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

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