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Within the algebraic setting of quantum field theory, a condition is givenwhich implies that the intersection of algebras generated by field operatorslocalized in wedge–shaped regions of two–dimensional Minkowski space isnon–trivial; in particular, there exist compactly localized operators in suchtheories which can be interpreted as local observables. The condition isbased on spectral (nuclearity) properties of the modular operators affiliatedwith wedge algebras and the vacuum state and is of interest in the algebraicapproach to the formfactor program, initiated by Schroer. It is illustratedhere in a simple class of examples.
There is growing evidence that algebraic quantum field theory [23] not onlyis useful in structural analysis but provides also a framework for the con-struction of models. Basic ingredients in this context are, on the one hand,the algebras affiliated with wedge shaped regions in Minkowski space, calledwedge algebras for short. On the other hand there enter the modular groupscorresponding to these algebras and the vacuum state by Tomita–Takesaki The wedge algebras are distinguished by the fact that the associated modular groups can be interpreted as unitary representations of specificPoincar´e transformations. This fact was established first by Bisognano andWichmann in the Wightman framework of quantum field theory [4] and,more recently, by Borchers in the algebraic setting [6], cf. also [22, 32]. Ittriggered attempts to construct families of such algebras directly within thealgebraic framework [9, 31].
A particularly interesting development was initiated by Schroer [35] who, starting from a given factorizing scattering matrix in two spacetime dimen-sions, recognized how one may reconstruct from these data a family of wedgealgebras satisfying locality. A complete construction of these algebras fora simple class of scattering matrices was given in [27]. These results area first important step in an algebraic approach to the formfactor program,i.e. the reconstruction of quantum fields from a scattering matrix [3, 26, 38];for more recent progress on this issue see also [1, 2, 17].
The second step in this approach consists in showing that, besides field operators localized in wedges, there appear also local observables, i.e. opera-tors which are localized in compact spacetime regions, such as double cones.
As any double cone in two dimensions is the intersection of two oppositewedges, local observables ought to be elements of the intersection of wedgealgebras. The question of whether these intersections are non–trivial turnedout to be a difficult one, however, and has not yet been settled. Some ideasas to how this problem may be tackled in models are discussed in [37].
It is the aim of the present letter to point out an alternative strategy for the proof of the non–triviality of the intersections of wedge algebras. Bycombining results scattered in the literature and casting them into a simplecondition, we will show that the non–triviality of these intersections can bededuced from spectral (nuclearity) properties of the modular operators oncertain specific subspaces of the Hilbert space. Thus the algebraic problemof determining intersections of wedge algebras amounts to a problem inspectral analysis which seems to be better tractable.
The subsequent section contains an abstract version of our nuclearity condition and a discussion of its consequences in a general algebraic setting.
In Section 3 these results are carried over to a family of theories with fac-torizing S–matrix in two–dimensional Minkowski space. It is shown thatcompactly localized operators exist in any theory complying with our con-dition. Section 4 illustrates the type of computations needed to verify thiscondition in a simple example. The article closes with a brief outlook.
2 Modular nuclearity and its consequences In this section we present our nuclearity condition in a general setting, ex-tracted from the more concrete structures in field theoretic models, anddiscuss its implications. We begin by introducing our notation and listingour assumptions.
(a) Let H be a Hilbert space and let U be a continuous unitary representa-tion of R2 acting on H. Choosing proper coordinates on R2, x = (x0, x1),the joint spectrum of the corresponding generators (P0, P1) of U is contained in the cone V+ = {(p0, p1) ∈ R2 : p0 ≥ |p1|} and there is an (up to a phaseunique) unit vector Ω ∈ H which is invariant under the action of U .
(b) There is a von Neumann algebra M ⊂ B(H) such that for each element x of the wedge W = {y ∈ R2 : |y0| + y1 < 0} the adjoint action of theunitaries U (x) induces endomorphisms of M, Moreover, Ω is cyclic and separating for M.
It is well known that, under these circumstances, the algebraic properties of M are strongly restricted. As a matter of fact, disregarding the trivialpossibility that H is one–dimensional and M = C, the following result hasbeen established in [28, Thm. 3].
Lemma 2.1. Under the preceding two conditions the algebra M is a factorof type III1 according to the classification of Connes.
