Some Aspects of q- and qp-Boson Calculus

  • M. R. Kibler
  • R. M. Asherova
  • Yu. F. Smirnov


The aim of the present paper is to continue the program of extending in the framework of q-deformations the main results of the work in ref. 1 on the SU2 unit tensor or Wigner operator (the matrix elements of which are coupling coefficients or 3 — jm symbols). A first part of this program was published in the proceedings of Symmetries in Science VI (see ref. 2) where the q-deformed Schwinger algebra was defined and where an algorithm, based on the method of complementary q-deformed algebras, was given for obtaining three-and four-term recursion relations for the Clebsch-Gordan coefficients (CGc’s) of Uq(su2) and Uq(su1,1). The algorithm was fully exploited in ref. 3 where the complementary of three quantum algebras in a q-deformation of the symplectic Lie algebra sp(8, ℝ) was used for producing 32 recursion relations.


Commutation Relation Hopf Algebra Recursion Relation Intermediate Form Quantum Algebra 
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  1. 1.
    M. Kibler and G. Grenet, J. Math. Phys. 21:422 (1980).MathSciNetADSMATHCrossRefGoogle Scholar
  2. 2.
    Yu.F. Smirnov and M.R. Kibler, Some aspects of q-boson calculus, in: Symmetries in Science VI: From the rotation group to quantum algebras, B. Gruber, ed., Plenum Press, New York (1993), p. 691.Google Scholar
  3. 3.
    M. Kibler, C. Campigotto and Yu.F. Smirnov, Recursion relations for Clebsch Gordan coefficients of Uq(su2) and Uq(su1,1), in: Proceedings of the International Workshop “Symmetry Methods in Physics, in Memory of Professor Ya.A. Smorodinsky”, A.N. Sissakian, G.S. Pogosyan and S.I. Vinitsky, eds., J.I.N.R., Dubna, Russia (1994), p. 246.Google Scholar
  4. 4.
    E. Wigner, Gruppentheorie and ihre Anwendungen auf die Quantenmechanik der Atomspektren, Vieweg & Sohn, Braunschweig (1931).Google Scholar
  5. 5.
    B.L. van der Waerden, Die Gruppentheoretische Methode in der Quantenmechanik, Springer, Berlin (1932).Google Scholar
  6. 6.
    G. Racah, Phys. Rev. 62:438 (1942).ADSCrossRefGoogle Scholar
  7. 7.
    S.D. Majumdar, Prog. Theor. Phys. 20:798 (1958).ADSCrossRefGoogle Scholar
  8. 8.
    A.P. Jucys and A.A. Bandzaitis, Theory of angular momentum in quantum mechanics, Mintis, Vilnius (1965).Google Scholar
  9. 9.
    L.C. Biedenharn and M. Tarlini, Lett. Math. Phys. 20:271 (1990).MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    M. Nomura, J. Phys. Soc. Jpn. 59:1954 (1990).MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    M. Nomura, J. Phys. Soc. Jpn. 59:2345 (1990).MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    V.A. Groza, I.I. Kachurik and A.U. Klimyk, J. Math. Phys. 31:2769 (1990).MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    Yu.F. Smirnov, V.N. Tolstoi and Yu.I. Kharitonov, Sov. J. Nucl. Phys. 53:593 (1991).MathSciNetGoogle Scholar
  14. 14.
    C. Quesne, q-bosons and irreducible tensors for q-algebras, in: Symmetries in Science VI: From the rotation group to quantum algebras, B. Gruber, ed., Plenum Press, New York (1993).Google Scholar
  15. 15.
    J. Shapiro, J. Math. Phys. 6:1680 (1965).ADSCrossRefGoogle Scholar
  16. 16.
    J.-L. Calais, Int. J. Quantum Chem. 2:715 (1968).ADSCrossRefGoogle Scholar
  17. 17.
    A.J. Macfarlane, J. Phys. A 22:4581 (1989).MathSciNetADSMATHCrossRefGoogle Scholar
  18. 18.
    L.C. Biedenharn, J. Phys. A 22:L873 (1989).MathSciNetADSMATHCrossRefGoogle Scholar
  19. 19.
    A. Sudbery, J. Phys. A: Math Gen. 23:L697 (1990).MathSciNetADSMATHCrossRefGoogle Scholar
  20. 20.
    N. Reshetikhin, Lett. Math. Phys. 20:331 (1990).MathSciNetADSMATHCrossRefGoogle Scholar
  21. 21.
    D.B. Fairlie and C.K. Zachos, Phys. Lett. B 256:43 (1991).MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    A. Schirrmacher, J. Wess and B. Zumino, Z. Phys. C 49:317 (1991).MathSciNetCrossRefGoogle Scholar
  23. 23.
    S.T. Vokos, J. Math. Phys. 32:2979 (1991).MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    R. Chakrabarti and R. Jagannathan, J. Phys. A: Math. Gen. 24:L711 (1991).MathSciNetADSMATHCrossRefGoogle Scholar
  25. 25.
    V.K. Dobrev, J. Math. Phys. 33:3419 (1992).MathSciNetADSMATHCrossRefGoogle Scholar
  26. 26.
    Yu.F. Smirnov and R.F. Wehrhahn, J. Phys. A: Math. Gen. 25:5563 (1992).MathSciNetADSMATHCrossRefGoogle Scholar
  27. 27.
    M.R. Kibler, Introduction to quantum algebras, in: Symmetry and Structural Properties of Condensed Matter, W. Florek, D. Lipinski and T. Lulek, eds., World Scientific, Singapore (1993), p. 445.Google Scholar
  28. 28.
    S. Meljanac and M. Milekovie, J. Phys. A: Math. Gen. 26:5177 (1993).ADSMATHCrossRefGoogle Scholar
  29. 29.
    C. Quesne, Phys. Lett. A 174:19 (1993).MathSciNetADSCrossRefGoogle Scholar
  30. 30.
    R. Chakrabarti and R. Jagannathan, J. Phys. A: Math. Gen. 27:2023 (1994).MathSciNetADSMATHCrossRefGoogle Scholar
  31. 31.
    R. Barbier, J. Meyer and M. Kibler, J. Phys. G: Nucl. Part. Phys. 17:L67 (1994).Google Scholar
  32. 32.
    R. Barbier, J. Meyer and M. Kibler, A qp-rotor model for rotational bands of superdeformed nuclei, Preprint LYCEN 9437, IPNL (1994).Google Scholar
  33. 33.
    Ya.A. Smorodinskii and L.A. Shelepin, Usp. Fiz. Nauk 15:1 (1972).MathSciNetGoogle Scholar
  34. 34.
    M. Kibler, G.-H. Lamot and Yu.F. Smirnov, to be published.Google Scholar

Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • M. R. Kibler
    • 1
  • R. M. Asherova
    • 1
  • Yu. F. Smirnov
    • 2
  1. 1.Institut de Physique Nucléaire de LyonIN2P3-CNRS et Université Claude BernardVilleurbanne CedexFrance
  2. 2.Institute of Nuclear PhysicsMoscow State UniversityMoscowRussia

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