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ANNALES DE L’ INSTITUT FOURIER

CHRISTIAN DUVALPIERRELECOMTEVALENTIN OVSIENKOConformally equivariant quantization:existence and uniqueness

Annales de l’institut Fourier, tome 49, no 6 (1999), p. 1999-2029.

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Ann. Inst. Fourier, Grenoble49, 6 (1999), 1999-2029

CONFORMALLY EQUIVARIANT QUANTIZATION:EXISTENCE AND UNIQUENESS

by C. DUVAL, P. LECOMTE, V. OVSIENKO

In Memory of Moshe Flato and Andre Lichnerowicz

1. Introduction.

The general problem of quantization is often understood as the questfor a correspondence between the smooth functions of a given symplecticmanifold, i.e., the classical observables, and (symmetric) operators on acertain associated Hilbert space, which are called the quantum observables.This correspondence must satisfy a number of additional properties thatheavily depend upon the standpoint of the authors and are, by no means,universal.

One guiding principle for the search of a quantization procedure isto impose further coherence with some natural symmetry of phase space.This constitutes the foundations of the "orbit method" [14], geometricquantization [17], [26], [15] in the presence of symmetries, Moyal-Weylquantization (see, e.g., [10]) defined by requiring invariance with respectto the linear symplectic group Sp(2n,IR) of R271.

1.1. Equivariant quantization problem.

In all previous examples, the symmetry group was a Lie subgroup ofthe group of all symplectomorphisms of the symplectic manifold. Here, we

Keywords: Quantization - Conformal structures - Modules of differential operators -Tensor densities — Casimir operators — Star-products.Math. classification: 58B99 - 17B56 - 17B65 - 47E05.

2000 C. DUVAL, P. LECOMTE, V. OVSIENKO

will confine considerations to the case of cotangent bundles, T*M, withtheir canonical (vertical) polarization. This polarization, together with theLiouville 1-form on T*M, should be preserved by the symmetry groupwhich naturally arises as the cotangent lift of a Lie group, G, acting on M.

We will thus look for an identification, as G-modules, between thespace, «?(M), of smooth functions on T*M that are polynomial on thefibers and the the space, T^M), of linear differential operators on M.

Note that the Moyal-Weyl quantization does not fit into this generalframework in which the symmetries of configuration space, M, play acentral role. Indeed, the action of Sp(2n,]R) on T*W1 does not descendto W.

Now, G-equivariance between G / H such that ^pa o(Pf^l ls given by an element ofG. This new approach significantly differs from the more usual one whichmakes use of connections to intrinsically define quantization proceduresand symbol calculus.

The G-structure we will consider in this article is the conformalstructure with G ==• S0(p +1,9+1) modeled on the pseudo-Riemannianmanifold 5^ x 5'9.

Our purpose is to show that there exists, actually, a canonicalisomorphism of S0(p + l,g + l)-modules between the space of symbols,5(M), and the space of differential operators, P(M).

Experience of other approaches to the quantization problem andof the geometrical study of differential equations prompts us to ratherconsider the space, P^^(M), of differential operators with arguments andvalues in the space of tensor densities of weights A and p, respectively. So,we will naturally need to study this space of differential operators as aS0(p + 1, q + l)-module. As a consequence, the S0(p + 1, q + l)-module ofsymbols will be twisted by the weight 6 = IJL — A, and denoted by Ss(M).

CONFORMALLY EQUIVARIANT QUANTIZATION 2001

Let us emphasize that equivariant quantization of G-structures hasalready been carried out in the case of projective structures, i.e. SL(n +1, Restructures, in the recent papers [22], [20]. As for conformal structures,a first step towards their equivariant quantization was taken in [9] in thecase of second order operators.

In the particular case n = 1, both conformal and projective structurescoincide. We refer to [6] for a thorough study of sl (2, R)-equivariantquantization and of the corresponding invariant star-product. See also [28]for a classic monography on the structures of sl(2, ]R)-module on the spaceof differential operators on the real line.

There exist various approaches to the quantization problem, however,our viewpoint put emphasis on the equivariance condition with respectto a (maximal) group, (7, of symmetry in the context of deformationquantization. Now, G-equivariance is the root of geometric quantization[26], [17], [15], Berezin quantization [2], [3], etc., but it seems to constitutea fairly new approach in the framework of symbol calculus, deformationtheory and semi-classical approximations dealt with in this work.

