  We study the mean-field approximation of Quantum Electrodynamics, by means of
a thermodynamic limit. The QED Hamiltonian is written in Coulomb gauge and does
not contain any normal-ordering or choice of bare electron/positron subspaces.
Neglecting photons, we define properly this Hamiltonian in a finite box
$[-L/2;L/2)^3$, with periodic boundary conditions and an ultraviolet cut-off
$\Lambda$. We then study the limit of the ground state (i.e. the vacuum) energy
and of the minimizers as $L$ goes to infinity, in the Hartree-Fock
approximation.
  In case with no external field, we prove that the energy per volume converges
and obtain in the limit a translation-invariant projector describing the free
Hartree-Fock vacuum. We also define the energy per unit volume of
translation-invariant states and prove that the free vacuum is the unique
minimizer of this energy.
  In the presence of an external field, we prove that the difference between
the minimum energy and the energy of the free vacuum converges as $L$ goes to
infinity. We obtain in the limit the so-called Bogoliubov-Dirac-Fock
functional. The Hartree-Fock (polarized) vacuum is a Hilbert-Schmidt
perturbation of the free vacuum and it minimizes the Bogoliubov-Dirac-Fock
energy.
