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Quantum reference frames for spacetime symmetries and large gauge transformations

Daan W. Janssen

TL;DR

The work develops an operational quantum reference frame (QRF) framework within algebraic quantum field theory (AQFT) to study observables and entropy under spacetime and gauge symmetries. It shows that coupling a QFT to a QRF can induce type reduction, making the $G$-invariant joint algebra $(\mathscr{A}(M)\otimes B(H_R))^G$ semifinite or finite when the QFT and QRF possess suitable thermal properties, via crossed-product and modular theory. It extends the approach to gauge theories on manifolds with boundaries, where edge-mode observables form a boundary QRF for large gauge transformations and lead to quantised boundary flux, enabling gluing of region algebras. These results connect AQFT, QRFs, and boundary gauge phenomena to provide a rigorous route to defining entropy in quantum gravity and to the consistent quantisation of edge degrees of freedom. Overall, the paper clarifies how invariant algebras constructed with QRFs can exhibit desirable thermodynamic and compositional properties relevant for quantum gravity and gauge theories on curved spacetimes.

Abstract

Symmetries are a central concept in our understanding of physics. In quantum theories, a quantum reference frame (QRF) can be used to distinguish between observables related by a symmetry. The framework of operational QRFs provides a means to describe observables in terms of their relation to a reference quantum system. We discuss a number of applications of QRFs in the context of quantum field theory on curved spacetimes: 1) A type reduction result for algebras arising from QFTs and QRFs with good thermal properties. 2) Quantisation of boundary electric fluxes and gluing procedures for quantum electromagnetism on spacetimes with boundaries.

Quantum reference frames for spacetime symmetries and large gauge transformations

TL;DR

The work develops an operational quantum reference frame (QRF) framework within algebraic quantum field theory (AQFT) to study observables and entropy under spacetime and gauge symmetries. It shows that coupling a QFT to a QRF can induce type reduction, making the -invariant joint algebra semifinite or finite when the QFT and QRF possess suitable thermal properties, via crossed-product and modular theory. It extends the approach to gauge theories on manifolds with boundaries, where edge-mode observables form a boundary QRF for large gauge transformations and lead to quantised boundary flux, enabling gluing of region algebras. These results connect AQFT, QRFs, and boundary gauge phenomena to provide a rigorous route to defining entropy in quantum gravity and to the consistent quantisation of edge degrees of freedom. Overall, the paper clarifies how invariant algebras constructed with QRFs can exhibit desirable thermodynamic and compositional properties relevant for quantum gravity and gauge theories on curved spacetimes.

Abstract

Symmetries are a central concept in our understanding of physics. In quantum theories, a quantum reference frame (QRF) can be used to distinguish between observables related by a symmetry. The framework of operational QRFs provides a means to describe observables in terms of their relation to a reference quantum system. We discuss a number of applications of QRFs in the context of quantum field theory on curved spacetimes: 1) A type reduction result for algebras arising from QFTs and QRFs with good thermal properties. 2) Quantisation of boundary electric fluxes and gluing procedures for quantum electromagnetism on spacetimes with boundaries.

Paper Structure

This paper contains 5 sections, 2 theorems, 2 equations, 1 figure.

Table of Contents

  1. Why (quantum) reference frames?
  2. Operational QRFs
  3. Invariant algebras and type reduction for QFT on curved spacetimes
  4. Gauge theories, boundaries and quantum reference frames
  5. Acknowledgements

Key Result

Theorem 1

There exists a Hilbert space $\mathcal{K}$ and a unitary map $V:\mathcal{H}_{\mathcal{R}}\to L^2(G)\otimes \mathcal{K}$ such that where $L^2(G)$ is defined w.r.t. the left Haar measure and $G$ acts on $L^2(G)$ via the left action $(g.\psi)(h)=\psi(g^{-1}h)$ for $g,h\in G$ and $\psi\in L^2(G)$.

Figures (1)

  • Figure 1: A Cauchy surface $\overline{\Sigma}$ is divided in $\overline{\Sigma}_1$ and $\overline{\Sigma}_2$. By treating the electromagnetic field on $\overline{\Sigma}_2$ as a QRF for $\overline{\Sigma}_1$, one can build large gauge invariant observables on $\overline{\Sigma}$ from edge mode observables through relativisation fewsterSemilocalObservablesEdge2025.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2