Classical limit for semi-relativistic Hartree systems

TL;DR

This paper proves that the semi-relativistic Hartree quantum model converges to the classical relativistic Vlasov-Poisson system in the limit, using Wigner transformation techniques, under certain initial conditions and interaction assumptions.

Contribution

It establishes the classical limit of the semi-relativistic Hartree model to the relativistic Vlasov-Poisson system, including attractive and repulsive interactions, with size constraints in the attractive case.

Findings

Convergence of quantum to classical dynamics proven

Applicable to both attractive and repulsive interactions

Uses Wigner transformation techniques for analysis

Abstract

We consider the three-dimensional semi-relativistic Hartree model for fast quantum mechanical particles moving in a self-consistent field. Under appropriate assumptions on the initial density matrix as a (fully) mixed quantum state we prove, using Wigner transformation techniques, that its classical limit yields the well known relativistic Vlasov-Poisson system. The result holds for the case of attractive and repulsive mean-field interaction, with an additional size constraint in the attractive case.