It immediately follows from this result that the algebras M(x) are factors of type III1 as well. Little is known, however, about the algebraic structureof the relative commutants M(x)′ ∩ M of M(x) in M, x ∈ W . Eventhe question of whether these relative commutants are non–trivial has notbeen settled in this general setting. Yet this question turns out to havean affirmative answer and, as a matter of fact, the algebraic structures arecompletely fixed if the inclusions are split, i.e. if for each x ∈ W thereis a factor N of type I∞ such that First, the split property implies that M is isomorphic to the unique [24] hyperfinite factor of type III1. We briefly recall here the argument: As Ω iscyclic and separating for M, and hence for M(x), this is also true for N . Itfollows that N , being of type I∞, is separable in the ultraweak topology andconsequently H is separable, cf. [21, Prop. 1.2]. Now, as U is continuous, Mis continuous from the inside, M = implies that M can be approximated from the inside by separable type I∞factors and therefore is hyperfinite, cf. [11, Prop. 3.1]. Knowing also that itis of type III1, the assertion follows.
Secondly, the split property implies that M(x)′ ∩ M, x ∈ W , is iso- morphic to the hyperfinite factor of type III1 as well. This can be seen asfollows [21]. On a separable Hilbert space H, any factor of type III has cyclicand separating vectors [34, Cor. 2.9.28]. Moreover, for any von Neumannalgebra on H with a cyclic and a separating vector there exists a dense Gδset of vectors which are both, cyclic and separating [20]. Now, taking intoaccount that N is isomorphic to B(H), the relative commutant M(x)′ ∩ Nof the type III factor M(x) in N is (anti)isomorphic to M(x) by Tomita–Takesaki theory. It is therefore of type III and has cyclic vectors in H. Thisholds a fortiori for M(x)′ ∩ M ⊃ M(x)′ ∩ N and, as Ω is separating forM, the relative commutant M(x)′ ∩ M has a dense Gδ set of cyclic andseparating vectors. But the intersection of a finite number of dense Gδ setsis non–empty. So we conclude that the triple M, M(x) and M(x)′ ∩ Mhas a joint cyclic and separating vector in H. The inclusion is thus astandard split inclusion according to the terminology in [21]. In particular,there is a spatial isomorphism mapping M(x) on H ⊗ H [19]. By taking commutants, we conclude that M(x)′ ∩ M isisomorphic to M(x)′ ⊗ M, x ∈ W . The statement about the algebraicstructure of the relative commutant then follows.
It seems difficult, however, to establish the existence of intermediate type I∞ factors N in the inclusions for concretely given {M, U, H}, andthis may be the reason why this strategy of establishing the non–triviality ofrelative commutants has been discarded in [37]. Yet the situation is actuallynot hopeless, the interesting point being that the existence of the desiredfactors can be derived from spectral properties of the modular operator ∆affiliated with the pair (M, Ω). Recalling that a linear map from a Banachspace into another one is said to be nuclear if it can be decomposed intoa series of maps of rank one whose norms are summable, we extract thefollowing pertinent condition from [12].
(c) Modular Nuclearity Condition: For any given x ∈ W the map is nuclear. Equivalently, since ∆1/4 is invertible, the image of the unit ballin M(x) under this map is a nuclear subset of H.
Since Ω is cyclic and separating for M and the algebras M(x), M both are factors, it follows from the modular nuclearity condition (c) that theinclusions M(x) ⊂ M, x ∈ W , are split [12, Thm. 3.3]. Conversely, if theseinclusions are split, the map has to be compact, at least. Thus a proofof the split property amounts to a spectral analysis of the operator ∆1/4on the subspaces M(x) Ω ⊂ H. This task is, as we shall see, manageable inconcrete applications. We summarize the results of the preceding discussionin the following proposition.
Proposition 2.2. Let {M, U, H} be a triple satisfying conditions (a), (b)and (c), stated above. Then, for x ∈ W ,(i) the inclusion M(x) ⊂ M is split; (ii) the relative commutant M(x)′ ∩ M is isomorphic to the unique hyper-finite type III1 factor. In particular, it has cyclic and separating vectors.
