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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 $\alpha_{\rm phys}\simeq \alpha(1+2\alpha/(3\pi)\log\Lambda)^{-1}$, 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.
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https://hal.archives-ouvertes.fr/hal-00200945
Contributor : Mathieu Lewin <>
Submitted on : Saturday, December 22, 2007 - 9:07:16 AM
Last modification on : Monday, January 25, 2021 - 2:36:02 PM

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Philippe Gravejat, Mathieu Lewin, Eric Séré. Ground State and Charge Renormalization in a Nonlinear Model of Relativistic Atoms. Communications in Mathematical Physics, Springer Verlag, 2009, 286 (1), pp.179-215. ⟨10.1007/s00220-008-0660-9⟩. ⟨hal-00200945⟩

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