On Feynman's handwritten notes on electromagnetism and the idea of introducing potentials before fields

TL;DR

This paper explores Feynman's 1963 idea of deriving Maxwell's equations from potentials first, developing a pedagogical approach that emphasizes potentials before fields, using axiomatic and covariant formulations.

Contribution

It introduces a novel pedagogical method to derive Maxwell's equations starting from potentials, emphasizing charge conservation and covariance in Minkowski spacetime.

Findings

Retarded potentials derived from charge conservation

Maxwell's equations obtained from wave equations for potentials

Covariant formulation in Minkowski spacetime

Abstract

In his recently discovered handwritten notes on "An alternate way to handle electrodynamics" dated on 1963, Richard P. Feynman speculated with the idea of getting the inhomogeneous Maxwell's equations for the electric and magnetic fields from the wave equation for the vector potential. With the aim of implementing this pedagogically interesting idea, we develop in this paper the approach of introducing the scalar and vector potentials before the electric and magnetic fields. We consider the charge conservation expressed through the continuity equation as a basic axiom and make a heuristic handle of this equation to obtain the retarded scalar and vector potentials, whose wave equations yield the homogeneous and inhomogeneous Maxwell's equations. We also show how this axiomatic-heuristic procedure to obtain Maxwell's equations can be formulated covariantly in the Minkowski spacetime.