Noether's first theorem and the energymomentum tensor ambiguity problem
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 energymomentum tensor" derived from the 4parameter translation of the Poincaré group does not correspond to what's widely accepted as the ``physical'' energymomentum 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 energymomentum tensor by adding suitable adhoc ``improvement" terms that will convert the canonical expression into the desired result. On the other hand, a less common but distinct method developed by BesselHagen considers gauge symmetries as well as coordinate symmetries when applying Noether's first theorem; this allows one to uniquely derive the accepted physical energymomentum tensor without the need for any adhoc improvement terms in theories with exactly gauge invariant actions. $\dots$ Using the converse of Noether's first theorem, we show that the BesselHagen 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 energymomentum tensor for the gravitational field in linearized gravity. Finally, we put the search for proper Noether energymomentum tensors into context with recent claims that Noether's theorem and its converse make statements on equivalence classes of symmetries and conservation laws$\dots$
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.10329
 Bibcode:
 2021arXiv210710329B
 Keywords:

 Physics  History and Philosophy of Physics;
 General Relativity and Quantum Cosmology;
 Mathematical Physics
 EPrint:
 30 pages