  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.
