Functional Methods in Quantum Field Theory

  • B. de Wit
Part of the NATO Advanced Study Institutes Series book series (NSSB, volume 85)

Abstract

In 1933 Dirac pointed out that it would be desirable to have a formulation of quantum mechanics in close correspondence to the Lagrangian method in classical mechanics, rather than to the more conventional Hamiltonian framework1). The Lagrangian method is based upon an action, defined as the time integral of the Lagrangian, and the principle of least action expresses the equations of motion in terms of a variational principle. The action is a relativistic invariant, and therefore the obvious advantage of this approach is that relativistic invariance is manifest at all stages. Feynman, in his pioneering work, fully developed this line of thought, and applied his methods to a large variety of problems 2,3). His work led to the notion of an integral over all paths, which is an integration in the space of functionals. Such integrations had actually been studied in the mathematical literature (for a review, see (4)).

Keywords

Covariance Propa Ghost Rine 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P.A.M. Dirac, Phys. Z. der Sowjetunion 3, reprinted in “Quantum Electrodynamics”, ed. J. Schwinger (Dover 1958); “The Principles of Quantum Mechanics” (The Clarendon Press, 1958).Google Scholar
  2. 2.
    R.P. Feynman, Rev. Mod. Phys. 20, 267 (1948).MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    R.P. Feynman and A.R. Hibbs, “Quantum Mechanics and Path Integrals” (Mc Graw-Hill 1965).Google Scholar
  4. 4.
    J.M. Gel’fand and A.M. Yaglom, J. Math. Phys. 1, 48 (1960).ADSMATHCrossRefGoogle Scholar
  5. 5.
    For a review, see M. Veltman, proc. Int. Symp. on Electron and Photon Interactions at High Energies, Bonn (1973) (North-Holland).Google Scholar
  6. 6.
    E.S. Abers and B.W. Lee, Phys. Rep. 9, 1 (1973).Google Scholar
  7. 7.
    M. Veltman, lectures at the Basko Polje Summer School, (1974).Google Scholar
  8. 8.
    S. Coleman, proc. of the 1975 International School of Subnuclear Physics “Ettore Majorana” (Acad. Press, 1975).Google Scholar
  9. 9.
    L.D. Faddeev, in Methods in Field Theory, eds. R. Balian and J. Zinn-Justin (North-Holland, 1976).Google Scholar
  10. 10.
    B.W. Lee, in Methods in Field Theory, eds. R. Balian and J. Zinn-Justin (North-Holland, 1976).Google Scholar
  11. 11.
    J.C. Taylor, Gauge Theories of Weak Interactions (Cambridge Univ. Press, 1976).Google Scholar
  12. 12.
    V.N. Popov, Functional Integrals in Quantum Field Theory and Statistical Physics (in Russian), (Atomizdat, 1976); CERN preprint TH.2424, 1977.Google Scholar
  13. 13.
    C. De Witt-Morette, A. Maheswari and B. Nelson, Phys. Rep. 50, 255 (1979).MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    L.D. Faddeev and A.A. Slavnov, Gauge fields: Introduction to Quantum Theory (Benjamin Cummings, 1980).MATHGoogle Scholar
  15. 15.
    C. Itzykson and J.B. Zuber, Quantum Field Theory (Mc Graw-Hill, 1980).Google Scholar
  16. 16.
    M.S. Marinov, Phys, Rep. 60, 1 (1980).MathSciNetGoogle Scholar
  17. 17.
    P. Ramond, Field Theory; a modern primer (Benjamin Cummings, 1981).Google Scholar
  18. 18.
    See for instance, H. Goldstein, Classical Mechanics (Addison-Wesley, 1950).Google Scholar
  19. 19.
    C.W. Bernard, Phys. Rev. D9, 3312 (1974).ADSGoogle Scholar
  20. 20.
    R.P. Feynman, Phys. Rev. 84, 108, (1951)MathSciNetADSMATHCrossRefGoogle Scholar
  21. 21.
    The effective action for the velocity-dependent potential was first obtained by T.D. Lee and C.N. Yang, Phys. Rev. 128, 885 (1962).MathSciNetADSMATHCrossRefGoogle Scholar
  22. 22.
    J. Schwinger, Proc. Nat. Acad. Sci. 44, 956 (1958)MathSciNetADSMATHCrossRefGoogle Scholar
  23. K. Symanzik, proc. Int. School “Enrico Fermi”, ed. R. Jost (Acad. Press, 1969);Google Scholar
