Kinetic theories: from curved space to flat space

TL;DR

This work addresses whether the equivalence between off-equilibrium states in flat space and gravitational perturbations of equilibrium states extends to microscopic particle dynamics. It develops a kinetic theory for spinless and spinning particles in a torsionful curved background using the Wigner function $W$ and shows that, by choosing a local inertial frame with vanishing spin connection, the curved-space kinetic equation maps to the flat-space Boltzmann equation and that torsion encodes vorticity via $T^{\hat{0}}_{ij}=2\epsilon^{ijk}\Omega^k$. For spinning particles, it derives the quantum kinetic equations in torsion, decomposes the Wigner function into $W=S+i\gamma^5P+V_a\gamma^a+A_a\gamma^5\gamma^a+\tfrac12\sigma^{ab}T_{ab}$, and, at $O(\partial_x)$, obtains expressions for the axial-vector $A^a$ and tensor $T^{ab}$ that reproduce spin polarization from shear and vorticity; an elastic-extension yields analogous results using the displacement-gradient tensor $\Sigma$ and velocity $U^a$. The paper also provides an order-of-magnitude polarization estimate in Dirac semimetals, highlighting potential experimental relevance. Overall, the work provides a unified geometric-kinetic framework to study off-equilibrium dynamics via curved-space perturbations with implications for relativistic heavy-ion physics and condensed-matter spintronics.

Abstract

We generalize the equivalence between off-equilibrium state and gravitational perturbation of equilibrium state from dynamics of macroscopic quantities to that of microscopic particles. We also generalize the equivalence to incorporate off-equilibrium state with vorticity by torsional perturbation to equilibrium state. The equivalence is achieved by mapping kinetic theories of spinless and spinning particles in torsional curved space to flat space through suitable choice of inertial frame that eliminates geodesic forces on particles. The equivalence has been shown for hydrodynamic and elastic regimes. In the latter case, we predict spin polarization induced by time-variation of shear strain in elastic materials. We also provide an order-of-magnitude estimate for the polarization in Dirac semi-metal.