  We investigate some foundational issues in the quantum theory of spin
transport, in the general case when the unperturbed Hamiltonian operator $H_0$
does not commute with the spin operator in view of Rashba interactions, as in
the typical models for the Quantum Spin Hall effect.
  A gapped periodic one-particle Hamiltonian $H_0$ is perturbed by adding a
constant electric field of intensity $\varepsilon \ll 1$ in the $j$-th
direction, and the linear response in terms of a $S$-current in the $i$-th
direction is computed, where $S$ is a generalized spin operator. We derive a
general formula for the spin conductivity that covers both the choice of the
conventional and of the proper spin current operator. We investigate the
independence of the spin conductivity from the choice of the fundamental cell
(Unit Cell Consistency), and we isolate a subclass of discrete periodic models
where the conventional and the proper $S$-conductivity agree, thus showing that
the controversy about the choice of the spin current operator is immaterial as
far as models in this class are concerned. As a consequence of the general
theory, we obtain that whenever the spin is (almost) conserved, the spin
conductivity is (approximately) equal to the spin-Chern number. The method
relies on the characterization of a non-equilibrium almost-stationary state
(NEASS), which well approximates the physical state of the system (in the sense
of space-adiabatic perturbation theory) and allows moreover to compute the
response of the adiabatic $S$-current as the trace per unit volume of the
$S$-current operator times the NEASS. This technique can be applied in a
general framework, which includes both discrete and continuum models.
