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The extended electrodynamic theory introduced by Aharonov and Bohm (after anearlier 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 wayin the standard covariant 4D formalism. This displays more clearly some of itsfeatures. The theory allows a very interesting consistent generalization of theMaxwell equations. In particular, the generalized field equations arecompatible with sources (classical, or more likely of quantum nature) for whichthe continuity/conservation equation $\partial_\mu j^\mu=0$ is not valideverywhere, 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 bya 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 microscopicviolation of the charge continuity condition is "censored" at the macroscopiclevel, although it has real consequences, because it generates a non-Maxwelliancomponent of the field. We consider possible applications of this formalism tocondensed-matter systems with macroscopic quantum tunneling. The extendedelectrodynamics can also be coupled to fractional quantum systems.

Covariant formulation of Aharonov-Bohm electrodynamics and its application to coherent tunnelling