  Semiclassical spin-coherent kinetic equations can be derived from quantum
theory with many different approaches (Liouville equation based approaches,
nonequilibrium Green's functions techniques, etc.). The collision integrals
turn out to be formally different, but coincide in textbook examples as well as
for systems where the spin-orbit coupling is only a small part of the kinetic
energy like in related studies on the spin Hall effect. In Dirac cone physics
(graphene, surface states of topological insulators like Bi_{1-x}Sb_x, Bi_2Te_3
etc.), where this coupling constitutes the entire kinetic energy, the
difference manifests itself in the precise value of the electron-hole coherence
originated quantum correction to the Drude conductivity $\sim e^2/h * \ell
k_F$. The leading correction is derived analytically for single and multilayer
graphene with general scalar impurities. The often neglected principal value
terms in the collision integral are important. Neglecting them yields a leading
correction of order $(\ell k_F)^{-1}$, whereas including them can give a
correction of order $(\ell k_F)^0$. The latter opens up a counterintuitive
scenario with finite electron-hole coherent effects at Fermi energies
arbitrarily far above the neutrality point regime, for example in the form of a
shift $\sim e^2/h$ that only depends on the dielectric constant. This residual
conductivity, possibly related to the one observed in recent experiments,
depends crucially on the approach and could offer a setting for experimentally
singling out one of the candidates. Concerning the different formalisms we
notice that the discrepancy between a density matrix approach and a Green's
function approach is removed if the Generalized Kadanoff-Baym Ansatz in the
latter is replaced by an anti-ordered version.
