Charge renormalisation in a mean-field approximation of QED

TL;DR

This paper analyzes a mean-field model of quantum electrodynamics to derive a charge renormalisation formula, revealing how vacuum polarization affects the non-relativistic limit and modifies the Hartree-Fock model.

Contribution

It establishes a charge renormalisation formula within the Bogoliubov-Dirac-Fock model near the non-relativistic limit, accounting for vacuum polarization effects.

Findings

Density of ground state is integrable despite not being trace-class.

Charge renormalisation formula holds close to the non-relativistic limit.

Vacuum polarization leads to an altered Hartree-Fock model.

Abstract

We study the Bogoliubov-Dirac-Fock (BDF) model, a no-photon, mean-field approxi- mation of quantum electrodynamics that allows to study relativistic electrons interacting with the vacuum. It is a variational model in which states are represented by Hilbert- Schmidt operators. We prove a charge renormalisation formula that holds close to the non-relativistic limit: the density of a ground state is shown to be integrable although such a state is known not to be trace-class. We prove that we can take the non-relativistic limit by keeping track of the vacuum polarisation. We get an altered Hartree-Fock model due to the screening effect.