  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.
