Existence of Ground State of an Electron in the BDF Approximation

TL;DR

This paper proves the existence of a ground state for an electron in the BDF model, showing that under certain conditions the electron can bind by polarizing the Dirac vacuum, and connects the relativistic model to the non-relativistic limit.

Contribution

It establishes the existence of minimizers in the BDF model for a single electron without external fields under small parameters, and analyzes its non-relativistic limit to the Choquard-Pekar ground state.

Findings

Existence of BDF minimizers for one electron under small coupling and cut-off.

Electron induces polarization in the Dirac vacuum enabling binding.

Non-relativistic limit converges to the Choquard-Pekar ground state.

Abstract

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 $\Lambda>0$: we consider the subspace of squared integrable functions made of the functions whose Fourier transform vanishes outside the ball $B(0,\La)$. We prove the existence of minimizers of the BDF-energy under the charge constraint of one electron and no external field provided that $\alpha,\La^{-1}$ and $\alpha\llo$ 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 $\alpha\llo$ fixed: after rescaling the electronic solution tends to the Choquard-Pekar ground state.