We conclude this section by noting that any triple {M, U, H} as in the preceding proposition can be used to construct a non–trivial Poincar´e co-variant net of local algebras on two–dimensional Minkowski space R2. Fol-lowing closely the discussion in [5, 6], we first note that the modular group∆is, s ∈ R, and the modular conjugation J affiliated with (M, Ω) can beinterpreted as representations of proper Lorentz transformations Λ (havingdeterminant one). More specifically, if Λ is any such transformation andΛ = (−1)σB(θ) its polar decomposition, where σ ∈ {0, 1} and B(θ) is aboost with rapidity θ ∈ R, one can show that defines a continuous (anti)unitary representation of the proper Poincar´egroup [5]. Moreover, Ω is invariant under the action of these operators and may thus be interpreted as a vacuum state. Setting R(ΛW + x) =U (x, Λ)MU (x, Λ)−1, one obtains a local (as a matter of fact, Haag–dual)Poincar´e covariant net of wedge algebras on R2. Denoting the double cones in R2 by Cx,y = (−W + x) ∩ (W + y), x − y ∈ W , the corresponding algebras R(Cx,y) = R(W + x)′ ∩ R(−W + y)′ = M(x)′ ∩ M(y) are non–trivial according to the preceding proposition. As was shown in [5],they form a local net on R2 which is relatively local to the wedge algebrasand transforms covariantly under the adjoint action of U (x, Λ). It may thusbe interpreted as a net of local observables. The vacuum vector Ω neednot be cyclic for the local algebras, however. In fact, thinking of theoriesexhibiting solitonic excitations of Ω which are localized in wedge regions,this may also not be expected in general.
We carry over now the results of the preceding section to the framework oftwo–dimensional models and indicate their significance for the formfactorprogram, i.e. the reconstruction of local observables and fields from a givenfactorizing scattering matrix.
For the sake of concreteness, we restrict attention here to the theory of a single massive particle with given two–particle scattering function S2, asconsidered in [35,36] and described in more detail in [27]. The Hilbert spaceof the theory is conveniently represented as the S2–symmetrized Fock spaceH = n. Here the subspace H0 consists of multiples of the vacuum vector Ω and, using the parameterization of the mass shell by the rapidityθ, the single particle space H1 can be identified with the space of square inte-grable functions θ → Ψ1(θ) with norm given by The elements of the n–particle space Hn are represented by square integrablefunctions θ1 . . . θn → Ψn(θ1, . . . , θn) which are S2–symmetric, Ψn(θ1, . . . , θi+1, θi, . . . , θn) = S2(θi−θi+1) Ψn(θ1, . . . , θi, θi+1, . . . , θn). (3.3) Here ζ → S2(ζ) is the scattering function which is continuous and boundedon the strip {ζ ∈ C : 0 ≤ Im ζ ≤ π}, analytic in its interior and satisfies, forθ ∈ R, the unitarity and crossing relations S2(θ)−1 = S2(θ) = S2(−θ) = S2(θ + iπ).
On H there acts a continuous unitary representation U of the proper or-thochronous Poincar´e group, given by Ψn(θ1 − θ, . . . , θn − θ). (3.5) It satisfies the relativistic spectrum condition, i.e. the joint spectrum of thegenerators P = (P0, P1) of the translations U (R2, 1) is contained in V+.
Moreover, there is an antiunitary operator J on H, representing the PCTsymmetry. It is given by J Ψ n(θ1, . . . , θn) = Ψn(θn, . . . , θ1).
As in the case of the bosonic and fermionic Fock spaces, one can define creation and annihilation operators z†(θ), z(θ) (in the sense of operatorvalued distributions) on the dense subspace D ⊂ H of vectors with a finiteparticle number. They are hermitian conjugates with respect to each otherand satisfy the Fadeev–Zamolodchikov relations z†(θ)z†(θ′) = S2(θ − θ′) z†(θ′)z†(θ), z(θ)z(θ′) = S2(θ − θ′) z(θ′)z(θ), z(θ)z†(θ′) = S2(θ′ − θ) z†(θ′)z(θ) + δ(θ − θ′) 1.
Their action on D is fixed by the equations (z†(θ1) . . . z†(θn) Ω, Ψ) = (n!)1/2 Ψn(θ1, . . . , θn), With the help of these creation and annihilation operators one can defineon D a field φ, setting and we adopt the convention that, both, z†( · ) and z( · ) are complex linearon the space of test functions.
It has been shown in [27] that φ transforms covariantly under the adjoint action of the proper orthochronous Poincar´e group, where fx,B(y) = f (B−1(y − x)), y ∈ R2. Moreover, φ is real, φ(f )∗ ⊃ φ(f ),and each vector in D is entire analytic for the operators φ(f ). Since Dis stable under their action, these operators are essentially selfadjoint onthis domain for real f . We mention as an aside that the fields φ(f ) arepolarization–free generators in the sense of [7].
