphism bundle End F = F ⊗ F ∗; (4) The Hilbert space H is realized as the Hilbert space L2 (F, h·, ·i, µ) of sections of F , i.e. (measurable) mappings σ : M → F such that π ¯ ◦ σ = idM and with finite norm induced by the inner product Z (σ, τ ) = hσ(u), τ (u)idµ(u) ; M

(5) The common invariant domain D for Q, P contains the set Sec0 F of smooth sections of F with compact support and P (X)Sec0 F ⊂ Sec0 F ; (6) The position operators Q(f ) have the usual form of the Schr¨ odinger representation Q(f )σ = f · σ ,

∀ f ∈ C ∞ (M, R) ,

σ ∈ Sec0 F .

5.3. Canonical representation of differentiable QBK r For local differentiable quantum Borel kinematics the formal calculations can be made precise. According to Sec. 5.2 it only remains to derive the representation of generalized momenta P (X) in a standard form. This is the content of Theorem 5.2 ([3, 14). Let (H, Q, P, R) be a local differentiable quantum Borel kinematics on M in a standard form. Then there is (1) a Hermitian connection ∇ with curvature R on F, i.e. a connection compatible with the inner product Xhσ, τ i = h∇X σ, τ i + hσ, ∇X τ i , (2) a covariantly constant self-adjoint section Φ of End F = F ⊗ F ∗ , the bundle of endomorphisms of F, such that for all X ∈ Xc (M ) and all σ ∈ Sec0 F i~ P (X)σ = −i~∇X σ + − I + Φ Q(divν X)σ , 2

σ ∈ D.

Moreover, R is a curvature 2-form on F satisfying the Bianchi identity DR = 0 , where D denotes the covariant differential defined by the connection ∇.

(5.5)

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The canonical form of generalized momenta (5.2) shows that by imposing Rcompatibility the quantum system on M is influenced by an external classical field on M . In general, this field is defined by a curvature 2-form on F , and thus a ˜ on the associated U(r)-principal bundle P˜ (M, U(r). We could think of curvature Ω ˜ this Ω as a classical gauge (Yang–Mills) field. The simplest example was provided in Sec. 4.1 for M = R3 , r = 1, where the 2-form eβ was interpreted as a coupling constant times a magnetostatic field on R3 , with the Bianchi identity corresponding to the Maxwell equation div B = 0. Up to a coupling constant the connection ∇ generalizes the notion of a vector potential. For M = R3 the global connection form α corresponds to the vector potential A and the 2-form β = dα to the magnetic field B = rot A of Maxwell’s theory. 5.4. Classification of differentiable QBK r ’s The canonical form given in Theorem 5.2 indicates that a classification of differentiable quantum Borel kinematics amounts to a classification of Hermitian Cr -bundles with connection over M and covariantly constant self-adjoint sections of the corresponding endomorphism bundle.k This is the content of the following theorem. Theorem 5.3. Two local differentiable quantum Borel kinematics (Hj , Qj , Pj , Rj ), j = 1, 2, in canonical form of Theorem 5.2 are equivalent, if and only if there is a strong, unitary, and connection (and thus curvature) preserving bundle isomorphism I : F1 → F2 mapping Φj into each other, i.e. ∇2 = I ◦ ∇1 ◦ I −1 ,

R2 = I ◦ R1 ◦ I −1 ,

Φ2 = I ◦ Φ1 ◦ I −1 .

Unfortunately, there are no general existence and classification theorems of Hermitian Cr -bundles with connection. Looking back to Sec. 4, there a rather big class of local differentiable QBKr ’s is constructively defined, however with Φ = 0. Hence even in these cases, the additional classification of covariantly constant self-adjoint sections has to be found as well. This last problem was solved only in certain special cases — elementary quantum Borel kinematics and type 0 or type U(1) QBKr ’s — described in the following sections. 5.5. Classification of elementary differentiable quantum Borel kinematics The problem of existence and classification of elementary, i.e. r = 1 local differentiable quantum Borel kinematics in terms of global geometrical properties (cohomology groups) of the underlying manifold M was completely solved [3, 14]. It is based on a theorem [20, 19] concerning existence and classification of complex line bundles with Hermitian connection. k The

zero section of End F always exists and is covariantly constant and self-adjoint.

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Theorem 5.4. Let M be a connected differentiable manifold and B ∈ Λ2 (M ) be a closed 2-form on M with dB = 0. Then there exists a complex line bundle F with Hermitian connection ∇ of curvature R = ~i B if and only if R satisfies the integrality condition Z Z 1 1 R= B∈Z 2πi σ 2π~ σ for all closed 2-surfaces σ in M. In terms of cohomology theory, the de Rham class of R/(2πi) = B/(2π~) has to be integral, 1 B ∈ H 2 (M, Z) . 2π~ Hence non-isomorphic equivalence classes of principal bundles over a manifold M with the structure group U(1) are labeled by elements of the second cohomology group H 2 (M, Z). The Lie algebra of U(1) coincides with the imaginary axis iR. Since U(1) is Abelian, the vector bundle End F = M × C is trivial. So the curvature R is a purely imaginary 2-form on M , R = Ω, where Ω is the curvature 2-form of connection Γ. If we put β=−

i~ R, e

β can be interpreted as the 2-form of external magnetic field on M . For an arbitrary 2-cycle σ of the singular homology on M , ∂σ = 0, we have Z exp Ω = 1. σ

This leads to the Dirac quantization condition on the magnetic field Z eg β = g , with = n~ , 2π σ where n ∈ Z. We may interpret this result that the 2-cycle σ — besides the usual R magnetic field satisfying σ β = 0 — encloses a Dirac magnetic monopole with quantized magnetic charge g. Furthermore, the various inequivalent choices of (F, h·, ·i, ∇) for fixed curvature R are parametrized by H 1 (M, U(1)) = π1 (M )∗ , where π1 (M )∗ denotes the group of characters of the fundamental group of M . We should emphasize that H 1 (M, U(1)) = π1 (M )∗ classifies pairs of Hermitian line bundles (F, h·, ·i) and compatible connections ∇. This implies that the curvature 2-forms of two equivalent complex line bundles with Hermitian connection are identical. The classification of complex line bundles F themselves ˇ — disregarding their connection — is given by elements of the Cech cohomology 1 2 ˇ H (M, U(1)) = H (M, Z): two complex line bundles are equivalent if and only if

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their Chern classes in H 2 (M, Z) coincide, i.e. the curvature 2-forms admissible on these two bundles are in the same integral de Rham cohomology class of H 2 (M, Z). Theorem 5.4 is the basis of a classification theorem that goes back to [3] for the flat (B = 0) case and was extended to the case of external fields B by [14]: Theorem 5.5. The equivalence classes of elementary local differentiable quantum Borel kinematics are in one-to-one correspondence to elements of the set H 2 (M, Z) × H 1 (M, U(1)) × R . For the proof it remains to classify the inequivalent choices of covariantly constant self-adjoint sections Φ of End F . For a line bundle the endomorphism bundle is actually trivial: As the transition functions ϕjk : Uj ∩ Uk → U(1) of F commute with complex numbers, the induced transition functions of End F = F ⊗ F ∗ become trivial, z 7→ ϕjk zϕ∗jk = z, hence End F = M × C. Thus the sections of this bundle correspond to complex functions on M . Furthermore, the induced connection on M is the trivial connection on M × C given by the Lie derivative. Thus covariantly constant self-adjoint sections Φ of End F are real multiples of the identity, Φ = ~c · idSec F ,

c ∈ R.

Obviously, c is not changed under strong bundle isomorphisms I, so each value of c ∈ R for a given Hermitian line bundle determines an inequivalent local differentiable quantum Borel kinematics. Finally let us note that elementary quantum Borel kinematics with c = 0 are, in terms of constructions of Sec. 4, described by generalized systems of imprimitivity (V, E) specified by the quadruple (P, U(1), Γ, id). 5.6. Classification of quantum Borel kinematics of type 0 The whole variety of quantizations could be read off the formula (5.5). In order to get a more transparent result we define a QBKr of type 0 [25] by Φ = ~c · idSec F ,

c ∈ R.

Then we obtain an identical formula for P (X) as in QBK1 [4]: i~ P (X)σ = −i~∇X σ + − + ~c (divµ X) · σ , σ ∈ D . 2

(5.6)

As proved in [25], on every smooth manifold M there exists a differentiable QBKr of type 0. Let us note that for r = 1, the type 0 QBK1 ’s classify all possible Borel quantizations [4]; this is not the case, however, for r > 1. Finally, a complete classification of QBKr ’s of type 0 was possible in the case of flat connection (R = 0) [4, 25, 33]. It turns out that it is essentially the question of the topology of M . The corresponding investigations can be summarized in

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Theorem 5.6. The set of classes of unitarily equivalent QBKr ’s of type 0 with flat connection on M can be bijectively mapped onto the set of pairs (L, c), where c ∈ R and L denotes the isomorphism class of flat Cr -bundles over M. Since there is a one-to-one correspondence between the isomorphism classes of flat Cr -bundles over M and flat U(r)-principal bundles over M , we can use Milnor’s Lemma. Lemma 5.1 ([24]). A U(r)-principal bundle over M admits a flat connection if and only if it is induced from the universal covering bundle of M by a homomorphism of the fundamental group π1 (M ) into U(r). Thus, disregarding the real constant c, the set of inequivalent quantizations of type 0 with Cr -valued wave functions is isomorphic to the set Hom(π1 (M ), U(r)) of (the conjugacy classes of) r-dimensional unitary representations of the fundamental group of M . In the case r = 1, i.e. of quantizations with complex-valued wave functions, the topological part of the classification reduces to Hom(π1 (M ), U(1)), i.e. to the set of one-dimensional unitary representations of π1 (M ) [3, 4, 12, 25].l Since the commutator subgroup Γ(π1 (M )) (generated by elements aba−1 b−1 ) belongs to the kernels of all such one-dimensional representations and since the singular homology group H1 (M, Z) is isomorphic to π1 (M )/Γ(π1 (M )) (the Hurewicz isomorphism), inequivalent QBK1 ’s are labeled by elements of the character group of H1 (M, Z). Finally, let us describe the general structure of the Abelian group H1 (M, Z) for compact M . It has a decomposition H1 (M, Z) = F ⊕ T , where the free Abelian group F is F = Z ⊕ · · · ⊕ Z (b1 terms), with b1 being the first Betti number of M , and the torsion Abelian group is T = Zτ1 ⊕ · · · ⊕ Zτk with Zτi being cyclic groups of orders τi (torsion coefficients) such that τi+1 /τi = positive integer. Thus the characters of H1 (M, Z) can be parametrized by (b1 + k)-tuples [e2πiθ1 , . . . , e2πiθb1 ; e2πim1 /τ1 , . . . , e2πimk /τk ] with the numbers θl ∈ [0, 1), l = 1, . . . , b1 , and mi = 0, 1, . . . , τi − 1, i = 1, . . . , k, classifying inequivalent quantum Borel 1-kinematics on M . 5.7. Elementary quantum Borel kinematics with vanishing external field It is remarkable that elementary quantum Borel kinematics with vanishing external magnetic field find application in quantum mechanics (Aharonov–Bohm effect [1]). l This

result was obtained independently also in the Feynman path integral approach [21] and in geometric quantization [20].

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According to Sec. 4.3 they can be labeled by one-dimensional unitary representations of the fundamental group π1 (M ), and consequently, as already explained in Sec. 4.9, inequivalent systems of imprimitivity with vanishing magnetic field are labeled by elements of the character group of H1 (M, Z). Quantum Borel kinematics in the case of trivial fibration P = M × U(1) was studied in detail in [11]. In this case a localized connection 1-form i(e/~)α can be defined globally on the whole manifold M ; the closed 1-form α represents the vector potential of the vanishing magnetic field. A covariant derivative on M × C1 has the form e ∇X = X − i α(X) , X ∈ X (M ) . ~ Two such covariant derivatives ∇1 , ∇2 are isomorphic if and only if there exists a function f : M → T 1 such that α2 = α1 − i

~ df . e f

Following the terminology of [11, 27], we say that the 1-forms (e/~)αj , j = 1, 2 are logarithmically cohomologous; λ = −i(df )/f is said to be logarithmically exact. A 1-form λ is logarithmically exact if and only if Z Z exp i λ = 1 , i.e. λ = 2πn, n ∈ Z , γ

γ

holds for all 1-cycles γ of the singular homology. If γ is a periodic cycle, i.e. pγ = ∂σ for some p ∈ Z and 2-cycle σ, then Z Z Z dλ = λ = p λ, 0= σ

pγ

γ

because a logarithmically exact form is closed. So it suffices to check independent, non-periodic 1-cycles. Their number is b1 (M ), the first Betti number of manifold M . Now, quantum Borel kinematics on M in the case of trivial fibration and of vanishing magnetic field is determined by a closed 1-form α. Its cohomology class is in turn — according to the de Rham Theorem — determined by b1 (M ) periods Z Φj = α. γj

For each j = 1, . . . , b1 (M ), Φj represents an external magnetic flux outside the manifold M and passing through the jth independent cycle γj . Two potentials α1 , α2 determine the same quantum Borel kinematics if e (1) (2) (Φ − Φj ) ∈ 2πZ for all j . ~ j We conclude that in the considered special case the family of all inequivalent quantum Borel kinematics can be parametrized by elements z = (z1 , . . . , zb1 (M) ) ∈ U(1) × · · · × U(1) ,

zj = exp(ieΦj /~) .

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5.8. Examples In the examples we concentrate on elementary quantum Borel kinematics. Example 5.1. M = R3 . Since H 2 (R3 , Z) = 0, only the trivial principal bundle exists over M . Every smooth function f : R3 → U(1) can be written in the form f = exp(ieλ/~) with λ being a smooth real function. Two 1-forms are logarithmically cohomologous if and only if they are cohomologous. Thus (an equivalence class of) quantum kinematics is obtained by taking (a cohomology class of) a closed 2-form β of magnetic field. There exists exactly one equivalence class of quantum Borel kinematics (i.e. with vanishing magnetic field) — the standard one with α = 0. Example 5.2. M = R3 \R is the 3-dimensional real space R3 with the x3 -axis excluded (Aharonov–Bohm configuration). Then H 2 (M, Z) = 0, and there exists only the trivial principal bundle over M . There exists exactly one (up to homology) independent cycle in M and it is non-periodic, i.e. H1 (M, Z) = Z, H 1 (M, R) = R. Inequivalent quantum Borel kinematics are labeled by z ∈ U(1), z = exp(ieΦ/~); Φ denotes the magnetic flux supported by the excluded line R. The corresponding vector potential 1-form α can be chosen in the form x2 x1 Φ α= − 2 dx1 + 2 dx2 . 2π x1 + x22 x1 + x22 From the point of view of quantum mechanics a charged particle cannot distinguish between the flux Φ = 2πn~/e, n ∈ Z, and the zero one. In fact, this is the effect discovered by Y. Aharonov and D. Bohm [1]. Let us note that the Aharonov–Bohm effect in the presence of two solenoids or n solenoids placed along a straight line was studied in [32]. Example 5.3. M be a compact orientable surface. It is known that, up to diffeomorphism, such M are classified by the Euler characteristic χ(M ) = 2 − 2p, where p = 0, 1, 2, . . . is the genus of M . For given p, we denote M = Kp ; it is modeled by a 2-sphere with p handles ([46], Chap. 9.3). Since H 2 (Kp , Z) = Z, there exist countably many principal bundles over Kp , labeled by n ∈ Z. According to our physical interpretation, n can by related to the magnetic charge of a Dirac monopole enclosed by the closed surface, g = (2π~/e)n. Further, H1 (Kp , Z) = Z2p ; thus inequivalent quantum Borel kinematics (with vanishing magnetic field) are labeled by elements of the dual group U(1)2p . Each quantum Borel kinematics depends on 2p external magnetic fluxes, each handle carrying two of them. On a 2-torus, generalized quantum kinematics was studied in [13], quantum Borel kinematics in [29]. Example 5.4. M = RP n = S n /{±} is a real projective space, n > 2. We can identify: S ∈ B(RP n ) ↔ S ∈ B(S n ) ,

S = −S; X ∈ X (RP n ) ↔ X ∈ B(S n ) ,

H 2 (RP n , Z) = Z2 ,

H1 (RP n ), Z) = Z2 .

X−u = −Xu ;

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ˇˇ H.-D. Doebner, P. S tov´ıˇ cek & J. Tolar Table 1. Examples of elementary quantum Borel kinematics.

H1 (M, Z) H 2 (M, Z)

Quantum system

M

π1 (M )

Spinless particle in R3

R3

{e}

0

Topological quantum numbers

0

—

Z

0

ϑ ∈ [0, 1)

0

Z

n∈Z

{e}

0

Z

n∈Z

S2

Z2

Z2

m ∈ Z2

R3 × SO(3)

Z2

Z2

Z2

m ∈ Z2

Symmetric top

S2

{e}

0

Z

n∈Z

Rotator with fixed axis

S1

Z

Z

0

ϑ ∈ [0, 1)

Particle on orientable surface of genus p

Kp

π1 (Kp )

Z 2p

Z

n ∈ Z, ϑ1 · · · ϑ2p ∈ [0, 1)

Aharonov–Bohm configuration

R3 \R

Z

Dirac’s monopole

R3 \O = R+ × S 2

{e}

2 distinguishable particles in R3

R3 × R+ × S 2 R3 × R+ × RP 2

2 indistinguishable particles in R3 Rigid body

Hence there exist two inequivalent principal bundles over M and two inequivalent QBK’s in mutually inequivalent fibrations. The QBK’s can be explicitly described in the following way: the Hilbert spaces H+ , H− are chosen as subspaces in L2 (S n , dµ) (with measure µ invariant under the transformation u → −u), ψ ∈ H± if and only if ψ(−u) = ±ψ(u); the two inequivalent systems of imprimitivity (c = 0) are defined for both signs + and − by the operators 1 E(S) = χS ·, P (X) = −i~ X + divµ X , S ∈ B(RP n ), X ∈ X (RP n )) 2 acting in H+ and H− , respectively. The real projective space RP n appears in quantum mechanics e.g. as a (topologically non-trivial) part of the effective configuration space of two indistinguishable point-like particles localized in the (n+1)-dimensional Euclidean space Rn+1 . The two cases with signs + and − correspond in quantum mechanics to the cases of bosonic and fermionic statistics, respectively. More details can be found in [9]; see also [21, 22]. It should be stressed that the case n = 2 presents unexpected features: in [22] it was found for a system of two particles in two dimensions that there is a continuous family of quantizations describing new statistics which interpolate between fermions and bosons. These anomalous or fractional statistics were later discovered independently by [16, 17] and by [34, 35], who actually coined the term ‘anyons’ for the corresponding particles. Acknowledgments ˇ are grateful to Prof. Dr. H. D. Doebner for the kind hospitalJ. T. and P. S. ity extended to them on various occasions at the Arnold Sommerfeld Institute in

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Clausthal. The authors thank A. B´ ona for typing the manuscript. Partial support of the Grant Agency of Czech Republic (contract No. 202/96/0218) is acknowledged. The list of references is by no means complete and we apologize to the authors of papers which have not been included. References [1] Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory”, Phys. Rev. 115 (1959), 485–491. [2] B. Angermann, “Global geometry and Schr¨ odinger quantization on manifolds”, Latin American Summer School of Physics at UNAM, Mexico, 1980. [3] B. Angermann and H.-D. Doebner, “Homotopy groups and quantization of localizable systems”, Proc. Xth Coll. Group Theoretical Methods in Physics, Physica 114A (1982), 433–439. [4] B. Angermann, H.-D. Doebner and J. Tolar, “Quantum kinematics on smooth manifolds”, pp. 171–208, in Non-linear Partial Differential Operators and Quantization Procedures, Lecture Notes in Mathematics, Vol. 1037, Springer-Verlag, Berlin, 1983. [5] F. Bayen, M. Flato, C. Fronsdal, A. Lichnérowicz and D. Sternheimer, “Deformation theory and quantization”, Ann. Phys. 111 (1978), 61–110, 111–152. [6] P. A. M. Dirac, “Quantized singularities in the electromagnetic field”, Proc. Roy. Soc. London A133 (1931), 60–72. [7] H.-D. Doebner, H. J. Elmers and W. F. Heidenreich, “On topological effects in quantum mechanics; The harmonic oscillator in the pointed plane”, J. Math. Phys. 30 (1989), 1053–1059. [8] H.-D. Doebner and G. A. Goldin, “On a general nonlinear Schr¨ odinger equation admitting diffusion currents”, Phys. Lett. A162 (1992), 397–401. ˇˇtov´ıˇcek and J. Tolar, “Quantizations of the system of two [9] H.-D. Doebner, P. S indistinguishable particles”, Czech. J. Phys. B32 (1982), 1240–1248. [10] H.-D. Doebner and J. Tolar, “Quantum mechanics on homogeneous spaces”, J. Math. Phys. 16 (1975), 975–984. [11] H.-D. Doebner and J. Tolar, “On global properties of quantum systems”, pp. 475–486, in Symmetries in Science (B. Gruber and R. S. Millman, eds.) Plenum Press, New York, 1980. [12] H.-D. Doebner and J. Tolar, “Symmetry and topology of the configuration space and quantization”, pp. 115–126, in Symmetries in Science II (B. Gruber and R. Lenczewski, eds.) Plenum, New York, 1986. [13] H.-D. Doebner and J. Tolar, “Quantum particle on a torus with an external magnetic field”, pp. 3–10, in Quantization and Coherent States Methods (S. T. Ali, I. M. Mladenov and A. Odzijewicz, eds.) World Scientific, Singapore, 1993. [14] M. Drees, “Zur Kinematik lokalisierter quantenmechanischer Systeme unter Ber¨ ucksichtigung innerer Freiheitsgrade und ¨ außerer Felder”, Ph.D. Thesis, Technical University, Clausthal, 1992. [15] P. Goddard and D. I. Olive, “Magnetic monopoles in gauge field theories”, Rep. Prog. Phys. 41 (1978), 1357–1437. [16] G. A. Goldin, R. Menikoff and D. H. Sharp, “Particle statistics from induced representations of a local current group”, J. Math. Phys. 21 (1980), 650–664. [17] G. A. Goldin, R. Menikoff and D. H. Sharp, “Representations of a local current algebra in non-simply connected space and the Aharonov–Bohm effect”, J. Math. Phys. 22 (1981), 1664–1668.

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[18] G. A. Goldin, R. Menikoff and D. H. Sharp, “Induced representations of the group of diffeomorphisms of R3 ”, J. Phys. A: Math. Gen. 16 (1983), 1827–1833. [19] W. Greub and H.-R. Petry, “Minimal coupling and complex line bundles”, J. Math. Phys. 16 (1975), 1347–1351. [20] B. Kostant, Quantization and unitary representations: Part I. Prequantization, Lecture Notes in Mathematics, Vol. 170, Springer-Verlag, Berlin, 1970, pp. 87–208. [21] M. G. G. Laidlaw and C. Morette-DeWitt, “Feynman functional integrals for systems of indistinguishable particles”, Phys. Rev. D3 (1970), 1375–1378. [22] J. M. Leinaas and J. Myrheim, “On the theory of identical particles”, Nuovo Cimento B37 (1977), 1–23. [23] G. W. Mackey, “Unitary representations of group extensions I”, Acta Math. 99 (1958), 265–311. [24] J. Milnor, “On the existence of a connection with curvature zero”, Comment. Math. Helv. 32 (1958), 215–223. [25] U. A. M¨ uller and H.D. Doebner, “Borel quantum kinematics of rank k on smooth manifolds”, J. Phys. A: Math. Gen. 26 (1993), 719–730. [26] P. Nattermann, “Dynamics in Borel quantization: Nonlinear Schr¨ odinger equations vs. master equations”, Ph.D. Thesis, Technical University, Clausthal, 1997. [27] I. E. Segal, “Quantization of nonlinear systems”, J. Math. Phys. 1 (1960), 468–488. [28] L. S. Schulman, “A path integral for spin”, Phys. Rev. 176 (1968), 1558–1569. [29] C. Schulte, “Quantum mechanics on the torus, Klein bottle and projective sphere”, pp. 313–323, in Symmetries in Science IX (B. Gruber ed.) Plenum Press, New York, 1997. ˇˇtov´ıˇcek, “Dirac monopole derived from representation theory”, Suppl. Rend. Circ. [30] P. S Mat. Palermo, Ser. II, No. 3 (1984), 301–306. ˇˇtov´ıˇcek, “Systems of imprimitivity for the group of diffeomorphisms I, II”, Ann. [31] P. S Global Anal. Geom. 5 (1987), 89–95; 6 (1988), 31–37. ˇˇtov´ıˇcek, “Scattering on a finite chain of vortices”, Duke Math. J. 76 (1994), [32] P. S 303–332. ˇˇtov´ıˇcek and J. Tolar, “Topology of the configuration manifold and quantum [33] P. S mechanics”, Acta Polytechnica (Prague) Ser. IV, No. 1 (1984), 37–75 (in Czech). [34] F. Wilczek, “Magnetic flux, angular momentum and statistics”, Phys. Rev. Lett. 48 (1982), 1144–1146. [35] F. Wilczek, “Quantum mechanics of fractional spin particles”, Phys. Rev. Lett. 49 (1982), 957–959. [36] T. T. Wu and C. N. Yang, “Concept of nonintegrable phase factors and global formulation of gauge fields”, Phys. Rev. D12 (1975), 3845–3857. [37] C. N. Yang, “Magnetic monopoles, fiber bundles, and gauge fields”, Ann. New York Acad. of Sci. 294 (1977), 86–97. [38] N. Bourbaki, Groupes et algèbres de Lie, Hermann, Paris, 1968. [39] C. Chevalley, Theory of Lie Groups I, Princeton University Press, Princeton, 1946. [40] E. B. Davies, Theory of Open Quantum Systems, Academic Press, London, 1976. [41] P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yeshiva University, New York, 1964. [42] P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, Oxford, 1958. [43] W. Greub, S. Halperin and R. Vanstone, Connections, Curvature and Cohomology, I – III, Academic Press, New York, 1973. [44] V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Inc., Englewood Cliffs, 1974.

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[45] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978. [46] M. W. Hirsch, Differential Topology, Springer-Verlag, Berlin, 1970. [47] A. A. Kirillov, Elements of Representation Theory, Springer-Verlag, Berlin, 1984. [48] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry I, InterscienceWiley, New York, 1963. [49] S. Lang, Introduction to Differentiable Manifolds, Interscience Publishers, New York, 1962. [50] G. W. Mackey, Induced Representations and Quantum Mechanics, W. A. Benjamin, Inc., New York, 1968. [51] K. Nomizu, Lie Groups and Differential Geometry, The Mathematical Society of Japan, 1956. [52] W. Pauli, Wellenmechanik, Handbuch der Physik Bd. 24, Teil 1, 120 (1933). [53] V. S. Varadarajan, Geometry of Quantum Theory I, Van Nostrand, Princeton, 1968. [54] V. S. Varadarajan, Geometry of Quantum Theory II. Quantum Theory of Covariant Systems, Van Nostrand Reinhold Co., New York, 1970.

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HEAT KERNEL ASYMPTOTICS OF OPERATORS WITH NON-LAPLACE PRINCIPAL PART

IVAN G. AVRAMIDI∗ and THOMAS BRANSON Department of Mathematics, The University of Iowa Iowa City, IA 52242, USA ∗ Department of Mathematics New Mexico Institute of Mining and Technology Socoro, NM 87801, USA

Received 12 May 1999 We consider second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold without boundary, working without the assumption of Laplace-like principal part −∇µ ∇µ . Our objective is to obtain information on the asymptotic expansions of the corresponding resolvent and the heat kernel. The heat kernel and the Green’s function are constructed explicitly in the leading order. The first two coefficients of the heat kernel asymptotic expansion are computed explicitly. A new semi-classical ansatz as well as the complete recursion system for the heat kernel of non-Laplace type operators is constructed. Some particular cases are studied in more detail.

1. Introduction The resolvent and the heat kernel of elliptic partial differential operators are of great importance in mathematical physics, differential geometry and quantum theory [17, 12, 8, 11, 19]. Of special interest in the study of elliptic operators are the socalled heat kernel asymptotics. It is well known [12] that for a second-order, elliptic, self-adjoint partial differential operator F with a positive definite leading symbol, acting on sections of a vector bundle over a compact, boundariless manifold M of dimension m, an asymptotic expansion of the following form is valid as t ↓ 0: TrL2 exp(−tF ) ∼ (4πt)−m/2

X (−t)k k≥0

k!

Ak .

The coefficients Ak are called the heat invariants, or heat kernel coefficients. They are spectral invariants of the operator F , and encode information about the asymptotic properties of the spectrum. Note that our normalization of the coefficients Ak differs by the factor (−1)k /k! from that used in [12]. This normalization has been used in previous works of one of the authors (see the book [5], the papers [2–4] and others). It has the advantage that for Laplace type operators with a potential (see 847

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I. G. Avramidi & T. Branson

R definition below), F = −∆+q, the numerical coefficient of the term M d vol(x) tr q k in Ak is equal to 1 for any k. An important subclass of the class of operators described above are the operators of Laplace type: those for which the leading symbol takes the form g µν ξµ ξν , where g µν is a non-degenerate, positive definite metric on the cotangent bundle of M . For such operators, the leading symbol naturally defines a Riemannian metric on M (the inverse gµν of the leading symbol). Alternatively, one may take the Riemannian metric as given, and produce the many natural operators of Laplace type which are so important in Physics. The assumption of Laplace type affords a considerable simplification in the study of spectral asymptotics. Partly as a result of this, much is known about the resolvent, the heat kernel, and the zeta function in this category of operators [12, 8, 2, 4, 5]. In particular, the heat kernel coefficients Ak for Laplace type operators are known explicitly up to k = 4 [2, 5] (for a review, see [4]). In this paper, we take a Riemannian metric as given, and study the most general class of second-order operators F , acting on sections of a vector bundle V, with positive definite leading symbol. That is, we drop the assumption of Laplace type, and assume only that σ2 (F ; x, ξ) = aµν (x)ξµ ξν , where aµν is a symmetric two-tensor valued in End(V). (We do not assume that aµν = g µν idV , nor that aµν is factored as g µν E for E a section of Aut(V).) We shall sometimes call these “NLT” (for “non-Laplace type”) operators. Of course, Laplace type is a special case, so to be more precise, NLT operators are operators that are not necessarily of Laplace type. NLT operators arise naturally in such areas of mathematical physics as quantum gauge field theory and quantum gravity [5, 11], differential geometry, classical continuum mechanics [18] and others. The most elementary examples that one can use to illustrate the class of operators in question are the weighted form Laplacians. If d is the exterior derivative and δ its formal adjoint, the form Laplacian is ∆ = δd + dδ; this differs from the Bochner Laplacian g µν ∇µ ∇ν by an operator of order zero, the so-called Bochner– Weitzenb¨ ock operator. If a, b are real constants, an operator of the form aδd + bdδ may be termed a weighted form Laplacian; such operators are elliptic if a 6= 0 6= b, and have positive definite leading symbol if a, b > 0. The non-Laplace type operators on differential forms have been studied extensively by many authors under the general name “nonminimal operators”, or “exotic operators” (see [13, 9, 14–16, 10, 1] and references therein). In particular, the paper [13] contains a complete discussion of the coefficients Ak of the weighted form Laplacian for all k and of A0 and A1 of the operator with a potential function (Theorems 1.2 and 1.3, pp. 2089–2090). In papers [14–16] a computer algorithm has been developed that employs the calculus of pseudodifferential operators and the coefficients A0 , A1 and A2 (more precisely, the local (non-itegrated) coefficients a0 , a1 and a2 ) have been computed.

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Heat Kernel Asymptotics of Non-Laplace Type Operators

849

The ζ-function for non-Laplace type operators acting on symmetric 2-tensors have been studied in [10] (for the origin of such operators, see [5, 11]). By restricting to maximally symmetric even dimensional manifolds the authors were able to compute the eigenvalues of such operators and evaluate the ζ-function at zero, ζ(0), (which is, in fact, equivalent to the coefficient Am/2 [12]) for a certain special case of the operator in Euclidean spaces and m-spheres in dimensions m = 2, 4, 6, 8, 10. In general, the study of spin-tensor quantum gauge fields in arbitrary gauge necessarily leads to non-Laplace type operators acting on sections of general tensorspinor bundles. It is precisely these operators that are of prime interest in the present paper. The main examples here are the symmetric 2-tensor bundle (gravitational field, spin 2) and the spin-vector bundle (gravitino field, spin 3/2) (for a discussion of such operators in gauge field theories, see [7]). Let us formulate our main result from the very beginning. Theorem 1.1. Let F : C ∞ (V) → C ∞ (V) be a self-adjoint elliptic second-order partial differential operator with a positive definite leading symbol, acting on sections of a tensor-spinor bundle V of fiber dimension d over a compact manifold M of dimension m without boundary. Let F = −aµν ∇µ ∇ν + q, where ∇ is a connection on the vector bundle V, aµν is a parallel symmetric two-tensor valued in End(V) and q is an endomorphism of the bundle V. The curvature of the connection ∇ is defined by [∇µ , ∇ν ]ϕ = Rα β µν T β α ϕ, where Rα βµν is the Riemann curvature tensor and T α β is given by the representation of so(m) which induces the bundle V. Let ξ ∈ T ∗ M be a cotangent vector, and let λi (ξ) = µi |ξ|2 , (i = 1, . . . , s), µi > 0, be the eigenvalues of the leading symbol, σ2 (F ) = aµν ξµ ξν , with the multiplicities di . Then the corresponding (orthogonal ) eigenspace projections have the form Πi (ξ) =

p X

1 ···µ2n Πµi(2n)

n=0

=

s X

1 ξµ · · · ξµ2n |ξ|2n 1

cik a(µ1 µ2 · · · aµ2k−3 µ2k−2 )

k=1

1 ξµ · · · ξµ2k−2 , |ξ|2k−2 1

where the numbers s and p depend on the structure of the bundle V and the leading symbol. The matrix of coefficients C := (cik ) is inverse to the (Vandermonde) matrix of powers M = (κkj ) := (µjk−1 ). Furthermore, the L2 trace of the heat kernel has the following asymptotics as t→0 TrL2 exp(−tF ) = (4πt)−m/2 [A0 − tA1 + O(t2 )] . The coefficients A0 and A1 are defined by s X −m/2 A0 = di µi vol(M ) , i=1

Z A1 =

d vol(x){trV (a0 q) + βR} , M

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where R is the scalar curvature, s X −m/2 a0 = µi hΠi i , i=1

hΠi i = Πi(0) =

s X k=1

cik

Γ(m/2)(2k − 2)! Γ(m/2 + k − 1)22k−2 (k − 1)!

× g(µ1 µ2 · · · gµ2k−3 µ2k−2 ) a(µ1 µ2 · · · aµ2k−3 µ2k−2 ) ,

(1.1)

and β is a constant defined by s 1 X −m/2 β=− µ trV (hΠi igµν aµν ) 6m i=1 i + − with κik

σik

1 12(m − 1) 1 2(m − 1)

X

[κik (3m − 2) + 4(m + 2)µi σik ]trV hΠi J˜α Πk J˜α i

1≤i,k≤s;i6=k

X

κik trV {[hJ˜ν Πi J˜α Πk i + hΠi J˜α Πk J˜ν i]T[να] } ,

1≤i,k≤s;i6=k

) ( −m/2 −m/2 −m/2+1 −m/2+1 + µk − µk µi µi Γ(m/2 − 1) =− , m +2 2Γ(m/2 + 1) (µi − µk ) (µi − µk )2 ( −m/2+1 −m/2+1 −m/2 −m/2 µ − µk µi + µk Γ(m/2 − 1) = 24 i + 8(m − 4) 8Γ(m/2 + 2) (µi − µk )3 (µi − µk )2 −m/2

−m/2−1

µi µ + 4(m − 2) + m(m − 2) i (µi − µk )2 (µi − µk ) hΠi J˜α Πk J˜β i =

X 1≤n,j≤s

) ,

Γ(m/2)(2n + 2j − 2)! Γ(m/2 + n + j − 1)22n+2j−2 (n + j − 1)!

× g(µ1 µ2 · · · gµ2n−3 µ2n−2 gν1 ν2 · · · gν2j−3 ν2j−2 gγδ) × a(µ1 µ2 · · · aµ2n−3 µ2n−2 aγ α aν1 ν2 · · · aν2j−3 ν2j−2 aδ) β . We shall prove this theorem in Sec. 9. Note that for Laplace type operators, when aµν = g µν IV , these formulas simplify considerably. We have then just one eigenvalue µ1 = 1 with multiplicity d1 = d = dim(V) and the projection Π1 = IV . Thus for Laplace type operators, a0 = hΠ1 i = IV , and β = −d/6, and we recover the well known result [12, 2] A0 = vol(M ) , Z d A1 = d vol(x) trV q − R . 6 M

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In the next section we shall give a detailed description of our operator class. We would like to stress from the beginning that the theory of NLT operators, despite being a theory of second order operators, is closely related to the theory of higher-order Laplace-like operators (for a discussion, see [3]); that is, operators with principal part a power of the Laplacian. In this sense, even the weighted form Laplacian example mentioned above does not capture the full flavor of the theory, since it (almost uniquely in this category) is accessible entirely through second-order methods (see [13, 16]). To contrast with the previous result mentioned above, it is worth stressing once again that: (i) there are many areas in both physics and mathematics where general secondorder non-Laplace type operators arise naturally, (ii) we are primarily interested not just in the weighted form Laplacians but in the general NLT operators (differential forms being a very particular case), (iii) our approach to computation of heat kernel asymptotics is completely different from that of the previous authors [13, 16]. Our method enables us to compute explicitly not just the heat trace asymptotics but also the heat kernel and resolvent in the leading order that describe the local off-diagonal behavior of the resolvent and the heat kernel. To best of our knowledge, such formulas for non-Laplace operators are presented here for the first time. Despite the importance of second-order operators with non-Laplace principal part in gauge field theory and quantum gravity (see, for example [11, 5]), their study is still quite new, and the available methodology is still underdeveloped in comparison with the Laplace type theory. In this paper, and in the more explicitly representation-theoretic treatment [6], we hope to lay the groundwork for a systematic attack on the spectral asymptotics of this larger class of operators.

2. Non-Laplace Type Differential Operators 2.1. General vector bundle setup Let M be a smooth compact manifold without boundary of dimension m, equipped with a (positive definite) Riemannian metric g. Let V be a smooth vector bundle over M , with End(V) ∼ = V ⊗ V ∗ the corresponding bundle of endomorphisms. Given any vector bundle V, we denote by C ∞ (M, V), or just C ∞ (V), its space of smooth sections. We assume that the vector bundle V is equipped with a Hermitian metric H. This naturally identifies the dual vector bundle V ∗ with V, and defines a natural L2 inner product, using the invariant Riemannian measure d vol(x) on the manifold M . The completion of C ∞ (M, V) in this norm defines the Hilbert space L2 (M, V) of square integrable sections.

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We denote by T M and T ∗ M the tangent and cotangent bundles of M . We assume given a connection ∇V on the vector bundle V. The covariant derivative on V is then a map ∇V : C ∞ (V) → C ∞ (T ∗ M ⊗ V)

(2.1)

which we assume to be compatible with the Hermitian metric on the vector bundle V, in the sense that ∇H = 0. Here the connection is given its unique natural extension to bundles in the tensor algebra over V and V ∗ ; in particular, to the bundle V ∗ ⊗ V ∗ of which H is a section. In fact, using the Levi–Civita connection ∇LC of the metric g together with ∇V , we naturally obtain connections on all bundles in the tensor algebra over V, V ∗ , T M , T ∗ M ; the resulting connection will usually be denoted just by ∇. It will usually be clear which bundle’s connection is being referred to, from the nature of the section being acted upon. We denote the curvature of ∇V (a section of T ∗ M ⊗ T ∗ M ⊗ V) by R: ϕ ∈ C ∞ (V) .

[∇α , ∇β ]ϕ = Rαβ ϕ ,

(2.2)

The formal adjoint of the covariant derivative of (2.1) is defined using the Riemannian metric and the Hermitian structure on V: (∇V )∗ : C ∞ (T ∗ M ⊗ V) → C ∞ (V) , ϕα 7→ −∇α ϕα . Let a ∈ C ∞ (T M ⊗ T M ) ⊗ End(V)) ,

b ∈ C ∞ (T M ⊗ V) ,

q ∈ C ∞ (End(V)) .

These sections define certain natural bundle maps by contraction, which, for simplicity, we denote by the same letters: a : T ∗M ⊗ V → T M ⊗ V , b : V → TM ⊗ V ,

(2.3)

q:V →V. Using these maps we can write the general second order operator F = ∇∗ (a∇) + b∇ + q ,

where aµν = aνµ .

(2.4)

Any formally self-adjoint second order operator may thus be written 1 ∇∗ (a∇) + (b∇ + (b∇)∗ ) + q , (2.5) 2 where we may assume that each aµν is Hermitian. (Here, if necessary, we clear the notation and redefine q and aµν .) In abstract index notation, any formally self-adjoint second order operator may be written F = −∇µ (aµν ∇ν ) + bµ ∇µ − ∇µ (b∗ )µ + q = −aµν ∇µ ∇ν + [bµ − (b∗ )µ − aνµ ;ν ]∇µ + q − (b∗ )µ ;µ .

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Hereafter we denote by “;” the covariant derivative. One may now restrict to the case when the endomorphism b∗ is anti-Hermitian, since the Hermitian part of b only contributes at order zero. With this, we have redefined b and q. Thus, henceforth, aµν = aνµ ,

b∗ = −b .

The formal adjoint to the operator F reads F ∗ = ∇∗ (a∗ ∇) + b∇ + (b∇)∗ + q ∗ . Hence, in addition to the other conditions posited so far, for the operator F to be formally self-adjoint the endomorphism q should be Hermitian. To sum up, the general formally self-adjoint second order operator is described by (2.5) with aµν = aνµ ,

(aµν )∗ = aµν ,

(bµ )∗ = −bµ ,

q∗ = q .

e = ∇ + A, with Let us consider the effect of a change of the connection, ∇ → ∇ a one form A valued in End(V. We have Fe = −∇µ aµν ∇ν +(bµ −Aν aνµ )∇µ +∇µ (bµ −aµν Aν )+bµAµ +Aµ bµ −Aµ aµν Aν +q . In many cases (but not always!) it is possible to choose A in such a way that Aµ aµν = bν . Then the first order part drops out. The point is whether the map a (2.3) is invertible, i.e. whether there is a solution, a−1 ∈ C ∞ (T ∗ M ⊗ T ∗ M ) ⊗ End(V)), to the equation aµν a−1 νλ = δλµ IV .

(2.6)

This can be put in another form. Let ei ∈ C ∞ (T ∗ M ⊗ V) be the basis in the space of one forms valued in V and e∗i ∈ C ∞ (T ∗ M ⊗ V ∗ ) be the adjoint basis in the space of one forms valued in V ∗ . Then Eq. (2.6) has a unique solution if and only if the bilinear form Bij = he∗j , aei i is nondegenerate, i.e. det Bij 6= 0. If this condition is satisfied then one can always redefine the connection in such a way, viz. Aµ = bν a−1 νµ , that Aµ aµν = bν and the first order terms are not present. In this paper we assume that this is the case, so that without loss of generality one can set the vector-endomorphism b to zero, b = 0. Moreover, we will assume that the tensor-endomorphism a is parallel, ∇a = 0. Thus the operator under consideration has the form F = −aµν ∇µ ∇ν + q ,

(2.7)

where aµν = aνµ ,

a∗µν = aµν ,

∇a = 0 ,

q∗ = q .

(2.8)

2.2. Tensor-spinor bundles We now restrict attention to operators acting on tensor-spinor bundles. These bundles may be characterized as those appearing as direct summands of iterated tensor products of the cotangent and spinor bundles. Alternatively, they may be

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described abstractly as bundles associated to representations of O(m), SO(m), Spin(m), or Pin(m), depending on how much structure we assume of our manifold. These are extremely interesting and important bundles, as they describe the fields in Euclidean quantum field theory. More general bundles appearing in field theory are actually tensor products of these with auxiliary bundles, usually carrying another (gauge) group structure. The connection on the tensor-spinor bundles is built in a canonical way from the Levi–Civita connection and its curvature is: Rµν = Rα βµν T β α , where Rα βµν is the Riemann curvature tensor, and T α β is determined by the representation of so(m) which induces the bundle V. T β α is a tensor-spinor constructed purely from Kronecker symbols, together with the fundamental tensor-spinor γ µ if spin structure is involved. We study in this paper a special class of second-order operators of the form (2.7) with the coefficient a built in a universal, polynomial way, using tensor product and contraction from the metric g and its inverse g ∗ , together with (if applicable) the volume form E and/or the fundamental tensor-spinor γ. Such a tensorendomorphism a is obviously parallel. (E is available given SO(m) or Spin(m) structure; γ is available given Spin(m) or Pin(m) structure.) We do not set any conditions on the endomorphism q, except that it should be Hermitian. An important subclass of this class of operators is the class of natural operators, when, in addition, q is also built from the geometric invariants only; i.e. from g, g ∗ , E, and γ, together with the Riemann curvature and its iterated covariant derivatives. By Weyl’s invariant theory and dimensional analysis (i.e., a check of the homogeneity of each term under uniform dilation of the metric), it is clear that while a must be built polynomially from g and g ∗ , together with E and/or γ if applicable, the endomorphism q must be a sum of terms linear in the curvature. However, we do not need the additional assumption that q is constructed from the curvature. In general, it could be any smooth endomorphism. 3. Leading Symbol of an NLT Operator Let us describe now more exactly the class of operators (2.7) we are working with. We have assumed the operator F to be self-adjoint leading to the conditions (2.8). Since we are going to study the heat kernel asymptotics of the operator F , we now require in addition that the leading symbol of the operator F , σ2 (F )(ξ) =: A(ξ) = aµν ξµ ξν ,

with ξ ∈ T ∗ M ,

be positive definite, i.e. we have ξ 6= 0 ⇒ A(ξ)

Hermitian and positive definite on V .

In particular, F is elliptic. Positive definiteness implies that the roots of the characteristic polynomial χa (ξ)(λ) := detV (A(ξ) − λ)

(3.1)

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are positive functions on M . Remark 3.1. An important point is that the eigenvalues λ1 , . . . , λs and their multiplicities d1 , . . . , ds are independent of the point x ∈ M . Indeed, all the bundles under consideration are vector bundles associated to the principal bundle S of spin frames via a finite-dimensional representation (ϕ, V ) of Spin(m): V = S ×ϕ V . Let V1 be the defining representation of SO(n). We get the section a, and the endomorphisms aµν ξµ ξν by “promoting” vectors a ∈ V1 ⊗V1 ⊗V ⊗V ∗ and aµν ξµ ξν ∈ V ⊗ V ∗ to sections of the associated bundles. In particular, the cited eigenvalues and multiplicities may be computed at the level of the representation (ϕ, V ). A useful way in which to perturb our operator F is to let it run through a one-parameter family F (ε) for which a(ε) = a + εb , where b is a section of T M ⊗ T M ⊗ End(V) which is built from g and g ∗ , and if applicable, E and/or γ. (This b is not to be confused with the b of Sec. 2.1.) To preserve formal self-adjointness, we need to assume that bµν ξµ ξν is self-adjoint on V for each ξ ∈ C ∞ (T ∗ M ). F = −(aµν + εbµν )∇µ ∇ν + q . For ε in some interval about 0, the symbol a(ε) remains positive definite. (This last statement does not even depend on the compactness of M ; by an argument analogous to Remark 3.1, this interval is independent of the point x ∈ M .) In particular, we might take perturbations about an operator with a leading symbol which is factored ; i.e., one for which aµν = g µν c , where c is a section of End(V) which is built invariantly from g and g ∗ , and if applicable, E and/or γ. As a special case of this, we could take c = IV ; that is, perturb an operator of Laplace type in NLT directions. In fact, in case V is associated to an irreducible representation of Spin(m), the endomorphism c must be a multiple of the identity IV by Schur’s Lemma. Given such a perturbation, one might hope that relevant spectral quantities could be expanded in powers of (very small) ε, or at least that one could work with the ε-variation of such quantities. Now consider the symmetric 2n-tensor quantity trV a(µ1 ν1 · · · aµn νn ) .

(3.2)

As usual, the parentheses denote complete symmetrization over all included indices. An index-free way of writing this is trV ∨n a ,

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where ∨n is the symmetrized tensor power defined by · · ∨ a} . ∨n a ≡ a | ∨ ·{z n

Remark 3.2. We claim that trV ∨n a = a(n) ∨n g ∗

(3.3)

with some constants a(n) . (Since ∇a and ∇g vanish, so must ∇a(n) .) Indeed, this quantity is built polynomially from g and g ∗ , and a priori possibly E and/or γ. Since it is a tensor, γ is not involved. Since E µ1 ···µm Eν1 ···νm = δ [µ1 [ν1 · · · δ µm ] νm ] , where the brackets denote antisymmetrization over enclosed indices, (3.2) is affine linear in E; that is, it has the form ϕ(µ1 ν1 ···µn νn ) + ψ (µ1 ν1 ···µn νn )[α1 ···αm ] Eα1 ···αm . The tensor ψ is constructed purely from the metric g ∗ and is symmetric in the first 2n indices and antisymmetric in the last m indices. It is clear that there are no such tensors, so ψ = 0. Similarly, the only symmetric 2n-tensor constructed from the metric is ∨n g ∗ , thus leading to (3.3). Contracting (3.3) with ξµ1 ξν1 · · · ξµn ξνn , we get trV An (ξ) = a(n) |ξ|2n . Furthermore, if we denote by trg the total trace of a symmetric 2n-tensor P , trg P ≡ gµ1 µ2 · · · gµ2n−1 µ2n P µ1 ···µ2n

(3.4)

then since trg ∨n g ∗ =

Γ(m/2 + n) 22n n! , Γ(m/2) (2n)!

we have from (3.3) a(n) =

Γ(m/2) (2n)! trV trg ∨n a . Γ(m/2 + n) 22n n!

It is clear that for an operator of Laplace type, when a = g ∗ ⊗ idV , each a(n) is just the fiber dimension d of V. Now let us consider the characteristic polynomial (3.1) in more detail. Applying the above remarks to the coefficients of χa (ξ)(λ), we find that χa (ξ) depends on ξ only through |ξ|2 . As a result, the dependence of the eigenvalues λi on ξ is only through |ξ|2 . Since A(ξ) is 2-homogeneous in ξ, the λi must be also: λi (ξ) = |ξ|2 µi ,

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for some positive real numbers µ1 , . . . , µs which are independent of the point (x, ξ) ∈ T ∗ M , and, in fact, independent of the specific Riemannian manifold (M, g), depending only on the representation (ϕ, V ) and the vector a ∈ V1 ⊗ V1 ⊗ V ⊗ V ∗ . Computing the trace trV A(ξ)n we get a sequence of equations, s X

di µni = a(n) ,

n∈N,

(3.5)

i=1

relating the eigenvalues, multiplicities, and the quantities a(n) . When n = 0, (3.5) is just a(0) = d; this was immediate from the definition of a(n) . Let Πi (ξ) be the orthogonal projection onto the λi -eigenspace. The Πi satisfy the conditions Π2i = Πi , (i 6= k) ,

Πi Πk = 0 s X

(3.6) Πi = IV ,

i=1

trV Πi = di . In contrast to the eigenvalues, the projections depend on the direction ξ/|ξ| of ξ, rather than on the magnitude |ξ|. In other words, they are 0-homogeneous in ξ. Furthermore, they are polynomial in ξ/|ξ|: Πi (ξ) =

2p X

Pi(n) (ξ) ,

(3.7)

n=0

for some p, where Pi(n) (ξ) =

1 1 ···µn ξµ · · · ξµn Πµi(n) . |ξ|n 1

Here the Πi(n) are some End(V )-valued symmetric n-tensors that do not depend on ξ. There is, however, quite a bit of ambiguity in the definition (3.7) of the homogeneous polynomials Pi(n) (ξ), since multiplication of an n-homogeneous polynomial q(ξ) by |ξ|2 produces an (n + 2)-homogeneous polynomial q˜(ξ), without changing the associated 0-homogeneous function q(ξ/|ξ|). We can remove the ambiguity by requiring that Pi(n) have no |ξ|2 factor. This is equivalent to requiring that Pi(n) (ξ) be a harmonic polynomial in ξ, which in turn is equivalent to requiring that its restriction to the unit ξ-sphere is an nth-order (End(V)-valued) spherical harmonic. Yet another equivalent formulation is to require that Πi(n) is trace free in all its indices. At any rate, with this requirement, we have uniquely defined quantities Πi(n) . Note that the explicit formula Πi =

(A − λ1 ) · · · (A − λi−1 )(A − λi+1 ) · · · (A − λs ) (λi − λ1 ) · · · (λi − λi−1 )(λi − λi+1 ) · · · (λi − λs )

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exhibits each projection as a polynomial of degree 2(s − 1). By homogeneity, Πi (ξ) =

(A(ξ/|ξ|) − µ1 ) · · · (A(ξ/|ξ|) − µi−1 )(A(ξ/|ξ|) − µi+1 ) · · · (A(ξ/|ξ|) − µs ) . (µi − µ1 ) · · · (µi − µi−1 )(µi − µi+1 ) · · · (µi − µs )

We can take this expression and expand as a product of homogeneous terms: Πi (ξ) =

s X

X ξµ · · · ξµ ξ = cik a(µ1 µ2 · · · aµ2k−3 µ2k−2 ) 1 2k−22k−2 , |ξ| |ξ| s

cik Ak−1

k=1

(3.8)

k=1

where cik are numerical constants depending only on the µj . Since A(ξ) is 2homogeneous it follows that all homogeneity orders are even, i.e. Pi(2n+1) (ξ) = Πi(2n+1) = 0, and therefore, Πi (ξ) = In turn, by writing A(ξ/|ξ|) = projections:

Ps

p X

Pi(2n) (ξ) .

(3.9)

n=0

i=0

µi Πi (ξ) we compute powers of A in terms of

X ξ µki Πi (ξ) . = |ξ| i=1 s

Ak

Substituting this into (3.8), we get s X

cik µjk−1 = δij .

k=1

In other words, the matrix of coefficients C := (cik ) is inverse to the (Vandermonde) matrix of powers M = (κkj ) := (µjk−1 ): (cik ) := (κkj )−1 .

(3.10)

Though the number s depends on the particular leading symbol a, there is an upper bound for s which depends only on the representation (ϕ, V ) to which the bundle V is associated. This is described in detail in [6]. In particular, in case (ϕ, V ) is irreducible, the algebra generated by restrictions to the unit ξ-sphere of equivariant leading symbols of all orders is commutative, and thus simultaneously diagonalizable. The resulting projections diagonalize equivariant leading symbols of any (not just second) order. The number of projections, i.e. the dimension of the algebra just described, may be described in terms of representation-theoretic parameters. For reducible (ϕ, V ), similar considerations are valid, but the algebra is not commutative. Now write the leading symbol in terms of projections A(ξ) = |ξ|

2

s X i=1

µi Πi (ξ) = |ξ|

2

p s X X i=1 n=0

µi Pi(2n) (ξ) .

(3.11)

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Since A(ξ) is 2-homogeneous, it is a sum of End(V)-valued spherical harmonics in ξ of degrees 2 and 0. Thus the spherical harmonics on the right of all other orders vanish: s X

µi Pi(2n) (ξ) =

s X

i=1

n 6= 0, 1 .

µi Πi(2n) = 0 ,

i=1

Furthermore, it is clear that the decomposition of A(ξ) into n = 2 and n = 0 contributions comes upon taking A(ξ) = aµν ξµ ξν = |ξ|2 b0 + bµν 2 ξµ ξν , µν where bµν 2 ξµ ξν is a second-order spherical harmonic in ξ; i.e., b2 is trace free in its two indices. This gives

aµν = g µν b0 + bµν 2 , where X 1 gµν aµν = µi Πi(0) , m i=1 s

b0 =

bµν 2 =

s X

µi Πµν i(2) .

i=1

4. Symmetric Two-Tensors It is instructive at this point to consider two examples: the bundle S 2 of symmetric two-tensors, and the subbundle S02 of trace-free symmetric two-tensors. In particular, in these examples, one begins to glimpse the differences and relations between the cases of reducible and irreducible V. We may compute with tensors valued in either complex or real tensor bundles; in fact, most of the following discussion holds in either setting. But for the sake of definiteness, and with a view toward applications, let us assume that all our tensors are real. First note that a basis of the 0-homogeneous symbols in our class is given by (X1 ϕ)αβ = gαβ ϕµ µ , (X2 ϕ)αβ =

1 µ ξ ξ(α ϕβ)µ , |ξ|2

(X3 ϕ)αβ =

1 ξα ξβ ϕµ µ , |ξ|2

(X4 ϕ)αβ =

1 gαβ ξ µ ξ ν ϕµν , |ξ|2

(X5 ϕ)αβ =

1 ξα ξβ ξ µ ξ ν ϕµν . |ξ|4

Note that in the inner product (ϕ, ψ) = ϕµν ψ µν , we have X1∗ = X1 ,

X2∗ = X2 ,

X3∗ = X4 ,

X5∗ = X5 .

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In addition, the traces of these endomorphisms are m(m − 1) , trS 2 IS 2 = 2 trS 2 X1 = m , trS 2 X2 = m + 1 , trS 2 X3 = trS 2 X4 = trS 2 X5 = 1 . The multiplication table of the algebra generated by IV and the Xi is shown in Table 1. The arrow indicates that the left factor is to be found in the leftmost column. For example, X3 X4 = mX5 . Table 1.

Multiplication table.

→

X1

X2

X3

X4

X5

X1

mX1

X4

X1

mX4

X4

X2

X3

1 (X2 + X5 ) 2

X3

X5

X5

X3

mX3

X5

X3

mX5

X5

X4

X1

X4

X1

X4

X4

X5

X3

X5

X3

X5

X5

The leading symbol of any even-order operator has the form A(ξ) = |ξ|2p (α0 IS 2 + α1 X1 + 2α2 X2 + α3 X3 + α4 X4 + α5 X5 ) , for some p and some numerical parameters αi . If A is the leading symbol of a secondorder operator, then α5 vanishes. If in addition A is self-adjoint, then α3 = α4 . Thus for a second-order self-adjoint leading symbol A, A(ξ) = |ξ|2 (α0 IS 2 + α1 X1 + 2α2 X2 + α3 (X3 + X4 )) .

(4.1)

If we further require that A be positive definite, we get additional inequality constraints on the αi ; these are discussed below. 4.1. Trace-free symmetric two-tensors Now consider the bundle S02 of trace-free symmetric two-tensors. The symbol 1 P := IS − X1 m is the self-adjoint projection onto S02 . Thus we may define a spanning set of symbols Yi on S02 by Yi = P Xi P .

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According to Table 1, Y1 = Y3 = Y4 = 0 , and Y2 = X2 −

1 1 (X3 + X4 ) + 2 X1 , m m

1 1 (X3 + X4 ) + 2 X1 . m m Clearly both Y2 and Y5 are self-adjoint. Together with the identity symbol IS02 , the symbols Y2 and Y5 form a basis of the symbol algebra on S02 . For purposes of explicit computation, it is useful to note that Y5 = X5 −

P X1 = X1 P = P X4 = X3 P = 0 .

(4.2)

We compute that Y2 Y5 = Y5 Y2 = Y52 =

m−1 Y5 , m

(4.3)

1 m−2 Y2 + Y5 . (4.4) 2 2m This points up a very important and useful fact: that the symbol algebra over S02 is commutative. In fact, as shown in [6], this is a general feature of the case of an irreducible bundle. In the case of a bundle, like S 2 , that is reducible under its structure group, the symbol algebra may be noncommutative. The commutativity of the symbol algebra over S02 significantly simplifies the computation of the projections. First note that all symbols will be simultaneously diagonalizable, so the discussion on this level is independent of the particular symbol A. From (4.3) we immediately obtain one projection, namely m Π3 = (4.5) Y5 . m−1 Y22 =

Since IS02 , Y2 , and Y5 form a basis of the symbol algebra, (4.4) shows that IS02 , Y2 , and Y22 are also a basis. Note that Y23 =

3m − 2 2 m − 1 Y2 − Y2 , 2m 2m

and 7m2 − 10m + 4 2 (3m − 2)(m − 1) Y2 − Y2 . 4m2 4m2 If Π := aIS02 + bY2 + cY22 , we thus get bc(m − 1) c2 (3m − 2)(m − 1) Π2 = a2 IS02 + 2ab − − Y2 m 4m2 bc(3m − 2) c2 (7m2 − 10m + 4) + 2ac + b2 + + Y22 . m 4m2 Y24 =

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There are 8 = 23 solutions (a, b, c) of the projection equation Π2 = Π. Two of these are 4(m − 1) 4m ,− (a2 , b2 , c2 ) := 0, m−2 m−2 and (a3 , b3 , c3 ) :=

0, −

m2 2m2 , (m − 1)(m − 2) (m − 1)(m − 2)

;

these in fact correspond to fundamental projections Π2 and Π3 . (Π3 is the same as the projection found by inspection in (4.5).) The remaining solutions correspond to the third fundamental projection Π1 := IS02 − Π2 − Π3 , and to 0, IS02 , Π2 + Π3 , Π1 + Π3 , and Π1 + Π2 . The fundamental projections have particularly simple expressions in terms of Y2 and Y5 : 4 Π2 = (−mY22 + (m − 1)Y2 ) = 2(Y2 − Y5 ) (4.6) m−2 Π3 =

m2 m (2Y22 − Y2 ) = Y5 (m − 1)(m − 2) m−1

Π1 = IS02 − 2Y2 +

m−2 Y5 . m−1

(4.7) (4.8)

The general self-adjoint second-order symbol A0 on S02 may be viewed as the compression P AP of a second-order symbol A on S 2 , which, in view of (4.1), is A0 (ξ) = |ξ|2 (α0 IS02 + 2α2 Y2 ) = |ξ|2 (µ1 Π1 + µ2 Π2 + µ3 Π3 ) 1 = |ξ|2 µ1 IS02 + 2(µ2 − µ1 )Y2 + [(m − 2)µ1 − 2(m − 1)µ2 + mµ3 ]Ya . m−1 From this we obtain the eigenvalues in terms of the parameters α0 and α2 : 2(m − 1) α2 . m Thus, self-adjoint positive definite second-order symbols are in one-to-one correspondence with choices of (α0 , α2 ) for which µ1 = α0 ,

µ2 = α0 + α2 ,

µ3 = α0 +

α0 , α0 + α2 , mα0 + 2(m − 1)α2 > 0 . Note that for m ≥ 2, the first and third conditions together imply the second one. Our projections have the following interpretation. After choosing the distinguished direction ξ, one can distinguish three subspaces of the trace-free symmetric two-tensors. First, there are tensors in the direction of the trace-free part (ξ ⊗ ξ)0 of ξ ⊗ ξ. This is clearly the range of Π3 , since Y5 ϕ is a scalar multiple of (ξ ⊗ ξ)0 . Next, there is the subspace consisting of tensors ξ ∨ ζ, where ζ ⊥ ξ. This is the range of Π2 , since for 1 λ 1 ζβ := ξ ϕβλ − 3 ξβ ξ λ ξ µ ϕλµ , |ξ| |ξ|

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we have (Y2 − Y5 )ϕ = ξ ∨ ζ ,

ζ λ ξλ = 0.

The remaining subspace is the range of Π1 ; it may be described as the space generated by tensors (ζ ∨ θ)0 , where ζ ⊥ ξ ⊥ θ. The traces of the projections are thus trV Π1 =

(m + 1)(m − 2) , 2

trV Π2 = m − 1 ,

trV Π3 = 1.

4.2. Symmetric two-tensors (not necessarily trace-free) To leave the irreducible setting, consider the bundle S 2 of symmetric two-tensors (unrestricted as to trace); this is equivariantly isomorphic to the direct sum of the trace-free symmetric two-tensors and the scalars. In fact, the direct sum decomposition is implemented by the projections P and IS 2 − P = (1/m)X1 . By noting that X1 = m(IS 2 − P ) and using Table 1, we obtain in addition to (4.2) 1 1 P X2 (IV − P ) = m X3 − X1 , m 1 1 (IV − P )X2 P = m X4 − X1 , m 1 X1 , m 1 (IV − P )X4 P = X4 − X1 , m 1 (IV − P )X2 (IV − P ) = (IV − P ) , m (IV − P )X3 (IV − P ) = (IV − P )X4 (IV − P ) = (IV − P ) . P X3 (IV − P ) = X3 −

(4.9)

Let Π1 , Π2 and Π3 be the fundamental projections defined by (4.6). Denoting 1 1 T = √ X3 − X1 , m m−1 1 1 T∗ = √ X4 − X1 m m−1 and using Eqs. (4.9), we obtain the decomposition A(ξ) := |ξ|2 [µ1 Π1 + µ2 Π2 + µ3 Π3 + κt(T + T ∗ ) + q(IS 2 − P )] , where κ and q are real constants defined by √ 2 κ = m−1 α2 + α3 , m q = α0 + mα1 +

2 α2 + 2α3 . m

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Note the useful relations T P = T Π1 = T Π2 = T Π3 = 0 , P T ∗ = Π1 T ∗ = Π2 T ∗ = Π3 T ∗ = 0 . T 2 = (T ∗ )2 = 0 ∗

T T = IV − P , Π3 T = T ,

(4.10) ∗

T T = Π3 .

T ∗ Π3 = T ∗ .

Observe also the obvious non-commutativity; the projections will now depend on the particular leading symbol we are studying. We see that for our new symbol on S 2 , the first two projections Π1 , Π2 computed above are untouched, and there are two more projections Z3 , Z4 onto one-dimensional subspaces. Π1 and Π2 are independent of the particular symbol we are diagonalizing, while and Z3 and Z4 depend on it. Z3 and Z4 take the form Z = aΠ3 + b(T + T ∗ ) + c(IV − P ) . By (4.10), we have Z 2 = (a2 + b2 )Π3 + b(a + c)(T + T ∗ ) + (b2 + c2 )(IS 2 − P ) . The projection equation Z 2 = Z gives (2a − 1)2 + (2b)2 = (2c − 1)2 + (2b)2 = 1 ,

b(a − c + 1) = 0 .

Aside from the solutions (a, b, c) = (0, 0, 0), (1, 0, 1) ,

(4.11)

all solutions have the form 1 1 1 a = (1 + cos θ) , b = sin θ , c = (1 − cos θ) , 2 2 2 where θ is an arbitrary real parameter; conversely, all choices of θ give solutions. The solutions (4.11) give 0- and 2-dimensional projections, so can be discarded from the present point of view, where we seek complementary one-dimensional projections. If we define 1 1 1 (a3 , b3 , c3 ) := (1 + cos θ), sin θ, (1 − cos θ) , 2 2 2 1 1 1 (a4 , b4 , c4 ) := (1 − cos θ), − sin θ, (1 + cos θ) , 2 2 2 we get a set of complementary projections Z3 and Z4 ; that is, we have Z3 Z4 = Z4 Z3 = 0. We still need to find a value of θ adapted to our given symbol A. Denote by ν3 and ν4 the eigenvalues of A in the ranges of Z3 and Z4 respectively: µ3 Π3 + κ(T + T ∗ ) + q(IV − P ) = ν3 Z3 + ν4 Z4 .

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We find that (ν3 + ν4 ) + (ν3 − ν4 ) cos θ = 2µ3 , (ν3 − ν4 ) sin θ = 2κ , (ν3 + ν4 ) − (ν3 − ν4 ) cos θ = 2q . From these equations we first determine the eigenvalues ν3 = ρ + ω ,

ν4 = ρ − ω ,

where 1 m (µ3 + q) = α0 + α1 + α2 + α3 , 2 2 2 µ3 − q ω 2 = κ2 + 2 ρ=

=

1 [−m2 α1 + 2(m − 2)α2 − 2mα3 ]2 + 4(m − 1)(2α2 + mα3 )2 . 4m2

The positivity of the leading symbol is translated now into the condition ω 2 < ρ2 , or κ2 < qµ3 . So, in addition to α0 > 0 and α0 + α2 > 0 we have (m − 1)(2α2 + mα3 )2 < (mα0 + m2 α1 + 2α2 + 2mα3 )[mα0 + 2(m − 1)α2 ]. The parameter θ is now determined by µ3 − q 1 = [−m2 α1 + 2(m − 2)α2 − 2mα3 ], 2ω 2mω √ κ m−1 sin θ = = (2α2 + mα3 ) . ω mω Therefore, the projections Z3 and Z4 have the form 1 κ 1 µ3 − q µ3 − q ∗ Z3,4 = 1± Π3 ± (T + T ) + 1∓ (IV − P ) , 2 2ω 2ω 2 2ω cos θ =

and indeed depend on the symbol A. 5. The Resolvent and the Heat Kernel Let F be a self-adjoint second-order partial differential operator on a compact manifold with positive definite leading symbol. (In particular, F is elliptic.) Then F has discrete real eigenvalue spectrum which is bounded below by some (possibly negative) real number c. If λ is a complex number with Re λ < c, then the resolvent (F − λI)−1 is well defined: if {(λj , ϕj )} is a spectral resolution, with the ϕj forming a complete orthonormal set in L2 (V), then (F − λI)−1 ϕj = (λj − λ)−1 ϕj .

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The resolvent is a bounded operator on L2 (V), and in fact is a compact operator by the Rellich Lemma, since it carries L2 (V) to the Sobolev section space L22 (V) continuously. The resolvent kernel is a section of the external tensor product of the vector bundles V and V ∗ over the product manifold M × M , and satisfies the equation (F − λI)G(λ|x, y) = δ(x, y) where δ(x, y) is the Dirac distribution (which in turn is the kernel function of the identity operator). Here and below, all differential operators will act in the first (x, as opposed to y) argument of any kernel functions to which they are applied. The resolvent is well-defined as long as the null space of F − λI vanishes; i.e., as long as λ is not one of the λj . As a consequence of self-adjointness, we have ¯ x) . G† (λ|x, y) = G(λ|y, Similarly, for t > 0 the heat operator U (t) = exp(−tF ) : L2 (M, V ) → L2 (M, V ) is well defined. U (t) is a smoothing operator; that is, it carries L2 sections to T sections in k∈R L2k = C ∞ . The kernel function of this operator, called the heat kernel, satisfies the equation (∂t + F )U (t|x, y) = 0 with the initial condition U (0+ |x, y) = δ(x, y) , and the self-adjointness condition U † (t|x, y) = U (t|y, x) . As is well known [12], the heat kernel and the resolvent kernel are related by the Laplace transform: Z ∞ G(λ) = dt etλ U (t) , 0

U (t) =

1 2πi

Z

c+i∞

dλ e−tλ G(λ) .

c−i∞

It is also well known [12] that the heat kernel U (t|x, y) is a smooth function near the diagonal {x = y} of M × M , with the diagonal values integrating to the functional trace: Z TrL2 exp(−tF ) = d vol(x) trV U (t|x, x) . M

Moreover, there is an asymptotic expansion of the heat kernel as t → 0+ , TrL2 exp(−tF ) ∼ (4πt)−m/2

X (−t)k k≥0

k!

Ak ,

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and a corresponding expansion of the resolvent as λ → −∞: X (−1)k TrL2 ∂λn G(λ) ∼ (4π)−m/2 Γ[(k − m)/2 + n + 1](−λ)m/2−k−n−1 Ak , k! k≥0

for n ≥ m/2. Here Ak are the global, or integrated heat coefficients, sometimes called the Minakshisundaram–Pleijel coefficients. There is additional information in the local asymptotic expansion of the heat kernel diagonal, X (−t)k U diag (t) := U (t|x, x) ∼ (4πt)−m/2 ak . (5.1) k! k≥0

The local heat coefficients ak integrate to the global ones: Z d vol(x) trV ak . Ak =

(5.2)

M

One can, in fact, get access to the local heat coefficients via a functional trace, by taking advantage of the principle that a function (or distribution) is determined by its integral against an arbitrary test section f ∈ C ∞ (End(V)): TrL2 f exp(−tF ) ∼ (4πt)−m/2

X (−t)k k≥0

We have

k!

Ak (f, F ) .

Z Ak = Ak (1, F ) ,

Ak (f, F ) =

d vol(x)f trV ak . M

The local heat coefficients ak have been calculated for Laplace type operators up to a4 [2]. For non-Laplace type operators, some of them are known only in the very specific case of differential form bundles [9]. We shall calculate below the coefficients A0 and A1 for NLT operators in terms of the projections introduced in the previous sections. Our tactic will be to construct an approximation to the heat kernel U (t|x, y), that is a parametrix. The important information in the parametrix for small t is carried by its values near the diagonal, since the heat kernel vanishes to order ∞ off the diagonal as t → 0+ . Since the heat and resolvent kernels are related by the Laplace transform, this is equivalent to studying an approximation to the resolvent kernel G(λ|x, y) near the diagonal for large negative Re λ. Let us stress here that our purpose is not to provide a rigorous construction of the resolvent with estimates; for this we rely on the standard references [12]. Rather, given that the existence of resolvent and heat parametrices is known, our aim is to compute various aspects of it; in particular, information on leading order terms sufficient to determine some of the heat kernel coefficients Ak . We shall employ the standard scaling device for the resolvent G(λ|x, y) and heat kernel U (t|x, y) when x → y, λ → −∞ and t → 0. This means that one introduces a small expansion parameter ε reflecting the fact that the points x and y are close

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to each other, the parameter t is small, and the parameter Re λ is negative and large. This can be done by fixing a point x0 , choosing normal coordinates at x0 (with gµν (x0 ) = δµν ), scaling according to x → xε = x0 + ε(x − x0 ) , t → tε = ε2 t ,

y → yε = x0 + ε(y − x0 ) , λ → λε = ε−2 λ ,

and expanding in an asymptotic series in ε. If one uses a local Fourier transform, then the corresponding momenta ξ ∈ T ∗ M are large and scale according to ξ → ξε = ε−1 ξ . This construction is standard [12]. This procedure can be done also in a completely covariant way [2, 3]. In the case of Laplace type operators, the most convenient form of the offdiagonal asymptotics as t → 0, among many equivalent forms, is [2, 3] σ X (−t)k U (t|x, y) ∼ (4πt)−m/2 exp − ∆1/2 bk (x, y) , 2t k!

(5.3)

k≥0

where σ = σ(x, y) = r2 (x, y)/2 is half the geodesic distance between x and y, and ∆ = ∆(x, y) = |g|−1/2 (x)|g|−1/2 (y) det(−∂µx ∂νy σ(x, y)) is the corresponding Van Vleck–Morette determinant. The functions bk (x, y) are called the off-diagonal heat coefficients. These coefficients satisfy certain differential recursion relations. Expanding each coefficient in a covariant Taylor series near the diagonal, one gets a recursively solvable system of algebraic equations on the Taylor coefficients [2]. The diagonal values give the local heat kernel coefficients ak (x) = bk (x, x). However, in the general case of non-Laplace type operators, it is very difficult to follow this approach, since the Ansatz for the off-diagonal heat kernel asymptotics (5.3) does not apply; the correct Ansatz would be much more complicated. For this reason, we employ the approach of pseudo-differential operators (or, roughly speaking, local Fourier transforms). An alternative approach, which generalizes the Ansatz (5.3), is developed in Sec. 10 6. Gaussian Integrals The Gaussian average of a function f (ξ) on Rm is Z dξ −|ξ|2 hf i ≡ e f (ξ) . m/2 Rm π The Gaussian average of an exponential function gives the generating function Z dξ |x|2 2 I0 (x) = exp(−|ξ| + iξ · x) = hexp(iξ · xt)i = exp − . m/2 4 Rm π

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Expansion of I0 (x) in a power series in x generates the Gaussian averages of polynomials h1i = 1 , hξµ ξν i =

hξµ i = 0 ,

1 gµν , 2

hξµ1 · · · ξµ2n+1 i = 0 , hξµ1 · · · ξµ2n i =

(2n)! g(µ µ · · · gµ2n−1 µ2n ) . 22n n! 1 2

Furthermore, using the relation Z ∞ 2 1 1 = ds sp−1 e−s|ξ| , |ξ|2p Γ(p) 0

Re p > 0

(6.1)

and analytic continuation, one can obtain the more general formulas ξµ1 · · · ξµ2n+1 = 0, |ξ|2p ξµ1 · · · ξµ2n Γ(m/2 + n − p) (2n)! g · · · gµ2n−1 µ2n ) = 2p |ξ| Γ(m/2 + n) 22n n! (µ1 µ2 for any p with Re p < n + m/2. This means, in particular, that ξµ1 · · · ξµ2n Γ(m/2 + n − p) ξµ1 · · · ξµ2n = , |ξ|2p Γ(m/2) |ξ|2n ξµ1 · · · ξµ2n Γ(m/2) (2n)! = g(µ µ · · · gµ2n−1 µ2n ) . |ξ|2n Γ(m/2 + n) 22n n! 1 2

(6.2)

(6.3)

Note that if a function f depends only on ξ/|ξ|, that is, if it is homogeneous of order 0, then the average introduced above is a constant multiple of the average over the unit (m − 1)-sphere S m−1 . We will also need to compute Fourier integrals of the form Z dξ ξµ · · · ξµ exp(iξ · x − |ξ|2 ) 1 2n 2n Iµ1 ...µ2n (x) = m/2 |ξ| π m R ξµ1 · · · ξµ2n = exp(iξ · x) . (6.4) |ξ|2n The trace of the symmetric 2n-form I2n over any two indices is 2(n − 1)-form I2n−2 : g µ2n−1 µ2n Iµ1 ···µ2n−2 µ2n−1 µ2n = Iµ1 ···µ2n−2 . Therefore, the total trace of the symmetric form I2n is I0 : |x|2 trg I2n (x) = I0 (x) = exp − . 4

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√ tξ one can show that this integral satisfies the equation x |x|2 . tm/2 ∂tn t−m/2 Iµ1 ···µ2n √ = ∂µ1 · · · ∂µ2n exp − 4t t √ Taking into account the obvious asymptotic condition limt→∞ [t−m/2 In (x/ t)] = 0, by multiple integration we get Z ∞ (−1)n |x|2 Iµ1 ···µ2n (x) = . ds(s − 1)n−1 s−m/2 ∂µ1 · · · ∂µ2n exp − (n − 1)! 1 4s By rescaling ξ →

By changing the variable, s = 1/u, we can rewrite this in the form Z 1 (−1)n |x|2 m/2−n−1 n−1 Iµ1 ···µ2n (x) = du u (1 − u) ∂µ1 · · · ∂µ2n exp − u . (n − 1)! 0 4 (6.5) If m is even this can be computed in elementary functions. In the case n < m/2, one can interchange the order of the integration and differentiation. Then by using the formula Z 1 Γ (a) Γ(b − a) du ua−1 (1 − u)b−a−1 ezu = 1 F1 (a; b; z) , Γ(b) 0 (with Re b > Re a > 0) where 1 F1 (a; b; z)

=

∞ X Γ(a + k)Γ(b) k z Γ(a)Γ(b + k)k! k=0

is the confluent hypergeometric function, we obtain

|x|2 ˜ Iµ1 ···µ2n (x) = (−1) ∂µ1 · · · ∂µ2n Φn − , 4 n

where X Γ(m/2 − n + k) ˜ n (z) = Γ(m/2 − n) 1 F1 m − n; m ; z = Φ zk Γ(m/2) 2 2 Γ(m/2 + k)k! ∞

k=0

(with n < m/2). If n ≥ m/2, then this formula cannot be applied directly. However, it is still valid if one does analytic continuation in the dimension m. The physical (integer) value of the dimension should be put after computing the derivatives. Alternatively, one can do the differentiation in (6.5) explicitly before the integration. This is equivalent to subtracting the first n terms of the Taylor series of the exponential. This subtraction is clearly harmless, since the 2nth derivatives of these terms vanish. However, this makes the integral finite, and justifies the interchange of the differentiation and integration. The result applies for any n. In this way one obtains |x|2 Iµ1 ···µ2n (x) = (−1)n ∂µ1 · · · ∂µ2n Φn − , (6.6) 4

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˜ n (z) by substracting the first n terms where Φn (z) is obtained from the function Φ of the power series: m Γ(m/2 − n) m X Γ(m/2 − n + k) k − n; ; z − z 1 F1 Γ(m/2) 2 2 Γ(m/2 + k)k! n−1

Φn (z) =

k=0

=

∞ X Γ(m/2 − n + k) k z . Γ(m/2 + k)k!

(6.7)

k=n

7. Leading Order Off-Diagonal Heat Kernel Asymptotics In general, the leading order resolvent and the heat kernel are determined by the leading symbol of the operator F : Z dξ iξ·(x−y) e [A(ξ) − λIV ]−1 , G0 (λ|x, y) = (2π)m Z dξ iξ·(x−y) U0 (t|x, y) = e exp[−tA(ξ)] , (2π)m where ξ ∈ T ∗ M , ξ · (x − y) = ξµ (xµ − y µ ), and dξ is Lebesgue measure on Rm . Here and everywhere below all integrals over ξ below will be over the whole Rm . Writing the leading symbol in terms of the projections from (3.11), it is not difficult to obtain s Z X dξ iξ·(x−y) Πi (ξ) G0 (λ|x, y) = e , (2π)m µi |ξ|2 − λ i=1 U0 (t|x, y) =

s Z X i=1

dξ iξ·(x−y)−tµi |ξ|2 e Πi (ξ) . (2π)m

The problem is now to compute these integrals. If we are only interested in traces, then by (3.6), we can easily compute s X |x − y|2 −m/2 trV U0 (t|x, y) = di (4πtµi ) exp − . 4tµi i=1 That is, the trace of the leading order term in the heat kernel asymptotics is a weighted linear combination of the scalar leading order heat kernel term with scaled times, t → µi t. Similarly, we can relate the trace of the resolvent at leading order to resolvents of Laplace type operators: s ! (m−2)/4 s X −λµi −λ −m/2 trV G0 (λ|x, y) = di (2πµi ) K(m−2)/2 |x − y| , |x − y|2 µi i=1 where Kp (z) is the modified Bessel function.

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Applying the methods described in the previous subsection we can compute the heat kernel and resolvent before taking the trace: s X (x − y) (4πtµi )−m/2 exp iξ · √ Πi (ξ) . U0 (t|x, y) = tµi i=1 Using the decomposition of the projections into spherical harmonics (3.9), we obtain X U0 (t|x, y) = (4πtµi )−m/2 cik a(µ1 µ2 · · · aµ2k−3 µ2k−2 ) (−tµi )k−1 1≤i,k≤s

|x − y|2 × ∂µ1 · · · ∂µ2k−2 Φk−1 − , 4tµi X

=

1≤i,k≤s

X

G0 (λ|x, y) =

|x − y|2 (4πtµi )−m/2 cik (tµi F0 )k−1 Φk−1 − , 4tµi

(7.1)

(4πµi )−m/2 cik (µi F0 )k−1

1≤i,k≤s

Z ×

∞

tλ −m/2+k−1

dt e t 0

|x − y|2 Φk−1 − , 4tµi

where F0 = aµν ∂µ ∂ν and the functions Φn (z) are defined in previous section; they are given by (6.7). We would like to stress that these formulas can be presented locally in a “covariantized” form. This is effectively achieved by replacing |x − y|2 by 2σ(x, y) and (x − y)µ by ∇µ σ(x, y) and adding a factor ∆1/2 (x, y) (for details, see [2]); the objects σ and ∆ being defined after equation (5.3). 8. The Heat Kernel Coefficient a0 The leading order term of the diagonal heat kernel asymptotics can now be written as s X −m/2 U0diag (t) := U0 (t|x, x) = (4πt)−m/2 µi hΠi i . i=1

This gives the coefficient a0 : a0 =

s X

−m/2

µi

hΠi i .

(8.1)

i=1

The Gaussian average is computed by using (6.3): hΠi i = Πi(0) =

s X k=1

cik

Γ(m/2)(2k − 2)! trg ∨k−1 a , Γ(m/2 + k − 1)22k−2 (k − 1)!

(8.2)

where ∨n is the symmetrized tensor power of symmetric form, trg is the total trace of a symmetric form defined in (3.4) and the constants cik are defined by (3.10).

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From this, we have the formula trV a0 =

s X di m/2

i=1

µi

for the trace of the heat coefficient. Note that if our bundle V is associated to an irreducible representation of Spin(m), the averages of the projections, being Spin(m)-invariant endomorphisms, are proportional to the identity, by Schur’s Lemma. The exact proportionality constant may be determined by taking the trace. We have: hΠi i =

di IV , d

1 X di IV . d i=1 µm/2 s

a0 =

i

These formulas point up a new feature of non-Laplace type operators; one which complicates life somewhat. Whereas the dimension dependence of the heat coefficients of Laplace type operators is isolated in the overall factor of (4π)−m/2 , the dimension dependence for NLT operators is more complicated. 9. The Heat Kernel Coefficient A1 As we have seen, even the computation of the leading order heat kernel requires significant effort in the case of non-Laplace principal part. This indicates that the calculation of higher-order coefficients will be a challenging task. In this paper we will compute the coefficient A1 . Since A1 is the integral of trV a1 by (5.2), it suffices to compute the local coefficient a1 modulo trace-free endomorphisms. By elementary invariant theory, a1 has the form X µ1 ···µk a1 = Q(k)µ1 ···µk qP(k) + H1 R + H2µν Rµν + H3µναβ Rµναβ (9.1) k≥0

where Q(k) , P(k) , and Hi are End(V)-valued tensors, and R, Rµν , and Rµναβ are the scalar, Ricci, and Riemann curvatures. For the trace we have a similar formula, µναβ Rµναβ trV a1 = trV (h0 q) + h1 R + hµν 2 Rµν + h3

where hi ≡ trV Hi (i = 1, 2, 3) are some tensors. By invariant theory, we may conclude that these tensors have the following form: X µ ···µ h0 = P(k)1 k Q(k)µ1 ···µk , k≥0

h1 = c1 ,

µν hµν , 2 = c2 g

hµναβ = c3 g µα g νβ + c4 g µβ g να + c5 g µν g αβ , 3 where ci are some constants.

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Note that for irreducible representations the endomorphisms Q0 , P0 , h0 and H1 are proportional to the identity endomorphism; in particular, h0 = c0 IV , H1 = (c1 /d)IV . Taking this into account, we obtain trV a1 = trV (h0 q) + βR , where β = c1 + c2 + c3 − c4 . Thus it suffices to compute the constant β and the endomorphism h0 . To compute h0 , we can employ a by now standard variational principle. If the operator F (ε) depends (in a suitably estimable way) on a parameter ε, then ∂ TrL2 exp(−tF ) = −t TrL2 (∂ε F ) exp(−tF ) . ∂ε If our variation comes from scaling q, q → εq , then (∂ε F ) = q. Expanding both sides in powers of t and comparing coefficients of like powers, we obtain Z ∂ε A1 = d vol(x)trV (a0 q) , M

so that

Z A1 =

d vol(x)[trV (a0 q) + βR], M

and, therefore, h0 = a0 , where a0 is given by (8.1) and (8.2). The computation of the constant β is considerably more difficult. It is clear the β is universal to our class of operators; thus we may compute it for any particular operator and manifold, or class of such. Accordingly, note that β is equal to the coefficient a1 when q = 0 and R = 1: β = trV a1 |q=0,R=1 . The following considerations will be completely local. Fix a point x0 , and compute in normal coordinates centered at x0 , with gµν (0) = δµν . Furthermore, impose by gauge transformation the Fock–Schwinger gauge for the connection 1-form A. We then have [gµν (x) − δµν ]xν = 0 ,

Aµ (x)xµ = 0 .

Further, we can expand all quantities in Taylor series about x0 and restrict our attention to terms linear in the curvature. This gives 1 gµν (x) = δµν − Rµανβ xα xβ + · · · , 3

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1 det gµν (x) = 1 − Rαβ xα xβ + · · · , 3 1 Aµ (x) = − Rµα xα + · · · . 2 Here and below, the dots denote higher-order terms in the curvature. Rµν is, of course, the curvature of the bundle V, given by Rµν = Rα βµν T β α . The Taylor expansion of the leading symbol section aµν is determined by the equation ∇µ aαβ = 0 . From this, we get 1 aµν (x) = aµν + aλ(µ Rν) αλβ xα xβ + · · · . 3 Here and below, we denote aµν (0) simply by aµν . By the above, the potential term q will only enter the calculation through q(0), which we denote simply by q: q(x) = q + · · · . Since the constant β is universal, we are free to compute in the case of a constant curvature metric, i.e. R (gµα gβν − gµβ gαν ) , m(m − 1) R = gµν , R = const . m

Rµναβ = Rµν

(9.2)

Now let us take the total symbol of our operator in normal coordinates, σ(F |x, ξ), and expand in a Taylor series: σ(F |x, ξ) = σL (F |0, ξ) + σ(F1 |x, ξ) + · · · , where σL (F |0, ξ) = aµν ξµ ξν ≡ A(ξ) , σ(F1 |x, ξ) = −X µν αβ xα xβ ξµ ξν + iY µ α xα ξµ + q , 1 X µν αβ = − aλ(µ Rν) (α|λ|β) , 3 Y µα =

2 µλ 1 a Rλα − [T σ ρ , aµν ]+ Rρ σαν , 3 2

and [A, B]+ = AB + BA denotes the anticommutator. There are many equivalent ways of constructing the heat kernel asympotics on the diagonal locally. Using the Volterra series Z 1 exp(−tF ) = exp(−tF0 ) − t dτ exp[−t(1 − τ )F0 ]F1 exp[−tτ F0 ] + · · · 0

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and the formula for the heat kernel on the diagonal Z dξ −iξ·x diag iξ·x U (t) = e exp(−tF )e , (2π)m x=0 we get Z U

diag

(t) =

dξ (2π)m

Z −tA(ξ) e −t

1

dτ e

−t(1−τ )A(ξ)

−tτ A(ξ) ˆ F1 e + ··· ,

0

where Fˆ1 = σ(F1 |i∂ξ , ξ) = X µν αβ ∂ξα ∂ξβ ξµ ξν − Y µ α ∂ξα ξµ + q . Finally, by scaling the integration variable ξ → t−1/2 ξ, we obtain the standard asymptotic expansion of the heat kernel on the diagonal (5.1), with the coefficients Z dξ −A(ξ) a0 = e , π m/2 Z Z 1 dξ a1 = dτ e−(1−τ )A(ξ) Fˆ1 e−τ A(ξ) . π m/2 0 In particular, we recover the formula for a0 derived in the previous section. To integrate by parts in this formula, we need to know how to differentiate the exponential eA . This can be done via the Duhamel formula Z τ α −τ A(ξ) = −2 ds e−(τ −s)A(ξ) J α (ξ)e−sA(ξ) , ∂ξ e 0

where J α (ξ) = aαβ ξβ . The contraction of this with ξ leads to a much simpler formula: ξα ∂ξα e−τ A(ξ) = −2τ A(ξ)e−τ A(ξ) . The exponential itself is computed by using the projections Πi : e−τ A(ξ) =

s X

e−τ µi |ξ| Πi (ξ) . 2

i=1

Note that when computing ξ-derivatives one can, using integration by parts, act in either direction. This trick can be used to avoid having to compute the second derivative of the exponential eA . First, we rewrite Fˆ1 in the form Fˆ1 = ∂ξα Xαβ ∂ξβ + ∂ξα Lα + Lα ∂ξα + Q ,

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where 1 Xαβ (ξ) = X µν αβ ξµ ξν = − aλ(µ Rν) (α|λ|β) ξµ ξν , 3 Lα (ξ) = Lµ α ξµ , 1 1 1 Lµ α = − aµλ Rλα + aλν Rµ λαν + Rρ σαν [T σ ρ , aµν ]+ , 4 12 4 1 Q = q − aµν Rµν . 6 Now, integrating by parts over ξ (and changing τ to 1 − τ in some places) we get Z Z 1 Z 1−τ Z τ dξ a1 = − 4 dτ ds ds2 e−(1−τ −s1 )A 1 π m/2 0 0 0 Z 1 Z τ × J α e−s1 A Xαβ e−(τ −s2 )A J β e−s2 A + 2 dτ ds 0

× [e Z +

−(τ −s)A α −sA

J e

1

Lα e

−(1−τ )A

dτ e−(1−τ )A Qe−τ A .

−e

−(1−τ )A

0

Lα e−(τ −s)A J α e−sA ] (9.3)

0

By separating the curvature factors in (9.3), one can compute each of the tensors Hi entering (9.1). Note that for a Laplace type operator (the case aµν = g µν I) we have A = |ξ|2 , J µ = ξ µ , Xαβ = −(1/3)Rµα ν β ξµ ξν , Lα = −(1/6)Rµα ξµ , Q = q − (1/6)RI. Therefore the first two terms (with X and L) vanish in the Laplace type case, and we get the well known result 1 a1 = q − RIV . 6 Note that the tensors Hi depend only on the leading symbol, i.e. on aµν . In principle, it is possible to compute them explicitly by using the representation of the eA in terms of projections and Gaussian averages. We shall not do this explicitly, but rather compute only the trace of a1 . The number of projections involved in this calculation is less by one. Computing in the case of constant curvature (9.2), and attaching a tilde to X, L and Q in this case, we have ˜ αβ (ξ) = − X ˜ α (ξ) = − L

R [Agαβ − J(α ξβ) ] , 3m(m − 1) R {¯ aξα − (3m − 2)Jα + 6[T[να] , J ν ]+ } , 12m(m − 1)

˜ =q− R a Q ¯, 6m

(9.4)

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where a ¯ = gµν aµν . Now taking the trace and changing s to τ − s in the second integral, we obtain Z Z 1 Z 1−τ Z τ dξ trV a1 = trV −4 dτ ds1 ds2 e−(τ −s2 )A π m/2 0 0 0 Z 1 Z τ ˜ αβ + 2 × J β e−(1−τ −s1 +s2 )A J α e−s1 A X dτ ds 0

0

˜ α + e−A Q ˜ . × [e−(1−s)A J α e−sA − e−sA J α e−(1−s)A ]L

(9.5)

In the second multiple, the s-integration may be accomplished explicitly, to give Z Z τ Z 1 Z 1−τ dξ dτ ds1 ds2 e−(τ −s2 )A trV a1 = trV −4 π m/2 0 0 0 ˜ αβ × J β e−(1−τ −s1 +s2 )A J α e−s1 A X Z 1 ˜ α + e−A Q ˜ . +2 dτ τ [e−τ A J α e−(1−τ )A − e−(1−τ )A J α e−τ A ]L

(9.6)

0

Now let us consider the different contributions separately. Q Contribution The Q contribution has the form Z s Z X dξ −A ˜ dξ −µi |ξ|2 ˜ trV e Q = tr e Πi Q . V m/2 π π m/2 i=1 By changing the integration variable ξ to (µi )−1/2 ξ, we obtain a Gaussian average: s X 1 1 −m/2 trV µi hΠi i q − a ¯R = trV a0 q − a ¯R . 6m 6m i=1 L Contribution This contribution has the form Z Z 1 dξ ˜α . 2 trV dτ τ [e−τ A J α e−(1−τ )A − e−(1−τ )A J α e−τ A ]L π m/2 0

(9.7)

Using (9.4), we get −

R trV 6m(m − 1)

Z

dξ π m/2

Z

1

dτ τ [e−τ A J α e−(1−τ )A − e−(1−τ )AJ α e−τ A ]

0

× (¯ aξα − (3m − 2)Jα + 6[T[να] , J ν ]+ ) .

(9.8)

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Noting that J α ξα = A, we see that the term in Lα proportional to ξα does not contribute. The term proportional to Jα gives Z Z 1 dξ (3m − 2) trV dτ τ [e−τ A J α e−(1−τ )A − e−(1−τ )AJ α e−τ A ]Jα . R 6m(m − 1) π m/2 0 This is computed by again introducing the projections Πi : Z Z 1 dξ (3m − 2) X dτ R 6m(m − 1) π m/2 0 1≤i,k≤s

× τ [e−[µi τ +µk (1−τ )]|ξ| − e−[µk τ +µi (1−τ )]|ξ| ]trV Πi J α Πk Jα . 2

2

By changing the integration variable ξ to [µi τ + µk (1 − τ )]−1/2 ξ, we may express this in terms of a Gaussian average: (3m − 2) X κik trV hΠi J α Πk Jα i , R 6m(m − 1) 1≤i,k≤s

where

Z

κik =

1

dτ τ {[µi τ + µk (1 − τ )]−(m+2)/2 − [µk τ + µi (1 − τ )]−(m+2)/2 } .

0

Clearly κik = −κki ; hence, for i = k κii = 0 . For i 6= k we compute κik

Γ(m/2 − 1) =− Γ(m/2 + 1)(µi − µk )

(

µ m −m/2 −m/2 µi + µk + i 2

−m/2+1

−m/2+1

− µk µi − µk

) . (9.9)

Similarly, the contribution of the term with T is computed to be Z Z 1 R dξ − trV dτ τ [e−τ A J α e−(1−τ )A − e−(1−τ )A J α e−τ A ][T[να] , J ν ]+ m(m − 1) π m/2 0 =− =−

R trV m(m − 1) R trV m(m − 1)

X

κik hΠi J α Πk [T[να] , J ν ]+ i

1≤i,k≤s

X

κik (hJ ν Πi J α Πk i + hΠi J α Πk J ν i)T[να] .

1≤i,k≤s

Thus the total contribution of L is X 1 R trV κik {(3m − 2)hΠi J α Πk Jα i 6m(m − 1) 1≤i,k≤s

− 6(hJ ν Πi J α Πk i + hΠi J α Πk J ν i)T[να] } .

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X Contribution This term has the form 4 trV R 3m(m − 1)

Z

Z

dξ

Z

1

dτ

π m/2

0

×J e

ds1 0

β −(1−τ −s1 +s2 )A α −s1 A

J e

Z

1−τ

τ

ds2 e−(τ −s2 )A

0

[Agαβ

− J(α ξβ) ] .

We consider first the term proportional to A. Because A commutes with the exponent, we have Z Z 1 Z 1−τ Z τ 4 dξ R trV dτ ds ds2 Ae−(τ −s2 +s1 )A 1 3m(m − 1) π m/2 0 0 0 × J α e−(1−τ −s1 +s2 )A Jα .

(9.10)

Introducing the projections again, we get Z τ X Z dξ Z 1 Z 1−τ 4 R trV dτ ds1 ds2 3m(m − 1) π m/2 0 0 0 1≤i,k≤s × e−[(τ −s2 +s1 )µi +(1−τ −s1 +s2 )µk ]|ξ| µi |ξ|2 Πi J α Πk Jα . 2

Scaling, ξ → [(τ − s2 + s1 )µi + (1 − τ − s1 + s2 )µk ]−1/2 ξ, we obtain R

4 trV 3m(m − 1)

X

ρik µi h|ξ|2 Πi J α Πk Jα i ,

1≤i,k≤s

where Z ρik =

Z

1

dτ 0

Z

1−τ

ds1 0

τ

ds2 [(τ − s2 + s1 )µi + (1 − τ − s1 + s2 )µk ]−(m+4)/2 .

0

By changing τ to 1 − τ and switching the roles of s1 and s2 , we see that ρik = ρki . For i = k, we easily obtain ρii =

1 −m/2−2 µ . 6 i

For i 6= k we compute that ρik =

Γ(m/2 − 1) Γ(m/2 + 2)(µi − µk )2 ( × (m −

−m/2 4)(µi

+

−m/2 µk )

−m/2+1

+2

µi

−m/2+1

− µk µi − µk

) .

(9.11)

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Furthermore, the term proportional to ξ in the X contribution is Z Z 1 Z 1−τ Z τ 4 dξ trV −R dτ ds1 ds2 3m(m − 1) π m/2 0 0 0 × e−(τ −s2 )A J β e−(1−τ −s1 +s2 )A J α e−s1 A J(α ξβ) . Remembering that J α ξα = A and changing the variables of integration, we have Z Z 1 4 dξ 1 trV −R dτ τ 2 Ae−τ A J α e−(1−τ )A Jα . m/2 3m(m − 1) 2 π 0 Introducing the projections Πi , this becomes −R

4 trV 3m(m − 1)

X Z

dξ π m/2

1≤i,k≤s

Z

1

dτ 0

2 1 × τ 2 µi e−[τ µi +(1−τ )µk ]|ξ| |ξ|2 Πi J α Πk Jα . 2

Finally, scaling ξ → [τ µi + (1 − τ )µk ]−1/2 ξ, we get −R

4 trV 3m(m − 1)

X

γik µi h|ξ|2 Πi J α Πk Jα i ,

1≤i,k≤s

where Z γik = 0

1

1 dτ τ 2 [τ µi + (1 − τ )µk ]−m/2−2 . 2

For i = k, γii =

1 −m/2−2 . µ 6 i

For i 6= k, we compute that γik =

Γ(m/2 − 1) 8Γ(m/2 + 2)(µi − µk ) − 4(m − 2)

−m/2

−m/2−1

− m(m − 2)µi −m/2+1

µ µi −8 i µi − µk

−m/2+1

− µk (µi − µk )2

Thus, the total X contribution is R

4 trV 3m(m − 1)

X

µi σik h|ξ|2 Πi J α Πk Jα i ,

1≤i,k≤s

where σik = ρik − γik .

.

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Clearly σii = 0 . For i 6= k, σik =

−m/2+1 −m/2+1 −m/2 −m/2 µ − µk µi + µk Γ(m/2 − 1) 24 i + 8(m − 4) 8Γ(m/2 + 2) (µi − µk )3 (µi − µk )2 −m/2 −m/2−1 µi µi . (9.12) + 4(m − 2) + m(m − 2) (µi − µk )2 (µi − µk )

Summing up the various contributions, we get the main result: trV a1 =

s X

−m/2

µi

trV hΠi iq + βR ,

i=1

where 1 X −m/2 1 µ trV hΠi i¯ a+ 6m i=1 i 6m(m − 1) s

β=−

X 1≤i,k≤s;i6=k

× κik (3m − 2) trV hΠi J α Πk Jα i − 6 trV [hJ ν Πi J α Πk i + hΠi J α Πk J ν i]T[να] +

4 3m(m − 1)

X

µi σik trV h|ξ|2 Πi J α Πk Jα i ,

1≤i,k≤s;i6=k

with the constants κik and σik given by (9.9) and (9.12). Next we need to compute the Gaussian averages. This can be easily done by using the formulas (6.2). First note that the radial part can be always separated by h|ξ|2p f (ξ/|ξ|)i =

Γ(m/2 + p) hf (ξ/|ξ|)i . Γ(m/2)

This gives m hΠi J˜α Πk J˜β i , 2 m hJ α Πk J β Πi i = hJ˜α Πk J˜β Πi i , 2 hΠi J α Πk J β i =

hξ|2 Πi J α Πk J β i =

m(m + 2) hΠi J˜α Πk J˜β i , 4

where Jα ξβ J˜α = = aαβ . |ξ| |ξ|

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This allows to simplify the result somewhat: 1 X −m/2 µ trV hΠi i¯ a 6m i=1 i s

β=− + −

1 12(m − 1) 1 2(m − 1)

X

[κik (3m − 2) + 4(m + 2)µi σik ]trV hΠi J˜α Πk J˜α i

1≤i,k≤s;i6=k

X

κik trV [hJ˜ν Πi J˜α Πk i + hΠi J˜α Πk J˜ν i]T[να] .

1≤i,k≤s;i6=k

Note that for irreducible representations the endomorphisms hΠi i, a ¯ = aµ µ , and α hΠi J Πk Jα i are proportional to the identity IV . Finally, for the sake of completeness, we list the averages explicitly. By using the formulas (6.3) we get X Γ(m/2)(2n + 2j − 2)! hΠi J˜α Πk J˜β i = Γ(m/2 + n + j − 1)22n+2j−2 (n + j − 1)! 1≤n,j≤s

× g(µ1 µ2 · · · gµ2n−3 µ2n−2 gν1 ν2 · · · gν2j−3 ν2j−2 gγδ) × a(µ1 µ2 · · · aµ2n−3 µ2n−2 aγ α aν1 ν2 · · · aν2j−3 ν2j−2 aδ) β . Denoting by ∨ the symmetric product of symmetric forms and by trg the total trace of a symmetric form, we can rewrite this in the compact form X Γ(m/2)(2n + 2j − 2)! hΠi J˜α Πk J˜β i = Γ(m/2 + n + j − 1)22n+2j−2 (n + j − 1)! 1≤n,j≤s

× trg [(∨n−1 a) ∨ a ˜α (∨j−1 a) ∨ a ˜β ] , where a ˜α is a vector defined by a ˜α = aµ α ∂µ . This completes the general calculation of heat kernel coefficient trV a1 . More explicit formulas may be obtained in particular cases by using explicit formulas for the projections. 10. The Covariant Semi-Classical Approximation In this section we show how the standard semi-classical method can be adapted to compute the small-t heat kernel asymptotics of a non-Laplace type operator. In treating the semi-classical approximation, we shall follow [19]. The object of study is the fundamental solution of the heat equation 1 ∂t + F U (t|x, x0 ) = 0 ε for a non-Laplace type operator F = −aµν ∇µ ∇ν + q

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with a singular initial condition U (t|x, x0 )|t→0 = δ(x, x0 ) .

(10.1)

Here ε is a small formal parameter. 0 Let σµ (x, x0 ) denote the tangent vector to the geodesic connecting the points x and x0 at the point x0 , the norm of which is equal to the length of that geodesic. In this section we describe a systematic method for constructing the local formal asymptotic solution of the heat equation as ε → 0. The initial condition suggests the following Ansatz: 1 U (t|x, x0 ) = J(t|x, x0 ) exp − S(t|x, x0 ) Ω(t|x, x0 ) , ε where S is a scalar function, J is another scalar function defined by 1 1 1 det − , J(t|x, x0 ) = p ∇µ ∇ν 0 S(t|x, x0 ) p 2πε det gµν (x) det gµν (x0 ) and Ω(t) has the expansion Ω(t|x, x0 ) ∼

X

εk φk (t|x, x0 )

k≥0

in powers of ε. The leading asymptotics as ε → 0 require (S˙ + aµν S;µ S;ν )φ0 = 0 where S˙ = ∂t S and S;µ ≡ ∇µ S. From this it follows that the function S is determined by the Hamilton–Jacobi equation detV [∂t S + aµν S;µ S;ν ] = 0 . Recalling the eigenvalues of the leading symbol,we have s Y [∂t S + µi g µν S;µ S;ν ]di = 0 . i=1

This has s different solutions, one for each eigenvalue, determined by 1 ∂t Si + g µν Si;µ Si;ν = 0 . µi

(10.2)

These solutions differ by scaling t → ti ≡ µi t, i.e. S(t|x, x0 ) = S0 (µi t|x, x0 ), where S0 is determined by the equation ∂t S0 + g µν S0;µ S0;ν = 0 . This is the Hamilton–Jacobi equation for a particle moving in a curved manifold. There is a Hamiltonian system that corresponds to each Hamilton–Jacobi equation (10.2). These Hamiltonian systems describe the geodesics parametrized by ti = tµi . The solution of this equation is given by the action along the geodesics connecting the points x and x0 and parameterized so that x(0) = x0 , x(t) = x. Thus S0 (t|x, x0 ) =

σ(x, x0 ) , 2t

Si (t|x, x0 ) =

σ(x, x0 ) , 2tµi

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where σ(x, x0 ) is half the square of the length of the geodesic. This means that Ji (t|x, x0 ) = (4πtµi ε)−m ∆(x, x0 ) , where ∆ is the Van Vleck–Morette determinant. As an immediate consequence of the Liouville Theorem, this satisfies the equation 1 ∂t Ji + 2∇µ (Si;µ Ji ) = 0 , µi or 1/2

(D + σµ µ )Ji

=0

where D = t∂t + σµ ∇µ . Here and below we denote the derivatives of σ simply by adding indices to it, i.e. σµ ≡ ∇µ σ, σµν ≡ ∇ν ∇µ σ, etc. Recall that D∆1/2 =

1 (m − σµ µ )∆1/2 . 2

(10.3)

Thus the semiclassical approximation as ε → 0 polarizes along different eigenvalues. That is, all quantities become dependent on the eigenvalue, and the total solution is the superposition of all particular solutions for all eigenvalues. Thus, our final Ansatz is: s X σ 0 −m/2 1/2 U (t|x, x ) = (4πtµi ε) ∆ exp − Ωi (t|x, x0 ) , 2tµ ε i i=1 Ωi (t|x, x0 ) ∼

X

tk εk φ(i)k (t|x, x0 ) .

k≥0

For the function Ωi (t), we get a transport equation 1 1 Ni + Li + M Ωi (t) = 0 , ε2 t2 εt where 1 Ni = 2µi

1 µν σ− a σµ σν , 2µi

Li = t∂t −

m 1 1 µν + ∆−1/2 aµν σµ ∇ν ∆1/2 + a σµν , 2 µi 2µi

M = ∆−1/2 F ∆1/2 . The initial condition for Ωi (t) is determined by the diagonal value of the heat kernel Ωi (0|x, x) = hΠi i ,

(10.4)

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where hΠi i is defined by (8.2). While for irreducible representations this is proportional to the identity matrix, for a general reducible bundle it is not. By using the expansion of Ωi (t), we get recursion relations Ni φ(i)0 = 0 , Ni φ(i)1 = −Liφ(i)0 ,

(10.5)

Ni φ(i)k = −(Li + k − 1)φ(i)k−1 − M φ(i)k−2 ,

k ≥ 2.

Note that Ni is just an endomorphism (i.e., has order zero as a differential operator). Now we have a quadratic form aµν σµ σν which can be expanded in terms of the same projections as before. The role of the covector argument ξ of each projection is now played by ∇σ, and we denote Pi ≡ Πi (∇σ) =

p X

1 ···µ2n Πµi(2n)

n=0

=

s X

σµ1 · · · σµ2n (2σ)n

cik a(µ1 µ2 · · · aµ2k−3 µ2k−2 )

k=1

aµν σµ σν = 2σ

s X

σµ1 · · · σµ2k−2 , (2σ)k−1

(10.6)

µi Pi .

i=0

Recall that ∇λ aµν = 0; this implies that the tensors Πi(n) (not Pi ) are covariantly constant: ∇Πi(n) = 0 . Note that this does not imply that the Pi are covariantly constant, since ∇σ is not. Now, by using the decomposition of the leading symbol, we observe important properties σ Pk Ni = Ni Pk = (µi − µk )Pk , 2µ2i Pi Ni = Ni Pi = 0 , (I − Pi )Ni = Ni (I − Pi ) = Ni , which imply that Ni =

X 0≤k6=i≤s

σ (µi − µk )Pk . 2µ2i

Next, decompose φ(i)k according to the projection Pi : φ(i)k = ψ(i)k + χ(i)k , where ψ(i)k = Pi φ(i)k ,

χ(i)k = (I − Pi )φ(i)k .

(10.7)

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Then the recursion (10.5) takes the form Ni χ(i)0 = 0 , Ni χ(i)1 = −Li (ψ(i)0 + χ(i)0 ) ,

(10.8)

Ni χ(i)k = −(Li + k − 1)(ψ(i)k−1 + χ(i)k−1 ) − M (ψ(i)k−2 + χ(i)k−2 ) . Now, by multiplying this recursion by Pn , n 6= i, and using (10.7), we obtain σ (µi − µn )Pn χ(i)0 = 0 , 2µ2i σ (µi − µn )Pn χ(i)1 = −Pn Li ψ(i)0 , 2µ2i (10.9) σ (µi − µn )Pn χ(i)k = −Pn Li ψ(i)k−1 − Pn (Li + k − 1)χ(i)k−1 2µ2i − Pn M (ψ(i)k−2 + χ(i)k−2 ) . This recursion determines χ(i)k algebraically in terms of ψ(i)k−1 , ψ(i)k−2 , χ(i)k−1 and χ(i)k−2 . In particular, we find that χ(i)0 = 0 , χ(i)1 = −

2 σ

X 1≤n6=i≤s

µ2i Pn Li ψ(i)0 . µi − µn

The recursion does not determine the ψ(i)k however. These are determined by another differential recursion that is obtained by multiplying (10.8) by Pi , Pi Li ψ(i)0 = 0 , (Pi Li + k)ψ(i)k = −Pi Li χ(i)k − Pi M (ψ(i)k−1 + χ(i)k−1 ) ,

k ≥ 1.

Now let us compute the operator Pi LPi entering this recursion. We have Pi Li Pi = t∂t + +

1 Pi ∆−1/2 aµν σµ Pi ∇ν ∆1/2 µi

1 1 (Pi aµν σµ Pi;ν + Pi aµν σµν Pi ) − mPi 2µi 2

(10.10)

where Pi;ν = ∇ν Pi . Further, noting that Pi aµν σµ σν Pi = 2µi σPi , we get Pi aµν σν Pi = µi σµ Pi . Taking into account the equation for the Van Vleck–Morette determinant (10.3), we obtain Pi Li Pi = Pi D + Ki ,

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where D = t∂t + σµ ∇µ is a first-order differential operator, and Ki =

1 Pi [2aµν σµ Pi;ν + (aµν − µi g µν )σµν ]Pi 2µi

is some endomorphism. Note that σµ = tdxµ /dt. Therefore, the operator D is expressed in terms of the operator of total differentiation along the geodesics, d . dt The operator D commutes with any function that depends only on “angular” co√ ordinates σµ / σ; in particular, it commutes with the projections: D=t

DPi = Pi D . Thus the recursion for the ψ’s takes the form (D + Ki )ψ(i)0 = 0 , (D + k + Ki )ψ(i)k = −Pi Li χ(i)k − Pi M (ψ(i)k−1 + χ(i)k−1 ) ,

k ≥ 1.

(10.11)

But these are exactly the transport equations along geodesics. They may be integrated with the appropriate initial conditions determined by (10.4). In particular, the first coefficient ψ(i)0 is Z t d ψ(i)0 = exp − Ki hΠi i , 0 τ where the integration is along the geodesic connecting the points x0 and x, parametrized so that x = x(τ ) that x(0) = x0 , x(t) = x. In the standard case, i.e. for a Laplace type operator, we have Ki = 0. Therefore, the first coefficient is just φ0 = I. Thus, the recursion relations determine the asymptotic solution completely. The algorithm is the following: at each step determine first the χ(i)k by (10.9), and then ψ(i)k by (10.11). 11. Concluding Remarks Let us summarize the results of this paper. We have studied in detail a general class of non-Laplace type operators, i.e. elliptic second-order partial differential operators acting on sections of a tensor-spinor vector bundle over a compact manifold without boundary. The only essential assumptions that have been made are: (i) the positivity of the leading symbol, aµν ξµ ξν > 0 (in the sense of endomorphisms) for ξ 6= 0, and (ii) the covariant constancy of the tensor aµν , i.e. ∇a = 0. We constructed the leading order resolvent and the heat kernel and computed the first two coefficients of the heat kernel asymptotic expansion explicitly. In the last section we developed

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an alternative approach for the off-diagonal heat kernel asymptotics for by making use of the general theory of the semi-classical approximation. This enabled us to construct a new ansatz for the heat kernel, as well as to find a complete set of recursion relations for the coefficients of the off-diagonal asymptotic expansion. This generalizes a well known ansatz for the heat kernel of Laplace type operators (e.g. see [2]). In contrast to the Laplace type case, the off-diagonal heat kernel for the non-Laplace type case exhibits some essentially new features, notably polarization along the different eigenvalues of the leading symbol. As an explicit example of a non-Laplace type operator we considered the most general second-order operator acting on the bundle of symmetric two-tensors. We computed the eigenvalues of the leading symbol, the multiplicities and the corresponding projections.

References [1] S. Alexandrov and D. Vassilevich, “Heat kernel for nonminimal operators on a Kaehler manifold”, J. Math. Phys. 37 (1996) 5715–5718. [2] I. G. Avramidi, “A covariant technique for the calculation of the one-loop efective action”, Nucl. Phys. B355 (1991) 712–754. [3] I. G. Avramidi, “Green functions of higher-order differential operators”, J. Math. Phys. 39 (1998) 2889–2909. [4] I. G. Avramidi, “Covariant techniques for computation of the heat kernel”, Rev. Math. Phys. 11 (1999) 947–980. [5] I. G. Avramidi, Heat Kernel and Quantum Gravity, Lecture Notes in Physics, Series Monographs, LNP: m64. Berlin, New York, Springer-Verlag, 2000. [6] I. G. Avramidi and T. Branson, “A discrete leading symbol and spectral asymptotics for natural differential operators”, Irvin Segal memorial volume (2001), to appear. [7] I. G. Avramidi and G. Esposito, “Gauge theories on manifolds with boundary”, Comm. Math. Phys. 200 (1999) 495–543. [8] N. Berline, E. Getzler and M. Vergne, Heat Kernels and Dirac Operators, Berlin, New York, Springer-Verlag, 1992. [9] T. P. Branson, P. B. Gilkey and A. Pierzchalski, “Heat equation asymptotics of elliptic differential operators with non-scalar leading symbol”, Math. Nachr. 166 (1994) 207–215. [10] H. T. Cho and R. Kantowski, “ζ-functions for nonminimal operators”, Phys. Rev. D52 (1995) 4588–4599. [11] B. D. De Witt, “The spacetime approach to quantum field theory”, pp. 383–738 in Relativity, Groups and Topology II, eds. B. S. De Witt and R. Stora, Amsterdam, North Holland, 1984. [12] P. B. Gilkey, Invariance Theory, the Heat Equation and the Atiyah–Singer Index Theorem, Boca Raton, CRC Press, 1995. [13] P. B. Gilkey, T. P. Branson and S. A. Fulling, “Heat equation asymptotics of ‘nonminimal’ operators on differential forms”, J. Math. Phys. 32 (1991) 2089–2091. [14] V. P. Gusynin, “Asymptotics of the heat kernel for nonminimal differential operators”, Ukrainian Math. Zh. 43 (1991) 1541–1551. [15] V. P. Gusinyn, “Heat kernel technique for nonminimal operators”, pp. 65–86 in Heat Kernel Techniques and Quantum Gravity, eds. S. A. Fulling, Proceedings, Winnipeg, Manitoba, 1994. Discourses in Mathematics and Its Applications, vol. 4, College Station, Texas, Texas A&M University, 1995.

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[16] V. P. Gusynin and V. V. Kornyak, “Complete computation of the De Witt–Seeley– Gilkey coefficient E4 for nonminimal operator on curved manifolds”, Fund. Appl. Math. 5 (1999) 649–674. [17] J. Hadamard, “Lectures on Cauchy’s problem”, in Linear Partial Differential Equations, New Haven, Yale University Press, 1923. [18] V. D. Kupradze, Potential Methods in the Theory of Elasticity, Jerusalem, Israel program of Scientific Translation, 1965. [19] V. P. Maslov and M. V. Fedoriuk, Semi-classical Approximation in Quantum Mechanics, Boston, D. Reidel Pub. Co., 1981.

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ON THE BEHAVIOR AT INFINITY OF THE FUNDAMENTAL SOLUTION OF TIME DEPENDENT ¨ SCHRODINGER EQUATION

KENJI YAJIMA∗ Department of Mathematical Sciences, University of Tokyo 3-8-1 Komaba, Meguro-ku, Tokyo 153-7815, Japan

Received 15 February 2000

Dedicated to the memory of Professor Tosio Kato We show that the asymptotic behavior at infinity of the fundamental solution of the initial value problem for the free Schr¨ odinger equation or of the harmonic oscillator at non-resonant time is stable under subquadratic perturbations. We also show that the same is true for the phase and the amplitude of the Fourier integral operator representing the propagator. 1991 Mathematical Subject Classification: Primary 35A08, 35Q40; Secondary 81Q20

1. Introduction We consider the Cauchy problem for the time dependent Schr¨ odinger equation i

∂u 1 = − ∆u + V (t, x)u, ∂t 2

(t, x) ∈ R1 × Rn ;

u(s, x) = φ(x),

x ∈ Rn

(1.1)

in the Hilbert space L2 (Rn ). We assume throughout the paper that V (t, x) is realvalued, smooth with respect to x and, for any α, ∂xα V (t, x) is continuous with respect to (t, x). Moreover, we assume that V is subquadratic or a subquadratic perturbation of (1/2)x2 , viz. it satisfies one of the following two conditions. ∂x2 V (t, x) is the Hessian matrix of V with respect to the x-variables. supt∈R1 |∂x2 V (t, x)| = o(1) as |x| → ∞. For |α| ≥ 3, sup(t,x)∈R1 ×Rn |∂xα V (t, x)| < ∞. (SQH) V (t, x) = 12 x2 + W (t, x) and W satisfies (SQ).

