Existence of ground state of an electron in the BDF approximation

The Bogoliubov-Dirac-Fock (BDF) model allows us to describe relativistic electrons interacting with the Dirac sea. It can be seen as a mean-field approximation of Quantum Electrodynamics (QED) where photons are neglected. This paper treats the case of an electron together with the Dirac sea in the absence of any external field. Such a system is described by its one-body density matrix, an infinite rank, self-adjoint operator. The parameters of the model are the coupling constant alpha > 0 and the ultraviolet cutoff Lambda > 0: we consider the subspace of squared integrable functions made of the functions whose Fourier transform vanishes outside the ball B(0, Lambda). We prove the existence of minimizers of the BDF energy under the charge constraint of one electron and no external field provided that alpha, Lambda(-1) and alpha log(Lambda) are sufficiently small. The interpretation is the following: in this regime the electron creates a polarization in the Dirac vacuum which allows it to bind. We then study the non-relativistic limit of such a system in which the speed of light tends to infinity (or equivalently alpha tends to zero) with alpha log(Lambda) fixed: after rescaling and translation the electronic solution tends to a Choquard-Pekar ground state.