Foundations of Physics

, Volume 25, Issue 1, pp 39–65 | Cite as

The unstable system in relativistic quantum mechanics

  • L. P. Horwitz
Invited Papers Dedicated to Jean-Pierre Vigier

Abstract

A soluble model for the relativistically covariant description of an unstable system is given in terms of relativistic quantum field theory, with a structure similar to Van Hove's generalization of the Lee model in the nonrelativistic theory. Since the Fock space for this model can be decomposed to sectors, it can be embedded in a one-particle Hilbert space in a spectral form similar to the Friedrichs model in the nonrelativistic theory. Several types of spectral models result, corresponding to physically motivated assumptions made in the framework of the field theory. For example, the continuous spectrum of the unperturbed problem may be (−∞, ∞) or semibounded. In the decay V → N+θ, the θ may have negative energy. In this case, the reaction corresponds to the crossed channel process V+\(\bar \theta \) → N.

Keywords

Relativistic Quantum Continuous Spectrum Process Versus Negative Energy Spectral Model 

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References

  1. 1.
    T. D. Lee,Phys. Rev. 95, 1329 (1954).Google Scholar
  2. 2.
    K. O. Friedrichs,Commun. Pure Appl. Math. 1, 361 (1950).Google Scholar
  3. 3.
    L. P. Horwitz and J.-P. Marchand,Rocky Mountain J. Math. 1, 225 (1971).Google Scholar
  4. 4.
    I. Segal and L. P. Horwitz,Helv. Phys. Acta 51, 685 (1978); W. Baumgartel,Math. Nachr. 75, 133 (1978); G. Parravicini, V. Gorini, and E. C. G. Sudarshan,J. Math. Phys. 21, 2208 (1980); T. Bailey and W. C. Schieve,Nuovo Cimento A 47, 231 (1978); A. Bohm,Quantum Mechanics: Foundations and Applications, Springer, Berlin (1986); A. Bohm, M. Gadella, and G. B. Mainland,Am. J. Phys. 57, 1103 (1989).Google Scholar
  5. 5.
    A. Bohm,The Rigged Hilbert Space and Quantum Mechanics (Springer Lecture Notes on Physics78, Berlin, 1978); I. M. Gel'fand and N. Ya. Vilenkin,Generalized Functions, Vol. 4 (Academic, New York, 1964).Google Scholar
  6. 6.
    I. Antoniou and S. Tasaki,J. Phys. A: Math. Gen. 26, 73 (1993); H. H. Hasegawa and W. C. Saphir,Phys. Lett. A 161, 471 (1992);Phys. Rev. A 46, 7401 (1992).Google Scholar
  7. 7.
    I. Antoniou and I. Prigogine,Physica A 192, 443 (1993), and references therein.Google Scholar
  8. 8.
    L. P. Horwitzs and C. Piron,Hev. Phys. Acta 46, 316 (1973); R. Fanchi,Phys. Rev. D 20, 3108 (1979); C. Dewdney, P. R. Holland, A. Kyprianides, and J. P. Vigier,Phys. Lett. A 113, 359 (1986);Phys. Lett. A 114, 444 (1986); A. Kyprianides,Phys. Rep. 155, 1 (1986).Google Scholar
  9. 9.
    E. C. G. Stueckelberg,Helv. Phys. Acta 14, 322, 588 (1941); J. Schwinger,Phys. Rev. 82, 664 (1951); R. P. Feynman,Rev. Mod. Phys. 20, 367 (1948);Phys. Rev. 80, 440 (1950).Google Scholar
  10. 10.
    L. P. Horwitz,Found. Phys. 22, 421 (1992).Google Scholar
  11. 11.
    See R. Arshansky and L. P. Horwitz,Found. Phys. 15, 701 (1985) for a discussion of localization in space and time, and the associated Newton-Wigner and Landau-Peierls operators. M. Usher and L. P. Horwitz,Found. Phys. Lett. 4, 289 (1991), have discussed the causal properties from the point of view taken by G. C. Hegerfeldt,Phys. Rev. D 10, 3320 (1974);Phys. Rev. Lett. 54, 2395 (1985);Nucl. Phys. B 6, 231 (1989).Google Scholar
  12. 12.
    L. P. Horwitz, W. C. Schieve, and C. Piron,Ann. Phys. 137, 306 (1981).Google Scholar
  13. 13.
    L. P. Horwitz, S. Shashoua, and W. C. Schieve,Physica A 161, 300 (1989).Google Scholar
  14. 14.
    L. Burakovsky and L. P. Horwitz,Physica A 201, 666 (1993);J. Phys. A: Math. Gen. 27, 2623 (1994). “Equilibrium relativistic mass distributions for indistinguishable events,”Found. Phys., to be published; L. Burakovksy and L. P. Horwitz, “Independence of specific heat on mass distribution for largec,” TAUP 2141-94.Google Scholar
  15. 15.
    L. Burakovsky, L. P. Horwitz, and W. C. Schieve, “Statistical mechanics of relativistic degenerate Fermi gas I. Cold adiabatic equation of state.” TAUP 2136-94; “Relativistic Bose-Einstein Condensation to Mass-Shell Limit,” TAUP 2149-94.Google Scholar
