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de Sitter Space and Positive Energy

  • T. O. Philips
  • E. P. Wigner
Chapter
Part of the The Scientific Papers book series (WIGNER, volume A / 3)

Summary

Even though the unitary representations of the pseudo-orthogonal groups O(p, q) (that is, of the groups of transformations that leave the form\(x_1^2 + x_2^2 + ... + x_p^2 - x_{p + 1}^2 - ... - x_{p + q}^2\) invariant) are known in principle,* the physical interpretation of the most important of them, of the ordinary de Sitter group O(4, 1), has not been adequately elucidated. The principal purpose of the present chapter is to contribute toward such elucidation, in particular, toward the understanding of how the positive nature of the energy can be incorporated into the interpretation. No attempt will be made at full generality nor at complete mathematical rigor. Nevertheless, a few problems of mathematical pathology will have to be discussed.

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References

  1. 1.
    L. H. Thomas, Ann. of Math. 42, 113 (1941).CrossRefMathSciNetGoogle Scholar
  2. 2.
    T. D. Newton, Ann. of Math. 51, 730 (1950).CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    J. Dixmier, Bull. Soc. Math. France 89, 9 (1961).zbMATHMathSciNetGoogle Scholar
  4. 4.
    V. Bargmann, Ann. of Math. 48, 568 (1947).CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    I. M. Gel’fand and M. A. Naimark, J. Phys. USSR 10, 93 (1946); Izv. Akad. Nauk SSSR 11, 411 (1947); Uspehi Mat. Nauk 9, 19 (1954).Google Scholar
  6. 6.
    A. Kihlberg, Ark. Pys. 30, 121 (1965).zbMATHMathSciNetGoogle Scholar
  7. 7.
    A. Kihlberg and S. Ström, Ark. Fys. 31, 491 (1966).zbMATHMathSciNetGoogle Scholar
  8. 8.
    I. M. Gel’fand, R. A. Minlos, and Z. Ya Shapiro, “Representations of the Rotation and Lorentz Groups.” Pergamon Press, London, 1963.Google Scholar
  9. 9.
    L. C. Biedenharn, J. N.yts, and N. Straumann, Ann. Inst. H. Poincaré A3, 13 (1965).zbMATHMathSciNetGoogle Scholar
  10. 10.
    R. Takahashi, Bull. Soc. Math. France 91, 289 (1963).zbMATHGoogle Scholar
  11. 11.
    Harish-Chandra, Proc. Nat. Acad. Sci. U.S.A. 37, 170, 362, 366, 691 (1951); Trans. Amer. Matlz. Soc. 75, 185 (1953); 76, 26, 234, 485 (1954); Proc. Nat. Acad. Sci. U.S.A. 40, 200, 1076, 1078 (1954).CrossRefMathSciNetGoogle Scholar
  12. 12.
    J. B. Ehrman, Princeton Dissertation, 1954; Proc. Cambridge Philos. Soc. 53, 290 (1957).zbMATHMathSciNetGoogle Scholar
  13. 13.
    F. A. Berezini, I. M. Gel’fand, M. I. Graev, and M. A. Naimark, Uspehi Mat. Nauk 11, 13 (1956).MathSciNetGoogle Scholar
  14. 14.
    J. Rosen and P. Roman, J. Mathematical Phys. 7, 2072 (1966).ADSCrossRefzbMATHGoogle Scholar
  15. 15.
    M. A. Melvin, Bull. Amer. Phys. Soc. 7, 493 (1962); 8, 356 (1963).Google Scholar
  16. 16.
    W. T. Sharp, Rept. 933, Atomic Energy of Canada, 1960.Google Scholar
  17. 17.
    D. W. Robinson, HeIv. Phys. Acta 35, 98 (1962).zbMATHGoogle Scholar
  18. 18.
    M. Levy-Nahas, J. Mathematical Phys. 8, 1211 (1967).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    G. Berendt, Acta Phys. Austriaca 25, 207 (1967).zbMATHGoogle Scholar
  20. 20.
    J. Rosen, Nuovo Cimento 35, 1234 (1965).CrossRefzbMATHGoogle Scholar
  21. 21.
    A. Esteve and P. G. Sona, Nuovo Cimento 32, 473 (1964).CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    I. E. Segal, Duke. Math. J. 18, 221 (1951).Google Scholar
  23. 23.
    E. Inönü and E. P. Wigner, Proc. Nat. Acad. Sci. U.S.A. 39, 510 (1953).ADSCrossRefzbMATHGoogle Scholar
  24. 24.
    E. Inönü, Comm. Fac. Sci. Univ. Ankara Ser. A 8, 83 (1956).Google Scholar
