Existence of global-in-time solutions to a generalized Dirac-Fock type evolution equation

TL;DR

This paper proves the existence of global-in-time solutions for a generalized Dirac-Fock evolution equation derived from quantum electrodynamics, modeling relativistic electrons including the Dirac sea within a Hartree-Fock framework.

Contribution

It establishes the first rigorous proof of global solutions for a Dirac-Fock type evolution equation using the Bogoliubov-Dirac-Fock formalism.

Findings

Existence of global-in-time solutions proven

Applicable to relativistic electrons in quantum electrodynamics

Utilizes Bogoliubov-Dirac-Fock formalism

Abstract

We consider a generalized Dirac-Fock type evolution equation deduced from no-photon Quantum Electrodynamics, which describes the self-consistent time-evolution of relativistic electrons, the observable ones as well as those filling up the Dirac sea. This equation has been originally introduced by Dirac in 1934 in a simplified form. Since we work in a Hartree-Fock type approximation, the elements describing the physical state of the electrons are infinite rank projectors. Using the Bogoliubov-Dirac-Fock formalism, introduced by Chaix-Iracane ({\it J. Phys. B.}, 22, 3791--3814, 1989), and recently established by Hainzl-Lewin-Sere, we prove the existence of global-in-time solutions of the considered evolution equation.