Proca equations derived from first principles

TL;DR

This paper derives Proca equations from fundamental principles, paralleling the derivation of Maxwell and Dirac equations, and highlights the necessity of potentials for transforming second-order equations into first-order form.

Contribution

It presents a novel derivation of Proca equations from first principles, extending the approach used for Maxwell and Dirac equations.

Findings

Proca equations can be derived from first principles.

Potentials are necessary to convert Klein-Gordon into Proca equations.

The derivation parallels that of Maxwell and Dirac equations.

Abstract

Gersten has shown how Maxwell equations can be derived from first principles, similar to those which have been used to obtain the Dirac relativistic electron equation. We show how Proca equations can be also deduced from first principles, similar to those which have been used to find Dirac and Maxwell equations. Contrary to Maxwell equations, it is necessary to introduce a potential in order to transform a second order differential equation, as the Klein-Gordon equation, into a first order differential equation, like Proca equations.