  16. 16.
    L. Van Hove,Physica 21, 901 (1955);22, 343 (1956);23, 441 (1957).Google Scholar
  17. 17.
    R. I. Arshansky and L. P. Horwitz.J. Math. Phys. 30, 66, 380, 213 (1989). See also R. Arshansky and L. P. Horwitz,Phys. Rev. D 29, 2860 (1984).Google Scholar
  18. 18.
    M. C. Land, R. I. Arshansky, and L. P. Horwitz,Found. Phys. 24, 1179 (1994).Google Scholar
  19. 19.
    M. C. Land and L. P. Horwitz, “The Zeeman effect for the relativistic bound state,” TAUP 2150-94.Google Scholar
  20. 20.
    L. P. Horwitz,J. Math. Phys. 34, 645 (1993).Google Scholar
  21. 21.
    R. Faibish and L. P. Horwitz,J. Group Theory Phys. 2, 41 (1994); “Dynamical groups of the relativistic Kepler problem and the harmonic oscillator.” Vol. II, p. 450,Anales de Fisica. Monografias, Vols. I, II, M. A. del Olmo, M. Santander, and J. M. Guilarte, eds., CIEMAT-RSEF. Madrid (1993).Google Scholar
  22. 22.
    E. P. Wigner and V. F. Weisskopf,Z. Phys. 63, 54 (1930);65, 18 (1930).Google Scholar
  23. 23.
    P. Exner,Open Quantum Systems and Feynman Integrals (Reidel, Boston, 1985); A. Bohm.Quantum Mechanics: Foundations and Applications (Springer, Berlin, 1986).Google Scholar
  24. 24.
    L. P. Horwitz and C. Piron,Helv. Phys. Acta 66, 693 (1993).Google Scholar
  25. 25.
    See, for example, J.-P. Marchand and L. P. Horwitz,Rocky Mountain J. Math. 1, 225 (1971).Google Scholar
  26. 26.
    N. Bleistein, R. Handelsman, L. P. Horwitz and H. Neumann,Nuovo Cimento A 41, 389 (1977).Google Scholar
  27. 27.
    E. Henley and W. Thirring,Elementary Quantum Field Theory (McGraw-Hill, New York, 1963).Google Scholar
  28. 28.
    B. Zumino, inLectures on Field Theory and the Many Body Problem, E. R. Caianiello, ed. (Academic Press, New York, 1961, p. 37).Google Scholar
  29. 29.
    L. Van Hove, second of Ref. 16..Google Scholar
  30. 30.
    D. Saad, L. P. Horwitz, and R. I. Arshansky,Found. Phys. 19, 1126 (1989); N. Shnerb and L. P. Horwitz,Phys. Rev. A 48, 4068 (1993).Google Scholar
  31. 31.
    J. Frastai and L. P. Horwitz, “Off-Shell Quantum Electrodynamics and Pauli-Villars Regularization,” TAUP 2138-94.Google Scholar
  32. 32.
    J. Schwinger,Phys. Rev. 82, 664 (1951).Google Scholar
  33. 33.
    W. Pauli and F. Villars,Rev. Mod. Phys. 21, 434 (1949).Google Scholar
  34. 34.
    See, for example, J. M. Jauch and F. Rohrlich,The Theory of Photons and Electrons, 2nd edn. (Springer, New York. 1976); C. Itzykson and J.-B. Zuber,Quantum Field Theory (McGraw-Hill, New York. 1980).Google Scholar
  35. 35.
    See, for example, L. de Broglie,Les incertitudes d'Heisenberg et l'interprétation probabiliste de la mécanique ondulatoire, preface by Georges Lochak (Gauthier-Villars, Paris, 1982). English translation: L. de Broglie,Heisenberg's Uncertainties and the Probabilistic Interpretation of Wave Mechanics, translated by Alwyn van der Merwe (Fundamental Theories of Physics, Vol. 40) (Kluwer Academic, Dordrecht, 1990).Google Scholar
  36. 36.
    K. Haller and R. B. Sohn,Phys. Rev. A 20, 1541 (1979); K. Haller,Acta Phys. Austriaca 42, 163 (1975); K. Haller,Phys. Rev. D 36, 1830 (1987).Google Scholar
  37. 37.
    L. P. Horwitz, C. Piron, and F. Reuse,Helv. Phys. Acta 48, 546 (1975); C. Piron and F. Reuse,Helv. Phys. Acta 51, 146 (1978); L. P. Horwitz and R. Ashansky,J. Phys. A: Math. Gen. 15, L659 (1982); A. Arensburg and L. P. Horwitz,Found. Phys. 22, 1025 (1992).Google Scholar
  38. 38.
    P. A. M. Dirac,The Principles of Quantum Mechanics, 3rd edn. (Oxford University Press, London, 1947).Google Scholar
  39. 39.
    G. Källén and W. Pauli,Mat. Fys. Medd. Dan. Vid. Selsk. 30, 1 (1955).Google Scholar
  40. 40.
    L. P. Horwitz and Y. Lavie,Phys. Rev. D 26, 819 (1982).Google Scholar
  41. 41.
    J. L. Pietenpol,Phys. Rev. 162, 1301 (1967); I. Antoniou, J. Levitan, and L. P. Horwitz,J. Phys. A: Math. Gen. 26, 6033 (1993).Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • L. P. Horwitz
    • 1
    • 2
  1. 1.School of Physics, Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv UniversityRamat AvivIsrael
  2. 2.Department of PhysicsBar Ilan UniversityRamat GanIsrael

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