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 ρ ph including both the external density of a nucleus and the self-consistent polarization of the Dirac sea, but no ‘real’ electron. We show that ρ ph admits an asymptotic expansion to any order in powers of the physical coupling constant α ph, provided that the ultraviolet cut-off behaves as \({\Lambda\sim e^{3\pi(1-Z_3)/2\alpha_{\rm ph}} \gg 1}\). The renormalization parameter 0 < Z 3 < 1 is defined by Z 3 = α ph/α, where α is the bare coupling constant. The coefficients of the expansion of ρ ph are independent of Z 3, as expected. The first order term gives rise to the well-known Uehling potential, whereas the higher order terms satisfy an explicit recursion relation.
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Communicated by I.M. Sigal
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Gravejat, P., Lewin, M. & Séré, É. Renormalization and Asymptotic Expansion of Dirac’s Polarized Vacuum. Commun. Math. Phys. 306, 1–33 (2011). https://doi.org/10.1007/s00220-011-1271-4
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DOI: https://doi.org/10.1007/s00220-011-1271-4