Historical research guide · 0993 · 2023
Aharonov–Bohm Electrodynamics in Material Media: A Scalar e.m. Field Cannot Cause Dissipation in a Medium
Fernando Minotti · Giovanni Modanese
In one minute
Fernando Minotti and Giovanni Modanese work inside extended electrodynamics — the version of Maxwell’s equations that follows from the Lagrangian Aharonov and Bohm wrote down, in which the electromagnetic field carries one extra piece, a scalar field. In ordinary Maxwell theory that piece can always be gauged away; here it cannot, because its source is the extra current, the amount by which charge fails to be conserved locally. Classical particles never do that; quantum systems, tunnelling junctions and molecular devices might. The authors ask what happens when such a scalar wave, travelling in vacuum, meets ordinary matter, and their answer is clean: it does not get in. In a transparent dielectric and in an ohmic conductor alike the scalar field goes to zero inside the medium, and so does the longitudinal electric field that would accompany it. What does cross is a gauge wave — pure potentials, no fields, no power, moving at the vacuum speed of light. That also dissolves a paradox in which a longitudinal wave in a conductor seemed to lose energy without decaying.
Connection
Chapter 10 is built on the vector potential and on the claim that a scalar or longitudinal electromagnetic mode is a real, usable thing; this paper is the careful version of that claim, because it says exactly what such a wave can and cannot do once it reaches matter, and it names the one kind of detector that could register it — a medium where charge is not locally conserved. For chapter 1 it is a model of how a strong idea stays checkable: the authors separate what the theory already gives from the experiment still to be built.
From the source
In the Aharonov–Bohm extension of Maxwell’s equations the electromagnetic field carries one further degree of freedom, a scalar field S built from the potentials as the time derivative of the scalar potential times the vacuum permittivity and permeability plus the divergence of the vector potential; because the gauge invariance of the Aharonov–Bohm Lagrangian is reduced, S cannot in general be set to zero the way it can in Maxwell theory.Section 1, Introduction; Section 2.1, Equation 1g; Appendix A, Equations A1 and A2
The only source of the scalar field is the extra current, the amount by which charge fails to be conserved locally; for classical particle-like sources that quantity is inconceivable, but for quantum sources — non-local Schrödinger potentials, macroscopic wavefunctions in tunnelling Josephson junctions, first-principles current calculations in molecular devices — the authors judge its generation rare but not excludable.Section 1, Introduction; Section 2.2, Equation 9
An electromagnetic scalar wave cannot propagate in a material medium in which local charge conservation holds: solving the equations for the potentials gives a zero Fourier amplitude for the scalar field, and hence a zero longitudinal electric field, both in a lossless dielectric and in an ohmic conductor over the frequency range where permittivity and conductivity are frequency-independent.Section 2.4; Abstract
The longitudinal mode that does exist inside the medium is a pure potential, or gauge, wave — zero electric, magnetic and scalar fields, and zero power — travelling at the speed of light in vacuum, with the Fourier amplitude of the vector potential times the angular frequency equal to the wave number times the amplitude of the scalar potential.Section 2.4; Abstract
This resolves an apparent dissipation paradox: assuming a longitudinal electric field in a conductor gives a dispersion relation with no decay at all, while the power dissipated per unit volume from an ohmic current has a positive time average — the resolution is that there is no longitudinal electric field in the medium in the first place, so nothing is dissipated.Section 2.3 and Section 2.4
A pure gauge wave can exchange power only with a medium in which the extra source is not zero, at a power per unit volume equal to minus the extra source times the scalar potential, which could show up as changes in currents and voltages in a circuit; the authors expect collective quantum effects to be required, and postpone the circuit design to future work.Section 4, Conclusions, closing paragraphs
The way in
https://doi.org/10.3390/sym15051119How to cite it
Fernando Minotti, Giovanni Modanese (2023) Aharonov–Bohm Electrodynamics in Material Media: A Scalar e.m. Field Cannot Cause Dissipation in a Medium doi:10.3390/sym15051119