Sources, Potentials and Fields in Lorenz and Coulomb Gauge: Cancellation of Instantaneous Interactions for Moving Point Charges

TL;DR

This paper demonstrates how instantaneous Coulomb interactions cancel out in the electric field calculations within classical electrodynamics, ensuring gauge invariance through Green function techniques and boundary term considerations.

Contribution

It provides a concise derivation showing the cancellation of instantaneous interactions in Coulomb gauge using retarded Green functions and boundary term analysis.

Findings

Instantaneous Coulomb interactions cancel in the electric field.

Retarded Green functions require careful boundary term treatment.

Derived integral representations of Lienard-Wiechert potentials in Coulomb gauge.

Abstract

We investigate the coupling of the electromagnetic sources (charge and current densities) to the scalar and vector potentials in classical electrodynamics, using Green function techniques. As is well known, the scalar potential shows an action-at-a-distance behavior in Coulomb gauge. The conundrum generated by the instantaneous interaction has intrigued physicists for a long time. Starting from the differential equations that couple the sources to the potentials, we here show in a concise derivation, using the retarded Green function, how the instantaneous interaction cancels in the calculation of the electric field. The time derivative of a specific additional term in the vector potential, present only in Coulomb gauge, yields a supplementary contribution to the electric field which cancels the gradient of the instantaneous Coulomb gauge scalar potential, as required by gauge invariance. This completely eliminates the contribution of the instantaneous interaction from the electric field. It turns out that a careful formulation of the retarded Green function, inspired by field theory, is required in order to correctly treat boundary terms in partial integrations. Finally, compact integral representations are derived for the Lienard-Wiechert potentials (scalar and vector) in Coulomb gauge which manifestly contain two compensating action-at-a-distance terms.