Chiral transport equation from the quantum Dirac Hamiltonian and the on-shell effective field theory

TL;DR

This paper derives a relativistic chiral transport equation for massless fermions using semiclassical diagonalization of the Dirac Hamiltonian, incorporating Berry curvature effects and confirming the invariance of anomalies at finite temperature and density.

Contribution

It introduces a novel derivation of the chiral transport equation that includes Berry curvature effects without requiring system state knowledge, applicable at high temperatures.

Findings

Berry connection modifies fermion equations of motion.

Dispersion relations are corrected by Berry curvature at first order in Planck's constant.

Axial and gauge anomalies remain unaltered at finite temperature and density.

Abstract

We derive the relativistic chiral transport equation for massless fermions and antifermions by performing a semiclassical Foldy-Wouthuysen diagonalization of the quantum Dirac Hamiltonian. The Berry connection naturally emerges in the diagonalization process to modify the classical equations of motion of a fermion in an electromagnetic field. We also see that the fermion and antifermion dispersion relations are corrected at first order in the Planck constant by the Berry curvature, as previously derived by Son and Yamamoto for the particular case of vanishing temperature. Our approach does not require knowledge of the state of the system, and thus it can also be applied at high temperature. We provide support for our result by an alternative computation using an effective field theory for fermions and antifermions: the on-shell effective field theory. In this formalism, the off-shell fermionic modes are integrated out to generate an effective Lagrangian for the quasi-on-shell fermions/antifermions. The dispersion relation at leading order exactly matches the result from the semiclassical diagonalization. From the transport equation, we explicitly show how the axial and gauge anomalies are not modified at finite temperature and density despite the incorporation of the new dispersion relation into the distribution function.