Letters in Mathematical Physics

, Volume 97, Issue 2, pp 165–183 | Cite as

A Formulation of Quantum Field Theory Realizing a Sea of Interacting Dirac Particles

  • Felix FinsterEmail author


In this survey article, we explain a few ideas behind the fermionic projector approach and summarize recent results which clarify the connection to quantum field theory. The fermionic projector is introduced, which describes the physical system by a collection of Dirac states, including the states of the Dirac sea. Formulating the interaction by an action principle for the fermionic projector, we obtain a consistent description of interacting quantum fields which reproduces the results of perturbative quantum field theory. We find a new mechanism for the generation of boson masses and obtain small corrections to the field equations which violate causality.

Mathematics Subject Classification (2010)

81-02 81T15 81T27 


Relativistic quantum theory Dirac sea fermionic projector 


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  1. 1.
    Bach V., Barbaroux J.-M., Helffer B., Siedentop H.: On the stability of the relativistic electron–positron field. Commun. Math. Phys. 201(2), 445–460 (1999)MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Bär, C., Fredenhagen, K. (eds): Quantum Field Theory on Curved Spacetimes. Lecture Notes in Physics, vol. 786. Springer, Berlin (2009)Google Scholar
  3. 3.
    Christensen S.M.: Vacuum expectation value of the stress tensor in an arbitrary curved background: the covariant point-separation method. Phys. Rev. D (3) 14(10), 2490–2501 (1976)MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Collins J.C.: Renormalization, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1984)Google Scholar
  5. 5.
    Deckert, D.-A., Dürr, D., Merkl, F., Schottenloher, M.: Time Evolution of the External Field Problem in QED. arXiv:0906.0046 [math-ph] (2009)Google Scholar
  6. 6.
    Dirac P.A.M.: A theory of electrons and protons. Proc. R. Soc. Lond. A 126, 360–365 (1930)ADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Dirac P.A.M.: Discussion of the infinite distribution of electrons in the theory of the positron. Proc. Cambridge Philos. Soc. 30, 150–163 (1934)CrossRefADSGoogle Scholar
  8. 8.
    Dirac, P.A.M.: Directions in physics. Wiley, New York (Five lectures delivered during a visit to Australia and New Zealand, August–September, 1975) 1978Google Scholar
  9. 9.
    Dürr H.-P., Heisenberg W., Mitter H., Schlieder S., Yamazaki K.: Zur Theorie der Elementarteilchen. Z. Naturf. 14a, 441–485 (1959)ADSGoogle Scholar
  10. 10.
    Dyson F.J.: The S matrix in quantum electrodynamics. Phys. Rev. 75, 1736–1755 (1949)MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Epstein H., Glaser V.: The role of locality in perturbation theory. Ann. Inst. H. Poincaré Section A (N.S.) 19, 211–295 (1973)MathSciNetGoogle Scholar
  12. 12.
    Feynman R.: The theory of positrons. Phys. Rev. 76, 749–759 (1949)MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. 13.
    Fierz H., Scharf G.: Particle interpretation for external field problems in QED. Helv. Phys. Acta 52(4), 437–453 (1979)MathSciNetGoogle Scholar
  14. 14.
    Finster, F.: The Principle of the Fermionic Projector, hep-th/0001048, hep-th/0202059, hep-th/0210121. AMS/IP Studies in Advanced Mathematics, vol. 35. American Mathematical Society, Providence (2006)Google Scholar
  15. 15.
    Finster, F.: An Action Principle for an Interacting Fermion System and its Analysis in the Continuum Limit. arXiv:0908.1542 [math-ph] (2009)Google Scholar
  16. 16.
