Gauge invariance of many-body Schr\"odinger equation with explicit Coulomb potential

TL;DR

This paper presents a simple argument demonstrating that the many-body Schrödinger equation with explicit Coulomb potential remains gauge invariant, even when eliminating the longitudinal electric field, by using the Coulomb gauge and minimal coupling.

Contribution

It provides a clear demonstration that gauge invariance is preserved in the many-body Schrödinger equation with Coulomb potential through the Coulomb gauge choice and minimal coupling, clarifying previous ambiguities.

Findings

Gauge invariance is maintained with Coulomb potential.

Longitudinal electric field elimination does not break gauge invariance.

Derived gauge-invariant forms of constitutive equations.

Abstract

A simple argument is presented which, based on the minimal coupling Lagrangian for a many-body system, keeps the gauge invariance of the many-body Schr\"odinger equation with explicit Coulomb potential. The elimination of longitudinal electric field does not necessarily lead to the breakdown of gauge invariance. The total time derivative term in the matter-EM field interaction in the Lagrangian is canceled out by the choice of Coulomb gauge. The remaining interaction is described by transverse vector potential and the longitudinal electric field which is the homogeneous solution of Gauss law. This leads directly to the gauge invariant forms of the linear and nonlinear constitutive equations. It is discussed how to reconcile this result with the cases of an isolated matter interacting with external charges.