On the non-uniqueness of the energy-momentum and spin currents

TL;DR

The paper addresses the non-uniqueness of local energy-momentum and spin currents arising from pseudogauge freedom. It applies Noether's second theorem to fix the super-potential, deriving a symmetric energy-momentum tensor and a vanishing spin tensor for free Dirac fields without relying on pseudogauge transformations. This yields a physically unique localization of densities consistent with general relativity and angular momentum algebra, matching Belinfante-type currents. The results clarify how boundary terms determine current localization and offer a robust framework for comparing angular-momentum decompositions in hadron physics and spin hydrodynamics.

Abstract

The macroscopic energy-momentum and spin densities of relativistic spin hydrodynamics are obtained from the ensemble average of their respective microscopic definitions (quantum operators). These microscopic definitions suffer from ambiguities, meaning that one may obtain different forms of symmetric energy-momentum tensor and spin tensor through pseudogauge transformations (or, in other words, Belinfante improvement procedure). However, this ambiguity may be fixed if we obtain these currents using Noether's second theorem instead of widely used Noether's first theorem. The second theorem fixes the super-potential determined by local symmetry, thereby selecting a unique physically consistent pseudogauge. In this article, we use Noether's second theorem to derive energy-momentum and spin currents without the need of pseudogauge transformations for free Dirac massive particles with spin one-half.