Existence of Ground State of an Electron in the BDF Approximation

The Bogoliubov-Dirac-Fock (BDF) model allows to describe relativistic electrons interacting with the Dirac sea. It can be seen as a mean-field approximation of Quantum Electro-dynamics (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 which is a compact pertubation of the negative spectral projector of the free Dirac operator. The parameters of the model are the coupling constant $\alpha >0$ and the ultraviolet cut-off $\mathrm{\Lambda }>0$: we consider the subspace of squared integrable functions made of the functions whose Fourier transform vanishes outside the ball $B(0,\text{La})$. We prove the existence of minimizers of the BDF-energy under the charge constraint of one electron and no external field provided that $\alpha ,{\text{La}}^{-1}$ and $\alpha \text{llo}$ 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 $\alpha \text{llo}$ fixed: after rescaling the electronic solution tends to the Choquard-Pekar ground state.