Electromagnetism, local covariance, the Aharonov-Bohm effect and Gauss' law

TL;DR

This paper quantizes the electromagnetic potential in a covariant manner on curved spacetimes, clarifies topological issues, and links the Aharonov-Bohm effect with Gauss' law through a Poisson bracket analysis.

Contribution

It introduces a covariant quantization framework for electromagnetism that incorporates topological aspects and relates the Aharonov-Bohm effect to Gauss' law, highlighting the non-local nature of the theory.

Findings

Establishes a Poisson bracket structure for local observables.

Shows the non-local behavior explained by Gauss' law.

Derives electric monopole charges from spacetime topology.

Abstract

We quantise the massless vector potential A of electromagnetism in the presence of a classical electromagnetic (background) current, j, in a generally covariant way on arbitrary globally hyperbolic spacetimes M. By carefully following general principles and procedures we clarify a number of topological issues. First we combine the interpretation of A as a connection on a principal U(1)-bundle with the perspective of general covariance to deduce a physical gauge equivalence relation, which is intimately related to the Aharonov-Bohm effect. By Peierls' method we subsequently find a Poisson bracket on the space of local, affine observables of the theory. This Poisson bracket is in general degenerate, leading to a quantum theory with non-local behaviour. We show that this non-local behaviour can be fully explained in terms of Gauss' law. Thus our analysis establishes a relationship, via the Poisson bracket, between the Aharonov-Bohm effect and Gauss' law (a relationship which seems to have gone unnoticed so far). Furthermore, we find a formula for the space of electric monopole charges in terms of the topology of the underlying spacetime. Because it costs little extra effort, we emphasise the cohomological perspective and derive our results for general p-form fields A (p < dim(M)), modulo exact fields. In conclusion we note that the theory is not locally covariant, in the sense of Brunetti-Fredenhagen-Verch. It is not possible to obtain such a theory by dividing out the centre of the algebras, nor is it physically desirable to do so. Instead we argue that electromagnetism forces us to weaken the axioms of the framework of local covariance, because the failure of locality is physically well-understood and should be accommodated.