  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.
