Generating Spin Polarization from Vorticity through Nonlocal Collisions

We derive the collision term in the Boltzmann equation using the equation of motion for the Wigner function of massive spin-$1/2$ particles. To next-to-lowest order in $\hslash$, it contains a nonlocal contribution, which is responsible for the conversion of orbital into spin angular momentum. In a proper choice of pseudogauge, the antisymmetric part of the energy-momentum tensor arises solely from this nonlocal contribution. We show that the collision term vanishes in global equilibrium and that the spin potential is, then, equal to the thermal vorticity. In the nonrelativistic limit, the equations of motion for the energy-momentum and spin tensors reduce to the well-known form for hydrodynamics for micropolar fluids.