Denoting the selfadjoint extensions of φ(f ), f real, by the same symbol, R(W + x) = {eiφ(f) : suppf ⊂ W + x}′′, where W denotes, as before, the wedge W = {y ∈ R2 : |y0| + y1 < 0}. Withthe help of the PCT operator J one can also define algebras correspondingto the opposite wedges, Now, given an arbitrary proper Lorentz transformation Λ with polar decomposition Λ = (−1)σB, σ ∈ {0, 1}, one obtains a representation of the proper Poincar´e group, setting U (x, Λ) = U (x, B)Jσ. It then follows fromthe covariance properties of the field that U (x, Λ) R(±W + y)U (x, Λ)−1 = R(±(−1)σW + Λy + x), taking into account that the wedge W is stable under the action of boosts.
So, by this construction, one arrives at a Poincar´e covariant net of wedgealgebras on two–dimensional Minkowski space.
It has been shown in [27] that this net is local, and that Ω is cyclic and separating for the wedge algebras (and hence fortheir commutants).
The triple {R(W ), U (R2, 1), H} satisfies conditions (a) and (b) given in the preceding section. More can be said by making use of modular theoryand certain specific domain properties of the field φ.
Proposition 3.1. Let R(W ) be the algebra defined above. Then (i) the modular group and conjugation affiliated with (R(W ), Ω) are givenby R ∋ λ → U (0, B(2πλ)) and J, respectively; (ii) R(W )′ = R(−W ) (Haag duality).
Proof. Let ∆W , JW be the modular operator and conjugation, respectively,affiliated with (R(W ), Ω). It follows from modular theory that any boostU (0, B) commutes with ∆W and JW since Ω is invariant and R(W ) is stable under its (adjoint) action. Hence λ → V (λ) = U (0, B(2πλ))∆−iλ is a continuous unitary representation of R with the latter properties. Moreover,V (λ) commutes with all boosts U (0, B) and, by a theorem of Borchers [5],also with all translations U (x, 1). Since the restriction of U to the properorthochronous Poincar´e group acts irreducibly on H1, one concludes thatV (λ) ↾ H1 = eiλc 1 for fixed real c and any λ ∈ R.
Now, for real f with suppf ⊂ W , φ(f ) is a selfadjoint operator affiliated with R(W ), and the same holds for φλ(f ) = V (λ)φ(f )V (λ)−1, λ ∈ R,because of the stability of R(W ) under the adjoint action of V (λ). So bothoperators commute with all elements of R(W )′. Since Ω is invariant underthe action of V (λ)−1 and since φ(f )Ω ∈ H1, the preceding result implies It will be shown below that the dense set of vectors R(W )′Ω is a core,both, for φ(f ) and φλ(f ). Hence φλ(f ) = eiλcφ(f ) which, in view of theselfadjointness of the field operators, is only possible if c = 0. This holdsfor any choice of f within the above limitations, so V (λ) acts trivially onR(W ). Taking also into account that Ω is cyclic for R(W ), one arrives atV (λ) = 1, λ ∈ R, from which the first part of statement (i) follows.
Similarly, modular theory and the theorem of Borchers mentioned above imply that the unitary operator I = JW J commutes with all Poincar´e trans-formations U (x, B) and, taking into account relation , one also has IR(W )I−1 ⊂ R(W ). Hence, putting φI (f ) = Iφ(f )I−1, one finds by thesame reasoning as in the preceding step that φI(f ) = φ(f ). Thus I = 1,proving the second part of statement (i). The statement about Haag dualitythen follows from the equalities R(W )′ = JW R(W )JW = JR(W )J = R(−W ).
It remains to prove the assertion that R(W )′Ω is a core for the selfadjoint operators φ(f ), φλ(f ) and φI(f ), respectively. To this end one makes useof bounds, given in [27], on the action of the field operators on n–particlestates Ψn. One has φ(f )Ψn ≤ cf (n+1)1/2 Ψn , where cf is some constantdepending only on f . Since the field operators change the particle numberat most by ±1, one can proceed from this estimate to corresponding boundsfor Ψ ∈ D, given by φ(f )Ψ ≤ 2cf (N + 1)1/2Ψ , where N is the particlenumber operator. Recalling that P0 denotes the (positive) generator of thetime translations, it is also clear that m (N + 1) ≤ (P0 + m1). So forΨ ∈ D ∩ D0, where D0 is the domain of P0, one arrives at the inequalities φ(f )Ψ ≤ 2cf (N + 1)1/2Ψ ≤ 2m−1/2cf (P0 + m1)1/2Ψ .
