The Bisognano-Wichmann property on nets of standard subspaces, some sufficient conditions
aa r X i v : . [ m a t h - ph ] F e b The Bisognano-Wichmann propertyon nets of standard subspaces,some sufficient conditions
Vincenzo Morinelli ∗ Dipartimento di Matematica, Universit`a di Roma Tor VergataVia della Ricerca Scientifica, 1, I-00133 Roma, Italyemail: [email protected] , Abstract
We discuss the Bisognano-Wichmann property for local Poincar´e covariant netsof standard subspaces. We provide a sufficient algebraic condition on the covariantrepresentation ensuring the Bisognano-Wichmann and the Duality properties withoutfurther assumptions on the net. We call it modularity condition. It holds for directintegrals of scalar massive and massless representations. We present a class of massivemodular covariant nets not satisfying the Bisognano-Wichmann property. Further-more, we give an outlook on the relation between the Bisognano-Wichmann propertyand the Split property in the standard subspace setting.
The Algebraic Quantum Field Theory (AQFT) is a very fruitful approach to study propertiesof quantum fields by using operator algebras. Models of local quantum field theories aredescribed by nets of von Neumann algebras on a fixed spacetime, satisfying basic relativisticand quantum assumptions. A local Poincar´e covariant net of von Neumann algebras on afixed Hilbert space H is a local isotonic map K ∋ O 7→ A ( O ) ⊂ B ( H ) from the set K ofopen, connected bounded regions of the 1 + 3 dimensional Minkowski spacetime R to vonNeumann algebras A ( O ) ⊂ B ( H ). It is assumed the existence of a unitary positive energyPoincar´e representation U on H acting covariantly on A and of a unique (up to a phase)normalized U -invariant vector Ω ∈ H which is cyclic for all the local algebras, namely thevacuum vector. The deep connection between the algebraic structure and the geometry ofthe model is a very fascinating fact. In [2, 3] Bisognano and Wichmann showed that anymodel coming from Wightman fields encloses within itself the information on its geometry.The authors proved that the modular operators related to algebras associated with wedge-shaped regions and the vacuum state have a geometrical meaning: they implement pure ∗ Supported by the ERC advanced grant 669240 QUEST “Quantum Algebraic Structures and Models”,and INdAM.
Bisognano-Wichmann property(B-W) : U (Λ W (2 πt )) = ∆ − it A ( W ) , Ω (1)for any wedge region W , where t Λ W ( t ) is the one-parameter group of boosts associatedwith the wedge W , and ∆ it A ( W ) , Ω is the modular group of the von Neumann algebra A ( W )generated by ∪ O⊂ W A ( O ) w.r.t. the vacuum vector Ω.It can be stated something more. A von Neumann algebra net A is said to be modularcovariant if ∆ it A ( W ) , Ω A ( O )∆ − it A ( W ) , Ω = A (Λ W ( − πt ) O ) , for any O ∈ K (2)i.e. the modular group associated with any wedge algebra implements the covariant actionof the associated one-parameter group of boosts on the local net. The modular covarianceproperty is introduced in [5], it is weaker than the B-W property and it ensures that thereexists a covariant unitary positive energy representation of the Poincar´e group generated bythe modular theory of the net algebras [18]. This marked one of the successes of the Tomita-Takesaki theory: once the algebra of the observables and the vacuum state are specified, themodular structure is determined and it has a geometrical meaning. Sufficient conditions onthese properties are given in [6, 7, 8, 32].The B-W property cares only about the modular theory of the algebraic model, whichis contained in the real structure of the net. This can be described by using real standardsubspaces of the Hilbert state space (cf. definition in Sect. 2.1). It is possible to characterizethe standard subspaces with an analogue of the Tomita-Takesaki modular theory whichcoincides with the von Neumann algebra Tomita-Takesaki theory when one considers vonNeumann algebras A ⊂ B ( H ) with a cyclic and separating vector Ω ∈ H and the subspaces H = A sa Ω ⊂ H .At this point it is natural to consider analogous nets of standard subspaces, which providea very fruitful approach to QFT. For instance, they have a key role in finding localizationproperties of Infinite Spin particles, cf. Ref. [24] or finding new models in low dimensionalquantum field theory [20, 25]. A geometrical approach to nets of standard subspaces is in[28].The B-W property is essential to give a canonical structure to the first quantizationnets, hence to the free fields (cf. [4]): given a particle, namely an irreducible, positiveenergy, unitary representation of the Poincar´e group, there is a canonical way to build upthe associated one-particle net of localized states and its second quantization free field. Wemake an analysis in the converse sense. Are those nets in some sense unique? At this level,the question is not well posed since there is some freeness in choosing the modular conjugationimplementing the position and time reflection. We can ask when a (unitary, positive energy)Poincar´e representation U is modular, i.e. any net of standard subspaces it actscovariantly on satisfies the B-W property (cf. Definition 3.3). In particular U wouldbe implemented by the modular operators. This would give an answer to the necessity toassume the B-W property instead of deducing it by the assumptions.An approach to this problem is to show the B-W property by exploiting the analyticityproperty of the wave functions as in [10, 26]. It is difficult to extend this analytic approach2o more general representations as infinite direct sums, direct integrals or to the masslesscase.We are going to present a purely algebraic argument giving a sufficient condition forthe modularity of a large family of Poincar´e representations as direct integrals of scalarrepresentations, including the massless case. The idea comes from the following facts: theLorentz group acts geometrically on the momentum space and one can check that on the