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

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.