Skip to main content
SpringerLink
Log in

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

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Reference

  1. V. Bach J. M. Barbaroux B. Helffer H. Siedentop (1999) ArticleTitleOn the Stability of the relativistic electronpositron field Comm. Math. Phys. 201 445–460 Occurrence Handle10.1007/s002200050562

    Article  Google Scholar 

  2. B. J. Bjorken S. D. Drell (1965) Relativistic quantum fields McGrawHill New York

    Google Scholar 

  3. P. Chaix D. Iracane (1989) ArticleTitleFrom quantum electrodynamics to mean field theory: I The Bo-go-liubov-Di-rac-Fock formalism. J. Phys. B. 22 3791–3814

    Google Scholar 

  4. P. Chaix D. Iracane L. Lions P. (1989) ArticleTitleFrom quantum electrodynamics to mean field theory: II Variational stability of the vacuum of quantum electrodynamics in the meanfield approximation. J. Phys. B. 22 3815–3828

    Google Scholar 

  5. È. Cancés C. Le Bris (1999) ArticleTitleOn the timedependent HartreeFock equations coupled with a classical nuclear dynamics Math Models Methods Appl. Sci. 9 IssueID(7 963–990 Occurrence Handle10.1142/S0218202599000440

    Article  Google Scholar 

  6. J.M. Chadam R.T. Glassey (1975) ArticleTitleGlobal existence of solutions to the Cauchy problem for timedependent Hartree equations J. Math. Phys. 16 1122–1230 Occurrence Handle10.1063/1.522642

    Article  Google Scholar 

  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)

  8. P.A.M. Dirac (1934) ArticleTitleDiscussion of the infinite distribution of electrons in the theory of the positron Proc. Camb. Philos. Soc. 30 150–163 Occurrence Handle1:CAS:528:DyaA2cXjs1SmsA%3D%3D

    CAS  Google Scholar 

  9. F.J. Dyson (1949) ArticleTitleThe S Matrix in Quantum Electrodynamics Phys. Rev. 75 IssueID11 1736–1755 Occurrence Handle10.1103/PhysRev.75.1736

    Article  Google Scholar 

  10. M. Escobedo L. Vega (1997) ArticleTitleA Semilinear Dirac Equation in Hs(R3) for s > 1. SIAM J Math. Anal. 28 IssueID(2 338–362 Occurrence Handle10.1137/S0036141095283017

    Article  Google Scholar 

  11. M. Esteban E. Séré (1999) ArticleTitleSolutions of the DiracFock equations for atoms and molecules Comm. Math. Phys. 203 IssueID3 499–530 Occurrence Handle10.1007/s002200050032

    Article  Google Scholar 

  12. M. Esteban E. Séré (2001) ArticleTitleNonrelativistic limit of the DiracFock equations, Ann Henri Poincaré 2 IssueID(5 941–961 Occurrence Handle10.1007/s00023-001-8600-7 Occurrence Handle1:CAS:528:DC%2BD3MXptVyqt78%3D

    Article  CAS  Google Scholar 

  13. M. Esteban V. Georgiev E. Séré (1996) ArticleTitleStationary solutions of the MaxwellDirac and the KleinGordonDirac equations Calc.Var. Part. Diff. Equ. 4 IssueID3 256–281

    Google Scholar 

  14. M. Flato J. Simon C.H. Taflin (1987) ArticleTitleOn global solutions of the Maxwell-Dirac equations Comm. Math. Phys. 112 IssueID(1 21–46 Occurrence Handle10.1007/BF01217678

    Article  Google Scholar 

  15. M. Flato J. Simon C.H. Taflin (1997) ArticleTitleAsymptotic completeness, global existence and the infrared problem for the Maxwell-Dirac equations Mem. Amer. Math. Soc. 127 IssueID(606 x+311

    Google Scholar 

  16. V. Georgiev (1991) ArticleTitleSmall amplitude solutions of the Maxwell-Dirac equations Indiana Univ. Math. J. 40 IssueID(3 845–883 Occurrence Handle10.1512/iumj.1991.40.40038

    Article  Google Scholar 

  17. R. Glauber W. Rarita P. Schwed (1960) ArticleTitleVacuum polarization effects on energy levels in mumesonic atoms Phys. Rev. 120 IssueID(2 609–613 Occurrence Handle10.1103/PhysRev.120.609 Occurrence Handle1:CAS:528:DyaF3MXhs1Cntw%3D%3D

