  We present the self-consistent Pauli equation, a semi-relativistic model for
charged spin-$1/2$-particles with self-interaction with the electromagnetic
field. The Pauli equation arises as the $O(1/c)$ approximation of the
relativistic Dirac equation. The fully relativistic self-consistent model is
the Dirac-Maxwell equation where the description of spin and the magnetic field
arises naturally. In the non-relativistic setting the correct self-consistent
equation is the Schr\"odinger-Poisson equation which does not describe spin and
the magnetic field and where the self-interaction is with the electric field
only.
  The Schr\"odinger-Poisson equation also arises as the mean field limit of the
$N$-body Schr\"odinger equation with Coulomb interaction. We propose that the
Pauli-Poisson equation arises as the mean field limit $N \rightarrow \infty$ of
the linear $N$-body Pauli equation with Coulomb interaction where one has to
pay extra attention to the fermionic nature of the Pauli equation.
  We present the semiclassical limit of the Pauli-Poisson equation by the
Wigner method to the Vlasov equation with Lorentz force coupled to the Poisson
equation which is also consistent with the hierarchy in $1/c$ of the
self-consistent Vlasov equation. This is a non-trivial extension of the
groundbreaking works by Lions & Paul and Markowich & Mauser, where we need
methods like magnetic Lieb-Thirring estimates.
