Gauge Invariance and Classical Electrodynamics

Abstract

A Lagrangian for deriving the Maxwell-Lorentz field equations is obtained by starting from a general tensor of the second rank ${\mathcal{G}}_{\mu \nu }$, which is expressed as the sum of a symmetric and an antisymmetric part ${\mathcal{G}}_{\mu \nu }=\lambda g_{\mu \nu }+F_{\mu \nu }$. The symmetric part is identified with the metric tensor and the antisymmetric part with the electromagnetic field. A relationship between this second rank tensor and the gauge invariant Ricci-Einstein tensor is established by means of the gauge invariant theories of Weyl and Eddington. This relationship leads directly to the Klein-Gordon relativistic wave equation for a point charge moving in an electromagnetic field provided the function $\lambda$ is properly chosen.