(SQ)

In what follows we say V is SQ or SQH if V satisfies the condition (SQ) or (SQH), respectively. ∗ Partly

supported by the Grant-in-Aid for Scientific Research, The Ministry of Education, Science, Sports and Culture, Japan #11304006. 891

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K. Yajima

It is well known under the condition that Eq. (1.1) generates a unique unitary propagator {U (t, s), −∞ < t, s < ∞} in L2 (Rn ) and u(t) = U (t, s)φ represents a unique solution of (1.1). The two-parameter family of unitary operators U (t, s) is strongly continuous and satisfies the semi-group properties: U (t, t) = 1 and U (t, r)U (r, s) = U (t, s). We denote by E(t, s, x, y) the distribution kernel of U (t, s): Z u(t, x) = U (t, s)φ(x) = E(t, s, x, y)φ(y)dy; E = E(t, s, x, y) is the fundamental solution (FDS for short) of the Cauchy problem (1.1). We denote by (x(t, s, y, k), p(t, s, y, k)) the solution of the Hamilton equations corresponding to (1.1) ( ( x˙ = p, x(s, s, y, k) = y, (1.2) p˙ = −∂x V (t, x), p(s, s, y, k) = k , where x˙ = dx/dt and p˙ = dp/dt. Most of the following results on the flow associated with (1.2) are well known. For 0 < < T , we set (1)

I,T = {(t, s) : 0 < |t − s| < T, |t − s − mπ| > , ∀m ∈ Z \ {0}} , (2)

I,T = {(t, s) : 0 < |t − s| < T, |t − s − (m + (1/2))π| > , ∀m ∈ Z} . 1. Let V be SQ or SQH. Then, there exists T1 > 0 such that for (t, s) with |t − s| < T1 and (x, y) ∈ R2n (respectively (y, ξ) ∈ R2n ) there exists a unique trajectory (x(r), p(r)) = (x(r, s, y, k), p(r, s, y, k)) of (1.2) such that x(t) = x (respectively p(t) = ξ) and x(s) = y [3]. 2. Let V be SQ and T > 0. Then, there exists R > 0 such that for (x, y) with x2 + y 2 ≥ R2 (respectively (y, ξ) with y 2 + ξ 2 ≥ R2 ) and (t, s) with 0 < |t − s| < T , there exists a unique trajectory (x(r), p(r)) of (1.2) such that x(t) = x (respectively p(t) = ξ) and x(s) = y [11]. 3. Let V be SQH and T > > 0. Then, there exists R > 0 such that for (x, y) (1) with x2 + y 2 ≥ R2 (respectively (y, ξ) with ξ 2 + y 2 ≥ R2 ) and (t, s) ∈ I,T (2)

(respectively (t, s) ∈ I,T ), there exists a unique trajectory (x(r), p(r)) of (1.2) such that x(t) = x (respectively p(t) = ξ) and x(s) = y ([7], see also Sec. 2). Using the trajectory (p(r), x(r)) such that x(t) = x (respectively p(t) = ξ) and x(s) = y, we set, for all (x, y) (respectively (y, ξ)) if |t − s| is small and otherwise for large x2 + y 2 (respectively ξ 2 + y 2 ), Z t 1 S(t, s, x, y) = p(r)2 − V (r, x(r)) dr , (1.3) 2 s Z t 1 respectively Φ(t, s, y, ξ) = x(t)ξ − p(r)2 − V (r, x(r)) dr . (1.4) 2 s

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On the Behavior at Infinity of the Fundamental Solution

893

The function S(t, s, x, y) (respectively Φ(t, s, y, ξ)) is a generating function of the Hamiltonian flow defined by (1.2) and (y, −∂y S) 7→ (x, ∂x S) (respectively (y, ∂y Φ) 7→ (∂ξ Φ, ξ)) is the corresponding canonical map. It is known ([7, 11, 14]) that E(t, s, x, y) is smooth with respect to (x, y) for all t 6= s if V is SQ and for t − s 6∈ πZ if V is SQH, and it may be written in the form: ˜

E(t, s, x, y) = eiS(t,s,x,y) e(t, s, x, y)

(1.5)

˜ s, x, y) = S(t, s, x, y) for all (x, y) if |t − s| is small and for large x2 + y 2 where S(t, otherwise (see also [4] and [12] where (1.5) is obtained for small |t − s| when V satisfies |∂xα V (t, x)| ≤ Cα for all |α| ≥ 2). Our first theorem is concerned with the asymptotic behavior at infinity x2 + y 2 → ∞ of S and e. Theorem 1.1. (1) Let V be SQ and T > 0. Then, for any 0 < ±(t − s) < T, E(t, s, x, y) is C ∞ with respect to (x, y) and may be written in the form E(t, s, x, y) =

e∓inπ/4 ˜ eiS(t,s,x,y) a(t, s, x, y) , (2π|t − s|)n/2

(1.6)

˜ s, x, y) = S(t, s, x, y) for all (x, y) if |t − s| is small, and for (x, y) with where S(t, x2 + y 2 ≥ R2 otherwise, R being a constant depending only on T . As x2 + y 2 → ∞, S and a satisfy (x − y)2 sup ∂xα ∂yβ S(t, s, x, y) − → 0, |α + β| ≥ 2 , (1.7) 2(t − s) 0 0 such that the following statements are satisfied whenever R2 = y 2 +k 2 ≥ C02 and s ∈ R1 : (i) The function `(t) attains a unique minimum `(a) in I1 and 2 101 2 `(a) + R2 |t − a|2 ≤ `(t) ≤ `(a) + R |t − a|2 , t ∈ I1 . (2.7) 5 100 (ii) The set {t ∈ [s − T, s + T ] : `(t) < R2 /10} consists at most of [8T ] + 1-number of disjoint relatively open subintervals for any T . Proof. Statement (1) is [11, Lemma 2.2]. For t-independent V statement (2) is proven in [14]. We prove here statement (2) for t-dependent V (t, x) = 12 x2 +W (t, x) by modifying the argument of [14]. We may assume s = 0 without losing generalities. Write p0 (t) = −y sin t + k cos t , Z t Z t x1 (t) = − sin(t − r)∂x W (r, x(r))dy, p1 (t) = − cos(t − r)∂x W (r, x(r))dy x0 (t) = y cos t + k sin t,

0

0

so that x(t) = x0 (t) + x1 (t) and p(t) = p0 (t) + p1 (t). We have x0 (t)2 + p0 (t)2 = y 2 + k 2 = R2 and by virtue of (2.5) Z T sup (x1 (t)2 + p1 (t)2 ) ≤ T |∂x W (t, x(t))|2 dt = o(R2 ) , (R → ∞) . (2.8) |t|≤T

0

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Let C0 be large enough. Then, (2.8) implies for (y, k) with R2 ≥ C02 that |x(t)2 + p(t)2 − R2 | ≤ R2 /103 for all |t| ≤ T and |x(t) · ∂x W (t, x(t))| ≤ R2 /103 when |x(t)|2 ≤ R2 /5. We apply these estimates to the right hand side of the virial identity: 1 d2 x(t)2 = x(t) ˙ 2 − ∂x V (t, x(t)) · x(t) 2 dt2 = p(t)2 + x(t)2 − (2x(t)2 + x(t) · ∂x W (t, x(t))) . It follows that `(t) = x(t)2 is strictly convex in the interval I1 : 99 2 2 2 2 2 101 2 1 d2 x(t)2 ≥ R ≥ R − R ≥ R , t ∈ I1 100 2 dt2 100 5 5 and, if `(a) is the minimum of `(t) in I1 , we have `(t) ≥ `(a) + (2/5)R2 (t − a)2 for all t ∈ I1 , which is the desired lower bound for `(t). The upper bound is obtained similarly. To prove the second statement, note that the lower bound of (2.7) implies I1 is in fact a bounded interval. The upper bound shows that `(t) ≤ R2 /5 when `(a) < R2 /10 and |t − a| < 1/4. Hence I1 has the length at least 1/4. Since the maximal intervals are disjoint and a connected component of {t ∈ [s−T, s+T ] : `(t) < R2 /10} is contained in a maximal interval I1 , the number of connected components is not larger than [8T ] + 1. Lemma 2.2. Suppose that lim|x|→∞ supt∈Rn |F (t, x)| = 0. Then, uniformly with Rt respect to t, s such that |t − s| ≤ T, s |F (r, x(r, s, y, k))|dr → 0 as R2 = y 2 + k 2 → ∞. Proof. We prove the lemma when V is SQH (and s = 0). The other case is proved in the proof of [11, Lemma 2.4]. For 0 < small, we split [0, T ] into two pieces: I1 = {0 ≤ t ≤ T : |x(t)|2 ≥ 2 R2 /10} and I2 =R{0 ≤ t ≤ T : |x(t)|2 ≤ 2 R2 /10}. It is obvious by the assumption that, as R → ∞, I1 |F (t, x(t, s, y, k))|dt → 0. If R is large enough, Lemma 2.1 implies that I2 consists of at most R [8T ] + 1 intervals and length of each interval does not exceed . It follows that I2 |F (t, x(t, s, y, k))|dt ≤ kF kL∞ ([8T ] + 1) and the lemma follows. Following [4], we set x ˜(t, s, y, k) = x(t, s, y, (t − s)−1 k) and p˜(t, s, y, k) = −1 (t − s)p(t, s, y, (t − s) k) for t 6= s. We have the following estimates: Lemma 2.3. (1) Let V be SQ. Then, as R2 = y 2 + k 2 → ∞, |t − s|−1 k∂yα ∂kβ (∂y x ˜(t) − 1)k → 0, |t − s|−1 k∂yα ∂kβ (∂y p˜(t))k → 0,

|t − s|−1 k∂yα ∂kβ (∂k x ˜(t) − 1)k → 0 , (2.9)

|t − s|−1 k∂yα ∂kβ (∂k p˜(t) − 1)k → 0

uniformly with respect to t, s with |t − s| ≤ T .

(2.10)

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(2) Let V is SQH. Then, as R2 = y 2 + k 2 → ∞, k∂yα ∂kβ (∂y x(t) − cos(t − s))k → 0,

k∂yα ∂kβ (∂k x(t) − sin(t − s))k → 0 ,

(2.11)

k∂yα ∂kβ (∂y p(t) + sin(t − s))k → 0,

k∂yα ∂kβ (∂k p(t) − cos(t − s))k → 0 ,

(2.12)

uniformly with respect to t, s with |t − s| ≤ T . When |t − s| ≤ π/2, the relations (2.11) and (2.12) remain true uniformly with respect (t, s) after the quantities on the left are divided by sin(t − s). Proof. We prove only (2.9) when s = 0. Proof for the others is similar. Differentiating (2.2), we obtain X Z t ∂yα ∂kβ (x(t) − y − tk) = − (t − r) κ,{(αj ,βj )}

0

Y β × Cκ,{(αj ,βj )} ∂xκ V (r, x(r)) ∂yαj ∂k j x(r) dr , (2.13) P P where j |αj | = |α| + 1, j |βj | = |β| and |κ| ≥ 2. It follows inductively by Gronwall’s inequality that |∂yα ∂kβ x(t, s, y, k)| ≤ Cαβ |t − s||β| for |α + β| ≥ 1, and applying this to (2.13), we obtain |∂yα ∂kβ (x(t) − y − tk)| ≤ C|t||β|+1

|α+β|+2 Z t X |κ|=2

|∂xκ V (r, x(r))|dr .

0

The integrals on the right converge to 0 as R → ∞ by Lemma 2.2 and (2.9) follows.

Now we invoke the following implicit function theorem. We refer the readers to [7] for the proof. We write for R ≥ 0, B≥R = {x ∈ Rn : |x| ≥ R} and B≤R = {x ∈ Rn : |x| ≤ R}. Lemma 2.4. Let F be a smooth map of Rn to itself. Assume that there exists a non-singular matrix A such that k∂x F (x) − Ak ≤ (πkA−1 k)−1 for x ∈ B≥R . Then (1) F is diffeomorphic from B≥R to its image, and F (B≥R ) ⊃ B≥ρ , where ρ = (kAk + (πkA−1 k)−1 )R + M,

M = sup |F (x) − Ax| . |x|=R

−1

(2) In addition, |F (x)| ≥ |x|/(2kA k) provided |x| ≥ 6M kA−1 k. (3) If F (x0 ) = 0 for some |x0 | ≥ R, then |F (x)| ≥ (2kA−1 k)−1 |x − x0 | for all x ∈ B≥R . When V is SQ, we apply Lemma 2.4 to the maps ˜ (1) (y, k) = (y, x Γ ˜(t, s, y, k)), t,s

˜ (2) (y, k) = (y, p˜(t, s, y, k)) Γ t,s

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and, when V is SQH, to (1)

Γt,s (y, k) = (y, x(t, s, y, k)),

(2)

Γt,s (y, k) = (y, p(t, s, y, k)) .

We obtain the following lemma. We write B≥R = {(y, k) : y 2 + k 2 ≥ R2 }. Lemma 2.5. (1) Let V be SQ. Then there exists R0 ≥ 0 depending only on T such ˜ (j) that for j = 1, 2 and for every t, s with |t − s| ≤ T, the map Γ t,s is diffeomorphic from B≥R to its image for any R ≥ R0 and (1)

˜ (B≥R ) ⊂ B √ B≥3R ⊂ Γ t,s ≥ 3R/6 ,

(2)

˜ (B≥R ) ⊂ B≥R/2 . B≥4R/3 ⊂ Γ t,s

(2.14)

˜ (j) (B≥R ) is unique in Rn × Rn . Moreover, there exThe inverse image of P ∈ Γ t,s ˜ (j) ists T1 > 0 such that Γ is a diffeomorphism on R2n onto R2n whenever 0 < t,s |t − s| < T1 . (2) Let V be SQH and 0 < < T . Then, there exist C3 > C4 > 0 and R0 ≥ 0 such (j) (j) that for j = 1, 2 and for any (t, s) ∈ I,T , Γt,s is diffeomorphic from B≥R to its image for any R ≥ R0 and (j)

B≥C3 R ⊂ Γt,s (B≥R ) ⊂ B≥C4 R .

(2.15)

(j)

The inverse image of P ∈ Γt,s (B≥R ) is unique in Rn × Rn . Moreover, there exists (j)

T1 > 0 such that Γt,s is a diffeomorphism of R2n onto R2n whenever |t − s| < T1 . Proof. When T1 > 0 is small, the lemma is well known and is a consequence of Lemma 2.3 and the Hadamard global implicit function √ We write X for theorem. ˜ (1) first. Let A = 1 0 . Then 2 ≤ kAk, kA−1 k ≤ (y, k) and prove the lemma for Γ t,s 1 1 √ 3 and, in virtue of (2.9) and (2.3), there exists a constant CR such that CR → 0 as R → ∞ and such that, for X 2 = y 2 + k 2 ≥ R2 ,

˜ (1)

˜ (1) (X) − AX| ≤ CR |X|

∂X Γt,s (X) − A ≤ CR , |Γ t,s whenever 0 < |t − s| ≤ T . Let R0 be such that CR < 10−10 for R ≥ R0 . The first ˜ (1) (B≥10R ) has a unique part of statement (1) follows from Lemma 2.4. Any w ∈ Γ t,s √ ˜ (1) (X)| ≤ 2R for any X ∈ B≤R , inverse in B≥R and |w| ≥ 10 3 R by (2.14). Since |Γ t,s

6

the inverse image of w is unique. Thus the statement on the inverse image holds if ˜ (2) R0 is replaced by 10R0 . The proof for Γ t,s goes entirely similarly by replacing A by A = 10 01 and by changing several constants in the formulae above. (1) 1 0 For proving statement (2) for Γt,s , we set A˜ = cos(t−s) sin(t−s) and apply √ 1 0 ˜ ≤ 2, kA˜−1 k ≤ 2/ sin , and , kAk Lemma 2.4. We have A˜−1 = − cot(t−s) cosec(t−s)

by virtue of (2.12)

∂z Γt,s (X) − A˜ ≤ (πkA˜−1 k)−1 ,

X ∈ B≥R

˜ ≤ for R ≥ R0 , if R0 ≥ 0 is large enough. We also have from (2.5) that |Γt,s (X)−AX| CR |X| for X ∈ B≥R with a constant CR such that CR → 0 as R → ∞. The (1) (2) statement (2) for Γt,s follows from Lemma 2.4. The proof for Γt,s is similar and is omitted.

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3. Action Integral Fix 0 < < T . It follows from Lemma 2.5 that for (t, s) with 0 < t − s ≤ T (1) (or (t, s) ∈ I,T if V is SQH) and for (x, y) with x2 + y 2 ≥ C˜32 R02 , there exists a unique trajectory (x(r), p(r)) = (x(r, s, y, k), p(r, s, y, k)) such that x = x(t, s, y, k), C˜3 = max(3, C3 ). Using this trajectory, we define for x2 + y 2 ≥ C˜32 R02 , Z t S(t, s, x, y) = (p(r, s, y, k)2 /2 − V (r, x(r, s, y, k)))dr , (3.1) s ( if V is SQ , |t − s|n/2 |det(∂k x(t, s, y, k))|−1/2 , (3.2) a0 (t, s, x, y) = −1/2 n/2 |sin(t − s)| |det(∂k x(t, s, y, k))| , if V is SQH , where x = x(t, s, y, k). Note that S and a0 are defined for all (x, y) ∈ R2n if 0 < t − s < T1 . The function S is a generating function of the Hamiltonian flow and satisfies the equations (∂x S)(t, s, x(t, s, y, k), y) = p(t, s, y, k) ,

(3.3)

(∂y S)(t, s, x(t, s, y, k), y) = −k ,

(3.4)

and the Hamilton–Jacobi equation 2 ∂S 1 ∂S + V (t, x) = 0 . + ∂t 2 ∂x For 0 < t − s < T1 , a0 satisfies the corresponding transport equation: n ∂a0 ∂S ∂a0 1 + + ∆x S − a0 = 0 , if V is SQ , ∂t ∂x ∂x 2 t−s 1 n ∂a0 ∂S ∂a0 + + ∆x S − a0 = 0 , if V is SQH . ∂t ∂x ∂x 2 tan(t − s)

(3.5)

(3.6) (3.7)

The following lemma is an obvious corollary of Lemma 2.3. Lemma 3.1. For any α, β, we have as x2 + y 2 → ∞, |(t − s)−1 ∂xα ∂yβ (a0 (t, s, x, y) − 1) | → 0 (1)

uniformly with respect to 0 < t − s ≤ T (or (t, s) ∈ I,T if V is SQH). The following lemma proves estimates (1.7) and (1.10) in Theorem 1.1. Lemma 3.2. For any α, β such that |α + β| ≥ 2, S(t, s, x, y) satisfies, as x2 + y 2 → ∞, 2 α β ∂ ∂ S(t, s, x, y) − (x − y) → 0, if V is SQ , (3.8) x y 2(t − s) 2 2 α β ∂x ∂y S(t, s, x, y) − (x + y ) cos(t − s) − 2xy → 0, if V is SQH , 2 sin(t − s) (1)

uniformly with respect to 0 < t − s ≤ T (or (t, s) ∈ I,T if V is SQH).

(3.9)

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Proof. We first prove (3.8) for s = 0 omitting the variable s. Other variables may as well be omitted if no confusion is feared. Denote by o(1) the functions of (t, y, k) (or (t, x, y)) which converge to 0 as y 2 + k 2 → ∞ (or x2 + y 2 → ∞) uniformly with respect to |t| ≤ T . Differentiate (3.3) by k and obtain (∂x2 S)(t, x ˜(t, y, k), y) = ˜(t, y, k))−1 . We have (∂k x ˜)−1 = 1 + to(1) and ∂k p˜ = 1 + to(1) t−1 ∂k p˜(t, y, k) · (∂k x in virtue of (2.9) and (2.10). It follows that ∂x2 S = t−1 + o(1). Differentiating ˜(t), y) = −t−1 · ∂k x ˜(t, y, k) and ∂x ∂y S = −t−1 + (3.4) by k, we have (∂x ∂y S)(t, x o(1). Differentiating (3.4) by y gives (∂y2 S)(t, x ˜(t), y) = −(∂x ∂y S)(t, x˜(t), y) · ∂y x ˜(t) and we obtain ∂y2 S = (t−1 + o(1))(1 + to(1)) = t−1 + o(1). Estimates for higher derivatives may be proved by further differentiating (3.3) and (3.4) by y and k and by applying Lemma 2.3. This proves (3.8). The estimate (3.9) may be proved similarly by applying the estimates (2.11) and (2.12) in place of (2.9) and (2.10). We omit the details. As remarked above S(t, s, x, y) is defined for all (x, y) ∈ Rn ×Rn if 0 < t−s < T1 , however, for larger t − s, S(t, s, x, y) is defined only outside the ball B≥C˜3 R0 . We show that it is possible to interpolate S(t, s, x, y) inside the ball in such a way that the resulting function is everywhere well approximated by the action integral of the free equation or of the harmonic oscillator respectively. We take χ ∈ C0∞ (Rn × Rn ) such that χ(x, y) = 1 for x2 + y 2 ≤ 1 and χ(x, y) = 0 for x2 + y 2 ≥ 4 and set for R > 2C˜3 R0 , χR (x, y) = χ(x/R, y/R). Define SR (t, s, x, y) = χR (x, y)

(x − y)2 + (1 − χR (x, y))S(t, s, x, y) , 2(t − s)

(3.10)

if V is SQ and, if V is SQH, SR (t, s, x, y) = χR (x, y)

(x2 + y 2 ) cos(t − s) − 2xy 2 sin(t − s)

+ (1 − χR (x, y))S(t, s, x, y) .

(3.11)

Lemma 3.3. Let α, β be such that 2 ≤ |α + β|. Then, if V is SQ, we have (x − y)2 ∂xα ∂yβ SR (t, s, x, y) − → 0 , (R → ∞) , (3.12) 2(t − s) uniformly with respect to (x, y) ∈ R2n and 0 < t − s ≤ T . If V is SQH, we have (x2 + y 2 ) cos(t − s) − 2xy ∂xα ∂yβ SR (t, s, x, y) − → 0 , (R → ∞) , (3.13) 2 sin(t − s) uniformly with respect to (x, y) ∈ R2n and (t, s) ∈ I,T . γ , |γ| ≥ 2 the Proof. We prove (3.12) first. Write X = (x, y) and differentiate by ∂X 2 2 both sides of SR (t, s, x, y) − (x − y) /2(t − s) = (1 − χR (x, y))(S(t, s, x, y) − (x−y) 2(t−s) ).

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By Leibniz rule, (x − y)2 γ ∂X SR (t, s, x, y) − 2(t − s) X γ γ−δ (x − y)2 δ . = ∂X (1 − χR )∂X S(t, s, x, y) − δ 2(t − s) 0≤δ≤γ

Due to (3.8) the summands with |δ| ≥ 2 on the right converge to zero as R → ∞ uniformly with respect to X ∈ R2n and 0 < t − s ≤ T . Note that, if |δ| ≤ 1, then γ−δ |γ − δ| ≥ 1 and ∂X (1 − χR ) is supported by {X : R ≤ |X| ≤ 2R} and that γ−δ (1 − χR )| ≤ CR−|γ−δ| . A simple computation with the identities (3.3) and |∂X (3.4) shows that for x = x(t, s, y, k), 1 (x − y)2 ∂x S(t, s, x, y) − =− (x(t, s, y, k) − y − (t − s)p(t, s, y, k)) 2(t − s) t−s and, by virtue of the integral equation (2.2), we have Z t (x − y)2 1 ∂x S(t, s, x, y) − (r − s)∂x V (r, x(r))dr . =− 2(t − s) t−s s Similarly, we have Z t (x − y)2 −1 ∂y S(t, s, x, y) − = (t − r)∂x V (r, x(r))dr . 2(t − s) t−s s It follows that Z t 2 ∂X S(t, s, x, y) − (x − y) ≤ |∂x V (r, x(r))|dr . 2(t − s) s

(3.14)

Using the integral equation (2.2), we likewise obtain for x = x(t, s, y, k) ) 2 Z t( Z r (x − y)2 1 S(t, s, x, y) − = k− ∂x V (τ, x(τ ))dτ − V (r, x(r)) dr 2(t − s) 2 s s 2 Z t 1 (t − s)k − (t − r)∂x V (r, x(r))dr 2(t − s) s 2 Z t Z t Z r 1 =− V (r, x(r))dr + ∂x V (τ, x(τ ))dτ dr 2 s s s Z t 2 1 − (t − r)∂x V (r, x(r))dr (3.15) 2(t − s) s −

and, hence, by Schwarz inequality, Z 2 S(t, s, x, y) − (x − y) ≤ |V (r, x(r))|dr 2(t − s) Z t 2 t−s + |∂x V (r, x(r))|dr . 2 s

(3.16)

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Now, let x2 +y 2 ≤ 4R2 , x = x(t, s, y, k) = x˜(t, s, y, (t−s)k) and r be in between s and t. Then, the second inclusion of (2.14) implies (t−s)2 k 2 +y 2 ≤ 48R2 and hence (r − s)2 k 2 + y 2 ≤ 48R2 . It follows by the first inclusion of (2.14) that x(r, s, y, k)2 + y 2 = x˜(r, s, y, (r − s)k)2 + y 2 ≤ 48 × 9R2 . Since limR→∞ sup|t|≤T,|x|≤R R−1 |∂x V (t, x)| = limR→∞ sup|t|≤T,|x|≤R R−2 |V (t, x)| = 0, we have 1 lim R→∞ R

Z s

t

1 |∂x V (r, x(r, s, y, k))|dr = 0 and lim 2 R→∞ R

Z

t

|V (r, x(r, s, y, k))|dr = 0 s

uniformly with respect to (y, k) such that x(t, s, y, k)2 + y 2 ≤ 4R2 . Combining this with (3.14) and (3.16), we obtain the lemma for SQ potentials. When V is SQH, we proceed similarly. It suffices to show that, as R → ∞, 2 2 1 ∂X S(t, s, x, y) − (x + y ) cos(t − s) − 2xy → 0 , (3.17) sup R x2 +y2 ≤4R2 2 sin(t − s) 1 (x2 + y 2 ) cos(t − s) − 2xy S(t, s, x, y) − sup (3.18) →0 R2 x2 +y2 ≤4R2 2 sin(t − s) uniformly with respect to (t, s) ∈ I,T . Here we wrote X = (x, y) as previously. We have, for x = x(t, s, y, k), (x2 + y 2 ) cos(t − s) − 2xy ∂x S(t, s, x, y) − 2 sin(t − s) Z t sin(s − r) = (3.19) ∂x W (r, x(r))dr , s sin(t − s) (x2 + y 2 ) cos(t − s) − 2xy ∂y S(t, s, x, y) − 2 sin(t − s) Z t sin(r − t) = (3.20) ∂x W (r, x(r))dr . s sin(t − s) Splitting x(t) = x0 (t) + x1 (t) and x(t) = p0 (t) + p1 (t) as in the proof of Lemma 2.1, we write S(t, s, x, y) − 1 = 2

Z

(x2 + y 2 ) cos(t − s) − 2xy 2 sin(t − s)

t

(p0 (r)2 − x0 (r)2 )dr − s

Z

(x0 (t)2 + y 2 ) cos(t − s) − 2x0 (t)y 2 sin(t − s)

t

(p0 (r)p1 (r) − x0 (r)x1 (r))dr −

+ s

1 + 2

Z

Z

t

(p1 (r) − x1 (r) )dr − 2

s

2

2x0 (t)x1 (t) cos(t − s) − 2x1 (t)y 2 sin(t − s)

t

W (r, x(r))dr + s

x1 (t)2 cos(t − s) . 2 sin(t − s)

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Here the first two terms cancel each other since they are both the action integrals of the trajectory (x0 (r), p0 (r)), s ≤ r ≤ t of the harmonic oscillator. Since x0 (r) = −p˙ 0 (r) and p1 (r) = x˙ 1 (r), we have by integration by parts that Z t Z t d (p0 (r)p1 (r) − x0 (r)x1 (r))dr = (p0 (r)x1 (r))dr = p0 (t)x1 (t) s s dr and the second two terms yield

(p0 (t) sin(t−s)−x0 (t) cos(t−s)+y)x1 (t) sin(t−s)

= 0. Thus,

(x2 + y 2 ) cos(t − s) − 2xy 2 sin(t − s) Z t Z t 1 x1 (t)2 cos(t − s) = (p1 (r)2 − x1 (r)2 )dr − W (r, x(r))dr + . (3.21) 2 s 2 sin(t − s) s

S(t, s, x, y) −

It follows that there exists a constant CT, such that for (t, s) ∈ I,T , 2 2 ∂X S(t, s, x, y) − (x + y ) cos(t − s) − 2xy 2 sin(t − s) Z t 1 ≤ |∂x W (r, x(r))|dr sin s 2 2 S(t, s, x, y) − (x + y ) cos(t − s) − 2xy 2 sin(t − s) Z t ≤ CT, (|∂x W (r, x(r))|2 + |W (r, x(r))|)dr .

(3.22)

(3.23)

s

Then the relations (3.17) and (3.18) follow virtually in the same way as in the case that V is subquadratic. This completes the proof. Remark 3.4. If in addition |∂x V (t, x)| = o(1) (respectively |∂x W (t, x)| = o(1)) as |x| → ∞ uniformly with respect to t ∈ R1 , then, by virtue of Lemma 2.2, the right hand side of (3.14) (respectively (3.22)) converges to zero as y 2 + k 2 → ∞. Since x2 (t, s, x, y) + y 2 → ∞ if and only if k 2 + y 2 → ∞ whenever t 6= s (respectively t − s 6∈ πZ), this implies that (1.7) (respectively (1.10)) is as well satisfied for |α + β| = 1 if V (respectively W ) is sublinear. Likewise, if V (respectively W ) is subconstant, (3.16) (respectively (3.23)) together with Lemma 2.2 implies that (1.7) (respectively (1.10)) holds for all α and β. This proves the statement in Remark 1.2. (2)

It follows also from Lemma 2.5 that for (t, s) with 0 < t − s ≤ T (or (t, s) ∈ I,T if V is SQH) and for (x, y) with y 2 + ξ 2 ≥ C˜32 R02 , there exists a unique trajectory (x(r), p(r)) of (1.2) such that ξ = p(t, s, y, k). Using this trajectory, we define for y 2 + ξ 2 ≥ C˜32 R02 , Z t Φ(t, s, ξ, y) = x(t, s, y, k) · ξ − (p(r, s, y, k)2 /2 − V (r, x(r, s, y, k)))dr . (3.24) s

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Here again we may set R0 = 0 and Φ is defined for all (x, y) if |t − s| < T1 is small. It is well known that Φ satisfies the Hamilton–Jacobi equation ∂Φ ∂Φ 1 2 = ξ + V t, (3.25) ∂t 2 ∂ξ and it is a generating function of the Hamiltonian flow (1.2): (∂ξ Φ)(t, s, y, p(t, s, y, k)) = x(t, s, y, k),

(∂y Φ)(t, s, y, p(t, s, y, k)) = k .

(3.26)

As is well known Φ(t, s, y, ξ) is the Legendre transform of S(t, s, x, y). By virtue of (1) (2) Lemma 3.3, for (t, s) such that 0 < |t − s| < T if V is SQ and (t, s) ∈ I,T ∩ I,T if V is SQH, the map Rn 3 x 7→ ξ = ∂x SR (t, s, x, y) ∈ Rn

(3.27)

is diffeomorphic and it has bounded derivatives with its inverse. We write xc (t, s, y, ξ) for the inverse of (3.27) and define ΦR (t, s, y, ξ) = xc (t, s, y, ξ) · ξ − SR (t, s, xc (t, s, y, ξ), y) .

(3.28)

Then, from the definition we have ∂y ΦR (t, s, y, ξ) = −(∂y SR )(t, s, xc (t, s, y, ξ), y) ,

∂ξ ΦR (t, s, y, ξ) = xc (t, s, y, ξ) . (3.29)

Since relations y 2 + k 2 → ∞, p(t, s, y, k)2 + y 2 → ∞ and x(t, s, y, k)2 + y 2 → ∞ are equivalent to each other by virtue of Lemma 2.5 and S = SR for large x2 + y 2 , we see from (3.3) and (3.27) that, if y 2 + ξ 2 is large enough and ξ = p(t, s, y, k), we have xc (t, s, y, ξ) = x(t, s, y, k) and Z t 1 2 ΦR (t, s, y, ξ) = x(t, s, y, k)ξ − p(r) − V (r, x(r)) dr = Φ(t, s, y, ξ) . (3.30) 2 s (0)

(0)

2

2

) sin(t−s)+2ξy We write Φf (t, s, y, ξ) = yξ + 12 (t − s)ξ 2 and Φos (t, s, y, ξ) = 12 (ξ +y cos(t−s) . Following lemma proves estimates (1.16) and (1.19) of Theorem 1.3.

Lemma 3.5. (1) Let V be SQ and T > 0. Then as y 2 + ξ 2 → ∞, (0)

sup 0 |α| + n, we see that I≥ρ (X) is C ∞ and, as ρ → ∞, Z 1 α |∂X I≥ρ (X)| ≤ CN α dz ≤ CN α µ(ρ)n+|α|−N → 0 . N −|α| |z−zc (X)|≥µ(ρ) |z − zc (X)|

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We then deal with I≤ρ (X). By Taylor’s formula 1 (M (z, X)(z − zc (X)), (z − zc (X))) , 2 Z 1 M (z, X) = (1 − σ)(∂z2 Φ)(σz + (1 − σ)zc (X), X)dσ . Ψ(z, X) =

(4.6) (4.7)

0

By condition (3), we have that β lim sup{k∂zα ∂X (M (z, X) − A)k : |z − zc (X)| ≤ ρ(X)} = 0 .

|X|→∞

(4.8)

We let (r, s) be the signature of A and denote E(r, s) = 1r ⊕ (−1s ). By standard diagonalization, we write as A = t P E(r, s)P,

det P = |det A|1/2 .

Note that the map G : S(n) 7→ S(n) defined by G(S) = t SA−1 S, S(n) being the space of n × n symmetric matrices, is a local diffeomorphism from a neighbourhood of A to another. Hence, we have M (z, X) = t G −1 (M (z, X))A−1 G −1 (M (z, X)) for (z, X) with large enough |X| and |z − zc (X)| ≤ ρ(X). Set G(z, X) = P A−1 G −1 (M (z, X)). It follows that t

G(z, X)E(r, s)G(z, X) = M (z, X)

and, by virtue of (4.8), that, for all k = 0, 1, . . . ,   k  X  β `k (ρ) ≡ sup k∂zα ∂X (G(z, X) − P )k : |X| ≥ ρ, |z − zc (X)| ≤ ρ(X) → 0 ,   |α|+|β|=0

(ρ → ∞) . For k = 0, 1, . . . , we choose functions ρk (ρ) such that the following two conditions are met: (a) 2ρk (ρ) ≤ inf{ρ(X) : |X| ≥ ρ} , (b) ρk (ρ) → ∞ and ρk (ρ)`k (ρ) → 0 ,

(ρ → ∞) .

Choose and fix k ≥ 0 arbitrarily and set µ(ρ) = ρk+n+4 (ρ) and Ω(ρ) = {(z, X) ∈ Rn × Rm : |X| ≥ ρ, |z − zc (X)| ≤ 2µ(ρ)}. Define z˜(z, X) = G(z, X)(z − zc (X)). Then k∂z z˜(z, X) − P k ≤ k∂z G(z, X)(z − zc (X))k + kG(z, X) − P k and, by virtue of condition (b), we see that there exists ρ∗ such that for |X| ≥ ρ∗ the map z → z˜(z, X) = G(z, X)(z − zc (X))

(4.9)

is a diffeomorphism from {z : |z − zc (X)| < 2µ(ρ)} to its image. Hereafter, we ˜ assume ρ > ρ∗ and denote by z(˜ z , X) the inverse map of (4.9). Ω(ρ) is the image

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of Ω(ρ) by the map (z, X) → (˜ z , X). A direct computation and the choice of µ(ρ) imply X α β sup{|∂X ∂z ∂z z˜(z, X)| : (z, X) ∈ Ω(ρ)} → 0 , (ρ → ∞) , 1≤|α|+|β|≤k+n+3

X

α β ˜ sup{|∂X ∂z˜ z(˜ z , X)| : (˜ z , X) ∈ Ω(ρ)} ≡ Ck < ∞ .

(4.10)

1≤|α|+|β|≤k+n+3

We change the variable z to z˜ in the integral I≤ρ (X) and write it in the following form: Z I≤ρ (X) = ei(E z˜,˜z)/2 f (˜ z , X)d˜ z , E = E(r, s) , (4.11) Rn

˜ where f (˜ z , X) = χρ (z(˜ z , X))|det ∂∂zz˜ (z(˜ z , X)))|−1 a(z(˜ z , X), X) is supported by Ω(ρ). Condition (4) on a and (4.10) produce X α β ˜ sup{|∂X ∂z˜ f (˜ z , X)| : (˜ z , X) ∈ Ω(ρ)} → 0 , (ρ → ∞) . (4.12) 1≤|α|+|β|≤k+n+3

R1 z , X)dσ into (4.11) and write Insert f (˜ z , X) = f (0, X) + z˜ · 0 (∂z˜f )(σ˜ Z I≤ρ (X) = ei(E z˜,˜z)/2 f (0, X)d˜ z Rn

Z −

(iE

−1

Z ∂z˜e

i(E z˜,˜ z )/2

)·

Rn

1

(∂z˜f )(σ˜ z , X)dσ d˜ z 0

(1)

(2)

≡ I≤ρ (X) + I≤ρ (X) . By integration by parts, we have, with vector notation for hE −1 ∂z˜, ∂z˜i, Z 1 Z (2) i(E z˜,˜ z )/2 −1 e σ(ihE ∂z˜, ∂z˜if )(X, σ˜ z )dσ d˜ z. I≤ρ (X) = Rn

0

Using the identity ei(E z˜,˜z)/2 = parts (n + 1)-times and obtain

1−i˜ z ·E −1 ∂z˜ 1+|˜ z |2

(

Z (2) I≤ρ (X)

n+1

ei(E z˜,˜z)/2

= Rn

Z ×

1

σ(ihE

−1

ei(E z˜,˜z)/2 , we apply integration by

1 − i˜ z · E −1 ∂z˜ 1 + |˜ z |2

† )n+1

∂z˜, ∂z˜if )(X, σ˜ z )dσ d˜ z,

0

where † stands for the real transpose. (4.12) guarantees that for |α| ≤ k, as ρ → ∞, ( † )n+1 Z 1 −1 ∂ 1 − i˜ z · E z˜ α (1 + |˜ z |)n+1 ∂X σ(ihE −1 ∂z˜, ∂z˜if )(X, σ˜ z )dσ → 0 2 1 + |˜ z| 0

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uniformly on {(X, z˜) : |X| ≥ ρ, z˜ ∈ Rn }. It follows by Lebesgue’s dominated convergence theorem that for |α| ≤ k α (2) sup |∂X Iρ (X)| → 0 ,

(ρ → ∞) .

|X|≥ρ

(4.13)

The point z˜ = 0 corresponds to z = zc (X) and an explicit computation yields: I≤ρ (X) = (2π)n/2 eiπσ(A)/4 a(zc (X), X)|det G(zc (X), X)|−1 . (1)

(4.14)

α Since zc (X) has bounded derivatives, we have for |α| ≤ k that ∂X (a(zc (X), X) − α a0 ) → 0 and k∂X {det G(zc (X), X) − det P }k → 0 as |X| → ∞. Thus, α |∂X (I≤ρ (X) − (2π)n/2 eiπσ(A)/4 a0 | det A|−1/2 )| → 0 , (1)

(|X| → ∞) .

(4.15)

Combining (4.5) with (4.13) and (4.15) and writing e = (2π)n/2 eiπσ(A)/4 a0 × |det A|−1/2 , we obtain for any |α| ≤ k that (1)

α α (I(X) − e0 )| ≤ sup |∂X (I≤ρ (X) − e0 )| sup |∂X

|X|≥ρ

|X|≥ρ

(2)

α α + sup |∂X I≤ρ (X)| + sup |∂X I≥ρ (X)| → 0 |X|≥ρ

X∈Rn

as ρ → ∞. Since k is arbitrary, the proof of the lemma is completed. 5. Proof of Theorems for Small t − s In this section we prove Theorem 1.1 and Theorem 1.3 for the case that V is SQ and 0 < t − s is sufficiently small. The proof for the case that V is SQH is similar and is omitted here. We first prove Theorem 1.1 and we briefly recall Fujiwara’s argument [4] for the construction of the FDS referring to [4] for the details. As was remarked earlier, S(t, s, x, y) and a0 (t, s, x, y) are defined globally on Rn ×Rn when 0 < t − s ≤ T1 . Using these functions, we define Z e−inπ/4 eiS(t,s,x,y) a0 (t, s, x, y)u(y)dy . (5.1) F0 (t, s)u(x) = (2π(t − s))n/2 Then, the family of operators F0 (t, s) is strongly differentiable in S(Rn ), strongly continuous in L2 (Rn ) and s−limt→s F0 (t, s) = I in L2 (Rn ). Moreover, as S satisfies the Hamilton–Jacobi equation (3.5) and a0 corresponding transport equation (3.6), the operator G(t, s) defined by G(t, s)φ = (i∂t + (1/2)∆ − V (t, x)) F0 (t, s)φ is again a family of oscillatory integrals Z e−inπ/4 G(t, s)φ(x) = eiS(t,s,x,y) ∆x a0 (t, s, x, y)φ(y)dy , 2(2π(t − s))n/2

(5.2)

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and G(t, s) is also strongly differentiable with respect to t, s in S(Rn ) and is strongly continuous in L2 (Rn ). It follows that F0 (t, s) may be solved by Duhamel formula: Z t F0 (t, s) = U (t, s) − i U (t, r)G(r, s)dr , s

and hence U (t, s) can be written in terms of F0 (t, s) and G(t, s) in the form Z t ∞ X L U (t, s) = (ΦG F0 )(t, s), (ΦG K)(t, s) = i K(t, r)G(r, s)dr . (5.3) s

L=0

It is obvious that the series (5.3) converges in the operator norm of L2 (Rn ). The proof that U (t, s) is the integral operator of the form (1.6) is based upon the following estimate of Kumanogo–Taniguchi type for the integral kernel of the product of the oscillatory integral operators of the form (5.1). For the proof we refer to [5]. Lemma 5.1. Let κm , m = 2, 3, . . . and AK , K = 0, 1, . . . be given constants. Suppose that real-valued smooth functions Sj (xj , xj−1 ) on R2n , j = 1, . . . , L and a smooth function a(x0 , . . . , xL ) on R(L+1)n satisfy the following conditions with positive constants t1 , . . . , tL : (a) For any m ≥ 2, X (xj − xj−1 )2 ∂xα ∂xβ S (x , x ) − j j j−1 ≤ tj κm , j j−1 2tj

j = 1, . . . , L .

|α+β|=m

(b) For any K = 0, 1, . . . and αj with |αj | ≤ K, j = 0, . . . , L, |∂xαLL · · · ∂xα00 a(xL , . . . , x0 )| ≤ AK . Then, there exists a constant δ > 0 such that the following statements are satisfied whenever t1 , . . . , tL satisfy TL = t1 + · · · + tL ≤ δ. We write S(xL , . . . , x0 ) = SL (xL , xL−1 ) + · · · + S1 (x1 , x0 ). (1) For any (xL , x0 ) ∈ R2n , there exists a unique set of points x∗L−1 , . . . , x∗1 ∈ Rn such that ∂xL−1 S(xL , x∗L−1 , . . . , x∗1 , x0 ) = · · · = ∂x1 S(xL , x∗L−1 , . . . , x∗1 , x0 ) = 0 . (2) The oscillatory integral with parameter ν ≥ 1 n/2 Z L L−1 Y Y −iν iνS(xL ,...,x0 ) I(xL , x0 ) = e a(xL , . . . , x0 ) dxj , (5.4) 2πtj R(L−1)n j=1 j=1 can be written in the form I(xL , x0 ) =

−iν 2πTL

n/2

ˆ

eiν S(xL ,x0 ) b(xL , x0 ) ,

ˆ L , x0 ) = SL (xL , x∗ ) + · · · + S1 (x∗ , x0 ) and b(xL , x0 ) where S(x 1 L−1 b(xL , x0 ; t1 , . . . , tL , ν) is a smooth bounded function of of (xL , x0 ).

=

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(3) For any m = 0, 1, . . . , there exist `(m) and K(m) and the constant Cm > 0 which depends only on constants {κ2 , . . . , κ`(m) } and K(m) such that for any αL and α0 such that |αL |, |α0 | ≤ m, L |∂xαLL ∂xα00 b(xL , x0 ; t1 , . . . , tL , ν)| ≤ Cm AK(m) .

The operator (ΦL G F )(t, s) can be written in the form Z t Z t Z t drL drL−1 · · · dr1 F0 (t, rL )G(rL , rL−1 ) · · · G(r2 , r1 )G(r1 , s) , s

rL

r2

and we apply Lemma 5.1 to the product IL = F0 (t, rL )G(rL , rL−1 ) · · · G(r2 , r1 )G(r1 , s) . Set Sj (x, y) = S(rj , rj−1 , x, y) for j = 1, . . . , L + 1, t = rL+1 and s = r0 , and L Y

a(xL+1 , . . . , x0 ) = 2−L a0 (rL+1 , rL , xL+1 , xL )

∆xj a0 (rj , rj−1 , xj , xj−1 ) .

j=1

The integral kernel of the operator IL is given by (5.4) with L being replaced by L + 1, ν = 1. It is easy to see that the conditions of Lemma 5.1 are satisfied with tj = rj − rj−1 . We let T (V ) < T1 be the corresponding δ of Lemma 5.1. By virtue of (3.3) and (3.4) and the uniqueness of the trajectory connecting x(t) = x and ˆ y) = S(t, s, x, y). Hence, IL has the kernel x(s) = y, we have S(x, e−inπ/4 eiS(t,s,x,y) bL (t, rL . . . , r1 , s, x, y) (2π(t − s))n/2 and bL satisfies the estimate |∂xα ∂yβ bL (t, rL , . . . , r1 , s, x, y)| ≤ CkL , for |α + β| ≤ k, where Ck is independent of L and t, rL , . . . , r1 , s, 0 < t − s ≤ T (V ). It is then obvious that (ΦL G F0 )(t, s) is the oscillatory integral operator of the form Z e−inπ/4 (ΦL F )(t, s)u(x) = eiS(t,s,x,y) aL (t, s, x, y)u(y)dy G 0 (2π(t − s))n/2 and that

Z aL (t, s, x, y) =

Z

t

t

drL−1 · · ·

drL s

Z

t

rL

dr1 bL (t, rL , . . . , r1 , s) , r2

satisfies |∂xα ∂yβ aL (t, s, x, y)| ≤ CkL |t − s|L /L!. It follows that a(t, s, x, y) ≡ P∞ L=0 aL (t, s, x, y) converges uniformly with derivatives with respect to (x, y) and that U (t, s) is an operator of the form (1.6). The estimate (1.7) has already been shown in (3.8) and we have only to prove (1.8). Due to the preceding argument, it suffices to show that, for any α, β, ∂xα ∂yβ bL (t, rL , . . . , r1 , s, x, y) converges to 0 as x2 +y 2 → ∞ for every fixed t, rL , . . . , r1 , s. This follows inductively from the following lemma because ∆x a0 (t, s, x, y) converges to 0 with derivatives as x2 + y 2 → ∞.

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Lemma 5.2. Let s < r < t, t − s < T (V ) and let K and L be the operators defined by Z Z iS(t,r,x,y) Ku(x) = e c(x, y)u(y)dy, Lu(x) = eiS(r,s,x,y) d(x, y)u(y)dy . Suppose that, for any α and β, ∂xα ∂yβ c(x, y) is bounded and ∂xα ∂yβ d(x, y) converges to 0 as x2 + y 2 → ∞. Then, the product KL is an integral operator of the form Z KLu(x) = eiS(t,s,x,y) e(x, y)u(y)dy , (5.5) where e ∈ C ∞ (R2n ) satisfies ∂xα ∂yβ e(x, y) → 0 as x2 + y 2 → ∞ for any α and β. P α β Proof. For m ∈ N, we write kakm = |α+β|≤m supx,y |∂x ∂y a(x, y)|. Then Lemma 5.1 implies that the product K(t, r)L(r, s) may be written in the form (5.5) and the amplitude e(x, y) satisfies the estimate kekm ≤ C(m)kckK(m) kdkK(m) where K(m) does not depend on c and d. Take χ ∈ C0∞ (Rn × Rn ) such that χ(x, y) = 1 for x2 + y 2 ≤ 1 and χ(x, y) = 0 for x2 + y 2 ≥ 4 as previously and set χR (x, y) = χ(x/R, y/R) for R > 0. We decompose as L = L≥R + L≤R : Z L≤R u(x) = eiS(r,s,x,y) (χR d)(x, y)u(y)dy , Z L≥R u(x) =

eiS(r,s,x,y) ((1 − χR )d)(x, y)u(y)dy .

Since L≤R has a compactly supported smooth integral kernel and K is a continuous operator in S(Rn ), KL≤R is a continuous operator from S 0 (Rn ) to S(Rn ). Hence, KL≤R has an S(R2n ) kernel by Schwartz’ kernel theorem and we have for any |α + β| ≤ m, limx2 +y2 →∞ |∂xα ∂yβ e(x, y)| ≤ C(m)kckK(m) k(1 − χR )dkK(m) for any R > 0. The right hand side converges to 0 as R → ∞ by assumption, which completes the proof of the lemma. For proving Theorem 1.3 for small |t − s|, we write by Fourier inversion formula Z Z 1 ixξ −izξ U (t, s)φ(x) = e e E(t, s, z, y)dz φ(y)dydξ (2π)n and apply Lemma 4.1 to the integral inside the braces: Z e−inπ/4 e−izξ+iS(t,s,z,y) a(t, s, z, y)dz . (2π(t − s))n/2 We set X = (y, ξ), a(z, X) = a(t, s, z, y) and Φ(z, X) = S(t, s, z, y) − zξ. By virtue of the small t − s part of Theorem 1.1, it is obvious that conditions (1) and (2) of Lemma 4.1 are satisfied. The point of stationary phase zc (y, ξ) is given by zc (y, ξ) = x(t, s, y, k) when ξ = p(t, s, y, k) by virtue of (3.3) and Lemma 2.3 implies that zc (y, ξ)2 + y 2 ≥ (L5 ρ)2 if ξ 2 + y 2 ≥ ρ2 and ρ is large, L5 being a constant.

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Hence, conditions (3) and (4) are satisfied with A = (t − s)−1 and a0 = 1. Thus, Theorem 1.3 for small |t − s| follows by applying Lemma 4.1 and remembering the Definition (3.24) of Φ(t, s, y, ξ). 6. Proof of Theorems for Large t − s In this section, we fix an arbitrarily large T > 0 and an arbitrarily small > 0 and first prove Theorem 1.1 for (t, s) such that 0 < t − s < T if V is SQ and such that (1) (t, s) ∈ I,T if V is SQH (we will abuse notation and write simply (t, s) ∈ I,T if (t, s) satisfies these conditions). Since Theorem 1.1 holds when 0 < t − s < T (V ) is small as has just been proved and the propagator U (t, s) enjoys the semi-group property U (t, s) = U (t, r)U (r, s), it suffices to show that if E(t, r, x, y) and E(r, s, x, y) satisfy the properties of Theorem 1.1, then so does E(t, s, x, y) whenever (t, s), (r, s), (t, r) ∈ I,T (we may of course assume 2 < T (V )). Let ER (t, r, x, y), ER (r, s, x, y) be the functions obtained from E(t, r) and E(r, s) by replacing S(t, r, x, y) and S(r, s, x, y) by SR (t, r, x, y) and SR (r, s, x, y), respectively, where SR (t, s, x, y) etc. are those defined by (3.10) and (3.11). Let UR (t, r) and UR (r, s) be the integral operators with the kernels ER (t, r) and ER (t, r) respectively. E(t, r)−ER (t, r) and E(r, s)−ER (r, s) are compactly supported smooth functions and U (t, r) and UR (r, s) are continuous both in S(Rn ) and in S 0 (Rn ). It follows that U (t, r) − UR (t, r)UR (r, s) = U (t, r)(U (r, s) − UR (r, s)) + (U (t, r) − UR (t, r))UR (r, s) is continuous from S 0 (Rn ) to S(Rn ) and, hence, is an integral operator with ker˜R (t, s, x, y) of the prodnel in S(R2n ). Thus, E(t, s, x, y) differs from the kernel E uct UR (t, r)UR (r, s) by an element in S(R2n ) and we have only to show that ˜R (t, s, x, y) satisfies the properties of Theorem 1.1 for sufficiently large R. We E begin with the following lemma. Lemma 6.1. Let s < r < t be fixed such that (t, s), (r, s), (t, r) ∈ I,T and let ΦR (x, z, y) = SR (t, r, x, z) + SR (r, s, z, y) . Then, there exists R1 ≥ C˜3 R0 such that the following statements are satisfied for ΦR , R ≥ R1 . (1) For some positive constant δ, | det ∂z2 ΦR (x, z, y)| > δ for all x, z, y ∈ Rn . (2) For any X = (x, y) ∈ R2n , there exists a unique z(X) such that ∂z ΦR (x, z, y) = 0. The function z(X) is smooth and has bounded derivatives. (3) There exist constants L1 and L2 such that x2 +z(X)2 ≥ L21 ρ2 and y 2 +z(X)2 ≥ −1 L22 ρ2 if x2 + y 2 ≥ ρ2 and ρ ≥ 2 max(L−1 1 , L2 , 1)R. −1 (4) If x2 + y 2 ≥ ρ2 and ρ ≥ 2 max(L−1 1 , L2 , 1)R, then ΦR (x, z(X), y) = S(t, s, x, y). Proof. Suppose first V is SQH. Write θ = r − s and τ = t − r. Then, by virtue of (3.13), we have k∂z2 ΦR − cot θ − cot τ k → 0 as R → ∞ uniformly with respect to

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(x, y, z). Since t − s 6∈ πZ, we have cot θ + cot τ 6= 0 and statement (1) follows for large enough R1 ≥ 2C˜3 R0 . By virtue of (3.13), the derivatives of ∂z ΦR are bounded. Hence, statement (1) and the Hadamard implicit function theorem (cf. [5]) imply that the map z → ∂z ΦR is a global diffeomorphism of Rn . In particular, the equation ∂z ΦR (x, z, y) = 0 for z has a unique solution z(X) and z(X) has bounded derivatives. Recall that s < r < t are fixed and t − s, t − r, r − s 6∈ πZ. It follows from the first inclusion relation of (2.15) that, for any X = (x, y) with X 2 ≥ ρ2 ≥ R02 there exists a unique k ∈ Rn such that x = x(t, s, y, k) and y 2 + k 2 ≥ (C3−1 ρ)2 . Write z˜ = x(r, s, y, k) and k˜ = p(r, s, y, k). It follows from the second relation of (2.15) that z˜2 + y 2 = x(r, s, y, k)2 + y 2 ≥ (L2 ρ)2 ,

L2 = C3−1 C4

(6.1)

˜ and, by virtue of (2.1), z˜2 + k˜2 ≥ CT−2 (y 2 +k 2 ) ≥ (C3−1 CT ρ)2 . Since x = x(t, r, z˜, k), again by the second relation of (2.15), we obtain x2 + z˜2 ≥ (L1 ρ)2 ,

L1 = C4 C3−1 CT .

(6.2)

−1 Thus, if ρ > 2 max(L−1 ˜2 + y 2 ≥ 4R2 and x2 + z˜2 ≥ 4R2 and 1 , L2 , 1)R, then z

∂z SR (t, r, x, z˜) + ∂z SR (r, s, z˜, y) = ∂z S(t, r, x, z˜) + ∂z S(r, s, z˜, y) = 0 because of the relations (3.3) and (3.4). This implies z(X) = z˜ and (6.1) and (6.2) imply statement (3). Statement (4) is obvious and the lemma is proved for this case. When V is SQ, the proof goes entirely similarly. By virtue of (3.12), we have, as R → ∞, k∂z2 ΦR − θ−1 − τ −1 k → 0 uniformly with respect (x, y, z) and the statement (1) follows. By virtue of the same relation (3.12), the derivatives of ∂z ΦR are bounded. Hence, the statement (1) and the Hadamard implicit function theorem imply that the map z → ∂z ΦR is a diffeomorphism of Rn . Thus, ∂z ΦR (x, z, y) = 0 has a unique solution z(X) and z(X) has bounded derivatives. If ρ ≥ C˜3 R0 and x2 + y 2 ≥ ρ2 , there exists a unique k such that x = x(t, s, y, k) 2 ˜ (1) and y 2 + (t − s)2 k 2 ≥ ρ9 by virtue of the first inclusion relation of (2.14) for Γ t,s . 2 2 2 ˜ Write z˜ = x(r, s, y, k) and k = p(r, s, y, k). We have clearly y + (r − s) k ≥ n o2 r−s ρ ˜ (1) and the second inclusion of (2.14) for Γ r,s implies t−s

3

( √ ! )2 3r−s ρ z˜ + y = x(r, s, y, k) + y ≥ ≡ (L2 ρ)2 . 6 t−s 3 2

2

2

We clearly have y 2 + k 2 ≥

2

ρ 3 max(1,(t−s)2 )

2 and the estimate (2.1) implies

z˜2 + (t − r)2 k˜2 ≥ min(1, (t − r)2 )(˜ z 2 + k˜2 ) ≥ min(1, (t − r)2 )CT−2 (y 2 + k 2 ) ≥

min(1, t − r) ρ max(1, t − s) 3CT

2 .

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˜ it then follows, again by virtue of the second inclusion of Since x = x(t, r, z˜, k), (1) ˜ (2.14) for Γt,r that !2 √ 2 3 min(1, t − r) ρ 2 2 ≡ (L1 ρ)2 . x + z˜ ≥ 6 max(1, t − s) 3CT −1 Hence, for ρ ≥ max(L−1 1 , L2 , 1)2R, we have

∂z SR (t, r, x, z˜) + ∂z SR (r, s, z˜, y) = ∂z S(t, r, x, z˜) + ∂z S(r, s, z˜, y) = 0 and, if we set z(X) = z˜, statements (3) and (4) follow as previously. The proof of the lemma is completed. Lemma 6.2. Let (t, s), (t, r) and (r, s) ∈ I,T and let R1 be as in Lemma 6.1. Let F (t, r) and G(r, s) be the oscillatory integral operators defined for R ≥ R1 respectively by Z F (t, r)u(x) = eiSR (t,r,x,y) c(x, y)u(y)dy , Z (6.3) G(r, s)u(x) = eiSR (r,s,x,y) d(x, y)u(y)dy . Suppose that c and d are smooth and satisfy for some constants c0 and d0 , and for any multi-indices α and β, |∂xα ∂yβ (c(x, y) − c0 )| → 0 and |∂xα ∂yβ (d(x, y) − d0 )| → 0 as x2 + y 2 → ∞. Then, the product F (t, r)G(r, s) may be expressed in the form Z F (t, r)G(r, s)u(x) = eiSR (t,s,x,y) e(x, y)u(y)dy . Here e(x, y) is smooth and satisfies for any α and β as x2 + y 2 → ∞, |∂xα ∂yβ (e(x, y) − e0 )| → 0 ,

(6.4)

where the constant e0 is given by ( einπ/4 (θτ /(θ + τ ))n/2 , if V is subquadratic , n/2 e0 = (2π) c0 d0 × ±inπ/4 (6.5) e (sin θ sin τ / sin(θ + τ ))n/2 , if V = (1/2)x2 + W , with θ = t − r, τ = r − s and ± is the sign of sin θ sin τ / sin(θ + τ ). Proof. We apply Lemma 4.1. Write X = (x, y) and set Φ(z, X) = SR (t, r, x, z) + SR (r, s, z, y) and a(z, X) = c(x, z)d(z, y). Then, the integral kernel of F (t, r)G(r, s) may be written Z K(X) = eiΦ(z,X) a(z, X)dz . Statement (1) of Lemma 6.1 implies condition (1) of Lemma 4.1 and condition (2) obviously holds by the assumption. We denote the critical point of the function z → Φ(z, X) by zc (X). Then, we have x2 + zc (X)2 ≥ L21 ρ2 and y 2 + zc (X)2 ≥ L2 ρ2

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for |X| ≥ ρ by statement (3) of Lemma 6.1. It follows that, if we set ρ(X) = 1 10 min(L1 , L2 )|X| and  1+ 1 if V is SQ , A = A(t, r, s) = θ τ (6.6)  cot θ + cot τ, if V is SQH , with θ = t − r and τ = r − s, then Lemma 3.2 imply condition (3) of Lemma 4.1. Notice that A is non-singular for t, r, s under consideration. Condition (4) is satisfied by virtue of statement (3) of Lemma 6.1 and the assumption on c and d. Since A is positive or negative definite and Φ(zc (X), X) = S(t, s, x, y) for large |X| by Lemma 6.1, Lemma 6.2 follows from Lemma 4.1. Proof of Theorem 1.1. Theorem 1.1 readily follows by applying the preceding Lemma 6.2 to F (t, r)G(r, s) = UR (t, r)UR (r, s). Indeed if E(t, r, x, y) and E(r, s, x, y) satisfy the properties in Theorem 1.1, then Lemma 6.2 implies that the integral kernel of UR (t, r)UR (r, s) is given by eiSR (t,s,x,y) e(x, y) and e(x, y) satisfies (6.4) with e0 being given, if V is SQ, by n/2 e−inπ/4 e−inπ/4 e−inπ/4 inπ/4 θτ e0 = (2π)n/2 e = , (6.7) θ+τ (2πθ)n/2 (2πτ )n/2 (2π(θ + τ ))n/2 and, if V is SQH and m1 π < θ < (m1 + 1)π and m2 π < τ < (m2 + 1)π, by n/2 −m1 −inπ/4 e i−m2 e−inπ/4 ±inπ/4 | sin θ|| sin τ | n/2 i e0 = (2π) e | sin(θ + τ )| (2π| sin θ|)n/2 (2π| sin τ |)n/2 =

i−(m1 +m2 ) e−inπ/2±inπ/4 , (2π| sin(θ + τ )|)n/2

(6.8)

where θ = t − r and τ = r − s and ± is the sign of sin θ sin τ / sin(θ + τ ). It is easy to see that sin θ sin τ / sin(θ+τ ) > 0 if and only if (m1 +m2 )π < θ+τ < (m1 +m2 +1)π and it is negative if and only (m1 + m2 + 1)π < θ + τ < (m1 + m2 + 2)π. Hence the right hand side of (6.8) is equal to   i−(m1 +m2 ) e−inπ/4    (2π|sin(θ + τ )|)n/2 , if (m1 + m2 )π < θ + τ < (m1 + m2 + 1)π ,    i−(m1 +m2 +1) e−inπ/4   , if (m1 + m2 + 1)π < θ + τ < (m1 + m2 + 2)π .  (2π|sin(θ + τ )|)n/2 Since SR (t, s, x, y) and S(t, s, x, y) differ only on the compact set x2 + y 2 ≤ 4R2 , this implies that E(t, s, x, y) satisfies the properties of Theorem 1.1 and the proof of Theorem 1.1 is completed. (1)

(2)

Proof of Theorem 1.3. Statement (1) and statement (2) for (t, s) ∈ I,T ∩ I,T may be proved in virtually the same way as in the proof of Theorem 1.1 by applying Lemma 4.1, and, we focus our attention only on the case that V is SQH and t−s is at or near resonant time (m + 1)π. We choose s < r < t in such a way that 0 < t − r
0 such that ξ 2 + x(r, s, y, k)2 ≥ (L3 ρ)2 ,

x(r, s, y, k)2 + y 2 ≥ (L4 ρ)2

(6.10)

whenever ξ 2 + y 2 ≥ ρ2 and ξ = p(t, s, y, k). Then, we have by (3.3) and (3.26) that ∂z Φ(t, r, x(r, s, y, k), ξ) = p(r, s, y, k) = ∂z S(r, s, x(r, s, y, k), y) . Since the point of stationary phase zc (y, ξ), viz. the solution of ∂z Φ(t, r, z, ξ) = ∂z SR (r, s, z, y) is unique and SR (r, s, z, y) = S(r, s, z, y) for large z 2 +y 2 , we see that zc (y, ξ) = x(r, s, y, k) and Lemma 6.10 together with Lemma 3.2 and Lemma 3.5 implies condition (3) of Lemma 4.1. Condition (4) is obviously satisfied by (6.10) since a(r, s, z, y) − 1 and b(t, r, y, ξ) − 1 both converges to 0 with derivatives as spatial variables tend to infinity. Thus, Lemma 4.1 implies that the integral in the parenthesis in (6.9) represents a smooth function and it can be written for large y 2 + ξ 2 in the form n/2 cos(t − r)|sin(r − s)| (2π)n/2 e−inπ/4 e−Φ(t,s,y,ξ) b(t, s, y, ξ) |cos(t − s)| where b(t, s, y, ξ) → 1 with derivatives as y 2 + ξ 2 → ∞. This completes the proof of Theorem 1.3.

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References [1] K. Asada and D. Fujiwara, “On some oscillatory integral transformation in L2 (Rn )”, Japan. J. Math. 4 (1978) 299–361. [2] W. Craig, T. Kappeler and W. Strauss, “Microlocal dispersive smoothing for the Schr¨ odinger equation”, Comm. Pure Appl. Math. 48 (1995). [3] D. Fujiwara, “A construction of the fundamental solution for the Schr¨ odinger equation”, J. d’Analyse Math. 35 (1979) 41–96. [4] D. Fujiwara, “Remarks on convergence of the Feynman path integrals”, Duke Math. J. 47 (1980) 559–600. [5] D. Fujiwara, Mathematical Theory of Feynman Path Integral, Time Slicing Approximation, Springer-Verlag, Tokyo, 1999 (in Japanese). [6] H. Kitada and H. Kumano-go, “A family of Fourier Integral operators and the fundamental solution for a Schr¨ odinger equation”, Osaka J. Math. 18 (1981) 291–360. [7] L. Kapitanski, I. Rodnianski and K. Yajima, “On the fundamental solution of a perturbed harmonic oscillator”, Topol. Methods Nonlinear Anal. 9 (1997) 77–106. [8] B. Simon, “Schr¨ odinger semigroups”, Bull. Amer. Math. Soc. 7 (1982) 447–526. [9] F. Treves, “Parametrices for a class of Schr¨ odinger equations”, Comm. Pure Appl. Math. 48 (1995) 13–78. [10] A. Weinstein, “A symbol class for some Schr¨ odinger equations on Rn ”, Amer. J. Math. 107 (1985) 1–21. [11] K. Yajima, “Smoothness and non-smoothness of the fundamental solution of time dependent Schr¨ odinger equations”, Commun. Math. Phys. 181 (1996) 605–629. [12] , “Schr¨ odinger evolution equation with magnetic fields”, J. d’Analyse Math. 56 (1991) 29–76. [13] , “Boundedness and continuity of the fundamental solution of time dependent Schr¨ odinger equation with singular potentials”, Tohoku Math. J. 50 (1998) 577–595. [14] , “On fundamental solution of time dependent Schr¨ odinger equations”, Contemp. Math. 217 (1998) 49–68. [15] S. Zelditch, “Reconstruction of singularities for solutions of Schr¨ odinger equations”, Commun. Math. Phys. 90 (1983) 1–26.

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BORN OPPENHEIMER APPROXIMATION FOR THE BROWN RAVENHALL EQUATION

VANIA SORDONI∗ Dipartimento di Matematica Universit` a di Bologna 40127 Bologna, Italy E-mail: [email protected]

Received 14 February 2000

In this paper we study the pseudo-relativistic Hamiltonian proposed by Brown and Ravenhall in the semiclassical limit when the mass ratio h2 of electronic to nuclear mass tends to zero. We show that the relativistic contribution of the nuclei on WKB-type expansions of the first energy levels are of order o(h2 ), as h → 0.

1. Introduction In this paper, we discuss the Born–Oppenheimer approximation for a multiparticles system (such as an atom or a molecule) composed by two kind of particles, some of them (that we call of type N as nuclei) much heavier than the other ones (that we call of type E as electrons). In the non relativistic case, the system is described by the multi-particles Schr¨ odinger operator and the Born– Oppenheimer expansion for polyatomic molecules has been widely investigated (see [2, 6, 10, 12, 13, 17, 21, 24, 26]). On the other hand, relativistic kinematic of spin 1/2-particles is usually described by the Dirac operator. But this operator is unbounded from below and therefore it is problematic to treat the multiparticles case. In particular, the multiparticles Dirac operator has no eigenfunctions corresponding to bound states. To overcome this difficulty, Brown–Ravenhall in [8] and Bethe– Salpeter in [5] proposed a new equation that essentially consists into summing up the projections of the one-particle Dirac equations to the subspaces of positive energy levels. More precisely, according to [4, 9, 11, 27, 28, 29, 30], we consider  B = Λ+ 

n+p+1 X

 Hj (Dzj ) + V˜  Λ+

j=1

∗ Investigation

supported by University of Bologna. Funds for selected research topics. 921

(1.1)

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acting on the Hilbert space     n+p+1 O H = Λ+  L2 (R3zj ; C4 ) ∼ Λ+ L2 (R3(n+p+1) ; C4 ⊗ C4 ⊗ · · · ⊗ C4  . | {z } j=1

n+p+1

Here zj ∈ R , j = 1, . . . , n + 1, denotes the coordinates of the particles of type N , zi ∈ R3 , i = n + 2, . . . , n + p + 1, the coordinates of the particles of type E, mj , j = 1, . . . , n + 1, and mi , i = n + 2, . . . , n + p + 1, are, respectively, their masses. Hj (Dzj ) stands for the free Dirac operator 3

Hj (Dzj ) = cαj · Dzj + mj c2 βj (1)

(2)

(1.2)

(3)

where c is the light speed, αj := (αj , αj , αj ) and βj = β are the four Dirac matrices ! ! (k) 0 σ I 0 2 α(k) = , k = 1, 2, 3 , β= σ(k) 0 0 −I2 ! ! ! 1 0 0 −i 0 1 (1) (2) (3) σ = , σ = , σ = 0 −1 i 0 1 0 and the subscript j indicates that they act on the jth component of the tensorial Qn+p+1 product. Moreover, Λ+ = j=1 Λ+ j is the Casimir projector, with Λ+ j = χ(0,+∞) (Hj (Dzj ))

(1.3)

and V˜ is a scalar potential depending only on zi − zj , i, j = 1, . . . , n + p + 1. The operator (1.1) is widely used to describe relativistic effects in atoms and molecules, see, e.g. [18, 19], and in this contest it is called no-pair operator. In this setting, all the particles are regarded as relativistic particles but one expect that, when the ratio h2 between the masses of the particles of type E and the masses of the particles of type N tends to zero, the particles of type N become non-relativistic. Following the strategy used for the Schr¨ odinger operators in [17], we show that, under suitable assumptions on the potential V˜ giving the interactions between the particles, the relativistic contributions of the particles of type N on WKB-type 2 expansions of the first energy levels of the system, are of order R ⊕ o(h ) as h → 0. More precisely, writing B = B(h) as a direct integral B = R3 BE (h)dE, we show that there is a bijection (modulo o(h2 )) between the lower part of the spectrum of BE and the spectrum of a Schrödinger type differential operator X P0E (h) = −h2 ∆x + λ1 (x) + h Aα (x)(hDx )α + h2 AE (x) |α|=1

acting on L

2

2 (R3n x ;C |

⊗ · · · ⊗ C ) or L2 (R3n ; C2 ⊗ · · · ⊗ C2 ⊕ C2 ⊗ · · · ⊗ C2 ) {z } | {z } | {z } 2

n+1

n+1

n+1

where x denote the coordinates of the N -type particles in the center of mass frame (see Theorems 5.2, 5.6, and 7.3 below). For simplicity, we start by studying the case of a system composed by two particles of type N and one particle of type E

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923

where the interactions between the particles are described by a regular potential. We remark that, even in this simple case, many technical difficulties come from the fact that the operator BE (h) is not a differential operator so that, in the coordinates (x, y) of the center of mass frame, we cannot write it, as in the Schrödinger case, as a sum of three terms, one acting in the x variable, one acting on the y variable and one depending only on the total energy E. The plan of the paper is the following: in Sec. 2 we reduce the study of the operator (2.1) to an operator in the coordinate system of the center of mass frame PE (h) acting on Pauli spinors; in Sec. 3 we give and discuss the main assumptions on the potential V ; in Sec. 4 we show that PE (h) is an h-admissible operator with operator-valued symbol; in Sec. 5 we show that the Grushin operator associated to PE (h) is invertible and we relate, in such a way, the spectrum of PE (h) to the spectrum of a suitable operator acting only on the variables of the particles of type N ; in Sec. 6 and 7 we extend the results to the case of Coulomb-type potential and to multi-particles systems respectively. 2. Reduction to Pauli Spinors and to the Center of Mass Frame Let us consider the operator (1.1) in the simplest case of two particles of type N and one particle of type E, i.e. B = Λ+ (H1 (Dz1 ) + H2 (Dz2 ) + H3 (Dz3 ))Λ+ + Λ+ V˜ (z2 − z1 , z3 − z1 , z2 − z3 )Λ+

(2.1)

acting on the Hilbert space   3 O H = Λ+  (L2 (R3zj ) ⊗ C4 ) ∼ Λ+ L2 (R9 ; C4 ⊗ C4 ⊗ C4 ) . j=1

It is well known that (2.1) may be reduced from four components (Dirac spinors) to a two components one (Pauli spinors). In the Fourier variables ζ1 , ζ2 , ζ3 , the free Dirac operators Hj and the Casimir projector Λ+ j become the 4 × 4 matrix multiplication operators Hj (ζj ) = cαj · ζj + mj c2 βj , Λ+ j (ζj ) =

1 Hj (ζj ) + , 2 2Ej (ζj )

where we have denoted by Ej (ζj ) = (c2 ζj2 + m2j c4 )1/2 a , the positive eigenvalue of Hj (ζj ). For fixed ζj , two orthonormal eigenvectors with eigenvalue Ej (ζj ) are ! (Ej (0) + Ej (ζj ))ek 1 , k = 1, 2 , Nj (ζj ) cζj · σj ek

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where e1 , e2 are the canonical unit vectors in R2 , Ej (0) = mj c2 , Nj (ζj ) = [2Ej (ζj )(Ej (ζj ) + Ej (0))]1/2 . Any spinor ψj (ζj ) in the positive spectral subspace of Hj (ζj ) can be written as ! (Ej (0) + Ej (ζj ))uj (ζj ) 1 ψj (ζj ) = , Nj (ζj ) cζj · σj uj (ζj ) with uj ∈ L2 (R3ζj ; C2 ). The results of [5, 27] show that (2.1) is unitary equivalent to the operator B = E1 (Dz1 ) + E2 (Dz2 ) + E3 (Dz3 ) ˜ (z2 − z1 , z3 − z1 , z2 − z3 , Dz1 , Dz2 , Dz3 ) +W acting on L2 (R9 , C 3 ), where we have set C = C2 , C 2 = C ⊗ C, C 3 = C ⊗ C ⊗ C. Here ˜ = A1 A2 A3 (V˜ + R1 V˜ R1 + R2 V˜ R2 + R3 V˜ R3 + R1 R2 V˜ R2 R1 W + R1 R3 V˜ R3 R1 + R2 R3 V˜ R3 R2 + R1 R2 R3 V˜ R3 R2 R1 )A3 A2 A1 , where Rj (Dzj ) =:

cσj · Dzj , Ej (0) + Ej (Dzj ) 2 −1/2

Aj (Dzj ) =: (1 + Rj (Dzj ) )

(2.2) =

Ej (0) + Ej (Dzj ) 2Ej (Dzj )

1/2 .

(2.3)

˜ by some conUp to a change of variables and a multiplication of the operator W stants, we can set the light speed c = 1 and the mass m3 = 1. Now, let us make a change of variables (z1 , z2 , z3 ) → (R, x, y) in order that R may be the position of the center of mass. We use the change of variables m1 z1 + m2 z2 + z3 R= , m1 + m2 + 1 x = z2 − z1 , y = z3 −

(2.4)

m1 z1 + m2 z2 , m1 + m2

m1 +m2 m1 +m2 1 +m2 whose Jacobian is one. Let us set h2 =: m so that 2m1 m2 , µ1 =: 2m2 , µ2 =: 2m1 µj 1 mj = h2 and µ1,2 =: µ1 + µ2 , K =: µ1,2 +h2 . After the change of variables given in (2.5), the operator (2.1) become 1 h2 Dx h2 Dy h2 Dx h2 Dy B(h) = 2 µ1 E h2 KDR − − + µ2 E h2 KDR + − h µ1 µ1,2 µ2 µ1,2

+ E(h2 KDR + Dy ) + W (x, y, hDx , Dy , DR ; h) ,

(2.5)

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where E(ξ) =

925

p 1 + ξ 2 and

W = A1 A2 A3 V + R1 V R1 + R2 V R2 + R3 V R3 + R1 R2 V R2 R1 + R1 R3 V R3 R2 + R2 R3 V R3 R2 + R1 R2 R3 V R3 R2 R1 )A3 A2 A1 . Here V =: V˜

(2.6)

x x + y, −y , x, 2µ1 2µ2

2 h2 D σj · h2 KDR + (−1)j h µDj x − µ1,2y , Rj = Rj (hDx , Dy , DR ; h) =: 2 h2 D 1 + E h2 KDR + (−1)j h µDj x − µ1,2y 1/2 2 h2 D 1 + E h2 KDR + (−1)j h µDj x − µ1,2y  , Aj = Aj (hDx , Dy , DR ; h) =:  2 h2 D 2E h2 KDR + (−1)j h µDj x − µ1,2y 

for j = 1, 2 and R3 = R3 (Dy , DR ; h) =:

σ3 · (h2 KDR + Dy ) , 1 + E(h2 KDR + Dy )

A3 = A3 (Dy , DR ; h) =:

1 + E(h2 KDR + Dy ) 2E(h2 KDR + Dy )

1/2 .

Since the Hamiltonian B(h) commutes with the total momentum, we can write (see [20, 22]) Z ⊕ ˜E (h)dE , B B(h) = R3

˜E (h) is obtained by substituting formally E to DR in B(h). where B ˜E (h) is unitary equivalent to B ˜E 0 (h) if |E| = |E 0 |. For later conveNotice that B nience, we prefer to work with the operator ˜E (h)eiµ1,2 hKE,yi − (1 + µ1,2 /h2 ) BE (h) =: e−iµ1,2 hKE,yi B given by BE (h) = Θ(hDx , h) + QE (x, hDx ; h) , QE (x; hDx ; h) = E(E + Dy ) − 1 + WE (x, hDx ; h) , where we have set ˜ Θ(hDx ; h) =: Θ(hD x ; h) + Θ1 (hDx ; h) , 2 2 µ1 h Dx µ2 h Dx ˜ Θ(hD ; h) =: E − 1 + E − 1 , x h2 µ1 h2 µ2 Θ1 (hDx ; h) = Θ1 (hDx , Dy ; h)

(2.7)

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2 2 h2 Dy h Dx h Dx E + −E µ1 µ1,2 µ1 2 2 µ2 h2 Dy h Dx h Dx + 2 E − −E , h µ2 µ1,2 µ2

µ1 =: 2 h

WE (x, hDx ; h) = WE (x, y, hDx , Dy ; h) =: A1 A2 (LE + R1 LE R1 + R2 LE R2 + R1 R2 LE R2 R1 )A2 A1 , LE (x) = LE (x, y, Dy ) =: AE3 (V + R3E V R3E )AE3 , and

2 h2 D σj · (−1)j h µDj x − µ1,2y , Rj = Rj (hDx ; h) =: 2 h2 D 1 + E (−1)j h µDj x − µ1,2y

j = 1, 2 ,

1/2 2 h2 D 1 + E (−1)j h µDj x − µ1,2y  Aj = Aj (hDx ; h) =:  , 2 h2 D 2E (−1)j h µDj x − µ1,2y 

R3E =:

σ3 · (E + Dy ) , 1 + E(E + Dy )

AE3 =:

1 + E(E + Dy ) 2E(E + Dy )

j = 1, 2

1/2 .

3. Assumptions We make the following assumptions on the potential V : (H1 ) (H2 )

V ∈ C 0 (R3x × R3y ) and, for any y ∈ R3 , x → V (x, y) ∈ C ∞ . For any α ∈ N 3 , there exists Cα > 0 such that |∂xα V (x, y)| ≤ Cα .

Under this assumptions, BE (h) is realized as a selfadjoint operator on L2 (R6 , C 3 ) with domain D(BE (h)) = H 1 (R6 ; C 3 ). Set now ˜ E (x) = Q ˜ E (x, y, Dy ) =: E(E + Dy ) − 1 + LE (x, y, Dy ) Q ˜ E (x) is unitary equivalent to Q ˜ 0 (x) and let us acting on L2 (R3y ; C). Notice that Q denote by ˜ 0 (x) λ1 (x) =: inf Sp Q given by the Min-Max principle. We suppose that λ1 (x) < lim inf V (x, y) . |y|→+∞

˜ 0 (x). By adding a This fact guarantees that λ1 (x) is in the discrete spectrum of Q constant, we may assume that lim inf |x|→+∞ λ1 (x) = 0. Here we are interesting in the spectrum of BE (h) in the interval I =] − ∞, b], with λ0 =: inf 3 λ1 (x) < b < 0 x∈R

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and we assume that: (H3 ) There exists a contractible open set Ω ⊂⊂ R3x such that λ−1 1 (I) ⊂⊂ Ω, and ˜ 0 (x))\λ1 (x) . b < inf Sp(Q x∈Ω

Notice that the multiplicity of λ1 (x) is always even. Actually, it is well known that ˜ 0 (x) commutes with the antilinear Kramers operator Q K : L2 (R3y ; C) → L2 (R3y ; C) ,

Ku = σ(2) Ju

where Ju = u¯. Hence if u(x, y) is an eigenfunction (normalized in L2 (R3y ; C)) of ˜ 0 (x) associated to λ1 (x), Ku(x, y) is still a normalized eigenfunction of Q ˜ 0 (x) Q 2 associated to the same eigenvalue. Moreover, since K = −1, u and Ku are linearly independent and the multiplicity of λ1 (x) is always even. We assume (H4 ) For x ∈ Ω, λ1 (x) is exactly of multiplicity 2. We have the following lemma. Lemma 3.1. Under the previous assumptions, λ1 ∈ C ∞ (Ω) and there exists u ∈ C ∞ (Ω; H 1 (R3y )) with ku(x)ky = 1 for any x ∈ Ω such that u(x) and Ku(x) generate the eigenspace associated to λ1 (x). Moreover, there exists w ∈ C ∞ (R3x ; H 1 (R3y )) such that kw(x)ky = 1 and w(x) = u(x) for x ∈ λ−1 1 (I). Here and in the following we have set ku(x)ky = ku(x)kL2 (R3y ) . ˜ 0 (x) is a continuous function of x, by the Min-Max principle also Proof. Since Q ˜ 0 (x))\λ1 (x) are continuous functions of x, for x ∈ Ω. Thanks λ1 (x) and inf Sp(Q to the gap assumption (H3 ), for any x ∈ Ω we can choose a contour Γρ (x) ⊂ C ˜ 0 (x)\λ1 (x) remains of center λ1 (x) and radius ρ > 0 small enough such that Sp Q outside Γ2ρ (x). Set Z ˜ ˜ 0 (x) − z)−1 dz Π(x) = (Q Γρ (x)

˜ 0 (x)\λ1 (x) remains outside Γρ (x0 ). Hence, and observe that, if kx0 − xk < ρ, Sp Q 0 if kx − xk < ρ we have Z ˜ ˜ 0) + ˜ 0 (x) − z)−1 − (Q ˜ 0 (x0 ) − z)−1 dz Π(x) = Π(x (Q Γρ (x0 )

˜ 0) + = Π(x

Z

Γρ (x0 )

˜ 0 (x) − z)−1 (L0 (x) − L0 (x0 ))(Q ˜ 0 (x0 ) − z)−1 dz (Q

˜ and this shows that the map x → Π(x) is C ∞ (R3x ; L(L2 (R3y )). Since Ω is contractible, the homotopy property of vector bundles (cfr: [7, 15]) ensures that the ˜ vector bundle E ⊂ Ω × L2 (R3y ) defined by Π(x) is trivial. Hence, taking into ac∞ count (H4 ), we can find u(x), u˜(x) ∈ C (Ω; H 1 (R3y )) with ku(x)ky = k˜ u(x)ky = 1 2 ˜ such that Span{u(x), u ˜(x)} = Π(x)(L (R3y )). Then also u(x) and Ku(x) generate the eigenspace associated to λ1 (x). Finally, since ˜ 0 (x)u(x), u(x)iy λ1 (x) = hQ

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(where h., .iy denotes the scalar product on L2 (R3y )) we obtain that λ1 ∈ C ∞ (Ω). Now, as in [17, Lemma 1.1], we can modify u(x) outside λ−1 1 (I) and complete the proof of the lemma. In the following we denote by u1 (x, y) =: u(x, y), u2 (x, y) =: Ku(x, y) the ˜ 0 (x) two (normalized in L2 (R3y )) smooth (with respect to x) eigenfunctions of Q associated to λ1 (x) and we set w1 (x, y) =: w(x, y), w2 (x, y) =: Kw(x, y). Moreover we define uEj (x, y) =: e−ihE,yi uj (x, y) and wjE (x, y) = e−ihE,yi wj (x, y) ,

j = 1, 2 .

4. A Class of h-Admissible Operator In this section, we show that BE (h) can be condidered as an h-admissible operator with operator-valued symbol. Let us start by giving the following definitions: ˜ k (R6 ; C 3 ) the space H k (R3x × R3y ; C 3 ) equipped Definition 4.1. We denote by H with the (h-dependent) norm Z 1/2 q k 2 2 kvkH˜ k (R6 ) =: |([hDx ]h + 1 + Dy + 1) v(x, y)| dxdy , where [hDx ]h =

p 1 + h2 (hDx )2 − 1 /h2 .

˜ k (R3 ; H1 ) the space Definition 4.2. Given an Hilbert space H1 we denote by H k 3 H (R ; H1 ) equipped with the (h-dependent) norm Z 1/2 k 2 kvkH˜ k (R3 ;H1 ) =: k([hDx ]h + 1) v(x)kH1 dx . Definition 4.3. Given two Hilbert space H1 , H2 , a family of bounded operators ˜ k (R3 ; H1 ) to L2 (R3 ; H2 ) is called h-admissible of order k (with weight ([ξ]h + from H 1)k ) if, for N sufficiently large, A(h) =

N X

hj Oph (aj (x, ξ; h)) + hN RN (h)

j=0

where RN (h) is uniformly bounded from L2 (R3 ; H1 ) to L2 (R3 ; H2 ), for 0 < h ≤ h0 , and aj ∈ C ∞ (T ∗ R3 ; L(H1 , H2 )) with α k∂x,ξ aj (x, ξ; h)kL(H1 ,H2 ) = O(([ξ]h + 1)k )

for all α ∈ N6 uniformly with respect to (x, ξ) ∈ T ∗ R3 and h, as h → 0. Here Oph (aj (x, ξ; h)) denotes the h-pseudodifferential operator which, for φ ∈ C0∞ (R3 , H1 ) is defined by the oscillatory integral Z 0 −3/2 Oph (aj (x, ξ; h))φ = (2πh) ei(x−x )ξ/h aj (x, ξ; h)φ(x0 )dx0 dξ .

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Observe that, for any α ∈ N3 ∂ξα [ξ]h = O([ξ]h + 1) uniformly in h and ξ. More precisely, ξj |∂ξj [ξ]h | = p ≤ |ξ| ≤ [ξ]h + 1 , 1 + h2 ξ 2 and |∂ξ2i ξj [ξ]h |

δi,j h2 ξi ξj = p − ≤ 1. 1 + h2 ξ 2 (1 + h2 ξ 2 )3/2

Hence, by Cauchy theorem, ∂ξα [ξ]h = O(1) for |α| ≥ 1. A slight generalization of the Calderon–Vaillancourt Theorem implies that Oph (aj ) extends uniquely to a ˜ k (R3 ; H1 ) to L2 (R3 ; H2 ). We shall call the formal series bounded map from H ∞ X

hj aj (x, ξ; h) = a(x, ξ; h)

j=0

the symbol σ(A(h)) of A. The principal symbol can be defined, as usual, by mean of equivalence class. Composition of h admissible operators (of weight ([ξ]h + 1)k ) A(h), B(h) induces the product of symbol (a # b)(x, ξ; h) = σ(A(h) ◦ B(h)) , which is given by (a # b)(x, ξ; h) =

X h|α| ∂ α a(x, ξ; h)∂xα b(x, ξ; h) , i|α| α! ξ 3

α∈N

assuming that the image of B matches the domain of A. Taking into account the previous definition here we want to show that ˜ 1 (R3x × R3y ; C 3 ) ∼ H ˜ 1 (R3x ; L2 (R3y ; C 3 )) ∩ BE (h) is a bounded operator from H 2 3 1 3 3 2 3 3 3 L (Rx ; H (Ry ; C )) to L (Rx × Ry ; C ) and its operator valued-symbol is an application from T ∗ R3x to L(H 1 (R3y ; C 3 ), L2 (R3y ; C 3 )). To do this, we need some precise estimates on Θ(ξ; h), Aj (ξ; h) and Rj (ξ; h), j = 1, 2, given by the following algebraic lemma. Lemma 4.4. (a) For any u ∈ H 1/2 (R3y ; C 3 ) we have hΘ(ξ; h)u, uiy ≥ 2 inf{µ1 , µ2 }[ξ]h kuky .

(4.1)

(b) We have kΘ1 (ξ; h) − S(ξ; h)kL(H 1 (R3y ),L2 (R3y )) = O(1) ,

(4.2)

kΘ1 (ξ; h) − S(ξ; h)kL(H 1 (R3y ),H −1 (R3y )) = O(h2 )

(4.3)

for some S(ξ; h) : H 1 (R3y ; C 3 ) → L2 (R3y ; C 3 ) with kS(ξ; h)kL(H 1 (R3y );L2 (R3y )) = O(h2 [ξ]h ) .

(4.4)

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Moreover k∂ξα Θ1 (ξ; h)kL(H 1 (R3y ),L2 (R3y )) = O(h) ,

for |α| ≥ 1 .

(4.5)

(c) We have k∂ξα Θ(ξ; h)kL(L2 (R3y )) = O([ξ]h + 1) , k∂ξα Θ(ξ; h)kL(L2 (R3y )) = O(1) ,

for |α| = 1

for |α| ≥ 2 .

(4.6) (4.7)

(d) We have, for j = 1, 2, kAj (ξ; h)kL(L2 (R3y )) = O(1) ,

(4.8)

kAj (ξ; h) − 1kL(H 1 (R3y ),H −1 (R3y )) = O(h2 ([ξ]h + h2 )) ,

(4.9)

k∂ξα Aj (ξ; h)kL(L2 (R3y )) = O(h) ,

for |α| = 1

k∂ξα Aj (ξ; h)kL(H 1 (R3y ),L2 (R3y )) = O(h2 ([ξ]h + h)) , k∂ξα Aj (ξ; h)kL(L2 (R3y )) = O(h2 ) ,

(4.10) for |α| = 1

for |α| ≥ 2 .

(4.11) (4.12)

(e) We have, for j = 1, 2, kRj (ξ; h)kL(L2 (R3y )) = O(1) ,

(4.13)

kRj (ξ; h)kL(H 1 (R3y ),L2 (R3y )) = O(h([ξ]h + h2 )1/2 ) ,

(4.14)

k∂ξα Rj (ξ; h)kL(L2 (R3y )) = O(h) ,

(4.15)

for |α| = 1

k∂ξα Rj (ξ; h)kL(H 1 (R3y ),L2 (R3y )) = O(h2 ([ξ]h + h)) , k∂ξα Rj (ξ; h)kL(L2 (R3y )) = O(h2 ) ,

for |α| ≥ 2 .

for |α| = 1

(4.16) (4.17)

Here O(1) means bounded uniformly with respect to h. 1 +µ2 Proof. (a) Let us observe that, since E is convex and µ2µ = 1, we have, for any 1 µ2 3 z, w ∈ R , z z µ1 E + w − 1 + µ2 E −w −1 µ1 µ2 z z ≥ inf{µ1 , µ2 } E +w +E −w −2 µ1 µ2

≥ 2 inf{µ1 , µ2 }(E(z) − 1) . ˆ η; h) the Fourier transform with respect to y of Θ(ξ, Dy ; h) and Denoting by Θ(ξ, applying the previous inequality inequality with z replaced by hξ and w replaced 2 by µh1,2η we get ˆ η; h) ≥ 2 inf{µ1 , µ2 }[ξ]h , Θ(ξ, and this proves (4.1).

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(b) Since, for any θ1 , θ2 ∈ R3 , |E(θ1 + θ2 ) − E(θ1 )| ≤ E(θ2 ) and E is even, we have 2 µ1 + µ2 h η ˆ |Θ1 (ξ, η; h)| ≤ . E 2 h µ1,2 On the other hand, if we set

  1 1 1 ˆ η; h) =:  −  hhξ, ηi , S(ξ, µ1,2 E hξ E µhξ2 µ1

we have ˆ η; h)| ≤ Ch2 [ξ]h |S(ξ,

p 1 + η2 ,

and, by Taylor expansion in the η variable, we get ˆ 1 (ξ, η; h) − S(ξ, ˆ η; h)| ≤ C 0 |h2 η|2 |Θ for some C, C 0 > 0. Hence, there exists C 00 ≥ max{µ1 + µ2 , C 0 } and C˜ > 0 such that 2 p ˆ 1 (ξ, ; h) − S(ξ, ˆ η; h)| ≤ C 00 1 E h η − 1 ≤ C˜ 1 + η 2 . |Θ 2 h µ1,2

h2 η µ1,2 −1 h4 (1+η2 )

E

is uniformly bounded with respect to h, (4.3) This implies (4.2). Since holds. Moreover, for (ξ, η) ∈ C6 with |Im ξ| + |Im η| < ε < 1 we have 2 + Z * 1 1 h2 η h η hξ ˆ |∂ξj Θ1 (ξ, η; h)| ≤ −t , ∇z ∂zj E dt h 0 µ1 µ1,2 µ1,2 2 + Z * 1 1 h2 η h η hξ + −t , ∇z ∂zj E dt ≤ Ch|η| , h 0 µ2 µ1,2 µ1,2 ˆ 1 (ξ, η; h). This proves and, by Cauchy Theorem, the same inequality holds for ∂ξα Θ (4.5). (c) We have 2 + Z 1 * h η hξ hξ ˆ η; h)| = 2 |∂ξj Θ(ξ, +t − (1 − t) , ξ dt ∇z ∂zj E µ µ µ 1,2 1 2 0 ≤ 2C|ξ| ≤ 2C([ξ]h + 1) , and one can check directly that, for any α ∈ N3 , |α| ≥ 2 ˆ η; h)| ≤ Cα . |∂ξα Θ(ξ, (d) (4.8), (4.10), (4.11), (4.12) follows from the definition of Aj . To prove (4.9), it is sufficient to observe that, for z ∈ R3 1/2 1 1 + √ 1 − ≤ C(E(z) − 1) , 2 2 1 + z2

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and then, applying this inequality to z =

hξ µj

±

h2 η µ1,2 ,

one can conclude that

ˆ η; h) . |1 − Aˆj (ξ, η; h)| ≤ Ch2 Θ(ξ,

(4.18)

Then, using (4.3) and (4.4), (4.9) follows. (e) (4.13), (4.15), (4.16), (4.17) follows directly by the definition of Rj . To prove (4.14), it is sufficient to observe that, by (4.18) ˆ j (ξ, η, h)|2 = |(1 − Aˆj (ξ, η; h)2 )Aˆj (ξ, η; h)−2 | ≤ 8|1 − Aˆj (ξ, η, h)| |R ˆ η, h) . ≤ 8Ch2 Θ(ξ, Then, using (4.9), (4.14) follows. Using Lemma 4.4 and assumptions (H1 ), (H2 ) we obtain ˜ 1 (R3 × R3 ; C 3 ) to L2 (R3 × Theorem 4.5. The operator BE (h) is bounded from H x y x R3y ; C 3 ) and its operator-valued symbol σ(BE (h))(x, ξ; h) is an application from T ∗ R3x to L(H 1 (R3y ; C 3 ), L2 (R3y ; C 3 )). Moreover (1)

(2)

σ(BE (h))(x, ξ; h) = pE (x, ξ; h) + h3 nE (x, ξ; h) + h4 nE (x, ξ; h) with (1)

α k∂x,ξ nE (x, ξ; h)kL(H 1 (R3y ),L2 (R3y )) = O([ξ]h + 1) , (2)

α k∂x,ξ nE (x, ξ; h)kL(L2 (R3y )) = O(1) ,

for any α ∈ N6 . Here pE (x, ξ; h) = Θ(ξ; h) + qE (x, ξ; h) , qE (x, ξ; h) = E(E + Dy ) − 1 + wE (x, ξ; h) , and wE (x, ξ; h) = A1 (ξ; h)A2 (ξ; h)(LE (x) + R1 (ξ; h)LE (x)R1 (ξ; h) + R2 (ξ; h)LE (x)R2 (ξ; h) + R1 (ξ; h)R2 (ξ; h)LE (x)R2 (ξ; h)R1 (ξ; h))A2 (ξ; h)A1 (ξ; h) . 5. The Grushin Operator Let us fix ζ ∈ C0∞ (Ω), 0 ≤ ζ ≤ 1, such that ζ = 1 in a neighborhood U of λ−1 1 (I) and set BEζ (h) = Θ(hDx , h) + QζE (x, hDx , h) , QζE (x, hDx ; h) = E(E + Dy ) − 1 + WEζ (x, hDx ; h)

(5.1)

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with WEζ = A1 A2 LζE + R1 LζE R1 + R2 LζE R2 + R1 R2 LζE R2 R1 A2 A1 and LζE (x) = ζ(x)LE (x) . Observe now that, if B is one of BE (h) or BEζ (h), we have / λ−1 σ(B)(x, ξ; h) ≥ 2 inf{µ1 , µ2 }[ξ]h + b + O(h3 ([ξ]h + 1)) if x ∈ 1 (I) σ(B)(x, ξ; h) ≥ C

if x ∈ λ−1 1 (I) and |ξ| ≥ c

for some C, c > 0. On the other hand, if x ∈ λ−1 1 (I) and |ξ| ≤ c, σ(B)(x, ξ; h) = ξ 2 + λ1 (x) + O(h2 ) . Hence, using Agmon-type estimates (cfr: [1, 14, 17, 21]), one can show that, if φ is a normalized eigenfunction of BE (h) or BEζ (h) associated to an eigenvalue in I, there exists a positive constant C > 0 independent of h such that: ked(x,U)/hφkH 1/2 (R6 ) ≤ C , where d denote the usual distance in the Agmon metric ds2 = max{λ1 (x) − b, 0}. Hence, it is easy to check that Sp BE (h) and Sp BEζ (h) coincide in I up to an error of order O(e−δ/h ) as h → 0 where 0 < δ < d(λ−1 1 (I), ∂U ). ˜ Let us denote by ΠE (x) the orthogonal projection onto F˜E (x) = Span{w1E (x), E w2 (x)} and by ˜ E (x) = ⊗wE (x)h., wE (x)iy + ⊗wE (x)h., wE (x)iy ΠE (x) = I ⊗ I ⊗ Π 1 1 2 2 the orthogonal projection onto FE (x) = Span{(ei ⊗ej ⊗wkE (x))i,j,k∈{1,2} }. Here and in the following, if w(x, y) ∈ L2 (R3x × R3y ; C) we denote by ⊗w the map ⊗w : L2 (R3x ; C 2 ) → L2 (R3x × R3y ; C 3 ) such that, for any v ∈ L2 (R3x ; C 2 ) (⊗w)(v)(x, y) = v(x) ⊗ w(x, y) and by h., wiy : L2 (R3x × R3y ; C 3 ) → L2 (R3x ; C 2 ) its adjoint. We have Lemma 5.1. For h small enough and E ∈ I, there exists a constant c > 0 such that ˆ E (x)(pζ (x, ξ; h) − E)Π ˆ E (x)u, uiy ≥ c([ξ]h + 1)kΠ ˆ E (x)uk2 hΠ y E ˆ E (x) =: 1 − ΠE (x). for any u ∈ H 1/2 (R3y ; C 3 ). Here we have set Π

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Proof. Let us observe that, for any ε ∈]0, 1[ ˆ E (x)(pζ (x, ξ; h) − E)Π ˆ E (x) ≥ Π ˆ E (x)(2 inf{µ1 , µ2 }(1 − ε)[ξ]h Π E ˆ E (x) . + εΘ(ξ; h) + qEζ (x, ξ; h) − E)Π 1/2 Observe now that, if x ∈ R3 \λ−1 (R3y ; C 3 ) there exists b0 ≥ b such 1 (I) and u ∈ H that

h(qEζ (x, ξ; h) − E)u, uiy ≥ (b0 − E)kuk2y . ˆ ζ ˆ For x ∈ λ−1 1 (I), let us write ΠE qE (x, ξ; h)ΠE (x) as a sum of the terms ˆ E A1 A2 Π ÊQ ˜ζ Π ˆ ˆ ˆ ˆ ˜ζ ˆ ˆ Γ1 =: Π E E A1 A2 ΠE + ΠE A1 A2 R1 ΠE QE ΠE R1 A1 A2 ΠE ˆ E A1 A2 R2 Π ÊQ ˜ζ Π ˆ ˆ ˆ ˆ ˜ζ ˆ ˆ +Π E E R2 A1 A2 ΠE + ΠE A1 A2 R1 R2 ΠE QE ΠE R2 R1 A1 A2 ΠE and ˆ E A1 A2 ΠE Q ˜ ζ ΠE A1 A2 Π Ê + Π ˆ E A1 A2 R1 ΠE Q ˜ ζ ΠE R1 A1 A2 Π Ê Γ2 =: Π E E ˆ E A1 A2 R2 ΠE Q ˜ ζ ΠE R2 A1 A2 Π Ê + Π ˆ E A1 A2 R1 R2 ΠE Q ˜ ζ ΠE R2 R1 A1 A2 Π Ê , +Π E E ˜ ζ = E(E + Dy ) − 1 + Lζ . Using (H3 ), we obtain where Q E E ˆ E A1 A2 Π ˆ E uk2 + |Π ˆ E A1 A2 R1 Π ˆ E uk2 hΓ1 u, uiy ≥ b0 kΠ y y ˆ E A1 A2 R2 Π ˆ E uk2y + kΠ ˆ E A1 A2 R1 R2 Π ˆ E uk2y + kΠ

ˆ E u, Π ˆ E uiy . − C 0 h2 hΘ(ξ; h)Π On the other hand, by (4.9) and (4.14), we have ! ˆ E uk2 ˆ E uk2 + kR1 Π |hΓ2 u, uiy | ≤ sup |λ1 (x)| k(A1 A2 − 1)Π y y x∈λ−1 1 (I)

ˆ E uk2 ≤ Ch2 hΘ(ξ; h)Π ˆ E u, Π ˆ E uiy . + 2kR2 Π y Moreover, since ˆ E uk2 , kΠE A1 A2 Rj Π ˆ E uk2 , kΠE A1 A2 R1 R2 Π ˆ E uk2 kΠE A1 A2 Π y y y are also terms of order ˆ E u, Π ˆ E uiy ) O(h2 hΘ(ξ; h)Π and recalling that A21 A22 (1 + R12 )(1 + R22 ) = 1, we get ˆ E (x)(pζ (x, ξ; h) − E)Π ˆ E (x)u, uiy hΠ E ˆ E (x)uk2 ≤ (2 inf{µ1 , µ2 }(1 − ε)[ξ]h + b − E)kΠ y ˆ E u, Π ˆ E uiy + (ε − Ch2 )hΘ(ξ; h)Π and, taking h sufficiently small, the lemma follows.

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Lemma 5.1 will be crucial for inverting the following Grushin operator associated to BEζ (h)  ζ  BE (h) − E ⊗w1E ⊗w2E   ζ E (h) =:  0 0  BE,E  h., w1 iy . E h., w2 iy 0 0 Using assumptions (H1 ), (H2 ) and the results of Lemma 4.4, one can check that ζ (h) is a selfadjoint operator in BE,E L2 (R3x × R3y ; C 3 ) ⊕ L2 (R3x ; C 2 ⊕ C 2 ) ∼ L2 (R3x ; L2 (R3y ; C 3 ) ⊕ C 2 ⊕ C 2 ) with domain H 1 (R3x × R3y ; C 3 ) ⊕ L2 (R3x ; C 2 ⊕ C 2 ). Moreover it is a pseu˜ 1 (R3x × R3y ; C 3 ) ⊕ L2 (R3x , C 2 ⊕ C 2 ) to dodifferential operator bounded from H 3 3 3 1 3 2 2 ˜ L(Rx × Ry ; C ) ⊕ H (Rx , C ⊕ C ) whose symbol (which is an application from T ∗ R3x to L(H 1 (R3y ; C 3 ) ⊕ C 2 ⊕ C 2 , L2 (R3y ; C 3 ) ⊕ C 2 ⊕ C 2 )) is given by ζ ζ σ(BE,E )(x, ξ; h) = σ1 (BE,E )(x, ξ; h) + h3 NE (x, ξ; h) ,

where

  ζ σ1 (BE,E )(x, ξ; h) =  

and

Θ(ξ; h) + qEζ (x, ξ; h) − E

⊗w1E (x)

⊗w2E (x)

h., w1E (x)iy

0

0

h., w2E (x)iy

0

0

  NE (x, ξ; h) =  

(1)

(2)

nE (x, ξ; h) + hnE (x, ξ; h) 0 0

0 0

   



 0 0 . 0 0

Our main result is the following theorem. ζ Theorem 5.2. The Grushin operator BE,E is invertible for all E ∈ I and, modulo ζ ∞ −1 O(h ), its inverse (BE,E ) is an h-admissible operator which is bounded from 2 6 3 1 3 2 2 ˜ ˜ 1 (R6 , C 3 )⊕L2 (R3 , C 2 ⊕C 2 ). Furthermore, writing L (R , C )⊕ H (Rx , C ⊕C ) to H x   + G G E,E E,E ζ , (BE,E )−1 =:  − GE,E G± E,E −1 the symbol of G± E,E , for x ∈ λ1 (I), is given by 2 ˜ σ(G± E,E ) = (E − Θ(ξ; h) − λ1 (x))1C⊕C + O(h ([ξ]h + 1))

and we have, for any E ∈ I, E ∈ Sp(BEζ )

if and only if

0 ∈ Sp(G± E,E ) .

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Moreover, if we denote by K2 =

(2) (σ1

⊗

(2) σ2 )J

˜2 = and by K

O K2 −K2 0

with

˜ 2 = −1, we have K ˜ 2 G± (x, hDx ; h) = G± (x, hDx ; h)K ˜2 . K E,E −E,E ˆ E (x)(pζ (x, ξ; h) − E)Π ˆ E (x) is invertible and, Proof. By Lemma 5.1, we have that Π E if we set ˆ E (x)(Π ˆ E (x)(pζ (x, ξ; h) − E)Π ˆ E (x))−1 Π ˆ E (x) XE (x, ξ; h) = Π E ζ one readily verifies that σ1 (BE,E )(x, ξ; h) is invertible with inverse mE (x, ξ; h) given by   TE (x, ξ; h)(⊗w1E (x)(.)) TE (x, ξ; h)(⊗w2E (x)(.)) XE (x, ξ; h)     (1,1) (1,2)  hSE (x, ξ; h)(.), w1E (x)iy , g (x, ξ; h) g (x, ξ; h) E,E E,E     (2,1) (2,2) E hSE (x, ξ; h)(.), w2 (x)iy gE,E (x, ξ; h) gE,E (x, ξ; h)

where TE (x, ξ; h) = 1 − XE (x, ξ; h)hE (x, ξ; h) , SE (x, ξ; h) = 1 − hE (x, ξ; h)XE (x, ξ; h) , hE (x, ξ; h) = Θ1 (ξ; h) + qE (x, ξ; h) and (k,l) ˜ h))δk,l − hhE (x, ξ; h)(⊗wlE (x)(.)), wkE (x)iy gE,E (x, ξ; h) = (E − Θ(ξ;

+ hhE (x, ξ; h)XE (x, ξ; h)hE (x, ξ; h)(⊗wlE (x)(.)), wkE (x)iy , where δk,l is the Kroneker index. Observe that α k∂x,ξ XE (x, ξ; h)kL(L2 (R3y )) = O(([ξ]h + 1)−1 ) , α k∂x,ξ XE (x, ξ; h)kL(L2 (R3y ),H 1 (R3y )) = O(1)

uniformly with respect to h. Setting ME = Oph (mE (x, ξ; h)) , one has that ME is bounded as a map from ˜ 1 (R3 , C 2 ⊕ C 2 ) D(ME ) = L2 (R6 ; C 3 ) ⊕ H x ζ ˜ 1 (R6 ; C 3 ) ⊕ L2 (R3 , C 2 ⊕ C 2 ). Thus the composition B ζ ME is ) = H to D(BE,E x E,E ζ well defined as a bounded operator in D(ME ) (viceversa ME BE,E is bounded on ζ D(BE,E )) and we shall apply the pseudodifferential calculus for operator-valued symbols to get ζ BE,E ME = 1 + hR

(5.2)

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where R is an h-admissible operator (depending on E an h) which on D(ME ) is uniformly bounded in h. Moreover, using the symbolic calculus, one can check that R = R1 + hR2 where the symbol of R1 is given by  ∂ξj pζE ∂xj XE ∂ξj pζE ∂xj TE (⊗w1E (.)) X  1  0 0 i j  0 0 while the symbol of R2 is given by  ∂ξ2i ξj pζE ∂x2i xj XE ∂ξ2i ξj pζE ∂x2i xj TE (⊗w1E (.)) 1 X  − 0 0 2 i,j  0 0

∂ξj pζE ∂xj TE (⊗w2E (.)) 0

   

0

∂ξ2i ξj pζE ∂x2i xj TE (⊗w2E (.)) 0

  . 

0

Notice that the pseudodifferential operator R2 is even bounded from L2 (R6 ; C 3 ) ⊕ L2 (R3x , C 2 ⊕ C 2 ) to D(ME ). To complete the proof we derive from (5.2) that " # N X ζ )−1 = ME 1 + (−h)k Rk + O(hN ) . (BE,E k=1 ζ One can check that (BE,E )−1 is an h-admissible operator which is bounded from 2 6 3 1 3 2 ˜ (Rx , C ⊕ C 2 ) to H ˜ 1 (R6 , C 3 ) ⊕ L2 (R3x , C 2 ⊕ C 2 ). Moreover, setting L (R , C ) ⊕ H   + G G E,E E,E ζ , )−1 =:  − (BE,E GE,E G± E,E

then σ(G± E,E )k,l (x, ξ; h) is given by (k,l)

gE,E (x, ξ; h) − " +h −

2

hX hSE ∂ξj pζE ∂xj (TE (⊗wlE (.))), wkE iy i j

1X hSE ∂ξ2i ξj pζE ∂x2i xj (TE (⊗wlE (.))), wkE iy 2 i,j

X hSE ∂ξi pζE ∂xi XE ∂ξj pζE ∂xj (TE (⊗wlE (.))), wkE iy i,j

X + h∂ξi (hE XE )∂xi (∂ξj pζE ∂xj TE (⊗wlE (.))), wkE iy

#

i,j

+ O(h3 ([ξ]h + 1)) .

(5.3)

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−1 Let us compute the symbol of σ(G± E,E ) for x ∈ λ1 (I). We have

Lemma 5.3. Under the previous assumptions, we have, for x ∈ λ−1 1 (I) 2 ˜ σ(G± E,E )k,l (x, ξ; h) = (E − Θ(ξ; h) − λ1 (x))δk,l − h F (ξ; h)

−

hX h∂ξj pE ∂xj (⊗uEl (.)), uEk (x)iy i j " 2

+h

1X 2 h∂ pE ∂x2i xj (⊗uEl (.), uEk (x)iy 2 i,j ξi ξj

X − h∂ξi pE ∂xi XE ∂ξj pE ∂xj (⊗uEl (.)), uEk iy

#

i,j

+ O(h2 ([ξ]h + h))

(5.4)

with F (ξ; h) =

1 h(Θ1 (ξ, h) − Θ1 (ξ, h)XE (x, ξ; h)Θ1 (ξ, h))wlE (x), wkE (x)iy = CO(1) . h2

Proof. If v ∈ C 2 , u ⊗ w(x) ∈ C 2 ⊗ H 1 (R3y ; C) and x ∈ λ−1 1 (I) hhE (x, ξ; h)(v ⊗ uEl (x)), u ⊗ w(x)iy = λ1 (x)hv, uihuEl (x), w(x)iy + hΘ1 (ξ; h)uEl (x), w(x)iy hv, ui + hLE (x)(A1 A2 − 1)uEl (x), A1 A2 w(x)iy hv, ui + hLE (x)uEl (x), (A1 A2 − 1)w(x)iy hv, ui + hLE (x)R1 A1 A2 (v ⊗ uEl (x)), R1 A1 A2 (u ⊗ w(x))iy + hLE (x)R2 A1 A2 (v ⊗ uEl (x)), R2 A1 A2 (u ⊗ w(x))iy + hLE (x)R1 R2 A1 A2 (v ⊗ uEl (x)), R1 R2 A1 A2 (u ⊗ w(x))iy . For the second term on the RHS of the previous expression we get, using (4.2), (4.3) |hΘ1 (ξ, h)uEl (x), w(x)iy | ≤ kΘ1 (ξ, h) − S(ξ; h)kL(H 1 (R3y ),H −1 (R3y )) kuEl (x)kH 1 (R3 ) kw(x)kH 1 (R3y ) + kS(ξ; h)kL(H 1 (R3y ),L2 (R3y )) kuEl (x)kH 1 (R3 ) kw(x)ky ≤ CE0 h2 ([ξ]h + 1) .

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For the third term, we have, taking into account (4.9), |hLE (x)(A1 (ξ; h)A2 (ξ; h) − 1)uEl (x), A1 (ξ; h)A2 (ξ; h)w(x)iy | ≤ C sup |V (x, y)| k(A1 (ξ; h)A2 (ξ; h) − 1)uEl (x)kH −1 (R3y ) x,y

× kA1 (ξ; h)A2 (ξ; h)w(x)kH 1 (R3y ) ≤ C 0 h2 ([ξ]h + h2 )kuEl (x)kH 1 (R3y ) kw(x)kH 1 (R3y ) ≤ CE h2 ([ξ]h + h2 ) . The fourth term can be handled in the same way. For the fifth term, taking into account (4.14), |hLE (x)A1 (ξ; h)A2 (ξ; h)R1 (ξ; h)(v ⊗ uEl (x)), A1 (ξ; h)A2 (ξ; h)R1 (ξ; h)(u ⊗ w(x))iy | ≤ C sup |V (x, y)| kR1 (ξ; h)(u ⊗ uEl (x))ky kR1 (ξ; h)(v ⊗ w(x))ky x,y

≤ C 0 h2 ([ξ]h + h2 )kuEl (x)kH 1 (R3y ) kw(x)kH 1 (R3y ) |v| |u| ≤ CE h2 ([ξ]h + h2 )|v| |u| . The other terms can be handled in the same way. Taking u ⊗ w(x) = u ⊗ uEk (x) or u ⊗ w(x) = XE (x, ξ; h)hE (x, ξ; h)(ν ⊗ uEk (x)) with ν ∈ C 2 , we obtain that (k,l) ˜ h) − λ1 (x))δk,l + h2 F (ξ; h) + O(h2 ([ξ]h + h2 )) . gE,E (x, ξ; h) = (E − Θ(ξ;

Using the same arguments one can check that, for x ∈ λ−1 1 (I) X 1 − hSE ∂ξj pE ∂xj TE (⊗uEl (.)), uEk (x)iy i j =−

1X h∂ξj pE ∂xj (⊗uEl (.)), uEk (x)iy + O(h2 ([ξ]h + 1)) , i j

1X hSE ∂ξ2i ξj pE ∂x2i xj TE (⊗uEl (.)), uEl (x)iy 2 i,j −

X hSE ∂ξi pE ∂xi XE ∂ξj pE ∂xj TE (⊗uEl (.)), uEk (x)iy i,j

X + h∂ξi (hE XE )∂xi ∂ξj pE ∂xj TE (⊗uEl (.)), uEk (x)iy i,j

1X 2 = h∂ pE ∂x2i xj (⊗uEl (.), uEk (x)iy 2 i,j ξi ξj −

X h∂ξi pE ∂xi XE ∂ξj pE ∂xj (⊗uEl (.)), uEk iy + O(h2 ([ξ]h + 1)) . i,j

This proves (5.4).

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Now we finish the proof of Theorem 5.2. The result of Lemma 5.4 shows the the −1 ± ˜ symbol of G± E,E , for x ∈ λ1 (I), is given by σ(GE,E ) = (E − Θ(ξ; h) − λ1 (x))1C⊕C 2 modulo rests of order O h ([ξ]h + 1) and, by construction, one has a natural bijection between Ker(BEζ − E) and Ker(G± E,E ). Moreover, one can easily check that ˜ 2 G± (x, hDx ; h) = G± (x, hDx ; h)K ˜2 . K E,E −E,E

As an application of Theorem 5.2, we shall prove that the eigenvalues of BE (h) can be approximated by those of the effective Hamiltonian. Theorem 5.4. Let P0E (h) denote the Dirichlet realization of P0E (h) = (−h2 ∆x + λ1 (x))1 + h

3 X (Bj (x)hDxj + hDxj Bj (x)) j=1

1 E 2 − EM (x) 2µ1,2

+ h2 C(x) + on Ω, where

"

Bj (x) = Bj∗ (x) =

C(x) =

3 X

(5.5) #

RehDxj u(x), u(x)iy

hDxj Ku(x), u(x)iy

hDxj Ku(x), u(x)iy

−RehDxj u(x), u(x)iy

kDxj u(x)k2y +

j=1

,

1 kDyj u(x)k2y 1C⊕C , 2µ1,2

" 3 1 X RehDyj u(x), u(x)iy M (x) = µ1,2 j=1 hDyj Ku(x), u(x)iy

hDyj Ku(x), u(x)iy −RehDyj u(x), u(x)iy

# .

Put I = (−∞, a(h)], where a(h) > λ0 is a function of h such that a(h) ≤ λ0 + ch

dist(a(h), Sp(P0E (h)) ≥ c−1 h2 ,

for some c > 0 .

Then, for h > 0 small enough, there exists a bijection b : Sp(BE (h)) ∩ I → Sp(P0E (h)) ∩ I such that |b(E) − E| = o(h2 )

for any E ∈ Sp(BE (h)) ∩ I .

(5.6)

Proof. We have seen before that µ ∈ Sp(BE (h)) ∩ I if there is E = µ + O(e−ε0 /h ) ± such that 0 ∈ Sp(G± E,E ). Moreover, the symbol of GE,E can be computed modulo terms of order o(h2 ). Actually, making the Taylor expansion with respect to ξ and taking into account Lemma 5.4 we obtain, for bounded ξ, 2h X 2 σ(G± ) (x, ξ; h) = E − ξ − λ (x) − ξj h∂xj ul (x), uk (x)iy k,l 1 E,E i j + h2 h(−∆x ul (x), uk (x)iy − h2 hFE ul (x), uk (x)iy ˜ E (x, 0; h)h2 FE )(⊗ul (x)(.)), ⊗uk (x)(.)iy + h2 hFE X X + h2 ξi ξj Ci,j (x, ξ, E, E) + O(h3 ) , i,j

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where Ci,j (., ., ., E) is holomorphic in E and s ! µ1,2 (Dy − E)2 1 4 FE (Dy ; h) = 4 1+h − 1 = 2 Θ1 (0, Dy − E; h) 2 h µ1,2 h and ˜ E = eihE,yi XE (x, 0; h)e−ihE,yi . X Now, let χ ∈ C0∞ (R), supp χ ⊂ [−δ, δ] with δ sufficiently small, and set χh,ε ∈ C0∞ (R3 ), χh,ε (η) = χ(|η|hε ), 0 < ε < 2. Observe that 1 2 (Dy − E) ul (x), uk (x) χh,ε (Dy ) FE − 2µ1,2 y ≤ C|hχ ˆh,ε (η)h4 (η − E)4 uˆl (x), uˆk (x)iy | ≤ CE h4−2ε kul (x)kH 1 (R3y ) kuk (x)kH 1 (R3y ) for some constant CE > 0 independent of h. Moreover, for u, v ∈ C 2 ˜ E h2 FE χh,ε (Dy )(u ⊗ ul (x)), v ⊗ uk (x)iy | |hFE X ˜ E h2 FE χh,ε (Dy )(u ⊗ ul (x))kH 1 kFE uk (x)kH −1 |v| ≤ kX ≤ Ckh2 FE χh,ε (Dy )ul (x)ky kuk (x)kH 1 |v||u| ≤ CE h2−ε kul (x)kH 1 kuk (x)kH 1 |u||v| . On the other hand, since the terms 1 2 ˜ E h2 FE (⊗ul (x)(.)), ⊗uk (x)(.)iy = O(1) FE − (Dy −E) ul (x), uk (x) , hFE X 2µ1,2 y uniformly in h, we have (1 − χh,ε (Dy )) FE −

1 (Dy − E)2 ul (x), uk (x) 2µ1,2 y

˜ E h2 FE (1 − χh,ε (Dy ))(⊗ul (x)(.)), ⊗uk (x)(.)iy = o(1) + hFE X as h → 0. In conclusion we get, for x ∈ λ−1 (I), 2 σ(G± E,h )k,l (x, ξ; h) = (E − ξ − λ1 (x))δk,l −

2h X ξj h∂xj ul (x), uk (x)iy i j

+ h2 h(−∆x ul (x), uk (x)iy − + h2

X

h2 h(Dy − E)2 ul (x), uk (x)iy 2µ1,2

k,l ξi ξj Ci,j (x, ξ, E; h) + o(h2 ) .

i,j

Recalling that u1 = u and u2 = Ku, the proof of the theorem can be done using exactly the same arguments of Proposition 1.5 in [17].