    Finster, F.: From discrete space-time to Minkowski space: Basic mechanisms, methods and perspectives. In: Fauser, B., Tolksdorf, J., Zeidler, E. (eds.) Quantum Field Theory. Birkhäuser Verlag, pp. 235–259 (2009). arXiv:0712.0685 [math-ph]Google Scholar
  17. 17.
    Finster, F.: Entanglement and second quantization in the framework of the fermionic projector. J. Phys. A Math. Theor. 43, 395302 (2010). arXiv:0911.0076 [math-ph]Google Scholar
  18. 18.
    Finster, F.: The fermionic projector, entanglement, and the collapse of the wave function. In: The Proceedings of DICE2010 (2011). arXiv:1011.2162 [quant-ph]Google Scholar
  19. 19.
    Finster, F., Grotz, A.: The causal perturbation expansion revisited: rescaling the interacting Dirac sea. J. Math. Phys. 51, 072301 (2010). arXiv:0901.0334 [math-ph]Google Scholar
  20. 20.
    Fulling S.A., Sweeny M., Wald R.M.: Singularity structure of the two-point function quantum field theory in curved spacetime. Commun. Math. Phys. 63(3), 257–264 (1978)MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. 21.
    Glimm J., Jaffe A.: Quantum Physics, a Functional Integral Point of View, 2nd edn. Springer, New York (1987)Google Scholar
  22. 22.
    Gravejat, P., Lewin, M., Séré, E.: Renormalization and Asymptotic Expansion of Dirac’s Polarized Vacuum (2010). arXiv:1004.1734v1Google Scholar
  23. 23.
    Hainzl C., Lewin M., Séré E.: Existence of a stable polarized vacuum in the Bogoliubov–Dirac–Fock approximation. Commun. Math. Phys. 257(3), 515–562 (2005) arXiv:math-ph/0403005ADSzbMATHCrossRefGoogle Scholar
  24. 24.
    Hainzl C., Lewin M., Séré E.: Self-consistent solution for the polarized vacuum in a no-photon QED model. J. Phys. A Math. Theor. 38(20), 4483–4499 (2005) arXiv:physics/0404047ADSzbMATHCrossRefGoogle Scholar
  25. 25.
    Hainzl C., Lewin M., Séré E., Solovej J.P.: A minimization method for relativistic electrons in a mean-field approximation of quantum electrodynamics. Phys. Rev. A 76, 052104 (2007) arXiv:0706.1486 [physics.atom-ph]ADSCrossRefGoogle Scholar
  26. 26.
    Heisenberg W.: Bemerkungen zur Diracschen Theorie des Positrons. Z. Phys. 90, 209–231 (1934)ADSzbMATHCrossRefGoogle Scholar
  27. 27.
    Klaus M.: Nonregularity of the Coulomb potential in quantum electrodynamics. Helv. Phys. Acta 53(1), 36–39 (1980)MathSciNetGoogle Scholar
  28. 28.
    Klaus M., Scharf G.: The regular external field problem in quantum electrodynamics. Helv Phys. Acta 50(6), 779–802 (1977)MathSciNetGoogle Scholar
  29. 29.
    Klaus M., Scharf G.: Vacuum polarization in Fock space. Helv. Phys. Acta 50(6), 803–814 (1977)MathSciNetGoogle Scholar
  30. 30.
    Nenciu G., Scharf G.: On regular external fields in quantum electrodynamics. Helv. Phys. Acta 51(3), 412–424 (1978)MathSciNetGoogle Scholar
  31. 31.
    Peskin M.E., Schroeder D.V.: An Introduction to Quantum Field Theory. Addison-Wesley, Reading (1995)Google Scholar
  32. 32.
    Radzikowski M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179(3), 529–553 (1996)MathSciNetADSzbMATHCrossRefGoogle Scholar
  33. 33.
    Scharf G.: Finite Quantum Electrodynamics. Texts and Monographs in Physics. Springer, Berlin (1989)Google Scholar
  34. 34.
    Schwinger J.: Quantum electrodynamics. I. A covariant formulation. Phys. Rev. 74, 1439–1461 (1948)MathSciNetADSzbMATHCrossRefGoogle Scholar
  35. 35.
    Serber R.: Linear modifications of the Maxwell field equations. Phys. Rev. 48, 49–54 (1935)ADSzbMATHCrossRefGoogle Scholar
  36. 36.
    Uehling E.A.: Polarization effects in the positron theory. Phys. Rev. 48, 55–63 (1935)ADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer 2011

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany

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