The Semi-Classical Limit from the Dirac Equation with Time-Dependent External Electromagnetic Field to Relativistic Vlasov Equations

TL;DR

This paper rigorously derives the relativistic Vlasov equations from the Dirac equation with time-dependent electromagnetic fields in the semi-classical limit, using Wigner transforms and a novel approach that avoids eigenspace projection.

Contribution

It introduces a new method for taking the semi-classical limit of the Dirac equation that handles less regular potentials and maintains relativistic effects like antimatter and spin.

Findings

Successfully passes to the limit in the full Wigner matrix equation.

Generalizes to less regular, time-dependent potentials.

Maintains relativistic effects such as antimatter and spin at the classical level.

Abstract

We prove the mathematically rigorous (semi-)classical limit $\hbar \to 0$ of the Dirac equation with time-dependent external electromagnetic field to relativistic Vlasov equations with Lorentz force for electrons and positrons. In this limit antimatter and spin remain as intrinsically relativistic effects on a classical level. Our global-in-time results use Wigner transforms and a Lagrange multiplier viewpoint of the matrix-valued Wigner equation. In particular, we pass to the limit in the ''full" Wigner matrix equation without projecting on the eigenspaces of the matrix-valued symbol of the Dirac operator. In the limit, the Lagrange multiplier maintains the constraint that the Wigner measure and the symbol of the Dirac operator commute and vanishes when projected on the electron or positron eigenspace. This is a different approach to the problem as discussed in [P. G\'erard, P. Markowich, N.J. Mauser, F. Poupaud: Comm. Pure Appl. Math. 50(4):323--379, 1997], where the limit is taken in the projected Wigner equation. By explicit calculation of the remainder term in the expansion of the Moyal product we are able to generalize to time-dependent potentials with much less regularity. We use uniform $L^2$ bounds for the Wigner transform, which are only possible for a special class of mixed states as initial data.