Fully relativistic spin torques and spin currents

In using the one-particle Dirac equation in the presence of an external electro-magnetic field an exact equation of motion for the density of the four-component Bargmann-Wigner polarization operator $T_{\mu }=(\vec{T},T_{4}) $ is presented, the various occuring terms of which can be viewed as the relativistic counterparts of \textit{ad hoc} defined non- relativistic spin-currents and spin-transfer torques. Based on the properties of the Berry phase the particle and the magnetization density can be formulated in terms of a instantanous resolvent $G(z;t)$ of the time dependent Dirac equation by means of contour integrations. The corresponding Greens function $G(\mathbf{r,r}^ {\prime }$, $z;t)$ can in turn be evaluated within a multiple scattering scheme by solving at each given time $t$ a ``quasi-stationary'' problem. In terms of this Greens function the time evolution of any single-particle density, i.e., also of $T_{\mu }=(\vec{T},T_{4})$ can be evaluated. As a first application the case of a single Fe atom is considered, for which very easily a comparison with a time- dependent first order perturbational scheme can be given.