Generalized Maxwell equations and charge conservation censorship

TL;DR

This paper reformulates Aharonov-Bohm electrodynamics to show how non-conserved charges can be effectively hidden in electromagnetic fields, with potential applications in systems with charge anomalies or quantum tunneling.

Contribution

It provides a new Lagrangian formulation that explicitly eliminates the scalar degree of freedom, revealing how charge non-conservation can be 'censored' by observable fields.

Findings

Generalized Maxwell equations include a non-local secondary current.

Non-conservation of charge is effectively hidden in the electromagnetic field.

Stationary solutions demonstrate charge censorship effects.

Abstract

The Aharonov-Bohm electrodynamics is a generalization of Maxwell theory with reduced gauge invariance. It allows to couple the electromagnetic field to a charge which is not locally conserved, and has an additional degree of freedom, the scalar field $S=\partial_\alpha A^\alpha$, usually interpreted as a longitudinal wave component. By re-formulating the theory in a compact Lagrangian formalism, we are able to eliminate $S$ explicitly from the dynamics and we obtain generalized Maxwell equation with interesting properties: they give $\partial_\mu F^{\mu \nu}$ as the (conserved) sum of the (possibly non-conserved) physical current density $j^\nu$, and a "secondary" current density $i^\nu$ which is a non-local function of $j^\nu$. This implies that any non-conservation of $j^\nu$ is effectively "censored" by the observable field $F^{\mu \nu}$, and yet it may have real physical consequences. We give examples of stationary solutions which display these properties. Possible applications are to systems where local charge conservation is violated due to anomalies of the ABJ kind or to macroscopic quantum tunnelling with currents which do not satisfy a local continuity equation.