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. 179215. http://dx.doi.org/10.1007/s0022000806609
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Article accepté pour publication ou publiéDate
2009Journal name
Communications in Mathematical PhysicsVolume
286Number
1Publisher
Springer
Pages
179215
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Show full item recordAbstract (EN)
We study the reduced BogoliubovDiracFock (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 meanfield approximation of Quantum Electrodynamics (QED) where photons and the socalled exchange term are neglected. A state of the system is described by its onebody density matrix, an infinite rank selfadjoint 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 electronpositron 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 cutoff. 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 meanfield approximation of nophoton QED.Subjects / Keywords
Analysis of PDEs; Physics; Mathematical Physics; MathematicsRelated items
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