Semi-local observables, edge modes and quantum reference frames in quantum electromagnetism: an algebraic approach

TL;DR

This paper develops an algebraic framework for semi-local quantum electromagnetism on bounded spacetimes, introducing quantum reference frames to handle gauge invariance and boundary effects, with implications for quantum gravity and entanglement.

Contribution

It introduces a novel algebraic approach to semi-local quantum electromagnetism on bounded spacetimes, incorporating quantum reference frames to address gauge invariance and boundary phenomena.

Findings

Decomposition of phase space into bulk and surface sectors.

Construction of a Weyl $C^{*}$-algebra of semi-local observables.

Use of quantum reference frames to achieve gauge invariance.

Abstract

Boundaries and corners of spacetime play a vital role in understanding physical concepts including entanglement entropy, the infrared problem in QFT and quantum gravity. Standard local quantum field theory struggles to accommodate such boundary-sensitive observables. In this paper we develop an algebraic framework for \emph{semi-local quantum electromagnetism} on finite Cauchy lenses: a class of compact spacetimes with boundaries and corner. At the classical level, we establish a decomposition of the reduced covariant phase space into bulk closed-loop and surface sectors and demonstrate how the covariant phase space approach relates to the Peierls bracket construction commonly used in perturbative algebraic quantum field theory. Upon quantisation, we obtain a Weyl $C^{*}$-algebra of semi-local observables transforming non-trivially under large gauge transformations (those with non-trivial boundary contribution). To recover gauge invariance, we invoke the notion of \emph{quantum reference frames} (QRFs) and construct a relativisation map, where we treat auxiliary surface degrees of freedom as QRFs for the large gauge transformations. The relativisation map is constructed directly on the level of $C^{*}$-algebras, making our construction state-independent. The QRF viewpoint on semi-local observables provides new tools for understanding gauge theories on manifolds with boundary, including the problem of gluing theories on Cauchy lenses with common boundaries.