q-Fock Space Representations of the q-Lorentz Algebra and Irreducible Tensors

  • M. Chaichian
  • J. A. De Azcárraga
  • F. Rodenas


We present the q-deformation of the Lorentz algebra, with Hopf structure, in terms of four independent harmonic oscillators. The explicit realization of the q-Fock space is given and the irreducible finite-dimensional representations of so(1,3)q are described and characterized by its two q-Casimir operators. The concept of irreducible q-Lorentz tensor is also introduced. The analysis is made for a real deformation parameter.


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  1. [1]
    V.G. Drinfel’d, Proc. Berkeley Int. Congress of Math., vol.1, 798 (1987).MathSciNetGoogle Scholar
  2. [2]
    L. D. Faddeev, N. Yu. Reshetikhin, L.A. Takhtajan, Algebra and Anal. 1, 178 (1989).MathSciNetGoogle Scholar
  3. [3]
    Proceedings of the 8th Int. Workshop on Mathematical Physics (Quantum Groups), Clausthal 1989, H.-D. Doebner and J.-D. Jenning (Eds.), Lect. Notes in Physics 370, Springer Verlag (1990).Google Scholar
  4. [4]
    Proceedings of the First Euler Math. Inst. Workshop on quantum groups (1990), P. Kulish Ed., Springer-Verlag, (1991).Google Scholar
  5. [5]
    Yu. I. Manin, Quantum groups and non-commutative geometry, Centre des Recherches Mathématiques, Montréal (1988).Google Scholar
  6. [6]
    S.L. Woronowicz, Publ. RIMS Kyoto Univ. 23, 117 (1987).MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    M. Schlieker, M. Scholl, Z. Phys. C47, 625 (1990).MathSciNetADSGoogle Scholar
  8. [8]
    U. Carow-Watamura, M. Schlieker, M. Scholl, S. Watamura, Z. Phys. C48, 159 (1990).MathSciNetGoogle Scholar
  9. [9]
    W.B. Schmidke, J. Wess and B. Zumino, Z. Phys. C52, 471 (1991).MathSciNetADSGoogle Scholar
  10. [10]
    D. Drabant, M. Schlieker, W. Weich and B. Zumino, Commun. Math. Phys. 147, 625 (1992).MathSciNetADSMATHCrossRefGoogle Scholar
  11. [11]
    O. Ogievetsky, W.B. Schmidke, J. Wess and B. Zumino, Six generator q-deformed Lorentz algebra, MPI-Ph/91-51,(1991); q-deformed Poincaré algebra, MPI-Ph/ 91-98, (1991).Google Scholar
  12. [12]
    V. K. Dobrev, Canonical q-deformations of Non-compact Lie (Super-) Algebras, Göttingen preprint (July/October 1991); q-deformations of non-compact Lie (Super-) Algebras: the examples of q-deformed Lorentz, Weyl, Poincaré and (Super-) conformal algebra, ICTP preprint IC/92/13, to appear in the Proc. of the II Wigner Symposium, Goslar (1991).Google Scholar
  13. [13]
    J. Lukierski, H. Ruegg, A. Nowicki and V.N. Tolstoy, Phys. Lett. B264, 331 (1991).MathSciNetADSGoogle Scholar
  14. J. Lukierski and A. Nowicki, Phys. Lett. B279, 299 (1992); see also these Proceedings.MathSciNetADSGoogle Scholar
  15. [14]
    O. Ogievetsky, M. Pillin, W.B. Schmidke, J. Wess and B. Zumino, q-Deformed Minkowski space, in the Proc. of the XIX Int. Colloquium on Group Theoretical Methods in Physics, Salamanca July 1992, J. Mateos, M. del Olmo and M. Santander (Eds.), to be published by CIEMAT-Real Soc. Española de Física.Google Scholar
  16. [15]
    M. Chaichian, J. A. De Azcárraga, P. Prešnajder and F. Rodenas, Phys. Lett. B291, 411 (1992).ADSGoogle Scholar
  17. [16]
    A. J. Macfarlane, J. Phys. A22, 4581 (1989).MathSciNetADSGoogle Scholar
  18. [17]
    L. C. Biedenharn, J. Phys. A22, L873 (1989).MathSciNetADSGoogle Scholar
  19. [18]
    J. Schwinger, On angular momentum, Report U.S. AEC NYO-3071 (unpublished). Reprinted in Quantum theory of angular momentum L.C. Biedenharn Ed., Acad. Press p. 229 (1965).Google Scholar
  20. [19]
    T. Hayashi, Commun. Math. Phys. 127, 129 (1990).ADSCrossRefGoogle Scholar
  21. [20]
    M. Chaichian and P. Kulish, Phys. Lett. B234, 72 (1990).MathSciNetADSGoogle Scholar
  22. [21]
    G. Gomez and G. Sierra, Phys. Lett B255, 51 (1991).MathSciNetADSGoogle Scholar
  23. [22]
    E. Celeghini, R. Giachetti, E. Sorace and M. Tarlini, J. Math. Phys. 32, 1159 (1991).MathSciNetADSMATHCrossRefGoogle Scholar
  24. [23]
    L. C. Biedenharn and M. Tarlini, Lett. in Math. Phys. 20, 271 (1990).MathSciNetADSMATHCrossRefGoogle Scholar
  25. [24]
    L. C. Biedenharn, A q-boson realization of the quantum group su(2) q and the theory of q-tensor operators, in Proc. of the Clausthal Summer Workshop on Math. Physics, H.-D. Doebner and J.-D. Jenning Eds., Springer-Verlag (1991).Google Scholar
  26. [25]
    L. C. Biedenharn and M. A. Lohe, Induced representations and tensor operators for quantum groups, in the First Euler Math. Workshop on Quantum Groups, P. Kulish Ed., Springer Verlag (1991).Google Scholar
  27. [26]
    Feng Pan, J. Phys. A24, L803 (1991).ADSGoogle Scholar
  28. [27]
    L. K. Hadjiivanov, R. R. Paunov. I. T. Todorov, J. Math. Phys. 33, 1379 (1992).MathSciNetADSCrossRefGoogle Scholar
  29. [28]
    V. Rittenberg and M. Scheunert. J. Math. Phys. 33 436 (1992).MathSciNetADSMATHCrossRefGoogle Scholar
  30. [29]
    C. Quesne, q-bosons and irreducible tensors for q-algebras, Bruxelles PNT /10/92 preprint (1992) and these Proceedings.Google Scholar
  31. [30]
    M. Nomura, J. Math. Phys. 30, 2397 (1989).MathSciNetADSMATHCrossRefGoogle Scholar
  32. [31]
    H. Ruegg. J. Math Phys. 31, 1085 (1990).MathSciNetADSMATHCrossRefGoogle Scholar
  33. [32]
    V.A. Groza, I.I. Kachurik and A.U. Klimyk, J. Math. Phys. 31, 2769 (1990).MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • M. Chaichian
    • 1
  • J. A. De Azcárraga
    • 2
  • F. Rodenas
    • 2
  1. 1.High Energy Physics Laboratory, Department of PhysicsUniversity of HelsinkiHelsinkiFinland
  2. 2.Departamento de Física Teórica and IFIC (CSIC)Universidad de ValenciaBurjassot, ValenciaSpain

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