Oscillating dipole with fractional quantum source in Aharonov-Bohm electrodynamics

Abstract: We show, in the case of a special dipolar source, that electromagnetic fields in fractional quantum mechanics have an unexpected space dependence: propagating fields may have non-transverse components, and the distinction between near-field zone and wave zone is blurred. We employ an extension of Maxwell theory, Aharonov-Bohm electrodynamics, which is compatible with currents $j^\nu$ conserved globally but not locally, we have derived in another work the field equation $\partial_\mu F^{\mu \nu}=j^\nu+i^\nu$, where $i^\nu$ is a non-local function of $j^\nu$, called "secondary current". Y.\ Wei has recently proved that the probability current in fractional quantum mechanics is in general not locally conserved. We compute this current for a Gaussian wave packet with fractional parameter $a=3/2$ and find that in a suitable limit it can be approximated by our simplified dipolar source. Currents which are not locally conserved may be present also in other quantum systems whose wave functions satisfy non-local equations. The combined electromagnetic effects of such sources and their secondary currents are very interesting both theoretically and for potential applications.