Spin-dependent distribution functions for relativistic hydrodynamics of spin-1/2 particles

TL;DR

This paper analyzes and simplifies phase-space spin density matrices for spin-1/2 particles in relativistic hydrodynamics, linking spin polarization vectors to the Pauli-Lubanski vector and clarifying tensor choices.

Contribution

It provides a detailed analysis and reduction of spin density matrices, establishing their relation to the Pauli-Lubanski vector and unifying different spin tensor forms.

Findings

Spin density matrices linear in Dirac spin operator

Spin polarization vectors match Pauli-Lubanski vector

Different spin tensor forms yield same Pauli-Lubanski vector

Abstract

Recently advocated expressions for the phase-space dependent spin-1/2 density matrices of particles and antiparticles are analyzed in detail and reduced to the forms linear in the Dirac spin operator. This allows for a natural determination of the spin polarization vectors of particles and antiparticles by the trace of products of the spin density matrices and the Pauli matrices. We demonstrate that the total spin polarization vector obtained in this way agrees with the Pauli-Lubanski four-vector, constructed from an appropriately chosen spin tensor and boosted to the particle rest frame. We further show that several forms of the spin tensor used in the literature give the same Pauli-Lubanski four-vector.