Advertisement

Letters in Mathematical Physics

, Volume 72, Issue 2, pp 99–113 | Cite as

Existence of Global-In-Time Solutions to a Generalized Dirac-Fock Type Evolution Equation

  • Christian Hainzl
  • Mathieu Lewinand
  • Christof Sparber
Article

Abstract

We consider a generalized DiracFock type evolution equation deduced from nophoton Quantum Electrodynamics, which describes the selfconsistent timeevolution of relativistic electrons, the observable ones as well as those filling up the Dirac sea. This equation has been originally introduced by Dirac in 1934 in a simplified form. Since we work in a Hartree-Fock type approximation, the elements describing the physical state of the electrons are infinite rank projectors. Using the Bogoliubov-Dirac-Fock formalism, introduced by ChaixIracane (J. Phys. B., 22, 37913814, 1989), and recently established by Hainzl-Lewin-Séré, we prove the existence of globalintime solutions of the considered evolution equation.

Keywords

QED vacuum polarization Dirac equation HartreeFock model semilinear evolution equations. 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reference

  1. 1.
    Bach, V., Barbaroux, J. M., Helffer, B., Siedentop, H. 1999On the Stability of the relativistic electronpositron fieldComm. Math. Phys.201445460CrossRefGoogle Scholar
  2. 2.
    Bjorken, B. J., Drell, S. D. 1965Relativistic quantum fieldsMcGrawHillNew YorkGoogle Scholar
  3. 3.
    Chaix, P., Iracane, D. 1989From quantum electrodynamics to mean field theory: IThe Bo-go-liubov-Di-rac-Fock formalism. J. Phys. B.2237913814Google Scholar
  4. 4.
    Chaix, P., Iracane, D., Lions P., L. 1989From quantum electrodynamics to mean field theory: IIVariational stability of the vacuum of quantum electrodynamics in the meanfield approximation. J. Phys. B.2238153828Google Scholar
  5. 5.
    Cancés, È., Le Bris, C. 1999On the timedependent HartreeFock equations coupled with a classical nuclear dynamicsMath Models Methods Appl. Sci.9963990CrossRefGoogle Scholar
  6. 6.
    Chadam, J.M., Glassey, R.T. 1975Global existence of solutions to the Cauchy problem for timedependent Hartree equationsJ. Math. Phys.1611221230CrossRefGoogle Scholar
  7. 7.
    Dirac P.A.M. 1934. Théorie du positron, Solvay report, 203212. Paris: GauthierVillars. XXV , 353 (reprinted in Selected papers on Quantum Electrodynamics, edited by J. Schwinger, Dover, 1958)Google Scholar
  8. 8.
    Dirac, P.A.M. 1934Discussion of the infinite distribution of electrons in the theory of the positronProc. Camb. Philos. Soc.30150163Google Scholar
  9. 9.
    Dyson, F.J. 1949The S Matrix in Quantum ElectrodynamicsPhys. Rev.7517361755CrossRefGoogle Scholar
  10. 10.
    Escobedo, M., Vega, L. 1997A Semilinear Dirac Equation in H s (R3) for s > 1. SIAM JMath. Anal.28338362CrossRefGoogle Scholar
  11. 11.
    Esteban, M., Séré, E. 1999Solutions of the DiracFock equations for atoms and moleculesComm. Math. Phys.203499530CrossRefGoogle Scholar
  12. 12.
    Esteban, M., Séré, E. 2001Nonrelativistic limit of the DiracFock equations, AnnHenri Poincaré2941961CrossRefGoogle Scholar
  13. 13.
    Esteban, M., Georgiev, V., Séré, E. 1996Stationary solutions of the MaxwellDirac and the KleinGordonDirac equationsCalc.Var. Part. Diff. Equ.4256281Google Scholar
  14. 14.
    Flato, M., Simon, J., Taflin, C.H. 1987On global solutions of the Maxwell-Dirac equationsComm. Math. Phys.1122146CrossRefGoogle Scholar
  15. 15.
    Flato, M., Simon, J., Taflin, C.H. 1997Asymptotic completeness, global existence and the infrared problem for the Maxwell-Dirac equationsMem. Amer. Math. Soc.127x+311Google Scholar
  16. 16.
