Mathematical Constraints on Gauge in Maxwellian Electrodynamics

TL;DR

This paper investigates the fundamental differences in gauge definitions within classical and quantum electrodynamics, emphasizing the role of variational principles and causality in establishing unique potentials versus gauge freedom.

Contribution

It demonstrates that variational principles and causality uniquely determine potentials in classical electrodynamics, contrasting with gauge transformations allowed when Maxwell equations are primary.

Findings

4-potentials are uniquely defined under variational principles

Gauge transformations are permitted when Maxwell equations are fundamental

Quantum physics shares the same gauge constraints as classical theory

Abstract

The structure of classical electrodynamics based on the variational principle together with causality and space-time homogeneity is analyzed. It is proved that in this case the 4-potentials are defined uniquely. On the other hand, the approach where Maxwell equations and the Lorentz law of force are regarded as cornerstones of the theory allows gauge transformations. For this reason, the two theories are not equivalent. A simple example substantiates this conclusion. Quantum physics is linked to the variational principle and it is proved that the same result holds for it. The compatibility of this conclusion with gauge invariance of the Lagrangian density is explained. Several alternative possibilities that may follow this work are pointed out.