massshell it is possible to pointwise reconstruct the action of a Lorentz transformation, sending awedge W to its causal complement W ′ just considering W -fixing transformations (see Remark4.1). With this hint we introduce the following condition on a (unitary, positive energy)representation U of the Poincar´e group, called modularity condition (cf. Definition 3.3).We ask that the von Neumann algebra, generated by translations and the Lorentz subgroupof transformations fixing W is large enough to enclose the transformations sending W onto W ′ . In Theorem 3.4 we show that the modularity condition is sufficient to prove that U ismodular. In particular, for any U -covariant standard subspace net H the quotient betweenthe two sides of (1) is the identity. This condition does neither depend on the net nor onmultiplicity of the representation and passes to some direct integrals. Our analysis holdsfor scalar Poincar´e representations in any spacetime R s , with s ≥
3, see Remark 4.6. Acomment on the massless finite helicity case is given in Remark 4.5. We rely on the ideathat the modular covariance has to be a natural assumption in every Quantum Field Theorysatisfying basic relativistic and quantum hypotheses.Known counterexamples to modular covariance seem very artificial as they imply a break-down of Poincar´e covariance, see [34, 19]. In Sect. 5, we give an explicit example of a massivePoincar´e covariant standard subspace net, which is modular covariant not satisfying the B-Wproperty. The massless case was treated in [24]. These kinds of general counterexamplesclarify what kind of settings may prevent the identification of the covariant representationwith the modular symmetries.The relation between the split and the modular covariance properties is an interestingproblem. In [14], Doplicher and Longo proved that if the dual net of local von Neumannalgebras associated with the scalar generalized free field with K¨allen-Lehmann measure µ has the split property, then µ is purely atomic and concentrated on isolated points. In Sect.6, we give an outlook in the standard subspace setting of the problem and of this result.The paper is organized as follows. In Sect. 2 we introduce the one-particle setting andwe recall relevant results on the real subspace structure, on Poincar´e representations and onnets of standard subspaces. In Sect. 3 we present our algebraic condition and its properties.We prove it for general scalar representations in Sect. 4. In Sect. 5 we show the massivecounterexample to the B-W property. In Sect.6 it is discussed the relation between the B-Wand the Split properties in this one-particle setting.This paper was reviewed in [27]. 3 One-particle net
Firstly, we recall some definitions and basic results on standard subspaces and (generalized)one-particle models. A linear, real, closed subspace H of a complex Hilbert space H is called cyclic if H + iH is dense in H , separating if H ∩ iH = { } and standard if it is cyclicand separating.Given a standard subspace H the associated Tomita operator S H is defined to be theclosed the anti-linear involution with domain H + iH , given by: S H : H + iH ∋ ξ + iη ξ − iη ∈ H + iH, ξ, η ∈ H, on the dense domain H + iH ⊂ H . The polar decomposition S H = J H ∆ / H defines the positive self-adjoint modular operator ∆ H and the anti-unitary modularconjugation J H . In particular, ∆ H is invertible and J H ∆ H J H = ∆ − H . If H is a real linear subspace of H , the symplectic complement of H is defined by H ′ ≡ { ξ ∈ H ; ℑ ( ξ, η ) = 0 , ∀ η ∈ H } = ( iH ) ⊥ R , where ⊥ R denotes the orthogonal in H viewed as a real Hilbert space with respect to thereal part of the scalar product on H . H ′ is a closed, real linear subspace of H . If H isstandard, then H = H ′′ . It is a fact that H is cyclic (resp. separating) iff H ′ is separating(resp. cyclic), thus H is standard iff H ′ is standard and we have S H ′ = S ∗ H . Fundamental properties of the modular operator and conjugation are∆ itH H = H, J H H = H ′ , t ∈ R . We shall call the one-parameter, strongly continuous group t ∆ itH , the modular group of H (cf. [30]).There is a 1-1 correspondence between Tomita operators and standard subspaces. Proposition 2.1. [22, 23].
The map H S H (3) is a bijection between the set of standard subspaces of H and the set of closed, densely defined,anti-linear involutions on H . The inverse of the map (3) is S ker( − S ) . Furthermore, this map is order-preserving, namely H ⊂ H ⇔ S H ⊂ S H , and we have S ∗ H = S H ′ .
4s a consequence,
Standardsubspaces H ⊂ H ←→ closed, dens. def.anti-linearinvolutions S ←→ ( J, ∆) pairs of ananti-unitaryinvolution and apositive self-adjointoperator on H s.t. J ∆ J = ∆ − Here is a basic result on the standard subspace modular theory.
Lemma 2.2. [22, 23].
Let
H, K ⊂ H be standard subspaces and U ∈ U ( H ) be a unitaryoperator on H such that U H = K . Then U ∆ H U ∗ = ∆ K and U J H U ∗ = J K . The following is the analogue of the Takesaki theorem for standard subspaces.
Lemma 2.3. [22, 23].
Let H ⊂ H be a standard subspace, and K ⊂ H be a closed, reallinear subspace of H .If ∆ itH K = K , ∀ t ∈ R , then K is a standard subspace of K ≡ K + iK and ∆ H | K is themodular operator of K on K . Moreover, if K is a cyclic subspace of H , then H = K . The following is the Borchers theorem in the standard subspace setting.
Theorem 2.4. [22, 23].
Let H ⊂ H be a standard subspace, and U ( t ) = e itP be a one-parameter unitary group on H with generator ± P > , such that U ( t ) H ⊂ H , ∀ t ≥ .Then, (cid:26) ∆ isH U ( t )∆ − isH = U ( e ∓ πs t ) J H U ( t ) J H = U ( − t ) ∀ t, s ∈ R . (4)In other words, the last theorem claims that if there is a one-parameter unitary group t U ( t ), with a positive generator, which properly translates a standard subspace, then (4)establishes the unique (up to multiplicity), positive energy representation of the translation-dilation group.The converse of the Borchers theorem can be stated in the following way. Theorem 2.5. [4].