It follows from this estimate by standard arguments that any core for P0 isalso a core for the field operators φ(f ). Since the unitary operators V (λ) andI in the preceding steps were shown to commute with the time translations,this domain property is also shared by the transformed field operators φλ(f )and φI(f ), respectively.
In order to complete the proof, one has only to show that R(W )′Ω ∩ D0 is a core for P0. Now R(W )′Ω is mapped into itself by all translations U (x),x ∈ −W . Hence, taking into account the invariance of Ω under translations,one finds that f (P )R(W )′Ω ⊂ R(W )′Ω ∩ D0 for any test function f withsuppf ⊂ −W . But this space of functions contains elements f such thatf (P ) is invertible. Hence (P0 ± i1)f (P )R(W )′Ω ⊂ (P0 ± i1)(R(W )′Ω ∩ D0)both are dense subspaces of H, proving the statement.
In view of the covariance properties of the net, it is apparent that anal- ogous statements hold for all wedge algebras. Thus the only point left openin this reconstruction of a relativistic quantum field theory from scatteringdata is the question of whether the wedge algebras contain operators whichcan be interpreted as observables localized in finite spacetime regions, such as the double cones Cx,y = (W + y) ∩ (−W + x), x − y ∈ W . By Einsteincausality, observables localized in Cx,y have to commute with all operatorslocalized in the adjacent wedges W + x and −W + y. They are thereforeelements of the algebra R(Cx,y) = R(W + x)′ ∩ R(−W + y)′ = R(−W + x) ∩ R(W + y). (3.19) It follows from the properties of the wedge algebras established thus far thatthe resulting map C → R(C) from double cones to von Neumann algebrasdefines a local and Poincar´e covariant net on Minkowski space. So if thetheory describes local observables, the algebras R(C) are to be non–trivial.
At this point the nuclearity condition formulated in Sec. 2 comes in.
Knowing by the preceding proposition the explicit form of the modularoperator affiliated with (R(W ), Ω) and taking into account the invarianceof Ω under spacetime translations, we are led to consider, for given x ∈ W ,the maps Within the present context one then has the following more concrete versionof Proposition Proposition 3.2. Let the maps be nuclear, x ∈ W . Then (i) the net of wedge algebras has the split property; (ii) for any open double cone C ⊂ R2 the corresponding algebra R(C) isisomorphic to the unique hyperfinite factor of type III1. In particular it hascyclic and separating vectors.
So in order to establish the existence of local operators in the theory, one needs an estimate of the size of the set of vectors i.e. the image of the unit ball R(W )1 under the action of the map . Webriefly indicate here the steps required in such an analysis which are similarto those carried out in [8] in an investigation of the Haag–Swieca compact-ness condition; a more detailed account of these results will be presentedelsewhere.
Making use of the localization properties of the operators A ∈ R(W ) and the analyticity properties of the scattering function S2, one can showthat the n–particle wave functions θ1, . . . θn → (AΩ)n(θ1, . . . θn) extend, in the sense of distributions, to analytic functions in the domain0 < Im θi < π, −δ < Im (θi − θk) < δ, where i, k = 1, . . . n and δ depends onthe domain of analyticity of the scattering function S2. Thus the functions θ1, . . .θn → (U (0, B(−iπ/2)) AΩ)n(θ1, . . . θn) = (AΩ)n(θ1 + iπ/2, . . . θn + iπ/2) are analytic in the domain −δ/n < Im θi < δ/n, i = 1, . . . n. As a matter offact, if A ∈ R(W )1, the family of these functions turns out to be uniformlybounded (normal) on this domain. Taking also into account that U is arepresentation of the Poincar´e group, one obtains for x ∈ W the equality U (0, B(−iπ/2)) U (x, 1) AΩ = ex1P0−x0P1 U (0, B(−iπ/2)) AΩ, so the n–particle components of the vectors have wave functions ofthe form θ1, . . . θn → (U (0, B(−iπ/2)) U (x, 1) AΩ)n(θ1, . . . θn) 1 ch(θk ) − x0 sh(θk )) (AΩ)n(θ1 + iπ/2, . . . θn + iπ/2).
