# Spin chemical potential for relativistic particles with spin 1/2

## TL;DR

This paper develops a relativistic semi-classical Wigner function framework for spin 1/2 particles, incorporating spin density matrices and exploring equilibrium states with spin chemical potentials, highlighting their momentum and space-time dependencies.

## Contribution

It introduces a consistent algebraic structure for the relativistic Wigner function that includes spin density information and analyzes different forms of spin chemical potentials in equilibrium.

## Key findings

- Scalar spin chemical potential should be momentum dependent
- Tensor spin chemical potential can depend on space-time coordinates
- Scalar and tensor spin chemical potentials can be related

## Abstract

We analyze algebraic structure of a relativistic semi-classical Wigner function of particles with spin 1/2 and show that it consistently includes information about the spin density matrix both in two-dimensional spin and four-dimensional spinor spaces. This result is subsequently used to explore various forms of equilibrium functions that differ by specific incorporation of spin chemical potential. We argue that a scalar spin chemical potential should be momentum dependent, while its tensor form may be a function of space-time coordinates only. This allows for the use of the tensor form in local thermodynamic relations. We furthermore show how scalar and tensor forms can be linked to each other.