Let H be a standard space in the Hilbert space H and U ( t ) = e itP aone-parameter group of unitaries on H satisfying: ∆ isH U ( t )∆ − isH = U ( e ∓ πs t ) and J H U ( t ) J H = U ( − t ) , ∀ t, s ∈ R The following are equivalent:1. U ( t ) H ⊂ H for t ≥ ;2. ± P is positive. .2 The Minkowski space and the Poincar´e group Let R be the Minkowski space, i.e. a four-dimensional real manifold endowed with themetric ( x, y ) = x y − X i =1 x i y i . In a 4-vector x = ( x , x , x , x ) x and { x i } i =1 , , are the time and space coordinates,respectively. The Minkowski space has a causal structure induced by the metric. The causalcomplement of a region O is given by O ′ = { x ∈ R : ( x − y ) < , ∀ y ∈ O } , where( x − y ) = ( x − y, x − y ) refers to the norm induced by the metric. A causally closed regionis such that O = O ′′ .We shall denote with P the Poincar´e group , i.e the inhomogeneous symmetry groupof R . It is the semidirect product of the Lorentz group L , the homogeneous Minkowskisymmetry group, and the R -translation group , i.e. P = R ⋊ L . It has four connectedcomponents, as L has four connected components, and we shall indicate with P ↑ + = R ⋊ L ↑ + the connected component of the identity. One usually refers to P ↑ + as the properortochronous (connected component of the) Poincar´e group . L ↑ + is not simply connected.The L ↑ + universal covering e L ↑ + is SL(2 , C ), thus the P ↑ + -universal covering e P ↑ + is isomorphicto R ⋊ SL(2 , C ). Let Λ : R ⋊ SL(2 , C ) ∋ ( a, A ) Λ( a, A ) ∈ P ↑ + be the covering map .Unitary positive energy representations of the (universal covering of the) Poincar´e groupbelong to three families: massive, massless finite helicity and massless infinite spin. Massiverepresentations are labelled by a mass parameter m ∈ (0 , + ∞ ) and a spin parameter s ∈ N ;massless finite helicity and infinite spin representations have zero mass and are labelled byan helicity parameter h ∈ Z and a couple ( κ, ǫ ) where κ ∈ R + is the radius and ǫ ∈ { , } the bose/fermi alternative, respectively.We shall indicate with P + the subgroup of P generated by P ↑ + and the space and timereflection θ . Consider the automorphism α of the Poincar´e group θ generated by the adjointaction of the θ -reflection. The proper Poincar´e group P + is generated as a semidirectproduct P ↑ + ⋊ α Z through the α -action. It is well known (see for example [33]) that any irreducible repre-sentation of the Poincar´e group U , except for finite helicity, extends to an (anti-)unitaryrepresentation of e U of e P ↑ + , i.e. e U ( g ) is ( is unitary g ∈ e P ↑ + is anti-unitary g ∈ e P ↓ + ˙ = θ · P ↑ + At this point it is necessary to fix some notations about regions and isometries of theMinkowski spacetime. A wedge-shaped region W ⊂ R is an open region of the form gW where g ∈ P ↑ + and W = { x ∈ R : | x | < x } . The set of wedges is denoted by W .Let W ⊂ W be the subset of wedges the form gW where g ∈ L ↑ + . Note that if W ∈ W (or W ∈ W ), then W ′ ∈ W (resp. W ′ ∈ W ). For every wedge region W ∈ W there exists aunique one-parameter group of Poincar´e boosts t Λ W ( t ) preserving W , i.e. Λ W ( t ) W = W for every t ∈ R . It is defined by the adjoint action of a g ∈ P ↑ + such that gW = W on Λ W ,6here Λ W ( t ) x = ( x cosh t + x sinh t, x sinh t + x cosh t, x , x ). Let t λ W ( t ) be the(unique, one-parameter group) lift to the covering group. We shall denote with W α ∈ W thewedge along x α -axis, i.e. W α = { x ∈ R : | x | < x α } with α = 1 , ,
3. Let t Λ α ( t ) and θ R α ( θ ) be respectively the boosts and the rotations of P ↑ + fixing W α and t λ j ( t ) and θ r j ( θ ) be the corresponding lifts to SL(2 , C ). Note that λ j ( t ) = e σ j t/ and r j ( θ ) = e iσ j θ/ ,where { σ i } i =1 , , are the Pauli matrices. Let U be a unitary positive energy representation of the Poincar´e group e P ↑ + on an Hilbertspace H . We shall call a U -covariant (or Poincar´e covariant) net of standard sub-spaces on wedges a map H : W ∋ W H ( W ) ⊂ H , associating to every wedge in R a closed real linear subspace of H , satisfying the followingproperties:1. Isotony: If W , W ∈ W and W ⊂ W then H ( W ) ⊂ H ( W );2. Poincar´e Covariance: U ( g ) H ( W ) = H ( gW ) , ∀ g ∈ e P ↑ + , ∀ W ∈ W ;3. Positivity of the energy: the joint spectrum of translations in U is contained in theforward light cone V + = { x ∈ R : x = ( x, x ) ≥ x ≥ } Reeh-Schlieder property (R-S): if W ∈ W , then H ( W ) is a cyclic subspace of H ;5. Twisted Locality: there exists a self-adjoint unitary operator Γ ∈ U ( e P ↑ + ) ′ , s.t.Γ H ( W ) = H ( W ) for any W ∈ W and if W ⊂ W ′ then BH ( W ) ⊂ H ( W ) ′ , with B = 1 + i Γ1 + i .
We shall indicate a U -covariant net of standard subspaces on wedges W H ( W ) sat-isfying 1.-5. with the couple ( U, H ) . This is the setting we are going to study the followingtwo properties:6.
Bisognano-Wichmann property : if W ∈ W , then U ( λ W (2 πt )) = ∆ − itH ( W ) , ∀ t ∈ R ; (5)7. Duality property : if W ∈ W , then H ( W ) ′ = BH ( W ′ ).Clearly, if U factors through P ↑ + then the two expressions of the B-W property (1) and (5)coincide.The relations between the modular theory of the wedge subspaces and the twisted oper-ator are expressed by the following proposition.7 roposition 2.6. The following hold [∆ H ( W ) , B ] = 0 J H ( W ) BJ H ( W ) = B ∗ Proof.
As Γ H ( W ) = H ( W ) then, by Lemma 2.2, Γ∆ H ( W ) Γ ∗ = ∆ H ( W ) and Γ J H ( W ) Γ ∗ = J H ( W ) . A straightforward computation concludes the argument. (cid:3) Proposition 2.7.
Wedge duality is consequence of the B-W property.
Proof.
By Proposition 2.6 ∆ H ( W ) = ∆ BH ( W ) and by covariance H ( W ′ ) = U (Λ W ( − πt )) H ( W ′ ) = ∆ itH ( W ) H ( W ′ ) . By twisted locality BH ( W ′ ) ⊂ H ( W ) ′ and Lemma 2.3 we get the thesis. (cid:3) It is possible to define closed real linear subspaces associated with bounded causallyclosed regions as follows H ( O ) ˙= \ W∋ W ⊃ O H ( W ) . (6)This defines a net of real subspaces on causally closed regions O H ( O ). Note that H ( O )is not necessarily cyclic. If H is a net satisfying 1.-7. assumptions and H ( O ) is cyclic, then H ( W ) = X O ⊂ W H ( O )by Lemma 2.3. If H ( O ) is cyclic for any double cone O , we say that the net O H ( O )satisfies the R-S property for double cones .In [4], Brunetti, Guido and Longo showed that there is a 1-1 correspondence between(anti-)unitary, positive energy representations of P + and covariant nets of standard subspacessatisfying the B-W property. For the sake of completeness, we recall their theorem and wepresent the proof in the fermionic case which is not contained in the original paper. Theorem 2.8. [4].
