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Reflections on Noether’s second theorem and the energy-momentum tensor

Adam Freese*

  • *Contact author: afreese@jlab.org
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Phys. Rev. D 113, 016011 – Published 9 January, 2026

DOI: https://doi.org/10.1103/wmz3-lrbz

Abstract

Through symmetry of the action under global spacetime translations, Noether’s first theorem infamously entails an energy-momentum tensor (EMT) that is neither symmetric nor gauge-invariant. In a prior work [Phys. Rev. D 106, 125012 (2022)], I had obtained a symmetric and gauge-invariant EMT by using Noether’s second theorem instead, with local spacetime translations as the symmetry group. However, the derivation therein was flawed, containing a faulty assumption about the transformation rule for spinor fields. In this work, I revisit the derivation of the previous work, both correcting the faulty step and simplifying the derivation for broader accessibility. The end result is an EMT for quantum chromodynamics that is gauge-invariant, but not symmetric.

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Noether’s theorems and the energy-momentum tensor in quantum gauge theories

Adam Freese
Phys. Rev. D 106, 125012 (2022)

Article Text

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Phys. Rev. D 113, 016011– Published 9 January, 2026
  • Received 30 June 2025
  • Accepted 23 December 2025
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