Exact expectation values in a boost-invariant fluid of Dirac fermions with finite spin density

TL;DR

This paper provides exact analytical and numerical results for expectation values, including spin polarization and related observables, in a boost-invariant, out-of-equilibrium Dirac fermion fluid with finite spin density, relevant for relativistic hydrodynamics.

Contribution

It derives an exact partition function and expectation values for a non-interacting Dirac fermion fluid with finite spin potential, advancing understanding of spin effects in relativistic fluids.

Findings

Analytic expression for the partition function at finite spin potential.

Thermodynamic relations hold in the system under consideration.

Both shear-induced polarization and the spin Hall effect are absent in boost-invariant systems.

Abstract

We study a boost-invariant, out-of-equilibrium fluid of non-interacting Dirac fermions with a finite canonical spin potential. After solving the Dirac equation in Milne coordinates, we exactly diagonalize the non-equilibrium density operator and compute the partition function and expectation values of relevant observables, including spin polarization, energy density, longitudinal and transverse pressures, spin density, and \emph{spin torque}, i.e. the source of spin non-conservation. We find an analytic expression for the partition function at finite spin potential, and show numerically that thermodynamic relations connecting it to thermodynamic functions hold in the system under consideration. We show that, in a boost-invariant system, both shear-induced polarization and the spin Hall effect are absent, and that a non-vanishing polarization can only arise from a finite spin potential in a free theory. We obtain an analytic expression for the spin polarization as a function of the spin potential in some particular cases, and otherwise compute numerically its exact expectation value at finite spin potential. Our results are discussed in the context of relativistic spin hydrodynamics and quark--gluon plasma phenomenology.