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

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