1211.3830 (Jérémy Sok)

Existence of Ground State of an Electron in the BDF Approximation    [PDF]

Jérémy Sok

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 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. We prove the existence of minimizers of the BDF-energy under the charge constraint of one electron assuming that the coupling constant {\alpha} and the quantity L = \alpha log(\Lambda) are small where \Lambda > 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 \alpha tends to zero) with L fixed: after rescaling the electronic solution tends to the Choquard-Pekar ground state.

View original: http://arxiv.org/abs/1211.3830