Existence of Ground State of an Electron in the BDF Approximation.
Résumé
The Bogoliubov-Dirac-Fock (BDF) model allows to describe relativistic elec- trons 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 absence of any exter- nal 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. We prove the existence of minimizers of the BDF-energy under the charge constraint of one electron assuming that the coupling constant α and the quantity L = α log(Λ) are small where Λ > 0 is the ultraviolet cut-off and chosen very large. We then study the non-relativistic limit of such a system in which the speed of light c tends to infinity (or equivalently α tends to zero) with L fixed: after rescaling the electronic solution tends to the Choquard-Pekar ground state.
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