There is a 1-1 correspondence between:a. (Anti-)unitary positive energy representations of e P + .b. Local nets of standard subspaces satisfying 1-7. Proof.
Consider the automorphism of the Poincar´e group e P + generated by the adjointaction of j , one of the two e P + elements implementing the x − x reflection. One can checkin e P + that j ( a, A ) j = ( j a, σ Aσ ) , ∀ ( a, A ) ∈ R ⋊ SL(2 , C ) (7)8nd r ( π ) λ ( t ) r ( π ) − = λ ( − t ) and r ( π ) j r ( π ) − = − j , (8)(e.g. cf. Appendix in [26]). Consider an (irreducible) fermionic representation U of e P + ,namely a e P + -representation which does not factor through P + . In particular U (2 π ) = − .Since U lifts to a representation of the Lie algebra of e P ↑ + on the G¨arding domain and byrelations (7) and (8) we get U ( R ( π )) K U ( R ( π )) ∗ = − K , (9) U ( R ( π )) J U ( R ( π )) ∗ = − J , (10)where U ( λ ( t )) = e iK t and J denotes U ( j ) (choose one of the two possible choices for U ( j )in e P + ). The anti-unitary operator J = U ( j ) and the self-adjoint operator ∆ W = e πK satisfy J ∆ W J = ∆ − W . Hence, it is possible to define an anti-unitary involution S W = J ∆ / W , and, by Proposition 2.1, a subspace H ( W ) associated with the W wedge. Clearly S W is the Tomita operator S H ( W ) of the subspace H ( W ). By covariance, a map of standardsubspaces W ∋ W H ( W ) ⊂ H is well defined. Indeed, for any wedge W , S H ( W ) is theTomita operator determining H ( W ), defined as S H ( W ) = U ( g ) S H ( W ) U ( g ) ∗ , with g ∈ P ↑ + such that gW = W . Note that S H ( W ) = S H ( r (2 π ) W ) and S H ( W ) is well defined. This clarifythe ambiguity in the choice of J . Furthermore, by covariance and relations (9) and (10), as S H ( W ) = J W ∆ / W then S H ( W ′ ) = − J W ∆ − / H ( W ) . It easily follows that H ( W ′ ) = iH ( W ) ′ . Thisensures twisted locality and duality as we can define Γ ˙= U (2 π ) ∈ U ( e P ↑ + ) ′ and B = − i i · = − i · is the twist operator, i.e. H ( W ′ ) = BH ( W ) ′ where W ∈ W . Positivity of the energy, Poincar´e covariance, B-W and R-S properties are ensured byconstruction. Isotony follows as in [4] by positivity of the energy and Theorem 2.5. (cid:3)
We define the following subgroups of e P ↑ + : • G W ˙= { A ∈ SL(2 , C ) : Λ( A ) W = W } , where W ∈ W . It is the subgroup of e L ↑ + elements fixing W through the covering homomorphism Λ. • G W = h G W , T i , with W ∈ W , where T is the R -translation group. G W is thegroup generated by G W and T . • For a general wedge W ∈ W , G W and G W are defined by the transitive action of P ↑ + on wedges.Let W ∈ W . Consider the strongly continuous map Z H ( W ) : R ∋ t ∆ itH ( W ) U ( λ W (2 πt )) . (11)It has to be the identity map if the B-W property (5) holds.9 roposition 3.1. Let ( U, H ) be a Poincar´e covariant net of standard subspaces. Then, forevery W ∈ W , the map t Z H ( W ) ( t ) defines a one-parameter group and Z H ( W ) ( t ) ∈ U ( G W ) ′ , ∀ t ∈ R . Proof. As P ↑ + acts transitively on wedges, there is no loss of generality if we fix W = W andconsider G W = G ⊂ e P ↑ + . As Λ ( t ) W = W for any t ∈ R then U ( λ ( t )) H ( W ) = H ( W )and, by Lemma 2.2, ∆ H ( W ) commutes with U ( λ ( t )). In particular, t Z H ( W ) ( t ) defines aunitary one-parameter group.By positivity of the energy and Theorem 2.4, ∆ − itH ( W ) has the same commutation relationsas boosts U ( λ (2 πt )) w.r.t. translations. Indeed, translations in x and x directions com-mute with ∆ itH ( W ) since they fix H ( W ), and with U ( λ ( t )). Translations along directionsv + = (1 , , ,
1) and v − = ( − , , ,
1) have, respectively, positive and negative generatorsand U ( t ) H ( W ) ⊂ H ( W ) for any t >
0. Then, by Theorem 2.4∆ isH ( W ) U ± ( t )∆ − isH ( W ) = U ± ( e ∓ πs t ) s, t ∈ R , as well as U ( λ W ( − πt )) U ± ( t ) U ( λ W (2 πt )) = U ± ( e ∓ πs t ) , s, t ∈ R where U ± ( t ) = U ( t · v ± ). Translations along x , x , v + and v − generate R translations andas a consequence Z H ( W ) ∈ U ( T ) ′ .Any element g ∈ G fixes the standard subspace H ( W ), hence by Lemma 2.2, U ( g )commutes with the modular operator ∆ H ( W ) . Furthermore, as g fixes W , then U ( g ) alsocommutes with U ( λ ). We conclude that Z H ( W ) ∈ U ( G ) ′ . (cid:3) Note that G = h r , λ , r (2 π ) i , where r (2 π ) is the 2 π rotation. Proposition 3.2.
Let ( U, H ) be a Poincar´e covariant net of standard subspaces. Let W ∈ W ,and r W ∈ e P ↑ + be such that Λ( r W ) W = W ′ . Assume that Z H ( W ) commutes with U ( r W ) , thenthe B-W and Duality properties hold. Proof.
