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Communications in Mathematical Physics

, Volume 306, Issue 1, pp 1–33 | Cite as

Renormalization and Asymptotic Expansion of Dirac’s Polarized Vacuum

  • Philippe Gravejat
  • Mathieu Lewin
  • Éric Séré
Article

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.

Keywords

Dirac Operator Lewin Quantum Electrodynamic Universal Constant Renormalization Parameter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© The Author(s) 2011

Authors and Affiliations

  • Philippe Gravejat
    • 1
  • Mathieu Lewin
    • 2
  • Éric Séré
    • 3
  1. 1.Centre de Mathématiques Laurent Schwartz (UMR 7640)École PolytechniquePalaiseau CedexFrance
  2. 2.CNRS & Laboratoire de Mathématiques (UMR 8088)Université de Cergy-PontoiseCergy-PontoiseFrance
  3. 3.Ceremade (UMR 7534), Université Paris-DauphinePlace du Maréchal de Lattre de TassignyParis Cedex 16France

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