Extension of the Conventional Framework of Local Quantum Field Theory and the Description of Resonances

  • Jerzy Lukierski
Conference paper
Part of the Acta Physica Austriaca book series (FEWBODY, volume 6/1969)


In last ten years many authors have attempted to formulate a consistent relativistic quantum field theory with unstable particles [1]. At present one can state only that the role of unstable particles was clarified in some soluble models: the lowest Lee model sectors [2], Zachariasen model [3]. The main difficulties in the construction of general relativistic theory with unstable particles appeared to be
  1. 1.

    to incorporate correctly the instability property into the theory,

  2. 2.

    to get the consistency with an experimental fact that in asymptotic space — time regions there are only observed free stable particles.


The aim of this lecture is to discuss the applicability to the description of strongly unstable objects the field operator ϕ(x;s), with continuous spectrum of asymptotic masses.


Field Operator Spectral Function Unstable Particle Elementary Object Asymptotic Field 
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  1. 1.
    The unstable particles were mainly studied in the framework of Ŝ-matrix theory, where the Peierl’s concept of complex pole on second Rieman sheet of the propagator has been used. Below we list some papers, from “classical” period 1958 — 1963, where there were made many attempts to find the place of unstable particles in general framework of relativistic local QFT: a) P. T. Matthews, A. Salam, Phys. Rev. 112, 283 (1958); 115, 1079 (1959).MathSciNetADSMATHGoogle Scholar
  2. b).
    M. Levy, Nuovo Cim. 13, 115 (1959); 14, 612 (1959).MATHCrossRefGoogle Scholar
  3. c).
    J. Schwinger, Ann.Phys. 9, 169 (1960).MathSciNetADSMATHCrossRefGoogle Scholar
  4. d).
    M. Ida, Progr. Theor. Phys. 24, 1135 (1960).MathSciNetADSMATHCrossRefGoogle Scholar
  5. e).
    M. Hama, S. Tanaka, Progr. Theor. Phys. 26, 829 (1961).MathSciNetADSMATHCrossRefGoogle Scholar
  6. f).
    P. J. Peebles, Phys. Rev. 128, 1412 (1962).MathSciNetADSMATHCrossRefGoogle Scholar
  7. g).
    J. McEvan, Phys. Rev. 132, 2353 (1963).MathSciNetADSCrossRefGoogle Scholar
  8. h).
    M. Veltman, Physica 29, 186 (1963).MathSciNetMATHCrossRefGoogle Scholar
  9. As basic paper for field-theoretic understanding of the properties of unstable state should be quoted N. S. KRYLOV and B. A. FOCK, JETF 17, 93 (1947). These ideas were elaborated further by L. A. Khalfin in his papers (see DAN SSSR and JETF; 1956–1969).Google Scholar
  10. 2.
    See, for exampleGoogle Scholar
  11. a).
    V. Glaser and G. Källén, Nucl. Phys. 2, 706 (1957).Google Scholar
  12. b).
    T. Okabayashi and Sato, S., Progr. Theor. Phys. 17, 30 (1957).MathSciNetADSCrossRefGoogle Scholar
  13. c).
    T. Goto, Progr. Theor. Phys. 21, 1 (1959).MathSciNetADSMATHCrossRefGoogle Scholar
  14. d).
    W. Jaus, Helv. Phys. Acta 39, 617 (1966).Google Scholar
  15. 3.
    See, for exampleGoogle Scholar
  16. a).
    M. Gell-Mann and Zachariasen, Phys. Rev. 124, 953 (1961).MathSciNetADSCrossRefGoogle Scholar
  17. b).
    C. R. Hagen, Nuovo Cim. 50A, 545 (1967).ADSCrossRefGoogle Scholar
  18. 4.a)
    A. L. Licht, Maryland Thesis 1963.Google Scholar
  19. b).
    A. L. Licht, Ann. of Phys. (N.Y.), 34, 161 (1965).MathSciNetADSMATHCrossRefGoogle Scholar
  20. c).
    See also A. S. Wightman, Cargese Summer School Lectures, Corsica, 1964.Google Scholar
  21. 5. a)
    S. Weinberg, Phys. Rev. 130, 776 (1963); 131, 440 (1963).MathSciNetADSCrossRefGoogle Scholar
  22. b).
