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

Baker, Mark Robert and Linnemann, Niels and Smeenk, Chris (2021) Noether's first theorem and the energy-momentum tensor ambiguity problem. The Physics and Philosophy of Noether's Theorems.

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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 Poincaré 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. Given these two distinct methods to obtain an energy-momentum tensor, the question arises as to whether one of these methods corresponds to a preferable application of Noether's first theorem. 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: We aim to identify clearly the limitations of this latter move, and develop our position by contrast with recent philosophical discussions about how symmetries relate to the representational capacities of our theories.


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Item Type: Published Article or Volume
Creators:
CreatorsEmailORCID
Baker, Mark Robertmrbaker@stfx.ca0000-0003-3641-0020
Linnemann, Nielsnielslinnemann@aol.com0000-0002-3695-8611
Smeenk, Chriscsmeenk2@uwo.ca0000-0002-6101-0048
Keywords: Noether energy-momentum tensor Bessel-Hagen conservation laws Noether theorems
Subjects: Specific Sciences > Physics > Classical Physics
General Issues > Structure of Theories
Specific Sciences > Physics > Symmetries/Invariances
Depositing User: Niels Linnemann
Date Deposited: 23 Jul 2021 14:00
Last Modified: 23 Jul 2021 14:00
Item ID: 19356
Journal or Publication Title: The Physics and Philosophy of Noether's Theorems
Publisher: Cambridge University Press
Subjects: Specific Sciences > Physics > Classical Physics
General Issues > Structure of Theories
Specific Sciences > Physics > Symmetries/Invariances
Date: 2021
URI: http://philsci-archive.pitt.edu/id/eprint/19356

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