Since, for x ∈ W , the exponential factor gives rise to a strong damping oflarge rapidities, it follows from the preceding results that the wave functionsform, for A ∈ R(W )1, a bounded subset of the space of test functionsS(Rn) and hence a nuclear subset of L2(Rn) = Hn. Moreover, taking intoaccount the spectral properties of (P0, P1), relation combined withthe estimate U (0, B(−iπ/2)) AΩ ≤ A following from modular theoryimplies (U (0, B(−iπ/2)) U (x, 1) AΩ)n ≤ en m(x1+|x0|), So these norms tend rapidly to 0 for large n ∈ N if x ∈ W . Combining thesefacts, one finds after a moments reflection that the sets are relativelycompact in H, implying that the maps are compact. So they canbe approximated with arbitrary precision by finite sums of maps of rankone. In order to prove that they are also nuclear, one needs more refinedestimates, however.
In order to illustrate the quantitative estimates needed for the proof thatthe map is nuclear, we consider here the case of trivial scattering,S2 = 1, i.e. the theory of a free massive Bose field φ. There the combinatorialproblems appearing in the analysis of the size of sets of the type havebeen settled in [15] and we shall make use of these results here.
We begin by recalling some well known facts: The restrictions of the field φ to the time zero plane are operator valued distributions on the domain D. These time zero fields, commonly denotedby ϕ and π, satisfy canonical equal time commutation relations. If smearedwith test functions h having support in the interval (−∞, 0), they generatethe von Neumann algebra R(W ) and, applying them to the vacuum vectorΩ, they create closed subspaces Lϕ(W ), Lπ(W ) of the single particle spaceH1 given by Lϕ(W ) = {θ → h(m sh(θ)) : supp h ⊂ (−∞, 0)}−, Lπ(W ) = {θ → ch(θ) h(m sh(θ)) : supp h ⊂ (−∞, 0)}−, where the tilde denotes Fourier transformation. We also consider the shifted subspaces Lϕ(W + x) = U (x, 1) Lϕ(W ) and Lπ(W + x) = U (x, 1) Lπ(W )and denote the corresponding orthogonal projections by Eϕ(W + x) andEπ(W + x), respectively. After these preparations we are in a position toapply the results in [15, Thm. 2.1] which we recall here for the convenienceof the reader in a form appropriate for the present investigation.
Lemma 4.1. Consider the theory with scattering function S2 = 1 and letEϕ(W + x) U (0, B(−iπ/2)) and Eπ(W + x) U (0, B(−iπ/2)) be trace classoperators with operator norms less than 1, x ∈ W . Then the sets arenuclear.
Thus the proof that the modular nuclearity condition is satisfied in thepresent theory reduces to a problem of spectral analysis in the single particlespace H1. We first turn to the task of providing estimates of the norms ofthe operators appearing in the lemma.
Let Φh ∈ Lϕ(W ) be a vector with wave function θ → Φh(θ) = h(m sh(θ)), where h is, as before, a test function with support in (−∞, 0). Because ofthese support properties, Φh lies in the domain of all boosts U(0, B(θ)) forcomplex θ with −π ≤ Im θ ≤ 0. Furthermore, as sh(θ + iπ) = sh(−θ),one has (U (0, B(−iπ))Φh)(θ) = h(m sh(−θ)) = Φh(−θ). But this implies U (0, B(−iπ)) Φh = Φh and consequently U(0, B(−iπ/2)) Φh ≤ Φh .
Making use now of the properties of the representation U , one obtains theestimate, x ∈ W , U (0, B(−iπ/2)) U (x, 1) Φh = ex1P0−x0P1 U(0, B(−iπ/2)) Φh ≤ em(x1+|x0|) U (0, B(−iπ/2)) Φh ≤ em(x1+|x0|) U(x, 1) Φh .
Since Φh was arbitrary within the above limitations and (x1 + |x0|) is nega-tive, this yields the norm estimate U (0, B(−iπ/2)) Eϕ(W +x) < 1, x ∈ W .
But the adjoint operator Eϕ(W + x) U (0, B(−iπ/2)) has the same norm, sothe desired bound follows. In a similar manner one can show that the oper-ator Eπ(W + x) U (0, B(−iπ/2)) also has norm less than 1.
