Advertisement

Communications in Mathematical Physics

, Volume 257, Issue 3, pp 515–562 | Cite as

Existence of a Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation

  • Christian HainzlEmail author
  • Mathieu Lewin
  • Éric Séré
Article

Abstract

According to Dirac’s ideas, the vacuum consists of infinitely many virtual electrons which completely fill up the negative part of the spectrum of the free Dirac operator D0. In the presence of an external field, these virtual particles react and the vacuum becomes polarized. In this paper, following Chaix and Iracane (J. Phys. B 22, 3791–3814 (1989)), we consider the Bogoliubov-Dirac-Fock model, which is derived from no-photon QED. The corresponding BDF-energy takes the polarization of the vacuum into account and is bounded from below. A BDF-stable vacuum is defined to be a minimizer of this energy. If it exists, such a minimizer is the solution of a self-consistent equation. We show the existence of a unique minimizer of the BDF-energy in the presence of an external electrostatic field, by means of a fixed-point approach. This minimizer is interpreted as the polarized vacuum.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics External Field 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aste, A., Baur, G., Hencken, K., Trautmann, D., Scharf, G.: Electron-positron pair production in the external electromagnetic field of colliding relativistic heavy ions. Eur. Phys. J. C 23(3), 545–550 (2002)Google Scholar
  2. 2.
    Avron, J., Seiler, R., Simon, B.: The index of a pair of projections. J. Funct. Anal. 120, 220–237 (1994)CrossRefGoogle Scholar
  3. 3.
    Bach, V.: Error bound for the Hartree-Fock energy of atoms and molecules. Commun. Math. Phys. 147, 527–548 (1992)Google Scholar
  4. 4.
    Bach, V., Barbaroux, J.-M., Helffer, B., Siedentop, H.: On the stability of the relativistic electron-positron field. Commun. Math. Phys. 201, 445–460 (1999)CrossRefGoogle Scholar
  5. 5.
    Bach, V., Lieb, E.H., Solovej, J.P.: Generalized Hartree-Fock theory and the Hubbard model. J. Statist. Phys. 76(1–2), 3–89 (1994)Google Scholar
  6. 6.
    Barbaroux, J.M., Esteban, M.J., Séré, E.: Some connections between Dirac-Fock and Electron-Positron Hartree-Fock. Ann. Henri Poincaré 6(1), 85–102 (2005)CrossRefGoogle Scholar
  7. 7.
    Barbaroux, J.-M., Farkas, W., Helffer, B., Siedentop, H.: On the Hartree-Fock equations of the electron/positron field. Commun. Math. Phys. 255, 131–159 (2005)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Cancès, E.: SCF algorithms for HF electronic calculations. Defranceschi, M. et al. (ed.), Mathematical models and methods for ab initio quantum chemistry. Lect. Notes Chem. 74, Berlin: Springer, 2000, pp. 17–43Google Scholar
  9. 9.
    Chaix, P., Iracane, D.: From quantum electrodynamics to mean field theory: I. The Bogoliubov-Dirac-Fock formalism. J. Phys. B 22(23), 3791–3814 (1989)Google Scholar
  10. 10.
    Chaix, P., Iracane, D., Lions, P.L.: From quantum electrodynamics to mean field theory: II. Variational stability of the vacuum of quantum electrodynamics in the mean-field approximation. J. Phys. B 22(23), 3815–3828 (1989)Google Scholar
  11. 11.
    Chaix, P.: Une Méthode de Champ Moyen Relativiste et Application à l’Etude du Vide de l’Electrodynamique Quantique. PhD Thesis, University Paris VI, 1990Google Scholar
  12. 12.
    Desclaux, J.P.: Relativistic Dirac-Fock expectation values for atoms with Z=1 to Z=120. Atomic Data and Nuclear Data Tables 12, 311–406 (1973)CrossRefGoogle Scholar
  13. 13.