  25. 25.
    E. Inönü, in “Group Theoretical Concepts and Methods in Elementary Particle Physics” ( F. Gürsey, ed.), p. 365. Gordon and Breach, New York, 1964.Google Scholar
  26. 26.
    E. J. Saletan, J. Mathematical Phys. 2 1 (1961).Google Scholar
  27. 27.
    H. Zassenhaus, Canad. Math. Bull. 1, 31, 101, 183 (1958).Google Scholar
  28. 28.
    E. P. Wigner, Ann. of Math. 40, 149 (1939).CrossRefMathSciNetGoogle Scholar
  29. 29.
    T. Takabayasi, Progr. Theoret. Phys. 36, 1074 (1966).zbMATHGoogle Scholar
  30. 30.
    E. P. Wigner, in “Group Theoretical Concepts and Methods in Elementary Particle Physics” ( F. Gürsey, ed.), p. 37. Gordon and Breach, New York, 1964.Google Scholar
  31. 31.
    N. Tarimer, Phys. Rev. 140B, 977 (1965).ADSCrossRefGoogle Scholar
  32. 32.
    E. P. Wigner, Geitt. Nachr. p. 546 (1932).Google Scholar
  33. 33.
    E. P. Wigner, “Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra,” pp. 329–333. Academic Press, New York, 1959.Google Scholar
  34. 34.
    J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, Phys. Rev. 140B, 74 (1965).ADSCrossRefGoogle Scholar
  35. 35.
    S. Treiman, Comments on Nuclear and Particle Physics 1, 89 (1967).Google Scholar
  36. 36.
    T. O. Philips, Localized States in de Sitter Space, Doctoral Dissertation, Princeton University, 1963.Google Scholar
  37. 37.
    A. Sankaranarayanan and R. H. Good, Phys. Rev. 140B, 509 (1965).ADSCrossRefMathSciNetGoogle Scholar
  38. 38.
    H. Bacry, Phys. Lett. 5, 37 (1963).ADSCrossRefGoogle Scholar
  39. 39.
    T. D. Newton and E. P. Wigner, Rev. Mod. Phys. 21, 400 (1949).ADSCrossRefzbMATHGoogle Scholar
  40. 40.
    G. W. Mackey, Proc. Nat. Acad. Sci. U.S.A. 35, 537 (1949).CrossRefzbMATHGoogle Scholar
  41. 41.
    L. H. Loomis, Duke Math. J. 27, 569 (1960).Google Scholar
  42. 42.
    A. S. Wightman, Rev. Mod. Phys. 34, 845 (1962).ADSCrossRefMathSciNetGoogle Scholar
  43. 43.
    T. O. Philips, Phys. Rev. 136B, 893 (1964).ADSCrossRefMathSciNetGoogle Scholar
  44. 44.
    J. Rosen, Nuovo Cimento 35, 1234 (1965).CrossRefzbMATHGoogle Scholar
  45. 45.
    G. Frobenius, Sitz. d. Kön. Preuss. Akad. p. 501 (1898).Google Scholar
  46. 46.
    I. Schur, Sitz. d. Kön. Preuss. Akad. p. 164 (1906).Google Scholar
  47. 47.
    F. Seitz, Ann. of Math. 37, 17 (1936).CrossRefMathSciNetGoogle Scholar
  48. 48.
    J. von Neumann, Ann. of Math. 50, 401 (1949).CrossRefzbMATHMathSciNetGoogle Scholar
  49. 49.
    F. I. Mautner, Ann. of Math. 51, 1 (1950); 52, 528 (1950); Proc. Amer. Math. Soc. 2, 490 (1951).CrossRefzbMATHMathSciNetGoogle Scholar
  50. 50.
    J. Dixmier, “Les Algèbres d’Operateurs dans l’Espace Hilbertien,” Chap. 2. Gauthier-Villars, Paris, 1957.Google Scholar
  51. 51.
    M. Naimark, “Normed Rings” (translated by L. F. Boron and P. Noordhoff), Sec. 41. Groningen, 1964.Google Scholar
  52. 52.
    C. Fronsdal, Rev. Mod. Phys. 37, 221 (1965).ADSCrossRefMathSciNetGoogle Scholar
  53. 53.
    F. Gürsey, in “Group Theoretical Concepts and Methods in Elementary Particle Physics” ( F. Gürsey, ed.), p. 365. Gordon and Breach, New York, 1964.Google Scholar
  54. 54.
    P. Roman and J. J. Aghassi, Nuovo Cimento 42, 193 (1966).Google Scholar
  55. 55.
    P. Roman and C. J. Koh, Nuovo Cimento 45A, 268 (1966).Google Scholar
  56. 56.
    L. Pukanszky, Math. Ann. 156, 96 (1964).CrossRefzbMATHMathSciNetGoogle Scholar
  57. 57.
    H. Bacry and J.-M. Lévy-Leblond, J. Mathematical Phys., to be published.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • T. O. Philips
  • E. P. Wigner

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