Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation

TL;DR

This paper proves the existence of a unique stable polarized vacuum in the Bogoliubov-Dirac-Fock model under external electrostatic fields, using a fixed-point method to find a minimizer of the BDF-energy.

Contribution

It establishes the existence and uniqueness of a polarized vacuum solution in the BDF model with external fields, advancing the mathematical understanding of vacuum polarization in QED.

Findings

Existence of a unique BDF vacuum minimizer under external fields

The BDF-energy is bounded from below and admits a minimizer

The minimizer solves a self-consistent equation for the polarized vacuum

Abstract

According to Dirac's ideas, the vacuum consists of infinitely many virtual electrons which completely fill up the negative part of the spectrum of the free Dirac operator $D^0$. In the presence of an external field, these virtual particles react and the vacuum becomes polarized. In this paper, following Chaix and Iracane ({\it J. Phys. B}, 22, 3791--3814, 1989), we consider the Bogoliubov-Dirac-Fock model, which is derived from no-photon QED. The corresponding BDF-energy takes the polarization of the vacuum into account and is bounded from below. A BDF-stable vacuum is defined to be a minimizer of this energy. If it exists, such a minimizer is solution of a self-consistent equation. We show the existence of a unique minimizer of the BDF-energy in the presence of an external electrostatic field, by means of a fixed-point approach. This minimizer is interpreted as the polarized vacuum.