Advertisement

Degenerate Representations of Non-Compact Lie Groups

  • R. Raczka
Conference paper
Part of the Acta Physica Austriaca book series (FEWBODY, volume 3/1966)

Abstract

Suppose, that we have selected a certain non-compact group G as the higher symmetry group of some physical system. Then to regard all consequences and restrictions which follow from the introduced symmetry we should examine the following questions:
  1. 1.

    What is the system of invariant operators which generate the ring of invariant operators? Have we the possibility of a reduction of this system if we have no convenient interpretation for these operators as physical observables?

     
  2. 2.

    What is the maximal system of commuting operators in the fixed representation space which we shall interpret as physical observables? What is the shape and range of their spectra?

     
  3. 3.

    What are the properties of the basic functions in the representation space, which we would like to identify with physical states?

     
  4. 4.

    What are the properties of the direct product of two representations (Clebsch-Gordan series) and two basic vectors (Clebsch-Gordan coefficients)?

     
  5. 5.

    What are the properties of the decomposition of a considered representation with respect to a maximal compact subgroup (which is often an initial symmetry group)?

     

Keywords

Invariant Operator Discrete Series Casimir Operator Maximal Compact Subgroup Cartan Subgroup 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    See e.g. I. M. Gelfand and M. A. Naimark, Trudi of Math. Steklov Institute 36 (1950)Google Scholar
  2. 2.
    See e.g. M. I. Graev, Trudi of Moscov Math. Soc. 7, 335 (1958)Google Scholar
  3. 3.
    I. M. Gelfand,Amer. Math. Soc. Transi. (Ser. 2) 37, 31–34 (1961+); I. M. Gelfand and M. I. Graev, Trudi of Moscov Math. Soc. 8, 321 (1959).Google Scholar
  4. 4.
    It turns out that a maximal set of invariant operators does not determine in general an irreducible representation. In ref. [16] it is shown that we must introduce the additional invariant operator with the discrete spectrum which splits a reducible representation space on irreducible parts.Google Scholar
  5. 5.
    See e.g. S. Helgason, chap. IX and X.Google Scholar
  6. 6.
    B. A. Rosenfeld, Dokl. Ak. Nauk. SSSR, 110, 23–27 (1956); A. S. Fedenko, Dokl. A.. Nauk SSSR 108, 1026–1028 (1956).Google Scholar
  7. 7.
    R. Raczka, N. Limic and J. Niederle preprint ICTP Trieste IC/66/2Google Scholar
  8. 8.
    If the metric tensor ga(X) on the homogeneous space X is induced by the Cartan metric tensor gik in the Lie algebra R of the group G, then the Laplace-Beltrami operator L\(X) is equal to the second order Casimir operator Q2= gikXiXk (see [5] chap. X)Google Scholar
  9. 9.
    Soo(m,n) denotes component of a unity of SO(m,n).Google Scholar
  10. 10.
    Here and elsewhere we use brackets for indices defined as follows \( % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada % WcaaqaaiaadggaaeaacaaIYaaaaaGaay5waiaaw2faaiabg2da9iaa % cUhadaqhaaWcbaWaaSaaaeaacaWGHbGaeyOeI0IaaGymaaqaaiaaik % daaaGaamyAaiaadAgacaWGHbGaeyypa0JaaGOmaiaadkhacqGHRaWk % caaIXaaabaWaaSaaaeaacaWGHbaabaGaaGOmaaaacaWGPbGaamOzai % aadggacqGH9aqpcaaIYaGaamOCaaaakmaacmaabaWaaSaaaeaacaWG % HbaabaGaaGOmaaaaaiaawUhacaGL9baacqGH9aqpcaGG7bWaa0baaS % qaamaalaaabaGaamyyaiabgUcaRiaaigdaaeaacaaIYaaaaiaadMga % caWGMbGaamyyaiabg2da9iaaikdacaWGYbGaey4kaSIaaGymaaqaam % aalaaabaGaamyyaaqaaiaaikdaaaGaamyAaiaadAgacaWGHbGaeyyp % a0JaaGOmaiaadkhaaaGccaWGYbGaeyypa0JaaGymaiaacYcacaaIYa % GaaiilaiablAcilbaa!6AE9! \left[ {\frac{a} {2}} \right] = \{ _{\frac{{a - 1}} {2}ifa = 2r + 1}^{\frac{a} {2}ifa = 2r} \left\{ {\frac{a} {2}} \right\} = \{ _{\frac{{a + 1}} {2}ifa = 2r + 1}^{\frac{a} {2}ifa = 2r} r = 1,2, \ldots \) Google Scholar
  11. 11.
    The measure \( % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabeY % 7aTnaabmaabaGaeuyQdCfacaGLOaGaayzkaaGaeyypa0ZaaqWaaeaa % ceWGNbGbaebadaqadaqaaiaadIeadaqhaaWcbaGaamyCaaqaaiaadc % haaaaakiaawIcacaGLPaaaaiaawEa7caGLiWoadaahaaWcbeqaaiab % gUcaRiaaigdacaGGVaGaaGOmaaaakiaadsgacqqHPoWvaaa!4AE1! d\mu \left( \Omega \right) = \left| {\bar g\left( {H_q^p } \right)} \right|^{ + 1/2} d\Omega \) is the Riemannian measure which is left invariant under the action of SOo(p,q) on\( % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaDa % aaleaacaWGXbaabaGaamiCaaaaaaa!38B2! H_q^p \) Google Scholar
  12. 12.
    E. L. Ince, Ordinary differential Equations (Dover Publications 1956 ).Google Scholar
  13. 13.
    V. Bargmann, Ann. of Math. 48, 568–640 (1947).MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    I. M. Gelfand and M. I. Graev, Trudi of Moscov, Math. Soc. 11 (1962).Google Scholar
  15. 15.
    J. Dixmier, Bull. Soc. Math. France, 89, 9–41 (1961)MATHGoogle Scholar
  16. 16.
    N. Limic. J. Niederle and R. Raczka, Continuous most degenerate representation of arbitrary noncompact SO(p,q) groups, ICTP (in print).Google Scholar
  17. 17.
    M. A. Naimark, Linear Representations of the Lorentz group (Pergamon Press 1964 ), p. 167.Google Scholar
  18. 19.
    B. A. Rosenfeld, Non-Euclidean Geometries, Moscow, 1955.Google Scholar
  19. 20.
    J. Fisher and R. Raczka, preprint ICTP, IC/66/8Google Scholar
  20. 21.
    N. Limic, J. Niederle and R. Raczka, Generalized Fourier transforms related to degenerate representations of S0(p,q) groups, ICTP Trieste, (in print)Google Scholar
  21. 22.
    R. Raczka and J.Fisher, Gelfand-Graev transforms related to degenerate representations of U(p,q) groups, ICTP, Trieste (in print).Google Scholar

Copyright information

© Springer-Verlag GmbH Wien 1966

Authors and Affiliations

  • R. Raczka
    • 1
    • 2
  1. 1.International Centre for Theoretical PhysicsTriesteItaly
  2. 2.Institute of Nuclear ResearchWarsawPoland

Personalised recommendations