A Microscopic Derivation of the Time-Dependent Hartree-Fock Equation with Coulomb Two-Body Interaction

TL;DR

This paper rigorously derives the time-dependent Hartree-Fock equation for a Fermi gas with Coulomb interactions in the mean-field limit, extending previous results to unbounded potentials and framing the dynamics as a superhamiltonian system.

Contribution

It provides a microscopic derivation of the Hartree-Fock equation with Coulomb interactions, including a novel superhamiltonian formulation and an Egorov-type theorem for observable dynamics.

Findings

Derivation of the Hartree-Fock equation for Coulomb interactions in the mean-field limit

Extension of previous work to unbounded interaction potentials

Representation of the dynamics as a superhamiltonian system

Abstract

We study the dynamics of a Fermi gas with a Coulomb interaction potential, and show that, in a mean-field limiting regime, the dynamics is described by the Hartree-Fock equation. This extends previous work of Bardos et al. to the case of unbounded interaction potentials. We also express the mean-field limit as a "superhamiltonian" system, and state our main result in terms of a Heisenberg-picture dynamics of observables. This is a Egorov-type theorem.