1.2. Quantizing equivariantly conformal structures.

We outline here the main results we have obtained, and describe thegeneral framework adopted in this article to answer the question raised inthe preceding section.

We review in Section 2 the structures of the spaces of symbolsSs(M) and of differential operators T>\^(M) as Vect(M)-modules. TheLie algebra, o(p + 1, q + 1), of conformal Killing vectors of a conformallyflat manifold (M^g) is then described. The restriction of the precedingVect(M)-act ions is also explicitly calculated.

Section 3 presents the main theorems which establish the existenceand the uniqueness of a o(p + 1,9 + 1)-equivariant quantization in thespecial and fundamental case A = /x. It turns out that the value A = ^ = ^guarantees that our quantization actually defines a star-product on T*M.This is precisely the value of the weights used in geometric quantization. Inthe general case, we again obtain a canonical isomorphism of o(p+1,^+1)-modules, except for an infinite series of values of 6 = fJ, — A, which we callresonances.

Section 4 is devoted to the algebra of invariants. One considers theaction of o(p + 1,9 + 1) and the Euclidean subalgebra e(p, q) on the space


of polynomials on r*R77'. Resorting to WeyPs theory of invariants, wecharacterize the commutant of o(p4-1,9+1) within the algebra of operatorson the latter space of polynomials: it is a commutative associative algebrawith two generators. The commutant ofe(p, q) has already been determinedin [9]. These algebras of conformal and Euclidean invariants play a crucialrole in our work and enable us to compute the Casimir operators C


2. Basic definitions and tools.

2.1. Differential operators on tensor densities.

Let us recall that a tensor density of degree A on manifold M is asmooth section of the line bundle ^\(M) == \AnT*M\(s)x over M. Thespace of tensor densities of degree A is naturally a DifF(M)- and Vect(M)-module. In this paper, we will consider the space J:'\(M) (or T\ in short) ofcomplex-valued smooth tensor densities, i.e., of the sections of A,\(M) (g)C.

The space "D\^ of linear differential operators

(2.1) A:^^^.

from A-densities to /^-densities on M is naturally a Diff(M)- and Vect(M)-module. These modules have been studied and classified in [8], [21], [22],[12], [II], [23], [9], [19].

There is a filtration P^ C V\^ C " • C P$^ C • • •, where themodule of zero-order operators P^ — T^-\ consists of multiplicationby (/^ — A)-densities. The higher-order modules are defined by induction:A e P^ if [A, /] G P^1 for every / C C°°(M).

2.2. Symbols with values in tensor densities.

Consider the space S = r(S(TM)) of contravariant symmetric tensorfields on M which is naturally a Diff(M)- and Vect(M)-module. We willdefine the space of ^-weighted symbols on T*M as the space of sections

(2.2) ^=r(^(rM)(g)A^(M)).

The space Ss is also, naturally, a DifF(M)- and Vect(M)-module.

Again, there is a filtration S^ C Sj C • • • C S^ C • • •, where


2.3. V\^ and Ss as Vect(M)-modules.

We will always assume M orientable and identify F\ to C°° (M) 0 Cby the choice of a volume form, Vol. on M.

It is clear from the definition ofV\^ that the corresponding Vect (Ab-action, Cx^^ is given by

(2.4) C^{A)=L^A-AL^

where X € Vect(M), and L^ is the standard Lie derivative of A-densitiesF\. Now, any A-density being represented by /[Vol^ for some / CC°°{M) 0 C, the Lie derivative L^ is thus given by

(2.5) L^( / )=X( / )+ADiv(X) /

where Div(X) = Lx(Vol)/Vol.

As to the Vect(M)-action, L6, on Ss, it reads

(2.6) L^(P) = Lx(P) + 6Div(X) P

where Lx denotes here the Lie derivative of contravariant tensors given bythe cotangent lift of X


As vector spaces, T>\^ and


a homogeneous space and thus admits a local action of S0(p +1,^+1) .The associated, locally denned, action of the Lie algebra o(p + l,q + 1)corresponds (if n = p + q ̂ 3) to that of the subalgebra of the vector fieldsX solutions of

Lxg = fgfor some / e C°°(M) depending upon X.