    Article  CAS  Google Scholar 

  18. Greiner W., Müller B., Rafelski J.1985. Quantum Electrodynamics of Strong Fields. Texts and Mongraphs in Physics. SpringerVerlag

  19. L. Gross (1966) ArticleTitleThe Cauchy problem for the coupled Maxwell and Dirac equations Comm. Pure Appl. Math. 19 1–15

    Google Scholar 

  20. C. Hainzl (2004) ArticleTitleOn the Vacuum Polarization Density caused by an External Field Ann. Henri Poincaré 5 1137–1157 Occurrence Handle10.1007/s00023-004-0194-4 Occurrence HandleMR2105322

    Article  MathSciNet  Google Scholar 

  21. Hainzl, C., Lewin, M., Séré, E.. Existence of a stable polarized vacuum in the BogoliubovDiracFock approximation, Comm. Math. Phys. to appear

  22. Hainzl, C., Lewin, M., Séré, E.. Selfconsistent solution for the polarized vacuum in a nophoton QED model, J. Phys. A. Math., Gen. to appear

  23. Hainzl C., Lewin M., Séré E.: in preparation

  24. C. Hainzl H. Siedentop (2003) ArticleTitleNonPerturbative Mass and Charge Renormalization in Relativistic nophoton Quantum Electrodynamics Comm. Math. Phys. 243 241–260 Occurrence Handle10.1007/s00220-003-0958-6

    Article  Google Scholar 

  25. W. Heisenberg (1934) ArticleTitleBemerkungen zur Diracschen Theorie des Positrons Zeits. f. Physik 90 209–223 Occurrence Handle10.1007/BF01333516 Occurrence Handle1:CAS:528:DyaA2MXisV2qsA%3D%3D

    Article  CAS  Google Scholar 

  26. M. Klaus G. Scharf (1977) ArticleTitleThe regular external field problem inquantum electrodynamics Helv. Phys. Acta 50 779–802 Occurrence Handle1:CAS:528:DyaE1cXkvFKhtr8%3D

    CAS  Google Scholar 

  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.

  28. L.D. Landau I. Pomeranchuk (1965) ArticleTitleOn point interactions in quantum electrodynamics Dokl. Akad. Nauk. SSSR 102 489–492

    Google Scholar 

  29. S. Machihara K. Nakanishi T. Ozawa (2003) ArticleTitleSmall global solutions and the nonrelativistic limit for the nonlinear Dirac equation. Rev Mat. Iberoam. 19 IssueID(1 179–194 Occurrence HandleMR1993419

    MathSciNet  Google Scholar 

  30. S.N.M. Ruijsenaars (1977) ArticleTitleOn Bogoliubov transformations for systems of relativistic charged particles J. Math. Phys. 18 IssueID(3 517–526 Occurrence Handle10.1063/1.523295

    Article  Google Scholar 

  31. Schweber S. S. 1994. QED and the men who made it: Dyson, Feynman, Schwinger and Tomonaga, Princeton University Press

  32. D. Shale W. Stinespring (1965) ArticleTitleSpinor representation of infinite orthogonal groups J. Math, Mech. 14 315–324

    Google Scholar 

  33. Simon B. 1979. Trace Ideals and their Applications. Vol 35 of London Mathematical Society Lecture Notes Series. Cambridge University Press

  34. Thaller B. 1992. The Dirac Equation, Springer Verlag

  35. A. Uehling E. (1935) ArticleTitlePolarization effects in the positron theory Phys. Rev. II. Ser. 48 55–63

    Google Scholar 

  36. V. Weisskopf (1936) ArticleTitleÜber die Elektrodynamik des Vakuums auf Grund der Quantentheorie des Elektrons Math.Fys. Medd, Danske Vid. Selsk 16 IssueID3 139

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

    Christian Hainzl & Mathieu Lewinand

  2. Department of Numerical Mathematics, University of Münster, Einsteinstrasse 62, D48149

    Christof Sparber

  3. Münster Wolfgang Pauli Institute Vienna c/o Faculty of Mathematics, Vienna University, Nordbergstrasse 15, A1090, Vienna, Austria

    Christof Sparber

Authors
  1. Christian Hainzl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mathieu Lewinand
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christof Sparber
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christof Sparber.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hainzl, C., Lewinand, M. & Sparber, C. Existence of Global-In-Time Solutions to a Generalized Dirac-Fock Type Evolution Equation. Lett Math Phys 72, 99–113 (2005). https://doi.org/10.1007/s11005-005-4377-9

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-005-4377-9

Keywords

Search

Navigation