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Remark 5.5. Notice that the Theorem 5.6 told us that (formally) the eigenvalues of BE (h) in I agrees, modulo o(h2 ), with those of the operator PE (h) = −h2 ∆x +

h2 D2 + QE (x) 2µ1,2 y

QE (x) = E(E + Dy ) − 1 + LE (x) . This means that the eigenvalues of BE (h) ∩ I coincides modulo o(h2 ) with the eigenvalues of the many body Hamiltonian PE (h) where the particles of type N are considered as non relativistic particles. So, the relativistic effects of the particles of type N are of order of o(h2 ). Remark 5.6. If the eigenfunction u(x) associated to λ1 (x) belongs to H 1+k (R3y ) for some k > 0 then, under the assumptions of Theorem 5.6, we obtain that |b(E) − E| = O(h2+p ) for any E ∈ Sp(BE (h)) ∩ I

(5.7)

where p = min{4k, 1/2}. Actually, in this case, let take χh,ε (Dy ) with ε = 2. Observe that, 1 (Dy − E)2 ul (x), uk (x) χh,2 (Dy ) FE − 2µ1,2 y ≤ CE hs kul (x)kH 1+k (R3y ) kuk (x)kH 1+k (R3y ) where s = min{4, 4k} and 1 2 (Dy − E) ul (x), uk (x) (1 − χh,2 (Dy )) FE − 2µ1,2 y ≤ CE h4k kul (x)kH 1+k (R3y ) kuk (x)kH 1+k (R3y ) for some constant CE > 0 independent of h. Finally ˜ E h2 FE (⊗ul (x)(.)), ⊗uk (x)(.)iy | |hFE X ≤ CE hr kul (x)kH 1+k (R3y ) kuk (x)kH 1+k (R3y ) where r = min{2, 4k}. Then using the same arguments of Theorem 5.6, (5.7) follows. 6. Coulomb-Type Potentials In this section, we want to show how one can extend the result of Theorem 5.6 in the case of Coulomb type potentials. Let us consider the operator BE (h) in (2.7) with a Coulomb type potential V (x, y) = −

e1 e2 e3 − + |y + x/2µ1 | |y − x/2µ2 | |x|

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and let us assume that the constants ej , j = 1, 2, 3, are positive and chosen in such a way that, for h sufficiently small, BE (h) can be realized as a selfadjoint operator in L2 (R6 ; C 3 ) with domain in D(BE (h)) = H 1 (R6 ; C 3 ) bounded from below (uniformly with respect to h) and such that spess (BE (h)) ⊂ [0, +∞[ (see: ([4, 11, 29, 30]). Moreover let us assume, as before, that (H3 ), (H4 ) holds. In such a situation, the ˜ E (x) can be removed using the techniques obstacle due to the non smoothness of Q of [17], that is, by replacing BE (h) in each coordinate path ω ∈ R3x , with a unitarily conjugated operator U (x)BE (h)U (x)−1 , where U (x) implements a diffeomorphism in R3y so that the singularities of the potential become x-independent. Actually, for a fixed x0 ∈ R3 \ {0}, there exists fj ∈ C0∞ (R3 ; R), j = 1, 2 with fj (−x/2µ1 ) = δj,1 , fj (x/2µ2 ) = δj,2 such that, if we set, for x in a neighborhood ω of x0 , F0 (x, s) = s −

(x − x0 ) (x − x0 ) f1 (s) + f2 (s) 2µ1 2µ2

one has V (., F0 (., y))(−∆y + 1)−1/2 ∈ C ∞ (ω, L2 (R3y )) . Hence, if U0 (x) : L2 (R3y ) → L2 (R3y ), denotes the unitary operator associated to the diffeomorphism F0 , such that (U0 (x)φ)(y) = φ(F0 (x, y))|det ∂y F0 (x, y)|1/2 , one can try to deal with the operator U0 (x)BEζ (h)U0 (x)−1 that is smooth with respect to the x-variable. Notice that, if we denote by K(hDx ; h) = K(hDx , Dy ; h) one of the operators Θ, Aj , Rj in (2.7) one has that U0 (x)K(hDx ; h)U0 (x)−1 is a formal pseudodifferential operator with a L(H 1 (R3y ); L2 (R3y ))-valued symbol of order 1 (and weight [ξ]h + 1): ∞ X

U0 (x) op1 (kj (x, ξ, y, y 0 , η; h))U0 (x)−1 hj ,

j=0

here op1 (kj (x, ξ, y, y 0 , η; h)) is defined by the oscillatory integral Z 0 op1 kj (x, ξ, y, y 0 , η; h)φ = (2π)−3/2 ei(y−y )η kj (x, ξ, y, y 0 , η; h)φ(y 0 )dy 0 dη . Moreover, its leading term is U0 (x) op1 (K(ξ + hm(x, y 0 )η; η; h))U0 (x)−1 where m(x, y 0 ) = f1 (G0 (x, y 0 ))/2µ1 − f2 (G0 (x, y 0 ))/2µ2 and G0 (x, y 0 ) is the inverse of F0 . The problem is that the quantity k∂ξα ∂xβ U0 (x) op1 (kj (x, ξ, y 0 , η; h))U0 (x)−1 kL(H 1 (R3y ;L2 (R3y )) behaves like ([ξ]h + 1)1+|β|−|α| and then, for this reason, we cannot regard U0 (x)K(hDx ; h)U0 (x)−1 (and hence U0 (x)BEζ (h)U0 (x)−1 ) as an h-admissible operator. As a consequence we are not able to invert the Grushin operator associated to U0 (x)BEζ (h)U0 (x)−1 in the class of h-admissible operator as in the case of

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C 0 -potential. On the other hand, we can reduce ourself to deal with formal pseudodifferential operators, and, as in [21] or [17], Sec. 3, one can construct, using a a partition of unity, a formal pseudodifferential operator G±,f E,E such that, for E ∈ I E ∈ Sp(BEζ )

if and only if

0 ∈ Sp(G±,f E,E ) .

−1 Moreover, the symbol of G±,f E,E , for x ∈ λ1 (I), is given by 2 ˜ σ(G± E,E ) = (E − Θ(ξ; h) − λ1 (x))1C⊕C + O(h ([ξ]h + 1)) .

Then, by slightly modifying the estimates in the proof of Lemma 5.4 and Theorem 5.6 one can show that Theorem 5.6 holds also in the case of Coulomb-type potentials. 7. Multiparticles System Let us consider the Brown–Ravenhall equation (1.1). The results of [5, 27] show that (1.1)0 is unitary equivalent to the operator B=

n+p+1 X

˜ Ej (Dzj ) + W

j=1

acting on L2 (R3(n+p+1) ; C n+1+p ) , Where we denote by C n+1+p = C n+1 ⊗ C p with C n+1 = C2 ⊗ C2 ⊗ · · · ⊗ C2 , C p = | {z } n+1

C2 ⊗ C2 ⊗ · · · ⊗ C2 . Moreover, here | {z } p

˜ = W

n+p+1 Y

Aj

V˜ +

j=1

n+p+1 X

i Y

i=1

k=1

! Rk

V˜

i Y

Rk

k=1

!! n+p+1 Y

Aj

j=1

and Rj , Aj are given by (2.3). In the following we suppose that the masses of the particles of type E are Pn+1 1 µ identical and all equal to 1, and we set mj = h2j , j = 1, . . . , n + 1, h2 = j=1 2m j and the light speed c = 1. As in the case treated in the previous sections, up to translations and unitary equivalences, we are lead to deal with the operator BE (h) = Θ(hDx , h) + QE (x, hDx ; h) , QE (x; hDx ; h) =

p X (E(E + Dyj ) − 1) + WE (x, hDx ; h) , j=1

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where x = (x1 , . . . , xn ) ∈ R3n , y = (y1 , . . . , yp ) ∈ R3p denotes respectively the coordinates of the particles of type N and E in the center of mass frame and ˜ Θ(hDx ; h) =: Θ(hD x ; h) + Θ1 (hDx ; h) ,     2 n+1 n n 2 X X X h h D D 1 xj xj ˜ + Θ(hD µj+1 E − µj  , x ; h) =: 2 µ1 E  h µ µ 1 j+1 j=1 j=1 j=1 Θ1 (hDx ; h) = Θ1 (hDx , Dy ; h)      n n 2 2 X X h Dx j h Dx j µ1  =: 2 E  + {h2 Dy }p  − E  h µ µ1 1 j=1 j=1 2 2 n X h Dxj h Dx j µj+1 2 E − {h Dy }p − E . + h2 µj+1 µj+1 j=1 Here we have set {h2 Dy }p =:

Pp i=1

h2 Dyi

Pn+1 j=1

µj . Moreover

WE (x, hDx ; h) = WE (x, y, hDx , Dy ; h) =

n+1 Y

Aj

LE +

j=1

LE (x) =

p Y

AEn+1+j

n+1 X

i Y

i=1

k=1

V +

j=1 i Y

×V

! Rk i Y

i=1

k=1

E Rn+1+k

Rk

k=1

p X

!!

LE

i Y

p Y

!! n+1 Y

Aj ,

j=1

! E Rn+1+k

AEn+1+j ,

j=1

k=1

where V = V (x, y) is just the potential W in the new coordinate system and Pn σ1 · ( j=1 h2 Dxj /µ1 + {h2 Dy }p ) P R1 = R1 (hDx ; h) =: , 1 + E( nj=1 h2 Dxj /µ1 + {h2 Dy }p ) Rj = Rj (hDx ; h) =: E Rn+1+i =:

σj · (h2 Dxj−1 /µj − {h2 Dy }p ) , 1 + E(h2 Dxj−1 /µj − {h2 Dy }p )

σn+1+i · (E + Dyi ) 1 + E(E + Dyi )

i = 1, . . . , p ,

Aj = Aj (hDx ; h) =: (1 + Rj2 )−1/2 , AEn+1+i =: (1 + (RiE )2 )−1/2 ,

j = 2, . . . , n + 1 ,

j = 1, . . . , n + 1 ,

i = 1, . . . , p .

On the potential V we make the same assumptions (H1 ), (H2 ) given in Sec. 3 3p 3n i.e. we suppose that V ∈ C 0 (R3n x × Ry ) an that for any y ∈ R , x → V (x, y) ∈

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C ∞ with |∂xα V (x, y)| = O(1) for any α ∈ N3n . Under this assumptions, BE (h) is realized as a selfadjoint operator on L2 (R3(n+p) , C n+1+p ) with domain D(BE (h)) = H 1 (R3(n+p) ; C n+1+p ). Set now ˜ E (x) = Q ˜ E (x, y, Dy ; h) =: Q

p X (E(E + Dyj ) − 1) + LE (x, y, Dy ; h) j=1

p ˜ ˜ acting on L2 (R3p y ; C ). As before, QE (x) is unitary equivalent to Q0 (x) and we set ˜ 0 (x) and we suppose that λ1 (x) < lim inf |y|→+∞ V (x, y). By λ1 (x) =: inf Sp Q eventually adding a constant, we may assume that lim inf |x|→+∞ λ1 (x) = 0. Let us choose b ∈ R such that λ0 =: inf x∈R3 λ1 (x) < b < 0 and let us set I =] − ∞, b]. We ˜ 0 (x) commutes with the antilinear assume, also that (H3 ) holds. The operator Q Kramers operator (2) p 2 3p p Kp : L2 (R3p Kp u = σ1 ⊗ σp(2) ⊗ · · · ⊗ σp(2) Ju y ; C ) → L (Ry ; C ) ,

where Ju = u¯. Since Kp2 = (−1)p , then, if p is odd and u(x) is an eigenfunction ˜ (normalized in L2 (R3p y ; C)) of Q0 (x) associated to λ1 (x) we always have that u and Kp u are linearly independent and the multiplicity of λ1 (x) is even. On the other hand, if p is even, the multiplicity of λ1 (x) may be lower than two. For this reason we assume that one of this two assumptions holds: (H4 )1 For x ∈ Ω, λ1 (x) is exactly of multiplicity 1. or (H4 )2 For x ∈ Ω, λ1 (x) is exactly of multiplicity 2. Under assumptions (H1 ), (H2 ), (H3 ) and (H4 )j with j = 1 or j = 2, an analogous of Lemma 3.1 holds. In order to prove that BE (h) is an h-admissible operator Pn (of weight [ξ]h = j=1 [ξj ]h ) with operator valued symbol we need an analogous of Lemma 4.4. The main point consists into proving the following lemma. Lemma 7.1. (a) There exists a constant C > 0 independent of h such that, for any u ∈ H 1/2 (R3p ; C n+1+p ) hΘ(ξ; h)u, uiy ≥ C[ξ]h kuky .

(7.1)

(b) We have 3p kΘ1 (ξ; h) − S(ξ; h)kL(H 1 (R3p = O(1) , 2 y ),L (Ry ))

(7.2)

2 kΘ1 (ξ; h) − S(ξ; h)kL(H 1 (R3p −1 (R3p )) = O(h ) y ),H y

(7.3)

n+1+p n+1+p for some S(ξ; h) : H 1 (R3p ) → L2 (R3p ) with y ;C y ;C 2 3p = O(h [ξ]h ) . kS(ξ; h)kL(H 1 (R3p 2 y );L (Ry )

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Proof. (a) Observe that we can write, for any zk , w ∈ R3 , k = 1, . . . , n ! Pn 1 1 1 µ1 zk j=1 zj + zk = −w + +w 2µk+1 2λk 2 µk+1 2λk µ1 +

where we have set λk = 1 2

n 1 X zj µj+1 w − , 2λk j=i µj+1 j6=k

Pn

j=1 j6=k

µj+1 . Hence, since E is convex and even, we have

µ1 zk −w −1 + E µk+1 2λk +

!

Pn E

j=1 zj

µ1

! −1

+w

n 1 X zj µj+1 E − w − 1 ≥ E (αk zk ) − 1 , 2λk j=1 µj+1 j6=k µ

j+1 ; j = 0, . . . , n, where we have set αk = ( 2µ1k+1 + 2λ1k ). Taking C > sup{ 12 , 2λ k k = 1, . . . , n} we get, that for some constant c > 0, " n ! !# Pn X zk j=1 zj Cn E −w −1 + E +w −1 µk+1 µ1

k=1

≥

n X

n X (E (αk zk ) − 1) ≥ c (E(zk ) − 1) .

k=1

k=1

ˆ η; h) the Fourier transform with respect to y of Θ(ξ, Dy ; h) and Denoting by Θ(ξ, applying the previous inequality inequality with zk replaced by hξk and w replaced by w = {h2 η}p we get ˆ η; h) ≥ C[ξ]h Θ(ξ, with a new constant C > 0 independent of h and this proves (7.1). (b) Since, for any x1 , x2 ∈ R3 , |E(x1 + x2 ) − E(x1 )| ≤ E(x2 ) we have   n+1 X 1 ˆ 1 (ξ, η; h)| ≤  |Θ µj  2 E({h2 η}p ) . h j=1 On the other hand, if we set ˆ η; h) = P 1 S(ξ, n+1 j=1

* Pn µj

j=1 P n

E

hξj

j=1

hξj

−

n X k=1

µ1

X hξ k , ηj p

E

hξk µk

we have ˆ η; h)| ≤ Ch2 [ξ]h |S(ξ,

p q X 1 + ηj2 , j=1

j=1

+ ,

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and, by Taylor expansion, we have ˆ 1 (ξ, η; h) − S(ξ, ˆ η; h)| ≤ C 0 {h2 η}2p |Θ for some C, C 0 > 0. Hence, there exists C 00 ≥ max{n + 1, C 0 } and C˜ > 0 such that p q X ˆ η; h)| ≤ C 00 1 (E({h2 η}p ) − 1) ≤ C˜ ˆ 1 (ξ, η; h) − S(ξ, 1 + ηj2 . |Θ h2 j=1

This proves (7.2) and (7.3). Using Lemma 7.1 and estimates for Aj an Rj analogous to that of Lemma 4.4 we can prove the following theorem. ˜ 1 (R3(n+p) ; Theorem 7.2. The operator BE (h) is a bounded operator from H n+1+p 2 3(n+p) n+1+p C ) to L (R ;C ) and its operator valued-symbol σ(BE (h))(x, ξ; h) n+1+p n+1+p is an application from T ∗ R3 to L H 1 (R3p )), L2 (R3p )). y ;C y ;C As in the diatomic case, we are able to invert the Grushin operator associated to BEζ (h) (the definition of BEζ (h) is analogous to (5.1)) ζ BE (h) − E Lj ζ,j BE,E (h) = , t Lj 0 where we have set n+1 3p n+1+p L1 : L2 (R3n ) → L2 (R3n ), x ;C x × Ry ; C

L1 (v)(x, y) = v(x) ⊗ w1E (x, y) , n+1 3p n+1+p L2 : L2 (R3n ⊕ C n+1 ) → L2 (R3n ⊕ C n+1+p ) , x ;C x × Ry ; C

L2 (v)(x, y) = (v(x) ⊗ w1E (x, y), v(x) ⊗ w2E (x, y) . Pp

Here w1E (x) = e−i j=1 hE,yj i w(x), w2E (x) = e−i for x ∈ λ−1 1 (I) as in Lemma 3.1.

Pp

j=1 hE,yj i

Ku(x) and w(x) = u(x)

Theorem 7.3. Under assumptions (H1 ), (H2 ), (H3 ) and (H4 )j with j = 1 or ζ,j j = 2, the Grushin operator BE,E is invertible for all E ∈ I and, modulo ζ,j −1 ∞ O(h ) its inverse (BE,E ) is an h-admissible operator which is bounded from 2 3(n+p) n+1+p 1 3n ˜ ˜ 1 (R3(n+p) , C n+1+p ) ⊕ L2 (R3n , C n+1 ) if L (R ,C ) ⊕ H (R , C n+1 ) to H 2 3(n+p) n+1+p 1 ˜ j = 1 and from L (R ,C ) ⊕ H (R3n , C n+1 ⊕ C n+1 ) to ˜ 1 (R3(n+p) , C n+1+p ) ⊕ L2 (R3n , C n+1 ⊕ C) H if j = 2. Furthermore, writing ζ,j −1 (BE,E ) =

"

GjE,E

G+,j E,E

G−,j E,E

G±,j E,E

# ,

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949

−1 the symbol of G±,j E,E , for x ∈ λ1 (I), is given by 2 ˜ σ(G±,j E,E ) = (E − Θ(ξ; h) − λ1 (x))1C n+1 + O(h ([ξ]h + 1))

if j = 1 ,

σ(G±,j E,E )

if j = 2

˜ h) − λ1 (x))1C n+1 ⊕C n+1 + O(h ([ξ]h + 1)) = (E − Θ(ξ; 2

and we have, for any E ∈ I, E ∈ Sp(BEζ ) (1)

iff (2)

0 ∈ Sp(G±,j E,E ) . (2)

(2)

Moreover, ¯ and by " if we denote by# Kn+1 = (σ1 ⊗ σ2 ⊗ · · · ⊗ σn+1 )J, Ju = u (1) O K (2) (j) n+1 Kn+1 = such that (Kn+1 )2 = (−1)n we have (1) −Kn+1 0 ± Kn+1 G± E,E (x, hDx ; h) = G−E,E (x, hDx ; h)Kn+1 (j)

(j)

for j = 1, 2 .

Proof. Taking into account Lemma 7.1, the proof of the theorem can be done following exactly the same kind of arguments of the proof of Theorem 5.2. Remark 7.4. We can obtain an analogous of Theorem 5.6. This told us that (formally) the eigenvalues of BE (h) in I agrees, modulo o(h2 ), with those of the operator PE (h) = −h2 ∆x + h2 p(Dy ) + QE (x) , QE (x) =

p X

E(E + Dyj ) − 1 + LE (x) ,

j=1

where p(Dy ) is a second order differential operator. The same result holds also for Coulomb-type potentials. Acknowledgments The author would like to gratefully acknowledge Christian Brouder for helpful discussions and enlightening conversations on the subject. References [1] S. Agmon, Lecture on Exponential Decay of Solutions of Second Order Elliptic Equations, Math. Notes 29, Princeton University Press, 1982. [2] P. Aventini and R. Seiler, “On the electronic spectrum of the diatomic molecule”, Comm. Math. Phys. 22 (1971) 269–279. [3] A. Balazard-Konlein, “Calcul fonctionnel pour des opérateurs h-admissibles a ` symbole opérateur et applications”, Ph.D. Thesis, University of Nantes 1985. [4] A. A. Balinsky and W. D. Evans, “Stability of one-electron molecules in the BrownRavenhall model”, Comm. Math. Phys. 202(2) (1999) 481–500. [5] H. A. Bethe and E. C. Salpeter, Quantum Theory of One and Two-Electron Atoms, Academic Press, New York, 1978.

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[6] M. Born and R. Oppenheimer, “Zur quantentheorie der Molekeln”, Annalen der Physik 84 (1927) 457. [7] R. Bott and L. W. Tu, “Differential Forms in Algebraic Topology”, Berlin, Heidelberg, New York, Springer, 1982. [8] G. E. Brown and D. G. Ravenhall, “On the interaction of two electrons”, Proc. R. Soc. London, A208 (1951) 552–559. [9] V. I. Burenkov and W. D. Evans “On the evaluation of the norm of an integral operator associated with the stability of one-electron atoms”, Proc. R. Soc. Edinb., Sect. A, Math. 128(5) (1998) 993–1005. [10] J. P. Combes, P. Duclos and R. Seiler, “The Born Oppenheimer approximation”, pp. 185–212 in Rigorous Atomic and Molecular Physics, eds. G. Velo and A. Wightman, New York Plenum 1981. [11] W. D. Evans, P. Perry and H. Siedentop, “The spectrum of relativistic one-electron atoms according to Bethe and Salpeter”, Comm. Math. Phys. 178 (1996) 733–746. [12] G. A. Hagedorn, “High order corrections to the time-independent Born-Oppenheimer approximation I: smooth potentials”, Ann. Inst. H. Poincarè A47 (1987) 1–16. [13] G. A. Hagedorn, “High order corrections to the time independent Born-Oppenheimer approximation II: Diatomic coulomb systems”, Comm. Math. Phys. 116 (1988) 23–44. [14] B. Helffer and J. Sj¨ ostrand, “Multiple wells in semiclassical limit I”, Comm. Partial Differential Equation, 9(4) (1984) 337–408. [15] D. Husemoller, Fiber Bundles, Berlin, Heidelberg, New York, Springer 1975. [16] T. Kato, Perturbation Theory for Linear Operator, Springer Verlag. [17] M. Klein, A. Martinez, R. Seiler and X. P. Wang, “On the Born-Oppenheimer Expansion for Polyatomic Molecules”, Comm. Math. Phys. 143(3) (1992) 607–639. [18] Y. Ishikawa and K. Koc, “Relativistic many-body perturbation theory based on the no-pair Dirac-Coulomb-Breit Hamiltonian: Relativistic correlation energies for the noble-gas sequence through RN (Z=86), the group-IIB atoms through Hg, and the ions of Ne isoelectronic sequence”, Phys. Rev. A50(6) (1994) 4733–4742. [19] H. J¨ orgen, Aa. Jensen, K. G. Dyall, T. Saue and K. F. Fraegri Jr. “Relativistic four-component multiconfigurational self-consistent-field theory for molecules: Formalism”, J. Chem. Phys. 104(11) (1996) 4083–4097. [20] R. T. Lewis, H. Siedentop and S. Vugalter,“The essential spectrum of relativistic multi-particle operators”, Ann. Inst. H. Poincarè (Phys. Theor.) 67(1) (1997) 1–28. [21] A. Martinez, “Développements asymptotiques et effet tunnel dans l’approximation de Born-Oppenheimer”, Ann. Inst. H. Poincarè 49(3) (1989) 239–257. [22] M. Reed and B. Simon, Methods of Modern Mathematical Physics, 4, Academic Press, New York, 1978. [23] D. Robert, Autour de l’Approximation Semi-Classique, Birk¨ auser, 1987. [24] R. Seiler, “Does the Born-Oppenheimer approximation work?”, Helv. Phys. Acta 46 (1973) 230–234. [25] B. Simon, “Semiclassical limit of low laying eigenvalues I, non degenerate minima: asymptotic expansion”, Ann. Inst. H. Poincarè 38(3) (1983) 295–307. [26] V. Sordoni, “Born-Oppenheimer expansion for diatomic molecules: excited states”, C. R. Acad. Sci., Serie I 320(9) (1995) 1091–1097. [27] J. Sucher, “Foundation of the relativistic theory of many-electron atoms”, Phys. Rev. A22(2) (1980) 348–362. [28] J. Sucher, “Relativistic many-electron Hamiltonians”, Phys. Scripta. 36 (1987) 271–281.

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[29] C. Tix, “Strict positivity of a relativistic Hamiltonian due to Brown and Ravenhall”, Bull. Math. Soc. 30(3) (1999) 283–290. [30] C. Tix, “Self-adjointness and spectral properties of a pseudo-relativistic Hamiltonian due to Brown and Ravenhall”, preprint 1997.

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THE PIN GROUPS IN PHYSICS: C, P AND T

´ MARCUS BERG and CECILE DeWITT-MORETTE Department of Physics and Center for Relativity, University of Texas, Austin, TX 78712, USA SHANGJR GWO Department of Physics, National Tsing Hua University, Hsinchu 30034, Taiwan ERIC KRAMER 290 Shelli Lane, Roswell, GA 30075 USA

Received 1 June 2000 Revised 22 December 2000

A simple, but not widely known, mathematical fact concerning the coverings of the full Lorentz group sheds light on parity and time reversal transformations of fermions. Whereas there is, up to an isomorphism, only one Spin group which double covers the orientation preserving Lorentz group, there are two essentially different groups, called Pin groups, which cover the full Lorentz group. Pin(1, 3) is to O(1, 3) what Spin(1, 3) is to SO(1, 3). The existence of two Pin groups offers a classification of fermions based on their properties under space or time reversal finer than the classification based on their properties under orientation preserving Lorentz transformations — provided one can design experiments that distinguish the two types of fermions. Many promising experimental setups give, for one reason or another, identical results for both types of fermions. These negative results are reported here because they are instructive. Two notable positive results show that the existence of two Pin groups is relevant to physics: • In a neutrinoless double beta decay, the neutrino emitted and reabsorbed in the course of the interaction can only be described in terms of Pin(3, 1). • If a space is topologically nontrivial, the vacuum expectation values of Fermi currents defined on this space can be totally different when described in terms of Pin(1, 3) and Pin(3, 1). Possibly more important than the two above predictions, the Pin groups provide a simple framework for the study of fermions; it makes possible clear definitions of intrinsic parities and time reversal; it clarifies colloquial, but literally meaningless, statements. Given the difference between the Pin group and the Spin group it is useful to distinguish their representations, as groups of transformations on “pinors” and “spinors”, respectively. The Pin(1, 3) and Pin(3, 1) fermions are twin-like particles whose behaviors differ only under space or time reversal. A section on Pin groups in arbitrary spacetime dimensions is included. 953

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Contents 0. Dictionary of Notation 1. Introduction 2. Background 2.1. As seen by physicists • Parity violation • Standard Model 2.2. As seen by Wigner 2.3. As seen by mathematicians 3. The Pin Groups in 3 Space, 1 Time Dimensions 3.1. The Pin groups 3.1.1. Pin(1, 3) 3.1.2. Pin(3, 1) 3.2. A Spin group is a subgroup of a Pin group 3.3. Pin group and Spin group representations on finite-dimensional spaces; classical fields. 3.3.1. Pinors 3.3.2. Spinors 3.3.3. Helicity 3.3.4. Massless spinors, massive pinors 3.3.5. Copinors, Dirac and Majorana adjoints 3.3.6. Charge conjugate pinors in Pin(1, 3) 3.3.7. Majorana pinors 3.3.8. Unitary and antiunitary transformations 3.3.9. Invariance of the Dirac equation under antiunitary transformations 3.3.10. CP T invariance 3.3.11. Charge conjugate pinors in Pin(3, 1) 3.4. Pin group and Spin group representations on infinite-dimensional spaces; quantum fields. 3.4.1. Particles, antiparticles 3.4.2. Fock space operators, unitary and antiunitary 3.4.3. Intrinsic parity 3.4.4. Majorana field operator 3.4.5. Majorana classical field vs. Majorana quantum field 3.4.6. CP T transformations 3.5. Bundles; Fermi fields on manifolds 3.5.1. Pinor coordinates 3.5.2. Dirac adjoint in Pin(1, 3) 3.5.3. Dirac adjoint in Pin(3, 1) 3.5.4. Pin structures 3.5.5. Fermi fields on topologically nontrivial manifolds 3.6. Bundle reduction 4. Search For Observable Differences 4.1. Computing observables with Pin(1, 3) and Pin(3, 1) 4.1.1. Trace theorems 4.1.2. Spin sums 4.2. Parity and the Particle Data Group publications 4.2.1. Parity conservation 4.3. Determining parity experimentally 4.3.1. Selection rules: Pion decay

955 959 960 960

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The Pin Groups in Physics: C, P and T

4.3.2. Selection rules: Three-fermion decay 4.3.3. Selection rules: Positronium 4.3.4. Decay rates; cross sections 4.4. Interference, reversing magnetic fields, reflection 4.4.1. Reversing magnetic fields 4.4.2. Reflection 4.5. Time reversal and Kramer’s degeneracy 4.6. Charge conjugation 4.6.1. Positronium 4.6.2. Neutrinoless double beta decay 5. The Pin Groups in s Space, t Time Dimensions 5.1. The difference between s + t even and s + t odd 5.1.1. The twisted map 5.2. Chirality 5.3. Construction of gamma matrices; periodicity modulo 8 5.3.1. Onsager construction of gamma matrices 5.3.2. Majorana pinors, Weyl–Majorana spinors 5.4. Conjugate and complex gamma matrices 5.5. The short exact sequence 11 → Spin(t, s) → Pin(t, s) → Z2 → 0 5.6. Grassman (superclassical) pinor fields 5.7. String theory and pin structures 6. Conclusion 6.1. Some facts 6.2. A tutorial 6.2.1. Parity 6.2.2. Time reversal 6.2.3. Charge conjugation 6.2.4. Wigner’s classification and classification by Pin groups 6.2.5. Fock space and one-particle states 6.3. Avenues to explore 7. Acknowledgments Appendices A. Induced transformations B. The isomorphism of M4 (R) and H ⊗ H C. Other double covers of the Lorentz group D. Collected calculations E. Collected references References

955

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998 999

1000 1000 1003 1004

1008 1009 1012 1013 1015 1015 1015

1017 1017 1018

1028

0. Dictionary of Notation The article proper begins with Sec. 1. We have occasionally changed some of our earlier notations to conform to the majority of users. As much as possible, we have tried to use the usual notation — but introducing different symbols for different objects when it is essential to distinguish them. For example, we distinguish the Spin group and the two Pin groups (sometimes still known as Spin groups) but we speak globally of spin 1/2 particles (lower case “s”) for all particles, whether they are represented by spinors

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or pinors of either type. This agrees with intuitive notion of spin as “the behavior of a field or a state under rotations” [88]. Our primary references are Peskin and Schroeder [87], Weinberg [115] and Choquet-Bruhat et al. [28, 30]. Groups Lorentz group O(1, 3) leaves ηαβ xα xβ invariant, ηαβ = diag(1, −1, −1, −1) with α, β ∈ {0, 1, 2, 3}, x0 = t O(3, 1) leaves ηˆαβ xα xβ invariant, ηˆαβ = diag(1, 1, 1, −1) with α, β ∈ {1, 2, 3, 4}, x4 = t Examples: Reverse

(Lα β ) ∈ O(1, 3)

ˆ α β ) ∈ O(3, 1) (L

1 space axis 3 space axes time axis

P (1) = diag(1, 1, 1, −1) P (3) = diag(1, −1, −1, −1) T = diag(−1, 1, 1, 1)

Pˆ (1) = diag(−1, 1, 1, 1) Pˆ (3) = diag(−1, −1, −1, 1) Tˆ = diag(1, 1, 1, −1)

Lα β (L−1 )β γ = δβα ,

LLT = 11 hence (L−1 )β γ = Lγ β .

Real Clifford algebra The Clifford algebra is a graded algebra C = C+ ⊕ C− . C+ is generated by even products of γα ’s, Λeven ∈ C+ C− is generated by odd products of γα ’s, Λodd ∈ C− . We choosea {γα , γβ } = 2ηαβ ,

{ˆ γα , γˆβ } = 2ˆ ηαβ .

We could also fix the signature of the metric and have {ˆ γα γˆβ } = −2ηαβ , but we prefer to associate γˆα with ηˆαβ rather than with −ηαβ . Pin groups β ΛL ∈ Pin(1, 3) ⇔ ΛL γα Λ−1 L = γβ L α

or

−1 α ΛL γ α Λ−1 ) β γβ L = (L

ˆ L ∈ Pin(3, 1) ⇔ Λ ˆ L γˆα Λ ˆ −1 = γˆβ L ˆβα Λ L

or

ˆ L γˆ α Λ ˆ −1 = (L ˆ −1 )α γˆ β . Λ L β

choice is {γα , γβ } = −2ηαβ , {ˆ γα , γ ˆβ } = −2ˆ ηαβ . Our choice implies that the norm N (vα γα ) = η(v, v) =: kvk2s,t . With the other choice N (Λ) = Λ(α(Λ))τ , where α(Λeven ) = Λeven , α(Λodd ) = −Λodd .

a Another

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Examples: ΛP (1) = ±γ0 γ1 γ2 ,

ˆ P (1) = ±ˆ Λ γ2 γˆ3 γˆ4 ,

ΛP (3) = ±γ0 ,

ˆ P (3) = ±ˆ Λ γ4 ,

ΛT = ±γ1 γ2 γ3 ,

ˆ T = ±ˆ Λ γ1 γˆ2 γˆ3 ,

Λ2P (1) = −11,

ˆ2 Λ 1, P (1) = +1

Λ2P (3) = +11,

ˆ2 Λ 1, P (3) = −1

Λ2T = +11,

ˆ 2 = −11 . Λ T

Spin group Spin(1, 3) ⊂ Pin(1, 3),

Spin(3, 1) ⊂ Pin(3, 1) .

A Spin group consists of elements ΛL for L such that det(Lβ α ) = 1. It consists of even elements (even products of γα ) of a Pin group. Pin(m, n) = Spin(m, n) n Z2

for m + n > 1

(n is a semidirect product, defined in Sec. 5.5). Group representations, unitary and antiunitary On finite-dimensional vector spaces, real or complex: Γα is a real or complex matrix representation of γα . ˆ α is a real or complex matrix representation of γˆα , Γ ˆ α = iΓα . Γ Chiral representation  0 0 0 0 Γ0 =  1 0 0 1  0 0  0 0 Γ2 =   0 i −i 0

1 0 0 0

 0 1 , 0 0  0 −i i 0 , 0 0 0 0



0 0  0 0 Γ1 =   0 −1 −1 0  0 0  0 0 Γ3 =   −1 0 0 1 

−1 0  0 −1 Γ5 = iΓ0 Γ1 Γ2 Γ3 =   0 0 0 0

0 0 1 0

 0 0 . 0 1

 1 0  0 0  1 0 0 −1   0 0 0 0 0 1 0 0

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Dirac representation, exchange Γ0  1 0 Γ0 =  0 0

above for

 0 0 0 1 0 0 . 0 −1 0 0 0 −1

Γ1 , Γ2 , Γ3 are the same as in the chiral representation. A Majorana (real) representation in Pin(3, 1):     0 0 0 −1 1 0 0 0   0 0 0 ˆ1 =  0 0 1 , ˆ2 =  0 1  Γ Γ  0 1 0   0 0 0 −1 0 −1 0 0 0 0 0 0 −1     0 0 −1 0 0 0 0 1   0 0 −1  0 1 0 ˆ3 =  0 ˆ4 =  0  ,  Γ Γ  −1   0 0 0 0 −1 0 0  0 −1 0 0 −1 0 0 0 Pauli matrices 0 σ1 = 1

1 0

,

σ2 =

0 −i i 0

,

σ3 =

1 0 0 −1

,

Dirac equation for a massive, charged particle (iΓα ∇α − m)ψ(x) = 0

ˆ ˆ α ∇α − m)ψ(x) (Γ =0

with ∇α = ∂α + iqAα

Dirac adjoint (see 3.5 for Dirac adjoints on general manifolds) ¯ ˆ4 . ψ¯ = ψ † Γ0 , ψˆ = ψˆ† Γ Charge conjugate (iΓα (∂α − iqAα ) − m)ψ c (x) = 0 ,

ˆ α (∂α − iqAα ) − m)ψˆc (x) = 0 (Γ

and ψ c = Cψ ∗ , ψˆc = Cˆψˆ∗ , where C Γ∗α C −1 = −Γα ,

ˆ ∗ Cˆ−1 = Γ ˆα . Cˆ Γ α

ˆ 2 and CC ∗ = 11, CˆCˆ∗ = 11. In the Dirac representation C = ±Γ2 , Cˆ = ±Γ Two-component fermions, Weyl fermions 1 P± = (11 ± Γ5 ) 2 P+ ψ = ϕR

chirality +1

P− ψ = ϕL

chirality −1 .

The antiunitary time reversal operator AT on ψ acts on the complex conjugate ψ ∗ , AT C −1 = ΛT .

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On infinite-dimensional Hilbert spaces of state vectors: Quantum fields (see 3.4) 1 X Ψ(x) = √ (a(p, s)ψ(p, s) + b† (p, s)ψ c (p, s)) Ω p,s ξ X (b(p, s)ψ(p, s) + a† (p, s)ψ c (p, s)) , Ψc (x) = √ Ω p,s where

P p,s

combines

P s

and

R

d4 p δ(p2 − m2 ), and ξ is a phase, |ξ| = 1.

States States are created by applying creation and annihilation operators on the vacuum state, e.g.  1 X †   Ψ(x)|0i = √ b (p, s)ψ c (p, s)|0i Ω p,s   hb|Ψ(x)|0i = ψ c (p, s) ψ, ψ c are called classical fields even when they are fermionic; the space of classical fields is the domain of the classical action (the classical action is not to be confused with its minimum value). Ψ(x)|0i is a linear superposition of one-antiparticle states of welldefined momentum p and spin polarization s.

1. Introduction A simple, but not widely known, mathematical fact concerning the coverings of the full Lorentz group sheds light on parity and time reversal transformations of fermions. Whereas there is, up to an isomorphism, only one Spin group which double covers the orientation preserving Lorentz group, there are two essentially different groups, called Pin groups, which cover the full Lorentz group. The name Pin is gaining acceptance because it is a useful name: Pin(1, 3) is to O(1, 3) what Spin(1, 3) is to SO(1, 3). The existence of two Pin groups explains several issues which we discuss in Sec 2. It offers a classification of fermions based on their properties under space or time reversal finer than the classification based on their properties under orientation preserving Lorentz transformations. For the convenience of the reader, we have divided this report into two parts: in the first part we present the Pin groups for three space dimensions and one time dimension; in the second part we present the Pin groups for s space dimensions and t time dimensions (with emphasis on the hyperbolic case t = 1). In Appendix E we have collected, by topic, references of related articles which we have consulted.

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2. Background 2.1. As seen by physicists Racah, in 1937 [90], and Yang and Tiomno, in 1950 [126], pointed out that under a space inversion four different transformations for fields of spin 1/2 are possible. Yang and Tiomno added, “The types of transformation properties to which the various known spin-1/2 fields belong are physical observables and could in principle be determined experimentally from their mutual interactions and their interactions with fields of integral spin”. They wrote down a list of all possible spinor interactions using the four types of spinors, and attempted to exclude some interactions based on the guiding principle of parity conservation. Fermi even scheduled a special session at a conference he organized in Chicago (September 1951) devoted to these ideas and to the experimental distinction between the different kinds of spinors [121]. Under the impact of the discovery of parity violation [62, 124], and the success of the Standard Model [97, 98, 114], the Yang and Tiomno paper fell by the wayside; its goal was rendered obsolete. Nevertheless the fact remains that there are four different kinds of spin-1/2 particles. Why has this fact, noted already in 1937, been largely ignored? a) The impact of parity violation. Right-left asymmetry. The angular distribution of electrons from the beta decay of the polarized Co60 nucleus, as well as other experiments involving weak interactions, are best interpreted in a theory of two-component spinors (see for instance T. D. Lee [63]). This theory distinguishes neutrinos, whose spins are antiparallel to their momenta (left-handed), from antineutrinos, whose spins are parallel to their momenta (righthanded). Neutrinos are emitted in β + decay, and antineutrinos in β − decays, such as Co60 decays. In a true (massless) two-component theory, antineutrinos whose spins are antiparallel to their momenta do not exist, so the theory is “maximally” parity-violating. The two-component versus the four-component fermion theory is central to the discussion of Spin and Pin, and to the analysis of parity, which can be found in Sec. 3. With the experimental evidence of at least one neutrino being massive, the massless two-component “maximally parity-violating” formalism has lost some of its absolute character in the Standard Model; it is therefore like conservation laws such as strangeness which were thought to be exact but are nonetheless useful in their range of validity. b) The impact of the Standard Model In the Standard Model, all spin-1/2 particles are chiral particles defined by the Weyl representation (the two-component theory) and the concept of intrinsic parity does not apply to a chiral particle. Stated in other words, since a left particle becomes a right particle under space inversion, how does one define its parity? Indeed, the Particle Data Group publications [84] do not attribute parity to leptons, presumably for this same reason. However, quarks and leptons are not “truly” two-component,

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since mass terms mix left and right. The mere circumstance that the quarks and leptons of the Standard Model are written in terms of chiral fermions certainly does not rule out the existence of two Pin groups.

2.2. As seen by Wigner In a fundamental paper [122], Wigner established the fact that relativistic invariance implies that physical states are represented by unitary representations of the Poincaré group, and simple systems by irreducible ones. In an article published in 1964 [123] Wigner, using an unpublished manuscript written much earlier with Bargmann and Wightman, analyzes the representations of the full Lorentz group (see Fig. 1). He recalls first the representations of the proper orthochronous Lorentz group (labelled 11 in Fig. 1), then includes space, time, and spacetime reflections. His analysis is anchored on SL(2, C) which is a covering group of the proper orthochronous Lorentz group. The group SL(2, C) is isomorphic, but not identical, to Spin↑ (3, 1) ∈ Spin(3, 1) (these covering groups are defined in Sec. 3.2). Adding reflections to SL(2, C), Wigner constructs four distinct covering groups. In the process of examining all the possibilities offered by adding reflections, Wigner constructs a multiplication table of reflection operators (Table 1 in [123]). Additional considerations eliminate some unwanted entries of the multiplication table. Wigner’s results explain the observations of Racah, Yang and Tiomno. Wigner contemplates the existence of a “whole group” as opposed to four distinct groups, but notes that it is not uniquely defined. Wigner’s work in the formalism of one-particle states has been extended to Fock space (see in particular [78–80] and references therein). An excellent presentation of Wigner’s work and its quantum field theory extension can be found in Moussa’s lecture notes [78], in which elementary methods for describing representation of the the Poincaré group are used, and the aim is to describe spin in particle physics in a natural way. Is there anything to be added to Wigner’s analysis? The answer is yes. Wigner’s interest in a “whole group” and his concern about it lacking a unique definition is taken care of in this report: there are two well-defined “whole groups”, namely the two Pin groups. In comparing our work with Wigner’s, one should keep in mind the following fact: Wigner works with quantum mechanical operators and their projective representations on one-particle states. We work with operators on Fock space. Therefore the phases in this report are not Wigner’s, but of course the phases in quantum mechanics and the phases in quantum field theory are related, since a representation on Fock space dictates a representation on a given one-particle state. Our discussion is structured as follows: • The Pin groups • Their representations on classical fields (not their projective representations on quantum one-particle states) • Their projective representations in quantum field theory.

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We can nevertheless establish correspondences between Wigner’s results and ours, namely • Wigner’s multiplication table (his Table 1) corresponds to the eight double covers of the Lorentz group listed here in Appendix C. Wigner’s Table 1 includes unwanted possibilities which he eventually excludes, the eight double covers include double covers other than the Pin groups. The double covers which are the Pin groups are called Cliffordian. See Chamblin [24] for discussion of the non-Cliffordian double covers. • After elimination of unwanted entries, Table 1 (modulo Wigner’s phases) corresponds to Pin group multiplications (our Eq. (9), which includes multipliˆ ∈ Pin(3, 1)). cation of elements Λ ∈ Pin(1, 3) and multiplication of elements Λ In brief Wigner works in quantum mechanics with four distinct groups based on SL(2, C); we work in quantum field theory with two groups Pin(1, 3) and Pin(3, 1). 2.3. As seen by mathematicians The earliest reference to Pin groups we know of is in the 1964 paper [5] of Atiyah, Bott and Shapiro on Clifford modules — a paper not likely to have come to the attention of physicists in those days. Moreover, the authors label both groups Pin(k), rather than Pin(k, 0) and Pin(0, k), so that the differences between the two groups is noticed only by a careful and motivated reader. Possibly detracting from the difference between the two Pin groups is Cartan’s book, Le¸cons sur la Théorie des Spineurs I [23]. We quote from the translation: Page 3. “Let φ be a quadratic form, φ(n − h, h), φ := x21 + x22 + · · · + x2n−h − x2n−h+1 − · · · − x2n

(1)

we shall assume, without any loss of generality, that n − h ≥ h”. There is no loss of generality in considering only O(s, t), with s ≥ t, but there is loss of generality in considering only Pin(s, t) with s ≥ t. It is little known that Cartan did distinguish spinors of the first and second kind, here identified as the two different kinds of pinors. Shortly after the Atiyah, Bott and Shapiro paper, Karoubi published in Annales Scientifiques de l’Ecole Normale Supérieure a long article on “Algèbres de Clifford et K-théorie” which contains a careful study of Pin(t, s) and Pin(s, t). However, it is not surprising that physicists did not relate Karoubi’s mathematical analysis to the experimental question of parity. When one of us (CD) could not figure out why there are different obstruction criteria for characterizing the manifolds which admit a Pin bundle, a letter from Y. Choquet-Bruhat paved the way for identifying not one but two Pin groups; a letter from S. Gutt gave us the construction of the two non-isomorphic Pin groups and a reference to Karoubi’s article. The reason for the different criteria became obvious; two groups, two bundles, each with its own criterion.

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The goal of this report is to clarify parity and related topics by defining them in terms of the Pin groups (Sec. 3.1), and to investigate the physical consequences of the fact that there are two Pin groups. 3. The Pin Groups in 3 Space, 1 Time Dimensions The title of this section could be “Basic Mathematics”. Here, we explain why there are two Pin groups, and we analyze their differences. For more information, see for instance [28]. Let O(1, 3) be the Lorentz group of transformations of (R4 , η) which leaves invariant the quadratic form ηαβ xα xβ , where (ηαβ ) := diag(1, −1, −1, −1)

(2)

and let O(3, 1) be the Lorentz group of transformations of (R4 , ηˆ), where (ˆ ηαβ ) := diag(1, 1, 1, −1) .

(3)

The Lorentz groups O(1, 3) and O(3, 1) are isomorphic; nevertheless we shall use different symbols for their elements because (Lα β ) ∈ O(1, 3) is not identical to ˆ α β ) ∈ O(3, 1) (see examples in the section on notation). (L The full Lorentz group consists of four components (Fig. 1).

1

PT

P

T

SO(1,3) O(1,3) Fig. 1.

Components of the Lorentz group.

Each component is labelled by a representative element: 11 the unit element, P the reversal of one or three space axes, T the reversal of the time axis. The component connected to 11 is called the proper orthochronous Lorentz group. The two components connected respectively to 11 and P T make up the subgroup of the Lorentz group consisting of orientation preserving transformations; i.e. the matrix of the transformation has determinant 1. If it changes the time orientation, it also changes the space orientation. The Pin groups entered physics by the requirement that the Dirac equation be invariant under Lorentz transformations. For the sake of clarity and brevity we

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proceed in the following order: 3.1 The Pin groups. 3.2 A Spin group as a subgroup of a Pin group. 3.3 Pin group and Spin group representations on finite-dimensional spaces; classical fields. 3.4 Pin group and Spin group representations on infinite-dimensional spaces; quantum fields. 3.5 Bundles; Fermi currents on topologically nontrivial manifolds. 3.6 Bundle reduction; massless and massive neutrinos. The distinction between Pin and Spin is not always recognized. A Spin group is a subgroup of a Pin group, but the expression “Spin group” is unfortunately still often used to mean the full group. The word “Pin” was originally a jokeb : Pin(n) is to O(n) what Spin(n) is to SO(n). 3.1. The Pin groups 3.1.1. Pin(1, 3) Let {γα } be the generators of a real Clifford algebra, such that {γα , γβ } = 2ηαβ 11 ,

ηαβ = diag(1, −1, −1, −1)

(4)

and let (Lα β ) ∈ O(1, 3). Pin(1, 3) consists of the invertible elements ΛL of the Clifford algebra such that β ΛL γα Λ−1 L = γβ L α

−1 α ΛL γ α Λ−1 ) β γβ L = (L

or equivalently

(5)

and such that ΛΛτ = ±11 . Here τ is the reversion, e.g. (γ0 γ1 γ2 )τ = γ2 γ1 γ0 . The two elements ±ΛL of the Pin group are said to cover the single element L of the Lorentz group (see Fig. 2). For future reference we solve Eq. (5) in a few cases. The solution is readily obtained when L is diagonal. For example, the reflection of 3 space axes in O(1, 3) is P = diag(1, −1, −1, −1). Hence   ΛP (3) γ0 Λ−1 P (3) = γ0 Λ

P (3) γi

Λ−1 P (3) = −γi ,

for i ∈ {1, 2, 3}

and the solution is ΛP (3) = ±γ0 . b The

joke has been attributed to J. P. Serre [5] but upon being asked, he did not confirm this.

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Pin(1,3) Spin(1,3) _ +

_1 +

1

_ +

ΛPT

Λ

P

PT

P

_ +

ΛT

T

SO(1,3) O(1,3) Fig. 2.

Double cover of the Lorentz group.

If L is an element of the proper orthochronous Lorentz group, 1 [γα , γβ ] θαβ ΛL = exp 8 where θαβ is an antisymmetric tensor made of boost and rotation generators. 3.1.2. Pin(3, 1) Let {ˆ γα } be the generators of a real Clifford algebra, such that {ˆ γα , γˆβ } = 2ˆ ηαβ 11 ,

ηˆαβ = diag(1, 1, 1, −1)

(6)

ˆ α β ) ∈ O(3, 1). and let (L ˆ L of the Clifford algebra such that Pin(3, 1) consists of the invertible elements Λ ˆ L γˆα Λ ˆ −1 = γˆβ L ˆβ α Λ L

ˆ −1 = (L ˆ L γˆ α Λ ˆ −1 )α γˆ β Λ β L

or equivalently

(7)

and such that ˆΛ ˆ τ = ±1 . Λ We summarize in the following table some results from solving Eqs. (5) and (7): ΛP (1) = ±γ0 γ1 γ2 ,

ˆ P (1) = ±ˆ Λ γ2 γˆ3 γˆ4 ,

ΛP (3) = ±γ0 ,

ˆ P (3) = ±ˆ Λ γ4 ,

ΛT = ±γ1 γ2 γ3 ,

(8)

ˆ T = ±ˆ Λ γ1 γˆ2 γˆ3 .

It follows that Λ2P (1) = −11,

ˆ2 Λ 1, P (1) = +1

Λ2P (3) = +11,

ˆ2 Λ 1, P (3) = −1

Λ2T = +11,

ˆ 2 = −11 , Λ T

(9)

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equations which can be used to distinguish Pin(3, 1) and Pin(3, 1). We note that P (3) is P (1) followed by a rotation; Λ2P (3) is Λ2P (1) followed by the effect of a 2π rotation on the pinor, hence Λ2P (3) = −Λ2P (1) . 3.2. A Spin group is a subgroup of a Pin group If (Lα β ) ∈ SO(1, 3) then ΛL ∈ Spin(1, 3). Spin(1, 3) consists of even elements (products of an even number of γα ) of Pin(1, 3). Spin(1, 3) is isomorphic to Spin(3, 1), but Pin(1, 3) is not isomorphic to Pin(3, 1). A simple but convincing argument that the two Pin groups are not isomorphic consists in writing the multiplication tables of the four generators of Pin(1, 0) and Pin(0, 1): (±1, ±γ)

with γ 2 = 1 is isomorphic to Z2 × Z2

(±1, ±ˆ γ)

with γˆ 2 = −1 is isomorphic to Z4 .

The proof for 1 time, 3 space dimensions is easier to carry out in terms of representations of the groups (see Sec. 3.3). We prove in Sec. 5.5 that Pin(1, 3) = Spin(1, 3) n Z2

(where n is a semidirect product)

Pin(3, 1) = Spin(3, 1) n Z2 and nevertheless Pin(1, 3) is not isomorphic to Pin(3, 1). The following is true for the Lie algebras of the Pin and Spin groups. The Lie algebras L(Pin(t, s))

and

L(Spin(t, s)) are identical.

The Lie algebras L(Spin(t, s))

and

L(Spin(s, t)) are isomorphic.

Since Spin(1, 3) and Spin(3, 1) are isomorphic, the differences between Pin(1, 3) and Pin(3, 1) appear only in discussions of space or time reversals. ΛP , ΛT are not in Spin(1, 3) but Λ2P and Λ2T are in Spin(1, 3) and can be used to identify a Spin group as a subgroup of either Pin(1, 3) or Pin(3, 1). We have come across confusion between the properties of parity and the properties of 2π and 4π rotations. Table 1 should clarify this confusion. Table 1. Parity and rotations in Pin(1, 3) vs. Pin(3, 1). Note that P (3) is the reversal of one axis P (1) together with a π rotation.

Let L = P (1) reverse 1 space axis

ˆ R(2π) = −11 then ΛR(2π) = Λ ˆ R(4π) = 11 then ΛR(4π) = Λ 2 ˆ 2 P (1) = 11 then Λ P (1) = −11, Λ

Let L = P (3) reverse 3 space axes

ˆ 2 P (3) = −11 then Λ2 P (3) = 11, Λ

Let L = R(2π) be a 2π rotation Let L = R(4π) be a 4π rotation

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ΛR belongs to the Spin group and does not distinguish ψ and ψˆ particles, whereas ΛP belongs to a Pin group without belonging to the Spin group. In a nonorientable space there is no fundamental difference between rotation and reflection, but in an orientable space there is. In brief: Spin↑ (s, t) Spin(s, t) Pin↑ (s, t) Pin(s, t)

double double double double

covers covers covers covers

Proper orthochronous Lorentz group Orientation preserving Lorentz group Orthochronous Lorentz group Full Lorentz group

SO↑ (s, t) SO(s, t) O↑ (s, t) O(s, t) .

We remark that it is the Spin↑ group which can be written in a 2 × 2 complex matrix representation: Spin↑ (1, 3) ' SL(2, C) . 3.3. Pin group and Spin group representations on finite dimensional spaces. Classical fields We use only real Clifford algebras because we are interested in real spacetimes, but we use real or complex matrix representations. Let Γα be a real or complex matrix representation of γα . ˆ α be a real or complex matrix representation of γˆα . Γ ˆ α = iΓα but this bijection does not define an algebra isomorphism. We can set Γ Indeed, let φ : Pin(1, 3) → Pin(3, 1) ; define ˆα φ(Γα ) = Γ

by

ˆ α = iΓα Γ

then φ(Γα ) φ(Γβ ) 6= φ(Γα Γβ ) . On the other hand, the elements of the Spin subgroups consist of even products of gamma matrices; the mapping ˆ αΓ ˆβ φ(Γα Γβ ) = Γ

by

ˆ αΓ ˆ β = −Γα Γβ Γ

maps Spin(1, 3) into itself; Spin(1, 3) and Spin(3, 1) are identical. 3.3.1. Pinors A representation of a Pin group on a vector space defines a pinor. For example, a fermion of mass m which satisfies the Dirac equation in an electromagnetic potential ˆ is a Dirac pinor ψ or ψ: (iΓα ∇α − m)ψ(x) = 0,

ψ(x) ∈ C4 ,

(10)

ˆ ˆ α ∇α − m)ψ(x) (Γ = 0,

ˆ ψ(x) ∈ C4 .

(11)

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where ∇α = ∂α + iqAα . Although they obey the same equation, ψ and ψˆ are different objects because they transform differently under space or time reversal. We use the same notation ΛL for an element of a Pin group and its matrix representation. For instance we write the pinor transformation ψ 7→ ψ 0 induced by a Lorentz transformation ψ 0 (Lα β xβ ) = ΛL ψ(xα ) .

(12)

3.3.2. Spinors The space of linear representations of Spin(1, 3) on C4 (similar property for Spin(3, 1)) splits into two spaces: S = S+ ⊕ S− S+ and S− are eigenspaces of the chirality operator Γ5 = iΓ0 Γ1 Γ2 Γ3 . Γ5 commutes with even elements of a Pin group, and anti-commutes with the odd elements. Let ϕ be an eigenspinor of Γ5 , let Γ+ be an even element of Pin(1, 3) and Γ− be an odd element of Pin(1, 3). Since Γ25 = 11, the eigenvalues of Γ5 are ±1: where λ ∈ {1, −1},

Γ5 ϕ = λϕ , thus Γ5 Γ+ ϕ = λΓ+ ϕ

and

Γ5 Γ− ϕ = −λΓ− ϕ .

Hence Γ+ : S+ → S+

and

Γ− : S− → S− .

Since the eigenvalues of Γ5 are ±1, the projection matrices 1 (11 ± Γ5 ) 2 project a 4-component ψ into two 2-component Weyl spinors ϕL and ϕR : P± =

ϕL =

1 (11 − Γ5 )ψ 2

1 (11 + Γ5 )ψ , 2 here L and R stand for left and right; the use of the words left and right is justified in the paragraph on helicity below. A representation adapted to the splitting S+ ⊕ S− is called a chiral representation; in the chiral representation Γ5 is block-diagonal: −11 0 Γ5 = . 0 11 ϕR =

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3.3.3. Helicity In terms of the momentum operator pµ = −i∂/∂xµ, the Dirac equation (10) with m = 0 is (Γ0 p0 + Γi pi ) ψ = 0,

i ∈ {1, 2, 3} .

When multiplied by Γ1 Γ2 Γ3 , this equation reads in the chiral representation (Γ5 p0 − (σi ⊗ 112 )pi )ψ = 0 where σi are the Pauli matrices (see p. 7). If ψ is a plane wave ψ(x, s) = u(p, s) exp(−ip · x) the spinor u(p, s) satisfies the equation (Γ5 − σi pi /p0 )u(p, s) = 0 . The helicity operator h = 12 σi pî (where pî = pi /|p| = pi /|p0 |) tells us if the spin of the particle is oriented along the direction of motion (“right-handed”, helicity eigenvalue +1/2), or oriented opposite to the direction of motion (“left-handed”, helicity eigenvalue −1/2). One often hears the phrase “a Weyl spinor cannot correspond to an eigenstate of parity”, but this is a meaningless statement, because the parity operator Λ P does not act on (2-component) spinors. In other words, only products of an even number of gamma matrices can be block-diagonalized; since ΛP is made of an odd number of gamma matrices, it cannot be block-diagonalized. Thus ΛP does not preserve the splitting S+ ⊕ S− , and is not an operator on the space of Weyl spinors. The fact that only left-handed neutrinos are emitted in Co60 disintegration is referred to as “parity is not conserved in beta decay”. Here “parity is not conserved” means that the interaction Hamiltonian does not commute with the space reversal operator. 3.3.4. Massless spinors, massive pinors The massless Dirac operator Γα ∇α changes the helicity of a Weyl fermion. The massive Dirac operator Γα ∇α + m is the sum of a helicity-changing and a helicityconserving operator; therefore a massive fermion can only be defined by the 4component Dirac representation.

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3.3.5. Copinors, Dirac and Majorana adjoints in Pin(1, 3) The representation ρ of Pin(1, 3) on C4 by ρ(γα ) = Γα which defines pinors as contravariant vectors is not the only useful representation. In order to make tensorial objects from spinorial objects, one needs to introduce covariant pinors, also called copinors. In the copinor representation ρ(γα ) = Γ−1 α is a right action on copinors. Let ψ be a pinor with components {ψ A } in a basis {eA }, ψ(x) = ψ A (x) eA ; by definition, the adjoint ψ¯ of the pinor ψ is the copinor ¯ ψ(x) = ψ¯A (x) eA such that the duality pairing Z ¯ ψi := dv(x)ψ¯A (x)ψ A (x), hψ,

dv(x) a volume element ,

spacetime

is invariant under Lorentz transformations: ¯ ψi = hΛψ, Λψi . hψ, We are therefore interested in the solutions of the equation ¯ −1 . Λψ = ψΛ

(13)

For Λ covering the proper orthochronous Lorentz group, the solutions of (13) most frequently used are the Dirac adjoint ψ¯ and the Majorana adjoint ψ¯M of a spinor ψ. The Dirac adjoint is ψ¯ = ψ † Γ0 ,

(14)

where a dagger stands for the complex conjugate transposed. There is another solution of Eq. (13), namely ψ¯M , called the Majorana adjoint or Majorana conjugate of ψ; it is defined by ψ¯M := ψ˜ T ˜ where ψ˜ is the transpose of ψ, and T defines the isomorphism of the group {Λ} with the group {Λ}. The qualifier “Majorana” has been used for different purposes: • A Majorana adjoint as defined above. • A Majorana representation is a representation by matrices all real or purely imaginary (see Notation, and Sec. 5.3). • A Majorana particle is identical to its antiparticle (see next section). The Majorana adjoint was introduced by Van Nieuwenhuizen [113] and we refer the reader to his article for the definition and uses of the Majorana adjoint in arbitrary dimensions. ˆ ∈ Pin(3, 1) is treated in Sec. 3.5, The Dirac adjoint for Λ ∈ Pin(1, 3) and for Λ after we have introduced pinor coordinates.

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3.3.6. Charge conjugate pinors in Pin(1, 3) The Dirac pinor ψ(x) satisfies the equation (iΓα (∂α + iqAα ) − m)ψ(x) = 0 .

(15)

The complex conjugate of this equation is (−iΓα ∗ (∂α − iqAα ) − m)ψ ∗ (x) = 0 ,

(16)

therefore we introduce a map C : C → C such that 4

4

C Γ∗α C −1 = −Γα

(17)

and we define the charge conjugate pinor as ψ c = Cψ ∗ ,

(18)

which is then a solution of (iΓα (∂α − iqAα ) − m)ψ c (x) = 0 .

(19)

∗

In the Dirac representation C = ±Γ2 so CC = 11, which is necessary for (ψ ) = ψ. The operation ψ → ψ c defined by (18) is an antiunitary operation which consists of two steps; take the complex conjugate of ψ, then apply a unitary matrix. A pinor and its charge conjugate have opposite eigenvalues of the parity operator ΛP . Let a pinor ψ be in an eigenstate of ΛP (3) , abbreviated to ΛP (reversal of 3 space axes); we have shown that ΛP = ±Γ0 . Using Eq. (17) we find Λ2P = 11 ,

ΛP ψ = λψ ,

c c

λ = ±1

∗

c

ΛP ψ = ΛP (Cψ ) = −C(Λ∗P ψ ∗ )

by (17)

= −λψ

by (18).

c

To summarize, if ψ is an eigenpinor of ΛP , ΛP ψ c = −λψ c .

ΛP ψ = λψ ,

(20)

We now compute ΛP ψ(p) which is needed in the section on intrinsic parity. Let Λ(p) be a 3-momentum boost, then we can write ΛP ψ(p, s) = ΛP Λ(p)Λ−1 P ΛP ψ(p0 , 0) = Λ(−p)ΛP ψ(p0 , 0) = Λ(−p) λ ψ(p0 , 0)

if the p = 0 pinor is an eigenstate of ΛP .

Thus, if the pinor at rest has ΛP eigenvalue λ, we may write ΛP ψ(p) = λ ψ(pP˜ )

(21) ˜ where pP = (p0 , −p). Thus all we need to require for the pinor to transform into something proportional to itself at the new spacetime point (x0 , −x) is that the pinor at rest (p = 0) is an eigenpinor of parity. The “eigenvalue” λ at nonzero momentum is the same as the eigenvalue for the pinor at rest.

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3.3.7. Majorana pinors By definition a Majorana pinor ψ M is such that (ψ M )c = ψ M . To be meaningful the property must remain satisfied under a parity transformation. In Pin(1, 3) ΛP ψ c = ΛP Cψ ∗ = −CΛ∗P ψ ∗ = −C(ΛP ψ)∗ = −(ΛP ψ)c , therefore the condition ψ c = ψ does not remain satisfied for the transformed pinor. On the other hand, in Pin(3, 1) ψˆc = ψˆ is form invariant under a parity transformation: ˆ c. ˆ P ψˆc = (Λ ˆ P ψ) Λ Conclusion: The classical field of a Majorana fermion can only be a section of a Pin(3, 1) bundle. Briefly, a Majorana pinor can only be a Pin(3, 1) pinor. Yang and Tiomno [126] and Berestetskii, Lifschitz and Pitaevskii [12] have also concluded that, of the four possible parities (±1, ±i), a Majorana pinor could be assigned only two. In these references, which bring out the particular status of Majorana particles (called “strictly neutral” in [12]), the four choices were not related to the existence of two Pin groups. Remark. In parity-asymmetric theories it may not be useful to require the Majorana condition to be invariant under parity. Remark. P. Van Nieuwenhuizen [113] defines a Majorana particle such that its Dirac adjoint is equal to its Majorana adjoint. According to his definition, a Majorana pinor is such that ψ c = ±ψ, rather than the more commonly used ψ c = ψ, or ψ c = −ψ, where one sticks to one choice. 3.3.8. Unitary and antiunitary transformations Motion reversal (sometimes called “time reversal”) is a transformation which changes t into −t but, if there is an electric charge, does not change its sign. Motion reversal is an antiunitary transformation. Both the antiunitary motion reversal and the unitary time reversal are useful, but they play different roles. For example,

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Maxwell’s equations in the presence of charge density ρ and current density J read either ∂B = 0, ∂t

∇· B = 0,

∂E ∇×B− = J, ∂t

∇ ·E = ρ,

∇×E+

(22)

or, if one wants to emphasize their relativistic invariance, dF = 0

or

δF + J = 0

or

µνρσ ∂ν Fρσ = 0 , ∂µ F µν = J ν ,

(23)

where F and J without indices are differential forms. If one is interested in motion reversal, Eqs. (22) are the appropriate Maxwell’s equations. On the other hand, if one works with the full Lorentz group, and uses Eqs. (23), the covariant current 4-vector Jα transforms under a Lorentz transformation L as follows: J → J0

such that

Jβ (x) = Jα0 (Lx)Lα β .

If the Lorentz transformation L is the time reversal (T α β ) = diag(−1, 1, 1, 1), then the charge density changes sign: J0 (x, t) = −J00 (x, −t) . 3.3.9. Invariance of the Dirac equation under antiunitary transformations We recall that the Dirac equation is invariant under a unitary transformation ΛL induced by a Lorentz transformation L if the new pinor ψ is related to the original pinor ψ by ψ 0 (Lx) = ΛL ψ(x)

with

β ΛL Γα Λ−1 L = Γβ L α .

Under an antiunitaryc transformation A the new pinor ψ 0 is related to the original one by ψ 0 (Lx) = AL ψ ∗ (x)

with

AL Γ∗α AL −1 = Γβ Lβ α .

This condition is equivalent to β (AL C −1 ) Γα (CA−1 L ) = Γβ L α .

Therefore AL C −1 carries out a unitary transformation which acts on pinors, rather than their complex conjugates. For example, if AT is the motion reversal, then AT C −1 = ΛT . c Here,

like in Eq. (18), the composite operation A is antiunitary, but it is carried out by a unitary matrix AL which acts on complex conjugate pinors, so A : ψ → AL ψ∗ .

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ΛT is made of an odd number of gamma matrices, and AT is made of an even number of matrices. The reason for introducing the antiunitary operation A performed by complex conjugation and the matrix AT is the requirement, necessary in a theory free of negative energy states, that the fourth component of the energy-momentum vector does not change sign under time reversal. 3.3.10. CP T invariance CP T invariance means invariance under the combined transformation of charge, parity and antiunitary time reversal. It follows from the above relation between unitary and antiunitary time reversal that CP T invariance is simply invariance under ΛP ΛT , where we emphasize that ΛT is unitary. The combination ΛP ΛT covers the component P T of the full Lorentz group, which together with the component connected to unity constitutes the component of the orientation preserving transformations (Determinant 1). Thus, CPT invariance is invariance under orientation preserving Lorentz transformations. For the consistency of a relativistic formalism it is advisable to derive first the equations in the framework of Lorentz transformations before investigating the transformations of interest in a specific context. With CP T , for example, it can be easier to work with ΛP ΛT than with CP T in the traditional sense. We will have more to say on CP T in the quantum field theory section. 3.3.11. Charge conjugate pinors in Pin(3, 1) ˆ The Dirac pinor ψ(x) in a Pin(3, 1) representation satisfies the equation ˆ ˆ α (∂α + iqAα ) − m)ψ(x) (Γ = 0.

(24)

The charge conjugate ψˆc of ψˆ must satisfy the equation ˆ α (∂α − iqAα ) − m)ψˆc (x) = 0 . (Γ

(25)

Therefore the map Cˆ such that ˆ ∗ Cˆ−1 = Γ ˆα Cˆ Γ α ˆ defines the charge conjugate ψˆc of ψ, ψˆc = Cˆψˆ∗ .

(26)

(27)

ˆ ˆ∗

∗

The requirement C C = 11 is indeed satisfied in Pin(3, 1), and CC = 11 is satisfied in Pin(1, 3). We also check that in Pin(3, 1), like in Pin(1, 3), a pinor and its charge conjugate have opposite eigenvalues of the parity operator ΛP . Let a pinor ψˆ be in ˆ P (3) ≡ Λ ˆ P = ±Γ ˆ 4 . Using Eq. (26) we find an eigenstate of Λ ˆ ψˆ , ˆ P ψˆ = λ Λ

ˆ 2 = −11 , Λ P

ˆ = ±i , λ

ˆ P ψˆc = Λ ˆ P (Cˆψˆ∗ ) = C( ˆΛ ˆ ∗ ψˆ∗ ) Λ P ˆ ψˆc , = −λ

ˆ ∗ = −Λ ˆP . since Λ P

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ˆ 2 = −11 imply opposite eigenvalues of the parity In conclusion, both Λ2P = 11 and Λ P operator for a pinor and its charge conjugate. We record the following: C Γ∗α C −1 = −Γα

In the Dirac representation C = ±Γ2 ,

ˆ ∗α Cˆ−1 = Γ ˆα Cˆ Γ

ˆ2 . In the Dirac representation Cˆ = ±Γ

(28)

3.4. Pin and Spin representations on infinite-dimensional spaces. Quantum fields 3.4.1. Particles, antiparticles In Sec. 3.3, charge conjugation meant electrical charge conjugation. Here the notion of charge conjugate fields is extended to “charges” other than electric: strong isospin, strangeness, etc., charge conjugate pairs are called antiparticles. Equations (28) used in defining electrical charge conjugation are now used in defining antiparticles. “The reason for antiparticles” is the title of a lecture given by Feynman as the first Dirac Memorial lecture in 1986. This is how Feynman introduced his lecture in honor of Dirac: “Dirac with his relativistic equation for the electron was the first to, as he put it, wed quantum mechanics and relativity together” and Feynman notes that the “crucial idea necessary” for achieving this is the existence of antiparticles. In the context of quantum field theory it can be shown that antiparticles are required by causality; an antiparticle gives rise to a contribution to the commutator of two fermion fields which exactly cancels the contribution from the particle at spacelike separation, as required by causality. Antiparticles are necessary; their existence is implied in systems invariant under CP T . The Dirac field operator Ψ acts on a Fock space of particle and antiparticle states. The free field decomposes into particle and antiparticle plane wave solutions of the Dirac equation: 1 X Ψ(x) = √ (a(p, s)ψ(x, p, s) + b† (p, s)ψ c (x, p, s)) , Ω p,s where Ω assigns a dimension to Ψ, and and where the mode functions

P p,s

ψ(x, p, s) = u(p, s) exp(−ip · x) ψ c (x, p, s) = v(p, s) exp(ip · x)

combines

P s

and

R

(29) d4 p δ(p2 − m2 ),

particle field (30) antiparticle field

are free particle and antiparticle solutions of the Dirac equation, with ψ c = Cψ ∗

and

C −1 Γα C = −Γ∗α .

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The operators a, a† , b, b† are annihilation and creation operators on the Fock space: a(p, s)

annihilates a particle of momentum p and spin s

b† (p, s)

creates an antiparticle of momentum p and spin s

and obvious definitions for a† and b. The usual form of the nonzero commutation relations are [a(p, s), a† (p0 , s0 )] = δ(p − p0 ) δs s0 , [b(p, s), b† (p0 , s0 )] = δ(p − p0 ) δs s0 . Recall that the relativistically invariant quantity is δ (3) (p0 − p)Ep . 3.4.2. Fock space operators, unitary and antiunitary In Sec. 3.3 we introduced three operators on the space of pinors ψ; the charge conjugation operator C, the space and time reversal unitary operators ΛP and ΛT . We also introduced the antiunitary operation ψ → ψ c = AT ψ ∗ such that AT C −1 = ΛT . The corresponding operators on Fock space are introduced below in Eqs. (31), (33), (34) and (42). Wigner has shown that a symmetry operator on the Hilbert space of states is either linear and unitary, or antilinear and antiunitary. A detailed proof can be found in Weinberg’s book [115, pp. 91–96]. If one requires the theory to be free of negative energy states, then the time reversal operator is antiunitary (see for instance the books by Lee [63] or Weinberg [115]). Let us start from the beginning. Let |αi and |βi be two state vectors, and ξ, η two complex numbers. An operator U is said to be linear if U (ξ|αi + η|βi) = ξU |αi + ηU |βi , isometric if hα|U † U |βi = hα | βi, and unitary if U † U = U U † = 11. An operator A is said to be antilinear if A(ξ|αi + η|βi) = ξ ∗ A|αi + η ∗ A|βi and antiunitary if hα|A† A|βi = hα | βi∗ . The charge conjugation operator UC on Fock space is by definition the unitary operator, UC−1 = UC† , on Fock space which exchanges particles and antiparticles: UC a(p, s)UC† = ξa b(p, s) ,

UC a† (p, s)UC† = ξa∗ b† (p, s) ,

UC b(p, s)UC† = ξb a(p, s) ,

UC b† (p, s)UC† = ξb∗ a† (p, s) ,

(31)

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where ξa and ξb are arbitrary phases at this stage. Hence the charge conjugate field operator Ψc of Ψ is Ψc = UC ΨUC† and 1 X (ξa b(p, s)ψ(p, s) + ξb∗ a† (p, s)ψ c (p, s)) . Ψc (x) = √ Ω p,s Ψ creates antiparticles and annihilates particles, Ψc creates particles and annihilates antiparticles. The matrix elements of the operator Ψ take their values in the space of classical fields ψ. For example, b† creates an antiparticle and hb |Ψ(a(p, s), b† (p, s))| 0i = ψbc (p, s) . Given ψ c = Cψ ∗ and the fact that C (an operator in Pin(1, 3)) is different from Cˆ (an operator in Pin(3, 1)) we reexpress Ψc in terms of C, and will later on express ˆ c in terms of C. ˆ With arguments suppressed Ψ 1 X Ψc = √ (ξa b(C −1 ψ c )∗ + ξb∗ a† Cψ ∗ ) Ω p,s ! 1 X c∗ ∗ † ∗ = C √ (ξa bψ + ξb a ψ ) Ω p,s

since CC ∗ = 11

=: ξ CΨ∗ , if we require ξb∗ = ξa := ξ, or equivalently ξa ξb = 1 .

(32)

We review how this fact is verified experimentally in Sec. 4.6. In the above expression for Ψc , the operator Ψ∗ is defined by h out |Ψ∗ | in i = (h out |Ψ| in i)∗ Remark. Note the difference between Ψ∗ and Ψ† defined by (h out |Ψ| in i)∗ = h in |Ψ† | out i . Remark. The operator UC on Fock space is unitary; it acts only on the creation and annihilation operators. On the other hand, the operator C on the space of pinors is antiunitary.

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3.4.3. Intrinsic parity In Sec. 3.3 we gave the transformation laws of classical fermion fields, Eq. (12), under Lorentz transformations L, and thus in particular under L = P (3), reflection of 3 axes; in this section we do not need P (1) and we abbreviate P (3) to P = diag(1, −1, −1, −1). We now determine the transformation law under P of the field operator 1 X c Ψ(x) = √ (a(p, s)ψp,s (x) + b† (p, s)ψp,s (x)) . Ω p,s Let UP be a unitary operator, UP† = UP−1 , such thatd UP a(p, s)UP† = ηa a(pP˜ , s)

(the components of pP˜ are (p0 , −p))

UP b(p, s)UP† = ηb b(pP˜ , s) .

(33) (34)

The requirement UP |0i = |0i fixes the values of ηa and ηb with respect to the vacuum. Remark. It is easy to convince oneself that spins do not change under a parity transformation. Intuitively one has a picture like Fig. 3. Another reason is that we want to add the spin operator s to an orbital angular momentum operator of the form r × p, which does not change sign under parity.

Fig. 3.

Spins do not change under reflection.

Remark. (U ηa a U −1 )∗ = U ηa∗ a† U −1 . [115] convention is UP a† (p, s)UP† = ηa a† (pP˜ , s), so it differs from ours by a complex conjugation. Our convention is the same as in Peskin and Schroeder [87]. d Weinberg’s

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There is obviously a relationship between ηa and ηb∗ so that Ψ(x) has a well-defined transformation under P . In order to relate UP to ΛP acting on ψ and ψ c , we proceed as in the section Particles, antiparticles; namely we look for a (complex) constant of proportionality such that UP Ψ(x, s)UP† ∝ ΛP Ψ(P x, s) ,

(35)

where the components of P x are (x0 , −x). We know that ψ and ψ c depend on x only through p · x, so when x 7→ P x, then p 7→ pP˜ ; the components of pP˜ = pP˜ −1 are (p0 , −p). Equation (35) is meaningful only in local quantum field theory, and so are Eqs. (42)–(44), which all have the same structure as (35). Given the definition (29) and (30) of Ψ and the Dirac Eqs. (15), (19), (24) and ˆ (25) satisfied by ψ(x), ψ(x) and their charge conjugates, we have for a particle at rest (omitting the spin label s) Γ0 u(p = 0) = u(p = 0) ,

−Γ0 v(p = 0) = v(p = 0) ,

ˆ 4u iΓ ˆ(p = 0) = u ˆ(p = 0) ,

ˆ 4 vˆ(p = 0) = vˆ(p = 0) . −iΓ

(36)

Therefore for a particle at rest, in momentum space u is an eigenpinor of Γ0 with eigenvalue 1, and v is an eigenpinor of Γ0 with eigenvalue −1. Similarily, u ˆ is an ˆ 4 with eigenvalue −i and vˆ is an eigenpinor of Γ ˆ 4 with eigenvalue i. eigenpinor of Γ We have established (see the Dictionary of Notation) that the space reversal operator ΛP (3) ∈ Pin(1, 3) is ±Γ0 , where the choice of pin structure dictates the sign. We have also computed in Eqs. (20) and (21) the action of ΛP on ψ(x) when ψ is an eigenpinor of ΛP at p = 0, with eigenvalue λ, ΛP u(pP˜ , s) exp(−ip · x) = λu(p, s) exp(−ip · x) . If we choose ΛP = Γ0 , it follows from (36) that ΛP u(pP˜ ) = u(p) = λu(p) , ΛP v(pP˜ ) = −v(p) = −λv(p) . We can now compute, omitting reference to s since s is not changed by UP , 1 X (ηa a(pP˜ )u(p)e−ip·x + ηb∗ b† (pP˜ )v(p)eip·x ) UP Ψ(x)UP−1 = √ Ω p ΛP X = √ (ηa a(pP˜ )u(pP˜ )e−ip·x − ηb∗ b† (pP˜ )v(pP˜ )eip·x ) . λ Ω p The sum over p is an integral, under the change of variable pP˜ 7→ p we have u(pP˜ ) exp(−ip · x) 7→ u(p) exp(−ip · P x) . Finally, ΛP X UP Ψ(x)UP−1 = √ (ηa a(p)u(p)e−ip·P x − ηb∗ b† (p)v(p)eip·P x ) . λ Ω p

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For this to be proportional to ΛP Ψ(P x), we must require ηa = −ηb∗ =: η, or equivalently ηa ηb = ηa (−ηa∗ ) = −1 .

(37)

This sign has been experimentally verified (see Sec. 4.3 on positronium). Thus we have UP Ψ(x)UP† = (η/λ)ΛP Ψ(P x) ,

ΛP = Γ0 .

(38)

ˆP = Γ ˆ0 . Λ

(39)

Similarly one finds for Pin(3, 1) † ˆ ˆ ˆ ˆ UP Ψ(x)U P = (η/λ)ΛP Ψ(P x) ,

ˆ P = −Γ ˆ 0 , the r.h.s. of (38) and (39) simply Remark. If we use ΛP = −Γ0 or Λ change sign. We summarize the results for the two Pin groups by UP Ψ(x, s)UP† = (η/λ)ΛP Ψ(P x, s)

Pin(1, 3)

ˆ Λ ˆ ˆ P Ψ(P ˆ x, s) UP Ψ(x, s)UP† = (η/λ)

Pin(3, 1)

Remark. UP2 Ψ(x)UP† 2 = (η/λ)2 Λ2P Ψ(x) = η 2 λ2 Λ2P Ψ(x) .

(40)

It has been argued that Eq. (40) must give η λ = −Ψ since a fermion changes sign under a rotation of 2π. However, on an orientable space, a 2π rotation (successive transformation by infinitesimal angles) is very different from the ˆ 2 = −1, discrete symmetry P . For example, as we have shown, Λ2P = 1 whereas Λ P ˆ but still ΛR(2π) = ΛR(2π) = −1. Thus we leave η to be determined. 2 2

Λ2P Ψ

The definition of intrinsic parity has changed throughout the years. When the canonical reference was Bjorken and Drell [14], the intrinsic parity of a field ψ was the eigenvalue of ΛP : ΛP ψ = λψ ,

for ψ in an eigenstate of ΛP .

This is the definition used in the fundamental paper of Tripp [111] entitled “Spin and Parity Determination of Elementary Particles”. A common reference nowadays is Peskin and Schroeder [87], where intrinsic parity η is defined by the field operator Ψ: UP Ψ(x)UP−1 = ηΛP Ψ(P x) ,

for ΛP u = u.

Since we want to allow for both Pin groups and use the modern view of field theory, we conclude from Eq. (38) that the most general quantity of interest is η/λ, i.e. η/λ is intrinsic parity ˆ The fact that we can have four possible or for Pin(3, 1) it would be denoted η/λ. 4 4 ˆ parities comes from (η/λ) = (η/λ) = 1.

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3.4.4. Majorana field operator By definition a Majorana field operator ΨM is such that ΨM (x) = (ΨM (x))c = ξ CΨM ∗ (x) ,

(41)

hence 1 X c (a(p, s)ψp,s (x) + a† (p, s)ψp,s (x)) . ΨM (x) = √ Ω p,s In other words, the creation/annihilation operators a and b are the same: a = b, thus ηa = ηb and Eq. (37) (which is ηa ηb = −1) implies η := ηa = ηb is imaginary. Putting everything together, we have for the field operator Pin(1, 3): λ is real: η/λ = −(η/λ)∗ . CΛ∗P = −ΛP C. UP (UC Ψ(t, x)UC−1 )UP−1 = (η/λ)ξΛP CΨ∗ (t, −x) = −(η/λ)ξCΛ∗P (Ψ(t, −x))∗ = +ξC((η/λ)ΛP Ψ(t, −x))∗ = UC (UP Ψ(t, x)UP−1 )UC−1 . ˆ is imaginary: η/λ ˆ = (η/λ) ˆ ∗ . CˆΛ ˆ∗ = Λ ˆ P C. ˆ Pin(3, 1): λ P ˆ Λ ˆ x)U −1 )U −1 = (η/λ)ξ ˆ ∗ (t, −x) ˆ P CˆΨ UP (UC Ψ(t, C P ˆ CˆΛ ˆ ∗ (Ψ(t, ˆ −x))∗ = (η/λ)ξ P ˆ Λ ˆ ˆ P Ψ(t, ˆ −x))∗ = ξ C((η/ λ) ˆ x)U −1 )U −1 . = UC (UP Ψ(t, P C Thus, in both cases, the phase η/λ makes sure that the two operations UC and UP commute on Fock space. That is, we can make statements about Ψc (such as the Majorana condition Ψc = Ψ) which are invariant under parity for both Pin groups. 3.4.5. Majorana classical field vs. Majorana quantum field We have established in Sec. 3.3 that the Majorana condition on a classical Dirac field ψc = ψ can be satisfied only by sections ψˆ of a Pin(3, 1) bundle. We have also established that the Majorana condition on a quantum Dirac field Ψ Ψc = Ψ

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ˆ On the other hand the classical can be satisfied by both types of operators Ψ and Ψ. field and field operator are of course related; the matrix elements of an operator Ψ take their values in the space of classical fields ψ. See for instance the subsection on Fock space operators, the equation h b |Ψ(a(p, s), b† (p, s))| 0 i = ψbc (p, s)

in Pin(1, 3).

ˆ a(p, s), ˆb† (p, s))| 0 i = ψˆbc (p, s) h b |Ψ(ˆ

in Pin(3, 1).

We also have

The nature of observed particles (i.e. excitations of the field) is dictated by the annihilation and creation operators. Since we do not observe the operator but its matrix elements, we observe the classical fields ψ. Hence the Majorana condition “particle identical to its antiparticle” needs to be implemented on the classical field ψ as well as the field operator Ψ — and we confirm the statements made by Yang and Tiomno, Beresteskii, Lifschitz, and Pitaevskii that a Majorana particle can only be a Pin(3, 1) particle. Remark. We have seen that η is necessarily imaginary for a Majorana field operator. This is an example of additional information which can be used to actually determine the Pin group through λ. We already showed above that a Majorana ˆ is imaginary, thus if we impose both Ψc = Ψ pinor must be a Pin(3, 1) pinor, i.e. λ ˆ of a Majorana particle is real. and ψ c = ψ the total intrinsic parity η/λ Remark. Weinberg [115] obtains an imaginary parity for a Majorana field operaˆ is imaginary as well as η, and we would expect tor. He works with Pin(3, 1), so λ ˆ to be real as in the previous remark. However, he redefines the parity the parity η/λ operator using other conserved quantities such as baryon number. See Sec. 4 for our discussion of parity and conserved quantities. Remark. Majorana fermions may be necessary in supersymmetric theories, such as eleven-dimensional N = 1 supergravity (see e.g. [26]). Indeed, in this theory, if the superpartner of the graviton — the gravitino — were not a Majorana fermion, the number of bosonic and fermionic degrees of freedom would not match. Even in the simplest N = 1 theory in four dimensions, the photino is a Majorana fermion [100]. 3.4.6. CP T transformations There are different equivalent formulations of the CP T theorem, combining charge conjugation, space and time reversal. See references in Appendix E: Collected References. We shall compute the effect of UC UP AT on the operator Ψ(x) where UC and UP are the unitary operators defined by Eqs. (31), (33) and (34), and AT is the antiunitary operator defined by AT Ψ(t, x)A−1 T = ζAT Ψ(−t, x) ,

(42)

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where AT is the operator on the space of pinors defined in Sec. 3.3: AT = ΛT C = ±(Γ1 Γ2 Γ3 )(±Γ2 ) = ∓(±Γ1 Γ3 )

since(Γ2 )2 = 11 .

The choice ζΛT C = −Γ1 Γ3 is the one used in the book by Peskin and Schroeder [87]. Equation (42) is the bridge connecting the Fock space and the space of pinors; so are the previously established relationships UC Ψ(x)UC−1 = ξCΨ∗ (x) ,

(43)

UP Ψ(x)UP−1 = (η/λ)ΛP Ψ(P x) .

(44)

In Pin(1, 3), ΛP = ±Γ0 and C = ±Γ2 , therefore Eqs. (42)–(44) give (UC UP AT )Ψ(t, x)(UC UP AT )−1 = ±ζ(η/λ)ξ Γ0 Γ1 Γ2 Γ3 Ψ(t, x) = (phases) Γ5 Ψ(t, x) = (phases) ΛP ΛT Ψ(t, x) . In conclusion, if CP T refers to an operator on one-particle states, we have established in Sec. 3.3 that CP T is the unitary operator ΛP ΛT which corresponds to orientation preserving Lorentz transformations (transformations of Determinant 1). If CP T is an operator on Fock space, it is the antiunitary transformation UC UP AT carried out by (phases) · ΛP ΛT . The determinant of (phases) · ΛP ΛT is 1. A similar result is obtained when working with Pin(3, 1). Invariance of a theory under CP T transformations implies the existence of antiparticles in the theory. 3.5. Bundles; Fermi fields on manifolds Given a representation ρ of a Pin group on a vector space V , we can construct a Pin bundle on a manifold, and define a pinor as a section of such a bundle. The same is true for the Spin group and spinors. The essence of bundle theorye is patching together trivial bundles (Manifold patch) × (Typical fibre) = Ui × V on the overlap of two manifold patches, Ui ∩ Uj . The fiber Vx at a point x in the manifold consists of all the pinors at this point. A map from the fiber at x to the typical fibre 4

ϕ i,x : Vx → V ,

x ∈ Ui

defines the coordinates of a pinor Ψ(x) for x ∈ Ui . But if x ∈ Ui ∩ Uj , the map 4 ϕj,x e We

: Vx → V ,

x ∈ Uj

use the notation of Choquet-Bruhat et al. [28] which is fairly standard.

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defines (probably different) coordinates for the same pinor Ψ(x). The patching is done by consistently choosing the maps 4 ϕi,x

4

◦ ϕ −1 j,x : V → V

which relate the coordinates of Ψ(x) for x ∈ Ui and x ∈ Uj . These maps are the transition functions gij (x) which act on V by the chosen representation on ρ of the Pin group on V 4

4

gij (x) = ϕi,x ◦ ϕ −1 j,x . The consistency condition is gik (x)gkj (x) = gij (x) . 3.5.1. Pinor coordinates See [28, p. 415]. The subtleties involved in defining the coordinates of a pinor arise from the fact that there is no unique choice of a transformation ΛL corresponding to a given Lorentz transformation L. Recall that if one wishes to define a vector v in a d-dimensional vector space by its coordinates, one says that v is an equivalence class of pairs (vi , ρi ) with vi ∈ Rd and ρi a linear frame in v, with the equivalence relation (ui , ρi ) ' (uj , ρj ) if and only if ui = Luj ,

ρj = Lρi ,

L ∈ GL(d) .

Similarly the coordinates of a pinor Ψ can be defined by an equivalence class of triples (Ψi , ρi , Λi ) with Ψi ∈ C4 , ρi an orthonormal frame, and Λi ∈ Pin(1, 3), with the equivalence relation Ψi = ΛL Ψj ,

ρj = Lρi ,

ΛL = (ΛL )i (Λ−1 L )j .

The four complex components of Ψ(x) in the Pin frame (ρi , Λi ) are the four complex numbers Ψi (x). In a similar fashion, the components of a copinor are defined by the equivalence class (Ψi , ρi , Λi ) with the equivalence relation Ψi = Ψj Λ−1 L ,

ρj = Lρi ,

ΛL = (ΛL )i (Λ−1 L )j .

3.5.2. Dirac adjoint in Pin(1, 3) Equipped with the definition of a pinor as an equivalence class of triples (Ψ(i) , ρ(i) , Λ(i) ) we can extend the definition of Dirac adjoint (14) from spinors

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to pinors [30, p. 36]. Let a(Λ) be a representation of Pin(1, 3) in Z2 = {1, −1}, such that a(Λ) = 1

for Λ covering orthochronous Lorentz transformations ,

a(Λ) = −1

otherwise .

Then ¯ (i) = a(Λ)Ψ† Γ0 Ψ (i) is such that ¯ (j) Λ−1 ¯ (i) = Ψ Ψ

when

Ψ(i) = ΛΨ(j) .

Proof. When ψ 7→ Λψ, then ψ † 7→ ψ † Λ† and ψ¯ 7→ aψ † Λ† Γ0 . †

(45) † Γ−1 0 Λ

−1

When is Λ Γ0 = Γ0 Λ ? Equivalently, when is Γ0 Λ = 11? −1 † We can check that Γ0 Λ Γ0 Λ commutes with all generators Γα in the basis of the Pin group, therefore it is a multiple of the unit matrix, † Γ−1 14 ; 0 Λ Γ0 Λ = a(Λ)1

by taking the determinant of both sides one obtains a4 (Λ) = 1 . Since a(Λ) takes discrete values, it is constant for Λ in any one connected component of the Pin group. We check that a(Λ) = 1 for Λ = 11 and Λ = Γ0 = Γ†0 ; hence a(Λ) = 1

for Λ ∈ components of Pin group labelled 11 and P ,

in other words for Λ covering the two components of the orthochronous Lorentz transformations. We check that a(Λ) = −1

otherwise.

3.5.3. Copinors, Dirac adjoints in Pin(3, 1) Following the same arguments as in the case of Pin(1, 3), one defines the Dirac adjoint ¯ ˆ4 , ψˆ := aψˆ† Γ where a is determined from the fact that ˆ −1 Λ ˆ †Γ ˆ 4Λ ˆ = a(Λ)1 ˆ 14 . Γ 4 ˆ † = −Γ ˆ 4 and Γ ˆ 2 = −1, we find that With Γ 4 4 ˆ =1 a(Λ)

ˆ in the component of Pin(3, 1) labelled 11 or P , for Λ

ˆ = −1 a(Λ)

otherwise,

For Dirac adjoints on hyperbolic manifolds, see for instance [30, p. 36].

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3.5.4. Pin structures We are now in a position to define a pin structure. Let H: Pin group → Lorentz group be the 2-to-1 homomorphism defined by β ΛL Γα Λ−1 L = Γβ L α ;

H(ΛL ) = L .

A pin structure over a (pseudo-)riemannian manifold M of signature (t, s) is a bundle of Pin frames over M together with its projection H over a given bundle of Lorentz frames over M . Two different Pin structures correspond to two different prescriptions for patching the pieces of the bundle of Pin groups (as a double cover of the Lorentz bundle) at the overlap of two patches on M (e.g. [30, p. 152]). 3.5.5. Fermi fields on topologically nontrivial manifolds The transition functions of a Pin(1, 3) bundle are elements of Pin(1, 3); the transition functions of a Pin(3, 1) bundle are elements of Pin(3, 1). The difference between Pin(1, 3) and Pin(3, 1) for pinors defined on a topologically nontrivial manifold is spectacular as we shall see shortly. But one should not conclude that the difference is topological: it is a group difference with topological implications which are fairly easy to display, and which were indeed the first ones to be analyzed. In chronological order, • Obstructions to the construction of Spin and Pin bundles [30, p. 134]. The criteria for obstruction are the nontriviality of some n-Stiefel–Whitney classes wn . For example, — A Pin(2, 0)-bundle can be constructed if w2 is trivial — A Pin(0, 2)-bundle can be constructed if w2 + w1 ∪ w1 is trivial • In supersymmetric Polyakov path integrals the contributing 2-surfaces depend on the choice of the Pin group [20]. • Quantized fermionic currents [36] on R(time) × (R × Klein Bottle) ≡ R2 × K2 . The Klein bottle alone would have been sufficient for displaying the difference between the Pin groups, but it was convenient to use earlier works done on 3 space, 1 time manifolds [38]. The Klein bottle K2 is an interesting manifold for displaying the difference between the Pin groups for the following reasons: • K2 is not orientable (the first Stiefel–Whitney class w1 is not trivial), Thus a Pin bundle, if it exists, is not reducible to a Spin bundle. The Klein bottle forces one to construct Fermi fields with nontrivial transformation laws under space inversion on at least one of the overlaps of the coordinate patches. • The Klein bottle admits both kinds of pinor fields, since both w2 and w1 ∪ w1 are trivial.

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We review briefly the results. In particular, we explain which types of currents (scalar, vector, . . .) can exist on R2 × K2 once a Pin group is chosen, and we review the explicit expectation values of those currents. To obtain a topology R2 × K2 we identify, in a cartesian coordinate system, the points (x0 , x1 , x2 , x3 )

with

(x0 , x1 , x2 + ma, x3 + (−1)m x3 + nb)

for all integers m, n. Schematically, one expresses the vacuum expectation values of all fermionic ¯ bilinears hΨ(x)AΨ(x)i in terms of vacuum expectation values of the chronologically ¯ ordered product hΨ(x)Ψ(x)i, i.e. in terms of the Feynman–Green function G(x, x0 ). The Feynman–Green’s function G is expressed in terms of the Feynman–Green function G of the Klein–Gordon operator. For a massless field G ∝ Γα ∂α G . G is infinite at the coincidence point x = x0 . Therefore we subtract the term which equals the Minkowski Feynman–Green function. This is a cheap and easy way to renormalize, but it is valid in this case. We find that G, or rather G renormalized to eliminate the infinity at the coincidence point, is in the case of Pin(1, 3) Gren =

i (2π)2

X

(−1)m (−(x0 − x00 )2 + (x1 − x01 )2 + (x2 − x02 + 2ma)2

m6=0,n6=0

+ (x3 − x03 + nb)2 )−1 +

X (−1)m (−iΓ0 Γ1 Γ2 )(−(x0 − x00 )2 m,n

!

+ (x1 − x01 )2 + (x2 − x02 + (2m + 1)a)2 + (x3 + x03 + nb)2 ))−1

.

The sum has been split into two sums, one with the contributions of 2m, and one with the contributions of 2m + 1 because of the factor (−1)m affecting x3 . The “renormalization” consists in removing the n = 0, m = 0 term from the first term since this term is, as in the Minkowski case, infinite at x = x0 . The remainder goes to zero as the Klein bottle becomes large (a, b → ∞); G should properly be treated as a distribution, but G is a distribution equivalent to a function. The term Γ0 Γ1 Γ2 implements on the fermion field the periodic reversal of the x3 coordinate. For Pin(3, 1), Gren =

1 (2π)2

X

(−(x0 − x00 )2 + (x1 − x01 )2 + (x2 − x02 + 2ma)2

m6=0,n6=0

+ (x3 − x03 + nb)2 )−1 +

X

Γ0 Γ1 Γ2 (−(x0 − x00 )2 + (x1 − x01 )2

m,n 02

! 03

2 −1

+ (x − x + (2m + 1)a) + (x + x + nb) ) 2

2

3

.

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The bilinear fermionic expectation values are expressed in terms of G by ¯ hΨAΨi ∝ tr(AΓα ∂α G) x=x0 for Pin(1, 3) and a similar expression for Pin(3, 1). In both cases the derivatives with respect to x0 and x1 vanish at the coincidence points, but for Pin(1, 3) the derivative w.r.t. x3 vanishes, for Pin(3, 1) the derivative w.r.t. x2 vanishes. It follows that for Pin(1, 3) the only nonvanishing current is a tensor with A = [Γ0 , Γ1 ], for Pin(3, 1) the only nonvanishing current is a pseudoscalar with A = Γ5 . We refer to [36] for the explicit expressions of the currents and their graphs. The two cases are totally different. Currents are observables and in principle one could measure them. While a spacetime with a Klein bottle topology, if it exists, would be difficult to probe, one could imagine solid-state systems for which the configuration space would be periodic like a Klein bottle. Pending such a situation, we searched for other observable differences in the Pin groups. This work, begun by two of us (SJG and EK) has been continued by MB. 3.6. Bundle reduction We recall briefly the essence of bundle reduction. Consider a principal Pin bundle over a manifold M (i.e. a bundle whose typical fiber is the Pin group) and a principal Spin bundle over the same manifold; or simply a G-bundle and an H-bundle, where H is a subgroup of G. Let the principal G-bundle be labeled (P, M, π, G), π : P → M ; and let the principal H-bundle be labeled (PH , M, πH , H). One says that the G-bundle is reducible to the H-bundle if PH ⊂ P , πH = π|PH . Alternatively: the G-bundle is reducible to the H-bundle if the G-bundle admits a family of local trivializations with H-valued transition functions. One says: the structure group G is reducible to H. A useful criterion: the G-bundle is reducible to an H-bundle if and only if the bundle P \ H (typical fibre G \ H, associated to P by the canonical left action of G on G \ H) admits a cross section. An example of a reducible bundle: the structure group GL(n, R) of the tangent bundle of the differentiable mainfold Rn is reducible to the identity. In other words, the tangent bundle is reducible to a trivial bundle. This does not mean that the action of GL(n, R) on the tangent bundle is without interest. A vector bundle with typical fiber V is said to be associated to a principal bundle G, if the transition functions act on V by a representation of G on V . Pinors are sections of vector bundles associated to a principal Pin bundle. For brevity we shall

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say “pinors are sections of a Pin-bundle”. The properties of a principal G-bundle induce corresponding properties on its associated bundles, such as reducibility. Massless pinors are sections of Pin bundles reducible to Spin bundles, Massive pinors are sections of Pin bundles not reducible to Spin bundles.

Consider a Pin bundle reducible to a Spin bundle, and an object, say a Lagrangian defined on the Pin bundle; let the inclusion map i : Spin bundle → Pin bundle . The pullback i∗ maps forms on the Pin bundle to forms on the Spin bundle; it maps the Pin bundle Lagrangian into a Spin bundle Lagrangian — which is likely to be different from the Lagrangian obtained by replacing pinors by spinors in the original Lagrangian. For example, symmetry breaking is responsible for introducing mass terms in a Lagrangian. Bundle reduction, the mathematical expression of symmetry breaking, yields the mass terms by pulling back the original Lagrangian into the subbundle. An obvious investigation is to apply bundle reduction to a massless neutrino Lagrangian defined on a Pin bundle in order to determine its pull back on a Spin bundle. But we are temporarily putting this project aside since this paper has been a long time on the drawing board and we wish to bring it to a closure. 4. Search for Observable Differences In Sec. 4, we investigate what the observable consequences of the mathematical issues discussed in Sec. 3 are. We are cautiously optimistic of finding experiments which can be used to select one Pin group over the other for a given particle. We have ruled out certain setups which seem attractive at first glance; we present them nevertheless because their failures are instructive. There are several promising ideas but it is too early to assess their chances of success. There is one iron-clad identification: the neutrino exchanged in neutrinoless double beta decay is a Pin(1, 3) particle. As means for selecting a Pin group we examine experiments involving parity in Sec. 4.3, time reversal in Sec. 4.5 and charge conjugation in Sec. 4.6. One conclusion which is easy to see is that Pin(3, 1) fermions cannot interact ¯ ψˆ where M is some matrix, because, as with Pin(1, 3) fermions via terms ψM mentioned by Berestetskii et al. [12] this term acquires an i under parity transformations and damps the exponential of the action. Of course, Pin(3, 1) and Pin(1, 3) ¯ˆ ˆ ¯ ψ ψN fermions could interact via nonrenormalizable four-fermion terms ψM ψ for some matrices M and N , but we do not consider such terms. 4.1. Computing observables with Pin(1, 3) and Pin(3, 1) One of the fundamental quantities one calculates in particle physics is the scattering cross section, or alternatively decay rate. It is almost always computed using pinors

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from Pin(1, 3). However, when using Pin(3, 1), there are some changes that could potentially affect observables. 4.1.1. Trace theorems Traces of 4n + 2 gamma matrices, where n = 0, 1, 2, . . . , are equal between the two Pin groups. Traces of 4n + 4 gamma matrices differ by a sign between the two groups. Therefore, linear combinations of the following traces, for instance, are different: tr(Γµ Γν ) = 4ηµν ˆ µΓ ˆ ν ) = 4ˆ tr(Γ ηµν = −4ηµν tr(Γµ Γν Γρ Γσ ) = 4(ηµν ηρσ − ηµρ ηνσ + ηµσ ηνρ )

(46)

ˆν Γ ˆρΓ ˆ σ ) = 4(ˆ ˆµΓ tr(Γ ηµν ηˆρσ − ηˆµρ ηˆνσ + ηˆµσ ηˆνρ ) = 4(ηµν ηρσ − ηµρ ηνσ + ηµσ ηνρ ) . For example, if A is proportional to tr Γµ Γν and B is proportional to tr Γµ Γν Γρ Γσ , ˆ in Pin(3, 1). then A + B in Pin(1, 3) corresponds to −Aˆ + B 4.1.2. Spin sums When computing unpolarized cross sections, one needs to sum over spin states. Let ˆ Ψ(x) in Pin(1, 3) be defined by (29) and Ψ(x) in Pin(3, 1) be defined similarly. With µ µ ˆ pµ as the case may be, we have /p = Γ pµ , or /p = Γ X X u(p, s)¯ u(p, s) = /p + m, uˆ(p, s)u ˆ¯(p, s) = −i/p + m s

X

s

v(p, s)¯ v (p, s) = /p − m,

s

X

vˆ(p, s)vˆ¯(p, s) = −i/p − m ,

s

if we use the normalizations u ¯(p, r)u(p, s) = 2mδrs ,

u ˆ¯(p, r)ˆ u(p, s) = 2mδrs

v¯(p, r)v(p, s) = −2mδrs ,

vˆ¯(p, r)ˆ v (p, s) = −2mδrs .

This is shown in Appendix D. 4.2. Parity and the particle data group publications In Sec. 3.4 we defined the intrinsic parity of a quantum field Ψ as η/λ, where the phase η comes from the Definition (31) of the unitary operator UP acting on the field operators, and the phase λ is the parity eigenvalue of the pinor u(p, s) (or uˆ(p, s)) in Eq. (36).

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Here in Sec. 4, P stands for P (3), reversal of the three space axes. See Eq. (8) ˆ P ∈ Pin(3, 1). We recall in Sec. 3.1 for the calculation of ΛP ∈ Pin(1, 3) and Λ Λ2P = 11 ,

ˆ 2 = −11 . Λ P

The phase η is usually a matter of convention, but as was shown in Sec. 3.4, η must be imaginary for a Majorana particle (a particle which is its own antiparticle). Thus there is at least one way of restricting the choice of phase η. In the Particle Data Group (PDG) publications, intrinsic parity is always real, so for us η/λ = ±1. The Pin group used in PDG publications is Pin(1, 3), and we infer that η = ±1 corresponds to the PDG convention. As can be seen in Eq. (36), the eigenvalue λ of the parity operator only takes on the values +1 and −i unless we change pin structure (see Sec. 3.5 for a discussion of pin structures, see also Eqs. (38) and (39) and the remark thereafter), which is only necessary on spaces with nontrivial topology (also Sec. 3.5). One reason many physicists discard parities ±i is the intuitive, but, in the case of fermions, faulty argument that two successive reflections bring us back to the original state: fermions change sign under 2π rotations. Recall that this is true for fermions of both pin groups — see Eqs. (8) and (9). As pointed out in the book by Bjorken and Drell [14], “four reflections return the spinor to itself in analogy with ˆ 4 = 1. a rotation through 4π radians”. Indeed (η/λ)4 = 1 and (η/λ) Finally, it is clear that intrinsic parity is a “relative” concept, i.e. one needs to define some particle to have, say, parity +1 to fix the number for another particle transforming under Pin(1, 3). In the PDG publications, three (composite) particles are chosen as “reference particles” and parities of other particles are determined by comparison with one of the three reference particles.f Since we have intrinsic parity as η/λ, defining a reference parity still leaves some freedom in specifying η and λ unless there are extra conditions such as that for a Majorana particle. Examples of this freedom are given in Sec. 4.3, for example in determining the intrinsic parity of a pion. The PDG defines

Reference Particles

f There

Particle

Intrinsic parity

Proton

+1

Neutron

+1

Λ

+1

is no logical necessity for having three reference particles, other than a convenience for analyzing experimental data within the bounds of the possibly approximate conservation laws of baryon number, lepton number and strangeness (or, as Weinberg proposes [115], electric charge, since strangeness conservation is now known to be approximate).

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These particles are not elementary particles, but the distinction seems to be insignificant; one finds experimentally that these composite particles have well-defined intrinsic parities, so they are just as good reference particles as any other. Furthermore, as far as the present set of elementary particles goes, confined quarks would be difficult to have as experimental references, and leptons are written as Weyl fermions in the Standard Model and as such cannot be acted upon by the parity operator (see Sec. 3.3). 4.2.1. Parity conservation In Appendix D, we briefly review how the observed angular distribution of scattered particles is used for concluding whether or not parity is conserved. When it is, one arrives at the expression for conservation of parity: (−1)ì ηa ηb = (−1)`f ηc ηd

(47)

which says that η is (multiplicatively) conserved in a parity-conserving interaction Hint , i.e. if Hint commutes with UP . We can use (47) to determine one unknown η, for example, using previously known or defined parities. Remark. When representing a particle by a classical field ψ, the eigenvalues λ of the parity operator ΛP identify the relevant Pin group. When representing a particle by a Fock state built by creation operators, parity is not identified by λ but by η/λ. Notice that η is the quantity which appears in the conservation law. Clearly the fact that intrinsic parity is multiplicatively rather than additively conserved is irrelevant, we could redefine η to be the exponential of another symbol. In the decay of Co60 , electrons are predominantly emitted in a certain direction, therefore Eq. (D.4) in the appendix is not satisfied, and the interaction is said to violate parity (conservation). 4.3. Determining parity experimentally There are two different broad approaches to determining intrinsic parity experimentally: by using selection rules, or by studying decay rates, cross sections and polarizations. 4.3.1. Selection rules: Pion decay The textbook example [46, 115] of determining intrinsic parity by a selection rule is the negative pion (π− ). It is reviewed in Appendix D, here we just give the result: ηπ ηd = (−1)ηn ηn where d is the deuteron captured by the pion, which then decays to two neutrons n. We study how the determination of the pion’s parity proceeds in Pin group language.

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First, the deuteron and the pion have integer spins, so they cannot have imaginary λ values. We show in the positronium example below that the parity of an s-wave bound state of two fermions a and b is ηa ηb (not just for positronium). Then the intrinsic parities of the pion and deuteron are ηπ and ηd = ηa ηb , respectively. If we assume the neutrons are Pin(1, 3) particles, then λp = λn = +1 so ηp = ηn = +1 by the reference parities. With these assumptions we find ηπ = −

ηn2 = −1 . ηd

ˆp = λ ˆ n = −i so ηp = ηn = If we assume the neutrons are Pin(3, 1) particles, then λ −i and we obtain the same result. From the explicit discussion in the appendix, we see that even this comparatively simple argument relies on input from various sources (orbital state of deuteron and π − d atom as a whole, fact that interaction is parity-conserving). The only three principles we invoked were angular momentum conservation, Fermi statistics and conservation of intrinsic parity. All of these are independent of the choice of Pin group. On general grounds we can therefore expect this experiment to be incapable of detecting a difference between the two Pin groups, but it is somewhat instructive.

4.3.2. Selection rules: Three-fermion decay Another example of a “selection rule” type argument can be found in the book by Sternberg [103]. There it is claimed that the following argument can determine the difference between the two Pin groups. It is mentioned that a fermion cannot decay into three fermions through a parity-conserving interaction in the Pin group for which λ, the eigenvalue of ΛP , is imaginary (in our conventions, this is Pin(3, 1)), because (±1)3 = ±1

whereas

(±i)3 = −(±i) .

There are two arguments that show why the conclusion “a Pin(3, 1) fermion cannot decay into three fermions” is too hasty. First, since intrinsic parity is η/λ, where η is the phase in the definition of UP and the quantity which appears in the parity conservation law, intrinsic parity is not directly related to Pin group through η unless there is an extra requirement on η, such as the Majorana condition. If η can be chosen real or imaginary by convention, we cannot determine the Pin group in this way. Second, just like a fermion with η = +1 can decay into three fermions of η = −1, −1 and +1, for example, a fermion of η = +i can decay into three fermions with η = +i, +i and −i. We cannot infer that three-fermion decay is always forbidden merely from i3 = −i.

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4.3.3. Selection rules: Positronium There is a beautiful experiment, first proposed by Wheeler [118], to verify experimentally the relation (37) for the phase of the parity operator from Sec. 3.4. We review the experiment in Appendix D for completeness. 4.3.4. Decay rates; cross sections There is a plethora of different accelerator experiments which are capable of determining the intrinsic parity of a particle. Actual examples include but are not limited to polarized target experiments, production experiments and electromagnetic decays. The methods for studying parity are sometimes similar to those used in determining spin, but there is no theoretical reason that we know of why the two should be related. We choose to concentrate on one particular experiment for definiteness. We have chosen the beautiful Σ0 -parity Steinberger experiment [3] from 1965. One could ask why we have chosen to analyze such an old experiment, given the immense progress that has been made in experimental particle physics during the last three decades. However, once a discrete attribute such as the intrinsic parity of a particle (or resonance) is determined to good accuracy, it is of course unattractive for experimentalists to construct a dedicated experiment to measure it again. The attribute might be measured as a “byproduct” of other experiments, but such determinations would, by the same token, be less straightforward for us to analyze here. Since parities of most particles were already measured in the 1960s, one finds that most of these dedicated experiments were done around 1965 or before. The experiment revolves around the electromagnetic decay Σ0 → Λ0 + e+ + e− where the parity of the Λ0 is chosen as one of the reference parities. The simple idea, put forward by Feinberg [44], is to measure the branching ratio for the above decay relative to the main decay mode Σ0 → γγ. The QED prediction

Σ

p

q

Λ e+

k

k2 k1 -

e Fig. 4.

Tree-level QED diagram for Σ0 → Λ0 + e+ + e− .

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is different for the hypotheses Σ0 -parity +1 and Σ0 -parity −1 (we prefer to not use the terms “odd” and “even” due to the possible existence of four parities). We shall show very briefly how this difference arises. Let us fix η, the phase in UP , to be η = 1 for now. The tree-level diagram is shown in Fig. 4. Using standard notation [87] we write down the matrix elements for the two hypotheses. Since we do not have an a priori ΛΣ vertex, we write down all possible bilinears, and determine the coefficients of each experimentally. It turns out that the dominant contribution is the tensor or pseudotensor: M+ = e2

iF 1 (¯ uΛ (q)Γµν k ν uΣ (p)) 2 (¯ u(k1 )Γµ v(k2 )) M k

M− = e2

1 iF (¯ uΛ (q)Γ5 Γµν k ν uΣ (p)) 2 (¯ u(k1 )Γµ v(k2 )) . M k

Here F is a form factor, and we have written Γµν = 2i [Γµ , Γν ] and used the average mass M = 12 (MΛ + MΣ ). There are also other terms contributing to the diagram, but the form factor F is sufficiently large for terms with other form factors to be neglected. The idea is that if, for example, M− is the correct matrix element, we can shift the Γ5 to the right and include it in uΣ . This means that λΣ , the eigenvalue of ΛP for the Σ particle, switches sign due to ΛP Γ5 = −Γ5 ΛP . Thus in the case of M− the relative parity of Λ and Σ would be −1. Since there is only one diagram (in this approximation), we immediately see that any phase will eventually be canceled when we take the absolute value squared. However, we summarize in Appendix D how this is manifested using the rules from Sec. 4.1, since a similar calculation may prove important in other settings. 4.4. Interference, reversing magnetic fields, reflection In Pin(1, 3) and Pin(3, 1), two successive parity transformations are given, ˆ 2 = −11. Hence if one could construct an experiment respectively, by Λ2P = 11 and Λ P corresponding to Fig. 5, one might be able to differentiate between the two types of pinor particles. A beam must somehow be split into two beams, one of which is inverted twice while the other is left unaffected. Then the two beams must be brought back together and allowed to interfere. Where Pin(1, 3) particles interfere constructively (Fig. 5a), Pin(3, 1) particles will interfere destructively (Fig. 5b). 4.4.1. Reversing magnetic fields Not knowing how to construct a space reversal apparatus (the boxes in Fig. 5) we considered the following experiment:

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a)

Ψ constructive

Ψ

interference

2

ΛP

+Ψ

b)

Ψ destructive

Ψ

interference

2

ΛP Fig. 5.

_

Ψ

A type of experiment which should give different results for the two types of pinors.

z

z b)

a)

s

s B1

y

1 0 0 1

1 0 0 1

y

x

x z c)

1 0 0 1

s

y

x B2 Fig. 6. A z-polarized electron (Fig. 6a) is placed for a while in a magnetic field B1 along the x-axis causing its spin to precess around the x-axis (Fig. 6b); later on a magnetic field B2 in the y-direction is switched on (Fig. 6c). One could also consider an electrically neutral spin 1/2 particle in a potential µs · B = µijk σij B k (i, j, k ∈ {1, 2, 3}) where µ is the gyromagnetic ratio and σij the spin angular momentum operator.

A particle beam is split in two parts: One part passes through a magnetic field in the x direction followed by a magnetic field in the y direction as in Fig. 6, the other part passes through a magnetic field in the x direction followed by a magnetic field in the −y direction. The two parts of the beam are then recombined and allowed to interfere. An explicit calculation by one of us (EK) showed that the interference of the two parts of the beam having experienced the two different magnetic field configurations is the same for a Pin(3, 1) beam and for a Pin(1, 3) beam. The explicit calculation consists in comparing the transition amplitudes for Pin (3, 1) and Pin (1, 3) electrons moving under the conditions described in Fig. 3. The

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system evolves according to the Dirac equation which reads (iΓα (∂α + iqAα ) − m)ψ = 0

for Pin(1, 3) particles ,

ˆ α (∂α + iqAα ) − m)ψˆ = 0 (Γ

for Pin (3, 1) particles .

ˆ α may be represented as the matrices −iΓα , the equations are the same. Two Since Γ equivalent initial states under the same evolution remain equivalent throughout all time. Thus, both parts of the beam in the Pin (3, 1) case evolve exactly the same way as their respective counterparts in the Pin (1, 3) case, the interference patterns produced in either case are identical. The same will hold true for any configuration in which the matter field is required to change continuously. 4.4.2. Reflection Since the previous experiment turned out not to give a parity transformation, EK went on to study the interference between two parts of a fermion beam passing through some medium M as shown in Fig. 7. The part which follows path 1 is transmitted directly through the medium, whereas the part which follows path 2 is reflected twice before passing through. Such an experiment could be achieved by passing neutrons through a magnetic crystal. Although the idea of reflection from a surface might bring to mind the idea of parity transformation, the reflections involved here have nothing to do with the ˆ P ). In other words, although the neutrons following parity transformation ΛP (or Λ path 2 are reflected twice, they do not undergo any parity transformations as called for in Fig. 5. Hence this setup also fails to realize Fig. 5. That reflection at a boundary does not produce any parity transformation is best seen by just solving the problem of reflection and transmission of a plane wave between two media, one with a free-particle Dirac Hamiltonian and one with a Hamiltonian consisting of the free-particle part plus a potential V , as shown schematically in Fig. 8. The solution of this problem can be found in several books [45]. It is done by matching coefficients of solutions and will not bring in any parity transformations; again the requirement of continuity is the stumbling block.

M

2

1

Fig. 7.

A reflection experiment intended to find a difference between two types of pinors.

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V= 0

V=0 Ψ ’’

Ψ’ Ψ

x z

Fig. 8.

Reflection and transmission of a pinor particle at an interface.

Thus reflection at a boundary cannot be used to distinguish between particles of different Pin groups either. 4.5. Time reversal and Kramers’ degeneracy In quantum mechanics, if one requires the Hamiltonian to be invariant under time reversal, then the time reversal operator is antiunitary. Indeed the Hamiltonian H generates time evolution: H| t i = i∂t | t i ; H is invariant under time reversal if there exists an antiunitary operator AT such that AT H ∗ A−1 T =H with the asterisk denoting complex conjugate; then AT | t i∗ satisfies the time reversed equation. Kramers has shown (see e.g. [115, p. 81] [96, p. 281] or [74, p. 408]; see also [6, p. 601]) that, given an antiunitary time reversal operator AT defined as in Eq. (42) with the matrix AT satisfying AT A∗T = −11 and such that it commutes with the Hamiltonian H of the system, an eigenstate (|ni) of H and the time reversed eigenstate AT (|ni∗ ) are two different states with the same energy. This degeneracy can be removed by adding an interaction which does not commute with AT . Can Kramers’ theorem provide a method for distinguishing Ψ-particles (with ˆ ˆ 2 = −11)? We have established the relationship Λ2T = 11) from Ψ-particles (with Λ T between the time reversal operator AT and the unitary time reversal operator ΛT in Sec. 3, namely AT C −1 = ΛT .

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Kramers’ theorem cannot be used to distinguish the two Pin groups (even though ˆ 2 = −11): Λ2T = 11 and Λ T ˆ T , respectively. Let the operators AT and AˆT correspond to ΛT and Λ ΛT = AT C −1 ,

ˆ T = AˆT Cˆ−1 , Λ

where CΓ∗α C −1 = −Γα

and

ˆ ∗ Cˆ−1 = Γ ˆα . CˆΓ α

A double time reversal is not produced by A2T but by AT A∗T since, according to the defining equation ψ 0 (T x) = AT ψ ∗ (x), and the field operator Ψ transforms according to Eq. (42) as −1 ∗ ∗ ∗ AT AT Ψ(x)A−1 T AT = ζAT (ζAT Ψ (x)) = AT AT Ψ(x) .