It is well known that a conformally flat manifold admits an atlas inwhich o(p + 1, q + 1) is generated by

X - 9x' ~ 9^_ 9 9

i j ~ ^9x3 x3^(2.14)xo = xiôx1xt = ̂ ^-^^

where i,j = 1,... ,n and Xi = g^x3. In the sequel, indices will be raisedand lowered by means of the (flat) metric g .

Let us introduce the following nested Lie subalgebras that will beconsidered below, namely

(2.15) o(p, q) C e(p, q) C ce(p, q) C o(p + 1, q + 1)

where o(p,q) is generated by the X^, the Euclidean subalgebra e(p,q) byXij and Xi and the Lie algebra ce(p, q) = e(p, q) x R by X^-, X, and XQ.

Remark 2.2. — It is worth noticing that the conformal Lie algebrao(p + l,g + 1) is maximal in the Lie algebra Vectpo^M71) of polynomialvector fields in the following sense: any bigger subalgebra of Vectpol^)necessarily coincides with Vectpol^). See [5] for a simple proof. Theuniqueness and the canonical character of our quantization proceduredefinitely originates from this maximality-property ofo(p+l, ̂ +1). See also[25] for a classification of a class of maximal Lie subalgebras ofVectpol^).

Remark 2.3. — From now on, we will use local coordinate systemsadapted to the flat conformal structure on M in which the generators ofo(p+1,94-1) retain the form (2.14). This flat conformal structure preciselycorresponds to a S0{p+1, q+ ̂ -structure on M (cf. Introduction) definedby the atlas of these adapted coordinate systems. Clearly, our formulae will


prove to be independent of the particular choice of an adapted coordinatesystem and to be globally denned.

2.6. Explicit formulae for the o(p + l,q-\- l)-actions.

As a first application of the preceding results, let us compute theaction of the conformal algebra on Sg and on V^ which is given by thefollowing two propositions.

PROPOSITION 2.4. — The action o f o ( p + l , ^ + l ) on Ss reads

^ = 9iL6^ = x,9j - XjQ, + ̂ - ̂ i

(C2A^ L^=^-^+n


3.1. Quantization in the case 6 = 0.

Let us consider first the special case 6 = ^ — X = 0 and use theshorthand notation \ = PA,A- Although this is definitely notthe most general case to start with, this zero value of the shift is of centralimportance to relate our conformally equivariant quantization to the moretraditional procedures such as geometric or deformation quantization.

THEOREM 3.1.— (i) There exists an isomorphism of o(p + 1,9 + 1)-modules

(3.1) QA:5-^A.

(ii) This isomorphism is unique provided the principal symbol bepreserved at each order, i.e., provided it reads Q\ = Id + A/A with nilpotentpart A/A :^-^5A-1.

Let us introduce a new operator on symbols that will eventuallyinsure the symmetry of the corresponding differential operators. DefineTn :


Recall that an associative operation */^ : S 0 • S[[ih}] is called astar-product [I], [7], [10] if it is of the form

ifi(3.6) P *, Q = PQ + ^ {P, Q} + 0(^2)

where { • , • } stands for the Poisson bracket on T*M, and is given by bi-differential operators at each order in h.

THEOREM 3.4. — The associative, conformally invariant, operation*^ denned by (3.5) is a star-product if and only if X = -

Let us emphasize that this theorem provides us precisely with thevalue of A used in geometric quantization and, in some sense, links thelatter to deformation quantization.

3.2. General formulation. Resonant values of 6.

In this section we formulate our result about the isomorphism of theo(p 4-1, q + l)-modules Ss and T>\^ in the general situation.

The discussion below mainly relies on the structure of the spectrumof the Casimir operator C\^ of (the o(p + 1, q -h l)-module) T)\^. Indeed,the Casimir operator C


If 6 (E S, the Casimir operator C\^ has "multiple" eigenvalues and,in some cases, is even not diagonalizable. The corresponding critical valuesof 6 are difficult to determine; however, they belong to

(3.10) So = {6k^t eS Q < ^ s - t ^ k - £ } .

THEOREM 3.6 (Resonant case).— I f n = p + q ^ ( 2 and 6 e S\So,then there exists an isomorphism (3.9) of o(p + 1, q + l)-modules which isunique provided the principal symbol be preserved at each order.

