The positronium and the dipositronium in a Hartree-Fock approximation of quantum electrodynamics

TL;DR

This paper proves the existence of bound states like positronium and dipositronium within a Hartree-Fock approximation of quantum electrodynamics using the Bogoliubov-Dirac-Fock model, highlighting new critical points of the energy functional.

Contribution

It demonstrates the existence of positronium, dipositronium, and related multi-particle states as critical points in the BDF model without external fields, advancing theoretical understanding.

Findings

Existence of para-positronium as a critical point.

Existence of dipositronium molecule as a critical point.

Generalization to states with multiple electrons and positrons.

Abstract

The Bogoliubov-Dirac-Fock (BDF) model is a no-photon approximation of quantum electrodynamics. It allows to study relativistic electrons in interaction with the Dirac sea. A state is fully characterized by its one-body density matrix, an infinite rank nonnegative projector. We prove the existence of the para-positronium, the bound state of an electron and a positron with antiparallel spins, in the BDF model represented by a critical point of the energy functional in the absence of external field. We also prove the existence of the dipositronium, a molecule made of two electrons and two positrons that also appears as a critical point. More generally, for any half integer $j\in \tfrac{1}{2}+\mathbb{Z}_+$, we prove the existence of a critical point of the energy functional made of $2j+1$ electrons and $2j+1$ positrons.