On the analytical formulation of classical electromagnetic fields

TL;DR

This paper reformulates the Lagrangian for covariant electromagnetic theory to address foundational objections, resulting in a complete derivation of Maxwell's equations and the observed energy-momentum tensor directly from Noether's theorem.

Contribution

It introduces a revised Lagrangian formulation that resolves existing issues and naturally derives symmetries and identities in electromagnetic theory.

Findings

Complete set of Maxwell's equations derived from the new Lagrangian

Observed energy-momentum tensor obtained directly from Noether's theorem

New symmetries and identities identified in the reformulated theory

Abstract

Three objections to the canonical analytical treatment of covariant electromagnetic theory are presented: (i) only half of Maxwell's equations are present upon variation of the fundamental Lagrangian; (ii) the trace of the canonical energy-momentum tensor is not equivalent to the trace of the observed energy-momentum tensor; (iii) the Belinfante symmetrization procedure exists separate from the analytical approach in order to obtain the known observed result. It is shown that the analytical construction from Noether's theorem is based on manipulations that were developed to obtain the compact forms of the theory presented by Minkowski and Einstein; presentations which were developed before the existence of Noether's theorem. By reformulating the fundamental Lagrangian, all of the objections are simultaneously relieved. Variation of the proposed Lagrangian yields the complete set of Maxwell's equations in the Euler-Lagrange equation of motion, and the observed energy-momentum tensor directly follows from Noether's theorem. Previously unavailable symmetries and identities that follow naturally from this procedure are also discussed.