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Communications in Mathematical Physics

, Volume 55, Issue 1, pp 1–28 | Cite as

All unitary ray representations of the conformal group SU(2,2) with positive energy

Article

Abstract

We find all those unitary irreducible representations of the ∞-sheeted covering group\(\tilde G\) of the conformal group SU(2,2)/ℤ4 which have positive energyP0≧0. They are all finite component field representations and are labelled by dimensiond and a finite dimensional irreducible representation (j1,j2) of the Lorentz group SL(2ℂ). They all decompose into a finite number of unitary irreducible representations of the Poincaré subgroup with dilations.

Keywords

Neural Network Field Representation Complex System Nonlinear Dynamics Finite Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Mack, G., Abdus Salam: Ann. Phys. (N.Y.)53, 174 (1969)Google Scholar
  2. 2.
    Segal, I.: MIT preprintGoogle Scholar
  3. 3.
    Lüscher, M., Mack, G.: Commun. math. Phys.41, 203 (1975)Google Scholar
  4. 4.
    Graev, M. L.: Dokl. Acad. Nauk SSR98, 517 (1954)Google Scholar
  5. 4a.
    Castell, L.: Nucl. Phys.B4, 343 (1967)Google Scholar
  6. 4b.
    Yao, T.: J. Math. Phys.8, 1931 (1967);9, 1615 (1968);Google Scholar
  7. 4c.
    Sternheimer, D.: J. Math. Pure Appl.47, 289 (1969) and references cited in 1Google Scholar
  8. 5.
    Rühl, W.: Commun. math. Phys.30, 287 (1973);34, 149 (1973); The canonical dimension of fields as the limit of noncanonical dimensions, preprint Kaiserslautern (March 1973)Google Scholar
  9. 6.
    Mack, G., Todorov, I.T.: J. Math. Phys.10, 2078 (1969)Google Scholar
  10. 7.
    Mack, G.: Osterwalder-Schrader positivity in conformal invariant quantum field theory. In: Lecture notes in physics, Vol. 37, (ed. H. Rollnik, K. Dietz), p. 66. Berlin-Heidelberg-New York: Springer 1975Google Scholar
  11. 8.
    Mack, G.: Commun. math. Phys.53, 155 (1977); Nucl. Phys.B 118, 445 (1977)Google Scholar
  12. 9.
    Dieudonné, I.: Treatise on analysis, Vol. III. New York: Academic Press 1972Google Scholar
  13. 10.
    Hermann, R.: Lie groups for physicists, Chap. 6, 7. New York: W. A. Benjamin 1966Google Scholar
  14. 11.
    Wigner, E.: Ann math.40, 149 (1939)Google Scholar
  15. 11a.
    Joos, H.: Forschr. Physik10, 65 (1965);Google Scholar
  16. 11b.
    Weinberg, S.: Phys. Rev.133, B 1318 (1964),134, B 882 (1964)Google Scholar
  17. 12.
    Kihlberg, A., Müller, V.F., Halbwachs, F.: Commun. math. Phys.3, 194 (1966)Google Scholar
  18. 13.
    Warner, G.: Harmonic analysis on semi-simple Lie groups, Vols. I, II. Berlin-Heidelberg-New York: Springer 1972Google Scholar
  19. 14.
    Wallach, N. R.: Harmonic analysis on homogeneous spaces. New York: Marcel Dekker 1973Google Scholar
  20. 15.
    Rose, M. E.: Elementary theory of angular momentum, Appendix I. New York: John Wiley 1957Google Scholar
  21. 16.
    Gelfand, I. M., Shilov, G. E.: Generalized functions, Vol. I. New York: Academic PressGoogle Scholar
  22. 17.
    Koller, K.: Commun. math. Phys.40, 15 (1975)Google Scholar
  23. 18.
    Dobrev, V. K., Mack, G., Petkova, V. B., Petrova, S. G., Todorov, I. T.: Elementary representations and intertwining operators for the generalized Lorentz group. Lecture notes in physics, Vol. 63. Berlin-Heidelberg-New York: Springer 1977Google Scholar
  24. 19.
    Kunze, R., Stein, E.: Amer. J. Math.82, 1 (1960);83, 723 (1961);89, 385 (1967)Google Scholar
  25. 19a.
    Knapp, A., Stein, E.: Ann. Math.93, 489 (1971);Google Scholar
  26. 19b.
    Schiffmann, G.: Bull. Soc. Math. France99, 3 (1971)Google Scholar
  27. 20.
    Neumark, M. A.: Lineare Darstellungen der Lorentzgruppe, §8, Satz 2, p. 110. Berlin: VEB dt. Verlag der Wissenschaften 1963Google Scholar
  28. 21.
    Nelson, E.: Analytic vectors, Ann. Math.70, 572 (1959)Google Scholar
  29. 22.
    Lüscher, M.: Analytic representations of simple Lie groups and their continuation to contractive representations of holomorphic Lie semi-groups, DESY 75/71 (1975)Google Scholar
  30. 23.
    Ferrara, S., Gatto, R., Grillo, A.: Phys. Rev.D9, 3564 (1975);Google Scholar
  31. 23a.
    Zaikov, R. P.: Bulg. J. Phys.2, 2 (1975)Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • G. Mack
    • 1
  1. 1.II. Institut für Theoretische Physik der UniversitätHamburg 50Germany

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