The Mean-Field Approximation in Quantum Electrodynamics. The no-photon case

TL;DR

This paper rigorously analyzes the mean-field approximation of Quantum Electrodynamics without photons, focusing on the thermodynamic limit and the properties of the vacuum state in a finite box with boundary conditions.

Contribution

It provides a mathematical framework for the mean-field limit of QED in the no-photon case, establishing the existence and uniqueness of the vacuum state and deriving the Bogoliubov-Dirac-Fock functional.

Findings

Energy per volume converges in the thermodynamic limit.

The free vacuum is the unique minimizer without external fields.

The polarized vacuum minimizes the Bogoliubov-Dirac-Fock energy in the presence of external fields.

Abstract

We study the mean-field approximation of Quantum Electrodynamics, by means of a thermodynamic limit. The QED Hamiltonian is written in Coulomb gauge and does not contain any normal-ordering or choice of bare electron/positron subspaces. Neglecting photons, we define properly this Hamiltonian in a finite box $[-L/2;L/2)^3$, with periodic boundary conditions and an ultraviolet cut-off $\Lambda$. We then study the limit of the ground state (i.e. the vacuum) energy and of the minimizers as $L$ goes to infinity, in the Hartree-Fock approximation. In case with no external field, we prove that the energy per volume converges and obtain in the limit a translation-invariant projector describing the free Hartree-Fock vacuum. We also define the energy per unit volume of translation-invariant states and prove that the free vacuum is the unique minimizer of this energy. In the presence of an external field, we prove that the difference between the minimum energy and the energy of the free vacuum converges as $L$ goes to infinity. We obtain in the limit the so-called Bogoliubov-Dirac-Fock functional. The Hartree-Fock (polarized) vacuum is a Hilbert-Schmidt perturbation of the free vacuum and it minimizes the Bogoliubov-Dirac-Fock energy.