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

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

2004-03-04

arxiv.org/abs/math-ph/0403005v3

Commun.Math.Phys. 257 (2005) 515-562

Physics

Mathematical Physics

final version, to appear in Commun. Math. Phys

Scientific paper

10.1007/s00220-005-1343-4

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.

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