On the causality of electrodynamics and the Aharonov-Bohm effect

Abstract

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 $A^{\mu }$, $\vE$, $\vB$ 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 $A^{\mu }$. Although the Lorentz condition is not satisfied for $A^{\mu }$, 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.