Complete and Consistent Chiral Transport from Wigner Function Formalism.

Recently there has been significant interest in understanding the macroscopic quantum transport in a many-body system of chiral fermions. A natural many-body framework for describing such a system which is generally out of equilibrium, is the transport equation for its phase space distribution function. There are however several subtle and confusing issues, particularly related to Lorentz invariance and frame dependence. In this paper, we derive a covariant form of chiral transport equation for massless fermions, by starting from the general Wigner function formalism and carrying out a complete and consistent semiclassical expansion up to $\hat{\mathbf{O}}(\hbar)$ order. This new result clarifies the Lorentz invariance and frame dependence issues associated with the $\hat{\mathbf{O}}(\hbar)$ correction. From the so-obtained chiral transport equation one also naturally derives as its consequences the 3D formulation of chiral kinetic theory as well as many peculiar phenomena of chiral fermions.