  The extended electrodynamic theory introduced by Aharonov and Bohm (after an
earlier attempt by Ohmura) and recently developed by Van Vlaenderen and Waser,
Hively and Giakos, can be re-written and solved in a simple and effective way
in the standard covariant 4D formalism. This displays more clearly some of its
features. The theory allows a very interesting consistent generalization of the
Maxwell equations. In particular, the generalized field equations are
compatible with sources (classical, or more likely of quantum nature) for which
the continuity/conservation equation $\partial_\mu j^\mu=0$ is not valid
everywhere, or is valid only as an average above a certain scale. And yet,
remarkably, in the end the observable $F^{\mu \nu}$ field is still generated by
a conserved effective source which we denote as $(j^\nu+i^\nu)$, being $i^\nu$
a suitable non-local function of $j^\nu$. This implies that any microscopic
violation of the charge continuity condition is "censored" at the macroscopic
level, although it has real consequences, because it generates a non-Maxwellian
component of the field. We consider possible applications of this formalism to
condensed-matter systems with macroscopic quantum tunneling. The extended
electrodynamics can also be coupled to fractional quantum systems.
