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Analytic states in quantum field theory on curved spacetimes

Alexander Strohmaier, Edward Witten

TL;DR

The paper develops a microlocal-analytic framework for quantum fields on real analytic curved spacetimes, introducing weak analytic states via high-energy localization properties encoded in the analytic wavefront set. By leveraging the FBI transform and uniform microlocalisation, it proves that analytic states satisfy the Reeh–Schlieder property and a timelike tube theorem, extending Borchers–Araki-type results from flat to curved backgrounds. The analysis also connects to tempered analyticity, microlocal spectrum conditions, and extensions to internal degrees of freedom, providing a robust structure for energy localization and causal restrictions in curved spacetimes. Overall, the work links microlocal analytic methods to foundational QFT properties in an analytic setting, with potential implications for interacting fields and Hadamard-type states.

Abstract

We discuss high energy properties of states for (possibly interacting) quantum fields in curved spacetimes. In particular, if the spacetime is real analytic, we show that an analogue of the timelike tube theorem and the Reeh-Schlieder property hold with respect to states satisfying a weak form of microlocal analyticity condition. The former means the von Neumann algebra of observables of a spacelike tube equals the von Neumann algebra of observables of a significantly bigger region, that is obtained by deforming the boundary of the tube in a timelike manner. This generalizes theorems by Borchers and Araki to curved spacetimes.

Analytic states in quantum field theory on curved spacetimes

TL;DR

The paper develops a microlocal-analytic framework for quantum fields on real analytic curved spacetimes, introducing weak analytic states via high-energy localization properties encoded in the analytic wavefront set. By leveraging the FBI transform and uniform microlocalisation, it proves that analytic states satisfy the Reeh–Schlieder property and a timelike tube theorem, extending Borchers–Araki-type results from flat to curved backgrounds. The analysis also connects to tempered analyticity, microlocal spectrum conditions, and extensions to internal degrees of freedom, providing a robust structure for energy localization and causal restrictions in curved spacetimes. Overall, the work links microlocal analytic methods to foundational QFT properties in an analytic setting, with potential implications for interacting fields and Hadamard-type states.

Abstract

We discuss high energy properties of states for (possibly interacting) quantum fields in curved spacetimes. In particular, if the spacetime is real analytic, we show that an analogue of the timelike tube theorem and the Reeh-Schlieder property hold with respect to states satisfying a weak form of microlocal analyticity condition. The former means the von Neumann algebra of observables of a spacelike tube equals the von Neumann algebra of observables of a significantly bigger region, that is obtained by deforming the boundary of the tube in a timelike manner. This generalizes theorems by Borchers and Araki to curved spacetimes.
Paper Structure (28 sections, 25 theorems, 149 equations, 1 figure)

This paper contains 28 sections, 25 theorems, 149 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Microlocal analysis and the FBI transform
  3. Families of test functions and distributions
  4. Uniform microlocalisation of a family
  5. Analytic wavefront sets
  6. The algebra of zero energy functions.
  7. Analytic states in Quantum Theory
  8. Quantum fields on spacetimes
  9. Quantum field theory
  10. Physical conditions on states
  11. Fields defined on compactly supported smooth test functions
  12. Tempered fields
  13. Wightman fields in Minkowski spacetime as an example
  14. The Free Klein-Gordon field as an example
  15. Relation to the microlocal spectrum conditions
  16. ...and 13 more sections

Key Result

Proposition 2.5

Let $(u_h)\in \mathcal{S}'_h(\mathbb{R}^d)$ be a polynomially bounded family of tempered distributions, and let $(x,\xi) \in \mathbb{R}^d \times \mathbb{R}^d$. Then $(x,-\xi) \notin \mu\mathrm{S}(u_h)$ if and only if there exists an $\epsilon>0$ such that the following holds. For all families $(f_h)

Figures (1)

  • Figure 1: The set $I_0(p,q,\gamma)$ with $\gamma$ contained in a timelike tube.

Theorems & Definitions (59)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • proof
  • ...and 49 more