The CPT-theorem in two-dimensional theories of local observables

Borchers, H. J.

doi:10.1007/bf02099011

Cited by 178 publications

(242 citation statements)

References 21 publications

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“…A2) and A3) are abstract characterizations of M as an algebra of observables localized in W R [11,17]. It should be mentioned that the assumptions A1)-A3) put strict constraints on the algebraic structure of M. The following result has been found in [47,26].…”

Section: A3)mentioning

confidence: 91%

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Construction of Quantum Field Theories with Factorizing S-Matrices

Lechner¹

2007

Commun. Math. Phys.

122

405

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A new approach to the construction of interacting quantum field theories on two-dimensional Minkowski space is discussed. In this program, models are obtained from a prescribed factorizing S-matrix in two steps. At first, quantum fields which are localized in infinitely extended, wedge-shaped regions of Minkowski space are constructed explicitly. In the second step, local observables are analyzed with operator-algebraic techniques, in particular by using the modular nuclearity condition of Buchholz, d'Antoni and Longo.Besides a model-independent result regarding the Reeh-Schlieder property of the vacuum in this framework, an infinite class of quantum field theoretic models with non-trivial interaction is constructed. This construction completes a program initiated by Schroer in a large family of theories, a particular example being the Sinh-Gordon model. The crucial problem of establishing the existence of local observables in these models is solved by verifying the modular nuclearity condition, which here amounts to a condition on analytic properties of form factors of observables localized in wedge regions.It is shown that the constructed models solve the inverse scattering problem for the considered class of S-matrices. Moreover, a proof of asymptotic completeness is obtained by explicitly computing total sets of scattering states. The structure of these collision states is found to be in agreement with the heuristic formulae underlying the Zamolodchikov-Faddeev algebra.

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Section: A3)mentioning

confidence: 91%

“…Keeping this structure of M in mind, let us recall how a net O → A(O) of local algebras can be constructed from the data (M, U, H) [11,17].…”

Section: Lemma 21 Consider a Triple (M U H) Satisfying The Assumptmentioning

confidence: 99%

Construction of Quantum Field Theories with Factorizing S-Matrices

Lechner¹

2007

Commun. Math. Phys.

122

405

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“…that we shall below describe in equations (6,7). We have used, only here, the notation udϕ ≡ ϕ(u) in order to provide resemblance with the classical context.…”

Section: O → A(o)mentioning

confidence: 99%

Notes for a Quantum Index Theorem

Longo¹

2001

Communications in Mathematical Physics

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We view DHR superselection sectors with finite statistics as Quantum Field Theory analogs of elliptic operators where KMS functionals play the role of the trace composed with the heat kernel regularization. We extend our local holomorphic dimension formula and prove an analogue of the index theorem in the Quantum Field Theory context. The analytic index is the Jones index, more precisely the minimal dimension, and, on a 4-dimensional spacetime, the DHR theorem gives the integrality of the index. We introduce the notion of holomorphic dimension; the geometric dimension is then defined as the part of the holomorphic dimension which is symmetric under charge conjugation. We apply the AHKT theory of chemical potential and we extend it to the low dimensional case, by using conformal field theory. Concerning Quantum Field Theory on curved spacetime, the geometry of the manifold enters in the expression for the dimension. If a quantum black hole is described by a spacetime with bifurcate Killing horizon and sectors are localizable on the horizon, the logarithm of the holomorphic dimension is proportional to the incremental free energy, due to the addition of the charge, and to the inverse temperature, hence to the surface gravity in the Hartle-Hawking KMS state. For this analysis we consider a conformal net obtained by restricting the field to the horizon ("holography"). Compared with our previous work on Rindler spacetime, this result differs inasmuch as it concerns true black hole spacetimes, like the Schwarzschild-Kruskal manifold, and pertains to the entropy of the black hole itself, rather than of the outside system. An outlook concerns a possible relation with supersymmetry and noncommutative geometry.

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“…By Borchers theorem [5], ∆ it has the same commutation relations (1.13) as U (t) with T (a), thus e itL commutes with T (a). Therefore if…”

Section: Case Where Further Symmetries Are Presentmentioning

confidence: 97%

Natural Energy Bounds in Quantum Thermodynamics

Guido

Longo

2001

Communications in Mathematical Physics

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Given a stationary state for a noncommutative flow, we study a boundedness condition, depending on a parameter β > 0, which is weaker than the KMS equilibrium condition at inverse temperature β. This condition is equivalent to a holomorphic property closely related to the one recently considered by Ruelle and D'Antoni-Zsido and shared by a natural class of non-equilibrium steady states. Our holomorphic property is stronger than the Ruelle's one and thus selects a restricted class of non-equilibrium steady states. We also introduce the complete boundedness condition and show this notion to be equivalent to the Pusz-Woronowicz complete passivity property, hence to the KMS condition.In Quantum Field Theory, the β-boundedness condition can be interpreted as the property that localized state vectors have energy density levels increasing β-subexponentially, a property which is similar in the form and weaker in the spirit than the modular compactness-nuclearity condition. In particular, for a Poincaré covariant net of C * -algebras on the Minkowski spacetime, the β-boundedness property, β ≥ 2π, for the boosts is shown to be equivalent to the Bisognano-Wichmann property. The Hawking temperature is thus minimal for a thermodynamical system in the background of a Rindler black hole within the class of β-holomorphic states. More generally, concerning the Killing evolution associated with a class of stationary quantum black holes, we characterize KMS thermal equilibrium states at Hawking temperature in terms of the boundedness property and the existence of a translation symmetry on the horizon.

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The CPT-theorem in two-dimensional theories of local observables

Cited by 178 publications

References 21 publications

Construction of Quantum Field Theories with Factorizing S-Matrices

Construction of Quantum Field Theories with Factorizing S-Matrices

Notes for a Quantum Index Theorem

Natural Energy Bounds in Quantum Thermodynamics

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