The Proca Field in Curved Spacetimes and its Zero Mass Limit

TL;DR

This paper studies the classical and quantum Proca field in curved spacetimes, focusing on the massless limit, and introduces a covariant, local approach that clarifies gauge choices and the emergence of Maxwell's equations.

Contribution

It develops a covariant, local method to analyze the massless limit of the Proca field, clarifying gauge equivalences and the conditions for recovering Maxwell's equations.

Findings

The massless limit exists only on a subset of observables.

The limiting procedure selects a gauge consistent with the Aharonov-Bohm effect.

Maxwell's equations can be recovered when the Lorenz constraint remains well behaved.

Abstract

We investigate the classical and quantum Proca field (a massive vector potential) of mass $m>0$ in arbitrary globally hyperbolic spacetimes and in the presence of external sources. We motivate a notion of continuity in the mass for families of observables $\left\{O_m\right\}_{m>0}$ and we investigate the massless limit $m\to0$. Our limiting procedure is local and covariant and it does not require a choice of reference state. We find that the limit exists only on a subset of observables, which automatically implements a gauge equivalence on the massless vector potential. For topologically non-trivial spacetimes, one may consider several inequivalent choices of gauge equivalence and our procedure selects the one which is expected from considerations involving the Aharonov-Bohm effect and Gauss' law. We note that the limiting theory does not automatically reproduce Maxwell's equation, but it can be imposed consistently when the external current is conserved. To recover the correct Maxwell dynamics from the limiting procedure would require an additional control on limits of states. We illustrate this only in the classical case, where the dynamics is recovered when the Lorenz constraint remains well behaved in the limit.