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

TL;DR

This paper revisits and corrects a previous derivation of a gauge-invariant energy-momentum tensor in quantum chromodynamics using Noether's second theorem, clarifying the properties of the resulting tensor.

Contribution

The work corrects a flawed derivation and simplifies the process of obtaining a gauge-invariant, non-symmetric energy-momentum tensor via Noether's second theorem.

Findings

Derived a gauge-invariant EMT for QCD

Corrected previous faulty assumptions in the derivation

Provided a clearer, more accessible derivation

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 [Phys. Rev. D 106, 125012 (2022)], 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.