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

Gravejat, Philippe; Lewin, Mathieu; Séré, Eric (2009), Ground State and Charge Renormalization in a Nonlinear Model of Relativistic Atoms, Communications in Mathematical Physics, 286, 1, p. 179-215. http://dx.doi.org/10.1007/s00220-008-0660-9

Type
Article accepté pour publication ou publié
Date
2009
Journal name
Communications in Mathematical Physics
Volume
286
Number
1
Publisher
Springer
Pages
179-215
Publication identifier
http://dx.doi.org/10.1007/s00220-008-0660-9
Metadata
Show full item record
Author(s)
Gravejat, Philippe
Lewin, Mathieu cc
Séré, Eric
Abstract (EN)
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.
Subjects / Keywords
Analysis of PDEs; Physics; Mathematical Physics; Mathematics

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