Two Comments on the Derivation of the Time-Dependent Hartree-Fock Equation

TL;DR

This paper revisits the derivation of the time-dependent Hartree-Fock equation for interacting fermions, extending its validity to arbitrary dimensions and reformulating the proof using a coherent state approach similar to bosonic systems.

Contribution

It provides two key comments: the derivation applies in any space dimension and introduces a new proof technique using particle-hole transformations.

Findings

Derivation valid in arbitrary space dimensions.

Reformulation of proof using particle-hole transformations.

Connection to coherent state methods for bosons.

Abstract

We revisit the derivation of the time-dependent Hartree-Fock equation for interacting fermions in a regime coupling a mean-field and a semiclassical scaling, contributing two comments to the result obtained in 2014 by Benedikter, Porta, and Schlein. First, the derivation holds in arbitrary space dimension. Second, by using an explicit formula for the unitary implementation of particle-hole transformations, we cast the proof in a form similar to the coherent state method of Rodnianski and Schlein for bosons.