Communications in Mathematical Physics

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

Renormalization and Asymptotic Expansion of Dirac’s Polarized Vacuum



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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Avron J., Seiler R., Simon B.: The index of a pair of projections. J. Funct. Anal. 120, 220–237 (1994)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bjorken J.D., Drell S.D.: Relativistic quantum fields. McGraw-Hill Book Co., New York (1965)MATHGoogle Scholar
  3. 3.
    Cancès É., Lewin M.: The dielectric permittivity of crystals in the reduced Hartree-Fock approximation. Arch. Rati. Mech. Anal. 197, 139–177 (2010)MATHCrossRefGoogle Scholar
  4. 4.
    Chaix P., Iracane D.: From quantum electrodynamics to mean field theory: I. The Bogoliubov-Dirac-Fock formalism. J. Phys. B 22, 3791–3814 (1989)ADSCrossRefGoogle Scholar
  5. 5.
    Dirac P.A.: The quantum theory of the electron. II. Proc. Royal Soc. London (A) 118, 351–361 (1928)ADSMATHCrossRefGoogle Scholar
  6. 6.
    Dirac P.A.: A theory of electrons and protons. Proc. Royal Soc. London (A) 126, 360–365 (1930)ADSMATHCrossRefGoogle Scholar
  7. 7.
    Dirac, P.A.: Theory of electrons and positrons. Nobel Lecture delivered at Stockholm, 1933Google Scholar
  8. 8.
    Dirac P.A.: Théorie du positron. Solvay report XXV, 203–212 (1934)Google Scholar
  9. 9.
    Dyson F.J.: The S matrix in quantum electrodynamics. Phys. Rev. 75(2), 1736–1755 (1949)MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Dyson F.J.: Divergence of Perturbation Theory in Quantum Electrodynamics. Phys. Rev. 85, 631–632 (1952)MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    Engel, E.: Relativistic Density Functional Theory: Foundations and Basic Formalism. Vol. ‘Relativistic Electronic Structure Theory, Part 1. Fundamentals’, Schwerdtfeger ed., Amsterdam: Elsevier, 2002, ch. 10, pp. 524–624Google Scholar
  12. 12.
    Engel E., Dreizler R.M.: Field-theoretical approach to a relativistic Thomas-Fermi-Dirac-Weizsäcker model. Phys. Rev. A 35, 3607–3618 (1987)ADSCrossRefGoogle Scholar
  13. 13.
    Gravejat P., Lewin M., Séré É.: Ground state and charge renormalization in a nonlinear model of relativistic atoms. Commun. Math. Phys. 286, 179–215 (2009)ADSMATHCrossRefGoogle Scholar
  14. 14.
    Greiner, W., Müller, B., Rafelski, J.: Quantum Electrodynamics of Strong Fields. First ed., Texts and Monographs in Physics. Berlin-Heidelberg-NewYork: Springer-Verlag, 1985Google Scholar
  15. 15.
    Hainzl C., Lewin M., Séré É.: Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation. Commun. Math. Phys. 257, 515–562 (2005)ADSMATHCrossRefGoogle Scholar
  16. 16.
    Hainzl C., Lewin M., Séré É.: Self-consistent solution for the polarized vacuum in a no-photon QED model. J. Phys. A 38, 4483–4499 (2005)MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    Hainzl C., Lewin M., Séré É.: Existence of atoms and molecules in the mean-field approximation of no-photon quantum electrodynamics. Arch. Rati. Mech. Anal. 192, 453–499 (2009)MATHCrossRefGoogle Scholar
  18. 18.
    Hainzl C., Lewin M., Séré É., Solovej J.P.: A minimization method for relativistic electrons in a mean-field approximation of quantum electrodynamics. Phys. Rev. A 76, 052104 (2007)ADSCrossRefGoogle Scholar
  19. 19.
    Hainzl C., Lewin M., Solovej J.P.: The mean-field approximation in quantum electrodynamics: the no-photon case. Comm. Pure Appl. Math. 60, 546–596 (2007)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Hainzl C., Siedentop H.: Non-perturbative mass and charge renormalization in relativistic no-photon quantum electrodynamics. Commun. Math. Phys. 243, 241–260 (2003)MathSciNetADSMATHCrossRefGoogle Scholar
  21. 21.
    Itzykson C., Zuber J.B.: Quantum field theory. McGraw-Hill International Book Co., New York (1980)Google Scholar
  22. 22.
    Landau L., Pomerančuk I.: On point interaction in quantum electrodynamics. Dokl. Akad. Nauk SSSR (N.S.) 102, 489–492 (1955)MathSciNetMATHGoogle Scholar
  23. 23.
    Landau, L.D.: On the quantum theory of fields. In: Niels Bohr and the development of physics, New York: McGraw-Hill Book Co., 1955, pp. 52–69Google Scholar
  24. 24.
    Lieb E.H., Siedentop H.: Renormalization of the regularized relativistic electron-positron field. Commun. Math. Phys. 213, 673–683 (2000)MathSciNetADSMATHCrossRefGoogle Scholar
  25. 25.
    Pauli W., Rose M.: Remarks on the polarization effects in the positron theory. Phys. Rev. II 49, 462–465 (1936)ADSMATHGoogle Scholar
  26. 26.
    Reinhard P.-G., Greiner W., Arenhövel H.: Electrons in strong external fields. Nucl. Phys. A 166, 173–197 (1971)ADSCrossRefGoogle Scholar
  27. 27.
    Seiler E., Simon B.: Bounds in the Yukawa 2 quantum field theory: upper bound on the pressure. Hamiltonian bound and linear lower bound. Commun. Math. Phys. 45, 99–114 (1975)MathSciNetADSCrossRefGoogle Scholar
  28. 28.
    Serber R.: Linear modifications in the Maxwell field equations. Phys. Rev. 48(2), 49–54 (1935)ADSMATHCrossRefGoogle Scholar
  29. 29.
    Shale D., Stinespring W.F.: Spinor representations of infinite orthogonal groups. J. Math. Mech. 14, 315–322 (1965)MathSciNetMATHGoogle Scholar
  30. 30.
    Simon, B.: Trace ideals and their applications. Vol. 35 of London Mathematical Society Lecture Note Series, Cambridge: Cambridge University Press, 1979Google Scholar
  31. 31.
    Solovej J.P.: Proof of the ionization conjecture in a reduced Hartree-Fock model. Invent. Math. 104, 291–311 (1991)MathSciNetADSMATHCrossRefGoogle Scholar
  32. 32.
    Thaller, B. The Dirac equation. Texts and Monographs in Physics. Berlin: Springer-Verlag, 1992Google Scholar
  33. 33.
    Uehling E.: Polarization effects in the positron theory. Phys. Rev. 48(2), 55–63 (1935)ADSMATHCrossRefGoogle Scholar

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

Personalised recommendations