Communications in Mathematical Physics

, Volume 257, Issue 3, pp 515–562

Existence of a Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation


DOI: 10.1007/s00220-005-1343-4

Cite this article as:
Hainzl, C., Lewin, M. & Séré, É. Commun. Math. Phys. (2005) 257: 515. doi: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 D0. In the presence of an external field, these virtual particles react and the vacuum becomes polarized. In this paper, following Chaix and Iracane (J. Phys. B22, 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 the 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Christian Hainzl
    • 1
    • 2
  • Mathieu Lewin
    • 1
  • Éric Séré
    • 1
  1. 1.CEREMADE, UMR CNRS 7534Université Paris IX DauphineParis, Cedex 16France
  2. 2.Laboratoire de MathématiquesOrsay CedexFrance