Berry phase in the phase space worldline representation: the axial anomaly and classical kinetic theory

TL;DR

This paper explores the Berry phase in phase space for Weyl and Dirac fermions, developing classical kinetic theories and analyzing the axial anomaly, revealing insights into their quantum and classical behaviors.

Contribution

It introduces a classical kinetic theory for Dirac fermions using a spinor construction and examines the axial anomaly within this framework.

Findings

Weyl fermion kinetic theory reduces in dimension, resembling quantum mechanics.

Dirac fermion kinetic theory maintains Lorentz covariance.

Under adiabatic approximation, the Dirac operator index vanishes.

Abstract

The Berry phase is analyzed for Weyl and Dirac fermions in a phase space representation of the worldline formalism. Kinetic theories are constructed for both at a classical level. Whereas the Weyl fermion case reduces in dimension, resembling a theory in quantum mechanics, the Dirac fermion case takes on a manifestly Lorentz covariant form. To achieve a classical kinetic theory for the non-Abelian Dirac fermion Berry phase a spinor construction of Barut and Zanghi is utilized. The axial anomaly is also studied at a quantum level. It is found that under an adiabatic approximation, which is necessary for facilitating a classical kinetic theory, the index of the Dirac operator for massless fermions vanishes. Even so, similarities of an axial rotation to an exact non-covariant Berry phase transform are drawn by application of the Fujikawa method to the Barut and Zanghi spinors on the worldline.