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The generalised principle of perturbative agreement and the thermal mass. (English) Zbl 1362.81064
In this technical paper the authors prove a conjecture of Hollands and Wald, the “principle of perturbative agreement” (PPA), in the case of a quantum field theory on a globally hyperbolic curved space-time consisting of scalar fields with quadratic interactions without derivative terms. In the rapidly developing area of formulating quantum field theory on curved space-times in a mathematically precise way, the algebraic quantum field theoretic approach seems to take a distinguished position. Using this language Hollands and Wald has recently formulated the PPA which roughly states that the perturbative and exact constructions of a field theoretic model given by the sum of a free and an exactly tractable interaction Lagrangian should agree (for the precise and rather technical formulation of the PPA cf. Sect. 3 and in particular Definition 3.3 in the article). The PPA in the case of scalar fields with quadratic interactions without derivative terms is proved through several steps (for the final result cf. Theorem 3.25 in the article). The proof rests on an application of the (classical and quantum) Möller maps which, again formally speaking, express the observables of an interacting field theory as formal power series of observables in the corresponding non-interacting theory. In Sect. 4 a more general PPA is introduced and proved by similar techniques (cf. Theorem 4.1 in the article). Finally, in Sect. 5 massive interacting KMS states on Minkowski space-time are reviewed and their construction is generalized to the massless case using methods developed by the authors.

MSC:
81T05 Axiomatic quantum field theory; operator algebras
81T20 Quantum field theory on curved space or space-time backgrounds
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
81R15 Operator algebra methods applied to problems in quantum theory
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