The map R ∋ t Z H ( W ) ( t )is a unitary, one-parameter, s.o.-continuous group by Proposition 3.1. Now by hypothesisand covariance Z H ( W ′ ) ( t ) = U ( r W ) Z H ( W ) ( t ) U ( r W ) ∗ = Z H ( W ) ( t )where Z H ( W ′ ) ( t ) = U ( λ W ( − πt ))∆ itH ( W ′ ) . We find that Z H ( W ) (2 t ) = Z H ( W ) ( t ) Z H ( W ) ( t ) = Z H ( W ) ( t ) Z H ( W ′ ) ( t ) = ∆ itH ( W ) ∆ itH ( W ′ ) Z H ( W ) (2 t ) is an automorphism of H ( W ) and∆ − itH ( W ) Z H ( W ) (2 t ) H ( W ) = ∆ itH ( W ′ ) H ( W ) ⇔ ∆ − itH ( W ) H ( W ) = ∆ itH ( W ′ ) H ( W ) ⇔ H ( W ) = ∆ itH ( W ′ ) H ( W ) ∀ t ∈ R . By locality H ( W ) ⊂ ( B H ( W ′ )) ′ , Lemma 2.2 and Proposition 2.6, we have∆ it ( BH ( W ′ )) ′ H ( W ) = ∆ − itBH ( W ′ ) H ( W ) = ∆ − itH ( W ′ ) H ( W ) = H ( W )and by Lemma 2.3 we conclude wedge duality, H ( W ) = ( B H ( W ′ )) ′ . Furthermore, by the last condition for any W ∈ W then∆ H ( W ) = ∆ − H ( W ′ ) and the B-W property follows since Z H ( W ) ( t ) = U ( r W ) Z H ( W ) ( t ) U ( r W ) ∗ = U ( λ W ( − πt ))∆ − itH ( W ) = Z H ( W ) ( − t )hence Z H ( W ) ( t ) = . (cid:3) Now we state the properties we are interested in.
Definition 3.3.
We shall say that a unitary, positive energy representation is modular iffor any U -covariant net of standard subspaces H , namely any couple ( U, H ), then the B-Wproperty holds.Let W ∈ W . A unitary, positive energy e P ↑ + -representation U satisfies the modularitycondition if for an element r W ∈ e P ↑ + such that Λ( r W ) W = W ′ we have that U ( r W ) ∈ U ( G W ) ′′ . (MC)Note that (MC) does neither depend on the choice of r W , nor of W . Indeed if e r W ∈ e P ↑ + is another element such that Λ( e r W ) W = W ′ then r W e r W ∈ G W and if (MC) holds for U ( r W ),then it holds for U ( e r W ). We conclude (MC) for any other wedge by the transitivity of the P ↑ + -action on wedge regions.Now, we prove that the representations satisfying (MC) are modular. Theorem 3.4.
Let U be a positive energy unitary representation of the Poincar´e group e P ↑ + .If the condition (MC) holds on U , then any local U -covariant net of standard subspaces,namely any pair ( U, H ) , satisfies the B-W and the Duality properties. roof. Let (
U, H ) be a U -covariant net of standard subspaces, then Z H ( W ) ∈ U ( G W ) ′ byProposition 3.1. Then by assumptions Z H ( W ) commutes with U ( r W ) where Λ( r W ) W = W ′ ,then we conclude the thesis by Proposition 3.2 (cid:3) Let U be a representation of e P ↑ + acting on a standard subspace net W ∋ W H ( W ) ⊂H . Assume that J geo,W is an anti-unitary operator extending U to a representation of f P + through W -reflection and assume that modular covariance holds. As in [18], the algebraic J H ( W ) implements the wedge W reflection and, up to a e P ↑ + element, the PCT operator (theproof in [18] can be straightforwardly adapted in the standard subspace net case). In thissetting, let K W be the W -boost generator on H , the following operators S geo,W ˙= J geo,W e − πK W , S alg,W ˙= J H ( W ) ∆ / H ( W ) , (12)can be called geometric and algebraic Tomita operators . Corollary 3.5.
With the assumptions of Proposition 3.2, S geo,W = CS alg,W , ∀ W ∈ W where C ∈ U ( e P ↑ + ) ′ Proof.
By duality, J geo,W and J H ( W ) both implement anti-unitarily U ( j W ), then J geo J H ( W ) ∈ U ( P ↑ + ) ′ . By the B-W property, e − πK W = ∆ / W , and we conclude. (cid:3) The B-W property and the e P + covariance do not imply that there is a unique net un-dergoing the U -action. The conjugation operator can differ from the geometric conjugationby a unitary in U ( P ↑ + ) ′ . For instance, given an irreducible, (anti-)unitary P + -representation U where U ( j W ) implements the W -reflection, we have two U -covariant nets, according tothe couples { ( U ( j W ) , e − πK W ) } W ∈W and { ( − U ( j W ) , e − πK W ) } W ∈W defining the wedge sub-spaces. If we just require e P ↑ + -covariance through a representation U , then for any modulusone complex number λ , the couples { ( λU ( j W ) , e − πK W ) } W ∈W define U -covariant standardsubspace nets. Direct sums
The modularity property easily extends to direct integrals and multiples of representationsas the following proposition shows.
Proposition 3.6.
Let U and { U x } x ∈ X be unitary positive energy representations of e P ↑ + satisfying (MC) .Let K be an Hilbert space, then (MC) holds for U ⊗ K ∈ B ( H ⊗ K ) .Let ( X, µ ) be a standard measure space. Assume that U x | G W and U y | G W are disjoint for µ -a.e. x = y in X . Then U = R X U x dµ ( x ) satisfies (MC) . roof. We can assume W = W . Let U ˙= U ⊗ K , since U ( G ) ′ = U ( G ) ′ ⊗ B ( K ) it followsthat U ( r ( π )) = U ( r ( π )) ⊗ U ( G ) ′ , hence U ( r ( π )) ∈ U ( G ) ′′ .For the second statement, let U ( a, Λ) = L Z X U x ( a, Λ) dµ ( x ) acting on H = L Z X H x dµ ( x ) . Then, by disjointness, U ( G ) ′′ = Z ⊕ X U x ( G ) ′′ dµ ( x )and U ( r ) = R ⊕ X U ( r ( π )) dµ ( x ) we have that U ( r ( π )) ∈ U ( G ) ′′ (cid:3) In this section we are going to show that (MC) holds for scalar representations, namelymassive or massless 0-spin/helicity representations. The scalar representations have thefollowing form( U m, ( a, A ) φ )( p ) = e iap φ (Λ( A ) − p ) , ( a, A ) ∈ R ⋊ e L ↑ + = e P ↑ + , where φ ∈ H m, ˙= L (Ω m , δ ( p − m ) θ ( p ) d p ) , and Ω m = { p = ( p , p ) ∈ R : p = p − p = m } with m ≥
0. We recall that U m, factorsthrough P ↑ + .Any momentum p ∈ Ω m is a point in the dual group of T i.e. a character. We recall that P ↑ + acts on Ω m -characters as dual action of the adjoint action of P ↑ + on T . Clearly, T actstrivially and L ↑ + acts geometrically on Ω m , i.e. ( a, Λ) · p = Λ − p with ( a, Λ) ∈ R ⋊ L ↑ + = P ↑ + .We start with the following remark. Remark . Fix p = ( p , p , p , p ) ∈ Ω m , m > R ( π ) p = ( p , p , − p , − p )can be obtained as a composition of a Λ -boost of parameter t p and a R -rotation of param-eter θ p as Λ ( t p ) R ( θ p )( p , p , p , p ) = Λ ( t p )( p , p , − p , p )= ( p , p , − p , − p ) . (13)Clearly t p and θ p depend on p . By (13), we deduce that G orbits on Ω m are not changed by R ( π ). With m = 0 an analogue argument holds for all the orbits except { ( p , , , p ) , p ≥ } and { ( p , , , − p ) , p ≥ } , i.e. there is no g ∈ G , such that g ( p , , , − p ) =( p , , , p ). On the other hand these orbits have null measure with the restriction of theLebesgue measure to ∂V + . This remark holds in R s with s > emma 4.2. Let f ∈ L ∞ (Ω m ) such that for every g ∈ G , f ( p ) = f ( gp ) for a.e. p ∈ Ω m .Then f ( p ) = h ( p ) for a.e. p ∈ Ω m where h ∈ L ∞ (Ω m ) is constant on any { g p } g ∈ G orbit. Proof.
