  In this paper we derive a fully relativistic kinetic theory for spin-1/2
particles and its coupling to Maxwell's equations, valid in the long
scale-length limit, where the fields vary on a scale much longer than the
localization of the particles; we work to first order in $\hbar$. Our starting
point is a Foldy-Wouthuysen (FW) transformation, applicable to this regime, of
the Dirac Hamiltonian. We derive the corresponding evolution equation for the
Wigner quasi-distribution in an external electromagnetic field. Using a
Lagrangian method we find expressions for the charge and current densities,
expressed as free and bound parts. It is furthermore found that the velocity is
non-trivially related to the momentum variable, with the difference depending
on the spin and the external electromagnetic fields. This fact that has
previously been discussed as "hidden momentum" and is due to that the FW
transformation maps pointlike particles to particle clouds for which the
prescription of minimal coupling is incorrect, as they have multipole moments.
We express energy and momentum conservation for the system of particles and the
electromagnetic field, and discuss our results in the context of the
Abraham-Minkowski dilemma.
