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We quantise the massless vector potential A of electromagnetism in thepresence of a classical electromagnetic (background) current, j, in a generallycovariant way on arbitrary globally hyperbolic spacetimes M. By carefullyfollowing general principles and procedures we clarify a number of topologicalissues. First we combine the interpretation of A as a connection on a principalU(1)-bundle with the perspective of general covariance to deduce a physicalgauge equivalence relation, which is intimately related to the Aharonov-Bohmeffect. By Peierls' method we subsequently find a Poisson bracket on the spaceof local, affine observables of the theory. This Poisson bracket is in generaldegenerate, leading to a quantum theory with non-local behaviour. We show thatthis non-local behaviour can be fully explained in terms of Gauss' law. Thusour analysis establishes a relationship, via the Poisson bracket, between theAharonov-Bohm effect and Gauss' law (a relationship which seems to have goneunnoticed so far). Furthermore, we find a formula for the space of electricmonopole charges in terms of the topology of the underlying spacetime. Becauseit costs little extra effort, we emphasise the cohomological perspective andderive our results for general p-form fields A (p < dim(M)), modulo exactfields. In conclusion we note that the theory is not locally covariant, in thesense of Brunetti-Fredenhagen-Verch. It is not possible to obtain such a theoryby dividing out the centre of the algebras, nor is it physically desirable todo so. Instead we argue that electromagnetism forces us to weaken the axioms ofthe framework of local covariance, because the failure of locality isphysically well-understood and should be accommodated.

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