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arXiv:1702.02026

Covariant Formulation of Aharonov-Bohm Electrodynamics and Its Application to Coherent Tunneling

Giovanni Modanese

Published Dec 28, 2017• 2 citations
arXivHTMLPDF

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.

DOI
10.1142/9789813232044_0023

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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 ∂μjμ=0\partial_\mu j^\mu=0∂μ​jμ=0 is not valid everywhere, or is valid only as an average above a certain scale. And yet, remarkably, in the end the observable FμνF^{\mu \nu}Fμν field is still generated by a conserved effective source which we denote as (jν+iν)(j^\nu+i^\nu)(jν+iν), being iνi^\nuiν a suitable non-local function of jνj^\nujν. 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.
We reconsider some well-known tunneling processes from the point of view of Aharonov-Bohm electrodynamics, a unique extension of Maxwell’s theory which admits charge-current sources that are not locally conserved. In particular we are interested into tunneling phenomena having relatively long range (otherwise the non-Maxwellian effects become irrelevant, especially at high frequency) and involving macroscopic wavefunctions and coherent matter, for which it makes sense to evaluate the classical e.m. field generated by the tunneling particles. For some condensed-matter systems, admitting discontinuities in the probability current is a possible way of formulating phenomenological models. In such cases, the Aharonov-Bohm theory offers a logically consistent approach and allows to derive observable consequences. Typical e.m. signatures of the failure of local conservation are at high frequency the generation of a longitudinal electric radiation field, and at low frequency a small effect of “missing” magnetic field. Possible causes of this failure are instant tunneling and phase slips in superconductors. For macroscopic quantum systems in which the phase-number uncertainty relation ΔNΔφ∼1 applies, the expectation value of the anomalous source I=∂tρ+∇·j has quantum fluctuations, thus becoming a random source of weak non-Maxwellian fields.
The problem of the relation between the Ahronov-Bohm effect and traditional postulates of electrodynamics, which claim that only electric and magnetic fields are observable, is resolved by denial of the statement about validity of the Maxwell equations for microscopic fields. We proceed from the idea that the Maxwell equations, as the generalization of experimental data, are valid only for averaged values. We show that microscopic electrodynamics should be based on postulation of the d'Alembert equations for four-vector of the electromagnetic field potential. The Lorentz condition is valid only for the averages and provides the implementation of the Maxwell equations for averages. This concept eliminates the problem of electromagnetic field quantization and provides the correctness of all known results of quantum electrodynamics. Therefore, the "virtuality" of the longitudinal and scalar photons has a formal mathematical character, conditioned by the Maxwell equations for averaged fields. The longitudinal and scalar photons provide not only the Coulomb interaction of charged particles, but also allow the electrical Aharonov-Bohm effect.
This paper presents a \emph{non-instant field model} for electrodynamics that permits a causal explanation of the \emph{Aharonov-Bohm effect} and a \emph{covariant quantization} of the respective Maxwell equations via the \emph{Gupta-Bleuler method}. Our model satisfies the following \emph{correspondence principle}: if AμA^μAμ, $\vE$, $\vB$ denote the four potential, the electric field and the magnetic field of the non-instant field model, then the respective classical quantities are $\A[A^μ]$, $\A[\vE]$, $\A[\vB]$, where $\A$ is a covariant time averaging operator. Here $\A[A^μ]$ is interpreted as the best possible measurement of the four potential AμA^μAμ. Although the Lorentz condition is not satisfied for AμA^μAμ, it is satisfied for $\A[A^μ]$. The latter fact means that the Lorentz condition does not hold for the quantized field but for its expectation value (cf. \emph{Gupta-Bleuler method} of quantization). Finally, we derive the energy conservation law of our field model and show that the field energy is quantized.
The extended Aharonov-Bohm electrodynamics has a simple formal structure and allows to couple the e.m. field also to currents which are not locally conserved, like those resulting from certain non-local effective quantum models of condensed matter. As it often happens in physics and mathematics when one tries to extend the validity of some equations or operations, new perspectives emerge in comparison to Maxwell theory, and some ''exotic'' phenomena are predicted. For the Aharonov-Bohm theory the main new feature is that the potentials AμA^μAμ become univocally defined and can be measured with probes in which the ''extra-current'' I=∂μjμI=\partial_μj^μI=∂μ​jμ is not zero at some points. As a consequence, it is possible in principle to detect pure gauge-waves with E=B=0\mathbf{E}=\mathbf{B}=0E=B=0, which would be regarded as non-physical in the Maxwell gauge-invariant theory with local current conservation. We discuss in detail the theoretical aspects of this phenomenon and propose an experimental realization of the detectors. A full treatment of wave propagation in anomalous media with extra-currents and of energy-momentum balance issues is also given.
Abstract The extended Aharonov–Bohm electrodynamics has a simple formal structure and allows to couple the e.m. field also to currents which are not locally conserved, like those resulting from certain non-local effective quantum models of condensed matter. As it often happens in physics and mathematics when one tries to extend the validity of some equations or operations, new perspectives emerge in comparison to Maxwell theory, and some “exotic” phenomena are predicted. For the Aharonov–Bohm theory the main new feature is that the potentials AμA^\mu Aμ A μ become univocally defined and can be measured with probes in which the “extra-current” I=∂μjμI=\partial _\mu j^\mu I=∂μ​jμ I = ∂ μ j μ is not zero at some points. As a consequence, it is possible in principle to detect pure gauge-waves with E=B=0{\textbf{E}}={\textbf{B}}=0E=B=0 E = B = 0 , which would be regarded as non-physical in the Maxwell gauge-invariant theory with local current conservation. We discuss in detail the theoretical aspects of this phenomenon and propose an experimental realization of the detectors. A full treatment of wave propagation in anomalous media with extra-currents and of energy–momentum balance issues is also given.
In this article, we start our discussion on the Aharonov-Bohm effect then raise a plausible interpretation within the quantum electrodynamics (QED) framework. We provide a quantum treatment of the source of the electromagnetic potential and argue that the underlying mechanism in AB effect can be viewed as interactions between electrons described by QED theory where the interactions are mediated by virtue photons. On further analysis, we claim that the framework of one particle quantum mechanics (OPQM) can be shown, in general, as a mathematically approximated model which can be reformulated from QED theory. In addition, the classical Maxwell equations are derived from QED scattering process while both classical electromagnetic fields and potentials serve as mathematical tools that are constructed to approximate the interactions among elementary particles described by QED physics, i.e. neither classical fields nor potentials represent any real entities of nature. At the conclusion of this article, we make a few remarks on the hypnosis of the existence of magnetic monopoles.