Any point p = ( p , p , p , p ) ∈ Ω m (except the massless null measure sets { ( p , , , p ) , p ≥ } and { ( p , , , − p ) , p ≥ } ), can be identified with a radius r = p + p , an an-gle θ ∈ [0 , π ] such that ( p , p ) = r (cos θ, sin θ ) and a parameter t ∈ R such that if( p , p ) = √ r + m (cosh t, sinh t ). In particular G orbits σ r are labelled by r ∈ R + . Oneach orbit (with its invariant G measure) the only positive measure set which is preservedby the G action is the full orbit (up to a set of measure zero). This fact ensures that G invariant functions have to be almost everywhere constant. Up to unitary equivalence, wecan decompose the Hilbert space as R R + L ( σ r , dµ r ) dr where dµ r is the G invariant measureon σ r . G representations on different orbits are inequivalent, then U ( G ) ′ = R R + ( f ( r ) · dr and we conclude. (cid:3) Proposition 4.3.
Let U be a unitary, positive energy, irreducible scalar representation ofthe Poicar´e group. Then U satisfies the modularity condition (MC) . Proof.
It is enough to consider W . Let Z ∈ U ( G ) ′ . Since U is a scalar representationthen the translation algebra T ′′ = { U ( a ) : a ∈ R } ′′ is a MASA and T ′′ = L ∞ (Ω m ). Indeed,the translation unitaries ( U ( a ) φ )( p ) = e iap φ ( p ) are multiplication operators and generate L ∞ (Ω m ) ultra-weakly. In particular Z ∈ T ′ = T ′′ , hence it is a multiplication operator Z = M f by f ∈ L ∞ (Ω). Furthermore, Z has to commute with U (Λ ) and U ( R ), hence ∀ t ∈ R and ∀ θ ∈ [0 , π ] U (Λ ( t )) ZU (Λ ( t )) ∗ = Z ⇔ f (Λ ( t ) − p ) = f ( p ) , a.e. p ∈ Ω m and U ( R ( θ )) ZU ( R ( θ )) ∗ = Z ⇔ f ( R ( θ ) − p ) = f ( p ) a.e. p ∈ Ω m . We can assume f ( p ) to be constant on any Λ and R orbit by Lemma 4.2.Now observe that any momentum on the hyperboloid p = ( p , p , p , p ) can be connectedto the R ( π ) p = ( p , p , − p , − p ) through a R -rotation and a Λ -boost as in Remark 4.1.It follows that f ( p ) = f ( R ( − π ) p ), for every p ∈ Ω m . As a consequence U ( R ( π )) ZU ( R ( π )) ∗ = Z as f ( R ( π ) − p ) = f ( p ) , ∀ p ∈ Ω m and we conclude. (cid:3) Now, we can state the theorem.
Theorem 4.4.
Let U = R [0 , + ∞ ) U m dµ ( m ) where { U m } are (finite or infinite) multiples ofthe scalar representation of mass m , then U satisfies (MC) . In particular if ( U, H ) is a U -covariant net of standard subspaces, then the Duality and the B-W properties hold. roof. Unitary representations of P ↑ + with different masses are disjoint, and they havedisjoint restrictions to G subgroup. The thesis becomes a consequence of Propositions 3.6,4.3 and Theorem 3.4 . (cid:3) Remark . Proposition 4.3 straightforwardly holds also for irreducible massless finite he-licity representations as they are induced from a one-dimensional representation of the littlegroup. As a consequence, an irreducible nonzero helicity representation U cannot act covari-antly on a net of standard subspaces on wedges H . Indeed, the B-W property must holdas in Proposition 4.3 and J W , the modular conjugation of H ( W ), would implement the j reflection on U (cf. [18]). In particular, the PT operator defined by Θ = J W U ( R ( π )) ex-tends U to an (anti-)unitary representation ˆ U of e P + and acts covariantly on H . On the otherhand nonzero helicity representations are not induced by a self-conjugate representation ofthe little group and do not extend to anti-unitary representations of e P + [33]. This shows acontradiction. Remark . Consider the R s spacetime with s ≥ U be a scalar (unitary, positiveenergy) representation of P ↑ + . The one-parameter group t Z H ( W ) ( t ) given in (11), isgenerated by the multiplication operator by a real function of the form f ( p + ... + p s ). Foreach value of the radius r = p + ... + p s there is a unique G -orbit on Ω m which is fixedby any transformation R ∈ P ↑ + such that RW = W ′ , for instance R ( π ). In particular, theanalysis of this section extends to any Minkowski spacetime R s with s ≥
3. It fails in 2 + 1spacetime dimensions as R ( π ) does not preserve G -orbits. Borchers, in [6], showed that a unitary, positive energy Poincar´e representation acting co-variantly on a modular covariant von Neumann algebra net A in the vacuum sector, can onlydiffer from the modular representation by a unitary representation of the Lorentz group. Theorem 5.1. [6].