ˆ 2 , the double antiunitary time reversal shows no such difference: Whereas Λ2T 6= Λ T AT A∗T = AˆT Aˆ∗T = −11 . ˆ Both Ψ- and Ψ-particles can be used to construct degenerate time-reversed pairs. It follows that Kramer’s degeneracy cannot be used to distinguish Ψ-particles and ˆ Ψ-particles. 4.6. Charge conjugation; positronium; neutrinoless double beta decay 4.6.1. Positronium Once more, positronium proves to be a useful test bed for discrete symmetries. In Appendix D we briefly review from our perspective the textbook example of how charge conjugation decides the lifetimes of two different positronium states. 4.6.2. Neutrinoless double beta decay Double beta decay occurs usually with the emission of two neutrinos. However, if the neutrino associated with a beta decay is reabsorbed to produce a second beta decay, then no neutrino is emitted, and the process is called neutrinoless double beta decay. Diagrams of neutrinoless double beta decay in two different reactions are given in Figs. 9 and 10. If the same neutrino is emitted and absorbed, it has to be a particle identical with its antiparticle, i.e. it has to be a Majorana particle. (Recall that a Majorana particle is one for which ψ c = ψ). The possible observation of neutrinoless double beta decay has been analyzed by Klapdor-Kleingrothaus [60]. The discussion in Sec. 3.4 indicates that a Majorana particle can only be a Pin(3, 1) particle. Thus the existence of Majorana neutrinos, if confirmed, would have some implication for the topology of the universe, namely that the universe is a manifold which can serve as a base for a Pin(3, 1)-bundle.

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d

u

_

W _

e

Majorana neutrino

e

_

_

W u

d Fig. 9.

Quark diagram for neutrinoless double beta decay (2n → 2p+2e− in 82 Se → 82 Kr+2e− ).

d

u W

+

e+

Majorana neutrino _

e+ W

+

s

_

u

Fig. 10. We note also another type of neutrinoless double beta decay: Quark diagram for K + → π − + 2e+ .

5. The Pin Group in s Space, t Time Dimensions In order to analyze the properties of the two Pin groups in arbitrary dimensions, we first review and simplify a few topics, treated in detail in [28, 30] and [39]. This section is organized as follows: 5.1 5.2 5.3 5.4 5.5 5.6 5.7

The difference between s + t even and s + t odd. Chirality. Construction of the gamma matrices. Periodicity modulo 8. Conjugate and complex gamma matrices. The short exact sequence 11 → Spin(t, s) → Pin(t, s) → Z2 → 0. Grassman (superclassical) pinor fields. String theory and spin structures.

In this section we will use the P (1) parity transformation, which reverses only one axis instead of three. 5.1. The difference between s + t even and s + t odd In this section s + t = d = 2p and s0 + t0 = d + 1 = 2p + 1. In brief: t + s = 2p = d • Only one irreducible faithful representation of the gamma matrices. • The center of Pin(t, s) is R+ 1 (multiples of the unit element). ×1

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t0 + s0 = 2p + 1 = d + 1. • Two inequivalent irreducible faithful representations of the gamma matrices. • The center of Pin(t0 , s0 ) is R+ 1 ∪ R+ ×1 × Γd+2 . 0 0 0 0 • The map Pin(t , s ) → O(t , s ) is not surjective. A key element in the proofs of some of the above statements is the construction of the generators for Pin(t0 , s0 ) given the generators {Γα } for Pin(t, s). We begin with s + t = d = 2p. The algebra over the reals generated by the (possibly complex) 2p × 2p matrices (Γα ) is a faithful representation of the Clifford algebra C(t, s). This representation is unique, modulo similarity transformations, and irreducible. The center (set of elements which commute with all elements) of Pin(t, s) is 1 R+ × 1. We now consider the case s0 + t0 = d + 1 = 2p + 1, with either s0 = s + 1 or t0 = t + 1. Set Γd+1 := Γ1 Γ2 · · · Γd

ˆ d+1 and similarly for Γ

(d even) .

(48)

We note (Γd+1 )2 = (−1)s+p 112p ,

ˆ d+1 )2 = (−1)t+p 112p , (Γ

d = 2p .

(49)

We also note that Γd+1 anticommutes with all Γα ∈ Pin(t, s), and that a similar ˆ d+1 . Therefore we can use kΓd+1 , where k is a phase, to statement holds for Γ construct a basis for Pin(t+1, s) and for Pin(t, s+1) as follows; similar construction ˆ d+1 . of Pin(s + 1, t) and of Pin(s, t + 1) can be done using Γ For k 2 = (−1)s+p ,

(kΓd+1 )2 = 11

(50)

the set of anticommuting matrices {Γα , kΓd+1 } generates Pin(t + 1, s). The two choices k = ±(−1)(s+p)/2 provide two inequivalent representations of the group Pin(t + 1, s). For k 2 = (−1)s+p+1 ,

(kΓd+1 )2 = −11

(51)

the set of anticommuting matrices {Γα , kΓd+1 } generates Pin(t, s + 1). The two choices k = ±i(−1)(s+p)/2 provide two inequivalent representations of Pin(t, s + 1). Similar results hold for Pin(s + 1, t) and Pin(s, t + 1). For s0 + t0 odd, the center of Pin(t0 , s0 ) consists of R+ 1 and R+ ×1 × Γd+2 . Indeed the product Γd+2 = Γ1 Γ2 · · · Γd Γd+1

with

commutes with all elements in {Γα , Γd+1 }.

Γd+1 = Γ1 Γ2 · · · Γd

(52)

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For s0 + t0 odd, the map Pin(t0 , s0 ) → O(t0 , s0 ) is not surjective. Namely, there is no element in Pin(t0 , s0 ) which maps into the element of O(t0 , s0 ) which reverses the axes. Indeed let (P T ) = diag(−1, . . . , −1), then there is no ΛP T satisfying β ΛP T Γα Λ−1 P T = Γβ (P T ) α = −Γα ,

for all Γα

(53)

since this would imply ΛP T Γd+2 + Γd+2 ΛP T = 0 , but Γd+2 commutes with all elements in Pin(t0 , s0 ), and ΛP T Γd+2 6= 0 since ΛP T and Γd+2 are invertible. 5.1.1. The twisted map To eliminate some of the differences between d odd and d even, one can introduce a map, sometimes called the twisted map, ˜ : Pin(t, s) → O(t, s) , H surjective in all dimensions, as follows. The Clifford algebra is a graded algebra C(t, s) = C+ (t, s) + C− (t, s)

(54)

where C+ is generated by even products of elements of the basis, and C− is generated by odd products. Let α(Λ+ ) = Λ+ ,

for Λ+ ∈ C+ (t, s) ,

α(Λ− ) = −Λ− ,

for Λ− ∈ C− (t, s) .

(55)

˜ defined by The map H, β α(ΛL )Γα Λ−1 L = Γβ L α

(56)

is surjective in all dimensions. ˜ α ) reverses the α-axis. The twisted map H ˜ seems deWe note also that H(Γ sirable, but Eq. (56) is not a similarity transformation and the invariance of the Dirac equation under Lorentz transformations requires the similarity transformation ΛL Γα Λ−1 = Γβ Lβ α . Attempts to find maps [58] ρ : Pin(t, s) → Pin(t, s) to recover a similarity transformation, i.e. ρ such that −1 α(ΛL )Γα Λ−1 , L = ρ(Λ)Γα (ρ(Λ))

obviously fail in odd dimensions, and are very awkward in even dimensions. We shall not work with the twisted map.

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5.2. Chirality In this paragraph the matrix Γd+1 = Γ1 Γ2 · · · Γd is used to define chirality for d even. Γd+1 is a linear operator on the space S of spinors. We recall (Γd+1 )2 = (−1)s+p 112p ,

ˆ d+1 )2 = (−1)t+p 112p . (Γ

(57)

The eigenvalue equation Γd+1 ψ = αψ implies (Γd+1 )2 ψ = α2 ψ = (−1)s+p ψ

by (57) ,

hence α2 = (−1)s+p . Thus α = ±1

if s + p is even ,

α = ±i

if s + p is odd (as in Sec. 3.3) .

One denotes by S+ the eigenspace with eigenvalue 1 or i, and by S− the eigenspace with eigenvalue −1 or −i. S = S+ ⊕ S− . Therefore the projection matrices  1    (112p ± Γd+1 ), 2 P± =  1   (112p ± iΓd+1 ), 2

s + p even (58) s + p odd

project a 2p -component pinor into two 2p−1 -component spinors. Γd+1 for s+p even, and iΓd+1 for s + p odd, are called chirality operators. A chiral basis is a basis adapted to the splitting S = S+ ⊕ S− . In a chiral basis ! −112p−1 0 Γd+1 = . 0 112p−1 Polchinski [88, App. B] gives a recursion construction of a chiral basis in terms of the Pauli matrices {σi }. Remark. For s + t even, Γα Γd+1 = −Γα Γd+1 , hence the matrix Γd+1 is a solution ΛL of ΛL Γα Λ−1 L = −Γα which implies (L) = diag(−1, . . . , −1). Since the dimension of space-time is even, this Lorentz transformation does not change the handedness of the system of coordinates.

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5.3. Construction of gamma matrices, periodicity modulo 8 Two useful mathematical references are the books by Gilbert and Murray [48] and by Porteous [89]. The study of Pin(t, s) for arbitrary t and s is considerably simplified by the fact that the groups depend not on s and t but on |s − t| modulo 8. Moreover, Pin(t, s) and Pin(s, t) are isomorphic for |s − t| = 0 modulo 4. The proof uses the isomorphism M4 (R) ' H ⊗R H ,

(59)

where M4 (R) is the algebra of real 4 × 4 matrices and H the quaternion algebra. This isomorphism is not trivial. It has been questioned on the grounds that the left hand side is real and the right hand side seems to be complex, given that a well known two-dimensional representation of the quaternion basis consists of the matrix 112 together with i times the Pauli matrices which cannot be all imaginary. This argument for the complexity of the quaternion algebra is obviously meaningless since complex representations of real algebras abound, as can be seen, for instance, in this paper. We prove the isomorphism in Appendix B because it is not easily available to non-specialists. In Sec. 5.1, we constructed Pin(t + 1, s) and Pin(t, s + 1) by adding kΓd+1 , with different values of k 2 , to the basis of Pin(t, s) with t + s = d = 2p. Now we combine t + s = 2p and t0 + s0 = d0 arbitrary. By tensoring the Clifford algebras C(t, s) and C(t0 , s0 ) one can obtain either one of the two Clifford algebras C(t + t0 , s + s0 ) or C(t + s0 , s + t0 ), depending on the sign of k 2 in kΓd+1 . C(t, s) ⊗k C(t0 , s0 ) = C(t + t0 , s + s0 )

for k 2 = 1

(60)

C(t, s) ⊗k C(t0 , s0 ) = C(t + s0 , s + t0 )

for k 2 = −1 .

(61)

Let {11, Γα} be a basis of C(t, s) and {110 , Γα0 } be a basis of C(t0 , s0 ), then the d + d0 elements {Γα ⊗ 110 , kΓd+1 ⊗ Γα0 } form a basis for their tensor product. Henceforth we abbreviate ⊗k to ⊗. Proof. Since Γα anticommutes with Γd+1 , the elements in this basis anticommute pairwise. Their squares are (Γα ⊗ 110 )2 = (Γα )2 ⊗ 110 = (11 ⊗ 110 )ηαα , (kΓd+1 ⊗ Γα0 )2 = k 2 (11 ⊗ 110 )ηα0 α0 ; the sign of k 2 determines the combination t + t0 or t + s0 in the tensor product C(t, s) ⊗ C(t0 , s0 ). In order to prove the periodicity of the Clifford algebra modulo 8, we prove C(0, s + 8) ' M16 (R) ⊗ C(0, s) , C(t, s) ' M2t (R) ⊗ C(0, s − t)

(62) for s > t .

(63)

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When tensoring Clifford algebras we can use either (60) or (61). Using (60) is easier but using (61) brings out interesting results. In brief, if we use (60) C(0, s + 8) ' C(0, 2) ⊗ C(0, s + 6) 4

' (⊗C(0, 2)) ⊗ C(0, s) ' M16 (R) ⊗ C(0, s)

since C(0, 2) ' M2 (R) .

But if we use (61) we bring out the quaternionic algebras since C(2, 0) is isomorphic to H; we have then enough information to construct the classification table. Using (61) we obtain C(0, s + 8) ' C(0, 2) ⊗ C(s + 6, 0) ' C(0, 2) ⊗ H ⊗ C(0, s + 4) ' · · · ' M2 (R) ⊗ H ⊗ M2 (R) ⊗ H ⊗ C(0, s) . The proof of (63) is analogous. With k 2 = 1, C(t, s) ' (⊗C(1, 1))t ⊗ C(0, s − t) = M2t (R) ⊗ C(0, s − t) . Example. Constructing gamma matrices from Pauli matrices 0 1 0 −i 1 0 σ1 = , σ2 = , σ3 = , 1 0 i 0 0 −1 σj σk = ijkl σl ,

(σi )2 = 112 ,

σ1 σ2 σ3 = i112 .

In Dimensions 2 and 3, the gamma matrices are 2×2 matrices and we can write down Table 2. Let a matrix in C be written M = m0 11+miΓi +mij Γi Γj +mΓ1 Γ2 · · · Γd ; M is an element of a real vector space; the isomorphism in the last column is dictated Table 2. Clifford alg.

Gamma matrices in 1, 2 and 3 dimensions.

Alg. generators

Vector space basis

dimR

Isomorphism

C(0, 1)

1, i

1, i

2

C

C(1, 0)

1, 10

1, 10

2

R⊕R

C(0, 2)

112 , iσ1 , iσ3

112 , iσ1 , iσ2 , iσ3

4

H

C(1, 1)

112 , (σ1 orσ3 ), iσ2

112 , σ1 , iσ2 , σ3

4

M2 (R)

C(2, 0)

112 , σ1 , σ3

112 , σ1 , iσ2 , σ3

4

M2 (R)

C(0, 3)

112 , iσ1 , iσ2 , iσ3

112 + 7 matrices

8

H⊕H

C(1, 2)

112 , σ2 , iσ1 , iσ3

112 + 7 matrices

8

H⊗C

C(2, 1)

112 , σ1 , σ3 , iσ2

112 + 7 matrices

8

M2 (R) ⊕ M2 (R)

C(3, 0)

112 , σ1 , σ2 , σ3

112 + 7 matrices

8

M2 (C)

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by the properties of the vector space basis. In dimensions higher than 3, Eqs. (60) and (61) can be used for constructing gamma matrices. We work out C(1, 3) and C(3, 1) explicitly: C(1, 3) ' C(1, 1) ⊗ C(0, 2) for k 2 = 1, k = ±1 the algebra generators are: σ1 ⊗ 112 , iσ2 ⊗ 112 , −kσ3 ⊗ (iσ1 ), −kσ3 ⊗ (iσ2 ) consist of 3 real matrices, and 1 imaginary one. C(1, 3) ' C(1, 1) ⊗ C(2, 0) for k 2 = −1, k = ±i the algebra generators are: σ1 ⊗ 112 , iσ2 ⊗ 112 , −kσ3 ⊗ σ1 , −kσ3 ⊗ σ3 consist of 2 real matrices, and 2 imaginary ones. C(3, 1) ' C(1, 1) ⊗ C(2, 0) for k 2 = 1, k = ±1 Changing the value of k in the previous basis yields 4 real matrices. This is a Majorana representation. C(3, 1) ' C(1, 1) ⊗ C(0, 2) for k 2 = −1, k = ±i Changing the value of k in the first basis yields 3 real matrices, and 1 imaginary one. C(1, 3) does not admit a real representation; C(3, 1) does admit a real representation. In Sec. 3.1, the label t for time is equal to 1 and the label s for space is equal to 3, therefore C(t, s) signals at a glance a metric of signature (+, −, −, −) and C(s, t) a metric of signature (+, +, +, −). Here t and s are arbitrary, and we shall use (m, n) rather than (t, s).g Table 3 lists the algebra isomorphisms of C(m, n) with d = m + n for all possible values of (m − n) mod 8. The vector space Mk (R) of k × k real matrices is abbreviated to R(k) and the space of k × k quaternionic matrices (the matrix elements are quaternions) is denoted H(k). For example the isomorphism M4 (R) ' H ⊗ H is abbreviated R(4) ' H(2) . This table is valid for both m − n > 0 and m − n < 0 since a negative number modulo 8 is equal to a positive number. Table 3. (m − n) mod 8

0

C(m, n)

R(2d/2 )

(m − n) mod 8

4

C(m, n)

H(2d/2−1 )

g C(n, m)

Algebra isomorphisms. 1

2

3

R(2d/2 )

C(2(d−1)/2 )

5

6

7

H(2(d−1)/2−1 ) ⊕ H(2(d−1)/2−1 )

H(2d/2−1 )

C(2(d−1)/2 )

R(2(d−1)/2 )

in Ref. [30] is C(m, n) in this report.

⊕

R(2(d−1)/2 )

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ˆ ∈ C(m, n), m > n, and Γ ∈ C(m, n), m < n, one can use Γ ˆ = iΓ Given Γ for verifying, for instance, C(0, 5) given C(5, 0) = H(2) ⊕ H(2); one finds C(0, 5) = H(2) ⊕ iH(2) = H(2) ⊗ C. Now H ⊗ C ' C(2), so C(0, 5) ' C(4) which is correct for −5 = 3 mod 8. From Table 3 we conclude the following: • For d even, the vector space isomorphisms are either with vector spaces of real matrices, or vector spaces of quaternionic matrices. • For n = 1, C(m, 1) admits a real representation for m = 1, 2, 3 mod 8 i.e. d = 2, 3, 4, 10, 11, 12 etc. • If C(m, 1) admits a real representation, C(1, m) admits a purely imaginary one. • C(m, n) and C(n, m) are not isomorphic unless m − n = 0 mod 4. Another technique for identifying the dimensions which admit real representations consists in assuming all the Γα ’s real, and seeing if it leads to a contradiction. For instance, let d = 4, and assume Γ1 , Γ2 , Γ3 , Γ4 to be real, with (Γj )2 = 11, j ∈ {1, 2, 3} and (Γ4 )2 = −11. The Γj are symmetric, and Γ4 is antisymmetric. The algebra generated by the Γα ’s consists of 10 symmetric matrices: 11, Γj , Γ4 Γj , Γ4 Γj Γk 6 antisymmetric matrices: Γ4 , Γj Γk , Γj Γk Γl , Γ1 Γ2 Γ3 Γ4 . These 16 matrices make a basis for M4 (R). There is no contradiction in having assumed the Γα ’s to be real. 5.3.1. Onsager construction of gamma matrices In the proof of (62) and (63) we have given a construction for a basis of C(t+t0 , s+s0 ) and C(t + s0 , s + t0 ) given a basis for C(t, s) and C(t0 , s0 ). It is worth mentioning another construction using the Onsager solution of the Ising model [83]; the explicit representation using this construction can be found in [39]. In particular, one sees by inspection which matrices are real, and which are imaginary both for d even and d odd. 5.3.2. Majorana pinors, Weyl–Majorana spinors A pinor is said to be Majorana if it is real (or purely imaginary). If a space S of Majorana pinors is of even dimension, it can be split into two eigenspaces of a chirality operator S = S+ ⊕ S− , then each eigenspace is a space of Weyl–Majorana spinors.

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Majorana pinors have been used to avoid confusion in charge conjugation. When a real representation is not available, one replaces a d-dimensional complex pinor ψ = ψ1 + iψ2 by a 2d-dimensional real pinor ! ψ1 11 0 ψ= and ψ∗ = ψ. 0 −11 ψ2 The 2d-dimensional representation is reducible. 5.4. Conjugate and complex gamma matrices The set of hermitian conjugate matrices {±Γ†α }, the set of inverse matrices {±Γ−1 α } and the set of complex conjugate matrices {Γ∗α } obey the same algebra as the set {±Γα } and the same normalization Γα Γ†α = ±11 (no summation). For d even there is only one irreducible faithful representation of the gamma matrices of dimension 2d/2 . Hence there are similarity transformations −1 † Γα H± = ±Γα H±

(Γ†α operates on bras)

(a)

−1 C± Γ∗α C± = ±Γα

(Γ∗α operates on kets)

(b) .

(64)

We shall not need the similarity transformation of inverse matrices. Remark. The similarity transformation on {Γ†α } does not imply that there is a similarity transformation on products Γ†α Γ†β . Indeed H−1 (Γα Γβ )† H = H−1 (Γ†β Γ†α )H = Γβ Γα 6= Γα Γβ . This explains the factor a(Λ) in the definition of copinor in Sec. 3.3. For d = 2p+1 odd there are two inequivalent irreducible faithful representations of the gamma matrices of dimension 2(d−1)/2 ; hence there may not exist matrices H± and C± satisfying those similarity transformations — in other words, we could have ( 1 for α = 0 −1 † α α H± Γα H± = (−1) Γα where (−1) = . (65) 0 for α ∈ {1, 2, 3} In Sec. 5.1 we gave a construction for a basis of Pin(t0 , s0 ), t0 + s0 = 2p + 1, given a basis of Pin(t, s), t + s = 2p. In this construction the first 2p elements were the same as in Pin(t, s), hence they satisfy (64a) or (64b) as the case may be. We need ˆ d+1 . to check only which equation is satisfied by the new element kΓd+1 , or kˆΓ ˆ d+1 : for d = 2p, Γd+1 := Γ1 Γ2 · · · Γd We recall (57) the properties of Γd+1 and Γ ˆ d+1 , and similarly for Γ Γ2d+1 = (−1)s+p 112p ,

ˆ 2d+1 = (−1)t+p 112p . Γ

The square of the new element, (kΓd+1 )2 , tells us which group we generate (Table 4). The two choices for k in their respective groups correspond to two different representations. We calculate the transformation under (64a) and (64b) of the new

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The Pin Groups in Physics: C, P and T Table 4. (kΓd+1 )2 11 −11

1009

Constructing a Pin group in odd dimensions. Group

k2

k

Pin(t + 1, s)

(−1)s+p

±(−1)(s+p)/2

Pin(t, s + 1)

−(−1)s+p

±i (−1)(s+p)/2

element kΓd+1 : −1 H± (kΓd+1 )H± = (−1)t Γ†d+1 , ( Γ∗d+1 if k ∗ = k , −1 C± (kΓd+1 )C± = −Γ∗d+1 if k ∗ = −k .

From this, and the requirement that (64a) and (64b) extend to Γd+1 , we read off if if if if

t is even, the only choice is H+ , t is odd, in particular if t = 1, the only choice is H− , s − t + 1 mod 4 = 2, the only choice is C+ , s − t + 1 mod 4 = 0, the only choice is C− .

For the corresponding Pin(t, s) transformations one finds ˆ ± = H∓ H

and

Cˆ± = C∓ .

The details of this calculation can be found in [39],h as well as properties of H and C. 5.5. The short exact sequence 11 → Spin(t, s) → Pin(t, s) → Z2 → 0 We shall prove that a Pin group is a semidirect product of a Spin group with Z2 , the group consisting of two elements {e, z}, where z 2 = e : Pin(t, s) = Spin(t, s) n Z2 ,

s+t>1

Pin(s, t) = Spin(s, t) n Z2 ,

s+t>1

where n is defined below. Spin(t, s) and Spin(s, t) are isomorphic, but a semidirect product “scrambles” the elements of its components and, as we know, Pin(t, s) is not necessarily isomorphic to Pin(s, t). As we have shown in Sec. 5.3, they are isomorphic only when s − t = 0 mod 4. The semidirect product construction does not work for Pin(0, 1), since Z2 n Z2 = Z2 × Z2 . h In

that paper Γd+1 is called the “orientation matrix” .

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a) First we prove general properties of semidirect products, then we apply them to Spin and Pin. Let G = H n Z2 ˆ=H ˆ n Z2 G ˆ where there is an isomorphism φ : H → H. Let t : (Z2 × H) → H be an element of the group of automorphisms of H indexed by Z2 . tz : H → H

by

h 7→ tz (h) = t(z, h) = zhz −1 .

Note that z is not in H but acts multiplicatively on H. The semidirect product G = H n Z2 is the space of ordered pairs (h, z) and (h, e), h ∈ H with the product law (h1 , z1 ) · (h2 , z2 ) = (h1 t(z1 , h2 ), z1 z2 ) ; here zi , i = 1, 2, is either e or z; when working with Z2 , it is difficult to be correct without being pedantic. If Z2 and H commute, t(zi , h) = h and the semidirect product becomes a direct product. As the simplest example of semidirect product scrambling, compare (see Sec. 3.2 and also Appendix C) Z3 × Z2

and

Z3 n Z2 = D3

where the dihedral group D3 is the group of symmetries of a regular triangle. Consider the short exact sequence 11 → Spin(t, s) → Pin(t, s) → Z2 → 0 . We shall show that if s + t is odd,

O(s, t) = SO(s, t) × Z2 ,

if s + t is even,

O(s, t) = SO(s, t) n Z2 .

Since a direct product is a special case of a semidirect product, we begin with (a, zi ) ∈ SO(s, t) n Z2 . The identification (ai , zi ) with ai zi =: gi ∈ O(s, t) makes sense because zi is not necessarily in SO(s, t), and because it makes the definition of the semidirect product consistent with the group product in O(s, t). Indeed (a1 , z1 ) · (a2 , z2 ) = (a1 z1 a2 z1−1 , z1 z2 ) ' a1 z1 a2 z1−1 z1 z2 = g1 g2 . The difference between s + t odd and s + t even stems from the fact that if s + t is odd we can choose Z2 = (11, −11)

because

− 11 ∈ / SO(s, t)

(s + t odd).

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Z2 commutes, then, with SO(s, t) and the semidirect product is simply a direct product. If s + t is even, −11 ∈ SO(s, t) so we cannot use −11 for z (since we require z ∈ / SO(s, t)). Other options for z ∈ / SO(s, t) are reflections. Reflections do not commute with all elements in SO(s, t) and the semidirect product does not reduce to a direct product. In general, G = H × Z2 if G has a central element z of order 2 which is not in H. ˆ=H ˆ n Z2 and an isomorphism b) Given G = H n Z2 , G ˆ, φ:H →H we shall prove that there is an isomorphism ˆ Φ:G→G if and only if, for every h ∈ H, Φ(h, zi ) = (φ(h), zî )

(66)

zi−1 = φ(zi hzi−1 ) . zî φ(h)ˆ

(67)

where zî is defined by It may be useful to refer to this diagram: nZ

2 H −−−−− →   φ y

G    Φ. y

ˆ −−−−−→ G ˆ H n Z2

Proof. Let (h1 , z) be identified with g1 = h1 z and (h2 , e) be identified with g2 = h2 . Φ is an algebra isomorphism if Φ(g1 )Φ(g2 ) = Φ(g1 g2 ). It is sufficient to choose g1 = (h1 , z) and g2 = (h2 , e) = h2 for identifying under which condition Φ is an isomorphism. If (66) is satisfied Φ((h, z)) = (φ(h), zˆ) ,

Φ((h, e)) = Φ(h) = φ(h)

then Φ((h1 , z)) Φ(h2 ) = (φ(h1 ), zˆ) φ(h2 ) ,

identified with

φ(h1 ) zˆ φ(h2 ) .

On the other hand Φ((h1 , z) · (h2 , e)) = Φ(h1 zh2 z −1 , ze)

identified to

= (φ(h1 zh2 z −1 ), zˆ)

assuming (66)

= (φ(h1 )φ(zh2 z −1 ), zˆ)

since φ is an isomorphism

φ(h1 )φ(zh2 z

−1

= φ(h1 ) zˆ φ(h2 )

) zˆ by the definition (67) of zî

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and thus Φ(g1 )Φ(g2 ) = Φ(g1 g2 ) .

(68)

We have proven that Eq. (66) implies Eq. (68). The converse follows by identification. 5.6. Grassman (superclassical ) pinor fields In several studies, pinors are sections of a supervector bundle associated to a principal Pin bundle by a representation (ρ, V ) of the Pin group where the typical fiber V is a supervector space. A supervector space is a linear space over the supernumbers. (A linear space is a module for which the ring of operators is a field, e.g. the real numbers or the complex numbers). Supernumbers are generated by a Grassman algebra; i.e. the generators of the algebra {ζ a } with a ∈ {1, . . . , N }, with possibly N = ∞, anticommute: ζ a ζ b = −ζ b ζ a and a supernumber z can be expressed in the form ζ = ζB + ζS where ζB is an ordinary complex number and ζS =

∞ X

ca1 ···an ζ an · · · ζ a1

n=1

the ca1 ···an being complex numbers, completely antisymmetric in the indices. It is often said (and we have done so in the past) that choosing representations of the Pin groups on supervector spaces is desirable for considering classical physics as the limit of quantum physics. In other words, if the anticommutator of a quantum field at two different causally related points goes to zero with Planck’s constant ~, then the classical pinor field at two different points anticommute when ~ = 0. But, as pointed out by Cartier, the anticommutator of the fermionic fields in the Lagrangian is not proportional to ~. Indeed, in QED, the electric current density j in terms of the electron field Ψ is (restoring ~ and c in this subsection): ¯ µΨ . jµ = ec ΨΓ The physical dimension of the current Jµ (t) =

Z jµ (x, t)d3 x

is [Jµ (t)] = eT −1 . Hence we can compare physical dimensions: Z ¯ µ Ψd3 x = eT −1 ec ΨΓ

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¯ µ Ψ] = L−4 . Thus the dimension of the field operator is which implies that [ΨΓ −2 [Ψ] = L and so [{Ψ(x), Ψ(y)}] = L−4 . The anticommutator does not have the same physical dimension as ~, which is [~] = M L2 T −1 . In order to have √ the anticommutator proportional to ~, it suffices to take the anticommutator of ~ Ψ. One important reason for treating classical pinors as supervector fields is functional integration: the functional integral needed to construct matrix elements of operators built with Fermi quantum fields is a functional integral over a space of functions with values in a Grassman algebra. In general it is convenient to have quantum fields and the corresponding classical fields taking their values in the same algebra. Having discussed the motivation for classically treating pinors as superclassical fields we refer the reader to the existing literature on supermanifolds [30, 37] and on the use of superclassical fields in studies aimed at comparing the two Pin groups [36, 39]. 5.7. String theory and pin structures The following remarks discuss string theory, where the Pin groups may be particularly relevant. Pin structures are defined in Sec. 3.5. The concept “string theory” now encompasses more objects than the onedimensional strings of the original string theories; those original theories emerge as different limits of modern string theory, or appear in duality relationships with other theories included in modern string theory. Of course, the original string theories are still of interest when viewed as different corners of the parameter space of modern string theory. We note briefly how spin structures enter into the Ramond–Neveu–Schwarz (RNS) formalism of closed superstrings in ten dimensions [49, 50, 88, 99]; the extension to pin structures follows the same pattern. One a priori problem in superstring theory in ten dimensions is the existence of a tachyon in the spectrum. It is solved by the projection on the space of states known as the Gliozzi–Scherk–Olive (GSO) projection: PGSO =

1 (1 + (−1)F ) 2

(69)

where F is the fermion number. This projection takes away the tachyon, and leaves an equal number of fermions and bosons, as required for a linear realization of supersymmetry. (It also solves other problems.) To show how the GSO projection involves spin structures, we study a torus diagram, which can represent the creation and annihilation of a pair of closed strings, as they move in time; a one-loop string diagram. If we carry a fermion field

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τ

τ+1

z 0

1

Fig. 11. Opposite sides of the parallelogram are identified, and the parallelogram has the topology of a torus.

around either one of the two nontrivial cycles of the torus, the spin structure dictates whether the fermion comes back to itself (periodic) or changes sign (antiperiodic). There are two cycles on the torus, so we have four combinations of “boundary conditions” for the functional integral, we label them (P, P ), (P, A), (A, P ), (A, A). The first letter refers to periodicity in z (see Fig. 11). For functional integrals of a single fermion we find, denoting by trA the trace in the antiperiodic sector, (P, P ) = q −1/48 trP (−1)F q L0 , (P, A) = q −1/48 trP q L0 , (A, P ) = q −1/48 trA (−1)F q L0 , (A, A) = q −1/48 trA q L0 , where L0 is the normal-ordered Hamiltonian, and q = exp(2πiτ ) where τ is the modular parameter on the torus. (We are not interested in the details here, just the (−1)F factors.) The general principle of modular invariance can be used as a guide for combining these four amplitudes. Here we simply add the four amplitudes; this amounts to inserting a factor (1 + (−1)F ) in both trA and trP . Inserting this factor is identical (up to the factor 12 ) to performing a GSO projection (69). Thus adding functional integral contributions from each spin structure (summing over spin structures), is a prescription which gives useful results, at least in weakly coupled string theory at the one-loop level. There is no reason to limit the above discussion to spin structures. In string theory one considers unoriented string diagrams (such as the Klein bottle) in addition to the torus diagram discussed above. The full Lorentz group is certainly relevant, and hence the pin structures. Criteria for the existence of pin structures on orientable and non-orientable manifolds with metrics of arbitrary signatures can be found in Karoubi [58]. The first thing one runs into is the criterion for isomorphicity: s−t = 0 mod 4. On a Minkowski string worldsheet, there is evidently only Pin(1, 1). On a Euclidean worldsheet the Pin groups are different: Pin(0, 2) and Pin(2, 0). Beyond the worldsheet, there are higher-dimensional hypersurfaces in string theory

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to which fermions may be restricted. Of these, checking the criterion we see that the two Pin groups are isomorphic only for 5+1-dimensional hypersurfaces and in 9+1-dimensional spacetime, alternatively 4-dimensional or 8-dimensional Euclidean hypersurfaces. In all other cases directly relevant to string theory (spatial dimensions 2, 3, 4, 6, 7, 8 and 10 of Minkowski space or 2, 3, 5, 6, 7, 9 and 11 dimensions of Euclidean space) the Pin groups are not isomorphic. There are already existing attempts in this direction in the literature. Chamblin [26] has mentioned one way of selecting pin structures in string theory. In a note on the 3D Ising model as a string theory [40], Distler pointed out that fermions used in open string theory make sense with Pin(0, 2) structure but not with Pin(2, 0) structure, since only Pin(0, 2) structure can be defined on any 2-manifold. The discussion takes place within his approach to the 3D Ising model, which in the continuum limit is equivalent to a certain unoriented string theory. The implication of the non-existence of pin structures on some 2-dimensional surfaces has been worked out for the Polyakov path integral of the NSR superstring action [20]. Finally, a paper by Dasgupta, Gaberdiel and Green discussed Pin groups in relation to breaking of O(16) to SO(16) symmetry in string theory [34]. 6. Conclusion 6.1. Some facts In 3+1 dimensions, there are two Pin groups, Pin(1, 3) and Pin(3, 1), which come into play in the analysis of time or space reversal. In principle the existence of two Pin groups provides a finer classification of fermions than one Pin group. Such a classification is useful only if one can design experiments which distinguish the two types of fermions. Many promising experimental setups give, for one reason or another, identical results for both types of fermions. These negative results are reported here because they are instructive. Two notable positive results show that the existence of two Pin groups is relevant to physics: • In a neutrinoless double beta decay, the neutrino emitted and reabsorbed in the course of the interaction can only be described in terms of Pin(3, 1). • If a space is topologically nontrivial, the vacuum expectation values of Fermi currents defined on this space can be totally different when described in terms of Pin(1, 3) and Pin(3, 1). • Only Pin(0, 2) can be used in open string theory [40]. The same conclusion applies to a 3D Ising model which is in the continuum limit equivalent to a certain unoriented string theory. 6.2. A tutorial The Pin groups are technically useful; they provide a simple framework for the study of fermions, in the context of the full Lorentz group.

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6.2.1. Parity The parity operator operates on the space of pinors. It cannot be defined on the space of spinors (Weyl fermions) for the following reason: the parity operator consists of an odd number of gamma matrices, whereas the elements of the Spin group consist of even numbers of gamma matrices. When there is no parity operator, there is no parity eigenspinor, therefore no parity eigenvalue can be assigned to a Weyl fermion. One often hears that no parity is assigned to Weyl fermions “because weak interactions do not conserve parity” but to say that an interaction does not conserve parity implies that a parity can be assigned to the initial state and to the final state. The statement is meaningless because the same word “parity” is used for two different concepts: “intrinsic parity” of a fermion (as in Sec. 3.4) and the “parity non-conservation of an interaction” (as in Sec. 3.3). The square of the parity operator does operate on the space of spinors. In ˆ2 Pin(1, 3), Λ2P (3) = +11, and in Pin(3, 1), Λ 1, hence it is meaningful to say P (3) = −1 if a Weyl fermion belongs to a subgroup of Pin(1, 3) or Pin(3, 1). 6.2.2. Time reversal There are two definitions of the time reversal operator: a unitary one and an antiunitary one, which serve different purposes. The unitary one is in the toolbox of the Lorentz group. The antiunitary one is used in motion reversal, an expression which Wigner credits to L¨ uders [123, p. 54]. Invariance under antiunitary time reversal is required so that quantum systems are free of negative energy states. 6.2.3. Charge conjugation Charge conjugation of pinors has nothing to do with the Lorentz group, nor does antiunitary time reversal; but the “CP T ” transformation on pinors correspons simply to an orientation preserving Lorentz transformation (transformation of determinant 1). 6.2.4. Wigner’s classification and classification by Pin groups To prefer one classification over another is in part a matter of taste, and in part a matter of its intended use. We prefer the classification by Pin groups because it is a straightforward consequence of the use of the full Lorentz group in physics. Wigner begins with SL(2, C) which is isomorphic, but not identical, to the covering group Spin↑ (3, 1) ⊂ Spin(3, 1). Then he combines it with reflections and constructs four different convering groups, but needs to discard representations which are not physically admissible. He raises the question of a “whole group” but notes that it is not uniquely defined in the context of his classification. We regret that his work precedes the identification of the two Pin groups. He would have made use of this fact in a more perceptive fashion than we have done so far.

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6.2.5. Fock space and one-particle states Operators on a Fock space and operators on a space of pinors are different objects. In Sec. 3.3 we present Pin group operators acting on the space of unquantized pinor fields (classical fields) ψ. In Sec. 3.4 we define charge conjugation, space and time reversal operators on quantum fields Ψ. The relationship between ψ and Ψ can be found in the dictionary of notation (Sec. 0). Equations (42), (43) and (44) provide the bridges between operators on Fock space and operators on a space of pinors. These equations are necessary in the analysis of the Pin groups in the quantum field theory of particle physics. 6.3. Avenues to explore This report is limited to the case where the inversion operators UP and UT take oneparticle states into other one-particle states of the same species. Inversions may act in a more complicated way than this on degenerate multiplets of one-particle states. This possibility was first suggested by Wigner [123]. Weinberg [115] explored generalized versions of the inversion operators, in which finite matrices appear in place of the inversion phases, but without making some of Wigner’s limiting assumptions. In the “Collected References” of Appendix E, under the heading “Carruther’s Theorem” we list basic references on the subject, in particular works of Moussa and Stora, which can be used as a starting point for investigating the role of the two Pin groups in the case of degenerate multiplets of one-particle states. Indeed, in Sec. 3.3 we learn to distinguish the Pin groups; in Sec. 3.4 we introduce the phases associated to projective representations of quantum field operators. Other investigations, such as the following, could reveal differences between Pin(t, s) and Pin(s, t) fermions: • Time or space reversal in the complex environment of atomic and molecular physics, dipole moments, etc. • Topologically nontrivial configuration spaces. • To first order, decay rates and cross sections computed in this report do not depend on the choice of Pin group, but given the trace and spin sum differences, it is not excluded that higher order contributions would be different. 7. Acknowledgments The first version was written by two of us (CD and SJG) in 1991 and then kept on the backburner while we analyzed situations in which one could observe experimentally the differences between the two Pin groups. EK joined us and worked out in detail the section on interference 4.4. Retrospectively, one can argue that the answers are obvious, but only an explicit calculation can be convincing, because the issues are subtle and the signs dictated by the choice of groups enter at various stages of the calculation.

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MB undertook the major project of attempting to make this work meaningful to experimental physicists. He investigated selection rules in positronium decay, three-fermion decay, positronium and decay rates, in particular Σ0 decay. The new team (MB and CD) has completely rewritten the previous version. In the course of nearly a decade, many colleagues have commented on this work. We thank them for their interest, but the list of their names would necessarily be incomplete, and serve little purpose other than name dropping. Special thanks are due to Steven Carlip whose suggestions vastly improved the first draft, to Yuval Ne’eman for his support during the adventures of the second version, and to Raymond Stora who read it seriously and noticed a number of issues requiring improvement. MB wishes to thank the Sweden-America Foundation for financial support. Appendix A. Induced Transformations We recall briefly the transformation laws induced by a Lorentz transformation L of spacetime. Let (M, g) ≡ M 1,3 ≡ M be a spacetime manifold with metric g of signature η = (1, −1, −1, −1). Let L map (M, g) into itself. Let Tx M and Ty M be the tangent spaces to M at x and y respectively, and Tx∗ M and Ty∗ M be their dual spaces (spaces of linear maps on the tangent spaces). Let V (x) ∈ Tx M and ω(x) ∈ Tx∗ M , V (x) is a contravariant vector, ω(x) is a covariant vector. In terms of components, the duality is X h ω(x), V (x) i = ωα (x) V α (x) ≡ ωα (x) V α (x) . α

Since L is a linear map, its derivative mapping L0 (x) is L itself, but is now a mapping from Tx M to Ty M . W (y) = L V (x) ,

(A.1)

W (Lx) = L V (x) . In terms of components W α (Lx) = Lα β V β (x) . The duality is used to determine the transformation properties of elements of the ˜ dual spaces. Let θ(y) ∈ Ty∗ M , then we define L(y) : θ(y) 7→ ω(x) by hω(x), V (x)i = hθ(y), W (y)i . ˜ Since L0 (x) is independent of x, then L(y) is independent of y. ˜ h(Lθ)(x), V (x)i = hθ(y), (LV )(y)i , ˜ ω(x) = Lθ(Lx) . In terms of components ωβ (x) = θα (Lx) Lα β .

(A.2)

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L

M

M y

x Fig. A.1.

Tx M

Lorentz transformation L.

L’(x)

Fig. A.2.

Tx M V(x)

Ty M W(y)

V(x)

Fig. A.3.

1019

Derivative mapping L0 (x).

L’(x)

Ty M W(y)

˜ ˜ ˜ Mapping L(y) between dual spaces. L(y) = L.

The same pattern applies to contravariant pinors (or just “pinors”) and covariant pinors (or “copinors”). Hence, under a Lorentz transformation L, a pinor ψ(x) becomes ψ 0 (Lx) such that ψ 0 (Lx) = ΛL ψ(x) , ¯ such that a copinor ψ¯0 (Lx) becomes ψ(x) ¯ ˜ L ψ¯0 (Lx) . ψ(x) =Λ Appendix B. The Isomorphism M4 (R R) ' H ⊗ H Let M4 (R) be the space of real 4 × 4 matrices, and H be the quaternion algebra. Let (1, i, j, k) be its basis with i2 = j 2 = k 2 = −1 and ij = k, jk = i, ki = j. Let (a1 , a2 , a3 , a4 ) be the coordinates of α ∈ H and (b1 , b2 , b3 , b4 ) the coordinates of β ∈ H. α = a1 + a2 i + a3 j + a4 k , α† = a1 − a2 i − a3 j − a4 k . As a vector space, H is a real 4-dimensional vector space. Let I :H→V4

by



 a1  a2   I (a1 + a2 i + a3 j + a4 k) =   a3  . a4

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Let α ⊗ β act linearly on H by (α ⊗ β)(γ) = (αL βR )(γ) := αγβ †

γ ∈ H.

We map H ⊗ H to the space of 4 × 4 real matrices M4 (R) by f : H ⊗ H → M4 (R) f (α ⊗ β) = M (α, β)

by

where M (α, β)I(γ) = I(αγβ † )

for all γ ∈ H .

It is straightforward to prove that f is an algebra isomorphism. One can construct explicitly the matrix M (α, β) for a pair of basis elements. Each matrix M (α, β) thus obtained can be written as a tensor product of a pair of matrices from the set {112 , σ1 , iσ2 , σ3 } (see example below). We recall the definitions of tensor products of algebras, and tensor products of matrices. Let {ei } and {eα } be bases for the real algebras A and B. Let c = ciα ei ⊗eα and d = djβ ej ⊗ eβ , then cd = ciα djβ (ei ej ⊗ eα eβ ) . Let a = (aij ) and b = (bα β ), then (a ⊗ b)IJ = aij bα β. Here I = (i, α), J = (j, β). There are two obvious choices for ordering the pairs. We choose ! a11 (b) a12 (b) (a ⊗ b) = . a21 (b) a22 (b) To prove that M (α, β) is a real matrix, we construct M (α, β) for all the elements in a basis of H ⊗ H, then extend the result by linearity. Let the basis for H ⊗ H consist of 11 ⊗ 11, 11 ⊗ i, 11 ⊗ j, 11 ⊗ k, i ⊗ 11, i ⊗ i, etc. Let γ = a + bi + cj + dk, then M (1, i)I(γ) = I(1γ(−i)) = I(b − ai − dj + ck) . Therefore 

0  −1 M (1, i) =   0 0

1 0 0 0

 0 0 0 0  = σ3 ⊗ iσ2 0 −1  1 0

(on the r.h.s. i =

√

−1).

A similar calculation for all the elements in the basis of H⊗H shows that M (α, β) ∈ M4 (R).

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Appendix C. Other Double Covers of the Lorentz Group After having identified Pin(3, 1) and Pin(1, 3) as two inequivalent covers of the Lorentz group, we have to review briefly the other double covers of the Lorentz group. We can look at Pin(s, t) and Pin(t, s) as extension of O(s, t) (equivalently O(t, s)) by Z2 in the short exact sequence 1 → Z2 → G → O(s, t) → 1,

Z2 = {1, −1} .

0

Two extensions G and G are equivalent if and only if G and G0 are isomorphic and the two sequences 1 → Z2 → G → O(s, t) → 1 ↓ ◦ oo ◦

↓

0

1 → Z2 → G → O(s, t) → 1 are made of two commutative diagrams: the two maps Z2 → G → G0 and Z2 → Z2 → G0 are identical (up to isomorphisms) — and the same property for the other diagram. There are eight double covers of the Lorentz group called Pinabc by Dabrowski [33] and characterized by Λ2P = a,

Λ2T = b,

(ΛP ΛT )2 = c,

a, b, c ∈ Z2 .

In the fourth column of Table C.1, we give the names of the corresponding finite groups with elements ±11, ±ΛP , ±ΛT . The dihedral group Dn is the group of symmetries of an n-sided regular polygon. Table C.1.

The eight double covers of the Lorentz group.

Λ2P

Λ2T

(ΛP ΛT )2

+1

+1

+1

+1

−1

−1

+1

−1

−1

−1

Group

ΛT ΛP = ±ΛP ΛT

Z2 ⊕ Z2 ⊕ Z2

+

−1

Z2 ⊕ Z4

+

−1

Z2 ⊕ Z4

+

+1

Z2 ⊕ Z4

+

−1

−1

quaternion

−

−1

+1

+1

dihedral generating Pin (1, 3)

−

+1

−1

+1

dihedral generating Pin (3, 1)

−

+1

+1

−1

dihedral

−

The following requirements identify the Pin groups, called cliffordian by Dabrowski [33]: ΛT ΛP = −ΛP ΛT , Λ2P 6= Λ2T .

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ˆP , Λ ˆ T to P, T ) By relating ΛP , ΛT ∈ Pin(s, t) to P, T ∈ O(s, t) respectively (or Λ we give the explicit pin structure (see Sec. 3.5). There is an extensive literature (see Appendix E) on pin structures, not only for Pin groups covering Lorentz groups, but also for Pin groups covering O(s, t) with arbitrary values of s and t. Appendix D. Collected Calculations In this appendix, we collect for reference some calculations that were either too long to have in the main body of the paper, or fairly standard and only slightly generalized to accommodate the two Pin groups. For each calculation we refer to the page where the relevant discussion can be found. D.1. Spin sums (p. 990) We compute the spin sums, using u(p, s) as an example. The Dirac equation (iΓα ∂α − m)ψ(x) = 0 gives (/p − m)u(p, s) = 0 .

(D.1)

With the given normalization, u is an eigenpinor of the spin sum with eigenvalue 2m; indeed ! X X u (p, r)¯ u(p, r) u(p, s) = u(p, r)(2mδrs ) r

r

= 2mu(p, s) . We can rewrite 2mu using (D.1) as 2mu(p, s) = (m + m)u(p, s) = (/p + m)u(p, s) . Thus we have the given spin sum for u. D.2. Parity conservation (p. 992) We review briefly how the observed angular distribution of scattered particles is used for concluding whether or not parity is conserved. To be specific, we study scattering of two fermions into two fermions. The argument is reviewed for a secondorder contribution to the S-matrix, but we could of course also have considered the full S-matrix. Let Z Z p0 , k0 d4 x Hint d4 y Hint p, k =: M(p, k, p0 , k0 ) . The “in” and “out” states |p, ki and hp0 , k0 | each have spin labels suppressed. The evolution of free states into states of the interacting theory is not relevant to the

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scattering process, so we take these states to be free states as is usual, in other words we only consider amputated Feynman diagrams. The external states are: |p, si ∼ a† (p, s)|0i . Now recall UP a(p, s)UP−1 = ηa a(−p, s) and ψ(p, s) = ΛP ψ(−p, s) with the fourmomentum being p = (p0 , p). Thus, we have under parity for our two-particle states UP |p, ki ∼ ηa∗ ηb∗ |−p, −ki hp0 , k0 |UP†

∼ h−p0 , −k0 | ηc ηd

provided UP |0i = |0i provided h0|UP† = h0| .

(D.2)

Now, if the operator UP commutes with the Hamiltonian, a parity transformation simply induces the following change in the S-matrix contribution: Z Z −p0 , −k0 ηc ηd d4 x Hint d4 xHint ηa∗ ηb∗ −p, −k . Under a parity transformation we now have M(p, k, p0 , k0 ) 7→ ηc ηd ηa∗ ηb∗ M(−p, −k, −p0 , −k0 ) . If the matrix element has some symmetry under inversion, we use this symmetry for reexpressing the right hand side to deduce a conservation rule for the intrinsic parities. For instance, when we decompose M(−p, −k, −p0 , −k0 ) into partial waves (spherical harmonics) labelled by `, the matrix element acquires a (−1)` due to the parity of the spherical harmonics Y`m : Y`m (−ˆ q ) = (−1)` Y`m (ˆ q) with qˆ the unit momentum transfer qˆ = (p − p0 )/|p − p0 |; this relation can be used in a relativistic theory as well as in non-relativistic quantum mechanics. Thus we can write M(p, k, p0 , k0 ) → ηc ηd ηa∗ ηb∗ M(−p, −k, −p0 , −k0 ) = ηc ηd ηa∗ ηb∗ (−1)ì (−1)`f M(p, k, p0 , k0 )

(D.3) (D.4)

or (−1)ì (−1)`f ηd∗ ηc∗ ηa ηb = 1 , or equivalently, (−1)ì ηa ηb = (−1)`f ηc ηd .

(D.5)

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D.3. Pion decay (p. 992) Consider pion capture by a deuteron followed by emission of two neutrons: π− + d → n + n . In this capture, the pionic “atom” is known [27] to be in the ` = 0 ground state. The pion has spin 0 and the deuteron spin 1, so the initial state has total angular momentum j = 1. There are two principles we can use for deducing the orbital angular momentum of the right-hand side • Angular momentum conservation (j = 1), • Antisymmetry of neutrons under exchange. The first yields the following possibilities for orbital quantum number ` of the neutrons, and total neutron spin s, consistent with j = 1: (a) ` = 1,

s = 0;

(b) ` = 0,

s = 1;

(c) ` = 1,

s = 1;

(d) ` = 2,

s = 1.

The wave function of two neutrons n1 and n2 (total spin s) in an `-state satisfies ψ(n1 , n2 ) = (−1)`+s+1 ψ(n2 , n1 ) . Since it is required that ψ(n1 , n2 ) = −ψ(n2 , n1 ) , the only option in the above table is (c). Now that we have ` for the two-neutron final state, it is a trivial matter to calculate the intrinsic parity of the pion. The orbital contribution is (−1)` = (−1), so the η-phases of the initial and final states are related by Eq. (47) since intrinsic parity is conserved: ηπ ηd = (−1)ηn ηn . D.4. Selection rules: Positronium (p. 994) The experiment revolves around positronium, the Coulomb bound state of an electron and a positron. To analyze it, we will use a common approach to bound states [87] which uses some nonrelativistic quantum mechanics in conjunction with quantum field theory. A bound state is created by letting the operator Z d3 p X B= ψ(p, se , sp ) a†p,se b†−p,sp (2π)3 s ,s e

p

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act on the vacuum state. Here, ψ is the Schr¨ odinger wavefunction obtained from solving the nonrelativistic Schrödinger equation in a Coulomb potential. The electron spin is se and the positron spin is sp , and we are working in the bound state CM system, hence the opposite momenta p and −p. To find the parity of the system, we compute the action of parity on the bound state operator B, using Eqs. (33) and (34) from the section on intrinsic parity in Sec. 3.4: Z d3 p X UP B UP−1 = ψ(p, se , sp )ηa∗ a†−p,se ηb∗ b†p,sp (2π)3 s ,s e

= (ηa ηb )∗

Z

p

d3 p X ψ(−p, se , sp )a†p,se b†−p,sp (2π)3 s ,s e

Z = (ηa ηb )

p

d3 p X (−1)` ψ(p, se , sp )a†p,se b†−p,sp (2π)3 s ,s e

p

`

= ηa ηb (−1) B . How can we measure this phase ηa ηb (−1)` ? The amplitude for annihilation into two photons is Z d3 p X M(B → 2γ) = ψ(p, se , sp ) M(p, se , −p, sp → 2γ) (2π)3 s ,s e

p

where the matrix element M(p, se , −p, sp → 2γ) is the ordinary field theory amplitude for a free electron and positron of momenta p and −p and spins se , sp . Nonrelativistically, we may think matrix element as being composed of R ∗ of this ψ2γ d3 x, and we can draw conclusions about a wavefunction overlap integral ψB ψ2γ from the photon part of the tree-level QED amplitude. In particular, since each photon vertex introduces a (transverse) polarization vector eµ , and the only other vector available is one photon momentum k, we can only form the following scalar or pseudoscalar combinations: + ψ2γ ∝ e1 · e2 , − ψ2γ ∝ k · (e1 × e2 ) .

If we denote by φ the angle between e1 and e2 , it is clear that the probability of the photons coming out polarized at φ = 90◦ is zero in the first (even) case and nonzero in the second (odd) case. Wu and Shaknov [125] performed the experiment with a 64 Cu positron source and found rate(φ = 90◦ ) = 2.04 ± 0.08 . rate(φ = 0◦ )

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If we write the wavefunction of the two-photon state as a product of spatial and spin wavefunctions ψ(space)ψ(spin), the spatial part is odd under inversion. Since parity is conserved in QED, this means that experiment dictates that for s-wave positronium (which is predominantly the case), ηa ηb = −1 . Thus the theoretical result (37) is on firm experimental ground. D.5. Cross sections of Σ0 → Λ0 + e+ + e− (p. 995) We square and sum the given matrix elements over polarizations, which introduces traces over the gamma matrices using the spin sums from Sec. 4.1. Σ-Λ trace in |M+ |2 : [tr(/q Γµν /p Γρσ ) + MΣ MΛ tr(Γµν Γρσ )]k µ k σ Σ-Λ trace in |M− |2 : − [tr(/q Γ5 Γµν /p Γ5 Γρσ ) + MΣ MΛ tr(Γ5 Γµν Γ5 Γρσ )]k µ k σ = [tr(/q Γµν /p Γρσ ) − MΣ MΛ tr(Γµν Γρσ )]k µ k σ (both of these are to be contracted with the electron-positron trace). We now see where the difference in the prediction comes in: a sign change in the MΣ MΛ term. We can study the decay rate as a function of the invariant mass of the electronpositron pair, and compare it to experiment. The Steinberger experiment yields a curve which to good accuracy agrees with the hypothesis Σ0 -parity +1. Now for the main question: would this prediction change if we allowed for four different hypotheses (±1, ±i) for the Σ0 parity? That is, what if Σ0 transformed under Pin(3, 1) instead of the previously assumed Pin(1, 3)? For all particles in Pin(1, 3), using the rules from Sec. 4.1, we compute the squared matrix element: 1 X |M+ |2 ∝ [tr(/q Γµν /p Γρσ ) + MΣ MΛ tr(Γµν Γρσ )] 4 0 0 ss rr

× k µ k σ [tr(/k1 Γν /k2 Γρ ) − m2e tr(Γν Γρ )] , 1 X |M− |2 ∝ [tr(/q Γµν /p Γρσ ) − MΣ MΛ tr(Γµν Γρσ )] 4 0 0 ss rr

× k µ k σ [tr(/k1 Γν /k2 Γρ ) − m2e tr(Γν Γρ )] . Similarly, we compute for Pin(3, 1): 1 X ˆ 2 ˆ µν /p Γ ˆ ρσ ) − MΣ MΛ tr(Γ ˆ µν Γ ˆ ρσ )] |M+ | ∝ [tr(/q Γ 4 0 0 ss rr

X ˆ ν /k2 Γ ˆ ρ ) − m2e tr(Γ ˆν Γ ˆ ρ )] ∝ 1 × k µ k σ [−tr(/k1 Γ |M+ |2 , 4 0 0 ss rr

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1 X ˆ 2 ˆ µν /p Γ ˆ ρσ ) + MΣ MΛ tr(Γ ˆ µν Γ ˆ ρσ )] |M− | ∝ [tr(/q Γ 4 0 0 ss rr

X ˆ ν /k2 Γ ˆ ρ ) − m2 tr(Γ ˆν Γ ˆ ρ )] ∝ 1 × k µ k σ [−tr(/k1 Γ |M− |2 , e 4 0 0 ss rr

where the constants of proportionality are everywhere the same. Thus we have shown how the difference in decay rates between Pin(1, 3) and Pin(3, 1) particles disappears in this calculation. D.6. Positronium (p. 999) We recall from Sec. 4.3 that the bound state operator is Z d3 p X B= ψ(p, se , sp )a†p,se b†−p,sp , (2π)3 s ,s e

p

so the action of UC on the bound state operator is Z d3 p X UC B UC−1 = ψ(p, se , sp )ξa∗ b†p,se ξb∗ a†−p,sp (2π)3 s ,s e

= −(ξa ξb )∗

Z

p

d3 p X ψ(p, se , sp )a†−p,sp b†p,se (2π)3 s ,s e

Z = (ξa ξb )

p

d3 p X (−1)`+s+1 ψ(p, se , sp )a†p,se b†−p,sp (2π)3 s ,s e

p

= ξa ξb (−1)`+s+1 B . We see that since C-parity, unlike P -parity, depends on the total positronium spin s, there will be different selection rules for the two spin states s = 0 (known as “para”-positronium) and s = 1 (“ortho”-positronium). To obtain these selection rules, we consider the C-parity of the final state of photons. Two photons have even C-parity whereas three photons have odd parity. C-parity of two photons: (−1)2 = +1 , C-parity of three photons: (−1)3 = −1 . Consider ` = 0. Decay to three photons is suppressed by a factor of order α (the fine-structure constant) compared to the amplitude for two photons, and we find

Spin state

C-parity

Exp. half-life

Indicates decay mode

Para (s = 0)

+ξa ξb

' 10−10 s

2 photons

Ortho (s = 1)

−ξa ξb

' 10−7 s

3 photons

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Thus we see agreement with experiment provided ξa ξb = 1, which verifies Eq. (32) of Sec. 3.4. Appendix E. Collected References So that the reader does not have to look up many different references, we have used — when possible — three basic references; Analysis, Manifolds and Physics Part I: Basics and Part II: 92 Applications by Y. Choquet-Bruhat and C. DeWittMorette [28, 30], and Introduction to Quantum Field Theory by M. E. Peskin and D. V. Schroeder [87]. The word Pin appears only in Part II: 92 Applications (8 entries in the index). Needless to say, the word Spin appears in all three references. Note that in this report we use the Pauli matrices commonly used in physics (see Sec. 0), not the ones used in the first two references [28, 30]). We would also like to mention some related work that was not directly used for this report, but which may be of interest to the reader depending on his or her specific interest in the Pin groups. • General discussion of the mathematics of the two Pin groups and/or CP T [31, 32, 52, 54, 57, 75, 92, 104, 105, 112, 119, 123]. • Superselection rules [106, 107, 108, 120, 121] (see also [1, 2, 8, 120]). • Pin structures [4, 24, 25, 29, 53]. • Spinors on non-trivial manifolds [11, 47, 51, 55]. • Solution of the Dirac equation in electromagnetic fields [9, 10]. • CP T theorem [42, 56, 68, 69, 85, 101]. • Carruther’s theorem [21, 22, 43, 61, 77, 79, 80, 102, 127]. • CP T and cosmology [93, 94, 95]. • Solar neutrinos [7]. • Interference and CP T in neutron physics [117]. • Majorana neutrinos and double lepton decay [35, 59, 67, 71, 109]. • Non-trivial manifolds in condensed matter physics [73, 76, 110]. • Phase factor observation [13, 91, 116]. • Space and time reversal in atomic and molecular processes [16, 17, 70, 72]. References [1] Y. Aharanov and L. Susskind, “Charge superselection rule”, Phys. Rev. 155 (1967) 1428–1431. [2] Y. Aharanov and L. Susskind, “Observability of the sign change of spinors under 2π rotations”, Phys. Rev. 158 (1967) 1237. [3] C. Alff, N. Gelfand, U. Nauenberg, M. Nussbaum, J. Schultz and J. Steinberger, “Determination of Σ–Λ relative parity”, Phys. Rev. 137(4B) (1965) 1105. [4] L. J. Alty and A. Chamblin, “Obstructions to pin structures on Kleinian manifolds”, J. Math. Phys. 37 (1996) 2001–2011. [5] M. F. Atiyah, R. Bott and A. Shapiro, “Clifford modules”, Topology 3 (Suppl. 1) (1964) 3–38.

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UNITARY REPRESENTATIONS OF NONCOMPACT QUANTUM GROUPS AT ROOTS OF UNITY

H. STEINACKER Institut f¨ ur Theoretische Physik, Ludwig–Maximilians–Universit¨ at M¨ unchen Theresienstr. 37, D-80333 M¨ unchen E-mail: [email protected]–muenchen.de

Received 11 July 2000 Noncompact forms of the Drinfeld–Jimbo quantum groups Uqfin (g) with Hi∗ = Hi , ∗ Xi± = si Xi∓ for si = ±1 are studied at roots of unity. This covers g = so(n, 2p), su(n, p), so∗ (2l), sp(n, p), sp(l, R), and exceptional cases. Finite dimensional unitary representations are found for all these forms, for even roots of unity. Their classical symmetry induced by the Frobenius map is determined, and the meaning of the extra quasi-classical generators appearing at even roots of unity is clarified. The unitary highest weight modules of the classical case are recovered in the limit q → 1.

1. Introduction Quantum groups allow to generalize the concept of symmetry, which has proved to be of great importance in physics. Up to this date, most of the work on quantum groups has been done for the compact case. However noncompact groups are important as well, for example the Lorentz group, or the Anti–de Sitter group SO(2, n) which has attracted much attention recently in the context of string theory [1]. We consider the Drinfeld–Jimbo quantized universal enveloping algebra Uqres (g) [2–4] corresponding to finite dimensional semisimple Lie algebras. In the q-deformed case, there are several possibilities to define real, in particular noncompact forms of these algebras. If q is real, the representation theory is largely parallel to the classical case, but more complicated; for some results in this case see [5, 6]. In the present paper, we consider instead the case where q is a root of unity, which provides additional structure that does not exist in the classical case. This turns out to be much simpler, rather than more difficult than the undeformed case. We study unitary representations of (a slight extension of) the so-called “finite” quantum group Uqfin (g) ⊂ Uqres (g) at roots of unity, with real structure of the form Hi∗ = Hi and Xi±∗ = si Xi∓ , where si = ±1. This covers so(n, 2p), su(n, p), so∗ (2l), sp(n, p), sp(l, R), as well as various forms for the exceptional groups. Even though this real form corresponds to a non-standard Hopf algebra ∗-structure, it is appropriate for our purpose, and leads to a large class of unitary representations. 1035

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Generalizing the method of [7], we find unitary representations for all these noncompact forms, provided q is an even roots of unity. It is shown that all of them can be related to unitary representations of the compact form in a simple way. As opposed to the classical case, they are finite dimensional, which means that the problem is a purely algebraic one. In many cases, they can be viewed as regularizations of classical, infinite dimensional representations. In particular, we show how almost all classical unitary highest weight modules (with the possible exception of a certain “small”, discrete set of highest weights) can be obtained as the limit q → 1 of unitary representations of Uqfin . In the example of the Anti–de Sitter group SO(2, 3), this was already studied for special cases in [8, 9], and more generally in [7]. Not all the representations found however have a classical limit in an obvious way; to understand this better is an interesting open problem. Moreover, it turns out that the unitary representations of Uqfin ⊂ Uqres (g) are very different from the ones studied in [10], where a different specialization of Uq (sl(2, R)) to roots of unity is considered, leading to an infinite dimensional algebra. This paper is organized as follows. After reviewing the definitions and basic concepts in Sec. 2, the unitary representations of the compact case are studied in Sec. 3, and the particular features appearing at roots of unity are discussed. In Sec. 4, the remarkable classical symmetry U (˜g) arising from Uqres (g) at roots of unity due to the Frobenius map [11, 12] is discussed, including the case of even roots of unity which turns out to be most important. The extra generators arising at even roots of unity which extend the classical universal enveloping algebra find a natural interpretation here. In Sec. 5, the noncompact forms are defined, and unitary representations are found for all of them in a rather simple way. It turns out that only a subgroup of the classical U (˜g) preserves the noncompact form, which is determined in Sec. 6. Finally in Sec. 7, the connection with the classical case is made, and it is shown how the classical unitary highest-weight representations are recovered in the limit q → 1. In the appendix, an explicit, self-contained approach to the classical symmetry arising from the Frobenius map is given including the case of even roots of unity, which was treated only implicitly in [12].

2. Definitions and Basic Properties (α ,α )

We first collect the basic definitions, in order to fix the notation. Let Aij = 2 (αji ,αjj ) be the Cartan matrix of a classical simple Lie algebra g of rank r, where ( , ) is the inner product in root space and {αi , i = 1, . . . , r} are the simple roots. The P positive roots will be denoted by Q+ , and ρ = 12 α∈Q+ α is the Weyl vector. For q ∈ C, the quantized universal enveloping algebra Uq (g) is the Hopf algebra with generators {Yi± , Ki , Ki−1 ; i = 1, . . . , r} and relations [2–4] [Ki , Kj ] = 0 ,

(2.1)

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Unitary Representations of Noncompact Quantum Groups

1037

Ki Yj± = q ±Aji Yj± Ki ,

(2.2)

Ki − Ki−1 Yi+ , Yj− = δi,j di , q − q −di

(2.3)

X 1 − Aji (Yi± )k Yj± (Yi± )1−Aji −k = 0, k q

1−Aji

k=0

i 6= j ,

(2.4)

i

where the di = (αi , αi )/2 are relatively prime, qi = q di , [n]qi =

n m

= qi

qin −qi−n qi −qi−1

[n]qi ! . [m]qi ![n − m]qi !

and (2.5)

We assume that q di 6= q −di . The comultiplication is defined by ∆(Ki ) = Ki ⊗ Ki , ∆(Yi+ ) = 1 ⊗ Yi+ + Yi+ ⊗ Ki , ∆(Yi− ) = Ki−1 ⊗ Yi− + Yi− ⊗ 1 .

(2.6)

Antipode and counit exist as well, but will not be needed. The Borel subalgebras Uq± (g) are defined in the obvious way. In this paper, q will always be a complex number, rather than a formal variable. Moreover, since we are mainly interested in representations, it is more intuitive to use the generators {Xi± , Hi } defined by Ki = q di Hi ,

Yi+ = Xi+ q Hi di /2 ,

Yi− = q −Hi di /2 Xi− ,

(2.7)

so that the relations take the more familiar form [Hi , Hj ] = 0 , Hi , Xj± = ±Aji Xj± ,

q di Hi − q −di Hi Xi+ , Xj− = δi,j = δi,j [Hi ]qi . q di − q −di

(2.8) (2.9) (2.10)

The comultiplication is now ∆(Hi ) = Hi ⊗ 1 + 1 ⊗ Hi , ∆(Xi± ) = Xi± ⊗ q di Hi /2 + q −di Hi /2 ⊗ Xi± .

(2.11)

The classical case is recovered for q = 1. Generators Xα± corresponding to the other positive roots α can be defined using the braid group action [13]; we will quote some properties as they are needed. A Poincare–Birkhoff–Witt (P.B.W.) basis is then given as classically in terms of ordered monomials of the raising and lowering operators corresponding to all positive respectively negative roots.

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If q is allowed to be a root of unity, we will instead consider the “restricted (X ± )k ±(k) specialization” Uqres (g) [13] with generators Xi = [k]iq ! for k ∈ N as well as Hi . i For generic q, i.e. q not a root of unity, this is the same as before. However if q is a root of unity, q = e2πin/m

(2.12)

with m and n relatively prime, then [k]q becomes 0 for certain k. Denote with M the smallest positive integer such that q 2M = 1, i.e. [M ]q = 0. Thus M = m if m is odd, and M = m/2 if m is even. In the first case q M = 1, and we will say that q is an “odd” root of unity. In the second case q M = −1, and q will be called “even”. More generally for qi = e2πidi n/m , let Mi be the the smallest integer such that [Mi ]qi = 0 .

(2.13)

Then Mi divides M ; similarly, we define Mα and dα for the other roots. Uqres (g) ±(M )

contains the additional generators Xi i , which have a well-defined coproduct, and thus are defined on tensor products of representations. Verma modules can also be defined in the usual way, for integral highest weights [14]. We will only consider these types of representations of Uqres (g). In particular, (Xi± )Mi = 0 in Uqres (g). Therefore Uqres (g) contains a remarkable sub-Hopf algebra ufin q (the “small quantum group”) generated by Xi± and Ki±1 . We prefer to slightly change the standard convention and define Uqfin by including the Hi as well, slightly abusing the name “finite”. This is a more intuitive generalization of the classical U (g) at least from a physical point of view, and poses no problem since q is a complex number here rather than a formal variable. ±(M ) ±(Mi ) The generators Xi i act as (graded) derivations on ufin , x]± . q by x → [Xi The right-hand side is indeed an element of ufin q , as can be seen from the commutation relations (A.14). Finally, we quote the following useful relation: [Xi+ , (Xi− )k ] = (Xi− )k−1 [k]qi [Hi − k + 1]qi .

(2.14)

3. Representations of Uqfin and Weight Space The Cartan generators can be evaluated on weights λ, such that hHi , λi =

(αi , λ) = (α∨ i , λ) , di

(3.1)

2α where as usual α∨ = (α,α) is the coroot of α. The fundamental weights Λi satisfy (Λi , α∨ ) = δ , therefore i,j j

hHi , Λj i = δij ,

(3.2)

and span the lattice of integral weights. The Weyl group W is defined as usual, and P D = { i ri Λi ; ri ∈ R≥0 } is the dominant Weyl chamber.

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It is well-known [15] that for generic q, the representation theory is essentially the same as in the classical case. In particular, the finite dimensional representations (=modules) of Uqres (g) are direct sums of irreducible representations Lres (λ), which are highest-weight representations with dominant integral highest weight λ. Their character X χ(Lres (λ)) = eλ dim Lres (λ)η e−η =: χ(λ) (3.3) η>0

is given by Weyls formula. Here Lres (λ)η is the weight space of Lres (λ) with weight λ − η. Irreducible highest weight representations of Uqfin are denoted by Lfin (λ). ˜ 3.1. Singlets, special points, and the dual algebra g One important feature at roots of unity is the existence of nontrivial onedimensional representations Lfin(λz ) of Uqfin , with weights X z i Mi Λi (3.4) λz = i

for zi ∈ Z; this follows from (2.14). There also exist similar representations with zi ∈ / Z which will be considered in Sec. 5, but for now we concentrate on the case of integral weights. These weights λz will be called special points. They span a lattice which is the weight lattice of a dual Lie algebra ˜g, rescaled by M . In particular, it contains the root lattice of ˜g, which is generated by the Mi αi or equivalently Mα α. Indeed, consider a second metric on weight space defined by [12] (αi , αj )d := (Mi αi , Mj αj ) ,

(3.5)

(αi , αj )d Mi A˜ij := 2 = Aij . (αj , αj )d Mj

(3.6)

Mi Aij = A˜ij Mj .

(3.7)

with associated matrix

In particular,

A˜ij is always a Cartan Matrix: it is clearly nondegenerate, and A˜ii = 2. To see that A˜ij ∈ −N0 for i 6= j, observe that by the definition of Mj , Mj dj is the smallest integer which is divisible by both M and dj . Similarly Aji di Mi is divisible by M Aji di Mi because Aji is an integer, and also by dj , since Aji di = Aij dj . Therefore M is j dj Mi ˜ an integer, equal to Aij = Aij . Mj

We shall determine A˜ij explicitly. In the simply laced case, all Mi are equal, therefore ( , )d is proportional to the Killing metric, and A˜ij = Aij . Thus the lattice of special points is nothing but the weight lattice rescaled by M , and ˜g = g. For Bn , Cn and F4 , there are roots with 2 different lengths ds = 1 and dl = 2. Again, if M is not divisible by 2, i.e. if q is odd, then clearly Mi = M is odd for all i, and A˜ij = Aij . On the other hand if q is even, then Mi = M/2 =: Ml if αi is long,

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d Mi and Mi = M =: Ms if αi is short. Thus M = dji and A˜ij = Aji , which means that j Ml αl are the short roots and Ms αs the long roots in the lattice of special points. Therefore the dual algebra of Bn is Cn and vice versa, while F4 remains F4 except that the roots change their role. For G2 , the roots have lengths ds = 1 and dl = 3. If M is not divisible by 3, then Mi = M for i = 1, 2, and again A˜ij = Aij . On the other hand if M is divisible by 3, let αl := α1 be the long simple root, and αs := α2 be the short one. Then Ml = M1 = M/3, Ms = M2 = M , and A˜ij = Aji . Thus the dual lattice is again of type G2 , but now Ms αs is the long root, and Ml αl the short one. To summarize, g˜ = g, except for Bñ = Cn and Cñ = Bn if q is even. For all cases, the Weyl group of ˜g is the same as that of g. In Sec. 4, we will see that in some sense, Uqres (g) contains indeed a classical algebra associated with the lattice of special points. The hyperplanes

Hαz := {λ; (λ, α∨ ) = Mα z} ,

(3.8)

where α is any root and z ∈ Z, divide weight space into simplices called alcoves. The alcove of dominant weights with the origin on its boundary is called the fundamental alcove. The reflections on these hyperplanes generate the affine Weyl group, which plays an important role in the representation theory at roots of unity. Notice that every special point is in some Hαz for every root α. To see this, we have to show that (Mi Λi , α∨ ) ∈ Mα Z for every root α. Since the Weyl group preserves the lattice P generated by Mi Λi , this follows from the fact that ( i zi Mi Λi , α∨ j ) ∈ Mj Z, for a suitable αj . In fact, the special points are the intersection points of a maximal number of hyperplanes. 3.2. Unitary representations of the compact form To define unitary representations, one first has to specify the real form of the algebra, or group in the classical case. A real form or ∗-structure is an antilinear involution (=anti-algebra map) on Uqres (g). In the classical case, the ∗ is acting on the complexified Lie algebra, and the real Lie algebra is by definition its eigenspace with eigenvalue −1. The interpretation of a real form at q 6= 1 is given by its classical limit. In this section, we only consider the compact form. It is defined by ∗ = θ where θ(Xi± ) = Xi∓ , θ(Hi ) = Hi is the Cartan–Weyl involution, thus (Xi± )∗ = Xi∓ ,

Hi∗ = Hi ,

(3.9)

extended as an antilinear anti-algebra map. This is consistent for q real and |q| = 1. A representation of Uqres (g) on a Hilbert space V is said to be unitary if the star is implemented as the adjoint on the Hilbert space, i.e. (v, x · w) = (x∗ · v, w) for any x ∈ Uqres (g) and v, w ∈ V . In particular, ( , ) is positive definite. In the classical case, this means precisely that the adjoint (=star) of a group element is

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its inverse. Since all unitary representations are completely reducible, we only need to consider irreducible ones. Unitary and unitarizable will be used synonymously. On unitary highest weight modules with (3.9) or (5.1), the inner product can be calculated recursively, descending from the highest weight state. In particular, it is unique up to normalization. Finite dimensional unitary representations of noncompact forms with the correct classical limit are possible only at roots of unity. Therefore we will concentrate on that case from now on, in particular q ∗ = q −1 . Even though (3.9) is then a “nonstandard” Hopf algebra ∗-structure,a it is appropriate for our purpose. All finite dimensional representations of Uqres (g) have integral weights, even at roots of unity. While this is not true for Uqfin any more, we nevertheless start with studying the unitary representations of Uqfin with integral weights. The following well-known fact [16] is useful: Theorem 3.1. Assume that λ is a dominant integral weight with (λ+ρ, α∨ ) ≤ Mα for all positive roots α. Then the highest weight representation Lres (λ) has the same character χ as in the classical case, given by Weyl’s character formula. In other words, λ + ρ is in the fundamental alcove. This follows from the strong linkage principle, which was first shown in [17]; for a more elementary approach, see ±(M ) [7]. Moreover, Lfin (λ) = Lres (λ) for these weights λ, since the Xi i act trivially. If the above bound is not satisfied, then the Verma module with highest weight λ contains additional highest weight submodules besides the classical ones. Now we can show the following: Theorem 3.2. Let λ be a dominant integral weight, and q = e2πin/m . Then Lfin (λ) is a unitary representation of the compact form (3.9) of Uqfin if the character of 0 Lfin (λ) is given by Weyl’s formula for all q 0 = e2πiϕ with 0 ≤ ϕ0 < n/m. In m particular, this holds if (λ + ρ, α∨ ) ≤ d 2ndα e + 1 for all positive roots α, where dce denotes the largest integer ≤ c for c ∈ R. Proof. Consider Lfin (λ) for all q 0 ∈ B := {e2πiϕ ; 0 ≤ ϕ < n/m}. If the character of Lfin (λ) is the same for all q ∈ B, one can identify the Lfin (λ) as vector spaces.b Their inner product matrix is smooth (in fact analytic) in q 0 , and positive definite at q 0 = 1 since we consider the compact case. This implies that all eigenvalues are positive on B: assume to the contrary that the matrix were not positive definite for some q 0 ∈ B. Then it would have a zero eigenvalue for some q0 ∈ B, which implies that its null space is a submodule of Lfin (λ). But this is impossible, since the Lfin (λ) are irreducible by definition. For q 0 = e2πin/m , some eigenvalues may vanish; but (S(x))∗ = S(x∗ ) here where S is the antipode, rather than (S(x))∗ = S −1 (x∗ ) for x ∈ Uqres (g). b Or even better, view them as trivial vector bundle over B, with local trivializations given in terms of the P.B.W. basis. a Since

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then Lfin (λ) is the quotient of lim q0 →q Lfin (λ) modulo its null space, which again q0 ∈B

has a positive definite inner product. m In particular, assume that (λ + ρ, α∨ ) ≤ d 2nd e + 1. Let Mα0 be the smallest α 2πi

m integer > 2dmα n , which is d 2nd e + 1. Then Mα0 is associated to q 0 := e 2dα Mα0 ∈ B as α defined in Sec. 2. Therefore by Theorem 3.1, Lfin (λ) at q 0 has the same character as m for q = 1, since (λ + ρ, α∨ ) ≤ d 2nd e + 1 = Mα0 . For all other roots of unity q 00 ∈ B, α the character is again the same since the associated Mα00 is larger than Mα0 . Thus the above argument applies.

For some highest weights λ on the boundary of the domain specified in Theorem 3.2, the character of the unitary representation Lfin (λ) is smaller than the classical one. The reason is that the generic representations develop null-submodules; this can be interpreted in the context of gauge theories, see [7]. One may ask if all the unitary representations have been found in Theorem 3.2. As will be discussed in Sec. 7, it is possible that there exist certain unitary representations with integral weights which do not even satisfy the first condition in Theorem 3.2, as suggested by the classcial noncompact case. This would have to be studied by different methods. Other unitary representations with integral and nonintegral weights will be obtained in Theorem 5.1, which however do not have a classical limit. ˜ 4. Frobenius Map and the Quasi-Classical Symmetry g The modules Lres (λ) = Lfin (λ) in Theorem 3.2 are irreducible representations of Uqfin . For larger λ, Lres (λ) decomposes into a direct sum of irreducible modules of Uqfin , which will be described now. This involves the special points introduced in Sec. 3.1. The basic observation is the following. Consider a highest-weight module P Uq−res (g) · vλz with highest weight λz = i zi Mi Λi and zi ∈ Z. From (2.14), it − follows that all Xi · vλz are highest weight vectors. Therefore Xi− · vλz = 0 in Lres (λz ), because it is irreducible by definition. Using the P.B.W. basis, one can see that any element of Uq−res (g) can be written as a sum of terms of the form −(M

)

−(Mβ )

(Xβ1 β1 )k1 · · · (XβN N )kN Uq−fin . It follows that all weights of Lres (λz ) have the P form λz0 = λz − i ni Mi αi with ni ∈ N. In other words, Lres (λz ) is a direct sum of one-dimensional representations Lfin (λz0 ) of Uqfin , since λz0 is a special point. ±(M )

However the “large” generators Xi i do act nontrivially, as we will see. Consider Lres (λz ) ⊗ Lres (λ0 ) for λz as above and integral λ0 with 0 ≤ (λ0 , α∨ i ) < Mi . Now the generators Yi± (2.6) are useful. Using the coproduct, one finds Yi± · (v ⊗ w) = v ⊗(Yi± · w) ,

(4.1)

and ±(Mi )

Yi

±(Mi )

· (v ⊗ w) = (Yi

· v) ⊗(KiMi · w)

(4.2)

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±(M )

i for v ∈ Lres (λz ) and w ∈ Lres (λ0 ), because Yi · w = 0 by the bound on λ0 . For res the same reason, L (λ0 ) is an irreducible representation of Uqfin . Together with (4.1) and (4.2), it follows that Lres (λz ) ⊗ Lres (λ0 ) is an irreducible highest weight module of Uqres (g), and we have verified [14]

Theorem 4.1. Let λz and λ0 be integral weights as above with 0 ≤ (λ0 , α∨ i ) < Mi , and λ = λ0 + λz . Then Lres (λ) = Lres (λ0 ) ⊗ Lres (λz ) .

(4.3)

In particular, Lres (λ) decomposes into a direct sum of irreducible representations P Lfin (λ − i ni Mi αi ) of Uqfin . Moreover, (4.1) and (4.2) show that Yi± commutes ±(M )

i with Yi KiMi on Lres (λ). We will now see that the latter generators acting on Lres (λz ) provide a representation of the classical universal enveloping algebra U (˜g) corresponding to the Cartan matrix A˜ij . This is the essence of a remarkable result of Lusztig [11, 12]. For odd roots of unity, it states that there is a surjective algebra homomorphism

Uqres (g) → U (g) , Yi± → 0 Ki → 1 ±(Mi )

Yi

˜± . →X i

(4.4)

˜ = g for odd roots of unity). It This is the so-called Frobenius map (recall that g is generalized to even roots of unity in [12]; unfortunately the results given there are not very explicit. Since this case is of central importance to us, we will give an elementary, self-contained approach, and show explicitly how the action of U (˜g) on ±(M ) Lres (λz ) is given in terms of the Xj j . The complications arise because at even roots of unity, Ki cannot be set to 1, while Ki2 must be, since [Yi+ , Yi− ] = Indeed, * + X Ki , zj Λj Mj = qizi Mi = ±1 .

Ki −Ki−1 . qi −qi−1

(4.5)

j

These extra, “quasiclassical” generators Ki in some cases anticommute with ±(M ) Xj j , and will extend the algebra U (˜g). They will play an important role in the noncompact case. Let ai ∈ {0, 1} such that ai + aj = 1 if A˜ij 6= 0 and i 6= j; this is always possible. ˜ i = K Mi , and Define K i ˜ ai , ˜ + = X +(Mi ) K X i i i 2

˜ − = X −(Mi ) K ˜ 1−ai q Mi , X i i i i ˜ i = [X ˜ +, X ˜ −] . H i i

(4.6)

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Then we can show the following: Theorem 4.2. For all special points λz , Lres (λz ) is an irreducible highest-weight ˜ i . If vz0 ∈ Lres (λz ) ˜ ± and H representation of the classical U (˜g), with generators X i P 0 ˜ i · vz0 = z 0 vz0 . Moreover, has weight j zj Mj Λj , then H i ˜ ±K ˜ j = sij K ˜jX ˜± , X i i

(4.7)

M M A qi i j ji

where sij = = q Mi Mj (αi ,αj ) = ±1. For dominant integral λ, Lres (λ) is a direct sum of such irreducible representations, by (4.2). ˜ ± ∈ U res (g) for the remaining roots This is proved in the appendix. Root vectors X q α ˜ α ˜ ∈ g˜ are then obtained as classically; see in particular (A.13). From Sec. 3.1, the ñ = Cn and Cñ = Bn if M is even, and ˜g = g otherwise. classical algebras are B This shows explicitly the refinements of (4.4) which arise for even roots of unity, in the most important case of finite dimensional representations. Notice that for odd roots of unity, Ki evaluates to 1 on the special points λz , and Theorem 4.2 essentially reduces to (4.4). The general, abstract result is given in [12]. To summarize the results of this section, any Lres (λ) for dominant integral λ is a direct sum of irreducible representations of Uqfin , which are related by an action of the classical U (˜g), extended by parity generators Ki for even roots of unity. In particular, this holds for unitary representations. 5. Noncompact Forms and Unitary Representations We first recall some concepts in the classical case, see e.g. [18, 19]. Consider a not necessarily compact semisimple Lie group G with real Lie algebra g. Let −σ be the conjugation on the complexification gC with respect to g extended as an involution, by which we mean an anti-linear anti-algebra map whose square lis the identity; one could equally well consider algebra maps. On the other hand, the compact form gK of gC is the eigenspace with eigenvalue −1 of the Cartan–Weyl involution θ. By a theorem of Cartan (see [18, Theorem 7.1]), one can assume that σ = φ ◦ θ, where φ is a linear automorphism of gK with φ2 = 1. Let k be the eigenspace of φ with eigenvalue +1, and p the eigenspace with eigenvalue −1. Then g = k ⊕ ip is the Cartan decomposition of g, and k is a maximal compact subalgebra. A root α is called compact if the corresponding root vector is in k. The star structure is then defined as ∗ = σ. Now there are two cases, depending on if φ is an inner automorphism or an outer automorphism [19]. In this work, we only consider the first type, which covers so(n, 2p), su(n, p), so∗ (2l), sp(n, p), sp(l, R), and various forms for the exceptional groups. We will find quantum versions and unitary representations for all them, even though not all of the representations will have a classical limit. The second type includes sl(l + 1, R), su∗ (l + 1), so(2l − 2p − 1, 2p + 1), and exceptional cases. Up to equivalence, the inner automorphisms of a simple Lie algebra of rank r are given by 2r “chief” inner automorphisms of the form φ(Hi ) = Hi , φ(Xi± ) = si Xi± ,

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for si = ±1 [19, Chap. 14]. They define the real forms Hi∗ = Hi , (Xi± )∗ = si Xi∓ ,

for si = ±1 .

(5.1)

They are not necessarily inequivalent; the compact case corresponds to all si = 1. It should however be noted that real forms which are equilvalent classically are not necessarily equivalent in the q-deformed case. For example, the real form (X ± )∗ = −X ± , H ∗ = −H for |q| = 1 of the “non-restricted” Uq (sl(2, R)) considered in [10] is classically equivalent to the form (X ± )∗ = −X ∓ , H ∗ = H, which is a special case of (5.1). Nevertheless, the first form has no unitary representations at roots of unity if imposed on Uqfin (sl(2)), while the second does. We consider Uqfin , which becomes a ∗-algebra for any of the forms (5.1) for q a root of unity. Now we allow non-integral weights as well (it should be noted that the weights must be integral if working with Uqres (g)). Then there exist one-dimensional P representations Lfin (λr ) of Uqfin with weight λr = i ri Mi Λi generalizing (3.4), where ri ∈ Q such that [ri Mi ]qi = 0, or equivalently q 2ri Mi di = 1 for all i. This follows immediately from (2.14). Explicitly, X m λr = pi Λ i (5.2) 2ndi i with pi ∈ Z. Let Lfin (λ) be a unitary representation of the compact form (such as in Theorem 3.2) with inner product ( , ), and consider Lfin (λ) ⊗ Lfin (λr ). This is again an irreducible representation of Uqfin , and we can define an inner product on it by (v ⊗ ρr , w ⊗ ρr ) := (v, w)

(5.3)

where ρr ∈ Lfin (λr ). It is positive definite by definition. Let us calculate the adjoint of Xi± on this Hilbert space: −ri Mi /2

(v ⊗ ρr , Xi± · (w ⊗ ρr )) = (v ⊗ ρr , Xi± · w ⊗ qi −ri Mi /2

= qi

ρr )

(v, Xi± · w) .

(5.4)

On the other hand, −ri Mi /2

(Xi∓ · (v ⊗ ρr ), w ⊗ ρr ) = (Xi∓ · v ⊗ qi −ri Mi /2

= (qi

ρr , w ⊗ ρr )

v, Xi± · w)

(5.5)

by unitarity of Lfin (λ). By definition, the inner product is antilinear in the first M r /2 argument. Now there are 2 cases: first, if qiMi ri = 1, then qi i i = ±1, and the M r /2 M r adjoint of Xi± becomes (Xi± )∗ = Xi∓ . Second, if qi i i = −1, then qi i i = ±i, and the adjoint of Xi± is (Xi± )∗ = −Xi∓ . Therefore we have proved

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Theorem 5.1. Let Lfin (λ) be a unitary representation of the compact form of Uqfin , P and Lfin (λr ) a one-dimensional representation of Uqfin with weight λr = i ri Mi Λi as in (5.2). Then Lfin (λ + λr ) = Lfin (λ) ⊗ Lfin (λr ) with inner product (5.3) is a unitary representation of the real form (5.1) of Uqfin , where si = qiMi ri = hKi , λr i = ±1. All unitary representations of that real form can be obtained in this way. The last statement follows since the noncompact representations can similarly be “shifted” back to the compact form. This explains the role of the extra, “quasiclassical” generators Ki at even roots of unity: they determine the real form of a representation. While the symmetric form of the coproduct (2.11) was useful in the proof, it is irrelevant for the result. For the remainder of this section we concentrate on the case of integral weights, i.e. λr = λz as in (3.4), and determine which of the classical noncompact forms actually occur in this way. In the simply laced case, Mi = M , and qiMi = −1 precisely if q is an even root of unity. Thus for odd roots of unity, si = 1 for all i, whereas for even roots of unity, si = (−1)zi , so that there are unitary representations for all the noncompact forms considered. In the non-simply laced case, consider first Bn , Cn and F4 . If q is odd, i.e. qsMs = 1 with odd Ms = M , then Mi = M , and si = 1 for all i. Therefore only the compact form occurs. For even q, one has to distinguish whether M = m/2 is even or odd. If M is odd, then Ml = Ms = M , therefore qlMl = 1 and qsMs = −1. This means that only those noncompact forms with si = 1 for αi a long root and si = (−1)zi for αi a short root occur. If M is even, then Ml = Ms /2, and qiMi = −1 for all i. Therefore si = (−1)zi for all i, and again all noncompact forms considered are realized (to recover the results in [7], notice that the conventions there are such that ds = 12 ). Finally consider G2 . If q is odd, then Ml = M/3 if M is a multiple of 3, and Ml = M otherwise. In either case, qiMi = 1 for all i, and si = 1 for all i. If q is even, then qiMi = −1 for all i, thus si = (−1)zi for all i, and again all noncompact forms considered are realized. The classical limit of these unitary representations will be discussed in Sec. 7. Notice that Theorem 5.1 also yields additional unitary representations of the compact form with generally non-integral weights, for qiMi ri = 1. We will see in Proposition 7.1 however that the distance of their weights from the origin becomes infinite as q approaches 1. In that sense, they are non-classical. 6. Reality-Preserving Algebra on Lres (λ) Consider a dominant integral weight λ0 such that Lfin(λ0 ) is a unitary representation of the compact form with 0 ≤ (λ0 , α∨ i ) < Mi for all i, and a special point λz . By Theorem 4.1, Lres (λ0 + λz ) = ⊕z0 (Lfin (λ0 ) ⊗ Lfin (λz0 )) is a direct sum of irreducible representations of Uqfin , where the Lfin (λz0 ) are one-dimensional components of Lres (λz ). These sectors are unitary representations of various real forms of Uqfin , ˜± according to Theorem 5.1. Moreover by Theorem 4.2, the “large” generators X i

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connect the various sectors with different z 0 . It is natural to ask which subalgebra of the classical U (˜g) connects only those sectors with the same real form. This will ˜ ± )2 always preserve the real be called reality-preserving algebra. Of course, the (X j form, but they do not form a closed algebra. Xα± ˜ as defined below Theorem 4.2 preserves the real form if and only if ˜ α± , Ki ] = 0 [X

for all i ,

(6.1)

q Mα (α,αi ) = 1

for all i .

(6.2)

or

This is equivalent to q Mα (α,β) = 1 for all roots β. Using the Weyl group, we can assume that α = αj is a simple root, since all other α satisfying (6.2) are then obtained as the image under the Weyl group of the simple ones. First consider the simply laced case. Then for any j, there is an i such that (αi , αj ) = −1, therefore q Mj (αi ,αj ) = 1 only if q is odd. But then all sectors are all compact. Therefore the reality-preserving algebra is g for odd q, and trivial otherwise. Next consider the non-simply laced case. If q is odd, then all forms are compact, and the reality-preserving algebra is clearly g. Thus assume q is even. For G2 , q Mj (αi ,αj ) = q −3Mj = −1 if i 6= j, and the reality-preserving algebra is trivial. For Bn , Cn and F4 , observe first that if Aij 6= 0 and dj ≥ di , then (αi , αj ) = −dj . Therefore q Mj (αi ,αj ) = q Mj max{di ,dj } . One has to distinguish M = m/2 even and odd. Assume M is even, so that Ml = Ms /2 = M/2. Then the only way that q Mj max{di ,dj } = 1 for all i 6= j with Aij 6= 0 is Mj = M and max{di , dj } = 2, i.e. j is short and is connected only to long nodes in the Dynkin diagram. The only case where this happens is Bn , which has one short simple root. By the Weyl group, it follows that the reality-preserving ˜ ± where αs are the short roots of Bn . Since q is even, algebra is generated by all X αs ˜ ± correspond precisely to the long the dual algebra ˜g of Bn is Cn , i.e. these X αs roots of ˜g. Now Cn has precisely n long roots which are all orthogonal, and the corresponding root vectors commute. Therefore for even M , the reality-preserving ˜ α± which commute with each other algebra for Bn is (su(2))n , generated by the X s res (on L (λ)). For Cn and F4 , it is trivial except for C2 ∼ = B2 . If M is odd, then Ml = Ms = M , thus q Mdl = 1, and q Mds = −1. Therefore if q M max{di ,dj } = 1 for all i 6= j with Aij 6= 0, then either j must be long, or j is connected only to long nodes in the Dynkin diagram. For Bn this holds for all j, for Cn this holds for the one long simple root, and for F4 this holds for the 2 long simple roots. Therefore the reality-preserving algebra for Bn is again Bn , with generators ˜ ± for all α. For Cn , it is (su(2))n with generators X ˜ ± which commute with each X α αl other, where αl are the long roots of Cn . For F4 , it is the algebra generated by all long roots, which is D4 .

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7. Non-Integral Weights and the Classical Limit In this section, we want to determine which of the unitary representations of Uqfin in Theorem 5.1 have a well-defined classical limit. The idea is to consider them as highest-weight modules in a suitable way with fixed highest weight, and let q approach 1. P m Λj as in (5.2), consider For dominant integral λ0 and λr = pj 2nd j Lfin (λ) = Lfin (λ0 ) ⊗ Lfin (λr ) with m (λ0 + ρ, α∨ ) ≤ + 1 for all α ∈ Q+ , 2ndα

(7.1)

where λ = λ0 + λr . According to Theorems 3.2 and 5.1, this is a unitary representation of a certain noncompact form determined by λr . We want to understand the location of the weights of Lfin (λ) in weight space, and in particular if they are close enough to the origin so that they can have a classical limit as a highest weight module. For q 6= 1 of course, they can always be viewed as highest weight modules. The bound (7.1) for Lfin (λ) being unitary can be stated more geometrically as follows. Divide weight space into alcoves separated by the hyperplanes n m o hzα := µ; (µ, α) = z (7.2) 2n for all roots α and z ∈ Z, similar as in Sec. 3.1. Then λ0 is in the fundamental alcove by (7.1), using the fact that (ρ, α∨ ) ≥ 1 for all positive roots α; the latter can be seen P using ρ = i Λi . By the Weyl group, all weights of Lfin(λ0 ) are therefore contained in the union of those alcoves which have the origin as corner, more precisely within a certain distance from its walls as determined by (7.1). Since the set of hyperplanes (7.2) is invariant under translations by λr , the weights of Lfin (λ) are contained in the union of those alcoves which have λr as corner. In particular, they are contained in a half-space with the origin on its boundary. Since the distance between parallel hyperplanes goes to infinity as q → 1, Lfin (λ) can have a classical limit only if λr and the origin belong to the same alcove. This puts a restriction on the possible real forms as determined by λr . To make this more precise, recall the definition of compact roots in Sec. 5, and the definition of the Coxeter labels ai which are the coefficients of the highest root P θ = j aj αj , and satisfy ai ≥ 1 for all i. P m pj 2nd Λj with pj ∈ Z, as in (5.2). It belongs Proposition 7.1. Consider λr = j to the closure of an alcove as defined above which also contains the origin as a corner, if and only if there is a set of simple roots (denoted again by αi ) such that αi0 has Coxeter label ai0 = 1 and pj = ±δj,i0 , hence m λr = ± Λi . (7.3) 2ndi0 0 In that case, the remaining r − 1 simple roots are compact with respect to the real form defined in Theorem 5.1.

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Proof. By using an element σ of the Weyl group if necessary, we can assume P m that λr is an anti-dominant weight. Then λr = − pj 2nd Λj with pj ∈ N, and j P −1 m m is a (λr , θ) = − pj aj 2n . This implies that (λr , θ) ≤ − 2n = (h−1 θ , θ) where hθ m hyperplane as defined in (7.2), and equality holds precisely if λr = − 2ndi Λi0 and 0 ai0 = 1 for some i0 . The desired set of simple roots is obtained by applying σ to the original simple roots. Then all (new) simple roots αj for j 6= i0 are compact according the definition in Theorem 5.1. We will always use this set of simple roots from now on. The corresponding generators in Uqres (g) could be obtained via the braid group action [13], but this is not needed since we will only make statements about the characters below. The corresponding real form is (Xi±0 )∗ = −Xi∓0 , (Xj± )∗ = Xj∓

and for j 6= i0 .

(7.4)

In the classical limit, the center of k is then one-dimensional and generated by an element of the Cartan subalgebra dual to Λi0 , which is orthogonal to the compact roots. Explicitly, this leads to the following cases: • • • • • •

i0 i0 i0 i0 i0 i0

= 1, 2, . . . , l for Al , corresponding to su(l + 1 − p, p) for all p = 1 for Bl , corresponding to so(2l − 1, 2) = 1 for Dl , corresponding to so(2l − 2, 2) = l or equivalently i0 = l − 1 for Dl , corresponding to so∗ (2l) = l for Cl , corresponding to sp(l, R) = 1 or equivalently i0 = 5 for E6 , and i = 6 for E7 ,

see for example [19, Table 14.1]. Not surprisingly, these are precisely the cases where highest weight modules exist in the classical limit, see [20, 21] and references therein. We will restrict ourselves to (7.4) from now on, and show how to recover the classical unitary highest weight representations from the Lfin (λ). To do that, we choose a minus sign in (7.3), since then it is possible to consider highest-weight modules Lfin (λ) with fixed highest weight λ independent of q, in particular as q → 1. The plus sign would correspond to lowest-weight modules in the classical limit. To make the connection with the literature on the classical case [20], consider the character χ(L(λ + zΛi0 ))e−zΛi0 for z ∈ R, where L(λ + zΛi0 ) is the classical irreducible highest weight module with highest weight λ + zΛi0 . It is independent of z for sufficiently negative z, which can be seen from the strong linkage principle P (see e.g. [22]): by writing λ = c0 Λi0 + j6=i0 nj Λj and noticing that the compact roots are orthogonal to Λi0 , if follows that for sufficiently negative z, all weights strongly linked to λ + zΛi0 are in the orbit of the compact Weyl group acting on λ + zΛi0 . The first reduction point z0 is the maximal value of z where this is no longer the case. Clearly L(λ + zΛi0 ) can only be unitary with respect to (7.4) if λ is a dominant integral weight with respect to k, i.e. nj ∈ N in the above notation.

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Provided this is the case, L(λ + zΛi0 ) is unitary [20] if and only if z ≤ z0 , or z is in a certain finite set of z > z0 . In the q-deformed case, we can show the following: Proposition 7.2. Let λ be a rational weight which is dominant integral with respect to k. If the first reduction point of L(λ + zΛi0 ) is at z ≥ 0 for q = 1, then there exists a series of roots of unity qk → 1 such that Lfin (λ) is unitary with respect to ∨ (7.4) for all q = qk . In particular, this holds if (λ+ρ, α∨ ) ≤ d(−λ, α∨ i0 )e+(λ, αi0 )+1 for all positive noncompact roots α. Of course, this generalizes to irrational λ which can be approximated by rational weights as above. Proof. Assume that λ is as required. Then there are m, n ∈ N such that (λ − m λr , α∨ i0 ) ∈ N where λr = − 2ndi Λi0 , thus λ − λr is dominant integral. For k ∈ N, 0

mk let mk := m + 2nkdi0 , λr,k := − 2nd Λi0 , and qk := e2πin/mk . We claim that for i 0

sufficiently large k, the character of Lfin (λ−λr,k ) is given by Weyls formula for all q 0 between 1 and qk . Then the first part of the proposition follows from Theorems 3.2 and 5.1. 0 0 n0 n 0 Let q 0 = e2πin /m with m 0 < m , with associated Mα as in Sec. 2. By the strong fin linkage principle [7, 17], the character of L (λ − λr,k ) can differ from χ(λ − λr,k ) only by the sum of classical characters χ(µn ) (3.3) with dominant µn , which are “strongly linked” to λ − λr,k by a series of reflections by hyperplanes Hαz 0 defined as in (3.8) using Mα0 , but shifted by −ρ. They again divide weight space into (shifted) alcoves, with corresponding special points for q 0 , also shifted by −ρ. Now ai0 = 1 implies that −ρ and −λr,k − ρ are in the same shifted alcove mk for any q 0 between 1 and qk , because (−λr,k , α∨ ) ≤ 2nd ≤ Mα0 for all positive α α. Moreover, the union of the alcoves which have −ρ as a corner is a convex set of weights, and invariant under the Weyl group action with center −ρ. Therefore all weights in that set which are strongly linked to −λr,k − ρ are obtained by the action of the classical Weyl group with center −ρ; in particular, −ρ is not. Thus if k is large enough, all dominant µn strongly linked to λ − λr,k can be obtained by reflections of λ − λr,k by those Hαz 0 which contain the special point Mi00 Λi0 − ρ. However using the assumption, the character of Lfin (λ − λr,k ) is not affected by 0 these µn : indeed, by a shift by − 2nm0 di Λi0 as in Sec. 5, Lfin (λ − λr,k ) can be related 0

to Lfin (λ + zΛi0 ) for z = 0

m 2ndi0

−

m0 2n0 di0

< 0. The special point Mi00 Λi0 − ρ is then 0

moved to (Mi00 − 2nm0 di )Λi0 − ρ, and is only relevant for large k if Mi00 = 2nm0 di , 0 0 when it becomes −ρ. However by the assumption on the first reduction point, the character of the classical L(λ + zΛi0 ) is not affected by the hyperplanes through −ρ for z < 0. Using the fact that Uq (g) is the same as U (g) as algebra over C[[q − 1]] [23], the character of Lfin (λ + zΛi0 ) is not affected by the hyperplanes through −ρ either. Combining all this, it follows that the character of Lfin (λ − λr,k ) is given by Weyls formula for all q 0 between 1 and qk .

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mk In particular, this holds if (λ − λr,k + ρ, α∨ ) ≤ d 2nd e + 1 for all positive nonα compact roots α, by Theorem 3.2. This is certainly satisfied for compact α if k is mk ∨ ∨ sufficiently large. Using 2nd = −(λ, α∨ i0 ) + (λ − λr,k , αi0 ) ∈ −(λ, αi0 ) + N and the i0 fact that di0 = 2 for the non-simply laced cases, this bound follows from the given condition.

Therefore we recover the classical results on unitary highest weight representations, except for the small, finite set of z > z0 , which we cannot address here. It is quite possible that there exist unitary representations of Uqfin corresponding to these remaining cases; this would have to be studied by other methods. By Theorem 5.1, they would correspond to additional unitary representations of the compact form, as was pointed out in Sec. 3. In the example of the Anti–de Sitter group, the highest-weight property corresponds to positivity of the energy [7], which is an important physical requirement. Notice that the unitary representations of noncompact forms of Uqfin in general have non-integral, but rational weights. Those with integral weights can have a m classical limit only if 2nd ∈ Z, in particular q must be an even root of unity. i 0

The question arises if and how the unitary representations of Uqfin in those cases where there exists no classical unitary highest weight representation might be related to other classical series of unitary representations, and how the latter may be obtained from the quantum case at roots of unity. The answer may be related to the fact that there do exist other types of unitary representations of the non-restricted specialization for |q| = 1, such as Uq (sl(2, R)) [10], as was mentioned in Sec. 5. This certainly deserves further investigation. Acknowledgments The author would like to thank Konrad Schm¨ udgen and David Vogan for pointing out some related references, and John Madore for useful comments. Appendix A We prove Theorem 4.2 by verifying the classical relations of the Chevalley basis ˜i. ˜ ± and H X i P ˜ i on the weights λz = To calculate H j zj Mj Λj , one can use the standard commutation relation Hi +(Mi ) −(Mi ) [Xi , Xi ]= + ufin (A.1) q , Mi q i

where the last term vanishes on Lres (λz ). We also need the following identity [12] which can be checked directly: If q 2M = 1, then aM + b a M 2 c(a+1)+M(ad−bc) b =q . (A.2) cM + d q d q c 1

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±(M ) ˜ +, X ˜ − ] = [X +(Mi ) , X −(Mi ) ]K ˜ i q Mi . Furthermore [Ki , Xi i ] = 0, and therefore [X i i i i i Using (A.1), (A.2) and (4.5), this evaluates on vλz0 to " # zi0 Mi z0 M 2 M 2 ˜ Hi · vλz0 = qi i i qi i · vλz0 = zi0 vλz0 , (A.3) Mi qi

as claimed. We next check that ˜ +, X ˜ −] = 0 [X i j

(A.4)

for i 6= j. This is clear if Aij = 0. Otherwise, one can write +(Mi )

Xi

˜ j = sij K ˜ j X +(Mi ) , K i

(A.5)

M M A qi i j ji

1−a sajii sij j

= q Mi Mj (αi ,αj ) = sji = ±1. Then where sij = Mi Mj (αi ,αj )2ai = 1, since ai = (1 − aj ) if Aij 6= 0 and i 6= j. Therefore q

=

˜ +X ˜ − = sai s1−aj X ˜ −X ˜+ = X ˜+ . ˜ −X X ji ij i j j i j i

(A.6)

˜i, X ˜ ± ] = ±A˜ji X ˜± , [H j j

(A.7)

Next, to verify

˜ i by replace again H ±(Mj )

[Hi ]qi Xj

H i

Mi qi

2

˜ i q Mi , and observe using (3.7) that K i

±(Mj )

= Xj

±(Mj )

[Hi ± Mj Aji ]qi = Xj

We first show ! 2 Hi M ˜ i q i X +(Mj ) K ˜ aj = X +(Mj ) K ˜ aj K j j i j j Mi q

i

Hi Mi

Using the above, this becomes ˜ji Mi 2 +(Mj ) ˜ aj Hi + A ˜ i sji q Mi = X +(Mj ) K ˜ aj Xj Kj K i j j Mi q i

[Hi ± Mi A˜ji ]qi .

(A.8)

!

2 ˜ i q Mi K i

+ A˜ji

.

(A.9)

qi

Hi Mi

!

2 ˜ i q Mi K i

+ A˜ji

.

qi

(A.10) Restricting on a weight λz0 , it remains to show ! 0 0 (zi + A˜ji )Mi z i Mi Mi2 zi0 Mi2 Mi2 zi0 Mi2 qi sji qi = q qi + A˜ji . Mi Mi q i q i

(A.11)

i

˜ M2A

˜ i, X ˜ −] = Now sji = qi i ji , and the claim follows from (A.2). The calculation for [H j ˜ − is completely analogous. −A˜ji X j Finally, the Serre relations are ˜ + , . . . , [X ˜ +, X ˜ + ] · · ·]1−A˜ji = 0 [X i i j (1 − A˜ji brackets) on Lres (λz ), and similarly for the negative roots.

(A.12)

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To prove this, consider a P.B.W. basis of Uq+res (g), which is given by the ex+(t )

+(t )

pressions XαN N · · · Xα1 1 where {α1 , . . . , αN } is an ordered basis of the positive ˜ = {˜ roots, obtained e.g. by the braid group action [12]. Let Q α = Mα α} be the set ˜ ˜ j ∈ Q, of roots of the lattice of special points. For k ∈ N such that β˜ = k α ˜i + α +(Mβ ) ˜ kai ˜ aj + + ˜ ˜ ˜ ˜ define Xβ˜ := Xβ Ki Kj generalizing (4.6), and Xβ˜ := 0 if β ∈ / Q. We claim that ˜ +, X ˜ + ] = cX ˜+ [X i α ˜ +β˜ β˜

(A.13)

i

if acting on Lres (λz ), for some constant c. This clearly implies the Serre relations. The proof is by induction on k, using the well-known commutation relations [16] X s−1 s−1 Xα+r Xα+s − q (αr ,αs ) Xα+s Xα+r = c(tr+1 , . . . , ts−1 )Xα+ts−1 · · · Xα+tr+1 (A.14) for r < s, with some constant c(tr+1 , . . . , ts−1 ). ˜ +X ˜ + (or the reversed form) as in the P.B.W. We want to order the expression X i β˜ basis, using (A.14). The leading term is a

˜ +X ˜+ , q Mi Mβ (αi ,β) sijj sajii X i β˜

(A.15)

˜ ai , X +(Mi ) ] = 0. We claim that the only other term on the rhs of (A.14) since [K i i ˜ + . This is so because only which may not vanish on Lres (λz ) is proportional to X α ˜ +β˜ i

+(M )

products of “large” generators Xα α are nonzero on Lres (λz ), and in fact only one “large” generator can occur on the rhs of (A.13), because only a simple (formal) pole in q can arise by the derivation property mentioned in Sec. 2. Moreover using 2 β˜ = Mβ β = kMi αi + Mj αj , it follows that q Mi Mβ (αi ,β) = q 2kMi di q Mi Mj (αj ,αi ) = 1−a q Mi Mj (αj ,αi ) = sij . Thus the overall coefficient in front of (A.15) is sij j sajii , which is 1 as above. This concludes the proof. References [1] J. Maldacena, “The Large N Limit of Superconformal Field Theories and Supergravity” Adv. Theor. Math. Phys. 2 (1998) 231. [2] V. Drinfeld, “Quantum Groups” p. 798 in Proceedings of the International Congress of Mathematicians, Berkeley, 1986, ed. A. M. Gleason, Amer. Math. Soc. Providence, RI. [3] L. D. Faddeev, N. Yu. Reshetikhin and L. A. Takhtajan, “Quantization of Lie Groups and Lie Algebras” Algebra Anal. 1 (1989) 178. [4] M. Jimbo, “A q–Difference Analogue of U (g) and the Yang–Baxter Equation” Lett. Math. Phys. 10 (1985) 63. [5] V. A. Groza, N. Z. Iorgov and A. U. Klimyk, “Representations of quantum algebra Uq (un,1 )” math/9805032; Klimyk A. and S. Pakuliak, “Representations of the quantum algebras Uq (ur,s ) and Uq (ur+s ) related to the quantum hyperboloid and sphere” J. Math. Phys. 33(6) (1992) 1987. [6] V. Guizzi, “A classification of unitary highest weight modules of the quantum analogue of the symmetric pairs (An , An−1 )”, J. Alg. 192, (1997) 202. [7] H. Steinacker, “Finite dimensional Unitary Representations of quantum Anti–de Sitter Groups at Roots of Unity” Comm. Math. Phys. 192 (1998) 687.

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[8] V. K. Dobrev and P. J. Moyan, Phys. Lett. 315B, 292 (1993); L. Dabrowski, V. K. Dobrev, R. Floreanini and V. Husain, “Positive energy representations of the conformal quantum algebra”, Phys. Lett. B302 (1993) 215–222. [9] M. Flato, L. K. Hadjiivanov and I. T. Todorov, “Quantum Deformations of Singletons and of Free Zero-Mass Fields” Found. Phys. 23(4) (1993) 571–586. [10] K. Schm¨ udgen, “Operator representations of Uq (sl2 (R))”, Lett. Math. Phys. 37 (1996) 211. [11] G. Lusztig, “Quantum groups at roots of 1” Geom. Ded. 35 (1990) 89. [12] G. Lusztig, Introduction to Quantum Groups, Progress in Mathematics Vol. 110, Birkhaeuser, 1993. [13] G. Lusztig, “On quantum groups”, J. Algebra 131 (1990) 466. [14] G. Lusztig, “Quantum deformations of certain simple modules over enveloping algebras”, Adv. Math. 70 (1988) 237. [15] M. Rosso, “Finite Dimensional Representations of the Quantum Analog of the Enveloping Algebra of a Complex Simple Lie Algebra” Comm. Math. Phys. 117 (1988) 581. [16] V. Chari and A. Pressley, A Guide to Quantum Groups”, Cambridge University Press, 1994. [17] H. H. Anderson, P. Polo and W. Kexin, Invent. Math. 104 (1991) 1. [18] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978. [19] J. F. Cornwell, Group Theory in Physics, Vol. II, Academic Press, 1984. [20] T. Enright, R. Howe and N. R. Wallach, “A classification of unitary highest weight modules”, in Representation Theory of Reductive Groups, ed. T. Trombi, Progress in Mathematics, Birkh¨ auser, Boston, 1982. [21] H. Garland and G. J. Zuckerman, “On unitarizable highest weight modules of Hermitian pairs”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(3) (1982) 877. [22] V. Kac and D. Kazhdan, “Structure of Representations with Highest Weight of infinite-dimensional Lie Algebras”, Adv. Math. 34 (1979) 97. [23] V. Drinfeld, “On Almost Cocommutative Hopf Algebras” Leningrad Math. J. 1(2) (1990) 321.

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RIGOROUS SEMICLASSICAL RESULTS FOR THE MAGNETIC RESPONSE OF AN ELECTRON GAS

MONIQUE COMBESCURE Laboratoire de Physique Th´ eorique∗ Universit´ e de Paris XI, Bˆ atiment 210, F-91405 ORSAY Cedex, France E-mail: [email protected] DIDIER ROBERT D´ epartement de Math´ ematiques, CNRS UMR 6629 Universit´ e de Nantes, 2 rue de la Houssini` ere F-44322 NANTES Cedex 03, France E-mail: [email protected]

Received 1 September 2000 Consider a free electron gas in a confining potential and a magnetic field in arbitrary dimensions. If this gas is in thermal equilibrium with a reservoir at temperature T > 0, one can study its orbital magnetic response (omitting the spin). One defines a conveniently “smeared out” magnetization M , and the corresponding magnetic susceptibility χ, which will be analyzed from a semiclassical point of view, namely when ~ (the Planck constant) is small compared to classical actions characterizing the system. Then various regimes of temperature T are studied where M and χ can be obtained in the form of suitable asymptotic ~-expansions. In particular when T is of the order of ~, oscillations “` a la de Haas-van Alphen” appear, that can be linked to the classical periodic orbits of the electronic motion.

1. Introduction The magnetic response theory for a free electron gas is an old problem considered by Landau [17], Fock [11] and Peierls [22]. The revival of interest in physics arose from the advances of recent experiments that made possible measurements of the magnetic response on small 2-dimensional electronic devices. These devices are so “pure” that the classical as well as quantum motion inside them can be considered as “ballistic”, i.e. is uniquely determined by the confining potential. (Taking into account impurities would consist in adding the random potential created by random point scatterers inside the material). The bi-dimensional structure of such electrongas is realized through semi-conductor heterostructures whose size, and shape can be controlled experimentally, together with the number N of confined electrons. The ∗ Unit´ e

Mixte de Recherche - CNRS - UMR 8627. 1055

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system being in contact with a reservoir at temperature T , and submitted to a magnetic field B perpendicular to the surface, the magnetic response can be measured: say, the magnetization M or magnetic susceptibility χ as a function of the thermodynamic parameters T , N , B [24]. These experiments manifest the sensitivity of the magnetic response to the integrable versus non-integrable character of the classical dynamics of one electron in the system. Numerical experiments on two-dimensional magnetic billards have confirmed this observation, and suggested that the quantum magnetic response is an experimentally accessible criterion for distinguishing classically integrable versus chaotic dynamics [19]. A number of theoretical studies have analyzed the magnetic response from a “semi-classical” point of view, namely as properties manifesting themselves in the limit when ~ (the Planck constant) is small compared to classical actions characterizing the system (say h a2 eB/c where a is a typical size of the system, e the charge of the electron, c the velocity of light and B the magnetic field size) [4]. In these studies, the Coulomb interactions between the electrons in the system are neglected, so that the system is a “free electron gas” to which the usual thermodynamic formalism is applied. The thermodynamic functions in the grand-canonical ensemble can be expressed through the density of states of the quantum Hamiltonian for one electron in the system. This quantum density of states, in the semi-classical limit, splits into a mean part and a strongly oscillating one, according to the well known semi-classical trace formula. This formula is known in mathematics as Poisson formula (Colin de Verdière [5], Duistermaat–Guillemin [10]) and in physics as the Gutzwiller trace formula in the chaotic case [14], or the Berry–Tabor trace formula in the integrable case [3]. This splitting provides a similar splitting in the magnetic response, which allows to understand the oscillations “` a la de Haas-van Alphen” of the magnetic susceptibility and their link with the classical periodic orbits of the electronic motion. The aim of the present paper is to reconsider these questions from a mathematical point of view, in the following two directions (for non-interacting electron gases in arbitrary dimension, and not necessarily homogeneous magnetic fields) — examine the regimes of temperature in which the magnetic response can be obtained semi-classically in the form of asymptotic ~ expansions. — investigate a “mesoscopic” regime of low temperatures where the periodic orbits of the classical one-electron dynamics manifest themselves as highly oscillating contributions to the magnetic response. In a recent work, Fournais [12] studies the semi-classics of the quantum current for a non-interacting gas of electrons in dimension n and temperature T , confined in a potential V and subject to a suitable magnetic field B. For fixed non-zero temperature T , he obtains a complete asymptotic expansion of the quantum current in small ~, and for zero temperature, he obtains the dominant contribution plus an error term under suitable assumptions. J. Butler [4] has recently reexamined this last case using a “semi-classical trace formula” by Petkov and Popov [21]. We recall that in all these studies, the spin of the electron is omitted so that only the orbital magnetic response is considered.

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Rigorous Semiclassical Results for the Magnetic Response

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The content of our paper is the following: • In Sec. 2 we consider the case when the temperature T is large compared to the Planck constant ~. We prove asymptotic expansion in ~ for the thermodynamical potential and we recover the Landau diamagnetic formula for 2-dimensional free electron gas. • In Sec. 3 we consider the case where the temperature T is of the same order as ~. Then we prove that the magnetization splits into two terms: an average part with a regular asymptotic expansion in ~ plus an oscillating part in ~ which is the contribution of the periodic orbits of the classical motion. • In Sec. 4 we come back to the regime T larger than ~ and prove that the contribution of non-zero periods of the classical motion is exponentially small in ~. 2. The Landau Magnetism We shall first give the notations and assumptions that will hold all along this paper. Given β > 0, we set 1 Fβ (x) = − Log(1 + e−βx ) , β

(2.1)

fβ (x) = Fβ0 (x) = (1 + eβx )−1 ,

(2.2)

fβ is related with the Fermi–Dirac distribution. These functions are meromorphic, with poles (or cuts for Fβ ) at 2k + 1 iπ, β

x=

k ∈ Z.

Let κ ∈ R be a real parameter (coupling constant with a magnetic field). We consider a family of Hamiltonians with magnetic fields given by Hκ (q, p) =

1 (p − κa(q))2 + V (q) , 2

(2.3)

where V : Rn 7→ R and a: Rn 7→ Rn are C ∞ functions satisfying the following properties. (H.1) (H.2) (H.3)

∀ q ∈ Rn ,

V (q) ≥ 1 ,

∀ q ∈ Rn , ∀ q ∈ Rn ,

|∂qα V (q)| ≤ Cα V (q)

|∂qα a(q)| ≤ Cα V (q)1/2

V (q) ≥ c0 (1 + |q|2 )s/2

some s, c0 > 0

(confinement assumption). b κ be the Weyl quantization of Hκ . The previous assumptions ensure Let now H b κ ) ⊂ [ε, ∞) is pure point for every b that Hκ is self-adjoint and its spectrum σ(H κ ∈ R where ε > 0 ([25]). Let us call (Ej )j∈N and (ϕj )j∈N the set of corresponding eigenvalues and eigenstates.

