Maxwell equations and the redundant gauge degree of freedom

TL;DR

This paper analyzes Maxwell equations in Fourier space, clarifies the gauge freedom of potentials, and discusses the physical origin of gauge degrees of freedom related to the photon’s masslessness, with implications for wave motion and causality.

Contribution

It provides an elementary perspective on gauge freedom in Maxwell equations, emphasizing the cancellation of causal contributions of potentials and linking gauge degrees of freedom to photon masslessness.

Findings

Explicit demonstration of potential cancellations in Fourier and space-time

Clarification of gauge freedom's physical origin from relativity and quantum mechanics

Separation of wave components relative to wave vector in Fourier space

Abstract

On transformation to the Fourier space $({\bf k}, \omega)$, the partial differential Maxwell equations simplify to algebraic equations, and the Helmholtz theorem of vector calculus reduces to vector algebraic projections. Maxwell equations and their solutions can then be separated readily into longitudinal and transverse components relative to the direction of the wave vector {\bf k}. The concepts of wave motion, causality, scalar and vector potentials and their gauge transformations in vacuum and in materials can also be discussed from an elementary perspective. In particular, the excessive freedom of choice associated with the gauge dependence of the scalar and the longitudinal vector potentials stands out with clarity in Fourier spaces. Since these potentials are introduced to represent the instantaneous longitudinal electric field, the actual cancellation in the latter of causal contributions arising from these potentials separately in most velocity gauges becomes an important issue. This cancellation is explicitly demonstrated both in the Fourier space, and for pedagogical reasons again in space-time. The physical origin of the gauge degree of freedom in the masslessness of the photon, the quantum of electromagnetic wave, is elucidated with the help of special relativity and quantum mechanics.