Renormalization and asymptotic expansion of Dirac's polarized vacuum

Abstract

We perform rigorously the charge renormalization of the so-called reduced Bogoliubov-Dirac-Fock (rBDF) model. This nonlinear theory, based on the Dirac operator, describes atoms and molecules while taking into account vacuum polarization effects. We consider the total physical density including both the external density of a nucleus and the self-consistent polarization of the Dirac sea, but no `real' electron. We show that it admits an asymptotic expansion to any order in powers of the physical coupling constant $\alphaph$, provided that the ultraviolet cut-off behaves as $Λ\sim e^{3π(1-Z_3)/2\alphaph}\gg1$. The renormalization parameter $0

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• The Dielectric Permittivity of Crystals in the reduced Hartree-Fock approximation
• Renormalization of the Regularized Relativistic Electron-Positron Field
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• Ground state properties of graphene in Hartree-Fock theory
• Derivation of the magnetic Euler-Heisenberg energy
• Existence of Ground State of an Electron in the BDF Approximation
• Effective quantum electrodynamics: One-dimensional model of the relativistic hydrogen-like atom
• The Pauli-Villars regularised free energy of Dirac's vacuum in purely magnetic fields
• Existence of minimizers for the Dirac-Fock model of crystals