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In the grand-canonical ensemble, the thermodynamic potential Ω is given by X b κ − µ)} Fβ (Ej − µ) = Tr{Fβ (H (2.4) Ω(β, µ, κ) = j∈N

where µ > 0 is the chemical potential, β = 1/kB T , kB being the Boltzmann constant, and T > 0 the temperature. (κ will be the size of the magnetic field). Furthermore, the mean-number of particles in the grand-canonical ensemble is given by b κ − µ)) . N (β, µ, κ) = Tr(fβ (H

(2.5)

In all that follows, we fix the chemical potential µ, which according to (2.5) implies that N (β, µ, κ) is large in the semiclassical régime so that we are appoaching the thermodynamical limit where canonical and grand canonical ensemble descriptions are equivalent. b − µ) and Using the functional calculus [25], it is not difficult to see that Fβ (H ∞ b fβ (H − µ) are trace-class and that the function: κ 7→ Ω(β, µ, κ) is C for |κ| ≤ κ0 . ∂ . We shall denote ∂κ = ∂κ Proposition 2.1. The function: κ 7→ Ω(β, µ, κ) is C ∞ on R. In particular we have bκ] . b κ − µ)∂κ H ∂κ Ω = Tr[fβ (H

(2.6)

Now we have the following definitions of magnetization M and magnetic susceptibility χ: bκ ] , b κ − µ)∂κ H M = ∂κ Ω = Tr[fβ (H χ = ∂κ M .

(2.7) (2.8)

bκ ) ⊂ [ε0 , +∞), we can draw a suitable curve Proof of Proposition 2.1. Since σ(H b κ ), with all branching points of Fβ (z −µ) Λ in the complex energy plane, around σ(H left outside. So using Cauchy formula we have Z b κ − µ) = 1 b κ − z)−1 . dz Fβ (z − µ)(H (2.9) Fβ (H 2iπ Λ Using Lebesgue convergence theorem and cyclicity of the trace, we get Z 1 bκ ] . b κ − z)−2 ∂κ H Tr dz[Fβ (z − µ)(H ∂κ Ω = − 2iπ Λ Integration by parts give Z 1 bκ . b κ − z)−1 ∂κ H Tr dz fβ (z − µ)(H ∂κ Ω = 2iπ Λ

(2.10)

(2.11)

This procedure can be easily iterated to prove that Ω is C ∞ -smooth in κ. Moreover in the semiclassical regime we can prove that the asymptotics for derivatives in κ of Ω can be computed using the following commutators formulas for derivatives of the resolvent. Starting from the well known identity b − z)−1 [H, b A]( b H b − z)−1 , b (H b − z)−1 ] = (H [A,

(2.12)

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we get b − z)−1 = ∂κ H( b H b − z)−2 − [∂κ H, b H]( b H b − z)−3 , ∂κ (H b H b − z)−4 b [H, b ∂κ H]]( − [H, b H b − z)−4 . b − z)−1 [H, b [H, b [H, b ∂κ H]]]( + (H

(2.13)

Each commutator gives one ~ and we can compute in the same way higher derivatives in κ. So using Cauchy formula we can compute asymptotics in ~ of derivatives in κ of Ω. Theorem 2.2. For any ε > 0 and κ0 > 0, Ω admits an asymptotic expansion in ~, uniform in κ for |κ| ≤ κ0 , and for β ≤ ~ε−2/3 . More explicitly, for any N ∈ N we have N X X (−1)k+1 j 3N ~ Ωjk + O(~N +1−n β 2 +k(n) ) (2.14) Ω = h−n k! 3j j=0 k≤

2

with

Z Ωjk =

(k)

R2n

dq dp djk (q, p)Fβ (Hκ − µ) ,

djk being a suitable linear combination of derivatives of Hκ with respect to q, p and k(n) a constant depending only on the dimension n (k(n) ≤ 2n + 1). In particular Z dq dp Fβ (Hκ (q, p) − µ) , (2.15) Ω00 = R2n

1 β 1 Ω22 − Ω23 = − 2 6 48π 2

Z dq dp R2n

κ2 kB(q)k2 −

jk

2 ∂jk V

cosh2 [ β2 (Hκ (q, p) − µ)]

where Bjk is the magnetic field X ∂aj ∂ak 2 Bjk , Bjk = − , kBk2 = ∂qk ∂qj j 0) comes from the fact that Fβ (x) = β k−1 F1 (βx), (k ≥ 1) so that ~j β k ≤ ~jε for k ≤ 3j 2 . Furthermore, the error term in (2.14) follows from (2.21). For the magnetization M , we start with formula (2.7) and we use the semiclassical expansion for the resolvant to get ! Z X 1 cκ Tr dz fβ (z − µ) ~j bbj (z)∂κ H (2.30) M= 2iπ Λ 0≤j≤N

−

~N +1 Tr 2iπ

Z

d c cκ − z)−1 R dz fβ (z − µ)(H N (z)∂κ Hκ .

(2.31)

Λ

Then using integration by parts we can prove that the first term in (2.30) is ~−n

X

X (−1)k+1 ~j ∂κ Ωj,k , k!

(2.32)

0≤j≤N k≤3j/2

and the second term in (2.30), using estimate (2.21), is O(~N +1−n β 3N/2+k(n) ) ,

(2.33)

uniformly in κ for |κ| ≤ κ0 . The same method can be used to prove semiclassical asymptotics for the susceptibility χ and also for higher order derivatives in κ of Ω.

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Now let us denote by Σκµ the energy surface at energy µ. Σκµ = {(q, p) ∈ R2n : Hκ (q, p) = µ}

(2.34)

and by dσµκ the Liouville measure on Σκµ dσµκ (q, p) =

dΣκµ |∇Hκ |

(2.35)

defined for any non-critical µ (i.e. ∇Hκ (q, p) 6= 0 on Σκµ ) (dΣκµ Lebesgue Euclidean measure). Then we have Corollary 2.3 (Landau Diamagnetism). Let χ be defined by (2.8), and µ be non-critical for H0 . Then for n = 2 and κ = 0 we have Z 1 χ=− kB(q)k2 dσµ0 (2.36) lim 24π 2 Σ0µ ~→0,β→∞,β≤~ε−2/3 which is nothing but Landau’s result of the diamagnetism for a 2-dimensional free electron gas. 3. A “Trace Formula” for the Magnetization In this section we shall consider a temperature regime β ∈ [ σ~0 , σ~1 ](0 < σ0 < σ1 ). In order to get information on the semi-classical limit of magnetization M , we shall use the Fourier inversion formula instead of Cauchy formula. First of all let us remark that fβ is not in the Schwartz space S(R) so it is more convenient to take its derivative, which is in S(R). We have explicitly fβ0 (x) =

−β . 4 cosh2 (βx/2)

So we have the following Fourier transform formula Z 1 πt/β 0 eitx . dt fβ (x) = − 2π R sinh(πt/β) So we can write b κ − µ) = h−1 fβ0 (H

Z

+∞

b

dt eit(Hκ −µ)/~

−∞

πt/σ , sinh πt/σ

(3.1)

(3.2)

where σ = β~ .

(3.3)

The parameter σ, which has the dimension of a time, will be important in what follows. It plays a role in the Kubo–Martin–Schwinger condition (see [8]). Because we cannot compute the semiclassical evolution for infinite time, we shall consider a “smeared out” magnetization defined as follows. Fix τ0 and τ : 0 < τ0 < τ ;

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given a C ∞ even function ρ such that ρ(t) ≡ 1 if |t| ≤ 1 , Z ρ(t) ≡ 0 if |t| ≥ 2

(3.4) ρ(t)dt = 1 .

and R

Let us define for τ > 0, ρτ (t) = ρ(t/τ ) and

( Mτ = Tr

(fσ ∗ ρ˜τ )

bκ − µ H ~

(3.5) !

) bκ ∂κ H

,

(3.6)

where g˜ denotes the inverse Fourier transform of g. Clearly Mτ → M when τ → ∞. However we shall not be able to let τ → ∞ in this paper (see discussion at the end of this section), and we will only obtain results for finite τ . Consider a cut-off function θ ∈ C0∞ (R) with supp θ ⊂ [−δ, δ], and θ ≡ 1 on δ δ [− 2 , 2 ]. It is a priori arbitrary but in the sequel, we will take δ so small that if µ is non-critical for Hκ , any λ ∈ ]µ − 3δ, µ + 3δ[ will remain so. Let us write the following decomposition Mτ = Mτ,θ + Mτ,1−θ where

! ( )  bκ − µ  H  bκ , b κ − µ)∂κ H  θ(H Mτ,θ = Tr (fσ ∗ ρ˜τ )   ~  ! ( )   bκ − µ H   b κ − µ)∂κ H bκ .  Mτ,1−θ = Tr (fσ ∗ ρ˜τ ) (1 − θ)(H  ~

(3.7)

(3.8)

We now prove b κ , and let σ1 > σ > σ0 > 0. Then Lemma 3.1. Let us assume (H.1–H.3) for H Mτ,1−θ has a complete asymptotic expansion in ~. Proof. (1 − θ)(x) is supported by the union of (−∞, −δ/2] and [δ/2, +∞), which ± . yields two contributions to Mτ,1−θ that we call Mτ,1−θ R∞ 0 Since fσ ∗ ρ˜τ = − x (fσ ∗ ρ˜τ )(y)dy is the primitive vanishing at +∞ * of a function in the Schwartz class S(R), we * have for every N , uniformly for σ > σ0 and ~∗ ∈ ]0, 1], ( ! ) bκ − µ H + b b κ ≤ CN ~N . (1 − θ) (Hκ − µ)∂κ H Tr (fσ ∗ ρ˜τ ) ~ − . We now consider Mτ,1−θ

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R +∞ 0 Since −∞ (fσ0 ∗ ρ˜τ )(y)dy = (ff σ · ρτ )(0) = 1, we clearly have, uniformly for any b κ − µ) ∩ (−∞, −δ/2], and any σ > σ0 : λ ∈ sp(H λ (3.9) fσ ∗ ρ˜τ = 1 + O(~N ) ~ b κ is semi-bounded from below, the contribution of 1 in (3.9) gives and, since H − b b κ } and obviously has a complete ~ expansion by the funcTr{(1 − θ) (Hκ − µ)∂ H b κ } for some θ1 ∈ C ∞ (R). b κ − µ)∂κ H tional calculus (in fact it is of the form Tr{θ1 (H 0

The next step is to decompose ρτ in order to isolate the neighborhood of t = 0 of the rest: ρτ = ρτ0 ρτ + (1 − ρτ0 )ρτ ≡ ρτ0 + ρ1,τ

(3.10)

since ρτ0 ρτ = ρτ0 if τ > 2τ0 . This yields ρ˜τ = ρ˜τ0 + ρ˜1,τ and correspondingly Mτ,θ = M0 + Mosc ,  ! ( ) bκ − µ  H  b κ − µ)∂κ H bκ ,  θ(H M0 = Tr (fσ ∗ ρ˜τ0 )    ~ ! ( )  bκ − µ  H   Mosc = Tr (fσ ∗ ρ˜1,τ ) b κ − µ)∂κ H bκ . θ(H   ~

(3.11)

(3.12)

Finally Mτ is decomposed into ¯ + Mosc , Mτ = M

(3.13)

where Mosc is given by (3.11), and ¯ = M0 + Mτ,1−θ . M

(3.14)

We will prove the following lemma. Lemma 3.2. Assume also µ is non-critical for Hκ . Then for τ0 small enough the classical flow induced by Hamiltonian Hκ on Σκµ has no periodic point of period ≤ τ20 and M0 admits an asymptotic expansion in ~, uniform for σ1 ≥ σ = β~ ≥ σ0 > 0. Proof. We have ( Z µ dλ Tr (fσ0 ∗ ρ˜τ0 ) M0 = +∞

Z

µ+3δ

=− µ

bκ − λ H ~

( dλ Tr (fσ0 ∗ ρ˜τ0 )

!

) b κ − µ)∂κ H bκ θ(H

bκ − λ H ~

!

) b κ − µ)∂κ H bκ θ(H

+ O(h∞ ) (3.15)

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using the fact that fσ0 ∗ ρ˜τ0 is in the space S(R) and the support property of the b κ , it will remain true for any cut-off function θ. Now, if µ is non-critical for H λ ∈ [µ, µ + 3δ] for small enough δ. We can rewrite the trace inside the integral in (3.15), using inverse Fourier transform, Z t 1 +∞ πt/σ b b b − µ)∂κ H} ρ (3.16) dt Tr{e−it(H−λ)/~ θ(H M0 = h −∞ sinh πt/σ τ0 b κ for simplicity). Now, using either WKB method (see (omitting the index κ of H for instance [25]), or the coherent state decomposition [6], a complete asymptotic expansion in ~ of M0 can be obtained, of the form M0 = ~−n+1 (C0 (λ) + ~C1 (λ) + · · · + hk Ck (λ) + · · ·) mod O(h∞ ) which can be further integrated with respect to λ on the interval [µ, µ+3δ], yielding the result. ¯ has a complete asymptotic Remark 3.3. Above lemmas therefore imply that M expansion in ~. It is, so to say, the analog of the (complete ~-expansion of) the “mean density of states” in the Gutzwiller trace formula. The other term Mosc will be the sum of highly oscillating terms, also in complete analogy with the oscillatory part of Gutzwiller trace formula. Before showing this now, let us remark that the ¯ has not yet been shown to reduce to the well known dominant ~ contribution to M “Landau diamagnetism”. This will be postponed to the end of this section. Proposition 3.4. Assume (H.1–H.3) together with (H.4) µ is non-critical for Hκ (|κ| < κ0 ). (H.5) On Σκµ , the set (Γµ )τ of classical periodic orbits denoted γ with period smaller than τ is such that the corresponding Poincaré maps Pγ do not have eigenvalue 1. Then for any σ1 > σ0 > 0 and for κ0 > 0 small enough, we have the following uniform asymptotics for β~ = σ ∈ [σ0 , σ1 ] and |κ| ≤ κ0 , ( ) X X (T ) im /2σ π ρ 1,τ γ γ k + ei(Sγ /~+νγ 2 ) d(k) +O(~∞ ) Mosc = γ ~ |det(1 − Pγ )|1/2 sinh(πTγ /σ) γ∈(Γµ )τ

k≥1

(3.17) where mγ =

R

Tγ∗

0

dt ∂κ Hκ (qt , pt ), Tγ∗ is the primitive period of orbit γ, Sγ (resp. νγ ) (k)

is the classical action (resp. Maslov index ) of orbit γ, and dγ are constants depending on orbit γ, on the function ρ1,τ , and on γ. Moreover the different orbits γ can be chosen such that they depend smoothly on the parameter κ and the asymptotic expansion holds uniformly in κ for |κ| small enough.

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Proof. Since ρ1,τ is supported away from zero, we can rewrite Mosc (defined in (3.12)) as the non-singular integral: Mosc

1 = h

Z

+∞

−∞

πt/σ dt cκ − µ)e−it(Hcκ −µ)/~ ∂κ H cκ } . ρ1,τ (t) Tr{θ(H t sinh πt/σ

(3.18)

It is not difficult to see that Mosc is a sum of terms like Z cκ −µ)/~ b −it(H Ag(t) , dt e L = Tr R

where g is a smooth function with compact support and A is a smooth, decreasing b is trace class. Now the method that we have fast enough observable such that A developed in [7] applies. Using a coherent states decomposition we have Z Z b z , U (t)ϕz ieitµ/~ , dt dzhAϕ L = (2π~)−n R

R2n

c

where U (t) = eit(Hκ /~) and ϕz denote the coherent state defined in [7]. Then using the semiclassical expansion for U (t)ϕz and the stationary phase theorem (see [7] for details) we get the result (3.17). Remark 3.5. At zero magnetic field (κ = 0) the term mγ is computed as follows. We have I I ∂κ H0 (qt , pt )dt = − a dq = −Φγ , mγ = γ

γ

where Φγ is the flux of the magnetic field through the closed curve γ. ¯ , and prove that the dominant contribution of the We now come back to M ~-expansion is indeed the well-known diamagnetic Landau term. Proposition 3.6. Assume (H.1–H.4). Then for any σ1 > 0 and for κ0 > 0, uniformly for β~ = σ ∈ ]0, σ1 [ we have, mod O(h∞ ), ¯ = −κ ~ M 24π 2

2−n

Z P0 µ

dσµ0 kB(q)k2 +

X

ck (µ, 0, σ, T )~k + O(~∞ ) .

(3.19)

k≥3−n

¯ results Proof. Whereas the existence of a complete asymptotic expansion for M immediately from Lemmas (3.1) and (3.2), the explicit calculus of the dominant contribution in (3.19) is not an immediate consequence. It will be done through the functional calculus. In the previous section we have shown that the coefficients of the asymptotic ~-expansion are regular functions of κ, and can be differentiated with respect to κ. Thus, according to (3.14), we shall obtain the dominant contribution to

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¯ from two different contributions Ω0 (λ)|λ=µ and Ωτ,1−θ (λ)|λ=µ by differentiating M with respect to κ:  ! " # bκ − λ  H  b κ − µ) ,  θ(H  Ω0 (λ) = Tr (Fβ ∗ ρ˜τ0 )   ~ (3.20) ! " #  bκ − λ  H   b κ − µ) . (1 − θ)(H  Ω0,1−θ (λ) = Tr (Fβ ∗ ρ˜τ )  ~ Let us consider the contribution of second derivatives in λ of Ω0 (λ): Ω000 (λ) := G(λ) = h−n c000 (λ) + h−n+2 c002 (λ) + O(h−n+3 )

(3.21)

and we shall identify c0 (λ) and c2 (λ) by the following trick (inspired from [25, Proposition V.8]): Take ϕ ∈ C0∞ (]µ − 3δ, µ + 3δ[) and integrate against (3.21); we get Z Z λ b − λ)]dλ , (3.22) Tr[ϕθ (H dλ ϕ(λ)G(λ) = h−1 ρ˜eff ~ where ρeff (t) = ρτ0 (t)

πt/σ , sinh(πt/σ)

ρ˜eff (λ) is its Fourier transform and ϕθ (E) := ϕ(E)θ(E + λ − µ) .

(3.23)

Then (3.22) follows from 1 G(λ) = h

Z

c cκ − µ)] . dt ρeff (t)Tr[e−it(Hκ −λ)/~ θ(H

Now (3.22) is rewritten as Z Z 1 cκ − λ~)] , dλ ρ˜eff (λ)Tr[ϕθ (H dλ ϕ(λ)G(λ) = 2π

(3.24)

which can be developed through Taylor’s formula (since integration variable λ is in a compact interval), as Z Z ∞ 1 X (−1)k ~k (k) c dλ λk ρ˜eff (λ)Tr[ϕθ (H dλ ϕ(λ)G(λ) = κ )] 2π k! k=0

=

∞ k k X k=0

i ~ (k) (k) c ρ (0)Tr[ϕθ (H κ )] . k! eff

(3.25)

Actually the term with k = 1 is absent since ρ0eff (0) = 0 (ρ is an even function, and so is ρeff ). We calculate the coefficients of ~−n and h2−n by the functional

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calculus, like in Sec. 2: Z Z −n ϕθ (H(q, p))dp dq dλ ϕ(λ)G(λ) = h ~2 − 12

Z

ϕ00θ (H(q, p))[κ2 kBk2 − 4V ]dq dp

~2 − ρ00eff (0) 2

Z

ϕ00θ [H(q, p)]dq

dp + O(h ) . 3

(3.26)

Clearly the first and third terms in (3.26) are independent on κ (through the change of variable p 7→ p − κa(q)), and we are left with lower order terms " # Z X h2−n 2 ∂j,k V dq dp ϕ00θ (H(q, p)) κ2 kBk2 − − 12 · 4π 2 1≤j,k≤n

−h2−n = 12 · 4π 2

Z

d2 dλ ϕθ (λ) 2 dλ

Z κ kBk − 2

Σκ λ

2

X

! 2 ∂j,k V

dσλκ (q, p) ,

1≤j,k≤n

(3.27) where we have used integration by parts: # "Z Z Z d2 00 κ G(q, p)ϕ ((Hκ (q, p))dq dp = dλ ϕ(λ) 2 P dσλ G(q, p) . κ dλ R2n λ Therefore since the above calculation holds for an arbitrary test function ϕ, we can identify the functions c0 and c2 (λ) appearing in (3.21), modulo κ-independent terms as:    c0 (λ) = 0 , Z (3.28) κ2  (λ) = −θ(λ) kBk2 dσλ (q, p) . c  2 P 2 12 · 4π λ We can do the same calculus for Ωτ,1−θ instead of Ω0 by replacing τ0 by τ and θ by 1 − θ. This yields a contribution to the magnetization which, added to that coming from c2 (λ) in (3.28) gives the dominant Landau term in (3.19). We shall extend now the above results to the magnetic susceptibility χ. The statement is the following Theorem 3.7. Let us assume (H.1–H.5) hold and σ = β~ ∈ [σ0 , σ1 ] where σ1 > σ0 > 0 are fixed. For χτ = χ ∗ ρτ , τ > 0, we have the decomposition ¯ + χosc χτ = χ with χ ¯=−

~2−n 24π 2

Z Pκ µ

dσµκ kB(q)k2 +

X k≥3−n

cχ,k (µ, κ, σ, T )~k + O(~∞ ) ,

(3.29)

(3.30)

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χosc =

X

( e

i(Sγ /~+νγ π 2)

X rγ m2γ /2σ ρ1,τ (Tγ ) k + d(k) χ,γ ~ 1/2 | det(1 − Pγ )| sinh(πTγ /σ)

1069

) + O(~∞ ) ,

k≥1

γ∈(Γµ )τ

(3.31) Tγ Tγ∗ , cχ,γ , dχ,γ

are where we have used the notations in Proposition 3.4 and rγ = smooth coefficients depending on the periodic orbit γ, on σ, and on function ρ. Proof. We use the same cut-off already introduced for the magnetization M . So we define in a natural way χ ¯ = ∂κ Mτ0 ,θ + ∂κ Mτ,1−θ , ¯. χosc = χτ − χ

(3.32) (3.33)

Compute first the term χτ0 ,θ = ∂κ Mτ0 ,θ . From the proof of Proposition 3.4 we get Z +∞ Z t 1 µ πt/σ ρ dλ dt χτ0 ,θ = − h 0 sinh(πt/σ) τ 0 −∞ c

cκ } . cκ − λ)∂κ H × ∂κ Tr{e− ~ (Hκ −λ) θ(H it

(3.34)

We compute derivative in the parameter κ with the following easy consequence of the Duhamel formula it c cκ } cκ − λ)∂κ H ∂κ Tr{e− ~ (Hκ −λ) θ(H

c

cκ ]} cκ − λ)∂κ H = Tr{e− ~ (Hκ −λ) ∂κ [θ(H Z t is c 1 cκ − it (H cκ −λ) Hκ H − is c c b ~ ~ ~ ds(e ∂κ Hκ e )e θ(H − λ)∂κ Hκ . + Tr i~ 0 (3.35) it

Then due to the support property of ρτ0 , the only stationary points corresponds to the period τ = 0 and the leading term in ~ is given by the first term. The term χτ,1−θ = ∂κ Mτ,1−θ is computed in the same way and the both terms combine to yield the asymptotic expansion of χ. ¯ For the term χosc we start from a formula like (3.34) replacing the time cut-off ρτ0 by the following ρ1,τ = ρτ (1 − ρτ0 ). Hence applying the methods of [7] we can compute with the stationary phase theorem the contributions of the periodic trajectories with period Tγ ∈ (Γµ )τ . Remark 3.8. In the so-called “mesoscopic regime” examined in this section (i.e. T = σk~B for some fixed σ having the dimension of time), and in the special ¯ and M1 to case of dimension 2, the dominant semi-classical contribution ML to M Mosc are of the same order (apart from highly oscillating factors). A comparison of the corresponding contributions χL and χ1 to the susceptibility is made in the physics literature, measuring a factor of 100 for χ1 /χL [24].

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Remark 3.9. Thus the magnetic response is a measurable quantity where the skeleton of the periodic orbits of the classical motion manifests itself clearly; we have investigated this effect rigorously and in great generality. Furthermore the oscillations in (3.17) are a generalization of the well-known de Haas-van Alphen oscillations of the magnetic response which are a result of the classical cyclotronic orbits demonstrated in dimensions 2 and 3, and which can be recovered from (3.17) in the limiting case where all classsical orbits are of cyclotronic nature (V = 0 or quadratic). Now, we want to comment about the fact that we have only been able to give semi-classical expansions for “smeared out magnetizations” Mτ instead of the true 0 one (τ = ∞). For non-zero temperature T 6= 0, the exponential decrease of ff β (t) when t → +∞ lets us expect that the Fourier inversion formula (3.1) combined with “trace formulas” will be enough to obtain Proposition 3.4 without the ρ˜τ which cuts off time at |t| ≤ τ . We expect that our method using semi-classical evolution estimates for coherent states [6] will allow to prove this for σ = β~ > σ0 > 0 with suitable assumptions on the classical flow. This is presently under study. However for T = 0, the cut-off ρ˜τ will be necessary to make the sum over periodic orbits finite and thus convergent, and we cannot expect to get rid of it. For the moment, using estimates proved in [6], we can see that it is sufficient to control the periods of the classical flow in the time interval [τ, c0 log( ~1 )]. In [6] we have proved that the semi-classical propagation of coherent states is valid in time interval [−c0 log( ~1 ), c0 log( ~1 )] for some c0 > 0. So we can write down the operator c

e− ~ (Hκ −λ) as a Fourier integral operator with a complex phase for |t| ≤ c0 log( ~1 ). So we have to compute two terms, (Hκ = H), ! ( ) Z 2t b − it (H−µ) b b ~ Rσ (t) , dt Tr e θ(H − µ)Aρ (3.36) F1 (~, σ) := c0 log( ~1 ) R ( " ) !# Z 2t it b b − µ)A b 1−ρ dt Tr e− ~ (H−µ) θ(H Rσ (t) , F2 (~, σ) := c0 log( ~1 ) R it

(3.37) b is some quantum observable and Rσ (t) = πt/σ . The term F1 is difficult where A sinh πt/σ to check and we have nothing to say about it here except that for each time, it is a Fourier integral with a known complex phase but it is difficult to control the stationary phase argument for large times. The term F2 is easily controlled because it contains the damping factor Rσ . More precisely we have Lemma 3.10. There exists C > 0 such that for every ~ ∈ ]0, 1] and σ > 0 we have easily 1 (3.38) ~(π/σ)c0 . F2 (~, σ) ≤ Cc0 log ~ So that F2 (~, σ) is negligible for

c0 σ

large enough.

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4. The Regime of Temperature ~1−ε ≤ T ≤ ~ 3 −ε 2

In Sec. 2 we have shown that the functional calculus applies to the thermodynamical functions in the grand-canonical ensemble and provided asymptotic expansions in 2 the semi-classical limit provided T ≥ ~ 3 −ε (some ε > 0). In Sec. 3 we have investigated a rather different temperature regime (called “mesoscopic”) where kB T = ~/σ (some σ > 0 but finite) where a splitting of the magnetic response into a “mean part” and an “oscillating part” appears in the semi-classical limit. In order to be complete, the “in-between regime” is now considered. Theorem 4.1. Assume (H.1–H.4). Then the magnetisation M = ∂κ Ω has for any 2 temperature T satisfying ~1−ε ≤ T ≤ ~ 3 −ε (some ε > 0) a complete asymptotic expansion in ~ obtained by taking the derivative in κ of the formal expansion in ~ for Ω given (2.14). Proof. As in Sec. 3 take τ0 > 0 so small that the classical flow induced by Hκ has no periodic point with non-zero period ∈ [−2τ0 , 2τ0 ], and take ρτ0 as in Sec. 3. Futhermore let θ ∈ C0∞ (R) be, as in Sec. 3 (θ ≡ 1 on [− 2δ , 2δ ], and ≡ 0 on R\ [−δ, δ]). We decompose M and M = Mθ + M1−θ

(4.1)

bκ} b κ − µ)θ(H bκ − µ)∂κ H Mθ = Tr{fβ (H

(4.2)

with

and similarly for M1−θ . Furthermore Z ∞ bκ} b κ − λ)θ(H bκ − µ)∂κ H dλ Tr{fβ0 (H Mθ = −

(4.3)

µ

Z =−

∞

dλ µ

1 h

Z

+∞

dt −∞

πt/σ c cκ } cκ − µ)∂κ H Tr{e−it(Hκ −λ) θ(H sinh πt/σ

= Mθ,ρ + Mθ,1−ρ ,

(4.4)

where we insert, inside the integral over t, the partition of unity 1 = ρτ0 (t) + (1 − ρτ0 )(t) , which yields, correspondingly a splitting of Mθ into the two contributions. Lemma 4.2. Assuming (H.1–H.3), M1−θ has a complete asymptotic expansion in ~. Proof. We can proceed as in the proof of Lemma 3.1, by splitting (1 − θ)(x) into the sum of two disjoint functions (1 − θ)± supported respectively in [δ, +∞) (for + sign) and (−∞, −δ]. Since fβ is the primitive vanishing at +∞ of a function in the Schwartz class of C ∞ functions of rapid decrease, we have cκ }| ≤ CN ~N (for any N ) cκ − µ)(1 − θ)+ (H cκ − µ)∂κ H |Tr{fβ (H

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and cκ }| ≤ CN ~N b − µ)(1 − θ)− (H b − µ)∂κ H |Tr{(1 − fβ )(H

(for any N ) .

cκ } cκ − µ)∂κ H Finally, we know, like in the proof of Lemma 3.1, that Tr{(1 − θ)− (H has a complete asymptotic ~ expansion by the functional calculus. Lemma 4.3. Assuming (H.1–H.3), one has, for ~1−ε ≤ T ≤ ~ 3 −ε 2

Mθ,1−ρ = O(e−c1 /~ ) ε

where c1 is a positive constant only depending on τ0 . Proof. Using (4.3), the support property of θ, and the exponential decrease of fβ0 , it is easy to show that Z µ+2δ √ cκ − λ)θ(H cκ − µ)∂ H cκ } + O(e−δ/ ~ ) . dλ Tr{fβ0 (H Mθ = − µ

Therefore

Z

µ+2δ

Mθ,1−ρ = − µ

dλ h−1

Z

+∞

−∞

dt(1 − ρτ0 (t))

πt/σ sinh πt/σ

c cκ } + O(e−δ/ cκ − µ)∂κ H × Tr{e−it(Hκ −λ)/~ θ(H

√

~

)

and since, in the considered temperature regime πt/σ −c0 |t|/~ε sinh(πt/σ) ≤ Ce we have, using the support property of 1 − ρτ0 : |Mθ,1−ρ | ≤ Ce−c1 /~ , ε

c1 being a positive constant depending on τ0 . Lemma 4.4. Assuming (H.1–H.4), then Mθ,ρ has a complete asymptotic expansion in ~. Proof. As is the previous section, we take δ so small that, if µ is non-critical for Hκ , then any λ ∈ [µ, µ + 2δ] is also non-critical for Hκ . Now, using the support property of ρτ0 , and either WKB method, or decomposition over coherent states, a complete asymptotic expansion can be obtained for Z +∞ πt/σ b b κ − µ)H bκ} Tr{e−it(Hκ −λ)/~ θ(H dt ρτ0 (t) h−1 sinh πtσ −∞ for any λ ∈ [µ, µ + 2δ]. Integrating with respect to λ in this interval yields the result.

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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

O. Agam, J. Phys. I (France) 4 (1994) 697–730. J. Avron, I. Herbst and B. Simon, 45(4), (1978) 847–883. M. V. Berry and M. Tabor, Proc. R. Soc. Land. A349 (1976) 101–134. J. D. Butler, “Semiclassical counting function with application to quantum current”, Orsay preprint, 1999. Y. Colin de Verdière, Composition Math. 27 (1973) 159–184. M. Combescure and D. Robert, Ann. Inst. Henri Poincaré 61 (1994) 443–483. M. Combescure, J. Ralston and D. Robert, Commun. Math. Phys. 202 (1999) 463–480. A. Connes, Non Commutative Geometry, InterEditions, Paris, 1990. M. Dauge and D. Robert, Lecture Notes in Mathematics 1256, Springer-Verlag, pp. 91–126, 1986. J. Duistermaat and V. Guillemin, Invent. Math. 29 (1975) 39–79. V. Fock, Z. Phys. 47 (1928) 446–50. S. Fournais, Commun. in P.D.E. 23 (1998) 601–628. J. P. Gazeau, private communication, 2000. M. Gutzwiller, J. Math. Phys. 12 (1971) 343-358; Chaos in Classical and Quantum Mechanics, Berlin-Heidelberg-New-York, Springer-Verlag, 1990. B. Helffer and D. Robert, Asymp. Anal. 3 (1990) 91–103. B. Helffer and J. Sj¨ ostrand, Ann. Inst. Henri Poincaré 52 (1990) 303–375. L. Landau, Z. Physik. 64 (1930) 629. J. H. van Leeuven, J. Phys. (Paris) 2 (1921) 361–64. L. P. Lévy, D. H. Reich, L. Pfeiffer and K. West, Physica B189 (1993) 204. D. Mailly, C. Chapelier and A. Benoit, Phys. Rev. Lett. 70 (1993) 2020. V. Petkov and G. Popov, Ann. Inst. Henri Poincaré 68 (1998) 17–83. R. E. Peierls, Z. Physik. 80 (1933) 763–791. S. Prado, M. de Aguiar, J. Keating and R. Egydio de Carvalho, J. Phys. A27 (1994) 6091–6106. K. Richter, D. Ullmo and R. Jalabert, Phys. Rep. 276 (1996) 1–83. D. Robert, Autour de l’approximation Semi-classique, Birkh¨ auser, Boston-BaselStuttgart, 1987. K. Tanaka, Ann. Phys. 268 (1998) 31–60.

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GROUND STATE OF THE MASSLESS NELSON MODEL WITHOUT INFRARED CUTOFF IN A NON-FOCK REPRESENTATION

ASAO ARAI∗ Department of Mathematics Hokkaido University, Sapporo 060-0810, Japan

Received 13 November 2000 We consider a model of quantum particles coupled to a massless quantum scalar field, called the massless Nelson model, in a non-Fock representation of the time-zero fields which satisfy the canonical commutation relations. We show that the model has a ground state for all values of the coupling constant even in the case where no infrared cutoff is made. The non-Fock representation used is inequivalent to the Fock one if no infrared cutoff is made. Keywords: Nelson’s model, massless quantum field, infrared divergence, ground state, canonical commutation relations, non-Fock representation.

1. Introduction We consider a system of N quantum particles (N ∈ N) moving on the d-dimensional Euclidean space Rd (d ∈ N) under the influence of an external potential V : RdN → R (Borel measurable) and coupled to a massless quantum scalar field. The model we discuss here is the so-called massless Nelson model [10]. The problem to which we address ourselves in this paper is that of existence of a ground state of the model without infrared cutoff. It has been shown that the massless Nelson model with infrared cutoff has a ground state [7, 14]. A natural question to be asked next is if the model without infrared cutoff has a ground state or not. We remark that, generally speaking, there is some subtlety on the existence of ground states of massless quantum field models, which is due to possible infrared divergences. Indeed, in a class of models which describe interactions of particles and massless quantum fields, it is proved or suggested that, if the time-zero fields are given by the Fock representation and no infrared cutoff is made in the interactions, then the models have no ground states, althogh it may depend on the strength of ∗ Supported

by the Grant-In-Aid No. 11440036 for Scientific Research from the Ministry of Education, Science, Sports and Culture, Japan. 1075

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the parameters contained in the Hamiltonians [2, 3, 6, 13]. On the other hand, the Pauli–Fierz model in non-relativistic quantum electrodynamics has a ground state in the Fock representation of time-zero fields even if no infrared cutoff is made [4, 8]. As for the massless Nelson model without infrared cutoff, L¨ orinczi, Minlos and Spohn [9] recently proved that, in the case d = 3 and N = 1, it has no ground state within the Fock space where the time-zero fields are given by the usual Fock representation of the canonical commutation relations (CCR) indexed by Sreal (Rd ), the space of real-valued, rapidly decreasing C ∞ -functions on Rd . The purpose of this paper is to point out that, if we consider the massless Nelson model in a non-Fock representation of the CCR for time-zero fields, then it has a ground state even in the case where no infrared cutoff is made. This new representation of the massless Nelson model is inequivalent to the Fock one if no infrared cutoff is made. This paper is organized as follows. In Sec. 2 we first briefly review the Nelson model in the standard form in which the time-zero fields are given by the Fock representation of the CCR indexed by Sreal (Rd ). We call it the “standard Nelson model”(SNM). Then we give an algebraic characterization of the Nelson model in a way independent of the choice of representations of the time-zero fields and derive rigorously field equations in a weak sense. To the author’s best knowledge, field equations of the Nelson model has not been rigorously established so far (only formal or classical ones are available). In Sec. 3 we define the Nelson model in a non-Fock representation of the timezero fields. In the last section, we prove that, under suitable hypotheses, the massless Nelson model introduced in Sec. 3 has a ground state even in the case where no infrared cutoff is made. 2. Algebraic Characterization and Field Equations of the Nelson Model In this section we first recall the SNM. Then we present an algebraic characterization of the Nelson model in an abstract manner. Finally we derive rigorously field equations of the abstract Nelson model. 2.1. The SNM The coordinate of the configuration space RdN of the N particles is denoted q = (q1 , . . . , qN ) ∈ RdN with qj := (qj1 , . . . , qjd ) ∈ Rd (j = 1, . . . , N ). Let Djµ (j = 1, . . . , N, µ = 1, . . . , d) be the generalized partial differential operator in the variable qjµ , so that the momentum operator of the jth particle is given by pj := (pj1 , . . . , pjd ) with pjµ := −iDjµ . The Hamiltonian of the particle system is then given by the Schr¨ odinger operator Hp :=

N X p2j +V , 2mj j=1

(2.1)

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Ground State of the Massless Nelson Model

1077

acting on L2 (RdN ), where mj > 0 is the mass of the jth particle and p2j = Pd 2 µ=1 pjµ = −∆qj is the d-dimensional generalized Laplacian in the variable qj . For a linear operator T , we denote its domain by D(T ). We assume the following: (H.1) The operator Hp is self-adjoint on its natural domain D(Hp ) TN j=1 D(∆qj ) ∩ D(V ) and bounded from below.

=

The Hilbert space for state vectors of the quantum scalar field of the SNM is given by Fb :=

n ∞ O M n=0

L2 (Rd ) ,

(2.2)

s

Nn the Boson Fock space over L2 (Rd ), where s L2 (Rd ) is the symmetric tensor prodN0 uct of L2 (Rd ) ( s L2 (Rd ) := C). Let ω be a nonnegative Borel measurable function on Rd such that 0 < ω(k) < ∞ for almost everywehre (a.e.) k ∈ Rd with respect to the d-dimensional Lebesgue measure and ω(k) = ω(−k)

a.e. k .

(2.3)

For a.e. k ∈ Rd , ω(k) physically means the energy of one free boson with momentum k. The function ω defines a nonnegative self-adjoint multiplication operator on L2 (Rd ) which is injective. Remark 2.1. A physical example for ω is given by p k ∈ Rd , ωm (k) := k 2 + m2 ,

(2.4)

with m ≥ 0 a constant denoting the mass of one boson. We are interested in the massless case m = 0: ω0 (k) = |k|. The free Hamiltonian of the quantum scalar field is defined by Hb := dΓ(ω) ,

(2.5)

the second quantization of ω [11, §X.7]. R We denote the annihilation operators on Fb by a(f ) = Rd a(k)f (k)∗ dk (f ∈ L2 (Rd )) [11, §X.7]. The symmetric operator 1 ΦS (f ) := √ (a(f )∗ + a(f )) , 2

(2.6)

called the Segal field operator, is essentially self-adjoint [11, §X.7]. We denote its closure by the same symbol. 0 (Rd ) be the set of real tempered distributions on Rd and, for each s ∈ R, Let Sreal 0 (Rd )|ω s fˆ ∈ L2 (Rd )} , Hωs := {f ∈ Sreal

(2.7)

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where fˆ(k) :=

1 (2π)d/2

Z

f (x)e−ikx dx

(2.8)

Rd

is the Fourier transform of f . −1/2 1/2 and g ∈ Hω , we can define For f ∈ Hω ! √ fˆ , πF (g) := ΦS (i ωˆ g) . φF (f ) := ΦS √ ω

(2.9)

Let (n) = 0 for all but finitely many n’s} , F0 := {ψ = {ψ (n) }∞ n=0 ∈ Fb |ψ −1/2

called the subspace of finite-particle vectors. Then, for all f ∈ Hω φF (f ) and πF (g) leave F0 invariant satisfying the CCR Z fˆ(k)∗ gˆ(k)dk , [φF (f ), πF (g)] = i

(2.10) 1/2

and g ∈ Hω ,

(2.11)

Rd

[φF (f ), φF (f 0 )] = 0,

[πF (g), πF (g 0 )] = 0, f, f 0 ∈ Hω−1/2 , g, g 0 ∈ Hω1/2 , −1/2 Hω , g

(2.12)

1/2 Hω }

∈ gives a representation of the on F0 . Namely, {φF (f ), πF (g)|f ∈ CCR. This representation is called the Fock representation of the CCR. In the SNM, φF (f ) and πF (g) are taken to be the time-zero fields. The Hilbert space for the SNM is H := L2 (RdN ) ⊗ Fb .

R⊕

(2.13)

As usual, we freely use the natural identification of H with RdN Fb dq, the constant fibre direct integral with base space (RdN , dq) and fibre Fb [12, §XIII.16]. R ⊕ In what follows, for notational simplicity, a decomposable operator AdN= ) RdN A(q)dq on H with fibre A(q) (which is an operator on Fb for each q ∈ R is denoted A(q) also. To describe an interaction between the particles and the quantum scalar field, 0 (Rd ), j = 1, . . . , N , which satisfy the following: we fix distributions ρj ∈ Sreal (H.2) For j = 1, . . . , N , ρj ∈ Hωs ,

s = −1/2, −1 .

(2.14)

The interaction of the particles and the quantum field in the SNM is given by the operator N X ρˆj ∗ (2.15) ΦS e−ikqj √ HIF := ω j=1

PN R acting in H. Formally we have HIF = j=1 Rd φF (qj − x)ρj (x)dx, where Z 1 p {a(k)∗ e−ikx + a(k)eikx }dk . φF (x) := 2(2π)d ω(k) Rd

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The total Hamiltonian of the SNM is defined by HSNM := H0 + λHIF ,

(2.16)

H0 := Hp + Hb

(2.17)

where

and λ ∈ R \ {0} denotes the coupling constant of the model. Proposition 2.1. Assume (H.1) and (H.2). Then HSNM is self-adjoint with D(HSNM ) = D(H0 ) and bounded from below. Moreover, HSNM is essentially selfadjoint on each core of H0 Proof. A simple application of [1, Proposition B.2] which employs the Kato– Rellich theorem (cf. [6]). In summary, the SNM is characterized in terms of the Hamiltonian HSNM and −1/2 1/2 the time-zero-fields {φF (f ), πF (g)|f ∈ Hω , g ∈ Hω }. 2.2. An abstract definition of the Nelson model As is seen above, the SNM uses the Fock representation of the CCR to give its time-zero fields and its Hamiltonain. We want to define the Nelson model in a way independent of the choice of representations of the CCR for time-zero fields. A natural manner for this is to use commutation relations fulfilled by observables, i.e., to find a possible Lie algebraic structure. We shall take commutation relations in a weak sense. We denote the inner product and the norm of a Hilbert space X by h · , · iX and k · kX respectively. But, if there is no danger of confusion, then we simply write them as h · , · i and k · k. Definition 2.1. Let A and B be densely defined linear operators on a Hilbert space X and D be a subspace of X such that D ⊂ D(A) ∩ D(B) ∩ D(A∗ ) ∩ D(B ∗ ). Then we define a quadratic form [A, B]D w by ∗ ∗ [A, B]D w (ψ, φ) := hA ψ, Bφi − hB ψ, Aφi,

ψ, φ ∈ D .

(“w” means “weak”.) Remark 2.2. If A and B are bounded on X with D(A) = D(B) = X , then, for all dense subspaces D of X , [A, B]D w (ψ, φ) = hψ, [A, B]φi for all ψ, φ ∈ D, where [A, B] := AB − BA (the usual commutator). Let F be a Hilbert space and K := L (R 2

dN

Z )⊗F =

⊕

RdN

F dq .

(2.18)

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Definition 2.2. Assume (H.1) and (H.2). Let s0 and s1 be real constants. A Nelson model is a set MNelson := {K, D, Lλ , {φ(f ), π(g)|f ∈ Hωs0 , g ∈ Hωs1 }} having the following properties: (N.1) D is a dense subspace of K. (N.2) Lλ is a symmetric operator on K and the operator HNM := Hp + Lλ

(2.19)

is self-adjoint (D(HNM ) := D(Hp ) ∩ D(Lλ )). We call HNM the Hamiltonian of the Nelson model MNelson . (N.3) The function ω is such that Sreal (Rd ) ⊂ Hωs0 ∩ Hωs0 +2 ∩ Hωs1 ∩ Hω1

(2.20)

and, for all j = 1, . . . , N, µ = 1, . . . , d, Dµ ρj ∈ Hωs0 ,

(2.21)

where Dµ (µ = 1, . . . , d) is the generalized partial differential operator in the µth variable xµ in x = (x1 , . . . , xd ) ∈ Rd . (N.4) {K, D, {φ(f ), π(f )|f ∈ Sreal (Rd )}} is a representation of the CCR indexed by Sreal (Rd ). Namely, for all f, g ∈ Sreal (Rd ), φ(f ) and π(f ) are self-adjoint operators on K with D ⊂ D(φ(f )π(g)) ∩ D(π(f )φ(g)) ∩ D(φ(f )φ(g)) ∩ D(π(f )π(g)) satisfying the CCR Z f (x)g(x)dx , (2.22) [φ(f ), π(g)] = i Rd

[φ(f ), φ(g)] = 0,

[π(f ), π(g)] = 0 ,

(2.23)

on D and the linearity: φ(af + bg) = aφ(f ) + bφ(g),

π(af + bg) = aπ(f ) + bπ(g),

a, b ∈ R ,

on D. (N.5) Let X = qjµ , pjµ , Y = φ(f ), π(f ), j = 1, . . . , N, µ = 1, . . . , d, f ∈ Sreal (Rd ). There exists a dense subspace E of K such that E ⊂ D(X) ∩ D(Y ) ∩ D(Lλ ) ∩ D(Hp )

(2.24)

and the following relations hold in the sense of quadratic form on E: [Lλ , qjµ ]Ew = 0,

[Lλ , pjµ ]Ew = iλφ(Dµ ρj (qj − ·)) ,

[Lλ , φ(f )]Ew = −iπ(f ) , [Lλ , π(f )]Ew = iφ(ω(−i∇)2 f ) + iλ

N X (ρj ∗ f )(qj ) , j=1

[X, Y

]Ew

= 0,

[Hp , Y

]Ew

= 0,

where ∇ = R(D1 , . . . , Dd ) and ρj ∗ f is the convolution of ρj and f : (ρj ∗ f )(x) := Rd ρj (x − y)f (y)dy.

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Remark 2.3. (i) We have (ω(−i∇)2 f )(x) =

1 (2π)d/2

1081

Z ω(k)2 fˆ(k)eikx dk, Rd

f ∈ S(Rd ) .

Hence (N.3) implies that ω(−i∇)2 f ∈ Hωs0 , so that φ(ω(−i∇)2 f ) is defined. (ii) It follows from (H.2) and (N.3) that, for all f ∈ Sreal (Rd ), ρj ∗f is a bounded continuous function on Rd , so that ρj ∗f (qj ) is a bounded self-adjoint multiplication operator with D(ρj ∗ f (qj )) = K. Let us show that the SNM is indeed a Nelson model in the above sense. We introduce Fω,fin := L{Ω0 , a(f1 )∗ · · · a(fn )∗ Ω0 |n ∈ N, fj ∈ D(ω), j = 1, . . . , n} ,

(2.25)

where Ω0 := {1, 0, 0, . . .} ∈ Fb is the Fock vacuum and L{· · ·} means the subspace algebraically spanned by all the vectors in the set {· · ·}, and O Fω,fin , (2.26) D0 := C0∞ (RdN ) where

alg

N alg

denotes algebraic tensor product.

Proposition 2.2. Assume (H.1) and (H.2). Suppose that Z |V (q)|2 dq < ∞, ∀R > 0 ,

(2.27)

|q|≤R

3/2

Sreal (Rd ) ⊂ Hω

−1/2

∩ Hω

,

(2.28)

and, for j = 1, . . . , N, µ = 1, . . . , d, Dµ ρj ∈ Hω−1/2 .

(2.29)

:= Hb + λHIF . LSNM λ

(2.30)

Let

Then , {φF (f ), πF (g)|f ∈ Hω−1/2 , g ∈ Hω1/2 }} MSNM := {H, D0 , LSNM λ is a Nelson model. Proof. (N.1) is obvious. (N.2) follows from Proposition 2.1. (N.3) with s0 = −1/2, s1 = 1/2 follows from the present assumption. (N.4) follows from (2.11) and (2.12). For (N.5), we take E = D0 . By condition (2.27), C0∞ (RdN ) ⊂ D(V ) so that C0∞ (RdN ) ⊂ D(Hp ). Hence (2.24) follows. The commutation relations in (N.5) follow from direct computations using (2.11), (2.12) and [Hb , a(f )] = −a(ωf ), on Fω,fin (f, ωf ∈ L2 (Rd )).

[Hb , a(f )∗ ] = a(ωf ) ,

(2.31)

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2.3. Field equations We derive field equations for the Nelson model MNelson in a weak sense. Definition 2.3. Let X be a Hilbert space and A(t) (t ∈ R) be a linear operator on X . Suppose that there exists a dense subspace D of X such that D ⊂ D(A(t)) for all t ∈ R. We say that A(t) is weakly differentiable on D if, for all ψ, φ ∈ D, the function hψ, A(t)φi is differentiable in t ∈ R. In that case we define a quadratic form w-dA(t)/dt|D on D by d dA(t) ψ, φ ∈ D . (ψ, φ) := hψ, A(t)φi, wdt D dt Definition 2.4. Let X be a Hilbert space and H be a self-adjoint operator on H. Let A be a delnsely defined linear operator on X . We say that A is in the set AH if it satisfies the following (i) and (ii): (i) There exists a dense subspace DA ⊂ D(H) such that, for all s ∈ R, eisH DA ⊂ D(A) ∩ D(A∗ ). (ii) For all ψ ∈ DA , the X -valued functions: s → AeisH ψ and s → A∗ eisH ψ are strongly continuous on R. For A ∈ AH , we set DA,H := L{eisH ψ|ψ ∈ DA , s ∈ R} .

(2.32)

Lemma 2.1. Let X be a Hilbert space and H be a self-adjoint operator on X . Let A be a densely defined linear operator on X and set A(t) := eitH Ae−itH ,

t ∈ R,

Assume that A ∈ AH . Then A(t) is weakly differentiable on DA,H and dA(t) A,H = i[H, A(t)]D . ww dt DA,H

(2.33)

Proof. Let ψ, φ ∈ DA,H . Then ψ, φ ∈ D(A(t)) ∩ D(A(t)∗ ) for all t ∈ R. For h ∈ R \ {0}, e−ihH − 1 A(t + h) − A(t) φ = e−itH ψ, Ae−i(t+h)H φ ψ, h h e−ihH − 1 ∗ φ . + A(t) ψ, h Since ψ ∈ D(H), it follows that e−ihH − 1 ψ = −iHψ . h→0 h lim

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It follows from Definition 2.4(ii) that AeisH φ is strongly continuous in s. Hence lim Ae−i(t+h)H φ = Ae−itH φ .

h→0

Hence

A(t + h) − A(t) φ = i e−itH Hψ, Ae−itH φ − ihA(t)∗ ψ, Hφi . lim ψ, h→0 h

Thus the desired result follows. The canonical Heisenberg operators of the Neson model MNelson (Definition 2.2) are defined as follows: qjµ (t) := eitHNM qjµ e−itHNM , φ(t, f ) := eitHNM φ(f )e−itHNM ,

pjµ (t) := eitHNM pjµ e−itHNM , π(t, f ) := eitHNM π(t, f )e−itHNM ,

(2.34) (2.35)

j = 1, . . . , N, µ = 1, . . . , d, f ∈ Sreal (Rd ), t ∈ R . For a self-adjoint operator T , Q(T ) denotes the form domain of the quadratic form associated with T . Theorem 2.1. Consider the Nelson model MNelson (Definition 2.2). Assume (H.1), (H.2) and the following (A.1)–(A.5) : (A.1) (2.27) holds and Hp is essentially self-adjoint on C0∞ (RdN ). (A.2) For all j = 1, . . . , N, µ = 1, . . . , d, the distributional partial derivative Djµ V in the variable qjµ is a Borel measurable function on RdN which is a.e. finite with respect to the Lebesgue measure. Moreover, C0∞ (RdN ) is a form core of the self-adjoint multiplication operator Djµ V and D(Hp ) ⊂ TN Td j=1 µ=1 Q(Djµ V ). (A.3) In addition to (2.24),     d d N N \ \  \ \ [D(pjµ qlν ) ∩ D(qlν pjµ )] ∩  Q(Djµ V ) . E ⊂ D(Hp ) ∩   j,l=1 µ,ν=1

j=1 µ=1

(A.4) For all s ∈ R, eisHNM leaves E invariant: eisHNM E ⊂ E. (A.5) For all j = 1, . . . , N, µ = 1, . . . , d and ψ ∈ E, the K-valued function qjµ eisHNM ψ is strongly continuous in s ∈ R. (A.6) For all f ∈ Sreal (Rd ), the K-valued functions φ(f )eisHNM ψ and π(f )eisHNM ψ are strongly continuous in s ∈ R. Then the Heisenberg operators qjµ (t), pjµ (t), φ(t, f ) and π(t, f ) (f ∈ Sreal (Rd )) are weakly differentiable on E and pjµ (t) dqjµ (t) = , (2.36) wdt E mj

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dpjµ (t) w= −Djµ V (q(t)) − λeitHNM φ(Dµ ρj (qj − ·))e−itHNM , dt

(2.37)

E

dφ(t, f ) = π(t, f ), wdt E w-

(2.38)

N X dπ(t, f ) 2 = −φ(t, ω(−i∇) f ) − λ (ρj ∗ f )(qj (t)) , dt E j=1

(2.39)

where qj (t) := (qj1 (t), . . . , qjd (t)), q(t) := (q1 (t), . . . , qN (t)). Proof. By (A.3) and the CCR [pjµ , qlν ] = −iδjl δµν

on D(pjµ qlν ) ∩ D(qlν pjµ ) ,

we have [Hp , qjµ ]Ew = −i

pjµ , mj

which, together with (N.5), implies that [HNM , qjµ ]Ew = −i

pjµ . mj

(2.40)

By this fact, (A.4) and (A.5), we can apply Lemma 2.1 to obtain (2.36). Let h0 := −

N X ∆qj . 2m j j=1

(2.41)

For all ψ ∈ D(h0 ), we have kpjµ ψk2 ≤ 2mj hψ, h0 ψi ≤ 2mj kψkkh0ψk ≤ εkh0 ψk2 +

m2j kψk2 , ε

(2.42)

where ε > 0 is arbitrary. By the self-adjointness of Hp and the closed graph theorem, there exist constants a ≥ 0, b ≥ 0 such that, for all ψ ∈ D(Hp ) = D(h0 ) ∩ D(V ), kh0 ψk + kV ψk ≤ akHp ψk + bkψk .

(2.43)

√ √ mj kpjµ ψk ≤ a εkHp ψk + b ε + √ kψk . ε

(2.44)

Hence, by (2.42),

Let ψ, φ ∈ C0∞ (RdN ). Then we have hHp ψ, pjµ φi − hpjµ ψ, Hp φi = hψ, i(Djµ V )φi .

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By (A.1), (A.2) and (2.44), we can extend this equality to all ψ, φ ∈ D(Hp ). Hence, in particular, we have [Hp , pjµ ]Ew = iDjµ V

(2.45)

in the sense of quadratic form on E. Hence, by (N.5), we have [HNM , pjµ ]Ew = iDjµ V + iλφ(Dµ ρj (qj − ·))

(2.46)

in the sense of quadratic form on E. By this fact, (A.4) and (A.6), we can apply Lemma 2.1 to obtain (2.37). Similarly (more easily) we can prove (2.38) and (2.39). Now we show that Theorem 2.1 can be applied to the SNM. Lemma 2.2. Let S be a self-adjoint operator on a Hilbert space X and A be a linear operator on X such that, for some α > 0, A is |S|α -bounded, i.e., D(|S|α ) ⊂ D(A) and kAψk ≤ ak |S|α ψk + bkψk,

ψ ∈ D(|S|α ) ,

where a, b ≥ 0 are constants. Then, for all ψ ∈ D(|S|α ), the function: t → AeitS ψ is strongly continuous. Proof. We have for all h ∈ R kAei(t+h)S ψ − AeitS ψk ≤ ak |S|α (eihS − 1)eitS ψk + bk(eihS − 1)ψk = ak(eihS − 1)|S|α ψk + bk(eihS − 1)ψk , which implies the strong continuity of the function: t → AeitS ψ. For a self-adjoint operator T on a Hilbert space which is bounded from below, we set E0 (T ) := inf σ(T ) ,

(2.47)

the ground state energy of T , where σ(T ) denotes the spectrum of T . Under hypotheses (H.1) and (H.2), we set ˆ SNM := HSNM − E0 (HSNM ) , H

(2.48)

which is nonnegative. Lemma 2.3. Assume (H.1) and (H.2). Suppose that Q(V ) ⊂ D(|q|) and there exist constants c1 , c2 ≥ 0 such that for all u ∈ Q(V ) k |q|uk ≤ c1 k |V |1/2 uk + c2 kuk .

(2.49)

Then, for all j = 1, . . . , N, µ = 1, . . . , d and ψ ∈ Q(HSNM ), the H-valued function qjµ eisHSNM ψ is strongly continuous in s ∈ R.

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Proof. By Proposition 2.1 and the closed graph theorem, there exist constants a1 , b1 ≥ 0 such that, for all ψ ∈ D(HSNM ) = D(H0 ), kHp ψk + kHb ψk ≤ a1 kHSNM ψk + b1 kψk .

(2.50)

By this inequality and (2.43), we have kV ψk ≤ aa1 kHSNM ψk + (ab1 + b)kψk , which, via an application [11, Theorem X.18], implies that Q(HSNM ) ⊂ Q(V ) and ˆ 1/2 φk + b2 kφk, k |V |1/2 φk ≤ a2 kH SNM

φ ∈ Q(HSNM ) ,

with a2 , b2 ≥ 0 being constants. Hence, by (2.49), we obtain ˆ 1/2 ψk + (c1 b2 + c2 )kψk, kqjµ ψk ≤ c1 a2 kH SNM

ψ ∈ Q(HSNM ) .

Hence we can apply Lemma 2.2 to obtain the desired result. Lemma 2.4. Assume (H.1) and (H.2). Suppose that Sreal (Rd ) ⊂ Hω−1 ∩ Hω1/2 .

(2.51)

Let ψ ∈ Q(HSNM ). Then, for all f ∈ Sreal (Rd ), the H-valued functions φF (f )eisHSNM ψ and πF (f )eisHSNM ψ are strongly continuous in s ∈ R. Proof. Using the well-known estimates ka(f )ψk ≤ kω −1/2 f kL2 (Rd ) kHb ψk , 1/2

(2.52)

ka(f )∗ ψk ≤ kω −1/2 f kL2 (Rd ) kHb ψk + kf kL2(Rd ) , 1/2

ψ ∈ D(Hb ), f, ω −1/2 f ∈ L2 (Rd ) , 1/2

we have kφF (f )ψk ≤

(2.53)

√ 1/2 2kω −1 fˆkL2 (Rd ) kHb ψk

1 + √ kω −1/2 fˆkL2 (Rd ) kψk, f ∈ Hω−1 ∩ Hω−1/2 , 2 √ 1/2 gkL2 (Rd ) kHb ψk kπF (g)ψk ≤ 2kˆ 1 + √ kω 1/2 gˆkL2 (Rd ) kψk, g ∈ L2 (Rd ) ∩ Hω1/2 . 2 By (2.50) and an application [11, Theorem X.18], we have 1/2 ˆ 1/2 ψk + b3 kψk, kHb ψk ≤ a3 kH SNM

ψ ∈ Q(HSNM ) ,

(2.54)

(2.55)

(2.56)

with a3 , b3 ≥ 0 being constants. It follows that, for all f ∈ Sreal (Rd ), φF (f ) and ˆ 1/2 -bounded. Hence an application of Lemma 2.2 yields the desired πF (f ) are H SNM result. We shall need the following condition too.

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(H.3) The function ω is such that Sreal (Rd ) ⊂ Hω−1 ∩ Hω3/2 .

(2.57)

Theorem 2.2. Assume (H.1)–(H.3), (A.1), (A.2), (2.29) and (2.49). Suppose that   d d N N \ \ \ \ [D(pjµ qlν ) ∩ D(qlν pjµ )] ∩  Q(Djµ V ) . (2.58) D(Hp ) ⊂ j,l=1 µ,ν=1

j=1 µ=1

Then the conclusion of Theorem 2.1 holds with HNM = HSNM , φ(f ) = φF (f ), π(f ) = πF (f ) and E = D(HSNM ). Proof. We need only check that the assumption of Theorem 2.1 is satisfied with HNM = HSNM , φ(f ) = φF (f ), π(f ) = πF (f ) and E = D(HSNM ). (A.3) and (A.4) follow from (2.58). (A.5) and (A.6) follow from Lemmas 2.3 and 2.4 respectively. 3. A Nelson Model in a Non-Fock Representation In this section we define a Nelson model whose time-zero fields are given by a nonFock representation of the CCR and show that it has a ground state even if no infrared cutoff is made. Under hypotheses (H.2) and (H.3), we can define for each f ∈ Sreal (Rd ) N Z X ρˆj (k)fˆ(k) e dk . (3.1) φ(f ) := φF (f ) + λ ω(k)2 d j=1 R We set π e(f ) := πF (f ) .

(3.2)

Proposition 3.1. Assume (H.2) and (H.3). Then: e ), π e(f )|f ∈ Sreal (Rd )}} is a representation of the CCR indexed by (i) {Fb , F0 , {φ(f d Sreal (R ). e ), π (ii) The representation {φ(f e(f )|f ∈ Sreal (Rd )}} is unitarily equivalent to the PN Fock representation {φF (f ), πF (f )|f ∈ Sreal (Rd )}} if and only if j=1 ρj ∈ −3/2

Hω

.

Proof. Part (i) is obvious. Part (ii) follows from an argument similar to that in [5, Chap. 1, §1-e]. We now consider a Nelson model whose time-zero fields are given by e ), π {φ(f e(f )|f ∈ Sreal (Rd )}. We introduce an L2 (Rd )-valued function G on RdN (G : RdN → L2 (Rd ), G(q) ∈ L2 (Rd ), q ∈ RdN ) by G(q)(k) :=

N ∗ X ρˆ (k) pj (e−iqj k − 1) ω(k) j=1

(3.3)

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and a function W (q) :=

N Z X j,l=1

Rd

ρˆj (k)ˆ ρl (k)∗ −iql k e dk, ω(k)2

q ∈ RdN .

(3.4)

By reality of ρj and (2.3), W is real-valued. We define a Hamiltonian by H := H0 + λΦS (G(q)) − λ2 W + c0 λ2 ,

(3.5)

where

PN

2

1

j=1 ρˆj c0 :=

2 2 ω

.

(3.6)

L (Rd )

Lemma 3.1. Assume (H.1) and (H.2). Then H is self-adjoint with D(H) = D(H0 ) and bounded from below. Moreover, H is essentially self-adjoint on each core of H0 . Proof. In the same way as in the case of HSNM , we can show that H 0 := H0 + λΦS (G(q)) is self-adjoint with D(H 0 ) = D(H0 ), bounded from below, and essentially self-adjoint on each core of H0 . Since −λ2 W + c0 λ2 is a bounded selfadjoint operator, the Kato-Rellich theorem yields the desired result. Let 2 2 LNF λ := Hb + λΦS (G(q)) − λ W + c0 λ ,

(3.7)

H = Hp + LNF λ .

(3.8)

so that

Proposition 3.2. Assume (H.1)–(H.3). Then e e(g)|f ∈ Hω−1 , g ∈ Hω1/2 }} MNF := {H, D0 , LNF λ , {φ(f ), π

(3.9)

is a Nelson model. Proof. Similar to the proof of Proposition 2.2. e ), π Proposition 3.3. Assume (H.1)–(H.3). Then {H, {φ(f e(f )|f ∈ Sreal (Rd )}} is d unitarily equivalent to {HSNM , {φF (f ), πF (f )|f ∈ Sreal (R )}} if and only if N X j=1

ρj ∈ Hω−3/2 .

(3.10)

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Proof. Suppose that there exists a unitary operator Y on H such that Y HY −1 = HSNM ,

(3.11)

e )Y −1 = φF (f ), Y π e (f )Y −1 = πF (f ), Y φ(f

f ∈ Sreal (Rd ) .

(3.12)

Then, by Proposition 3.1, (3.10) holds. Conversely, suppose that (3.10) holds. Then we can define a unitary operator U by !! PN ˆj ∗ j=1 ρ . (3.13) U := exp −iλΦS i ω 3/2 It is easy to check that (3.11) and (3.12) hold with Y = U . Proposition 3.3 shows that, under (H.1)–(H.3) and the condition that −3/2 Hω ,

PN j=1

ρj ∈

the Nelson model MNF is equivalent to the SNM. In this case, under suitable additional conditions, HSNM has a ground state [7] and so does H. Thus we are interested in the case N X

ρj 6∈ Hω−3/2 .

(3.14)

j=1

This condition is called an infrared singularity condition. In this paper, we say that the Nelson model MNelson has no infrared cutoff if (3.14) holds. Under condition (3.14), the Nelson model MNF is not unitarily equivalent to the SNM. Note that PN √ (3.14) and the natural condition j=1 ρj / ω ∈ L2 (Rd ) imply ess.inf k∈Rd ω(k) = 0

(3.15)

(“ess.inf” means essential infimum), i.e., the quantum scalar field under consideration is “massless”. e ), π Remark 3.1. In the new representation Π := {φ(f e(f )|f ∈ Sreal (Rd )} of timezero fields, also other observables than the Hamiltonian may take different forms from those in the Fock representation. Observables in the Fock representation are made of the position operator q, the momentum operator p, the annihilation and the creation operators a(·) and a(·)∗ . Corresponding to this structure, observables in the new representation Π are made of q, p and the operators * PN ∗ + ρˆj f λ √ , j=1 e a(f ) := a(f ) + √ ω ω 2 2 d L (R )

√ defined for all f satisfying f / ω ∈ L2 (Rd ) and their conjugates e a(f )∗ . Note that, d under (H.2) and (H.3), {e a(f )|f ∈ S(R )} is unitarily equivalent to {a(f )|f ∈ S(Rd )} if and only if (3.10) holds. It follows that an observable of the form O(q, p, a, a∗ ) in the Fock representation takes, in the non-Fock representation Π, the form O(q, p, e a, e a∗ ) (with possible restrictions due to infrared divergences if (3.14) holds). In particular the observables of particles do not change.

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As an example, consider the local number operator which is defined by NΛ := R dΓ(χΛ ) = Λ a(k)∗ a(k)dk in the Fock representation, where Λ ⊂ Rd is a Borel set and χΛ is the characteristic function of Λ. In the non-Fock representation Π with (3.14), the operator corresponding to NΛ can be defined for all sets Λ such that Λ ⊂ {k ∈ Rd |ω(k) ≥ δ} with some δ > 0. For such sets Λ, the local number operator in the representation Π is given by   Z PN N X | j=1 ρˆj (k)|2 χ λ2 Λ ∗ e  ρ ˆ dk . (3.16) + NΛ := NΛ + λΦS j 2 Λ ω(k)3 ω 3/2 j=1 eΛ = eΛ is N A formal expression for N

R Λ

e a(k)∗ e a(k)dk, where

λ X ρˆj (k)∗ . e a(k) := a(k) + √ 2 j=1 ω(k)3/2 N

If Λ ⊃ {k ∈ Rd |ω(k) ≤ δ} with some δ > 0 and (3.14) holds, then the second term of the right hand side of (3.16) is not well defined, while the third term of the right eΛ is not well defined. hand side of (3.16) is divergent. Hence, for such sets Λ, N 4. Existence of a Ground State of the Nelson Model MNF Without Infrared Cutoff Let T be a self-adjoint operator on a Hilbert space and bounded from below. We say that T has a ground state if there exists a non-zero vector ψ ∈ D(T ) such that T ψ = E0 (T )ψ. In this case ψ is called a ground state of T . We show below that, under suitable conditions, the Hamiltonian H of the Nelson model MNF has a ground state even in the case where it has no infrared cutoff (i.e., (3.14) holds). We formulate some additional conditions: (H.4) The function ω is continuous on Rd satisfying (3.15) and the following conditions: Dµ ω ∈ L∞ (Rd ),

µ = 1, . . . , d,

lim ω(k) = ∞ ,

|k|→∞

(H.5) For j = 1, . . . , N , Z

(H.6) For all R > 0, that

Rd

R |q|≤R

|k|2 |ˆ ρj (k)|2 dk < ∞ . ω(k)3

(4.1)

|V (q)|dq < ∞ and there exist constants c1 , c2 ≥ 0 such

|q|2 ≤ c1 V (q) + c2 ,

a.e. q ∈ RdN .

(4.2)

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An infrared-cutoff Hamiltonian of the SNM is defined by N X χω≥σ ρˆj ∗ ΦS e−ikqj √ , HSNM,σ := H0 + λ ω j=1

1091

(4.3)

where σ > 0 is an infrared cutoff parameter and χS is a characteristic function of the set S. Under (H.2), we can define for all σ > 0 a unitary operator !! PN ˆj j=1 χω≥σ ρ . (4.4) Uσ := exp −iλΦS i ω 3/2 Theorem 4.1. Assume (H.1), (H.2) and (H.4)–(H.6). Then H has a ground state ψ0 which has the following property: there exists a sequence {φσn }∞ n=1 of unit vectors in D(H0 ) such that σn > 0, n ∈ N, limn→∞ σn = 0, each φσn is a ground state of HSNM,σn and φσn = ψ0 , w- lim Uσ−1 n n→∞

(4.5)

where “w-lim” means weak limit. Remark 4.1. Consider the physical case ω = ω0 ((2.4) with m = 0). Hence (H.4) holds. Assume (H.2) with ω = ω0 and that N X ρˆj (·) 6∈ L2 (Rd ) . 3/2 |k| j=1

Then (3.14) and (H.5) hold with ω = ω0 . Hence Theorem 4.1 holds in the physical case without infrared cutoff. Remark 4.2. Assume (3.14). Then we have w- lim Uσ = 0 . σ→0

This is an expression of infrared divergence. Hence the relation φσn = ψ0 in Theorem 4.1 suggests that w-limn→∞ φσn = 0. limn→∞ Uσ−1 n

(4.6) w-

The idea of proof of Theorem 4.1 is to apply [7, Theorem 1]. Let K := Hp − λ2 W + c0 λ2 . Lemma 4.1. Assume (H.1) and (H.2). Then K is self-adjoint and bounded from below. Proof. This follows from the boundedness of λ2 W and a simple application of the Kato–Rellich theorem. Let ˆ := K − E0 (K) . K

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ˆ + i)−1 is compact. Lemma 4.2. Assume (H.1), (H.2) and (H.6). Then (K Proof. By (H.6) and the boundedness of W , we have lim {V (q) − λ2 W (q) + c0 λ2 } = ∞ .

|q|→∞

Hence, applying [12, p. 249, Theorem XIII.67], we obtain the desired result. We denote by B(L2 (RdN )) the set of bounded linear operators on L2 (RdN ). For each k ∈ Rd , we define a linear operator on L2 (RdN ) as follows: ( G(·)(k) , ω(k) > 0 , v(k) := (4.7) 0, ω(k) = 0 . Remark 4.3. By the original assumption for ω, the Lebesgue measure of the set {k ∈ Rd |ω(k) = 0} is equal to zero. Hence, v(k) = G(·)(k) a.e. k ∈ Rd . Lemma 4.3. Assume (H.2). Then, for a.e. k ∈ Rd , v(k) ∈ B(L2 (RdN )) with kv(k)k ≤ 2

N X |ˆ ρj (k)| p . ω(k) j=1

Proof. This easily follows from (3.3). The following lemma immediately follows from Lemma 4.3. Lemma 4.4. Assume (H.1) and (H.2). Then, for a.e. k ∈ Rd , ˆ + 1)−1/2 ∈ B(L2 (RdN ), v(k)(K

ˆ + 1)−1/2 v(k) ∈ B(L2 (RdN )) . (K

Lemma 4.5. Assume (H.1) and (H.2). Then, for all u1 , u2 ∈ L2 (RdN ), the functions ˆ + 1)−1/2 u1 ), (u2 , (K ˆ + 1)−1/2 v(·)u1 ) , (u2 , v(·)(K on Rd are measurable. Moreover, Z 1 ˆ + R)−1/2 k2 + k(K ˆ + R)−1/2 v(k)k2 )dk < ∞ (kv(k)(K C(R) := Rd ω(k) and lim C(R) = 0 .

R→∞

Proof. We have ˆ + 1)−1/2 u1 ) = (u2 , v(k)(K

N X ρˆj (k)∗ p gj (k) ω(k) j=1

(4.8)

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with

Z gj (k) =

Rd

1093

ˆ + 1)−1/2 u1 )(q)dq . u2 (q)∗ (e−iqj k − 1)((K

ˆ + 1)−1/2 u1 ) is It is easy to see that gj is continuous on Rd . Hence (u2 , v(·)(K −1/2 ˆ + 1) v(·)u1 ) is measurable. measurable. Similarly we can show that (u2 , (K We have Z N Z 8N X 1 |ˆ ρj (k)|2 2 2 kv(k)k dk ≤ dk . C(R) ≤ R Rd ω(k) R j=1 Rd ω(k)2 Hence C(R) < ∞ and (4.8) holds. ˆ 1/2 ). Then ψ ∈ D(|qj |) and, Lemma 4.6. Assume (H.1) and (H.6). Let ψ ∈ D(K for all j = 1, . . . , N, ˆ 1/2 ψk + (c1 λ2 kW k + c1 c0 λ2 + c1 E0 (K) + c2 )kψk2 . k |qj |ψk ≤ c1 kK

(4.9)

In particular, ˆ + 1)−1/2 Qj := |qj |(K

(4.10)

is bounded. Proof. Let ψ ∈ D(K). Then, using (H.6), one can easily prove (4.9). Since D(K) ˆ 1/2 ) and ˆ 1/2 , it follows from a limiting argument that D(|qj |) ⊃ D(K is a core of K 1/2 ˆ ). (4.9) holds for all ψ ∈ D(K Lemma 4.7. Assume (H.1), (H.2), (H.5) and (H.6). Then Z 1 ˆ + 1)−1/2 k2 dk < ∞ . kv(k)(K 2 Rd ω(k) Proof. By the elementary inequality |e−iqj k − 1| ≤ |qj | |k| we have for all u ∈ L2 (RdN ) kv(k)uk ≤

N X |ˆ ρj (k)| |k| p k |qj |uk . ω(k) j=1

Hence ˆ + 1)−1/2 k ≤ kv(k)(K

N X |ˆ ρj (k)| |k| p kQj k , ω(k) j=1

which, by (H.5), implies (4.11). Proof of Theorem 4.1. We have H = K ⊗ I + I ⊗ Hb + HI (v)

(4.11)

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with λ HI (v) := √ 2

Z

(v(k)a(k)∗ + v(k)∗ a(k))dk .

Rd

Hence H is exactly of the same form as the Hamiltonians considered in [7]. Lemmas 4.1–4.7 show that H satisfies all the assumptions of [7, Theorem 1]. Thus we can apply [7, Theorem 1] to conclude that H has a ground state ψ0 . From the proof of [7, Theorem 1], we see that ψ0 is a weak limit of a sequence {ψσn }∞ n=1 of unit vectors in D(H0 ) such that σn > 0, limn→∞ σn = 0 and each ψσn is a ground state of the infrared-cutoff version Hσn of H: Hσ := K ⊗ I + I ⊗ Hb + HI (vσ ),

σ > 0,

where vσ (k) := v(k)χω≥σ (k). By Proposition 3.3, φσn := Uσn ψσn is a ground state of HSNM,σn with kφσn k = 1. Hence (4.5) holds. References [1] A. Arai, “Scaling limit for quantum systems of nonrelativistic particles interacting with a Bose field”, Hokkaido University Preprint Series in Mathematics #59, 1989. [2] A. Arai, M. Hirokawa and F. Hiroshima, “On the absence of eigenvectors of Hamiltonians in a class of massless quantum field models without infrared cutoff”, J. Funct. Anal. 168 (1999) 470–497. [3] A. Arai and M. Hirokawa, “Ground states of a general class of quantum field Hamiltonians”, Rev. Math. Phys. 12 (2000) 1085–1135. [4] V. Bach, J. Fr¨ ohlich and I. M. Sigal, “Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field”, Commun. Math. Phys. 207 (1998) 249–290. [5] G. G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, John Wiley & Sons, 1972. [6] J. Fr¨ ohlich, “On the infrared problem in a model of scalar electrons and massless scalar bosons”, Ann. Inst. Henri Poincaré 19 (1973) 1–103. [7] C. Gérard, “On the existence of ground states for massless Pauli–Fierz Hamiltonians”, Ann. Henri Poincaré 1 (2000) 443–459. [8] M. Griesemer, E. H. Lieb and M. Loss, “Ground states in non-relativistic quantum electrodynamics”, preprint, 2000. [9] J. L¨ orinczi, R. A. Minlos and H. Spohn, “The infrared behaviour in Nelson’s model of a quantum particle coupled to a massless scalar field”, preprint, 2000. [10] E. Nelson, “Interaction of nonrelativistic particles with a quantized scalar field”, J. Math. Phys. 5 (1964) 1190–1197. [11] M. Reed and B. Simon, Methods of Modern Mathematical Physics Vol. II : Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975. [12] M. Reed and B. Simon, Methods of Modern Mathematical Physics Vol. IV : Analysis of Operators, Academic Press, New York, 1978. [13] H. Spohn, “Ground state(s) of the spin-boson Hamiltonian”, Commun. Math. Phys. 123 (1989) 277–304. [14] H. Spohn, “Ground state of a quantum particle coupled to a scalar Bose field”, Lett. Math. Phys. 44 (1998) 9–16.

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STOCHASTIC ADAMS THEOREM FOR A GENERAL COMPACT MANIFOLD

´ ´ REMI LEANDRE D´ epartement de Math´ ematiques, Universit´ e de Nancy I 54000 Vandoeuvre-les-Nancy, France Mathematical Science Research Institute, 1000 Centennial Drive Berkeley, California, USA

Received 30 March 1999 We give a stochastic analoguous of the theorem of Adams, which says that the Hochschild cohomology is equal to the cohomology of the based smooth loop space. The key tools are the stochastic Chen iterated integrals as well as Driver’s flow.

0. Introduction Let us begin by giving the long term goal of this paper. Let M be a compact orientable manifold endowed with a circle action. It is a periodic group of diffeomorphisms φs of M . It is spanned by a vector field X, called the Killing vector field. The fixed point set of this circle action, that is the subset of M such that φs x = x for all s, coincides with the zero set of X. It is a submanifold of M . The topology of the fixed point set is related to the topology of the full manifold. Let us consider namely an equivariantly closed form. It is a form µ such that (d + iX )µ = 0. It is not in general a form with a given degree. If we consider forms which are invariant under the circle action, we get a complex by considering (d + iX )µ: namely the Lie derivative of a form is given by (d + iX )2 µ. If (d + iX )µ = 0, the integral of µ over the big manifold is equal to the integral of µ over the fixed point set modulo a universal factor which depends only on the circle action. See [6, 16]. We can see that by using the following considerations [9, 10]. Since there is a Riemannian structure over the manifold, X can be considered as a one form. We get Z Z µ= exp[−t(d + iX )X] ∧ µ = cht (µ) . (0.1) M

M

The localization formula arises when we do t → ∞, because iX X = |X|2 . Let us consider the free loop space L∞ (M ) of the manifold. It is the space of smooth maps γ· from the circle into M . There is a natural circle action by rotating 1095

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the loop. The fixed point set of this circle action is the manifold itself. The Killing vector field X ∞ is the vector over a loop which to γs associates its speed d/dsγs (Let us recall that the tangent space of a loop is the space of smooth sections over the loop of the tangent bundle of M .) Let us consider a complex bundle ξ over M . We deduce an infinite dimensional bundle over the loop space by taking the smooth section over the loop γ of ξ. Bismut has introduced to it an equivariant cohomology class, and a representative of it called ch∞ (ξ), which is a formal series of forms of finite degree over the loop space, which satisfies formally to the formula: Z exp[−t(d + iX ∞ )X ∞ ] ∧ ch∞ (ξ) = (Ind D+ (ξ)) = cht (ch∞ (ξ)) (0.2) C L∞ (M)

C is a constant written in term of infinite diverging product. Ind D+ (ξ) is the index of the Dirac operator D+ (ξ) tensorized by ξ. For that, we have supposed that the manifold is spin, such that the free loop is orientable [2, 47]. The index theorem should be a localization formula over the free loop space, as it was pioneered by [2], when there is no auxiliary bundle. Bismut has given a probabilistic interpretation of this fact by using measure theory. Let us recall that there are now a lot of simpler proofs of the Index theorem than the proof of Bismut (see [5, 18, 28, 30, 37]). The reader interested by short time asymptotics of heat kernels can see the survey of Kusuoka [34], Léandre [36] and Watanabe [56]. Getzler–Jones–Petrack [23] have given an algebraic treatment of this fact. They remark that Bismut’s Chern character is more or less an infinite sum of Chen forms. The equivariant exterior derivative over the loop space corresponds to the cyclic coboundary operator, when we consider Chen forms and average them under the circle action. [33] remark that the equivariant cohomology of the free loop space is equal to the cohomology of the manifold. [22] constructs a current over the free loop space, called Witten’s current, such that (0.2) is true. It is purely algebaically defined. There is no requirement of measure theory. The far goal of this work is then the following. We remark that Getzler’s theory of current is similar to the theory of distribution of white noise analysis [26], if we suppose that the Chen forms over the loop space play an analoguous role of the Wiener chaos over the flat Brownian motion. Our goal is to develope a current theory over the loop space in Watanabe’s sense [55], such that the localization formula (0.2) is still true. For that, Jones–Léandre [31] have introduced a tangent space over the free Brownian bridge of the manifold. Let us recall namely that in infinite dimensional analysis, it is known since [24] that the tangent space of a Banach space is a smaller Hilbert space. [38] and [21] remark that the tangent space introduced by [31] is nothing else than the tangent space introduced by Bismut in [7], in order to get intrinsic integration by parts formulas over the Brownian motion of a compact Riemannian manifold. [31] performed an Lp theory of forms over the loop space such that the Bismut–Chern character belongs to all the Lp .

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A scalar Sobolev Calculus over the free loop space is performed in [38] and [39]. [40] does a Sobolev Calculus over the algebraic space corresponding to the Chen forms: it is shown that the Hochschild coboundary is continuous over the intersection of Sobolev spaces. [40] produces a Sobolev Calculus over the set of forms of the free loop space such that the exterior derivative is continuous over this set of forms. It is strongly related to the theory of Stochastic anticipative integrals, because the Lie bracket of two vector fields is not a vector field. The map stochastic Chen iterated integral is continuous. Adams [1] has shown that for the smooth based loop space, the Hochschild cohomology is equal to the cohomology of the based loop space, by using the map deterministic Chen integral. Chen [12] has shown that this result remains true for the free loop space. [41] had shown that this result is still true for the free Brownian bridge of an homogeneous manifold. [41] used deeply the Albeverio–Hoegh–Krohn quasi-invariance formula over loop group in order to give a stochastic interpretation of the path fibration property of the path space with fiber the based loop space, which is used in the deterministic context in order to proof this property. The purpose of this paper is to give a stochastic interpretation of the result of Adams, by using Driver’s flow (see [14, 15, 19, 27, 45, 49]). Let us give a brief outlook of the paper. Over the based loop space, we consider the Brownian bridge measure and the Jones–Léandre tangent space [31] which was given in a preliminary form by Bismut [7], and which is almost surely defined, because the parallel transport over a Brownian path is almost surely defined. This allows to define a space of forms which belong to all the Lp , because the tangent space is an Hilbert space and because there is by definition a measure over the canonical space of the Brownian bridge. In order to define the exterior derivative of a r-form, we have to consider Lie brackets of vector fields, and we have to take the covariant derivative of the parallel transport: this implies in particular that the tangent space of the Brownian bridge is not stable by Lie bracket [38]. This shows us that the definition of the stochastic exterior derivative is related to some anticipative Stratonovitch integrals [40]. By introducing the trivial connection over the based loop space, which is parallelisable, we can define the covariant derivative of length k of an r-form. It is given by kernels σr (s1 , . . . , sr ; t1 , . . . , tk ). We suppose that over all connected components of the complement subset of the diagonals of [0, 1]r × [0, 1]k , we have: kσr (s1 , . . . , sr ; t1 , . . . , tk ) − σr (s01 , . . . , s0r ; t01 , . . . , t0k )kLp X q Xq 0 0 ≤ Cp,k (σ) |si − si | + |tj − tj | .

(0.3)

If this condition is checked for all k and all p, we say that the corresponding form is smooth in the Nualart–Pardoux sense and belongs to N · P∞− . This regularity condition over the kernels of a r-form smooth in the Nualart–Pardoux sense allows to define a stochastic exterior derivative which acts continuously over N · P∞− . We

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can produce an example of forms smooth in the Nualart–Pardoux sense by using stochastic Chen iterated integrals, which are almost surely defined, because they use Stratonovitch stochastic integrals [31, 40, 41]. The stochastic exterior derivative over a stochastic Chen form corresponds to an algebraic operator called Hochschild coboundary. There are not so many Chen forms as stochastic form which are smooth in the Nualart–Pardoux sense. It is however tempting to say that both cohomology groups are equal. The main theorem of this work is the following: Theorem 0.1 (Stochastic Adams theorem). Let us suppose that the manifold is l (Lx (M )) simply connected. Then the stochastic cohomology groups of degree l H∞− of forms, which are smooth in the Nualart–Pardoux sense over the based loop space Lx (M ), is equal to the Hochschild cohomology. Let us remark that in this statement, we consider a generalized Hochschild complex with criteria of convergence in the manner of Connes cyclic cohomology [13]. This Sobolev Hochschild complex was studied extensively in [40]. The Theorem 0.1 was already proved in [41] for a homogeneous manifold. For smooth loops, there are a lot of proofs of this theorem, and [41] has given a stochastic interpretation for a homogeneous manifold of the proof of Getzler–Jones–Petrack [23] of this well know result of algebraic topology. Let us recall the articulation of this proof: (a) [23] consider the Hochschild complex for the smooth based path space. They show that its cohomology groups are trivial, except in degree 0 where they are C. Since the based path space is contractible, the cohomology groups of the based path space are trivial. Therefore the map deterministic Chen iterated integral realizes an isomorphism in cohomology for the based path space. In the stochastic case, [40] and [41] show, by using the Clark–Ocone formula, that the stochastic cohomology groups of the based path space are trivial, which correspond to a stochastic generalization from the fact that the Brownian path is in some sense contractible. So [40] and [41] show that the first part of the proof of [23] has a stochastic counterpart. (b) The second part of the proof of [23] is to remark that the based path space is a fibration with fiber the based loop space and basis the manifold, a fact which is known since a long time in algebraic topology. It is the difficult part in the stochastic adaptation of the proof of [23]. In [41], in the case of an homogeneous manifold, it is shown that this property remains true in some sense by studying convenient functional spaces of forms and by using the Albeverio–Hoegh–Krohn quasi-invariance formulas. The heart of the present work is to show that this property remains true in the general case by studying Driver’s flow [15]. (c) When there is a fibration, there is associated a spectral sequence. The Zeeman comparison theorem allows to state the theorem for smooth loops in [23]. The same argument remains valid in order to finish the proof of Theorem 0.1. It is the point where the hypothesis M simply connected is important.

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The main novelty of the present work is the proof of the point (b) in the general case. It is the purpose of Driver’s flow to show that the Brownian bridge going from x to y is isomorphic to the Brownian bridge going from x to x. This means that there is a natural transformation from the Brownian bridge of paths joining x to x to the Brownian bridge joining x to y which keeps the measure of the Brownian bridge quasi-invariant. C. Cross has started the study of differential properties of Driver’s flow [14]. A careful analysis allows to study in a deeper way the infinitesimal properties of Driver’s flow and to state the following theorem which is the main part in the proof of Theorem 0.1.: Theorem 0.2. Let U be a small open convex subset of M and let PU be the set of Brownian paths starting from x and arriving in U. We can define the stochastic cohomology groups of degree l for forms smooth in the Nualart–Pardoux sense over PU . Let us denote them by H∞−,U (Px (M )). Then l l (Px (M )) = H∞− (Lx (M )) . H∞−,U

(0.4)

1. Algebraic Properties and Statement of the Main Theorem Let M be a compact Riemannian simply connected manifold of dimension d. Let ∆ be the Laplace–Beltrami operator. Let pt (x, y) be the associated heat kernel. Let dP1,x be the law of the Brownian bridge starting from x and returning in time 1 to x. Let dP1x be the law of the Brownian motion starting from x. The time interval is [0, 1]. We consider 2 infinite dimensional spaces: (1) The path space Px (M ): it is the space of continuous functions γs from [0, 1] into M endowed with the measure dP1x such that γ0 = x. (2) The based loop space Lx (M ): it is the space of continuous functions γs from the circle S 1 (that is [0, 1] with 0 and 1 identified) into M such that γ0 = x endowed with the measure dP1,x . Let τt be the parallel transport from γ0 to γt for the Levi–Civita connection over M . It is almost surely defined. These 2 infinite dimensional curved spaces are endowed with different tangent bundles. (1) For the path space, a tangent vector is of the shape Xs = τs Hs where the path Hs takes its values in the linear space Tγ0 (M ), and has bounded energy. Moreover, since we consider the based loop space, we have H0 = 0. We take as Hilbert structure Z 1 2 kd/dsHs k2 ds . (1.1) kXk = 0

(2) For the based loop space, we suppose moreover that X1 = 0, and we take the same Hilbert structure. Over the path space, we define the connection ∇. For Xs = τs Hs , we have (∇X)s = τs ∇Hs

(1.2)

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where ∇Hs is the H-derivative of Hs . Let us recall briefly the notion of H-derivative. If Hs is deterministic over the path space or over the loop space, we get the integration by parts for a cylindrical formula: E[hdF, Xi] = E[F div X] , where

Z

Z

1

hτS d/dsHs , δγs i + 1/2

div X = 0

(1.3)

1

hSXs , δγs i .

(1.4)

0

S is the Ricci tensor of the Levi–Civita connection, and δ denotes the curved Itô integral. Let us recall what is the curved Itˆ o integral. For that we introduce dγs,flat = −1 with values in τs dγs , which is the Stratonovitch differential of a semi-martingale R1 Tx (M ). If Hs is a predictible process with values in Tγs (M ), 0 hHs , δγs i is equal R1 by definition to 0 hτs−1 Hs , δγs,flat i. dF exists for cylindrical functionals and can be extended by continuity. dF can be seen as a one form over the path space or over the based loop space. dr∇ F is a r-cotensor defined by induction over the path space or over the loop space. We get X r F (X , . . . , X ) = hd(d F (X , . . . , X )), X i − dr∇ F dr+1 1 r+1 1 r r+1 ∇ ∇ × (X1 , . . . , Xi−1 , ∇Xr+1 Xi , Xi+1 , . . . , Xr ) .

(1.5)

It can be extended continuously because we have integration by parts (see [38] and [39] for analoguous considerations). dr∇ F is given by a kernel k r (s1 , . . . , sr ): ZZ 1 k r (s1 , . . . , sr )d/dsHs−1 · · · d/dsHsrr ds1 · · · dsr , (1.6) dr∇ F (X1 , . . . , Xr ) = k r (s1 , . . . , sr ) is almost surely an element of (Tx (M )∗ )⊗r . If we work over the loop space, we have the same formula with the extra-condition: Z (1.7) k r (s1 , . . . , sr )dsi = 0 , for i = 1, . . . , r. Let σ be an r-form over the path space or over the loop space. σ is given by a kernel σr : ZZZ σr (s1 , . . . , sr )d/dsHs11 · · · d/dsHsrr ds1 · · · dsr . (1.8) σ(X1 , . . . , Xr ) = R1 The antisymmetry condition says that 0 σr (s1 , . . . , sr )d/dsHsσ11 · · · d/dsHsσrr R1 ds1 · · · dsr is almost surely equal to (−1)sign σ 0 σ(s1 , . . . , sr )d/dsHs1 · · · d/dsHsr ds1 · · · dsr for a deterministic permutation σ of (1, . . . , r). In the case where M is one-dimensional, this means that almost surely σ(s1 , . . . , sr ) = (−1)sign σ σ(sσ1 , . . . , sσr ) . This property does not imply that σ(s1 , . . . , sr ) belongs to Λr (Tx (M )). In such a case all the stochastic forms over the loop space should have a degree smaller than the dimension the case, property which is obviously wrong. If we R work over the loop space, we have moreover σr (s1 , . . . , sr )dsi = 0 for i = 1, . . . , r. We are now ready to define the Nualart–Pardoux spaces of forms. Let ∇l σ be the

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covariant derivative of order l of a r-form. It has kernels σr (s1 , . . . , sr ; t1 , . . . , tl ) which belong almost surely to (Tx (M )∗ )⊗r ⊗ (Tx (M )∗ )⊗l . We suppose that over all connected components of the complement subset of the diagonals of [0, 1]r × [0, 1]l , we have kσr (s1 , . . . , sr ; t1 , . . . , t0l ) − σr (s01 , . . . , s0r ; t01 , . . . , t0l0 )kLp X q Xq ≤ Cp,l (σ) |si − s0i | + |tj − t0j | ,

(1.9)

for l0 ≤ l and we suppose that 0 (σ) . kσj (s1 , . . . , sr ; t1 , . . . , tl0 )kLp ≤ Cp,l

(1.10)

0 (σ) = We call the Nualart–Pardoux norms of order l in Lp the quantity Cp,l (σ)+Cp,l kσkp,l . If kσkp,l < ∞, we say that σ ∈ (N · P )p,l (Path) or σ ∈ (N · P )p,l (Loop).

Definition 1.1. We say that an r-form σ is smooth in the Nualart–Pardoux sense if σ ∈ (N · P )p,l for all p, l. In this case, we say that σ ∈ (N · P )∞− (Path) or that σ ∈ (N · P )∞− (Loop). Let us recall what is the meaning of ∇τt : let Xt be a section of the pull-back bundle of the tangent bundle by the evaluation map et : γ → γt . Let ∇t be the pull-back of the Levi–Civita connection by this evaluation map. We get by definition of ∇τt ∇t (τt Ht ) = (∇τt )Ht + τt ∇Ht .

(1.11)

In particular, if Ht = v is deterministic, (1.11) reduces to ∇tX (τt v) = (∇X τt )v where X is a vector field over the based path space or the based loop space. In some sense, τt corresponds in a section over the path space of the bundle (e∗0 T (M ))∗ ⊗ e∗t T (M ) where e∗t T (M ) is the pullback bundle over the path space by the evaluation map et of the tangent space of M . Moreover, see [7] and [38] (4.64) and [4] for a preliminary form, we have Z t τs−1 R(dγs , Xs )τs , (1.12) ∇X τt = τt 0

where R is the curvature tensor. Let Xt = τt Ht and let Xt0 = τt Ht0 be two vector fields over the path space or the loop space. We Z t 0 τs−1 R(dγs , Xs0 )τs Ht + antisymmetry . (1.13) [X, X ]t = τt ∇X 0 Ht + τt 0

Let us explain this formula: [X, X 0 ] is a section over γ· of the tangent bundle of M . Since the Levi–Civita connection is without torsion, we have [X, X 0 ]t = ∇tX 0 Xt − ∇tX Xt0 . (1.11) and (1.12) allow to conclude. Therefore the tangent space of the path space or of the loop space is not stable by Lie bracket.

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Let us recall that for a n − 1 form, its exterior derivative is defined by X X (−1)i−1 hdσ(X1 , . . . , Xi−1 , Xi+1 , Xn ), Xi i + (−1)i+j σ dσ(X1 , . . . , Xn ) = i<j

× ([Xi , Xj ], . . . , Xi−1 , Xi+1 , . . . , Xj−1 , Xj+1 , . . . , Xn ) . (1.14) Let us recall (see [40, Théoreme I.3] for an analoguous theorem): Theorem 1.2. d is continuous over the set of forms of finite degree smooth in the Nualart–Pardoux sense over the path space or over the loop space. p (N · P )(Path) be the associated cohomology group with complex Let H∞− coefficients of order p over the path space. We get by [40, Théoreme I.7] and [41, Théoreme I.2] the following theorem which reflects that the path space retracts over the constant paths. p 0 (N ·P )(Path) = 0. If p = 0, H∞− (N ·P )(Path) = C. Theorem 1.3. If p > 0, H∞−

We get an inclusion from the based loop space into the path space. We deduce a restriction map i∗ from the set of forms over the path space to the set of forms over the based loop space. Let us recall how we restrict a functional F which belongs to all the Sobolev spaces over Ps (M ), and which is almost surely defined, to Lx (M ). F is the limit in all the Sobolev spaces of a sequence Fn of cylindrical functionals. Let p an even positive integer. We introduce the measure µn,m : f → EPath [|Fn −Fm |p f (γ1 )]. Its density is ELoop [|Fn −Fm |p ]p1 (x, y). By using integration by parts formulas and the Malliavin Calculus [48], we deduce that the density to this measure tends to 0, because Fn is a Cauchy sequence in all Sobolev spaces. Since the heat kernel does not vanish, we deduce that ELoop [|Fn − Fm |p ] tends to 0 when n → ∞ and m → ∞. We refer to [39, Theorem 3.9] for more precisions. Therefore if σr (s1 , . . . , sr ) belongs to all the Nualart–Pardoux spaces, it belongs to all the Sobolev spaces, and we can restrict the kernel σr (s1 , . . . , sr ) of the form σ over Px (M ) to Lx (M ). Let us recall ([40, Théoreme II.7] and [41, p. 326]) that i∗ is continuous from the set of forms smooth in the Nualart–Pardoux sense over the path space onto the set of forms smooth in the Nualart–Pardoux sense over the loop space. It is not clear in converse that we can extend a form over the loop space into a form over the path space. It is the purpose of the next theorem which is one of the main step in the proof of Theorem 0.1, which will be proved in the last part of this work: Theorem 1.4. i∗ is a surjection from (N · P )∞− (Path) over (N · P )∞− (Loop). Let us recall the following definition which is related to the cobar construction of forms over the manifold. Let Ω(M ) be the set of forms over the manifold and let ˜ n = ω1 ⊗ · · · ⊗ Ω· (M ) be the set of differential forms over M of degree > 0. Let ω ωn ⊗ ωn+1 an element of Ω· (M )⊗n ⊗ Ω(M ): we suppose here it a simple tensor

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product, but we can suppose that ω ˜ n belongs to the tensor product of Hilbert spaces associated to the Hilbert Sobolev space of kth order involved with the operator P ˜ n is ni=1 (deg ωi − 1) + deg ωn+1 . d∗ d + dd∗ + 2 over Ω(M ). The total degree of ω We suppose that the total degree of ω ˜ n is given. The corresponding Sobolev norm P ω ñ = ω ˜ with total degree l (that is is denoted by k · k2,k . We consider a series each element ω ˜ n of Ω· (M )⊗n ⊗ Ω(M ) has a degree l), and we introduce the family of semi-norms X zn √ k˜ ωn k2,k . ω) = (1.15) φk,z (˜ n! We get for the norm φk,z a Banach space Bk,z (Ω(M )). Since φk,z increase when z or k → ∞, we consider the intersection of these Banach spaces for the Frechet ω ) < ∞ for all k and z > 0, we say that ω ˜ is topology called B∞− (Ω(M )). If φk,z (˜ smooth. We can do the same considerations if we remove the last term Ω(M ) in the tensor product. The Hochschild boundary for the based path space is given by bp = b0,p + b1,p where ˜ n = dω1 ⊗ · · · ⊗ ωn+1 b0,p ω X + (−1)i−1 ω1 ⊗ · · · ⊗ dωi ⊗ · · · ⊗ ωn+1 ,

(1.16)

1 0, then clearly λ ∈ Σ+ (k) (recall that Σ+ (k) is the positive spectrum of DN,(k) ). The multiplicity of e in the subspace {ψ : Qψ = (k + c)ψ} is bounded by the multiplicity of λ in the set Σ+ (k). If λ = 0, then by Theorem 8.3 the eigenvalue 0 belongs to the spectrum of DN,(k) if and only if m 6= k. In this case the multiplicity of the 0-eigenvalue is |m − k|, and the eigenspinor is contained in the subspace {ψ : σ(ν)ψ = [sgn(m − k)]ψ}. Hence, by (33) we obtain e = [sgn(m − k)](k + c) − 12 and the multiplicity of this eigenvalue ˜ M,(k) is at most |m − k|. This shows that Spec DM is included in the union of D given in (20) with multiplicity. For the converse statement, for any fixed k ∈ Z we start with an eigenspace of DN,(k) with eigenvalue λ. If λ = 0, then the same space is also an eigenspace of σ(ν) with eigenvalue sgn(m − k) by Theorem 8.3. Hence by (33) the lift of this eigenspace to ΨM is an ˜ M,(k) with eigenvalue e = [sgn(m − k)](k + c) − 1 . eigenspace of D 2 If λ > 0, then for any element ξ of this eigenspace we form p −λ ± λ2 + (k + c)2 σ(ν)ξ χ± := ξ + k+c + if k + c 6= 0 and χ+ := ξ, χ− := σ(ν)ξ if k + c = 0. The sets {χ− j } and {χj } are both linearly independent as ξ’s run through a linearly independent set {ξj } ⊂ P Ker(DN,(k) − λ). For, if j cj χ+ j = 0 with some constants cj then X X c j χ+ cj ξ j . 0 = (1 + λ−1 DN,(k) ) j = 2 j

j

±

˜M,(k) with eigenIt is easypto check from (33) that χ ◦ φ is an eigenspinor of D 1 2 2 value e = ± λ + (k + c) − 2 . This completes the proof of Theorem 8.1. Appendix A. Spinor Bundles on S2 (and S3 ) and the Aharonov Casher Theorem We shall now construct spinor bundles Ψ on S2 . First we choose coordinates. Let S2+ = S2 \ {S} and S2− = S2 \ {N }, where N and S are the north and south poles respectively. Consider the stereographic projections z± : S2± → C defined by !  2  −4z 4 − |z (ω) (ω)| − −  2    4 + |z− (ω)|2 , − 4 + |z− (ω)|2 , for ω ∈ S− , ω=  2  4z 4 − |z (ω) (ω)|  + +  , , for ω ∈ S2+  4 + |z+ (ω)|2 4 + |z+ (ω)|2

September 19, 2001 10:36 WSPC/148-RMP

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where we have identified S2 ⊂ C × R. Note that for ω ∈ S2− ∩ S2+ we have z− (ω) = −4z+ (ω)−1 . With the above choice the maps z± are orientation preserving when we choose the standard orientations of S2 and C (strictly speaking z− is a stereographic projection followed by a reflection). If we use the metric −2 1 2 2 dzd¯ z ds = 1 + |z| 4 on C both maps z± are also isometries. Appendix A.1. Spinor bundles on S2 and S3 Corresponding to each n ∈ Z we define a spinor bundle Ψn on S2 by the following properties: + − • There are open subsets Ψ± n ⊂ Ψn such that Ψn = Ψn ∪ Ψn . (±) 2 2 • There are diffeomorphisms φ± : Ψn → S± × C . • If η ∈ Ψn and φ± (η) = (ω± , u± ) then

ω+ = ω− where

Un (z) =

|z| z

and u− = Un (z+ (ω+ ))W(z+ (ω+ ))u+

n and W(z) =

z|z|−1 0

0 z¯|z|−1

(A.1)

∈ SU (2) .

• If α = a(z)d¯ z + a(z)dz is a real one-form on C then the Clifford multiplication σ on Ψn is defined by ! ! a(z± (ω)) 0 1 ∗ 2 u± , φ± (σ(z± (α))η) = ω, 1 + |z± (ω)| 4 0 a(z± (ω)) ∗ (α) are the pull-backs of the one-form α to S2± . when φ± (η) = (ω, u± ). Here z± Note that it is the Clifford multiplication relative to the metric ds2 which is being used on C2 .

It is fairly easy to check that this really defines a spinor bundle Ψn on S2 . In ∗ ˜ |S2± = z± (α± ), where particular, we notice that if α ˜ is a one-form on S2 then α z −2 . Thus we see that the α± = a± (z)dz + a± (z)dz satisfies a+ (z) = 4a− (−4z −1 )¯ Clifford multiplication transforms consistently between Ψ(±) , i.e. ! a− (z− ) 0 1 2 1 + |z− | 4 0 a− (z− ) ! 0 a+ (z+ ) 1 2 W(z+ )∗ = 1 + |z+ | W(z+ ) 4 0 a+ (z+ ) where z± = z± (ω).


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Proposition A.1. If Ψ is a spinor bundle over S2 then Ψ is diffeomorphic to Ψn for some n ∈ Z. Proof. We just sketch this standard argument. Since any vector bundle on C or on S2± is trivial we easily see that any spinor bundle on S2 is of the form described above with Un (z+ (ω)) replaced by a general function U : S2 \ {N, S} → U (1). Let −n be the degree of the map U (i.e. the degree when we restrict to e.g. the equatorial circle). Then S2 \ {N, S} 3 ω 7→ U(ω)Un∗ (z+ (ω)) ∈ U (1) is a map of degree 0. We may therefore find two functions U± : S2 \ {N, S} → U (1) with U+ equal 1 near N and likewise for S such that U(ω)Un∗ (z+ (ω)) = U+ (ω)U− (ω)∗ . If we now use U± to change the coordinates on the fibers of Ψ over S2± respectively we see that the transformation matrix U will be replaced by Un ◦ z+ . Remark A.2. A similar argument immediately proves that on S3 there is only one Spinc bundle. This bundle is in fact trivial, since we can find a global orthonormal frame (e1 , e2 , e3 ) := (u1 , u2 , n) on S3 (see Sec. 7). The Clifford multiplication on S3 × C2 is defined by σ(ej ) := σj , where σj is the jth Pauli matrix. Appendix A.2. Spinc connections on Ψn We first describe Spinc connections on C with respect to the metric ds2 = (1 + 1 2 −2 dzd¯ z . On C we consider the trivial spinor bundle, i.e. Ψ = C × C2 . 4 |z| ) z with Since the metric ds2 is conformally equivalent to the standard metric dzd¯ the conformal factor Ω(z) = (1 + 14 |z|2 )−1 we may use the results from Sec. 4. The spinor sections with respect to the standard metric on C are given in terms of a (real) one-form α = a(z)d¯ z + a(z)dz as follows. The covariant derivative along a (real) vector field X = ξ(z)∂z + ξ(z)∂z¯ is ξ(z)(∂z − ia(z)) + ξ(z)(∂z¯ − ia(z)) .

(A.2)

We may calculate the covariant derivative of spinors in the conformal metric ds2 from Proposition 4.2 using ! 0 ξ(z) 1 ∗ z) = σ(X ) = σ(ξ(z)dz + ξ(z)d¯ 2 ξ(z) 0 and 1 σ(dΩ) = − 2

−2 1 2 1 + |z| 4

0

z¯

z

0

! .

According to Proposition 4.2 the covariant derivative of spinors in the metric ds2 is ! 1 0 i Im(zξ(z)) . ∇α X = ξ(z)(∂z − ia(z)) + ξ(z)(∂z¯ − ia(z)) − (4 + |z|2 ) 0 −1


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Now assume that we have a covariant derivative ∇ on Ψn . Let η be a section in ˜ a vectorfield on S2 . For ω ∈ S2 write Ψn and X ± φ± (η(ω)) = (ω, u± (z± (ω)))

˜ = (ω, X± (z± (ω))) . and (z± )∗ X(ω)

Then there exists one-forms α± on C such that α

φ± (∇X˜ η(ω)) = (ω, ∇X±± u± (z± (ω))) .

(A.3)

Note that by (A.1) we must have n |z+ (ω)| α− α W(z+ (ω))∇X++ u+ (z+ (ω)) , ∇X− u− (z− (ω)) = z+ (ω) i.e. ∇X−− [(Un Wu+ )(−4z −1 )] = Un (−4z −1 )W(−4z −1 )(∇X++ u+ )(−4z −1 ) . α

α

(A.4)

With X± = 2 Re[ξ± (z)∂z ] we have the relation ξ+ (z) = 14 z 2 ξ− (−4z −1 ). A straightz ] then (A.4) implies that forward calculation then shows that if α± = 2 Re[a± (z)d¯ n z −2 a+ (−4z −1) + i z¯−1 . (A.5) a− (z) = 4¯ 2 Conversely, for any choice of functions a± on C satisfying (A.5) the relation (A.3) will define a Spinc connection on S2 . ∗ (dα± ). Using Stokes law we then have that It is easy to see that β|S2± = z± Z Z Z dz in d¯ z ∗ ∗ − = 2πn , (A.6) β= z− (α− ) + z+ (α+ ) = 2 z ¯ z 2 S C |z|=2 where C is the equatiorial curve oriented appropriately, corresponding to the circle |z| = 2 being oriented counterclockwise. Note that Z Z 4¯ z −2 a+ (−4z −1 )d¯ z=− a+ (z)d¯ z. |z|=2

|z|=2

Proposition A.3. For any closed 2-form β on S2 with connection on Ψn such that β is the magnetic 2-form.

R

β = 2πn there is a Spinc

Proof. We must show that there are one-forms α± on C satisfying (A.5) such ∗ (dα± ). We construct them explicitly. As before we will write α± = that β|S2± = z± z ] with a± defined below. 2 Re[a± (z)d¯ −1 ∗ ) (β). We define Let β˜± = (z± R h+ (z) := π −1 C log |z − z 0 |2 β˜+ (z 0 ) , (A.7) i a+ (z) := ∂z¯h+ (z) 4 and n z −2 a+ (−4z −1 ) + i z¯−1 a− (z) := 4¯ 2


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according to (A.5). A simple calculation shows that a− (z) is smooth on C, i.e. the singularities apparently present in (A.5) exactly cancel each other. For this arguR ment we use that C β+ = 2πn and that β+ is the pushforward of a (smooth) 2 form on S2 . z = −2iβ˜+ (z), which From the definition of h+ we see that (∂z¯∂z h(z))dz ∧ d¯ ˜ ˜ implies dα+ = β+ . Finally dα− = β− follows from the relation (A.5) and that ∗ ∗ (dα− ) = z+ (dα+ ) on S2− ∩ S2+ . z− Appendix A.3. The Dirac operator on Ψn Let D be the Dirac operator corresponding to the Spinc connection ∇ on a spinor bundle Ψn on S2 with Chern number n. We may then for any spinor field η in Ψn write ˜ = (ω, D± u± (z± (ω))) φ± (Dη(ω)) for ω ∈ S2± . Here D± are the Dirac operators on C corresponding to the metric ds2 . According to Theorem 4.3 we can express D± in terms of the standard Dirac operators on C corresponding to the standard covariant derivative (A.2). The standard Dirac operators are ! 0 ∂z − ia± (z) . (A.8) −2i 0 ∂z¯ − ia± (z) Finally, we give the proof of the Aharonov–Casher theorem stated in Sec. 8. Proof of Theorem 8.3. Let η ∈ ker D and define u± : C → C2 by φ± η(ω) = (ω, u± (z± (ω))), for ω ∈ S2± . Then D± u± = 0. According to Theorem 4.3 we then have that v± (z) = Ω(z)1/2 u± (z), where as before Ω(z) = (1 + 14 |z|2 )−1 , are in the kernel of the standard Dirac operators (A.8). By (A.1) the spinors v± satisfy the transformation property Ω(z)−1/2 v− (z) = Ω(−4z −1 )−1/2 U(−4z −1 )W(−4z −1 )v+ (−4z −1 ) .

(A.9)

Clearly the map from η to (v− , v+ ) with the above transformation property is a linear isomorphism. It turns out to be fairly simple to characterize the elements v± in the kernel of the standard Dirac operators. Since σ3 anticommutes with the standard Dirac operator, we can consider the cases σ3 v± = v± and σ3 v± = −v± separately. Assume for definiteness that σ3 v± = v± , i.e. [∂z¯ − ia± (z)]v± = 0 . We write v± in the form v± (z) = f± (z)e− 4 h± (z) 1

where h+ is defined in (A.7) and h− (z) := h+ (−4z −1 ) + 2n log |z|2 .

(A.10)


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One easily computes that ∂z¯h± (z) = ia± (z), and using (A.10) we obtain that ∂z¯f± (z) = 0, i.e. these are analytic functions. Finally (A.9) gives the following relation between f− and f+ f− (z) = 2(−z)n−1 f+ (−4z −1 ) . Hence f− is an analytic function, bounded by a constant times |z|n−1 at infinity. Then n ≥ 1 and f− is a polynomial of degree at most n − 1. A basis in the kernel of the Dirac operator is obtained by choosing f− (z) = 1, z, . . . z n−1 , and the dimension is n. Similar argument shows that if σ3 v± = −v± , then n ≤ −1, and the dimension of the space of such zero modes is |n|. In particular all zero modes have definite spin and only one of the two eigenspaces of σ3 = σ(ν) can accomodate zero modes, depending on the sign of n. Recalling that n is the total flux divided by 2π, we have completed the proof of Theorem 8.3. Acknowledgments L. Erd˝ os was supported by the N.S.F. grant DMS-9970323. J. P. Solovej was supported in parts by the EU TMR-grant FMRX-CT 96-0001 by MaPhySto — Centre for Mathematical Physics and Stochastics, funded by a grant from The Danish National Research Foundation and by a grant from the Danish Natural Science Research Council. References [1] [2] [3] [4] [5] [6] [7] [8]

[9] [10] [11]

[12]

M. Loss and H.-T. Yau, Commun. Math. Phys. 104 (1986) 283. J. Fr¨ ohlich, E. H. Lieb and M. Loss, Commun. Math. Phys. 104 (1986) 251. E. H. Lieb and M. Loss, Commun. Math. Phys. 104 (1986) 271. Elton, M. Daniel, J. Phys. A33(41) (2000) 7297. C. Adam, B. Muratori and C. Nash, Phys. Rev. D60 (1999) 125001. N. Hitchin, Adv. Math. 14 (1974) 1. C. B¨ ar, Geom. Func. Anal. 6 (1996) 899. P. B. Gilkey, J. V. Leahy and J. Park, Spinors, Spectral Geometry, and Riemannian Submersions. Lecture Notes Series, 40, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1998. Can also be found online at http://www.emc.dk/EMIS/monographs/GLP/ A. Moroianu, Commun. Math. Phys. 193 (1998) 661. C. Adam, B. Muratori and C. Nash, “Degeneracy of zero modes of the Dirac operator in three dimensions”, Phys. Rev. D(3)62 (2000) 085026. L. Erd˝ os and J. P. Solovej, “On the kernel of Spinc Dirac operators on S3 and R3 ”, in Differential Equations and Mathematical Physics (Birmingham, AL, 1999) III, AMS/IP Stud. Adr. Math., 16, Amer. Math. Soc. Providence, RI, 2000. Y. Aharonov and A. Casher, Phys. Rev. A(3) 19 (1979) 2461.


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COHERENT STATES AND THE QUANTIZATION OF (1 + 1)-DIMENSIONAL YANG MILLS THEORY

BRIAN C. HALL Department of Mathematics, University of Notre Dame Notre Dame, IN 46556 USA E-mail: [email protected]

Received 4 December 2000

This paper discusses the canonical quantization of (1+1)-dimensional Yang–Mills theory on a spacetime cylinder from the point of view of coherent states, or equivalently, the Segal–Bargmann transform. Before gauge symmetry is imposed, the coherent states are simply ordinary coherent states labeled by points in an infinite-dimensional linear phase space. Gauge symmetry is imposed by projecting the original coherent states onto the gauge-invariant subspace, using a suitable regularization procedure. We obtain in this way a new family of “reduced” coherent states labeled by points in the reduced phase space, which in this case is simply the cotangent bundle of the structure group K. The main result explained here, obtained originally in a joint work of the author with B. Driver, is this: The reduced coherent states are precisely those associated to the generalized Segal–Bargmann transform for K, as introduced by the author from a different point of view. This result agrees with that of K. Wren, who uses a different method of implementing the gauge symmetry. The coherent states also provide a rigorous way of making sense out of the quantum Hamiltonian for the unreduced system. Various related issues are discussed, including the complex structure on the reduced phase space and the question of whether quantization commutes with reduction.

Contents 1. Introduction 2. Classical Yang–Mills Theory on a Spacetime Cylinder 3. Formal and Semiformal Quantization 4. The Segal–Bargmann Transform to the Rescue 5. Coherent States: from AC to KC 6. Identification of T ∗(K) with KC 7. Reduction of the Laplacian 8. Does Quantization Commute with Reduction? 9. Notes Acknowledgments References 1281

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1. Introduction The quantization of Yang–Mills theory is an important example of the quantization of reduced Hamiltonian systems. This paper concerns the simplest non-trivial case of quantized Yang–Mills theory, namely, pure Yang–Mills on a spacetime cylinder. The main result described here is from a joint work [1] with B. Driver. However, I also discuss a number of related conceptual points, and the emphasis here is on the ideas rather than the mathematical technicalities. Driver and I use as our main tool the Segal–Bargmann transform, or equivalently, coherent states. We reach two main conclusions. First, upon reduction the ordinary coherent states on the space of connections become the generalized coherent states in the sense of [2] on the finite-dimensional compact structure group. Second, coherent states provide a way to make rigorous the generally accepted idea that upon reduction the Laplacian for the infinite-dimensional space of connections becomes the Laplacian on the structure group. In the rest of the introduction I give a schematic description of the paper. More details are found in the body of the paper and in [1]. See also [3] for additional exposition. Driver and I use the canonical quantization approach rather than the pathintegral approach, and we work in the temporal gauge. As stated, we assume that spacetime is a cylinder, namely, S 1 × R. We fix a compact connected structure group K, which I will assume here is simple connected, with Lie algebra k. The configuration space for the classical theory is the space A of k-valued connection 1-forms over the spatial circle. The gauge group G, consisting of maps of the spatial circle into K, acts naturally on A. The based gauge group G0 , consisting of gauge transformations that equal the identity at one fixed point in the spatial circle, acts freely on A, and the quotient A/G0 is simply the compact structure group K. This reflects that in this simple case the only gauge-invariant quantity is the holonomy of a connection around the spatial circle. Meanwhile we have the complexification of A, namely, AC := A + iA, which is identifiable with the cotangent bundle of A and is the phase space for the unreduced system. We have also the complexification KC of the structure group K, which is identifiable with the cotangent bundle of K. Here KC is the unique simply connected complex Lie group whose Lie algebra is k + ik. One defines in the obvious way the based complexified gauge group G0,C , which acts holomorphically on AC . The quotient AC /G0,C is KC . This is the reduced phase space for the theory. Now we have the ordinary Segal–Bargmann transform for A, which maps from an L2 space of functions on A to an L2 space of holomorphic functions on AC . Much more recently there is a generalized Segal–Bargmann transform for K [2], which maps from an L2 space of functions on K to an L2 space of holomorphic functions on KC . The gist of [1] is that the ordinary Segal–Bargmann transform for A, when restricted to the gauge-invariant subspace is precisely the generalized Segal–Bargmann transform for A/G0 = K. To say the same thing in the language of coherent states, taking the ordinary coherent states for A and projecting them onto the gauge-invariant subspace gives the generalized coherent states for K, in


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the sense of [2]. So [1] gives a new way of understanding the generalized Segal– Bargmann transform (or generalized coherent states) for a compact Lie group K. Another purpose for [1] is to understand the Hamiltonian for the Yang–Mills theory, which at the unreduced level is a multiple of the Laplacian ∆A for A. (The usual curvature term is zero in this case, since there cannot be any curvature on the one-dimensional space manifold S 1 .) The difficulty lies in making sense of ∆A as a reasonable operator in the quantum Hilbert space. However, the Segal– Bargmann transform for A is expressible in terms of ∆A , and it is well-defined. The Segal–Bargmann transform for K is expressed in a precisely parallel way in terms of the Laplacian for K. Theorem 5.2 of [1] (see Theorem 4.4 below) states that the Segal–Bargmann transform for A becomes the generalized Segal–Bargmann transform for K when restricted to the gauge-invariant subspace. This is formally equivalent to the following generally accepted principle. On the gauge-invariant subspace , ∆A reduces to ∆K .

