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

Alexander Strohmaier

TL;DR

The paper develops an analytic framework in which the stress-energy tensor is realized as a local field, encoded as a connection on the moduli space of globally hyperbolic metrics. This structure yields a local time-slice property and, under a central holonomy condition, unitary implementability of local metric changes and Killing flows; it also provides a concrete Klein-Gordon example in pure Hadamard states, showing smooth, Shale-criterion–compliant implementability of the metric-perturbation scattering map in Fock space. The Klein-Gordon field is shown to be a quantum field theory with a stress-energy tensor within this framework, with Hadamard-state microlocal analysis ensuring well-behaved parameter dependence and differentiability of the associated unitaries. By blending covariant AQFT methods with scattering-theory techniques, the work generalizes earlier Dimock–Wald results and offers a robust approach to stress-energy localization in curved spacetimes.

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 is a consequence 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 in which the ground state is Hadamard, 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.

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

TL;DR

The paper develops an analytic framework in which the stress-energy tensor is realized as a local field, encoded as a connection on the moduli space of globally hyperbolic metrics. This structure yields a local time-slice property and, under a central holonomy condition, unitary implementability of local metric changes and Killing flows; it also provides a concrete Klein-Gordon example in pure Hadamard states, showing smooth, Shale-criterion–compliant implementability of the metric-perturbation scattering map in Fock space. The Klein-Gordon field is shown to be a quantum field theory with a stress-energy tensor within this framework, with Hadamard-state microlocal analysis ensuring well-behaved parameter dependence and differentiability of the associated unitaries. By blending covariant AQFT methods with scattering-theory techniques, the work generalizes earlier Dimock–Wald results and offers a robust approach to stress-energy localization in curved spacetimes.

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 is a consequence 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 in which the ground state is Hadamard, 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.

Paper Structure

This paper contains 23 sections, 27 theorems, 175 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Mathematical setting and formulation of results
  4. Discussion of the conditions
  5. Possible applications
  6. Conventions
  7. Properties of theories with stress energy tensors
  8. Time reversal
  9. The local time-slice property
  10. Local implementability of Killing flows by unitaries
  11. The example of the Klein-Gordon field
  12. Pure quasifree states and Fock representations
  13. Pure Hadamard states
  14. Frequency splitting as an example
  15. Parameter dependent Hadamard states
  16. ...and 8 more sections

Key Result

Lemma 7

Assume that $\Sigma \subset M$ is any spacelike Cauchy hypersurface and $C \subset \Sigma$ is an open subset. Let $\mathcal{O}$ be any open neighborhood of $\overline{C}$ in $M$. Then, for any open subset $\mathcal{U}$ of compact closure contained in $D^+(C)$ or $D^-(C)$ there exists an open subset

Figures (3)

  • Figure 1: Illustration of the proof of Lemma \ref{['sevenlemma']}
  • Figure 2: Illustration of the proof of Theorem \ref{['lts']}
  • Figure 3: The sets $\mathcal{O},\mathcal{U},\mathcal{W}$ and the cover $\mathcal{U}_\pm,\mathcal{U}_0,\mathcal{W}$.

Theorems & Definitions (61)

  • Definition 1
  • Definition 3
  • Remark 4
  • Definition 5
  • Definition 6
  • Lemma 7
  • proof
  • Theorem 8
  • proof
  • Definition 9
  • ...and 51 more