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

November 27, 2012

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.

open access link
Commun.Math.Phys.328: 625-667, 2014

@article{Sanders:2012sf,
author = "Sanders, Ko and Dappiaggi, Claudio and Hack, Thomas-Paul",
title = "{Electromagnetism, Local Covariance, the Aharonov-Bohm
Effect and Gauss' Law}",
journal = "Commun. Math. Phys.",
volume = "328",
year = "2014",
pages = "625-667",
doi = "10.1007/s00220-014-1989-x",
eprint = "1211.6420",
archivePrefix = "arXiv",
primaryClass = "math-ph",
SLACcitation = "%%CITATION = ARXIV:1211.6420;%%"
}

Keywords:

QFT on curved spacetimes, locally covariant QFT, electromagnetism