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Using the phase-space formulation of quantum mechanics, we derive afour-component Wigner equation for a system composed of spin-1/2 fermions(typically, electrons) including the Zeeman effect and the spin-orbit coupling.This Wigner equation is coupled to the appropriate Maxwell equations to form aself-consistent mean-field model. A set of semiclassical Vlasov equations withspin effects is obtained by expanding the full quantum model to first order inthe Planck constant. The corresponding hydrodynamic equations are derived bytaking velocity moments of the phase-space distribution function. A simpleclosure relation is proposed to obtain a closed set of hydrodynamic equations.

philosophical transactions - royal society. mathematical, physical and engineering sciences/philosophical transactions - royal society. mathematical, physical and engineering sciences

Phase space methods for the spin dynamics in condensed matter systems