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

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. Gérard, 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.

💡 Research Summary

The paper establishes a mathematically rigorous semiclassical limit (ℏ → 0) for the Dirac equation with time‑dependent external electromagnetic fields, showing that the quantum dynamics converge globally in time to a relativistic Vlasov system describing electrons and positrons under the Lorentz force. The authors depart from the traditional approach of projecting the matrix‑valued Wigner equation onto the eigenspaces of the Dirac symbol. Instead, they treat the full Wigner matrix equation and introduce a matrix‑valued Lagrange multiplier Y to enforce the commutation constraint