    M. Scadron and S. Weinberg, Phys. Rev. 133B, 1589 (1964).MathSciNetADSCrossRefGoogle Scholar
  23. c).
    M. Scadron and J. Wright, Phys. Rev. 135B, 202 (1964).MathSciNetADSCrossRefGoogle Scholar
  24. 6.
    Free objects with continuous mass spectrum were discussed from a group-theoretical point of view by F. Lurcat, Phys. Rev. 173, 1461 (1968).ADSCrossRefGoogle Scholar
  25. 7.
    See also J. Lukierski, “Field Theory Describing Interacting Two-Particle Subsystems, I. General Formalism”, Nuovo Cim.; in press.Google Scholar
  26. 8.
    Lukierski, J., Proceedings of Balaton Seminar on Hadrons, September 1968, publ. Acta Phys. Hungarica, 1969.Google Scholar
  27. 9.
    See, for exampleGoogle Scholar
  28. a).
    R. Jost, “General Theory of Quantized Fields”, Providence, R. I. 1965.Google Scholar
  29. b).
    R. F. Streater and A. S. Wightman, “PCT, Spin and Statistics, and All That”, Bejamin Inc., New York 1965.Google Scholar
  30. 10. a)
    H. Lehmann, K. Symanzik and W. Zimmermann, Nuovo Cim. 1, 205 (1955).MathSciNetMATHCrossRefGoogle Scholar
  31. b).
    K. Hepp, Comm. Math. Phys., 1, 95 (1965).MathSciNetADSMATHCrossRefGoogle Scholar
  32. 11.
    See [6]: also H. Joos, Fortschr. d. Phys. 10, 65–146 (1962).ADSMATHCrossRefGoogle Scholar
  33. A. J. McFarlane, Rev. Mod. Phys. 34, 41 (1962).ADSCrossRefGoogle Scholar
  34. A. J. McFarlane, Journ. Math. Phys. 4, 490 (1963).ADSCrossRefGoogle Scholar
  35. 12.
    See [4a-b] and [7]. The proof of generalized LSZ asymptotic condition has been given by W. KARWOWSKI and N. SZNAJDER, Acta Phys. Polon., in press: see also W. KARWOWSKI, J. Lukierski and N. SZNAJDER, Nuovo Cim., to be published.Google Scholar
  36. 13.
    L. Streit, Helv. Phys. Acta 39, 65 (1965).Google Scholar
  37. 14.
    R. A. Brandt and O. W. Greenberg, University of Maryland, preprint No. 817, June 1968.Google Scholar
  38. 15.
    The notion of generalized free field was introduced by O. W. Greenberg, Ann. Phys. 16, 158 (1961).ADSMATHCrossRefGoogle Scholar
  39. See also G. F. Dell’Antonio, Journ. Math. Phys. 2, 759 (1961).MathSciNetADSCrossRefGoogle Scholar
  40. 16.
    See J. Lukierski, Bull. Acad. Sei. Polon. 16, 343 (1968).MATHGoogle Scholar
  41. 17.
    R. Jacob and R. G. Sachs, Phys. Rev. 121, 350 (1961).MathSciNetADSCrossRefGoogle Scholar
  42. 18.
    F. Zachariasen, Phys. Rev. 121, 1851 (1961).MathSciNetADSMATHCrossRefGoogle Scholar
  43. 19.
    W. Thirring, Phys. Rev. 126, 1209 (1962).MathSciNetADSCrossRefGoogle Scholar
  44. 20.
    See [12], the second reference.Google Scholar
  45. 21.
    See, for example, F. R. Halpern, Phys. Rev. 137, B1587 (1965).ADSCrossRefGoogle Scholar
  46. 22.
    M. Cini and S. Fubini, Ann. Phys. 10, 362 (1960).MathSciNetADSGoogle Scholar
  47. 23.
    B. W. Lee and R. F. Sawyer, Phys. Rev. 121, 2266 (1962).MathSciNetADSCrossRefGoogle Scholar
  48. 24.
    It should be mentioned, that in order to get the Regge behaviour the spectral function Λ2(s) should tend to zero for large values of s → ∞, but can be non-integrable.Google Scholar
  49. 25.
    See, for example, J. Harte, Phys. Rev. 171, 1825 (1968), and the references given therein.ADSCrossRefGoogle Scholar
  50. 26. a)
    L. D. Faddeev, Mathematical Problems of Quantum Theory of Scattering of a Three-Particle State, publ. Steklov Institute, No. 39 (1963).Google Scholar
  51. b).
    C. Lovelace, Phys. Rev. 135B, 1225 (1964).MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag / Wien 1969

Authors and Affiliations

  • Jerzy Lukierski
    • 1
  1. 1.Institute of Theoretical PhysicsUniversity of WroclawWroclawPoland

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