Derivation of Maxwell’s equations from the gauge invariance of classical mechanics

Abstract

The Lagrangian for a single classical charged particle is made form invariant under the addition of a total time derivative by adding an interaction Lagrangian which involves compensating fields. The compensating fields are the vector and scalar potentials of the electromagnetic field which couple to the current and charge densities, respectively. To insure form invariance of the Lagrangian, the vector and scalar potentials must undergo the usual gauge transformations of electromagnetism. The electric and magnetic fields, which are gauge invariant, are obtained by examining the equation of motion for the charged particle. Faraday’s law and the condition that there are no magnetic monopoles are obtained from the expressions for the electric and magnetic fields in terms of the potentials. The simplest possible gauge-invariant Lagrangian which is quadratic in the electric and magnetic fields is constructed. From the principle of least action Gauss’ law and the Ampère–Maxwell law are obtained.