Ground State and Charge Renormalization in a Nonlinear Model of Relativistic Atoms

TL;DR

This paper investigates the reduced Bogoliubov-Dirac-Fock model for relativistic electrons interacting with the Dirac sea, establishing existence of minimizers, charge renormalization, and bounds on electron binding in a mean-field QED approximation.

Contribution

It proves the existence of minimizers for the reduced BDF energy with charge constraints, characterizes charge renormalization, and estimates the maximum electrons bound by a nucleus.

Findings

Existence of minimizers for various charges and coupling constants.

Derivation of the effective charge renormalization formula.

Bound on the maximum number of electrons as 2Z in the nonrelativistic limit.

Abstract

We study the reduced Bogoliubov-Dirac-Fock (BDF) energy which allows to describe relativistic electrons interacting with the Dirac sea, in an external electrostatic potential. The model can be seen as a mean-field approximation of Quantum Electrodynamics (QED) where photons and the so-called exchange term are neglected. A state of the system is described by its one-body density matrix, an infinite rank self-adjoint operator which is a compact perturbation of the negative spectral projector of the free Dirac operator (the Dirac sea). We study the minimization of the reduced BDF energy under a charge constraint. We prove the existence of minimizers for a large range of values of the charge, and any positive value of the coupling constant $\alpha$. Our result covers neutral and positively charged molecules, provided that the positive charge is not large enough to create electron-positron pairs. We also prove that the density of any minimizer is an $L^1$ function and compute the effective charge of the system, recovering the usual renormalization of charge: the physical coupling constant is related to $\alpha$ by the formula $\alpha_{\rm phys}\simeq \alpha(1+2\alpha/(3\pi)\log\Lambda)^{-1}$, where $\Lambda$ is the ultraviolet cut-off. We eventually prove an estimate on the highest number of electrons which can be bound by a nucleus of charge $Z$. In the nonrelativistic limit, we obtain that this number is $\leq 2Z$, recovering a result of Lieb. This work is based on a series of papers by Hainzl, Lewin, Sere and Solovej on the mean-field approximation of no-photon QED.