Noether's first theorem and the energy-momentum tensor ambiguity problem

TL;DR

This paper examines the ambiguity in deriving the physical energy-momentum tensor via Noether's theorem, proposing that gauge symmetries uniquely determine it without ad-hoc modifications, especially in electrodynamics and gravity.

Contribution

It demonstrates that Bessel-Hagen type transformations uniquely derive the physical energy-momentum tensor, resolving ambiguity without ad-hoc improvements in gauge-invariant theories.

Findings

Bessel-Hagen transformations are uniquely selected in electrodynamics.

The approach clarifies energy-momentum tensor definition in linearized gravity.

It contextualizes Noether's theorem in relation to symmetry classes and conservation laws.

Abstract

Noether's theorems are widely praised as some of the most beautiful and useful results in physics. However, if one reads the majority of standard texts and literature on the application of Noether's first theorem to field theory, one immediately finds that the ``canonical Noether energy-momentum tensor" derived from the 4-parameter translation of the Poincar\'e group does not correspond to what's widely accepted as the ``physical'' energy-momentum tensor for central theories such as electrodynamics. This gives the impression that Noether's first theorem is in some sense not working. In recognition of this issue, common practice is to ``improve" the canonical Noether energy-momentum tensor by adding suitable ad-hoc ``improvement" terms that will convert the canonical expression into the desired result. On the other hand, a less common but distinct method developed by Bessel-Hagen considers gauge symmetries as well as coordinate symmetries when applying Noether's first theorem; this allows one to uniquely derive the accepted physical energy-momentum tensor without the need for any ad-hoc improvement terms in theories with exactly gauge invariant actions. $\dots$ Using the converse of Noether's first theorem, we show that the Bessel-Hagen type transformations are uniquely selected in the case of electrodynamics, which powerfully dissolves the methodological ambiguity at hand. We then go on to consider how this line of argument applies to a variety of other cases, including in particular the challenge of defining an energy-momentum tensor for the gravitational field in linearized gravity. Finally, we put the search for proper Noether energy-momentum tensors into context with recent claims that Noether's theorem and its converse make statements on equivalence classes of symmetries and conservation laws$\dots$