Nonlinear PDE models in semi-relativistic quantum physics

Abstract

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.