Renormalization and Asymptotic Expansion of Dirac’s Polarized Vacuum

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• 作者：1. Centre de Mathématiques Laurent Schwartz (UMR 7640) ; école Polytechnique ; 91128 Palaiseau Cedex ; France2. CNRS & Laboratoire de Mathématiques (UMR 8088) ; Université de Cergy-Pontoise ; 95000 Cergy-Pontoise ; France3. Ceremade (UMR 7534) ; Université Paris-Dauphine ; Place du Maréchal de Lattre de Tassigny ; 75775 Paris Cedex 16 ; France
• 刊名：Communications in Mathematical Physics
• 出版年：2011
• 出版时间：August 2011
• 年：2011
• 卷：306
• 期：1
• 页码：1-33
• 全文大小：462.4 KB
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• 作者单位：http://www.springerlink.com/content/6771k21157558165/
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mathematical and Computational Physics
  Quantum Physics
  Quantum Computing, Information and Physics
  Complexity
  Statistical Physics
  Relativity and Cosmology
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0916

文摘

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 L ~ e3p(1-Z₃)/2a_ph >> 1{\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.