(1)

Driver and I wish to interpret Theorem 5.2 of [1] as a rigorous version of this principle, which does not make mathematical sense as written. (See Sec. 3.) Thus the Hamiltonian for the reduced system becomes a multiple of ∆K . I discuss three additional points. First, I consider the question of finding the “right” complex structure on the reduced phase space T ∗ (K). Although having such a complex structure is necessary in order to construct a Segal–Bargmann transform, it is not a priori obvious what the correct complex structure is. I explain in Sec. 6 how a complex structure on the reduced phase space arises naturally out of the reduction process, and show that this complex structure is the same as the one previously considered at an “intrinsic” level. Second, I discuss why, even at a formal level, ∆A should go to ∆K on the invariant subspace. For Yang–Mills theory in higher dimensions, K. Gaw¸edzki [4] has shown that the reduced and the unreduced Laplacians do not agree (even at a formal level) when applied to gauge-invariant functions. So there is something geometrically special about the (1+1)-dimensional case, as discussed in Sec. 7. Finally, I consider the possibility of doing things in the opposite order, namely, first passing to the reduced phase space KC , and then constructing coherent states by means of geometric quantization. It turns out that the two procedures give the same answer, provided that one includes as part of the geometric quantization the “half-form correction”. Thus one may say that in this case, “quantization commutes with reduction”. It is unlikely that such a result holds (even formally) for higher-dimensional Yang–Mills theory. I have tried to emphasize the concepts rather than the mathematical technicalities. Some of the subtleties that I have glossed over elsewhere are discussed in Sec. 9. 2. Classical Yang Mills Theory on a Spacetime Cylinder Yang–Mills theory on a spacetime cylinder is an exactly solvable model. (See for example [10].) Nevertheless, I believe that there are things to learn here, both classically and quantum mechanically, by comparing what happens before gauge


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symmetry is imposed to what happens afterward. I begin with the classical theory, borrowing heavily from the treatment of Landsman [9, Sec. IV.3.6]. We work on the spacetime manifold S 1 × R, with S 1 being space and R time. Fix a connected compact Lie group K, the structure group, which for simplicity I take to be simply connected, and fix an Ad-invariant inner product on the Lie algebra k of K. We work in the temporal gauge, which has the advantage of allowing the classical Yang–Mills equations to be put into Hamiltonian form. The temporal gauge is only a partial gauge-fixing, leaving still a large gauge group G, namely the group of mappings of the space manifold S 1 into the structure group K. Note that the gauge group is just a loop group in this case. I will concern myself only with the based gauge group G0 , consisting of maps of S 1 into K that equal the identity at one fixed point in S 1 . This group acts freely on A. The remaining gauge symmetry can easily be added later. In the temporal gauge, the Yang–Mills equations have a configuration space A consisting of connections on the space manifold. The connections are 1-forms with values in the Lie algebra k. Since our space manifold is one-dimensional, we may think of the connections as k-valued functions. There is a natural norm on A given by Z 1 2 |A(τ )|2 dτ , A ∈ A . kAk = 0

Here S 1 is the interval [0, 1] with ends identified, and |A(τ )|2 is computed using the inner product on k. The norm allows us to define a distance function d(A, B) := kA − Bk . The gauge group G0 acts on A by dg −1 g . (2) dτ τ The map A → gAg −1 is linear, invertible, and norm-preserving, hence a “rotation” of A. The map A → g · A is affine and satisfies (g · A)τ = gτ Aτ gτ−1 −

d(g · A, g · B) = d(A, B) . So a gauge transformation is a combination of a rotation and a translation of A. The phase space of the theory is the cotangent bundle of A, T ∗(A) ∼ = A + A. The action of G0 on A extends in a natural way to an action on A + A given by g · (A, P ) = (g · A, gP g −1 ) . Note that the translation part of (2) affects only the “position” A and not the “momentum” P . The Yang–Mills equations take place in the phase space A + A and have three parts. First we have a dynamical part. The equations of motion are just Hamilton’s equations, for the Hamiltonian function 1 H(A, P ) = kP k2 . 2


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Normally there would be another term involving the curvature of A, but that term is necessarily zero in this case, since S 1 is one-dimensional. Thus the solutions of Hamilton’s equations are embarrassingly easy to write down: the general solution is (A(t), P (t)) = (A0 + tP0 , P0 ) . This is just free motion in A. Observe that the Hamiltonian H is invariant under the action of G0 on A + A. Second we have a constraint part. This says that the solutions (trajectories in A+ A) have to lie in a certain set, which I will denote J −1 (0), which is “the zero set of the moment mapping for the action of G0 ”. I will not repeat here the formulas, which may be found for example in [1, Sec. 2] or [9, Sec. IV.3.6]. This constraint is of a simple sort, in that J −1 (0) is invariant under the dynamics and under the action of G0 on A + A. So the constraint does not alter the dynamics, it merely restricts us to certain special solutions of the original equations of motion. Third we have a philosophical part. This says that the only functions on phase space that are physically observable are the gauge-invariant ones. The last two points together say that we may think of the dynamics as taking place in J −1 (0)/G0 , which is the same as T ∗(A/G0 ). The process of restricting to J −1 (0) and then dividing by the action of G0 is called Marsden–Weinstein or symplectic reduction. Since the Hamiltonian function H is G0 -invariant, it makes sense as a function on J −1 (0)/G0 . Now, we are in a very simple situation, with the space manifold being just a circle. In this case two connections are gauge-equivalent if and only if they have the same holonomy around the spatial circle. So the orbits of G0 are labeled by the holonomy h(A) of a connection A around the circle, where for A ∈ A, h(A) is an element of the structure group K. It is easily seen that any x ∈ K can be the holonomy of some A, and so we have A/G0 ∼ =K. Thus J −1 (0)/G0 ∼ = T ∗(K) . = T ∗(A/G0 ) ∼ After the reduction, the dynamics become geodesic motion in K, where explicitly the geodesics in K may be written as γ(t) = xetX with x ∈ K and X ∈ k. We require one last discussion before turning to the quantum theory. We may think of A + A as the complex vector space AC = A + iA, in the same way that we think of T ∗(R) ∼ = R + R as C. We may then think of elements of AC as functions (or 1-forms) with values in the complexified Lie algebra kC = k + ik. The action of G0 extends to an action on AC by dg −1 g , dτ τ where Z : [0, 1] → kC . Note that the translation part in the real direction; that dg −1 gτ is in A. One can think of elements of AC as complex connections and is, dτ (g · Z)τ = gτ Zτ gτ−1 −


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thus define their holonomy. But the holonomy now takes values in the complexified group KC , where KC is the unique simply connected complex Lie group with Lie algebra k + ik. (For example, if K = SU (n) then KC = SL(n; C).) The complexified based gauge group G0,C is then the group of based loops with values in KC . The same reasoning as on A shows that the only G0,C -invariant quantity on AC is the holonomy; so AC /G0,C = KC . It turns out that restricting to the zero set of the moment mapping and then dividing out by the action of G0 gives the same result as working on the whole phase space and then dividing out by the action of G0,C . Thus J −1 (0)/G0 = AC /G0,C = KC . On the other hand, we have already said that J −1 (0)/G0 is identifiable with T ∗(K). So we have a natural identification KC ∼ = T ∗(K) . This is explained in detail in Sec. 6 and the resulting identification is given there explicitly. 3. Formal and Semiformal Quantization In this section we will see what is involved in trying to quantize this system. This discussion will set the stage for the entrance of the Segal–Bargmann transform and the coherent states in the next two sections. Let us first try to quantize our Yang–Mills example at a purely formal level, that is, without worrying too much whether our formulas make sense. I want to do the quantization before the reduction by G0 . If we did the reduction before the quantization, then we would have a finite-dimensional system, which is easily quantized. So it is of interest to do the quantization first and see if this gives the same result. See [10], where quantization is done after the reduction, and [11], where quantization is done before the reduction. Since our system has a configuration space A, we may formally take our unreduced quantum Hilbert space to be L2 (A, DA) , where DA is the fictitious Lebesgue measure on A. The quantization of the constraint equation (see [1, Sec. 2]) then imposes the condition that our “wave functions” be G0 -invariant. Note that the quantization of the second part of the classical theory (the constraint) automatically incorporates the third part as well (the G0 -invariance). So we want the reduced (physical) quantum Hilbert space to be L2 (A, DA)G0 := {f ∈ L2 (A, DA)|f (g · A) = f (A) , ∀ g, A} .


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Recall that in our example, in which space is a circle, two connections are G0 -equivalent if and only if they have the same holonomy around the spatial circle. That means that a G0 -invariant function must be of the form f (A) = φ(h(A)) ,

(3)

where h(A) ∈ K is the holonomy of A and where φ is a function on the structure group K. Furthermore, as we shall see more clearly in the next section, it is reasonable to think that for a function of the form (3), integrating |f (A)|2 with respect to DA is the same as integrating |φ(g)|2 with respect to a multiple of the Haar measure on K. Thus (formally) L2 (A, DA)G0 ∼ = L2 (K, C · dg) ,

(4)

for some (infinite) constant C. This is our physical Hilbert space. Next we consider the Hamiltonian. Formally quantizing the function 12 kP k2 in the usual way gives ∞ 2 X

ˆ = − ~ ∆A = − ~ H 2 2 2

k=1

∂2 , ∂x2k

(5)

where the xk ’s are coordinates with respect to an orthonormal basis of A. We must ˆ acts on the G0 -invariant subspace. In light of what now try to determine how H happens when performing the reduction before the quantization, it is reasonable to guess that on the invariant subspace ∆A reduces to ∆K , that is, ∆A [φ(h(A))] = (∆K φ)(h(A)) .

(6)

(See also [6, 11]. See Sec. 7 for a discussion of why (6) is formally correct.) If we accept this and if we ignore the infinite constant C in (4) then we conclude that our quantum Hilbert space is L2 (K, dx) and our Hamiltonian is ˆ = − ~ ∆K . H 2 This concludes the formal quantization of our system. We now begin to consider how to make this mathematically precise. One approach is to forget about the measure theory (i.e. the Hilbert space) and to try to prove (6). As it turns out, the answer is basis-dependent–choosing different bases in (5) will give different answers. Another way of saying this is that the matrix of second derivatives of a function f of the form (3) is in general not of trace class. However, if one uses the most obvious sort of basis, then indeed it turns out that (5) is true. See the appendix of [1]. Even without the problem of basis-dependence, the above approach is unsatisˆ as an operator in some Hilbert space. fying because we would like to define H Since Lebesgue measure DA does not actually exist, one reasonable procedure is 2


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to “approximate” DA by a Gaussian measure dPs (A) with large variance s. This means that Ps is formally given by the expression dPs (A) = cs e−kAk

2

/2s

DA ,

where cs is supposed to be a normalization constant that makes the total integral one. Formally as s → ∞ we get back a multiple of Lebesgue measure DA. The measure Ps does exist rigorously, provided that one allows sufficiently non-smooth connections. There is good news and bad news about this approach. First the good news. (1) Even though the connections in the support of Ps are not smooth, the holonomy of such a connection makes sense, as the solution to a stochastic differential equation. (2) If we define the gauge-invariant subspace to be L2 (A, Ps )G0 = {f | for all g ∈ G0 , f (g · A) = f (A) for Ps -almost every A} , then the Gross ergodicity theorem [12] asserts that L2 (A, Ps )G0 is precisely what we expect, namely, the space of functions of the form f (A) = φ(h(A)), with φ a function on K. (3) There is a natural dense subspace of L2 (A, Ps ) on which ∆A is unambiguously defined, consisting of smooth cylinder functions. Here a cylinder function is one which depends on only finitely many of the infinitely many variables in A. See [1, Definition 4.2]. Note that the map which takes f (A) to f (g · A) is not unitary, because the measure Ps is not invariant under the action of G0 . Driver and I wish to avoid “unitarizing” the action of G0 , because if we did unitarize then there would be no gauge-invariant subspace. (See [13].) Instead of unitarizing the action for a fixed value of s, we will eventually let s tend to infinity, at which point unitarity will be formally recovered. The bad news about this approach is that ∆A is not a closable operator, and that functions of the holonomy are not cylinder functions. The non-closability of ∆A means that if we approximate φ(h(A)) by cylinder functions, then the value of ∆A φ(h(A)) depends on the choice of approximating sequence. So we still have a major problem in making mathematical sense out of the Hamiltonian in our quantum theory. 4. The Segal Bargmann Transform to the Rescue In this section I will explain how the Segal–Bargmann transform can be used to make sense out of the quantization of the Hamiltonian. At the same time, we will see how the generalized Segal–Bargmann transform for the structure group K arises from the restriction of the ordinary Segal–Bargmann transform for the gaugeinvariant subspace. Although it is technically easier to describe the quantization in terms of the Segal–Bargmann transform, there is a formally equivalent description in terms of coherent states, as I will explain in the next section. See [14–17] and also [18, 19] for results on the ordinary Segal–Bargmann transform.


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Let me explain the normalization of the Segal–Bargmann transform that I wish to use, first for the finite-dimensional space Rd . Let H(Cd ) denote the space of holomorphic (complex analytic) functions on Cd . For any positive constant ~, define C~ : L2 (Rd , dx) → H(Cd ) by the formula C~ f (z) = (2π~)−d/2

Z

e−(z−x)

2

/2~

f (x) dx ,

Rd

z ∈ Cd .

(7)

Here (z − x)2 means Σ(zk − xk )2 . If we restrict attention to z ∈ Rd , then this is the standard expression for the solution of the heat equation ∂u/∂~ = 12 ∆u, at time ~ and with initial condition f . Thus we may write C~ f = analytic continuation of e~∆/2 f . Here e~∆/2 f is just a mnemonic for the solution of the heat equation with initial condition f , and the analytic continuation is in the space variable (analytic continuation from Rd to Cd ). Because ~ is playing the role of time in the heat equation, it is tempting call this parameter t instead of ~; this is what we do in [1]. Now let ν~ be the measure on Cd given by dν~ (z) = (π~)−d/2 e−(Im z)

2

/~

dz ,

where dz refers to the 2d-dimensional Lebesgue measure on Cd . Theorem 4.1 (Segal Bargmann Transform). For each positive value of ~, C~ is a unitary map of L2 (Rd , dx) onto HL2 (Cd , ν~ ), where HL2 denotes the space of entire holomorphic functions on Cd that are square-integrable with respect to ν~ . This is not quite the form of the transform given by either Segal or Bargmann. Comparing to Bargmann’s map A (and taking ~ = 1 since that is what Bargmann does) we have z −d/4 −z 2 /4 e Af √ . C1 f (z) = (4π) 2 √ The factor of √2 accounts for the difference between Bargmann’s convention that z = (x + ip)/ 2 and my convention that z = x + ip, √ which is preferable for me because on a more general manifold, the map z → z/ 2 does not make sense. The 2 factor of e−z /4 has the effect of converting from the fully Gaussian measure in [14] to the measure ν~ , which is Gaussian in the imaginary directions but constant in the real directions. (See [2, Appendix].) The C~ form of the Segal–Bargmann transform has the advantage of making explicit the symmetries of position-space. The measure dx on Rd and the measure ν~ on Cd are both invariant under rotations and translations of x-space, and the transform commutes with rotations and translations of x-space. Since a gauge transformation is just a combination of a rotation and a translation, this property of C~ will be useful.


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On the other hand, as it stands this form of the Segal–Bargmann transform does not permit taking the infinite-dimensional limit, as we must do if we want to quantize A, since neither dx nor ν~ makes sense when the dimension tends to infinity. Fortunately, it is not too hard to fix this problem by adding a little bit of Gaussian-ness to our measures in the x-directions. It turns out that if we do this correctly, then we can keep the same formula for the Segal–Bargmann transform while making a small change in the measures and still have a unitary map. (See [1, Sec. 3], [20] or [21].) Theorem 4.2. For all s > ~/2, let Ps denote the measure on Rd given by dPs (x) = (2πs)−d/2 e−x

2

/2s

dx

and let Ms,~ denote the measure on Cd given by dMs,~ (x + ip) = (π~)−d/2 (πr)−d/2 e−x

2

/r −p2 /~

e

,

where r = 2s − ~. Then the map Ss,~ : L2 (Rd , Ps ) → H(Cd ) given by Ss,~ f = analytic continuation of e~∆/2 f is a unitary map of L2 (Rd , Ps ) onto HL2 (Cd , Ms,~ ). If we multiply the measures on both sides by (2πs)d/2 and then let s → ∞ we recover the C~ version of the transform. On the other hand, for any finite value of s it is possible to let d → ∞ to get a transform that is applicable to our gauge-theory example. So we consider L2 (A, Ps ), where Ps is the Gaussian measure on A described in Sec. 3, which is just the infinite-dimensional limit of the measures Ps on Rd . We consider also the Gaussian measure Ms,~ on AC that is the infinite-dimensional limit of the corresponding measures on Cd . We then work with cylinder functions in L2 (A, Ps ), that is, functions that depend on only finitely many of the infinitely many variables in A. (See [1, Definition 4.2].) On such functions the Segal–Bargmann transform makes sense, since on such functions ∆A reduces to the Laplacian for some finite-dimensional space. It then follows from Theorem 4.2 that the Segal–Bargmann transform Ss,~ is an isometric map of the space of cylinder functions in L2 (A, Ps ) into HL2 (AC , Ms,~ ). This transform extends by continuity to a unitary map of L2 (A, Ps ) onto HL2 (AC , Ms,~ ). Recall that ∆A by itself is a non-closable operator as a map of L2 (A, Ps ) to itself. Considering e~∆A /2 as a map from L2 (A, Ps ) to itself will not help matters. But by considering e~∆A /2 followed by analytic continuation, as a map from L2 (A, Ps ) to HL2 (AC , Ms,~ ), we get a map which is not only closable but continuous (even isometric). It then makes perfect sense to apply this operator (the Segal–Bargmann transform) to functions of the holonomy.


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The following theorem summarizes the above discussion. (See [1, Theorem 4.3].) Theorem 4.3. For all s > ~/2 the map Ss,~ given by Ss,~ f = analytic continuation of e~∆A /2 f makes sense and is isometric on cylinder functions, and extends by continuity to a unitary map of L2 (A, Ps ) onto HL2 (AC , Ms,~ ). We are now ready to state the main result (Theorem 5.2) of [1]. Theorem 4.4. Suppose f ∈ L2 (A, Ps ) is of the form f (A) = φ(h(A)) where φ is a function on K. Then there exists a unique holomorphic function Φ on KC such that Ss,~ f (C) = Φ(hC (C)) . The function Φ is given by Φ = analytic continuation e~∆K /2 φ . Note that in light of the definition of Ss,~ , this result says that on the gaugeinvariant subspace, e~∆A /2 (followed by analytic continuation) reduces to e~∆K /2 (followed by analytic continuation). Thus Theorem 4.4 is a formally equivalent to the principle (1) with which we started. The s = ~ case of this result is essentially due to Gross and Malliavin [5]. See also [22, Sec. 2.5] for more on the s = ~ case. Now, the gauge-invariant subspace L2 (A, Ps )G0 consists of functions of the form f (A) = φ(h(A)), with φ a function on K. It may be shown that Z Z 2 |φ(h(A))| dPs (A) = |φ(x)|2 ρs (x)dx , A

K

where ρs is the heat kernel at the identity on K at time s. Similarly, Z Z 2 |Φ(hC (Z))| dMs,~ (Z) = |Φ(g)|2 µs,~ (g)dg , AC

KC

where µs,~ is a suitable heat kernel on KC and dg is Haar measure on KC . So the gauge-invariant subspace on the real side is identifiable with L2 (K, ρs (x)dx) and on the complex side with HL2 (KC , µs,~ (g)dg). So we have the following commutative diagram in which all maps are unitary. L2 (A, Ps )G0 x y

e~∆A /2

−−−−−→

HL2 (AC , Ms,~ )G0 x y

(8)

~∆K /2

e

L2 (K, ρs (x)dx) −−−−−→ HL2 (KC , µs,~ (g)dg) . The horizontal maps contain an implicit analytic continuation. This result embodies a rigorous version of the principle (1) and also shows that a form of the Segal–Bargmann transform for A can descend to a Segal–Bargmann


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transform for A/G0 = K. But so far we still have the regularization parameter s, which we are supposed to remove by letting it tend to infinity. On the full space L2 (A, Ps ) or HL2 (AC , Ms,~ ) the limit s → ∞ does not exist; this was the point of putting in the s in the first place. But on the gauge-invariant subspaces, identified with functions on K or KC , the limit does exist. As s → ∞, the heat kernel measure ρs on K converges to normalized Haar measure on K. This confirms our earlier conjecture that the fictitious Lebesgue measure on A (formally the s → ∞ limit of Ps ) pushes forward to the Haar measure on K. Meanwhile, the measure µs,~ converges as s → ∞ to a certain measure I call ν~ , which coincides with the “K-averaged heat kernel measure” of [2]. So taking the limit in the bottom line of (8) gives a unitary map L2 (K, dx)

e~∆K /2

−−−−−→ HL2 (KC , ν~ ) .

(9)

This supports the expected conclusion that our reduced quantum Hilbert space is L2 (K, dx) and that the quantum Hamiltonian is (−~2 /2)∆K . It further shows that the generalized Segal–Bargmann transform for K, as given in (9), arises naturally from the ordinary Segal–Bargmann transform for the space of connections, upon restriction to the gauge-invariant subspace. The transform in (9) is precisely the K-invariant form C~ of the transform, as previously constructed in [2] from a purely finite-dimensional point of view. 5. Coherent States: from AC to KC Let us now reformulate the results of the last section in terms of coherent states. Described in this way, our results are in the spirit of the proposal of J. Klauder [23, 24] on how to quantize systems with constraints. (See also [25].) Klauder and B.-S. Skagerstam [26] think of coherent states as a collection of states ψα in some Hilbert space H, labeled by points α in some parameter space X. They assume that there is a “resolution of the identity” Z |ψα ihψα |dν(α) (10) I= X

for some measure ν on X. One may then define a “coherent state transform”, that is, a linear map C : H → L2 (X, ν) given by taking the inner product of a vector in H with each of the coherent states: C(v)(α) = hψα |vi . The resolution of the identity implies that Z Z 2 |hψα |vi| dν(α) = hv|ψα ihψα |vidν(α) X

X

Z |ψα ihψα |dν(α)|vi

= hv| X

= hv|vi .


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Thus the resolution of the identity (10) is equivalent to the statement that C is an isometric linear map. Note that C is only isometric but not unitary. In all the interesting cases the image of C is a proper subspace of L2 (X, ν), which may be characterized by a certain reproducing kernel condition. Although the resolution of the identity looks on the surface like an orthonormal basis expansion, it is in fact quite different. The coherent states are typically non-orthogonal and “overcomplete”. The overcompleteness is reflected in the fact that C does not map onto L2 (X, ν). As an example, consider the finite-dimensional Segal–Bargmann transform, in my normalization. The coherent states are then the states ψz ∈ L2 (Rd , dx) given by ψz (x) = (2π~)−n/2 e−(¯z−x)

2

/2~

,

z ∈ Cn .

This means that the coherent state transform is given by Z 2 (2π~)−n/2 e−(z−x) /2~ f (x)dx , (C~ f )(z) = hψz |f iL2 (Rd ,dx) = Rd

as above. In this case the parameter space X is Cd and the measure on X is the measure ν~ of the last section. The overcompleteness of the coherent states means here that the image of C~ is not all of L2 (Cd , ν~ ), but only the holomorphic subspace. Next consider what happens to a set of coherent states under reduction. Suppose we have a set of coherent states in a Hilbert space H, satisfying a resolution of the identity (10). Then suppose that V is a closed subspace of H and that P is the orthogonal projection onto V . Since P 2 = P ∗ = P , (10) gives Z |P ψα ihP ψα |dν(α) . P = P IP = X

Thus by projecting each coherent state into V we get a resolution of the identity (and hence a coherent state transform) for the subspace V . The “reduced coherent states” are the projections P ψα of the original coherent states into the subspace V . Note that at the moment the parameter space for the coherent states, and the measure on it, are unchanged by the projection. However, it may happen that certain sets of distinct coherent states become the same after the projection is applied. In that case we may reduce (or “collapse”) the parameter space X by identifying any two parameters α and β for which P ψα = P ψβ . The measure ν ˜ then pushes forward to a measure ν˜ on the reduced parameter space X. This is what happens in our Yang–Mills case. Initially we have coherent states ψZ ∈ L2 (A, Ps ) labeled by points Z in the phase space AC . These states satisfy Ss,~ f (Z) = hψZ |f iL2 (A,Ps ) ,

(11)

I suppress the dependence of ψZ on s and ~. We want to project the ψZ ’s onto the gauge-invariant subspace, that is, onto the space of functions of the form φ(h(A)).


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The projection amounts to the same thing as restricting attention in (11) to f ’s of the form f (A) = φ(h(A)). For such f ’s, Theorem 4.4 tells us that hψZ |f i = [Ss,~ (φ ◦ h)](Z) = Φ(hC (Z)) , where Φ is the analytic continuation to KC of e~∆K /2 φ. We see then that for f in the invariant subspace, the right side of (11) depends only on the holonomy of Z. This says that if we have two different coherent states ψZ and ψW such that hC (Z) = hC (W ), then upon projection into the gauge-invariant subspace, they will become equal. Thus the parameter space for the coherent states “collapses” from AC to AC /G0,C = KC . If we identify the gauge-invariant subspace with L2 (K, ρs ) as in the previous section, then the reduced coherent states are the vectors ψ˜g ∈ L2 (K, ρs ) given by ρ~ (gx−1 ) , ψ˜g (x) = ρs (x) so that, as required, we have

Z

hψ˜g |φiL2 (K,ρs ) = K

g ∈ KC ,

ρ~ (gx−1 ) φ(x)ρs (x)dx = Φ(g) . ρs (x)

−1

continuation of the heat kernel from K to KC , Here ρ~ (gx ) refers to the analytic R and for g ∈ K the convolution K ρ~ (gx−1 )φ(x)dx is nothing but (e~∆K /2 φ)(g). The ψ˜g ’s satisfy a resolution of the identity with respect to the measure µs,~ on KC . This measure is the one which is naturally induced from the measure Ms,~ on AC , upon reduction from AC to KC . That is, µs,~ is the “push-forward” of Ms,~ from AC to KC , under the map hC . Now, as s → ∞, ρs (x) converges to the constant function 1. Thus we obtain in the limit coherent states χg ∈ L2 (K, dx) given by ρ~ (gx−1 ) = ρ~ (gx−1 ) , s→∞ ρs (x)

χg (x) := lim

g ∈ KC .

(12)

These satisfy the following resolution of the identity: Z |χg ihχg |dν~ (g) , I= KC

where ν~ = lims→∞ µs,~ . The measure ν~ coincides with the “K-averaged heat kernel measure” of [2]. Although we are “supposed to” let s → ∞, we get a well-defined family of coherent states for any s > ~/2. The case s = ~, as well as the limiting case s → ∞, had previously been described in [2]. For other values of s we get something new, which I investigate from a finite-dimensional point of view in [20]. Let me compare the above results to those in the paper of Wren [6], which motivated Driver and me to develop our paper [1]. Wren uses the“Rieffel induction” method proposed by Landsman [7], applied to this same problem of Yang–Mills


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theory on a spacetime cylinder. The commutative case was considered previously by Landsman and Wren in [8]. Wren uses a fixed Gaussian measure and a “unitarized” action of the gauge group. In this approach there is no gauge-invariant subspace (see [13]) and so an integration over the gauge group is used to define a reduced Hilbert space, which substitutes for the gauge-invariant subspace. Wren shows that the reduced Hilbert space can be identified with L2 (K, dx) and further shows that under the reduction map the ordinary coherent states map precisely to the coherent states χg in (12). So the appearance of these coherent states in [1] was expected on the basis of Wren’s results. The paper [1] set out to understand better two issues raised by [6]. First, because in [6] there is no true gauge-invariant subspace to project onto, the resolution of the identity for the classical coherent states does not survive the reduction. That is, Rieffel induction does not tell you what measure to use on KC in order to get a resolution of the identity. Of course, the relevant measure had already been described in [2], but it would be nice not to have to know this a head of time. By contrast, in our approach the measure ν~ arises naturally by pushing forward the Gaussian measure Ms,~ to KC and then letting s tend to infinity. Second, the calculation in [6] concerning the reduction of the Hamiltonian is non-rigorous, mainly because the unconstrained Hamiltonian is not well-defined. Driver and I used the Segal–Bargmann transform in order to get some form of the Hamiltonian ∆A to make rigorous sense. Finally, let me mention that the generalized coherent states on K do not fall into the framework of Perelomov [27], because there does not seem to be in the compact group case anything analogous to the irreducible unitary representation of the Heisenberg group on L2 (Rd ). 6. Identification of T ∗(K) with KC I am thinking of K as the configuration space for the reduced classical Yang–Mills theory and of KC as the corresponding phase space. For this to be sensible, there should be an identification of KC with the standard phase space over K, namely the cotangent bundle T ∗(K). In this section I will explain how such an identification comes out of the reduction process. The resulting identification coincides with the one constructed in [28, 29] from an intrinsic point of view. Let us see what comes out of the reduction process. From the symplectic point of view we have the Marsden–Weinstein symplectic quotient J −1 (0)/G0 . Since the action of G0 on AC = T ∗(A) arises from an action of G0 on the configuration space A, general principles tell us that J −1 (0)/G0 coincides with T ∗(A/G0 ) = T ∗(K). On the other hand, from the complex point of view we may analytically continue the action of G0 on AC to get an action of G0,C on AC . Dividing out by this action gives AC /G0,C = KC . But in this case there is a natural identification of J −1 (0)/G0 with AC /G0,C : each orbit of G0,C intersects J −1 (0) in precisely one G0 -orbit. This may be seen from [9, Sec. IV.3.6].


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This result is not a coincidence. In general, given a K¨ ahler manifold M (in our example AC ) and an action of a group G that preserves both the complex and symplectic structure of M , we may analytically continue to get an action of GC on M , an action which preserves the complex but not the symplectic structure of M . Then if J −1 (0) is the moment mapping for the action of G, one expects that J −1 (0)/G = M/GC .

(13)

This would mean that for each orbit O of GC in M the intersection of O with J −1 (0) is precisely a single G-orbit. Now, (13) is not actually true in general, but only with various provisos and qualifications. (See [30, 31] and the notes in Sec. 9.) Still, this is an important idea and in our case it works out exactly. Putting everything together we have the following identifications: AC /G0,C = J −1 (0)/G0 = T ∗(A/G0 ) x x y y T ∗(K)

KC

If one does the calculations, one obtains the following explicit identification of T ∗(K) with KC . (See Proposition 3.6.8 of Landsman [9, Sec. IV.3.6]. Landsman uses slightly different conventions.) First, use left-translation to trivialize the cotangent bundle, so that T ∗(K) ∼ = K × k∗ . Then use the inner product on k to identify K × k∗ with K × k. Finally map from K × k to KC by the map Φ(x, Y ) = xeiY ,

x ∈ K, Y ∈ k .

(14)

The map Φ is a diffeomorphism of K × k onto KC , and Φ−1 is called the polar decomposition of KC . Of course, one could simply write down (14) directly at the finite-dimensional level, and indeed this is what I do in [28, Sec. 3]. However, it is interesting that this same identification comes out naturally from the reduction process (along with the Segal–Bargmann transform). To illustrate the identification of KC with T ∗(K), consider the case K = SU (n), in which case KC = SL(n; C). Then for any g in SL(n; C) we may use the standard polar decomposition for matrices to write g = xp with x unitary and p positive. Since det g = 1 it follows that det x = 1 and det p = 1 (since det x has absolute value one and det p is real and positive). In particular, x ∈ SU (n). Then p has a unique self-adjoint logarithm ξ, which has trace zero. Letting Y = ξ/i we have g = xeiY , where Y is skew and has trace zero, i.e. Y is in su(n). Thus we see that SL(n; C) decomposes as SU (n) × su(n) ∼ = T ∗ (SU (n)) as in (14).


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Now in [28, Sec. 3] (see also [29]) I argued from an intrinsic, finite-dimensional point of view that the above identification of T ∗(K) with KC was natural. The argument was based on the notion of “adapted complex structures” [32–35]. There is a good reason that the reduction argument gives the same identification as the adapted complex structures do. Suppose X is a finite-dimensional compact Riemannian manifold such that T ∗(X) has a global adapted complex structure, and suppose G is a compact Lie group which acts freely and isometrically on X. Then a result of R. Aguilar [36] says that T ∗(X/G) has a global adapted complex structure and that this complex structure coincides with the one inherited from T ∗(X) by means of reduction. We have the same sort of situation here, with X = A and G = G0 . Of course, G0 is not compact and A is neither compact nor finite-dimensional, but nevertheless what happens is reasonable in light of Aguilar’s result. 7. Reduction of the Laplacian Why should ∆A correspond to ∆K on gauge-invariant functions? Let us strip away the infinite-dimensional technicalities and consider the analogous question in finitely many dimensions. Suppose X is a finite-dimensional connected Riemannian manifold and suppose G is a Lie group that acts by isometries on X. For simplicity I will assume that G is compact and that G acts freely on X. Then X/G is again a manifold, and it has a unique Riemannian metric such that the quotient map q : X → X/G is a Riemannian submersion. This means that at each point x ∈ X, the differential of q is an isometry when restricted to the orthogonal complement of the tangent space to the G-orbit through x. Given this “submersion” metric on X/G we may consider the Laplace–Beltrami operator ∆X/G . For a smooth function f on X/G we may ask whether (∆X/G f ) ◦ q coincides with ∆X (f ◦ q). This amounts to asking whether ∆X and ∆X/G agree on the G-invariant subspace of C ∞ (X). Since ∆X commutes with isometries, it will at least preserve the G-invariant subspace. The answer in general is no, ∆X and ∆X/G do not agree on C ∞ (X)G . For example, consider SO(2) acting on R2 \ {0} by rotations. The quotient manifold is diffeomorphic to (0, ∞), with the point r ∈ (0, ∞) corresponding to the orbit x2 + y 2 = r2 in R2 \ {0}. The submersion metric on (0, ∞) is the usual metric on (0, ∞) as a subset of R. So the Laplace–Beltrami operator on (0, ∞) is just d2 /dr2 . On the other hand, the formula for the two-dimensional Laplacian on radial functions is 2 p 2 d f (r) 1 df (r) ∂2 ∂ 2 2 + 2 f( x + y ) = + , (15) ∂x2 ∂y dr2 r dr √ 2 2 r=

x +y

which differs from d2 f /dr2 by a first-order term. The source of the trouble is the discrepancy between the intrinsic volume measure dr on (0, ∞) and the push-forward of the volume measure from R2 \ {0}, which is 2πr dr.


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In general, each G-orbit in X inherits a natural Riemannian metric from X, and we may compute the total volume of this orbit with respect to the associated Riemannian volume measure. The expression Vol(G · x) is a function on X/G, and it measures the discrepancy between the intrinsic volume measure on X/G and the push-forward of the volume measure on X. The two Laplacians on C ∞ (X)G will be related by the formula ∆X = ∆X/G + ∇(log Vol(G · x)) · ∇ .

(16)

(The gradient may be thought of as that for X/G, although this coincides in a natural sense with that for X, on G-invariant functions.) Formula (15) is a special case of (16) with volume factor 2πr. We have arrived at the following conclusion. The Laplacians ∆X and ∆X/G agree on G-invariant functions if and only if all the G-orbits have the same volume. Let us return, then, to the case of A/G0 . By considering the appendix of [1] it is easily seen that the metric on K that makes the map h : A → K a Riemannian submersion is simply the bi-invariant metric on K induced by the inner product on k. (We use on A the metric coming from the L2 norm as in Sec. 2.) So in light of (16) the statement that ∆A and ∆K agree on the G0 -invariant subspace is formally equivalent to the statement that all the G0 -orbits have the same volume. In this case (for connections on a circle) it can be shown that there exist isometries of A that map any G0 -orbit to any other. Thus formally all the G0 -orbits should have the same volume. The relevant isometries come from extending the gauge action (2) of G0 (the loop group over K) on A to an action of the path group over K, given by the same formula. This action of the path group on A is isometric. If gτ is a path in K with g0 = e but with g1 arbitrary, then g changes holonomies according to the formula h(g · A) = h(A)g1−1 . Thus the path group permutes the G0 -orbits (labeled by the holonomy), and any G0 -orbit can be mapped to any other by an element of the path group. To look at the problem in another way, to see that ∆A matches up with ∆K on gauge-invariant functions we need to show that the (fictitious) volume measure DA on A pushes forward to a constant multiple of the Haar measure on K. After all, the density of the push-forward of DA with respect to the Haar measure should be the volume factor, which we want to show is constant. If we accept the Gaussian measures Ps as an approximation to DA, we note that these push forward to the measures ρs (x)dx, which indeed converge to Haar measure as s tends to infinity. It should be emphasized that these arguments apply only in our (1 + 1)dimensional example. If A is the space of connections over some space manifold M with dimension at least two, then ∆A and ∆A/G0 will not coincide (even formally) on gauge-invariant functions, as shown by K. Gaw¸edzki [4].


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8. Does Quantization Commute with Reduction? When quantizing a reduced Hamiltonian system such as Yang–Mills theory, one may ask whether the quantization should be done before or after the reduction. If we were very optimistic, we might hope that it doesn’t matter, that one gets the same answer either way. If this were so, we could say that quantization commutes with reduction. Of course the question of whether quantization commutes with reduction may well depend on the system being quantized and on how one interprets the question. I want to consider this question from the point of view of geometric quantization [3, 37] and I want specifically to compare the Segal–Bargmann space obtained by first quantizing AC and then reducing by G0 to the one obtained by directly quantizing KC . In geometric quantization [38] one begins with a symplectic manifold (M, ω) and constructs over M a Hermitian complex line bundle L with connection, whose curvature form is iω/2π~. If M is a cotangent bundle then such a bundle exists and may be taken to be topologically and Hermitianly trivial (though the connection is necessarily non-trivial). The “prequantum Hilbert space” is then the space of sections of L which are square-integrable with respect to the symplectic volume measure on M . To obtain the “quantum Hilbert space” one picks a “polarization” and restricts to the space of square-integrable polarized sections of L. If M is a K¨ ahler manifold, i.e. it has a complex structure which is compatible in a natural sense with ω, then there is a natural K¨ ahler polarization. In that case L may be given the structure of a holomorphic line bundle and the quantum Hilbert space becomes the space of square-integrable holomorphic sections of L. In the case M = Cd the resulting bundle is holomorphically trivial. So by choosing a nowhere vanishing holomorphic section, the space of holomorphic sections of L may be identified with the space of holomorphic functions on Cd . This nowhere vanishing section will not, however, have constant norm. This means that the inner product on the space of holomorphic functions will be an L2 inner product with respect to a measure which is Lebesgue measure multiplied by the norm-squared of the trivializing section. Working this out we get simply the Segal– Bargmann space, with different normalizations of the space coming from different choices of the trivializing section. In summary: applying geometric quantization to ahler polarization, yields the Segal–Bargmann space. Cd , using a K¨ To apply geometric quantization to the infinite-dimensional space AC we may try to quantize Cd and then let d tend to infinity. For this to make sense with my normalization, we need to add the additional parameter s. So we obtain the Segal–Bargmann space HL2 (AC , Ms,~ ). We then want to reduce by the action of G0 , which amounts to restricting to the space of functions in HL2 (AC , Ms,~ ) that are G0 -invariant, and thus by analyticity, G0,C -invariant. The resulting space is identifiable with HL2 (KC , µs,~ ). Finally, letting s tend to infinity we obtain HL2 (KC , ν~ ). It is therefore reasonable to say that HL2 (KC , ν~ ) is the space obtained by quantizing AC and then reducing by G0 .


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Alternatively, we may do the reduction first, obtaining the symplectic manifold ahler manifold by identifying T ∗(K) with KC T ∗(K). This may be made into a K¨ using the polar decomposition, as described in Sec. 6 and in [28, Sec. 3]. We may then apply geometric quantization directly to KC ∼ = T ∗(K). I do this calculation in [29] and find that geometric quantization yields the space HL2 (KC , γ~ ), where γ~ and ν~ are related by the formula dν~ (g) = a~ u(g)dγ~ (g) .

(17)

Here a~ is an irrelevant constant and u is a function which is non-constant except when K is commutative. So it seems that quantizing KC directly does not yield the same answer as quantizing AC first and then reducing by G0 . However, this is not the end of the story. One can quantize KC taking into account the “half-form correction” (also known as the“metaplectic correction”). This “corrected” quantization yields an extra factor in the measure, a factor that coincides precisely with the factor u(g) in (17)! (See [3, 37].) On the other hand, in the Cn case the half-form correction does not affect the Hilbert space, so even with the half-form correction we would get HL2 (AC , Ms,~ ) and then after reduction HL2 (KC , ν~ ). So our conclusion is the following: In this example, if we use geometric quantization with a K¨ ahler polarization and the half-form correction, quantization does in fact commute with reduction. Let me conclude by mentioning a related setting in which one can ask whether quantization commutes with reduction. In an influential paper [39], V. Guillemin and S. Sternberg consider the geometric quantization of a compact K¨ ahler manifold M . They assume then that there is an action of a compact group G on M that preserves the complex structure and the symplectic structure of M and they consider as well the Marsden–Weinstein quotient M G := J −1 (0)/G, where J is the moment mapping for the action of G. Under certain conditions they show that there is a natural invertible linear map between on the one hand the G-invariant subspace of the quantum Hilbert space over M and on the other hand the quantum Hilbert space over M G . They interpret this result as a form of quantization commuting with reduction. However, Guillemin and Sternberg do not show that this invertible linear map is unitary, and indeed there seems to be no reason that it should be in general. So in their setting we may say that quantization fails to commute unitarily with reduction. Dan Freed [40] has suggested to me that inclusion of the halfform correction in the quantization might make the map unitary, and indeed our Yang–Mills example seems to confirm this. (It was Freed’s suggestion that led me to work out that u is just the half-form correction.) After all, upon inclusion of the half-form correction we get the same measure (except for an irrelevant overall constant) and therefore the same inner product whether quantizing before or after the reduction. Nevertheless, I do not believe that one will get a unitary correspondence in general, even with half-forms. We are left, then, with the following open question.


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Given a Kähler manifold M with an action of a group G, under what conditions on M and G will quantization commute unitarily with reduction? Although the question may be considered with or without the half-form correction, what little evidence there is so far suggests that the answer is more likely to be yes if the half-form correction is included. 9. Notes Section 2. One should say something about the degree of smoothness assumed on the connections and gauge transformations. Although it does not matter so much at the classical level, it seems natural to take the space of connections to be the Hilbert space of square-integrable connections. This amounts to completing A with respect to the natural norm, the one which appears in the formula for the classical Hamiltonian. We may then take the gauge group to be the largest group whose action on A makes sense. This is the group of “finite-energy” gauge transformations, namely, the ones for which kg −1 dgk is finite. It is easily shown that in our example of a spatial circle, two square-integrable connections are related by a finite-energy based gauge transformation if and only if they have the same holonomy. In the quantized theory we will be forced to consider a larger space of connections. The geodesics in K (in the reduced dynamics) are relative to the bi-invariant Riemannian metric determined by the chosen Ad-invariant inner product on the Lie algebra. Section 3. The measure Ps is a Gaussian measure, about which there is an extensive theory. For example, see [41–43]. The distinctive feature of Gaussian measures on infinite-dimensional spaces is the presence of two different spaces, a Hilbert space H whose norm enters the formal expression for the measure, and a larger topological vector space B on which the measure lives. Although one should think of the Gaussian measure as being canonically associated to H, the measure lives on B and H is a measure-zero subspace. In our example H is the space of square-integrable connections and B is a suitable space of distributional connections. Since the elements of B are highly non-smooth, the holonomy must be defined as the solution of a stochastic differential equation. If one glosses over questions of smoothness, the Gross ergodicity theorem [12] sounds trivial. But we have just said that we must enlarge the space of connections in order for the measure Ps to exist. Unfortunately, we may not correspondingly enlarge the gauge group without losing the quasi-invariance of the measure Ps under the action of G0 , without which the definition of L2 (A, Ps )G0 does not make sense. In the end two connections with the same holonomy need not be G0 -equivalent, because the would-be gauge transformation is not smooth enough to be in G0 . It was the “J-perp” theorem, which arose as a corollary of Gross’s proof of the


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ergodicity theorem, which led him to suggest to me to look for an analog of the Segal–Bargmann transform on K. Section 4. Driver and I define the holomorphic subspace of L2 (AC , Ms,~ ) to be the L2 closure of the space of holomorphic cylinder functions. An important question then is whether a function of the form F (Z) = Φ(hC (Z)), with Φ ∈ HL2 (KC , µs,~ ), is in this holomorphic subspace. The answer is yes, but the proof that we give is indirect. I am defining HL2 (AC , Ms,~ )G0 to be the image of L2 (A, Ps )G0 under the Segal–Bargmann transform. Certainly every element of HL2 (AC , Ms,~ )G0 is actual invariant under the action of G0 on AC . The converse is probably true as well, namely that every element of HL2 (AC , Ms,~ ) which is G0 -invariant is in the image of L2 (A, Ps )G0 , but we have not proved this. Section 5. Except when s = ~ the coherent states ψZ in L2 (A, Ps ) are nonnormalizable states. When s = ~, the coherent states ψZ are normalizable states provided that Z is a square-integrable (complex) connection [22, Sec. 2.3]. But even then the measure M~,~ does not live on the space of square-integrable connections, and so it is a bit delicate to formulate the resolution of the identity. This shows that it is technically easier to formulate things in terms of the Segal–Bargmann transform in stead of the coherent states. Nevertheless, we may continue to think of unitarity for the Segal–Bargmann transform as formally equivalent to a resolution of the identity for the coherent states. Section 6. There are several obstructions to (13) holding in general. One needs some condition to guarantee that the analytic continuation of the G-action exists globally. Even when it does, one needs to worry about the possibility of “unstable points,” that is points whose GC -orbit does not intersect the zero set of the moment mapping, and also about the possibility that the GC -orbits may not be closed. In the case of a cotangent bundle of a compact Riemannian manifold whose cotangent bundle admits a global adapted complex structure, none of these problems actually arises. See [36, Sec. 7]. Section 8. I jumping to conclusions about the correct action of the gauge group G0 on the Segal–Bargmann space HL2 (AC , Ms,~ ). One should properly use geometric quantization to determine this action. To do this, we restrict first to the finite-dimensional space HL2 (Cd , ν~ ) and then consider the action of the group of rotations and translations of Rd on this space. Going through the calculations, one finds that with my normalization these rotations and translations act in the obvious way, namely, by composing a function in HL2 (Cd , ν~ ) with the rotation or translation. Note that this holds only for rotations and translations in the x-directions. Now we have said that the action of G0 on A consists just of a rotation and a translation. So, taking HL2 (AC , Ms,~ ) as the best approximation of HL2 (Cd , ν~ )


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when d = ∞, it is reasonable to say that the action of G0 on HL2 (AC , Ms,~ ) should be just F (Z) → F (g −1 · Z) There is a large body of work extending the results of [39]; see for example the survey article of Sjamaar [44]. Acknowledgments The idea of deriving the generalized Segal–Bargmann transform from the infinitedimensional ordinary Segal–Bargmann transform is due to L. Gross and P. Malliavin [5]. However, [5] is a paper on stochastic analysis, and it was not intended to be about Yang–Mills theory. What I am here calling the gauge group G0 they call the loop group, and its action in [5] is not unitary. To apply the approach of Gross and Malliavin in the Yang–Mills setting, Driver and I modified that approach so as to make the action of G0 unitary. (More precisely, we take a certain limit under which the action of G0 becomes formally unitary.) The idea that the generalized coherent states for K could be obtained from the ordinary coherent states for A by reduction is due to K. Wren [6]. Wren uses the “Rieffel induction” approach proposed by N. Landsman [7] and carried out in the abelian case by Landsman and Wren [8]. See also the exposition in the book of Landsman [9, Sec. IV.3.7]. I describe in Sec. 5 the relationship of our results to those of Wren. I am indebted to Bruce Driver for clarifying to me many aspects of what is discussed here. I also acknowledge valuable discussions with Andrew Dancer, and I thank Dan Freed for a valuable suggestion regarding the half-form correction. References [1] [2] [3] [4] [5]

[6] [7] [8] [9] [10] [11] [12] [13]

[14]

B. K. Driver and B. C. Hall, Comm. Math. Phys. 201 (1999) 249. B. C. Hall, J. Funct. Anal. 122 (1994) 103. B. C. Hall, Bull. (N.S.) Amer. Math. Soc. 38 (2001) 43. K. Gaw¸edzki, Phys. Rev. D26 (1982) 3593. L. Gross and P. Malliavin, “Hall’s transform and the Segal–Bargmann map”, pp. 73–116 in Itˆ o’s Stochastic Calculus and Probability Theory, eds. M. Fukushima et al., Springer-Verlag, New York/Berlin, 1996. K. K. Wren, Nucl. Phys. B521 (1998) 472. N. P. Landsman, J. Geom. Phys. 15 (1995) 285. N. P. Landsman and K. K. Wren, Nucl. Phys. B502 (1997) 537. N. P. Landsman, Mathematical Topics between Classical and Quantum Mechanics, Springer-Verlag, New York/Berlin, 1998. S. Rajeev, Phys. Lett. B212 (1988) 203. J. Dimock, Rev. Math. Phys. 8 (1996) 85. L. Gross, J. Funct. Anal. 112 (1993) 373. B. K. Driver and B. C. Hall, “The energy representation has non-zero fixed vectors”, pp. 143–155 in Stochastic Processes, Physicsand Geometry: New Interplays. II: A Volume in Honor of Sergio Albeverio, eds. F. Gesztesy et al., Amer. Math. Soc., Providence, RI, 2000. V. Bargmann, Comm. Pure Appl. Math. 14 (1961) 187.


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[15] I. E. Segal, “Mathematical problems of relativistic physics”, pp. 73–84 in Lectures in Applied Mathematics. Proceedings of the Summer Seminar, Boulder, Colorado, 1960, Vol. II, ed. M. Kac, American Math. Soc, Providence, R.I., 1963. [16] I. E. Segal, Illinois J. Math. 6 (1962) 500. [17] I. E. Segal, “The complex wave representation of the free Boson field”, pp. 321–343 in Topics in functional analysis. Essays dedicated to M. G. Krein on the occasion of his 70th birthday, eds. I. Gohberg and M. Kac, Advances in Mathematics Supplementary Studies, Vol. 3, Academic Press, New York, 1978. [18] J. Baez, I. E. Segal and Z. Zhou, Introduction to Algebraic and Constructive Quantum Field Theory, Princeton Univ. Press, Princeton, NJ, 1992. [19] B. C. Hall, “Holomorphic methods in analysis and mathematical physics”, pp. 1–59 in First Summer School in Analysis and Mathematical Physics, eds. S. Pérez-Esteva and C. Villegas-Blas, Contemp. Math., Vol. 260, Amer. Math. Soc., Providence, R.I., 2000. [20] B. C. Hall, Can. J. Math. 51 (1999) 816. [21] A. N. Sengupta, “The two-parameter Segal–Bargmann transform”, (preprint). [22] B. C. Hall and A. N. Sengupta, J. Funct. Anal. 152 (1998) 220. [23] J. R. Klauder, Ann. Physics 254 (1997) 419. [24] J. R. Klauder, Nuclear Phys. B547 (1999) 397. [25] J. Govaerts and J. R. Klauder, Ann. Physics 274 (1999) 251. [26] J. R. Klauder and B.-S. Skagerstam (Eds.), Coherent States. Applications in Physics and Mathematical Physics, World Scientific, Singapore, 1985. [27] A. Perelomov, Generalized Coherent States and Their Applications, Springer-Verlag, New York/Berlin, 1986. [28] B. C. Hall, Comm. Math. Phys. 184 (1997) 233. [29] B. C. Hall, “Quantum mechanics in phase space”, pp. 47–62 in Perspectives on Quantization, eds. L. Coburn and M. Rieffel, Contemp. Math., Vol. 214, Amer. Math. Soc., Providence, RI, 1998. [30] F. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Mathematical Notes 31, Princeton Univ. Press, Princeton, N.J., 1984. [31] D. Mumford, J. Fogarty and F. Kirwan, Geometric Invariant Theory, Third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete 34, Springer-Verlag, New York/Berlin, 1994. [32] V. Guillemin and M. Stenzel, J. Differential Geom. 34 (1991) 561. [33] V. Guillemin and M. Stenzel, J. Differential Geom. 35 (1992) 627. [34] L. Lempert and R. Sz˝ oke, Math. Ann. 290 (1991) 689. [35] R. Sz˝ oke, Math. Ann. 291 (1991) 409. [36] R. M. Aguilar, “Symplectic reduction and new global unbounded solutions of the homogeneous complex Monge–Ampère equation”, (preprint). [37] B. C. Hall, “Geometric quantization and the generalized Segal–Bargmann transform for Lie groups of compact type”, (preprint). [xxx.lanl.gov, quant-ph/0012105]. [38] N. J. M. Woodhouse, Geometric Quantization, Second edition, Oxford Univ. Press, New York, 1992. [39] V. Guillemin and S. Sternberg, Invent. Math. 67 (1982) 515. [40] D. Freed, personal communication. [41] L. Gross, “Abstract Wiener spaces”, pp. 31–42 in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probablility, Vol. II, Univ. of California Press, 1967. [42] H.-H. Kuo, Guassian Measures in Banach Spaces, Lecture Notes in Mathematics, Vol. 463, Springer-Verlag, New York/Berlin, 1975.


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[43] J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, Second edition, Springer-Verlag, New York/Berlin, 1987. [44] R. Sjamaar, Bull. (N.S.) Amer. Math. Soc. 33 (1996) 327.


00087


A STOCHASTIC APPROACH TO THE EULER POINCARE CHARACTERISTIC OF A QUOTIENT OF A LOOP GROUP

´ ´ REMI LEANDRE Institut Elie Cartan, D´ epartement de Math´ ematiques, Universit´ e Nancy I 54000, Vandoeuvre-les-Nancy, France

Received 8 December 1999

1. Introduction Let us consider the free loop space L(M ) of a compact manifold: it is endowed with a natural circle action. The fixed point set of this circle action is the manifold itself. In finite dimension, when there is a circle action over a manifold, there are two types of relations between the fixed point set and the total space. There are the Berline–Vergne localization formulas [10] and the Lefschetz formulas [12, 39]. In the first case, we localize the integrals over the fixed point set; in the second case, using the method of Taubes [39], we localize the operators over the fixed point set. Over the loop space, the Berline–Vergne localization formulas give the index theorem for the Dirac operator over a manifold [8, 13, 14]. If we localize an operator, this gives the Index theorem over the loop space, which gives quantities which are very important for algebraic topology as noticed by Witten [20, 35, 42]. The problem with the Lefschetz formula over the loop space is that it is purely hypothetical. Taubes [39] gives a rigorous construction for a limit model. He considers as model of the free loop space the family of flat loops on the tangent space. He considers a measure over the flat loop space, which lives over random distributions, and whose choice comes from quantum field theory. So we cannot curve its model. There is another way to attack the problem of defining the Dirac operator over the loop space and other geometrical operators: it is the way followed by Léandre– Roan [32], Jones–Léandre [24] and Léandre [28] (See in [29] the survey of Léandre). We consider the Brownian bridge measure. There is a tangent Hilbert space, which allows to state integration by parts formulas over the free loop space. [32] meets the problem that the tangent Hilbert space is not stable by Lie brackets; a connection allows to compute a regularized stochastic exterior derivative and to compute its adjoint. This gives a rigorous approach to the Euler–Poincaré characteristic of Dixon–Harve–Vafa–Witten of an orbifold [15]. [28] meets the same problem for the Dolbeault operator over the loop space of a Kaehler manifold, with the natural 1307


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polarization over the loop space which results from the polarization of the Kaehler manifold. This gives a rigorous approach to Hirzebruch’s genus of level N [21]. On the other hand, the analysis over a loop group is simpler than the analysis over the path space of a general manifold [3, 18]. But the Witten genus of a group is trivial, because the tangent bundle of a group is trivial. The idea is therefore to consider the quotient of a loop group L(G) by a compact subgroup H of G. (LG)/H inherites clearly a circle action. This leads to: Conjecture 1.1 (Bismut). Over (LG)/H, there exists a Dirac operator and its equivariant index is equal to the Witten genus of G/H. In [31], there is a stochastic approach to this conjecture. We are concerned in the present paper with the following conjecture: Conjecture 1.2. The regularized Euler–Poincaré characteristic of (LG)/H is equal to the Euler–Poincaré characteristic of G/H. We define an operator over forms over (LG)/H, and we deform it in order to arrive to an operator at the manner of Taubes. The idea is to consider over LG, an H-invariant operator acting over H-invariant forms over LG. In order to deform the operator, there are a lot of short time asymptotics in this paper: the reader interested by that can see the survey of Léandre [27], the survey of Kusuoka [25] and of Watanabe [41]. The reader interested by further developments about analysis over loop space can see the survey of Léandre [30]. 2. Construction of the Regularized de Rham Operator Let G be a compact simply connected Lie group. We consider the Killing metric over its Lie algebra Lie G. We consider the Brownian motion over G starting from g. It is the solution of the following stochastic differential equation starting from g: dgs = gs dBs ,

(2.1)

where Bs is a Brownian motion over Lie G and where d denotes the Stratonovitch differential. It has a heat kernel pt (g, g 0 ) with respect to the Haar measure dπ(g) over G. Over the free loop group of G called L(G) of continuous loops, we consider the following measure dµG,1 = dπ(g) ⊗ dP1,g ,

(2.2)

where P1,g is the law of the Brownian bridge starting from g and arriving at g in time 1. For a cylindrical functional F (gs1 , . . . , gsr ), s1 < s2 < · · · < sr , we have a more explicit description of µG,1 in terms of the heat kernel: Z Z E[F (gs1 , . . . , gsr )] = C −1 · · · ps2 −s1 (g1 , g2 )ps3 −s2 (g2 , g3 ) · · · × psr −s1 (gr , g1 )F (g1 , . . . , gr )dπ(g1 ) · · · dπ(gr ) .

(2.3)


00087

A Stochastic Approach to the Euler–Poincare Characteristic

1309

R The normalizing constant C is equal to p1 (g, g)dπ(g) = p1 (e, e), the last equality resulting from the symmetry of the heat kernel on the group. sr − s1 denotes the length of the segment on the circle with summits sr and s1 , or in other words 1 − sr + s1 . We consider as tangent space at an element g. of the loop group the set of X. where Xs = gs Ks such that K0 = K1 and where the path K. in the Lie algebra of G is of finite energy. We introduce the following Hilbert structure: Z 1 Z 1 |Ks |2 ds + |Ks0 |2 ds . (2.4) kX. kg. = 0

0

In the sequel, we are concerned by (LG)/H, where H is a compact Lie subgroup of G. This means that we identify two loops gs1 and gs2 if there exists an h ∈ H such that gs1 = gs2 h. Let us compute the orthogonal of vector fields of the type gs f where f is in the Lie algebra of H. They are of 3 types: we introduce for that a orthonormal basis ei of the Lie algebra of G, such that the first vectors are a basis of the Lie algebra of H and the last are a basis of the orthogonal complement of the Lie algebra of H. We say that such a basis satisfies the condition (*). (i) For n > 0 and ei belonging to the Lie algebra of G: √ cos(2πns)ei = Xn (ei )(g. )s , (2.5) gs 2 √ Cn2 + 1 C is equal to 4π 2 . (ii) For n < 0 and ei belonginging to the Lie algebra of G: √ sin(2πns)ei , Xn (ei )(g. )s = gs 2 √ Cn2 + 1

(2.6)

C is equal to 4π 2 . (iii) For n = 0 and ei belonging to the orthogonal complement of the Lie algebra of H: Xn (ei )(g. )s = gs ei ,

(2.7)

L(G) clearly inherites of an H action ψh : g. → g. h which preserves the measure since h−1 dBs h is still a Brownian motion in the Lie algebra of G. It acts over the vector fields Xn (ei )(g. ). . The lift over vector fields ψh∗ , which preserves the metric, is given by: ψh∗ Xn (ei )(g. )s = Xn (h−1 ei h)(g. h)s .

(2.8)

This means nothing else that gs αs ei h = gs h(h−1 αs ei h)

(2.9)

for αs a deteministic finite energy function. We consider the map p : LG → (LG)/H. The vectors Xn (ei )(g. ). span a Hilbert space H+ for n > 0, an Hilbert space H− for n < 0 and a space E for n = 0. We complexified these Hilbert spaces. By definition, p∗ T ((LG)/H) = H+ + H− + E, the sum being direct. We consider the Fermionic Fock space Λ(p∗ T ((LG)/H))).


1310

00087

R. L´ eandre

If we consider an L2 section of the Fermionic Fock space, we can average it in order to get an L2 section invariant under the H-action. Definition 2.1. The space of L2 forms over (LG)/H coincides with the space of L2 sections of Λ(p∗ T ((LG)/H))) which are invariant under the H-action. Let I = ((n1 , i1 ), . . . , (nr , ir )) be a finite set. XI (g. ) denotes the ordered wedge product of Xnj (eij )(g. ). P We consider as core E the set of finite combinations σ = FI (g. )XI (g. ) where FI (g. ) is a cylindrical functional. If σ belongs to E, we average it under the H-action: Z X (2.10) σinv = ψh∗ FI (g. )XI (g. )dπ(h) . H

We get a set of forms invariant under the H-action, and which is dense in the set of L2 sections of Λ(p∗ (T ((LG)/H))) which are invariant under the H action. Definition 2.2. Einv is the core constituted of H-invariant forms of the type σinv . Over p∗ T ((LG)/H)), we choose a connection. If ei is deterministic, it is given by ∇. Xn (ei )(g. ). = 0 .

(2.11)

Since ψh∗ Xn (ei )(g. ). = Xn (hei h−1 )(g. ). , the connection is invariant under the H action, because hei h−1 is deterministic if ei is deterministic. Moreover the connection preserves the metric. Definition 2.3. If σ ∈ E, we put dr,A σ: X An Xn (ei )(g. ). ∧ ∇Xn (ei )(g. ) σ , dr,A σ =

(2.12)

where Xn (ei )(g; ) is the orthogonal basis of p∗ T ((LG)/H) given by (2.5)–(2.7) and An is a sequence of deterministic numbers strictly positive which are smaller than |n|1/2− when n → ∞. The criterium over the A(n) is done in order that the series (2.12) converges in the L2 space of L2 sections of the Fermionic Fock bundle Λ(p∗ T (L(G)/H))). The series converges namely because for a cylindrical functional F , hdF, Xn (ei )(g. ). i has a behaviour in |n|−1 when |n| → ∞. This operator is invariant under the H action, because hei h−1 satisfy still (*), if ei is a basis of the Lie algebra of G satisfying (*). dr,A is well defined over E, and therefore over Einv , and apply an element of Einv over an L2 H-invariant section of Λ(p∗ T ((LG)/H))). Let us recall this property [3]: if ei is deterministic, we have for all cylindrical functionals the following integration by part formula: E[hdF, Xn (ei )(g; ). i] = E[F div Xn (ei )(g. ). ] , where divXn (ei )(g. ). belongs to all the Lp .

(2.13)


00087


1311

This allows to define d∗r,A over E, which is invariant under the H-action because dr,A is invariant under the H-action: X d∗r,A σ = (−∇Xn (ei )(g. ). . + div Xn (ei )(g. ). )An iXn (ei )(g. ). σ , (2.14) where the sum is finite because we suppose that σ belongs to E. This allows to define, since d∗r,A is invariant under the H-action, d∗r,A over Einv . Definition 2.4. The regularized de Rham operator over (LG)/H is the closure over Einv of dr,A + d∗r,A . Namely, dr,A + d∗r,A is symmetric over the core Einv , which is dense in the set of H-invariant L2 forms over (LG)/H. It is therefore closable. The introduction of numbers An is classical in quantum field theory (See [6, 22, 23]). Conjecture 2.5. The index of the regularized de Rham operator is equal to the Euler–Poincaré characteristic of G/H. 3. Limit Theorems We consider over the group the equation dg,s = g,s (dBs )

(3.1)

starting from g where Bs is a Brownian motion over the Lie algebra of G. It has a heat kernel ps, (g, g 0 ). Moreover, p1, (g, g) = p1, (e, e) clearly. Over the loop group, we consider the probability measure dµG, = dπ(g)⊗dP (g). P (g) is the law of the diffusion g,. starting from g constrained to come back to g. We consider a new Hilbert structure for vector fields of the type Xs = gs Ks . It is given by Z 1 (|Ks |2 + −2 |d/dsKs |2 )ds . (3.2) kX. k2g. , = 0

An orthogonal basis of the tangent space is given over H+ √ cos[2πns]ei , (3.3) Xn, (ei )s = g,s 2 √ Cn2 −2 + 1 where ei is an orthonormal basis of the Lie algebra of G and given over H− √ sin[2πns]ei , (3.4) Xn, (ei )s = g,s 2 √ Cn2 −2 + 1 where ei is an orthogonal basis of the Lie algebra of G. Over E, the basis is given by X0 (ei )s = g,s ei where ei belongs to the orthogonal of Lie(H). This means that we don’t do any rescaling over E. We identify Xn, (ei ). with Xn (ei ). . Let us compute the divergence of a vector field Xn, (ei ). for n > 0 and µG, . For that, we consider cos[2πns] λ ei = exp λ √ g,s cn2 −2 + 1


1312

00087

R. L´ eandre

λ where λ is a small real parameter. We consider the Eq. (2.1) starting from e. g,. g,. is the solution of a differential equation starting from g,0 : λ λ λ −1 λ λ −1 λ ) = g,s g,s ((g,s ) dBs (g,s ) + (g,s ) dg,s . d(g,s g,s

(3.5)

λ is equivalent to the original law, if we don’t fix It follows that the law of g,. g,. the starting point and the end point, that is if we consider the free path group, because the Haar measure is invariant under rotation and because the law of λ −1 λ ) dBs (g,s ) is still the law of a Brownian motion. By differentiating in λ = 0, (g,s we get infinitisimal integration by parts formulas. By using the quasi-sure analysis [4], this integration by parts formula can be desintegrated over the loop group. For cylindrical functionals F , we get

(3.6) E[hdF, Xn, (ei ). .i] = E[F div Xn, (ei ). ] R 1 1 (ei ). behaves in small time in C 0 hsin[2πns]ei , dBs i for n > 0, in where div Xn, small time because the law of g,. has an equivalent in g exp[B. + O(2 )] where B. is a flat Brownian bridge over the Lie algebra of G. This result is got by using Bismut’s procedure for short time asymptotics of heat kernels [11, 26, 27]. We have supposed that g,. starts from g. By the same argument, we show that in law for n > 0 Z 1 hcos[2πns]ei , dBs i . (3.7) div Xn, (ei ). → C 0

So we have the following lemma. Lemma 3.1. In law, when → 0, we have over the loop group, if n > 0, Z 1 div Xn, (ei ). → −C hsin[2πns]ei , dBs i

(3.8)

0

and if n < 0, Z

1

div Xn (ei ). →

Chcos[2πns]ei , dBs i ,

(3.9)

0

where Bs is a flat Brownian bridge over the Lie algebra of G. Let us remark that if we consider the vector field X0 (ei ). = g,s ei , its divergence is equal to 0, because the probability law P (g) is equal to the probability law of P (gg 0 ) by the transformation g. → g. g 0 . This leads to the introduction of a limit model, according Taubes. We consider the bundle Vg of ge, e belonging to the orthogonal of the Lie algebra of Lie H. We consider the set of continuous paths gflat,. in this bundle such that gflat,0 is equal to 0. Over this set of paths, we consider the measure dπ(g)⊗dP1,g,flat , where dPi,g,flat is the law of the flat Brownian bridge starting from 0 in Vg . It is the same as gBs1 (g) where Bs1 (g) is a Brownian bridge in the orthogonal of Lie H. The


00087


1313

tangent space is the space Xl,t of path in Vg with finite energy. As Hilbert norm, we take Z 1 kd/dsXl,s k2 dt = kXl,. k21 . (3.10) kXl,0 k2 + 0

√ R. An orthogonal basis is given by Xn,l (ei ). = −g 2 0 sin[2πns]ei ds for n > 0, by √ R. g q2 0 cos[2πns]ei ds = Xn,l (ei ). if n < 0 and by X0 (ei ). = gei if n = 0. For forms, we take the Hilbert structure Z 1 Z 1 kXl,t k2 dt + kd/dtXl,t k2 dt = kXl k22 . (3.11) 0

0

The complexified cotangent space for the Hilbert structure k . k22 has an orthonormal basis √ cos[2πns] ei for n > 0 , and X n,l (ei )s = g 2 √ Cn2 + 1 √ sin[2πns] ei X n,l (ei )s = g 2 √ Cn2 + 1

for n < 0 .

For n = 0, we don’t write the corresponding form. We introduce the following limit bundle Ξl = Λ(H− + H+ + E) = Λ(H− ) ⊗ Λ(H+ ) ⊗ Λ(E) ,

(3.12)

where the complexified vector are considered as forms with the Hilbert structure (3.11) by identifying Xn,l (ei ). with X n,l (ei ). . This bundle inherits a natural Haction, which transform X n,l (ei ) over g into X n,l (h−1 ei h) over gh. We say nothing else that gαs ei h = gh(h−1 αs ei h) for αs a smooth deterministic function. We consider the Bosonic Fock space B(Vg ) over gflat,. in Vg and we consider the bundle over G B(Vg ) ⊗ Ξl .

(3.13)

Over this limit supersymmetric Fock bundle, we have a connection, which acts as follow: we take the derivative of ei in X n,l (ei ) for the trivial connection. This means that ∇. X n,l (ei ). = 0 if ei is deterministic. It acts over the Wick polynomial : hgei αi , gflat,. i : by taking the derivative of ei and of g in the following formula, but not of dB.1 (g). Namely for αs a smooth path, we have Z 1 hgei αs , dgflat,s i hgei α. , gflat,. i = 0

Z

1

hgei αs , gdBs (g)i

= 0

Z

1

hei αs , dBs (g)i

= 0

(3.14)


1314

00087

R. L´ eandre

whose derivative along a vector field ge is equal to 0. Namely, the derivaR1 R1 tive of hgei α. , gflat,. i is equal to 0 hgeei αs , gdBs (g)i + 0 hgei αs , gedBs,flat i = R1 R1 heei αs , dBs (g)i + 0 hei αs , edBs (g)i. We embed the Lie group in a special 0 orthogonal group such that the elements of the Lie algebra become antisymmetric matrices and such that he, f i becomes CT r(ef t ). We deduce the relation hef, ki = −hf, eki and therefore that the equality (3.14) holds. This induces a connection over the bundle over G B(Vg ) ⊗ Ξl . To take the derivative of a Wick product, we proceed by taking the derivative of each term of the Wick product and by summing the expression which are got, as if it was a traditional product and a traditional derivation. This bundle inherits clearly an Haction, which is compatible with the connection, because h−1 ei h is still a element of the orthogonal of the Lie algebra of H or equivalently h−1 dBs (g)h is still a bridge in the orthogonal of the Lie algebra of H. Over the functional RBrownian 1 hge α , gdB i s s (g)i, the action of h ∈ H is defined by 0 Z 1 Z 1 Z 1 hghei αs , gdBs (g)hi = hhei h−1 αs , dBs (g)i = hei αs , h−1 dBs (g)hi . 0

0

0

This H-action over the supersymmetric Fock bundle acts by isometries over the L2 sections of this bundle in infinite dimensional Hilbert spaces over G. The connection is a unitary connection over it. The infinite dimensional part of the limit de Rham operator is built of the following operator and its adjoint. If we consider a local orthonormal basis gei of Vg which depends only on g for ei detreministic, we have X d∞,l = c(n)X n,l (ei ). ∧ ∇Xn,l (ei ). (3.15) i,n

for a family of deterministic constants c(n) 6= 0 satisfying c(n)2 < C|n|k + 1. We suppose that c(−n) = c(n) in order to simplify the exposure. It consists in a family parametrized by G of operators highly studied in probability (See [6, 7, 37]). This operator is clearly invariant under the H-action, and acts Rover the core El we will de1 fine later. Namely, if n > 0, Xn,l (ei ). .hgej α, gflat,. i = C 0 hgej αs , g sin[2πns]ei dsi, which is quickly decreasing because αs is chosen smooth. This means that the infinitesimal variation of dBs (g) under Xn,l (ei ). is −C sin[2πns]ei ds if ei belongs to the orthogonal of the Lie algebra of H if n > 0, an anologuous result being true for n < 0. The covariant derivative of a fermion X n,l (ei ). with ei deterministic along a vector field Xn,l (ei ). , n 6= 0 is equal to 0. The adjoint of d∞,l , denoted by d∗∞,l is defined by the following formula X ∗ ∂ ∞,l = c(n)(−∇Xn,l (ei ). + div Xn,l (ei ). )iXn,l (ei ). . (3.16) i,n

We put D∞,l = d∞,l + d∗∞,l .

(3.17)


00087


1315

This operator is clearly invariant under the H-action and defined over the core El we will define later. Namely, a vector field Xn,l (ei ). is transformed under the Haction into Xn,l (h−1 ei h) and the system of vectors h−1 ei h is still a basis of the orthogonal of the Lie algebra of H. The finite dimensional part of the limit de Rham operator is constructed as follow. We add a finite dimensional df,l operator: X X0,l (ei ). ∧ ∇X0 (ei ). , (3.18) df,l = i

which operates over the bundle B(Vg ) ⊗ Ξl (ei runs over an orthonormal basis of the orthogonal of the Lie algebra of H). We compute its adjoint: with the notations of (2.27), we have X −∇X0 (ei ). iX0 (ei ). + div X0 (ei ). iX0 (ei ). . (3.19) d∗f,l = i

In order to summarize us, they are two parts in the limit de Rham operator Dl = df,l + df,l + d∞,l + d∗∞,l .

(3.20)

There is a finite dimensional part which acts over the bundle B(Vg )⊗Ξl and infinite dimensional part which acts over the Brownian bridge paths in the fiber of the linear bundle Vg , and which consists in a family parametrized by G of Gaussian de Rham operators highly studied in probability (See [6, 7]). This limit de Rham operator acts over the following core: it is define as finite combinations of F (g) : hgαi eli , dgs,flat i : X I,l where αi is smooth deterministic and X I,l a wedge product of X n,l (ei ) and F a smooth functional over G. We can average under the limit H-action, and we got a core El,inv of L2 sections of Ξl , which are invariant under the H-action. El,inv is dense in the set of L2 sections of Ξl which are invariant under the H-action. The main remark is the following: if the analysis of Dl acting over El does not give any information, because the bundle Ξl is trivial, the analysis over El,inv will give information over G/H. Theorem 3.2. The index of the limit de Rham operator over the set of H-invariant sections of Ξl is equal to the Euler–Poincaré characteristic of G/H χ(G/H) if the numbers c(n) are not equal to 0. Proof. Let us compute ∆l = Dl2 . The infinite dimensional part and the finite dimensional part of the limit de Rham operator anticommute. Therefore, we have ∆l = (df,l + d∗f,l )2 + (d∞,l + d∗∞,l )2 = ∆f,l + ∆∞,l .

(3.21)

Let us consider NB,c the second Rquantized Bosonic number operator which 1 counts the number of Bosons, a Boson 0 hgei cos[2πns], dgflat,s i being counted with R 1 the multiplicity c(n)2 and a Boson 0 hgei sin[2πns], dgflat,s i being counted with multiplicity c(n)2 . In order to understand this operator, let us denote by hn (s) = sin[2πns] if n < w0 and by cos[2πns] if n > 0. A Wick product : hgeik hnk , gflat,. i :


1316

00087

R. L´ eandre

P is an eigenvector associated to NB,c with the eigenvalue c(nk )2 . Let us introduce NF,c the second quantized Fermionic number operator: a fermion X n,l (ei ) is counted with the multiplicity c(n)2 . In order to understand, we have the same formula than before, Wick products being replaced by exterior products of the fermion X n,l (ei ). Following [7] and [6], ∆∞,l = NB,C + NF,c . Therefore, we can diagonalize ∆∞,l over El,inv , because the H-action is compatible with the decomposition of NB,c and NF,c in finite dimensional eigenspaces. The kernel of ∆∞,l coincide with H-invariant sections of Λ(Vg ) over G, which depends only on G, because the numbers c(n) are all different of 0. But df,l +d∗f,l over this space of H-invariant sections can be seen as an operator homotopic to the de Rham operator over G/H (and not equal because we don’t take the pullback of the Levi–Civita connection over G/H). Therefore, if we use the invariance of the index under homotopy of an elliptic operator: Ind(df,l + d∗f,l ) = χ(G/H) ,

(3.22)

where the index is computed over H-invariant section of Λ(V. ). Let A = ((i1 , n1 ), . . . , (ir , nr )). We put Y Z 1 hg,s ei hn (s), dg,s i = ξ(A) = (i,n)∈A

0

Y Z (i,n)∈A

1

hei hn (s), dBs i ,

(3.23)

0

−1 dg,s . Let B of the same type of A, with the restriction that the where dBs = g,s eij belongs to the orthogonal of the Lie algebra of H and C of the same type of A, the eij being in the Lie algebra of H. We denote XB,C,. the wedge products of Xn (ei ). for (i, n) belonging to B and belonging to C. If D = (eij ) where eij belonging to the orthogonal of the Lie algebra of H, we denote by X0,D,. the wedge product of X0 (eij ). for eij in D. As core E , we choose the set of finite combinations X (3.24) σ= Fi (g,0 )λA,B,C,D ξ(A)XB,C,. X0,D,. ,

where the Fi are a finite collection of smooths functionals over G. The core E is invariant under the H-action. E,inv is the core which is got from E by averaging σ under the H-action. We define Bismut’s dilatation B which acts over ξ(A) by Y Z 1 hg,s ei hn (s), dg,s idt , (3.25) B ξ(A) = {αi } 0 where {αi } = 1 if ei belongs to the orthogonal of the Lie alebra of H and equal 1/2 if ei belongs to the Lie algebra of H. Bismut’s dilatation acts over wedge products XC,D,. by multiplying each elements of the wedge product of the type Xn (ei ) with ei belonging to the Lie algebra of H by .


00087


Definition 3.3. Bismut’s dilatation over the core E is defined by X Fi (g,0 )λ(A, B, C, D, i)B ξ(A)B (XB,C,. ) ∧ X0,D,. ⊗ . B σ(g,. ) =

1317

(3.26)

Lemma 3.4. Bismut’s dilatation are well defined over E . Proof. We suppose in order to simplify that = 1. The result will follow of the following property: if X λ(A, B, C, D)ξ(A)XB,C,. ∧ X0,D,. = 0

(3.27)

A,B,C,D

for the based loop group of loops starting from g, then R 1 λ(A, B, C, D) = 0. Let us introduce the random variables ζ(i, n) = 0 hei cos[2πns], dBs i for n ≥ 0 R1 and ζ(i, n) = 0 hei sin[2πns], dBs i for n < 0. We suppose in order to simplify the exposure that the diffusion g. starts from e. We consider the measure over G × RN µ: F → E[f (g1 , ζ(., .)] ,

(3.28)

where we consider |n| ≤ C. Let us show that µ has an absolutely continuous part which has a strictly positive density in (e, ζ) for some convenient ζ. This will show the result. gs is the linear diffusion starting from e solution of the differential equation dgs = gs dBs where Bs is a Brownian motion in the Lie algebra of G. It is enough to show the following property: let us replace formally dBs by hs ds with the L2 topology over h. . g1 is replaced by g1 (h. ) and ζ(., .) by ζ(., .)(h. ). The property says that the application Ψ: h. → (g1 (h. ), ζ(., .)(h. ) is a submersion from a small h. such that g1 (h. ) = e. By the positivity theorem of [9] (See [2] for an abstract version), the density part of µ in (e, ζ(., .)(h. ) is strictly positive. If we don’t suppose that n can be equal to zero, it is clearly a submersion in h. = 0. The only problem in order that Ψ is a submersion in h = 0 is n = 0. But, we can perturb a little bit h. in order to get a submersion, because there are multiple iterated integrals which appear in the expression of g1 (h. ). Let us precise a little bit this statement, after imbedding the Lie group in an orthogonal matrix group. Let us denote by Dgs (h. )(h0. ) the expression of the derivative of gs (h. ) in the direction h0. . It is the solution of the differential equation dDgs (h. )(h0. ) = Dgs (h. )(h0. )hs ds + gs (h. )h0s ds .

(3.29)

We can solve this equation by the method of variation of constants. We get Z 1 gs (h. )h0s gs−1 (h. )dsg1 (h. ) . (3.30) Dg1 (h. )(h0. ) = 0

It is enough to choose a small h such that g1 (h) = e and such that s → gs (h. ).gs−1 (h. ) is not proportional to s → cos[2πns] and to s → sin[2πns] for |n| ≤ n0 in order to prove the assertion for (g1 (h), ζ(., .)).


1318

00087

R. L´ eandre

We would like to compute the limit of B σ(g,. ) when → 0 in order to justify the appellation “limit model” which was used previously. Definition 3.5. We say that a sequence of sections σ(g,. , ) of the type X FB,C,D (g,. , )XB,C,. ∧ X0,D,. σ(g,. , ) =

(3.31)

B,C,D

tends in law to the section σ(gflat,. ) of the limit model defined by X ξB,C,D (gflat,. )X B,C,l,. ∧ Y 0,D,l,. σ(gflat,. ) =

(3.32)

B,C,D,

if the family of random variables FB,C,D (g,. , ) in L2 (N ) tends in law to the family P of random variable ξB,C,D (gflat,. ) and if E[ FI,J,K (g,. , )|2 ] remains bounded. In this definition, it is supposed that the random variables FB,C,D (g,. , ) tends in law to zero if there is in XB,C,i,. a Xn (ei ). considered as vector or a X n,l (ei ). considered as a form at the limit associated to an element ei of the Lie algebra of H. Proposition 3.6. If σ(g,. ) is a section of the type (3.26), B σ(g,. ) tends in law to a section of the limit model σl . Proof. It will result from the following observation. The contribution which is normalized and which comes from the loop close of the constant loops can be treated by the following constatation (See [11] and [26]): in law, when → 0 g,. = g exp[B. + O(2 )] for a Brownian bridge B. over the Lie algebra of G. This shows that, in law: Y Z 1 hgei hn (s), dgflat,s i B ξ(A) → i,n∈A

(3.33)

(3.34)

0

if all the ei belong to the orthogonal of the Lie algebra of H and tends to 0 in law in the others cases. Moreover the H-action is compatible with Bismut’s dilatation and tends to the limit H-action over the limit model. We define the regularized de Rham operator over the core E by D = d,∞ + d∗,∞ , where d,∞ =

X n6=0,i

Xn, (ei ). ∧ ∇Xn, (ei ). +

X fj

(3.35)

X0 (fj ). ∧ ∇X0 (fj ). ,

(3.36)


00087


1319

where the fj run over the orthogonal of the Lie algebra of H. This shows X −∇Xn, (ei ). iXn (ei ). + div Xn, (ei ). iXn (ei ). d∗,∞ = n6=0,i

+

X

−∇X0 (fj ). iX0 (fj ). + div X0 (fj ). iX0 (fj ).

(3.37)

j

This operator is clearly invariant under the H-action. Let us remark that D operates over the core E . Namely: Z 1 Xm, (ej ). · hg,s hn (s)ei , dg,s i 0

Z

1

hg,s hn (s)ei , g,s h−m (s)ej ids

= O(1/m) + C(, m) 0

= O(1/n) + C(, m, n)δm,−n .

(3.38)

We have the following theorem. Theorem 3.7. When → 0, we have in law if σ(g,. ) is a section belonging to E D B σ(g,. ) → Dl σl

(3.39)

for some c(n) 6= 0, σ(g,. ) and σl being given as in Proposition 3.6. Proof. The treatment of the covariant derivative of B (XB,C,. ) and X0,D,. is trivial, with the chosen connections. The treatment of div Xn, (ei ). is given at Lemma 3.1. It remains to remark that div Xn, (ei ). at the limit does not appear, because there is an interior product in D , iXn (ei ). , and that the corresponding forms are multiplied by and disappear at the limit if ei belongs to the Lie algebra of H. It remains to consider Xn, (ei ). · B ξ(A) or equivalently

Z Xn, (ei ). ·

1

hg,s hm (s)ej , δg,s i/{αj } .

(3.40)

(3.41)

0

Let us consider first n 6= 0 and ei belonging to the orthogonal of Lie H. It tends in law to Z 1 − hghm (s)ej , gh−n (s)ei i , (3.42) 0

R1

which is the derivative of 0 hghm (s)ej , dgflat,s i along Xn,l (ei ). . If n 6= 0 and ei belongs to LieH, Xn, (ei ). = O() and Z 1 Xn, (ei ). · hg,s hm (s)ej , dg,s i/1/2 = O(1/2 ) 0

(3.43)


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and tends to zero, which is the derivative of the limit of this expression, which is 0. If n = 0 and if ej belongs to the orthogonal of the Lie algebra of H: Z 1 X0 (ei ). · hg,s hm (s)ej , δg,s i/ 0

Z

Z

1

hg,0 ei hm (s)ej , δg,s i/ +

= 0

1

hg,s hm (s)ej , δg,s ei i/

which tends in law to Z 1 Z 1 hgei hm (s)ej , gdBs i + hghm (s)ej , gdBs ei i = 0 0

(3.44)

0

(3.45)

0

R1 which is the derivative of 0 hghm (s)ej , dgflat,s i in the direction gei . If ej belongs to the Lie algebra of h, it tends to 0, by the same argument. We have considered the limit in law of each component of the section σ(g,. ), but the collection of each components converges in law. The last property of Proposition 3.5. is clearly checked. Since D commute with the H-action, which is compatible with Bismut’s dilatation, Theorem 2.7 remains true for an element σ(g,. ) of E, inv . We get at the limit an element of El,inv , where Dl acts. References [1] S. Aida, “Sobolev Spaces over loop groups”, J. Funct. Anal. 127 (1995) 155–172. [2] S. Aida, S. Kusuoka and D. Stroock, “On the support of Wiener functionals,” pp. 3–35 in Asymptotics Problems in Probability Theory: Wiener Functionals and Asymptotics, eds. K. D. Elworthy and N. Ikeda, Longman Scientific 284, 1993. [3] H. Airault and P. Malliavin, Integration on Loop Groups, Publication Paris VI, 1991. [4] H. Airault and P. Malliavin, Quasi-sure Analysis, Publication Paris VI, 1991. [5] S. Albévério and R. Hoegh-Krohn, “The energy representation of Sobolev Lie groups”, Compositio Math. 36 (1978) 37–52. [6] A. Arai, “A path integral representation of the index of Kaehler-Dirac operators on an infinite dimensional space”, J. Funct. Anal. 82 (1989) 330–369. [7] A. Arai and I. Mitoma, “De Rham-Hodge-Kodaira decomposition in infinite dimension”, Math. Anna. 291 (1991) 51–73. [8] M. Atiyah, “Circular symmetry and stationary phase approximation”, pp. 43–59 in Conference in Honour of L. Schwartz, Astérisque 131, 1985. [9] G. Ben Arous and R. Léandre, “Décroissance exponentielle du noyau de la chaleur sur la diagonale, II”, Probab. Theory Related Fields 90 (1991) 377–402. [10] N. Berline and M. Vergne, “Zéros d’un champ de vecteur et classes caractéristiques équivariantes”, Duke Math. J. 50 (1983) 539–548. [11] J. M. Bismut, Large Deviations and the Malliavin Calculus, Progress in Math. 45, Birkhauser 1984. [12] J. M. Bismut, “The Lefschetz fixed point formulas”, J. Funct. Anal. 57 (1984) 329– 348. [13] J. M. Bismut, “Index theorem and equivariant cohomology on the loop space”, Comm. Math. Phys. 98 (1985) 213–237.


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[14] J. M. Bismut, “Localisation formulas, superconnections and the index theorem for families”, Comm. Math. Phys. 103 (1986) 127–166. [15] L. Dixon, J. Harvey, C. Vafa and E. Witten, “Strings on orbifold”, Nuclear Phys. B261 (1985) 678–686. [16] S. Fang and J. Franchi, “De Rham-Hodge-Kodaira operator on loop groups”, J. Funct. Anal. 148 (1997) 391–407. [17] P. B. Gilkey, Invariance Theory, the Heat Equation and the Atiyah-Singer Theorem, Publish and Perish, 1984. [18] L. Gross, “Logarithmic Sobolev inequalities on loop groups”, J. Funct. Anal. 102 (1991) 268–313. [19] L. Gross, “Uniqueness of ground states for Schrodinger operators over loop groups”, J. Funct. Anal. 112 (1993) 373–441. [20] F. Hirzebruch, T. Berger and R. Jung, Manifolds and Modular Forms, Aspect of mathematics 20, Vieweg 1993. [21] F. Hirzebruch, “Elliptic genera genera of level N for complex manifolds”, pp. 37–63 in Differential Geometrical Methods in Theoretical Physics, eds. K. Bleuler and M. Werner, Kluwer, 1988. [22] R. Hoegh-Krohn, “Relativistic quantum statistical mechanics in 2-dimensional space time”, Comm. Math. Phys. 38 (1974) 195–224. [23] A. Jaffe, A. Lesniewski and J. Weitsman, “Index of a family of Dirac operators on loop space”, Comm. Math. Phys. 112 (1987) 75–88. [24] J. D. S. Jones and R. Léandre, “A stochastic approach to the Dirac operator over the free loop space”, pp. 253–282 in Loop Spaces and Groups of Diffeomorphisms, Proceedings of the Steklov Institute 217, 1997. [25] S. Kusuoka, “More recent theory of Malliavin calculus”, Sugaku 5(2) (1992) 155–173. [26] R. Léandre, “Intégration dans la fibre associée a une diffusion dégénérée”, Probab. Theory Related Fields 76 (1987) 341–358. [27] R. Léandre, “Applications quantitatives et qualitatives du Calcul de Malliavin”, pp. 109–133 in French-Japanese Seminar, eds. M. Métivier and S. Watanabe, Lecture Notes Math. 1322, 1988. English translation, pp. 173–197 in Geometry of Random Motion, eds. R. Durrett and M. Pinsky, Contemporary Maths. 73, 1988. [28] R. Léandre, “Brownian motion over a Kaehler manifold and elliptic genera of level N”, pp. 193–219 in Stochastic Analysis and Applications in Physics, eds. R. Sénéor and L. Streit, Nato Asie Series 449, 1994. [29] R. Léandre, “Cover of the Brownian bridge and stochastic symplectic action”, Rev. Math. Phys. 12(1) (2000) 91–137. [30] R. Léandre, “Analysis over loop space and topology”, to be published in Math. Notes. [31] R. Léandre, “Quotient of a loop group and Witten genus”, to be published in J. Math. Phys. [32] R. Léandre and S. S. Roan, “A stochastic approach to the Euler-Poincar´ e number of the loop space of a developable orbifold”, J. Geom. Phys. 16 (1995) 71–98. [33] P. Malliavin, Stochastic Analysis, Springer, 1997. [34] T. Nishimura, “Exterior bundle over a complex abstract Wiener space” (preprint), 1990. [35] G. Segal, “Elliptic cohomology”, pp. 187–201 in Séminaire Bourbaki 695, Astérisque 161–162, 1988. [36] I. Shigekawa, “Transformations of Brownian motion on a Riemannian symmetric space”, Z. Wahrsch. Verw. Gebiete 65 (1984) 493–522. [37] I. Shigekawa, “De Rham-Hodge-Kodaira’s decomposition on an abstract Wiener space”, J. Math. Kyoto Univ. 26 (1986) 191–202.


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[38] I. Shigekawa, “Differential calculus on a based loop group”, pp. 375–398 in New Trends in Stochastic Analysis, eds. K. D. Elworthy, S. Kusuoka and I. Shigekawa, World Scientific, 1997. [39] C. Taubes, “S 1 actions and elliptic genera”, Comm. Math. Phys. 122 (1989) 455–526. [40] T. Uglanov, “Surface integrals and differential equations on an infinite dimensional space”, Sov. Math. Dok. 20(4) (1979) 917–920. [41] S. Watanabe, “Stochastic analysis and its applications”, Sugaku 5(1) (1992) 51–71. [42] E. Witten, “The index of the Dirac operator in loop space”, pp. 161–181 in Elliptic Curves and Modular Forms in Algebraic Topology, ed. P. S. Landweber, Lecture Note Math. 1326, 1988.

October 19, 2001 17:49 WSPC/148-RMP

00105


RENORMALIZATION GROUP, HIDDEN SYMMETRIES AND APPROXIMATE WARD IDENTITIES IN THE XY Z MODEL

G. BENFATTO and V. MASTROPIETRO Dipartimento di Matematica, Universit` a di Roma “Tor Vergata” Via della Ricerca Scientifica, I-00133, Roma [email protected] [email protected]

Received 15 February 2001 Revised 17 May 2001 Using renormalization group methods, we study the Heisenberg–Ising XY Z chain in an external magnetic field directed as the z axis, in the case of small coupling J3 in the z direction. In particular, we focus our attention on the asymptotic behaviour of the spin correlation function in the direction of the magnetic field and the singularities of its Fourier transform. An expansion for the ground state energy and the effective potential is derived, which is convergent if the running coupling constants are small enough. Moreover, by using hidden symmetries of the model, we show that this condition is indeed verified, if J3 is small enough, and we derive an expansion for the spin correlation function. We also prove, by means of an approximate Ward identity, that a critical index, related with the asymptotic behaviour of the correlation function, is exactly vanishing, together with other properties, so obtaining a rather detailed description of the XY Z correlation function.

1. Introduction 1.1. If (Sx1 , Sx2 , Sx3 ) = 12 (σx1 , σx2 , σx3 ), for i = 1, 2, . . . , L, σiα , α = 1, 2, 3, being the Pauli matrices, the Hamiltonian of the Heisenberg–Ising XY Z chain is given by H =−

L−1 X

1 2 3 [J1 Sx1 Sx+1 + J2 Sx2 Sx+1 + J3 Sx3 Sx+1 + hSx3 ] − hSL3 + UL1 ,

(1.1)

x=1

where the last term, to be fixed later, depends on the boundary conditions. The space-time spin correlation function at temperature β −1 is given by α α α α Ωα L,β (x) = hSx S0 iL,β − hSx iL,β hS0 iL,β ,

(1.2)

where x = (x, x0 ), Sxα = eHx0 Sxα e−Hx0 and h·iL,β = Tr[e−βH .]/Tr[e−βH ] denotes the expectation in the grand canonical ensemble. We shall use also the notation Ωα (x) ≡ limL,β→∞ Ωα L,β (x). 1323


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The Hamiltonian (1.1) can be written [14] as a fermionic interacting spinless Hamiltonian. In fact, it is easy to check that the operators "x−1 # Y ± 3 (−σy ) σx± (1.3) ax ≡ y=1

are a set of anticommuting operators and that, if σx± = (σx1 ± iσx2 )/2, we can write σx− = e−iπ

Px−1 y=1

− a+ y ay a− , x

iπ σx+ = a+ xe

Px−1 y=1

− a+ y ay

,

− σx3 = 2a+ x ax − 1 .

(1.4)

Hence, if we fix the units so that J1 + J2 = 2 and we introduce the anisotropy u = (J1 − J2 )/(J1 + J2 ), we get L−1 X 1 u + + − + − − − − [a+ H= x ax+1 + ax+1 ax ] − [ax ax+1 + ax+1 ax ] 2 2 x=1 L X 1 1 1 + − − + − a − h + UL2 , a − a − a − a − J3 a+ x x x x x+1 x+1 2 2 2 x=1

(1.5)

where UL2 is the boundary term in the new variables. We choose it so that the fermionic Hamiltonian (1.5) coincides with the Hamiltonian of a fermion system on the lattice with periodic boundary conditions, that is we put UL2 equal to the term ± in the first sum in the r.h.s. of (1.5) with x = L and a± L+1 = a1 (in [14] this choice for the XY chain is called “c-cyclic”). It is easy to see that this choice corresponds to fix the boundary conditions for the spin variables so that 1 + iπN − − iπN + e σ1 + σL e σ1 ] UL1 = − [σL 2 J3 3 3 u + iπN + − iπN − e σ1 + σL e σ1 ] − σL σ1 , − [σL 2 4

(1.6)

PL 1 where N = x=1 a+ x ax . Strictly speaking, with this choice UL does not look really like a boundary term, because N depends on all the spins of the chain. However [(−1)N , H] = 0; hence the Hilbert space splits up in two subspaces on which (−1)N is equal to 1 or to −1 and on each of these subspaces UL1 really depends only on the boundary spins. One expects that, in the L → ∞ limit, the correlation functions are independent on the boundary term, but we shall not face here this problem. 1.2. The Heisenberg XY Z chain has been the subject of a very active research over many years with a variety of methods. A first class of results is based on the exact solutions. If one of the three parameters is vanishing (e.g. J3 = 0), the model is called XY chain. Its solution is based on the fact that the Hamiltonian, in the fermionic form (1.5), is quadratic in the fermionic fields, so that it can be diagonalized (see [14, 15]) by a Bogoliubov transformation. If u = 0, we get the free Fermi gas with Fermi momentum pF = arccos(−h); if |u| > 0, it turns out that the energy spectrum has a gap at pF .


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Renormalization Group, Hidden Symmetries and Approximate Ward Identities

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The equal time correlation functions Ωα (x, 0) were explicitly calculated in [18] (even at finite L and β), in the case h = 0, that is pF = π/2. Note that, while Ω3 (x) coincides with the correlation function of the density in the fermionic representation of the model, Ω1 (x) and Ω2 (x) are given by quite complicated expressions. It turns out, for example, that, if |u| < 1, Ω3 (x, 0) is of the following form: πx α|x| F (−|x| log α, |x|) , α = (1 − |u|)/(1 + |u|) , (1.7) Ω3 (x, 0) = − 2 2 sin2 π x 2 where F (γ, n) is a bounded function, such that, if γ ≤ 1, F (γ, n) = 1 + O(γ log γ) + O(1/n), while, if γ ≥ 1 and n ≥ 2γ, F (γ, n) = π/2 + O(1/γ). For |h| > 0, it is not possible to get a so explicit expression for Ω3 (x, 0). However, it is not difficult to prove that, if |u| < sin pF , |Ω3 (x, 0)| ≤ α|x| and, if x 6= 0 and |ux| ≤ 1 1 sin2 (pF x)[1 + O(|ux| log|ux|) + O(1/|x|)] . (1.8) π 2 x2 Note that, if u = 0, a very easy calculation shows that Ω3 (x, 0) = −(π 2 x2 )−2 sin2 (pF x). We want to stress that the only case in which the correlation functions and their asymptotic behaviour can be computed explicitly in a rigorous way is just the J3 = 0 case. If two parameters are equal (e.g. J1 = J2 ), but J3 6= 0, the model is called XXZ model. In the case h = 0, it was solved in [24] via the Bethe-ansatz, in the sense that the Hamiltonian was diagonalized. However, it was not possible till now to obtain the correlation functions from the exact solution. Such solution is a particular case of the general solution of the XY Z model by Baxter [2], but again only in the case of zero magnetic field. The ground state energy has been computed and it has been proved that there is a gap in the spectrum, which, if J1 − J2 and J3 are not too large, is given approximately by (see [12]) π 2µ |J12 − J22 | sin µ |J1 | (1.9) ∆ = 8π µ 16(J12 − J32 ) Ω3 (x, 0) = −

with cos µ = −J3 /J1 . The solution is based on the fact that the XY Z chain with periodic boundary conditions is equivalent to the eight vertex model, in the sense that H is proportional to the logarithmic derivative with respect to a parameter of the eight vertex transfer matrix, if a suitable identification of the parameters is done, see [2, 21]. The eight vertex model is obtained by putting arrows in a suitable way on a two-dimensional lattice with M rows, L columns and periodic boundary conditions. There are eight allowed vertices, and with each of them an energy is associated in a suitable way (there are four different values of the energy). With the above choice of the parameters and T −Tc < 0 and small, u = O(|T −Tc |), so that the critical temperature of the eight vertex model corresponds to no anisotropy in the XY Z chain. Moreover, see [11], the correlation function Cx between two vertical arrows in a row, separated


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by x vertices, is given, in the limit M → ∞, by Cx = hS03 Sx3 i. However, an explicit expression for the correlation functions cannot be derived for the XY Z or the eight vertex model. In [11] the correlation length of Cx was computed heuristically under some physical assumptions (an exact computation is difficult because it does not depend only on the largest and the next to the largest eigenvalues). The result is π ξ −1 = (T − Tc ) 2µ , if ξ is the correlation length. One sees that the critical index of the correlation length is non-universal. Another interesting observation is that the XY Z model is equivalent to two interpenetrating two-dimensional Ising lattices with nearest-neighbor coupling, interacting via a four spins coupling (which is proportional to J3 ). The four spin correlation function is identical to Cx . In the decoupling limit J3 = 0 the two Ising lattices are independent and one can see that the Ising model solution can be reduced to the diagonalization, via a Bogoliubov transformation, of a quadratic Fermi Hamiltonian, see [15]. Recent new results using the properties of the transfer matrix can be found in [9], in which an integro-difference equation for the correlation function of the XXZ chain is obtained. It is however not clear how to deduce the physical properties of the correlation function from this equation. 1.3. Since it is very difficult to extract detailed information on the behaviour of the correlation functions from the above exact solutions, the XY Z model has been studied by quantum field theory methods, see [12]. The idea is to approximate the fermionic Hamiltonian (1.5) by the Hamiltonian of the massive Thirring model, describing a massive relativistic spinning particle on the continuum d = 1 space interacting with a local current-current potential (for a heuristic justification of this approximation, see [1]). As a relativistic field theory, the massive Thirring model is plagued by ultraviolet divergences, which were absent in the original model, defined on a lattice; one can heuristically remove this problem by introducing “by hand” an ultraviolet cut-off. A way to introduce it could be to consider a short-ranged instead of a local potential; if J1 = J2 , this means that we have approximated the XXZ-chain with the Luttinger model, whose correlation functions can be explicitly computed, see [5, 19]. The Luttinger model is defined in terms of two fields ψx,ω , ω = ±1, and one expects that, if |h| < 1 and J3 is small enough, the large distance asymptotic behaviour of Ω3 (x) is qualitatively similar to that of the truncated correlation of P σ , if some “reasonable” the operator ρx = ψx+ ψx− , where ψxσ = ω exp(iσωpF x)ψx,ω relationship between the parameters of the two models is assumed. One can make for instance the substitutions λ → −J3 and p−1 0 → a = 1, if λ is the coupling in the Luttinger model, a is the chain step and p−1 0 is the potential range. Moreover, one expects that it is possible to choose a constant ν of order J3 , so that h = h0 + ν and pF = arccos(J3 − h0 ), see Sec. 1.4 below. Of course such identification is completely arbitrary, but one can hope that for large distances the function Ω3 (x) has something to do with the truncated


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correlation of ρx , which can be obtained by the general formula (2.5) of [5], based on the exact solution of [19]. There is apparently a problem, since the expectation of ρx is infinite; however, it is possible to see that there exists the limit, as ε1 , ε2 → 0+ , + ψ− , and it is natural to of [hρx,ε1 ρy,ε2 i − hρx,ε1 ihρy,ε2 i], where ρx,ε = ψ(x,x 0 +ε) (x,x0 ) take this quantity, let us call it G(x − y), as the truncated correlation of ρx . Let us define v0 = sin pF ; from (2.5) of [5] (by inserting a missing (−εi εj ) in the last sum), it follows that, for |x| → ∞ G(x) ' [1 + λa1 (λ)]

cos(2pF x) (v0 x0 )2 − x2 + 2 , 2 1+λa (λ) 3 2π [(v0 x0 )2 + x2 ]2 +x ]

2π 2 [(v0∗ x0 )2

(1.10)

where v0∗ = v0 [1 + λa2 (λ)] and ai (λ), i = 1, 2, 3, are bounded functions. Note that, in the second term in the r.h.s. of (1.10), the bare Fermi velocity v0 appears, instead of the renormalized one, v0∗ , as one could maybe expect. In the physical literature, it is more usual the introduction of other ultraviolet cutoffs, such that the resulting model is not exactly soluble, even if J1 = J2 ; however, it can be studied heuristically, see [12], and the resulting density-density correlation function is more or less of the form (1.10). If J1 6= J2 , there is no soluble model suitable for a similar analysis of the large distance behaviour of Ω3 (x). However, one can guess that the asymptotic behaviour is still of the form (1.10), if 1 |x| 1/|u|α , for some α. We shall prove that this is indeed true, with α = 1 + O(J3 ). 1.4. In this paper we develop a rigorous renormalization group analysis for the XY Z Hamiltonian in its fermionic form (some “not optimal” bounds for the correlation function Ω3 (x) were already found in [17]). As we said before, Ω3 (x) can be obtained from the exact solution only in the case J3 = 0, when the fermionic theory is a noninteracting one. In particular, if x = (x, 0) and |ux| 1, (1.8) and a more detailed analysis of the “small” terms in the r.h.s. (in order to prove that their derivatives of order n decay as |x|−n ), show that Ω3 (x, 0) is a sum of “oscillating” functions with frequencies (npF )/π mod 1, n = 0, ±1, where pF = arccos(−h); this means that its Fourier transform has to be a smooth function, even for u = 0, in the neighborhood of any momentum k 6= 0, ±2pF . These frequencies are proportional to pF , so they depend only on the external magnetic field h. If J3 6= 0, a similar property is satisfied for the leading terms in the asymptotic behaviour, as we shall prove, but the value of pF depends in general also on u and J3 . For example, if u = 0, the Hamiltonian (1.5) is equal, up to a constant, to the Hamiltonian of a free fermion gas with Fermi momentum pF = arccos(J3 − h) plus an interaction term proportional to J3 . As it is well known, the interaction modifies the Fermi momentum of the system by terms of order J3 and it is convenient (see [3], for example), in order to study the interacting model, to fix the Fermi momentum to an interaction independent value, by adding a counterterm to the Hamiltonian. We proceed here in a similar way, that is we fix pF and h0 so that h = h0 − ν ,

cos pF = J3 − h0 ,

(1.11)


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and we look for a value of ν, depending on u, J3 , h0 , such that, as in the J3 = 0 case, the leading terms in the asymptotic behaviour of Ω3L,β (x) can be represented as a sum of oscillating functions with frequencies (npF )/π mod 1, n = 0, ±1. As we shall see, we can realize this program only if J3 is small enough and it turns out that ν is of order J3 . It follows that we can only consider magnetic fields such that |h| < 1. Moreover, it is clear that the equation h = h0 − ν(u, J3 , h0 ) can be inverted, once the function ν(u, J3 , h0 ) has been determined, so that pF is indeed a function of the parameters appearing in the original model. If J1 = J2 , it is conjectured, on the base of heuristic calculations, that to fix pF is equivalent to the impose the condition that, in the limit L, β → ∞, the density is fixed (“Luttinger Theorem”) to the free model value ρ = pF /π. Remembering that ρ − 12 is the magnetization in the 3-direction for the original spin variables, this would mean that to fix pF is equivalent to fix the magnetization in the 3 direction, by suitably choosing the magnetic field. If J1 6= J2 , there is in any case no simple relation between pF and the mean magnetization, as one can see directly in the case J3 = 0, where one can do explicit calculations. The only exception is the case pF = π/2, where one can see that, in the limit L → ∞, ν = J3 (so that h = 0 by (1.11)) and hSx3 i = 0. This last property easily follows from the observation that, if one choose h = 0 in the original Hamiltonian (1.1), then the expectation of Sx3 has to be equal to zero, by symmetry reasons, up to terms which go to 0 for L → ∞. Our main achievement is an expansion of Ω3L,β (x), which provides a very detailed and explicit description of it. We state in the following theorem some of its properties, but we stress that many other interesting properties of Ω3L,β (x) can be extracted from the expansion. 1.5 Theorem. Suppose that Eq. (1.11) are satisfied and that v0 = sin pF ≥ v¯0 > 0, for some value of v¯0 fixed once for all, and let us define a0 = min{pF /2, (π−pF )/2}; then the following is true. (a) There exists a constant ε, such that, if (u, J3 ) ∈ A, with a0 √ , |J3 | ≤ ε , (1.12) A = (u, J3 ) : |u| ≤ 8(1 + 2) it is possible to choose ν, so that |ν| ≤ c|J3 |, for some constant c independent of L, β, u, J3 , pF , and the spin correlation function Ω3L,β (x) is a bounded (uniformly in L, β, pF and (u, J3 ) ∈ A) function of x = (x, x0 ), x = 1, . . . , L, x0 ∈ [0, β], periodic in x and x0 of period L and β respectively, continuous as a function of x0 . (b) We can write 3,b 3,c Ω3L,β (x) = cos(2pF x)Ω3,a L,β (x) + ΩL,β (x) + ΩL,β (x) ,

(1.13)

with Ω3,i L,β (x), i = a, b, c, continuous bounded functions, which are infinitely times differentiable as functions of x0 , if i = a, b. Moreover, there exist two constants η1 and η2 of the form


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η1 = a1 J3 + O(J32 ) ,

η2 = −a2 J3 + O(J32 ) ,

1329

(1.14)

a1 and a2 being positive constants, uniformly bounded in L, β, pF and (u, J3 ) ∈ A, such that the following is true. Let us define πx β πx0 L sin , sin (1.15) d(x) = π L π β and suppose that |d(x)| ≥ 1. Then, given any positive integers n and N, there exist positive constants ϑ < 1 and Cn,N, , independent of L, β, pF and (u, J3 ) ∈ A, so that, for any integers n0 , n1 ≥ 0 and putting n = n0 + n1 , |∂xn00 ∂¯xn1 Ω3,a L,β (x)| ≤

1 Cn,N , |d(x)|2+2η1 +n 1 + [∆|d(x)|]N

1 Cn,N , |d(x)|2+n 1 + [∆|d(x)|]N 1 C0,N 1 (∆|d(x)|)ϑ 3,c + , |ΩL,β (x)| ≤ |d(x)|2 |d(x)|ϑ |d(x)|min{0,2η1 } 1 + [∆|d(x)|]N

|∂xn00 ∂¯xn1 Ω3,b L,β (x)| ≤

where ∂¯x denotes the discrete derivative and p ∆ = max |u|1+η2 , (v0 β)−2 + L−2 .

(1.16) (1.17) (1.18)

(1.19)

(c) There exist the limits Ω3,i (x) = limL,β→∞ Ω3,i L,β (x), x ∈ Z × R; they satisfy the bounds (1.16), with |x| in place of |d(x)|. Moreover, Ω3,a (x) and Ω3,b (x) are even functions of x and there exists a constant δ ∗ , of order J3 , such that, if 1 ≤ |x| ≤ ∆−1 and v0∗ = v0 (1 + δ ∗ ), given any N > 0 1 + A1 (x) , + (v0∗ x0 )2 ]1+η1 2 x0 − (x/v0∗ )2 1 + A (x) , Ω3,b (x) = 2 2π 2 [x2 + (v0∗ x0 )2 ] x2 + (v0∗ x0 )2 1 1/2 + |J3 | + (∆|x|) |Ai (x)| ≤ CN , 1 + |x|N

Ω3,a (x) =

2π 2 [x2

(1.20)

(1.21)

for some constant CN . The function Ω3,a (x) is the restriction to Z × R of a function on R2 , satisfying the symmetry relation x (1.22) Ω3,a (x, x0 ) = Ω3,a x0 v0∗ , ∗ . v0 ˆ 3 (k), k = (k, k0 ) ∈ [−π, π] × R1 , the Fourier transform of Ω3 (x). For any (d) Let Ω ˆ 3 (k) is uniformly bounded as u → 0; moreover, fixed k with k 6= (0, 0), (±2pF , 0), Ω


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for some constant c2 ,

ˆ 3 (0, 0)| ≤ c2 1 + |J3 | log 1 , |Ω ∆

(1.23) 1 − ∆2η1 . |Ω (±2pF , 0)| ≤ c2 2η1 ˆ 3 (k)| ≤ c2 [1 + |J3 | log|k|−1 ] near k = (0, 0), and, at Finally, if u = 0, |Ω k = (±2pF , 0), it is singular only if J3 < 0; in this case it diverges as |k − (±2pF , 0)|2η1 /|η1 |. ˆ its Fourier transform. For any fixed k 6= (e) Let G(x) = Ω3 (x, 0) and G(k) ˆ is uniformly bounded as u → 0, together with its first derivative; 0, ±2pF , G(k) moreover ˆ ≤ c2 , |∂k G(0)| (1.24) 2η1 ˆ ). |∂k G(±2p F )| ≤ c2 (1 + ∆ ˆ3

ˆ has a first order discontinuity at k = 0, with a jump equal Finally, if u = 0, ∂k G(k) to 1 + O(J3 ), and, at k = ±2pF , it is singular only if J3 < 0; in this case it diverges as |k − (±2pF )|2η1 . 1.6 Remarks. (a) The above theorem holds for any magnetic field h such that sin pF > 0; remember that the exact solution given in [2] is valid only for h = 0. Moreover u has not to be very small, but we only need a bound of order 1 on its value, see (1.12); the only perturbative parameter is J3 . However the interesting (and more difficult) case is when also u is small. (b) A naive estimate of ε is ε = c(sin pF )α , with c, α positive numbers; in other words we must take smaller and smaller J3 for pF closer and closer to 0 or π, i.e. for magnetic fields of size close to 1. It is unclear at the moment if this is only a technical problem or a property of the model. (c) If J1 6= J2 and J3 6= 0, one can distinguish, like in the J3 = 0 case (1.7), two different regimes in the asymptotic behaviour of the correlation function Ω3 (x), discriminated by an intrinsic length ξ, which is approximately given by the inverse of spectral gap, whose size, is of order |u|1+η2 , see (1.19), in agreement with (1.9), found by the exact solution. If 1 |x| ξ, the bounds for the correlation function are the same as in the gapless J1 = J2 case; if ξ |x|, there is a faster than any power decay with rate of order ξ −1 . In the first region we can obtain the exact large distance asymptotic behaviour of Ω3 (x), see (1.20) and (1.21); in the second region only an upper bound is obtained. Note that, even in the J3 = 0 case, it is not so easy to obtain a more precise result, if h 6= 0, see Sec. 1.2. The spin interaction in the z direction has the effect that the gap becomes anomalous, in the sense that it acquires a critical index η2 ; the ratio between the “renormalized” and the “bare” gap is very small or very large, if u is small, depending on the sign of J3 .


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(d) It is useful to compare the expression for the large distance behaviour of Ω3 (x) in the case u = 0 with its analogous for the Luttinger model, see Sec. 1.3. A first difference is that, while in the Luttinger model the Fermi momentum is independent of the interaction, in the XY Z model in general it is changed nontrivially by the interaction, unless the magnetic external field is zero, i.e. pF = π2 . The reason is that the Luttinger model has special parity properties which are not satisfied by the XY Z chain (except if the magnetic field is vanishing). (e) Another peculiar property of the Luttinger model correlation function is that it depends on pF only through the factor cos(2pF x); this is true not only for the asymptotic behaviour (1.10), but also for the complete expression given in [5], and is due to a special symmetry of the Luttinger model (the Fermi momentum disappears from the Hamiltonian if a suitable redefinition of the fermionic fields is done, see [5]). This property is of course not true in the XY Z model and in fact the dependence on pF of Ω3 (x) is very complicated. However we prove that Ω3 (x) can be written as sum of three terms, see (1.13), and the first two terms are very similar to the two terms in the r.h.s. of (1.10). In particular, the functions Ω3,a (x) and Ω3,b (x) have the same power decay as the analogous functions in the Luttinger model and are “free of oscillations”, in the sense that each derivative increases the decay power of one unit, see (1.16) and (1.17). This is not true for the third term Ω3,c (x), which does not satisfy a similar bound, because of the presence of oscillating contributions. However we can prove that such term, if u = 0, is negligible for large distances, see (1.18) (note that ϑ is J3 and u independent, unlike η1 ). Of course this is true only for small J3 and it could be that Ω3,c (x) plays an important role for larger J3 . If we compare, in the case u = 0, the functions Ω3,a (x) and Ω3,b (x), see (1.20), with the corresponding ones in the Luttinger model, see (1.10), we see that they differ essentially for the non-oscillating functions Ai (x), containing terms of higher order in our expansion. However, this difference is not important in the case of Ω3,a (x), which also satisfies the same symmetry property (1.22) as the analogue in the Luttinger model, of course with different values of v0∗ ; note that the validity of (1.22) allows to interpret v0∗ as the renormalized Fermi velocity. Guided by the analogy with the Luttinger model, one would like to prove a similar property for Ω3b (x) with v0 replacing v0∗ ; such property holds in fact for the Luttinger model, see (1.10). However we were not able to prove a similar properties for A2 (x), and this has some influence on our results, see below. (f) Another important property of the Luttinger model correlation function is the fact that the “not oscillating term”, that is the term corresponding to Ω3,b (x), does not acquire a critical index, contrary to what happens for the term corresponding to cos(2pF x)Ω3,a (x). Hence one is naturally led to the conjecture that the critical index of Ω3,b L,β (x) is still vanishing, see for instance [22]. In our expansion, the critical index of Ω3,b (x) is represented as a convergent series, but, even if an explicit computation of the first order term gives a vanishing result, it is not obvious that this is true at any order. However, due to some hidden symmetries


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of the model (i.e. symmetries approximately enjoyed by the relevant part of the effective interaction), we can prove a suitable approximate Ward identity, implying that all the coefficients of the series are indeed vanishing. ˆ (g) The above properties can be used to study the Fourier transform G(k) of 3 ˆ the equal time correlation function G(x) = Ω (x, 0). If J3 = 0, G(k) is bounded together with its first order derivative up to u = 0; in fact, the possible logarithmic ˆ is changed by the parity divergence at k = ±2pF and k = 0 (if u = 0) of ∂ G(k) properties of G(x) in a first order discontinuity. ˆ behaves near k = ±2pF in a completely different way. In fact it If J3 6= 0, ∂ G(k) is bounded and continuous if J3 > 0, while it has a power like singularity, if u = 0 and J3 < 0, see item (e) of Theorem 1.5. This is due to the fact that the critical index η1 , characterizing the asymptotic behaviour of Ω3,a (x), has the same sign of J3 (note that η1 has nothing to do with the critical index η related with the two point fermionic Schwinger function, which is O((J3 )2 )). ˆ On the other hand, the behaviour of ∂ G(k) near k = 0 is the same for the Luttinger model, the XY Z model and the free fermionic gas (J1 = J2 , J3 = 0) (see also [22] for a heuristic explanation). This is due to the vanishing of the critical index related with Ω3,b (x) and to the parity properties of the leading terms, which change, as in the J3 = 0 case, the apparent dimensional logarithmic divergence in a first order discontinuity. (h) If u = 0, the (two-dimensional) Fourier transform can be singular only at k = (0, 0) and k = (±2pF , 0). If J3 = 0, the singularity is logarithmic at k = (±2pF , 0); if J3 6= 0, the singularity is removed if J3 > 0, while it is enhanced to a power like singularity if J3 < 0, see item (d) in the Theorem 1.5. Hence, the singularity at k = (±2pF , 0) is of the same type as in the Luttinger model, see (1.10). However, we can not conclude that the same is true for the Fourier transform at k = 0, which is bounded in the Luttinger model, while we can not exclude a logarithmic divergence. In order to get such a stronger result, it would be sufficient to prove that the function Ω3,b (x) is odd in the exchange of (x, x0 ) with (x0 v, x/v), for some v; this property is true for the leading term corresponding to Ω3,b (x) in (1.10), with v = v0 , but seems impossible to prove on the base of our expansion. We can only see this symmetry for the leading term, with v = v0∗ (or any other value v differing for terms of order J3 , since the substitution of v0∗ with v would not affect the bound (1.20)), but this is only sufficient to prove that the singularity has to be of order J3 , at least. (i) If u = 0, the critical indices and ν can be computed with any prefixed precision; we write explicitly in the theorem only the first order for simplicity. However, if u 6= 0, they are not fixed uniquely; for what concerns ν, this means that, in the gapped case, the system is insensitive to variations of the magnetic field much smaller than the gap size. (j) There is no reason to restrict the analysis to a nearest-neighbor Hamiltonian like (1.1); it will be clear in the following that our results still holds for non-nearestneighbor spin Hamiltonians; see also [23].


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(k) The same techniques could perhaps be used to study Ω1L,β (x) and Ω2L,β (x), however this problem is more difficult, as one has to study the average of the exponential of the sum of fermionic density operators, see (1.4). In the J3 = 0 case the evaluation of Ω1L,β (x) and Ω2L,β (x) was done in [18]. 1.7. In order to prove Theorem 1.5, we use the well known representation of Ω3x−y in terms of a Grassmanian integral, that is R R R DψeA(ψ) ρx ρy DψeA(ψ) ρx DψeA(ψ) ρy 3 R R R − · , (1.25) Ωx−y = DψeA(ψ) DψeA(ψ) DψeA(ψ) where ψx± are elements of a Grassmanian algebra, Dψ is the usual Lebesgue measure on the algebra, A(ψ) is the action corresponding to (1.5) and ρx = ψx+ ψx− . Of course, in order to give a meaning to (1.25), we have to regularize the model so that the Grassmanian algebra is finite dimensional, hence we introduce an ultraviolet cutoff also in the time variable (in the space variable such cut-off is provided by the lattice, while the finite volume and temperature provide natural infrared cutoffs); see Sec. 2.1 for the precise definitions. This procedure allows to write expansions for the physical quantities, which satisfy uniform bounds in the various cutoffs and admit a well defined limit as the ultraviolet cutoff is removed and as L, β go to infinity. For pedagogicalR reasons we begin our analysis not directly from (1.25) but from the normalization DψeA(ψ) , which isRmuch easier to study; the expansion for Ω3x will be clearer once the expansion for DψeA(ψ) is understood. The simplest way to evaluate such Grassmanian integral is to write 1 N

Z

Z DψeA(ψ) =

P (dψ)e−V(ψ) =

∞ Z X n=0

P (dψ)

[−V(ψ)]n , n!

