Nuclearity and Thermal States in Conformal Field Theory

Abstract

We introduce a new type of spectral density condition, that we call L^2-nuclearity. One formulation concerns lowest weight unitary representations of SL(2,R) and turns out to be equivalent to the existence of characters. A second formulation concerns inclusions of local observable von Neumann algebras in Quantum Field Theory. We show the two formulations to agree in chiral Conformal QFT and, starting from the trace class condition for the semigroup generated by the conformal Hamiltonian L_0, we infer and naturally estimate the Buchholz-Wichmann nuclearity condition and the (distal) split property. As a corollary, if L_0 is log-elliptic, the Buchholz-Junglas set up is realized and so there exists a beta-KMS state for the translation dynamics on the net of C*-algebras for every inverse temperature beta>0. We include further discussions on higher dimensional spacetimes. In particular, we verify that L^2-nuclearity is satisfied for the scalar, massless Klein-Gordon field.

37 pages, minor corrections

References in corpus (9)

• Modular Structure and Duality in Conformal Quantum Field Theory
• Modular localization and Wigner particles
• Extensions of Conformal Nets and Superselection Structures
• An Algebraic Spin and Statistics Theorem
• Notes for a Quantum Index Theorem
• Modular Nuclearity and Localization
• Extension of the structure theorem of Borchers and its application to half-sided modular inclusions
• Noncommutative Spectral Invariants and Black Hole Entropy
• Two-dimensional models as testing ground for principles and concepts of local quantum physics

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• The Bisognano-Wichmann property on nets of standard subspaces, some sufficient conditions
• Scale and Möbius covariance in two-dimensional Haag-Kastler net
• Modular Structure and Inclusions of Twisted Araki-Woods Algebras
• Split property for free massless finite helicity fields
• Some Results On Relative Entropy in Quantum Field Theory
• Local equivalence of representations of Diff${}^{+}(S^1)$ corresponding to different highest weights
• Rotational KMS states and type I conformal nets
• Asymptotic morphisms and superselection theory in the scaling limit II: analysis of some models