The proofs of Theorems 3.5 and 3.6 both consist mainly in showingthat C\^ is diagonalizable. In the case of Theorem 3.5, this is quiteimmediate to prove, whereas the resonant case is much more involved.

Remark 3.7. — The isomorphisms of o(p + 1, q + l)-modules obtainedin the above results are in fact decribed in terms of local operators: theyare expressed by differential formulae in the coordinates ( x ' ) and (\[ih] defined by

(3.11) Qx^=Qx^oIn

as a natural generalization of (3.3).

Let us recall that if A + ^ = 1, there exists, for compactly-supporteddensities, a Vect(M)-invariant pairing T\ (g) J^ -^ C defined by

(3.12) (^0^^ / Tp^.J M


We can then formulate the important

COROLLARY 3.11.— Assume 6 ^ Eo and A+/x = 1. The quantization

(3.13) P=Q^^,^P)

of any symbol P € Sg is a symmetric (formally self-adjoint) operator.

Proof. — Let us denote by A* the adjoint of A e V\^-\ with respectto the pairing (3.12). Consider the symmetric operator

Symm(Q^;,(P)) = J (QA,^(P) + (QA,^(P))*)

which exists whenever A+/^ = 1. Notice that it has the same principal sym-bol as Q\^n(P). Now, the map Symm(


Remark 3.12. — So far, we only needed a conformal class of metricsto define a conformally equivariant quantization map. But, in the currentconstruction, we definitely make a particular choice of metric, ^, in thelatter class to express the operator P.

In the case 6 = 0, which is relevant for quantum mechanics, theoperator P admits a prolongation as a (formally) self-adjoint operatoron the Hilbert space F^_ (the completion of the space of compactlysupported half-densities with Hermitian inner product (3.12)). It will betherefore legitimate to call P the quantum Hamiltonian associated withthe Hamiltonian P 6 S ^ Pol(T*R71). This quantum Hamiltonian is thena (formally) self-adjoint operator on the space L^M, |Vol^|).

4. Conformally invariant operators.

The space C[x1,..., a171, ̂ i , . . . , ^n} °f polynomials on T*W1 is nat-urally a module over the Lie algebra, Vectpoi(M77'), of polynomial vectorfields on R71. This module structure is induced by the Vec^R71) action onT*R71. But, we will rather consider, as in Section 2.4, the deformed action(2.12) depending on a parameter 6-, we will henceforth denote this modulebyC^1,...,^!,...,^.

DEFINITION 4.1. — We denote byEnddiff(C[;r1,... .a;71,^,... ,^]) thesubspace

C\x1 x-f 6 -Q- -°- -9- -9-!L ' ' " ' ^i-'-^^i,..., ̂ , Q ^ ' " ' 9^\

of polynomial differential operators on C[.r1,..., re71, ̂ i,..., ̂ ].

DEFINITION 4.2.— To any Lie algebra Q C Vectpo^M71) we associateits commutant, Q-, as the Lie subalgebra ofEnddiff(C[:r1,..., x71, ̂ i , . . . , ̂ n])of those operators that commute with Q.

This classical notion of commutant has first been considered in thecontext of differential operators by Kirillov [16].

4.1. Algebra of Euclidean invariants.

To work out a conformally equivariant quantization map, we need tostudy first equivariance with respect to the Euclidean subalgebra e(p,^).To this end, we will introduce the commutant e(p,^)1.


Let us recall the structure of e(p, q)1 which has been shown [9] to bethe associative algebra generated by the operators

( A n p eic ^ € 9 i n rp 9 9(4.1) R=^, E=^^r T =^^

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

..^ ^ ., 9 - 3 9 .,

2014 c. DUVAL, P. LECOMTE, V. OVSIENKO

COROLLARY 4.4. — The commutant ce(p, q)' m Enddiff(C[.r1,...,^n,^ • • • ^n]


more tractable, let us rather deal with the principal symbol of the operatorn

S ^[We,r,d,p,^^.]; it is of the form1=1 'l

2^Ee+ lRS+lDdGg- lA^(4.10) - 2d(E6 R^1 D^-1 Gg A^ - 2 E^2 R^ D^-1 Gg A^)

+ 4^(2 E64-1 R^ D^ G^1 A^-1 - E6 R^4-1 D^1 Gg A^-1).