Let A be a local quantum field theory von Neumann algebra net in thevacuum sector undergoing two different representations of the Poincar´e group. Let U be therepresentation implemented by wedge modular operators and U be the second representation.Then there exists a unitary representation of the Lorentz Group G (Λ) defined G (Λ) = U ( a, Λ) U ( a, Λ) ∗ . Furthermore, G (Λ) commutes with U ( a, Λ ′ ) for all a ∈ R , Λ , Λ ′ ∈ L ↑ + and the G (Λ) adjoint action on A implements automorphisms of local algebras, i.e. maps any local algebrainto itself. With this hint it is possible to build up Poincar´e covariant nets picturing the abovesituation: modular covariance without the B-W property . We are going to make explicitcomputations on this kind of counterexamples in order to understand what may prevent theB-W property. Here, we study the massive case. The massless case can be found in [24].15onsider U m,s , the m -mass, s -spin, unitary, irreducible representation of the Poincar´egroup e P ↑ + and H : W H ( W ) its canonical net of standard subspaces. Let p ∈ Ω m , A p ˙= q p ∼ /m , where p ∼ = p · + P i =1 p i σ i is the SL(2 , C ) element implementing the boostsending the point q m = ( m, , ,
0) to p . An explicit form of U m,s is the following( U m,s ( a, A ) φ )( p ) = e iap D s ( A − p AA Λ − p ) φ (Λ( A ) − p ) , where D s is the s -spin representation of SU(2) on the 2 s + 1 dimensional Hilbert space h s and φ ∈ H m,s ˙= h s ⊗ L (Ω m , δ ( p − m ) θ ( p ) d p )= C s +1 ⊗ L (Ω m , δ ( p − m ) θ ( p ) d p ) . Let V be a real unitary, nontrivial, SL(2 , C )-representation on an Hilbert space K , i.e.there exists an anti-unitary involution J on the Hilbert space K , commuting with V suchthat the real vector space K ⊂ K of J -fixed vectors, is a standard subspace and V (SL(2 , C )) K = K. In particular
J K = K and ∆ K = 1.We can define the following net of standard subspaces, K ⊗ H : W ∋ W → K ⊗ H ( W ) ⊂ K ⊗ H . We can see two Poincar´e representations acting covariantly on K ⊗ H : U I ( a, A ) ≡ ⊗ U m,s ( a, A ) A ∈ SL(2 , C ) , a ∈ R and U V ( a, A ) ≡ V ( A ) ⊗ U m,s ( a, A ) A ∈ SL(2 , C ) , a ∈ R .U I is implemented by K ⊗ H modular operators and the B-W property holds w.r.t. U I (cf.Lemma 2.6 in [24]). Note that for s = 0, U I satisfies the condition (MC), hence the B-Wproperty also by Theorem 3.4. U V decomposes in a direct sum of infinitely many inequivalent representations of mass m ,i.e. infinitely many spins appear. Indeed, consider the Lorentz transformation A p = q p ∼ /m and the unitary operator on H ⊗ K W : K ⊗ H ∋ ( p φ ( p )) (cid:0) p ( V ( A − p ) ⊗ C s +1 ) φ ( p ) (cid:1) ∈ K ⊗ H (14)as K ⊗ H = L (Ω m , K ⊗ C s +1 , δ ( p − m ) θ ( p ) d p ) . We can define a unitarily equivalentrepresentation U ′ V = W U V W ∗ as follows:( U ′ V ( a, A ) φ )( p ) = e ia · p ( V ( A − p AA Λ( A ) − p ) ⊗ D s ( A − p AA Λ( A ) − p )) φ (Λ( A − p )) . It is easy to see that A − p AA Λ( A ) − p ∈ Stab( m, , ,
0) = SU(2).As we are interested in the disintegration of U V it is not restrictive to assume that V is irreducible. Unitary irreducible representations of SL(2 , C ), denoted by V ρ,n , are labelled16y pairs of numbers ( ρ, n ) such that ρ ∈ R and n ∈ Z + . The restriction of V ρ,n to SU(2)decomposes in L + ∞ s = n/ D s (see, for example, [31, 12]). Thus, if V = V ρ,n is an irreducibleSL(2 , C ) representation then U V ≃ ∞ M i = n s + i M j = | s − i | U m,j (15)since U ′ V , hence U V , decomposes according to the ( V ⊗ D s ) | SU(2) decomposition into irre-ducible representations.Note that any representation class appears with finite multiplicity and U V does not satisfythe condition (MC). Furthermore, we can conclude that having modular covariance withoutthe B-W property requires to the “wrong” representation the presence of an infinite familyof inequivalent Poincar´e representations possibly with finite multiplicity. U I cannot act covariantly on the U V -canonical net H V . We can see it explicitly. Let W be the wedge in x direction and W = gW where g ∈ L ↑ + and W = W . If U I ( g ) H V ( W ) = H V ( W ) then U I ( g )∆ − itH V ( W ) U I ( g ) ∗ = ∆ − itH V ( W ) where ∆ H V ( W ) and ∆ H V ( W ) are the modularoperator of H V ( W ) and H V ( W ), respectively. As U I ( g )∆ − itH V ( W ) U I ( g ) ∗ == (1 ⊗ U m,s ( g ))( V (Λ W (2 πt )) ⊗ U m,s (Λ W (2 πt )))(1 ⊗ U m,s ( g )) ∗ = V (Λ W (2 πt )) ⊗ U m,s ( g Λ W (2 πt )) g − )but ∆ − itH V ( W ) = V ( g Λ W (2 πt ) g − ) ⊗ U m,s ( g Λ W (2 πt )) g − ), we get the contradiction unless V is trivial.The modular covariance and the identification of the geometric and the algebraic PCToperators imply the uniqueness of the covariant representation. Proposition 5.2.
With the assumptions of Theorem 5.1 assume that the algebraic PCToperator defined in (12) implements the U -PCT operator too. Then U = U . Proof.
