Classical version of the non-Abelian gauge anomaly

Abstract

We show that a version of the covariant gauge anomaly for a 3+1 dimensional chiral fermion interacting with a non-Abelian gauge field can be obtained from the classical Hamiltonian flow of its probability distribution in phase space. The only quantum input needed is the Berry phase that arises from the direction of the spin being slaved to the particle's momentum.

Classical chiral kinetic theory and anomalies in even space-time dimensions

“…How to do this is explained in physicist's language in [13,14] and also in [7]. Briefly stated, a representation of G with highest weight Λ determines a Lie algebra element α Λ that lies in the Cartan subalgebra.…”

“…It exhibited explicitly in [7] for the case G = SU (3), that the accuracy of the classical approximation to the symmetrized trace is again greatly increased if the classical trace integral is actually taken over the orbit corresponding to the Weyl-shifted weight Λ → Λ + ρ, where the Weyl vector ρ is half the sum of the positive roots.…”

Wigner Translations and the Observer Dependence of the Position of Massless Spinning Particles

“…This shift is related to the optical spin Hall effect [4][5][6][7][8] and to the observer dependence of the location of massless particles [9]. It gives rise to the unusual Lorentz covariance properties found [10,11] in the chiral kinetic theory approach to anomalous conservation laws [12][13][14] and is also the source of the difficulty of obtaining a conventionally-covariant classical mechanics for a massless spinning particle in a gravitational field [15,16].…”

Chiral Langevin theory for non-Abelian plasmas

“…For the latter, we started with the linearized non-Abelian Boltzmann-Vlasov equation with Berry curvature corrections and integrated out (semi)hard degrees of freedom. Alternatively, one should also be able to arrive at the same Langevintype equation starting from a different classical transport theory [58] with Berry curvature corrections [59] where the trajectories of a particle are specified by x, p, and the nonAbelian charge Q a (known as the Wong equations [60]). Such a derivation was done in the case without anomalous parity-violating effects in Ref.…”