Quantum reference frames, measurement schemes and the type of local algebras in quantum field theory

TL;DR

This paper develops an operational framework combining relativistic quantum measurement theory with quantum reference frames, analyzing the algebraic structure of local quantum field measurements relative to symmetries and thermal properties.

Contribution

It introduces a new framework for local quantum field measurements relative to quantum reference frames, characterizes the resulting algebraic structures, and generalizes recent mathematical results.

Findings

Invariant algebra admits a semifinite trace under thermal conditions

Conditions for the algebra to be a type II_1 factor are established

Provides a refined operational understanding of quantum reference frames in QFT

Abstract

We develop an operational framework, combining relativistic quantum measurement theory with quantum reference frames (QRFs), in which local measurements of a quantum field on a background with symmetries are performed relative to a QRF. This yields a joint algebra of quantum-field and reference-frame observables that is invariant under the natural action of the group of spacetime isometries. For the appropriate class of quantum reference frames, this algebra is parameterised in terms of crossed products. Provided that the quantum field has good thermal properties (expressed by the existence of a KMS state at some nonzero temperature), one can use modular theory to show that the invariant algebra admits a semifinite trace. If furthermore the quantum reference frame has good thermal behaviour (expressed in terms of the properties of a KMS weight) at the same temperature, this trace is finite. We give precise conditions for the invariant algebra of physical observables to be a type $II_1$ factor. Our results build upon recent work of Chandrasekaran, Longo, Penington and Witten [JHEP $\mathbf{2023}$, 82 (2023)], providing both a significant mathematical generalisation of these findings and a refined operational understanding of their model.