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

TL;DR

The paper shows that Noether's second theorem, applied to local spacetime translations, yields a symmetric and gauge/BRST-invariant energy-momentum tensor (EMT) for gauge theories, avoiding the need for Belinfante-type improvements. It derives a general EMT formula $T^{\mu}{}_{\nu}=\sum_l \mathscr{D}^{\mu}{}_{\nu}[\Psi_l]-\delta^{\mu}{}_{\nu}\mathscr{L}$, and demonstrates its application to simple theories (scalar, vector, spinor) and to QED and QCD with gauge-fixing and ghosts. The resulting EMTs are identical to the Belinfante-improved forms and reproduce the standard electromagnetic and Dirac results in appropriate limits, while maintaining BRST invariance in QCD. The approach provides a conceptually clean path to symmetric EMTs and suggests new ways to interpret the quark–gluon contributions to momentum and angular momentum. It also clarifies historical ambiguities by highlighting the correct transformation properties under local translations and the role of surface terms in Noether's second theorem.

Abstract

Noether's first and second theorems both imply conserved currents that can be identified as an energy-momentum tensor (EMT). The first theorem identifies the EMT as the conserved current associated with global spacetime translations, while the second theorem identifies it as a conserved current associated with local spacetime translations. This work obtains an EMT for quantum electrodynamics and quantum chromodynamics through the second theorem, which is automatically symmetric in its indices and invariant under the expected symmetries (e.g., BRST invariance) without the need for introducing an ad hoc improvement procedure.

Figures (1)

• Figure 1: Depiction of a local spacetime translation.