(1.26)

where N is a suitable constant, P (dψ) is the Grassmanian measure generated by the quadratic terms of (1.5) for u = 0, to be called the free measure, while V(ψ) contains the other terms together with the counterterm for the chemical potential, in agreement with the discussion of Sec. 1.4. In other words we are considering as reference model the isotropic XY model with Hamiltonian (1.5) with J3 = u = 0 and h = − cos pF . R If Q(ψ) is a monomial in the Grassmanian variables, it is easy to see that P (dψ)Q(ψ) is given by the anticommutative Wick rule; the corresponding propagator has two singularities in momentum space, as L, β → ∞, at k = (±pF , 0). As a consequence, one can see that the r.h.s. of (1.26) is, when the ultraviolet cutoff is removed, a series convergent for |u|, |J3 | ≤ εL,β , with εL,β →L,β→∞ 0, so that we cannot control the zero temperature infinite volume limit by the trivial perturbative expansion. If u 6= 0, the failure of the above expansion is quite clear, since we are expanding around the isotropic XY model, so considering the anisotropy, which is a sort of


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mass, as a perturbation. We could instead include the anisotropy in the free measure by writing Z Z ∞ Z n X ˜ [−V(ψ)] 1 ˜ , (1.27) P˜ (dψ) DψeA(ψ) = P˜ (dψ)e−V(ψ) = N n! n=0 where P˜ (dψ) is the Grassmanian measure generated by all the quadratic terms of (1.5). In this case the propagator in momentum space corresponding to P˜ (dψ) has no singularities, for L, β → ∞; in fact, it is easy to see, by a Bogoliubov transformation, that there is an O(u) mass term which plays the role of an infrared cutoff, so that one can indeed prove the convergence of the r.h.s. of (1.27) in the L, β → ∞ limit. However convergence holds only for |J3 | ≤ εu , with εu →u→0 0, i.e. it is not uniform in the anisotropy. Then also this expansion fails in providing results in the critical region of parameters we are interested in. The point is that the evaluation of the correlation functions in the critical region by simple power series in J3 cannot work (even if one introduces a counterterm in order to fix the singularities of the correlation functions at the same point when J3 = 0 or J3 6= 0), as the J3 = 0 theory in not analytically close to the J3 6= 0 theory; this is clear if one looks, for instance, to the gap (1.9), which cannot be expanded in a power series of J3 for u small enough. We have then to set-up a much more complicated procedure to evaluate the correlation function (1.25) and the partition function (1.26). This procedure is based on (Wilsonian) Renormalization group as implemented in [4]. The idea is to take as a reference model the isotropic XY model like in (1.26), by considering u and J3 as perturbations. However, we do not simply expand in power series of u and J3 as in (1.26). The first step (see Sec. 2) is to decompose the measure Q P (dψ) as a product of independent measures P (dψ) = 1h=−∞ P (dψ (h) ), where the momentum space propagator corresponding to P (dψ (h) ) is not singular, but O(γ −h ), for L, β → ∞, γ being a fixed scaling parameter greater than 1. This decomposition is realized by slicing in a smooth way the momentum space, so that ψ (h) , if h ≤ 0, depends only on the momenta between γ h−1 and γ h+1 . Then we integrate each field iteratively, starting from ψ (1) , so obtaining a sequence of Pj effective potentials V (h) in the following way. We write, if ψ (≤j) = h=−∞ ψ (h) , Z Y Z 1 (≤+1) −V(ψ) ) = P (dψ (h) )e−V(ψ P (dψ)e Z =

h=−∞

P (dψ (≤0) )e−V

(0)

(ψ (≤0) )

(1.28)

where V (0) (ψ) can be written as a sum (finite, since we work with a finite algebra) of monomials in the Grassmanian variables, with coefficients which are perturbative finite expansions converging, uniformly in L and β, to well defined power series, as the ultraviolet cutoff is removed. According to renormalization group, one has to identify in the effective potential the relevant, marginal and irrelevant terms. We write then V (0) = LV (0) + RV (0) ,


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where R = 1 − L and L is a linear operator, called localization operator, whose role is to extract from V (0) some local relevant or marginal terms, so that the remainder is irrelevant. It turns out that all the monomials with six Grassmanian variables or more are irrelevant, while the terms quartic in ψ and similar to the original interaction (but with a different coupling) are marginal. The relevant terms are all quadratic in the Grassmanian variables; more exactly there are terms like ψ +(≤0) ψ −(≤0) , representing the shift in the chemical potential, and terms like ψ +(≤0) ψ +(≤0) or ψ −(≤0) ψ −(≤0) , which behave as mass terms. Note that the definition of relevant terms is not simply based on power counting arguments but also on momentum conservation considerations, i.e. the power counting must be improved with respect to the trivial one, as we shall see in Sec. 2. The relevant terms corresponding to the shift of the chemical potential are controlled by choosing in a suitable way the counterterm ν so that they are smaller and smaller at each Renormalization Group iteration. On the contrary, the relevant mass terms must be included in the reference free measure, which then acquires a mass. Among the marginal terms there are also quadratic terms of the form ψ +(≤0) ∂ψ −(≤0) , related with the wave function renormalization, which must be also included in the reference measure; then we write (1.28) as Z 0 ≤0 0 ≤0 P (dψ (≤0) )e−LV (ψ )−RV (ψ ) =

1 N

Z

0 ≤0 ¯ 0 ≤0 P¯ (dψ (≤0) )e−LV (ψ )−RV (ψ )

(1.29)

where P¯ (dψ (h) ) is the new reference measure, obtained by absorbing in the old one the terms in LV (0) which are quadratic in the Grassmanian variables and are related with the mass and the wave function renormalization. We then integrate the field ψ (0) and the procedure is iterated, so that at each step we get new contributions to the mass and wave function renormalization and the field ψ (h) is integrated by a measure with mass σh and wave function renormalization Zh ; the iteration stops as soon as σh , which at the beginning is of size |u| ≤ 1, becomes of order γ h , that is the same order of the momenta contributing to ψ (h) . If we ∗ call h∗ the corresponding value of h, the integration of the field ψ (≤h ) can be performed in a single step, since σh∗ acts as an infrared cutoff on the momentum ∗ scale γ h . Of course h∗ → −∞ as u → 0, but the dependence of the effective mass σh∗ on u and J3 is highly non-trivial. In fact σh∗ /u tends to 0 or ∞, as u → 0, depending on the sign of J3 ; this result can be expressed in terms of a critical index, see (1.19). Note that the inclusion of the mass term in the reference measure means essentially that we have to perform a different Bogoliubov transformation at each integration step (up to h = h∗ ), instead of a single one as in (1.28), in order to take into account the anomalous dependence of the effective mass on u. Also the dependence of the wave function renormalization Zh∗ on u and J3 is 2 ∗ non-trivial; it turns out that Zh∗ ' γ cJ3 |h | , with c > 0, so it diverges as u → 0.


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This result is strictly related with the fact that, for u = 0, Zh ' γ cJ3 |h| , implying an anomalous asymptotic behaviour of the field correlation function. This iterative procedure allows to write the effective potential on any scale as a sum of monomials in the Grassmanian variables, with coefficients which are perturbative expansions (well defined as the ultraviolet cutoff is removed, uniformly in L and β) in terms of a few running coupling constants and renormalization constants. The running coupling constants are λh , which is the effective coupling of the interaction between fermions, νh , related with the chemical potential renormalization, and δh , related with the shift of the Fermi velocity. The renormalization constants are σh and Zh . The running coupling constants and the renormalization constants verify a recursive equation called Beta function. In Sec. 3 we prove that such expansions can be controlled, uniformly in L and β (even if the renormalization constants are diverging or go to 0 as h → −∞), if the running coupling constants are small enough, see Theorem 3.12. Section 3 is the more technical section of the paper; the expansion is written as sum over trees and we use determinant bounds for the fermionic expectations. The proof of the convergence requires some care as the power counting has to be improved. Moreover we pay attention to perform all the estimates taking finite L, β; this requires some care, as the preceding analysis of similar problems were not so careful about this point. In Sec. 5 we build an expansion for the correlation function in the direction of the magnetic field Ω3x , which is very similar to the previous one. The idea is to note that 2

Ω3x−y = [∂ 2 S(φ)/∂φx ∂φy ]|φ=0 , where φx is a bosonic external field and Z R + − S(φ) = P (dψ)e−V(ψ)+ dxφx ψx ψx . e

(1.30)

Hence, the previous analysis can be applied, by adding a new term to the interaction. As we shall see, this implies that we have to introduce two new renormalization (1) (2) constants, Zh and Zh , related respectively with the oscillating and non-oscillating part of the correlation function (i.e. the first two addenda in (1.13)). We prove the convergence also for this expansion and careful estimates on the Fourier transform are obtained, always under the hypothesis that the running coupling constants are small enough, see Theorem 5.8. Once the convergence problems of the renormalized expansions are solved, one has still to face two main problems: the first one is to show that the running coupling constants indeed remain small if J3 is small enough; the second one is to prove that (2) the ratio between Zh and Zh , both diverging as h → −∞ (which is an important property for u small) is close to one. This last property is not essential to prove the convergence of the series, but it is crucial to obtain the correct asymptotic behaviour of the correlation function as it is related to the vanishing of a critical index appearing in the non-oscillating part of the correlation function.


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Both problems are solved in Secs. 4 and 7 by a careful analysis of the cancellations arising in our expansions as a consequence of some symmetry properties. (2) We write the beta function governing the flow of λh , δh and of Zh /Zh as a sum of several terms, and we show that only one term is really crucial, while the other ones have a little effect on the flow in absence of the first one, if the finite counterterm ν is chosen in a proper way. On the other hand, one recognizes that such crucial contribution to the Beta function of the XY Z model is coinciding with the Beta function obtained by applying the same Renormalization group analysis to the Luttinger model. For such model many properties are true, like local gauge invariance and exact solubility (thanks to the possibility of representing its Hamiltonian as a quadratic bosonic one, [19]); these properties are not enjoyed by the XY Z Hamiltonian but the model is close, in a Renormalization group sense, to a model enjoying them. Note that, despite the fact that the Luttinger model Hamiltonian is formally gauge invariant, the ultraviolet and infrared cutoffs introduced to perform our Renormalization group analysis have the effect that gauge invariance is lost even in that model. Nevertheless in Sec. 7 we can derive an approximate Ward identity (approximate as the gauge invariance is only approximately true), which tells us that in the Luttinger model (2)

Zh = 1 + O(λ) . Zh

(1.31)

Note that the formal Ward identity obtained in absence of cutoffs would give exactly one in the r.h.s. of (1.31). This result is obtained by considering a tree expansion also for the corrections to the formal Ward identity (the bounds for the corrections are only sketched and more details will be published elsewhere). In Sec. 5 we show how to use (1.31) to prove that the critical index related with the asymptotic behaviour of the leading non-oscillating part of the XY Z model correlation function (the second term in the r.h.s. of (1.13)) is exactly vanishing. By using another important property of the Luttinger model, i.e. its exact solubility, it was proved in [6, 7, 10] that the beta function of the Luttinger model for the running coupling constants is vanishing; this means that the crucial contribution to the XY Z beta function is vanishing. This result is used in Sec. 4 to prove that the running coupling constants are small for any h. Finally in Sec. 7 we complete the proof of the main theorem, deriving the correlation function properties listed in the main theorem. In particular we prove, for J3 small enough (1) upper bounds for the asymptotic behaviour, see (1.16)–(1.18); (2) a rather explicit expression for the asymptotic behaviour for distances smaller than the inverse of the gap, see (1.20) and (1.21); (3) bounds for the Fourier transform of the correlation function and its derivatives in the limit L = β = ∞.


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2. Multiscale Decomposition and Anomalous Integration 2.1. The Hamiltonian (1.5) can be written, if UL2 is chosen as explained in Sec. 1.1 and (1.11) are used, in the following way (by neglecting a constant term): X 1 + − + − − (cos pF + ν)a+ H= x ax − [ax ax+1 + ax+1 ax ] 2 x∈Λ

u + − + − − + − a + a a ] + λ(a a )(a a ) , − [a+ x x x+1 x x+1 x+1 2 x x+1

(2.1)

where Λ is an interval of L points on the one-dimensional lattice of step one, which will chosen equal to (−[L/2], [(L − 1)/2]), the fermionic field a± x satisfies periodic boundary conditions and λ = −J3 .

(2.2)

The Hamiltonian (2.1) will be considered as a perturbation of the Hamiltonian H0 of a system of free fermions in Λ with unit mass and chemical potential µ = 1 − cos pF (u = J3 = ν = 0); pF is the Fermi momentum. This system will have, at zero temperature, density ρ = pF /π, corresponding to magnetization ρ − 1/2 in the 3-direction for the original spin system. Since pF is not uniquely defined at finite volume, we choose it so that 2π 1 lim pF = πρ . (2.3) nF + , nF ∈ N , pF = L→∞ L 2 This means, in particular, that pF is not an allowed momentum of the fermions. x0 H ± −Hx0 ax e , with We consider also the operators a± x =e x = (x, x0 ) ,

−β/2 ≤ x0 ≤ β/2 ,

(2.4)

for some β > 0; on x0 , which we shall call the time variable, antiperiodic boundary conditions are imposed. Many interesting physical properties of the fermionic system at inverse temperature β can be expressed in terms of the Schwinger functions, that is the truncated expectations in the Grand Canonical Ensemble of the time order product of the field a± x at different space-time points. There is of course a relation between these functions and the expectations of some suitable observables in the spin system. However, by looking at (1.4), one sees that this relation is simple enough only in the case of the truncated expectations of the time order product of the fermionic density − operator ρx = a+ x ax at different space-time points, which we shall call the density Schwinger functions; they coincide with the truncated expectations of the time order product of the operator Sx3 = ex0 H Sx3 e−Hx0 at different space-time points. As it is well known, the Schwinger functions can be written as power series in λ and u, convergent for |λ|, |u| ≤ εβ , for some constant εβ (the only trivial bound of εβ goes to zero, as β → ∞). This power expansion is constructed in the usual way


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in terms of Feynman graphs, by using as free propagator the function g L,β (x − y) =

+ Tr[e−βH0 T(a− 1 X −ik(x−y) x ay )] = e Tr[e−βH0 ] L k∈DL

e−τ e(k) e−(β+τ )e(k) 1(τ > 0) − 1(τ ≤ 0) , (2.5) 1 + e−βe(k) 1 + e−βe(k) P + where T is the time order product, N = x∈Λ a+ x ax , τ = x0 − y0 , 1(E) denotes the indicator function (1(E) = 1, if E is true, 1(E) = 0 otherwise), ·

e(k) = cos pF − cos k ,

(2.6)

and DL ≡ {k = 2πn/L, n ∈ Z, −[L/2] ≤ n ≤ [(L − 1)/2]}. It is also well known that, if x0 6= y0 , g L,β (x − y) = limM→∞ g L,β,M (x − y), where g L,β,M (x − y) =

1 Lβ

X k∈DL,β

e−ik·(x−y) , −ik0 + cos pF − cos k

(2.7)

k = (k, k0 ), k · x = k0 x0 + kx, DL,β ≡ DL × Dβ , Dβ ≡ {k0 = 2(n + 1/2)π/β, n ∈ Z, −M ≤ n ≤ M − 1}. Note that g L,β,M (x − y) is real, ∀ M . Hence, if we introduce a finite set of Grassmanian variables {ˆ a± k }, one for each k ∈ DL,β , and a linear functional P (da) on the generated Grassmanian algebra, such that Z 1 ˆ ˆ , (2.8) ˆ+ G(k) = P (da)ˆ a− k1 a k2 = Lβδk1 ,k2 G(k1 ) , −ik0 + cos pF − cos k we have 1 M→∞ Lβ lim

X

ˆ = lim e−ik·(x−y) G(k)

Z

M→∞

k∈DL,β

+ L,β (x; y) , P (da)a− x ay ≡ g

where the Grassmanian field ax is defined by 1 X ± ±ik·x a ˆk e . a± x = Lβ

(2.9)

(2.10)

k∈DL,β

The “Gaussian measure” P (da) has a simple representation in terms of the Q − “Lebesgue Grassmanian measure” k∈DL,β da+ k dak , defined as the linear functional on the Grassmanian algebra, such that, given a monomial Q(a− , a+ ) in the variables Q − + + − + ˆk a ˆk , up to a a− k , ak , k ∈ DL,β , its value is 0, except in the case Q(a , a ) = ka permutation of the variables. In this case the value of the functional is determined, by using the anticommuting properties of the variables, by the condition   Z Y Y −  da+ a ˆ− ˆ+ (2.11) k dak ka k = 1. k∈DL,β

k∈DL,β


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We have P (da) =

( Y

) ˆ k )ˆ (Lβ G a+ ˆ− ka k

( exp −

k

X

) ˆ k )−1 a (Lβ G ˆ+ ˆ− ka k

.

(2.12)

k

2 2 −zˆ ak a ˆk a+ = 1 − zˆ a+ ˆk , for any complex z. Note that, since (ˆ a− k ) = (ˆ k ) = 0, e ka We remark that the ultraviolet cutoff M on the k0 variable was introduced so that the Grassmanian algebra is finite; this implies that the Grassmanian integration is indeed a simple algebraic operation and all quantities that appear in the calculations are finite sums. However, M does not play any essential role in this paper, since all bounds will be uniform with respect to M and they easily imply the existence of the limit. The only problem is that, if x1 = y1 , the propagator (2.5) has a first order discontinuity at x0 − y0 = 0, where it has to be defined as the limit from the left, while limM→∞ g L,β,M (0, 0) = [g L,β (0, 0+ ) + g L,β (0, 0− )]/2. One could take care of this problem, by adding to the r.h.s. of (2.10) √ a factor exp(iδM k0 ), where δM is a suitable positive constant proportional to β/ M , and by leaving unchanged (2.8); then the r.h.s. of (2.7) is multiplied by exp(2iδM k0 ) and it is easy to see that the new propagator has the right value in x = 0 for M → ∞. In order to simplify the notation, we shall neglect this minor problem in the following and we shall not stress the dependence on M of the various quantities we shall study. By using standard arguments (see, for example, [20], where a different regularization of the propagator is used), one can show that the partition function and the Schwinger functions can be calculated as expectations of suitable functions of the Grassmanian field with respect to the “Gaussian measure” P (da). In particular the partition function Tr[e−βH ] is equal to ZL,β Tr[e−βH0 ], with Z (2.13) ZL,β = P (da)e−V(a) , +

where V(a) = uVu (a) + λVλ (a) + νN (a) , Z β/2 X Z β/2 + − − dx0 dy0 vλ (x − y)a+ Vλ (a) = x ay ay ax , x,y∈Λ

N (a) =

XZ

x∈Λ

Vu (a) =

−β/2

β/2 −β/2

X Z

x,y∈Λ

−β/2

(2.14)

− dx0 a+ x ax ,

Z

β/2

−β/2

dx0

β/2

−β/2

+ − − dy0 vu (x − y)[a+ x ay − ax ay ]

where 1 1 δ1,|x−y| δ(x0 − y0 ) , vu (x − y) = δx,y+1 δ(x0 − y0 ) . (2.15) 2 2 Note that the parameter ν has been introduced in order to fix the singularities of the interacting propagator to the values of the free model, that is k = (0, ±pF ). vλ (x − y) =


00105


1341

Hence ν is a function of λ, u, pF , which has to be fixed so that the perturbation expansion is convergent (uniformly in L, β). This choice of ν has also the effect of fixing the singularities of the spin correlation function Fourier transform, as we explained in the introduction, see Sec. 1.4. Note that, if pF = π/2, one can prove that ν = −λ, by using simple symmetry properties of our expansion; this implies, by using (1.11), that h = 0. If u = 0, it is conjectured, on the base of heuristic calculations, that this condition is equivalent to the condition that, in the limit L, β → ∞, the density is fixed (“Luttinger Theorem”) to the free model value ρ = pF /π. If u 6= 0, there is no simple relation between the value of pF and the density, as one can see directly in the case λ = 0, where one can do explicit calculations. 2.2. We shall begin our analysis by rewriting the potential V(a) as V(a) = V (1) (a) + uVu (a) + δ ∗ Vδ (a) ,

(2.16)

V (1) (a) = λVλ (a) + νN (a) − δ ∗ Vδ (a) ,

(2.17)

where

and Vδ (a) =

1 X e(k)ˆ a+ ˆ− ka k . Lβ

(2.18)

k

δ ∗ is an arbitrary parameter, to be fixed later, of modulus smaller than 1/2; its introduction is not really necessary, but allows to simplify the discussion of the spin correlation function asymptotic behaviour. In terms of the Fermionic system, it will describe the modification of the Fermi velocity due to the interaction. Afterwards we “move” the terms uVu (a) and δ ∗ Vδ (a) from the interaction to the Gaussian measure. In order to describe the properties of the new Gaussian measure, it is convenient to introduce a new set of Grassmanian variables ˆbσk,ω , + , by defining ω = ±1, k ∈ DL,β ω = {k ∈ DL,β : ωk > 0} ∪ {k ∈ DL,β : k = 0, ωk0 > 0} , DL,β

(2.19)

ˆbσ = a ˆσω k,ω ωk ,

(2.20)

so that, by using (2.10) aσx =

1 Lβ

X

ˆbσω eiσωk·x . k,ω

(2.21)

+ k∈DL,β , ω=±1

It is easy to see that ZL,β = e

−Lβt1

Z

˜ (1) (b)

P (db)e−V

,

(2.22)


1342

00105


with V˜ (1) (b) = V (1) (a), where a has to be interpreted as the r.h.s. of (2.21),      Y  2 Y (Lβ) + ˆ− ˆ b b P (db) =  −k02 − (1 + δ ∗ )2 e(k)2 − u2 sin2 k ω=±1 k,ω k,ω    + k∈DL,β

· exp

T (k) =

t1 = −

  

1 −  Lβ 

X X + k∈DL,β

ˆb+ Tω,ω0 (k)ˆb− k,ω k,ω

ω,ω 0

    

,

−ik0 + (1 + δ ∗ )e(k)

iu sin k

−iu sin k

−ik0 − (1 + δ ∗ )e(k)

1 Lβ

X + k∈DL,β

log

(2.23) ! ,

k02 + (1 + δ ∗ )e(k)2 + u2 sin2 k . k02 + e(k)2

(2.24)

(2.25)

Note that t1 is uniformly bounded as L, β → ∞, if |δ ∗ | ≤ 1/2, as we are supposing. For λ = ν = δ ∗ = 0, it represents the free energy for lattice site of H − H0 . 2.3. For λ = ν = 0, all the properties of the model can be analyzed in terms of the Grassmanian measure (2.23). In particular, we have Z P (db)aσx1 aσy2 =

1 Lβ

X

[e−ik(x−y) T −1 (k)−σ1 ,σ2 − eik(x−y) T −1 (k)−σ2 ,σ1 ] ,

(2.26)

+ k∈DL,β

where T −1 (k) denotes the inverse of the matrix T (k). This matrix is defined for any k ∈ DL,β and satisfies the symmetry relation T −1 (k)−σ2 ,σ1 = −T −1 (−k)−σ1 ,σ2 , so that we can write (2.26) also in the form Z 1 X −ik(x−y) −1 e T (k)−σ1 ,σ2 . P (db)aσx1 aσy2 = Lβ

(2.27)

(2.28)

k∈DL,β

If λ 6= 0, we shall study the model, for λ small, in terms of a perturbative expansion, based on a multiscale decomposition of the measure (2.23), by using the methods introduced in [3] and extended in various other papers [6, 7, 16]. In order to discuss the structure of the expansion, it is convenient to explain first how it works in the case of the free energy for site of H − H0 EL,β = −

1 log ZL,β . Lβ

(2.29)


00105


1343

Let T 1 be the one-dimensional torus, kk − k 0 kT 1 the usual distance between k and k 0 in T 1 and kkk = kk − 0k. We introduce a scaling parameter γ > 1 and a positive function χ(k0 ) ∈ C ∞ (T 1 × R), k0 = (k 0 , k0 ), such that ( 1 if |k0 | < t0 ≡ a0 v0∗ /γ , (2.30) χ(k0 ) = χ(−k0 ) = 0 if |k0 | < a0 v0∗ , where

q k02 + (v0∗ kk 0 kT 1 )2 ,

(2.31)

a0 = min{pF /2, (π − pF )/2} ,

(2.32)

v0∗ = v0 (1 + δ ∗ ) ,

(2.33)

|k0 | =

v0 = sin pF .

In order to give a well defined meaning to the definition (2.30), v0∗ > 0 has to be positive. Hence we shall suppose that v0 ≥ v¯0 > 0 ,

|δ ∗ | ≤

1 , 2

(2.34)

where v¯0 is fixed once for all. All our results will be uniform in v0 , under the conditions (2.34), but we shall not stress this fact anymore in the following. The definition (2.30) is such that the supports of χ(k − pF , k0 ) and χ(k + pF , k0 ) are disjoint and the C ∞ function on T 1 × R fˆ1 (k) ≡ 1 − χ(k − pF , k0 ) − χ(k + pF , k0 )

(2.35)

is equal to 0, if [v0∗ k(|k| − pF )kT 1 ]2 + k02 < t20 . We define also, for any integer h ≤ 0, fh (k0 ) = χ(γ −h k0 ) − χ(γ −h+1 k0 ) ;

(2.36)

¯ < 0, we have, for any h χ(k0 ) =

0 X

¯

fh (k0 ) + χ(γ −h k0 ) .

(2.37)

¯ h=h+1

Note that, if h ≤ 0, fh (k0 ) = 0 for |k0 | < t0 γ h−1 or |k0 | > t0 γ h+1 , and fh (k0 ) = 1, if |k0 | = t0 γ h , so that fh1 (k0 )fh2 (k0 ) = 0 ,

if |h1 − h2 | > 1 .

(2.38)

We finally define, for any h ≤ 0: fˆh (k) = fh (k − pF , k0 ) + fh (k + pF , k0 ) .

(2.39)

This definition implies that, if h ≤ 0, the support of fˆh (k) is the union of two − + h disjoint sets, A+ h and Ah . In Ah , k is strictly positive and kk − pF kT 1 ≤ a0 γ ≤ a0 , − h while, in Ah , k is strictly negative and kk + pF kT 1 ≤ a0 γ .


1344

00105


The label h is called p the scale or frequency label. Note that, if k ∈ DL,β , then |k ± (pF , 0)| ≥ (πβ −1 )2 + (v0∗ πL−1 )2 , by (2.3) and the definition of DL,β . Therefore q n o (2.40) fˆh (k) = 0 ∀ h < hL,β = min h : t0 γ h+1 > (πβ −1 )2 + (v0∗ πL−1 )2 , and, if k ∈ DL,β , the definitions (2.35) and (2.39), together with the identity (2.37), imply that 1 X

1=

fˆh (k) .

(2.41)

h=hL,β

We now introduce, for each scale label h, such that hL,β ≤ h ≤ 1, a set of (h)σ Grassmanian variables bk,ω and a corresponding Gaussian measure P (db(h) ), such that, if h = 1, then k ∈ DL,β and Z 1 (1)−σ (1)σ (2.42) P (db(1) )bk1 ,ω1 1 bk2 ,ω22 = Lβσ1 δσ1 ,σ2 δk1 ,k2 T −1 (k1 )ω1 ,ω2 fˆ1 (k1 ) , 2 + and while, if h ≤ 0, then k ∈ DL,β Z (h)−σ (h)σ P (db(h) )bk1 ,ω1 1 bk2 ,ω22 = Lβσ1 δσ1 ,σ2 δk1 ,k2 T −1 (k1 )ω1 ,ω2 fh (k1 − pF , k0 ) .

(2.43)

The support properties of the r.h.s. of (2.42) and (2.43) allow to impose the condition (h)σ

if fˆh (k) = 0 .

bk,ω = 0 ,

By using (2.26) and (2.27), it is easy to see that Z 1 X X Z 1 ω1 (h)σ2 ω2 by,ω2 , P (db(h) )b(h)σ P (db)aσx1 aσy2 = x,ω1

(2.44)

(2.45)

h=hL,β ω1 ,ω2

where, if h ≤ 0, (h)σ = bx,ω

1 Lβ

X

ˆb(h)σ eiσk·x , k,ω

(2.46)

+ k∈DL,β

+ . Note that this while, if h = 1, a similar definition is used, with DL,β in place of DL,β different definition, which is at the origin of the factor 1/2 in the r.h.s. of (2.42), is R (1)− (h)+ not really necessary, but implies that P (db(1) )bx,ω1 by,ω2 is bounded for M → ∞, a R P (1)σ ω (h)σ ω property which should otherwise be true only for ω1 ,ω2 P (db(1) )bx,ω11 1 by,ω22 2 . In the following, we shall use this property in order to simplify the discussion in some minor points. The identity (2.45), as it is well known, implies that, if F (a) is any function of the variables aσx , then   Z Z Y 1 1 X P (db(h) )F  a(h)  , (2.47) P (da)F (a) = h=hL,β

h=hL,β


00105


where ax(h)σ =

X

(h)σω bx,ω .

1345

(2.48)

ω=±1

It is now convenient to introduce a variable which measures the distance of 0 = the momentum from the Fermi surface, by putting k = k 0 + pF , with k 0 ∈ DL 0 {k = 2(n + 1/2)π/L, n ∈ Z, −[L/2] ≤ n ≤ [(L − 1)/2]}. Moreover, we rename the Grassmanian variables, by defining X 0 1 (h)σ (h)σ (h)σ (h)σ = eiσk x ψˆk0 ,ω , (2.49) ψˆk0 ,ω = ˆbk0 +pF ,ω , ψx,ω Lβ 0 0 k ∈DL,β

0 0 = DL × Dβ , k0 = (k 0 , k0 ) and pF = (pF , 0). Note that, by (2.44), where DL,β (h)σ ψˆk0 ,ω = 0 if fˆh (k0 + pF ) = 0 .

(2.50)

The definition (2.49) allows to write (2.48) in the form X (h)σω eiσωpF x ψx,ω . ax(h)σ =

(2.51)

ω

The measure P (db(h) ) can be thought in a natural way as a measure on the (h)σ variables ψx,ω , that we shall denote P (dψ (h) ). Then, (2.43) and (2.49) imply that, if h ≤ 1, Z 1 (h)−σ (h)σ (k01 ) , (2.52) P (dψ (h) )ψˆk0 ,ω1 1 ψˆk0 ,ω22 = 1 − δh,1 Lβσ1 δσ1 ,σ2 δk01 ,k02 g˜ω(h) 1 ,ω2 1 2 2 where, if f1 (k0 ) ≡ fˆ1 (k0 + pF ), g˜(h) (k0 ) =

fh (k0 ) −k02 − E(k 0 )2 − u2 sin2 (k 0 + pF ) ×

−ik0 − E(k 0 )

−iu sin(k 0 + pF )

iu sin(k 0 + pF )

−ik0 + E(k 0 )

! ,

E(k 0 ) = v0∗ sin k 0 + (1 + δ ∗ )(1 − cos k 0 ) cos pF .

(2.53) (2.54)

In the following we shall use also the notation (≤h)σ = ψx,ω

h X h0 =hL,β

0

(h )σ ψx,ω ,

P (dψ (≤h) ) =

h Y

0

P (dψ (h ) ) ,

(2.55)

h0 =hL,β

which allows to write the identity (2.47) as Z Z P (da)F (a) = P (dψ (≤1) )F˜ (ψ (≤1) ) ,

(2.56)

P where F˜ (ψ (≤1) ) is obtained from F ( h a(h) ), by using (2.51). We note that the sum over k0 in (2.49) can be thought as a finite sum for any (h)σ M , if h ≤ 0, because of the support properties of ψˆk0 ,ω . Hence, all quantities that


1346

00105


we shall calculate will depend on M only trough the propagator g˜(1) (k0 ), if M is large enough. 2.4. If we apply (2.56) to ZL,β and we use (2.29) and (2.22), we get Z (1) (≤1) ) , e−LβEL,β = e−Lβt1 P (dψ (≤1) )e−V (ψ where



V (1) (ψ (≤1) ) = λVλ 

1 X





a(h)  + νN 

h=hL,β



1 X



a(h)  − δ ∗ Vδ 

h=hL,β

(2.57)

1 X

 a(h)  .

h=hL,β

(2.58) Let us now perform the integration over ψ (1) ; we get Z ˜ ¯ (0) (≤0) ) , e−LβEL,β = e−Lβ(E1 +t1 ) P (dψ (≤0) )e−V (ψ e

¯ (0) (ψ (≤0) )−Lβ E ˜1 −V

Z =

P (dψ (+1) )e−V

(1)

(ψ (≤0) +ψ (+1)

V¯ (0) (0) = 0 ,

).

(2.59) (2.60)

It is easy to see that V¯ (0) (ψ (≤0) ) can be written in the form V¯ (0) (ψ (≤0) ) =

∞ X

X 1 2n (Lβ) σ,ω n=1

X

2n Y

k01 ,...,k02n i=1

ˆ (0) (k01 , . . . , k02n−1 )δ ·W 2n,σ,ω

(≤0)σ ψˆk0 ,ωi i i

2n X

! σi (k0i

+ pF ) ,

(2.61)

i=1

where σ = (σ1 , . . . , σ2n ), ω = (ω1 , . . . , ω2n ) and we used the notation X δk,2πn , δ(k0 ) = βδk0 ,0 . δ(k) = δ(k)δ(k0 ) , δ(k) = L

(2.62)

n∈Z

˜ ˆ (0) (k0 , . . . , k0 As we shall prove in Sec. 3, the kernels W 2n−1 ), as well as E1 , are 2n,σ,ω 1 expressed as power series of λ, ν, convergent for ε ≡ Max(|λ|, |ν|) ≤ ε0 , for ε0 small ˜1 | ≤ Cε enough. Moreover there exists a constant C, such that, uniformly in L, β, |E (0) n max(1,n−1) ˆ . and |W2n,σ,ω | ≤ C ε We remark that the conservation of momentum and the support property (2.50) (≤0)σ of ψˆk0 ,ω imply that, if n = 1, only the terms with σ1 + σ2 = 0 contribute to the sum in (2.61). Let us now define k∗ = (k, −k0 ). It is possible to show, by using the symmetries of the interaction and of the covariance g˜(1) (k0 ), that 1 (0) ˆ n,σ,ω (k∗1 , . . . , k∗n−1 ) = (−1) 2 W 1

= (−1) 2

Pn i=1

Pn i=1

σi ω1

(0) ˆ n,σ,ω [W (k1 , . . . , kn−1 )]∗

σi ωi

(0) ˆ n,−σ,−ω W (k1 , . . . , kn−1 ) .

(2.63)


00105


1347

2.5. The integration of the fields of scale h ≤ 0 is performed iteratively. We define a sequence of positive constants Zh , h = hL,β , . . . , 0, a sequence of effective potentials V (h) (ψ), a sequence of constants Eh and a sequence of functions σh (k0 ), such that Z0 = 1 , and e

−LβEL,β

Z =

σ0 (k0 ) = u sin(k 0 + pF ) ,

˜ 1 + t1 , E0 = E

PZh ,σh ,Ch (dψ (≤h) )e−V

(h)

√ ( Zh ψ (≤h) )−LβEh

(2.64)

V (h) (0) = 0 ,

,

(2.65)

where PZh ,σh ,Ch (dψ (≤h) ) =

(≤h)− Y dψˆk(≤h)+ dψˆk0 ,ω 0 ,ω

Y

N (k0 )

−1 k0 :Ch (k0 )>0 ω=±1

  1 · exp −  Lβ

X

Ch (k0 )Zh

ω,ω 0 =±1

−1 k0 :Ch (k0 )>0

N (k0 ) =

 

X

(≤h)+ (h+1) (≤h)− ψˆk0 ,ω Tω,ω0 ψˆk0 ,ω0 , 

Ch (k0 )Zh 2 [k0 + E(k 0 )2 + σh (k0 )2 ]1/2 , Lβ h X

Ch (k0 )−1 =

(2.66)

(2.67)

fj (k0 ) ,

(2.68)

j=hL,β

and the 2 × 2 matrix Th (k0 ) is given by Th (k0 ) =

!

−ik0 + E(k 0 )

iσh−1 (k0 )

−iσh−1 (k0 )

−ik0 − E(k 0 )

.

(2.69)

We shall also prove that the V (h) can be represented as V (h) (ψ (≤h) ) =

∞ X

1 2n (Lβ) n=1

X

2n Y

k01 ,...,k02n , i=1 σ,ω

(≤h)σ ψˆk0 ,ωi i

ˆ (h) (k01 , . . . , k02n−1 )δ ·W 2n,σ,ω

i

2n X

! σi (k0i

+ pF ) ,

(2.70)

i=1

ˆ (h) with the kernels W 2n,σ,ω verifying the symmetry relations 1 (h) ˆ n,σ,ω (k∗1 , . . . , k∗n−1 ) = (−1) 2 W 1

= (−1) 2

Pn i=1

Pn i=1

σi ωi

(h) ˆ n,σ,ω [W (k1 , . . . , kn−1 )]∗

σi ωi

ˆ (h) W n,−σ,−ω (k1 , . . . , kn−1 ) .

(2.71)

The previous claims are true for h = 0, by (2.59), (2.61), (2.64) and (2.53). In order to prove them for any h ≥ hL,β , we must explain how V (h−1) (ψ) is calculated,


1348

00105


given V (h) (ψ). It is convenient, for reasons which will be clear below, to split V (h) as LV (h) + RV (h) , where R = 1 − L and L, the localization operator, is a linear operator on functions of the form (2.70), defined in the following way by its action ˆ (h) . on the kernels W 2n,σ,ω (1) If 2n = 4, then ¯ ++ , k ¯ ++ , k ¯ ++ ) , ˆ (h) (k ˆ (h) (k01 , k02 , k03 ) = W LW 4,σ,ω 4,σ,ω where ¯ ηη0 = k

π 0π η ,η . L β

(2.72)

(2.73)

Note that this definition depends on the the field variables order in the r.h.s. of P P P (2.70), if 4i=1 σi 6= 0. In fact, since σ4 k04 = − 3i=1 σi k0i − pF 4i=1 σi (modulo P ¯ ++ for i = 1, 2, 3, k0 = k ¯ ++ only if 4 σi = 0. This is apparently (2π, 0)), if k0i = k 4 i=1 a problem, because the representation (2.70) is not uniquely defined (the terms which differ by a common permutation of the σ and ω indices are equivalent). However, it is easy to see, by using the anticommuting property of the field variables, that the contribution to LV (h) of the terms with 2n = 4 is equal to 0, unless, after a suitable permutation of the fields, σ = (+, −, +, −), ω = (+1, −1, −1, +1). The previous discussion implies that we are free to change the order of the field variables as we like, before applying the definition (2.72); this freedom will be useful in the construction of the main expansion in Sec. 3. (2) If 2n = 2 and, possibly after a suitable permutation of the fields, σ = (+, −) (σ1 + σ2 = 0, by the remark following (2.62)), then X ˆ (h) (k ¯ ηη0 ) ˆ (h) (k0 ) = 1 W LW 2,σ,ω 2,σ,ω 4 0 η,η =±1

E(k 0 ) L 0β b L + aL ∗ + η k0 , · 1 + δω1 ,ω2 η π v0 π

(2.74)

cos pF π π L + bL sin = 0 . 1 − cos v0 L π L

(2.75)

where aL

π L sin = 1 , π L

In order to better understand this definition, note that, if L = β = ∞, # " ˆ (h) ˆ (h) ∂W E(k 0 ) ∂ W 2,σ,ω 2,σ,ω (h) (h) 0 ˆ ˆ (0) + k0 (0) . LW2,σ,ω (k ) = W2,σ,ω (0) + δω1 ,ω2 v0∗ ∂k 0 ∂k0

(2.76)

0 ˆ Hence, LW 2,σ,ω (k ) has to be understood as a discrete version of the Taylor expansion up to order 1. Since aL = 1 + O(L−2 ) and bL = O(L−2 ), this property would be true also if aL = 1 and bL = 0; however the choice (2.75) has the advantage to ˆ (h) (k0 ) = LW ˆ (h) (k0 ). share with (2.76) another important property, that is L2 W 2,σ,ω 2,σ,ω (3) In all the other cases (h)

h ˆ 2n,σ,ω (k01 , . . . , k02n−1 ) = 0 . LW

(2.77)


00105


1349

By (2.72) and the remark following (2.76), the operator L satisfies the relation RL = 0 .

(2.78)

By using the anticommuting properties of the Grassmanian variables (see discussion in item (1) above) and the symmetry relations (2.71), we can write LV (h) in the following way: (≤h)

LV (h) (ψ (≤h) ) = γ h nh Fν(≤h) + sh Fσ(≤h) + zh Fζ

(≤h)

+ ah Fα(≤h) + lh Fλ

,

(2.79)

where nh , sh , zh , ah and lh are real numbers and X ω X (≤h) (≤h)+ (≤h)− = ψˆk0 ,ω ψˆk0 ,ω , Fν Lβ 0 0 ω=±1 (≤h)

Fσ

(≤h)

Fα

(≤h)

Fζ

(≤h)

Fλ

k ∈DL,β

X

iω = (Lβ) ω=±1 = = =

X

ω (Lβ) ω=±1 X

1 (Lβ) ω=±1 1 (Lβ)4

X

(≤h)+ (≤h)− ψˆk0 ,ω ψˆk0 ,−ω ,

0 k0 ∈DL,β

X

E(k 0 ) ˆ(≤h)+ ˆ(≤h)− ψk0 ,ω ψk0 ,ω , v0∗

0 k0 ∈DL,β

X

(2.80)

(≤h)+ (≤h)− (−ik0 )ψˆk0 ,ω ψˆk0 ,ω ,

0 k0 ∈DL,β

X

0 k01 ,...,k04 ∈DL,β

(≤h)+ (≤h)− (≤h)+ (≤h)− ψˆk0 ,+1 ψˆk0 ,−1 ψˆk0 ,−1 ψˆk0 ,+1 1

2

3

4

× δ(k01 − k02 + k03 − k04 ) . By using (2.72) and (2.74), it is easy to see that, if ε ≡ max{|λ|, |ν|}, l0 = 4λ sin2 (pF + π/L) + O(ε2 ) , s0 = O(uε) ,

a0 = −δ ∗ v0 + cδ0 λ1 + O(ε2 ) ,

z0 = O(ε2 ) ,

n0 = ν + O(ε) ,

(2.81)

where cδ0 is a constant, bounded uniformly in L, β. We now renormalize the free measure PZh ,σh ,Ch (dψ (≤h) ), by adding to it part of the r.h.s. of (2.79). We get Z (h) √ (≤h) ) PZh ,σh ,Ch (dψ (≤h) )e−V ( Zh ψ = e−Lβth

Z

√ ˜ (h) ( Zh ψ (≤h) )

PZ˜h−1 ,σh−1 ,Ch (dψ (≤h) )e−V

,

(2.82)

where PZ˜h−1 ,σh−1 ,Ch (dψ (≤h) ) is obtained from PZh ,σh ,Ch (dψ (≤h) ) by substituting Zh with Z˜h−1 (k0 ) = Zh [1 + Ch−1 (k0 )zh ]

(2.83)

and σh (k0 ) with σh−1 (k0 ) =

Zh [σh (k0 ) + Ch−1 (k0 )sh ] ; Z˜h−1 (k0 )

(2.84)


1350

00105


moreover V˜ (h)

p p Zh ψ (≤h) = V (h) Zh ψ (≤h) − sh Zh Fσ(≤h) + v0∗ Fα(≤h) ]

(≤h)

− zh Zh [Fζ

(2.85)

and the factor exp(−Lβth ) in (2.82) takes into account the different normalization of the two measures, so that 2 0 2 0 2 X 1 −1 0 2 k0 + E(k ) + σh−1 (k ) log [1 + zh Ch (k )] . (2.86) th = − Lβ 0 −1 0 k02 + E(k 0 )2 + σh (k0 )2 k :Ch (k )>0

Note that LV˜ (h) (ψ) = γ h nh Fν(≤h) + (ah − zh v0∗ )Fα(≤h) + lh Fλ

(≤h)

.

(2.87)

The r.h.s of (2.82) can be written as Z Z √ (≤h) ˜ (h) −Lβth (≤h−1) ) ) PZh−1 ,σh−1 ,f˜−1 (dψ (h) )e−V ( Zh ψ , PZh−1 ,σh−1 ,Ch−1 (dψ e h

(2.88) where

" f˜h (k0 ) = Zh−1

Zh−1 = Zh (1 + zh ) ,

# −1 Ch−1 (k0 ) Ch−1 (k0 ) − . Zh−1 Z˜h−1 (k0 )

(2.89)

Note that f˜h (k0 ) has the same support of fh (k0 ); in fact, by using (2.38), it is easy to see that zh fh+1 (k0 ) 0 0 ˜ . (2.90) fh (k ) = fh (k ) 1 + 1 + zh fh (k0 ) Moreover, by (2.49), Z

(h)

(h)+

(h)− ψy,ω0 = PZh−1 ,σh−1 ,f˜−1 (dψ (h) )ψx,ω

gω,ω0 (x − y)

,

(2.91)

1 X −ik0 (x−y) ˜ 0 −1 0 e fh (k )[Th (k )]ω,ω0 , Lβ 0

(2.92)

h

Zh−1

where (h)

gω,ω0 (x − y) =

k

and Th−1 (k0 ) is the inverse of the Th (k0 ) defined in (2.69). Th−1 (k0 ) is well defined on the support of f˜h (k0 ) and, if we set Ah (k0 ) = det Th (k0 ) = −k02 − E(k 0 )2 − [σh−1 (k0 )]2 , then 1 Th−1 (k0 ) = Ah (k0 )

−ik0 − E(k 0 )

−iσh−1 (k0 )

iσh−1 (k0 )

−ik0 + E(k 0 )

(2.93)

! .

(2.94)


00105


1351

(h)

The propagator gω,ω0 (x) is an antiperiodic function of x and x0 , of period L and β, respectively. Its large distance behaviour is given by the following lemma (see also [8]), where we use the definitions

dL (x) =

L sin π

σh ≡ σh (0) , πx πx0 β , dβ (x0 ) = sin , L π β

d(x − y) = (dL (x − y), dβ (x0 − y0 )) .

(2.95) (2.96) (2.97)

2.6 Lemma. Let us suppose that hL,β ≤ h ≤ 0 and |zh | ≤

1 , 2

|sh | ≤

1 |σh | , 2

|δ ∗ | ≤

1 . 2

(2.98)

We can write (h)

(h)

(h)

(h) (x − y) = gL,ω (x − y) + r1 (x − y) + r2 (x − y) , gω,ω

(2.99)

where (h)

gL,ω (x − y) =

0 1 X e−ik (x−y) ˜ 0 fh (k ) . Lβ 0 −ik0 + ωv0∗ k 0

(2.100)

k

Moreover, given the positive integers N, n0 , n1 and putting n = n0 + n1 , there exist a constant CN,n such that γ 2h+n , 1 + (γ h |d(x − y))|N h 2 σ γ h+n n0 ¯n1 (h) , |∂x0 ∂x r2 (x − y)| ≤ CN,n h γ 1 + (γ h |d(x − y)|)N h σ γ h+n n0 ¯n1 (h) , |∂x0 ∂x gω,−ω (x − y)| ≤ CN,n h γ 1 + (γ h |d(x − y)|)N (h)

|∂xn00 ∂¯xn1 r1 (x − y)| ≤ CN,n

(2.101)

(2.102)

where ∂¯x denotes the discrete derivative. (h) Note that gL,ω (x − y) coincides, in the limit β → ∞, with the propagator “at scale γ h ” of the Luttinger model, see [5], with f˜h in place of fh . This remark will be crucial for studying the renormalization group flow in [4]. 2.7 Proof of Lemma 2.6. By using (2.38), it is easy to see that σh (k0 ) = σh (0) on the support of fh (k0 ); hence, by (2.83) and (2.84), we have σh−1 (k0 ) =

σh + Ch (k0 )−1 sh , 1 + zh Ch (k0 )−1

(2.103)

implying, together with (2.98), that there exist two constants c1 , c2 such that: c1 |σh | ≤ |σh−1 (k0 )| ≤ c2 |σh | .

(2.104)


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00105


Let us now consider two integers N0 , N1 ≥ 0, such that N = N0 + N1 , and note that (h)

dL (x − y)N1 dβ (x0 − y0 )N0 gω,ω0 (x − y) = e−iπ(xL ×

−1

N1 +x0 β −1 N0 )

(−i)N0 +N1

1 X −ik0 (x−y) N1 N0 ˜ 0 −1 0 e ∂k0 ∂k0 [fh (k )[Th (k )]ω,ω0 ] , Lβ 0

(2.105)

k

where ∂k0 and ∂k0 denote the discrete derivatives. If ω = ω 0 , the decomposition (2.99) is related to the following identity: 1 1 1 −1 0 − + [Th (k )]ω,ω = −ik0 + ωv0∗ k 0 −ik0 + ωE(k 0 ) −ik0 + ωv0∗ k 0 1 ik0 + ωE(k 0 ) − . (2.106) + 2 k0 + E(k 0 )2 + [σh−1 (k0 )]2 −ik0 + ωE(k 0 ) The bounds (2.101) and (2.102) easily follow from (2.98) and (2.104), the support properties of fh (k0 ) and the observation that f˜h (k0 ) and σh (k0 ) are smooth functions of k0 in R2 , in the support of fh (k0 ), so that the discrete derivatives can be bounded as the continuous derivatives. The main point is of course the fact that, on the support of fh (k0 ), | − ik0 + ωE(k 0 )|, | − ik0 + ωv0∗ k 0 | and p k02 + E(k 0 )2 + [σh−1 (k0 )]2 are of order γ h . 2.8. We now rescale the field so that p p Zh ψ (≤h) = Vˆ (h) Zh−1 ψ (≤h) ; V˜ (h)

(2.107)

it follows that (≤h)

LVˆ (h) (ψ) = γ h νh Fν(≤h) + δh Fα(≤h) + λh Fλ where νh =

Zh nh , Zh−1

δh =

Zh (ah − v0∗ zh ) , Zh−1

λh =

,

Zh Zh−1

(2.108) 2 lh .

We call the set ~vh = (νh , δh , λh ) the running coupling constants. If we now define √ (h−1) ˜h ( Zh−1 ψ (≤h−1) )−Lβ E e−V Z √ (≤h) ˆ (h) ) = PZh−1 ,σh−1 ,f˜−1 (dψ (h) )e−V ( Zh−1 ψ , h

(2.109)

(2.110)

p it is easy to see that V (h−1) ( Zh−1 ψ (≤h−1) ) is of the form (2.70) and that Eh−1 = Eh + th + E˜h .

(2.111)


00105


1353

It is sufficient to use the well known identity p ˜h + V (h−1) Zh−1 ψ (≤h−1) Lβ E =

∞ X p 1 (−1)n+1 EhT,n Vˆ (h) Zh−1 ψ (≤h) , n! n=1

(2.112)

−1 gω,ω0 , where EhT,n denotes the truncated expectation of order n with propagator Zh−1 (≤h) (≤h−1) (h) =ψ +ψ . see (2.91), and observe that ψ Moreover, the symmetry relations (2.71) are still satisfied, because the symmetry properties of the free measure are not modified by the renormalization procedure, so that the effective potential on scale h has the same symmetries as the effective potential on scale 0. Let us now define E˜hL,β , so that Z √ ˆ (hL,β ) ( Zh ˜ −V ψ (hL,β ) ) L,β −1 . e−Lβ EhL,β = PZh −1 ,σh −1 ,f˜−1 (dψ (hL,β ) )e (h)

L,β

L,β

hL,β

(2.113) We have EL,β =

1 X

˜ h + th ] . [E

(2.114)

h=hL,β

Note that the above procedure allows us to write the running coupling constants ~vh , h ≤ 0, in terms of ~vh0 , 0 ≥ h0 ≥ h + 1, and λ, ν, u: ~ vh+1 , . . . , ~v0 , λ, ν, u, δ ∗ ) . ~vh = β(~

(2.115)

~ vh+1 , . . . , ~v0 , λ, ν, u, δ ∗ ) is called the Beta function. The function β(~ 2.9. Let us now explain the main motivations of the integration procedure discussed above. In a renormalization group approach one has to identify the relevant, marginal and irrelevant interactions. By a power counting argument one sees that the terms bilinear in the fields are relevant, hence one should extract from them the relevant and marginal local contributions by a Taylor expansion of the kernel up to order 1 in the external momenta. Since σ1 + σ2 = 0 by the remark following (2.62), we have to consider only two kinds of bilinear terms: those with ω1 = ω2 and those with ω1 = −ω2 . It turns out that, for the bilinear terms with ω1 = −ω2 , a Taylor expansion up to order 0 is sufficient; the reason is that the Feynman graphs contributing to such terms contain at least one non-diagonal propagator and, by Lemma 2.6, such propagators are smaller than the diagonal ones by a factor σh γ −h ; as we shall see, this is sufficient to improve the power counting by 1. The previous discussion implies that the regularization of the bilinear terms produces four local terms. One of them, that proportional to Fν , is relevant; it reflects the renormalization of the Fermi momentum and is faced in a standard way [3], by fixing properly the counterterm ν in the Hamiltonian, i.e. by fixing properly


1354

00105


the chemical potential, so that the corresponding running coupling νh goes to 0 for h → −∞. The term proportional to Fα is marginal, but, as we shall see, stays bounded and of order λ as h → −∞, if δ ∗ is of order λ; hence the convergence of the flow is not related to the exact value of δ ∗ . However, in order to get a detailed description of the spin correlation function asymptotic behaviour, it is convenient to choose δ ∗ so that δh → 0 as h → −∞. This choice implies that v0∗ = v0 (1 + δ ∗ ) is the “effective” Fermi velocity of the fermion system. The other two terms are marginal, but have to be treated in different ways. The term proportional to Fζ is absorbed in the free measure and produces a field renormalization, as in the Luttinger liquid (which is indeed obtained for u = 0). The term proportional to Fσ , related to the presence of a gap in the spectrum, is also absorbed in the free measure, since there is no free parameter in the Hamiltonian to control its flow, as for Fζ . This operation can be seen as the application of a sequence of different Bogoliubov transformations at each integration step, to compare with the single Bogoliubov transformation that it is sufficient to see a gap O(u) at the Fermi surface, in the XY model (λ = 0). It turns out that the gap is deeply renormalized by the interaction, since σh is a sort of “mass terms” with a non-trivial renormalization group flow. Let us now consider the quartic terms, which are all marginal. Since there are many of them, depending on the labels ωi and σi of each field, their renormalization group flow seems difficult to study. However, as we have explained in Sec. 2.5, the running couplings corresponding to the quartic terms are all exactly equal to 0 for trivial reasons, unless, after a suitable permutation of the fields, σ = (+, −, +, −), ω = (+1, −1, −1, +1). Hence, by a Taylor expansion of the kernel up to order 0 in the external momenta, all quartic terms can be regularized, by introducing only one running coupling, λh . As in the Luttinger liquid [6, 7], the flow of λh and δh can be controlled by using some cancellations, due to the fact that the Beta function is “close” (for small u) to the Luttinger model Beta function. In Lemma 2.6 we write the propagator as the Luttinger model propagator plus a remainder, so that the Beta function is equal to the Luttinger model Beta function plus a “remainder”, which is small if σh γ −h is small. Let us define ¯ ¯:0≥h ¯ ≥ h} . h∗ = inf{h : 0 ≥ h ≥ hL,β , a0 v0∗ γ h−1 ≥ 4|σh¯ |, ∀ h

(2.116)

Of course this definition is meaningful only if a0 v0∗ γ −1 ≥ 4|σ0 | = 4|u|v0 (see (2.64)), that is if a0 (1 + δ ∗ ) . (2.117) |u| ≤ 4γ If the condition (2.117) is not satisfied, we shall put h∗ = 1. Lemma 2.6, (2.86) and the definition of h∗ easily imply the following lemma.


00105


1355

2.10 Lemma. If h > h∗ ≥ 0 and the conditions (2.98) are satisfied, there is a constant C such that |th | ≤ Cγ 2h .

(2.118)

Moreover, given the positive integers N, n0 , n1 and putting n = n0 + n1 , there exist a constant CN,n such that (h)

|∂xn00 ∂¯xn1 gω,ω0 (x; y)| ≤ CN,n

1+

γ h+n . − y)|)N

(2.119)

(γ h |d(x

2.11. In Sec. 3 we will see that, using the above lemmas and assuming that the running coupling constants are bounded, the integration of the field ψ (h) in (2.88) is well defined in the limit L, β → ∞, for 0 ≥ h > h∗ . The integration of the scales from h∗ to hL,β will be performed “in a single step”. This is possible because we shall prove in Sec. 3 that the integration in the r.h.s. in (2.82) is well defined in the limit L, β → ∞, for h = h∗ . In order to do that, we shall use the following lemma, whose proof is similar to the proof of Lemma 2.6. ∗

¯, 2.12 Lemma. Assume that h∗ is finite uniformly in L, β, so that |σh∗ −1 γ −h | ≥ κ for a suitable constant κ ¯ and define Z

(≤h∗ )

g¯ω,ω0 (x − y) Zh∗ −1

≡

∗

(≤h PZ˜h∗ −1 ,σh∗ −1 ,Ch∗ (dψ (≤h ) )ψx,ω

∗

)−

(≤h∗ )+

ψy,ω0

.

(2.120)

Then, given the positive integers N, n0 , n1 and putting n = n0 + n1 , there exist a constant CN,n such that (≤h∗ )

|∂xn00 ∂¯xn1 gω,ω0 (x; y)| ≤ CN,n

∗

γ h +n . ∗ h 1 + (γ |d(x − y)|)N

(2.121)

2.13. Comparing Lemmas 2.10 and 2.12, we see that the propagator of the integration of all the scales between h∗ and hL,β has the same bound as the propagator of the integration of a single scale greater than h∗ ; this property is used to perform the ∗ integration of all the scales ≤ h∗ in a single step. In fact 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(σh γ −h ) terms), while for momenta smaller than γ h it is ∗ a “massive” theory with mass O(γ h ). 3. Analyticity of the Effective Potential 3.1. We want to study the expansion of the effective potential, which follows from the renormalization procedure discussed in Sec. 2. In order to do that, we find (≤h)σ it convenient to write V (h) , h ≤ 1, in terms of the variables ψx,ω . The two contributions to V (1) (ψ (≤1) ), see (2.58) and (2.14), become


1356

00105


λVλ (ψ ≤1 ) =

XZ

dxdyλvλ (x − y)eipF x(σ1 +σ4 )+ipF y(σ2 +σ3 )

σ (≤1)σ

(≤1)σ

(≤1)σ1 (≤1)σ2 · ψx,σ ψy,σ2 ψy,−σ33 ψx,−σ44 , 1 X Z (≤1)σ1 (≤1)σ2 ψx,−σ2 , dxeipF x(σ1 +σ2 ) νψx,σ νN (ψ ≤1 ) = 1

(3.1)

σ1 ,σ2

R β/2 R P where dx is a shorthand for x∈Λ −β/2 dx0 . If we define X P2n 0 1 (h) e−i r=1 σr kr xr W2n,σ,ω (x1 , . . . , x2n ) = 2n (Lβ) 0 0 k1 ,...,k2n

ˆ (h) (k0 , . . . , k0 ×W 1 2n−1 )δ 2n,σ,ω

2n X

! σi (k0i

+ pF ) ,

(3.2)

i ψx(≤h)σ W2n,σ,ω (x1 , . . . , x2n ) . i ,ωi

(3.3)

i=1

we can write (2.70) as V (h) (ψ (≤h) ) =

∞ XZ X

dx1 · · · dx2n

" 2n Y

n=1 σ,ω

# (h)

i=1

Note that (h)

W2n,σ,ω (x1 + x, . . . , x2n + x) = eipF x

P2n r=1

σr

(h)

W2n,σ,ω (x1 , . . . , x2n ) ,

(h)

hence W2n,σ,ω (x1 , . . . , x2n ) is translation invariant if and only if

P2n r=1

(3.4)

σr = 0.

(≤h)σ ψx,ω

) in terms of the variables is obtained The representation of LV (ψ by substituting in the r.h.s. of (2.79) the x-space representations of the definitions (2.80). We have X Z (≤h) (≤h)+ (≤h)− = ω dxψx,ω ψx,ω , Fν (h)

ω=±1 (≤h) Fσ

=

X

Z

(≤h)

(≤h)

Fζ

i cos pF ¯2 (≤h)− (≤h)+ ¯ (≤h)− dxψx,ω ∂1 ψx,ω ∂1 ψx,ω + 2v0 ω=±1 Z X i cos pF ¯2 (≤h)+ (≤h)− (≤h)+ ¯ iω dx −∂1 ψx,ω + ∂1 ψx,ω ψx,ω , = 2v0 ω=±1 X Z X Z (≤h)+ (≤h)− (≤h)+ (≤h)− = ψx,ω , dxψx,ω ∂0 ψx,ω = − dx∂0 ψx,ω =

X

ω=±1

Z

(≤h) Fλ

(≤h)−

(≤h)+ dxψx,ω ψx,−ω ,

iω

ω=±1

Fα

(≤h)

=

Z

iω

ω=±1

(≤h)+ (≤h)− (≤h)+ (≤h)− dxψx,+1 ψx,−1 ψx,−1 ψx,+1

,

(3.5)


00105


1357

where ∂0 is the derivative w.r.t. x0 , ∂¯1 is the symmetric discrete derivative w.r.t. x, that is, given a function f (x), ∂¯1 f (x) = [f (x + 1, x0 ) − f (x − 1, x0 )]/2 ,

(3.6)

and ∂¯12 (which is not the square of ∂¯1 , but has the same properties) is defined by the equation ∂¯12 f (x) = f (x + 1, x0 ) + f (x − 1, x0 ) − 2f (x, x0 ) .

(3.7)

Let us now discuss the action of the operator L and R = 1 − L on the effective potential in the x-space representation, by considering the terms for which L = 6 0. (1) If 2n = 4, by (2.72), Z Z 4 4 Y Y (≤h)σi i ψxi ,ωi = dxW (x) [Gσi (xi − x4 )ψx(≤h)σ ], (3.8) L dxW (x) 4 ,ωi i=1

i=1 (h)

where x = (x1 , . . . , x4 ), W (x) = W4,σ,ω (x1 , x2 , x3 , x4 ) and ¯

x

Gσ (x) = eiσk++ x = eiσπ( L +

x0 β

)

.

(3.9)

Note that, as we have discussed in Sec. 2.5, the r.h.s. of (3.8) is always equal to 0, unless, after a suitable permutation of the fields, σ = (+, −, +, −), Q4 ω = (+1, −1, −1, +1). In this last case the function W (x) i=1 Gσi (xi − x4 ) = W (x)G+ (x1 − x2 + x3 − x4 ) is translation invariant and periodic in the space and time components of all variables xk , of period L and β, respectively. It follows that (≤h)σ the quantities Gσi (xi − x4 )ψx4 ,ωi i in the r.h.s. of (3.8) can be substituted with (≤h)σi Gσi (xi − xk )ψxk ,ωi , k = 1, 2, 3. Hence we have four equivalent representations of the localization operation, which differ by the choice of the localization point. The freedom in the choice of the localization point will be useful in the following. If the localization point is chosen as in (3.8), we have " 4 # Z Z 4 4 Y Y Y (≤h)σi (≤h)σi (≤h)σi ψxi ,ωi = dxW (x) ψxi ,ωi − Gσi (xi − x4 )ψx4 ,ωi . R dxW (x) i=1

i=1

i=1

(3.10) The term in square brackets in the above equation can be written as 1 2 3 4 ψx(≤h)σ Dx1,1(≤h)σ ψx(≤h)σ ψx(≤h)σ 1 ,ω1 2 ,ω2 3 ,x4 ,ω3 4 ,ω4 1 2 3 4 + Gσ3 (x3 − x4 )ψx(≤h)σ Dx1,1(≤h)σ ψx(≤h)σ ψx(≤h)σ 1 ,ω1 2 ,x4 ,ω2 4 ,ω3 4 ,ω4 1 2 3 4 + Gσ3 (x3 − x4 )Gσ2 (x2 − x4 )Dx1,1(≤h)σ ψx(≤h)σ ψx(≤h)σ ψx(≤h)σ , 1 ,x4 ,ω1 4 ,ω2 4 ,ω3 4 ,ω4

(3.11)

where 1,1(≤h)σ (≤h)σ (≤h)σ = ψy,ω − Gσ (y − x)ψx,ω . Dy,x,ω

(3.12)

Similar expressions can be written, if the localization point is chosen in a different way.


1358

00105


Note that the decomposition (3.11) corresponds to the following identity: ˆ (k0 , k0 , k0 ) − W ˆ (k0 , k0 , k ¯ ++ )] ˆ (k0 , k0 , k0 ) = [W RW 2 3 2 3 2 τ,P 1 τ,P 1 τ,P 1 (h)

(h)

(h)

¯ ++ ) − W ¯ ++ , k ¯ ++ )] ˆ (k01 , k ˆ (k01 , k02 , k + [W τ,P τ,P (h)

(h)

¯ ++ , k ¯ ++ ) − W ¯ ¯ ¯ ˆ (k ˆ (k0 , k + [W τ,P 1 τ,P ++ , k++ , k++ )] , (h)

(h)

(3.13)

¯ ++ . and that the ith term in the r.h.s. of (3.13) is equal to 0 for k0i = k 1,1(≤h)σ is antiperiodic in the space and time components of x and The field Dy,x,ω y, of period L and β, and is equal to 0 if x = y modulo (L, β). This means that it is dimensionally equivalent to the product of d(x, y) (see (2.97)) and the derivative of the field, so that the bound of its contraction with another field variable on a scale 0 (≤h)σ h0 < h will produce a “gain” γ −(h−h ) with respect to the contraction of ψy,ω . If we insert (3.11) in the r.h.s. of (3.10), we can decompose the l.h.s in the sum of three terms, which differ from the term which R acts on mainly because one ψ (≤h) field is substituted with a D1,1(≤h) field and some of the other ψ (≤h) fields are “translated” in the localization point. All three terms share the property that the field whose x coordinate is equal to the localization point is not affected by the action of R. In our approach, the regularization effect of R will be exploited trough the decomposition (3.11). However, for reasons that will become clear in the following, it is convenient to start the analysis by using another representation of the expression (≤h)σ resulting from the insertion of (3.11) in (3.10). If ψxi ≡ ψxi ,ωi i , we can write, if the localization point is x4 , Z 4 Y ψxi W (x) R dx i=1

Z =

dx

4 Y

Z ψxi W (x) − δ(x3 − x4 ) dy3 W (x1 , x2 , y3 , x4 )Gσ3 (y3 − x4 )

i=1

Z +

dx

4 Y

Z ψxi δ(x3 − x4 )

i=1

Z − δ(x2 − x4 ) Z + "

dx

4 Y

dy3 W (x1 , x2 , y3 , x4 )Gσ3 (y3 − x4 )

dy2 W (x1 , y2 , y3 , x4 )Gσ3 (y3 − x4 )Gσ2 (y2 − x4 ) Z

ψxi δ(x2 − x4 )δ(x3 − x4 )

Z dy2

dy3

i=1

· W (x1 , y2 , y3 , x4 )Gσ3 (y3 − x4 ) · Gσ2 (y2 − x4 ) Z − δ(x1 − x4 )

dy1 W (y1 , y2 , y3 , x4 )

3 Y i=1

# Gσi (yi − x4 ) ,

(3.14)


00105


where δ(x) is the antiperiodic delta function, that is X 0 1 eiσk x . δ(x) = Lβ 0 0

1359

(3.15)

k ∈DL,β

Similar expressions are obtained, if the localization point is chosen in a different way. In the new representation, the action of R is seen as the decomposition of the original term in the sum of three terms, which are still of the form (3.3), but with a different kernel, containing suitable delta functions. (2) If 2n = 2 and, possibly after a suitable permutation of the fields, σ = (+, −), ω1 = ω2 = ω, by (2.74), Z (h) ψx(≤h)− L dx1 dx2 W2,σ,ω (x1 − x2 )ψx(≤h)+ 1 ,ω 2 ,ω Z (h)

dx1 dx2 W2,σ,ω (x1 − x2 )ψx(≤h)+ Tx1(≤h)− 1 ,ω 2 ,x1 ,ω

= Z

(h)

dx1 dx2 W2,σ,ω (x1 − x2 )Tx1(≤h)+ ψx(≤h)− , 1 ,x2 ,ω 2 ,ω

=

(3.16)

with 1(≤h)σ (≤h)σ = ψx,ω cβ (y0 − x0 )[cL (y − x) + bL dL (y − x)] Ty,x,ω i cos pF ¯2 (≤h)σ (≤h)σ + ∂1 ψx,ω + ∂¯1 ψx,ω cβ (y0 − x0 )aL dL (y − x) 2v0 (≤h)σ dβ (y0 − x0 )cL (y − x) , + ∂0 ψx,ω

(3.17)

where dL (x) and dβ (x0 ) are defined as in (2.96) and cL (x) = cos(πxL−1 ) ,

cβ (x0 ) = cos(πx0 β −1 ) .

(3.18)

As in the item (1), we define the localization point as the x coordinate of the field which is left unchanged L. We are free to choose it equal to x1 or x2 . This freedom affects also the action of R, which can be written as Z (h) ψx(≤h)− R dx1 dx2 W2,σ,ω (x1 − x2 )ψx(≤h)+ 1 ,ω 2 ,ω Z (h)

dx1 dx2 W2,σ,ω (x1 − x2 )ψx(≤h)+ Dx2(≤h)− 1 ,ω 2 ,x1 ,ω

= Z

(h)

(3.19)

2(≤h)σ (≤h)σ 1(≤h)σ = ψy,ω − Ty,x,ω . Dy,x,ω

(3.20)

=

dx1 dx2 W2,σ,ω (x1 − x2 )Dx2(≤h)+ ψ (≤h)− , 1 ,x2 ,ω x2 ,ω

with Hence the effect of R can be described as the replacement of a ψ (≤h)σ field with a D2(≤h)σ field, with a gain in the bounds (see discussion in item (1) above) of a 0 factor γ −2(h−h ) .


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Also in this case, it is possible to write the regularized term in the form (3.3). We get Z (h) (≤h)+ (≤h)− ψy,ω R dxdyW2,σ,ω (x − y)ψx,ω Z =

(h) (≤h)+ (≤h)− dxdyψx,ω ψy,ω W2,σ,ω (x − y) − δ(y − x) Z (h)

dzW2,σ,ω (x − z)cβ (z0 − x0 )[cL (z − x) + bL dL (z − x)]

×

i cos pF ¯2 ∂1 δ(y − x) − −∂¯1 δ(y − x) + 2v0 Z (h) × dzW2,σ,ω (x − z)cβ (z0 − x0 )aL dL (z − x) Z + ∂0 δ(y − x)

(h) dzW2,σ,ω (x − z)dβ (z0 − x0 )cL (z − x) .

(3.21)

(3) If 2n = 2 and, possibly after a suitable permutation of the fields, σ = (+, −), ω1 = −ω2 = ω, by (2.74), Z (h) (≤h)− ψx2 ,−ω L dx1 dx2 W2,σ,ω (x1 − x2 )ψx(≤h)+ 1 ,ω Z (h)

0(≤h)−

(h)

(≤h)−

dx1 dx2 W2,σ,ω (x1 − x2 )ψx(≤h)+ Tx2 ,x1 ,−ω 1 ,ω

= Z =

dx1 dx2 W2,σ,ω (x1 − x2 )Tx0(≤h)+ ψx2 ,−ω , 1 ,x2 ,ω

(3.22)

where 0(≤h)σ (≤h)σ = cβ (y0 − x0 )cL (y − x)ψx,ω . Ty,x,ω

Therefore

(3.23)

Z R

(h)

(≤h)−

ψx2 ,−ω dx1 dx2 W2,σ,ω (x1 − x2 )ψx(≤h)+ 1 ,ω Z (h)

1,2(≤h)−

(h)

(≤h)−

dx1 dx2 W2,σ,ω (x1 − x2 )ψx(≤h)+ Dx2 ,x1 ,−ω 1 ,ω

= Z =

dx1 dx2 W2,σ,ω (x1 − x2 )Dx1,2(≤h)+ ψx2 ,−ω , 1 ,x2 ,ω

(3.24)

where 1,2(≤h)σ (≤h)σ 0(≤h)σ = ψy,ω − Ty,x,ω . Dy,x,ω

(3.25)

Hence the effect of R can be described as the replacement of a ψ (≤h)σ field with a D1,2(≤h)σ field, with a gain in the bounds (see discussion in item (1) above) of a


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0

factor γ −(h−h ) . As before, we can also write Z (h) (≤h)+ (≤h)− ψy,−ω R dxdyW2,σ,ω (x − y)ψx,ω Z (≤h)+ (≤h)− dxdyψx,ω ψy,−ω

=

Z ×

(h) dzW2,σ,ω (x

(h) · W2,σ,ω (x − y) − δ(y − x)

− z)cβ (z0 − x0 )cL (z − x) .

(3.26)

3.2. By using iteratively the “single scale expansion” (2.112), starting from Vˆ (1) = √ V (1) , we can write the effective potential V (h) ( Zh ψ (≤h) ), for h ≤ 0, in terms of a tree expansion, similar to that described, for example, in [6]. 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 (see Fig. 1), 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. It is then easy to see that 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. (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 an integer taking values in [h, 2], and we represent any

Fig. 1.


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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. 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. Finally, if there is only one endpoint, its scale must be equal to +2 or h + 2. (3) With each endpoint v of scale hv = +2 we associate one of the two contributions to V (1) (ψ (≤1) ), written as in (3.1) and a set xv of space-time points (the corresponding integration variables), two for λVλ (ψ (≤1) ), one for νN (ψ (≤1) ); we shall say that the endpoint is of type λ or ν, respectively. With each endpoint v of scale hv ≤ 1 we associate one of the four local terms that we obtain if we write LV (hv −1) (see (2.108)) by using the expressions (3.5) (there are four terms since Fα is the sum of two different local terms), and one space-time point xv ; we shall say that the endpoint is of type ν, δ1 , δ2 , λ, with an obvious correspondence with the different terms. Given a vertex v, which is not an endpoint, xv will denote the family of all space-time points associated with one of the endpoints following v. Moreover, we impose the constraint that, if v is an endpoint and xv is a single space-time point (that is the corresponding term is local), hv = hv0 + 1, if v 0 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) The trees containing only the root and an endpoint of scale h + 1 will be called the trivial trees; note that they do not belong to Th,1 , if h ≤ 0, and can be associated with the four terms in the local part of Vˆ (h) . (6) 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 . If hv ≤ 0, one of the field variables belonging to Iv carries also a discrete derivative ∂¯1m , m ∈ {1, 2}, if the corresponding local term is of type δm , see (3.5). Hence we can associate with each field label f an integer m(f ) ∈ {0, 1, 2}, denoting the order of the discrete derivative. Note that m(f ) is not uniquely determined, (≤h −1) in (3.5); since we are free to use the first or the second representation of Fα v we shall use this freedom in the following.


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By using (2.112), it is not hard to see that, if h ≤ 0, the effective potential can be written in the following way: V (h)

∞ X X p p ˜h+1 = Zh ψ (≤h) + Lβ E V (h) τ, Zh ψ (≤h) ,

(3.27)

n=1 τ ∈Th,n

where, if v0 is the first vertex of τ and τ1 , . . . , τs (s = sv0 ) are the subtrees of τ √ with root v0 , V (h) (τ, Zh ψ (≤h) ) is defined inductively by the relation p V (h) τ, Zh ψ (≤h) p p (−1)s+1 T ¯ (h+1) Eh+1 V τ1 , Zh ψ (≤h+1) ; . . . ; V¯ (h+1) τs , Zh ψ (≤h+1) , s! (3.28) √ and V¯ (h+1) (τi , Zh ψ (≤h+1) ) √ (a) is equal to RVˆ (h+1) (τi , Zh ψ (≤h+1) ) if the subtree τi is not trivial (see (2.107) for the definition of Vˆ (h) ); (b) if τi is trivial and h ≤ −1, it is equal to one of the terms in the r.h.s. of (2.108) with scale h + 1 or, if h = 0, to one of the terms contributing to Vˆ (1) (ψ ≤1 ). If h = 0, the r.h.s. of (3.28) can be written more explicitly in the following way. Given τ ∈ T0,n , there are n endpoints of scale 2 and only another one vertex, v0 , of scale 1; let us call v1 , . . . , vn the endpoints. We choose, in any set Ivi , a subset Qvi S and we define Pv0 = i Qvi ; then we can write (recall that Z0 = 1) p X (0) V (τ, Pv0 ) , (3.29) V (0) τ, Z0 ψ (≤0) = =

V (0) (τ, Pv0 ) = (1)

Kτ,Pv (xv0 ) = 0

p |Pv0 | Z0

Z

Pv0

dxv0 ψ˜≤0 (Pv0 )Kτ,Pv (xv0 ) , (1)

0

n Y 1 T ˜(1) E1 [ψ (Pv1 \Qv1 ), . . . , ψ˜(1) (Pvn \Qvn )] Kv(2) (xvi ) , i n! i=1

where we used the definitions ψ˜(h) (Pv ) =

Y

m(f ) (h)σ(f ) ψx(f ),ω(f ) , ∂¯1

(3.30)

(3.31)

(3.32)

f ∈Pv

(xvi ) = e Kv(2) i

ipF

×

(

P f ∈Iv

i

x(f )σ(f )

λvλ (x − y)

if vi is of type λ and xvi = (x, y) ,

ν

if vi is of type ν ,

(3.33)

and we suppose that the order of the (anticommuting) field variables in (3.32) is suitable chosen in order to fix the sign as in (3.31). ˜1 , Note that the terms with Pv0 6= ∅ in the r.h.s. of (3.29) contribute to Lβ E √ (0) (≤0) ). while the others contribute to V ( Z0 ψ


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p The potential Vˆ (0) ( Z−1 ψ (≤0) ), needed to iterate the previous procedure, is obtained, as explained in Secs. 2.5 and 2.8, by decomposing V (0) in the sum of LV (0) and RV (0) , by moving afterwards some local terms to the free measure p and finally by rescaling the fields variables. The representation we get for V (−1) ( Z−1 ψ (≤−1) ) depends on the representation we use for RV (0) (τ, Pv0 ). We choose to use that based on (3.14), (3.21) and (3.26), where the regularization is seen, for each term in the r.h.s. of (3.29) with Pv0 6= ∅, as a modification of the kernel Z (0) (1) (3.34) Wτ,Pv (xPv0 ) = d(xv0 \xPv0 )Kτ,Pv (xv0 ) , 0

where xPv0 = RV

S

(0)

f ∈Pv0

0

x(f ). In order to remember this choice, we write

p |Pv0 | Z (1) (τ, Pv0 ) = Z0 dxv0 ψ˜(≤0) (Pv0 )[RKτ,Pv (xv0 )] . 0

(3.35)

It is then easy to get, by iteration of the previous procedure, a simple expression √ for V (h) (τ, Zh ψ (≤h) ), for any τ ∈ Th,n . 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 S and v1 , . . . , vsv are the 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 S 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. Given τ ∈ Th,n , there are many possible choices of the subsets Pv , v ∈ τ , compatible with all the constraints; we shall denote Pτ the family of all these choices and P the elements of Pτ . Then we can write X p V (h) (τ, P) ; (3.36) V (h) τ, Zh ψ (≤h) = P∈Pτ

V (h) (τ, P) can be represented as in (3.30), that is as p |Pv0 | Z (h+1) V (h) (τ, P) = Zh dxv0 ψ˜(≤h) (Pv0 )Kτ,P (xv0 ) ,

(3.37)

(h+1)

with Kτ,P (xv0 ) defined inductively (recall that hv0 = h + 1) by the equation, valid for any v ∈ τ which is not an endpoint, (h ) Kτ,Pv (xv )

1 = sv !

Z hv Zhv −1

|P2v | Y sv

[Kv(hi v +1) (xvi )]

i=1

· E˜hTv [ψ˜(hv ) (Pv1 \Qv1 ), . . . , ψ˜(hv ) (Pvsv \Qvsv )] ,

(3.38)

where E˜hT denotes the truncated expectation with propagator g (h) (without the scaling factor Zh−1 , which is present in the definition of EhT used in (2.112)) and


00105


Z1 ≡ 1. Moreover, if v is an endpoint otherwise  λhv −1        iωδhv −1 (hv ) Kv (xv ) =     ωγ hv −1 νhv −1    (h ) Kv v

(hv )

and hv = 2, Kv

1365

(xv ) is defined by (3.33),

if v is of type λ , if v is of type δ1 , d2 and ω(f ) = ω for both f ∈ Iv ,

(3.39)

if v is of type ν and ω(f ) = ω for both f ∈ Iv .

(hv ) RKτi ,P , i

= where τ1 , . . . , τsv are the subtrees of τ with If v is not an endpoint, root v, Pi = {Pv , v ∈ τi } and the action of R is defined using the representation (3.14), (3.21) and (3.26) of the regularization operation, seen as a modification of the kernel Z (h ) (h ) (3.40) Wτ,Pv (xPv ) = d(xv \xPv )Kτ,Pv (xv ) , S where xPv = f ∈Pv x(f ). Finally we suppose again that the order of the (anticommuting) field variables is suitable chosen in order to fix the sign as in (3.37). We remark that the definitions (3.14), (3.21) and (3.26) of R are sufficient, even if they are restricted to external fields with m(f ) = 0, because we can use the freedom in the definition of m(f ), see item (6) above, so that the external fields of v have always m(f ) = 0, if v is a vertex where the R operation is acting on. This last claim follows from the observation that, since the truncated expectation in (3.38) vanishes if sv > 1 and Pvi \Qv1 = ∅ for some i, at least one of the fields associated with the endpoints of type δ1 or δ2 , the only ones which have fields with m(f ) > 0, has to be an internal field; hence, if one of the two fields is external, we can put m(f ) = 0 for it. If sv = 1 the previous argument should not work, but in this case the only vertex immediately following v can be an endpoint of type δ1 or δ2 only if v = v0 , see item (2) above; however this is not a problem since the action of R on a local term is equal to 0. P (h ) Note also that the kernel Kτ,Pv (xv ) is translation invariant, if f ∈Pv σ(f ) = 0; in general, it satisfies the relation (h )

Kτ,Pv (xv + x) = eipF x

P

f ∈Pv

σ(f )

(h )

Kτ,Pv (xv ) .

(3.41)

There is a simple interpretation of V (h) (τ, P) as the sum of a family GP of connected Feynman graphs build with single scale propagators of different scales, connecting the space-time points associated with the endpoints of the tree. A graph g ∈ GP is build by contracting, for any v ∈ τ , all the internal fields in couples in all possible ways, by using the propagator g hv , so that we get a connected Feynman graph, if we represent as single points all the clusters associated with the vertices immediately following v. These graphs have the property that the set of lines connecting the endpoints of the cluster Lv and having scale h0 ≥ hv is a connected subgraph; by the way this property is indeed another constraint on the possible choices of P. We shall call these graphs compatible with P.


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3.3. The representation (3.37) of V (h) (τ, P) is based on the choice of representing the regularization as acting on the kernels. If we use instead the representation of R based on (3.10), (3.11), (3.19) and (3.24), some field variables have to be substituted with new ones, depending on two space-time points and containing possibly some derivatives. As we shall see, these new variables allow to get the right dimensional bounds, at the price of making much more involved the combinatorics. Hence, it is convenient to introduce a label rv (f ) to keep trace of the regularization in the vertices of the tree where f is associated with an external field and the action of R turns out to be non-trivial, that is R = 6 1. There are many vertices, where R = 1 by definition, that is the vertices with more than 4 external fields, the endpoints and v0 . For these vertices all external fields will be associated with a label rv (f ) = 0. Moreover, since LR = 0, the action of R is trivial even in most trivial vertices v with |Pv | ≤ 4. This happens if the vertex (trivial or not) v˜ immediately following v has the same number of external fields as v, since then the kernels associated with v and v˜ are identical, up to a rescaling constant. In particular, this remark implies that, given the non-trivial vertex v and the non-trivial vertex v 0 immediately preceding v on the tree, there are at most two vertices v¯, such that v 0 < v¯ ≤ v and the action of R is non-trivial. For the same reason, given an endpoint v of scale hv = +2 of type λ (hence not local), there are at most two vertices between v and the non-trivial vertex v 0 immediately preceding v, where the action of R is nontrivial. Since the number of endpoints is n and the number of non-trivial vertices is bounded by n − 1, the number of vertices where the action of R is non-trivial is bounded by 2(2n − 1). Let us now consider one of these vertices, which all have 4 or 2 external fields. If |Pv | = 2 and the ω indices of the external fields are equal, we keep trace of the regularization by labeling the field variable, which is substituted with a D2 field, see (3.19), with rv (f ) = 2 and the other with rv (f ) = 0. In principle we are free to decide which variable is labeled with rv (f ) = 2, that is how we fix the localization point; we make a choice in the following way. If there is no non-trivial vertex v 0 such that v0 ≤ v 0 < v, we make an arbitrary choice, otherwise we put rv (f ) = 2 for the field which is an internal field in the nearest non-trivial vertex preceding v. In other words, we try to avoid that a field affected by the regularization stays external in the vertices preceding v. If |Pv | = 2 and the ω indices of the external fields are different, we label the field variable, which is substituted with a D1,2 field, see (3.24), with rv (f ) = 1 and the other with rv (f ) = 0; which variable is labeled with rv (f ) = 1 is decided as in the previous case. If |Pv | = 4, first of all we choose the localization point in the following way. If there is a vertex v 0 such that v0 ≤ v 0 < v and Pv0 contains one and only one f ∈ Pv , we chose x(f ) as the localization point in v; in the other cases, we make an arbitrary choice. After that, we split the kernel associated with v into three terms as in (3.14); then we distinguish the three terms by putting rv (f ) = 1 for the


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external field which is substituted with a D1,1(≤h) field, when the delta functions are eliminated, and rv (f ) = 0 for the others. The previous definitions imply that, given f ∈ Iv0 , it is possible that there are many different vertices in the tree, such that rv (f ) 6= 0, that is many vertices where the corresponding field variable appears as an external field and the action of R is non-trivial. As a consequence, the expressions given in Sec. 3.1 for the regularized potentials would not be sufficient and we should consider more general expressions, containing as external fields more general variables. Even worse, there is the risk that field derivatives of arbitrary order have to be considered; this event would produce “bad” factorials in the bounds. Fortunately, we can prove that this phenomenon can be easily controlled, thanks to our choice of the localization point, see above, by a more careful analysis of the regularization procedure, that we shall keep trace of by changing the definition of the rv (f ) labels. Let us suppose first that |Pv | = 4 and that there is f ∈ Pv , such that rv¯ (f ) 6= 0 for some v¯ > v. We want to show that the action of R on v is indeed trivial; hence we can put rv (f ) = 0 for all f ∈ Pv , in agreement with the fact that the contribution to the effective potential associated with v is dimensionally irrelevant. First of all, note that it is not possible that |Pv¯ | = 2, as a consequence of the choice of the localization point in the vertices with two external fields, see above. On the other hand, if |Pv¯ | = 4, the fact that the action of R in the vertex v is equal to the identity follows from the observation following (3.13) and the definition (2.72). Let us now consider the vertices v with Pv = (f1 , f2 ). We can exclude as before that rv¯ (fi ) 6= 0 for i = 1 or i = 2 or both and |Pv¯ | = 2. The same conclusion can be reached, if there is no vertex v¯ > v, such that |Pv¯ | = 4, the action of R on v¯ is non-trivial and both f1 and f2 belong to the set of its external fields; this claim easily follows from the criterion for the choice of the localization point in the vertices with 4 external fields. If, on the contrary, f1 and f2 are both labels of external fields of a vertex v¯ > v, such that |Pv¯ | = 4 and the action of R is non-trivial, we have to distinguish two possibilities. If there is a non-trivial vertex v 0 such that v0 ≤ v 0 < v, and one of the external fields of v, let us say of label f1 , is an internal field, our choice of the localization points imply that both rv (f1 ) and rv¯ (f1 ) are different from 0, while rv (f2 ) = rv¯ (f2 ) = 0. If there is no non-trivial vertex v 0 < v with the previous property, that is if f1 and f2 are both labels of external fields down to v0 (hence all vertices between v and v0 are trivial) or they become together labels of internal fields in some vertex v 0 < v, we are still free to choose as we want the localization points in v and v¯; we decide to choose them equal. The previous discussion implies that, as a consequence of our prescriptions, a field variable can be affected by the regularization only once, except in the case considered in the last paragraph. However, also in this case, it is easy to see that everything works as we did not apply to the variable with label f1 the regularization in the vertex v¯. In fact, the first or second order zero (modulo (L, β)) in the difference x(f1 ) − x(f2 ), related to the regularization in the vertex v, see Sec. 3.1,


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cancels the contribution of the term proportional to the delta function, related with the regularization of v¯, see (3.14). This apparent lack of regularization in v¯ is compensated by the fact that x(f1 ) − x(f2 ) is of order γ −hv¯ , hence smaller than the factor γ −hv sufficient for the regularization of v (together with the improving effect of the field derivative). Hence there is a gain with respect to the usual bound of a factor γ −(hv¯ −hv ) , sufficient to regularize the vertex v¯. 3.4. There is in principle another problem. Let us suppose that we decide to represent all the non-trivial R operations as acting on the field variables. Let us suppose also that the field variable with label f is substituted, by the action of R on the 1,i 2 or a Dy,x field, where y = x(f ) and x = x(f 0 ) is the correvertex v, with a Dy,x sponding localization point. At first sight it seems possible that even the variable with label f 0 can be substituted with a D1,i or a D2 field by the action of R on a vertex v¯ > v. If this happens, the point x(f 0 ) can not be considered as fixed and there is an “interference” between the two regularization operations, or even more than two, since this phenomenon could involve an ordered chain of vertices. This interference would not produce bad factorials in the bounds, but would certainly make more involved our expansion. However, we can show that, thanks to our localization prescription, this problem is not really present. Let us suppose first that |Pv | = 2. In this case, if the field with label f 0 is external in some vertex v¯ > v, with |Pv¯ | equal to 2 or 4, we are sure that x(f 0 ) is the localization point in v¯, see Sec. 3.3, hence the corresponding filed can not be affected by the action of R on v¯. The same conclusion can be reached, if |Pv | = 4 and |Pv¯ | = 2. If |Pv | = |Pv¯ | = 4 and the field with label f 0 is substituted, by the action of R on the vertex v¯, with a D1,i , we know that the same can not be true for the field with label f , since the action of R on v is trivial. The previous discussion implies that the field with label f 0 can be affected by the regularization (if |Pv | = |Pv¯ | = 4) only by changing its x label, but this is not a source of any problem. 1,i(≤h)σ

3.5. In this section we want to discuss the representation of the fields Dy,x,ω , i = 2(≤h)σ 1, 2, and Dy,x,ω introduced in Sec. 3.1, which allows to exploit the regularization properties of the R operation. In order to do that, we extend the definition of the (≤h)σ fields ψx,ω to R2 , by using (2.49); we get functions with values in the Grassmanian algebra, antiperiodic in x0 and x with periods β and L, respectively. Let us choose a family of positive functions χη,η0 (x), η, η 0 ∈ {−1, 0, +1}, on R2 , such that ( 1 , if |x − η| ≤ 1/4 and |x0 − η 0 | ≤ 1/4 , χη,η0 (x) = 0 , if |x − η| ≥ 3/4 or |x0 − η 0 | ≥ 3/4 , (3.42) X χη,η0 (x) = 1 if x ∈ [−1, 1] × [−1, 1] . η,η 0


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1369

˜ ) > 0, where x ˜ = (x/L, x0 /β) and Given x, y ∈ Λ × [−β/2, β/2], if χη,η0 (˜ y−x ˜ = (y/L, y0/β), we can define y ¯ = y − (ηL, η 0 β), so that |x0 − y¯0 | ≤ 3β/4 and y 0 1,1(≤h)σ 1,1(≤h)σ = (−1)η+η Dy¯ ,x,ω and we |x − y¯| ≤ 3L/4. We see immediately that Dy,x,ω can write 1,1(≤h)σ

Dy¯ ,x,ω

(≤h)σ

= [ψy¯ ,ω

(≤h)σ (≤h)σ − ψx,ω ] + [1 − Gσ (¯ y − x)]ψx,ω .

(3.43)

It is easy to see that, if |y0 | ≤ 3β/4 and |y| ≤ 3L/4, 1 − Gσ (y) =

1¯ 1¯ h1 (˜ y)dL (y) + h y)dβ (y0 ) , 2 (˜ L β

˜ = (y/L, y0/β) , y

(3.44)

¯ i (y), i = 1, 2, are suitable functions, uniformly smooth in L and β. Moreover where h Z 1 (≤h)σ (≤h)σ (≤h)σ = (¯ y − x) · dt∂ψξ(t),ω , ξ(t) = x + t(¯ y − x) , (3.45) ψy¯ ,ω − ψx,ω 0

where ∂ = (∂1 , ∂0 ) is the gradient, and it is easy to see that, if |y0 | ≤ 3β/4 and |y| ≤ 3L/4, ¯ 4 (˜ ¯ 3 (˜ y)dL (y), h y)dβ (y0 )) , y = (h

(3.46)

¯ i (y), i = 3, 4, are other suitable functions, uniformly smooth in L and β. where h Hence we can write X 1 1 1,1(≤h)σ (≤h)σ ˜ )dL (y − x) + h2,η,η0 (˜ ˜ )dβ (y0 − x0 ) ψx,ω h1,η,η0 (˜ = y, x y, x Dy,x,ω L β 0 η,η

Z

1

(≤h)σ

˜ )dL (y − x) + h3,η,η0 (˜ y, x

dt∂1 ψξ(t),ω

0

Z +h

4,η,η 0

˜ )dβ (y0 − x0 ) (˜ y, x 0

1

(≤h)σ dt∂0 ψξ(t),ω

,

(3.47)

where 0 ¯ i ((¯ ˜ ) = (−1)η+η χη,η0 (˜ ˜ )h y, x y−x y − x)/L, (¯ y0 − x0 )/β) , hi,η,η0 (˜

i = 1, 4 ,

(3.48)

are smooth functions with support in the region {|y−x−ηL| ≤ 3L/4, |y0 −x0 −η 0 β| ≤ 3β/4}, such that their derivatives of order n are bounded by a constant (depending on n) times γ nhL,β . 1,2(≤h)σ 2(≤h)σ A similar expression is valid for Dy,x,ω . Let us now consider Dy,x,ω , see (3.20). We can write 0 2(≤h)σ (≤h)σ ˜ y2(≤h)σ ˜ )dL (y − x)∂¯12 ψx,ω = (−1)η+η D y−x , Dy,x,ω ¯ ,x,ω + h(˜

(3.49)

where h(y − x) is a uniformly smooth function and (≤h)σ (≤h)σ (≤h)σ ˜ y2(≤h)σ − ψx,ω − (¯ y − x) · ∂ψx,ω D ¯ ,x,ω = ψy ¯ ,ω (≤h)σ − ψx,ω {[cβ (¯ y0 − x0 )cL (¯ y − x) − 1] + bL cβ (¯ y0 − x0 )dL (¯ y − x)} (≤h)σ {[cβ (¯ y0 − x0 ) − 1]dL (¯ y − x) + [dL (¯ y − x) − (¯ y − x)]} − ∂¯1 ψx,ω


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G. Benfatto & V. Mastropietro (≤h)σ (≤h)σ − (¯ y − x)[∂¯1 ψx,ω − ∂1 ψx,ω ] (≤h)σ [dβ (¯ y0 − x0 )cL (y − x) − (¯ y0 − x0 )] . − ∂0 ψx,ω

(3.50)

Note that iσ (≤h)σ (≤h)σ − ∂1 ψx,ω = ∂¯1 ψx,ω Lβ (≤h)σ

behaves dimensionally as ∂13 ψx,ω

X

0

eiσk x (sin k 0 − k 0 )ψˆk0 ,ω

(h)σ

(3.51)

0 k0 ∈DL,β

, hence we shall define

(≤h)σ (≤h)σ (≤h)σ = ∂¯1 ψx,ω − ∂1 ψx,ω . ∂¯13 ψx,ω

(3.52)

It is now easy to show that there exist functions hn,η,η0 (y, x), with n = (n1 , . . . , n6 ), and hi,j,η,η0 (y, x), i, j = 0, 1, smooth uniformly in L and β, such that ( X X 2(≤h)σ ˜ )dL (y − x)n1 dβ (y0 − x0 )n2 hn,η,η0 (˜ y, x Dy,x,ω = η,η 0

n

(≤h)σ × L−n3 β −n4 ∂¯1n5 ∂0n6 ψx,ω +

X

˜) hi,j,η,η0 (˜ y, x

i,j

Z

)

1

× di (y − x)di (y − x)

dt(1 − 0

(≤h)σ t)∂i ∂j ψξ(t),ω

,

(3.53)

the sum over n being constrained by the conditions n1 + n2 ≤ 2 ,

3≥

6 X

ni ≥ 2 .

(3.54)

i=3

3.6. In order to exploit the regularization properties of formulas like (3.47) or (3.53), one has to prove that the “zeros” dL (y−x) and dβ (y0 −x0 ) give a contribution to the 0 bounds of order γ −h , with h0 ≥ h, if h is the scale at which the zero was produced by the action of R. In Sec. 3.7 we shall realize this task by “distributing” the zeros along a path connecting a family of space-time points associated with a subset of field variables. Let x0 = x, x1 , . . . , xn = y be the family of points connected by the path; it is easy to show that dL (y − x) =

n X

dL (xr − xr−1 )e−i L (xr +xr−1 −xn −x0 ) . π

(3.55)

r=1

A similar expression is valid for dβ (y0 − x0 ). It can happen that one of the terms in the r.h.s. of (3.55) or the analogous expansion for dβ (y0 − x0 ) depends on the same space-time points as the integration variables in the r.h.s. of a term like (3.21) or (3.26). We want to study the effect of this event. Let us call W (x − y) the kernel appearing in the l.h.s. of (3.21) or


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1371

(3.26), WR (x − y) its regularization, that is the quantity appearing in braces in the corresponding r.h.s., and let us define Z (≤h)+ (≤h)− ψy,ω WR (x − y) In1 ,n2 = dxdyψx,ω × [e−iπ L dL (y − x)]n1 [e−iπ y

y0 β

dβ (y0 − x0 )]n2 .

(3.56)

In the following we shall meet such expressions for values of n1 and n2 , such that 1 ≤ n1 + n2 ≤ 2. If W (x − y) is the kernel appearing in the l.h.s. of (3.26), it is easy to see that, if n1 + n2 ≥ 1, Z (≤h)+ (≤h)− ψy,ω W (x − y) In1 ,n2 = dxdyψx,ω × [e−iπ L dL (y − x)]n1 [e−iπ y

y0 β

dβ (y0 − x0 )]n2 ,

(3.57)

that is the presence of the zeros simply erases the effect of the regularization. Let us now suppose that W (x − y) is the kernel appearing in the l.h.s. of (3.21) and WR (x − y) its regularization. We have Z (≤h)+ 1,3(≤h)− − cβ (y0 − x0 ) I1,0 = dxdyψx,ω W (x − y)dL (y − x) Dy,x,ω × Z I0,1 =

1 ¯2 −iπ x (≤h)− i cos pF ¯ −iπ x (≤h)− Lψ Lψ ∂1 (e ∂ ) + (e ) , 1 x,ω x,ω 2 v0

(≤h)+ 1,4(≤h)− dxdyψx,ω W (x − y)dβ (y0 − x0 )Dy,x,ω ,

(3.58) (3.59)

where 1,3(≤h)− (≤h)− (≤h)− = e−iπ L ψy,ω − cβ (y0 − x0 )e−iπ L ψx,ω . Dy,x,ω y

1,4(≤h)− =e Dy,x,ω

y −iπ β0

x

(≤h)− ψy,ω − cL (y − x)e−iπ

x0 β

(≤h)− ψx,ω .

(3.60) (3.61)

Moreover Z y (≤h)+ (≤h)− W (x − y)dL (y − x) dL (y − x)e−2iπ L ψy,ω I2,0 = dxdyψx,ω

I0,2

cβ (y0 − x0 ) ¯ −2iπ x (≤h)− Lψ − ) ∂1 (e x,ω aL x 1 ¯2 −2iπ x (≤h)− i cos pF −2iπ L (≤h)− Lψ ψx,ω + ∂1 (e ) , e + x,ω v0 2 Z y0 (≤h)+ (≤h)− = dxdyψx,ω W (x − y)dβ (y0 − x0 )2 e−2iπ β ψy,ω , Z

I1,1 =

y

(≤h)+ dxdyψx,ω W (x − y)dL (y − x)dβ (y0 − x0 )e−iπ L −iπ

(3.62) (3.63) y0 β

(≤h)− ψy,ω .

(3.64)


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Note that no cancellations are possible for x = y modulo (L, β) between the various terms contributing to In1 ,n2 ; hence they will be bounded separately. 1,3(≤h)− 1,4(≤h)− and Dy,x,ω have a zero of first order for Note also that the fields Dy,x,ω x = y modulo (L, β) and can be represented by expressions analogous to the r.h.s. of (3.47). Moreover, the terms contributing to I0,1 and I1,0 and containing these fields can also be written in a form analogous to (3.26). Finally, we want to stress the fact that the integrands in the previous expressions of In1 ,n2 , 1 ≤ n1 + n2 ≤ 2, have a zero of order at most two for x = y modulo (L, β), that is a zero of order not higher of the zero introduced in the r.h.s. of (3.56). As it will be more clear in Sec. 3.7, this property would be lost if one uses the representation (3.19) of the regularization operation, before performing the “decomposition of the zeros”; one should get in this case a zero of order four and the iteration of the procedure of decomposition of the zeros would produce zeros of arbitrary order and, as a consequence, bad combinatorial factors in the bounds. 3.7. We are now ready to describe in more detail our expansion. First of all, we insert the decomposition (3.14) of V (h) (τ, ψ (≤h) ) in the vertices with |Pv | = 4, by following the prescription for the choice of the localization point described in Sec. 3.3. The discussion of Sec. 3.3 allows also to define a new label r(f ), to be called the R-label, for any f ∈ Iv0 , by putting (i) r(f ) = 0, if rv (f ) = 0 for any v such that f ∈ Pv ; (ii) r(f ) = (i, v), if there exists one and only one vertex v, such that f ∈ Pv and rv (f ) = i 6= 0; (iii) r(f ) = (2, v, v¯), if there are two vertices v and v¯, such that v < v¯, f ∈ Pv ⊂ Pv¯ , |Pv | = 2, |Pv¯ | = 4, rv (f ) = 2, rv¯ (f ) = 1; see discussion in the last two paragraphs of Sec. 3.3. Then, we can write V (h) τ,

X p Zh ψ (≤h) = V (h) (τ, P, r) ,

(3.65)

P∈Pτ ,r

where r = {r(f ), f ∈ Iv0 } and the sum over r must be understood as the sum over the possible choices of r compatible with P. We can also write p |Pv0 | Z (h) (3.66) dxv0 Kτ,P,r (xv0 )ψ˜(≤h) (Pv0 ) , V (h) (τ, P, r) = Zh (h)

with Kτ,P,r (xv0 ) defined inductively as in (3.38). Let us consider first the action of R on V (h) (τ, P, r). We can write for RV (h) (τ, P, r) an expression similar to (3.66), if we continue to use for the R operation the representation based on (3.14), (3.21) and (3.26), which affects the kernels leaving the fields unchanged. We shall use the notation Z (h) (h) (3.67) RV (τ, P, r) = dxv0 ψ˜(≤h) (Pv0 )[RKτ,P,r (xv0 )] .


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1373

Moreover, we define r0 so that r0 (f ) = r(f ) except for the field labels f ∈ Pv0 , for which r0 (f ) takes into account also the regularization acting on v0 . However, we can use for the R operation also the representation based on (3.10), (3.11), (3.19) and (3.24), which can be derived from the previous one by integrating the δ-functions; the effect is to replace one of the external fields with one of the fields D1,i(≤h)σ , i = 1, 2 or D2(≤h)σ . We can describe the result by writing Z (h) (h) (3.68) RV (τ, P, r) = dxv0 [Rψ˜(≤h) (Pv0 )]Kτ,P,r (xv0 ) . The discussion in Secs. 3.3 and 3.5 implies that there is a finite set Av0 , such that X Y q (f ) (≤h)σ(f ) n (α) n (α) hα (˜ xPv0 )dL1 dβ 2 [∂ˆjαα(f ) ψ]xα (f ),ω(f ) , (3.69) [Rψ˜(≤h) (Pv0 )] = f ∈Pv0

α∈Av0

˜ Pv0 = (L−1 xPv0 , β −1 x0Pv0 ), dL1 where x

n (α)

n (α)

and dβ 2

are powers of the functions (2.96), with argument the difference of two points belonging to xPv0 , and ∂ˆjq , q = 0, 1, 2, j = 1, . . . , mq , is a family of operators acting on the field variables, which are dimensionally equivalent to derivatives of order q. In particular m0 = 1, ∂ˆ10 is the identity and the action of R is trivial, that is |Av0 | = 1, hα = 1, n1 (α) = n2 (α) = 0 and qα (f ) = 0 for any f ∈ Pv0 , except in the following cases. (1) If |Pv0 | = 4 and r(f ) = 0 for any f ∈ Pv0 , there is f¯ ∈ Pv0 , such that the action of R over the fields consists in replacing one of the field variables with 1,1(≤h)σ field, where y = x(f¯) and x = x(f ) for some other f ∈ Pv0 , see a Dy,x,ω (3.11); moreover, one or two of the other fields change their space-time point. We 1,1(≤h)σ in the representation (3.47); the resulting expression is of the form write Dy,x,ω (3.69), with Av0 consisting of four different terms, such that dL = dL (y − x), dβ = dβ (y0 − x0 ), n1 (α) + n2 (α) = 1 and, for all f 6= f¯, qα (f ) = 0, while qα (f¯) = 1. Moreover, if f 6= f¯, xα (f ) is a single point belonging to xPv0 , not necessarily coinciding with x(f ), while, if f = f¯, xα (f ) is equal to x or to the couple (x, y) (≤h)σ(f¯) (using the previous definitions). The precise values of xα (f¯) and [∂ˆj1α (f¯) ψ]xα (f¯),ω(f) ¯, together with the functions hα , can be deduced from (3.47). (2) If Pv0 = (f1 , f2 ) and ω(f1 ) = ω(f2 ), the action of R consists in replacing one 2(≤h)σ of the external fields, of label, let us say, f1 , with a Dy,x,ω field, where y = x(f1 ) and x = x(f2 ), if f2 is the second field label. By using the representation (3.53) 2(≤h)σ of Dy,x,ω , we get an expression of the form (3.69) consisting of many different terms, such that dL = dL (y − x), dβ = dβ (y0 − x0 ), n1 (α) + n2 (α) ≤ 2, qα (f1 ) = 2, (≤h)σ(f1 ) , together qα (f2 ) = 0, xα (f2 ) = x(f2 ). The values of xα (f1 ) and [∂ˆj2α (f1 ) ψ]xα (f1 ),ω(f 1) with the functions hα , can be deduced from (3.53). (3) If Pv0 = (f1 , f2 ) and ω(f1 ) = −ω(f2 ), the action of R consists in replacing 1,2(≤h)σ field, where one of the external fields, of label, let us say, f1 , with a Dy,x,ω y = x(f1 ) and x = x(f2 ), if f2 is the second field label. By using the analogous 1,2(≤h)σ of the representation (3.47) for Dy,x,ω , we get an expression of the form (3.69)


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consisting of four different terms, such that n1 (α)+ n2 (α) = 1, qα (f1 ) = 1, qα (f2 ) = 0, xα (f2 ) = x(f2 ). √ Let us now consider the action of L on V (h) (τ, Zh ψ (≤h) ). We get an expansion similar to that based on (3.68), that we can write, by using (2.79) and (3.65) and translation invariance, in the form p (≤h) LV (h) τ, Zh ψ (≤h) = γ h nh (τ )Zh Fν(≤h) + sh (τ )Zh Fσ(≤h) + zh (τ )Zh Fζ (≤h)

+ ah (τ )Zh Fα(≤h) + lh (τ )Zh2 Fλ where γ −h nh (τ ) = Lβ

sh (τ ) =

zh (τ ) =

1 Lβ 1 Lβ Z ×

ah (τ ) =

1 Lβ × 1 Lβ

(3.70)

Z

X

(h)

xPv0 )Kτ,P,r (xv0 ) , dxv0 h1 (˜

P∈Pτ ,r Pv0 =(f1 ,f2 ),ω(f1 )=ω(f2 )=+1

X

Z (h)

xPv0 )Kτ,P,r (xv0 ) , dxv0 h2 (˜

P∈Pτ ,r Pv0 =(f1 ,f2 ),ω(f1 )=−ω(f2 )=+1

X P∈Pτ ,r Pv0 =(f1 ,f2 ),ω(f1 )=ω(f2 )=+1 (h)

dxv0 h3 (˜ xPv0 )dβ (x(f2 ) − x(f1 ))Kτ,P,r (xv0 ) ,

Z

lh (τ ) =

,

(3.71)

X P∈Pτ ,r Pv0 =(f1 ,f2 ),ω(f1 )=ω(f2 )=+1 (h)

dxv0 h4 (˜ xPv0 )dL (x(f2 ) − x(f1 ))Kτ,P,r (xv0 ) , Z X (h) xPv0 )Kτ,P,r (xv0 ) , dxv0 h5 (˜ P∈Pτ ,r |Pv0 |=4,σ=(+,−,+,−),ω=(+1,−1,−1,+1)

xPv0 ), i = 1, . . . , 5, being bounded functions, whose expressions can be deduced hi (˜ from (3.8), (3.16) and (3.22), also taking into account the permutations needed to order the field variables as in the r.h.s. of (3.70). The constants nh , sh , zh , ah and lh , which characterize the local part of the effective potential, can be obtained from (3.71) by summing over n ≥ 1 and τ ∈ Th,n . ˜h+1 appearing in the l.h.s. of (3.27) can be written in the form Finally, the constant E ∞ X X ˜h+1 (τ ) , ˜ (3.72) E Eh+1 = n=1 τ ∈Th,n

where ˜h+1 (τ ) = 1 E Lβ

X Z P∈Pτ ,r Pv0 =∅

(h)

dxv0 Kτ,P,r (xv0 ) .

(3.73)


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3.8. We want now to iterate the previous procedure, by using Eq. (3.38), in order to suitably take into account the non-trivial R operations in the vertices v 6= v0 . We shall focus our discussion on RV (h) (τ, P, r), but the following analysis applies ˜h+1 (τ ). also to LV (h) (τ, P, r) and E Let us consider the truncated expectation in the r.h.s. of (3.38) and let us put s = sv , Pi ≡ Pvi \Qvi . Moreover we order in an arbitrary way the sets Pi± ≡ {f ∈ S Pi , σ(f ) = ±}, we call fij± their elements and we define x(i) = f ∈P − x(f ), y(i) = Ps Pi s S − + − + i=1 |Pi | = i=1 |Pi | ≡ n, f ∈P + x(f ), xij = x(fi,j ), yij = x(fi,j ). Note that i

otherwise the truncated expectation vanishes. A couple l ≡ (fij− , fi+0 j 0 ) ≡ (fl− , fl+ ) will be called a line joining the fields with labels fij− , fi+0 j 0 and ω indices ωl− , ωl+ and connecting the points xl ≡ xi,j and yl ≡ yi0 j 0 , the endpoints of l; moreover we shall put ml ≡ m(fl− ) + m(fl+ ). Then, it is well known (see [6, 13], for example) that, up to a sign, if s > 1, E˜hT (ψ˜(h) (P1 ), . . . , ψ˜(h) (Ps )) =

XY T

Z

(h) ∂¯1ml gω− ,ω+ (xl − yl ) l

l∈T

dPT (t) det Gh,T (t) ,

(3.74)

l

where T is a set of lines forming an anchored tree graph between the clusters of points x(i) ∪ y(i) , that is T is a set of lines, which becomes a tree graph if one identifies all the points in the same cluster. Moreover t = {ti,i0 ∈ [0, 1], 1 ≤ i, i0 ≤ s}, dPT (t) is a probability measure with support on a set of t such that ti,i0 = ui · ui0 for some family of vectors ui ∈ Rs of unit norm. Finally Gh,T (t) is a (n − s + 1) × (n − s + 1) −

+

¯m(fij )+m(fi0 j0 ) g (h) (xij − yi0 j 0 ) matrix, whose elements are given by Gh,T ij,i0 j 0 = ti,i0 ∂1 ω − ,ω + l

l

with (fij− , fi+0 j 0 ) not belonging to T . If s = 1, the sum over T is empty, but we shall still use Eq. (3.74), by interpreting the r.h.s. as 1, if P1 is empty (which is possible, for s = 1), and as det Gh (1) otherwise. Inserting (3.74) in the r.h.s. of (3.38) (with v = v0 ) we obtain, up to a sign, Z Z 1 p |Pv0 | X Zh dxv0 dPTv0 (t)[Rψ˜(≤h) (Pv0 )] RV (h) (τ, P, r) = sv0 !  ·

Tv0

Y

l∈Tv0

r ×



(h+1) ∂¯1ml gω− ,ω+ (xl l l

Zh+1 Zh

− yl ) det Gh+1,Tv0 (t)

|Pv0 | sv0

Y

[Kv(h+2) (xvi )] . i

(3.75)

i=1

Let us now consider the contribution to the r.h.s. of (3.75) of one of the terms in the representation (3.69) of Rψ˜(≤h) (Pv0 ) with n1 (α)+ n2 (α) > 0. For each choice of n (α)

Tv0 , we decompose the factors dL1

n (α)

(y − x) and dβ 2

(y0 − x0 ), by using Eq. (3.55)


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and the analogous equation for dβ (y0 − x0 ), with x0 = x, xn = y and the other points xr , r = 1, . . . , n − 1, chosen in the following way. Let us consider the unique subset (l1 , . . . , lm ) of Tv0 , which selects a path joining the cluster containing x0 with the cluster containing xn , if one identifies all the points in the same cluster. Let (¯ vi−1 , v¯i ), i = 1, m, the couple of vertices whose clusters of points are joined by li . We shall put x2i−1 , i = 1, m, equal to the endpoint of li belonging to xv¯i−1 and x2i equal to the endpoint of li belonging to xv¯i . This definition implies that there are two points of the sequence xr , r = 0, . . . , n = 2m + 1, possibly coinciding, in any set xv¯i , i = 0, . . . , m; these two points are the space-time points of two different fields belonging to Pv¯i . Since n ≤ 2sv0 − 1, this decomposition will produce a finite number of different terms (≤ (2sv0 − 1)2 , since n1 (α) + n2 (α) ≤ 2), that we shall distinguish with a label α0 belonging to a set Bv0 , depending on α ∈ Av0 and Tv0 . These terms can be described in the following way. Each term is obtained from the one chosen in the r.h.s. of (3.75) by adding a factor exp{iπL−1 n1 (α)(x + y) + iπβ −1 n2 (α)(x0 + y0 )}. Moreover each propagator b (l) (h+1) gω− ,ω+ (xl − yl ) is multiplied by a factor djα00 (l) (xl , yl ), where dbj , d = 0, 1, 2, j = l

α

l

1, . . . , mb is a family of functions so defined. If b = 0, m0 = 1 and d01 = 1. If b = 1, mb = 2 and j distinguishes, up to the sign, the two functions e−i L (xl +yl ) dL (xl − yl ) , π

e−i β (x0,l +y0,l ) dβ (xl0 − yl0 ) . π

(3.76)

If b = 2, j distinguishes the three possibilities, obtained by taking the product of two factors equal to one of the terms in (3.76). Finally each one of the vertices b (v ) v1 , . . . , vsv0 is multiplied by a similar factor djα00 (vii ) (xi , yi ). α Note that the definitions were chosen so that |dbj (x, y)| ≤ |d(x − y)|b . Moreover there is the constraint that sv0 X X bα0 (l) + bα0 (vi ) = n1 (α) + n2 (α) . (3.77) i=1

l∈Tv0

The previous discussion implies that (3.75) can be written in the form RV (h) (τ, P, r) =

Z Z 1 p |Pv0 | X X X Zh xPv0 ) dxv0 dPTv0 (t) · hα (˜ sv0 ! 0 α∈Av0 Tv0 α ∈Bv0





Y

Y



q (f ) (≤h)σ(f ) (∂ˆjαα(f ) ψ)xα (f ),ω(f )   · f ∈Pv0 l∈Tv0

r · det G

h+1,Tv0

(t)

Zh+1 Zh

|Pv0 |

" sv Y0

b (l) (h+1) djα00 (l) (xl , yl )∂¯1ml gω− ,ω+ (xl α l

− yl )

l

# b (v ) djα00 (vii ) (xi , yi )Kv(h+2) (xvi ) i α

,

i=1

xPv0 ) has be redefined in order to absorb the factor where the function hα (˜ exp{iπL−1 n1 (α)(x + y) + iπβ −1 n2 (α)(x0 + y0 )} .

(3.78)


00105


1377

3.9. We are now ready to begin the iteration of the previous procedure, by considering those among the vertices v1 , . . . , vsv0 , where the action of R is non-trivial. It turns out that we can not simply repeat the arguments used for v0 , but we have to consider some new situations and introduce some new prescriptions, which will be however sufficient to complete the iteration up to the endpoints, without any new problem. Let us select a term in the r.h.s. of (3.78) and one of the vertices immediately following v0 , let us say v¯, where the action of R is non-trivial. We have to consider a few different cases. (A) Suppose that b(¯ v ) = 0 (we shall omit the dependence on α and α0 ). In this case the action of R is exploited following essentially the same procedure as for v0 . If R is different from the identity, we move its action on the external fields of v¯, by using the analogous of (3.69), by taking into account that some of the external fields of v¯ are internal fields of v0 , hence they are involved in the calculation of the truncated expectation (3.74). This means that, if f is the label of an internal field q(f ) with q(f ) > 0, the corresponding (non-trivial) ∂ˆj(f ) operator acts on the quantities b(l)

(h+1)

in the r.h.s. of (3.78), which depend on f , that is dj(l) (xl , yl )gω− ,ω+ (xl − yl ) or l

l

the matrix elements of det Gh+1,Tv0 , which are obtained by contracting the field q(f ) with label f with another internal field. For example, if x(f ) = xl and ∂ˆj(f ) is the operator associated with the third term in the r.h.s. of (3.47), we must substitute b(l) (h+1) dj(l) (xl , yl )∂¯1ml gω− ,ω+ (xl − yl ) with l

Z 0

1

l

b(l)

(h+1)

dt∂1 [dj(l) (ξ(t) − yl )∂¯1ml gω− ,ω+ (ξ(t) − yl )] , l

(3.79)

l

¯ l being defined in terms of xl as y¯ xl − x0 ), for some x0 ∈ xv¯ , x with ξ(t) = x0 + t(¯ ¯ l and xl are equivalent representation is defined in terms of y in Sec. 3.5 (that is x of the same point on the space-time torus). There is apparently another problem, related to the possibilities that the operaq(f ) tors ∂ˆj(f ) related with the action of R on v¯ do not commute with the functions hα and the field variables introduced by the action of R on v0 . However, the discussion in Sec. 3.4 implies that this can not happen, because of our prescription for the choice of the localization points. This argument is of general validity, hence we will not consider anymore this problem in the following. (B) If b(¯ v ) > 0, we shall proceed in a different way, in order to avoid growing powers of the factors dL and dβ , which should produce at the end bad combinatorial factors in the bounds. We need to distinguish four different cases. (B1) If |Pv¯ | = 4, we do not use the decomposition (3.47) for the field changed by the action of R in a D1,1 field, but we simply write it as the sum of the two terms in the r.h.s. of (3.12) (in some cases the second term does not really contributes, b(¯ v) because the argument of the factor dj(¯v ) is the same as the argument of the delta function in the representation (3.14) of the R action, but this is not true in general). We still get a representation of the form (3.69) for [Rψ˜(≤h) (Pv¯ )], but with the


1378

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property that q(f ) = 0 for any α ∈ Av¯ and any f ∈ Pv¯ . This procedure works, because we do not need to exploit the regularization property of R in this case, as the following analysis will make clear. (B2) If |Pv¯ | = 2, and the ω-labels of the external fields are different, the action of R, after the insertion of the zero, is indeed trivial, as explained in Sec. 3.6, see (3.57). Hence we do not make any change in the external fields. v ) = 2, the (B3) If |Pv¯ | = 2, the ω-labels of the external fields are equal and b(¯ b(¯ v) presence of the factor dj(¯v ) does not allow to use for the action of R on the external fields the representation (3.69), because that factor depends on the space-time labels of the external fields. However, we can use the representation following from (3.62)– (3.64), by considering the different terms in the r.h.s. as different contributions (in any case no cancellations among such terms are possible). Note that this representation has the same properties of the representation (3.69) and can be written exactly in the same form, by suitable defining the various quantities. In particular, it is still true that n1 (α) + n2 (α) ≤ 2. Of course, we have to take also into account that some of the external fields of v¯ are internal fields of v0 , but this can be done exactly as in item (A). (B4) Finally, if |Pv¯ | = 2, the ω-labels of the external fields are equal and bv¯ = 1, we use for the action of R on the external fields the representation following from Eqs. (3.58) and (3.59), after writing for the fields D1,3 and D1,4 the analogous of the decomposition (3.47). b(v) The above procedure can be iterated, by decomposing the factors dj(v) coming from the previous steps of the iteration along the spanning tree associated with the clusters Lv , up to the endpoints. The final result can be described in the following way. Let us call a zero each factor equal to one of the two terms in (3.76). Each zero produced by the action of R on the vertex v is distributed along a tree graph Sv on the set xv , obtained by putting together an anchored tree graph Tv¯ for each non-trivial vertex v¯ ≥ v and adding a line for the couple of space-time points belonging to the set xv¯ for each (not local) endpoint v¯ ≥ v with hv¯ = 2 of type λ or u. At the end we have many terms, which are characterized, for what concerns the zeros, by a tree graph T on the set xv0 and not more than two zeros on each line l ∈ T ; the very important fact that there are at most two zeros on each line follows from the considerations in item (B) of Sec. 3.9. 3.10. The final result can be written in the following way: p |Pv0 | X X Z dxv0 Wτ,P,r,T,α (xv0 ) RV (h) (τ, P, r) = Zh

·

  Y 

f ∈Pv0

T ∈T α∈AT

q (f )

(≤h)σ(f )

[∂ˆjαα(f ) ψ]xα (f ),ω(f )

  

,

(3.80)


00105


1379

where Wτ,P,r,T,α (xv0 ) " xv0 ) = hα (˜

# "

Y

|Pv |/2

(Zhv /Zhv −1 )

v not e.p.

