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Starting from the Keldysh theory, for a general low energy $N$-bandHamiltonian in the clean limit, we perform a manifestly $\smash{U(1) \timesSU(N)}$ gauge invariant semiclassical expansion. A generalized Berry curvaturetensor is shown to control a redistribution of spectral weights. Newexpressions for certain physical quantities ensue, establishing the limits of apreviously 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 anomalousHall conductivity and chiral magnetic effect, in all generality. Thedemonstrated flexibility and efficiency of our formalism derives from theinsulation it provides from the underlying complexity of the quantum kinetics,notwithstanding its rigorous connection to this deeper level.

Semiclassical kinetic theory for systems with non-trivial quantum geometry and the expectation value of physical quantities