  Starting from the Keldysh theory, for a general low energy $N$-band
Hamiltonian in the clean limit, we perform a manifestly $\smash{U(1) \times
SU(N)}$ gauge invariant semiclassical expansion. A generalized Berry curvature
tensor is shown to control a redistribution of spectral weights. New
expressions for certain physical quantities ensue, establishing the limits of a
previously proposed correction to the density of states. In the two-band case,
we derive a completely general semiclassical kinetic theory including all
$O(\hbar)$ quantum corrections. As an application, we show how one can recover,
out of a single simple calculation, the chiral anomaly, intrinsic anomalous
Hall conductivity and chiral magnetic effect, in all generality. The
demonstrated flexibility and efficiency of our formalism derives from the
insulation it provides from the underlying complexity of the quantum kinetics,
notwithstanding its rigorous connection to this deeper level.