    Georgiev, V. 1991Small amplitude solutions of the Maxwell-Dirac equationsIndiana Univ. Math. J.40845883CrossRefGoogle Scholar
  17. 17.
    Glauber, R., Rarita, W., Schwed, P. 1960Vacuum polarization effects on energy levels in mumesonic atomsPhys. Rev.120609613CrossRefGoogle Scholar
  18. 18.
    Greiner W., Müller B., Rafelski J.1985. Quantum Electrodynamics of Strong Fields. Texts and Mongraphs in Physics. SpringerVerlagGoogle Scholar
  19. 19.
    Gross, L. 1966The Cauchy problem for the coupled Maxwell and Dirac equationsComm. Pure Appl. Math.19115Google Scholar
  20. 20.
    Hainzl, C. 2004On the Vacuum Polarization Density caused by an External FieldAnn. Henri Poincaré511371157CrossRefMathSciNetGoogle Scholar
  21. 21.
    Hainzl, C., Lewin, M., Séré, E.. Existence of a stable polarized vacuum in the BogoliubovDiracFock approximation, Comm. Math. Phys. to appearGoogle Scholar
  22. 22.
    Hainzl, C., Lewin, M., Séré, E.. Selfconsistent solution for the polarized vacuum in a nophoton QED model, J. Phys. A. Math., Gen. to appearGoogle Scholar
  23. 23.
    Hainzl C., Lewin M., Séré E.: in preparationGoogle Scholar
  24. 24.
    Hainzl, C., Siedentop, H. 2003NonPerturbative Mass and Charge Renormalization in Relativistic nophoton Quantum ElectrodynamicsComm. Math. Phys.243241260CrossRefGoogle Scholar
  25. 25.
    Heisenberg, W. 1934Bemerkungen zur Diracschen Theorie des PositronsZeits. f. Physik90209223CrossRefGoogle Scholar
  26. 26.
    Klaus, M., Scharf, G. 1977The regular external field problem inquantum electrodynamicsHelv. Phys. Acta50779802Google Scholar
  27. 27.
    Landau, L.D.1965. On the Quantum Theory of Fields, Pergamon Press, Oxford 1955. Reprinted in emph Collected papers of L.D. Landau, D. Ter Haar, (eds.) Pergamon Press.Google Scholar
  28. 28.
    Landau, L.D., Pomeranchuk, I. 1965On point interactions in quantum electrodynamicsDokl. Akad. Nauk. SSSR102489492Reprinted in emph Collected papers of L.D. Landau, D. Ter Haar, (eds.) Pergamon PressGoogle Scholar
  29. 29.
    Machihara, S., Nakanishi, K., Ozawa, T. 2003Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation. RevMat. Iberoam.19179194MathSciNetGoogle Scholar
  30. 30.
    Ruijsenaars, S.N.M. 1977On Bogoliubov transformations for systems of relativistic charged particlesJ. Math. Phys.18517526CrossRefGoogle Scholar
  31. 31.
    Schweber S. S. 1994. QED and the men who made it: Dyson, Feynman, Schwinger and Tomonaga, Princeton University PressGoogle Scholar
  32. 32.
    Shale, D., Stinespring, W. 1965Spinor representation of infinite orthogonal groupsJ. Math, Mech.14315324Google Scholar
  33. 33.
    Simon B. 1979. Trace Ideals and their Applications. Vol 35 of London Mathematical Society Lecture Notes Series. Cambridge University PressGoogle Scholar
  34. 34.
    Thaller B. 1992. The Dirac Equation, Springer VerlagGoogle Scholar
  35. 35.
    Uehling E., A. 1935Polarization effects in the positron theoryPhys. Rev. II. Ser.485563Google Scholar
  36. 36.
    Weisskopf, V. 1936Über die Elektrodynamik des Vakuums auf Grund der Quantentheorie des ElektronsMath.Fys. Medd, Danske Vid. Selsk16139reprinted in Selected papers on Quantum Electrodynamics, J. Schwinger, (ed.) Dover, 1958Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Christian Hainzl
    • 1
  • Mathieu Lewinand
    • 1
  • Christof Sparber
    • 2
    • 3
  1. 1.Department of MathematicsUniversity of CopenhagenCopenhagenDenmark
  2. 2.Department of Numerical MathematicsUniversity of Münster
  3. 3.Münster Wolfgang Pauli Institute Vienna c/o Faculty of MathematicsVienna UniversityViennaAustria

Personalised recommendations