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

TL;DR

This paper reformulates Aharonov-Bohm electrodynamics covariantly, revealing its ability to handle non-conserved sources and its implications for quantum tunneling and condensed matter systems.

Contribution

It presents a covariant formulation of extended electrodynamics that accommodates non-conserved sources and explores its applications to quantum tunneling and fractional quantum systems.

Findings

The theory allows sources with non-conserved current to generate observable fields.

Microscopic charge violations are effectively 'censored' at the macroscopic level.

Potential applications to condensed matter systems with quantum tunneling are discussed.

Abstract

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.