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

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 ${\alpha }_{\mathrm{phys}}\simeq \alpha (1+2\alpha /(3\pi )\log \Lambda {)}^{-1}$, 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 $\le 2Z$, 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.

arXiv comment37 pages, 1 figure

References in corpus (6)

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• Derivation of the magnetic Euler-Heisenberg energy
• A new definition of the Dirac-Fock ground state
• Existence of Ground State of an Electron in the BDF Approximation
• Effective quantum electrodynamics: One-dimensional model of the relativistic hydrogen-like atom
• Relativistic electrons coupled with Newtonian nuclear dynamics
• Charge renormalisation in a mean-field approximation of QED
• Existence of minimizers for the Dirac-Fock model of crystals
• The Pauli-Villars regularised free energy of Dirac's vacuum in purely magnetic fields
• The positronium in a mean-field approximation of quantum electrodynamics
• The positronium and the dipositronium in a Hartree-Fock approximation of quantum electrodynamics
• Absence of binding in a mean-field approximation of quantum electrodynamics