An analytic approach to the stress energy tensor in quantum field theory

Abstract

We discuss a framework for quantum fields in curved spacetimes that possess a stress energy tensor as a connection one form on a suitable moduli space of metrics. In generic spacetimes the existence of such a tensor is thought to be a replacement for the existence of symmetries that the Minkowski theory relies on. It is shown that the local time-slice property and the implementability of local isometries are consequences of the existence of a stress energy tensor that is a local field. We prove that the Klein-Gordon field, in an irreducible Fock representation determined by a quasifree Hadamard state, is an example. In this example we show that the scattering matrix for compactly supported metric perturbations exists in the Fock space and is smooth on a dense set with respect to the perturbation parameter. This generalises results by Dimock and Wald. As a tool we also establish the precise microlocal properties of parameter dependent fundamental solutions.

49 pages, typos and some sign mistakes corrected, fixed technical inaccuracy in Lemma 33, restructured proof of Lemma 8, missing term in Remark 27 fixed. Definition 7 is explained better