The Semiclassical Limit of the 2D Dirac--Hartree Equation with Periodic Potentials

TL;DR

This paper rigorously derives relativistic Vlasov equations from the semiclassical limit of the 2D Dirac--Hartree equation with periodic potentials, clarifying the transition from quantum to classical relativistic dynamics.

Contribution

It provides a rigorous derivation of band-resolved Vlasov equations from the Dirac--Hartree model in the semiclassical regime, including both massive and massless cases.

Findings

Massless case exhibits ballistic propagation with constant speed.

Massive case shows relativistic velocity behavior.

Relativistic Vlasov--Poisson equation emerges as a limit.

Abstract

We study the semiclassical limit of the two-dimensional Dirac--Hartree equation in the presence of a periodic external potential. The spinor dynamics are formulated using the matrix-valued Wigner transform together with spectral projectors onto the positive and negative energy bands. Under suitable assumptions on the initial data and the potentials, we rigorously derive Vlasov-type transport equations describing the evolution of the band-resolved phase-space densities in both the massive and massless regimes. In the massless case, the limiting dynamics propagate ballistically with constant speed, while in the massive case the velocity is relativistic. Our analysis justifies the emergence of relativistic Vlasov equations from Dirac--Hartree dynamics in the semiclassical regime. As a corollary, we recover the relativistic Vlasov--Poisson equation from the Dirac equation with a regularized Coulomb interaction when the regularization vanishes together with the semiclassical parameter.