Using gauge invariance to symmetrize the energy-momentum tensor of electrodynamics

Abstract

It is shown that using Noether's Theorem explicitly employing gauge invariance for variations of the electromagnetic four-potential $A^\mu$ straightforwardly ensures that the resulting electromagnetic energy-momentum tensor is symmetric. The Belinfante symmetrization procedure is not necessary. The method is based on Bessel-Hagen's 1921 clarification of Noether's original procedure, suggesting that the symmetry problem arises from an incomplete implementation of Noether's Theorem. The derivation addresses in some detail where the usual application of Noether's Theorem falls short, what the Belinfante procedure actually does to fix the problem, and why the usual unsymmetric canonical energy-momentum tensor can only be used for extracting four-momentum conservation based on translational invariance, but will provide meaningless results when applied to rotations or boosts, unless modified appropriately.