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A von Neumann algebraic approach to Quantum Theory on curved spacetime

Louis E Labuschagne, W Adam Majewski

TL;DR

The paper develops a von Neumann algebraic formulation of AQFT on curved spacetime, replacing a global Hilbert space with local standard-forms to obtain a locally covariant, tensorial framework. It demonstrates how Killing local flows lift to automorphisms of local algebras and establishes conditions for quantum Lie derivatives, while constructing representations via Araki–Woods factors from quasi-free, particularly Hadamard, states. The work also integrates Orlicz-space techniques to accommodate unbounded field observables and discusses implications for semiclassical gravity through well-defined stress-energy expectations. Collectively, the results provide a robust, entanglement-capable, locally covariant AQFT on globally hyperbolic spacetimes, with a clear path toward including broader observables and applications to curved-background dynamics.

Abstract

By extending the method developed in our recent paper \cite{LM} we present the AQFT framework in terms of von Neumann algebras. In particular, this approach allows for a locally covariant categorical description of AQFT which moreover satisfies the additivity property and provides a natural and intrinsic framework for a description of entanglement. Turning to dynamical aspects of QFT we show that Killing local flows may be lifted to the algebraic setting in curved space-time. Furthermore, conditions under which quantum Lie derivatives of such local flows exist are provided. The central question that then emerges is how such quantum local flows might be described in interesting representations. We show that quasi-free representations of Weyl algebras fit the presented framework perfectly. Finally, the problem of enlarging the set of observables is discussed. We point out the usefulness of Orlicz space techniques to encompass unbounded field operators. In particular, a well-defined framework within which one can manipulate such operators is necessary for the correct presentation of (semiclassical) Einstein's equation.

A von Neumann algebraic approach to Quantum Theory on curved spacetime

TL;DR

The paper develops a von Neumann algebraic formulation of AQFT on curved spacetime, replacing a global Hilbert space with local standard-forms to obtain a locally covariant, tensorial framework. It demonstrates how Killing local flows lift to automorphisms of local algebras and establishes conditions for quantum Lie derivatives, while constructing representations via Araki–Woods factors from quasi-free, particularly Hadamard, states. The work also integrates Orlicz-space techniques to accommodate unbounded field observables and discusses implications for semiclassical gravity through well-defined stress-energy expectations. Collectively, the results provide a robust, entanglement-capable, locally covariant AQFT on globally hyperbolic spacetimes, with a clear path toward including broader observables and applications to curved-background dynamics.

Abstract

By extending the method developed in our recent paper \cite{LM} we present the AQFT framework in terms of von Neumann algebras. In particular, this approach allows for a locally covariant categorical description of AQFT which moreover satisfies the additivity property and provides a natural and intrinsic framework for a description of entanglement. Turning to dynamical aspects of QFT we show that Killing local flows may be lifted to the algebraic setting in curved space-time. Furthermore, conditions under which quantum Lie derivatives of such local flows exist are provided. The central question that then emerges is how such quantum local flows might be described in interesting representations. We show that quasi-free representations of Weyl algebras fit the presented framework perfectly. Finally, the problem of enlarging the set of observables is discussed. We point out the usefulness of Orlicz space techniques to encompass unbounded field operators. In particular, a well-defined framework within which one can manipulate such operators is necessary for the correct presentation of (semiclassical) Einstein's equation.

Paper Structure

This paper contains 27 sections, 34 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The standard form of von Neumann algebras
  4. VN-SF and VN-MA-SF are tensorial categories
  5. The structure of both VN-SF and VN-MA-SF allow for entanglement
  6. Quantum Killing vector fields for AQFT
  7. Killing local flows on compacta of Lorentzian manifolds
  8. Killing flows on local algebras: an illustrative example based on the Klein-Gordon equation
  9. Quantum Killing Lie derivatives
  10. Generators of one-parameter local groups of contractions
  11. Existence of Quantum Killing Lie derivatives
  12. Von Neumann algebraic representations of local algebras
  13. The selection of appropriate von Neumann algebraic representations
  14. Constructing Araki-Woods factors
  15. The behaviour of $\{{\mathcal{M}}(\mathcal{O}):\mathcal{O}\in\mathcal{K}(M,g)\}$ with respect to local isometries
  16. ...and 12 more sections

Key Result

Theorem 3.2

The standard form of a von Neumann algebra ${\mathcal{M}}$ is unique in the sense that if are two standard forms, and $\alpha:\pi({\mathcal{M}})\to\widetilde{\pi}(\widetilde{{\mathcal{M}}})$ is a $*$-isomorphism, then there exists a unique unitary operator $u:H\to\widetilde{H}$ such that

Theorems & Definitions (76)

  • Definition 3.1
  • Theorem 3.2: haag-stdfm
  • Definition 3.3
  • Remark 3.4
  • Definition 3.5
  • Lemma 3.6
  • proof
  • Proposition 3.7: Proposition 2a, IC
  • Proposition 3.8: Proposition 3g, IC
  • Proposition 3.9: Prop 2c, IC
  • ...and 66 more