    Dirac, P.A.M.: Théorie du positron. Solvay report, Paris: Gauthier-Villars, XXV, 353 S., 1934, pp. 203–212Google Scholar
  14. 14.
    Dirac, P.A.M.: Discussion of the infinite distribution of electrons in the theory of the positron. Proc. Camb. Philos. Soc. 30, 150–163 (1934)Google Scholar
  15. 15.
    Esteban, M.J., Séré, E.: Solutions of the Dirac-Fock Equations for atoms and molecules. Commun. Math. Phys. 203, 499–530 (1999)CrossRefGoogle Scholar
  16. 16.
    Esteban, M.J., Séré, E.: Nonrelativistic limit of the Dirac-Fock equations. Ann. Henri Poincaré 2(5), 941–961 (2001)CrossRefGoogle Scholar
  17. 17.
    Esteban, M.J., Séré, E.: A max-min principle for the ground state of the Dirac-Fock functional. Contemp. Math. 307, 135–139 (2002)Google Scholar
  18. 18.
    Foldy, L.L., Eriksen, E.: Some physical consequences of vacuum polarization. Phys. Rev. 95(4), 1048–1051 (1954)CrossRefGoogle Scholar
  19. 19.
    French, J.D., Weisskopf, V.F.: The electromagnetic shift of energy levels. Phys. Rev., II. Ser. 75, 1240–1248 (1949)Google Scholar
  20. 20.
    Fierz, H., Scharf, G.: Particle interpretation for external field problems in QED. Helv. Phys. Acta 52, 437–453 (1979)Google Scholar
  21. 21.
    Furry, W.H.: On bound states and scattering in positron theory. Phys. Rev. 81(1), 115–124 (1951)CrossRefGoogle Scholar
  22. 22.
    Furry, W.H.: A symmetry theorem in the positron theory. Phys. Rev. 51, 125–129 (1937)CrossRefGoogle Scholar
  23. 23.
    Furry, W.H., Oppenheimer, J.R.: On the theory of the electron and positive. Phys. Rev., II. Ser. 45, 245–262 (1934)Google Scholar
  24. 24.
    Glauber, R., Rarita, W., Schwed, P.: Vacuum polarization effects on energy levels in μ-mesonic atoms. Phys. Rev. 120(2), 609–613 (1960)CrossRefGoogle Scholar
  25. 25.
    Gorceix, O. Indelicato, P., Desclaux, J.P.: Multiconfiguration Dirac-Fock studies of two-electron ions: I. Electron-electron interaction. J. Phys. B: At. Mol. Phys. 20, 639–649 (1987)Google Scholar
  26. 26.
    Grant, I.P.: Relativistic calculation of atomic structures. Adv. Phys. 19, 747–811 (1970)Google Scholar
  27. 27.
    Hainzl, C.: On the vacuum polarization density caused by an external field. Ann. Henri Poincaré 5, 1137–1157 (2004)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Hainzl, C., Lewin, M., Séré, E.: Self-consistent solution for the polarized vacuum in a no-photon QED model. To appear in J. Phys. A: Math. and Gen.Google Scholar
  29. 29.
    Hainzl, C., Siedentop, H.: Non-perturbative mass and charge renormalization in relativistic no-photon quantum electrodynamics. Commun. Math. Phys. 243, 241–260 (2003)CrossRefGoogle Scholar
  30. 30.
    Helffer, B., Siedentop, H.: Form perturbation of the second quantized Dirac field. Mathematical Physics Electronic Journal 4, paper 4, 1998Google Scholar
  31. 31.
    Heisenberg, W.: Bemerkungen zur Diracschen Theorie des Positrons. Z. Phys. 90, 209–231 (1934)CrossRefGoogle Scholar
  32. 32.
    Hundertmark, D., Röhrl, N., Siedentop, H.: The sharp bound on the stability of the relativistic electron-positron field in Hartree-Fock approximation. Commun. Math. Phys. 211(3), 629–642 (2000)CrossRefGoogle Scholar
  33. 33.
    Itzykson, C., Zuber, J.-B.: Quantum Field Theory. New York: McGraw-Hill, 1980Google Scholar
  34. 34.
    Kim, Y.K.: Relativistic self-consistent Field theory for closed-shell atoms. Phys. Rev. 154, 17–39 (1967)CrossRefGoogle Scholar
  35. 35.
    Klaus, M.: Non-regularity of the Coulomb potential in quantum electrodynamics. Helv. Phys. Acta 53, 36–39 (1980)Google Scholar
