Nonlinear PDE Models in Semi-relativistic Quantum Physics

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 & ouml;dinger-Poisson equation which does not describe spin and the magnetic field and where the self-interaction is with the electric field only. The Schr & ouml;dinger-Poisson equation also arises as the mean field limit of the N-body Schr & ouml;dinger equation with Coulomb interaction. We propose that the Pauli- Poisson equation arises as the mean field limit N ? 8 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.