Oscillating dipole with fractional quantum source in Aharonov-Bohm electrodynamics
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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 jνjν conserved globally but not locally; we have derived in another work the field equation ∂μFμν=jν+iν∂μFμν=jν+iν, where iνiν is a non-local function of jνjν, 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=3/2a=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.