  36. 36.
    Klaus, M., Scharf, G.: The regular external field problem in quantum electrodynamics. Helv. Phys. Acta 50, 779–802 (1977)Google Scholar
  37. 37.
    Klaus, M., Scharf, G.: Vacuum polarization in fock space. Helv. Phys. Acta 50, 803–814 (1977)Google Scholar
  38. 38.
    Lieb, E.H.: Variational principle for many-fermion systems. Phys. Rev. Lett. 46, 457–459 (1981)CrossRefGoogle Scholar
  39. 39.
    Lieb, E.H., Siedentop, H.: Renormalization of the regularized relativistic electron-positron field. Commun. Math. Phys. 213(3), 673–683 (2000)CrossRefGoogle Scholar
  40. 40.
    Lieb, E.H., Simon, B.: The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys. 53, 185–194 (1977)CrossRefGoogle Scholar
  41. 41.
    Lindgren, I., Rosen, A.: Relativistic self-consistent field calculations. Case Stud. At. Phys. 4, 93–149 (1974)Google Scholar
  42. 42.
    Mittleman, M.H.: Theory of Relativistic effects on atoms: Configuration-space Hamiltonian. Phys. Rev. A 24(3), 1167–1175 (1981)CrossRefGoogle Scholar
  43. 43.
    Mohr, P.J., Plunien, G., Soff, G.: QED corrections in heavy atoms. Phys. Rep. 293, 227–369 (1998)CrossRefGoogle Scholar
  44. 44.
    Paturel, E.: Solutions of the Dirac-Fock equations without projector. Ann. Henri Poincaré 1(6), 1123–1157 (2000)Google Scholar
  45. 45.
    Pauli, W., Rose, M.E.: Remarks on the polarization effects in the positron theory. Phys. Rev II 49, 462–465 (1936)CrossRefGoogle Scholar
  46. 46.
    Reinhardt, J., Müller, B., Greiner, W.: Theory of positron production in heavy-ion collision. Phys. Rev. A 24(1), 103–128 (1981)CrossRefGoogle Scholar
  47. 47.
    Ruijsenaars, S.N.M.: On Bogoliubov transformations for systems of relativistic charged particles. J. Math. Phys. 18(3), 517–526 (1977)CrossRefGoogle Scholar
  48. 48.
    Simon, B.: Trace Ideals and their Applications. Vol. 35 of London Mathematical Society Lecture Notes Series. Cambridge: Cambridge University Press, 1979Google Scholar
  49. 49.
    Scharf, G., Seipp, H.P.: Charged vacuum, spontaneous positron production and all that. Phys. Lett. 108B(3), 196–198 (1982)Google Scholar
  50. 50.
    Seipp, H.P.: On the S-operator for the external field problem of QED. Helv. Phys. Acta 55, 1–28 (1982)Google Scholar
  51. 51.
    Serber, R.: Linear modifications in the Maxwell field equations. Phys. Rev., II. Ser. 48, 49–54 (1935)Google Scholar
  52. 52.
    Shale, D., Stinespring, W.F.: Spinor representations of infinite orthogonal groups. J. Math. Mech. 14, 315–322 (1965)Google Scholar
  53. 53.
    Swirles, B.: The relativistic self-consistent field. Proc. Roy. Soc. A 152, 625–649 (1935)Google Scholar
  54. 54.
    Thaller, B.: The Dirac Equation. Berlin-Heidelberg-New York: Springer Verlag, 1992Google Scholar
  55. 55.
    Uehling, E.A.: Polarization effects in the positron theory. Phys. Rev., II. Ser. 48, 55–63 (1935)Google Scholar
  56. 56.
    Weisskopf, V.: Über die Elektrodynamik des Vakuums auf Grund der Quantentheorie des Elektrons. Math.-Fys. Medd., Danske Vid. Selsk. 16(6), 1–39 (1936)Google Scholar
  57. 57.
    Wolkowisky, J.H.: Existence of solutions of the Hartree equations for N electrons. An application of the Schauder-Tychonoff theorem. Indiana Univ. Math. J. 22, 551–568 (1972–73)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Christian Hainzl
    • 1
    • 2
    Email author
  • Mathieu Lewin
    • 1
  • Éric Séré
    • 1
  1. 1.CEREMADE, UMR CNRS 7534Université Paris IX DauphineParis, Cedex 16France
  2. 2.Laboratoire de MathématiquesOrsay CedexFrance

Personalised recommendations