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

TL;DR

This paper demonstrates that using Noether's second theorem can uniquely determine energy-momentum and spin currents in relativistic spin hydrodynamics, avoiding ambiguities associated with pseudogauge transformations.

Contribution

It introduces a method based on Noether's second theorem to derive unique energy-momentum and spin currents without pseudogauge ambiguities for free Dirac particles.

Findings

Noether's second theorem fixes the super-potential uniquely.

The method applies to free Dirac particles with spin one-half.

It avoids the need for pseudogauge transformations.

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.