It remains to establish the trace class property of these operators. To this end we consider the restriction of the operator U (0, B(−iπ/2)) U (x, 1),x ∈ W , to the subspaces Lϕ(W ) and Lπ(W ), respectively. Let, as before,Φh ∈ Lϕ(W ), then (U (0, B(−iπ/2)) U (x, 1) Φh)(θ) = ex1p0(θ) − x0p1(θ) Φh(θ + iπ/2).
Making use of the analyticity and boundedness properties of θ → Φh(θ) andthe fact that Φh(θ + iπ) = Φh(−θ), one can represent Φh(θ + iπ/2) by aCauchy integral, Next, for x ∈ W , let Xϕ be the operator on H1 with kernel Being the sum of products of multiplication operators in rapidity space,respectively its dual space, which are bounded and rapidly decreasing, itis apparent that Xϕ is of trace class. Moreover, by the preceding results,U (0, B(−iπ/2)) Eϕ(W + x) = Xϕ Eϕ(W ) U (x, 1)−1. Since the trace classoperators form a *–ideal in B(H1), if follows that U (0, B(−iπ/2)) Eϕ(W +x)and its adjoint Eϕ(W + x) U (0, B(−iπ/2)) are of trace class.
By a similar argument one can also establish the trace class property of Eπ(W + x) U (0, B(−iπ/2)), the only difference being that for vectorsΦh ∈ Lπ(W ) with wave functions θ → Φh(θ) = ch(θ) h(m sh(θ)) one nowhas Φh(θ + iπ) = −Φh(−θ). As a consequence, the sum in relation turns into a difference, but this does not affect the conclusions. So thefollowing statement has been proven.
Proposition 4.2. In the theory with scattering function S2 = 1, the setsand corresponding maps We thus have verified in the present model the modular nuclearity con- dition for wedge algebras with all of its consequences. In particular, thewedge algebras have the split property. Although the latter fact was knownbefore [29], there did not yet exist a proof in the literature.
By similar arguments one can also treat the theory with scattering func- tion S2 = −1, i.e. the theory of a (non–local) free massive Fermi field. Thereone expects that the sets are somewhat smaller than in the presentcase because of the Pauli principle. It is even more challenging, however, toprovide a quantitative estimate of the size of the sets in theories withgeneric scattering function. This problem will be tackled elsewhere.
Within the algebraic setting of quantum field theory, we have presented amethod which allows one to decide whether algebras affiliated with wedgeshaped regions in two–dimensional Minkowski space contain compactly lo-calized operators. This method seems to be particularly useful for provingthe existence of local operators in theories with factorizing S–matrix. It isthus complementary to the formfactor program, where one tries to exhibitsuch operators explicitly by solving an infinite system of equations.
The upshot of the present investigation is the insight that the basic al- gebraic problem of checking locality, which amounts to computing relativecommutants, can be replaced by an analysis of spectral properties of repre-sentations of the Poincar´e group. There exist other methods by which thecrucial intermediate step in our argument, the proof of the split property ofwedge algebras, could be accomplished [10,13,16,18,19,30]. But the presentapproach requires less a priori information about the underlying theory andalso seems better managable in concrete applications. Moreover, in viewof the fact that it relies only on the modular structure, it is applicable totheories on arbitrary spacetime manifolds.
It is apparent, however, that the split property of wedge algebras is in general an unnecessarily strong requirement if one is merely interested in theexistence of compactly localized operators. As a matter of fact, it followsfrom an argument of Araki that it cannot hold in more than two spacetimedimensions, cf. [10, Sec. 2]. It would therefore be desirable to establish lessstringent conditions which still imply that the relative commutant fixed by agiven inclusion of von Neumann algebras is non–trivial. The present resultsseem to suggest that this information is encoded in spectral properties of thecorresponding modular operators, but a clarification of this point requiressome further analysis.
An appropriately weakened condition which would allow one to establish the existence of local operators in non–local algebras also in higher dimen-sions would have several interesting applications. This existence problemwas recently met in the context of theories of massless particles with in-finite spin [33], for example. It also appears in the algebraic approach tothe construction of theories of particles with anyonic statistics [32] and theconstruction of nets of wedge algebras from information on the modulardata [6, 14, 25, 39, 40]. A solution of this problem would thus be a majorstep in the algebraic approach to constructive problems in local quantumphysics.
We would like to thank the Deutsche Forschungsgemeinschaft for financialsupport.
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