Ground State and Charge Renormalization in a Nonlinear Model of Relativistic Atoms |
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Authors: | Philippe Gravejat Mathieu Lewin and éric Séré |
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Institution: | (1) CEREMADE, UMR 7534, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France;(2) CNRS & Laboratoire de Mathématiques UMR 8088, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France |
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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 α. 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 α by the formula αphys ≃ α(1 + 2α/(3π) log Λ)−1, where Λ 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 ≤ 2Z, recovering a result of Lieb. This work is based on a series of papers by Hainzl, Lewin, Séré and Solovej on the mean-field
approximation of no-photon QED. |
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