With the notations of Theorem 5.1, we consider the representation U ( a, Λ) = G (Λ) U ( a, Λ) of e P ↑ + on H . G fixes any subspace H ( W ) = A ( W )Ω and in particular J H ( W ) G (Λ) J H ( W ) = G (Λ), ∀ Λ ∈ L ↑ + . It follows that J H ( W ) has the correct commutationrelations with U if and only if G = 1. (cid:3) We have seen that given a Poincar´e covariant net with the B-W property (
U, H ) wecan find a second covariant Poincar´e representation e U if the commutant U ( e P ↑ + ) ′ is largeenough, i.e. when the Poincar´e representation implemented by the modular operators hasinfinite multiplicity. Furthermore, there are no conditions on the spin of e U . In particular,counterexamples can produce wrong relations between spin and statistics. For instanceassuming U I ( r (2 π )) = and V ( r (2 π )) = − where r (2 π ) is the 2 π -rotation. The (MC)condition does not hold for U V and a wrong spin-statistics relation may arise whenever theB-W property fails.As we said in the introduction known counterexamples to modular covariance are veryartificial and not all the basic relativistic and quantum assumptions are respected. It is17nteresting to look for natural counterexamples to modular covariance, in the class of repre-sentations excluded by this discussion, if they exist. In [15] it is shown that assuming finiteone-particle degeneracy, then the Spin-Statistics Theorem holds. This accords with the anal-ysis obtained by Mund in [26]. We expect that an algebraic proof for the B-W property canbe established, without assuming finite multiplicity of sub-representations. Now we come to an outlook on the relation between the split and the B-W properties.
Definition 6.1. [14] Let (
N ⊂ M , Ω) be a standard inclusion of von Neumann algebras,i.e. Ω is a cyclic and separating vector for
N, M and N ′ ∩ M .A standard inclusion ( N ⊂ M , Ω) is split if there exists a type I factor F such that N ⊂ F ⊂ M .A Poincar´e covariant net ( A , U, Ω) satisfies the split property if the von Neumann algebrainclusion ( A ( O ) ⊂ A ( O ) , Ω) is split, for every compact inclusion of bounded causally closedregions O ⋐ O .In a natural way one can define the split property for an inclusion of subspaces by secondquantization: let K ⊂ H ⊂ H be an inclusion of standard subspaces of an Hilbert space H such that K ′ ∩ H is standard, then the inclusion K ⊂ H is split if its second quantizationinclusion R ( K ) ⊂ R ( H ) is standard split w.r.t. the vacuum vector Ω ∈ e H . The secondquantization respects the lattice structure (cf. [1]). Here, we just deal with the Bosonicsecond quantization (see, for example, [21]).We want a coherent first quantization version of the split property. We assume H and K to be factor subspaces, i.e. H ∩ H ′ = { } = K ∩ K ′ . We need the following theorem. Theorem 6.2. [16].
Let H be a standard subspace and R ( H ) be its second quantization. • Second quantization factors are type I if and only if ∆ H | [0 , is a trace class operatorwhere ∆ H | [0 , is the restriction of the H -modular operator ∆ H to the spectral subspacerelative to the interval [0 , • Second quantization factors which are not type I are type III.
The canonical intermediate factor, for a standard split inclusion of von Neumann algebras N ⊂ M is F = N ∨ J N J = M ∩ J M J where J is the modular conjugation associated with the relative commutant algebra ( N ′ ∩ M, Ω), cf. [14]. The standard inclusion ( N ⊂ M, Ω) is split iff F is type I. A canonicalintermediate subspace can be analogously defined for standard split subspace inclusions K ⊂ H : F = K + J K = H ∩ J H.
Second quantization of modular operators were computed in [13, 21, 17].Consider the following proposition: 18 roposition 6.3.
Let K ⊂ H be an inclusion of standard subspaces, such that K ′ ∩ H isstandard. Let J be the modular conjugation of the symplectic relative complement K ′ ∩ H and ∆ = ∆ F be the modular operator of the intermediate subspace F = K + J K . Assumethat F is a factor. Then the following statements are equivalent.1. R ( K ) ⊂ R ( H ) is a split inclusion.2. the operator ∆ | [0 , is trace class. Proof. (1 . ⇒ . ) If the inclusion is split, then the intermediate canonical factor R ( K ) ∨ R ( J K ) is the second quantization of K + J K . In particular by Theorem 6.2 the thesis holds.(2 . ⇒ . ) As ∆ | [0 , is trace class, then the second quantization of F is an intermediate typeI factor between R ( K ) and R ( H ) by Theorem 6.2. Then the split property holds for theinclusion R ( H ) ⊂ R ( K ). (cid:3) Let U be an (anti-)unitary representation of P + . We shall say that U is split if the canonicalnet associated with U satisfies the split property on bounded causally closed regions (definedthrough equation (6)), i.e. the inclusion H ( O ) ⊂ H ( O )is split for every O ⋐ O as above.Scalar free fields satisfy Haag duality, thus (6) holds (see, for example, [29]). Furthermore,any scalar irreducible representation is split (cf. [11, 9]). The following theorem is the firstquantization analogue of Theorem 10.2 in [14]. Theorem 6.4.
Let U be an (anti-)unitary representation of P + , direct integral of scalarrepresentations. If U has the split property then U = R + ∞ U m dµ ( m ) where µ is purelyatomic on isolated points and for each mass there can only be a finite multiple of U m, . Proof.
As the B-W property holds, the net disintegrates according to the representation: H = Z + ∞ H m dµ ( m )where H m is the canonical net associated with U m . Fix a couple of bounded and causally closed regions O ⋐ O ⊂ R . Irreducible componentssatisfy the split property. In particular by Proposition 6.3, the restriction of the modularoperator of the intermediate standard subspace F , defined as F = H ( O ) + J H ( O ) = Z + ∞ H m ( O ) + J m H m ( O ) m dµ ( m )= Z + ∞ F m dµ ( m ) , to the spectral subset [0 , µ has to be purely atomicand for each isolated mass (no finite accumulation point) there can only be a finite multipleof the scalar representation. (cid:3) orollary 6.5. Let ( U, H ) be a Poincar´e covariant net of standard subspaces and U be a P ↑ + -split representation. Assume that U is a direct integral of scalar representations. Thenthe B-W and the duality properties hold. Proof.
By Theorem 6.4 the disintegration of the covariant Poincar´e representation is purelyatomic on masses, concentrated on isolated points and for each mass there can only be afinite multiple of the scalar representation. The disintegration satisfies the condition (MC)and the thesis follows by Theorem 3.4. (cid:3)
It remains an old interesting challenge to build up a more general bridge between theSplit and the B-W properties. We expect this analysis to be generalized to finite multiplesof spinorial representations.
Acknowledgements
I thank Roberto Longo for suggesting me the problem and useful comments and M. Bischofffor discussing Proposition 6.3. I also thank Sergio Ciamprone and Yoh Tanimoto for com-ments.
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