Canonical and phenomenological formulations of spin hydrodynamics

TL;DR

This paper compares two formulations of relativistic spin hydrodynamics, demonstrating their equivalence through a pseudogauge transformation when the canonical approach is suitably modified, with implications for entropy production.

Contribution

It establishes the conditions under which canonical and phenomenological spin hydrodynamics frameworks are equivalent, highlighting the need for tensor modifications.

Findings

The two frameworks are equivalent via a pseudogauge transformation with tensor modification.

Entropy production positivity is consistent across both frameworks after modification.

Canonical approach requires a divergence-free tensor addition for equivalence.

Abstract

Two formulations of relativistic hydrodynamics of particles with spin 1/2 are compared. The first approach, dubbed the canonical one, uses expressions for the energy-momentum and spin tensors that have properties that follow a direct application of Noether's theorem, which yields a totally antisymmetric spin tensor. The other one is based on a simplified form of the spin tensor and is commonly used in the current literature under the name of a phenomenological approach. We show that these two frameworks are equivalent, i.e., they can be directly connected by a suitably defined pseudogauge transformation, only if the first framework is initially improved by a suitable modification of the energy-momentum tensor (addition of a divergence-free term that cannot be interpreted as a pseudogauge). Our analysis uses arguments related to the positivity of entropy production. The latter turns out to be equivalent for the improved canonical and phenomenological frameworks.