Duality and absence of locally generated superselection sectors for CCR-type algebras

Driessler, W.

doi:10.1007/bf01200052

Cited by 32 publications

(36 citation statements)

References 11 publications

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“…It is obvious that local quasiequivalence then holds for S. Also, local definiteness holds in this case, as was proved by Wightman [72]. If Haag-duality holds in the vacuum representation (which, as indicated above, can be assumed to hold quite generally), then it does not follow automatically that all pure states locally quasiequivalent to ω 0 will also have GNS representations fulfilling Haag-duality; however, it follows once some regularity conditions are satisfied which have been checked in certain quantum field models [19,61]. So far there seems to be no general physically motivated criterion enforcing local primarity of a quantum field theory in algebraic formulation in Minkowski spacetime.…”

Section: Local Definitenessmentioning

confidence: 86%

Continuity of Symplectically Adjoint Maps and the Algebraic Structure of Hadamard Vacuum Representations for Quantum Fields on Curved Spacetime

Verch

1997

Rev. Math. Phys.

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We derive for a pair of operators on a symplectic space which are adjoints of each other with respect to the symplectic form (that is, they are sympletically adjoint) that, if they are bounded for some scalar product on the symplectic space dominating the symplectic form, then they are bounded with respect to a one-parametric family of scalar products canonically associated with the initially given one, among them being its "purification". As a typical example we consider a scalar field on a globally hyperbolic spacetime governed by the Klein-Gordon equation; the classical system is described by a symplectic space and the temporal evolution by symplectomorphisms (which are symplectically adjoint to their inverses). A natural scalar product is that inducing the classical energy norm, and an application of the above result yields that its "purification" induces on the one-particle space of the quantized system a topology which coincides with that given by the two-point functions of quasifree Hadamard states. These findings will be shown to lead to new results concerning the structure of the local (von Neumann) observable-algebras in representations of quasifree Hadamard states of the Klein-Gordon field in an arbitrary globally hyperbolic spacetime, such as local definiteness, local primarity and Haag-duality (and also split-and type III 1properties). A brief review of this circle of notions, as well as of properties of Hadamard states, forms part of the article. * Supported by a Von Neumann Fellowship of the Operator Algebras Network, EC Human Capital and Mobility Programme. 1 in a globally hyperbolic spacetime with respect to a certain topology on the Cauchy-data space. (Here, ∇ is the covariant derivative of the metric g on the spacetime, and r an arbitrary realvalued, smooth function.) The Cauchy-data space is a symplectic space on which the said temporal evolution is realized by symplectomorphisms. It turns out that the classical "energy-norm" of solutions of (1.1), which is given by a scalar product µ 0 on the Cauchy-data space, and the topology relevant for the required continuity statement (the "Hadamard oneparticle space norm"), induced by a scalar product µ 1 on the Cauchy-data space, are precisely in the relation for which our result on relative µ 0 − µ 1 continuity of symplectically adjoint maps applies. Since the continuity of the Cauchy-data evolution in the classical energy norm, i.e. µ 0 , is well-known, the desired continuity in the µ 1 -topology follows.The argument just described may be viewed as the prime example of application of the relative continuity result. In fact, the relation between µ 0 and µ 1

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Section: Local Definitenessmentioning

confidence: 86%

Continuity of Symplectically Adjoint Maps and the Algebraic Structure of Hadamard Vacuum Representations for Quantum Fields on Curved Spacetime

Verch

1997

Rev. Math. Phys.

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“…For applications of the split property to the construction of local current algebras and a quantum version of Noether's theorem, see the publications [13][14][15]. Some implications relating to the superselection structure of models are discussed in [10,16,17].…”

Section: Introductionmentioning

confidence: 98%

Causal independence and the energy-level density of states in local quantum field theory

1986

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Abstract. Within the general framework of local quantum field theory a physically motivated condition on the energy-level density of well-localized states is proposed and discussed. It is shown that any model satisfying this condition obeys a strong form of the principle of causal (statistical) independence, which manifests itself in a specific algebraic structure of the local algebras ("split property"). It is also shown that the proposed condition holds in a free field theory.

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“…Indeed, the results for interacting theories [13], [14] are restricted to two space-time dimensions and rely on the corresponding properties for free fields. The proof of duality for free Bose fields was first given by Araki and falls naturally into two parts.…”

Section: Introductionmentioning

confidence: 99%

Abstract Twisted Duality for Quantum Free Fermi Fields

Foit¹

1983

Publ. Res. Inst. Math. Sci.

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Duality and absence of locally generated superselection sectors for CCR-type algebras

Cited by 32 publications

References 11 publications

Continuity of Symplectically Adjoint Maps and the Algebraic Structure of Hadamard Vacuum Representations for Quantum Fields on Curved Spacetime

Continuity of Symplectically Adjoint Maps and the Algebraic Structure of Hadamard Vacuum Representations for Quantum Fields on Curved Spacetime

Causal independence and the energy-level density of states in local quantum field theory

Abstract Twisted Duality for Quantum Free Fermi Fields

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