We then seek the linear combinations of the monomials We^,d,^ of fixeddegree (4.9), for which the previous expression is identically zero. Resortingto Theorem 4.3, we immediately get g = 0 since the first term in (4.10) isclearly independent of the others. The same is true for the next two terms,yielding d = 0 and £ = 0. D

4.3. Casimir operator C


Proof. — The matrix realization of the o(p+l, 9+1 ̂ generators (2.14)is given by

/ 0 -V2ei 0'(4.14) Xi = 0 0 0

\^N 0 0,(e,e\-e^ 0 0\ /O 0 0'

(4.15) X^ - 0 0 0 , Xo = 0 -1 0V 0 0 O/ \0 0 1,/ 0 0 ^2e,\

(4.16) X, = -^le\ O O .\ 0 0 0 /

where (e,) is the canonical basis of R" and {e\ = g(e,)) its dual basisassociated with the metric g.

A simple calculation yields the basis (X° = B^X/)) of o(p +1, q +1)dual to (Xa) with respect to the Killing metric (4.12). One gets

(4.17)

Xi = -Jff^

X^ = g^g^Xki

XQ = -Xo

x1 = -J^-Using the o(p + l,q + l)-action, L6, on Sg given in (2.16), one showsimmediately that the Casimir operator

(4.18) C, = ̂ VL^L^ - (L^)2 - \^L^ - J


PROPOSITION 4.8. — The Casimir operator of the o(p+l, q-}-l)-actionon "D\^ is of the form

(4.21) Cx^ = Cs + Go - 2(nA + £)D.

Proof. — The explicit formula for C\^ is obviously obtained by replac-ing L6 by C^ in (4.18). Applying then (2.17) and (2.18) to that expressionimmediately leads to the result (4.21).

5. Proofs of the main theorems.

Throughout this section we use, for convenience, the local identifica-tion (2.9) ofV\^ and Ss.

5.1. Diagonalization of the Casimir operator C^.

We have already mentioned that we will study the diagonalizationof the Casimir operators Cs and C\^. Here, we understand that anendomorphism of an infinite-dimensional space is diagonalizable if anyelement of the latter is a (finite) sum of eigenvectors of the former.

Let us recall that [


COROLLARY 5.2. — The spectrum of the Casimir operator C


PROPOSITION 5.4 (Generic case).— If 6 ^ E, then the eigenvalueproblem (5.7) for the Casimir operator C\^ has a solution if and only if7 = 7M for some k, s (as given by (5.4)). The corresponding eigenvectorsare uniquely determined by their principal symbols, arbitrarily taken inS{k,s),6'

Proof. — The highest degree component of the eigenvalue equation(5.7) is just the eigenvalue equation for C


At last, D(Q) is harmonic because [T,D] = 0 and one furthermoreeasily checks that G(Q) € 5(A;-25+i,o),

CONFORMALLY EQUI VARIANT QUANTIZATION 2021

5.3. Proof of Theorems 3.5 and 3.6.

Let us show that the diagonalization of the Casimir operator actuallyleads to the determination of a unique isomorphism ofo(p-t-l, g+l)-modulesQ\^ '' ^6 —> ̂ \^- Looking for a map Q\^ such that

^ ̂ Vx^

(5.12) 5^ Q^


5.3.2. Proof of Theorem 3.6.

This proof is built on the same pattern as the previous one. But,since 6 has resonant values, we must resort to Proposition 5.6 instead ofProposition 5.4. This is done at the expense of some preparation due to thefact that the uniqueness of the eigenvector P^g is guaranteed only within^(k, s } , 6 '

The above proof of the equivariance property (5.14) should now becompleted with the help of

LEMMA 5.10. — For every X 6 o{p + 1, q + 1), one has L^S^^s)^ C^{k,s),6 an(^ ^X^^{k,s),6 C 0.


Proof.—In the expression (3.8), both factors in the first term,k — £ -\-1 — s and k + i — 2(5 + t) + n — 1, are non-negative in view of(5.15) and (5.2). Also (5.15) yields kt - £s ^ -£(s -1) so that the secondterm is bounded from below by {s - t)(k - £ + 1), which is non-negative.We have just shown that 6 ^ 0.

