Global equilibrium at order $\mathcal{O}(\hbar^2)$: pseudo-gauge ambiguity, Maxwell relations, and thermodynamic consistency

Abstract

We study massive spin-$1/2$ Boltzmann particles in global thermodynamic equilibrium with rotation and acceleration, consistently including quantum corrections up to order $\mathcal{O}(\hbar^2)$. For such a system, the fluid velocity becomes an independent thermodynamic variable and fundamental thermodynamic relations have to be extended. Employing the Wigner-function formalism, we show that the standard mass-shell condition $k^2 = m^2$ is modified at order $\mathcal{O}(\hbar^2)$ by contributions quadratic in the thermal vorticity. Incorporating this modified mass-shell constraint, we derive the (net) particle-number current, energy-momentum tensor, spin-current tensor, and associated thermodynamic quantities in the kinetic, canonical, and de Groot--van Leeuwen--van Weert (GLW) pseudo-gauges. We explicitly demonstrate that, while global quantities, like the total particle number or total energy, are independent of the choice of pseudo-gauge, local quantities, like the particle-number density or the energy density, are pseudo-gauge dependent. We then examine the thermodynamic relations in each pseudo-gauge and demonstrate that thermodynamic consistency requires that the thermodynamic Maxwell relations are satisfied. We find that these relations hold in the kinetic-theory pseudo-gauge, whereas they are in general violated in the canonical and GLW pseudo-gauges. These findings suggest a preference for using the kinetic-theory pseudo-gauge in applications such as spin hydrodynamics.