  24. K. Osterwalder and R. Schrader, Comm. Math. Phys. 31, 83 (1973).MathSciNetADSMATHCrossRefGoogle Scholar
  25. 23.
    R.P. Feynman, Phys. Rev. 91, 1291 (1953); Statistical Mechanics: A Set of Lectures (Benjamin, 1972).MathSciNetADSMATHCrossRefGoogle Scholar
  26. 24.
    See for instance, A. Fetter and J. Walecka, Quantum Theory of Many-Particle Systems (Mc Graw-Hill, 1965).Google Scholar
  27. 25.
    G. Jona-Lasinio, Nuovo Cim. 34, 1790 (1964).CrossRefGoogle Scholar
  28. 26.
    J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127, 965 (1962).MathSciNetADSMATHCrossRefGoogle Scholar
  29. 27.
    S. Coleman and E. Weinberg, Phys. Rev. D7, 1888 (1973).ADSGoogle Scholar
  30. 28.
    R. Jackiw, Phys. Rev. D9, 1686 (1974).ADSGoogle Scholar
  31. 29.
    G. ’t Hooft and M. Veltman, Nucl. Phys. B44, 189 (1972);ADSCrossRefGoogle Scholar
  32. C.G. Bollini and J.J. Giambiagi, Phys. Lett. 40B, 566 (1972)ADSGoogle Scholar
  33. J.F. Ashmore, Lett. Nuovo Cim. 4, 289 (1972).CrossRefGoogle Scholar
  34. G.M. Cicuta and E. Montaldi, Lett. Nuovo Cim. 4, 329 (1972).CrossRefGoogle Scholar
  35. 30.
    S. Weinberg, Phys. Rev. Lett. 36, 294 (1976)ADSCrossRefGoogle Scholar
  36. A.D. Linde, Zh. Eksp. Teor. Fiz. Pis. Red. 23, 73 (1976) (JEPT Lett. 23, 64 (1976)).ADSGoogle Scholar
  37. 31.
    L. Dolan and R. Jackiw, Phys. Rev. D9, 3320 (1974).ADSGoogle Scholar
  38. 32.
    S. Weinberg, Phys. Rev. D9, 3357 (1974).ADSGoogle Scholar
  39. 33.
    D. Kirzhnits and A. Linde, Phys. Lett. 42B, 471 (1972).ADSGoogle Scholar
  40. 34.
    A.D. Linde, Rep. Prog. Phys. 42, 389, (1979).ADSCrossRefGoogle Scholar
  41. 35.
    G. ‘t Hooft, this volume.Google Scholar
  42. 36.
    J. Iliopoulos and N. Papanicolaou, Nucl. Phys. Bill, 209 (1976).Google Scholar
  43. 37.
    F.A. Berezin, The Method of Second Quantization (Acad. Press, (1966).MATHGoogle Scholar
  44. 38.
    B. de Wit, Phys. Rev. D12, 1628 (1975).ADSGoogle Scholar
  45. 39.
    R. Arnowitt, P. Nath and B. Zumino, Phys. Lett. 56B, 81 (1975).MathSciNetADSGoogle Scholar
  46. 40.
    A. De Rujula, this volume.Google Scholar
  47. 41.
    E.C.G. Stueckelberg, Helv. Phys. Acta 11, 225 (1938).MATHGoogle Scholar
  48. 42.
    P. van Nieuwenhuizen, Phys. Rep. 68, 189 (1981).Google Scholar
  49. 43.
    L.D. Faddeev and V.N. Popov, Phys. Lett. 25B, 29 (1976).ADSGoogle Scholar
  50. 44.
    N.K. Nielsen, Nucl. Phys. B140, 499 (1978)ADSCrossRefGoogle Scholar
  51. R.E. Kallosh, Nucl. Phys. B141, 141 (1978).MathSciNetADSCrossRefGoogle Scholar
  52. 45.
    R.P. Feynman, In Relativistic Theories of Gravitation, (Pergamon Press 1964); Acta Phys. Polonica 24, 697 (1963).Google Scholar
  53. 46.
    B.S. De Witt. Phvs. Rev. Lett. 12, 742 (1964); Phys. Rev. 162, 1195, 1239 (1967).ADSCrossRefGoogle Scholar
  54. M. Veltman, Nucl. Phys. B21, 288 (1970)ADSGoogle Scholar
  55. E.S. Fradkin and J.V. Tyutin, Phys. Rev. D2, 2841 (1970)MathSciNetADSGoogle Scholar
  56. G.’t Hooft, Nucl. Phys. B33, 173 (1971);ADSCrossRefGoogle Scholar
  57. G.‘t Hooft and M. Veltman, Nucl. Phys. B50, 318 (1972);CrossRefGoogle Scholar
  58. B.W. Lee and J. Zinn-Justin, Phys. Rev. D7, 1049 (1973).ADSGoogle Scholar
  59. 47.
    C. Fronsdal, Phys. Rev. D18, 3624 (1978).ADSGoogle Scholar
  60. 48.
    T.L. Curtright, Phys. Lett. 85B, 219 (1979).ADSGoogle Scholar
  61. 49.
    B. de Wit and D.Z. Freedman, Phys. Rev. D21, 358 (1980).ADSGoogle Scholar
  62. 50.
    J. Fang and C. Fronsdal, Phys. Rev. D18, 3630 (1978).ADSGoogle Scholar
  63. 51.
    C. Aragone and S. Deser, Phys. Rev. D21, 352 (1980).MathSciNetADSGoogle Scholar
  64. 52.
    M.A. Namazie and D. Storey, Nucl. Phys. B157, 170 (1979); P.K. Townsend, Phys. Lett. 88B, 97 (1979)MathSciNetADSCrossRefGoogle Scholar
  65. W. Siegel, Phys. Lett. 93B, 170 (1980).ADSGoogle Scholar

Copyright information

© Plenum Press, New York 1982

Authors and Affiliations

  • B. de Wit
    • 1
  1. 1.Nikhef-HAmsterdamThe Netherlands

Personalised recommendations