Now, if s = t, one has n 6 = k + £ + n - ( 2 t - l ^ k because of (5.2).The result follows since k > 0 in (3.8). D

5.5. Proof of Theorem 3.4:Conformally invariant star-product.

Throughout this section we will only consider the case X = fi. Let usgive an explicit expression for the quantization map (3.3) up to the secondorder in h.

PROPOSITION 5.13.—If Pk,s ^ ^(k,s) ls a homogeneous polynomial(see (5.6)) with k > 2, then

(i) if s > 0, the quantization map is of the form(5.16)

Q^s) = PM + J (D(P.,) + ^a^_A):^) OO(PM)) ^ 0^

(ii) if s = 0 (i.e., the harmonic case), one has

( ^\ | L, _ 1 \(5.17) Qx-APk,o)=Pk,o+ih .., . D(Pfc,o)) + 0(fi2).U -\- Z^K, — l ) yProof. — (i) An eigenvector of the Casimir operator C\ \, with prin-

cipal symbol P/e^, is of the form

P = Pk,s + Pk-i,s + PA;-I,S-I + terms of degree ^ k - 2.

(See the formula (4.21) and Lemma 5.5.)

The eigenvalue problem (5.7) therefore leads to ^k,sPk,s = 7-Pfc,^where 7^^ is the eigenvalue of the Casimir operator Co given by (5.4), andto

^k-l,sPk-l,s + 7fe-l,5-A-l,5-l =7(^-1,5 + Pfc-l,s-l)

(5.18) + 2(nA + k - l)D(P^) - Go(P^).


In order to solve this equation for P/c-i s o^d Pfc-i s-i^ one needs tointroduce the projectors

T-rk-l,s RO — Qk-l,s-l , TT^-1,5 ^0 — Qk-l,s111 = ———————•——— and 11.2 = ————————:——Qk-l,s — ^/c-l,s-l ^-l,s-l — Qk-l,s

from the space x[[ih]] of the form

Q(P) = p + ih(aD(P) + ^Go(P)) + 0(^2),


the associative product * : S 0 «S —^


6. Quantizing second-order polynomials.

This problem has first been solved in [9]. It was proved that ifn = p + q ^ 2, there exists an isomorphism of o(p + 1,^ + l)-modulesQ2^ : s] ̂ ̂ where 6 = ̂ - A. Provided

f e n ^P n + 2 i n + l n+2}[b'L) b9L\n'~~2r^^~^^'~n~]'This result is clearly consistent with the general Theorem 3.5. Moreover,the latter guarantees the uniqueness of such an isomorphisms under thefurther condition that the principal symbol be preserved at each order.

6.1. Explicit formulae.

In the non-resonant case, the explicit formula for the unique isomor-phism has also been computed in [9]. One has

(6.2) Qi^ = Id + 71 Go + 72D + 73


r*^71. In our framework, the Weyl quantization map, Sweyb retains thevery elegant form

/ ih \Qweyl = e x p l — D l

(6.5)= Id+^D-^+O^3)

2 8where the divergence operator D is as in (4.2). (See, e.g., [10] p. 87.)

6.2. Study of the resonant modules.

For the sake of completeness, let us study in some more detail theparticular modules of differential operators corresponding to the resonances(6.1). It has been shown [9] that, for each resonant value of


Let us put A == IJL = - and apply, using (6.4), the construction ofthe quantum Hamiltonian (3.14) spelled out in Section 3.3. In doing so, werecover a result obtained in [9], namely

n2

^ ^-^^^(n-lXn^)^

where Rg stands for the scalar curvature of (M,g). The operator (6.7) istherefore a natural candidate for the quantum Hamiltonian of the geodesicflow on a pseudo-Riemannian manifold.

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Manuscrit recu Ie 19 fevrier 1999,accepte Ie 22 juin 1999.

C.DUVAL,Universite de la Mediterranee et CPT-CNRSLuminy Case 907F-13228 Marseille cedex [email protected]&P. LECOMTE,Universite de LiegeInstitut de MathematiquesSart TilmanGde Traverse, 12 (B 37)B-4000 [email protected]&V. OVSIENKOCNRS, Centre de physique theoriqueCPT-CNRSLuminy Case 907F-13288 Marseille cedex [email protected]

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