(

Y

×

v not

" ×

Y

l∈Tv

1 s ! e.p. v

·

n Y

# b (v ∗ ) djαα (vi∗ ) (xi , yi )Kvh∗i (xvi∗ ) i i

i=1

Z dPTv (tv ) · det Ghαv ,Tv (tv ) #)

− + ˆqα (fl ) ˆqα (fl )

b (l) (h ) ∂j (f − ) ∂j (f + ) [djαα(l) (xl , yl )∂¯1ml gω−v,ω+ (xl α α l l l l

− yl )]

,

(3.81)

T is the set of the tree graphs on xv0 , obtained by putting together an anchored tree graph Tv for each non-trivial vertex v and adding a line (which will be by definition the only element of Tv ) for the couple of space-time points belonging to the set xv for each (not local) endpoint v with hv = 2 of type λ or u; AT is a set of indices which allows to distinguish the different terms produced by the non-trivial R operations and the iterative decomposition of the zeros; v1∗ , . . . , vn∗ are the endpoints of τ , fl− and fl+ are the labels of the two fields forming the line l, “e.p.” is an abbreviation of “endpoint”. Moreover Ghαv ,Tv (tv ) is obtained from the matrix Ghv ,Tv (tv ), associated with the vertex v and Tv , see (3.74), by substituting − m(fij )+m(fi+0 j0 ) (hv ) gω− ,ω+ (xij l l

v ,Tv ¯ Ghij,i 0 j 0 = tv,i,i0 ∂1

− yi0 j 0 )

with − + − + qα (fij ) qα (fij ) m(fij )+m(fi0 j0 ) (hv ) ˆ ¯ gω− ,ω+ (xij − ∂ + ∂1 α (f ) jα (f )

v ,Tv ˆ Ghα,ij,i 0 j 0 = tv,i,i0 ∂ j

ij

ij

l

− yi0 j 0 ) .

(3.82)

l

Finally, ∂ˆjq , q = 0, 1, 2, 3, j = 1, . . . , mq , is a family of operators, implicitly defined in the previous sections, which are dimensionally equivalent to derivatives of order q (f ) q; for each α ∈ AT , there is an operator ∂ˆjαα(f ) associated with each f ∈ Iv0 . It would be very difficult to give a precise description of the various contributions to the sum over AT , but fortunately we only need to know some very general properties, which easily follows from the discussion in the previous sections. (1) There is a constant C such that, ∀ T ∈ Tτ , |AT | ≤ C n and, ∀ α ∈ AT , xv0 )| ≤ C n . |ha (˜ (2) For any α ∈ AT , the following inequality is satisfied #" # " Y Y Y hα (f )qα (f ) −hα (l)bα (l) γ γ γ −z(Pv ) , (3.83) ≤ f ∈Iv0

l∈T

v not e.p.

where hα (f ) = hv0 − 1 if f ∈ Pv0 , otherwise it is the scale of the vertex where the field with label f is contracted; hα (l) = hv , if l ∈ Tv and


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00105


z(Pv ) =

 1,     1,  2,     0,

if |Pv | = 4 , if |Pv | = 2 and if |Pv | = 2 and

P f ∈Pv

ω(f ) 6= 0 ,

f ∈Pv

ω(f ) = 0 ,

P

(3.84)

otherwise .

3.11. In order to prove (3.83), let us suppose first that there is no vertex with two external fields and equal ω indices; hence qα (f ) ≤ 1, ∀ f ∈ Iv0 , and bα (l) ≤ 1, ∀ l ∈ T . Let us choose f ∈ Iv0 , such that qα (f ) = 1; by analyzing the procedure described in Secs. 3.8 and 3.9, one can easily see that there are three vertices v 0 < v ≤ v¯ and a line l ∈ Tv¯ , such that (i) (ii) (iii) (iv)

the field with label f is affected by the action of R on the vertex v; hv0 = hα (f ) and bα (l) = 1; if v ≤ v˜ < v¯ and ˜l ∈ Tv˜ , then bα (˜l) = 0; if v 0 < v˜ ≤ v¯ and f 6= f˜ ∈ Pv˜ , then qα (f˜) = 0.

(ii) follows from the definition of hα (f ) and from the remark that the zero produced by the action of R on v is moved by the process of distribution of the zeros along T in some vertex v¯ ≥ v. The property (iii) characterizes v¯; in fact the procedure described in item (B1) and (B2) of Sec. 3.9 guarantees that no zero can be produced by the action of R in the vertices between v and v¯, if the zero in v¯ “originated” from the regularization in v. (iv) follows from the previous remark and from the fact that the action of R is trivial in all the vertices between v 0 and v, see Sec. 3.3. The previous considerations imply that we can associate each factor γ hα (f ) in the l.h.s. of (3.83) with a factor γ −bα (l) , by forming disjoint pairs; with each pair we can associate two vertices v 0 and v¯ and the path on τ containing all the vertices v 0 < v˜ ≤ v¯. Since each vertex with four external fields or two external fields and different ω indices certainly belongs to one of these paths, the inequality (3.83) then follows from the trivial identity Y γ −1 . (3.85) γ −(bα (l)−hα (f )) = γ −(hv¯ −hv0 ) = v 0 h

Moreover, we suppose that the condition (2.117) is satisfied, so that h∗ ≥ 0. We shall prove the following theorem. 3.12 Theorem. Let h > h∗ ≥ 0, with h∗ defined by (2.116). If the bounds (2.98) are satisfied and, for some constants c1 , Z h0 ≤ ec1 ε2h , sup σh0 ≤ ec1 εh , (3.88) sup h0 >h Zh0 −1 h0 >h σh0 −1 there exists a constant ε¯ (depending on c1 ) such that, if εh ≤ ε¯, then, for a suitable constant c0 , independent of c1 , as well as of u, L and β, X XX X Z X dxv0 |Wτ,P,r,T,α (xv0 )| τ ∈Th,n

P |Pv0 |=2m

r

T ∈T

α∈AT qα (Pv0 )=k

≤ Lβγ −hDk (Pv0 ) (c0 εh )n ,

(3.89)

where Dk (Pv0 ) = −2 + m + k . Moreover

X

[|nh (τ )| + |zh (τ )| + |ah (τ )| + |lh (τ )|] ≤ (c0 εh )n ,

(3.90)

(3.91)

τ ∈Th,n

X

|sh (τ )| ≤ |σh |(c0 εh )n ,

(3.92)

˜h+1 (τ )| ≤ γ 2h (c0 εh )n . |E

(3.93)

τ ∈Th,n

X τ ∈Th,n


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3.13. An important role in the proof of Theorem 3.12 plays the estimation of det Ghαv ,Tv (tv ), that we shall now discuss, by referring to Secs. 3.8 and 3.10 for the notation. From now on C will denote a generic constant independent of u, L and β. Given a vertex v which is not an endpoint and an anchored tree graph Tv (empty, if v is trivial), we consider the set of internal fields which do not belong to the any line of Tv and the corresponding sets P˜ σ,ω of field labels with σ(f ) = σ S S and ω(f ) = ω. The sets ω P˜ −,ω and ω P˜ +,ω label the rows and the columns, respectively, of the matrix Ghαv ,Tv (tv ), hence they contain the same number of elements; however, |P˜ −,ω | can be different from |P˜ +,ω |, if h ≤ 0. We introduce an integer ρ(Tv ), that we put equal to 1, if |P˜ −,ω | 6= |P˜ +,ω |, equal to 0 otherwise. We want to prove that ρ(Tv ) P sv |σhv | hv ,Tv (tv )| ≤ C i=1 |Pvi |−|Pv |−2(sv −1) |det Gα γ hv ·γ

hv 2

(

Psv

· γ −hv

i=1

|Pvi |−|Pv |−2(sv −1)) hv

γ

Psv

i=1 [qα (Pvi \Qvi )+m(Pvi \Qvi )]

P

+ − + − l∈Tv [qα (fl )+qα (fl )+m(fl )+m(fl )]

.

(3.94)

In order to prove this inequality, we shall suppose, for simplicity, that all the S q(f ) m(f ) acting on the fields with field label f ∈ σ,ω P˜ σ,ω are operators ∂ˆj(f ) and ∂ˆ1 equal to the identity. It is very easy to modify the following argument, in order q(f ) m(f ) gives a contribution to the bound to prove that each operator ∂ˆj(f ) or ∂ˆ1 proportional to γ hv q(f ) or γ hv m(f ) , so proving (3.94) in the general case. The proof is based on the well known Gram–Hadamard inequality, stating that, if M is a square matrix with elements Mij of the form Mij = hAi , Bj i, where Ai , Bj are vectors in a Hilbert space with scalar product h·, ·i, then Y kAi k · kBi k . (3.95) |det M | ≤ i

where k · k is the norm induced by the scalar product. Let H = Rs ⊗ H0 , where H0 is the Hilbert space of complex four-dimensional 0 , with vectors F (k0 ) = (F1 (k0 ), . . . , F4 (k0 )), Fi (k0 ) being a function on the set DL,β scalar product hF, Gi =

4 X 1 X ∗ 0 Fi (k )Gi (k0 ) . βL 0 i=1

(3.96)

k

If hv ≤ 0, it is easy to verify that (h )

v v ,Tv (xij − yi0 j 0 ) Ghij,i 0 j 0 = ti,i0 g − ω ,ω + l

l

(h ) (h ) = ui ⊗ Ax(fv − ),ω(f − ) , ui0 ⊗ Bx(fv + ij

ij

),ω(fi+0 j0 ) i0 j 0

,

(3.97)


00105


where ui ∈ Rs , i = 1, . . . , s, are the vectors such that ti,i0 = ui · ui0 , and q f˜h (k0 ) 0 (h) Ax,ω (k0 ) = eik x p −Ah (k0 )  0 0   (−ik0 + E(k ), 0, −iσh−1 (k ), 0) , if ω = +1 , ·   (0, iσ 0 if ω = −1 , h−1 (k ), 0, σh−1 ) , (h)

Bx,ω

q f˜h (k0 ) 0 = eik y p −Ah (k0 ) ( (1, 1, 0, 0) , · (0, 0, 1, (ik0 − E(k 0 ))/σh−1 ) ,

1383

(3.98)

if ω = +1 , if ω = −1 .

Let us now define n+ = |P˜ −,+ |, m+ = |P˜ +,+ |, m = |P˜ −,+ | + |P˜ −,− | = |P˜ +,+ | + |; by using (3.95) and (3.98), it is easy to see, by proceeding as in Sec. 2.7, |P that, if the conditions (2.98) hold, hv m−m+ γ |det Ghαv ,Tv (tv )| ≤ C m γ hv n+ |σhv |m−n+ |σhv | ˜ +,−

m+ −n+ |σhv | =C γ . (3.99) γ hv Pv P Pv |Pvi | − |Pv | − 2(sv − 1) and si=1 qα (Pvi /Qvi ) − l∈Tv [qα (fl+ ) + Since 2m = si=1 qα (fl− )] = 0, we get the inequality (3.94), if m+ ≥ n+ , by using (2.116). The case (h) m+ < n+ can be treated in a similar way, by exchanging the definitions of Ax,ω (k0 ) (h) and Bx,ω (k0 ). If hv = 0, the inequality (3.95) can not be directly applied, because of the k0−1 behaviour of the ultraviolet propagator for k0 → ∞; we would not get bounds uniform in the ultraviolet cutoff M (see (2.7)). However, we can make a further multiscale expansion of g (+1) (x), by an obvious smooth partition of the interval {|k0 | ≥ 1}, and we can modify the trees by putting the endpoints on scale h = M ; see [6] for a similar procedure. It is easy to see that there is no relevant or marginal term on any scale hv > 0, except for those which are obtained by contracting two fields associated with the same space-time point in a vertex located between an endpoint and the first non-trivial vertex following it. However, the sum on the scale of this type of term (which is not absolutely convergent for M → ∞) can be controlled by using the explicit expression of the single scale propagator, since there is indeed no divergence, but only a discontinuity at x0 = 0 for x = 0. Hence, we can reduce again the calculation to the bound (3.95); we shall omit the details, which are of the same type of those used below for the infrared part of the model. m hv m


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3.14 Proof of Theorem 3.12. By using (3.81) and (3.94) we get Z dxv0 |Wτ,P,r,T,α (xv0 )| P sv (Zhv /Zhv −1 )|Pv |/2 · C i=1 |Pvi |−|Pv |−2(sv −1)

Y

≤ C Jτ,P,r,T,α n

v not e.p.

· ·γ

|σhv | γ hv

γ

hv 2

(

P sv

P

i=1

|Pvi |−|Pv |−2(sv −1))


−hv

where

ρ(Tv )

P sv

i=1 [qα (Pvi \Qvi )+m(Pvi \Qvi )]

· γ hv

,

(3.100)

" # ( n Y Y 1 bα (vi∗ ) hi dxv0 djα (v∗ ) (xi , yi )Kv∗ (xvi∗ ) · i i s ! v not e.p. v

Z

Jτ,P,r,T,α =

i=1

" ×

Y

l∈Tv

#) qα (fl− ) ˆqα (fl+ ) bα (l) ml (hv ) ˆ ˆ ∂j (f − ) ∂j (f + ) [djα (l) (xl , yl )∂1 gω− ,ω+ (xl − yl )] . α l α l l l

In Sec. 3.15 we will prove that " Y 1 2(sv −1) hv nν (v) C γ Jτ,P,r,T,α ≤ C n Lβ(εh )n s v! v not e.p. ·γ

−hv (sv −1) hv

P

γ

(3.101)

! Y σh P v · γ −hv l∈Tv bα (l) γ hv ¯

l∈Tv


# ,

(3.102)

where nν (v) is the number of vertices of type ν with scale hv +1 and T¯v is the subset of the lines of Tv corresponding to non-diagonal propagators, that is propagators with different ω indices. It is easy to see that X v not e.p.

hv

sv X

qα (Pvi \Qvi ) + hqα (Pv0 ) =

X

hα (f )qα (f )

(3.103)

f ∈Iv0

i=1

and, by using also the remark after (3.86), that ! ) ( sv¯ sv¯ X X 1 X |Pv¯i | − |Pv¯ | − 2(sv¯−1 ) + nν (¯ v) + m(Pv¯i \Qv¯i ) 2 i=1 i=1 v ¯≥v

=

X 1 (|Iv | − |Pv |) + m(Iv \Pv ) + nν (¯ v ) − 2(nv − 1) 2 v ¯≥v

1 = − |Pv | + 2 . 2

(3.104)


00105


1385

By inserting (3.102) in (3.100) and using (3.83), (3.103) and (3.104), we find Z Y Y |σh | v · dxv0 |Wτ,P,r,T,α (xv0 )| ≤ C n Lβεnh γ −hDk (Pv0 ) h v γ v not e.p. ·

v∈V2

|Pv | 1 Psi=1 v |P |−|P | vi v C (Zhv /Zhv −1 )|Pv |/2 γ −[−2+ 2 +z(Pv )] , sv !

(3.105)

where V2 is the set of vertices, which are not endpoints, such that ρ(Tv ) + |T˜v | > 0, while the vertices v¯ > v do not enjoy this property. Let us now consider a vertex v, which is not an endpoint, such that |Pv | = 2 P and f ∈Pv ω(f ) 6= 0. We want to show that there is a vertex v¯ ≥ v, such that v¯ ∈ V2 . In order to prove this claim, we note that, if v ∗ is an endpoint, then P P f ∈Pv∗ σ(f )ω(f ) = 0, while f ∈Pv σ(f )ω(f ) 6= 0. Since all diagonal propagators join two fields with equal ω indices and opposite σ indices, given any Feynman graph connecting the endpoints of the cluster Lv , at least one of its lines has to be a non-diagonal propagator, so that at least one of the vertices v¯ ≥ v must belong to V2 . Moreover, if v ∈ V2 , |σh | |σhv | h−hv |σh | |σhv | γ = h ≤ h γ (h−hv )(1−c1 εh ) ≤ Cγ (h−hv )(1/2) , h v γ γ |σh | γ

(3.106)

if εh ≤ ε¯ and ε¯ ≤ 1/(2c1 ). We have used the second inequality in (3.88) and the a0 h γ , if h ≥ h∗ . definition (2.116), implying that |σh | ≤ 4γ It follows that Y Y |σh | 1 v n ≤ C γ − 2 z˜(Pv ) , (3.107) γ hv v not e.p. v∈V2

where

( z˜(Pv ) =

1,

if |Pv | = 2 and

0,

otherwise ,

P f ∈Pv

ω(f ) 6= 0 ,

(3.108)

so that 1 z˜(Pv ) |Pv | + z(Pv ) + ≥ , ∀ v not e.p. . 2 2 2 Hence (3.105) can be changed in Z Y dxv0 |Wτ,P,r,T,α (xv0 )| ≤ C n Lβεnh γ −hDk (Pv0 ) · −2 +

(3.109)

v not e.p.

·

|Pv | z ˜(Pv ) 1 Psi=1 v |P |−|P | vi v C (Zhv /Zhv −1 )|Pv |/2 γ −[−2+ 2 +z(Pv )+ 2 ] . sv !

(3.110)

In order to complete the proof of the bound (3.89), we have to perform the sums in the r.h.s. of (3.89). The number of unlabeled trees is ≤ 4n ; fixed an unlabeled tree, the number of terms in the sum over the various labels of the tree is bounded


1386

00105


by C n , except the sums over the scale labels and the sets P. The number of addenda in the sums over α and r is again bounded by C n , since the action of R can be nontrivial at most two times between two consecutive non-trivial vertices (see Sec. 3.3) and the number of non-trivial vertices is of order n. Regarding the sum over T , it is empty if sv = 1. If sv > 1 and Nvi ≡ |Pvi |−|Qvi |, the number of anchored trees with di lines branching from the vertex vi can be bounded, by using Caley’s formula, by (sv − 2)! d Nvd11 · · · Nvssvv ; (d1 − 1)! · · · (dsv − 1)! P sv Q P hence the number of addenda in T ∈T is bounded by v not e.p. sv !C i=1 |Pvi |−|Pv | . In order to bound the sums over the scale labels and P we first use the inequality, following from (3.109) and the first inequality in (3.88), if c1 ε2h ≤ 1/16, Y |Pv | z ˜(Pv ) (Zhv /Zhv −1 )|Pv |/2 γ −[−2+ 2 +z(Pv )+ 2 ] v not e.p.

≤

" Y

#" γ

1 − 40 (hv˜ −hv˜0 )

#

Y

γ

− |P40v |

,

(3.111)

v not e.p.

v ˜

where v˜ are the non-trivial vertices, and v˜0 is the non-trivial vertex immediately 1 preceding v˜ or the root. The factors γ − 40 (hv˜ −hv˜0 ) in the r.h.s. of (3.111) allow to bound the sums over the scale labels by C n . Finally the sum over P can be bound by using the following combinatorial inequality, trivial for γ large enough, but valid for any γ > 1 (see [6, Sec. 3]). Let Psv pvi for all v ∈ τ which are not {pv , v ∈ τ } a set of integers such that pv ≤ i=1 endpoints; then Y X pv γ − 40 ≤ C n . (3.112) v not e.p. pv

It follows that

X

Y

v not e.p. P |Pv0 |=2m

γ−

|Pv | 40

≤

Y

X

pv

γ − 40 ≤ C n .

(3.113)

v not e.p. pv

The proof of the bounds (3.91) and (3.93) is very similar. For the terms contributing to nh one gets a bound like (3.89), with m = 1 and k = 0, but the factor γ −hDk (Pv0 ) = γ h is compensated by the factor γ −h appearing in the definition of nh (τ ), see (3.71). For the terms contributing to zh and ah Dk (Pv0 ) = 0 (m = k = 1), as well as for those contributing to lh (m = 2, k = 0). Finally, for the terms con˜h+1 , Dk (Pv0 ) = 2. For the terms contributing to sh , Dk (Pv0 ) = −1, tributing to E but each term has also at least one small factor |σh |γ −h in its bound, since |V2 | ≥ 1, see (3.106); so we get the bound (3.92). 3.15 Proof of (3.102). We shall refer to the definitions and the discussion in Secs. 3.7 and 3.9. Let us consider the factor in the r.h.s. of (3.101) associated with


00105


1387

0

the line l ∈ Tv and let us suppose that xl ∈ x(i) , yl ∈ x(i ) . By using (3.47) and (3.53) and the similar expressions for the other difference fields produced by the regularization, we can write −

+

q (fl ) ˆqα (fl ) bα (l) (h ) [d (xl , yl )∂¯1ml gω−v,ω+ (xl − yl )] ∂ ∂ˆj α(f − ) j (f + ) jα (l) α

α

l

Z

l

l

Z

1

dtl

= 0

0

1

−

l

+

q (fl ) ˜qα (fl ) dsl ∂˜j α(f − ∂ ) j (f + ) α

l

α

l

b (l) (h ) × [djαα(l) (x0l (tl ), yl0 (sl ))∂¯1ml gω−v,ω+ (x0l (tl ) − yl0 (sl ))] , l

(3.114)

l

where, depending on α, there are essentially two different possibilities for the operators ∂˜jqαα and the space-time points x0l (tl ), yl0 (sl ). Let us consider, for example, fl− ; then the first possibility is that ∂˜jqαα is a derivative of order qα and ˜ l + tl (¯ ˜l ) , xl − x x0l (tl ) = x

˜ l ∈ x(i) , for some x

(3.115)

¯ l and ¯ l being defined in terms of xl as y¯ is defined in terms of y in Sec. 3.5 (that is x x xl are equivalent representation of the same point on the space-time torus). The second possibility is that ∂˜jqαα is a local operator of the form L−n1 β −n2 ∂¯1n3 ∂0n4 , with P4 ˜ l ∈ x(i) . Note that, by (2.40), L−n1 β −n2 ≤ qα ≤ i=1 ni ≤ qα + 1, and x0l (tl ) = x γ hL,β (n1 +n2 ) ≤ γ hv (n1 +n2 ) . By proceeding as in the proof of Lemma 2.6 and using (2.105) it is very easy to show that, for any N > 1, ˜qα (fl− ) ˜qα (fl+ ) bα (l) 0 m (h ) ∂j (f − ) ∂j (f + ) [djα (l) (xl (tl ), yl0 (sl ))∂¯1 l gω−v,ω+ (x0l (tl ) − yl0 (sl ))] α

α

l

≤C

γ

l

l

l

hv [1+qα (fl+ )+qα (fl− )+m(fl− )+m(fl+ )−bα (l)]

1 + [γ hv |d(x0l (tl ) − yl0 (sl ))|]N

|σhv | γ hv

ρl ,

(3.116)

where d(x) is defined in (2.97) and ρl = 1 if ω(fl− ) 6= ω(fl+ ), ρl = 0 otherwise. We used here the fact that, if hv = +1, then qα (fl− ) = qα (fl+ ) = 0, which allows to avoid the problems connected with the singularity of the time derivatives of the 0 (sl ) = 0. scale 1 propagator at x0l,0 (tl ) − yl,0 Let us now consider the contribution of the endpoints to the r.h.s. of (3.101) and recall (see Sec. 3.10) that Tvi∗ is empty, if |xvi∗ | = 1, hence bα (vi∗ ) = 0, while, if xvi∗ = (xi , yi ), Tvi∗ contains the line li connecting xi with yi and hvi∗ = 2. By using (3.33) and (3.39), we get, if hi ≡ hvi∗ and Sν ≡ {i : vi∗ is of type ν}, " # n Y bα (vi∗ ) djα (v∗ ) (xi , yi )Kvh∗i (xvi∗ ) i i i=1

≤ C n εnh

Y i:|xv∗ |=2 i

Y 1 γ (hi −1) . N [1 + |d(xi − yi )|] i∈Sν

(3.117)


1388

00105


Let us now remark that, after the insertion of the bounds (3.116) and (3.117) in R the r.h.s. of (3.101), by possibly changing the constant C, we can substitute dx v0 , R Q P , with the real integral over which is there a shorthand for x∈xv dx 0 x∈Λ 0

(TL,β )|xv0 | , where TL,β is the space-time torus [−L/2, L/2]×[−β/2, β/2]. Moreover, Eq. (3.115) can be thought, and we shall do that, as defining an interval on TL,β , when tl spans the interval [0, 1]; this is possible thanks to the introduction of the partition (3.42) in Sec. 3.5. Hence, in order to complete the proof of (3.102), we have to show that, fixed ¯ ∈ xv0 , the interpolation parameters associated with the regularization a point x operations and an integer N ≥ 3, Z Y Y Y 1 d(xv0 \¯ x) ≤ Cγ −hv (sv −1) , (3.118) 0 0 h N v 1 + [γ |d(x (t ) − y (s ))|] l l Ξ l l v∈τ v∈τ l∈Tv

where Ξ denotes the subset of (TL,β )|xv0 \¯x| satisfying all the constraints associated with the interpolated points of the form (3.115). S Let us call T˜ = v T˜v , where T˜v is the set of lines connecting x0l (tl ) with yl0 (sl ), for any l ∈ Tv . T˜ is not a tree in general; however, for any v, T˜v is still an anchored tree graph between the clusters of points x(i) , i = 1, . . . , sv . Hence, the proof of (3.118) becomes trivial, if we can show that Y x) = drl , (3.119) d(xv0 \¯ l∈T˜

where rl = x0l (tl ) − yl0 (sl ). In order to prove (3.119), we can proceed, for example, as in [7]. Let us consider first a vertex v with |Tv | > 0, which is maximal with respect to the tree order; hence either v is a non-local endpoint with hv = 2 or it is a non-trivial vertex with no vertex v 0 with |Tv0 | > 0 following it. In this case T˜v = Tv , that is no line depends on the interpolation parameters, and T˜v is a tree on the set xv , so that we get immediately the identity Y x(v) drl , (3.120) dxv = d¯ l∈T˜v

¯ (v) is an arbitrary point of xv . If we use (3.120) for the family S0 of all where x maximal vertices with |Tv | > 0, we get   Y Y d¯ drl  . x(v) (3.121) dxv0 = v∈S0

l∈T˜v

Let us now consider a line ¯l ∈ T˜, which connects two clusters of points xv1 and xv2 , with vi ∈ S0 , i = 1, 2. By (3.115) x¯l − y¯l0 (s¯l ) , r¯l = x¯0l (t¯l ) − yl0 (s¯l ) = t¯l x¯l + (1 − t¯l )¯

(3.122)


00105


1389

implying that ¯ (v1 ) − r¯l ¯ (v1 ) = r¯l + x x ¯ ¯l ) + y¯l0 (s¯l ) . = r¯l + t¯l (¯ x(v1 ) − x¯l ) + (1 − t¯l )(¯ x(v1 ) − x

(3.123)

¯ ¯l ) both x(v1 ) − x¯l ) and (¯ x(v1 ) − x Since y¯l0 (s¯l ) depends only on the variables xv2 and (¯ depend only on {rl , l ∈ T˜v1 }, we get   2 Y 2 Y Y Y d¯ drl  = dr¯l d¯ x(v2 ) drl . (3.124) x(vi ) i=1

i=1 l∈T˜v

l∈T˜vi

i

By iterating this procedure, one gets (3.119). 3.16. As we have discussed in Sec. 2.13, it is not necessary to perform the scale decomposition of the Grassmanian integration up to the last scale hL,β , but we can ˜h∗ , so that stop it to the scale h∗ , defined in (2.116). Hence, we redefine E Z √ (≤h∗ ) ∗ ˆ (h∗ ) ˜ ) , (3.125) e−Lβ Eh∗ = PZh∗ −1 ,σh∗ −1 ,Ch∗ (dψ (≤h ) )e−V ( Zh∗ −1 ψ implying that EL,β =

1 X

˜ h + th ] . [E

(3.126)

h=h∗

Thanks to Lemma 2.12, we can proceed as in the proof of Theorem 3.12 to prove the following theorem. 3.17 Theorem. There exists a constant ε¯ such that, if εh∗ ≤ ε¯ and, for h = h∗ , (2.98) holds and the bounds (3.88) are satisfied, then X ˜h∗ (τ )| ≤ γ 2h∗ (Cεh∗ )n . |E (3.127) τ ∈Th∗ −1,n

3.18. Theorems 3.12 and 3.17, together with (3.126) and (2.118), imply that the expansion defining EL,β is convergent, uniformly in L, β. With some more work (essentially trivial, but cumbersome to describe) one can also prove that limL,β→∞ EL,β does exist. 4. The Flow of the Running Coupling Constants 4.1. The convergence of the expansion for the effective potential is proved by Theorems 3.12 and 3.17 under the hypothesis that, uniformly in h ≥ h∗ , the running coupling constants are small enough and the bounds (2.98) and (3.88) are satisfied. In this section we prove that, if |λ| is small enough and ν is properly chosen, the above conditions are indeed verified.


1390

00105


Let us consider first the bounds in (2.98). They immediately follow from (3.91) and (3.92), by a simple inductive argument, if the bounds (3.88) are verified and εh ≤ ε¯0 ≤ ε¯ ,

for h > h∗ ,

(4.1)

with ε¯0 small enough. Let us now consider the bounds (3.88). By (2.83) and (2.84), the first of (2.89) and the third of (2.98), we get Zh−1 = 1 + zh , Zh

(4.2)

sh /σh − zh σh−1 = 1+ . σh 1 + zh

(4.3)

By explicit calculation of the lower order non-zero terms contributing to zh and sh /σh , one can prove that zh = b1 λ2h + O(ε3h ) , sh /σh = −b2 λh +

b1 > 0 ,

O(ε2h ) ,

(4.4)

b2 > 0 ,

which imply (3.88), if ε¯0 is small enough, with a suitable constant c1 depending on the constant c0 appearing in Theorem 3.12, since the value of c0 is independent of c1 . Equation (4.2) and the definitions (2.109) allow to get the following representation of the Beta function in terms of the tree expansions (3.71):   2 ∞ X X 1 −λh+1 (zh2 + 2zh ) + lh (τ ) , (4.5) λh = λh+1 + 1 + zh n=2 τ ∈Th,n

 δh = δh+1 +

∞ X X

1  −δh+1 zh + cδ0 λ1 δh,0 + 1 + zh n=2 

νh = γνh+1 +

 (ah (τ ) − zh (τ )) ,

(4.6)

τ ∈Th,n ∞ X X

1  −γνh+1 zh + cνh γ h λh+1 + 1 + zh n=2

 nh (τ ) ,

(4.7)

τ ∈Th,n

where we have extracted the terms of first order in the running couplings and we have extended to h = +1 the definition of λh and δh , so that, see (2.81), λ1 = 4λ sin2 (pF + π/L) ,

δ1 = −v0 δ ∗ .

(4.8)

Note that the first order term proportional to λh+1 in the equation for νh is of size γ h , while the similar term in the equation for δh is equal to zero, if h < 0; moreover the constants cν0 and cλh are bounded uniformly in L, β. Hence, if we put ~ah = (δh , λh ), the Beta function can be written, if condition (4.1) is satisfied, with ε¯0 small enough, in the form


00105


βhλ ,

βhδ

1391

λh−1 = λh + βhλ (~ah , νh ; . . . ; ~a1 , ν1 ; u, δ ∗ ) ,

(4.9)

δh−1 = δh + βhδ (~ah , νh ; . . . ; ~a1 , ν1 ; u, δ ∗ ) ,

(4.10)

νh−1 = γνh + βhν (~ah , νh ; . . . ; ~a1 , ν1 ; u, δ ∗ ) ,

(4.11)

βhν

and are functions of ~ah , νh , . . . , ~a1 , ν1 , u, which can be easily where bounded, by using Theorem (3.12), if the condition (4.1) is verified. Note that these functions depend on ~ah , νh , . . . , ~a1 , ν1 , u, directly trough the endpoints of the trees, indirectly trough zh and the quantities Zh0 /Zh0 −1 and σh0 −1 (k0 ), h < h0 ≤ 0, appearing in the tree expansions. Let us define ¯ h = sup |λk | . (4.12) µh = sup max{|λk |, |δk |} , λ k≥h

k≥h

We want to prove the following lemma. 4.2 Lemma. Suppose that u satisfies the condition (2.117) and let us consider ˜ ≤ h ≤ 1, satisfying the Eq. (4.11) for fixed values of ~ah , Zh−1 and σh−1 (k0 ), h conditions µh ≤ ε¯1 ≤ ε¯0 , a0 γ h−1 ≥ 4|σh | , γ −c0 µh ≤

σh−1 ≤ γ +c0 µh , σh

(4.13) (4.14) (4.15)

2 Zh−1 ≤ γ +c0 µh , (4.16) Zh for some constant c0 . Then, if ε¯0 is small enough, there exist some constants ε¯1 , η, ¯ ˜ ≤ ¯h ≤ 0, such that ε¯1 ≤ ε¯0 , 0 < η < 1, γ 0 , c1 , B, and a family of intervals I (h) , h ¯ ¯ ¯ ¯ ¯ 0 (h) (h+1) (h) 0 h , |I | ≤ c1 ε¯1 (γ ) and, if ν = ν1 ∈ I (h) , 1 < γ < γ, I ⊂ I 1 ¯ ¯ ≤ h ≤ 1. (4.17) |νh | ≤ B ε¯1 [γ − 2 (h−h) + γ ηh ] ≤ ε¯0 , h

γ −c0 µh ≤ 2

4.3 Proof. Let us consider (4.11), for fixed values of ~ah , Zh /Zh−1 (hence of zh ) ˜ ≤ h ≤ 1, satisfying (4.13)–(4.16). and σh−1 (k0 ), h ¯ ≤ h ≤ 1 and ε¯0 is small enough, the r.h.s. of (4.11) Note that, if |νh | ≤ ε¯0 for h ¯ is well defined for h = h and we can write, by using (4.7), = γνh¯ + bh¯ + rh¯ , νh−1 ¯

(4.18)

¯ cνh−1 γ h−1 λh¯ ¯

and rh¯ collects all terms of second or higher order in ε¯0 . where bh¯ = Note also that, in the tree expansion of nh (τ ), the dependence on νh , . . . , ν1 appears only in the endpoints of the trees and there is no contribution from the trees with n ≥ 2 endpoints, which are only of type ν or δ, because of the support properties of the single scale propagators. It follows, by using (3.91) and (4.14)–(4.16), that |rh¯ | ≤ c2 µh¯ ε¯0 .

(4.19)


1392

00105


Let us now fix a positive constant c, consider the intervals bh bh (h) − c¯ ε1 , − + c¯ ε1 = − J γ−1 γ −1 ¯

(4.20) ¯

and suppose that there is an interval I (h) such that, if ν1 spans I (h) , then νh¯ spans ¯ ¯ ¯ ≤ h ≤ 1. Let us call J˜(h) the interval spanned the interval J (h+1) and |νh | ≤ ε¯0 for h ¯ when ν1 spans I (h) . Equation (4.18) can be written in the form by νh−1 ¯ bh¯ bh¯ = γ ν + + (4.21) + rh¯ . νh−1 ¯ ¯ h γ−1 γ−1 Hence, by using also the definition of bh and (4.19), we see that bh+1 bh¯ ¯ + = γ min min¯ νh−1 νh¯ + ¯ ¯ (h+1) γ−1 γ−1 ν1 ∈I (h) νh ¯ ∈J γ (bh¯ − bh+1 ) + min¯ rh¯ + ¯ γ−1 ν1 ∈I (h) ¯

≤ −γc¯ ε1 + c2 ε¯1 ε¯0 + c3 γ h ε¯1 , for some constant c3 . In a similar way we can show that bh¯ ¯ + ≥ γc¯ ε1 − c2 ε¯1 ε¯0 − c3 γ h ε¯1 . max¯ νh−1 ¯ ( h) γ − 1 ν1 ∈I

(4.22)

(4.23)

¯

It follows that, if c is large enough and ε¯0 is small enough, J (h) is strictly contained ¯ in J˜(h) . On the other hand, it is obvious that there is a one to one correspondence ¯ ¯ − 1 ≤ h ≤ 1. Hence there is an interval I (h−1) ⊂ between ν1 and the sequence νh , h ¯ ¯ ¯ spans the interval J (h) and, if ε¯1 is I (h) , such that, if ν1 spans I (h−1) , then νh−1 ¯ ¯ − 1 ≤ h ≤ 1. small enough, |νh | ≤ ε¯0 for h The previous calculations also imply that the inductive hypothesis is verified ¯ = 0, so that we have proved that there exists a decreasing family of intervals for h ¯ ˜ ¯ (h) h ≤ 0, such that, if ν = ν1 ∈ I (h) , then the sequence νh is well defined for I ,h≤¯ ¯ and satisfies the bound |νh | ≤ ε¯0 . h≥h ¯ The bound on the size of I (h) easily follows (4.18) and (4.19). Let us denote by ¯ 0 ¯ νh and νh , h ≤ h ≤ 1, the sequences corresponding to ν1 , ν10 ∈ I (h) . We have 0 = γ(νh − νh0 ) + rh − rh0 , νh−1 − νh−1

(4.24)

where rh0 is a shorthand for the value taken from rh in correspondence of the sequence νh0 . Let us now observe that rh − rh0 is equal to γzh−1 (1 + zh−1 )−1 (νh0 − νh ) plus a sum of terms, associated with trees, containing at least one endpoint of type ν, with a difference νk − νk0 , k ≥ h, in place of the corresponding running coupling, and one endpoint of type λ. Then, if |νk − νk0 | ≤ |νh − νh0 |, k ≥ h, we have |νh − νh0 | ≤

0 | |νh−1 − νh−1 + C ε¯1 |νh − νh0 | . γ

(4.25)


00105


1393

On the other hand, if h = 1, this bound implies that |ν1 − ν10 | ≤ |ν0 − ν00 |, if ε¯1 is small enough; hence it allows to show inductively that, given any γ 0 , such that 1 < γ 0 < γ, if ε¯1 is small enough, then ¯

|ν1 − ν10 | ≤ γ 0(h−1) |νh¯ − νh¯0 | .

(4.26)

¯

¯

ε1 , Since, by definition, if ν1 spans I (h) , then νh¯ spans the interval J (h+1) , of size 2c¯ ¯ ¯ ε1 γ 0(h−1) . the size of I (h) is bounded by 2c¯ In order to complete the proof of Lemma 4.2, we have still to prove the bound ¯ ¯ ≤ h ≤ 0 and ν1 ∈ I (h) , (4.17). Note that, if we iterate (4.11), we can write, if h # " 1 X γ k−2 βkν (νk , . . . , ν1 ) , (4.27) νh = γ −h+1 ν1 + k=h+1

where now the functions βνk are thought as functions of νk , . . . , ν1 only. ¯ in (4.27), we get the following identity: If we put h = h ν1 = −

1 X

¯

γ k−2 βkν (νk , . . . , ν1 ) + γ h−1 νh¯ .

(4.28)

¯ k=h+1

(4.27) and (4.28) are equivalent to νh = −γ −h

h X

¯

γ k−1 βkν (νk , . . . , ν1 ) + γ −(h−h) νh¯ ,

¯ < h ≤ 1. h

(4.29)

¯ k=h+1

The discussion following (4.18) implies that |βkν (νk , . . . , ν1 )| ≤ Cµk ,

(4.30)

if ε¯0 is small enough. However this bound it is not sufficient and we have to analyze in more detail the structure of the functions βhν , by looking in particular to the trees in the expansion of nh (τ ), which have no endpoint of type ν. Let us suppose that, given a tree with this property, we decompose the propagators by using (2.99); we get a family of C n different contributions, which can be bounded as before, by using an argument similar0 to that used in Sec. 3.13. However, the terms containing only (h ) the propagators gL,ω cancel out, for simple parity properties. On the other hand, (h )

(h )

v (the the terms containing at least one propagator r2 v or two propagators gω,−ω n number of such propagators has to be even) can be bounded by (Cεh ) (|σh |/γ h )2 , by using (2.101) and (3.106). Analogously the terms with at least one propagator (h ) r1 v can be bounded by (Cεh )n γ ηh , with some positive η < 1. In fact, for these terms, by using (2.101), the bound can be improved by a factor γ hv ≤ γ ηh γ η(hv −h) , for any positive η ≤ 1, and the bad factor γ η(hv −h) can be controlled by the sum over the scales, if η is small enough, thanks to (3.111). Finally, the parity properties of the propagators imply that the only term linear in the running couplings, which contributes to νh , is of order γ h . Hence, we can write


1394

00105


βhν

= µh

1 X

ν νk β˜h,k γ −2η(k−h) + µh εh

k=h

|σh | γh

2 βˆhν + γ ηh µh Rhν ,

(4.31)

ν | ≤ C. where |Rhν |, |βˆhν |, |β˜h,k −2η(k−h) in the r.h.s. of (4.31) follows from the simple remark that The factor γ the bound over all the trees contributing to νh , which have at least one endpoint of 0 fixed scale k > h, can be improved by a factor γ −η (k−h) , with η 0 positive but small enough. It is sufficient to use again (3.111), which allows to extract such factor from the r.h.s. before performing the sum over the scale indices, and to choose η 0 = 2η, which is possible if η is small enough. ¯ < h ≤ 1, satisfying (4.29) can Let us now observe that the sequence νh , h (n) ¯ < h ≤ 1, n ≥ 0, be obtained as the limit as n → ∞ of the sequence {νh }, h ¯ (h+1) and defined recursively in the following way: parameterized by νh¯ ∈ J (0)

νh = 0 , (n) νh

= −γ

−h

h X

(n−1)

γ k−1 βkν (νk

(n−1)

, . . . , ν1

¯

) + γ −(h−h) νh¯ ,

n ≥ 1.

(4.32)

¯ k=h+1

In fact, it is easy to show inductively, by using (4.30), that, if ε¯1 is small enough, (n) |νh | ≤ C ε¯1 ≤ ε¯0 , so that (4.32) is meaningful, and (n)

(n−1)

max |νh − νh

h∗ 1 it (n−1) (n−1) (n−2) (n−2) , . . . , ν1 ) − βkν (νk , . . . , ν1 ) can follows trivially by the fact that βkν (νk be written as a sum of terms in which there are at least one endpoint of type ν, with − νhn−2 , h0 ≥ k, in place of the corresponding running coupling, a difference νhn−1 0 0 (n) and one endpoint of type λ. Then νh converges as n → ∞, for ¯h < h ≤ 1, to a limit νh , satisfying (4.29) and the bound |νh | ≤ ε¯0 , if ε¯1 is small enough. Since the solution of Eq. (4.29) is unique, it must coincide with the previous one. Conditions (4.14) and (4.15) imply that |σ¯ | |σh | h−h |σh | ¯ ¯ γ = hh¯ ≤ Cγ −(h−h)(1−c0 ε¯1 ) . γh γ |σh¯ | Hence, if ε¯1 is small enough, by (4.31), " 1 # X 1 ¯ |νm |γ −2η(m−k) + ε¯0 γ − 2 (h−h) + γ ηk . |βkν | ≤ C ε¯1

(4.34)

(4.35)

m=k

Hence, it is easy to show that there exists a constant c¯ such that " h 1 X X (n) (n−1) −(h−m) (n−1) −2η(m−h) |νm |γ + |νm |γ |νh | ≤ c¯ε¯1 ¯ m=h+1

m=h+1

# + ε¯0 γ

¯ − 12 (h−h)

+γ

ηh

+γ

¯ −(h−h)

.

(4.36)


00105


1395

Let us now suppose that, for some constant cn−1 , ¯

(n−1) | ≤ cn−1 ε¯1 [γ − 2 (h−h) + γ ηh ] ≤ ε¯0 , |νm 1

(4.37)

(0) νm

= 0, if ε¯1 is small enough. Then, it is easy to which is true for n = 1, since (n) verify that the same bound is verified by νm , if cn−1 is substituted with cn = c¯(1 + c4 cn−1 ε¯1 ) ,

(4.38)

where c4 is a suitable constant. Hence, we can easily prove the bound (4.17) for (n) νh = limn→∞ νh , for ε¯1 small enough. 4.4. Let us now consider Eq. (4.9) and (4.10), for a fixed, arbitrary, sequence νh , ¯ ≤ h ≤ 1, satisfying the bound (4.17). In order to study the corresponding flow, h we compare our model with an approximate model, obtained by putting u = ν = 0 (k) and by substituting all the propagators with the Luttinger propagator gL,ω (x; y), 0 see (2.100). It is easy to see that, in this model, σh (k ) = νh = 0, for any h ≤ 1, so that the flow of the running couplings is described only by the equations λh−1 = λh + βhλ,L (~ah , . . . , ~a1 ; δ ∗ ) , (L)

(L)

(L)

(L)

δh−1 = δh

(L)

(L)

+ βhδ,L (~ah , . . . , ~a1 ; δ ∗ ) , (L)

(L)

(4.39)

where the functions βhλ,L and βhδ,L can be represented as in (4.5) and (4.6), by suitably changing the definition of the trees and of the related quantities lh (τ ), ah (τ ), zh (τ ), which we shall distinguish by a superscript L. Of course Theorem 3.12 applies also to the new model, which differs from the well known Luttinger model only because the space variables are restricted to the unit lattice, instead of the real axis. Let us define, for α = λ, δ, rhα (~ah , νh ; . . . ; ~a1 , ν1 ; u, δ ∗ ) = βhα (~ah , νh ; . . . ; ~a1 , ν1 ; u, δ ∗ ) − βhα,L (~ah , . . . , ~a1 ; δ ∗ ) .

(4.40)

βhα,L

is calculated at the values of Note that, in the r.h.s. of (4.40), the function ~ah0 , h ≤ h0 ≤ 1, which are the solutions of Eqs. (4.9) and (4.10); these values are of course different from those satisfying Eqs. (4.39). We shall prove the following lemma. ¯ ≤ 4.5 Lemma. Suppose that u satisfies the condition (2.117), the sequence νh , h h ≤ 1, satisfies the bound (4.17) and δ ∗ satisfies the condition |−δ ∗ v0 + cδ0 λ1 | ≤ |λ1 | ,

(4.41)

cδ0 being the constant appearing in the r.h.s. of (4.6). Then, if η is defined as in Lemma 4.2 and µh ≤ ε¯0 (hence (4.1) is satisfied) and ε¯0 is small enough, ¯ ¯ 2 [γ − 12 (h−h) + γ ηh ] , |rhλ | + |rhδ | ≤ C λ h

|r1λ | ≤ Cλ21 ,

¯ ≤ h ≤ 0; h

|r1δ | ≤ C|λ1 | .

(4.42) (4.43)


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4.6 Sketch of the proof. Note that all trees with n ≥ 2 endpoints, contributing to the expansions in the r.h.s. of Eqs. (4.5)–(4.7), may have an endpoint of type ν or δ only if there are at least two endpoints of type λ; this claim follows from the definition of localization and the support properties of the single scale propagators. The bound (4.43) is an easy consequence of this remark, Eqs. (4.5) and (4.6), condition (4.41) and Theorem 3.12. We then consider h ≤ 0 and we define ∆zh = zh − zhL =

ZL Zh−1 − h−1 . Zh ZhL

(4.44)

Remember that all quantities in (4.44) have to be considered as functions of the same running couplings. Suppose now that ¯

|∆zk | ≤ c0 µ2k [γ − 2 (k−h) + γ ηk ] , 1

h < k ≤ 0.

(4.45)

We want to prove that this bound is verified also for k = h, together with (4.42). Since the proof will also imply that (4.45) is verified for k = 0, we shall achieve the proof of Lemma 4.5. By using the decomposition (2.99) of the propagator, it is easy to see that rhα =

3 X

rhα,i ,

(4.46)

i=1

where the quantities rhα,i are defined in the following way. (1) rhα,1 is obtained from βhα by restricting the sum over the trees in the r.h.s. of (4.5) and (4.6) to those containing at least one endpoint of type ν. (2) rhα,2 is obtained from βhα by restricting the sum over the trees to those containing no endpoint of type ν, and by substituting, in each term contributing to the expansions appearing in the r.h.s. of (4.5) and (4.6), at least one propagator (h0 ) (h0 ) with a propagator of type r1 or r2 (see (2.99)), h ≤ h0 ≤ 1. Note that zh and all the ratios Zk /Zk−1 , k > h, appearing in the expansions are left unchanged. (3) rhα,3 is obtained by subtracting βhα,L from the expression we get, if we substitute all propagators appearing in the expansions contributing to βhα with Luttinger propagators and if we eliminate all trees containing endpoints of type ν. By using (4.17), (2.101) and (4.34), it is easy to prove that rhα,1 and rhα,2 satisfy a bound like (4.42). The main point is the remark, already used in the proof of 0 Lemma 4.2, that there is an improvement of order γ −η (k−h) , 0 < η 0 < 1, in the bound of the sum over the trees with a vertex of fixed scale k > h. One has also to use a trick similar to that of Sec. 3.13, in order to keep the bound (3.94) on the determinants, after the decomposition of the propagators. Finally, one has to use the remark made at the beginning of this section in order to justify the presence of ¯ 2 , instead of ε2 , in the r.h.s. of (4.42). λ h h In order to prove that a bound like (4.42) is satisfied also by rhα,3 , one must first prove that the bound in (4.45) is valid for k = h, with the same constant c0 . This result can be achieved by decomposing ∆zh in a way similar to that used for rhα ;


00105


1397

let us call ∆i zh the three corresponding terms. By proceeding as before, we can show that ¯ ¯ 2 [γ − 12 (h−h) + γ ηh ] . (4.47) |∆1 zh | + |∆2 zh | ≤ C λ h P∞ P Let us now consider ∆3 zh ; we can write ∆3 zh = n=2 τ ∈Th,n ∆3 zh (τ ), with P ∆3 zh (τ ) = 0, if τ contains endpoints of type ν, and ∆3 zh (τ ) = v∈τ z¯h (τ, v), where z¯h (τ, v) = 0, if v is an endpoint, otherwise z¯h (τ, v) is obtained from zh (τ ) by selecting a family V vertices, which are not endpoints, containing v, and by substituting, for each v 0 ∈ V , the factor Zhv0 /Zhv0 −1 with Zhv0 /Zhv0 −1 − ZhLv0 /ZhLv0 −1 . By using (4.2), we have

|Zhv /Zhv −1 − ZhLv /ZhLv −1 | ≤ C|∆zhv |2 ;

(4.48)

hence it is easy to show that ∆3 zh can be bounded as in the proof of Theorem 3.12, by adding a sum over the non-trivial vertices (whose number is proportional to n) and, for each term of this sum, a factor ¯ ¯2 [γ − 12 (h−h) + γ ηh ]γ η(hv˜ −h) (hv˜ − hv˜0 ) , Cc0 λ h

(4.49)

where v˜ is the non-trivial vertex corresponding to the selected term and v˜0 is the non-trivial vertex immediately preceding v˜ or the root. Hence, we get ¯

¯ 2 [γ − 2 (h−h) + γ ηh ] , |∆3 zh | ≤ Cc0 ε2h λ h 1

(4.50)

implying, together with (4.47), the bound (4.45) for k = h, if ε¯0 is small enough and c0 is large enough. Given this result, it is possible to prove in the same manner that rhα,3 satisfies a bound like (4.42). This completes the proof of Lemma 4.5. 4.7. Lemma 4.5 allows to reduce the study of running couplings flow to the same problem for the flow (4.39). This one, in its turn, can be reduced to the study of the beta function for the Luttinger model, see [5]. This model is exactly solvable, see [19], and the Schwinger functions can be exactly computed, see [5]. It is then possible to show, see [5, 6, 7, 10], that there exists ε¯ > 0, such that, if |~ah | ≤ ε¯, 0

|β¯hα,L (~ah , . . . , ~ah )| ≤ Cµ2h γ η h ,

(4.51)

where β¯hα,L (~ah , . . . , ~a1 ), α = λ, δ, denote the analogous of the functions βhα,L (~ah , . . . , ~a1 ) for this model and 0 < η 0 < 1. Note that in the l.h.s. of (4.51) all running couplings ~ak , h ≤ k ≤ 1, are put equal to ~ah and that ~ah can take any value such that |~ah | ≤ ε¯, since ~ah is a continuous function of ~a0 and ~ah = ~a0 +O(µ2h ), see [6]. We argue now that a bound like (4.51) is valid also for the functions βhα,L . In fact the Luttinger model differs from our approximate model only because the space coordinates take values on the real axis, instead of the unit lattice. This implies, in particular, that we have to introduce a scale decomposition with a scale index h going up to +∞. However, as it has been shown in[10], the effective potential


1398

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on scale h = 0 is well defined; on the other hand, it differs from the effective potential on scale h = 0 of our approximate model only for the non-local part of the interaction. In terms of the representation (2.61) of V (0) (ψ (≤0) ), this difference is the same we would get, by changing the kernels of the non-local terms (without qualitatively affecting their bounds) and the delta function, which in the Luttinger model is defined as Lβδk,0 δk0 ,0 , instead of as in (2.62). Note that the difference of the two delta functions has no effect on the local part of V (0) (ψ (≤0) ), because of the support properties of ψˆ(≤0) , but it slightly affects the non-local terms on any scale, hence it affects the beta function; however, it is easy to show that this is a negligible phenomenon. Let us consider in fact a particular tree τ and a vertex v ∈ τ of scale hv with 2n external fields of space momenta P2n kr0 , r = 1, . . . , 2n; the conservation of momentum implies that r=1 σr kr0 = 2πm, with m = 0 in the continuous model, but m arbitrary integer for the lattice model. On the other hand, kr0 is of order γ hv for any r, hence m can be different from 0 only if n is of order γ hv . Since the number of endpoints following a vertex with 2n external fields is greater or equal to n − 1 and there is a small factor (of order µh ) associated with each endpoint, we get an improvement, in the bound of the terms with |m| > 0, with respect to the others, of a factor exp(−Cγ −hv ). Hence, by using the usual arguments, it is easy to show that the difference between the two beta functions is of order µ2h γ ηh . The previous considerations prove the following, very important, lemma. 4.8 Lemma. There are ε¯0 and η 0 > 0, such that, if |µh | ≤ ε¯0 , α = λ, δ and h ≤ 0, ¯ 2 γ η0 h . (4.52) |βhα,L (~ah , . . . , ~ah )| ≤ C λ h We are now ready to prove the following main theorem on the running couplings flow. 4.9 Theorem. If u 6= 0 satisfies the condition (2.117) and δ ∗ satisfies the condition (4.41), there exist ε¯3 and a finite integer h∗ ≤ 0, such that, if |λ1 | ≤ ε¯3 and ν ∗ ∗ belongs to a suitable interval I (h ) , of size smaller than c|λ1 |γ 0h for some constants c and γ 0 , 1 < γ 0 < γ, then the running coupling constants are well defined for h∗ − 1 ≤ h ≤ 0 and h∗ satisfies the definition (2.116). Moreover, there exist positive constants ci , i = 1, . . . , 5, such that |λh − λ1 | ≤ c1 |λ1 |3/2 , γ λ1 c2 h 0, |δh | ≤ C|λ1 |3/2 [γ −η(h−h

∗

)

+ γ ηh ] .

(4.57)

4.10 Proof. We shall proceed by induction. Equations (4.5), (4.6) and Lemma 4.2 imply that, if λ1 is small enough, there exists an interval I (0) , whose size is of order λ1 , such that, if ν ∈ I (0) , then the bound (4.17) is satisfied, together with |λ0 − λ1 | ≤ C|λ1 |2 ,

|δ0 − δ1 | = |δ0 | ≤ C|λ1 | .

(4.58)

¯≤h≤0 Let us now suppose that the solution of (4.9)–(4.11) is well defined for h ¯ and satisfies the conditions (4.14)–(4.17), for any ν belonging to an interval I (h) , ¯ see (4.14) and defined as in Lemma 4.2. This implies, in particular, that h∗ ≤ h, (2.116). Suppose also that there exists a constant c0 , such that ¯ ¯ ≤ 2|λ1 | . λ h

(4.59)

We want to prove that all these conditions are verified also if ¯h is substituted ¯ − 1, if λ1 is small enough. The induction will be stopped as soon as the with h ¯ condition (4.14) is violated for some ν ∈ I (h) . We shall put ν equal to one of these ¯ + 1. values, so defining h∗ as equal to h The fact that the condition on ν1 and the bound (4.17) are verified also if ¯h − 1 takes the place of ¯ h, follows from Lemma 4.2, since the condition (4.13) follows from (4.59), if λ1 is small enough. There is apparently a problem in using this lemma, since in its proof we used the hypothesis that the values of ~ah , Zh−1 and σh−1 (k0 ), ¯ ≤ h ≤ 1, are independent of ν1 . This is not true for the full flow, but the proof h of Lemma 4.5 can be easily extended to cover this case. In fact, the only part of the proof, where we use the fact that ~ah is constant, is the identity (4.24), which should be corrected by adding to the r.h.s. the difference bh − b0h. However, since λ1 is independent of ν1 , it is not hard to prove that |bh − b0h | ≤ C|νh − νh0 | and that the bound on rh − rh0 does not change (qualitatively), if we take into account also the dependence on ν1 of the various quantities, before considered as constant. Hence, the bound (4.25) is left unchanged. The conditions (4.15) and (4.16) follow immediately from (4.59) and (4.2)–(4.4). ¯ is substituted with Hence, we still have to show only that (4.59) is verified also if h ¯ h − 1, if λ1 is small enough. By using (4.39) and (4.40), we have, if α = λ, δ, = αh¯ + βh¯α,L (~ah¯ , . . . , ~ah¯ ) + αh−1 ¯

1 X

α α Dh,k ah¯ , νh¯ ; . . . ; ~a1 , ν1 ; u) , ¯ + rh ¯ (~

(4.60)

¯ k=h+1

where α = βhα,L (~ah , . . . , ~ah , ~ak , ~ak+1 , . . . , ~a1 ) Dh,k

− βhα,L (~ah , . . . , ~ah , ~ah , ~ak+1 , . . . , ~a1 ) .

(4.61)


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00105


α On the other hand, it is easy to see that Dh,k admits a tree expansion similar to that α,L of βh (~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 η is the same constant of Lemmas 4.2 and 4.5 and h ≤ 0, α ¯ h |γ −η(k−h) |~ak − ~ah | . | ≤ C|λ |Dh,k

(4.62)

¯ ≤ h ≤ 0 and that there exists a constant c0 , such that Let us now suppose that h ¯

|~ak−1 − ~ak | ≤ c0 |λ1 |3/2 [γ − 2 (k−h) + γ ϑk ] , 1

h0

k :C0 (k )>0

(≤0)+ (≤0)− · C0 (k0 )(−ik0 + ωv0∗ k 0 )ψˆk0 ,ω ψˆk0 ,ω ,

(L)

Z

V (L) (ψ (≤0) ) = λ0

(≤0)+

TL,β (L)

+ δ0

(≤0)−

(≤0)+

(5.30)

(≤0)−

dxψx,+1 ψx,−1 ψx,−1 ψx,+1 Z

X

iω TL,β

ω=±1

(≤h)+ (≤h)− dxψx,ω ∂x ψx,ω ,

(5.31)

where NL (k0 ) = C0 (k0 )(Lβ)−1 [k02 + (v0∗ k 0 )2 ]1/2 , λ0 and δ0 have the role of the running couplings on scale 0 of the original model, but are not necessarily equal to them, TL,β is the (continuous, as in Sec. 3.15) torus [0, L] × [0, β] and ψ (≤0) is the (continuous) Grassmanian field on TL,β with antiperiodic boundary conditions. Moreover, the interaction with the external field B (1) (ψ (≤1) , φ) is substituted with the corresponding expression on scale 0, deprived of the irrelevant terms, that is X Z (0) (≤0) (≤h)σ (≤h)σ (≤h)σ (≤h)−σ , φ) = ψx,−σ + ψx,σ ψx,σ ) . (5.32) dxφ(x)(e2iσpF x ψx,σ B (ψ (L)

(L)

σ=±1 (2,L)

(2,L)

(L)

(L)

(2)

(2)

We shall call Zh , zh , Zh and zh the analogous of Zh , zh , Zh and zh for this approximate Luttinger model. (2,L) (L) (2) We want to compare the flow of Zh /Zh with the flow of Zh /Zh ; hence we write (2)

(2) Zh−1 Zh = [1 + β (2) (~ah , νh ; . . . ; ~a1 , ν1 ; u, δ ∗ )] , Zh−1 Zh (2,L)

Zh−1

(L) Zh−1

(2,L)

=

Zh

(L) Zh

[1 + β (2,L) (~ah , . . . ; ~a0 , δ ∗ )] , (L)

(L)

(5.33)

(5.34)


00105


1407

(L)

where ah are the running couplings in the approximated Luttinger model (by (L) (2) symmetry νh = 0, since ν = 0, see Sec. 4.4), 1 + β (2) = (1 + zh )/(1 + zh ) and (2,L) (L) 1 + β (2,L) = (1 + zh )/(1 + zh ). The Luttinger model has a special symmetry, the local gauge invariance, which allows to prove many Ward identities. As we shall prove in Sec. 7, the approximate Luttinger model satisfies some approximate version of these identities and one of (L) them implies that, if |δ∗ + (δ0 /v0 )| ≤ 1/2, (L)

γ −C|λ0

|

(2,L)

≤

Zh

(L) Zh

(L)

≤ γ C|λ0

|

.

(5.35)

By proceeding as in the proof of (4.51) (see [6, Sec. 7]), one can show that (5.35) implies that there exists ε¯ > 0 and η 0 < 1, such that, if |~ah | ≤ ε¯, (2,L)

|βh

0

(~ah , . . . , ~ah , δ ∗ )| ≤ Cµ2h γ η h .

(5.36)

We note that the analogous bound (4.51) was obtained in [6] by a comparison with the exact solution of the Luttinger model; this was possible, thanks to the proof given in [10] that the effective potential on scale 0 is well defined also in the Luttinger model, a non-trivial result because of the ultraviolet problem. This procedure would be much harder in the case of the bound (5.36), because the density is not well defined in the Luttinger model, see Sec. 1.3. In any case, the bound (5.35), whose proof is relatively simple, allows to get very easily the same result. One can also show, as in the proof of Lemma 4.5, that |β (2) (~ah , νh ; . . . ; ~a1 , ν1 ; u, δ ∗ ) − β (2,L) (~ah , , . . . ; ~a0 , δ ∗ )| ¯ 2 [γ − 12 (h−h∗ ) + γ ηh ] , ≤ Cλ h

(5.37)

for any h ≥ h∗ and for some η < 1. Note that, in (5.37), β (2,L) is evaluated at the values of the running couplings ~ah of the original model; this is meaningful, since in (5.36) ~ah can take any value (L) such that |~ah | ≤ ε¯; this follows from the remark, already used in Sec. 4.7, that ~ah (L) (L) (L) is a continuous function of ~a0 and ~ah = ~a0 + O(µ2h ), see also [6]. By using (5.36) and (5.37) and proceeding as in the proof of Theorem 4.9, one can easily prove the following theorem. 5.6 Theorem. If the hypotheses of Theorem 5.4 are verified, there exists a positive constant c1 , independent of u, L, β, such that γ −c1 |λ1 | ≤

(2)

Zh ≤ γ c1 |λ1 | . Zh

(5.38)

5.7. We are now ready to study the expansion of the correlation function Ω3L,β (x), which follows from (5.21) and the considerations of Sec. 5.2. 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


1408

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by hx,y the scale of the smallest cluster containing both special endpoints. Finally 2 2 will denote the family of all trees belonging to Th,n , such that the two special Th,n,l (1) endpoints are both of type Z , if l = 1, both of type Z (2) , if l = 2, one of type Z (1) and the other of type Z (2) , if l = 3. If we extract from the expansion the contribution of the trees with one special endpoint and no normal endpoints, we can write ( 1 (1) X X (Zh∨h0 )2 (h) (h0 ) 3 [gσ,σ (−σx)g−σ,−σ (−σx) e2iσpF x · ΩL,β (x) = Zh−1 Zh0 −1 0 ∗ σ=±1 h,h =h

(2)

(Zh∨h0 )2 (h) (h0 ) [−gσ,σ (−σx)gσ,σ (σx) Zh−1 Zh0 −1  ) ! 1  (1) 2 X 0 Zh (h) (h ) (h) G1,L,β (x) + g−1,+1 (−σx)g+1,−1 (σx)] +  Z h ∗ (h0 )

(h)

− g+1,−1 (−σx)g−1,+1 (−σx)] +

h=h

(2) Zh

+



!2 (h)

G2,L,β (x) +

Zh

(1) (2)  Zh Zh (h) G (x) , 3,L,β  Zh2

(h∗ )

(5.39) (≤h∗ )

where h ∨ h0 = max{h, h0 } and gω1 ,ω2 (x) has to be understood as gω1 ,ω2 (x); moreover, (h)

Gl,L,β (x) =

∞ X

h−1 X

X

X

X X

(h,h )

Gl,L,βr (x, τ, P, r, T, α) ,

(5.40)

n=1 hr =h∗ −1 τ ∈T 2

P∈Pτ ,r T ∈T α∈AT hr ,n,l hx,y =h Pv0 =∅

ˆ v0 denotes the set of space-time points associated with the normal where, if x endpoints and ix = i, if the corresponding special endpoint is of type Z (i) , (h,h )

Gl,L,βr (x, τ, P, r, T, α) " ! (iy ) # Z (i ) Y Z hy Z h Zhxx Zh   (Zhv /Zhv −1 )|Pv |/2 · dˆ xv0 ) xv0 hα (ˆ = (i ) (i ) Zhx −1 Zh x Zhy −1 Zh y v not e.p. " ·

n Y

#( b (v ∗ ) (h ) djαα(vi∗ ) (xi , yi )Kv∗ i (xvi∗ ) i i

i=1

" ·

Y

v not

1 s ! e.p. v

Z dPTv (tv ) · det Ghαv ,Tv (tv )

#) qα (fl− ) ˆqα (fl+ ) bα (l) ml (hv ) ¯ ˆ . ∂j (f − ) ∂j (f + ) [djα (l) (xl , yl )∂1 gω− ,ω+ (xl − yl )] α

l∈Tv

Y

l

α

l

l

(5.41)

l

In the r.h.s. of (5.41) all quantities are defined as in Sec. 3, except the kernels (h ) Kv∗ i (xvi∗ ) associated with the special endpoints. If v is one of these endpoints, xv i is always a single point and Kv(hv ) (xv ) = eipF xv

P f ∈Iv

σ(f )

.

(5.42)


00105


1409

We want to prove the following theorem. 5.8 Theorem. Suppose that the conditions of Theorem 5.4 are verified, that ε¯4 is defined as in that theorem and that δ ∗ is chosen so that condition (4.57) is satisfied. Then, there exist positive constants ϑ < 1 and ε¯5 ≤ ε¯4 , independent of u, L, β, such √ that, if |λ1 | ≤ ε¯5 and γ ≥ 1 + 2, given any integer N ≥ 0, |G1,L,β (x)| + |G2,L,β (x)| + γ −ϑh |G3,L,β (x)| ≤ CN |λ1 | (h)

(h)

γ 2h

(h)

for a suitable constant CN . Moreover, if h ≤ 0, we can write (h)

(h)

¯ G1,L,β (x) = cos(2pF x)G 1,L,β (x) +

X

1+

(h)

[γ h |d(x)|]N

=

¯ (h) (x) G 2,L,β

+

(h) s2,L,β (x)

+

(5.43)

(h)

eipF σx s1,σ,L,β (x) + r1,L,β (x) ,

σ=±1 (h) G2,L,β (x)

,

(5.44)

(h) r2,L,β (x) ,

so that ¯ (h) (−x) , ¯ (h) (x) = G G l,L,β l,L,β (h)

(h)

l = 1, 2 ,

(5.45)

γ ϑh , 1 + [γ h |d(x)|]N

(5.46)

|r1,L,β (x)| + |r2,L,β (x)| ≤ CN |λ1 |γ 2h

and, if we define Dm0 ,m1 = ∂0m0 ∂¯1m1 , given any integers m0 , m1 ≥ 0, there exists a constant CN,m0 ,m1 , such that X

2h h(m0 +m1 ) ¯ (h) (x)| ≤ CN,m0 ,m1 |λ1 | γ γ |Dm0 ,m1 G , l,L,β 1 + [γ h |d(x)|]N l=1,2 X (h) (h) |Dm0 ,m1 s1,σ,L,β (x)| + |Dm0 ,m1 s2,L,β (x)|

(5.47)

σ=±1

≤ CN,m0 ,m1 |λ1 |

γ 2h γ h(m0 +m1 ) −ϑ(h−h∗ ) [γ + γ ϑh ] . 1 + [γ h |d(x)|]N

(5.48)

(h) ¯ (h) (x), r(h) (x), s(h) Ω3L,β (x), as well as the functions G l,L,β l,L,β 1,σ,L,β (x) and s2,L,β (x) converge, as L, β → ∞, to continuous bounded functions on Z × R, that we shall de¯ (h) (x), r(h) (x), s(h) (x) and s(h) (x), respectively. G ¯ (h) (x) and G ¯ (h) (x) note Ω3 (x), G 1,σ 2 1 2 l l are the restrictions to Z × R of two even functions on R2 satisfying the bound (5.47) with the continuous derivative ∂1 in place of the discrete one and |x| in place of |d(x)|. ¯ (h) (x), as a function on R2 , satisfies the symmetry relation Finally, G 1 (h) (h) ∗ x ¯ ¯ G1 (x, x0 ) = G1 x0 v0 , ∗ . (5.49) v0

5.9 Proof. As in the proof of Theorem 5.4, we shall try to mimic as much as possible the proof of the bound (3.110), by only remarking the relevant differences. Since D0 (Pv0 ) + lv0 = 0, if the integral in the r.h.s. of (5.41) were over the set


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00105

G. Benfatto & V. Mastropietro (h,h )

of variables xv0 \x, we should get for Gl,L,βr (x, τ, P, r, T, α) the same bound we (l)

derived in Sec. 5.3 for zh (τ, P, r, T, α). However, in this case, we have to perform the integration over the set xv0 by keeping fixed two points (x and y), instead of one; hence we have to modify the bound (3.102) in a way different from what we did in the proof of Theorem 5.4. Let us call v¯0 the higher vertex v ∈ τ , such that both x and y belong to xv ; by the definition of h, it is a non-trivial vertex and its scale is equal to h. Moreover, given the tree graph T on xv0 , let us call Tx,y its subtree connecting the points of S xv¯0 and T˜x,y = v≥¯v0 T˜v , T˜v being defined as Sec. 3.15, after (3.118). We want to bound d(x − y) in terms of the distances between the points connected by the lines l ∈ T˜x,y . Let us call v¯(i) , i = 1, . . . , sv¯0 the non-trivial vertices or endpoints following v¯0 . The definition of v¯0 implies that sv¯0 > 1 and that x and y belong to two different sets xv¯(i) ; note also that T˜v¯0 is an anchored tree graph between the sets of points xv¯(i) . Hence there is an integer r, a family l1 , . . . , lr of lines belonging to T˜v¯0 and a family v (1) , . . . , v (r+1) of vertices to be chosen among v¯(1) , . . . , v¯(sv¯0 ) , such that 1 ≤ r ≤ sv¯0 − 1 and |d(x − y)| ≤

r X

|d(x0lj − yl0j )| +

j=1

≤

X

r+1 X

|d(x(j) − y(j) )|

j=1

|d(x0l − yl0 )| +

r+1 X

|d(x(j) − y(j) )| ,

(5.50)

j=1

l∈T˜v¯0

where x(1) = x, y(r+1) = y, x0lj and yl0j are defined as in (3.114) and, finally, the couple of points (x0lj , yl0j ) coincide, up to the order, with the couple (y(j) , xj+1 ). If no propagator associated with a line l ∈ T˜x,y is affected by the regularization, we can iterate in an obvious way the previous considerations, so getting the bound X |d(x0l − yl0 )| . (5.51) |d(x − y)| ≤ l∈T˜x,y

However, this is not in general true and we have to consider in more detail the subsequent steps of the iteration. Let us consider one of the vertices xv(j) ; if x(j) = y(j) , there is nothing to do. Hence we shall suppose that x(j) 6= y(j) and we shall say that the propagators associated with the lines lj , if 1 ≤ j ≤ r, and lj−1 , if 2 ≤ j ≤ r + 1, are linked to v (j) . There are two different cases to consider. (1) x(j) and y(j) belong to two different non-trivial vertices or endpoints following v (j) and the propagators linked to v (j) are not affected by action of R on the vertex v (j) or some trivial vertex v, such that v¯0 < v < v (j) . In this case, we iterate the previous procedure without any change.


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1411

(2) One of the propagators linked to v (j) is affected by action of R on the vertex v (j) or some trivial vertex v, such that v¯0 < v < v (j) ; note that, if there are two linked propagators, only one may have this property, as a consequence of the regularization procedure described in Sec. 3. This means that x(j) or y(j) , let us ˜ l , xl ∈ xv(j) say x(j) , is of the form (3.115), with tl 6= 1, that is there are two points x ¯ l ∈ R2 , coinciding with xl modulo (L, β), such that and a point x ˜ l + tl (¯ ˜l ) , xl − x x(j) = x

|¯ xl − xl | ≤ 3L/4, |¯ xl,0 − xl,0 | ≤ 3β/4 .

(5.52)

xl − By using (2.96) and (5.52), the fact that 0 ≤ |tl | ≤ 1 and the remark that d(¯ ˜ l ) = d(xl − x ˜ l ), we get x √ ˜ l )| . xl − y(j) )| + 2|d(xl − x (5.53) |d(x(j) − y(j) )| ≤ |d(˜ ˜ l )|, by proceeding as in the proof of We can now bound |d(˜ xl − y(j) )| and |d(xl − x ˜ l , xl and y(j) all belong to v (j) . We get (5.50), since the points x   rj X X √   |d(x0l − yl0 )| + |d(x0(m) − y0(m) )| , (5.54) |d(x(j) −y(j) )| ≤ (1+ 2)  m=1

l∈T˜v(j)

where 2 ≤ rj ≤ sv(j) and the points x0(m) , y0(m) are endpoints of propagators linked to some non-trivial vertex or endpoint following v (j) . By iterating the previous procedure we get, instead of (5.51), the bound X √ (1 + 2)pl |d(x0l − yl0 )| , (5.55) |d(x − y)| ≤ l∈T˜x,y

where, if l ∈ Tvl , pl is an integer less or equal to the number of non-trivial vertices v such that v¯0 ≤ v < vl ; note that pl ≤ hvl − h .

(5.56)

√ 2.

(5.57)

Let us now suppose that γ ≥1+

Since there are at most 2n + 1 lines in T , (5.55)–(5.57) imply that there exists at least one line l ∈ Tx,y , such that γ h |d(x − y)| . (5.58) 2n + 1 It follows that, given any N ≥ 0, for the corresponding propagator we can use, instead of the bound (3.116), the following one: ˜qα (fl− ) ˜qα (fl+ ) bα (l) 0 (h ) ∂j (f − ) ∂j (f + ) [djα (l) (xl (tl ), yl0 (sl ))∂¯1ml gω−v,ω+ (x0l (tl ) − yl0 (sl ))] γ hvl |d(x0l − yl0 )| ≥

α

α

l

≤

γ

l

l

l

hv [1+qα (fl+ )+qα (fl− )+m(fl− )+m(fl+ )−bα (l)]

1 + [γ hv |d(x0l (tl ) − yl0 (sl ))|]3

|σhv | γ hv

ρl

CN (2n + 1)N . 1 + [γ h |d(x − y)|]N (5.59)


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00105


For all others propagators we use again the bound (3.116) with N = 3 and we proceed as in Sec. 3.15, recalling that we have to substitute in (3.118) d(xv0 \¯ x) with dˆ xv0 . This implies that, in the r.h.s. of (3.119), one has to eliminate one drl factor and, of course, this can be done in an arbitrary way. We choose to eliminate the integration over the rl corresponding to a propagator of scale h (there is at least one of them), so that the bound (3.118) is improved by a factor γ 2h . At the end, we get (h,h )

|Gl,L,βr (x, τ, P, r, T, α)| ≤ (Cεh )n CN (2n + 1)N ·

!

(i )



Y γ   · h N (i ) (i ) 1 + [γ d(x)] Zhx −1 Zh x Zhy −1 Zh y v not e.p. 2h

(i )

Zhxx Zh

Zhyy Zh

|Pv | z ˜(Pv ,lv ) 1 Psi=1 v |Pvi |−|Pv | C · (Zhv /Zhv −1 )|Pv |/2 γ −[−2+ 2 +lv +z(Pv ,lv )+ 2 ] . sv ! (5.60)

We can now perform as in Sec. 3.14 the various sums in the r.h.s. of (5.40). There are some differences in the sum over the scale labels, but they can be easily (i ) (i ) treated. First of all, one has to take care of the factors (Zhxx Zh )/(Zhx −1 Zh x ) and (i )

(i )

(Zhyy Zh )/(Zhy −1 Zh y ). However, by using (5.29) and (5.38), it is easy to see that these factors have the only effect to add to the final bound a factor γ C|λ1 |(hv −hv0 ) 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 analogous of the bound (3.111), 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 the differences hv − hv0 . (h) The previous considerations are sufficient to get the bound (5.43) for G1,L,β (x) (h)

(h)

and G2,L,β (x). In order to explain the factor γ ϑh multiplying G3,L,β (x), one has to note that the trees whose normal endpoints are all of scale lower than 2 give (h) no contribution to G3,L,β (x). In fact, these endpoints have the property that P f ∈Pv σ(f ) = 0, while this condition is satisfied from one of the special endpoints (h)

but not from the other, in any tree contributing to G3,L,β (x). It follows, since any propagator couples two fields with different σ indices, that it is possible to produce (h) a non-zero contribution to G3,L,β (x), only if there is at least one endpoint of scale 2; this allows to extract from the bound a factor γ ϑh , with 0 < ϑ < 1, as remarked many times before. (h) (h) We now want to show that G1,L,β (x) and G2,L,β (x) can be decomposed as in (5.44), so that the bounds (5.46), (5.47) and (5.45) are satisfied. To begin with, (h) (h) we define ri,L,β (x), i = 1, 2, by using the definition (5.40) of Gi,L,β (x), with the constraint that the sum is restricted to the trees, which contain at least one endpoint (+1) (+1) of scale hv = 2; this implies, in particular, that Gi,L,β (x) − ri,L,β (x) = 0. Moreover,


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1413

in the remaining trees, we decompose the propagators in the following way: (h)

(h)

(h)

gω,ω0 (x) = g¯ω,ω0 (x) + δgω,ω0 (x) ,

(5.61)

where g¯ω,ω0 (x) is defined by putting, in the r.h.s. of (2.94), (v0∗ k 0 ) in place of E(k 0 ), (h)

(h)

(h)

and we absorb in ri,L,β (x) the terms containing at least one propagator δgω,ω0 (x), which is of size γ 2h . The substitution of (v0∗ k 0 ) in place of E(k 0 ) is done also in the definition of the R operator, so producing other “corrections”, to be added to (h) (h) ri,L,β (x). An argument similar to that used for G3,L,β (x) easily allows to prove the bound (5.46). P (h) (h) σ=±1 exp(iσpF x)s1,σ,L,β (x) and s2,L,β (x) will denote the sum of the trees (h)

(h)

(h)

(h)

contributing to G1,L,β (x) − r1,L,β (x) and G2,L,β (x) − r2,L,β (x), respectively, which have at least one endpoint of type ν or δ. (h) Let us now consider the “leading” contribution to G2,L,β (x), which is defined ¯ (h) (x) and is obtained by using again (5.40), by the second of Eqs. (5.44) as G 2,L,β

but with the constraint that the sum over the trees is restricted to those having only endpoints with scale hv ≤ 1 and only normal endpoints of type λ. Moreover (h) we have to use everywhere the propagator g¯ω,ω0 (x), which has well defined parity properties in the x variables; it is odd, if ω = ω 0 , and even, if ω = −ω 0 . P Note that all the normal endpoints with hv ≤ 1 are such that f ∈Iv σ(f ) = 0 and that this property is true also for the special endpoints, which have to be of type Z (2) ; hence there is no oscillating factor in the kernels associated with the endpoints, which are suitable constants (the associated effective potential terms ¯ (h) (x) is given, up to a are local). It follows that any graph contributing to G 2,L,β constant, by an integral over the product of an even number of propagators (we are using here the fact that there is no endpoint of type ν or δ). Moreover, since P all the endpoints satisfy also the condition f ∈Iv σ(f )ω(f ) = 0, which is violated by the set of two lines connected by a non-diagonal propagator, the number of non-diagonal propagators has to be even. These remarks immediately imply that ¯ (h) (−x). ¯ (h) (x) = G G 2,L,β 2,L,β ¯ (h) (x), we observe that, since the In order to prove the bound (5.47) for G 2,L,β P propagators only couple fields with different σ indices and f ∈Iv σ(f ) = 0, given ¯ (h) (x) and any v ∈ τ , we must have any tree τ contributing to G 2,L,β

X

σ(f ) = 0 .

(5.62)

f ∈Pv

Let us now consider the vertex v¯0 , defined as in Sec. 5.9, that is the higher vertex v ∈ τ , such that both x and y = 0 belong to xv , and let vx be the vertex immediately following v¯0 , such that x ∈ vx . We can associate with vx a contribution to B h (ψ (≤h) , φ) (recall that h is the scale of v¯0 and hence the scale of the external fields of vx ), with m = 1 and 2n = Pvx (see (5.6)), whose kernel is of the form, thanks to (5.62)


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00105


B(x; y1 , . . . , y2n ) =

1 (Lβ)2n+1

X

eipx−i

P2n r=1

σr k0r yr

p,k01 ,...,k02n

ˆ · B(p; k01 , . . . , k02n−1 )δ

2n X

! σr k0r

−p .

(5.63)

r=1 (h)

¯ If we apply the differential operator ∂0m0 to G 2,L,β (x), this operator acts on B(x; y1 , . . . , y2n ), so that its Fourier transform is multiplied by (ip0 )m0 ; since P2n p0 = r=1 σr kr0 and the external fields of vx are contracted on a scale smaller or equal to h, it is easy to see that there is an improvement on the bound of ¯ (h) (x), with respect to the bound of G ¯ (h) (x), of a factor cm0 γ hm0 , for a ∂0 G 2,L,β 2,L,β ¯ (+1) (x) = 0, so that we suitable constant cm0 . We are using here the fact that G i,L,β

can suppose h ≤ 0, otherwise we would be involved with the singularity of the scale (1) 1 propagator gω− ,ω+ (xl − yl ) at xl − yl = 0, which allows to get uniform bounds on l

l

the derivatives only for |xl − yl | bounded below, a condition not verified in general. ¯ (h) (x) (see (3.6) for the definition of ∂¯1 ). By using Let us now consider ∂¯1m1 G 2,L,β (2.62) and the conservation of the spatial momentum, we find that ∂¯1m1 acts on B(x; y1 , . . . , y2n ), so that its Fourier transform is multiplied by sin(px)m1 , with P2n p = r=1 σr kr0 +2πm, where m is an arbitrary integer and p is chosen so that |p| ≤ π. If m = 0, we proceed as in the case of the time derivative, otherwise we note that P 0 the support properties of the external fields, see Sec. 2.2, implies that | 2n r=1 σr kr | ≤ h −h 2na0 γ ; hence, if |m| > 0, 2n ≥ (π/a0 )γ . Since the number of endpoints following vx is proportional to 2n and each endpoint carries a small factor of order λ1 , it is clear that, if λ1 is small enough, we get an improvement in the bound of the terms ¯ (h) (x), of a with |m| > 0, with respect to the corresponding contributions to G 2,L,β −h hm1 , for some constant cm1 . In the same manner, we factor exp(−Cγ ) ≤ cm1 γ ¯ (h) (x). can treat the operator Dm0 ,m1 , so proving the bound (5.47) for Dm0 ,m1 G 2,L,β (h)

(h)

Let us now consider G1,L,β (x − y) − r1,L,β (x − y). In this case the kernels of the two special endpoints x and y are equal to exp(2iσx pF x) and exp(2iσy pF y), respectively. However, since the propagators couple fields with different σ indices P and all the other endpoints satisfy the condition f ∈Iv σ(f ) = 0, σx = −σy and we can write (h)

(h)

G1,L,β (x − y) − r1,L,β (x − y) =

1 X 2iσpF (x−y) ¯ (h) (h) e [G1,σ (x − y) + 2s1,σ,L,β (x − y)] , 2 σ=±1

(5.64)

¯ (h) (x); in particular it is an even ¯ (h) (x) having the same properties as G with G 1,σ 2 ¯ (h) (x− function of x and satisfies the bound (5.47). Moreover, it is easy to see that G 1,+

¯ (h) (x − y), hence G ¯ (h) (y − x) is independent of σ ¯ (h) (y − x) = G y) is equal to G 1,− 1,− 1,σ ¯ (h) (x − y) satisfying and we get the decomposition in the first line of (5.44), with G 1

(5.47) and (5.45).


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1415

The bound (5.48) is proved in the same way as the bound (5.47). The factor ∗ [γ −ϑ(h−h ) + γ ϑh ] in the r.h.s. comes from the fact that the trees contributing (h) (h) to s1,σ,L,β (x) and s2,L,β (x) have at least one vertex of type ν or δ, whose running constants satisfy (4.17) and (4.57). (h) (h) Note that s1,σ,L,β (x) and s2,L,β (x) are not even functions of x and that (h)

s1,σ,L,β (x) is not independent of σ. In order to complete the proof of Theorem 5.8, we observe that all the functions appearing in the r.h.s. of (5.39), as well as those defined in (5.44), clearly converge, as L, β → ∞, and that their limits can be represented in the same way as the finite L and β quantities, by substituting all the propagators with the corresponding limits. This follows from the tree structure of our expansions and some straightforward but lengthy standard arguments; we shall omit the details. (h) (h) Let us consider, in particular, the limits Gi (x) of the functions Gi,L,β (x). Their tree expansions contain only trees with endpoints of scale hv ≤ 1, which are associated with local terms of type λ or of the form (5.13) and (5.14), whose ψ fields are of scale less or equal to 0. The support properties of the field Fourier transform imply that the local terms of type λ can be rewritten by substituting the sum over the corresponding lattice space point with a continuous integral over R1 . We can of course use these new expressions to build the expansions, since the propagators of scale h ≤ 0, in the limit L, β → ∞, are well defined smooth functions on R2 . For the same reason, the tree expansions are well defined also if the space points associated with the special endpoints vary over R1 , instead of Z1 ; therefore there (h) is a natural way to extend to R2 the functions Gi (x), which of course satisfy the bound (5.47), with the continuous derivative ∂1 in place of the discrete one and |x| in place of |d(x)|, as well as the analogous of identity (5.45). (h) The function G1,L,β (x) satisfies also another symmetry relation, related with a (h)

remarkable property of the propagators g¯ω,ω0 , see (5.61), appearing in its expansion, that is x (h) (h) ∗ g−ω,−ω v0 x0 , ∗ , g¯ω,ω (x, x0 ) = −iω¯ v0 (5.65) x (h) (h) ∗ g−ω,+ω v0 x0 , ∗ . g¯ω,−ω (x, x0 ) = −¯ v0 (h)

On the other hand, each tree contributing to G1,L,β (x) with n normal endpoints (which are all of type λ) can be written as a sum of Feynman graphs (if we use the representation of the regularization operator as acting on the kernels, see Sec. 3), built by using 4n + 4 ψ fields, 2n + 2 with ω = +1 and 2n + 2 with ω = −1, (h) (h) hence containing the same number of propagators g¯+1,+1 and g¯−1,−1 and, by the argument used in the proof of (5.45), an even number of non-diagonal propagators. Then, by using (5.65), we can easily show that the value of any graph, calculated at (x, x0 ), is equal to the value at (v0∗ x0 , x/v0∗ ) of the graph with the same structure but opposite values for the ω-indices of all propagators, which implies (5.49).


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6. Proof of Theorem 1.5 6.1. Theorem 3.12 and the analysis performed in Secs. 4 and 5 imply immediately the statements in item (a) of Theorem 1.5, except the continuity of Ω3L,β (x) in x0 = 0, which will be briefly discussed below. Hence, from now on we shall suppose that all parameters are chosen as in item (a). Let us define η = logγ (1 + z ∗ ) ,

z ∗ = z[h∗ /2] ,

(6.1)

zh being defined as in (4.2). The analysis performed in Sec. 4 allows to show (we omit the details) that there exists a positive ϑ < 1, such that ∗

|zh − zh+1 | ≤ Cλ21 [γ −ϑ(h−h ) + γ ϑh ] ,

h∗ ≤ h ≤ 0 .

(6.2)

We can write 0 X

logγ Zh =

∗

logγ [1 + z + (z

h0

∗

− z )] = −ηh +

h0 =h+1

0 X

rh0 .

(6.3)

h0 =h+1

Ph−1 On the other hand, if h > [h∗ /2], thanks to (6.2), |rh | ≤ C h0 =[h∗ /2] |zh0 −zh0 +1 | ≤ ∗ Cλ21 γ ϑh and, if h ≤ [h∗ /2], |rh | ≤ Cλ21 γ −ϑ(h−h ) ; it follows that ∗

|rh | ≤ Cλ21 [γ −ϑ(h−h ) + γ ϑh ] .

(6.4)

Hence, if we define ch =

γ −ηh , Zh−1

(6.5)

we get immediately the bound |ch − 1| ≤ Cλ21 .

(6.6)

In a similar way, if we define (1)

η˜1 = logγ (1 + z[h∗ /2] ) ,

(1)

ch =

γ −˜η1 h (1)

,

(6.7)

Zh

(1)

zh being defined by (5.18), we get the bound (1)

|ch − 1| ≤ C|λ1 | .

(6.8) (2)

Bounds similar to (6.7) and (6.8) are valid also for the constants Zh , but in this case Theorem 5.6 implies a stronger result; if we define (2)

(2)

ch =

Zh , Zh−1

(6.9)

then (2)

|ch − 1| ≤ C|λ1 | .

(6.10)


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1417

Let us now consider the terms in the first three lines of the r.h.s. of (5.39) and let us call Ω3,0 L,β their sum; we can write 3,0 ¯ 3,0 Ω3,0 L,β (x) = ΩL,β (x) + δΩL,β (x) ,

where

¯ 3,0 Ω L,β

is obtained from

(6.11)

by restricting the sums over h and h0 to the

Ω3,0 L,β

(h)

(h)

values ≤ 0 and by substituting the propagators gω,ω0 with the propagators g¯ω,ω0 , defined in (5.61). By using the symmetry relations (h)

(h)

(h)

gω,ω (−x, −x0 ) = g¯+,+ (ωx, x0 ) , g¯ω,ω (x, x0 ) = −¯ (h)

(h)

(6.12)

(h)

g+,− (x) , g¯ω,−ω (x) = g¯ω,−ω (−x) = ω¯ it is easy to show that we can write ¯ 1,L,β (x) + Ω ¯ 2,L,β (x) , ¯ 3,0 (x) = cos(2pF x)Ω Ω

(6.13)

L,β

X

¯ 1,L,β (x) = 2 Ω

(1)

h∗ ≤h,h0 ≤0

(Zh∨h0 )2 (h) (h0 ) [¯ g+,+ (x, x0 )¯ g+,+ (−x, x0 ) 0 Zh−1 Zh −1 (h0 )

(h)

g+,− (x, x0 )] , + g¯+,− (x, x0 )¯ " X (Z (2) 0 )2 X (h) (h0 ) h∨h ¯ g¯+,+ (ωx, x0 )¯ g+,+ (ωx, x0 ) Ω2,L,β (x) = Zh−1 Zh0 −1 ω 0 h,h ≤0

(6.14)

#

(h) (h0 ) g+,− (x, x0 ) − 2¯ g+,− (x, x0 )¯

.

(6.15) (+1)

(+1)

By using (5.39), (5.44) and (6.13) and the fact that Gi,L,β (x) − ri,L,β (x) = 0 for i = 1, 2, we can decompose Ω3L,β as in (1.14), by defining ! 0 (1) 2 X Z 3,a h ¯ (h) (x) , ¯ 1,L,β (x) + G (6.16) ΩL,β (x) = Ω 1,L,β Z h ∗ h=h

Ω3,b L,β (x)

0 X

¯ 2,L,β (x) + =Ω

h=h∗

(2)

Zh Zh

!2 ¯ (h) (x) , G 2,L,β

(6.17)

 ! 1  (1) 2 X Zh (h) 3,c 3,0 r1,L,β (x) ΩL,β (x) = δΩL,β (x) +  Z h ∗ h=h

(2)

+

sL,β (x) =

Zh Zh

!2

 0  X X h=h∗



σ=±1

(h) r2,L,β (x)

 (1) (2)  Zh Zh (h) + G (x) + sL,β (x) , 3,L,β  Zh2 (1)

e2iσpF x

Zh Zh

!2

(2)

(h)

s1,σ,L,β (x) +

Zh Zh

!2

(6.18)

  (h) s2,L,β (x) .  (6.19)


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3,b Theorem 5.8 implies that Ω3,a L,β (x), ΩL,β (x) and sL,β (x) are smooth functions of x0 , essentially because their expansions do not contain any graph with a propagator of scale +1 (this propagator has a discontinuity at x0 = 0). The function Ω3,c L,β (x) is not differentiable at x0 = 0, but it is in any case continuous, since all graphs contributing to it have a Fourier transform decaying at least as k0−2 as k0 → ∞.

6.2. We want now to prove the bounds in item (b) of Theorem 1.5. To start with, ¯ 1,L,β (x) defined in (6.14) and note that it can be written we consider the function Ω in the form ! 0 (1) 2 X Z h ¯ (h) (x) , ¯ 1,L,β (x) = Ω (6.20) Ω 1,L,β Z h ∗ h=h

¯ (h) (x) Ω 1,L,β

with that is

¯ (h) (x), see (5.47), satisfying a bound similar to that proved for G 1,L,β (h)

¯ |Dm0 ,m1 Ω 1,L,β (x)| ≤ CN,m0 ,m1

γ 2h γ h(m0 +m1 ) . 1 + [γ h |d(x)|]N

(6.21)

This claim easily follows from Lemma 2.6, together with (6.5) and (6.6). Hence we can write, by using (5.47), (6.6), (6.8) and (6.21), given any positive integers n0 , n1 and putting n = n0 + n1 , |∂xn00 ∂¯xn1 Ω3,a L,β (x)| ≤ CN,n

0 X h=h∗

≤

γ (2+2η1 +n)h [1 + (γ h |d(x)|)N ]

CN,n HN,2+2η1 +n (|d(x)|) , |d(x)|2+2η1 +n

(6.22)

where η1 = η − η˜1 , HN,α (r) =

0 X h=h∗

(6.23)

(γ h r)α . 1 + (γ h r)N

(6.24)

By using the second of the definitions (2.2), the definition (4.8) and the bounds (4.16) and (5.29), one can see that the constant η1 can be represented as in (1.15). On the other hand, it is easy to see that, if α ≥ 1/2 and N − α ≥ 1, there exists a constant CN,α such that HN,α (r) ≤

CN,α , 1 + (∆r)N −α

∗

∆ = γh .

(6.25)

The definition (2.40), the first of definitions (2.33), the second bound in (2.34) and the bound (4.56) easily imply that ∆ can be represented as in (1.20), with η2 satisfying the second of Eqs. (1.15). By using (6.22) and (6.25), one immediately gets the bound (1.17). A similar procedure allows to get also the bound (1.18), by using (6.10).


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1419

Let us now consider Ω3,c L,β (x). By using (5.43) and (5.46), as well as the remark (h)

(h)

that one gains a factor γ h in the bound of gω,ω0 (x) − g¯ω,ω0 (x) with respect to the (h)

bound of g¯ω,ω0 (x), we get |Ω3,c L,β (x) − sL,β (x)| ≤

HN,2+2η1 +ϑ (|d(x)|) HN,2+ϑ (|d(x)|) CN + , |d(x)|2 |d(x)|ϑ+2η1 |d(x)|ϑ

(6.26)

for some positive ϑ < 1. ∗ The bound of sL,β (x) is slightly different, because of the γ −ϑ(h−h ) in the r.h.s. of (5.48). We get, in addition to a term of the same form as the r.h.s. of (6.26), another term of the form CN ϑ HN,2+2η1 −ϑ (|d(x)|) (∆|d(x)|) + H (|d(x)|) . (6.27) N,2−ϑ |d(x)|2 |d(x)|2η1 The bounds (6.26) and (6.27) immediately imply (1.19), if λ is so small that, for example, 2|η1 | ≤ ϑ/2. 6.3. We want now to prove the statements in item (c) of Theorem 1.5. The existence of the limit as L, β → ∞ of all functions follows from Theorem 5.8. The claim that Ω3,a (x) and Ω3,b (x) are even as functions of x follows from (5.45) and (6.14)–(6.18). Moreover Ω3,a (x) and Ω3,b (x) are the restriction to Z × R of two functions on R2 , that we shall denote by the same symbols, and Ω3,a (x) satisfies the symmetry ¯ 1,L,β (x), as it is easy to check by relation (1.23), since this is true for limL,β→∞ Ω (h) ¯ using (5.65), and for G1 (x), see (5.49). ¯ i (x) = In order to prove (1.21), we suppose that |x| ≥ 1 and we put Ω ¯ i,L,β (x); then we define Ω ˜ i (x), i = 1, 2, as the functions which are oblimL,β→∞ Ω tained by making in the r.h.s. of (6.14) and (6.15), evaluated in the limit L, β → ∞, the substitutions (1)

(2)

(Zh∨h0 )2 → [x2 + (v0∗ x0 )2 ]−η1 , Zh−1 Zh0 −1

(Zh∨h0 )2 → 1. Zh−1 Zh0 −1

(6.28)

Note that the choice of x2 + (v0∗ x0 )2 , instead of x2 + x20 , which is equivalent for what concerns the following arguments, was done only in order to have a function ¯ 1 (x) in the exchange of (x, x0 ) ˜ 1 (x) satisfying the same symmetry relation as Ω Ω ∗ ∗ with (v0 x0 , x/v0 ). It is easy to see that ˜ 1 (x)| ¯ 1 (x) − Ω |Ω ≤

CN |x|2+2η1

X h∗ ≤h,h0 ≤0

0

γ h |x| γ h |x| h N 1 + (γ |x|) 1 + (γ h0 |x|)N

η1 x2 + x20 h η h0 η h∨h0 −2˜ η1 ch ch0 · 2 (γ |x|) (γ |x|) (γ |x|) − 1 . (1) 2 x + (v0∗ x0 )2 (c 0 ) h∨h

(6.29)


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00105


Note that, if r > 0 and α ∈ R |rα − 1| ≤ |α log r|(rα + r−α ) ;

(6.30)

Hence, by using (6.6), (6.8), (6.25) and (1.15), we get |J3 | CN . 2+2η 1 |x| 1 + (∆|x|)N

(6.31)

|J3 | CN . |x|2 1 + (∆|x|)N

(6.32)

2 1 gL (x/v0∗ , x0 )gL (−x/v0∗ , x0 ) , ∗ 2 η 1 + (v0 x0 ) ] (v0∗ )2

(6.33)

˜ 1 (x)| ≤ ¯ 1 (x) − Ω |Ω In the same way, one can show that

˜ 2 (x)| ≤ ¯ 2 (x) − Ω |Ω Let us now define Ω∗1 (x) = Ω∗2 (x) =

[x2

X 1 gL (ωx/v0∗ , x0 )gL (ωx/v0∗ , x0 ) , (v0∗ )2 ω=±1

where 1 gL (x) = (2π)2

Z dkeikx

χ0 (k) , −ik0 + k

(6.34)

(6.35)

χ0 (k) being a smooth function of k, which is equal to 1, if |k| ≤ t0 , and equal to 0, if |k| ≥ γt0 (see Sec. 2.3) for the definition of t0 ). ˜ i (x) by making in the It is easy to check that Ω∗i (x), i = 1, 2, is obtained from Ω (h) L, β = ∞ expression of the propagators g¯ω,ω0 (x), which are evaluated from (2.92), if h∗ < h ≤ 0, and (2.121), if h = h∗ , the following substitutions: σh−1 (k0 ) → 0 ,

f˜h (k0 ) → fh (k0 ) .

(6.36)

Hence, by using also the remark that, by (2.116) and (4.54), |σh /γ h | ≤ Cγ −(h−h it is easy to show that

∗

)/2

,

˜ 1 (x)| |Ω∗1 (x) − Ω ≤

CN HN,1 (∆|x|)[λ21 HN,1 (∆|x|) + (∆|x|)1/2 HN,1/2 (∆|x|)] . |x|2+2η1

(6.37)

In a similar way, one can show also that ˜ 2 (x)| |Ω∗2 (x) − Ω ≤

CN HN,1 (∆|x|)[λ21 HN,1 (∆|x|) + (∆|x|)1/2 HN,1/2 (∆|x|)] . |x|2

(6.38)

Moreover, by an explicit calculation, one finds that, if |x| ≥ 1, gL (x) =

x0 − ix F (x) , 2π|x|2

(6.39)


00105


1421

where F (x) is a smooth function of x, satisfying the bound |F (x) − 1| ≤

CN . 1 + |x|N

(6.40)

¯ 1 (x)| The bounds (6.31) and (6.32), the similar bounds satisfied by |Ω3,a (x) − Ω 3,b ¯ and |Ω (x) − Ω2 (x)| and Eqs. (6.37)–(6.40) allow to prove very easily (1.21) and (1.22). 6.4. We still have to prove the statements in items (d) and (e) of Theorem 1.5. By using (1.14), (6.18) and (6.19), we see that X 1 3 3,a ˆ ˆ Ω (k + 2σpF , k0 ) + sˆ1,σ (k + 2σpF , k0 ) Ω (k) = 2 σ=±1 c ˆ 3,b (k) + sˆ2 (k) + δΩ +Ω where we used the definitions ! 0 (1) 2 X Zh (h) s1,σ (x) , s1,σ (x) = Z h ∗ h=h

3,c

(k) ,

s2 (x) =

0 X h=h∗

δΩ3,c (x) = Ω3,c (x) − s(x) .

(6.41)

(2)

Zh Zh

!2 s2 (h)(x) ,

(6.42) (6.43)

Since any graph contributing to the expansion of Ω3,a (x − y) has only two ˆ 3,a (k) has support on a set of value propagators of scale ≤ 0 connected to x or y, Ω ˆ 3,a (k) by thinking Ω3,a (x) of k such that |k| ≤ 2γt0 < π; hence we can calculate Ω as a function on R2 . Let us suppose that |k| > 0 and |k| ≥ |k|/2; then Z Z ˆ 3,a (k) = dxeikx Ω3,a (x) = i dx[eikx − 1]∂x Ω3,a (x) , (6.44) Ω k since Ω3,a (x), by (1.17), is a smooth function of fast decrease as |x| → ∞. If |k| < |k|/2, it has to be true that |k0 | ≥ |k|/2 and we write a similar identity, with k0 in place of k and ∂x0 in place of ∂x . In both case we can write, by using (1.17), Z Z |x| dx ˆ 3,a (k)| ≤ C + C dx . (6.45) |Ω 3+2η1 |k| |x|≥|k|−1 1 + |x|3+2η1 1 + |x| −1 |x|≤|k| A even better bound can be proved for |ˆ s1,σ (k)|, σ = ±1, by using (5.48). Hence, ˆ 3,a (k)| + |ˆ s1,σ (k)| ≤ C|k|−1 for |k| ≥ 1 and uniformly for u → 0, |Ω 1 − |k|2η1 1 ˆ 3,a |Ω (k)| + |ˆ s1,σ (k)| ≤ C 1 + , 0 < |k| ≤ 1 . (6.46) 2 2η1 This bound is divergent for |k| → 0, if η1 < 0, that is if J3 < 0; however, if u 6= 0 and |k| ≤ ∆, we easily get from (1.17) (with n = 0) the better bound 1 − ∆2η1 1 ˆ 3,a |Ω (k)| + |ˆ s1,σ (k)| ≤ C 1 + . (6.47) 2 2η1


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00105


In a similar way, by using (1.18), one can prove that ˆ 3,b (k)| + |ˆ s2 (k)| ≤ C[1 + log|k|−1 ] , |Ω

0 < |k| ≤ 1 ,

ˆ 3,b (k)| + |ˆ s2 (k)| ≤ C[1 + log ∆−1 ] . |Ω

(6.48) (6.49)

However, a more careful analysis of the Fourier transform of the leading contribution to Ω3,b (x), given by Ω∗2 (x) (see (6.34)), which takes into account the oddness in the exchange (x, x0 ) → (x0 v0∗ , x/v0∗ ) , ˆ ∗2 (k)| ≤ C. One can show that a similar bound is satisfied by the shows that |Ω ˜ 2 (x) and proportional to σh /γ h . Fourier transform of the terms contributing to Ω Therefore, in the bounds (6.48) and (6.49), we can multiply by J3 both log|k|−1 and log ∆−1 . ˆ 3,c (k). By using (6.26), we see immediately that, Let us now consider δΩ uniformly in k and u, c |δΩ

3,c

(k) − sˆ(k)| ≤ C .

(6.50)

The bounds (6.46)–(6.50), together with the positivity of the leading term in (1.21) and the remark after (6.49), immediately imply all the claims in item (d) of Theorem 1.5. Let us now consider G(x) ≡ Ω3 (x, 0), x ∈ Z. It is easy to see, by using the previous results and the fact that also s1,σ (x) and s2 (x) are even functions of x, that G(x) can be written in the form X e2iσpF x G1,σ (x) + G2 (x) + δG(x) , (6.51) G(x) = σ=±1

where G1,σ (x) and G2 (x) are the restrictions to Z of some even smooth functions on R, satisfying, for any integers n, N ≥ 0, the bounds |∂xn G1,σ (x)| ≤ |∂xn G2 (x)| ≤

[1 + 1+

Cn,N 2+n+2η 1 ][1 |x|

Cn,N 2+n |x| [1 +

+ (∆|x|)N ]

(∆|x|)N ]

,

,

(6.52) (6.53)

while δG(x) satisfies the bound |δG(x)| ≤

C , [1 + |x|2+ϑ ][1 + (∆|x|)N ]

(6.54)

with some ϑ > 0. These properties immediately imply that, uniformly in k and u, ˆ ˆ ≤C. |G(k)| + |∂k δ G(k)|

(6.55)


00105


ˆ 1,σ (k) and note that, if |k| > 0, Let us now consider ∂k G Z 1 ˆ dx[eikx − 1]∂x [xG1,σ (x)] ∂k G1,σ (k) = − k Z 1 dx[eikx − 1]∂x [xG1,σ (x)] =− k |x|≥|k|−1 Z 1 dx[eikx − 1 − ikx]∂x [xG1,σ (x)] , − k |x|≤|k|−1

1423

(6.56)

where we used the fact that ∂x [xG1,σ (x)] is an even function of x, since G1,σ (x) ˆ 1,σ (k)| ≤ C|k|−1 , while, if 0 < |k| ≤ 1, is even, see (5.45). Hence, if |k| ≥ 1, |∂k G uniformly in u, ˆ 1,σ (k)| ≤ C[1 + |k|2η1 ] . |∂k G

(6.57)

In a similar way, we can prove that, uniformly in k and u, ˆ 2 (k)| ≤ C . |∂k G

(6.58)

The bound (6.57) is divergent for k → 0, if J3 < 0; however, if |u| > 0 and |k| ≤ ∆, one can get a better bound, by using the identity Z ˆ 1,σ (k) = i dxeikx [xG1,σ (x)] ∂k G |x|≥∆−1

Z

+i |x|≤∆−1

dx[eikx − 1][xG1,σ (x)] ,

(6.59)

together with (6.52). One finds ˆ 1,σ (k)| ≤ C[1 + ∆2η1 ] . |∂k G

(6.60)

The bounds (6.55), (6.58) and (6.60), together with the identity (6.51), imply ˆ at u = 0 and k = 0, ±2pF (1.24). The statements about the discontinuities of ∂k G(k) follow from an explicit calculation involving the leading contribution, obtained by putting A1 (x) = A2 (x) = 0 in (1.21). 7. Proof of the Approximate Ward Identity (5.35) (L)

7.1. In this section we prove the relation (5.35) between the quantities Zh and (2,L) Zh , related to the approximate Luttinger model defined by (5.30) and (5.31). First of all, we move from the interaction to the free measure (4.30) the term (L) proportional to δ0 and we redefine correspondingly the interaction. This can be realized by slightly changing the free measure normalization (which has no effect on (L) the problem we are studying), by putting δ0 = 0 in (4.31) and by substituting, in (L) (4.30), v0∗ with v¯0 (k0 ) = v0∗ + δ0 C0−1 (k0 ). However, since C0−1 (k0 ) = 1 on all scales (L) (2,L) (L) and Zh may be modified only by a factor γ C|λ0 | , if we substitute h < 0, Zh


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00105


v¯0 (k0 ) with v¯0 ≡ v¯0 (0). It follows that it is sufficient to prove the bound (5.35) for the corresponding free measure ( (≤0)− Y Y dψˆk(≤0)+ X dψˆk0 ,ω 0 ,ω 1 X (L) (≤0) · exp − )= P (dψ 0 NL (k ) Lβ ω=±1 0 −1 0 −1 0 0 ω=±1 k :C0 (k )>0

k :C0 (k )>0

)

(≤0)+ (≤0)− × C0 (k0 )(−ik0 + ω¯ v0 k 0 )ψˆk0 ,ω ψˆk0 ,ω ,

(7.1)

by using as interaction the function Z (L) (≤0)+ (≤0)− (≤0)+ (≤0)− (L) (≤0) ) = λ0 dxψx,+1 ψx,−1 ψx,−1 ψx,+1 . V (ψ

(7.2)

TL,β

Let us consider, instead of the free measure (7.1), the corresponding measure with infrared cutoff on scale h, h ≤ 0, given by ( [h,0]− Y dψˆk[h,0]+ X Y dψˆk0 ,ω 0 ,ω 1 X (L,h) [h,0] · exp − (dψ )= P 0 NL (k ) Lβ ω=±1 0 −1 0 −1 0 0 ω=±1 k :Ch,0 (k )>0

k :Ch,0 (k )>0

) 0

0

[h,0]+ [h,0]− × Ch,0 (k )(−ik0 + ω¯ v0 k )ψˆk0 ,ω ψˆk0 ,ω

,

(7.3)

P0 −1 = k=h fk . where Ch,0 We will find convenient to write the above integration in terms of the space-time field variables; if we put Y

Dψ [h,0] =

[h,0]− Y dψˆk[h,0]+ dψˆk0 ,ω 0 ,ω

−1 k0 :Ch,0 (k0 )>0

we can rewrite (7.3) as P

(L,h)

(dψ

[h,0]

"

) = Dψ

[h,0]

exp −

ω=±1

where [h,0]σ = Dω[h,0] ψx,ω

[h,0]

Dω

1 Lβ

X

0

,

(7.4)

#

XZ ω

NL (k0 )

TL,β

[h,0]+ [h,0] [h,0]− dxψx,ω Dω ψx,ω

,

eiσk x Ch,0 (k0 )(iσk0 − ωσ¯ v0 k 0 )ψˆk0 ,ω . [h,0]σ

(7.5)

(7.6)

−1 k0 :Ch,0 (k0 )>0

has to be thought as a “regularization” of the linear differential operator

∂ ∂ . (7.7) + iω¯ v0 ∂x0 ∂x Let us now introduce the external field variables φσx,ω , x ∈ TL,β , ω = ±1, Dω =

[h,0]σ

antiperiodic in x0 and x and anticommuting with themselves and ψx,ω , and let us define Z (L) [h,0] (7.8) U (φ) = − log P (L,h) (dψ [h,0] )e−V (ψ +φ) .


00105


1425

If we perform the gauge transformation [h,0]σ [h,0]σ → eiσαx ψx,ω , ψx,ω

and we define (e

−iα

φ)σx,ω

=

e−iσαx φσx,ω ,

Z U (φ) = − log

−

XZ

(7.9)

we get (

P (L,h) (dψ [h,0] ) exp

−V (L) (ψ [h,0] + e−iα φ) )

[h,0]+ iαx [h,0] −iαx (e Dω e dxψx,ω

−

[h,0]− Dω[h,0] )ψx,ω

.

(7.10)

ω

Since U (φ) is independent of α, the functional derivative of the r.h.s. of (7.10) w.r.t. αx is equal to 0 for any x ∈ TL,β . Hence, we find the following identity: Z X ∂U − 1 ∂U (L,h) [h,0] −V (L) (ψ [h,0] +φ) + φ + (dψ )T e P −φ+ = 0, x,ω x,ω x,ω Z(φ) ∂φ+ ∂φ− x,ω x,ω ω (7.11) where

Z Z(φ) =

P (L,h) (dψ [h,0] )e−V

(L)

(ψ [h,0] +φ)

,

(7.12)

[h,0]+ [h,0]− [h,0]+ [h,0]− [Dω[h,0] ψx,ω ] + [Dω[h,0] ψx,ω ]ψx,ω Tx,ω = ψx,ω

=

1 X −ipx ˆ[h,0],+ e ψk,ω (Lβ)2 p,k

[h,0],− · [Ch,0 (p + k)Dω (p + k) − Ch,0 (k)Dω (k)]ψˆp+k,ω ,

v0 k . Dω (k) = −ik0 + ω¯

(7.13) (7.14)

Moreover, the sum over p and k in (7.13) is restricted to the momenta of the form p = (2πn/L, 2πm/β) and k = (2π(n + 1/2)/L, 2π(m + 1/2)/β), with n and m integers, such that |p|, |p0 |, |k0 |, |k| are all smaller or equal to π and satisfy the −1 −1 (p + k) > 0, Ch,0 (k) > 0. constraints Ch,0 Note that (7.13) can be rewritten as [h,0]+ [h,0]− ψx,ω ] + δTx,ω , Tx,ω = Dω [ψx,ω

where δTx,ω =

(7.15)

1 X −ipx ˆ[h,0],+ e ψk,ω (Lβ)2 p,k

[h,0],−

· {[Ch,0 (p + k) − 1]Dω (p + k) − [Ch,0 (k) − 1]Dω (k)}ψˆp+k,ω .

(7.16)

It follows that, if Ch,0 is substituted with 1, that is if we consider the formal [h,0]+ [h,0]− theory without any ultraviolet and infrared cutoff, Tx,ω = Dω [ψx,ω ψx,ω ] and we would get the usual Ward identities. As we shall see, the presence of the cutoffs make the analysis a bit more involved and adds some corrections to the Ward


1426

00105


identities, which however, for λ0 small enough, can be controlled by the same type of multiscale analysis, that we used in Sec. 5. 7.2. Let us introduce a new external field Jx , x ∈ TL,β , periodic in x0 and x and commuting with the fields φσ and ψ [h,0]σ , and let us consider the functional Z R P (L) [h,0] [h,0]+ [h,0]− (7.17) W(φ, J) = −log P (L,h) (dψ [h,0] )e−V (ψ +φ)+ dxJx ω ψx,ω ψx,ω . We also define the functions ∂2 ∂2 U (φ)| = Σh,ω (x − y) = φ=0 − − W(φ, J)|φ=J=0 , ∂φ+ ∂φ+ x,ω ∂φy,ω x,ω ∂φy,ω Γh,ω (x; y, z) =

∂ ∂2 W(φ, J)|φ=J=0 . + ∂Jx ∂φy,ω ∂φ− z,ω

(7.18) (7.19)

These functions have here the role of the self-energy and the vertex part in the usual treatment of the Ward identities. However, they do not coincide with them, because the corresponding Feynman graphs expansions are not restricted to the one particle irreducible graphs. However, their Fourier transforms at zero external momenta, which are the interesting quantities in the limit L, β → ∞, are the same; in fact, because of the support properties of the fermion fields, the propagators vanish at zero momentum, hence the one particle reducible graphs give no contribution at that quantities. In the language of this paper, if we did not perform any free measure regularization, Σh,ω (x − y) would coincide with the kernel of the contribution to the effective potential on scale h − 1 with two external fields, that is the function (h−1) W2,(+,−),(ω,ω) of Eq. (3.3). Analogously, 1 + Γh,ω (x; y, z) would coincide with the (h−1)

kernel B1,2,(+,−),(ω,ω) of Eq. (5.6). Note that Γh,ω (x; y, z) =

X

Γh,ω,˜ω (x; y, z) ,

(7.20)

ω ˜

P [h,0]+ [h,0]− where Γh,ω,˜ω (x; y, z) is defined as in (7.17), by substituting Jx ω ψx,ω ψx,ω [h,0]+ [h,0]− with Jx ψx,˜ω ψx,˜ω . − If we derive the l.h.s. of (7.11) with respect to φ+ y,ω and to φz,ω and we put φ = 0, we get 0 = −δ(x − y)Σh,ω (x − z) + δ(x − z)Σh,ω (y − x) +T # *" X ∂V ∂V ∂2V [h,0]+ [h,0]− − [h,0]+ [Dω˜ (ψx,˜ω ψx,˜ω )+δTx,˜ω ] , ; − [h,0]+ [h,0]− [h,0]− ∂ψy,ω ∂ψz,ω ∂ψy,ω ∂ψz,ω ω ˜ (7.21) where h·; ·iT denotes the truncated expectation w.r.t. the measure Z(0)−1 P (L,h) (L) [h,0] (dψ [h,0] )e−V (ψ ) .


00105


1427

By using the definitions (7.18) and (7.19), Eq. (7.21) can be rewritten as 0 = −δ(x − y)Σh,ω (x − z) + δ(x − z)Σh,ω (y − x) X Dx,˜ω Γh,ω,˜ω (x; y, z) − ∆h,ω (x; y, z) , −

(7.22)

ω ˜

where

* ∆h,ω (x; y, z) =

+T X ∂2V ∂V ∂V δTx,˜ω . ; + − − + − ∂ψy,ω ∂ψz,ω ∂ψy,ω ∂ψz,ω ω ˜

(7.23)

In terms of the Fourier transforms, defined so that, in agreement with (3.2) and (5.9), 1 X −ik(x−y) ˆ Σh,ω (k) , e (7.24) Σh,ω (x − y) = Lβ k

Γh,ω,˜ω (x; y, z) =

1 X ip(x−z) −ik(y−z) ˆ Γh,ω,˜ω (p, k) , e e (Lβ)2

(7.25)

1 X ip(x−z) −ik(y−z) ˆ ∆h,ω (p, k) , e e (Lβ)2

(7.26)

p,k

∆h,ω (x; y, z) =

p,k

(7.22) can be written as ˆ h,ω (k) − ˆ h,ω (k − p) − Σ 0=Σ

X

ˆ h,ω,˜ω (p, k) + ∆ ˆ h,ω (p, k) . (−ip0 + ω ˜ v¯0 p)Γ

(7.27)

ω ˜

Let us now define 1 (2) Z˜h = 1 − 4 i Z˜h = 1 + 4

X

¯ η,η0 ) , ˆ h,ω (¯ Γ pη0 , k

(7.28)

βˆ ¯ η0 Σ h,ω (kη,η 0 ) , π

(7.29)

η,η 0 =±1

X η,η 0 =±1

¯ η,η0 as in (2.73). ¯ η0 is defined as in (5.11) and k where p ¯ η,η0 , multiply both sides by (iη 0 β)/(2π) ¯ η0 and k = k If we put in (7.27) p = p 0 and sum over η, η , we get (2) (2) Z˜h = Z˜h + δ Z˜h ,

(7.30)

where (2)

δ Z˜h =

1 4

X η,η 0 =±1

¯ η,η0 ) ˆ h,ω (¯ ∆ pη0 , k . −i¯ pη 0 0

(7.31)

(2) 7.3. The considerations preceding (7.21) suggest that Z˜h and Z˜h are “almost (L) (2,L) equal” to the quantities Zh and Zh , related to the full approximate Luttinger (2) model and defined analogously to Zh and Zh for the original model, on the base


1428

00105


of a multiscale analysis. In order to clarify this point, we consider the measure (L) [h,0] (

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